LMFDB - Embedded newform 2151.2.a.h.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.a (trivial)

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:	yes
Analytic conductor:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 15 x^{10} + 47 x^{9} + 75 x^{8} - 256 x^{7} - 134 x^{6} + 571 x^{5} + 23 x^{4} + \cdots - 31 $ x^12 - 3x^11 - 15x^10 + 47x^9 + 75x^8 - 256x^7 - 134x^6 + 571x^5 + 23x^4 - 479x^3 + 129x^2 + 88*x - 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.496586$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.496586 q^{2} -1.75340 q^{4} -1.64136 q^{5} -3.34365 q^{7} -1.86389 q^{8} +O(q^{10})$	\(q+0.496586 q^{2} -1.75340 q^{4} -1.64136 q^{5} -3.34365 q^{7} -1.86389 q^{8} -0.815074 q^{10} -0.269109 q^{11} -6.61052 q^{13} -1.66041 q^{14} +2.58123 q^{16} -2.98958 q^{17} +5.96737 q^{19} +2.87796 q^{20} -0.133636 q^{22} -8.36700 q^{23} -2.30595 q^{25} -3.28269 q^{26} +5.86276 q^{28} +9.38349 q^{29} -0.868715 q^{31} +5.00957 q^{32} -1.48458 q^{34} +5.48812 q^{35} +11.6343 q^{37} +2.96331 q^{38} +3.05930 q^{40} -2.76855 q^{41} +7.87141 q^{43} +0.471857 q^{44} -4.15493 q^{46} -7.11859 q^{47} +4.17998 q^{49} -1.14510 q^{50} +11.5909 q^{52} -11.7092 q^{53} +0.441704 q^{55} +6.23218 q^{56} +4.65971 q^{58} -4.39528 q^{59} +7.35792 q^{61} -0.431391 q^{62} -2.67477 q^{64} +10.8502 q^{65} +4.25328 q^{67} +5.24194 q^{68} +2.72532 q^{70} -2.21595 q^{71} +5.74357 q^{73} +5.77741 q^{74} -10.4632 q^{76} +0.899807 q^{77} -3.09991 q^{79} -4.23671 q^{80} -1.37482 q^{82} +0.729733 q^{83} +4.90697 q^{85} +3.90883 q^{86} +0.501589 q^{88} +10.9912 q^{89} +22.1032 q^{91} +14.6707 q^{92} -3.53499 q^{94} -9.79457 q^{95} -7.37397 q^{97} +2.07572 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8} - 15 q^{11} + 7 q^{13} + 6 q^{14} + 21 q^{16} + 3 q^{17} + 10 q^{19} + 4 q^{20} + 23 q^{22} - 20 q^{23} + 19 q^{25} + 10 q^{26} + 34 q^{28} - 2 q^{29} + 10 q^{31} - 26 q^{32} + 12 q^{34} - 7 q^{35} + 30 q^{37} + 3 q^{38} + 25 q^{40} + 28 q^{41} + 48 q^{43} - 25 q^{44} + 22 q^{46} - 13 q^{47} + 19 q^{49} - 12 q^{50} + 24 q^{52} + 2 q^{53} + 8 q^{55} + 7 q^{56} + 42 q^{58} + 14 q^{59} + 14 q^{61} - 8 q^{62} + 9 q^{64} + 35 q^{65} + 52 q^{67} - 3 q^{68} - 33 q^{70} + 7 q^{71} + 14 q^{73} + 13 q^{74} - 12 q^{76} + 6 q^{77} + 15 q^{79} + 8 q^{80} - 61 q^{82} - 29 q^{83} + 8 q^{85} + 9 q^{86} + 11 q^{88} + 71 q^{89} + 13 q^{91} - 2 q^{92} - 22 q^{94} - 2 q^{95} - 2 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8 - 15 * q^11 + 7 * q^13 + 6 * q^14 + 21 * q^16 + 3 * q^17 + 10 * q^19 + 4 * q^20 + 23 * q^22 - 20 * q^23 + 19 * q^25 + 10 * q^26 + 34 * q^28 - 2 * q^29 + 10 * q^31 - 26 * q^32 + 12 * q^34 - 7 * q^35 + 30 * q^37 + 3 * q^38 + 25 * q^40 + 28 * q^41 + 48 * q^43 - 25 * q^44 + 22 * q^46 - 13 * q^47 + 19 * q^49 - 12 * q^50 + 24 * q^52 + 2 * q^53 + 8 * q^55 + 7 * q^56 + 42 * q^58 + 14 * q^59 + 14 * q^61 - 8 * q^62 + 9 * q^64 + 35 * q^65 + 52 * q^67 - 3 * q^68 - 33 * q^70 + 7 * q^71 + 14 * q^73 + 13 * q^74 - 12 * q^76 + 6 * q^77 + 15 * q^79 + 8 * q^80 - 61 * q^82 - 29 * q^83 + 8 * q^85 + 9 * q^86 + 11 * q^88 + 71 * q^89 + 13 * q^91 - 2 * q^92 - 22 * q^94 - 2 * q^95 - 2 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	0.496586	0.351139	0.175570	−	0.984467i	$-0.443823\pi$
$2$	0.496586	0.351139	0.175570	+	0.984467i	$0.443823\pi$
$3$	0	0
$3$	0	0
$4$	−1.75340	−0.876701
$5$	−1.64136	−0.734037	−0.367018	−	0.930214i	$-0.619621\pi$
$5$	−1.64136	−0.734037	−0.367018	+	0.930214i	$0.619621\pi$
$6$	0	0
$7$	−3.34365	−1.26378	−0.631890	−	0.775058i	$-0.717721\pi$
$7$	−3.34365	−1.26378	−0.631890	+	0.775058i	$0.717721\pi$
$8$	−1.86389	−0.658983
$9$	0	0
$10$	−0.815074	−0.257749
$11$	−0.269109	−0.0811395	−0.0405698	−	0.999177i	$-0.512917\pi$
$11$	−0.269109	−0.0811395	−0.0405698	+	0.999177i	$0.512917\pi$
$12$	0	0
$13$	−6.61052	−1.83343	−0.916714	−	0.399545i	$-0.869168\pi$
$13$	−6.61052	−1.83343	−0.916714	+	0.399545i	$0.869168\pi$
$14$	−1.66041	−0.443763
$15$	0	0
$16$	2.58123	0.645307
$17$	−2.98958	−0.725080	−0.362540	−	0.931968i	$-0.618090\pi$
$17$	−2.98958	−0.725080	−0.362540	+	0.931968i	$0.618090\pi$
$18$	0	0
$19$	5.96737	1.36901	0.684504	−	0.729009i	$-0.260019\pi$
$19$	5.96737	1.36901	0.684504	+	0.729009i	$0.260019\pi$
$20$	2.87796	0.643531
$21$	0	0
$22$	−0.133636	−0.0284913
$23$	−8.36700	−1.74464	−0.872320	−	0.488935i	$-0.837385\pi$
$23$	−8.36700	−1.74464	−0.872320	+	0.488935i	$0.837385\pi$
$24$	0	0
$25$	−2.30595	−0.461190
$26$	−3.28269	−0.643788
$27$	0	0
$28$	5.86276	1.10796
$29$	9.38349	1.74247	0.871236	−	0.490865i	$-0.163319\pi$
$29$	9.38349	1.74247	0.871236	+	0.490865i	$0.163319\pi$
$30$	0	0
$31$	−0.868715	−0.156026	−0.0780129	−	0.996952i	$-0.524858\pi$
$31$	−0.868715	−0.156026	−0.0780129	+	0.996952i	$0.524858\pi$
$32$	5.00957	0.885576
$33$	0	0
$34$	−1.48458	−0.254604
$35$	5.48812	0.927661
$36$	0	0
$37$	11.6343	1.91266	0.956330	−	0.292288i	$-0.0944166\pi$
$37$	11.6343	1.91266	0.956330	+	0.292288i	$0.0944166\pi$
$38$	2.96331	0.480712
$39$	0	0
$40$	3.05930	0.483718
$41$	−2.76855	−0.432375	−0.216187	−	0.976352i	$-0.569362\pi$
$41$	−2.76855	−0.432375	−0.216187	+	0.976352i	$0.569362\pi$
$42$	0	0
$43$	7.87141	1.20038	0.600190	−	0.799858i	$-0.295092\pi$
$43$	7.87141	1.20038	0.600190	+	0.799858i	$0.295092\pi$
$44$	0.471857	0.0711351
$45$	0	0
$46$	−4.15493	−0.612611
$47$	−7.11859	−1.03835	−0.519177	−	0.854667i	$-0.673761\pi$
$47$	−7.11859	−1.03835	−0.519177	+	0.854667i	$0.673761\pi$
$48$	0	0
$49$	4.17998	0.597140
$50$	−1.14510	−0.161942
$51$	0	0
$52$	11.5909	1.60737
$53$	−11.7092	−1.60838	−0.804190	−	0.594372i	$-0.797400\pi$
$53$	−11.7092	−1.60838	−0.804190	+	0.594372i	$0.797400\pi$
$54$	0	0
$55$	0.441704	0.0595594
$56$	6.23218	0.832810
$57$	0	0
$58$	4.65971	0.611850
$59$	−4.39528	−0.572216	−0.286108	−	0.958197i	$-0.592362\pi$
$59$	−4.39528	−0.572216	−0.286108	+	0.958197i	$0.592362\pi$
$60$	0	0
$61$	7.35792	0.942085	0.471043	−	0.882110i	$-0.343878\pi$
$61$	7.35792	0.942085	0.471043	+	0.882110i	$0.343878\pi$
$62$	−0.431391	−0.0547868
$63$	0	0
$64$	−2.67477	−0.334346
$65$	10.8502	1.34580
$66$	0	0
$67$	4.25328	0.519620	0.259810	−	0.965660i	$-0.416340\pi$
$67$	4.25328	0.519620	0.259810	+	0.965660i	$0.416340\pi$
$68$	5.24194	0.635679
$69$	0	0
$70$	2.72532	0.325738
$71$	−2.21595	−0.262985	−0.131492	−	0.991317i	$-0.541977\pi$
$71$	−2.21595	−0.262985	−0.131492	+	0.991317i	$0.541977\pi$
$72$	0	0
$73$	5.74357	0.672234	0.336117	−	0.941820i	$-0.390886\pi$
$73$	5.74357	0.672234	0.336117	+	0.941820i	$0.390886\pi$
$74$	5.77741	0.671610
$75$	0	0
$76$	−10.4632	−1.20021
$77$	0.899807	0.102543
$78$	0	0
$79$	−3.09991	−0.348767	−0.174383	−	0.984678i	$-0.555793\pi$
$79$	−3.09991	−0.348767	−0.174383	+	0.984678i	$0.555793\pi$
$80$	−4.23671	−0.473679
$81$	0	0
$82$	−1.37482	−0.151824
$83$	0.729733	0.0800986	0.0400493	−	0.999198i	$-0.487248\pi$
$83$	0.729733	0.0800986	0.0400493	+	0.999198i	$0.487248\pi$
$84$	0	0
$85$	4.90697	0.532236
$86$	3.90883	0.421500
$87$	0	0
$88$	0.501589	0.0534696
$89$	10.9912	1.16506	0.582532	−	0.812808i	$-0.302062\pi$
$89$	10.9912	1.16506	0.582532	+	0.812808i	$0.302062\pi$
$90$	0	0
$91$	22.1032	2.31705
$92$	14.6707	1.52953
$93$	0	0
$94$	−3.53499	−0.364607
$95$	−9.79457	−1.00490
$96$	0	0
$97$	−7.37397	−0.748713	−0.374357	−	0.927285i	$-0.622136\pi$
$97$	−7.37397	−0.748713	−0.374357	+	0.927285i	$0.622136\pi$
$98$	2.07572	0.209679
$99$	0	0
$100$	4.04326	0.404326
$101$	12.2281	1.21674	0.608372	−	0.793652i	$-0.291823\pi$
$101$	12.2281	1.21674	0.608372	+	0.793652i	$0.291823\pi$
$102$	0	0
$103$	10.0897	0.994172	0.497086	−	0.867701i	$-0.334403\pi$
$103$	10.0897	0.994172	0.497086	+	0.867701i	$0.334403\pi$
$104$	12.3212	1.20820
$105$	0	0
$106$	−5.81461	−0.564765
$107$	−0.886796	−0.0857298	−0.0428649	−	0.999081i	$-0.513649\pi$
$107$	−0.886796	−0.0857298	−0.0428649	+	0.999081i	$0.513649\pi$
$108$	0	0
$109$	−14.7806	−1.41573	−0.707863	−	0.706350i	$-0.750341\pi$
$109$	−14.7806	−1.41573	−0.707863	+	0.706350i	$0.750341\pi$
$110$	0.219344	0.0209136
$111$	0	0
$112$	−8.63071	−0.815525
$113$	−4.96145	−0.466734	−0.233367	−	0.972389i	$-0.574974\pi$
$113$	−4.96145	−0.466734	−0.233367	+	0.972389i	$0.574974\pi$
$114$	0	0
$115$	13.7332	1.28063
$116$	−16.4530	−1.52763
$117$	0	0
$118$	−2.18263	−0.200928
$119$	9.99611	0.916342
$120$	0	0
$121$	−10.9276	−0.993416
$122$	3.65384	0.330803
$123$	0	0
$124$	1.52321	0.136788
$125$	11.9917	1.07257
$126$	0	0
$127$	5.29527	0.469879	0.234939	−	0.972010i	$-0.424511\pi$
$127$	5.29527	0.469879	0.234939	+	0.972010i	$0.424511\pi$
$128$	−11.3474	−1.00298
$129$	0	0
$130$	5.38806	0.472564
$131$	5.58271	0.487763	0.243882	−	0.969805i	$-0.421579\pi$
$131$	5.58271	0.487763	0.243882	+	0.969805i	$0.421579\pi$
$132$	0	0
$133$	−19.9528	−1.73012
$134$	2.11212	0.182459
$135$	0	0
$136$	5.57224	0.477816
$137$	−14.2397	−1.21658	−0.608292	−	0.793713i	$-0.708145\pi$
$137$	−14.2397	−1.21658	−0.608292	+	0.793713i	$0.708145\pi$
$138$	0	0
$139$	15.7594	1.33670	0.668349	−	0.743848i	$-0.267001\pi$
$139$	15.7594	1.33670	0.668349	+	0.743848i	$0.267001\pi$
$140$	−9.62288	−0.813282
$141$	0	0
$142$	−1.10041	−0.0923443
$143$	1.77895	0.148763
$144$	0	0
$145$	−15.4017	−1.27904
$146$	2.85217	0.236048
$147$	0	0
$148$	−20.3995	−1.67683
$149$	6.97244	0.571205	0.285602	−	0.958348i	$-0.407806\pi$
$149$	6.97244	0.571205	0.285602	+	0.958348i	$0.407806\pi$
$150$	0	0
$151$	−8.75428	−0.712413	−0.356207	−	0.934407i	$-0.615930\pi$
$151$	−8.75428	−0.712413	−0.356207	+	0.934407i	$0.615930\pi$
$152$	−11.1225	−0.902153
$153$	0	0
$154$	0.446831	0.0360067
$155$	1.42587	0.114529
$156$	0	0
$157$	−3.96994	−0.316835	−0.158418	−	0.987372i	$-0.550639\pi$
$157$	−3.96994	−0.316835	−0.158418	+	0.987372i	$0.550639\pi$
$158$	−1.53937	−0.122466
$159$	0	0
$160$	−8.22249	−0.650045
$161$	27.9763	2.20484
$162$	0	0
$163$	13.5521	1.06148	0.530741	−	0.847534i	$-0.321914\pi$
$163$	13.5521	1.06148	0.530741	+	0.847534i	$0.321914\pi$
$164$	4.85438	0.379064
$165$	0	0
$166$	0.362375	0.0281258
$167$	16.6748	1.29034	0.645169	−	0.764040i	$-0.276787\pi$
$167$	16.6748	1.29034	0.645169	+	0.764040i	$0.276787\pi$
$168$	0	0
$169$	30.6989	2.36146
$170$	2.43673	0.186889
$171$	0	0
$172$	−13.8018	−1.05237
$173$	3.98230	0.302769	0.151384	−	0.988475i	$-0.451627\pi$
$173$	3.98230	0.302769	0.151384	+	0.988475i	$0.451627\pi$
$174$	0	0
$175$	7.71028	0.582842
$176$	−0.694632	−0.0523599
$177$	0	0
$178$	5.45807	0.409100
$179$	20.6900	1.54645	0.773223	−	0.634134i	$-0.218643\pi$
$179$	20.6900	1.54645	0.773223	+	0.634134i	$0.218643\pi$
$180$	0	0
$181$	10.0415	0.746376	0.373188	−	0.927756i	$-0.378265\pi$
$181$	10.0415	0.746376	0.373188	+	0.927756i	$0.378265\pi$
$182$	10.9761	0.813606
$183$	0	0
$184$	15.5951	1.14969
$185$	−19.0960	−1.40396
$186$	0	0
$187$	0.804525	0.0588327
$188$	12.4818	0.910326
$189$	0	0
$190$	−4.86384	−0.352860
$191$	−25.4238	−1.83960	−0.919801	−	0.392385i	$-0.871650\pi$
$191$	−25.4238	−1.83960	−0.919801	+	0.392385i	$0.871650\pi$
$192$	0	0
$193$	−18.4380	−1.32720	−0.663599	−	0.748088i	$-0.730972\pi$
$193$	−18.4380	−1.32720	−0.663599	+	0.748088i	$0.730972\pi$
$194$	−3.66181	−0.262902
$195$	0	0
$196$	−7.32918	−0.523513
$197$	−24.5037	−1.74582	−0.872910	−	0.487881i	$-0.837770\pi$
$197$	−24.5037	−1.74582	−0.872910	+	0.487881i	$0.837770\pi$
$198$	0	0
$199$	−13.1626	−0.933072	−0.466536	−	0.884502i	$-0.654498\pi$
$199$	−13.1626	−0.933072	−0.466536	+	0.884502i	$0.654498\pi$
$200$	4.29803	0.303916
$201$	0	0
$202$	6.07231	0.427246
$203$	−31.3751	−2.20210
$204$	0	0
$205$	4.54418	0.317379
$206$	5.01042	0.349093
$207$	0	0
$208$	−17.0632	−1.18312
$209$	−1.60587	−0.111081
$210$	0	0
$211$	24.8058	1.70770	0.853851	−	0.520517i	$-0.174261\pi$
$211$	24.8058	1.70770	0.853851	+	0.520517i	$0.174261\pi$
$212$	20.5309	1.41007
$213$	0	0
$214$	−0.440370	−0.0301031
$215$	−12.9198	−0.881123
$216$	0	0
$217$	2.90468	0.197182
$218$	−7.33984	−0.497117
$219$	0	0
$220$	−0.774486	−0.0522158
$221$	19.7627	1.32938
$222$	0	0
$223$	−11.0643	−0.740919	−0.370459	−	0.928849i	$-0.620800\pi$
$223$	−11.0643	−0.740919	−0.370459	+	0.928849i	$0.620800\pi$
$224$	−16.7502	−1.11917
$225$	0	0
$226$	−2.46379	−0.163889
$227$	−14.9368	−0.991390	−0.495695	−	0.868497i	$-0.665087\pi$
$227$	−14.9368	−0.991390	−0.495695	+	0.868497i	$0.665087\pi$
$228$	0	0
$229$	−1.78447	−0.117921	−0.0589605	−	0.998260i	$-0.518779\pi$
$229$	−1.78447	−0.117921	−0.0589605	+	0.998260i	$0.518779\pi$
$230$	6.81973	0.449679
$231$	0	0
$232$	−17.4898	−1.14826
$233$	−0.991738	−0.0649709	−0.0324855	−	0.999472i	$-0.510342\pi$
$233$	−0.991738	−0.0649709	−0.0324855	+	0.999472i	$0.510342\pi$
$234$	0	0
$235$	11.6842	0.762190
$236$	7.70669	0.501663
$237$	0	0
$238$	4.96392	0.321763
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	26.5537	1.71048	0.855238	−	0.518236i	$-0.173411\pi$
$241$	26.5537	1.71048	0.855238	+	0.518236i	$0.173411\pi$
$242$	−5.42648	−0.348827
$243$	0	0
$244$	−12.9014	−0.825927
$245$	−6.86083	−0.438323
$246$	0	0
$247$	−39.4474	−2.50998
$248$	1.61919	0.102818
$249$	0	0
$250$	5.95489	0.376620
$251$	−19.1833	−1.21084	−0.605420	−	0.795906i	$-0.706995\pi$
$251$	−19.1833	−1.21084	−0.605420	+	0.795906i	$0.706995\pi$
$252$	0	0
$253$	2.25164	0.141559
$254$	2.62955	0.164993
$255$	0	0
$256$	−0.285413	−0.0178383
$257$	−11.9647	−0.746339	−0.373169	−	0.927763i	$-0.621729\pi$
$257$	−11.9647	−0.746339	−0.373169	+	0.927763i	$0.621729\pi$
$258$	0	0
$259$	−38.9009	−2.41718
$260$	−19.0248	−1.17987
$261$	0	0
$262$	2.77229	0.171273
$263$	19.9677	1.23126	0.615630	−	0.788035i	$-0.288902\pi$
$263$	19.9677	1.23126	0.615630	+	0.788035i	$0.288902\pi$
$264$	0	0
$265$	19.2189	1.18061
$266$	−9.90826	−0.607514
$267$	0	0
$268$	−7.45771	−0.455552
$269$	6.80087	0.414657	0.207328	−	0.978271i	$-0.433523\pi$
$269$	6.80087	0.414657	0.207328	+	0.978271i	$0.433523\pi$
$270$	0	0
$271$	2.36513	0.143671	0.0718356	−	0.997416i	$-0.477114\pi$
$271$	2.36513	0.143671	0.0718356	+	0.997416i	$0.477114\pi$
$272$	−7.71679	−0.467899
$273$	0	0
$274$	−7.07125	−0.427190
$275$	0.620553	0.0374207
$276$	0	0
$277$	−7.49350	−0.450241	−0.225120	−	0.974331i	$-0.572277\pi$
$277$	−7.49350	−0.450241	−0.225120	+	0.974331i	$0.572277\pi$
$278$	7.82591	0.469367
$279$	0	0
$280$	−10.2292	−0.611313
$281$	−16.6170	−0.991288	−0.495644	−	0.868526i	$-0.665068\pi$
$281$	−16.6170	−0.991288	−0.495644	+	0.868526i	$0.665068\pi$
$282$	0	0
$283$	1.84033	0.109396	0.0546980	−	0.998503i	$-0.482580\pi$
$283$	1.84033	0.109396	0.0546980	+	0.998503i	$0.482580\pi$
$284$	3.88545	0.230559
$285$	0	0
$286$	0.883402	0.0522367
$287$	9.25705	0.546426
$288$	0	0
$289$	−8.06240	−0.474259
$290$	−7.64824	−0.449120
$291$	0	0
$292$	−10.0708	−0.589348
$293$	14.4967	0.846907	0.423454	−	0.905918i	$-0.360818\pi$
$293$	14.4967	0.846907	0.423454	+	0.905918i	$0.360818\pi$
$294$	0	0
$295$	7.21422	0.420028
$296$	−21.6849	−1.26041
$297$	0	0
$298$	3.46242	0.200572
$299$	55.3102	3.19867
$300$	0	0
$301$	−26.3192	−1.51702
$302$	−4.34725	−0.250156
$303$	0	0
$304$	15.4031	0.883429
$305$	−12.0770	−0.691525
$306$	0	0
$307$	−14.6500	−0.836120	−0.418060	−	0.908419i	$-0.637290\pi$
$307$	−14.6500	−0.836120	−0.418060	+	0.908419i	$0.637290\pi$
$308$	−1.57772	−0.0898992
$309$	0	0
$310$	0.708067	0.0402155
$311$	−26.3243	−1.49272	−0.746358	−	0.665544i	$-0.768199\pi$
$311$	−26.3243	−1.49272	−0.746358	+	0.665544i	$0.768199\pi$
$312$	0	0
$313$	23.0303	1.30175	0.650874	−	0.759186i	$-0.274403\pi$
$313$	23.0303	1.30175	0.650874	+	0.759186i	$0.274403\pi$
$314$	−1.97141	−0.111253
$315$	0	0
$316$	5.43538	0.305764
$317$	9.08972	0.510529	0.255265	−	0.966871i	$-0.417837\pi$
$317$	9.08972	0.510529	0.255265	+	0.966871i	$0.417837\pi$
$318$	0	0
$319$	−2.52519	−0.141383
$320$	4.39025	0.245423
$321$	0	0
$322$	13.8926	0.774206
$323$	−17.8399	−0.992640
$324$	0	0
$325$	15.2435	0.845558
$326$	6.72977	0.372728
$327$	0	0
$328$	5.16026	0.284928
$329$	23.8021	1.31225
$330$	0	0
$331$	10.7730	0.592138	0.296069	−	0.955167i	$-0.404324\pi$
$331$	10.7730	0.592138	0.296069	+	0.955167i	$0.404324\pi$
$332$	−1.27952	−0.0702226
$333$	0	0
$334$	8.28049	0.453088
$335$	−6.98114	−0.381421
$336$	0	0
$337$	−11.9965	−0.653492	−0.326746	−	0.945112i	$-0.605952\pi$
$337$	−11.9965	−0.653492	−0.326746	+	0.945112i	$0.605952\pi$
$338$	15.2446	0.829200
$339$	0	0
$340$	−8.60389	−0.466612
$341$	0.233779	0.0126599
$342$	0	0
$343$	9.42916	0.509127
$344$	−14.6714	−0.791030
$345$	0	0
$346$	1.97755	0.106314
$347$	6.87065	0.368836	0.184418	−	0.982848i	$-0.440960\pi$
$347$	6.87065	0.368836	0.184418	+	0.982848i	$0.440960\pi$
$348$	0	0
$349$	18.8946	1.01140	0.505702	−	0.862708i	$-0.331234\pi$
$349$	18.8946	1.01140	0.505702	+	0.862708i	$0.331234\pi$
$350$	3.82881	0.204659
$351$	0	0
$352$	−1.34812	−0.0718552
$353$	27.1280	1.44388	0.721939	−	0.691957i	$-0.243251\pi$
$353$	27.1280	1.44388	0.721939	+	0.691957i	$0.243251\pi$
$354$	0	0
$355$	3.63716	0.193041
$356$	−19.2720	−1.02141
$357$	0	0
$358$	10.2744	0.543018
$359$	−5.50625	−0.290609	−0.145304	−	0.989387i	$-0.546416\pi$
$359$	−5.50625	−0.290609	−0.145304	+	0.989387i	$0.546416\pi$
$360$	0	0
$361$	16.6094	0.874181
$362$	4.98645	0.262082
$363$	0	0
$364$	−38.7559	−2.03136
$365$	−9.42725	−0.493445
$366$	0	0
$367$	−16.1366	−0.842325	−0.421163	−	0.906985i	$-0.638378\pi$
$367$	−16.1366	−0.842325	−0.421163	+	0.906985i	$0.638378\pi$
$368$	−21.5971	−1.12583
$369$	0	0
$370$	−9.48278	−0.492986
$371$	39.1514	2.03264
$372$	0	0
$373$	19.3887	1.00391	0.501953	−	0.864895i	$-0.332615\pi$
$373$	19.3887	1.00391	0.501953	+	0.864895i	$0.332615\pi$
$374$	0.399515	0.0206585
$375$	0	0
$376$	13.2682	0.684258
$377$	−62.0297	−3.19469
$378$	0	0
$379$	16.6773	0.856656	0.428328	−	0.903623i	$-0.359103\pi$
$379$	16.6773	0.856656	0.428328	+	0.903623i	$0.359103\pi$
$380$	17.1738	0.880999
$381$	0	0
$382$	−12.6251	−0.645956
$383$	9.22468	0.471359	0.235680	−	0.971831i	$-0.424268\pi$
$383$	9.22468	0.471359	0.235680	+	0.971831i	$0.424268\pi$
$384$	0	0
$385$	−1.47690	−0.0752700
$386$	−9.15606	−0.466031
$387$	0	0
$388$	12.9295	0.656398
$389$	0.902772	0.0457724	0.0228862	−	0.999738i	$-0.492714\pi$
$389$	0.902772	0.0457724	0.0228862	+	0.999738i	$0.492714\pi$
$390$	0	0
$391$	25.0138	1.26500
$392$	−7.79100	−0.393505
$393$	0	0
$394$	−12.1682	−0.613026
$395$	5.08805	0.256008
$396$	0	0
$397$	3.08846	0.155005	0.0775027	−	0.996992i	$-0.475305\pi$
$397$	3.08846	0.155005	0.0775027	+	0.996992i	$0.475305\pi$
$398$	−6.53636	−0.327638
$399$	0	0
$400$	−5.95218	−0.297609
$401$	−8.93832	−0.446358	−0.223179	−	0.974777i	$-0.571643\pi$
$401$	−8.93832	−0.446358	−0.223179	+	0.974777i	$0.571643\pi$
$402$	0	0
$403$	5.74265	0.286062
$404$	−21.4408	−1.06672
$405$	0	0
$406$	−15.5804	−0.773243
$407$	−3.13089	−0.155192
$408$	0	0
$409$	30.0844	1.48758	0.743789	−	0.668414i	$-0.233027\pi$
$409$	30.0844	1.48758	0.743789	+	0.668414i	$0.233027\pi$
$410$	2.25657	0.111444
$411$	0	0
$412$	−17.6914	−0.871592
$413$	14.6963	0.723156
$414$	0	0
$415$	−1.19775	−0.0587953
$416$	−33.1159	−1.62364
$417$	0	0
$418$	−0.797454	−0.0390048
$419$	−7.31528	−0.357375	−0.178688	−	0.983906i	$-0.557185\pi$
$419$	−7.31528	−0.357375	−0.178688	+	0.983906i	$0.557185\pi$
$420$	0	0
$421$	−7.85674	−0.382914	−0.191457	−	0.981501i	$-0.561321\pi$
$421$	−7.85674	−0.382914	−0.191457	+	0.981501i	$0.561321\pi$
$422$	12.3182	0.599641
$423$	0	0
$424$	21.8246	1.05990
$425$	6.89382	0.334400
$426$	0	0
$427$	−24.6023	−1.19059
$428$	1.55491	0.0751594
$429$	0	0
$430$	−6.41579	−0.309397
$431$	−22.2906	−1.07370	−0.536850	−	0.843678i	$-0.680386\pi$
$431$	−22.2906	−1.07370	−0.536850	+	0.843678i	$0.680386\pi$
$432$	0	0
$433$	14.7391	0.708316	0.354158	−	0.935186i	$-0.384767\pi$
$433$	14.7391	0.708316	0.354158	+	0.935186i	$0.384767\pi$
$434$	1.44242	0.0692384
$435$	0	0
$436$	25.9164	1.24117
$437$	−49.9290	−2.38843
$438$	0	0
$439$	−34.2668	−1.63546	−0.817732	−	0.575599i	$-0.804769\pi$
$439$	−34.2668	−1.63546	−0.817732	+	0.575599i	$0.804769\pi$
$440$	−0.823287	−0.0392487
$441$	0	0
$442$	9.81387	0.466798
$443$	−28.9427	−1.37511	−0.687554	−	0.726133i	$-0.741316\pi$
$443$	−28.9427	−1.37511	−0.687554	+	0.726133i	$0.741316\pi$
$444$	0	0
$445$	−18.0405	−0.855200
$446$	−5.49436	−0.260166
$447$	0	0
$448$	8.94349	0.422540
$449$	34.7930	1.64198	0.820992	−	0.570939i	$-0.193421\pi$
$449$	34.7930	1.64198	0.820992	+	0.570939i	$0.193421\pi$
$450$	0	0
$451$	0.745043	0.0350827
$452$	8.69942	0.409186
$453$	0	0
$454$	−7.41740	−0.348116
$455$	−36.2793	−1.70080
$456$	0	0
$457$	15.4833	0.724279	0.362140	−	0.932124i	$-0.382046\pi$
$457$	15.4833	0.724279	0.362140	+	0.932124i	$0.382046\pi$
$458$	−0.886142	−0.0414067
$459$	0	0
$460$	−24.0799	−1.12273
$461$	−4.77826	−0.222546	−0.111273	−	0.993790i	$-0.535493\pi$
$461$	−4.77826	−0.222546	−0.111273	+	0.993790i	$0.535493\pi$
$462$	0	0
$463$	8.41048	0.390868	0.195434	−	0.980717i	$-0.437388\pi$
$463$	8.41048	0.390868	0.195434	+	0.980717i	$0.437388\pi$
$464$	24.2209	1.12443
$465$	0	0
$466$	−0.492483	−0.0228138
$467$	−13.8733	−0.641979	−0.320990	−	0.947083i	$-0.604015\pi$
$467$	−13.8733	−0.641979	−0.320990	+	0.947083i	$0.604015\pi$
$468$	0	0
$469$	−14.2215	−0.656686
$470$	5.80218	0.267635
$471$	0	0
$472$	8.19230	0.377081
$473$	−2.11827	−0.0973982
$474$	0	0
$475$	−13.7604	−0.631372
$476$	−17.5272	−0.803358
$477$	0	0
$478$	−0.496586	−0.0227133
$479$	−9.94353	−0.454331	−0.227166	−	0.973856i	$-0.572946\pi$
$479$	−9.94353	−0.454331	−0.227166	+	0.973856i	$0.572946\pi$
$480$	0	0
$481$	−76.9085	−3.50672
$482$	13.1862	0.600615
$483$	0	0
$484$	19.1604	0.870929
$485$	12.1033	0.549583
$486$	0	0
$487$	−0.196396	−0.00889956	−0.00444978	−	0.999990i	$-0.501416\pi$
$487$	−0.196396	−0.00889956	−0.00444978	+	0.999990i	$0.501416\pi$
$488$	−13.7143	−0.620818
$489$	0	0
$490$	−3.40699	−0.153912
$491$	28.4326	1.28315	0.641573	−	0.767062i	$-0.278282\pi$
$491$	28.4326	1.28315	0.641573	+	0.767062i	$0.278282\pi$
$492$	0	0
$493$	−28.0527	−1.26343
$494$	−19.5890	−0.881351
$495$	0	0
$496$	−2.24235	−0.100684
$497$	7.40935	0.332355
$498$	0	0
$499$	−27.4842	−1.23036	−0.615181	−	0.788386i	$-0.710917\pi$
$499$	−27.4842	−1.23036	−0.615181	+	0.788386i	$0.710917\pi$
$500$	−21.0262	−0.940321
$501$	0	0
$502$	−9.52615	−0.425173
$503$	39.7183	1.77095	0.885477	−	0.464684i	$-0.153832\pi$
$503$	39.7183	1.77095	0.885477	+	0.464684i	$0.153832\pi$
$504$	0	0
$505$	−20.0707	−0.893135
$506$	1.11813	0.0497070
$507$	0	0
$508$	−9.28473	−0.411943
$509$	18.5614	0.822721	0.411360	−	0.911473i	$-0.365054\pi$
$509$	18.5614	0.822721	0.411360	+	0.911473i	$0.365054\pi$
$510$	0	0
$511$	−19.2045	−0.849556
$512$	22.5531	0.996714
$513$	0	0
$514$	−5.94151	−0.262069
$515$	−16.5609	−0.729759
$516$	0	0
$517$	1.91568	0.0842515
$518$	−19.3176	−0.848767
$519$	0	0
$520$	−20.2236	−0.886862
$521$	33.1905	1.45410	0.727052	−	0.686583i	$-0.240890\pi$
$521$	33.1905	1.45410	0.727052	+	0.686583i	$0.240890\pi$
$522$	0	0
$523$	14.5181	0.634831	0.317415	−	0.948287i	$-0.397185\pi$
$523$	14.5181	0.634831	0.317415	+	0.948287i	$0.397185\pi$
$524$	−9.78873	−0.427623
$525$	0	0
$526$	9.91567	0.432344
$527$	2.59709	0.113131
$528$	0	0
$529$	47.0067	2.04377
$530$	9.54385	0.414558
$531$	0	0
$532$	34.9852	1.51680
$533$	18.3015	0.792728
$534$	0	0
$535$	1.45555	0.0629288
$536$	−7.92762	−0.342421
$537$	0	0
$538$	3.37722	0.145602
$539$	−1.12487	−0.0484516
$540$	0	0
$541$	23.2103	0.997888	0.498944	−	0.866634i	$-0.333721\pi$
$541$	23.2103	0.997888	0.498944	+	0.866634i	$0.333721\pi$
$542$	1.17449	0.0504486
$543$	0	0
$544$	−14.9765	−0.642113
$545$	24.2603	1.03920
$546$	0	0
$547$	−0.169946	−0.00726637	−0.00363318	−	0.999993i	$-0.501156\pi$
$547$	−0.169946	−0.00726637	−0.00363318	+	0.999993i	$0.501156\pi$
$548$	24.9680	1.06658
$549$	0	0
$550$	0.308158	0.0131399
$551$	55.9947	2.38546
$552$	0	0
$553$	10.3650	0.440764
$554$	−3.72116	−0.158097
$555$	0	0
$556$	−27.6326	−1.17188
$557$	37.4968	1.58879	0.794395	−	0.607402i	$-0.207788\pi$
$557$	37.4968	1.58879	0.794395	+	0.607402i	$0.207788\pi$
$558$	0	0
$559$	−52.0341	−2.20081
$560$	14.1661	0.598626
$561$	0	0
$562$	−8.25178	−0.348080
$563$	−20.2653	−0.854080	−0.427040	−	0.904233i	$-0.640444\pi$
$563$	−20.2653	−0.854080	−0.427040	+	0.904233i	$0.640444\pi$
$564$	0	0
$565$	8.14351	0.342600
$566$	0.913880	0.0384132
$567$	0	0
$568$	4.13028	0.173303
$569$	33.1147	1.38824	0.694120	−	0.719859i	$-0.255794\pi$
$569$	33.1147	1.38824	0.694120	+	0.719859i	$0.255794\pi$
$570$	0	0
$571$	−12.2061	−0.510808	−0.255404	−	0.966834i	$-0.582208\pi$
$571$	−12.2061	−0.510808	−0.255404	+	0.966834i	$0.582208\pi$
$572$	−3.11922	−0.130421
$573$	0	0
$574$	4.59692	0.191872
$575$	19.2939	0.804610
$576$	0	0
$577$	13.5982	0.566099	0.283049	−	0.959105i	$-0.408654\pi$
$577$	13.5982	0.566099	0.283049	+	0.959105i	$0.408654\pi$
$578$	−4.00367	−0.166531
$579$	0	0
$580$	27.0053	1.12133
$581$	−2.43997	−0.101227
$582$	0	0
$583$	3.15105	0.130503
$584$	−10.7054	−0.442991
$585$	0	0
$586$	7.19886	0.297382
$587$	25.0563	1.03419	0.517093	−	0.855929i	$-0.327014\pi$
$587$	25.0563	1.03419	0.517093	+	0.855929i	$0.327014\pi$
$588$	0	0
$589$	−5.18394	−0.213600
$590$	3.58248	0.147488
$591$	0	0
$592$	30.0307	1.23425
$593$	−38.0505	−1.56255	−0.781274	−	0.624189i	$-0.785430\pi$
$593$	−38.0505	−1.56255	−0.781274	+	0.624189i	$0.785430\pi$
$594$	0	0
$595$	−16.4072	−0.672629
$596$	−12.2255	−0.500776
$597$	0	0
$598$	27.4663	1.12318
$599$	−41.1858	−1.68281	−0.841404	−	0.540407i	$-0.818270\pi$
$599$	−41.1858	−1.68281	−0.841404	+	0.540407i	$0.818270\pi$
$600$	0	0
$601$	1.74488	0.0711752	0.0355876	−	0.999367i	$-0.488670\pi$
$601$	1.74488	0.0711752	0.0355876	+	0.999367i	$0.488670\pi$
$602$	−13.0698	−0.532683
$603$	0	0
$604$	15.3498	0.624574
$605$	17.9361	0.729204
$606$	0	0
$607$	0.530257	0.0215225	0.0107612	−	0.999942i	$-0.496575\pi$
$607$	0.530257	0.0215225	0.0107612	+	0.999942i	$0.496575\pi$
$608$	29.8939	1.21236
$609$	0	0
$610$	−5.99725	−0.242822
$611$	47.0576	1.90375
$612$	0	0
$613$	20.8056	0.840330	0.420165	−	0.907448i	$-0.361972\pi$
$613$	20.8056	0.840330	0.420165	+	0.907448i	$0.361972\pi$
$614$	−7.27499	−0.293595
$615$	0	0
$616$	−1.67714	−0.0675738
$617$	−11.7009	−0.471061	−0.235530	−	0.971867i	$-0.575683\pi$
$617$	−11.7009	−0.471061	−0.235530	+	0.971867i	$0.575683\pi$
$618$	0	0
$619$	3.30686	0.132914	0.0664569	−	0.997789i	$-0.478830\pi$
$619$	3.30686	0.132914	0.0664569	+	0.997789i	$0.478830\pi$
$620$	−2.50013	−0.100407
$621$	0	0
$622$	−13.0723	−0.524151
$623$	−36.7507	−1.47238
$624$	0	0
$625$	−8.15286	−0.326114
$626$	11.4365	0.457094
$627$	0	0
$628$	6.96090	0.277770
$629$	−34.7816	−1.38683
$630$	0	0
$631$	−46.7330	−1.86041	−0.930206	−	0.367039i	$-0.880372\pi$
$631$	−46.7330	−1.86041	−0.930206	+	0.367039i	$0.880372\pi$
$632$	5.77787	0.229831
$633$	0	0
$634$	4.51382	0.179267
$635$	−8.69142	−0.344908
$636$	0	0
$637$	−27.6318	−1.09481
$638$	−1.25397	−0.0496452
$639$	0	0
$640$	18.6251	0.736223
$641$	0.551427	0.0217801	0.0108900	−	0.999941i	$-0.496534\pi$
$641$	0.551427	0.0217801	0.0108900	+	0.999941i	$0.496534\pi$
$642$	0	0
$643$	47.7387	1.88263	0.941316	−	0.337527i	$-0.109591\pi$
$643$	47.7387	1.88263	0.941316	+	0.337527i	$0.109591\pi$
$644$	−49.0537	−1.93299
$645$	0	0
$646$	−8.85905	−0.348555
$647$	−26.3258	−1.03498	−0.517488	−	0.855691i	$-0.673133\pi$
$647$	−26.3258	−1.03498	−0.517488	+	0.855691i	$0.673133\pi$
$648$	0	0
$649$	1.18281	0.0464294
$650$	7.56971	0.296908
$651$	0	0
$652$	−23.7623	−0.930603
$653$	−14.0418	−0.549498	−0.274749	−	0.961516i	$-0.588595\pi$
$653$	−14.0418	−0.549498	−0.274749	+	0.961516i	$0.588595\pi$
$654$	0	0
$655$	−9.16321	−0.358036
$656$	−7.14625	−0.279014
$657$	0	0
$658$	11.8198	0.460782
$659$	23.5245	0.916383	0.458192	−	0.888853i	$-0.348497\pi$
$659$	23.5245	0.916383	0.458192	+	0.888853i	$0.348497\pi$
$660$	0	0
$661$	1.32971	0.0517197	0.0258598	−	0.999666i	$-0.491768\pi$
$661$	1.32971	0.0517197	0.0258598	+	0.999666i	$0.491768\pi$
$662$	5.34972	0.207923
$663$	0	0
$664$	−1.36014	−0.0527836
$665$	32.7496	1.26997
$666$	0	0
$667$	−78.5117	−3.03999
$668$	−29.2377	−1.13124
$669$	0	0
$670$	−3.46673	−0.133932
$671$	−1.98009	−0.0764404
$672$	0	0
$673$	39.6413	1.52806	0.764031	−	0.645180i	$-0.223218\pi$
$673$	39.6413	1.52806	0.764031	+	0.645180i	$0.223218\pi$
$674$	−5.95730	−0.229466
$675$	0	0
$676$	−53.8276	−2.07029
$677$	47.7532	1.83530	0.917652	−	0.397385i	$-0.130082\pi$
$677$	47.7532	1.83530	0.917652	+	0.397385i	$0.130082\pi$
$678$	0	0
$679$	24.6560	0.946209
$680$	−9.14603	−0.350734
$681$	0	0
$682$	0.116091	0.00444537
$683$	−7.00151	−0.267905	−0.133953	−	0.990988i	$-0.542767\pi$
$683$	−7.00151	−0.267905	−0.133953	+	0.990988i	$0.542767\pi$
$684$	0	0
$685$	23.3725	0.893017
$686$	4.68239	0.178774
$687$	0	0
$688$	20.3179	0.774613
$689$	77.4037	2.94885
$690$	0	0
$691$	3.07726	0.117064	0.0585322	−	0.998286i	$-0.481358\pi$
$691$	3.07726	0.117064	0.0585322	+	0.998286i	$0.481358\pi$
$692$	−6.98257	−0.265438
$693$	0	0
$694$	3.41187	0.129513
$695$	−25.8668	−0.981185
$696$	0	0
$697$	8.27681	0.313506
$698$	9.38277	0.355143
$699$	0	0
$700$	−13.5192	−0.510979
$701$	−29.8523	−1.12751	−0.563753	−	0.825943i	$-0.690643\pi$
$701$	−29.8523	−1.12751	−0.563753	+	0.825943i	$0.690643\pi$
$702$	0	0
$703$	69.4259	2.61845
$704$	0.719806	0.0271287
$705$	0	0
$706$	13.4714	0.507002
$707$	−40.8865	−1.53770
$708$	0	0
$709$	26.3252	0.988665	0.494332	−	0.869273i	$-0.335413\pi$
$709$	26.3252	0.988665	0.494332	+	0.869273i	$0.335413\pi$
$710$	1.80616	0.0677841
$711$	0	0
$712$	−20.4863	−0.767758
$713$	7.26854	0.272209
$714$	0	0
$715$	−2.91989	−0.109198
$716$	−36.2780	−1.35577
$717$	0	0
$718$	−2.73432	−0.102044
$719$	0.608546	0.0226949	0.0113475	−	0.999936i	$-0.496388\pi$
$719$	0.608546	0.0226949	0.0113475	+	0.999936i	$0.496388\pi$
$720$	0	0
$721$	−33.7366	−1.25642
$722$	8.24801	0.306959
$723$	0	0
$724$	−17.6067	−0.654349
$725$	−21.6379	−0.803610
$726$	0	0
$727$	−35.9371	−1.33283	−0.666417	−	0.745579i	$-0.732173\pi$
$727$	−35.9371	−1.33283	−0.666417	+	0.745579i	$0.732173\pi$
$728$	−41.1979	−1.52690
$729$	0	0
$730$	−4.68143	−0.173268
$731$	−23.5322	−0.870371
$732$	0	0
$733$	−6.30248	−0.232787	−0.116394	−	0.993203i	$-0.537133\pi$
$733$	−6.30248	−0.232787	−0.116394	+	0.993203i	$0.537133\pi$
$734$	−8.01322	−0.295773
$735$	0	0
$736$	−41.9151	−1.54501
$737$	−1.14460	−0.0421618
$738$	0	0
$739$	−50.8884	−1.87196	−0.935980	−	0.352054i	$-0.885483\pi$
$739$	−50.8884	−1.87196	−0.935980	+	0.352054i	$0.885483\pi$
$740$	33.4829	1.23086
$741$	0	0
$742$	19.4420	0.713739
$743$	22.1034	0.810896	0.405448	−	0.914118i	$-0.367115\pi$
$743$	22.1034	0.810896	0.405448	+	0.914118i	$0.367115\pi$
$744$	0	0
$745$	−11.4443	−0.419285
$746$	9.62813	0.352511
$747$	0	0
$748$	−1.41066	−0.0515787
$749$	2.96513	0.108344
$750$	0	0
$751$	−33.5841	−1.22550	−0.612751	−	0.790276i	$-0.709937\pi$
$751$	−33.5841	−1.22550	−0.612751	+	0.790276i	$0.709937\pi$
$752$	−18.3747	−0.670056
$753$	0	0
$754$	−30.8031	−1.12178
$755$	14.3689	0.522938
$756$	0	0
$757$	2.30914	0.0839271	0.0419635	−	0.999119i	$-0.486639\pi$
$757$	2.30914	0.0839271	0.0419635	+	0.999119i	$0.486639\pi$
$758$	8.28172	0.300805
$759$	0	0
$760$	18.2560	0.662214
$761$	42.7590	1.55001	0.775006	−	0.631953i	$-0.217747\pi$
$761$	42.7590	1.55001	0.775006	+	0.631953i	$0.217747\pi$
$762$	0	0
$763$	49.4212	1.78917
$764$	44.5782	1.61278
$765$	0	0
$766$	4.58084	0.165513
$767$	29.0551	1.04912
$768$	0	0
$769$	−19.1193	−0.689460	−0.344730	−	0.938702i	$-0.612030\pi$
$769$	−19.1193	−0.689460	−0.344730	+	0.938702i	$0.612030\pi$
$770$	−0.733409	−0.0264302
$771$	0	0
$772$	32.3293	1.16356
$773$	0.570335	0.0205135	0.0102568	−	0.999947i	$-0.496735\pi$
$773$	0.570335	0.0205135	0.0102568	+	0.999947i	$0.496735\pi$
$774$	0	0
$775$	2.00321	0.0719575
$776$	13.7442	0.493389
$777$	0	0
$778$	0.448304	0.0160725
$779$	−16.5209	−0.591924
$780$	0	0
$781$	0.596333	0.0213385
$782$	12.4215	0.444192
$783$	0	0
$784$	10.7895	0.385338
$785$	6.51608	0.232569
$786$	0	0
$787$	−4.22596	−0.150639	−0.0753196	−	0.997159i	$-0.523998\pi$
$787$	−4.22596	−0.150639	−0.0753196	+	0.997159i	$0.523998\pi$
$788$	42.9649	1.53056
$789$	0	0
$790$	2.52665	0.0898943
$791$	16.5893	0.589849
$792$	0	0
$793$	−48.6397	−1.72724
$794$	1.53368	0.0544284
$795$	0	0
$796$	23.0793	0.818025
$797$	−7.47275	−0.264698	−0.132349	−	0.991203i	$-0.542252\pi$
$797$	−7.47275	−0.264698	−0.132349	+	0.991203i	$0.542252\pi$
$798$	0	0
$799$	21.2816	0.752890
$800$	−11.5518	−0.408418
$801$	0	0
$802$	−4.43864	−0.156734
$803$	−1.54565	−0.0545448
$804$	0	0
$805$	−45.9191	−1.61843
$806$	2.85172	0.100448
$807$	0	0
$808$	−22.7918	−0.801813
$809$	5.06211	0.177974	0.0889872	−	0.996033i	$-0.471637\pi$
$809$	5.06211	0.177974	0.0889872	+	0.996033i	$0.471637\pi$
$810$	0	0
$811$	24.2852	0.852770	0.426385	−	0.904542i	$-0.359787\pi$
$811$	24.2852	0.852770	0.426385	+	0.904542i	$0.359787\pi$
$812$	55.0132	1.93058
$813$	0	0
$814$	−1.55475	−0.0544941
$815$	−22.2438	−0.779167
$816$	0	0
$817$	46.9716	1.64333
$818$	14.9395	0.522347
$819$	0	0
$820$	−7.96777	−0.278247
$821$	−4.35047	−0.151832	−0.0759162	−	0.997114i	$-0.524188\pi$
$821$	−4.35047	−0.151832	−0.0759162	+	0.997114i	$0.524188\pi$
$822$	0	0
$823$	9.09439	0.317011	0.158505	−	0.987358i	$-0.449333\pi$
$823$	9.09439	0.317011	0.158505	+	0.987358i	$0.449333\pi$
$824$	−18.8061	−0.655143
$825$	0	0
$826$	7.29795	0.253928
$827$	−3.43968	−0.119609	−0.0598047	−	0.998210i	$-0.519048\pi$
$827$	−3.43968	−0.119609	−0.0598047	+	0.998210i	$0.519048\pi$
$828$	0	0
$829$	−29.9447	−1.04002	−0.520011	−	0.854160i	$-0.674072\pi$
$829$	−29.9447	−1.04002	−0.520011	+	0.854160i	$0.674072\pi$
$830$	−0.594787	−0.0206453
$831$	0	0
$832$	17.6816	0.613000
$833$	−12.4964	−0.432974
$834$	0	0
$835$	−27.3694	−0.947156
$836$	2.81574	0.0973845
$837$	0	0
$838$	−3.63267	−0.125488
$839$	18.7978	0.648971	0.324486	−	0.945891i	$-0.394809\pi$
$839$	18.7978	0.648971	0.324486	+	0.945891i	$0.394809\pi$
$840$	0	0
$841$	59.0500	2.03621
$842$	−3.90154	−0.134456
$843$	0	0
$844$	−43.4946	−1.49714
$845$	−50.3879	−1.73340
$846$	0	0
$847$	36.5380	1.25546
$848$	−30.2240	−1.03790
$849$	0	0
$850$	3.42337	0.117421
$851$	−97.3439	−3.33690
$852$	0	0
$853$	−36.6902	−1.25625	−0.628125	−	0.778113i	$-0.716177\pi$
$853$	−36.6902	−1.25625	−0.628125	+	0.778113i	$0.716177\pi$
$854$	−12.2171	−0.418062
$855$	0	0
$856$	1.65289	0.0564945
$857$	35.3354	1.20704	0.603518	−	0.797350i	$-0.293765\pi$
$857$	35.3354	1.20704	0.603518	+	0.797350i	$0.293765\pi$
$858$	0	0
$859$	38.6041	1.31715	0.658577	−	0.752513i	$-0.271159\pi$
$859$	38.6041	1.31715	0.658577	+	0.752513i	$0.271159\pi$
$860$	22.6536	0.772481
$861$	0	0
$862$	−11.0692	−0.377018
$863$	−4.56559	−0.155415	−0.0777073	−	0.996976i	$-0.524760\pi$
$863$	−4.56559	−0.155415	−0.0777073	+	0.996976i	$0.524760\pi$
$864$	0	0
$865$	−6.53637	−0.222243
$866$	7.31923	0.248718
$867$	0	0
$868$	−5.09307	−0.172870
$869$	0.834214	0.0282988
$870$	0	0
$871$	−28.1163	−0.952686
$872$	27.5494	0.932940
$873$	0	0
$874$	−24.7940	−0.838670
$875$	−40.0959	−1.35549
$876$	0	0
$877$	10.5535	0.356367	0.178183	−	0.983997i	$-0.442978\pi$
$877$	10.5535	0.356367	0.178183	+	0.983997i	$0.442978\pi$
$878$	−17.0164	−0.574276
$879$	0	0
$880$	1.14014	0.0384341
$881$	−14.7172	−0.495836	−0.247918	−	0.968781i	$-0.579746\pi$
$881$	−14.7172	−0.495836	−0.247918	+	0.968781i	$0.579746\pi$
$882$	0	0
$883$	36.2785	1.22087	0.610434	−	0.792067i	$-0.290995\pi$
$883$	36.2785	1.22087	0.610434	+	0.792067i	$0.290995\pi$
$884$	−34.6519	−1.16547
$885$	0	0
$886$	−14.3725	−0.482854
$887$	10.2048	0.342643	0.171321	−	0.985215i	$-0.445196\pi$
$887$	10.2048	0.342643	0.171321	+	0.985215i	$0.445196\pi$
$888$	0	0
$889$	−17.7055	−0.593824
$890$	−8.95864	−0.300294
$891$	0	0
$892$	19.4001	0.649565
$893$	−42.4792	−1.42151
$894$	0	0
$895$	−33.9597	−1.13515
$896$	37.9417	1.26754
$897$	0	0
$898$	17.2777	0.576565
$899$	−8.15158	−0.271870
$900$	0	0
$901$	35.0056	1.16620
$902$	0.369977	0.0123189
$903$	0	0
$904$	9.24758	0.307570
$905$	−16.4816	−0.547868
$906$	0	0
$907$	−24.5955	−0.816680	−0.408340	−	0.912830i	$-0.633892\pi$
$907$	−24.5955	−0.816680	−0.408340	+	0.912830i	$0.633892\pi$
$908$	26.1902	0.869153
$909$	0	0
$910$	−18.0158	−0.597217
$911$	9.82170	0.325407	0.162704	−	0.986675i	$-0.447979\pi$
$911$	9.82170	0.325407	0.162704	+	0.986675i	$0.447979\pi$
$912$	0	0
$913$	−0.196378	−0.00649916
$914$	7.68879	0.254323
$915$	0	0
$916$	3.12889	0.103382
$917$	−18.6666	−0.616425
$918$	0	0
$919$	48.7389	1.60775	0.803875	−	0.594799i	$-0.202768\pi$
$919$	48.7389	1.60775	0.803875	+	0.594799i	$0.202768\pi$
$920$	−25.5972	−0.843914
$921$	0	0
$922$	−2.37282	−0.0781445
$923$	14.6486	0.482164
$924$	0	0
$925$	−26.8280	−0.882100
$926$	4.17652	0.137249
$927$	0	0
$928$	47.0073	1.54309
$929$	18.9214	0.620790	0.310395	−	0.950608i	$-0.399539\pi$
$929$	18.9214	0.620790	0.310395	+	0.950608i	$0.399539\pi$
$930$	0	0
$931$	24.9434	0.817489
$932$	1.73892	0.0569601
$933$	0	0
$934$	−6.88928	−0.225424
$935$	−1.32051	−0.0431854
$936$	0	0
$937$	23.3624	0.763215	0.381607	−	0.924324i	$-0.375371\pi$
$937$	23.3624	0.763215	0.381607	+	0.924324i	$0.375371\pi$
$938$	−7.06217	−0.230588
$939$	0	0
$940$	−20.4870	−0.668213
$941$	−44.5578	−1.45254	−0.726271	−	0.687408i	$-0.758748\pi$
$941$	−44.5578	−1.45254	−0.726271	+	0.687408i	$0.758748\pi$
$942$	0	0
$943$	23.1645	0.754338
$944$	−11.3452	−0.369255
$945$	0	0
$946$	−1.05190	−0.0342003
$947$	29.4394	0.956653	0.478326	−	0.878182i	$-0.341244\pi$
$947$	29.4394	0.956653	0.478326	+	0.878182i	$0.341244\pi$
$948$	0	0
$949$	−37.9680	−1.23249
$950$	−6.83324	−0.221699
$951$	0	0
$952$	−18.6316	−0.603854
$953$	−5.59794	−0.181335	−0.0906675	−	0.995881i	$-0.528900\pi$
$953$	−5.59794	−0.181335	−0.0906675	+	0.995881i	$0.528900\pi$
$954$	0	0
$955$	41.7295	1.35034
$956$	1.75340	0.0567091
$957$	0	0
$958$	−4.93781	−0.159533
$959$	47.6127	1.53749
$960$	0	0
$961$	−30.2453	−0.975656
$962$	−38.1916	−1.23135
$963$	0	0
$964$	−46.5594	−1.49958
$965$	30.2634	0.974213
$966$	0	0
$967$	−6.56435	−0.211095	−0.105548	−	0.994414i	$-0.533660\pi$
$967$	−6.56435	−0.211095	−0.105548	+	0.994414i	$0.533660\pi$
$968$	20.3678	0.654645
$969$	0	0
$970$	6.01033	0.192980
$971$	39.7350	1.27516	0.637578	−	0.770386i	$-0.279936\pi$
$971$	39.7350	1.27516	0.637578	+	0.770386i	$0.279936\pi$
$972$	0	0
$973$	−52.6940	−1.68929
$974$	−0.0975275	−0.00312498
$975$	0	0
$976$	18.9925	0.607934
$977$	35.9087	1.14882	0.574411	−	0.818567i	$-0.305231\pi$
$977$	35.9087	1.14882	0.574411	+	0.818567i	$0.305231\pi$
$978$	0	0
$979$	−2.95783	−0.0945328
$980$	12.0298	0.384278
$981$	0	0
$982$	14.1192	0.450563
$983$	−16.9181	−0.539603	−0.269802	−	0.962916i	$-0.586958\pi$
$983$	−16.9181	−0.539603	−0.269802	+	0.962916i	$0.586958\pi$
$984$	0	0
$985$	40.2194	1.28150
$986$	−13.9306	−0.443640
$987$	0	0
$988$	69.1671	2.20050
$989$	−65.8601	−2.09423
$990$	0	0
$991$	28.7435	0.913067	0.456533	−	0.889706i	$-0.349091\pi$
$991$	28.7435	0.913067	0.456533	+	0.889706i	$0.349091\pi$
$992$	−4.35189	−0.138173
$993$	0	0
$994$	3.67938	0.116703
$995$	21.6045	0.684909
$996$	0	0
$997$	10.9633	0.347210	0.173605	−	0.984815i	$-0.444458\pi$
$997$	10.9633	0.347210	0.173605	+	0.984815i	$0.444458\pi$
$998$	−13.6483	−0.432028
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.h.1.8		12
3.2	odd	2		717.2.a.g.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.g.1.5	✓	12	3.2	odd	2
2151.2.a.h.1.8		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+0.496586 q^{2} -1.75340 q^{4} -1.64136 q^{5} -3.34365 q^{7} -1.86389 q^{8} +O(q^{10})\)	\(q+0.496586 q^{2} -1.75340 q^{4} -1.64136 q^{5} -3.34365 q^{7} -1.86389 q^{8} -0.815074 q^{10} -0.269109 q^{11} -6.61052 q^{13} -1.66041 q^{14} +2.58123 q^{16} -2.98958 q^{17} +5.96737 q^{19} +2.87796 q^{20} -0.133636 q^{22} -8.36700 q^{23} -2.30595 q^{25} -3.28269 q^{26} +5.86276 q^{28} +9.38349 q^{29} -0.868715 q^{31} +5.00957 q^{32} -1.48458 q^{34} +5.48812 q^{35} +11.6343 q^{37} +2.96331 q^{38} +3.05930 q^{40} -2.76855 q^{41} +7.87141 q^{43} +0.471857 q^{44} -4.15493 q^{46} -7.11859 q^{47} +4.17998 q^{49} -1.14510 q^{50} +11.5909 q^{52} -11.7092 q^{53} +0.441704 q^{55} +6.23218 q^{56} +4.65971 q^{58} -4.39528 q^{59} +7.35792 q^{61} -0.431391 q^{62} -2.67477 q^{64} +10.8502 q^{65} +4.25328 q^{67} +5.24194 q^{68} +2.72532 q^{70} -2.21595 q^{71} +5.74357 q^{73} +5.77741 q^{74} -10.4632 q^{76} +0.899807 q^{77} -3.09991 q^{79} -4.23671 q^{80} -1.37482 q^{82} +0.729733 q^{83} +4.90697 q^{85} +3.90883 q^{86} +0.501589 q^{88} +10.9912 q^{89} +22.1032 q^{91} +14.6707 q^{92} -3.53499 q^{94} -9.79457 q^{95} -7.37397 q^{97} +2.07572 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8} - 15 q^{11} + 7 q^{13} + 6 q^{14} + 21 q^{16} + 3 q^{17} + 10 q^{19} + 4 q^{20} + 23 q^{22} - 20 q^{23} + 19 q^{25} + 10 q^{26} + 34 q^{28} - 2 q^{29} + 10 q^{31} - 26 q^{32} + 12 q^{34} - 7 q^{35} + 30 q^{37} + 3 q^{38} + 25 q^{40} + 28 q^{41} + 48 q^{43} - 25 q^{44} + 22 q^{46} - 13 q^{47} + 19 q^{49} - 12 q^{50} + 24 q^{52} + 2 q^{53} + 8 q^{55} + 7 q^{56} + 42 q^{58} + 14 q^{59} + 14 q^{61} - 8 q^{62} + 9 q^{64} + 35 q^{65} + 52 q^{67} - 3 q^{68} - 33 q^{70} + 7 q^{71} + 14 q^{73} + 13 q^{74} - 12 q^{76} + 6 q^{77} + 15 q^{79} + 8 q^{80} - 61 q^{82} - 29 q^{83} + 8 q^{85} + 9 q^{86} + 11 q^{88} + 71 q^{89} + 13 q^{91} - 2 q^{92} - 22 q^{94} - 2 q^{95} - 2 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8 - 15 * q^11 + 7 * q^13 + 6 * q^14 + 21 * q^16 + 3 * q^17 + 10 * q^19 + 4 * q^20 + 23 * q^22 - 20 * q^23 + 19 * q^25 + 10 * q^26 + 34 * q^28 - 2 * q^29 + 10 * q^31 - 26 * q^32 + 12 * q^34 - 7 * q^35 + 30 * q^37 + 3 * q^38 + 25 * q^40 + 28 * q^41 + 48 * q^43 - 25 * q^44 + 22 * q^46 - 13 * q^47 + 19 * q^49 - 12 * q^50 + 24 * q^52 + 2 * q^53 + 8 * q^55 + 7 * q^56 + 42 * q^58 + 14 * q^59 + 14 * q^61 - 8 * q^62 + 9 * q^64 + 35 * q^65 + 52 * q^67 - 3 * q^68 - 33 * q^70 + 7 * q^71 + 14 * q^73 + 13 * q^74 - 12 * q^76 + 6 * q^77 + 15 * q^79 + 8 * q^80 - 61 * q^82 - 29 * q^83 + 8 * q^85 + 9 * q^86 + 11 * q^88 + 71 * q^89 + 13 * q^91 - 2 * q^92 - 22 * q^94 - 2 * q^95 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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