LMFDB - Embedded newform 1520.2.q.n.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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.1500534351369.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 13x^{6} - 18x^{5} + 147x^{4} - 156x^{3} + 369x^{2} + 180x + 144 $ x^8 - x^7 + 13x^6 - 18x^5 + 147x^4 - 156x^3 + 369x^2 + 180x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
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			$-1.72425 + 2.98649i$ of defining polynomial
Character	$\chi$	$=$	1520.961
Dual form			1520.2.q.n.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.72425 + 2.98649i) q^{3} +(-0.500000 + 0.866025i) q^{5} +4.44850 q^{7} +(-4.44607 - 7.70083i) q^{9} +O(q^{10})$	\(q+(-1.72425 + 2.98649i) q^{3} +(-0.500000 + 0.866025i) q^{5} +4.44850 q^{7} +(-4.44607 - 7.70083i) q^{9} +0.918574 q^{11} +(1.76496 + 3.05701i) q^{13} +(-1.72425 - 2.98649i) q^{15} +(1.94607 - 3.37070i) q^{17} +(4.22425 + 1.07504i) q^{19} +(-7.67032 + 13.2854i) q^{21} +(3.44607 + 5.96878i) q^{23} +(-0.500000 - 0.866025i) q^{25} +20.3191 q^{27} +(0.0407130 + 0.0705170i) q^{29} +3.00485 q^{31} +(-1.58385 + 2.74331i) q^{33} +(-2.22425 + 3.85251i) q^{35} +6.44850 q^{37} -12.1729 q^{39} +(0.224250 - 0.388412i) q^{41} +(-1.90536 + 3.30018i) q^{43} +8.89215 q^{45} +(-2.40779 - 4.17041i) q^{47} +12.7891 q^{49} +(6.71104 + 11.6239i) q^{51} +(2.76739 + 4.79326i) q^{53} +(-0.459287 + 0.795508i) q^{55} +(-10.4943 + 10.7620i) q^{57} +(-1.04071 + 1.80257i) q^{59} +(-5.20861 - 9.02158i) q^{61} +(-19.7784 - 34.2571i) q^{63} -3.52993 q^{65} +(-4.44365 - 7.69663i) q^{67} -23.7676 q^{69} +(-1.46171 + 2.53176i) q^{71} +(8.07326 - 13.9833i) q^{73} +3.44850 q^{75} +4.08628 q^{77} +(-6.44850 + 11.1691i) q^{79} +(-21.6969 + 37.5802i) q^{81} -5.16770 q^{83} +(1.94607 + 3.37070i) q^{85} -0.280798 q^{87} +(1.26011 + 2.18258i) q^{89} +(7.85144 + 13.5991i) q^{91} +(-5.18111 + 8.97395i) q^{93} +(-3.04314 + 3.12079i) q^{95} +(-3.27575 + 5.67377i) q^{97} +(-4.08405 - 7.07378i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{3} - 4 q^{5} + 6 q^{7} - 13 q^{9}+O(q^{10}) $ 8 * q + q^3 - 4 * q^5 + 6 * q^7 - 13 * q^9	$ 8 q + q^{3} - 4 q^{5} + 6 q^{7} - 13 q^{9} + 8 q^{11} - q^{13} + q^{15} - 7 q^{17} + 19 q^{19} - 24 q^{21} + 5 q^{23} - 4 q^{25} + 28 q^{27} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 22 q^{37} - 38 q^{39} - 13 q^{41} + 7 q^{43} + 26 q^{45} + 10 q^{47} - 2 q^{49} + 16 q^{51} - 4 q^{55} - 25 q^{57} - 8 q^{59} - 11 q^{61} - 24 q^{63} + 2 q^{65} - 20 q^{67} - 26 q^{69} - 5 q^{71} + 12 q^{73} - 2 q^{75} + 18 q^{77} - 22 q^{79} - 40 q^{81} - 26 q^{83} - 7 q^{85} + 12 q^{87} + 9 q^{89} + 18 q^{91} - 34 q^{93} - 17 q^{95} - 41 q^{97} - 10 q^{99}+O(q^{100}) $ 8 * q + q^3 - 4 * q^5 + 6 * q^7 - 13 * q^9 + 8 * q^11 - q^13 + q^15 - 7 * q^17 + 19 * q^19 - 24 * q^21 + 5 * q^23 - 4 * q^25 + 28 * q^27 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 22 * q^37 - 38 * q^39 - 13 * q^41 + 7 * q^43 + 26 * q^45 + 10 * q^47 - 2 * q^49 + 16 * q^51 - 4 * q^55 - 25 * q^57 - 8 * q^59 - 11 * q^61 - 24 * q^63 + 2 * q^65 - 20 * q^67 - 26 * q^69 - 5 * q^71 + 12 * q^73 - 2 * q^75 + 18 * q^77 - 22 * q^79 - 40 * q^81 - 26 * q^83 - 7 * q^85 + 12 * q^87 + 9 * q^89 + 18 * q^91 - 34 * q^93 - 17 * q^95 - 41 * q^97 - 10 * 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
$3$	−1.72425	+	2.98649i	−0.995496	+	1.72425i	−0.415646	+	0.909526i	$0.636445\pi$
$3$	−1.72425	+	2.98649i	−0.995496	+	1.72425i	−0.579850	+	0.814723i	$0.696889\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	4.44850			1.68137			0.840687	−	0.541521i	$-0.182151\pi$
$7$	4.44850			1.68137			0.840687	+	0.541521i	$0.182151\pi$
$8$		0			0
$9$	−4.44607	−	7.70083i	−1.48202	−	2.56694i
$10$		0			0
$11$	0.918574			0.276960			0.138480	−	0.990365i	$-0.455778\pi$
$11$	0.918574			0.276960			0.138480	+	0.990365i	$0.455778\pi$
$12$		0			0
$13$	1.76496	+	3.05701i	0.489513	+	0.847861i	0.999927	−	0.0120678i	$-0.00384138\pi$
$13$	1.76496	+	3.05701i	0.489513	+	0.847861i	−0.510415	+	0.859928i	$0.670508\pi$
$14$		0			0
$15$	−1.72425	−	2.98649i	−0.445199	−	0.771108i
$16$		0			0
$17$	1.94607	−	3.37070i	0.471992	−	0.817515i	−0.527494	−	0.849559i	$-0.676868\pi$
$17$	1.94607	−	3.37070i	0.471992	−	0.817515i	0.999486	+	0.0320439i	$0.0102016\pi$
$18$		0			0
$19$	4.22425	+	1.07504i	0.969109	+	0.246631i
$19$	4.22425	+	1.07504i	0.969109	+	0.246631i
$20$		0			0
$21$	−7.67032	+	13.2854i	−1.67380	+	2.89911i
$22$		0			0
$23$	3.44607	+	5.96878i	0.718556	+	1.24458i	0.961572	+	0.274554i	$0.0885300\pi$
$23$	3.44607	+	5.96878i	0.718556	+	1.24458i	−0.243016	+	0.970022i	$0.578137\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	20.3191			3.91041
$28$		0			0
$29$	0.0407130	+	0.0705170i	0.00756022	+	0.0130947i	0.869781	−	0.493438i	$-0.164260\pi$
$29$	0.0407130	+	0.0705170i	0.00756022	+	0.0130947i	−0.862221	+	0.506533i	$0.830927\pi$
$30$		0			0
$31$	3.00485			0.539687			0.269843	−	0.962904i	$-0.413028\pi$
$31$	3.00485			0.539687			0.269843	+	0.962904i	$0.413028\pi$
$32$		0			0
$33$	−1.58385	+	2.74331i	−0.275713	+	0.477549i
$34$		0			0
$35$	−2.22425	+	3.85251i	−0.375967	+	0.651194i
$36$		0			0
$37$	6.44850			1.06013			0.530063	−	0.847958i	$-0.322168\pi$
$37$	6.44850			1.06013			0.530063	+	0.847958i	$0.322168\pi$
$38$		0			0
$39$	−12.1729			−1.94923
$40$		0			0
$41$	0.224250	−	0.388412i	0.0350219	−	0.0606598i	−0.847983	−	0.530023i	$-0.822183\pi$
$41$	0.224250	−	0.388412i	0.0350219	−	0.0606598i	0.883005	+	0.469363i	$0.155517\pi$
$42$		0			0
$43$	−1.90536	+	3.30018i	−0.290565	+	0.503273i	−0.973943	−	0.226791i	$-0.927177\pi$
$43$	−1.90536	+	3.30018i	−0.290565	+	0.503273i	0.683379	+	0.730064i	$0.260510\pi$
$44$		0			0
$45$	8.89215			1.32556
$46$		0			0
$47$	−2.40779	−	4.17041i	−0.351212	−	0.608317i	0.635250	−	0.772306i	$-0.280897\pi$
$47$	−2.40779	−	4.17041i	−0.351212	−	0.608317i	−0.986462	+	0.163990i	$0.947564\pi$
$48$		0			0
$49$	12.7891			1.82702
$50$		0			0
$51$	6.71104	+	11.6239i	0.939733	+	1.62767i
$52$		0			0
$53$	2.76739	+	4.79326i	0.380130	+	0.658404i	0.991081	−	0.133264i	$-0.0425459\pi$
$53$	2.76739	+	4.79326i	0.380130	+	0.658404i	−0.610951	+	0.791669i	$0.709213\pi$
$54$		0			0
$55$	−0.459287	+	0.795508i	−0.0619302	+	0.107266i
$56$		0			0
$57$	−10.4943	+	10.7620i	−1.39000	+	1.42547i
$58$		0			0
$59$	−1.04071	+	1.80257i	−0.135489	+	0.234674i	−0.925784	−	0.378052i	$-0.876594\pi$
$59$	−1.04071	+	1.80257i	−0.135489	+	0.234674i	0.790295	+	0.612727i	$0.209927\pi$
$60$		0			0
$61$	−5.20861	−	9.02158i	−0.666894	−	1.15510i	−0.978768	−	0.204971i	$-0.934290\pi$
$61$	−5.20861	−	9.02158i	−0.666894	−	1.15510i	0.311873	−	0.950124i	$-0.399043\pi$
$62$		0			0
$63$	−19.7784	−	34.2571i	−2.49184	−	4.31599i
$64$		0			0
$65$	−3.52993			−0.437833
$66$		0			0
$67$	−4.44365	−	7.69663i	−0.542878	−	0.940293i	−0.998737	−	0.0502406i	$-0.984001\pi$
$67$	−4.44365	−	7.69663i	−0.542878	−	0.940293i	0.455859	−	0.890052i	$-0.349332\pi$
$68$		0			0
$69$	−23.7676			−2.86128
$70$		0			0
$71$	−1.46171	+	2.53176i	−0.173473	+	0.300465i	−0.939632	−	0.342187i	$-0.888832\pi$
$71$	−1.46171	+	2.53176i	−0.173473	+	0.300465i	0.766159	+	0.642652i	$0.222166\pi$
$72$		0			0
$73$	8.07326	−	13.9833i	0.944904	−	1.63662i	0.188960	−	0.981985i	$-0.439488\pi$
$73$	8.07326	−	13.9833i	0.944904	−	1.63662i	0.755944	−	0.654636i	$-0.227178\pi$
$74$		0			0
$75$	3.44850			0.398198
$76$		0			0
$77$	4.08628			0.465674
$78$		0			0
$79$	−6.44850	+	11.1691i	−0.725513	+	1.25663i	0.233250	+	0.972417i	$0.425064\pi$
$79$	−6.44850	+	11.1691i	−0.725513	+	1.25663i	−0.958763	+	0.284208i	$0.908269\pi$
$80$		0			0
$81$	−21.6969	+	37.5802i	−2.41077	+	4.17558i
$82$		0			0
$83$	−5.16770			−0.567229			−0.283614	−	0.958938i	$-0.591534\pi$
$83$	−5.16770			−0.567229			−0.283614	+	0.958938i	$0.591534\pi$
$84$		0			0
$85$	1.94607	+	3.37070i	0.211081	+	0.365604i
$86$		0			0
$87$	−0.280798			−0.0301047
$88$		0			0
$89$	1.26011	+	2.18258i	0.133572	+	0.231353i	0.925051	−	0.379843i	$-0.124022\pi$
$89$	1.26011	+	2.18258i	0.133572	+	0.231353i	−0.791479	+	0.611196i	$0.790689\pi$
$90$		0			0
$91$	7.85144	+	13.5991i	0.823054	+	1.42557i
$92$		0			0
$93$	−5.18111	+	8.97395i	−0.537256	+	0.930555i
$94$		0			0
$95$	−3.04314	+	3.12079i	−0.312219	+	0.320186i
$96$		0			0
$97$	−3.27575	+	5.67377i	−0.332602	+	0.576084i	−0.983021	−	0.183492i	$-0.941260\pi$
$97$	−3.27575	+	5.67377i	−0.332602	+	0.576084i	0.650419	+	0.759575i	$0.274593\pi$
$98$		0			0
$99$	−4.08405	−	7.07378i	−0.410462	−	0.710942i
$100$		0			0
$101$	−4.47600	−	7.75266i	−0.445379	−	0.771418i	0.552700	−	0.833380i	$-0.313598\pi$
$101$	−4.47600	−	7.75266i	−0.445379	−	0.771418i	−0.998079	+	0.0619619i	$0.980264\pi$
$102$		0			0
$103$	4.89215			0.482038			0.241019	−	0.970520i	$-0.422518\pi$
$103$	4.89215			0.482038			0.241019	+	0.970520i	$0.422518\pi$
$104$		0			0
$105$	−7.67032	−	13.2854i	−0.748547	−	1.29652i
$106$		0			0
$107$	−9.58493			−0.926610			−0.463305	−	0.886199i	$-0.653337\pi$
$107$	−9.58493			−0.926610			−0.463305	+	0.886199i	$0.653337\pi$
$108$		0			0
$109$	4.54314	−	7.86895i	0.435154	−	0.753708i	−0.562154	−	0.827032i	$-0.690027\pi$
$109$	4.54314	−	7.86895i	0.435154	−	0.753708i	0.997308	+	0.0733239i	$0.0233607\pi$
$110$		0			0
$111$	−11.1188	+	19.2584i	−1.05535	+	1.82792i
$112$		0			0
$113$	−8.33580			−0.784166			−0.392083	−	0.919930i	$-0.628245\pi$
$113$	−8.33580			−0.784166			−0.392083	+	0.919930i	$0.628245\pi$
$114$		0			0
$115$	−6.89215			−0.642696
$116$		0			0
$117$	15.6943	−	27.1833i	1.45094	−	2.51310i
$118$		0			0
$119$	8.65711	−	14.9946i	0.793596	−	1.37455i
$120$		0			0
$121$	−10.1562			−0.923293
$122$		0			0
$123$	0.773325	+	1.33944i	0.0697284	+	0.120773i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	2.26496	+	3.92303i	0.200983	+	0.348113i	0.948845	−	0.315741i	$-0.102253\pi$
$127$	2.26496	+	3.92303i	0.200983	+	0.348113i	−0.747862	+	0.663854i	$0.768920\pi$
$128$		0			0
$129$	−6.57064	−	11.3807i	−0.578512	−	1.00201i
$130$		0			0
$131$	−6.67275	+	11.5575i	−0.583001	+	1.00979i	0.412121	+	0.911129i	$0.364788\pi$
$131$	−6.67275	+	11.5575i	−0.583001	+	1.00979i	−0.995121	+	0.0986577i	$0.968545\pi$
$132$		0			0
$133$	18.7916	+	4.78232i	1.62944	+	0.414680i
$134$		0			0
$135$	−10.1595	+	17.5968i	−0.874394	+	1.51449i
$136$		0			0
$137$	−1.37544	−	2.38233i	−0.117511	−	0.203536i	0.801269	−	0.598304i	$-0.204158\pi$
$137$	−1.37544	−	2.38233i	−0.117511	−	0.203536i	−0.918781	+	0.394768i	$0.870825\pi$
$138$		0			0
$139$	5.76011	+	9.97681i	0.488566	+	0.846222i	0.999914	−	0.0131524i	$-0.00418668\pi$
$139$	5.76011	+	9.97681i	0.488566	+	0.846222i	−0.511347	+	0.859374i	$0.670853\pi$
$140$		0			0
$141$	16.6065			1.39852
$142$		0			0
$143$	1.62125	+	2.80809i	0.135576	+	0.234824i
$144$		0			0
$145$	−0.0814260			−0.00676206
$146$		0			0
$147$	−22.0517	+	38.1946i	−1.81879	+	3.15024i
$148$		0			0
$149$	0.900314	−	1.55939i	0.0737566	−	0.127750i	−0.826788	−	0.562513i	$-0.809835\pi$
$149$	0.900314	−	1.55939i	0.0737566	−	0.127750i	0.900545	+	0.434763i	$0.143168\pi$
$150$		0			0
$151$	21.3671			1.73883			0.869414	−	0.494084i	$-0.164497\pi$
$151$	21.3671			1.73883			0.869414	+	0.494084i	$0.164497\pi$
$152$		0			0
$153$	−34.6096			−2.79802
$154$		0			0
$155$	−1.50242	+	2.60228i	−0.120678	+	0.209020i
$156$		0			0
$157$	−6.38379	+	11.0570i	−0.509482	+	0.882448i	0.490458	+	0.871465i	$0.336829\pi$
$157$	−6.38379	+	11.0570i	−0.509482	+	0.882448i	−0.999940	+	0.0109833i	$0.996504\pi$
$158$		0			0
$159$	−19.0867			−1.51367
$160$		0			0
$161$	15.3299	+	26.5521i	1.20816	+	2.09260i
$162$		0			0
$163$	3.71960			0.291341			0.145671	−	0.989333i	$-0.453466\pi$
$163$	3.71960			0.291341			0.145671	+	0.989333i	$0.453466\pi$
$164$		0			0
$165$	−1.58385	−	2.74331i	−0.123303	−	0.213566i
$166$		0			0
$167$	−12.0398	−	20.8536i	−0.931669	−	1.61370i	−0.780468	−	0.625196i	$-0.785019\pi$
$167$	−12.0398	−	20.8536i	−0.931669	−	1.61370i	−0.151202	−	0.988503i	$-0.548314\pi$
$168$		0			0
$169$	0.269813	−	0.467329i	0.0207548	−	0.0359484i
$170$		0			0
$171$	−10.5026	−	37.3099i	−0.803156	−	2.85316i
$172$		0			0
$173$	−6.21940	+	10.7723i	−0.472852	+	0.819004i	−0.999517	−	0.0310688i	$-0.990109\pi$
$173$	−6.21940	+	10.7723i	−0.472852	+	0.819004i	0.526665	+	0.850073i	$0.323442\pi$
$174$		0			0
$175$	−2.22425	−	3.85251i	−0.168137	−	0.291223i
$176$		0			0
$177$	−3.58890	−	6.21615i	−0.269758	−	0.467235i
$178$		0			0
$179$	4.72930			0.353484			0.176742	−	0.984257i	$-0.443444\pi$
$179$	4.72930			0.353484			0.176742	+	0.984257i	$0.443444\pi$
$180$		0			0
$181$	2.20861	+	3.82543i	0.164165	+	0.284342i	0.936358	−	0.351046i	$-0.114174\pi$
$181$	2.20861	+	3.82543i	0.164165	+	0.284342i	−0.772194	+	0.635387i	$0.780840\pi$
$182$		0			0
$183$	35.9238			2.65556
$184$		0			0
$185$	−3.22425	+	5.58456i	−0.237052	+	0.410585i
$186$		0			0
$187$	1.78761	−	3.09624i	0.130723	−	0.226419i
$188$		0			0
$189$	90.3894			6.57486
$190$		0			0
$191$	−19.0696			−1.37982			−0.689912	−	0.723893i	$-0.742351\pi$
$191$	−19.0696			−1.37982			−0.689912	+	0.723893i	$0.742351\pi$
$192$		0			0
$193$	3.42100	−	5.92534i	0.246249	−	0.426516i	−0.716233	−	0.697861i	$-0.754135\pi$
$193$	3.42100	−	5.92534i	0.246249	−	0.426516i	0.962482	+	0.271346i	$0.0874686\pi$
$194$		0			0
$195$	6.08647	−	10.5421i	0.435861	−	0.754934i
$196$		0			0
$197$	−12.4053			−0.883845			−0.441922	−	0.897053i	$-0.645703\pi$
$197$	−12.4053			−0.883845			−0.441922	+	0.897053i	$0.645703\pi$
$198$		0			0
$199$	5.01321	+	8.68314i	0.355377	+	0.615531i	0.987182	−	0.159596i	$-0.0510191\pi$
$199$	5.01321	+	8.68314i	0.355377	+	0.615531i	−0.631805	+	0.775127i	$0.717686\pi$
$200$		0			0
$201$	30.6478			2.16173
$202$		0			0
$203$	0.181112	+	0.313695i	0.0127116	+	0.0220171i
$204$		0			0
$205$	0.224250	+	0.388412i	0.0156623	+	0.0271279i
$206$		0			0
$207$	30.6430	−	53.0752i	2.12984	−	3.68898i
$208$		0			0
$209$	3.88029	+	0.987505i	0.268405	+	0.0683071i
$210$		0			0
$211$	3.37281	−	5.84188i	0.232194	−	0.402172i	−0.726259	−	0.687421i	$-0.758743\pi$
$211$	3.37281	−	5.84188i	0.232194	−	0.402172i	0.958454	+	0.285249i	$0.0920762\pi$
$212$		0			0
$213$	−5.04071	−	8.73077i	−0.345384	−	0.598223i
$214$		0			0
$215$	−1.90536	−	3.30018i	−0.129945	−	0.225071i
$216$		0			0
$217$	13.3671			0.907416
$218$		0			0
$219$	27.8406	+	48.2214i	1.88130	+	3.25850i
$220$		0			0
$221$	13.7390			0.924185
$222$		0			0
$223$	−1.07064	+	1.85440i	−0.0716953	+	0.124180i	−0.899644	−	0.436623i	$-0.856174\pi$
$223$	−1.07064	+	1.85440i	−0.0716953	+	0.124180i	0.827949	+	0.560803i	$0.189508\pi$
$224$		0			0
$225$	−4.44607	+	7.70083i	−0.296405	+	0.513388i
$226$		0			0
$227$	−2.94015			−0.195145			−0.0975723	−	0.995228i	$-0.531108\pi$
$227$	−2.94015			−0.195145			−0.0975723	+	0.995228i	$0.531108\pi$
$228$		0			0
$229$	−8.50835			−0.562248			−0.281124	−	0.959672i	$-0.590707\pi$
$229$	−8.50835			−0.562248			−0.281124	+	0.959672i	$0.590707\pi$
$230$		0			0
$231$	−7.04576	+	12.2036i	−0.463577	+	0.802939i
$232$		0			0
$233$	13.6390	−	23.6235i	0.893524	−	1.54763i	0.0579029	−	0.998322i	$-0.481559\pi$
$233$	13.6390	−	23.6235i	0.893524	−	1.54763i	0.835621	−	0.549306i	$-0.185108\pi$
$234$		0			0
$235$	4.81557			0.314133
$236$		0			0
$237$	−22.2376	−	38.5167i	−1.44449	−	2.50193i
$238$		0			0
$239$	4.99515			0.323109			0.161555	−	0.986864i	$-0.448349\pi$
$239$	4.99515			0.323109			0.161555	+	0.986864i	$0.448349\pi$
$240$		0			0
$241$	−4.97358	−	8.61448i	−0.320376	−	0.554908i	0.660189	−	0.751099i	$-0.270476\pi$
$241$	−4.97358	−	8.61448i	−0.320376	−	0.554908i	−0.980566	+	0.196191i	$0.937143\pi$
$242$		0			0
$243$	−44.3433	−	76.8048i	−2.84462	−	4.92703i
$244$		0			0
$245$	−6.39457	+	11.0757i	−0.408534	+	0.707602i
$246$		0			0
$247$	4.16924	+	14.8110i	0.265282	+	0.942399i
$248$		0			0
$249$	8.91041	−	15.4333i	0.564674	−	0.978044i
$250$		0			0
$251$	−11.6978	−	20.2612i	−0.738360	−	1.27888i	−0.953233	−	0.302235i	$-0.902267\pi$
$251$	−11.6978	−	20.2612i	−0.738360	−	1.27888i	0.214873	−	0.976642i	$-0.431066\pi$
$252$		0			0
$253$	3.16547	+	5.48276i	0.199012	+	0.344698i
$254$		0			0
$255$	−13.4221			−0.840523
$256$		0			0
$257$	12.6261	+	21.8690i	0.787594	+	1.36415i	0.927437	+	0.373980i	$0.122007\pi$
$257$	12.6261	+	21.8690i	0.787594	+	1.36415i	−0.139842	+	0.990174i	$0.544660\pi$
$258$		0			0
$259$	28.6861			1.78247
$260$		0			0
$261$	0.362026	−	0.627048i	0.0224089	−	0.0388133i
$262$		0			0
$263$	10.9128	−	18.9016i	0.672914	−	1.16552i	−0.304160	−	0.952621i	$-0.598376\pi$
$263$	10.9128	−	18.9016i	0.672914	−	1.16552i	0.977074	−	0.212900i	$-0.0682910\pi$
$264$		0			0
$265$	−5.53478			−0.339999
$266$		0			0
$267$	−8.69100			−0.531880
$268$		0			0
$269$	4.68354	−	8.11212i	0.285560	−	0.494605i	−0.687185	−	0.726483i	$-0.741154\pi$
$269$	4.68354	−	8.11212i	0.285560	−	0.494605i	0.972745	+	0.231878i	$0.0744870\pi$
$270$		0			0
$271$	−2.70619	+	4.68725i	−0.164389	+	0.284730i	−0.936438	−	0.350833i	$-0.885899\pi$
$271$	−2.70619	+	4.68725i	−0.164389	+	0.284730i	0.772049	+	0.635563i	$0.219232\pi$
$272$		0			0
$273$	−54.1513			−3.27739
$274$		0			0
$275$	−0.459287	−	0.795508i	−0.0276960	−	0.0479710i
$276$		0			0
$277$	9.45859			0.568312			0.284156	−	0.958778i	$-0.408287\pi$
$277$	9.45859			0.568312			0.284156	+	0.958778i	$0.408287\pi$
$278$		0			0
$279$	−13.3598	−	23.1398i	−0.799830	−	1.38535i
$280$		0			0
$281$	8.56821	+	14.8406i	0.511137	+	0.885315i	0.999917	+	0.0129077i	$0.00410877\pi$
$281$	8.56821	+	14.8406i	0.511137	+	0.885315i	−0.488780	+	0.872407i	$0.662558\pi$
$282$		0			0
$283$	−11.5576	+	20.0184i	−0.687029	+	1.18997i	0.285765	+	0.958300i	$0.407752\pi$
$283$	−11.5576	+	20.0184i	−0.687029	+	1.18997i	−0.972794	+	0.231670i	$0.925581\pi$
$284$		0			0
$285$	−4.07306	−	14.4693i	−0.241267	−	0.857088i
$286$		0			0
$287$	0.997575	−	1.72785i	0.0588850	−	0.101992i
$288$		0			0
$289$	0.925587	+	1.60316i	0.0544463	+	0.0943038i
$290$		0			0
$291$	−11.2964	−	19.5660i	−0.662208	−	1.14698i
$292$		0			0
$293$	9.05500			0.528999			0.264499	−	0.964386i	$-0.414793\pi$
$293$	9.05500			0.528999			0.264499	+	0.964386i	$0.414793\pi$
$294$		0			0
$295$	−1.04071	−	1.80257i	−0.0605927	−	0.104950i
$296$		0			0
$297$	18.6646			1.08303
$298$		0			0
$299$	−12.1644	+	21.0693i	−0.703485	+	1.21847i
$300$		0			0
$301$	−8.47600	+	14.6809i	−0.488549	+	0.846191i
$302$		0			0
$303$	30.8710			1.77349
$304$		0			0
$305$	10.4172			0.596489
$306$		0			0
$307$	−10.4271	+	18.0603i	−0.595107	+	1.03076i	0.398424	+	0.917201i	$0.369557\pi$
$307$	−10.4271	+	18.0603i	−0.595107	+	1.03076i	−0.993532	+	0.113555i	$0.963776\pi$
$308$		0			0
$309$	−8.43529	+	14.6103i	−0.479867	+	0.831154i
$310$		0			0
$311$	−3.02157			−0.171338			−0.0856689	−	0.996324i	$-0.527303\pi$
$311$	−3.02157			−0.171338			−0.0856689	+	0.996324i	$0.527303\pi$
$312$		0			0
$313$	2.71104	+	4.69565i	0.153237	+	0.265414i	0.932416	−	0.361388i	$-0.117697\pi$
$313$	2.71104	+	4.69565i	0.153237	+	0.265414i	−0.779179	+	0.626802i	$0.784364\pi$
$314$		0			0
$315$	39.5567			2.22877
$316$		0			0
$317$	7.70025	+	13.3372i	0.432489	+	0.749093i	0.997087	−	0.0762730i	$-0.0243021\pi$
$317$	7.70025	+	13.3372i	0.432489	+	0.749093i	−0.564598	+	0.825366i	$0.690969\pi$
$318$		0			0
$319$	0.0373979	+	0.0647751i	0.00209388	+	0.00362671i
$320$		0			0
$321$	16.5268	−	28.6253i	0.922436	−	1.59771i
$322$		0			0
$323$	11.8443	−	12.1466i	0.659037	−	0.675853i
$324$		0			0
$325$	1.76496	−	3.05701i	0.0979025	−	0.169572i
$326$		0			0
$327$	15.6670	+	27.1361i	0.866388	+	1.50063i
$328$		0			0
$329$	−10.7110	−	18.5521i	−0.590519	−	1.02281i
$330$		0			0
$331$	9.09113			0.499693			0.249847	−	0.968285i	$-0.419620\pi$
$331$	9.09113			0.499693			0.249847	+	0.968285i	$0.419620\pi$
$332$		0			0
$333$	−28.6705	−	49.6588i	−1.57113	−	2.72128i
$334$		0			0
$335$	8.88730			0.485565
$336$		0			0
$337$	−12.1778	+	21.0926i	−0.663367	+	1.14899i	0.316358	+	0.948640i	$0.397540\pi$
$337$	−12.1778	+	21.0926i	−0.663367	+	1.14899i	−0.979725	+	0.200345i	$0.935794\pi$
$338$		0			0
$339$	14.3730	−	24.8948i	0.780634	−	1.35210i
$340$		0			0
$341$	2.76018			0.149472
$342$		0			0
$343$	25.7530			1.39053
$344$		0			0
$345$	11.8838	−	20.5833i	0.639802	−	1.10817i
$346$		0			0
$347$	1.09949	−	1.90437i	0.0590236	−	0.102232i	−0.835004	−	0.550244i	$-0.814535\pi$
$347$	1.09949	−	1.90437i	0.0590236	−	0.102232i	0.894027	+	0.448012i	$0.147868\pi$
$348$		0			0
$349$	−29.6962			−1.58960			−0.794802	−	0.606869i	$-0.792425\pi$
$349$	−29.6962			−1.58960			−0.794802	+	0.606869i	$0.792425\pi$
$350$		0			0
$351$	35.8624	+	62.1155i	1.91419	+	3.31548i
$352$		0			0
$353$	−7.38340			−0.392979			−0.196489	−	0.980506i	$-0.562954\pi$
$353$	−7.38340			−0.392979			−0.196489	+	0.980506i	$0.562954\pi$
$354$		0			0
$355$	−1.46171	−	2.53176i	−0.0775796	−	0.134372i
$356$		0			0
$357$	29.8540	+	51.7087i	1.58004	+	2.73672i
$358$		0			0
$359$	−7.29157	+	12.6294i	−0.384835	+	0.666553i	−0.991746	−	0.128216i	$-0.959075\pi$
$359$	−7.29157	+	12.6294i	−0.384835	+	0.666553i	0.606912	+	0.794769i	$0.292408\pi$
$360$		0			0
$361$	16.6886	+	9.08249i	0.878346	+	0.478026i
$362$		0			0
$363$	17.5119	−	30.3314i	0.919134	−	1.59199i
$364$		0			0
$365$	8.07326	+	13.9833i	0.422574	+	0.731919i
$366$		0			0
$367$	2.60211	+	4.50699i	0.135829	+	0.235263i	0.925914	−	0.377735i	$-0.123297\pi$
$367$	2.60211	+	4.50699i	0.135829	+	0.235263i	−0.790085	+	0.612998i	$0.789964\pi$
$368$		0			0
$369$	−3.98812			−0.207613
$370$		0			0
$371$	12.3107	+	21.3228i	0.639141	+	1.10702i
$372$		0			0
$373$	22.2781			1.15352			0.576758	−	0.816915i	$-0.304318\pi$
$373$	22.2781			1.15352			0.576758	+	0.816915i	$0.304318\pi$
$374$		0			0
$375$	−1.72425	+	2.98649i	−0.0890399	+	0.154222i
$376$		0			0
$377$	−0.143714	+	0.248920i	−0.00740164	+	0.0128200i
$378$		0			0
$379$	−29.5950			−1.52019			−0.760097	−	0.649810i	$-0.774849\pi$
$379$	−29.5950			−1.52019			−0.760097	+	0.649810i	$0.774849\pi$
$380$		0			0
$381$	−15.6214			−0.800311
$382$		0			0
$383$	8.19101	−	14.1872i	0.418541	−	0.724934i	−0.577252	−	0.816566i	$-0.695875\pi$
$383$	8.19101	−	14.1872i	0.418541	−	0.724934i	0.995793	+	0.0916318i	$0.0292083\pi$
$384$		0			0
$385$	−2.04314	+	3.53882i	−0.104128	+	0.180355i
$386$		0			0
$387$	33.8855			1.72250
$388$		0			0
$389$	12.5350	+	21.7112i	0.635548	+	1.10080i	0.986399	+	0.164370i	$0.0525593\pi$
$389$	12.5350	+	21.7112i	0.635548	+	1.10080i	−0.350850	+	0.936432i	$0.614107\pi$
$390$		0			0
$391$	26.8253			1.35661
$392$		0			0
$393$	−23.0110	−	39.8562i	−1.16075	−	2.01048i
$394$		0			0
$395$	−6.44850	−	11.1691i	−0.324459	−	0.561980i
$396$		0			0
$397$	12.2793	−	21.2683i	0.616278	−	1.06742i	−0.373881	−	0.927477i	$-0.621973\pi$
$397$	12.2793	−	21.2683i	0.616278	−	1.06742i	0.990159	−	0.139948i	$-0.0446935\pi$
$398$		0			0
$399$	−46.6837	+	47.8749i	−2.33711	+	2.39674i
$400$		0			0
$401$	−9.27244	+	16.0603i	−0.463043	+	0.802015i	−0.999111	−	0.0421606i	$-0.986576\pi$
$401$	−9.27244	+	16.0603i	−0.463043	+	0.802015i	0.536068	+	0.844175i	$0.319909\pi$
$402$		0			0
$403$	5.30345	+	9.18584i	0.264184	+	0.457579i
$404$		0			0
$405$	−21.6969	−	37.5802i	−1.07813	−	1.86737i
$406$		0			0
$407$	5.92342			0.293613
$408$		0			0
$409$	3.28322	+	5.68671i	0.162345	+	0.281190i	0.935709	−	0.352772i	$-0.114761\pi$
$409$	3.28322	+	5.68671i	0.162345	+	0.281190i	−0.773364	+	0.633962i	$0.781428\pi$
$410$		0			0
$411$	9.48638			0.467929
$412$		0			0
$413$	−4.62961	+	8.01872i	−0.227808	+	0.394576i
$414$		0			0
$415$	2.58385	−	4.47536i	0.126836	−	0.219687i
$416$		0			0
$417$	−39.7275			−1.94546
$418$		0			0
$419$	−21.6448			−1.05742			−0.528708	−	0.848804i	$-0.677323\pi$
$419$	−21.6448			−1.05742			−0.528708	+	0.848804i	$0.677323\pi$
$420$		0			0
$421$	0.194324	−	0.336579i	0.00947078	−	0.0164039i	−0.861251	−	0.508180i	$-0.830319\pi$
$421$	0.194324	−	0.336579i	0.00947078	−	0.0164039i	0.870722	+	0.491776i	$0.163652\pi$
$422$		0			0
$423$	−21.4104	+	37.0839i	−1.04101	+	1.80308i
$424$		0			0
$425$	−3.89215			−0.188797
$426$		0			0
$427$	−23.1705	−	40.1325i	−1.12130	−	1.94215i
$428$		0			0
$429$	−11.1818			−0.539860
$430$		0			0
$431$	−1.42936	−	2.47573i	−0.0688499	−	0.119252i	0.829545	−	0.558439i	$-0.188600\pi$
$431$	−1.42936	−	2.47573i	−0.0688499	−	0.119252i	−0.898395	+	0.439188i	$0.855266\pi$
$432$		0			0
$433$	−15.2326	−	26.3836i	−0.732032	−	1.26792i	−0.956013	−	0.293323i	$-0.905239\pi$
$433$	−15.2326	−	26.3836i	−0.732032	−	1.26792i	0.223981	−	0.974593i	$-0.428095\pi$
$434$		0			0
$435$	0.140399	−	0.243178i	0.00673161	−	0.0116595i
$436$		0			0
$437$	8.14040	+	28.9183i	0.389408	+	1.38335i
$438$		0			0
$439$	−5.52527	+	9.57005i	−0.263707	+	0.456754i	−0.967224	−	0.253925i	$-0.918279\pi$
$439$	−5.52527	+	9.57005i	−0.263707	+	0.456754i	0.703517	+	0.710678i	$0.251612\pi$
$440$		0			0
$441$	−56.8615	−	98.4870i	−2.70769	−	4.68986i
$442$		0			0
$443$	19.7484	+	34.2053i	0.938276	+	1.62514i	0.768684	+	0.639628i	$0.220912\pi$
$443$	19.7484	+	34.2053i	0.938276	+	1.62514i	0.169592	+	0.985514i	$0.445755\pi$
$444$		0			0
$445$	−2.52023			−0.119470
$446$		0			0
$447$	3.10473	+	5.37756i	0.146849	+	0.254350i
$448$		0			0
$449$	−36.3604			−1.71596			−0.857978	−	0.513687i	$-0.828279\pi$
$449$	−36.3604			−1.71596			−0.857978	+	0.513687i	$0.828279\pi$
$450$		0			0
$451$	0.205990	−	0.356785i	0.00969969	−	0.0168004i
$452$		0			0
$453$	−36.8422	+	63.8125i	−1.73100	+	2.99817i
$454$		0			0
$455$	−15.7029			−0.736162
$456$		0			0
$457$	13.0145			0.608795			0.304397	−	0.952545i	$-0.401545\pi$
$457$	13.0145			0.608795			0.304397	+	0.952545i	$0.401545\pi$
$458$		0			0
$459$	39.5424	−	68.4895i	1.84568	−	3.19682i
$460$		0			0
$461$	2.56579	−	4.44408i	0.119501	−	0.206981i	−0.800069	−	0.599908i	$-0.795204\pi$
$461$	2.56579	−	4.44408i	0.119501	−	0.206981i	0.919570	+	0.392926i	$0.128537\pi$
$462$		0			0
$463$	3.78915			0.176097			0.0880483	−	0.996116i	$-0.471937\pi$
$463$	3.78915			0.176097			0.0880483	+	0.996116i	$0.471937\pi$
$464$		0			0
$465$	−5.18111	−	8.97395i	−0.240268	−	0.416157i
$466$		0			0
$467$	29.4339			1.36204			0.681020	−	0.732265i	$-0.261537\pi$
$467$	29.4339			1.36204			0.681020	+	0.732265i	$0.261537\pi$
$468$		0			0
$469$	−19.7676	−	34.2384i	−0.912782	−	1.58098i
$470$		0			0
$471$	−22.0145	−	38.1302i	−1.01437	−	1.75695i
$472$		0			0
$473$	−1.75022	+	3.03146i	−0.0804750	+	0.139387i
$474$		0			0
$475$	−1.18111	−	4.19583i	−0.0541931	−	0.192518i
$476$		0			0
$477$	24.6080	−	42.6223i	1.12672	−	1.95154i
$478$		0			0
$479$	−9.78303	−	16.9447i	−0.446998	−	0.774223i	0.551191	−	0.834379i	$-0.314174\pi$
$479$	−9.78303	−	16.9447i	−0.446998	−	0.774223i	−0.998189	+	0.0601560i	$0.980840\pi$
$480$		0			0
$481$	11.3814	+	19.7131i	0.518945	+	0.898840i
$482$		0			0
$483$	−105.730			−4.81088
$484$		0			0
$485$	−3.27575	−	5.67377i	−0.148744	−	0.257632i
$486$		0			0
$487$	4.62808			0.209718			0.104859	−	0.994487i	$-0.466561\pi$
$487$	4.62808			0.209718			0.104859	+	0.994487i	$0.466561\pi$
$488$		0			0
$489$	−6.41351	+	11.1085i	−0.290029	+	0.502345i
$490$		0			0
$491$	−4.05743	+	7.02767i	−0.183109	+	0.317154i	−0.942938	−	0.332969i	$-0.891949\pi$
$491$	−4.05743	+	7.02767i	−0.183109	+	0.317154i	0.759829	+	0.650123i	$0.225283\pi$
$492$		0			0
$493$	0.316922			0.0142735
$494$		0			0
$495$	8.16810			0.367129
$496$		0			0
$497$	−6.50242	+	11.2625i	−0.291674	+	0.505194i
$498$		0			0
$499$	22.0625	−	38.2133i	0.987652	−	1.71066i	0.358151	−	0.933664i	$-0.383407\pi$
$499$	22.0625	−	38.2133i	0.987652	−	1.71066i	0.629501	−	0.777000i	$-0.283259\pi$
$500$		0			0
$501$	83.0386			3.70989
$502$		0			0
$503$	−18.3083	−	31.7109i	−0.816326	−	1.41392i	−0.908372	−	0.418164i	$-0.862674\pi$
$503$	−18.3083	−	31.7109i	−0.816326	−	1.41392i	0.0920456	−	0.995755i	$-0.470659\pi$
$504$		0			0
$505$	8.95200			0.398359
$506$		0			0
$507$	0.930449	+	1.61158i	0.0413227	+	0.0715730i
$508$		0			0
$509$	−4.31315	−	7.47059i	−0.191177	−	0.331128i	0.754464	−	0.656342i	$-0.227897\pi$
$509$	−4.31315	−	7.47059i	−0.191177	−	0.331128i	−0.945641	+	0.325214i	$0.894564\pi$
$510$		0			0
$511$	35.9139	−	62.2047i	1.58874	−	2.75177i
$512$		0			0
$513$	85.8328	+	21.8438i	3.78961	+	0.964429i
$514$		0			0
$515$	−2.44607	+	4.23673i	−0.107787	+	0.186692i
$516$		0			0
$517$	−2.21173	−	3.83083i	−0.0972718	−	0.168480i
$518$		0			0
$519$	−21.4476	−	37.1483i	−0.941445	−	1.63063i
$520$		0			0
$521$	−13.2878			−0.582149			−0.291075	−	0.956700i	$-0.594013\pi$
$521$	−13.2878			−0.582149			−0.291075	+	0.956700i	$0.594013\pi$
$522$		0			0
$523$	−13.5037	−	23.3891i	−0.590475	−	1.02273i	−0.994168	−	0.107839i	$-0.965607\pi$
$523$	−13.5037	−	23.3891i	−0.590475	−	1.02273i	0.403693	−	0.914895i	$-0.367726\pi$
$524$		0			0
$525$	15.3406			0.669521
$526$		0			0
$527$	5.84766	−	10.1284i	0.254728	−	0.441202i
$528$		0			0
$529$	−12.2509	+	21.2191i	−0.532646	+	0.922570i
$530$		0			0
$531$	18.5084			0.803194
$532$		0			0
$533$	1.58317			0.0685747
$534$		0			0
$535$	4.79246	−	8.30079i	0.207196	−	0.358874i
$536$		0			0
$537$	−8.15449	+	14.1240i	−0.351892	+	0.609495i
$538$		0			0
$539$	11.7478			0.506013
$540$		0			0
$541$	−19.3587	−	33.5303i	−0.832296	−	1.44158i	−0.896213	−	0.443624i	$-0.853693\pi$
$541$	−19.3587	−	33.5303i	−0.832296	−	1.44158i	0.0639171	−	0.997955i	$-0.479641\pi$
$542$		0			0
$543$	−15.2328			−0.653702
$544$		0			0
$545$	4.54314	+	7.86895i	0.194607	+	0.337069i
$546$		0			0
$547$	0.574151	+	0.994459i	0.0245489	+	0.0425200i	0.878039	−	0.478589i	$-0.158852\pi$
$547$	0.574151	+	0.994459i	0.0245489	+	0.0425200i	−0.853490	+	0.521109i	$0.825518\pi$
$548$		0			0
$549$	−46.3158	+	80.2212i	−1.97671	+	3.42376i
$550$		0			0
$551$	0.0961732	+	0.341650i	0.00409712	+	0.0145548i
$552$		0			0
$553$	−28.6861	+	49.6859i	−1.21986	+	2.11286i
$554$		0			0
$555$	−11.1188	−	19.2584i	−0.471968	−	0.817472i
$556$		0			0
$557$	−18.4441	−	31.9461i	−0.781501	−	1.35360i	−0.931067	−	0.364848i	$-0.881121\pi$
$557$	−18.4441	−	31.9461i	−0.781501	−	1.35360i	0.149565	−	0.988752i	$-0.452213\pi$
$558$		0			0
$559$	−13.4516			−0.568941
$560$		0			0
$561$	6.16458	+	10.6774i	0.260269	+	0.450799i
$562$		0			0
$563$	9.52993			0.401638			0.200819	−	0.979628i	$-0.435640\pi$
$563$	9.52993			0.401638			0.200819	+	0.979628i	$0.435640\pi$
$564$		0			0
$565$	4.16790	−	7.21901i	0.175345	−	0.303706i
$566$		0			0
$567$	−96.5188	+	167.175i	−4.05341	+	7.02071i
$568$		0			0
$569$	−13.9454			−0.584620			−0.292310	−	0.956324i	$-0.594424\pi$
$569$	−13.9454			−0.584620			−0.292310	+	0.956324i	$0.594424\pi$
$570$		0			0
$571$	−30.6197			−1.28139			−0.640697	−	0.767794i	$-0.721354\pi$
$571$	−30.6197			−1.28139			−0.640697	+	0.767794i	$0.721354\pi$
$572$		0			0
$573$	32.8807	−	56.9510i	1.37361	−	2.37916i
$574$		0			0
$575$	3.44607	−	5.96878i	0.143711	−	0.248915i
$576$		0			0
$577$	42.4106			1.76558			0.882788	−	0.469771i	$-0.155664\pi$
$577$	42.4106			1.76558			0.882788	+	0.469771i	$0.155664\pi$
$578$		0			0
$579$	11.7973	+	20.4335i	0.490280	+	0.849189i
$580$		0			0
$581$	−22.9885			−0.953725
$582$		0			0
$583$	2.54205	+	4.40296i	0.105281	+	0.182352i
$584$		0			0
$585$	15.6943	+	27.1833i	0.648880	+	1.12389i
$586$		0			0
$587$	1.25310	−	2.17043i	0.0517210	−	0.0895833i	−0.839006	−	0.544122i	$-0.816863\pi$
$587$	1.25310	−	2.17043i	0.0517210	−	0.0895833i	0.890727	+	0.454539i	$0.150196\pi$
$588$		0			0
$589$	12.6932	+	3.23034i	0.523016	+	0.133104i
$590$		0			0
$591$	21.3899	−	37.0484i	0.879864	−	1.52397i
$592$		0			0
$593$	−0.556350	−	0.963627i	−0.0228466	−	0.0395714i	0.854376	−	0.519655i	$-0.173940\pi$
$593$	−0.556350	−	0.963627i	−0.0228466	−	0.0395714i	−0.877223	+	0.480084i	$0.840606\pi$
$594$		0			0
$595$	8.65711	+	14.9946i	0.354907	+	0.614717i
$596$		0			0
$597$	−34.5761			−1.41511
$598$		0			0
$599$	15.4727	+	26.7995i	0.632197	+	1.09500i	0.987102	+	0.160095i	$0.0511799\pi$
$599$	15.4727	+	26.7995i	0.632197	+	1.09500i	−0.354905	+	0.934902i	$0.615487\pi$
$600$		0			0
$601$	−6.75612			−0.275588			−0.137794	−	0.990461i	$-0.544001\pi$
$601$	−6.75612			−0.275588			−0.137794	+	0.990461i	$0.544001\pi$
$602$		0			0
$603$	−39.5136	+	68.4396i	−1.60912	+	2.78707i
$604$		0			0
$605$	5.07811	−	8.79555i	0.206455	−	0.357590i
$606$		0			0
$607$	24.6958			1.00237			0.501187	−	0.865339i	$-0.332897\pi$
$607$	24.6958			1.00237			0.501187	+	0.865339i	$0.332897\pi$
$608$		0			0
$609$	−1.24913			−0.0506172
$610$		0			0
$611$	8.49931	−	14.7212i	0.343845	−	0.595557i
$612$		0			0
$613$	8.39107	−	14.5338i	0.338912	−	0.587013i	−0.645316	−	0.763916i	$-0.723274\pi$
$613$	8.39107	−	14.5338i	0.338912	−	0.587013i	0.984228	+	0.176902i	$0.0566077\pi$
$614$		0			0
$615$	−1.54665			−0.0623670
$616$		0			0
$617$	2.21239	+	3.83197i	0.0890673	+	0.154269i	0.907117	−	0.420878i	$-0.138278\pi$
$617$	2.21239	+	3.83197i	0.0890673	+	0.154269i	−0.818050	+	0.575147i	$0.804945\pi$
$618$		0			0
$619$	−30.0787			−1.20897			−0.604483	−	0.796618i	$-0.706620\pi$
$619$	−30.0787			−1.20897			−0.604483	+	0.796618i	$0.706620\pi$
$620$		0			0
$621$	70.0210	+	121.280i	2.80985	+	4.86680i
$622$		0			0
$623$	5.60561	+	9.70920i	0.224584	+	0.388991i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−9.63975	+	9.88572i	−0.384975	+	0.394798i
$628$		0			0
$629$	12.5493	−	21.7360i	0.500372	−	0.866669i
$630$		0			0
$631$	7.77905	+	13.4737i	0.309679	+	0.536380i	0.978292	−	0.207230i	$-0.0664450\pi$
$631$	7.77905	+	13.4737i	0.309679	+	0.536380i	−0.668613	+	0.743611i	$0.733112\pi$
$632$		0			0
$633$	11.6311	+	20.1457i	0.462297	+	0.800721i
$634$		0			0
$635$	−4.52993			−0.179765
$636$		0			0
$637$	22.5724	+	39.0965i	0.894350	+	1.54906i
$638$		0			0
$639$	25.9955			1.02837
$640$		0			0
$641$	6.30345	−	10.9179i	0.248971	−	0.431231i	−0.714269	−	0.699871i	$-0.753241\pi$
$641$	6.30345	−	10.9179i	0.248971	−	0.431231i	0.963241	+	0.268640i	$0.0865742\pi$
$642$		0			0
$643$	−1.98086	+	3.43095i	−0.0781176	+	0.135304i	−0.902438	−	0.430820i	$-0.858224\pi$
$643$	−1.98086	+	3.43095i	−0.0781176	+	0.135304i	0.824320	+	0.566124i	$0.191558\pi$
$644$		0			0
$645$	13.1413			0.517437
$646$		0			0
$647$	5.09776			0.200413			0.100207	−	0.994967i	$-0.468050\pi$
$647$	5.09776			0.200413			0.100207	+	0.994967i	$0.468050\pi$
$648$		0			0
$649$	−0.955972	+	1.65579i	−0.0375252	+	0.0649955i
$650$		0			0
$651$	−23.0482	+	39.9206i	−0.903329	+	1.56461i
$652$		0			0
$653$	18.1413			0.709923			0.354962	−	0.934881i	$-0.384494\pi$
$653$	18.1413			0.709923			0.354962	+	0.934881i	$0.384494\pi$
$654$		0			0
$655$	−6.67275	−	11.5575i	−0.260726	−	0.451590i
$656$		0			0
$657$	−143.577			−5.60148
$658$		0			0
$659$	−4.61378	−	7.99130i	−0.179727	−	0.311297i	0.762060	−	0.647507i	$-0.224188\pi$
$659$	−4.61378	−	7.99130i	−0.179727	−	0.311297i	−0.941787	+	0.336210i	$0.890855\pi$
$660$		0			0
$661$	−19.7317	−	34.1763i	−0.767475	−	1.32930i	−0.938928	−	0.344113i	$-0.888180\pi$
$661$	−19.7317	−	34.1763i	−0.767475	−	1.32930i	0.171454	−	0.985192i	$-0.445154\pi$
$662$		0			0
$663$	−23.6895	+	41.0314i	−0.920022	+	1.59353i
$664$		0			0
$665$	−13.5374	+	13.8828i	−0.524958	+	0.538353i
$666$		0			0
$667$	−0.280600	+	0.486014i	−0.0108649	+	0.0188185i
$668$		0			0
$669$	−3.69210	−	6.39490i	−0.142745	−	0.247241i
$670$		0			0
$671$	−4.78450	−	8.28699i	−0.184703	−	0.319916i
$672$		0			0
$673$	29.9863			1.15589			0.577944	−	0.816076i	$-0.303855\pi$
$673$	29.9863			1.15589			0.577944	+	0.816076i	$0.303855\pi$
$674$		0			0
$675$	−10.1595	−	17.5968i	−0.391041	−	0.677302i
$676$		0			0
$677$	16.0343			0.616250			0.308125	−	0.951346i	$-0.400299\pi$
$677$	16.0343			0.616250			0.308125	+	0.951346i	$0.400299\pi$
$678$		0			0
$679$	−14.5722	+	25.2397i	−0.559229	+	0.968613i
$680$		0			0
$681$	5.06955	−	8.78072i	0.194266	−	0.336478i
$682$		0			0
$683$	−41.0268			−1.56985			−0.784923	−	0.619593i	$-0.787298\pi$
$683$	−41.0268			−1.56985			−0.784923	+	0.619593i	$0.787298\pi$
$684$		0			0
$685$	2.75087			0.105105
$686$		0			0
$687$	14.6705	−	25.4101i	0.559715	−	0.969455i
$688$		0			0
$689$	−9.76867	+	16.9198i	−0.372157	+	0.644594i
$690$		0			0
$691$	12.0602			0.458793			0.229397	−	0.973333i	$-0.426325\pi$
$691$	12.0602			0.458793			0.229397	+	0.973333i	$0.426325\pi$
$692$		0			0
$693$	−18.1679	−	31.4677i	−0.690141	−	1.19536i
$694$		0			0
$695$	−11.5202			−0.436987
$696$		0			0
$697$	−0.872814	−	1.51176i	−0.0330602	−	0.0572619i
$698$		0			0
$699$	47.0343	+	81.4657i	1.77900	+	3.08132i
$700$		0			0
$701$	11.4219	−	19.7833i	0.431398	−	0.747204i	−0.565596	−	0.824683i	$-0.691354\pi$
$701$	11.4219	−	19.7833i	0.431398	−	0.747204i	0.996994	+	0.0774788i	$0.0246870\pi$
$702$		0			0
$703$	27.2401	+	6.93240i	1.02738	+	0.261461i
$704$		0			0
$705$	−8.30325	+	14.3817i	−0.312719	+	0.541644i
$706$		0			0
$707$	−19.9115	−	34.4877i	−0.748848	−	1.29704i
$708$		0			0
$709$	−0.770010	−	1.33370i	−0.0289183	−	0.0500880i	0.851204	−	0.524835i	$-0.175873\pi$
$709$	−0.770010	−	1.33370i	−0.0289183	−	0.0500880i	−0.880122	+	0.474747i	$0.842540\pi$
$710$		0			0
$711$	114.682			4.30091
$712$		0			0
$713$	10.3549	+	17.9353i	0.387795	+	0.671681i
$714$		0			0
$715$	−3.24250			−0.121263
$716$		0			0
$717$	−8.61289	+	14.9180i	−0.321654	+	0.557121i
$718$		0			0
$719$	14.8133	−	25.6574i	0.552444	−	0.956861i	−0.445653	−	0.895206i	$-0.647029\pi$
$719$	14.8133	−	25.6574i	0.552444	−	0.956861i	0.998097	−	0.0616557i	$-0.0196381\pi$
$720$		0			0
$721$	21.7627			0.810486
$722$		0			0
$723$	34.3027			1.27573
$724$		0			0
$725$	0.0407130	−	0.0705170i	0.00151204	−	0.00261894i
$726$		0			0
$727$	8.74251	−	15.1425i	0.324242	−	0.561603i	−0.657117	−	0.753789i	$-0.728224\pi$
$727$	8.74251	−	15.1425i	0.324242	−	0.561603i	0.981359	+	0.192186i	$0.0615575\pi$
$728$		0			0
$729$	175.654			6.50570
$730$		0			0
$731$	7.41595	+	12.8448i	0.274289	+	0.475082i
$732$		0			0
$733$	−14.7557			−0.545015			−0.272508	−	0.962154i	$-0.587853\pi$
$733$	−14.7557			−0.545015			−0.272508	+	0.962154i	$0.587853\pi$
$734$		0			0
$735$	−22.0517	−	38.1946i	−0.813389	−	1.40883i
$736$		0			0
$737$	−4.08182	−	7.06992i	−0.150356	−	0.260424i
$738$		0			0
$739$	−2.79866	+	4.84743i	−0.102950	+	0.178315i	−0.912899	−	0.408186i	$-0.866162\pi$
$739$	−2.79866	+	4.84743i	−0.102950	+	0.178315i	0.809949	+	0.586501i	$0.199495\pi$
$740$		0			0
$741$	−51.4216	−	13.0864i	−1.88902	−	0.480742i
$742$		0			0
$743$	−11.7984	+	20.4354i	−0.432841	+	0.749703i	−0.997117	−	0.0758835i	$-0.975822\pi$
$743$	−11.7984	+	20.4354i	−0.432841	+	0.749703i	0.564275	+	0.825587i	$0.309156\pi$
$744$		0			0
$745$	0.900314	+	1.55939i	0.0329850	+	0.0571316i
$746$		0			0
$747$	22.9760	+	39.7956i	0.840647	+	1.45604i
$748$		0			0
$749$	−42.6385			−1.55798
$750$		0			0
$751$	14.9064	+	25.8187i	0.543944	+	0.942138i	0.998673	+	0.0515081i	$0.0164028\pi$
$751$	14.9064	+	25.8187i	0.543944	+	0.942138i	−0.454729	+	0.890630i	$0.650264\pi$
$752$		0			0
$753$	80.6799			2.94014
$754$		0			0
$755$	−10.6835	+	18.5044i	−0.388814	+	0.673445i
$756$		0			0
$757$	13.5048	−	23.3911i	0.490842	−	0.850163i	−0.509102	−	0.860706i	$-0.670023\pi$
$757$	13.5048	−	23.3911i	0.490842	−	0.850163i	0.999944	+	0.0105425i	$0.00335585\pi$
$758$		0			0
$759$	−21.8323			−0.792461
$760$		0			0
$761$	−33.9286			−1.22991			−0.614956	−	0.788561i	$-0.710826\pi$
$761$	−33.9286			−1.22991			−0.614956	+	0.788561i	$0.710826\pi$
$762$		0			0
$763$	20.2101	−	35.0050i	0.731656	−	1.26727i
$764$		0			0
$765$	17.3048	−	29.9728i	0.625656	−	1.08367i
$766$		0			0
$767$	−7.34728			−0.265295
$768$		0			0
$769$	−17.1023	−	29.6220i	−0.616725	−	1.06820i	−0.990079	−	0.140510i	$-0.955126\pi$
$769$	−17.1023	−	29.6220i	−0.616725	−	1.06820i	0.373355	−	0.927689i	$-0.378207\pi$
$770$		0			0
$771$	−87.0822			−3.13619
$772$		0			0
$773$	−2.94716	−	5.10463i	−0.106002	−	0.183601i	0.808145	−	0.588983i	$-0.200472\pi$
$773$	−2.94716	−	5.10463i	−0.106002	−	0.183601i	−0.914147	+	0.405382i	$0.867138\pi$
$774$		0			0
$775$	−1.50242	−	2.60228i	−0.0539687	−	0.0934765i
$776$		0			0
$777$	−49.4621	+	85.6708i	−1.77444	+	3.07342i
$778$		0			0
$779$	1.36485	−	1.39967i	0.0489007	−	0.0501484i
$780$		0			0
$781$	−1.34269	+	2.32561i	−0.0480452	+	0.0832168i
$782$		0			0
$783$	0.827251	+	1.43284i	0.0295635	+	0.0512055i
$784$		0			0
$785$	−6.38379	−	11.0570i	−0.227847	−	0.394643i
$786$		0			0
$787$	−31.7055			−1.13018			−0.565090	−	0.825029i	$-0.691159\pi$
$787$	−31.7055			−1.13018			−0.565090	+	0.825029i	$0.691159\pi$
$788$		0			0
$789$	37.6329	+	65.1821i	1.33977	+	2.32054i
$790$		0			0
$791$	−37.0818			−1.31848

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)

\(f(q)\)	\(=\)	\(q+(-1.72425 + 2.98649i) q^{3} +(-0.500000 + 0.866025i) q^{5} +4.44850 q^{7} +(-4.44607 - 7.70083i) q^{9} +O(q^{10})\)	\(q+(-1.72425 + 2.98649i) q^{3} +(-0.500000 + 0.866025i) q^{5} +4.44850 q^{7} +(-4.44607 - 7.70083i) q^{9} +0.918574 q^{11} +(1.76496 + 3.05701i) q^{13} +(-1.72425 - 2.98649i) q^{15} +(1.94607 - 3.37070i) q^{17} +(4.22425 + 1.07504i) q^{19} +(-7.67032 + 13.2854i) q^{21} +(3.44607 + 5.96878i) q^{23} +(-0.500000 - 0.866025i) q^{25} +20.3191 q^{27} +(0.0407130 + 0.0705170i) q^{29} +3.00485 q^{31} +(-1.58385 + 2.74331i) q^{33} +(-2.22425 + 3.85251i) q^{35} +6.44850 q^{37} -12.1729 q^{39} +(0.224250 - 0.388412i) q^{41} +(-1.90536 + 3.30018i) q^{43} +8.89215 q^{45} +(-2.40779 - 4.17041i) q^{47} +12.7891 q^{49} +(6.71104 + 11.6239i) q^{51} +(2.76739 + 4.79326i) q^{53} +(-0.459287 + 0.795508i) q^{55} +(-10.4943 + 10.7620i) q^{57} +(-1.04071 + 1.80257i) q^{59} +(-5.20861 - 9.02158i) q^{61} +(-19.7784 - 34.2571i) q^{63} -3.52993 q^{65} +(-4.44365 - 7.69663i) q^{67} -23.7676 q^{69} +(-1.46171 + 2.53176i) q^{71} +(8.07326 - 13.9833i) q^{73} +3.44850 q^{75} +4.08628 q^{77} +(-6.44850 + 11.1691i) q^{79} +(-21.6969 + 37.5802i) q^{81} -5.16770 q^{83} +(1.94607 + 3.37070i) q^{85} -0.280798 q^{87} +(1.26011 + 2.18258i) q^{89} +(7.85144 + 13.5991i) q^{91} +(-5.18111 + 8.97395i) q^{93} +(-3.04314 + 3.12079i) q^{95} +(-3.27575 + 5.67377i) q^{97} +(-4.08405 - 7.07378i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{3} - 4 q^{5} + 6 q^{7} - 13 q^{9}+O(q^{10}) \) 8 * q + q^3 - 4 * q^5 + 6 * q^7 - 13 * q^9	\( 8 q + q^{3} - 4 q^{5} + 6 q^{7} - 13 q^{9} + 8 q^{11} - q^{13} + q^{15} - 7 q^{17} + 19 q^{19} - 24 q^{21} + 5 q^{23} - 4 q^{25} + 28 q^{27} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 22 q^{37} - 38 q^{39} - 13 q^{41} + 7 q^{43} + 26 q^{45} + 10 q^{47} - 2 q^{49} + 16 q^{51} - 4 q^{55} - 25 q^{57} - 8 q^{59} - 11 q^{61} - 24 q^{63} + 2 q^{65} - 20 q^{67} - 26 q^{69} - 5 q^{71} + 12 q^{73} - 2 q^{75} + 18 q^{77} - 22 q^{79} - 40 q^{81} - 26 q^{83} - 7 q^{85} + 12 q^{87} + 9 q^{89} + 18 q^{91} - 34 q^{93} - 17 q^{95} - 41 q^{97} - 10 q^{99}+O(q^{100}) \) 8 * q + q^3 - 4 * q^5 + 6 * q^7 - 13 * q^9 + 8 * q^11 - q^13 + q^15 - 7 * q^17 + 19 * q^19 - 24 * q^21 + 5 * q^23 - 4 * q^25 + 28 * q^27 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 22 * q^37 - 38 * q^39 - 13 * q^41 + 7 * q^43 + 26 * q^45 + 10 * q^47 - 2 * q^49 + 16 * q^51 - 4 * q^55 - 25 * q^57 - 8 * q^59 - 11 * q^61 - 24 * q^63 + 2 * q^65 - 20 * q^67 - 26 * q^69 - 5 * q^71 + 12 * q^73 - 2 * q^75 + 18 * q^77 - 22 * q^79 - 40 * q^81 - 26 * q^83 - 7 * q^85 + 12 * q^87 + 9 * q^89 + 18 * q^91 - 34 * q^93 - 17 * q^95 - 41 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more