LMFDB - Embedded newform 2352.2.b.k.1567.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(1567,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2352.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.7
Root			$-1.60021 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	2352.1567
Dual form			2352.2.b.k.1567.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.21486i q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.21486i q^{5} +1.00000 q^{9} -1.36710i q^{11} -2.93015i q^{13} -3.21486i q^{15} +6.91037i q^{17} +7.35449 q^{19} -3.62774i q^{23} -5.33530 q^{25} -1.00000 q^{27} -1.11185 q^{29} +8.70319 q^{31} +1.36710i q^{33} +7.63792 q^{37} +2.93015i q^{39} -0.833147i q^{41} +4.82362i q^{43} +3.21486i q^{45} +2.95367 q^{47} -6.91037i q^{51} -4.57794 q^{53} +4.39502 q^{55} -7.35449 q^{57} -14.0985 q^{59} +11.0618i q^{61} +9.42002 q^{65} -12.1545i q^{67} +3.62774i q^{69} +12.6249i q^{71} -6.49011i q^{73} +5.33530 q^{75} +7.79367i q^{79} +1.00000 q^{81} +8.87502 q^{83} -22.2159 q^{85} +1.11185 q^{87} +12.6146i q^{89} -8.70319 q^{93} +23.6436i q^{95} +13.8811i q^{97} -1.36710i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 8 * q^9	$ 8 q - 8 q^{3} + 8 q^{9} - 24 q^{25} - 8 q^{27} + 16 q^{29} + 32 q^{31} + 16 q^{47} + 16 q^{53} + 64 q^{55} - 48 q^{59} - 16 q^{65} + 24 q^{75} + 8 q^{81} - 64 q^{85} - 16 q^{87} - 32 q^{93}+O(q^{100}) $ 8 * q - 8 * q^3 + 8 * q^9 - 24 * q^25 - 8 * q^27 + 16 * q^29 + 32 * q^31 + 16 * q^47 + 16 * q^53 + 64 * q^55 - 48 * q^59 - 16 * q^65 + 24 * q^75 + 8 * q^81 - 64 * q^85 - 16 * q^87 - 32 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.21486i	1.43773i	0.695151	+	0.718864i	$0.255338\pi$
$5$	3.21486i	1.43773i	−0.695151	+	0.718864i	$0.744662\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 1.36710i	− 0.412195i	−0.978531	−	0.206098i	$-0.933924\pi$
$11$	− 1.36710i	− 0.412195i	0.978531	−	0.206098i	$-0.0660765\pi$
$12$	0	0
$13$	− 2.93015i	− 0.812678i	−0.913722	−	0.406339i	$-0.866805\pi$
$13$	− 2.93015i	− 0.812678i	0.913722	−	0.406339i	$-0.133195\pi$
$14$	0	0
$15$	− 3.21486i	− 0.830072i
$16$	0	0
$17$	6.91037i	1.67601i	0.545661	+	0.838006i	$0.316279\pi$
$17$	6.91037i	1.67601i	−0.545661	+	0.838006i	$0.683721\pi$
$18$	0	0
$19$	7.35449	1.68724	0.843618	−	0.536943i	$-0.180421\pi$
$19$	7.35449	1.68724	0.843618	+	0.536943i	$0.180421\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3.62774i	− 0.756436i	−0.925717	−	0.378218i	$-0.876537\pi$
$23$	− 3.62774i	− 0.756436i	0.925717	−	0.378218i	$-0.123463\pi$
$24$	0	0
$25$	−5.33530	−1.06706
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.11185	−0.206466	−0.103233	−	0.994657i	$-0.532919\pi$
$29$	−1.11185	−0.206466	−0.103233	+	0.994657i	$0.532919\pi$
$30$	0	0
$31$	8.70319	1.56314	0.781569	−	0.623819i	$-0.214420\pi$
$31$	8.70319	1.56314	0.781569	+	0.623819i	$0.214420\pi$
$32$	0	0
$33$	1.36710i	0.237981i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.63792	1.25567	0.627833	−	0.778348i	$-0.283942\pi$
$37$	7.63792	1.25567	0.627833	+	0.778348i	$0.283942\pi$
$38$	0	0
$39$	2.93015i	0.469200i
$40$	0	0
$41$	− 0.833147i	− 0.130116i	−0.997881	−	0.0650579i	$-0.979277\pi$
$41$	− 0.833147i	− 0.130116i	0.997881	−	0.0650579i	$-0.0207232\pi$
$42$	0	0
$43$	4.82362i	0.735595i	0.929906	+	0.367797i	$0.119888\pi$
$43$	4.82362i	0.735595i	−0.929906	+	0.367797i	$0.880112\pi$
$44$	0	0
$45$	3.21486i	0.479243i
$46$	0	0
$47$	2.95367	0.430837	0.215418	−	0.976522i	$-0.430888\pi$
$47$	2.95367	0.430837	0.215418	+	0.976522i	$0.430888\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 6.91037i	− 0.967646i
$52$	0	0
$53$	−4.57794	−0.628829	−0.314414	−	0.949286i	$-0.601808\pi$
$53$	−4.57794	−0.628829	−0.314414	+	0.949286i	$0.601808\pi$
$54$	0	0
$55$	4.39502	0.592625
$56$	0	0
$57$	−7.35449	−0.974127
$58$	0	0
$59$	−14.0985	−1.83546	−0.917732	−	0.397200i	$-0.869982\pi$
$59$	−14.0985	−1.83546	−0.917732	+	0.397200i	$0.869982\pi$
$60$	0	0
$61$	11.0618i	1.41632i	0.706052	+	0.708160i	$0.250475\pi$
$61$	11.0618i	1.41632i	−0.706052	+	0.708160i	$0.749525\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.42002	1.16841
$66$	0	0
$67$	− 12.1545i	− 1.48490i	−0.669900	−	0.742452i	$-0.733663\pi$
$67$	− 12.1545i	− 1.48490i	0.669900	−	0.742452i	$-0.266337\pi$
$68$	0	0
$69$	3.62774i	0.436728i
$70$	0	0
$71$	12.6249i	1.49830i	0.662402	+	0.749148i	$0.269537\pi$
$71$	12.6249i	1.49830i	−0.662402	+	0.749148i	$0.730463\pi$
$72$	0	0
$73$	− 6.49011i	− 0.759610i	−0.925067	−	0.379805i	$-0.875991\pi$
$73$	− 6.49011i	− 0.759610i	0.925067	−	0.379805i	$-0.124009\pi$
$74$	0	0
$75$	5.33530	0.616068
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.79367i	0.876856i	0.898766	+	0.438428i	$0.144465\pi$
$79$	7.79367i	0.876856i	−0.898766	+	0.438428i	$0.855535\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.87502	0.974159	0.487080	−	0.873358i	$-0.338062\pi$
$83$	8.87502	0.974159	0.487080	+	0.873358i	$0.338062\pi$
$84$	0	0
$85$	−22.2159	−2.40965
$86$	0	0
$87$	1.11185	0.119203
$88$	0	0
$89$	12.6146i	1.33715i	0.743646	+	0.668574i	$0.233095\pi$
$89$	12.6146i	1.33715i	−0.743646	+	0.668574i	$0.766905\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.70319	−0.902478
$94$	0	0
$95$	23.6436i	2.42579i
$96$	0	0
$97$	13.8811i	1.40942i	0.709497	+	0.704708i	$0.248922\pi$
$97$	13.8811i	1.40942i	−0.709497	+	0.704708i	$0.751078\pi$
$98$	0	0
$99$	− 1.36710i	− 0.137398i
$100$	0	0
$101$	− 1.65891i	− 0.165068i	−0.996588	−	0.0825338i	$-0.973699\pi$
$101$	− 1.65891i	− 0.165068i	0.996588	−	0.0825338i	$-0.0263013\pi$
$102$	0	0
$103$	1.30791	0.128872	0.0644359	−	0.997922i	$-0.479475\pi$
$103$	1.30791	0.128872	0.0644359	+	0.997922i	$0.479475\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.15204i	− 0.691414i	−0.938343	−	0.345707i	$-0.887639\pi$
$107$	− 7.15204i	− 0.691414i	0.938343	−	0.345707i	$-0.112361\pi$
$108$	0	0
$109$	−15.0443	−1.44098	−0.720491	−	0.693464i	$-0.756083\pi$
$109$	−15.0443	−1.44098	−0.720491	+	0.693464i	$0.756083\pi$
$110$	0	0
$111$	−7.63792	−0.724959
$112$	0	0
$113$	−15.4530	−1.45369	−0.726846	−	0.686800i	$-0.759015\pi$
$113$	−15.4530	−1.45369	−0.726846	+	0.686800i	$0.759015\pi$
$114$	0	0
$115$	11.6627	1.08755
$116$	0	0
$117$	− 2.93015i	− 0.270893i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.13104	0.830095
$122$	0	0
$123$	0.833147i	0.0751224i
$124$	0	0
$125$	− 1.07795i	− 0.0964149i
$126$	0	0
$127$	2.16478i	0.192094i	0.995377	+	0.0960468i	$0.0306198\pi$
$127$	2.16478i	0.192094i	−0.995377	+	0.0960468i	$0.969380\pi$
$128$	0	0
$129$	− 4.82362i	− 0.424696i
$130$	0	0
$131$	−4.97877	−0.434997	−0.217499	−	0.976061i	$-0.569790\pi$
$131$	−4.97877	−0.434997	−0.217499	+	0.976061i	$0.569790\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 3.21486i	− 0.276691i
$136$	0	0
$137$	−3.68980	−0.315241	−0.157620	−	0.987500i	$-0.550382\pi$
$137$	−3.68980	−0.315241	−0.157620	+	0.987500i	$0.550382\pi$
$138$	0	0
$139$	−12.0577	−1.02272	−0.511360	−	0.859367i	$-0.670858\pi$
$139$	−12.0577	−1.02272	−0.511360	+	0.859367i	$0.670858\pi$
$140$	0	0
$141$	−2.95367	−0.248744
$142$	0	0
$143$	−4.00580	−0.334982
$144$	0	0
$145$	− 3.57445i	− 0.296842i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.30262	0.270561	0.135280	−	0.990807i	$-0.456806\pi$
$149$	3.30262	0.270561	0.135280	+	0.990807i	$0.456806\pi$
$150$	0	0
$151$	18.9511i	1.54222i	0.636701	+	0.771111i	$0.280299\pi$
$151$	18.9511i	1.54222i	−0.636701	+	0.771111i	$0.719701\pi$
$152$	0	0
$153$	6.91037i	0.558671i
$154$	0	0
$155$	27.9795i	2.24737i
$156$	0	0
$157$	0.886436i	0.0707453i	0.999374	+	0.0353726i	$0.0112618\pi$
$157$	0.886436i	0.0707453i	−0.999374	+	0.0353726i	$0.988738\pi$
$158$	0	0
$159$	4.57794	0.363055
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 19.3538i	− 1.51591i	−0.652310	−	0.757953i	$-0.726200\pi$
$163$	− 19.3538i	− 1.51591i	0.652310	−	0.757953i	$-0.273800\pi$
$164$	0	0
$165$	−4.39502	−0.342152
$166$	0	0
$167$	25.4855	1.97213	0.986065	−	0.166360i	$-0.0532014\pi$
$167$	25.4855	1.97213	0.986065	+	0.166360i	$0.0532014\pi$
$168$	0	0
$169$	4.41421	0.339555
$170$	0	0
$171$	7.35449	0.562412
$172$	0	0
$173$	21.6924i	1.64925i	0.565683	+	0.824623i	$0.308613\pi$
$173$	21.6924i	1.64925i	−0.565683	+	0.824623i	$0.691387\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.0985	1.05971
$178$	0	0
$179$	− 12.2824i	− 0.918032i	−0.888428	−	0.459016i	$-0.848202\pi$
$179$	− 12.2824i	− 0.918032i	0.888428	−	0.459016i	$-0.151798\pi$
$180$	0	0
$181$	2.74444i	0.203993i	0.994785	+	0.101996i	$0.0325230\pi$
$181$	2.74444i	0.203993i	−0.994785	+	0.101996i	$0.967477\pi$
$182$	0	0
$183$	− 11.0618i	− 0.817713i
$184$	0	0
$185$	24.5548i	1.80531i
$186$	0	0
$187$	9.44716	0.690844
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.8857i	1.65595i	0.560766	+	0.827974i	$0.310507\pi$
$191$	22.8857i	1.65595i	−0.560766	+	0.827974i	$0.689493\pi$
$192$	0	0
$193$	14.6163	1.05211	0.526053	−	0.850452i	$-0.323671\pi$
$193$	14.6163	1.05211	0.526053	+	0.850452i	$0.323671\pi$
$194$	0	0
$195$	−9.42002	−0.674581
$196$	0	0
$197$	24.5430	1.74861	0.874307	−	0.485374i	$-0.161316\pi$
$197$	24.5430	1.74861	0.874307	+	0.485374i	$0.161316\pi$
$198$	0	0
$199$	6.04659	0.428631	0.214316	−	0.976764i	$-0.431248\pi$
$199$	6.04659	0.428631	0.214316	+	0.976764i	$0.431248\pi$
$200$	0	0
$201$	12.1545i	0.857309i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.67845	0.187071
$206$	0	0
$207$	− 3.62774i	− 0.252145i
$208$	0	0
$209$	− 10.0543i	− 0.695471i
$210$	0	0
$211$	− 9.33513i	− 0.642657i	−0.946968	−	0.321328i	$-0.895871\pi$
$211$	− 9.33513i	− 0.642657i	0.946968	−	0.321328i	$-0.104129\pi$
$212$	0	0
$213$	− 12.6249i	− 0.865042i
$214$	0	0
$215$	−15.5072	−1.05758
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.49011i	0.438561i
$220$	0	0
$221$	20.2484	1.36206
$222$	0	0
$223$	−5.48794	−0.367500	−0.183750	−	0.982973i	$-0.558824\pi$
$223$	−5.48794	−0.367500	−0.183750	+	0.982973i	$0.558824\pi$
$224$	0	0
$225$	−5.33530	−0.355687
$226$	0	0
$227$	1.03813	0.0689028	0.0344514	−	0.999406i	$-0.489032\pi$
$227$	1.03813	0.0689028	0.0344514	+	0.999406i	$0.489032\pi$
$228$	0	0
$229$	15.6039i	1.03113i	0.856850	+	0.515566i	$0.172418\pi$
$229$	15.6039i	1.03113i	−0.856850	+	0.515566i	$0.827582\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.21561	0.603734	0.301867	−	0.953350i	$-0.402390\pi$
$233$	9.21561	0.603734	0.301867	+	0.953350i	$0.402390\pi$
$234$	0	0
$235$	9.49562i	0.619426i
$236$	0	0
$237$	− 7.79367i	− 0.506253i
$238$	0	0
$239$	− 1.54809i	− 0.100137i	−0.998746	−	0.0500687i	$-0.984056\pi$
$239$	− 1.54809i	− 0.100137i	0.998746	−	0.0500687i	$-0.0159440\pi$
$240$	0	0
$241$	3.68527i	0.237389i	0.992931	+	0.118695i	$0.0378709\pi$
$241$	3.68527i	0.237389i	−0.992931	+	0.118695i	$0.962129\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 21.5498i	− 1.37118i
$248$	0	0
$249$	−8.87502	−0.562431
$250$	0	0
$251$	−26.1940	−1.65335	−0.826676	−	0.562679i	$-0.809771\pi$
$251$	−26.1940	−1.65335	−0.826676	+	0.562679i	$0.809771\pi$
$252$	0	0
$253$	−4.95947	−0.311799
$254$	0	0
$255$	22.2159	1.39121
$256$	0	0
$257$	0.833147i	0.0519703i	0.999662	+	0.0259851i	$0.00827226\pi$
$257$	0.833147i	0.0519703i	−0.999662	+	0.0259851i	$0.991728\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.11185	−0.0688220
$262$	0	0
$263$	− 5.04212i	− 0.310911i	−0.987843	−	0.155455i	$-0.950316\pi$
$263$	− 5.04212i	− 0.310911i	0.987843	−	0.155455i	$-0.0496845\pi$
$264$	0	0
$265$	− 14.7174i	− 0.904085i
$266$	0	0
$267$	− 12.6146i	− 0.772002i
$268$	0	0
$269$	4.85594i	0.296072i	0.988982	+	0.148036i	$0.0472951\pi$
$269$	4.85594i	0.296072i	−0.988982	+	0.148036i	$0.952705\pi$
$270$	0	0
$271$	22.2488	1.35152	0.675759	−	0.737122i	$-0.263816\pi$
$271$	22.2488	1.35152	0.675759	+	0.737122i	$0.263816\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.29388i	0.439837i
$276$	0	0
$277$	0.670606	0.0402928	0.0201464	−	0.999797i	$-0.493587\pi$
$277$	0.670606	0.0402928	0.0201464	+	0.999797i	$0.493587\pi$
$278$	0	0
$279$	8.70319	0.521046
$280$	0	0
$281$	25.5971	1.52700	0.763499	−	0.645810i	$-0.223480\pi$
$281$	25.5971	1.52700	0.763499	+	0.645810i	$0.223480\pi$
$282$	0	0
$283$	−11.1115	−0.660510	−0.330255	−	0.943892i	$-0.607135\pi$
$283$	−11.1115	−0.660510	−0.330255	+	0.943892i	$0.607135\pi$
$284$	0	0
$285$	− 23.6436i	− 1.40053i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−30.7533	−1.80902
$290$	0	0
$291$	− 13.8811i	− 0.813727i
$292$	0	0
$293$	− 19.2818i	− 1.12645i	−0.826303	−	0.563226i	$-0.809560\pi$
$293$	− 19.2818i	− 1.12645i	0.826303	−	0.563226i	$-0.190440\pi$
$294$	0	0
$295$	− 45.3245i	− 2.63890i
$296$	0	0
$297$	1.36710i	0.0793270i
$298$	0	0
$299$	−10.6298	−0.614738
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.65891i	0.0953019i
$304$	0	0
$305$	−35.5622	−2.03628
$306$	0	0
$307$	−26.0058	−1.48423	−0.742115	−	0.670273i	$-0.766177\pi$
$307$	−26.0058	−1.48423	−0.742115	+	0.670273i	$0.766177\pi$
$308$	0	0
$309$	−1.30791	−0.0744042
$310$	0	0
$311$	−19.0772	−1.08177	−0.540885	−	0.841096i	$-0.681911\pi$
$311$	−19.0772	−1.08177	−0.540885	+	0.841096i	$0.681911\pi$
$312$	0	0
$313$	− 21.3583i	− 1.20724i	−0.797272	−	0.603620i	$-0.793724\pi$
$313$	− 21.3583i	− 1.20724i	0.797272	−	0.603620i	$-0.206276\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.74131	0.434795	0.217398	−	0.976083i	$-0.430243\pi$
$317$	7.74131	0.434795	0.217398	+	0.976083i	$0.430243\pi$
$318$	0	0
$319$	1.52001i	0.0851043i
$320$	0	0
$321$	7.15204i	0.399188i
$322$	0	0
$323$	50.8223i	2.82783i
$324$	0	0
$325$	15.6332i	0.867176i
$326$	0	0
$327$	15.0443	0.831951
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0.675988i	0.0371557i	0.999827	+	0.0185778i	$0.00591385\pi$
$331$	0.675988i	0.0371557i	−0.999827	+	0.0185778i	$0.994086\pi$
$332$	0	0
$333$	7.63792	0.418555
$334$	0	0
$335$	39.0748	2.13489
$336$	0	0
$337$	−0.176755	−0.00962847	−0.00481423	−	0.999988i	$-0.501532\pi$
$337$	−0.176755	−0.00962847	−0.00481423	+	0.999988i	$0.501532\pi$
$338$	0	0
$339$	15.4530	0.839290
$340$	0	0
$341$	− 11.8981i	− 0.644318i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−11.6627	−0.627896
$346$	0	0
$347$	6.22637i	0.334249i	0.985936	+	0.167125i	$0.0534482\pi$
$347$	6.22637i	0.334249i	−0.985936	+	0.167125i	$0.946552\pi$
$348$	0	0
$349$	17.4414i	0.933616i	0.884359	+	0.466808i	$0.154596\pi$
$349$	17.4414i	0.933616i	−0.884359	+	0.466808i	$0.845404\pi$
$350$	0	0
$351$	2.93015i	0.156400i
$352$	0	0
$353$	− 8.70218i	− 0.463170i	−0.972815	−	0.231585i	$-0.925609\pi$
$353$	− 8.70218i	− 0.463170i	0.972815	−	0.231585i	$-0.0743912\pi$
$354$	0	0
$355$	−40.5871	−2.15414
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 0.764263i	− 0.0403362i	−0.999797	−	0.0201681i	$-0.993580\pi$
$359$	− 0.764263i	− 0.0403362i	0.999797	−	0.0201681i	$-0.00642015\pi$
$360$	0	0
$361$	35.0886	1.84677
$362$	0	0
$363$	−9.13104	−0.479256
$364$	0	0
$365$	20.8648	1.09211
$366$	0	0
$367$	−26.5162	−1.38413	−0.692067	−	0.721833i	$-0.743300\pi$
$367$	−26.5162	−1.38413	−0.692067	+	0.721833i	$0.743300\pi$
$368$	0	0
$369$	− 0.833147i	− 0.0433719i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−20.1310	−1.04235	−0.521173	−	0.853451i	$-0.674505\pi$
$373$	−20.1310	−1.04235	−0.521173	+	0.853451i	$0.674505\pi$
$374$	0	0
$375$	1.07795i	0.0556652i
$376$	0	0
$377$	3.25790i	0.167790i
$378$	0	0
$379$	− 10.2608i	− 0.527061i	−0.964651	−	0.263531i	$-0.915113\pi$
$379$	− 10.2608i	− 0.527061i	0.964651	−	0.263531i	$-0.0848870\pi$
$380$	0	0
$381$	− 2.16478i	− 0.110905i
$382$	0	0
$383$	2.34315	0.119729	0.0598646	−	0.998207i	$-0.480933\pi$
$383$	2.34315	0.119729	0.0598646	+	0.998207i	$0.480933\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.82362i	0.245198i
$388$	0	0
$389$	24.0440	1.21908	0.609540	−	0.792755i	$-0.291354\pi$
$389$	24.0440	1.21908	0.609540	+	0.792755i	$0.291354\pi$
$390$	0	0
$391$	25.0690	1.26780
$392$	0	0
$393$	4.97877	0.251146
$394$	0	0
$395$	−25.0555	−1.26068
$396$	0	0
$397$	− 18.3526i	− 0.921088i	−0.887637	−	0.460544i	$-0.847654\pi$
$397$	− 18.3526i	− 0.921088i	0.887637	−	0.460544i	$-0.152346\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.96972	−0.148301	−0.0741503	−	0.997247i	$-0.523624\pi$
$401$	−2.96972	−0.148301	−0.0741503	+	0.997247i	$0.523624\pi$
$402$	0	0
$403$	− 25.5016i	− 1.27033i
$404$	0	0
$405$	3.21486i	0.159748i
$406$	0	0
$407$	− 10.4418i	− 0.517580i
$408$	0	0
$409$	− 23.9502i	− 1.18426i	−0.805842	−	0.592131i	$-0.798287\pi$
$409$	− 23.9502i	− 1.18426i	0.805842	−	0.592131i	$-0.201713\pi$
$410$	0	0
$411$	3.68980	0.182004
$412$	0	0
$413$	0	0
$414$	0	0
$415$	28.5319i	1.40058i
$416$	0	0
$417$	12.0577	0.590467
$418$	0	0
$419$	36.3065	1.77369	0.886844	−	0.462069i	$-0.152893\pi$
$419$	36.3065	1.77369	0.886844	+	0.462069i	$0.152893\pi$
$420$	0	0
$421$	−22.6274	−1.10279	−0.551396	−	0.834243i	$-0.685905\pi$
$421$	−22.6274	−1.10279	−0.551396	+	0.834243i	$0.685905\pi$
$422$	0	0
$423$	2.95367	0.143612
$424$	0	0
$425$	− 36.8689i	− 1.78841i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.00580	0.193402
$430$	0	0
$431$	− 30.6040i	− 1.47414i	−0.675816	−	0.737071i	$-0.736208\pi$
$431$	− 30.6040i	− 1.47414i	0.675816	−	0.737071i	$-0.263792\pi$
$432$	0	0
$433$	14.4650i	0.695146i	0.937653	+	0.347573i	$0.112994\pi$
$433$	14.4650i	0.695146i	−0.937653	+	0.347573i	$0.887006\pi$
$434$	0	0
$435$	3.57445i	0.171382i
$436$	0	0
$437$	− 26.6802i	− 1.27629i
$438$	0	0
$439$	9.10981	0.434788	0.217394	−	0.976084i	$-0.430244\pi$
$439$	9.10981	0.434788	0.217394	+	0.976084i	$0.430244\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.83319i	0.134609i	0.997732	+	0.0673045i	$0.0214399\pi$
$443$	2.83319i	0.134609i	−0.997732	+	0.0673045i	$0.978560\pi$
$444$	0	0
$445$	−40.5542	−1.92245
$446$	0	0
$447$	−3.30262	−0.156208
$448$	0	0
$449$	10.3630	0.489058	0.244529	−	0.969642i	$-0.421367\pi$
$449$	10.3630	0.489058	0.244529	+	0.969642i	$0.421367\pi$
$450$	0	0
$451$	−1.13899	−0.0536331
$452$	0	0
$453$	− 18.9511i	− 0.890402i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.65951	−0.311519	−0.155759	−	0.987795i	$-0.549782\pi$
$457$	−6.65951	−0.311519	−0.155759	+	0.987795i	$0.549782\pi$
$458$	0	0
$459$	− 6.91037i	− 0.322549i
$460$	0	0
$461$	1.71311i	0.0797873i	0.999204	+	0.0398936i	$0.0127019\pi$
$461$	1.71311i	0.0797873i	−0.999204	+	0.0398936i	$0.987298\pi$
$462$	0	0
$463$	− 24.1733i	− 1.12343i	−0.827331	−	0.561715i	$-0.810142\pi$
$463$	− 24.1733i	− 1.12343i	0.827331	−	0.561715i	$-0.189858\pi$
$464$	0	0
$465$	− 27.9795i	− 1.29752i
$466$	0	0
$467$	−37.2654	−1.72444	−0.862220	−	0.506535i	$-0.830926\pi$
$467$	−37.2654	−1.72444	−0.862220	+	0.506535i	$0.830926\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 0.886436i	− 0.0408448i
$472$	0	0
$473$	6.59436	0.303209
$474$	0	0
$475$	−39.2385	−1.80038
$476$	0	0
$477$	−4.57794	−0.209610
$478$	0	0
$479$	−21.8596	−0.998790	−0.499395	−	0.866374i	$-0.666444\pi$
$479$	−21.8596	−0.998790	−0.499395	+	0.866374i	$0.666444\pi$
$480$	0	0
$481$	− 22.3803i	− 1.02045i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−44.6259	−2.02636
$486$	0	0
$487$	− 3.37269i	− 0.152831i	−0.997076	−	0.0764155i	$-0.975652\pi$
$487$	− 3.37269i	− 0.152831i	0.997076	−	0.0764155i	$-0.0243475\pi$
$488$	0	0
$489$	19.3538i	0.875208i
$490$	0	0
$491$	− 9.68824i	− 0.437224i	−0.975812	−	0.218612i	$-0.929847\pi$
$491$	− 9.68824i	− 0.437224i	0.975812	−	0.218612i	$-0.0701529\pi$
$492$	0	0
$493$	− 7.68332i	− 0.346039i
$494$	0	0
$495$	4.39502	0.197542
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 13.7401i	− 0.615089i	−0.951534	−	0.307545i	$-0.900493\pi$
$499$	− 13.7401i	− 0.615089i	0.951534	−	0.307545i	$-0.0995074\pi$
$500$	0	0
$501$	−25.4855	−1.13861
$502$	0	0
$503$	1.92715	0.0859273	0.0429637	−	0.999077i	$-0.486320\pi$
$503$	1.92715	0.0859273	0.0429637	+	0.999077i	$0.486320\pi$
$504$	0	0
$505$	5.33316	0.237322
$506$	0	0
$507$	−4.41421	−0.196042
$508$	0	0
$509$	− 14.1621i	− 0.627723i	−0.949469	−	0.313862i	$-0.898377\pi$
$509$	− 14.1621i	− 0.627723i	0.949469	−	0.313862i	$-0.101623\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−7.35449	−0.324709
$514$	0	0
$515$	4.20473i	0.185283i
$516$	0	0
$517$	− 4.03795i	− 0.177589i
$518$	0	0
$519$	− 21.6924i	− 0.952193i
$520$	0	0
$521$	21.6026i	0.946427i	0.880948	+	0.473213i	$0.156906\pi$
$521$	21.6026i	0.946427i	−0.880948	+	0.473213i	$0.843094\pi$
$522$	0	0
$523$	−13.9650	−0.610648	−0.305324	−	0.952249i	$-0.598765\pi$
$523$	−13.9650	−0.610648	−0.305324	+	0.952249i	$0.598765\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	60.1423i	2.61984i
$528$	0	0
$529$	9.83952	0.427805
$530$	0	0
$531$	−14.0985	−0.611821
$532$	0	0
$533$	−2.44125	−0.105742
$534$	0	0
$535$	22.9928	0.994064
$536$	0	0
$537$	12.2824i	0.530026i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3.88005	0.166816	0.0834081	−	0.996515i	$-0.473419\pi$
$541$	3.88005	0.166816	0.0834081	+	0.996515i	$0.473419\pi$
$542$	0	0
$543$	− 2.74444i	− 0.117775i
$544$	0	0
$545$	− 48.3652i	− 2.07174i
$546$	0	0
$547$	35.1640i	1.50351i	0.659445	+	0.751753i	$0.270791\pi$
$547$	35.1640i	1.50351i	−0.659445	+	0.751753i	$0.729209\pi$
$548$	0	0
$549$	11.0618i	0.472107i
$550$	0	0
$551$	−8.17712	−0.348357
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 24.5548i	− 1.04229i
$556$	0	0
$557$	42.4346	1.79801	0.899006	−	0.437936i	$-0.144290\pi$
$557$	42.4346	1.79801	0.899006	+	0.437936i	$0.144290\pi$
$558$	0	0
$559$	14.1339	0.597802
$560$	0	0
$561$	−9.44716	−0.398859
$562$	0	0
$563$	16.5488	0.697447	0.348724	−	0.937226i	$-0.386615\pi$
$563$	16.5488	0.697447	0.348724	+	0.937226i	$0.386615\pi$
$564$	0	0
$565$	− 49.6790i	− 2.09001i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24.2566	−1.01689	−0.508446	−	0.861094i	$-0.669780\pi$
$569$	−24.2566	−1.01689	−0.508446	+	0.861094i	$0.669780\pi$
$570$	0	0
$571$	− 4.62563i	− 0.193576i	−0.995305	−	0.0967882i	$-0.969143\pi$
$571$	− 4.62563i	− 0.193576i	0.995305	−	0.0967882i	$-0.0308569\pi$
$572$	0	0
$573$	− 22.8857i	− 0.956062i
$574$	0	0
$575$	19.3551i	0.807163i
$576$	0	0
$577$	− 26.7907i	− 1.11531i	−0.830072	−	0.557656i	$-0.811701\pi$
$577$	− 26.7907i	− 1.11531i	0.830072	−	0.557656i	$-0.188299\pi$
$578$	0	0
$579$	−14.6163	−0.607434
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.25850i	0.259200i
$584$	0	0
$585$	9.42002	0.389470
$586$	0	0
$587$	−27.8591	−1.14987	−0.574933	−	0.818200i	$-0.694972\pi$
$587$	−27.8591	−1.14987	−0.574933	+	0.818200i	$0.694972\pi$
$588$	0	0
$589$	64.0075	2.63738
$590$	0	0
$591$	−24.5430	−1.00956
$592$	0	0
$593$	1.56906i	0.0644334i	0.999481	+	0.0322167i	$0.0102567\pi$
$593$	1.56906i	0.0644334i	−0.999481	+	0.0322167i	$0.989743\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.04659	−0.247470
$598$	0	0
$599$	40.1156i	1.63908i	0.573021	+	0.819540i	$0.305771\pi$
$599$	40.1156i	1.63908i	−0.573021	+	0.819540i	$0.694229\pi$
$600$	0	0
$601$	− 24.7827i	− 1.01091i	−0.862854	−	0.505453i	$-0.831325\pi$
$601$	− 24.7827i	− 1.01091i	0.862854	−	0.505453i	$-0.168675\pi$
$602$	0	0
$603$	− 12.1545i	− 0.494968i
$604$	0	0
$605$	29.3550i	1.19345i
$606$	0	0
$607$	4.66240	0.189241	0.0946205	−	0.995513i	$-0.469836\pi$
$607$	4.66240	0.189241	0.0946205	+	0.995513i	$0.469836\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 8.65470i	− 0.350132i
$612$	0	0
$613$	28.5817	1.15440	0.577202	−	0.816601i	$-0.304144\pi$
$613$	28.5817	1.15440	0.577202	+	0.816601i	$0.304144\pi$
$614$	0	0
$615$	−2.67845	−0.108005
$616$	0	0
$617$	10.1524	0.408720	0.204360	−	0.978896i	$-0.434489\pi$
$617$	10.1524	0.408720	0.204360	+	0.978896i	$0.434489\pi$
$618$	0	0
$619$	8.64022	0.347280	0.173640	−	0.984809i	$-0.444447\pi$
$619$	8.64022	0.347280	0.173640	+	0.984809i	$0.444447\pi$
$620$	0	0
$621$	3.62774i	0.145576i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−23.2111	−0.928442
$626$	0	0
$627$	10.0543i	0.401530i
$628$	0	0
$629$	52.7809i	2.10451i
$630$	0	0
$631$	− 5.46211i	− 0.217443i	−0.994072	−	0.108722i	$-0.965324\pi$
$631$	− 5.46211i	− 0.217443i	0.994072	−	0.108722i	$-0.0346757\pi$
$632$	0	0
$633$	9.33513i	0.371038i
$634$	0	0
$635$	−6.95947	−0.276178
$636$	0	0
$637$	0	0
$638$	0	0
$639$	12.6249i	0.499432i
$640$	0	0
$641$	1.42822	0.0564114	0.0282057	−	0.999602i	$-0.491021\pi$
$641$	1.42822	0.0564114	0.0282057	+	0.999602i	$0.491021\pi$
$642$	0	0
$643$	24.6036	0.970270	0.485135	−	0.874439i	$-0.338771\pi$
$643$	24.6036	0.970270	0.485135	+	0.874439i	$0.338771\pi$
$644$	0	0
$645$	15.5072	0.610597
$646$	0	0
$647$	−20.2785	−0.797229	−0.398614	−	0.917119i	$-0.630509\pi$
$647$	−20.2785	−0.797229	−0.398614	+	0.917119i	$0.630509\pi$
$648$	0	0
$649$	19.2740i	0.756570i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	42.8841	1.67818	0.839092	−	0.543990i	$-0.183087\pi$
$653$	42.8841	1.67818	0.839092	+	0.543990i	$0.183087\pi$
$654$	0	0
$655$	− 16.0060i	− 0.625407i
$656$	0	0
$657$	− 6.49011i	− 0.253203i
$658$	0	0
$659$	− 31.4717i	− 1.22596i	−0.790098	−	0.612981i	$-0.789970\pi$
$659$	− 31.4717i	− 1.22596i	0.790098	−	0.612981i	$-0.210030\pi$
$660$	0	0
$661$	− 12.3154i	− 0.479015i	−0.970895	−	0.239507i	$-0.923014\pi$
$661$	− 12.3154i	− 0.479015i	0.970895	−	0.239507i	$-0.0769859\pi$
$662$	0	0
$663$	−20.2484	−0.786384
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.03351i	0.156178i
$668$	0	0
$669$	5.48794	0.212176
$670$	0	0
$671$	15.1226	0.583801
$672$	0	0
$673$	37.1864	1.43343	0.716716	−	0.697365i	$-0.245645\pi$
$673$	37.1864	1.43343	0.716716	+	0.697365i	$0.245645\pi$
$674$	0	0
$675$	5.33530	0.205356
$676$	0	0
$677$	35.6046i	1.36840i	0.729296	+	0.684198i	$0.239848\pi$
$677$	35.6046i	1.36840i	−0.729296	+	0.684198i	$0.760152\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.03813	−0.0397810
$682$	0	0
$683$	− 32.2056i	− 1.23231i	−0.787623	−	0.616157i	$-0.788689\pi$
$683$	− 32.2056i	− 1.23231i	0.787623	−	0.616157i	$-0.211311\pi$
$684$	0	0
$685$	− 11.8622i	− 0.453230i
$686$	0	0
$687$	− 15.6039i	− 0.595325i
$688$	0	0
$689$	13.4141i	0.511035i
$690$	0	0
$691$	38.6168	1.46905	0.734527	−	0.678580i	$-0.237404\pi$
$691$	38.6168	1.46905	0.734527	+	0.678580i	$0.237404\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 38.7637i	− 1.47039i
$696$	0	0
$697$	5.75736	0.218076
$698$	0	0
$699$	−9.21561	−0.348566
$700$	0	0
$701$	30.8725	1.16604	0.583018	−	0.812459i	$-0.301872\pi$
$701$	30.8725	1.16604	0.583018	+	0.812459i	$0.301872\pi$
$702$	0	0
$703$	56.1730	2.11861
$704$	0	0
$705$	− 9.49562i	− 0.357626i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	39.8452	1.49642	0.748209	−	0.663463i	$-0.230914\pi$
$709$	39.8452	1.49642	0.748209	+	0.663463i	$0.230914\pi$
$710$	0	0
$711$	7.79367i	0.292285i
$712$	0	0
$713$	− 31.5729i	− 1.18241i
$714$	0	0
$715$	− 12.8781i	− 0.481613i
$716$	0	0
$717$	1.54809i	0.0578144i
$718$	0	0
$719$	−2.09266	−0.0780431	−0.0390216	−	0.999238i	$-0.512424\pi$
$719$	−2.09266	−0.0780431	−0.0390216	+	0.999238i	$0.512424\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 3.68527i	− 0.137057i
$724$	0	0
$725$	5.93207	0.220312
$726$	0	0
$727$	−21.5164	−0.798001	−0.399000	−	0.916951i	$-0.630643\pi$
$727$	−21.5164	−0.798001	−0.399000	+	0.916951i	$0.630643\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−33.3330	−1.23287
$732$	0	0
$733$	7.72856i	0.285461i	0.989762	+	0.142730i	$0.0455882\pi$
$733$	7.72856i	0.285461i	−0.989762	+	0.142730i	$0.954412\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.6163	−0.612070
$738$	0	0
$739$	− 16.1063i	− 0.592481i	−0.955113	−	0.296240i	$-0.904267\pi$
$739$	− 16.1063i	− 0.592481i	0.955113	−	0.296240i	$-0.0957329\pi$
$740$	0	0
$741$	21.5498i	0.791651i
$742$	0	0
$743$	− 29.3296i	− 1.07600i	−0.842945	−	0.537999i	$-0.819180\pi$
$743$	− 29.3296i	− 1.07600i	0.842945	−	0.537999i	$-0.180820\pi$
$744$	0	0
$745$	10.6174i	0.388993i
$746$	0	0
$747$	8.87502	0.324720
$748$	0	0
$749$	0	0
$750$	0	0
$751$	13.5991i	0.496238i	0.968730	+	0.248119i	$0.0798125\pi$
$751$	13.5991i	0.496238i	−0.968730	+	0.248119i	$0.920188\pi$
$752$	0	0
$753$	26.1940	0.954563
$754$	0	0
$755$	−60.9252	−2.21730
$756$	0	0
$757$	−2.59688	−0.0943851	−0.0471926	−	0.998886i	$-0.515027\pi$
$757$	−2.59688	−0.0943851	−0.0471926	+	0.998886i	$0.515027\pi$
$758$	0	0
$759$	4.95947	0.180017
$760$	0	0
$761$	− 17.2920i	− 0.626833i	−0.949616	−	0.313416i	$-0.898526\pi$
$761$	− 17.2920i	− 0.626833i	0.949616	−	0.313416i	$-0.101474\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−22.2159	−0.803216
$766$	0	0
$767$	41.3106i	1.49164i
$768$	0	0
$769$	41.4025i	1.49301i	0.665378	+	0.746507i	$0.268270\pi$
$769$	41.4025i	1.49301i	−0.665378	+	0.746507i	$0.731730\pi$
$770$	0	0
$771$	− 0.833147i	− 0.0300051i
$772$	0	0
$773$	− 29.4669i	− 1.05985i	−0.848044	−	0.529925i	$-0.822220\pi$
$773$	− 29.4669i	− 1.05985i	0.848044	−	0.529925i	$-0.177780\pi$
$774$	0	0
$775$	−46.4341	−1.66796
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 6.12738i	− 0.219536i
$780$	0	0
$781$	17.2594	0.617591
$782$	0	0
$783$	1.11185	0.0397344
$784$	0	0
$785$	−2.84976	−0.101712
$786$	0	0
$787$	54.1841	1.93146	0.965728	−	0.259557i	$-0.0835765\pi$
$787$	54.1841	1.93146	0.965728	+	0.259557i	$0.0835765\pi$
$788$	0	0
$789$	5.04212i	0.179504i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	32.4128	1.15101
$794$	0	0
$795$	14.7174i	0.521974i
$796$	0	0
$797$	− 0.512549i	− 0.0181554i	−0.999959	−	0.00907771i	$-0.997110\pi$
$797$	− 0.512549i	− 0.0181554i	0.999959	−	0.00907771i	$-0.00288956\pi$
$798$	0	0
$799$	20.4110i	0.722088i
$800$	0	0
$801$	12.6146i	0.445716i
$802$	0	0
$803$	−8.87261	−0.313108
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 4.85594i	− 0.170937i
$808$	0	0
$809$	−53.1407	−1.86833	−0.934164	−	0.356844i	$-0.883853\pi$
$809$	−53.1407	−1.86833	−0.934164	+	0.356844i	$0.883853\pi$
$810$	0	0
$811$	−10.8127	−0.379687	−0.189843	−	0.981814i	$-0.560798\pi$
$811$	−10.8127	−0.379687	−0.189843	+	0.981814i	$0.560798\pi$
$812$	0	0
$813$	−22.2488	−0.780300
$814$	0	0
$815$	62.2196	2.17946
$816$	0	0
$817$	35.4753i	1.24112i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.70078	−0.0942579	−0.0471290	−	0.998889i	$-0.515007\pi$
$821$	−2.70078	−0.0942579	−0.0471290	+	0.998889i	$0.515007\pi$
$822$	0	0
$823$	12.6132i	0.439670i	0.975537	+	0.219835i	$0.0705519\pi$
$823$	12.6132i	0.439670i	−0.975537	+	0.219835i	$0.929448\pi$
$824$	0	0
$825$	− 7.29388i	− 0.253940i
$826$	0	0
$827$	2.95803i	0.102861i	0.998677	+	0.0514304i	$0.0163780\pi$
$827$	2.95803i	0.102861i	−0.998677	+	0.0514304i	$0.983622\pi$
$828$	0	0
$829$	15.8101i	0.549106i	0.961572	+	0.274553i	$0.0885300\pi$
$829$	15.8101i	0.549106i	−0.961572	+	0.274553i	$0.911470\pi$
$830$	0	0
$831$	−0.670606	−0.0232631
$832$	0	0
$833$	0	0
$834$	0	0
$835$	81.9324i	2.83539i
$836$	0	0
$837$	−8.70319	−0.300826
$838$	0	0
$839$	−5.13130	−0.177152	−0.0885761	−	0.996069i	$-0.528232\pi$
$839$	−5.13130	−0.177152	−0.0885761	+	0.996069i	$0.528232\pi$
$840$	0	0
$841$	−27.7638	−0.957372
$842$	0	0
$843$	−25.5971	−0.881612
$844$	0	0
$845$	14.1911i	0.488187i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11.1115	0.381345
$850$	0	0
$851$	− 27.7084i	− 0.949831i
$852$	0	0
$853$	− 45.1627i	− 1.54634i	−0.634198	−	0.773170i	$-0.718670\pi$
$853$	− 45.1627i	− 1.54634i	0.634198	−	0.773170i	$-0.281330\pi$
$854$	0	0
$855$	23.6436i	0.808596i
$856$	0	0
$857$	1.47725i	0.0504618i	0.999682	+	0.0252309i	$0.00803210\pi$
$857$	1.47725i	0.0504618i	−0.999682	+	0.0252309i	$0.991968\pi$
$858$	0	0
$859$	−3.08183	−0.105151	−0.0525753	−	0.998617i	$-0.516743\pi$
$859$	−3.08183	−0.105151	−0.0525753	+	0.998617i	$0.516743\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 46.4000i	− 1.57948i	−0.613445	−	0.789738i	$-0.710217\pi$
$863$	− 46.4000i	− 1.57948i	0.613445	−	0.789738i	$-0.289783\pi$
$864$	0	0
$865$	−69.7381	−2.37117
$866$	0	0
$867$	30.7533	1.04444
$868$	0	0
$869$	10.6547	0.361436
$870$	0	0
$871$	−35.6144	−1.20675
$872$	0	0
$873$	13.8811i	0.469806i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−50.4774	−1.70450	−0.852251	−	0.523133i	$-0.824763\pi$
$877$	−50.4774	−1.70450	−0.852251	+	0.523133i	$0.824763\pi$
$878$	0	0
$879$	19.2818i	0.650357i
$880$	0	0
$881$	3.66920i	0.123619i	0.998088	+	0.0618093i	$0.0196870\pi$
$881$	3.66920i	0.123619i	−0.998088	+	0.0618093i	$0.980313\pi$
$882$	0	0
$883$	18.0393i	0.607071i	0.952820	+	0.303535i	$0.0981671\pi$
$883$	18.0393i	0.607071i	−0.952820	+	0.303535i	$0.901833\pi$
$884$	0	0
$885$	45.3245i	1.52357i
$886$	0	0
$887$	12.8996	0.433126	0.216563	−	0.976269i	$-0.430515\pi$
$887$	12.8996	0.433126	0.216563	+	0.976269i	$0.430515\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 1.36710i	− 0.0457995i
$892$	0	0
$893$	21.7227	0.726924
$894$	0	0
$895$	39.4863	1.31988
$896$	0	0
$897$	10.6298	0.354919
$898$	0	0
$899$	−9.67667	−0.322735
$900$	0	0
$901$	− 31.6353i	− 1.05392i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.82299	−0.293286
$906$	0	0
$907$	0.765170i	0.0254071i	0.999919	+	0.0127035i	$0.00404377\pi$
$907$	0.765170i	0.0254071i	−0.999919	+	0.0127035i	$0.995956\pi$
$908$	0	0
$909$	− 1.65891i	− 0.0550226i
$910$	0	0
$911$	18.7415i	0.620934i	0.950584	+	0.310467i	$0.100485\pi$
$911$	18.7415i	0.620934i	−0.950584	+	0.310467i	$0.899515\pi$
$912$	0	0
$913$	− 12.1330i	− 0.401544i
$914$	0	0
$915$	35.5622	1.17565
$916$	0	0
$917$	0	0
$918$	0	0
$919$	21.0460i	0.694242i	0.937820	+	0.347121i	$0.112841\pi$
$919$	21.0460i	0.694242i	−0.937820	+	0.347121i	$0.887159\pi$
$920$	0	0
$921$	26.0058	0.856920
$922$	0	0
$923$	36.9928	1.21763
$924$	0	0
$925$	−40.7506	−1.33987
$926$	0	0
$927$	1.30791	0.0429573
$928$	0	0
$929$	− 32.7286i	− 1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$929$	− 32.7286i	− 1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	19.0772	0.624561
$934$	0	0
$935$	30.3713i	0.993246i
$936$	0	0
$937$	− 2.53824i	− 0.0829207i	−0.999140	−	0.0414603i	$-0.986799\pi$
$937$	− 2.53824i	− 0.0829207i	0.999140	−	0.0414603i	$-0.0132010\pi$
$938$	0	0
$939$	21.3583i	0.697001i
$940$	0	0
$941$	− 46.1741i	− 1.50523i	−0.658459	−	0.752617i	$-0.728791\pi$
$941$	− 46.1741i	− 1.50523i	0.658459	−	0.752617i	$-0.271209\pi$
$942$	0	0
$943$	−3.02244	−0.0984242
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.51338i	− 0.0491782i	−0.999698	−	0.0245891i	$-0.992172\pi$
$947$	− 1.51338i	− 0.0491782i	0.999698	−	0.0245891i	$-0.00782775\pi$
$948$	0	0
$949$	−19.0170	−0.617318
$950$	0	0
$951$	−7.74131	−0.251029
$952$	0	0
$953$	−43.1132	−1.39657	−0.698287	−	0.715818i	$-0.746054\pi$
$953$	−43.1132	−1.39657	−0.698287	+	0.715818i	$0.746054\pi$
$954$	0	0
$955$	−73.5741	−2.38080
$956$	0	0
$957$	− 1.52001i	− 0.0491350i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	44.7454	1.44340
$962$	0	0
$963$	− 7.15204i	− 0.230471i
$964$	0	0
$965$	46.9894i	1.51264i
$966$	0	0
$967$	22.5218i	0.724253i	0.932129	+	0.362127i	$0.117949\pi$
$967$	22.5218i	0.724253i	−0.932129	+	0.362127i	$0.882051\pi$
$968$	0	0
$969$	− 50.8223i	− 1.63265i
$970$	0	0
$971$	24.4894	0.785902	0.392951	−	0.919559i	$-0.371454\pi$
$971$	24.4894	0.785902	0.392951	+	0.919559i	$0.371454\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 15.6332i	− 0.500665i
$976$	0	0
$977$	16.6382	0.532302	0.266151	−	0.963931i	$-0.414248\pi$
$977$	16.6382	0.532302	0.266151	+	0.963931i	$0.414248\pi$
$978$	0	0
$979$	17.2454	0.551166
$980$	0	0
$981$	−15.0443	−0.480327
$982$	0	0
$983$	−5.35617	−0.170835	−0.0854177	−	0.996345i	$-0.527222\pi$
$983$	−5.35617	−0.170835	−0.0854177	+	0.996345i	$0.527222\pi$
$984$	0	0
$985$	78.9021i	2.51403i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17.4988	0.556430
$990$	0	0
$991$	4.67605i	0.148540i	0.997238	+	0.0742698i	$0.0236626\pi$
$991$	4.67605i	0.148540i	−0.997238	+	0.0742698i	$0.976337\pi$
$992$	0	0
$993$	− 0.675988i	− 0.0214518i
$994$	0	0
$995$	19.4389i	0.616255i
$996$	0	0
$997$	− 0.906930i	− 0.0287228i	−0.999897	−	0.0143614i	$-0.995428\pi$
$997$	− 0.906930i	− 0.0287228i	0.999897	−	0.0143614i	$-0.00457153\pi$
$998$	0	0
$999$	−7.63792	−0.241653

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.b.k.1567.7	yes	8
3.2	odd	2		7056.2.b.x.1567.2		8
4.3	odd	2		2352.2.b.l.1567.7	yes	8
7.2	even	3		2352.2.bl.r.31.4		8
7.3	odd	6		2352.2.bl.q.607.4		8
7.4	even	3		2352.2.bl.t.607.1		8
7.5	odd	6		2352.2.bl.o.31.1		8
7.6	odd	2		2352.2.b.l.1567.2	yes	8
12.11	even	2		7056.2.b.w.1567.2		8
21.20	even	2		7056.2.b.w.1567.7		8
28.3	even	6		2352.2.bl.r.607.4		8
28.11	odd	6		2352.2.bl.o.607.1		8
28.19	even	6		2352.2.bl.t.31.1		8
28.23	odd	6		2352.2.bl.q.31.4		8
28.27	even	2	inner	2352.2.b.k.1567.2	✓	8
84.83	odd	2		7056.2.b.x.1567.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.2.b.k.1567.2	✓	8	28.27	even	2	inner
2352.2.b.k.1567.7	yes	8	1.1	even	1	trivial
2352.2.b.l.1567.2	yes	8	7.6	odd	2
2352.2.b.l.1567.7	yes	8	4.3	odd	2
2352.2.bl.o.31.1		8	7.5	odd	6
2352.2.bl.o.607.1		8	28.11	odd	6
2352.2.bl.q.31.4		8	28.23	odd	6
2352.2.bl.q.607.4		8	7.3	odd	6
2352.2.bl.r.31.4		8	7.2	even	3
2352.2.bl.r.607.4		8	28.3	even	6
2352.2.bl.t.31.1		8	28.19	even	6
2352.2.bl.t.607.1		8	7.4	even	3
7056.2.b.w.1567.2		8	12.11	even	2
7056.2.b.w.1567.7		8	21.20	even	2
7056.2.b.x.1567.2		8	3.2	odd	2
7056.2.b.x.1567.7		8	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.21486i q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.21486i q^{5} +1.00000 q^{9} -1.36710i q^{11} -2.93015i q^{13} -3.21486i q^{15} +6.91037i q^{17} +7.35449 q^{19} -3.62774i q^{23} -5.33530 q^{25} -1.00000 q^{27} -1.11185 q^{29} +8.70319 q^{31} +1.36710i q^{33} +7.63792 q^{37} +2.93015i q^{39} -0.833147i q^{41} +4.82362i q^{43} +3.21486i q^{45} +2.95367 q^{47} -6.91037i q^{51} -4.57794 q^{53} +4.39502 q^{55} -7.35449 q^{57} -14.0985 q^{59} +11.0618i q^{61} +9.42002 q^{65} -12.1545i q^{67} +3.62774i q^{69} +12.6249i q^{71} -6.49011i q^{73} +5.33530 q^{75} +7.79367i q^{79} +1.00000 q^{81} +8.87502 q^{83} -22.2159 q^{85} +1.11185 q^{87} +12.6146i q^{89} -8.70319 q^{93} +23.6436i q^{95} +13.8811i q^{97} -1.36710i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 8 * q^9	\( 8 q - 8 q^{3} + 8 q^{9} - 24 q^{25} - 8 q^{27} + 16 q^{29} + 32 q^{31} + 16 q^{47} + 16 q^{53} + 64 q^{55} - 48 q^{59} - 16 q^{65} + 24 q^{75} + 8 q^{81} - 64 q^{85} - 16 q^{87} - 32 q^{93}+O(q^{100}) \) 8 * q - 8 * q^3 + 8 * q^9 - 24 * q^25 - 8 * q^27 + 16 * q^29 + 32 * q^31 + 16 * q^47 + 16 * q^53 + 64 * q^55 - 48 * q^59 - 16 * q^65 + 24 * q^75 + 8 * q^81 - 64 * q^85 - 16 * q^87 - 32 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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