LMFDB - Embedded newform 400.4.c.l.49.1

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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 100)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +26.0000i q^{7} +26.0000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +26.0000i q^{7} +26.0000 q^{9} -45.0000 q^{11} +44.0000i q^{13} -117.000i q^{17} -91.0000 q^{19} +26.0000 q^{21} +18.0000i q^{23} -53.0000i q^{27} -144.000 q^{29} -26.0000 q^{31} +45.0000i q^{33} +214.000i q^{37} +44.0000 q^{39} -459.000 q^{41} +460.000i q^{43} -468.000i q^{47} -333.000 q^{49} -117.000 q^{51} +558.000i q^{53} +91.0000i q^{57} -72.0000 q^{59} -118.000 q^{61} +676.000i q^{63} +251.000i q^{67} +18.0000 q^{69} -108.000 q^{71} +299.000i q^{73} -1170.00i q^{77} -898.000 q^{79} +649.000 q^{81} -927.000i q^{83} +144.000i q^{87} -351.000 q^{89} -1144.00 q^{91} +26.0000i q^{93} -386.000i q^{97} -1170.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 52 q^{9}+O(q^{10}) $ 2 * q + 52 * q^9	$ 2 q + 52 q^{9} - 90 q^{11} - 182 q^{19} + 52 q^{21} - 288 q^{29} - 52 q^{31} + 88 q^{39} - 918 q^{41} - 666 q^{49} - 234 q^{51} - 144 q^{59} - 236 q^{61} + 36 q^{69} - 216 q^{71} - 1796 q^{79} + 1298 q^{81} - 702 q^{89} - 2288 q^{91} - 2340 q^{99}+O(q^{100}) $ 2 * q + 52 * q^9 - 90 * q^11 - 182 * q^19 + 52 * q^21 - 288 * q^29 - 52 * q^31 + 88 * q^39 - 918 * q^41 - 666 * q^49 - 234 * q^51 - 144 * q^59 - 236 * q^61 + 36 * q^69 - 216 * q^71 - 1796 * q^79 + 1298 * q^81 - 702 * q^89 - 2288 * q^91 - 2340 * 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
$3$	− 1.00000i	− 0.192450i	−0.995360	−	0.0962250i	$-0.969323\pi$
$3$	− 1.00000i	− 0.192450i	0.995360	−	0.0962250i	$-0.0306768\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	26.0000i	1.40387i	0.712242	+	0.701934i	$0.247680\pi$
$7$	26.0000i	1.40387i	−0.712242	+	0.701934i	$0.752320\pi$
$8$	0	0
$9$	26.0000	0.962963
$10$	0	0
$11$	−45.0000	−1.23346	−0.616728	−	0.787177i	$-0.711542\pi$
$11$	−45.0000	−1.23346	−0.616728	+	0.787177i	$0.711542\pi$
$12$	0	0
$13$	44.0000i	0.938723i	0.883006	+	0.469362i	$0.155516\pi$
$13$	44.0000i	0.938723i	−0.883006	+	0.469362i	$0.844484\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 117.000i	− 1.66922i	−0.550845	−	0.834608i	$-0.685694\pi$
$17$	− 117.000i	− 1.66922i	0.550845	−	0.834608i	$-0.314306\pi$
$18$	0	0
$19$	−91.0000	−1.09878	−0.549390	−	0.835566i	$-0.685140\pi$
$19$	−91.0000	−1.09878	−0.549390	+	0.835566i	$0.685140\pi$
$20$	0	0
$21$	26.0000	0.270175
$22$	0	0
$23$	18.0000i	0.163185i	0.996666	+	0.0815926i	$0.0260006\pi$
$23$	18.0000i	0.163185i	−0.996666	+	0.0815926i	$0.973999\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 53.0000i	− 0.377772i
$28$	0	0
$29$	−144.000	−0.922073	−0.461037	−	0.887381i	$-0.652522\pi$
$29$	−144.000	−0.922073	−0.461037	+	0.887381i	$0.652522\pi$
$30$	0	0
$31$	−26.0000	−0.150637	−0.0753184	−	0.997160i	$-0.523997\pi$
$31$	−26.0000	−0.150637	−0.0753184	+	0.997160i	$0.523997\pi$
$32$	0	0
$33$	45.0000i	0.237379i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	214.000i	0.950848i	0.879757	+	0.475424i	$0.157705\pi$
$37$	214.000i	0.950848i	−0.879757	+	0.475424i	$0.842295\pi$
$38$	0	0
$39$	44.0000	0.180657
$40$	0	0
$41$	−459.000	−1.74838	−0.874192	−	0.485580i	$-0.838608\pi$
$41$	−459.000	−1.74838	−0.874192	+	0.485580i	$0.838608\pi$
$42$	0	0
$43$	460.000i	1.63138i	0.578489	+	0.815690i	$0.303642\pi$
$43$	460.000i	1.63138i	−0.578489	+	0.815690i	$0.696358\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 468.000i	− 1.45244i	−0.687461	−	0.726221i	$-0.741275\pi$
$47$	− 468.000i	− 1.45244i	0.687461	−	0.726221i	$-0.258725\pi$
$48$	0	0
$49$	−333.000	−0.970845
$50$	0	0
$51$	−117.000	−0.321241
$52$	0	0
$53$	558.000i	1.44617i	0.690757	+	0.723087i	$0.257277\pi$
$53$	558.000i	1.44617i	−0.690757	+	0.723087i	$0.742723\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	91.0000i	0.211460i
$58$	0	0
$59$	−72.0000	−0.158875	−0.0794373	−	0.996840i	$-0.525312\pi$
$59$	−72.0000	−0.158875	−0.0794373	+	0.996840i	$0.525312\pi$
$60$	0	0
$61$	−118.000	−0.247678	−0.123839	−	0.992302i	$-0.539521\pi$
$61$	−118.000	−0.247678	−0.123839	+	0.992302i	$0.539521\pi$
$62$	0	0
$63$	676.000i	1.35187i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	251.000i	0.457680i	0.973464	+	0.228840i	$0.0734932\pi$
$67$	251.000i	0.457680i	−0.973464	+	0.228840i	$0.926507\pi$
$68$	0	0
$69$	18.0000	0.0314050
$70$	0	0
$71$	−108.000	−0.180525	−0.0902623	−	0.995918i	$-0.528771\pi$
$71$	−108.000	−0.180525	−0.0902623	+	0.995918i	$0.528771\pi$
$72$	0	0
$73$	299.000i	0.479388i	0.970849	+	0.239694i	$0.0770471\pi$
$73$	299.000i	0.479388i	−0.970849	+	0.239694i	$0.922953\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1170.00i	− 1.73161i
$78$	0	0
$79$	−898.000	−1.27890	−0.639449	−	0.768834i	$-0.720837\pi$
$79$	−898.000	−1.27890	−0.639449	+	0.768834i	$0.720837\pi$
$80$	0	0
$81$	649.000	0.890261
$82$	0	0
$83$	− 927.000i	− 1.22592i	−0.790113	−	0.612961i	$-0.789978\pi$
$83$	− 927.000i	− 1.22592i	0.790113	−	0.612961i	$-0.210022\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	144.000i	0.177453i
$88$	0	0
$89$	−351.000	−0.418044	−0.209022	−	0.977911i	$-0.567028\pi$
$89$	−351.000	−0.418044	−0.209022	+	0.977911i	$0.567028\pi$
$90$	0	0
$91$	−1144.00	−1.31784
$92$	0	0
$93$	26.0000i	0.0289900i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 386.000i	− 0.404045i	−0.979381	−	0.202022i	$-0.935249\pi$
$97$	− 386.000i	− 0.404045i	0.979381	−	0.202022i	$-0.0647514\pi$
$98$	0	0
$99$	−1170.00	−1.18777
$100$	0	0
$101$	−954.000	−0.939867	−0.469933	−	0.882702i	$-0.655722\pi$
$101$	−954.000	−0.939867	−0.469933	+	0.882702i	$0.655722\pi$
$102$	0	0
$103$	772.000i	0.738519i	0.929326	+	0.369259i	$0.120389\pi$
$103$	772.000i	0.738519i	−0.929326	+	0.369259i	$0.879611\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1197.00i	1.08148i	0.841190	+	0.540740i	$0.181856\pi$
$107$	1197.00i	1.08148i	−0.841190	+	0.540740i	$0.818144\pi$
$108$	0	0
$109$	802.000	0.704749	0.352375	−	0.935859i	$-0.385374\pi$
$109$	802.000	0.704749	0.352375	+	0.935859i	$0.385374\pi$
$110$	0	0
$111$	214.000	0.182991
$112$	0	0
$113$	1143.00i	0.951543i	0.879569	+	0.475772i	$0.157831\pi$
$113$	1143.00i	0.951543i	−0.879569	+	0.475772i	$0.842169\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1144.00i	0.903956i
$118$	0	0
$119$	3042.00	2.34336
$120$	0	0
$121$	694.000	0.521412
$122$	0	0
$123$	459.000i	0.336477i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2374.00i	− 1.65873i	−0.558709	−	0.829364i	$-0.688703\pi$
$127$	− 2374.00i	− 1.65873i	0.558709	−	0.829364i	$-0.311297\pi$
$128$	0	0
$129$	460.000	0.313959
$130$	0	0
$131$	1260.00	0.840357	0.420178	−	0.907442i	$-0.361968\pi$
$131$	1260.00	0.840357	0.420178	+	0.907442i	$0.361968\pi$
$132$	0	0
$133$	− 2366.00i	− 1.54254i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	891.000i	0.555644i	0.960633	+	0.277822i	$0.0896126\pi$
$137$	891.000i	0.555644i	−0.960633	+	0.277822i	$0.910387\pi$
$138$	0	0
$139$	389.000	0.237371	0.118685	−	0.992932i	$-0.462132\pi$
$139$	389.000	0.237371	0.118685	+	0.992932i	$0.462132\pi$
$140$	0	0
$141$	−468.000	−0.279523
$142$	0	0
$143$	− 1980.00i	− 1.15787i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	333.000i	0.186839i
$148$	0	0
$149$	−1296.00	−0.712567	−0.356283	−	0.934378i	$-0.615956\pi$
$149$	−1296.00	−0.712567	−0.356283	+	0.934378i	$0.615956\pi$
$150$	0	0
$151$	2710.00	1.46051	0.730254	−	0.683176i	$-0.239402\pi$
$151$	2710.00	1.46051	0.730254	+	0.683176i	$0.239402\pi$
$152$	0	0
$153$	− 3042.00i	− 1.60739i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1846.00i	0.938388i	0.883095	+	0.469194i	$0.155455\pi$
$157$	1846.00i	0.938388i	−0.883095	+	0.469194i	$0.844545\pi$
$158$	0	0
$159$	558.000	0.278316
$160$	0	0
$161$	−468.000	−0.229090
$162$	0	0
$163$	− 1475.00i	− 0.708779i	−0.935098	−	0.354389i	$-0.884689\pi$
$163$	− 1475.00i	− 0.708779i	0.935098	−	0.354389i	$-0.115311\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1476.00i	− 0.683930i	−0.939713	−	0.341965i	$-0.888908\pi$
$167$	− 1476.00i	− 0.683930i	0.939713	−	0.341965i	$-0.111092\pi$
$168$	0	0
$169$	261.000	0.118798
$170$	0	0
$171$	−2366.00	−1.05809
$172$	0	0
$173$	1368.00i	0.601197i	0.953751	+	0.300599i	$0.0971864\pi$
$173$	1368.00i	0.601197i	−0.953751	+	0.300599i	$0.902814\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	72.0000i	0.0305754i
$178$	0	0
$179$	−1503.00	−0.627595	−0.313797	−	0.949490i	$-0.601601\pi$
$179$	−1503.00	−0.627595	−0.313797	+	0.949490i	$0.601601\pi$
$180$	0	0
$181$	3770.00	1.54819	0.774094	−	0.633071i	$-0.218206\pi$
$181$	3770.00	1.54819	0.774094	+	0.633071i	$0.218206\pi$
$182$	0	0
$183$	118.000i	0.0476656i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5265.00i	2.05890i
$188$	0	0
$189$	1378.00	0.530343
$190$	0	0
$191$	4122.00	1.56156	0.780779	−	0.624808i	$-0.214823\pi$
$191$	4122.00	1.56156	0.780779	+	0.624808i	$0.214823\pi$
$192$	0	0
$193$	− 1963.00i	− 0.732123i	−0.930591	−	0.366062i	$-0.880706\pi$
$193$	− 1963.00i	− 0.732123i	0.930591	−	0.366062i	$-0.119294\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2934.00i	− 1.06111i	−0.847650	−	0.530555i	$-0.821983\pi$
$197$	− 2934.00i	− 1.06111i	0.847650	−	0.530555i	$-0.178017\pi$
$198$	0	0
$199$	1412.00	0.502985	0.251493	−	0.967859i	$-0.419079\pi$
$199$	1412.00	0.502985	0.251493	+	0.967859i	$0.419079\pi$
$200$	0	0
$201$	251.000	0.0880805
$202$	0	0
$203$	− 3744.00i	− 1.29447i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	468.000i	0.157141i
$208$	0	0
$209$	4095.00	1.35530
$210$	0	0
$211$	−3419.00	−1.11552	−0.557758	−	0.830004i	$-0.688338\pi$
$211$	−3419.00	−1.11552	−0.557758	+	0.830004i	$0.688338\pi$
$212$	0	0
$213$	108.000i	0.0347420i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 676.000i	− 0.211474i
$218$	0	0
$219$	299.000	0.0922582
$220$	0	0
$221$	5148.00	1.56693
$222$	0	0
$223$	100.000i	0.0300291i	0.999887	+	0.0150146i	$0.00477946\pi$
$223$	100.000i	0.0300291i	−0.999887	+	0.0150146i	$0.995221\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4212.00i	1.23154i	0.787925	+	0.615771i	$0.211156\pi$
$227$	4212.00i	1.23154i	−0.787925	+	0.615771i	$0.788844\pi$
$228$	0	0
$229$	3484.00	1.00537	0.502684	−	0.864470i	$-0.332346\pi$
$229$	3484.00	1.00537	0.502684	+	0.864470i	$0.332346\pi$
$230$	0	0
$231$	−1170.00	−0.333248
$232$	0	0
$233$	918.000i	0.258112i	0.991637	+	0.129056i	$0.0411948\pi$
$233$	918.000i	0.258112i	−0.991637	+	0.129056i	$0.958805\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	898.000i	0.246124i
$238$	0	0
$239$	3744.00	1.01330	0.506651	−	0.862151i	$-0.330883\pi$
$239$	3744.00	1.01330	0.506651	+	0.862151i	$0.330883\pi$
$240$	0	0
$241$	−4231.00	−1.13088	−0.565441	−	0.824789i	$-0.691294\pi$
$241$	−4231.00	−1.13088	−0.565441	+	0.824789i	$0.691294\pi$
$242$	0	0
$243$	− 2080.00i	− 0.549103i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4004.00i	− 1.03145i
$248$	0	0
$249$	−927.000	−0.235929
$250$	0	0
$251$	2925.00	0.735555	0.367778	−	0.929914i	$-0.380119\pi$
$251$	2925.00	0.735555	0.367778	+	0.929914i	$0.380119\pi$
$252$	0	0
$253$	− 810.000i	− 0.201282i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000i	0.00436891i	0.999998	+	0.00218445i	$0.000695334\pi$
$257$	18.0000i	0.00436891i	−0.999998	+	0.00218445i	$0.999305\pi$
$258$	0	0
$259$	−5564.00	−1.33487
$260$	0	0
$261$	−3744.00	−0.887923
$262$	0	0
$263$	6786.00i	1.59104i	0.605929	+	0.795518i	$0.292801\pi$
$263$	6786.00i	1.59104i	−0.605929	+	0.795518i	$0.707199\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	351.000i	0.0804526i
$268$	0	0
$269$	−7632.00	−1.72986	−0.864928	−	0.501896i	$-0.832636\pi$
$269$	−7632.00	−1.72986	−0.864928	+	0.501896i	$0.832636\pi$
$270$	0	0
$271$	−650.000	−0.145700	−0.0728500	−	0.997343i	$-0.523209\pi$
$271$	−650.000	−0.145700	−0.0728500	+	0.997343i	$0.523209\pi$
$272$	0	0
$273$	1144.00i	0.253619i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3232.00i	0.701054i	0.936553	+	0.350527i	$0.113998\pi$
$277$	3232.00i	0.701054i	−0.936553	+	0.350527i	$0.886002\pi$
$278$	0	0
$279$	−676.000	−0.145058
$280$	0	0
$281$	4446.00	0.943865	0.471933	−	0.881635i	$-0.343557\pi$
$281$	4446.00	0.943865	0.471933	+	0.881635i	$0.343557\pi$
$282$	0	0
$283$	− 2483.00i	− 0.521551i	−0.965399	−	0.260776i	$-0.916022\pi$
$283$	− 2483.00i	− 0.521551i	0.965399	−	0.260776i	$-0.0839783\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 11934.0i	− 2.45450i
$288$	0	0
$289$	−8776.00	−1.78628
$290$	0	0
$291$	−386.000	−0.0777585
$292$	0	0
$293$	− 4050.00i	− 0.807521i	−0.914865	−	0.403760i	$-0.867703\pi$
$293$	− 4050.00i	− 0.807521i	0.914865	−	0.403760i	$-0.132297\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2385.00i	0.465965i
$298$	0	0
$299$	−792.000	−0.153186
$300$	0	0
$301$	−11960.0	−2.29024
$302$	0	0
$303$	954.000i	0.180877i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2321.00i	0.431487i	0.976450	+	0.215743i	$0.0692175\pi$
$307$	2321.00i	0.431487i	−0.976450	+	0.215743i	$0.930783\pi$
$308$	0	0
$309$	772.000	0.142128
$310$	0	0
$311$	3258.00	0.594033	0.297016	−	0.954872i	$-0.404008\pi$
$311$	3258.00	0.594033	0.297016	+	0.954872i	$0.404008\pi$
$312$	0	0
$313$	3626.00i	0.654804i	0.944885	+	0.327402i	$0.106173\pi$
$313$	3626.00i	0.654804i	−0.944885	+	0.327402i	$0.893827\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3852.00i	− 0.682492i	−0.939974	−	0.341246i	$-0.889151\pi$
$317$	− 3852.00i	− 0.682492i	0.939974	−	0.341246i	$-0.110849\pi$
$318$	0	0
$319$	6480.00	1.13734
$320$	0	0
$321$	1197.00	0.208131
$322$	0	0
$323$	10647.0i	1.83410i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 802.000i	− 0.135629i
$328$	0	0
$329$	12168.0	2.03904
$330$	0	0
$331$	−7553.00	−1.25423	−0.627115	−	0.778926i	$-0.715765\pi$
$331$	−7553.00	−1.25423	−0.627115	+	0.778926i	$0.715765\pi$
$332$	0	0
$333$	5564.00i	0.915632i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	109.000i	0.0176190i	0.999961	+	0.00880951i	$0.00280419\pi$
$337$	109.000i	0.0176190i	−0.999961	+	0.00880951i	$0.997196\pi$
$338$	0	0
$339$	1143.00	0.183125
$340$	0	0
$341$	1170.00	0.185804
$342$	0	0
$343$	260.000i	0.0409291i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2835.00i	− 0.438590i	−0.975659	−	0.219295i	$-0.929624\pi$
$347$	− 2835.00i	− 0.438590i	0.975659	−	0.219295i	$-0.0703757\pi$
$348$	0	0
$349$	−2990.00	−0.458599	−0.229299	−	0.973356i	$-0.573644\pi$
$349$	−2990.00	−0.458599	−0.229299	+	0.973356i	$0.573644\pi$
$350$	0	0
$351$	2332.00	0.354624
$352$	0	0
$353$	9126.00i	1.37600i	0.725711	+	0.688000i	$0.241511\pi$
$353$	9126.00i	1.37600i	−0.725711	+	0.688000i	$0.758489\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3042.00i	− 0.450980i
$358$	0	0
$359$	−9594.00	−1.41045	−0.705226	−	0.708983i	$-0.749154\pi$
$359$	−9594.00	−1.41045	−0.705226	+	0.708983i	$0.749154\pi$
$360$	0	0
$361$	1422.00	0.207319
$362$	0	0
$363$	− 694.000i	− 0.100346i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9764.00i	1.38876i	0.719606	+	0.694382i	$0.244322\pi$
$367$	9764.00i	1.38876i	−0.719606	+	0.694382i	$0.755678\pi$
$368$	0	0
$369$	−11934.0	−1.68363
$370$	0	0
$371$	−14508.0	−2.03024
$372$	0	0
$373$	6722.00i	0.933115i	0.884491	+	0.466558i	$0.154506\pi$
$373$	6722.00i	0.933115i	−0.884491	+	0.466558i	$0.845494\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6336.00i	− 0.865572i
$378$	0	0
$379$	−13537.0	−1.83469	−0.917347	−	0.398089i	$-0.869674\pi$
$379$	−13537.0	−1.83469	−0.917347	+	0.398089i	$0.869674\pi$
$380$	0	0
$381$	−2374.00	−0.319222
$382$	0	0
$383$	8658.00i	1.15510i	0.816355	+	0.577550i	$0.195991\pi$
$383$	8658.00i	1.15510i	−0.816355	+	0.577550i	$0.804009\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11960.0i	1.57096i
$388$	0	0
$389$	8874.00	1.15663	0.578316	−	0.815813i	$-0.303710\pi$
$389$	8874.00	1.15663	0.578316	+	0.815813i	$0.303710\pi$
$390$	0	0
$391$	2106.00	0.272391
$392$	0	0
$393$	− 1260.00i	− 0.161727i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5876.00i	− 0.742841i	−0.928465	−	0.371421i	$-0.878871\pi$
$397$	− 5876.00i	− 0.742841i	0.928465	−	0.371421i	$-0.121129\pi$
$398$	0	0
$399$	−2366.00	−0.296863
$400$	0	0
$401$	−1755.00	−0.218555	−0.109277	−	0.994011i	$-0.534854\pi$
$401$	−1755.00	−0.218555	−0.109277	+	0.994011i	$0.534854\pi$
$402$	0	0
$403$	− 1144.00i	− 0.141406i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 9630.00i	− 1.17283i
$408$	0	0
$409$	−4589.00	−0.554796	−0.277398	−	0.960755i	$-0.589472\pi$
$409$	−4589.00	−0.554796	−0.277398	+	0.960755i	$0.589472\pi$
$410$	0	0
$411$	891.000	0.106934
$412$	0	0
$413$	− 1872.00i	− 0.223039i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 389.000i	− 0.0456820i
$418$	0	0
$419$	5409.00	0.630661	0.315330	−	0.948982i	$-0.397885\pi$
$419$	5409.00	0.630661	0.315330	+	0.948982i	$0.397885\pi$
$420$	0	0
$421$	12116.0	1.40261	0.701304	−	0.712863i	$-0.252602\pi$
$421$	12116.0	1.40261	0.701304	+	0.712863i	$0.252602\pi$
$422$	0	0
$423$	− 12168.0i	− 1.39865i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3068.00i	− 0.347707i
$428$	0	0
$429$	−1980.00	−0.222833
$430$	0	0
$431$	−9126.00	−1.01992	−0.509958	−	0.860199i	$-0.670339\pi$
$431$	−9126.00	−1.01992	−0.509958	+	0.860199i	$0.670339\pi$
$432$	0	0
$433$	629.000i	0.0698102i	0.999391	+	0.0349051i	$0.0111129\pi$
$433$	629.000i	0.0698102i	−0.999391	+	0.0349051i	$0.988887\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1638.00i	− 0.179305i
$438$	0	0
$439$	4472.00	0.486189	0.243094	−	0.970003i	$-0.421838\pi$
$439$	4472.00	0.486189	0.243094	+	0.970003i	$0.421838\pi$
$440$	0	0
$441$	−8658.00	−0.934888
$442$	0	0
$443$	3393.00i	0.363897i	0.983308	+	0.181948i	$0.0582404\pi$
$443$	3393.00i	0.363897i	−0.983308	+	0.181948i	$0.941760\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1296.00i	0.137134i
$448$	0	0
$449$	5031.00	0.528792	0.264396	−	0.964414i	$-0.414827\pi$
$449$	5031.00	0.528792	0.264396	+	0.964414i	$0.414827\pi$
$450$	0	0
$451$	20655.0	2.15655
$452$	0	0
$453$	− 2710.00i	− 0.281075i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6487.00i	0.664002i	0.943279	+	0.332001i	$0.107724\pi$
$457$	6487.00i	0.664002i	−0.943279	+	0.332001i	$0.892276\pi$
$458$	0	0
$459$	−6201.00	−0.630584
$460$	0	0
$461$	2700.00	0.272780	0.136390	−	0.990655i	$-0.456450\pi$
$461$	2700.00	0.272780	0.136390	+	0.990655i	$0.456450\pi$
$462$	0	0
$463$	2932.00i	0.294302i	0.989114	+	0.147151i	$0.0470102\pi$
$463$	2932.00i	0.294302i	−0.989114	+	0.147151i	$0.952990\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 15660.0i	− 1.55173i	−0.630898	−	0.775866i	$-0.717314\pi$
$467$	− 15660.0i	− 1.55173i	0.630898	−	0.775866i	$-0.282686\pi$
$468$	0	0
$469$	−6526.00	−0.642522
$470$	0	0
$471$	1846.00	0.180593
$472$	0	0
$473$	− 20700.0i	− 2.01223i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	14508.0i	1.39261i
$478$	0	0
$479$	10134.0	0.966669	0.483334	−	0.875436i	$-0.339426\pi$
$479$	10134.0	0.966669	0.483334	+	0.875436i	$0.339426\pi$
$480$	0	0
$481$	−9416.00	−0.892583
$482$	0	0
$483$	468.000i	0.0440885i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1898.00i	0.176605i	0.996094	+	0.0883025i	$0.0281442\pi$
$487$	1898.00i	0.176605i	−0.996094	+	0.0883025i	$0.971856\pi$
$488$	0	0
$489$	−1475.00	−0.136405
$490$	0	0
$491$	6300.00	0.579053	0.289526	−	0.957170i	$-0.406502\pi$
$491$	6300.00	0.579053	0.289526	+	0.957170i	$0.406502\pi$
$492$	0	0
$493$	16848.0i	1.53914i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2808.00i	− 0.253433i
$498$	0	0
$499$	18044.0	1.61876	0.809379	−	0.587286i	$-0.199804\pi$
$499$	18044.0	1.61876	0.809379	+	0.587286i	$0.199804\pi$
$500$	0	0
$501$	−1476.00	−0.131622
$502$	0	0
$503$	6876.00i	0.609514i	0.952430	+	0.304757i	$0.0985753\pi$
$503$	6876.00i	0.609514i	−0.952430	+	0.304757i	$0.901425\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 261.000i	− 0.0228628i
$508$	0	0
$509$	4806.00	0.418511	0.209256	−	0.977861i	$-0.432896\pi$
$509$	4806.00	0.418511	0.209256	+	0.977861i	$0.432896\pi$
$510$	0	0
$511$	−7774.00	−0.672997
$512$	0	0
$513$	4823.00i	0.415089i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	21060.0i	1.79152i
$518$	0	0
$519$	1368.00	0.115700
$520$	0	0
$521$	7749.00	0.651612	0.325806	−	0.945437i	$-0.394364\pi$
$521$	7749.00	0.651612	0.325806	+	0.945437i	$0.394364\pi$
$522$	0	0
$523$	− 8153.00i	− 0.681655i	−0.940126	−	0.340828i	$-0.889293\pi$
$523$	− 8153.00i	− 0.681655i	0.940126	−	0.340828i	$-0.110707\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3042.00i	0.251445i
$528$	0	0
$529$	11843.0	0.973371
$530$	0	0
$531$	−1872.00	−0.152990
$532$	0	0
$533$	− 20196.0i	− 1.64125i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1503.00i	0.120781i
$538$	0	0
$539$	14985.0	1.19749
$540$	0	0
$541$	−10576.0	−0.840476	−0.420238	−	0.907414i	$-0.638053\pi$
$541$	−10576.0	−0.840476	−0.420238	+	0.907414i	$0.638053\pi$
$542$	0	0
$543$	− 3770.00i	− 0.297949i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7553.00i	0.590389i	0.955437	+	0.295195i	$0.0953845\pi$
$547$	7553.00i	0.590389i	−0.955437	+	0.295195i	$0.904615\pi$
$548$	0	0
$549$	−3068.00	−0.238505
$550$	0	0
$551$	13104.0	1.01316
$552$	0	0
$553$	− 23348.0i	− 1.79540i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 13500.0i	− 1.02695i	−0.858103	−	0.513477i	$-0.828357\pi$
$557$	− 13500.0i	− 1.02695i	0.858103	−	0.513477i	$-0.171643\pi$
$558$	0	0
$559$	−20240.0	−1.53141
$560$	0	0
$561$	5265.00	0.396236
$562$	0	0
$563$	− 23184.0i	− 1.73550i	−0.496997	−	0.867752i	$-0.665564\pi$
$563$	− 23184.0i	− 1.73550i	0.496997	−	0.867752i	$-0.334436\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	16874.0i	1.24981i
$568$	0	0
$569$	8055.00	0.593468	0.296734	−	0.954960i	$-0.404103\pi$
$569$	8055.00	0.593468	0.296734	+	0.954960i	$0.404103\pi$
$570$	0	0
$571$	−3068.00	−0.224854	−0.112427	−	0.993660i	$-0.535862\pi$
$571$	−3068.00	−0.224854	−0.112427	+	0.993660i	$0.535862\pi$
$572$	0	0
$573$	− 4122.00i	− 0.300522i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 12419.0i	− 0.896031i	−0.894026	−	0.448015i	$-0.852131\pi$
$577$	− 12419.0i	− 0.896031i	0.894026	−	0.448015i	$-0.147869\pi$
$578$	0	0
$579$	−1963.00	−0.140897
$580$	0	0
$581$	24102.0	1.72103
$582$	0	0
$583$	− 25110.0i	− 1.78379i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12393.0i	− 0.871403i	−0.900091	−	0.435702i	$-0.856500\pi$
$587$	− 12393.0i	− 0.871403i	0.900091	−	0.435702i	$-0.143500\pi$
$588$	0	0
$589$	2366.00	0.165517
$590$	0	0
$591$	−2934.00	−0.204211
$592$	0	0
$593$	23751.0i	1.64475i	0.568946	+	0.822375i	$0.307351\pi$
$593$	23751.0i	1.64475i	−0.568946	+	0.822375i	$0.692649\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 1412.00i	− 0.0967995i
$598$	0	0
$599$	−11610.0	−0.791939	−0.395970	−	0.918264i	$-0.629591\pi$
$599$	−11610.0	−0.791939	−0.395970	+	0.918264i	$0.629591\pi$
$600$	0	0
$601$	26675.0	1.81048	0.905238	−	0.424905i	$-0.139693\pi$
$601$	26675.0	1.81048	0.905238	+	0.424905i	$0.139693\pi$
$602$	0	0
$603$	6526.00i	0.440728i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17264.0i	1.15441i	0.816601	+	0.577203i	$0.195856\pi$
$607$	17264.0i	1.15441i	−0.816601	+	0.577203i	$0.804144\pi$
$608$	0	0
$609$	−3744.00	−0.249121
$610$	0	0
$611$	20592.0	1.36344
$612$	0	0
$613$	− 26026.0i	− 1.71481i	−0.514640	−	0.857406i	$-0.672074\pi$
$613$	− 26026.0i	− 1.71481i	0.514640	−	0.857406i	$-0.327926\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 5022.00i	− 0.327679i	−0.986487	−	0.163840i	$-0.947612\pi$
$617$	− 5022.00i	− 0.327679i	0.986487	−	0.163840i	$-0.0523880\pi$
$618$	0	0
$619$	7820.00	0.507774	0.253887	−	0.967234i	$-0.418291\pi$
$619$	7820.00	0.507774	0.253887	+	0.967234i	$0.418291\pi$
$620$	0	0
$621$	954.000	0.0616469
$622$	0	0
$623$	− 9126.00i	− 0.586879i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4095.00i	− 0.260827i
$628$	0	0
$629$	25038.0	1.58717
$630$	0	0
$631$	−15002.0	−0.946466	−0.473233	−	0.880937i	$-0.656913\pi$
$631$	−15002.0	−0.946466	−0.473233	+	0.880937i	$0.656913\pi$
$632$	0	0
$633$	3419.00i	0.214681i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 14652.0i	− 0.911355i
$638$	0	0
$639$	−2808.00	−0.173838
$640$	0	0
$641$	−918.000	−0.0565660	−0.0282830	−	0.999600i	$-0.509004\pi$
$641$	−918.000	−0.0565660	−0.0282830	+	0.999600i	$0.509004\pi$
$642$	0	0
$643$	23452.0i	1.43835i	0.694831	+	0.719173i	$0.255479\pi$
$643$	23452.0i	1.43835i	−0.694831	+	0.719173i	$0.744521\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20556.0i	1.24906i	0.781002	+	0.624528i	$0.214709\pi$
$647$	20556.0i	1.24906i	−0.781002	+	0.624528i	$0.785291\pi$
$648$	0	0
$649$	3240.00	0.195965
$650$	0	0
$651$	−676.000	−0.0406982
$652$	0	0
$653$	30654.0i	1.83703i	0.395381	+	0.918517i	$0.370613\pi$
$653$	30654.0i	1.83703i	−0.395381	+	0.918517i	$0.629387\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7774.00i	0.461633i
$658$	0	0
$659$	8919.00	0.527215	0.263608	−	0.964630i	$-0.415088\pi$
$659$	8919.00	0.527215	0.263608	+	0.964630i	$0.415088\pi$
$660$	0	0
$661$	−22912.0	−1.34822	−0.674110	−	0.738631i	$-0.735473\pi$
$661$	−22912.0	−1.34822	−0.674110	+	0.738631i	$0.735473\pi$
$662$	0	0
$663$	− 5148.00i	− 0.301556i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2592.00i	− 0.150469i
$668$	0	0
$669$	100.000	0.00577911
$670$	0	0
$671$	5310.00	0.305500
$672$	0	0
$673$	− 1222.00i	− 0.0699920i	−0.999387	−	0.0349960i	$-0.988858\pi$
$673$	− 1222.00i	− 0.0699920i	0.999387	−	0.0349960i	$-0.0111419\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 144.000i	− 0.00817484i	−0.999992	−	0.00408742i	$-0.998699\pi$
$677$	− 144.000i	− 0.00817484i	0.999992	−	0.00408742i	$-0.00130107\pi$
$678$	0	0
$679$	10036.0	0.567226
$680$	0	0
$681$	4212.00	0.237011
$682$	0	0
$683$	12519.0i	0.701356i	0.936496	+	0.350678i	$0.114049\pi$
$683$	12519.0i	0.701356i	−0.936496	+	0.350678i	$0.885951\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3484.00i	− 0.193483i
$688$	0	0
$689$	−24552.0	−1.35756
$690$	0	0
$691$	−11873.0	−0.653647	−0.326824	−	0.945085i	$-0.605978\pi$
$691$	−11873.0	−0.653647	−0.326824	+	0.945085i	$0.605978\pi$
$692$	0	0
$693$	− 30420.0i	− 1.66748i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	53703.0i	2.91843i
$698$	0	0
$699$	918.000	0.0496737
$700$	0	0
$701$	−3060.00	−0.164871	−0.0824355	−	0.996596i	$-0.526270\pi$
$701$	−3060.00	−0.164871	−0.0824355	+	0.996596i	$0.526270\pi$
$702$	0	0
$703$	− 19474.0i	− 1.04477i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 24804.0i	− 1.31945i
$708$	0	0
$709$	−4004.00	−0.212092	−0.106046	−	0.994361i	$-0.533819\pi$
$709$	−4004.00	−0.212092	−0.106046	+	0.994361i	$0.533819\pi$
$710$	0	0
$711$	−23348.0	−1.23153
$712$	0	0
$713$	− 468.000i	− 0.0245817i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3744.00i	− 0.195010i
$718$	0	0
$719$	10314.0	0.534975	0.267488	−	0.963561i	$-0.413807\pi$
$719$	10314.0	0.534975	0.267488	+	0.963561i	$0.413807\pi$
$720$	0	0
$721$	−20072.0	−1.03678
$722$	0	0
$723$	4231.00i	0.217638i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9872.00i	0.503621i	0.967777	+	0.251810i	$0.0810259\pi$
$727$	9872.00i	0.503621i	−0.967777	+	0.251810i	$0.918974\pi$
$728$	0	0
$729$	15443.0	0.784586
$730$	0	0
$731$	53820.0	2.72313
$732$	0	0
$733$	7436.00i	0.374700i	0.982293	+	0.187350i	$0.0599898\pi$
$733$	7436.00i	0.374700i	−0.982293	+	0.187350i	$0.940010\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 11295.0i	− 0.564527i
$738$	0	0
$739$	−16900.0	−0.841240	−0.420620	−	0.907237i	$-0.638187\pi$
$739$	−16900.0	−0.841240	−0.420620	+	0.907237i	$0.638187\pi$
$740$	0	0
$741$	−4004.00	−0.198503
$742$	0	0
$743$	23058.0i	1.13851i	0.822160	+	0.569257i	$0.192769\pi$
$743$	23058.0i	1.13851i	−0.822160	+	0.569257i	$0.807231\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 24102.0i	− 1.18052i
$748$	0	0
$749$	−31122.0	−1.51826
$750$	0	0
$751$	8224.00	0.399598	0.199799	−	0.979837i	$-0.435971\pi$
$751$	8224.00	0.399598	0.199799	+	0.979837i	$0.435971\pi$
$752$	0	0
$753$	− 2925.00i	− 0.141558i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7696.00i	0.369506i	0.982785	+	0.184753i	$0.0591485\pi$
$757$	7696.00i	0.369506i	−0.982785	+	0.184753i	$0.940852\pi$
$758$	0	0
$759$	−810.000	−0.0387367
$760$	0	0
$761$	−6363.00	−0.303099	−0.151550	−	0.988450i	$-0.548426\pi$
$761$	−6363.00	−0.303099	−0.151550	+	0.988450i	$0.548426\pi$
$762$	0	0
$763$	20852.0i	0.989375i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3168.00i	− 0.149139i
$768$	0	0
$769$	−8333.00	−0.390762	−0.195381	−	0.980727i	$-0.562594\pi$
$769$	−8333.00	−0.390762	−0.195381	+	0.980727i	$0.562594\pi$
$770$	0	0
$771$	18.0000	0.000840797 0
$772$	0	0
$773$	32760.0i	1.52431i	0.647392	+	0.762157i	$0.275860\pi$
$773$	32760.0i	1.52431i	−0.647392	+	0.762157i	$0.724140\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5564.00i	0.256895i
$778$	0	0
$779$	41769.0	1.92109
$780$	0	0
$781$	4860.00	0.222669
$782$	0	0
$783$	7632.00i	0.348334i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 43732.0i	− 1.98078i	−0.138286	−	0.990392i	$-0.544159\pi$
$787$	− 43732.0i	− 1.98078i	0.138286	−	0.990392i	$-0.455841\pi$
$788$	0	0
$789$	6786.00	0.306195
$790$	0	0
$791$	−29718.0	−1.33584
$792$	0	0
$793$	− 5192.00i	− 0.232501i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 16866.0i	− 0.749591i	−0.927107	−	0.374796i	$-0.877713\pi$
$797$	− 16866.0i	− 0.749591i	0.927107	−	0.374796i	$-0.122287\pi$
$798$	0	0
$799$	−54756.0	−2.42444
$800$	0	0
$801$	−9126.00	−0.402561
$802$	0	0
$803$	− 13455.0i	− 0.591303i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7632.00i	0.332911i
$808$	0	0
$809$	−16146.0	−0.701685	−0.350842	−	0.936434i	$-0.614105\pi$
$809$	−16146.0	−0.701685	−0.350842	+	0.936434i	$0.614105\pi$
$810$	0	0
$811$	−32444.0	−1.40476	−0.702382	−	0.711801i	$-0.747880\pi$
$811$	−32444.0	−1.40476	−0.702382	+	0.711801i	$0.747880\pi$
$812$	0	0
$813$	650.000i	0.0280400i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 41860.0i	− 1.79253i
$818$	0	0
$819$	−29744.0	−1.26903
$820$	0	0
$821$	−2574.00	−0.109419	−0.0547096	−	0.998502i	$-0.517423\pi$
$821$	−2574.00	−0.109419	−0.0547096	+	0.998502i	$0.517423\pi$
$822$	0	0
$823$	27604.0i	1.16916i	0.811338	+	0.584578i	$0.198740\pi$
$823$	27604.0i	1.16916i	−0.811338	+	0.584578i	$0.801260\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11655.0i	0.490065i	0.969515	+	0.245033i	$0.0787987\pi$
$827$	11655.0i	0.490065i	−0.969515	+	0.245033i	$0.921201\pi$
$828$	0	0
$829$	−33428.0	−1.40049	−0.700243	−	0.713905i	$-0.746925\pi$
$829$	−33428.0	−1.40049	−0.700243	+	0.713905i	$0.746925\pi$
$830$	0	0
$831$	3232.00	0.134918
$832$	0	0
$833$	38961.0i	1.62055i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1378.00i	0.0569064i
$838$	0	0
$839$	−17712.0	−0.728827	−0.364414	−	0.931237i	$-0.618731\pi$
$839$	−17712.0	−0.728827	−0.364414	+	0.931237i	$0.618731\pi$
$840$	0	0
$841$	−3653.00	−0.149781
$842$	0	0
$843$	− 4446.00i	− 0.181647i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	18044.0i	0.731994i
$848$	0	0
$849$	−2483.00	−0.100373
$850$	0	0
$851$	−3852.00	−0.155164
$852$	0	0
$853$	− 10270.0i	− 0.412237i	−0.978527	−	0.206118i	$-0.933917\pi$
$853$	− 10270.0i	− 0.412237i	0.978527	−	0.206118i	$-0.0660832\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 38709.0i	− 1.54291i	−0.636284	−	0.771455i	$-0.719529\pi$
$857$	− 38709.0i	− 1.54291i	0.636284	−	0.771455i	$-0.280471\pi$
$858$	0	0
$859$	15509.0	0.616019	0.308009	−	0.951383i	$-0.400337\pi$
$859$	15509.0	0.616019	0.308009	+	0.951383i	$0.400337\pi$
$860$	0	0
$861$	−11934.0	−0.472369
$862$	0	0
$863$	− 15912.0i	− 0.627637i	−0.949483	−	0.313819i	$-0.898392\pi$
$863$	− 15912.0i	− 0.627637i	0.949483	−	0.313819i	$-0.101608\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8776.00i	0.343770i
$868$	0	0
$869$	40410.0	1.57746
$870$	0	0
$871$	−11044.0	−0.429635
$872$	0	0
$873$	− 10036.0i	− 0.389080i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10972.0i	0.422461i	0.977436	+	0.211230i	$0.0677470\pi$
$877$	10972.0i	0.422461i	−0.977436	+	0.211230i	$0.932253\pi$
$878$	0	0
$879$	−4050.00	−0.155407
$880$	0	0
$881$	−18738.0	−0.716571	−0.358286	−	0.933612i	$-0.616639\pi$
$881$	−18738.0	−0.716571	−0.358286	+	0.933612i	$0.616639\pi$
$882$	0	0
$883$	21367.0i	0.814334i	0.913354	+	0.407167i	$0.133483\pi$
$883$	21367.0i	0.814334i	−0.913354	+	0.407167i	$0.866517\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20124.0i	0.761779i	0.924621	+	0.380889i	$0.124382\pi$
$887$	20124.0i	0.761779i	−0.924621	+	0.380889i	$0.875618\pi$
$888$	0	0
$889$	61724.0	2.32864
$890$	0	0
$891$	−29205.0	−1.09810
$892$	0	0
$893$	42588.0i	1.59592i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	792.000i	0.0294806i
$898$	0	0
$899$	3744.00	0.138898
$900$	0	0
$901$	65286.0	2.41398
$902$	0	0
$903$	11960.0i	0.440757i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23132.0i	0.846842i	0.905933	+	0.423421i	$0.139171\pi$
$907$	23132.0i	0.846842i	−0.905933	+	0.423421i	$0.860829\pi$
$908$	0	0
$909$	−24804.0	−0.905057
$910$	0	0
$911$	−31212.0	−1.13513	−0.567563	−	0.823330i	$-0.692114\pi$
$911$	−31212.0	−1.13513	−0.567563	+	0.823330i	$0.692114\pi$
$912$	0	0
$913$	41715.0i	1.51212i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	32760.0i	1.17975i
$918$	0	0
$919$	−6994.00	−0.251045	−0.125523	−	0.992091i	$-0.540061\pi$
$919$	−6994.00	−0.251045	−0.125523	+	0.992091i	$0.540061\pi$
$920$	0	0
$921$	2321.00	0.0830397
$922$	0	0
$923$	− 4752.00i	− 0.169463i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	20072.0i	0.711166i
$928$	0	0
$929$	−19422.0	−0.685915	−0.342958	−	0.939351i	$-0.611429\pi$
$929$	−19422.0	−0.685915	−0.342958	+	0.939351i	$0.611429\pi$
$930$	0	0
$931$	30303.0	1.06675
$932$	0	0
$933$	− 3258.00i	− 0.114322i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 11699.0i	− 0.407887i	−0.978983	−	0.203943i	$-0.934624\pi$
$937$	− 11699.0i	− 0.407887i	0.978983	−	0.203943i	$-0.0653758\pi$
$938$	0	0
$939$	3626.00	0.126017
$940$	0	0
$941$	−42948.0	−1.48785	−0.743924	−	0.668264i	$-0.767038\pi$
$941$	−42948.0	−1.48785	−0.743924	+	0.668264i	$0.767038\pi$
$942$	0	0
$943$	− 8262.00i	− 0.285310i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 3816.00i	− 0.130943i	−0.997854	−	0.0654717i	$-0.979145\pi$
$947$	− 3816.00i	− 0.130943i	0.997854	−	0.0654717i	$-0.0208552\pi$
$948$	0	0
$949$	−13156.0	−0.450012
$950$	0	0
$951$	−3852.00	−0.131346
$952$	0	0
$953$	− 43407.0i	− 1.47544i	−0.675109	−	0.737718i	$-0.735903\pi$
$953$	− 43407.0i	− 1.47544i	0.675109	−	0.737718i	$-0.264097\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 6480.00i	− 0.218881i
$958$	0	0
$959$	−23166.0	−0.780051
$960$	0	0
$961$	−29115.0	−0.977309
$962$	0	0
$963$	31122.0i	1.04143i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 43216.0i	− 1.43716i	−0.695445	−	0.718580i	$-0.744793\pi$
$967$	− 43216.0i	− 1.43716i	0.695445	−	0.718580i	$-0.255207\pi$
$968$	0	0
$969$	10647.0	0.352973
$970$	0	0
$971$	47619.0	1.57381	0.786903	−	0.617076i	$-0.211683\pi$
$971$	47619.0	1.57381	0.786903	+	0.617076i	$0.211683\pi$
$972$	0	0
$973$	10114.0i	0.333237i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4671.00i	0.152957i	0.997071	+	0.0764783i	$0.0243676\pi$
$977$	4671.00i	0.152957i	−0.997071	+	0.0764783i	$0.975632\pi$
$978$	0	0
$979$	15795.0	0.515639
$980$	0	0
$981$	20852.0	0.678647
$982$	0	0
$983$	− 9054.00i	− 0.293772i	−0.989153	−	0.146886i	$-0.953075\pi$
$983$	− 9054.00i	− 0.293772i	0.989153	−	0.146886i	$-0.0469250\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 12168.0i	− 0.392413i
$988$	0	0
$989$	−8280.00	−0.266217
$990$	0	0
$991$	−8126.00	−0.260475	−0.130238	−	0.991483i	$-0.541574\pi$
$991$	−8126.00	−0.260475	−0.130238	+	0.991483i	$0.541574\pi$
$992$	0	0
$993$	7553.00i	0.241377i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 38468.0i	− 1.22196i	−0.791646	−	0.610980i	$-0.790776\pi$
$997$	− 38468.0i	− 1.22196i	0.791646	−	0.610980i	$-0.209224\pi$
$998$	0	0
$999$	11342.0	0.359204

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.c.l.49.1		2
4.3	odd	2		100.4.c.b.49.2		2
5.2	odd	4		400.4.a.i.1.1		1
5.3	odd	4		400.4.a.l.1.1		1
5.4	even	2	inner	400.4.c.l.49.2		2
12.11	even	2		900.4.d.a.649.1		2
20.3	even	4		100.4.a.b.1.1	✓	1
20.7	even	4		100.4.a.c.1.1	yes	1
20.19	odd	2		100.4.c.b.49.1		2
40.3	even	4		1600.4.a.bc.1.1		1
40.13	odd	4		1600.4.a.y.1.1		1
40.27	even	4		1600.4.a.x.1.1		1
40.37	odd	4		1600.4.a.bd.1.1		1
60.23	odd	4		900.4.a.c.1.1		1
60.47	odd	4		900.4.a.p.1.1		1
60.59	even	2		900.4.d.a.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.a.b.1.1	✓	1	20.3	even	4
100.4.a.c.1.1	yes	1	20.7	even	4
100.4.c.b.49.1		2	20.19	odd	2
100.4.c.b.49.2		2	4.3	odd	2
400.4.a.i.1.1		1	5.2	odd	4
400.4.a.l.1.1		1	5.3	odd	4
400.4.c.l.49.1		2	1.1	even	1	trivial
400.4.c.l.49.2		2	5.4	even	2	inner
900.4.a.c.1.1		1	60.23	odd	4
900.4.a.p.1.1		1	60.47	odd	4
900.4.d.a.649.1		2	12.11	even	2
900.4.d.a.649.2		2	60.59	even	2
1600.4.a.x.1.1		1	40.27	even	4
1600.4.a.y.1.1		1	40.13	odd	4
1600.4.a.bc.1.1		1	40.3	even	4
1600.4.a.bd.1.1		1	40.37	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +26.0000i q^{7} +26.0000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +26.0000i q^{7} +26.0000 q^{9} -45.0000 q^{11} +44.0000i q^{13} -117.000i q^{17} -91.0000 q^{19} +26.0000 q^{21} +18.0000i q^{23} -53.0000i q^{27} -144.000 q^{29} -26.0000 q^{31} +45.0000i q^{33} +214.000i q^{37} +44.0000 q^{39} -459.000 q^{41} +460.000i q^{43} -468.000i q^{47} -333.000 q^{49} -117.000 q^{51} +558.000i q^{53} +91.0000i q^{57} -72.0000 q^{59} -118.000 q^{61} +676.000i q^{63} +251.000i q^{67} +18.0000 q^{69} -108.000 q^{71} +299.000i q^{73} -1170.00i q^{77} -898.000 q^{79} +649.000 q^{81} -927.000i q^{83} +144.000i q^{87} -351.000 q^{89} -1144.00 q^{91} +26.0000i q^{93} -386.000i q^{97} -1170.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 52 q^{9}+O(q^{10}) \) 2 * q + 52 * q^9	\( 2 q + 52 q^{9} - 90 q^{11} - 182 q^{19} + 52 q^{21} - 288 q^{29} - 52 q^{31} + 88 q^{39} - 918 q^{41} - 666 q^{49} - 234 q^{51} - 144 q^{59} - 236 q^{61} + 36 q^{69} - 216 q^{71} - 1796 q^{79} + 1298 q^{81} - 702 q^{89} - 2288 q^{91} - 2340 q^{99}+O(q^{100}) \) 2 * q + 52 * q^9 - 90 * q^11 - 182 * q^19 + 52 * q^21 - 288 * q^29 - 52 * q^31 + 88 * q^39 - 918 * q^41 - 666 * q^49 - 234 * q^51 - 144 * q^59 - 236 * q^61 + 36 * q^69 - 216 * q^71 - 1796 * q^79 + 1298 * q^81 - 702 * q^89 - 2288 * q^91 - 2340 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more