LMFDB - Embedded newform 6348.2.a.s.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6348,2,Mod(1,6348)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6348, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6348.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6348 = 2^{2} \cdot 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6348.a (trivial)

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:	yes
Analytic conductor:	$50.6890352031$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 14x^{8} + 65x^{7} + 57x^{6} - 354x^{5} - 46x^{4} + 714x^{3} - 74x^{2} - 323x - 23 $ x^10 - 4x^9 - 14x^8 + 65x^7 + 57x^6 - 354x^5 - 46x^4 + 714x^3 - 74x^2 - 323*x - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.66501$ of defining polynomial
Character	$\chi$	$=$	6348.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.49042 q^{5} -4.04500 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.49042 q^{5} -4.04500 q^{7} +1.00000 q^{9} -4.82545 q^{11} +4.29523 q^{13} +2.49042 q^{15} +4.51136 q^{17} -4.43007 q^{19} -4.04500 q^{21} +1.20218 q^{25} +1.00000 q^{27} -0.499143 q^{29} +5.11451 q^{31} -4.82545 q^{33} -10.0737 q^{35} -10.6715 q^{37} +4.29523 q^{39} -7.60394 q^{41} -7.58427 q^{43} +2.49042 q^{45} -7.67725 q^{47} +9.36202 q^{49} +4.51136 q^{51} -7.63771 q^{53} -12.0174 q^{55} -4.43007 q^{57} +2.82837 q^{59} +1.58800 q^{61} -4.04500 q^{63} +10.6969 q^{65} +12.0203 q^{67} +3.79169 q^{71} -7.45343 q^{73} +1.20218 q^{75} +19.5190 q^{77} -11.6432 q^{79} +1.00000 q^{81} +5.30658 q^{83} +11.2352 q^{85} -0.499143 q^{87} +11.1462 q^{89} -17.3742 q^{91} +5.11451 q^{93} -11.0327 q^{95} +7.13353 q^{97} -4.82545 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9	$ 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9} - 11 q^{11} - 2 q^{15} - 13 q^{17} - 18 q^{19} - 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} - 11 q^{33} - 13 q^{35} - 5 q^{37} + 24 q^{41} - 40 q^{43} - 2 q^{45} - 9 q^{47} + 5 q^{49} - 13 q^{51} - 6 q^{53} - 14 q^{55} - 18 q^{57} + 28 q^{59} - 39 q^{61} - 11 q^{63} - 14 q^{65} - 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} - 33 q^{79} + 10 q^{81} - 29 q^{83} - 21 q^{85} + 5 q^{87} - 17 q^{89} - 36 q^{91} + 15 q^{93} - 42 q^{95} - 46 q^{97} - 11 q^{99}+O(q^{100}) $ 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9 - 11 * q^11 - 2 * q^15 - 13 * q^17 - 18 * q^19 - 11 * q^21 + 10 * q^25 + 10 * q^27 + 5 * q^29 + 15 * q^31 - 11 * q^33 - 13 * q^35 - 5 * q^37 + 24 * q^41 - 40 * q^43 - 2 * q^45 - 9 * q^47 + 5 * q^49 - 13 * q^51 - 6 * q^53 - 14 * q^55 - 18 * q^57 + 28 * q^59 - 39 * q^61 - 11 * q^63 - 14 * q^65 - 32 * q^67 - 33 * q^71 - 50 * q^73 + 10 * q^75 + 19 * q^77 - 33 * q^79 + 10 * q^81 - 29 * q^83 - 21 * q^85 + 5 * q^87 - 17 * q^89 - 36 * q^91 + 15 * q^93 - 42 * q^95 - 46 * q^97 - 11 * q^99

Display 100 coefficients 10 coefficients

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
$5$	2.49042	1.11375	0.556874	−	0.830597i	$-0.312000\pi$
$5$	2.49042	1.11375	0.556874	+	0.830597i	$0.312000\pi$
$6$	0	0
$7$	−4.04500	−1.52887	−0.764433	−	0.644703i	$-0.776981\pi$
$7$	−4.04500	−1.52887	−0.764433	+	0.644703i	$0.776981\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.82545	−1.45493	−0.727465	−	0.686145i	$-0.759301\pi$
$11$	−4.82545	−1.45493	−0.727465	+	0.686145i	$0.759301\pi$
$12$	0	0
$13$	4.29523	1.19128	0.595641	−	0.803251i	$-0.296898\pi$
$13$	4.29523	1.19128	0.595641	+	0.803251i	$0.296898\pi$
$14$	0	0
$15$	2.49042	0.643023
$16$	0	0
$17$	4.51136	1.09417	0.547083	−	0.837079i	$-0.315738\pi$
$17$	4.51136	1.09417	0.547083	+	0.837079i	$0.315738\pi$
$18$	0	0
$19$	−4.43007	−1.01633	−0.508164	−	0.861260i	$-0.669676\pi$
$19$	−4.43007	−1.01633	−0.508164	+	0.861260i	$0.669676\pi$
$20$	0	0
$21$	−4.04500	−0.882691
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	1.20218	0.240437
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.499143	−0.0926885	−0.0463442	−	0.998926i	$-0.514757\pi$
$29$	−0.499143	−0.0926885	−0.0463442	+	0.998926i	$0.514757\pi$
$30$	0	0
$31$	5.11451	0.918593	0.459296	−	0.888283i	$-0.348102\pi$
$31$	5.11451	0.918593	0.459296	+	0.888283i	$0.348102\pi$
$32$	0	0
$33$	−4.82545	−0.840004
$34$	0	0
$35$	−10.0737	−1.70277
$36$	0	0
$37$	−10.6715	−1.75438	−0.877189	−	0.480146i	$-0.840584\pi$
$37$	−10.6715	−1.75438	−0.877189	+	0.480146i	$0.840584\pi$
$38$	0	0
$39$	4.29523	0.687787
$40$	0	0
$41$	−7.60394	−1.18754	−0.593768	−	0.804636i	$-0.702360\pi$
$41$	−7.60394	−1.18754	−0.593768	+	0.804636i	$0.702360\pi$
$42$	0	0
$43$	−7.58427	−1.15659	−0.578295	−	0.815828i	$-0.696282\pi$
$43$	−7.58427	−1.15659	−0.578295	+	0.815828i	$0.696282\pi$
$44$	0	0
$45$	2.49042	0.371250
$46$	0	0
$47$	−7.67725	−1.11984	−0.559921	−	0.828546i	$-0.689169\pi$
$47$	−7.67725	−1.11984	−0.559921	+	0.828546i	$0.689169\pi$
$48$	0	0
$49$	9.36202	1.33743
$50$	0	0
$51$	4.51136	0.631717
$52$	0	0
$53$	−7.63771	−1.04912	−0.524560	−	0.851373i	$-0.675770\pi$
$53$	−7.63771	−1.04912	−0.524560	+	0.851373i	$0.675770\pi$
$54$	0	0
$55$	−12.0174	−1.62043
$56$	0	0
$57$	−4.43007	−0.586777
$58$	0	0
$59$	2.82837	0.368222	0.184111	−	0.982905i	$-0.441059\pi$
$59$	2.82837	0.368222	0.184111	+	0.982905i	$0.441059\pi$
$60$	0	0
$61$	1.58800	0.203322	0.101661	−	0.994819i	$-0.467584\pi$
$61$	1.58800	0.203322	0.101661	+	0.994819i	$0.467584\pi$
$62$	0	0
$63$	−4.04500	−0.509622
$64$	0	0
$65$	10.6969	1.32679
$66$	0	0
$67$	12.0203	1.46851	0.734257	−	0.678871i	$-0.237531\pi$
$67$	12.0203	1.46851	0.734257	+	0.678871i	$0.237531\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.79169	0.449990	0.224995	−	0.974360i	$-0.427763\pi$
$71$	3.79169	0.449990	0.224995	+	0.974360i	$0.427763\pi$
$72$	0	0
$73$	−7.45343	−0.872358	−0.436179	−	0.899860i	$-0.643669\pi$
$73$	−7.45343	−0.872358	−0.436179	+	0.899860i	$0.643669\pi$
$74$	0	0
$75$	1.20218	0.138816
$76$	0	0
$77$	19.5190	2.22439
$78$	0	0
$79$	−11.6432	−1.30997	−0.654984	−	0.755643i	$-0.727325\pi$
$79$	−11.6432	−1.30997	−0.654984	+	0.755643i	$0.727325\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.30658	0.582473	0.291236	−	0.956651i	$-0.405933\pi$
$83$	5.30658	0.582473	0.291236	+	0.956651i	$0.405933\pi$
$84$	0	0
$85$	11.2352	1.21863
$86$	0	0
$87$	−0.499143	−0.0535137
$88$	0	0
$89$	11.1462	1.18149	0.590746	−	0.806857i	$-0.298833\pi$
$89$	11.1462	1.18149	0.590746	+	0.806857i	$0.298833\pi$
$90$	0	0
$91$	−17.3742	−1.82131
$92$	0	0
$93$	5.11451	0.530350
$94$	0	0
$95$	−11.0327	−1.13193
$96$	0	0
$97$	7.13353	0.724300	0.362150	−	0.932120i	$-0.382043\pi$
$97$	7.13353	0.724300	0.362150	+	0.932120i	$0.382043\pi$
$98$	0	0
$99$	−4.82545	−0.484976
$100$	0	0
$101$	−5.28560	−0.525937	−0.262969	−	0.964804i	$-0.584702\pi$
$101$	−5.28560	−0.525937	−0.262969	+	0.964804i	$0.584702\pi$
$102$	0	0
$103$	19.1553	1.88742	0.943712	−	0.330769i	$-0.107308\pi$
$103$	19.1553	1.88742	0.943712	+	0.330769i	$0.107308\pi$
$104$	0	0
$105$	−10.0737	−0.983096
$106$	0	0
$107$	−11.0318	−1.06649	−0.533243	−	0.845962i	$-0.679027\pi$
$107$	−11.0318	−1.06649	−0.533243	+	0.845962i	$0.679027\pi$
$108$	0	0
$109$	−1.74350	−0.166997	−0.0834985	−	0.996508i	$-0.526609\pi$
$109$	−1.74350	−0.166997	−0.0834985	+	0.996508i	$0.526609\pi$
$110$	0	0
$111$	−10.6715	−1.01289
$112$	0	0
$113$	−17.2752	−1.62511	−0.812555	−	0.582884i	$-0.801924\pi$
$113$	−17.2752	−1.62511	−0.812555	+	0.582884i	$0.801924\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.29523	0.397094
$118$	0	0
$119$	−18.2484	−1.67283
$120$	0	0
$121$	12.2850	1.11682
$122$	0	0
$123$	−7.60394	−0.685624
$124$	0	0
$125$	−9.45815	−0.845963
$126$	0	0
$127$	−14.2893	−1.26797	−0.633985	−	0.773345i	$-0.718582\pi$
$127$	−14.2893	−1.26797	−0.633985	+	0.773345i	$0.718582\pi$
$128$	0	0
$129$	−7.58427	−0.667758
$130$	0	0
$131$	−9.94393	−0.868805	−0.434402	−	0.900719i	$-0.643040\pi$
$131$	−9.94393	−0.868805	−0.434402	+	0.900719i	$0.643040\pi$
$132$	0	0
$133$	17.9196	1.55383
$134$	0	0
$135$	2.49042	0.214341
$136$	0	0
$137$	−8.14309	−0.695711	−0.347855	−	0.937548i	$-0.613090\pi$
$137$	−8.14309	−0.695711	−0.347855	+	0.937548i	$0.613090\pi$
$138$	0	0
$139$	−10.1303	−0.859242	−0.429621	−	0.903009i	$-0.641353\pi$
$139$	−10.1303	−0.859242	−0.429621	+	0.903009i	$0.641353\pi$
$140$	0	0
$141$	−7.67725	−0.646541
$142$	0	0
$143$	−20.7264	−1.73323
$144$	0	0
$145$	−1.24307	−0.103232
$146$	0	0
$147$	9.36202	0.772166
$148$	0	0
$149$	6.63376	0.543459	0.271730	−	0.962374i	$-0.412404\pi$
$149$	6.63376	0.543459	0.271730	+	0.962374i	$0.412404\pi$
$150$	0	0
$151$	−8.49892	−0.691632	−0.345816	−	0.938302i	$-0.612398\pi$
$151$	−8.49892	−0.691632	−0.345816	+	0.938302i	$0.612398\pi$
$152$	0	0
$153$	4.51136	0.364722
$154$	0	0
$155$	12.7373	1.02308
$156$	0	0
$157$	9.37642	0.748319	0.374160	−	0.927364i	$-0.377931\pi$
$157$	9.37642	0.748319	0.374160	+	0.927364i	$0.377931\pi$
$158$	0	0
$159$	−7.63771	−0.605710
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−4.84855	−0.379768	−0.189884	−	0.981807i	$-0.560811\pi$
$163$	−4.84855	−0.379768	−0.189884	+	0.981807i	$0.560811\pi$
$164$	0	0
$165$	−12.0174	−0.935553
$166$	0	0
$167$	−13.3557	−1.03349	−0.516747	−	0.856138i	$-0.672857\pi$
$167$	−13.3557	−1.03349	−0.516747	+	0.856138i	$0.672857\pi$
$168$	0	0
$169$	5.44897	0.419151
$170$	0	0
$171$	−4.43007	−0.338776
$172$	0	0
$173$	10.1954	0.775142	0.387571	−	0.921840i	$-0.373314\pi$
$173$	10.1954	0.775142	0.387571	+	0.921840i	$0.373314\pi$
$174$	0	0
$175$	−4.86283	−0.367596
$176$	0	0
$177$	2.82837	0.212593
$178$	0	0
$179$	6.08336	0.454692	0.227346	−	0.973814i	$-0.426995\pi$
$179$	6.08336	0.454692	0.227346	+	0.973814i	$0.426995\pi$
$180$	0	0
$181$	−4.64361	−0.345157	−0.172578	−	0.984996i	$-0.555210\pi$
$181$	−4.64361	−0.345157	−0.172578	+	0.984996i	$0.555210\pi$
$182$	0	0
$183$	1.58800	0.117388
$184$	0	0
$185$	−26.5764	−1.95394
$186$	0	0
$187$	−21.7694	−1.59193
$188$	0	0
$189$	−4.04500	−0.294230
$190$	0	0
$191$	−11.8060	−0.854250	−0.427125	−	0.904193i	$-0.640473\pi$
$191$	−11.8060	−0.854250	−0.427125	+	0.904193i	$0.640473\pi$
$192$	0	0
$193$	22.3608	1.60956	0.804781	−	0.593572i	$-0.202283\pi$
$193$	22.3608	1.60956	0.804781	+	0.593572i	$0.202283\pi$
$194$	0	0
$195$	10.6969	0.766022
$196$	0	0
$197$	9.05748	0.645319	0.322659	−	0.946515i	$-0.395423\pi$
$197$	9.05748	0.645319	0.322659	+	0.946515i	$0.395423\pi$
$198$	0	0
$199$	−3.60978	−0.255891	−0.127945	−	0.991781i	$-0.540838\pi$
$199$	−3.60978	−0.255891	−0.127945	+	0.991781i	$0.540838\pi$
$200$	0	0
$201$	12.0203	0.847847
$202$	0	0
$203$	2.01903	0.141708
$204$	0	0
$205$	−18.9370	−1.32262
$206$	0	0
$207$	0	0
$208$	0	0
$209$	21.3771	1.47869
$210$	0	0
$211$	−2.45599	−0.169078	−0.0845388	−	0.996420i	$-0.526942\pi$
$211$	−2.45599	−0.169078	−0.0845388	+	0.996420i	$0.526942\pi$
$212$	0	0
$213$	3.79169	0.259802
$214$	0	0
$215$	−18.8880	−1.28815
$216$	0	0
$217$	−20.6882	−1.40441
$218$	0	0
$219$	−7.45343	−0.503656
$220$	0	0
$221$	19.3773	1.30346
$222$	0	0
$223$	−16.7373	−1.12081	−0.560406	−	0.828218i	$-0.689355\pi$
$223$	−16.7373	−1.12081	−0.560406	+	0.828218i	$0.689355\pi$
$224$	0	0
$225$	1.20218	0.0801456
$226$	0	0
$227$	−13.6388	−0.905238	−0.452619	−	0.891704i	$-0.649510\pi$
$227$	−13.6388	−0.905238	−0.452619	+	0.891704i	$0.649510\pi$
$228$	0	0
$229$	−6.76866	−0.447286	−0.223643	−	0.974671i	$-0.571795\pi$
$229$	−6.76866	−0.447286	−0.223643	+	0.974671i	$0.571795\pi$
$230$	0	0
$231$	19.5190	1.28425
$232$	0	0
$233$	−10.8212	−0.708921	−0.354461	−	0.935071i	$-0.615335\pi$
$233$	−10.8212	−0.708921	−0.354461	+	0.935071i	$0.615335\pi$
$234$	0	0
$235$	−19.1196	−1.24722
$236$	0	0
$237$	−11.6432	−0.756310
$238$	0	0
$239$	19.1644	1.23964	0.619820	−	0.784744i	$-0.287206\pi$
$239$	19.1644	1.23964	0.619820	+	0.784744i	$0.287206\pi$
$240$	0	0
$241$	−9.95386	−0.641185	−0.320592	−	0.947217i	$-0.603882\pi$
$241$	−9.95386	−0.641185	−0.320592	+	0.947217i	$0.603882\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	23.3153	1.48956
$246$	0	0
$247$	−19.0282	−1.21073
$248$	0	0
$249$	5.30658	0.336291
$250$	0	0
$251$	11.4013	0.719643	0.359821	−	0.933021i	$-0.382838\pi$
$251$	11.4013	0.719643	0.359821	+	0.933021i	$0.382838\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	11.2352	0.703574
$256$	0	0
$257$	26.1644	1.63209	0.816045	−	0.577988i	$-0.196162\pi$
$257$	26.1644	1.63209	0.816045	+	0.577988i	$0.196162\pi$
$258$	0	0
$259$	43.1661	2.68221
$260$	0	0
$261$	−0.499143	−0.0308962
$262$	0	0
$263$	−29.3436	−1.80940	−0.904701	−	0.426046i	$-0.859906\pi$
$263$	−29.3436	−1.80940	−0.904701	+	0.426046i	$0.859906\pi$
$264$	0	0
$265$	−19.0211	−1.16846
$266$	0	0
$267$	11.1462	0.682135
$268$	0	0
$269$	−28.2655	−1.72338	−0.861689	−	0.507437i	$-0.830593\pi$
$269$	−28.2655	−1.72338	−0.861689	+	0.507437i	$0.830593\pi$
$270$	0	0
$271$	−0.113125	−0.00687185	−0.00343592	−	0.999994i	$-0.501094\pi$
$271$	−0.113125	−0.00687185	−0.00343592	+	0.999994i	$0.501094\pi$
$272$	0	0
$273$	−17.3742	−1.05153
$274$	0	0
$275$	−5.80108	−0.349818
$276$	0	0
$277$	−3.96519	−0.238245	−0.119123	−	0.992880i	$-0.538008\pi$
$277$	−3.96519	−0.238245	−0.119123	+	0.992880i	$0.538008\pi$
$278$	0	0
$279$	5.11451	0.306198
$280$	0	0
$281$	−16.6368	−0.992469	−0.496234	−	0.868189i	$-0.665284\pi$
$281$	−16.6368	−0.992469	−0.496234	+	0.868189i	$0.665284\pi$
$282$	0	0
$283$	−16.9251	−1.00609	−0.503046	−	0.864259i	$-0.667788\pi$
$283$	−16.9251	−1.00609	−0.503046	+	0.864259i	$0.667788\pi$
$284$	0	0
$285$	−11.0327	−0.653523
$286$	0	0
$287$	30.7579	1.81558
$288$	0	0
$289$	3.35237	0.197198
$290$	0	0
$291$	7.13353	0.418175
$292$	0	0
$293$	20.4715	1.19596	0.597980	−	0.801511i	$-0.295970\pi$
$293$	20.4715	1.19596	0.597980	+	0.801511i	$0.295970\pi$
$294$	0	0
$295$	7.04382	0.410107
$296$	0	0
$297$	−4.82545	−0.280001
$298$	0	0
$299$	0	0
$300$	0	0
$301$	30.6784	1.76827
$302$	0	0
$303$	−5.28560	−0.303650
$304$	0	0
$305$	3.95478	0.226450
$306$	0	0
$307$	16.8237	0.960180	0.480090	−	0.877219i	$-0.340604\pi$
$307$	16.8237	0.960180	0.480090	+	0.877219i	$0.340604\pi$
$308$	0	0
$309$	19.1553	1.08970
$310$	0	0
$311$	−29.7791	−1.68862	−0.844308	−	0.535858i	$-0.819988\pi$
$311$	−29.7791	−1.68862	−0.844308	+	0.535858i	$0.819988\pi$
$312$	0	0
$313$	−30.2184	−1.70804	−0.854022	−	0.520236i	$-0.825844\pi$
$313$	−30.2184	−1.70804	−0.854022	+	0.520236i	$0.825844\pi$
$314$	0	0
$315$	−10.0737	−0.567591
$316$	0	0
$317$	−5.24217	−0.294429	−0.147215	−	0.989105i	$-0.547031\pi$
$317$	−5.24217	−0.294429	−0.147215	+	0.989105i	$0.547031\pi$
$318$	0	0
$319$	2.40859	0.134855
$320$	0	0
$321$	−11.0318	−0.615736
$322$	0	0
$323$	−19.9856	−1.11203
$324$	0	0
$325$	5.16365	0.286428
$326$	0	0
$327$	−1.74350	−0.0964158
$328$	0	0
$329$	31.0545	1.71209
$330$	0	0
$331$	18.9047	1.03909	0.519547	−	0.854442i	$-0.326101\pi$
$331$	18.9047	1.03909	0.519547	+	0.854442i	$0.326101\pi$
$332$	0	0
$333$	−10.6715	−0.584793
$334$	0	0
$335$	29.9356	1.63556
$336$	0	0
$337$	−12.1959	−0.664351	−0.332175	−	0.943218i	$-0.607783\pi$
$337$	−12.1959	−0.664351	−0.332175	+	0.943218i	$0.607783\pi$
$338$	0	0
$339$	−17.2752	−0.938258
$340$	0	0
$341$	−24.6798	−1.33649
$342$	0	0
$343$	−9.55435	−0.515886
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.39256	0.450536	0.225268	−	0.974297i	$-0.427674\pi$
$347$	8.39256	0.450536	0.225268	+	0.974297i	$0.427674\pi$
$348$	0	0
$349$	21.8713	1.17074	0.585371	−	0.810766i	$-0.300949\pi$
$349$	21.8713	1.17074	0.585371	+	0.810766i	$0.300949\pi$
$350$	0	0
$351$	4.29523	0.229262
$352$	0	0
$353$	20.5871	1.09574	0.547869	−	0.836564i	$-0.315439\pi$
$353$	20.5871	1.09574	0.547869	+	0.836564i	$0.315439\pi$
$354$	0	0
$355$	9.44289	0.501176
$356$	0	0
$357$	−18.2484	−0.965810
$358$	0	0
$359$	−21.3819	−1.12849	−0.564246	−	0.825606i	$-0.690833\pi$
$359$	−21.3819	−1.12849	−0.564246	+	0.825606i	$0.690833\pi$
$360$	0	0
$361$	0.625537	0.0329230
$362$	0	0
$363$	12.2850	0.644795
$364$	0	0
$365$	−18.5622	−0.971588
$366$	0	0
$367$	11.1339	0.581182	0.290591	−	0.956847i	$-0.406148\pi$
$367$	11.1339	0.581182	0.290591	+	0.956847i	$0.406148\pi$
$368$	0	0
$369$	−7.60394	−0.395845
$370$	0	0
$371$	30.8945	1.60396
$372$	0	0
$373$	13.3999	0.693819	0.346909	−	0.937899i	$-0.387231\pi$
$373$	13.3999	0.693819	0.346909	+	0.937899i	$0.387231\pi$
$374$	0	0
$375$	−9.45815	−0.488417
$376$	0	0
$377$	−2.14393	−0.110418
$378$	0	0
$379$	−0.543914	−0.0279390	−0.0139695	−	0.999902i	$-0.504447\pi$
$379$	−0.543914	−0.0279390	−0.0139695	+	0.999902i	$0.504447\pi$
$380$	0	0
$381$	−14.2893	−0.732063
$382$	0	0
$383$	−2.51598	−0.128560	−0.0642802	−	0.997932i	$-0.520475\pi$
$383$	−2.51598	−0.128560	−0.0642802	+	0.997932i	$0.520475\pi$
$384$	0	0
$385$	48.6104	2.47741
$386$	0	0
$387$	−7.58427	−0.385530
$388$	0	0
$389$	−12.5044	−0.633997	−0.316998	−	0.948426i	$-0.602675\pi$
$389$	−12.5044	−0.633997	−0.316998	+	0.948426i	$0.602675\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−9.94393	−0.501605
$394$	0	0
$395$	−28.9966	−1.45898
$396$	0	0
$397$	22.5511	1.13181	0.565903	−	0.824472i	$-0.308528\pi$
$397$	22.5511	1.13181	0.565903	+	0.824472i	$0.308528\pi$
$398$	0	0
$399$	17.9196	0.897104
$400$	0	0
$401$	−1.43392	−0.0716066	−0.0358033	−	0.999359i	$-0.511399\pi$
$401$	−1.43392	−0.0716066	−0.0358033	+	0.999359i	$0.511399\pi$
$402$	0	0
$403$	21.9680	1.09430
$404$	0	0
$405$	2.49042	0.123750
$406$	0	0
$407$	51.4946	2.55249
$408$	0	0
$409$	2.10444	0.104058	0.0520291	−	0.998646i	$-0.483431\pi$
$409$	2.10444	0.104058	0.0520291	+	0.998646i	$0.483431\pi$
$410$	0	0
$411$	−8.14309	−0.401669
$412$	0	0
$413$	−11.4407	−0.562962
$414$	0	0
$415$	13.2156	0.648729
$416$	0	0
$417$	−10.1303	−0.496084
$418$	0	0
$419$	−14.7519	−0.720679	−0.360339	−	0.932821i	$-0.617339\pi$
$419$	−14.7519	−0.720679	−0.360339	+	0.932821i	$0.617339\pi$
$420$	0	0
$421$	−2.37889	−0.115940	−0.0579700	−	0.998318i	$-0.518463\pi$
$421$	−2.37889	−0.115940	−0.0579700	+	0.998318i	$0.518463\pi$
$422$	0	0
$423$	−7.67725	−0.373281
$424$	0	0
$425$	5.42348	0.263078
$426$	0	0
$427$	−6.42345	−0.310853
$428$	0	0
$429$	−20.7264	−1.00068
$430$	0	0
$431$	17.4060	0.838418	0.419209	−	0.907890i	$-0.362307\pi$
$431$	17.4060	0.838418	0.419209	+	0.907890i	$0.362307\pi$
$432$	0	0
$433$	−19.7578	−0.949500	−0.474750	−	0.880121i	$-0.657461\pi$
$433$	−19.7578	−0.949500	−0.474750	+	0.880121i	$0.657461\pi$
$434$	0	0
$435$	−1.24307	−0.0596009
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−18.3530	−0.875940	−0.437970	−	0.898989i	$-0.644302\pi$
$439$	−18.3530	−0.875940	−0.437970	+	0.898989i	$0.644302\pi$
$440$	0	0
$441$	9.36202	0.445810
$442$	0	0
$443$	−1.28120	−0.0608715	−0.0304358	−	0.999537i	$-0.509689\pi$
$443$	−1.28120	−0.0608715	−0.0304358	+	0.999537i	$0.509689\pi$
$444$	0	0
$445$	27.7586	1.31589
$446$	0	0
$447$	6.63376	0.313766
$448$	0	0
$449$	9.47075	0.446952	0.223476	−	0.974709i	$-0.428260\pi$
$449$	9.47075	0.446952	0.223476	+	0.974709i	$0.428260\pi$
$450$	0	0
$451$	36.6925	1.72778
$452$	0	0
$453$	−8.49892	−0.399314
$454$	0	0
$455$	−43.2690	−2.02848
$456$	0	0
$457$	20.9465	0.979835	0.489917	−	0.871769i	$-0.337027\pi$
$457$	20.9465	0.979835	0.489917	+	0.871769i	$0.337027\pi$
$458$	0	0
$459$	4.51136	0.210572
$460$	0	0
$461$	7.43095	0.346094	0.173047	−	0.984914i	$-0.444639\pi$
$461$	7.43095	0.346094	0.173047	+	0.984914i	$0.444639\pi$
$462$	0	0
$463$	−27.4134	−1.27401	−0.637005	−	0.770860i	$-0.719827\pi$
$463$	−27.4134	−1.27401	−0.637005	+	0.770860i	$0.719827\pi$
$464$	0	0
$465$	12.7373	0.590677
$466$	0	0
$467$	37.4148	1.73135	0.865675	−	0.500607i	$-0.166890\pi$
$467$	37.4148	1.73135	0.865675	+	0.500607i	$0.166890\pi$
$468$	0	0
$469$	−48.6221	−2.24516
$470$	0	0
$471$	9.37642	0.432042
$472$	0	0
$473$	36.5976	1.68276
$474$	0	0
$475$	−5.32576	−0.244363
$476$	0	0
$477$	−7.63771	−0.349707
$478$	0	0
$479$	15.9439	0.728494	0.364247	−	0.931302i	$-0.381326\pi$
$479$	15.9439	0.728494	0.364247	+	0.931302i	$0.381326\pi$
$480$	0	0
$481$	−45.8363	−2.08996
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.7655	0.806689
$486$	0	0
$487$	15.1551	0.686742	0.343371	−	0.939200i	$-0.388431\pi$
$487$	15.1551	0.686742	0.343371	+	0.939200i	$0.388431\pi$
$488$	0	0
$489$	−4.84855	−0.219259
$490$	0	0
$491$	35.2929	1.59275	0.796373	−	0.604806i	$-0.206749\pi$
$491$	35.2929	1.59275	0.796373	+	0.604806i	$0.206749\pi$
$492$	0	0
$493$	−2.25181	−0.101417
$494$	0	0
$495$	−12.0174	−0.540142
$496$	0	0
$497$	−15.3374	−0.687975
$498$	0	0
$499$	−24.3864	−1.09169	−0.545843	−	0.837888i	$-0.683790\pi$
$499$	−24.3864	−1.09169	−0.545843	+	0.837888i	$0.683790\pi$
$500$	0	0
$501$	−13.3557	−0.596688
$502$	0	0
$503$	33.3097	1.48521	0.742603	−	0.669732i	$-0.233591\pi$
$503$	33.3097	1.48521	0.742603	+	0.669732i	$0.233591\pi$
$504$	0	0
$505$	−13.1634	−0.585762
$506$	0	0
$507$	5.44897	0.241997
$508$	0	0
$509$	19.9637	0.884877	0.442438	−	0.896799i	$-0.354114\pi$
$509$	19.9637	0.884877	0.442438	+	0.896799i	$0.354114\pi$
$510$	0	0
$511$	30.1491	1.33372
$512$	0	0
$513$	−4.43007	−0.195592
$514$	0	0
$515$	47.7046	2.10212
$516$	0	0
$517$	37.0462	1.62929
$518$	0	0
$519$	10.1954	0.447528
$520$	0	0
$521$	41.2380	1.80667	0.903334	−	0.428938i	$-0.141112\pi$
$521$	41.2380	1.80667	0.903334	+	0.428938i	$0.141112\pi$
$522$	0	0
$523$	10.3238	0.451429	0.225715	−	0.974193i	$-0.427528\pi$
$523$	10.3238	0.451429	0.225715	+	0.974193i	$0.427528\pi$
$524$	0	0
$525$	−4.86283	−0.212231
$526$	0	0
$527$	23.0734	1.00509
$528$	0	0
$529$	0	0
$530$	0	0
$531$	2.82837	0.122741
$532$	0	0
$533$	−32.6607	−1.41469
$534$	0	0
$535$	−27.4739	−1.18780
$536$	0	0
$537$	6.08336	0.262516
$538$	0	0
$539$	−45.1760	−1.94587
$540$	0	0
$541$	−34.4965	−1.48312	−0.741559	−	0.670887i	$-0.765913\pi$
$541$	−34.4965	−1.48312	−0.741559	+	0.670887i	$0.765913\pi$
$542$	0	0
$543$	−4.64361	−0.199276
$544$	0	0
$545$	−4.34204	−0.185993
$546$	0	0
$547$	−10.2697	−0.439102	−0.219551	−	0.975601i	$-0.570459\pi$
$547$	−10.2697	−0.439102	−0.219551	+	0.975601i	$0.570459\pi$
$548$	0	0
$549$	1.58800	0.0677741
$550$	0	0
$551$	2.21124	0.0942019
$552$	0	0
$553$	47.0969	2.00277
$554$	0	0
$555$	−26.5764	−1.12811
$556$	0	0
$557$	30.3008	1.28389	0.641943	−	0.766752i	$-0.278129\pi$
$557$	30.3008	1.28389	0.641943	+	0.766752i	$0.278129\pi$
$558$	0	0
$559$	−32.5762	−1.37782
$560$	0	0
$561$	−21.7694	−0.919103
$562$	0	0
$563$	−31.8282	−1.34140	−0.670699	−	0.741730i	$-0.734006\pi$
$563$	−31.8282	−1.34140	−0.670699	+	0.741730i	$0.734006\pi$
$564$	0	0
$565$	−43.0224	−1.80997
$566$	0	0
$567$	−4.04500	−0.169874
$568$	0	0
$569$	−1.99250	−0.0835301	−0.0417651	−	0.999127i	$-0.513298\pi$
$569$	−1.99250	−0.0835301	−0.0417651	+	0.999127i	$0.513298\pi$
$570$	0	0
$571$	−10.1012	−0.422720	−0.211360	−	0.977408i	$-0.567789\pi$
$571$	−10.1012	−0.422720	−0.211360	+	0.977408i	$0.567789\pi$
$572$	0	0
$573$	−11.8060	−0.493201
$574$	0	0
$575$	0	0
$576$	0	0
$577$	10.8171	0.450324	0.225162	−	0.974321i	$-0.427709\pi$
$577$	10.8171	0.450324	0.225162	+	0.974321i	$0.427709\pi$
$578$	0	0
$579$	22.3608	0.929281
$580$	0	0
$581$	−21.4651	−0.890523
$582$	0	0
$583$	36.8554	1.52640
$584$	0	0
$585$	10.6969	0.442263
$586$	0	0
$587$	22.8340	0.942460	0.471230	−	0.882010i	$-0.343810\pi$
$587$	22.8340	0.942460	0.471230	+	0.882010i	$0.343810\pi$
$588$	0	0
$589$	−22.6576	−0.933592
$590$	0	0
$591$	9.05748	0.372575
$592$	0	0
$593$	−13.2828	−0.545460	−0.272730	−	0.962091i	$-0.587927\pi$
$593$	−13.2828	−0.545460	−0.272730	+	0.962091i	$0.587927\pi$
$594$	0	0
$595$	−45.4463	−1.86312
$596$	0	0
$597$	−3.60978	−0.147739
$598$	0	0
$599$	15.3103	0.625563	0.312781	−	0.949825i	$-0.398739\pi$
$599$	15.3103	0.625563	0.312781	+	0.949825i	$0.398739\pi$
$600$	0	0
$601$	−21.9980	−0.897319	−0.448659	−	0.893703i	$-0.648098\pi$
$601$	−21.9980	−0.897319	−0.448659	+	0.893703i	$0.648098\pi$
$602$	0	0
$603$	12.0203	0.489505
$604$	0	0
$605$	30.5948	1.24386
$606$	0	0
$607$	1.89755	0.0770192	0.0385096	−	0.999258i	$-0.487739\pi$
$607$	1.89755	0.0770192	0.0385096	+	0.999258i	$0.487739\pi$
$608$	0	0
$609$	2.01903	0.0818153
$610$	0	0
$611$	−32.9755	−1.33405
$612$	0	0
$613$	20.4130	0.824474	0.412237	−	0.911077i	$-0.364748\pi$
$613$	20.4130	0.824474	0.412237	+	0.911077i	$0.364748\pi$
$614$	0	0
$615$	−18.9370	−0.763613
$616$	0	0
$617$	20.3286	0.818397	0.409199	−	0.912445i	$-0.365808\pi$
$617$	20.3286	0.818397	0.409199	+	0.912445i	$0.365808\pi$
$618$	0	0
$619$	−35.0078	−1.40708	−0.703540	−	0.710655i	$-0.748399\pi$
$619$	−35.0078	−1.40708	−0.703540	+	0.710655i	$0.748399\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−45.0863	−1.80634
$624$	0	0
$625$	−29.5657	−1.18263
$626$	0	0
$627$	21.3771	0.853719
$628$	0	0
$629$	−48.1428	−1.91958
$630$	0	0
$631$	30.9163	1.23076	0.615380	−	0.788231i	$-0.289003\pi$
$631$	30.9163	1.23076	0.615380	+	0.788231i	$0.289003\pi$
$632$	0	0
$633$	−2.45599	−0.0976170
$634$	0	0
$635$	−35.5863	−1.41220
$636$	0	0
$637$	40.2120	1.59326
$638$	0	0
$639$	3.79169	0.149997
$640$	0	0
$641$	49.5692	1.95787	0.978933	−	0.204183i	$-0.0654539\pi$
$641$	49.5692	1.95787	0.978933	+	0.204183i	$0.0654539\pi$
$642$	0	0
$643$	36.1135	1.42418	0.712089	−	0.702089i	$-0.247749\pi$
$643$	36.1135	1.42418	0.712089	+	0.702089i	$0.247749\pi$
$644$	0	0
$645$	−18.8880	−0.743715
$646$	0	0
$647$	14.4639	0.568634	0.284317	−	0.958730i	$-0.408233\pi$
$647$	14.4639	0.568634	0.284317	+	0.958730i	$0.408233\pi$
$648$	0	0
$649$	−13.6482	−0.535737
$650$	0	0
$651$	−20.6882	−0.810834
$652$	0	0
$653$	−41.5005	−1.62404	−0.812021	−	0.583629i	$-0.801632\pi$
$653$	−41.5005	−1.62404	−0.812021	+	0.583629i	$0.801632\pi$
$654$	0	0
$655$	−24.7645	−0.967631
$656$	0	0
$657$	−7.45343	−0.290786
$658$	0	0
$659$	23.7534	0.925300	0.462650	−	0.886541i	$-0.346899\pi$
$659$	23.7534	0.925300	0.462650	+	0.886541i	$0.346899\pi$
$660$	0	0
$661$	−44.8627	−1.74496	−0.872479	−	0.488651i	$-0.837489\pi$
$661$	−44.8627	−1.74496	−0.872479	+	0.488651i	$0.837489\pi$
$662$	0	0
$663$	19.3773	0.752552
$664$	0	0
$665$	44.6274	1.73058
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−16.7373	−0.647101
$670$	0	0
$671$	−7.66282	−0.295820
$672$	0	0
$673$	−44.4524	−1.71351	−0.856757	−	0.515721i	$-0.827524\pi$
$673$	−44.4524	−1.71351	−0.856757	+	0.515721i	$0.827524\pi$
$674$	0	0
$675$	1.20218	0.0462721
$676$	0	0
$677$	−24.5726	−0.944401	−0.472200	−	0.881491i	$-0.656540\pi$
$677$	−24.5726	−0.944401	−0.472200	+	0.881491i	$0.656540\pi$
$678$	0	0
$679$	−28.8551	−1.10736
$680$	0	0
$681$	−13.6388	−0.522639
$682$	0	0
$683$	−36.1598	−1.38362	−0.691808	−	0.722082i	$-0.743186\pi$
$683$	−36.1598	−1.38362	−0.691808	+	0.722082i	$0.743186\pi$
$684$	0	0
$685$	−20.2797	−0.774847
$686$	0	0
$687$	−6.76866	−0.258241
$688$	0	0
$689$	−32.8057	−1.24980
$690$	0	0
$691$	−29.9939	−1.14102	−0.570511	−	0.821290i	$-0.693255\pi$
$691$	−29.9939	−1.14102	−0.570511	+	0.821290i	$0.693255\pi$
$692$	0	0
$693$	19.5190	0.741464
$694$	0	0
$695$	−25.2287	−0.956980
$696$	0	0
$697$	−34.3041	−1.29936
$698$	0	0
$699$	−10.8212	−0.409296
$700$	0	0
$701$	46.3504	1.75063	0.875315	−	0.483553i	$-0.160654\pi$
$701$	46.3504	1.75063	0.875315	+	0.483553i	$0.160654\pi$
$702$	0	0
$703$	47.2753	1.78302
$704$	0	0
$705$	−19.1196	−0.720085
$706$	0	0
$707$	21.3803	0.804088
$708$	0	0
$709$	−1.63325	−0.0613379	−0.0306690	−	0.999530i	$-0.509764\pi$
$709$	−1.63325	−0.0613379	−0.0306690	+	0.999530i	$0.509764\pi$
$710$	0	0
$711$	−11.6432	−0.436656
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−51.6174	−1.93038
$716$	0	0
$717$	19.1644	0.715707
$718$	0	0
$719$	4.48947	0.167429	0.0837145	−	0.996490i	$-0.473322\pi$
$719$	4.48947	0.167429	0.0837145	+	0.996490i	$0.473322\pi$
$720$	0	0
$721$	−77.4830	−2.88562
$722$	0	0
$723$	−9.95386	−0.370188
$724$	0	0
$725$	−0.600061	−0.0222857
$726$	0	0
$727$	26.1640	0.970369	0.485184	−	0.874412i	$-0.338753\pi$
$727$	26.1640	0.970369	0.485184	+	0.874412i	$0.338753\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−34.2154	−1.26550
$732$	0	0
$733$	−32.8269	−1.21249	−0.606244	−	0.795279i	$-0.707325\pi$
$733$	−32.8269	−1.21249	−0.606244	+	0.795279i	$0.707325\pi$
$734$	0	0
$735$	23.3153	0.859999
$736$	0	0
$737$	−58.0034	−2.13658
$738$	0	0
$739$	−10.7605	−0.395832	−0.197916	−	0.980219i	$-0.563417\pi$
$739$	−10.7605	−0.395832	−0.197916	+	0.980219i	$0.563417\pi$
$740$	0	0
$741$	−19.0282	−0.699017
$742$	0	0
$743$	13.7384	0.504013	0.252006	−	0.967726i	$-0.418910\pi$
$743$	13.7384	0.504013	0.252006	+	0.967726i	$0.418910\pi$
$744$	0	0
$745$	16.5208	0.605277
$746$	0	0
$747$	5.30658	0.194158
$748$	0	0
$749$	44.6237	1.63052
$750$	0	0
$751$	−9.50573	−0.346869	−0.173435	−	0.984845i	$-0.555486\pi$
$751$	−9.50573	−0.346869	−0.173435	+	0.984845i	$0.555486\pi$
$752$	0	0
$753$	11.4013	0.415486
$754$	0	0
$755$	−21.1659	−0.770305
$756$	0	0
$757$	40.9235	1.48739	0.743695	−	0.668519i	$-0.233071\pi$
$757$	40.9235	1.48739	0.743695	+	0.668519i	$0.233071\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21.1113	0.765284	0.382642	−	0.923897i	$-0.375014\pi$
$761$	21.1113	0.765284	0.382642	+	0.923897i	$0.375014\pi$
$762$	0	0
$763$	7.05245	0.255316
$764$	0	0
$765$	11.2352	0.406209
$766$	0	0
$767$	12.1485	0.438656
$768$	0	0
$769$	25.0850	0.904589	0.452295	−	0.891869i	$-0.350606\pi$
$769$	25.0850	0.904589	0.452295	+	0.891869i	$0.350606\pi$
$770$	0	0
$771$	26.1644	0.942288
$772$	0	0
$773$	26.5857	0.956221	0.478110	−	0.878300i	$-0.341322\pi$
$773$	26.5857	0.956221	0.478110	+	0.878300i	$0.341322\pi$
$774$	0	0
$775$	6.14858	0.220863
$776$	0	0
$777$	43.1661	1.54857
$778$	0	0
$779$	33.6860	1.20693
$780$	0	0
$781$	−18.2966	−0.654704
$782$	0	0
$783$	−0.499143	−0.0178379
$784$	0	0
$785$	23.3512	0.833440
$786$	0	0
$787$	−23.4023	−0.834203	−0.417102	−	0.908860i	$-0.636954\pi$
$787$	−23.4023	−0.834203	−0.417102	+	0.908860i	$0.636954\pi$
$788$	0	0
$789$	−29.3436	−1.04466
$790$	0	0
$791$	69.8780	2.48458
$792$	0	0
$793$	6.82082	0.242214
$794$	0	0
$795$	−19.0211	−0.674609
$796$	0	0
$797$	11.2883	0.399851	0.199926	−	0.979811i	$-0.435930\pi$
$797$	11.2883	0.399851	0.199926	+	0.979811i	$0.435930\pi$
$798$	0	0
$799$	−34.6348	−1.22529
$800$	0	0
$801$	11.1462	0.393831
$802$	0	0
$803$	35.9662	1.26922
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−28.2655	−0.994992
$808$	0	0
$809$	34.4656	1.21175	0.605873	−	0.795562i	$-0.292824\pi$
$809$	34.4656	1.21175	0.605873	+	0.795562i	$0.292824\pi$
$810$	0	0
$811$	−23.6348	−0.829930	−0.414965	−	0.909837i	$-0.636206\pi$
$811$	−23.6348	−0.829930	−0.414965	+	0.909837i	$0.636206\pi$
$812$	0	0
$813$	−0.113125	−0.00396746
$814$	0	0
$815$	−12.0749	−0.422966
$816$	0	0
$817$	33.5989	1.17548
$818$	0	0
$819$	−17.3742	−0.607103
$820$	0	0
$821$	−11.0461	−0.385513	−0.192756	−	0.981247i	$-0.561743\pi$
$821$	−11.0461	−0.385513	−0.192756	+	0.981247i	$0.561743\pi$
$822$	0	0
$823$	41.3786	1.44237	0.721184	−	0.692744i	$-0.243598\pi$
$823$	41.3786	1.44237	0.721184	+	0.692744i	$0.243598\pi$
$824$	0	0
$825$	−5.80108	−0.201968
$826$	0	0
$827$	−33.3095	−1.15828	−0.579142	−	0.815227i	$-0.696612\pi$
$827$	−33.3095	−1.15828	−0.579142	+	0.815227i	$0.696612\pi$
$828$	0	0
$829$	−26.7169	−0.927917	−0.463959	−	0.885857i	$-0.653571\pi$
$829$	−26.7169	−0.927917	−0.463959	+	0.885857i	$0.653571\pi$
$830$	0	0
$831$	−3.96519	−0.137551
$832$	0	0
$833$	42.2354	1.46337
$834$	0	0
$835$	−33.2612	−1.15105
$836$	0	0
$837$	5.11451	0.176783
$838$	0	0
$839$	−8.68892	−0.299975	−0.149987	−	0.988688i	$-0.547923\pi$
$839$	−8.68892	−0.299975	−0.149987	+	0.988688i	$0.547923\pi$
$840$	0	0
$841$	−28.7509	−0.991409
$842$	0	0
$843$	−16.6368	−0.573002
$844$	0	0
$845$	13.5702	0.466830
$846$	0	0
$847$	−49.6928	−1.70747
$848$	0	0
$849$	−16.9251	−0.580868
$850$	0	0
$851$	0	0
$852$	0	0
$853$	25.0612	0.858078	0.429039	−	0.903286i	$-0.358852\pi$
$853$	25.0612	0.858078	0.429039	+	0.903286i	$0.358852\pi$
$854$	0	0
$855$	−11.0327	−0.377311
$856$	0	0
$857$	23.9910	0.819516	0.409758	−	0.912194i	$-0.365613\pi$
$857$	23.9910	0.819516	0.409758	+	0.912194i	$0.365613\pi$
$858$	0	0
$859$	−7.59647	−0.259188	−0.129594	−	0.991567i	$-0.541367\pi$
$859$	−7.59647	−0.259188	−0.129594	+	0.991567i	$0.541367\pi$
$860$	0	0
$861$	30.7579	1.04823
$862$	0	0
$863$	3.71488	0.126456	0.0632280	−	0.997999i	$-0.479860\pi$
$863$	3.71488	0.126456	0.0632280	+	0.997999i	$0.479860\pi$
$864$	0	0
$865$	25.3908	0.863314
$866$	0	0
$867$	3.35237	0.113852
$868$	0	0
$869$	56.1840	1.90591
$870$	0	0
$871$	51.6299	1.74941
$872$	0	0
$873$	7.13353	0.241433
$874$	0	0
$875$	38.2582	1.29336
$876$	0	0
$877$	33.3497	1.12614	0.563070	−	0.826409i	$-0.309620\pi$
$877$	33.3497	1.12614	0.563070	+	0.826409i	$0.309620\pi$
$878$	0	0
$879$	20.4715	0.690488
$880$	0	0
$881$	−24.3273	−0.819606	−0.409803	−	0.912174i	$-0.634403\pi$
$881$	−24.3273	−0.819606	−0.409803	+	0.912174i	$0.634403\pi$
$882$	0	0
$883$	−28.1205	−0.946331	−0.473166	−	0.880974i	$-0.656889\pi$
$883$	−28.1205	−0.946331	−0.473166	+	0.880974i	$0.656889\pi$
$884$	0	0
$885$	7.04382	0.236775
$886$	0	0
$887$	24.4526	0.821037	0.410519	−	0.911852i	$-0.365348\pi$
$887$	24.4526	0.821037	0.410519	+	0.911852i	$0.365348\pi$
$888$	0	0
$889$	57.8002	1.93856
$890$	0	0
$891$	−4.82545	−0.161659
$892$	0	0
$893$	34.0108	1.13813
$894$	0	0
$895$	15.1501	0.506413
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.55287	−0.0851430
$900$	0	0
$901$	−34.4565	−1.14791
$902$	0	0
$903$	30.6784	1.02091
$904$	0	0
$905$	−11.5645	−0.384418
$906$	0	0
$907$	7.78161	0.258384	0.129192	−	0.991620i	$-0.458762\pi$
$907$	7.78161	0.258384	0.129192	+	0.991620i	$0.458762\pi$
$908$	0	0
$909$	−5.28560	−0.175312
$910$	0	0
$911$	34.9065	1.15650	0.578252	−	0.815858i	$-0.303735\pi$
$911$	34.9065	1.15650	0.578252	+	0.815858i	$0.303735\pi$
$912$	0	0
$913$	−25.6067	−0.847457
$914$	0	0
$915$	3.95478	0.130741
$916$	0	0
$917$	40.2232	1.32829
$918$	0	0
$919$	17.9522	0.592187	0.296093	−	0.955159i	$-0.404316\pi$
$919$	17.9522	0.592187	0.296093	+	0.955159i	$0.404316\pi$
$920$	0	0
$921$	16.8237	0.554360
$922$	0	0
$923$	16.2862	0.536065
$924$	0	0
$925$	−12.8291	−0.421817
$926$	0	0
$927$	19.1553	0.629141
$928$	0	0
$929$	21.0040	0.689119	0.344560	−	0.938764i	$-0.388028\pi$
$929$	21.0040	0.689119	0.344560	+	0.938764i	$0.388028\pi$
$930$	0	0
$931$	−41.4744	−1.35927
$932$	0	0
$933$	−29.7791	−0.974923
$934$	0	0
$935$	−54.2148	−1.77301
$936$	0	0
$937$	−24.6484	−0.805229	−0.402615	−	0.915370i	$-0.631899\pi$
$937$	−24.6484	−0.805229	−0.402615	+	0.915370i	$0.631899\pi$
$938$	0	0
$939$	−30.2184	−0.986140
$940$	0	0
$941$	46.6557	1.52093	0.760467	−	0.649377i	$-0.224970\pi$
$941$	46.6557	1.52093	0.760467	+	0.649377i	$0.224970\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−10.0737	−0.327699
$946$	0	0
$947$	25.0345	0.813510	0.406755	−	0.913537i	$-0.366660\pi$
$947$	25.0345	0.813510	0.406755	+	0.913537i	$0.366660\pi$
$948$	0	0
$949$	−32.0142	−1.03922
$950$	0	0
$951$	−5.24217	−0.169989
$952$	0	0
$953$	−33.1164	−1.07275	−0.536373	−	0.843981i	$-0.680206\pi$
$953$	−33.1164	−1.07275	−0.536373	+	0.843981i	$0.680206\pi$
$954$	0	0
$955$	−29.4018	−0.951420
$956$	0	0
$957$	2.40859	0.0778587
$958$	0	0
$959$	32.9388	1.06365
$960$	0	0
$961$	−4.84180	−0.156187
$962$	0	0
$963$	−11.0318	−0.355496
$964$	0	0
$965$	55.6876	1.79265
$966$	0	0
$967$	12.0764	0.388352	0.194176	−	0.980967i	$-0.437797\pi$
$967$	12.0764	0.388352	0.194176	+	0.980967i	$0.437797\pi$
$968$	0	0
$969$	−19.9856	−0.642032
$970$	0	0
$971$	−49.4960	−1.58840	−0.794201	−	0.607655i	$-0.792110\pi$
$971$	−49.4960	−1.58840	−0.794201	+	0.607655i	$0.792110\pi$
$972$	0	0
$973$	40.9771	1.31367
$974$	0	0
$975$	5.16365	0.165369
$976$	0	0
$977$	−21.8547	−0.699194	−0.349597	−	0.936900i	$-0.613682\pi$
$977$	−21.8547	−0.699194	−0.349597	+	0.936900i	$0.613682\pi$
$978$	0	0
$979$	−53.7854	−1.71899
$980$	0	0
$981$	−1.74350	−0.0556657
$982$	0	0
$983$	7.24236	0.230995	0.115498	−	0.993308i	$-0.463154\pi$
$983$	7.24236	0.230995	0.115498	+	0.993308i	$0.463154\pi$
$984$	0	0
$985$	22.5569	0.718723
$986$	0	0
$987$	31.0545	0.988475
$988$	0	0
$989$	0	0
$990$	0	0
$991$	56.2306	1.78622	0.893111	−	0.449836i	$-0.148517\pi$
$991$	56.2306	1.78622	0.893111	+	0.449836i	$0.148517\pi$
$992$	0	0
$993$	18.9047	0.599921
$994$	0	0
$995$	−8.98987	−0.284998
$996$	0	0
$997$	−62.6162	−1.98307	−0.991537	−	0.129823i	$-0.958559\pi$
$997$	−62.6162	−1.98307	−0.991537	+	0.129823i	$0.958559\pi$
$998$	0	0
$999$	−10.6715	−0.337630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6348.2.a.s.1.8		10
23.9	even	11		276.2.i.a.265.1	yes	20
23.18	even	11		276.2.i.a.25.1	✓	20
23.22	odd	2		6348.2.a.t.1.3		10
69.32	odd	22		828.2.q.c.541.2		20
69.41	odd	22		828.2.q.c.577.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.i.a.25.1	✓	20	23.18	even	11
276.2.i.a.265.1	yes	20	23.9	even	11
828.2.q.c.541.2		20	69.32	odd	22
828.2.q.c.577.2		20	69.41	odd	22
6348.2.a.s.1.8		10	1.1	even	1	trivial
6348.2.a.t.1.3		10	23.22	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 6348 = 2^{2} \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6348.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.49042 q^{5} -4.04500 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.49042 q^{5} -4.04500 q^{7} +1.00000 q^{9} -4.82545 q^{11} +4.29523 q^{13} +2.49042 q^{15} +4.51136 q^{17} -4.43007 q^{19} -4.04500 q^{21} +1.20218 q^{25} +1.00000 q^{27} -0.499143 q^{29} +5.11451 q^{31} -4.82545 q^{33} -10.0737 q^{35} -10.6715 q^{37} +4.29523 q^{39} -7.60394 q^{41} -7.58427 q^{43} +2.49042 q^{45} -7.67725 q^{47} +9.36202 q^{49} +4.51136 q^{51} -7.63771 q^{53} -12.0174 q^{55} -4.43007 q^{57} +2.82837 q^{59} +1.58800 q^{61} -4.04500 q^{63} +10.6969 q^{65} +12.0203 q^{67} +3.79169 q^{71} -7.45343 q^{73} +1.20218 q^{75} +19.5190 q^{77} -11.6432 q^{79} +1.00000 q^{81} +5.30658 q^{83} +11.2352 q^{85} -0.499143 q^{87} +11.1462 q^{89} -17.3742 q^{91} +5.11451 q^{93} -11.0327 q^{95} +7.13353 q^{97} -4.82545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9	\( 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9} - 11 q^{11} - 2 q^{15} - 13 q^{17} - 18 q^{19} - 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} - 11 q^{33} - 13 q^{35} - 5 q^{37} + 24 q^{41} - 40 q^{43} - 2 q^{45} - 9 q^{47} + 5 q^{49} - 13 q^{51} - 6 q^{53} - 14 q^{55} - 18 q^{57} + 28 q^{59} - 39 q^{61} - 11 q^{63} - 14 q^{65} - 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} - 33 q^{79} + 10 q^{81} - 29 q^{83} - 21 q^{85} + 5 q^{87} - 17 q^{89} - 36 q^{91} + 15 q^{93} - 42 q^{95} - 46 q^{97} - 11 q^{99}+O(q^{100}) \) 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9 - 11 * q^11 - 2 * q^15 - 13 * q^17 - 18 * q^19 - 11 * q^21 + 10 * q^25 + 10 * q^27 + 5 * q^29 + 15 * q^31 - 11 * q^33 - 13 * q^35 - 5 * q^37 + 24 * q^41 - 40 * q^43 - 2 * q^45 - 9 * q^47 + 5 * q^49 - 13 * q^51 - 6 * q^53 - 14 * q^55 - 18 * q^57 + 28 * q^59 - 39 * q^61 - 11 * q^63 - 14 * q^65 - 32 * q^67 - 33 * q^71 - 50 * q^73 + 10 * q^75 + 19 * q^77 - 33 * q^79 + 10 * q^81 - 29 * q^83 - 21 * q^85 + 5 * q^87 - 17 * q^89 - 36 * q^91 + 15 * q^93 - 42 * q^95 - 46 * q^97 - 11 * q^99

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