LMFDB - Embedded newform 1900.2.c.g.1749.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.c (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:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.4227136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 22x^{2} + 9 $ x^6 + 9x^4 + 22x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.5
Root			$2.19869i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.2.c.g.1749.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.19869i q^{3} -1.19869i q^{7} +1.56314 q^{9} +O(q^{10})$	$q+1.19869i q^{3} -1.19869i q^{7} +1.56314 q^{9} -5.86718 q^{11} -0.364448i q^{13} -1.19869i q^{17} +1.00000 q^{19} +1.43686 q^{21} -8.23163i q^{23} +5.46980i q^{27} +7.86718 q^{29} +7.30404 q^{31} -7.03293i q^{33} -7.13828i q^{37} +0.436861 q^{39} +2.43686 q^{41} -7.39738i q^{43} +13.7014i q^{47} +5.56314 q^{49} +1.43686 q^{51} -7.39738i q^{53} +1.19869i q^{57} -12.8606 q^{59} -1.30404 q^{61} -1.87372i q^{63} -11.9330i q^{67} +9.86718 q^{69} +2.12628 q^{71} +2.50273i q^{73} +7.03293i q^{77} +7.74090 q^{79} -1.86718 q^{81} -3.02093i q^{83} +9.43032i q^{87} +5.68942 q^{89} -0.436861 q^{91} +8.75529i q^{93} -1.09334i q^{97} -9.17122 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{9}+O(q^{10}) $ 6 * q - 2 * q^9	$ 6 q - 2 q^{9} - 2 q^{11} + 6 q^{19} + 20 q^{21} + 14 q^{29} + 22 q^{31} + 14 q^{39} + 26 q^{41} + 22 q^{49} + 20 q^{51} + 12 q^{59} + 14 q^{61} + 26 q^{69} - 10 q^{71} + 36 q^{79} + 22 q^{81} - 14 q^{91}+O(q^{100}) $ 6 * q - 2 * q^9 - 2 * q^11 + 6 * q^19 + 20 * q^21 + 14 * q^29 + 22 * q^31 + 14 * q^39 + 26 * q^41 + 22 * q^49 + 20 * q^51 + 12 * q^59 + 14 * q^61 + 26 * q^69 - 10 * q^71 + 36 * q^79 + 22 * q^81 - 14 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.19869i	0.692065i	0.938223	+	0.346032i	$0.112471\pi$
$3$	1.19869i	0.692065i	−0.938223	+	0.346032i	$0.887529\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.19869i	− 0.453063i	−0.974004	−	0.226531i	$-0.927261\pi$
$7$	− 1.19869i	− 0.453063i	0.974004	−	0.226531i	$-0.0727386\pi$
$8$	0	0
$9$	1.56314	0.521046
$10$	0	0
$11$	−5.86718	−1.76902	−0.884510	−	0.466521i	$-0.845507\pi$
$11$	−5.86718	−1.76902	−0.884510	+	0.466521i	$0.845507\pi$
$12$	0	0
$13$	− 0.364448i	− 0.101080i	−0.998722	−	0.0505399i	$-0.983906\pi$
$13$	− 0.364448i	− 0.101080i	0.998722	−	0.0505399i	$-0.0160942\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.19869i	− 0.290725i	−0.989378	−	0.145363i	$-0.953565\pi$
$17$	− 1.19869i	− 0.290725i	0.989378	−	0.145363i	$-0.0464349\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.43686	0.313549
$22$	0	0
$23$	− 8.23163i	− 1.71641i	−0.513305	−	0.858206i	$-0.671579\pi$
$23$	− 8.23163i	− 1.71641i	0.513305	−	0.858206i	$-0.328421\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.46980i	1.05266i
$28$	0	0
$29$	7.86718	1.46090	0.730449	−	0.682967i	$-0.239311\pi$
$29$	7.86718	1.46090	0.730449	+	0.682967i	$0.239311\pi$
$30$	0	0
$31$	7.30404	1.31184	0.655922	−	0.754829i	$-0.272280\pi$
$31$	7.30404	1.31184	0.655922	+	0.754829i	$0.272280\pi$
$32$	0	0
$33$	− 7.03293i	− 1.22428i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.13828i	− 1.17353i	−0.809759	−	0.586763i	$-0.800402\pi$
$37$	− 7.13828i	− 1.17353i	0.809759	−	0.586763i	$-0.199598\pi$
$38$	0	0
$39$	0.436861	0.0699537
$40$	0	0
$41$	2.43686	0.380574	0.190287	−	0.981729i	$-0.439058\pi$
$41$	2.43686	0.380574	0.190287	+	0.981729i	$0.439058\pi$
$42$	0	0
$43$	− 7.39738i	− 1.12809i	−0.825744	−	0.564045i	$-0.809244\pi$
$43$	− 7.39738i	− 1.12809i	0.825744	−	0.564045i	$-0.190756\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	13.7014i	1.99856i	0.0379717	+	0.999279i	$0.487910\pi$
$47$	13.7014i	1.99856i	−0.0379717	+	0.999279i	$0.512090\pi$
$48$	0	0
$49$	5.56314	0.794734
$50$	0	0
$51$	1.43686	0.201201
$52$	0	0
$53$	− 7.39738i	− 1.01611i	−0.861325	−	0.508054i	$-0.830365\pi$
$53$	− 7.39738i	− 1.01611i	0.861325	−	0.508054i	$-0.169635\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.19869i	0.158771i
$58$	0	0
$59$	−12.8606	−1.67431	−0.837156	−	0.546964i	$-0.815783\pi$
$59$	−12.8606	−1.67431	−0.837156	+	0.546964i	$0.815783\pi$
$60$	0	0
$61$	−1.30404	−0.166965	−0.0834825	−	0.996509i	$-0.526604\pi$
$61$	−1.30404	−0.166965	−0.0834825	+	0.996509i	$0.526604\pi$
$62$	0	0
$63$	− 1.87372i	− 0.236067i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 11.9330i	− 1.45785i	−0.684592	−	0.728927i	$-0.740019\pi$
$67$	− 11.9330i	− 1.45785i	0.684592	−	0.728927i	$-0.259981\pi$
$68$	0	0
$69$	9.86718	1.18787
$70$	0	0
$71$	2.12628	0.252343	0.126171	−	0.992008i	$-0.459731\pi$
$71$	2.12628	0.252343	0.126171	+	0.992008i	$0.459731\pi$
$72$	0	0
$73$	2.50273i	0.292922i	0.989216	+	0.146461i	$0.0467883\pi$
$73$	2.50273i	0.292922i	−0.989216	+	0.146461i	$0.953212\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.03293i	0.801477i
$78$	0	0
$79$	7.74090	0.870919	0.435460	−	0.900208i	$-0.356586\pi$
$79$	7.74090	0.870919	0.435460	+	0.900208i	$0.356586\pi$
$80$	0	0
$81$	−1.86718	−0.207464
$82$	0	0
$83$	− 3.02093i	− 0.331590i	−0.986160	−	0.165795i	$-0.946981\pi$
$83$	− 3.02093i	− 0.331590i	0.986160	−	0.165795i	$-0.0530191\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.43032i	1.01104i
$88$	0	0
$89$	5.68942	0.603077	0.301539	−	0.953454i	$-0.402500\pi$
$89$	5.68942	0.603077	0.301539	+	0.953454i	$0.402500\pi$
$90$	0	0
$91$	−0.436861	−0.0457954
$92$	0	0
$93$	8.75529i	0.907881i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.09334i	− 0.111012i	−0.998458	−	0.0555061i	$-0.982323\pi$
$97$	− 1.09334i	− 0.111012i	0.998458	−	0.0555061i	$-0.0176772\pi$
$98$	0	0
$99$	−9.17122	−0.921742
$100$	0	0
$101$	10.8672	1.08132	0.540662	−	0.841240i	$-0.318174\pi$
$101$	10.8672	1.08132	0.540662	+	0.841240i	$0.318174\pi$
$102$	0	0
$103$	− 9.74090i	− 0.959799i	−0.877323	−	0.479900i	$-0.840673\pi$
$103$	− 9.74090i	− 0.959799i	0.877323	−	0.479900i	$-0.159327\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.3634i	− 1.58191i	−0.611877	−	0.790953i	$-0.709585\pi$
$107$	− 16.3634i	− 1.58191i	0.611877	−	0.790953i	$-0.290415\pi$
$108$	0	0
$109$	−9.29749	−0.890538	−0.445269	−	0.895397i	$-0.646892\pi$
$109$	−9.29749	−0.890538	−0.445269	+	0.895397i	$0.646892\pi$
$110$	0	0
$111$	8.55660	0.812156
$112$	0	0
$113$	− 2.96707i	− 0.279118i	−0.990214	−	0.139559i	$-0.955432\pi$
$113$	− 2.96707i	− 0.279118i	0.990214	−	0.139559i	$-0.0445685\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 0.569683i	− 0.0526672i
$118$	0	0
$119$	−1.43686	−0.131717
$120$	0	0
$121$	23.4238	2.12943
$122$	0	0
$123$	2.92104i	0.263382i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.3579i	1.00785i	0.863747	+	0.503926i	$0.168111\pi$
$127$	11.3579i	1.00785i	−0.863747	+	0.503926i	$0.831889\pi$
$128$	0	0
$129$	8.86718	0.780711
$130$	0	0
$131$	−1.56314	−0.136572	−0.0682861	−	0.997666i	$-0.521753\pi$
$131$	−1.56314	−0.136572	−0.0682861	+	0.997666i	$0.521753\pi$
$132$	0	0
$133$	− 1.19869i	− 0.103940i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.8397i	1.35328i	0.736315	+	0.676639i	$0.236564\pi$
$137$	15.8397i	1.35328i	−0.736315	+	0.676639i	$0.763436\pi$
$138$	0	0
$139$	19.8606	1.68456	0.842278	−	0.539043i	$-0.181214\pi$
$139$	19.8606	1.68456	0.842278	+	0.539043i	$0.181214\pi$
$140$	0	0
$141$	−16.4238	−1.38313
$142$	0	0
$143$	2.13828i	0.178812i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.66849i	0.550007i
$148$	0	0
$149$	−2.68942	−0.220326	−0.110163	−	0.993914i	$-0.535137\pi$
$149$	−2.68942	−0.220326	−0.110163	+	0.993914i	$0.535137\pi$
$150$	0	0
$151$	17.1647	1.39684	0.698421	−	0.715688i	$-0.253887\pi$
$151$	17.1647	1.39684	0.698421	+	0.715688i	$0.253887\pi$
$152$	0	0
$153$	− 1.87372i	− 0.151481i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.3634i	1.30594i	0.757384	+	0.652969i	$0.226477\pi$
$157$	16.3634i	1.30594i	−0.757384	+	0.652969i	$0.773523\pi$
$158$	0	0
$159$	8.86718	0.703213
$160$	0	0
$161$	−9.86718	−0.777643
$162$	0	0
$163$	− 8.28549i	− 0.648970i	−0.945891	−	0.324485i	$-0.894809\pi$
$163$	− 8.28549i	− 0.648970i	0.945891	−	0.324485i	$-0.105191\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.49727i	0.580156i	0.957003	+	0.290078i	$0.0936813\pi$
$167$	7.49727i	0.580156i	−0.957003	+	0.290078i	$0.906319\pi$
$168$	0	0
$169$	12.8672	0.989783
$170$	0	0
$171$	1.56314	0.119536
$172$	0	0
$173$	− 14.7464i	− 1.12114i	−0.828105	−	0.560572i	$-0.810581\pi$
$173$	− 14.7464i	− 1.12114i	0.828105	−	0.560572i	$-0.189419\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 15.4159i	− 1.15873i
$178$	0	0
$179$	−13.2526	−0.990543	−0.495271	−	0.868738i	$-0.664931\pi$
$179$	−13.2526	−0.990543	−0.495271	+	0.868738i	$0.664931\pi$
$180$	0	0
$181$	16.7344	1.24385	0.621927	−	0.783075i	$-0.286350\pi$
$181$	16.7344	1.24385	0.621927	+	0.783075i	$0.286350\pi$
$182$	0	0
$183$	− 1.56314i	− 0.115551i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7.03293i	0.514299i
$188$	0	0
$189$	6.55660	0.476922
$190$	0	0
$191$	−4.13282	−0.299041	−0.149520	−	0.988759i	$-0.547773\pi$
$191$	−4.13282	−0.299041	−0.149520	+	0.988759i	$0.547773\pi$
$192$	0	0
$193$	7.08680i	0.510119i	0.966925	+	0.255060i	$0.0820951\pi$
$193$	7.08680i	0.510119i	−0.966925	+	0.255060i	$0.917905\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 20.4753i	− 1.45880i	−0.684087	−	0.729401i	$-0.739799\pi$
$197$	− 20.4753i	− 1.45880i	0.684087	−	0.729401i	$-0.260201\pi$
$198$	0	0
$199$	−8.74090	−0.619626	−0.309813	−	0.950798i	$-0.600266\pi$
$199$	−8.74090	−0.619626	−0.309813	+	0.950798i	$0.600266\pi$
$200$	0	0
$201$	14.3040	1.00893
$202$	0	0
$203$	− 9.43032i	− 0.661878i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 12.8672i	− 0.894331i
$208$	0	0
$209$	−5.86718	−0.405841
$210$	0	0
$211$	−18.7344	−1.28973	−0.644863	−	0.764298i	$-0.723086\pi$
$211$	−18.7344	−1.28973	−0.644863	+	0.764298i	$0.723086\pi$
$212$	0	0
$213$	2.54875i	0.174638i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 8.75529i	− 0.594348i
$218$	0	0
$219$	−3.00000	−0.202721
$220$	0	0
$221$	−0.436861	−0.0293864
$222$	0	0
$223$	4.27110i	0.286014i	0.989722	+	0.143007i	$0.0456772\pi$
$223$	4.27110i	0.286014i	−0.989722	+	0.143007i	$0.954323\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.5631i	1.43120i	0.698512	+	0.715598i	$0.253846\pi$
$227$	21.5631i	1.43120i	−0.698512	+	0.715598i	$0.746154\pi$
$228$	0	0
$229$	−27.4172	−1.81178	−0.905891	−	0.423511i	$-0.860797\pi$
$229$	−27.4172	−1.81178	−0.905891	+	0.423511i	$0.860797\pi$
$230$	0	0
$231$	−8.43032	−0.554674
$232$	0	0
$233$	7.86718i	0.515396i	0.966226	+	0.257698i	$0.0829640\pi$
$233$	7.86718i	0.515396i	−0.966226	+	0.257698i	$0.917036\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.27895i	0.602732i
$238$	0	0
$239$	−9.30404	−0.601828	−0.300914	−	0.953651i	$-0.597292\pi$
$239$	−9.30404	−0.601828	−0.300914	+	0.953651i	$0.597292\pi$
$240$	0	0
$241$	−21.9056	−1.41106	−0.705531	−	0.708679i	$-0.749291\pi$
$241$	−21.9056	−1.41106	−0.705531	+	0.708679i	$0.749291\pi$
$242$	0	0
$243$	14.1712i	0.909084i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 0.364448i	− 0.0231893i
$248$	0	0
$249$	3.62116	0.229482
$250$	0	0
$251$	−7.43032	−0.468997	−0.234499	−	0.972116i	$-0.575345\pi$
$251$	−7.43032	−0.468997	−0.234499	+	0.972116i	$0.575345\pi$
$252$	0	0
$253$	48.2964i	3.03637i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5.05387i	− 0.315251i	−0.987499	−	0.157626i	$-0.949616\pi$
$257$	− 5.05387i	− 0.315251i	0.987499	−	0.157626i	$-0.0503840\pi$
$258$	0	0
$259$	−8.55660	−0.531681
$260$	0	0
$261$	12.2975	0.761196
$262$	0	0
$263$	− 17.7673i	− 1.09558i	−0.836617	−	0.547789i	$-0.815470\pi$
$263$	− 17.7673i	− 1.09558i	0.836617	−	0.547789i	$-0.184530\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.81986i	0.417368i
$268$	0	0
$269$	−9.17776	−0.559578	−0.279789	−	0.960062i	$-0.590264\pi$
$269$	−9.17776	−0.559578	−0.279789	+	0.960062i	$0.590264\pi$
$270$	0	0
$271$	−20.7278	−1.25912	−0.629562	−	0.776950i	$-0.716766\pi$
$271$	−20.7278	−1.25912	−0.629562	+	0.776950i	$0.716766\pi$
$272$	0	0
$273$	− 0.523661i	− 0.0316934i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 24.5895i	− 1.47744i	−0.674012	−	0.738721i	$-0.735430\pi$
$277$	− 24.5895i	− 1.47744i	0.674012	−	0.738721i	$-0.264570\pi$
$278$	0	0
$279$	11.4172	0.683532
$280$	0	0
$281$	3.73436	0.222773	0.111386	−	0.993777i	$-0.464471\pi$
$281$	3.73436	0.222773	0.111386	+	0.993777i	$0.464471\pi$
$282$	0	0
$283$	− 13.0318i	− 0.774663i	−0.921941	−	0.387332i	$-0.873397\pi$
$283$	− 13.0318i	− 0.774663i	0.921941	−	0.387332i	$-0.126603\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.92104i	− 0.172424i
$288$	0	0
$289$	15.5631	0.915479
$290$	0	0
$291$	1.31058	0.0768277
$292$	0	0
$293$	19.9001i	1.16258i	0.813698	+	0.581288i	$0.197451\pi$
$293$	19.9001i	1.16258i	−0.813698	+	0.581288i	$0.802549\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 32.0923i	− 1.86218i
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	−8.86718	−0.511096
$302$	0	0
$303$	13.0264i	0.748347i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 8.02093i	− 0.457779i	−0.973452	−	0.228889i	$-0.926491\pi$
$307$	− 8.02093i	− 0.457779i	0.973452	−	0.228889i	$-0.0735094\pi$
$308$	0	0
$309$	11.6763	0.664243
$310$	0	0
$311$	−16.1778	−0.917357	−0.458678	−	0.888602i	$-0.651677\pi$
$311$	−16.1778	−0.917357	−0.458678	+	0.888602i	$0.651677\pi$
$312$	0	0
$313$	− 7.13828i	− 0.403480i	−0.979439	−	0.201740i	$-0.935340\pi$
$313$	− 7.13828i	− 0.403480i	0.979439	−	0.201740i	$-0.0646595\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3.96052i	− 0.222445i	−0.993796	−	0.111223i	$-0.964523\pi$
$317$	− 3.96052i	− 0.222445i	0.993796	−	0.111223i	$-0.0354766\pi$
$318$	0	0
$319$	−46.1581	−2.58436
$320$	0	0
$321$	19.6146	1.09478
$322$	0	0
$323$	− 1.19869i	− 0.0666970i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 11.1448i	− 0.616310i
$328$	0	0
$329$	16.4238	0.905472
$330$	0	0
$331$	−5.94852	−0.326960	−0.163480	−	0.986547i	$-0.552272\pi$
$331$	−5.94852	−0.326960	−0.163480	+	0.986547i	$0.552272\pi$
$332$	0	0
$333$	− 11.1581i	− 0.611462i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8.90666i	− 0.485176i	−0.970129	−	0.242588i	$-0.922004\pi$
$337$	− 8.90666i	− 0.485176i	0.970129	−	0.242588i	$-0.0779964\pi$
$338$	0	0
$339$	3.55660	0.193168
$340$	0	0
$341$	−42.8541	−2.32068
$342$	0	0
$343$	− 15.0593i	− 0.813127i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.5961i	0.998290i	0.866519	+	0.499145i	$0.166352\pi$
$347$	18.5961i	0.998290i	−0.866519	+	0.499145i	$0.833648\pi$
$348$	0	0
$349$	14.3040	0.765678	0.382839	−	0.923815i	$-0.374946\pi$
$349$	14.3040	0.765678	0.382839	+	0.923815i	$0.374946\pi$
$350$	0	0
$351$	1.99346	0.106403
$352$	0	0
$353$	− 12.7673i	− 0.679534i	−0.940510	−	0.339767i	$-0.889652\pi$
$353$	− 12.7673i	− 0.679534i	0.940510	−	0.339767i	$-0.110348\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.72235i	− 0.0911566i
$358$	0	0
$359$	−2.69596	−0.142287	−0.0711437	−	0.997466i	$-0.522665\pi$
$359$	−2.69596	−0.142287	−0.0711437	+	0.997466i	$0.522665\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	28.0779i	1.47371i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3.43140i	− 0.179118i	−0.995982	−	0.0895589i	$-0.971454\pi$
$367$	− 3.43140i	− 0.179118i	0.995982	−	0.0895589i	$-0.0285457\pi$
$368$	0	0
$369$	3.80915	0.198297
$370$	0	0
$371$	−8.86718	−0.460361
$372$	0	0
$373$	15.5697i	0.806168i	0.915163	+	0.403084i	$0.132062\pi$
$373$	15.5697i	0.806168i	−0.915163	+	0.403084i	$0.867938\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.86718i	− 0.147667i
$378$	0	0
$379$	0.164672	0.00845864	0.00422932	−	0.999991i	$-0.498654\pi$
$379$	0.164672	0.00845864	0.00422932	+	0.999991i	$0.498654\pi$
$380$	0	0
$381$	−13.6146	−0.697498
$382$	0	0
$383$	22.3514i	1.14210i	0.820915	+	0.571051i	$0.193464\pi$
$383$	22.3514i	1.14210i	−0.820915	+	0.571051i	$0.806536\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 11.5631i	− 0.587787i
$388$	0	0
$389$	9.34243	0.473680	0.236840	−	0.971549i	$-0.423888\pi$
$389$	9.34243	0.473680	0.236840	+	0.971549i	$0.423888\pi$
$390$	0	0
$391$	−9.86718	−0.499005
$392$	0	0
$393$	− 1.87372i	− 0.0945167i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 4.83533i	− 0.242678i	−0.992611	−	0.121339i	$-0.961281\pi$
$397$	− 4.83533i	− 0.242678i	0.992611	−	0.121339i	$-0.0387188\pi$
$398$	0	0
$399$	1.43686	0.0719330
$400$	0	0
$401$	37.4622	1.87077	0.935386	−	0.353629i	$-0.115053\pi$
$401$	37.4622	1.87077	0.935386	+	0.353629i	$0.115053\pi$
$402$	0	0
$403$	− 2.66194i	− 0.132601i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	41.8816i	2.07599i
$408$	0	0
$409$	−9.42377	−0.465976	−0.232988	−	0.972480i	$-0.574850\pi$
$409$	−9.42377	−0.465976	−0.232988	+	0.972480i	$0.574850\pi$
$410$	0	0
$411$	−18.9869	−0.936555
$412$	0	0
$413$	15.4159i	0.758568i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	23.8068i	1.16582i
$418$	0	0
$419$	−27.4303	−1.34006	−0.670029	−	0.742335i	$-0.733718\pi$
$419$	−27.4303	−1.34006	−0.670029	+	0.742335i	$0.733718\pi$
$420$	0	0
$421$	18.4303	0.898239	0.449119	−	0.893472i	$-0.351738\pi$
$421$	18.4303	0.898239	0.449119	+	0.893472i	$0.351738\pi$
$422$	0	0
$423$	21.4172i	1.04134i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.56314i	0.0756456i
$428$	0	0
$429$	−2.56314	−0.123750
$430$	0	0
$431$	−15.5117	−0.747170	−0.373585	−	0.927596i	$-0.621872\pi$
$431$	−15.5117	−0.747170	−0.373585	+	0.927596i	$0.621872\pi$
$432$	0	0
$433$	− 31.8870i	− 1.53239i	−0.642607	−	0.766196i	$-0.722147\pi$
$433$	− 31.8870i	− 1.53239i	0.642607	−	0.766196i	$-0.277853\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8.23163i	− 0.393772i
$438$	0	0
$439$	−14.7278	−0.702920	−0.351460	−	0.936203i	$-0.614315\pi$
$439$	−14.7278	−0.702920	−0.351460	+	0.936203i	$0.614315\pi$
$440$	0	0
$441$	8.69596	0.414093
$442$	0	0
$443$	− 6.46087i	− 0.306965i	−0.988151	−	0.153483i	$-0.950951\pi$
$443$	− 6.46087i	− 0.306965i	0.988151	−	0.153483i	$-0.0490489\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 3.22378i	− 0.152480i
$448$	0	0
$449$	−21.2910	−1.00478	−0.502391	−	0.864641i	$-0.667546\pi$
$449$	−21.2910	−1.00478	−0.502391	+	0.864641i	$0.667546\pi$
$450$	0	0
$451$	−14.2975	−0.673243
$452$	0	0
$453$	20.5751i	0.966705i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 19.6465i	− 0.919023i	−0.888172	−	0.459512i	$-0.848024\pi$
$457$	− 19.6465i	− 0.919023i	0.888172	−	0.459512i	$-0.151976\pi$
$458$	0	0
$459$	6.55660	0.306036
$460$	0	0
$461$	1.16467	0.0542442	0.0271221	−	0.999632i	$-0.491366\pi$
$461$	1.16467	0.0542442	0.0271221	+	0.999632i	$0.491366\pi$
$462$	0	0
$463$	5.15375i	0.239515i	0.992803	+	0.119758i	$0.0382117\pi$
$463$	5.15375i	0.239515i	−0.992803	+	0.119758i	$0.961788\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4.17122i	− 0.193021i	−0.995332	−	0.0965104i	$-0.969232\pi$
$467$	− 4.17122i	− 0.193021i	0.995332	−	0.0965104i	$-0.0307681\pi$
$468$	0	0
$469$	−14.3040	−0.660499
$470$	0	0
$471$	−19.6146	−0.903794
$472$	0	0
$473$	43.4018i	1.99561i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 11.5631i	− 0.529440i
$478$	0	0
$479$	2.35552	0.107626	0.0538132	−	0.998551i	$-0.482862\pi$
$479$	2.35552	0.107626	0.0538132	+	0.998551i	$0.482862\pi$
$480$	0	0
$481$	−2.60153	−0.118620
$482$	0	0
$483$	− 11.8277i	− 0.538179i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	19.9056i	0.902008i	0.892522	+	0.451004i	$0.148934\pi$
$487$	19.9056i	0.902008i	−0.892522	+	0.451004i	$0.851066\pi$
$488$	0	0
$489$	9.93175	0.449129
$490$	0	0
$491$	7.90557	0.356773	0.178387	−	0.983960i	$-0.442912\pi$
$491$	7.90557	0.356773	0.178387	+	0.983960i	$0.442912\pi$
$492$	0	0
$493$	− 9.43032i	− 0.424720i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.54875i	− 0.114327i
$498$	0	0
$499$	35.7147	1.59881	0.799405	−	0.600792i	$-0.205148\pi$
$499$	35.7147	1.59881	0.799405	+	0.600792i	$0.205148\pi$
$500$	0	0
$501$	−8.98691	−0.401506
$502$	0	0
$503$	30.7991i	1.37327i	0.727004	+	0.686633i	$0.240912\pi$
$503$	30.7991i	1.37327i	−0.727004	+	0.686633i	$0.759088\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	15.4238i	0.684994i
$508$	0	0
$509$	−17.5631	−0.778472	−0.389236	−	0.921138i	$-0.627261\pi$
$509$	−17.5631	−0.778472	−0.389236	+	0.921138i	$0.627261\pi$
$510$	0	0
$511$	3.00000	0.132712
$512$	0	0
$513$	5.46980i	0.241497i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 80.3887i	− 3.53549i
$518$	0	0
$519$	17.6763	0.775905
$520$	0	0
$521$	−24.6081	−1.07810	−0.539050	−	0.842274i	$-0.681217\pi$
$521$	−24.6081	−1.07810	−0.539050	+	0.842274i	$0.681217\pi$
$522$	0	0
$523$	43.7147i	1.91151i	0.294162	+	0.955756i	$0.404960\pi$
$523$	43.7147i	1.91151i	−0.294162	+	0.955756i	$0.595040\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8.75529i	− 0.381386i
$528$	0	0
$529$	−44.7597	−1.94607
$530$	0	0
$531$	−20.1030	−0.872394
$532$	0	0
$533$	− 0.888109i	− 0.0384683i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 15.8857i	− 0.685520i
$538$	0	0
$539$	−32.6399	−1.40590
$540$	0	0
$541$	−8.38538	−0.360516	−0.180258	−	0.983619i	$-0.557693\pi$
$541$	−8.38538	−0.360516	−0.180258	+	0.983619i	$0.557693\pi$
$542$	0	0
$543$	20.0593i	0.860828i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	26.3040i	1.12468i	0.826906	+	0.562340i	$0.190099\pi$
$547$	26.3040i	1.12468i	−0.826906	+	0.562340i	$0.809901\pi$
$548$	0	0
$549$	−2.03839	−0.0869965
$550$	0	0
$551$	7.86718	0.335153
$552$	0	0
$553$	− 9.27895i	− 0.394581i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.66740i	0.113021i	0.998402	+	0.0565107i	$0.0179975\pi$
$557$	2.66740i	0.113021i	−0.998402	+	0.0565107i	$0.982002\pi$
$558$	0	0
$559$	−2.69596	−0.114027
$560$	0	0
$561$	−8.43032	−0.355928
$562$	0	0
$563$	− 37.7751i	− 1.59203i	−0.605276	−	0.796016i	$-0.706937\pi$
$563$	− 37.7751i	− 1.59203i	0.605276	−	0.796016i	$-0.293063\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.23817i	0.0939943i
$568$	0	0
$569$	−38.8606	−1.62912	−0.814561	−	0.580078i	$-0.803022\pi$
$569$	−38.8606	−1.62912	−0.814561	+	0.580078i	$0.803022\pi$
$570$	0	0
$571$	4.17122	0.174560	0.0872800	−	0.996184i	$-0.472183\pi$
$571$	4.17122	0.174560	0.0872800	+	0.996184i	$0.472183\pi$
$572$	0	0
$573$	− 4.95398i	− 0.206955i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	39.3413i	1.63780i	0.573935	+	0.818901i	$0.305416\pi$
$577$	39.3413i	1.63780i	−0.573935	+	0.818901i	$0.694584\pi$
$578$	0	0
$579$	−8.49489	−0.353035
$580$	0	0
$581$	−3.62116	−0.150231
$582$	0	0
$583$	43.4018i	1.79752i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 17.8672i	− 0.737457i	−0.929537	−	0.368729i	$-0.879793\pi$
$587$	− 17.8672i	− 0.737457i	0.929537	−	0.368729i	$-0.120207\pi$
$588$	0	0
$589$	7.30404	0.300958
$590$	0	0
$591$	24.5435	1.00959
$592$	0	0
$593$	13.0803i	0.537142i	0.963260	+	0.268571i	$0.0865514\pi$
$593$	13.0803i	0.537142i	−0.963260	+	0.268571i	$0.913449\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 10.4776i	− 0.428821i
$598$	0	0
$599$	−10.7278	−0.438326	−0.219163	−	0.975688i	$-0.570333\pi$
$599$	−10.7278	−0.438326	−0.219163	+	0.975688i	$0.570333\pi$
$600$	0	0
$601$	39.8026	1.62358	0.811791	−	0.583948i	$-0.198493\pi$
$601$	39.8026	1.62358	0.811791	+	0.583948i	$0.198493\pi$
$602$	0	0
$603$	− 18.6530i	− 0.759609i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	43.0318i	1.74661i	0.487175	+	0.873304i	$0.338027\pi$
$607$	43.0318i	1.74661i	−0.487175	+	0.873304i	$0.661973\pi$
$608$	0	0
$609$	11.3040	0.458063
$610$	0	0
$611$	4.99346	0.202014
$612$	0	0
$613$	15.4722i	0.624915i	0.949932	+	0.312458i	$0.101152\pi$
$613$	15.4722i	0.624915i	−0.949932	+	0.312458i	$0.898848\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.5237i	1.43013i	0.699059	+	0.715064i	$0.253603\pi$
$617$	35.5237i	1.43013i	−0.699059	+	0.715064i	$0.746397\pi$
$618$	0	0
$619$	15.0449	0.604707	0.302354	−	0.953196i	$-0.402228\pi$
$619$	15.0449	0.604707	0.302354	+	0.953196i	$0.402228\pi$
$620$	0	0
$621$	45.0253	1.80680
$622$	0	0
$623$	− 6.81986i	− 0.273232i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 7.03293i	− 0.280868i
$628$	0	0
$629$	−8.55660	−0.341174
$630$	0	0
$631$	35.6399	1.41880	0.709402	−	0.704805i	$-0.248965\pi$
$631$	35.6399	1.41880	0.709402	+	0.704805i	$0.248965\pi$
$632$	0	0
$633$	− 22.4567i	− 0.892574i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.02748i	− 0.0803315i
$638$	0	0
$639$	3.32367	0.131482
$640$	0	0
$641$	6.98691	0.275966	0.137983	−	0.990435i	$-0.455938\pi$
$641$	6.98691	0.275966	0.137983	+	0.990435i	$0.455938\pi$
$642$	0	0
$643$	− 3.74744i	− 0.147785i	−0.997266	−	0.0738924i	$-0.976458\pi$
$643$	− 3.74744i	− 0.147785i	0.997266	−	0.0738924i	$-0.0235421\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	34.1658i	1.34319i	0.740916	+	0.671597i	$0.234391\pi$
$647$	34.1658i	1.34319i	−0.740916	+	0.671597i	$0.765609\pi$
$648$	0	0
$649$	75.4556	2.96189
$650$	0	0
$651$	10.4949	0.411327
$652$	0	0
$653$	13.5446i	0.530041i	0.964243	+	0.265020i	$0.0853787\pi$
$653$	13.5446i	0.530041i	−0.964243	+	0.265020i	$0.914621\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.91211i	0.152626i
$658$	0	0
$659$	22.6334	0.881671	0.440836	−	0.897588i	$-0.354682\pi$
$659$	22.6334	0.881671	0.440836	+	0.897588i	$0.354682\pi$
$660$	0	0
$661$	−3.96161	−0.154089	−0.0770443	−	0.997028i	$-0.524548\pi$
$661$	−3.96161	−0.154089	−0.0770443	+	0.997028i	$0.524548\pi$
$662$	0	0
$663$	− 0.523661i	− 0.0203373i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 64.7597i	− 2.50750i
$668$	0	0
$669$	−5.11973	−0.197940
$670$	0	0
$671$	7.65102	0.295365
$672$	0	0
$673$	− 28.9605i	− 1.11635i	−0.829725	−	0.558173i	$-0.811503\pi$
$673$	− 28.9605i	− 1.11635i	0.829725	−	0.558173i	$-0.188497\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.7518i	0.951290i	0.879637	+	0.475645i	$0.157785\pi$
$677$	24.7518i	0.951290i	−0.879637	+	0.475645i	$0.842215\pi$
$678$	0	0
$679$	−1.31058	−0.0502955
$680$	0	0
$681$	−25.8475	−0.990480
$682$	0	0
$683$	− 40.0593i	− 1.53283i	−0.642347	−	0.766414i	$-0.722039\pi$
$683$	− 40.0593i	− 1.53283i	0.642347	−	0.766414i	$-0.277961\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 32.8648i	− 1.25387i
$688$	0	0
$689$	−2.69596	−0.102708
$690$	0	0
$691$	35.2162	1.33969	0.669843	−	0.742503i	$-0.266361\pi$
$691$	35.2162	1.33969	0.669843	+	0.742503i	$0.266361\pi$
$692$	0	0
$693$	10.9935i	0.417607i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 2.92104i	− 0.110642i
$698$	0	0
$699$	−9.43032	−0.356687
$700$	0	0
$701$	−38.1283	−1.44008	−0.720042	−	0.693930i	$-0.755878\pi$
$701$	−38.1283	−1.44008	−0.720042	+	0.693930i	$0.755878\pi$
$702$	0	0
$703$	− 7.13828i	− 0.269225i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 13.0264i	− 0.489908i
$708$	0	0
$709$	6.08789	0.228635	0.114318	−	0.993444i	$-0.463532\pi$
$709$	6.08789	0.228635	0.114318	+	0.993444i	$0.463532\pi$
$710$	0	0
$711$	12.1001	0.453789
$712$	0	0
$713$	− 60.1241i	− 2.25167i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 11.1527i	− 0.416504i
$718$	0	0
$719$	12.1647	0.453666	0.226833	−	0.973934i	$-0.427163\pi$
$719$	12.1647	0.453666	0.226833	+	0.973934i	$0.427163\pi$
$720$	0	0
$721$	−11.6763	−0.434849
$722$	0	0
$723$	− 26.2580i	− 0.976546i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 25.6859i	− 0.952639i	−0.879272	−	0.476320i	$-0.841971\pi$
$727$	− 25.6859i	− 0.952639i	0.879272	−	0.476320i	$-0.158029\pi$
$728$	0	0
$729$	−22.5884	−0.836609
$730$	0	0
$731$	−8.86718	−0.327964
$732$	0	0
$733$	43.4502i	1.60487i	0.596741	+	0.802434i	$0.296462\pi$
$733$	43.4502i	1.60487i	−0.596741	+	0.802434i	$0.703538\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	70.0133i	2.57897i
$738$	0	0
$739$	43.4172	1.59713	0.798564	−	0.601910i	$-0.205593\pi$
$739$	43.4172	1.59713	0.798564	+	0.601910i	$0.205593\pi$
$740$	0	0
$741$	0.436861	0.0160485
$742$	0	0
$743$	− 26.6709i	− 0.978459i	−0.872155	−	0.489230i	$-0.837278\pi$
$743$	− 26.6709i	− 0.978459i	0.872155	−	0.489230i	$-0.162722\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4.72214i	− 0.172774i
$748$	0	0
$749$	−19.6146	−0.716703
$750$	0	0
$751$	−1.52674	−0.0557114	−0.0278557	−	0.999612i	$-0.508868\pi$
$751$	−1.52674	−0.0557114	−0.0278557	+	0.999612i	$0.508868\pi$
$752$	0	0
$753$	− 8.90666i	− 0.324577i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 9.38191i	− 0.340991i	−0.985358	−	0.170496i	$-0.945463\pi$
$757$	− 9.38191i	− 0.340991i	0.985358	−	0.170496i	$-0.0545369\pi$
$758$	0	0
$759$	−57.8925	−2.10136
$760$	0	0
$761$	−6.70251	−0.242966	−0.121483	−	0.992594i	$-0.538765\pi$
$761$	−6.70251	−0.242966	−0.121483	+	0.992594i	$0.538765\pi$
$762$	0	0
$763$	11.1448i	0.403470i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.68703i	0.169239i
$768$	0	0
$769$	36.0637	1.30049	0.650245	−	0.759724i	$-0.274666\pi$
$769$	36.0637	1.30049	0.650245	+	0.759724i	$0.274666\pi$
$770$	0	0
$771$	6.05802	0.218174
$772$	0	0
$773$	11.6763i	0.419968i	0.977705	+	0.209984i	$0.0673413\pi$
$773$	11.6763i	0.419968i	−0.977705	+	0.209984i	$0.932659\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 10.2567i	− 0.367958i
$778$	0	0
$779$	2.43686	0.0873096
$780$	0	0
$781$	−12.4753	−0.446400
$782$	0	0
$783$	43.0318i	1.53783i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	43.1880i	1.53949i	0.638354	+	0.769743i	$0.279616\pi$
$787$	43.1880i	1.53949i	−0.638354	+	0.769743i	$0.720384\pi$
$788$	0	0
$789$	21.2975	0.758211
$790$	0	0
$791$	−3.55660	−0.126458
$792$	0	0
$793$	0.475254i	0.0168768i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.1658i	1.03310i	0.856256	+	0.516552i	$0.172785\pi$
$797$	29.1658i	1.03310i	−0.856256	+	0.516552i	$0.827215\pi$
$798$	0	0
$799$	16.4238	0.581031
$800$	0	0
$801$	8.89335	0.314231
$802$	0	0
$803$	− 14.6840i	− 0.518186i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 11.0013i	− 0.387264i
$808$	0	0
$809$	−2.43032	−0.0854454	−0.0427227	−	0.999087i	$-0.513603\pi$
$809$	−2.43032	−0.0854454	−0.0427227	+	0.999087i	$0.513603\pi$
$810$	0	0
$811$	−42.2779	−1.48458	−0.742288	−	0.670081i	$-0.766259\pi$
$811$	−42.2779	−1.48458	−0.742288	+	0.670081i	$0.766259\pi$
$812$	0	0
$813$	− 24.8462i	− 0.871396i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 7.39738i	− 0.258802i
$818$	0	0
$819$	−0.682874	−0.0238616
$820$	0	0
$821$	52.5136	1.83274	0.916369	−	0.400334i	$-0.131106\pi$
$821$	52.5136	1.83274	0.916369	+	0.400334i	$0.131106\pi$
$822$	0	0
$823$	36.3150i	1.26586i	0.774209	+	0.632930i	$0.218148\pi$
$823$	36.3150i	1.26586i	−0.774209	+	0.632930i	$0.781852\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	46.3150i	1.61053i	0.592916	+	0.805264i	$0.297977\pi$
$827$	46.3150i	1.61053i	−0.592916	+	0.805264i	$0.702023\pi$
$828$	0	0
$829$	−20.0899	−0.697750	−0.348875	−	0.937169i	$-0.613436\pi$
$829$	−20.0899	−0.697750	−0.348875	+	0.937169i	$0.613436\pi$
$830$	0	0
$831$	29.4753	1.02249
$832$	0	0
$833$	− 6.66849i	− 0.231049i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	39.9516i	1.38093i
$838$	0	0
$839$	10.5930	0.365711	0.182855	−	0.983140i	$-0.441466\pi$
$839$	10.5930	0.365711	0.182855	+	0.983140i	$0.441466\pi$
$840$	0	0
$841$	32.8925	1.13422
$842$	0	0
$843$	4.47634i	0.154173i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 28.0779i	− 0.964767i
$848$	0	0
$849$	15.6212	0.536117
$850$	0	0
$851$	−58.7597	−2.01426
$852$	0	0
$853$	− 40.0648i	− 1.37179i	−0.727700	−	0.685896i	$-0.759410\pi$
$853$	− 40.0648i	− 1.37179i	0.727700	−	0.685896i	$-0.240590\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 38.3019i	− 1.30837i	−0.756336	−	0.654183i	$-0.773012\pi$
$857$	− 38.3019i	− 1.30837i	0.756336	−	0.654183i	$-0.226988\pi$
$858$	0	0
$859$	39.9056	1.36156	0.680780	−	0.732488i	$-0.261641\pi$
$859$	39.9056	1.36156	0.680780	+	0.732488i	$0.261641\pi$
$860$	0	0
$861$	3.50143	0.119328
$862$	0	0
$863$	− 23.3315i	− 0.794214i	−0.917772	−	0.397107i	$-0.870014\pi$
$863$	− 23.3315i	− 0.794214i	0.917772	−	0.397107i	$-0.129986\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	18.6554i	0.633571i
$868$	0	0
$869$	−45.4172	−1.54067
$870$	0	0
$871$	−4.34898	−0.147359
$872$	0	0
$873$	− 1.70905i	− 0.0578426i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 10.4105i	− 0.351537i	−0.984432	−	0.175768i	$-0.943759\pi$
$877$	− 10.4105i	− 0.351537i	0.984432	−	0.175768i	$-0.0562410\pi$
$878$	0	0
$879$	−23.8541	−0.804578
$880$	0	0
$881$	−23.7662	−0.800704	−0.400352	−	0.916361i	$-0.631112\pi$
$881$	−23.7662	−0.800704	−0.400352	+	0.916361i	$0.631112\pi$
$882$	0	0
$883$	− 28.3908i	− 0.955428i	−0.878515	−	0.477714i	$-0.841466\pi$
$883$	− 28.3908i	− 0.955428i	0.878515	−	0.477714i	$-0.158534\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.5644i	1.09341i	0.837326	+	0.546703i	$0.184117\pi$
$887$	32.5644i	1.09341i	−0.837326	+	0.546703i	$0.815883\pi$
$888$	0	0
$889$	13.6146	0.456620
$890$	0	0
$891$	10.9551	0.367008
$892$	0	0
$893$	13.7014i	0.458501i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3.59607i	− 0.120069i
$898$	0	0
$899$	57.4622	1.91647
$900$	0	0
$901$	−8.86718	−0.295409
$902$	0	0
$903$	− 10.6290i	− 0.353711i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 57.2109i	− 1.89966i	−0.312771	−	0.949829i	$-0.601257\pi$
$907$	− 57.2109i	− 1.89966i	0.312771	−	0.949829i	$-0.398743\pi$
$908$	0	0
$909$	16.9869	0.563420
$910$	0	0
$911$	−42.0617	−1.39357	−0.696783	−	0.717282i	$-0.745386\pi$
$911$	−42.0617	−1.39357	−0.696783	+	0.717282i	$0.745386\pi$
$912$	0	0
$913$	17.7243i	0.586590i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.87372i	0.0618757i
$918$	0	0
$919$	−5.68287	−0.187461	−0.0937304	−	0.995598i	$-0.529879\pi$
$919$	−5.68287	−0.187461	−0.0937304	+	0.995598i	$0.529879\pi$
$920$	0	0
$921$	9.61462	0.316813
$922$	0	0
$923$	− 0.774918i	− 0.0255067i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 15.2264i	− 0.500100i
$928$	0	0
$929$	30.9289	1.01474	0.507372	−	0.861727i	$-0.330617\pi$
$929$	30.9289	1.01474	0.507372	+	0.861727i	$0.330617\pi$
$930$	0	0
$931$	5.56314	0.182325
$932$	0	0
$933$	− 19.3921i	− 0.634870i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 44.5357i	− 1.45492i	−0.686152	−	0.727458i	$-0.740701\pi$
$937$	− 44.5357i	− 1.45492i	0.686152	−	0.727458i	$-0.259299\pi$
$938$	0	0
$939$	8.55660	0.279234
$940$	0	0
$941$	−2.08134	−0.0678498	−0.0339249	−	0.999424i	$-0.510801\pi$
$941$	−2.08134	−0.0678498	−0.0339249	+	0.999424i	$0.510801\pi$
$942$	0	0
$943$	− 20.0593i	− 0.653221i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.7189i	1.71313i	0.516036	+	0.856567i	$0.327407\pi$
$947$	52.7189i	1.71313i	−0.516036	+	0.856567i	$0.672593\pi$
$948$	0	0
$949$	0.912115	0.0296085
$950$	0	0
$951$	4.74744	0.153946
$952$	0	0
$953$	− 3.22071i	− 0.104329i	−0.998639	−	0.0521645i	$-0.983388\pi$
$953$	− 3.22071i	− 0.104329i	0.998639	−	0.0521645i	$-0.0166120\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 55.3293i	− 1.78854i
$958$	0	0
$959$	18.9869	0.613119
$960$	0	0
$961$	22.3490	0.720935
$962$	0	0
$963$	− 25.5782i	− 0.824247i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.65956i	0.0855256i	0.999085	+	0.0427628i	$0.0136160\pi$
$967$	2.65956i	0.0855256i	−0.999085	+	0.0427628i	$0.986384\pi$
$968$	0	0
$969$	1.43686	0.0461586
$970$	0	0
$971$	35.1263	1.12726	0.563628	−	0.826029i	$-0.309405\pi$
$971$	35.1263	1.12726	0.563628	+	0.826029i	$0.309405\pi$
$972$	0	0
$973$	− 23.8068i	− 0.763210i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.1668i	0.581209i	0.956843	+	0.290604i	$0.0938563\pi$
$977$	18.1668i	0.581209i	−0.956843	+	0.290604i	$0.906144\pi$
$978$	0	0
$979$	−33.3808	−1.06686
$980$	0	0
$981$	−14.5333	−0.464012
$982$	0	0
$983$	− 48.4556i	− 1.54549i	−0.634714	−	0.772747i	$-0.718882\pi$
$983$	− 48.4556i	− 1.54549i	0.634714	−	0.772747i	$-0.281118\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	19.6870i	0.626645i
$988$	0	0
$989$	−60.8925	−1.93627
$990$	0	0
$991$	29.8157	0.947127	0.473563	−	0.880760i	$-0.342967\pi$
$991$	29.8157	0.947127	0.473563	+	0.880760i	$0.342967\pi$
$992$	0	0
$993$	− 7.13044i	− 0.226278i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	29.6894i	0.940273i	0.882594	+	0.470137i	$0.155795\pi$
$997$	29.6894i	0.940273i	−0.882594	+	0.470137i	$0.844205\pi$
$998$	0	0
$999$	39.0449	1.23533

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.c.g.1749.5		6
5.2	odd	4		1900.2.a.f.1.3	✓	3
5.3	odd	4		1900.2.a.h.1.1	yes	3
5.4	even	2	inner	1900.2.c.g.1749.2		6
20.3	even	4		7600.2.a.bj.1.3		3
20.7	even	4		7600.2.a.by.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.2.a.f.1.3	✓	3	5.2	odd	4
1900.2.a.h.1.1	yes	3	5.3	odd	4
1900.2.c.g.1749.2		6	5.4	even	2	inner
1900.2.c.g.1749.5		6	1.1	even	1	trivial
7600.2.a.bj.1.3		3	20.3	even	4
7600.2.a.by.1.1		3	20.7	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.19869i q^{3} -1.19869i q^{7} +1.56314 q^{9} +O(q^{10})\)	\(q+1.19869i q^{3} -1.19869i q^{7} +1.56314 q^{9} -5.86718 q^{11} -0.364448i q^{13} -1.19869i q^{17} +1.00000 q^{19} +1.43686 q^{21} -8.23163i q^{23} +5.46980i q^{27} +7.86718 q^{29} +7.30404 q^{31} -7.03293i q^{33} -7.13828i q^{37} +0.436861 q^{39} +2.43686 q^{41} -7.39738i q^{43} +13.7014i q^{47} +5.56314 q^{49} +1.43686 q^{51} -7.39738i q^{53} +1.19869i q^{57} -12.8606 q^{59} -1.30404 q^{61} -1.87372i q^{63} -11.9330i q^{67} +9.86718 q^{69} +2.12628 q^{71} +2.50273i q^{73} +7.03293i q^{77} +7.74090 q^{79} -1.86718 q^{81} -3.02093i q^{83} +9.43032i q^{87} +5.68942 q^{89} -0.436861 q^{91} +8.75529i q^{93} -1.09334i q^{97} -9.17122 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{9}+O(q^{10}) \) 6 * q - 2 * q^9	\( 6 q - 2 q^{9} - 2 q^{11} + 6 q^{19} + 20 q^{21} + 14 q^{29} + 22 q^{31} + 14 q^{39} + 26 q^{41} + 22 q^{49} + 20 q^{51} + 12 q^{59} + 14 q^{61} + 26 q^{69} - 10 q^{71} + 36 q^{79} + 22 q^{81} - 14 q^{91}+O(q^{100}) \) 6 * q - 2 * q^9 - 2 * q^11 + 6 * q^19 + 20 * q^21 + 14 * q^29 + 22 * q^31 + 14 * q^39 + 26 * q^41 + 22 * q^49 + 20 * q^51 + 12 * q^59 + 14 * q^61 + 26 * q^69 - 10 * q^71 + 36 * q^79 + 22 * q^81 - 14 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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