LMFDB - Embedded newform 520.2.d.c.209.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [520,2,Mod(209,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, 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("520.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 520 = 2^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	520.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:	$4.15222090511$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} + 2x^{8} + 16x^{7} - 15x^{6} - 40x^{5} - 75x^{4} + 400x^{3} + 250x^{2} - 2500x + 3125 $ x^10 - 4x^9 + 2x^8 + 16x^7 - 15x^6 - 40x^5 - 75x^4 + 400x^3 + 250x^2 - 2500*x + 3125
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.6
Root			$1.43589 + 1.71412i$ of defining polynomial
Character	$\chi$	$=$	520.209
Dual form			520.2.d.c.209.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.16232i q^{3} +(-1.43589 - 1.71412i) q^{5} +4.64901i q^{7} +1.64901 q^{9} +O(q^{10})$	$q+1.16232i q^{3} +(-1.43589 - 1.71412i) q^{5} +4.64901i q^{7} +1.64901 q^{9} -3.25486 q^{11} +1.00000i q^{13} +(1.99236 - 1.66896i) q^{15} -0.871779i q^{17} -6.93023 q^{19} -5.40364 q^{21} +6.03410i q^{23} +(-0.876443 + 4.92259i) q^{25} +5.40364i q^{27} -3.63573 q^{29} +7.78672 q^{31} -3.78319i q^{33} +(7.96899 - 6.67547i) q^{35} +8.39257i q^{37} -1.16232 q^{39} -6.44153 q^{41} +6.01882i q^{43} +(-2.36780 - 2.82662i) q^{45} -8.64901i q^{47} -14.6133 q^{49} +1.01329 q^{51} -4.53186i q^{53} +(4.67363 + 5.57924i) q^{55} -8.05513i q^{57} -3.51131 q^{59} +14.8434 q^{61} +7.66629i q^{63} +(1.71412 - 1.43589i) q^{65} -9.49618i q^{67} -7.01355 q^{69} +8.59031 q^{71} -3.31109i q^{73} +(-5.72161 - 1.01871i) q^{75} -15.1319i q^{77} +5.18113 q^{79} -1.33371 q^{81} +1.66230i q^{83} +(-1.49434 + 1.25178i) q^{85} -4.22588i q^{87} +10.6001 q^{89} -4.64901 q^{91} +9.05066i q^{93} +(9.95104 + 11.8793i) q^{95} +9.62267i q^{97} -5.36732 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 4 q^{5} - 30 q^{9}+O(q^{10}) $ 10 * q - 4 * q^5 - 30 * q^9	$ 10 q - 4 q^{5} - 30 q^{9} + 16 q^{11} + 4 q^{15} - 36 q^{19} + 4 q^{21} + 12 q^{25} - 4 q^{29} - 8 q^{31} + 18 q^{35} - 4 q^{39} - 24 q^{41} + 36 q^{45} - 38 q^{49} - 4 q^{51} + 32 q^{55} - 28 q^{59} + 20 q^{61} + 4 q^{65} - 72 q^{69} + 36 q^{71} + 62 q^{75} - 16 q^{79} + 18 q^{81} + 8 q^{85} + 12 q^{89} + 52 q^{95} - 68 q^{99}+O(q^{100}) $ 10 * q - 4 * q^5 - 30 * q^9 + 16 * q^11 + 4 * q^15 - 36 * q^19 + 4 * q^21 + 12 * q^25 - 4 * q^29 - 8 * q^31 + 18 * q^35 - 4 * q^39 - 24 * q^41 + 36 * q^45 - 38 * q^49 - 4 * q^51 + 32 * q^55 - 28 * q^59 + 20 * q^61 + 4 * q^65 - 72 * q^69 + 36 * q^71 + 62 * q^75 - 16 * q^79 + 18 * q^81 + 8 * q^85 + 12 * q^89 + 52 * q^95 - 68 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/520\mathbb{Z}\right)^\times$.

$n$	$41$	$261$	$391$	$417$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			1.16232i			0.671065i	0.942029	+	0.335533i	$0.108916\pi$
$3$			1.16232i			0.671065i	−0.942029	+	0.335533i	$0.891084\pi$
$4$		0			0
$5$	−1.43589	−	1.71412i	−0.642149	−	0.766580i
$5$	−1.43589	−	1.71412i	−0.642149	−	0.766580i
$6$		0			0
$7$			4.64901i			1.75716i	0.477593	+	0.878581i	$0.341509\pi$
$7$			4.64901i			1.75716i	−0.477593	+	0.878581i	$0.658491\pi$
$8$		0			0
$9$	1.64901			0.549672
$10$		0			0
$11$	−3.25486			−0.981379			−0.490689	−	0.871335i	$-0.663255\pi$
$11$	−3.25486			−0.981379			−0.490689	+	0.871335i	$0.663255\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$		0			0
$15$	1.99236	−	1.66896i	0.514425	−	0.430924i
$16$		0			0
$17$		−	0.871779i		−	0.211437i	−0.994396	−	0.105719i	$-0.966286\pi$
$17$		−	0.871779i		−	0.211437i	0.994396	−	0.105719i	$-0.0337143\pi$
$18$		0			0
$19$	−6.93023			−1.58990			−0.794952	−	0.606673i	$-0.792504\pi$
$19$	−6.93023			−1.58990			−0.794952	+	0.606673i	$0.792504\pi$
$20$		0			0
$21$	−5.40364			−1.17917
$22$		0			0
$23$			6.03410i			1.25820i	0.777326	+	0.629098i	$0.216576\pi$
$23$			6.03410i			1.25820i	−0.777326	+	0.629098i	$0.783424\pi$
$24$		0			0
$25$	−0.876443	+	4.92259i	−0.175289	+	0.984517i
$26$		0			0
$27$			5.40364i			1.03993i
$28$		0			0
$29$	−3.63573			−0.675138			−0.337569	−	0.941301i	$-0.609605\pi$
$29$	−3.63573			−0.675138			−0.337569	+	0.941301i	$0.609605\pi$
$30$		0			0
$31$	7.78672			1.39854			0.699268	−	0.714859i	$-0.253509\pi$
$31$	7.78672			1.39854			0.699268	+	0.714859i	$0.253509\pi$
$32$		0			0
$33$		−	3.78319i		−	0.658569i
$34$		0			0
$35$	7.96899	−	6.67547i	1.34700	−	1.12836i
$36$		0			0
$37$			8.39257i			1.37973i	0.723938	+	0.689865i	$0.242330\pi$
$37$			8.39257i			1.37973i	−0.723938	+	0.689865i	$0.757670\pi$
$38$		0			0
$39$	−1.16232			−0.186120
$40$		0			0
$41$	−6.44153			−1.00600			−0.502999	−	0.864287i	$-0.667770\pi$
$41$	−6.44153			−1.00600			−0.502999	+	0.864287i	$0.667770\pi$
$42$		0			0
$43$			6.01882i			0.917861i	0.888472	+	0.458930i	$0.151767\pi$
$43$			6.01882i			0.917861i	−0.888472	+	0.458930i	$0.848233\pi$
$44$		0			0
$45$	−2.36780	−	2.82662i	−0.352971	−	0.421367i
$46$		0			0
$47$		−	8.64901i		−	1.26159i	−0.775950	−	0.630794i	$-0.782729\pi$
$47$		−	8.64901i		−	1.26159i	0.775950	−	0.630794i	$-0.217271\pi$
$48$		0			0
$49$	−14.6133			−2.08762
$50$		0			0
$51$	1.01329			0.141888
$52$		0			0
$53$		−	4.53186i		−	0.622499i	−0.950328	−	0.311249i	$-0.899253\pi$
$53$		−	4.53186i		−	0.622499i	0.950328	−	0.311249i	$-0.100747\pi$
$54$		0			0
$55$	4.67363	+	5.57924i	0.630192	+	0.752305i
$56$		0			0
$57$		−	8.05513i		−	1.06693i
$58$		0			0
$59$	−3.51131			−0.457133			−0.228567	−	0.973528i	$-0.573404\pi$
$59$	−3.51131			−0.457133			−0.228567	+	0.973528i	$0.573404\pi$
$60$		0			0
$61$	14.8434			1.90051			0.950253	−	0.311478i	$-0.100824\pi$
$61$	14.8434			1.90051			0.950253	+	0.311478i	$0.100824\pi$
$62$		0			0
$63$			7.66629i			0.965862i
$64$		0			0
$65$	1.71412	−	1.43589i	0.212611	−	0.178100i
$66$		0			0
$67$		−	9.49618i		−	1.16014i	−0.814565	−	0.580072i	$-0.803025\pi$
$67$		−	9.49618i		−	1.16014i	0.814565	−	0.580072i	$-0.196975\pi$
$68$		0			0
$69$	−7.01355			−0.844332
$70$		0			0
$71$	8.59031			1.01948			0.509741	−	0.860328i	$-0.329741\pi$
$71$	8.59031			1.01948			0.509741	+	0.860328i	$0.329741\pi$
$72$		0			0
$73$		−	3.31109i		−	0.387534i	−0.981048	−	0.193767i	$-0.937929\pi$
$73$		−	3.31109i		−	0.387534i	0.981048	−	0.193767i	$-0.0620706\pi$
$74$		0			0
$75$	−5.72161	−	1.01871i	−0.660675	−	0.117630i
$76$		0			0
$77$		−	15.1319i		−	1.72444i
$78$		0			0
$79$	5.18113			0.582923			0.291462	−	0.956583i	$-0.405858\pi$
$79$	5.18113			0.582923			0.291462	+	0.956583i	$0.405858\pi$
$80$		0			0
$81$	−1.33371			−0.148189
$82$		0			0
$83$			1.66230i			0.182461i	0.995830	+	0.0912306i	$0.0290800\pi$
$83$			1.66230i			0.182461i	−0.995830	+	0.0912306i	$0.970920\pi$
$84$		0			0
$85$	−1.49434	+	1.25178i	−0.162084	+	0.135774i
$86$		0			0
$87$		−	4.22588i		−	0.453062i
$88$		0			0
$89$	10.6001			1.12360			0.561802	−	0.827272i	$-0.310108\pi$
$89$	10.6001			1.12360			0.561802	+	0.827272i	$0.310108\pi$
$90$		0			0
$91$	−4.64901			−0.487349
$92$		0			0
$93$			9.05066i			0.938509i
$94$		0			0
$95$	9.95104	+	11.8793i	1.02096	+	1.21879i
$96$		0			0
$97$			9.62267i			0.977034i	0.872554	+	0.488517i	$0.162462\pi$
$97$			9.62267i			0.977034i	−0.872554	+	0.488517i	$0.837538\pi$
$98$		0			0
$99$	−5.36732			−0.539436
$100$		0			0
$101$	19.3662			1.92701			0.963506	−	0.267688i	$-0.0862596\pi$
$101$	19.3662			1.92701			0.963506	+	0.267688i	$0.0862596\pi$
$102$		0			0
$103$		−	5.33213i		−	0.525390i	−0.964879	−	0.262695i	$-0.915389\pi$
$103$		−	5.33213i		−	0.525390i	0.964879	−	0.262695i	$-0.0846113\pi$
$104$		0			0
$105$	7.75903	+	9.26251i	0.757203	+	0.903928i
$106$		0			0
$107$			5.61913i			0.543222i	0.962407	+	0.271611i	$0.0875564\pi$
$107$			5.61913i			0.543222i	−0.962407	+	0.271611i	$0.912444\pi$
$108$		0			0
$109$	−2.68669			−0.257338			−0.128669	−	0.991688i	$-0.541070\pi$
$109$	−2.68669			−0.257338			−0.128669	+	0.991688i	$0.541070\pi$
$110$		0			0
$111$	−9.75485			−0.925889
$112$		0			0
$113$			2.71695i			0.255589i	0.991801	+	0.127795i	$0.0407898\pi$
$113$			2.71695i			0.255589i	−0.991801	+	0.127795i	$0.959210\pi$
$114$		0			0
$115$	10.3432	−	8.66430i	0.964508	−	0.807950i
$116$		0			0
$117$			1.64901i			0.152451i
$118$		0			0
$119$	4.05291			0.371530
$120$		0			0
$121$	−0.405858			−0.0368962
$122$		0			0
$123$		−	7.48712i		−	0.675090i
$124$		0			0
$125$	9.69640	−	5.56596i	0.867272	−	0.497834i
$126$		0			0
$127$		−	3.61913i		−	0.321146i	−0.987024	−	0.160573i	$-0.948666\pi$
$127$		−	3.61913i		−	0.321146i	0.987024	−	0.160573i	$-0.0513342\pi$
$128$		0			0
$129$	−6.99578			−0.615944
$130$		0			0
$131$	14.1452			1.23587			0.617936	−	0.786229i	$-0.287969\pi$
$131$	14.1452			1.23587			0.617936	+	0.786229i	$0.287969\pi$
$132$		0			0
$133$		−	32.2187i		−	2.79372i
$134$		0			0
$135$	9.26251	−	7.75903i	0.797190	−	0.667791i
$136$		0			0
$137$			1.30202i			0.111239i	0.998452	+	0.0556197i	$0.0177135\pi$
$137$			1.30202i			0.111239i	−0.998452	+	0.0556197i	$0.982287\pi$
$138$		0			0
$139$	−6.07752			−0.515489			−0.257744	−	0.966213i	$-0.582979\pi$
$139$	−6.07752			−0.515489			−0.257744	+	0.966213i	$0.582979\pi$
$140$		0			0
$141$	10.0529			0.846608
$142$		0			0
$143$		−	3.25486i		−	0.272185i
$144$		0			0
$145$	5.22051	+	6.23209i	0.433539	+	0.517547i
$146$		0			0
$147$		−	16.9854i		−	1.40093i
$148$		0			0
$149$	−12.0863			−0.990151			−0.495075	−	0.868850i	$-0.664860\pi$
$149$	−12.0863			−0.990151			−0.495075	+	0.868850i	$0.664860\pi$
$150$		0			0
$151$	−14.9526			−1.21682			−0.608412	−	0.793622i	$-0.708193\pi$
$151$	−14.9526			−1.21682			−0.608412	+	0.793622i	$0.708193\pi$
$152$		0			0
$153$		−	1.43758i		−	0.116221i
$154$		0			0
$155$	−11.1809	−	13.3474i	−0.898069	−	1.07209i
$156$		0			0
$157$			0.887061i			0.0707952i	0.999373	+	0.0353976i	$0.0112698\pi$
$157$			0.887061i			0.0707952i	−0.999373	+	0.0353976i	$0.988730\pi$
$158$		0			0
$159$	5.26746			0.417737
$160$		0			0
$161$	−28.0526			−2.21086
$162$		0			0
$163$		−	1.42851i		−	0.111889i	−0.998434	−	0.0559447i	$-0.982183\pi$
$163$		−	1.42851i		−	0.111889i	0.998434	−	0.0559447i	$-0.0178171\pi$
$164$		0			0
$165$	−6.48486	+	5.43224i	−0.504845	+	0.422900i
$166$		0			0
$167$			14.9116i			1.15390i	0.816781	+	0.576948i	$0.195757\pi$
$167$			14.9116i			1.15390i	−0.816781	+	0.576948i	$0.804243\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	−11.4280			−0.873925
$172$		0			0
$173$		−	9.18113i		−	0.698029i	−0.937117	−	0.349014i	$-0.886516\pi$
$173$		−	9.18113i		−	0.698029i	0.937117	−	0.349014i	$-0.113484\pi$
$174$		0			0
$175$	−22.8852	−	4.07460i	−1.72996	−	0.308011i
$176$		0			0
$177$		−	4.08126i		−	0.306766i
$178$		0			0
$179$	10.7076			0.800325			0.400163	−	0.916444i	$-0.368954\pi$
$179$	10.7076			0.800325			0.400163	+	0.916444i	$0.368954\pi$
$180$		0			0
$181$	19.3753			1.44015			0.720077	−	0.693894i	$-0.244106\pi$
$181$	19.3753			1.44015			0.720077	+	0.693894i	$0.244106\pi$
$182$		0			0
$183$			17.2528i			1.27536i
$184$		0			0
$185$	14.3859	−	12.0508i	1.05767	−	0.885993i
$186$		0			0
$187$			2.83752i			0.207500i
$188$		0			0
$189$	−25.1216			−1.82733
$190$		0			0
$191$	14.3928			1.04143			0.520714	−	0.853731i	$-0.325666\pi$
$191$	14.3928			1.04143			0.520714	+	0.853731i	$0.325666\pi$
$192$		0			0
$193$		−	3.34725i		−	0.240940i	−0.992717	−	0.120470i	$-0.961560\pi$
$193$		−	3.34725i		−	0.240940i	0.992717	−	0.120470i	$-0.0384402\pi$
$194$		0			0
$195$	1.66896	+	1.99236i	0.119517	+	0.142676i
$196$		0			0
$197$			18.3529i			1.30759i	0.756670	+	0.653796i	$0.226825\pi$
$197$			18.3529i			1.30759i	−0.756670	+	0.653796i	$0.773175\pi$
$198$		0			0
$199$	−18.4792			−1.30995			−0.654977	−	0.755649i	$-0.727322\pi$
$199$	−18.4792			−1.30995			−0.654977	+	0.755649i	$0.727322\pi$
$200$		0			0
$201$	11.0376			0.778532
$202$		0			0
$203$		−	16.9026i		−	1.18633i
$204$		0			0
$205$	9.24933	+	11.0416i	0.646001	+	0.771178i
$206$		0			0
$207$			9.95032i			0.691595i
$208$		0			0
$209$	22.5569			1.56030
$210$		0			0
$211$	−17.5641			−1.20916			−0.604582	−	0.796543i	$-0.706660\pi$
$211$	−17.5641			−1.20916			−0.604582	+	0.796543i	$0.706660\pi$
$212$		0			0
$213$			9.98468i			0.684139i
$214$		0			0
$215$	10.3170	−	8.64235i	0.703613	−	0.589404i
$216$		0			0
$217$			36.2006i			2.45746i
$218$		0			0
$219$	3.84854			0.260061
$220$		0			0
$221$	0.871779			0.0586422
$222$		0			0
$223$			14.5963i			0.977442i	0.872440	+	0.488721i	$0.162536\pi$
$223$			14.5963i			0.977442i	−0.872440	+	0.488721i	$0.837464\pi$
$224$		0			0
$225$	−1.44527	+	8.11742i	−0.0963512	+	0.541161i
$226$		0			0
$227$			2.68891i			0.178469i	0.996011	+	0.0892345i	$0.0284421\pi$
$227$			2.68891i			0.178469i	−0.996011	+	0.0892345i	$0.971558\pi$
$228$		0			0
$229$	0.241160			0.0159363			0.00796814	−	0.999968i	$-0.497464\pi$
$229$	0.241160			0.0159363			0.00796814	+	0.999968i	$0.497464\pi$
$230$		0			0
$231$	17.5881			1.15721
$232$		0			0
$233$		−	17.7965i		−	1.16589i	−0.812513	−	0.582943i	$-0.801901\pi$
$233$		−	17.7965i		−	1.16589i	0.812513	−	0.582943i	$-0.198099\pi$
$234$		0			0
$235$	−14.8255	+	12.4190i	−0.967108	+	0.810128i
$236$		0			0
$237$			6.02213i			0.391179i
$238$		0			0
$239$	−19.0473			−1.23207			−0.616035	−	0.787719i	$-0.711262\pi$
$239$	−19.0473			−1.23207			−0.616035	+	0.787719i	$0.711262\pi$
$240$		0			0
$241$	14.6266			0.942184			0.471092	−	0.882084i	$-0.343860\pi$
$241$	14.6266			0.942184			0.471092	+	0.882084i	$0.343860\pi$
$242$		0			0
$243$			14.6607i			0.940486i
$244$		0			0
$245$	20.9831	+	25.0491i	1.34056	+	1.60033i
$246$		0			0
$247$		−	6.93023i		−	0.440960i
$248$		0			0
$249$	−1.93212			−0.122443
$250$		0			0
$251$	0.154526			0.00975361			0.00487680	−	0.999988i	$-0.498448\pi$
$251$	0.154526			0.00975361			0.00487680	+	0.999988i	$0.498448\pi$
$252$		0			0
$253$		−	19.6402i		−	1.23477i
$254$		0			0
$255$	−1.45497	−	1.73690i	−0.0911135	−	0.108769i
$256$		0			0
$257$		−	4.05291i		−	0.252814i	−0.991978	−	0.126407i	$-0.959656\pi$
$257$		−	4.05291i		−	0.252814i	0.991978	−	0.126407i	$-0.0403445\pi$
$258$		0			0
$259$	−39.0172			−2.42441
$260$		0			0
$261$	−5.99537			−0.371104
$262$		0			0
$263$			16.7321i			1.03174i	0.856666	+	0.515872i	$0.172532\pi$
$263$			16.7321i			1.03174i	−0.856666	+	0.515872i	$0.827468\pi$
$264$		0			0
$265$	−7.76817	+	6.50725i	−0.477195	+	0.399737i
$266$		0			0
$267$			12.3206i			0.754011i
$268$		0			0
$269$	−20.0155			−1.22037			−0.610183	−	0.792260i	$-0.708904\pi$
$269$	−20.0155			−1.22037			−0.610183	+	0.792260i	$0.708904\pi$
$270$		0			0
$271$	10.0319			0.609393			0.304697	−	0.952449i	$-0.401445\pi$
$271$	10.0319			0.609393			0.304697	+	0.952449i	$0.401445\pi$
$272$		0			0
$273$		−	5.40364i		−	0.327043i
$274$		0			0
$275$	2.85270	−	16.0223i	0.172024	−	0.966184i
$276$		0			0
$277$			16.2227i			0.974729i	0.873199	+	0.487364i	$0.162042\pi$
$277$			16.2227i			0.974729i	−0.873199	+	0.487364i	$0.837958\pi$
$278$		0			0
$279$	12.8404			0.768736
$280$		0			0
$281$	−10.9057			−0.650581			−0.325290	−	0.945614i	$-0.605462\pi$
$281$	−10.9057			−0.650581			−0.325290	+	0.945614i	$0.605462\pi$
$282$		0			0
$283$		−	3.33213i		−	0.198074i	−0.995084	−	0.0990372i	$-0.968424\pi$
$283$		−	3.33213i		−	0.198074i	0.995084	−	0.0990372i	$-0.0315763\pi$
$284$		0			0
$285$	−13.8075	+	11.5663i	−0.817886	+	0.685127i
$286$		0			0
$287$		−	29.9468i		−	1.76770i
$288$		0			0
$289$	16.2400			0.955294
$290$		0			0
$291$	−11.1846			−0.655653
$292$		0			0
$293$			14.4191i			0.842375i	0.906974	+	0.421188i	$0.138387\pi$
$293$			14.4191i			0.842375i	−0.906974	+	0.421188i	$0.861613\pi$
$294$		0			0
$295$	5.04185	+	6.01882i	0.293548	+	0.350429i
$296$		0			0
$297$		−	17.5881i		−	1.02057i
$298$		0			0
$299$	−6.03410			−0.348961
$300$		0			0
$301$	−27.9816			−1.61283
$302$		0			0
$303$			22.5097i			1.29315i
$304$		0			0
$305$	−21.3135	−	25.4435i	−1.22041	−	1.45689i
$306$		0			0
$307$		−	16.9643i		−	0.968205i	−0.875011	−	0.484103i	$-0.839146\pi$
$307$		−	16.9643i		−	0.968205i	0.875011	−	0.484103i	$-0.160854\pi$
$308$		0			0
$309$	6.19763			0.352571
$310$		0			0
$311$	−14.2072			−0.805618			−0.402809	−	0.915284i	$-0.631966\pi$
$311$	−14.2072			−0.805618			−0.402809	+	0.915284i	$0.631966\pi$
$312$		0			0
$313$			29.1627i			1.64837i	0.566318	+	0.824187i	$0.308367\pi$
$313$			29.1627i			1.64837i	−0.566318	+	0.824187i	$0.691633\pi$
$314$		0			0
$315$	13.1410	−	11.0080i	0.740410	−	0.620228i
$316$		0			0
$317$			19.9604i			1.12109i	0.828125	+	0.560543i	$0.189408\pi$
$317$			19.9604i			1.12109i	−0.828125	+	0.560543i	$0.810592\pi$
$318$		0			0
$319$	11.8338			0.662566
$320$		0			0
$321$	−6.53123			−0.364537
$322$		0			0
$323$			6.04163i			0.336165i
$324$		0			0
$325$	−4.92259	−	0.876443i	−0.273056	−	0.0486163i
$326$		0			0
$327$		−	3.12279i		−	0.172691i
$328$		0			0
$329$	40.2094			2.21682
$330$		0			0
$331$	−4.20169			−0.230946			−0.115473	−	0.993311i	$-0.536838\pi$
$331$	−4.20169			−0.230946			−0.115473	+	0.993311i	$0.536838\pi$
$332$		0			0
$333$			13.8395i			0.758399i
$334$		0			0
$335$	−16.2776	+	13.6355i	−0.889342	+	0.744985i
$336$		0			0
$337$		−	24.1964i		−	1.31806i	−0.752116	−	0.659030i	$-0.770967\pi$
$337$		−	24.1964i		−	1.31806i	0.752116	−	0.659030i	$-0.229033\pi$
$338$		0			0
$339$	−3.15796			−0.171517
$340$		0			0
$341$	−25.3447			−1.37249
$342$		0			0
$343$		−	35.3945i		−	1.91112i
$344$		0			0
$345$	10.0707	+	12.0221i	0.542187	+	0.647247i
$346$		0			0
$347$		−	19.9058i		−	1.06860i	−0.845294	−	0.534301i	$-0.820575\pi$
$347$		−	19.9058i		−	1.06860i	0.845294	−	0.534301i	$-0.179425\pi$
$348$		0			0
$349$	25.4413			1.36184			0.680920	−	0.732358i	$-0.261580\pi$
$349$	25.4413			1.36184			0.680920	+	0.732358i	$0.261580\pi$
$350$		0			0
$351$	−5.40364			−0.288425
$352$		0			0
$353$		−	29.6778i		−	1.57959i	−0.613371	−	0.789795i	$-0.710187\pi$
$353$		−	29.6778i		−	1.57959i	0.613371	−	0.789795i	$-0.289813\pi$
$354$		0			0
$355$	−12.3347	−	14.7249i	−0.654660	−	0.781514i
$356$		0			0
$357$			4.71078i			0.249321i
$358$		0			0
$359$	0.556889			0.0293915			0.0146957	−	0.999892i	$-0.495322\pi$
$359$	0.556889			0.0293915			0.0146957	+	0.999892i	$0.495322\pi$
$360$		0			0
$361$	29.0280			1.52779
$362$		0			0
$363$		−	0.471736i		−	0.0247597i
$364$		0			0
$365$	−5.67562	+	4.75436i	−0.297076	+	0.248855i
$366$		0			0
$367$			12.4185i			0.648240i	0.946016	+	0.324120i	$0.105068\pi$
$367$			12.4185i			0.648240i	−0.946016	+	0.324120i	$0.894932\pi$
$368$		0			0
$369$	−10.6222			−0.552969
$370$		0			0
$371$	21.0687			1.09383
$372$		0			0
$373$			23.2188i			1.20222i	0.799166	+	0.601111i	$0.205275\pi$
$373$			23.2188i			1.20222i	−0.799166	+	0.601111i	$0.794725\pi$
$374$		0			0
$375$	6.46942	+	11.2703i	0.334079	+	0.581996i
$376$		0			0
$377$		−	3.63573i		−	0.187250i
$378$		0			0
$379$	23.5763			1.21104			0.605518	−	0.795832i	$-0.292966\pi$
$379$	23.5763			1.21104			0.605518	+	0.795832i	$0.292966\pi$
$380$		0			0
$381$	4.20659			0.215510
$382$		0			0
$383$			25.0874i			1.28191i	0.767580	+	0.640953i	$0.221461\pi$
$383$			25.0874i			1.28191i	−0.767580	+	0.640953i	$0.778539\pi$
$384$		0			0
$385$	−25.9380	+	21.7278i	−1.32192	+	1.10735i
$386$		0			0
$387$			9.92512i			0.504522i
$388$		0			0
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$	5.26040			0.266030
$392$		0			0
$393$			16.4412i			0.829350i
$394$		0			0
$395$	−7.43954	−	8.88111i	−0.374324	−	0.446857i
$396$		0			0
$397$		−	39.1073i		−	1.96274i	−0.192135	−	0.981369i	$-0.561541\pi$
$397$		−	39.1073i		−	1.96274i	0.192135	−	0.981369i	$-0.438459\pi$
$398$		0			0
$399$	37.4484			1.87477
$400$		0			0
$401$	−15.3473			−0.766405			−0.383203	−	0.923664i	$-0.625179\pi$
$401$	−15.3473			−0.766405			−0.383203	+	0.923664i	$0.625179\pi$
$402$		0			0
$403$			7.78672i			0.387884i
$404$		0			0
$405$	1.91505	+	2.28614i	0.0951598	+	0.113599i
$406$		0			0
$407$		−	27.3167i		−	1.35404i
$408$		0			0
$409$	−1.09376			−0.0540830			−0.0270415	−	0.999634i	$-0.508609\pi$
$409$	−1.09376			−0.0540830			−0.0270415	+	0.999634i	$0.508609\pi$
$410$		0			0
$411$	−1.51337			−0.0746489
$412$		0			0
$413$		−	16.3241i		−	0.803257i
$414$		0			0
$415$	2.84939	−	2.38688i	0.139871	−	0.117167i
$416$		0			0
$417$		−	7.06402i		−	0.345927i
$418$		0			0
$419$	−21.1567			−1.03357			−0.516787	−	0.856114i	$-0.672872\pi$
$419$	−21.1567			−1.03357			−0.516787	+	0.856114i	$0.672872\pi$
$420$		0			0
$421$	8.43025			0.410865			0.205433	−	0.978671i	$-0.434140\pi$
$421$	8.43025			0.410865			0.205433	+	0.978671i	$0.434140\pi$
$422$		0			0
$423$		−	14.2624i		−	0.693459i
$424$		0			0
$425$	4.29141	+	0.764064i	0.208164	+	0.0370626i
$426$		0			0
$427$			69.0073i			3.33950i
$428$		0			0
$429$	3.78319			0.182654
$430$		0			0
$431$	12.3038			0.592654			0.296327	−	0.955087i	$-0.404238\pi$
$431$	12.3038			0.592654			0.296327	+	0.955087i	$0.404238\pi$
$432$		0			0
$433$			0.781453i			0.0375543i	0.999824	+	0.0187771i	$0.00597730\pi$
$433$			0.781453i			0.0375543i	−0.999824	+	0.0187771i	$0.994023\pi$
$434$		0			0
$435$	−7.24368	+	6.06789i	−0.347308	+	0.290933i
$436$		0			0
$437$		−	41.8177i		−	2.00041i
$438$		0			0
$439$	−19.2949			−0.920894			−0.460447	−	0.887687i	$-0.652311\pi$
$439$	−19.2949			−0.920894			−0.460447	+	0.887687i	$0.652311\pi$
$440$		0			0
$441$	−24.0976			−1.14751
$442$		0			0
$443$			12.0489i			0.572459i	0.958161	+	0.286229i	$0.0924019\pi$
$443$			12.0489i			0.572459i	−0.958161	+	0.286229i	$0.907598\pi$
$444$		0			0
$445$	−15.2205	−	18.1698i	−0.721521	−	0.861332i
$446$		0			0
$447$		−	14.0482i		−	0.664456i
$448$		0			0
$449$	−6.12796			−0.289196			−0.144598	−	0.989490i	$-0.546189\pi$
$449$	−6.12796			−0.289196			−0.144598	+	0.989490i	$0.546189\pi$
$450$		0			0
$451$	20.9663			0.987265
$452$		0			0
$453$		−	17.3797i		−	0.816568i
$454$		0			0
$455$	6.67547	+	7.96899i	0.312951	+	0.373592i
$456$		0			0
$457$		−	4.50973i		−	0.210956i	−0.994422	−	0.105478i	$-0.966363\pi$
$457$		−	4.50973i		−	0.210956i	0.994422	−	0.105478i	$-0.0336373\pi$
$458$		0			0
$459$	4.71078			0.219880
$460$		0			0
$461$	−14.5853			−0.679305			−0.339652	−	0.940551i	$-0.610309\pi$
$461$	−14.5853			−0.679305			−0.339652	+	0.940551i	$0.610309\pi$
$462$		0			0
$463$			16.3337i			0.759092i	0.925173	+	0.379546i	$0.123920\pi$
$463$			16.3337i			0.759092i	−0.925173	+	0.379546i	$0.876080\pi$
$464$		0			0
$465$	15.5139	−	12.9957i	0.719442	−	0.602663i
$466$		0			0
$467$			8.05223i			0.372613i	0.982492	+	0.186306i	$0.0596517\pi$
$467$			8.05223i			0.372613i	−0.982492	+	0.186306i	$0.940348\pi$
$468$		0			0
$469$	44.1479			2.03856
$470$		0			0
$471$	−1.03105			−0.0475082
$472$		0			0
$473$		−	19.5904i		−	0.900769i
$474$		0			0
$475$	6.07395	−	34.1146i	0.278692	−	1.56529i
$476$		0			0
$477$		−	7.47310i		−	0.342170i
$478$		0			0
$479$	−1.42467			−0.0650950			−0.0325475	−	0.999470i	$-0.510362\pi$
$479$	−1.42467			−0.0650950			−0.0325475	+	0.999470i	$0.510362\pi$
$480$		0			0
$481$	−8.39257			−0.382668
$482$		0			0
$483$		−	32.6061i		−	1.48363i
$484$		0			0
$485$	16.4944	−	13.8171i	0.748974	−	0.627402i
$486$		0			0
$487$		−	29.2547i		−	1.32566i	−0.748771	−	0.662829i	$-0.769356\pi$
$487$		−	29.2547i		−	1.32566i	0.748771	−	0.662829i	$-0.230644\pi$
$488$		0			0
$489$	1.66038			0.0750851
$490$		0			0
$491$	−5.01329			−0.226246			−0.113123	−	0.993581i	$-0.536085\pi$
$491$	−5.01329			−0.226246			−0.113123	+	0.993581i	$0.536085\pi$
$492$		0			0
$493$			3.16955i			0.142749i
$494$		0			0
$495$	7.70688	+	9.20025i	0.346398	+	0.413521i
$496$		0			0
$497$			39.9365i			1.79140i
$498$		0			0
$499$	22.4731			1.00603			0.503017	−	0.864276i	$-0.332223\pi$
$499$	22.4731			1.00603			0.503017	+	0.864276i	$0.332223\pi$
$500$		0			0
$501$	−17.3321			−0.774340
$502$		0			0
$503$		−	11.3321i		−	0.505275i	−0.967561	−	0.252637i	$-0.918702\pi$
$503$		−	11.3321i		−	0.505275i	0.967561	−	0.252637i	$-0.0812979\pi$
$504$		0			0
$505$	−27.8078	−	33.1961i	−1.23743	−	1.47721i
$506$		0			0
$507$		−	1.16232i		−	0.0516204i
$508$		0			0
$509$	41.8078			1.85310			0.926548	−	0.376176i	$-0.122761\pi$
$509$	41.8078			1.85310			0.926548	+	0.376176i	$0.122761\pi$
$510$		0			0
$511$	15.3933			0.680960
$512$		0			0
$513$		−	37.4484i		−	1.65339i
$514$		0			0
$515$	−9.13993	+	7.65635i	−0.402753	+	0.337379i
$516$		0			0
$517$			28.1514i			1.23810i
$518$		0			0
$519$	10.6714			0.468423
$520$		0			0
$521$	31.5289			1.38131			0.690654	−	0.723186i	$-0.257323\pi$
$521$	31.5289			1.38131			0.690654	+	0.723186i	$0.257323\pi$
$522$		0			0
$523$			16.9061i			0.739252i	0.929181	+	0.369626i	$0.120514\pi$
$523$			16.9061i			0.739252i	−0.929181	+	0.369626i	$0.879486\pi$
$524$		0			0
$525$	4.73598	−	26.5999i	0.206695	−	1.16091i
$526$		0			0
$527$		−	6.78830i		−	0.295703i
$528$		0			0
$529$	−13.4103			−0.583058
$530$		0			0
$531$	−5.79020			−0.251273
$532$		0			0
$533$		−	6.44153i		−	0.279014i
$534$		0			0
$535$	9.63189	−	8.06846i	0.416423	−	0.348830i
$536$		0			0
$537$			12.4457i			0.537070i
$538$		0			0
$539$	47.5644			2.04875
$540$		0			0
$541$	−19.7022			−0.847062			−0.423531	−	0.905882i	$-0.639209\pi$
$541$	−19.7022			−0.847062			−0.423531	+	0.905882i	$0.639209\pi$
$542$		0			0
$543$			22.5203i			0.966437i
$544$		0			0
$545$	3.85779	+	4.60532i	0.165249	+	0.197270i
$546$		0			0
$547$			30.4572i			1.30226i	0.758968	+	0.651128i	$0.225704\pi$
$547$			30.4572i			1.30226i	−0.758968	+	0.651128i	$0.774296\pi$
$548$		0			0
$549$	24.4770			1.04465
$550$		0			0
$551$	25.1964			1.07340
$552$		0			0
$553$			24.0872i			1.02429i
$554$		0			0
$555$	14.0069	+	16.7210i	0.594559	+	0.709768i
$556$		0			0
$557$		−	4.15826i		−	0.176191i	−0.996112	−	0.0880956i	$-0.971922\pi$
$557$		−	4.15826i		−	0.176191i	0.996112	−	0.0880956i	$-0.0280781\pi$
$558$		0			0
$559$	−6.01882			−0.254569
$560$		0			0
$561$	−3.29811			−0.139246
$562$		0			0
$563$		−	9.64096i		−	0.406318i	−0.979146	−	0.203159i	$-0.934879\pi$
$563$		−	9.64096i		−	0.406318i	0.979146	−	0.203159i	$-0.0651208\pi$
$564$		0			0
$565$	4.65719	−	3.90124i	0.195929	−	0.164126i
$566$		0			0
$567$		−	6.20042i		−	0.260393i
$568$		0			0
$569$	−20.4684			−0.858078			−0.429039	−	0.903286i	$-0.641148\pi$
$569$	−20.4684			−0.858078			−0.429039	+	0.903286i	$0.641148\pi$
$570$		0			0
$571$	6.35557			0.265973			0.132986	−	0.991118i	$-0.457543\pi$
$571$	6.35557			0.265973			0.132986	+	0.991118i	$0.457543\pi$
$572$		0			0
$573$			16.7291i			0.698867i
$574$		0			0
$575$	−29.7034	−	5.28854i	−1.23872	−	0.220547i
$576$		0			0
$577$			8.19416i			0.341127i	0.985347	+	0.170564i	$0.0545588\pi$
$577$			8.19416i			0.341127i	−0.985347	+	0.170564i	$0.945441\pi$
$578$		0			0
$579$	3.89057			0.161687
$580$		0			0
$581$	−7.72806			−0.320614
$582$		0			0
$583$			14.7506i			0.610907i
$584$		0			0
$585$	2.82662	−	2.36780i	0.116866	−	0.0978966i
$586$		0			0
$587$		−	11.2208i		−	0.463131i	−0.972819	−	0.231565i	$-0.925615\pi$
$587$		−	11.2208i		−	0.463131i	0.972819	−	0.231565i	$-0.0743847\pi$
$588$		0			0
$589$	−53.9638			−2.22354
$590$		0			0
$591$	−21.3320			−0.877480
$592$		0			0
$593$		−	14.2346i		−	0.584546i	−0.956335	−	0.292273i	$-0.905588\pi$
$593$		−	14.2346i		−	0.584546i	0.956335	−	0.292273i	$-0.0944116\pi$
$594$		0			0
$595$	−5.81954	−	6.94720i	−0.238578	−	0.284807i
$596$		0			0
$597$		−	21.4787i		−	0.879064i
$598$		0			0
$599$	34.1018			1.39336			0.696682	−	0.717381i	$-0.254659\pi$
$599$	34.1018			1.39336			0.696682	+	0.717381i	$0.254659\pi$
$600$		0			0
$601$	24.5296			1.00058			0.500292	−	0.865857i	$-0.333226\pi$
$601$	24.5296			1.00058			0.500292	+	0.865857i	$0.333226\pi$
$602$		0			0
$603$		−	15.6593i		−	0.637698i
$604$		0			0
$605$	0.582767	+	0.695691i	0.0236928	+	0.0282838i
$606$		0			0
$607$		−	24.8113i		−	1.00706i	−0.863978	−	0.503530i	$-0.832034\pi$
$607$		−	24.8113i		−	1.00706i	0.863978	−	0.503530i	$-0.167966\pi$
$608$		0			0
$609$	19.6462			0.796103
$610$		0			0
$611$	8.64901			0.349902
$612$		0			0
$613$			18.7164i			0.755949i	0.925816	+	0.377975i	$0.123379\pi$
$613$			18.7164i			0.755949i	−0.925816	+	0.377975i	$0.876621\pi$
$614$		0			0
$615$	−12.8338	+	10.7507i	−0.517511	+	0.433509i
$616$		0			0
$617$			38.6488i			1.55594i	0.628301	+	0.777970i	$0.283751\pi$
$617$			38.6488i			1.55594i	−0.628301	+	0.777970i	$0.716249\pi$
$618$		0			0
$619$	11.9452			0.480120			0.240060	−	0.970758i	$-0.422833\pi$
$619$	11.9452			0.480120			0.240060	+	0.970758i	$0.422833\pi$
$620$		0			0
$621$	−32.6061			−1.30844
$622$		0			0
$623$			49.2798i			1.97435i
$624$		0			0
$625$	−23.4637	−	8.62873i	−0.938548	−	0.345149i
$626$		0			0
$627$			26.2184i			1.04706i
$628$		0			0
$629$	7.31647			0.291727
$630$		0			0
$631$	−7.80907			−0.310874			−0.155437	−	0.987846i	$-0.549679\pi$
$631$	−7.80907			−0.310874			−0.155437	+	0.987846i	$0.549679\pi$
$632$		0			0
$633$		−	20.4151i		−	0.811428i
$634$		0			0
$635$	−6.20365	+	5.19668i	−0.246184	+	0.206224i
$636$		0			0
$637$		−	14.6133i		−	0.579002i
$638$		0			0
$639$	14.1655			0.560380
$640$		0			0
$641$	−3.12485			−0.123424			−0.0617120	−	0.998094i	$-0.519656\pi$
$641$	−3.12485			−0.123424			−0.0617120	+	0.998094i	$0.519656\pi$
$642$		0			0
$643$		−	10.6402i		−	0.419609i	−0.977743	−	0.209804i	$-0.932717\pi$
$643$		−	10.6402i		−	0.419609i	0.977743	−	0.209804i	$-0.0672827\pi$
$644$		0			0
$645$	10.0452	+	11.9916i	0.395528	+	0.472170i
$646$		0			0
$647$			19.1320i			0.752155i	0.926588	+	0.376078i	$0.122727\pi$
$647$			19.1320i			0.752155i	−0.926588	+	0.376078i	$0.877273\pi$
$648$		0			0
$649$	11.4288			0.448621
$650$		0			0
$651$	−42.0766			−1.64911
$652$		0			0
$653$		−	42.5252i		−	1.66414i	−0.554670	−	0.832070i	$-0.687155\pi$
$653$		−	42.5252i		−	1.66414i	0.554670	−	0.832070i	$-0.312845\pi$
$654$		0			0
$655$	−20.3109	−	24.2466i	−0.793614	−	0.947394i
$656$		0			0
$657$		−	5.46004i		−	0.213016i
$658$		0			0
$659$	1.14298			0.0445243			0.0222621	−	0.999752i	$-0.492913\pi$
$659$	1.14298			0.0445243			0.0222621	+	0.999752i	$0.492913\pi$
$660$		0			0
$661$	49.0073			1.90616			0.953082	−	0.302711i	$-0.0978917\pi$
$661$	49.0073			1.90616			0.953082	+	0.302711i	$0.0978917\pi$
$662$		0			0
$663$			1.01329i			0.0393527i
$664$		0			0
$665$	−55.2269	+	46.2625i	−2.14161	+	1.79398i
$666$		0			0
$667$		−	21.9383i		−	0.849456i
$668$		0			0
$669$	−16.9656			−0.655927
$670$		0			0
$671$	−48.3134			−1.86512
$672$		0			0
$673$			8.44491i			0.325527i	0.986665	+	0.162764i	$0.0520408\pi$
$673$			8.44491i			0.325527i	−0.986665	+	0.162764i	$0.947959\pi$
$674$		0			0
$675$	−26.5999	−	4.73598i	−1.02383	−	0.182288i
$676$		0			0
$677$		−	15.7211i		−	0.604211i	−0.953274	−	0.302106i	$-0.902310\pi$
$677$		−	15.7211i		−	0.604211i	0.953274	−	0.302106i	$-0.0976895\pi$
$678$		0			0
$679$	−44.7359			−1.71681
$680$		0			0
$681$	−3.12537			−0.119764
$682$		0			0
$683$		−	26.1823i		−	1.00184i	−0.865495	−	0.500918i	$-0.832996\pi$
$683$		−	26.1823i		−	1.00184i	0.865495	−	0.500918i	$-0.167004\pi$
$684$		0			0
$685$	2.23183	−	1.86956i	0.0852739	−	0.0714323i
$686$		0			0
$687$			0.280305i			0.0106943i
$688$		0			0
$689$	4.53186			0.172650
$690$		0			0
$691$	34.2659			1.30354			0.651768	−	0.758418i	$-0.274028\pi$
$691$	34.2659			1.30354			0.651768	+	0.758418i	$0.274028\pi$
$692$		0			0
$693$		−	24.9527i		−	0.947877i
$694$		0			0
$695$	8.72665	+	10.4176i	0.331021	+	0.395163i
$696$		0			0
$697$			5.61559i			0.212706i
$698$		0			0
$699$	20.6852			0.782385
$700$		0			0
$701$	−10.7662			−0.406633			−0.203316	−	0.979113i	$-0.565172\pi$
$701$	−10.7662			−0.406633			−0.203316	+	0.979113i	$0.565172\pi$
$702$		0			0
$703$		−	58.1624i		−	2.19364i
$704$		0			0
$705$	−14.4349	−	17.2319i	−0.543649	−	0.648992i
$706$		0			0
$707$			90.0339i			3.38607i
$708$		0			0
$709$	15.8972			0.597034			0.298517	−	0.954404i	$-0.403508\pi$
$709$	15.8972			0.597034			0.298517	+	0.954404i	$0.403508\pi$
$710$		0			0
$711$	8.54377			0.320416
$712$		0			0
$713$			46.9858i			1.75963i
$714$		0			0
$715$	−5.57924	+	4.67363i	−0.208652	+	0.174784i
$716$		0			0
$717$		−	22.1391i		−	0.826799i
$718$		0			0
$719$	27.1956			1.01422			0.507112	−	0.861880i	$-0.330713\pi$
$719$	27.1956			1.01422			0.507112	+	0.861880i	$0.330713\pi$
$720$		0			0
$721$	24.7891			0.923196
$722$		0			0
$723$			17.0008i			0.632267i
$724$		0			0
$725$	3.18651	−	17.8972i	0.118344	−	0.664685i
$726$		0			0
$727$		−	42.6226i		−	1.58078i	−0.612602	−	0.790392i	$-0.709877\pi$
$727$		−	42.6226i		−	1.58078i	0.612602	−	0.790392i	$-0.290123\pi$
$728$		0			0
$729$	−21.0415			−0.779317
$730$		0			0
$731$	5.24708			0.194070
$732$		0			0
$733$			26.6852i			0.985639i	0.870131	+	0.492820i	$0.164034\pi$
$733$			26.6852i			0.985639i	−0.870131	+	0.492820i	$0.835966\pi$
$734$		0			0
$735$	−29.1150	+	24.3891i	−1.07392	+	0.899605i
$736$		0			0
$737$			30.9088i			1.13854i
$738$		0			0
$739$	8.12634			0.298932			0.149466	−	0.988767i	$-0.452245\pi$
$739$	8.12634			0.298932			0.149466	+	0.988767i	$0.452245\pi$
$740$		0			0
$741$	8.05513			0.295913
$742$		0			0
$743$			17.4905i			0.641665i	0.947136	+	0.320833i	$0.103963\pi$
$743$			17.4905i			0.641665i	−0.947136	+	0.320833i	$0.896037\pi$
$744$		0			0
$745$	17.3546	+	20.7175i	0.635825	+	0.759029i
$746$		0			0
$747$			2.74116i			0.100294i
$748$		0			0
$749$	−26.1234			−0.954530
$750$		0			0
$751$	−50.2377			−1.83320			−0.916600	−	0.399806i	$-0.869078\pi$
$751$	−50.2377			−1.83320			−0.916600	+	0.399806i	$0.869078\pi$
$752$		0			0
$753$			0.179609i			0.00654531i
$754$		0			0
$755$	21.4702	+	25.6306i	0.781382	+	0.932792i
$756$		0			0
$757$			21.4761i			0.780560i	0.920696	+	0.390280i	$0.127622\pi$
$757$			21.4761i			0.780560i	−0.920696	+	0.390280i	$0.872378\pi$
$758$		0			0
$759$	22.8281			0.828609
$760$		0			0
$761$	−23.0495			−0.835544			−0.417772	−	0.908552i	$-0.637189\pi$
$761$	−23.0495			−0.835544			−0.417772	+	0.908552i	$0.637189\pi$
$762$		0			0
$763$		−	12.4905i		−	0.452185i
$764$		0			0
$765$	−2.46418	+	2.06420i	−0.0890928	+	0.0746313i
$766$		0			0
$767$		−	3.51131i		−	0.126786i
$768$		0			0
$769$	−51.1473			−1.84442			−0.922210	−	0.386689i	$-0.873619\pi$
$769$	−51.1473			−1.84442			−0.922210	+	0.386689i	$0.873619\pi$
$770$		0			0
$771$	4.71078			0.169655
$772$		0			0
$773$			21.8083i			0.784391i	0.919882	+	0.392196i	$0.128284\pi$
$773$			21.8083i			0.784391i	−0.919882	+	0.392196i	$0.871716\pi$
$774$		0			0
$775$	−6.82462	+	38.3308i	−0.245148	+	1.37688i
$776$		0			0
$777$		−	45.3504i		−	1.62694i
$778$		0			0
$779$	44.6413			1.59944
$780$		0			0
$781$	−27.9603			−1.00050
$782$		0			0
$783$		−	19.6462i		−	0.702097i
$784$		0			0
$785$	1.52053	−	1.27372i	0.0542702	−	0.0454611i
$786$		0			0
$787$			7.66629i			0.273274i	0.990621	+	0.136637i	$0.0436294\pi$
$787$			7.66629i			0.273274i	−0.990621	+	0.136637i	$0.956371\pi$
$788$		0			0
$789$	−19.4480			−0.692367
$790$		0			0
$791$	−12.6311			−0.449112
$792$		0			0
$793$			14.8434i			0.527106i
$794$		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:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.16232i q^{3} +(-1.43589 - 1.71412i) q^{5} +4.64901i q^{7} +1.64901 q^{9} +O(q^{10})\)	\(q+1.16232i q^{3} +(-1.43589 - 1.71412i) q^{5} +4.64901i q^{7} +1.64901 q^{9} -3.25486 q^{11} +1.00000i q^{13} +(1.99236 - 1.66896i) q^{15} -0.871779i q^{17} -6.93023 q^{19} -5.40364 q^{21} +6.03410i q^{23} +(-0.876443 + 4.92259i) q^{25} +5.40364i q^{27} -3.63573 q^{29} +7.78672 q^{31} -3.78319i q^{33} +(7.96899 - 6.67547i) q^{35} +8.39257i q^{37} -1.16232 q^{39} -6.44153 q^{41} +6.01882i q^{43} +(-2.36780 - 2.82662i) q^{45} -8.64901i q^{47} -14.6133 q^{49} +1.01329 q^{51} -4.53186i q^{53} +(4.67363 + 5.57924i) q^{55} -8.05513i q^{57} -3.51131 q^{59} +14.8434 q^{61} +7.66629i q^{63} +(1.71412 - 1.43589i) q^{65} -9.49618i q^{67} -7.01355 q^{69} +8.59031 q^{71} -3.31109i q^{73} +(-5.72161 - 1.01871i) q^{75} -15.1319i q^{77} +5.18113 q^{79} -1.33371 q^{81} +1.66230i q^{83} +(-1.49434 + 1.25178i) q^{85} -4.22588i q^{87} +10.6001 q^{89} -4.64901 q^{91} +9.05066i q^{93} +(9.95104 + 11.8793i) q^{95} +9.62267i q^{97} -5.36732 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{5} - 30 q^{9}+O(q^{10}) \) 10 * q - 4 * q^5 - 30 * q^9	\( 10 q - 4 q^{5} - 30 q^{9} + 16 q^{11} + 4 q^{15} - 36 q^{19} + 4 q^{21} + 12 q^{25} - 4 q^{29} - 8 q^{31} + 18 q^{35} - 4 q^{39} - 24 q^{41} + 36 q^{45} - 38 q^{49} - 4 q^{51} + 32 q^{55} - 28 q^{59} + 20 q^{61} + 4 q^{65} - 72 q^{69} + 36 q^{71} + 62 q^{75} - 16 q^{79} + 18 q^{81} + 8 q^{85} + 12 q^{89} + 52 q^{95} - 68 q^{99}+O(q^{100}) \) 10 * q - 4 * q^5 - 30 * q^9 + 16 * q^11 + 4 * q^15 - 36 * q^19 + 4 * q^21 + 12 * q^25 - 4 * q^29 - 8 * q^31 + 18 * q^35 - 4 * q^39 - 24 * q^41 + 36 * q^45 - 38 * q^49 - 4 * q^51 + 32 * q^55 - 28 * q^59 + 20 * q^61 + 4 * q^65 - 72 * q^69 + 36 * q^71 + 62 * q^75 - 16 * q^79 + 18 * q^81 + 8 * q^85 + 12 * q^89 + 52 * q^95 - 68 * q^99

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