LMFDB - Embedded newform 5328.2.e.g.2591.3

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			2591.3
Character	$\chi$	$=$	5328.2591
Dual form			5328.2.e.g.2591.22

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.87880i q^{5} -1.02706i q^{7} +2.36524 q^{11} +4.27524 q^{13} -4.53288i q^{17} -5.45103i q^{19} +2.13121 q^{23} -3.28750 q^{25} +7.35138i q^{29} -7.28234i q^{31} -2.95671 q^{35} +1.00000 q^{37} -2.15844i q^{41} -11.3753i q^{43} -5.71996 q^{47} +5.94514 q^{49} +6.21282i q^{53} -6.80906i q^{55} -6.49046 q^{59} +5.33255 q^{61} -12.3076i q^{65} +15.8813i q^{67} +13.9297 q^{71} +11.3473 q^{73} -2.42925i q^{77} +12.8673i q^{79} -2.04396 q^{83} -13.0492 q^{85} +17.0077i q^{89} -4.39095i q^{91} -15.6924 q^{95} +6.00245 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 2.87880i	− 1.28744i	−0.765262	−	0.643720i	$-0.777390\pi$
$5$	− 2.87880i	− 1.28744i	0.765262	−	0.643720i	$-0.222610\pi$
$6$	0	0
$7$	− 1.02706i	− 0.388194i	−0.980982	−	0.194097i	$-0.937822\pi$
$7$	− 1.02706i	− 0.388194i	0.980982	−	0.194097i	$-0.0621776\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.36524	0.713147	0.356573	−	0.934267i	$-0.383945\pi$
$11$	2.36524	0.713147	0.356573	+	0.934267i	$0.383945\pi$
$12$	0	0
$13$	4.27524	1.18574	0.592869	−	0.805299i	$-0.297995\pi$
$13$	4.27524	1.18574	0.592869	+	0.805299i	$0.297995\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.53288i	− 1.09938i	−0.835367	−	0.549692i	$-0.814745\pi$
$17$	− 4.53288i	− 1.09938i	0.835367	−	0.549692i	$-0.185255\pi$
$18$	0	0
$19$	− 5.45103i	− 1.25055i	−0.780404	−	0.625276i	$-0.784987\pi$
$19$	− 5.45103i	− 1.25055i	0.780404	−	0.625276i	$-0.215013\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.13121	0.444388	0.222194	−	0.975002i	$-0.428678\pi$
$23$	2.13121	0.444388	0.222194	+	0.975002i	$0.428678\pi$
$24$	0	0
$25$	−3.28750	−0.657499
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.35138i	1.36512i	0.730831	+	0.682558i	$0.239133\pi$
$29$	7.35138i	1.36512i	−0.730831	+	0.682558i	$0.760867\pi$
$30$	0	0
$31$	− 7.28234i	− 1.30795i	−0.756517	−	0.653974i	$-0.773101\pi$
$31$	− 7.28234i	− 1.30795i	0.756517	−	0.653974i	$-0.226899\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.95671	−0.499776
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.15844i	− 0.337092i	−0.985694	−	0.168546i	$-0.946093\pi$
$41$	− 2.15844i	− 0.337092i	0.985694	−	0.168546i	$-0.0539071\pi$
$42$	0	0
$43$	− 11.3753i	− 1.73472i	−0.497679	−	0.867361i	$-0.665814\pi$
$43$	− 11.3753i	− 1.73472i	0.497679	−	0.867361i	$-0.334186\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.71996	−0.834342	−0.417171	−	0.908828i	$-0.636978\pi$
$47$	−5.71996	−0.834342	−0.417171	+	0.908828i	$0.636978\pi$
$48$	0	0
$49$	5.94514	0.849306
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.21282i	0.853397i	0.904394	+	0.426699i	$0.140323\pi$
$53$	6.21282i	0.853397i	−0.904394	+	0.426699i	$0.859677\pi$
$54$	0	0
$55$	− 6.80906i	− 0.918133i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.49046	−0.844985	−0.422493	−	0.906366i	$-0.638845\pi$
$59$	−6.49046	−0.844985	−0.422493	+	0.906366i	$0.638845\pi$
$60$	0	0
$61$	5.33255	0.682763	0.341382	−	0.939925i	$-0.389105\pi$
$61$	5.33255	0.682763	0.341382	+	0.939925i	$0.389105\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 12.3076i	− 1.52657i
$66$	0	0
$67$	15.8813i	1.94021i	0.242678	+	0.970107i	$0.421974\pi$
$67$	15.8813i	1.94021i	−0.242678	+	0.970107i	$0.578026\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.9297	1.65315	0.826576	−	0.562825i	$-0.190286\pi$
$71$	13.9297	1.65315	0.826576	+	0.562825i	$0.190286\pi$
$72$	0	0
$73$	11.3473	1.32810	0.664048	−	0.747690i	$-0.268837\pi$
$73$	11.3473	1.32810	0.664048	+	0.747690i	$0.268837\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.42925i	− 0.276839i
$78$	0	0
$79$	12.8673i	1.44769i	0.689963	+	0.723845i	$0.257627\pi$
$79$	12.8673i	1.44769i	−0.689963	+	0.723845i	$0.742373\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.04396	−0.224354	−0.112177	−	0.993688i	$-0.535782\pi$
$83$	−2.04396	−0.224354	−0.112177	+	0.993688i	$0.535782\pi$
$84$	0	0
$85$	−13.0492	−1.41539
$86$	0	0
$87$	0	0
$88$	0	0
$89$	17.0077i	1.80281i	0.432978	+	0.901405i	$0.357463\pi$
$89$	17.0077i	1.80281i	−0.432978	+	0.901405i	$0.642537\pi$
$90$	0	0
$91$	− 4.39095i	− 0.460296i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−15.6924	−1.61001
$96$	0	0
$97$	6.00245	0.609456	0.304728	−	0.952439i	$-0.401434\pi$
$97$	6.00245	0.609456	0.304728	+	0.952439i	$0.401434\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 18.9877i	− 1.88935i	−0.328008	−	0.944675i	$-0.606377\pi$
$101$	− 18.9877i	− 1.88935i	0.328008	−	0.944675i	$-0.393623\pi$
$102$	0	0
$103$	− 11.7063i	− 1.15346i	−0.816936	−	0.576728i	$-0.804329\pi$
$103$	− 11.7063i	− 1.15346i	0.816936	−	0.576728i	$-0.195671\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.42913	0.138159	0.0690794	−	0.997611i	$-0.477994\pi$
$107$	1.42913	0.138159	0.0690794	+	0.997611i	$0.477994\pi$
$108$	0	0
$109$	−8.94035	−0.856330	−0.428165	−	0.903701i	$-0.640840\pi$
$109$	−8.94035	−0.856330	−0.428165	+	0.903701i	$0.640840\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.6194i	1.75157i	0.482704	+	0.875783i	$0.339655\pi$
$113$	18.6194i	1.75157i	−0.482704	+	0.875783i	$0.660345\pi$
$114$	0	0
$115$	− 6.13533i	− 0.572123i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.65555	−0.426774
$120$	0	0
$121$	−5.40564	−0.491422
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 4.92996i	− 0.440949i
$126$	0	0
$127$	− 12.5922i	− 1.11738i	−0.829378	−	0.558688i	$-0.811305\pi$
$127$	− 12.5922i	− 1.11738i	0.829378	−	0.558688i	$-0.188695\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.33975	−0.204424	−0.102212	−	0.994763i	$-0.532592\pi$
$131$	−2.33975	−0.204424	−0.102212	+	0.994763i	$0.532592\pi$
$132$	0	0
$133$	−5.59855	−0.485456
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.62741i	− 0.480782i	−0.970676	−	0.240391i	$-0.922724\pi$
$137$	− 5.62741i	− 0.480782i	0.970676	−	0.240391i	$-0.0772757\pi$
$138$	0	0
$139$	14.0276i	1.18981i	0.803796	+	0.594905i	$0.202810\pi$
$139$	14.0276i	1.18981i	−0.803796	+	0.594905i	$0.797190\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.1120	0.845606
$144$	0	0
$145$	21.1632	1.75750
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 17.7929i	− 1.45765i	−0.684699	−	0.728826i	$-0.740066\pi$
$149$	− 17.7929i	− 1.45765i	0.684699	−	0.728826i	$-0.259934\pi$
$150$	0	0
$151$	− 0.553778i	− 0.0450659i	−0.999746	−	0.0225329i	$-0.992827\pi$
$151$	− 0.553778i	− 0.0450659i	0.999746	−	0.0225329i	$-0.00717306\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−20.9644	−1.68390
$156$	0	0
$157$	22.2917	1.77907	0.889535	−	0.456867i	$-0.151028\pi$
$157$	22.2917	1.77907	0.889535	+	0.456867i	$0.151028\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2.18889i	− 0.172509i
$162$	0	0
$163$	− 11.0998i	− 0.869402i	−0.900575	−	0.434701i	$-0.856854\pi$
$163$	− 11.0998i	− 0.869402i	0.900575	−	0.434701i	$-0.143146\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.4119	−1.65690	−0.828449	−	0.560064i	$-0.810777\pi$
$167$	−21.4119	−1.65690	−0.828449	+	0.560064i	$0.810777\pi$
$168$	0	0
$169$	5.27769	0.405976
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.15296i	− 0.0876582i	−0.999039	−	0.0438291i	$-0.986044\pi$
$173$	− 1.15296i	− 0.0876582i	0.999039	−	0.0438291i	$-0.0139557\pi$
$174$	0	0
$175$	3.37647i	0.255237i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.81773	−0.210607	−0.105303	−	0.994440i	$-0.533581\pi$
$179$	−2.81773	−0.210607	−0.105303	+	0.994440i	$0.533581\pi$
$180$	0	0
$181$	3.48649	0.259149	0.129574	−	0.991570i	$-0.458639\pi$
$181$	3.48649	0.259149	0.129574	+	0.991570i	$0.458639\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.87880i	− 0.211654i
$186$	0	0
$187$	− 10.7213i	− 0.784022i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.27089	−0.670818	−0.335409	−	0.942073i	$-0.608874\pi$
$191$	−9.27089	−0.670818	−0.335409	+	0.942073i	$0.608874\pi$
$192$	0	0
$193$	−16.0154	−1.15281	−0.576406	−	0.817163i	$-0.695545\pi$
$193$	−16.0154	−1.15281	−0.576406	+	0.817163i	$0.695545\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.61938i	− 0.186623i	−0.995637	−	0.0933114i	$-0.970255\pi$
$197$	− 2.61938i	− 0.186623i	0.995637	−	0.0933114i	$-0.0297452\pi$
$198$	0	0
$199$	− 8.27789i	− 0.586804i	−0.955989	−	0.293402i	$-0.905213\pi$
$199$	− 8.27789i	− 0.586804i	0.955989	−	0.293402i	$-0.0947875\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.55034	0.529930
$204$	0	0
$205$	−6.21372	−0.433985
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 12.8930i	− 0.891826i
$210$	0	0
$211$	− 14.3354i	− 0.986892i	−0.869776	−	0.493446i	$-0.835737\pi$
$211$	− 14.3354i	− 0.986892i	0.869776	−	0.493446i	$-0.164263\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−32.7473	−2.23335
$216$	0	0
$217$	−7.47943	−0.507737
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 19.3791i	− 1.30358i
$222$	0	0
$223$	− 1.85369i	− 0.124132i	−0.998072	−	0.0620662i	$-0.980231\pi$
$223$	− 1.85369i	− 0.124132i	0.998072	−	0.0620662i	$-0.0197690\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.60609	−0.571207	−0.285603	−	0.958348i	$-0.592194\pi$
$227$	−8.60609	−0.571207	−0.285603	+	0.958348i	$0.592194\pi$
$228$	0	0
$229$	−4.30067	−0.284196	−0.142098	−	0.989853i	$-0.545385\pi$
$229$	−4.30067	−0.284196	−0.142098	+	0.989853i	$0.545385\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	25.8191i	1.69147i	0.533606	+	0.845733i	$0.320837\pi$
$233$	25.8191i	1.69147i	−0.533606	+	0.845733i	$0.679163\pi$
$234$	0	0
$235$	16.4666i	1.07416i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.05918	−0.585990	−0.292995	−	0.956114i	$-0.594652\pi$
$239$	−9.05918	−0.585990	−0.292995	+	0.956114i	$0.594652\pi$
$240$	0	0
$241$	−5.63802	−0.363177	−0.181588	−	0.983375i	$-0.558124\pi$
$241$	−5.63802	−0.363177	−0.181588	+	0.983375i	$0.558124\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 17.1149i	− 1.09343i
$246$	0	0
$247$	− 23.3045i	− 1.48283i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−25.8188	−1.62967	−0.814835	−	0.579693i	$-0.803173\pi$
$251$	−25.8188	−1.62967	−0.814835	+	0.579693i	$0.803173\pi$
$252$	0	0
$253$	5.04083	0.316914
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.00912i	− 0.374839i	−0.982280	−	0.187419i	$-0.939988\pi$
$257$	− 6.00912i	− 0.374839i	0.982280	−	0.187419i	$-0.0600123\pi$
$258$	0	0
$259$	− 1.02706i	− 0.0638187i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.38164	−0.578497	−0.289248	−	0.957254i	$-0.593405\pi$
$263$	−9.38164	−0.578497	−0.289248	+	0.957254i	$0.593405\pi$
$264$	0	0
$265$	17.8855	1.09870
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 23.5675i	− 1.43694i	−0.695560	−	0.718468i	$-0.744843\pi$
$269$	− 23.5675i	− 1.43694i	0.695560	−	0.718468i	$-0.255157\pi$
$270$	0	0
$271$	6.54371i	0.397502i	0.980050	+	0.198751i	$0.0636885\pi$
$271$	6.54371i	0.397502i	−0.980050	+	0.198751i	$0.936311\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.77572	−0.468893
$276$	0	0
$277$	−22.6298	−1.35969	−0.679847	−	0.733354i	$-0.737954\pi$
$277$	−22.6298	−1.35969	−0.679847	+	0.733354i	$0.737954\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	24.2015i	1.44374i	0.692028	+	0.721871i	$0.256718\pi$
$281$	24.2015i	1.44374i	−0.692028	+	0.721871i	$0.743282\pi$
$282$	0	0
$283$	18.8597i	1.12109i	0.828124	+	0.560545i	$0.189408\pi$
$283$	18.8597i	1.12109i	−0.828124	+	0.560545i	$0.810592\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.21686	−0.130857
$288$	0	0
$289$	−3.54696	−0.208645
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 21.3325i	− 1.24626i	−0.782119	−	0.623129i	$-0.785861\pi$
$293$	− 21.3325i	− 1.24626i	0.782119	−	0.623129i	$-0.214139\pi$
$294$	0	0
$295$	18.6847i	1.08787i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.11144	0.526928
$300$	0	0
$301$	−11.6832	−0.673409
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 15.3514i	− 0.879016i
$306$	0	0
$307$	16.6176i	0.948416i	0.880413	+	0.474208i	$0.157265\pi$
$307$	16.6176i	0.948416i	−0.880413	+	0.474208i	$0.842735\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−27.0819	−1.53567	−0.767836	−	0.640647i	$-0.778666\pi$
$311$	−27.0819	−1.53567	−0.767836	+	0.640647i	$0.778666\pi$
$312$	0	0
$313$	−2.34550	−0.132575	−0.0662877	−	0.997801i	$-0.521116\pi$
$313$	−2.34550	−0.132575	−0.0662877	+	0.997801i	$0.521116\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.00052i	0.505520i	0.967529	+	0.252760i	$0.0813382\pi$
$317$	9.00052i	0.505520i	−0.967529	+	0.252760i	$0.918662\pi$
$318$	0	0
$319$	17.3878i	0.973529i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−24.7088	−1.37484
$324$	0	0
$325$	−14.0548	−0.779622
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.87477i	0.323886i
$330$	0	0
$331$	21.6315i	1.18897i	0.804106	+	0.594486i	$0.202645\pi$
$331$	21.6315i	1.18897i	−0.804106	+	0.594486i	$0.797355\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	45.7192	2.49791
$336$	0	0
$337$	−7.38100	−0.402069	−0.201034	−	0.979584i	$-0.564430\pi$
$337$	−7.38100	−0.402069	−0.201034	+	0.979584i	$0.564430\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 17.2245i	− 0.932758i
$342$	0	0
$343$	− 13.2955i	− 0.717889i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.3594	−0.609806	−0.304903	−	0.952383i	$-0.598624\pi$
$347$	−11.3594	−0.609806	−0.304903	+	0.952383i	$0.598624\pi$
$348$	0	0
$349$	22.2679	1.19197	0.595986	−	0.802995i	$-0.296761\pi$
$349$	22.2679	1.19197	0.595986	+	0.802995i	$0.296761\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.71263i	0.304052i	0.988376	+	0.152026i	$0.0485798\pi$
$353$	5.71263i	0.304052i	−0.988376	+	0.152026i	$0.951420\pi$
$354$	0	0
$355$	− 40.1008i	− 2.12833i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.27312	0.489416	0.244708	−	0.969597i	$-0.421308\pi$
$359$	9.27312	0.489416	0.244708	+	0.969597i	$0.421308\pi$
$360$	0	0
$361$	−10.7137	−0.563878
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 32.6665i	− 1.70984i
$366$	0	0
$367$	− 16.1345i	− 0.842212i	−0.907011	−	0.421106i	$-0.861642\pi$
$367$	− 16.1345i	− 0.842212i	0.907011	−	0.421106i	$-0.138358\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.38097	0.331283
$372$	0	0
$373$	29.8355	1.54482	0.772412	−	0.635122i	$-0.219050\pi$
$373$	29.8355	1.54482	0.772412	+	0.635122i	$0.219050\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	31.4289i	1.61867i
$378$	0	0
$379$	− 23.8268i	− 1.22390i	−0.790897	−	0.611949i	$-0.790386\pi$
$379$	− 23.8268i	− 1.22390i	0.790897	−	0.611949i	$-0.209614\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.71328	−0.240837	−0.120419	−	0.992723i	$-0.538424\pi$
$383$	−4.71328	−0.240837	−0.120419	+	0.992723i	$0.538424\pi$
$384$	0	0
$385$	−6.99334	−0.356414
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.70388i	0.289198i	0.989490	+	0.144599i	$0.0461893\pi$
$389$	5.70388i	0.289198i	−0.989490	+	0.144599i	$0.953811\pi$
$390$	0	0
$391$	− 9.66052i	− 0.488553i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	37.0425	1.86381
$396$	0	0
$397$	3.97401	0.199450	0.0997248	−	0.995015i	$-0.468204\pi$
$397$	3.97401	0.199450	0.0997248	+	0.995015i	$0.468204\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 0.545971i	− 0.0272645i	−0.999907	−	0.0136322i	$-0.995661\pi$
$401$	− 0.545971i	− 0.0272645i	0.999907	−	0.0136322i	$-0.00433941\pi$
$402$	0	0
$403$	− 31.1338i	− 1.55088i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.36524	0.117241
$408$	0	0
$409$	31.6840	1.56667	0.783337	−	0.621597i	$-0.213516\pi$
$409$	31.6840	1.56667	0.783337	+	0.621597i	$0.213516\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.66611i	0.328018i
$414$	0	0
$415$	5.88416i	0.288842i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.6764	1.15667	0.578334	−	0.815800i	$-0.303703\pi$
$419$	23.6764	1.15667	0.578334	+	0.815800i	$0.303703\pi$
$420$	0	0
$421$	1.50446	0.0733230	0.0366615	−	0.999328i	$-0.488328\pi$
$421$	1.50446	0.0733230	0.0366615	+	0.999328i	$0.488328\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	14.9018i	0.722844i
$426$	0	0
$427$	− 5.47687i	− 0.265045i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.1023	−1.20914	−0.604568	−	0.796554i	$-0.706654\pi$
$431$	−25.1023	−1.20914	−0.604568	+	0.796554i	$0.706654\pi$
$432$	0	0
$433$	9.45794	0.454519	0.227260	−	0.973834i	$-0.427023\pi$
$433$	9.45794	0.454519	0.227260	+	0.973834i	$0.427023\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 11.6173i	− 0.555730i
$438$	0	0
$439$	30.0714i	1.43523i	0.696441	+	0.717614i	$0.254766\pi$
$439$	30.0714i	1.43523i	−0.696441	+	0.717614i	$0.745234\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.2455	1.00940	0.504702	−	0.863293i	$-0.331602\pi$
$443$	21.2455	1.00940	0.504702	+	0.863293i	$0.331602\pi$
$444$	0	0
$445$	48.9617	2.32101
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.851207i	0.0401710i	0.999798	+	0.0200855i	$0.00639384\pi$
$449$	0.851207i	0.0401710i	−0.999798	+	0.0200855i	$0.993606\pi$
$450$	0	0
$451$	− 5.10523i	− 0.240396i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−12.6407	−0.592604
$456$	0	0
$457$	−2.50459	−0.117160	−0.0585799	−	0.998283i	$-0.518657\pi$
$457$	−2.50459	−0.117160	−0.0585799	+	0.998283i	$0.518657\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	38.7362i	1.80412i	0.431607	+	0.902062i	$0.357947\pi$
$461$	38.7362i	1.80412i	−0.431607	+	0.902062i	$0.642053\pi$
$462$	0	0
$463$	33.2210i	1.54391i	0.635678	+	0.771955i	$0.280721\pi$
$463$	33.2210i	1.54391i	−0.635678	+	0.771955i	$0.719279\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.3008	1.03196	0.515978	−	0.856602i	$-0.327429\pi$
$467$	22.3008	1.03196	0.515978	+	0.856602i	$0.327429\pi$
$468$	0	0
$469$	16.3111	0.753179
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 26.9054i	− 1.23711i
$474$	0	0
$475$	17.9202i	0.822236i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−29.6945	−1.35678	−0.678388	−	0.734704i	$-0.737321\pi$
$479$	−29.6945	−1.35678	−0.678388	+	0.734704i	$0.737321\pi$
$480$	0	0
$481$	4.27524	0.194934
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 17.2799i	− 0.784638i
$486$	0	0
$487$	− 30.7849i	− 1.39500i	−0.716585	−	0.697499i	$-0.754296\pi$
$487$	− 30.7849i	− 1.39500i	0.716585	−	0.697499i	$-0.245704\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.40366	0.0633462	0.0316731	−	0.999498i	$-0.489916\pi$
$491$	1.40366	0.0633462	0.0316731	+	0.999498i	$0.489916\pi$
$492$	0	0
$493$	33.3229	1.50079
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 14.3067i	− 0.641743i
$498$	0	0
$499$	− 0.829761i	− 0.0371452i	−0.999828	−	0.0185726i	$-0.994088\pi$
$499$	− 0.829761i	− 0.0371452i	0.999828	−	0.0185726i	$-0.00591218\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	36.6457	1.63395	0.816975	−	0.576674i	$-0.195650\pi$
$503$	36.6457	1.63395	0.816975	+	0.576674i	$0.195650\pi$
$504$	0	0
$505$	−54.6619	−2.43242
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.3601i	0.459205i	0.973284	+	0.229603i	$0.0737426\pi$
$509$	10.3601i	0.459205i	−0.973284	+	0.229603i	$0.926257\pi$
$510$	0	0
$511$	− 11.6544i	− 0.515559i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−33.7001	−1.48500
$516$	0	0
$517$	−13.5291	−0.595008
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.63324i	0.378229i	0.981955	+	0.189115i	$0.0605618\pi$
$521$	8.63324i	0.378229i	−0.981955	+	0.189115i	$0.939438\pi$
$522$	0	0
$523$	− 29.4217i	− 1.28652i	−0.765648	−	0.643260i	$-0.777582\pi$
$523$	− 29.4217i	− 1.28652i	0.765648	−	0.643260i	$-0.222418\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−33.0100	−1.43794
$528$	0	0
$529$	−18.4579	−0.802519
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 9.22785i	− 0.399703i
$534$	0	0
$535$	− 4.11417i	− 0.177871i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	14.0617	0.605680
$540$	0	0
$541$	10.9121	0.469148	0.234574	−	0.972098i	$-0.424631\pi$
$541$	10.9121	0.469148	0.234574	+	0.972098i	$0.424631\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	25.7375i	1.10247i
$546$	0	0
$547$	12.0554i	0.515452i	0.966218	+	0.257726i	$0.0829731\pi$
$547$	12.0554i	0.515452i	−0.966218	+	0.257726i	$0.917027\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	40.0726	1.70715
$552$	0	0
$553$	13.2156	0.561984
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.07427i	0.215004i	0.994205	+	0.107502i	$0.0342852\pi$
$557$	5.07427i	0.215004i	−0.994205	+	0.107502i	$0.965715\pi$
$558$	0	0
$559$	− 48.6323i	− 2.05693i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−38.3978	−1.61828	−0.809138	−	0.587619i	$-0.800066\pi$
$563$	−38.3978	−1.61828	−0.809138	+	0.587619i	$0.800066\pi$
$564$	0	0
$565$	53.6016	2.25504
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 18.8422i	− 0.789905i	−0.918701	−	0.394953i	$-0.870761\pi$
$569$	− 18.8422i	− 0.789905i	0.918701	−	0.394953i	$-0.129239\pi$
$570$	0	0
$571$	− 41.8434i	− 1.75109i	−0.483134	−	0.875546i	$-0.660502\pi$
$571$	− 41.8434i	− 1.75109i	0.483134	−	0.875546i	$-0.339498\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.00635	−0.292185
$576$	0	0
$577$	36.9994	1.54030	0.770152	−	0.637860i	$-0.220180\pi$
$577$	36.9994	1.54030	0.770152	+	0.637860i	$0.220180\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.09928i	0.0870928i
$582$	0	0
$583$	14.6948i	0.608597i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.9499	0.699599	0.349799	−	0.936825i	$-0.386250\pi$
$587$	16.9499	0.699599	0.349799	+	0.936825i	$0.386250\pi$
$588$	0	0
$589$	−39.6962	−1.63565
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 0.652514i	− 0.0267956i	−0.999910	−	0.0133978i	$-0.995735\pi$
$593$	− 0.652514i	− 0.0267956i	0.999910	−	0.0133978i	$-0.00426477\pi$
$594$	0	0
$595$	13.4024i	0.549445i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.2320	1.23525	0.617624	−	0.786474i	$-0.288095\pi$
$599$	30.2320	1.23525	0.617624	+	0.786474i	$0.288095\pi$
$600$	0	0
$601$	−12.7079	−0.518365	−0.259183	−	0.965828i	$-0.583453\pi$
$601$	−12.7079	−0.518365	−0.259183	+	0.965828i	$0.583453\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.5618i	0.632675i
$606$	0	0
$607$	− 5.79033i	− 0.235022i	−0.993072	−	0.117511i	$-0.962508\pi$
$607$	− 5.79033i	− 0.235022i	0.993072	−	0.117511i	$-0.0374916\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−24.4542	−0.989311
$612$	0	0
$613$	−33.8865	−1.36866	−0.684332	−	0.729171i	$-0.739906\pi$
$613$	−33.8865	−1.36866	−0.684332	+	0.729171i	$0.739906\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.0465i	0.525234i	0.964900	+	0.262617i	$0.0845856\pi$
$617$	13.0465i	0.525234i	−0.964900	+	0.262617i	$0.915414\pi$
$618$	0	0
$619$	27.5775i	1.10843i	0.832373	+	0.554216i	$0.186982\pi$
$619$	27.5775i	1.10843i	−0.832373	+	0.554216i	$0.813018\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	17.4680	0.699839
$624$	0	0
$625$	−30.6298	−1.22519
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 4.53288i	− 0.180738i
$630$	0	0
$631$	28.3239i	1.12756i	0.825926	+	0.563778i	$0.190653\pi$
$631$	28.3239i	1.12756i	−0.825926	+	0.563778i	$0.809347\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−36.2504	−1.43855
$636$	0	0
$637$	25.4169	1.00705
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.3097i	0.920679i	0.887743	+	0.460340i	$0.152272\pi$
$641$	23.3097i	0.920679i	−0.887743	+	0.460340i	$0.847728\pi$
$642$	0	0
$643$	− 48.4259i	− 1.90973i	−0.297037	−	0.954866i	$-0.595999\pi$
$643$	− 48.4259i	− 1.90973i	0.297037	−	0.954866i	$-0.404001\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.4978	−0.412711	−0.206355	−	0.978477i	$-0.566160\pi$
$647$	−10.4978	−0.412711	−0.206355	+	0.978477i	$0.566160\pi$
$648$	0	0
$649$	−15.3515	−0.602599
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.16836i	0.163121i	0.996668	+	0.0815603i	$0.0259903\pi$
$653$	4.16836i	0.163121i	−0.996668	+	0.0815603i	$0.974010\pi$
$654$	0	0
$655$	6.73566i	0.263184i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	31.1564	1.21368	0.606840	−	0.794824i	$-0.292437\pi$
$659$	31.1564	1.21368	0.606840	+	0.794824i	$0.292437\pi$
$660$	0	0
$661$	−14.1687	−0.551098	−0.275549	−	0.961287i	$-0.588860\pi$
$661$	−14.1687	−0.551098	−0.275549	+	0.961287i	$0.588860\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.1171i	0.624995i
$666$	0	0
$667$	15.6673i	0.606642i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12.6128	0.486911
$672$	0	0
$673$	−9.21047	−0.355037	−0.177519	−	0.984117i	$-0.556807\pi$
$673$	−9.21047	−0.355037	−0.177519	+	0.984117i	$0.556807\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.74329i	0.220733i	0.993891	+	0.110366i	$0.0352024\pi$
$677$	5.74329i	0.220733i	−0.993891	+	0.110366i	$0.964798\pi$
$678$	0	0
$679$	− 6.16490i	− 0.236587i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−0.889438	−0.0340334	−0.0170167	−	0.999855i	$-0.505417\pi$
$683$	−0.889438	−0.0340334	−0.0170167	+	0.999855i	$0.505417\pi$
$684$	0	0
$685$	−16.2002	−0.618978
$686$	0	0
$687$	0	0
$688$	0	0
$689$	26.5613i	1.01191i
$690$	0	0
$691$	− 11.5513i	− 0.439431i	−0.975564	−	0.219716i	$-0.929487\pi$
$691$	− 11.5513i	− 0.439431i	0.975564	−	0.219716i	$-0.0705129\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	40.3828	1.53181
$696$	0	0
$697$	−9.78394	−0.370593
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 27.1402i	− 1.02507i	−0.858666	−	0.512536i	$-0.828706\pi$
$701$	− 27.1402i	− 1.02507i	0.858666	−	0.512536i	$-0.171294\pi$
$702$	0	0
$703$	− 5.45103i	− 0.205589i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−19.5016	−0.733434
$708$	0	0
$709$	21.5338	0.808718	0.404359	−	0.914600i	$-0.367495\pi$
$709$	21.5338	0.808718	0.404359	+	0.914600i	$0.367495\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 15.5202i	− 0.581237i
$714$	0	0
$715$	− 29.1104i	− 1.08867i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	46.2358	1.72431	0.862153	−	0.506649i	$-0.169116\pi$
$719$	46.2358	1.72431	0.862153	+	0.506649i	$0.169116\pi$
$720$	0	0
$721$	−12.0231	−0.447765
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 24.1676i	− 0.897563i
$726$	0	0
$727$	13.8632i	0.514158i	0.966390	+	0.257079i	$0.0827600\pi$
$727$	13.8632i	0.514158i	−0.966390	+	0.257079i	$0.917240\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−51.5630	−1.90713
$732$	0	0
$733$	−5.03981	−0.186150	−0.0930748	−	0.995659i	$-0.529670\pi$
$733$	−5.03981	−0.186150	−0.0930748	+	0.995659i	$0.529670\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	37.5632i	1.38366i
$738$	0	0
$739$	51.1408i	1.88124i	0.339455	+	0.940622i	$0.389757\pi$
$739$	51.1408i	1.88124i	−0.339455	+	0.940622i	$0.610243\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35.8041	1.31353	0.656763	−	0.754097i	$-0.271925\pi$
$743$	35.8041	1.31353	0.656763	+	0.754097i	$0.271925\pi$
$744$	0	0
$745$	−51.2222	−1.87664
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.46780i	− 0.0536324i
$750$	0	0
$751$	− 35.8576i	− 1.30846i	−0.756294	−	0.654232i	$-0.772992\pi$
$751$	− 35.8576i	− 1.30846i	0.756294	−	0.654232i	$-0.227008\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.59422	−0.0580195
$756$	0	0
$757$	21.6543	0.787038	0.393519	−	0.919317i	$-0.371258\pi$
$757$	21.6543	0.787038	0.393519	+	0.919317i	$0.371258\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.31194i	0.337557i	0.985654	+	0.168779i	$0.0539823\pi$
$761$	9.31194i	0.337557i	−0.985654	+	0.168779i	$0.946018\pi$
$762$	0	0
$763$	9.18231i	0.332422i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−27.7483	−1.00193
$768$	0	0
$769$	12.0434	0.434295	0.217147	−	0.976139i	$-0.430325\pi$
$769$	12.0434	0.434295	0.217147	+	0.976139i	$0.430325\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13.9486i	0.501697i	0.968026	+	0.250848i	$0.0807096\pi$
$773$	13.9486i	0.501697i	−0.968026	+	0.250848i	$0.919290\pi$
$774$	0	0
$775$	23.9407i	0.859974i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.7657	−0.421550
$780$	0	0
$781$	32.9471	1.17894
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 64.1733i	− 2.29044i
$786$	0	0
$787$	− 33.2065i	− 1.18368i	−0.806054	−	0.591842i	$-0.798401\pi$
$787$	− 33.2065i	− 1.18368i	0.806054	−	0.591842i	$-0.201599\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	19.1233	0.679947
$792$	0	0
$793$	22.7980	0.809579
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 20.2949i	− 0.718882i	−0.933168	−	0.359441i	$-0.882967\pi$
$797$	− 20.2949i	− 0.718882i	0.933168	−	0.359441i	$-0.117033\pi$
$798$	0	0
$799$	25.9279i	0.917262i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	26.8390	0.947127
$804$	0	0
$805$	−6.30138	−0.222095
$806$	0	0
$807$	0	0
$808$	0	0
$809$	37.1277i	1.30534i	0.757643	+	0.652670i	$0.226351\pi$
$809$	37.1277i	1.30534i	−0.757643	+	0.652670i	$0.773649\pi$
$810$	0	0
$811$	− 21.6318i	− 0.759596i	−0.925070	−	0.379798i	$-0.875994\pi$
$811$	− 21.6318i	− 0.759596i	0.925070	−	0.379798i	$-0.124006\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−31.9541	−1.11930
$816$	0	0
$817$	−62.0073	−2.16936
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.8195i	0.482302i	0.970488	+	0.241151i	$0.0775249\pi$
$821$	13.8195i	0.482302i	−0.970488	+	0.241151i	$0.922475\pi$
$822$	0	0
$823$	25.4121i	0.885809i	0.896569	+	0.442904i	$0.146052\pi$
$823$	25.4121i	0.885809i	−0.896569	+	0.442904i	$0.853948\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	21.8297	0.759093	0.379547	−	0.925173i	$-0.376080\pi$
$827$	21.8297	0.759093	0.379547	+	0.925173i	$0.376080\pi$
$828$	0	0
$829$	−7.92058	−0.275093	−0.137546	−	0.990495i	$-0.543922\pi$
$829$	−7.92058	−0.275093	−0.137546	+	0.990495i	$0.543922\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 26.9486i	− 0.933713i
$834$	0	0
$835$	61.6405i	2.13316i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9.21862	0.318262	0.159131	−	0.987257i	$-0.449131\pi$
$839$	9.21862	0.318262	0.159131	+	0.987257i	$0.449131\pi$
$840$	0	0
$841$	−25.0428	−0.863544
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 15.1934i	− 0.522670i
$846$	0	0
$847$	5.55194i	0.190767i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.13121	0.0730570
$852$	0	0
$853$	−27.6984	−0.948376	−0.474188	−	0.880424i	$-0.657258\pi$
$853$	−27.6984	−0.948376	−0.474188	+	0.880424i	$0.657258\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	51.4736i	1.75830i	0.476542	+	0.879152i	$0.341890\pi$
$857$	51.4736i	1.75830i	−0.476542	+	0.879152i	$0.658110\pi$
$858$	0	0
$859$	10.6349i	0.362858i	0.983404	+	0.181429i	$0.0580722\pi$
$859$	10.6349i	0.362858i	−0.983404	+	0.181429i	$0.941928\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	46.4827	1.58229	0.791145	−	0.611629i	$-0.209486\pi$
$863$	46.4827	1.58229	0.791145	+	0.611629i	$0.209486\pi$
$864$	0	0
$865$	−3.31915	−0.112855
$866$	0	0
$867$	0	0
$868$	0	0
$869$	30.4344i	1.03242i
$870$	0	0
$871$	67.8965i	2.30059i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.06338	−0.171174
$876$	0	0
$877$	28.3937	0.958787	0.479393	−	0.877600i	$-0.340857\pi$
$877$	28.3937	0.958787	0.479393	+	0.877600i	$0.340857\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 45.0162i	− 1.51663i	−0.651885	−	0.758317i	$-0.726022\pi$
$881$	− 45.0162i	− 1.51663i	0.651885	−	0.758317i	$-0.273978\pi$
$882$	0	0
$883$	34.3701i	1.15665i	0.815808	+	0.578323i	$0.196293\pi$
$883$	34.3701i	1.15665i	−0.815808	+	0.578323i	$0.803707\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5.36799	−0.180239	−0.0901197	−	0.995931i	$-0.528725\pi$
$887$	−5.36799	−0.180239	−0.0901197	+	0.995931i	$0.528725\pi$
$888$	0	0
$889$	−12.9330	−0.433759
$890$	0	0
$891$	0	0
$892$	0	0
$893$	31.1796i	1.04339i
$894$	0	0
$895$	8.11167i	0.271143i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	53.5353	1.78550
$900$	0	0
$901$	28.1620	0.938211
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 10.0369i	− 0.333639i
$906$	0	0
$907$	36.7659i	1.22079i	0.792097	+	0.610395i	$0.208989\pi$
$907$	36.7659i	1.22079i	−0.792097	+	0.610395i	$0.791011\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	13.8479	0.458801	0.229401	−	0.973332i	$-0.426323\pi$
$911$	13.8479	0.458801	0.229401	+	0.973332i	$0.426323\pi$
$912$	0	0
$913$	−4.83446	−0.159997
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.40307i	0.0793563i
$918$	0	0
$919$	− 22.7337i	− 0.749916i	−0.927042	−	0.374958i	$-0.877657\pi$
$919$	− 22.7337i	− 0.749916i	0.927042	−	0.374958i	$-0.122343\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	59.5528	1.96021
$924$	0	0
$925$	−3.28750	−0.108092
$926$	0	0
$927$	0	0
$928$	0	0
$929$	5.19793i	0.170539i	0.996358	+	0.0852693i	$0.0271751\pi$
$929$	5.19793i	0.170539i	−0.996358	+	0.0852693i	$0.972825\pi$
$930$	0	0
$931$	− 32.4071i	− 1.06210i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−30.8646	−1.00938
$936$	0	0
$937$	60.0023	1.96019	0.980095	−	0.198528i	$-0.0636162\pi$
$937$	60.0023	1.96019	0.980095	+	0.198528i	$0.0636162\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 17.4345i	− 0.568347i	−0.958773	−	0.284174i	$-0.908281\pi$
$941$	− 17.4345i	− 0.568347i	0.958773	−	0.284174i	$-0.0917192\pi$
$942$	0	0
$943$	− 4.60009i	− 0.149800i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.3773	1.44207	0.721034	−	0.692900i	$-0.243667\pi$
$947$	44.3773	1.44207	0.721034	+	0.692900i	$0.243667\pi$
$948$	0	0
$949$	48.5123	1.57477
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16.2713i	0.527080i	0.964649	+	0.263540i	$0.0848900\pi$
$953$	16.2713i	0.527080i	−0.964649	+	0.263540i	$0.915110\pi$
$954$	0	0
$955$	26.6890i	0.863637i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.77971	−0.186637
$960$	0	0
$961$	−22.0325	−0.710726
$962$	0	0
$963$	0	0
$964$	0	0
$965$	46.1051i	1.48418i
$966$	0	0
$967$	16.5912i	0.533535i	0.963761	+	0.266768i	$0.0859557\pi$
$967$	16.5912i	0.533535i	−0.963761	+	0.266768i	$0.914044\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	21.5980	0.693114	0.346557	−	0.938029i	$-0.387351\pi$
$971$	21.5980	0.693114	0.346557	+	0.938029i	$0.387351\pi$
$972$	0	0
$973$	14.4073	0.461876
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18.9602i	− 0.606590i	−0.952897	−	0.303295i	$-0.901913\pi$
$977$	− 18.9602i	− 0.606590i	0.952897	−	0.303295i	$-0.0980867\pi$
$978$	0	0
$979$	40.2272i	1.28567i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−46.4106	−1.48027	−0.740135	−	0.672459i	$-0.765238\pi$
$983$	−46.4106	−1.48027	−0.740135	+	0.672459i	$0.765238\pi$
$984$	0	0
$985$	−7.54066	−0.240265
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 24.2433i	− 0.770891i
$990$	0	0
$991$	− 11.2986i	− 0.358913i	−0.983766	−	0.179457i	$-0.942566\pi$
$991$	− 11.2986i	− 0.358913i	0.983766	−	0.179457i	$-0.0574339\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−23.8304	−0.755475
$996$	0	0
$997$	−51.9321	−1.64471	−0.822353	−	0.568977i	$-0.807339\pi$
$997$	−51.9321	−1.64471	−0.822353	+	0.568977i	$0.807339\pi$
$998$	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.2591.3	✓	24
3.2	odd	2	inner	5328.2.e.g.2591.21	yes	24
4.3	odd	2	inner	5328.2.e.g.2591.4	yes	24
12.11	even	2	inner	5328.2.e.g.2591.22	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5328.2.e.g.2591.3	✓	24	1.1	even	1	trivial
5328.2.e.g.2591.4	yes	24	4.3	odd	2	inner
5328.2.e.g.2591.21	yes	24	3.2	odd	2	inner
5328.2.e.g.2591.22	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)

\(f(q)\)	\(=\)	\(q-2.87880i q^{5} -1.02706i q^{7} +2.36524 q^{11} +4.27524 q^{13} -4.53288i q^{17} -5.45103i q^{19} +2.13121 q^{23} -3.28750 q^{25} +7.35138i q^{29} -7.28234i q^{31} -2.95671 q^{35} +1.00000 q^{37} -2.15844i q^{41} -11.3753i q^{43} -5.71996 q^{47} +5.94514 q^{49} +6.21282i q^{53} -6.80906i q^{55} -6.49046 q^{59} +5.33255 q^{61} -12.3076i q^{65} +15.8813i q^{67} +13.9297 q^{71} +11.3473 q^{73} -2.42925i q^{77} +12.8673i q^{79} -2.04396 q^{83} -13.0492 q^{85} +17.0077i q^{89} -4.39095i q^{91} -15.6924 q^{95} +6.00245 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 16 q^{13} + 24 q^{37} - 16 q^{49} + 48 q^{61} + 48 q^{73} + 56 q^{85} - 8 q^{97}+O(q^{100}) \) 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