LMFDB - Embedded newform 2592.3.h.a.1457.29

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,3,Mod(1457,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2592.1457");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2592.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$70.6268845222$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.29
Character	$\chi$	$=$	2592.1457
Dual form			2592.3.h.a.1457.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.06253 q^{5} -1.44096 q^{7} +O(q^{10})$	$q+3.06253 q^{5} -1.44096 q^{7} +17.6558 q^{11} +9.23764i q^{13} +4.69563i q^{17} -16.8874i q^{19} +39.0706i q^{23} -15.6209 q^{25} +15.2070 q^{29} +48.7030 q^{31} -4.41298 q^{35} -14.1853i q^{37} -10.1435i q^{41} +22.3460i q^{43} +42.5709i q^{47} -46.9236 q^{49} -71.2201 q^{53} +54.0715 q^{55} +69.9985 q^{59} +103.643i q^{61} +28.2906i q^{65} +12.1430i q^{67} -112.880i q^{71} +84.0137 q^{73} -25.4413 q^{77} -45.9651 q^{79} +51.5768 q^{83} +14.3805i q^{85} -105.051i q^{89} -13.3111i q^{91} -51.7181i q^{95} -38.8643 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{7}+O(q^{10}) $ 44 * q - 4 * q^7	$ 44 q - 4 q^{7} + 144 q^{25} - 4 q^{31} + 144 q^{49} - 92 q^{55} - 8 q^{73} - 4 q^{79} + 4 q^{97}+O(q^{100}) $ 44 * q - 4 * q^7 + 144 * q^25 - 4 * q^31 + 144 * q^49 - 92 * q^55 - 8 * q^73 - 4 * q^79 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.06253	0.612506	0.306253	−	0.951950i	$-0.400925\pi$
$5$	3.06253	0.612506	0.306253	+	0.951950i	$0.400925\pi$
$6$	0	0
$7$	−1.44096	−0.205851	−0.102926	−	0.994689i	$-0.532820\pi$
$7$	−1.44096	−0.205851	−0.102926	+	0.994689i	$0.532820\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	17.6558	1.60508	0.802538	−	0.596601i	$-0.203483\pi$
$11$	17.6558	1.60508	0.802538	+	0.596601i	$0.203483\pi$
$12$	0	0
$13$	9.23764i	0.710588i	0.934755	+	0.355294i	$0.115619\pi$
$13$	9.23764i	0.710588i	−0.934755	+	0.355294i	$0.884381\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.69563i	0.276213i	0.990417	+	0.138107i	$0.0441017\pi$
$17$	4.69563i	0.276213i	−0.990417	+	0.138107i	$0.955898\pi$
$18$	0	0
$19$	− 16.8874i	− 0.888810i	−0.895826	−	0.444405i	$-0.853415\pi$
$19$	− 16.8874i	− 0.888810i	0.895826	−	0.444405i	$-0.146585\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	39.0706i	1.69872i	0.527813	+	0.849361i	$0.323012\pi$
$23$	39.0706i	1.69872i	−0.527813	+	0.849361i	$0.676988\pi$
$24$	0	0
$25$	−15.6209	−0.624836
$26$	0	0
$27$	0	0
$28$	0	0
$29$	15.2070	0.524379	0.262190	−	0.965016i	$-0.415555\pi$
$29$	15.2070	0.524379	0.262190	+	0.965016i	$0.415555\pi$
$30$	0	0
$31$	48.7030	1.57107	0.785533	−	0.618820i	$-0.212389\pi$
$31$	48.7030	1.57107	0.785533	+	0.618820i	$0.212389\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.41298	−0.126085
$36$	0	0
$37$	− 14.1853i	− 0.383386i	−0.981455	−	0.191693i	$-0.938602\pi$
$37$	− 14.1853i	− 0.383386i	0.981455	−	0.191693i	$-0.0613977\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 10.1435i	− 0.247404i	−0.992319	−	0.123702i	$-0.960523\pi$
$41$	− 10.1435i	− 0.247404i	0.992319	−	0.123702i	$-0.0394766\pi$
$42$	0	0
$43$	22.3460i	0.519675i	0.965652	+	0.259837i	$0.0836689\pi$
$43$	22.3460i	0.519675i	−0.965652	+	0.259837i	$0.916331\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	42.5709i	0.905763i	0.891571	+	0.452882i	$0.149604\pi$
$47$	42.5709i	0.905763i	−0.891571	+	0.452882i	$0.850396\pi$
$48$	0	0
$49$	−46.9236	−0.957625
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−71.2201	−1.34378	−0.671888	−	0.740653i	$-0.734516\pi$
$53$	−71.2201	−1.34378	−0.671888	+	0.740653i	$0.734516\pi$
$54$	0	0
$55$	54.0715	0.983119
$56$	0	0
$57$	0	0
$58$	0	0
$59$	69.9985	1.18642	0.593208	−	0.805049i	$-0.297861\pi$
$59$	69.9985	1.18642	0.593208	+	0.805049i	$0.297861\pi$
$60$	0	0
$61$	103.643i	1.69907i	0.527534	+	0.849534i	$0.323117\pi$
$61$	103.643i	1.69907i	−0.527534	+	0.849534i	$0.676883\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	28.2906i	0.435239i
$66$	0	0
$67$	12.1430i	0.181239i	0.995886	+	0.0906197i	$0.0288848\pi$
$67$	12.1430i	0.181239i	−0.995886	+	0.0906197i	$0.971115\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 112.880i	− 1.58985i	−0.606705	−	0.794927i	$-0.707509\pi$
$71$	− 112.880i	− 1.58985i	0.606705	−	0.794927i	$-0.292491\pi$
$72$	0	0
$73$	84.0137	1.15087	0.575436	−	0.817847i	$-0.304832\pi$
$73$	84.0137	1.15087	0.575436	+	0.817847i	$0.304832\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−25.4413	−0.330407
$78$	0	0
$79$	−45.9651	−0.581837	−0.290918	−	0.956748i	$-0.593961\pi$
$79$	−45.9651	−0.581837	−0.290918	+	0.956748i	$0.593961\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	51.5768	0.621408	0.310704	−	0.950507i	$-0.399435\pi$
$83$	51.5768	0.621408	0.310704	+	0.950507i	$0.399435\pi$
$84$	0	0
$85$	14.3805i	0.169182i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 105.051i	− 1.18035i	−0.807275	−	0.590176i	$-0.799058\pi$
$89$	− 105.051i	− 1.18035i	0.807275	−	0.590176i	$-0.200942\pi$
$90$	0	0
$91$	− 13.3111i	− 0.146275i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 51.7181i	− 0.544401i
$96$	0	0
$97$	−38.8643	−0.400663	−0.200331	−	0.979728i	$-0.564202\pi$
$97$	−38.8643	−0.400663	−0.200331	+	0.979728i	$0.564202\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−113.144	−1.12024	−0.560118	−	0.828413i	$-0.689244\pi$
$101$	−113.144	−1.12024	−0.560118	+	0.828413i	$0.689244\pi$
$102$	0	0
$103$	−30.4098	−0.295241	−0.147621	−	0.989044i	$-0.547161\pi$
$103$	−30.4098	−0.295241	−0.147621	+	0.989044i	$0.547161\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	32.1820	0.300767	0.150383	−	0.988628i	$-0.451949\pi$
$107$	32.1820	0.300767	0.150383	+	0.988628i	$0.451949\pi$
$108$	0	0
$109$	111.285i	1.02096i	0.859890	+	0.510480i	$0.170532\pi$
$109$	111.285i	1.02096i	−0.859890	+	0.510480i	$0.829468\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	104.060i	0.920884i	0.887690	+	0.460442i	$0.152309\pi$
$113$	104.060i	0.920884i	−0.887690	+	0.460442i	$0.847691\pi$
$114$	0	0
$115$	119.655i	1.04048i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 6.76620i	− 0.0568589i
$120$	0	0
$121$	190.728	1.57627
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−124.403	−0.995222
$126$	0	0
$127$	215.952	1.70041	0.850203	−	0.526455i	$-0.176479\pi$
$127$	215.952	1.70041	0.850203	+	0.526455i	$0.176479\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	69.3808	0.529625	0.264812	−	0.964300i	$-0.414690\pi$
$131$	69.3808	0.529625	0.264812	+	0.964300i	$0.414690\pi$
$132$	0	0
$133$	24.3340i	0.182963i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	86.2214i	0.629354i	0.949199	+	0.314677i	$0.101896\pi$
$137$	86.2214i	0.629354i	−0.949199	+	0.314677i	$0.898104\pi$
$138$	0	0
$139$	12.9183i	0.0929373i	0.998920	+	0.0464686i	$0.0147968\pi$
$139$	12.9183i	0.0929373i	−0.998920	+	0.0464686i	$0.985203\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	163.098i	1.14055i
$144$	0	0
$145$	46.5719	0.321185
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−89.1486	−0.598313	−0.299156	−	0.954204i	$-0.596705\pi$
$149$	−89.1486	−0.598313	−0.299156	+	0.954204i	$0.596705\pi$
$150$	0	0
$151$	225.047	1.49038	0.745189	−	0.666854i	$-0.232359\pi$
$151$	225.047	1.49038	0.745189	+	0.666854i	$0.232359\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	149.155	0.962287
$156$	0	0
$157$	161.672i	1.02976i	0.857263	+	0.514878i	$0.172163\pi$
$157$	161.672i	1.02976i	−0.857263	+	0.514878i	$0.827837\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 56.2991i	− 0.349684i
$162$	0	0
$163$	181.252i	1.11197i	0.831191	+	0.555987i	$0.187660\pi$
$163$	181.252i	1.11197i	−0.831191	+	0.555987i	$0.812340\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.83241i	0.0528887i	0.999650	+	0.0264443i	$0.00841847\pi$
$167$	8.83241i	0.0528887i	−0.999650	+	0.0264443i	$0.991582\pi$
$168$	0	0
$169$	83.6660	0.495065
$170$	0	0
$171$	0	0
$172$	0	0
$173$	51.2824	0.296430	0.148215	−	0.988955i	$-0.452647\pi$
$173$	51.2824	0.296430	0.148215	+	0.988955i	$0.452647\pi$
$174$	0	0
$175$	22.5091	0.128623
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.7809	0.0658149	0.0329074	−	0.999458i	$-0.489523\pi$
$179$	11.7809	0.0658149	0.0329074	+	0.999458i	$0.489523\pi$
$180$	0	0
$181$	− 215.066i	− 1.18821i	−0.804387	−	0.594105i	$-0.797506\pi$
$181$	− 215.066i	− 1.18821i	0.804387	−	0.594105i	$-0.202494\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 43.4428i	− 0.234826i
$186$	0	0
$187$	82.9052i	0.443343i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 64.2197i	− 0.336229i	−0.985767	−	0.168114i	$-0.946232\pi$
$191$	− 64.2197i	− 0.336229i	0.985767	−	0.168114i	$-0.0537678\pi$
$192$	0	0
$193$	201.268	1.04284	0.521421	−	0.853300i	$-0.325402\pi$
$193$	201.268	1.04284	0.521421	+	0.853300i	$0.325402\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	159.855	0.811448	0.405724	−	0.913996i	$-0.367019\pi$
$197$	159.855	0.811448	0.405724	+	0.913996i	$0.367019\pi$
$198$	0	0
$199$	−252.515	−1.26892	−0.634461	−	0.772955i	$-0.718778\pi$
$199$	−252.515	−1.26892	−0.634461	+	0.772955i	$0.718778\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−21.9127	−0.107944
$204$	0	0
$205$	− 31.0649i	− 0.151536i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 298.161i	− 1.42661i
$210$	0	0
$211$	345.001i	1.63508i	0.575873	+	0.817539i	$0.304662\pi$
$211$	345.001i	1.63508i	−0.575873	+	0.817539i	$0.695338\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	68.4353i	0.318304i
$216$	0	0
$217$	−70.1791	−0.323406
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−43.3765	−0.196274
$222$	0	0
$223$	−80.9474	−0.362993	−0.181496	−	0.983392i	$-0.558094\pi$
$223$	−80.9474	−0.362993	−0.181496	+	0.983392i	$0.558094\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−165.633	−0.729659	−0.364829	−	0.931074i	$-0.618873\pi$
$227$	−165.633	−0.729659	−0.364829	+	0.931074i	$0.618873\pi$
$228$	0	0
$229$	− 10.0886i	− 0.0440549i	−0.999757	−	0.0220274i	$-0.992988\pi$
$229$	− 10.0886i	− 0.0440549i	0.999757	−	0.0220274i	$-0.00701212\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	219.759i	0.943170i	0.881821	+	0.471585i	$0.156318\pi$
$233$	219.759i	0.943170i	−0.881821	+	0.471585i	$0.843682\pi$
$234$	0	0
$235$	130.375i	0.554785i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 46.5130i	− 0.194615i	−0.995254	−	0.0973076i	$-0.968977\pi$
$239$	− 46.5130i	− 0.194615i	0.995254	−	0.0973076i	$-0.0310231\pi$
$240$	0	0
$241$	473.803	1.96599	0.982995	−	0.183634i	$-0.0587862\pi$
$241$	473.803	1.96599	0.982995	+	0.183634i	$0.0587862\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−143.705	−0.586551
$246$	0	0
$247$	156.000	0.631577
$248$	0	0
$249$	0	0
$250$	0	0
$251$	151.118	0.602065	0.301033	−	0.953614i	$-0.402669\pi$
$251$	151.118	0.602065	0.301033	+	0.953614i	$0.402669\pi$
$252$	0	0
$253$	689.824i	2.72658i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 300.498i	− 1.16925i	−0.811303	−	0.584626i	$-0.801241\pi$
$257$	− 300.498i	− 1.16925i	0.811303	−	0.584626i	$-0.198759\pi$
$258$	0	0
$259$	20.4404i	0.0789204i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 49.2004i	− 0.187074i	−0.995616	−	0.0935369i	$-0.970183\pi$
$263$	− 49.2004i	− 0.187074i	0.995616	−	0.0935369i	$-0.0298173\pi$
$264$	0	0
$265$	−218.114	−0.823071
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−151.922	−0.564767	−0.282383	−	0.959302i	$-0.591125\pi$
$269$	−151.922	−0.564767	−0.282383	+	0.959302i	$0.591125\pi$
$270$	0	0
$271$	−114.040	−0.420811	−0.210405	−	0.977614i	$-0.567478\pi$
$271$	−114.040	−0.420811	−0.210405	+	0.977614i	$0.567478\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−275.800	−1.00291
$276$	0	0
$277$	131.081i	0.473217i	0.971605	+	0.236608i	$0.0760359\pi$
$277$	131.081i	0.473217i	−0.971605	+	0.236608i	$0.923964\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 174.139i	− 0.619711i	−0.950784	−	0.309856i	$-0.899719\pi$
$281$	− 174.139i	− 0.619711i	0.950784	−	0.309856i	$-0.100281\pi$
$282$	0	0
$283$	118.115i	0.417369i	0.977983	+	0.208685i	$0.0669182\pi$
$283$	118.115i	0.417369i	−0.977983	+	0.208685i	$0.933082\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.6164i	0.0509283i
$288$	0	0
$289$	266.951	0.923706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−32.7323	−0.111714	−0.0558571	−	0.998439i	$-0.517789\pi$
$293$	−32.7323	−0.111714	−0.0558571	+	0.998439i	$0.517789\pi$
$294$	0	0
$295$	214.373	0.726687
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−360.920	−1.20709
$300$	0	0
$301$	− 32.1997i	− 0.106976i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	317.410i	1.04069i
$306$	0	0
$307$	309.352i	1.00766i	0.863802	+	0.503831i	$0.168077\pi$
$307$	309.352i	1.00766i	−0.863802	+	0.503831i	$0.831923\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	126.641i	0.407206i	0.979054	+	0.203603i	$0.0652651\pi$
$311$	126.641i	0.407206i	−0.979054	+	0.203603i	$0.934735\pi$
$312$	0	0
$313$	197.265	0.630241	0.315120	−	0.949052i	$-0.397955\pi$
$313$	197.265	0.630241	0.315120	+	0.949052i	$0.397955\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	191.400	0.603785	0.301892	−	0.953342i	$-0.402382\pi$
$317$	191.400	0.603785	0.301892	+	0.953342i	$0.402382\pi$
$318$	0	0
$319$	268.492	0.841668
$320$	0	0
$321$	0	0
$322$	0	0
$323$	79.2969	0.245501
$324$	0	0
$325$	− 144.300i	− 0.444001i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 61.3429i	− 0.186452i
$330$	0	0
$331$	− 375.392i	− 1.13411i	−0.823679	−	0.567057i	$-0.808082\pi$
$331$	− 375.392i	− 1.13411i	0.823679	−	0.567057i	$-0.191918\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	37.1884i	0.111010i
$336$	0	0
$337$	23.3081	0.0691635	0.0345818	−	0.999402i	$-0.488990\pi$
$337$	23.3081	0.0691635	0.0345818	+	0.999402i	$0.488990\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	859.893	2.52168
$342$	0	0
$343$	138.222	0.402980
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−66.9756	−0.193013	−0.0965067	−	0.995332i	$-0.530767\pi$
$347$	−66.9756	−0.193013	−0.0965067	+	0.995332i	$0.530767\pi$
$348$	0	0
$349$	− 346.531i	− 0.992925i	−0.868058	−	0.496463i	$-0.834632\pi$
$349$	− 346.531i	− 0.992925i	0.868058	−	0.496463i	$-0.165368\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	588.364i	1.66675i	0.552705	+	0.833377i	$0.313596\pi$
$353$	588.364i	1.66675i	−0.552705	+	0.833377i	$0.686404\pi$
$354$	0	0
$355$	− 345.697i	− 0.973796i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 325.201i	− 0.905854i	−0.891548	−	0.452927i	$-0.850380\pi$
$359$	− 325.201i	− 0.905854i	0.891548	−	0.452927i	$-0.149620\pi$
$360$	0	0
$361$	75.8162	0.210017
$362$	0	0
$363$	0	0
$364$	0	0
$365$	257.295	0.704917
$366$	0	0
$367$	205.397	0.559664	0.279832	−	0.960049i	$-0.409721\pi$
$367$	205.397	0.559664	0.279832	+	0.960049i	$0.409721\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	102.625	0.276618
$372$	0	0
$373$	380.451i	1.01998i	0.860182	+	0.509988i	$0.170350\pi$
$373$	380.451i	1.01998i	−0.860182	+	0.509988i	$0.829650\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	140.477i	0.372617i
$378$	0	0
$379$	122.536i	0.323315i	0.986847	+	0.161658i	$0.0516840\pi$
$379$	122.536i	0.323315i	−0.986847	+	0.161658i	$0.948316\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 162.192i	− 0.423477i	−0.977326	−	0.211738i	$-0.932087\pi$
$383$	− 162.192i	− 0.423477i	0.977326	−	0.211738i	$-0.0679125\pi$
$384$	0	0
$385$	−77.9148	−0.202376
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−258.066	−0.663408	−0.331704	−	0.943383i	$-0.607624\pi$
$389$	−258.066	−0.663408	−0.331704	+	0.943383i	$0.607624\pi$
$390$	0	0
$391$	−183.461	−0.469210
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−140.770	−0.356379
$396$	0	0
$397$	214.037i	0.539137i	0.962981	+	0.269568i	$0.0868810\pi$
$397$	214.037i	0.539137i	−0.962981	+	0.269568i	$0.913119\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	270.214i	0.673850i	0.941531	+	0.336925i	$0.109387\pi$
$401$	270.214i	0.673850i	−0.941531	+	0.336925i	$0.890613\pi$
$402$	0	0
$403$	449.901i	1.11638i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 250.453i	− 0.615363i
$408$	0	0
$409$	11.6548	0.0284958	0.0142479	−	0.999898i	$-0.495465\pi$
$409$	11.6548	0.0284958	0.0142479	+	0.999898i	$0.495465\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−100.865	−0.244225
$414$	0	0
$415$	157.956	0.380616
$416$	0	0
$417$	0	0
$418$	0	0
$419$	272.082	0.649361	0.324681	−	0.945824i	$-0.394743\pi$
$419$	272.082	0.649361	0.324681	+	0.945824i	$0.394743\pi$
$420$	0	0
$421$	501.154i	1.19039i	0.803582	+	0.595195i	$0.202925\pi$
$421$	501.154i	1.19039i	−0.803582	+	0.595195i	$0.797075\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 73.3499i	− 0.172588i
$426$	0	0
$427$	− 149.345i	− 0.349755i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	701.705i	1.62809i	0.580805	+	0.814043i	$0.302738\pi$
$431$	701.705i	1.62809i	−0.580805	+	0.814043i	$0.697262\pi$
$432$	0	0
$433$	−387.204	−0.894236	−0.447118	−	0.894475i	$-0.647550\pi$
$433$	−387.204	−0.894236	−0.447118	+	0.894475i	$0.647550\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	659.800	1.50984
$438$	0	0
$439$	324.269	0.738653	0.369327	−	0.929300i	$-0.379588\pi$
$439$	324.269	0.738653	0.369327	+	0.929300i	$0.379588\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.1018	−0.0228032	−0.0114016	−	0.999935i	$-0.503629\pi$
$443$	−10.1018	−0.0228032	−0.0114016	+	0.999935i	$0.503629\pi$
$444$	0	0
$445$	− 321.723i	− 0.722973i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 71.1875i	− 0.158547i	−0.996853	−	0.0792734i	$-0.974740\pi$
$449$	− 71.1875i	− 0.158547i	0.996853	−	0.0792734i	$-0.0252600\pi$
$450$	0	0
$451$	− 179.093i	− 0.397102i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 40.7655i	− 0.0895946i
$456$	0	0
$457$	−334.849	−0.732711	−0.366355	−	0.930475i	$-0.619395\pi$
$457$	−334.849	−0.732711	−0.366355	+	0.930475i	$0.619395\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	811.172	1.75959	0.879796	−	0.475351i	$-0.157679\pi$
$461$	811.172	1.75959	0.879796	+	0.475351i	$0.157679\pi$
$462$	0	0
$463$	102.320	0.220995	0.110497	−	0.993876i	$-0.464756\pi$
$463$	102.320	0.220995	0.110497	+	0.993876i	$0.464756\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−211.289	−0.452439	−0.226220	−	0.974076i	$-0.572637\pi$
$467$	−211.289	−0.452439	−0.226220	+	0.974076i	$0.572637\pi$
$468$	0	0
$469$	− 17.4976i	− 0.0373084i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	394.537i	0.834117i
$474$	0	0
$475$	263.796i	0.555361i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	462.267i	0.965066i	0.875878	+	0.482533i	$0.160283\pi$
$479$	462.267i	0.965066i	−0.875878	+	0.482533i	$0.839717\pi$
$480$	0	0
$481$	131.038	0.272429
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−119.023	−0.245408
$486$	0	0
$487$	581.940	1.19495	0.597474	−	0.801888i	$-0.296171\pi$
$487$	581.940	1.19495	0.597474	+	0.801888i	$0.296171\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−578.179	−1.17755	−0.588777	−	0.808296i	$-0.700390\pi$
$491$	−578.179	−1.17755	−0.588777	+	0.808296i	$0.700390\pi$
$492$	0	0
$493$	71.4064i	0.144841i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	162.655i	0.327273i
$498$	0	0
$499$	798.964i	1.60113i	0.599246	+	0.800565i	$0.295467\pi$
$499$	798.964i	1.60113i	−0.599246	+	0.800565i	$0.704533\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 252.819i	− 0.502622i	−0.967906	−	0.251311i	$-0.919138\pi$
$503$	− 252.819i	− 0.502622i	0.967906	−	0.251311i	$-0.0808616\pi$
$504$	0	0
$505$	−346.507	−0.686152
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−461.382	−0.906448	−0.453224	−	0.891397i	$-0.649726\pi$
$509$	−461.382	−0.906448	−0.453224	+	0.891397i	$0.649726\pi$
$510$	0	0
$511$	−121.060	−0.236909
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−93.1311	−0.180837
$516$	0	0
$517$	751.624i	1.45382i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 765.575i	− 1.46943i	−0.678374	−	0.734717i	$-0.737315\pi$
$521$	− 765.575i	− 1.46943i	0.678374	−	0.734717i	$-0.262685\pi$
$522$	0	0
$523$	− 1007.67i	− 1.92671i	−0.268222	−	0.963357i	$-0.586436\pi$
$523$	− 1007.67i	− 1.92671i	0.268222	−	0.963357i	$-0.413564\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	228.691i	0.433949i
$528$	0	0
$529$	−997.512	−1.88566
$530$	0	0
$531$	0	0
$532$	0	0
$533$	93.7024	0.175802
$534$	0	0
$535$	98.5584	0.184221
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−828.476	−1.53706
$540$	0	0
$541$	− 367.842i	− 0.679930i	−0.940438	−	0.339965i	$-0.889585\pi$
$541$	− 367.842i	− 0.679930i	0.940438	−	0.339965i	$-0.110415\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	340.813i	0.625344i
$546$	0	0
$547$	− 674.370i	− 1.23285i	−0.787413	−	0.616426i	$-0.788580\pi$
$547$	− 674.370i	− 1.23285i	0.787413	−	0.616426i	$-0.211420\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 256.806i	− 0.466073i
$552$	0	0
$553$	66.2338	0.119772
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−552.503	−0.991927	−0.495963	−	0.868344i	$-0.665185\pi$
$557$	−552.503	−0.991927	−0.495963	+	0.868344i	$0.665185\pi$
$558$	0	0
$559$	−206.424	−0.369274
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−144.946	−0.257453	−0.128726	−	0.991680i	$-0.541089\pi$
$563$	−144.946	−0.257453	−0.128726	+	0.991680i	$0.541089\pi$
$564$	0	0
$565$	318.687i	0.564047i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 624.449i	− 1.09745i	−0.836003	−	0.548725i	$-0.815114\pi$
$569$	− 624.449i	− 1.09745i	0.836003	−	0.548725i	$-0.184886\pi$
$570$	0	0
$571$	− 522.294i	− 0.914700i	−0.889287	−	0.457350i	$-0.848799\pi$
$571$	− 522.294i	− 0.914700i	0.889287	−	0.457350i	$-0.151201\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 610.318i	− 1.06142i
$576$	0	0
$577$	17.0045	0.0294706	0.0147353	−	0.999891i	$-0.495309\pi$
$577$	17.0045	0.0294706	0.0147353	+	0.999891i	$0.495309\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−74.3201	−0.127918
$582$	0	0
$583$	−1257.45	−2.15686
$584$	0	0
$585$	0	0
$586$	0	0
$587$	122.224	0.208218	0.104109	−	0.994566i	$-0.466801\pi$
$587$	122.224	0.208218	0.104109	+	0.994566i	$0.466801\pi$
$588$	0	0
$589$	− 822.467i	− 1.39638i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 811.371i	− 1.36825i	−0.729366	−	0.684124i	$-0.760185\pi$
$593$	− 811.371i	− 1.36825i	0.729366	−	0.684124i	$-0.239815\pi$
$594$	0	0
$595$	− 20.7217i	− 0.0348264i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1003.26i	− 1.67489i	−0.546518	−	0.837447i	$-0.684047\pi$
$599$	− 1003.26i	− 1.67489i	0.546518	−	0.837447i	$-0.315953\pi$
$600$	0	0
$601$	408.811	0.680218	0.340109	−	0.940386i	$-0.389536\pi$
$601$	408.811	0.680218	0.340109	+	0.940386i	$0.389536\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	584.112	0.965474
$606$	0	0
$607$	−635.975	−1.04774	−0.523868	−	0.851800i	$-0.675511\pi$
$607$	−635.975	−1.04774	−0.523868	+	0.851800i	$0.675511\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−393.254	−0.643624
$612$	0	0
$613$	− 1029.22i	− 1.67899i	−0.543367	−	0.839495i	$-0.682851\pi$
$613$	− 1029.22i	− 1.67899i	0.543367	−	0.839495i	$-0.317149\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 532.843i	− 0.863603i	−0.901969	−	0.431801i	$-0.857878\pi$
$617$	− 532.843i	− 0.863603i	0.901969	−	0.431801i	$-0.142122\pi$
$618$	0	0
$619$	89.0355i	0.143838i	0.997410	+	0.0719188i	$0.0229123\pi$
$619$	89.0355i	0.143838i	−0.997410	+	0.0719188i	$0.977088\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	151.375i	0.242977i
$624$	0	0
$625$	9.53535	0.0152566
$626$	0	0
$627$	0	0
$628$	0	0
$629$	66.6087	0.105896
$630$	0	0
$631$	−8.45150	−0.0133938	−0.00669691	−	0.999978i	$-0.502132\pi$
$631$	−8.45150	−0.0133938	−0.00669691	+	0.999978i	$0.502132\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	661.358	1.04151
$636$	0	0
$637$	− 433.464i	− 0.680477i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 621.134i	− 0.969008i	−0.874789	−	0.484504i	$-0.839000\pi$
$641$	− 621.134i	− 0.969008i	0.874789	−	0.484504i	$-0.161000\pi$
$642$	0	0
$643$	− 931.443i	− 1.44859i	−0.689490	−	0.724295i	$-0.742165\pi$
$643$	− 931.443i	− 1.44859i	0.689490	−	0.724295i	$-0.257835\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	168.057i	0.259749i	0.991530	+	0.129874i	$0.0414574\pi$
$647$	168.057i	0.259749i	−0.991530	+	0.129874i	$0.958543\pi$
$648$	0	0
$649$	1235.88	1.90429
$650$	0	0
$651$	0	0
$652$	0	0
$653$	121.170	0.185559	0.0927796	−	0.995687i	$-0.470425\pi$
$653$	121.170	0.185559	0.0927796	+	0.995687i	$0.470425\pi$
$654$	0	0
$655$	212.481	0.324398
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−163.245	−0.247716	−0.123858	−	0.992300i	$-0.539527\pi$
$659$	−163.245	−0.247716	−0.123858	+	0.992300i	$0.539527\pi$
$660$	0	0
$661$	− 238.641i	− 0.361030i	−0.983572	−	0.180515i	$-0.942224\pi$
$661$	− 238.641i	− 0.361030i	0.983572	−	0.180515i	$-0.0577765\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	74.5237i	0.112066i
$666$	0	0
$667$	594.146i	0.890774i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1829.91i	2.72713i
$672$	0	0
$673$	−190.214	−0.282636	−0.141318	−	0.989964i	$-0.545134\pi$
$673$	−190.214	−0.282636	−0.141318	+	0.989964i	$0.545134\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1256.32	−1.85571	−0.927857	−	0.372937i	$-0.878351\pi$
$677$	−1256.32	−1.85571	−0.927857	+	0.372937i	$0.878351\pi$
$678$	0	0
$679$	56.0018	0.0824769
$680$	0	0
$681$	0	0
$682$	0	0
$683$	415.707	0.608648	0.304324	−	0.952569i	$-0.401569\pi$
$683$	415.707	0.608648	0.304324	+	0.952569i	$0.401569\pi$
$684$	0	0
$685$	264.056i	0.385483i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 657.906i	− 0.954871i
$690$	0	0
$691$	609.942i	0.882695i	0.897336	+	0.441348i	$0.145499\pi$
$691$	609.942i	0.882695i	−0.897336	+	0.441348i	$0.854501\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	39.5626i	0.0569247i
$696$	0	0
$697$	47.6303	0.0683362
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−922.743	−1.31632	−0.658162	−	0.752877i	$-0.728666\pi$
$701$	−922.743	−1.31632	−0.658162	+	0.752877i	$0.728666\pi$
$702$	0	0
$703$	−239.552	−0.340757
$704$	0	0
$705$	0	0
$706$	0	0
$707$	163.036	0.230602
$708$	0	0
$709$	261.447i	0.368755i	0.982856	+	0.184377i	$0.0590268\pi$
$709$	261.447i	0.368755i	−0.982856	+	0.184377i	$0.940973\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1902.86i	2.66880i
$714$	0	0
$715$	499.493i	0.698592i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 995.870i	− 1.38508i	−0.721381	−	0.692538i	$-0.756492\pi$
$719$	− 995.870i	− 1.38508i	0.721381	−	0.692538i	$-0.243508\pi$
$720$	0	0
$721$	43.8193	0.0607757
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−237.547	−0.327651
$726$	0	0
$727$	−818.322	−1.12561	−0.562807	−	0.826588i	$-0.690279\pi$
$727$	−818.322	−1.12561	−0.562807	+	0.826588i	$0.690279\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−104.928	−0.143541
$732$	0	0
$733$	− 657.950i	− 0.897613i	−0.893629	−	0.448807i	$-0.851849\pi$
$733$	− 657.950i	− 0.897613i	0.893629	−	0.448807i	$-0.148151\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	214.396i	0.290903i
$738$	0	0
$739$	− 971.543i	− 1.31467i	−0.753597	−	0.657336i	$-0.771683\pi$
$739$	− 971.543i	− 1.31467i	0.753597	−	0.657336i	$-0.228317\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 899.657i	− 1.21084i	−0.795905	−	0.605422i	$-0.793004\pi$
$743$	− 899.657i	− 1.21084i	0.795905	−	0.605422i	$-0.206996\pi$
$744$	0	0
$745$	−273.020	−0.366470
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−46.3730	−0.0619132
$750$	0	0
$751$	−797.369	−1.06174	−0.530871	−	0.847453i	$-0.678135\pi$
$751$	−797.369	−1.06174	−0.530871	+	0.847453i	$0.678135\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	689.213	0.912865
$756$	0	0
$757$	− 577.348i	− 0.762680i	−0.924435	−	0.381340i	$-0.875463\pi$
$757$	− 577.348i	− 0.762680i	0.924435	−	0.381340i	$-0.124537\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	689.287i	0.905765i	0.891570	+	0.452882i	$0.149604\pi$
$761$	689.287i	0.905765i	−0.891570	+	0.452882i	$0.850396\pi$
$762$	0	0
$763$	− 160.357i	− 0.210166i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	646.621i	0.843052i
$768$	0	0
$769$	−1505.38	−1.95759	−0.978793	−	0.204850i	$-0.934329\pi$
$769$	−1505.38	−1.95759	−0.978793	+	0.204850i	$0.934329\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	584.466	0.756101	0.378051	−	0.925785i	$-0.376595\pi$
$773$	584.466	0.756101	0.378051	+	0.925785i	$0.376595\pi$
$774$	0	0
$775$	−760.786	−0.981659
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−171.298	−0.219895
$780$	0	0
$781$	− 1992.98i	− 2.55184i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	495.125i	0.630732i
$786$	0	0
$787$	− 257.351i	− 0.327003i	−0.986543	−	0.163501i	$-0.947721\pi$
$787$	− 257.351i	− 0.327003i	0.986543	−	0.163501i	$-0.0522788\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 149.946i	− 0.189565i
$792$	0	0
$793$	−957.418	−1.20734
$794$	0	0
$795$	0	0
$796$	0	0
$797$	893.657	1.12128	0.560638	−	0.828061i	$-0.310556\pi$
$797$	893.657	1.12128	0.560638	+	0.828061i	$0.310556\pi$
$798$	0	0
$799$	−199.897	−0.250184
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1483.33	1.84724
$804$	0	0
$805$	− 172.418i	− 0.214184i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1243.52i	1.53711i	0.639784	+	0.768554i	$0.279024\pi$
$809$	1243.52i	1.53711i	−0.639784	+	0.768554i	$0.720976\pi$
$810$	0	0
$811$	− 257.659i	− 0.317705i	−0.987302	−	0.158852i	$-0.949221\pi$
$811$	− 257.659i	− 0.317705i	0.987302	−	0.158852i	$-0.0507794\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	555.089i	0.681091i
$816$	0	0
$817$	377.366	0.461892
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−997.784	−1.21533	−0.607664	−	0.794194i	$-0.707893\pi$
$821$	−997.784	−1.21533	−0.607664	+	0.794194i	$0.707893\pi$
$822$	0	0
$823$	318.613	0.387137	0.193568	−	0.981087i	$-0.437994\pi$
$823$	318.613	0.387137	0.193568	+	0.981087i	$0.437994\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	279.151	0.337547	0.168773	−	0.985655i	$-0.446019\pi$
$827$	279.151	0.337547	0.168773	+	0.985655i	$0.446019\pi$
$828$	0	0
$829$	444.345i	0.536001i	0.963419	+	0.268000i	$0.0863628\pi$
$829$	444.345i	0.536001i	−0.963419	+	0.268000i	$0.913637\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 220.336i	− 0.264509i
$834$	0	0
$835$	27.0495i	0.0323946i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 113.742i	− 0.135569i	−0.997700	−	0.0677845i	$-0.978407\pi$
$839$	− 113.742i	− 0.135569i	0.997700	−	0.0677845i	$-0.0215930\pi$
$840$	0	0
$841$	−609.747	−0.725027
$842$	0	0
$843$	0	0
$844$	0	0
$845$	256.230	0.303230
$846$	0	0
$847$	−274.832	−0.324477
$848$	0	0
$849$	0	0
$850$	0	0
$851$	554.227	0.651265
$852$	0	0
$853$	733.694i	0.860133i	0.902797	+	0.430067i	$0.141510\pi$
$853$	733.694i	0.860133i	−0.902797	+	0.430067i	$0.858490\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 956.875i	− 1.11654i	−0.829659	−	0.558270i	$-0.811465\pi$
$857$	− 956.875i	− 1.11654i	0.829659	−	0.558270i	$-0.188535\pi$
$858$	0	0
$859$	− 825.873i	− 0.961435i	−0.876876	−	0.480717i	$-0.840376\pi$
$859$	− 825.873i	− 0.961435i	0.876876	−	0.480717i	$-0.159624\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	945.401i	1.09548i	0.836648	+	0.547741i	$0.184512\pi$
$863$	945.401i	1.09548i	−0.836648	+	0.547741i	$0.815488\pi$
$864$	0	0
$865$	157.054	0.181565
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−811.552	−0.933892
$870$	0	0
$871$	−112.173	−0.128787
$872$	0	0
$873$	0	0
$874$	0	0
$875$	179.259	0.204868
$876$	0	0
$877$	− 847.547i	− 0.966416i	−0.875506	−	0.483208i	$-0.839472\pi$
$877$	− 847.547i	− 0.966416i	0.875506	−	0.483208i	$-0.160528\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	716.539i	0.813324i	0.913579	+	0.406662i	$0.133307\pi$
$881$	716.539i	0.813324i	−0.913579	+	0.406662i	$0.866693\pi$
$882$	0	0
$883$	915.122i	1.03638i	0.855266	+	0.518189i	$0.173394\pi$
$883$	915.122i	1.03638i	−0.855266	+	0.518189i	$0.826606\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 57.1916i	− 0.0644776i	−0.999480	−	0.0322388i	$-0.989736\pi$
$887$	− 57.1916i	− 0.0644776i	0.999480	−	0.0322388i	$-0.0102637\pi$
$888$	0	0
$889$	−311.177	−0.350031
$890$	0	0
$891$	0	0
$892$	0	0
$893$	718.911	0.805051
$894$	0	0
$895$	36.0793	0.0403120
$896$	0	0
$897$	0	0
$898$	0	0
$899$	740.627	0.823834
$900$	0	0
$901$	− 334.423i	− 0.371169i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 658.647i	− 0.727786i
$906$	0	0
$907$	− 1539.82i	− 1.69770i	−0.528631	−	0.848852i	$-0.677294\pi$
$907$	− 1539.82i	− 1.69770i	0.528631	−	0.848852i	$-0.322706\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 888.075i	− 0.974835i	−0.873169	−	0.487418i	$-0.837939\pi$
$911$	− 888.075i	− 0.974835i	0.873169	−	0.487418i	$-0.162061\pi$
$912$	0	0
$913$	910.632	0.997406
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−99.9749	−0.109024
$918$	0	0
$919$	−1135.32	−1.23539	−0.617695	−	0.786418i	$-0.711933\pi$
$919$	−1135.32	−1.23539	−0.617695	+	0.786418i	$0.711933\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1042.74	1.12973
$924$	0	0
$925$	221.587i	0.239553i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1215.41i	− 1.30830i	−0.756367	−	0.654148i	$-0.773027\pi$
$929$	− 1215.41i	− 1.30830i	0.756367	−	0.654148i	$-0.226973\pi$
$930$	0	0
$931$	792.418i	0.851147i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	253.900i	0.271551i
$936$	0	0
$937$	−551.201	−0.588262	−0.294131	−	0.955765i	$-0.595030\pi$
$937$	−551.201	−0.588262	−0.294131	+	0.955765i	$0.595030\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1369.24	−1.45509	−0.727543	−	0.686062i	$-0.759338\pi$
$941$	−1369.24	−1.45509	−0.727543	+	0.686062i	$0.759338\pi$
$942$	0	0
$943$	396.314	0.420270
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−924.838	−0.976597	−0.488299	−	0.872677i	$-0.662382\pi$
$947$	−924.838	−0.976597	−0.488299	+	0.872677i	$0.662382\pi$
$948$	0	0
$949$	776.089i	0.817796i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 279.545i	− 0.293331i	−0.989186	−	0.146666i	$-0.953146\pi$
$953$	− 279.545i	− 0.293331i	0.989186	−	0.146666i	$-0.0468541\pi$
$954$	0	0
$955$	− 196.675i	− 0.205942i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 124.242i	− 0.129553i
$960$	0	0
$961$	1410.99	1.46825
$962$	0	0
$963$	0	0
$964$	0	0
$965$	616.391	0.638747
$966$	0	0
$967$	−2.56721	−0.00265482	−0.00132741	−	0.999999i	$-0.500423\pi$
$967$	−2.56721	−0.00265482	−0.00132741	+	0.999999i	$0.500423\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1053.61	1.08508	0.542540	−	0.840030i	$-0.317463\pi$
$971$	1053.61	1.08508	0.542540	+	0.840030i	$0.317463\pi$
$972$	0	0
$973$	− 18.6147i	− 0.0191313i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	788.865i	0.807436i	0.914884	+	0.403718i	$0.132282\pi$
$977$	788.865i	0.807436i	−0.914884	+	0.403718i	$0.867718\pi$
$978$	0	0
$979$	− 1854.77i	− 1.89455i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 544.960i	− 0.554385i	−0.960814	−	0.277192i	$-0.910596\pi$
$983$	− 544.960i	− 0.554385i	0.960814	−	0.277192i	$-0.0894039\pi$
$984$	0	0
$985$	489.561	0.497017
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−873.072	−0.882782
$990$	0	0
$991$	−830.022	−0.837560	−0.418780	−	0.908088i	$-0.637542\pi$
$991$	−830.022	−0.837560	−0.418780	+	0.908088i	$0.637542\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−773.336	−0.777222
$996$	0	0
$997$	− 1282.56i	− 1.28642i	−0.765690	−	0.643210i	$-0.777602\pi$
$997$	− 1282.56i	− 1.28642i	0.765690	−	0.643210i	$-0.222398\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.3.h.a.1457.29		44
3.2	odd	2	inner	2592.3.h.a.1457.16		44
4.3	odd	2		648.3.h.a.485.9		44
8.3	odd	2		648.3.h.a.485.35		44
8.5	even	2	inner	2592.3.h.a.1457.15		44
9.2	odd	6		864.3.n.a.17.15		44
9.4	even	3		864.3.n.a.305.8		44
9.5	odd	6		288.3.n.a.209.10		44
9.7	even	3		288.3.n.a.113.13		44
12.11	even	2		648.3.h.a.485.36		44
24.5	odd	2	inner	2592.3.h.a.1457.30		44
24.11	even	2		648.3.h.a.485.10		44
36.7	odd	6		72.3.j.a.5.11	yes	44
36.11	even	6		216.3.j.a.125.12		44
36.23	even	6		72.3.j.a.29.3	yes	44
36.31	odd	6		216.3.j.a.197.20		44
72.5	odd	6		288.3.n.a.209.13		44
72.11	even	6		216.3.j.a.125.20		44
72.13	even	6		864.3.n.a.305.15		44
72.29	odd	6		864.3.n.a.17.8		44
72.43	odd	6		72.3.j.a.5.3	✓	44
72.59	even	6		72.3.j.a.29.11	yes	44
72.61	even	6		288.3.n.a.113.10		44
72.67	odd	6		216.3.j.a.197.12		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.j.a.5.3	✓	44	72.43	odd	6
72.3.j.a.5.11	yes	44	36.7	odd	6
72.3.j.a.29.3	yes	44	36.23	even	6
72.3.j.a.29.11	yes	44	72.59	even	6
216.3.j.a.125.12		44	36.11	even	6
216.3.j.a.125.20		44	72.11	even	6
216.3.j.a.197.12		44	72.67	odd	6
216.3.j.a.197.20		44	36.31	odd	6
288.3.n.a.113.10		44	72.61	even	6
288.3.n.a.113.13		44	9.7	even	3
288.3.n.a.209.10		44	9.5	odd	6
288.3.n.a.209.13		44	72.5	odd	6
648.3.h.a.485.9		44	4.3	odd	2
648.3.h.a.485.10		44	24.11	even	2
648.3.h.a.485.35		44	8.3	odd	2
648.3.h.a.485.36		44	12.11	even	2
864.3.n.a.17.8		44	72.29	odd	6
864.3.n.a.17.15		44	9.2	odd	6
864.3.n.a.305.8		44	9.4	even	3
864.3.n.a.305.15		44	72.13	even	6
2592.3.h.a.1457.15		44	8.5	even	2	inner
2592.3.h.a.1457.16		44	3.2	odd	2	inner
2592.3.h.a.1457.29		44	1.1	even	1	trivial
2592.3.h.a.1457.30		44	24.5	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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2592.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.06253 q^{5} -1.44096 q^{7} +O(q^{10})\)	\(q+3.06253 q^{5} -1.44096 q^{7} +17.6558 q^{11} +9.23764i q^{13} +4.69563i q^{17} -16.8874i q^{19} +39.0706i q^{23} -15.6209 q^{25} +15.2070 q^{29} +48.7030 q^{31} -4.41298 q^{35} -14.1853i q^{37} -10.1435i q^{41} +22.3460i q^{43} +42.5709i q^{47} -46.9236 q^{49} -71.2201 q^{53} +54.0715 q^{55} +69.9985 q^{59} +103.643i q^{61} +28.2906i q^{65} +12.1430i q^{67} -112.880i q^{71} +84.0137 q^{73} -25.4413 q^{77} -45.9651 q^{79} +51.5768 q^{83} +14.3805i q^{85} -105.051i q^{89} -13.3111i q^{91} -51.7181i q^{95} -38.8643 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{7}+O(q^{10}) \) 44 * q - 4 * q^7	\( 44 q - 4 q^{7} + 144 q^{25} - 4 q^{31} + 144 q^{49} - 92 q^{55} - 8 q^{73} - 4 q^{79} + 4 q^{97}+O(q^{100}) \) 44 * q - 4 * q^7 + 144 * q^25 - 4 * q^31 + 144 * q^49 - 92 * q^55 - 8 * q^73 - 4 * q^79 + 4 * q^97

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