LMFDB - Embedded newform 5328.2.e.g.2591.5

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			2591.5
Character	$\chi$	$=$	5328.2591
Dual form			5328.2.e.g.2591.20

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.18234i q^{5} -1.23564i q^{7} +4.50980 q^{11} -3.37538 q^{13} +4.35211i q^{17} -0.552888i q^{19} -1.18481 q^{23} +0.237388 q^{25} -9.20998i q^{29} -4.52853i q^{31} -2.69659 q^{35} +1.00000 q^{37} -1.49330i q^{41} +4.20762i q^{43} +6.35326 q^{47} +5.47320 q^{49} -1.40320i q^{53} -9.84192i q^{55} -4.72905 q^{59} -1.08004 q^{61} +7.36622i q^{65} -13.7429i q^{67} -8.96501 q^{71} +9.82648 q^{73} -5.57249i q^{77} -12.0673i q^{79} +0.0657560 q^{83} +9.49778 q^{85} +6.63713i q^{89} +4.17075i q^{91} -1.20659 q^{95} +6.76853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 2.18234i	− 0.975973i	−0.872851	−	0.487986i	$-0.837732\pi$
$5$	− 2.18234i	− 0.975973i	0.872851	−	0.487986i	$-0.162268\pi$
$6$	0	0
$7$	− 1.23564i	− 0.467028i	−0.972353	−	0.233514i	$-0.924978\pi$
$7$	− 1.23564i	− 0.467028i	0.972353	−	0.233514i	$-0.0750224\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.50980	1.35976	0.679878	−	0.733325i	$-0.262033\pi$
$11$	4.50980	1.35976	0.679878	+	0.733325i	$0.262033\pi$
$12$	0	0
$13$	−3.37538	−0.936161	−0.468080	−	0.883686i	$-0.655054\pi$
$13$	−3.37538	−0.936161	−0.468080	+	0.883686i	$0.655054\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.35211i	1.05554i	0.849387	+	0.527771i	$0.176972\pi$
$17$	4.35211i	1.05554i	−0.849387	+	0.527771i	$0.823028\pi$
$18$	0	0
$19$	− 0.552888i	− 0.126841i	−0.997987	−	0.0634206i	$-0.979799\pi$
$19$	− 0.552888i	− 0.126841i	0.997987	−	0.0634206i	$-0.0202009\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.18481	−0.247050	−0.123525	−	0.992341i	$-0.539420\pi$
$23$	−1.18481	−0.247050	−0.123525	+	0.992341i	$0.539420\pi$
$24$	0	0
$25$	0.237388	0.0474776
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 9.20998i	− 1.71025i	−0.518422	−	0.855125i	$-0.673480\pi$
$29$	− 9.20998i	− 1.71025i	0.518422	−	0.855125i	$-0.326520\pi$
$30$	0	0
$31$	− 4.52853i	− 0.813348i	−0.913573	−	0.406674i	$-0.866688\pi$
$31$	− 4.52853i	− 0.813348i	0.913573	−	0.406674i	$-0.133312\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.69659	−0.455806
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1.49330i	− 0.233214i	−0.993178	−	0.116607i	$-0.962798\pi$
$41$	− 1.49330i	− 0.233214i	0.993178	−	0.116607i	$-0.0372018\pi$
$42$	0	0
$43$	4.20762i	0.641656i	0.947138	+	0.320828i	$0.103961\pi$
$43$	4.20762i	0.641656i	−0.947138	+	0.320828i	$0.896039\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.35326	0.926718	0.463359	−	0.886171i	$-0.346644\pi$
$47$	6.35326	0.926718	0.463359	+	0.886171i	$0.346644\pi$
$48$	0	0
$49$	5.47320	0.781885
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.40320i	− 0.192744i	−0.995345	−	0.0963720i	$-0.969276\pi$
$53$	− 1.40320i	− 0.192744i	0.995345	−	0.0963720i	$-0.0307238\pi$
$54$	0	0
$55$	− 9.84192i	− 1.32708i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.72905	−0.615670	−0.307835	−	0.951440i	$-0.599605\pi$
$59$	−4.72905	−0.615670	−0.307835	+	0.951440i	$0.599605\pi$
$60$	0	0
$61$	−1.08004	−0.138285	−0.0691423	−	0.997607i	$-0.522026\pi$
$61$	−1.08004	−0.138285	−0.0691423	+	0.997607i	$0.522026\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.36622i	0.913667i
$66$	0	0
$67$	− 13.7429i	− 1.67896i	−0.543392	−	0.839479i	$-0.682860\pi$
$67$	− 13.7429i	− 1.67896i	0.543392	−	0.839479i	$-0.317140\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.96501	−1.06395	−0.531976	−	0.846760i	$-0.678550\pi$
$71$	−8.96501	−1.06395	−0.531976	+	0.846760i	$0.678550\pi$
$72$	0	0
$73$	9.82648	1.15010	0.575051	−	0.818117i	$-0.304982\pi$
$73$	9.82648	1.15010	0.575051	+	0.818117i	$0.304982\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.57249i	− 0.635044i
$78$	0	0
$79$	− 12.0673i	− 1.35768i	−0.734288	−	0.678838i	$-0.762484\pi$
$79$	− 12.0673i	− 1.35768i	0.734288	−	0.678838i	$-0.237516\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.0657560	0.00721766	0.00360883	−	0.999993i	$-0.498851\pi$
$83$	0.0657560	0.00721766	0.00360883	+	0.999993i	$0.498851\pi$
$84$	0	0
$85$	9.49778	1.03018
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.63713i	0.703534i	0.936088	+	0.351767i	$0.114419\pi$
$89$	6.63713i	0.703534i	−0.936088	+	0.351767i	$0.885581\pi$
$90$	0	0
$91$	4.17075i	0.437213i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.20659	−0.123793
$96$	0	0
$97$	6.76853	0.687240	0.343620	−	0.939109i	$-0.388347\pi$
$97$	6.76853	0.687240	0.343620	+	0.939109i	$0.388347\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.46961i	0.345239i	0.984989	+	0.172620i	$0.0552231\pi$
$101$	3.46961i	0.345239i	−0.984989	+	0.172620i	$0.944777\pi$
$102$	0	0
$103$	− 3.84578i	− 0.378936i	−0.981887	−	0.189468i	$-0.939324\pi$
$103$	− 3.84578i	− 0.378936i	0.981887	−	0.189468i	$-0.0606763\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.2686	−1.76610	−0.883048	−	0.469282i	$-0.844513\pi$
$107$	−18.2686	−1.76610	−0.883048	+	0.469282i	$0.844513\pi$
$108$	0	0
$109$	11.5354	1.10490	0.552448	−	0.833547i	$-0.313694\pi$
$109$	11.5354	1.10490	0.552448	+	0.833547i	$0.313694\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 10.6050i	− 0.997637i	−0.866707	−	0.498818i	$-0.833767\pi$
$113$	− 10.6050i	− 0.997637i	0.866707	−	0.498818i	$-0.166233\pi$
$114$	0	0
$115$	2.58566i	0.241114i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.37764	0.492967
$120$	0	0
$121$	9.33830	0.848937
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.4298i	− 1.02231i
$126$	0	0
$127$	7.96434i	0.706721i	0.935487	+	0.353360i	$0.114961\pi$
$127$	7.96434i	0.706721i	−0.935487	+	0.353360i	$0.885039\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.50748	−0.830672	−0.415336	−	0.909668i	$-0.636336\pi$
$131$	−9.50748	−0.830672	−0.415336	+	0.909668i	$0.636336\pi$
$132$	0	0
$133$	−0.683170	−0.0592383
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.2628i	0.962248i	0.876652	+	0.481124i	$0.159771\pi$
$137$	11.2628i	0.962248i	−0.876652	+	0.481124i	$0.840229\pi$
$138$	0	0
$139$	1.52683i	0.129504i	0.997901	+	0.0647520i	$0.0206256\pi$
$139$	1.52683i	0.129504i	−0.997901	+	0.0647520i	$0.979374\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−15.2223	−1.27295
$144$	0	0
$145$	−20.0993	−1.66916
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1.92572i	− 0.157761i	−0.996884	−	0.0788805i	$-0.974865\pi$
$149$	− 1.92572i	− 0.157761i	0.996884	−	0.0788805i	$-0.0251346\pi$
$150$	0	0
$151$	− 6.54903i	− 0.532953i	−0.963841	−	0.266476i	$-0.914141\pi$
$151$	− 6.54903i	− 0.532953i	0.963841	−	0.266476i	$-0.0858594\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.88280	−0.793805
$156$	0	0
$157$	−0.892522	−0.0712310	−0.0356155	−	0.999366i	$-0.511339\pi$
$157$	−0.892522	−0.0712310	−0.0356155	+	0.999366i	$0.511339\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.46400i	0.115379i
$162$	0	0
$163$	3.19931i	0.250589i	0.992120	+	0.125295i	$0.0399876\pi$
$163$	3.19931i	0.250589i	−0.992120	+	0.125295i	$0.960012\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.7181	−1.21630	−0.608150	−	0.793822i	$-0.708088\pi$
$167$	−15.7181	−1.21630	−0.608150	+	0.793822i	$0.708088\pi$
$168$	0	0
$169$	−1.60684	−0.123603
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 24.0907i	− 1.83158i	−0.401656	−	0.915791i	$-0.631565\pi$
$173$	− 24.0907i	− 1.83158i	0.401656	−	0.915791i	$-0.368435\pi$
$174$	0	0
$175$	− 0.293326i	− 0.0221733i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.61397	−0.344864	−0.172432	−	0.985021i	$-0.555163\pi$
$179$	−4.61397	−0.344864	−0.172432	+	0.985021i	$0.555163\pi$
$180$	0	0
$181$	−21.3152	−1.58435	−0.792174	−	0.610295i	$-0.791051\pi$
$181$	−21.3152	−1.58435	−0.792174	+	0.610295i	$0.791051\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.18234i	− 0.160449i
$186$	0	0
$187$	19.6271i	1.43528i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.99226	−0.650657	−0.325329	−	0.945601i	$-0.605475\pi$
$191$	−8.99226	−0.650657	−0.325329	+	0.945601i	$0.605475\pi$
$192$	0	0
$193$	−14.9430	−1.07562	−0.537810	−	0.843066i	$-0.680748\pi$
$193$	−14.9430	−1.07562	−0.537810	+	0.843066i	$0.680748\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.2496i	0.801499i	0.916188	+	0.400750i	$0.131250\pi$
$197$	11.2496i	0.801499i	−0.916188	+	0.400750i	$0.868750\pi$
$198$	0	0
$199$	− 12.4353i	− 0.881516i	−0.897626	−	0.440758i	$-0.854710\pi$
$199$	− 12.4353i	− 0.881516i	0.897626	−	0.440758i	$-0.145290\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−11.3802	−0.798734
$204$	0	0
$205$	−3.25889	−0.227611
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 2.49341i	− 0.172473i
$210$	0	0
$211$	− 20.0333i	− 1.37915i	−0.724215	−	0.689574i	$-0.757798\pi$
$211$	− 20.0333i	− 1.37915i	0.724215	−	0.689574i	$-0.242202\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.18246	0.626238
$216$	0	0
$217$	−5.59563	−0.379856
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 14.6900i	− 0.988156i
$222$	0	0
$223$	15.2697i	1.02253i	0.859422	+	0.511267i	$0.170824\pi$
$223$	15.2697i	1.02253i	−0.859422	+	0.511267i	$0.829176\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.74081	0.513776	0.256888	−	0.966441i	$-0.417303\pi$
$227$	7.74081	0.513776	0.256888	+	0.966441i	$0.417303\pi$
$228$	0	0
$229$	26.3871	1.74371	0.871856	−	0.489763i	$-0.162917\pi$
$229$	26.3871	1.74371	0.871856	+	0.489763i	$0.162917\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 0.807326i	− 0.0528897i	−0.999650	−	0.0264448i	$-0.991581\pi$
$233$	− 0.807326i	− 0.0528897i	0.999650	−	0.0264448i	$-0.00841863\pi$
$234$	0	0
$235$	− 13.8650i	− 0.904452i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−25.6622	−1.65995	−0.829976	−	0.557799i	$-0.811646\pi$
$239$	−25.6622	−1.65995	−0.829976	+	0.557799i	$0.811646\pi$
$240$	0	0
$241$	11.4585	0.738109	0.369055	−	0.929408i	$-0.379682\pi$
$241$	11.4585	0.738109	0.369055	+	0.929408i	$0.379682\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 11.9444i	− 0.763098i
$246$	0	0
$247$	1.86620i	0.118744i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.1648	0.957192	0.478596	−	0.878035i	$-0.341146\pi$
$251$	15.1648	0.957192	0.478596	+	0.878035i	$0.341146\pi$
$252$	0	0
$253$	−5.34326	−0.335928
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.64575i	− 0.289794i	−0.989447	−	0.144897i	$-0.953715\pi$
$257$	− 4.64575i	− 0.289794i	0.989447	−	0.144897i	$-0.0462851\pi$
$258$	0	0
$259$	− 1.23564i	− 0.0767789i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.38959	0.147349	0.0736743	−	0.997282i	$-0.476527\pi$
$263$	2.38959	0.147349	0.0736743	+	0.997282i	$0.476527\pi$
$264$	0	0
$265$	−3.06225	−0.188113
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.55467i	0.582559i	0.956638	+	0.291279i	$0.0940809\pi$
$269$	9.55467i	0.582559i	−0.956638	+	0.291279i	$0.905919\pi$
$270$	0	0
$271$	5.66987i	0.344420i	0.985060	+	0.172210i	$0.0550907\pi$
$271$	5.66987i	0.344420i	−0.985060	+	0.172210i	$0.944909\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.07057	0.0645579
$276$	0	0
$277$	−2.32650	−0.139786	−0.0698928	−	0.997555i	$-0.522266\pi$
$277$	−2.32650	−0.139786	−0.0698928	+	0.997555i	$0.522266\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 18.6773i	− 1.11419i	−0.830448	−	0.557096i	$-0.811915\pi$
$281$	− 18.6773i	− 1.11419i	0.830448	−	0.557096i	$-0.188085\pi$
$282$	0	0
$283$	− 2.65304i	− 0.157707i	−0.996886	−	0.0788533i	$-0.974874\pi$
$283$	− 2.65304i	− 0.157707i	0.996886	−	0.0788533i	$-0.0251259\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.84518	−0.108918
$288$	0	0
$289$	−1.94084	−0.114167
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 18.1773i	− 1.06193i	−0.847394	−	0.530965i	$-0.821830\pi$
$293$	− 18.1773i	− 1.06193i	0.847394	−	0.530965i	$-0.178170\pi$
$294$	0	0
$295$	10.3204i	0.600877i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.99918	0.231278
$300$	0	0
$301$	5.19910	0.299671
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.35701i	0.134962i
$306$	0	0
$307$	− 25.1776i	− 1.43696i	−0.695546	−	0.718482i	$-0.744837\pi$
$307$	− 25.1776i	− 1.43696i	0.695546	−	0.718482i	$-0.255163\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.8346	−0.614375	−0.307188	−	0.951649i	$-0.599388\pi$
$311$	−10.8346	−0.614375	−0.307188	+	0.951649i	$0.599388\pi$
$312$	0	0
$313$	−7.81994	−0.442009	−0.221005	−	0.975273i	$-0.570934\pi$
$313$	−7.81994	−0.442009	−0.221005	+	0.975273i	$0.570934\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.7656i	0.885484i	0.896649	+	0.442742i	$0.145994\pi$
$317$	15.7656i	0.885484i	−0.896649	+	0.442742i	$0.854006\pi$
$318$	0	0
$319$	− 41.5352i	− 2.32552i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.40623	0.133886
$324$	0	0
$325$	−0.801273	−0.0444466
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 7.85034i	− 0.432803i
$330$	0	0
$331$	− 35.0371i	− 1.92581i	−0.269833	−	0.962907i	$-0.586968\pi$
$331$	− 35.0371i	− 1.92581i	0.269833	−	0.962907i	$-0.413032\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−29.9916	−1.63862
$336$	0	0
$337$	10.8682	0.592030	0.296015	−	0.955183i	$-0.404342\pi$
$337$	10.8682	0.592030	0.296015	+	0.955183i	$0.404342\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 20.4228i	− 1.10595i
$342$	0	0
$343$	− 15.4124i	− 0.832190i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.507924	0.0272668	0.0136334	−	0.999907i	$-0.495660\pi$
$347$	0.507924	0.0272668	0.0136334	+	0.999907i	$0.495660\pi$
$348$	0	0
$349$	12.0237	0.643614	0.321807	−	0.946805i	$-0.395710\pi$
$349$	12.0237	0.643614	0.321807	+	0.946805i	$0.395710\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 6.91684i	− 0.368146i	−0.982913	−	0.184073i	$-0.941072\pi$
$353$	− 6.91684i	− 0.368146i	0.982913	−	0.184073i	$-0.0589284\pi$
$354$	0	0
$355$	19.5647i	1.03839i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.93817	0.260626	0.130313	−	0.991473i	$-0.458402\pi$
$359$	4.93817	0.260626	0.130313	+	0.991473i	$0.458402\pi$
$360$	0	0
$361$	18.6943	0.983911
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 21.4447i	− 1.12247i
$366$	0	0
$367$	33.8053i	1.76462i	0.470667	+	0.882311i	$0.344013\pi$
$367$	33.8053i	1.76462i	−0.470667	+	0.882311i	$0.655987\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.73385	−0.0900168
$372$	0	0
$373$	−3.48220	−0.180302	−0.0901508	−	0.995928i	$-0.528735\pi$
$373$	−3.48220	−0.180302	−0.0901508	+	0.995928i	$0.528735\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	31.0871i	1.60107i
$378$	0	0
$379$	19.5201i	1.00268i	0.865250	+	0.501340i	$0.167159\pi$
$379$	19.5201i	1.00268i	−0.865250	+	0.501340i	$0.832841\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−35.0875	−1.79289	−0.896444	−	0.443156i	$-0.853859\pi$
$383$	−35.0875	−1.79289	−0.896444	+	0.443156i	$0.853859\pi$
$384$	0	0
$385$	−12.1611	−0.619785
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.00971i	0.152598i	0.997085	+	0.0762992i	$0.0243104\pi$
$389$	3.00971i	0.152598i	−0.997085	+	0.0762992i	$0.975690\pi$
$390$	0	0
$391$	− 5.15642i	− 0.260771i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−26.3349	−1.32506
$396$	0	0
$397$	14.5037	0.727921	0.363961	−	0.931414i	$-0.381424\pi$
$397$	14.5037	0.727921	0.363961	+	0.931414i	$0.381424\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.68497i	0.0841433i	0.999115	+	0.0420716i	$0.0133958\pi$
$401$	1.68497i	0.0841433i	−0.999115	+	0.0420716i	$0.986604\pi$
$402$	0	0
$403$	15.2855i	0.761424i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.50980	0.223543
$408$	0	0
$409$	17.5215	0.866384	0.433192	−	0.901302i	$-0.357387\pi$
$409$	17.5215	0.866384	0.433192	+	0.901302i	$0.357387\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.84341i	0.287535i
$414$	0	0
$415$	− 0.143502i	− 0.00704424i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14.9519	−0.730449	−0.365224	−	0.930920i	$-0.619008\pi$
$419$	−14.9519	−0.730449	−0.365224	+	0.930920i	$0.619008\pi$
$420$	0	0
$421$	−30.4563	−1.48435	−0.742176	−	0.670205i	$-0.766206\pi$
$421$	−30.4563	−1.48435	−0.742176	+	0.670205i	$0.766206\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.03314i	0.0501145i
$426$	0	0
$427$	1.33454i	0.0645827i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.0995	1.49801	0.749005	−	0.662564i	$-0.230532\pi$
$431$	31.0995	1.49801	0.749005	+	0.662564i	$0.230532\pi$
$432$	0	0
$433$	12.5962	0.605336	0.302668	−	0.953096i	$-0.402123\pi$
$433$	12.5962	0.605336	0.302668	+	0.953096i	$0.402123\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.655067i	0.0313361i
$438$	0	0
$439$	− 8.08647i	− 0.385946i	−0.981204	−	0.192973i	$-0.938187\pi$
$439$	− 8.08647i	− 0.385946i	0.981204	−	0.192973i	$-0.0618130\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−38.3790	−1.82344	−0.911721	−	0.410809i	$-0.865246\pi$
$443$	−38.3790	−1.82344	−0.911721	+	0.410809i	$0.865246\pi$
$444$	0	0
$445$	14.4845	0.686630
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 0.0489300i	− 0.00230915i	−0.999999	−	0.00115457i	$-0.999632\pi$
$449$	− 0.0489300i	− 0.00230915i	0.999999	−	0.00115457i	$-0.000367512\pi$
$450$	0	0
$451$	− 6.73448i	− 0.317114i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9.10199	0.426708
$456$	0	0
$457$	34.2024	1.59992	0.799960	−	0.600053i	$-0.204854\pi$
$457$	34.2024	1.59992	0.799960	+	0.600053i	$0.204854\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 36.4899i	− 1.69950i	−0.527183	−	0.849752i	$-0.676752\pi$
$461$	− 36.4899i	− 1.69950i	0.527183	−	0.849752i	$-0.323248\pi$
$462$	0	0
$463$	28.9733i	1.34650i	0.739414	+	0.673251i	$0.235102\pi$
$463$	28.9733i	1.34650i	−0.739414	+	0.673251i	$0.764898\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.0475	0.696315	0.348158	−	0.937436i	$-0.386807\pi$
$467$	15.0475	0.696315	0.348158	+	0.937436i	$0.386807\pi$
$468$	0	0
$469$	−16.9812	−0.784120
$470$	0	0
$471$	0	0
$472$	0	0
$473$	18.9755i	0.872495i
$474$	0	0
$475$	− 0.131249i	− 0.00602211i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.00280	0.137201	0.0686007	−	0.997644i	$-0.478147\pi$
$479$	3.00280	0.137201	0.0686007	+	0.997644i	$0.478147\pi$
$480$	0	0
$481$	−3.37538	−0.153904
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 14.7712i	− 0.670728i
$486$	0	0
$487$	− 1.45189i	− 0.0657914i	−0.999459	−	0.0328957i	$-0.989527\pi$
$487$	− 1.45189i	− 0.0657914i	0.999459	−	0.0328957i	$-0.0104729\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.6514	0.706339	0.353169	−	0.935559i	$-0.385104\pi$
$491$	15.6514	0.706339	0.353169	+	0.935559i	$0.385104\pi$
$492$	0	0
$493$	40.0828	1.80524
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.0775i	0.496895i
$498$	0	0
$499$	− 33.4544i	− 1.49763i	−0.662782	−	0.748813i	$-0.730624\pi$
$499$	− 33.4544i	− 1.49763i	0.662782	−	0.748813i	$-0.269376\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.83589	0.260210	0.130105	−	0.991500i	$-0.458469\pi$
$503$	5.83589	0.260210	0.130105	+	0.991500i	$0.458469\pi$
$504$	0	0
$505$	7.57187	0.336944
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11.1972i	0.496307i	0.968721	+	0.248153i	$0.0798237\pi$
$509$	11.1972i	0.496307i	−0.968721	+	0.248153i	$0.920176\pi$
$510$	0	0
$511$	− 12.1420i	− 0.537130i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.39280	−0.369831
$516$	0	0
$517$	28.6519	1.26011
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.4453i	0.808105i	0.914736	+	0.404053i	$0.132399\pi$
$521$	18.4453i	0.808105i	−0.914736	+	0.404053i	$0.867601\pi$
$522$	0	0
$523$	− 16.1246i	− 0.705080i	−0.935797	−	0.352540i	$-0.885318\pi$
$523$	− 16.1246i	− 0.705080i	0.935797	−	0.352540i	$-0.114682\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19.7086	0.858522
$528$	0	0
$529$	−21.5962	−0.938966
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.04045i	0.218326i
$534$	0	0
$535$	39.8684i	1.72366i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	24.6830	1.06317
$540$	0	0
$541$	32.5280	1.39849	0.699244	−	0.714883i	$-0.253520\pi$
$541$	32.5280	1.39849	0.699244	+	0.714883i	$0.253520\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 25.1743i	− 1.07835i
$546$	0	0
$547$	− 8.67730i	− 0.371015i	−0.982643	−	0.185507i	$-0.940607\pi$
$547$	− 8.67730i	− 0.371015i	0.982643	−	0.185507i	$-0.0593929\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.09208	−0.216930
$552$	0	0
$553$	−14.9108	−0.634073
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.40358i	0.271328i	0.990755	+	0.135664i	$0.0433168\pi$
$557$	6.40358i	0.271328i	−0.990755	+	0.135664i	$0.956683\pi$
$558$	0	0
$559$	− 14.2023i	− 0.600693i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.18571	0.302841	0.151421	−	0.988469i	$-0.451615\pi$
$563$	7.18571	0.302841	0.151421	+	0.988469i	$0.451615\pi$
$564$	0	0
$565$	−23.1438	−0.973666
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.2585i	0.430060i	0.976607	+	0.215030i	$0.0689850\pi$
$569$	10.2585i	0.430060i	−0.976607	+	0.215030i	$0.931015\pi$
$570$	0	0
$571$	− 29.2821i	− 1.22542i	−0.790308	−	0.612709i	$-0.790080\pi$
$571$	− 29.2821i	− 1.22542i	0.790308	−	0.612709i	$-0.209920\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.281260	−0.0117293
$576$	0	0
$577$	−16.3893	−0.682296	−0.341148	−	0.940010i	$-0.610816\pi$
$577$	−16.3893	−0.682296	−0.341148	+	0.940010i	$0.610816\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 0.0812508i	− 0.00337085i
$582$	0	0
$583$	− 6.32814i	− 0.262085i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.6627	1.55450	0.777252	−	0.629189i	$-0.216613\pi$
$587$	37.6627	1.55450	0.777252	+	0.629189i	$0.216613\pi$
$588$	0	0
$589$	−2.50377	−0.103166
$590$	0	0
$591$	0	0
$592$	0	0
$593$	38.5714i	1.58394i	0.610562	+	0.791969i	$0.290944\pi$
$593$	38.5714i	1.58394i	−0.610562	+	0.791969i	$0.709056\pi$
$594$	0	0
$595$	− 11.7358i	− 0.481122i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−14.1033	−0.576247	−0.288123	−	0.957593i	$-0.593031\pi$
$599$	−14.1033	−0.576247	−0.288123	+	0.957593i	$0.593031\pi$
$600$	0	0
$601$	−26.4866	−1.08041	−0.540205	−	0.841534i	$-0.681653\pi$
$601$	−26.4866	−1.08041	−0.540205	+	0.841534i	$0.681653\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 20.3794i	− 0.828539i
$606$	0	0
$607$	− 12.3882i	− 0.502822i	−0.967880	−	0.251411i	$-0.919105\pi$
$607$	− 12.3882i	− 0.502822i	0.967880	−	0.251411i	$-0.0808945\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.4446	−0.867557
$612$	0	0
$613$	28.6781	1.15830	0.579149	−	0.815222i	$-0.303385\pi$
$613$	28.6781	1.15830	0.579149	+	0.815222i	$0.303385\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.1371i	0.931466i	0.884925	+	0.465733i	$0.154209\pi$
$617$	23.1371i	0.931466i	−0.884925	+	0.465733i	$0.845791\pi$
$618$	0	0
$619$	33.5827i	1.34980i	0.737907	+	0.674902i	$0.235814\pi$
$619$	33.5827i	1.34980i	−0.737907	+	0.674902i	$0.764186\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.20109	0.328570
$624$	0	0
$625$	−23.7567	−0.950268
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.35211i	0.173530i
$630$	0	0
$631$	− 21.3319i	− 0.849208i	−0.905379	−	0.424604i	$-0.860413\pi$
$631$	− 21.3319i	− 0.849208i	0.905379	−	0.424604i	$-0.139587\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	17.3809	0.689740
$636$	0	0
$637$	−18.4741	−0.731970
$638$	0	0
$639$	0	0
$640$	0	0
$641$	14.2087i	0.561209i	0.959824	+	0.280604i	$0.0905349\pi$
$641$	14.2087i	0.561209i	−0.959824	+	0.280604i	$0.909465\pi$
$642$	0	0
$643$	− 2.97282i	− 0.117236i	−0.998280	−	0.0586182i	$-0.981331\pi$
$643$	− 2.97282i	− 0.117236i	0.998280	−	0.0586182i	$-0.0186695\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	43.5445	1.71191	0.855956	−	0.517049i	$-0.172969\pi$
$647$	43.5445	1.71191	0.855956	+	0.517049i	$0.172969\pi$
$648$	0	0
$649$	−21.3271	−0.837162
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.0959i	0.395085i	0.980294	+	0.197543i	$0.0632960\pi$
$653$	10.0959i	0.395085i	−0.980294	+	0.197543i	$0.936704\pi$
$654$	0	0
$655$	20.7486i	0.810714i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.32666	0.129588	0.0647942	−	0.997899i	$-0.479361\pi$
$659$	3.32666	0.129588	0.0647942	+	0.997899i	$0.479361\pi$
$660$	0	0
$661$	−27.5317	−1.07086	−0.535430	−	0.844579i	$-0.679851\pi$
$661$	−27.5317	−1.07086	−0.535430	+	0.844579i	$0.679851\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.49091i	0.0578150i
$666$	0	0
$667$	10.9121i	0.422517i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.87075	−0.188033
$672$	0	0
$673$	−16.1999	−0.624462	−0.312231	−	0.950006i	$-0.601076\pi$
$673$	−16.1999	−0.624462	−0.312231	+	0.950006i	$0.601076\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23.0184i	0.884670i	0.896850	+	0.442335i	$0.145850\pi$
$677$	23.0184i	0.884670i	−0.896850	+	0.442335i	$0.854150\pi$
$678$	0	0
$679$	− 8.36347i	− 0.320960i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21.5353	0.824026	0.412013	−	0.911178i	$-0.364826\pi$
$683$	21.5353	0.824026	0.412013	+	0.911178i	$0.364826\pi$
$684$	0	0
$685$	24.5793	0.939128
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.73632i	0.180439i
$690$	0	0
$691$	− 37.5267i	− 1.42758i	−0.700359	−	0.713791i	$-0.746977\pi$
$691$	− 37.5267i	− 1.42758i	0.700359	−	0.713791i	$-0.253023\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.33206	0.126392
$696$	0	0
$697$	6.49900	0.246167
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 5.50003i	− 0.207733i	−0.994591	−	0.103867i	$-0.966878\pi$
$701$	− 5.50003i	− 0.207733i	0.994591	−	0.103867i	$-0.0331215\pi$
$702$	0	0
$703$	− 0.552888i	− 0.0208526i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.28719	0.161236
$708$	0	0
$709$	33.3651	1.25305	0.626526	−	0.779401i	$-0.284476\pi$
$709$	33.3651	1.25305	0.626526	+	0.779401i	$0.284476\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.36545i	0.200938i
$714$	0	0
$715$	33.2202i	1.24236i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7.96155	0.296916	0.148458	−	0.988919i	$-0.452569\pi$
$719$	7.96155	0.296916	0.148458	+	0.988919i	$0.452569\pi$
$720$	0	0
$721$	−4.75200	−0.176974
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.18634i	− 0.0811985i
$726$	0	0
$727$	38.3533i	1.42244i	0.702968	+	0.711222i	$0.251858\pi$
$727$	38.3533i	1.42244i	−0.702968	+	0.711222i	$0.748142\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−18.3120	−0.677294
$732$	0	0
$733$	40.6875	1.50283	0.751413	−	0.659833i	$-0.229373\pi$
$733$	40.6875	1.50283	0.751413	+	0.659833i	$0.229373\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 61.9776i	− 2.28297i
$738$	0	0
$739$	23.1444i	0.851380i	0.904869	+	0.425690i	$0.139969\pi$
$739$	23.1444i	0.851380i	−0.904869	+	0.425690i	$0.860031\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.60498	−0.168940	−0.0844701	−	0.996426i	$-0.526920\pi$
$743$	−4.60498	−0.168940	−0.0844701	+	0.996426i	$0.526920\pi$
$744$	0	0
$745$	−4.20257	−0.153970
$746$	0	0
$747$	0	0
$748$	0	0
$749$	22.5735i	0.824816i
$750$	0	0
$751$	− 40.1504i	− 1.46511i	−0.680708	−	0.732555i	$-0.738328\pi$
$751$	− 40.1504i	− 1.46511i	0.680708	−	0.732555i	$-0.261672\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−14.2922	−0.520147
$756$	0	0
$757$	−43.3039	−1.57391	−0.786954	−	0.617011i	$-0.788343\pi$
$757$	−43.3039	−1.57391	−0.786954	+	0.617011i	$0.788343\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	50.6204i	1.83499i	0.397751	+	0.917493i	$0.369791\pi$
$761$	50.6204i	1.83499i	−0.397751	+	0.917493i	$0.630209\pi$
$762$	0	0
$763$	− 14.2537i	− 0.516017i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	15.9623	0.576366
$768$	0	0
$769$	−45.3086	−1.63387	−0.816935	−	0.576729i	$-0.804329\pi$
$769$	−45.3086	−1.63387	−0.816935	+	0.576729i	$0.804329\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 41.7432i	− 1.50140i	−0.660644	−	0.750699i	$-0.729717\pi$
$773$	− 41.7432i	− 1.50140i	0.660644	−	0.750699i	$-0.270283\pi$
$774$	0	0
$775$	− 1.07502i	− 0.0386158i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.825627	−0.0295812
$780$	0	0
$781$	−40.4304	−1.44671
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.94779i	0.0695195i
$786$	0	0
$787$	− 8.00343i	− 0.285291i	−0.989774	−	0.142646i	$-0.954439\pi$
$787$	− 8.00343i	− 0.285291i	0.989774	−	0.142646i	$-0.0455610\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13.1040	−0.465924
$792$	0	0
$793$	3.64553	0.129457
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19.8459i	− 0.702977i	−0.936192	−	0.351488i	$-0.885676\pi$
$797$	− 19.8459i	− 0.702977i	0.936192	−	0.351488i	$-0.114324\pi$
$798$	0	0
$799$	27.6501i	0.978189i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	44.3155	1.56386
$804$	0	0
$805$	3.19494	0.112607
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 16.3325i	− 0.574220i	−0.957898	−	0.287110i	$-0.907305\pi$
$809$	− 16.3325i	− 0.574220i	0.957898	−	0.287110i	$-0.0926945\pi$
$810$	0	0
$811$	− 1.53456i	− 0.0538858i	−0.999637	−	0.0269429i	$-0.991423\pi$
$811$	− 1.53456i	− 0.0538858i	0.999637	−	0.0269429i	$-0.00857723\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.98199	0.244568
$816$	0	0
$817$	2.32634	0.0813883
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 8.40723i	− 0.293414i	−0.989180	−	0.146707i	$-0.953132\pi$
$821$	− 8.40723i	− 0.293414i	0.989180	−	0.146707i	$-0.0468675\pi$
$822$	0	0
$823$	− 28.4061i	− 0.990173i	−0.868844	−	0.495087i	$-0.835136\pi$
$823$	− 28.4061i	− 0.990173i	0.868844	−	0.495087i	$-0.164864\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30.4070	−1.05736	−0.528678	−	0.848822i	$-0.677312\pi$
$827$	−30.4070	−1.05736	−0.528678	+	0.848822i	$0.677312\pi$
$828$	0	0
$829$	41.3887	1.43749	0.718745	−	0.695274i	$-0.244717\pi$
$829$	41.3887	1.43749	0.718745	+	0.695274i	$0.244717\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	23.8199i	0.825312i
$834$	0	0
$835$	34.3022i	1.18708i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.6553	1.05834	0.529170	−	0.848516i	$-0.322503\pi$
$839$	30.6553	1.05834	0.529170	+	0.848516i	$0.322503\pi$
$840$	0	0
$841$	−55.8236	−1.92495
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.50668i	0.120633i
$846$	0	0
$847$	− 11.5388i	− 0.396477i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.18481	−0.0406148
$852$	0	0
$853$	6.78842	0.232431	0.116215	−	0.993224i	$-0.462924\pi$
$853$	6.78842	0.232431	0.116215	+	0.993224i	$0.462924\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.6455i	1.35427i	0.735861	+	0.677133i	$0.236778\pi$
$857$	39.6455i	1.35427i	−0.735861	+	0.677133i	$0.763222\pi$
$858$	0	0
$859$	44.1955i	1.50793i	0.656915	+	0.753965i	$0.271861\pi$
$859$	44.1955i	1.50793i	−0.656915	+	0.753965i	$0.728139\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16.6561	0.566980	0.283490	−	0.958975i	$-0.408508\pi$
$863$	16.6561	0.566980	0.283490	+	0.958975i	$0.408508\pi$
$864$	0	0
$865$	−52.5741	−1.78757
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 54.4211i	− 1.84611i
$870$	0	0
$871$	46.3873i	1.57177i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−14.1231	−0.477447
$876$	0	0
$877$	39.2125	1.32411	0.662056	−	0.749454i	$-0.269684\pi$
$877$	39.2125	1.32411	0.662056	+	0.749454i	$0.269684\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 31.4981i	− 1.06120i	−0.847623	−	0.530600i	$-0.821967\pi$
$881$	− 31.4981i	− 1.06120i	0.847623	−	0.530600i	$-0.178033\pi$
$882$	0	0
$883$	− 15.6180i	− 0.525588i	−0.964852	−	0.262794i	$-0.915356\pi$
$883$	− 15.6180i	− 0.525588i	0.964852	−	0.262794i	$-0.0846439\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−23.9018	−0.802544	−0.401272	−	0.915959i	$-0.631432\pi$
$887$	−23.9018	−0.802544	−0.401272	+	0.915959i	$0.631432\pi$
$888$	0	0
$889$	9.84105	0.330058
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 3.51264i	− 0.117546i
$894$	0	0
$895$	10.0693i	0.336578i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−41.7076	−1.39103
$900$	0	0
$901$	6.10686	0.203449
$902$	0	0
$903$	0	0
$904$	0	0
$905$	46.5171i	1.54628i
$906$	0	0
$907$	17.6400i	0.585727i	0.956154	+	0.292863i	$0.0946081\pi$
$907$	17.6400i	0.585727i	−0.956154	+	0.292863i	$0.905392\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−14.0211	−0.464540	−0.232270	−	0.972651i	$-0.574615\pi$
$911$	−14.0211	−0.464540	−0.232270	+	0.972651i	$0.574615\pi$
$912$	0	0
$913$	0.296547	0.00981426
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.7478i	0.387947i
$918$	0	0
$919$	− 10.5677i	− 0.348598i	−0.984693	−	0.174299i	$-0.944234\pi$
$919$	− 10.5677i	− 0.348598i	0.984693	−	0.174299i	$-0.0557659\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	30.2603	0.996029
$924$	0	0
$925$	0.237388	0.00780527
$926$	0	0
$927$	0	0
$928$	0	0
$929$	12.9352i	0.424389i	0.977227	+	0.212195i	$0.0680611\pi$
$929$	12.9352i	0.424389i	−0.977227	+	0.212195i	$0.931939\pi$
$930$	0	0
$931$	− 3.02606i	− 0.0991752i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	42.8331	1.40079
$936$	0	0
$937$	−33.3145	−1.08834	−0.544168	−	0.838976i	$-0.683155\pi$
$937$	−33.3145	−1.08834	−0.544168	+	0.838976i	$0.683155\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	9.67059i	0.315252i	0.987499	+	0.157626i	$0.0503841\pi$
$941$	9.67059i	0.315252i	−0.987499	+	0.157626i	$0.949616\pi$
$942$	0	0
$943$	1.76928i	0.0576156i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.38345	−0.0774516	−0.0387258	−	0.999250i	$-0.512330\pi$
$947$	−2.38345	−0.0774516	−0.0387258	+	0.999250i	$0.512330\pi$
$948$	0	0
$949$	−33.1681	−1.07668
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 7.99617i	− 0.259021i	−0.991578	−	0.129511i	$-0.958659\pi$
$953$	− 7.99617i	− 0.259021i	0.991578	−	0.129511i	$-0.0413407\pi$
$954$	0	0
$955$	19.6242i	0.635023i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13.9168	0.449397
$960$	0	0
$961$	10.4924	0.338465
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32.6107i	1.04977i
$966$	0	0
$967$	13.8330i	0.444839i	0.974951	+	0.222420i	$0.0713955\pi$
$967$	13.8330i	0.444839i	−0.974951	+	0.222420i	$0.928604\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	37.2497	1.19540	0.597700	−	0.801720i	$-0.296081\pi$
$971$	37.2497	1.19540	0.597700	+	0.801720i	$0.296081\pi$
$972$	0	0
$973$	1.88661	0.0604819
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.34921i	0.0431651i	0.999767	+	0.0215826i	$0.00687047\pi$
$977$	1.34921i	0.0431651i	−0.999767	+	0.0215826i	$0.993130\pi$
$978$	0	0
$979$	29.9321i	0.956634i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	45.2031	1.44175	0.720877	−	0.693063i	$-0.243739\pi$
$983$	45.2031	1.44175	0.720877	+	0.693063i	$0.243739\pi$
$984$	0	0
$985$	24.5504	0.782241
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 4.98523i	− 0.158521i
$990$	0	0
$991$	− 13.0405i	− 0.414245i	−0.978315	−	0.207123i	$-0.933590\pi$
$991$	− 13.0405i	− 0.414245i	0.978315	−	0.207123i	$-0.0664099\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−27.1381	−0.860336
$996$	0	0
$997$	14.7383	0.466766	0.233383	−	0.972385i	$-0.425020\pi$
$997$	14.7383	0.466766	0.233383	+	0.972385i	$0.425020\pi$
$998$	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.2591.5	✓	24
3.2	odd	2	inner	5328.2.e.g.2591.19	yes	24
4.3	odd	2	inner	5328.2.e.g.2591.6	yes	24
12.11	even	2	inner	5328.2.e.g.2591.20	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5328.2.e.g.2591.5	✓	24	1.1	even	1	trivial
5328.2.e.g.2591.6	yes	24	4.3	odd	2	inner
5328.2.e.g.2591.19	yes	24	3.2	odd	2	inner
5328.2.e.g.2591.20	yes	24	12.11	even	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 $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q-2.18234i q^{5} -1.23564i q^{7} +4.50980 q^{11} -3.37538 q^{13} +4.35211i q^{17} -0.552888i q^{19} -1.18481 q^{23} +0.237388 q^{25} -9.20998i q^{29} -4.52853i q^{31} -2.69659 q^{35} +1.00000 q^{37} -1.49330i q^{41} +4.20762i q^{43} +6.35326 q^{47} +5.47320 q^{49} -1.40320i q^{53} -9.84192i q^{55} -4.72905 q^{59} -1.08004 q^{61} +7.36622i q^{65} -13.7429i q^{67} -8.96501 q^{71} +9.82648 q^{73} -5.57249i q^{77} -12.0673i q^{79} +0.0657560 q^{83} +9.49778 q^{85} +6.63713i q^{89} +4.17075i q^{91} -1.20659 q^{95} +6.76853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 16 q^{13} + 24 q^{37} - 16 q^{49} + 48 q^{61} + 48 q^{73} + 56 q^{85} - 8 q^{97}+O(q^{100}) \) 24 * q + 16 * q^13 + 24 * q^37 - 16 * q^49 + 48 * q^61 + 48 * q^73 + 56 * q^85 - 8 * q^97

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