LMFDB - Embedded newform 1520.2.d.e.609.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(609,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("1520.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.d (of order $2$, degree $1$, not 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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1520.609
Dual form			1520.2.d.e.609.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214i q^{3} +(0.707107 - 2.12132i) q^{5} +4.41421i q^{7} +2.82843 q^{9} +O(q^{10})$	$q+0.414214i q^{3} +(0.707107 - 2.12132i) q^{5} +4.41421i q^{7} +2.82843 q^{9} +1.41421 q^{11} +5.82843i q^{13} +(0.878680 + 0.292893i) q^{15} -1.00000i q^{17} -1.00000 q^{19} -1.82843 q^{21} -0.757359i q^{23} +(-4.00000 - 3.00000i) q^{25} +2.41421i q^{27} +0.171573 q^{29} -6.24264 q^{31} +0.585786i q^{33} +(9.36396 + 3.12132i) q^{35} +8.48528i q^{37} -2.41421 q^{39} -4.24264 q^{41} -1.75736i q^{43} +(2.00000 - 6.00000i) q^{45} -12.4853 q^{49} +0.414214 q^{51} +5.48528i q^{53} +(1.00000 - 3.00000i) q^{55} -0.414214i q^{57} +6.89949 q^{59} +14.2426 q^{61} +12.4853i q^{63} +(12.3640 + 4.12132i) q^{65} +4.75736i q^{67} +0.313708 q^{69} +13.4142 q^{71} +11.4853i q^{73} +(1.24264 - 1.65685i) q^{75} +6.24264i q^{77} -6.48528 q^{79} +7.48528 q^{81} -14.4853i q^{83} +(-2.12132 - 0.707107i) q^{85} +0.0710678i q^{87} -7.07107 q^{89} -25.7279 q^{91} -2.58579i q^{93} +(-0.707107 + 2.12132i) q^{95} +0.343146i q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 12 q^{15} - 4 q^{19} + 4 q^{21} - 16 q^{25} + 12 q^{29} - 8 q^{31} + 12 q^{35} - 4 q^{39} + 8 q^{45} - 16 q^{49} - 4 q^{51} + 4 q^{55} - 12 q^{59} + 40 q^{61} + 24 q^{65} - 44 q^{69} + 48 q^{71} - 12 q^{75} + 8 q^{79} - 4 q^{81} - 52 q^{91} + 16 q^{99}+O(q^{100}) $ 4 * q + 12 * q^15 - 4 * q^19 + 4 * q^21 - 16 * q^25 + 12 * q^29 - 8 * q^31 + 12 * q^35 - 4 * q^39 + 8 * q^45 - 16 * q^49 - 4 * q^51 + 4 * q^55 - 12 * q^59 + 40 * q^61 + 24 * q^65 - 44 * q^69 + 48 * q^71 - 12 * q^75 + 8 * q^79 - 4 * q^81 - 52 * q^91 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$			0.414214i			0.239146i	0.992825	+	0.119573i	$0.0381526\pi$
$3$			0.414214i			0.239146i	−0.992825	+	0.119573i	$0.961847\pi$
$4$		0			0
$5$	0.707107	−	2.12132i	0.316228	−	0.948683i
$5$	0.707107	−	2.12132i	0.316228	−	0.948683i
$6$		0			0
$7$			4.41421i			1.66842i	0.551450	+	0.834208i	$0.314075\pi$
$7$			4.41421i			1.66842i	−0.551450	+	0.834208i	$0.685925\pi$
$8$		0			0
$9$	2.82843			0.942809
$10$		0			0
$11$	1.41421			0.426401			0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421			0.426401			0.213201	+	0.977008i	$0.431611\pi$
$12$		0			0
$13$			5.82843i			1.61651i	0.588829	+	0.808257i	$0.299589\pi$
$13$			5.82843i			1.61651i	−0.588829	+	0.808257i	$0.700411\pi$
$14$		0			0
$15$	0.878680	+	0.292893i	0.226874	+	0.0756247i
$16$		0			0
$17$		−	1.00000i		−	0.242536i	−0.992620	−	0.121268i	$-0.961304\pi$
$17$		−	1.00000i		−	0.242536i	0.992620	−	0.121268i	$-0.0386960\pi$
$18$		0			0
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$		0			0
$21$	−1.82843			−0.398996
$22$		0			0
$23$		−	0.757359i		−	0.157920i	−0.996878	−	0.0789602i	$-0.974840\pi$
$23$		−	0.757359i		−	0.157920i	0.996878	−	0.0789602i	$-0.0251600\pi$
$24$		0			0
$25$	−4.00000	−	3.00000i	−0.800000	−	0.600000i
$26$		0			0
$27$			2.41421i			0.464616i
$28$		0			0
$29$	0.171573			0.0318603			0.0159301	−	0.999873i	$-0.494929\pi$
$29$	0.171573			0.0318603			0.0159301	+	0.999873i	$0.494929\pi$
$30$		0			0
$31$	−6.24264			−1.12121			−0.560606	−	0.828083i	$-0.689432\pi$
$31$	−6.24264			−1.12121			−0.560606	+	0.828083i	$0.689432\pi$
$32$		0			0
$33$			0.585786i			0.101972i
$34$		0			0
$35$	9.36396	+	3.12132i	1.58280	+	0.527599i
$36$		0			0
$37$			8.48528i			1.39497i	0.716599	+	0.697486i	$0.245698\pi$
$37$			8.48528i			1.39497i	−0.716599	+	0.697486i	$0.754302\pi$
$38$		0			0
$39$	−2.41421			−0.386584
$40$		0			0
$41$	−4.24264			−0.662589			−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264			−0.662589			−0.331295	+	0.943527i	$0.607485\pi$
$42$		0			0
$43$		−	1.75736i		−	0.267995i	−0.990982	−	0.133997i	$-0.957219\pi$
$43$		−	1.75736i		−	0.267995i	0.990982	−	0.133997i	$-0.0427814\pi$
$44$		0			0
$45$	2.00000	−	6.00000i	0.298142	−	0.894427i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−12.4853			−1.78361
$50$		0			0
$51$	0.414214			0.0580015
$52$		0			0
$53$			5.48528i			0.753461i	0.926323	+	0.376731i	$0.122952\pi$
$53$			5.48528i			0.753461i	−0.926323	+	0.376731i	$0.877048\pi$
$54$		0			0
$55$	1.00000	−	3.00000i	0.134840	−	0.404520i
$56$		0			0
$57$		−	0.414214i		−	0.0548639i
$58$		0			0
$59$	6.89949			0.898238			0.449119	−	0.893472i	$-0.351738\pi$
$59$	6.89949			0.898238			0.449119	+	0.893472i	$0.351738\pi$
$60$		0			0
$61$	14.2426			1.82358			0.911792	−	0.410653i	$-0.134699\pi$
$61$	14.2426			1.82358			0.911792	+	0.410653i	$0.134699\pi$
$62$		0			0
$63$			12.4853i			1.57300i
$64$		0			0
$65$	12.3640	+	4.12132i	1.53356	+	0.511187i
$66$		0			0
$67$			4.75736i			0.581204i	0.956844	+	0.290602i	$0.0938555\pi$
$67$			4.75736i			0.581204i	−0.956844	+	0.290602i	$0.906144\pi$
$68$		0			0
$69$	0.313708			0.0377661
$70$		0			0
$71$	13.4142			1.59197			0.795987	−	0.605314i	$-0.206952\pi$
$71$	13.4142			1.59197			0.795987	+	0.605314i	$0.206952\pi$
$72$		0			0
$73$			11.4853i			1.34425i	0.740437	+	0.672125i	$0.234618\pi$
$73$			11.4853i			1.34425i	−0.740437	+	0.672125i	$0.765382\pi$
$74$		0			0
$75$	1.24264	−	1.65685i	0.143488	−	0.191317i
$76$		0			0
$77$			6.24264i			0.711415i
$78$		0			0
$79$	−6.48528			−0.729651			−0.364826	−	0.931076i	$-0.618871\pi$
$79$	−6.48528			−0.729651			−0.364826	+	0.931076i	$0.618871\pi$
$80$		0			0
$81$	7.48528			0.831698
$82$		0			0
$83$		−	14.4853i		−	1.58997i	−0.606632	−	0.794983i	$-0.707480\pi$
$83$		−	14.4853i		−	1.58997i	0.606632	−	0.794983i	$-0.292520\pi$
$84$		0			0
$85$	−2.12132	−	0.707107i	−0.230089	−	0.0766965i
$86$		0			0
$87$			0.0710678i			0.00761927i
$88$		0			0
$89$	−7.07107			−0.749532			−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107			−0.749532			−0.374766	+	0.927119i	$0.622277\pi$
$90$		0			0
$91$	−25.7279			−2.69702
$92$		0			0
$93$		−	2.58579i		−	0.268134i
$94$		0			0
$95$	−0.707107	+	2.12132i	−0.0725476	+	0.217643i
$96$		0			0
$97$			0.343146i			0.0348412i	0.999848	+	0.0174206i	$0.00554543\pi$
$97$			0.343146i			0.0348412i	−0.999848	+	0.0174206i	$0.994455\pi$
$98$		0			0
$99$	4.00000			0.402015
$100$		0			0
$101$	13.0711			1.30062			0.650310	−	0.759669i	$-0.274639\pi$
$101$	13.0711			1.30062			0.650310	+	0.759669i	$0.274639\pi$
$102$		0			0
$103$		−	4.24264i		−	0.418040i	−0.977911	−	0.209020i	$-0.932973\pi$
$103$		−	4.24264i		−	0.418040i	0.977911	−	0.209020i	$-0.0670273\pi$
$104$		0			0
$105$	−1.29289	+	3.87868i	−0.126173	+	0.378520i
$106$		0			0
$107$			19.7279i			1.90717i	0.301124	+	0.953585i	$0.402638\pi$
$107$			19.7279i			1.90717i	−0.301124	+	0.953585i	$0.597362\pi$
$108$		0			0
$109$	17.9706			1.72127			0.860634	−	0.509224i	$-0.170068\pi$
$109$	17.9706			1.72127			0.860634	+	0.509224i	$0.170068\pi$
$110$		0			0
$111$	−3.51472			−0.333602
$112$		0			0
$113$		−	10.2426i		−	0.963547i	−0.876296	−	0.481773i	$-0.839993\pi$
$113$		−	10.2426i		−	0.963547i	0.876296	−	0.481773i	$-0.160007\pi$
$114$		0			0
$115$	−1.60660	−	0.535534i	−0.149816	−	0.0499388i
$116$		0			0
$117$			16.4853i			1.52406i
$118$		0			0
$119$	4.41421			0.404650
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$		−	1.75736i		−	0.158456i
$124$		0			0
$125$	−9.19239	+	6.36396i	−0.822192	+	0.569210i
$126$		0			0
$127$		−	2.48528i		−	0.220533i	−0.993902	−	0.110267i	$-0.964830\pi$
$127$		−	2.48528i		−	0.220533i	0.993902	−	0.110267i	$-0.0351704\pi$
$128$		0			0
$129$	0.727922			0.0640900
$130$		0			0
$131$	16.9706			1.48272			0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706			1.48272			0.741362	+	0.671105i	$0.234180\pi$
$132$		0			0
$133$		−	4.41421i		−	0.382761i
$134$		0			0
$135$	5.12132	+	1.70711i	0.440773	+	0.146924i
$136$		0			0
$137$		−	13.0000i		−	1.11066i	−0.831628	−	0.555332i	$-0.812591\pi$
$137$		−	13.0000i		−	1.11066i	0.831628	−	0.555332i	$-0.187409\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			8.24264i			0.689284i
$144$		0			0
$145$	0.121320	−	0.363961i	0.0100751	−	0.0302253i
$146$		0			0
$147$		−	5.17157i		−	0.426544i
$148$		0			0
$149$	−17.6569			−1.44651			−0.723253	−	0.690583i	$-0.757354\pi$
$149$	−17.6569			−1.44651			−0.723253	+	0.690583i	$0.757354\pi$
$150$		0			0
$151$	10.4853			0.853280			0.426640	−	0.904422i	$-0.359697\pi$
$151$	10.4853			0.853280			0.426640	+	0.904422i	$0.359697\pi$
$152$		0			0
$153$		−	2.82843i		−	0.228665i
$154$		0			0
$155$	−4.41421	+	13.2426i	−0.354558	+	1.06367i
$156$		0			0
$157$		−	11.6569i		−	0.930318i	−0.885227	−	0.465159i	$-0.845997\pi$
$157$		−	11.6569i		−	0.930318i	0.885227	−	0.465159i	$-0.154003\pi$
$158$		0			0
$159$	−2.27208			−0.180188
$160$		0			0
$161$	3.34315			0.263477
$162$		0			0
$163$			10.2426i			0.802266i	0.916020	+	0.401133i	$0.131383\pi$
$163$			10.2426i			0.802266i	−0.916020	+	0.401133i	$0.868617\pi$
$164$		0			0
$165$	1.24264	+	0.414214i	0.0967394	+	0.0322465i
$166$		0			0
$167$		−	18.2426i		−	1.41166i	−0.708382	−	0.705829i	$-0.750575\pi$
$167$		−	18.2426i		−	1.41166i	0.708382	−	0.705829i	$-0.249425\pi$
$168$		0			0
$169$	−20.9706			−1.61312
$170$		0			0
$171$	−2.82843			−0.216295
$172$		0			0
$173$			0.485281i			0.0368953i	0.999830	+	0.0184476i	$0.00587240\pi$
$173$			0.485281i			0.0368953i	−0.999830	+	0.0184476i	$0.994128\pi$
$174$		0			0
$175$	13.2426	−	17.6569i	1.00105	−	1.33473i
$176$		0			0
$177$			2.85786i			0.214810i
$178$		0			0
$179$	−11.6569			−0.871274			−0.435637	−	0.900122i	$-0.643477\pi$
$179$	−11.6569			−0.871274			−0.435637	+	0.900122i	$0.643477\pi$
$180$		0			0
$181$	−8.48528			−0.630706			−0.315353	−	0.948974i	$-0.602123\pi$
$181$	−8.48528			−0.630706			−0.315353	+	0.948974i	$0.602123\pi$
$182$		0			0
$183$			5.89949i			0.436103i
$184$		0			0
$185$	18.0000	+	6.00000i	1.32339	+	0.441129i
$186$		0			0
$187$		−	1.41421i		−	0.103418i
$188$		0			0
$189$	−10.6569			−0.775172
$190$		0			0
$191$	−12.5563			−0.908546			−0.454273	−	0.890863i	$-0.650101\pi$
$191$	−12.5563			−0.908546			−0.454273	+	0.890863i	$0.650101\pi$
$192$		0			0
$193$		−	0.343146i		−	0.0247002i	−0.999924	−	0.0123501i	$-0.996069\pi$
$193$		−	0.343146i		−	0.0247002i	0.999924	−	0.0123501i	$-0.00393125\pi$
$194$		0			0
$195$	−1.70711	+	5.12132i	−0.122248	+	0.366745i
$196$		0			0
$197$			11.7574i			0.837677i	0.908061	+	0.418839i	$0.137563\pi$
$197$			11.7574i			0.837677i	−0.908061	+	0.418839i	$0.862437\pi$
$198$		0			0
$199$	9.24264			0.655193			0.327597	−	0.944818i	$-0.393761\pi$
$199$	9.24264			0.655193			0.327597	+	0.944818i	$0.393761\pi$
$200$		0			0
$201$	−1.97056			−0.138993
$202$		0			0
$203$			0.757359i			0.0531562i
$204$		0			0
$205$	−3.00000	+	9.00000i	−0.209529	+	0.628587i
$206$		0			0
$207$		−	2.14214i		−	0.148889i
$208$		0			0
$209$	−1.41421			−0.0978232
$210$		0			0
$211$	19.7279			1.35813			0.679063	−	0.734080i	$-0.262386\pi$
$211$	19.7279			1.35813			0.679063	+	0.734080i	$0.262386\pi$
$212$		0			0
$213$			5.55635i			0.380715i
$214$		0			0
$215$	−3.72792	−	1.24264i	−0.254242	−	0.0847474i
$216$		0			0
$217$		−	27.5563i		−	1.87065i
$218$		0			0
$219$	−4.75736			−0.321473
$220$		0			0
$221$	5.82843			0.392062
$222$		0			0
$223$			15.1716i			1.01596i	0.861368	+	0.507982i	$0.169608\pi$
$223$			15.1716i			1.01596i	−0.861368	+	0.507982i	$0.830392\pi$
$224$		0			0
$225$	−11.3137	−	8.48528i	−0.754247	−	0.565685i
$226$		0			0
$227$		−	16.7574i		−	1.11223i	−0.831107	−	0.556113i	$-0.812292\pi$
$227$		−	16.7574i		−	1.11223i	0.831107	−	0.556113i	$-0.187708\pi$
$228$		0			0
$229$	−14.9706			−0.989283			−0.494641	−	0.869097i	$-0.664701\pi$
$229$	−14.9706			−0.989283			−0.494641	+	0.869097i	$0.664701\pi$
$230$		0			0
$231$	−2.58579			−0.170132
$232$		0			0
$233$		−	24.9706i		−	1.63588i	−0.575306	−	0.817938i	$-0.695117\pi$
$233$		−	24.9706i		−	1.63588i	0.575306	−	0.817938i	$-0.304883\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	2.68629i		−	0.174493i
$238$		0			0
$239$	−6.89949			−0.446291			−0.223146	−	0.974785i	$-0.571633\pi$
$239$	−6.89949			−0.446291			−0.223146	+	0.974785i	$0.571633\pi$
$240$		0			0
$241$	8.97056			0.577845			0.288922	−	0.957353i	$-0.406703\pi$
$241$	8.97056			0.577845			0.288922	+	0.957353i	$0.406703\pi$
$242$		0			0
$243$			10.3431i			0.663513i
$244$		0			0
$245$	−8.82843	+	26.4853i	−0.564028	+	1.69208i
$246$		0			0
$247$		−	5.82843i		−	0.370854i
$248$		0			0
$249$	6.00000			0.380235
$250$		0			0
$251$	−3.55635			−0.224475			−0.112237	−	0.993681i	$-0.535802\pi$
$251$	−3.55635			−0.224475			−0.112237	+	0.993681i	$0.535802\pi$
$252$		0			0
$253$		−	1.07107i		−	0.0673375i
$254$		0			0
$255$	0.292893	−	0.878680i	0.0183417	−	0.0550251i
$256$		0			0
$257$		−	20.7279i		−	1.29297i	−0.762926	−	0.646486i	$-0.776238\pi$
$257$		−	20.7279i		−	1.29297i	0.762926	−	0.646486i	$-0.223762\pi$
$258$		0			0
$259$	−37.4558			−2.32739
$260$		0			0
$261$	0.485281			0.0300382
$262$		0			0
$263$			26.9706i			1.66308i	0.555468	+	0.831538i	$0.312539\pi$
$263$			26.9706i			1.66308i	−0.555468	+	0.831538i	$0.687461\pi$
$264$		0			0
$265$	11.6360	+	3.87868i	0.714796	+	0.238265i
$266$		0			0
$267$		−	2.92893i		−	0.179248i
$268$		0			0
$269$	16.6274			1.01379			0.506896	−	0.862007i	$-0.330793\pi$
$269$	16.6274			1.01379			0.506896	+	0.862007i	$0.330793\pi$
$270$		0			0
$271$	27.2426			1.65487			0.827436	−	0.561560i	$-0.189798\pi$
$271$	27.2426			1.65487			0.827436	+	0.561560i	$0.189798\pi$
$272$		0			0
$273$		−	10.6569i		−	0.644982i
$274$		0			0
$275$	−5.65685	−	4.24264i	−0.341121	−	0.255841i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	−17.6569			−1.05709
$280$		0			0
$281$	4.24264			0.253095			0.126547	−	0.991961i	$-0.459610\pi$
$281$	4.24264			0.253095			0.126547	+	0.991961i	$0.459610\pi$
$282$		0			0
$283$		−	32.1421i		−	1.91065i	−0.295555	−	0.955326i	$-0.595504\pi$
$283$		−	32.1421i		−	1.91065i	0.295555	−	0.955326i	$-0.404496\pi$
$284$		0			0
$285$	−0.878680	−	0.292893i	−0.0520485	−	0.0173495i
$286$		0			0
$287$		−	18.7279i		−	1.10547i
$288$		0			0
$289$	16.0000			0.941176
$290$		0			0
$291$	−0.142136			−0.00833214
$292$		0			0
$293$		−	5.48528i		−	0.320454i	−0.987080	−	0.160227i	$-0.948777\pi$
$293$		−	5.48528i		−	0.320454i	0.987080	−	0.160227i	$-0.0512226\pi$
$294$		0			0
$295$	4.87868	−	14.6360i	0.284048	−	0.852143i
$296$		0			0
$297$			3.41421i			0.198113i
$298$		0			0
$299$	4.41421			0.255281
$300$		0			0
$301$	7.75736			0.447127
$302$		0			0
$303$			5.41421i			0.311038i
$304$		0			0
$305$	10.0711	−	30.2132i	0.576668	−	1.73000i
$306$		0			0
$307$		−	17.6569i		−	1.00773i	−0.863782	−	0.503865i	$-0.831911\pi$
$307$		−	17.6569i		−	1.00773i	0.863782	−	0.503865i	$-0.168089\pi$
$308$		0			0
$309$	1.75736			0.0999727
$310$		0			0
$311$	4.75736			0.269765			0.134883	−	0.990862i	$-0.456934\pi$
$311$	4.75736			0.269765			0.134883	+	0.990862i	$0.456934\pi$
$312$		0			0
$313$		−	7.97056i		−	0.450523i	−0.974298	−	0.225261i	$-0.927676\pi$
$313$		−	7.97056i		−	0.450523i	0.974298	−	0.225261i	$-0.0723236\pi$
$314$		0			0
$315$	26.4853	+	8.82843i	1.49228	+	0.497426i
$316$		0			0
$317$			9.48528i			0.532746i	0.963870	+	0.266373i	$0.0858254\pi$
$317$			9.48528i			0.532746i	−0.963870	+	0.266373i	$0.914175\pi$
$318$		0			0
$319$	0.242641			0.0135853
$320$		0			0
$321$	−8.17157			−0.456093
$322$		0			0
$323$			1.00000i			0.0556415i
$324$		0			0
$325$	17.4853	−	23.3137i	0.969909	−	1.29321i
$326$		0			0
$327$			7.44365i			0.411635i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	19.2426			1.05767			0.528836	−	0.848724i	$-0.322629\pi$
$331$	19.2426			1.05767			0.528836	+	0.848724i	$0.322629\pi$
$332$		0			0
$333$			24.0000i			1.31519i
$334$		0			0
$335$	10.0919	+	3.36396i	0.551378	+	0.183793i
$336$		0			0
$337$		−	14.1005i		−	0.768103i	−0.923312	−	0.384052i	$-0.874528\pi$
$337$		−	14.1005i		−	0.768103i	0.923312	−	0.384052i	$-0.125472\pi$
$338$		0			0
$339$	4.24264			0.230429
$340$		0			0
$341$	−8.82843			−0.478086
$342$		0			0
$343$		−	24.2132i		−	1.30739i
$344$		0			0
$345$	0.221825	−	0.665476i	0.0119427	−	0.0358280i
$346$		0			0
$347$		−	5.51472i		−	0.296046i	−0.988984	−	0.148023i	$-0.952709\pi$
$347$		−	5.51472i		−	0.296046i	0.988984	−	0.148023i	$-0.0472909\pi$
$348$		0			0
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$		0			0
$351$	−14.0711			−0.751058
$352$		0			0
$353$		−	2.51472i		−	0.133845i	−0.997758	−	0.0669225i	$-0.978682\pi$
$353$		−	2.51472i		−	0.133845i	0.997758	−	0.0669225i	$-0.0213180\pi$
$354$		0			0
$355$	9.48528	−	28.4558i	0.503426	−	1.51028i
$356$		0			0
$357$			1.82843i			0.0967706i
$358$		0			0
$359$	−22.7574			−1.20109			−0.600544	−	0.799592i	$-0.705049\pi$
$359$	−22.7574			−1.20109			−0.600544	+	0.799592i	$0.705049\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$		−	3.72792i		−	0.195665i
$364$		0			0
$365$	24.3640	+	8.12132i	1.27527	+	0.425089i
$366$		0			0
$367$		−	25.4558i		−	1.32878i	−0.747384	−	0.664392i	$-0.768691\pi$
$367$		−	25.4558i		−	1.32878i	0.747384	−	0.664392i	$-0.231309\pi$
$368$		0			0
$369$	−12.0000			−0.624695
$370$		0			0
$371$	−24.2132			−1.25709
$372$		0			0
$373$			9.00000i			0.466002i	0.972476	+	0.233001i	$0.0748546\pi$
$373$			9.00000i			0.466002i	−0.972476	+	0.233001i	$0.925145\pi$
$374$		0			0
$375$	−2.63604	−	3.80761i	−0.136124	−	0.196624i
$376$		0			0
$377$			1.00000i			0.0515026i
$378$		0			0
$379$	11.2426			0.577496			0.288748	−	0.957405i	$-0.406761\pi$
$379$	11.2426			0.577496			0.288748	+	0.957405i	$0.406761\pi$
$380$		0			0
$381$	1.02944			0.0527397
$382$		0			0
$383$		−	12.2426i		−	0.625570i	−0.949824	−	0.312785i	$-0.898738\pi$
$383$		−	12.2426i		−	0.625570i	0.949824	−	0.312785i	$-0.101262\pi$
$384$		0			0
$385$	13.2426	+	4.41421i	0.674907	+	0.224969i
$386$		0			0
$387$		−	4.97056i		−	0.252668i
$388$		0			0
$389$	−22.9289			−1.16254			−0.581272	−	0.813710i	$-0.697445\pi$
$389$	−22.9289			−1.16254			−0.581272	+	0.813710i	$0.697445\pi$
$390$		0			0
$391$	−0.757359			−0.0383013
$392$		0			0
$393$			7.02944i			0.354588i
$394$		0			0
$395$	−4.58579	+	13.7574i	−0.230736	+	0.692208i
$396$		0			0
$397$		−	24.0000i		−	1.20453i	−0.798298	−	0.602263i	$-0.794266\pi$
$397$		−	24.0000i		−	1.20453i	0.798298	−	0.602263i	$-0.205734\pi$
$398$		0			0
$399$	1.82843			0.0915358
$400$		0			0
$401$	−25.4142			−1.26913			−0.634563	−	0.772871i	$-0.718820\pi$
$401$	−25.4142			−1.26913			−0.634563	+	0.772871i	$0.718820\pi$
$402$		0			0
$403$		−	36.3848i		−	1.81245i
$404$		0			0
$405$	5.29289	−	15.8787i	0.263006	−	0.789018i
$406$		0			0
$407$			12.0000i			0.594818i
$408$		0			0
$409$	−25.2132			−1.24671			−0.623356	−	0.781938i	$-0.714231\pi$
$409$	−25.2132			−1.24671			−0.623356	+	0.781938i	$0.714231\pi$
$410$		0			0
$411$	5.38478			0.265611
$412$		0			0
$413$			30.4558i			1.49863i
$414$		0			0
$415$	−30.7279	−	10.2426i	−1.50837	−	0.502791i
$416$		0			0
$417$		−	4.97056i		−	0.243410i
$418$		0			0
$419$	19.4142			0.948446			0.474223	−	0.880405i	$-0.342729\pi$
$419$	19.4142			0.948446			0.474223	+	0.880405i	$0.342729\pi$
$420$		0			0
$421$	19.4853			0.949655			0.474827	−	0.880079i	$-0.342511\pi$
$421$	19.4853			0.949655			0.474827	+	0.880079i	$0.342511\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.00000	+	4.00000i	−0.145521	+	0.194029i
$426$		0			0
$427$			62.8701i			3.04250i
$428$		0			0
$429$	−3.41421			−0.164840
$430$		0			0
$431$	6.38478			0.307544			0.153772	−	0.988106i	$-0.450858\pi$
$431$	6.38478			0.307544			0.153772	+	0.988106i	$0.450858\pi$
$432$		0			0
$433$		−	9.55635i		−	0.459249i	−0.973279	−	0.229624i	$-0.926250\pi$
$433$		−	9.55635i		−	0.459249i	0.973279	−	0.229624i	$-0.0737498\pi$
$434$		0			0
$435$	0.150758	+	0.0502525i	0.00722827	+	0.00240942i
$436$		0			0
$437$			0.757359i			0.0362294i
$438$		0			0
$439$	−5.75736			−0.274784			−0.137392	−	0.990517i	$-0.543872\pi$
$439$	−5.75736			−0.274784			−0.137392	+	0.990517i	$0.543872\pi$
$440$		0			0
$441$	−35.3137			−1.68161
$442$		0			0
$443$		−	4.24264i		−	0.201574i	−0.994908	−	0.100787i	$-0.967864\pi$
$443$		−	4.24264i		−	0.201574i	0.994908	−	0.100787i	$-0.0321361\pi$
$444$		0			0
$445$	−5.00000	+	15.0000i	−0.237023	+	0.711068i
$446$		0			0
$447$		−	7.31371i		−	0.345927i
$448$		0			0
$449$	−17.3137			−0.817084			−0.408542	−	0.912739i	$-0.633963\pi$
$449$	−17.3137			−0.817084			−0.408542	+	0.912739i	$0.633963\pi$
$450$		0			0
$451$	−6.00000			−0.282529
$452$		0			0
$453$			4.34315i			0.204059i
$454$		0			0
$455$	−18.1924	+	54.5772i	−0.852872	+	2.55862i
$456$		0			0
$457$			3.00000i			0.140334i	0.997535	+	0.0701670i	$0.0223532\pi$
$457$			3.00000i			0.140334i	−0.997535	+	0.0701670i	$0.977647\pi$
$458$		0			0
$459$	2.41421			0.112686
$460$		0			0
$461$	−3.55635			−0.165636			−0.0828178	−	0.996565i	$-0.526392\pi$
$461$	−3.55635			−0.165636			−0.0828178	+	0.996565i	$0.526392\pi$
$462$		0			0
$463$		−	14.1421i		−	0.657241i	−0.944462	−	0.328620i	$-0.893416\pi$
$463$		−	14.1421i		−	0.657241i	0.944462	−	0.328620i	$-0.106584\pi$
$464$		0			0
$465$	−5.48528	−	1.82843i	−0.254374	−	0.0847913i
$466$		0			0
$467$			0.727922i			0.0336842i	0.999858	+	0.0168421i	$0.00536126\pi$
$467$			0.727922i			0.0336842i	−0.999858	+	0.0168421i	$0.994639\pi$
$468$		0			0
$469$	−21.0000			−0.969690
$470$		0			0
$471$	4.82843			0.222482
$472$		0			0
$473$		−	2.48528i		−	0.114273i
$474$		0			0
$475$	4.00000	+	3.00000i	0.183533	+	0.137649i
$476$		0			0
$477$			15.5147i			0.710370i
$478$		0			0
$479$	−31.1127			−1.42158			−0.710788	−	0.703407i	$-0.751661\pi$
$479$	−31.1127			−1.42158			−0.710788	+	0.703407i	$0.751661\pi$
$480$		0			0
$481$	−49.4558			−2.25499
$482$		0			0
$483$			1.38478i			0.0630095i
$484$		0			0
$485$	0.727922	+	0.242641i	0.0330532	+	0.0110177i
$486$		0			0
$487$			37.7990i			1.71284i	0.516283	+	0.856418i	$0.327315\pi$
$487$			37.7990i			1.71284i	−0.516283	+	0.856418i	$0.672685\pi$
$488$		0			0
$489$	−4.24264			−0.191859
$490$		0			0
$491$	33.5563			1.51438			0.757188	−	0.653197i	$-0.226572\pi$
$491$	33.5563			1.51438			0.757188	+	0.653197i	$0.226572\pi$
$492$		0			0
$493$		−	0.171573i		−	0.00772725i
$494$		0			0
$495$	2.82843	−	8.48528i	0.127128	−	0.381385i
$496$		0			0
$497$			59.2132i			2.65608i
$498$		0			0
$499$	−25.7574			−1.15306			−0.576529	−	0.817077i	$-0.695593\pi$
$499$	−25.7574			−1.15306			−0.576529	+	0.817077i	$0.695593\pi$
$500$		0			0
$501$	7.55635			0.337593
$502$		0			0
$503$		−	14.2721i		−	0.636361i	−0.948030	−	0.318180i	$-0.896928\pi$
$503$		−	14.2721i		−	0.636361i	0.948030	−	0.318180i	$-0.103072\pi$
$504$		0			0
$505$	9.24264	−	27.7279i	0.411292	−	1.23388i
$506$		0			0
$507$		−	8.68629i		−	0.385772i
$508$		0			0
$509$	−28.9706			−1.28410			−0.642049	−	0.766664i	$-0.721915\pi$
$509$	−28.9706			−1.28410			−0.642049	+	0.766664i	$0.721915\pi$
$510$		0			0
$511$	−50.6985			−2.24277
$512$		0			0
$513$		−	2.41421i		−	0.106590i
$514$		0			0
$515$	−9.00000	−	3.00000i	−0.396587	−	0.132196i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−0.201010			−0.00882337
$520$		0			0
$521$	23.3137			1.02139			0.510696	−	0.859761i	$-0.329388\pi$
$521$	23.3137			1.02139			0.510696	+	0.859761i	$0.329388\pi$
$522$		0			0
$523$			2.27208i			0.0993510i	0.998765	+	0.0496755i	$0.0158187\pi$
$523$			2.27208i			0.0993510i	−0.998765	+	0.0496755i	$0.984181\pi$
$524$		0			0
$525$	7.31371	+	5.48528i	0.319196	+	0.239397i
$526$		0			0
$527$			6.24264i			0.271934i
$528$		0			0
$529$	22.4264			0.975061
$530$		0			0
$531$	19.5147			0.846867
$532$		0			0
$533$		−	24.7279i		−	1.07109i
$534$		0			0
$535$	41.8492	+	13.9497i	1.80930	+	0.603100i
$536$		0			0
$537$		−	4.82843i		−	0.208362i
$538$		0			0
$539$	−17.6569			−0.760535
$540$		0			0
$541$	9.75736			0.419502			0.209751	−	0.977755i	$-0.432735\pi$
$541$	9.75736			0.419502			0.209751	+	0.977755i	$0.432735\pi$
$542$		0			0
$543$		−	3.51472i		−	0.150831i
$544$		0			0
$545$	12.7071	−	38.1213i	0.544313	−	1.63294i
$546$		0			0
$547$		−	17.3137i		−	0.740281i	−0.928976	−	0.370140i	$-0.879310\pi$
$547$		−	17.3137i		−	0.740281i	0.928976	−	0.370140i	$-0.120690\pi$
$548$		0			0
$549$	40.2843			1.71929
$550$		0			0
$551$	−0.171573			−0.00730925
$552$		0			0
$553$		−	28.6274i		−	1.21736i
$554$		0			0
$555$	−2.48528	+	7.45584i	−0.105494	+	0.316483i
$556$		0			0
$557$		−	16.0000i		−	0.677942i	−0.940797	−	0.338971i	$-0.889921\pi$
$557$		−	16.0000i		−	0.677942i	0.940797	−	0.338971i	$-0.110079\pi$
$558$		0			0
$559$	10.2426			0.433218
$560$		0			0
$561$	0.585786			0.0247319
$562$		0			0
$563$		−	4.97056i		−	0.209484i	−0.994499	−	0.104742i	$-0.966598\pi$
$563$		−	4.97056i		−	0.209484i	0.994499	−	0.104742i	$-0.0334017\pi$
$564$		0			0
$565$	−21.7279	−	7.24264i	−0.914101	−	0.304700i
$566$		0			0
$567$			33.0416i			1.38762i
$568$		0			0
$569$	28.2843			1.18574			0.592869	−	0.805299i	$-0.297995\pi$
$569$	28.2843			1.18574			0.592869	+	0.805299i	$0.297995\pi$
$570$		0			0
$571$	2.24264			0.0938516			0.0469258	−	0.998898i	$-0.485058\pi$
$571$	2.24264			0.0938516			0.0469258	+	0.998898i	$0.485058\pi$
$572$		0			0
$573$		−	5.20101i		−	0.217275i
$574$		0			0
$575$	−2.27208	+	3.02944i	−0.0947522	+	0.126336i
$576$		0			0
$577$			20.3137i			0.845671i	0.906207	+	0.422835i	$0.138965\pi$
$577$			20.3137i			0.845671i	−0.906207	+	0.422835i	$0.861035\pi$
$578$		0			0
$579$	0.142136			0.00590695
$580$		0			0
$581$	63.9411			2.65272
$582$		0			0
$583$			7.75736i			0.321277i
$584$		0			0
$585$	34.9706	+	11.6569i	1.44585	+	0.481952i
$586$		0			0
$587$			30.2426i			1.24825i	0.781326	+	0.624124i	$0.214544\pi$
$587$			30.2426i			1.24825i	−0.781326	+	0.624124i	$0.785456\pi$
$588$		0			0
$589$	6.24264			0.257224
$590$		0			0
$591$	−4.87006			−0.200327
$592$		0			0
$593$			22.0000i			0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$			22.0000i			0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$		0			0
$595$	3.12132	−	9.36396i	0.127962	−	0.383885i
$596$		0			0
$597$			3.82843i			0.156687i
$598$		0			0
$599$	−15.2132			−0.621595			−0.310797	−	0.950476i	$-0.600596\pi$
$599$	−15.2132			−0.621595			−0.310797	+	0.950476i	$0.600596\pi$
$600$		0			0
$601$	−20.2426			−0.825715			−0.412857	−	0.910796i	$-0.635469\pi$
$601$	−20.2426			−0.825715			−0.412857	+	0.910796i	$0.635469\pi$
$602$		0			0
$603$			13.4558i			0.547964i
$604$		0			0
$605$	−6.36396	+	19.0919i	−0.258732	+	0.776195i
$606$		0			0
$607$			3.17157i			0.128730i	0.997926	+	0.0643651i	$0.0205022\pi$
$607$			3.17157i			0.128730i	−0.997926	+	0.0643651i	$0.979498\pi$
$608$		0			0
$609$	−0.313708			−0.0127121
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	11.6985i		−	0.472497i	−0.971693	−	0.236249i	$-0.924082\pi$
$613$		−	11.6985i		−	0.472497i	0.971693	−	0.236249i	$-0.0759180\pi$
$614$		0			0
$615$	−3.72792	−	1.24264i	−0.150324	−	0.0501081i
$616$		0			0
$617$			4.48528i			0.180571i	0.995916	+	0.0902853i	$0.0287779\pi$
$617$			4.48528i			0.180571i	−0.995916	+	0.0902853i	$0.971222\pi$
$618$		0			0
$619$	7.75736			0.311795			0.155897	−	0.987773i	$-0.450173\pi$
$619$	7.75736			0.311795			0.155897	+	0.987773i	$0.450173\pi$
$620$		0			0
$621$	1.82843			0.0733723
$622$		0			0
$623$		−	31.2132i		−	1.25053i
$624$		0			0
$625$	7.00000	+	24.0000i	0.280000	+	0.960000i
$626$		0			0
$627$		−	0.585786i		−	0.0233941i
$628$		0			0
$629$	8.48528			0.338330
$630$		0			0
$631$	38.9706			1.55139			0.775697	−	0.631106i	$-0.217399\pi$
$631$	38.9706			1.55139			0.775697	+	0.631106i	$0.217399\pi$
$632$		0			0
$633$			8.17157i			0.324791i
$634$		0			0
$635$	−5.27208	−	1.75736i	−0.209216	−	0.0697387i
$636$		0			0
$637$		−	72.7696i		−	2.88323i
$638$		0			0
$639$	37.9411			1.50093
$640$		0			0
$641$	−18.0416			−0.712602			−0.356301	−	0.934371i	$-0.615962\pi$
$641$	−18.0416			−0.712602			−0.356301	+	0.934371i	$0.615962\pi$
$642$		0			0
$643$			14.4853i			0.571244i	0.958342	+	0.285622i	$0.0922001\pi$
$643$			14.4853i			0.571244i	−0.958342	+	0.285622i	$0.907800\pi$
$644$		0			0
$645$	0.514719	−	1.54416i	0.0202670	−	0.0608011i
$646$		0			0
$647$		−	18.7574i		−	0.737428i	−0.929543	−	0.368714i	$-0.879798\pi$
$647$		−	18.7574i		−	0.737428i	0.929543	−	0.368714i	$-0.120202\pi$
$648$		0			0
$649$	9.75736			0.383010
$650$		0			0
$651$	11.4142			0.447358
$652$		0			0
$653$		−	28.9706i		−	1.13371i	−0.823819	−	0.566853i	$-0.808161\pi$
$653$		−	28.9706i		−	1.13371i	0.823819	−	0.566853i	$-0.191839\pi$
$654$		0			0
$655$	12.0000	−	36.0000i	0.468879	−	1.40664i
$656$		0			0
$657$			32.4853i			1.26737i
$658$		0			0
$659$	18.8995			0.736220			0.368110	−	0.929782i	$-0.380005\pi$
$659$	18.8995			0.736220			0.368110	+	0.929782i	$0.380005\pi$
$660$		0			0
$661$	−18.4558			−0.717849			−0.358925	−	0.933367i	$-0.616856\pi$
$661$	−18.4558			−0.717849			−0.358925	+	0.933367i	$0.616856\pi$
$662$		0			0
$663$			2.41421i			0.0937603i
$664$		0			0
$665$	−9.36396	−	3.12132i	−0.363119	−	0.121040i
$666$		0			0
$667$		−	0.129942i		−	0.00503139i
$668$		0			0
$669$	−6.28427			−0.242964
$670$		0			0
$671$	20.1421			0.777579
$672$		0			0
$673$		−	12.0000i		−	0.462566i	−0.972887	−	0.231283i	$-0.925708\pi$
$673$		−	12.0000i		−	0.462566i	0.972887	−	0.231283i	$-0.0742923\pi$
$674$		0			0
$675$	7.24264	−	9.65685i	0.278769	−	0.371692i
$676$		0			0
$677$			40.9411i			1.57350i	0.617275	+	0.786748i	$0.288237\pi$
$677$			40.9411i			1.57350i	−0.617275	+	0.786748i	$0.711763\pi$
$678$		0			0
$679$	−1.51472			−0.0581296
$680$		0			0
$681$	6.94113			0.265985
$682$		0			0
$683$		−	12.0000i		−	0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$		−	12.0000i		−	0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$		0			0
$685$	−27.5772	−	9.19239i	−1.05367	−	0.351223i
$686$		0			0
$687$		−	6.20101i		−	0.236583i
$688$		0			0
$689$	−31.9706			−1.21798
$690$		0			0
$691$	−22.4853			−0.855380			−0.427690	−	0.903925i	$-0.640673\pi$
$691$	−22.4853			−0.855380			−0.427690	+	0.903925i	$0.640673\pi$
$692$		0			0
$693$			17.6569i			0.670728i
$694$		0			0
$695$	−8.48528	+	25.4558i	−0.321865	+	0.965595i
$696$		0			0
$697$			4.24264i			0.160701i
$698$		0			0
$699$	10.3431			0.391214
$700$		0			0
$701$	16.9706			0.640969			0.320485	−	0.947254i	$-0.396154\pi$
$701$	16.9706			0.640969			0.320485	+	0.947254i	$0.396154\pi$
$702$		0			0
$703$		−	8.48528i		−	0.320028i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			57.6985i			2.16997i
$708$		0			0
$709$	34.0000			1.27690			0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000			1.27690			0.638448	+	0.769665i	$0.279577\pi$
$710$		0			0
$711$	−18.3431			−0.687922
$712$		0			0
$713$			4.72792i			0.177062i
$714$		0			0
$715$	17.4853	+	5.82843i	0.653912	+	0.217971i
$716$		0			0
$717$		−	2.85786i		−	0.106729i
$718$		0			0
$719$	−5.10051			−0.190217			−0.0951084	−	0.995467i	$-0.530320\pi$
$719$	−5.10051			−0.190217			−0.0951084	+	0.995467i	$0.530320\pi$
$720$		0			0
$721$	18.7279			0.697464
$722$		0			0
$723$			3.71573i			0.138189i
$724$		0			0
$725$	−0.686292	−	0.514719i	−0.0254882	−	0.0191162i
$726$		0			0
$727$			9.72792i			0.360789i	0.983594	+	0.180394i	$0.0577374\pi$
$727$			9.72792i			0.360789i	−0.983594	+	0.180394i	$0.942263\pi$
$728$		0			0
$729$	18.1716			0.673021
$730$		0			0
$731$	−1.75736			−0.0649983
$732$		0			0
$733$			12.0000i			0.443230i	0.975134	+	0.221615i	$0.0711328\pi$
$733$			12.0000i			0.443230i	−0.975134	+	0.221615i	$0.928867\pi$
$734$		0			0
$735$	−10.9706	−	3.65685i	−0.404655	−	0.134885i
$736$		0			0
$737$			6.72792i			0.247826i
$738$		0			0
$739$	1.27208			0.0467941			0.0233971	−	0.999726i	$-0.492552\pi$
$739$	1.27208			0.0467941			0.0233971	+	0.999726i	$0.492552\pi$
$740$		0			0
$741$	2.41421			0.0886884
$742$		0			0
$743$		−	6.72792i		−	0.246824i	−0.992356	−	0.123412i	$-0.960616\pi$
$743$		−	6.72792i		−	0.246824i	0.992356	−	0.123412i	$-0.0393836\pi$
$744$		0			0
$745$	−12.4853	+	37.4558i	−0.457425	+	1.37228i
$746$		0			0
$747$		−	40.9706i		−	1.49903i
$748$		0			0
$749$	−87.0833			−3.18195
$750$		0			0
$751$	26.7279			0.975316			0.487658	−	0.873035i	$-0.337851\pi$
$751$	26.7279			0.975316			0.487658	+	0.873035i	$0.337851\pi$
$752$		0			0
$753$		−	1.47309i		−	0.0536823i
$754$		0			0
$755$	7.41421	−	22.2426i	0.269831	−	0.809493i
$756$		0			0
$757$		−	12.3431i		−	0.448619i	−0.974518	−	0.224310i	$-0.927987\pi$
$757$		−	12.3431i		−	0.448619i	0.974518	−	0.224310i	$-0.0720127\pi$
$758$		0			0
$759$	0.443651			0.0161035
$760$		0			0
$761$	43.9706			1.59393			0.796966	−	0.604024i	$-0.206437\pi$
$761$	43.9706			1.59393			0.796966	+	0.604024i	$0.206437\pi$
$762$		0			0
$763$			79.3259i			2.87179i
$764$		0			0
$765$	−6.00000	−	2.00000i	−0.216930	−	0.0723102i
$766$		0			0
$767$			40.2132i			1.45201i
$768$		0			0
$769$	36.4558			1.31463			0.657316	−	0.753615i	$-0.271692\pi$
$769$	36.4558			1.31463			0.657316	+	0.753615i	$0.271692\pi$
$770$		0			0
$771$	8.58579			0.309210
$772$		0			0
$773$			13.9706i			0.502486i	0.967924	+	0.251243i	$0.0808394\pi$
$773$			13.9706i			0.502486i	−0.967924	+	0.251243i	$0.919161\pi$
$774$		0			0
$775$	24.9706	+	18.7279i	0.896969	+	0.672727i
$776$		0			0
$777$		−	15.5147i		−	0.556587i
$778$		0			0
$779$	4.24264			0.152008
$780$		0			0
$781$	18.9706			0.678820
$782$		0			0
$783$			0.414214i			0.0148028i
$784$		0			0
$785$	−24.7279	−	8.24264i	−0.882577	−	0.294192i
$786$		0			0
$787$		−	41.1838i		−	1.46804i	−0.679126	−	0.734021i	$-0.737641\pi$
$787$		−	41.1838i		−	1.46804i	0.679126	−	0.734021i	$-0.262359\pi$
$788$		0			0
$789$	−11.1716			−0.397719
$790$		0			0
$791$	45.2132			1.60760
$792$		0			0
$793$			83.0122i			2.94785i
$794$		0			0
$795$	−1.60660	+	4.81981i	−0.0569803	+	0.170941i
$796$		0			0
$797$			47.4853i			1.68201i	0.541023	+	0.841007i	$0.318037\pi$
$797$			47.4853i			1.68201i	−0.541023	+	0.841007i	$0.681963\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−20.0000			−0.706665
$802$		0			0
$803$			16.2426i			0.573190i
$804$		0			0
$805$	2.36396	−	7.09188i	0.0833187	−	0.249956i
$806$		0			0
$807$			6.88730i			0.242444i
$808$		0			0
$809$	28.7990			1.01252			0.506259	−	0.862381i	$-0.331028\pi$
$809$	28.7990			1.01252			0.506259	+	0.862381i	$0.331028\pi$
$810$		0			0
$811$	−52.6985			−1.85049			−0.925247	−	0.379365i	$-0.876142\pi$
$811$	−52.6985			−1.85049			−0.925247	+	0.379365i	$0.876142\pi$
$812$		0			0
$813$			11.2843i			0.395757i
$814$		0			0
$815$	21.7279	+	7.24264i	0.761096	+	0.253699i
$816$		0			0
$817$			1.75736i			0.0614822i
$818$		0			0
$819$	−72.7696			−2.54277
$820$		0			0
$821$	6.34315			0.221377			0.110689	−	0.993855i	$-0.464694\pi$
$821$	6.34315			0.221377			0.110689	+	0.993855i	$0.464694\pi$
$822$		0			0
$823$			15.7279i			0.548241i	0.961695	+	0.274120i	$0.0883867\pi$
$823$			15.7279i			0.548241i	−0.961695	+	0.274120i	$0.911613\pi$
$824$		0			0
$825$	1.75736	−	2.34315i	0.0611834	−	0.0815779i
$826$		0			0
$827$		−	20.6985i		−	0.719757i	−0.932999	−	0.359878i	$-0.882818\pi$
$827$		−	20.6985i		−	0.719757i	0.932999	−	0.359878i	$-0.117182\pi$
$828$		0			0
$829$	−10.4558			−0.363146			−0.181573	−	0.983377i	$-0.558119\pi$
$829$	−10.4558			−0.363146			−0.181573	+	0.983377i	$0.558119\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			12.4853i			0.432589i
$834$		0			0
$835$	−38.6985	−	12.8995i	−1.33922	−	0.446405i
$836$		0			0
$837$		−	15.0711i		−	0.520932i
$838$		0			0
$839$	22.6274			0.781185			0.390593	−	0.920564i	$-0.372270\pi$
$839$	22.6274			0.781185			0.390593	+	0.920564i	$0.372270\pi$
$840$		0			0
$841$	−28.9706			−0.998985
$842$		0			0
$843$			1.75736i			0.0605267i
$844$		0			0
$845$	−14.8284	+	44.4853i	−0.510113	+	1.53034i
$846$		0			0
$847$		−	39.7279i		−	1.36507i
$848$		0			0
$849$	13.3137			0.456925
$850$		0			0
$851$	6.42641			0.220294
$852$		0			0
$853$		−	27.9411i		−	0.956686i	−0.878173	−	0.478343i	$-0.841238\pi$
$853$		−	27.9411i		−	0.956686i	0.878173	−	0.478343i	$-0.158762\pi$
$854$		0			0
$855$	−2.00000	+	6.00000i	−0.0683986	+	0.205196i
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	4.72792			0.161315			0.0806573	−	0.996742i	$-0.474298\pi$
$859$	4.72792			0.161315			0.0806573	+	0.996742i	$0.474298\pi$
$860$		0			0
$861$	7.75736			0.264370
$862$		0			0
$863$		−	0.727922i		−	0.0247788i	−0.999923	−	0.0123894i	$-0.996056\pi$
$863$		−	0.727922i		−	0.0247788i	0.999923	−	0.0123894i	$-0.00394376\pi$
$864$		0			0
$865$	1.02944	+	0.343146i	0.0350019	+	0.0116673i
$866$		0			0
$867$			6.62742i			0.225079i
$868$		0			0
$869$	−9.17157			−0.311124
$870$		0			0
$871$	−27.7279			−0.939525
$872$		0			0
$873$			0.970563i			0.0328486i
$874$		0			0
$875$	−28.0919	−	40.5772i	−0.949679	−	1.37176i
$876$		0			0
$877$			18.8579i			0.636785i	0.947959	+	0.318392i	$0.103143\pi$
$877$			18.8579i			0.636785i	−0.947959	+	0.318392i	$0.896857\pi$
$878$		0			0
$879$	2.27208			0.0766353
$880$		0			0
$881$	−39.5980			−1.33409			−0.667045	−	0.745018i	$-0.732441\pi$
$881$	−39.5980			−1.33409			−0.667045	+	0.745018i	$0.732441\pi$
$882$		0			0
$883$		−	14.4853i		−	0.487469i	−0.969842	−	0.243734i	$-0.921628\pi$
$883$		−	14.4853i		−	0.487469i	0.969842	−	0.243734i	$-0.0783725\pi$
$884$		0			0
$885$	6.06245	+	2.02082i	0.203787	+	0.0679290i
$886$		0			0
$887$		−	45.2132i		−	1.51811i	−0.651026	−	0.759055i	$-0.725661\pi$
$887$		−	45.2132i		−	1.51811i	0.651026	−	0.759055i	$-0.274339\pi$
$888$		0			0
$889$	10.9706			0.367941
$890$		0			0
$891$	10.5858			0.354637
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−8.24264	+	24.7279i	−0.275521	+	0.826563i
$896$		0			0
$897$			1.82843i			0.0610494i
$898$		0			0
$899$	−1.07107			−0.0357221
$900$		0			0
$901$	5.48528			0.182741
$902$		0			0
$903$			3.21320i			0.106929i
$904$		0			0
$905$	−6.00000	+	18.0000i	−0.199447	+	0.598340i
$906$		0			0
$907$			38.6985i			1.28496i	0.766302	+	0.642481i	$0.222095\pi$
$907$			38.6985i			1.28496i	−0.766302	+	0.642481i	$0.777905\pi$
$908$		0			0
$909$	36.9706			1.22624
$910$		0			0
$911$	−40.2843			−1.33468			−0.667339	−	0.744754i	$-0.732567\pi$
$911$	−40.2843			−1.33468			−0.667339	+	0.744754i	$0.732567\pi$
$912$		0			0
$913$		−	20.4853i		−	0.677964i
$914$		0			0
$915$	12.5147	+	4.17157i	0.413724	+	0.137908i
$916$		0			0
$917$			74.9117i			2.47380i
$918$		0			0
$919$	−6.21320			−0.204955			−0.102477	−	0.994735i	$-0.532677\pi$
$919$	−6.21320			−0.204955			−0.102477	+	0.994735i	$0.532677\pi$
$920$		0			0
$921$	7.31371			0.240995
$922$		0			0
$923$			78.1838i			2.57345i
$924$		0			0
$925$	25.4558	−	33.9411i	0.836983	−	1.11598i
$926$		0			0
$927$		−	12.0000i		−	0.394132i
$928$		0			0
$929$	12.1716			0.399336			0.199668	−	0.979864i	$-0.436014\pi$
$929$	12.1716			0.399336			0.199668	+	0.979864i	$0.436014\pi$
$930$		0			0
$931$	12.4853			0.409189
$932$		0			0
$933$			1.97056i			0.0645133i
$934$		0			0
$935$	−3.00000	−	1.00000i	−0.0981105	−	0.0327035i
$936$		0			0
$937$		−	27.0000i		−	0.882052i	−0.897494	−	0.441026i	$-0.854615\pi$
$937$		−	27.0000i		−	0.882052i	0.897494	−	0.441026i	$-0.145385\pi$
$938$		0			0
$939$	3.30152			0.107741
$940$		0			0
$941$	53.1421			1.73238			0.866192	−	0.499711i	$-0.166561\pi$
$941$	53.1421			1.73238			0.866192	+	0.499711i	$0.166561\pi$
$942$		0			0
$943$			3.21320i			0.104636i
$944$		0			0
$945$	−7.53553	+	22.6066i	−0.245131	+	0.735393i
$946$		0			0
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	−66.9411			−2.17300
$950$		0			0
$951$	−3.92893			−0.127404
$952$		0			0
$953$			18.7279i			0.606657i	0.952886	+	0.303328i	$0.0980979\pi$
$953$			18.7279i			0.606657i	−0.952886	+	0.303328i	$0.901902\pi$
$954$		0			0
$955$	−8.87868	+	26.6360i	−0.287307	+	0.861922i
$956$		0			0
$957$			0.100505i			0.00324887i
$958$		0			0
$959$	57.3848			1.85305
$960$		0			0
$961$	7.97056			0.257115
$962$		0			0
$963$			55.7990i			1.79810i
$964$		0			0
$965$	−0.727922	−	0.242641i	−0.0234326	−	0.00781088i
$966$		0			0
$967$			43.1127i			1.38641i	0.720740	+	0.693205i	$0.243802\pi$
$967$			43.1127i			1.38641i	−0.720740	+	0.693205i	$0.756198\pi$
$968$		0			0
$969$	−0.414214			−0.0133065
$970$		0			0
$971$	−41.6569			−1.33683			−0.668416	−	0.743788i	$-0.733027\pi$
$971$	−41.6569			−1.33683			−0.668416	+	0.743788i	$0.733027\pi$
$972$		0			0
$973$		−	52.9706i		−	1.69816i
$974$		0			0
$975$	9.65685	+	7.24264i	0.309267	+	0.231950i
$976$		0			0
$977$		−	0.242641i		−	0.00776276i	−0.999992	−	0.00388138i	$-0.998765\pi$
$977$		−	0.242641i		−	0.00776276i	0.999992	−	0.00388138i	$-0.00123549\pi$
$978$		0			0
$979$	−10.0000			−0.319601
$980$		0			0
$981$	50.8284			1.62283
$982$		0			0
$983$			51.4558i			1.64119i	0.571513	+	0.820593i	$0.306357\pi$
$983$			51.4558i			1.64119i	−0.571513	+	0.820593i	$0.693643\pi$
$984$		0			0
$985$	24.9411	+	8.31371i	0.794690	+	0.264897i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.33095			−0.0423218
$990$		0			0
$991$	−13.7574			−0.437017			−0.218508	−	0.975835i	$-0.570119\pi$
$991$	−13.7574			−0.437017			−0.218508	+	0.975835i	$0.570119\pi$
$992$		0			0
$993$			7.97056i			0.252938i
$994$		0			0
$995$	6.53553	−	19.6066i	0.207190	−	0.621571i
$996$		0			0
$997$		−	48.7279i		−	1.54323i	−0.636091	−	0.771614i	$-0.719450\pi$
$997$		−	48.7279i		−	1.54323i	0.636091	−	0.771614i	$-0.280550\pi$
$998$		0			0
$999$	−20.4853			−0.648126

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.d.e.609.3		4
4.3	odd	2		190.2.b.a.39.3	yes	4
5.2	odd	4		7600.2.a.v.1.2		2
5.3	odd	4		7600.2.a.bg.1.1		2
5.4	even	2	inner	1520.2.d.e.609.2		4
12.11	even	2		1710.2.d.c.1369.1		4
20.3	even	4		950.2.a.g.1.2		2
20.7	even	4		950.2.a.f.1.1		2
20.19	odd	2		190.2.b.a.39.2	✓	4
60.23	odd	4		8550.2.a.bn.1.1		2
60.47	odd	4		8550.2.a.cb.1.2		2
60.59	even	2		1710.2.d.c.1369.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.a.39.2	✓	4	20.19	odd	2
190.2.b.a.39.3	yes	4	4.3	odd	2
950.2.a.f.1.1		2	20.7	even	4
950.2.a.g.1.2		2	20.3	even	4
1520.2.d.e.609.2		4	5.4	even	2	inner
1520.2.d.e.609.3		4	1.1	even	1	trivial
1710.2.d.c.1369.1		4	12.11	even	2
1710.2.d.c.1369.3		4	60.59	even	2
7600.2.a.v.1.2		2	5.2	odd	4
7600.2.a.bg.1.1		2	5.3	odd	4
8550.2.a.bn.1.1		2	60.23	odd	4
8550.2.a.cb.1.2		2	60.47	odd	4

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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.414214i q^{3} +(0.707107 - 2.12132i) q^{5} +4.41421i q^{7} +2.82843 q^{9} +O(q^{10})\)	\(q+0.414214i q^{3} +(0.707107 - 2.12132i) q^{5} +4.41421i q^{7} +2.82843 q^{9} +1.41421 q^{11} +5.82843i q^{13} +(0.878680 + 0.292893i) q^{15} -1.00000i q^{17} -1.00000 q^{19} -1.82843 q^{21} -0.757359i q^{23} +(-4.00000 - 3.00000i) q^{25} +2.41421i q^{27} +0.171573 q^{29} -6.24264 q^{31} +0.585786i q^{33} +(9.36396 + 3.12132i) q^{35} +8.48528i q^{37} -2.41421 q^{39} -4.24264 q^{41} -1.75736i q^{43} +(2.00000 - 6.00000i) q^{45} -12.4853 q^{49} +0.414214 q^{51} +5.48528i q^{53} +(1.00000 - 3.00000i) q^{55} -0.414214i q^{57} +6.89949 q^{59} +14.2426 q^{61} +12.4853i q^{63} +(12.3640 + 4.12132i) q^{65} +4.75736i q^{67} +0.313708 q^{69} +13.4142 q^{71} +11.4853i q^{73} +(1.24264 - 1.65685i) q^{75} +6.24264i q^{77} -6.48528 q^{79} +7.48528 q^{81} -14.4853i q^{83} +(-2.12132 - 0.707107i) q^{85} +0.0710678i q^{87} -7.07107 q^{89} -25.7279 q^{91} -2.58579i q^{93} +(-0.707107 + 2.12132i) q^{95} +0.343146i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 12 q^{15} - 4 q^{19} + 4 q^{21} - 16 q^{25} + 12 q^{29} - 8 q^{31} + 12 q^{35} - 4 q^{39} + 8 q^{45} - 16 q^{49} - 4 q^{51} + 4 q^{55} - 12 q^{59} + 40 q^{61} + 24 q^{65} - 44 q^{69} + 48 q^{71} - 12 q^{75} + 8 q^{79} - 4 q^{81} - 52 q^{91} + 16 q^{99}+O(q^{100}) \) 4 * q + 12 * q^15 - 4 * q^19 + 4 * q^21 - 16 * q^25 + 12 * q^29 - 8 * q^31 + 12 * q^35 - 4 * q^39 + 8 * q^45 - 16 * q^49 - 4 * q^51 + 4 * q^55 - 12 * q^59 + 40 * q^61 + 24 * q^65 - 44 * q^69 + 48 * q^71 - 12 * q^75 + 8 * q^79 - 4 * q^81 - 52 * q^91 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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