LMFDB - Embedded newform 5328.2.e.g.2591.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5328,2,Mod(2591,5328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5328.2591"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5328 = 2^{4} \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5328.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$42.5442941969$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.18
Character	$\chi$	$=$	5328.2591
Dual form			5328.2.e.g.2591.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.16614i q^{5} +4.00867i q^{7} +1.34249 q^{11} +2.84621 q^{13} -4.41620i q^{17} -7.02429i q^{19} +4.58186 q^{23} +3.64012 q^{25} -2.59711i q^{29} +5.66144i q^{31} -4.67467 q^{35} +1.00000 q^{37} -9.38782i q^{41} -9.95268i q^{43} +12.2309 q^{47} -9.06942 q^{49} -9.53352i q^{53} +1.56553i q^{55} -12.3636 q^{59} +10.1703 q^{61} +3.31908i q^{65} +1.48229i q^{67} -0.505614 q^{71} +1.93868 q^{73} +5.38160i q^{77} -4.99743i q^{79} +2.33694 q^{83} +5.14991 q^{85} -3.64412i q^{89} +11.4095i q^{91} +8.19131 q^{95} -2.74531 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 16 q^{13} + 24 q^{37} - 16 q^{49} + 48 q^{61} + 48 q^{73} + 56 q^{85} - 8 q^{97}+O(q^{100}) $ 24 * q + 16 * q^13 + 24 * q^37 - 16 * q^49 + 48 * q^61 + 48 * q^73 + 56 * q^85 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5328\mathbb{Z}\right)^\times$.

$n$	$1297$	$1333$	$1999$	$2369$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.16614i	0.521514i	0.965405	+	0.260757i	$0.0839721\pi$
$5$	1.16614i	0.521514i	−0.965405	+	0.260757i	$0.916028\pi$
$6$	0	0
$7$	4.00867i	1.51513i	0.652757	+	0.757567i	$0.273612\pi$
$7$	4.00867i	1.51513i	−0.652757	+	0.757567i	$0.726388\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.34249	0.404776	0.202388	−	0.979305i	$-0.435130\pi$
$11$	1.34249	0.404776	0.202388	+	0.979305i	$0.435130\pi$
$12$	0	0
$13$	2.84621	0.789396	0.394698	−	0.918811i	$-0.370849\pi$
$13$	2.84621	0.789396	0.394698	+	0.918811i	$0.370849\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.41620i	− 1.07109i	−0.844508	−	0.535543i	$-0.820107\pi$
$17$	− 4.41620i	− 1.07109i	0.844508	−	0.535543i	$-0.179893\pi$
$18$	0	0
$19$	− 7.02429i	− 1.61148i	−0.592267	−	0.805742i	$-0.701767\pi$
$19$	− 7.02429i	− 1.61148i	0.592267	−	0.805742i	$-0.298233\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.58186	0.955383	0.477692	−	0.878528i	$-0.341474\pi$
$23$	4.58186	0.955383	0.477692	+	0.878528i	$0.341474\pi$
$24$	0	0
$25$	3.64012	0.728023
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.59711i	− 0.482272i	−0.970491	−	0.241136i	$-0.922480\pi$
$29$	− 2.59711i	− 0.482272i	0.970491	−	0.241136i	$-0.0775199\pi$
$30$	0	0
$31$	5.66144i	1.01682i	0.861114	+	0.508412i	$0.169767\pi$
$31$	5.66144i	1.01682i	−0.861114	+	0.508412i	$0.830233\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.67467	−0.790163
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 9.38782i	− 1.46613i	−0.680158	−	0.733066i	$-0.738089\pi$
$41$	− 9.38782i	− 1.46613i	0.680158	−	0.733066i	$-0.261911\pi$
$42$	0	0
$43$	− 9.95268i	− 1.51777i	−0.651225	−	0.758885i	$-0.725745\pi$
$43$	− 9.95268i	− 1.51777i	0.651225	−	0.758885i	$-0.274255\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.2309	1.78407	0.892033	−	0.451971i	$-0.149279\pi$
$47$	12.2309	1.78407	0.892033	+	0.451971i	$0.149279\pi$
$48$	0	0
$49$	−9.06942	−1.29563
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9.53352i	− 1.30953i	−0.755832	−	0.654765i	$-0.772768\pi$
$53$	− 9.53352i	− 1.30953i	0.755832	−	0.654765i	$-0.227232\pi$
$54$	0	0
$55$	1.56553i	0.211096i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.3636	−1.60960	−0.804802	−	0.593544i	$-0.797728\pi$
$59$	−12.3636	−1.60960	−0.804802	+	0.593544i	$0.797728\pi$
$60$	0	0
$61$	10.1703	1.30218	0.651088	−	0.759002i	$-0.274313\pi$
$61$	10.1703	1.30218	0.651088	+	0.759002i	$0.274313\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.31908i	0.411681i
$66$	0	0
$67$	1.48229i	0.181091i	0.995892	+	0.0905454i	$0.0288610\pi$
$67$	1.48229i	0.181091i	−0.995892	+	0.0905454i	$0.971139\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.505614	−0.0600054	−0.0300027	−	0.999550i	$-0.509552\pi$
$71$	−0.505614	−0.0600054	−0.0300027	+	0.999550i	$0.509552\pi$
$72$	0	0
$73$	1.93868	0.226906	0.113453	−	0.993543i	$-0.463809\pi$
$73$	1.93868	0.226906	0.113453	+	0.993543i	$0.463809\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.38160i	0.613290i
$78$	0	0
$79$	− 4.99743i	− 0.562255i	−0.959670	−	0.281128i	$-0.909292\pi$
$79$	− 4.99743i	− 0.562255i	0.959670	−	0.281128i	$-0.0907084\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.33694	0.256512	0.128256	−	0.991741i	$-0.459062\pi$
$83$	2.33694	0.256512	0.128256	+	0.991741i	$0.459062\pi$
$84$	0	0
$85$	5.14991	0.558586
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 3.64412i	− 0.386276i	−0.981172	−	0.193138i	$-0.938133\pi$
$89$	− 3.64412i	− 0.386276i	0.981172	−	0.193138i	$-0.0618666\pi$
$90$	0	0
$91$	11.4095i	1.19604i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.19131	0.840411
$96$	0	0
$97$	−2.74531	−0.278744	−0.139372	−	0.990240i	$-0.544508\pi$
$97$	−2.74531	−0.278744	−0.139372	+	0.990240i	$0.544508\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 15.6445i	− 1.55669i	−0.627840	−	0.778343i	$-0.716061\pi$
$101$	− 15.6445i	− 1.55669i	0.627840	−	0.778343i	$-0.283939\pi$
$102$	0	0
$103$	− 5.37152i	− 0.529272i	−0.964348	−	0.264636i	$-0.914748\pi$
$103$	− 5.37152i	− 0.529272i	0.964348	−	0.264636i	$-0.0852517\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.3000	1.38243	0.691215	−	0.722650i	$-0.257076\pi$
$107$	14.3000	1.38243	0.691215	+	0.722650i	$0.257076\pi$
$108$	0	0
$109$	−17.1868	−1.64620	−0.823101	−	0.567896i	$-0.807758\pi$
$109$	−17.1868	−1.64620	−0.823101	+	0.567896i	$0.807758\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 14.6579i	− 1.37890i	−0.724335	−	0.689448i	$-0.757853\pi$
$113$	− 14.6579i	− 1.37890i	0.724335	−	0.689448i	$-0.242147\pi$
$114$	0	0
$115$	5.34309i	0.498246i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	17.7031	1.62284
$120$	0	0
$121$	−9.19772	−0.836156
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.0756i	0.901188i
$126$	0	0
$127$	11.8301i	1.04975i	0.851180	+	0.524874i	$0.175888\pi$
$127$	11.8301i	1.04975i	−0.851180	+	0.524874i	$0.824112\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.67144	0.408145	0.204073	−	0.978956i	$-0.434582\pi$
$131$	4.67144	0.408145	0.204073	+	0.978956i	$0.434582\pi$
$132$	0	0
$133$	28.1581	2.44161
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.2826i	1.90374i	0.306509	+	0.951868i	$0.400839\pi$
$137$	22.2826i	1.90374i	−0.306509	+	0.951868i	$0.599161\pi$
$138$	0	0
$139$	17.3731i	1.47357i	0.676129	+	0.736783i	$0.263656\pi$
$139$	17.3731i	1.47357i	−0.676129	+	0.736783i	$0.736344\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.82101	0.319529
$144$	0	0
$145$	3.02860	0.251511
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.5738i	1.35778i	0.734240	+	0.678890i	$0.237539\pi$
$149$	16.5738i	1.35778i	−0.734240	+	0.678890i	$0.762461\pi$
$150$	0	0
$151$	− 0.0872396i	− 0.00709945i	−0.999994	−	0.00354973i	$-0.998870\pi$
$151$	− 0.0872396i	− 0.00709945i	0.999994	−	0.00354973i	$-0.00112992\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.60203	−0.530288
$156$	0	0
$157$	−14.6005	−1.16524	−0.582622	−	0.812743i	$-0.697973\pi$
$157$	−14.6005	−1.16524	−0.582622	+	0.812743i	$0.697973\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	18.3671i	1.44753i
$162$	0	0
$163$	16.9118i	1.32463i	0.749225	+	0.662316i	$0.230426\pi$
$163$	16.9118i	1.32463i	−0.749225	+	0.662316i	$0.769574\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.72806	−0.211104	−0.105552	−	0.994414i	$-0.533661\pi$
$167$	−2.72806	−0.211104	−0.105552	+	0.994414i	$0.533661\pi$
$168$	0	0
$169$	−4.89910	−0.376854
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.86031i	0.217465i	0.994071	+	0.108733i	$0.0346792\pi$
$173$	2.86031i	0.217465i	−0.994071	+	0.108733i	$0.965321\pi$
$174$	0	0
$175$	14.5920i	1.10305i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.2570	1.36459	0.682297	−	0.731075i	$-0.260981\pi$
$179$	18.2570	1.36459	0.682297	+	0.731075i	$0.260981\pi$
$180$	0	0
$181$	−7.54572	−0.560869	−0.280434	−	0.959873i	$-0.590479\pi$
$181$	−7.54572	−0.560869	−0.280434	+	0.959873i	$0.590479\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.16614i	0.0857363i
$186$	0	0
$187$	− 5.92871i	− 0.433550i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.5884	1.56208	0.781041	−	0.624480i	$-0.214689\pi$
$191$	21.5884	1.56208	0.781041	+	0.624480i	$0.214689\pi$
$192$	0	0
$193$	26.3816	1.89899	0.949494	−	0.313785i	$-0.101597\pi$
$193$	26.3816	1.89899	0.949494	+	0.313785i	$0.101597\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.208985i	0.0148896i	0.999972	+	0.00744479i	$0.00236977\pi$
$197$	0.208985i	0.0148896i	−0.999972	+	0.00744479i	$0.997630\pi$
$198$	0	0
$199$	22.3636i	1.58531i	0.609670	+	0.792655i	$0.291302\pi$
$199$	22.3636i	1.58531i	−0.609670	+	0.792655i	$0.708698\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	10.4110	0.730706
$204$	0	0
$205$	10.9475	0.764608
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 9.43005i	− 0.652290i
$210$	0	0
$211$	− 6.94402i	− 0.478046i	−0.971014	−	0.239023i	$-0.923173\pi$
$211$	− 6.94402i	− 0.478046i	0.971014	−	0.239023i	$-0.0768271\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.6062	0.791537
$216$	0	0
$217$	−22.6948	−1.54063
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 12.5694i	− 0.845510i
$222$	0	0
$223$	15.8908i	1.06413i	0.846705	+	0.532063i	$0.178583\pi$
$223$	15.8908i	1.06413i	−0.846705	+	0.532063i	$0.821417\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.82468	−0.253853	−0.126926	−	0.991912i	$-0.540511\pi$
$227$	−3.82468	−0.253853	−0.126926	+	0.991912i	$0.540511\pi$
$228$	0	0
$229$	−6.94895	−0.459199	−0.229600	−	0.973285i	$-0.573742\pi$
$229$	−6.94895	−0.459199	−0.229600	+	0.973285i	$0.573742\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 16.4831i	− 1.07984i	−0.841715	−	0.539922i	$-0.818454\pi$
$233$	− 16.4831i	− 1.07984i	0.841715	−	0.539922i	$-0.181546\pi$
$234$	0	0
$235$	14.2630i	0.930415i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.1783	−1.30523	−0.652614	−	0.757691i	$-0.726328\pi$
$239$	−20.1783	−1.30523	−0.652614	+	0.757691i	$0.726328\pi$
$240$	0	0
$241$	10.1370	0.652982	0.326491	−	0.945200i	$-0.394134\pi$
$241$	10.1370	0.652982	0.326491	+	0.945200i	$0.394134\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 10.5762i	− 0.675690i
$246$	0	0
$247$	− 19.9926i	− 1.27210i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.05238	0.255784	0.127892	−	0.991788i	$-0.459179\pi$
$251$	4.05238	0.255784	0.127892	+	0.991788i	$0.459179\pi$
$252$	0	0
$253$	6.15110	0.386717
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5.49371i	− 0.342688i	−0.985211	−	0.171344i	$-0.945189\pi$
$257$	− 5.49371i	− 0.342688i	0.985211	−	0.171344i	$-0.0548110\pi$
$258$	0	0
$259$	4.00867i	0.249087i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.6937	−0.782725	−0.391363	−	0.920237i	$-0.627996\pi$
$263$	−12.6937	−0.782725	−0.391363	+	0.920237i	$0.627996\pi$
$264$	0	0
$265$	11.1174	0.682938
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 28.4339i	− 1.73364i	−0.498618	−	0.866822i	$-0.666159\pi$
$269$	− 28.4339i	− 1.73364i	0.498618	−	0.866822i	$-0.333841\pi$
$270$	0	0
$271$	27.8687i	1.69290i	0.532464	+	0.846452i	$0.321266\pi$
$271$	27.8687i	1.69290i	−0.532464	+	0.846452i	$0.678734\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.88682	0.294687
$276$	0	0
$277$	15.6408	0.939762	0.469881	−	0.882730i	$-0.344297\pi$
$277$	15.6408	0.939762	0.469881	+	0.882730i	$0.344297\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 32.3514i	− 1.92992i	−0.262388	−	0.964962i	$-0.584510\pi$
$281$	− 32.3514i	− 1.92992i	0.262388	−	0.964962i	$-0.415490\pi$
$282$	0	0
$283$	− 11.5218i	− 0.684899i	−0.939536	−	0.342450i	$-0.888743\pi$
$283$	− 11.5218i	− 0.684899i	0.939536	−	0.342450i	$-0.111257\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	37.6327	2.22139
$288$	0	0
$289$	−2.50280	−0.147224
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 15.3695i	− 0.897896i	−0.893558	−	0.448948i	$-0.851799\pi$
$293$	− 15.3695i	− 0.897896i	0.893558	−	0.448948i	$-0.148201\pi$
$294$	0	0
$295$	− 14.4177i	− 0.839430i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.0409	0.754176
$300$	0	0
$301$	39.8970	2.29962
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.8600i	0.679103i
$306$	0	0
$307$	2.25797i	0.128869i	0.997922	+	0.0644346i	$0.0205244\pi$
$307$	2.25797i	0.128869i	−0.997922	+	0.0644346i	$0.979476\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0.726693	0.0412070	0.0206035	−	0.999788i	$-0.493441\pi$
$311$	0.726693	0.0412070	0.0206035	+	0.999788i	$0.493441\pi$
$312$	0	0
$313$	−12.9640	−0.732768	−0.366384	−	0.930464i	$-0.619404\pi$
$313$	−12.9640	−0.732768	−0.366384	+	0.930464i	$0.619404\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 17.6775i	− 0.992870i	−0.868074	−	0.496435i	$-0.834642\pi$
$317$	− 17.6775i	− 0.992870i	0.868074	−	0.496435i	$-0.165358\pi$
$318$	0	0
$319$	− 3.48660i	− 0.195212i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−31.0207	−1.72604
$324$	0	0
$325$	10.3605	0.574699
$326$	0	0
$327$	0	0
$328$	0	0
$329$	49.0298i	2.70310i
$330$	0	0
$331$	− 31.2247i	− 1.71626i	−0.513429	−	0.858132i	$-0.671625\pi$
$331$	− 31.2247i	− 1.71626i	0.513429	−	0.858132i	$-0.328375\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.72856	−0.0944414
$336$	0	0
$337$	23.0252	1.25426	0.627132	−	0.778913i	$-0.284229\pi$
$337$	23.0252	1.25426	0.627132	+	0.778913i	$0.284229\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.60043i	0.411587i
$342$	0	0
$343$	− 8.29563i	− 0.447922i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.97408	0.320705	0.160353	−	0.987060i	$-0.448737\pi$
$347$	5.97408	0.320705	0.160353	+	0.987060i	$0.448737\pi$
$348$	0	0
$349$	8.84191	0.473297	0.236648	−	0.971595i	$-0.423951\pi$
$349$	8.84191	0.473297	0.236648	+	0.971595i	$0.423951\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.1213i	1.17740i	0.808352	+	0.588700i	$0.200360\pi$
$353$	22.1213i	1.17740i	−0.808352	+	0.588700i	$0.799640\pi$
$354$	0	0
$355$	− 0.589617i	− 0.0312936i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.60658	−0.190348	−0.0951740	−	0.995461i	$-0.530341\pi$
$359$	−3.60658	−0.190348	−0.0951740	+	0.995461i	$0.530341\pi$
$360$	0	0
$361$	−30.3407	−1.59688
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.26078i	0.118334i
$366$	0	0
$367$	10.2407i	0.534558i	0.963619	+	0.267279i	$0.0861246\pi$
$367$	10.2407i	0.534558i	−0.963619	+	0.267279i	$0.913875\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	38.2167	1.98411
$372$	0	0
$373$	−10.6323	−0.550522	−0.275261	−	0.961370i	$-0.588764\pi$
$373$	−10.6323	−0.550522	−0.275261	+	0.961370i	$0.588764\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 7.39192i	− 0.380703i
$378$	0	0
$379$	26.0491i	1.33805i	0.743238	+	0.669027i	$0.233289\pi$
$379$	26.0491i	1.33805i	−0.743238	+	0.669027i	$0.766711\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7.53555	−0.385049	−0.192524	−	0.981292i	$-0.561667\pi$
$383$	−7.53555	−0.385049	−0.192524	+	0.981292i	$0.561667\pi$
$384$	0	0
$385$	−6.27570	−0.319839
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 5.11609i	− 0.259396i	−0.991554	−	0.129698i	$-0.958599\pi$
$389$	− 5.11609i	− 0.259396i	0.991554	−	0.129698i	$-0.0414008\pi$
$390$	0	0
$391$	− 20.2344i	− 1.02330i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.82771	0.293224
$396$	0	0
$397$	33.8129	1.69702	0.848509	−	0.529180i	$-0.177501\pi$
$397$	33.8129	1.69702	0.848509	+	0.529180i	$0.177501\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	29.1090i	1.45364i	0.686830	+	0.726818i	$0.259002\pi$
$401$	29.1090i	1.45364i	−0.686830	+	0.726818i	$0.740998\pi$
$402$	0	0
$403$	16.1136i	0.802677i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.34249	0.0665448
$408$	0	0
$409$	−1.66162	−0.0821616	−0.0410808	−	0.999156i	$-0.513080\pi$
$409$	−1.66162	−0.0821616	−0.0410808	+	0.999156i	$0.513080\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 49.5616i	− 2.43876i
$414$	0	0
$415$	2.72519i	0.133775i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.0140	0.489215	0.244608	−	0.969622i	$-0.421341\pi$
$419$	10.0140	0.489215	0.244608	+	0.969622i	$0.421341\pi$
$420$	0	0
$421$	−23.2129	−1.13133	−0.565664	−	0.824636i	$-0.691380\pi$
$421$	−23.2129	−1.13133	−0.565664	+	0.824636i	$0.691380\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 16.0755i	− 0.779775i
$426$	0	0
$427$	40.7694i	1.97297i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.34780	−0.257594	−0.128797	−	0.991671i	$-0.541112\pi$
$431$	−5.34780	−0.257594	−0.128797	+	0.991671i	$0.541112\pi$
$432$	0	0
$433$	−6.99342	−0.336082	−0.168041	−	0.985780i	$-0.553744\pi$
$433$	−6.99342	−0.336082	−0.168041	+	0.985780i	$0.553744\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 32.1843i	− 1.53958i
$438$	0	0
$439$	− 19.9139i	− 0.950440i	−0.879867	−	0.475220i	$-0.842369\pi$
$439$	− 19.9139i	− 0.950440i	0.879867	−	0.475220i	$-0.157631\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.8096	−1.46381	−0.731905	−	0.681407i	$-0.761368\pi$
$443$	−30.8096	−1.46381	−0.731905	+	0.681407i	$0.761368\pi$
$444$	0	0
$445$	4.24956	0.201448
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 18.4992i	− 0.873033i	−0.899696	−	0.436516i	$-0.856212\pi$
$449$	− 18.4992i	− 0.873033i	0.899696	−	0.436516i	$-0.143788\pi$
$450$	0	0
$451$	− 12.6031i	− 0.593455i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−13.3051	−0.623752
$456$	0	0
$457$	−22.9827	−1.07508	−0.537542	−	0.843237i	$-0.680647\pi$
$457$	−22.9827	−1.07508	−0.537542	+	0.843237i	$0.680647\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.74394i	0.174373i	0.996192	+	0.0871863i	$0.0277875\pi$
$461$	3.74394i	0.174373i	−0.996192	+	0.0871863i	$0.972212\pi$
$462$	0	0
$463$	40.1040i	1.86379i	0.362729	+	0.931895i	$0.381845\pi$
$463$	40.1040i	1.86379i	−0.362729	+	0.931895i	$0.618155\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.62171	−0.306416	−0.153208	−	0.988194i	$-0.548961\pi$
$467$	−6.62171	−0.306416	−0.153208	+	0.988194i	$0.548961\pi$
$468$	0	0
$469$	−5.94202	−0.274377
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 13.3614i	− 0.614357i
$474$	0	0
$475$	− 25.5692i	− 1.17320i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.71121	−0.215261	−0.107630	−	0.994191i	$-0.534326\pi$
$479$	−4.71121	−0.215261	−0.107630	+	0.994191i	$0.534326\pi$
$480$	0	0
$481$	2.84621	0.129776
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 3.20142i	− 0.145369i
$486$	0	0
$487$	− 3.40262i	− 0.154188i	−0.997024	−	0.0770938i	$-0.975436\pi$
$487$	− 3.40262i	− 0.154188i	0.997024	−	0.0770938i	$-0.0245641\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.9099	0.718005	0.359002	−	0.933337i	$-0.383117\pi$
$491$	15.9099	0.718005	0.359002	+	0.933337i	$0.383117\pi$
$492$	0	0
$493$	−11.4694	−0.516554
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.02684i	− 0.0909162i
$498$	0	0
$499$	3.69949i	0.165612i	0.996566	+	0.0828060i	$0.0263882\pi$
$499$	3.69949i	0.165612i	−0.996566	+	0.0828060i	$0.973612\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−42.4218	−1.89150	−0.945748	−	0.324902i	$-0.894669\pi$
$503$	−42.4218	−1.89150	−0.945748	+	0.324902i	$0.894669\pi$
$504$	0	0
$505$	18.2437	0.811833
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.8398i	1.10101i	0.834833	+	0.550503i	$0.185564\pi$
$509$	24.8398i	1.10101i	−0.834833	+	0.550503i	$0.814436\pi$
$510$	0	0
$511$	7.77154i	0.343793i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.26395	0.276022
$516$	0	0
$517$	16.4199	0.722147
$518$	0	0
$519$	0	0
$520$	0	0
$521$	35.3606i	1.54918i	0.632466	+	0.774588i	$0.282043\pi$
$521$	35.3606i	1.54918i	−0.632466	+	0.774588i	$0.717957\pi$
$522$	0	0
$523$	− 33.0019i	− 1.44307i	−0.692377	−	0.721536i	$-0.743436\pi$
$523$	− 33.0019i	− 1.44307i	0.692377	−	0.721536i	$-0.256564\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.0020	1.08911
$528$	0	0
$529$	−2.00658	−0.0872425
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 26.7197i	− 1.15736i
$534$	0	0
$535$	16.6758i	0.720956i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12.1756	−0.524441
$540$	0	0
$541$	−17.8016	−0.765352	−0.382676	−	0.923883i	$-0.624997\pi$
$541$	−17.8016	−0.765352	−0.382676	+	0.923883i	$0.624997\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 20.0423i	− 0.858516i
$546$	0	0
$547$	− 17.5620i	− 0.750898i	−0.926843	−	0.375449i	$-0.877489\pi$
$547$	− 17.5620i	− 0.750898i	0.926843	−	0.375449i	$-0.122511\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−18.2429	−0.777173
$552$	0	0
$553$	20.0331	0.851892
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 16.4699i	− 0.697853i	−0.937150	−	0.348927i	$-0.886546\pi$
$557$	− 16.4699i	− 0.697853i	0.937150	−	0.348927i	$-0.113454\pi$
$558$	0	0
$559$	− 28.3274i	− 1.19812i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−0.346649	−0.0146095	−0.00730477	−	0.999973i	$-0.502325\pi$
$563$	−0.346649	−0.0146095	−0.00730477	+	0.999973i	$0.502325\pi$
$564$	0	0
$565$	17.0931	0.719113
$566$	0	0
$567$	0	0
$568$	0	0
$569$	43.6278i	1.82897i	0.404617	+	0.914486i	$0.367405\pi$
$569$	43.6278i	1.82897i	−0.404617	+	0.914486i	$0.632595\pi$
$570$	0	0
$571$	1.00467i	0.0420443i	0.999779	+	0.0210222i	$0.00669206\pi$
$571$	1.00467i	0.0420443i	−0.999779	+	0.0210222i	$0.993308\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	16.6785	0.695542
$576$	0	0
$577$	−9.13403	−0.380254	−0.190127	−	0.981759i	$-0.560890\pi$
$577$	−9.13403	−0.380254	−0.190127	+	0.981759i	$0.560890\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9.36800i	0.388650i
$582$	0	0
$583$	− 12.7987i	− 0.530067i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.6672	0.729202	0.364601	−	0.931164i	$-0.381205\pi$
$587$	17.6672	0.729202	0.364601	+	0.931164i	$0.381205\pi$
$588$	0	0
$589$	39.7676	1.63860
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 21.2677i	− 0.873360i	−0.899617	−	0.436680i	$-0.856154\pi$
$593$	− 21.2677i	− 0.873360i	0.899617	−	0.436680i	$-0.143846\pi$
$594$	0	0
$595$	20.6443i	0.846332i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	37.7241	1.54136	0.770682	−	0.637220i	$-0.219916\pi$
$599$	37.7241	1.54136	0.770682	+	0.637220i	$0.219916\pi$
$600$	0	0
$601$	−35.9456	−1.46625	−0.733127	−	0.680092i	$-0.761940\pi$
$601$	−35.9456	−1.46625	−0.733127	+	0.680092i	$0.761940\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 10.7258i	− 0.436067i
$606$	0	0
$607$	− 9.28867i	− 0.377015i	−0.982072	−	0.188508i	$-0.939635\pi$
$607$	− 9.28867i	− 0.377015i	0.982072	−	0.188508i	$-0.0603651\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	34.8118	1.40833
$612$	0	0
$613$	6.56733	0.265252	0.132626	−	0.991166i	$-0.457659\pi$
$613$	6.56733	0.265252	0.132626	+	0.991166i	$0.457659\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.8117i	0.837849i	0.908021	+	0.418924i	$0.137593\pi$
$617$	20.8117i	0.837849i	−0.908021	+	0.418924i	$0.862407\pi$
$618$	0	0
$619$	− 7.65243i	− 0.307577i	−0.988104	−	0.153788i	$-0.950853\pi$
$619$	− 7.65243i	− 0.307577i	0.988104	−	0.153788i	$-0.0491474\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.6081	0.585260
$624$	0	0
$625$	6.45104	0.258042
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 4.41620i	− 0.176085i
$630$	0	0
$631$	7.62949i	0.303725i	0.988402	+	0.151863i	$0.0485271\pi$
$631$	7.62949i	0.303725i	−0.988402	+	0.151863i	$0.951473\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.7955	−0.547458
$636$	0	0
$637$	−25.8135	−1.02277
$638$	0	0
$639$	0	0
$640$	0	0
$641$	11.7916i	0.465740i	0.972508	+	0.232870i	$0.0748117\pi$
$641$	11.7916i	0.465740i	−0.972508	+	0.232870i	$0.925188\pi$
$642$	0	0
$643$	20.6975i	0.816228i	0.912931	+	0.408114i	$0.133813\pi$
$643$	20.6975i	0.816228i	−0.912931	+	0.408114i	$0.866187\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−46.3745	−1.82317	−0.911585	−	0.411111i	$-0.865141\pi$
$647$	−46.3745	−1.82317	−0.911585	+	0.411111i	$0.865141\pi$
$648$	0	0
$649$	−16.5980	−0.651529
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 19.3115i	− 0.755717i	−0.925863	−	0.377859i	$-0.876661\pi$
$653$	− 19.3115i	− 0.755717i	0.925863	−	0.377859i	$-0.123339\pi$
$654$	0	0
$655$	5.44755i	0.212853i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.0856	0.587650	0.293825	−	0.955859i	$-0.405072\pi$
$659$	15.0856	0.587650	0.293825	+	0.955859i	$0.405072\pi$
$660$	0	0
$661$	13.5682	0.527743	0.263871	−	0.964558i	$-0.415001\pi$
$661$	13.5682	0.527743	0.263871	+	0.964558i	$0.415001\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	32.8362i	1.27333i
$666$	0	0
$667$	− 11.8996i	− 0.460754i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	13.6536	0.527090
$672$	0	0
$673$	16.2478	0.626308	0.313154	−	0.949702i	$-0.398614\pi$
$673$	16.2478	0.626308	0.313154	+	0.949702i	$0.398614\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	41.6863i	1.60213i	0.598575	+	0.801067i	$0.295734\pi$
$677$	41.6863i	1.60213i	−0.598575	+	0.801067i	$0.704266\pi$
$678$	0	0
$679$	− 11.0050i	− 0.422335i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.5001	1.24358	0.621791	−	0.783184i	$-0.286406\pi$
$683$	32.5001	1.24358	0.621791	+	0.783184i	$0.286406\pi$
$684$	0	0
$685$	−25.9847	−0.992824
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 27.1344i	− 1.03374i
$690$	0	0
$691$	25.0047i	0.951223i	0.879656	+	0.475611i	$0.157773\pi$
$691$	25.0047i	0.951223i	−0.879656	+	0.475611i	$0.842227\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.2595	−0.768485
$696$	0	0
$697$	−41.4585	−1.57035
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 16.0129i	− 0.604797i	−0.953182	−	0.302399i	$-0.902213\pi$
$701$	− 16.0129i	− 0.604797i	0.953182	−	0.302399i	$-0.0977873\pi$
$702$	0	0
$703$	− 7.02429i	− 0.264926i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	62.7136	2.35859
$708$	0	0
$709$	−28.5162	−1.07095	−0.535474	−	0.844551i	$-0.679867\pi$
$709$	−28.5162	−1.07095	−0.535474	+	0.844551i	$0.679867\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	25.9399i	0.971458i
$714$	0	0
$715$	4.45583i	0.166639i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	40.3190	1.50365	0.751823	−	0.659365i	$-0.229175\pi$
$719$	40.3190	1.50365	0.751823	+	0.659365i	$0.229175\pi$
$720$	0	0
$721$	21.5326	0.801918
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 9.45379i	− 0.351105i
$726$	0	0
$727$	16.6756i	0.618465i	0.950987	+	0.309232i	$0.100072\pi$
$727$	16.6756i	0.618465i	−0.950987	+	0.309232i	$0.899928\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−43.9530	−1.62566
$732$	0	0
$733$	−25.2222	−0.931602	−0.465801	−	0.884890i	$-0.654234\pi$
$733$	−25.2222	−0.931602	−0.465801	+	0.884890i	$0.654234\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.98996i	0.0733013i
$738$	0	0
$739$	25.5135i	0.938531i	0.883057	+	0.469265i	$0.155481\pi$
$739$	25.5135i	0.938531i	−0.883057	+	0.469265i	$0.844519\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−48.8281	−1.79133	−0.895665	−	0.444729i	$-0.853300\pi$
$743$	−48.8281	−1.79133	−0.895665	+	0.444729i	$0.853300\pi$
$744$	0	0
$745$	−19.3274	−0.708101
$746$	0	0
$747$	0	0
$748$	0	0
$749$	57.3238i	2.09457i
$750$	0	0
$751$	− 39.4690i	− 1.44024i	−0.693847	−	0.720122i	$-0.744086\pi$
$751$	− 39.4690i	− 1.44024i	0.693847	−	0.720122i	$-0.255914\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.101734	0.00370246
$756$	0	0
$757$	45.9630	1.67055	0.835277	−	0.549830i	$-0.185308\pi$
$757$	45.9630	1.67055	0.835277	+	0.549830i	$0.185308\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 27.9458i	− 1.01304i	−0.862229	−	0.506518i	$-0.830932\pi$
$761$	− 27.9458i	− 1.01304i	0.862229	−	0.506518i	$-0.169068\pi$
$762$	0	0
$763$	− 68.8964i	− 2.49422i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−35.1894	−1.27061
$768$	0	0
$769$	−38.1841	−1.37696	−0.688478	−	0.725258i	$-0.741721\pi$
$769$	−38.1841	−1.37696	−0.688478	+	0.725258i	$0.741721\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	33.6749i	1.21120i	0.795768	+	0.605602i	$0.207068\pi$
$773$	33.6749i	1.21120i	−0.795768	+	0.605602i	$0.792932\pi$
$774$	0	0
$775$	20.6083i	0.740272i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−65.9428	−2.36265
$780$	0	0
$781$	−0.678782	−0.0242887
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 17.0262i	− 0.607690i
$786$	0	0
$787$	− 19.3409i	− 0.689430i	−0.938707	−	0.344715i	$-0.887976\pi$
$787$	− 19.3409i	− 0.689430i	0.938707	−	0.344715i	$-0.112024\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	58.7585	2.08921
$792$	0	0
$793$	28.9468	1.02793
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.1355i	1.49251i	0.665658	+	0.746257i	$0.268151\pi$
$797$	42.1355i	1.49251i	−0.665658	+	0.746257i	$0.731849\pi$
$798$	0	0
$799$	− 54.0142i	− 1.91089i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.60267	0.0918461
$804$	0	0
$805$	−21.4187	−0.754909
$806$	0	0
$807$	0	0
$808$	0	0
$809$	40.6822i	1.43031i	0.698966	+	0.715155i	$0.253644\pi$
$809$	40.6822i	1.43031i	−0.698966	+	0.715155i	$0.746356\pi$
$810$	0	0
$811$	8.76803i	0.307887i	0.988080	+	0.153944i	$0.0491974\pi$
$811$	8.76803i	0.307887i	−0.988080	+	0.153944i	$0.950803\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−19.7215	−0.690814
$816$	0	0
$817$	−69.9105	−2.44586
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33.0419i	1.15317i	0.817037	+	0.576586i	$0.195615\pi$
$821$	33.0419i	1.15317i	−0.817037	+	0.576586i	$0.804385\pi$
$822$	0	0
$823$	1.43182i	0.0499103i	0.999689	+	0.0249551i	$0.00794429\pi$
$823$	1.43182i	0.0499103i	−0.999689	+	0.0249551i	$0.992056\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−29.7337	−1.03394	−0.516972	−	0.856003i	$-0.672941\pi$
$827$	−29.7337	−1.03394	−0.516972	+	0.856003i	$0.672941\pi$
$828$	0	0
$829$	37.9062	1.31654	0.658268	−	0.752784i	$-0.271289\pi$
$829$	37.9062	1.31654	0.658268	+	0.752784i	$0.271289\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	40.0524i	1.38773i
$834$	0	0
$835$	− 3.18130i	− 0.110094i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14.4510	−0.498903	−0.249452	−	0.968387i	$-0.580250\pi$
$839$	−14.4510	−0.498903	−0.249452	+	0.968387i	$0.580250\pi$
$840$	0	0
$841$	22.2550	0.767414
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 5.71304i	− 0.196535i
$846$	0	0
$847$	− 36.8706i	− 1.26689i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.58186	0.157064
$852$	0	0
$853$	19.8979	0.681290	0.340645	−	0.940192i	$-0.389355\pi$
$853$	19.8979	0.681290	0.340645	+	0.940192i	$0.389355\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 13.9876i	− 0.477808i	−0.971043	−	0.238904i	$-0.923212\pi$
$857$	− 13.9876i	− 0.477808i	0.971043	−	0.238904i	$-0.0767882\pi$
$858$	0	0
$859$	35.8221i	1.22224i	0.791540	+	0.611118i	$0.209280\pi$
$859$	35.8221i	1.22224i	−0.791540	+	0.611118i	$0.790720\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.7357	−0.399489	−0.199744	−	0.979848i	$-0.564011\pi$
$863$	−11.7357	−0.399489	−0.199744	+	0.979848i	$0.564011\pi$
$864$	0	0
$865$	−3.33552	−0.113411
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 6.70901i	− 0.227588i
$870$	0	0
$871$	4.21891i	0.142952i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−40.3897	−1.36542
$876$	0	0
$877$	15.0689	0.508840	0.254420	−	0.967094i	$-0.418115\pi$
$877$	15.0689	0.508840	0.254420	+	0.967094i	$0.418115\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	52.9000i	1.78225i	0.453762	+	0.891123i	$0.350082\pi$
$881$	52.9000i	1.78225i	−0.453762	+	0.891123i	$0.649918\pi$
$882$	0	0
$883$	− 9.07882i	− 0.305527i	−0.988263	−	0.152763i	$-0.951183\pi$
$883$	− 9.07882i	− 0.305527i	0.988263	−	0.152763i	$-0.0488172\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25.7694	−0.865251	−0.432625	−	0.901574i	$-0.642413\pi$
$887$	−25.7694	−0.865251	−0.432625	+	0.901574i	$0.642413\pi$
$888$	0	0
$889$	−47.4228	−1.59051
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 85.9137i	− 2.87499i
$894$	0	0
$895$	21.2902i	0.711654i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	14.7034	0.490386
$900$	0	0
$901$	−42.1019	−1.40262
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 8.79937i	− 0.292501i
$906$	0	0
$907$	3.97021i	0.131829i	0.997825	+	0.0659144i	$0.0209964\pi$
$907$	3.97021i	0.131829i	−0.997825	+	0.0659144i	$0.979004\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	29.5236	0.978159	0.489080	−	0.872239i	$-0.337333\pi$
$911$	29.5236	0.978159	0.489080	+	0.872239i	$0.337333\pi$
$912$	0	0
$913$	3.13731	0.103830
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.7262i	0.618395i
$918$	0	0
$919$	25.1329i	0.829058i	0.910036	+	0.414529i	$0.136054\pi$
$919$	25.1329i	0.829058i	−0.910036	+	0.414529i	$0.863946\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.43908	−0.0473680
$924$	0	0
$925$	3.64012	0.119686
$926$	0	0
$927$	0	0
$928$	0	0
$929$	33.8785i	1.11152i	0.831343	+	0.555760i	$0.187572\pi$
$929$	33.8785i	1.11152i	−0.831343	+	0.555760i	$0.812428\pi$
$930$	0	0
$931$	63.7063i	2.08789i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.91370	0.226102
$936$	0	0
$937$	6.54124	0.213693	0.106846	−	0.994276i	$-0.465925\pi$
$937$	6.54124	0.213693	0.106846	+	0.994276i	$0.465925\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	13.8114i	0.450239i	0.974331	+	0.225119i	$0.0722772\pi$
$941$	13.8114i	0.450239i	−0.974331	+	0.225119i	$0.927723\pi$
$942$	0	0
$943$	− 43.0137i	− 1.40072i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.9294	−0.940081	−0.470040	−	0.882645i	$-0.655761\pi$
$947$	−28.9294	−0.940081	−0.470040	+	0.882645i	$0.655761\pi$
$948$	0	0
$949$	5.51790	0.179118
$950$	0	0
$951$	0	0
$952$	0	0
$953$	59.5067i	1.92761i	0.266607	+	0.963805i	$0.414097\pi$
$953$	59.5067i	1.92761i	−0.266607	+	0.963805i	$0.585903\pi$
$954$	0	0
$955$	25.1751i	0.814647i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−89.3238	−2.88441
$960$	0	0
$961$	−1.05191	−0.0339325
$962$	0	0
$963$	0	0
$964$	0	0
$965$	30.7646i	0.990348i
$966$	0	0
$967$	− 5.90752i	− 0.189973i	−0.995479	−	0.0949865i	$-0.969719\pi$
$967$	− 5.90752i	− 0.189973i	0.995479	−	0.0949865i	$-0.0302808\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.0521	0.707684	0.353842	−	0.935305i	$-0.384875\pi$
$971$	22.0521	0.707684	0.353842	+	0.935305i	$0.384875\pi$
$972$	0	0
$973$	−69.6430	−2.23265
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18.1374i	− 0.580266i	−0.956986	−	0.290133i	$-0.906300\pi$
$977$	− 18.1374i	− 0.580266i	0.956986	−	0.290133i	$-0.0936996\pi$
$978$	0	0
$979$	− 4.89220i	− 0.156355i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.87097	−0.123465	−0.0617323	−	0.998093i	$-0.519663\pi$
$983$	−3.87097	−0.123465	−0.0617323	+	0.998093i	$0.519663\pi$
$984$	0	0
$985$	−0.243706	−0.00776512
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 45.6018i	− 1.45005i
$990$	0	0
$991$	− 34.3205i	− 1.09023i	−0.838362	−	0.545114i	$-0.816486\pi$
$991$	− 34.3205i	− 1.09023i	0.838362	−	0.545114i	$-0.183514\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−26.0790	−0.826761
$996$	0	0
$997$	50.1024	1.58676	0.793380	−	0.608726i	$-0.208319\pi$
$997$	50.1024	1.58676	0.793380	+	0.608726i	$0.208319\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5328.2.e.g.2591.18	yes	24
3.2	odd	2	inner	5328.2.e.g.2591.8	yes	24
4.3	odd	2	inner	5328.2.e.g.2591.17	yes	24
12.11	even	2	inner	5328.2.e.g.2591.7	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5328.2.e.g.2591.7	✓	24	12.11	even	2	inner
5328.2.e.g.2591.8	yes	24	3.2	odd	2	inner
5328.2.e.g.2591.17	yes	24	4.3	odd	2	inner
5328.2.e.g.2591.18	yes	24	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5328.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.16614i q^{5} +4.00867i q^{7} +1.34249 q^{11} +2.84621 q^{13} -4.41620i q^{17} -7.02429i q^{19} +4.58186 q^{23} +3.64012 q^{25} -2.59711i q^{29} +5.66144i q^{31} -4.67467 q^{35} +1.00000 q^{37} -9.38782i q^{41} -9.95268i q^{43} +12.2309 q^{47} -9.06942 q^{49} -9.53352i q^{53} +1.56553i q^{55} -12.3636 q^{59} +10.1703 q^{61} +3.31908i q^{65} +1.48229i q^{67} -0.505614 q^{71} +1.93868 q^{73} +5.38160i q^{77} -4.99743i q^{79} +2.33694 q^{83} +5.14991 q^{85} -3.64412i q^{89} +11.4095i q^{91} +8.19131 q^{95} -2.74531 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 16 q^{13} + 24 q^{37} - 16 q^{49} + 48 q^{61} + 48 q^{73} + 56 q^{85} - 8 q^{97}+O(q^{100}) \) 24 * q + 16 * q^13 + 24 * q^37 - 16 * q^49 + 48 * q^61 + 48 * q^73 + 56 * q^85 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more