LMFDB - Embedded newform 5700.2.f.n.3649.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5700,2,Mod(3649,5700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5700, 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("5700.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5700.f (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:	$45.5147291521$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.3
Root			$-1.30278i$ of defining polynomial
Character	$\chi$	$=$	5700.3649
Dual form			5700.2.f.n.3649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.60555i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.60555i q^{7} -1.00000 q^{9} -4.60555 q^{11} -4.60555i q^{13} +2.00000i q^{17} +1.00000 q^{19} +2.60555 q^{21} +2.00000i q^{23} -1.00000i q^{27} -2.60555 q^{29} +4.00000 q^{31} -4.60555i q^{33} +3.39445i q^{37} +4.60555 q^{39} +6.60555 q^{41} -10.6056i q^{43} +6.00000i q^{47} +0.211103 q^{49} -2.00000 q^{51} +1.00000i q^{57} -5.21110 q^{59} -7.21110 q^{61} +2.60555i q^{63} +4.00000i q^{67} -2.00000 q^{69} -9.21110 q^{71} -6.00000i q^{73} +12.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +11.2111i q^{83} -2.60555i q^{87} -6.60555 q^{89} -12.0000 q^{91} +4.00000i q^{93} +16.6056i q^{97} +4.60555 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 4 q^{11} + 4 q^{19} - 4 q^{21} + 4 q^{29} + 16 q^{31} + 4 q^{39} + 12 q^{41} - 28 q^{49} - 8 q^{51} + 8 q^{59} - 8 q^{69} - 8 q^{71} - 32 q^{79} + 4 q^{81} - 12 q^{89} - 48 q^{91} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 - 4 * q^11 + 4 * q^19 - 4 * q^21 + 4 * q^29 + 16 * q^31 + 4 * q^39 + 12 * q^41 - 28 * q^49 - 8 * q^51 + 8 * q^59 - 8 * q^69 - 8 * q^71 - 32 * q^79 + 4 * q^81 - 12 * q^89 - 48 * q^91 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5700\mathbb{Z}\right)^\times$.

$n$	$1901$	$2851$	$3877$	$4201$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.60555i	− 0.984806i	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	− 2.60555i	− 0.984806i	0.870367	−	0.492403i	$-0.163881\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.60555	−1.38863	−0.694313	−	0.719673i	$-0.744292\pi$
$11$	−4.60555	−1.38863	−0.694313	+	0.719673i	$0.744292\pi$
$12$	0	0
$13$	− 4.60555i	− 1.27735i	−0.769477	−	0.638675i	$-0.779483\pi$
$13$	− 4.60555i	− 1.27735i	0.769477	−	0.638675i	$-0.220517\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.60555	0.568578
$22$	0	0
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−2.60555	−0.483839	−0.241919	−	0.970296i	$-0.577777\pi$
$29$	−2.60555	−0.483839	−0.241919	+	0.970296i	$0.577777\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	− 4.60555i	− 0.801724i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.39445i	0.558044i	0.960285	+	0.279022i	$0.0900102\pi$
$37$	3.39445i	0.558044i	−0.960285	+	0.279022i	$0.909990\pi$
$38$	0	0
$39$	4.60555	0.737478
$40$	0	0
$41$	6.60555	1.03161	0.515807	−	0.856705i	$-0.327492\pi$
$41$	6.60555	1.03161	0.515807	+	0.856705i	$0.327492\pi$
$42$	0	0
$43$	− 10.6056i	− 1.61733i	−0.588268	−	0.808666i	$-0.700190\pi$
$43$	− 10.6056i	− 1.61733i	0.588268	−	0.808666i	$-0.299810\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	0.211103	0.0301575
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000i	0.132453i
$58$	0	0
$59$	−5.21110	−0.678428	−0.339214	−	0.940709i	$-0.610161\pi$
$59$	−5.21110	−0.678428	−0.339214	+	0.940709i	$0.610161\pi$
$60$	0	0
$61$	−7.21110	−0.923287	−0.461644	−	0.887066i	$-0.652740\pi$
$61$	−7.21110	−0.923287	−0.461644	+	0.887066i	$0.652740\pi$
$62$	0	0
$63$	2.60555i	0.328269i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	−9.21110	−1.09316	−0.546578	−	0.837408i	$-0.684070\pi$
$71$	−9.21110	−1.09316	−0.546578	+	0.837408i	$0.684070\pi$
$72$	0	0
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.0000i	1.36753i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.2111i	1.23058i	0.788301	+	0.615289i	$0.210961\pi$
$83$	11.2111i	1.23058i	−0.788301	+	0.615289i	$0.789039\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.60555i	− 0.279344i
$88$	0	0
$89$	−6.60555	−0.700187	−0.350094	−	0.936715i	$-0.613850\pi$
$89$	−6.60555	−0.700187	−0.350094	+	0.936715i	$0.613850\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	0	0
$93$	4.00000i	0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.6056i	1.68604i	0.537884	+	0.843019i	$0.319224\pi$
$97$	16.6056i	1.68604i	−0.537884	+	0.843019i	$0.680776\pi$
$98$	0	0
$99$	4.60555	0.462875
$100$	0	0
$101$	−7.21110	−0.717532	−0.358766	−	0.933428i	$-0.616802\pi$
$101$	−7.21110	−0.717532	−0.358766	+	0.933428i	$0.616802\pi$
$102$	0	0
$103$	− 18.4222i	− 1.81519i	−0.419843	−	0.907597i	$-0.637915\pi$
$103$	− 18.4222i	− 1.81519i	0.419843	−	0.907597i	$-0.362085\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.4222i	1.78094i	0.455040	+	0.890471i	$0.349625\pi$
$107$	18.4222i	1.78094i	−0.455040	+	0.890471i	$0.650375\pi$
$108$	0	0
$109$	11.2111	1.07383	0.536914	−	0.843637i	$-0.319590\pi$
$109$	11.2111	1.07383	0.536914	+	0.843637i	$0.319590\pi$
$110$	0	0
$111$	−3.39445	−0.322187
$112$	0	0
$113$	1.21110i	0.113931i	0.998376	+	0.0569655i	$0.0181425\pi$
$113$	1.21110i	0.113931i	−0.998376	+	0.0569655i	$0.981858\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.60555i	0.425783i
$118$	0	0
$119$	5.21110	0.477701
$120$	0	0
$121$	10.2111	0.928282
$122$	0	0
$123$	6.60555i	0.595603i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	18.4222i	1.63471i	0.576137	+	0.817353i	$0.304559\pi$
$127$	18.4222i	1.63471i	−0.576137	+	0.817353i	$0.695441\pi$
$128$	0	0
$129$	10.6056	0.933767
$130$	0	0
$131$	−15.3944	−1.34502	−0.672510	−	0.740088i	$-0.734784\pi$
$131$	−15.3944	−1.34502	−0.672510	+	0.740088i	$0.734784\pi$
$132$	0	0
$133$	− 2.60555i	− 0.225930i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.4222i	− 1.06130i	−0.847591	−	0.530650i	$-0.821948\pi$
$137$	− 12.4222i	− 1.06130i	0.847591	−	0.530650i	$-0.178052\pi$
$138$	0	0
$139$	−9.21110	−0.781276	−0.390638	−	0.920544i	$-0.627745\pi$
$139$	−9.21110	−0.781276	−0.390638	+	0.920544i	$0.627745\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	21.2111i	1.77376i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.211103i	0.0174114i
$148$	0	0
$149$	−16.4222	−1.34536	−0.672680	−	0.739934i	$-0.734857\pi$
$149$	−16.4222	−1.34536	−0.672680	+	0.739934i	$0.734857\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.2111i	1.21398i	0.794710	+	0.606989i	$0.207623\pi$
$157$	15.2111i	1.21398i	−0.794710	+	0.606989i	$0.792377\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.21110	0.410692
$162$	0	0
$163$	17.0278i	1.33372i	0.745184	+	0.666858i	$0.232361\pi$
$163$	17.0278i	1.33372i	−0.745184	+	0.666858i	$0.767639\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 6.78890i	− 0.525341i	−0.964886	−	0.262670i	$-0.915397\pi$
$167$	− 6.78890i	− 0.525341i	0.964886	−	0.262670i	$-0.0846032\pi$
$168$	0	0
$169$	−8.21110	−0.631623
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	9.21110i	0.700307i	0.936692	+	0.350154i	$0.113871\pi$
$173$	9.21110i	0.700307i	−0.936692	+	0.350154i	$0.886129\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 5.21110i	− 0.391690i
$178$	0	0
$179$	21.2111	1.58539	0.792696	−	0.609617i	$-0.208677\pi$
$179$	21.2111	1.58539	0.792696	+	0.609617i	$0.208677\pi$
$180$	0	0
$181$	−0.788897	−0.0586383	−0.0293191	−	0.999570i	$-0.509334\pi$
$181$	−0.788897	−0.0586383	−0.0293191	+	0.999570i	$0.509334\pi$
$182$	0	0
$183$	− 7.21110i	− 0.533060i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 9.21110i	− 0.673583i
$188$	0	0
$189$	−2.60555	−0.189526
$190$	0	0
$191$	−5.81665	−0.420878	−0.210439	−	0.977607i	$-0.567489\pi$
$191$	−5.81665	−0.420878	−0.210439	+	0.977607i	$0.567489\pi$
$192$	0	0
$193$	0.605551i	0.0435885i	0.999762	+	0.0217943i	$0.00693788\pi$
$193$	0.605551i	0.0435885i	−0.999762	+	0.0217943i	$0.993062\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.0000i	0.997459i	0.866758	+	0.498729i	$0.166200\pi$
$197$	14.0000i	0.997459i	−0.866758	+	0.498729i	$0.833800\pi$
$198$	0	0
$199$	−17.2111	−1.22006	−0.610031	−	0.792377i	$-0.708843\pi$
$199$	−17.2111	−1.22006	−0.610031	+	0.792377i	$0.708843\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	6.78890i	0.476487i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.00000i	− 0.139010i
$208$	0	0
$209$	−4.60555	−0.318573
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	− 9.21110i	− 0.631134i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 10.4222i	− 0.707505i
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	9.21110	0.619606
$222$	0	0
$223$	10.4222i	0.697922i	0.937137	+	0.348961i	$0.113466\pi$
$223$	10.4222i	0.697922i	−0.937137	+	0.348961i	$0.886534\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 13.2111i	− 0.876852i	−0.898767	−	0.438426i	$-0.855536\pi$
$227$	− 13.2111i	− 0.876852i	0.898767	−	0.438426i	$-0.144464\pi$
$228$	0	0
$229$	3.21110	0.212196	0.106098	−	0.994356i	$-0.466164\pi$
$229$	3.21110	0.212196	0.106098	+	0.994356i	$0.466164\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	16.4222i	1.07585i	0.842991	+	0.537927i	$0.180792\pi$
$233$	16.4222i	1.07585i	−0.842991	+	0.537927i	$0.819208\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	−25.8167	−1.66994	−0.834970	−	0.550295i	$-0.814515\pi$
$239$	−25.8167	−1.66994	−0.834970	+	0.550295i	$0.814515\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4.60555i	− 0.293044i
$248$	0	0
$249$	−11.2111	−0.710475
$250$	0	0
$251$	−6.18335	−0.390289	−0.195145	−	0.980774i	$-0.562518\pi$
$251$	−6.18335	−0.390289	−0.195145	+	0.980774i	$0.562518\pi$
$252$	0	0
$253$	− 9.21110i	− 0.579097i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.78890i	0.173967i	0.996210	+	0.0869833i	$0.0277227\pi$
$257$	2.78890i	0.173967i	−0.996210	+	0.0869833i	$0.972277\pi$
$258$	0	0
$259$	8.84441	0.549565
$260$	0	0
$261$	2.60555	0.161280
$262$	0	0
$263$	− 7.21110i	− 0.444656i	−0.974972	−	0.222328i	$-0.928634\pi$
$263$	− 7.21110i	− 0.444656i	0.974972	−	0.222328i	$-0.0713655\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 6.60555i	− 0.404253i
$268$	0	0
$269$	17.3944	1.06056	0.530279	−	0.847823i	$-0.322087\pi$
$269$	17.3944	1.06056	0.530279	+	0.847823i	$0.322087\pi$
$270$	0	0
$271$	11.6333	0.706673	0.353337	−	0.935496i	$-0.385047\pi$
$271$	11.6333	0.706673	0.353337	+	0.935496i	$0.385047\pi$
$272$	0	0
$273$	− 12.0000i	− 0.726273i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	18.2389	1.08804	0.544020	−	0.839073i	$-0.316902\pi$
$281$	18.2389	1.08804	0.544020	+	0.839073i	$0.316902\pi$
$282$	0	0
$283$	18.6056i	1.10599i	0.833186	+	0.552993i	$0.186514\pi$
$283$	18.6056i	1.10599i	−0.833186	+	0.552993i	$0.813486\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 17.2111i	− 1.01594i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	−16.6056	−0.973435
$292$	0	0
$293$	− 19.6333i	− 1.14699i	−0.819209	−	0.573495i	$-0.805587\pi$
$293$	− 19.6333i	− 1.14699i	0.819209	−	0.573495i	$-0.194413\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.60555i	0.267241i
$298$	0	0
$299$	9.21110	0.532692
$300$	0	0
$301$	−27.6333	−1.59276
$302$	0	0
$303$	− 7.21110i	− 0.414267i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 11.6333i	− 0.663948i	−0.943289	−	0.331974i	$-0.892285\pi$
$307$	− 11.6333i	− 0.663948i	0.943289	−	0.331974i	$-0.107715\pi$
$308$	0	0
$309$	18.4222	1.04800
$310$	0	0
$311$	16.6056	0.941614	0.470807	−	0.882236i	$-0.343963\pi$
$311$	16.6056	0.941614	0.470807	+	0.882236i	$0.343963\pi$
$312$	0	0
$313$	− 0.788897i	− 0.0445911i	−0.999751	−	0.0222956i	$-0.992903\pi$
$313$	− 0.788897i	− 0.0445911i	0.999751	−	0.0222956i	$-0.00709749\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6.42221i	− 0.360707i	−0.983602	−	0.180353i	$-0.942276\pi$
$317$	− 6.42221i	− 0.360707i	0.983602	−	0.180353i	$-0.0577242\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	−18.4222	−1.02823
$322$	0	0
$323$	2.00000i	0.111283i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.2111i	0.619975i
$328$	0	0
$329$	15.6333	0.861892
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	− 3.39445i	− 0.186015i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.02776i	0.164932i	0.996594	+	0.0824662i	$0.0262797\pi$
$337$	3.02776i	0.164932i	−0.996594	+	0.0824662i	$0.973720\pi$
$338$	0	0
$339$	−1.21110	−0.0657781
$340$	0	0
$341$	−18.4222	−0.997618
$342$	0	0
$343$	− 18.7889i	− 1.01451i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.6333i	1.16134i	0.814140	+	0.580668i	$0.197209\pi$
$347$	21.6333i	1.16134i	−0.814140	+	0.580668i	$0.802791\pi$
$348$	0	0
$349$	−34.8444	−1.86518	−0.932589	−	0.360939i	$-0.882456\pi$
$349$	−34.8444	−1.86518	−0.932589	+	0.360939i	$0.882456\pi$
$350$	0	0
$351$	−4.60555	−0.245826
$352$	0	0
$353$	− 0.422205i	− 0.0224717i	−0.999937	−	0.0112359i	$-0.996423\pi$
$353$	− 0.422205i	− 0.0224717i	0.999937	−	0.0112359i	$-0.00357656\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.21110i	0.275801i
$358$	0	0
$359$	13.8167	0.729215	0.364608	−	0.931161i	$-0.381203\pi$
$359$	13.8167	0.729215	0.364608	+	0.931161i	$0.381203\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	10.2111i	0.535944i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	25.0278i	1.30644i	0.757169	+	0.653219i	$0.226582\pi$
$367$	25.0278i	1.30644i	−0.757169	+	0.653219i	$0.773418\pi$
$368$	0	0
$369$	−6.60555	−0.343871
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 16.6056i	− 0.859803i	−0.902876	−	0.429901i	$-0.858548\pi$
$373$	− 16.6056i	− 0.859803i	0.902876	−	0.429901i	$-0.141452\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−26.4222	−1.35722	−0.678609	−	0.734500i	$-0.737417\pi$
$379$	−26.4222	−1.35722	−0.678609	+	0.734500i	$0.737417\pi$
$380$	0	0
$381$	−18.4222	−0.943798
$382$	0	0
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.6056i	0.539110i
$388$	0	0
$389$	−36.4222	−1.84668	−0.923340	−	0.383984i	$-0.874552\pi$
$389$	−36.4222	−1.84668	−0.923340	+	0.383984i	$0.874552\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	− 15.3944i	− 0.776547i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7.57779i	− 0.380319i	−0.981753	−	0.190159i	$-0.939100\pi$
$397$	− 7.57779i	− 0.380319i	0.981753	−	0.190159i	$-0.0609005\pi$
$398$	0	0
$399$	2.60555	0.130441
$400$	0	0
$401$	25.3944	1.26814	0.634069	−	0.773276i	$-0.281383\pi$
$401$	25.3944	1.26814	0.634069	+	0.773276i	$0.281383\pi$
$402$	0	0
$403$	− 18.4222i	− 0.917675i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 15.6333i	− 0.774914i
$408$	0	0
$409$	12.7889	0.632370	0.316185	−	0.948698i	$-0.397598\pi$
$409$	12.7889	0.632370	0.316185	+	0.948698i	$0.397598\pi$
$410$	0	0
$411$	12.4222	0.612742
$412$	0	0
$413$	13.5778i	0.668120i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 9.21110i	− 0.451070i
$418$	0	0
$419$	7.02776	0.343328	0.171664	−	0.985156i	$-0.445086\pi$
$419$	7.02776	0.343328	0.171664	+	0.985156i	$0.445086\pi$
$420$	0	0
$421$	−8.42221	−0.410473	−0.205237	−	0.978712i	$-0.565796\pi$
$421$	−8.42221	−0.410473	−0.205237	+	0.978712i	$0.565796\pi$
$422$	0	0
$423$	− 6.00000i	− 0.291730i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.7889i	0.909258i
$428$	0	0
$429$	−21.2111	−1.02408
$430$	0	0
$431$	−9.21110	−0.443683	−0.221842	−	0.975083i	$-0.571207\pi$
$431$	−9.21110	−0.443683	−0.221842	+	0.975083i	$0.571207\pi$
$432$	0	0
$433$	16.6056i	0.798012i	0.916948	+	0.399006i	$0.130645\pi$
$433$	16.6056i	0.798012i	−0.916948	+	0.399006i	$0.869355\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00000i	0.0956730i
$438$	0	0
$439$	−34.4222	−1.64288	−0.821441	−	0.570293i	$-0.806830\pi$
$439$	−34.4222	−1.64288	−0.821441	+	0.570293i	$0.806830\pi$
$440$	0	0
$441$	−0.211103	−0.0100525
$442$	0	0
$443$	− 18.8444i	− 0.895325i	−0.894203	−	0.447662i	$-0.852257\pi$
$443$	− 18.8444i	− 0.895325i	0.894203	−	0.447662i	$-0.147743\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 16.4222i	− 0.776744i
$448$	0	0
$449$	18.2389	0.860745	0.430372	−	0.902651i	$-0.358382\pi$
$449$	18.2389	0.860745	0.430372	+	0.902651i	$0.358382\pi$
$450$	0	0
$451$	−30.4222	−1.43253
$452$	0	0
$453$	− 12.0000i	− 0.563809i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.0555i	1.49949i	0.661725	+	0.749747i	$0.269825\pi$
$457$	32.0555i	1.49949i	−0.661725	+	0.749747i	$0.730175\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	−22.8444	−1.06397	−0.531985	−	0.846754i	$-0.678554\pi$
$461$	−22.8444	−1.06397	−0.531985	+	0.846754i	$0.678554\pi$
$462$	0	0
$463$	− 33.0278i	− 1.53493i	−0.641091	−	0.767465i	$-0.721518\pi$
$463$	− 33.0278i	− 1.53493i	0.641091	−	0.767465i	$-0.278482\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.8444i	1.05711i	0.848898	+	0.528557i	$0.177267\pi$
$467$	22.8444i	1.05711i	−0.848898	+	0.528557i	$0.822733\pi$
$468$	0	0
$469$	10.4222	0.481253
$470$	0	0
$471$	−15.2111	−0.700891
$472$	0	0
$473$	48.8444i	2.24587i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−28.6056	−1.30702	−0.653510	−	0.756917i	$-0.726704\pi$
$479$	−28.6056	−1.30702	−0.653510	+	0.756917i	$0.726704\pi$
$480$	0	0
$481$	15.6333	0.712817
$482$	0	0
$483$	5.21110i	0.237113i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0.366692i	0.0166164i	0.999965	+	0.00830821i	$0.00264462\pi$
$487$	0.366692i	0.0166164i	−0.999965	+	0.00830821i	$0.997355\pi$
$488$	0	0
$489$	−17.0278	−0.770022
$490$	0	0
$491$	−12.2389	−0.552332	−0.276166	−	0.961110i	$-0.589064\pi$
$491$	−12.2389	−0.552332	−0.276166	+	0.961110i	$0.589064\pi$
$492$	0	0
$493$	− 5.21110i	− 0.234696i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.0000i	1.07655i
$498$	0	0
$499$	27.6333	1.23704	0.618518	−	0.785770i	$-0.287733\pi$
$499$	27.6333	1.23704	0.618518	+	0.785770i	$0.287733\pi$
$500$	0	0
$501$	6.78890	0.303306
$502$	0	0
$503$	− 12.7889i	− 0.570229i	−0.958494	−	0.285114i	$-0.907968\pi$
$503$	− 12.7889i	− 0.570229i	0.958494	−	0.285114i	$-0.0920316\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 8.21110i	− 0.364668i
$508$	0	0
$509$	35.4500	1.57129	0.785646	−	0.618676i	$-0.212331\pi$
$509$	35.4500	1.57129	0.785646	+	0.618676i	$0.212331\pi$
$510$	0	0
$511$	−15.6333	−0.691577
$512$	0	0
$513$	− 1.00000i	− 0.0441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 27.6333i	− 1.21531i
$518$	0	0
$519$	−9.21110	−0.404323
$520$	0	0
$521$	−4.18335	−0.183276	−0.0916379	−	0.995792i	$-0.529210\pi$
$521$	−4.18335	−0.183276	−0.0916379	+	0.995792i	$0.529210\pi$
$522$	0	0
$523$	25.2111i	1.10240i	0.834372	+	0.551202i	$0.185831\pi$
$523$	25.2111i	1.10240i	−0.834372	+	0.551202i	$0.814169\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.00000i	0.348485i
$528$	0	0
$529$	19.0000	0.826087
$530$	0	0
$531$	5.21110	0.226143
$532$	0	0
$533$	− 30.4222i	− 1.31773i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	21.2111i	0.915327i
$538$	0	0
$539$	−0.972244	−0.0418775
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	− 0.788897i	− 0.0338548i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 22.7889i	− 0.974383i	−0.873295	−	0.487191i	$-0.838021\pi$
$547$	− 22.7889i	− 0.974383i	0.873295	−	0.487191i	$-0.161979\pi$
$548$	0	0
$549$	7.21110	0.307762
$550$	0	0
$551$	−2.60555	−0.111000
$552$	0	0
$553$	20.8444i	0.886394i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.42221i	− 0.187375i	−0.995602	−	0.0936874i	$-0.970135\pi$
$557$	− 4.42221i	− 0.187375i	0.995602	−	0.0936874i	$-0.0298654\pi$
$558$	0	0
$559$	−48.8444	−2.06590
$560$	0	0
$561$	9.21110	0.388893
$562$	0	0
$563$	− 31.6333i	− 1.33318i	−0.745422	−	0.666592i	$-0.767752\pi$
$563$	− 31.6333i	− 1.33318i	0.745422	−	0.666592i	$-0.232248\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.60555i	− 0.109423i
$568$	0	0
$569$	−39.4500	−1.65383	−0.826914	−	0.562328i	$-0.809906\pi$
$569$	−39.4500	−1.65383	−0.826914	+	0.562328i	$0.809906\pi$
$570$	0	0
$571$	7.63331	0.319444	0.159722	−	0.987162i	$-0.448940\pi$
$571$	7.63331	0.319444	0.159722	+	0.987162i	$0.448940\pi$
$572$	0	0
$573$	− 5.81665i	− 0.242994i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 23.2111i	− 0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$577$	− 23.2111i	− 0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$578$	0	0
$579$	−0.605551	−0.0251659
$580$	0	0
$581$	29.2111	1.21188
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.4222i	0.512719i	0.966581	+	0.256360i	$0.0825231\pi$
$587$	12.4222i	0.512719i	−0.966581	+	0.256360i	$0.917477\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−14.0000	−0.575883
$592$	0	0
$593$	3.57779i	0.146922i	0.997298	+	0.0734612i	$0.0234045\pi$
$593$	3.57779i	0.146922i	−0.997298	+	0.0734612i	$0.976595\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 17.2111i	− 0.704404i
$598$	0	0
$599$	−36.8444	−1.50542	−0.752711	−	0.658351i	$-0.771254\pi$
$599$	−36.8444	−1.50542	−0.752711	+	0.658351i	$0.771254\pi$
$600$	0	0
$601$	4.78890	0.195343	0.0976716	−	0.995219i	$-0.468861\pi$
$601$	4.78890	0.195343	0.0976716	+	0.995219i	$0.468861\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 23.6333i	− 0.959246i	−0.877475	−	0.479623i	$-0.840773\pi$
$607$	− 23.6333i	− 0.959246i	0.877475	−	0.479623i	$-0.159227\pi$
$608$	0	0
$609$	−6.78890	−0.275100
$610$	0	0
$611$	27.6333	1.11792
$612$	0	0
$613$	− 4.78890i	− 0.193422i	−0.995313	−	0.0967109i	$-0.969168\pi$
$613$	− 4.78890i	− 0.193422i	0.995313	−	0.0967109i	$-0.0308322\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.0000i	1.36879i	0.729112	+	0.684394i	$0.239933\pi$
$617$	34.0000i	1.36879i	−0.729112	+	0.684394i	$0.760067\pi$
$618$	0	0
$619$	4.36669	0.175512	0.0877561	−	0.996142i	$-0.472030\pi$
$619$	4.36669	0.175512	0.0877561	+	0.996142i	$0.472030\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	17.2111i	0.689548i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4.60555i	− 0.183928i
$628$	0	0
$629$	−6.78890	−0.270691
$630$	0	0
$631$	36.8444	1.46675	0.733376	−	0.679823i	$-0.237943\pi$
$631$	36.8444	1.46675	0.733376	+	0.679823i	$0.237943\pi$
$632$	0	0
$633$	− 4.00000i	− 0.158986i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 0.972244i	− 0.0385217i
$638$	0	0
$639$	9.21110	0.364386
$640$	0	0
$641$	35.4500	1.40019	0.700095	−	0.714050i	$-0.253141\pi$
$641$	35.4500	1.40019	0.700095	+	0.714050i	$0.253141\pi$
$642$	0	0
$643$	− 5.39445i	− 0.212736i	−0.994327	−	0.106368i	$-0.966078\pi$
$643$	− 5.39445i	− 0.212736i	0.994327	−	0.106368i	$-0.0339222\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.36669i	− 0.0930443i	−0.998917	−	0.0465221i	$-0.985186\pi$
$647$	− 2.36669i	− 0.0930443i	0.998917	−	0.0465221i	$-0.0148138\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	10.4222	0.408478
$652$	0	0
$653$	− 42.0000i	− 1.64359i	−0.569785	−	0.821794i	$-0.692974\pi$
$653$	− 42.0000i	− 1.64359i	0.569785	−	0.821794i	$-0.307026\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.00000i	0.234082i
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−31.2111	−1.21397	−0.606986	−	0.794713i	$-0.707621\pi$
$661$	−31.2111	−1.21397	−0.606986	+	0.794713i	$0.707621\pi$
$662$	0	0
$663$	9.21110i	0.357730i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 5.21110i	− 0.201775i
$668$	0	0
$669$	−10.4222	−0.402946
$670$	0	0
$671$	33.2111	1.28210
$672$	0	0
$673$	− 36.6056i	− 1.41104i	−0.708690	−	0.705520i	$-0.750713\pi$
$673$	− 36.6056i	− 1.41104i	0.708690	−	0.705520i	$-0.249287\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.4222i	0.400558i	0.979739	+	0.200279i	$0.0641848\pi$
$677$	10.4222i	0.400558i	−0.979739	+	0.200279i	$0.935815\pi$
$678$	0	0
$679$	43.2666	1.66042
$680$	0	0
$681$	13.2111	0.506251
$682$	0	0
$683$	48.0000i	1.83667i	0.395805	+	0.918334i	$0.370466\pi$
$683$	48.0000i	1.83667i	−0.395805	+	0.918334i	$0.629534\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3.21110i	0.122511i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−42.0555	−1.59987	−0.799934	−	0.600089i	$-0.795132\pi$
$691$	−42.0555	−1.59987	−0.799934	+	0.600089i	$0.795132\pi$
$692$	0	0
$693$	− 12.0000i	− 0.455842i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	13.2111i	0.500406i
$698$	0	0
$699$	−16.4222	−0.621145
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	3.39445i	0.128024i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.7889i	0.706629i
$708$	0	0
$709$	25.6333	0.962679	0.481340	−	0.876534i	$-0.340150\pi$
$709$	25.6333	0.962679	0.481340	+	0.876534i	$0.340150\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	8.00000i	0.299602i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 25.8167i	− 0.964141i
$718$	0	0
$719$	−16.2389	−0.605607	−0.302804	−	0.953053i	$-0.597923\pi$
$719$	−16.2389	−0.605607	−0.302804	+	0.953053i	$0.597923\pi$
$720$	0	0
$721$	−48.0000	−1.78761
$722$	0	0
$723$	− 10.0000i	− 0.371904i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 17.3944i	− 0.645124i	−0.946548	−	0.322562i	$-0.895456\pi$
$727$	− 17.3944i	− 0.645124i	0.946548	−	0.322562i	$-0.104544\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	21.2111	0.784521
$732$	0	0
$733$	24.4222i	0.902055i	0.892510	+	0.451027i	$0.148942\pi$
$733$	24.4222i	0.902055i	−0.892510	+	0.451027i	$0.851058\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 18.4222i	− 0.678591i
$738$	0	0
$739$	−38.4222	−1.41338	−0.706692	−	0.707521i	$-0.749813\pi$
$739$	−38.4222	−1.41338	−0.706692	+	0.707521i	$0.749813\pi$
$740$	0	0
$741$	4.60555	0.169189
$742$	0	0
$743$	20.0000i	0.733729i	0.930274	+	0.366864i	$0.119569\pi$
$743$	20.0000i	0.733729i	−0.930274	+	0.366864i	$0.880431\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 11.2111i	− 0.410193i
$748$	0	0
$749$	48.0000	1.75388
$750$	0	0
$751$	40.8444	1.49043	0.745217	−	0.666822i	$-0.232346\pi$
$751$	40.8444	1.49043	0.745217	+	0.666822i	$0.232346\pi$
$752$	0	0
$753$	− 6.18335i	− 0.225334i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 36.0555i	− 1.31046i	−0.755429	−	0.655230i	$-0.772572\pi$
$757$	− 36.0555i	− 1.31046i	0.755429	−	0.655230i	$-0.227428\pi$
$758$	0	0
$759$	9.21110	0.334342
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	− 29.2111i	− 1.05751i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24.0000i	0.866590i
$768$	0	0
$769$	20.4222	0.736444	0.368222	−	0.929738i	$-0.379967\pi$
$769$	20.4222	0.736444	0.368222	+	0.929738i	$0.379967\pi$
$770$	0	0
$771$	−2.78890	−0.100440
$772$	0	0
$773$	43.2666i	1.55619i	0.628145	+	0.778096i	$0.283814\pi$
$773$	43.2666i	1.55619i	−0.628145	+	0.778096i	$0.716186\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.84441i	0.317291i
$778$	0	0
$779$	6.60555	0.236668
$780$	0	0
$781$	42.4222	1.51799
$782$	0	0
$783$	2.60555i	0.0931148i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 19.6333i	− 0.699852i	−0.936777	−	0.349926i	$-0.886207\pi$
$787$	− 19.6333i	− 0.699852i	0.936777	−	0.349926i	$-0.113793\pi$
$788$	0	0
$789$	7.21110	0.256722
$790$	0	0
$791$	3.15559	0.112200
$792$	0	0
$793$	33.2111i	1.17936i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	17.2111i	0.609649i	0.952409	+	0.304824i	$0.0985977\pi$
$797$	17.2111i	0.609649i	−0.952409	+	0.304824i	$0.901402\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	6.60555	0.233396
$802$	0	0
$803$	27.6333i	0.975158i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	17.3944i	0.612314i
$808$	0	0
$809$	48.4222	1.70243	0.851217	−	0.524814i	$-0.175865\pi$
$809$	48.4222	1.70243	0.851217	+	0.524814i	$0.175865\pi$
$810$	0	0
$811$	−40.8444	−1.43424	−0.717121	−	0.696949i	$-0.754540\pi$
$811$	−40.8444	−1.43424	−0.717121	+	0.696949i	$0.754540\pi$
$812$	0	0
$813$	11.6333i	0.407998i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 10.6056i	− 0.371041i
$818$	0	0
$819$	12.0000	0.419314
$820$	0	0
$821$	43.2111	1.50808	0.754039	−	0.656830i	$-0.228103\pi$
$821$	43.2111	1.50808	0.754039	+	0.656830i	$0.228103\pi$
$822$	0	0
$823$	47.8167i	1.66678i	0.552683	+	0.833392i	$0.313604\pi$
$823$	47.8167i	1.66678i	−0.552683	+	0.833392i	$0.686396\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 27.6333i	− 0.960904i	−0.877021	−	0.480452i	$-0.840473\pi$
$827$	− 27.6333i	− 0.960904i	0.877021	−	0.480452i	$-0.159527\pi$
$828$	0	0
$829$	−15.2111	−0.528303	−0.264152	−	0.964481i	$-0.585092\pi$
$829$	−15.2111	−0.528303	−0.264152	+	0.964481i	$0.585092\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	0.422205i	0.0146285i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4.00000i	− 0.138260i
$838$	0	0
$839$	−27.6333	−0.954008	−0.477004	−	0.878901i	$-0.658277\pi$
$839$	−27.6333	−0.954008	−0.477004	+	0.878901i	$0.658277\pi$
$840$	0	0
$841$	−22.2111	−0.765900
$842$	0	0
$843$	18.2389i	0.628180i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 26.6056i	− 0.914178i
$848$	0	0
$849$	−18.6056	−0.638541
$850$	0	0
$851$	−6.78890	−0.232720
$852$	0	0
$853$	− 29.6333i	− 1.01463i	−0.861762	−	0.507313i	$-0.830639\pi$
$853$	− 29.6333i	− 1.01463i	0.861762	−	0.507313i	$-0.169361\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.42221i	0.219378i	0.993966	+	0.109689i	$0.0349855\pi$
$857$	6.42221i	0.219378i	−0.993966	+	0.109689i	$0.965014\pi$
$858$	0	0
$859$	−8.84441	−0.301767	−0.150884	−	0.988552i	$-0.548212\pi$
$859$	−8.84441	−0.301767	−0.150884	+	0.988552i	$0.548212\pi$
$860$	0	0
$861$	17.2111	0.586553
$862$	0	0
$863$	− 6.42221i	− 0.218614i	−0.994008	−	0.109307i	$-0.965137\pi$
$863$	− 6.42221i	− 0.218614i	0.994008	−	0.109307i	$-0.0348632\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	36.8444	1.24986
$870$	0	0
$871$	18.4222	0.624213
$872$	0	0
$873$	− 16.6056i	− 0.562013i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27.0278i	0.912662i	0.889810	+	0.456331i	$0.150837\pi$
$877$	27.0278i	0.912662i	−0.889810	+	0.456331i	$0.849163\pi$
$878$	0	0
$879$	19.6333	0.662215
$880$	0	0
$881$	59.2111	1.99487	0.997436	−	0.0715590i	$-0.0227974\pi$
$881$	59.2111	1.99487	0.997436	+	0.0715590i	$0.0227974\pi$
$882$	0	0
$883$	− 27.8167i	− 0.936105i	−0.883701	−	0.468052i	$-0.844956\pi$
$883$	− 27.8167i	− 0.936105i	0.883701	−	0.468052i	$-0.155044\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 20.8444i	− 0.699887i	−0.936771	−	0.349943i	$-0.886201\pi$
$887$	− 20.8444i	− 0.699887i	0.936771	−	0.349943i	$-0.113799\pi$
$888$	0	0
$889$	48.0000	1.60987
$890$	0	0
$891$	−4.60555	−0.154292
$892$	0	0
$893$	6.00000i	0.200782i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.21110i	0.307550i
$898$	0	0
$899$	−10.4222	−0.347600
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 27.6333i	− 0.919579i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	30.0555i	0.997977i	0.866609	+	0.498988i	$0.166295\pi$
$907$	30.0555i	0.997977i	−0.866609	+	0.498988i	$0.833705\pi$
$908$	0	0
$909$	7.21110	0.239177
$910$	0	0
$911$	10.4222	0.345303	0.172652	−	0.984983i	$-0.444767\pi$
$911$	10.4222	0.345303	0.172652	+	0.984983i	$0.444767\pi$
$912$	0	0
$913$	− 51.6333i	− 1.70881i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	40.1110i	1.32458i
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	11.6333	0.383331
$922$	0	0
$923$	42.4222i	1.39634i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	18.4222i	0.605065i
$928$	0	0
$929$	45.6333	1.49718	0.748590	−	0.663033i	$-0.230731\pi$
$929$	45.6333	1.49718	0.748590	+	0.663033i	$0.230731\pi$
$930$	0	0
$931$	0.211103	0.00691861
$932$	0	0
$933$	16.6056i	0.543641i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 19.2111i	− 0.627599i	−0.949489	−	0.313800i	$-0.898398\pi$
$937$	− 19.2111i	− 0.627599i	0.949489	−	0.313800i	$-0.101602\pi$
$938$	0	0
$939$	0.788897	0.0257447
$940$	0	0
$941$	−30.2389	−0.985759	−0.492879	−	0.870098i	$-0.664056\pi$
$941$	−30.2389	−0.985759	−0.492879	+	0.870098i	$0.664056\pi$
$942$	0	0
$943$	13.2111i	0.430213i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 7.21110i	− 0.234329i	−0.993113	−	0.117165i	$-0.962619\pi$
$947$	− 7.21110i	− 0.234329i	0.993113	−	0.117165i	$-0.0373805\pi$
$948$	0	0
$949$	−27.6333	−0.897015
$950$	0	0
$951$	6.42221	0.208254
$952$	0	0
$953$	− 5.21110i	− 0.168804i	−0.996432	−	0.0844021i	$-0.973102\pi$
$953$	− 5.21110i	− 0.168804i	0.996432	−	0.0844021i	$-0.0268980\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	12.0000i	0.387905i
$958$	0	0
$959$	−32.3667	−1.04518
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 18.4222i	− 0.593647i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 26.2389i	− 0.843785i	−0.906646	−	0.421892i	$-0.861366\pi$
$967$	− 26.2389i	− 0.843785i	0.906646	−	0.421892i	$-0.138634\pi$
$968$	0	0
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	24.0000i	0.769405i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.42221i	0.205465i	0.994709	+	0.102732i	$0.0327585\pi$
$977$	6.42221i	0.205465i	−0.994709	+	0.102732i	$0.967242\pi$
$978$	0	0
$979$	30.4222	0.972298
$980$	0	0
$981$	−11.2111	−0.357943
$982$	0	0
$983$	9.21110i	0.293789i	0.989152	+	0.146894i	$0.0469277\pi$
$983$	9.21110i	0.293789i	−0.989152	+	0.146894i	$0.953072\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	15.6333i	0.497614i
$988$	0	0
$989$	21.2111	0.674474
$990$	0	0
$991$	50.4222	1.60171	0.800857	−	0.598856i	$-0.204378\pi$
$991$	50.4222	1.60171	0.800857	+	0.598856i	$0.204378\pi$
$992$	0	0
$993$	− 8.00000i	− 0.253872i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 61.6333i	− 1.95195i	−0.217891	−	0.975973i	$-0.569918\pi$
$997$	− 61.6333i	− 1.95195i	0.217891	−	0.975973i	$-0.430082\pi$
$998$	0	0
$999$	3.39445	0.107396

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5700.2.f.n.3649.3		4
5.2	odd	4		5700.2.a.u.1.2		2
5.3	odd	4		1140.2.a.e.1.1	✓	2
5.4	even	2	inner	5700.2.f.n.3649.2		4
15.8	even	4		3420.2.a.i.1.1		2
20.3	even	4		4560.2.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.e.1.1	✓	2	5.3	odd	4
3420.2.a.i.1.1		2	15.8	even	4
4560.2.a.bl.1.2		2	20.3	even	4
5700.2.a.u.1.2		2	5.2	odd	4
5700.2.f.n.3649.2		4	5.4	even	2	inner
5700.2.f.n.3649.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5700.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.60555i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.60555i q^{7} -1.00000 q^{9} -4.60555 q^{11} -4.60555i q^{13} +2.00000i q^{17} +1.00000 q^{19} +2.60555 q^{21} +2.00000i q^{23} -1.00000i q^{27} -2.60555 q^{29} +4.00000 q^{31} -4.60555i q^{33} +3.39445i q^{37} +4.60555 q^{39} +6.60555 q^{41} -10.6056i q^{43} +6.00000i q^{47} +0.211103 q^{49} -2.00000 q^{51} +1.00000i q^{57} -5.21110 q^{59} -7.21110 q^{61} +2.60555i q^{63} +4.00000i q^{67} -2.00000 q^{69} -9.21110 q^{71} -6.00000i q^{73} +12.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +11.2111i q^{83} -2.60555i q^{87} -6.60555 q^{89} -12.0000 q^{91} +4.00000i q^{93} +16.6056i q^{97} +4.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 4 q^{11} + 4 q^{19} - 4 q^{21} + 4 q^{29} + 16 q^{31} + 4 q^{39} + 12 q^{41} - 28 q^{49} - 8 q^{51} + 8 q^{59} - 8 q^{69} - 8 q^{71} - 32 q^{79} + 4 q^{81} - 12 q^{89} - 48 q^{91} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 - 4 * q^11 + 4 * q^19 - 4 * q^21 + 4 * q^29 + 16 * q^31 + 4 * q^39 + 12 * q^41 - 28 * q^49 - 8 * q^51 + 8 * q^59 - 8 * q^69 - 8 * q^71 - 32 * q^79 + 4 * q^81 - 12 * q^89 - 48 * q^91 + 4 * q^99

Introduction

L-functions

Modular forms

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Database

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