LMFDB - Embedded newform 825.2.k.e.782.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(518,825)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("825.518");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.k (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			782.2
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	825.782
Dual form			825.2.k.e.518.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 1.22474i) q^{2} +(1.22474 - 1.22474i) q^{3} -1.00000i q^{4} -3.00000i q^{6} +(2.44949 + 2.44949i) q^{7} +(1.22474 + 1.22474i) q^{8} -3.00000i q^{9} +O(q^{10})$	\(q+(1.22474 - 1.22474i) q^{2} +(1.22474 - 1.22474i) q^{3} -1.00000i q^{4} -3.00000i q^{6} +(2.44949 + 2.44949i) q^{7} +(1.22474 + 1.22474i) q^{8} -3.00000i q^{9} -1.00000i q^{11} +(-1.22474 - 1.22474i) q^{12} +(-2.44949 + 2.44949i) q^{13} +6.00000 q^{14} +5.00000 q^{16} +(4.89898 - 4.89898i) q^{17} +(-3.67423 - 3.67423i) q^{18} +2.00000i q^{19} +6.00000 q^{21} +(-1.22474 - 1.22474i) q^{22} +(-4.89898 - 4.89898i) q^{23} +3.00000 q^{24} +6.00000i q^{26} +(-3.67423 - 3.67423i) q^{27} +(2.44949 - 2.44949i) q^{28} -6.00000 q^{29} +4.00000 q^{31} +(3.67423 - 3.67423i) q^{32} +(-1.22474 - 1.22474i) q^{33} -12.0000i q^{34} -3.00000 q^{36} +(2.44949 + 2.44949i) q^{38} +6.00000i q^{39} +6.00000i q^{41} +(7.34847 - 7.34847i) q^{42} +(-7.34847 + 7.34847i) q^{43} -1.00000 q^{44} -12.0000 q^{46} +(-4.89898 + 4.89898i) q^{47} +(6.12372 - 6.12372i) q^{48} +5.00000i q^{49} -12.0000i q^{51} +(2.44949 + 2.44949i) q^{52} +(-4.89898 - 4.89898i) q^{53} -9.00000 q^{54} +6.00000i q^{56} +(2.44949 + 2.44949i) q^{57} +(-7.34847 + 7.34847i) q^{58} +2.00000 q^{61} +(4.89898 - 4.89898i) q^{62} +(7.34847 - 7.34847i) q^{63} +1.00000i q^{64} -3.00000 q^{66} +(2.44949 + 2.44949i) q^{67} +(-4.89898 - 4.89898i) q^{68} -12.0000 q^{69} -12.0000i q^{71} +(3.67423 - 3.67423i) q^{72} +(2.44949 - 2.44949i) q^{73} +2.00000 q^{76} +(2.44949 - 2.44949i) q^{77} +(7.34847 + 7.34847i) q^{78} +10.0000i q^{79} -9.00000 q^{81} +(7.34847 + 7.34847i) q^{82} +(-7.34847 - 7.34847i) q^{83} -6.00000i q^{84} +18.0000i q^{86} +(-7.34847 + 7.34847i) q^{87} +(1.22474 - 1.22474i) q^{88} +12.0000 q^{89} -12.0000 q^{91} +(-4.89898 + 4.89898i) q^{92} +(4.89898 - 4.89898i) q^{93} +12.0000i q^{94} -9.00000i q^{96} +(-9.79796 - 9.79796i) q^{97} +(6.12372 + 6.12372i) q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 24 q^{14} + 20 q^{16} + 24 q^{21} + 12 q^{24} - 24 q^{29} + 16 q^{31} - 12 q^{36} - 4 q^{44} - 48 q^{46} - 36 q^{54} + 8 q^{61} - 12 q^{66} - 48 q^{69} + 8 q^{76} - 36 q^{81} + 48 q^{89} - 48 q^{91} - 12 q^{99}+O(q^{100}) $ 4 * q + 24 * q^14 + 20 * q^16 + 24 * q^21 + 12 * q^24 - 24 * q^29 + 16 * q^31 - 12 * q^36 - 4 * q^44 - 48 * q^46 - 36 * q^54 + 8 * q^61 - 12 * q^66 - 48 * q^69 + 8 * q^76 - 36 * q^81 + 48 * q^89 - 48 * q^91 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.22474	−	1.22474i	0.866025	−	0.866025i	−0.126004	−	0.992030i	$-0.540215\pi$
$2$	1.22474	−	1.22474i	0.866025	−	0.866025i	0.992030	+	0.126004i	$0.0402153\pi$
$3$	1.22474	−	1.22474i	0.707107	−	0.707107i
$3$	1.22474	−	1.22474i	0.707107	−	0.707107i
$4$		−	1.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		−	3.00000i		−	1.22474i
$7$	2.44949	+	2.44949i	0.925820	+	0.925820i	0.997433	−	0.0716124i	$-0.0228145\pi$
$7$	2.44949	+	2.44949i	0.925820	+	0.925820i	−0.0716124	+	0.997433i	$0.522814\pi$
$8$	1.22474	+	1.22474i	0.433013	+	0.433013i
$9$		−	3.00000i		−	1.00000i
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$	−1.22474	−	1.22474i	−0.353553	−	0.353553i
$13$	−2.44949	+	2.44949i	−0.679366	+	0.679366i	−0.959857	−	0.280491i	$-0.909503\pi$
$13$	−2.44949	+	2.44949i	−0.679366	+	0.679366i	0.280491	+	0.959857i	$0.409503\pi$
$14$	6.00000			1.60357
$15$		0			0
$16$	5.00000			1.25000
$17$	4.89898	−	4.89898i	1.18818	−	1.18818i	0.210606	−	0.977571i	$-0.432456\pi$
$17$	4.89898	−	4.89898i	1.18818	−	1.18818i	0.977571	−	0.210606i	$-0.0675437\pi$
$18$	−3.67423	−	3.67423i	−0.866025	−	0.866025i
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$	6.00000			1.30931
$22$	−1.22474	−	1.22474i	−0.261116	−	0.261116i
$23$	−4.89898	−	4.89898i	−1.02151	−	1.02151i	−0.999764	−	0.0217443i	$-0.993078\pi$
$23$	−4.89898	−	4.89898i	−1.02151	−	1.02151i	−0.0217443	−	0.999764i	$-0.506922\pi$
$24$	3.00000			0.612372
$25$		0			0
$26$			6.00000i			1.17670i
$27$	−3.67423	−	3.67423i	−0.707107	−	0.707107i
$28$	2.44949	−	2.44949i	0.462910	−	0.462910i
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$	3.67423	−	3.67423i	0.649519	−	0.649519i
$33$	−1.22474	−	1.22474i	−0.213201	−	0.213201i
$34$		−	12.0000i		−	2.05798i
$35$		0			0
$36$	−3.00000			−0.500000
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	2.44949	+	2.44949i	0.397360	+	0.397360i
$39$			6.00000i			0.960769i
$40$		0			0
$41$			6.00000i			0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$			6.00000i			0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	7.34847	−	7.34847i	1.13389	−	1.13389i
$43$	−7.34847	+	7.34847i	−1.12063	+	1.12063i	−0.128984	+	0.991647i	$0.541172\pi$
$43$	−7.34847	+	7.34847i	−1.12063	+	1.12063i	−0.991647	+	0.128984i	$0.958828\pi$
$44$	−1.00000			−0.150756
$45$		0			0
$46$	−12.0000			−1.76930
$47$	−4.89898	+	4.89898i	−0.714590	+	0.714590i	−0.967492	−	0.252902i	$-0.918615\pi$
$47$	−4.89898	+	4.89898i	−0.714590	+	0.714590i	0.252902	+	0.967492i	$0.418615\pi$
$48$	6.12372	−	6.12372i	0.883883	−	0.883883i
$49$			5.00000i			0.714286i
$50$		0			0
$51$		−	12.0000i		−	1.68034i
$52$	2.44949	+	2.44949i	0.339683	+	0.339683i
$53$	−4.89898	−	4.89898i	−0.672927	−	0.672927i	0.285463	−	0.958390i	$-0.407853\pi$
$53$	−4.89898	−	4.89898i	−0.672927	−	0.672927i	−0.958390	+	0.285463i	$0.907853\pi$
$54$	−9.00000			−1.22474
$55$		0			0
$56$			6.00000i			0.801784i
$57$	2.44949	+	2.44949i	0.324443	+	0.324443i
$58$	−7.34847	+	7.34847i	−0.964901	+	0.964901i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$	4.89898	−	4.89898i	0.622171	−	0.622171i
$63$	7.34847	−	7.34847i	0.925820	−	0.925820i
$64$			1.00000i			0.125000i
$65$		0			0
$66$	−3.00000			−0.369274
$67$	2.44949	+	2.44949i	0.299253	+	0.299253i	0.840721	−	0.541468i	$-0.182131\pi$
$67$	2.44949	+	2.44949i	0.299253	+	0.299253i	−0.541468	+	0.840721i	$0.682131\pi$
$68$	−4.89898	−	4.89898i	−0.594089	−	0.594089i
$69$	−12.0000			−1.44463
$70$		0			0
$71$		−	12.0000i		−	1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$		−	12.0000i		−	1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$	3.67423	−	3.67423i	0.433013	−	0.433013i
$73$	2.44949	−	2.44949i	0.286691	−	0.286691i	−0.549079	−	0.835770i	$-0.685021\pi$
$73$	2.44949	−	2.44949i	0.286691	−	0.286691i	0.835770	+	0.549079i	$0.185021\pi$
$74$		0			0
$75$		0			0
$76$	2.00000			0.229416
$77$	2.44949	−	2.44949i	0.279145	−	0.279145i
$78$	7.34847	+	7.34847i	0.832050	+	0.832050i
$79$			10.0000i			1.12509i	0.826767	+	0.562544i	$0.190177\pi$
$79$			10.0000i			1.12509i	−0.826767	+	0.562544i	$0.809823\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$	7.34847	+	7.34847i	0.811503	+	0.811503i
$83$	−7.34847	−	7.34847i	−0.806599	−	0.806599i	0.177518	−	0.984118i	$-0.443193\pi$
$83$	−7.34847	−	7.34847i	−0.806599	−	0.806599i	−0.984118	+	0.177518i	$0.943193\pi$
$84$		−	6.00000i		−	0.654654i
$85$		0			0
$86$			18.0000i			1.94099i
$87$	−7.34847	+	7.34847i	−0.787839	+	0.787839i
$88$	1.22474	−	1.22474i	0.130558	−	0.130558i
$89$	12.0000			1.27200			0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000			1.27200			0.635999	+	0.771690i	$0.280588\pi$
$90$		0			0
$91$	−12.0000			−1.25794
$92$	−4.89898	+	4.89898i	−0.510754	+	0.510754i
$93$	4.89898	−	4.89898i	0.508001	−	0.508001i
$94$			12.0000i			1.23771i
$95$		0			0
$96$		−	9.00000i		−	0.918559i
$97$	−9.79796	−	9.79796i	−0.994832	−	0.994832i	0.00515471	−	0.999987i	$-0.498359\pi$
$97$	−9.79796	−	9.79796i	−0.994832	−	0.994832i	−0.999987	+	0.00515471i	$0.998359\pi$
$98$	6.12372	+	6.12372i	0.618590	+	0.618590i
$99$	−3.00000			−0.301511
$100$		0			0
$101$			18.0000i			1.79107i	0.444994	+	0.895533i	$0.353206\pi$
$101$			18.0000i			1.79107i	−0.444994	+	0.895533i	$0.646794\pi$
$102$	−14.6969	−	14.6969i	−1.45521	−	1.45521i
$103$	−2.44949	+	2.44949i	−0.241355	+	0.241355i	−0.817411	−	0.576055i	$-0.804591\pi$
$103$	−2.44949	+	2.44949i	−0.241355	+	0.241355i	0.576055	+	0.817411i	$0.304591\pi$
$104$	−6.00000			−0.588348
$105$		0			0
$106$	−12.0000			−1.16554
$107$	−2.44949	+	2.44949i	−0.236801	+	0.236801i	−0.815524	−	0.578723i	$-0.803551\pi$
$107$	−2.44949	+	2.44949i	−0.236801	+	0.236801i	0.578723	+	0.815524i	$0.303551\pi$
$108$	−3.67423	+	3.67423i	−0.353553	+	0.353553i
$109$			10.0000i			0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$			10.0000i			0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$		0			0
$111$		0			0
$112$	12.2474	+	12.2474i	1.15728	+	1.15728i
$113$	4.89898	+	4.89898i	0.460857	+	0.460857i	0.898936	−	0.438079i	$-0.144341\pi$
$113$	4.89898	+	4.89898i	0.460857	+	0.460857i	−0.438079	+	0.898936i	$0.644341\pi$
$114$	6.00000			0.561951
$115$		0			0
$116$			6.00000i			0.557086i
$117$	7.34847	+	7.34847i	0.679366	+	0.679366i
$118$		0			0
$119$	24.0000			2.20008
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	2.44949	−	2.44949i	0.221766	−	0.221766i
$123$	7.34847	+	7.34847i	0.662589	+	0.662589i
$124$		−	4.00000i		−	0.359211i
$125$		0			0
$126$		−	18.0000i		−	1.60357i
$127$	−2.44949	−	2.44949i	−0.217357	−	0.217357i	0.590027	−	0.807384i	$-0.299117\pi$
$127$	−2.44949	−	2.44949i	−0.217357	−	0.217357i	−0.807384	+	0.590027i	$0.799117\pi$
$128$	8.57321	+	8.57321i	0.757772	+	0.757772i
$129$			18.0000i			1.58481i
$130$		0			0
$131$		−	12.0000i		−	1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$		−	12.0000i		−	1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$	−1.22474	+	1.22474i	−0.106600	+	0.106600i
$133$	−4.89898	+	4.89898i	−0.424795	+	0.424795i
$134$	6.00000			0.518321
$135$		0			0
$136$	12.0000			1.02899
$137$	−9.79796	+	9.79796i	−0.837096	+	0.837096i	−0.988476	−	0.151380i	$-0.951628\pi$
$137$	−9.79796	+	9.79796i	−0.837096	+	0.837096i	0.151380	+	0.988476i	$0.451628\pi$
$138$	−14.6969	+	14.6969i	−1.25109	+	1.25109i
$139$		−	22.0000i		−	1.86602i	−0.359856	−	0.933008i	$-0.617174\pi$
$139$		−	22.0000i		−	1.86602i	0.359856	−	0.933008i	$-0.382826\pi$
$140$		0			0
$141$			12.0000i			1.01058i
$142$	−14.6969	−	14.6969i	−1.23334	−	1.23334i
$143$	2.44949	+	2.44949i	0.204837	+	0.204837i
$144$		−	15.0000i		−	1.25000i
$145$		0			0
$146$		−	6.00000i		−	0.496564i
$147$	6.12372	+	6.12372i	0.505076	+	0.505076i
$148$		0			0
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$	−2.44949	+	2.44949i	−0.198680	+	0.198680i
$153$	−14.6969	−	14.6969i	−1.18818	−	1.18818i
$154$		−	6.00000i		−	0.483494i
$155$		0			0
$156$	6.00000			0.480384
$157$	−9.79796	−	9.79796i	−0.781962	−	0.781962i	0.198199	−	0.980162i	$-0.436491\pi$
$157$	−9.79796	−	9.79796i	−0.781962	−	0.781962i	−0.980162	+	0.198199i	$0.936491\pi$
$158$	12.2474	+	12.2474i	0.974355	+	0.974355i
$159$	−12.0000			−0.951662
$160$		0			0
$161$		−	24.0000i		−	1.89146i
$162$	−11.0227	+	11.0227i	−0.866025	+	0.866025i
$163$	12.2474	−	12.2474i	0.959294	−	0.959294i	−0.0399091	−	0.999203i	$-0.512707\pi$
$163$	12.2474	−	12.2474i	0.959294	−	0.959294i	0.999203	+	0.0399091i	$0.0127068\pi$
$164$	6.00000			0.468521
$165$		0			0
$166$	−18.0000			−1.39707
$167$	−2.44949	+	2.44949i	−0.189547	+	0.189547i	−0.795500	−	0.605953i	$-0.792792\pi$
$167$	−2.44949	+	2.44949i	−0.189547	+	0.189547i	0.605953	+	0.795500i	$0.292792\pi$
$168$	7.34847	+	7.34847i	0.566947	+	0.566947i
$169$			1.00000i			0.0769231i
$170$		0			0
$171$	6.00000			0.458831
$172$	7.34847	+	7.34847i	0.560316	+	0.560316i
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$			18.0000i			1.36458i
$175$		0			0
$176$		−	5.00000i		−	0.376889i
$177$		0			0
$178$	14.6969	−	14.6969i	1.10158	−	1.10158i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$	−14.6969	+	14.6969i	−1.08941	+	1.08941i
$183$	2.44949	−	2.44949i	0.181071	−	0.181071i
$184$		−	12.0000i		−	0.884652i
$185$		0			0
$186$		−	12.0000i		−	0.879883i
$187$	−4.89898	−	4.89898i	−0.358249	−	0.358249i
$188$	4.89898	+	4.89898i	0.357295	+	0.357295i
$189$		−	18.0000i		−	1.30931i
$190$		0			0
$191$			24.0000i			1.73658i	0.496058	+	0.868290i	$0.334780\pi$
$191$			24.0000i			1.73658i	−0.496058	+	0.868290i	$0.665220\pi$
$192$	1.22474	+	1.22474i	0.0883883	+	0.0883883i
$193$	7.34847	−	7.34847i	0.528954	−	0.528954i	−0.391306	−	0.920261i	$-0.627977\pi$
$193$	7.34847	−	7.34847i	0.528954	−	0.528954i	0.920261	+	0.391306i	$0.127977\pi$
$194$	−24.0000			−1.72310
$195$		0			0
$196$	5.00000			0.357143
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$	−3.67423	+	3.67423i	−0.261116	+	0.261116i
$199$			4.00000i			0.283552i	0.989899	+	0.141776i	$0.0452813\pi$
$199$			4.00000i			0.283552i	−0.989899	+	0.141776i	$0.954719\pi$
$200$		0			0
$201$	6.00000			0.423207
$202$	22.0454	+	22.0454i	1.55111	+	1.55111i
$203$	−14.6969	−	14.6969i	−1.03152	−	1.03152i
$204$	−12.0000			−0.840168
$205$		0			0
$206$			6.00000i			0.418040i
$207$	−14.6969	+	14.6969i	−1.02151	+	1.02151i
$208$	−12.2474	+	12.2474i	−0.849208	+	0.849208i
$209$	2.00000			0.138343
$210$		0			0
$211$	26.0000			1.78991			0.894957	−	0.446153i	$-0.147206\pi$
$211$	26.0000			1.78991			0.894957	+	0.446153i	$0.147206\pi$
$212$	−4.89898	+	4.89898i	−0.336463	+	0.336463i
$213$	−14.6969	−	14.6969i	−1.00702	−	1.00702i
$214$			6.00000i			0.410152i
$215$		0			0
$216$		−	9.00000i		−	0.612372i
$217$	9.79796	+	9.79796i	0.665129	+	0.665129i
$218$	12.2474	+	12.2474i	0.829502	+	0.829502i
$219$		−	6.00000i		−	0.405442i
$220$		0			0
$221$			24.0000i			1.61441i
$222$		0			0
$223$	−17.1464	+	17.1464i	−1.14821	+	1.14821i	−0.161305	+	0.986905i	$0.551570\pi$
$223$	−17.1464	+	17.1464i	−1.14821	+	1.14821i	−0.986905	+	0.161305i	$0.948430\pi$
$224$	18.0000			1.20268
$225$		0			0
$226$	12.0000			0.798228
$227$	−12.2474	+	12.2474i	−0.812892	+	0.812892i	−0.985066	−	0.172175i	$-0.944921\pi$
$227$	−12.2474	+	12.2474i	−0.812892	+	0.812892i	0.172175	+	0.985066i	$0.444921\pi$
$228$	2.44949	−	2.44949i	0.162221	−	0.162221i
$229$		−	10.0000i		−	0.660819i	−0.943838	−	0.330409i	$-0.892813\pi$
$229$		−	10.0000i		−	0.660819i	0.943838	−	0.330409i	$-0.107187\pi$
$230$		0			0
$231$		−	6.00000i		−	0.394771i
$232$	−7.34847	−	7.34847i	−0.482451	−	0.482451i
$233$	4.89898	+	4.89898i	0.320943	+	0.320943i	0.849129	−	0.528186i	$-0.177128\pi$
$233$	4.89898	+	4.89898i	0.320943	+	0.320943i	−0.528186	+	0.849129i	$0.677128\pi$
$234$	18.0000			1.17670
$235$		0			0
$236$		0			0
$237$	12.2474	+	12.2474i	0.795557	+	0.795557i
$238$	29.3939	−	29.3939i	1.90532	−	1.90532i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$	−1.22474	+	1.22474i	−0.0787296	+	0.0787296i
$243$	−11.0227	+	11.0227i	−0.707107	+	0.707107i
$244$		−	2.00000i		−	0.128037i
$245$		0			0
$246$	18.0000			1.14764
$247$	−4.89898	−	4.89898i	−0.311715	−	0.311715i
$248$	4.89898	+	4.89898i	0.311086	+	0.311086i
$249$	−18.0000			−1.14070
$250$		0			0
$251$			12.0000i			0.757433i	0.925513	+	0.378717i	$0.123635\pi$
$251$			12.0000i			0.757433i	−0.925513	+	0.378717i	$0.876365\pi$
$252$	−7.34847	−	7.34847i	−0.462910	−	0.462910i
$253$	−4.89898	+	4.89898i	−0.307996	+	0.307996i
$254$	−6.00000			−0.376473
$255$		0			0
$256$	19.0000			1.18750
$257$	9.79796	−	9.79796i	0.611180	−	0.611180i	−0.332074	−	0.943253i	$-0.607748\pi$
$257$	9.79796	−	9.79796i	0.611180	−	0.611180i	0.943253	+	0.332074i	$0.107748\pi$
$258$	22.0454	+	22.0454i	1.37249	+	1.37249i
$259$		0			0
$260$		0			0
$261$			18.0000i			1.11417i
$262$	−14.6969	−	14.6969i	−0.907980	−	0.907980i
$263$	17.1464	+	17.1464i	1.05729	+	1.05729i	0.998256	+	0.0590383i	$0.0188034\pi$
$263$	17.1464	+	17.1464i	1.05729	+	1.05729i	0.0590383	+	0.998256i	$0.481197\pi$
$264$		−	3.00000i		−	0.184637i
$265$		0			0
$266$			12.0000i			0.735767i
$267$	14.6969	−	14.6969i	0.899438	−	0.899438i
$268$	2.44949	−	2.44949i	0.149626	−	0.149626i
$269$	−24.0000			−1.46331			−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000			−1.46331			−0.731653	+	0.681677i	$0.761251\pi$
$270$		0			0
$271$	2.00000			0.121491			0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000			0.121491			0.0607457	+	0.998153i	$0.480652\pi$
$272$	24.4949	−	24.4949i	1.48522	−	1.48522i
$273$	−14.6969	+	14.6969i	−0.889499	+	0.889499i
$274$			24.0000i			1.44989i
$275$		0			0
$276$			12.0000i			0.722315i
$277$	−12.2474	−	12.2474i	−0.735878	−	0.735878i	0.235900	−	0.971777i	$-0.424196\pi$
$277$	−12.2474	−	12.2474i	−0.735878	−	0.735878i	−0.971777	+	0.235900i	$0.924196\pi$
$278$	−26.9444	−	26.9444i	−1.61602	−	1.61602i
$279$		−	12.0000i		−	0.718421i
$280$		0			0
$281$		−	30.0000i		−	1.78965i	−0.446417	−	0.894825i	$-0.647300\pi$
$281$		−	30.0000i		−	1.78965i	0.446417	−	0.894825i	$-0.352700\pi$
$282$	14.6969	+	14.6969i	0.875190	+	0.875190i
$283$	12.2474	−	12.2474i	0.728035	−	0.728035i	−0.242193	−	0.970228i	$-0.577867\pi$
$283$	12.2474	−	12.2474i	0.728035	−	0.728035i	0.970228	+	0.242193i	$0.0778667\pi$
$284$	−12.0000			−0.712069
$285$		0			0
$286$	6.00000			0.354787
$287$	−14.6969	+	14.6969i	−0.867533	+	0.867533i
$288$	−11.0227	−	11.0227i	−0.649519	−	0.649519i
$289$		−	31.0000i		−	1.82353i
$290$		0			0
$291$	−24.0000			−1.40690
$292$	−2.44949	−	2.44949i	−0.143346	−	0.143346i
$293$	−9.79796	−	9.79796i	−0.572403	−	0.572403i	0.360396	−	0.932799i	$-0.382641\pi$
$293$	−9.79796	−	9.79796i	−0.572403	−	0.572403i	−0.932799	+	0.360396i	$0.882641\pi$
$294$	15.0000			0.874818
$295$		0			0
$296$		0			0
$297$	−3.67423	+	3.67423i	−0.213201	+	0.213201i
$298$	7.34847	−	7.34847i	0.425685	−	0.425685i
$299$	24.0000			1.38796
$300$		0			0
$301$	−36.0000			−2.07501
$302$	12.2474	−	12.2474i	0.704761	−	0.704761i
$303$	22.0454	+	22.0454i	1.26648	+	1.26648i
$304$			10.0000i			0.573539i
$305$		0			0
$306$	−36.0000			−2.05798
$307$	−17.1464	−	17.1464i	−0.978598	−	0.978598i	0.0211774	−	0.999776i	$-0.493259\pi$
$307$	−17.1464	−	17.1464i	−0.978598	−	0.978598i	−0.999776	+	0.0211774i	$0.993259\pi$
$308$	−2.44949	−	2.44949i	−0.139573	−	0.139573i
$309$			6.00000i			0.341328i
$310$		0			0
$311$			24.0000i			1.36092i	0.732787	+	0.680458i	$0.238219\pi$
$311$			24.0000i			1.36092i	−0.732787	+	0.680458i	$0.761781\pi$
$312$	−7.34847	+	7.34847i	−0.416025	+	0.416025i
$313$	−4.89898	+	4.89898i	−0.276907	+	0.276907i	−0.831873	−	0.554966i	$-0.812731\pi$
$313$	−4.89898	+	4.89898i	−0.276907	+	0.276907i	0.554966	+	0.831873i	$0.312731\pi$
$314$	−24.0000			−1.35440
$315$		0			0
$316$	10.0000			0.562544
$317$	4.89898	−	4.89898i	0.275154	−	0.275154i	−0.556017	−	0.831171i	$-0.687671\pi$
$317$	4.89898	−	4.89898i	0.275154	−	0.275154i	0.831171	+	0.556017i	$0.187671\pi$
$318$	−14.6969	+	14.6969i	−0.824163	+	0.824163i
$319$			6.00000i			0.335936i
$320$		0			0
$321$			6.00000i			0.334887i
$322$	−29.3939	−	29.3939i	−1.63806	−	1.63806i
$323$	9.79796	+	9.79796i	0.545173	+	0.545173i
$324$			9.00000i			0.500000i
$325$		0			0
$326$		−	30.0000i		−	1.66155i
$327$	12.2474	+	12.2474i	0.677285	+	0.677285i
$328$	−7.34847	+	7.34847i	−0.405751	+	0.405751i
$329$	−24.0000			−1.32316
$330$		0			0
$331$	−28.0000			−1.53902			−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000			−1.53902			−0.769510	+	0.638635i	$0.779499\pi$
$332$	−7.34847	+	7.34847i	−0.403300	+	0.403300i
$333$		0			0
$334$			6.00000i			0.328305i
$335$		0			0
$336$	30.0000			1.63663
$337$	22.0454	+	22.0454i	1.20089	+	1.20089i	0.973896	+	0.226994i	$0.0728897\pi$
$337$	22.0454	+	22.0454i	1.20089	+	1.20089i	0.226994	+	0.973896i	$0.427110\pi$
$338$	1.22474	+	1.22474i	0.0666173	+	0.0666173i
$339$	12.0000			0.651751
$340$		0			0
$341$		−	4.00000i		−	0.216612i
$342$	7.34847	−	7.34847i	0.397360	−	0.397360i
$343$	4.89898	−	4.89898i	0.264520	−	0.264520i
$344$	−18.0000			−0.970495
$345$		0			0
$346$		0			0
$347$	−12.2474	+	12.2474i	−0.657477	+	0.657477i	−0.954783	−	0.297305i	$-0.903912\pi$
$347$	−12.2474	+	12.2474i	−0.657477	+	0.657477i	0.297305	+	0.954783i	$0.403912\pi$
$348$	7.34847	+	7.34847i	0.393919	+	0.393919i
$349$			2.00000i			0.107058i	0.998566	+	0.0535288i	$0.0170469\pi$
$349$			2.00000i			0.107058i	−0.998566	+	0.0535288i	$0.982953\pi$
$350$		0			0
$351$	18.0000			0.960769
$352$	−3.67423	−	3.67423i	−0.195837	−	0.195837i
$353$	−24.4949	−	24.4949i	−1.30373	−	1.30373i	−0.925856	−	0.377875i	$-0.876655\pi$
$353$	−24.4949	−	24.4949i	−1.30373	−	1.30373i	−0.377875	−	0.925856i	$-0.623345\pi$
$354$		0			0
$355$		0			0
$356$		−	12.0000i		−	0.635999i
$357$	29.3939	−	29.3939i	1.55569	−	1.55569i
$358$		0			0
$359$	24.0000			1.26667			0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000			1.26667			0.633336	+	0.773877i	$0.281685\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$	12.2474	−	12.2474i	0.643712	−	0.643712i
$363$	−1.22474	+	1.22474i	−0.0642824	+	0.0642824i
$364$			12.0000i			0.628971i
$365$		0			0
$366$		−	6.00000i		−	0.313625i
$367$	2.44949	+	2.44949i	0.127862	+	0.127862i	0.768142	−	0.640280i	$-0.221182\pi$
$367$	2.44949	+	2.44949i	0.127862	+	0.127862i	−0.640280	+	0.768142i	$0.721182\pi$
$368$	−24.4949	−	24.4949i	−1.27688	−	1.27688i
$369$	18.0000			0.937043
$370$		0			0
$371$		−	24.0000i		−	1.24602i
$372$	−4.89898	−	4.89898i	−0.254000	−	0.254000i
$373$	17.1464	−	17.1464i	0.887808	−	0.887808i	−0.106504	−	0.994312i	$-0.533966\pi$
$373$	17.1464	−	17.1464i	0.887808	−	0.887808i	0.994312	+	0.106504i	$0.0339657\pi$
$374$	−12.0000			−0.620505
$375$		0			0
$376$	−12.0000			−0.618853
$377$	14.6969	−	14.6969i	0.756931	−	0.756931i
$378$	−22.0454	−	22.0454i	−1.13389	−	1.13389i
$379$			20.0000i			1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$			20.0000i			1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$		0			0
$381$	−6.00000			−0.307389
$382$	29.3939	+	29.3939i	1.50392	+	1.50392i
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$	21.0000			1.07165
$385$		0			0
$386$		−	18.0000i		−	0.916176i
$387$	22.0454	+	22.0454i	1.12063	+	1.12063i
$388$	−9.79796	+	9.79796i	−0.497416	+	0.497416i
$389$	12.0000			0.608424			0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000			0.608424			0.304212	+	0.952604i	$0.401607\pi$
$390$		0			0
$391$	−48.0000			−2.42746
$392$	−6.12372	+	6.12372i	−0.309295	+	0.309295i
$393$	−14.6969	−	14.6969i	−0.741362	−	0.741362i
$394$		0			0
$395$		0			0
$396$			3.00000i			0.150756i
$397$	4.89898	+	4.89898i	0.245873	+	0.245873i	0.819274	−	0.573402i	$-0.194377\pi$
$397$	4.89898	+	4.89898i	0.245873	+	0.245873i	−0.573402	+	0.819274i	$0.694377\pi$
$398$	4.89898	+	4.89898i	0.245564	+	0.245564i
$399$			12.0000i			0.600751i
$400$		0			0
$401$		−	24.0000i		−	1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$		−	24.0000i		−	1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$	7.34847	−	7.34847i	0.366508	−	0.366508i
$403$	−9.79796	+	9.79796i	−0.488071	+	0.488071i
$404$	18.0000			0.895533
$405$		0			0
$406$	−36.0000			−1.78665
$407$		0			0
$408$	14.6969	−	14.6969i	0.727607	−	0.727607i
$409$		−	10.0000i		−	0.494468i	−0.968956	−	0.247234i	$-0.920478\pi$
$409$		−	10.0000i		−	0.494468i	0.968956	−	0.247234i	$-0.0795217\pi$
$410$		0			0
$411$			24.0000i			1.18383i
$412$	2.44949	+	2.44949i	0.120678	+	0.120678i
$413$		0			0
$414$			36.0000i			1.76930i
$415$		0			0
$416$			18.0000i			0.882523i
$417$	−26.9444	−	26.9444i	−1.31947	−	1.31947i
$418$	2.44949	−	2.44949i	0.119808	−	0.119808i
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$	31.8434	−	31.8434i	1.55011	−	1.55011i
$423$	14.6969	+	14.6969i	0.714590	+	0.714590i
$424$		−	12.0000i		−	0.582772i
$425$		0			0
$426$	−36.0000			−1.74421
$427$	4.89898	+	4.89898i	0.237078	+	0.237078i
$428$	2.44949	+	2.44949i	0.118401	+	0.118401i
$429$	6.00000			0.289683
$430$		0			0
$431$		−	24.0000i		−	1.15604i	−0.816023	−	0.578020i	$-0.803826\pi$
$431$		−	24.0000i		−	1.15604i	0.816023	−	0.578020i	$-0.196174\pi$
$432$	−18.3712	−	18.3712i	−0.883883	−	0.883883i
$433$	19.5959	−	19.5959i	0.941720	−	0.941720i	−0.0566731	−	0.998393i	$-0.518049\pi$
$433$	19.5959	−	19.5959i	0.941720	−	0.941720i	0.998393	+	0.0566731i	$0.0180493\pi$
$434$	24.0000			1.15204
$435$		0			0
$436$	10.0000			0.478913
$437$	9.79796	−	9.79796i	0.468700	−	0.468700i
$438$	−7.34847	−	7.34847i	−0.351123	−	0.351123i
$439$		−	26.0000i		−	1.24091i	−0.784241	−	0.620456i	$-0.786947\pi$
$439$		−	26.0000i		−	1.24091i	0.784241	−	0.620456i	$-0.213053\pi$
$440$		0			0
$441$	15.0000			0.714286
$442$	29.3939	+	29.3939i	1.39812	+	1.39812i
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$		0			0
$446$			42.0000i			1.98876i
$447$	7.34847	−	7.34847i	0.347571	−	0.347571i
$448$	−2.44949	+	2.44949i	−0.115728	+	0.115728i
$449$	12.0000			0.566315			0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000			0.566315			0.283158	+	0.959073i	$0.408618\pi$
$450$		0			0
$451$	6.00000			0.282529
$452$	4.89898	−	4.89898i	0.230429	−	0.230429i
$453$	12.2474	−	12.2474i	0.575435	−	0.575435i
$454$			30.0000i			1.40797i
$455$		0			0
$456$			6.00000i			0.280976i
$457$	−22.0454	−	22.0454i	−1.03124	−	1.03124i	−0.999496	−	0.0317447i	$-0.989894\pi$
$457$	−22.0454	−	22.0454i	−1.03124	−	1.03124i	−0.0317447	−	0.999496i	$-0.510106\pi$
$458$	−12.2474	−	12.2474i	−0.572286	−	0.572286i
$459$	−36.0000			−1.68034
$460$		0			0
$461$		−	6.00000i		−	0.279448i	−0.990190	−	0.139724i	$-0.955378\pi$
$461$		−	6.00000i		−	0.279448i	0.990190	−	0.139724i	$-0.0446215\pi$
$462$	−7.34847	−	7.34847i	−0.341882	−	0.341882i
$463$	−2.44949	+	2.44949i	−0.113837	+	0.113837i	−0.761731	−	0.647893i	$-0.775650\pi$
$463$	−2.44949	+	2.44949i	−0.113837	+	0.113837i	0.647893	+	0.761731i	$0.275650\pi$
$464$	−30.0000			−1.39272
$465$		0			0
$466$	12.0000			0.555889
$467$	−24.4949	+	24.4949i	−1.13349	+	1.13349i	−0.143896	+	0.989593i	$0.545963\pi$
$467$	−24.4949	+	24.4949i	−1.13349	+	1.13349i	−0.989593	+	0.143896i	$0.954037\pi$
$468$	7.34847	−	7.34847i	0.339683	−	0.339683i
$469$			12.0000i			0.554109i
$470$		0			0
$471$	−24.0000			−1.10586
$472$		0			0
$473$	7.34847	+	7.34847i	0.337883	+	0.337883i
$474$	30.0000			1.37795
$475$		0			0
$476$		−	24.0000i		−	1.10004i
$477$	−14.6969	+	14.6969i	−0.672927	+	0.672927i
$478$		0			0
$479$	24.0000			1.09659			0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000			1.09659			0.548294	+	0.836286i	$0.315277\pi$
$480$		0			0
$481$		0			0
$482$	−12.2474	+	12.2474i	−0.557856	+	0.557856i
$483$	−29.3939	−	29.3939i	−1.33747	−	1.33747i
$484$			1.00000i			0.0454545i
$485$		0			0
$486$			27.0000i			1.22474i
$487$	−17.1464	−	17.1464i	−0.776979	−	0.776979i	0.202337	−	0.979316i	$-0.435146\pi$
$487$	−17.1464	−	17.1464i	−0.776979	−	0.776979i	−0.979316	+	0.202337i	$0.935146\pi$
$488$	2.44949	+	2.44949i	0.110883	+	0.110883i
$489$		−	30.0000i		−	1.35665i
$490$		0			0
$491$			12.0000i			0.541552i	0.962642	+	0.270776i	$0.0872803\pi$
$491$			12.0000i			0.541552i	−0.962642	+	0.270776i	$0.912720\pi$
$492$	7.34847	−	7.34847i	0.331295	−	0.331295i
$493$	−29.3939	+	29.3939i	−1.32383	+	1.32383i
$494$	−12.0000			−0.539906
$495$		0			0
$496$	20.0000			0.898027
$497$	29.3939	−	29.3939i	1.31850	−	1.31850i
$498$	−22.0454	+	22.0454i	−0.987878	+	0.987878i
$499$			16.0000i			0.716258i	0.933672	+	0.358129i	$0.116585\pi$
$499$			16.0000i			0.716258i	−0.933672	+	0.358129i	$0.883415\pi$
$500$		0			0
$501$			6.00000i			0.268060i
$502$	14.6969	+	14.6969i	0.655956	+	0.655956i
$503$	17.1464	+	17.1464i	0.764521	+	0.764521i	0.977136	−	0.212615i	$-0.0681979\pi$
$503$	17.1464	+	17.1464i	0.764521	+	0.764521i	−0.212615	+	0.977136i	$0.568198\pi$
$504$	18.0000			0.801784
$505$		0			0
$506$			12.0000i			0.533465i
$507$	1.22474	+	1.22474i	0.0543928	+	0.0543928i
$508$	−2.44949	+	2.44949i	−0.108679	+	0.108679i
$509$	−24.0000			−1.06378			−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000			−1.06378			−0.531891	+	0.846813i	$0.678518\pi$
$510$		0			0
$511$	12.0000			0.530849
$512$	6.12372	−	6.12372i	0.270633	−	0.270633i
$513$	7.34847	−	7.34847i	0.324443	−	0.324443i
$514$		−	24.0000i		−	1.05859i
$515$		0			0
$516$	18.0000			0.792406
$517$	4.89898	+	4.89898i	0.215457	+	0.215457i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	24.0000i		−	1.05146i	−0.850652	−	0.525730i	$-0.823792\pi$
$521$		−	24.0000i		−	1.05146i	0.850652	−	0.525730i	$-0.176208\pi$
$522$	22.0454	+	22.0454i	0.964901	+	0.964901i
$523$	7.34847	−	7.34847i	0.321326	−	0.321326i	−0.527950	−	0.849276i	$-0.677039\pi$
$523$	7.34847	−	7.34847i	0.321326	−	0.321326i	0.849276	+	0.527950i	$0.177039\pi$
$524$	−12.0000			−0.524222
$525$		0			0
$526$	42.0000			1.83129
$527$	19.5959	−	19.5959i	0.853612	−	0.853612i
$528$	−6.12372	−	6.12372i	−0.266501	−	0.266501i
$529$			25.0000i			1.08696i
$530$		0			0
$531$		0			0
$532$	4.89898	+	4.89898i	0.212398	+	0.212398i
$533$	−14.6969	−	14.6969i	−0.636595	−	0.636595i
$534$		−	36.0000i		−	1.55787i
$535$		0			0
$536$			6.00000i			0.259161i
$537$		0			0
$538$	−29.3939	+	29.3939i	−1.26726	+	1.26726i
$539$	5.00000			0.215365
$540$		0			0
$541$	−10.0000			−0.429934			−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000			−0.429934			−0.214967	+	0.976621i	$0.568964\pi$
$542$	2.44949	−	2.44949i	0.105215	−	0.105215i
$543$	12.2474	−	12.2474i	0.525588	−	0.525588i
$544$		−	36.0000i		−	1.54349i
$545$		0			0
$546$			36.0000i			1.54066i
$547$	−12.2474	−	12.2474i	−0.523663	−	0.523663i	0.395013	−	0.918676i	$-0.370740\pi$
$547$	−12.2474	−	12.2474i	−0.523663	−	0.523663i	−0.918676	+	0.395013i	$0.870740\pi$
$548$	9.79796	+	9.79796i	0.418548	+	0.418548i
$549$		−	6.00000i		−	0.256074i
$550$		0			0
$551$		−	12.0000i		−	0.511217i
$552$	−14.6969	−	14.6969i	−0.625543	−	0.625543i
$553$	−24.4949	+	24.4949i	−1.04163	+	1.04163i
$554$	−30.0000			−1.27458
$555$		0			0
$556$	−22.0000			−0.933008
$557$	19.5959	−	19.5959i	0.830306	−	0.830306i	−0.157253	−	0.987558i	$-0.550264\pi$
$557$	19.5959	−	19.5959i	0.830306	−	0.830306i	0.987558	+	0.157253i	$0.0502637\pi$
$558$	−14.6969	−	14.6969i	−0.622171	−	0.622171i
$559$		−	36.0000i		−	1.52264i
$560$		0			0
$561$	−12.0000			−0.506640
$562$	−36.7423	−	36.7423i	−1.54988	−	1.54988i
$563$	−7.34847	−	7.34847i	−0.309701	−	0.309701i	0.535092	−	0.844794i	$-0.320277\pi$
$563$	−7.34847	−	7.34847i	−0.309701	−	0.309701i	−0.844794	+	0.535092i	$0.820277\pi$
$564$	12.0000			0.505291
$565$		0			0
$566$		−	30.0000i		−	1.26099i
$567$	−22.0454	−	22.0454i	−0.925820	−	0.925820i
$568$	14.6969	−	14.6969i	0.616670	−	0.616670i
$569$	−18.0000			−0.754599			−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000			−0.754599			−0.377300	+	0.926091i	$0.623147\pi$
$570$		0			0
$571$	14.0000			0.585882			0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000			0.585882			0.292941	+	0.956131i	$0.405366\pi$
$572$	2.44949	−	2.44949i	0.102418	−	0.102418i
$573$	29.3939	+	29.3939i	1.22795	+	1.22795i
$574$			36.0000i			1.50261i
$575$		0			0
$576$	3.00000			0.125000
$577$	14.6969	+	14.6969i	0.611842	+	0.611842i	0.943426	−	0.331584i	$-0.107583\pi$
$577$	14.6969	+	14.6969i	0.611842	+	0.611842i	−0.331584	+	0.943426i	$0.607583\pi$
$578$	−37.9671	−	37.9671i	−1.57922	−	1.57922i
$579$		−	18.0000i		−	0.748054i
$580$		0			0
$581$		−	36.0000i		−	1.49353i
$582$	−29.3939	+	29.3939i	−1.21842	+	1.21842i
$583$	−4.89898	+	4.89898i	−0.202895	+	0.202895i
$584$	6.00000			0.248282
$585$		0			0
$586$	−24.0000			−0.991431
$587$	−4.89898	+	4.89898i	−0.202203	+	0.202203i	−0.800943	−	0.598741i	$-0.795668\pi$
$587$	−4.89898	+	4.89898i	−0.202203	+	0.202203i	0.598741	+	0.800943i	$0.295668\pi$
$588$	6.12372	−	6.12372i	0.252538	−	0.252538i
$589$			8.00000i			0.329634i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.6969	+	14.6969i	0.603531	+	0.603531i	0.941248	−	0.337717i	$-0.109655\pi$
$593$	14.6969	+	14.6969i	0.603531	+	0.603531i	−0.337717	+	0.941248i	$0.609655\pi$
$594$			9.00000i			0.369274i
$595$		0			0
$596$		−	6.00000i		−	0.245770i
$597$	4.89898	+	4.89898i	0.200502	+	0.200502i
$598$	29.3939	−	29.3939i	1.20201	−	1.20201i
$599$	−12.0000			−0.490307			−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000			−0.490307			−0.245153	+	0.969484i	$0.578838\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$	−44.0908	+	44.0908i	−1.79701	+	1.79701i
$603$	7.34847	−	7.34847i	0.299253	−	0.299253i
$604$		−	10.0000i		−	0.406894i
$605$		0			0
$606$	54.0000			2.19360
$607$	12.2474	+	12.2474i	0.497109	+	0.497109i	0.910537	−	0.413428i	$-0.135669\pi$
$607$	12.2474	+	12.2474i	0.497109	+	0.497109i	−0.413428	+	0.910537i	$0.635669\pi$
$608$	7.34847	+	7.34847i	0.298020	+	0.298020i
$609$	−36.0000			−1.45879
$610$		0			0
$611$		−	24.0000i		−	0.970936i
$612$	−14.6969	+	14.6969i	−0.594089	+	0.594089i
$613$	−2.44949	+	2.44949i	−0.0989340	+	0.0989340i	−0.754841	−	0.655907i	$-0.772286\pi$
$613$	−2.44949	+	2.44949i	−0.0989340	+	0.0989340i	0.655907	+	0.754841i	$0.272286\pi$
$614$	−42.0000			−1.69498
$615$		0			0
$616$	6.00000			0.241747
$617$	14.6969	−	14.6969i	0.591676	−	0.591676i	−0.346408	−	0.938084i	$-0.612599\pi$
$617$	14.6969	−	14.6969i	0.591676	−	0.591676i	0.938084	+	0.346408i	$0.112599\pi$
$618$	7.34847	+	7.34847i	0.295599	+	0.295599i
$619$			4.00000i			0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$			4.00000i			0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$		0			0
$621$			36.0000i			1.44463i
$622$	29.3939	+	29.3939i	1.17859	+	1.17859i
$623$	29.3939	+	29.3939i	1.17764	+	1.17764i
$624$			30.0000i			1.20096i
$625$		0			0
$626$			12.0000i			0.479616i
$627$	2.44949	−	2.44949i	0.0978232	−	0.0978232i
$628$	−9.79796	+	9.79796i	−0.390981	+	0.390981i
$629$		0			0
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$	−12.2474	+	12.2474i	−0.487177	+	0.487177i
$633$	31.8434	−	31.8434i	1.26566	−	1.26566i
$634$		−	12.0000i		−	0.476581i
$635$		0			0
$636$			12.0000i			0.475831i
$637$	−12.2474	−	12.2474i	−0.485262	−	0.485262i
$638$	7.34847	+	7.34847i	0.290929	+	0.290929i
$639$	−36.0000			−1.42414
$640$		0			0
$641$		−	12.0000i		−	0.473972i	−0.971513	−	0.236986i	$-0.923841\pi$
$641$		−	12.0000i		−	0.473972i	0.971513	−	0.236986i	$-0.0761595\pi$
$642$	7.34847	+	7.34847i	0.290021	+	0.290021i
$643$	12.2474	−	12.2474i	0.482992	−	0.482992i	−0.423094	−	0.906086i	$-0.639056\pi$
$643$	12.2474	−	12.2474i	0.482992	−	0.482992i	0.906086	+	0.423094i	$0.139056\pi$
$644$	−24.0000			−0.945732
$645$		0			0
$646$	24.0000			0.944267
$647$	19.5959	−	19.5959i	0.770395	−	0.770395i	−0.207780	−	0.978175i	$-0.566624\pi$
$647$	19.5959	−	19.5959i	0.770395	−	0.770395i	0.978175	+	0.207780i	$0.0666240\pi$
$648$	−11.0227	−	11.0227i	−0.433013	−	0.433013i
$649$		0			0
$650$		0			0
$651$	24.0000			0.940634
$652$	−12.2474	−	12.2474i	−0.479647	−	0.479647i
$653$	29.3939	+	29.3939i	1.15027	+	1.15027i	0.986498	+	0.163773i	$0.0523666\pi$
$653$	29.3939	+	29.3939i	1.15027	+	1.15027i	0.163773	+	0.986498i	$0.447633\pi$
$654$	30.0000			1.17309
$655$		0			0
$656$			30.0000i			1.17130i
$657$	−7.34847	−	7.34847i	−0.286691	−	0.286691i
$658$	−29.3939	+	29.3939i	−1.14589	+	1.14589i
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$	34.0000			1.32245			0.661223	−	0.750189i	$-0.270038\pi$
$661$	34.0000			1.32245			0.661223	+	0.750189i	$0.270038\pi$
$662$	−34.2929	+	34.2929i	−1.33283	+	1.33283i
$663$	29.3939	+	29.3939i	1.14156	+	1.14156i
$664$		−	18.0000i		−	0.698535i
$665$		0			0
$666$		0			0
$667$	29.3939	+	29.3939i	1.13814	+	1.13814i
$668$	2.44949	+	2.44949i	0.0947736	+	0.0947736i
$669$			42.0000i			1.62381i
$670$		0			0
$671$		−	2.00000i		−	0.0772091i
$672$	22.0454	−	22.0454i	0.850420	−	0.850420i
$673$	2.44949	−	2.44949i	0.0944209	−	0.0944209i	−0.658319	−	0.752739i	$-0.728732\pi$
$673$	2.44949	−	2.44949i	0.0944209	−	0.0944209i	0.752739	+	0.658319i	$0.228732\pi$
$674$	54.0000			2.08000
$675$		0			0
$676$	1.00000			0.0384615
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$	14.6969	−	14.6969i	0.564433	−	0.564433i
$679$		−	48.0000i		−	1.84207i
$680$		0			0
$681$			30.0000i			1.14960i
$682$	−4.89898	−	4.89898i	−0.187592	−	0.187592i
$683$	14.6969	+	14.6969i	0.562363	+	0.562363i	0.929978	−	0.367615i	$-0.119826\pi$
$683$	14.6969	+	14.6969i	0.562363	+	0.562363i	−0.367615	+	0.929978i	$0.619826\pi$
$684$		−	6.00000i		−	0.229416i
$685$		0			0
$686$		−	12.0000i		−	0.458162i
$687$	−12.2474	−	12.2474i	−0.467269	−	0.467269i
$688$	−36.7423	+	36.7423i	−1.40079	+	1.40079i
$689$	24.0000			0.914327
$690$		0			0
$691$	28.0000			1.06517			0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000			1.06517			0.532585	+	0.846376i	$0.321221\pi$
$692$		0			0
$693$	−7.34847	−	7.34847i	−0.279145	−	0.279145i
$694$			30.0000i			1.13878i
$695$		0			0
$696$	−18.0000			−0.682288
$697$	29.3939	+	29.3939i	1.11337	+	1.11337i
$698$	2.44949	+	2.44949i	0.0927146	+	0.0927146i
$699$	12.0000			0.453882
$700$		0			0
$701$			18.0000i			0.679851i	0.940452	+	0.339925i	$0.110402\pi$
$701$			18.0000i			0.679851i	−0.940452	+	0.339925i	$0.889598\pi$
$702$	22.0454	−	22.0454i	0.832050	−	0.832050i
$703$		0			0
$704$	1.00000			0.0376889
$705$		0			0
$706$	−60.0000			−2.25813
$707$	−44.0908	+	44.0908i	−1.65821	+	1.65821i
$708$		0			0
$709$		−	46.0000i		−	1.72757i	−0.503864	−	0.863783i	$-0.668089\pi$
$709$		−	46.0000i		−	1.72757i	0.503864	−	0.863783i	$-0.331911\pi$
$710$		0			0
$711$	30.0000			1.12509
$712$	14.6969	+	14.6969i	0.550791	+	0.550791i
$713$	−19.5959	−	19.5959i	−0.733873	−	0.733873i
$714$		−	72.0000i		−	2.69453i
$715$		0			0
$716$		0			0
$717$		0			0
$718$	29.3939	−	29.3939i	1.09697	−	1.09697i
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$	−12.0000			−0.446903
$722$	18.3712	−	18.3712i	0.683704	−	0.683704i
$723$	−12.2474	+	12.2474i	−0.455488	+	0.455488i
$724$		−	10.0000i		−	0.371647i
$725$		0			0
$726$			3.00000i			0.111340i
$727$	22.0454	+	22.0454i	0.817619	+	0.817619i	0.985763	−	0.168144i	$-0.0537772\pi$
$727$	22.0454	+	22.0454i	0.817619	+	0.817619i	−0.168144	+	0.985763i	$0.553777\pi$
$728$	−14.6969	−	14.6969i	−0.544705	−	0.544705i
$729$			27.0000i			1.00000i
$730$		0			0
$731$			72.0000i			2.66302i
$732$	−2.44949	−	2.44949i	−0.0905357	−	0.0905357i
$733$	−22.0454	+	22.0454i	−0.814266	+	0.814266i	−0.985270	−	0.171005i	$-0.945299\pi$
$733$	−22.0454	+	22.0454i	−0.814266	+	0.814266i	0.171005	+	0.985270i	$0.445299\pi$
$734$	6.00000			0.221464
$735$		0			0
$736$	−36.0000			−1.32698
$737$	2.44949	−	2.44949i	0.0902281	−	0.0902281i
$738$	22.0454	−	22.0454i	0.811503	−	0.811503i
$739$			38.0000i			1.39785i	0.715194	+	0.698926i	$0.246338\pi$
$739$			38.0000i			1.39785i	−0.715194	+	0.698926i	$0.753662\pi$
$740$		0			0
$741$	−12.0000			−0.440831
$742$	−29.3939	−	29.3939i	−1.07908	−	1.07908i
$743$	−12.2474	−	12.2474i	−0.449315	−	0.449315i	0.445812	−	0.895127i	$-0.352915\pi$
$743$	−12.2474	−	12.2474i	−0.449315	−	0.449315i	−0.895127	+	0.445812i	$0.852915\pi$
$744$	12.0000			0.439941
$745$		0			0
$746$		−	42.0000i		−	1.53773i
$747$	−22.0454	+	22.0454i	−0.806599	+	0.806599i
$748$	−4.89898	+	4.89898i	−0.179124	+	0.179124i
$749$	−12.0000			−0.438470
$750$		0			0
$751$	4.00000			0.145962			0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000			0.145962			0.0729810	+	0.997333i	$0.476749\pi$
$752$	−24.4949	+	24.4949i	−0.893237	+	0.893237i
$753$	14.6969	+	14.6969i	0.535586	+	0.535586i
$754$		−	36.0000i		−	1.31104i
$755$		0			0
$756$	−18.0000			−0.654654
$757$	9.79796	+	9.79796i	0.356113	+	0.356113i	0.862378	−	0.506265i	$-0.168974\pi$
$757$	9.79796	+	9.79796i	0.356113	+	0.356113i	−0.506265	+	0.862378i	$0.668974\pi$
$758$	24.4949	+	24.4949i	0.889695	+	0.889695i
$759$			12.0000i			0.435572i
$760$		0			0
$761$		−	42.0000i		−	1.52250i	−0.648459	−	0.761249i	$-0.724586\pi$
$761$		−	42.0000i		−	1.52250i	0.648459	−	0.761249i	$-0.275414\pi$
$762$	−7.34847	+	7.34847i	−0.266207	+	0.266207i
$763$	−24.4949	+	24.4949i	−0.886775	+	0.886775i
$764$	24.0000			0.868290
$765$		0			0
$766$		0			0
$767$		0			0
$768$	23.2702	−	23.2702i	0.839689	−	0.839689i
$769$			22.0000i			0.793340i	0.917961	+	0.396670i	$0.129834\pi$
$769$			22.0000i			0.793340i	−0.917961	+	0.396670i	$0.870166\pi$
$770$		0			0
$771$		−	24.0000i		−	0.864339i
$772$	−7.34847	−	7.34847i	−0.264477	−	0.264477i
$773$	−19.5959	−	19.5959i	−0.704816	−	0.704816i	0.260624	−	0.965440i	$-0.416072\pi$
$773$	−19.5959	−	19.5959i	−0.704816	−	0.704816i	−0.965440	+	0.260624i	$0.916072\pi$
$774$	54.0000			1.94099
$775$		0			0
$776$		−	24.0000i		−	0.861550i
$777$		0			0
$778$	14.6969	−	14.6969i	0.526911	−	0.526911i
$779$	−12.0000			−0.429945
$780$		0			0
$781$	−12.0000			−0.429394
$782$	−58.7878	+	58.7878i	−2.10225	+	2.10225i
$783$	22.0454	+	22.0454i	0.787839	+	0.787839i
$784$			25.0000i			0.892857i
$785$		0			0
$786$	−36.0000			−1.28408
$787$	−2.44949	−	2.44949i	−0.0873149	−	0.0873149i	0.662100	−	0.749415i	$-0.269665\pi$
$787$	−2.44949	−	2.44949i	−0.0873149	−	0.0873149i	−0.749415	+	0.662100i	$0.769665\pi$
$788$		0			0
$789$	42.0000			1.49524
$790$		0			0
$791$			24.0000i			0.853342i
$792$	−3.67423	−	3.67423i	−0.130558	−	0.130558i
$793$	−4.89898	+	4.89898i	−0.173968	+	0.173968i
$794$	12.0000			0.425864
$795$		0			0
$796$	4.00000			0.141776
$797$	−14.6969	+	14.6969i	−0.520592	+	0.520592i	−0.917750	−	0.397158i	$-0.869996\pi$
$797$	−14.6969	+	14.6969i	−0.520592	+	0.520592i	0.397158	+	0.917750i	$0.369996\pi$
$798$	14.6969	+	14.6969i	0.520266	+	0.520266i
$799$			48.0000i			1.69812i
$800$		0			0
$801$		−	36.0000i		−	1.27200i
$802$	−29.3939	−	29.3939i	−1.03793	−	1.03793i
$803$	−2.44949	−	2.44949i	−0.0864406	−	0.0864406i
$804$		−	6.00000i		−	0.211604i
$805$		0			0
$806$			24.0000i			0.845364i
$807$	−29.3939	+	29.3939i	−1.03471	+	1.03471i
$808$	−22.0454	+	22.0454i	−0.775555	+	0.775555i
$809$	42.0000			1.47664			0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000			1.47664			0.738321	+	0.674450i	$0.235619\pi$
$810$		0			0
$811$	−2.00000			−0.0702295			−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000			−0.0702295			−0.0351147	+	0.999383i	$0.511180\pi$
$812$	−14.6969	+	14.6969i	−0.515761	+	0.515761i
$813$	2.44949	−	2.44949i	0.0859074	−	0.0859074i
$814$		0			0
$815$		0			0
$816$		−	60.0000i		−	2.10042i
$817$	−14.6969	−	14.6969i	−0.514181	−	0.514181i
$818$	−12.2474	−	12.2474i	−0.428222	−	0.428222i
$819$			36.0000i			1.25794i
$820$		0			0
$821$			18.0000i			0.628204i	0.949389	+	0.314102i	$0.101703\pi$
$821$			18.0000i			0.628204i	−0.949389	+	0.314102i	$0.898297\pi$
$822$	29.3939	+	29.3939i	1.02523	+	1.02523i
$823$	22.0454	−	22.0454i	0.768455	−	0.768455i	−0.209380	−	0.977834i	$-0.567144\pi$
$823$	22.0454	−	22.0454i	0.768455	−	0.768455i	0.977834	+	0.209380i	$0.0671445\pi$
$824$	−6.00000			−0.209020
$825$		0			0
$826$		0			0
$827$	−7.34847	+	7.34847i	−0.255531	+	0.255531i	−0.823234	−	0.567702i	$-0.807832\pi$
$827$	−7.34847	+	7.34847i	−0.255531	+	0.255531i	0.567702	+	0.823234i	$0.307832\pi$
$828$	14.6969	+	14.6969i	0.510754	+	0.510754i
$829$			50.0000i			1.73657i	0.496064	+	0.868286i	$0.334778\pi$
$829$			50.0000i			1.73657i	−0.496064	+	0.868286i	$0.665222\pi$
$830$		0			0
$831$	−30.0000			−1.04069
$832$	−2.44949	−	2.44949i	−0.0849208	−	0.0849208i
$833$	24.4949	+	24.4949i	0.848698	+	0.848698i
$834$	−66.0000			−2.28539
$835$		0			0
$836$		−	2.00000i		−	0.0691714i
$837$	−14.6969	−	14.6969i	−0.508001	−	0.508001i
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$	12.2474	−	12.2474i	0.422075	−	0.422075i
$843$	−36.7423	−	36.7423i	−1.26547	−	1.26547i
$844$		−	26.0000i		−	0.894957i
$845$		0			0
$846$	36.0000			1.23771
$847$	−2.44949	−	2.44949i	−0.0841655	−	0.0841655i
$848$	−24.4949	−	24.4949i	−0.841158	−	0.841158i
$849$		−	30.0000i		−	1.02960i
$850$		0			0
$851$		0			0
$852$	−14.6969	+	14.6969i	−0.503509	+	0.503509i
$853$	2.44949	−	2.44949i	0.0838689	−	0.0838689i	−0.663928	−	0.747797i	$-0.731112\pi$
$853$	2.44949	−	2.44949i	0.0838689	−	0.0838689i	0.747797	+	0.663928i	$0.231112\pi$
$854$	12.0000			0.410632
$855$		0			0
$856$	−6.00000			−0.205076
$857$	4.89898	−	4.89898i	0.167346	−	0.167346i	−0.618466	−	0.785812i	$-0.712245\pi$
$857$	4.89898	−	4.89898i	0.167346	−	0.167346i	0.785812	+	0.618466i	$0.212245\pi$
$858$	7.34847	−	7.34847i	0.250873	−	0.250873i
$859$			4.00000i			0.136478i	0.997669	+	0.0682391i	$0.0217381\pi$
$859$			4.00000i			0.136478i	−0.997669	+	0.0682391i	$0.978262\pi$
$860$		0			0
$861$			36.0000i			1.22688i
$862$	−29.3939	−	29.3939i	−1.00116	−	1.00116i
$863$	29.3939	+	29.3939i	1.00058	+	1.00058i	1.00000		0.000579374i	$0.000184421\pi$
$863$	29.3939	+	29.3939i	1.00058	+	1.00058i	0.000579374		1.00000i	$0.499816\pi$
$864$	−27.0000			−0.918559
$865$		0			0
$866$		−	48.0000i		−	1.63111i
$867$	−37.9671	−	37.9671i	−1.28943	−	1.28943i
$868$	9.79796	−	9.79796i	0.332564	−	0.332564i
$869$	10.0000			0.339227
$870$		0			0
$871$	−12.0000			−0.406604
$872$	−12.2474	+	12.2474i	−0.414751	+	0.414751i
$873$	−29.3939	+	29.3939i	−0.994832	+	0.994832i
$874$		−	24.0000i		−	0.811812i
$875$		0			0
$876$	−6.00000			−0.202721
$877$	7.34847	+	7.34847i	0.248140	+	0.248140i	0.820207	−	0.572067i	$-0.193858\pi$
$877$	7.34847	+	7.34847i	0.248140	+	0.248140i	−0.572067	+	0.820207i	$0.693858\pi$
$878$	−31.8434	−	31.8434i	−1.07466	−	1.07466i
$879$	−24.0000			−0.809500
$880$		0			0
$881$		−	36.0000i		−	1.21287i	−0.795133	−	0.606435i	$-0.792599\pi$
$881$		−	36.0000i		−	1.21287i	0.795133	−	0.606435i	$-0.207401\pi$
$882$	18.3712	−	18.3712i	0.618590	−	0.618590i
$883$	2.44949	−	2.44949i	0.0824319	−	0.0824319i	−0.664689	−	0.747121i	$-0.731436\pi$
$883$	2.44949	−	2.44949i	0.0824319	−	0.0824319i	0.747121	+	0.664689i	$0.231436\pi$
$884$	24.0000			0.807207
$885$		0			0
$886$		0			0
$887$	−7.34847	+	7.34847i	−0.246737	+	0.246737i	−0.819630	−	0.572893i	$-0.805821\pi$
$887$	−7.34847	+	7.34847i	−0.246737	+	0.246737i	0.572893	+	0.819630i	$0.305821\pi$
$888$		0			0
$889$		−	12.0000i		−	0.402467i
$890$		0			0
$891$			9.00000i			0.301511i
$892$	17.1464	+	17.1464i	0.574105	+	0.574105i
$893$	−9.79796	−	9.79796i	−0.327876	−	0.327876i
$894$		−	18.0000i		−	0.602010i
$895$		0			0
$896$			42.0000i			1.40312i
$897$	29.3939	−	29.3939i	0.981433	−	0.981433i
$898$	14.6969	−	14.6969i	0.490443	−	0.490443i
$899$	−24.0000			−0.800445
$900$		0			0
$901$	−48.0000			−1.59911
$902$	7.34847	−	7.34847i	0.244677	−	0.244677i
$903$	−44.0908	+	44.0908i	−1.46725	+	1.46725i
$904$			12.0000i			0.399114i
$905$		0			0
$906$		−	30.0000i		−	0.996683i
$907$	7.34847	+	7.34847i	0.244002	+	0.244002i	0.818503	−	0.574502i	$-0.194804\pi$
$907$	7.34847	+	7.34847i	0.244002	+	0.244002i	−0.574502	+	0.818503i	$0.694804\pi$
$908$	12.2474	+	12.2474i	0.406446	+	0.406446i
$909$	54.0000			1.79107
$910$		0			0
$911$		−	48.0000i		−	1.59031i	−0.606406	−	0.795155i	$-0.707389\pi$
$911$		−	48.0000i		−	1.59031i	0.606406	−	0.795155i	$-0.292611\pi$
$912$	12.2474	+	12.2474i	0.405554	+	0.405554i
$913$	−7.34847	+	7.34847i	−0.243199	+	0.243199i
$914$	−54.0000			−1.78616
$915$		0			0
$916$	−10.0000			−0.330409
$917$	29.3939	−	29.3939i	0.970671	−	0.970671i
$918$	−44.0908	+	44.0908i	−1.45521	+	1.45521i
$919$			26.0000i			0.857661i	0.903385	+	0.428830i	$0.141074\pi$
$919$			26.0000i			0.857661i	−0.903385	+	0.428830i	$0.858926\pi$
$920$		0			0
$921$	−42.0000			−1.38395
$922$	−7.34847	−	7.34847i	−0.242009	−	0.242009i
$923$	29.3939	+	29.3939i	0.967511	+	0.967511i
$924$	−6.00000			−0.197386
$925$		0			0
$926$			6.00000i			0.197172i
$927$	7.34847	+	7.34847i	0.241355	+	0.241355i
$928$	−22.0454	+	22.0454i	−0.723676	+	0.723676i
$929$	36.0000			1.18112			0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000			1.18112			0.590561	+	0.806993i	$0.298907\pi$
$930$		0			0
$931$	−10.0000			−0.327737
$932$	4.89898	−	4.89898i	0.160471	−	0.160471i
$933$	29.3939	+	29.3939i	0.962312	+	0.962312i
$934$			60.0000i			1.96326i
$935$		0			0
$936$			18.0000i			0.588348i
$937$	−12.2474	−	12.2474i	−0.400107	−	0.400107i	0.478164	−	0.878271i	$-0.341302\pi$
$937$	−12.2474	−	12.2474i	−0.400107	−	0.400107i	−0.878271	+	0.478164i	$0.841302\pi$
$938$	14.6969	+	14.6969i	0.479872	+	0.479872i
$939$			12.0000i			0.391605i
$940$		0			0
$941$			6.00000i			0.195594i	0.995206	+	0.0977972i	$0.0311797\pi$
$941$			6.00000i			0.195594i	−0.995206	+	0.0977972i	$0.968820\pi$
$942$	−29.3939	+	29.3939i	−0.957704	+	0.957704i
$943$	29.3939	−	29.3939i	0.957196	−	0.957196i
$944$		0			0
$945$		0			0
$946$	18.0000			0.585230
$947$	−4.89898	+	4.89898i	−0.159195	+	0.159195i	−0.782210	−	0.623015i	$-0.785908\pi$
$947$	−4.89898	+	4.89898i	−0.159195	+	0.159195i	0.623015	+	0.782210i	$0.285908\pi$
$948$	12.2474	−	12.2474i	0.397779	−	0.397779i
$949$			12.0000i			0.389536i
$950$		0			0
$951$		−	12.0000i		−	0.389127i
$952$	29.3939	+	29.3939i	0.952661	+	0.952661i
$953$	14.6969	+	14.6969i	0.476081	+	0.476081i	0.903876	−	0.427795i	$-0.140710\pi$
$953$	14.6969	+	14.6969i	0.476081	+	0.476081i	−0.427795	+	0.903876i	$0.640710\pi$
$954$			36.0000i			1.16554i
$955$		0			0
$956$		0			0
$957$	7.34847	+	7.34847i	0.237542	+	0.237542i
$958$	29.3939	−	29.3939i	0.949673	−	0.949673i
$959$	−48.0000			−1.55000
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$	7.34847	+	7.34847i	0.236801	+	0.236801i
$964$			10.0000i			0.322078i
$965$		0			0
$966$	−72.0000			−2.31656
$967$	36.7423	+	36.7423i	1.18155	+	1.18155i	0.979342	+	0.202213i	$0.0648132\pi$
$967$	36.7423	+	36.7423i	1.18155	+	1.18155i	0.202213	+	0.979342i	$0.435187\pi$
$968$	−1.22474	−	1.22474i	−0.0393648	−	0.0393648i
$969$	24.0000			0.770991
$970$		0			0
$971$			12.0000i			0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$971$			12.0000i			0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$972$	11.0227	+	11.0227i	0.353553	+	0.353553i
$973$	53.8888	−	53.8888i	1.72759	−	1.72759i
$974$	−42.0000			−1.34577
$975$		0			0
$976$	10.0000			0.320092
$977$	19.5959	−	19.5959i	0.626929	−	0.626929i	−0.320365	−	0.947294i	$-0.603806\pi$
$977$	19.5959	−	19.5959i	0.626929	−	0.626929i	0.947294	+	0.320365i	$0.103806\pi$
$978$	−36.7423	−	36.7423i	−1.17489	−	1.17489i
$979$		−	12.0000i		−	0.383522i
$980$		0			0
$981$	30.0000			0.957826
$982$	14.6969	+	14.6969i	0.468998	+	0.468998i
$983$	9.79796	+	9.79796i	0.312506	+	0.312506i	0.845880	−	0.533373i	$-0.179076\pi$
$983$	9.79796	+	9.79796i	0.312506	+	0.312506i	−0.533373	+	0.845880i	$0.679076\pi$
$984$			18.0000i			0.573819i
$985$		0			0
$986$			72.0000i			2.29295i
$987$	−29.3939	+	29.3939i	−0.935617	+	0.935617i
$988$	−4.89898	+	4.89898i	−0.155857	+	0.155857i
$989$	72.0000			2.28947
$990$		0			0
$991$	−52.0000			−1.65183			−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000			−1.65183			−0.825917	+	0.563791i	$0.809342\pi$
$992$	14.6969	−	14.6969i	0.466628	−	0.466628i
$993$	−34.2929	+	34.2929i	−1.08825	+	1.08825i
$994$		−	72.0000i		−	2.28370i
$995$		0			0
$996$			18.0000i			0.570352i
$997$	−26.9444	−	26.9444i	−0.853337	−	0.853337i	0.137205	−	0.990543i	$-0.456188\pi$
$997$	−26.9444	−	26.9444i	−0.853337	−	0.853337i	−0.990543	+	0.137205i	$0.956188\pi$
$998$	19.5959	+	19.5959i	0.620298	+	0.620298i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.k.e.782.2	yes	4
3.2	odd	2		825.2.k.d.782.1	yes	4
5.2	odd	4		825.2.k.d.518.2	yes	4
5.3	odd	4		825.2.k.d.518.1	✓	4
5.4	even	2	inner	825.2.k.e.782.1	yes	4
15.2	even	4	inner	825.2.k.e.518.1	yes	4
15.8	even	4	inner	825.2.k.e.518.2	yes	4
15.14	odd	2		825.2.k.d.782.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.k.d.518.1	✓	4	5.3	odd	4
825.2.k.d.518.2	yes	4	5.2	odd	4
825.2.k.d.782.1	yes	4	3.2	odd	2
825.2.k.d.782.2	yes	4	15.14	odd	2
825.2.k.e.518.1	yes	4	15.2	even	4	inner
825.2.k.e.518.2	yes	4	15.8	even	4	inner
825.2.k.e.782.1	yes	4	5.4	even	2	inner
825.2.k.e.782.2	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 1.22474i) q^{2} +(1.22474 - 1.22474i) q^{3} -1.00000i q^{4} -3.00000i q^{6} +(2.44949 + 2.44949i) q^{7} +(1.22474 + 1.22474i) q^{8} -3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 - 1.22474i) q^{2} +(1.22474 - 1.22474i) q^{3} -1.00000i q^{4} -3.00000i q^{6} +(2.44949 + 2.44949i) q^{7} +(1.22474 + 1.22474i) q^{8} -3.00000i q^{9} -1.00000i q^{11} +(-1.22474 - 1.22474i) q^{12} +(-2.44949 + 2.44949i) q^{13} +6.00000 q^{14} +5.00000 q^{16} +(4.89898 - 4.89898i) q^{17} +(-3.67423 - 3.67423i) q^{18} +2.00000i q^{19} +6.00000 q^{21} +(-1.22474 - 1.22474i) q^{22} +(-4.89898 - 4.89898i) q^{23} +3.00000 q^{24} +6.00000i q^{26} +(-3.67423 - 3.67423i) q^{27} +(2.44949 - 2.44949i) q^{28} -6.00000 q^{29} +4.00000 q^{31} +(3.67423 - 3.67423i) q^{32} +(-1.22474 - 1.22474i) q^{33} -12.0000i q^{34} -3.00000 q^{36} +(2.44949 + 2.44949i) q^{38} +6.00000i q^{39} +6.00000i q^{41} +(7.34847 - 7.34847i) q^{42} +(-7.34847 + 7.34847i) q^{43} -1.00000 q^{44} -12.0000 q^{46} +(-4.89898 + 4.89898i) q^{47} +(6.12372 - 6.12372i) q^{48} +5.00000i q^{49} -12.0000i q^{51} +(2.44949 + 2.44949i) q^{52} +(-4.89898 - 4.89898i) q^{53} -9.00000 q^{54} +6.00000i q^{56} +(2.44949 + 2.44949i) q^{57} +(-7.34847 + 7.34847i) q^{58} +2.00000 q^{61} +(4.89898 - 4.89898i) q^{62} +(7.34847 - 7.34847i) q^{63} +1.00000i q^{64} -3.00000 q^{66} +(2.44949 + 2.44949i) q^{67} +(-4.89898 - 4.89898i) q^{68} -12.0000 q^{69} -12.0000i q^{71} +(3.67423 - 3.67423i) q^{72} +(2.44949 - 2.44949i) q^{73} +2.00000 q^{76} +(2.44949 - 2.44949i) q^{77} +(7.34847 + 7.34847i) q^{78} +10.0000i q^{79} -9.00000 q^{81} +(7.34847 + 7.34847i) q^{82} +(-7.34847 - 7.34847i) q^{83} -6.00000i q^{84} +18.0000i q^{86} +(-7.34847 + 7.34847i) q^{87} +(1.22474 - 1.22474i) q^{88} +12.0000 q^{89} -12.0000 q^{91} +(-4.89898 + 4.89898i) q^{92} +(4.89898 - 4.89898i) q^{93} +12.0000i q^{94} -9.00000i q^{96} +(-9.79796 - 9.79796i) q^{97} +(6.12372 + 6.12372i) q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 24 q^{14} + 20 q^{16} + 24 q^{21} + 12 q^{24} - 24 q^{29} + 16 q^{31} - 12 q^{36} - 4 q^{44} - 48 q^{46} - 36 q^{54} + 8 q^{61} - 12 q^{66} - 48 q^{69} + 8 q^{76} - 36 q^{81} + 48 q^{89} - 48 q^{91} - 12 q^{99}+O(q^{100}) \) 4 * q + 24 * q^14 + 20 * q^16 + 24 * q^21 + 12 * q^24 - 24 * q^29 + 16 * q^31 - 12 * q^36 - 4 * q^44 - 48 * q^46 - 36 * q^54 + 8 * q^61 - 12 * q^66 - 48 * q^69 + 8 * q^76 - 36 * q^81 + 48 * q^89 - 48 * q^91 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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