LMFDB - Embedded newform 760.2.d.c.609.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [760,2,Mod(609,760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(760, 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("760.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 760 = 2^{3} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	760.d (of order $2$, degree $1$, 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:	$6.06863055362$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	760.609
Dual form			760.2.d.c.609.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421i q^{3} +(2.12132 + 0.707107i) q^{5} -2.41421i q^{7} -2.82843 q^{9} +O(q^{10})$	$q-2.41421i q^{3} +(2.12132 + 0.707107i) q^{5} -2.41421i q^{7} -2.82843 q^{9} +1.41421 q^{11} -1.82843i q^{13} +(1.70711 - 5.12132i) q^{15} +1.00000i q^{17} +1.00000 q^{19} -5.82843 q^{21} -5.24264i q^{23} +(4.00000 + 3.00000i) q^{25} -0.414214i q^{27} -3.82843 q^{29} +3.41421 q^{31} -3.41421i q^{33} +(1.70711 - 5.12132i) q^{35} +5.17157i q^{37} -4.41421 q^{39} -7.07107 q^{41} -2.24264i q^{43} +(-6.00000 - 2.00000i) q^{45} +8.00000i q^{47} +1.17157 q^{49} +2.41421 q^{51} +1.82843i q^{53} +(3.00000 + 1.00000i) q^{55} -2.41421i q^{57} -14.4142 q^{59} +3.41421 q^{61} +6.82843i q^{63} +(1.29289 - 3.87868i) q^{65} -6.07107i q^{67} -12.6569 q^{69} +3.07107 q^{71} -13.8284i q^{73} +(7.24264 - 9.65685i) q^{75} -3.41421i q^{77} +0.828427 q^{79} -9.48528 q^{81} +2.48528i q^{83} +(-0.707107 + 2.12132i) q^{85} +9.24264i q^{87} +3.75736 q^{89} -4.41421 q^{91} -8.24264i q^{93} +(2.12132 + 0.707107i) q^{95} +7.65685i q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 4 q^{15} + 4 q^{19} - 12 q^{21} + 16 q^{25} - 4 q^{29} + 8 q^{31} + 4 q^{35} - 12 q^{39} - 24 q^{45} + 16 q^{49} + 4 q^{51} + 12 q^{55} - 52 q^{59} + 8 q^{61} + 8 q^{65} - 28 q^{69} - 16 q^{71} + 12 q^{75} - 8 q^{79} - 4 q^{81} + 32 q^{89} - 12 q^{91} - 16 q^{99}+O(q^{100}) $ 4 * q + 4 * q^15 + 4 * q^19 - 12 * q^21 + 16 * q^25 - 4 * q^29 + 8 * q^31 + 4 * q^35 - 12 * q^39 - 24 * q^45 + 16 * q^49 + 4 * q^51 + 12 * q^55 - 52 * q^59 + 8 * q^61 + 8 * q^65 - 28 * q^69 - 16 * q^71 + 12 * q^75 - 8 * q^79 - 4 * q^81 + 32 * q^89 - 12 * q^91 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/760\mathbb{Z}\right)^\times$.

$n$	$191$	$381$	$401$	$457$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	2.41421i		−	1.39385i	−0.717146	−	0.696923i	$-0.754552\pi$
$3$		−	2.41421i		−	1.39385i	0.717146	−	0.696923i	$-0.245448\pi$
$4$		0			0
$5$	2.12132	+	0.707107i	0.948683	+	0.316228i
$5$	2.12132	+	0.707107i	0.948683	+	0.316228i
$6$		0			0
$7$		−	2.41421i		−	0.912487i	−0.889855	−	0.456243i	$-0.849195\pi$
$7$		−	2.41421i		−	0.912487i	0.889855	−	0.456243i	$-0.150805\pi$
$8$		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
$13$		−	1.82843i		−	0.507114i	−0.967320	−	0.253557i	$-0.918399\pi$
$13$		−	1.82843i		−	0.507114i	0.967320	−	0.253557i	$-0.0816006\pi$
$14$		0			0
$15$	1.70711	−	5.12132i	0.440773	−	1.32232i
$16$		0			0
$17$			1.00000i			0.242536i	0.992620	+	0.121268i	$0.0386960\pi$
$17$			1.00000i			0.242536i	−0.992620	+	0.121268i	$0.961304\pi$
$18$		0			0
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$		0			0
$21$	−5.82843			−1.27187
$22$		0			0
$23$		−	5.24264i		−	1.09317i	−0.837405	−	0.546583i	$-0.815928\pi$
$23$		−	5.24264i		−	1.09317i	0.837405	−	0.546583i	$-0.184072\pi$
$24$		0			0
$25$	4.00000	+	3.00000i	0.800000	+	0.600000i
$26$		0			0
$27$		−	0.414214i		−	0.0797154i
$28$		0			0
$29$	−3.82843			−0.710921			−0.355461	−	0.934691i	$-0.615676\pi$
$29$	−3.82843			−0.710921			−0.355461	+	0.934691i	$0.615676\pi$
$30$		0			0
$31$	3.41421			0.613211			0.306605	−	0.951837i	$-0.400807\pi$
$31$	3.41421			0.613211			0.306605	+	0.951837i	$0.400807\pi$
$32$		0			0
$33$		−	3.41421i		−	0.594338i
$34$		0			0
$35$	1.70711	−	5.12132i	0.288554	−	0.865661i
$36$		0			0
$37$			5.17157i			0.850201i	0.905146	+	0.425101i	$0.139761\pi$
$37$			5.17157i			0.850201i	−0.905146	+	0.425101i	$0.860239\pi$
$38$		0			0
$39$	−4.41421			−0.706840
$40$		0			0
$41$	−7.07107			−1.10432			−0.552158	−	0.833740i	$-0.686195\pi$
$41$	−7.07107			−1.10432			−0.552158	+	0.833740i	$0.686195\pi$
$42$		0			0
$43$		−	2.24264i		−	0.341999i	−0.985271	−	0.171000i	$-0.945300\pi$
$43$		−	2.24264i		−	0.341999i	0.985271	−	0.171000i	$-0.0546997\pi$
$44$		0			0
$45$	−6.00000	−	2.00000i	−0.894427	−	0.298142i
$46$		0			0
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	1.17157			0.167368
$50$		0			0
$51$	2.41421			0.338058
$52$		0			0
$53$			1.82843i			0.251154i	0.992084	+	0.125577i	$0.0400782\pi$
$53$			1.82843i			0.251154i	−0.992084	+	0.125577i	$0.959922\pi$
$54$		0			0
$55$	3.00000	+	1.00000i	0.404520	+	0.134840i
$56$		0			0
$57$		−	2.41421i		−	0.319770i
$58$		0			0
$59$	−14.4142			−1.87657			−0.938285	−	0.345862i	$-0.887587\pi$
$59$	−14.4142			−1.87657			−0.938285	+	0.345862i	$0.887587\pi$
$60$		0			0
$61$	3.41421			0.437145			0.218573	−	0.975821i	$-0.429860\pi$
$61$	3.41421			0.437145			0.218573	+	0.975821i	$0.429860\pi$
$62$		0			0
$63$			6.82843i			0.860301i
$64$		0			0
$65$	1.29289	−	3.87868i	0.160364	−	0.481091i
$66$		0			0
$67$		−	6.07107i		−	0.741699i	−0.928693	−	0.370849i	$-0.879067\pi$
$67$		−	6.07107i		−	0.741699i	0.928693	−	0.370849i	$-0.120933\pi$
$68$		0			0
$69$	−12.6569			−1.52371
$70$		0			0
$71$	3.07107			0.364469			0.182234	−	0.983255i	$-0.441667\pi$
$71$	3.07107			0.364469			0.182234	+	0.983255i	$0.441667\pi$
$72$		0			0
$73$		−	13.8284i		−	1.61849i	−0.587468	−	0.809247i	$-0.699875\pi$
$73$		−	13.8284i		−	1.61849i	0.587468	−	0.809247i	$-0.300125\pi$
$74$		0			0
$75$	7.24264	−	9.65685i	0.836308	−	1.11508i
$76$		0			0
$77$		−	3.41421i		−	0.389086i
$78$		0			0
$79$	0.828427			0.0932053			0.0466027	−	0.998914i	$-0.485161\pi$
$79$	0.828427			0.0932053			0.0466027	+	0.998914i	$0.485161\pi$
$80$		0			0
$81$	−9.48528			−1.05392
$82$		0			0
$83$			2.48528i			0.272795i	0.990654	+	0.136398i	$0.0435524\pi$
$83$			2.48528i			0.272795i	−0.990654	+	0.136398i	$0.956448\pi$
$84$		0			0
$85$	−0.707107	+	2.12132i	−0.0766965	+	0.230089i
$86$		0			0
$87$			9.24264i			0.990915i
$88$		0			0
$89$	3.75736			0.398279			0.199140	−	0.979971i	$-0.436185\pi$
$89$	3.75736			0.398279			0.199140	+	0.979971i	$0.436185\pi$
$90$		0			0
$91$	−4.41421			−0.462735
$92$		0			0
$93$		−	8.24264i		−	0.854722i
$94$		0			0
$95$	2.12132	+	0.707107i	0.217643	+	0.0725476i
$96$		0			0
$97$			7.65685i			0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$97$			7.65685i			0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$98$		0			0
$99$	−4.00000			−0.402015
$100$		0			0
$101$	15.8995			1.58206			0.791029	−	0.611778i	$-0.209545\pi$
$101$	15.8995			1.58206			0.791029	+	0.611778i	$0.209545\pi$
$102$		0			0
$103$			9.89949i			0.975426i	0.873004	+	0.487713i	$0.162169\pi$
$103$			9.89949i			0.975426i	−0.873004	+	0.487713i	$0.837831\pi$
$104$		0			0
$105$	−12.3640	−	4.12132i	−1.20660	−	0.402200i
$106$		0			0
$107$		−	2.41421i		−	0.233391i	−0.993168	−	0.116695i	$-0.962770\pi$
$107$		−	2.41421i		−	0.233391i	0.993168	−	0.116695i	$-0.0372301\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$		0			0
$111$	12.4853			1.18505
$112$		0			0
$113$			13.0711i			1.22962i	0.788674	+	0.614811i	$0.210768\pi$
$113$			13.0711i			1.22962i	−0.788674	+	0.614811i	$0.789232\pi$
$114$		0			0
$115$	3.70711	−	11.1213i	0.345689	−	1.03707i
$116$		0			0
$117$			5.17157i			0.478112i
$118$		0			0
$119$	2.41421			0.221311
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$			17.0711i			1.53925i
$124$		0			0
$125$	6.36396	+	9.19239i	0.569210	+	0.822192i
$126$		0			0
$127$			11.1716i			0.991317i	0.868518	+	0.495658i	$0.165073\pi$
$127$			11.1716i			0.991317i	−0.868518	+	0.495658i	$0.834927\pi$
$128$		0			0
$129$	−5.41421			−0.476695
$130$		0			0
$131$	21.6569			1.89217			0.946084	−	0.323921i	$-0.105001\pi$
$131$	21.6569			1.89217			0.946084	+	0.323921i	$0.105001\pi$
$132$		0			0
$133$		−	2.41421i		−	0.209339i
$134$		0			0
$135$	0.292893	−	0.878680i	0.0252082	−	0.0756247i
$136$		0			0
$137$			16.3137i			1.39377i	0.717181	+	0.696887i	$0.245432\pi$
$137$			16.3137i			1.39377i	−0.717181	+	0.696887i	$0.754568\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$	19.3137			1.62651
$142$		0			0
$143$		−	2.58579i		−	0.216234i
$144$		0			0
$145$	−8.12132	−	2.70711i	−0.674439	−	0.224813i
$146$		0			0
$147$		−	2.82843i		−	0.233285i
$148$		0			0
$149$	17.6569			1.44651			0.723253	−	0.690583i	$-0.242646\pi$
$149$	17.6569			1.44651			0.723253	+	0.690583i	$0.242646\pi$
$150$		0			0
$151$	11.1716			0.909130			0.454565	−	0.890714i	$-0.349795\pi$
$151$	11.1716			0.909130			0.454565	+	0.890714i	$0.349795\pi$
$152$		0			0
$153$		−	2.82843i		−	0.228665i
$154$		0			0
$155$	7.24264	+	2.41421i	0.581743	+	0.193914i
$156$		0			0
$157$			0.343146i			0.0273860i	0.999906	+	0.0136930i	$0.00435876\pi$
$157$			0.343146i			0.0273860i	−0.999906	+	0.0136930i	$0.995641\pi$
$158$		0			0
$159$	4.41421			0.350070
$160$		0			0
$161$	−12.6569			−0.997500
$162$		0			0
$163$		−	2.92893i		−	0.229412i	−0.993400	−	0.114706i	$-0.963407\pi$
$163$		−	2.92893i		−	0.229412i	0.993400	−	0.114706i	$-0.0365925\pi$
$164$		0			0
$165$	2.41421	−	7.24264i	0.187946	−	0.563839i
$166$		0			0
$167$			23.2132i			1.79629i	0.439698	+	0.898146i	$0.355086\pi$
$167$			23.2132i			1.79629i	−0.439698	+	0.898146i	$0.644914\pi$
$168$		0			0
$169$	9.65685			0.742835
$170$		0			0
$171$	−2.82843			−0.216295
$172$		0			0
$173$		−	18.8284i		−	1.43150i	−0.698357	−	0.715749i	$-0.746085\pi$
$173$		−	18.8284i		−	1.43150i	0.698357	−	0.715749i	$-0.253915\pi$
$174$		0			0
$175$	7.24264	−	9.65685i	0.547492	−	0.729990i
$176$		0			0
$177$			34.7990i			2.61565i
$178$		0			0
$179$	22.9706			1.71690			0.858450	−	0.512897i	$-0.171428\pi$
$179$	22.9706			1.71690			0.858450	+	0.512897i	$0.171428\pi$
$180$		0			0
$181$	0.485281			0.0360707			0.0180353	−	0.999837i	$-0.494259\pi$
$181$	0.485281			0.0360707			0.0180353	+	0.999837i	$0.494259\pi$
$182$		0			0
$183$		−	8.24264i		−	0.609314i
$184$		0			0
$185$	−3.65685	+	10.9706i	−0.268857	+	0.806572i
$186$		0			0
$187$			1.41421i			0.103418i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	−11.2426			−0.813489			−0.406744	−	0.913542i	$-0.633336\pi$
$191$	−11.2426			−0.813489			−0.406744	+	0.913542i	$0.633336\pi$
$192$		0			0
$193$		−	7.65685i		−	0.551152i	−0.961279	−	0.275576i	$-0.911131\pi$
$193$		−	7.65685i		−	0.551152i	0.961279	−	0.275576i	$-0.0888686\pi$
$194$		0			0
$195$	−9.36396	−	3.12132i	−0.670567	−	0.223522i
$196$		0			0
$197$			20.7279i			1.47680i	0.674361	+	0.738402i	$0.264419\pi$
$197$			20.7279i			1.47680i	−0.674361	+	0.738402i	$0.735581\pi$
$198$		0			0
$199$	−16.0711			−1.13925			−0.569624	−	0.821905i	$-0.692911\pi$
$199$	−16.0711			−1.13925			−0.569624	+	0.821905i	$0.692911\pi$
$200$		0			0
$201$	−14.6569			−1.03381
$202$		0			0
$203$			9.24264i			0.648706i
$204$		0			0
$205$	−15.0000	−	5.00000i	−1.04765	−	0.349215i
$206$		0			0
$207$			14.8284i			1.03065i
$208$		0			0
$209$	1.41421			0.0978232
$210$		0			0
$211$	−24.8995			−1.71415			−0.857076	−	0.515190i	$-0.827721\pi$
$211$	−24.8995			−1.71415			−0.857076	+	0.515190i	$0.827721\pi$
$212$		0			0
$213$		−	7.41421i		−	0.508014i
$214$		0			0
$215$	1.58579	−	4.75736i	0.108150	−	0.324449i
$216$		0			0
$217$		−	8.24264i		−	0.559547i
$218$		0			0
$219$	−33.3848			−2.25593
$220$		0			0
$221$	1.82843			0.122993
$222$		0			0
$223$		−	3.17157i		−	0.212384i	−0.994346	−	0.106192i	$-0.966134\pi$
$223$		−	3.17157i		−	0.212384i	0.994346	−	0.106192i	$-0.0338659\pi$
$224$		0			0
$225$	−11.3137	−	8.48528i	−0.754247	−	0.565685i
$226$		0			0
$227$		−	17.2426i		−	1.14443i	−0.820102	−	0.572217i	$-0.806083\pi$
$227$		−	17.2426i		−	1.14443i	0.820102	−	0.572217i	$-0.193917\pi$
$228$		0			0
$229$	26.9706			1.78226			0.891132	−	0.453743i	$-0.149912\pi$
$229$	26.9706			1.78226			0.891132	+	0.453743i	$0.149912\pi$
$230$		0			0
$231$	−8.24264			−0.542326
$232$		0			0
$233$			2.34315i			0.153505i	0.997050	+	0.0767523i	$0.0244551\pi$
$233$			2.34315i			0.153505i	−0.997050	+	0.0767523i	$0.975545\pi$
$234$		0			0
$235$	−5.65685	+	16.9706i	−0.369012	+	1.10704i
$236$		0			0
$237$		−	2.00000i		−	0.129914i
$238$		0			0
$239$	−2.27208			−0.146969			−0.0734843	−	0.997296i	$-0.523412\pi$
$239$	−2.27208			−0.146969			−0.0734843	+	0.997296i	$0.523412\pi$
$240$		0			0
$241$	5.65685			0.364390			0.182195	−	0.983262i	$-0.441680\pi$
$241$	5.65685			0.364390			0.182195	+	0.983262i	$0.441680\pi$
$242$		0			0
$243$			21.6569i			1.38929i
$244$		0			0
$245$	2.48528	+	0.828427i	0.158779	+	0.0529263i
$246$		0			0
$247$		−	1.82843i		−	0.116340i
$248$		0			0
$249$	6.00000			0.380235
$250$		0			0
$251$	−27.5563			−1.73934			−0.869671	−	0.493632i	$-0.835669\pi$
$251$	−27.5563			−1.73934			−0.869671	+	0.493632i	$0.835669\pi$
$252$		0			0
$253$		−	7.41421i		−	0.466128i
$254$		0			0
$255$	5.12132	+	1.70711i	0.320710	+	0.106903i
$256$		0			0
$257$			3.27208i			0.204107i	0.994779	+	0.102053i	$0.0325412\pi$
$257$			3.27208i			0.204107i	−0.994779	+	0.102053i	$0.967459\pi$
$258$		0			0
$259$	12.4853			0.775798
$260$		0			0
$261$	10.8284			0.670263
$262$		0			0
$263$			4.34315i			0.267810i	0.990994	+	0.133905i	$0.0427517\pi$
$263$			4.34315i			0.267810i	−0.990994	+	0.133905i	$0.957248\pi$
$264$		0			0
$265$	−1.29289	+	3.87868i	−0.0794218	+	0.238265i
$266$		0			0
$267$		−	9.07107i		−	0.555140i
$268$		0			0
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$	−21.3848			−1.29903			−0.649516	−	0.760348i	$-0.725029\pi$
$271$	−21.3848			−1.29903			−0.649516	+	0.760348i	$0.725029\pi$
$272$		0			0
$273$			10.6569i			0.644982i
$274$		0			0
$275$	5.65685	+	4.24264i	0.341121	+	0.255841i
$276$		0			0
$277$			19.3137i			1.16045i	0.814457	+	0.580224i	$0.197035\pi$
$277$			19.3137i			1.16045i	−0.814457	+	0.580224i	$0.802965\pi$
$278$		0			0
$279$	−9.65685			−0.578141
$280$		0			0
$281$	−12.2426			−0.730335			−0.365167	−	0.930942i	$-0.618988\pi$
$281$	−12.2426			−0.730335			−0.365167	+	0.930942i	$0.618988\pi$
$282$		0			0
$283$		−	26.4853i		−	1.57439i	−0.616706	−	0.787193i	$-0.711533\pi$
$283$		−	26.4853i		−	1.57439i	0.616706	−	0.787193i	$-0.288467\pi$
$284$		0			0
$285$	1.70711	−	5.12132i	0.101120	−	0.303361i
$286$		0			0
$287$			17.0711i			1.00767i
$288$		0			0
$289$	16.0000			0.941176
$290$		0			0
$291$	18.4853			1.08363
$292$		0			0
$293$			28.7990i			1.68245i	0.540681	+	0.841227i	$0.318166\pi$
$293$			28.7990i			1.68245i	−0.540681	+	0.841227i	$0.681834\pi$
$294$		0			0
$295$	−30.5772	−	10.1924i	−1.78027	−	0.593424i
$296$		0			0
$297$		−	0.585786i		−	0.0339908i
$298$		0			0
$299$	−9.58579			−0.554360
$300$		0			0
$301$	−5.41421			−0.312070
$302$		0			0
$303$		−	38.3848i		−	2.20515i
$304$		0			0
$305$	7.24264	+	2.41421i	0.414712	+	0.138237i
$306$		0			0
$307$		−	24.2843i		−	1.38598i	−0.720949	−	0.692988i	$-0.756294\pi$
$307$		−	24.2843i		−	1.38598i	0.720949	−	0.692988i	$-0.243706\pi$
$308$		0			0
$309$	23.8995			1.35959
$310$		0			0
$311$	14.7574			0.836813			0.418407	−	0.908260i	$-0.362589\pi$
$311$	14.7574			0.836813			0.418407	+	0.908260i	$0.362589\pi$
$312$		0			0
$313$			4.65685i			0.263221i	0.991302	+	0.131610i	$0.0420148\pi$
$313$			4.65685i			0.263221i	−0.991302	+	0.131610i	$0.957985\pi$
$314$		0			0
$315$	−4.82843	+	14.4853i	−0.272051	+	0.816153i
$316$		0			0
$317$		−	32.7990i		−	1.84217i	−0.389356	−	0.921087i	$-0.627302\pi$
$317$		−	32.7990i		−	1.84217i	0.389356	−	0.921087i	$-0.372698\pi$
$318$		0			0
$319$	−5.41421			−0.303138
$320$		0			0
$321$	−5.82843			−0.325311
$322$		0			0
$323$			1.00000i			0.0556415i
$324$		0			0
$325$	5.48528	−	7.31371i	0.304269	−	0.405692i
$326$		0			0
$327$		−	12.0711i		−	0.667532i
$328$		0			0
$329$	19.3137			1.06480
$330$		0			0
$331$	−17.3848			−0.955554			−0.477777	−	0.878481i	$-0.658557\pi$
$331$	−17.3848			−0.955554			−0.477777	+	0.878481i	$0.658557\pi$
$332$		0			0
$333$		−	14.6274i		−	0.801578i
$334$		0			0
$335$	4.29289	−	12.8787i	0.234546	−	0.703637i
$336$		0			0
$337$		−	12.7279i		−	0.693334i	−0.937988	−	0.346667i	$-0.887313\pi$
$337$		−	12.7279i		−	0.693334i	0.937988	−	0.346667i	$-0.112687\pi$
$338$		0			0
$339$	31.5563			1.71391
$340$		0			0
$341$	4.82843			0.261474
$342$		0			0
$343$		−	19.7279i		−	1.06521i
$344$		0			0
$345$	−26.8492	−	8.94975i	−1.44551	−	0.481838i
$346$		0			0
$347$			20.8284i			1.11813i	0.829124	+	0.559064i	$0.188840\pi$
$347$			20.8284i			1.11813i	−0.829124	+	0.559064i	$0.811160\pi$
$348$		0			0
$349$	16.6274			0.890045			0.445023	−	0.895519i	$-0.353196\pi$
$349$	16.6274			0.890045			0.445023	+	0.895519i	$0.353196\pi$
$350$		0			0
$351$	−0.757359			−0.0404248
$352$		0			0
$353$			26.1127i			1.38984i	0.719088	+	0.694919i	$0.244560\pi$
$353$			26.1127i			1.38984i	−0.719088	+	0.694919i	$0.755440\pi$
$354$		0			0
$355$	6.51472	+	2.17157i	0.345765	+	0.115255i
$356$		0			0
$357$		−	5.82843i		−	0.308473i
$358$		0			0
$359$	−8.07107			−0.425975			−0.212987	−	0.977055i	$-0.568319\pi$
$359$	−8.07107			−0.425975			−0.212987	+	0.977055i	$0.568319\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$			21.7279i			1.14042i
$364$		0			0
$365$	9.77817	−	29.3345i	0.511813	−	1.53544i
$366$		0			0
$367$			17.4558i			0.911188i	0.890188	+	0.455594i	$0.150573\pi$
$367$			17.4558i			0.911188i	−0.890188	+	0.455594i	$0.849427\pi$
$368$		0			0
$369$	20.0000			1.04116
$370$		0			0
$371$	4.41421			0.229175
$372$		0			0
$373$		−	5.97056i		−	0.309144i	−0.987982	−	0.154572i	$-0.950600\pi$
$373$		−	5.97056i		−	0.309144i	0.987982	−	0.154572i	$-0.0493999\pi$
$374$		0			0
$375$	22.1924	−	15.3640i	1.14601	−	0.793392i
$376$		0			0
$377$			7.00000i			0.360518i
$378$		0			0
$379$	−33.3848			−1.71486			−0.857430	−	0.514600i	$-0.827940\pi$
$379$	−33.3848			−1.71486			−0.857430	+	0.514600i	$0.827940\pi$
$380$		0			0
$381$	26.9706			1.38174
$382$		0			0
$383$		−	4.72792i		−	0.241586i	−0.992678	−	0.120793i	$-0.961456\pi$
$383$		−	4.72792i		−	0.241586i	0.992678	−	0.120793i	$-0.0385437\pi$
$384$		0			0
$385$	2.41421	−	7.24264i	0.123040	−	0.369119i
$386$		0			0
$387$			6.34315i			0.322440i
$388$		0			0
$389$	11.8995			0.603328			0.301664	−	0.953414i	$-0.402458\pi$
$389$	11.8995			0.603328			0.301664	+	0.953414i	$0.402458\pi$
$390$		0			0
$391$	5.24264			0.265132
$392$		0			0
$393$		−	52.2843i		−	2.63739i
$394$		0			0
$395$	1.75736	+	0.585786i	0.0884223	+	0.0294741i
$396$		0			0
$397$		−	22.6274i		−	1.13564i	−0.823154	−	0.567819i	$-0.807787\pi$
$397$		−	22.6274i		−	1.13564i	0.823154	−	0.567819i	$-0.192213\pi$
$398$		0			0
$399$	−5.82843			−0.291786
$400$		0			0
$401$	−25.8995			−1.29336			−0.646680	−	0.762762i	$-0.723843\pi$
$401$	−25.8995			−1.29336			−0.646680	+	0.762762i	$0.723843\pi$
$402$		0			0
$403$		−	6.24264i		−	0.310968i
$404$		0			0
$405$	−20.1213	−	6.70711i	−0.999836	−	0.333279i
$406$		0			0
$407$			7.31371i			0.362527i
$408$		0			0
$409$	−5.41421			−0.267716			−0.133858	−	0.991001i	$-0.542737\pi$
$409$	−5.41421			−0.267716			−0.133858	+	0.991001i	$0.542737\pi$
$410$		0			0
$411$	39.3848			1.94271
$412$		0			0
$413$			34.7990i			1.71235i
$414$		0			0
$415$	−1.75736	+	5.27208i	−0.0862654	+	0.258796i
$416$		0			0
$417$		−	9.65685i		−	0.472898i
$418$		0			0
$419$	−15.2132			−0.743214			−0.371607	−	0.928390i	$-0.621193\pi$
$419$	−15.2132			−0.743214			−0.371607	+	0.928390i	$0.621193\pi$
$420$		0			0
$421$	−15.1421			−0.737983			−0.368991	−	0.929433i	$-0.620297\pi$
$421$	−15.1421			−0.737983			−0.368991	+	0.929433i	$0.620297\pi$
$422$		0			0
$423$		−	22.6274i		−	1.10018i
$424$		0			0
$425$	−3.00000	+	4.00000i	−0.145521	+	0.194029i
$426$		0			0
$427$		−	8.24264i		−	0.398889i
$428$		0			0
$429$	−6.24264			−0.301398
$430$		0			0
$431$	−2.58579			−0.124553			−0.0622765	−	0.998059i	$-0.519836\pi$
$431$	−2.58579			−0.124553			−0.0622765	+	0.998059i	$0.519836\pi$
$432$		0			0
$433$			9.07107i			0.435928i	0.975957	+	0.217964i	$0.0699415\pi$
$433$			9.07107i			0.435928i	−0.975957	+	0.217964i	$0.930059\pi$
$434$		0			0
$435$	−6.53553	+	19.6066i	−0.313355	+	0.940065i
$436$		0			0
$437$		−	5.24264i		−	0.250790i
$438$		0			0
$439$	5.27208			0.251623			0.125811	−	0.992054i	$-0.459847\pi$
$439$	5.27208			0.251623			0.125811	+	0.992054i	$0.459847\pi$
$440$		0			0
$441$	−3.31371			−0.157796
$442$		0			0
$443$			15.5563i			0.739104i	0.929210	+	0.369552i	$0.120489\pi$
$443$			15.5563i			0.739104i	−0.929210	+	0.369552i	$0.879511\pi$
$444$		0			0
$445$	7.97056	+	2.65685i	0.377841	+	0.125947i
$446$		0			0
$447$		−	42.6274i		−	2.01621i
$448$		0			0
$449$	−31.9411			−1.50739			−0.753697	−	0.657222i	$-0.771732\pi$
$449$	−31.9411			−1.50739			−0.753697	+	0.657222i	$0.771732\pi$
$450$		0			0
$451$	−10.0000			−0.470882
$452$		0			0
$453$		−	26.9706i		−	1.26719i
$454$		0			0
$455$	−9.36396	−	3.12132i	−0.438989	−	0.146330i
$456$		0			0
$457$		−	6.31371i		−	0.295343i	−0.989036	−	0.147671i	$-0.952822\pi$
$457$		−	6.31371i		−	0.295343i	0.989036	−	0.147671i	$-0.0471778\pi$
$458$		0			0
$459$	0.414214			0.0193338
$460$		0			0
$461$	−32.7279			−1.52429			−0.762146	−	0.647406i	$-0.775854\pi$
$461$	−32.7279			−1.52429			−0.762146	+	0.647406i	$0.775854\pi$
$462$		0			0
$463$			6.14214i			0.285449i	0.989762	+	0.142725i	$0.0455863\pi$
$463$			6.14214i			0.285449i	−0.989762	+	0.142725i	$0.954414\pi$
$464$		0			0
$465$	5.82843	−	17.4853i	0.270287	−	0.810861i
$466$		0			0
$467$		−	2.10051i		−	0.0971998i	−0.998818	−	0.0485999i	$-0.984524\pi$
$467$		−	2.10051i		−	0.0971998i	0.998818	−	0.0485999i	$-0.0154759\pi$
$468$		0			0
$469$	−14.6569			−0.676791
$470$		0			0
$471$	0.828427			0.0381719
$472$		0			0
$473$		−	3.17157i		−	0.145829i
$474$		0			0
$475$	4.00000	+	3.00000i	0.183533	+	0.137649i
$476$		0			0
$477$		−	5.17157i		−	0.236790i
$478$		0			0
$479$	17.4558			0.797578			0.398789	−	0.917043i	$-0.369431\pi$
$479$	17.4558			0.797578			0.398789	+	0.917043i	$0.369431\pi$
$480$		0			0
$481$	9.45584			0.431149
$482$		0			0
$483$			30.5563i			1.39036i
$484$		0			0
$485$	−5.41421	+	16.2426i	−0.245847	+	0.737540i
$486$		0			0
$487$			12.8284i			0.581312i	0.956828	+	0.290656i	$0.0938734\pi$
$487$			12.8284i			0.581312i	−0.956828	+	0.290656i	$0.906127\pi$
$488$		0			0
$489$	−7.07107			−0.319765
$490$		0			0
$491$	26.2426			1.18431			0.592157	−	0.805823i	$-0.298277\pi$
$491$	26.2426			1.18431			0.592157	+	0.805823i	$0.298277\pi$
$492$		0			0
$493$		−	3.82843i		−	0.172424i
$494$		0			0
$495$	−8.48528	−	2.82843i	−0.381385	−	0.127128i
$496$		0			0
$497$		−	7.41421i		−	0.332573i
$498$		0			0
$499$	−33.0711			−1.48046			−0.740232	−	0.672351i	$-0.765284\pi$
$499$	−33.0711			−1.48046			−0.740232	+	0.672351i	$0.765284\pi$
$500$		0			0
$501$	56.0416			2.50376
$502$		0			0
$503$		−	13.1005i		−	0.584123i	−0.956400	−	0.292061i	$-0.905659\pi$
$503$		−	13.1005i		−	0.584123i	0.956400	−	0.292061i	$-0.0943411\pi$
$504$		0			0
$505$	33.7279	+	11.2426i	1.50087	+	0.500291i
$506$		0			0
$507$		−	23.3137i		−	1.03540i
$508$		0			0
$509$	1.65685			0.0734388			0.0367194	−	0.999326i	$-0.488309\pi$
$509$	1.65685			0.0734388			0.0367194	+	0.999326i	$0.488309\pi$
$510$		0			0
$511$	−33.3848			−1.47686
$512$		0			0
$513$		−	0.414214i		−	0.0182880i
$514$		0			0
$515$	−7.00000	+	21.0000i	−0.308457	+	0.925371i
$516$		0			0
$517$			11.3137i			0.497576i
$518$		0			0
$519$	−45.4558			−1.99529
$520$		0			0
$521$	−28.0000			−1.22670			−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000			−1.22670			−0.613351	+	0.789810i	$0.710179\pi$
$522$		0			0
$523$		−	22.2132i		−	0.971316i	−0.874149	−	0.485658i	$-0.838580\pi$
$523$		−	22.2132i		−	0.971316i	0.874149	−	0.485658i	$-0.161420\pi$
$524$		0			0
$525$	−23.3137	−	17.4853i	−1.01749	−	0.763120i
$526$		0			0
$527$			3.41421i			0.148725i
$528$		0			0
$529$	−4.48528			−0.195012
$530$		0			0
$531$	40.7696			1.76925
$532$		0			0
$533$			12.9289i			0.560014i
$534$		0			0
$535$	1.70711	−	5.12132i	0.0738047	−	0.221414i
$536$		0			0
$537$		−	55.4558i		−	2.39310i
$538$		0			0
$539$	1.65685			0.0713658
$540$		0			0
$541$	−18.0416			−0.775670			−0.387835	−	0.921729i	$-0.626777\pi$
$541$	−18.0416			−0.775670			−0.387835	+	0.921729i	$0.626777\pi$
$542$		0			0
$543$		−	1.17157i		−	0.0502770i
$544$		0			0
$545$	10.6066	+	3.53553i	0.454337	+	0.151446i
$546$		0			0
$547$			43.9411i			1.87879i	0.342841	+	0.939393i	$0.388611\pi$
$547$			43.9411i			1.87879i	−0.342841	+	0.939393i	$0.611389\pi$
$548$		0			0
$549$	−9.65685			−0.412144
$550$		0			0
$551$	−3.82843			−0.163096
$552$		0			0
$553$		−	2.00000i		−	0.0850487i
$554$		0			0
$555$	26.4853	+	8.82843i	1.12424	+	0.374746i
$556$		0			0
$557$		−	20.6863i		−	0.876506i	−0.898852	−	0.438253i	$-0.855597\pi$
$557$		−	20.6863i		−	0.876506i	0.898852	−	0.438253i	$-0.144403\pi$
$558$		0			0
$559$	−4.10051			−0.173433
$560$		0			0
$561$	3.41421			0.144148
$562$		0			0
$563$			11.0294i			0.464835i	0.972616	+	0.232418i	$0.0746636\pi$
$563$			11.0294i			0.464835i	−0.972616	+	0.232418i	$0.925336\pi$
$564$		0			0
$565$	−9.24264	+	27.7279i	−0.388841	+	1.16652i
$566$		0			0
$567$			22.8995i			0.961688i
$568$		0			0
$569$	−24.9706			−1.04682			−0.523410	−	0.852081i	$-0.675340\pi$
$569$	−24.9706			−1.04682			−0.523410	+	0.852081i	$0.675340\pi$
$570$		0			0
$571$	−11.6985			−0.489566			−0.244783	−	0.969578i	$-0.578717\pi$
$571$	−11.6985			−0.489566			−0.244783	+	0.969578i	$0.578717\pi$
$572$		0			0
$573$			27.1421i			1.13388i
$574$		0			0
$575$	15.7279	−	20.9706i	0.655900	−	0.874533i
$576$		0			0
$577$		−	25.0000i		−	1.04076i	−0.853934	−	0.520382i	$-0.825790\pi$
$577$		−	25.0000i		−	1.04076i	0.853934	−	0.520382i	$-0.174210\pi$
$578$		0			0
$579$	−18.4853			−0.768222
$580$		0			0
$581$	6.00000			0.248922
$582$		0			0
$583$			2.58579i			0.107092i
$584$		0			0
$585$	−3.65685	+	10.9706i	−0.151192	+	0.453577i
$586$		0			0
$587$			4.38478i			0.180979i	0.995897	+	0.0904895i	$0.0288432\pi$
$587$			4.38478i			0.180979i	−0.995897	+	0.0904895i	$0.971157\pi$
$588$		0			0
$589$	3.41421			0.140680
$590$		0			0
$591$	50.0416			2.05844
$592$		0			0
$593$			42.0000i			1.72473i	0.506284	+	0.862367i	$0.331019\pi$
$593$			42.0000i			1.72473i	−0.506284	+	0.862367i	$0.668981\pi$
$594$		0			0
$595$	5.12132	+	1.70711i	0.209954	+	0.0699846i
$596$		0			0
$597$			38.7990i			1.58794i
$598$		0			0
$599$	2.44365			0.0998449			0.0499224	−	0.998753i	$-0.484103\pi$
$599$	2.44365			0.0998449			0.0499224	+	0.998753i	$0.484103\pi$
$600$		0			0
$601$	31.5563			1.28721			0.643605	−	0.765358i	$-0.277438\pi$
$601$	31.5563			1.28721			0.643605	+	0.765358i	$0.277438\pi$
$602$		0			0
$603$			17.1716i			0.699281i
$604$		0			0
$605$	−19.0919	−	6.36396i	−0.776195	−	0.258732i
$606$		0			0
$607$			13.5147i			0.548546i	0.961652	+	0.274273i	$0.0884371\pi$
$607$			13.5147i			0.548546i	−0.961652	+	0.274273i	$0.911563\pi$
$608$		0			0
$609$	22.3137			0.904197
$610$		0			0
$611$	14.6274			0.591762
$612$		0			0
$613$		−	22.7279i		−	0.917972i	−0.888443	−	0.458986i	$-0.848213\pi$
$613$		−	22.7279i		−	0.917972i	0.888443	−	0.458986i	$-0.151787\pi$
$614$		0			0
$615$	−12.0711	+	36.2132i	−0.486752	+	1.46026i
$616$		0			0
$617$			7.79899i			0.313976i	0.987601	+	0.156988i	$0.0501783\pi$
$617$			7.79899i			0.313976i	−0.987601	+	0.156988i	$0.949822\pi$
$618$		0			0
$619$	30.3848			1.22127			0.610634	−	0.791913i	$-0.290915\pi$
$619$	30.3848			1.22127			0.610634	+	0.791913i	$0.290915\pi$
$620$		0			0
$621$	−2.17157			−0.0871422
$622$		0			0
$623$		−	9.07107i		−	0.363425i
$624$		0			0
$625$	7.00000	+	24.0000i	0.280000	+	0.960000i
$626$		0			0
$627$		−	3.41421i		−	0.136351i
$628$		0			0
$629$	−5.17157			−0.206204
$630$		0			0
$631$	−38.9706			−1.55139			−0.775697	−	0.631106i	$-0.782601\pi$
$631$	−38.9706			−1.55139			−0.775697	+	0.631106i	$0.782601\pi$
$632$		0			0
$633$			60.1127i			2.38927i
$634$		0			0
$635$	−7.89949	+	23.6985i	−0.313482	+	0.940446i
$636$		0			0
$637$		−	2.14214i		−	0.0848745i
$638$		0			0
$639$	−8.68629			−0.343624
$640$		0			0
$641$	−39.2132			−1.54883			−0.774414	−	0.632679i	$-0.781955\pi$
$641$	−39.2132			−1.54883			−0.774414	+	0.632679i	$0.781955\pi$
$642$		0			0
$643$		−	31.1716i		−	1.22929i	−0.788805	−	0.614643i	$-0.789300\pi$
$643$		−	31.1716i		−	1.22929i	0.788805	−	0.614643i	$-0.210700\pi$
$644$		0			0
$645$	−11.4853	−	3.82843i	−0.452233	−	0.150744i
$646$		0			0
$647$			12.0711i			0.474563i	0.971441	+	0.237281i	$0.0762563\pi$
$647$			12.0711i			0.474563i	−0.971441	+	0.237281i	$0.923744\pi$
$648$		0			0
$649$	−20.3848			−0.800172
$650$		0			0
$651$	−19.8995			−0.779923
$652$		0			0
$653$			32.2843i			1.26338i	0.775221	+	0.631691i	$0.217639\pi$
$653$			32.2843i			1.26338i	−0.775221	+	0.631691i	$0.782361\pi$
$654$		0			0
$655$	45.9411	+	15.3137i	1.79507	+	0.598356i
$656$		0			0
$657$			39.1127i			1.52593i
$658$		0			0
$659$	8.89949			0.346675			0.173338	−	0.984862i	$-0.444545\pi$
$659$	8.89949			0.346675			0.173338	+	0.984862i	$0.444545\pi$
$660$		0			0
$661$	43.4853			1.69138			0.845691	−	0.533673i	$-0.179189\pi$
$661$	43.4853			1.69138			0.845691	+	0.533673i	$0.179189\pi$
$662$		0			0
$663$		−	4.41421i		−	0.171434i
$664$		0			0
$665$	1.70711	−	5.12132i	0.0661988	−	0.198596i
$666$		0			0
$667$			20.0711i			0.777155i
$668$		0			0
$669$	−7.65685			−0.296031
$670$		0			0
$671$	4.82843			0.186399
$672$		0			0
$673$		−	28.0000i		−	1.07932i	−0.841883	−	0.539660i	$-0.818553\pi$
$673$		−	28.0000i		−	1.07932i	0.841883	−	0.539660i	$-0.181447\pi$
$674$		0			0
$675$	1.24264	−	1.65685i	0.0478293	−	0.0637723i
$676$		0			0
$677$		−	21.3431i		−	0.820284i	−0.912022	−	0.410142i	$-0.865479\pi$
$677$		−	21.3431i		−	0.820284i	0.912022	−	0.410142i	$-0.134521\pi$
$678$		0			0
$679$	18.4853			0.709400
$680$		0			0
$681$	−41.6274			−1.59517
$682$		0			0
$683$		−	23.3137i		−	0.892074i	−0.895014	−	0.446037i	$-0.852835\pi$
$683$		−	23.3137i		−	0.892074i	0.895014	−	0.446037i	$-0.147165\pi$
$684$		0			0
$685$	−11.5355	+	34.6066i	−0.440750	+	1.32225i
$686$		0			0
$687$		−	65.1127i		−	2.48420i
$688$		0			0
$689$	3.34315			0.127364
$690$		0			0
$691$	7.45584			0.283634			0.141817	−	0.989893i	$-0.454706\pi$
$691$	7.45584			0.283634			0.141817	+	0.989893i	$0.454706\pi$
$692$		0			0
$693$			9.65685i			0.366834i
$694$		0			0
$695$	8.48528	+	2.82843i	0.321865	+	0.107288i
$696$		0			0
$697$		−	7.07107i		−	0.267836i
$698$		0			0
$699$	5.65685			0.213962
$700$		0			0
$701$	32.9706			1.24528			0.622640	−	0.782508i	$-0.286060\pi$
$701$	32.9706			1.24528			0.622640	+	0.782508i	$0.286060\pi$
$702$		0			0
$703$			5.17157i			0.195050i
$704$		0			0
$705$	40.9706	+	13.6569i	1.54304	+	0.514347i
$706$		0			0
$707$		−	38.3848i		−	1.44361i
$708$		0			0
$709$	−10.6863			−0.401332			−0.200666	−	0.979660i	$-0.564311\pi$
$709$	−10.6863			−0.401332			−0.200666	+	0.979660i	$0.564311\pi$
$710$		0			0
$711$	−2.34315			−0.0878748
$712$		0			0
$713$		−	17.8995i		−	0.670341i
$714$		0			0
$715$	1.82843	−	5.48528i	0.0683793	−	0.205138i
$716$		0			0
$717$			5.48528i			0.204852i
$718$		0			0
$719$	−30.4142			−1.13426			−0.567129	−	0.823629i	$-0.691946\pi$
$719$	−30.4142			−1.13426			−0.567129	+	0.823629i	$0.691946\pi$
$720$		0			0
$721$	23.8995			0.890064
$722$		0			0
$723$		−	13.6569i		−	0.507904i
$724$		0			0
$725$	−15.3137	−	11.4853i	−0.568737	−	0.426553i
$726$		0			0
$727$		−	1.78680i		−	0.0662686i	−0.999451	−	0.0331343i	$-0.989451\pi$
$727$		−	1.78680i		−	0.0662686i	0.999451	−	0.0331343i	$-0.0105489\pi$
$728$		0			0
$729$	23.8284			0.882534
$730$		0			0
$731$	2.24264			0.0829471
$732$		0			0
$733$			10.6274i			0.392533i	0.980551	+	0.196266i	$0.0628817\pi$
$733$			10.6274i			0.392533i	−0.980551	+	0.196266i	$0.937118\pi$
$734$		0			0
$735$	2.00000	−	6.00000i	0.0737711	−	0.221313i
$736$		0			0
$737$		−	8.58579i		−	0.316262i
$738$		0			0
$739$	−7.41421			−0.272736			−0.136368	−	0.990658i	$-0.543543\pi$
$739$	−7.41421			−0.272736			−0.136368	+	0.990658i	$0.543543\pi$
$740$		0			0
$741$	−4.41421			−0.162160
$742$		0			0
$743$		−	14.2426i		−	0.522512i	−0.965270	−	0.261256i	$-0.915863\pi$
$743$		−	14.2426i		−	0.522512i	0.965270	−	0.261256i	$-0.0841366\pi$
$744$		0			0
$745$	37.4558	+	12.4853i	1.37228	+	0.457425i
$746$		0			0
$747$		−	7.02944i		−	0.257194i
$748$		0			0
$749$	−5.82843			−0.212966
$750$		0			0
$751$	29.7574			1.08586			0.542931	−	0.839777i	$-0.317315\pi$
$751$	29.7574			1.08586			0.542931	+	0.839777i	$0.317315\pi$
$752$		0			0
$753$			66.5269i			2.42438i
$754$		0			0
$755$	23.6985	+	7.89949i	0.862476	+	0.287492i
$756$		0			0
$757$		−	16.3431i		−	0.594002i	−0.954877	−	0.297001i	$-0.904014\pi$
$757$		−	16.3431i		−	0.594002i	0.954877	−	0.297001i	$-0.0959864\pi$
$758$		0			0
$759$	−17.8995			−0.649711
$760$		0			0
$761$	−4.02944			−0.146067			−0.0730335	−	0.997329i	$-0.523268\pi$
$761$	−4.02944			−0.146067			−0.0730335	+	0.997329i	$0.523268\pi$
$762$		0			0
$763$		−	12.0711i		−	0.437002i
$764$		0			0
$765$	2.00000	−	6.00000i	0.0723102	−	0.216930i
$766$		0			0
$767$			26.3553i			0.951636i
$768$		0			0
$769$	30.7990			1.11064			0.555320	−	0.831637i	$-0.312596\pi$
$769$	30.7990			1.11064			0.555320	+	0.831637i	$0.312596\pi$
$770$		0			0
$771$	7.89949			0.284493
$772$		0			0
$773$		−	9.00000i		−	0.323708i	−0.986815	−	0.161854i	$-0.948253\pi$
$773$		−	9.00000i		−	0.323708i	0.986815	−	0.161854i	$-0.0517473\pi$
$774$		0			0
$775$	13.6569	+	10.2426i	0.490569	+	0.367927i
$776$		0			0
$777$		−	30.1421i		−	1.08134i
$778$		0			0
$779$	−7.07107			−0.253347
$780$		0			0
$781$	4.34315			0.155410
$782$		0			0
$783$			1.58579i			0.0566714i
$784$		0			0
$785$	−0.242641	+	0.727922i	−0.00866022	+	0.0259807i
$786$		0			0
$787$		−	22.7574i		−	0.811212i	−0.914048	−	0.405606i	$-0.867060\pi$
$787$		−	22.7574i		−	0.811212i	0.914048	−	0.405606i	$-0.132940\pi$
$788$		0			0
$789$	10.4853			0.373286
$790$		0			0
$791$	31.5563			1.12201
$792$		0			0
$793$		−	6.24264i		−	0.221683i
$794$		0			0
$795$	9.36396	+	3.12132i	0.332105	+	0.110702i
$796$		0			0
$797$			35.1421i

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:	\( N \)	\(=\)	\( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	760.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.41421i q^{3} +(2.12132 + 0.707107i) q^{5} -2.41421i q^{7} -2.82843 q^{9} +O(q^{10})\)	\(q-2.41421i q^{3} +(2.12132 + 0.707107i) q^{5} -2.41421i q^{7} -2.82843 q^{9} +1.41421 q^{11} -1.82843i q^{13} +(1.70711 - 5.12132i) q^{15} +1.00000i q^{17} +1.00000 q^{19} -5.82843 q^{21} -5.24264i q^{23} +(4.00000 + 3.00000i) q^{25} -0.414214i q^{27} -3.82843 q^{29} +3.41421 q^{31} -3.41421i q^{33} +(1.70711 - 5.12132i) q^{35} +5.17157i q^{37} -4.41421 q^{39} -7.07107 q^{41} -2.24264i q^{43} +(-6.00000 - 2.00000i) q^{45} +8.00000i q^{47} +1.17157 q^{49} +2.41421 q^{51} +1.82843i q^{53} +(3.00000 + 1.00000i) q^{55} -2.41421i q^{57} -14.4142 q^{59} +3.41421 q^{61} +6.82843i q^{63} +(1.29289 - 3.87868i) q^{65} -6.07107i q^{67} -12.6569 q^{69} +3.07107 q^{71} -13.8284i q^{73} +(7.24264 - 9.65685i) q^{75} -3.41421i q^{77} +0.828427 q^{79} -9.48528 q^{81} +2.48528i q^{83} +(-0.707107 + 2.12132i) q^{85} +9.24264i q^{87} +3.75736 q^{89} -4.41421 q^{91} -8.24264i q^{93} +(2.12132 + 0.707107i) q^{95} +7.65685i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 4 q^{15} + 4 q^{19} - 12 q^{21} + 16 q^{25} - 4 q^{29} + 8 q^{31} + 4 q^{35} - 12 q^{39} - 24 q^{45} + 16 q^{49} + 4 q^{51} + 12 q^{55} - 52 q^{59} + 8 q^{61} + 8 q^{65} - 28 q^{69} - 16 q^{71} + 12 q^{75} - 8 q^{79} - 4 q^{81} + 32 q^{89} - 12 q^{91} - 16 q^{99}+O(q^{100}) \) 4 * q + 4 * q^15 + 4 * q^19 - 12 * q^21 + 16 * q^25 - 4 * q^29 + 8 * q^31 + 4 * q^35 - 12 * q^39 - 24 * q^45 + 16 * q^49 + 4 * q^51 + 12 * q^55 - 52 * q^59 + 8 * q^61 + 8 * q^65 - 28 * q^69 - 16 * q^71 + 12 * q^75 - 8 * q^79 - 4 * q^81 + 32 * q^89 - 12 * q^91 - 16 * q^99

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