LMFDB - Embedded newform 2352.2.b.k.1567.8

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			1567.8
Root			$-0.662827 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	2352.1567
Dual form			2352.2.b.k.1567.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +4.29725i q^{5} +1.00000 q^{9} -5.06262i q^{11} +3.37849i q^{13} -4.29725i q^{15} +2.76652i q^{17} -4.70319 q^{19} +4.83120i q^{23} -13.4663 q^{25} -1.00000 q^{27} +2.46054 q^{29} +5.69764 q^{31} +5.06262i q^{33} -2.33530 q^{37} -3.37849i q^{39} -5.14822i q^{41} +13.0199i q^{43} +4.29725i q^{45} -5.35449 q^{47} -2.76652i q^{51} -4.22371 q^{53} +21.7553 q^{55} +4.70319 q^{57} -9.60498 q^{59} -3.87698i q^{61} -14.5182 q^{65} +4.76342i q^{67} -4.83120i q^{69} -12.3181i q^{71} +11.5102i q^{73} +13.4663 q^{75} -13.9166i q^{79} +1.00000 q^{81} -7.32191 q^{83} -11.8884 q^{85} -2.46054 q^{87} -14.0448i q^{89} -5.69764 q^{93} -20.2108i q^{95} -14.5716i q^{97} -5.06262i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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

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
$5$	4.29725i	1.92179i	0.276916	+	0.960894i	$0.410688\pi$
$5$	4.29725i	1.92179i	−0.276916	+	0.960894i	$0.589312\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 5.06262i	− 1.52644i	−0.646141	−	0.763218i	$-0.723618\pi$
$11$	− 5.06262i	− 1.52644i	0.646141	−	0.763218i	$-0.276382\pi$
$12$	0	0
$13$	3.37849i	0.937025i	0.883457	+	0.468513i	$0.155210\pi$
$13$	3.37849i	0.937025i	−0.883457	+	0.468513i	$0.844790\pi$
$14$	0	0
$15$	− 4.29725i	− 1.10954i
$16$	0	0
$17$	2.76652i	0.670978i	0.942044	+	0.335489i	$0.108902\pi$
$17$	2.76652i	0.670978i	−0.942044	+	0.335489i	$0.891098\pi$
$18$	0	0
$19$	−4.70319	−1.07898	−0.539492	−	0.841990i	$-0.681384\pi$
$19$	−4.70319	−1.07898	−0.539492	+	0.841990i	$0.681384\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.83120i	1.00737i	0.863886	+	0.503687i	$0.168024\pi$
$23$	4.83120i	1.00737i	−0.863886	+	0.503687i	$0.831976\pi$
$24$	0	0
$25$	−13.4663	−2.69327
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.46054	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$29$	2.46054	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$30$	0	0
$31$	5.69764	1.02333	0.511663	−	0.859186i	$-0.329030\pi$
$31$	5.69764	1.02333	0.511663	+	0.859186i	$0.329030\pi$
$32$	0	0
$33$	5.06262i	0.881288i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.33530	−0.383921	−0.191961	−	0.981403i	$-0.561485\pi$
$37$	−2.33530	−0.383921	−0.191961	+	0.981403i	$0.561485\pi$
$38$	0	0
$39$	− 3.37849i	− 0.540992i
$40$	0	0
$41$	− 5.14822i	− 0.804018i	−0.915636	−	0.402009i	$-0.868312\pi$
$41$	− 5.14822i	− 0.804018i	0.915636	−	0.402009i	$-0.131688\pi$
$42$	0	0
$43$	13.0199i	1.98552i	0.120117	+	0.992760i	$0.461673\pi$
$43$	13.0199i	1.98552i	−0.120117	+	0.992760i	$0.538327\pi$
$44$	0	0
$45$	4.29725i	0.640596i
$46$	0	0
$47$	−5.35449	−0.781033	−0.390517	−	0.920596i	$-0.627704\pi$
$47$	−5.35449	−0.781033	−0.390517	+	0.920596i	$0.627704\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 2.76652i	− 0.387390i
$52$	0	0
$53$	−4.22371	−0.580171	−0.290085	−	0.957001i	$-0.593684\pi$
$53$	−4.22371	−0.580171	−0.290085	+	0.957001i	$0.593684\pi$
$54$	0	0
$55$	21.7553	2.93349
$56$	0	0
$57$	4.70319	0.622952
$58$	0	0
$59$	−9.60498	−1.25046	−0.625231	−	0.780440i	$-0.714995\pi$
$59$	−9.60498	−1.25046	−0.625231	+	0.780440i	$0.714995\pi$
$60$	0	0
$61$	− 3.87698i	− 0.496397i	−0.968709	−	0.248198i	$-0.920162\pi$
$61$	− 3.87698i	− 0.496397i	0.968709	−	0.248198i	$-0.0798385\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.5182	−1.80076
$66$	0	0
$67$	4.76342i	0.581944i	0.956732	+	0.290972i	$0.0939787\pi$
$67$	4.76342i	0.581944i	−0.956732	+	0.290972i	$0.906021\pi$
$68$	0	0
$69$	− 4.83120i	− 0.581608i
$70$	0	0
$71$	− 12.3181i	− 1.46189i	−0.682437	−	0.730944i	$-0.739080\pi$
$71$	− 12.3181i	− 1.46189i	0.682437	−	0.730944i	$-0.260920\pi$
$72$	0	0
$73$	11.5102i	1.34716i	0.739113	+	0.673581i	$0.235245\pi$
$73$	11.5102i	1.34716i	−0.739113	+	0.673581i	$0.764755\pi$
$74$	0	0
$75$	13.4663	1.55496
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 13.9166i	− 1.56574i	−0.622185	−	0.782870i	$-0.713755\pi$
$79$	− 13.9166i	− 1.56574i	0.622185	−	0.782870i	$-0.286245\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.32191	−0.803685	−0.401842	−	0.915709i	$-0.631630\pi$
$83$	−7.32191	−0.803685	−0.401842	+	0.915709i	$0.631630\pi$
$84$	0	0
$85$	−11.8884	−1.28948
$86$	0	0
$87$	−2.46054	−0.263798
$88$	0	0
$89$	− 14.0448i	− 1.48874i	−0.667765	−	0.744372i	$-0.732749\pi$
$89$	− 14.0448i	− 1.48874i	0.667765	−	0.744372i	$-0.267251\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.69764	−0.590818
$94$	0	0
$95$	− 20.2108i	− 2.07358i
$96$	0	0
$97$	− 14.5716i	− 1.47952i	−0.672868	−	0.739762i	$-0.734938\pi$
$97$	− 14.5716i	− 1.47952i	0.672868	−	0.739762i	$-0.265062\pi$
$98$	0	0
$99$	− 5.06262i	− 0.508812i
$100$	0	0
$101$	13.1087i	1.30436i	0.758063	+	0.652181i	$0.226146\pi$
$101$	13.1087i	1.30436i	−0.758063	+	0.652181i	$0.773854\pi$
$102$	0	0
$103$	−0.209698	−0.0206622	−0.0103311	−	0.999947i	$-0.503289\pi$
$103$	−0.209698	−0.0206622	−0.0103311	+	0.999947i	$0.503289\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6.42657i	− 0.621280i	−0.950528	−	0.310640i	$-0.899457\pi$
$107$	− 6.42657i	− 0.621280i	0.950528	−	0.310640i	$-0.100543\pi$
$108$	0	0
$109$	0.940024	0.0900379	0.0450190	−	0.998986i	$-0.485665\pi$
$109$	0.940024	0.0900379	0.0450190	+	0.998986i	$0.485665\pi$
$110$	0	0
$111$	2.33530	0.221657
$112$	0	0
$113$	1.09821	0.103311	0.0516554	−	0.998665i	$-0.483550\pi$
$113$	1.09821	0.103311	0.0516554	+	0.998665i	$0.483550\pi$
$114$	0	0
$115$	−20.7609	−1.93596
$116$	0	0
$117$	3.37849i	0.312342i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−14.6301	−1.33001
$122$	0	0
$123$	5.14822i	0.464200i
$124$	0	0
$125$	− 36.3820i	− 3.25411i
$126$	0	0
$127$	− 5.22625i	− 0.463755i	−0.972745	−	0.231877i	$-0.925513\pi$
$127$	− 5.22625i	− 0.463755i	0.972745	−	0.231877i	$-0.0744868\pi$
$128$	0	0
$129$	− 13.0199i	− 1.14634i
$130$	0	0
$131$	−0.875015	−0.0764504	−0.0382252	−	0.999269i	$-0.512170\pi$
$131$	−0.875015	−0.0764504	−0.0382252	+	0.999269i	$0.512170\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 4.29725i	− 0.369848i
$136$	0	0
$137$	0.236838	0.0202344	0.0101172	−	0.999949i	$-0.496780\pi$
$137$	0.236838	0.0202344	0.0101172	+	0.999949i	$0.496780\pi$
$138$	0	0
$139$	3.00555	0.254927	0.127464	−	0.991843i	$-0.459316\pi$
$139$	3.00555	0.254927	0.127464	+	0.991843i	$0.459316\pi$
$140$	0	0
$141$	5.35449	0.450930
$142$	0	0
$143$	17.1040	1.43031
$144$	0	0
$145$	10.5736i	0.878087i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.8017	−1.21260	−0.606299	−	0.795237i	$-0.707347\pi$
$149$	−14.8017	−1.21260	−0.606299	+	0.795237i	$0.707347\pi$
$150$	0	0
$151$	12.3654i	1.00628i	0.864205	+	0.503140i	$0.167822\pi$
$151$	12.3654i	1.00628i	−0.864205	+	0.503140i	$0.832178\pi$
$152$	0	0
$153$	2.76652i	0.223659i
$154$	0	0
$155$	24.4842i	1.96662i
$156$	0	0
$157$	10.8906i	0.869164i	0.900632	+	0.434582i	$0.143104\pi$
$157$	10.8906i	0.869164i	−0.900632	+	0.434582i	$0.856896\pi$
$158$	0	0
$159$	4.22371	0.334962
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 1.51024i	− 0.118291i	−0.998249	−	0.0591455i	$-0.981162\pi$
$163$	− 1.51024i	− 0.118291i	0.998249	−	0.0591455i	$-0.0188376\pi$
$164$	0	0
$165$	−21.7553	−1.69365
$166$	0	0
$167$	−10.3333	−0.799612	−0.399806	−	0.916600i	$-0.630922\pi$
$167$	−10.3333	−0.799612	−0.399806	+	0.916600i	$0.630922\pi$
$168$	0	0
$169$	1.58579	0.121984
$170$	0	0
$171$	−4.70319	−0.359662
$172$	0	0
$173$	− 3.35642i	− 0.255184i	−0.991827	−	0.127592i	$-0.959275\pi$
$173$	− 3.35642i	− 0.255184i	0.991827	−	0.127592i	$-0.0407248\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.60498	0.721954
$178$	0	0
$179$	− 13.2589i	− 0.991018i	−0.868603	−	0.495509i	$-0.834982\pi$
$179$	− 13.2589i	− 0.991018i	0.868603	−	0.495509i	$-0.165018\pi$
$180$	0	0
$181$	11.8519i	0.880946i	0.897766	+	0.440473i	$0.145189\pi$
$181$	11.8519i	0.880946i	−0.897766	+	0.440473i	$0.854811\pi$
$182$	0	0
$183$	3.87698i	0.286595i
$184$	0	0
$185$	− 10.0354i	− 0.737816i
$186$	0	0
$187$	14.0058	1.02421
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.34660i	0.0974368i	0.998813	+	0.0487184i	$0.0155137\pi$
$191$	1.34660i	0.0974368i	−0.998813	+	0.0487184i	$0.984486\pi$
$192$	0	0
$193$	−26.1154	−1.87982	−0.939912	−	0.341416i	$-0.889094\pi$
$193$	−26.1154	−1.87982	−0.939912	+	0.341416i	$0.889094\pi$
$194$	0	0
$195$	14.5182	1.03967
$196$	0	0
$197$	−7.49083	−0.533699	−0.266850	−	0.963738i	$-0.585983\pi$
$197$	−7.49083	−0.533699	−0.266850	+	0.963738i	$0.585983\pi$
$198$	0	0
$199$	−4.49349	−0.318535	−0.159267	−	0.987235i	$-0.550913\pi$
$199$	−4.49349	−0.318535	−0.159267	+	0.987235i	$0.550913\pi$
$200$	0	0
$201$	− 4.76342i	− 0.335986i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	22.1232	1.54515
$206$	0	0
$207$	4.83120i	0.335792i
$208$	0	0
$209$	23.8104i	1.64700i
$210$	0	0
$211$	− 5.93122i	− 0.408322i	−0.978937	−	0.204161i	$-0.934553\pi$
$211$	− 5.93122i	− 0.408322i	0.978937	−	0.204161i	$-0.0654466\pi$
$212$	0	0
$213$	12.3181i	0.844022i
$214$	0	0
$215$	−55.9498	−3.81575
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 11.5102i	− 0.777784i
$220$	0	0
$221$	−9.34665	−0.628724
$222$	0	0
$223$	−20.6163	−1.38057	−0.690286	−	0.723537i	$-0.742515\pi$
$223$	−20.6163	−1.38057	−0.690286	+	0.723537i	$0.742515\pi$
$224$	0	0
$225$	−13.4663	−0.897756
$226$	0	0
$227$	−20.4911	−1.36004	−0.680021	−	0.733193i	$-0.738029\pi$
$227$	−20.4911	−1.36004	−0.680021	+	0.733193i	$0.738029\pi$
$228$	0	0
$229$	29.0409i	1.91908i	0.281576	+	0.959539i	$0.409143\pi$
$229$	29.0409i	1.91908i	−0.281576	+	0.959539i	$0.590857\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.7364	1.16195	0.580975	−	0.813922i	$-0.302672\pi$
$233$	17.7364	1.16195	0.580975	+	0.813922i	$0.302672\pi$
$234$	0	0
$235$	− 23.0096i	− 1.50098i
$236$	0	0
$237$	13.9166i	0.903981i
$238$	0	0
$239$	20.4248i	1.32117i	0.750750	+	0.660586i	$0.229692\pi$
$239$	20.4248i	1.32117i	−0.750750	+	0.660586i	$0.770308\pi$
$240$	0	0
$241$	− 3.25742i	− 0.209829i	−0.994481	−	0.104915i	$-0.966543\pi$
$241$	− 3.25742i	− 0.209829i	0.994481	−	0.104915i	$-0.0334569\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 15.8897i	− 1.01104i
$248$	0	0
$249$	7.32191	0.464007
$250$	0	0
$251$	−3.95633	−0.249721	−0.124861	−	0.992174i	$-0.539848\pi$
$251$	−3.95633	−0.249721	−0.124861	+	0.992174i	$0.539848\pi$
$252$	0	0
$253$	24.4585	1.53769
$254$	0	0
$255$	11.8884	0.744481
$256$	0	0
$257$	5.14822i	0.321137i	0.987025	+	0.160569i	$0.0513328\pi$
$257$	5.14822i	0.321137i	−0.987025	+	0.160569i	$0.948667\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.46054	0.152304
$262$	0	0
$263$	− 0.0785021i	− 0.00484065i	−0.999997	−	0.00242032i	$-0.999230\pi$
$263$	− 0.0785021i	− 0.00484065i	0.999997	−	0.00242032i	$-0.000770414\pi$
$264$	0	0
$265$	− 18.1503i	− 1.11497i
$266$	0	0
$267$	14.0448i	0.859527i
$268$	0	0
$269$	0.883296i	0.0538555i	0.999637	+	0.0269277i	$0.00857240\pi$
$269$	0.883296i	0.0538555i	−0.999637	+	0.0269277i	$0.991428\pi$
$270$	0	0
$271$	19.3084	1.17290	0.586451	−	0.809984i	$-0.300524\pi$
$271$	19.3084	1.17290	0.586451	+	0.809984i	$0.300524\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	68.1749i	4.11110i
$276$	0	0
$277$	16.9327	1.01739	0.508694	−	0.860948i	$-0.330129\pi$
$277$	16.9327	1.01739	0.508694	+	0.860948i	$0.330129\pi$
$278$	0	0
$279$	5.69764	0.341109
$280$	0	0
$281$	5.05417	0.301507	0.150753	−	0.988571i	$-0.451830\pi$
$281$	5.05417	0.301507	0.150753	+	0.988571i	$0.451830\pi$
$282$	0	0
$283$	19.1156	1.13631	0.568153	−	0.822923i	$-0.307658\pi$
$283$	19.1156	1.13631	0.568153	+	0.822923i	$0.307658\pi$
$284$	0	0
$285$	20.2108i	1.19718i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	9.34639	0.549788
$290$	0	0
$291$	14.5716i	0.854204i
$292$	0	0
$293$	− 2.37837i	− 0.138946i	−0.997584	−	0.0694730i	$-0.977868\pi$
$293$	− 2.37837i	− 0.138946i	0.997584	−	0.0694730i	$-0.0221318\pi$
$294$	0	0
$295$	− 41.2750i	− 2.40312i
$296$	0	0
$297$	5.06262i	0.293763i
$298$	0	0
$299$	−16.3222	−0.943936
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 13.1087i	− 0.753073i
$304$	0	0
$305$	16.6604	0.953969
$306$	0	0
$307$	−4.89599	−0.279429	−0.139714	−	0.990192i	$-0.544618\pi$
$307$	−4.89599	−0.279429	−0.139714	+	0.990192i	$0.544618\pi$
$308$	0	0
$309$	0.209698	0.0119293
$310$	0	0
$311$	−10.4800	−0.594266	−0.297133	−	0.954836i	$-0.596030\pi$
$311$	−10.4800	−0.594266	−0.297133	+	0.954836i	$0.596030\pi$
$312$	0	0
$313$	9.60172i	0.542722i	0.962478	+	0.271361i	$0.0874737\pi$
$313$	9.60172i	0.542722i	−0.962478	+	0.271361i	$0.912526\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.7934	−0.943214	−0.471607	−	0.881809i	$-0.656326\pi$
$317$	−16.7934	−0.943214	−0.471607	+	0.881809i	$0.656326\pi$
$318$	0	0
$319$	− 12.4568i	− 0.697446i
$320$	0	0
$321$	6.42657i	0.358696i
$322$	0	0
$323$	− 13.0114i	− 0.723976i
$324$	0	0
$325$	− 45.4960i	− 2.52366i
$326$	0	0
$327$	−0.940024	−0.0519834
$328$	0	0
$329$	0	0
$330$	0	0
$331$	26.8362i	1.47505i	0.675318	+	0.737526i	$0.264006\pi$
$331$	26.8362i	1.47505i	−0.675318	+	0.737526i	$0.735994\pi$
$332$	0	0
$333$	−2.33530	−0.127974
$334$	0	0
$335$	−20.4696	−1.11837
$336$	0	0
$337$	23.0827	1.25739	0.628697	−	0.777651i	$-0.283589\pi$
$337$	23.0827	1.25739	0.628697	+	0.777651i	$0.283589\pi$
$338$	0	0
$339$	−1.09821	−0.0596465
$340$	0	0
$341$	− 28.8450i	− 1.56204i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	20.7609	1.11773
$346$	0	0
$347$	− 1.30690i	− 0.0701580i	−0.999385	−	0.0350790i	$-0.988832\pi$
$347$	− 1.30690i	− 0.0701580i	0.999385	−	0.0350790i	$-0.0111683\pi$
$348$	0	0
$349$	26.5489i	1.42113i	0.703633	+	0.710564i	$0.251560\pi$
$349$	26.5489i	1.42113i	−0.703633	+	0.710564i	$0.748440\pi$
$350$	0	0
$351$	− 3.37849i	− 0.180331i
$352$	0	0
$353$	16.8893i	0.898928i	0.893298	+	0.449464i	$0.148385\pi$
$353$	16.8893i	0.898928i	−0.893298	+	0.449464i	$0.851615\pi$
$354$	0	0
$355$	52.9339	2.80944
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 33.5901i	− 1.77282i	−0.462904	−	0.886408i	$-0.653193\pi$
$359$	− 33.5901i	− 1.77282i	0.462904	−	0.886408i	$-0.346807\pi$
$360$	0	0
$361$	3.11995	0.164208
$362$	0	0
$363$	14.6301	0.767880
$364$	0	0
$365$	−49.4620	−2.58896
$366$	0	0
$367$	7.35978	0.384178	0.192089	−	0.981378i	$-0.438474\pi$
$367$	7.35978	0.384178	0.192089	+	0.981378i	$0.438474\pi$
$368$	0	0
$369$	− 5.14822i	− 0.268006i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3.63008	0.187958	0.0939791	−	0.995574i	$-0.470041\pi$
$373$	3.63008	0.187958	0.0939791	+	0.995574i	$0.470041\pi$
$374$	0	0
$375$	36.3820i	1.87876i
$376$	0	0
$377$	8.31293i	0.428138i
$378$	0	0
$379$	− 13.6647i	− 0.701908i	−0.936393	−	0.350954i	$-0.885857\pi$
$379$	− 13.6647i	− 0.701908i	0.936393	−	0.350954i	$-0.114143\pi$
$380$	0	0
$381$	5.22625i	0.267749i
$382$	0	0
$383$	13.6569	0.697833	0.348916	−	0.937154i	$-0.386550\pi$
$383$	13.6569	0.697833	0.348916	+	0.937154i	$0.386550\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	13.0199i	0.661840i
$388$	0	0
$389$	26.9080	1.36429	0.682144	−	0.731218i	$-0.261048\pi$
$389$	26.9080	1.36429	0.682144	+	0.731218i	$0.261048\pi$
$390$	0	0
$391$	−13.3656	−0.675927
$392$	0	0
$393$	0.875015	0.0441387
$394$	0	0
$395$	59.8031	3.00902
$396$	0	0
$397$	− 36.7242i	− 1.84314i	−0.388216	−	0.921568i	$-0.626909\pi$
$397$	− 36.7242i	− 1.84314i	0.388216	−	0.921568i	$-0.373091\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−27.6816	−1.38235	−0.691176	−	0.722686i	$-0.742907\pi$
$401$	−27.6816	−1.38235	−0.691176	+	0.722686i	$0.742907\pi$
$402$	0	0
$403$	19.2494i	0.958883i
$404$	0	0
$405$	4.29725i	0.213532i
$406$	0	0
$407$	11.8227i	0.586032i
$408$	0	0
$409$	− 8.42820i	− 0.416747i	−0.978049	−	0.208374i	$-0.933183\pi$
$409$	− 8.42820i	− 0.416747i	0.978049	−	0.208374i	$-0.0668170\pi$
$410$	0	0
$411$	−0.236838	−0.0116824
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 31.4641i	− 1.54451i
$416$	0	0
$417$	−3.00555	−0.147182
$418$	0	0
$419$	18.3029	0.894154	0.447077	−	0.894495i	$-0.352465\pi$
$419$	18.3029	0.894154	0.447077	+	0.894495i	$0.352465\pi$
$420$	0	0
$421$	22.6274	1.10279	0.551396	−	0.834243i	$-0.314095\pi$
$421$	22.6274	1.10279	0.551396	+	0.834243i	$0.314095\pi$
$422$	0	0
$423$	−5.35449	−0.260344
$424$	0	0
$425$	− 37.2549i	− 1.80713i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−17.1040	−0.825789
$430$	0	0
$431$	4.44906i	0.214304i	0.994243	+	0.107152i	$0.0341731\pi$
$431$	4.44906i	0.214304i	−0.994243	+	0.107152i	$0.965827\pi$
$432$	0	0
$433$	− 1.66205i	− 0.0798730i	−0.999202	−	0.0399365i	$-0.987284\pi$
$433$	− 1.66205i	− 0.0798730i	0.999202	−	0.0399365i	$-0.0127156\pi$
$434$	0	0
$435$	− 10.5736i	− 0.506964i
$436$	0	0
$437$	− 22.7220i	− 1.08694i
$438$	0	0
$439$	−18.7551	−0.895130	−0.447565	−	0.894251i	$-0.647709\pi$
$439$	−18.7551	−0.895130	−0.447565	+	0.894251i	$0.647709\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	33.0314i	1.56937i	0.619896	+	0.784684i	$0.287175\pi$
$443$	33.0314i	1.56937i	−0.619896	+	0.784684i	$0.712825\pi$
$444$	0	0
$445$	60.3539	2.86105
$446$	0	0
$447$	14.8017	0.700094
$448$	0	0
$449$	9.29441	0.438630	0.219315	−	0.975654i	$-0.429618\pi$
$449$	9.29441	0.438630	0.219315	+	0.975654i	$0.429618\pi$
$450$	0	0
$451$	−26.0635	−1.22728
$452$	0	0
$453$	− 12.3654i	− 0.580976i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−27.4448	−1.28381	−0.641906	−	0.766784i	$-0.721856\pi$
$457$	−27.4448	−1.28381	−0.641906	+	0.766784i	$0.721856\pi$
$458$	0	0
$459$	− 2.76652i	− 0.129130i
$460$	0	0
$461$	− 4.28209i	− 0.199437i	−0.995016	−	0.0997183i	$-0.968206\pi$
$461$	− 4.28209i	− 0.199437i	0.995016	−	0.0997183i	$-0.0317942\pi$
$462$	0	0
$463$	16.1278i	0.749521i	0.927122	+	0.374760i	$0.122275\pi$
$463$	16.1278i	0.749521i	−0.927122	+	0.374760i	$0.877725\pi$
$464$	0	0
$465$	− 24.4842i	− 1.13543i
$466$	0	0
$467$	−27.5403	−1.27441	−0.637207	−	0.770692i	$-0.719910\pi$
$467$	−27.5403	−1.27441	−0.637207	+	0.770692i	$0.719910\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 10.8906i	− 0.501812i
$472$	0	0
$473$	65.9149	3.03077
$474$	0	0
$475$	63.3347	2.90600
$476$	0	0
$477$	−4.22371	−0.193390
$478$	0	0
$479$	19.5509	0.893304	0.446652	−	0.894708i	$-0.352616\pi$
$479$	19.5509	0.893304	0.446652	+	0.894708i	$0.352616\pi$
$480$	0	0
$481$	− 7.88980i	− 0.359744i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	62.6179	2.84333
$486$	0	0
$487$	37.7917i	1.71250i	0.516558	+	0.856252i	$0.327213\pi$
$487$	37.7917i	1.71250i	−0.516558	+	0.856252i	$0.672787\pi$
$488$	0	0
$489$	1.51024i	0.0682954i
$490$	0	0
$491$	29.2605i	1.32051i	0.751042	+	0.660254i	$0.229551\pi$
$491$	29.2605i	1.32051i	−0.751042	+	0.660254i	$0.770449\pi$
$492$	0	0
$493$	6.80713i	0.306578i
$494$	0	0
$495$	21.7553	0.977829
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12.6422i	0.565944i	0.959128	+	0.282972i	$0.0913203\pi$
$499$	12.6422i	0.565944i	−0.959128	+	0.282972i	$0.908680\pi$
$500$	0	0
$501$	10.3333	0.461656
$502$	0	0
$503$	−27.0714	−1.20706	−0.603528	−	0.797342i	$-0.706239\pi$
$503$	−27.0714	−1.20706	−0.603528	+	0.797342i	$0.706239\pi$
$504$	0	0
$505$	−56.3312	−2.50671
$506$	0	0
$507$	−1.58579	−0.0704272
$508$	0	0
$509$	− 11.6983i	− 0.518520i	−0.965808	−	0.259260i	$-0.916521\pi$
$509$	− 11.6983i	− 0.518520i	0.965808	−	0.259260i	$-0.0834786\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.70319	0.207651
$514$	0	0
$515$	− 0.901124i	− 0.0397083i
$516$	0	0
$517$	27.1077i	1.19220i
$518$	0	0
$519$	3.35642i	0.147330i
$520$	0	0
$521$	7.20643i	0.315719i	0.987462	+	0.157860i	$0.0504594\pi$
$521$	7.20643i	0.315719i	−0.987462	+	0.157860i	$0.949541\pi$
$522$	0	0
$523$	17.7145	0.774602	0.387301	−	0.921953i	$-0.373407\pi$
$523$	17.7145	0.774602	0.387301	+	0.921953i	$0.373407\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.7626i	0.686630i
$528$	0	0
$529$	−0.340485	−0.0148037
$530$	0	0
$531$	−9.60498	−0.416821
$532$	0	0
$533$	17.3932	0.753385
$534$	0	0
$535$	27.6166	1.19397
$536$	0	0
$537$	13.2589i	0.572165i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	23.1180	0.993921	0.496961	−	0.867773i	$-0.334449\pi$
$541$	23.1180	0.993921	0.496961	+	0.867773i	$0.334449\pi$
$542$	0	0
$543$	− 11.8519i	− 0.508615i
$544$	0	0
$545$	4.03951i	0.173034i
$546$	0	0
$547$	− 16.0524i	− 0.686351i	−0.939271	−	0.343176i	$-0.888497\pi$
$547$	− 16.0524i	− 0.686351i	0.939271	−	0.343176i	$-0.111503\pi$
$548$	0	0
$549$	− 3.87698i	− 0.165466i
$550$	0	0
$551$	−11.5724	−0.493001
$552$	0	0
$553$	0	0
$554$	0	0
$555$	10.0354i	0.425978i
$556$	0	0
$557$	−12.5808	−0.533067	−0.266533	−	0.963826i	$-0.585878\pi$
$557$	−12.5808	−0.533067	−0.266533	+	0.963826i	$0.585878\pi$
$558$	0	0
$559$	−43.9877	−1.86048
$560$	0	0
$561$	−14.0058	−0.591325
$562$	0	0
$563$	−36.5948	−1.54229	−0.771144	−	0.636660i	$-0.780315\pi$
$563$	−36.5948	−1.54229	−0.771144	+	0.636660i	$0.780315\pi$
$564$	0	0
$565$	4.71927i	0.198541i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24.4989	−1.02705	−0.513524	−	0.858075i	$-0.671660\pi$
$569$	−24.4989	−1.02705	−0.513524	+	0.858075i	$0.671660\pi$
$570$	0	0
$571$	32.7924i	1.37232i	0.727451	+	0.686159i	$0.240705\pi$
$571$	32.7924i	1.37232i	−0.727451	+	0.686159i	$0.759295\pi$
$572$	0	0
$573$	− 1.34660i	− 0.0562552i
$574$	0	0
$575$	− 65.0586i	− 2.71313i
$576$	0	0
$577$	19.2140i	0.799888i	0.916540	+	0.399944i	$0.130970\pi$
$577$	19.2140i	0.799888i	−0.916540	+	0.399944i	$0.869030\pi$
$578$	0	0
$579$	26.1154	1.08532
$580$	0	0
$581$	0	0
$582$	0	0
$583$	21.3830i	0.885594i
$584$	0	0
$585$	−14.5182	−0.600255
$586$	0	0
$587$	−24.1451	−0.996573	−0.498286	−	0.867012i	$-0.666037\pi$
$587$	−24.1451	−0.996573	−0.498286	+	0.867012i	$0.666037\pi$
$588$	0	0
$589$	−26.7971	−1.10415
$590$	0	0
$591$	7.49083	0.308131
$592$	0	0
$593$	− 2.54582i	− 0.104544i	−0.998633	−	0.0522722i	$-0.983354\pi$
$593$	− 2.54582i	− 0.104544i	0.998633	−	0.0522722i	$-0.0166463\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.49349	0.183906
$598$	0	0
$599$	14.9899i	0.612469i	0.951956	+	0.306234i	$0.0990691\pi$
$599$	14.9899i	0.612469i	−0.951956	+	0.306234i	$0.900931\pi$
$600$	0	0
$601$	24.3343i	0.992618i	0.868146	+	0.496309i	$0.165312\pi$
$601$	24.3343i	0.992618i	−0.868146	+	0.496309i	$0.834688\pi$
$602$	0	0
$603$	4.76342i	0.193981i
$604$	0	0
$605$	− 62.8691i	− 2.55599i
$606$	0	0
$607$	−8.91288	−0.361763	−0.180881	−	0.983505i	$-0.557895\pi$
$607$	−8.91288	−0.361763	−0.180881	+	0.983505i	$0.557895\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 18.0901i	− 0.731848i
$612$	0	0
$613$	−17.1748	−0.693685	−0.346842	−	0.937923i	$-0.612746\pi$
$613$	−17.1748	−0.693685	−0.346842	+	0.937923i	$0.612746\pi$
$614$	0	0
$615$	−22.1232	−0.892094
$616$	0	0
$617$	35.9980	1.44922	0.724612	−	0.689157i	$-0.242019\pi$
$617$	35.9980	1.44922	0.724612	+	0.689157i	$0.242019\pi$
$618$	0	0
$619$	4.11123	0.165244	0.0826222	−	0.996581i	$-0.473671\pi$
$619$	4.11123	0.165244	0.0826222	+	0.996581i	$0.473671\pi$
$620$	0	0
$621$	− 4.83120i	− 0.193869i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	89.0108	3.56043
$626$	0	0
$627$	− 23.8104i	− 0.950897i
$628$	0	0
$629$	− 6.46065i	− 0.257603i
$630$	0	0
$631$	34.8970i	1.38923i	0.719383	+	0.694613i	$0.244425\pi$
$631$	34.8970i	1.38923i	−0.719383	+	0.694613i	$0.755575\pi$
$632$	0	0
$633$	5.93122i	0.235745i
$634$	0	0
$635$	22.4585	0.891239
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 12.3181i	− 0.487296i
$640$	0	0
$641$	7.32735	0.289413	0.144707	−	0.989475i	$-0.453776\pi$
$641$	7.32735	0.289413	0.144707	+	0.989475i	$0.453776\pi$
$642$	0	0
$643$	−9.24275	−0.364498	−0.182249	−	0.983252i	$-0.558338\pi$
$643$	−9.24275	−0.364498	−0.182249	+	0.983252i	$0.558338\pi$
$644$	0	0
$645$	55.9498	2.20302
$646$	0	0
$647$	15.1803	0.596798	0.298399	−	0.954441i	$-0.403547\pi$
$647$	15.1803	0.596798	0.298399	+	0.954441i	$0.403547\pi$
$648$	0	0
$649$	48.6263i	1.90875i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.12849	−0.0832943	−0.0416471	−	0.999132i	$-0.513261\pi$
$653$	−2.12849	−0.0832943	−0.0416471	+	0.999132i	$0.513261\pi$
$654$	0	0
$655$	− 3.76016i	− 0.146922i
$656$	0	0
$657$	11.5102i	0.449054i
$658$	0	0
$659$	− 13.0520i	− 0.508436i	−0.967147	−	0.254218i	$-0.918182\pi$
$659$	− 13.0520i	− 0.508436i	0.967147	−	0.254218i	$-0.0818180\pi$
$660$	0	0
$661$	19.2786i	0.749851i	0.927055	+	0.374926i	$0.122332\pi$
$661$	19.2786i	0.749851i	−0.927055	+	0.374926i	$0.877668\pi$
$662$	0	0
$663$	9.34665	0.362994
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.8874i	0.460281i
$668$	0	0
$669$	20.6163	0.797073
$670$	0	0
$671$	−19.6277	−0.757718
$672$	0	0
$673$	−7.08216	−0.272997	−0.136499	−	0.990640i	$-0.543585\pi$
$673$	−7.08216	−0.272997	−0.136499	+	0.990640i	$0.543585\pi$
$674$	0	0
$675$	13.4663	0.518320
$676$	0	0
$677$	36.5042i	1.40297i	0.712685	+	0.701485i	$0.247479\pi$
$677$	36.5042i	1.40297i	−0.712685	+	0.701485i	$0.752521\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	20.4911	0.785220
$682$	0	0
$683$	− 30.1206i	− 1.15253i	−0.817261	−	0.576267i	$-0.804509\pi$
$683$	− 30.1206i	− 1.15253i	0.817261	−	0.576267i	$-0.195491\pi$
$684$	0	0
$685$	1.01775i	0.0388863i
$686$	0	0
$687$	− 29.0409i	− 1.10798i
$688$	0	0
$689$	− 14.2698i	− 0.543635i
$690$	0	0
$691$	−39.8113	−1.51449	−0.757247	−	0.653129i	$-0.773456\pi$
$691$	−39.8113	−1.51449	−0.757247	+	0.653129i	$0.773456\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.9156i	0.489916i
$696$	0	0
$697$	14.2426	0.539478
$698$	0	0
$699$	−17.7364	−0.670852
$700$	0	0
$701$	28.0795	1.06055	0.530275	−	0.847826i	$-0.322089\pi$
$701$	28.0795	1.06055	0.530275	+	0.847826i	$0.322089\pi$
$702$	0	0
$703$	10.9834	0.414245
$704$	0	0
$705$	23.0096i	0.866591i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−36.9475	−1.38759	−0.693797	−	0.720170i	$-0.744064\pi$
$709$	−36.9475	−1.38759	−0.693797	+	0.720170i	$0.744064\pi$
$710$	0	0
$711$	− 13.9166i	− 0.521913i
$712$	0	0
$713$	27.5264i	1.03087i
$714$	0	0
$715$	73.5002i	2.74875i
$716$	0	0
$717$	− 20.4248i	− 0.762779i
$718$	0	0
$719$	−18.7090	−0.697728	−0.348864	−	0.937173i	$-0.613432\pi$
$719$	−18.7090	−0.697728	−0.348864	+	0.937173i	$0.613432\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.25742i	0.121145i
$724$	0	0
$725$	−33.1345	−1.23059
$726$	0	0
$727$	31.2078	1.15743	0.578716	−	0.815529i	$-0.303554\pi$
$727$	31.2078	1.15743	0.578716	+	0.815529i	$0.303554\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−36.0198	−1.33224
$732$	0	0
$733$	− 4.06897i	− 0.150291i	−0.997173	−	0.0751455i	$-0.976058\pi$
$733$	− 4.06897i	− 0.150291i	0.997173	−	0.0751455i	$-0.0239421\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.1154	0.888301
$738$	0	0
$739$	39.9025i	1.46784i	0.679237	+	0.733919i	$0.262311\pi$
$739$	39.9025i	1.46784i	−0.679237	+	0.733919i	$0.737689\pi$
$740$	0	0
$741$	15.8897i	0.583722i
$742$	0	0
$743$	41.7530i	1.53177i	0.642979	+	0.765884i	$0.277698\pi$
$743$	41.7530i	1.53177i	−0.642979	+	0.765884i	$0.722302\pi$
$744$	0	0
$745$	− 63.6064i	− 2.33036i
$746$	0	0
$747$	−7.32191	−0.267895
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 9.09927i	− 0.332037i	−0.986123	−	0.166019i	$-0.946909\pi$
$751$	− 9.09927i	− 0.332037i	0.986123	−	0.166019i	$-0.0530911\pi$
$752$	0	0
$753$	3.95633	0.144177
$754$	0	0
$755$	−53.1371	−1.93386
$756$	0	0
$757$	−0.902155	−0.0327894	−0.0163947	−	0.999866i	$-0.505219\pi$
$757$	−0.902155	−0.0327894	−0.0163947	+	0.999866i	$0.505219\pi$
$758$	0	0
$759$	−24.4585	−0.887787
$760$	0	0
$761$	23.7781i	0.861955i	0.902363	+	0.430978i	$0.141831\pi$
$761$	23.7781i	0.861955i	−0.902363	+	0.430978i	$0.858169\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−11.8884	−0.429826
$766$	0	0
$767$	− 32.4503i	− 1.17171i
$768$	0	0
$769$	16.3028i	0.587895i	0.955822	+	0.293948i	$0.0949691\pi$
$769$	16.3028i	0.587895i	−0.955822	+	0.293948i	$0.905031\pi$
$770$	0	0
$771$	− 5.14822i	− 0.185409i
$772$	0	0
$773$	25.0881i	0.902358i	0.892434	+	0.451179i	$0.148996\pi$
$773$	25.0881i	0.902358i	−0.892434	+	0.451179i	$0.851004\pi$
$774$	0	0
$775$	−76.7264	−2.75609
$776$	0	0
$777$	0	0
$778$	0	0
$779$	24.2131i	0.867523i
$780$	0	0
$781$	−62.3618	−2.23148
$782$	0	0
$783$	−2.46054	−0.0879327
$784$	0	0
$785$	−46.7996	−1.67035
$786$	0	0
$787$	4.47130	0.159385	0.0796924	−	0.996820i	$-0.474606\pi$
$787$	4.47130	0.159385	0.0796924	+	0.996820i	$0.474606\pi$
$788$	0	0
$789$	0.0785021i	0.00279475i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	13.0984	0.465136
$794$	0	0
$795$	18.1503i	0.643725i
$796$	0	0
$797$	6.62320i	0.234606i	0.993096	+	0.117303i	$0.0374248\pi$
$797$	6.62320i	0.234606i	−0.993096	+	0.117303i	$0.962575\pi$
$798$	0	0
$799$	− 14.8133i	− 0.524056i
$800$	0	0
$801$	− 14.0448i	− 0.496248i
$802$	0	0
$803$	58.2715	2.05636
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 0.883296i	− 0.0310935i
$808$	0	0
$809$	−8.37044	−0.294289	−0.147144	−	0.989115i	$-0.547008\pi$
$809$	−8.37044	−0.294289	−0.147144	+	0.989115i	$0.547008\pi$
$810$	0	0
$811$	1.20944	0.0424692	0.0212346	−	0.999775i	$-0.493240\pi$
$811$	1.20944	0.0424692	0.0212346	+	0.999775i	$0.493240\pi$
$812$	0	0
$813$	−19.3084	−0.677176
$814$	0	0
$815$	6.48988	0.227330
$816$	0	0
$817$	− 61.2351i	− 2.14235i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	51.2519	1.78871	0.894353	−	0.447363i	$-0.147637\pi$
$821$	51.2519	1.78871	0.894353	+	0.447363i	$0.147637\pi$
$822$	0	0
$823$	− 31.5546i	− 1.09992i	−0.835189	−	0.549962i	$-0.814642\pi$
$823$	− 31.5546i	− 1.09992i	0.835189	−	0.549962i	$-0.185358\pi$
$824$	0	0
$825$	− 68.1749i	− 2.37355i
$826$	0	0
$827$	23.4800i	0.816480i	0.912875	+	0.408240i	$0.133857\pi$
$827$	23.4800i	0.816480i	−0.912875	+	0.408240i	$0.866143\pi$
$828$	0	0
$829$	17.2639i	0.599599i	0.954002	+	0.299800i	$0.0969199\pi$
$829$	17.2639i	0.599599i	−0.954002	+	0.299800i	$0.903080\pi$
$830$	0	0
$831$	−16.9327	−0.587389
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 44.4046i	− 1.53668i
$836$	0	0
$837$	−5.69764	−0.196939
$838$	0	0
$839$	37.4781	1.29389	0.646943	−	0.762538i	$-0.276047\pi$
$839$	37.4781	1.29389	0.646943	+	0.762538i	$0.276047\pi$
$840$	0	0
$841$	−22.9457	−0.791232
$842$	0	0
$843$	−5.05417	−0.174075
$844$	0	0
$845$	6.81452i	0.234427i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−19.1156	−0.656046
$850$	0	0
$851$	− 11.2823i	− 0.386753i
$852$	0	0
$853$	9.50115i	0.325313i	0.986683	+	0.162657i	$0.0520063\pi$
$853$	9.50115i	0.325313i	−0.986683	+	0.162657i	$0.947994\pi$
$854$	0	0
$855$	− 20.2108i	− 0.691193i
$856$	0	0
$857$	32.7797i	1.11973i	0.828583	+	0.559866i	$0.189147\pi$
$857$	32.7797i	1.11973i	−0.828583	+	0.559866i	$0.810853\pi$
$858$	0	0
$859$	−5.37306	−0.183327	−0.0916633	−	0.995790i	$-0.529218\pi$
$859$	−5.37306	−0.183327	−0.0916633	+	0.995790i	$0.529218\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 29.6824i	− 1.01040i	−0.863003	−	0.505200i	$-0.831419\pi$
$863$	− 29.6824i	− 1.01040i	0.863003	−	0.505200i	$-0.168581\pi$
$864$	0	0
$865$	14.4234	0.490409
$866$	0	0
$867$	−9.34639	−0.317420
$868$	0	0
$869$	−70.4544	−2.39000
$870$	0	0
$871$	−16.0932	−0.545296
$872$	0	0
$873$	− 14.5716i	− 0.493175i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−30.3242	−1.02398	−0.511988	−	0.858993i	$-0.671091\pi$
$877$	−30.3242	−1.02398	−0.511988	+	0.858993i	$0.671091\pi$
$878$	0	0
$879$	2.37837i	0.0802205i
$880$	0	0
$881$	16.5012i	0.555939i	0.960590	+	0.277969i	$0.0896614\pi$
$881$	16.5012i	0.555939i	−0.960590	+	0.277969i	$0.910339\pi$
$882$	0	0
$883$	22.5909i	0.760245i	0.924936	+	0.380122i	$0.124118\pi$
$883$	22.5909i	0.760245i	−0.924936	+	0.380122i	$0.875882\pi$
$884$	0	0
$885$	41.2750i	1.38744i
$886$	0	0
$887$	38.6036	1.29618	0.648090	−	0.761563i	$-0.275568\pi$
$887$	38.6036	1.29618	0.648090	+	0.761563i	$0.275568\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 5.06262i	− 0.169604i
$892$	0	0
$893$	25.1832	0.842723
$894$	0	0
$895$	56.9769	1.90453
$896$	0	0
$897$	16.3222	0.544981
$898$	0	0
$899$	14.0193	0.467570
$900$	0	0
$901$	− 11.6849i	− 0.389282i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−50.9307	−1.69299
$906$	0	0
$907$	36.6743i	1.21775i	0.793266	+	0.608875i	$0.208379\pi$
$907$	36.6743i	1.21775i	−0.793266	+	0.608875i	$0.791621\pi$
$908$	0	0
$909$	13.1087i	0.434787i
$910$	0	0
$911$	− 52.6835i	− 1.74548i	−0.488185	−	0.872740i	$-0.662341\pi$
$911$	− 52.6835i	− 1.74548i	0.488185	−	0.872740i	$-0.337659\pi$
$912$	0	0
$913$	37.0680i	1.22677i
$914$	0	0
$915$	−16.6604	−0.550775
$916$	0	0
$917$	0	0
$918$	0	0
$919$	5.11618i	0.168767i	0.996433	+	0.0843836i	$0.0268921\pi$
$919$	5.11618i	0.168767i	−0.996433	+	0.0843836i	$0.973108\pi$
$920$	0	0
$921$	4.89599	0.161328
$922$	0	0
$923$	41.6166	1.36983
$924$	0	0
$925$	31.4480	1.03400
$926$	0	0
$927$	−0.209698	−0.00688738
$928$	0	0
$929$	35.3694i	1.16043i	0.814462	+	0.580217i	$0.197032\pi$
$929$	35.3694i	1.16043i	−0.814462	+	0.580217i	$0.802968\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	10.4800	0.343100
$934$	0	0
$935$	60.1864i	1.96831i
$936$	0	0
$937$	− 23.6290i	− 0.771924i	−0.922515	−	0.385962i	$-0.873870\pi$
$937$	− 23.6290i	− 0.771924i	0.922515	−	0.385962i	$-0.126130\pi$
$938$	0	0
$939$	− 9.60172i	− 0.313340i
$940$	0	0
$941$	− 19.2187i	− 0.626510i	−0.949669	−	0.313255i	$-0.898581\pi$
$941$	− 19.2187i	− 0.626510i	0.949669	−	0.313255i	$-0.101419\pi$
$942$	0	0
$943$	24.8721	0.809947
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 27.8159i	− 0.903894i	−0.892045	−	0.451947i	$-0.850730\pi$
$947$	− 27.8159i	− 0.903894i	0.892045	−	0.451947i	$-0.149270\pi$
$948$	0	0
$949$	−38.8870	−1.26232
$950$	0	0
$951$	16.7934	0.544565
$952$	0	0
$953$	56.8087	1.84021	0.920107	−	0.391668i	$-0.128102\pi$
$953$	56.8087	1.84021	0.920107	+	0.391668i	$0.128102\pi$
$954$	0	0
$955$	−5.78669	−0.187253
$956$	0	0
$957$	12.4568i	0.402671i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.46310	0.0471967
$962$	0	0
$963$	− 6.42657i	− 0.207093i
$964$	0	0
$965$	− 112.224i	− 3.61262i
$966$	0	0
$967$	20.3860i	0.655570i	0.944752	+	0.327785i	$0.106302\pi$
$967$	20.3860i	0.655570i	−0.944752	+	0.327785i	$0.893698\pi$
$968$	0	0
$969$	13.0114i	0.417987i
$970$	0	0
$971$	−11.2287	−0.360347	−0.180174	−	0.983635i	$-0.557666\pi$
$971$	−11.2287	−0.360347	−0.180174	+	0.983635i	$0.557666\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	45.4960i	1.45704i
$976$	0	0
$977$	−12.1833	−0.389778	−0.194889	−	0.980825i	$-0.562435\pi$
$977$	−12.1833	−0.389778	−0.194889	+	0.980825i	$0.562435\pi$
$978$	0	0
$979$	−71.1033	−2.27247
$980$	0	0
$981$	0.940024	0.0300126
$982$	0	0
$983$	9.06374	0.289088	0.144544	−	0.989498i	$-0.453828\pi$
$983$	9.06374	0.289088	0.144544	+	0.989498i	$0.453828\pi$
$984$	0	0
$985$	− 32.1899i	− 1.02566i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−62.9018	−2.00016
$990$	0	0
$991$	− 5.37158i	− 0.170634i	−0.996354	−	0.0853170i	$-0.972810\pi$
$991$	− 5.37158i	− 0.170634i	0.996354	−	0.0853170i	$-0.0271903\pi$
$992$	0	0
$993$	− 26.8362i	− 0.851622i
$994$	0	0
$995$	− 19.3096i	− 0.612157i
$996$	0	0
$997$	− 14.3440i	− 0.454278i	−0.973862	−	0.227139i	$-0.927063\pi$
$997$	− 14.3440i	− 0.454278i	0.973862	−	0.227139i	$-0.0729372\pi$
$998$	0	0
$999$	2.33530	0.0738857

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.b.k.1567.8	yes	8
3.2	odd	2		7056.2.b.x.1567.1		8
4.3	odd	2		2352.2.b.l.1567.8	yes	8
7.2	even	3		2352.2.bl.t.31.4		8
7.3	odd	6		2352.2.bl.o.607.4		8
7.4	even	3		2352.2.bl.r.607.1		8
7.5	odd	6		2352.2.bl.q.31.1		8
7.6	odd	2		2352.2.b.l.1567.1	yes	8
12.11	even	2		7056.2.b.w.1567.1		8
21.20	even	2		7056.2.b.w.1567.8		8
28.3	even	6		2352.2.bl.t.607.4		8
28.11	odd	6		2352.2.bl.q.607.1		8
28.19	even	6		2352.2.bl.r.31.1		8
28.23	odd	6		2352.2.bl.o.31.4		8
28.27	even	2	inner	2352.2.b.k.1567.1	✓	8
84.83	odd	2		7056.2.b.x.1567.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.2.b.k.1567.1	✓	8	28.27	even	2	inner
2352.2.b.k.1567.8	yes	8	1.1	even	1	trivial
2352.2.b.l.1567.1	yes	8	7.6	odd	2
2352.2.b.l.1567.8	yes	8	4.3	odd	2
2352.2.bl.o.31.4		8	28.23	odd	6
2352.2.bl.o.607.4		8	7.3	odd	6
2352.2.bl.q.31.1		8	7.5	odd	6
2352.2.bl.q.607.1		8	28.11	odd	6
2352.2.bl.r.31.1		8	28.19	even	6
2352.2.bl.r.607.1		8	7.4	even	3
2352.2.bl.t.31.4		8	7.2	even	3
2352.2.bl.t.607.4		8	28.3	even	6
7056.2.b.w.1567.1		8	12.11	even	2
7056.2.b.w.1567.8		8	21.20	even	2
7056.2.b.x.1567.1		8	3.2	odd	2
7056.2.b.x.1567.8		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.29725i q^{5} +1.00000 q^{9} -5.06262i q^{11} +3.37849i q^{13} -4.29725i q^{15} +2.76652i q^{17} -4.70319 q^{19} +4.83120i q^{23} -13.4663 q^{25} -1.00000 q^{27} +2.46054 q^{29} +5.69764 q^{31} +5.06262i q^{33} -2.33530 q^{37} -3.37849i q^{39} -5.14822i q^{41} +13.0199i q^{43} +4.29725i q^{45} -5.35449 q^{47} -2.76652i q^{51} -4.22371 q^{53} +21.7553 q^{55} +4.70319 q^{57} -9.60498 q^{59} -3.87698i q^{61} -14.5182 q^{65} +4.76342i q^{67} -4.83120i q^{69} -12.3181i q^{71} +11.5102i q^{73} +13.4663 q^{75} -13.9166i q^{79} +1.00000 q^{81} -7.32191 q^{83} -11.8884 q^{85} -2.46054 q^{87} -14.0448i q^{89} -5.69764 q^{93} -20.2108i q^{95} -14.5716i q^{97} -5.06262i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 8 q^{9} - 24 q^{25} - 8 q^{27} + 16 q^{29} + 32 q^{31} + 16 q^{47} + 16 q^{53} + 64 q^{55} - 48 q^{59} - 16 q^{65} + 24 q^{75} + 8 q^{81} - 64 q^{85} - 16 q^{87} - 32 q^{93}+O(q^{100}) \) 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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