LMFDB - Embedded newform 825.2.k.e.782.1

Newspace parameters

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

Self dual:	no
Analytic conductor:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
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			782.1
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	825.782
Dual form			825.2.k.e.518.1

$q$-expansion

$f(q)$	$=$	$q+(-1.22474 + 1.22474i) q^{2} +(-1.22474 + 1.22474i) q^{3} -1.00000i q^{4} -3.00000i q^{6} +(-2.44949 - 2.44949i) q^{7} +(-1.22474 - 1.22474i) q^{8} -3.00000i q^{9} +O(q^{10})$	\(q+(-1.22474 + 1.22474i) q^{2} +(-1.22474 + 1.22474i) q^{3} -1.00000i q^{4} -3.00000i q^{6} +(-2.44949 - 2.44949i) q^{7} +(-1.22474 - 1.22474i) q^{8} -3.00000i q^{9} -1.00000i q^{11} +(1.22474 + 1.22474i) q^{12} +(2.44949 - 2.44949i) q^{13} +6.00000 q^{14} +5.00000 q^{16} +(-4.89898 + 4.89898i) q^{17} +(3.67423 + 3.67423i) q^{18} +2.00000i q^{19} +6.00000 q^{21} +(1.22474 + 1.22474i) q^{22} +(4.89898 + 4.89898i) q^{23} +3.00000 q^{24} +6.00000i q^{26} +(3.67423 + 3.67423i) q^{27} +(-2.44949 + 2.44949i) q^{28} -6.00000 q^{29} +4.00000 q^{31} +(-3.67423 + 3.67423i) q^{32} +(1.22474 + 1.22474i) q^{33} -12.0000i q^{34} -3.00000 q^{36} +(-2.44949 - 2.44949i) q^{38} +6.00000i q^{39} +6.00000i q^{41} +(-7.34847 + 7.34847i) q^{42} +(7.34847 - 7.34847i) q^{43} -1.00000 q^{44} -12.0000 q^{46} +(4.89898 - 4.89898i) q^{47} +(-6.12372 + 6.12372i) q^{48} +5.00000i q^{49} -12.0000i q^{51} +(-2.44949 - 2.44949i) q^{52} +(4.89898 + 4.89898i) q^{53} -9.00000 q^{54} +6.00000i q^{56} +(-2.44949 - 2.44949i) q^{57} +(7.34847 - 7.34847i) q^{58} +2.00000 q^{61} +(-4.89898 + 4.89898i) q^{62} +(-7.34847 + 7.34847i) q^{63} +1.00000i q^{64} -3.00000 q^{66} +(-2.44949 - 2.44949i) q^{67} +(4.89898 + 4.89898i) q^{68} -12.0000 q^{69} -12.0000i q^{71} +(-3.67423 + 3.67423i) q^{72} +(-2.44949 + 2.44949i) q^{73} +2.00000 q^{76} +(-2.44949 + 2.44949i) q^{77} +(-7.34847 - 7.34847i) q^{78} +10.0000i q^{79} -9.00000 q^{81} +(-7.34847 - 7.34847i) q^{82} +(7.34847 + 7.34847i) q^{83} -6.00000i q^{84} +18.0000i q^{86} +(7.34847 - 7.34847i) q^{87} +(-1.22474 + 1.22474i) q^{88} +12.0000 q^{89} -12.0000 q^{91} +(4.89898 - 4.89898i) q^{92} +(-4.89898 + 4.89898i) q^{93} +12.0000i q^{94} -9.00000i q^{96} +(9.79796 + 9.79796i) q^{97} +(-6.12372 - 6.12372i) q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	−0.992030	−	0.126004i	$-0.959785\pi$
$2$	−1.22474	+	1.22474i	−0.866025	+	0.866025i	0.126004	+	0.992030i	$0.459785\pi$
$3$	−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	0.0716124	−	0.997433i	$-0.477186\pi$
$7$	−2.44949	−	2.44949i	−0.925820	−	0.925820i	−0.997433	+	0.0716124i	$0.977186\pi$
$8$	−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	−0.280491	−	0.959857i	$-0.590497\pi$
$13$	2.44949	−	2.44949i	0.679366	−	0.679366i	0.959857	+	0.280491i	$0.0904971\pi$
$14$	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	$0.567544\pi$
$17$	−4.89898	+	4.89898i	−1.18818	+	1.18818i	−0.977571	+	0.210606i	$0.932456\pi$
$18$	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	$0.00692196\pi$
$23$	4.89898	+	4.89898i	1.02151	+	1.02151i	0.0217443	+	0.999764i	$0.493078\pi$
$24$	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	$-0.458828\pi$
$43$	7.34847	−	7.34847i	1.12063	−	1.12063i	0.991647	−	0.128984i	$-0.0411717\pi$
$44$	−1.00000			−0.150756
$45$		0			0
$46$	−12.0000			−1.76930
$47$	4.89898	−	4.89898i	0.714590	−	0.714590i	−0.252902	−	0.967492i	$-0.581385\pi$
$47$	4.89898	−	4.89898i	0.714590	−	0.714590i	0.967492	+	0.252902i	$0.0813851\pi$
$48$	−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	0.958390	−	0.285463i	$-0.0921474\pi$
$53$	4.89898	+	4.89898i	0.672927	+	0.672927i	−0.285463	+	0.958390i	$0.592147\pi$
$54$	−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	0.541468	−	0.840721i	$-0.317869\pi$
$67$	−2.44949	−	2.44949i	−0.299253	−	0.299253i	−0.840721	+	0.541468i	$0.817869\pi$
$68$	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	−0.835770	−	0.549079i	$-0.814979\pi$
$73$	−2.44949	+	2.44949i	−0.286691	+	0.286691i	0.549079	+	0.835770i	$0.314979\pi$
$74$		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	0.984118	−	0.177518i	$-0.0568069\pi$
$83$	7.34847	+	7.34847i	0.806599	+	0.806599i	−0.177518	+	0.984118i	$0.556807\pi$
$84$		−	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	0.999987	−	0.00515471i	$-0.00164080\pi$
$97$	9.79796	+	9.79796i	0.994832	+	0.994832i	−0.00515471	+	0.999987i	$0.501641\pi$
$98$	−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	−0.576055	−	0.817411i	$-0.695409\pi$
$103$	2.44949	−	2.44949i	0.241355	−	0.241355i	0.817411	+	0.576055i	$0.195409\pi$
$104$	−6.00000			−0.588348
$105$		0			0
$106$	−12.0000			−1.16554
$107$	2.44949	−	2.44949i	0.236801	−	0.236801i	−0.578723	−	0.815524i	$-0.696449\pi$
$107$	2.44949	−	2.44949i	0.236801	−	0.236801i	0.815524	+	0.578723i	$0.196449\pi$
$108$	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	0.438079	−	0.898936i	$-0.355659\pi$
$113$	−4.89898	−	4.89898i	−0.460857	−	0.460857i	−0.898936	+	0.438079i	$0.855659\pi$
$114$	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	0.807384	−	0.590027i	$-0.200883\pi$
$127$	2.44949	+	2.44949i	0.217357	+	0.217357i	−0.590027	+	0.807384i	$0.700883\pi$
$128$	−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	−0.151380	−	0.988476i	$-0.548372\pi$
$137$	9.79796	−	9.79796i	0.837096	−	0.837096i	0.988476	+	0.151380i	$0.0483716\pi$
$138$	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	0.980162	−	0.198199i	$-0.0635094\pi$
$157$	9.79796	+	9.79796i	0.781962	+	0.781962i	−0.198199	+	0.980162i	$0.563509\pi$
$158$	−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	−0.999203	−	0.0399091i	$-0.987293\pi$
$163$	−12.2474	+	12.2474i	−0.959294	+	0.959294i	0.0399091	+	0.999203i	$0.487293\pi$
$164$	6.00000			0.468521
$165$		0			0
$166$	−18.0000			−1.39707
$167$	2.44949	−	2.44949i	0.189547	−	0.189547i	−0.605953	−	0.795500i	$-0.707208\pi$
$167$	2.44949	−	2.44949i	0.189547	−	0.189547i	0.795500	+	0.605953i	$0.207208\pi$
$168$	−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	−0.920261	−	0.391306i	$-0.872023\pi$
$193$	−7.34847	+	7.34847i	−0.528954	+	0.528954i	0.391306	+	0.920261i	$0.372023\pi$
$194$	−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	$-0.448430\pi$
$223$	17.1464	−	17.1464i	1.14821	−	1.14821i	0.986905	−	0.161305i	$-0.0515703\pi$
$224$	18.0000			1.20268
$225$		0			0
$226$	12.0000			0.798228
$227$	12.2474	−	12.2474i	0.812892	−	0.812892i	−0.172175	−	0.985066i	$-0.555079\pi$
$227$	12.2474	−	12.2474i	0.812892	−	0.812892i	0.985066	+	0.172175i	$0.0550794\pi$
$228$	−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	0.528186	−	0.849129i	$-0.322872\pi$
$233$	−4.89898	−	4.89898i	−0.320943	−	0.320943i	−0.849129	+	0.528186i	$0.822872\pi$
$234$	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	−0.943253	−	0.332074i	$-0.892252\pi$
$257$	−9.79796	+	9.79796i	−0.611180	+	0.611180i	0.332074	+	0.943253i	$0.392252\pi$
$258$	−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	$-0.981197\pi$
$263$	−17.1464	−	17.1464i	−1.05729	−	1.05729i	−0.0590383	−	0.998256i	$-0.518803\pi$
$264$		−	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	0.971777	−	0.235900i	$-0.0758036\pi$
$277$	12.2474	+	12.2474i	0.735878	+	0.735878i	−0.235900	+	0.971777i	$0.575804\pi$
$278$	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	−0.970228	−	0.242193i	$-0.922133\pi$
$283$	−12.2474	+	12.2474i	−0.728035	+	0.728035i	0.242193	+	0.970228i	$0.422133\pi$
$284$	−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	0.932799	−	0.360396i	$-0.117359\pi$
$293$	9.79796	+	9.79796i	0.572403	+	0.572403i	−0.360396	+	0.932799i	$0.617359\pi$
$294$	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	0.999776	−	0.0211774i	$-0.00674148\pi$
$307$	17.1464	+	17.1464i	0.978598	+	0.978598i	−0.0211774	+	0.999776i	$0.506741\pi$
$308$	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	−0.554966	−	0.831873i	$-0.687269\pi$
$313$	4.89898	−	4.89898i	0.276907	−	0.276907i	0.831873	+	0.554966i	$0.187269\pi$
$314$	−24.0000			−1.35440
$315$		0			0
$316$	10.0000			0.562544
$317$	−4.89898	+	4.89898i	−0.275154	+	0.275154i	−0.831171	−	0.556017i	$-0.812329\pi$
$317$	−4.89898	+	4.89898i	−0.275154	+	0.275154i	0.556017	+	0.831171i	$0.312329\pi$
$318$	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	$-0.927110\pi$
$337$	−22.0454	−	22.0454i	−1.20089	−	1.20089i	−0.226994	−	0.973896i	$-0.572890\pi$
$338$	−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	−0.297305	−	0.954783i	$-0.596088\pi$
$347$	12.2474	−	12.2474i	0.657477	−	0.657477i	0.954783	+	0.297305i	$0.0960878\pi$
$348$	−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	$0.123345\pi$
$353$	24.4949	+	24.4949i	1.30373	+	1.30373i	0.377875	+	0.925856i	$0.376655\pi$
$354$		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	0.640280	−	0.768142i	$-0.278818\pi$
$367$	−2.44949	−	2.44949i	−0.127862	−	0.127862i	−0.768142	+	0.640280i	$0.778818\pi$
$368$	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	−0.994312	−	0.106504i	$-0.966034\pi$
$373$	−17.1464	+	17.1464i	−0.887808	+	0.887808i	0.106504	+	0.994312i	$0.466034\pi$
$374$	−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	0.573402	−	0.819274i	$-0.305623\pi$
$397$	−4.89898	−	4.89898i	−0.245873	−	0.245873i	−0.819274	+	0.573402i	$0.805623\pi$
$398$	−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	−0.998393	−	0.0566731i	$-0.981951\pi$
$433$	−19.5959	+	19.5959i	−0.941720	+	0.941720i	0.0566731	+	0.998393i	$0.481951\pi$
$434$	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	$0.0101063\pi$
$457$	22.0454	+	22.0454i	1.03124	+	1.03124i	0.0317447	+	0.999496i	$0.489894\pi$
$458$	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	−0.647893	−	0.761731i	$-0.724350\pi$
$463$	2.44949	−	2.44949i	0.113837	−	0.113837i	0.761731	+	0.647893i	$0.224350\pi$
$464$	−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	$-0.454037\pi$
$467$	24.4949	−	24.4949i	1.13349	−	1.13349i	0.989593	−	0.143896i	$-0.0459630\pi$
$468$	−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	0.979316	−	0.202337i	$-0.0648537\pi$
$487$	17.1464	+	17.1464i	0.776979	+	0.776979i	−0.202337	+	0.979316i	$0.564854\pi$
$488$	−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	0.212615	−	0.977136i	$-0.431802\pi$
$503$	−17.1464	−	17.1464i	−0.764521	−	0.764521i	−0.977136	+	0.212615i	$0.931802\pi$
$504$	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	−0.849276	−	0.527950i	$-0.822961\pi$
$523$	−7.34847	+	7.34847i	−0.321326	+	0.321326i	0.527950	+	0.849276i	$0.322961\pi$
$524$	−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	0.918676	−	0.395013i	$-0.129260\pi$
$547$	12.2474	+	12.2474i	0.523663	+	0.523663i	−0.395013	+	0.918676i	$0.629260\pi$
$548$	−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	−0.987558	−	0.157253i	$-0.949736\pi$
$557$	−19.5959	+	19.5959i	−0.830306	+	0.830306i	0.157253	+	0.987558i	$0.449736\pi$
$558$	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	0.844794	−	0.535092i	$-0.179723\pi$
$563$	7.34847	+	7.34847i	0.309701	+	0.309701i	−0.535092	+	0.844794i	$0.679723\pi$
$564$	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	0.331584	−	0.943426i	$-0.392417\pi$
$577$	−14.6969	−	14.6969i	−0.611842	−	0.611842i	−0.943426	+	0.331584i	$0.892417\pi$
$578$	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	−0.598741	−	0.800943i	$-0.704332\pi$
$587$	4.89898	−	4.89898i	0.202203	−	0.202203i	0.800943	+	0.598741i	$0.204332\pi$
$588$	−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	0.337717	−	0.941248i	$-0.390345\pi$
$593$	−14.6969	−	14.6969i	−0.603531	−	0.603531i	−0.941248	+	0.337717i	$0.890345\pi$
$594$			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	0.413428	−	0.910537i	$-0.364331\pi$
$607$	−12.2474	−	12.2474i	−0.497109	−	0.497109i	−0.910537	+	0.413428i	$0.864331\pi$
$608$	−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	−0.655907	−	0.754841i	$-0.727714\pi$
$613$	2.44949	−	2.44949i	0.0989340	−	0.0989340i	0.754841	+	0.655907i	$0.227714\pi$
$614$	−42.0000			−1.69498
$615$		0			0
$616$	6.00000			0.241747
$617$	−14.6969	+	14.6969i	−0.591676	+	0.591676i	−0.938084	−	0.346408i	$-0.887401\pi$
$617$	−14.6969	+	14.6969i	−0.591676	+	0.591676i	0.346408	+	0.938084i	$0.387401\pi$
$618$	−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	−0.906086	−	0.423094i	$-0.860944\pi$
$643$	−12.2474	+	12.2474i	−0.482992	+	0.482992i	0.423094	+	0.906086i	$0.360944\pi$
$644$	−24.0000			−0.945732
$645$		0			0
$646$	24.0000			0.944267
$647$	−19.5959	+	19.5959i	−0.770395	+	0.770395i	−0.978175	−	0.207780i	$-0.933376\pi$
$647$	−19.5959	+	19.5959i	−0.770395	+	0.770395i	0.207780	+	0.978175i	$0.433376\pi$
$648$	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	$-0.947633\pi$
$653$	−29.3939	−	29.3939i	−1.15027	−	1.15027i	−0.163773	−	0.986498i	$-0.552367\pi$
$654$	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	−0.752739	−	0.658319i	$-0.771268\pi$
$673$	−2.44949	+	2.44949i	−0.0944209	+	0.0944209i	0.658319	+	0.752739i	$0.271268\pi$
$674$	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	0.367615	−	0.929978i	$-0.380174\pi$
$683$	−14.6969	−	14.6969i	−0.562363	−	0.562363i	−0.929978	+	0.367615i	$0.880174\pi$
$684$		−	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	0.168144	−	0.985763i	$-0.446223\pi$
$727$	−22.0454	−	22.0454i	−0.817619	−	0.817619i	−0.985763	+	0.168144i	$0.946223\pi$
$728$	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	−0.171005	−	0.985270i	$-0.554701\pi$
$733$	22.0454	−	22.0454i	0.814266	−	0.814266i	0.985270	+	0.171005i	$0.0547013\pi$
$734$	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	0.895127	−	0.445812i	$-0.147085\pi$
$743$	12.2474	+	12.2474i	0.449315	+	0.449315i	−0.445812	+	0.895127i	$0.647085\pi$
$744$	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	0.506265	−	0.862378i	$-0.331026\pi$
$757$	−9.79796	−	9.79796i	−0.356113	−	0.356113i	−0.862378	+	0.506265i	$0.831026\pi$
$758$	−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	0.965440	−	0.260624i	$-0.0839283\pi$
$773$	19.5959	+	19.5959i	0.704816	+	0.704816i	−0.260624	+	0.965440i	$0.583928\pi$
$774$	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	0.749415	−	0.662100i	$-0.230335\pi$
$787$	2.44949	+	2.44949i	0.0873149	+	0.0873149i	−0.662100	+	0.749415i	$0.730335\pi$
$788$		0			0

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

Related objects

Downloads

Learn more