LMFDB - Embedded newform 960.3.e.c.511.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(511,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("960.511");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$26.1581053786$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.85100625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{6} + x^{5} + 3x^{4} + 2x^{3} - 8x^{2} - 8x + 16 $ x^8 - x^7 - 2x^6 + x^5 + 3x^4 + 2x^3 - 8x^2 - 8*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.7
Root			$-1.34966 - 0.422403i$ of defining polynomial
Character	$\chi$	$=$	960.511
Dual form			960.3.e.c.511.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +2.23607 q^{5} -12.3959i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +2.23607 q^{5} -12.3959i q^{7} -3.00000 q^{9} +11.0403i q^{11} -2.82009 q^{13} +3.87298i q^{15} +6.52606 q^{17} -27.9928i q^{19} +21.4703 q^{21} +7.90421i q^{23} +5.00000 q^{25} -5.19615i q^{27} -50.7169 q^{29} -36.3467i q^{31} -19.1224 q^{33} -27.7181i q^{35} +18.9279 q^{37} -4.88453i q^{39} +5.30410 q^{41} -45.5870i q^{43} -6.70820 q^{45} -11.7246i q^{47} -104.658 q^{49} +11.3035i q^{51} -41.1680 q^{53} +24.6869i q^{55} +48.4849 q^{57} +10.7008i q^{59} -56.1297 q^{61} +37.1877i q^{63} -6.30590 q^{65} -16.1709i q^{67} -13.6905 q^{69} -66.1617i q^{71} +15.6330 q^{73} +8.66025i q^{75} +136.855 q^{77} -123.057i q^{79} +9.00000 q^{81} -99.6700i q^{83} +14.5927 q^{85} -87.8443i q^{87} +101.083 q^{89} +34.9575i q^{91} +62.9543 q^{93} -62.5937i q^{95} +127.293 q^{97} -33.1209i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} - 16 q^{13} + 48 q^{21} + 40 q^{25} - 64 q^{29} + 112 q^{37} - 16 q^{41} - 56 q^{49} - 352 q^{53} + 144 q^{57} + 176 q^{61} - 80 q^{65} + 96 q^{69} - 240 q^{73} + 288 q^{77} + 72 q^{81} - 160 q^{85} + 80 q^{89} - 144 q^{93} + 432 q^{97}+O(q^{100}) $ 8 * q - 24 * q^9 - 16 * q^13 + 48 * q^21 + 40 * q^25 - 64 * q^29 + 112 * q^37 - 16 * q^41 - 56 * q^49 - 352 * q^53 + 144 * q^57 + 176 * q^61 - 80 * q^65 + 96 * q^69 - 240 * q^73 + 288 * q^77 + 72 * q^81 - 160 * q^85 + 80 * q^89 - 144 * q^93 + 432 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	− 12.3959i	− 1.77084i	−0.464789	−	0.885422i	$-0.653870\pi$
$7$	− 12.3959i	− 1.77084i	0.464789	−	0.885422i	$-0.346130\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	11.0403i	1.00366i	0.864965	+	0.501832i	$0.167341\pi$
$11$	11.0403i	1.00366i	−0.864965	+	0.501832i	$0.832659\pi$
$12$	0	0
$13$	−2.82009	−0.216930	−0.108465	−	0.994100i	$-0.534593\pi$
$13$	−2.82009	−0.216930	−0.108465	+	0.994100i	$0.534593\pi$
$14$	0	0
$15$	3.87298i	0.258199i
$16$	0	0
$17$	6.52606	0.383886	0.191943	−	0.981406i	$-0.438521\pi$
$17$	6.52606	0.383886	0.191943	+	0.981406i	$0.438521\pi$
$18$	0	0
$19$	− 27.9928i	− 1.47330i	−0.676273	−	0.736651i	$-0.736406\pi$
$19$	− 27.9928i	− 1.47330i	0.676273	−	0.736651i	$-0.263594\pi$
$20$	0	0
$21$	21.4703	1.02240
$22$	0	0
$23$	7.90421i	0.343661i	0.985126	+	0.171831i	$0.0549682\pi$
$23$	7.90421i	0.343661i	−0.985126	+	0.171831i	$0.945032\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−50.7169	−1.74886	−0.874429	−	0.485153i	$-0.838764\pi$
$29$	−50.7169	−1.74886	−0.874429	+	0.485153i	$0.838764\pi$
$30$	0	0
$31$	− 36.3467i	− 1.17247i	−0.810140	−	0.586236i	$-0.800609\pi$
$31$	− 36.3467i	− 1.17247i	0.810140	−	0.586236i	$-0.199391\pi$
$32$	0	0
$33$	−19.1224	−0.579466
$34$	0	0
$35$	− 27.7181i	− 0.791945i
$36$	0	0
$37$	18.9279	0.511566	0.255783	−	0.966734i	$-0.417667\pi$
$37$	18.9279	0.511566	0.255783	+	0.966734i	$0.417667\pi$
$38$	0	0
$39$	− 4.88453i	− 0.125244i
$40$	0	0
$41$	5.30410	0.129368	0.0646842	−	0.997906i	$-0.479396\pi$
$41$	5.30410	0.129368	0.0646842	+	0.997906i	$0.479396\pi$
$42$	0	0
$43$	− 45.5870i	− 1.06016i	−0.847947	−	0.530081i	$-0.822162\pi$
$43$	− 45.5870i	− 1.06016i	0.847947	−	0.530081i	$-0.177838\pi$
$44$	0	0
$45$	−6.70820	−0.149071
$46$	0	0
$47$	− 11.7246i	− 0.249460i	−0.992191	−	0.124730i	$-0.960194\pi$
$47$	− 11.7246i	− 0.249460i	0.992191	−	0.124730i	$-0.0398064\pi$
$48$	0	0
$49$	−104.658	−2.13589
$50$	0	0
$51$	11.3035i	0.221637i
$52$	0	0
$53$	−41.1680	−0.776755	−0.388378	−	0.921500i	$-0.626964\pi$
$53$	−41.1680	−0.776755	−0.388378	+	0.921500i	$0.626964\pi$
$54$	0	0
$55$	24.6869i	0.448853i
$56$	0	0
$57$	48.4849	0.850612
$58$	0	0
$59$	10.7008i	0.181370i	0.995880	+	0.0906848i	$0.0289056\pi$
$59$	10.7008i	0.181370i	−0.995880	+	0.0906848i	$0.971094\pi$
$60$	0	0
$61$	−56.1297	−0.920159	−0.460080	−	0.887878i	$-0.652179\pi$
$61$	−56.1297	−0.920159	−0.460080	+	0.887878i	$0.652179\pi$
$62$	0	0
$63$	37.1877i	0.590281i
$64$	0	0
$65$	−6.30590	−0.0970139
$66$	0	0
$67$	− 16.1709i	− 0.241357i	−0.992692	−	0.120679i	$-0.961493\pi$
$67$	− 16.1709i	− 0.241357i	0.992692	−	0.120679i	$-0.0385071\pi$
$68$	0	0
$69$	−13.6905	−0.198413
$70$	0	0
$71$	− 66.1617i	− 0.931855i	−0.884823	−	0.465928i	$-0.845721\pi$
$71$	− 66.1617i	− 0.931855i	0.884823	−	0.465928i	$-0.154279\pi$
$72$	0	0
$73$	15.6330	0.214150	0.107075	−	0.994251i	$-0.465851\pi$
$73$	15.6330	0.214150	0.107075	+	0.994251i	$0.465851\pi$
$74$	0	0
$75$	8.66025i	0.115470i
$76$	0	0
$77$	136.855	1.77733
$78$	0	0
$79$	− 123.057i	− 1.55768i	−0.627223	−	0.778840i	$-0.715809\pi$
$79$	− 123.057i	− 1.55768i	0.627223	−	0.778840i	$-0.284191\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 99.6700i	− 1.20084i	−0.799684	−	0.600422i	$-0.794999\pi$
$83$	− 99.6700i	− 1.20084i	0.799684	−	0.600422i	$-0.205001\pi$
$84$	0	0
$85$	14.5927	0.171679
$86$	0	0
$87$	− 87.8443i	− 1.00970i
$88$	0	0
$89$	101.083	1.13576	0.567881	−	0.823110i	$-0.307763\pi$
$89$	101.083	1.13576	0.567881	+	0.823110i	$0.307763\pi$
$90$	0	0
$91$	34.9575i	0.384148i
$92$	0	0
$93$	62.9543	0.676927
$94$	0	0
$95$	− 62.5937i	− 0.658881i
$96$	0	0
$97$	127.293	1.31230	0.656151	−	0.754630i	$-0.272183\pi$
$97$	127.293	1.31230	0.656151	+	0.754630i	$0.272183\pi$
$98$	0	0
$99$	− 33.1209i	− 0.334555i
$100$	0	0
$101$	94.3535	0.934193	0.467096	−	0.884206i	$-0.345300\pi$
$101$	94.3535	0.934193	0.467096	+	0.884206i	$0.345300\pi$
$102$	0	0
$103$	31.8455i	0.309180i	0.987979	+	0.154590i	$0.0494056\pi$
$103$	31.8455i	0.309180i	−0.987979	+	0.154590i	$0.950594\pi$
$104$	0	0
$105$	48.0091	0.457230
$106$	0	0
$107$	33.7912i	0.315805i	0.987455	+	0.157903i	$0.0504732\pi$
$107$	33.7912i	0.315805i	−0.987455	+	0.157903i	$0.949527\pi$
$108$	0	0
$109$	83.4266	0.765382	0.382691	−	0.923876i	$-0.374997\pi$
$109$	83.4266	0.765382	0.382691	+	0.923876i	$0.374997\pi$
$110$	0	0
$111$	32.7842i	0.295353i
$112$	0	0
$113$	−111.796	−0.989342	−0.494671	−	0.869080i	$-0.664711\pi$
$113$	−111.796	−0.989342	−0.494671	+	0.869080i	$0.664711\pi$
$114$	0	0
$115$	17.6744i	0.153690i
$116$	0	0
$117$	8.46026	0.0723099
$118$	0	0
$119$	− 80.8964i	− 0.679802i
$120$	0	0
$121$	−0.888544	−0.00734334
$122$	0	0
$123$	9.18697i	0.0746908i
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	− 16.6855i	− 0.131382i	−0.997840	−	0.0656909i	$-0.979075\pi$
$127$	− 16.6855i	− 0.131382i	0.997840	−	0.0656909i	$-0.0209251\pi$
$128$	0	0
$129$	78.9589	0.612085
$130$	0	0
$131$	− 196.418i	− 1.49937i	−0.661794	−	0.749686i	$-0.730204\pi$
$131$	− 196.418i	− 1.49937i	0.661794	−	0.749686i	$-0.269796\pi$
$132$	0	0
$133$	−346.995	−2.60899
$134$	0	0
$135$	− 11.6190i	− 0.0860663i
$136$	0	0
$137$	−117.127	−0.854942	−0.427471	−	0.904029i	$-0.640595\pi$
$137$	−117.127	−0.854942	−0.427471	+	0.904029i	$0.640595\pi$
$138$	0	0
$139$	− 187.238i	− 1.34704i	−0.739170	−	0.673519i	$-0.764782\pi$
$139$	− 187.238i	− 1.34704i	0.739170	−	0.673519i	$-0.235218\pi$
$140$	0	0
$141$	20.3076	0.144026
$142$	0	0
$143$	− 31.1346i	− 0.217725i
$144$	0	0
$145$	−113.406	−0.782113
$146$	0	0
$147$	− 181.274i	− 1.23315i
$148$	0	0
$149$	50.2274	0.337096	0.168548	−	0.985693i	$-0.446092\pi$
$149$	50.2274	0.337096	0.168548	+	0.985693i	$0.446092\pi$
$150$	0	0
$151$	213.160i	1.41166i	0.708382	+	0.705829i	$0.249425\pi$
$151$	213.160i	1.41166i	−0.708382	+	0.705829i	$0.750575\pi$
$152$	0	0
$153$	−19.5782	−0.127962
$154$	0	0
$155$	− 81.2736i	− 0.524346i
$156$	0	0
$157$	−203.918	−1.29884	−0.649419	−	0.760431i	$-0.724988\pi$
$157$	−203.918	−1.29884	−0.649419	+	0.760431i	$0.724988\pi$
$158$	0	0
$159$	− 71.3051i	− 0.448460i
$160$	0	0
$161$	97.9798	0.608570
$162$	0	0
$163$	215.898i	1.32452i	0.749272	+	0.662262i	$0.230404\pi$
$163$	215.898i	1.32452i	−0.749272	+	0.662262i	$0.769596\pi$
$164$	0	0
$165$	−42.7590	−0.259145
$166$	0	0
$167$	255.029i	1.52712i	0.645737	+	0.763560i	$0.276550\pi$
$167$	255.029i	1.52712i	−0.645737	+	0.763560i	$0.723450\pi$
$168$	0	0
$169$	−161.047	−0.952942
$170$	0	0
$171$	83.9783i	0.491101i
$172$	0	0
$173$	235.426	1.36084	0.680421	−	0.732822i	$-0.261797\pi$
$173$	235.426	1.36084	0.680421	+	0.732822i	$0.261797\pi$
$174$	0	0
$175$	− 61.9795i	− 0.354169i
$176$	0	0
$177$	−18.5343	−0.104714
$178$	0	0
$179$	− 102.669i	− 0.573572i	−0.957995	−	0.286786i	$-0.907413\pi$
$179$	− 102.669i	− 0.573572i	0.957995	−	0.286786i	$-0.0925869\pi$
$180$	0	0
$181$	56.8222	0.313935	0.156967	−	0.987604i	$-0.449828\pi$
$181$	56.8222	0.313935	0.156967	+	0.987604i	$0.449828\pi$
$182$	0	0
$183$	− 97.2195i	− 0.531254i
$184$	0	0
$185$	42.3242	0.228779
$186$	0	0
$187$	72.0498i	0.385293i
$188$	0	0
$189$	−64.4110	−0.340799
$190$	0	0
$191$	− 158.493i	− 0.829808i	−0.909865	−	0.414904i	$-0.863815\pi$
$191$	− 158.493i	− 0.829808i	0.909865	−	0.414904i	$-0.136185\pi$
$192$	0	0
$193$	−156.732	−0.812084	−0.406042	−	0.913854i	$-0.633091\pi$
$193$	−156.732	−0.812084	−0.406042	+	0.913854i	$0.633091\pi$
$194$	0	0
$195$	− 10.9221i	− 0.0560110i
$196$	0	0
$197$	−260.127	−1.32044	−0.660221	−	0.751072i	$-0.729537\pi$
$197$	−260.127	−1.32044	−0.660221	+	0.751072i	$0.729537\pi$
$198$	0	0
$199$	14.0326i	0.0705157i	0.999378	+	0.0352579i	$0.0112253\pi$
$199$	14.0326i	0.0705157i	−0.999378	+	0.0352579i	$0.988775\pi$
$200$	0	0
$201$	28.0089	0.139348
$202$	0	0
$203$	628.682i	3.09696i
$204$	0	0
$205$	11.8603	0.0578553
$206$	0	0
$207$	− 23.7126i	− 0.114554i
$208$	0	0
$209$	309.049	1.47870
$210$	0	0
$211$	74.4941i	0.353052i	0.984296	+	0.176526i	$0.0564860\pi$
$211$	74.4941i	0.353052i	−0.984296	+	0.176526i	$0.943514\pi$
$212$	0	0
$213$	114.595	0.538007
$214$	0	0
$215$	− 101.936i	− 0.474119i
$216$	0	0
$217$	−450.550	−2.07627
$218$	0	0
$219$	27.0771i	0.123640i
$220$	0	0
$221$	−18.4041	−0.0832763
$222$	0	0
$223$	− 159.996i	− 0.717471i	−0.933439	−	0.358736i	$-0.883208\pi$
$223$	− 159.996i	− 0.717471i	0.933439	−	0.358736i	$-0.116792\pi$
$224$	0	0
$225$	−15.0000	−0.0666667
$226$	0	0
$227$	175.978i	0.775236i	0.921820	+	0.387618i	$0.126702\pi$
$227$	175.978i	0.775236i	−0.921820	+	0.387618i	$0.873298\pi$
$228$	0	0
$229$	114.170	0.498560	0.249280	−	0.968431i	$-0.419806\pi$
$229$	114.170	0.498560	0.249280	+	0.968431i	$0.419806\pi$
$230$	0	0
$231$	237.039i	1.02614i
$232$	0	0
$233$	260.062	1.11615	0.558073	−	0.829792i	$-0.311541\pi$
$233$	260.062	1.11615	0.558073	+	0.829792i	$0.311541\pi$
$234$	0	0
$235$	− 26.2170i	− 0.111562i
$236$	0	0
$237$	213.140	0.899327
$238$	0	0
$239$	140.089i	0.586147i	0.956090	+	0.293073i	$0.0946780\pi$
$239$	140.089i	0.586147i	−0.956090	+	0.293073i	$0.905322\pi$
$240$	0	0
$241$	105.920	0.439503	0.219752	−	0.975556i	$-0.429475\pi$
$241$	105.920	0.439503	0.219752	+	0.975556i	$0.429475\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−234.023	−0.955197
$246$	0	0
$247$	78.9419i	0.319603i
$248$	0	0
$249$	172.633	0.693307
$250$	0	0
$251$	− 167.879i	− 0.668839i	−0.942424	−	0.334420i	$-0.891460\pi$
$251$	− 167.879i	− 0.668839i	0.942424	−	0.334420i	$-0.108540\pi$
$252$	0	0
$253$	−87.2650	−0.344921
$254$	0	0
$255$	25.2753i	0.0991190i
$256$	0	0
$257$	198.849	0.773732	0.386866	−	0.922136i	$-0.373558\pi$
$257$	198.849	0.773732	0.386866	+	0.922136i	$0.373558\pi$
$258$	0	0
$259$	− 234.629i	− 0.905903i
$260$	0	0
$261$	152.151	0.582953
$262$	0	0
$263$	480.528i	1.82710i	0.406722	+	0.913552i	$0.366672\pi$
$263$	480.528i	1.82710i	−0.406722	+	0.913552i	$0.633328\pi$
$264$	0	0
$265$	−92.0545	−0.347376
$266$	0	0
$267$	175.081i	0.655733i
$268$	0	0
$269$	291.496	1.08363	0.541815	−	0.840498i	$-0.317737\pi$
$269$	291.496	1.08363	0.541815	+	0.840498i	$0.317737\pi$
$270$	0	0
$271$	174.063i	0.642299i	0.947029	+	0.321150i	$0.104069\pi$
$271$	174.063i	0.642299i	−0.947029	+	0.321150i	$0.895931\pi$
$272$	0	0
$273$	−60.5482	−0.221788
$274$	0	0
$275$	55.2016i	0.200733i
$276$	0	0
$277$	50.5203	0.182384	0.0911918	−	0.995833i	$-0.470932\pi$
$277$	50.5203	0.182384	0.0911918	+	0.995833i	$0.470932\pi$
$278$	0	0
$279$	109.040i	0.390824i
$280$	0	0
$281$	−66.0514	−0.235058	−0.117529	−	0.993069i	$-0.537497\pi$
$281$	−66.0514	−0.235058	−0.117529	+	0.993069i	$0.537497\pi$
$282$	0	0
$283$	116.934i	0.413196i	0.978426	+	0.206598i	$0.0662392\pi$
$283$	116.934i	0.413196i	−0.978426	+	0.206598i	$0.933761\pi$
$284$	0	0
$285$	108.415	0.380405
$286$	0	0
$287$	− 65.7491i	− 0.229091i
$288$	0	0
$289$	−246.411	−0.852631
$290$	0	0
$291$	220.478i	0.757658i
$292$	0	0
$293$	68.3732	0.233356	0.116678	−	0.993170i	$-0.462776\pi$
$293$	68.3732	0.233356	0.116678	+	0.993170i	$0.462776\pi$
$294$	0	0
$295$	23.9277i	0.0811110i
$296$	0	0
$297$	57.3672	0.193155
$298$	0	0
$299$	− 22.2905i	− 0.0745503i
$300$	0	0
$301$	−565.092	−1.87738
$302$	0	0
$303$	163.425i	0.539356i
$304$	0	0
$305$	−125.510	−0.411508
$306$	0	0
$307$	− 369.497i	− 1.20357i	−0.798657	−	0.601786i	$-0.794456\pi$
$307$	− 369.497i	− 1.20357i	0.798657	−	0.601786i	$-0.205544\pi$
$308$	0	0
$309$	−55.1580	−0.178505
$310$	0	0
$311$	303.446i	0.975712i	0.872924	+	0.487856i	$0.162221\pi$
$311$	303.446i	0.975712i	−0.872924	+	0.487856i	$0.837779\pi$
$312$	0	0
$313$	297.693	0.951097	0.475549	−	0.879689i	$-0.342250\pi$
$313$	297.693	0.951097	0.475549	+	0.879689i	$0.342250\pi$
$314$	0	0
$315$	83.1542i	0.263982i
$316$	0	0
$317$	−264.678	−0.834948	−0.417474	−	0.908689i	$-0.637084\pi$
$317$	−264.678	−0.834948	−0.417474	+	0.908689i	$0.637084\pi$
$318$	0	0
$319$	− 559.931i	− 1.75527i
$320$	0	0
$321$	−58.5280	−0.182330
$322$	0	0
$323$	− 182.682i	− 0.565580i
$324$	0	0
$325$	−14.1004	−0.0433859
$326$	0	0
$327$	144.499i	0.441893i
$328$	0	0
$329$	−145.337	−0.441754
$330$	0	0
$331$	473.426i	1.43029i	0.698976	+	0.715145i	$0.253639\pi$
$331$	473.426i	1.43029i	−0.698976	+	0.715145i	$0.746361\pi$
$332$	0	0
$333$	−56.7838	−0.170522
$334$	0	0
$335$	− 36.1593i	− 0.107938i
$336$	0	0
$337$	29.7588	0.0883051	0.0441526	−	0.999025i	$-0.485941\pi$
$337$	29.7588	0.0883051	0.0441526	+	0.999025i	$0.485941\pi$
$338$	0	0
$339$	− 193.636i	− 0.571197i
$340$	0	0
$341$	401.278	1.17677
$342$	0	0
$343$	689.937i	2.01148i
$344$	0	0
$345$	−30.6129	−0.0887330
$346$	0	0
$347$	306.190i	0.882391i	0.897411	+	0.441195i	$0.145445\pi$
$347$	306.190i	0.882391i	−0.897411	+	0.441195i	$0.854555\pi$
$348$	0	0
$349$	−649.149	−1.86002	−0.930012	−	0.367528i	$-0.880204\pi$
$349$	−649.149	−1.86002	−0.930012	+	0.367528i	$0.880204\pi$
$350$	0	0
$351$	14.6536i	0.0417481i
$352$	0	0
$353$	−275.547	−0.780587	−0.390293	−	0.920691i	$-0.627626\pi$
$353$	−275.547	−0.780587	−0.390293	+	0.920691i	$0.627626\pi$
$354$	0	0
$355$	− 147.942i	− 0.416738i
$356$	0	0
$357$	140.117	0.392484
$358$	0	0
$359$	− 507.672i	− 1.41413i	−0.707149	−	0.707065i	$-0.750019\pi$
$359$	− 507.672i	− 1.41413i	0.707149	−	0.707065i	$-0.249981\pi$
$360$	0	0
$361$	−422.594	−1.17062
$362$	0	0
$363$	− 1.53900i	− 0.00423968i
$364$	0	0
$365$	34.9564	0.0957709
$366$	0	0
$367$	62.7671i	0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$367$	62.7671i	0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$368$	0	0
$369$	−15.9123	−0.0431228
$370$	0	0
$371$	510.315i	1.37551i
$372$	0	0
$373$	272.776	0.731302	0.365651	−	0.930752i	$-0.380846\pi$
$373$	272.776	0.731302	0.365651	+	0.930752i	$0.380846\pi$
$374$	0	0
$375$	19.3649i	0.0516398i
$376$	0	0
$377$	143.026	0.379379
$378$	0	0
$379$	376.828i	0.994270i	0.867673	+	0.497135i	$0.165615\pi$
$379$	376.828i	0.994270i	−0.867673	+	0.497135i	$0.834385\pi$
$380$	0	0
$381$	28.9001	0.0758533
$382$	0	0
$383$	412.206i	1.07625i	0.842864	+	0.538127i	$0.180868\pi$
$383$	412.206i	1.07625i	−0.842864	+	0.538127i	$0.819132\pi$
$384$	0	0
$385$	306.016	0.794848
$386$	0	0
$387$	136.761i	0.353387i
$388$	0	0
$389$	161.289	0.414623	0.207312	−	0.978275i	$-0.433529\pi$
$389$	161.289	0.414623	0.207312	+	0.978275i	$0.433529\pi$
$390$	0	0
$391$	51.5834i	0.131927i
$392$	0	0
$393$	340.206	0.865663
$394$	0	0
$395$	− 275.163i	− 0.696615i
$396$	0	0
$397$	186.505	0.469785	0.234893	−	0.972021i	$-0.424526\pi$
$397$	186.505	0.469785	0.234893	+	0.972021i	$0.424526\pi$
$398$	0	0
$399$	− 601.014i	− 1.50630i
$400$	0	0
$401$	239.061	0.596162	0.298081	−	0.954541i	$-0.403653\pi$
$401$	239.061	0.596162	0.298081	+	0.954541i	$0.403653\pi$
$402$	0	0
$403$	102.501i	0.254344i
$404$	0	0
$405$	20.1246	0.0496904
$406$	0	0
$407$	208.970i	0.513441i
$408$	0	0
$409$	47.8016	0.116874	0.0584372	−	0.998291i	$-0.481388\pi$
$409$	47.8016	0.116874	0.0584372	+	0.998291i	$0.481388\pi$
$410$	0	0
$411$	− 202.870i	− 0.493601i
$412$	0	0
$413$	132.646	0.321177
$414$	0	0
$415$	− 222.869i	− 0.537033i
$416$	0	0
$417$	324.306	0.777713
$418$	0	0
$419$	− 239.009i	− 0.570428i	−0.958464	−	0.285214i	$-0.907935\pi$
$419$	− 239.009i	− 0.570428i	0.958464	−	0.285214i	$-0.0920647\pi$
$420$	0	0
$421$	257.592	0.611857	0.305929	−	0.952054i	$-0.401033\pi$
$421$	257.592	0.611857	0.305929	+	0.952054i	$0.401033\pi$
$422$	0	0
$423$	35.1738i	0.0831532i
$424$	0	0
$425$	32.6303	0.0767772
$426$	0	0
$427$	695.779i	1.62946i
$428$	0	0
$429$	53.9268	0.125703
$430$	0	0
$431$	− 343.164i	− 0.796205i	−0.917341	−	0.398103i	$-0.869669\pi$
$431$	− 343.164i	− 0.796205i	0.917341	−	0.398103i	$-0.130331\pi$
$432$	0	0
$433$	−234.760	−0.542171	−0.271085	−	0.962555i	$-0.587383\pi$
$433$	−234.760	−0.542171	−0.271085	+	0.962555i	$0.587383\pi$
$434$	0	0
$435$	− 196.426i	− 0.451553i
$436$	0	0
$437$	221.261	0.506317
$438$	0	0
$439$	374.473i	0.853013i	0.904484	+	0.426507i	$0.140256\pi$
$439$	374.473i	0.853013i	−0.904484	+	0.426507i	$0.859744\pi$
$440$	0	0
$441$	313.975	0.711962
$442$	0	0
$443$	108.557i	0.245050i	0.992465	+	0.122525i	$0.0390992\pi$
$443$	108.557i	0.245050i	−0.992465	+	0.122525i	$0.960901\pi$
$444$	0	0
$445$	226.028	0.507929
$446$	0	0
$447$	86.9963i	0.194623i
$448$	0	0
$449$	−431.511	−0.961050	−0.480525	−	0.876981i	$-0.659554\pi$
$449$	−431.511	−0.961050	−0.480525	+	0.876981i	$0.659554\pi$
$450$	0	0
$451$	58.5589i	0.129842i
$452$	0	0
$453$	−369.205	−0.815021
$454$	0	0
$455$	78.1674i	0.171796i
$456$	0	0
$457$	219.747	0.480847	0.240424	−	0.970668i	$-0.422714\pi$
$457$	219.747	0.480847	0.240424	+	0.970668i	$0.422714\pi$
$458$	0	0
$459$	− 33.9104i	− 0.0738789i
$460$	0	0
$461$	−223.434	−0.484673	−0.242337	−	0.970192i	$-0.577914\pi$
$461$	−223.434	−0.484673	−0.242337	+	0.970192i	$0.577914\pi$
$462$	0	0
$463$	740.855i	1.60012i	0.599921	+	0.800059i	$0.295198\pi$
$463$	740.855i	1.60012i	−0.599921	+	0.800059i	$0.704802\pi$
$464$	0	0
$465$	140.770	0.302731
$466$	0	0
$467$	− 249.381i	− 0.534007i	−0.963696	−	0.267004i	$-0.913966\pi$
$467$	− 249.381i	− 0.534007i	0.963696	−	0.267004i	$-0.0860336\pi$
$468$	0	0
$469$	−200.454	−0.427406
$470$	0	0
$471$	− 353.196i	− 0.749885i
$472$	0	0
$473$	503.294	1.06405
$474$	0	0
$475$	− 139.964i	− 0.294661i
$476$	0	0
$477$	123.504	0.258918
$478$	0	0
$479$	− 210.915i	− 0.440324i	−0.975463	−	0.220162i	$-0.929341\pi$
$479$	− 210.915i	− 0.440324i	0.975463	−	0.220162i	$-0.0706587\pi$
$480$	0	0
$481$	−53.3784	−0.110974
$482$	0	0
$483$	169.706i	0.351358i
$484$	0	0
$485$	284.636	0.586879
$486$	0	0
$487$	− 710.541i	− 1.45902i	−0.683972	−	0.729508i	$-0.739749\pi$
$487$	− 710.541i	− 1.45902i	0.683972	−	0.729508i	$-0.260251\pi$
$488$	0	0
$489$	−373.946	−0.764715
$490$	0	0
$491$	697.876i	1.42134i	0.703528	+	0.710668i	$0.251607\pi$
$491$	697.876i	1.42134i	−0.703528	+	0.710668i	$0.748393\pi$
$492$	0	0
$493$	−330.982	−0.671363
$494$	0	0
$495$	− 74.0607i	− 0.149618i
$496$	0	0
$497$	−820.135	−1.65017
$498$	0	0
$499$	− 875.602i	− 1.75471i	−0.479838	−	0.877357i	$-0.659305\pi$
$499$	− 875.602i	− 1.75471i	0.479838	−	0.877357i	$-0.340695\pi$
$500$	0	0
$501$	−441.723	−0.881683
$502$	0	0
$503$	− 142.849i	− 0.283995i	−0.989867	−	0.141997i	$-0.954648\pi$
$503$	− 142.849i	− 0.283995i	0.989867	−	0.141997i	$-0.0453524\pi$
$504$	0	0
$505$	210.981	0.417784
$506$	0	0
$507$	− 278.942i	− 0.550181i
$508$	0	0
$509$	147.662	0.290102	0.145051	−	0.989424i	$-0.453665\pi$
$509$	147.662	0.290102	0.145051	+	0.989424i	$0.453665\pi$
$510$	0	0
$511$	− 193.785i	− 0.379227i
$512$	0	0
$513$	−145.455	−0.283537
$514$	0	0
$515$	71.2087i	0.138269i
$516$	0	0
$517$	129.443	0.250374
$518$	0	0
$519$	407.769i	0.785682i
$520$	0	0
$521$	−348.592	−0.669082	−0.334541	−	0.942381i	$-0.608581\pi$
$521$	−348.592	−0.669082	−0.334541	+	0.942381i	$0.608581\pi$
$522$	0	0
$523$	370.317i	0.708063i	0.935233	+	0.354032i	$0.115189\pi$
$523$	370.317i	0.708063i	−0.935233	+	0.354032i	$0.884811\pi$
$524$	0	0
$525$	107.352	0.204479
$526$	0	0
$527$	− 237.201i	− 0.450096i
$528$	0	0
$529$	466.523	0.881897
$530$	0	0
$531$	− 32.1024i	− 0.0604565i
$532$	0	0
$533$	−14.9580	−0.0280638
$534$	0	0
$535$	75.5593i	0.141232i
$536$	0	0
$537$	177.829	0.331152
$538$	0	0
$539$	− 1155.46i	− 2.14371i
$540$	0	0
$541$	279.719	0.517041	0.258520	−	0.966006i	$-0.416765\pi$
$541$	279.719	0.517041	0.258520	+	0.966006i	$0.416765\pi$
$542$	0	0
$543$	98.4190i	0.181250i
$544$	0	0
$545$	186.548	0.342289
$546$	0	0
$547$	− 387.716i	− 0.708804i	−0.935093	−	0.354402i	$-0.884685\pi$
$547$	− 387.716i	− 0.708804i	0.935093	−	0.354402i	$-0.115315\pi$
$548$	0	0
$549$	168.389	0.306720
$550$	0	0
$551$	1419.71i	2.57660i
$552$	0	0
$553$	−1525.40	−2.75841
$554$	0	0
$555$	73.3076i	0.132086i
$556$	0	0
$557$	−43.5564	−0.0781983	−0.0390991	−	0.999235i	$-0.512449\pi$
$557$	−43.5564	−0.0781983	−0.0390991	+	0.999235i	$0.512449\pi$
$558$	0	0
$559$	128.559i	0.229981i
$560$	0	0
$561$	−124.794	−0.222449
$562$	0	0
$563$	− 361.646i	− 0.642355i	−0.947019	−	0.321178i	$-0.895921\pi$
$563$	− 361.646i	− 0.642355i	0.947019	−	0.321178i	$-0.104079\pi$
$564$	0	0
$565$	−249.983	−0.442447
$566$	0	0
$567$	− 111.563i	− 0.196760i
$568$	0	0
$569$	888.559	1.56161	0.780807	−	0.624772i	$-0.214808\pi$
$569$	888.559	1.56161	0.780807	+	0.624772i	$0.214808\pi$
$570$	0	0
$571$	− 447.745i	− 0.784142i	−0.919935	−	0.392071i	$-0.871759\pi$
$571$	− 447.745i	− 0.784142i	0.919935	−	0.392071i	$-0.128241\pi$
$572$	0	0
$573$	274.519	0.479090
$574$	0	0
$575$	39.5211i	0.0687323i
$576$	0	0
$577$	1069.90	1.85425	0.927124	−	0.374756i	$-0.122273\pi$
$577$	1069.90	1.85425	0.927124	+	0.374756i	$0.122273\pi$
$578$	0	0
$579$	− 271.468i	− 0.468857i
$580$	0	0
$581$	−1235.50	−2.12651
$582$	0	0
$583$	− 454.508i	− 0.779602i
$584$	0	0
$585$	18.9177	0.0323380
$586$	0	0
$587$	− 129.637i	− 0.220847i	−0.993885	−	0.110424i	$-0.964779\pi$
$587$	− 129.637i	− 0.220847i	0.993885	−	0.110424i	$-0.0352208\pi$
$588$	0	0
$589$	−1017.44	−1.72741
$590$	0	0
$591$	− 450.553i	− 0.762357i
$592$	0	0
$593$	892.757	1.50549	0.752746	−	0.658311i	$-0.228729\pi$
$593$	892.757	1.50549	0.752746	+	0.658311i	$0.228729\pi$
$594$	0	0
$595$	− 180.890i	− 0.304017i
$596$	0	0
$597$	−24.3052	−0.0407123
$598$	0	0
$599$	− 1030.62i	− 1.72057i	−0.509816	−	0.860284i	$-0.670286\pi$
$599$	− 1030.62i	− 1.72057i	0.509816	−	0.860284i	$-0.329714\pi$
$600$	0	0
$601$	−815.961	−1.35767	−0.678836	−	0.734289i	$-0.737515\pi$
$601$	−815.961	−1.35767	−0.678836	+	0.734289i	$0.737515\pi$
$602$	0	0
$603$	48.5128i	0.0804525i
$604$	0	0
$605$	−1.98684	−0.00328404
$606$	0	0
$607$	842.678i	1.38827i	0.719847	+	0.694133i	$0.244212\pi$
$607$	842.678i	1.38827i	−0.719847	+	0.694133i	$0.755788\pi$
$608$	0	0
$609$	−1088.91	−1.78803
$610$	0	0
$611$	33.0644i	0.0541152i
$612$	0	0
$613$	731.088	1.19264	0.596320	−	0.802747i	$-0.296629\pi$
$613$	731.088	1.19264	0.596320	+	0.802747i	$0.296629\pi$
$614$	0	0
$615$	20.5427i	0.0334028i
$616$	0	0
$617$	−919.609	−1.49045	−0.745226	−	0.666812i	$-0.767658\pi$
$617$	−919.609	−1.49045	−0.745226	+	0.666812i	$0.767658\pi$
$618$	0	0
$619$	688.974i	1.11304i	0.830833	+	0.556522i	$0.187865\pi$
$619$	688.974i	1.11304i	−0.830833	+	0.556522i	$0.812135\pi$
$620$	0	0
$621$	41.0715	0.0661377
$622$	0	0
$623$	− 1253.01i	− 2.01126i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	535.288i	0.853729i
$628$	0	0
$629$	123.525	0.196383
$630$	0	0
$631$	− 418.968i	− 0.663975i	−0.943284	−	0.331987i	$-0.892281\pi$
$631$	− 418.968i	− 0.663975i	0.943284	−	0.331987i	$-0.107719\pi$
$632$	0	0
$633$	−129.027	−0.203835
$634$	0	0
$635$	− 37.3099i	− 0.0587557i
$636$	0	0
$637$	295.146	0.463337
$638$	0	0
$639$	198.485i	0.310618i
$640$	0	0
$641$	−47.2426	−0.0737014	−0.0368507	−	0.999321i	$-0.511733\pi$
$641$	−47.2426	−0.0737014	−0.0368507	+	0.999321i	$0.511733\pi$
$642$	0	0
$643$	710.880i	1.10557i	0.833325	+	0.552784i	$0.186435\pi$
$643$	710.880i	1.10557i	−0.833325	+	0.552784i	$0.813565\pi$
$644$	0	0
$645$	176.558	0.273733
$646$	0	0
$647$	− 468.195i	− 0.723641i	−0.932248	−	0.361820i	$-0.882155\pi$
$647$	− 468.195i	− 0.723641i	0.932248	−	0.361820i	$-0.117845\pi$
$648$	0	0
$649$	−118.140	−0.182034
$650$	0	0
$651$	− 780.375i	− 1.19873i
$652$	0	0
$653$	−551.066	−0.843900	−0.421950	−	0.906619i	$-0.638654\pi$
$653$	−551.066	−0.843900	−0.421950	+	0.906619i	$0.638654\pi$
$654$	0	0
$655$	− 439.203i	− 0.670540i
$656$	0	0
$657$	−46.8989	−0.0713834
$658$	0	0
$659$	− 158.259i	− 0.240151i	−0.992765	−	0.120075i	$-0.961686\pi$
$659$	− 158.259i	− 0.240151i	0.992765	−	0.120075i	$-0.0383136\pi$
$660$	0	0
$661$	−92.4953	−0.139932	−0.0699662	−	0.997549i	$-0.522289\pi$
$661$	−92.4953	−0.139932	−0.0699662	+	0.997549i	$0.522289\pi$
$662$	0	0
$663$	− 31.8768i	− 0.0480796i
$664$	0	0
$665$	−775.905	−1.16678
$666$	0	0
$667$	− 400.877i	− 0.601015i
$668$	0	0
$669$	277.121	0.414232
$670$	0	0
$671$	− 619.690i	− 0.923532i
$672$	0	0
$673$	956.062	1.42060	0.710299	−	0.703900i	$-0.248560\pi$
$673$	956.062	1.42060	0.710299	+	0.703900i	$0.248560\pi$
$674$	0	0
$675$	− 25.9808i	− 0.0384900i
$676$	0	0
$677$	−1116.67	−1.64944	−0.824719	−	0.565543i	$-0.808667\pi$
$677$	−1116.67	−1.64944	−0.824719	+	0.565543i	$0.808667\pi$
$678$	0	0
$679$	− 1577.92i	− 2.32388i
$680$	0	0
$681$	−304.804	−0.447583
$682$	0	0
$683$	826.776i	1.21051i	0.796033	+	0.605254i	$0.206928\pi$
$683$	826.776i	1.21051i	−0.796033	+	0.605254i	$0.793072\pi$
$684$	0	0
$685$	−261.904	−0.382342
$686$	0	0
$687$	197.749i	0.287844i
$688$	0	0
$689$	116.097	0.168501
$690$	0	0
$691$	− 965.432i	− 1.39715i	−0.715536	−	0.698576i	$-0.753818\pi$
$691$	− 965.432i	− 1.39715i	0.715536	−	0.698576i	$-0.246182\pi$
$692$	0	0
$693$	−410.564	−0.592444
$694$	0	0
$695$	− 418.677i	− 0.602414i
$696$	0	0
$697$	34.6149	0.0496627
$698$	0	0
$699$	450.441i	0.644408i
$700$	0	0
$701$	1109.94	1.58337	0.791686	−	0.610928i	$-0.209203\pi$
$701$	1109.94	1.58337	0.791686	+	0.610928i	$0.209203\pi$
$702$	0	0
$703$	− 529.845i	− 0.753691i
$704$	0	0
$705$	45.4092	0.0644102
$706$	0	0
$707$	− 1169.60i	− 1.65431i
$708$	0	0
$709$	964.244	1.36001	0.680003	−	0.733210i	$-0.261979\pi$
$709$	964.244	1.36001	0.680003	+	0.733210i	$0.261979\pi$
$710$	0	0
$711$	369.170i	0.519226i
$712$	0	0
$713$	287.292	0.402934
$714$	0	0
$715$	− 69.6191i	− 0.0973694i
$716$	0	0
$717$	−242.641	−0.338412
$718$	0	0
$719$	190.820i	0.265396i	0.991157	+	0.132698i	$0.0423641\pi$
$719$	190.820i	0.265396i	−0.991157	+	0.132698i	$0.957636\pi$
$720$	0	0
$721$	394.754	0.547509
$722$	0	0
$723$	183.459i	0.253747i
$724$	0	0
$725$	−253.585	−0.349772
$726$	0	0
$727$	202.134i	0.278039i	0.990290	+	0.139019i	$0.0443951\pi$
$727$	202.134i	0.278039i	−0.990290	+	0.139019i	$0.955605\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 297.503i	− 0.406981i
$732$	0	0
$733$	962.435	1.31301	0.656504	−	0.754322i	$-0.272034\pi$
$733$	962.435	1.31301	0.656504	+	0.754322i	$0.272034\pi$
$734$	0	0
$735$	− 405.340i	− 0.551483i
$736$	0	0
$737$	178.532	0.242242
$738$	0	0
$739$	− 932.112i	− 1.26132i	−0.776061	−	0.630658i	$-0.782785\pi$
$739$	− 932.112i	− 1.26132i	0.776061	−	0.630658i	$-0.217215\pi$
$740$	0	0
$741$	−136.731	−0.184523
$742$	0	0
$743$	1153.70i	1.55276i	0.630266	+	0.776379i	$0.282946\pi$
$743$	1153.70i	1.55276i	−0.630266	+	0.776379i	$0.717054\pi$
$744$	0	0
$745$	112.312	0.150754
$746$	0	0
$747$	299.010i	0.400281i
$748$	0	0
$749$	418.872	0.559242
$750$	0	0
$751$	204.359i	0.272116i	0.990701	+	0.136058i	$0.0434434\pi$
$751$	204.359i	0.272116i	−0.990701	+	0.136058i	$0.956557\pi$
$752$	0	0
$753$	290.774	0.386155
$754$	0	0
$755$	476.641i	0.631313i
$756$	0	0
$757$	216.739	0.286314	0.143157	−	0.989700i	$-0.454275\pi$
$757$	216.739	0.286314	0.143157	+	0.989700i	$0.454275\pi$
$758$	0	0
$759$	− 151.147i	− 0.199140i
$760$	0	0
$761$	1324.78	1.74085	0.870424	−	0.492303i	$-0.163845\pi$
$761$	1324.78	1.74085	0.870424	+	0.492303i	$0.163845\pi$
$762$	0	0
$763$	− 1034.15i	− 1.35537i
$764$	0	0
$765$	−43.7782	−0.0572264
$766$	0	0
$767$	− 30.1772i	− 0.0393444i
$768$	0	0
$769$	444.088	0.577488	0.288744	−	0.957406i	$-0.406762\pi$
$769$	444.088	0.577488	0.288744	+	0.957406i	$0.406762\pi$
$770$	0	0
$771$	344.417i	0.446714i
$772$	0	0
$773$	751.987	0.972817	0.486408	−	0.873732i	$-0.338307\pi$
$773$	751.987	0.972817	0.486408	+	0.873732i	$0.338307\pi$
$774$	0	0
$775$	− 181.733i	− 0.234495i
$776$	0	0
$777$	406.389	0.523023
$778$	0	0
$779$	− 148.476i	− 0.190599i
$780$	0	0
$781$	730.446	0.935271
$782$	0	0
$783$	263.533i	0.336568i
$784$	0	0
$785$	−455.974	−0.580858
$786$	0	0
$787$	442.296i	0.562002i	0.959707	+	0.281001i	$0.0906665\pi$
$787$	442.296i	0.562002i	−0.959707	+	0.281001i	$0.909334\pi$
$788$	0	0
$789$	−832.299	−1.05488
$790$	0	0
$791$	1385.81i	1.75197i
$792$	0	0
$793$	158.291	0.199610
$794$	0	0
$795$	− 159.443i	− 0.200557i
$796$	0	0
$797$	56.2072	0.0705235	0.0352618	−	0.999378i	$-0.488774\pi$
$797$	56.2072	0.0705235	0.0352618	+	0.999378i	$0.488774\pi$
$798$	0	0
$799$	− 76.5155i	− 0.0957641i
$800$	0	0
$801$	−303.249	−0.378588
$802$	0	0
$803$	172.593i	0.214935i
$804$	0	0
$805$	219.090	0.272161
$806$	0	0
$807$	504.887i	0.625634i
$808$	0	0
$809$	1522.16	1.88153	0.940765	−	0.339060i	$-0.110109\pi$
$809$	1522.16	1.88153	0.940765	+	0.339060i	$0.110109\pi$
$810$	0	0
$811$	− 930.734i	− 1.14764i	−0.818982	−	0.573819i	$-0.805461\pi$
$811$	− 930.734i	− 1.14764i	0.818982	−	0.573819i	$-0.194539\pi$
$812$	0	0
$813$	−301.486	−0.370832
$814$	0	0
$815$	482.762i	0.592346i
$816$	0	0
$817$	−1276.10	−1.56194
$818$	0	0
$819$	− 104.873i	− 0.128049i
$820$	0	0
$821$	−349.814	−0.426083	−0.213041	−	0.977043i	$-0.568337\pi$
$821$	−349.814	−0.426083	−0.213041	+	0.977043i	$0.568337\pi$
$822$	0	0
$823$	61.2187i	0.0743849i	0.999308	+	0.0371924i	$0.0118414\pi$
$823$	61.2187i	0.0743849i	−0.999308	+	0.0371924i	$0.988159\pi$
$824$	0	0
$825$	−95.6119	−0.115893
$826$	0	0
$827$	− 46.2063i	− 0.0558721i	−0.999610	−	0.0279361i	$-0.991107\pi$
$827$	− 46.2063i	− 0.0558721i	0.999610	−	0.0279361i	$-0.00889348\pi$
$828$	0	0
$829$	−223.832	−0.270002	−0.135001	−	0.990845i	$-0.543104\pi$
$829$	−223.832	−0.270002	−0.135001	+	0.990845i	$0.543104\pi$
$830$	0	0
$831$	87.5037i	0.105299i
$832$	0	0
$833$	−683.007	−0.819937
$834$	0	0
$835$	570.262i	0.682949i
$836$	0	0
$837$	−188.863	−0.225642
$838$	0	0
$839$	− 361.794i	− 0.431220i	−0.976480	−	0.215610i	$-0.930826\pi$
$839$	− 361.794i	− 0.431220i	0.976480	−	0.215610i	$-0.0691740\pi$
$840$	0	0
$841$	1731.20	2.05851
$842$	0	0
$843$	− 114.404i	− 0.135711i
$844$	0	0
$845$	−360.112	−0.426168
$846$	0	0
$847$	11.0143i	0.0130039i
$848$	0	0
$849$	−202.536	−0.238559
$850$	0	0
$851$	149.610i	0.175805i
$852$	0	0
$853$	844.503	0.990039	0.495019	−	0.868882i	$-0.335161\pi$
$853$	844.503	0.990039	0.495019	+	0.868882i	$0.335161\pi$
$854$	0	0
$855$	187.781i	0.219627i
$856$	0	0
$857$	−1389.51	−1.62137	−0.810685	−	0.585482i	$-0.800905\pi$
$857$	−1389.51	−1.62137	−0.810685	+	0.585482i	$0.800905\pi$
$858$	0	0
$859$	− 1205.45i	− 1.40332i	−0.712512	−	0.701660i	$-0.752442\pi$
$859$	− 1205.45i	− 1.40332i	0.712512	−	0.701660i	$-0.247558\pi$
$860$	0	0
$861$	113.881	0.132266
$862$	0	0
$863$	258.868i	0.299963i	0.988689	+	0.149981i	$0.0479214\pi$
$863$	258.868i	0.299963i	−0.988689	+	0.149981i	$0.952079\pi$
$864$	0	0
$865$	526.428	0.608587
$866$	0	0
$867$	− 426.796i	− 0.492267i
$868$	0	0
$869$	1358.58	1.56339
$870$	0	0
$871$	45.6035i	0.0523576i
$872$	0	0
$873$	−381.880	−0.437434
$874$	0	0
$875$	− 138.590i	− 0.158389i
$876$	0	0
$877$	156.268	0.178185	0.0890926	−	0.996023i	$-0.471603\pi$
$877$	156.268	0.178185	0.0890926	+	0.996023i	$0.471603\pi$
$878$	0	0
$879$	118.426i	0.134728i
$880$	0	0
$881$	−1343.58	−1.52507	−0.762533	−	0.646950i	$-0.776044\pi$
$881$	−1343.58	−1.52507	−0.762533	+	0.646950i	$0.776044\pi$
$882$	0	0
$883$	− 149.478i	− 0.169284i	−0.996411	−	0.0846420i	$-0.973025\pi$
$883$	− 149.478i	− 0.169284i	0.996411	−	0.0846420i	$-0.0269747\pi$
$884$	0	0
$885$	−41.4440	−0.0468294
$886$	0	0
$887$	− 1532.07i	− 1.72725i	−0.504134	−	0.863626i	$-0.668188\pi$
$887$	− 1532.07i	− 1.72725i	0.504134	−	0.863626i	$-0.331812\pi$
$888$	0	0
$889$	−206.832	−0.232656
$890$	0	0
$891$	99.3628i	0.111518i
$892$	0	0
$893$	−328.204	−0.367529
$894$	0	0
$895$	− 229.576i	− 0.256509i
$896$	0	0
$897$	38.6084	0.0430417
$898$	0	0
$899$	1843.39i	2.05049i
$900$	0	0
$901$	−268.665	−0.298186
$902$	0	0
$903$	− 978.767i	− 1.08391i
$904$	0	0
$905$	127.058	0.140396
$906$	0	0
$907$	1245.02i	1.37268i	0.727280	+	0.686341i	$0.240784\pi$
$907$	1245.02i	1.37268i	−0.727280	+	0.686341i	$0.759216\pi$
$908$	0	0
$909$	−283.060	−0.311398
$910$	0	0
$911$	− 173.681i	− 0.190649i	−0.995446	−	0.0953245i	$-0.969611\pi$
$911$	− 173.681i	− 0.190649i	0.995446	−	0.0953245i	$-0.0303889\pi$
$912$	0	0
$913$	1100.39	1.20524
$914$	0	0
$915$	− 217.389i	− 0.237584i
$916$	0	0
$917$	−2434.78	−2.65515
$918$	0	0
$919$	874.426i	0.951498i	0.879581	+	0.475749i	$0.157823\pi$
$919$	874.426i	0.951498i	−0.879581	+	0.475749i	$0.842177\pi$
$920$	0	0
$921$	639.987	0.694882
$922$	0	0
$923$	186.582i	0.202147i
$924$	0	0
$925$	94.6397	0.102313
$926$	0	0
$927$	− 95.5365i	− 0.103060i
$928$	0	0
$929$	−1564.05	−1.68358	−0.841792	−	0.539803i	$-0.818499\pi$
$929$	−1564.05	−1.68358	−0.841792	+	0.539803i	$0.818499\pi$
$930$	0	0
$931$	2929.68i	3.14681i
$932$	0	0
$933$	−525.584	−0.563327
$934$	0	0
$935$	161.108i	0.172308i
$936$	0	0
$937$	958.621	1.02308	0.511538	−	0.859261i	$-0.329076\pi$
$937$	958.621	1.02308	0.511538	+	0.859261i	$0.329076\pi$
$938$	0	0
$939$	515.620i	0.549116i
$940$	0	0
$941$	752.357	0.799529	0.399765	−	0.916618i	$-0.369092\pi$
$941$	752.357	0.799529	0.399765	+	0.916618i	$0.369092\pi$
$942$	0	0
$943$	41.9247i	0.0444589i
$944$	0	0
$945$	−144.027	−0.152410
$946$	0	0
$947$	− 1013.16i	− 1.06986i	−0.844895	−	0.534932i	$-0.820337\pi$
$947$	− 1013.16i	− 1.06986i	0.844895	−	0.534932i	$-0.179663\pi$
$948$	0	0
$949$	−44.0863	−0.0464555
$950$	0	0
$951$	− 458.436i	− 0.482057i
$952$	0	0
$953$	−21.5482	−0.0226109	−0.0113054	−	0.999936i	$-0.503599\pi$
$953$	−21.5482	−0.0226109	−0.0113054	+	0.999936i	$0.503599\pi$
$954$	0	0
$955$	− 354.402i	− 0.371102i
$956$	0	0
$957$	969.828	1.01340
$958$	0	0
$959$	1451.90i	1.51397i
$960$	0	0
$961$	−360.079	−0.374692
$962$	0	0
$963$	− 101.373i	− 0.105268i
$964$	0	0
$965$	−350.464	−0.363175
$966$	0	0
$967$	303.965i	0.314338i	0.987572	+	0.157169i	$0.0502367\pi$
$967$	303.965i	0.314338i	−0.987572	+	0.157169i	$0.949763\pi$
$968$	0	0
$969$	316.415	0.326538
$970$	0	0
$971$	356.162i	0.366799i	0.983038	+	0.183399i	$0.0587101\pi$
$971$	356.162i	0.366799i	−0.983038	+	0.183399i	$0.941290\pi$
$972$	0	0
$973$	−2320.99	−2.38539
$974$	0	0
$975$	− 24.4227i	− 0.0250489i
$976$	0	0
$977$	−1845.09	−1.88852	−0.944262	−	0.329194i	$-0.893223\pi$
$977$	−1845.09	−1.88852	−0.944262	+	0.329194i	$0.893223\pi$
$978$	0	0
$979$	1115.99i	1.13993i
$980$	0	0
$981$	−250.280	−0.255127
$982$	0	0
$983$	− 289.444i	− 0.294449i	−0.989103	−	0.147225i	$-0.952966\pi$
$983$	− 289.444i	− 0.294449i	0.989103	−	0.147225i	$-0.0470340\pi$
$984$	0	0
$985$	−581.662	−0.590519
$986$	0	0
$987$	− 251.731i	− 0.255047i
$988$	0	0
$989$	360.329	0.364337
$990$	0	0
$991$	− 441.980i	− 0.445994i	−0.974819	−	0.222997i	$-0.928416\pi$
$991$	− 441.980i	− 0.445994i	0.974819	−	0.222997i	$-0.0715840\pi$
$992$	0	0
$993$	−819.998	−0.825778
$994$	0	0
$995$	31.3779i	0.0315356i
$996$	0	0
$997$	1045.42	1.04856	0.524282	−	0.851545i	$-0.324334\pi$
$997$	1045.42	1.04856	0.524282	+	0.851545i	$0.324334\pi$
$998$	0	0
$999$	− 98.3525i	− 0.0984509i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.e.c.511.7		8
3.2	odd	2		2880.3.e.j.2431.1		8
4.3	odd	2	inner	960.3.e.c.511.4		8
8.3	odd	2		60.3.c.a.31.2	yes	8
8.5	even	2		60.3.c.a.31.1	✓	8
12.11	even	2		2880.3.e.j.2431.4		8
24.5	odd	2		180.3.c.b.91.8		8
24.11	even	2		180.3.c.b.91.7		8
40.3	even	4		300.3.f.b.199.10		16
40.13	odd	4		300.3.f.b.199.8		16
40.19	odd	2		300.3.c.d.151.7		8
40.27	even	4		300.3.f.b.199.7		16
40.29	even	2		300.3.c.d.151.8		8
40.37	odd	4		300.3.f.b.199.9		16
120.29	odd	2		900.3.c.u.451.1		8
120.53	even	4		900.3.f.f.199.9		16
120.59	even	2		900.3.c.u.451.2		8
120.77	even	4		900.3.f.f.199.8		16
120.83	odd	4		900.3.f.f.199.7		16
120.107	odd	4		900.3.f.f.199.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.c.a.31.1	✓	8	8.5	even	2
60.3.c.a.31.2	yes	8	8.3	odd	2
180.3.c.b.91.7		8	24.11	even	2
180.3.c.b.91.8		8	24.5	odd	2
300.3.c.d.151.7		8	40.19	odd	2
300.3.c.d.151.8		8	40.29	even	2
300.3.f.b.199.7		16	40.27	even	4
300.3.f.b.199.8		16	40.13	odd	4
300.3.f.b.199.9		16	40.37	odd	4
300.3.f.b.199.10		16	40.3	even	4
900.3.c.u.451.1		8	120.29	odd	2
900.3.c.u.451.2		8	120.59	even	2
900.3.f.f.199.7		16	120.83	odd	4
900.3.f.f.199.8		16	120.77	even	4
900.3.f.f.199.9		16	120.53	even	4
900.3.f.f.199.10		16	120.107	odd	4
960.3.e.c.511.4		8	4.3	odd	2	inner
960.3.e.c.511.7		8	1.1	even	1	trivial
2880.3.e.j.2431.1		8	3.2	odd	2
2880.3.e.j.2431.4		8	12.11	even	2

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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +2.23607 q^{5} -12.3959i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +2.23607 q^{5} -12.3959i q^{7} -3.00000 q^{9} +11.0403i q^{11} -2.82009 q^{13} +3.87298i q^{15} +6.52606 q^{17} -27.9928i q^{19} +21.4703 q^{21} +7.90421i q^{23} +5.00000 q^{25} -5.19615i q^{27} -50.7169 q^{29} -36.3467i q^{31} -19.1224 q^{33} -27.7181i q^{35} +18.9279 q^{37} -4.88453i q^{39} +5.30410 q^{41} -45.5870i q^{43} -6.70820 q^{45} -11.7246i q^{47} -104.658 q^{49} +11.3035i q^{51} -41.1680 q^{53} +24.6869i q^{55} +48.4849 q^{57} +10.7008i q^{59} -56.1297 q^{61} +37.1877i q^{63} -6.30590 q^{65} -16.1709i q^{67} -13.6905 q^{69} -66.1617i q^{71} +15.6330 q^{73} +8.66025i q^{75} +136.855 q^{77} -123.057i q^{79} +9.00000 q^{81} -99.6700i q^{83} +14.5927 q^{85} -87.8443i q^{87} +101.083 q^{89} +34.9575i q^{91} +62.9543 q^{93} -62.5937i q^{95} +127.293 q^{97} -33.1209i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} - 16 q^{13} + 48 q^{21} + 40 q^{25} - 64 q^{29} + 112 q^{37} - 16 q^{41} - 56 q^{49} - 352 q^{53} + 144 q^{57} + 176 q^{61} - 80 q^{65} + 96 q^{69} - 240 q^{73} + 288 q^{77} + 72 q^{81} - 160 q^{85} + 80 q^{89} - 144 q^{93} + 432 q^{97}+O(q^{100}) \) 8 * q - 24 * q^9 - 16 * q^13 + 48 * q^21 + 40 * q^25 - 64 * q^29 + 112 * q^37 - 16 * q^41 - 56 * q^49 - 352 * q^53 + 144 * q^57 + 176 * q^61 - 80 * q^65 + 96 * q^69 - 240 * q^73 + 288 * q^77 + 72 * q^81 - 160 * q^85 + 80 * q^89 - 144 * q^93 + 432 * q^97

Introduction

L-functions

Modular forms

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Database

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