LMFDB - Embedded newform 400.4.c.f.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.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:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.4.c.f.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.00000i q^{3} -34.0000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q+6.00000i q^{3} -34.0000i q^{7} -9.00000 q^{9} -16.0000 q^{11} +58.0000i q^{13} +70.0000i q^{17} +4.00000 q^{19} +204.000 q^{21} +134.000i q^{23} +108.000i q^{27} +242.000 q^{29} -100.000 q^{31} -96.0000i q^{33} +438.000i q^{37} -348.000 q^{39} -138.000 q^{41} -178.000i q^{43} +22.0000i q^{47} -813.000 q^{49} -420.000 q^{51} +162.000i q^{53} +24.0000i q^{57} -268.000 q^{59} +250.000 q^{61} +306.000i q^{63} +422.000i q^{67} -804.000 q^{69} +852.000 q^{71} +306.000i q^{73} +544.000i q^{77} -456.000 q^{79} -891.000 q^{81} -434.000i q^{83} +1452.00i q^{87} +726.000 q^{89} +1972.00 q^{91} -600.000i q^{93} -1378.00i q^{97} +144.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} - 32 q^{11} + 8 q^{19} + 408 q^{21} + 484 q^{29} - 200 q^{31} - 696 q^{39} - 276 q^{41} - 1626 q^{49} - 840 q^{51} - 536 q^{59} + 500 q^{61} - 1608 q^{69} + 1704 q^{71} - 912 q^{79} - 1782 q^{81} + 1452 q^{89} + 3944 q^{91} + 288 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 - 32 * q^11 + 8 * q^19 + 408 * q^21 + 484 * q^29 - 200 * q^31 - 696 * q^39 - 276 * q^41 - 1626 * q^49 - 840 * q^51 - 536 * q^59 + 500 * q^61 - 1608 * q^69 + 1704 * q^71 - 912 * q^79 - 1782 * q^81 + 1452 * q^89 + 3944 * q^91 + 288 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	6.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	6.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 34.0000i	− 1.83583i	−0.396780	−	0.917914i	$-0.629872\pi$
$7$	− 34.0000i	− 1.83583i	0.396780	−	0.917914i	$-0.370128\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−16.0000	−0.438562	−0.219281	−	0.975662i	$-0.570371\pi$
$11$	−16.0000	−0.438562	−0.219281	+	0.975662i	$0.570371\pi$
$12$	0	0
$13$	58.0000i	1.23741i	0.785624	+	0.618704i	$0.212342\pi$
$13$	58.0000i	1.23741i	−0.785624	+	0.618704i	$0.787658\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	70.0000i	0.998676i	0.866407	+	0.499338i	$0.166423\pi$
$17$	70.0000i	0.998676i	−0.866407	+	0.499338i	$0.833577\pi$
$18$	0	0
$19$	4.00000	0.0482980	0.0241490	−	0.999708i	$-0.492312\pi$
$19$	4.00000	0.0482980	0.0241490	+	0.999708i	$0.492312\pi$
$20$	0	0
$21$	204.000	2.11983
$22$	0	0
$23$	134.000i	1.21482i	0.794387	+	0.607412i	$0.207792\pi$
$23$	134.000i	1.21482i	−0.794387	+	0.607412i	$0.792208\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	108.000i	0.769800i
$28$	0	0
$29$	242.000	1.54960	0.774798	−	0.632209i	$-0.217852\pi$
$29$	242.000	1.54960	0.774798	+	0.632209i	$0.217852\pi$
$30$	0	0
$31$	−100.000	−0.579372	−0.289686	−	0.957122i	$-0.593551\pi$
$31$	−100.000	−0.579372	−0.289686	+	0.957122i	$0.593551\pi$
$32$	0	0
$33$	− 96.0000i	− 0.506408i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	438.000i	1.94613i	0.230534	+	0.973064i	$0.425953\pi$
$37$	438.000i	1.94613i	−0.230534	+	0.973064i	$0.574047\pi$
$38$	0	0
$39$	−348.000	−1.42884
$40$	0	0
$41$	−138.000	−0.525658	−0.262829	−	0.964842i	$-0.584656\pi$
$41$	−138.000	−0.525658	−0.262829	+	0.964842i	$0.584656\pi$
$42$	0	0
$43$	− 178.000i	− 0.631273i	−0.948880	−	0.315637i	$-0.897782\pi$
$43$	− 178.000i	− 0.631273i	0.948880	−	0.315637i	$-0.102218\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	22.0000i	0.0682772i	0.999417	+	0.0341386i	$0.0108688\pi$
$47$	22.0000i	0.0682772i	−0.999417	+	0.0341386i	$0.989131\pi$
$48$	0	0
$49$	−813.000	−2.37026
$50$	0	0
$51$	−420.000	−1.15317
$52$	0	0
$53$	162.000i	0.419857i	0.977717	+	0.209928i	$0.0673231\pi$
$53$	162.000i	0.419857i	−0.977717	+	0.209928i	$0.932677\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	24.0000i	0.0557698i
$58$	0	0
$59$	−268.000	−0.591367	−0.295683	−	0.955286i	$-0.595547\pi$
$59$	−268.000	−0.591367	−0.295683	+	0.955286i	$0.595547\pi$
$60$	0	0
$61$	250.000	0.524741	0.262371	−	0.964967i	$-0.415496\pi$
$61$	250.000	0.524741	0.262371	+	0.964967i	$0.415496\pi$
$62$	0	0
$63$	306.000i	0.611942i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	422.000i	0.769485i	0.923024	+	0.384743i	$0.125710\pi$
$67$	422.000i	0.769485i	−0.923024	+	0.384743i	$0.874290\pi$
$68$	0	0
$69$	−804.000	−1.40276
$70$	0	0
$71$	852.000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	852.000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	306.000i	0.490611i	0.969446	+	0.245305i	$0.0788882\pi$
$73$	306.000i	0.490611i	−0.969446	+	0.245305i	$0.921112\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	544.000i	0.805124i
$78$	0	0
$79$	−456.000	−0.649418	−0.324709	−	0.945814i	$-0.605266\pi$
$79$	−456.000	−0.649418	−0.324709	+	0.945814i	$0.605266\pi$
$80$	0	0
$81$	−891.000	−1.22222
$82$	0	0
$83$	− 434.000i	− 0.573948i	−0.957938	−	0.286974i	$-0.907351\pi$
$83$	− 434.000i	− 0.573948i	0.957938	−	0.286974i	$-0.0926493\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1452.00i	1.78932i
$88$	0	0
$89$	726.000	0.864672	0.432336	−	0.901712i	$-0.357689\pi$
$89$	726.000	0.864672	0.432336	+	0.901712i	$0.357689\pi$
$90$	0	0
$91$	1972.00	2.27167
$92$	0	0
$93$	− 600.000i	− 0.669001i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1378.00i	− 1.44242i	−0.692717	−	0.721210i	$-0.743586\pi$
$97$	− 1378.00i	− 1.44242i	0.692717	−	0.721210i	$-0.256414\pi$
$98$	0	0
$99$	144.000	0.146187
$100$	0	0
$101$	126.000	0.124133	0.0620667	−	0.998072i	$-0.480231\pi$
$101$	126.000	0.124133	0.0620667	+	0.998072i	$0.480231\pi$
$102$	0	0
$103$	1262.00i	1.20727i	0.797262	+	0.603634i	$0.206281\pi$
$103$	1262.00i	1.20727i	−0.797262	+	0.603634i	$0.793719\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	510.000i	0.460781i	0.973098	+	0.230390i	$0.0740003\pi$
$107$	510.000i	0.460781i	−0.973098	+	0.230390i	$0.926000\pi$
$108$	0	0
$109$	−26.0000	−0.0228472	−0.0114236	−	0.999935i	$-0.503636\pi$
$109$	−26.0000	−0.0228472	−0.0114236	+	0.999935i	$0.503636\pi$
$110$	0	0
$111$	−2628.00	−2.24720
$112$	0	0
$113$	1242.00i	1.03396i	0.855997	+	0.516980i	$0.172944\pi$
$113$	1242.00i	1.03396i	−0.855997	+	0.516980i	$0.827056\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 522.000i	− 0.412469i
$118$	0	0
$119$	2380.00	1.83340
$120$	0	0
$121$	−1075.00	−0.807663
$122$	0	0
$123$	− 828.000i	− 0.606978i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 978.000i	− 0.683334i	−0.939821	−	0.341667i	$-0.889008\pi$
$127$	− 978.000i	− 0.683334i	0.939821	−	0.341667i	$-0.110992\pi$
$128$	0	0
$129$	1068.00	0.728931
$130$	0	0
$131$	912.000	0.608258	0.304129	−	0.952631i	$-0.401635\pi$
$131$	912.000	0.608258	0.304129	+	0.952631i	$0.401635\pi$
$132$	0	0
$133$	− 136.000i	− 0.0886669i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	926.000i	0.577471i	0.957409	+	0.288735i	$0.0932348\pi$
$137$	926.000i	0.577471i	−0.957409	+	0.288735i	$0.906765\pi$
$138$	0	0
$139$	516.000	0.314867	0.157434	−	0.987530i	$-0.449678\pi$
$139$	516.000	0.314867	0.157434	+	0.987530i	$0.449678\pi$
$140$	0	0
$141$	−132.000	−0.0788398
$142$	0	0
$143$	− 928.000i	− 0.542680i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4878.00i	− 2.73694i
$148$	0	0
$149$	958.000	0.526728	0.263364	−	0.964697i	$-0.415168\pi$
$149$	958.000	0.526728	0.263364	+	0.964697i	$0.415168\pi$
$150$	0	0
$151$	−332.000	−0.178926	−0.0894628	−	0.995990i	$-0.528515\pi$
$151$	−332.000	−0.178926	−0.0894628	+	0.995990i	$0.528515\pi$
$152$	0	0
$153$	− 630.000i	− 0.332892i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1022.00i	0.519519i	0.965673	+	0.259759i	$0.0836433\pi$
$157$	1022.00i	0.519519i	−0.965673	+	0.259759i	$0.916357\pi$
$158$	0	0
$159$	−972.000	−0.484809
$160$	0	0
$161$	4556.00	2.23021
$162$	0	0
$163$	926.000i	0.444969i	0.974936	+	0.222484i	$0.0714166\pi$
$163$	926.000i	0.444969i	−0.974936	+	0.222484i	$0.928583\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	654.000i	0.303042i	0.988454	+	0.151521i	$0.0484171\pi$
$167$	654.000i	0.303042i	−0.988454	+	0.151521i	$0.951583\pi$
$168$	0	0
$169$	−1167.00	−0.531179
$170$	0	0
$171$	−36.0000	−0.0160993
$172$	0	0
$173$	− 1294.00i	− 0.568676i	−0.958724	−	0.284338i	$-0.908226\pi$
$173$	− 1294.00i	− 0.568676i	0.958724	−	0.284338i	$-0.0917738\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1608.00i	− 0.682851i
$178$	0	0
$179$	−2836.00	−1.18420	−0.592102	−	0.805863i	$-0.701702\pi$
$179$	−2836.00	−1.18420	−0.592102	+	0.805863i	$0.701702\pi$
$180$	0	0
$181$	1742.00	0.715369	0.357685	−	0.933842i	$-0.383566\pi$
$181$	1742.00	0.715369	0.357685	+	0.933842i	$0.383566\pi$
$182$	0	0
$183$	1500.00i	0.605919i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1120.00i	− 0.437981i
$188$	0	0
$189$	3672.00	1.41322
$190$	0	0
$191$	−4460.00	−1.68960	−0.844802	−	0.535079i	$-0.820282\pi$
$191$	−4460.00	−1.68960	−0.844802	+	0.535079i	$0.820282\pi$
$192$	0	0
$193$	− 3782.00i	− 1.41054i	−0.708939	−	0.705270i	$-0.750826\pi$
$193$	− 3782.00i	− 1.41054i	0.708939	−	0.705270i	$-0.249174\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4474.00i	− 1.61807i	−0.587762	−	0.809034i	$-0.699991\pi$
$197$	− 4474.00i	− 1.61807i	0.587762	−	0.809034i	$-0.300009\pi$
$198$	0	0
$199$	3608.00	1.28525	0.642624	−	0.766182i	$-0.277846\pi$
$199$	3608.00	1.28525	0.642624	+	0.766182i	$0.277846\pi$
$200$	0	0
$201$	−2532.00	−0.888525
$202$	0	0
$203$	− 8228.00i	− 2.84479i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1206.00i	− 0.404941i
$208$	0	0
$209$	−64.0000	−0.0211817
$210$	0	0
$211$	256.000	0.0835250	0.0417625	−	0.999128i	$-0.486703\pi$
$211$	256.000	0.0835250	0.0417625	+	0.999128i	$0.486703\pi$
$212$	0	0
$213$	5112.00i	1.64445i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3400.00i	1.06363i
$218$	0	0
$219$	−1836.00	−0.566509
$220$	0	0
$221$	−4060.00	−1.23577
$222$	0	0
$223$	5158.00i	1.54890i	0.632634	+	0.774451i	$0.281974\pi$
$223$	5158.00i	1.54890i	−0.632634	+	0.774451i	$0.718026\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2226.00i	− 0.650858i	−0.945566	−	0.325429i	$-0.894491\pi$
$227$	− 2226.00i	− 0.650858i	0.945566	−	0.325429i	$-0.105509\pi$
$228$	0	0
$229$	−2086.00	−0.601951	−0.300975	−	0.953632i	$-0.597312\pi$
$229$	−2086.00	−0.601951	−0.300975	+	0.953632i	$0.597312\pi$
$230$	0	0
$231$	−3264.00	−0.929677
$232$	0	0
$233$	− 5718.00i	− 1.60772i	−0.594819	−	0.803860i	$-0.702776\pi$
$233$	− 5718.00i	− 1.60772i	0.594819	−	0.803860i	$-0.297224\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2736.00i	− 0.749883i
$238$	0	0
$239$	−3624.00	−0.980825	−0.490412	−	0.871491i	$-0.663154\pi$
$239$	−3624.00	−0.980825	−0.490412	+	0.871491i	$0.663154\pi$
$240$	0	0
$241$	−82.0000	−0.0219174	−0.0109587	−	0.999940i	$-0.503488\pi$
$241$	−82.0000	−0.0219174	−0.0109587	+	0.999940i	$0.503488\pi$
$242$	0	0
$243$	− 2430.00i	− 0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	232.000i	0.0597644i
$248$	0	0
$249$	2604.00	0.662738
$250$	0	0
$251$	5040.00	1.26742	0.633709	−	0.773571i	$-0.281532\pi$
$251$	5040.00	1.26742	0.633709	+	0.773571i	$0.281532\pi$
$252$	0	0
$253$	− 2144.00i	− 0.532775i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2310.00i	0.560676i	0.959901	+	0.280338i	$0.0904466\pi$
$257$	2310.00i	0.560676i	−0.959901	+	0.280338i	$0.909553\pi$
$258$	0	0
$259$	14892.0	3.57276
$260$	0	0
$261$	−2178.00	−0.516532
$262$	0	0
$263$	4110.00i	0.963625i	0.876274	+	0.481813i	$0.160021\pi$
$263$	4110.00i	0.963625i	−0.876274	+	0.481813i	$0.839979\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4356.00i	0.998438i
$268$	0	0
$269$	−746.000	−0.169087	−0.0845435	−	0.996420i	$-0.526943\pi$
$269$	−746.000	−0.169087	−0.0845435	+	0.996420i	$0.526943\pi$
$270$	0	0
$271$	4596.00	1.03021	0.515105	−	0.857127i	$-0.327753\pi$
$271$	4596.00	1.03021	0.515105	+	0.857127i	$0.327753\pi$
$272$	0	0
$273$	11832.0i	2.62310i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2206.00i	0.478504i	0.970957	+	0.239252i	$0.0769023\pi$
$277$	2206.00i	0.478504i	−0.970957	+	0.239252i	$0.923098\pi$
$278$	0	0
$279$	900.000	0.193124
$280$	0	0
$281$	8278.00	1.75738	0.878691	−	0.477392i	$-0.158418\pi$
$281$	8278.00	1.75738	0.878691	+	0.477392i	$0.158418\pi$
$282$	0	0
$283$	− 1178.00i	− 0.247438i	−0.992317	−	0.123719i	$-0.960518\pi$
$283$	− 1178.00i	− 0.247438i	0.992317	−	0.123719i	$-0.0394821\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4692.00i	0.965017i
$288$	0	0
$289$	13.0000	0.00264604
$290$	0	0
$291$	8268.00	1.66556
$292$	0	0
$293$	106.000i	0.0211351i	0.999944	+	0.0105676i	$0.00336382\pi$
$293$	106.000i	0.0211351i	−0.999944	+	0.0105676i	$0.996636\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1728.00i	− 0.337605i
$298$	0	0
$299$	−7772.00	−1.50323
$300$	0	0
$301$	−6052.00	−1.15891
$302$	0	0
$303$	756.000i	0.143337i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8134.00i	1.51216i	0.654482	+	0.756078i	$0.272887\pi$
$307$	8134.00i	1.51216i	−0.654482	+	0.756078i	$0.727113\pi$
$308$	0	0
$309$	−7572.00	−1.39403
$310$	0	0
$311$	4396.00	0.801525	0.400763	−	0.916182i	$-0.368745\pi$
$311$	4396.00	0.801525	0.400763	+	0.916182i	$0.368745\pi$
$312$	0	0
$313$	4826.00i	0.871507i	0.900066	+	0.435753i	$0.143518\pi$
$313$	4826.00i	0.871507i	−0.900066	+	0.435753i	$0.856482\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 7026.00i	− 1.24486i	−0.782677	−	0.622428i	$-0.786146\pi$
$317$	− 7026.00i	− 1.24486i	0.782677	−	0.622428i	$-0.213854\pi$
$318$	0	0
$319$	−3872.00	−0.679594
$320$	0	0
$321$	−3060.00	−0.532064
$322$	0	0
$323$	280.000i	0.0482341i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 156.000i	− 0.0263817i
$328$	0	0
$329$	748.000	0.125345
$330$	0	0
$331$	−8808.00	−1.46263	−0.731316	−	0.682038i	$-0.761094\pi$
$331$	−8808.00	−1.46263	−0.731316	+	0.682038i	$0.761094\pi$
$332$	0	0
$333$	− 3942.00i	− 0.648710i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 5602.00i	− 0.905520i	−0.891632	−	0.452760i	$-0.850439\pi$
$337$	− 5602.00i	− 0.905520i	0.891632	−	0.452760i	$-0.149561\pi$
$338$	0	0
$339$	−7452.00	−1.19391
$340$	0	0
$341$	1600.00	0.254090
$342$	0	0
$343$	15980.0i	2.51557i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6634.00i	− 1.02632i	−0.858294	−	0.513158i	$-0.828475\pi$
$347$	− 6634.00i	− 1.02632i	0.858294	−	0.513158i	$-0.171525\pi$
$348$	0	0
$349$	−3198.00	−0.490501	−0.245251	−	0.969460i	$-0.578870\pi$
$349$	−3198.00	−0.490501	−0.245251	+	0.969460i	$0.578870\pi$
$350$	0	0
$351$	−6264.00	−0.952557
$352$	0	0
$353$	− 5230.00i	− 0.788569i	−0.918988	−	0.394284i	$-0.870992\pi$
$353$	− 5230.00i	− 0.788569i	0.918988	−	0.394284i	$-0.129008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	14280.0i	2.11702i
$358$	0	0
$359$	−312.000	−0.0458683	−0.0229342	−	0.999737i	$-0.507301\pi$
$359$	−312.000	−0.0458683	−0.0229342	+	0.999737i	$0.507301\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	0	0
$363$	− 6450.00i	− 0.932609i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10790.0i	1.53470i	0.641231	+	0.767348i	$0.278424\pi$
$367$	10790.0i	1.53470i	−0.641231	+	0.767348i	$0.721576\pi$
$368$	0	0
$369$	1242.00	0.175219
$370$	0	0
$371$	5508.00	0.770785
$372$	0	0
$373$	− 4190.00i	− 0.581635i	−0.956778	−	0.290818i	$-0.906073\pi$
$373$	− 4190.00i	− 0.581635i	0.956778	−	0.290818i	$-0.0939273\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14036.0i	1.91748i
$378$	0	0
$379$	−6980.00	−0.946012	−0.473006	−	0.881059i	$-0.656831\pi$
$379$	−6980.00	−0.946012	−0.473006	+	0.881059i	$0.656831\pi$
$380$	0	0
$381$	5868.00	0.789047
$382$	0	0
$383$	− 13962.0i	− 1.86273i	−0.364089	−	0.931364i	$-0.618620\pi$
$383$	− 13962.0i	− 1.86273i	0.364089	−	0.931364i	$-0.381380\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1602.00i	0.210424i
$388$	0	0
$389$	−3810.00	−0.496593	−0.248296	−	0.968684i	$-0.579871\pi$
$389$	−3810.00	−0.496593	−0.248296	+	0.968684i	$0.579871\pi$
$390$	0	0
$391$	−9380.00	−1.21321
$392$	0	0
$393$	5472.00i	0.702356i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9158.00i	1.15775i	0.815416	+	0.578875i	$0.196508\pi$
$397$	9158.00i	1.15775i	−0.815416	+	0.578875i	$0.803492\pi$
$398$	0	0
$399$	816.000	0.102384
$400$	0	0
$401$	4866.00	0.605976	0.302988	−	0.952994i	$-0.402016\pi$
$401$	4866.00	0.605976	0.302988	+	0.952994i	$0.402016\pi$
$402$	0	0
$403$	− 5800.00i	− 0.716920i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7008.00i	− 0.853498i
$408$	0	0
$409$	−13486.0	−1.63042	−0.815208	−	0.579169i	$-0.803377\pi$
$409$	−13486.0	−1.63042	−0.815208	+	0.579169i	$0.803377\pi$
$410$	0	0
$411$	−5556.00	−0.666806
$412$	0	0
$413$	9112.00i	1.08565i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3096.00i	0.363577i
$418$	0	0
$419$	5628.00	0.656195	0.328098	−	0.944644i	$-0.393593\pi$
$419$	5628.00	0.656195	0.328098	+	0.944644i	$0.393593\pi$
$420$	0	0
$421$	7938.00	0.918942	0.459471	−	0.888193i	$-0.348039\pi$
$421$	7938.00	0.918942	0.459471	+	0.888193i	$0.348039\pi$
$422$	0	0
$423$	− 198.000i	− 0.0227591i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 8500.00i	− 0.963334i
$428$	0	0
$429$	5568.00	0.626633
$430$	0	0
$431$	−1916.00	−0.214131	−0.107066	−	0.994252i	$-0.534145\pi$
$431$	−1916.00	−0.214131	−0.107066	+	0.994252i	$0.534145\pi$
$432$	0	0
$433$	− 16510.0i	− 1.83238i	−0.400746	−	0.916189i	$-0.631249\pi$
$433$	− 16510.0i	− 1.83238i	0.400746	−	0.916189i	$-0.368751\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	536.000i	0.0586736i
$438$	0	0
$439$	−1256.00	−0.136550	−0.0682752	−	0.997667i	$-0.521750\pi$
$439$	−1256.00	−0.136550	−0.0682752	+	0.997667i	$0.521750\pi$
$440$	0	0
$441$	7317.00	0.790087
$442$	0	0
$443$	12222.0i	1.31080i	0.755282	+	0.655400i	$0.227500\pi$
$443$	12222.0i	1.31080i	−0.755282	+	0.655400i	$0.772500\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5748.00i	0.608213i
$448$	0	0
$449$	5946.00	0.624965	0.312482	−	0.949924i	$-0.398840\pi$
$449$	5946.00	0.624965	0.312482	+	0.949924i	$0.398840\pi$
$450$	0	0
$451$	2208.00	0.230534
$452$	0	0
$453$	− 1992.00i	− 0.206606i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1258.00i	− 0.128768i	−0.997925	−	0.0643838i	$-0.979492\pi$
$457$	− 1258.00i	− 0.128768i	0.997925	−	0.0643838i	$-0.0205082\pi$
$458$	0	0
$459$	−7560.00	−0.768781
$460$	0	0
$461$	16422.0	1.65911	0.829554	−	0.558426i	$-0.188595\pi$
$461$	16422.0	1.65911	0.829554	+	0.558426i	$0.188595\pi$
$462$	0	0
$463$	− 2658.00i	− 0.266799i	−0.991062	−	0.133399i	$-0.957411\pi$
$463$	− 2658.00i	− 0.266799i	0.991062	−	0.133399i	$-0.0425893\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3686.00i	0.365241i	0.983183	+	0.182621i	$0.0584580\pi$
$467$	3686.00i	0.365241i	−0.983183	+	0.182621i	$0.941542\pi$
$468$	0	0
$469$	14348.0	1.41264
$470$	0	0
$471$	−6132.00	−0.599889
$472$	0	0
$473$	2848.00i	0.276852i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1458.00i	− 0.139952i
$478$	0	0
$479$	88.0000	0.00839420	0.00419710	−	0.999991i	$-0.498664\pi$
$479$	88.0000	0.00839420	0.00419710	+	0.999991i	$0.498664\pi$
$480$	0	0
$481$	−25404.0	−2.40816
$482$	0	0
$483$	27336.0i	2.57522i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 14714.0i	− 1.36911i	−0.728963	−	0.684553i	$-0.759997\pi$
$487$	− 14714.0i	− 1.36911i	0.728963	−	0.684553i	$-0.240003\pi$
$488$	0	0
$489$	−5556.00	−0.513806
$490$	0	0
$491$	7344.00	0.675010	0.337505	−	0.941324i	$-0.390417\pi$
$491$	7344.00	0.675010	0.337505	+	0.941324i	$0.390417\pi$
$492$	0	0
$493$	16940.0i	1.54754i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 28968.0i	− 2.61447i
$498$	0	0
$499$	1604.00	0.143898	0.0719488	−	0.997408i	$-0.477078\pi$
$499$	1604.00	0.143898	0.0719488	+	0.997408i	$0.477078\pi$
$500$	0	0
$501$	−3924.00	−0.349923
$502$	0	0
$503$	− 14802.0i	− 1.31210i	−0.754715	−	0.656052i	$-0.772225\pi$
$503$	− 14802.0i	− 1.31210i	0.754715	−	0.656052i	$-0.227775\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 7002.00i	− 0.613353i
$508$	0	0
$509$	22514.0	1.96054	0.980271	−	0.197660i	$-0.0633342\pi$
$509$	22514.0	1.96054	0.980271	+	0.197660i	$0.0633342\pi$
$510$	0	0
$511$	10404.0	0.900677
$512$	0	0
$513$	432.000i	0.0371799i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 352.000i	− 0.0299438i
$518$	0	0
$519$	7764.00	0.656651
$520$	0	0
$521$	−6710.00	−0.564243	−0.282121	−	0.959379i	$-0.591038\pi$
$521$	−6710.00	−0.564243	−0.282121	+	0.959379i	$0.591038\pi$
$522$	0	0
$523$	− 7930.00i	− 0.663011i	−0.943453	−	0.331505i	$-0.892443\pi$
$523$	− 7930.00i	− 0.663011i	0.943453	−	0.331505i	$-0.107557\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 7000.00i	− 0.578605i
$528$	0	0
$529$	−5789.00	−0.475795
$530$	0	0
$531$	2412.00	0.197122
$532$	0	0
$533$	− 8004.00i	− 0.650454i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 17016.0i	− 1.36740i
$538$	0	0
$539$	13008.0	1.03951
$540$	0	0
$541$	4918.00	0.390834	0.195417	−	0.980720i	$-0.437394\pi$
$541$	4918.00	0.390834	0.195417	+	0.980720i	$0.437394\pi$
$542$	0	0
$543$	10452.0i	0.826037i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 3922.00i	− 0.306568i	−0.988182	−	0.153284i	$-0.951015\pi$
$547$	− 3922.00i	− 0.306568i	0.988182	−	0.153284i	$-0.0489849\pi$
$548$	0	0
$549$	−2250.00	−0.174914
$550$	0	0
$551$	968.000	0.0748424
$552$	0	0
$553$	15504.0i	1.19222i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 17786.0i	− 1.35299i	−0.736446	−	0.676496i	$-0.763497\pi$
$557$	− 17786.0i	− 1.35299i	0.736446	−	0.676496i	$-0.236503\pi$
$558$	0	0
$559$	10324.0	0.781143
$560$	0	0
$561$	6720.00	0.505737
$562$	0	0
$563$	− 20266.0i	− 1.51707i	−0.651633	−	0.758535i	$-0.725916\pi$
$563$	− 20266.0i	− 1.51707i	0.651633	−	0.758535i	$-0.274084\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	30294.0i	2.24379i
$568$	0	0
$569$	−13358.0	−0.984177	−0.492088	−	0.870545i	$-0.663766\pi$
$569$	−13358.0	−0.984177	−0.492088	+	0.870545i	$0.663766\pi$
$570$	0	0
$571$	−16360.0	−1.19903	−0.599514	−	0.800364i	$-0.704639\pi$
$571$	−16360.0	−1.19903	−0.599514	+	0.800364i	$0.704639\pi$
$572$	0	0
$573$	− 26760.0i	− 1.95099i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15574.0i	1.12366i	0.827251	+	0.561832i	$0.189903\pi$
$577$	15574.0i	1.12366i	−0.827251	+	0.561832i	$0.810097\pi$
$578$	0	0
$579$	22692.0	1.62875
$580$	0	0
$581$	−14756.0	−1.05367
$582$	0	0
$583$	− 2592.00i	− 0.184133i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6654.00i	0.467870i	0.972252	+	0.233935i	$0.0751604\pi$
$587$	6654.00i	0.467870i	−0.972252	+	0.233935i	$0.924840\pi$
$588$	0	0
$589$	−400.000	−0.0279825
$590$	0	0
$591$	26844.0	1.86838
$592$	0	0
$593$	− 17742.0i	− 1.22863i	−0.789062	−	0.614314i	$-0.789433\pi$
$593$	− 17742.0i	− 1.22863i	0.789062	−	0.614314i	$-0.210567\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21648.0i	1.48408i
$598$	0	0
$599$	15840.0	1.08048	0.540238	−	0.841512i	$-0.318334\pi$
$599$	15840.0	1.08048	0.540238	+	0.841512i	$0.318334\pi$
$600$	0	0
$601$	−3002.00	−0.203751	−0.101875	−	0.994797i	$-0.532484\pi$
$601$	−3002.00	−0.203751	−0.101875	+	0.994797i	$0.532484\pi$
$602$	0	0
$603$	− 3798.00i	− 0.256495i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 23610.0i	− 1.57875i	−0.613912	−	0.789374i	$-0.710405\pi$
$607$	− 23610.0i	− 1.57875i	0.613912	−	0.789374i	$-0.289595\pi$
$608$	0	0
$609$	49368.0	3.28488
$610$	0	0
$611$	−1276.00	−0.0844868
$612$	0	0
$613$	23850.0i	1.57144i	0.618583	+	0.785720i	$0.287707\pi$
$613$	23850.0i	1.57144i	−0.618583	+	0.785720i	$0.712293\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5334.00i	0.348037i	0.984742	+	0.174018i	$0.0556752\pi$
$617$	5334.00i	0.348037i	−0.984742	+	0.174018i	$0.944325\pi$
$618$	0	0
$619$	−2164.00	−0.140515	−0.0702573	−	0.997529i	$-0.522382\pi$
$619$	−2164.00	−0.140515	−0.0702573	+	0.997529i	$0.522382\pi$
$620$	0	0
$621$	−14472.0	−0.935171
$622$	0	0
$623$	− 24684.0i	− 1.58739i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 384.000i	− 0.0244585i
$628$	0	0
$629$	−30660.0	−1.94355
$630$	0	0
$631$	25220.0	1.59111	0.795557	−	0.605879i	$-0.207179\pi$
$631$	25220.0	1.59111	0.795557	+	0.605879i	$0.207179\pi$
$632$	0	0
$633$	1536.00i	0.0964463i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 47154.0i	− 2.93298i
$638$	0	0
$639$	−7668.00	−0.474713
$640$	0	0
$641$	−12306.0	−0.758280	−0.379140	−	0.925339i	$-0.623780\pi$
$641$	−12306.0	−0.758280	−0.379140	+	0.925339i	$0.623780\pi$
$642$	0	0
$643$	27414.0i	1.68134i	0.541547	+	0.840671i	$0.317839\pi$
$643$	27414.0i	1.68134i	−0.541547	+	0.840671i	$0.682161\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 21834.0i	− 1.32671i	−0.748304	−	0.663356i	$-0.769131\pi$
$647$	− 21834.0i	− 1.32671i	0.748304	−	0.663356i	$-0.230869\pi$
$648$	0	0
$649$	4288.00	0.259351
$650$	0	0
$651$	−20400.0	−1.22817
$652$	0	0
$653$	− 23998.0i	− 1.43815i	−0.694931	−	0.719077i	$-0.744565\pi$
$653$	− 23998.0i	− 1.43815i	0.694931	−	0.719077i	$-0.255435\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2754.00i	− 0.163537i
$658$	0	0
$659$	−32004.0	−1.89180	−0.945902	−	0.324452i	$-0.894820\pi$
$659$	−32004.0	−1.89180	−0.945902	+	0.324452i	$0.894820\pi$
$660$	0	0
$661$	−8526.00	−0.501699	−0.250849	−	0.968026i	$-0.580710\pi$
$661$	−8526.00	−0.501699	−0.250849	+	0.968026i	$0.580710\pi$
$662$	0	0
$663$	− 24360.0i	− 1.42694i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	32428.0i	1.88248i
$668$	0	0
$669$	−30948.0	−1.78852
$670$	0	0
$671$	−4000.00	−0.230132
$672$	0	0
$673$	8178.00i	0.468408i	0.972187	+	0.234204i	$0.0752484\pi$
$673$	8178.00i	0.468408i	−0.972187	+	0.234204i	$0.924752\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16646.0i	0.944989i	0.881334	+	0.472495i	$0.156646\pi$
$677$	16646.0i	0.944989i	−0.881334	+	0.472495i	$0.843354\pi$
$678$	0	0
$679$	−46852.0	−2.64803
$680$	0	0
$681$	13356.0	0.751546
$682$	0	0
$683$	22446.0i	1.25750i	0.777608	+	0.628750i	$0.216433\pi$
$683$	22446.0i	1.25750i	−0.777608	+	0.628750i	$0.783567\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 12516.0i	− 0.695073i
$688$	0	0
$689$	−9396.00	−0.519534
$690$	0	0
$691$	−35336.0	−1.94536	−0.972681	−	0.232147i	$-0.925425\pi$
$691$	−35336.0	−1.94536	−0.972681	+	0.232147i	$0.925425\pi$
$692$	0	0
$693$	− 4896.00i	− 0.268375i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 9660.00i	− 0.524962i
$698$	0	0
$699$	34308.0	1.85643
$700$	0	0
$701$	3482.00	0.187608	0.0938041	−	0.995591i	$-0.470097\pi$
$701$	3482.00	0.187608	0.0938041	+	0.995591i	$0.470097\pi$
$702$	0	0
$703$	1752.00i	0.0939942i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4284.00i	− 0.227887i
$708$	0	0
$709$	19402.0	1.02773	0.513863	−	0.857872i	$-0.328214\pi$
$709$	19402.0	1.02773	0.513863	+	0.857872i	$0.328214\pi$
$710$	0	0
$711$	4104.00	0.216473
$712$	0	0
$713$	− 13400.0i	− 0.703834i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 21744.0i	− 1.13256i
$718$	0	0
$719$	−9896.00	−0.513294	−0.256647	−	0.966505i	$-0.582618\pi$
$719$	−9896.00	−0.513294	−0.256647	+	0.966505i	$0.582618\pi$
$720$	0	0
$721$	42908.0	2.21633
$722$	0	0
$723$	− 492.000i	− 0.0253080i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	494.000i	0.0252014i	0.999921	+	0.0126007i	$0.00401104\pi$
$727$	494.000i	0.0252014i	−0.999921	+	0.0126007i	$0.995989\pi$
$728$	0	0
$729$	−9477.00	−0.481481
$730$	0	0
$731$	12460.0	0.630437
$732$	0	0
$733$	9282.00i	0.467720i	0.972270	+	0.233860i	$0.0751357\pi$
$733$	9282.00i	0.467720i	−0.972270	+	0.233860i	$0.924864\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 6752.00i	− 0.337467i
$738$	0	0
$739$	−3252.00	−0.161877	−0.0809383	−	0.996719i	$-0.525792\pi$
$739$	−3252.00	−0.161877	−0.0809383	+	0.996719i	$0.525792\pi$
$740$	0	0
$741$	−1392.00	−0.0690100
$742$	0	0
$743$	4710.00i	0.232561i	0.993216	+	0.116281i	$0.0370972\pi$
$743$	4710.00i	0.232561i	−0.993216	+	0.116281i	$0.962903\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3906.00i	0.191316i
$748$	0	0
$749$	17340.0	0.845914
$750$	0	0
$751$	−25764.0	−1.25185	−0.625927	−	0.779882i	$-0.715279\pi$
$751$	−25764.0	−1.25185	−0.625927	+	0.779882i	$0.715279\pi$
$752$	0	0
$753$	30240.0i	1.46349i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	30094.0i	1.44489i	0.691426	+	0.722447i	$0.256983\pi$
$757$	30094.0i	1.44489i	−0.691426	+	0.722447i	$0.743017\pi$
$758$	0	0
$759$	12864.0	0.615196
$760$	0	0
$761$	22362.0	1.06521	0.532603	−	0.846365i	$-0.321214\pi$
$761$	22362.0	1.06521	0.532603	+	0.846365i	$0.321214\pi$
$762$	0	0
$763$	884.000i	0.0419436i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 15544.0i	− 0.731762i
$768$	0	0
$769$	30398.0	1.42546	0.712731	−	0.701438i	$-0.247458\pi$
$769$	30398.0	1.42546	0.712731	+	0.701438i	$0.247458\pi$
$770$	0	0
$771$	−13860.0	−0.647413
$772$	0	0
$773$	1290.00i	0.0600234i	0.999550	+	0.0300117i	$0.00955445\pi$
$773$	1290.00i	0.0600234i	−0.999550	+	0.0300117i	$0.990446\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	89352.0i	4.12546i
$778$	0	0
$779$	−552.000	−0.0253883
$780$	0	0
$781$	−13632.0	−0.624573
$782$	0	0
$783$	26136.0i	1.19288i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14.0000i	0 0.000634112i	1.00000		0.000317056i	$0.000100922\pi$
$787$	14.0000i	0 0.000634112i	−1.00000		0.000317056i	$0.999899\pi$
$788$	0	0
$789$	−24660.0	−1.11270
$790$	0	0
$791$	42228.0	1.89817
$792$	0	0
$793$	14500.0i	0.649319i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38814.0i	1.72505i	0.506017	+	0.862523i	$0.331117\pi$
$797$	38814.0i	1.72505i	−0.506017	+	0.862523i	$0.668883\pi$
$798$	0	0
$799$	−1540.00	−0.0681868
$800$	0	0
$801$	−6534.00	−0.288224
$802$	0	0
$803$	− 4896.00i	− 0.215163i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 4476.00i	− 0.195245i
$808$	0	0
$809$	−27402.0	−1.19086	−0.595428	−	0.803408i	$-0.703018\pi$
$809$	−27402.0	−1.19086	−0.595428	+	0.803408i	$0.703018\pi$
$810$	0	0
$811$	28576.0	1.23729	0.618643	−	0.785672i	$-0.287683\pi$
$811$	28576.0	1.23729	0.618643	+	0.785672i	$0.287683\pi$
$812$	0	0
$813$	27576.0i	1.18958i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 712.000i	− 0.0304893i
$818$	0	0
$819$	−17748.0	−0.757223
$820$	0	0
$821$	31762.0	1.35018	0.675092	−	0.737733i	$-0.264104\pi$
$821$	31762.0	1.35018	0.675092	+	0.737733i	$0.264104\pi$
$822$	0	0
$823$	− 20506.0i	− 0.868523i	−0.900787	−	0.434261i	$-0.857009\pi$
$823$	− 20506.0i	− 0.868523i	0.900787	−	0.434261i	$-0.142991\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13014.0i	0.547208i	0.961842	+	0.273604i	$0.0882158\pi$
$827$	13014.0i	0.547208i	−0.961842	+	0.273604i	$0.911784\pi$
$828$	0	0
$829$	22790.0	0.954800	0.477400	−	0.878686i	$-0.341579\pi$
$829$	22790.0	0.954800	0.477400	+	0.878686i	$0.341579\pi$
$830$	0	0
$831$	−13236.0	−0.552529
$832$	0	0
$833$	− 56910.0i	− 2.36712i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 10800.0i	− 0.446001i
$838$	0	0
$839$	23696.0	0.975062	0.487531	−	0.873106i	$-0.337898\pi$
$839$	23696.0	0.975062	0.487531	+	0.873106i	$0.337898\pi$
$840$	0	0
$841$	34175.0	1.40125
$842$	0	0
$843$	49668.0i	2.02925i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	36550.0i	1.48273i
$848$	0	0
$849$	7068.00	0.285716
$850$	0	0
$851$	−58692.0	−2.36420
$852$	0	0
$853$	5306.00i	0.212982i	0.994314	+	0.106491i	$0.0339616\pi$
$853$	5306.00i	0.212982i	−0.994314	+	0.106491i	$0.966038\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21054.0i	0.839196i	0.907710	+	0.419598i	$0.137829\pi$
$857$	21054.0i	0.839196i	−0.907710	+	0.419598i	$0.862171\pi$
$858$	0	0
$859$	7364.00	0.292499	0.146249	−	0.989248i	$-0.453280\pi$
$859$	7364.00	0.292499	0.146249	+	0.989248i	$0.453280\pi$
$860$	0	0
$861$	−28152.0	−1.11431
$862$	0	0
$863$	− 17226.0i	− 0.679467i	−0.940522	−	0.339733i	$-0.889663\pi$
$863$	− 17226.0i	− 0.679467i	0.940522	−	0.339733i	$-0.110337\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	78.0000i	0.00305539i
$868$	0	0
$869$	7296.00	0.284810
$870$	0	0
$871$	−24476.0	−0.952167
$872$	0	0
$873$	12402.0i	0.480807i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 21202.0i	− 0.816352i	−0.912903	−	0.408176i	$-0.866165\pi$
$877$	− 21202.0i	− 0.816352i	0.912903	−	0.408176i	$-0.133835\pi$
$878$	0	0
$879$	−636.000	−0.0244047
$880$	0	0
$881$	−29490.0	−1.12774	−0.563872	−	0.825862i	$-0.690689\pi$
$881$	−29490.0	−1.12774	−0.563872	+	0.825862i	$0.690689\pi$
$882$	0	0
$883$	− 2570.00i	− 0.0979472i	−0.998800	−	0.0489736i	$-0.984405\pi$
$883$	− 2570.00i	− 0.0979472i	0.998800	−	0.0489736i	$-0.0155950\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36334.0i	1.37540i	0.725997	+	0.687698i	$0.241379\pi$
$887$	36334.0i	1.37540i	−0.725997	+	0.687698i	$0.758621\pi$
$888$	0	0
$889$	−33252.0	−1.25448
$890$	0	0
$891$	14256.0	0.536020
$892$	0	0
$893$	88.0000i	0.00329766i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 46632.0i	− 1.73578i
$898$	0	0
$899$	−24200.0	−0.897792
$900$	0	0
$901$	−11340.0	−0.419301
$902$	0	0
$903$	− 36312.0i	− 1.33819i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 12474.0i	− 0.456662i	−0.973584	−	0.228331i	$-0.926673\pi$
$907$	− 12474.0i	− 0.456662i	0.973584	−	0.228331i	$-0.0733268\pi$
$908$	0	0
$909$	−1134.00	−0.0413778
$910$	0	0
$911$	41132.0	1.49590	0.747949	−	0.663756i	$-0.231039\pi$
$911$	41132.0	1.49590	0.747949	+	0.663756i	$0.231039\pi$
$912$	0	0
$913$	6944.00i	0.251712i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 31008.0i	− 1.11666i
$918$	0	0
$919$	−38416.0	−1.37892	−0.689460	−	0.724324i	$-0.742152\pi$
$919$	−38416.0	−1.37892	−0.689460	+	0.724324i	$0.742152\pi$
$920$	0	0
$921$	−48804.0	−1.74609
$922$	0	0
$923$	49416.0i	1.76224i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 11358.0i	− 0.402423i
$928$	0	0
$929$	−41302.0	−1.45864	−0.729319	−	0.684174i	$-0.760163\pi$
$929$	−41302.0	−1.45864	−0.729319	+	0.684174i	$0.760163\pi$
$930$	0	0
$931$	−3252.00	−0.114479
$932$	0	0
$933$	26376.0i	0.925521i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26150.0i	0.911722i	0.890051	+	0.455861i	$0.150669\pi$
$937$	26150.0i	0.911722i	−0.890051	+	0.455861i	$0.849331\pi$
$938$	0	0
$939$	−28956.0	−1.00633
$940$	0	0
$941$	35254.0	1.22130	0.610652	−	0.791899i	$-0.290907\pi$
$941$	35254.0	1.22130	0.610652	+	0.791899i	$0.290907\pi$
$942$	0	0
$943$	− 18492.0i	− 0.638582i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18550.0i	0.636530i	0.948002	+	0.318265i	$0.103100\pi$
$947$	18550.0i	0.636530i	−0.948002	+	0.318265i	$0.896900\pi$
$948$	0	0
$949$	−17748.0	−0.607086
$950$	0	0
$951$	42156.0	1.43744
$952$	0	0
$953$	17322.0i	0.588788i	0.955684	+	0.294394i	$0.0951177\pi$
$953$	17322.0i	0.588788i	−0.955684	+	0.294394i	$0.904882\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 23232.0i	− 0.784727i
$958$	0	0
$959$	31484.0	1.06014
$960$	0	0
$961$	−19791.0	−0.664328
$962$	0	0
$963$	− 4590.00i	− 0.153594i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	35190.0i	1.17025i	0.810942	+	0.585126i	$0.198955\pi$
$967$	35190.0i	1.17025i	−0.810942	+	0.585126i	$0.801045\pi$
$968$	0	0
$969$	−1680.00	−0.0556960
$970$	0	0
$971$	40696.0	1.34500	0.672501	−	0.740096i	$-0.265220\pi$
$971$	40696.0	1.34500	0.672501	+	0.740096i	$0.265220\pi$
$972$	0	0
$973$	− 17544.0i	− 0.578042i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 44306.0i	− 1.45084i	−0.688304	−	0.725422i	$-0.741645\pi$
$977$	− 44306.0i	− 1.45084i	0.688304	−	0.725422i	$-0.258355\pi$
$978$	0	0
$979$	−11616.0	−0.379212
$980$	0	0
$981$	234.000	0.00761574
$982$	0	0
$983$	18798.0i	0.609932i	0.952363	+	0.304966i	$0.0986451\pi$
$983$	18798.0i	0.609932i	−0.952363	+	0.304966i	$0.901355\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4488.00i	0.144736i
$988$	0	0
$989$	23852.0	0.766885
$990$	0	0
$991$	−2468.00	−0.0791106	−0.0395553	−	0.999217i	$-0.512594\pi$
$991$	−2468.00	−0.0791106	−0.0395553	+	0.999217i	$0.512594\pi$
$992$	0	0
$993$	− 52848.0i	− 1.68890i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	61086.0i	1.94043i	0.242237	+	0.970217i	$0.422119\pi$
$997$	61086.0i	1.94043i	−0.242237	+	0.970217i	$0.577881\pi$
$998$	0	0
$999$	−47304.0	−1.49813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.c.f.49.2		2
4.3	odd	2		200.4.c.c.49.1		2
5.2	odd	4		80.4.a.e.1.1		1
5.3	odd	4		400.4.a.e.1.1		1
5.4	even	2	inner	400.4.c.f.49.1		2
12.11	even	2		1800.4.f.j.649.2		2
15.2	even	4		720.4.a.bd.1.1		1
20.3	even	4		200.4.a.i.1.1		1
20.7	even	4		40.4.a.a.1.1	✓	1
20.19	odd	2		200.4.c.c.49.2		2
40.3	even	4		1600.4.a.j.1.1		1
40.13	odd	4		1600.4.a.br.1.1		1
40.27	even	4		320.4.a.l.1.1		1
40.37	odd	4		320.4.a.c.1.1		1
60.23	odd	4		1800.4.a.bi.1.1		1
60.47	odd	4		360.4.a.h.1.1		1
60.59	even	2		1800.4.f.j.649.1		2
80.27	even	4		1280.4.d.p.641.2		2
80.37	odd	4		1280.4.d.a.641.1		2
80.67	even	4		1280.4.d.p.641.1		2
80.77	odd	4		1280.4.d.a.641.2		2
140.27	odd	4		1960.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.a.a.1.1	✓	1	20.7	even	4
80.4.a.e.1.1		1	5.2	odd	4
200.4.a.i.1.1		1	20.3	even	4
200.4.c.c.49.1		2	4.3	odd	2
200.4.c.c.49.2		2	20.19	odd	2
320.4.a.c.1.1		1	40.37	odd	4
320.4.a.l.1.1		1	40.27	even	4
360.4.a.h.1.1		1	60.47	odd	4
400.4.a.e.1.1		1	5.3	odd	4
400.4.c.f.49.1		2	5.4	even	2	inner
400.4.c.f.49.2		2	1.1	even	1	trivial
720.4.a.bd.1.1		1	15.2	even	4
1280.4.d.a.641.1		2	80.37	odd	4
1280.4.d.a.641.2		2	80.77	odd	4
1280.4.d.p.641.1		2	80.67	even	4
1280.4.d.p.641.2		2	80.27	even	4
1600.4.a.j.1.1		1	40.3	even	4
1600.4.a.br.1.1		1	40.13	odd	4
1800.4.a.bi.1.1		1	60.23	odd	4
1800.4.f.j.649.1		2	60.59	even	2
1800.4.f.j.649.2		2	12.11	even	2
1960.4.a.h.1.1		1	140.27	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.00000i q^{3} -34.0000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+6.00000i q^{3} -34.0000i q^{7} -9.00000 q^{9} -16.0000 q^{11} +58.0000i q^{13} +70.0000i q^{17} +4.00000 q^{19} +204.000 q^{21} +134.000i q^{23} +108.000i q^{27} +242.000 q^{29} -100.000 q^{31} -96.0000i q^{33} +438.000i q^{37} -348.000 q^{39} -138.000 q^{41} -178.000i q^{43} +22.0000i q^{47} -813.000 q^{49} -420.000 q^{51} +162.000i q^{53} +24.0000i q^{57} -268.000 q^{59} +250.000 q^{61} +306.000i q^{63} +422.000i q^{67} -804.000 q^{69} +852.000 q^{71} +306.000i q^{73} +544.000i q^{77} -456.000 q^{79} -891.000 q^{81} -434.000i q^{83} +1452.00i q^{87} +726.000 q^{89} +1972.00 q^{91} -600.000i q^{93} -1378.00i q^{97} +144.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} - 32 q^{11} + 8 q^{19} + 408 q^{21} + 484 q^{29} - 200 q^{31} - 696 q^{39} - 276 q^{41} - 1626 q^{49} - 840 q^{51} - 536 q^{59} + 500 q^{61} - 1608 q^{69} + 1704 q^{71} - 912 q^{79} - 1782 q^{81} + 1452 q^{89} + 3944 q^{91} + 288 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 - 32 * q^11 + 8 * q^19 + 408 * q^21 + 484 * q^29 - 200 * q^31 - 696 * q^39 - 276 * q^41 - 1626 * q^49 - 840 * q^51 - 536 * q^59 + 500 * q^61 - 1608 * q^69 + 1704 * q^71 - 912 * q^79 - 1782 * q^81 + 1452 * q^89 + 3944 * q^91 + 288 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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