LMFDB - Embedded newform 1900.2.c.e.1749.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.2.c.e.1749.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205i q^{3} -2.00000i q^{7} -4.46410 q^{9} +O(q^{10})$	$q+2.73205i q^{3} -2.00000i q^{7} -4.46410 q^{9} -3.46410 q^{11} -2.73205i q^{13} +3.46410i q^{17} -1.00000 q^{19} +5.46410 q^{21} -3.46410i q^{23} -4.00000i q^{27} -3.46410 q^{29} -1.46410 q^{31} -9.46410i q^{33} -6.73205i q^{37} +7.46410 q^{39} -6.00000 q^{41} -4.92820i q^{43} -12.9282i q^{47} +3.00000 q^{49} -9.46410 q^{51} -10.7321i q^{53} -2.73205i q^{57} -6.92820 q^{59} +12.3923 q^{61} +8.92820i q^{63} -6.73205i q^{67} +9.46410 q^{69} -2.53590 q^{71} -0.535898i q^{73} +6.92820i q^{77} -2.92820 q^{79} -2.46410 q^{81} +3.46410i q^{83} -9.46410i q^{87} +15.4641 q^{89} -5.46410 q^{91} -4.00000i q^{93} +16.5885i q^{97} +15.4641 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 4 q^{19} + 8 q^{21} + 8 q^{31} + 16 q^{39} - 24 q^{41} + 12 q^{49} - 24 q^{51} + 8 q^{61} + 24 q^{69} - 24 q^{71} + 16 q^{79} + 4 q^{81} + 48 q^{89} - 8 q^{91} + 48 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 - 4 * q^19 + 8 * q^21 + 8 * q^31 + 16 * q^39 - 24 * q^41 + 12 * q^49 - 24 * q^51 + 8 * q^61 + 24 * q^69 - 24 * q^71 + 16 * q^79 + 4 * q^81 + 48 * q^89 - 8 * q^91 + 48 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.73205i	1.57735i	0.614810	+	0.788675i	$0.289233\pi$
$3$	2.73205i	1.57735i	−0.614810	+	0.788675i	$0.710767\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	−4.46410	−1.48803
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	− 2.73205i	− 0.757735i	−0.925451	−	0.378867i	$-0.876314\pi$
$13$	− 2.73205i	− 0.757735i	0.925451	−	0.378867i	$-0.123686\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.46410i	0.840168i	0.907485	+	0.420084i	$0.137999\pi$
$17$	3.46410i	0.840168i	−0.907485	+	0.420084i	$0.862001\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	5.46410	1.19236
$22$	0	0
$23$	− 3.46410i	− 0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$	− 3.46410i	− 0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	−1.46410	−0.262960	−0.131480	−	0.991319i	$-0.541973\pi$
$31$	−1.46410	−0.262960	−0.131480	+	0.991319i	$0.541973\pi$
$32$	0	0
$33$	− 9.46410i	− 1.64749i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.73205i	− 1.10674i	−0.832935	−	0.553371i	$-0.813341\pi$
$37$	− 6.73205i	− 1.10674i	0.832935	−	0.553371i	$-0.186659\pi$
$38$	0	0
$39$	7.46410	1.19521
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 4.92820i	− 0.751544i	−0.926712	−	0.375772i	$-0.877378\pi$
$43$	− 4.92820i	− 0.751544i	0.926712	−	0.375772i	$-0.122622\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 12.9282i	− 1.88577i	−0.333115	−	0.942886i	$-0.608100\pi$
$47$	− 12.9282i	− 1.88577i	0.333115	−	0.942886i	$-0.391900\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	−9.46410	−1.32524
$52$	0	0
$53$	− 10.7321i	− 1.47416i	−0.675805	−	0.737080i	$-0.736204\pi$
$53$	− 10.7321i	− 1.47416i	0.675805	−	0.737080i	$-0.263796\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.73205i	− 0.361869i
$58$	0	0
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	12.3923	1.58667	0.793336	−	0.608784i	$-0.208342\pi$
$61$	12.3923	1.58667	0.793336	+	0.608784i	$0.208342\pi$
$62$	0	0
$63$	8.92820i	1.12485i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 6.73205i	− 0.822451i	−0.911534	−	0.411225i	$-0.865101\pi$
$67$	− 6.73205i	− 0.822451i	0.911534	−	0.411225i	$-0.134899\pi$
$68$	0	0
$69$	9.46410	1.13934
$70$	0	0
$71$	−2.53590	−0.300956	−0.150478	−	0.988613i	$-0.548081\pi$
$71$	−2.53590	−0.300956	−0.150478	+	0.988613i	$0.548081\pi$
$72$	0	0
$73$	− 0.535898i	− 0.0627222i	−0.999508	−	0.0313611i	$-0.990016\pi$
$73$	− 0.535898i	− 0.0627222i	0.999508	−	0.0313611i	$-0.00998418\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.92820i	0.789542i
$78$	0	0
$79$	−2.92820	−0.329449	−0.164724	−	0.986340i	$-0.552673\pi$
$79$	−2.92820	−0.329449	−0.164724	+	0.986340i	$0.552673\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	3.46410i	0.380235i	0.981761	+	0.190117i	$0.0608868\pi$
$83$	3.46410i	0.380235i	−0.981761	+	0.190117i	$0.939113\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 9.46410i	− 1.01466i
$88$	0	0
$89$	15.4641	1.63919	0.819596	−	0.572942i	$-0.194198\pi$
$89$	15.4641	1.63919	0.819596	+	0.572942i	$0.194198\pi$
$90$	0	0
$91$	−5.46410	−0.572793
$92$	0	0
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.5885i	1.68430i	0.539241	+	0.842151i	$0.318711\pi$
$97$	16.5885i	1.68430i	−0.539241	+	0.842151i	$0.681289\pi$
$98$	0	0
$99$	15.4641	1.55420
$100$	0	0
$101$	−16.3923	−1.63110	−0.815548	−	0.578690i	$-0.803564\pi$
$101$	−16.3923	−1.63110	−0.815548	+	0.578690i	$0.803564\pi$
$102$	0	0
$103$	13.6603i	1.34598i	0.739649	+	0.672992i	$0.234991\pi$
$103$	13.6603i	1.34598i	−0.739649	+	0.672992i	$0.765009\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.66025i	0.547197i	0.961844	+	0.273599i	$0.0882140\pi$
$107$	5.66025i	0.547197i	−0.961844	+	0.273599i	$0.911786\pi$
$108$	0	0
$109$	14.3923	1.37853	0.689266	−	0.724508i	$-0.257933\pi$
$109$	14.3923	1.37853	0.689266	+	0.724508i	$0.257933\pi$
$110$	0	0
$111$	18.3923	1.74572
$112$	0	0
$113$	12.5885i	1.18422i	0.805856	+	0.592111i	$0.201705\pi$
$113$	12.5885i	1.18422i	−0.805856	+	0.592111i	$0.798295\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	12.1962i	1.12753i
$118$	0	0
$119$	6.92820	0.635107
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 16.3923i	− 1.47804i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 20.5885i	− 1.82693i	−0.406917	−	0.913465i	$-0.633396\pi$
$127$	− 20.5885i	− 1.82693i	0.406917	−	0.913465i	$-0.366604\pi$
$128$	0	0
$129$	13.4641	1.18545
$130$	0	0
$131$	−18.9282	−1.65376	−0.826882	−	0.562375i	$-0.809888\pi$
$131$	−18.9282	−1.65376	−0.826882	+	0.562375i	$0.809888\pi$
$132$	0	0
$133$	2.00000i	0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 17.3205i	− 1.47979i	−0.672722	−	0.739895i	$-0.734875\pi$
$137$	− 17.3205i	− 1.47979i	0.672722	−	0.739895i	$-0.265125\pi$
$138$	0	0
$139$	−11.4641	−0.972372	−0.486186	−	0.873855i	$-0.661612\pi$
$139$	−11.4641	−0.972372	−0.486186	+	0.873855i	$0.661612\pi$
$140$	0	0
$141$	35.3205	2.97452
$142$	0	0
$143$	9.46410i	0.791428i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	8.19615i	0.676007i
$148$	0	0
$149$	4.39230	0.359832	0.179916	−	0.983682i	$-0.442417\pi$
$149$	4.39230	0.359832	0.179916	+	0.983682i	$0.442417\pi$
$150$	0	0
$151$	−8.39230	−0.682956	−0.341478	−	0.939890i	$-0.610927\pi$
$151$	−8.39230	−0.682956	−0.341478	+	0.939890i	$0.610927\pi$
$152$	0	0
$153$	− 15.4641i	− 1.25020i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 23.4641i	− 1.87264i	−0.351149	−	0.936320i	$-0.614209\pi$
$157$	− 23.4641i	− 1.87264i	0.351149	−	0.936320i	$-0.385791\pi$
$158$	0	0
$159$	29.3205	2.32527
$160$	0	0
$161$	−6.92820	−0.546019
$162$	0	0
$163$	7.07180i	0.553906i	0.960883	+	0.276953i	$0.0893246\pi$
$163$	7.07180i	0.553906i	−0.960883	+	0.276953i	$0.910675\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.7321i	0.830471i	0.909714	+	0.415236i	$0.136301\pi$
$167$	10.7321i	0.830471i	−0.909714	+	0.415236i	$0.863699\pi$
$168$	0	0
$169$	5.53590	0.425838
$170$	0	0
$171$	4.46410	0.341378
$172$	0	0
$173$	3.80385i	0.289201i	0.989490	+	0.144601i	$0.0461897\pi$
$173$	3.80385i	0.289201i	−0.989490	+	0.144601i	$0.953810\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 18.9282i	− 1.42273i
$178$	0	0
$179$	−20.7846	−1.55351	−0.776757	−	0.629800i	$-0.783137\pi$
$179$	−20.7846	−1.55351	−0.776757	+	0.629800i	$0.783137\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	33.8564i	2.50274i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 12.0000i	− 0.877527i
$188$	0	0
$189$	−8.00000	−0.581914
$190$	0	0
$191$	6.92820	0.501307	0.250654	−	0.968077i	$-0.419354\pi$
$191$	6.92820	0.501307	0.250654	+	0.968077i	$0.419354\pi$
$192$	0	0
$193$	− 7.80385i	− 0.561733i	−0.959747	−	0.280867i	$-0.909378\pi$
$193$	− 7.80385i	− 0.561733i	0.959747	−	0.280867i	$-0.0906219\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 0.928203i	− 0.0661317i	−0.999453	−	0.0330659i	$-0.989473\pi$
$197$	− 0.928203i	− 0.0661317i	0.999453	−	0.0330659i	$-0.0105271\pi$
$198$	0	0
$199$	−26.9282	−1.90889	−0.954445	−	0.298387i	$-0.903551\pi$
$199$	−26.9282	−1.90889	−0.954445	+	0.298387i	$0.903551\pi$
$200$	0	0
$201$	18.3923	1.29729
$202$	0	0
$203$	6.92820i	0.486265i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	15.4641i	1.07483i
$208$	0	0
$209$	3.46410	0.239617
$210$	0	0
$211$	19.3205	1.33008	0.665039	−	0.746808i	$-0.268415\pi$
$211$	19.3205	1.33008	0.665039	+	0.746808i	$0.268415\pi$
$212$	0	0
$213$	− 6.92820i	− 0.474713i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.92820i	0.198779i
$218$	0	0
$219$	1.46410	0.0989348
$220$	0	0
$221$	9.46410	0.636624
$222$	0	0
$223$	− 9.66025i	− 0.646898i	−0.946246	−	0.323449i	$-0.895158\pi$
$223$	− 9.66025i	− 0.646898i	0.946246	−	0.323449i	$-0.104842\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.19615i	− 0.543998i	−0.962298	−	0.271999i	$-0.912315\pi$
$227$	− 8.19615i	− 0.543998i	0.962298	−	0.271999i	$-0.0876847\pi$
$228$	0	0
$229$	−17.4641	−1.15406	−0.577030	−	0.816723i	$-0.695788\pi$
$229$	−17.4641	−1.15406	−0.577030	+	0.816723i	$0.695788\pi$
$230$	0	0
$231$	−18.9282	−1.24538
$232$	0	0
$233$	− 0.928203i	− 0.0608086i	−0.999538	−	0.0304043i	$-0.990321\pi$
$233$	− 0.928203i	− 0.0608086i	0.999538	−	0.0304043i	$-0.00967948\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	13.8564	0.896296	0.448148	−	0.893959i	$-0.352084\pi$
$239$	13.8564	0.896296	0.448148	+	0.893959i	$0.352084\pi$
$240$	0	0
$241$	−11.8564	−0.763738	−0.381869	−	0.924216i	$-0.624719\pi$
$241$	−11.8564	−0.763738	−0.381869	+	0.924216i	$0.624719\pi$
$242$	0	0
$243$	− 18.7321i	− 1.20166i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.73205i	0.173836i
$248$	0	0
$249$	−9.46410	−0.599763
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	12.0000i	0.754434i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.1244i	0.943431i	0.881751	+	0.471716i	$0.156365\pi$
$257$	15.1244i	0.943431i	−0.881751	+	0.471716i	$0.843635\pi$
$258$	0	0
$259$	−13.4641	−0.836619
$260$	0	0
$261$	15.4641	0.957204
$262$	0	0
$263$	8.53590i	0.526346i	0.964749	+	0.263173i	$0.0847690\pi$
$263$	8.53590i	0.526346i	−0.964749	+	0.263173i	$0.915231\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	42.2487i	2.58558i
$268$	0	0
$269$	−10.3923	−0.633630	−0.316815	−	0.948487i	$-0.602613\pi$
$269$	−10.3923	−0.633630	−0.316815	+	0.948487i	$0.602613\pi$
$270$	0	0
$271$	−2.39230	−0.145322	−0.0726611	−	0.997357i	$-0.523149\pi$
$271$	−2.39230	−0.145322	−0.0726611	+	0.997357i	$0.523149\pi$
$272$	0	0
$273$	− 14.9282i	− 0.903496i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 18.3923i	− 1.10509i	−0.833484	−	0.552543i	$-0.813657\pi$
$277$	− 18.3923i	− 1.10509i	0.833484	−	0.552543i	$-0.186343\pi$
$278$	0	0
$279$	6.53590	0.391294
$280$	0	0
$281$	0.928203	0.0553720	0.0276860	−	0.999617i	$-0.491186\pi$
$281$	0.928203	0.0553720	0.0276860	+	0.999617i	$0.491186\pi$
$282$	0	0
$283$	18.3923i	1.09331i	0.837358	+	0.546655i	$0.184099\pi$
$283$	18.3923i	1.09331i	−0.837358	+	0.546655i	$0.815901\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000i	0.708338i
$288$	0	0
$289$	5.00000	0.294118
$290$	0	0
$291$	−45.3205	−2.65674
$292$	0	0
$293$	− 5.66025i	− 0.330676i	−0.986237	−	0.165338i	$-0.947129\pi$
$293$	− 5.66025i	− 0.330676i	0.986237	−	0.165338i	$-0.0528714\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	13.8564i	0.804030i
$298$	0	0
$299$	−9.46410	−0.547323
$300$	0	0
$301$	−9.85641	−0.568114
$302$	0	0
$303$	− 44.7846i	− 2.57281i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	7.80385i	0.445389i	0.974888	+	0.222695i	$0.0714853\pi$
$307$	7.80385i	0.445389i	−0.974888	+	0.222695i	$0.928515\pi$
$308$	0	0
$309$	−37.3205	−2.12309
$310$	0	0
$311$	−1.60770	−0.0911640	−0.0455820	−	0.998961i	$-0.514514\pi$
$311$	−1.60770	−0.0911640	−0.0455820	+	0.998961i	$0.514514\pi$
$312$	0	0
$313$	10.7846i	0.609582i	0.952419	+	0.304791i	$0.0985866\pi$
$313$	10.7846i	0.609582i	−0.952419	+	0.304791i	$0.901413\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3.12436i	− 0.175481i	−0.996143	−	0.0877406i	$-0.972035\pi$
$317$	− 3.12436i	− 0.175481i	0.996143	−	0.0877406i	$-0.0279647\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	−15.4641	−0.863122
$322$	0	0
$323$	− 3.46410i	− 0.192748i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	39.3205i	2.17443i
$328$	0	0
$329$	−25.8564	−1.42551
$330$	0	0
$331$	−22.2487	−1.22290	−0.611450	−	0.791283i	$-0.709413\pi$
$331$	−22.2487	−1.22290	−0.611450	+	0.791283i	$0.709413\pi$
$332$	0	0
$333$	30.0526i	1.64687i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.2679i	0.940645i	0.882495	+	0.470323i	$0.155862\pi$
$337$	17.2679i	0.940645i	−0.882495	+	0.470323i	$0.844138\pi$
$338$	0	0
$339$	−34.3923	−1.86793
$340$	0	0
$341$	5.07180	0.274653
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 31.1769i	− 1.67366i	−0.547459	−	0.836832i	$-0.684405\pi$
$347$	− 31.1769i	− 1.67366i	0.547459	−	0.836832i	$-0.315595\pi$
$348$	0	0
$349$	−20.9282	−1.12026	−0.560131	−	0.828404i	$-0.689249\pi$
$349$	−20.9282	−1.12026	−0.560131	+	0.828404i	$0.689249\pi$
$350$	0	0
$351$	−10.9282	−0.583304
$352$	0	0
$353$	1.60770i	0.0855690i	0.999084	+	0.0427845i	$0.0136229\pi$
$353$	1.60770i	0.0855690i	−0.999084	+	0.0427845i	$0.986377\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	18.9282i	1.00179i
$358$	0	0
$359$	8.53590	0.450507	0.225254	−	0.974300i	$-0.427679\pi$
$359$	8.53590	0.450507	0.225254	+	0.974300i	$0.427679\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.73205i	0.143395i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3.85641i	− 0.201303i	−0.994922	−	0.100651i	$-0.967907\pi$
$367$	− 3.85641i	− 0.201303i	0.994922	−	0.100651i	$-0.0320927\pi$
$368$	0	0
$369$	26.7846	1.39435
$370$	0	0
$371$	−21.4641	−1.11436
$372$	0	0
$373$	− 16.5885i	− 0.858918i	−0.903086	−	0.429459i	$-0.858704\pi$
$373$	− 16.5885i	− 0.858918i	0.903086	−	0.429459i	$-0.141296\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.46410i	0.487426i
$378$	0	0
$379$	−14.9282	−0.766810	−0.383405	−	0.923580i	$-0.625249\pi$
$379$	−14.9282	−0.766810	−0.383405	+	0.923580i	$0.625249\pi$
$380$	0	0
$381$	56.2487	2.88171
$382$	0	0
$383$	6.33975i	0.323946i	0.986795	+	0.161973i	$0.0517857\pi$
$383$	6.33975i	0.323946i	−0.986795	+	0.161973i	$0.948214\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	22.0000i	1.11832i
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	− 51.7128i	− 2.60857i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	19.4641i	0.976875i	0.872599	+	0.488438i	$0.162433\pi$
$397$	19.4641i	0.976875i	−0.872599	+	0.488438i	$0.837567\pi$
$398$	0	0
$399$	−5.46410	−0.273547
$400$	0	0
$401$	38.7846	1.93681	0.968405	−	0.249381i	$-0.0802271\pi$
$401$	38.7846	1.93681	0.968405	+	0.249381i	$0.0802271\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	23.3205i	1.15595i
$408$	0	0
$409$	14.3923	0.711654	0.355827	−	0.934552i	$-0.384199\pi$
$409$	14.3923	0.711654	0.355827	+	0.934552i	$0.384199\pi$
$410$	0	0
$411$	47.3205	2.33415
$412$	0	0
$413$	13.8564i	0.681829i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 31.3205i	− 1.53377i
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	0	0
$423$	57.7128i	2.80609i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 24.7846i	− 1.19941i
$428$	0	0
$429$	−25.8564	−1.24836
$430$	0	0
$431$	−33.4641	−1.61191	−0.805955	−	0.591977i	$-0.798347\pi$
$431$	−33.4641	−1.61191	−0.805955	+	0.591977i	$0.798347\pi$
$432$	0	0
$433$	− 22.3397i	− 1.07358i	−0.843716	−	0.536790i	$-0.819637\pi$
$433$	− 22.3397i	− 1.07358i	0.843716	−	0.536790i	$-0.180363\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.46410i	0.165710i
$438$	0	0
$439$	31.7128	1.51357	0.756785	−	0.653664i	$-0.226769\pi$
$439$	31.7128	1.51357	0.756785	+	0.653664i	$0.226769\pi$
$440$	0	0
$441$	−13.3923	−0.637729
$442$	0	0
$443$	13.6077i	0.646521i	0.946310	+	0.323261i	$0.104779\pi$
$443$	13.6077i	0.646521i	−0.946310	+	0.323261i	$0.895221\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	12.0000i	0.567581i
$448$	0	0
$449$	3.46410	0.163481	0.0817405	−	0.996654i	$-0.473952\pi$
$449$	3.46410	0.163481	0.0817405	+	0.996654i	$0.473952\pi$
$450$	0	0
$451$	20.7846	0.978709
$452$	0	0
$453$	− 22.9282i	− 1.07726i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 34.7846i	− 1.62716i	−0.581456	−	0.813578i	$-0.697517\pi$
$457$	− 34.7846i	− 1.62716i	0.581456	−	0.813578i	$-0.302483\pi$
$458$	0	0
$459$	13.8564	0.646762
$460$	0	0
$461$	−21.7128	−1.01127	−0.505633	−	0.862749i	$-0.668741\pi$
$461$	−21.7128	−1.01127	−0.505633	+	0.862749i	$0.668741\pi$
$462$	0	0
$463$	4.53590i	0.210801i	0.994430	+	0.105401i	$0.0336125\pi$
$463$	4.53590i	0.210801i	−0.994430	+	0.105401i	$0.966388\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.3205i	1.35679i	0.734697	+	0.678396i	$0.237324\pi$
$467$	29.3205i	1.35679i	−0.734697	+	0.678396i	$0.762676\pi$
$468$	0	0
$469$	−13.4641	−0.621714
$470$	0	0
$471$	64.1051	2.95381
$472$	0	0
$473$	17.0718i	0.784962i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	47.9090i	2.19360i
$478$	0	0
$479$	22.3923	1.02313	0.511565	−	0.859244i	$-0.329066\pi$
$479$	22.3923	1.02313	0.511565	+	0.859244i	$0.329066\pi$
$480$	0	0
$481$	−18.3923	−0.838617
$482$	0	0
$483$	− 18.9282i	− 0.861263i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.1962i	0.552660i	0.961063	+	0.276330i	$0.0891183\pi$
$487$	12.1962i	0.552660i	−0.961063	+	0.276330i	$0.910882\pi$
$488$	0	0
$489$	−19.3205	−0.873704
$490$	0	0
$491$	18.9282	0.854218	0.427109	−	0.904200i	$-0.359532\pi$
$491$	18.9282	0.854218	0.427109	+	0.904200i	$0.359532\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.07180i	0.227501i
$498$	0	0
$499$	−23.4641	−1.05040	−0.525199	−	0.850980i	$-0.676009\pi$
$499$	−23.4641	−1.05040	−0.525199	+	0.850980i	$0.676009\pi$
$500$	0	0
$501$	−29.3205	−1.30994
$502$	0	0
$503$	15.4641i	0.689510i	0.938693	+	0.344755i	$0.112038\pi$
$503$	15.4641i	0.689510i	−0.938693	+	0.344755i	$0.887962\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	15.1244i	0.671696i
$508$	0	0
$509$	5.32051	0.235827	0.117914	−	0.993024i	$-0.462379\pi$
$509$	5.32051	0.235827	0.117914	+	0.993024i	$0.462379\pi$
$510$	0	0
$511$	−1.07180	−0.0474135
$512$	0	0
$513$	4.00000i	0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	44.7846i	1.96962i
$518$	0	0
$519$	−10.3923	−0.456172
$520$	0	0
$521$	−14.7846	−0.647726	−0.323863	−	0.946104i	$-0.604982\pi$
$521$	−14.7846	−0.647726	−0.323863	+	0.946104i	$0.604982\pi$
$522$	0	0
$523$	− 37.3731i	− 1.63421i	−0.576489	−	0.817105i	$-0.695578\pi$
$523$	− 37.3731i	− 1.63421i	0.576489	−	0.817105i	$-0.304422\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 5.07180i	− 0.220931i
$528$	0	0
$529$	11.0000	0.478261
$530$	0	0
$531$	30.9282	1.34217
$532$	0	0
$533$	16.3923i	0.710030i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 56.7846i	− 2.45044i
$538$	0	0
$539$	−10.3923	−0.447628
$540$	0	0
$541$	−29.1769	−1.25441	−0.627207	−	0.778853i	$-0.715802\pi$
$541$	−29.1769	−1.25441	−0.627207	+	0.778853i	$0.715802\pi$
$542$	0	0
$543$	38.2487i	1.64141i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 6.73205i	− 0.287842i	−0.989589	−	0.143921i	$-0.954029\pi$
$547$	− 6.73205i	− 0.287842i	0.989589	−	0.143921i	$-0.0459711\pi$
$548$	0	0
$549$	−55.3205	−2.36102
$550$	0	0
$551$	3.46410	0.147576
$552$	0	0
$553$	5.85641i	0.249040i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 5.32051i	− 0.225437i	−0.993627	−	0.112719i	$-0.964044\pi$
$557$	− 5.32051i	− 0.225437i	0.993627	−	0.112719i	$-0.0359559\pi$
$558$	0	0
$559$	−13.4641	−0.569471
$560$	0	0
$561$	32.7846	1.38417
$562$	0	0
$563$	44.1962i	1.86265i	0.364195	+	0.931323i	$0.381344\pi$
$563$	44.1962i	1.86265i	−0.364195	+	0.931323i	$0.618656\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.92820i	0.206965i
$568$	0	0
$569$	29.3205	1.22918	0.614590	−	0.788847i	$-0.289322\pi$
$569$	29.3205	1.22918	0.614590	+	0.788847i	$0.289322\pi$
$570$	0	0
$571$	18.3923	0.769694	0.384847	−	0.922980i	$-0.374254\pi$
$571$	18.3923	0.769694	0.384847	+	0.922980i	$0.374254\pi$
$572$	0	0
$573$	18.9282i	0.790737i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.78461i	0.282447i	0.989978	+	0.141223i	$0.0451036\pi$
$577$	6.78461i	0.282447i	−0.989978	+	0.141223i	$0.954896\pi$
$578$	0	0
$579$	21.3205	0.886050
$580$	0	0
$581$	6.92820	0.287430
$582$	0	0
$583$	37.1769i	1.53971i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 43.1769i	− 1.78210i	−0.453903	−	0.891051i	$-0.649969\pi$
$587$	− 43.1769i	− 1.78210i	0.453903	−	0.891051i	$-0.350031\pi$
$588$	0	0
$589$	1.46410	0.0603273
$590$	0	0
$591$	2.53590	0.104313
$592$	0	0
$593$	31.8564i	1.30819i	0.756414	+	0.654093i	$0.226949\pi$
$593$	31.8564i	1.30819i	−0.756414	+	0.654093i	$0.773051\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 73.5692i	− 3.01099i
$598$	0	0
$599$	20.7846	0.849236	0.424618	−	0.905373i	$-0.360408\pi$
$599$	20.7846	0.849236	0.424618	+	0.905373i	$0.360408\pi$
$600$	0	0
$601$	24.6410	1.00513	0.502564	−	0.864540i	$-0.332390\pi$
$601$	24.6410	1.00513	0.502564	+	0.864540i	$0.332390\pi$
$602$	0	0
$603$	30.0526i	1.22383i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	39.9090i	1.61985i	0.586530	+	0.809927i	$0.300494\pi$
$607$	39.9090i	1.61985i	−0.586530	+	0.809927i	$0.699506\pi$
$608$	0	0
$609$	−18.9282	−0.767010
$610$	0	0
$611$	−35.3205	−1.42891
$612$	0	0
$613$	6.39230i	0.258183i	0.991633	+	0.129091i	$0.0412061\pi$
$613$	6.39230i	0.258183i	−0.991633	+	0.129091i	$0.958794\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.60770i	− 0.0647234i	−0.999476	−	0.0323617i	$-0.989697\pi$
$617$	− 1.60770i	− 0.0647234i	0.999476	−	0.0323617i	$-0.0103028\pi$
$618$	0	0
$619$	2.39230	0.0961549	0.0480774	−	0.998844i	$-0.484691\pi$
$619$	2.39230	0.0961549	0.0480774	+	0.998844i	$0.484691\pi$
$620$	0	0
$621$	−13.8564	−0.556038
$622$	0	0
$623$	− 30.9282i	− 1.23911i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	9.46410i	0.377960i
$628$	0	0
$629$	23.3205	0.929850
$630$	0	0
$631$	11.4641	0.456379	0.228189	−	0.973617i	$-0.426719\pi$
$631$	11.4641	0.456379	0.228189	+	0.973617i	$0.426719\pi$
$632$	0	0
$633$	52.7846i	2.09800i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 8.19615i	− 0.324743i
$638$	0	0
$639$	11.3205	0.447832
$640$	0	0
$641$	−24.9282	−0.984605	−0.492302	−	0.870424i	$-0.663845\pi$
$641$	−24.9282	−0.984605	−0.492302	+	0.870424i	$0.663845\pi$
$642$	0	0
$643$	− 14.3923i	− 0.567577i	−0.958887	−	0.283789i	$-0.908409\pi$
$643$	− 14.3923i	− 0.567577i	0.958887	−	0.283789i	$-0.0915914\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 15.4641i	− 0.607957i	−0.952679	−	0.303978i	$-0.901685\pi$
$647$	− 15.4641i	− 0.607957i	0.952679	−	0.303978i	$-0.0983150\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	27.4641i	1.07475i	0.843342	+	0.537377i	$0.180585\pi$
$653$	27.4641i	1.07475i	−0.843342	+	0.537377i	$0.819415\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.39230i	0.0933327i
$658$	0	0
$659$	−8.78461	−0.342200	−0.171100	−	0.985254i	$-0.554732\pi$
$659$	−8.78461	−0.342200	−0.171100	+	0.985254i	$0.554732\pi$
$660$	0	0
$661$	5.21539	0.202855	0.101428	−	0.994843i	$-0.467659\pi$
$661$	5.21539	0.202855	0.101428	+	0.994843i	$0.467659\pi$
$662$	0	0
$663$	25.8564i	1.00418i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.0000i	0.464642i
$668$	0	0
$669$	26.3923	1.02039
$670$	0	0
$671$	−42.9282	−1.65722
$672$	0	0
$673$	− 50.7321i	− 1.95558i	−0.209594	−	0.977788i	$-0.567214\pi$
$673$	− 50.7321i	− 1.95558i	0.209594	−	0.977788i	$-0.432786\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 16.9808i	− 0.652624i	−0.945262	−	0.326312i	$-0.894194\pi$
$677$	− 16.9808i	− 0.652624i	0.945262	−	0.326312i	$-0.105806\pi$
$678$	0	0
$679$	33.1769	1.27321
$680$	0	0
$681$	22.3923	0.858075
$682$	0	0
$683$	− 17.6603i	− 0.675751i	−0.941191	−	0.337875i	$-0.890292\pi$
$683$	− 17.6603i	− 0.675751i	0.941191	−	0.337875i	$-0.109708\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 47.7128i	− 1.82036i
$688$	0	0
$689$	−29.3205	−1.11702
$690$	0	0
$691$	51.1769	1.94686	0.973431	−	0.228981i	$-0.0735395\pi$
$691$	51.1769	1.94686	0.973431	+	0.228981i	$0.0735395\pi$
$692$	0	0
$693$	− 30.9282i	− 1.17487i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 20.7846i	− 0.787273i
$698$	0	0
$699$	2.53590	0.0959165
$700$	0	0
$701$	−26.5359	−1.00225	−0.501124	−	0.865376i	$-0.667080\pi$
$701$	−26.5359	−1.00225	−0.501124	+	0.865376i	$0.667080\pi$
$702$	0	0
$703$	6.73205i	0.253904i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	32.7846i	1.23299i
$708$	0	0
$709$	30.7846	1.15614	0.578070	−	0.815987i	$-0.303806\pi$
$709$	30.7846	1.15614	0.578070	+	0.815987i	$0.303806\pi$
$710$	0	0
$711$	13.0718	0.490231
$712$	0	0
$713$	5.07180i	0.189940i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	37.8564i	1.41377i
$718$	0	0
$719$	−17.3205	−0.645946	−0.322973	−	0.946408i	$-0.604682\pi$
$719$	−17.3205	−0.645946	−0.322973	+	0.946408i	$0.604682\pi$
$720$	0	0
$721$	27.3205	1.01747
$722$	0	0
$723$	− 32.3923i	− 1.20468i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 0.143594i	− 0.00532559i	−0.999996	−	0.00266279i	$-0.999152\pi$
$727$	− 0.143594i	− 0.00532559i	0.999996	−	0.00266279i	$-0.000847595\pi$
$728$	0	0
$729$	43.7846	1.62165
$730$	0	0
$731$	17.0718	0.631423
$732$	0	0
$733$	10.7846i	0.398339i	0.979965	+	0.199169i	$0.0638244\pi$
$733$	10.7846i	0.398339i	−0.979965	+	0.199169i	$0.936176\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	23.3205i	0.859022i
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	−7.46410	−0.274201
$742$	0	0
$743$	20.8756i	0.765853i	0.923779	+	0.382927i	$0.125084\pi$
$743$	20.8756i	0.765853i	−0.923779	+	0.382927i	$0.874916\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 15.4641i	− 0.565802i
$748$	0	0
$749$	11.3205	0.413642
$750$	0	0
$751$	−25.4641	−0.929198	−0.464599	−	0.885521i	$-0.653802\pi$
$751$	−25.4641	−0.929198	−0.464599	+	0.885521i	$0.653802\pi$
$752$	0	0
$753$	− 65.5692i	− 2.38948i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	0	0
$759$	−32.7846	−1.19001
$760$	0	0
$761$	19.6077	0.710778	0.355389	−	0.934718i	$-0.384348\pi$
$761$	19.6077	0.710778	0.355389	+	0.934718i	$0.384348\pi$
$762$	0	0
$763$	− 28.7846i	− 1.04207i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.9282i	0.683458i
$768$	0	0
$769$	30.5359	1.10115	0.550576	−	0.834785i	$-0.314408\pi$
$769$	30.5359	1.10115	0.550576	+	0.834785i	$0.314408\pi$
$770$	0	0
$771$	−41.3205	−1.48812
$772$	0	0
$773$	22.0526i	0.793175i	0.917997	+	0.396588i	$0.129806\pi$
$773$	22.0526i	0.793175i	−0.917997	+	0.396588i	$0.870194\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 36.7846i	− 1.31964i
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	8.78461	0.314338
$782$	0	0
$783$	13.8564i	0.495188i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	12.1962i	0.434746i	0.976089	+	0.217373i	$0.0697488\pi$
$787$	12.1962i	0.434746i	−0.976089	+	0.217373i	$0.930251\pi$
$788$	0	0
$789$	−23.3205	−0.830232
$790$	0	0
$791$	25.1769	0.895188
$792$	0	0
$793$	− 33.8564i	− 1.20228i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.66025i	0.200496i	0.994962	+	0.100248i	$0.0319637\pi$
$797$	5.66025i	0.200496i	−0.994962	+	0.100248i	$0.968036\pi$
$798$	0	0
$799$	44.7846	1.58437
$800$	0	0
$801$	−69.0333	−2.43917
$802$	0	0
$803$	1.85641i	0.0655112i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 28.3923i	− 0.999456i
$808$	0	0
$809$	−9.71281	−0.341484	−0.170742	−	0.985316i	$-0.554617\pi$
$809$	−9.71281	−0.341484	−0.170742	+	0.985316i	$0.554617\pi$
$810$	0	0
$811$	15.6077	0.548060	0.274030	−	0.961721i	$-0.411643\pi$
$811$	15.6077	0.548060	0.274030	+	0.961721i	$0.411643\pi$
$812$	0	0
$813$	− 6.53590i	− 0.229224i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.92820i	0.172416i
$818$	0	0
$819$	24.3923	0.852336
$820$	0	0
$821$	−16.1436	−0.563415	−0.281708	−	0.959500i	$-0.590901\pi$
$821$	−16.1436	−0.563415	−0.281708	+	0.959500i	$0.590901\pi$
$822$	0	0
$823$	− 39.5692i	− 1.37930i	−0.724145	−	0.689648i	$-0.757765\pi$
$823$	− 39.5692i	− 1.37930i	0.724145	−	0.689648i	$-0.242235\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.6603i	1.03139i	0.856773	+	0.515694i	$0.172466\pi$
$827$	29.6603i	1.03139i	−0.856773	+	0.515694i	$0.827534\pi$
$828$	0	0
$829$	−8.24871	−0.286490	−0.143245	−	0.989687i	$-0.545754\pi$
$829$	−8.24871	−0.286490	−0.143245	+	0.989687i	$0.545754\pi$
$830$	0	0
$831$	50.2487	1.74311
$832$	0	0
$833$	10.3923i	0.360072i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.85641i	0.202427i
$838$	0	0
$839$	−18.9282	−0.653474	−0.326737	−	0.945115i	$-0.605949\pi$
$839$	−18.9282	−0.653474	−0.326737	+	0.945115i	$0.605949\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	2.53590i	0.0873410i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	0	0
$849$	−50.2487	−1.72453
$850$	0	0
$851$	−23.3205	−0.799417
$852$	0	0
$853$	− 44.6410i	− 1.52848i	−0.644932	−	0.764240i	$-0.723114\pi$
$853$	− 44.6410i	− 1.52848i	0.644932	−	0.764240i	$-0.276886\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.80385i	− 0.129937i	−0.997887	−	0.0649685i	$-0.979305\pi$
$857$	− 3.80385i	− 0.129937i	0.997887	−	0.0649685i	$-0.0206947\pi$
$858$	0	0
$859$	−47.7128	−1.62794	−0.813970	−	0.580907i	$-0.802698\pi$
$859$	−47.7128	−1.62794	−0.813970	+	0.580907i	$0.802698\pi$
$860$	0	0
$861$	−32.7846	−1.11730
$862$	0	0
$863$	− 37.2679i	− 1.26862i	−0.773081	−	0.634308i	$-0.781285\pi$
$863$	− 37.2679i	− 1.26862i	0.773081	−	0.634308i	$-0.218715\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.6603i	0.463927i
$868$	0	0
$869$	10.1436	0.344098
$870$	0	0
$871$	−18.3923	−0.623199
$872$	0	0
$873$	− 74.0526i	− 2.50630i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 0.980762i	− 0.0331180i	−0.999863	−	0.0165590i	$-0.994729\pi$
$877$	− 0.980762i	− 0.0331180i	0.999863	−	0.0165590i	$-0.00527113\pi$
$878$	0	0
$879$	15.4641	0.521591
$880$	0	0
$881$	−0.679492	−0.0228927	−0.0114463	−	0.999934i	$-0.503644\pi$
$881$	−0.679492	−0.0228927	−0.0114463	+	0.999934i	$0.503644\pi$
$882$	0	0
$883$	− 22.0000i	− 0.740359i	−0.928960	−	0.370179i	$-0.879296\pi$
$883$	− 22.0000i	− 0.740359i	0.928960	−	0.370179i	$-0.120704\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 23.4115i	− 0.786083i	−0.919521	−	0.393041i	$-0.871423\pi$
$887$	− 23.4115i	− 0.786083i	0.919521	−	0.393041i	$-0.128577\pi$
$888$	0	0
$889$	−41.1769	−1.38103
$890$	0	0
$891$	8.53590	0.285963
$892$	0	0
$893$	12.9282i	0.432626i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 25.8564i	− 0.863320i
$898$	0	0
$899$	5.07180	0.169154
$900$	0	0
$901$	37.1769	1.23854
$902$	0	0
$903$	− 26.9282i	− 0.896114i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 7.41154i	− 0.246096i	−0.992401	−	0.123048i	$-0.960733\pi$
$907$	− 7.41154i	− 0.246096i	0.992401	−	0.123048i	$-0.0392670\pi$
$908$	0	0
$909$	73.1769	2.42713
$910$	0	0
$911$	14.5359	0.481596	0.240798	−	0.970575i	$-0.422591\pi$
$911$	14.5359	0.481596	0.240798	+	0.970575i	$0.422591\pi$
$912$	0	0
$913$	− 12.0000i	− 0.397142i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	37.8564i	1.25013i
$918$	0	0
$919$	−44.4974	−1.46783	−0.733917	−	0.679239i	$-0.762310\pi$
$919$	−44.4974	−1.46783	−0.733917	+	0.679239i	$0.762310\pi$
$920$	0	0
$921$	−21.3205	−0.702535
$922$	0	0
$923$	6.92820i	0.228045i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 60.9808i	− 2.00287i
$928$	0	0
$929$	47.5692	1.56070	0.780348	−	0.625346i	$-0.215042\pi$
$929$	47.5692	1.56070	0.780348	+	0.625346i	$0.215042\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	0	0
$933$	− 4.39230i	− 0.143798i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 51.1769i	− 1.67188i	−0.548823	−	0.835938i	$-0.684924\pi$
$937$	− 51.1769i	− 1.67188i	0.548823	−	0.835938i	$-0.315076\pi$
$938$	0	0
$939$	−29.4641	−0.961525
$940$	0	0
$941$	52.6410	1.71605	0.858024	−	0.513610i	$-0.171692\pi$
$941$	52.6410	1.71605	0.858024	+	0.513610i	$0.171692\pi$
$942$	0	0
$943$	20.7846i	0.676840i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 6.67949i	− 0.217054i	−0.994093	−	0.108527i	$-0.965387\pi$
$947$	− 6.67949i	− 0.217054i	0.994093	−	0.108527i	$-0.0346134\pi$
$948$	0	0
$949$	−1.46410	−0.0475267
$950$	0	0
$951$	8.53590	0.276795
$952$	0	0
$953$	60.5885i	1.96265i	0.192350	+	0.981326i	$0.438389\pi$
$953$	60.5885i	1.96265i	−0.192350	+	0.981326i	$0.561611\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	32.7846i	1.05978i
$958$	0	0
$959$	−34.6410	−1.11862
$960$	0	0
$961$	−28.8564	−0.930852
$962$	0	0
$963$	− 25.2679i	− 0.814248i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	45.3205i	1.45741i	0.684828	+	0.728705i	$0.259877\pi$
$967$	45.3205i	1.45741i	−0.684828	+	0.728705i	$0.740123\pi$
$968$	0	0
$969$	9.46410	0.304031
$970$	0	0
$971$	−14.5359	−0.466479	−0.233240	−	0.972419i	$-0.574933\pi$
$971$	−14.5359	−0.466479	−0.233240	+	0.972419i	$0.574933\pi$
$972$	0	0
$973$	22.9282i	0.735044i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 32.1962i	− 1.03005i	−0.857176	−	0.515023i	$-0.827783\pi$
$977$	− 32.1962i	− 1.03005i	0.857176	−	0.515023i	$-0.172217\pi$
$978$	0	0
$979$	−53.5692	−1.71208
$980$	0	0
$981$	−64.2487	−2.05130
$982$	0	0
$983$	− 2.44486i	− 0.0779790i	−0.999240	−	0.0389895i	$-0.987586\pi$
$983$	− 2.44486i	− 0.0779790i	0.999240	−	0.0389895i	$-0.0124139\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 70.6410i	− 2.24853i
$988$	0	0
$989$	−17.0718	−0.542852
$990$	0	0
$991$	−8.39230	−0.266590	−0.133295	−	0.991076i	$-0.542556\pi$
$991$	−8.39230	−0.266590	−0.133295	+	0.991076i	$0.542556\pi$
$992$	0	0
$993$	− 60.7846i	− 1.92894i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 30.3923i	− 0.962534i	−0.876574	−	0.481267i	$-0.840177\pi$
$997$	− 30.3923i	− 0.962534i	0.876574	−	0.481267i	$-0.159823\pi$
$998$	0	0
$999$	−26.9282	−0.851971

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.c.e.1749.4		4
5.2	odd	4		380.2.a.d.1.2	✓	2
5.3	odd	4		1900.2.a.d.1.1		2
5.4	even	2	inner	1900.2.c.e.1749.1		4
15.2	even	4		3420.2.a.h.1.2		2
20.3	even	4		7600.2.a.bf.1.2		2
20.7	even	4		1520.2.a.l.1.1		2
40.27	even	4		6080.2.a.bj.1.2		2
40.37	odd	4		6080.2.a.z.1.1		2
95.37	even	4		7220.2.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.a.d.1.2	✓	2	5.2	odd	4
1520.2.a.l.1.1		2	20.7	even	4
1900.2.a.d.1.1		2	5.3	odd	4
1900.2.c.e.1749.1		4	5.4	even	2	inner
1900.2.c.e.1749.4		4	1.1	even	1	trivial
3420.2.a.h.1.2		2	15.2	even	4
6080.2.a.z.1.1		2	40.37	odd	4
6080.2.a.bj.1.2		2	40.27	even	4
7220.2.a.h.1.1		2	95.37	even	4
7600.2.a.bf.1.2		2	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.73205i q^{3} -2.00000i q^{7} -4.46410 q^{9} +O(q^{10})\)	\(q+2.73205i q^{3} -2.00000i q^{7} -4.46410 q^{9} -3.46410 q^{11} -2.73205i q^{13} +3.46410i q^{17} -1.00000 q^{19} +5.46410 q^{21} -3.46410i q^{23} -4.00000i q^{27} -3.46410 q^{29} -1.46410 q^{31} -9.46410i q^{33} -6.73205i q^{37} +7.46410 q^{39} -6.00000 q^{41} -4.92820i q^{43} -12.9282i q^{47} +3.00000 q^{49} -9.46410 q^{51} -10.7321i q^{53} -2.73205i q^{57} -6.92820 q^{59} +12.3923 q^{61} +8.92820i q^{63} -6.73205i q^{67} +9.46410 q^{69} -2.53590 q^{71} -0.535898i q^{73} +6.92820i q^{77} -2.92820 q^{79} -2.46410 q^{81} +3.46410i q^{83} -9.46410i q^{87} +15.4641 q^{89} -5.46410 q^{91} -4.00000i q^{93} +16.5885i q^{97} +15.4641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 4 q^{19} + 8 q^{21} + 8 q^{31} + 16 q^{39} - 24 q^{41} + 12 q^{49} - 24 q^{51} + 8 q^{61} + 24 q^{69} - 24 q^{71} + 16 q^{79} + 4 q^{81} + 48 q^{89} - 8 q^{91} + 48 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 - 4 * q^19 + 8 * q^21 + 8 * q^31 + 16 * q^39 - 24 * q^41 + 12 * q^49 - 24 * q^51 + 8 * q^61 + 24 * q^69 - 24 * q^71 + 16 * q^79 + 4 * q^81 + 48 * q^89 - 8 * q^91 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more