LMFDB - Embedded newform 1840.3.c.b.1151.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(1151,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1840.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.c (of order $2$, degree $1$, 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:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.15
Character	$\chi$	$=$	1840.1151
Dual form			1840.3.c.b.1151.42

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.27428i q^{3} +2.23607 q^{5} -5.54665i q^{7} -1.72088 q^{9} +O(q^{10})$	$q-3.27428i q^{3} +2.23607 q^{5} -5.54665i q^{7} -1.72088 q^{9} +15.9311i q^{11} +3.25742 q^{13} -7.32150i q^{15} -8.34132 q^{17} +0.0817557i q^{19} -18.1612 q^{21} +4.79583i q^{23} +5.00000 q^{25} -23.8338i q^{27} -0.456414 q^{29} -49.9673i q^{31} +52.1628 q^{33} -12.4027i q^{35} -39.7383 q^{37} -10.6657i q^{39} -9.83926 q^{41} -81.7624i q^{43} -3.84801 q^{45} -53.1943i q^{47} +18.2347 q^{49} +27.3118i q^{51} -51.5988 q^{53} +35.6230i q^{55} +0.267691 q^{57} -35.9267i q^{59} +28.3815 q^{61} +9.54512i q^{63} +7.28380 q^{65} -78.4902i q^{67} +15.7029 q^{69} +79.3143i q^{71} +22.6053 q^{73} -16.3714i q^{75} +88.3641 q^{77} -39.8514i q^{79} -93.5265 q^{81} -37.2534i q^{83} -18.6518 q^{85} +1.49443i q^{87} -78.4622 q^{89} -18.0677i q^{91} -163.607 q^{93} +0.182811i q^{95} +83.5307 q^{97} -27.4155i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 120 q^{9}+O(q^{10}) $ 56 * q - 120 * q^9	$ 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) $ 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 3.27428i	− 1.09143i	−0.837972	−	0.545713i	$-0.816259\pi$
$3$	− 3.27428i	− 1.09143i	0.837972	−	0.545713i	$-0.183741\pi$
$4$	0	0
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	− 5.54665i	− 0.792378i	−0.918169	−	0.396189i	$-0.870333\pi$
$7$	− 5.54665i	− 0.792378i	0.918169	−	0.396189i	$-0.129667\pi$
$8$	0	0
$9$	−1.72088	−0.191209
$10$	0	0
$11$	15.9311i	1.44828i	0.689652	+	0.724140i	$0.257763\pi$
$11$	15.9311i	1.44828i	−0.689652	+	0.724140i	$0.742237\pi$
$12$	0	0
$13$	3.25742	0.250570	0.125285	−	0.992121i	$-0.460015\pi$
$13$	3.25742	0.250570	0.125285	+	0.992121i	$0.460015\pi$
$14$	0	0
$15$	− 7.32150i	− 0.488100i
$16$	0	0
$17$	−8.34132	−0.490666	−0.245333	−	0.969439i	$-0.578897\pi$
$17$	−8.34132	−0.490666	−0.245333	+	0.969439i	$0.578897\pi$
$18$	0	0
$19$	0.0817557i	0.00430293i	0.999998	+	0.00215147i	$0.000684834\pi$
$19$	0.0817557i	0.00430293i	−0.999998	+	0.00215147i	$0.999315\pi$
$20$	0	0
$21$	−18.1612	−0.864821
$22$	0	0
$23$	4.79583i	0.208514i
$23$	4.79583i	0.208514i
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 23.8338i	− 0.882735i
$28$	0	0
$29$	−0.456414	−0.0157384	−0.00786921	−	0.999969i	$-0.502505\pi$
$29$	−0.456414	−0.0157384	−0.00786921	+	0.999969i	$0.502505\pi$
$30$	0	0
$31$	− 49.9673i	− 1.61185i	−0.592019	−	0.805924i	$-0.701669\pi$
$31$	− 49.9673i	− 1.61185i	0.592019	−	0.805924i	$-0.298331\pi$
$32$	0	0
$33$	52.1628	1.58069
$34$	0	0
$35$	− 12.4027i	− 0.354362i
$36$	0	0
$37$	−39.7383	−1.07401	−0.537004	−	0.843580i	$-0.680444\pi$
$37$	−39.7383	−1.07401	−0.537004	+	0.843580i	$0.680444\pi$
$38$	0	0
$39$	− 10.6657i	− 0.273479i
$40$	0	0
$41$	−9.83926	−0.239982	−0.119991	−	0.992775i	$-0.538287\pi$
$41$	−9.83926	−0.239982	−0.119991	+	0.992775i	$0.538287\pi$
$42$	0	0
$43$	− 81.7624i	− 1.90145i	−0.310033	−	0.950726i	$-0.600340\pi$
$43$	− 81.7624i	− 1.90145i	0.310033	−	0.950726i	$-0.399660\pi$
$44$	0	0
$45$	−3.84801	−0.0855113
$46$	0	0
$47$	− 53.1943i	− 1.13179i	−0.824476	−	0.565897i	$-0.808530\pi$
$47$	− 53.1943i	− 1.13179i	0.824476	−	0.565897i	$-0.191470\pi$
$48$	0	0
$49$	18.2347	0.372137
$50$	0	0
$51$	27.3118i	0.535525i
$52$	0	0
$53$	−51.5988	−0.973562	−0.486781	−	0.873524i	$-0.661829\pi$
$53$	−51.5988	−0.973562	−0.486781	+	0.873524i	$0.661829\pi$
$54$	0	0
$55$	35.6230i	0.647691i
$56$	0	0
$57$	0.267691	0.00469633
$58$	0	0
$59$	− 35.9267i	− 0.608928i	−0.952524	−	0.304464i	$-0.901523\pi$
$59$	− 35.9267i	− 0.608928i	0.952524	−	0.304464i	$-0.0984773\pi$
$60$	0	0
$61$	28.3815	0.465270	0.232635	−	0.972564i	$-0.425265\pi$
$61$	28.3815	0.465270	0.232635	+	0.972564i	$0.425265\pi$
$62$	0	0
$63$	9.54512i	0.151510i
$64$	0	0
$65$	7.28380	0.112059
$66$	0	0
$67$	− 78.4902i	− 1.17150i	−0.810493	−	0.585748i	$-0.800801\pi$
$67$	− 78.4902i	− 1.17150i	0.810493	−	0.585748i	$-0.199199\pi$
$68$	0	0
$69$	15.7029	0.227578
$70$	0	0
$71$	79.3143i	1.11710i	0.829470	+	0.558552i	$0.188643\pi$
$71$	79.3143i	1.11710i	−0.829470	+	0.558552i	$0.811357\pi$
$72$	0	0
$73$	22.6053	0.309662	0.154831	−	0.987941i	$-0.450517\pi$
$73$	22.6053	0.309662	0.154831	+	0.987941i	$0.450517\pi$
$74$	0	0
$75$	− 16.3714i	− 0.218285i
$76$	0	0
$77$	88.3641	1.14759
$78$	0	0
$79$	− 39.8514i	− 0.504448i	−0.967669	−	0.252224i	$-0.918838\pi$
$79$	− 39.8514i	− 0.504448i	0.967669	−	0.252224i	$-0.0811620\pi$
$80$	0	0
$81$	−93.5265	−1.15465
$82$	0	0
$83$	− 37.2534i	− 0.448837i	−0.974493	−	0.224418i	$-0.927952\pi$
$83$	− 37.2534i	− 0.448837i	0.974493	−	0.224418i	$-0.0720482\pi$
$84$	0	0
$85$	−18.6518	−0.219433
$86$	0	0
$87$	1.49443i	0.0171773i
$88$	0	0
$89$	−78.4622	−0.881597	−0.440799	−	0.897606i	$-0.645305\pi$
$89$	−78.4622	−0.881597	−0.440799	+	0.897606i	$0.645305\pi$
$90$	0	0
$91$	− 18.0677i	− 0.198547i
$92$	0	0
$93$	−163.607	−1.75921
$94$	0	0
$95$	0.182811i	0.00192433i
$96$	0	0
$97$	83.5307	0.861141	0.430570	−	0.902557i	$-0.358312\pi$
$97$	83.5307	0.861141	0.430570	+	0.902557i	$0.358312\pi$
$98$	0	0
$99$	− 27.4155i	− 0.276924i
$100$	0	0
$101$	−169.481	−1.67803	−0.839017	−	0.544105i	$-0.816869\pi$
$101$	−169.481	−1.67803	−0.839017	+	0.544105i	$0.816869\pi$
$102$	0	0
$103$	− 94.3939i	− 0.916446i	−0.888837	−	0.458223i	$-0.848486\pi$
$103$	− 94.3939i	− 0.916446i	0.888837	−	0.458223i	$-0.151514\pi$
$104$	0	0
$105$	−40.6098	−0.386760
$106$	0	0
$107$	48.1755i	0.450238i	0.974331	+	0.225119i	$0.0722771\pi$
$107$	48.1755i	0.450238i	−0.974331	+	0.225119i	$0.927723\pi$
$108$	0	0
$109$	43.1331	0.395716	0.197858	−	0.980231i	$-0.436601\pi$
$109$	43.1331	0.395716	0.197858	+	0.980231i	$0.436601\pi$
$110$	0	0
$111$	130.114i	1.17220i
$112$	0	0
$113$	78.8284	0.697596	0.348798	−	0.937198i	$-0.386590\pi$
$113$	78.8284	0.697596	0.348798	+	0.937198i	$0.386590\pi$
$114$	0	0
$115$	10.7238i	0.0932505i
$116$	0	0
$117$	−5.60563	−0.0479113
$118$	0	0
$119$	46.2664i	0.388793i
$120$	0	0
$121$	−132.800	−1.09752
$122$	0	0
$123$	32.2164i	0.261922i
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	− 228.862i	− 1.80207i	−0.433750	−	0.901033i	$-0.642810\pi$
$127$	− 228.862i	− 1.80207i	0.433750	−	0.901033i	$-0.357190\pi$
$128$	0	0
$129$	−267.713	−2.07529
$130$	0	0
$131$	108.313i	0.826819i	0.910545	+	0.413410i	$0.135662\pi$
$131$	108.313i	0.826819i	−0.910545	+	0.413410i	$0.864338\pi$
$132$	0	0
$133$	0.453470	0.00340955
$134$	0	0
$135$	− 53.2941i	− 0.394771i
$136$	0	0
$137$	26.0150	0.189891	0.0949453	−	0.995482i	$-0.469732\pi$
$137$	26.0150	0.189891	0.0949453	+	0.995482i	$0.469732\pi$
$138$	0	0
$139$	189.469i	1.36308i	0.731780	+	0.681541i	$0.238690\pi$
$139$	189.469i	1.36308i	−0.731780	+	0.681541i	$0.761310\pi$
$140$	0	0
$141$	−174.173	−1.23527
$142$	0	0
$143$	51.8942i	0.362896i
$144$	0	0
$145$	−1.02057	−0.00703844
$146$	0	0
$147$	− 59.7055i	− 0.406160i
$148$	0	0
$149$	171.608	1.15173	0.575866	−	0.817544i	$-0.304665\pi$
$149$	171.608	1.15173	0.575866	+	0.817544i	$0.304665\pi$
$150$	0	0
$151$	− 172.229i	− 1.14059i	−0.821441	−	0.570294i	$-0.806829\pi$
$151$	− 172.229i	− 1.14059i	0.821441	−	0.570294i	$-0.193171\pi$
$152$	0	0
$153$	14.3544	0.0938198
$154$	0	0
$155$	− 111.730i	− 0.720840i
$156$	0	0
$157$	95.9747	0.611304	0.305652	−	0.952143i	$-0.401126\pi$
$157$	95.9747	0.611304	0.305652	+	0.952143i	$0.401126\pi$
$158$	0	0
$159$	168.949i	1.06257i
$160$	0	0
$161$	26.6008	0.165222
$162$	0	0
$163$	175.576i	1.07715i	0.842577	+	0.538576i	$0.181037\pi$
$163$	175.576i	1.07715i	−0.842577	+	0.538576i	$0.818963\pi$
$164$	0	0
$165$	116.640	0.706906
$166$	0	0
$167$	111.843i	0.669717i	0.942268	+	0.334859i	$0.108689\pi$
$167$	111.843i	0.669717i	−0.942268	+	0.334859i	$0.891311\pi$
$168$	0	0
$169$	−158.389	−0.937214
$170$	0	0
$171$	− 0.140692i	0 0.000822760i
$172$	0	0
$173$	−244.124	−1.41112	−0.705560	−	0.708650i	$-0.749304\pi$
$173$	−244.124	−1.41112	−0.705560	+	0.708650i	$0.749304\pi$
$174$	0	0
$175$	− 27.7332i	− 0.158476i
$176$	0	0
$177$	−117.634	−0.664599
$178$	0	0
$179$	− 51.8575i	− 0.289707i	−0.989453	−	0.144853i	$-0.953729\pi$
$179$	− 51.8575i	− 0.289707i	0.989453	−	0.144853i	$-0.0462711\pi$
$180$	0	0
$181$	198.871	1.09874	0.549368	−	0.835580i	$-0.314868\pi$
$181$	198.871	1.09874	0.549368	+	0.835580i	$0.314868\pi$
$182$	0	0
$183$	− 92.9288i	− 0.507807i
$184$	0	0
$185$	−88.8575	−0.480311
$186$	0	0
$187$	− 132.886i	− 0.710622i
$188$	0	0
$189$	−132.198	−0.699460
$190$	0	0
$191$	− 234.857i	− 1.22962i	−0.788676	−	0.614808i	$-0.789233\pi$
$191$	− 234.857i	− 1.22962i	0.788676	−	0.614808i	$-0.210767\pi$
$192$	0	0
$193$	−103.605	−0.536814	−0.268407	−	0.963306i	$-0.586497\pi$
$193$	−103.605	−0.536814	−0.268407	+	0.963306i	$0.586497\pi$
$194$	0	0
$195$	− 23.8492i	− 0.122304i
$196$	0	0
$197$	69.7257	0.353937	0.176969	−	0.984216i	$-0.443371\pi$
$197$	69.7257	0.353937	0.176969	+	0.984216i	$0.443371\pi$
$198$	0	0
$199$	208.186i	1.04616i	0.852283	+	0.523081i	$0.175218\pi$
$199$	208.186i	1.04616i	−0.852283	+	0.523081i	$0.824782\pi$
$200$	0	0
$201$	−256.999	−1.27860
$202$	0	0
$203$	2.53157i	0.0124708i
$204$	0	0
$205$	−22.0012	−0.107323
$206$	0	0
$207$	− 8.25306i	− 0.0398698i
$208$	0	0
$209$	−1.30246	−0.00623186
$210$	0	0
$211$	− 221.696i	− 1.05069i	−0.850889	−	0.525345i	$-0.823936\pi$
$211$	− 221.696i	− 1.05069i	0.850889	−	0.525345i	$-0.176064\pi$
$212$	0	0
$213$	259.697	1.21923
$214$	0	0
$215$	− 182.826i	− 0.850355i
$216$	0	0
$217$	−277.151	−1.27719
$218$	0	0
$219$	− 74.0160i	− 0.337973i
$220$	0	0
$221$	−27.1712	−0.122946
$222$	0	0
$223$	− 289.660i	− 1.29892i	−0.760395	−	0.649461i	$-0.774995\pi$
$223$	− 289.660i	− 1.29892i	0.760395	−	0.649461i	$-0.225005\pi$
$224$	0	0
$225$	−8.60441	−0.0382418
$226$	0	0
$227$	109.581i	0.482735i	0.970434	+	0.241367i	$0.0775958\pi$
$227$	109.581i	0.482735i	−0.970434	+	0.241367i	$0.922404\pi$
$228$	0	0
$229$	113.606	0.496098	0.248049	−	0.968748i	$-0.420211\pi$
$229$	113.606	0.496098	0.248049	+	0.968748i	$0.420211\pi$
$230$	0	0
$231$	− 289.328i	− 1.25250i
$232$	0	0
$233$	−211.736	−0.908738	−0.454369	−	0.890814i	$-0.650135\pi$
$233$	−211.736	−0.908738	−0.454369	+	0.890814i	$0.650135\pi$
$234$	0	0
$235$	− 118.946i	− 0.506154i
$236$	0	0
$237$	−130.484	−0.550567
$238$	0	0
$239$	68.0002i	0.284520i	0.989829	+	0.142260i	$0.0454369\pi$
$239$	68.0002i	0.284520i	−0.989829	+	0.142260i	$0.954563\pi$
$240$	0	0
$241$	37.6321	0.156150	0.0780750	−	0.996947i	$-0.475123\pi$
$241$	37.6321	0.156150	0.0780750	+	0.996947i	$0.475123\pi$
$242$	0	0
$243$	91.7270i	0.377477i
$244$	0	0
$245$	40.7741	0.166425
$246$	0	0
$247$	0.266312i	0.00107819i
$248$	0	0
$249$	−121.978	−0.489872
$250$	0	0
$251$	363.846i	1.44958i	0.688967	+	0.724792i	$0.258064\pi$
$251$	363.846i	1.44958i	−0.688967	+	0.724792i	$0.741936\pi$
$252$	0	0
$253$	−76.4028	−0.301987
$254$	0	0
$255$	61.0710i	0.239494i
$256$	0	0
$257$	−230.778	−0.897968	−0.448984	−	0.893540i	$-0.648214\pi$
$257$	−230.778	−0.897968	−0.448984	+	0.893540i	$0.648214\pi$
$258$	0	0
$259$	220.414i	0.851020i
$260$	0	0
$261$	0.785435	0.00300933
$262$	0	0
$263$	− 56.7048i	− 0.215608i	−0.994172	−	0.107804i	$-0.965618\pi$
$263$	− 56.7048i	− 0.215608i	0.994172	−	0.107804i	$-0.0343819\pi$
$264$	0	0
$265$	−115.378	−0.435390
$266$	0	0
$267$	256.907i	0.962198i
$268$	0	0
$269$	−123.426	−0.458832	−0.229416	−	0.973328i	$-0.573682\pi$
$269$	−123.426	−0.458832	−0.229416	+	0.973328i	$0.573682\pi$
$270$	0	0
$271$	147.267i	0.543421i	0.962379	+	0.271710i	$0.0875893\pi$
$271$	147.267i	0.543421i	−0.962379	+	0.271710i	$0.912411\pi$
$272$	0	0
$273$	−59.1587	−0.216699
$274$	0	0
$275$	79.6555i	0.289656i
$276$	0	0
$277$	291.526	1.05244	0.526220	−	0.850348i	$-0.323609\pi$
$277$	291.526	1.05244	0.526220	+	0.850348i	$0.323609\pi$
$278$	0	0
$279$	85.9878i	0.308200i
$280$	0	0
$281$	−416.475	−1.48212	−0.741059	−	0.671440i	$-0.765676\pi$
$281$	−416.475	−1.48212	−0.741059	+	0.671440i	$0.765676\pi$
$282$	0	0
$283$	238.094i	0.841322i	0.907218	+	0.420661i	$0.138202\pi$
$283$	238.094i	0.841322i	−0.907218	+	0.420661i	$0.861798\pi$
$284$	0	0
$285$	0.598575	0.00210026
$286$	0	0
$287$	54.5749i	0.190156i
$288$	0	0
$289$	−219.422	−0.759247
$290$	0	0
$291$	− 273.502i	− 0.939871i
$292$	0	0
$293$	−170.639	−0.582384	−0.291192	−	0.956665i	$-0.594052\pi$
$293$	−170.639	−0.582384	−0.291192	+	0.956665i	$0.594052\pi$
$294$	0	0
$295$	− 80.3346i	− 0.272321i
$296$	0	0
$297$	379.699	1.27845
$298$	0	0
$299$	15.6220i	0.0522476i
$300$	0	0
$301$	−453.507	−1.50667
$302$	0	0
$303$	554.929i	1.83145i
$304$	0	0
$305$	63.4629	0.208075
$306$	0	0
$307$	250.870i	0.817166i	0.912721	+	0.408583i	$0.133977\pi$
$307$	250.870i	0.817166i	−0.912721	+	0.408583i	$0.866023\pi$
$308$	0	0
$309$	−309.072	−1.00023
$310$	0	0
$311$	42.4888i	0.136620i	0.997664	+	0.0683099i	$0.0217607\pi$
$311$	42.4888i	0.136620i	−0.997664	+	0.0683099i	$0.978239\pi$
$312$	0	0
$313$	−217.839	−0.695970	−0.347985	−	0.937500i	$-0.613134\pi$
$313$	−217.839	−0.695970	−0.347985	+	0.937500i	$0.613134\pi$
$314$	0	0
$315$	21.3435i	0.0677573i
$316$	0	0
$317$	342.466	1.08033	0.540167	−	0.841558i	$-0.318361\pi$
$317$	342.466	1.08033	0.540167	+	0.841558i	$0.318361\pi$
$318$	0	0
$319$	− 7.27118i	− 0.0227937i
$320$	0	0
$321$	157.740	0.491401
$322$	0	0
$323$	− 0.681951i	− 0.00211130i
$324$	0	0
$325$	16.2871	0.0501141
$326$	0	0
$327$	− 141.230i	− 0.431895i
$328$	0	0
$329$	−295.050	−0.896808
$330$	0	0
$331$	− 512.542i	− 1.54847i	−0.632901	−	0.774233i	$-0.718136\pi$
$331$	− 512.542i	− 1.54847i	0.632901	−	0.774233i	$-0.281864\pi$
$332$	0	0
$333$	68.3849	0.205360
$334$	0	0
$335$	− 175.509i	− 0.523909i
$336$	0	0
$337$	265.729	0.788514	0.394257	−	0.919000i	$-0.371002\pi$
$337$	265.729	0.788514	0.394257	+	0.919000i	$0.371002\pi$
$338$	0	0
$339$	− 258.106i	− 0.761374i
$340$	0	0
$341$	796.033	2.33441
$342$	0	0
$343$	− 372.927i	− 1.08725i
$344$	0	0
$345$	35.1127	0.101776
$346$	0	0
$347$	− 377.565i	− 1.08808i	−0.839058	−	0.544041i	$-0.816893\pi$
$347$	− 377.565i	− 1.08808i	0.839058	−	0.544041i	$-0.183107\pi$
$348$	0	0
$349$	−12.9040	−0.0369742	−0.0184871	−	0.999829i	$-0.505885\pi$
$349$	−12.9040	−0.0369742	−0.0184871	+	0.999829i	$0.505885\pi$
$350$	0	0
$351$	− 77.6367i	− 0.221187i
$352$	0	0
$353$	−170.714	−0.483610	−0.241805	−	0.970325i	$-0.577739\pi$
$353$	−170.714	−0.483610	−0.241805	+	0.970325i	$0.577739\pi$
$354$	0	0
$355$	177.352i	0.499584i
$356$	0	0
$357$	151.489	0.424338
$358$	0	0
$359$	208.501i	0.580782i	0.956908	+	0.290391i	$0.0937854\pi$
$359$	208.501i	0.580782i	−0.956908	+	0.290391i	$0.906215\pi$
$360$	0	0
$361$	360.993	0.999981
$362$	0	0
$363$	434.823i	1.19786i
$364$	0	0
$365$	50.5470	0.138485
$366$	0	0
$367$	− 396.381i	− 1.08006i	−0.841647	−	0.540028i	$-0.818414\pi$
$367$	− 396.381i	− 1.08006i	0.841647	−	0.540028i	$-0.181586\pi$
$368$	0	0
$369$	16.9322	0.0458867
$370$	0	0
$371$	286.200i	0.771429i
$372$	0	0
$373$	293.122	0.785850	0.392925	−	0.919570i	$-0.371463\pi$
$373$	293.122	0.785850	0.392925	+	0.919570i	$0.371463\pi$
$374$	0	0
$375$	− 36.6075i	− 0.0976200i
$376$	0	0
$377$	−1.48673	−0.00394358
$378$	0	0
$379$	672.370i	1.77406i	0.461709	+	0.887032i	$0.347237\pi$
$379$	672.370i	1.77406i	−0.461709	+	0.887032i	$0.652763\pi$
$380$	0	0
$381$	−749.359	−1.96682
$382$	0	0
$383$	42.3003i	0.110445i	0.998474	+	0.0552223i	$0.0175867\pi$
$383$	42.3003i	0.110445i	−0.998474	+	0.0552223i	$0.982413\pi$
$384$	0	0
$385$	197.588	0.513216
$386$	0	0
$387$	140.703i	0.363575i
$388$	0	0
$389$	546.090	1.40383	0.701915	−	0.712260i	$-0.252328\pi$
$389$	546.090	1.40383	0.701915	+	0.712260i	$0.252328\pi$
$390$	0	0
$391$	− 40.0036i	− 0.102311i
$392$	0	0
$393$	354.648	0.902411
$394$	0	0
$395$	− 89.1104i	− 0.225596i
$396$	0	0
$397$	725.486	1.82742	0.913711	−	0.406365i	$-0.133204\pi$
$397$	725.486	1.82742	0.913711	+	0.406365i	$0.133204\pi$
$398$	0	0
$399$	− 1.48479i	− 0.00372127i
$400$	0	0
$401$	−571.493	−1.42517	−0.712585	−	0.701586i	$-0.752476\pi$
$401$	−571.493	−1.42517	−0.712585	+	0.701586i	$0.752476\pi$
$402$	0	0
$403$	− 162.764i	− 0.403881i
$404$	0	0
$405$	−209.132	−0.516374
$406$	0	0
$407$	− 633.074i	− 1.55546i
$408$	0	0
$409$	484.879	1.18552	0.592762	−	0.805378i	$-0.298038\pi$
$409$	484.879	1.18552	0.592762	+	0.805378i	$0.298038\pi$
$410$	0	0
$411$	− 85.1803i	− 0.207251i
$412$	0	0
$413$	−199.273	−0.482501
$414$	0	0
$415$	− 83.3012i	− 0.200726i
$416$	0	0
$417$	620.372	1.48770
$418$	0	0
$419$	− 92.3825i	− 0.220483i	−0.993905	−	0.110242i	$-0.964838\pi$
$419$	− 92.3825i	− 0.220483i	0.993905	−	0.110242i	$-0.0351625\pi$
$420$	0	0
$421$	258.028	0.612894	0.306447	−	0.951888i	$-0.400860\pi$
$421$	258.028	0.612894	0.306447	+	0.951888i	$0.400860\pi$
$422$	0	0
$423$	91.5411i	0.216409i
$424$	0	0
$425$	−41.7066	−0.0981332
$426$	0	0
$427$	− 157.422i	− 0.368670i
$428$	0	0
$429$	169.916	0.396074
$430$	0	0
$431$	− 743.178i	− 1.72431i	−0.506643	−	0.862156i	$-0.669114\pi$
$431$	− 743.178i	− 1.72431i	0.506643	−	0.862156i	$-0.330886\pi$
$432$	0	0
$433$	696.705	1.60902	0.804509	−	0.593940i	$-0.202428\pi$
$433$	696.705	1.60902	0.804509	+	0.593940i	$0.202428\pi$
$434$	0	0
$435$	3.34164i	0.00768193i
$436$	0	0
$437$	−0.392087	−0.000897224 0
$438$	0	0
$439$	− 493.025i	− 1.12306i	−0.827455	−	0.561532i	$-0.810212\pi$
$439$	− 493.025i	− 1.12306i	0.827455	−	0.561532i	$-0.189788\pi$
$440$	0	0
$441$	−31.3798	−0.0711560
$442$	0	0
$443$	692.367i	1.56290i	0.623965	+	0.781452i	$0.285521\pi$
$443$	692.367i	1.56290i	−0.623965	+	0.781452i	$0.714479\pi$
$444$	0	0
$445$	−175.447	−0.394262
$446$	0	0
$447$	− 561.892i	− 1.25703i
$448$	0	0
$449$	661.351	1.47294	0.736471	−	0.676469i	$-0.236491\pi$
$449$	661.351	1.47294	0.736471	+	0.676469i	$0.236491\pi$
$450$	0	0
$451$	− 156.750i	− 0.347561i
$452$	0	0
$453$	−563.925	−1.24487
$454$	0	0
$455$	− 40.4007i	− 0.0887927i
$456$	0	0
$457$	−474.346	−1.03796	−0.518978	−	0.854788i	$-0.673687\pi$
$457$	−474.346	−1.03796	−0.518978	+	0.854788i	$0.673687\pi$
$458$	0	0
$459$	198.806i	0.433128i
$460$	0	0
$461$	−60.1197	−0.130412	−0.0652058	−	0.997872i	$-0.520770\pi$
$461$	−60.1197	−0.130412	−0.0652058	+	0.997872i	$0.520770\pi$
$462$	0	0
$463$	7.99765i	0.0172735i	0.999963	+	0.00863677i	$0.00274920\pi$
$463$	7.99765i	0.0172735i	−0.999963	+	0.00863677i	$0.997251\pi$
$464$	0	0
$465$	−365.836	−0.786743
$466$	0	0
$467$	− 591.353i	− 1.26628i	−0.774037	−	0.633140i	$-0.781766\pi$
$467$	− 591.353i	− 1.26628i	0.774037	−	0.633140i	$-0.218234\pi$
$468$	0	0
$469$	−435.357	−0.928267
$470$	0	0
$471$	− 314.248i	− 0.667193i
$472$	0	0
$473$	1302.56	2.75384
$474$	0	0
$475$	0.408779i	0 0.000860587i
$476$	0	0
$477$	88.7954	0.186154
$478$	0	0
$479$	− 137.064i	− 0.286145i	−0.989712	−	0.143073i	$-0.954302\pi$
$479$	− 137.064i	− 0.286145i	0.989712	−	0.143073i	$-0.0456983\pi$
$480$	0	0
$481$	−129.444	−0.269115
$482$	0	0
$483$	− 87.0983i	− 0.180328i
$484$	0	0
$485$	186.780	0.385114
$486$	0	0
$487$	668.321i	1.37232i	0.727449	+	0.686161i	$0.240706\pi$
$487$	668.321i	1.37232i	−0.727449	+	0.686161i	$0.759294\pi$
$488$	0	0
$489$	574.883	1.17563
$490$	0	0
$491$	− 347.552i	− 0.707845i	−0.935275	−	0.353922i	$-0.884848\pi$
$491$	− 347.552i	− 0.707845i	0.935275	−	0.353922i	$-0.115152\pi$
$492$	0	0
$493$	3.80710	0.00772231
$494$	0	0
$495$	− 61.3030i	− 0.123844i
$496$	0	0
$497$	439.928	0.885168
$498$	0	0
$499$	− 105.570i	− 0.211563i	−0.994389	−	0.105781i	$-0.966266\pi$
$499$	− 105.570i	− 0.211563i	0.994389	−	0.105781i	$-0.0337344\pi$
$500$	0	0
$501$	366.204	0.730946
$502$	0	0
$503$	445.252i	0.885192i	0.896721	+	0.442596i	$0.145942\pi$
$503$	445.252i	0.885192i	−0.896721	+	0.442596i	$0.854058\pi$
$504$	0	0
$505$	−378.972	−0.750440
$506$	0	0
$507$	518.610i	1.02290i
$508$	0	0
$509$	646.086	1.26933	0.634663	−	0.772789i	$-0.281139\pi$
$509$	646.086	1.26933	0.634663	+	0.772789i	$0.281139\pi$
$510$	0	0
$511$	− 125.384i	− 0.245369i
$512$	0	0
$513$	1.94855	0.00379835
$514$	0	0
$515$	− 211.071i	− 0.409847i
$516$	0	0
$517$	847.443	1.63916
$518$	0	0
$519$	799.328i	1.54013i
$520$	0	0
$521$	213.777	0.410320	0.205160	−	0.978728i	$-0.434228\pi$
$521$	213.777	0.410320	0.205160	+	0.978728i	$0.434228\pi$
$522$	0	0
$523$	452.763i	0.865703i	0.901465	+	0.432851i	$0.142493\pi$
$523$	452.763i	0.865703i	−0.901465	+	0.432851i	$0.857507\pi$
$524$	0	0
$525$	−90.8062	−0.172964
$526$	0	0
$527$	416.793i	0.790879i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	0	0
$531$	61.8257i	0.116433i
$532$	0	0
$533$	−32.0506	−0.0601324
$534$	0	0
$535$	107.724i	0.201353i
$536$	0	0
$537$	−169.796	−0.316193
$538$	0	0
$539$	290.499i	0.538959i
$540$	0	0
$541$	−889.669	−1.64449	−0.822245	−	0.569133i	$-0.807279\pi$
$541$	−889.669	−1.64449	−0.822245	+	0.569133i	$0.807279\pi$
$542$	0	0
$543$	− 651.160i	− 1.19919i
$544$	0	0
$545$	96.4485	0.176970
$546$	0	0
$547$	277.828i	0.507912i	0.967216	+	0.253956i	$0.0817319\pi$
$547$	277.828i	0.507912i	−0.967216	+	0.253956i	$0.918268\pi$
$548$	0	0
$549$	−48.8411	−0.0889638
$550$	0	0
$551$	− 0.0373145i	0 6.77214e-5i
$552$	0	0
$553$	−221.042	−0.399713
$554$	0	0
$555$	290.944i	0.524223i
$556$	0	0
$557$	801.445	1.43886	0.719430	−	0.694565i	$-0.244403\pi$
$557$	801.445	1.43886	0.719430	+	0.694565i	$0.244403\pi$
$558$	0	0
$559$	− 266.334i	− 0.476448i
$560$	0	0
$561$	−435.107	−0.775591
$562$	0	0
$563$	348.311i	0.618669i	0.950953	+	0.309335i	$0.100106\pi$
$563$	348.311i	0.618669i	−0.950953	+	0.309335i	$0.899894\pi$
$564$	0	0
$565$	176.266	0.311975
$566$	0	0
$567$	518.758i	0.914918i
$568$	0	0
$569$	666.590	1.17151	0.585756	−	0.810487i	$-0.300798\pi$
$569$	666.590	1.17151	0.585756	+	0.810487i	$0.300798\pi$
$570$	0	0
$571$	754.563i	1.32148i	0.750616	+	0.660738i	$0.229757\pi$
$571$	754.563i	1.32148i	−0.750616	+	0.660738i	$0.770243\pi$
$572$	0	0
$573$	−768.986	−1.34203
$574$	0	0
$575$	23.9792i	0.0417029i
$576$	0	0
$577$	−560.346	−0.971136	−0.485568	−	0.874199i	$-0.661387\pi$
$577$	−560.346	−0.971136	−0.485568	+	0.874199i	$0.661387\pi$
$578$	0	0
$579$	339.232i	0.585893i
$580$	0	0
$581$	−206.632	−0.355648
$582$	0	0
$583$	− 822.025i	− 1.40999i
$584$	0	0
$585$	−12.5346	−0.0214266
$586$	0	0
$587$	771.270i	1.31392i	0.753926	+	0.656959i	$0.228158\pi$
$587$	771.270i	1.31392i	−0.753926	+	0.656959i	$0.771842\pi$
$588$	0	0
$589$	4.08511	0.00693567
$590$	0	0
$591$	− 228.301i	− 0.386296i
$592$	0	0
$593$	800.711	1.35027	0.675136	−	0.737693i	$-0.264085\pi$
$593$	800.711	1.35027	0.675136	+	0.737693i	$0.264085\pi$
$594$	0	0
$595$	103.455i	0.173873i
$596$	0	0
$597$	681.660	1.14181
$598$	0	0
$599$	− 652.145i	− 1.08872i	−0.838851	−	0.544361i	$-0.816772\pi$
$599$	− 652.145i	− 1.08872i	0.838851	−	0.544361i	$-0.183228\pi$
$600$	0	0
$601$	−881.047	−1.46597	−0.732984	−	0.680245i	$-0.761873\pi$
$601$	−881.047	−1.46597	−0.732984	+	0.680245i	$0.761873\pi$
$602$	0	0
$603$	135.072i	0.224001i
$604$	0	0
$605$	−296.949	−0.490825
$606$	0	0
$607$	748.754i	1.23353i	0.787146	+	0.616766i	$0.211558\pi$
$607$	748.754i	1.23353i	−0.787146	+	0.616766i	$0.788442\pi$
$608$	0	0
$609$	8.28905	0.0136109
$610$	0	0
$611$	− 173.276i	− 0.283594i
$612$	0	0
$613$	−553.968	−0.903700	−0.451850	−	0.892094i	$-0.649236\pi$
$613$	−553.968	−0.903700	−0.451850	+	0.892094i	$0.649236\pi$
$614$	0	0
$615$	72.0382i	0.117135i
$616$	0	0
$617$	563.793	0.913765	0.456883	−	0.889527i	$-0.348966\pi$
$617$	563.793	0.913765	0.456883	+	0.889527i	$0.348966\pi$
$618$	0	0
$619$	791.288i	1.27833i	0.769069	+	0.639166i	$0.220720\pi$
$619$	791.288i	1.27833i	−0.769069	+	0.639166i	$0.779280\pi$
$620$	0	0
$621$	114.303	0.184063
$622$	0	0
$623$	435.202i	0.698558i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	4.26461i	0.00680160i
$628$	0	0
$629$	331.470	0.526979
$630$	0	0
$631$	464.907i	0.736778i	0.929672	+	0.368389i	$0.120091\pi$
$631$	464.907i	0.736778i	−0.929672	+	0.368389i	$0.879909\pi$
$632$	0	0
$633$	−725.893	−1.14675
$634$	0	0
$635$	− 511.752i	− 0.805909i
$636$	0	0
$637$	59.3981	0.0932466
$638$	0	0
$639$	− 136.491i	− 0.213600i
$640$	0	0
$641$	−151.822	−0.236851	−0.118426	−	0.992963i	$-0.537785\pi$
$641$	−151.822	−0.236851	−0.118426	+	0.992963i	$0.537785\pi$
$642$	0	0
$643$	− 116.626i	− 0.181379i	−0.995879	−	0.0906893i	$-0.971093\pi$
$643$	− 116.626i	− 0.181379i	0.995879	−	0.0906893i	$-0.0289070\pi$
$644$	0	0
$645$	−598.624	−0.928099
$646$	0	0
$647$	− 778.680i	− 1.20352i	−0.798675	−	0.601762i	$-0.794465\pi$
$647$	− 778.680i	− 1.20352i	0.798675	−	0.601762i	$-0.205535\pi$
$648$	0	0
$649$	572.352	0.881899
$650$	0	0
$651$	907.468i	1.39396i
$652$	0	0
$653$	727.986	1.11483	0.557417	−	0.830233i	$-0.311793\pi$
$653$	727.986	1.11483	0.557417	+	0.830233i	$0.311793\pi$
$654$	0	0
$655$	242.196i	0.369765i
$656$	0	0
$657$	−38.9010	−0.0592101
$658$	0	0
$659$	283.174i	0.429702i	0.976647	+	0.214851i	$0.0689266\pi$
$659$	283.174i	0.429702i	−0.976647	+	0.214851i	$0.931073\pi$
$660$	0	0
$661$	153.245	0.231838	0.115919	−	0.993259i	$-0.463019\pi$
$661$	153.245	0.231838	0.115919	+	0.993259i	$0.463019\pi$
$662$	0	0
$663$	88.9659i	0.134187i
$664$	0	0
$665$	1.01399	0.00152480
$666$	0	0
$667$	− 2.18889i	− 0.00328169i
$668$	0	0
$669$	−948.426	−1.41768
$670$	0	0
$671$	452.148i	0.673842i
$672$	0	0
$673$	961.420	1.42856	0.714279	−	0.699861i	$-0.246755\pi$
$673$	961.420	1.42856	0.714279	+	0.699861i	$0.246755\pi$
$674$	0	0
$675$	− 119.169i	− 0.176547i
$676$	0	0
$677$	346.176	0.511339	0.255669	−	0.966764i	$-0.417704\pi$
$677$	346.176	0.511339	0.255669	+	0.966764i	$0.417704\pi$
$678$	0	0
$679$	− 463.315i	− 0.682349i
$680$	0	0
$681$	358.798	0.526869
$682$	0	0
$683$	− 544.789i	− 0.797641i	−0.917029	−	0.398820i	$-0.869420\pi$
$683$	− 544.789i	− 0.797641i	0.917029	−	0.398820i	$-0.130580\pi$
$684$	0	0
$685$	58.1713	0.0849216
$686$	0	0
$687$	− 371.979i	− 0.541454i
$688$	0	0
$689$	−168.079	−0.243946
$690$	0	0
$691$	804.592i	1.16439i	0.813050	+	0.582194i	$0.197806\pi$
$691$	804.592i	1.16439i	−0.813050	+	0.582194i	$0.802194\pi$
$692$	0	0
$693$	−152.064	−0.219429
$694$	0	0
$695$	423.664i	0.609589i
$696$	0	0
$697$	82.0724	0.117751
$698$	0	0
$699$	693.282i	0.991820i
$700$	0	0
$701$	−19.3174	−0.0275569	−0.0137785	−	0.999905i	$-0.504386\pi$
$701$	−19.3174	−0.0275569	−0.0137785	+	0.999905i	$0.504386\pi$
$702$	0	0
$703$	− 3.24883i	− 0.00462138i
$704$	0	0
$705$	−389.462	−0.552429
$706$	0	0
$707$	940.053i	1.32964i
$708$	0	0
$709$	522.045	0.736311	0.368156	−	0.929764i	$-0.379989\pi$
$709$	522.045	0.736311	0.368156	+	0.929764i	$0.379989\pi$
$710$	0	0
$711$	68.5795i	0.0964550i
$712$	0	0
$713$	239.635	0.336093
$714$	0	0
$715$	116.039i	0.162292i
$716$	0	0
$717$	222.651	0.310532
$718$	0	0
$719$	9.07275i	0.0126186i	0.999980	+	0.00630928i	$0.00200832\pi$
$719$	9.07275i	0.0126186i	−0.999980	+	0.00630928i	$0.997992\pi$
$720$	0	0
$721$	−523.569	−0.726171
$722$	0	0
$723$	− 123.218i	− 0.170426i
$724$	0	0
$725$	−2.28207	−0.00314768
$726$	0	0
$727$	607.855i	0.836114i	0.908421	+	0.418057i	$0.137289\pi$
$727$	607.855i	0.836114i	−0.908421	+	0.418057i	$0.862711\pi$
$728$	0	0
$729$	−541.399	−0.742660
$730$	0	0
$731$	682.007i	0.932978i
$732$	0	0
$733$	−275.769	−0.376219	−0.188110	−	0.982148i	$-0.560236\pi$
$733$	−275.769	−0.376219	−0.188110	+	0.982148i	$0.560236\pi$
$734$	0	0
$735$	− 133.506i	− 0.181640i
$736$	0	0
$737$	1250.43	1.69665
$738$	0	0
$739$	− 857.012i	− 1.15969i	−0.814726	−	0.579846i	$-0.803113\pi$
$739$	− 857.012i	− 1.15969i	0.814726	−	0.579846i	$-0.196887\pi$
$740$	0	0
$741$	0.871980	0.00117676
$742$	0	0
$743$	154.660i	0.208155i	0.994569	+	0.104078i	$0.0331891\pi$
$743$	154.660i	0.208155i	−0.994569	+	0.104078i	$0.966811\pi$
$744$	0	0
$745$	383.727	0.515070
$746$	0	0
$747$	64.1088i	0.0858216i
$748$	0	0
$749$	267.212	0.356759
$750$	0	0
$751$	− 1134.78i	− 1.51103i	−0.655132	−	0.755514i	$-0.727387\pi$
$751$	− 1134.78i	− 1.51103i	0.655132	−	0.755514i	$-0.272613\pi$
$752$	0	0
$753$	1191.33	1.58211
$754$	0	0
$755$	− 385.115i	− 0.510087i
$756$	0	0
$757$	12.0948	0.0159772	0.00798861	−	0.999968i	$-0.497457\pi$
$757$	12.0948	0.0159772	0.00798861	+	0.999968i	$0.497457\pi$
$758$	0	0
$759$	250.164i	0.329597i
$760$	0	0
$761$	−798.754	−1.04961	−0.524806	−	0.851222i	$-0.675862\pi$
$761$	−798.754	−1.04961	−0.524806	+	0.851222i	$0.675862\pi$
$762$	0	0
$763$	− 239.244i	− 0.313557i
$764$	0	0
$765$	32.0975	0.0419575
$766$	0	0
$767$	− 117.028i	− 0.152579i
$768$	0	0
$769$	892.060	1.16003	0.580013	−	0.814607i	$-0.303047\pi$
$769$	892.060	1.16003	0.580013	+	0.814607i	$0.303047\pi$
$770$	0	0
$771$	755.630i	0.980065i
$772$	0	0
$773$	−1516.32	−1.96160	−0.980801	−	0.195009i	$-0.937526\pi$
$773$	−1516.32	−1.96160	−0.980801	+	0.195009i	$0.937526\pi$
$774$	0	0
$775$	− 249.836i	− 0.322370i
$776$	0	0
$777$	721.697	0.928825
$778$	0	0
$779$	− 0.804416i	− 0.00103263i
$780$	0	0
$781$	−1263.56	−1.61788
$782$	0	0
$783$	10.8781i	0.0138929i
$784$	0	0
$785$	214.606	0.273383
$786$	0	0
$787$	394.861i	0.501729i	0.968022	+	0.250864i	$0.0807148\pi$
$787$	394.861i	0.501729i	−0.968022	+	0.250864i	$0.919285\pi$
$788$	0	0
$789$	−185.667	−0.235320
$790$	0	0
$791$	− 437.233i	− 0.552760i
$792$	0	0
$793$	92.4503	0.116583
$794$	0	0
$795$	377.781i	0.475196i
$796$	0	0
$797$	−594.538	−0.745970	−0.372985	−	0.927837i	$-0.621666\pi$
$797$	−594.538	−0.745970	−0.372985	+	0.927837i	$0.621666\pi$
$798$	0	0
$799$	443.711i	0.555333i
$800$	0	0
$801$	135.024	0.168569
$802$	0	0
$803$	360.127i	0.448477i
$804$	0	0
$805$	59.4811	0.0738896
$806$	0	0
$807$	404.130i	0.500781i
$808$	0	0
$809$	216.390	0.267479	0.133739	−	0.991017i	$-0.457301\pi$
$809$	216.390	0.267479	0.133739	+	0.991017i	$0.457301\pi$
$810$	0	0
$811$	− 902.530i	− 1.11286i	−0.830894	−	0.556430i	$-0.812171\pi$
$811$	− 902.530i	− 1.11286i	0.830894	−	0.556430i	$-0.187829\pi$
$812$	0	0
$813$	482.193	0.593103
$814$	0	0
$815$	392.599i	0.481717i
$816$	0	0
$817$	6.68455	0.00818182
$818$	0	0
$819$	31.0924i	0.0379639i
$820$	0	0
$821$	−387.249	−0.471679	−0.235840	−	0.971792i	$-0.575784\pi$
$821$	−387.249	−0.471679	−0.235840	+	0.971792i	$0.575784\pi$
$822$	0	0
$823$	− 1295.92i	− 1.57463i	−0.616552	−	0.787315i	$-0.711471\pi$
$823$	− 1295.92i	− 1.57463i	0.616552	−	0.787315i	$-0.288529\pi$
$824$	0	0
$825$	260.814	0.316138
$826$	0	0
$827$	381.340i	0.461112i	0.973059	+	0.230556i	$0.0740545\pi$
$827$	381.340i	0.461112i	−0.973059	+	0.230556i	$0.925946\pi$
$828$	0	0
$829$	412.915	0.498088	0.249044	−	0.968492i	$-0.419884\pi$
$829$	412.915	0.498088	0.249044	+	0.968492i	$0.419884\pi$
$830$	0	0
$831$	− 954.537i	− 1.14866i
$832$	0	0
$833$	−152.102	−0.182595
$834$	0	0
$835$	250.088i	0.299507i
$836$	0	0
$837$	−1190.91	−1.42283
$838$	0	0
$839$	− 438.433i	− 0.522566i	−0.965262	−	0.261283i	$-0.915854\pi$
$839$	− 438.433i	− 0.522566i	0.965262	−	0.261283i	$-0.0841455\pi$
$840$	0	0
$841$	−840.792	−0.999752
$842$	0	0
$843$	1363.65i	1.61762i
$844$	0	0
$845$	−354.169	−0.419135
$846$	0	0
$847$	736.592i	0.869649i
$848$	0	0
$849$	779.586	0.918240
$850$	0	0
$851$	− 190.578i	− 0.223946i
$852$	0	0
$853$	490.825	0.575410	0.287705	−	0.957719i	$-0.407108\pi$
$853$	490.825	0.575410	0.287705	+	0.957719i	$0.407108\pi$
$854$	0	0
$855$	− 0.314597i	0 0.000367949i
$856$	0	0
$857$	−193.777	−0.226111	−0.113055	−	0.993589i	$-0.536064\pi$
$857$	−193.777	−0.226111	−0.113055	+	0.993589i	$0.536064\pi$
$858$	0	0
$859$	− 1096.36i	− 1.27632i	−0.769905	−	0.638159i	$-0.779696\pi$
$859$	− 1096.36i	− 1.27632i	0.769905	−	0.638159i	$-0.220304\pi$
$860$	0	0
$861$	178.693	0.207541
$862$	0	0
$863$	− 322.234i	− 0.373388i	−0.982418	−	0.186694i	$-0.940223\pi$
$863$	− 322.234i	− 0.373388i	0.982418	−	0.186694i	$-0.0597773\pi$
$864$	0	0
$865$	−545.877	−0.631072
$866$	0	0
$867$	718.449i	0.828661i
$868$	0	0
$869$	634.876	0.730583
$870$	0	0
$871$	− 255.675i	− 0.293542i
$872$	0	0
$873$	−143.746	−0.164658
$874$	0	0
$875$	− 62.0134i	− 0.0708724i
$876$	0	0
$877$	−998.753	−1.13883	−0.569414	−	0.822051i	$-0.692830\pi$
$877$	−998.753	−1.13883	−0.569414	+	0.822051i	$0.692830\pi$
$878$	0	0
$879$	558.718i	0.635629i
$880$	0	0
$881$	−352.852	−0.400513	−0.200256	−	0.979744i	$-0.564177\pi$
$881$	−352.852	−0.400513	−0.200256	+	0.979744i	$0.564177\pi$
$882$	0	0
$883$	− 812.969i	− 0.920690i	−0.887740	−	0.460345i	$-0.847726\pi$
$883$	− 812.969i	− 0.920690i	0.887740	−	0.460345i	$-0.152274\pi$
$884$	0	0
$885$	−263.038	−0.297218
$886$	0	0
$887$	1566.18i	1.76570i	0.469654	+	0.882851i	$0.344379\pi$
$887$	1566.18i	1.76570i	−0.469654	+	0.882851i	$0.655621\pi$
$888$	0	0
$889$	−1269.42	−1.42792
$890$	0	0
$891$	− 1489.98i	− 1.67225i
$892$	0	0
$893$	4.34894	0.00487003
$894$	0	0
$895$	− 115.957i	− 0.129561i
$896$	0	0
$897$	51.1508	0.0570243
$898$	0	0
$899$	22.8058i	0.0253679i
$900$	0	0
$901$	430.402	0.477694
$902$	0	0
$903$	1484.91i	1.64442i
$904$	0	0
$905$	444.690	0.491370
$906$	0	0
$907$	1034.35i	1.14041i	0.821503	+	0.570204i	$0.193136\pi$
$907$	1034.35i	1.14041i	−0.821503	+	0.570204i	$0.806864\pi$
$908$	0	0
$909$	291.657	0.320855
$910$	0	0
$911$	− 55.4856i	− 0.0609062i	−0.999536	−	0.0304531i	$-0.990305\pi$
$911$	− 55.4856i	− 0.0609062i	0.999536	−	0.0304531i	$-0.00969502\pi$
$912$	0	0
$913$	593.488	0.650041
$914$	0	0
$915$	− 207.795i	− 0.227098i
$916$	0	0
$917$	600.775	0.655153
$918$	0	0
$919$	− 320.357i	− 0.348594i	−0.984693	−	0.174297i	$-0.944235\pi$
$919$	− 320.357i	− 0.348594i	0.984693	−	0.174297i	$-0.0557652\pi$
$920$	0	0
$921$	821.417	0.891876
$922$	0	0
$923$	258.360i	0.279913i
$924$	0	0
$925$	−198.691	−0.214802
$926$	0	0
$927$	162.441i	0.175233i
$928$	0	0
$929$	955.047	1.02804	0.514019	−	0.857779i	$-0.328156\pi$
$929$	955.047	1.02804	0.514019	+	0.857779i	$0.328156\pi$
$930$	0	0
$931$	1.49079i	0.00160128i
$932$	0	0
$933$	139.120	0.149110
$934$	0	0
$935$	− 297.143i	− 0.317800i
$936$	0	0
$937$	−388.002	−0.414089	−0.207045	−	0.978331i	$-0.566385\pi$
$937$	−388.002	−0.414089	−0.207045	+	0.978331i	$0.566385\pi$
$938$	0	0
$939$	713.264i	0.759600i
$940$	0	0
$941$	1727.70	1.83602	0.918010	−	0.396556i	$-0.129795\pi$
$941$	1727.70	1.83602	0.918010	+	0.396556i	$0.129795\pi$
$942$	0	0
$943$	− 47.1874i	− 0.0500397i
$944$	0	0
$945$	−295.603	−0.312808
$946$	0	0
$947$	− 468.279i	− 0.494487i	−0.968953	−	0.247243i	$-0.920475\pi$
$947$	− 468.279i	− 0.494487i	0.968953	−	0.247243i	$-0.0795247\pi$
$948$	0	0
$949$	73.6349	0.0775921
$950$	0	0
$951$	− 1121.33i	− 1.17910i
$952$	0	0
$953$	1510.31	1.58480	0.792398	−	0.610004i	$-0.208832\pi$
$953$	1510.31	1.58480	0.792398	+	0.610004i	$0.208832\pi$
$954$	0	0
$955$	− 525.156i	− 0.549901i
$956$	0	0
$957$	−23.8078	−0.0248776
$958$	0	0
$959$	− 144.296i	− 0.150465i
$960$	0	0
$961$	−1535.73	−1.59805
$962$	0	0
$963$	− 82.9043i	− 0.0860896i
$964$	0	0
$965$	−231.668	−0.240071
$966$	0	0
$967$	231.998i	0.239915i	0.992779	+	0.119958i	$0.0382758\pi$
$967$	231.998i	0.239915i	−0.992779	+	0.119958i	$0.961724\pi$
$968$	0	0
$969$	−2.23290	−0.00230433
$970$	0	0
$971$	− 444.200i	− 0.457467i	−0.973489	−	0.228733i	$-0.926542\pi$
$971$	− 444.200i	− 0.457467i	0.973489	−	0.228733i	$-0.0734584\pi$
$972$	0	0
$973$	1050.91	1.08008
$974$	0	0
$975$	− 53.3284i	− 0.0546958i
$976$	0	0
$977$	−229.920	−0.235333	−0.117666	−	0.993053i	$-0.537541\pi$
$977$	−229.920	−0.235333	−0.117666	+	0.993053i	$0.537541\pi$
$978$	0	0
$979$	− 1249.99i	− 1.27680i
$980$	0	0
$981$	−74.2269	−0.0756645
$982$	0	0
$983$	677.659i	0.689379i	0.938717	+	0.344689i	$0.112016\pi$
$983$	677.659i	0.689379i	−0.938717	+	0.344689i	$0.887984\pi$
$984$	0	0
$985$	155.911	0.158286
$986$	0	0
$987$	966.075i	0.978799i
$988$	0	0
$989$	392.119	0.396480
$990$	0	0
$991$	− 233.527i	− 0.235648i	−0.993035	−	0.117824i	$-0.962408\pi$
$991$	− 233.527i	− 0.235648i	0.993035	−	0.117824i	$-0.0375918\pi$
$992$	0	0
$993$	−1678.20	−1.69004
$994$	0	0
$995$	465.519i	0.467858i
$996$	0	0
$997$	300.205	0.301108	0.150554	−	0.988602i	$-0.451894\pi$
$997$	300.205	0.301108	0.150554	+	0.988602i	$0.451894\pi$
$998$	0	0
$999$	947.116i	0.948064i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.c.b.1151.15	✓	56
4.3	odd	2	inner	1840.3.c.b.1151.42	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.3.c.b.1151.15	✓	56	1.1	even	1	trivial
1840.3.c.b.1151.42	yes	56	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.27428i q^{3} +2.23607 q^{5} -5.54665i q^{7} -1.72088 q^{9} +O(q^{10})\)	\(q-3.27428i q^{3} +2.23607 q^{5} -5.54665i q^{7} -1.72088 q^{9} +15.9311i q^{11} +3.25742 q^{13} -7.32150i q^{15} -8.34132 q^{17} +0.0817557i q^{19} -18.1612 q^{21} +4.79583i q^{23} +5.00000 q^{25} -23.8338i q^{27} -0.456414 q^{29} -49.9673i q^{31} +52.1628 q^{33} -12.4027i q^{35} -39.7383 q^{37} -10.6657i q^{39} -9.83926 q^{41} -81.7624i q^{43} -3.84801 q^{45} -53.1943i q^{47} +18.2347 q^{49} +27.3118i q^{51} -51.5988 q^{53} +35.6230i q^{55} +0.267691 q^{57} -35.9267i q^{59} +28.3815 q^{61} +9.54512i q^{63} +7.28380 q^{65} -78.4902i q^{67} +15.7029 q^{69} +79.3143i q^{71} +22.6053 q^{73} -16.3714i q^{75} +88.3641 q^{77} -39.8514i q^{79} -93.5265 q^{81} -37.2534i q^{83} -18.6518 q^{85} +1.49443i q^{87} -78.4622 q^{89} -18.0677i q^{91} -163.607 q^{93} +0.182811i q^{95} +83.5307 q^{97} -27.4155i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 120 q^{9}+O(q^{10}) \) 56 * q - 120 * q^9	\( 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) \) 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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