LMFDB - Embedded newform 2312.2.b.n.577.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,2,Mod(577,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2312.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.b (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:	$18.4614129473$
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} + 32x^{10} + 380x^{8} + 2000x^{6} + 4068x^{4} + 800x^{2} + 32 $ x^12 + 32x^10 + 380x^8 + 2000x^6 + 4068x^4 + 800*x^2 + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.7
Root			$0.234385i$ of defining polynomial
Character	$\chi$	$=$	2312.577
Dual form			2312.2.b.n.577.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.234385i q^{3} +3.47903i q^{5} -2.44081i q^{7} +2.94506 q^{9} +O(q^{10})$	$q+0.234385i q^{3} +3.47903i q^{5} -2.44081i q^{7} +2.94506 q^{9} -4.07208i q^{11} +1.01887 q^{13} -0.815432 q^{15} +3.89756 q^{19} +0.572090 q^{21} -7.42338i q^{23} -7.10364 q^{25} +1.39343i q^{27} +4.22995i q^{29} -8.45645i q^{31} +0.954435 q^{33} +8.49166 q^{35} -2.97179i q^{37} +0.238809i q^{39} -8.24419i q^{41} -6.19876 q^{43} +10.2460i q^{45} -4.51641 q^{47} +1.04243 q^{49} -3.27521 q^{53} +14.1669 q^{55} +0.913529i q^{57} +0.629667 q^{59} -4.77602i q^{61} -7.18835i q^{63} +3.54469i q^{65} -2.29290 q^{67} +1.73993 q^{69} -7.58866i q^{71} -3.48670i q^{73} -1.66499i q^{75} -9.93919 q^{77} +13.5391i q^{79} +8.50859 q^{81} +17.1811 q^{83} -0.991437 q^{87} +15.4657 q^{89} -2.48688i q^{91} +1.98207 q^{93} +13.5597i q^{95} -0.684286i q^{97} -11.9925i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 28 q^{9}+O(q^{10}) $ 12 * q - 28 * q^9	$ 12 q - 28 q^{9} + 24 q^{13} - 8 q^{15} + 8 q^{19} + 16 q^{21} - 20 q^{25} + 24 q^{33} - 32 q^{35} - 8 q^{43} + 24 q^{47} - 36 q^{49} - 8 q^{53} + 56 q^{55} + 40 q^{59} + 40 q^{67} + 56 q^{69} - 80 q^{77} + 60 q^{81} + 24 q^{83} - 24 q^{87} + 48 q^{89} - 40 q^{93}+O(q^{100}) $ 12 * q - 28 * q^9 + 24 * q^13 - 8 * q^15 + 8 * q^19 + 16 * q^21 - 20 * q^25 + 24 * q^33 - 32 * q^35 - 8 * q^43 + 24 * q^47 - 36 * q^49 - 8 * q^53 + 56 * q^55 + 40 * q^59 + 40 * q^67 + 56 * q^69 - 80 * q^77 + 60 * q^81 + 24 * q^83 - 24 * q^87 + 48 * q^89 - 40 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$1737$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0.234385i	0.135322i	0.997708	+	0.0676611i	$0.0215537\pi$
$3$	0.234385i	0.135322i	−0.997708	+	0.0676611i	$0.978446\pi$
$4$	0	0
$5$	3.47903i	1.55587i	0.628346	+	0.777934i	$0.283732\pi$
$5$	3.47903i	1.55587i	−0.628346	+	0.777934i	$0.716268\pi$
$6$	0	0
$7$	− 2.44081i	− 0.922540i	−0.887260	−	0.461270i	$-0.847394\pi$
$7$	− 2.44081i	− 0.922540i	0.887260	−	0.461270i	$-0.152606\pi$
$8$	0	0
$9$	2.94506	0.981688
$10$	0	0
$11$	− 4.07208i	− 1.22778i	−0.789392	−	0.613889i	$-0.789604\pi$
$11$	− 4.07208i	− 1.22778i	0.789392	−	0.613889i	$-0.210396\pi$
$12$	0	0
$13$	1.01887	0.282585	0.141292	−	0.989968i	$-0.454874\pi$
$13$	1.01887	0.282585	0.141292	+	0.989968i	$0.454874\pi$
$14$	0	0
$15$	−0.815432	−0.210544
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	3.89756	0.894161	0.447080	−	0.894494i	$-0.352464\pi$
$19$	3.89756	0.894161	0.447080	+	0.894494i	$0.352464\pi$
$20$	0	0
$21$	0.572090	0.124840
$22$	0	0
$23$	− 7.42338i	− 1.54788i	−0.633258	−	0.773941i	$-0.718283\pi$
$23$	− 7.42338i	− 1.54788i	0.633258	−	0.773941i	$-0.281717\pi$
$24$	0	0
$25$	−7.10364	−1.42073
$26$	0	0
$27$	1.39343i	0.268167i
$28$	0	0
$29$	4.22995i	0.785482i	0.919649	+	0.392741i	$0.128473\pi$
$29$	4.22995i	0.785482i	−0.919649	+	0.392741i	$0.871527\pi$
$30$	0	0
$31$	− 8.45645i	− 1.51882i	−0.650610	−	0.759412i	$-0.725487\pi$
$31$	− 8.45645i	− 1.51882i	0.650610	−	0.759412i	$-0.274513\pi$
$32$	0	0
$33$	0.954435	0.166146
$34$	0	0
$35$	8.49166	1.43535
$36$	0	0
$37$	− 2.97179i	− 0.488560i	−0.969705	−	0.244280i	$-0.921448\pi$
$37$	− 2.97179i	− 0.488560i	0.969705	−	0.244280i	$-0.0785516\pi$
$38$	0	0
$39$	0.238809i	0.0382400i
$40$	0	0
$41$	− 8.24419i	− 1.28753i	−0.765225	−	0.643763i	$-0.777372\pi$
$41$	− 8.24419i	− 1.28753i	0.765225	−	0.643763i	$-0.222628\pi$
$42$	0	0
$43$	−6.19876	−0.945302	−0.472651	−	0.881250i	$-0.656703\pi$
$43$	−6.19876	−0.945302	−0.472651	+	0.881250i	$0.656703\pi$
$44$	0	0
$45$	10.2460i	1.52738i
$46$	0	0
$47$	−4.51641	−0.658786	−0.329393	−	0.944193i	$-0.606844\pi$
$47$	−4.51641	−0.658786	−0.329393	+	0.944193i	$0.606844\pi$
$48$	0	0
$49$	1.04243	0.148919
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.27521	−0.449885	−0.224942	−	0.974372i	$-0.572219\pi$
$53$	−3.27521	−0.449885	−0.224942	+	0.974372i	$0.572219\pi$
$54$	0	0
$55$	14.1669	1.91026
$56$	0	0
$57$	0.913529i	0.121000i
$58$	0	0
$59$	0.629667	0.0819757	0.0409878	−	0.999160i	$-0.486950\pi$
$59$	0.629667	0.0819757	0.0409878	+	0.999160i	$0.486950\pi$
$60$	0	0
$61$	− 4.77602i	− 0.611507i	−0.952111	−	0.305753i	$-0.901092\pi$
$61$	− 4.77602i	− 0.611507i	0.952111	−	0.305753i	$-0.0989083\pi$
$62$	0	0
$63$	− 7.18835i	− 0.905647i
$64$	0	0
$65$	3.54469i	0.439665i
$66$	0	0
$67$	−2.29290	−0.280122	−0.140061	−	0.990143i	$-0.544730\pi$
$67$	−2.29290	−0.280122	−0.140061	+	0.990143i	$0.544730\pi$
$68$	0	0
$69$	1.73993	0.209463
$70$	0	0
$71$	− 7.58866i	− 0.900608i	−0.892875	−	0.450304i	$-0.851316\pi$
$71$	− 7.58866i	− 0.900608i	0.892875	−	0.450304i	$-0.148684\pi$
$72$	0	0
$73$	− 3.48670i	− 0.408088i	−0.978962	−	0.204044i	$-0.934591\pi$
$73$	− 3.48670i	− 0.408088i	0.978962	−	0.204044i	$-0.0654085\pi$
$74$	0	0
$75$	− 1.66499i	− 0.192256i
$76$	0	0
$77$	−9.93919	−1.13268
$78$	0	0
$79$	13.5391i	1.52327i	0.648005	+	0.761636i	$0.275604\pi$
$79$	13.5391i	1.52327i	−0.648005	+	0.761636i	$0.724396\pi$
$80$	0	0
$81$	8.50859	0.945399
$82$	0	0
$83$	17.1811	1.88587	0.942934	−	0.332979i	$-0.108054\pi$
$83$	17.1811	1.88587	0.942934	+	0.332979i	$0.108054\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.991437	−0.106293
$88$	0	0
$89$	15.4657	1.63936	0.819678	−	0.572824i	$-0.194152\pi$
$89$	15.4657	1.63936	0.819678	+	0.572824i	$0.194152\pi$
$90$	0	0
$91$	− 2.48688i	− 0.260696i
$92$	0	0
$93$	1.98207	0.205531
$94$	0	0
$95$	13.5597i	1.39120i
$96$	0	0
$97$	− 0.684286i	− 0.0694787i	−0.999396	−	0.0347394i	$-0.988940\pi$
$97$	− 0.684286i	− 0.0694787i	0.999396	−	0.0347394i	$-0.0110601\pi$
$98$	0	0
$99$	− 11.9925i	− 1.20530i
$100$	0	0
$101$	15.5509	1.54737	0.773687	−	0.633568i	$-0.218410\pi$
$101$	15.5509	1.54737	0.773687	+	0.633568i	$0.218410\pi$
$102$	0	0
$103$	−13.2156	−1.30217	−0.651086	−	0.759004i	$-0.725686\pi$
$103$	−13.2156	−1.30217	−0.651086	+	0.759004i	$0.725686\pi$
$104$	0	0
$105$	1.99032i	0.194235i
$106$	0	0
$107$	16.6987i	1.61433i	0.590327	+	0.807164i	$0.298999\pi$
$107$	16.6987i	1.61433i	−0.590327	+	0.807164i	$0.701001\pi$
$108$	0	0
$109$	− 2.90444i	− 0.278195i	−0.990279	−	0.139097i	$-0.955580\pi$
$109$	− 2.90444i	− 0.278195i	0.990279	−	0.139097i	$-0.0444201\pi$
$110$	0	0
$111$	0.696544	0.0661131
$112$	0	0
$113$	16.8056i	1.58094i	0.612502	+	0.790469i	$0.290163\pi$
$113$	16.8056i	1.58094i	−0.612502	+	0.790469i	$0.709837\pi$
$114$	0	0
$115$	25.8261	2.40830
$116$	0	0
$117$	3.00065	0.277410
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.58185	−0.507441
$122$	0	0
$123$	1.93231	0.174231
$124$	0	0
$125$	− 7.31862i	− 0.654597i
$126$	0	0
$127$	15.0837	1.33846	0.669231	−	0.743054i	$-0.266623\pi$
$127$	15.0837	1.33846	0.669231	+	0.743054i	$0.266623\pi$
$128$	0	0
$129$	− 1.45290i	− 0.127920i
$130$	0	0
$131$	0.298156i	0.0260500i	0.999915	+	0.0130250i	$0.00414611\pi$
$131$	0.298156i	0.0260500i	−0.999915	+	0.0130250i	$0.995854\pi$
$132$	0	0
$133$	− 9.51320i	− 0.824899i
$134$	0	0
$135$	−4.84780	−0.417232
$136$	0	0
$137$	10.9294	0.933764	0.466882	−	0.884320i	$-0.345377\pi$
$137$	10.9294	0.933764	0.466882	+	0.884320i	$0.345377\pi$
$138$	0	0
$139$	18.5276i	1.57149i	0.618552	+	0.785744i	$0.287720\pi$
$139$	18.5276i	1.57149i	−0.618552	+	0.785744i	$0.712280\pi$
$140$	0	0
$141$	− 1.05858i	− 0.0891484i
$142$	0	0
$143$	− 4.14894i	− 0.346952i
$144$	0	0
$145$	−14.7161	−1.22211
$146$	0	0
$147$	0.244331i	0.0201521i
$148$	0	0
$149$	−11.2176	−0.918985	−0.459492	−	0.888182i	$-0.651969\pi$
$149$	−11.2176	−0.918985	−0.459492	+	0.888182i	$0.651969\pi$
$150$	0	0
$151$	−6.37551	−0.518832	−0.259416	−	0.965766i	$-0.583530\pi$
$151$	−6.37551	−0.518832	−0.259416	+	0.965766i	$0.583530\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	29.4202	2.36309
$156$	0	0
$157$	0.745606	0.0595058	0.0297529	−	0.999557i	$-0.490528\pi$
$157$	0.745606	0.0595058	0.0297529	+	0.999557i	$0.490528\pi$
$158$	0	0
$159$	− 0.767660i	− 0.0608794i
$160$	0	0
$161$	−18.1191	−1.42798
$162$	0	0
$163$	− 0.404545i	− 0.0316864i	−0.999874	−	0.0158432i	$-0.994957\pi$
$163$	− 0.404545i	− 0.0316864i	0.999874	−	0.0158432i	$-0.00504326\pi$
$164$	0	0
$165$	3.32051i	0.258501i
$166$	0	0
$167$	3.94754i	0.305469i	0.988267	+	0.152735i	$0.0488080\pi$
$167$	3.94754i	0.305469i	−0.988267	+	0.152735i	$0.951192\pi$
$168$	0	0
$169$	−11.9619	−0.920146
$170$	0	0
$171$	11.4786	0.877787
$172$	0	0
$173$	− 7.91589i	− 0.601834i	−0.953650	−	0.300917i	$-0.902707\pi$
$173$	− 7.91589i	− 0.601834i	0.953650	−	0.300917i	$-0.0972928\pi$
$174$	0	0
$175$	17.3386i	1.31068i
$176$	0	0
$177$	0.147585i	0.0110931i
$178$	0	0
$179$	12.4466	0.930301	0.465151	−	0.885232i	$-0.346000\pi$
$179$	12.4466	0.930301	0.465151	+	0.885232i	$0.346000\pi$
$180$	0	0
$181$	− 7.58572i	− 0.563842i	−0.959438	−	0.281921i	$-0.909028\pi$
$181$	− 7.58572i	− 0.563842i	0.959438	−	0.281921i	$-0.0909716\pi$
$182$	0	0
$183$	1.11943	0.0827505
$184$	0	0
$185$	10.3390	0.760135
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3.40111	0.247394
$190$	0	0
$191$	13.3487	0.965880	0.482940	−	0.875654i	$-0.339569\pi$
$191$	13.3487	0.965880	0.482940	+	0.875654i	$0.339569\pi$
$192$	0	0
$193$	− 13.0727i	− 0.940994i	−0.882401	−	0.470497i	$-0.844075\pi$
$193$	− 13.0727i	− 0.940994i	0.882401	−	0.470497i	$-0.155925\pi$
$194$	0	0
$195$	−0.830822	−0.0594964
$196$	0	0
$197$	5.07143i	0.361324i	0.983545	+	0.180662i	$0.0578241\pi$
$197$	5.07143i	0.361324i	−0.983545	+	0.180662i	$0.942176\pi$
$198$	0	0
$199$	− 8.71042i	− 0.617465i	−0.951149	−	0.308733i	$-0.900095\pi$
$199$	− 8.71042i	− 0.617465i	0.951149	−	0.308733i	$-0.0999049\pi$
$200$	0	0
$201$	− 0.537421i	− 0.0379067i
$202$	0	0
$203$	10.3245	0.724639
$204$	0	0
$205$	28.6818	2.00322
$206$	0	0
$207$	− 21.8623i	− 1.51954i
$208$	0	0
$209$	− 15.8712i	− 1.09783i
$210$	0	0
$211$	− 3.59337i	− 0.247378i	−0.992321	−	0.123689i	$-0.960528\pi$
$211$	− 3.59337i	− 0.247378i	0.992321	−	0.123689i	$-0.0394724\pi$
$212$	0	0
$213$	1.77867	0.121872
$214$	0	0
$215$	− 21.5657i	− 1.47077i
$216$	0	0
$217$	−20.6406	−1.40118
$218$	0	0
$219$	0.817231	0.0552234
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−19.0303	−1.27437	−0.637183	−	0.770713i	$-0.719900\pi$
$223$	−19.0303	−1.27437	−0.637183	+	0.770713i	$0.719900\pi$
$224$	0	0
$225$	−20.9207	−1.39471
$226$	0	0
$227$	− 2.93980i	− 0.195121i	−0.995230	−	0.0975607i	$-0.968896\pi$
$227$	− 2.93980i	− 0.195121i	0.995230	−	0.0975607i	$-0.0311040\pi$
$228$	0	0
$229$	−28.1470	−1.86001	−0.930004	−	0.367550i	$-0.880197\pi$
$229$	−28.1470	−1.86001	−0.930004	+	0.367550i	$0.880197\pi$
$230$	0	0
$231$	− 2.32960i	− 0.153276i
$232$	0	0
$233$	16.5092i	1.08155i	0.841166	+	0.540777i	$0.181870\pi$
$233$	16.5092i	1.08155i	−0.841166	+	0.540777i	$0.818130\pi$
$234$	0	0
$235$	− 15.7127i	− 1.02498i
$236$	0	0
$237$	−3.17337	−0.206133
$238$	0	0
$239$	−14.6235	−0.945918	−0.472959	−	0.881084i	$-0.656814\pi$
$239$	−14.6235	−0.945918	−0.472959	+	0.881084i	$0.656814\pi$
$240$	0	0
$241$	− 11.0756i	− 0.713443i	−0.934211	−	0.356721i	$-0.883895\pi$
$241$	− 11.0756i	− 0.713443i	0.934211	−	0.356721i	$-0.116105\pi$
$242$	0	0
$243$	6.17459i	0.396100i
$244$	0	0
$245$	3.62666i	0.231699i
$246$	0	0
$247$	3.97112	0.252676
$248$	0	0
$249$	4.02699i	0.255200i
$250$	0	0
$251$	−10.9608	−0.691840	−0.345920	−	0.938264i	$-0.612433\pi$
$251$	−10.9608	−0.691840	−0.345920	+	0.938264i	$0.612433\pi$
$252$	0	0
$253$	−30.2286	−1.90046
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.99288	−0.124312	−0.0621562	−	0.998066i	$-0.519798\pi$
$257$	−1.99288	−0.124312	−0.0621562	+	0.998066i	$0.519798\pi$
$258$	0	0
$259$	−7.25359	−0.450716
$260$	0	0
$261$	12.4575i	0.771098i
$262$	0	0
$263$	1.77310	0.109334	0.0546671	−	0.998505i	$-0.482590\pi$
$263$	1.77310	0.109334	0.0546671	+	0.998505i	$0.482590\pi$
$264$	0	0
$265$	− 11.3946i	− 0.699962i
$266$	0	0
$267$	3.62492i	0.221842i
$268$	0	0
$269$	19.8963i	1.21310i	0.795047	+	0.606548i	$0.207446\pi$
$269$	19.8963i	1.21310i	−0.795047	+	0.606548i	$0.792554\pi$
$270$	0	0
$271$	0.752424	0.0457065	0.0228532	−	0.999739i	$-0.492725\pi$
$271$	0.752424	0.0457065	0.0228532	+	0.999739i	$0.492725\pi$
$272$	0	0
$273$	0.582887	0.0352780
$274$	0	0
$275$	28.9266i	1.74434i
$276$	0	0
$277$	− 16.8064i	− 1.00980i	−0.863179	−	0.504899i	$-0.831530\pi$
$277$	− 16.8064i	− 1.00980i	0.863179	−	0.504899i	$-0.168470\pi$
$278$	0	0
$279$	− 24.9048i	− 1.49101i
$280$	0	0
$281$	−1.06300	−0.0634130	−0.0317065	−	0.999497i	$-0.510094\pi$
$281$	−1.06300	−0.0634130	−0.0317065	+	0.999497i	$0.510094\pi$
$282$	0	0
$283$	− 18.1657i	− 1.07984i	−0.841717	−	0.539920i	$-0.818455\pi$
$283$	− 18.1657i	− 1.07984i	0.841717	−	0.539920i	$-0.181545\pi$
$284$	0	0
$285$	−3.17819	−0.188260
$286$	0	0
$287$	−20.1225	−1.18779
$288$	0	0
$289$	0	0
$290$	0	0
$291$	0.160386	0.00940202
$292$	0	0
$293$	6.47478	0.378260	0.189130	−	0.981952i	$-0.439433\pi$
$293$	6.47478	0.378260	0.189130	+	0.981952i	$0.439433\pi$
$294$	0	0
$295$	2.19063i	0.127543i
$296$	0	0
$297$	5.67418	0.329249
$298$	0	0
$299$	− 7.56348i	− 0.437407i
$300$	0	0
$301$	15.1300i	0.872079i
$302$	0	0
$303$	3.64490i	0.209394i
$304$	0	0
$305$	16.6159	0.951424
$306$	0	0
$307$	−14.7940	−0.844341	−0.422170	−	0.906516i	$-0.638732\pi$
$307$	−14.7940	−0.844341	−0.422170	+	0.906516i	$0.638732\pi$
$308$	0	0
$309$	− 3.09754i	− 0.176213i
$310$	0	0
$311$	9.57132i	0.542739i	0.962475	+	0.271370i	$0.0874766\pi$
$311$	9.57132i	0.542739i	−0.962475	+	0.271370i	$0.912523\pi$
$312$	0	0
$313$	5.83519i	0.329824i	0.986308	+	0.164912i	$0.0527340\pi$
$313$	5.83519i	0.329824i	−0.986308	+	0.164912i	$0.947266\pi$
$314$	0	0
$315$	25.0085	1.40907
$316$	0	0
$317$	23.3261i	1.31013i	0.755574	+	0.655063i	$0.227358\pi$
$317$	23.3261i	1.31013i	−0.755574	+	0.655063i	$0.772642\pi$
$318$	0	0
$319$	17.2247	0.964398
$320$	0	0
$321$	−3.91394	−0.218455
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−7.23771	−0.401476
$326$	0	0
$327$	0.680757	0.0376460
$328$	0	0
$329$	11.0237i	0.607757i
$330$	0	0
$331$	31.5997	1.73688	0.868439	−	0.495796i	$-0.165124\pi$
$331$	31.5997	1.73688	0.868439	+	0.495796i	$0.165124\pi$
$332$	0	0
$333$	− 8.75212i	− 0.479613i
$334$	0	0
$335$	− 7.97705i	− 0.435833i
$336$	0	0
$337$	− 0.430314i	− 0.0234407i	−0.999931	−	0.0117203i	$-0.996269\pi$
$337$	− 0.430314i	− 0.0234407i	0.999931	−	0.0117203i	$-0.00373078\pi$
$338$	0	0
$339$	−3.93898	−0.213936
$340$	0	0
$341$	−34.4354	−1.86478
$342$	0	0
$343$	− 19.6301i	− 1.05992i
$344$	0	0
$345$	6.05326i	0.325897i
$346$	0	0
$347$	16.0827i	0.863366i	0.902025	+	0.431683i	$0.142080\pi$
$347$	16.0827i	0.863366i	−0.902025	+	0.431683i	$0.857920\pi$
$348$	0	0
$349$	12.9133	0.691235	0.345617	−	0.938376i	$-0.387670\pi$
$349$	12.9133	0.691235	0.345617	+	0.938376i	$0.387670\pi$
$350$	0	0
$351$	1.41973i	0.0757798i
$352$	0	0
$353$	−1.09118	−0.0580775	−0.0290388	−	0.999578i	$-0.509245\pi$
$353$	−1.09118	−0.0580775	−0.0290388	+	0.999578i	$0.509245\pi$
$354$	0	0
$355$	26.4012	1.40123
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.45185	−0.182182	−0.0910909	−	0.995843i	$-0.529035\pi$
$359$	−3.45185	−0.182182	−0.0910909	+	0.995843i	$0.529035\pi$
$360$	0	0
$361$	−3.80906	−0.200477
$362$	0	0
$363$	− 1.30830i	− 0.0686681i
$364$	0	0
$365$	12.1303	0.634931
$366$	0	0
$367$	22.8839i	1.19453i	0.802043	+	0.597266i	$0.203746\pi$
$367$	22.8839i	1.19453i	−0.802043	+	0.597266i	$0.796254\pi$
$368$	0	0
$369$	− 24.2797i	− 1.26395i
$370$	0	0
$371$	7.99418i	0.415037i
$372$	0	0
$373$	−18.7107	−0.968805	−0.484403	−	0.874845i	$-0.660963\pi$
$373$	−18.7107	−0.968805	−0.484403	+	0.874845i	$0.660963\pi$
$374$	0	0
$375$	1.71537	0.0885815
$376$	0	0
$377$	4.30978i	0.221965i
$378$	0	0
$379$	− 31.8894i	− 1.63805i	−0.573759	−	0.819024i	$-0.694515\pi$
$379$	− 31.8894i	− 1.63805i	0.573759	−	0.819024i	$-0.305485\pi$
$380$	0	0
$381$	3.53540i	0.181124i
$382$	0	0
$383$	−16.1848	−0.827003	−0.413501	−	0.910503i	$-0.635694\pi$
$383$	−16.1848	−0.827003	−0.413501	+	0.910503i	$0.635694\pi$
$384$	0	0
$385$	− 34.5787i	− 1.76229i
$386$	0	0
$387$	−18.2557	−0.927992
$388$	0	0
$389$	37.3359	1.89301	0.946503	−	0.322695i	$-0.104589\pi$
$389$	37.3359	1.89301	0.946503	+	0.322695i	$0.104589\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−0.0698834	−0.00352515
$394$	0	0
$395$	−47.1031	−2.37001
$396$	0	0
$397$	27.3879i	1.37456i	0.726393	+	0.687280i	$0.241195\pi$
$397$	27.3879i	1.37456i	−0.726393	+	0.687280i	$0.758805\pi$
$398$	0	0
$399$	2.22975	0.111627
$400$	0	0
$401$	− 5.02491i	− 0.250932i	−0.992098	−	0.125466i	$-0.959957\pi$
$401$	− 5.02491i	− 0.250932i	0.992098	−	0.125466i	$-0.0400426\pi$
$402$	0	0
$403$	− 8.61605i	− 0.429196i
$404$	0	0
$405$	29.6016i	1.47092i
$406$	0	0
$407$	−12.1014	−0.599844
$408$	0	0
$409$	−18.2881	−0.904289	−0.452144	−	0.891945i	$-0.649341\pi$
$409$	−18.2881	−0.904289	−0.452144	+	0.891945i	$0.649341\pi$
$410$	0	0
$411$	2.56169i	0.126359i
$412$	0	0
$413$	− 1.53690i	− 0.0756259i
$414$	0	0
$415$	59.7735i	2.93416i
$416$	0	0
$417$	−4.34258	−0.212657
$418$	0	0
$419$	1.10530i	0.0539973i	0.999635	+	0.0269986i	$0.00859498\pi$
$419$	1.10530i	0.0539973i	−0.999635	+	0.0269986i	$0.991405\pi$
$420$	0	0
$421$	22.3412	1.08885	0.544423	−	0.838811i	$-0.316749\pi$
$421$	22.3412	1.08885	0.544423	+	0.838811i	$0.316749\pi$
$422$	0	0
$423$	−13.3011	−0.646722
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−11.6574	−0.564140
$428$	0	0
$429$	0.972449	0.0469503
$430$	0	0
$431$	− 0.891134i	− 0.0429244i	−0.999770	−	0.0214622i	$-0.993168\pi$
$431$	− 0.891134i	− 0.0429244i	0.999770	−	0.0214622i	$-0.00683216\pi$
$432$	0	0
$433$	2.21402	0.106399	0.0531994	−	0.998584i	$-0.483058\pi$
$433$	2.21402	0.106399	0.0531994	+	0.998584i	$0.483058\pi$
$434$	0	0
$435$	− 3.44924i	− 0.165378i
$436$	0	0
$437$	− 28.9330i	− 1.38405i
$438$	0	0
$439$	− 11.3608i	− 0.542222i	−0.962548	−	0.271111i	$-0.912609\pi$
$439$	− 11.3608i	− 0.542222i	0.962548	−	0.271111i	$-0.0873911\pi$
$440$	0	0
$441$	3.07003	0.146192
$442$	0	0
$443$	7.02968	0.333990	0.166995	−	0.985958i	$-0.446594\pi$
$443$	7.02968	0.333990	0.166995	+	0.985958i	$0.446594\pi$
$444$	0	0
$445$	53.8055i	2.55062i
$446$	0	0
$447$	− 2.62925i	− 0.124359i
$448$	0	0
$449$	6.70480i	0.316419i	0.987406	+	0.158210i	$0.0505722\pi$
$449$	6.70480i	0.316419i	−0.987406	+	0.158210i	$0.949428\pi$
$450$	0	0
$451$	−33.5710	−1.58080
$452$	0	0
$453$	− 1.49432i	− 0.0702095i
$454$	0	0
$455$	8.65192	0.405608
$456$	0	0
$457$	−28.3215	−1.32482	−0.662412	−	0.749140i	$-0.730467\pi$
$457$	−28.3215	−1.32482	−0.662412	+	0.749140i	$0.730467\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.77507	−0.315547	−0.157773	−	0.987475i	$-0.550432\pi$
$461$	−6.77507	−0.315547	−0.157773	+	0.987475i	$0.550432\pi$
$462$	0	0
$463$	4.64870	0.216043	0.108022	−	0.994149i	$-0.465548\pi$
$463$	4.64870	0.216043	0.108022	+	0.994149i	$0.465548\pi$
$464$	0	0
$465$	6.89566i	0.319779i
$466$	0	0
$467$	25.8709	1.19716	0.598582	−	0.801061i	$-0.295731\pi$
$467$	25.8709	1.19716	0.598582	+	0.801061i	$0.295731\pi$
$468$	0	0
$469$	5.59653i	0.258424i
$470$	0	0
$471$	0.174759i	0.00805246i
$472$	0	0
$473$	25.2419i	1.16062i
$474$	0	0
$475$	−27.6868	−1.27036
$476$	0	0
$477$	−9.64570	−0.441646
$478$	0	0
$479$	6.96996i	0.318466i	0.987241	+	0.159233i	$0.0509020\pi$
$479$	6.96996i	0.318466i	−0.987241	+	0.159233i	$0.949098\pi$
$480$	0	0
$481$	− 3.02788i	− 0.138060i
$482$	0	0
$483$	− 4.24684i	− 0.193238i
$484$	0	0
$485$	2.38065	0.108100
$486$	0	0
$487$	− 1.17446i	− 0.0532201i	−0.999646	−	0.0266100i	$-0.991529\pi$
$487$	− 1.17446i	− 0.0532201i	0.999646	−	0.0266100i	$-0.00847124\pi$
$488$	0	0
$489$	0.0948192	0.00428787
$490$	0	0
$491$	7.16315	0.323269	0.161634	−	0.986851i	$-0.448324\pi$
$491$	7.16315	0.323269	0.161634	+	0.986851i	$0.448324\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	41.7224	1.87528
$496$	0	0
$497$	−18.5225	−0.830847
$498$	0	0
$499$	25.9621i	1.16222i	0.813823	+	0.581112i	$0.197382\pi$
$499$	25.9621i	1.16222i	−0.813823	+	0.581112i	$0.802618\pi$
$500$	0	0
$501$	−0.925243	−0.0413368
$502$	0	0
$503$	1.17649i	0.0524569i	0.999656	+	0.0262285i	$0.00834973\pi$
$503$	1.17649i	0.0524569i	−0.999656	+	0.0262285i	$0.991650\pi$
$504$	0	0
$505$	54.1021i	2.40751i
$506$	0	0
$507$	− 2.80369i	− 0.124516i
$508$	0	0
$509$	4.06345	0.180109	0.0900547	−	0.995937i	$-0.471296\pi$
$509$	4.06345	0.180109	0.0900547	+	0.995937i	$0.471296\pi$
$510$	0	0
$511$	−8.51039	−0.376478
$512$	0	0
$513$	5.43099i	0.239784i
$514$	0	0
$515$	− 45.9774i	− 2.02601i
$516$	0	0
$517$	18.3912i	0.808843i
$518$	0	0
$519$	1.85537	0.0814416
$520$	0	0
$521$	11.1308i	0.487650i	0.969819	+	0.243825i	$0.0784022\pi$
$521$	11.1308i	0.487650i	−0.969819	+	0.243825i	$0.921598\pi$
$522$	0	0
$523$	−23.0411	−1.00752	−0.503758	−	0.863845i	$-0.668050\pi$
$523$	−23.0411	−1.00752	−0.503758	+	0.863845i	$0.668050\pi$
$524$	0	0
$525$	−4.06392	−0.177364
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−32.1065	−1.39594
$530$	0	0
$531$	1.85441	0.0804745
$532$	0	0
$533$	− 8.39978i	− 0.363835i
$534$	0	0
$535$	−58.0954	−2.51168
$536$	0	0
$537$	2.91729i	0.125890i
$538$	0	0
$539$	− 4.24488i	− 0.182840i
$540$	0	0
$541$	7.59601i	0.326578i	0.986578	+	0.163289i	$0.0522103\pi$
$541$	7.59601i	0.326578i	−0.986578	+	0.163289i	$0.947790\pi$
$542$	0	0
$543$	1.77798	0.0763004
$544$	0	0
$545$	10.1046	0.432835
$546$	0	0
$547$	18.5334i	0.792433i	0.918157	+	0.396216i	$0.129677\pi$
$547$	18.5334i	0.792433i	−0.918157	+	0.396216i	$0.870323\pi$
$548$	0	0
$549$	− 14.0657i	− 0.600309i
$550$	0	0
$551$	16.4865i	0.702347i
$552$	0	0
$553$	33.0465	1.40528
$554$	0	0
$555$	2.42330i	0.102863i
$556$	0	0
$557$	10.6192	0.449952	0.224976	−	0.974364i	$-0.427770\pi$
$557$	10.6192	0.449952	0.224976	+	0.974364i	$0.427770\pi$
$558$	0	0
$559$	−6.31575	−0.267128
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.47916	−0.188774	−0.0943871	−	0.995536i	$-0.530089\pi$
$563$	−4.47916	−0.188774	−0.0943871	+	0.995536i	$0.530089\pi$
$564$	0	0
$565$	−58.4671	−2.45973
$566$	0	0
$567$	− 20.7679i	− 0.872169i
$568$	0	0
$569$	42.2877	1.77279	0.886395	−	0.462929i	$-0.153202\pi$
$569$	42.2877	1.77279	0.886395	+	0.462929i	$0.153202\pi$
$570$	0	0
$571$	− 12.5555i	− 0.525430i	−0.964873	−	0.262715i	$-0.915382\pi$
$571$	− 12.5555i	− 0.525430i	0.964873	−	0.262715i	$-0.0846179\pi$
$572$	0	0
$573$	3.12874i	0.130705i
$574$	0	0
$575$	52.7330i	2.19912i
$576$	0	0
$577$	39.5199	1.64524	0.822618	−	0.568594i	$-0.192513\pi$
$577$	39.5199	1.64524	0.822618	+	0.568594i	$0.192513\pi$
$578$	0	0
$579$	3.06405	0.127338
$580$	0	0
$581$	− 41.9358i	− 1.73979i
$582$	0	0
$583$	13.3369i	0.552359i
$584$	0	0
$585$	10.4393i	0.431613i
$586$	0	0
$587$	34.1613	1.40999	0.704993	−	0.709214i	$-0.250950\pi$
$587$	34.1613	1.40999	0.704993	+	0.709214i	$0.250950\pi$
$588$	0	0
$589$	− 32.9595i	− 1.35807i
$590$	0	0
$591$	−1.18867	−0.0488953
$592$	0	0
$593$	−19.5955	−0.804690	−0.402345	−	0.915488i	$-0.631805\pi$
$593$	−19.5955	−0.804690	−0.402345	+	0.915488i	$0.631805\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.04159	0.0835568
$598$	0	0
$599$	−34.4419	−1.40726	−0.703629	−	0.710567i	$-0.748438\pi$
$599$	−34.4419	−1.40726	−0.703629	+	0.710567i	$0.748438\pi$
$600$	0	0
$601$	8.28073i	0.337778i	0.985635	+	0.168889i	$0.0540180\pi$
$601$	8.28073i	0.337778i	−0.985635	+	0.168889i	$0.945982\pi$
$602$	0	0
$603$	−6.75272	−0.274992
$604$	0	0
$605$	− 19.4194i	− 0.789512i
$606$	0	0
$607$	20.4124i	0.828515i	0.910160	+	0.414257i	$0.135959\pi$
$607$	20.4124i	0.828515i	−0.910160	+	0.414257i	$0.864041\pi$
$608$	0	0
$609$	2.41991i	0.0980598i
$610$	0	0
$611$	−4.60165	−0.186163
$612$	0	0
$613$	−29.3921	−1.18714	−0.593568	−	0.804784i	$-0.702281\pi$
$613$	−29.3921	−1.18714	−0.593568	+	0.804784i	$0.702281\pi$
$614$	0	0
$615$	6.72258i	0.271080i
$616$	0	0
$617$	20.3962i	0.821120i	0.911834	+	0.410560i	$0.134667\pi$
$617$	20.3962i	0.821120i	−0.911834	+	0.410560i	$0.865333\pi$
$618$	0	0
$619$	− 15.1158i	− 0.607557i	−0.952743	−	0.303779i	$-0.901752\pi$
$619$	− 15.1158i	− 0.607557i	0.952743	−	0.303779i	$-0.0982483\pi$
$620$	0	0
$621$	10.3440	0.415090
$622$	0	0
$623$	− 37.7488i	− 1.51237i
$624$	0	0
$625$	−10.0565	−0.402261
$626$	0	0
$627$	3.71996	0.148561
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−31.5824	−1.25728	−0.628639	−	0.777698i	$-0.716388\pi$
$631$	−31.5824	−1.25728	−0.628639	+	0.777698i	$0.716388\pi$
$632$	0	0
$633$	0.842231	0.0334757
$634$	0	0
$635$	52.4767i	2.08247i
$636$	0	0
$637$	1.06211	0.0420823
$638$	0	0
$639$	− 22.3491i	− 0.884116i
$640$	0	0
$641$	− 40.4674i	− 1.59837i	−0.601088	−	0.799183i	$-0.705266\pi$
$641$	− 40.4674i	− 1.59837i	0.601088	−	0.799183i	$-0.294734\pi$
$642$	0	0
$643$	− 4.48760i	− 0.176974i	−0.996077	−	0.0884868i	$-0.971797\pi$
$643$	− 4.48760i	− 0.176974i	0.996077	−	0.0884868i	$-0.0282031\pi$
$644$	0	0
$645$	5.05467	0.199027
$646$	0	0
$647$	−11.0201	−0.433243	−0.216622	−	0.976256i	$-0.569504\pi$
$647$	−11.0201	−0.433243	−0.216622	+	0.976256i	$0.569504\pi$
$648$	0	0
$649$	− 2.56406i	− 0.100648i
$650$	0	0
$651$	− 4.83785i	− 0.189610i
$652$	0	0
$653$	− 42.9742i	− 1.68171i	−0.541260	−	0.840855i	$-0.682053\pi$
$653$	− 42.9742i	− 1.68171i	0.541260	−	0.840855i	$-0.317947\pi$
$654$	0	0
$655$	−1.03729	−0.0405305
$656$	0	0
$657$	− 10.2686i	− 0.400615i
$658$	0	0
$659$	−38.9296	−1.51648	−0.758240	−	0.651975i	$-0.773941\pi$
$659$	−38.9296	−1.51648	−0.758240	+	0.651975i	$0.773941\pi$
$660$	0	0
$661$	−0.487631	−0.0189666	−0.00948332	−	0.999955i	$-0.503019\pi$
$661$	−0.487631	−0.0189666	−0.00948332	+	0.999955i	$0.503019\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	33.0967	1.28344
$666$	0	0
$667$	31.4005	1.21583
$668$	0	0
$669$	− 4.46042i	− 0.172450i
$670$	0	0
$671$	−19.4483	−0.750795
$672$	0	0
$673$	30.1670i	1.16285i	0.813599	+	0.581426i	$0.197505\pi$
$673$	30.1670i	1.16285i	−0.813599	+	0.581426i	$0.802495\pi$
$674$	0	0
$675$	− 9.89845i	− 0.380992i
$676$	0	0
$677$	− 44.5961i	− 1.71397i	−0.515344	−	0.856983i	$-0.672336\pi$
$677$	− 44.5961i	− 1.71397i	0.515344	−	0.856983i	$-0.327664\pi$
$678$	0	0
$679$	−1.67021	−0.0640970
$680$	0	0
$681$	0.689045	0.0264043
$682$	0	0
$683$	− 43.9131i	− 1.68029i	−0.542365	−	0.840143i	$-0.682471\pi$
$683$	− 43.9131i	− 1.68029i	0.542365	−	0.840143i	$-0.317529\pi$
$684$	0	0
$685$	38.0238i	1.45281i
$686$	0	0
$687$	− 6.59724i	− 0.251700i
$688$	0	0
$689$	−3.33703	−0.127131
$690$	0	0
$691$	5.44313i	0.207066i	0.994626	+	0.103533i	$0.0330148\pi$
$691$	5.44313i	0.207066i	−0.994626	+	0.103533i	$0.966985\pi$
$692$	0	0
$693$	−29.2715	−1.11193
$694$	0	0
$695$	−64.4579	−2.44503
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−3.86951	−0.146358
$700$	0	0
$701$	17.0023	0.642166	0.321083	−	0.947051i	$-0.395953\pi$
$701$	17.0023	0.642166	0.321083	+	0.947051i	$0.395953\pi$
$702$	0	0
$703$	− 11.5827i	− 0.436851i
$704$	0	0
$705$	3.68283	0.138703
$706$	0	0
$707$	− 37.9569i	− 1.42752i
$708$	0	0
$709$	29.5109i	1.10830i	0.832415	+	0.554152i	$0.186957\pi$
$709$	29.5109i	1.10830i	−0.832415	+	0.554152i	$0.813043\pi$
$710$	0	0
$711$	39.8736i	1.49538i
$712$	0	0
$713$	−62.7754	−2.35096
$714$	0	0
$715$	14.4343	0.539811
$716$	0	0
$717$	− 3.42754i	− 0.128004i
$718$	0	0
$719$	1.70185i	0.0634685i	0.999496	+	0.0317342i	$0.0101030\pi$
$719$	1.70185i	0.0634685i	−0.999496	+	0.0317342i	$0.989897\pi$
$720$	0	0
$721$	32.2568i	1.20131i
$722$	0	0
$723$	2.59596	0.0965447
$724$	0	0
$725$	− 30.0480i	− 1.11596i
$726$	0	0
$727$	3.14595	0.116677	0.0583384	−	0.998297i	$-0.481420\pi$
$727$	3.14595	0.116677	0.0583384	+	0.998297i	$0.481420\pi$
$728$	0	0
$729$	24.0785	0.891798
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−32.0919	−1.18534	−0.592670	−	0.805445i	$-0.701926\pi$
$733$	−32.0919	−1.18534	−0.592670	+	0.805445i	$0.701926\pi$
$734$	0	0
$735$	−0.850034	−0.0313540
$736$	0	0
$737$	9.33686i	0.343928i
$738$	0	0
$739$	33.9003	1.24704	0.623521	−	0.781807i	$-0.285702\pi$
$739$	33.9003	1.24704	0.623521	+	0.781807i	$0.285702\pi$
$740$	0	0
$741$	0.930770i	0.0341927i
$742$	0	0
$743$	1.55976i	0.0572221i	0.999591	+	0.0286110i	$0.00910842\pi$
$743$	1.55976i	0.0572221i	−0.999591	+	0.0286110i	$0.990892\pi$
$744$	0	0
$745$	− 39.0265i	− 1.42982i
$746$	0	0
$747$	50.5994	1.85133
$748$	0	0
$749$	40.7585	1.48928
$750$	0	0
$751$	15.9140i	0.580710i	0.956919	+	0.290355i	$0.0937734\pi$
$751$	15.9140i	0.580710i	−0.956919	+	0.290355i	$0.906227\pi$
$752$	0	0
$753$	− 2.56905i	− 0.0936214i
$754$	0	0
$755$	− 22.1806i	− 0.807234i
$756$	0	0
$757$	2.62492	0.0954042	0.0477021	−	0.998862i	$-0.484810\pi$
$757$	2.62492	0.0954042	0.0477021	+	0.998862i	$0.484810\pi$
$758$	0	0
$759$	− 7.08513i	− 0.257174i
$760$	0	0
$761$	5.34842	0.193880	0.0969401	−	0.995290i	$-0.469094\pi$
$761$	5.34842	0.193880	0.0969401	+	0.995290i	$0.469094\pi$
$762$	0	0
$763$	−7.08919	−0.256646
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.641551	0.0231651
$768$	0	0
$769$	10.0160	0.361186	0.180593	−	0.983558i	$-0.442198\pi$
$769$	10.0160	0.361186	0.180593	+	0.983558i	$0.442198\pi$
$770$	0	0
$771$	− 0.467101i	− 0.0168222i
$772$	0	0
$773$	−9.92221	−0.356877	−0.178438	−	0.983951i	$-0.557105\pi$
$773$	−9.92221	−0.356877	−0.178438	+	0.983951i	$0.557105\pi$
$774$	0	0
$775$	60.0716i	2.15783i
$776$	0	0
$777$	− 1.70013i	− 0.0609920i
$778$	0	0
$779$	− 32.1322i	− 1.15125i
$780$	0	0
$781$	−30.9016	−1.10575
$782$	0	0
$783$	−5.89416	−0.210640
$784$	0	0
$785$	2.59398i	0.0925832i
$786$	0	0
$787$	− 39.3603i	− 1.40304i	−0.712649	−	0.701521i	$-0.752505\pi$
$787$	− 39.3603i	− 1.40304i	0.712649	−	0.701521i	$-0.247495\pi$
$788$	0	0
$789$	0.415589i	0.0147953i
$790$	0	0
$791$	41.0193	1.45848
$792$	0	0
$793$	− 4.86616i	− 0.172802i
$794$	0	0
$795$	2.67071	0.0947204
$796$	0	0
$797$	−17.3163	−0.613376	−0.306688	−	0.951810i	$-0.599221\pi$
$797$	−17.3163	−0.613376	−0.306688	+	0.951810i	$0.599221\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	45.5474	1.60934
$802$	0	0
$803$	−14.1981	−0.501042
$804$	0	0
$805$	− 63.0368i	− 2.22175i
$806$	0	0
$807$	−4.66339	−0.164159
$808$	0	0
$809$	− 13.0044i	− 0.457212i	−0.973519	−	0.228606i	$-0.926583\pi$
$809$	− 13.0044i	− 0.457212i	0.973519	−	0.228606i	$-0.0734167\pi$
$810$	0	0
$811$	46.8252i	1.64426i	0.569302	+	0.822128i	$0.307213\pi$
$811$	46.8252i	1.64426i	−0.569302	+	0.822128i	$0.692787\pi$
$812$	0	0
$813$	0.176357i	0.00618511i
$814$	0	0
$815$	1.40742	0.0492999
$816$	0	0
$817$	−24.1600	−0.845252
$818$	0	0
$819$	− 7.32402i	− 0.255922i
$820$	0	0
$821$	− 9.66168i	− 0.337195i	−0.985685	−	0.168598i	$-0.946076\pi$
$821$	− 9.66168i	− 0.337195i	0.985685	−	0.168598i	$-0.0539238\pi$
$822$	0	0
$823$	30.0975i	1.04913i	0.851370	+	0.524566i	$0.175772\pi$
$823$	30.0975i	1.04913i	−0.851370	+	0.524566i	$0.824228\pi$
$824$	0	0
$825$	−6.77996	−0.236048
$826$	0	0
$827$	22.8235i	0.793649i	0.917894	+	0.396825i	$0.129888\pi$
$827$	22.8235i	0.793649i	−0.917894	+	0.396825i	$0.870112\pi$
$828$	0	0
$829$	−17.8881	−0.621279	−0.310640	−	0.950528i	$-0.600543\pi$
$829$	−17.8881	−0.621279	−0.310640	+	0.950528i	$0.600543\pi$
$830$	0	0
$831$	3.93916	0.136648
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−13.7336	−0.475270
$836$	0	0
$837$	11.7835	0.407298
$838$	0	0
$839$	− 2.30008i	− 0.0794077i	−0.999211	−	0.0397039i	$-0.987359\pi$
$839$	− 2.30008i	− 0.0794077i	0.999211	−	0.0397039i	$-0.0126415\pi$
$840$	0	0
$841$	11.1075	0.383018
$842$	0	0
$843$	− 0.249150i	− 0.00858119i
$844$	0	0
$845$	− 41.6158i	− 1.43163i
$846$	0	0
$847$	13.6243i	0.468135i
$848$	0	0
$849$	4.25777	0.146126
$850$	0	0
$851$	−22.0607	−0.756233
$852$	0	0
$853$	− 4.26593i	− 0.146063i	−0.997330	−	0.0730313i	$-0.976733\pi$
$853$	− 4.26593i	− 0.146063i	0.997330	−	0.0730313i	$-0.0232673\pi$
$854$	0	0
$855$	39.9342i	1.36572i
$856$	0	0
$857$	− 8.19993i	− 0.280104i	−0.990144	−	0.140052i	$-0.955273\pi$
$857$	− 8.19993i	− 0.280104i	0.990144	−	0.140052i	$-0.0447270\pi$
$858$	0	0
$859$	−40.2295	−1.37261	−0.686307	−	0.727312i	$-0.740769\pi$
$859$	−40.2295	−1.37261	−0.686307	+	0.727312i	$0.740769\pi$
$860$	0	0
$861$	− 4.71642i	− 0.160735i
$862$	0	0
$863$	40.4898	1.37829	0.689145	−	0.724624i	$-0.257986\pi$
$863$	40.4898	1.37829	0.689145	+	0.724624i	$0.257986\pi$
$864$	0	0
$865$	27.5396	0.936375
$866$	0	0
$867$	0	0
$868$	0	0
$869$	55.1325	1.87024
$870$	0	0
$871$	−2.33617	−0.0791581
$872$	0	0
$873$	− 2.01527i	− 0.0682064i
$874$	0	0
$875$	−17.8634	−0.603892
$876$	0	0
$877$	5.16474i	0.174401i	0.996191	+	0.0872005i	$0.0277921\pi$
$877$	5.16474i	0.174401i	−0.996191	+	0.0872005i	$0.972208\pi$
$878$	0	0
$879$	1.51759i	0.0511871i
$880$	0	0
$881$	− 35.0344i	− 1.18034i	−0.807279	−	0.590170i	$-0.799061\pi$
$881$	− 35.0344i	− 1.18034i	0.807279	−	0.590170i	$-0.200939\pi$
$882$	0	0
$883$	42.5114	1.43062	0.715312	−	0.698805i	$-0.246285\pi$
$883$	42.5114	1.43062	0.715312	+	0.698805i	$0.246285\pi$
$884$	0	0
$885$	−0.513451	−0.0172595
$886$	0	0
$887$	− 14.7056i	− 0.493766i	−0.969045	−	0.246883i	$-0.920594\pi$
$887$	− 14.7056i	− 0.493766i	0.969045	−	0.246883i	$-0.0794064\pi$
$888$	0	0
$889$	− 36.8165i	− 1.23479i
$890$	0	0
$891$	− 34.6477i	− 1.16074i
$892$	0	0
$893$	−17.6030	−0.589060
$894$	0	0
$895$	43.3020i	1.44743i
$896$	0	0
$897$	1.77277	0.0591910
$898$	0	0
$899$	35.7704	1.19301
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−3.54625	−0.118012
$904$	0	0
$905$	26.3909	0.877264
$906$	0	0
$907$	− 4.44781i	− 0.147687i	−0.997270	−	0.0738435i	$-0.976473\pi$
$907$	− 4.44781i	− 0.147687i	0.997270	−	0.0738435i	$-0.0235265\pi$
$908$	0	0
$909$	45.7984	1.51904
$910$	0	0
$911$	8.51570i	0.282138i	0.990000	+	0.141069i	$0.0450539\pi$
$911$	8.51570i	0.282138i	−0.990000	+	0.141069i	$0.954946\pi$
$912$	0	0
$913$	− 69.9628i	− 2.31543i
$914$	0	0
$915$	3.89452i	0.128749i
$916$	0	0
$917$	0.727744	0.0240322
$918$	0	0
$919$	54.9607	1.81299	0.906493	−	0.422221i	$-0.138749\pi$
$919$	54.9607	1.81299	0.906493	+	0.422221i	$0.138749\pi$
$920$	0	0
$921$	− 3.46750i	− 0.114258i
$922$	0	0
$923$	− 7.73188i	− 0.254498i
$924$	0	0
$925$	21.1106i	0.694111i
$926$	0	0
$927$	−38.9208	−1.27833
$928$	0	0
$929$	26.4612i	0.868165i	0.900873	+	0.434083i	$0.142927\pi$
$929$	26.4612i	0.868165i	−0.900873	+	0.434083i	$0.857073\pi$
$930$	0	0
$931$	4.06294	0.133158
$932$	0	0
$933$	−2.24337	−0.0734447
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6.40806	0.209342	0.104671	−	0.994507i	$-0.466621\pi$
$937$	6.40806	0.209342	0.104671	+	0.994507i	$0.466621\pi$
$938$	0	0
$939$	−1.36768	−0.0446326
$940$	0	0
$941$	8.01704i	0.261348i	0.991425	+	0.130674i	$0.0417141\pi$
$941$	8.01704i	0.261348i	−0.991425	+	0.130674i	$0.958286\pi$
$942$	0	0
$943$	−61.1997	−1.99294
$944$	0	0
$945$	11.8326i	0.384913i
$946$	0	0
$947$	15.4062i	0.500633i	0.968164	+	0.250317i	$0.0805347\pi$
$947$	15.4062i	0.500633i	−0.968164	+	0.250317i	$0.919465\pi$
$948$	0	0
$949$	− 3.55251i	− 0.115319i
$950$	0	0
$951$	−5.46730	−0.177289
$952$	0	0
$953$	20.9794	0.679589	0.339795	−	0.940500i	$-0.389642\pi$
$953$	20.9794	0.679589	0.339795	+	0.940500i	$0.389642\pi$
$954$	0	0
$955$	46.4406i	1.50278i
$956$	0	0
$957$	4.03721i	0.130505i
$958$	0	0
$959$	− 26.6767i	− 0.861435i
$960$	0	0
$961$	−40.5115	−1.30682
$962$	0	0
$963$	49.1789i	1.58477i
$964$	0	0
$965$	45.4803	1.46406
$966$	0	0
$967$	30.2493	0.972752	0.486376	−	0.873750i	$-0.338319\pi$
$967$	30.2493	0.972752	0.486376	+	0.873750i	$0.338319\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−40.3805	−1.29587	−0.647936	−	0.761694i	$-0.724368\pi$
$971$	−40.3805	−1.29587	−0.647936	+	0.761694i	$0.724368\pi$
$972$	0	0
$973$	45.2223	1.44976
$974$	0	0
$975$	− 1.69641i	− 0.0543286i
$976$	0	0
$977$	−43.9012	−1.40452	−0.702261	−	0.711919i	$-0.747826\pi$
$977$	−43.9012	−1.40452	−0.702261	+	0.711919i	$0.747826\pi$
$978$	0	0
$979$	− 62.9774i	− 2.01277i
$980$	0	0
$981$	− 8.55376i	− 0.273101i
$982$	0	0
$983$	43.9128i	1.40060i	0.713849	+	0.700300i	$0.246950\pi$
$983$	43.9128i	1.40060i	−0.713849	+	0.700300i	$0.753050\pi$
$984$	0	0
$985$	−17.6437	−0.562173
$986$	0	0
$987$	−2.58379	−0.0822430
$988$	0	0
$989$	46.0157i	1.46322i
$990$	0	0
$991$	47.7621i	1.51721i	0.651548	+	0.758607i	$0.274120\pi$
$991$	47.7621i	1.51721i	−0.651548	+	0.758607i	$0.725880\pi$
$992$	0	0
$993$	7.40651i	0.235038i
$994$	0	0
$995$	30.3038	0.960695
$996$	0	0
$997$	− 1.51536i	− 0.0479918i	−0.999712	−	0.0239959i	$-0.992361\pi$
$997$	− 1.51536i	− 0.0479918i	0.999712	−	0.0239959i	$-0.00763887\pi$
$998$	0	0
$999$	4.14100	0.131015

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.b.n.577.7		12
17.4	even	4		2312.2.a.w.1.7		12
17.5	odd	16		136.2.n.c.9.2	✓	12
17.10	odd	16		136.2.n.c.121.2	yes	12
17.13	even	4		2312.2.a.w.1.6		12
17.16	even	2	inner	2312.2.b.n.577.6		12
51.5	even	16		1224.2.bq.c.145.1		12
51.44	even	16		1224.2.bq.c.937.1		12
68.27	even	16		272.2.v.f.257.2		12
68.39	even	16		272.2.v.f.145.2		12
68.47	odd	4		4624.2.a.bt.1.7		12
68.55	odd	4		4624.2.a.bt.1.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.n.c.9.2	✓	12	17.5	odd	16
136.2.n.c.121.2	yes	12	17.10	odd	16
272.2.v.f.145.2		12	68.39	even	16
272.2.v.f.257.2		12	68.27	even	16
1224.2.bq.c.145.1		12	51.5	even	16
1224.2.bq.c.937.1		12	51.44	even	16
2312.2.a.w.1.6		12	17.13	even	4
2312.2.a.w.1.7		12	17.4	even	4
2312.2.b.n.577.6		12	17.16	even	2	inner
2312.2.b.n.577.7		12	1.1	even	1	trivial
4624.2.a.bt.1.6		12	68.55	odd	4
4624.2.a.bt.1.7		12	68.47	odd	4

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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.234385i q^{3} +3.47903i q^{5} -2.44081i q^{7} +2.94506 q^{9} +O(q^{10})\)	\(q+0.234385i q^{3} +3.47903i q^{5} -2.44081i q^{7} +2.94506 q^{9} -4.07208i q^{11} +1.01887 q^{13} -0.815432 q^{15} +3.89756 q^{19} +0.572090 q^{21} -7.42338i q^{23} -7.10364 q^{25} +1.39343i q^{27} +4.22995i q^{29} -8.45645i q^{31} +0.954435 q^{33} +8.49166 q^{35} -2.97179i q^{37} +0.238809i q^{39} -8.24419i q^{41} -6.19876 q^{43} +10.2460i q^{45} -4.51641 q^{47} +1.04243 q^{49} -3.27521 q^{53} +14.1669 q^{55} +0.913529i q^{57} +0.629667 q^{59} -4.77602i q^{61} -7.18835i q^{63} +3.54469i q^{65} -2.29290 q^{67} +1.73993 q^{69} -7.58866i q^{71} -3.48670i q^{73} -1.66499i q^{75} -9.93919 q^{77} +13.5391i q^{79} +8.50859 q^{81} +17.1811 q^{83} -0.991437 q^{87} +15.4657 q^{89} -2.48688i q^{91} +1.98207 q^{93} +13.5597i q^{95} -0.684286i q^{97} -11.9925i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 28 q^{9}+O(q^{10}) \) 12 * q - 28 * q^9	\( 12 q - 28 q^{9} + 24 q^{13} - 8 q^{15} + 8 q^{19} + 16 q^{21} - 20 q^{25} + 24 q^{33} - 32 q^{35} - 8 q^{43} + 24 q^{47} - 36 q^{49} - 8 q^{53} + 56 q^{55} + 40 q^{59} + 40 q^{67} + 56 q^{69} - 80 q^{77} + 60 q^{81} + 24 q^{83} - 24 q^{87} + 48 q^{89} - 40 q^{93}+O(q^{100}) \) 12 * q - 28 * q^9 + 24 * q^13 - 8 * q^15 + 8 * q^19 + 16 * q^21 - 20 * q^25 + 24 * q^33 - 32 * q^35 - 8 * q^43 + 24 * q^47 - 36 * q^49 - 8 * q^53 + 56 * q^55 + 40 * q^59 + 40 * q^67 + 56 * q^69 - 80 * q^77 + 60 * q^81 + 24 * q^83 - 24 * q^87 + 48 * q^89 - 40 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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