LMFDB - Embedded newform 6292.2.a.r.1.2

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:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.477260$ 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)

$f(q)$	$=$	$q-0.618034 q^{3} +0.954520 q^{5} +1.02811 q^{7} -2.61803 q^{9} +O(q^{10})$	$q-0.618034 q^{3} +0.954520 q^{5} +1.02811 q^{7} -2.61803 q^{9} -1.00000 q^{13} -0.589926 q^{15} -6.97110 q^{17} -1.66351 q^{19} -0.635406 q^{21} -1.56182 q^{23} -4.08889 q^{25} +3.47214 q^{27} +4.30838 q^{29} -4.83673 q^{31} +0.981350 q^{35} -4.47214 q^{37} +0.618034 q^{39} -2.29892 q^{41} +10.2981 q^{43} -2.49897 q^{45} +8.97110 q^{47} -5.94299 q^{49} +4.30838 q^{51} +8.27002 q^{53} +1.02811 q^{57} +10.8881 q^{59} +12.5503 q^{61} -2.69162 q^{63} -0.954520 q^{65} +12.1162 q^{67} +0.965256 q^{69} -1.54445 q^{71} +6.95324 q^{73} +2.52707 q^{75} +6.36459 q^{79} +5.70820 q^{81} -14.9703 q^{83} -6.65406 q^{85} -2.66272 q^{87} -5.79125 q^{89} -1.02811 q^{91} +2.98926 q^{93} -1.58786 q^{95} +5.39727 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 2 q^{5} + 6 q^{7} - 6 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 2 * q^5 + 6 * q^7 - 6 * q^9	$ 4 q + 2 q^{3} - 2 q^{5} + 6 q^{7} - 6 q^{9} - 4 q^{13} + 4 q^{15} + 8 q^{17} - 8 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} - 4 q^{27} + 14 q^{29} - 2 q^{31} - 2 q^{35} - 2 q^{39} - 10 q^{41} + 8 q^{43} + 8 q^{45} + 14 q^{49} + 14 q^{51} - 2 q^{53} + 6 q^{57} + 4 q^{59} + 10 q^{61} - 14 q^{63} + 2 q^{65} - 8 q^{67} - 4 q^{69} + 6 q^{71} + 20 q^{73} - 6 q^{75} + 26 q^{79} - 4 q^{81} + 10 q^{83} - 32 q^{85} + 22 q^{87} - 6 q^{91} + 14 q^{93} + 36 q^{95} - 6 q^{97}+O(q^{100}) $ 4 * q + 2 * q^3 - 2 * q^5 + 6 * q^7 - 6 * q^9 - 4 * q^13 + 4 * q^15 + 8 * q^17 - 8 * q^19 - 2 * q^21 + 2 * q^23 + 8 * q^25 - 4 * q^27 + 14 * q^29 - 2 * q^31 - 2 * q^35 - 2 * q^39 - 10 * q^41 + 8 * q^43 + 8 * q^45 + 14 * q^49 + 14 * q^51 - 2 * q^53 + 6 * q^57 + 4 * q^59 + 10 * q^61 - 14 * q^63 + 2 * q^65 - 8 * q^67 - 4 * q^69 + 6 * q^71 + 20 * q^73 - 6 * q^75 + 26 * q^79 - 4 * q^81 + 10 * q^83 - 32 * q^85 + 22 * q^87 - 6 * q^91 + 14 * q^93 + 36 * q^95 - 6 * 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
$3$	−0.618034	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$3$	−0.618034	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$4$	0	0
$5$	0.954520	0.426874	0.213437	−	0.976957i	$-0.431534\pi$
$5$	0.954520	0.426874	0.213437	+	0.976957i	$0.431534\pi$
$6$	0	0
$7$	1.02811	0.388588	0.194294	−	0.980943i	$-0.437758\pi$
$7$	1.02811	0.388588	0.194294	+	0.980943i	$0.437758\pi$
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.589926	−0.152318
$16$	0	0
$17$	−6.97110	−1.69074	−0.845370	−	0.534181i	$-0.820620\pi$
$17$	−6.97110	−1.69074	−0.845370	+	0.534181i	$0.820620\pi$
$18$	0	0
$19$	−1.66351	−0.381636	−0.190818	−	0.981625i	$-0.561114\pi$
$19$	−1.66351	−0.381636	−0.190818	+	0.981625i	$0.561114\pi$
$20$	0	0
$21$	−0.635406	−0.138657
$22$	0	0
$23$	−1.56182	−0.325661	−0.162831	−	0.986654i	$-0.552062\pi$
$23$	−1.56182	−0.325661	−0.162831	+	0.986654i	$0.552062\pi$
$24$	0	0
$25$	−4.08889	−0.817778
$26$	0	0
$27$	3.47214	0.668213
$28$	0	0
$29$	4.30838	0.800046	0.400023	−	0.916505i	$-0.369002\pi$
$29$	4.30838	0.800046	0.400023	+	0.916505i	$0.369002\pi$
$30$	0	0
$31$	−4.83673	−0.868702	−0.434351	−	0.900744i	$-0.643022\pi$
$31$	−4.83673	−0.868702	−0.434351	+	0.900744i	$0.643022\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.981350	0.165878
$36$	0	0
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	0	0
$39$	0.618034	0.0989646
$40$	0	0
$41$	−2.29892	−0.359031	−0.179515	−	0.983755i	$-0.557453\pi$
$41$	−2.29892	−0.359031	−0.179515	+	0.983755i	$0.557453\pi$
$42$	0	0
$43$	10.2981	1.57045	0.785225	−	0.619211i	$-0.212547\pi$
$43$	10.2981	1.57045	0.785225	+	0.619211i	$0.212547\pi$
$44$	0	0
$45$	−2.49897	−0.372524
$46$	0	0
$47$	8.97110	1.30857	0.654285	−	0.756248i	$-0.272970\pi$
$47$	8.97110	1.30857	0.654285	+	0.756248i	$0.272970\pi$
$48$	0	0
$49$	−5.94299	−0.848999
$50$	0	0
$51$	4.30838	0.603294
$52$	0	0
$53$	8.27002	1.13597	0.567987	−	0.823037i	$-0.307722\pi$
$53$	8.27002	1.13597	0.567987	+	0.823037i	$0.307722\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.02811	0.136176
$58$	0	0
$59$	10.8881	1.41750	0.708752	−	0.705458i	$-0.249259\pi$
$59$	10.8881	1.41750	0.708752	+	0.705458i	$0.249259\pi$
$60$	0	0
$61$	12.5503	1.60690	0.803450	−	0.595372i	$-0.202995\pi$
$61$	12.5503	1.60690	0.803450	+	0.595372i	$0.202995\pi$
$62$	0	0
$63$	−2.69162	−0.339113
$64$	0	0
$65$	−0.954520	−0.118394
$66$	0	0
$67$	12.1162	1.48023	0.740115	−	0.672480i	$-0.234771\pi$
$67$	12.1162	1.48023	0.740115	+	0.672480i	$0.234771\pi$
$68$	0	0
$69$	0.965256	0.116203
$70$	0	0
$71$	−1.54445	−0.183292	−0.0916460	−	0.995792i	$-0.529213\pi$
$71$	−1.54445	−0.183292	−0.0916460	+	0.995792i	$0.529213\pi$
$72$	0	0
$73$	6.95324	0.813815	0.406908	−	0.913469i	$-0.366607\pi$
$73$	6.95324	0.813815	0.406908	+	0.913469i	$0.366607\pi$
$74$	0	0
$75$	2.52707	0.291801
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.36459	0.716073	0.358036	−	0.933708i	$-0.383446\pi$
$79$	6.36459	0.716073	0.358036	+	0.933708i	$0.383446\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	−14.9703	−1.64321	−0.821603	−	0.570061i	$-0.806920\pi$
$83$	−14.9703	−1.64321	−0.821603	+	0.570061i	$0.806920\pi$
$84$	0	0
$85$	−6.65406	−0.721734
$86$	0	0
$87$	−2.66272	−0.285474
$88$	0	0
$89$	−5.79125	−0.613871	−0.306936	−	0.951730i	$-0.599304\pi$
$89$	−5.79125	−0.613871	−0.306936	+	0.951730i	$0.599304\pi$
$90$	0	0
$91$	−1.02811	−0.107775
$92$	0	0
$93$	2.98926	0.309972
$94$	0	0
$95$	−1.58786	−0.162911
$96$	0	0
$97$	5.39727	0.548010	0.274005	−	0.961728i	$-0.411652\pi$
$97$	5.39727	0.548010	0.274005	+	0.961728i	$0.411652\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.57919	0.555150	0.277575	−	0.960704i	$-0.410469\pi$
$101$	5.57919	0.555150	0.277575	+	0.960704i	$0.410469\pi$
$102$	0	0
$103$	−7.88678	−0.777107	−0.388554	−	0.921426i	$-0.627025\pi$
$103$	−7.88678	−0.777107	−0.388554	+	0.921426i	$0.627025\pi$
$104$	0	0
$105$	−0.606508	−0.0591891
$106$	0	0
$107$	−3.03757	−0.293653	−0.146826	−	0.989162i	$-0.546906\pi$
$107$	−3.03757	−0.293653	−0.146826	+	0.989162i	$0.546906\pi$
$108$	0	0
$109$	15.6057	1.49476	0.747378	−	0.664399i	$-0.231312\pi$
$109$	15.6057	1.49476	0.747378	+	0.664399i	$0.231312\pi$
$110$	0	0
$111$	2.76393	0.262341
$112$	0	0
$113$	4.61803	0.434428	0.217214	−	0.976124i	$-0.430303\pi$
$113$	4.61803	0.434428	0.217214	+	0.976124i	$0.430303\pi$
$114$	0	0
$115$	−1.49079	−0.139017
$116$	0	0
$117$	2.61803	0.242037
$118$	0	0
$119$	−7.16705	−0.657002
$120$	0	0
$121$	0	0
$122$	0	0
$123$	1.42081	0.128110
$124$	0	0
$125$	−8.67553	−0.775963
$126$	0	0
$127$	−12.9609	−1.15009	−0.575045	−	0.818122i	$-0.695015\pi$
$127$	−12.9609	−1.15009	−0.575045	+	0.818122i	$0.695015\pi$
$128$	0	0
$129$	−6.36459	−0.560371
$130$	0	0
$131$	16.6627	1.45583	0.727914	−	0.685668i	$-0.240490\pi$
$131$	16.6627	1.45583	0.727914	+	0.685668i	$0.240490\pi$
$132$	0	0
$133$	−1.71027	−0.148299
$134$	0	0
$135$	3.31422	0.285243
$136$	0	0
$137$	2.71894	0.232295	0.116147	−	0.993232i	$-0.462946\pi$
$137$	2.71894	0.232295	0.116147	+	0.993232i	$0.462946\pi$
$138$	0	0
$139$	12.0649	1.02333	0.511665	−	0.859185i	$-0.329029\pi$
$139$	12.0649	1.02333	0.511665	+	0.859185i	$0.329029\pi$
$140$	0	0
$141$	−5.54445	−0.466927
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.11243	0.341519
$146$	0	0
$147$	3.67297	0.302942
$148$	0	0
$149$	20.3536	1.66743	0.833714	−	0.552196i	$-0.186210\pi$
$149$	20.3536	1.66743	0.833714	+	0.552196i	$0.186210\pi$
$150$	0	0
$151$	−8.61676	−0.701222	−0.350611	−	0.936521i	$-0.614026\pi$
$151$	−8.61676	−0.701222	−0.350611	+	0.936521i	$0.614026\pi$
$152$	0	0
$153$	18.2506	1.47547
$154$	0	0
$155$	−4.61676	−0.370827
$156$	0	0
$157$	0.360493	0.0287705	0.0143852	−	0.999897i	$-0.495421\pi$
$157$	0.360493	0.0287705	0.0143852	+	0.999897i	$0.495421\pi$
$158$	0	0
$159$	−5.11115	−0.405341
$160$	0	0
$161$	−1.60572	−0.126548
$162$	0	0
$163$	7.69162	0.602454	0.301227	−	0.953552i	$-0.402604\pi$
$163$	7.69162	0.602454	0.301227	+	0.953552i	$0.402604\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.6073	1.05296	0.526482	−	0.850186i	$-0.323511\pi$
$167$	13.6073	1.05296	0.526482	+	0.850186i	$0.323511\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.35514	0.333046
$172$	0	0
$173$	−9.02732	−0.686334	−0.343167	−	0.939274i	$-0.611500\pi$
$173$	−9.02732	−0.686334	−0.343167	+	0.939274i	$0.611500\pi$
$174$	0	0
$175$	−4.20382	−0.317779
$176$	0	0
$177$	−6.72919	−0.505797
$178$	0	0
$179$	−19.8512	−1.48375	−0.741876	−	0.670537i	$-0.766064\pi$
$179$	−19.8512	−1.48375	−0.741876	+	0.670537i	$0.766064\pi$
$180$	0	0
$181$	1.06870	0.0794356	0.0397178	−	0.999211i	$-0.487354\pi$
$181$	1.06870	0.0794356	0.0397178	+	0.999211i	$0.487354\pi$
$182$	0	0
$183$	−7.75651	−0.573377
$184$	0	0
$185$	−4.26874	−0.313844
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3.56973	0.259660
$190$	0	0
$191$	−12.2700	−0.887828	−0.443914	−	0.896069i	$-0.646410\pi$
$191$	−12.2700	−0.887828	−0.443914	+	0.896069i	$0.646410\pi$
$192$	0	0
$193$	22.3163	1.60636	0.803180	−	0.595737i	$-0.203140\pi$
$193$	22.3163	1.60636	0.803180	+	0.595737i	$0.203140\pi$
$194$	0	0
$195$	0.589926	0.0422455
$196$	0	0
$197$	5.77595	0.411519	0.205760	−	0.978603i	$-0.434034\pi$
$197$	5.77595	0.411519	0.205760	+	0.978603i	$0.434034\pi$
$198$	0	0
$199$	−2.40137	−0.170229	−0.0851143	−	0.996371i	$-0.527126\pi$
$199$	−2.40137	−0.170229	−0.0851143	+	0.996371i	$0.527126\pi$
$200$	0	0
$201$	−7.48823	−0.528179
$202$	0	0
$203$	4.42948	0.310888
$204$	0	0
$205$	−2.19436	−0.153261
$206$	0	0
$207$	4.08889	0.284198
$208$	0	0
$209$	0	0
$210$	0	0
$211$	2.25216	0.155045	0.0775226	−	0.996991i	$-0.475299\pi$
$211$	2.25216	0.155045	0.0775226	+	0.996991i	$0.475299\pi$
$212$	0	0
$213$	0.954520	0.0654026
$214$	0	0
$215$	9.82977	0.670385
$216$	0	0
$217$	−4.97268	−0.337568
$218$	0	0
$219$	−4.29734	−0.290387
$220$	0	0
$221$	6.97110	0.468927
$222$	0	0
$223$	−19.8426	−1.32876	−0.664379	−	0.747396i	$-0.731304\pi$
$223$	−19.8426	−1.32876	−0.664379	+	0.747396i	$0.731304\pi$
$224$	0	0
$225$	10.7049	0.713657
$226$	0	0
$227$	23.4371	1.55557	0.777786	−	0.628529i	$-0.216343\pi$
$227$	23.4371	1.55557	0.777786	+	0.628529i	$0.216343\pi$
$228$	0	0
$229$	−18.4491	−1.21915	−0.609575	−	0.792729i	$-0.708660\pi$
$229$	−18.4491	−1.21915	−0.609575	+	0.792729i	$0.708660\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.65406	0.435922	0.217961	−	0.975958i	$-0.430059\pi$
$233$	6.65406	0.435922	0.217961	+	0.975958i	$0.430059\pi$
$234$	0	0
$235$	8.56310	0.558595
$236$	0	0
$237$	−3.93354	−0.255511
$238$	0	0
$239$	−27.5495	−1.78203	−0.891015	−	0.453975i	$-0.850006\pi$
$239$	−27.5495	−1.78203	−0.891015	+	0.453975i	$0.850006\pi$
$240$	0	0
$241$	−18.2973	−1.17864	−0.589318	−	0.807901i	$-0.700603\pi$
$241$	−18.2973	−1.17864	−0.589318	+	0.807901i	$0.700603\pi$
$242$	0	0
$243$	−13.9443	−0.894525
$244$	0	0
$245$	−5.67271	−0.362416
$246$	0	0
$247$	1.66351	0.105847
$248$	0	0
$249$	9.25216	0.586332
$250$	0	0
$251$	10.8194	0.682912	0.341456	−	0.939898i	$-0.389080\pi$
$251$	10.8194	0.682912	0.341456	+	0.939898i	$0.389080\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	4.11243	0.257531
$256$	0	0
$257$	−13.4014	−0.835955	−0.417977	−	0.908457i	$-0.637261\pi$
$257$	−13.4014	−0.835955	−0.417977	+	0.908457i	$0.637261\pi$
$258$	0	0
$259$	−4.59784	−0.285696
$260$	0	0
$261$	−11.2795	−0.698182
$262$	0	0
$263$	22.2419	1.37150	0.685748	−	0.727839i	$-0.259475\pi$
$263$	22.2419	1.37150	0.685748	+	0.727839i	$0.259475\pi$
$264$	0	0
$265$	7.89390	0.484918
$266$	0	0
$267$	3.57919	0.219043
$268$	0	0
$269$	−8.05543	−0.491148	−0.245574	−	0.969378i	$-0.578976\pi$
$269$	−8.05543	−0.491148	−0.245574	+	0.969378i	$0.578976\pi$
$270$	0	0
$271$	18.9327	1.15008	0.575041	−	0.818124i	$-0.304986\pi$
$271$	18.9327	1.15008	0.575041	+	0.818124i	$0.304986\pi$
$272$	0	0
$273$	0.635406	0.0384565
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.77516	−0.407080	−0.203540	−	0.979067i	$-0.565245\pi$
$277$	−6.77516	−0.407080	−0.203540	+	0.979067i	$0.565245\pi$
$278$	0	0
$279$	12.6627	0.758097
$280$	0	0
$281$	2.44892	0.146090	0.0730451	−	0.997329i	$-0.476728\pi$
$281$	2.44892	0.146090	0.0730451	+	0.997329i	$0.476728\pi$
$282$	0	0
$283$	−28.7276	−1.70768	−0.853840	−	0.520536i	$-0.825732\pi$
$283$	−28.7276	−1.70768	−0.853840	+	0.520536i	$0.825732\pi$
$284$	0	0
$285$	0.981350	0.0581301
$286$	0	0
$287$	−2.36354	−0.139515
$288$	0	0
$289$	31.5963	1.85860
$290$	0	0
$291$	−3.33570	−0.195542
$292$	0	0
$293$	32.0546	1.87265	0.936326	−	0.351132i	$-0.114203\pi$
$293$	32.0546	1.87265	0.936326	+	0.351132i	$0.114203\pi$
$294$	0	0
$295$	10.3929	0.605096
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.56182	0.0903222
$300$	0	0
$301$	10.5876	0.610259
$302$	0	0
$303$	−3.44813	−0.198090
$304$	0	0
$305$	11.9795	0.685944
$306$	0	0
$307$	15.2708	0.871551	0.435776	−	0.900055i	$-0.356474\pi$
$307$	15.2708	0.871551	0.435776	+	0.900055i	$0.356474\pi$
$308$	0	0
$309$	4.87430	0.277289
$310$	0	0
$311$	22.3247	1.26592	0.632958	−	0.774186i	$-0.281841\pi$
$311$	22.3247	1.26592	0.632958	+	0.774186i	$0.281841\pi$
$312$	0	0
$313$	−31.7942	−1.79712	−0.898558	−	0.438854i	$-0.855384\pi$
$313$	−31.7942	−1.79712	−0.898558	+	0.438854i	$0.855384\pi$
$314$	0	0
$315$	−2.56921	−0.144758
$316$	0	0
$317$	12.1944	0.684904	0.342452	−	0.939535i	$-0.388743\pi$
$317$	12.1944	0.684904	0.342452	+	0.939535i	$0.388743\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	1.87732	0.104782
$322$	0	0
$323$	11.5965	0.645248
$324$	0	0
$325$	4.08889	0.226811
$326$	0	0
$327$	−9.64486	−0.533362
$328$	0	0
$329$	9.22326	0.508495
$330$	0	0
$331$	2.59248	0.142496	0.0712479	−	0.997459i	$-0.477302\pi$
$331$	2.59248	0.142496	0.0712479	+	0.997459i	$0.477302\pi$
$332$	0	0
$333$	11.7082	0.641606
$334$	0	0
$335$	11.5652	0.631872
$336$	0	0
$337$	9.69162	0.527936	0.263968	−	0.964531i	$-0.414969\pi$
$337$	9.69162	0.527936	0.263968	+	0.964531i	$0.414969\pi$
$338$	0	0
$339$	−2.85410	−0.155014
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−13.3068	−0.718500
$344$	0	0
$345$	0.921356	0.0496042
$346$	0	0
$347$	−9.08511	−0.487715	−0.243857	−	0.969811i	$-0.578413\pi$
$347$	−9.08511	−0.487715	−0.243857	+	0.969811i	$0.578413\pi$
$348$	0	0
$349$	5.56815	0.298056	0.149028	−	0.988833i	$-0.452385\pi$
$349$	5.56815	0.298056	0.149028	+	0.988833i	$0.452385\pi$
$350$	0	0
$351$	−3.47214	−0.185329
$352$	0	0
$353$	−9.39494	−0.500042	−0.250021	−	0.968240i	$-0.580438\pi$
$353$	−9.39494	−0.500042	−0.250021	+	0.968240i	$0.580438\pi$
$354$	0	0
$355$	−1.47420	−0.0782426
$356$	0	0
$357$	4.42948	0.234433
$358$	0	0
$359$	−2.99081	−0.157849	−0.0789244	−	0.996881i	$-0.525149\pi$
$359$	−2.99081	−0.157849	−0.0789244	+	0.996881i	$0.525149\pi$
$360$	0	0
$361$	−16.2327	−0.854354
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.63701	0.347397
$366$	0	0
$367$	−28.7406	−1.50025	−0.750123	−	0.661298i	$-0.770006\pi$
$367$	−28.7406	−1.50025	−0.750123	+	0.661298i	$0.770006\pi$
$368$	0	0
$369$	6.01865	0.313318
$370$	0	0
$371$	8.50248	0.441427
$372$	0	0
$373$	−4.42948	−0.229350	−0.114675	−	0.993403i	$-0.536583\pi$
$373$	−4.42948	−0.229350	−0.114675	+	0.993403i	$0.536583\pi$
$374$	0	0
$375$	5.36177	0.276881
$376$	0	0
$377$	−4.30838	−0.221893
$378$	0	0
$379$	0.688800	0.0353813	0.0176906	−	0.999844i	$-0.494369\pi$
$379$	0.688800	0.0353813	0.0176906	+	0.999844i	$0.494369\pi$
$380$	0	0
$381$	8.01025	0.410377
$382$	0	0
$383$	−10.4759	−0.535294	−0.267647	−	0.963517i	$-0.586246\pi$
$383$	−10.4759	−0.535294	−0.267647	+	0.963517i	$0.586246\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−26.9609	−1.37050
$388$	0	0
$389$	12.4131	0.629367	0.314684	−	0.949197i	$-0.398102\pi$
$389$	12.4131	0.629367	0.314684	+	0.949197i	$0.398102\pi$
$390$	0	0
$391$	10.8876	0.550609
$392$	0	0
$393$	−10.2981	−0.519472
$394$	0	0
$395$	6.07513	0.305673
$396$	0	0
$397$	13.0835	0.656644	0.328322	−	0.944566i	$-0.393517\pi$
$397$	13.0835	0.656644	0.328322	+	0.944566i	$0.393517\pi$
$398$	0	0
$399$	1.05701	0.0529165
$400$	0	0
$401$	31.7090	1.58347	0.791735	−	0.610865i	$-0.209178\pi$
$401$	31.7090	1.58347	0.791735	+	0.610865i	$0.209178\pi$
$402$	0	0
$403$	4.83673	0.240935
$404$	0	0
$405$	5.44859	0.270743
$406$	0	0
$407$	0	0
$408$	0	0
$409$	23.4379	1.15893	0.579464	−	0.814998i	$-0.303262\pi$
$409$	23.4379	1.15893	0.579464	+	0.814998i	$0.303262\pi$
$410$	0	0
$411$	−1.68040	−0.0828879
$412$	0	0
$413$	11.1941	0.550826
$414$	0	0
$415$	−14.2895	−0.701442
$416$	0	0
$417$	−7.45651	−0.365147
$418$	0	0
$419$	8.11164	0.396280	0.198140	−	0.980174i	$-0.436510\pi$
$419$	8.11164	0.396280	0.198140	+	0.980174i	$0.436510\pi$
$420$	0	0
$421$	−20.8587	−1.01659	−0.508295	−	0.861183i	$-0.669724\pi$
$421$	−20.8587	−1.01659	−0.508295	+	0.861183i	$0.669724\pi$
$422$	0	0
$423$	−23.4866	−1.14196
$424$	0	0
$425$	28.5041	1.38265
$426$	0	0
$427$	12.9031	0.624423
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.7573	0.662666	0.331333	−	0.943514i	$-0.392502\pi$
$431$	13.7573	0.662666	0.331333	+	0.943514i	$0.392502\pi$
$432$	0	0
$433$	33.6921	1.61914	0.809569	−	0.587025i	$-0.199701\pi$
$433$	33.6921	1.61914	0.809569	+	0.587025i	$0.199701\pi$
$434$	0	0
$435$	−2.54162	−0.121862
$436$	0	0
$437$	2.59811	0.124284
$438$	0	0
$439$	−8.07355	−0.385330	−0.192665	−	0.981265i	$-0.561713\pi$
$439$	−8.07355	−0.385330	−0.192665	+	0.981265i	$0.561713\pi$
$440$	0	0
$441$	15.5590	0.740903
$442$	0	0
$443$	7.69730	0.365710	0.182855	−	0.983140i	$-0.441466\pi$
$443$	7.69730	0.365710	0.182855	+	0.983140i	$0.441466\pi$
$444$	0	0
$445$	−5.52786	−0.262046
$446$	0	0
$447$	−12.5792	−0.594975
$448$	0	0
$449$	22.2174	1.04850	0.524252	−	0.851563i	$-0.324345\pi$
$449$	22.2174	1.04850	0.524252	+	0.851563i	$0.324345\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	5.32545	0.250211
$454$	0	0
$455$	−0.981350	−0.0460064
$456$	0	0
$457$	25.0079	1.16982	0.584910	−	0.811098i	$-0.301130\pi$
$457$	25.0079	1.16982	0.584910	+	0.811098i	$0.301130\pi$
$458$	0	0
$459$	−24.2046	−1.12977
$460$	0	0
$461$	4.41162	0.205470	0.102735	−	0.994709i	$-0.467241\pi$
$461$	4.41162	0.205470	0.102735	+	0.994709i	$0.467241\pi$
$462$	0	0
$463$	−14.9123	−0.693035	−0.346517	−	0.938044i	$-0.612636\pi$
$463$	−14.9123	−0.693035	−0.346517	+	0.938044i	$0.612636\pi$
$464$	0	0
$465$	2.85331	0.132319
$466$	0	0
$467$	−0.235580	−0.0109013	−0.00545066	−	0.999985i	$-0.501735\pi$
$467$	−0.235580	−0.0109013	−0.00545066	+	0.999985i	$0.501735\pi$
$468$	0	0
$469$	12.4568	0.575200
$470$	0	0
$471$	−0.222797	−0.0102659
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6.80193	0.312094
$476$	0	0
$477$	−21.6512	−0.991340
$478$	0	0
$479$	−35.2881	−1.61236	−0.806178	−	0.591673i	$-0.798468\pi$
$479$	−35.2881	−1.61236	−0.806178	+	0.591673i	$0.798468\pi$
$480$	0	0
$481$	4.47214	0.203912
$482$	0	0
$483$	0.992388	0.0451552
$484$	0	0
$485$	5.15180	0.233931
$486$	0	0
$487$	0.751127	0.0340368	0.0170184	−	0.999855i	$-0.494583\pi$
$487$	0.751127	0.0340368	0.0170184	+	0.999855i	$0.494583\pi$
$488$	0	0
$489$	−4.75368	−0.214969
$490$	0	0
$491$	20.3068	0.916433	0.458216	−	0.888841i	$-0.348488\pi$
$491$	20.3068	0.916433	0.458216	+	0.888841i	$0.348488\pi$
$492$	0	0
$493$	−30.0341	−1.35267
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.58786	−0.0712251
$498$	0	0
$499$	−14.5503	−0.651360	−0.325680	−	0.945480i	$-0.605593\pi$
$499$	−14.5503	−0.651360	−0.325680	+	0.945480i	$0.605593\pi$
$500$	0	0
$501$	−8.40977	−0.375721
$502$	0	0
$503$	11.4754	0.511664	0.255832	−	0.966721i	$-0.417651\pi$
$503$	11.4754	0.511664	0.255832	+	0.966721i	$0.417651\pi$
$504$	0	0
$505$	5.32545	0.236979
$506$	0	0
$507$	−0.618034	−0.0274479
$508$	0	0
$509$	−8.97344	−0.397741	−0.198870	−	0.980026i	$-0.563727\pi$
$509$	−8.97344	−0.397741	−0.198870	+	0.980026i	$0.563727\pi$
$510$	0	0
$511$	7.14868	0.316239
$512$	0	0
$513$	−5.77595	−0.255014
$514$	0	0
$515$	−7.52809	−0.331727
$516$	0	0
$517$	0	0
$518$	0	0
$519$	5.57919	0.244899
$520$	0	0
$521$	−21.5024	−0.942039	−0.471020	−	0.882123i	$-0.656114\pi$
$521$	−21.5024	−0.942039	−0.471020	+	0.882123i	$0.656114\pi$
$522$	0	0
$523$	20.2214	0.884221	0.442110	−	0.896961i	$-0.354230\pi$
$523$	20.2214	0.884221	0.442110	+	0.896961i	$0.354230\pi$
$524$	0	0
$525$	2.59811	0.113391
$526$	0	0
$527$	33.7173	1.46875
$528$	0	0
$529$	−20.5607	−0.893945
$530$	0	0
$531$	−28.5053	−1.23702
$532$	0	0
$533$	2.29892	0.0995773
$534$	0	0
$535$	−2.89942	−0.125353
$536$	0	0
$537$	12.2687	0.529435
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12.9719	0.557705	0.278853	−	0.960334i	$-0.410046\pi$
$541$	12.9719	0.557705	0.278853	+	0.960334i	$0.410046\pi$
$542$	0	0
$543$	−0.660491	−0.0283444
$544$	0	0
$545$	14.8960	0.638073
$546$	0	0
$547$	−9.57052	−0.409206	−0.204603	−	0.978845i	$-0.565590\pi$
$547$	−9.57052	−0.409206	−0.204603	+	0.978845i	$0.565590\pi$
$548$	0	0
$549$	−32.8571	−1.40231
$550$	0	0
$551$	−7.16705	−0.305326
$552$	0	0
$553$	6.54349	0.278258
$554$	0	0
$555$	2.63823	0.111987
$556$	0	0
$557$	−1.06383	−0.0450759	−0.0225379	−	0.999746i	$-0.507175\pi$
$557$	−1.06383	−0.0450759	−0.0225379	+	0.999746i	$0.507175\pi$
$558$	0	0
$559$	−10.2981	−0.435564
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.65406	−0.280435	−0.140218	−	0.990121i	$-0.544780\pi$
$563$	−6.65406	−0.280435	−0.140218	+	0.990121i	$0.544780\pi$
$564$	0	0
$565$	4.40801	0.185446
$566$	0	0
$567$	5.86865	0.246460
$568$	0	0
$569$	24.4192	1.02371	0.511854	−	0.859073i	$-0.328959\pi$
$569$	24.4192	1.02371	0.511854	+	0.859073i	$0.328959\pi$
$570$	0	0
$571$	4.77674	0.199900	0.0999501	−	0.994992i	$-0.468132\pi$
$571$	4.77674	0.199900	0.0999501	+	0.994992i	$0.468132\pi$
$572$	0	0
$573$	7.58329	0.316796
$574$	0	0
$575$	6.38610	0.266319
$576$	0	0
$577$	41.4876	1.72715	0.863576	−	0.504218i	$-0.168219\pi$
$577$	41.4876	1.72715	0.863576	+	0.504218i	$0.168219\pi$
$578$	0	0
$579$	−13.7922	−0.573184
$580$	0	0
$581$	−15.3911	−0.638530
$582$	0	0
$583$	0	0
$584$	0	0
$585$	2.49897	0.103320
$586$	0	0
$587$	−22.7838	−0.940389	−0.470195	−	0.882563i	$-0.655816\pi$
$587$	−22.7838	−0.940389	−0.470195	+	0.882563i	$0.655816\pi$
$588$	0	0
$589$	8.04597	0.331528
$590$	0	0
$591$	−3.56973	−0.146839
$592$	0	0
$593$	−2.20540	−0.0905650	−0.0452825	−	0.998974i	$-0.514419\pi$
$593$	−2.20540	−0.0905650	−0.0452825	+	0.998974i	$0.514419\pi$
$594$	0	0
$595$	−6.84109	−0.280457
$596$	0	0
$597$	1.48413	0.0607413
$598$	0	0
$599$	−8.60732	−0.351686	−0.175843	−	0.984418i	$-0.556265\pi$
$599$	−8.60732	−0.351686	−0.175843	+	0.984418i	$0.556265\pi$
$600$	0	0
$601$	30.3155	1.23659	0.618297	−	0.785945i	$-0.287823\pi$
$601$	30.3155	1.23659	0.618297	+	0.785945i	$0.287823\pi$
$602$	0	0
$603$	−31.7206	−1.29176
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.62703	−0.147216	−0.0736082	−	0.997287i	$-0.523451\pi$
$607$	−3.62703	−0.147216	−0.0736082	+	0.997287i	$0.523451\pi$
$608$	0	0
$609$	−2.73757	−0.110932
$610$	0	0
$611$	−8.97110	−0.362932
$612$	0	0
$613$	23.0452	0.930786	0.465393	−	0.885104i	$-0.345913\pi$
$613$	23.0452	0.930786	0.465393	+	0.885104i	$0.345913\pi$
$614$	0	0
$615$	1.35619	0.0546869
$616$	0	0
$617$	34.7344	1.39835	0.699177	−	0.714948i	$-0.253550\pi$
$617$	34.7344	1.39835	0.699177	+	0.714948i	$0.253550\pi$
$618$	0	0
$619$	11.7946	0.474063	0.237032	−	0.971502i	$-0.423825\pi$
$619$	11.7946	0.474063	0.237032	+	0.971502i	$0.423825\pi$
$620$	0	0
$621$	−5.42284	−0.217611
$622$	0	0
$623$	−5.95403	−0.238543
$624$	0	0
$625$	12.1635	0.486540
$626$	0	0
$627$	0	0
$628$	0	0
$629$	31.1757	1.24306
$630$	0	0
$631$	35.4946	1.41302	0.706508	−	0.707705i	$-0.250269\pi$
$631$	35.4946	1.41302	0.706508	+	0.707705i	$0.250269\pi$
$632$	0	0
$633$	−1.39191	−0.0553235
$634$	0	0
$635$	−12.3714	−0.490944
$636$	0	0
$637$	5.94299	0.235470
$638$	0	0
$639$	4.04341	0.159955
$640$	0	0
$641$	−13.3620	−0.527766	−0.263883	−	0.964555i	$-0.585003\pi$
$641$	−13.3620	−0.527766	−0.263883	+	0.964555i	$0.585003\pi$
$642$	0	0
$643$	39.5497	1.55969	0.779843	−	0.625975i	$-0.215299\pi$
$643$	39.5497	1.55969	0.779843	+	0.625975i	$0.215299\pi$
$644$	0	0
$645$	−6.07513	−0.239208
$646$	0	0
$647$	−18.0202	−0.708447	−0.354223	−	0.935161i	$-0.615255\pi$
$647$	−18.0202	−0.708447	−0.354223	+	0.935161i	$0.615255\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	3.07329	0.120452
$652$	0	0
$653$	6.25141	0.244636	0.122318	−	0.992491i	$-0.460967\pi$
$653$	6.25141	0.244636	0.122318	+	0.992491i	$0.460967\pi$
$654$	0	0
$655$	15.9049	0.621456
$656$	0	0
$657$	−18.2038	−0.710199
$658$	0	0
$659$	−1.93512	−0.0753814	−0.0376907	−	0.999289i	$-0.512000\pi$
$659$	−1.93512	−0.0753814	−0.0376907	+	0.999289i	$0.512000\pi$
$660$	0	0
$661$	−49.0230	−1.90678	−0.953388	−	0.301748i	$-0.902430\pi$
$661$	−49.0230	−1.90678	−0.953388	+	0.301748i	$0.902430\pi$
$662$	0	0
$663$	−4.30838	−0.167324
$664$	0	0
$665$	−1.63249	−0.0633052
$666$	0	0
$667$	−6.72890	−0.260544
$668$	0	0
$669$	12.2634	0.474130
$670$	0	0
$671$	0	0
$672$	0	0
$673$	21.5198	0.829528	0.414764	−	0.909929i	$-0.363864\pi$
$673$	21.5198	0.829528	0.414764	+	0.909929i	$0.363864\pi$
$674$	0	0
$675$	−14.1972	−0.546450
$676$	0	0
$677$	−3.23351	−0.124274	−0.0621370	−	0.998068i	$-0.519792\pi$
$677$	−3.23351	−0.124274	−0.0621370	+	0.998068i	$0.519792\pi$
$678$	0	0
$679$	5.54898	0.212950
$680$	0	0
$681$	−14.4849	−0.555063
$682$	0	0
$683$	32.4736	1.24257	0.621283	−	0.783586i	$-0.286612\pi$
$683$	32.4736	1.24257	0.621283	+	0.783586i	$0.286612\pi$
$684$	0	0
$685$	2.59528	0.0991607
$686$	0	0
$687$	11.4022	0.435020
$688$	0	0
$689$	−8.27002	−0.315063
$690$	0	0
$691$	10.1461	0.385974	0.192987	−	0.981201i	$-0.438182\pi$
$691$	10.1461	0.385974	0.192987	+	0.981201i	$0.438182\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.5162	0.436833
$696$	0	0
$697$	16.0260	0.607028
$698$	0	0
$699$	−4.11243	−0.155546
$700$	0	0
$701$	33.9898	1.28378	0.641888	−	0.766799i	$-0.278152\pi$
$701$	33.9898	1.28378	0.641888	+	0.766799i	$0.278152\pi$
$702$	0	0
$703$	7.43946	0.280585
$704$	0	0
$705$	−5.29228	−0.199319
$706$	0	0
$707$	5.73601	0.215725
$708$	0	0
$709$	−32.2414	−1.21085	−0.605426	−	0.795901i	$-0.706997\pi$
$709$	−32.2414	−1.21085	−0.605426	+	0.795901i	$0.706997\pi$
$710$	0	0
$711$	−16.6627	−0.624901
$712$	0	0
$713$	7.55409	0.282903
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.0265	0.635867
$718$	0	0
$719$	29.0848	1.08468	0.542339	−	0.840160i	$-0.317539\pi$
$719$	29.0848	1.08468	0.542339	+	0.840160i	$0.317539\pi$
$720$	0	0
$721$	−8.10846	−0.301975
$722$	0	0
$723$	11.3084	0.420563
$724$	0	0
$725$	−17.6165	−0.654260
$726$	0	0
$727$	−3.93288	−0.145863	−0.0729313	−	0.997337i	$-0.523235\pi$
$727$	−3.93288	−0.145863	−0.0729313	+	0.997337i	$0.523235\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	−71.7893	−2.65522
$732$	0	0
$733$	20.8390	0.769705	0.384852	−	0.922978i	$-0.374252\pi$
$733$	20.8390	0.769705	0.384852	+	0.922978i	$0.374252\pi$
$734$	0	0
$735$	3.50593	0.129318
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−44.5752	−1.63973	−0.819863	−	0.572560i	$-0.805950\pi$
$739$	−44.5752	−1.63973	−0.819863	+	0.572560i	$0.805950\pi$
$740$	0	0
$741$	−1.02811	−0.0377685
$742$	0	0
$743$	−43.5471	−1.59759	−0.798794	−	0.601604i	$-0.794528\pi$
$743$	−43.5471	−1.59759	−0.798794	+	0.601604i	$0.794528\pi$
$744$	0	0
$745$	19.4279	0.711782
$746$	0	0
$747$	39.1928	1.43399
$748$	0	0
$749$	−3.12295	−0.114110
$750$	0	0
$751$	9.92043	0.362002	0.181001	−	0.983483i	$-0.442066\pi$
$751$	9.92043	0.362002	0.181001	+	0.983483i	$0.442066\pi$
$752$	0	0
$753$	−6.68673	−0.243678
$754$	0	0
$755$	−8.22487	−0.299334
$756$	0	0
$757$	41.9041	1.52303	0.761515	−	0.648147i	$-0.224456\pi$
$757$	41.9041	1.52303	0.761515	+	0.648147i	$0.224456\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−29.6979	−1.07655	−0.538274	−	0.842770i	$-0.680924\pi$
$761$	−29.6979	−1.07655	−0.538274	+	0.842770i	$0.680924\pi$
$762$	0	0
$763$	16.0444	0.580845
$764$	0	0
$765$	17.4205	0.629841
$766$	0	0
$767$	−10.8881	−0.393145
$768$	0	0
$769$	12.9498	0.466982	0.233491	−	0.972359i	$-0.424985\pi$
$769$	12.9498	0.466982	0.233491	+	0.972359i	$0.424985\pi$
$770$	0	0
$771$	8.28250	0.298287
$772$	0	0
$773$	20.1263	0.723893	0.361947	−	0.932199i	$-0.382112\pi$
$773$	20.1263	0.723893	0.361947	+	0.932199i	$0.382112\pi$
$774$	0	0
$775$	19.7769	0.710406
$776$	0	0
$777$	2.84162	0.101943
$778$	0	0
$779$	3.82429	0.137019
$780$	0	0
$781$	0	0
$782$	0	0
$783$	14.9593	0.534601
$784$	0	0
$785$	0.344098	0.0122814
$786$	0	0
$787$	−18.2616	−0.650956	−0.325478	−	0.945550i	$-0.605525\pi$
$787$	−18.2616	−0.650956	−0.325478	+	0.945550i	$0.605525\pi$
$788$	0	0
$789$	−13.7463	−0.489380
$790$	0	0
$791$	4.74784	0.168814
$792$	0	0
$793$	−12.5503	−0.445674
$794$	0	0
$795$	−4.87870	−0.173030
$796$	0	0
$797$	−24.2268	−0.858156	−0.429078	−	0.903267i	$-0.641162\pi$
$797$	−24.2268	−0.858156	−0.429078	+	0.903267i	$0.641162\pi$
$798$	0	0
$799$	−62.5385	−2.21245
$800$	0	0
$801$	15.1617	0.535712
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−1.53269	−0.0540202
$806$	0	0
$807$	4.97853	0.175252
$808$	0	0
$809$	−26.8860	−0.945262	−0.472631	−	0.881260i	$-0.656696\pi$
$809$	−26.8860	−0.945262	−0.472631	+	0.881260i	$0.656696\pi$
$810$	0	0
$811$	31.5477	1.10779	0.553894	−	0.832587i	$-0.313141\pi$
$811$	31.5477	1.10779	0.553894	+	0.832587i	$0.313141\pi$
$812$	0	0
$813$	−11.7011	−0.410375
$814$	0	0
$815$	7.34181	0.257172
$816$	0	0
$817$	−17.1311	−0.599341
$818$	0	0
$819$	2.69162	0.0940529
$820$	0	0
$821$	−20.5385	−0.716797	−0.358399	−	0.933569i	$-0.616677\pi$
$821$	−20.5385	−0.716797	−0.358399	+	0.933569i	$0.616677\pi$
$822$	0	0
$823$	16.7821	0.584986	0.292493	−	0.956268i	$-0.405515\pi$
$823$	16.7821	0.584986	0.292493	+	0.956268i	$0.405515\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−0.878109	−0.0305348	−0.0152674	−	0.999883i	$-0.504860\pi$
$827$	−0.878109	−0.0305348	−0.0152674	+	0.999883i	$0.504860\pi$
$828$	0	0
$829$	−35.3354	−1.22725	−0.613624	−	0.789598i	$-0.710289\pi$
$829$	−35.3354	−1.22725	−0.613624	+	0.789598i	$0.710289\pi$
$830$	0	0
$831$	4.18728	0.145255
$832$	0	0
$833$	41.4292	1.43544
$834$	0	0
$835$	12.9884	0.449483
$836$	0	0
$837$	−16.7938	−0.580478
$838$	0	0
$839$	19.7634	0.682310	0.341155	−	0.940007i	$-0.389182\pi$
$839$	19.7634	0.682310	0.341155	+	0.940007i	$0.389182\pi$
$840$	0	0
$841$	−10.4379	−0.359927
$842$	0	0
$843$	−1.51352	−0.0521282
$844$	0	0
$845$	0.954520	0.0328365
$846$	0	0
$847$	0	0
$848$	0	0
$849$	17.7546	0.609338
$850$	0	0
$851$	6.98466	0.239431
$852$	0	0
$853$	37.7371	1.29209	0.646046	−	0.763298i	$-0.276421\pi$
$853$	37.7371	1.29209	0.646046	+	0.763298i	$0.276421\pi$
$854$	0	0
$855$	4.15706	0.142169
$856$	0	0
$857$	−1.16025	−0.0396333	−0.0198166	−	0.999804i	$-0.506308\pi$
$857$	−1.16025	−0.0396333	−0.0198166	+	0.999804i	$0.506308\pi$
$858$	0	0
$859$	4.01922	0.137134	0.0685670	−	0.997647i	$-0.478157\pi$
$859$	4.01922	0.137134	0.0685670	+	0.997647i	$0.478157\pi$
$860$	0	0
$861$	1.46075	0.0497821
$862$	0	0
$863$	14.0273	0.477495	0.238747	−	0.971082i	$-0.423263\pi$
$863$	14.0273	0.477495	0.238747	+	0.971082i	$0.423263\pi$
$864$	0	0
$865$	−8.61676	−0.292978
$866$	0	0
$867$	−19.5276	−0.663191
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−12.1162	−0.410542
$872$	0	0
$873$	−14.1302	−0.478236
$874$	0	0
$875$	−8.91938	−0.301530
$876$	0	0
$877$	−38.2004	−1.28994	−0.644968	−	0.764210i	$-0.723129\pi$
$877$	−38.2004	−1.28994	−0.644968	+	0.764210i	$0.723129\pi$
$878$	0	0
$879$	−19.8109	−0.668204
$880$	0	0
$881$	−12.2726	−0.413474	−0.206737	−	0.978397i	$-0.566284\pi$
$881$	−12.2726	−0.413474	−0.206737	+	0.978397i	$0.566284\pi$
$882$	0	0
$883$	6.86786	0.231122	0.115561	−	0.993300i	$-0.463133\pi$
$883$	6.86786	0.231122	0.115561	+	0.993300i	$0.463133\pi$
$884$	0	0
$885$	−6.42314	−0.215912
$886$	0	0
$887$	−41.0820	−1.37940	−0.689699	−	0.724097i	$-0.742257\pi$
$887$	−41.0820	−1.37940	−0.689699	+	0.724097i	$0.742257\pi$
$888$	0	0
$889$	−13.3252	−0.446912
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−14.9236	−0.499398
$894$	0	0
$895$	−18.9484	−0.633375
$896$	0	0
$897$	−0.965256	−0.0322290
$898$	0	0
$899$	−20.8385	−0.695002
$900$	0	0
$901$	−57.6512	−1.92064
$902$	0	0
$903$	−6.54349	−0.217754
$904$	0	0
$905$	1.02009	0.0339090
$906$	0	0
$907$	17.3278	0.575361	0.287680	−	0.957726i	$-0.407116\pi$
$907$	17.3278	0.575361	0.287680	+	0.957726i	$0.407116\pi$
$908$	0	0
$909$	−14.6065	−0.484467
$910$	0	0
$911$	48.7252	1.61434	0.807170	−	0.590320i	$-0.200998\pi$
$911$	48.7252	1.61434	0.807170	+	0.590320i	$0.200998\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−7.40374	−0.244760
$916$	0	0
$917$	17.1311	0.565718
$918$	0	0
$919$	−37.8655	−1.24907	−0.624534	−	0.780998i	$-0.714711\pi$
$919$	−37.8655	−1.24907	−0.624534	+	0.780998i	$0.714711\pi$
$920$	0	0
$921$	−9.43788	−0.310989
$922$	0	0
$923$	1.54445	0.0508361
$924$	0	0
$925$	18.2861	0.601243
$926$	0	0
$927$	20.6479	0.678164
$928$	0	0
$929$	−13.5603	−0.444898	−0.222449	−	0.974944i	$-0.571405\pi$
$929$	−13.5603	−0.444898	−0.222449	+	0.974944i	$0.571405\pi$
$930$	0	0
$931$	9.88625	0.324009
$932$	0	0
$933$	−13.7974	−0.451707
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−9.80219	−0.320223	−0.160112	−	0.987099i	$-0.551185\pi$
$937$	−9.80219	−0.320223	−0.160112	+	0.987099i	$0.551185\pi$
$938$	0	0
$939$	19.6499	0.641251
$940$	0	0
$941$	17.6041	0.573878	0.286939	−	0.957949i	$-0.407362\pi$
$941$	17.6041	0.573878	0.286939	+	0.957949i	$0.407362\pi$
$942$	0	0
$943$	3.59049	0.116923
$944$	0	0
$945$	3.40738	0.110842
$946$	0	0
$947$	−47.6869	−1.54962	−0.774808	−	0.632197i	$-0.782153\pi$
$947$	−47.6869	−1.54962	−0.774808	+	0.632197i	$0.782153\pi$
$948$	0	0
$949$	−6.95324	−0.225712
$950$	0	0
$951$	−7.53653	−0.244389
$952$	0	0
$953$	46.1074	1.49357	0.746783	−	0.665068i	$-0.231598\pi$
$953$	46.1074	1.49357	0.746783	+	0.665068i	$0.231598\pi$
$954$	0	0
$955$	−11.7120	−0.378991
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.79536	0.0902670
$960$	0	0
$961$	−7.60604	−0.245356
$962$	0	0
$963$	7.95245	0.256264
$964$	0	0
$965$	21.3013	0.685714
$966$	0	0
$967$	−31.5106	−1.01331	−0.506657	−	0.862148i	$-0.669119\pi$
$967$	−31.5106	−1.01331	−0.506657	+	0.862148i	$0.669119\pi$
$968$	0	0
$969$	−7.16705	−0.230239
$970$	0	0
$971$	3.30992	0.106220	0.0531102	−	0.998589i	$-0.483087\pi$
$971$	3.30992	0.106220	0.0531102	+	0.998589i	$0.483087\pi$
$972$	0	0
$973$	12.4040	0.397654
$974$	0	0
$975$	−2.52707	−0.0809311
$976$	0	0
$977$	−53.1982	−1.70196	−0.850980	−	0.525198i	$-0.823991\pi$
$977$	−53.1982	−1.70196	−0.850980	+	0.525198i	$0.823991\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−40.8563	−1.30444
$982$	0	0
$983$	−39.8501	−1.27102	−0.635511	−	0.772092i	$-0.719210\pi$
$983$	−39.8501	−1.27102	−0.635511	+	0.772092i	$0.719210\pi$
$984$	0	0
$985$	5.51326	0.175667
$986$	0	0
$987$	−5.70029	−0.181442
$988$	0	0
$989$	−16.0838	−0.511435
$990$	0	0
$991$	16.4387	0.522192	0.261096	−	0.965313i	$-0.415916\pi$
$991$	16.4387	0.522192	0.261096	+	0.965313i	$0.415916\pi$
$992$	0	0
$993$	−1.60224	−0.0508456
$994$	0	0
$995$	−2.29216	−0.0726662
$996$	0	0
$997$	−32.6969	−1.03552	−0.517760	−	0.855526i	$-0.673234\pi$
$997$	−32.6969	−1.03552	−0.517760	+	0.855526i	$0.673234\pi$
$998$	0	0
$999$	−15.5279	−0.491280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6292.2.a.r.1.2	yes	4
11.10	odd	2		6292.2.a.q.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6292.2.a.q.1.2	✓	4	11.10	odd	2
6292.2.a.r.1.2	yes	4	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)

\(f(q)\)	\(=\)	\(q-0.618034 q^{3} +0.954520 q^{5} +1.02811 q^{7} -2.61803 q^{9} +O(q^{10})\)	\(q-0.618034 q^{3} +0.954520 q^{5} +1.02811 q^{7} -2.61803 q^{9} -1.00000 q^{13} -0.589926 q^{15} -6.97110 q^{17} -1.66351 q^{19} -0.635406 q^{21} -1.56182 q^{23} -4.08889 q^{25} +3.47214 q^{27} +4.30838 q^{29} -4.83673 q^{31} +0.981350 q^{35} -4.47214 q^{37} +0.618034 q^{39} -2.29892 q^{41} +10.2981 q^{43} -2.49897 q^{45} +8.97110 q^{47} -5.94299 q^{49} +4.30838 q^{51} +8.27002 q^{53} +1.02811 q^{57} +10.8881 q^{59} +12.5503 q^{61} -2.69162 q^{63} -0.954520 q^{65} +12.1162 q^{67} +0.965256 q^{69} -1.54445 q^{71} +6.95324 q^{73} +2.52707 q^{75} +6.36459 q^{79} +5.70820 q^{81} -14.9703 q^{83} -6.65406 q^{85} -2.66272 q^{87} -5.79125 q^{89} -1.02811 q^{91} +2.98926 q^{93} -1.58786 q^{95} +5.39727 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 2 q^{5} + 6 q^{7} - 6 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 2 * q^5 + 6 * q^7 - 6 * q^9	\( 4 q + 2 q^{3} - 2 q^{5} + 6 q^{7} - 6 q^{9} - 4 q^{13} + 4 q^{15} + 8 q^{17} - 8 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} - 4 q^{27} + 14 q^{29} - 2 q^{31} - 2 q^{35} - 2 q^{39} - 10 q^{41} + 8 q^{43} + 8 q^{45} + 14 q^{49} + 14 q^{51} - 2 q^{53} + 6 q^{57} + 4 q^{59} + 10 q^{61} - 14 q^{63} + 2 q^{65} - 8 q^{67} - 4 q^{69} + 6 q^{71} + 20 q^{73} - 6 q^{75} + 26 q^{79} - 4 q^{81} + 10 q^{83} - 32 q^{85} + 22 q^{87} - 6 q^{91} + 14 q^{93} + 36 q^{95} - 6 q^{97}+O(q^{100}) \) 4 * q + 2 * q^3 - 2 * q^5 + 6 * q^7 - 6 * q^9 - 4 * q^13 + 4 * q^15 + 8 * q^17 - 8 * q^19 - 2 * q^21 + 2 * q^23 + 8 * q^25 - 4 * q^27 + 14 * q^29 - 2 * q^31 - 2 * q^35 - 2 * q^39 - 10 * q^41 + 8 * q^43 + 8 * q^45 + 14 * q^49 + 14 * q^51 - 2 * q^53 + 6 * q^57 + 4 * q^59 + 10 * q^61 - 14 * q^63 + 2 * q^65 - 8 * q^67 - 4 * q^69 + 6 * q^71 + 20 * q^73 - 6 * q^75 + 26 * q^79 - 4 * q^81 + 10 * q^83 - 32 * q^85 + 22 * q^87 - 6 * q^91 + 14 * q^93 + 36 * q^95 - 6 * q^97

Introduction

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