LMFDB - Embedded newform 6292.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6292,2,Mod(1,6292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6292, 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("6292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6292 = 2^{2} \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6292.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:	$50.2418729518$
Analytic rank:	$0$
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} - 2x^{9} - 19x^{8} + 40x^{7} + 106x^{6} - 244x^{5} - 154x^{4} + 488x^{3} - 107x^{2} - 138x + 33 $ x^10 - 2x^9 - 19x^8 + 40x^7 + 106x^6 - 244x^5 - 154x^4 + 488x^3 - 107x^2 - 138*x + 33
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.05668$ of defining polynomial
Character	$\chi$	$=$	6292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.05668 q^{3} -1.91704 q^{5} +4.03552 q^{7} +6.34327 q^{9} +O(q^{10})$	$q-3.05668 q^{3} -1.91704 q^{5} +4.03552 q^{7} +6.34327 q^{9} -1.00000 q^{13} +5.85977 q^{15} +7.69559 q^{17} -1.94557 q^{19} -12.3353 q^{21} -1.72149 q^{23} -1.32495 q^{25} -10.2193 q^{27} +10.1932 q^{29} +9.09082 q^{31} -7.73626 q^{35} +11.8745 q^{37} +3.05668 q^{39} +5.28047 q^{41} +1.49422 q^{43} -12.1603 q^{45} +7.84750 q^{47} +9.28542 q^{49} -23.5229 q^{51} -10.5312 q^{53} +5.94699 q^{57} -1.12995 q^{59} -8.65043 q^{61} +25.5984 q^{63} +1.91704 q^{65} +1.55023 q^{67} +5.26202 q^{69} +11.3395 q^{71} -3.99516 q^{73} +4.04995 q^{75} -9.44413 q^{79} +12.2072 q^{81} -4.29408 q^{83} -14.7528 q^{85} -31.1574 q^{87} -4.54207 q^{89} -4.03552 q^{91} -27.7877 q^{93} +3.72975 q^{95} +13.1118 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9	$ 10 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9} - 10 q^{13} - 4 q^{15} + 20 q^{17} + 14 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 10 q^{27} + 16 q^{29} - 2 q^{31} - 20 q^{35} + 2 q^{39} + 22 q^{41} - 12 q^{43} - 8 q^{45} + 6 q^{47} - 2 q^{49} - 8 q^{51} + 6 q^{53} + 12 q^{57} - 4 q^{59} + 24 q^{61} + 68 q^{63} + 6 q^{65} - 6 q^{67} - 2 q^{69} - 8 q^{71} - 6 q^{73} - 18 q^{75} - 24 q^{79} + 10 q^{81} + 22 q^{83} + 34 q^{85} - 4 q^{87} - 42 q^{89} - 4 q^{91} - 38 q^{93} + 14 q^{97}+O(q^{100}) $ 10 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9 - 10 * q^13 - 4 * q^15 + 20 * q^17 + 14 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 10 * q^27 + 16 * q^29 - 2 * q^31 - 20 * q^35 + 2 * q^39 + 22 * q^41 - 12 * q^43 - 8 * q^45 + 6 * q^47 - 2 * q^49 - 8 * q^51 + 6 * q^53 + 12 * q^57 - 4 * q^59 + 24 * q^61 + 68 * q^63 + 6 * q^65 - 6 * q^67 - 2 * q^69 - 8 * q^71 - 6 * q^73 - 18 * q^75 - 24 * q^79 + 10 * q^81 + 22 * q^83 + 34 * q^85 - 4 * q^87 - 42 * q^89 - 4 * q^91 - 38 * q^93 + 14 * q^97

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
$3$	−3.05668	−1.76477	−0.882386	−	0.470526i	$-0.844064\pi$
$3$	−3.05668	−1.76477	−0.882386	+	0.470526i	$0.844064\pi$
$4$	0	0
$5$	−1.91704	−0.857327	−0.428663	−	0.903464i	$-0.641015\pi$
$5$	−1.91704	−0.857327	−0.428663	+	0.903464i	$0.641015\pi$
$6$	0	0
$7$	4.03552	1.52528	0.762642	−	0.646821i	$-0.223902\pi$
$7$	4.03552	1.52528	0.762642	+	0.646821i	$0.223902\pi$
$8$	0	0
$9$	6.34327	2.11442
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	5.85977	1.51299
$16$	0	0
$17$	7.69559	1.86645	0.933227	−	0.359287i	$-0.116980\pi$
$17$	7.69559	1.86645	0.933227	+	0.359287i	$0.116980\pi$
$18$	0	0
$19$	−1.94557	−0.446345	−0.223173	−	0.974779i	$-0.571641\pi$
$19$	−1.94557	−0.446345	−0.223173	+	0.974779i	$0.571641\pi$
$20$	0	0
$21$	−12.3353	−2.69178
$22$	0	0
$23$	−1.72149	−0.358955	−0.179477	−	0.983762i	$-0.557441\pi$
$23$	−1.72149	−0.358955	−0.179477	+	0.983762i	$0.557441\pi$
$24$	0	0
$25$	−1.32495	−0.264991
$26$	0	0
$27$	−10.2193	−1.96670
$28$	0	0
$29$	10.1932	1.89283	0.946416	−	0.322949i	$-0.104674\pi$
$29$	10.1932	1.89283	0.946416	+	0.322949i	$0.104674\pi$
$30$	0	0
$31$	9.09082	1.63276	0.816380	−	0.577516i	$-0.195978\pi$
$31$	9.09082	1.63276	0.816380	+	0.577516i	$0.195978\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7.73626	−1.30767
$36$	0	0
$37$	11.8745	1.95216	0.976080	−	0.217411i	$-0.0697612\pi$
$37$	11.8745	1.95216	0.976080	+	0.217411i	$0.0697612\pi$
$38$	0	0
$39$	3.05668	0.489460
$40$	0	0
$41$	5.28047	0.824670	0.412335	−	0.911032i	$-0.364713\pi$
$41$	5.28047	0.824670	0.412335	+	0.911032i	$0.364713\pi$
$42$	0	0
$43$	1.49422	0.227866	0.113933	−	0.993488i	$-0.463655\pi$
$43$	1.49422	0.227866	0.113933	+	0.993488i	$0.463655\pi$
$44$	0	0
$45$	−12.1603	−1.81275
$46$	0	0
$47$	7.84750	1.14468	0.572338	−	0.820018i	$-0.306037\pi$
$47$	7.84750	1.14468	0.572338	+	0.820018i	$0.306037\pi$
$48$	0	0
$49$	9.28542	1.32649
$50$	0	0
$51$	−23.5229	−3.29387
$52$	0	0
$53$	−10.5312	−1.44658	−0.723289	−	0.690546i	$-0.757370\pi$
$53$	−10.5312	−1.44658	−0.723289	+	0.690546i	$0.757370\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.94699	0.787698
$58$	0	0
$59$	−1.12995	−0.147107	−0.0735536	−	0.997291i	$-0.523434\pi$
$59$	−1.12995	−0.147107	−0.0735536	+	0.997291i	$0.523434\pi$
$60$	0	0
$61$	−8.65043	−1.10757	−0.553787	−	0.832658i	$-0.686818\pi$
$61$	−8.65043	−1.10757	−0.553787	+	0.832658i	$0.686818\pi$
$62$	0	0
$63$	25.5984	3.22509
$64$	0	0
$65$	1.91704	0.237780
$66$	0	0
$67$	1.55023	0.189391	0.0946954	−	0.995506i	$-0.469812\pi$
$67$	1.55023	0.189391	0.0946954	+	0.995506i	$0.469812\pi$
$68$	0	0
$69$	5.26202	0.633473
$70$	0	0
$71$	11.3395	1.34575	0.672875	−	0.739756i	$-0.265059\pi$
$71$	11.3395	1.34575	0.672875	+	0.739756i	$0.265059\pi$
$72$	0	0
$73$	−3.99516	−0.467598	−0.233799	−	0.972285i	$-0.575116\pi$
$73$	−3.99516	−0.467598	−0.233799	+	0.972285i	$0.575116\pi$
$74$	0	0
$75$	4.04995	0.467648
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.44413	−1.06255	−0.531274	−	0.847200i	$-0.678286\pi$
$79$	−9.44413	−1.06255	−0.531274	+	0.847200i	$0.678286\pi$
$80$	0	0
$81$	12.2072	1.35636
$82$	0	0
$83$	−4.29408	−0.471337	−0.235668	−	0.971834i	$-0.575728\pi$
$83$	−4.29408	−0.471337	−0.235668	+	0.971834i	$0.575728\pi$
$84$	0	0
$85$	−14.7528	−1.60016
$86$	0	0
$87$	−31.1574	−3.34042
$88$	0	0
$89$	−4.54207	−0.481458	−0.240729	−	0.970592i	$-0.577387\pi$
$89$	−4.54207	−0.481458	−0.240729	+	0.970592i	$0.577387\pi$
$90$	0	0
$91$	−4.03552	−0.423037
$92$	0	0
$93$	−27.7877	−2.88145
$94$	0	0
$95$	3.72975	0.382664
$96$	0	0
$97$	13.1118	1.33130	0.665650	−	0.746264i	$-0.268154\pi$
$97$	13.1118	1.33130	0.665650	+	0.746264i	$0.268154\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.70677	−0.169830	−0.0849151	−	0.996388i	$-0.527062\pi$
$101$	−1.70677	−0.169830	−0.0849151	+	0.996388i	$0.527062\pi$
$102$	0	0
$103$	−3.79534	−0.373966	−0.186983	−	0.982363i	$-0.559871\pi$
$103$	−3.79534	−0.373966	−0.186983	+	0.982363i	$0.559871\pi$
$104$	0	0
$105$	23.6472	2.30773
$106$	0	0
$107$	2.32323	0.224595	0.112298	−	0.993675i	$-0.464179\pi$
$107$	2.32323	0.224595	0.112298	+	0.993675i	$0.464179\pi$
$108$	0	0
$109$	−1.21294	−0.116179	−0.0580894	−	0.998311i	$-0.518501\pi$
$109$	−1.21294	−0.116179	−0.0580894	+	0.998311i	$0.518501\pi$
$110$	0	0
$111$	−36.2966	−3.44512
$112$	0	0
$113$	0.779652	0.0733435	0.0366717	−	0.999327i	$-0.488324\pi$
$113$	0.779652	0.0733435	0.0366717	+	0.999327i	$0.488324\pi$
$114$	0	0
$115$	3.30016	0.307741
$116$	0	0
$117$	−6.34327	−0.586435
$118$	0	0
$119$	31.0557	2.84687
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−16.1407	−1.45536
$124$	0	0
$125$	12.1252	1.08451
$126$	0	0
$127$	−13.1263	−1.16477	−0.582387	−	0.812912i	$-0.697881\pi$
$127$	−13.1263	−1.16477	−0.582387	+	0.812912i	$0.697881\pi$
$128$	0	0
$129$	−4.56735	−0.402132
$130$	0	0
$131$	−5.55362	−0.485222	−0.242611	−	0.970124i	$-0.578004\pi$
$131$	−5.55362	−0.485222	−0.242611	+	0.970124i	$0.578004\pi$
$132$	0	0
$133$	−7.85140	−0.680803
$134$	0	0
$135$	19.5908	1.68611
$136$	0	0
$137$	−8.63886	−0.738068	−0.369034	−	0.929416i	$-0.620311\pi$
$137$	−8.63886	−0.738068	−0.369034	+	0.929416i	$0.620311\pi$
$138$	0	0
$139$	−17.4689	−1.48170	−0.740848	−	0.671673i	$-0.765576\pi$
$139$	−17.4689	−1.48170	−0.740848	+	0.671673i	$0.765576\pi$
$140$	0	0
$141$	−23.9873	−2.02009
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−19.5408	−1.62278
$146$	0	0
$147$	−28.3825	−2.34095
$148$	0	0
$149$	6.17762	0.506091	0.253045	−	0.967454i	$-0.418568\pi$
$149$	6.17762	0.506091	0.253045	+	0.967454i	$0.418568\pi$
$150$	0	0
$151$	9.69951	0.789335	0.394668	−	0.918824i	$-0.370860\pi$
$151$	9.69951	0.789335	0.394668	+	0.918824i	$0.370860\pi$
$152$	0	0
$153$	48.8152	3.94647
$154$	0	0
$155$	−17.4275	−1.39981
$156$	0	0
$157$	−20.5219	−1.63783	−0.818913	−	0.573917i	$-0.805423\pi$
$157$	−20.5219	−1.63783	−0.818913	+	0.573917i	$0.805423\pi$
$158$	0	0
$159$	32.1906	2.55288
$160$	0	0
$161$	−6.94709	−0.547507
$162$	0	0
$163$	16.6299	1.30256	0.651279	−	0.758839i	$-0.274233\pi$
$163$	16.6299	1.30256	0.651279	+	0.758839i	$0.274233\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.4561	−0.886499	−0.443249	−	0.896398i	$-0.646174\pi$
$167$	−11.4561	−0.886499	−0.443249	+	0.896398i	$0.646174\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−12.3413	−0.943763
$172$	0	0
$173$	−6.70518	−0.509785	−0.254893	−	0.966969i	$-0.582040\pi$
$173$	−6.70518	−0.509785	−0.254893	+	0.966969i	$0.582040\pi$
$174$	0	0
$175$	−5.34688	−0.404186
$176$	0	0
$177$	3.45390	0.259611
$178$	0	0
$179$	4.70488	0.351659	0.175830	−	0.984421i	$-0.443739\pi$
$179$	4.70488	0.351659	0.175830	+	0.984421i	$0.443739\pi$
$180$	0	0
$181$	13.1147	0.974808	0.487404	−	0.873177i	$-0.337944\pi$
$181$	13.1147	0.974808	0.487404	+	0.873177i	$0.337944\pi$
$182$	0	0
$183$	26.4416	1.95462
$184$	0	0
$185$	−22.7640	−1.67364
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−41.2401	−2.99978
$190$	0	0
$191$	−19.2994	−1.39646	−0.698228	−	0.715876i	$-0.746028\pi$
$191$	−19.2994	−1.39646	−0.698228	+	0.715876i	$0.746028\pi$
$192$	0	0
$193$	11.7163	0.843358	0.421679	−	0.906745i	$-0.361441\pi$
$193$	11.7163	0.843358	0.421679	+	0.906745i	$0.361441\pi$
$194$	0	0
$195$	−5.85977	−0.419627
$196$	0	0
$197$	12.6349	0.900203	0.450101	−	0.892977i	$-0.351388\pi$
$197$	12.6349	0.900203	0.450101	+	0.892977i	$0.351388\pi$
$198$	0	0
$199$	−18.6343	−1.32095	−0.660476	−	0.750847i	$-0.729646\pi$
$199$	−18.6343	−1.32095	−0.660476	+	0.750847i	$0.729646\pi$
$200$	0	0
$201$	−4.73855	−0.334232
$202$	0	0
$203$	41.1349	2.88711
$204$	0	0
$205$	−10.1229	−0.707012
$206$	0	0
$207$	−10.9198	−0.758982
$208$	0	0
$209$	0	0
$210$	0	0
$211$	6.98415	0.480809	0.240405	−	0.970673i	$-0.422720\pi$
$211$	6.98415	0.480809	0.240405	+	0.970673i	$0.422720\pi$
$212$	0	0
$213$	−34.6612	−2.37494
$214$	0	0
$215$	−2.86448	−0.195356
$216$	0	0
$217$	36.6862	2.49042
$218$	0	0
$219$	12.2119	0.825204
$220$	0	0
$221$	−7.69559	−0.517661
$222$	0	0
$223$	−10.0358	−0.672048	−0.336024	−	0.941853i	$-0.609082\pi$
$223$	−10.0358	−0.672048	−0.336024	+	0.941853i	$0.609082\pi$
$224$	0	0
$225$	−8.40454	−0.560302
$226$	0	0
$227$	16.6403	1.10445	0.552227	−	0.833694i	$-0.313778\pi$
$227$	16.6403	1.10445	0.552227	+	0.833694i	$0.313778\pi$
$228$	0	0
$229$	8.45467	0.558700	0.279350	−	0.960189i	$-0.409881\pi$
$229$	8.45467	0.558700	0.279350	+	0.960189i	$0.409881\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	25.9604	1.70072	0.850360	−	0.526201i	$-0.176384\pi$
$233$	25.9604	1.70072	0.850360	+	0.526201i	$0.176384\pi$
$234$	0	0
$235$	−15.0440	−0.981361
$236$	0	0
$237$	28.8676	1.87516
$238$	0	0
$239$	−15.0354	−0.972559	−0.486280	−	0.873803i	$-0.661646\pi$
$239$	−15.0354	−0.972559	−0.486280	+	0.873803i	$0.661646\pi$
$240$	0	0
$241$	1.47284	0.0948738	0.0474369	−	0.998874i	$-0.484895\pi$
$241$	1.47284	0.0948738	0.0474369	+	0.998874i	$0.484895\pi$
$242$	0	0
$243$	−6.65571	−0.426964
$244$	0	0
$245$	−17.8005	−1.13723
$246$	0	0
$247$	1.94557	0.123794
$248$	0	0
$249$	13.1256	0.831802
$250$	0	0
$251$	−0.293870	−0.0185489	−0.00927446	−	0.999957i	$-0.502952\pi$
$251$	−0.293870	−0.0185489	−0.00927446	+	0.999957i	$0.502952\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	45.0944	2.82392
$256$	0	0
$257$	7.92046	0.494065	0.247032	−	0.969007i	$-0.420545\pi$
$257$	7.92046	0.494065	0.247032	+	0.969007i	$0.420545\pi$
$258$	0	0
$259$	47.9199	2.97760
$260$	0	0
$261$	64.6583	4.00225
$262$	0	0
$263$	9.93258	0.612469	0.306235	−	0.951956i	$-0.400931\pi$
$263$	9.93258	0.612469	0.306235	+	0.951956i	$0.400931\pi$
$264$	0	0
$265$	20.1888	1.24019
$266$	0	0
$267$	13.8836	0.849664
$268$	0	0
$269$	18.1467	1.10642	0.553211	−	0.833041i	$-0.313402\pi$
$269$	18.1467	1.10642	0.553211	+	0.833041i	$0.313402\pi$
$270$	0	0
$271$	−7.68320	−0.466721	−0.233361	−	0.972390i	$-0.574972\pi$
$271$	−7.68320	−0.466721	−0.233361	+	0.972390i	$0.574972\pi$
$272$	0	0
$273$	12.3353	0.746565
$274$	0	0
$275$	0	0
$276$	0	0
$277$	11.5967	0.696777	0.348388	−	0.937350i	$-0.386729\pi$
$277$	11.5967	0.696777	0.348388	+	0.937350i	$0.386729\pi$
$278$	0	0
$279$	57.6655	3.45234
$280$	0	0
$281$	−9.44728	−0.563578	−0.281789	−	0.959476i	$-0.590928\pi$
$281$	−9.44728	−0.563578	−0.281789	+	0.959476i	$0.590928\pi$
$282$	0	0
$283$	24.5732	1.46073	0.730363	−	0.683059i	$-0.239351\pi$
$283$	24.5732	1.46073	0.730363	+	0.683059i	$0.239351\pi$
$284$	0	0
$285$	−11.4006	−0.675315
$286$	0	0
$287$	21.3094	1.25786
$288$	0	0
$289$	42.2221	2.48365
$290$	0	0
$291$	−40.0785	−2.34944
$292$	0	0
$293$	9.39343	0.548770	0.274385	−	0.961620i	$-0.411526\pi$
$293$	9.39343	0.548770	0.274385	+	0.961620i	$0.411526\pi$
$294$	0	0
$295$	2.16617	0.126119
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.72149	0.0995561
$300$	0	0
$301$	6.02995	0.347561
$302$	0	0
$303$	5.21705	0.299712
$304$	0	0
$305$	16.5832	0.949553
$306$	0	0
$307$	−17.3084	−0.987843	−0.493921	−	0.869507i	$-0.664437\pi$
$307$	−17.3084	−0.987843	−0.493921	+	0.869507i	$0.664437\pi$
$308$	0	0
$309$	11.6011	0.659965
$310$	0	0
$311$	10.3719	0.588138	0.294069	−	0.955784i	$-0.404990\pi$
$311$	10.3719	0.588138	0.294069	+	0.955784i	$0.404990\pi$
$312$	0	0
$313$	−13.5882	−0.768048	−0.384024	−	0.923323i	$-0.625462\pi$
$313$	−13.5882	−0.768048	−0.384024	+	0.923323i	$0.625462\pi$
$314$	0	0
$315$	−49.0731	−2.76496
$316$	0	0
$317$	12.7488	0.716043	0.358022	−	0.933713i	$-0.383451\pi$
$317$	12.7488	0.716043	0.358022	+	0.933713i	$0.383451\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−7.10137	−0.396360
$322$	0	0
$323$	−14.9723	−0.833083
$324$	0	0
$325$	1.32495	0.0734952
$326$	0	0
$327$	3.70757	0.205029
$328$	0	0
$329$	31.6687	1.74595
$330$	0	0
$331$	−5.83145	−0.320525	−0.160263	−	0.987074i	$-0.551234\pi$
$331$	−5.83145	−0.320525	−0.160263	+	0.987074i	$0.551234\pi$
$332$	0	0
$333$	75.3233	4.12769
$334$	0	0
$335$	−2.97185	−0.162370
$336$	0	0
$337$	22.0566	1.20150	0.600751	−	0.799437i	$-0.294868\pi$
$337$	22.0566	1.20150	0.600751	+	0.799437i	$0.294868\pi$
$338$	0	0
$339$	−2.38314	−0.129435
$340$	0	0
$341$	0	0
$342$	0	0
$343$	9.22286	0.497987
$344$	0	0
$345$	−10.0875	−0.543094
$346$	0	0
$347$	−20.5648	−1.10398	−0.551989	−	0.833851i	$-0.686131\pi$
$347$	−20.5648	−1.10398	−0.551989	+	0.833851i	$0.686131\pi$
$348$	0	0
$349$	−15.1495	−0.810934	−0.405467	−	0.914110i	$-0.632891\pi$
$349$	−15.1495	−0.810934	−0.405467	+	0.914110i	$0.632891\pi$
$350$	0	0
$351$	10.2193	0.545465
$352$	0	0
$353$	−26.7554	−1.42405	−0.712023	−	0.702156i	$-0.752221\pi$
$353$	−26.7554	−1.42405	−0.712023	+	0.702156i	$0.752221\pi$
$354$	0	0
$355$	−21.7383	−1.15375
$356$	0	0
$357$	−94.9272	−5.02408
$358$	0	0
$359$	24.9185	1.31515	0.657573	−	0.753391i	$-0.271583\pi$
$359$	24.9185	1.31515	0.657573	+	0.753391i	$0.271583\pi$
$360$	0	0
$361$	−15.2147	−0.800776
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.65888	0.400884
$366$	0	0
$367$	−33.8519	−1.76705	−0.883526	−	0.468382i	$-0.844837\pi$
$367$	−33.8519	−1.76705	−0.883526	+	0.468382i	$0.844837\pi$
$368$	0	0
$369$	33.4954	1.74370
$370$	0	0
$371$	−42.4990	−2.20644
$372$	0	0
$373$	−6.26432	−0.324354	−0.162177	−	0.986762i	$-0.551852\pi$
$373$	−6.26432	−0.324354	−0.162177	+	0.986762i	$0.551852\pi$
$374$	0	0
$375$	−37.0628	−1.91391
$376$	0	0
$377$	−10.1932	−0.524977
$378$	0	0
$379$	−24.7016	−1.26884	−0.634419	−	0.772989i	$-0.718761\pi$
$379$	−24.7016	−1.26884	−0.634419	+	0.772989i	$0.718761\pi$
$380$	0	0
$381$	40.1229	2.05556
$382$	0	0
$383$	−20.6711	−1.05624	−0.528121	−	0.849169i	$-0.677103\pi$
$383$	−20.6711	−1.05624	−0.528121	+	0.849169i	$0.677103\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.47824	0.481806
$388$	0	0
$389$	29.4917	1.49529	0.747645	−	0.664098i	$-0.231184\pi$
$389$	29.4917	1.49529	0.747645	+	0.664098i	$0.231184\pi$
$390$	0	0
$391$	−13.2478	−0.669972
$392$	0	0
$393$	16.9756	0.856306
$394$	0	0
$395$	18.1048	0.910951
$396$	0	0
$397$	37.6342	1.88881	0.944404	−	0.328788i	$-0.106640\pi$
$397$	37.6342	1.88881	0.944404	+	0.328788i	$0.106640\pi$
$398$	0	0
$399$	23.9992	1.20146
$400$	0	0
$401$	1.84186	0.0919782	0.0459891	−	0.998942i	$-0.485356\pi$
$401$	1.84186	0.0919782	0.0459891	+	0.998942i	$0.485356\pi$
$402$	0	0
$403$	−9.09082	−0.452846
$404$	0	0
$405$	−23.4018	−1.16284
$406$	0	0
$407$	0	0
$408$	0	0
$409$	25.2509	1.24857	0.624287	−	0.781195i	$-0.285389\pi$
$409$	25.2509	1.24857	0.624287	+	0.781195i	$0.285389\pi$
$410$	0	0
$411$	26.4062	1.30252
$412$	0	0
$413$	−4.55995	−0.224380
$414$	0	0
$415$	8.23193	0.404090
$416$	0	0
$417$	53.3969	2.61485
$418$	0	0
$419$	33.2261	1.62320	0.811601	−	0.584212i	$-0.198597\pi$
$419$	33.2261	1.62320	0.811601	+	0.584212i	$0.198597\pi$
$420$	0	0
$421$	−15.2943	−0.745401	−0.372700	−	0.927952i	$-0.621568\pi$
$421$	−15.2943	−0.745401	−0.372700	+	0.927952i	$0.621568\pi$
$422$	0	0
$423$	49.7788	2.42033
$424$	0	0
$425$	−10.1963	−0.494593
$426$	0	0
$427$	−34.9090	−1.68936
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0.977673	0.0470929	0.0235464	−	0.999723i	$-0.492504\pi$
$431$	0.977673	0.0470929	0.0235464	+	0.999723i	$0.492504\pi$
$432$	0	0
$433$	−17.7868	−0.854781	−0.427390	−	0.904067i	$-0.640567\pi$
$433$	−17.7868	−0.854781	−0.427390	+	0.904067i	$0.640567\pi$
$434$	0	0
$435$	59.7299	2.86383
$436$	0	0
$437$	3.34928	0.160218
$438$	0	0
$439$	−7.68850	−0.366952	−0.183476	−	0.983024i	$-0.558735\pi$
$439$	−7.68850	−0.366952	−0.183476	+	0.983024i	$0.558735\pi$
$440$	0	0
$441$	58.8999	2.80476
$442$	0	0
$443$	17.0418	0.809680	0.404840	−	0.914388i	$-0.367327\pi$
$443$	17.0418	0.809680	0.404840	+	0.914388i	$0.367327\pi$
$444$	0	0
$445$	8.70733	0.412767
$446$	0	0
$447$	−18.8830	−0.893135
$448$	0	0
$449$	16.2068	0.764847	0.382423	−	0.923987i	$-0.375090\pi$
$449$	16.2068	0.764847	0.382423	+	0.923987i	$0.375090\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−29.6483	−1.39300
$454$	0	0
$455$	7.73626	0.362681
$456$	0	0
$457$	3.82030	0.178706	0.0893531	−	0.996000i	$-0.471520\pi$
$457$	3.82030	0.178706	0.0893531	+	0.996000i	$0.471520\pi$
$458$	0	0
$459$	−78.6434	−3.67076
$460$	0	0
$461$	6.29895	0.293371	0.146686	−	0.989183i	$-0.453139\pi$
$461$	6.29895	0.293371	0.146686	+	0.989183i	$0.453139\pi$
$462$	0	0
$463$	−9.59614	−0.445971	−0.222985	−	0.974822i	$-0.571580\pi$
$463$	−9.59614	−0.445971	−0.222985	+	0.974822i	$0.571580\pi$
$464$	0	0
$465$	53.2701	2.47034
$466$	0	0
$467$	−19.3712	−0.896392	−0.448196	−	0.893935i	$-0.647933\pi$
$467$	−19.3712	−0.896392	−0.448196	+	0.893935i	$0.647933\pi$
$468$	0	0
$469$	6.25598	0.288875
$470$	0	0
$471$	62.7288	2.89039
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.57780	0.118277
$476$	0	0
$477$	−66.8025	−3.05868
$478$	0	0
$479$	−19.5927	−0.895214	−0.447607	−	0.894230i	$-0.647724\pi$
$479$	−19.5927	−0.895214	−0.447607	+	0.894230i	$0.647724\pi$
$480$	0	0
$481$	−11.8745	−0.541432
$482$	0	0
$483$	21.2350	0.966226
$484$	0	0
$485$	−25.1358	−1.14136
$486$	0	0
$487$	−29.9092	−1.35532	−0.677658	−	0.735377i	$-0.737005\pi$
$487$	−29.9092	−1.35532	−0.677658	+	0.735377i	$0.737005\pi$
$488$	0	0
$489$	−50.8323	−2.29872
$490$	0	0
$491$	24.6210	1.11113	0.555565	−	0.831473i	$-0.312502\pi$
$491$	24.6210	1.11113	0.555565	+	0.831473i	$0.312502\pi$
$492$	0	0
$493$	78.4428	3.53289
$494$	0	0
$495$	0	0
$496$	0	0
$497$	45.7608	2.05265
$498$	0	0
$499$	19.9781	0.894345	0.447172	−	0.894448i	$-0.352431\pi$
$499$	19.9781	0.894345	0.447172	+	0.894448i	$0.352431\pi$
$500$	0	0
$501$	35.0176	1.56447
$502$	0	0
$503$	29.3525	1.30877	0.654383	−	0.756163i	$-0.272929\pi$
$503$	29.3525	1.30877	0.654383	+	0.756163i	$0.272929\pi$
$504$	0	0
$505$	3.27195	0.145600
$506$	0	0
$507$	−3.05668	−0.135752
$508$	0	0
$509$	17.9852	0.797178	0.398589	−	0.917130i	$-0.369500\pi$
$509$	17.9852	0.797178	0.398589	+	0.917130i	$0.369500\pi$
$510$	0	0
$511$	−16.1225	−0.713219
$512$	0	0
$513$	19.8824	0.877828
$514$	0	0
$515$	7.27583	0.320611
$516$	0	0
$517$	0	0
$518$	0	0
$519$	20.4956	0.899655
$520$	0	0
$521$	20.6354	0.904054	0.452027	−	0.892004i	$-0.350701\pi$
$521$	20.6354	0.904054	0.452027	+	0.892004i	$0.350701\pi$
$522$	0	0
$523$	−18.7650	−0.820538	−0.410269	−	0.911965i	$-0.634565\pi$
$523$	−18.7650	−0.820538	−0.410269	+	0.911965i	$0.634565\pi$
$524$	0	0
$525$	16.3437	0.713296
$526$	0	0
$527$	69.9592	3.04747
$528$	0	0
$529$	−20.0365	−0.871152
$530$	0	0
$531$	−7.16759	−0.311047
$532$	0	0
$533$	−5.28047	−0.228722
$534$	0	0
$535$	−4.45373	−0.192552
$536$	0	0
$537$	−14.3813	−0.620599
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−0.258752	−0.0111246	−0.00556230	−	0.999985i	$-0.501771\pi$
$541$	−0.258752	−0.0111246	−0.00556230	+	0.999985i	$0.501771\pi$
$542$	0	0
$543$	−40.0874	−1.72031
$544$	0	0
$545$	2.32526	0.0996032
$546$	0	0
$547$	7.81749	0.334252	0.167126	−	0.985936i	$-0.446551\pi$
$547$	7.81749	0.334252	0.167126	+	0.985936i	$0.446551\pi$
$548$	0	0
$549$	−54.8720	−2.34188
$550$	0	0
$551$	−19.8317	−0.844857
$552$	0	0
$553$	−38.1120	−1.62069
$554$	0	0
$555$	69.5820	2.95359
$556$	0	0
$557$	44.1391	1.87023	0.935116	−	0.354342i	$-0.115295\pi$
$557$	44.1391	1.87023	0.935116	+	0.354342i	$0.115295\pi$
$558$	0	0
$559$	−1.49422	−0.0631988
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−33.2932	−1.40314	−0.701571	−	0.712600i	$-0.747518\pi$
$563$	−33.2932	−1.40314	−0.701571	+	0.712600i	$0.747518\pi$
$564$	0	0
$565$	−1.49462	−0.0628793
$566$	0	0
$567$	49.2625	2.06883
$568$	0	0
$569$	−18.4393	−0.773016	−0.386508	−	0.922286i	$-0.626319\pi$
$569$	−18.4393	−0.773016	−0.386508	+	0.922286i	$0.626319\pi$
$570$	0	0
$571$	−36.6013	−1.53172	−0.765859	−	0.643009i	$-0.777686\pi$
$571$	−36.6013	−1.53172	−0.765859	+	0.643009i	$0.777686\pi$
$572$	0	0
$573$	58.9920	2.46443
$574$	0	0
$575$	2.28089	0.0951197
$576$	0	0
$577$	20.5866	0.857032	0.428516	−	0.903534i	$-0.359037\pi$
$577$	20.5866	0.857032	0.428516	+	0.903534i	$0.359037\pi$
$578$	0	0
$579$	−35.8129	−1.48833
$580$	0	0
$581$	−17.3289	−0.718922
$582$	0	0
$583$	0	0
$584$	0	0
$585$	12.1603	0.502767
$586$	0	0
$587$	8.61478	0.355570	0.177785	−	0.984069i	$-0.443107\pi$
$587$	8.61478	0.355570	0.177785	+	0.984069i	$0.443107\pi$
$588$	0	0
$589$	−17.6869	−0.728775
$590$	0	0
$591$	−38.6209	−1.58865
$592$	0	0
$593$	−41.8529	−1.71869	−0.859346	−	0.511395i	$-0.829129\pi$
$593$	−41.8529	−1.71869	−0.859346	+	0.511395i	$0.829129\pi$
$594$	0	0
$595$	−59.5351	−2.44070
$596$	0	0
$597$	56.9591	2.33118
$598$	0	0
$599$	24.9481	1.01935	0.509676	−	0.860366i	$-0.329765\pi$
$599$	24.9481	1.01935	0.509676	+	0.860366i	$0.329765\pi$
$600$	0	0
$601$	17.2852	0.705078	0.352539	−	0.935797i	$-0.385318\pi$
$601$	17.2852	0.705078	0.352539	+	0.935797i	$0.385318\pi$
$602$	0	0
$603$	9.83352	0.400452
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.6313	0.756220	0.378110	−	0.925761i	$-0.376574\pi$
$607$	18.6313	0.756220	0.378110	+	0.925761i	$0.376574\pi$
$608$	0	0
$609$	−125.736	−5.09509
$610$	0	0
$611$	−7.84750	−0.317476
$612$	0	0
$613$	−2.66210	−0.107521	−0.0537607	−	0.998554i	$-0.517121\pi$
$613$	−2.66210	−0.107521	−0.0537607	+	0.998554i	$0.517121\pi$
$614$	0	0
$615$	30.9423	1.24772
$616$	0	0
$617$	33.3263	1.34167	0.670833	−	0.741608i	$-0.265937\pi$
$617$	33.3263	1.34167	0.670833	+	0.741608i	$0.265937\pi$
$618$	0	0
$619$	23.8053	0.956817	0.478408	−	0.878137i	$-0.341214\pi$
$619$	23.8053	0.956817	0.478408	+	0.878137i	$0.341214\pi$
$620$	0	0
$621$	17.5924	0.705957
$622$	0	0
$623$	−18.3296	−0.734360
$624$	0	0
$625$	−16.6197	−0.664789
$626$	0	0
$627$	0	0
$628$	0	0
$629$	91.3815	3.64362
$630$	0	0
$631$	−3.30656	−0.131632	−0.0658161	−	0.997832i	$-0.520965\pi$
$631$	−3.30656	−0.131632	−0.0658161	+	0.997832i	$0.520965\pi$
$632$	0	0
$633$	−21.3483	−0.848519
$634$	0	0
$635$	25.1637	0.998591
$636$	0	0
$637$	−9.28542	−0.367902
$638$	0	0
$639$	71.9295	2.84549
$640$	0	0
$641$	3.08399	0.121810	0.0609051	−	0.998144i	$-0.480601\pi$
$641$	3.08399	0.121810	0.0609051	+	0.998144i	$0.480601\pi$
$642$	0	0
$643$	11.2265	0.442728	0.221364	−	0.975191i	$-0.428949\pi$
$643$	11.2265	0.442728	0.221364	+	0.975191i	$0.428949\pi$
$644$	0	0
$645$	8.75579	0.344759
$646$	0	0
$647$	27.0195	1.06225	0.531123	−	0.847295i	$-0.321770\pi$
$647$	27.0195	1.06225	0.531123	+	0.847295i	$0.321770\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−112.138	−4.39503
$652$	0	0
$653$	1.23740	0.0484232	0.0242116	−	0.999707i	$-0.492292\pi$
$653$	1.23740	0.0484232	0.0242116	+	0.999707i	$0.492292\pi$
$654$	0	0
$655$	10.6465	0.415994
$656$	0	0
$657$	−25.3423	−0.988699
$658$	0	0
$659$	36.9162	1.43805	0.719026	−	0.694984i	$-0.244588\pi$
$659$	36.9162	1.43805	0.719026	+	0.694984i	$0.244588\pi$
$660$	0	0
$661$	−47.6262	−1.85244	−0.926222	−	0.376979i	$-0.876963\pi$
$661$	−47.6262	−1.85244	−0.926222	+	0.376979i	$0.876963\pi$
$662$	0	0
$663$	23.5229	0.913555
$664$	0	0
$665$	15.0515	0.583671
$666$	0	0
$667$	−17.5475	−0.679441
$668$	0	0
$669$	30.6762	1.18601
$670$	0	0
$671$	0	0
$672$	0	0
$673$	3.87401	0.149332	0.0746661	−	0.997209i	$-0.476211\pi$
$673$	3.87401	0.149332	0.0746661	+	0.997209i	$0.476211\pi$
$674$	0	0
$675$	13.5401	0.521158
$676$	0	0
$677$	−41.4986	−1.59492	−0.797459	−	0.603372i	$-0.793823\pi$
$677$	−41.4986	−1.59492	−0.797459	+	0.603372i	$0.793823\pi$
$678$	0	0
$679$	52.9129	2.03061
$680$	0	0
$681$	−50.8639	−1.94911
$682$	0	0
$683$	−8.22654	−0.314780	−0.157390	−	0.987537i	$-0.550308\pi$
$683$	−8.22654	−0.314780	−0.157390	+	0.987537i	$0.550308\pi$
$684$	0	0
$685$	16.5611	0.632765
$686$	0	0
$687$	−25.8432	−0.985979
$688$	0	0
$689$	10.5312	0.401208
$690$	0	0
$691$	−20.0880	−0.764184	−0.382092	−	0.924124i	$-0.624796\pi$
$691$	−20.0880	−0.764184	−0.382092	+	0.924124i	$0.624796\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	33.4887	1.27030
$696$	0	0
$697$	40.6363	1.53921
$698$	0	0
$699$	−79.3524	−3.00138
$700$	0	0
$701$	−11.6975	−0.441809	−0.220905	−	0.975295i	$-0.570901\pi$
$701$	−11.6975	−0.441809	−0.220905	+	0.975295i	$0.570901\pi$
$702$	0	0
$703$	−23.1028	−0.871338
$704$	0	0
$705$	45.9846	1.73188
$706$	0	0
$707$	−6.88771	−0.259039
$708$	0	0
$709$	−41.6803	−1.56534	−0.782668	−	0.622439i	$-0.786142\pi$
$709$	−41.6803	−1.56534	−0.782668	+	0.622439i	$0.786142\pi$
$710$	0	0
$711$	−59.9066	−2.24667
$712$	0	0
$713$	−15.6497	−0.586086
$714$	0	0
$715$	0	0
$716$	0	0
$717$	45.9584	1.71635
$718$	0	0
$719$	−5.35005	−0.199523	−0.0997616	−	0.995011i	$-0.531808\pi$
$719$	−5.35005	−0.199523	−0.0997616	+	0.995011i	$0.531808\pi$
$720$	0	0
$721$	−15.3162	−0.570404
$722$	0	0
$723$	−4.50199	−0.167431
$724$	0	0
$725$	−13.5055	−0.501583
$726$	0	0
$727$	−27.6271	−1.02463	−0.512316	−	0.858797i	$-0.671212\pi$
$727$	−27.6271	−1.02463	−0.512316	+	0.858797i	$0.671212\pi$
$728$	0	0
$729$	−16.2774	−0.602865
$730$	0	0
$731$	11.4989	0.425302
$732$	0	0
$733$	35.7408	1.32012	0.660059	−	0.751214i	$-0.270531\pi$
$733$	35.7408	1.32012	0.660059	+	0.751214i	$0.270531\pi$
$734$	0	0
$735$	54.4104	2.00696
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−6.09319	−0.224141	−0.112071	−	0.993700i	$-0.535748\pi$
$739$	−6.09319	−0.224141	−0.112071	+	0.993700i	$0.535748\pi$
$740$	0	0
$741$	−5.94699	−0.218468
$742$	0	0
$743$	49.5653	1.81837	0.909186	−	0.416389i	$-0.136705\pi$
$743$	49.5653	1.81837	0.909186	+	0.416389i	$0.136705\pi$
$744$	0	0
$745$	−11.8428	−0.433885
$746$	0	0
$747$	−27.2385	−0.996605
$748$	0	0
$749$	9.37545	0.342572
$750$	0	0
$751$	−43.6026	−1.59108	−0.795540	−	0.605901i	$-0.792813\pi$
$751$	−43.6026	−1.59108	−0.795540	+	0.605901i	$0.792813\pi$
$752$	0	0
$753$	0.898266	0.0327346
$754$	0	0
$755$	−18.5944	−0.676718
$756$	0	0
$757$	31.6498	1.15033	0.575166	−	0.818037i	$-0.304937\pi$
$757$	31.6498	1.15033	0.575166	+	0.818037i	$0.304937\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.6373	−1.03810	−0.519051	−	0.854743i	$-0.673715\pi$
$761$	−28.6373	−1.03810	−0.519051	+	0.854743i	$0.673715\pi$
$762$	0	0
$763$	−4.89485	−0.177205
$764$	0	0
$765$	−93.5807	−3.38342
$766$	0	0
$767$	1.12995	0.0408002
$768$	0	0
$769$	26.5020	0.955688	0.477844	−	0.878445i	$-0.341418\pi$
$769$	26.5020	0.955688	0.477844	+	0.878445i	$0.341418\pi$
$770$	0	0
$771$	−24.2103	−0.871912
$772$	0	0
$773$	23.2917	0.837746	0.418873	−	0.908045i	$-0.362425\pi$
$773$	23.2917	0.837746	0.418873	+	0.908045i	$0.362425\pi$
$774$	0	0
$775$	−12.0449	−0.432666
$776$	0	0
$777$	−146.476	−5.25478
$778$	0	0
$779$	−10.2735	−0.368088
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−104.167	−3.72264
$784$	0	0
$785$	39.3413	1.40415
$786$	0	0
$787$	−51.2465	−1.82674	−0.913370	−	0.407131i	$-0.866529\pi$
$787$	−51.2465	−1.82674	−0.913370	+	0.407131i	$0.866529\pi$
$788$	0	0
$789$	−30.3607	−1.08087
$790$	0	0
$791$	3.14630	0.111870
$792$	0	0
$793$	8.65043	0.307186
$794$	0	0
$795$	−61.7107	−2.18865
$796$	0	0
$797$	8.87292	0.314295	0.157148	−	0.987575i	$-0.449770\pi$
$797$	8.87292	0.314295	0.157148	+	0.987575i	$0.449770\pi$
$798$	0	0
$799$	60.3911	2.13648
$800$	0	0
$801$	−28.8115	−1.01801
$802$	0	0
$803$	0	0
$804$	0	0
$805$	13.3179	0.469393
$806$	0	0
$807$	−55.4685	−1.95258
$808$	0	0
$809$	0.858804	0.0301939	0.0150970	−	0.999886i	$-0.495194\pi$
$809$	0.858804	0.0301939	0.0150970	+	0.999886i	$0.495194\pi$
$810$	0	0
$811$	−53.2622	−1.87029	−0.935144	−	0.354268i	$-0.884730\pi$
$811$	−53.2622	−1.87029	−0.935144	+	0.354268i	$0.884730\pi$
$812$	0	0
$813$	23.4851	0.823657
$814$	0	0
$815$	−31.8803	−1.11672
$816$	0	0
$817$	−2.90712	−0.101707
$818$	0	0
$819$	−25.5984	−0.894480
$820$	0	0
$821$	16.7139	0.583319	0.291659	−	0.956522i	$-0.405793\pi$
$821$	16.7139	0.583319	0.291659	+	0.956522i	$0.405793\pi$
$822$	0	0
$823$	6.85754	0.239039	0.119519	−	0.992832i	$-0.461865\pi$
$823$	6.85754	0.239039	0.119519	+	0.992832i	$0.461865\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.6278	1.51709	0.758544	−	0.651622i	$-0.225911\pi$
$827$	43.6278	1.51709	0.758544	+	0.651622i	$0.225911\pi$
$828$	0	0
$829$	−46.8034	−1.62555	−0.812774	−	0.582579i	$-0.802044\pi$
$829$	−46.8034	−1.62555	−0.812774	+	0.582579i	$0.802044\pi$
$830$	0	0
$831$	−35.4473	−1.22965
$832$	0	0
$833$	71.4568	2.47583
$834$	0	0
$835$	21.9618	0.760019
$836$	0	0
$837$	−92.9017	−3.21115
$838$	0	0
$839$	−37.9001	−1.30846	−0.654228	−	0.756297i	$-0.727006\pi$
$839$	−37.9001	−1.30846	−0.654228	+	0.756297i	$0.727006\pi$
$840$	0	0
$841$	74.9017	2.58282
$842$	0	0
$843$	28.8773	0.994586
$844$	0	0
$845$	−1.91704	−0.0659482
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−75.1123	−2.57785
$850$	0	0
$851$	−20.4418	−0.700737
$852$	0	0
$853$	46.5765	1.59475	0.797375	−	0.603484i	$-0.206221\pi$
$853$	46.5765	1.59475	0.797375	+	0.603484i	$0.206221\pi$
$854$	0	0
$855$	23.6588	0.809113
$856$	0	0
$857$	32.4524	1.10855	0.554277	−	0.832332i	$-0.312995\pi$
$857$	32.4524	1.10855	0.554277	+	0.832332i	$0.312995\pi$
$858$	0	0
$859$	−19.6831	−0.671580	−0.335790	−	0.941937i	$-0.609003\pi$
$859$	−19.6831	−0.671580	−0.335790	+	0.941937i	$0.609003\pi$
$860$	0	0
$861$	−65.1360	−2.21983
$862$	0	0
$863$	−11.5880	−0.394459	−0.197230	−	0.980357i	$-0.563194\pi$
$863$	−11.5880	−0.394459	−0.197230	+	0.980357i	$0.563194\pi$
$864$	0	0
$865$	12.8541	0.437053
$866$	0	0
$867$	−129.059	−4.38308
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−1.55023	−0.0525275
$872$	0	0
$873$	83.1715	2.81493
$874$	0	0
$875$	48.9315	1.65419
$876$	0	0
$877$	−37.8964	−1.27967	−0.639836	−	0.768512i	$-0.720998\pi$
$877$	−37.8964	−1.27967	−0.639836	+	0.768512i	$0.720998\pi$
$878$	0	0
$879$	−28.7127	−0.968454
$880$	0	0
$881$	45.1113	1.51984	0.759919	−	0.650018i	$-0.225239\pi$
$881$	45.1113	1.51984	0.759919	+	0.650018i	$0.225239\pi$
$882$	0	0
$883$	−2.83536	−0.0954175	−0.0477087	−	0.998861i	$-0.515192\pi$
$883$	−2.83536	−0.0954175	−0.0477087	+	0.998861i	$0.515192\pi$
$884$	0	0
$885$	−6.62126	−0.222571
$886$	0	0
$887$	8.23810	0.276608	0.138304	−	0.990390i	$-0.455835\pi$
$887$	8.23810	0.276608	0.138304	+	0.990390i	$0.455835\pi$
$888$	0	0
$889$	−52.9716	−1.77661
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−15.2679	−0.510920
$894$	0	0
$895$	−9.01945	−0.301487
$896$	0	0
$897$	−5.26202	−0.175694
$898$	0	0
$899$	92.6647	3.09054
$900$	0	0
$901$	−81.0441	−2.69997
$902$	0	0
$903$	−18.4316	−0.613366
$904$	0	0
$905$	−25.1414	−0.835729
$906$	0	0
$907$	8.67101	0.287916	0.143958	−	0.989584i	$-0.454017\pi$
$907$	8.67101	0.287916	0.143958	+	0.989584i	$0.454017\pi$
$908$	0	0
$909$	−10.8265	−0.359093
$910$	0	0
$911$	−19.9437	−0.660766	−0.330383	−	0.943847i	$-0.607178\pi$
$911$	−19.9437	−0.660766	−0.330383	+	0.943847i	$0.607178\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−50.6896	−1.67575
$916$	0	0
$917$	−22.4117	−0.740101
$918$	0	0
$919$	23.0222	0.759433	0.379717	−	0.925103i	$-0.376021\pi$
$919$	23.0222	0.759433	0.379717	+	0.925103i	$0.376021\pi$
$920$	0	0
$921$	52.9062	1.74332
$922$	0	0
$923$	−11.3395	−0.373244
$924$	0	0
$925$	−15.7332	−0.517304
$926$	0	0
$927$	−24.0749	−0.790722
$928$	0	0
$929$	−15.2695	−0.500975	−0.250488	−	0.968120i	$-0.580591\pi$
$929$	−15.2695	−0.500975	−0.250488	+	0.968120i	$0.580591\pi$
$930$	0	0
$931$	−18.0655	−0.592072
$932$	0	0
$933$	−31.7036	−1.03793
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−13.6801	−0.446909	−0.223454	−	0.974714i	$-0.571733\pi$
$937$	−13.6801	−0.446909	−0.223454	+	0.974714i	$0.571733\pi$
$938$	0	0
$939$	41.5346	1.35543
$940$	0	0
$941$	−1.49651	−0.0487849	−0.0243924	−	0.999702i	$-0.507765\pi$
$941$	−1.49651	−0.0487849	−0.0243924	+	0.999702i	$0.507765\pi$
$942$	0	0
$943$	−9.09025	−0.296019
$944$	0	0
$945$	79.0590	2.57179
$946$	0	0
$947$	24.0642	0.781981	0.390991	−	0.920395i	$-0.372132\pi$
$947$	24.0642	0.781981	0.390991	+	0.920395i	$0.372132\pi$
$948$	0	0
$949$	3.99516	0.129688
$950$	0	0
$951$	−38.9689	−1.26365
$952$	0	0
$953$	−2.98295	−0.0966271	−0.0483135	−	0.998832i	$-0.515385\pi$
$953$	−2.98295	−0.0966271	−0.0483135	+	0.998832i	$0.515385\pi$
$954$	0	0
$955$	36.9977	1.19722
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−34.8623	−1.12576
$960$	0	0
$961$	51.6430	1.66590
$962$	0	0
$963$	14.7369	0.474890
$964$	0	0
$965$	−22.4606	−0.723033
$966$	0	0
$967$	9.37940	0.301621	0.150811	−	0.988563i	$-0.451812\pi$
$967$	9.37940	0.301621	0.150811	+	0.988563i	$0.451812\pi$
$968$	0	0
$969$	45.7656	1.47020
$970$	0	0
$971$	−7.77376	−0.249472	−0.124736	−	0.992190i	$-0.539808\pi$
$971$	−7.77376	−0.249472	−0.124736	+	0.992190i	$0.539808\pi$
$972$	0	0
$973$	−70.4962	−2.26000
$974$	0	0
$975$	−4.04995	−0.129702
$976$	0	0
$977$	−17.7902	−0.569160	−0.284580	−	0.958652i	$-0.591854\pi$
$977$	−17.7902	−0.569160	−0.284580	+	0.958652i	$0.591854\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−7.69401	−0.245651
$982$	0	0
$983$	59.4265	1.89541	0.947706	−	0.319144i	$-0.103395\pi$
$983$	59.4265	1.89541	0.947706	+	0.319144i	$0.103395\pi$
$984$	0	0
$985$	−24.2217	−0.771768
$986$	0	0
$987$	−96.8011	−3.08121
$988$	0	0
$989$	−2.57228	−0.0817937
$990$	0	0
$991$	37.0689	1.17753	0.588767	−	0.808303i	$-0.299614\pi$
$991$	37.0689	1.17753	0.588767	+	0.808303i	$0.299614\pi$
$992$	0	0
$993$	17.8248	0.565654
$994$	0	0
$995$	35.7228	1.13249
$996$	0	0
$997$	37.8116	1.19751	0.598753	−	0.800934i	$-0.295663\pi$
$997$	37.8116	1.19751	0.598753	+	0.800934i	$0.295663\pi$
$998$	0	0
$999$	−121.349	−3.83932

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6292.2.a.v.1.1	yes	10
11.10	odd	2		6292.2.a.u.1.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6292.2.a.u.1.1	✓	10	11.10	odd	2
6292.2.a.v.1.1	yes	10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6292 = 2^{2} \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6292.a (trivial)

\(f(q)\)	\(=\)	\(q-3.05668 q^{3} -1.91704 q^{5} +4.03552 q^{7} +6.34327 q^{9} +O(q^{10})\)	\(q-3.05668 q^{3} -1.91704 q^{5} +4.03552 q^{7} +6.34327 q^{9} -1.00000 q^{13} +5.85977 q^{15} +7.69559 q^{17} -1.94557 q^{19} -12.3353 q^{21} -1.72149 q^{23} -1.32495 q^{25} -10.2193 q^{27} +10.1932 q^{29} +9.09082 q^{31} -7.73626 q^{35} +11.8745 q^{37} +3.05668 q^{39} +5.28047 q^{41} +1.49422 q^{43} -12.1603 q^{45} +7.84750 q^{47} +9.28542 q^{49} -23.5229 q^{51} -10.5312 q^{53} +5.94699 q^{57} -1.12995 q^{59} -8.65043 q^{61} +25.5984 q^{63} +1.91704 q^{65} +1.55023 q^{67} +5.26202 q^{69} +11.3395 q^{71} -3.99516 q^{73} +4.04995 q^{75} -9.44413 q^{79} +12.2072 q^{81} -4.29408 q^{83} -14.7528 q^{85} -31.1574 q^{87} -4.54207 q^{89} -4.03552 q^{91} -27.7877 q^{93} +3.72975 q^{95} +13.1118 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9	\( 10 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9} - 10 q^{13} - 4 q^{15} + 20 q^{17} + 14 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 10 q^{27} + 16 q^{29} - 2 q^{31} - 20 q^{35} + 2 q^{39} + 22 q^{41} - 12 q^{43} - 8 q^{45} + 6 q^{47} - 2 q^{49} - 8 q^{51} + 6 q^{53} + 12 q^{57} - 4 q^{59} + 24 q^{61} + 68 q^{63} + 6 q^{65} - 6 q^{67} - 2 q^{69} - 8 q^{71} - 6 q^{73} - 18 q^{75} - 24 q^{79} + 10 q^{81} + 22 q^{83} + 34 q^{85} - 4 q^{87} - 42 q^{89} - 4 q^{91} - 38 q^{93} + 14 q^{97}+O(q^{100}) \) 10 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9 - 10 * q^13 - 4 * q^15 + 20 * q^17 + 14 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 10 * q^27 + 16 * q^29 - 2 * q^31 - 20 * q^35 + 2 * q^39 + 22 * q^41 - 12 * q^43 - 8 * q^45 + 6 * q^47 - 2 * q^49 - 8 * q^51 + 6 * q^53 + 12 * q^57 - 4 * q^59 + 24 * q^61 + 68 * q^63 + 6 * q^65 - 6 * q^67 - 2 * q^69 - 8 * q^71 - 6 * q^73 - 18 * q^75 - 24 * q^79 + 10 * q^81 + 22 * q^83 + 34 * q^85 - 4 * q^87 - 42 * q^89 - 4 * q^91 - 38 * q^93 + 14 * q^97

Introduction

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