LMFDB - Embedded newform 6292.2.a.r.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:	$4$
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			1.1
Root			$2.09529$ 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)

$f(q)$	$=$	$q-0.618034 q^{3} -4.19059 q^{5} +4.20796 q^{7} -2.61803 q^{9} +O(q^{10})$	$q-0.618034 q^{3} -4.19059 q^{5} +4.20796 q^{7} -2.61803 q^{9} -1.00000 q^{13} +2.58993 q^{15} +6.49897 q^{17} -6.80862 q^{19} -2.60066 q^{21} +4.79789 q^{23} +12.5610 q^{25} +3.47214 q^{27} -4.01658 q^{29} -2.87147 q^{31} -17.6338 q^{35} -4.47214 q^{37} +0.618034 q^{39} -9.40928 q^{41} +7.11828 q^{43} +10.9711 q^{45} -4.49897 q^{47} +10.7069 q^{49} -4.01658 q^{51} +1.91032 q^{53} +4.20796 q^{57} +4.52835 q^{59} -5.31422 q^{61} -11.0166 q^{63} +4.19059 q^{65} -11.6441 q^{67} -2.96526 q^{69} +6.78051 q^{71} -14.8418 q^{73} -7.76314 q^{75} +4.39934 q^{79} +5.70820 q^{81} +8.78997 q^{83} -27.2345 q^{85} +2.48238 q^{87} +1.31911 q^{89} -4.20796 q^{91} +1.77467 q^{93} +28.5321 q^{95} -19.5776 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−4.19059	−1.87409	−0.937044	−	0.349211i	$-0.886449\pi$
$5$	−4.19059	−1.87409	−0.937044	+	0.349211i	$0.886449\pi$
$6$	0	0
$7$	4.20796	1.59046	0.795230	−	0.606308i	$-0.207350\pi$
$7$	4.20796	1.59046	0.795230	+	0.606308i	$0.207350\pi$
$8$	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
$15$	2.58993	0.668716
$16$	0	0
$17$	6.49897	1.57623	0.788115	−	0.615528i	$-0.211057\pi$
$17$	6.49897	1.57623	0.788115	+	0.615528i	$0.211057\pi$
$18$	0	0
$19$	−6.80862	−1.56201	−0.781003	−	0.624528i	$-0.785291\pi$
$19$	−6.80862	−1.56201	−0.781003	+	0.624528i	$0.785291\pi$
$20$	0	0
$21$	−2.60066	−0.567511
$22$	0	0
$23$	4.79789	1.00043	0.500214	−	0.865902i	$-0.333255\pi$
$23$	4.79789	1.00043	0.500214	+	0.865902i	$0.333255\pi$
$24$	0	0
$25$	12.5610	2.51221
$26$	0	0
$27$	3.47214	0.668213
$28$	0	0
$29$	−4.01658	−0.745861	−0.372930	−	0.927859i	$-0.621647\pi$
$29$	−4.01658	−0.745861	−0.372930	+	0.927859i	$0.621647\pi$
$30$	0	0
$31$	−2.87147	−0.515732	−0.257866	−	0.966181i	$-0.583019\pi$
$31$	−2.87147	−0.515732	−0.257866	+	0.966181i	$0.583019\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−17.6338	−2.98066
$36$	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
$41$	−9.40928	−1.46948	−0.734742	−	0.678347i	$-0.762697\pi$
$41$	−9.40928	−1.46948	−0.734742	+	0.678347i	$0.762697\pi$
$42$	0	0
$43$	7.11828	1.08553	0.542764	−	0.839886i	$-0.317378\pi$
$43$	7.11828	1.08553	0.542764	+	0.839886i	$0.317378\pi$
$44$	0	0
$45$	10.9711	1.63548
$46$	0	0
$47$	−4.49897	−0.656242	−0.328121	−	0.944636i	$-0.606415\pi$
$47$	−4.49897	−0.656242	−0.328121	+	0.944636i	$0.606415\pi$
$48$	0	0
$49$	10.7069	1.52956
$50$	0	0
$51$	−4.01658	−0.562434
$52$	0	0
$53$	1.91032	0.262402	0.131201	−	0.991356i	$-0.458117\pi$
$53$	1.91032	0.262402	0.131201	+	0.991356i	$0.458117\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.20796	0.557358
$58$	0	0
$59$	4.52835	0.589541	0.294771	−	0.955568i	$-0.404757\pi$
$59$	4.52835	0.589541	0.294771	+	0.955568i	$0.404757\pi$
$60$	0	0
$61$	−5.31422	−0.680417	−0.340208	−	0.940350i	$-0.610498\pi$
$61$	−5.31422	−0.680417	−0.340208	+	0.940350i	$0.610498\pi$
$62$	0	0
$63$	−11.0166	−1.38796
$64$	0	0
$65$	4.19059	0.519778
$66$	0	0
$67$	−11.6441	−1.42255	−0.711275	−	0.702914i	$-0.751882\pi$
$67$	−11.6441	−1.42255	−0.711275	+	0.702914i	$0.751882\pi$
$68$	0	0
$69$	−2.96526	−0.356975
$70$	0	0
$71$	6.78051	0.804699	0.402349	−	0.915486i	$-0.368194\pi$
$71$	6.78051	0.804699	0.402349	+	0.915486i	$0.368194\pi$
$72$	0	0
$73$	−14.8418	−1.73710	−0.868550	−	0.495602i	$-0.834948\pi$
$73$	−14.8418	−1.73710	−0.868550	+	0.495602i	$0.834948\pi$
$74$	0	0
$75$	−7.76314	−0.896410
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.39934	0.494964	0.247482	−	0.968892i	$-0.420397\pi$
$79$	4.39934	0.494964	0.247482	+	0.968892i	$0.420397\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	8.78997	0.964825	0.482412	−	0.875944i	$-0.339761\pi$
$83$	8.78997	0.964825	0.482412	+	0.875944i	$0.339761\pi$
$84$	0	0
$85$	−27.2345	−2.95399
$86$	0	0
$87$	2.48238	0.266140
$88$	0	0
$89$	1.31911	0.139826	0.0699129	−	0.997553i	$-0.477728\pi$
$89$	1.31911	0.139826	0.0699129	+	0.997553i	$0.477728\pi$
$90$	0	0
$91$	−4.20796	−0.441114
$92$	0	0
$93$	1.77467	0.184025
$94$	0	0
$95$	28.5321	2.92733
$96$	0	0
$97$	−19.5776	−1.98781	−0.993903	−	0.110262i	$-0.964831\pi$
$97$	−19.5776	−1.98781	−0.993903	+	0.110262i	$0.964831\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.18474	0.117886	0.0589431	−	0.998261i	$-0.481227\pi$
$101$	1.18474	0.117886	0.0589431	+	0.998261i	$0.481227\pi$
$102$	0	0
$103$	15.1228	1.49010	0.745049	−	0.667010i	$-0.232426\pi$
$103$	15.1228	1.49010	0.745049	+	0.667010i	$0.232426\pi$
$104$	0	0
$105$	10.8983	1.06357
$106$	0	0
$107$	9.21791	0.891129	0.445564	−	0.895250i	$-0.353003\pi$
$107$	9.21791	0.891129	0.445564	+	0.895250i	$0.353003\pi$
$108$	0	0
$109$	−6.18931	−0.592828	−0.296414	−	0.955059i	$-0.595791\pi$
$109$	−6.18931	−0.592828	−0.296414	+	0.955059i	$0.595791\pi$
$110$	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
$115$	−20.1060	−1.87489
$116$	0	0
$117$	2.61803	0.242037
$118$	0	0
$119$	27.3474	2.50693
$120$	0	0
$121$	0	0
$122$	0	0
$123$	5.81526	0.524344
$124$	0	0
$125$	−31.6852	−2.83401
$126$	0	0
$127$	−4.63589	−0.411369	−0.205685	−	0.978618i	$-0.565942\pi$
$127$	−4.63589	−0.411369	−0.205685	+	0.978618i	$0.565942\pi$
$128$	0	0
$129$	−4.39934	−0.387340
$130$	0	0
$131$	11.5176	1.00630	0.503149	−	0.864199i	$-0.332175\pi$
$131$	11.5176	1.00630	0.503149	+	0.864199i	$0.332175\pi$
$132$	0	0
$133$	−28.6504	−2.48431
$134$	0	0
$135$	−14.5503	−1.25229
$136$	0	0
$137$	3.93354	0.336065	0.168032	−	0.985781i	$-0.446259\pi$
$137$	3.93354	0.336065	0.168032	+	0.985781i	$0.446259\pi$
$138$	0	0
$139$	−7.30095	−0.619259	−0.309629	−	0.950857i	$-0.600205\pi$
$139$	−7.30095	−0.619259	−0.309629	+	0.950857i	$0.600205\pi$
$140$	0	0
$141$	2.78051	0.234162
$142$	0	0
$143$	0	0
$144$	0	0
$145$	16.8318	1.39781
$146$	0	0
$147$	−6.61724	−0.545781
$148$	0	0
$149$	13.2432	1.08492	0.542462	−	0.840080i	$-0.317492\pi$
$149$	13.2432	1.08492	0.542462	+	0.840080i	$0.317492\pi$
$150$	0	0
$151$	8.03316	0.653730	0.326865	−	0.945071i	$-0.394008\pi$
$151$	8.03316	0.653730	0.326865	+	0.945071i	$0.394008\pi$
$152$	0	0
$153$	−17.0145	−1.37554
$154$	0	0
$155$	12.0332	0.966527
$156$	0	0
$157$	−2.06870	−0.165100	−0.0825500	−	0.996587i	$-0.526306\pi$
$157$	−2.06870	−0.165100	−0.0825500	+	0.996587i	$0.526306\pi$
$158$	0	0
$159$	−1.18064	−0.0936310
$160$	0	0
$161$	20.1893	1.59114
$162$	0	0
$163$	16.0166	1.25452	0.627258	−	0.778812i	$-0.284177\pi$
$163$	16.0166	1.25452	0.627258	+	0.778812i	$0.284177\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.3927	0.958976	0.479488	−	0.877548i	$-0.340822\pi$
$167$	12.3927	0.958976	0.479488	+	0.877548i	$0.340822\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	17.8252	1.36313
$172$	0	0
$173$	−1.91695	−0.145743	−0.0728716	−	0.997341i	$-0.523216\pi$
$173$	−1.91695	−0.145743	−0.0728716	+	0.997341i	$0.523216\pi$
$174$	0	0
$175$	52.8563	3.99556
$176$	0	0
$177$	−2.79868	−0.210361
$178$	0	0
$179$	17.3791	1.29898	0.649488	−	0.760372i	$-0.274983\pi$
$179$	17.3791	1.29898	0.649488	+	0.760372i	$0.274983\pi$
$180$	0	0
$181$	−1.36049	−0.101125	−0.0505623	−	0.998721i	$-0.516101\pi$
$181$	−1.36049	−0.101125	−0.0505623	+	0.998721i	$0.516101\pi$
$182$	0	0
$183$	3.28437	0.242788
$184$	0	0
$185$	18.7409	1.37786
$186$	0	0
$187$	0	0
$188$	0	0
$189$	14.6106	1.06277
$190$	0	0
$191$	−5.91032	−0.427656	−0.213828	−	0.976871i	$-0.568593\pi$
$191$	−5.91032	−0.427656	−0.213828	+	0.976871i	$0.568593\pi$
$192$	0	0
$193$	−22.0245	−1.58536	−0.792678	−	0.609641i	$-0.791314\pi$
$193$	−22.0245	−1.58536	−0.792678	+	0.609641i	$0.791314\pi$
$194$	0	0
$195$	−2.58993	−0.185468
$196$	0	0
$197$	23.6405	1.68431	0.842157	−	0.539233i	$-0.181286\pi$
$197$	23.6405	1.68431	0.842157	+	0.539233i	$0.181286\pi$
$198$	0	0
$199$	22.1096	1.56731	0.783653	−	0.621199i	$-0.213354\pi$
$199$	22.1096	1.56731	0.783653	+	0.621199i	$0.213354\pi$
$200$	0	0
$201$	7.19643	0.507597
$202$	0	0
$203$	−16.9016	−1.18626
$204$	0	0
$205$	39.4304	2.75394
$206$	0	0
$207$	−12.5610	−0.873052
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−12.4325	−0.855889	−0.427944	−	0.903805i	$-0.640762\pi$
$211$	−12.4325	−0.855889	−0.427944	+	0.903805i	$0.640762\pi$
$212$	0	0
$213$	−4.19059	−0.287134
$214$	0	0
$215$	−29.8298	−2.03437
$216$	0	0
$217$	−12.0830	−0.820251
$218$	0	0
$219$	9.17273	0.619836
$220$	0	0
$221$	−6.49897	−0.437168
$222$	0	0
$223$	−8.33776	−0.558338	−0.279169	−	0.960242i	$-0.590059\pi$
$223$	−8.33776	−0.558338	−0.279169	+	0.960242i	$0.590059\pi$
$224$	0	0
$225$	−32.8852	−2.19235
$226$	0	0
$227$	−17.4371	−1.15734	−0.578669	−	0.815562i	$-0.696428\pi$
$227$	−17.4371	−1.15734	−0.578669	+	0.815562i	$0.696428\pi$
$228$	0	0
$229$	4.56054	0.301369	0.150684	−	0.988582i	$-0.451852\pi$
$229$	4.56054	0.301369	0.150684	+	0.988582i	$0.451852\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	27.2345	1.78419	0.892095	−	0.451847i	$-0.149235\pi$
$233$	27.2345	1.78419	0.892095	+	0.451847i	$0.149235\pi$
$234$	0	0
$235$	18.8533	1.22985
$236$	0	0
$237$	−2.71894	−0.176614
$238$	0	0
$239$	0.605229	0.0391490	0.0195745	−	0.999808i	$-0.493769\pi$
$239$	0.605229	0.0391490	0.0195745	+	0.999808i	$0.493769\pi$
$240$	0	0
$241$	−4.82727	−0.310952	−0.155476	−	0.987840i	$-0.549691\pi$
$241$	−4.82727	−0.310952	−0.155476	+	0.987840i	$0.549691\pi$
$242$	0	0
$243$	−13.9443	−0.894525
$244$	0	0
$245$	−44.8683	−2.86653
$246$	0	0
$247$	6.80862	0.433222
$248$	0	0
$249$	−5.43250	−0.344271
$250$	0	0
$251$	6.88885	0.434820	0.217410	−	0.976080i	$-0.430239\pi$
$251$	6.88885	0.434820	0.217410	+	0.976080i	$0.430239\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	16.8318	1.05405
$256$	0	0
$257$	11.1096	0.692996	0.346498	−	0.938051i	$-0.387371\pi$
$257$	11.1096	0.692996	0.346498	+	0.938051i	$0.387371\pi$
$258$	0	0
$259$	−18.8186	−1.16933
$260$	0	0
$261$	10.5155	0.650896
$262$	0	0
$263$	12.7024	0.783261	0.391630	−	0.920123i	$-0.371911\pi$
$263$	12.7024	0.783261	0.391630	+	0.920123i	$0.371911\pi$
$264$	0	0
$265$	−8.00536	−0.491765
$266$	0	0
$267$	−0.815257	−0.0498929
$268$	0	0
$269$	−4.12491	−0.251500	−0.125750	−	0.992062i	$-0.540134\pi$
$269$	−4.12491	−0.251500	−0.125750	+	0.992062i	$0.540134\pi$
$270$	0	0
$271$	7.42793	0.451215	0.225607	−	0.974218i	$-0.427563\pi$
$271$	7.42793	0.451215	0.225607	+	0.974218i	$0.427563\pi$
$272$	0	0
$273$	2.60066	0.157399
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−14.3495	−0.862175	−0.431088	−	0.902310i	$-0.641870\pi$
$277$	−14.3495	−0.862175	−0.431088	+	0.902310i	$0.641870\pi$
$278$	0	0
$279$	7.51762	0.450068
$280$	0	0
$281$	10.0232	0.597935	0.298968	−	0.954263i	$-0.403358\pi$
$281$	10.0232	0.597935	0.298968	+	0.954263i	$0.403358\pi$
$282$	0	0
$283$	−4.21666	−0.250655	−0.125327	−	0.992115i	$-0.539998\pi$
$283$	−4.21666	−0.250655	−0.125327	+	0.992115i	$0.539998\pi$
$284$	0	0
$285$	−17.6338	−1.04454
$286$	0	0
$287$	−39.5939	−2.33715
$288$	0	0
$289$	25.2366	1.48450
$290$	0	0
$291$	12.0996	0.709293
$292$	0	0
$293$	17.8339	1.04187	0.520934	−	0.853597i	$-0.325584\pi$
$293$	17.8339	1.04187	0.520934	+	0.853597i	$0.325584\pi$
$294$	0	0
$295$	−18.9765	−1.10485
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.79789	−0.277469
$300$	0	0
$301$	29.9534	1.72649
$302$	0	0
$303$	−0.732211	−0.0420644
$304$	0	0
$305$	22.2697	1.27516
$306$	0	0
$307$	19.2013	1.09588	0.547939	−	0.836518i	$-0.315413\pi$
$307$	19.2013	1.09588	0.547939	+	0.836518i	$0.315413\pi$
$308$	0	0
$309$	−9.34643	−0.531700
$310$	0	0
$311$	1.74423	0.0989060	0.0494530	−	0.998776i	$-0.484252\pi$
$311$	1.74423	0.0989060	0.0494530	+	0.998776i	$0.484252\pi$
$312$	0	0
$313$	22.0860	1.24838	0.624188	−	0.781274i	$-0.285430\pi$
$313$	22.0860	1.24838	0.624188	+	0.781274i	$0.285430\pi$
$314$	0	0
$315$	46.1660	2.60116
$316$	0	0
$317$	−29.4304	−1.65298	−0.826489	−	0.562953i	$-0.809665\pi$
$317$	−29.4304	−1.65298	−0.826489	+	0.562953i	$0.809665\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−5.69698	−0.317974
$322$	0	0
$323$	−44.2490	−2.46208
$324$	0	0
$325$	−12.5610	−0.696760
$326$	0	0
$327$	3.82520	0.211534
$328$	0	0
$329$	−18.9315	−1.04373
$330$	0	0
$331$	32.7125	1.79804	0.899020	−	0.437908i	$-0.144280\pi$
$331$	32.7125	1.79804	0.899020	+	0.437908i	$0.144280\pi$
$332$	0	0
$333$	11.7082	0.641606
$334$	0	0
$335$	48.7955	2.66598
$336$	0	0
$337$	18.0166	0.981426	0.490713	−	0.871321i	$-0.336736\pi$
$337$	18.0166	0.981426	0.490713	+	0.871321i	$0.336736\pi$
$338$	0	0
$339$	−2.85410	−0.155014
$340$	0	0
$341$	0	0
$342$	0	0
$343$	15.5986	0.842245
$344$	0	0
$345$	12.4262	0.669002
$346$	0	0
$347$	−28.9149	−1.55223	−0.776116	−	0.630590i	$-0.782813\pi$
$347$	−28.9149	−1.55223	−0.776116	+	0.630590i	$0.782813\pi$
$348$	0	0
$349$	−3.97140	−0.212584	−0.106292	−	0.994335i	$-0.533898\pi$
$349$	−3.97140	−0.212584	−0.106292	+	0.994335i	$0.533898\pi$
$350$	0	0
$351$	−3.47214	−0.185329
$352$	0	0
$353$	−6.96574	−0.370749	−0.185375	−	0.982668i	$-0.559350\pi$
$353$	−6.96574	−0.370749	−0.185375	+	0.982668i	$0.559350\pi$
$354$	0	0
$355$	−28.4143	−1.50808
$356$	0	0
$357$	−16.9016	−0.894528
$358$	0	0
$359$	31.0597	1.63927	0.819634	−	0.572888i	$-0.194177\pi$
$359$	31.0597	1.63927	0.819634	+	0.572888i	$0.194177\pi$
$360$	0	0
$361$	27.3573	1.43986
$362$	0	0
$363$	0	0
$364$	0	0
$365$	62.1958	3.25548
$366$	0	0
$367$	−1.80044	−0.0939823	−0.0469912	−	0.998895i	$-0.514963\pi$
$367$	−1.80044	−0.0939823	−0.0469912	+	0.998895i	$0.514963\pi$
$368$	0	0
$369$	24.6338	1.28239
$370$	0	0
$371$	8.03854	0.417340
$372$	0	0
$373$	16.9016	0.875132	0.437566	−	0.899186i	$-0.355841\pi$
$373$	16.9016	0.875132	0.437566	+	0.899186i	$0.355841\pi$
$374$	0	0
$375$	19.5825	1.01124
$376$	0	0
$377$	4.01658	0.206864
$378$	0	0
$379$	25.1997	1.29442	0.647212	−	0.762310i	$-0.275935\pi$
$379$	25.1997	1.29442	0.647212	+	0.762310i	$0.275935\pi$
$380$	0	0
$381$	2.86514	0.146786
$382$	0	0
$383$	26.0038	1.32873	0.664365	−	0.747408i	$-0.268702\pi$
$383$	26.0038	1.32873	0.664365	+	0.747408i	$0.268702\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−18.6359	−0.947316
$388$	0	0
$389$	−31.1770	−1.58074	−0.790368	−	0.612632i	$-0.790111\pi$
$389$	−31.1770	−1.58074	−0.790368	+	0.612632i	$0.790111\pi$
$390$	0	0
$391$	31.1813	1.57691
$392$	0	0
$393$	−7.11828	−0.359070
$394$	0	0
$395$	−18.4358	−0.927606
$396$	0	0
$397$	12.3329	0.618969	0.309485	−	0.950904i	$-0.399843\pi$
$397$	12.3329	0.618969	0.309485	+	0.950904i	$0.399843\pi$
$398$	0	0
$399$	17.7069	0.886455
$400$	0	0
$401$	−11.4172	−0.570146	−0.285073	−	0.958506i	$-0.592018\pi$
$401$	−11.4172	−0.570146	−0.285073	+	0.958506i	$0.592018\pi$
$402$	0	0
$403$	2.87147	0.143038
$404$	0	0
$405$	−23.9207	−1.18863
$406$	0	0
$407$	0	0
$408$	0	0
$409$	25.8671	1.27904	0.639522	−	0.768773i	$-0.279132\pi$
$409$	25.8671	1.27904	0.639522	+	0.768773i	$0.279132\pi$
$410$	0	0
$411$	−2.43106	−0.119915
$412$	0	0
$413$	19.0551	0.937641
$414$	0	0
$415$	−36.8352	−1.80817
$416$	0	0
$417$	4.51224	0.220965
$418$	0	0
$419$	10.5408	0.514953	0.257477	−	0.966285i	$-0.417109\pi$
$419$	10.5408	0.514953	0.257477	+	0.966285i	$0.417109\pi$
$420$	0	0
$421$	5.33080	0.259808	0.129904	−	0.991527i	$-0.458533\pi$
$421$	5.33080	0.259808	0.129904	+	0.991527i	$0.458533\pi$
$422$	0	0
$423$	11.7784	0.572688
$424$	0	0
$425$	81.6337	3.95982
$426$	0	0
$427$	−22.3620	−1.08217
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.0066	0.626508	0.313254	−	0.949669i	$-0.398581\pi$
$431$	13.0066	0.626508	0.313254	+	0.949669i	$0.398581\pi$
$432$	0	0
$433$	−5.03960	−0.242188	−0.121094	−	0.992641i	$-0.538640\pi$
$433$	−5.03960	−0.242188	−0.121094	+	0.992641i	$0.538640\pi$
$434$	0	0
$435$	−10.4026	−0.498769
$436$	0	0
$437$	−32.6670	−1.56267
$438$	0	0
$439$	37.0178	1.76676	0.883382	−	0.468653i	$-0.155261\pi$
$439$	37.0178	1.76676	0.883382	+	0.468653i	$0.155261\pi$
$440$	0	0
$441$	−28.0311	−1.33481
$442$	0	0
$443$	37.0666	1.76109	0.880544	−	0.473964i	$-0.157177\pi$
$443$	37.0666	1.76109	0.880544	+	0.473964i	$0.157177\pi$
$444$	0	0
$445$	−5.52786	−0.262046
$446$	0	0
$447$	−8.18474	−0.387125
$448$	0	0
$449$	3.60224	0.170000	0.0850002	−	0.996381i	$-0.472911\pi$
$449$	3.60224	0.170000	0.0850002	+	0.996381i	$0.472911\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−4.96477	−0.233265
$454$	0	0
$455$	17.6338	0.826686
$456$	0	0
$457$	−11.0079	−0.514927	−0.257463	−	0.966288i	$-0.582887\pi$
$457$	−11.0079	−0.514927	−0.257463	+	0.966288i	$0.582887\pi$
$458$	0	0
$459$	22.5653	1.05326
$460$	0	0
$461$	−25.2444	−1.17575	−0.587875	−	0.808952i	$-0.700035\pi$
$461$	−25.2444	−1.17575	−0.587875	+	0.808952i	$0.700035\pi$
$462$	0	0
$463$	38.2173	1.77611	0.888054	−	0.459740i	$-0.152057\pi$
$463$	38.2173	1.77611	0.888054	+	0.459740i	$0.152057\pi$
$464$	0	0
$465$	−7.43690	−0.344878
$466$	0	0
$467$	6.12412	0.283391	0.141695	−	0.989910i	$-0.454745\pi$
$467$	6.12412	0.283391	0.141695	+	0.989910i	$0.454745\pi$
$468$	0	0
$469$	−48.9978	−2.26251
$470$	0	0
$471$	1.27853	0.0589113
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−85.5233	−3.92408
$476$	0	0
$477$	−5.00128	−0.228993
$478$	0	0
$479$	12.2324	0.558914	0.279457	−	0.960158i	$-0.409846\pi$
$479$	12.2324	0.558914	0.279457	+	0.960158i	$0.409846\pi$
$480$	0	0
$481$	4.47214	0.203912
$482$	0	0
$483$	−12.4777	−0.567754
$484$	0	0
$485$	82.0417	3.72532
$486$	0	0
$487$	−27.4036	−1.24178	−0.620888	−	0.783899i	$-0.713228\pi$
$487$	−27.4036	−1.24178	−0.620888	+	0.783899i	$0.713228\pi$
$488$	0	0
$489$	−9.89879	−0.447639
$490$	0	0
$491$	−8.59859	−0.388049	−0.194025	−	0.980997i	$-0.562154\pi$
$491$	−8.59859	−0.388049	−0.194025	+	0.980997i	$0.562154\pi$
$492$	0	0
$493$	−26.1036	−1.17565
$494$	0	0
$495$	0	0
$496$	0	0
$497$	28.5321	1.27984
$498$	0	0
$499$	3.31422	0.148365	0.0741825	−	0.997245i	$-0.476365\pi$
$499$	3.31422	0.148365	0.0741825	+	0.997245i	$0.476365\pi$
$500$	0	0
$501$	−7.65911	−0.342184
$502$	0	0
$503$	−31.3640	−1.39845	−0.699225	−	0.714902i	$-0.746471\pi$
$503$	−31.3640	−1.39845	−0.699225	+	0.714902i	$0.746471\pi$
$504$	0	0
$505$	−4.96477	−0.220929
$506$	0	0
$507$	−0.618034	−0.0274479
$508$	0	0
$509$	27.0423	1.19863	0.599315	−	0.800513i	$-0.295440\pi$
$509$	27.0423	1.19863	0.599315	+	0.800513i	$0.295440\pi$
$510$	0	0
$511$	−62.4536	−2.76279
$512$	0	0
$513$	−23.6405	−1.04375
$514$	0	0
$515$	−63.3736	−2.79258
$516$	0	0
$517$	0	0
$518$	0	0
$519$	1.18474	0.0520044
$520$	0	0
$521$	32.3778	1.41850	0.709249	−	0.704958i	$-0.249034\pi$
$521$	32.3778	1.41850	0.709249	+	0.704958i	$0.249034\pi$
$522$	0	0
$523$	20.9721	0.917045	0.458522	−	0.888683i	$-0.348379\pi$
$523$	20.9721	0.917045	0.458522	+	0.888683i	$0.348379\pi$
$524$	0	0
$525$	−32.6670	−1.42570
$526$	0	0
$527$	−18.6616	−0.812912
$528$	0	0
$529$	0.0197061	0.000856787 0
$530$	0	0
$531$	−11.8554	−0.514480
$532$	0	0
$533$	9.40928	0.407561
$534$	0	0
$535$	−38.6284	−1.67005
$536$	0	0
$537$	−10.7409	−0.463503
$538$	0	0
$539$	0	0
$540$	0	0
$541$	9.79204	0.420993	0.210496	−	0.977595i	$-0.432492\pi$
$541$	9.79204	0.420993	0.210496	+	0.977595i	$0.432492\pi$
$542$	0	0
$543$	0.840831	0.0360835
$544$	0	0
$545$	25.9368	1.11101
$546$	0	0
$547$	−30.9016	−1.32126	−0.660629	−	0.750713i	$-0.729710\pi$
$547$	−30.9016	−1.32126	−0.660629	+	0.750713i	$0.729710\pi$
$548$	0	0
$549$	13.9128	0.593785
$550$	0	0
$551$	27.3474	1.16504
$552$	0	0
$553$	18.5122	0.787220
$554$	0	0
$555$	−11.5825	−0.491650
$556$	0	0
$557$	−20.8936	−0.885290	−0.442645	−	0.896697i	$-0.645960\pi$
$557$	−20.8936	−0.885290	−0.442645	+	0.896697i	$0.645960\pi$
$558$	0	0
$559$	−7.11828	−0.301071
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.2345	−1.14780	−0.573898	−	0.818927i	$-0.694570\pi$
$563$	−27.2345	−1.14780	−0.573898	+	0.818927i	$0.694570\pi$
$564$	0	0
$565$	−19.3523	−0.814156
$566$	0	0
$567$	24.0199	1.00874
$568$	0	0
$569$	8.23325	0.345156	0.172578	−	0.984996i	$-0.444790\pi$
$569$	8.23325	0.345156	0.172578	+	0.984996i	$0.444790\pi$
$570$	0	0
$571$	32.9315	1.37814	0.689070	−	0.724695i	$-0.258019\pi$
$571$	32.9315	1.37814	0.689070	+	0.724695i	$0.258019\pi$
$572$	0	0
$573$	3.65278	0.152597
$574$	0	0
$575$	60.2664	2.51328
$576$	0	0
$577$	18.9419	0.788563	0.394282	−	0.918990i	$-0.370993\pi$
$577$	18.9419	0.788563	0.394282	+	0.918990i	$0.370993\pi$
$578$	0	0
$579$	13.6119	0.565690
$580$	0	0
$581$	36.9878	1.53451
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−10.9711	−0.453599
$586$	0	0
$587$	−4.63258	−0.191207	−0.0956036	−	0.995419i	$-0.530478\pi$
$587$	−4.63258	−0.191207	−0.0956036	+	0.995419i	$0.530478\pi$
$588$	0	0
$589$	19.5508	0.805576
$590$	0	0
$591$	−14.6106	−0.601000
$592$	0	0
$593$	34.2743	1.40748	0.703738	−	0.710459i	$-0.251513\pi$
$593$	34.2743	1.40748	0.703738	+	0.710459i	$0.251513\pi$
$594$	0	0
$595$	−114.602	−4.69821
$596$	0	0
$597$	−13.6645	−0.559249
$598$	0	0
$599$	−40.4058	−1.65094	−0.825469	−	0.564448i	$-0.809089\pi$
$599$	−40.4058	−1.65094	−0.825469	+	0.564448i	$0.809089\pi$
$600$	0	0
$601$	−24.3155	−0.991849	−0.495924	−	0.868366i	$-0.665171\pi$
$601$	−24.3155	−0.991849	−0.495924	+	0.868366i	$0.665171\pi$
$602$	0	0
$603$	30.4846	1.24143
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−14.8451	−0.602544	−0.301272	−	0.953538i	$-0.597411\pi$
$607$	−14.8451	−0.602544	−0.301272	+	0.953538i	$0.597411\pi$
$608$	0	0
$609$	10.4458	0.423284
$610$	0	0
$611$	4.49897	0.182009
$612$	0	0
$613$	24.2598	0.979843	0.489922	−	0.871767i	$-0.337025\pi$
$613$	24.2598	0.979843	0.489922	+	0.871767i	$0.337025\pi$
$614$	0	0
$615$	−24.3693	−0.982667
$616$	0	0
$617$	−19.6098	−0.789461	−0.394730	−	0.918797i	$-0.629162\pi$
$617$	−19.6098	−0.789461	−0.394730	+	0.918797i	$0.629162\pi$
$618$	0	0
$619$	−5.14209	−0.206678	−0.103339	−	0.994646i	$-0.532953\pi$
$619$	−5.14209	−0.206678	−0.103339	+	0.994646i	$0.532953\pi$
$620$	0	0
$621$	16.6589	0.668499
$622$	0	0
$623$	5.55078	0.222387
$624$	0	0
$625$	69.9743	2.79897
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−29.0643	−1.15887
$630$	0	0
$631$	17.6300	0.701841	0.350921	−	0.936405i	$-0.385869\pi$
$631$	17.6300	0.701841	0.350921	+	0.936405i	$0.385869\pi$
$632$	0	0
$633$	7.68371	0.305400
$634$	0	0
$635$	19.4271	0.770942
$636$	0	0
$637$	−10.7069	−0.424224
$638$	0	0
$639$	−17.7516	−0.702243
$640$	0	0
$641$	−30.0119	−1.18540	−0.592699	−	0.805424i	$-0.701938\pi$
$641$	−30.0119	−1.18540	−0.592699	+	0.805424i	$0.701938\pi$
$642$	0	0
$643$	−19.1890	−0.756740	−0.378370	−	0.925654i	$-0.623515\pi$
$643$	−19.1890	−0.756740	−0.378370	+	0.925654i	$0.623515\pi$
$644$	0	0
$645$	18.4358	0.725909
$646$	0	0
$647$	−3.79947	−0.149372	−0.0746862	−	0.997207i	$-0.523796\pi$
$647$	−3.79947	−0.149372	−0.0746862	+	0.997207i	$0.523796\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	7.46773	0.292684
$652$	0	0
$653$	34.6929	1.35764	0.678818	−	0.734306i	$-0.262492\pi$
$653$	34.6929	1.35764	0.678818	+	0.734306i	$0.262492\pi$
$654$	0	0
$655$	−48.2656	−1.88589
$656$	0	0
$657$	38.8563	1.51593
$658$	0	0
$659$	−21.3010	−0.829767	−0.414884	−	0.909874i	$-0.636178\pi$
$659$	−21.3010	−0.829767	−0.414884	+	0.909874i	$0.636178\pi$
$660$	0	0
$661$	46.4820	1.80794	0.903971	−	0.427595i	$-0.140639\pi$
$661$	46.4820	1.80794	0.903971	+	0.427595i	$0.140639\pi$
$662$	0	0
$663$	4.01658	0.155991
$664$	0	0
$665$	120.062	4.65581
$666$	0	0
$667$	−19.2711	−0.746180
$668$	0	0
$669$	5.15302	0.199227
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−30.3952	−1.17165	−0.585824	−	0.810438i	$-0.699229\pi$
$673$	−30.3952	−1.17165	−0.585824	+	0.810438i	$0.699229\pi$
$674$	0	0
$675$	43.6136	1.67869
$676$	0	0
$677$	30.0663	1.15554	0.577771	−	0.816199i	$-0.303923\pi$
$677$	30.0663	1.15554	0.577771	+	0.816199i	$0.303923\pi$
$678$	0	0
$679$	−82.3818	−3.16152
$680$	0	0
$681$	10.7767	0.412964
$682$	0	0
$683$	18.5396	0.709397	0.354699	−	0.934981i	$-0.384583\pi$
$683$	18.5396	0.709397	0.354699	+	0.934981i	$0.384583\pi$
$684$	0	0
$685$	−16.4838	−0.629815
$686$	0	0
$687$	−2.81857	−0.107535
$688$	0	0
$689$	−1.91032	−0.0727773
$690$	0	0
$691$	−20.4379	−0.777493	−0.388746	−	0.921345i	$-0.627092\pi$
$691$	−20.4379	−0.777493	−0.388746	+	0.921345i	$0.627092\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	30.5953	1.16055
$696$	0	0
$697$	−61.1506	−2.31624
$698$	0	0
$699$	−16.8318	−0.636639
$700$	0	0
$701$	39.1349	1.47810	0.739052	−	0.673649i	$-0.235274\pi$
$701$	39.1349	1.47810	0.739052	+	0.673649i	$0.235274\pi$
$702$	0	0
$703$	30.4491	1.14841
$704$	0	0
$705$	−11.6520	−0.438839
$706$	0	0
$707$	4.98535	0.187493
$708$	0	0
$709$	−49.3553	−1.85358	−0.926789	−	0.375583i	$-0.877442\pi$
$709$	−49.3553	−1.85358	−0.926789	+	0.375583i	$0.877442\pi$
$710$	0	0
$711$	−11.5176	−0.431944
$712$	0	0
$713$	−13.7770	−0.515953
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.374052	−0.0139692
$718$	0	0
$719$	−41.4454	−1.54565	−0.772827	−	0.634617i	$-0.781158\pi$
$719$	−41.4454	−1.54565	−0.772827	+	0.634617i	$0.781158\pi$
$720$	0	0
$721$	63.6363	2.36994
$722$	0	0
$723$	2.98342	0.110955
$724$	0	0
$725$	−50.4524	−1.87375
$726$	0	0
$727$	−26.9425	−0.999242	−0.499621	−	0.866244i	$-0.666527\pi$
$727$	−26.9425	−0.999242	−0.499621	+	0.866244i	$0.666527\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	46.2614	1.71104
$732$	0	0
$733$	15.2299	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$733$	15.2299	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$734$	0	0
$735$	27.7301	1.02284
$736$	0	0
$737$	0	0
$738$	0	0
$739$	44.2834	1.62899	0.814496	−	0.580169i	$-0.197014\pi$
$739$	44.2834	1.62899	0.814496	+	0.580169i	$0.197014\pi$
$740$	0	0
$741$	−4.20796	−0.154583
$742$	0	0
$743$	48.4914	1.77898	0.889488	−	0.456958i	$-0.151061\pi$
$743$	48.4914	1.77898	0.889488	+	0.456958i	$0.151061\pi$
$744$	0	0
$745$	−55.4968	−2.03324
$746$	0	0
$747$	−23.0124	−0.841981
$748$	0	0
$749$	38.7886	1.41730
$750$	0	0
$751$	−23.3794	−0.853127	−0.426563	−	0.904458i	$-0.640276\pi$
$751$	−23.3794	−0.853127	−0.426563	+	0.904458i	$0.640276\pi$
$752$	0	0
$753$	−4.25754	−0.155153
$754$	0	0
$755$	−33.6637	−1.22515
$756$	0	0
$757$	−32.5566	−1.18329	−0.591645	−	0.806199i	$-0.701521\pi$
$757$	−32.5566	−1.18329	−0.591645	+	0.806199i	$0.701521\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.5733	0.673282	0.336641	−	0.941633i	$-0.390709\pi$
$761$	18.5733	0.673282	0.336641	+	0.941633i	$0.390709\pi$
$762$	0	0
$763$	−26.0444	−0.942869
$764$	0	0
$765$	71.3008	2.57789
$766$	0	0
$767$	−4.52835	−0.163509
$768$	0	0
$769$	−0.520252	−0.0187608	−0.00938038	−	0.999956i	$-0.502986\pi$
$769$	−0.520252	−0.0187608	−0.00938038	+	0.999956i	$0.502986\pi$
$770$	0	0
$771$	−6.86609	−0.247276
$772$	0	0
$773$	−43.2935	−1.55716	−0.778580	−	0.627546i	$-0.784059\pi$
$773$	−43.2935	−1.55716	−0.778580	+	0.627546i	$0.784059\pi$
$774$	0	0
$775$	−36.0687	−1.29562
$776$	0	0
$777$	11.6305	0.417242
$778$	0	0
$779$	64.0643	2.29534
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−13.9461	−0.498394
$784$	0	0
$785$	8.66906	0.309412
$786$	0	0
$787$	11.8584	0.422705	0.211353	−	0.977410i	$-0.432213\pi$
$787$	11.8584	0.422705	0.211353	+	0.977410i	$0.432213\pi$
$788$	0	0
$789$	−7.85049	−0.279485
$790$	0	0
$791$	19.4325	0.690940
$792$	0	0
$793$	5.31422	0.188714
$794$	0	0
$795$	4.94758	0.175473
$796$	0	0
$797$	−9.07818	−0.321566	−0.160783	−	0.986990i	$-0.551402\pi$
$797$	−9.07818	−0.321566	−0.160783	+	0.986990i	$0.551402\pi$
$798$	0	0
$799$	−29.2386	−1.03439
$800$	0	0
$801$	−3.45349	−0.122023
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−84.6051	−2.98194
$806$	0	0
$807$	2.54934	0.0897409
$808$	0	0
$809$	−26.5993	−0.935181	−0.467590	−	0.883945i	$-0.654878\pi$
$809$	−26.5993	−0.935181	−0.467590	+	0.883945i	$0.654878\pi$
$810$	0	0
$811$	32.2983	1.13415	0.567074	−	0.823667i	$-0.308075\pi$
$811$	32.2983	1.13415	0.567074	+	0.823667i	$0.308075\pi$
$812$	0	0
$813$	−4.59072	−0.161003
$814$	0	0
$815$	−67.1189	−2.35107
$816$	0	0
$817$	−48.4657	−1.69560
$818$	0	0
$819$	11.0166	0.384951
$820$	0	0
$821$	12.7614	0.445375	0.222688	−	0.974890i	$-0.428517\pi$
$821$	12.7614	0.445375	0.222688	+	0.974890i	$0.428517\pi$
$822$	0	0
$823$	−24.3788	−0.849792	−0.424896	−	0.905242i	$-0.639689\pi$
$823$	−24.3788	−0.849792	−0.424896	+	0.905242i	$0.639689\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.59403	−0.124977	−0.0624883	−	0.998046i	$-0.519904\pi$
$827$	−3.59403	−0.124977	−0.0624883	+	0.998046i	$0.519904\pi$
$828$	0	0
$829$	17.0436	0.591948	0.295974	−	0.955196i	$-0.404356\pi$
$829$	17.0436	0.591948	0.295974	+	0.955196i	$0.404356\pi$
$830$	0	0
$831$	8.86845	0.307643
$832$	0	0
$833$	69.5839	2.41094
$834$	0	0
$835$	−51.9327	−1.79721
$836$	0	0
$837$	−9.97015	−0.344619
$838$	0	0
$839$	13.4037	0.462749	0.231374	−	0.972865i	$-0.425678\pi$
$839$	13.4037	0.462749	0.231374	+	0.972865i	$0.425678\pi$
$840$	0	0
$841$	−12.8671	−0.443692
$842$	0	0
$843$	−6.19469	−0.213356
$844$	0	0
$845$	−4.19059	−0.144161
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.60604	0.0894391
$850$	0	0
$851$	−21.4568	−0.735530
$852$	0	0
$853$	−2.20920	−0.0756416	−0.0378208	−	0.999285i	$-0.512042\pi$
$853$	−2.20920	−0.0756416	−0.0378208	+	0.999285i	$0.512042\pi$
$854$	0	0
$855$	−74.6981	−2.55462
$856$	0	0
$857$	3.52093	0.120273	0.0601363	−	0.998190i	$-0.480846\pi$
$857$	3.52093	0.120273	0.0601363	+	0.998190i	$0.480846\pi$
$858$	0	0
$859$	−22.9209	−0.782052	−0.391026	−	0.920380i	$-0.627880\pi$
$859$	−22.9209	−0.782052	−0.391026	+	0.920380i	$0.627880\pi$
$860$	0	0
$861$	24.4704	0.833948
$862$	0	0
$863$	−26.0962	−0.888324	−0.444162	−	0.895946i	$-0.646499\pi$
$863$	−26.0962	−0.888324	−0.444162	+	0.895946i	$0.646499\pi$
$864$	0	0
$865$	8.03316	0.273136
$866$	0	0
$867$	−15.5970	−0.529704
$868$	0	0
$869$	0	0
$870$	0	0
$871$	11.6441	0.394544
$872$	0	0
$873$	51.2548	1.73471
$874$	0	0
$875$	−133.330	−4.50737
$876$	0	0
$877$	10.5348	0.355734	0.177867	−	0.984055i	$-0.443080\pi$
$877$	10.5348	0.355734	0.177867	+	0.984055i	$0.443080\pi$
$878$	0	0
$879$	−11.0220	−0.371762
$880$	0	0
$881$	−39.2127	−1.32111	−0.660555	−	0.750778i	$-0.729679\pi$
$881$	−39.2127	−1.32111	−0.660555	+	0.750778i	$0.729679\pi$
$882$	0	0
$883$	14.7289	0.495666	0.247833	−	0.968803i	$-0.420281\pi$
$883$	14.7289	0.495666	0.247833	+	0.968803i	$0.420281\pi$
$884$	0	0
$885$	11.7281	0.394236
$886$	0	0
$887$	−19.7509	−0.663169	−0.331585	−	0.943426i	$-0.607583\pi$
$887$	−19.7509	−0.663169	−0.331585	+	0.943426i	$0.607583\pi$
$888$	0	0
$889$	−19.5077	−0.654266
$890$	0	0
$891$	0	0
$892$	0	0
$893$	30.6318	1.02505
$894$	0	0
$895$	−72.8287	−2.43439
$896$	0	0
$897$	2.96526	0.0990070
$898$	0	0
$899$	11.5335	0.384664
$900$	0	0
$901$	12.4151	0.413607
$902$	0	0
$903$	−18.5122	−0.616049
$904$	0	0
$905$	5.70127	0.189517
$906$	0	0
$907$	37.9082	1.25872	0.629361	−	0.777113i	$-0.283317\pi$
$907$	37.9082	1.25872	0.629361	+	0.777113i	$0.283317\pi$
$908$	0	0
$909$	−3.10170	−0.102877
$910$	0	0
$911$	−6.65635	−0.220535	−0.110267	−	0.993902i	$-0.535171\pi$
$911$	−6.65635	−0.220535	−0.110267	+	0.993902i	$0.535171\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−13.7634	−0.455005
$916$	0	0
$917$	48.4657	1.60048
$918$	0	0
$919$	−14.8559	−0.490050	−0.245025	−	0.969517i	$-0.578796\pi$
$919$	−14.8559	−0.490050	−0.245025	+	0.969517i	$0.578796\pi$
$920$	0	0
$921$	−11.8671	−0.391033
$922$	0	0
$923$	−6.78051	−0.223183
$924$	0	0
$925$	−56.1746	−1.84701
$926$	0	0
$927$	−39.5921	−1.30038
$928$	0	0
$929$	−40.0365	−1.31355	−0.656777	−	0.754085i	$-0.728081\pi$
$929$	−40.0365	−1.31355	−0.656777	+	0.754085i	$0.728081\pi$
$930$	0	0
$931$	−72.8994	−2.38918
$932$	0	0
$933$	−1.07799	−0.0352918
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.7388	−0.873520	−0.436760	−	0.899578i	$-0.643874\pi$
$937$	−26.7388	−0.873520	−0.436760	+	0.899578i	$0.643874\pi$
$938$	0	0
$939$	−13.6499	−0.445448
$940$	0	0
$941$	−24.7713	−0.807522	−0.403761	−	0.914864i	$-0.632297\pi$
$941$	−24.7713	−0.807522	−0.403761	+	0.914864i	$0.632297\pi$
$942$	0	0
$943$	−45.1447	−1.47011
$944$	0	0
$945$	−61.2270	−1.99172
$946$	0	0
$947$	5.72945	0.186182	0.0930911	−	0.995658i	$-0.470325\pi$
$947$	5.72945	0.186182	0.0930911	+	0.995658i	$0.470325\pi$
$948$	0	0
$949$	14.8418	0.481785
$950$	0	0
$951$	18.1890	0.589819
$952$	0	0
$953$	13.5582	0.439194	0.219597	−	0.975591i	$-0.429526\pi$
$953$	13.5582	0.439194	0.219597	+	0.975591i	$0.429526\pi$
$954$	0	0
$955$	24.7677	0.801464
$956$	0	0
$957$	0	0
$958$	0	0
$959$	16.5522	0.534497
$960$	0	0
$961$	−22.7546	−0.734021
$962$	0	0
$963$	−24.1328	−0.777668
$964$	0	0
$965$	92.2954	2.97110
$966$	0	0
$967$	54.4549	1.75115	0.875576	−	0.483081i	$-0.160482\pi$
$967$	54.4549	1.75115	0.875576	+	0.483081i	$0.160482\pi$
$968$	0	0
$969$	27.3474	0.878525
$970$	0	0
$971$	−37.8509	−1.21469	−0.607347	−	0.794436i	$-0.707766\pi$
$971$	−37.8509	−1.21469	−0.607347	+	0.794436i	$0.707766\pi$
$972$	0	0
$973$	−30.7221	−0.984906
$974$	0	0
$975$	7.76314	0.248620
$976$	0	0
$977$	−8.10679	−0.259359	−0.129680	−	0.991556i	$-0.541395\pi$
$977$	−8.10679	−0.259359	−0.129680	+	0.991556i	$0.541395\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	16.2038	0.517348
$982$	0	0
$983$	44.6141	1.42297	0.711484	−	0.702703i	$-0.248024\pi$
$983$	44.6141	1.42297	0.711484	+	0.702703i	$0.248024\pi$
$984$	0	0
$985$	−99.0674	−3.15655
$986$	0	0
$987$	11.7003	0.372424
$988$	0	0
$989$	34.1527	1.08599
$990$	0	0
$991$	29.1581	0.926237	0.463118	−	0.886296i	$-0.346730\pi$
$991$	29.1581	0.926237	0.463118	+	0.886296i	$0.346730\pi$
$992$	0	0
$993$	−20.2174	−0.641580
$994$	0	0
$995$	−92.6521	−2.93727
$996$	0	0
$997$	−23.6212	−0.748092	−0.374046	−	0.927410i	$-0.622030\pi$
$997$	−23.6212	−0.748092	−0.374046	+	0.927410i	$0.622030\pi$
$998$	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.1.1	yes	4
11.10	odd	2		6292.2.a.q.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6292.2.a.q.1.1	✓	4	11.10	odd	2
6292.2.a.r.1.1	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)

\(f(q)\)	\(=\)	\(q-0.618034 q^{3} -4.19059 q^{5} +4.20796 q^{7} -2.61803 q^{9} +O(q^{10})\)	\(q-0.618034 q^{3} -4.19059 q^{5} +4.20796 q^{7} -2.61803 q^{9} -1.00000 q^{13} +2.58993 q^{15} +6.49897 q^{17} -6.80862 q^{19} -2.60066 q^{21} +4.79789 q^{23} +12.5610 q^{25} +3.47214 q^{27} -4.01658 q^{29} -2.87147 q^{31} -17.6338 q^{35} -4.47214 q^{37} +0.618034 q^{39} -9.40928 q^{41} +7.11828 q^{43} +10.9711 q^{45} -4.49897 q^{47} +10.7069 q^{49} -4.01658 q^{51} +1.91032 q^{53} +4.20796 q^{57} +4.52835 q^{59} -5.31422 q^{61} -11.0166 q^{63} +4.19059 q^{65} -11.6441 q^{67} -2.96526 q^{69} +6.78051 q^{71} -14.8418 q^{73} -7.76314 q^{75} +4.39934 q^{79} +5.70820 q^{81} +8.78997 q^{83} -27.2345 q^{85} +2.48238 q^{87} +1.31911 q^{89} -4.20796 q^{91} +1.77467 q^{93} +28.5321 q^{95} -19.5776 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 2 q^{5} + 6 q^{7} - 6 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

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Database

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