LMFDB - Embedded newform 6240.2.a.bp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -3.37228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -3.37228 q^{7} +1.00000 q^{9} -0.627719 q^{11} -1.00000 q^{13} -1.00000 q^{15} -5.37228 q^{17} -6.74456 q^{19} -3.37228 q^{21} +7.37228 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.74456 q^{29} -0.627719 q^{33} +3.37228 q^{35} -9.37228 q^{37} -1.00000 q^{39} -8.11684 q^{41} +10.7446 q^{43} -1.00000 q^{45} +4.00000 q^{47} +4.37228 q^{49} -5.37228 q^{51} -1.37228 q^{53} +0.627719 q^{55} -6.74456 q^{57} -4.00000 q^{59} +2.62772 q^{61} -3.37228 q^{63} +1.00000 q^{65} -9.48913 q^{67} +7.37228 q^{69} +11.3723 q^{71} +10.0000 q^{73} +1.00000 q^{75} +2.11684 q^{77} -3.37228 q^{79} +1.00000 q^{81} +9.48913 q^{83} +5.37228 q^{85} +8.74456 q^{87} -9.37228 q^{89} +3.37228 q^{91} +6.74456 q^{95} +5.37228 q^{97} -0.627719 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} - 7 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} - 2 q^{19} - q^{21} + 9 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} - 7 q^{33} + q^{35} - 13 q^{37} - 2 q^{39} + q^{41} + 10 q^{43} - 2 q^{45} + 8 q^{47} + 3 q^{49} - 5 q^{51} + 3 q^{53} + 7 q^{55} - 2 q^{57} - 8 q^{59} + 11 q^{61} - q^{63} + 2 q^{65} + 4 q^{67} + 9 q^{69} + 17 q^{71} + 20 q^{73} + 2 q^{75} - 13 q^{77} - q^{79} + 2 q^{81} - 4 q^{83} + 5 q^{85} + 6 q^{87} - 13 q^{89} + q^{91} + 2 q^{95} + 5 q^{97} - 7 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 - 7 * q^11 - 2 * q^13 - 2 * q^15 - 5 * q^17 - 2 * q^19 - q^21 + 9 * q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 - 7 * q^33 + q^35 - 13 * q^37 - 2 * q^39 + q^41 + 10 * q^43 - 2 * q^45 + 8 * q^47 + 3 * q^49 - 5 * q^51 + 3 * q^53 + 7 * q^55 - 2 * q^57 - 8 * q^59 + 11 * q^61 - q^63 + 2 * q^65 + 4 * q^67 + 9 * q^69 + 17 * q^71 + 20 * q^73 + 2 * q^75 - 13 * q^77 - q^79 + 2 * q^81 - 4 * q^83 + 5 * q^85 + 6 * q^87 - 13 * q^89 + q^91 + 2 * q^95 + 5 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.37228	−1.27460	−0.637301	−	0.770615i	$-0.719949\pi$
$7$	−3.37228	−1.27460	−0.637301	+	0.770615i	$0.719949\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.627719	−0.189264	−0.0946322	−	0.995512i	$-0.530167\pi$
$11$	−0.627719	−0.189264	−0.0946322	+	0.995512i	$0.530167\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−5.37228	−1.30297	−0.651485	−	0.758662i	$-0.725854\pi$
$17$	−5.37228	−1.30297	−0.651485	+	0.758662i	$0.725854\pi$
$18$	0	0
$19$	−6.74456	−1.54731	−0.773654	−	0.633608i	$-0.781573\pi$
$19$	−6.74456	−1.54731	−0.773654	+	0.633608i	$0.781573\pi$
$20$	0	0
$21$	−3.37228	−0.735892
$22$	0	0
$23$	7.37228	1.53723	0.768613	−	0.639713i	$-0.220947\pi$
$23$	7.37228	1.53723	0.768613	+	0.639713i	$0.220947\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.74456	1.62382	0.811912	−	0.583779i	$-0.198427\pi$
$29$	8.74456	1.62382	0.811912	+	0.583779i	$0.198427\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−0.627719	−0.109272
$34$	0	0
$35$	3.37228	0.570020
$36$	0	0
$37$	−9.37228	−1.54079	−0.770397	−	0.637565i	$-0.779942\pi$
$37$	−9.37228	−1.54079	−0.770397	+	0.637565i	$0.779942\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−8.11684	−1.26764	−0.633819	−	0.773481i	$-0.718514\pi$
$41$	−8.11684	−1.26764	−0.633819	+	0.773481i	$0.718514\pi$
$42$	0	0
$43$	10.7446	1.63853	0.819265	−	0.573415i	$-0.194382\pi$
$43$	10.7446	1.63853	0.819265	+	0.573415i	$0.194382\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	−5.37228	−0.752270
$52$	0	0
$53$	−1.37228	−0.188497	−0.0942487	−	0.995549i	$-0.530045\pi$
$53$	−1.37228	−0.188497	−0.0942487	+	0.995549i	$0.530045\pi$
$54$	0	0
$55$	0.627719	0.0846416
$56$	0	0
$57$	−6.74456	−0.893339
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	2.62772	0.336445	0.168222	−	0.985749i	$-0.446197\pi$
$61$	2.62772	0.336445	0.168222	+	0.985749i	$0.446197\pi$
$62$	0	0
$63$	−3.37228	−0.424868
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−9.48913	−1.15928	−0.579641	−	0.814872i	$-0.696807\pi$
$67$	−9.48913	−1.15928	−0.579641	+	0.814872i	$0.696807\pi$
$68$	0	0
$69$	7.37228	0.887518
$70$	0	0
$71$	11.3723	1.34964	0.674821	−	0.737982i	$-0.264221\pi$
$71$	11.3723	1.34964	0.674821	+	0.737982i	$0.264221\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.11684	0.241237
$78$	0	0
$79$	−3.37228	−0.379411	−0.189706	−	0.981841i	$-0.560753\pi$
$79$	−3.37228	−0.379411	−0.189706	+	0.981841i	$0.560753\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.48913	1.04157	0.520783	−	0.853689i	$-0.325640\pi$
$83$	9.48913	1.04157	0.520783	+	0.853689i	$0.325640\pi$
$84$	0	0
$85$	5.37228	0.582706
$86$	0	0
$87$	8.74456	0.937516
$88$	0	0
$89$	−9.37228	−0.993460	−0.496730	−	0.867905i	$-0.665466\pi$
$89$	−9.37228	−0.993460	−0.496730	+	0.867905i	$0.665466\pi$
$90$	0	0
$91$	3.37228	0.353511
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.74456	0.691978
$96$	0	0
$97$	5.37228	0.545473	0.272736	−	0.962089i	$-0.412071\pi$
$97$	5.37228	0.545473	0.272736	+	0.962089i	$0.412071\pi$
$98$	0	0
$99$	−0.627719	−0.0630881
$100$	0	0
$101$	16.7446	1.66615	0.833073	−	0.553163i	$-0.186579\pi$
$101$	16.7446	1.66615	0.833073	+	0.553163i	$0.186579\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	3.37228	0.329101
$106$	0	0
$107$	−0.627719	−0.0606839	−0.0303419	−	0.999540i	$-0.509660\pi$
$107$	−0.627719	−0.0606839	−0.0303419	+	0.999540i	$0.509660\pi$
$108$	0	0
$109$	12.7446	1.22071	0.610354	−	0.792129i	$-0.291027\pi$
$109$	12.7446	1.22071	0.610354	+	0.792129i	$0.291027\pi$
$110$	0	0
$111$	−9.37228	−0.889578
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−7.37228	−0.687469
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	18.1168	1.66077
$120$	0	0
$121$	−10.6060	−0.964179
$122$	0	0
$123$	−8.11684	−0.731871
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	10.7446	0.946006
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	22.7446	1.97220
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	11.3723	0.964584	0.482292	−	0.876010i	$-0.339804\pi$
$139$	11.3723	0.964584	0.482292	+	0.876010i	$0.339804\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	0.627719	0.0524925
$144$	0	0
$145$	−8.74456	−0.726196
$146$	0	0
$147$	4.37228	0.360620
$148$	0	0
$149$	−14.8614	−1.21749	−0.608747	−	0.793364i	$-0.708328\pi$
$149$	−14.8614	−1.21749	−0.608747	+	0.793364i	$0.708328\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	−5.37228	−0.434323
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	−1.37228	−0.108829
$160$	0	0
$161$	−24.8614	−1.95935
$162$	0	0
$163$	0.627719	0.0491667	0.0245834	−	0.999698i	$-0.492174\pi$
$163$	0.627719	0.0491667	0.0245834	+	0.999698i	$0.492174\pi$
$164$	0	0
$165$	0.627719	0.0488678
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.74456	−0.515770
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	−3.37228	−0.254921
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−2.74456	−0.205138	−0.102569	−	0.994726i	$-0.532706\pi$
$179$	−2.74456	−0.205138	−0.102569	+	0.994726i	$0.532706\pi$
$180$	0	0
$181$	8.11684	0.603320	0.301660	−	0.953416i	$-0.402459\pi$
$181$	8.11684	0.603320	0.301660	+	0.953416i	$0.402459\pi$
$182$	0	0
$183$	2.62772	0.194247
$184$	0	0
$185$	9.37228	0.689064
$186$	0	0
$187$	3.37228	0.246606
$188$	0	0
$189$	−3.37228	−0.245297
$190$	0	0
$191$	−20.2337	−1.46406	−0.732029	−	0.681273i	$-0.761427\pi$
$191$	−20.2337	−1.46406	−0.732029	+	0.681273i	$0.761427\pi$
$192$	0	0
$193$	17.6060	1.26731	0.633653	−	0.773618i	$-0.281555\pi$
$193$	17.6060	1.26731	0.633653	+	0.773618i	$0.281555\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	7.48913	0.533578	0.266789	−	0.963755i	$-0.414037\pi$
$197$	7.48913	0.533578	0.266789	+	0.963755i	$0.414037\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−9.48913	−0.669311
$202$	0	0
$203$	−29.4891	−2.06973
$204$	0	0
$205$	8.11684	0.566905
$206$	0	0
$207$	7.37228	0.512409
$208$	0	0
$209$	4.23369	0.292850
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	11.3723	0.779216
$214$	0	0
$215$	−10.7446	−0.732773
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	5.37228	0.361379
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	2.74456	0.182163	0.0910815	−	0.995843i	$-0.470968\pi$
$227$	2.74456	0.182163	0.0910815	+	0.995843i	$0.470968\pi$
$228$	0	0
$229$	−0.744563	−0.0492021	−0.0246010	−	0.999697i	$-0.507832\pi$
$229$	−0.744563	−0.0492021	−0.0246010	+	0.999697i	$0.507832\pi$
$230$	0	0
$231$	2.11684	0.139278
$232$	0	0
$233$	6.86141	0.449506	0.224753	−	0.974416i	$-0.427843\pi$
$233$	6.86141	0.449506	0.224753	+	0.974416i	$0.427843\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	−3.37228	−0.219053
$238$	0	0
$239$	19.3723	1.25309	0.626544	−	0.779386i	$-0.284469\pi$
$239$	19.3723	1.25309	0.626544	+	0.779386i	$0.284469\pi$
$240$	0	0
$241$	−20.7446	−1.33627	−0.668137	−	0.744038i	$-0.732908\pi$
$241$	−20.7446	−1.33627	−0.668137	+	0.744038i	$0.732908\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.37228	−0.279335
$246$	0	0
$247$	6.74456	0.429146
$248$	0	0
$249$	9.48913	0.601349
$250$	0	0
$251$	6.51087	0.410963	0.205481	−	0.978661i	$-0.434124\pi$
$251$	6.51087	0.410963	0.205481	+	0.978661i	$0.434124\pi$
$252$	0	0
$253$	−4.62772	−0.290942
$254$	0	0
$255$	5.37228	0.336425
$256$	0	0
$257$	11.4891	0.716672	0.358336	−	0.933593i	$-0.383344\pi$
$257$	11.4891	0.716672	0.358336	+	0.933593i	$0.383344\pi$
$258$	0	0
$259$	31.6060	1.96390
$260$	0	0
$261$	8.74456	0.541275
$262$	0	0
$263$	−17.4891	−1.07843	−0.539213	−	0.842170i	$-0.681278\pi$
$263$	−17.4891	−1.07843	−0.539213	+	0.842170i	$0.681278\pi$
$264$	0	0
$265$	1.37228	0.0842986
$266$	0	0
$267$	−9.37228	−0.573574
$268$	0	0
$269$	−31.7228	−1.93417	−0.967087	−	0.254446i	$-0.918107\pi$
$269$	−31.7228	−1.93417	−0.967087	+	0.254446i	$0.918107\pi$
$270$	0	0
$271$	−1.25544	−0.0762624	−0.0381312	−	0.999273i	$-0.512140\pi$
$271$	−1.25544	−0.0762624	−0.0381312	+	0.999273i	$0.512140\pi$
$272$	0	0
$273$	3.37228	0.204100
$274$	0	0
$275$	−0.627719	−0.0378529
$276$	0	0
$277$	−18.2337	−1.09556	−0.547778	−	0.836624i	$-0.684526\pi$
$277$	−18.2337	−1.09556	−0.547778	+	0.836624i	$0.684526\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.4891	0.924004	0.462002	−	0.886879i	$-0.347131\pi$
$281$	15.4891	0.924004	0.462002	+	0.886879i	$0.347131\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	0	0
$285$	6.74456	0.399513
$286$	0	0
$287$	27.3723	1.61573
$288$	0	0
$289$	11.8614	0.697730
$290$	0	0
$291$	5.37228	0.314929
$292$	0	0
$293$	−15.2554	−0.891232	−0.445616	−	0.895224i	$-0.647015\pi$
$293$	−15.2554	−0.891232	−0.445616	+	0.895224i	$0.647015\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−0.627719	−0.0364239
$298$	0	0
$299$	−7.37228	−0.426350
$300$	0	0
$301$	−36.2337	−2.08848
$302$	0	0
$303$	16.7446	0.961950
$304$	0	0
$305$	−2.62772	−0.150463
$306$	0	0
$307$	0.627719	0.0358258	0.0179129	−	0.999840i	$-0.494298\pi$
$307$	0.627719	0.0358258	0.0179129	+	0.999840i	$0.494298\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	22.7446	1.28973	0.644863	−	0.764298i	$-0.276914\pi$
$311$	22.7446	1.28973	0.644863	+	0.764298i	$0.276914\pi$
$312$	0	0
$313$	15.4891	0.875497	0.437749	−	0.899097i	$-0.355776\pi$
$313$	15.4891	0.875497	0.437749	+	0.899097i	$0.355776\pi$
$314$	0	0
$315$	3.37228	0.190007
$316$	0	0
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	−5.48913	−0.307332
$320$	0	0
$321$	−0.627719	−0.0350358
$322$	0	0
$323$	36.2337	2.01610
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	12.7446	0.704776
$328$	0	0
$329$	−13.4891	−0.743680
$330$	0	0
$331$	12.2337	0.672424	0.336212	−	0.941786i	$-0.390854\pi$
$331$	12.2337	0.672424	0.336212	+	0.941786i	$0.390854\pi$
$332$	0	0
$333$	−9.37228	−0.513598
$334$	0	0
$335$	9.48913	0.518446
$336$	0	0
$337$	7.48913	0.407959	0.203979	−	0.978975i	$-0.434612\pi$
$337$	7.48913	0.407959	0.203979	+	0.978975i	$0.434612\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	0	0
$341$	0	0
$342$	0	0
$343$	8.86141	0.478471
$344$	0	0
$345$	−7.37228	−0.396910
$346$	0	0
$347$	9.88316	0.530556	0.265278	−	0.964172i	$-0.414536\pi$
$347$	9.88316	0.530556	0.265278	+	0.964172i	$0.414536\pi$
$348$	0	0
$349$	−7.48913	−0.400884	−0.200442	−	0.979706i	$-0.564238\pi$
$349$	−7.48913	−0.400884	−0.200442	+	0.979706i	$0.564238\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	31.7228	1.68843	0.844217	−	0.536001i	$-0.180066\pi$
$353$	31.7228	1.68843	0.844217	+	0.536001i	$0.180066\pi$
$354$	0	0
$355$	−11.3723	−0.603578
$356$	0	0
$357$	18.1168	0.958845
$358$	0	0
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	26.4891	1.39416
$362$	0	0
$363$	−10.6060	−0.556669
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	22.7446	1.18726	0.593628	−	0.804739i	$-0.297695\pi$
$367$	22.7446	1.18726	0.593628	+	0.804739i	$0.297695\pi$
$368$	0	0
$369$	−8.11684	−0.422546
$370$	0	0
$371$	4.62772	0.240259
$372$	0	0
$373$	12.5109	0.647789	0.323894	−	0.946093i	$-0.395008\pi$
$373$	12.5109	0.647789	0.323894	+	0.946093i	$0.395008\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−8.74456	−0.450368
$378$	0	0
$379$	21.4891	1.10382	0.551911	−	0.833903i	$-0.313899\pi$
$379$	21.4891	1.10382	0.551911	+	0.833903i	$0.313899\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	0	0
$383$	2.74456	0.140241	0.0701203	−	0.997539i	$-0.477662\pi$
$383$	2.74456	0.140241	0.0701203	+	0.997539i	$0.477662\pi$
$384$	0	0
$385$	−2.11684	−0.107884
$386$	0	0
$387$	10.7446	0.546177
$388$	0	0
$389$	−20.7446	−1.05179	−0.525896	−	0.850549i	$-0.676270\pi$
$389$	−20.7446	−1.05179	−0.525896	+	0.850549i	$0.676270\pi$
$390$	0	0
$391$	−39.6060	−2.00296
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	3.37228	0.169678
$396$	0	0
$397$	−17.3723	−0.871890	−0.435945	−	0.899973i	$-0.643586\pi$
$397$	−17.3723	−0.871890	−0.435945	+	0.899973i	$0.643586\pi$
$398$	0	0
$399$	22.7446	1.13865
$400$	0	0
$401$	−8.51087	−0.425013	−0.212506	−	0.977160i	$-0.568163\pi$
$401$	−8.51087	−0.425013	−0.212506	+	0.977160i	$0.568163\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	5.88316	0.291617
$408$	0	0
$409$	−23.7228	−1.17302	−0.586509	−	0.809943i	$-0.699498\pi$
$409$	−23.7228	−1.17302	−0.586509	+	0.809943i	$0.699498\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	13.4891	0.663756
$414$	0	0
$415$	−9.48913	−0.465803
$416$	0	0
$417$	11.3723	0.556903
$418$	0	0
$419$	−24.2337	−1.18389	−0.591947	−	0.805977i	$-0.701640\pi$
$419$	−24.2337	−1.18389	−0.591947	+	0.805977i	$0.701640\pi$
$420$	0	0
$421$	26.2337	1.27855	0.639276	−	0.768977i	$-0.279234\pi$
$421$	26.2337	1.27855	0.639276	+	0.768977i	$0.279234\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	−5.37228	−0.260594
$426$	0	0
$427$	−8.86141	−0.428834
$428$	0	0
$429$	0.627719	0.0303065
$430$	0	0
$431$	34.9783	1.68484	0.842422	−	0.538819i	$-0.181129\pi$
$431$	34.9783	1.68484	0.842422	+	0.538819i	$0.181129\pi$
$432$	0	0
$433$	−30.4674	−1.46417	−0.732084	−	0.681214i	$-0.761452\pi$
$433$	−30.4674	−1.46417	−0.732084	+	0.681214i	$0.761452\pi$
$434$	0	0
$435$	−8.74456	−0.419270
$436$	0	0
$437$	−49.7228	−2.37856
$438$	0	0
$439$	−15.6060	−0.744832	−0.372416	−	0.928066i	$-0.621471\pi$
$439$	−15.6060	−0.744832	−0.372416	+	0.928066i	$0.621471\pi$
$440$	0	0
$441$	4.37228	0.208204
$442$	0	0
$443$	12.8614	0.611064	0.305532	−	0.952182i	$-0.401166\pi$
$443$	12.8614	0.611064	0.305532	+	0.952182i	$0.401166\pi$
$444$	0	0
$445$	9.37228	0.444289
$446$	0	0
$447$	−14.8614	−0.702920
$448$	0	0
$449$	−10.6277	−0.501553	−0.250777	−	0.968045i	$-0.580686\pi$
$449$	−10.6277	−0.501553	−0.250777	+	0.968045i	$0.580686\pi$
$450$	0	0
$451$	5.09509	0.239919
$452$	0	0
$453$	8.00000	0.375873
$454$	0	0
$455$	−3.37228	−0.158095
$456$	0	0
$457$	−0.116844	−0.00546573	−0.00273287	−	0.999996i	$-0.500870\pi$
$457$	−0.116844	−0.00546573	−0.00273287	+	0.999996i	$0.500870\pi$
$458$	0	0
$459$	−5.37228	−0.250757
$460$	0	0
$461$	−16.1168	−0.750636	−0.375318	−	0.926896i	$-0.622467\pi$
$461$	−16.1168	−0.750636	−0.375318	+	0.926896i	$0.622467\pi$
$462$	0	0
$463$	12.6277	0.586860	0.293430	−	0.955981i	$-0.405203\pi$
$463$	12.6277	0.586860	0.293430	+	0.955981i	$0.405203\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.88316	0.0871421	0.0435710	−	0.999050i	$-0.486127\pi$
$467$	1.88316	0.0871421	0.0435710	+	0.999050i	$0.486127\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	−6.74456	−0.310115
$474$	0	0
$475$	−6.74456	−0.309462
$476$	0	0
$477$	−1.37228	−0.0628324
$478$	0	0
$479$	10.1168	0.462250	0.231125	−	0.972924i	$-0.425759\pi$
$479$	10.1168	0.462250	0.231125	+	0.972924i	$0.425759\pi$
$480$	0	0
$481$	9.37228	0.427339
$482$	0	0
$483$	−24.8614	−1.13123
$484$	0	0
$485$	−5.37228	−0.243943
$486$	0	0
$487$	40.8614	1.85161	0.925804	−	0.378005i	$-0.123390\pi$
$487$	40.8614	1.85161	0.925804	+	0.378005i	$0.123390\pi$
$488$	0	0
$489$	0.627719	0.0283864
$490$	0	0
$491$	−32.2337	−1.45469	−0.727343	−	0.686274i	$-0.759245\pi$
$491$	−32.2337	−1.45469	−0.727343	+	0.686274i	$0.759245\pi$
$492$	0	0
$493$	−46.9783	−2.11579
$494$	0	0
$495$	0.627719	0.0282139
$496$	0	0
$497$	−38.3505	−1.72026
$498$	0	0
$499$	−22.7446	−1.01819	−0.509093	−	0.860711i	$-0.670019\pi$
$499$	−22.7446	−1.01819	−0.509093	+	0.860711i	$0.670019\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	−36.4674	−1.62600	−0.813000	−	0.582264i	$-0.802167\pi$
$503$	−36.4674	−1.62600	−0.813000	+	0.582264i	$0.802167\pi$
$504$	0	0
$505$	−16.7446	−0.745123
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	28.1168	1.24626	0.623129	−	0.782119i	$-0.285861\pi$
$509$	28.1168	1.24626	0.623129	+	0.782119i	$0.285861\pi$
$510$	0	0
$511$	−33.7228	−1.49181
$512$	0	0
$513$	−6.74456	−0.297780
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−2.51087	−0.110428
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	−40.9783	−1.79529	−0.897645	−	0.440719i	$-0.854723\pi$
$521$	−40.9783	−1.79529	−0.897645	+	0.440719i	$0.854723\pi$
$522$	0	0
$523$	0.233688	0.0102185	0.00510923	−	0.999987i	$-0.498374\pi$
$523$	0.233688	0.0102185	0.00510923	+	0.999987i	$0.498374\pi$
$524$	0	0
$525$	−3.37228	−0.147178
$526$	0	0
$527$	0	0
$528$	0	0
$529$	31.3505	1.36307
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	8.11684	0.351580
$534$	0	0
$535$	0.627719	0.0271386
$536$	0	0
$537$	−2.74456	−0.118437
$538$	0	0
$539$	−2.74456	−0.118217
$540$	0	0
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	0	0
$543$	8.11684	0.348327
$544$	0	0
$545$	−12.7446	−0.545917
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	0	0
$549$	2.62772	0.112148
$550$	0	0
$551$	−58.9783	−2.51256
$552$	0	0
$553$	11.3723	0.483599
$554$	0	0
$555$	9.37228	0.397831
$556$	0	0
$557$	−31.7228	−1.34414	−0.672069	−	0.740488i	$-0.734594\pi$
$557$	−31.7228	−1.34414	−0.672069	+	0.740488i	$0.734594\pi$
$558$	0	0
$559$	−10.7446	−0.454447
$560$	0	0
$561$	3.37228	0.142378
$562$	0	0
$563$	−27.6060	−1.16345	−0.581726	−	0.813384i	$-0.697623\pi$
$563$	−27.6060	−1.16345	−0.581726	+	0.813384i	$0.697623\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	−3.37228	−0.141623
$568$	0	0
$569$	−40.9783	−1.71790	−0.858949	−	0.512061i	$-0.828882\pi$
$569$	−40.9783	−1.71790	−0.858949	+	0.512061i	$0.828882\pi$
$570$	0	0
$571$	31.6060	1.32267	0.661334	−	0.750091i	$-0.269990\pi$
$571$	31.6060	1.32267	0.661334	+	0.750091i	$0.269990\pi$
$572$	0	0
$573$	−20.2337	−0.845274
$574$	0	0
$575$	7.37228	0.307445
$576$	0	0
$577$	37.3723	1.55583	0.777914	−	0.628370i	$-0.216278\pi$
$577$	37.3723	1.55583	0.777914	+	0.628370i	$0.216278\pi$
$578$	0	0
$579$	17.6060	0.731679
$580$	0	0
$581$	−32.0000	−1.32758
$582$	0	0
$583$	0.861407	0.0356758
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	24.2337	1.00023	0.500116	−	0.865959i	$-0.333291\pi$
$587$	24.2337	1.00023	0.500116	+	0.865959i	$0.333291\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	7.48913	0.308061
$592$	0	0
$593$	−11.7228	−0.481398	−0.240699	−	0.970600i	$-0.577377\pi$
$593$	−11.7228	−0.481398	−0.240699	+	0.970600i	$0.577377\pi$
$594$	0	0
$595$	−18.1168	−0.742718
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.4674	0.999710	0.499855	−	0.866109i	$-0.333387\pi$
$599$	24.4674	0.999710	0.499855	+	0.866109i	$0.333387\pi$
$600$	0	0
$601$	16.3505	0.666952	0.333476	−	0.942759i	$-0.391778\pi$
$601$	16.3505	0.666952	0.333476	+	0.942759i	$0.391778\pi$
$602$	0	0
$603$	−9.48913	−0.386427
$604$	0	0
$605$	10.6060	0.431194
$606$	0	0
$607$	12.2337	0.496550	0.248275	−	0.968690i	$-0.420136\pi$
$607$	12.2337	0.496550	0.248275	+	0.968690i	$0.420136\pi$
$608$	0	0
$609$	−29.4891	−1.19496
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	24.3505	0.983509	0.491754	−	0.870734i	$-0.336356\pi$
$613$	24.3505	0.983509	0.491754	+	0.870734i	$0.336356\pi$
$614$	0	0
$615$	8.11684	0.327303
$616$	0	0
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	0	0
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	7.37228	0.295839
$622$	0	0
$623$	31.6060	1.26627
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.23369	0.169077
$628$	0	0
$629$	50.3505	2.00761
$630$	0	0
$631$	42.9783	1.71094	0.855469	−	0.517855i	$-0.173269\pi$
$631$	42.9783	1.71094	0.855469	+	0.517855i	$0.173269\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	−16.0000	−0.634941
$636$	0	0
$637$	−4.37228	−0.173236
$638$	0	0
$639$	11.3723	0.449880
$640$	0	0
$641$	20.5109	0.810131	0.405065	−	0.914288i	$-0.367249\pi$
$641$	20.5109	0.810131	0.405065	+	0.914288i	$0.367249\pi$
$642$	0	0
$643$	3.13859	0.123774	0.0618870	−	0.998083i	$-0.480288\pi$
$643$	3.13859	0.123774	0.0618870	+	0.998083i	$0.480288\pi$
$644$	0	0
$645$	−10.7446	−0.423067
$646$	0	0
$647$	−10.3505	−0.406921	−0.203461	−	0.979083i	$-0.565219\pi$
$647$	−10.3505	−0.406921	−0.203461	+	0.979083i	$0.565219\pi$
$648$	0	0
$649$	2.51087	0.0985605
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.4891	−1.07573	−0.537866	−	0.843030i	$-0.680769\pi$
$653$	−27.4891	−1.07573	−0.537866	+	0.843030i	$0.680769\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	0	0
$663$	5.37228	0.208642
$664$	0	0
$665$	−22.7446	−0.881996
$666$	0	0
$667$	64.4674	2.49619
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	−1.64947	−0.0636770
$672$	0	0
$673$	35.2554	1.35900	0.679499	−	0.733677i	$-0.262197\pi$
$673$	35.2554	1.35900	0.679499	+	0.733677i	$0.262197\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	28.1168	1.08062	0.540309	−	0.841467i	$-0.318307\pi$
$677$	28.1168	1.08062	0.540309	+	0.841467i	$0.318307\pi$
$678$	0	0
$679$	−18.1168	−0.695261
$680$	0	0
$681$	2.74456	0.105172
$682$	0	0
$683$	16.2337	0.621165	0.310582	−	0.950546i	$-0.399476\pi$
$683$	16.2337	0.621165	0.310582	+	0.950546i	$0.399476\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	−0.744563	−0.0284068
$688$	0	0
$689$	1.37228	0.0522798
$690$	0	0
$691$	−30.7446	−1.16958	−0.584789	−	0.811185i	$-0.698823\pi$
$691$	−30.7446	−1.16958	−0.584789	+	0.811185i	$0.698823\pi$
$692$	0	0
$693$	2.11684	0.0804123
$694$	0	0
$695$	−11.3723	−0.431375
$696$	0	0
$697$	43.6060	1.65169
$698$	0	0
$699$	6.86141	0.259522
$700$	0	0
$701$	−9.76631	−0.368869	−0.184434	−	0.982845i	$-0.559045\pi$
$701$	−9.76631	−0.368869	−0.184434	+	0.982845i	$0.559045\pi$
$702$	0	0
$703$	63.2119	2.38408
$704$	0	0
$705$	−4.00000	−0.150649
$706$	0	0
$707$	−56.4674	−2.12367
$708$	0	0
$709$	37.2119	1.39752	0.698762	−	0.715354i	$-0.253735\pi$
$709$	37.2119	1.39752	0.698762	+	0.715354i	$0.253735\pi$
$710$	0	0
$711$	−3.37228	−0.126470
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−0.627719	−0.0234753
$716$	0	0
$717$	19.3723	0.723471
$718$	0	0
$719$	−24.4674	−0.912479	−0.456240	−	0.889857i	$-0.650804\pi$
$719$	−24.4674	−0.912479	−0.456240	+	0.889857i	$0.650804\pi$
$720$	0	0
$721$	−26.9783	−1.00472
$722$	0	0
$723$	−20.7446	−0.771499
$724$	0	0
$725$	8.74456	0.324765
$726$	0	0
$727$	−25.7228	−0.954006	−0.477003	−	0.878902i	$-0.658277\pi$
$727$	−25.7228	−0.954006	−0.477003	+	0.878902i	$0.658277\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−57.7228	−2.13496
$732$	0	0
$733$	25.6060	0.945778	0.472889	−	0.881122i	$-0.343211\pi$
$733$	25.6060	0.945778	0.472889	+	0.881122i	$0.343211\pi$
$734$	0	0
$735$	−4.37228	−0.161274
$736$	0	0
$737$	5.95650	0.219411
$738$	0	0
$739$	−47.2119	−1.73672	−0.868360	−	0.495935i	$-0.834825\pi$
$739$	−47.2119	−1.73672	−0.868360	+	0.495935i	$0.834825\pi$
$740$	0	0
$741$	6.74456	0.247768
$742$	0	0
$743$	−21.7228	−0.796933	−0.398466	−	0.917183i	$-0.630457\pi$
$743$	−21.7228	−0.796933	−0.398466	+	0.917183i	$0.630457\pi$
$744$	0	0
$745$	14.8614	0.544480
$746$	0	0
$747$	9.48913	0.347189
$748$	0	0
$749$	2.11684	0.0773478
$750$	0	0
$751$	11.3723	0.414980	0.207490	−	0.978237i	$-0.433470\pi$
$751$	11.3723	0.414980	0.207490	+	0.978237i	$0.433470\pi$
$752$	0	0
$753$	6.51087	0.237269
$754$	0	0
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−50.2337	−1.82577	−0.912887	−	0.408212i	$-0.866152\pi$
$757$	−50.2337	−1.82577	−0.912887	+	0.408212i	$0.866152\pi$
$758$	0	0
$759$	−4.62772	−0.167976
$760$	0	0
$761$	47.4891	1.72148	0.860740	−	0.509045i	$-0.170001\pi$
$761$	47.4891	1.72148	0.860740	+	0.509045i	$0.170001\pi$
$762$	0	0
$763$	−42.9783	−1.55592
$764$	0	0
$765$	5.37228	0.194235
$766$	0	0
$767$	4.00000	0.144432
$768$	0	0
$769$	44.9783	1.62196	0.810979	−	0.585076i	$-0.198935\pi$
$769$	44.9783	1.62196	0.810979	+	0.585076i	$0.198935\pi$
$770$	0	0
$771$	11.4891	0.413771
$772$	0	0
$773$	−44.7446	−1.60935	−0.804675	−	0.593716i	$-0.797660\pi$
$773$	−44.7446	−1.60935	−0.804675	+	0.593716i	$0.797660\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	31.6060	1.13386
$778$	0	0
$779$	54.7446	1.96143
$780$	0	0
$781$	−7.13859	−0.255439
$782$	0	0
$783$	8.74456	0.312505
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	1.48913	0.0530816	0.0265408	−	0.999648i	$-0.491551\pi$
$787$	1.48913	0.0530816	0.0265408	+	0.999648i	$0.491551\pi$
$788$	0	0
$789$	−17.4891	−0.622629
$790$	0	0
$791$	6.74456	0.239809
$792$	0	0
$793$	−2.62772	−0.0933130
$794$	0	0
$795$	1.37228	0.0486698
$796$	0	0
$797$	29.3723	1.04042	0.520210	−	0.854039i	$-0.325854\pi$
$797$	29.3723	1.04042	0.520210	+	0.854039i	$0.325854\pi$
$798$	0	0
$799$	−21.4891	−0.760231
$800$	0	0
$801$	−9.37228	−0.331153
$802$	0	0
$803$	−6.27719	−0.221517
$804$	0	0
$805$	24.8614	0.876249
$806$	0	0
$807$	−31.7228	−1.11670
$808$	0	0
$809$	50.4674	1.77434	0.887169	−	0.461444i	$-0.152669\pi$
$809$	50.4674	1.77434	0.887169	+	0.461444i	$0.152669\pi$
$810$	0	0
$811$	−54.7446	−1.92234	−0.961171	−	0.275954i	$-0.911006\pi$
$811$	−54.7446	−1.92234	−0.961171	+	0.275954i	$0.911006\pi$
$812$	0	0
$813$	−1.25544	−0.0440301
$814$	0	0
$815$	−0.627719	−0.0219880
$816$	0	0
$817$	−72.4674	−2.53531
$818$	0	0
$819$	3.37228	0.117837
$820$	0	0
$821$	51.3288	1.79139	0.895693	−	0.444672i	$-0.146680\pi$
$821$	51.3288	1.79139	0.895693	+	0.444672i	$0.146680\pi$
$822$	0	0
$823$	42.9783	1.49813	0.749064	−	0.662498i	$-0.230504\pi$
$823$	42.9783	1.49813	0.749064	+	0.662498i	$0.230504\pi$
$824$	0	0
$825$	−0.627719	−0.0218544
$826$	0	0
$827$	22.9783	0.799032	0.399516	−	0.916726i	$-0.369178\pi$
$827$	22.9783	0.799032	0.399516	+	0.916726i	$0.369178\pi$
$828$	0	0
$829$	−42.4674	−1.47495	−0.737476	−	0.675373i	$-0.763983\pi$
$829$	−42.4674	−1.47495	−0.737476	+	0.675373i	$0.763983\pi$
$830$	0	0
$831$	−18.2337	−0.632520
$832$	0	0
$833$	−23.4891	−0.813850
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	0	0
$838$	0	0
$839$	19.8397	0.684941	0.342471	−	0.939529i	$-0.388736\pi$
$839$	19.8397	0.684941	0.342471	+	0.939529i	$0.388736\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	0	0
$843$	15.4891	0.533474
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	35.7663	1.22895
$848$	0	0
$849$	12.0000	0.411839
$850$	0	0
$851$	−69.0951	−2.36855
$852$	0	0
$853$	−0.116844	−0.00400066	−0.00200033	−	0.999998i	$-0.500637\pi$
$853$	−0.116844	−0.00400066	−0.00200033	+	0.999998i	$0.500637\pi$
$854$	0	0
$855$	6.74456	0.230659
$856$	0	0
$857$	−6.62772	−0.226399	−0.113199	−	0.993572i	$-0.536110\pi$
$857$	−6.62772	−0.226399	−0.113199	+	0.993572i	$0.536110\pi$
$858$	0	0
$859$	−56.0733	−1.91320	−0.956599	−	0.291408i	$-0.905876\pi$
$859$	−56.0733	−1.91320	−0.956599	+	0.291408i	$0.905876\pi$
$860$	0	0
$861$	27.3723	0.932845
$862$	0	0
$863$	−32.2337	−1.09725	−0.548624	−	0.836069i	$-0.684848\pi$
$863$	−32.2337	−1.09725	−0.548624	+	0.836069i	$0.684848\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	0	0
$867$	11.8614	0.402834
$868$	0	0
$869$	2.11684	0.0718090
$870$	0	0
$871$	9.48913	0.321527
$872$	0	0
$873$	5.37228	0.181824
$874$	0	0
$875$	3.37228	0.114004
$876$	0	0
$877$	42.4674	1.43402	0.717011	−	0.697062i	$-0.245510\pi$
$877$	42.4674	1.43402	0.717011	+	0.697062i	$0.245510\pi$
$878$	0	0
$879$	−15.2554	−0.514553
$880$	0	0
$881$	27.7228	0.934005	0.467003	−	0.884256i	$-0.345334\pi$
$881$	27.7228	0.934005	0.467003	+	0.884256i	$0.345334\pi$
$882$	0	0
$883$	25.4891	0.857777	0.428889	−	0.903357i	$-0.358905\pi$
$883$	25.4891	0.857777	0.428889	+	0.903357i	$0.358905\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	38.1168	1.27984	0.639919	−	0.768442i	$-0.278968\pi$
$887$	38.1168	1.27984	0.639919	+	0.768442i	$0.278968\pi$
$888$	0	0
$889$	−53.9565	−1.80964
$890$	0	0
$891$	−0.627719	−0.0210294
$892$	0	0
$893$	−26.9783	−0.902793
$894$	0	0
$895$	2.74456	0.0917406
$896$	0	0
$897$	−7.37228	−0.246153
$898$	0	0
$899$	0	0
$900$	0	0
$901$	7.37228	0.245606
$902$	0	0
$903$	−36.2337	−1.20578
$904$	0	0
$905$	−8.11684	−0.269813
$906$	0	0
$907$	21.2554	0.705775	0.352888	−	0.935666i	$-0.385200\pi$
$907$	21.2554	0.705775	0.352888	+	0.935666i	$0.385200\pi$
$908$	0	0
$909$	16.7446	0.555382
$910$	0	0
$911$	−14.7446	−0.488509	−0.244255	−	0.969711i	$-0.578543\pi$
$911$	−14.7446	−0.488509	−0.244255	+	0.969711i	$0.578543\pi$
$912$	0	0
$913$	−5.95650	−0.197131
$914$	0	0
$915$	−2.62772	−0.0868697
$916$	0	0
$917$	−13.4891	−0.445450
$918$	0	0
$919$	53.0951	1.75145	0.875723	−	0.482814i	$-0.160385\pi$
$919$	53.0951	1.75145	0.875723	+	0.482814i	$0.160385\pi$
$920$	0	0
$921$	0.627719	0.0206840
$922$	0	0
$923$	−11.3723	−0.374323
$924$	0	0
$925$	−9.37228	−0.308159
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	39.0951	1.28267	0.641334	−	0.767262i	$-0.278381\pi$
$929$	39.0951	1.28267	0.641334	+	0.767262i	$0.278381\pi$
$930$	0	0
$931$	−29.4891	−0.966467
$932$	0	0
$933$	22.7446	0.744624
$934$	0	0
$935$	−3.37228	−0.110285
$936$	0	0
$937$	41.2119	1.34634	0.673168	−	0.739490i	$-0.264933\pi$
$937$	41.2119	1.34634	0.673168	+	0.739490i	$0.264933\pi$
$938$	0	0
$939$	15.4891	0.505469
$940$	0	0
$941$	−20.3505	−0.663408	−0.331704	−	0.943383i	$-0.607624\pi$
$941$	−20.3505	−0.663408	−0.331704	+	0.943383i	$0.607624\pi$
$942$	0	0
$943$	−59.8397	−1.94865
$944$	0	0
$945$	3.37228	0.109700
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	−14.0000	−0.453981
$952$	0	0
$953$	−23.8832	−0.773651	−0.386826	−	0.922153i	$-0.626428\pi$
$953$	−23.8832	−0.773651	−0.386826	+	0.922153i	$0.626428\pi$
$954$	0	0
$955$	20.2337	0.654747
$956$	0	0
$957$	−5.48913	−0.177438
$958$	0	0
$959$	6.74456	0.217793
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−0.627719	−0.0202280
$964$	0	0
$965$	−17.6060	−0.566756
$966$	0	0
$967$	29.9565	0.963336	0.481668	−	0.876354i	$-0.340031\pi$
$967$	29.9565	0.963336	0.481668	+	0.876354i	$0.340031\pi$
$968$	0	0
$969$	36.2337	1.16399
$970$	0	0
$971$	−45.7228	−1.46731	−0.733657	−	0.679519i	$-0.762188\pi$
$971$	−45.7228	−1.46731	−0.733657	+	0.679519i	$0.762188\pi$
$972$	0	0
$973$	−38.3505	−1.22946
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	−42.4674	−1.35865	−0.679326	−	0.733837i	$-0.737728\pi$
$977$	−42.4674	−1.35865	−0.679326	+	0.733837i	$0.737728\pi$
$978$	0	0
$979$	5.88316	0.188026
$980$	0	0
$981$	12.7446	0.406903
$982$	0	0
$983$	−22.5109	−0.717985	−0.358993	−	0.933340i	$-0.616880\pi$
$983$	−22.5109	−0.717985	−0.358993	+	0.933340i	$0.616880\pi$
$984$	0	0
$985$	−7.48913	−0.238623
$986$	0	0
$987$	−13.4891	−0.429364
$988$	0	0
$989$	79.2119	2.51879
$990$	0	0
$991$	48.8614	1.55213	0.776067	−	0.630651i	$-0.217212\pi$
$991$	48.8614	1.55213	0.776067	+	0.630651i	$0.217212\pi$
$992$	0	0
$993$	12.2337	0.388224
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−58.2337	−1.84428	−0.922140	−	0.386857i	$-0.873561\pi$
$997$	−58.2337	−1.84428	−0.922140	+	0.386857i	$0.873561\pi$
$998$	0	0
$999$	−9.37228	−0.296526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.bp.1.1	yes	2
4.3	odd	2		6240.2.a.bj.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bj.1.2	✓	2	4.3	odd	2
6240.2.a.bp.1.1	yes	2	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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -3.37228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -3.37228 q^{7} +1.00000 q^{9} -0.627719 q^{11} -1.00000 q^{13} -1.00000 q^{15} -5.37228 q^{17} -6.74456 q^{19} -3.37228 q^{21} +7.37228 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.74456 q^{29} -0.627719 q^{33} +3.37228 q^{35} -9.37228 q^{37} -1.00000 q^{39} -8.11684 q^{41} +10.7446 q^{43} -1.00000 q^{45} +4.00000 q^{47} +4.37228 q^{49} -5.37228 q^{51} -1.37228 q^{53} +0.627719 q^{55} -6.74456 q^{57} -4.00000 q^{59} +2.62772 q^{61} -3.37228 q^{63} +1.00000 q^{65} -9.48913 q^{67} +7.37228 q^{69} +11.3723 q^{71} +10.0000 q^{73} +1.00000 q^{75} +2.11684 q^{77} -3.37228 q^{79} +1.00000 q^{81} +9.48913 q^{83} +5.37228 q^{85} +8.74456 q^{87} -9.37228 q^{89} +3.37228 q^{91} +6.74456 q^{95} +5.37228 q^{97} -0.627719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} - 7 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} - 2 q^{19} - q^{21} + 9 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} - 7 q^{33} + q^{35} - 13 q^{37} - 2 q^{39} + q^{41} + 10 q^{43} - 2 q^{45} + 8 q^{47} + 3 q^{49} - 5 q^{51} + 3 q^{53} + 7 q^{55} - 2 q^{57} - 8 q^{59} + 11 q^{61} - q^{63} + 2 q^{65} + 4 q^{67} + 9 q^{69} + 17 q^{71} + 20 q^{73} + 2 q^{75} - 13 q^{77} - q^{79} + 2 q^{81} - 4 q^{83} + 5 q^{85} + 6 q^{87} - 13 q^{89} + q^{91} + 2 q^{95} + 5 q^{97} - 7 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 - 7 * q^11 - 2 * q^13 - 2 * q^15 - 5 * q^17 - 2 * q^19 - q^21 + 9 * q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 - 7 * q^33 + q^35 - 13 * q^37 - 2 * q^39 + q^41 + 10 * q^43 - 2 * q^45 + 8 * q^47 + 3 * q^49 - 5 * q^51 + 3 * q^53 + 7 * q^55 - 2 * q^57 - 8 * q^59 + 11 * q^61 - q^63 + 2 * q^65 + 4 * q^67 + 9 * q^69 + 17 * q^71 + 20 * q^73 + 2 * q^75 - 13 * q^77 - q^79 + 2 * q^81 - 4 * q^83 + 5 * q^85 + 6 * q^87 - 13 * q^89 + q^91 + 2 * q^95 + 5 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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