LMFDB - Embedded newform 624.2.d.h.287.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,2,Mod(287,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("624.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	624.d (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:	$4.98266508613$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.4
Root			$0.517638i$ of defining polynomial
Character	$\chi$	$=$	624.287
Dual form			624.2.d.h.287.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +2.44949i q^{5} -4.24264i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+1.73205 q^{3} +2.44949i q^{5} -4.24264i q^{7} +3.00000 q^{9} +3.46410 q^{11} -1.00000 q^{13} +4.24264i q^{15} -4.89898i q^{17} +4.24264i q^{19} -7.34847i q^{21} +3.46410 q^{23} -1.00000 q^{25} +5.19615 q^{27} +9.79796i q^{29} -4.24264i q^{31} +6.00000 q^{33} +10.3923 q^{35} -10.0000 q^{37} -1.73205 q^{39} +2.44949i q^{41} +8.48528i q^{43} +7.34847i q^{45} +3.46410 q^{47} -11.0000 q^{49} -8.48528i q^{51} -9.79796i q^{53} +8.48528i q^{55} +7.34847i q^{57} -3.46410 q^{59} -4.00000 q^{61} -12.7279i q^{63} -2.44949i q^{65} -12.7279i q^{67} +6.00000 q^{69} -10.3923 q^{71} -2.00000 q^{73} -1.73205 q^{75} -14.6969i q^{77} +9.00000 q^{81} -3.46410 q^{83} +12.0000 q^{85} +16.9706i q^{87} +7.34847i q^{89} +4.24264i q^{91} -7.34847i q^{93} -10.3923 q^{95} -2.00000 q^{97} +10.3923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} - 4 q^{13} - 4 q^{25} + 24 q^{33} - 40 q^{37} - 44 q^{49} - 16 q^{61} + 24 q^{69} - 8 q^{73} + 36 q^{81} + 48 q^{85} - 8 q^{97}+O(q^{100}) $ 4 * q + 12 * q^9 - 4 * q^13 - 4 * q^25 + 24 * q^33 - 40 * q^37 - 44 * q^49 - 16 * q^61 + 24 * q^69 - 8 * q^73 + 36 * q^81 + 48 * q^85 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.73205	1.00000
$3$	1.73205	1.00000
$4$	0	0
$5$	2.44949i	1.09545i	0.836660	+	0.547723i	$0.184505\pi$
$5$	2.44949i	1.09545i	−0.836660	+	0.547723i	$0.815495\pi$
$6$	0	0
$7$	− 4.24264i	− 1.60357i	−0.597614	−	0.801784i	$-0.703885\pi$
$7$	− 4.24264i	− 1.60357i	0.597614	−	0.801784i	$-0.296115\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	4.24264i	1.09545i
$16$	0	0
$17$	− 4.89898i	− 1.18818i	−0.804400	−	0.594089i	$-0.797513\pi$
$17$	− 4.89898i	− 1.18818i	0.804400	−	0.594089i	$-0.202487\pi$
$18$	0	0
$19$	4.24264i	0.973329i	0.873589	+	0.486664i	$0.161786\pi$
$19$	4.24264i	0.973329i	−0.873589	+	0.486664i	$0.838214\pi$
$20$	0	0
$21$	− 7.34847i	− 1.60357i
$22$	0	0
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.19615	1.00000
$28$	0	0
$29$	9.79796i	1.81944i	0.415227	+	0.909718i	$0.363702\pi$
$29$	9.79796i	1.81944i	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	− 4.24264i	− 0.762001i	−0.924575	−	0.381000i	$-0.875580\pi$
$31$	− 4.24264i	− 0.762001i	0.924575	−	0.381000i	$-0.124420\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	10.3923	1.75662
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	−1.73205	−0.277350
$40$	0	0
$41$	2.44949i	0.382546i	0.981537	+	0.191273i	$0.0612616\pi$
$41$	2.44949i	0.382546i	−0.981537	+	0.191273i	$0.938738\pi$
$42$	0	0
$43$	8.48528i	1.29399i	0.762493	+	0.646997i	$0.223975\pi$
$43$	8.48528i	1.29399i	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	7.34847i	1.09545i
$46$	0	0
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	−11.0000	−1.57143
$50$	0	0
$51$	− 8.48528i	− 1.18818i
$52$	0	0
$53$	− 9.79796i	− 1.34585i	−0.739709	−	0.672927i	$-0.765037\pi$
$53$	− 9.79796i	− 1.34585i	0.739709	−	0.672927i	$-0.234963\pi$
$54$	0	0
$55$	8.48528i	1.14416i
$56$	0	0
$57$	7.34847i	0.973329i
$58$	0	0
$59$	−3.46410	−0.450988	−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410	−0.450988	−0.225494	+	0.974245i	$0.572400\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	− 12.7279i	− 1.60357i
$64$	0	0
$65$	− 2.44949i	− 0.303822i
$66$	0	0
$67$	− 12.7279i	− 1.55496i	−0.628906	−	0.777482i	$-0.716497\pi$
$67$	− 12.7279i	− 1.55496i	0.628906	−	0.777482i	$-0.283503\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	−10.3923	−1.23334	−0.616670	−	0.787222i	$-0.711519\pi$
$71$	−10.3923	−1.23334	−0.616670	+	0.787222i	$0.711519\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	−1.73205	−0.200000
$76$	0	0
$77$	− 14.6969i	− 1.67487i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−3.46410	−0.380235	−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410	−0.380235	−0.190117	+	0.981761i	$0.560887\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	0	0
$87$	16.9706i	1.81944i
$88$	0	0
$89$	7.34847i	0.778936i	0.921040	+	0.389468i	$0.127341\pi$
$89$	7.34847i	0.778936i	−0.921040	+	0.389468i	$0.872659\pi$
$90$	0	0
$91$	4.24264i	0.444750i
$92$	0	0
$93$	− 7.34847i	− 0.762001i
$94$	0	0
$95$	−10.3923	−1.06623
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	10.3923	1.04447
$100$	0	0
$101$	4.89898i	0.487467i	0.969842	+	0.243733i	$0.0783722\pi$
$101$	4.89898i	0.487467i	−0.969842	+	0.243733i	$0.921628\pi$
$102$	0	0
$103$	− 8.48528i	− 0.836080i	−0.908429	−	0.418040i	$-0.862717\pi$
$103$	− 8.48528i	− 0.836080i	0.908429	−	0.418040i	$-0.137283\pi$
$104$	0	0
$105$	18.0000	1.75662
$106$	0	0
$107$	13.8564	1.33955	0.669775	−	0.742564i	$-0.266391\pi$
$107$	13.8564	1.33955	0.669775	+	0.742564i	$0.266391\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−17.3205	−1.64399
$112$	0	0
$113$	− 9.79796i	− 0.921714i	−0.887474	−	0.460857i	$-0.847542\pi$
$113$	− 9.79796i	− 0.921714i	0.887474	−	0.460857i	$-0.152458\pi$
$114$	0	0
$115$	8.48528i	0.791257i
$116$	0	0
$117$	−3.00000	−0.277350
$118$	0	0
$119$	−20.7846	−1.90532
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.24264i	0.382546i
$124$	0	0
$125$	9.79796i	0.876356i
$126$	0	0
$127$	8.48528i	0.752947i	0.926427	+	0.376473i	$0.122863\pi$
$127$	8.48528i	0.752947i	−0.926427	+	0.376473i	$0.877137\pi$
$128$	0	0
$129$	14.6969i	1.29399i
$130$	0	0
$131$	−13.8564	−1.21064	−0.605320	−	0.795982i	$-0.706955\pi$
$131$	−13.8564	−1.21064	−0.605320	+	0.795982i	$0.706955\pi$
$132$	0	0
$133$	18.0000	1.56080
$134$	0	0
$135$	12.7279i	1.09545i
$136$	0	0
$137$	12.2474i	1.04637i	0.852219	+	0.523185i	$0.175256\pi$
$137$	12.2474i	1.04637i	−0.852219	+	0.523185i	$0.824744\pi$
$138$	0	0
$139$	− 16.9706i	− 1.43942i	−0.694273	−	0.719712i	$-0.744274\pi$
$139$	− 16.9706i	− 1.43942i	0.694273	−	0.719712i	$-0.255726\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	−3.46410	−0.289683
$144$	0	0
$145$	−24.0000	−1.99309
$146$	0	0
$147$	−19.0526	−1.57143
$148$	0	0
$149$	2.44949i	0.200670i	0.994954	+	0.100335i	$0.0319915\pi$
$149$	2.44949i	0.200670i	−0.994954	+	0.100335i	$0.968009\pi$
$150$	0	0
$151$	4.24264i	0.345261i	0.984987	+	0.172631i	$0.0552267\pi$
$151$	4.24264i	0.345261i	−0.984987	+	0.172631i	$0.944773\pi$
$152$	0	0
$153$	− 14.6969i	− 1.18818i
$154$	0	0
$155$	10.3923	0.834730
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	− 16.9706i	− 1.34585i
$160$	0	0
$161$	− 14.6969i	− 1.15828i
$162$	0	0
$163$	− 12.7279i	− 0.996928i	−0.866910	−	0.498464i	$-0.833898\pi$
$163$	− 12.7279i	− 0.996928i	0.866910	−	0.498464i	$-0.166102\pi$
$164$	0	0
$165$	14.6969i	1.14416i
$166$	0	0
$167$	−24.2487	−1.87642	−0.938211	−	0.346064i	$-0.887518\pi$
$167$	−24.2487	−1.87642	−0.938211	+	0.346064i	$0.887518\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	12.7279i	0.973329i
$172$	0	0
$173$	14.6969i	1.11739i	0.829374	+	0.558694i	$0.188697\pi$
$173$	14.6969i	1.11739i	−0.829374	+	0.558694i	$0.811303\pi$
$174$	0	0
$175$	4.24264i	0.320713i
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	−6.92820	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$179$	−6.92820	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$180$	0	0
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	0	0
$183$	−6.92820	−0.512148
$184$	0	0
$185$	− 24.4949i	− 1.80090i
$186$	0	0
$187$	− 16.9706i	− 1.24101i
$188$	0	0
$189$	− 22.0454i	− 1.60357i
$190$	0	0
$191$	17.3205	1.25327	0.626634	−	0.779314i	$-0.284432\pi$
$191$	17.3205	1.25327	0.626634	+	0.779314i	$0.284432\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	− 4.24264i	− 0.303822i
$196$	0	0
$197$	7.34847i	0.523557i	0.965128	+	0.261778i	$0.0843089\pi$
$197$	7.34847i	0.523557i	−0.965128	+	0.261778i	$0.915691\pi$
$198$	0	0
$199$	− 8.48528i	− 0.601506i	−0.953702	−	0.300753i	$-0.902762\pi$
$199$	− 8.48528i	− 0.601506i	0.953702	−	0.300753i	$-0.0972379\pi$
$200$	0	0
$201$	− 22.0454i	− 1.55496i
$202$	0	0
$203$	41.5692	2.91759
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	10.3923	0.722315
$208$	0	0
$209$	14.6969i	1.01661i
$210$	0	0
$211$	16.9706i	1.16830i	0.811645	+	0.584151i	$0.198572\pi$
$211$	16.9706i	1.16830i	−0.811645	+	0.584151i	$0.801428\pi$
$212$	0	0
$213$	−18.0000	−1.23334
$214$	0	0
$215$	−20.7846	−1.41750
$216$	0	0
$217$	−18.0000	−1.22192
$218$	0	0
$219$	−3.46410	−0.234082
$220$	0	0
$221$	4.89898i	0.329541i
$222$	0	0
$223$	4.24264i	0.284108i	0.989859	+	0.142054i	$0.0453707\pi$
$223$	4.24264i	0.284108i	−0.989859	+	0.142054i	$0.954629\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	−17.3205	−1.14960	−0.574801	−	0.818293i	$-0.694921\pi$
$227$	−17.3205	−1.14960	−0.574801	+	0.818293i	$0.694921\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	− 25.4558i	− 1.67487i
$232$	0	0
$233$	14.6969i	0.962828i	0.876493	+	0.481414i	$0.159877\pi$
$233$	14.6969i	0.962828i	−0.876493	+	0.481414i	$0.840123\pi$
$234$	0	0
$235$	8.48528i	0.553519i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.3923	0.672222	0.336111	−	0.941822i	$-0.390888\pi$
$239$	10.3923	0.672222	0.336111	+	0.941822i	$0.390888\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	15.5885	1.00000
$244$	0	0
$245$	− 26.9444i	− 1.72141i
$246$	0	0
$247$	− 4.24264i	− 0.269953i
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	−17.3205	−1.09326	−0.546630	−	0.837374i	$-0.684090\pi$
$251$	−17.3205	−1.09326	−0.546630	+	0.837374i	$0.684090\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	20.7846	1.30158
$256$	0	0
$257$	24.4949i	1.52795i	0.645246	+	0.763975i	$0.276755\pi$
$257$	24.4949i	1.52795i	−0.645246	+	0.763975i	$0.723245\pi$
$258$	0	0
$259$	42.4264i	2.63625i
$260$	0	0
$261$	29.3939i	1.81944i
$262$	0	0
$263$	20.7846	1.28163	0.640817	−	0.767694i	$-0.278596\pi$
$263$	20.7846	1.28163	0.640817	+	0.767694i	$0.278596\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	12.7279i	0.778936i
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	4.24264i	0.257722i	0.991663	+	0.128861i	$0.0411321\pi$
$271$	4.24264i	0.257722i	−0.991663	+	0.128861i	$0.958868\pi$
$272$	0	0
$273$	7.34847i	0.444750i
$274$	0	0
$275$	−3.46410	−0.208893
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	− 12.7279i	− 0.762001i
$280$	0	0
$281$	12.2474i	0.730622i	0.930886	+	0.365311i	$0.119037\pi$
$281$	12.2474i	0.730622i	−0.930886	+	0.365311i	$0.880963\pi$
$282$	0	0
$283$	8.48528i	0.504398i	0.967675	+	0.252199i	$0.0811537\pi$
$283$	8.48528i	0.504398i	−0.967675	+	0.252199i	$0.918846\pi$
$284$	0	0
$285$	−18.0000	−1.06623
$286$	0	0
$287$	10.3923	0.613438
$288$	0	0
$289$	−7.00000	−0.411765
$290$	0	0
$291$	−3.46410	−0.203069
$292$	0	0
$293$	2.44949i	0.143101i	0.997437	+	0.0715504i	$0.0227947\pi$
$293$	2.44949i	0.143101i	−0.997437	+	0.0715504i	$0.977205\pi$
$294$	0	0
$295$	− 8.48528i	− 0.494032i
$296$	0	0
$297$	18.0000	1.04447
$298$	0	0
$299$	−3.46410	−0.200334
$300$	0	0
$301$	36.0000	2.07501
$302$	0	0
$303$	8.48528i	0.487467i
$304$	0	0
$305$	− 9.79796i	− 0.561029i
$306$	0	0
$307$	− 21.2132i	− 1.21070i	−0.795959	−	0.605351i	$-0.793033\pi$
$307$	− 21.2132i	− 1.21070i	0.795959	−	0.605351i	$-0.206967\pi$
$308$	0	0
$309$	− 14.6969i	− 0.836080i
$310$	0	0
$311$	24.2487	1.37502	0.687509	−	0.726176i	$-0.258704\pi$
$311$	24.2487	1.37502	0.687509	+	0.726176i	$0.258704\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	31.1769	1.75662
$316$	0	0
$317$	− 31.8434i	− 1.78850i	−0.447566	−	0.894251i	$-0.647709\pi$
$317$	− 31.8434i	− 1.78850i	0.447566	−	0.894251i	$-0.352291\pi$
$318$	0	0
$319$	33.9411i	1.90034i
$320$	0	0
$321$	24.0000	1.33955
$322$	0	0
$323$	20.7846	1.15649
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−24.2487	−1.34096
$328$	0	0
$329$	− 14.6969i	− 0.810268i
$330$	0	0
$331$	− 29.6985i	− 1.63238i	−0.577786	−	0.816188i	$-0.696083\pi$
$331$	− 29.6985i	− 1.63238i	0.577786	−	0.816188i	$-0.303917\pi$
$332$	0	0
$333$	−30.0000	−1.64399
$334$	0	0
$335$	31.1769	1.70338
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	− 16.9706i	− 0.921714i
$340$	0	0
$341$	− 14.6969i	− 0.795884i
$342$	0	0
$343$	16.9706i	0.916324i
$344$	0	0
$345$	14.6969i	0.791257i
$346$	0	0
$347$	24.2487	1.30174	0.650870	−	0.759190i	$-0.274404\pi$
$347$	24.2487	1.30174	0.650870	+	0.759190i	$0.274404\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−5.19615	−0.277350
$352$	0	0
$353$	− 2.44949i	− 0.130373i	−0.997873	−	0.0651866i	$-0.979236\pi$
$353$	− 2.44949i	− 0.130373i	0.997873	−	0.0651866i	$-0.0207643\pi$
$354$	0	0
$355$	− 25.4558i	− 1.35106i
$356$	0	0
$357$	−36.0000	−1.90532
$358$	0	0
$359$	24.2487	1.27980	0.639899	−	0.768459i	$-0.278976\pi$
$359$	24.2487	1.27980	0.639899	+	0.768459i	$0.278976\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.73205	0.0909091
$364$	0	0
$365$	− 4.89898i	− 0.256424i
$366$	0	0
$367$	− 33.9411i	− 1.77171i	−0.463960	−	0.885856i	$-0.653572\pi$
$367$	− 33.9411i	− 1.77171i	0.463960	−	0.885856i	$-0.346428\pi$
$368$	0	0
$369$	7.34847i	0.382546i
$370$	0	0
$371$	−41.5692	−2.15817
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	16.9706i	0.876356i
$376$	0	0
$377$	− 9.79796i	− 0.504621i
$378$	0	0
$379$	− 12.7279i	− 0.653789i	−0.945061	−	0.326895i	$-0.893998\pi$
$379$	− 12.7279i	− 0.653789i	0.945061	−	0.326895i	$-0.106002\pi$
$380$	0	0
$381$	14.6969i	0.752947i
$382$	0	0
$383$	10.3923	0.531022	0.265511	−	0.964108i	$-0.414459\pi$
$383$	10.3923	0.531022	0.265511	+	0.964108i	$0.414459\pi$
$384$	0	0
$385$	36.0000	1.83473
$386$	0	0
$387$	25.4558i	1.29399i
$388$	0	0
$389$	− 4.89898i	− 0.248388i	−0.992258	−	0.124194i	$-0.960365\pi$
$389$	− 4.89898i	− 0.248388i	0.992258	−	0.124194i	$-0.0396345\pi$
$390$	0	0
$391$	− 16.9706i	− 0.858238i
$392$	0	0
$393$	−24.0000	−1.21064
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	31.1769	1.56080
$400$	0	0
$401$	12.2474i	0.611608i	0.952094	+	0.305804i	$0.0989253\pi$
$401$	12.2474i	0.611608i	−0.952094	+	0.305804i	$0.901075\pi$
$402$	0	0
$403$	4.24264i	0.211341i
$404$	0	0
$405$	22.0454i	1.09545i
$406$	0	0
$407$	−34.6410	−1.71709
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	21.2132i	1.04637i
$412$	0	0
$413$	14.6969i	0.723189i
$414$	0	0
$415$	− 8.48528i	− 0.416526i
$416$	0	0
$417$	− 29.3939i	− 1.43942i
$418$	0	0
$419$	13.8564	0.676930	0.338465	−	0.940979i	$-0.390092\pi$
$419$	13.8564	0.676930	0.338465	+	0.940979i	$0.390092\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	10.3923	0.505291
$424$	0	0
$425$	4.89898i	0.237635i
$426$	0	0
$427$	16.9706i	0.821263i
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	−3.46410	−0.166860	−0.0834300	−	0.996514i	$-0.526587\pi$
$431$	−3.46410	−0.166860	−0.0834300	+	0.996514i	$0.526587\pi$
$432$	0	0
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	0	0
$435$	−41.5692	−1.99309
$436$	0	0
$437$	14.6969i	0.703050i
$438$	0	0
$439$	− 8.48528i	− 0.404980i	−0.979284	−	0.202490i	$-0.935097\pi$
$439$	− 8.48528i	− 0.404980i	0.979284	−	0.202490i	$-0.0649034\pi$
$440$	0	0
$441$	−33.0000	−1.57143
$442$	0	0
$443$	13.8564	0.658338	0.329169	−	0.944271i	$-0.393231\pi$
$443$	13.8564	0.658338	0.329169	+	0.944271i	$0.393231\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	0	0
$447$	4.24264i	0.200670i
$448$	0	0
$449$	− 7.34847i	− 0.346796i	−0.984852	−	0.173398i	$-0.944525\pi$
$449$	− 7.34847i	− 0.346796i	0.984852	−	0.173398i	$-0.0554746\pi$
$450$	0	0
$451$	8.48528i	0.399556i
$452$	0	0
$453$	7.34847i	0.345261i
$454$	0	0
$455$	−10.3923	−0.487199
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	− 25.4558i	− 1.18818i
$460$	0	0
$461$	12.2474i	0.570421i	0.958465	+	0.285210i	$0.0920634\pi$
$461$	12.2474i	0.570421i	−0.958465	+	0.285210i	$0.907937\pi$
$462$	0	0
$463$	12.7279i	0.591517i	0.955263	+	0.295758i	$0.0955723\pi$
$463$	12.7279i	0.591517i	−0.955263	+	0.295758i	$0.904428\pi$
$464$	0	0
$465$	18.0000	0.834730
$466$	0	0
$467$	−31.1769	−1.44270	−0.721348	−	0.692573i	$-0.756477\pi$
$467$	−31.1769	−1.44270	−0.721348	+	0.692573i	$0.756477\pi$
$468$	0	0
$469$	−54.0000	−2.49349
$470$	0	0
$471$	−6.92820	−0.319235
$472$	0	0
$473$	29.3939i	1.35153i
$474$	0	0
$475$	− 4.24264i	− 0.194666i
$476$	0	0
$477$	− 29.3939i	− 1.34585i
$478$	0	0
$479$	10.3923	0.474837	0.237418	−	0.971408i	$-0.423699\pi$
$479$	10.3923	0.474837	0.237418	+	0.971408i	$0.423699\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	0	0
$483$	− 25.4558i	− 1.15828i
$484$	0	0
$485$	− 4.89898i	− 0.222451i
$486$	0	0
$487$	4.24264i	0.192252i	0.995369	+	0.0961262i	$0.0306452\pi$
$487$	4.24264i	0.192252i	−0.995369	+	0.0961262i	$0.969355\pi$
$488$	0	0
$489$	− 22.0454i	− 0.996928i
$490$	0	0
$491$	−38.1051	−1.71966	−0.859830	−	0.510581i	$-0.829431\pi$
$491$	−38.1051	−1.71966	−0.859830	+	0.510581i	$0.829431\pi$
$492$	0	0
$493$	48.0000	2.16181
$494$	0	0
$495$	25.4558i	1.14416i
$496$	0	0
$497$	44.0908i	1.97774i
$498$	0	0
$499$	29.6985i	1.32949i	0.747072	+	0.664743i	$0.231459\pi$
$499$	29.6985i	1.32949i	−0.747072	+	0.664743i	$0.768541\pi$
$500$	0	0
$501$	−42.0000	−1.87642
$502$	0	0
$503$	34.6410	1.54457	0.772283	−	0.635278i	$-0.219115\pi$
$503$	34.6410	1.54457	0.772283	+	0.635278i	$0.219115\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	1.73205	0.0769231
$508$	0	0
$509$	− 31.8434i	− 1.41143i	−0.708495	−	0.705716i	$-0.750625\pi$
$509$	− 31.8434i	− 1.41143i	0.708495	−	0.705716i	$-0.249375\pi$
$510$	0	0
$511$	8.48528i	0.375367i
$512$	0	0
$513$	22.0454i	0.973329i
$514$	0	0
$515$	20.7846	0.915879
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	25.4558i	1.11739i
$520$	0	0
$521$	− 19.5959i	− 0.858513i	−0.903183	−	0.429256i	$-0.858776\pi$
$521$	− 19.5959i	− 0.858513i	0.903183	−	0.429256i	$-0.141224\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	7.34847i	0.320713i
$526$	0	0
$527$	−20.7846	−0.905392
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	−10.3923	−0.450988
$532$	0	0
$533$	− 2.44949i	− 0.106099i
$534$	0	0
$535$	33.9411i	1.46740i
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−38.1051	−1.64130
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	27.7128	1.18927
$544$	0	0
$545$	− 34.2929i	− 1.46894i
$546$	0	0
$547$	33.9411i	1.45122i	0.688107	+	0.725609i	$0.258442\pi$
$547$	33.9411i	1.45122i	−0.688107	+	0.725609i	$0.741558\pi$
$548$	0	0
$549$	−12.0000	−0.512148
$550$	0	0
$551$	−41.5692	−1.77091
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 42.4264i	− 1.80090i
$556$	0	0
$557$	26.9444i	1.14167i	0.821065	+	0.570835i	$0.193380\pi$
$557$	26.9444i	1.14167i	−0.821065	+	0.570835i	$0.806620\pi$
$558$	0	0
$559$	− 8.48528i	− 0.358889i
$560$	0	0
$561$	− 29.3939i	− 1.24101i
$562$	0	0
$563$	−20.7846	−0.875967	−0.437983	−	0.898983i	$-0.644307\pi$
$563$	−20.7846	−0.875967	−0.437983	+	0.898983i	$0.644307\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	0	0
$567$	− 38.1838i	− 1.60357i
$568$	0	0
$569$	− 44.0908i	− 1.84838i	−0.381929	−	0.924192i	$-0.624740\pi$
$569$	− 44.0908i	− 1.84838i	0.381929	−	0.924192i	$-0.375260\pi$
$570$	0	0
$571$	− 25.4558i	− 1.06529i	−0.846338	−	0.532647i	$-0.821197\pi$
$571$	− 25.4558i	− 1.06529i	0.846338	−	0.532647i	$-0.178803\pi$
$572$	0	0
$573$	30.0000	1.25327
$574$	0	0
$575$	−3.46410	−0.144463
$576$	0	0
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	0	0
$579$	−38.1051	−1.58359
$580$	0	0
$581$	14.6969i	0.609732i
$582$	0	0
$583$	− 33.9411i	− 1.40570i
$584$	0	0
$585$	− 7.34847i	− 0.303822i
$586$	0	0
$587$	−24.2487	−1.00085	−0.500426	−	0.865779i	$-0.666823\pi$
$587$	−24.2487	−1.00085	−0.500426	+	0.865779i	$0.666823\pi$
$588$	0	0
$589$	18.0000	0.741677
$590$	0	0
$591$	12.7279i	0.523557i
$592$	0	0
$593$	− 31.8434i	− 1.30765i	−0.756646	−	0.653825i	$-0.773163\pi$
$593$	− 31.8434i	− 1.30765i	0.756646	−	0.653825i	$-0.226837\pi$
$594$	0	0
$595$	− 50.9117i	− 2.08718i
$596$	0	0
$597$	− 14.6969i	− 0.601506i
$598$	0	0
$599$	6.92820	0.283079	0.141539	−	0.989933i	$-0.454795\pi$
$599$	6.92820	0.283079	0.141539	+	0.989933i	$0.454795\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	− 38.1838i	− 1.55496i
$604$	0	0
$605$	2.44949i	0.0995859i
$606$	0	0
$607$	16.9706i	0.688814i	0.938820	+	0.344407i	$0.111920\pi$
$607$	16.9706i	0.688814i	−0.938820	+	0.344407i	$0.888080\pi$
$608$	0	0
$609$	72.0000	2.91759
$610$	0	0
$611$	−3.46410	−0.140143
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	−10.3923	−0.419058
$616$	0	0
$617$	31.8434i	1.28197i	0.767555	+	0.640983i	$0.221473\pi$
$617$	31.8434i	1.28197i	−0.767555	+	0.640983i	$0.778527\pi$
$618$	0	0
$619$	21.2132i	0.852631i	0.904575	+	0.426315i	$0.140189\pi$
$619$	21.2132i	0.852631i	−0.904575	+	0.426315i	$0.859811\pi$
$620$	0	0
$621$	18.0000	0.722315
$622$	0	0
$623$	31.1769	1.24908
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	25.4558i	1.01661i
$628$	0	0
$629$	48.9898i	1.95335i
$630$	0	0
$631$	− 4.24264i	− 0.168897i	−0.996428	−	0.0844484i	$-0.973087\pi$
$631$	− 4.24264i	− 0.168897i	0.996428	−	0.0844484i	$-0.0269128\pi$
$632$	0	0
$633$	29.3939i	1.16830i
$634$	0	0
$635$	−20.7846	−0.824812
$636$	0	0
$637$	11.0000	0.435836
$638$	0	0
$639$	−31.1769	−1.23334
$640$	0	0
$641$	4.89898i	0.193498i	0.995309	+	0.0967490i	$0.0308444\pi$
$641$	4.89898i	0.193498i	−0.995309	+	0.0967490i	$0.969156\pi$
$642$	0	0
$643$	29.6985i	1.17119i	0.810602	+	0.585597i	$0.199140\pi$
$643$	29.6985i	1.17119i	−0.810602	+	0.585597i	$0.800860\pi$
$644$	0	0
$645$	−36.0000	−1.41750
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−31.1769	−1.22192
$652$	0	0
$653$	29.3939i	1.15027i	0.818058	+	0.575136i	$0.195051\pi$
$653$	29.3939i	1.15027i	−0.818058	+	0.575136i	$0.804949\pi$
$654$	0	0
$655$	− 33.9411i	− 1.32619i
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−38.1051	−1.48436	−0.742182	−	0.670198i	$-0.766209\pi$
$659$	−38.1051	−1.48436	−0.742182	+	0.670198i	$0.766209\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	8.48528i	0.329541i
$664$	0	0
$665$	44.0908i	1.70977i
$666$	0	0
$667$	33.9411i	1.31421i
$668$	0	0
$669$	7.34847i	0.284108i
$670$	0	0
$671$	−13.8564	−0.534921
$672$	0	0
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	0	0
$675$	−5.19615	−0.200000
$676$	0	0
$677$	− 29.3939i	− 1.12970i	−0.825194	−	0.564849i	$-0.808934\pi$
$677$	− 29.3939i	− 1.12970i	0.825194	−	0.564849i	$-0.191066\pi$
$678$	0	0
$679$	8.48528i	0.325635i
$680$	0	0
$681$	−30.0000	−1.14960
$682$	0	0
$683$	10.3923	0.397650	0.198825	−	0.980035i	$-0.436287\pi$
$683$	10.3923	0.397650	0.198825	+	0.980035i	$0.436287\pi$
$684$	0	0
$685$	−30.0000	−1.14624
$686$	0	0
$687$	17.3205	0.660819
$688$	0	0
$689$	9.79796i	0.373273i
$690$	0	0
$691$	12.7279i	0.484193i	0.970252	+	0.242096i	$0.0778351\pi$
$691$	12.7279i	0.484193i	−0.970252	+	0.242096i	$0.922165\pi$
$692$	0	0
$693$	− 44.0908i	− 1.67487i
$694$	0	0
$695$	41.5692	1.57681
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	25.4558i	0.962828i
$700$	0	0
$701$	44.0908i	1.66529i	0.553809	+	0.832644i	$0.313174\pi$
$701$	44.0908i	1.66529i	−0.553809	+	0.832644i	$0.686826\pi$
$702$	0	0
$703$	− 42.4264i	− 1.60014i
$704$	0	0
$705$	14.6969i	0.553519i
$706$	0	0
$707$	20.7846	0.781686
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 14.6969i	− 0.550405i
$714$	0	0
$715$	− 8.48528i	− 0.317332i
$716$	0	0
$717$	18.0000	0.672222
$718$	0	0
$719$	−10.3923	−0.387568	−0.193784	−	0.981044i	$-0.562076\pi$
$719$	−10.3923	−0.387568	−0.193784	+	0.981044i	$0.562076\pi$
$720$	0	0
$721$	−36.0000	−1.34071
$722$	0	0
$723$	17.3205	0.644157
$724$	0	0
$725$	− 9.79796i	− 0.363887i
$726$	0	0
$727$	− 25.4558i	− 0.944105i	−0.881570	−	0.472052i	$-0.843513\pi$
$727$	− 25.4558i	− 0.944105i	0.881570	−	0.472052i	$-0.156487\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	41.5692	1.53749
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	− 46.6690i	− 1.72141i
$736$	0	0
$737$	− 44.0908i	− 1.62411i
$738$	0	0
$739$	− 21.2132i	− 0.780340i	−0.920743	−	0.390170i	$-0.872416\pi$
$739$	− 21.2132i	− 0.780340i	0.920743	−	0.390170i	$-0.127584\pi$
$740$	0	0
$741$	− 7.34847i	− 0.269953i
$742$	0	0
$743$	−3.46410	−0.127086	−0.0635428	−	0.997979i	$-0.520240\pi$
$743$	−3.46410	−0.127086	−0.0635428	+	0.997979i	$0.520240\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	−10.3923	−0.380235
$748$	0	0
$749$	− 58.7878i	− 2.14806i
$750$	0	0
$751$	42.4264i	1.54816i	0.633087	+	0.774081i	$0.281788\pi$
$751$	42.4264i	1.54816i	−0.633087	+	0.774081i	$0.718212\pi$
$752$	0	0
$753$	−30.0000	−1.09326
$754$	0	0
$755$	−10.3923	−0.378215
$756$	0	0
$757$	4.00000	0.145382	0.0726912	−	0.997354i	$-0.476841\pi$
$757$	4.00000	0.145382	0.0726912	+	0.997354i	$0.476841\pi$
$758$	0	0
$759$	20.7846	0.754434
$760$	0	0
$761$	− 22.0454i	− 0.799145i	−0.916702	−	0.399573i	$-0.869159\pi$
$761$	− 22.0454i	− 0.799145i	0.916702	−	0.399573i	$-0.130841\pi$
$762$	0	0
$763$	59.3970i	2.15031i
$764$	0	0
$765$	36.0000	1.30158
$766$	0	0
$767$	3.46410	0.125081
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	42.4264i	1.52795i
$772$	0	0
$773$	− 51.4393i	− 1.85014i	−0.379794	−	0.925071i	$-0.624005\pi$
$773$	− 51.4393i	− 1.85014i	0.379794	−	0.925071i	$-0.375995\pi$
$774$	0	0
$775$	4.24264i	0.152400i
$776$	0	0
$777$	73.4847i	2.63625i
$778$	0	0
$779$	−10.3923	−0.372343
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	50.9117i	1.81944i
$784$	0	0
$785$	− 9.79796i	− 0.349704i
$786$	0	0
$787$	− 21.2132i	− 0.756169i	−0.925771	−	0.378085i	$-0.876583\pi$
$787$	− 21.2132i	− 0.756169i	0.925771	−	0.378085i	$-0.123417\pi$
$788$	0	0
$789$	36.0000	1.28163
$790$	0	0
$791$	−41.5692	−1.47803
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	41.5692	1.47431
$796$	0	0
$797$	− 14.6969i	− 0.520592i	−0.965529	−	0.260296i	$-0.916180\pi$
$797$	− 14.6969i	− 0.520592i	0.965529	−	0.260296i	$-0.0838202\pi$
$798$	0	0
$799$	− 16.9706i	− 0.600375i
$800$	0	0
$801$	22.0454i	0.778936i
$802$	0	0
$803$	−6.92820	−0.244491
$804$	0	0
$805$	36.0000	1.26883
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 9.79796i	− 0.344478i	−0.985055	−	0.172239i	$-0.944900\pi$
$809$	− 9.79796i	− 0.344478i	0.985055	−	0.172239i	$-0.0551001\pi$
$810$	0	0
$811$	29.6985i	1.04285i	0.853296	+	0.521427i	$0.174600\pi$
$811$	29.6985i	1.04285i	−0.853296	+	0.521427i	$0.825400\pi$
$812$	0	0
$813$	7.34847i	0.257722i
$814$	0	0
$815$	31.1769	1.09208
$816$	0	0
$817$	−36.0000	−1.25948
$818$	0	0
$819$	12.7279i	0.444750i
$820$	0	0
$821$	7.34847i	0.256463i	0.991744	+	0.128232i	$0.0409301\pi$
$821$	7.34847i	0.256463i	−0.991744	+	0.128232i	$0.959070\pi$
$822$	0	0
$823$	8.48528i	0.295778i	0.989004	+	0.147889i	$0.0472479\pi$
$823$	8.48528i	0.295778i	−0.989004	+	0.147889i	$0.952752\pi$
$824$	0	0
$825$	−6.00000	−0.208893
$826$	0	0
$827$	45.0333	1.56596	0.782981	−	0.622046i	$-0.213698\pi$
$827$	45.0333	1.56596	0.782981	+	0.622046i	$0.213698\pi$
$828$	0	0
$829$	−56.0000	−1.94496	−0.972480	−	0.232986i	$-0.925151\pi$
$829$	−56.0000	−1.94496	−0.972480	+	0.232986i	$0.925151\pi$
$830$	0	0
$831$	17.3205	0.600842
$832$	0	0
$833$	53.8888i	1.86714i
$834$	0	0
$835$	− 59.3970i	− 2.05552i
$836$	0	0
$837$	− 22.0454i	− 0.762001i
$838$	0	0
$839$	10.3923	0.358782	0.179391	−	0.983778i	$-0.442587\pi$
$839$	10.3923	0.358782	0.179391	+	0.983778i	$0.442587\pi$
$840$	0	0
$841$	−67.0000	−2.31034
$842$	0	0
$843$	21.2132i	0.730622i
$844$	0	0
$845$	2.44949i	0.0842650i
$846$	0	0
$847$	− 4.24264i	− 0.145779i
$848$	0	0
$849$	14.6969i	0.504398i
$850$	0	0
$851$	−34.6410	−1.18748
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	−31.1769	−1.06623
$856$	0	0
$857$	− 34.2929i	− 1.17142i	−0.810520	−	0.585711i	$-0.800815\pi$
$857$	− 34.2929i	− 1.17142i	0.810520	−	0.585711i	$-0.199185\pi$
$858$	0	0
$859$	− 50.9117i	− 1.73708i	−0.495615	−	0.868542i	$-0.665057\pi$
$859$	− 50.9117i	− 1.73708i	0.495615	−	0.868542i	$-0.334943\pi$
$860$	0	0
$861$	18.0000	0.613438
$862$	0	0
$863$	17.3205	0.589597	0.294798	−	0.955559i	$-0.404747\pi$
$863$	17.3205	0.589597	0.294798	+	0.955559i	$0.404747\pi$
$864$	0	0
$865$	−36.0000	−1.22404
$866$	0	0
$867$	−12.1244	−0.411765
$868$	0	0
$869$	0	0
$870$	0	0
$871$	12.7279i	0.431269i
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	41.5692	1.40530
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	0	0
$879$	4.24264i	0.143101i
$880$	0	0
$881$	− 24.4949i	− 0.825254i	−0.910900	−	0.412627i	$-0.864611\pi$
$881$	− 24.4949i	− 0.825254i	0.910900	−	0.412627i	$-0.135389\pi$
$882$	0	0
$883$	− 25.4558i	− 0.856657i	−0.903623	−	0.428329i	$-0.859103\pi$
$883$	− 25.4558i	− 0.856657i	0.903623	−	0.428329i	$-0.140897\pi$
$884$	0	0
$885$	− 14.6969i	− 0.494032i
$886$	0	0
$887$	−45.0333	−1.51207	−0.756035	−	0.654531i	$-0.772866\pi$
$887$	−45.0333	−1.51207	−0.756035	+	0.654531i	$0.772866\pi$
$888$	0	0
$889$	36.0000	1.20740
$890$	0	0
$891$	31.1769	1.04447
$892$	0	0
$893$	14.6969i	0.491814i
$894$	0	0
$895$	− 16.9706i	− 0.567263i
$896$	0	0
$897$	−6.00000	−0.200334
$898$	0	0
$899$	41.5692	1.38641
$900$	0	0
$901$	−48.0000	−1.59911
$902$	0	0
$903$	62.3538	2.07501
$904$	0	0
$905$	39.1918i	1.30278i
$906$	0	0
$907$	− 8.48528i	− 0.281749i	−0.990027	−	0.140875i	$-0.955009\pi$
$907$	− 8.48528i	− 0.281749i	0.990027	−	0.140875i	$-0.0449914\pi$
$908$	0	0
$909$	14.6969i	0.487467i
$910$	0	0
$911$	−24.2487	−0.803396	−0.401698	−	0.915772i	$-0.631580\pi$
$911$	−24.2487	−0.803396	−0.401698	+	0.915772i	$0.631580\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	− 16.9706i	− 0.561029i
$916$	0	0
$917$	58.7878i	1.94134i
$918$	0	0
$919$	16.9706i	0.559807i	0.960028	+	0.279904i	$0.0903025\pi$
$919$	16.9706i	0.559807i	−0.960028	+	0.279904i	$0.909697\pi$
$920$	0	0
$921$	− 36.7423i	− 1.21070i
$922$	0	0
$923$	10.3923	0.342067
$924$	0	0
$925$	10.0000	0.328798
$926$	0	0
$927$	− 25.4558i	− 0.836080i
$928$	0	0
$929$	12.2474i	0.401826i	0.979609	+	0.200913i	$0.0643908\pi$
$929$	12.2474i	0.401826i	−0.979609	+	0.200913i	$0.935609\pi$
$930$	0	0
$931$	− 46.6690i	− 1.52952i
$932$	0	0
$933$	42.0000	1.37502
$934$	0	0
$935$	41.5692	1.35946
$936$	0	0
$937$	−16.0000	−0.522697	−0.261349	−	0.965244i	$-0.584167\pi$
$937$	−16.0000	−0.522697	−0.261349	+	0.965244i	$0.584167\pi$
$938$	0	0
$939$	3.46410	0.113047
$940$	0	0
$941$	− 7.34847i	− 0.239553i	−0.992801	−	0.119777i	$-0.961782\pi$
$941$	− 7.34847i	− 0.239553i	0.992801	−	0.119777i	$-0.0382178\pi$
$942$	0	0
$943$	8.48528i	0.276319i
$944$	0	0
$945$	54.0000	1.75662
$946$	0	0
$947$	38.1051	1.23825	0.619125	−	0.785292i	$-0.287487\pi$
$947$	38.1051	1.23825	0.619125	+	0.785292i	$0.287487\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	0	0
$951$	− 55.1543i	− 1.78850i
$952$	0	0
$953$	14.6969i	0.476081i	0.971255	+	0.238040i	$0.0765050\pi$
$953$	14.6969i	0.476081i	−0.971255	+	0.238040i	$0.923495\pi$
$954$	0	0
$955$	42.4264i	1.37289i
$956$	0	0
$957$	58.7878i	1.90034i
$958$	0	0
$959$	51.9615	1.67793
$960$	0	0
$961$	13.0000	0.419355
$962$	0	0
$963$	41.5692	1.33955
$964$	0	0
$965$	− 53.8888i	− 1.73474i
$966$	0	0
$967$	4.24264i	0.136434i	0.997671	+	0.0682171i	$0.0217310\pi$
$967$	4.24264i	0.136434i	−0.997671	+	0.0682171i	$0.978269\pi$
$968$	0	0
$969$	36.0000	1.15649
$970$	0	0
$971$	−24.2487	−0.778178	−0.389089	−	0.921200i	$-0.627210\pi$
$971$	−24.2487	−0.778178	−0.389089	+	0.921200i	$0.627210\pi$
$972$	0	0
$973$	−72.0000	−2.30821
$974$	0	0
$975$	1.73205	0.0554700
$976$	0	0
$977$	− 17.1464i	− 0.548563i	−0.961649	−	0.274281i	$-0.911560\pi$
$977$	− 17.1464i	− 0.548563i	0.961649	−	0.274281i	$-0.0884400\pi$
$978$	0	0
$979$	25.4558i	0.813572i
$980$	0	0
$981$	−42.0000	−1.34096
$982$	0	0
$983$	−17.3205	−0.552438	−0.276219	−	0.961095i	$-0.589082\pi$
$983$	−17.3205	−0.552438	−0.276219	+	0.961095i	$0.589082\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	− 25.4558i	− 0.810268i
$988$	0	0
$989$	29.3939i	0.934671i
$990$	0	0
$991$	33.9411i	1.07818i	0.842250	+	0.539088i	$0.181231\pi$
$991$	33.9411i	1.07818i	−0.842250	+	0.539088i	$0.818769\pi$
$992$	0	0
$993$	− 51.4393i	− 1.63238i
$994$	0	0
$995$	20.7846	0.658916
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	0	0
$999$	−51.9615	−1.64399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.2.d.h.287.4	yes	4
3.2	odd	2	inner	624.2.d.h.287.1	✓	4
4.3	odd	2	inner	624.2.d.h.287.2	yes	4
8.3	odd	2		2496.2.d.k.1535.3		4
8.5	even	2		2496.2.d.k.1535.1		4
12.11	even	2	inner	624.2.d.h.287.3	yes	4
24.5	odd	2		2496.2.d.k.1535.4		4
24.11	even	2		2496.2.d.k.1535.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.2.d.h.287.1	✓	4	3.2	odd	2	inner
624.2.d.h.287.2	yes	4	4.3	odd	2	inner
624.2.d.h.287.3	yes	4	12.11	even	2	inner
624.2.d.h.287.4	yes	4	1.1	even	1	trivial
2496.2.d.k.1535.1		4	8.5	even	2
2496.2.d.k.1535.2		4	24.11	even	2
2496.2.d.k.1535.3		4	8.3	odd	2
2496.2.d.k.1535.4		4	24.5	odd	2

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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +2.44949i q^{5} -4.24264i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{3} +2.44949i q^{5} -4.24264i q^{7} +3.00000 q^{9} +3.46410 q^{11} -1.00000 q^{13} +4.24264i q^{15} -4.89898i q^{17} +4.24264i q^{19} -7.34847i q^{21} +3.46410 q^{23} -1.00000 q^{25} +5.19615 q^{27} +9.79796i q^{29} -4.24264i q^{31} +6.00000 q^{33} +10.3923 q^{35} -10.0000 q^{37} -1.73205 q^{39} +2.44949i q^{41} +8.48528i q^{43} +7.34847i q^{45} +3.46410 q^{47} -11.0000 q^{49} -8.48528i q^{51} -9.79796i q^{53} +8.48528i q^{55} +7.34847i q^{57} -3.46410 q^{59} -4.00000 q^{61} -12.7279i q^{63} -2.44949i q^{65} -12.7279i q^{67} +6.00000 q^{69} -10.3923 q^{71} -2.00000 q^{73} -1.73205 q^{75} -14.6969i q^{77} +9.00000 q^{81} -3.46410 q^{83} +12.0000 q^{85} +16.9706i q^{87} +7.34847i q^{89} +4.24264i q^{91} -7.34847i q^{93} -10.3923 q^{95} -2.00000 q^{97} +10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} - 4 q^{13} - 4 q^{25} + 24 q^{33} - 40 q^{37} - 44 q^{49} - 16 q^{61} + 24 q^{69} - 8 q^{73} + 36 q^{81} + 48 q^{85} - 8 q^{97}+O(q^{100}) \) 4 * q + 12 * q^9 - 4 * q^13 - 4 * q^25 + 24 * q^33 - 40 * q^37 - 44 * q^49 - 16 * q^61 + 24 * q^69 - 8 * q^73 + 36 * q^81 + 48 * q^85 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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