LMFDB - Embedded newform 2312.2.a.w.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,2,Mod(1,2312)]

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, 0]))

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

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

chi := DirichletCharacter("2312.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.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:	$18.4614129473$
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)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.234385$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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
$3$	−0.234385	−0.135322	−0.0676611	−	0.997708i	$-0.521554\pi$
$3$	−0.234385	−0.135322	−0.0676611	+	0.997708i	$0.521554\pi$
$4$	0	0
$5$	−3.47903	−1.55587	−0.777934	−	0.628346i	$-0.783732\pi$
$5$	−3.47903	−1.55587	−0.777934	+	0.628346i	$0.783732\pi$
$6$	0	0
$7$	−2.44081	−0.922540	−0.461270	−	0.887260i	$-0.652606\pi$
$7$	−2.44081	−0.922540	−0.461270	+	0.887260i	$0.652606\pi$
$8$	0	0
$9$	−2.94506	−0.981688
$10$	0	0
$11$	−4.07208	−1.22778	−0.613889	−	0.789392i	$-0.710396\pi$
$11$	−4.07208	−1.22778	−0.613889	+	0.789392i	$0.710396\pi$
$12$	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	$-0.647536\pi$
$19$	−3.89756	−0.894161	−0.447080	+	0.894494i	$0.647536\pi$
$20$	0	0
$21$	0.572090	0.124840
$22$	0	0
$23$	−7.42338	−1.54788	−0.773941	−	0.633258i	$-0.781717\pi$
$23$	−7.42338	−1.54788	−0.773941	+	0.633258i	$0.781717\pi$
$24$	0	0
$25$	7.10364	1.42073
$26$	0	0
$27$	1.39343	0.268167
$28$	0	0
$29$	−4.22995	−0.785482	−0.392741	−	0.919649i	$-0.628473\pi$
$29$	−4.22995	−0.785482	−0.392741	+	0.919649i	$0.628473\pi$
$30$	0	0
$31$	8.45645	1.51882	0.759412	−	0.650610i	$-0.225487\pi$
$31$	8.45645	1.51882	0.759412	+	0.650610i	$0.225487\pi$
$32$	0	0
$33$	0.954435	0.166146
$34$	0	0
$35$	8.49166	1.43535
$36$	0	0
$37$	2.97179	0.488560	0.244280	−	0.969705i	$-0.421448\pi$
$37$	2.97179	0.488560	0.244280	+	0.969705i	$0.421448\pi$
$38$	0	0
$39$	−0.238809	−0.0382400
$40$	0	0
$41$	−8.24419	−1.28753	−0.643763	−	0.765225i	$-0.722628\pi$
$41$	−8.24419	−1.28753	−0.643763	+	0.765225i	$0.722628\pi$
$42$	0	0
$43$	6.19876	0.945302	0.472651	−	0.881250i	$-0.343297\pi$
$43$	6.19876	0.945302	0.472651	+	0.881250i	$0.343297\pi$
$44$	0	0
$45$	10.2460	1.52738
$46$	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	$-0.427781\pi$
$53$	3.27521	0.449885	0.224942	+	0.974372i	$0.427781\pi$
$54$	0	0
$55$	14.1669	1.91026
$56$	0	0
$57$	0.913529	0.121000
$58$	0	0
$59$	−0.629667	−0.0819757	−0.0409878	−	0.999160i	$-0.513050\pi$
$59$	−0.629667	−0.0819757	−0.0409878	+	0.999160i	$0.513050\pi$
$60$	0	0
$61$	−4.77602	−0.611507	−0.305753	−	0.952111i	$-0.598908\pi$
$61$	−4.77602	−0.611507	−0.305753	+	0.952111i	$0.598908\pi$
$62$	0	0
$63$	7.18835	0.905647
$64$	0	0
$65$	−3.54469	−0.439665
$66$	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
$71$	7.58866	0.900608	0.450304	−	0.892875i	$-0.351316\pi$
$71$	7.58866	0.900608	0.450304	+	0.892875i	$0.351316\pi$
$72$	0	0
$73$	3.48670	0.408088	0.204044	−	0.978962i	$-0.434591\pi$
$73$	3.48670	0.408088	0.204044	+	0.978962i	$0.434591\pi$
$74$	0	0
$75$	−1.66499	−0.192256
$76$	0	0
$77$	9.93919	1.13268
$78$	0	0
$79$	13.5391	1.52327	0.761636	−	0.648005i	$-0.224396\pi$
$79$	13.5391	1.52327	0.761636	+	0.648005i	$0.224396\pi$
$80$	0	0
$81$	8.50859	0.945399
$82$	0	0
$83$	−17.1811	−1.88587	−0.942934	−	0.332979i	$-0.891946\pi$
$83$	−17.1811	−1.88587	−0.942934	+	0.332979i	$0.891946\pi$
$84$	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
$91$	−2.48688	−0.260696
$92$	0	0
$93$	−1.98207	−0.205531
$94$	0	0
$95$	13.5597	1.39120
$96$	0	0
$97$	0.684286	0.0694787	0.0347394	−	0.999396i	$-0.488940\pi$
$97$	0.684286	0.0694787	0.0347394	+	0.999396i	$0.488940\pi$
$98$	0	0
$99$	11.9925	1.20530
$100$	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
$105$	−1.99032	−0.194235
$106$	0	0
$107$	−16.6987	−1.61433	−0.807164	−	0.590327i	$-0.798999\pi$
$107$	−16.6987	−1.61433	−0.807164	+	0.590327i	$0.798999\pi$
$108$	0	0
$109$	−2.90444	−0.278195	−0.139097	−	0.990279i	$-0.544420\pi$
$109$	−2.90444	−0.278195	−0.139097	+	0.990279i	$0.544420\pi$
$110$	0	0
$111$	−0.696544	−0.0661131
$112$	0	0
$113$	16.8056	1.58094	0.790469	−	0.612502i	$-0.209837\pi$
$113$	16.8056	1.58094	0.790469	+	0.612502i	$0.209837\pi$
$114$	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
$125$	−7.31862	−0.654597
$126$	0	0
$127$	−15.0837	−1.33846	−0.669231	−	0.743054i	$-0.733377\pi$
$127$	−15.0837	−1.33846	−0.669231	+	0.743054i	$0.733377\pi$
$128$	0	0
$129$	−1.45290	−0.127920
$130$	0	0
$131$	−0.298156	−0.0260500	−0.0130250	−	0.999915i	$-0.504146\pi$
$131$	−0.298156	−0.0260500	−0.0130250	+	0.999915i	$0.504146\pi$
$132$	0	0
$133$	9.51320	0.824899
$134$	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
$139$	−18.5276	−1.57149	−0.785744	−	0.618552i	$-0.787720\pi$
$139$	−18.5276	−1.57149	−0.785744	+	0.618552i	$0.787720\pi$
$140$	0	0
$141$	1.05858	0.0891484
$142$	0	0
$143$	−4.14894	−0.346952
$144$	0	0
$145$	14.7161	1.22211
$146$	0	0
$147$	0.244331	0.0201521
$148$	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	$-0.416470\pi$
$151$	6.37551	0.518832	0.259416	+	0.965766i	$0.416470\pi$
$152$	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
$159$	−0.767660	−0.0608794
$160$	0	0
$161$	18.1191	1.42798
$162$	0	0
$163$	−0.404545	−0.0316864	−0.0158432	−	0.999874i	$-0.505043\pi$
$163$	−0.404545	−0.0316864	−0.0158432	+	0.999874i	$0.505043\pi$
$164$	0	0
$165$	−3.32051	−0.258501
$166$	0	0
$167$	−3.94754	−0.305469	−0.152735	−	0.988267i	$-0.548808\pi$
$167$	−3.94754	−0.305469	−0.152735	+	0.988267i	$0.548808\pi$
$168$	0	0
$169$	−11.9619	−0.920146
$170$	0	0
$171$	11.4786	0.877787
$172$	0	0
$173$	7.91589	0.601834	0.300917	−	0.953650i	$-0.402707\pi$
$173$	7.91589	0.601834	0.300917	+	0.953650i	$0.402707\pi$
$174$	0	0
$175$	−17.3386	−1.31068
$176$	0	0
$177$	0.147585	0.0110931
$178$	0	0
$179$	−12.4466	−0.930301	−0.465151	−	0.885232i	$-0.654000\pi$
$179$	−12.4466	−0.930301	−0.465151	+	0.885232i	$0.654000\pi$
$180$	0	0
$181$	−7.58572	−0.563842	−0.281921	−	0.959438i	$-0.590972\pi$
$181$	−7.58572	−0.563842	−0.281921	+	0.959438i	$0.590972\pi$
$182$	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
$193$	−13.0727	−0.940994	−0.470497	−	0.882401i	$-0.655925\pi$
$193$	−13.0727	−0.940994	−0.470497	+	0.882401i	$0.655925\pi$
$194$	0	0
$195$	0.830822	0.0594964
$196$	0	0
$197$	5.07143	0.361324	0.180662	−	0.983545i	$-0.442176\pi$
$197$	5.07143	0.361324	0.180662	+	0.983545i	$0.442176\pi$
$198$	0	0
$199$	8.71042	0.617465	0.308733	−	0.951149i	$-0.400095\pi$
$199$	8.71042	0.617465	0.308733	+	0.951149i	$0.400095\pi$
$200$	0	0
$201$	0.537421	0.0379067
$202$	0	0
$203$	10.3245	0.724639
$204$	0	0
$205$	28.6818	2.00322
$206$	0	0
$207$	21.8623	1.51954
$208$	0	0
$209$	15.8712	1.09783
$210$	0	0
$211$	−3.59337	−0.247378	−0.123689	−	0.992321i	$-0.539472\pi$
$211$	−3.59337	−0.247378	−0.123689	+	0.992321i	$0.539472\pi$
$212$	0	0
$213$	−1.77867	−0.121872
$214$	0	0
$215$	−21.5657	−1.47077
$216$	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	$-0.280100\pi$
$223$	19.0303	1.27437	0.637183	+	0.770713i	$0.280100\pi$
$224$	0	0
$225$	−20.9207	−1.39471
$226$	0	0
$227$	−2.93980	−0.195121	−0.0975607	−	0.995230i	$-0.531104\pi$
$227$	−2.93980	−0.195121	−0.0975607	+	0.995230i	$0.531104\pi$
$228$	0	0
$229$	28.1470	1.86001	0.930004	−	0.367550i	$-0.119803\pi$
$229$	28.1470	1.86001	0.930004	+	0.367550i	$0.119803\pi$
$230$	0	0
$231$	−2.32960	−0.153276
$232$	0	0
$233$	−16.5092	−1.08155	−0.540777	−	0.841166i	$-0.681870\pi$
$233$	−16.5092	−1.08155	−0.540777	+	0.841166i	$0.681870\pi$
$234$	0	0
$235$	15.7127	1.02498
$236$	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
$241$	11.0756	0.713443	0.356721	−	0.934211i	$-0.383895\pi$
$241$	11.0756	0.713443	0.356721	+	0.934211i	$0.383895\pi$
$242$	0	0
$243$	−6.17459	−0.396100
$244$	0	0
$245$	3.62666	0.231699
$246$	0	0
$247$	−3.97112	−0.252676
$248$	0	0
$249$	4.02699	0.255200
$250$	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	$-0.480202\pi$
$257$	1.99288	0.124312	0.0621562	+	0.998066i	$0.480202\pi$
$258$	0	0
$259$	−7.25359	−0.450716
$260$	0	0
$261$	12.4575	0.771098
$262$	0	0
$263$	−1.77310	−0.109334	−0.0546671	−	0.998505i	$-0.517410\pi$
$263$	−1.77310	−0.109334	−0.0546671	+	0.998505i	$0.517410\pi$
$264$	0	0
$265$	−11.3946	−0.699962
$266$	0	0
$267$	−3.62492	−0.221842
$268$	0	0
$269$	−19.8963	−1.21310	−0.606548	−	0.795047i	$-0.707446\pi$
$269$	−19.8963	−1.21310	−0.606548	+	0.795047i	$0.707446\pi$
$270$	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
$275$	−28.9266	−1.74434
$276$	0	0
$277$	16.8064	1.00980	0.504899	−	0.863179i	$-0.331530\pi$
$277$	16.8064	1.00980	0.504899	+	0.863179i	$0.331530\pi$
$278$	0	0
$279$	−24.9048	−1.49101
$280$	0	0
$281$	1.06300	0.0634130	0.0317065	−	0.999497i	$-0.489906\pi$
$281$	1.06300	0.0634130	0.0317065	+	0.999497i	$0.489906\pi$
$282$	0	0
$283$	−18.1657	−1.07984	−0.539920	−	0.841717i	$-0.681545\pi$
$283$	−18.1657	−1.07984	−0.539920	+	0.841717i	$0.681545\pi$
$284$	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
$295$	2.19063	0.127543
$296$	0	0
$297$	−5.67418	−0.329249
$298$	0	0
$299$	−7.56348	−0.437407
$300$	0	0
$301$	−15.1300	−0.872079
$302$	0	0
$303$	−3.64490	−0.209394
$304$	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
$309$	3.09754	0.176213
$310$	0	0
$311$	−9.57132	−0.542739	−0.271370	−	0.962475i	$-0.587477\pi$
$311$	−9.57132	−0.542739	−0.271370	+	0.962475i	$0.587477\pi$
$312$	0	0
$313$	5.83519	0.329824	0.164912	−	0.986308i	$-0.447266\pi$
$313$	5.83519	0.329824	0.164912	+	0.986308i	$0.447266\pi$
$314$	0	0
$315$	−25.0085	−1.40907
$316$	0	0
$317$	23.3261	1.31013	0.655063	−	0.755574i	$-0.272642\pi$
$317$	23.3261	1.31013	0.655063	+	0.755574i	$0.272642\pi$
$318$	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
$329$	11.0237	0.607757
$330$	0	0
$331$	−31.5997	−1.73688	−0.868439	−	0.495796i	$-0.834876\pi$
$331$	−31.5997	−1.73688	−0.868439	+	0.495796i	$0.834876\pi$
$332$	0	0
$333$	−8.75212	−0.479613
$334$	0	0
$335$	7.97705	0.435833
$336$	0	0
$337$	0.430314	0.0234407	0.0117203	−	0.999931i	$-0.496269\pi$
$337$	0.430314	0.0234407	0.0117203	+	0.999931i	$0.496269\pi$
$338$	0	0
$339$	−3.93898	−0.213936
$340$	0	0
$341$	−34.4354	−1.86478
$342$	0	0
$343$	19.6301	1.05992
$344$	0	0
$345$	−6.05326	−0.325897
$346$	0	0
$347$	16.0827	0.863366	0.431683	−	0.902025i	$-0.357920\pi$
$347$	16.0827	0.863366	0.431683	+	0.902025i	$0.357920\pi$
$348$	0	0
$349$	−12.9133	−0.691235	−0.345617	−	0.938376i	$-0.612330\pi$
$349$	−12.9133	−0.691235	−0.345617	+	0.938376i	$0.612330\pi$
$350$	0	0
$351$	1.41973	0.0757798
$352$	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	$-0.470965\pi$
$359$	3.45185	0.182182	0.0910909	+	0.995843i	$0.470965\pi$
$360$	0	0
$361$	−3.80906	−0.200477
$362$	0	0
$363$	−1.30830	−0.0686681
$364$	0	0
$365$	−12.1303	−0.634931
$366$	0	0
$367$	22.8839	1.19453	0.597266	−	0.802043i	$-0.296254\pi$
$367$	22.8839	1.19453	0.597266	+	0.802043i	$0.296254\pi$
$368$	0	0
$369$	24.2797	1.26395
$370$	0	0
$371$	−7.99418	−0.415037
$372$	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
$377$	−4.30978	−0.221965
$378$	0	0
$379$	31.8894	1.63805	0.819024	−	0.573759i	$-0.194515\pi$
$379$	31.8894	1.63805	0.819024	+	0.573759i	$0.194515\pi$
$380$	0	0
$381$	3.53540	0.181124
$382$	0	0
$383$	16.1848	0.827003	0.413501	−	0.910503i	$-0.364306\pi$
$383$	16.1848	0.827003	0.413501	+	0.910503i	$0.364306\pi$
$384$	0	0
$385$	−34.5787	−1.76229
$386$	0	0
$387$	−18.2557	−0.927992
$388$	0	0
$389$	−37.3359	−1.89301	−0.946503	−	0.322695i	$-0.895411\pi$
$389$	−37.3359	−1.89301	−0.946503	+	0.322695i	$0.895411\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0.0698834	0.00352515
$394$	0	0
$395$	−47.1031	−2.37001
$396$	0	0
$397$	27.3879	1.37456	0.687280	−	0.726393i	$-0.258805\pi$
$397$	27.3879	1.37456	0.687280	+	0.726393i	$0.258805\pi$
$398$	0	0
$399$	−2.22975	−0.111627
$400$	0	0
$401$	−5.02491	−0.250932	−0.125466	−	0.992098i	$-0.540043\pi$
$401$	−5.02491	−0.250932	−0.125466	+	0.992098i	$0.540043\pi$
$402$	0	0
$403$	8.61605	0.429196
$404$	0	0
$405$	−29.6016	−1.47092
$406$	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
$411$	−2.56169	−0.126359
$412$	0	0
$413$	1.53690	0.0756259
$414$	0	0
$415$	59.7735	2.93416
$416$	0	0
$417$	4.34258	0.212657
$418$	0	0
$419$	1.10530	0.0539973	0.0269986	−	0.999635i	$-0.491405\pi$
$419$	1.10530	0.0539973	0.0269986	+	0.999635i	$0.491405\pi$
$420$	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
$431$	−0.891134	−0.0429244	−0.0214622	−	0.999770i	$-0.506832\pi$
$431$	−0.891134	−0.0429244	−0.0214622	+	0.999770i	$0.506832\pi$
$432$	0	0
$433$	−2.21402	−0.106399	−0.0531994	−	0.998584i	$-0.516942\pi$
$433$	−2.21402	−0.106399	−0.0531994	+	0.998584i	$0.516942\pi$
$434$	0	0
$435$	−3.44924	−0.165378
$436$	0	0
$437$	28.9330	1.38405
$438$	0	0
$439$	11.3608	0.542222	0.271111	−	0.962548i	$-0.412609\pi$
$439$	11.3608	0.542222	0.271111	+	0.962548i	$0.412609\pi$
$440$	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
$445$	−53.8055	−2.55062
$446$	0	0
$447$	2.62925	0.124359
$448$	0	0
$449$	6.70480	0.316419	0.158210	−	0.987406i	$-0.449428\pi$
$449$	6.70480	0.316419	0.158210	+	0.987406i	$0.449428\pi$
$450$	0	0
$451$	33.5710	1.58080
$452$	0	0
$453$	−1.49432	−0.0702095
$454$	0	0
$455$	8.65192	0.405608
$456$	0	0
$457$	28.3215	1.32482	0.662412	−	0.749140i	$-0.269533\pi$
$457$	28.3215	1.32482	0.662412	+	0.749140i	$0.269533\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.77507	0.315547	0.157773	−	0.987475i	$-0.449568\pi$
$461$	6.77507	0.315547	0.157773	+	0.987475i	$0.449568\pi$
$462$	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
$465$	6.89566	0.319779
$466$	0	0
$467$	−25.8709	−1.19716	−0.598582	−	0.801061i	$-0.704269\pi$
$467$	−25.8709	−1.19716	−0.598582	+	0.801061i	$0.704269\pi$
$468$	0	0
$469$	5.59653	0.258424
$470$	0	0
$471$	−0.174759	−0.00805246
$472$	0	0
$473$	−25.2419	−1.16062
$474$	0	0
$475$	−27.6868	−1.27036
$476$	0	0
$477$	−9.64570	−0.441646
$478$	0	0
$479$	−6.96996	−0.318466	−0.159233	−	0.987241i	$-0.550902\pi$
$479$	−6.96996	−0.318466	−0.159233	+	0.987241i	$0.550902\pi$
$480$	0	0
$481$	3.02788	0.138060
$482$	0	0
$483$	−4.24684	−0.193238
$484$	0	0
$485$	−2.38065	−0.108100
$486$	0	0
$487$	−1.17446	−0.0532201	−0.0266100	−	0.999646i	$-0.508471\pi$
$487$	−1.17446	−0.0532201	−0.0266100	+	0.999646i	$0.508471\pi$
$488$	0	0
$489$	0.0948192	0.00428787
$490$	0	0
$491$	−7.16315	−0.323269	−0.161634	−	0.986851i	$-0.551676\pi$
$491$	−7.16315	−0.323269	−0.161634	+	0.986851i	$0.551676\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−41.7224	−1.87528
$496$	0	0
$497$	−18.5225	−0.830847
$498$	0	0
$499$	25.9621	1.16222	0.581112	−	0.813823i	$-0.302618\pi$
$499$	25.9621	1.16222	0.581112	+	0.813823i	$0.302618\pi$
$500$	0	0
$501$	0.925243	0.0413368
$502$	0	0
$503$	1.17649	0.0524569	0.0262285	−	0.999656i	$-0.491650\pi$
$503$	1.17649	0.0524569	0.0262285	+	0.999656i	$0.491650\pi$
$504$	0	0
$505$	−54.1021	−2.40751
$506$	0	0
$507$	2.80369	0.124516
$508$	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
$513$	−5.43099	−0.239784
$514$	0	0
$515$	45.9774	2.02601
$516$	0	0
$517$	18.3912	0.808843
$518$	0	0
$519$	−1.85537	−0.0814416
$520$	0	0
$521$	11.1308	0.487650	0.243825	−	0.969819i	$-0.421598\pi$
$521$	11.1308	0.487650	0.243825	+	0.969819i	$0.421598\pi$
$522$	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
$533$	−8.39978	−0.363835
$534$	0	0
$535$	58.0954	2.51168
$536$	0	0
$537$	2.91729	0.125890
$538$	0	0
$539$	4.24488	0.182840
$540$	0	0
$541$	−7.59601	−0.326578	−0.163289	−	0.986578i	$-0.552210\pi$
$541$	−7.59601	−0.326578	−0.163289	+	0.986578i	$0.552210\pi$
$542$	0	0
$543$	1.77798	0.0763004
$544$	0	0
$545$	10.1046	0.432835
$546$	0	0
$547$	−18.5334	−0.792433	−0.396216	−	0.918157i	$-0.629677\pi$
$547$	−18.5334	−0.792433	−0.396216	+	0.918157i	$0.629677\pi$
$548$	0	0
$549$	14.0657	0.600309
$550$	0	0
$551$	16.4865	0.702347
$552$	0	0
$553$	−33.0465	−1.40528
$554$	0	0
$555$	2.42330	0.102863
$556$	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	$-0.469911\pi$
$563$	4.47916	0.188774	0.0943871	+	0.995536i	$0.469911\pi$
$564$	0	0
$565$	−58.4671	−2.45973
$566$	0	0
$567$	−20.7679	−0.872169
$568$	0	0
$569$	−42.2877	−1.77279	−0.886395	−	0.462929i	$-0.846798\pi$
$569$	−42.2877	−1.77279	−0.886395	+	0.462929i	$0.846798\pi$
$570$	0	0
$571$	−12.5555	−0.525430	−0.262715	−	0.964873i	$-0.584618\pi$
$571$	−12.5555	−0.525430	−0.262715	+	0.964873i	$0.584618\pi$
$572$	0	0
$573$	−3.12874	−0.130705
$574$	0	0
$575$	−52.7330	−2.19912
$576$	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
$581$	41.9358	1.73979
$582$	0	0
$583$	−13.3369	−0.552359
$584$	0	0
$585$	10.4393	0.431613
$586$	0	0
$587$	−34.1613	−1.40999	−0.704993	−	0.709214i	$-0.749050\pi$
$587$	−34.1613	−1.40999	−0.704993	+	0.709214i	$0.749050\pi$
$588$	0	0
$589$	−32.9595	−1.35807
$590$	0	0
$591$	−1.18867	−0.0488953
$592$	0	0
$593$	19.5955	0.804690	0.402345	−	0.915488i	$-0.368195\pi$
$593$	19.5955	0.804690	0.402345	+	0.915488i	$0.368195\pi$
$594$	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
$601$	8.28073	0.337778	0.168889	−	0.985635i	$-0.445982\pi$
$601$	8.28073	0.337778	0.168889	+	0.985635i	$0.445982\pi$
$602$	0	0
$603$	6.75272	0.274992
$604$	0	0
$605$	−19.4194	−0.789512
$606$	0	0
$607$	−20.4124	−0.828515	−0.414257	−	0.910160i	$-0.635959\pi$
$607$	−20.4124	−0.828515	−0.414257	+	0.910160i	$0.635959\pi$
$608$	0	0
$609$	−2.41991	−0.0980598
$610$	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
$615$	−6.72258	−0.271080
$616$	0	0
$617$	−20.3962	−0.821120	−0.410560	−	0.911834i	$-0.634667\pi$
$617$	−20.3962	−0.821120	−0.410560	+	0.911834i	$0.634667\pi$
$618$	0	0
$619$	−15.1158	−0.607557	−0.303779	−	0.952743i	$-0.598248\pi$
$619$	−15.1158	−0.607557	−0.303779	+	0.952743i	$0.598248\pi$
$620$	0	0
$621$	−10.3440	−0.415090
$622$	0	0
$623$	−37.7488	−1.51237
$624$	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	$-0.283612\pi$
$631$	31.5824	1.25728	0.628639	+	0.777698i	$0.283612\pi$
$632$	0	0
$633$	0.842231	0.0334757
$634$	0	0
$635$	52.4767	2.08247
$636$	0	0
$637$	−1.06211	−0.0420823
$638$	0	0
$639$	−22.3491	−0.884116
$640$	0	0
$641$	40.4674	1.59837	0.799183	−	0.601088i	$-0.205266\pi$
$641$	40.4674	1.59837	0.799183	+	0.601088i	$0.205266\pi$
$642$	0	0
$643$	4.48760	0.176974	0.0884868	−	0.996077i	$-0.471797\pi$
$643$	4.48760	0.176974	0.0884868	+	0.996077i	$0.471797\pi$
$644$	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
$649$	2.56406	0.100648
$650$	0	0
$651$	4.83785	0.189610
$652$	0	0
$653$	−42.9742	−1.68171	−0.840855	−	0.541260i	$-0.817947\pi$
$653$	−42.9742	−1.68171	−0.840855	+	0.541260i	$0.817947\pi$
$654$	0	0
$655$	1.03729	0.0405305
$656$	0	0
$657$	−10.2686	−0.400615
$658$	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	$-0.496981\pi$
$661$	0.487631	0.0189666	0.00948332	+	0.999955i	$0.496981\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−33.0967	−1.28344
$666$	0	0
$667$	31.4005	1.21583
$668$	0	0
$669$	−4.46042	−0.172450
$670$	0	0
$671$	19.4483	0.750795
$672$	0	0
$673$	30.1670	1.16285	0.581426	−	0.813599i	$-0.302495\pi$
$673$	30.1670	1.16285	0.581426	+	0.813599i	$0.302495\pi$
$674$	0	0
$675$	9.89845	0.380992
$676$	0	0
$677$	44.5961	1.71397	0.856983	−	0.515344i	$-0.172336\pi$
$677$	44.5961	1.71397	0.856983	+	0.515344i	$0.172336\pi$
$678$	0	0
$679$	−1.67021	−0.0640970
$680$	0	0
$681$	0.689045	0.0264043
$682$	0	0
$683$	43.9131	1.68029	0.840143	−	0.542365i	$-0.182471\pi$
$683$	43.9131	1.68029	0.840143	+	0.542365i	$0.182471\pi$
$684$	0	0
$685$	−38.0238	−1.45281
$686$	0	0
$687$	−6.59724	−0.251700
$688$	0	0
$689$	3.33703	0.127131
$690$	0	0
$691$	5.44313	0.207066	0.103533	−	0.994626i	$-0.466985\pi$
$691$	5.44313	0.207066	0.103533	+	0.994626i	$0.466985\pi$
$692$	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
$703$	−11.5827	−0.436851
$704$	0	0
$705$	−3.68283	−0.138703
$706$	0	0
$707$	−37.9569	−1.42752
$708$	0	0
$709$	−29.5109	−1.10830	−0.554152	−	0.832415i	$-0.686957\pi$
$709$	−29.5109	−1.10830	−0.554152	+	0.832415i	$0.686957\pi$
$710$	0	0
$711$	−39.8736	−1.49538
$712$	0	0
$713$	−62.7754	−2.35096
$714$	0	0
$715$	14.4343	0.539811
$716$	0	0
$717$	3.42754	0.128004
$718$	0	0
$719$	−1.70185	−0.0634685	−0.0317342	−	0.999496i	$-0.510103\pi$
$719$	−1.70185	−0.0634685	−0.0317342	+	0.999496i	$0.510103\pi$
$720$	0	0
$721$	32.2568	1.20131
$722$	0	0
$723$	−2.59596	−0.0965447
$724$	0	0
$725$	−30.0480	−1.11596
$726$	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	$-0.298074\pi$
$733$	32.0919	1.18534	0.592670	+	0.805445i	$0.298074\pi$
$734$	0	0
$735$	−0.850034	−0.0313540
$736$	0	0
$737$	9.33686	0.343928
$738$	0	0
$739$	−33.9003	−1.24704	−0.623521	−	0.781807i	$-0.714298\pi$
$739$	−33.9003	−1.24704	−0.623521	+	0.781807i	$0.714298\pi$
$740$	0	0
$741$	0.930770	0.0341927
$742$	0	0
$743$	−1.55976	−0.0572221	−0.0286110	−	0.999591i	$-0.509108\pi$
$743$	−1.55976	−0.0572221	−0.0286110	+	0.999591i	$0.509108\pi$
$744$	0	0
$745$	39.0265	1.42982
$746$	0	0
$747$	50.5994	1.85133
$748$	0	0
$749$	40.7585	1.48928
$750$	0	0
$751$	−15.9140	−0.580710	−0.290355	−	0.956919i	$-0.593773\pi$
$751$	−15.9140	−0.580710	−0.290355	+	0.956919i	$0.593773\pi$
$752$	0	0
$753$	2.56905	0.0936214
$754$	0	0
$755$	−22.1806	−0.807234
$756$	0	0
$757$	−2.62492	−0.0954042	−0.0477021	−	0.998862i	$-0.515190\pi$
$757$	−2.62492	−0.0954042	−0.0477021	+	0.998862i	$0.515190\pi$
$758$	0	0
$759$	−7.08513	−0.257174
$760$	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
$771$	−0.467101	−0.0168222
$772$	0	0
$773$	9.92221	0.356877	0.178438	−	0.983951i	$-0.442895\pi$
$773$	9.92221	0.356877	0.178438	+	0.983951i	$0.442895\pi$
$774$	0	0
$775$	60.0716	2.15783
$776$	0	0
$777$	1.70013	0.0609920
$778$	0	0
$779$	32.1322	1.15125
$780$	0	0
$781$	−30.9016	−1.10575
$782$	0	0
$783$	−5.89416	−0.210640
$784$	0	0
$785$	−2.59398	−0.0925832
$786$	0	0
$787$	39.3603	1.40304	0.701521	−	0.712649i	$-0.252505\pi$
$787$	39.3603	1.40304	0.701521	+	0.712649i	$0.252505\pi$
$788$	0	0
$789$	0.415589	0.0147953
$790$	0	0
$791$	−41.0193	−1.45848
$792$	0	0
$793$	−4.86616	−0.172802
$794$	0	0
$795$	2.67071	0.0947204
$796$	0	0
$797$	17.3163	0.613376	0.306688	−	0.951810i	$-0.400779\pi$
$797$	17.3163	0.613376	0.306688	+	0.951810i	$0.400779\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−45.5474	−1.60934
$802$	0	0
$803$	−14.1981	−0.501042
$804$	0	0
$805$	−63.0368	−2.22175
$806$	0	0
$807$	4.66339	0.164159
$808$	0	0
$809$	−13.0044	−0.457212	−0.228606	−	0.973519i	$-0.573417\pi$
$809$	−13.0044	−0.457212	−0.228606	+	0.973519i	$0.573417\pi$
$810$	0	0
$811$	−46.8252	−1.64426	−0.822128	−	0.569302i	$-0.807213\pi$
$811$	−46.8252	−1.64426	−0.822128	+	0.569302i	$0.807213\pi$
$812$	0	0
$813$	−0.176357	−0.00618511
$814$	0	0
$815$	1.40742	0.0492999
$816$	0	0
$817$	−24.1600	−0.845252
$818$	0	0
$819$	7.32402	0.255922
$820$	0	0
$821$	9.66168	0.337195	0.168598	−	0.985685i	$-0.446076\pi$
$821$	9.66168	0.337195	0.168598	+	0.985685i	$0.446076\pi$
$822$	0	0
$823$	30.0975	1.04913	0.524566	−	0.851370i	$-0.324228\pi$
$823$	30.0975	1.04913	0.524566	+	0.851370i	$0.324228\pi$
$824$	0	0
$825$	6.77996	0.236048
$826$	0	0
$827$	22.8235	0.793649	0.396825	−	0.917894i	$-0.370112\pi$
$827$	22.8235	0.793649	0.396825	+	0.917894i	$0.370112\pi$
$828$	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
$839$	−2.30008	−0.0794077	−0.0397039	−	0.999211i	$-0.512641\pi$
$839$	−2.30008	−0.0794077	−0.0397039	+	0.999211i	$0.512641\pi$
$840$	0	0
$841$	−11.1075	−0.383018
$842$	0	0
$843$	−0.249150	−0.00858119
$844$	0	0
$845$	41.6158	1.43163
$846$	0	0
$847$	−13.6243	−0.468135
$848$	0	0
$849$	4.25777	0.146126
$850$	0	0
$851$	−22.0607	−0.756233
$852$	0	0
$853$	4.26593	0.146063	0.0730313	−	0.997330i	$-0.476733\pi$
$853$	4.26593	0.146063	0.0730313	+	0.997330i	$0.476733\pi$
$854$	0	0
$855$	−39.9342	−1.36572
$856$	0	0
$857$	−8.19993	−0.280104	−0.140052	−	0.990144i	$-0.544727\pi$
$857$	−8.19993	−0.280104	−0.140052	+	0.990144i	$0.544727\pi$
$858$	0	0
$859$	40.2295	1.37261	0.686307	−	0.727312i	$-0.259231\pi$
$859$	40.2295	1.37261	0.686307	+	0.727312i	$0.259231\pi$
$860$	0	0
$861$	−4.71642	−0.160735
$862$	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
$873$	−2.01527	−0.0682064
$874$	0	0
$875$	17.8634	0.603892
$876$	0	0
$877$	5.16474	0.174401	0.0872005	−	0.996191i	$-0.472208\pi$
$877$	5.16474	0.174401	0.0872005	+	0.996191i	$0.472208\pi$
$878$	0	0
$879$	−1.51759	−0.0511871
$880$	0	0
$881$	35.0344	1.18034	0.590170	−	0.807279i	$-0.299061\pi$
$881$	35.0344	1.18034	0.590170	+	0.807279i	$0.299061\pi$
$882$	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
$887$	14.7056	0.493766	0.246883	−	0.969045i	$-0.420594\pi$
$887$	14.7056	0.493766	0.246883	+	0.969045i	$0.420594\pi$
$888$	0	0
$889$	36.8165	1.23479
$890$	0	0
$891$	−34.6477	−1.16074
$892$	0	0
$893$	17.6030	0.589060
$894$	0	0
$895$	43.3020	1.44743
$896$	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
$907$	−4.44781	−0.147687	−0.0738435	−	0.997270i	$-0.523527\pi$
$907$	−4.44781	−0.147687	−0.0738435	+	0.997270i	$0.523527\pi$
$908$	0	0
$909$	−45.7984	−1.51904
$910$	0	0
$911$	8.51570	0.282138	0.141069	−	0.990000i	$-0.454946\pi$
$911$	8.51570	0.282138	0.141069	+	0.990000i	$0.454946\pi$
$912$	0	0
$913$	69.9628	2.31543
$914$	0	0
$915$	−3.89452	−0.128749
$916$	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
$921$	3.46750	0.114258
$922$	0	0
$923$	7.73188	0.254498
$924$	0	0
$925$	21.1106	0.694111
$926$	0	0
$927$	38.9208	1.27833
$928$	0	0
$929$	26.4612	0.868165	0.434083	−	0.900873i	$-0.357073\pi$
$929$	26.4612	0.868165	0.434083	+	0.900873i	$0.357073\pi$
$930$	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	$-0.533379\pi$
$937$	−6.40806	−0.209342	−0.104671	+	0.994507i	$0.533379\pi$
$938$	0	0
$939$	−1.36768	−0.0446326
$940$	0	0
$941$	8.01704	0.261348	0.130674	−	0.991425i	$-0.458286\pi$
$941$	8.01704	0.261348	0.130674	+	0.991425i	$0.458286\pi$
$942$	0	0
$943$	61.1997	1.99294
$944$	0	0
$945$	11.8326	0.384913
$946$	0	0
$947$	−15.4062	−0.500633	−0.250317	−	0.968164i	$-0.580535\pi$
$947$	−15.4062	−0.500633	−0.250317	+	0.968164i	$0.580535\pi$
$948$	0	0
$949$	3.55251	0.115319
$950$	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
$955$	−46.4406	−1.50278
$956$	0	0
$957$	−4.03721	−0.130505
$958$	0	0
$959$	−26.6767	−0.861435
$960$	0	0
$961$	40.5115	1.30682
$962$	0	0
$963$	49.1789	1.58477
$964$	0	0
$965$	45.4803	1.46406
$966$	0	0
$967$	−30.2493	−0.972752	−0.486376	−	0.873750i	$-0.661681\pi$
$967$	−30.2493	−0.972752	−0.486376	+	0.873750i	$0.661681\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	40.3805	1.29587	0.647936	−	0.761694i	$-0.275632\pi$
$971$	40.3805	1.29587	0.647936	+	0.761694i	$0.275632\pi$
$972$	0	0
$973$	45.2223	1.44976
$974$	0	0
$975$	−1.69641	−0.0543286
$976$	0	0
$977$	43.9012	1.40452	0.702261	−	0.711919i	$-0.252174\pi$
$977$	43.9012	1.40452	0.702261	+	0.711919i	$0.252174\pi$
$978$	0	0
$979$	−62.9774	−2.01277
$980$	0	0
$981$	8.55376	0.273101
$982$	0	0
$983$	−43.9128	−1.40060	−0.700300	−	0.713849i	$-0.746950\pi$
$983$	−43.9128	−1.40060	−0.700300	+	0.713849i	$0.746950\pi$
$984$	0	0
$985$	−17.6437	−0.562173
$986$	0	0
$987$	−2.58379	−0.0822430
$988$	0	0
$989$	−46.0157	−1.46322
$990$	0	0
$991$	−47.7621	−1.51721	−0.758607	−	0.651548i	$-0.774120\pi$
$991$	−47.7621	−1.51721	−0.758607	+	0.651548i	$0.774120\pi$
$992$	0	0
$993$	7.40651	0.235038
$994$	0	0
$995$	−30.3038	−0.960695
$996$	0	0
$997$	−1.51536	−0.0479918	−0.0239959	−	0.999712i	$-0.507639\pi$
$997$	−1.51536	−0.0479918	−0.0239959	+	0.999712i	$0.507639\pi$
$998$	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.a.w.1.6		12
4.3	odd	2		4624.2.a.bt.1.7		12
17.3	odd	16		136.2.n.c.9.2	✓	12
17.4	even	4		2312.2.b.n.577.7		12
17.6	odd	16		136.2.n.c.121.2	yes	12
17.13	even	4		2312.2.b.n.577.6		12
17.16	even	2	inner	2312.2.a.w.1.7		12
51.20	even	16		1224.2.bq.c.145.1		12
51.23	even	16		1224.2.bq.c.937.1		12
68.3	even	16		272.2.v.f.145.2		12
68.23	even	16		272.2.v.f.257.2		12
68.67	odd	2		4624.2.a.bt.1.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.n.c.9.2	✓	12	17.3	odd	16
136.2.n.c.121.2	yes	12	17.6	odd	16
272.2.v.f.145.2		12	68.3	even	16
272.2.v.f.257.2		12	68.23	even	16
1224.2.bq.c.145.1		12	51.20	even	16
1224.2.bq.c.937.1		12	51.23	even	16
2312.2.a.w.1.6		12	1.1	even	1	trivial
2312.2.a.w.1.7		12	17.16	even	2	inner
2312.2.b.n.577.6		12	17.13	even	4
2312.2.b.n.577.7		12	17.4	even	4
4624.2.a.bt.1.6		12	68.67	odd	2
4624.2.a.bt.1.7		12	4.3	odd	2

Overview	Random
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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.a (trivial)

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

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