LMFDB - Embedded newform 1520.2.q.p.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(881,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1520.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.q (of order $3$, degree $2$, 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:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
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} - x^{9} + 13x^{8} - 2x^{7} + 125x^{6} - 48x^{5} + 345x^{4} - 202x^{3} + 788x^{2} - 416x + 256 $ x^10 - x^9 + 13x^8 - 2x^7 + 125x^6 - 48x^5 + 345x^4 - 202x^3 + 788x^2 - 416x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$1.69277 - 2.93196i$ of defining polynomial
Character	$\chi$	$=$	1520.961
Dual form			1520.2.q.p.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.69277 + 2.93196i) q^{3} +(0.500000 - 0.866025i) q^{5} -2.29919 q^{7} +(-4.23092 - 7.32817i) q^{9} +O(q^{10})$	\(q+(-1.69277 + 2.93196i) q^{3} +(0.500000 - 0.866025i) q^{5} -2.29919 q^{7} +(-4.23092 - 7.32817i) q^{9} -5.21642 q^{11} +(-0.116253 - 0.201356i) q^{13} +(1.69277 + 2.93196i) q^{15} +(-2.19923 + 3.80917i) q^{17} +(2.13900 - 3.79798i) q^{19} +(3.89199 - 6.74113i) q^{21} +(0.643136 + 1.11395i) q^{23} +(-0.500000 - 0.866025i) q^{25} +18.4912 q^{27} +(3.31046 + 5.73388i) q^{29} -0.286273 q^{31} +(8.83018 - 15.2943i) q^{33} +(-1.14959 + 1.99116i) q^{35} +9.07026 q^{37} +0.787155 q^{39} +(-2.14959 + 3.72321i) q^{41} +(5.53994 - 9.59546i) q^{43} -8.46184 q^{45} +(2.57651 + 4.46265i) q^{47} -1.71373 q^{49} +(-7.44555 - 12.8961i) q^{51} +(-3.54942 - 6.14778i) q^{53} +(-2.60821 + 4.51755i) q^{55} +(7.51470 + 12.7006i) q^{57} +(3.24377 - 5.61838i) q^{59} +(-4.47153 - 7.74492i) q^{61} +(9.72769 + 16.8489i) q^{63} -0.232505 q^{65} +(5.09926 + 8.83218i) q^{67} -4.35472 q^{69} +(7.36352 - 12.7540i) q^{71} +(-0.282901 + 0.489999i) q^{73} +3.38553 q^{75} +11.9935 q^{77} +(-2.18328 + 3.78156i) q^{79} +(-18.6086 + 32.2311i) q^{81} +6.67867 q^{83} +(2.19923 + 3.80917i) q^{85} -22.4153 q^{87} +(-2.21463 - 3.83586i) q^{89} +(0.267287 + 0.462955i) q^{91} +(0.484593 - 0.839340i) q^{93} +(-2.21965 - 3.75142i) q^{95} +(1.75616 - 3.04175i) q^{97} +(22.0703 + 38.2268i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{3} + 5 q^{5} - 10 q^{7} - 10 q^{9}+O(q^{10}) $ 10 * q - q^3 + 5 * q^5 - 10 * q^7 - 10 * q^9	$ 10 q - q^{3} + 5 q^{5} - 10 q^{7} - 10 q^{9} - 2 q^{11} - 5 q^{13} + q^{15} + 5 q^{17} + 4 q^{19} - 4 q^{21} + 11 q^{23} - 5 q^{25} + 20 q^{27} + 3 q^{29} - 12 q^{31} + 15 q^{33} - 5 q^{35} + 14 q^{37} + 2 q^{39} - 15 q^{41} + 11 q^{43} - 20 q^{45} + 6 q^{47} - 8 q^{49} - 20 q^{51} - 2 q^{53} - q^{55} + 25 q^{57} + 23 q^{59} - 4 q^{61} + 30 q^{63} - 10 q^{65} + 10 q^{67} - 2 q^{69} - 10 q^{71} + 2 q^{75} + 50 q^{77} + 5 q^{79} - 25 q^{81} - 22 q^{83} - 5 q^{85} + 44 q^{87} - 18 q^{91} - 7 q^{95} - 9 q^{97} + 63 q^{99}+O(q^{100}) $ 10 * q - q^3 + 5 * q^5 - 10 * q^7 - 10 * q^9 - 2 * q^11 - 5 * q^13 + q^15 + 5 * q^17 + 4 * q^19 - 4 * q^21 + 11 * q^23 - 5 * q^25 + 20 * q^27 + 3 * q^29 - 12 * q^31 + 15 * q^33 - 5 * q^35 + 14 * q^37 + 2 * q^39 - 15 * q^41 + 11 * q^43 - 20 * q^45 + 6 * q^47 - 8 * q^49 - 20 * q^51 - 2 * q^53 - q^55 + 25 * q^57 + 23 * q^59 - 4 * q^61 + 30 * q^63 - 10 * q^65 + 10 * q^67 - 2 * q^69 - 10 * q^71 + 2 * q^75 + 50 * q^77 + 5 * q^79 - 25 * q^81 - 22 * q^83 - 5 * q^85 + 44 * q^87 - 18 * q^91 - 7 * q^95 - 9 * q^97 + 63 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$3$	−1.69277	+	2.93196i	−0.977319	+	1.69277i	−0.305261	+	0.952269i	$0.598744\pi$
$3$	−1.69277	+	2.93196i	−0.977319	+	1.69277i	−0.672058	+	0.740498i	$0.734589\pi$
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	−2.29919			−0.869012			−0.434506	−	0.900669i	$-0.643077\pi$
$7$	−2.29919			−0.869012			−0.434506	+	0.900669i	$0.643077\pi$
$8$		0			0
$9$	−4.23092	−	7.32817i	−1.41031	−	2.44272i
$10$		0			0
$11$	−5.21642			−1.57281			−0.786405	−	0.617712i	$-0.788060\pi$
$11$	−5.21642			−1.57281			−0.786405	+	0.617712i	$0.788060\pi$
$12$		0			0
$13$	−0.116253	−	0.201356i	−0.0322427	−	0.0558460i	0.849454	−	0.527663i	$-0.176932\pi$
$13$	−0.116253	−	0.201356i	−0.0322427	−	0.0558460i	−0.881696	+	0.471817i	$0.843598\pi$
$14$		0			0
$15$	1.69277	+	2.93196i	0.437071	+	0.757028i
$16$		0			0
$17$	−2.19923	+	3.80917i	−0.533390	+	0.923859i	0.465849	+	0.884864i	$0.345749\pi$
$17$	−2.19923	+	3.80917i	−0.533390	+	0.923859i	−0.999239	+	0.0389952i	$0.987584\pi$
$18$		0			0
$19$	2.13900	−	3.79798i	0.490720	−	0.871317i
$19$	2.13900	−	3.79798i	0.490720	−	0.871317i
$20$		0			0
$21$	3.89199	−	6.74113i	0.849302	−	1.47103i
$22$		0			0
$23$	0.643136	+	1.11395i	0.134103	+	0.232274i	0.925255	−	0.379347i	$-0.123851\pi$
$23$	0.643136	+	1.11395i	0.134103	+	0.232274i	−0.791151	+	0.611620i	$0.790518\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	18.4912			3.55864
$28$		0			0
$29$	3.31046	+	5.73388i	0.614737	+	1.06476i	0.990431	+	0.138012i	$0.0440711\pi$
$29$	3.31046	+	5.73388i	0.614737	+	1.06476i	−0.375694	+	0.926744i	$0.622596\pi$
$30$		0			0
$31$	−0.286273			−0.0514161			−0.0257081	−	0.999669i	$-0.508184\pi$
$31$	−0.286273			−0.0514161			−0.0257081	+	0.999669i	$0.508184\pi$
$32$		0			0
$33$	8.83018	−	15.2943i	1.53714	−	2.66240i
$34$		0			0
$35$	−1.14959	+	1.99116i	−0.194317	+	0.336567i
$36$		0			0
$37$	9.07026			1.49114			0.745571	−	0.666427i	$-0.232177\pi$
$37$	9.07026			1.49114			0.745571	+	0.666427i	$0.232177\pi$
$38$		0			0
$39$	0.787155			0.126046
$40$		0			0
$41$	−2.14959	+	3.72321i	−0.335710	+	0.581467i	−0.983621	−	0.180249i	$-0.942310\pi$
$41$	−2.14959	+	3.72321i	−0.335710	+	0.581467i	0.647911	+	0.761716i	$0.275643\pi$
$42$		0			0
$43$	5.53994	−	9.59546i	0.844833	−	1.46329i	−0.0409333	−	0.999162i	$-0.513033\pi$
$43$	5.53994	−	9.59546i	0.844833	−	1.46329i	0.885766	−	0.464132i	$-0.153634\pi$
$44$		0			0
$45$	−8.46184			−1.26142
$46$		0			0
$47$	2.57651	+	4.46265i	0.375823	+	0.650945i	0.990450	−	0.137873i	$-0.0440266\pi$
$47$	2.57651	+	4.46265i	0.375823	+	0.650945i	−0.614627	+	0.788818i	$0.710693\pi$
$48$		0			0
$49$	−1.71373			−0.244818
$50$		0			0
$51$	−7.44555	−	12.8961i	−1.04259	−	1.80581i
$52$		0			0
$53$	−3.54942	−	6.14778i	−0.487551	−	0.844463i	0.512347	−	0.858779i	$-0.328776\pi$
$53$	−3.54942	−	6.14778i	−0.487551	−	0.844463i	−0.999898	+	0.0143160i	$0.995443\pi$
$54$		0			0
$55$	−2.60821	+	4.51755i	−0.351691	+	0.609146i
$56$		0			0
$57$	7.51470	+	12.7006i	0.995347	+	1.68223i
$58$		0			0
$59$	3.24377	−	5.61838i	0.422304	−	0.731451i	−0.573861	−	0.818953i	$-0.694555\pi$
$59$	3.24377	−	5.61838i	0.422304	−	0.731451i	0.996164	+	0.0875017i	$0.0278883\pi$
$60$		0			0
$61$	−4.47153	−	7.74492i	−0.572521	−	0.991635i	−0.996306	−	0.0858726i	$-0.972632\pi$
$61$	−4.47153	−	7.74492i	−0.572521	−	0.991635i	0.423785	−	0.905763i	$-0.360701\pi$
$62$		0			0
$63$	9.72769	+	16.8489i	1.22557	+	2.12276i
$64$		0			0
$65$	−0.232505			−0.0288387
$66$		0			0
$67$	5.09926	+	8.83218i	0.622974	+	1.07902i	0.988929	+	0.148390i	$0.0474091\pi$
$67$	5.09926	+	8.83218i	0.622974	+	1.07902i	−0.365955	+	0.930633i	$0.619258\pi$
$68$		0			0
$69$	−4.35472			−0.524247
$70$		0			0
$71$	7.36352	−	12.7540i	0.873889	−	1.51362i	0.0159484	−	0.999873i	$-0.494923\pi$
$71$	7.36352	−	12.7540i	0.873889	−	1.51362i	0.857941	−	0.513748i	$-0.171743\pi$
$72$		0			0
$73$	−0.282901	+	0.489999i	−0.0331111	+	0.0573501i	−0.882106	−	0.471051i	$-0.843875\pi$
$73$	−0.282901	+	0.489999i	−0.0331111	+	0.0573501i	0.848995	+	0.528401i	$0.177208\pi$
$74$		0			0
$75$	3.38553			0.390928
$76$		0			0
$77$	11.9935			1.36679
$78$		0			0
$79$	−2.18328	+	3.78156i	−0.245639	+	0.425459i	−0.962311	−	0.271951i	$-0.912331\pi$
$79$	−2.18328	+	3.78156i	−0.245639	+	0.425459i	0.716672	+	0.697410i	$0.245664\pi$
$80$		0			0
$81$	−18.6086	+	32.2311i	−2.06762	+	3.58123i
$82$		0			0
$83$	6.67867			0.733080			0.366540	−	0.930402i	$-0.380542\pi$
$83$	6.67867			0.733080			0.366540	+	0.930402i	$0.380542\pi$
$84$		0			0
$85$	2.19923	+	3.80917i	0.238539	+	0.413163i
$86$		0			0
$87$	−22.4153			−2.40318
$88$		0			0
$89$	−2.21463	−	3.83586i	−0.234751	−	0.406600i	0.724450	−	0.689328i	$-0.242094\pi$
$89$	−2.21463	−	3.83586i	−0.234751	−	0.406600i	−0.959200	+	0.282728i	$0.908761\pi$
$90$		0			0
$91$	0.267287	+	0.462955i	0.0280193	+	0.0485308i
$92$		0			0
$93$	0.484593	−	0.839340i	0.0502500	−	0.0870355i
$94$		0			0
$95$	−2.21965	−	3.75142i	−0.227731	−	0.384888i
$96$		0			0
$97$	1.75616	−	3.04175i	0.178311	−	0.308843i	−0.762991	−	0.646409i	$-0.776270\pi$
$97$	1.75616	−	3.04175i	0.178311	−	0.308843i	0.941302	+	0.337565i	$0.109603\pi$
$98$		0			0
$99$	22.0703	+	38.2268i	2.21814	+	3.84194i
$100$		0			0
$101$	7.15963	+	12.4008i	0.712409	+	1.23393i	0.963950	+	0.266083i	$0.0857294\pi$
$101$	7.15963	+	12.4008i	0.712409	+	1.23393i	−0.251541	+	0.967847i	$0.580937\pi$
$102$		0			0
$103$	−19.0649			−1.87852			−0.939258	−	0.343211i	$-0.888485\pi$
$103$	−19.0649			−1.87852			−0.939258	+	0.343211i	$0.888485\pi$
$104$		0			0
$105$	−3.89199	−	6.74113i	−0.379820	−	0.657867i
$106$		0			0
$107$	0.712658			0.0688952			0.0344476	−	0.999407i	$-0.489033\pi$
$107$	0.712658			0.0688952			0.0344476	+	0.999407i	$0.489033\pi$
$108$		0			0
$109$	3.47423	−	6.01754i	0.332771	−	0.576376i	−0.650283	−	0.759692i	$-0.725350\pi$
$109$	3.47423	−	6.01754i	0.332771	−	0.576376i	0.983054	+	0.183316i	$0.0586831\pi$
$110$		0			0
$111$	−15.3538	+	26.5936i	−1.45732	+	2.52415i
$112$		0			0
$113$	17.1125			1.60980			0.804902	−	0.593407i	$-0.202218\pi$
$113$	17.1125			1.60980			0.804902	+	0.593407i	$0.202218\pi$
$114$		0			0
$115$	1.28627			0.119946
$116$		0			0
$117$	−0.983712	+	1.70384i	−0.0909442	+	0.157520i
$118$		0			0
$119$	5.05644	−	8.75800i	0.463523	−	0.802845i
$120$		0			0
$121$	16.2110			1.47373
$122$		0			0
$123$	−7.27753	−	12.6050i	−0.656192	−	1.13656i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	0.191120	+	0.331029i	0.0169591	+	0.0293741i	0.874380	−	0.485241i	$-0.161268\pi$
$127$	0.191120	+	0.331029i	0.0169591	+	0.0293741i	−0.857421	+	0.514615i	$0.827935\pi$
$128$		0			0
$129$	18.7557	+	32.4857i	1.65134	+	2.86021i
$130$		0			0
$131$	1.76406	−	3.05544i	0.154127	−	0.266955i	−0.778614	−	0.627503i	$-0.784077\pi$
$131$	1.76406	−	3.05544i	0.154127	−	0.266955i	0.932741	+	0.360548i	$0.117410\pi$
$132$		0			0
$133$	−4.91797	+	8.73229i	−0.426442	+	0.757185i
$134$		0			0
$135$	9.24562	−	16.0139i	0.795737	−	1.37826i
$136$		0			0
$137$	−0.975668	−	1.68991i	−0.0833570	−	0.144378i	0.821333	−	0.570449i	$-0.193231\pi$
$137$	−0.975668	−	1.68991i	−0.0833570	−	0.144378i	−0.904690	+	0.426071i	$0.859898\pi$
$138$		0			0
$139$	−4.20548	−	7.28410i	−0.356704	−	0.617829i	0.630704	−	0.776023i	$-0.282766\pi$
$139$	−4.20548	−	7.28410i	−0.356704	−	0.617829i	−0.987408	+	0.158194i	$0.949433\pi$
$140$		0			0
$141$	−17.4458			−1.46920
$142$		0			0
$143$	0.606423	+	1.05035i	0.0507116	+	0.0878351i
$144$		0			0
$145$	6.62092			0.549837
$146$		0			0
$147$	2.90094	−	5.02458i	0.239266	−	0.414420i
$148$		0			0
$149$	3.94411	−	6.83140i	0.323114	−	0.559650i	−0.658015	−	0.753005i	$-0.728603\pi$
$149$	3.94411	−	6.83140i	0.323114	−	0.559650i	0.981129	+	0.193355i	$0.0619368\pi$
$150$		0			0
$151$	−12.3188			−1.00249			−0.501247	−	0.865304i	$-0.667125\pi$
$151$	−12.3188			−1.00249			−0.501247	+	0.865304i	$0.667125\pi$
$152$		0			0
$153$	37.2190			3.00898
$154$		0			0
$155$	−0.143136	+	0.247920i	−0.0114970	+	0.0199134i
$156$		0			0
$157$	5.85061	−	10.1335i	0.466929	−	0.808745i	−0.532357	−	0.846520i	$-0.678694\pi$
$157$	5.85061	−	10.1335i	0.466929	−	0.808745i	0.999286	+	0.0377747i	$0.0120269\pi$
$158$		0			0
$159$	24.0334			1.90597
$160$		0			0
$161$	−1.47869	−	2.56117i	−0.116537	−	0.201849i
$162$		0			0
$163$	2.98968			0.234170			0.117085	−	0.993122i	$-0.462645\pi$
$163$	2.98968			0.234170			0.117085	+	0.993122i	$0.462645\pi$
$164$		0			0
$165$	−8.83018	−	15.2943i	−0.687429	−	1.19066i
$166$		0			0
$167$	1.68129	+	2.91208i	0.130102	+	0.225343i	0.923716	−	0.383079i	$-0.125136\pi$
$167$	1.68129	+	2.91208i	0.130102	+	0.225343i	−0.793614	+	0.608422i	$0.791803\pi$
$168$		0			0
$169$	6.47297	−	11.2115i	0.497921	−	0.862424i
$170$		0			0
$171$	−36.8822	+	0.394018i	−2.82045	+	0.0301313i
$172$		0			0
$173$	0.926712	−	1.60511i	0.0704566	−	0.122034i	−0.828645	−	0.559775i	$-0.810888\pi$
$173$	0.926712	−	1.60511i	0.0704566	−	0.122034i	0.899101	+	0.437740i	$0.144221\pi$
$174$		0			0
$175$	1.14959	+	1.99116i	0.0869012	+	0.150517i
$176$		0			0
$177$	10.9819	+	19.0212i	0.825451	+	1.42972i
$178$		0			0
$179$	18.9406			1.41569			0.707843	−	0.706370i	$-0.249668\pi$
$179$	18.9406			1.41569			0.707843	+	0.706370i	$0.249668\pi$
$180$		0			0
$181$	−6.27285	−	10.8649i	−0.466258	−	0.807582i	0.533000	−	0.846115i	$-0.321065\pi$
$181$	−6.27285	−	10.8649i	−0.466258	−	0.807582i	−0.999257	+	0.0385335i	$0.987731\pi$
$182$		0			0
$183$	30.2770			2.23814
$184$		0			0
$185$	4.53513	−	7.85507i	0.333429	−	0.577516i
$186$		0			0
$187$	11.4721	−	19.8702i	0.838922	−	1.45305i
$188$		0			0
$189$	−42.5149			−3.09250
$190$		0			0
$191$	−6.68472			−0.483690			−0.241845	−	0.970315i	$-0.577752\pi$
$191$	−6.68472			−0.483690			−0.241845	+	0.970315i	$0.577752\pi$
$192$		0			0
$193$	4.51808	−	7.82554i	0.325218	−	0.563294i	−0.656338	−	0.754467i	$-0.727896\pi$
$193$	4.51808	−	7.82554i	0.325218	−	0.563294i	0.981556	+	0.191172i	$0.0612289\pi$
$194$		0			0
$195$	0.393577	−	0.681696i	0.0281847	−	0.0488173i
$196$		0			0
$197$	−16.7458			−1.19309			−0.596543	−	0.802581i	$-0.703460\pi$
$197$	−16.7458			−1.19309			−0.596543	+	0.802581i	$0.703460\pi$
$198$		0			0
$199$	−2.01555	−	3.49103i	−0.142878	−	0.247472i	0.785701	−	0.618606i	$-0.212302\pi$
$199$	−2.01555	−	3.49103i	−0.142878	−	0.247472i	−0.928579	+	0.371134i	$0.878969\pi$
$200$		0			0
$201$	−34.5274			−2.43538
$202$		0			0
$203$	−7.61137	−	13.1833i	−0.534214	−	0.925285i
$204$		0			0
$205$	2.14959	+	3.72321i	0.150134	+	0.260040i
$206$		0			0
$207$	5.44212	−	9.42603i	0.378253	−	0.655154i
$208$		0			0
$209$	−11.1579	+	19.8119i	−0.771809	+	1.37042i
$210$		0			0
$211$	−3.19344	+	5.53120i	−0.219846	+	0.380784i	−0.954761	−	0.297376i	$-0.903889\pi$
$211$	−3.19344	+	5.53120i	−0.219846	+	0.380784i	0.734915	+	0.678159i	$0.237222\pi$
$212$		0			0
$213$	24.9295	+	43.1791i	1.70814	+	2.95858i
$214$		0			0
$215$	−5.53994	−	9.59546i	−0.377821	−	0.654405i
$216$		0			0
$217$	0.658196			0.0446812
$218$		0			0
$219$	−0.957772	−	1.65891i	−0.0647202	−	0.112099i
$220$		0			0
$221$	1.02266			0.0687918
$222$		0			0
$223$	−6.44142	+	11.1569i	−0.431349	+	0.747118i	−0.996990	−	0.0775333i	$-0.975296\pi$
$223$	−6.44142	+	11.1569i	−0.431349	+	0.747118i	0.565641	+	0.824652i	$0.308629\pi$
$224$		0			0
$225$	−4.23092	+	7.32817i	−0.282061	+	0.488545i
$226$		0			0
$227$	−19.2498			−1.27766			−0.638828	−	0.769350i	$-0.720580\pi$
$227$	−19.2498			−1.27766			−0.638828	+	0.769350i	$0.720580\pi$
$228$		0			0
$229$	−19.3443			−1.27831			−0.639153	−	0.769080i	$-0.720715\pi$
$229$	−19.3443			−1.27831			−0.639153	+	0.769080i	$0.720715\pi$
$230$		0			0
$231$	−20.3023	+	35.1645i	−1.33579	+	2.31366i
$232$		0			0
$233$	−4.63365	+	8.02572i	−0.303561	+	0.525783i	−0.976940	−	0.213515i	$-0.931509\pi$
$233$	−4.63365	+	8.02572i	−0.303561	+	0.525783i	0.673379	+	0.739297i	$0.264842\pi$
$234$		0			0
$235$	5.15303			0.336147
$236$		0			0
$237$	−7.39158	−	12.8026i	−0.480135	−	0.831618i
$238$		0			0
$239$	−3.20995			−0.207635			−0.103817	−	0.994596i	$-0.533106\pi$
$239$	−3.20995			−0.207635			−0.103817	+	0.994596i	$0.533106\pi$
$240$		0			0
$241$	1.19291	+	2.06617i	0.0768419	+	0.133094i	0.901886	−	0.431975i	$-0.142183\pi$
$241$	1.19291	+	2.06617i	0.0768419	+	0.133094i	−0.825044	+	0.565069i	$0.808850\pi$
$242$		0			0
$243$	−35.2632	−	61.0777i	−2.26214	−	3.91814i
$244$		0			0
$245$	−0.856864	+	1.48413i	−0.0547430	+	0.0948177i
$246$		0			0
$247$	−1.01341	+	0.0108264i	−0.0644817	+	0.000688868i
$248$		0			0
$249$	−11.3054	+	19.5816i	−0.716453	+	1.24093i
$250$		0			0
$251$	−0.257596	−	0.446170i	−0.0162593	−	0.0281620i	0.857781	−	0.514015i	$-0.171842\pi$
$251$	−0.257596	−	0.446170i	−0.0162593	−	0.0281620i	−0.874041	+	0.485853i	$0.838509\pi$
$252$		0			0
$253$	−3.35487	−	5.81080i	−0.210919	−	0.365322i
$254$		0			0
$255$	−14.8911			−0.932517
$256$		0			0
$257$	−3.87373	−	6.70950i	−0.241637	−	0.418527i	0.719544	−	0.694447i	$-0.244351\pi$
$257$	−3.87373	−	6.70950i	−0.241637	−	0.418527i	−0.961181	+	0.275920i	$0.911018\pi$
$258$		0			0
$259$	−20.8542			−1.29582
$260$		0			0
$261$	28.0126	−	48.5192i	1.73394	−	3.00326i
$262$		0			0
$263$	15.7196	−	27.2272i	0.969315	−	1.67890i	0.271769	−	0.962363i	$-0.412391\pi$
$263$	15.7196	−	27.2272i	0.969315	−	1.67890i	0.697546	−	0.716540i	$-0.254275\pi$
$264$		0			0
$265$	−7.09885			−0.436079
$266$		0			0
$267$	14.9954			0.917705
$268$		0			0
$269$	6.14663	−	10.6463i	0.374767	−	0.649115i	−0.615525	−	0.788117i	$-0.711056\pi$
$269$	6.14663	−	10.6463i	0.374767	−	0.649115i	0.990292	+	0.139002i	$0.0443894\pi$
$270$		0			0
$271$	6.15091	−	10.6537i	0.373641	−	0.647165i	−0.616482	−	0.787369i	$-0.711442\pi$
$271$	6.15091	−	10.6537i	0.373641	−	0.647165i	0.990123	+	0.140204i	$0.0447758\pi$
$272$		0			0
$273$	−1.80982			−0.109535
$274$		0			0
$275$	2.60821	+	4.51755i	0.157281	+	0.272419i
$276$		0			0
$277$	16.3814			0.984264			0.492132	−	0.870521i	$-0.336218\pi$
$277$	16.3814			0.984264			0.492132	+	0.870521i	$0.336218\pi$
$278$		0			0
$279$	1.21120	+	2.09786i	0.0725125	+	0.125595i
$280$		0			0
$281$	2.81293	+	4.87213i	0.167805	+	0.290647i	0.937648	−	0.347587i	$-0.112999\pi$
$281$	2.81293	+	4.87213i	0.167805	+	0.290647i	−0.769843	+	0.638234i	$0.779665\pi$
$282$		0			0
$283$	12.2775	−	21.2653i	0.729823	−	1.26409i	−0.227135	−	0.973863i	$-0.572936\pi$
$283$	12.2775	−	21.2653i	0.729823	−	1.26409i	0.956958	−	0.290227i	$-0.0937310\pi$
$284$		0			0
$285$	14.7564	−	0.157644i	0.874091	−	0.00933805i
$286$		0			0
$287$	4.94233	−	8.56036i	0.291736	−	0.505302i
$288$		0			0
$289$	−1.17318	−	2.03201i	−0.0690108	−	0.119530i
$290$		0			0
$291$	5.94553	+	10.2980i	0.348533	+	0.603677i
$292$		0			0
$293$	0.437953			0.0255855			0.0127928	−	0.999918i	$-0.495928\pi$
$293$	0.437953			0.0255855			0.0127928	+	0.999918i	$0.495928\pi$
$294$		0			0
$295$	−3.24377	−	5.61838i	−0.188860	−	0.327115i
$296$		0			0
$297$	−96.4581			−5.59707
$298$		0			0
$299$	0.149533	−	0.258998i	0.00864770	−	0.0149783i
$300$		0			0
$301$	−12.7374	+	22.0618i	−0.734170	+	1.27162i
$302$		0			0
$303$	−48.4783			−2.78501
$304$		0			0
$305$	−8.94306			−0.512078
$306$		0			0
$307$	−8.56269	+	14.8310i	−0.488699	+	0.846451i	−0.999915	−	0.0130008i	$-0.995862\pi$
$307$	−8.56269	+	14.8310i	−0.488699	+	0.846451i	0.511217	+	0.859452i	$0.329195\pi$
$308$		0			0
$309$	32.2724	−	55.8974i	1.83591	−	3.17989i
$310$		0			0
$311$	31.6381			1.79403			0.897016	−	0.441999i	$-0.145730\pi$
$311$	31.6381			1.79403			0.897016	+	0.441999i	$0.145730\pi$
$312$		0			0
$313$	7.71574	+	13.3641i	0.436119	+	0.755381i	0.997386	−	0.0722540i	$-0.0230192\pi$
$313$	7.71574	+	13.3641i	0.436119	+	0.755381i	−0.561267	+	0.827635i	$0.689686\pi$
$314$		0			0
$315$	19.4554			1.09619
$316$		0			0
$317$	−4.24022	−	7.34427i	−0.238154	−	0.412495i	0.722030	−	0.691861i	$-0.243209\pi$
$317$	−4.24022	−	7.34427i	−0.238154	−	0.412495i	−0.960185	+	0.279366i	$0.909876\pi$
$318$		0			0
$319$	−17.2687	−	29.9103i	−0.966864	−	1.67466i
$320$		0			0
$321$	−1.20636	+	2.08948i	−0.0673327	+	0.116624i
$322$		0			0
$323$	9.76302	+	16.5004i	0.543229	+	0.918109i
$324$		0			0
$325$	−0.116253	+	0.201356i	−0.00644854	+	0.0111692i
$326$		0			0
$327$	11.7621	+	20.3726i	0.650447	+	1.12661i
$328$		0			0
$329$	−5.92390	−	10.2605i	−0.326595	−	0.565679i
$330$		0			0
$331$	7.39022			0.406203			0.203102	−	0.979158i	$-0.434898\pi$
$331$	7.39022			0.406203			0.203102	+	0.979158i	$0.434898\pi$
$332$		0			0
$333$	−38.3755	−	66.4684i	−2.10297	−	3.64244i
$334$		0			0
$335$	10.1985			0.557205
$336$		0			0
$337$	18.2139	−	31.5475i	0.992177	−	1.71850i	0.387972	−	0.921671i	$-0.373176\pi$
$337$	18.2139	−	31.5475i	0.992177	−	1.71850i	0.604204	−	0.796829i	$-0.293491\pi$
$338$		0			0
$339$	−28.9674	+	50.1730i	−1.57329	+	2.72502i
$340$		0			0
$341$	1.49332			0.0808678
$342$		0			0
$343$	20.0345			1.08176
$344$		0			0
$345$	−2.17736	+	3.77130i	−0.117225	+	0.203040i
$346$		0			0
$347$	−2.04868	+	3.54842i	−0.109979	+	0.190489i	−0.915761	−	0.401722i	$-0.868412\pi$
$347$	−2.04868	+	3.54842i	−0.109979	+	0.190489i	0.805783	+	0.592212i	$0.201745\pi$
$348$		0			0
$349$	−27.4105			−1.46725			−0.733626	−	0.679554i	$-0.762173\pi$
$349$	−27.4105			−1.46725			−0.733626	+	0.679554i	$0.762173\pi$
$350$		0			0
$351$	−2.14966	−	3.72332i	−0.114740	−	0.198736i
$352$		0			0
$353$	−18.0481			−0.960605			−0.480302	−	0.877103i	$-0.659473\pi$
$353$	−18.0481			−0.960605			−0.480302	+	0.877103i	$0.659473\pi$
$354$		0			0
$355$	−7.36352	−	12.7540i	−0.390815	−	0.676912i
$356$		0			0
$357$	17.1187	+	29.6505i	0.906020	+	1.56927i
$358$		0			0
$359$	6.76679	−	11.7204i	0.357138	−	0.618580i	−0.630344	−	0.776316i	$-0.717086\pi$
$359$	6.76679	−	11.7204i	0.357138	−	0.618580i	0.987481	+	0.157736i	$0.0504194\pi$
$360$		0			0
$361$	−9.84936	−	16.2478i	−0.518388	−	0.855146i
$362$		0			0
$363$	−27.4415	+	47.5300i	−1.44030	+	2.49468i
$364$		0			0
$365$	0.282901	+	0.489999i	0.0148077	+	0.0256477i
$366$		0			0
$367$	15.2090	+	26.3428i	0.793906	+	1.37509i	0.923532	+	0.383522i	$0.125289\pi$
$367$	15.2090	+	26.3428i	0.793906	+	1.37509i	−0.129626	+	0.991563i	$0.541378\pi$
$368$		0			0
$369$	36.3791			1.89382
$370$		0			0
$371$	8.16080	+	14.1349i	0.423687	+	0.733848i
$372$		0			0
$373$	20.6468			1.06905			0.534525	−	0.845153i	$-0.320491\pi$
$373$	20.6468			1.06905			0.534525	+	0.845153i	$0.320491\pi$
$374$		0			0
$375$	1.69277	−	2.93196i	0.0874141	−	0.151406i
$376$		0			0
$377$	0.769699	−	1.33316i	0.0396415	−	0.0686612i
$378$		0			0
$379$	27.5790			1.41664			0.708319	−	0.705893i	$-0.249454\pi$
$379$	27.5790			1.41664			0.708319	+	0.705893i	$0.249454\pi$
$380$		0			0
$381$	−1.29409			−0.0662980
$382$		0			0
$383$	−4.19855	+	7.27210i	−0.214536	+	0.371587i	−0.953129	−	0.302564i	$-0.902157\pi$
$383$	−4.19855	+	7.27210i	−0.214536	+	0.371587i	0.738593	+	0.674152i	$0.235491\pi$
$384$		0			0
$385$	5.99677	−	10.3867i	0.305624	−	0.529356i
$386$		0			0
$387$	−93.7562			−4.76589
$388$		0			0
$389$	5.54246	+	9.59983i	0.281014	+	0.486731i	0.971635	−	0.236487i	$-0.0759959\pi$
$389$	5.54246	+	9.59983i	0.281014	+	0.486731i	−0.690621	+	0.723217i	$0.742663\pi$
$390$		0			0
$391$	−5.65761			−0.286118
$392$		0			0
$393$	5.97229	+	10.3443i	0.301262	+	0.521801i
$394$		0			0
$395$	2.18328	+	3.78156i	0.109853	+	0.190271i
$396$		0			0
$397$	18.0199	−	31.2114i	0.904394	−	1.56646i	0.0826655	−	0.996577i	$-0.473657\pi$
$397$	18.0199	−	31.2114i	0.904394	−	1.56646i	0.821729	−	0.569879i	$-0.193010\pi$
$398$		0			0
$399$	−17.2777	−	29.2010i	−0.864968	−	1.46188i
$400$		0			0
$401$	3.09515	−	5.36095i	0.154564	−	0.267713i	−0.778336	−	0.627848i	$-0.783936\pi$
$401$	3.09515	−	5.36095i	0.154564	−	0.267713i	0.932900	+	0.360135i	$0.117269\pi$
$402$		0			0
$403$	0.0332800	+	0.0576426i	0.00165779	+	0.00287138i
$404$		0			0
$405$	18.6086	+	32.2311i	0.924669	+	1.60157i
$406$		0			0
$407$	−47.3143			−2.34528
$408$		0			0
$409$	−14.1432	−	24.4967i	−0.699335	−	1.21128i	−0.968697	−	0.248244i	$-0.920146\pi$
$409$	−14.1432	−	24.4967i	−0.699335	−	1.21128i	0.269363	−	0.963039i	$-0.413187\pi$
$410$		0			0
$411$	6.60632			0.325866
$412$		0			0
$413$	−7.45805	+	12.9177i	−0.366987	+	0.635640i
$414$		0			0
$415$	3.33934	−	5.78390i	0.163922	−	0.283921i
$416$		0			0
$417$	28.4756			1.39445
$418$		0			0
$419$	23.9433			1.16971			0.584854	−	0.811139i	$-0.301152\pi$
$419$	23.9433			1.16971			0.584854	+	0.811139i	$0.301152\pi$
$420$		0			0
$421$	−3.33251	+	5.77207i	−0.162417	+	0.281314i	−0.935735	−	0.352704i	$-0.885262\pi$
$421$	−3.33251	+	5.77207i	−0.162417	+	0.281314i	0.773318	+	0.634018i	$0.218595\pi$
$422$		0			0
$423$	21.8021	−	37.7623i	1.06005	−	1.83606i
$424$		0			0
$425$	4.39845			0.213356
$426$		0			0
$427$	10.2809	+	17.8070i	0.497528	+	0.861743i
$428$		0			0
$429$	−4.10613			−0.198246
$430$		0			0
$431$	12.0568	+	20.8829i	0.580754	+	1.00590i	0.995390	+	0.0959080i	$0.0305755\pi$
$431$	12.0568	+	20.8829i	0.580754	+	1.00590i	−0.414636	+	0.909987i	$0.636091\pi$
$432$		0			0
$433$	2.02923	+	3.51474i	0.0975188	+	0.168907i	0.910657	−	0.413163i	$-0.135576\pi$
$433$	2.02923	+	3.51474i	0.0975188	+	0.168907i	−0.813138	+	0.582071i	$0.802243\pi$
$434$		0			0
$435$	−11.2077	+	19.4123i	−0.537367	+	0.930747i
$436$		0			0
$437$	5.60641	−	0.0598942i	0.268191	−	0.00286513i
$438$		0			0
$439$	9.21029	−	15.9527i	0.439583	−	0.761381i	−0.558074	−	0.829791i	$-0.688459\pi$
$439$	9.21029	−	15.9527i	0.439583	−	0.761381i	0.997657	+	0.0684106i	$0.0217928\pi$
$440$		0			0
$441$	7.25064	+	12.5585i	0.345269	+	0.598023i
$442$		0			0
$443$	1.71375	+	2.96830i	0.0814226	+	0.141028i	0.903861	−	0.427826i	$-0.140720\pi$
$443$	1.71375	+	2.96830i	0.0814226	+	0.141028i	−0.822439	+	0.568854i	$0.807387\pi$
$444$		0			0
$445$	−4.42926			−0.209967
$446$		0			0
$447$	13.3529	+	23.1279i	0.631572	+	1.09391i
$448$		0			0
$449$	−5.30746			−0.250475			−0.125237	−	0.992127i	$-0.539969\pi$
$449$	−5.30746			−0.250475			−0.125237	+	0.992127i	$0.539969\pi$
$450$		0			0
$451$	11.2132	−	19.4218i	0.528008	−	0.914537i
$452$		0			0
$453$	20.8529	−	36.1184i	0.979757	−	1.69699i
$454$		0			0
$455$	0.534574			0.0250612
$456$		0			0
$457$	−3.75416			−0.175612			−0.0878062	−	0.996138i	$-0.527986\pi$
$457$	−3.75416			−0.175612			−0.0878062	+	0.996138i	$0.527986\pi$
$458$		0			0
$459$	−40.6664	+	70.4363i	−1.89815	+	3.28769i
$460$		0			0
$461$	−17.8776	+	30.9649i	−0.832643	+	1.44218i	0.0632916	+	0.997995i	$0.479840\pi$
$461$	−17.8776	+	30.9649i	−0.832643	+	1.44218i	−0.895935	+	0.444185i	$0.853493\pi$
$462$		0			0
$463$	−32.9059			−1.52927			−0.764634	−	0.644465i	$-0.777080\pi$
$463$	−32.9059			−1.52927			−0.764634	+	0.644465i	$0.777080\pi$
$464$		0			0
$465$	−0.484593	−	0.839340i	−0.0224725	−	0.0389235i
$466$		0			0
$467$	17.9580			0.830999			0.415499	−	0.909593i	$-0.363607\pi$
$467$	17.9580			0.830999			0.415499	+	0.909593i	$0.363607\pi$
$468$		0			0
$469$	−11.7242	−	20.3069i	−0.541372	−	0.937683i
$470$		0			0
$471$	19.8074	+	34.3075i	0.912678	+	1.58080i
$472$		0			0
$473$	−28.8986	+	50.0539i	−1.32876	+	2.30148i
$474$		0			0
$475$	−4.35865	+	0.0465641i	−0.199989	+	0.00213651i
$476$		0			0
$477$	−30.0347	+	52.0215i	−1.37519	+	2.38190i
$478$		0			0
$479$	−12.5142	−	21.6752i	−0.571787	−	0.990364i	−0.996383	−	0.0849806i	$-0.972917\pi$
$479$	−12.5142	−	21.6752i	−0.571787	−	0.990364i	0.424596	−	0.905383i	$-0.360416\pi$
$480$		0			0
$481$	−1.05444	−	1.82635i	−0.0480784	−	0.0832742i
$482$		0			0
$483$	10.0123			0.455577
$484$		0			0
$485$	−1.75616	−	3.04175i	−0.0797430	−	0.138119i
$486$		0			0
$487$	−15.4665			−0.700855			−0.350428	−	0.936590i	$-0.613964\pi$
$487$	−15.4665			−0.700855			−0.350428	+	0.936590i	$0.613964\pi$
$488$		0			0
$489$	−5.06084	+	8.76563i	−0.228859	+	0.396395i
$490$		0			0
$491$	2.88154	−	4.99097i	0.130042	−	0.225239i	−0.793651	−	0.608374i	$-0.791822\pi$
$491$	2.88154	−	4.99097i	0.130042	−	0.225239i	0.923693	+	0.383135i	$0.125155\pi$
$492$		0			0
$493$	−29.1218			−1.31158
$494$		0			0
$495$	44.1405			1.98397
$496$		0			0
$497$	−16.9301	+	29.3239i	−0.759420	+	1.31535i
$498$		0			0
$499$	12.4539	−	21.5708i	0.557512	−	0.965640i	−0.440191	−	0.897904i	$-0.645089\pi$
$499$	12.4539	−	21.5708i	0.557512	−	0.965640i	0.997703	−	0.0677354i	$-0.0215774\pi$
$500$		0			0
$501$	−11.3841			−0.508605
$502$		0			0
$503$	16.7754	+	29.0559i	0.747978	+	1.29554i	0.948790	+	0.315907i	$0.102309\pi$
$503$	16.7754	+	29.0559i	0.747978	+	1.29554i	−0.200812	+	0.979630i	$0.564358\pi$
$504$		0			0
$505$	14.3193			0.637198
$506$		0			0
$507$	21.9145	+	37.9570i	0.973255	+	1.68573i
$508$		0			0
$509$	9.48949	+	16.4363i	0.420614	+	0.728525i	0.996000	−	0.0893573i	$-0.0284813\pi$
$509$	9.48949	+	16.4363i	0.420614	+	0.728525i	−0.575386	+	0.817882i	$0.695148\pi$
$510$		0			0
$511$	0.650444	−	1.12660i	0.0287739	−	0.0498379i
$512$		0			0
$513$	39.5528	−	70.2295i	1.74630	−	3.10071i
$514$		0			0
$515$	−9.53243	+	16.5107i	−0.420049	+	0.727546i
$516$		0			0
$517$	−13.4402	−	23.2791i	−0.591098	−	1.02381i
$518$		0			0
$519$	3.13741	+	5.43416i	0.137717	+	0.238533i
$520$		0			0
$521$	34.4750			1.51038			0.755189	−	0.655507i	$-0.227545\pi$
$521$	34.4750			1.51038			0.755189	+	0.655507i	$0.227545\pi$
$522$		0			0
$523$	−5.64257	−	9.77322i	−0.246732	−	0.427353i	0.715885	−	0.698218i	$-0.246024\pi$
$523$	−5.64257	−	9.77322i	−0.246732	−	0.427353i	−0.962617	+	0.270865i	$0.912690\pi$
$524$		0			0
$525$	−7.78398			−0.339721
$526$		0			0
$527$	0.629579	−	1.09046i	0.0274249	−	0.0475013i
$528$		0			0
$529$	10.6728	−	18.4857i	0.464033	−	0.803728i
$530$		0			0
$531$	−54.8966			−2.38231
$532$		0			0
$533$	0.999585			0.0432968
$534$		0			0
$535$	0.356329	−	0.617180i	0.0154054	−	0.0266830i
$536$		0			0
$537$	−32.0620	+	55.5330i	−1.38358	+	2.39643i
$538$		0			0
$539$	8.93952			0.385052
$540$		0			0
$541$	3.86609	+	6.69627i	0.166216	+	0.287895i	0.937087	−	0.349097i	$-0.113512\pi$
$541$	3.86609	+	6.69627i	0.166216	+	0.287895i	−0.770870	+	0.636992i	$0.780178\pi$
$542$		0			0
$543$	42.4739			1.82273
$544$		0			0
$545$	−3.47423	−	6.01754i	−0.148820	−	0.257763i
$546$		0			0
$547$	8.48302	+	14.6930i	0.362708	+	0.628228i	0.988406	−	0.151837i	$-0.0485190\pi$
$547$	8.48302	+	14.6930i	0.362708	+	0.628228i	−0.625698	+	0.780066i	$0.715186\pi$
$548$		0			0
$549$	−37.8374	+	65.5363i	−1.61486	+	2.79702i
$550$		0			0
$551$	28.8583	−	0.308297i	1.22940	−	0.0131339i
$552$		0			0
$553$	5.01978	−	8.69452i	0.213463	−	0.369729i
$554$		0			0
$555$	15.3538	+	26.5936i	0.651734	+	1.12884i
$556$		0			0
$557$	2.88450	+	4.99611i	0.122220	+	0.211692i	0.920643	−	0.390406i	$-0.127665\pi$
$557$	2.88450	+	4.99611i	0.122220	+	0.211692i	−0.798423	+	0.602097i	$0.794332\pi$
$558$		0			0
$559$	−2.57613			−0.108959
$560$		0			0
$561$	38.8391	+	67.2713i	1.63979	+	2.84020i
$562$		0			0
$563$	36.3722			1.53291			0.766453	−	0.642301i	$-0.222020\pi$
$563$	36.3722			1.53291			0.766453	+	0.642301i	$0.222020\pi$
$564$		0			0
$565$	8.55623	−	14.8198i	0.359963	−	0.623475i
$566$		0			0
$567$	42.7847	−	74.1053i	1.79679	−	3.11213i
$568$		0			0
$569$	−45.9450			−1.92611			−0.963057	−	0.269298i	$-0.913208\pi$
$569$	−45.9450			−1.92611			−0.963057	+	0.269298i	$0.913208\pi$
$570$		0			0
$571$	−28.7409			−1.20277			−0.601385	−	0.798960i	$-0.705384\pi$
$571$	−28.7409			−1.20277			−0.601385	+	0.798960i	$0.705384\pi$
$572$		0			0
$573$	11.3157	−	19.5993i	0.472719	−	0.818774i
$574$		0			0
$575$	0.643136	−	1.11395i	0.0268206	−	0.0464547i
$576$		0			0
$577$	−3.39938			−0.141518			−0.0707591	−	0.997493i	$-0.522542\pi$
$577$	−3.39938			−0.141518			−0.0707591	+	0.997493i	$0.522542\pi$
$578$		0			0
$579$	15.2961	+	26.4936i	0.635684	+	1.10104i
$580$		0			0
$581$	−15.3555			−0.637055
$582$		0			0
$583$	18.5153	+	32.0694i	0.766824	+	1.32818i
$584$		0			0
$585$	0.983712	+	1.70384i	0.0406715	+	0.0704451i
$586$		0			0
$587$	−8.77194	+	15.1935i	−0.362057	+	0.627101i	−0.988299	−	0.152528i	$-0.951259\pi$
$587$	−8.77194	+	15.1935i	−0.362057	+	0.627101i	0.626242	+	0.779628i	$0.284592\pi$
$588$		0			0
$589$	−0.612338	+	1.08726i	−0.0252309	+	0.0447998i
$590$		0			0
$591$	28.3467	−	49.0979i	1.16603	−	2.01962i
$592$		0			0
$593$	14.8445	+	25.7114i	0.609590	+	1.05584i	0.991308	+	0.131562i	$0.0419994\pi$
$593$	14.8445	+	25.7114i	0.609590	+	1.05584i	−0.381718	+	0.924279i	$0.624667\pi$
$594$		0			0
$595$	−5.05644	−	8.75800i	−0.207294	−	0.359043i
$596$		0			0
$597$	13.6474			0.558551
$598$		0			0
$599$	−18.2305	−	31.5762i	−0.744879	−	1.29017i	−0.950251	−	0.311484i	$-0.899174\pi$
$599$	−18.2305	−	31.5762i	−0.744879	−	1.29017i	0.205373	−	0.978684i	$-0.434159\pi$
$600$		0			0
$601$	−1.25119			−0.0510372			−0.0255186	−	0.999674i	$-0.508124\pi$
$601$	−1.25119			−0.0510372			−0.0255186	+	0.999674i	$0.508124\pi$
$602$		0			0
$603$	43.1491	−	74.7365i	1.75717	−	3.04351i
$604$		0			0
$605$	8.10551	−	14.0392i	0.329536	−	0.570773i
$606$		0			0
$607$	−21.1279			−0.857554			−0.428777	−	0.903410i	$-0.641055\pi$
$607$	−21.1279			−0.857554			−0.428777	+	0.903410i	$0.641055\pi$
$608$		0			0
$609$	51.5371			2.08839
$610$		0			0
$611$	0.599053	−	1.03759i	0.0242351	−	0.0419764i
$612$		0			0
$613$	18.0094	−	31.1932i	0.727393	−	1.25988i	−0.230588	−	0.973052i	$-0.574065\pi$
$613$	18.0094	−	31.1932i	0.727393	−	1.25988i	0.957981	−	0.286831i	$-0.0926017\pi$
$614$		0			0
$615$	−14.5551			−0.586916
$616$		0			0
$617$	10.1693	+	17.6138i	0.409401	+	0.709103i	0.994823	−	0.101626i	$-0.0324044\pi$
$617$	10.1693	+	17.6138i	0.409401	+	0.709103i	−0.585422	+	0.810729i	$0.699071\pi$
$618$		0			0
$619$	8.04168			0.323222			0.161611	−	0.986855i	$-0.448331\pi$
$619$	8.04168			0.323222			0.161611	+	0.986855i	$0.448331\pi$
$620$		0			0
$621$	11.8924	+	20.5982i	0.477225	+	0.826579i
$622$		0			0
$623$	5.09186	+	8.81936i	0.204001	+	0.353340i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−39.1998	−	66.2514i	−1.56549	−	2.64583i
$628$		0			0
$629$	−19.9475	+	34.5502i	−0.795361	+	1.37760i
$630$		0			0
$631$	−17.3979	−	30.1340i	−0.692599	−	1.19962i	−0.970983	−	0.239147i	$-0.923132\pi$
$631$	−17.3979	−	30.1340i	−0.692599	−	1.19962i	0.278385	−	0.960470i	$-0.410201\pi$
$632$		0			0
$633$	−10.8115	−	18.7261i	−0.429719	−	0.744295i
$634$		0			0
$635$	0.382240			0.0151687
$636$		0			0
$637$	0.199225	+	0.345068i	0.00789360	+	0.0136721i
$638$		0			0
$639$	−124.618			−4.92981
$640$		0			0
$641$	13.8100	−	23.9196i	0.545461	−	0.944766i	−0.453117	−	0.891451i	$-0.649688\pi$
$641$	13.8100	−	23.9196i	0.545461	−	0.944766i	0.998578	−	0.0533147i	$-0.0169786\pi$
$642$		0			0
$643$	−5.96205	+	10.3266i	−0.235120	+	0.407240i	−0.959308	−	0.282363i	$-0.908882\pi$
$643$	−5.96205	+	10.3266i	−0.235120	+	0.407240i	0.724187	+	0.689603i	$0.242215\pi$
$644$		0			0
$645$	37.5113			1.47701
$646$		0			0
$647$	−25.5124			−1.00300			−0.501498	−	0.865159i	$-0.667218\pi$
$647$	−25.5124			−1.00300			−0.501498	+	0.865159i	$0.667218\pi$
$648$		0			0
$649$	−16.9209	+	29.3078i	−0.664203	+	1.15043i
$650$		0			0
$651$	−1.11417	+	1.92980i	−0.0436678	+	0.0756349i
$652$		0			0
$653$	18.5410			0.725565			0.362782	−	0.931874i	$-0.381827\pi$
$653$	18.5410			0.725565			0.362782	+	0.931874i	$0.381827\pi$
$654$		0			0
$655$	−1.76406	−	3.05544i	−0.0689276	−	0.119386i
$656$		0			0
$657$	4.78773			0.186787
$658$		0			0
$659$	−4.42424	−	7.66301i	−0.172344	−	0.298509i	0.766895	−	0.641773i	$-0.221801\pi$
$659$	−4.42424	−	7.66301i	−0.172344	−	0.298509i	−0.939239	+	0.343264i	$0.888467\pi$
$660$		0			0
$661$	14.3789	+	24.9050i	0.559275	+	0.968692i	0.997557	+	0.0698549i	$0.0222536\pi$
$661$	14.3789	+	24.9050i	0.559275	+	0.968692i	−0.438283	+	0.898837i	$0.644413\pi$
$662$		0			0
$663$	−1.73113	+	2.99841i	−0.0672315	+	0.116448i
$664$		0			0
$665$	5.10340	+	8.62523i	0.197901	+	0.334472i
$666$		0			0
$667$	−4.25815	+	7.37534i	−0.164876	+	0.285574i
$668$		0			0
$669$	−21.8076	−	37.7719i	−0.843132	−	1.46035i
$670$		0			0
$671$	23.3254	+	40.4007i	0.900466	+	1.55965i
$672$		0			0
$673$	−47.2200			−1.82020			−0.910098	−	0.414393i	$-0.863994\pi$
$673$	−47.2200			−1.82020			−0.910098	+	0.414393i	$0.863994\pi$
$674$		0			0
$675$	−9.24562	−	16.0139i	−0.355864	−	0.616375i
$676$		0			0
$677$	−21.2638			−0.817236			−0.408618	−	0.912706i	$-0.633989\pi$
$677$	−21.2638			−0.817236			−0.408618	+	0.912706i	$0.633989\pi$
$678$		0			0
$679$	−4.03774	+	6.99357i	−0.154954	+	0.268388i
$680$		0			0
$681$	32.5855	−	56.4397i	1.24868	−	2.16277i
$682$		0			0
$683$	14.9535			0.572180			0.286090	−	0.958203i	$-0.407644\pi$
$683$	14.9535			0.572180			0.286090	+	0.958203i	$0.407644\pi$
$684$		0			0
$685$	−1.95134			−0.0745567
$686$		0			0
$687$	32.7453	−	56.7166i	1.24931	−	2.16387i
$688$		0			0
$689$	−0.825260	+	1.42939i	−0.0314399	+	0.0544555i
$690$		0			0
$691$	14.1119			0.536842			0.268421	−	0.963302i	$-0.413498\pi$
$691$	14.1119			0.536842			0.268421	+	0.963302i	$0.413498\pi$
$692$		0			0
$693$	−50.7437	−	87.8907i	−1.92759	−	3.33869i
$694$		0			0
$695$	−8.41095			−0.319046
$696$		0			0
$697$	−9.45489	−	16.3763i	−0.358129	−	0.620298i
$698$		0			0
$699$	−15.6874	−	27.1714i	−0.593352	−	1.02772i
$700$		0			0
$701$	10.1936	−	17.6559i	0.385008	−	0.666854i	−0.606762	−	0.794884i	$-0.707532\pi$
$701$	10.1936	−	17.6559i	0.385008	−	0.666854i	0.991770	+	0.128030i	$0.0408652\pi$
$702$		0			0
$703$	19.4013	−	34.4487i	0.731733	−	1.29926i
$704$		0			0
$705$	−8.72288	+	15.1085i	−0.328523	+	0.569018i
$706$		0			0
$707$	−16.4613	−	28.5119i	−0.619092	−	1.07230i
$708$		0			0
$709$	−13.1360	−	22.7522i	−0.493332	−	0.854477i	0.506638	−	0.862159i	$-0.330888\pi$
$709$	−13.1360	−	22.7522i	−0.493332	−	0.854477i	−0.999970	+	0.00768201i	$0.997555\pi$
$710$		0			0
$711$	36.9492			1.38570
$712$		0			0
$713$	−0.184113	−	0.318892i	−0.00689507	−	0.0119426i
$714$		0			0
$715$	1.21285			0.0453578
$716$		0			0
$717$	5.43370	−	9.41145i	0.202925	−	0.351477i
$718$		0			0
$719$	15.5443	−	26.9235i	0.579703	−	1.00408i	−0.415810	−	0.909452i	$-0.636502\pi$
$719$	15.5443	−	26.9235i	0.579703	−	1.00408i	0.995513	−	0.0946239i	$-0.0301649\pi$
$720$		0			0
$721$	43.8337			1.63245
$722$		0			0
$723$	−8.07725			−0.300396
$724$		0			0
$725$	3.31046	−	5.73388i	0.122947	−	0.212951i
$726$		0			0
$727$	−9.26593	+	16.0491i	−0.343654	+	0.595227i	−0.985108	−	0.171935i	$-0.944998\pi$
$727$	−9.26593	+	16.0491i	−0.343654	+	0.595227i	0.641454	+	0.767161i	$0.278332\pi$
$728$		0			0
$729$	127.118			4.70807
$730$		0			0
$731$	24.3672	+	42.2051i	0.901252	+	1.56101i
$732$		0			0
$733$	−5.60306			−0.206954			−0.103477	−	0.994632i	$-0.532997\pi$
$733$	−5.60306			−0.206954			−0.103477	+	0.994632i	$0.532997\pi$
$734$		0			0
$735$	−2.90094	−	5.02458i	−0.107003	−	0.185334i
$736$		0			0
$737$	−26.5999	−	46.0723i	−0.979819	−	1.69710i
$738$		0			0
$739$	−3.45647	+	5.98678i	−0.127148	+	0.220227i	−0.922571	−	0.385828i	$-0.873916\pi$
$739$	−3.45647	+	5.98678i	−0.127148	+	0.220227i	0.795422	+	0.606056i	$0.207249\pi$
$740$		0			0
$741$	1.68372	−	2.98960i	0.0618531	−	0.109826i
$742$		0			0
$743$	−13.7812	+	23.8698i	−0.505585	+	0.875698i	0.494394	+	0.869238i	$0.335390\pi$
$743$	−13.7812	+	23.8698i	−0.505585	+	0.875698i	−0.999979	+	0.00646075i	$0.997943\pi$
$744$		0			0
$745$	−3.94411	−	6.83140i	−0.144501	−	0.250283i
$746$		0			0
$747$	−28.2569	−	48.9425i	−1.03387	−	1.79071i
$748$		0			0
$749$	−1.63854			−0.0598708
$750$		0			0
$751$	−13.5924	−	23.5428i	−0.495995	−	0.859089i	0.503994	−	0.863707i	$-0.331863\pi$
$751$	−13.5924	−	23.5428i	−0.495995	−	0.859089i	−0.999989	+	0.00461793i	$0.998530\pi$
$752$		0			0
$753$	1.74420			0.0635623
$754$		0			0
$755$	−6.15942	+	10.6684i	−0.224164	+	0.388264i
$756$		0			0
$757$	−17.8013	+	30.8327i	−0.646998	+	1.12063i	0.336838	+	0.941562i	$0.390642\pi$
$757$	−17.8013	+	30.8327i	−0.646998	+	1.12063i	−0.983836	+	0.179071i	$0.942691\pi$
$758$		0			0
$759$	22.7160			0.824540
$760$		0			0
$761$	−10.3170			−0.373992			−0.186996	−	0.982361i	$-0.559875\pi$
$761$	−10.3170			−0.373992			−0.186996	+	0.982361i	$0.559875\pi$
$762$		0			0
$763$	−7.98791	+	13.8355i	−0.289182	+	0.500878i
$764$		0			0
$765$	18.6095	−	32.2326i	0.672828	−	1.16537i
$766$		0			0
$767$	−1.50839			−0.0544648
$768$		0			0
$769$	−4.57335	−	7.92127i	−0.164919	−	0.285648i	0.771707	−	0.635978i	$-0.219403\pi$
$769$	−4.57335	−	7.92127i	−0.164919	−	0.285648i	−0.936627	+	0.350329i	$0.886070\pi$
$770$		0			0
$771$	26.2293			0.944625
$772$		0			0
$773$	18.3662	+	31.8112i	0.660587	+	1.14417i	0.980462	+	0.196710i	$0.0630258\pi$
$773$	18.3662	+	31.8112i	0.660587	+	1.14417i	−0.319875	+	0.947460i	$0.603641\pi$
$774$		0			0
$775$	0.143136	+	0.247920i	0.00514161	+	0.00890554i
$776$		0			0
$777$	35.3014	−	61.1438i	1.26643	−	2.19352i
$778$		0			0
$779$	9.54270	+	16.1281i	0.341903	+	0.577848i
$780$		0			0
$781$	−38.4112	+	66.5302i	−1.37446	+	2.38064i
$782$		0			0
$783$	61.2145	+	106.027i	2.18763	+	3.78908i
$784$		0			0
$785$	−5.85061	−	10.1335i	−0.208817	−	0.361682i
$786$		0			0
$787$	44.3784			1.58192			0.790959	−	0.611870i	$-0.209582\pi$
$787$	44.3784			1.58192			0.790959	+	0.611870i	$0.209582\pi$
$788$		0			0
$789$	53.2194	+	92.1787i	1.89466	+	3.28165i
$790$		0			0
$791$	−39.3448

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.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.69277 + 2.93196i) q^{3} +(0.500000 - 0.866025i) q^{5} -2.29919 q^{7} +(-4.23092 - 7.32817i) q^{9} +O(q^{10})\)	\(q+(-1.69277 + 2.93196i) q^{3} +(0.500000 - 0.866025i) q^{5} -2.29919 q^{7} +(-4.23092 - 7.32817i) q^{9} -5.21642 q^{11} +(-0.116253 - 0.201356i) q^{13} +(1.69277 + 2.93196i) q^{15} +(-2.19923 + 3.80917i) q^{17} +(2.13900 - 3.79798i) q^{19} +(3.89199 - 6.74113i) q^{21} +(0.643136 + 1.11395i) q^{23} +(-0.500000 - 0.866025i) q^{25} +18.4912 q^{27} +(3.31046 + 5.73388i) q^{29} -0.286273 q^{31} +(8.83018 - 15.2943i) q^{33} +(-1.14959 + 1.99116i) q^{35} +9.07026 q^{37} +0.787155 q^{39} +(-2.14959 + 3.72321i) q^{41} +(5.53994 - 9.59546i) q^{43} -8.46184 q^{45} +(2.57651 + 4.46265i) q^{47} -1.71373 q^{49} +(-7.44555 - 12.8961i) q^{51} +(-3.54942 - 6.14778i) q^{53} +(-2.60821 + 4.51755i) q^{55} +(7.51470 + 12.7006i) q^{57} +(3.24377 - 5.61838i) q^{59} +(-4.47153 - 7.74492i) q^{61} +(9.72769 + 16.8489i) q^{63} -0.232505 q^{65} +(5.09926 + 8.83218i) q^{67} -4.35472 q^{69} +(7.36352 - 12.7540i) q^{71} +(-0.282901 + 0.489999i) q^{73} +3.38553 q^{75} +11.9935 q^{77} +(-2.18328 + 3.78156i) q^{79} +(-18.6086 + 32.2311i) q^{81} +6.67867 q^{83} +(2.19923 + 3.80917i) q^{85} -22.4153 q^{87} +(-2.21463 - 3.83586i) q^{89} +(0.267287 + 0.462955i) q^{91} +(0.484593 - 0.839340i) q^{93} +(-2.21965 - 3.75142i) q^{95} +(1.75616 - 3.04175i) q^{97} +(22.0703 + 38.2268i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{3} + 5 q^{5} - 10 q^{7} - 10 q^{9}+O(q^{10}) \) 10 * q - q^3 + 5 * q^5 - 10 * q^7 - 10 * q^9	\( 10 q - q^{3} + 5 q^{5} - 10 q^{7} - 10 q^{9} - 2 q^{11} - 5 q^{13} + q^{15} + 5 q^{17} + 4 q^{19} - 4 q^{21} + 11 q^{23} - 5 q^{25} + 20 q^{27} + 3 q^{29} - 12 q^{31} + 15 q^{33} - 5 q^{35} + 14 q^{37} + 2 q^{39} - 15 q^{41} + 11 q^{43} - 20 q^{45} + 6 q^{47} - 8 q^{49} - 20 q^{51} - 2 q^{53} - q^{55} + 25 q^{57} + 23 q^{59} - 4 q^{61} + 30 q^{63} - 10 q^{65} + 10 q^{67} - 2 q^{69} - 10 q^{71} + 2 q^{75} + 50 q^{77} + 5 q^{79} - 25 q^{81} - 22 q^{83} - 5 q^{85} + 44 q^{87} - 18 q^{91} - 7 q^{95} - 9 q^{97} + 63 q^{99}+O(q^{100}) \) 10 * q - q^3 + 5 * q^5 - 10 * q^7 - 10 * q^9 - 2 * q^11 - 5 * q^13 + q^15 + 5 * q^17 + 4 * q^19 - 4 * q^21 + 11 * q^23 - 5 * q^25 + 20 * q^27 + 3 * q^29 - 12 * q^31 + 15 * q^33 - 5 * q^35 + 14 * q^37 + 2 * q^39 - 15 * q^41 + 11 * q^43 - 20 * q^45 + 6 * q^47 - 8 * q^49 - 20 * q^51 - 2 * q^53 - q^55 + 25 * q^57 + 23 * q^59 - 4 * q^61 + 30 * q^63 - 10 * q^65 + 10 * q^67 - 2 * q^69 - 10 * q^71 + 2 * q^75 + 50 * q^77 + 5 * q^79 - 25 * q^81 - 22 * q^83 - 5 * q^85 + 44 * q^87 - 18 * q^91 - 7 * q^95 - 9 * q^97 + 63 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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