LMFDB - Embedded newform 3800.2.d.q.3649.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (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:	$30.3431527681$
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} + 24x^{10} + 194x^{8} + 618x^{6} + 733x^{4} + 286x^{2} + 9 $ x^12 + 24x^10 + 194x^8 + 618x^6 + 733x^4 + 286*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.12
Root			$3.26143i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.q.3649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.26143i q^{3} -4.07225i q^{7} -7.63693 q^{9} +O(q^{10})$	$q+3.26143i q^{3} -4.07225i q^{7} -7.63693 q^{9} -0.786366 q^{11} +1.07974i q^{13} +1.90793i q^{17} +1.00000 q^{19} +13.2813 q^{21} -1.41383i q^{23} -15.1230i q^{27} +7.26439 q^{29} +2.22003 q^{31} -2.56468i q^{33} -9.14283i q^{37} -3.52151 q^{39} -6.11703 q^{41} +8.40634i q^{43} -3.56468i q^{47} -9.58319 q^{49} -6.22257 q^{51} -8.57472i q^{53} +3.26143i q^{57} +13.4043 q^{59} +12.7768 q^{61} +31.0994i q^{63} +5.10008i q^{67} +4.61112 q^{69} +1.65535 q^{71} +10.3302i q^{73} +3.20228i q^{77} +16.2256 q^{79} +26.4119 q^{81} +9.35104i q^{83} +23.6923i q^{87} -3.10419 q^{89} +4.39698 q^{91} +7.24046i q^{93} -4.55874i q^{97} +6.00542 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} + 6 q^{11} + 12 q^{19} + 30 q^{21} - 18 q^{29} + 10 q^{31} - 24 q^{39} + 6 q^{41} - 44 q^{49} + 66 q^{51} + 18 q^{61} + 22 q^{69} + 38 q^{71} + 32 q^{79} + 52 q^{81} - 28 q^{89} + 84 q^{91} + 12 q^{99}+O(q^{100}) $ 12 * q - 12 * q^9 + 6 * q^11 + 12 * q^19 + 30 * q^21 - 18 * q^29 + 10 * q^31 - 24 * q^39 + 6 * q^41 - 44 * q^49 + 66 * q^51 + 18 * q^61 + 22 * q^69 + 38 * q^71 + 32 * q^79 + 52 * q^81 - 28 * q^89 + 84 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.26143i	1.88299i	0.337031	+	0.941494i	$0.390577\pi$
$3$	3.26143i	1.88299i	−0.337031	+	0.941494i	$0.609423\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.07225i	− 1.53916i	−0.638548	−	0.769582i	$-0.720464\pi$
$7$	− 4.07225i	− 1.53916i	0.638548	−	0.769582i	$-0.279536\pi$
$8$	0	0
$9$	−7.63693	−2.54564
$10$	0	0
$11$	−0.786366	−0.237098	−0.118549	−	0.992948i	$-0.537824\pi$
$11$	−0.786366	−0.237098	−0.118549	+	0.992948i	$0.537824\pi$
$12$	0	0
$13$	1.07974i	0.299467i	0.988726	+	0.149733i	$0.0478416\pi$
$13$	1.07974i	0.299467i	−0.988726	+	0.149733i	$0.952158\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.90793i	0.462741i	0.972866	+	0.231370i	$0.0743209\pi$
$17$	1.90793i	0.462741i	−0.972866	+	0.231370i	$0.925679\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	13.2813	2.89823
$22$	0	0
$23$	− 1.41383i	− 0.294805i	−0.989077	−	0.147402i	$-0.952909\pi$
$23$	− 1.41383i	− 0.294805i	0.989077	−	0.147402i	$-0.0470912\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 15.1230i	− 2.91042i
$28$	0	0
$29$	7.26439	1.34896	0.674482	−	0.738291i	$-0.264367\pi$
$29$	7.26439	1.34896	0.674482	+	0.738291i	$0.264367\pi$
$30$	0	0
$31$	2.22003	0.398728	0.199364	−	0.979925i	$-0.436112\pi$
$31$	2.22003	0.398728	0.199364	+	0.979925i	$0.436112\pi$
$32$	0	0
$33$	− 2.56468i	− 0.446453i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9.14283i	− 1.50307i	−0.659692	−	0.751536i	$-0.729313\pi$
$37$	− 9.14283i	− 1.50307i	0.659692	−	0.751536i	$-0.270687\pi$
$38$	0	0
$39$	−3.52151	−0.563893
$40$	0	0
$41$	−6.11703	−0.955319	−0.477660	−	0.878545i	$-0.658515\pi$
$41$	−6.11703	−0.955319	−0.477660	+	0.878545i	$0.658515\pi$
$42$	0	0
$43$	8.40634i	1.28195i	0.767560	+	0.640977i	$0.221471\pi$
$43$	8.40634i	1.28195i	−0.767560	+	0.640977i	$0.778529\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.56468i	− 0.519962i	−0.965614	−	0.259981i	$-0.916284\pi$
$47$	− 3.56468i	− 0.519962i	0.965614	−	0.259981i	$-0.0837163\pi$
$48$	0	0
$49$	−9.58319	−1.36903
$50$	0	0
$51$	−6.22257	−0.871335
$52$	0	0
$53$	− 8.57472i	− 1.17783i	−0.808195	−	0.588914i	$-0.799555\pi$
$53$	− 8.57472i	− 1.17783i	0.808195	−	0.588914i	$-0.200445\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.26143i	0.431987i
$58$	0	0
$59$	13.4043	1.74509	0.872543	−	0.488537i	$-0.162469\pi$
$59$	13.4043	1.74509	0.872543	+	0.488537i	$0.162469\pi$
$60$	0	0
$61$	12.7768	1.63590	0.817950	−	0.575289i	$-0.195110\pi$
$61$	12.7768	1.63590	0.817950	+	0.575289i	$0.195110\pi$
$62$	0	0
$63$	31.0994i	3.91816i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.10008i	0.623073i	0.950234	+	0.311537i	$0.100844\pi$
$67$	5.10008i	0.623073i	−0.950234	+	0.311537i	$0.899156\pi$
$68$	0	0
$69$	4.61112	0.555114
$70$	0	0
$71$	1.65535	0.196454	0.0982268	−	0.995164i	$-0.468683\pi$
$71$	1.65535	0.196454	0.0982268	+	0.995164i	$0.468683\pi$
$72$	0	0
$73$	10.3302i	1.20906i	0.796581	+	0.604532i	$0.206640\pi$
$73$	10.3302i	1.20906i	−0.796581	+	0.604532i	$0.793360\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.20228i	0.364933i
$78$	0	0
$79$	16.2256	1.82553	0.912764	−	0.408488i	$-0.133944\pi$
$79$	16.2256	1.82553	0.912764	+	0.408488i	$0.133944\pi$
$80$	0	0
$81$	26.4119	2.93465
$82$	0	0
$83$	9.35104i	1.02641i	0.858266	+	0.513205i	$0.171542\pi$
$83$	9.35104i	1.02641i	−0.858266	+	0.513205i	$0.828458\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	23.6923i	2.54008i
$88$	0	0
$89$	−3.10419	−0.329044	−0.164522	−	0.986373i	$-0.552608\pi$
$89$	−3.10419	−0.329044	−0.164522	+	0.986373i	$0.552608\pi$
$90$	0	0
$91$	4.39698	0.460929
$92$	0	0
$93$	7.24046i	0.750801i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.55874i	− 0.462870i	−0.972850	−	0.231435i	$-0.925658\pi$
$97$	− 4.55874i	− 0.462870i	0.972850	−	0.231435i	$-0.0743421\pi$
$98$	0	0
$99$	6.00542	0.603567
$100$	0	0
$101$	0.743401	0.0739712	0.0369856	−	0.999316i	$-0.488224\pi$
$101$	0.743401	0.0739712	0.0369856	+	0.999316i	$0.488224\pi$
$102$	0	0
$103$	− 10.3710i	− 1.02188i	−0.859616	−	0.510941i	$-0.829297\pi$
$103$	− 10.3710i	− 1.02188i	0.859616	−	0.510941i	$-0.170703\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.1400i	1.65698i	0.560002	+	0.828491i	$0.310800\pi$
$107$	17.1400i	1.65698i	−0.560002	+	0.828491i	$0.689200\pi$
$108$	0	0
$109$	17.1878	1.64629	0.823144	−	0.567832i	$-0.192218\pi$
$109$	17.1878	1.64629	0.823144	+	0.567832i	$0.192218\pi$
$110$	0	0
$111$	29.8187	2.83027
$112$	0	0
$113$	− 1.05696i	− 0.0994300i	−0.998763	−	0.0497150i	$-0.984169\pi$
$113$	− 1.05696i	− 0.0994300i	0.998763	−	0.0497150i	$-0.0158313\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 8.24592i	− 0.762336i
$118$	0	0
$119$	7.76956	0.712234
$120$	0	0
$121$	−10.3816	−0.943784
$122$	0	0
$123$	− 19.9503i	− 1.79885i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.22444i	0.463594i	0.972764	+	0.231797i	$0.0744606\pi$
$127$	5.22444i	0.463594i	−0.972764	+	0.231797i	$0.925539\pi$
$128$	0	0
$129$	−27.4167	−2.41390
$130$	0	0
$131$	3.00267	0.262344	0.131172	−	0.991360i	$-0.458126\pi$
$131$	3.00267	0.262344	0.131172	+	0.991360i	$0.458126\pi$
$132$	0	0
$133$	− 4.07225i	− 0.353109i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	23.0922i	1.97290i	0.164072	+	0.986448i	$0.447537\pi$
$137$	23.0922i	1.97290i	−0.164072	+	0.986448i	$0.552463\pi$
$138$	0	0
$139$	11.9630	1.01469	0.507343	−	0.861744i	$-0.330628\pi$
$139$	11.9630	1.01469	0.507343	+	0.861744i	$0.330628\pi$
$140$	0	0
$141$	11.6259	0.979082
$142$	0	0
$143$	− 0.849074i	− 0.0710031i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 31.2549i	− 2.57786i
$148$	0	0
$149$	−23.5411	−1.92856	−0.964282	−	0.264879i	$-0.914668\pi$
$149$	−23.5411	−1.92856	−0.964282	+	0.264879i	$0.914668\pi$
$150$	0	0
$151$	10.6136	0.863721	0.431860	−	0.901940i	$-0.357857\pi$
$151$	10.6136	0.863721	0.431860	+	0.901940i	$0.357857\pi$
$152$	0	0
$153$	− 14.5707i	− 1.17797i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 13.9997i	− 1.11730i	−0.829405	−	0.558648i	$-0.811320\pi$
$157$	− 13.9997i	− 1.11730i	0.829405	−	0.558648i	$-0.188680\pi$
$158$	0	0
$159$	27.9659	2.21784
$160$	0	0
$161$	−5.75748	−0.453753
$162$	0	0
$163$	− 10.6662i	− 0.835442i	−0.908575	−	0.417721i	$-0.862829\pi$
$163$	− 10.6662i	− 0.835442i	0.908575	−	0.417721i	$-0.137171\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.97856i	0.307870i	0.988081	+	0.153935i	$0.0491947\pi$
$167$	3.97856i	0.307870i	−0.988081	+	0.153935i	$0.950805\pi$
$168$	0	0
$169$	11.8342	0.910320
$170$	0	0
$171$	−7.63693	−0.584010
$172$	0	0
$173$	22.1310i	1.68259i	0.540577	+	0.841295i	$0.318206\pi$
$173$	22.1310i	1.68259i	−0.540577	+	0.841295i	$0.681794\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	43.7171i	3.28598i
$178$	0	0
$179$	−4.80086	−0.358833	−0.179417	−	0.983773i	$-0.557421\pi$
$179$	−4.80086	−0.358833	−0.179417	+	0.983773i	$0.557421\pi$
$180$	0	0
$181$	−16.4333	−1.22148	−0.610740	−	0.791831i	$-0.709128\pi$
$181$	−16.4333	−1.22148	−0.610740	+	0.791831i	$0.709128\pi$
$182$	0	0
$183$	41.6706i	3.08038i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.50033i	− 0.109715i
$188$	0	0
$189$	−61.5846	−4.47962
$190$	0	0
$191$	5.45151	0.394457	0.197229	−	0.980358i	$-0.436806\pi$
$191$	5.45151	0.394457	0.197229	+	0.980358i	$0.436806\pi$
$192$	0	0
$193$	− 16.3968i	− 1.18026i	−0.807306	−	0.590132i	$-0.799076\pi$
$193$	− 16.3968i	− 1.18026i	0.807306	−	0.590132i	$-0.200924\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 19.2546i	− 1.37183i	−0.727681	−	0.685916i	$-0.759402\pi$
$197$	− 19.2546i	− 1.37183i	0.727681	−	0.685916i	$-0.240598\pi$
$198$	0	0
$199$	−7.95572	−0.563966	−0.281983	−	0.959419i	$-0.590992\pi$
$199$	−7.95572	−0.563966	−0.281983	+	0.959419i	$0.590992\pi$
$200$	0	0
$201$	−16.6335	−1.17324
$202$	0	0
$203$	− 29.5824i	− 2.07628i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	10.7973i	0.750468i
$208$	0	0
$209$	−0.786366	−0.0543941
$210$	0	0
$211$	18.2270	1.25480	0.627400	−	0.778697i	$-0.284119\pi$
$211$	18.2270	1.25480	0.627400	+	0.778697i	$0.284119\pi$
$212$	0	0
$213$	5.39880i	0.369920i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 9.04049i	− 0.613709i
$218$	0	0
$219$	−33.6914	−2.27665
$220$	0	0
$221$	−2.06007	−0.138576
$222$	0	0
$223$	− 0.430994i	− 0.0288615i	−0.999896	−	0.0144307i	$-0.995406\pi$
$223$	− 0.430994i	− 0.0288615i	0.999896	−	0.0144307i	$-0.00459361\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.63334i	0.506643i	0.967382	+	0.253321i	$0.0815230\pi$
$227$	7.63334i	0.506643i	−0.967382	+	0.253321i	$0.918477\pi$
$228$	0	0
$229$	−20.3914	−1.34750	−0.673750	−	0.738959i	$-0.735318\pi$
$229$	−20.3914	−1.34750	−0.673750	+	0.738959i	$0.735318\pi$
$230$	0	0
$231$	−10.4440	−0.687165
$232$	0	0
$233$	− 20.6308i	− 1.35157i	−0.737099	−	0.675785i	$-0.763805\pi$
$233$	− 20.6308i	− 1.35157i	0.737099	−	0.675785i	$-0.236195\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	52.9188i	3.43744i
$238$	0	0
$239$	4.61247	0.298356	0.149178	−	0.988810i	$-0.452337\pi$
$239$	4.61247	0.298356	0.149178	+	0.988810i	$0.452337\pi$
$240$	0	0
$241$	3.92363	0.252744	0.126372	−	0.991983i	$-0.459667\pi$
$241$	3.92363	0.252744	0.126372	+	0.991983i	$0.459667\pi$
$242$	0	0
$243$	40.7714i	2.61549i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.07974i	0.0687024i
$248$	0	0
$249$	−30.4978	−1.93272
$250$	0	0
$251$	23.9491	1.51165	0.755827	−	0.654771i	$-0.227235\pi$
$251$	23.9491	1.51165	0.755827	+	0.654771i	$0.227235\pi$
$252$	0	0
$253$	1.11179i	0.0698978i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.05331i	− 0.564730i	−0.959307	−	0.282365i	$-0.908881\pi$
$257$	− 9.05331i	− 0.564730i	0.959307	−	0.282365i	$-0.0911189\pi$
$258$	0	0
$259$	−37.2319	−2.31348
$260$	0	0
$261$	−55.4776	−3.43398
$262$	0	0
$263$	15.4483i	0.952583i	0.879287	+	0.476291i	$0.158019\pi$
$263$	15.4483i	0.952583i	−0.879287	+	0.476291i	$0.841981\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 10.1241i	− 0.619586i
$268$	0	0
$269$	4.44718	0.271150	0.135575	−	0.990767i	$-0.456712\pi$
$269$	4.44718	0.271150	0.135575	+	0.990767i	$0.456712\pi$
$270$	0	0
$271$	23.8511	1.44885	0.724425	−	0.689354i	$-0.242105\pi$
$271$	23.8511	1.44885	0.724425	+	0.689354i	$0.242105\pi$
$272$	0	0
$273$	14.3404i	0.867923i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 6.86751i	− 0.412629i	−0.978486	−	0.206314i	$-0.933853\pi$
$277$	− 6.86751i	− 0.412629i	0.978486	−	0.206314i	$-0.0661470\pi$
$278$	0	0
$279$	−16.9542	−1.01502
$280$	0	0
$281$	26.7419	1.59529	0.797645	−	0.603128i	$-0.206079\pi$
$281$	26.7419	1.59529	0.797645	+	0.603128i	$0.206079\pi$
$282$	0	0
$283$	− 17.4698i	− 1.03847i	−0.854632	−	0.519235i	$-0.826217\pi$
$283$	− 17.4698i	− 1.03847i	0.854632	−	0.519235i	$-0.173783\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.9101i	1.47039i
$288$	0	0
$289$	13.3598	0.785871
$290$	0	0
$291$	14.8680	0.871579
$292$	0	0
$293$	− 14.8968i	− 0.870282i	−0.900362	−	0.435141i	$-0.856698\pi$
$293$	− 14.8968i	− 0.870282i	0.900362	−	0.435141i	$-0.143302\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	11.8922i	0.690057i
$298$	0	0
$299$	1.52658	0.0882843
$300$	0	0
$301$	34.2327	1.97314
$302$	0	0
$303$	2.42455i	0.139287i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 11.2895i	− 0.644326i	−0.946684	−	0.322163i	$-0.895590\pi$
$307$	− 11.2895i	− 0.644326i	0.946684	−	0.322163i	$-0.104410\pi$
$308$	0	0
$309$	33.8242	1.92419
$310$	0	0
$311$	−5.65634	−0.320741	−0.160371	−	0.987057i	$-0.551269\pi$
$311$	−5.65634	−0.320741	−0.160371	+	0.987057i	$0.551269\pi$
$312$	0	0
$313$	− 29.9564i	− 1.69324i	−0.532201	−	0.846618i	$-0.678635\pi$
$313$	− 29.9564i	− 1.69324i	0.532201	−	0.846618i	$-0.321365\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 11.7992i	− 0.662706i	−0.943507	−	0.331353i	$-0.892495\pi$
$317$	− 11.7992i	− 0.662706i	0.943507	−	0.331353i	$-0.107505\pi$
$318$	0	0
$319$	−5.71247	−0.319837
$320$	0	0
$321$	−55.9008	−3.12008
$322$	0	0
$323$	1.90793i	0.106160i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	56.0567i	3.09994i
$328$	0	0
$329$	−14.5163	−0.800307
$330$	0	0
$331$	−21.7197	−1.19382	−0.596912	−	0.802307i	$-0.703606\pi$
$331$	−21.7197	−1.19382	−0.596912	+	0.802307i	$0.703606\pi$
$332$	0	0
$333$	69.8231i	3.82628i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	32.7221i	1.78249i	0.453523	+	0.891245i	$0.350167\pi$
$337$	32.7221i	1.78249i	−0.453523	+	0.891245i	$0.649833\pi$
$338$	0	0
$339$	3.44718	0.187225
$340$	0	0
$341$	−1.74575	−0.0945378
$342$	0	0
$343$	10.5194i	0.567994i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.5890i	1.53474i	0.641206	+	0.767369i	$0.278435\pi$
$347$	28.5890i	1.53474i	−0.641206	+	0.767369i	$0.721565\pi$
$348$	0	0
$349$	−22.4911	−1.20392	−0.601961	−	0.798525i	$-0.705614\pi$
$349$	−22.4911	−1.20392	−0.601961	+	0.798525i	$0.705614\pi$
$350$	0	0
$351$	16.3290	0.871576
$352$	0	0
$353$	− 21.2381i	− 1.13039i	−0.824957	−	0.565196i	$-0.808801\pi$
$353$	− 21.2381i	− 1.13039i	0.824957	−	0.565196i	$-0.191199\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	25.3399i	1.34113i
$358$	0	0
$359$	5.30705	0.280095	0.140048	−	0.990145i	$-0.455274\pi$
$359$	5.30705	0.280095	0.140048	+	0.990145i	$0.455274\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 33.8590i	− 1.77713i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.12291i	− 0.0586156i	−0.999570	−	0.0293078i	$-0.990670\pi$
$367$	− 1.12291i	− 0.0586156i	0.999570	−	0.0293078i	$-0.00933030\pi$
$368$	0	0
$369$	46.7153	2.43190
$370$	0	0
$371$	−34.9184	−1.81287
$372$	0	0
$373$	3.99673i	0.206943i	0.994632	+	0.103471i	$0.0329950\pi$
$373$	3.99673i	0.206943i	−0.994632	+	0.103471i	$0.967005\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.84368i	0.403970i
$378$	0	0
$379$	−3.36846	−0.173026	−0.0865132	−	0.996251i	$-0.527572\pi$
$379$	−3.36846	−0.173026	−0.0865132	+	0.996251i	$0.527572\pi$
$380$	0	0
$381$	−17.0392	−0.872942
$382$	0	0
$383$	17.8476i	0.911969i	0.889988	+	0.455985i	$0.150713\pi$
$383$	17.8476i	0.911969i	−0.889988	+	0.455985i	$0.849287\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 64.1986i	− 3.26340i
$388$	0	0
$389$	19.1690	0.971907	0.485953	−	0.873985i	$-0.338472\pi$
$389$	19.1690	0.971907	0.485953	+	0.873985i	$0.338472\pi$
$390$	0	0
$391$	2.69750	0.136418
$392$	0	0
$393$	9.79298i	0.493991i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 9.04085i	− 0.453747i	−0.973924	−	0.226874i	$-0.927150\pi$
$397$	− 9.04085i	− 0.453747i	0.973924	−	0.226874i	$-0.0728504\pi$
$398$	0	0
$399$	13.2813	0.664899
$400$	0	0
$401$	−15.0634	−0.752230	−0.376115	−	0.926573i	$-0.622740\pi$
$401$	−15.0634	−0.752230	−0.376115	+	0.926573i	$0.622740\pi$
$402$	0	0
$403$	2.39706i	0.119406i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.18961i	0.356376i
$408$	0	0
$409$	7.22091	0.357051	0.178525	−	0.983935i	$-0.442867\pi$
$409$	7.22091	0.357051	0.178525	+	0.983935i	$0.442867\pi$
$410$	0	0
$411$	−75.3135	−3.71494
$412$	0	0
$413$	− 54.5855i	− 2.68597i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	39.0164i	1.91064i
$418$	0	0
$419$	21.0968	1.03065	0.515323	−	0.856996i	$-0.327672\pi$
$419$	21.0968	1.03065	0.515323	+	0.856996i	$0.327672\pi$
$420$	0	0
$421$	5.50321	0.268210	0.134105	−	0.990967i	$-0.457184\pi$
$421$	5.50321	0.268210	0.134105	+	0.990967i	$0.457184\pi$
$422$	0	0
$423$	27.2232i	1.32364i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 52.0303i	− 2.51792i
$428$	0	0
$429$	2.76920	0.133698
$430$	0	0
$431$	−10.9748	−0.528640	−0.264320	−	0.964435i	$-0.585147\pi$
$431$	−10.9748	−0.528640	−0.264320	+	0.964435i	$0.585147\pi$
$432$	0	0
$433$	14.2313i	0.683915i	0.939715	+	0.341957i	$0.111090\pi$
$433$	14.2313i	0.683915i	−0.939715	+	0.341957i	$0.888910\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.41383i	− 0.0676329i
$438$	0	0
$439$	−12.0446	−0.574859	−0.287429	−	0.957802i	$-0.592801\pi$
$439$	−12.0446	−0.574859	−0.287429	+	0.957802i	$0.592801\pi$
$440$	0	0
$441$	73.1861	3.48505
$442$	0	0
$443$	11.1718i	0.530789i	0.964140	+	0.265394i	$0.0855022\pi$
$443$	11.1718i	0.530789i	−0.964140	+	0.265394i	$0.914498\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 76.7777i	− 3.63146i
$448$	0	0
$449$	9.07650	0.428347	0.214173	−	0.976796i	$-0.431294\pi$
$449$	9.07650	0.428347	0.214173	+	0.976796i	$0.431294\pi$
$450$	0	0
$451$	4.81023	0.226505
$452$	0	0
$453$	34.6154i	1.62638i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.6795i	0.780235i	0.920765	+	0.390118i	$0.127566\pi$
$457$	16.6795i	0.780235i	−0.920765	+	0.390118i	$0.872434\pi$
$458$	0	0
$459$	28.8536	1.34677
$460$	0	0
$461$	−31.2657	−1.45619	−0.728095	−	0.685476i	$-0.759594\pi$
$461$	−31.2657	−1.45619	−0.728095	+	0.685476i	$0.759594\pi$
$462$	0	0
$463$	− 10.1802i	− 0.473116i	−0.971617	−	0.236558i	$-0.923981\pi$
$463$	− 10.1802i	− 0.473116i	0.971617	−	0.236558i	$-0.0760193\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 33.0731i	− 1.53044i	−0.643769	−	0.765220i	$-0.722630\pi$
$467$	− 33.0731i	− 1.53044i	0.643769	−	0.765220i	$-0.277370\pi$
$468$	0	0
$469$	20.7688	0.959012
$470$	0	0
$471$	45.6590	2.10385
$472$	0	0
$473$	− 6.61046i	− 0.303949i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	65.4845i	2.99833i
$478$	0	0
$479$	−7.80955	−0.356827	−0.178414	−	0.983956i	$-0.557097\pi$
$479$	−7.80955	−0.356827	−0.178414	+	0.983956i	$0.557097\pi$
$480$	0	0
$481$	9.87191	0.450120
$482$	0	0
$483$	− 18.7776i	− 0.854412i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.37584i	0.198288i	0.995073	+	0.0991442i	$0.0316105\pi$
$487$	4.37584i	0.198288i	−0.995073	+	0.0991442i	$0.968389\pi$
$488$	0	0
$489$	34.7871	1.57313
$490$	0	0
$491$	42.0821	1.89914	0.949568	−	0.313560i	$-0.101522\pi$
$491$	42.0821	1.89914	0.949568	+	0.313560i	$0.101522\pi$
$492$	0	0
$493$	13.8599i	0.624220i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6.74098i	− 0.302374i
$498$	0	0
$499$	8.59546	0.384786	0.192393	−	0.981318i	$-0.438375\pi$
$499$	8.59546	0.384786	0.192393	+	0.981318i	$0.438375\pi$
$500$	0	0
$501$	−12.9758	−0.579716
$502$	0	0
$503$	6.30722i	0.281225i	0.990065	+	0.140612i	$0.0449071\pi$
$503$	6.30722i	0.281225i	−0.990065	+	0.140612i	$0.955093\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	38.5963i	1.71412i
$508$	0	0
$509$	31.0352	1.37561	0.687805	−	0.725896i	$-0.258574\pi$
$509$	31.0352	1.37561	0.687805	+	0.725896i	$0.258574\pi$
$510$	0	0
$511$	42.0673	1.86095
$512$	0	0
$513$	− 15.1230i	− 0.667697i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.80314i	0.123282i
$518$	0	0
$519$	−72.1787	−3.16830
$520$	0	0
$521$	32.8924	1.44104	0.720522	−	0.693432i	$-0.243902\pi$
$521$	32.8924	1.44104	0.720522	+	0.693432i	$0.243902\pi$
$522$	0	0
$523$	− 37.6201i	− 1.64501i	−0.568757	−	0.822506i	$-0.692575\pi$
$523$	− 37.6201i	− 1.64501i	0.568757	−	0.822506i	$-0.307425\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.23565i	0.184508i
$528$	0	0
$529$	21.0011	0.913090
$530$	0	0
$531$	−102.367	−4.44236
$532$	0	0
$533$	− 6.60482i	− 0.286087i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 15.6577i	− 0.675679i
$538$	0	0
$539$	7.53590	0.324594
$540$	0	0
$541$	−33.6736	−1.44774	−0.723871	−	0.689936i	$-0.757639\pi$
$541$	−33.6736	−1.44774	−0.723871	+	0.689936i	$0.757639\pi$
$542$	0	0
$543$	− 53.5962i	− 2.30003i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 14.0295i	− 0.599859i	−0.953961	−	0.299930i	$-0.903037\pi$
$547$	− 14.0295i	− 0.599859i	0.953961	−	0.299930i	$-0.0969632\pi$
$548$	0	0
$549$	−97.5754	−4.16442
$550$	0	0
$551$	7.26439	0.309474
$552$	0	0
$553$	− 66.0748i	− 2.80979i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.5434i	0.700968i	0.936569	+	0.350484i	$0.113983\pi$
$557$	16.5434i	0.700968i	−0.936569	+	0.350484i	$0.886017\pi$
$558$	0	0
$559$	−9.07669	−0.383903
$560$	0	0
$561$	4.89322	0.206592
$562$	0	0
$563$	36.3727i	1.53293i	0.642288	+	0.766463i	$0.277985\pi$
$563$	36.3727i	1.53293i	−0.642288	+	0.766463i	$0.722015\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 107.556i	− 4.51691i
$568$	0	0
$569$	−21.8270	−0.915035	−0.457517	−	0.889201i	$-0.651261\pi$
$569$	−21.8270	−0.915035	−0.457517	+	0.889201i	$0.651261\pi$
$570$	0	0
$571$	41.2781	1.72743	0.863717	−	0.503977i	$-0.168130\pi$
$571$	41.2781	1.72743	0.863717	+	0.503977i	$0.168130\pi$
$572$	0	0
$573$	17.7797i	0.742758i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 23.6240i	− 0.983479i	−0.870742	−	0.491739i	$-0.836361\pi$
$577$	− 23.6240i	− 0.983479i	0.870742	−	0.491739i	$-0.163639\pi$
$578$	0	0
$579$	53.4769	2.22242
$580$	0	0
$581$	38.0798	1.57981
$582$	0	0
$583$	6.74287i	0.279261i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.48730i	− 0.102662i	−0.998682	−	0.0513309i	$-0.983654\pi$
$587$	− 2.48730i	− 0.102662i	0.998682	−	0.0513309i	$-0.0163463\pi$
$588$	0	0
$589$	2.22003	0.0914746
$590$	0	0
$591$	62.7975	2.58314
$592$	0	0
$593$	− 11.6543i	− 0.478587i	−0.970947	−	0.239293i	$-0.923084\pi$
$593$	− 11.6543i	− 0.478587i	0.970947	−	0.239293i	$-0.0769157\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 25.9470i	− 1.06194i
$598$	0	0
$599$	0.149185	0.00609555	0.00304777	−	0.999995i	$-0.499030\pi$
$599$	0.149185	0.00609555	0.00304777	+	0.999995i	$0.499030\pi$
$600$	0	0
$601$	29.2954	1.19498	0.597492	−	0.801875i	$-0.296164\pi$
$601$	29.2954	1.19498	0.597492	+	0.801875i	$0.296164\pi$
$602$	0	0
$603$	− 38.9489i	− 1.58612i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.55466i	0.103691i	0.998655	+	0.0518453i	$0.0165103\pi$
$607$	2.55466i	0.103691i	−0.998655	+	0.0518453i	$0.983490\pi$
$608$	0	0
$609$	96.4809	3.90960
$610$	0	0
$611$	3.84894	0.155711
$612$	0	0
$613$	− 42.6703i	− 1.72344i	−0.507387	−	0.861718i	$-0.669389\pi$
$613$	− 42.6703i	− 1.72344i	0.507387	−	0.861718i	$-0.330611\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.8058i	0.797353i	0.917092	+	0.398676i	$0.130530\pi$
$617$	19.8058i	0.797353i	−0.917092	+	0.398676i	$0.869470\pi$
$618$	0	0
$619$	23.6180	0.949287	0.474643	−	0.880178i	$-0.342577\pi$
$619$	23.6180	0.949287	0.474643	+	0.880178i	$0.342577\pi$
$620$	0	0
$621$	−21.3814	−0.858007
$622$	0	0
$623$	12.6410i	0.506453i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 2.56468i	− 0.102423i
$628$	0	0
$629$	17.4439	0.695533
$630$	0	0
$631$	33.3007	1.32568	0.662841	−	0.748760i	$-0.269351\pi$
$631$	33.3007	1.32568	0.662841	+	0.748760i	$0.269351\pi$
$632$	0	0
$633$	59.4462i	2.36277i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 10.3474i	− 0.409979i
$638$	0	0
$639$	−12.6418	−0.500100
$640$	0	0
$641$	−37.5897	−1.48470	−0.742352	−	0.670010i	$-0.766290\pi$
$641$	−37.5897	−1.48470	−0.742352	+	0.670010i	$0.766290\pi$
$642$	0	0
$643$	− 2.61556i	− 0.103148i	−0.998669	−	0.0515739i	$-0.983576\pi$
$643$	− 2.61556i	− 0.103148i	0.998669	−	0.0515739i	$-0.0164238\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 10.3725i	− 0.407784i	−0.978993	−	0.203892i	$-0.934641\pi$
$647$	− 10.3725i	− 0.407784i	0.978993	−	0.203892i	$-0.0653591\pi$
$648$	0	0
$649$	−10.5407	−0.413757
$650$	0	0
$651$	29.4849	1.15561
$652$	0	0
$653$	0.949620i	0.0371615i	0.999827	+	0.0185808i	$0.00591478\pi$
$653$	0.949620i	0.0371615i	−0.999827	+	0.0185808i	$0.994085\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 78.8913i	− 3.07784i
$658$	0	0
$659$	−32.1009	−1.25047	−0.625237	−	0.780435i	$-0.714998\pi$
$659$	−32.1009	−1.25047	−0.625237	+	0.780435i	$0.714998\pi$
$660$	0	0
$661$	−32.8796	−1.27887	−0.639434	−	0.768846i	$-0.720831\pi$
$661$	−32.8796	−1.27887	−0.639434	+	0.768846i	$0.720831\pi$
$662$	0	0
$663$	− 6.71879i	− 0.260936i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 10.2707i	− 0.397681i
$668$	0	0
$669$	1.40566	0.0543458
$670$	0	0
$671$	−10.0472	−0.387869
$672$	0	0
$673$	25.9466i	1.00017i	0.865977	+	0.500084i	$0.166698\pi$
$673$	25.9466i	1.00017i	−0.865977	+	0.500084i	$0.833302\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 47.0807i	− 1.80946i	−0.425987	−	0.904729i	$-0.640073\pi$
$677$	− 47.0807i	− 1.80946i	0.425987	−	0.904729i	$-0.359927\pi$
$678$	0	0
$679$	−18.5643	−0.712433
$680$	0	0
$681$	−24.8956	−0.954002
$682$	0	0
$683$	29.9429i	1.14573i	0.819648	+	0.572867i	$0.194169\pi$
$683$	29.9429i	1.14573i	−0.819648	+	0.572867i	$0.805831\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 66.5050i	− 2.53733i
$688$	0	0
$689$	9.25850	0.352721
$690$	0	0
$691$	−30.4205	−1.15725	−0.578625	−	0.815594i	$-0.696411\pi$
$691$	−30.4205	−1.15725	−0.578625	+	0.815594i	$0.696411\pi$
$692$	0	0
$693$	− 24.4556i	− 0.928990i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 11.6709i	− 0.442065i
$698$	0	0
$699$	67.2859	2.54499
$700$	0	0
$701$	25.0769	0.947141	0.473570	−	0.880756i	$-0.342965\pi$
$701$	25.0769	0.947141	0.473570	+	0.880756i	$0.342965\pi$
$702$	0	0
$703$	− 9.14283i	− 0.344828i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 3.02731i	− 0.113854i
$708$	0	0
$709$	−13.5057	−0.507219	−0.253610	−	0.967307i	$-0.581618\pi$
$709$	−13.5057	−0.507219	−0.253610	+	0.967307i	$0.581618\pi$
$710$	0	0
$711$	−123.914	−4.64714
$712$	0	0
$713$	− 3.13875i	− 0.117547i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15.0433i	0.561801i
$718$	0	0
$719$	2.27969	0.0850179	0.0425090	−	0.999096i	$-0.486465\pi$
$719$	2.27969	0.0850179	0.0425090	+	0.999096i	$0.486465\pi$
$720$	0	0
$721$	−42.2331	−1.57284
$722$	0	0
$723$	12.7967i	0.475913i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 0.925937i	− 0.0343411i	−0.999853	−	0.0171705i	$-0.994534\pi$
$727$	− 0.925937i	− 0.0343411i	0.999853	−	0.0171705i	$-0.00546582\pi$
$728$	0	0
$729$	−53.7374	−1.99028
$730$	0	0
$731$	−16.0387	−0.593212
$732$	0	0
$733$	20.9455i	0.773639i	0.922155	+	0.386820i	$0.126426\pi$
$733$	20.9455i	0.773639i	−0.922155	+	0.386820i	$0.873574\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4.01053i	− 0.147730i
$738$	0	0
$739$	−36.9385	−1.35880	−0.679402	−	0.733766i	$-0.737761\pi$
$739$	−36.9385	−1.35880	−0.679402	+	0.733766i	$0.737761\pi$
$740$	0	0
$741$	−3.52151	−0.129366
$742$	0	0
$743$	− 48.3612i	− 1.77420i	−0.461577	−	0.887100i	$-0.652716\pi$
$743$	− 48.3612i	− 1.77420i	0.461577	−	0.887100i	$-0.347284\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 71.4132i	− 2.61287i
$748$	0	0
$749$	69.7981	2.55037
$750$	0	0
$751$	−19.9871	−0.729340	−0.364670	−	0.931137i	$-0.618818\pi$
$751$	−19.9871	−0.729340	−0.364670	+	0.931137i	$0.618818\pi$
$752$	0	0
$753$	78.1084i	2.84643i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	36.3938i	1.32276i	0.750053	+	0.661378i	$0.230028\pi$
$757$	36.3938i	1.32276i	−0.750053	+	0.661378i	$0.769972\pi$
$758$	0	0
$759$	−3.62603	−0.131617
$760$	0	0
$761$	−24.2163	−0.877839	−0.438920	−	0.898526i	$-0.644639\pi$
$761$	−24.2163	−0.877839	−0.438920	+	0.898526i	$0.644639\pi$
$762$	0	0
$763$	− 69.9928i	− 2.53391i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.4732i	0.522596i
$768$	0	0
$769$	−30.7013	−1.10712	−0.553559	−	0.832810i	$-0.686731\pi$
$769$	−30.7013	−1.10712	−0.553559	+	0.832810i	$0.686731\pi$
$770$	0	0
$771$	29.5267	1.06338
$772$	0	0
$773$	21.9986i	0.791234i	0.918416	+	0.395617i	$0.129469\pi$
$773$	21.9986i	0.791234i	−0.918416	+	0.395617i	$0.870531\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 121.429i	− 4.35625i
$778$	0	0
$779$	−6.11703	−0.219165
$780$	0	0
$781$	−1.30171	−0.0465788
$782$	0	0
$783$	− 109.859i	− 3.92606i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	20.9830i	0.747962i	0.927436	+	0.373981i	$0.122007\pi$
$787$	20.9830i	0.747962i	−0.927436	+	0.373981i	$0.877993\pi$
$788$	0	0
$789$	−50.3835	−1.79370
$790$	0	0
$791$	−4.30418	−0.153039
$792$	0	0
$793$	13.7957i	0.489898i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.1150i	1.03131i	0.856798	+	0.515653i	$0.172451\pi$
$797$	29.1150i	1.03131i	−0.856798	+	0.515653i	$0.827549\pi$
$798$	0	0
$799$	6.80115	0.240607
$800$	0	0
$801$	23.7065	0.837628
$802$	0	0
$803$	− 8.12336i	− 0.286667i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.5042i	0.510571i
$808$	0	0
$809$	47.5139	1.67050	0.835250	−	0.549871i	$-0.185323\pi$
$809$	47.5139	1.67050	0.835250	+	0.549871i	$0.185323\pi$
$810$	0	0
$811$	11.7957	0.414204	0.207102	−	0.978319i	$-0.433597\pi$
$811$	11.7957	0.414204	0.207102	+	0.978319i	$0.433597\pi$
$812$	0	0
$813$	77.7886i	2.72817i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.40634i	0.294101i
$818$	0	0
$819$	−33.5794	−1.17336
$820$	0	0
$821$	28.0549	0.979123	0.489562	−	0.871969i	$-0.337157\pi$
$821$	28.0549	0.979123	0.489562	+	0.871969i	$0.337157\pi$
$822$	0	0
$823$	16.7826i	0.585006i	0.956265	+	0.292503i	$0.0944882\pi$
$823$	16.7826i	0.585006i	−0.956265	+	0.292503i	$0.905512\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 12.1984i	− 0.424180i	−0.977250	−	0.212090i	$-0.931973\pi$
$827$	− 12.1984i	− 0.424180i	0.977250	−	0.212090i	$-0.0680271\pi$
$828$	0	0
$829$	−3.30978	−0.114954	−0.0574768	−	0.998347i	$-0.518306\pi$
$829$	−3.30978	−0.114954	−0.0574768	+	0.998347i	$0.518306\pi$
$830$	0	0
$831$	22.3979	0.776975
$832$	0	0
$833$	− 18.2840i	− 0.633505i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 33.5735i	− 1.16047i
$838$	0	0
$839$	27.9925	0.966408	0.483204	−	0.875508i	$-0.339473\pi$
$839$	27.9925	0.966408	0.483204	+	0.875508i	$0.339473\pi$
$840$	0	0
$841$	23.7714	0.819704
$842$	0	0
$843$	87.2169i	3.00391i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	42.2766i	1.45264i
$848$	0	0
$849$	56.9764	1.95543
$850$	0	0
$851$	−12.9265	−0.443113
$852$	0	0
$853$	− 11.5191i	− 0.394408i	−0.980362	−	0.197204i	$-0.936814\pi$
$853$	− 11.5191i	− 0.394408i	0.980362	−	0.197204i	$-0.0631861\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 5.40616i	− 0.184671i	−0.995728	−	0.0923355i	$-0.970567\pi$
$857$	− 5.40616i	− 0.184671i	0.995728	−	0.0923355i	$-0.0294332\pi$
$858$	0	0
$859$	17.4306	0.594725	0.297363	−	0.954765i	$-0.403893\pi$
$859$	17.4306	0.594725	0.297363	+	0.954765i	$0.403893\pi$
$860$	0	0
$861$	−81.2424	−2.76873
$862$	0	0
$863$	− 8.83048i	− 0.300593i	−0.988641	−	0.150297i	$-0.951977\pi$
$863$	− 8.83048i	− 0.300593i	0.988641	−	0.150297i	$-0.0480229\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	43.5721i	1.47979i
$868$	0	0
$869$	−12.7593	−0.432829
$870$	0	0
$871$	−5.50677	−0.186590
$872$	0	0
$873$	34.8148i	1.17830i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0.840573i	0.0283841i	0.999899	+	0.0141921i	$0.00451763\pi$
$877$	0.840573i	0.0283841i	−0.999899	+	0.0141921i	$0.995482\pi$
$878$	0	0
$879$	48.5850	1.63873
$880$	0	0
$881$	6.78562	0.228613	0.114307	−	0.993446i	$-0.463535\pi$
$881$	6.78562	0.228613	0.114307	+	0.993446i	$0.463535\pi$
$882$	0	0
$883$	− 10.0799i	− 0.339217i	−0.985512	−	0.169608i	$-0.945750\pi$
$883$	− 10.0799i	− 0.339217i	0.985512	−	0.169608i	$-0.0542503\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.41650i	0.316175i	0.987425	+	0.158088i	$0.0505328\pi$
$887$	9.41650i	0.316175i	−0.987425	+	0.158088i	$0.949467\pi$
$888$	0	0
$889$	21.2752	0.713548
$890$	0	0
$891$	−20.7694	−0.695801
$892$	0	0
$893$	− 3.56468i	− 0.119287i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.97883i	0.166238i
$898$	0	0
$899$	16.1271	0.537870
$900$	0	0
$901$	16.3600	0.545029
$902$	0	0
$903$	111.647i	3.71540i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23.2639i	0.772466i	0.922401	+	0.386233i	$0.126224\pi$
$907$	23.2639i	0.772466i	−0.922401	+	0.386233i	$0.873776\pi$
$908$	0	0
$909$	−5.67730	−0.188304
$910$	0	0
$911$	15.4610	0.512245	0.256123	−	0.966644i	$-0.417555\pi$
$911$	15.4610	0.512245	0.256123	+	0.966644i	$0.417555\pi$
$912$	0	0
$913$	− 7.35335i	− 0.243360i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 12.2276i	− 0.403791i
$918$	0	0
$919$	−20.1775	−0.665594	−0.332797	−	0.942998i	$-0.607992\pi$
$919$	−20.1775	−0.665594	−0.332797	+	0.942998i	$0.607992\pi$
$920$	0	0
$921$	36.8199	1.21326
$922$	0	0
$923$	1.78735i	0.0588314i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	79.2023i	2.60134i
$928$	0	0
$929$	−11.3148	−0.371227	−0.185613	−	0.982623i	$-0.559427\pi$
$929$	−11.3148	−0.371227	−0.185613	+	0.982623i	$0.559427\pi$
$930$	0	0
$931$	−9.58319	−0.314076
$932$	0	0
$933$	− 18.4477i	− 0.603952i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.8943i	0.551911i	0.961170	+	0.275956i	$0.0889943\pi$
$937$	16.8943i	0.551911i	−0.961170	+	0.275956i	$0.911006\pi$
$938$	0	0
$939$	97.7007	3.18834
$940$	0	0
$941$	−15.0272	−0.489871	−0.244936	−	0.969539i	$-0.578767\pi$
$941$	−15.0272	−0.489871	−0.244936	+	0.969539i	$0.578767\pi$
$942$	0	0
$943$	8.64847i	0.281633i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.208895i	0.00678818i	0.999994	+	0.00339409i	$0.00108037\pi$
$947$	0.208895i	0.00678818i	−0.999994	+	0.00339409i	$0.998920\pi$
$948$	0	0
$949$	−11.1540	−0.362075
$950$	0	0
$951$	38.4821	1.24787
$952$	0	0
$953$	18.0875i	0.585910i	0.956126	+	0.292955i	$0.0946387\pi$
$953$	18.0875i	0.585910i	−0.956126	+	0.292955i	$0.905361\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 18.6308i	− 0.602249i
$958$	0	0
$959$	94.0370	3.03661
$960$	0	0
$961$	−26.0715	−0.841016
$962$	0	0
$963$	− 130.897i	− 4.21808i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40.5540i	1.30413i	0.758164	+	0.652064i	$0.226097\pi$
$967$	40.5540i	1.30413i	−0.758164	+	0.652064i	$0.773903\pi$
$968$	0	0
$969$	−6.22257	−0.199898
$970$	0	0
$971$	−51.7597	−1.66105	−0.830523	−	0.556984i	$-0.811959\pi$
$971$	−51.7597	−1.66105	−0.830523	+	0.556984i	$0.811959\pi$
$972$	0	0
$973$	− 48.7161i	− 1.56177i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.6104i	0.755364i	0.925935	+	0.377682i	$0.123279\pi$
$977$	23.6104i	0.755364i	−0.925935	+	0.377682i	$0.876721\pi$
$978$	0	0
$979$	2.44103	0.0780158
$980$	0	0
$981$	−131.262	−4.19086
$982$	0	0
$983$	34.0532i	1.08613i	0.839691	+	0.543065i	$0.182736\pi$
$983$	34.0532i	1.08613i	−0.839691	+	0.543065i	$0.817264\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 47.3437i	− 1.50697i
$988$	0	0
$989$	11.8852	0.377926
$990$	0	0
$991$	42.7936	1.35938	0.679692	−	0.733497i	$-0.262113\pi$
$991$	42.7936	1.35938	0.679692	+	0.733497i	$0.262113\pi$
$992$	0	0
$993$	− 70.8373i	− 2.24796i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	30.1937i	0.956243i	0.878294	+	0.478122i	$0.158682\pi$
$997$	30.1937i	0.956243i	−0.878294	+	0.478122i	$0.841318\pi$
$998$	0	0
$999$	−138.267	−4.37458

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.q.3649.12		12
5.2	odd	4		3800.2.a.bc.1.6	yes	6
5.3	odd	4		3800.2.a.ba.1.1	✓	6
5.4	even	2	inner	3800.2.d.q.3649.1		12
20.3	even	4		7600.2.a.cl.1.6		6
20.7	even	4		7600.2.a.ch.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.ba.1.1	✓	6	5.3	odd	4
3800.2.a.bc.1.6	yes	6	5.2	odd	4
3800.2.d.q.3649.1		12	5.4	even	2	inner
3800.2.d.q.3649.12		12	1.1	even	1	trivial
7600.2.a.ch.1.1		6	20.7	even	4
7600.2.a.cl.1.6		6	20.3	even	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.26143i q^{3} -4.07225i q^{7} -7.63693 q^{9} +O(q^{10})\)	\(q+3.26143i q^{3} -4.07225i q^{7} -7.63693 q^{9} -0.786366 q^{11} +1.07974i q^{13} +1.90793i q^{17} +1.00000 q^{19} +13.2813 q^{21} -1.41383i q^{23} -15.1230i q^{27} +7.26439 q^{29} +2.22003 q^{31} -2.56468i q^{33} -9.14283i q^{37} -3.52151 q^{39} -6.11703 q^{41} +8.40634i q^{43} -3.56468i q^{47} -9.58319 q^{49} -6.22257 q^{51} -8.57472i q^{53} +3.26143i q^{57} +13.4043 q^{59} +12.7768 q^{61} +31.0994i q^{63} +5.10008i q^{67} +4.61112 q^{69} +1.65535 q^{71} +10.3302i q^{73} +3.20228i q^{77} +16.2256 q^{79} +26.4119 q^{81} +9.35104i q^{83} +23.6923i q^{87} -3.10419 q^{89} +4.39698 q^{91} +7.24046i q^{93} -4.55874i q^{97} +6.00542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} + 6 q^{11} + 12 q^{19} + 30 q^{21} - 18 q^{29} + 10 q^{31} - 24 q^{39} + 6 q^{41} - 44 q^{49} + 66 q^{51} + 18 q^{61} + 22 q^{69} + 38 q^{71} + 32 q^{79} + 52 q^{81} - 28 q^{89} + 84 q^{91} + 12 q^{99}+O(q^{100}) \) 12 * q - 12 * q^9 + 6 * q^11 + 12 * q^19 + 30 * q^21 - 18 * q^29 + 10 * q^31 - 24 * q^39 + 6 * q^41 - 44 * q^49 + 66 * q^51 + 18 * q^61 + 22 * q^69 + 38 * q^71 + 32 * q^79 + 52 * q^81 - 28 * q^89 + 84 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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