LMFDB - Embedded newform 4400.2.a.bx.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,0,0,-3,0,6,0,3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$35.1341768894$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2200)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.36147$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.36147 q^{3} +0.576535 q^{7} +8.29947 q^{9} +1.00000 q^{11} +3.72294 q^{13} -6.51454 q^{17} -1.00000 q^{19} -1.93800 q^{21} -4.36147 q^{23} -17.8140 q^{27} +2.42347 q^{29} -2.15307 q^{31} -3.36147 q^{33} +3.21507 q^{37} -12.5145 q^{39} +5.93800 q^{41} +4.93800 q^{43} -12.6609 q^{47} -6.66761 q^{49} +21.8984 q^{51} +8.02241 q^{53} +3.36147 q^{57} +0.660941 q^{59} -10.4525 q^{61} +4.78493 q^{63} +5.87601 q^{67} +14.6609 q^{69} +15.2308 q^{71} -12.5765 q^{73} +0.576535 q^{77} -12.0844 q^{79} +34.9828 q^{81} +7.45254 q^{83} -8.14640 q^{87} +11.4525 q^{89} +2.14640 q^{91} +7.23748 q^{93} -14.6543 q^{97} +8.29947 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{7} + 6 q^{9} + 3 q^{11} - 3 q^{13} - 3 q^{17} - 3 q^{19} + 6 q^{21} - 6 q^{23} - 18 q^{27} + 12 q^{29} + 3 q^{31} - 3 q^{33} + 12 q^{37} - 21 q^{39} + 6 q^{41} + 3 q^{43} - 12 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^7 + 6 * q^9 + 3 * q^11 - 3 * q^13 - 3 * q^17 - 3 * q^19 + 6 * q^21 - 6 * q^23 - 18 * q^27 + 12 * q^29 + 3 * q^31 - 3 * q^33 + 12 * q^37 - 21 * q^39 + 6 * q^41 + 3 * q^43 - 12 * q^47 + 6 * q^49 + 9 * q^51 - 9 * q^53 + 3 * q^57 - 24 * q^59 - 3 * q^61 + 12 * q^63 - 6 * q^67 + 18 * q^69 + 15 * q^71 - 33 * q^73 - 3 * q^77 - 15 * q^79 + 27 * q^81 - 6 * q^83 - 15 * q^87 + 6 * q^89 - 3 * q^91 - 9 * q^93 - 18 * q^97 + 6 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.36147	−1.94074	−0.970372	−	0.241614i	$-0.922323\pi$
$3$	−3.36147	−1.94074	−0.970372	+	0.241614i	$0.922323\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.576535	0.217910	0.108955	−	0.994047i	$-0.465250\pi$
$7$	0.576535	0.217910	0.108955	+	0.994047i	$0.465250\pi$
$8$	0	0
$9$	8.29947	2.76649
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.72294	1.03256	0.516279	−	0.856421i	$-0.327317\pi$
$13$	3.72294	1.03256	0.516279	+	0.856421i	$0.327317\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.51454	−1.58001	−0.790004	−	0.613102i	$-0.789921\pi$
$17$	−6.51454	−1.58001	−0.790004	+	0.613102i	$0.789921\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−1.93800	−0.422907
$22$	0	0
$23$	−4.36147	−0.909429	−0.454715	−	0.890637i	$-0.650259\pi$
$23$	−4.36147	−0.909429	−0.454715	+	0.890637i	$0.650259\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−17.8140	−3.42831
$28$	0	0
$29$	2.42347	0.450026	0.225013	−	0.974356i	$-0.427758\pi$
$29$	2.42347	0.450026	0.225013	+	0.974356i	$0.427758\pi$
$30$	0	0
$31$	−2.15307	−0.386703	−0.193351	−	0.981130i	$-0.561936\pi$
$31$	−2.15307	−0.386703	−0.193351	+	0.981130i	$0.561936\pi$
$32$	0	0
$33$	−3.36147	−0.585157
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.21507	0.528554	0.264277	−	0.964447i	$-0.414867\pi$
$37$	3.21507	0.528554	0.264277	+	0.964447i	$0.414867\pi$
$38$	0	0
$39$	−12.5145	−2.00393
$40$	0	0
$41$	5.93800	0.927360	0.463680	−	0.886003i	$-0.346529\pi$
$41$	5.93800	0.927360	0.463680	+	0.886003i	$0.346529\pi$
$42$	0	0
$43$	4.93800	0.753038	0.376519	−	0.926409i	$-0.377121\pi$
$43$	4.93800	0.753038	0.376519	+	0.926409i	$0.377121\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.6609	−1.84679	−0.923394	−	0.383853i	$-0.874597\pi$
$47$	−12.6609	−1.84679	−0.923394	+	0.383853i	$0.874597\pi$
$48$	0	0
$49$	−6.66761	−0.952515
$50$	0	0
$51$	21.8984	3.06639
$52$	0	0
$53$	8.02241	1.10196	0.550981	−	0.834518i	$-0.314254\pi$
$53$	8.02241	1.10196	0.550981	+	0.834518i	$0.314254\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.36147	0.445237
$58$	0	0
$59$	0.660941	0.0860472	0.0430236	−	0.999074i	$-0.486301\pi$
$59$	0.660941	0.0860472	0.0430236	+	0.999074i	$0.486301\pi$
$60$	0	0
$61$	−10.4525	−1.33831	−0.669155	−	0.743122i	$-0.733344\pi$
$61$	−10.4525	−1.33831	−0.669155	+	0.743122i	$0.733344\pi$
$62$	0	0
$63$	4.78493	0.602845
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.87601	0.717869	0.358934	−	0.933363i	$-0.383140\pi$
$67$	5.87601	0.717869	0.358934	+	0.933363i	$0.383140\pi$
$68$	0	0
$69$	14.6609	1.76497
$70$	0	0
$71$	15.2308	1.80756	0.903782	−	0.427993i	$-0.140779\pi$
$71$	15.2308	1.80756	0.903782	+	0.427993i	$0.140779\pi$
$72$	0	0
$73$	−12.5765	−1.47197	−0.735986	−	0.676997i	$-0.763281\pi$
$73$	−12.5765	−1.47197	−0.735986	+	0.676997i	$0.763281\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.576535	0.0657022
$78$	0	0
$79$	−12.0844	−1.35960	−0.679801	−	0.733397i	$-0.737934\pi$
$79$	−12.0844	−1.35960	−0.679801	+	0.733397i	$0.737934\pi$
$80$	0	0
$81$	34.9828	3.88698
$82$	0	0
$83$	7.45254	0.818023	0.409011	−	0.912529i	$-0.365874\pi$
$83$	7.45254	0.818023	0.409011	+	0.912529i	$0.365874\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.14640	−0.873386
$88$	0	0
$89$	11.4525	1.21397	0.606983	−	0.794714i	$-0.292379\pi$
$89$	11.4525	1.21397	0.606983	+	0.794714i	$0.292379\pi$
$90$	0	0
$91$	2.14640	0.225004
$92$	0	0
$93$	7.23748	0.750491
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.6543	−1.48792	−0.743958	−	0.668226i	$-0.767054\pi$
$97$	−14.6543	−1.48792	−0.743958	+	0.668226i	$0.767054\pi$
$98$	0	0
$99$	8.29947	0.834128
$100$	0	0
$101$	17.5145	1.74276	0.871381	−	0.490607i	$-0.163225\pi$
$101$	17.5145	1.74276	0.871381	+	0.490607i	$0.163225\pi$
$102$	0	0
$103$	13.4392	1.32420	0.662102	−	0.749413i	$-0.269664\pi$
$103$	13.4392	1.32420	0.662102	+	0.749413i	$0.269664\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.4459	1.39654	0.698268	−	0.715837i	$-0.253955\pi$
$107$	14.4459	1.39654	0.698268	+	0.715837i	$0.253955\pi$
$108$	0	0
$109$	−3.99333	−0.382492	−0.191246	−	0.981542i	$-0.561253\pi$
$109$	−3.99333	−0.382492	−0.191246	+	0.981542i	$0.561253\pi$
$110$	0	0
$111$	−10.8073	−1.02579
$112$	0	0
$113$	−5.21507	−0.490592	−0.245296	−	0.969448i	$-0.578885\pi$
$113$	−5.21507	−0.490592	−0.245296	+	0.969448i	$0.578885\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	30.8984	2.85656
$118$	0	0
$119$	−3.75586	−0.344299
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−19.9604	−1.79977
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−13.3615	−1.18564	−0.592819	−	0.805335i	$-0.701985\pi$
$127$	−13.3615	−1.18564	−0.592819	+	0.805335i	$0.701985\pi$
$128$	0	0
$129$	−16.5989	−1.46146
$130$	0	0
$131$	−14.6543	−1.28035	−0.640175	−	0.768229i	$-0.721138\pi$
$131$	−14.6543	−1.28035	−0.640175	+	0.768229i	$0.721138\pi$
$132$	0	0
$133$	−0.576535	−0.0499919
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.29947	0.196457	0.0982286	−	0.995164i	$-0.468682\pi$
$137$	2.29947	0.196457	0.0982286	+	0.995164i	$0.468682\pi$
$138$	0	0
$139$	−1.43013	−0.121302	−0.0606511	−	0.998159i	$-0.519318\pi$
$139$	−1.43013	−0.121302	−0.0606511	+	0.998159i	$0.519318\pi$
$140$	0	0
$141$	42.5594	3.58414
$142$	0	0
$143$	3.72294	0.311328
$144$	0	0
$145$	0	0
$146$	0	0
$147$	22.4130	1.84859
$148$	0	0
$149$	−9.87601	−0.809074	−0.404537	−	0.914522i	$-0.632567\pi$
$149$	−9.87601	−0.809074	−0.404537	+	0.914522i	$0.632567\pi$
$150$	0	0
$151$	7.21507	0.587154	0.293577	−	0.955935i	$-0.405154\pi$
$151$	7.21507	0.587154	0.293577	+	0.955935i	$0.405154\pi$
$152$	0	0
$153$	−54.0672	−4.37108
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−23.1979	−1.85139	−0.925697	−	0.378267i	$-0.876520\pi$
$157$	−23.1979	−1.85139	−0.925697	+	0.378267i	$0.876520\pi$
$158$	0	0
$159$	−26.9671	−2.13863
$160$	0	0
$161$	−2.51454	−0.198173
$162$	0	0
$163$	−6.88267	−0.539093	−0.269546	−	0.962987i	$-0.586874\pi$
$163$	−6.88267	−0.539093	−0.269546	+	0.962987i	$0.586874\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.6900	1.36889	0.684447	−	0.729062i	$-0.260044\pi$
$167$	17.6900	1.36889	0.684447	+	0.729062i	$0.260044\pi$
$168$	0	0
$169$	0.860264	0.0661741
$170$	0	0
$171$	−8.29947	−0.634677
$172$	0	0
$173$	−4.44588	−0.338014	−0.169007	−	0.985615i	$-0.554056\pi$
$173$	−4.44588	−0.338014	−0.169007	+	0.985615i	$0.554056\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.22173	−0.166996
$178$	0	0
$179$	−4.70053	−0.351334	−0.175667	−	0.984450i	$-0.556208\pi$
$179$	−4.70053	−0.351334	−0.175667	+	0.984450i	$0.556208\pi$
$180$	0	0
$181$	−17.8984	−1.33038	−0.665189	−	0.746675i	$-0.731649\pi$
$181$	−17.8984	−1.33038	−0.665189	+	0.746675i	$0.731649\pi$
$182$	0	0
$183$	35.1359	2.59732
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.51454	−0.476390
$188$	0	0
$189$	−10.2704	−0.747061
$190$	0	0
$191$	−7.99333	−0.578377	−0.289189	−	0.957272i	$-0.593385\pi$
$191$	−7.99333	−0.578377	−0.289189	+	0.957272i	$0.593385\pi$
$192$	0	0
$193$	11.7520	0.845928	0.422964	−	0.906146i	$-0.360990\pi$
$193$	11.7520	0.845928	0.422964	+	0.906146i	$0.360990\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.2666	−1.65767	−0.828837	−	0.559491i	$-0.810997\pi$
$197$	−23.2666	−1.65767	−0.828837	+	0.559491i	$0.810997\pi$
$198$	0	0
$199$	−7.86934	−0.557843	−0.278921	−	0.960314i	$-0.589977\pi$
$199$	−7.86934	−0.557843	−0.278921	+	0.960314i	$0.589977\pi$
$200$	0	0
$201$	−19.7520	−1.39320
$202$	0	0
$203$	1.39721	0.0980651
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−36.1979	−2.51593
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−2.23081	−0.153575	−0.0767876	−	0.997047i	$-0.524466\pi$
$211$	−2.23081	−0.153575	−0.0767876	+	0.997047i	$0.524466\pi$
$212$	0	0
$213$	−51.1979	−3.50802
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.24132	−0.0842662
$218$	0	0
$219$	42.2756	2.85672
$220$	0	0
$221$	−24.2532	−1.63145
$222$	0	0
$223$	−4.66094	−0.312120	−0.156060	−	0.987748i	$-0.549879\pi$
$223$	−4.66094	−0.312120	−0.156060	+	0.987748i	$0.549879\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.15974	−0.0769744	−0.0384872	−	0.999259i	$-0.512254\pi$
$227$	−1.15974	−0.0769744	−0.0384872	+	0.999259i	$0.512254\pi$
$228$	0	0
$229$	10.0844	0.666396	0.333198	−	0.942857i	$-0.391872\pi$
$229$	10.0844	0.666396	0.333198	+	0.942857i	$0.391872\pi$
$230$	0	0
$231$	−1.93800	−0.127511
$232$	0	0
$233$	−9.29947	−0.609229	−0.304614	−	0.952476i	$-0.598528\pi$
$233$	−9.29947	−0.609229	−0.304614	+	0.952476i	$0.598528\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	40.6214	2.63864
$238$	0	0
$239$	5.13066	0.331875	0.165937	−	0.986136i	$-0.446935\pi$
$239$	5.13066	0.331875	0.165937	+	0.986136i	$0.446935\pi$
$240$	0	0
$241$	−23.1135	−1.48887	−0.744435	−	0.667695i	$-0.767281\pi$
$241$	−23.1135	−1.48887	−0.744435	+	0.667695i	$0.767281\pi$
$242$	0	0
$243$	−64.1516	−4.11533
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.72294	−0.236885
$248$	0	0
$249$	−25.0515	−1.58757
$250$	0	0
$251$	−1.06866	−0.0674534	−0.0337267	−	0.999431i	$-0.510738\pi$
$251$	−1.06866	−0.0674534	−0.0337267	+	0.999431i	$0.510738\pi$
$252$	0	0
$253$	−4.36147	−0.274203
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.5079	−0.780220	−0.390110	−	0.920768i	$-0.627563\pi$
$257$	−12.5079	−0.780220	−0.390110	+	0.920768i	$0.627563\pi$
$258$	0	0
$259$	1.85360	0.115177
$260$	0	0
$261$	20.1135	1.24499
$262$	0	0
$263$	−5.64520	−0.348098	−0.174049	−	0.984737i	$-0.555685\pi$
$263$	−5.64520	−0.348098	−0.174049	+	0.984737i	$0.555685\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−38.4974	−2.35600
$268$	0	0
$269$	10.3772	0.632710	0.316355	−	0.948641i	$-0.397541\pi$
$269$	10.3772	0.632710	0.316355	+	0.948641i	$0.397541\pi$
$270$	0	0
$271$	−18.1201	−1.10072	−0.550360	−	0.834927i	$-0.685510\pi$
$271$	−18.1201	−1.10072	−0.550360	+	0.834927i	$0.685510\pi$
$272$	0	0
$273$	−7.21507	−0.436676
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.9537	0.658147	0.329073	−	0.944304i	$-0.393264\pi$
$277$	10.9537	0.658147	0.329073	+	0.944304i	$0.393264\pi$
$278$	0	0
$279$	−17.8693	−1.06981
$280$	0	0
$281$	−2.77160	−0.165340	−0.0826699	−	0.996577i	$-0.526345\pi$
$281$	−2.77160	−0.165340	−0.0826699	+	0.996577i	$0.526345\pi$
$282$	0	0
$283$	6.80734	0.404655	0.202327	−	0.979318i	$-0.435150\pi$
$283$	6.80734	0.404655	0.202327	+	0.979318i	$0.435150\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.42347	0.202081
$288$	0	0
$289$	25.4392	1.49642
$290$	0	0
$291$	49.2599	2.88767
$292$	0	0
$293$	18.6123	1.08734	0.543670	−	0.839299i	$-0.317034\pi$
$293$	18.6123	1.08734	0.543670	+	0.839299i	$0.317034\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−17.8140	−1.03367
$298$	0	0
$299$	−16.2375	−0.939037
$300$	0	0
$301$	2.84693	0.164094
$302$	0	0
$303$	−58.8746	−3.38226
$304$	0	0
$305$	0	0
$306$	0	0
$307$	25.8231	1.47380	0.736901	−	0.676001i	$-0.236289\pi$
$307$	25.8231	1.47380	0.736901	+	0.676001i	$0.236289\pi$
$308$	0	0
$309$	−45.1755	−2.56994
$310$	0	0
$311$	−18.8760	−1.07036	−0.535180	−	0.844738i	$-0.679756\pi$
$311$	−18.8760	−1.07036	−0.535180	+	0.844738i	$0.679756\pi$
$312$	0	0
$313$	15.6147	0.882594	0.441297	−	0.897361i	$-0.354518\pi$
$313$	15.6147	0.882594	0.441297	+	0.897361i	$0.354518\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.1001	−1.52210	−0.761048	−	0.648696i	$-0.775315\pi$
$317$	−27.1001	−1.52210	−0.761048	+	0.648696i	$0.775315\pi$
$318$	0	0
$319$	2.42347	0.135688
$320$	0	0
$321$	−48.5594	−2.71032
$322$	0	0
$323$	6.51454	0.362479
$324$	0	0
$325$	0	0
$326$	0	0
$327$	13.4235	0.742319
$328$	0	0
$329$	−7.29947	−0.402433
$330$	0	0
$331$	9.93800	0.546242	0.273121	−	0.961980i	$-0.411944\pi$
$331$	9.93800	0.546242	0.273121	+	0.961980i	$0.411944\pi$
$332$	0	0
$333$	26.6834	1.46224
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−4.30614	−0.234570	−0.117285	−	0.993098i	$-0.537419\pi$
$337$	−4.30614	−0.234570	−0.117285	+	0.993098i	$0.537419\pi$
$338$	0	0
$339$	17.5303	0.952114
$340$	0	0
$341$	−2.15307	−0.116595
$342$	0	0
$343$	−7.87985	−0.425472
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.5765	1.31934	0.659669	−	0.751556i	$-0.270697\pi$
$347$	24.5765	1.31934	0.659669	+	0.751556i	$0.270697\pi$
$348$	0	0
$349$	−13.1240	−0.702511	−0.351256	−	0.936280i	$-0.614245\pi$
$349$	−13.1240	−0.702511	−0.351256	+	0.936280i	$0.614245\pi$
$350$	0	0
$351$	−66.3204	−3.53992
$352$	0	0
$353$	28.4459	1.51402	0.757011	−	0.653403i	$-0.226659\pi$
$353$	28.4459	1.51402	0.757011	+	0.653403i	$0.226659\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	12.6252	0.668196
$358$	0	0
$359$	−35.6280	−1.88038	−0.940188	−	0.340657i	$-0.889350\pi$
$359$	−35.6280	−1.88038	−0.940188	+	0.340657i	$0.889350\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−3.36147	−0.176431
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.0844	−0.683000	−0.341500	−	0.939882i	$-0.610935\pi$
$367$	−13.0844	−0.683000	−0.341500	+	0.939882i	$0.610935\pi$
$368$	0	0
$369$	49.2823	2.56553
$370$	0	0
$371$	4.62520	0.240128
$372$	0	0
$373$	−31.1226	−1.61147	−0.805733	−	0.592280i	$-0.798228\pi$
$373$	−31.1226	−1.61147	−0.805733	+	0.592280i	$0.798228\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.02241	0.464678
$378$	0	0
$379$	−10.9537	−0.562656	−0.281328	−	0.959612i	$-0.590775\pi$
$379$	−10.9537	−0.562656	−0.281328	+	0.959612i	$0.590775\pi$
$380$	0	0
$381$	44.9142	2.30102
$382$	0	0
$383$	27.8298	1.42203	0.711017	−	0.703175i	$-0.248235\pi$
$383$	27.8298	1.42203	0.711017	+	0.703175i	$0.248235\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	40.9828	2.08327
$388$	0	0
$389$	−21.4459	−1.08735	−0.543675	−	0.839296i	$-0.682967\pi$
$389$	−21.4459	−1.08735	−0.543675	+	0.839296i	$0.682967\pi$
$390$	0	0
$391$	28.4130	1.43690
$392$	0	0
$393$	49.2599	2.48483
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.2666	0.615641	0.307820	−	0.951445i	$-0.400400\pi$
$397$	12.2666	0.615641	0.307820	+	0.951445i	$0.400400\pi$
$398$	0	0
$399$	1.93800	0.0970215
$400$	0	0
$401$	16.9380	0.845844	0.422922	−	0.906166i	$-0.361005\pi$
$401$	16.9380	0.845844	0.422922	+	0.906166i	$0.361005\pi$
$402$	0	0
$403$	−8.01574	−0.399293
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.21507	0.159365
$408$	0	0
$409$	−39.6767	−1.96189	−0.980943	−	0.194296i	$-0.937758\pi$
$409$	−39.6767	−1.96189	−0.980943	+	0.194296i	$0.937758\pi$
$410$	0	0
$411$	−7.72960	−0.381273
$412$	0	0
$413$	0.381055	0.0187505
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.80734	0.235417
$418$	0	0
$419$	−40.0276	−1.95548	−0.977739	−	0.209824i	$-0.932711\pi$
$419$	−40.0276	−1.95548	−0.977739	+	0.209824i	$0.932711\pi$
$420$	0	0
$421$	−6.13307	−0.298908	−0.149454	−	0.988769i	$-0.547752\pi$
$421$	−6.13307	−0.298908	−0.149454	+	0.988769i	$0.547752\pi$
$422$	0	0
$423$	−105.079	−5.10912
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.02625	−0.291631
$428$	0	0
$429$	−12.5145	−0.604208
$430$	0	0
$431$	6.67427	0.321488	0.160744	−	0.986996i	$-0.448611\pi$
$431$	6.67427	0.321488	0.160744	+	0.986996i	$0.448611\pi$
$432$	0	0
$433$	6.62802	0.318522	0.159261	−	0.987236i	$-0.449089\pi$
$433$	6.62802	0.318522	0.159261	+	0.987236i	$0.449089\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.36147	0.208637
$438$	0	0
$439$	−13.4063	−0.639847	−0.319924	−	0.947443i	$-0.603657\pi$
$439$	−13.4063	−0.639847	−0.319924	+	0.947443i	$0.603657\pi$
$440$	0	0
$441$	−55.3376	−2.63513
$442$	0	0
$443$	−39.9208	−1.89670	−0.948348	−	0.317232i	$-0.897247\pi$
$443$	−39.9208	−1.89670	−0.948348	+	0.317232i	$0.897247\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	33.1979	1.57021
$448$	0	0
$449$	21.8207	1.02978	0.514891	−	0.857256i	$-0.327832\pi$
$449$	21.8207	1.02978	0.514891	+	0.857256i	$0.327832\pi$
$450$	0	0
$451$	5.93800	0.279610
$452$	0	0
$453$	−24.2532	−1.13952
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.00667	−0.327758	−0.163879	−	0.986480i	$-0.552401\pi$
$457$	−7.00667	−0.327758	−0.163879	+	0.986480i	$0.552401\pi$
$458$	0	0
$459$	116.050	5.41675
$460$	0	0
$461$	13.0911	0.609712	0.304856	−	0.952398i	$-0.401392\pi$
$461$	13.0911	0.609712	0.304856	+	0.952398i	$0.401392\pi$
$462$	0	0
$463$	29.7811	1.38404	0.692022	−	0.721876i	$-0.256720\pi$
$463$	29.7811	1.38404	0.692022	+	0.721876i	$0.256720\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.9671	0.970241	0.485120	−	0.874447i	$-0.338776\pi$
$467$	20.9671	0.970241	0.485120	+	0.874447i	$0.338776\pi$
$468$	0	0
$469$	3.38772	0.156430
$470$	0	0
$471$	77.9790	3.59308
$472$	0	0
$473$	4.93800	0.227050
$474$	0	0
$475$	0	0
$476$	0	0
$477$	66.5818	3.04857
$478$	0	0
$479$	−19.2466	−0.879397	−0.439699	−	0.898145i	$-0.644915\pi$
$479$	−19.2466	−0.879397	−0.439699	+	0.898145i	$0.644915\pi$
$480$	0	0
$481$	11.9695	0.545762
$482$	0	0
$483$	8.45254	0.384604
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−14.2771	−0.646955	−0.323478	−	0.946236i	$-0.604852\pi$
$487$	−14.2771	−0.646955	−0.323478	+	0.946236i	$0.604852\pi$
$488$	0	0
$489$	23.1359	1.04624
$490$	0	0
$491$	5.18215	0.233867	0.116933	−	0.993140i	$-0.462694\pi$
$491$	5.18215	0.233867	0.116933	+	0.993140i	$0.462694\pi$
$492$	0	0
$493$	−15.7878	−0.711045
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.78109	0.393886
$498$	0	0
$499$	0.315215	0.0141110	0.00705549	−	0.999975i	$-0.497754\pi$
$499$	0.315215	0.0141110	0.00705549	+	0.999975i	$0.497754\pi$
$500$	0	0
$501$	−59.4644	−2.65668
$502$	0	0
$503$	27.6147	1.23128	0.615639	−	0.788028i	$-0.288898\pi$
$503$	27.6147	1.23128	0.615639	+	0.788028i	$0.288898\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.89175	−0.128427
$508$	0	0
$509$	−9.71627	−0.430666	−0.215333	−	0.976541i	$-0.569084\pi$
$509$	−9.71627	−0.430666	−0.215333	+	0.976541i	$0.569084\pi$
$510$	0	0
$511$	−7.25081	−0.320757
$512$	0	0
$513$	17.8140	0.786508
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12.6609	−0.556828
$518$	0	0
$519$	14.9447	0.655998
$520$	0	0
$521$	−16.5527	−0.725187	−0.362593	−	0.931947i	$-0.618109\pi$
$521$	−16.5527	−0.725187	−0.362593	+	0.931947i	$0.618109\pi$
$522$	0	0
$523$	3.30998	0.144735	0.0723677	−	0.997378i	$-0.476944\pi$
$523$	3.30998	0.144735	0.0723677	+	0.997378i	$0.476944\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.0263	0.610993
$528$	0	0
$529$	−3.97759	−0.172939
$530$	0	0
$531$	5.48546	0.238049
$532$	0	0
$533$	22.1068	0.957552
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.8007	0.681850
$538$	0	0
$539$	−6.66761	−0.287194
$540$	0	0
$541$	3.09774	0.133182	0.0665911	−	0.997780i	$-0.478788\pi$
$541$	3.09774	0.133182	0.0665911	+	0.997780i	$0.478788\pi$
$542$	0	0
$543$	60.1650	2.58193
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2.22414	−0.0950975	−0.0475487	−	0.998869i	$-0.515141\pi$
$547$	−2.22414	−0.0950975	−0.0475487	+	0.998869i	$0.515141\pi$
$548$	0	0
$549$	−86.7506	−3.70242
$550$	0	0
$551$	−2.42347	−0.103243
$552$	0	0
$553$	−6.96708	−0.296270
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.7968	−1.22016	−0.610080	−	0.792339i	$-0.708863\pi$
$557$	−28.7968	−1.22016	−0.610080	+	0.792339i	$0.708863\pi$
$558$	0	0
$559$	18.3839	0.777555
$560$	0	0
$561$	21.8984	0.924552
$562$	0	0
$563$	−25.3128	−1.06681	−0.533404	−	0.845861i	$-0.679087\pi$
$563$	−25.3128	−1.06681	−0.533404	+	0.845861i	$0.679087\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	20.1688	0.847011
$568$	0	0
$569$	−19.4235	−0.814274	−0.407137	−	0.913367i	$-0.633473\pi$
$569$	−19.4235	−0.814274	−0.407137	+	0.913367i	$0.633473\pi$
$570$	0	0
$571$	−39.4354	−1.65032	−0.825159	−	0.564900i	$-0.808915\pi$
$571$	−39.4354	−1.65032	−0.825159	+	0.564900i	$0.808915\pi$
$572$	0	0
$573$	26.8693	1.12248
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−39.6543	−1.65083	−0.825415	−	0.564527i	$-0.809059\pi$
$577$	−39.6543	−1.65083	−0.825415	+	0.564527i	$0.809059\pi$
$578$	0	0
$579$	−39.5040	−1.64173
$580$	0	0
$581$	4.29665	0.178255
$582$	0	0
$583$	8.02241	0.332254
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.49880	−0.309508	−0.154754	−	0.987953i	$-0.549459\pi$
$587$	−7.49880	−0.309508	−0.154754	+	0.987953i	$0.549459\pi$
$588$	0	0
$589$	2.15307	0.0887157
$590$	0	0
$591$	78.2098	3.21712
$592$	0	0
$593$	−18.0935	−0.743010	−0.371505	−	0.928431i	$-0.621158\pi$
$593$	−18.0935	−0.743010	−0.371505	+	0.928431i	$0.621158\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	26.4525	1.08263
$598$	0	0
$599$	11.8364	0.483623	0.241812	−	0.970323i	$-0.422258\pi$
$599$	11.8364	0.483623	0.241812	+	0.970323i	$0.422258\pi$
$600$	0	0
$601$	3.05149	0.124473	0.0622364	−	0.998061i	$-0.480177\pi$
$601$	3.05149	0.124473	0.0622364	+	0.998061i	$0.480177\pi$
$602$	0	0
$603$	48.7678	1.98598
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−45.5040	−1.84695	−0.923476	−	0.383657i	$-0.874665\pi$
$607$	−45.5040	−1.84695	−0.923476	+	0.383657i	$0.874665\pi$
$608$	0	0
$609$	−4.69668	−0.190319
$610$	0	0
$611$	−47.1359	−1.90691
$612$	0	0
$613$	23.3615	0.943561	0.471780	−	0.881716i	$-0.343612\pi$
$613$	23.3615	0.943561	0.471780	+	0.881716i	$0.343612\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.0329	−0.484427	−0.242214	−	0.970223i	$-0.577873\pi$
$617$	−12.0329	−0.484427	−0.242214	+	0.970223i	$0.577873\pi$
$618$	0	0
$619$	−7.67668	−0.308552	−0.154276	−	0.988028i	$-0.549304\pi$
$619$	−7.67668	−0.308552	−0.154276	+	0.988028i	$0.549304\pi$
$620$	0	0
$621$	77.6953	3.11780
$622$	0	0
$623$	6.60279	0.264535
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.36147	0.134244
$628$	0	0
$629$	−20.9447	−0.835119
$630$	0	0
$631$	−9.28614	−0.369675	−0.184838	−	0.982769i	$-0.559176\pi$
$631$	−9.28614	−0.369675	−0.184838	+	0.982769i	$0.559176\pi$
$632$	0	0
$633$	7.49880	0.298050
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−24.8231	−0.983527
$638$	0	0
$639$	126.408	5.00061
$640$	0	0
$641$	11.2795	0.445512	0.222756	−	0.974874i	$-0.428495\pi$
$641$	11.2795	0.445512	0.222756	+	0.974874i	$0.428495\pi$
$642$	0	0
$643$	2.76776	0.109150	0.0545748	−	0.998510i	$-0.482620\pi$
$643$	2.76776	0.109150	0.0545748	+	0.998510i	$0.482620\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−14.3996	−0.566108	−0.283054	−	0.959104i	$-0.591347\pi$
$647$	−14.3996	−0.566108	−0.283054	+	0.959104i	$0.591347\pi$
$648$	0	0
$649$	0.660941	0.0259442
$650$	0	0
$651$	4.17266	0.163539
$652$	0	0
$653$	−15.9471	−0.624057	−0.312029	−	0.950073i	$-0.601009\pi$
$653$	−15.9471	−0.624057	−0.312029	+	0.950073i	$0.601009\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−104.379	−4.07220
$658$	0	0
$659$	−0.560792	−0.0218453	−0.0109227	−	0.999940i	$-0.503477\pi$
$659$	−0.560792	−0.0218453	−0.0109227	+	0.999940i	$0.503477\pi$
$660$	0	0
$661$	−36.6609	−1.42595	−0.712973	−	0.701192i	$-0.752652\pi$
$661$	−36.6609	−1.42595	−0.712973	+	0.701192i	$0.752652\pi$
$662$	0	0
$663$	81.5264	3.16622
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−10.5699	−0.409267
$668$	0	0
$669$	15.6676	0.605745
$670$	0	0
$671$	−10.4525	−0.403516
$672$	0	0
$673$	−20.3681	−0.785134	−0.392567	−	0.919723i	$-0.628413\pi$
$673$	−20.3681	−0.785134	−0.392567	+	0.919723i	$0.628413\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.23081	−0.354769	−0.177384	−	0.984142i	$-0.556764\pi$
$677$	−9.23081	−0.354769	−0.177384	+	0.984142i	$0.556764\pi$
$678$	0	0
$679$	−8.44870	−0.324231
$680$	0	0
$681$	3.89842	0.149388
$682$	0	0
$683$	−31.7034	−1.21310	−0.606548	−	0.795047i	$-0.707446\pi$
$683$	−31.7034	−1.21310	−0.606548	+	0.795047i	$0.707446\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−33.8984	−1.29331
$688$	0	0
$689$	29.8669	1.13784
$690$	0	0
$691$	22.6080	0.860050	0.430025	−	0.902817i	$-0.358505\pi$
$691$	22.6080	0.860050	0.430025	+	0.902817i	$0.358505\pi$
$692$	0	0
$693$	4.78493	0.181765
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−38.6834	−1.46524
$698$	0	0
$699$	31.2599	1.18236
$700$	0	0
$701$	36.1092	1.36383	0.681913	−	0.731433i	$-0.261148\pi$
$701$	36.1092	1.36383	0.681913	+	0.731433i	$0.261148\pi$
$702$	0	0
$703$	−3.21507	−0.121259
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.0977	0.379765
$708$	0	0
$709$	18.0582	0.678188	0.339094	−	0.940752i	$-0.389879\pi$
$709$	18.0582	0.678188	0.339094	+	0.940752i	$0.389879\pi$
$710$	0	0
$711$	−100.294	−3.76133
$712$	0	0
$713$	9.39055	0.351679
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−17.2466	−0.644084
$718$	0	0
$719$	−23.6319	−0.881320	−0.440660	−	0.897674i	$-0.645256\pi$
$719$	−23.6319	−0.881320	−0.440660	+	0.897674i	$0.645256\pi$
$720$	0	0
$721$	7.74817	0.288557
$722$	0	0
$723$	77.6953	2.88952
$724$	0	0
$725$	0	0
$726$	0	0
$727$	40.0382	1.48493	0.742466	−	0.669883i	$-0.233656\pi$
$727$	40.0382	1.48493	0.742466	+	0.669883i	$0.233656\pi$
$728$	0	0
$729$	110.695	4.09982
$730$	0	0
$731$	−32.1688	−1.18981
$732$	0	0
$733$	−15.4883	−0.572073	−0.286036	−	0.958219i	$-0.592338\pi$
$733$	−15.4883	−0.572073	−0.286036	+	0.958219i	$0.592338\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.87601	0.216445
$738$	0	0
$739$	46.3867	1.70636	0.853181	−	0.521615i	$-0.174670\pi$
$739$	46.3867	1.70636	0.853181	+	0.521615i	$0.174670\pi$
$740$	0	0
$741$	12.5145	0.459733
$742$	0	0
$743$	−18.7453	−0.687700	−0.343850	−	0.939025i	$-0.611731\pi$
$743$	−18.7453	−0.687700	−0.343850	+	0.939025i	$0.611731\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	61.8522	2.26305
$748$	0	0
$749$	8.32855	0.304319
$750$	0	0
$751$	18.6504	0.680564	0.340282	−	0.940323i	$-0.389477\pi$
$751$	18.6504	0.680564	0.340282	+	0.940323i	$0.389477\pi$
$752$	0	0
$753$	3.59228	0.130910
$754$	0	0
$755$	0	0
$756$	0	0
$757$	28.4921	1.03556	0.517782	−	0.855513i	$-0.326758\pi$
$757$	28.4921	1.03556	0.517782	+	0.855513i	$0.326758\pi$
$758$	0	0
$759$	14.6609	0.532158
$760$	0	0
$761$	−14.8918	−0.539826	−0.269913	−	0.962885i	$-0.586995\pi$
$761$	−14.8918	−0.539826	−0.269913	+	0.962885i	$0.586995\pi$
$762$	0	0
$763$	−2.30230	−0.0833487
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.46064	0.0888486
$768$	0	0
$769$	26.2175	0.945426	0.472713	−	0.881216i	$-0.343275\pi$
$769$	26.2175	0.945426	0.472713	+	0.881216i	$0.343275\pi$
$770$	0	0
$771$	42.0448	1.51421
$772$	0	0
$773$	37.7744	1.35865	0.679326	−	0.733837i	$-0.262272\pi$
$773$	37.7744	1.35865	0.679326	+	0.733837i	$0.262272\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−6.23081	−0.223529
$778$	0	0
$779$	−5.93800	−0.212751
$780$	0	0
$781$	15.2308	0.545001
$782$	0	0
$783$	−43.1716	−1.54283
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−32.2441	−1.14938	−0.574690	−	0.818371i	$-0.694877\pi$
$787$	−32.2441	−1.14938	−0.574690	+	0.818371i	$0.694877\pi$
$788$	0	0
$789$	18.9762	0.675569
$790$	0	0
$791$	−3.00667	−0.106905
$792$	0	0
$793$	−38.9142	−1.38188
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.77827	−0.204677	−0.102338	−	0.994750i	$-0.532632\pi$
$797$	−5.77827	−0.204677	−0.102338	+	0.994750i	$0.532632\pi$
$798$	0	0
$799$	82.4802	2.91794
$800$	0	0
$801$	95.0501	3.35843
$802$	0	0
$803$	−12.5765	−0.443816
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−34.8827	−1.22793
$808$	0	0
$809$	−33.9656	−1.19417	−0.597084	−	0.802179i	$-0.703674\pi$
$809$	−33.9656	−1.19417	−0.597084	+	0.802179i	$0.703674\pi$
$810$	0	0
$811$	0.412955	0.0145008	0.00725041	−	0.999974i	$-0.497692\pi$
$811$	0.412955	0.0145008	0.00725041	+	0.999974i	$0.497692\pi$
$812$	0	0
$813$	60.9103	2.13622
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4.93800	−0.172759
$818$	0	0
$819$	17.8140	0.622472
$820$	0	0
$821$	6.17025	0.215343	0.107672	−	0.994187i	$-0.465661\pi$
$821$	6.17025	0.215343	0.107672	+	0.994187i	$0.465661\pi$
$822$	0	0
$823$	34.3061	1.19584	0.597918	−	0.801557i	$-0.295995\pi$
$823$	34.3061	1.19584	0.597918	+	0.801557i	$0.295995\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30.6108	−1.06444	−0.532222	−	0.846605i	$-0.678643\pi$
$827$	−30.6108	−1.06444	−0.532222	+	0.846605i	$0.678643\pi$
$828$	0	0
$829$	3.44064	0.119498	0.0597492	−	0.998213i	$-0.480970\pi$
$829$	3.44064	0.119498	0.0597492	+	0.998213i	$0.480970\pi$
$830$	0	0
$831$	−36.8207	−1.27730
$832$	0	0
$833$	43.4364	1.50498
$834$	0	0
$835$	0	0
$836$	0	0
$837$	38.3548	1.32574
$838$	0	0
$839$	11.7611	0.406038	0.203019	−	0.979175i	$-0.434925\pi$
$839$	11.7611	0.406038	0.203019	+	0.979175i	$0.434925\pi$
$840$	0	0
$841$	−23.1268	−0.797476
$842$	0	0
$843$	9.31665	0.320882
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.576535	0.0198100
$848$	0	0
$849$	−22.8827	−0.785331
$850$	0	0
$851$	−14.0224	−0.480682
$852$	0	0
$853$	18.5279	0.634382	0.317191	−	0.948362i	$-0.397260\pi$
$853$	18.5279	0.634382	0.317191	+	0.948362i	$0.397260\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.7348	1.56227	0.781136	−	0.624361i	$-0.214640\pi$
$857$	45.7348	1.56227	0.781136	+	0.624361i	$0.214640\pi$
$858$	0	0
$859$	−13.0606	−0.445621	−0.222810	−	0.974862i	$-0.571523\pi$
$859$	−13.0606	−0.445621	−0.222810	+	0.974862i	$0.571523\pi$
$860$	0	0
$861$	−11.5079	−0.392187
$862$	0	0
$863$	26.7520	0.910649	0.455325	−	0.890326i	$-0.349523\pi$
$863$	26.7520	0.910649	0.455325	+	0.890326i	$0.349523\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−85.5131	−2.90418
$868$	0	0
$869$	−12.0844	−0.409935
$870$	0	0
$871$	21.8760	0.741240
$872$	0	0
$873$	−121.623	−4.11631
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−35.3023	−1.19207	−0.596037	−	0.802957i	$-0.703259\pi$
$877$	−35.3023	−1.19207	−0.596037	+	0.802957i	$0.703259\pi$
$878$	0	0
$879$	−62.5646	−2.11025
$880$	0	0
$881$	31.7282	1.06895	0.534475	−	0.845185i	$-0.320510\pi$
$881$	31.7282	1.06895	0.534475	+	0.845185i	$0.320510\pi$
$882$	0	0
$883$	29.9380	1.00749	0.503747	−	0.863851i	$-0.331954\pi$
$883$	29.9380	1.00749	0.503747	+	0.863851i	$0.331954\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.66094	0.223653	0.111826	−	0.993728i	$-0.464330\pi$
$887$	6.66094	0.223653	0.111826	+	0.993728i	$0.464330\pi$
$888$	0	0
$889$	−7.70335	−0.258362
$890$	0	0
$891$	34.9828	1.17197
$892$	0	0
$893$	12.6609	0.423682
$894$	0	0
$895$	0	0
$896$	0	0
$897$	54.5818	1.82243
$898$	0	0
$899$	−5.21789	−0.174026
$900$	0	0
$901$	−52.2623	−1.74111
$902$	0	0
$903$	−9.56987	−0.318465
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−45.0157	−1.49472	−0.747362	−	0.664418i	$-0.768680\pi$
$907$	−45.0157	−1.49472	−0.747362	+	0.664418i	$0.768680\pi$
$908$	0	0
$909$	145.361	4.82133
$910$	0	0
$911$	25.0739	0.830735	0.415368	−	0.909654i	$-0.363653\pi$
$911$	25.0739	0.830735	0.415368	+	0.909654i	$0.363653\pi$
$912$	0	0
$913$	7.45254	0.246643
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.44870	−0.279001
$918$	0	0
$919$	−22.9184	−0.756009	−0.378004	−	0.925804i	$-0.623390\pi$
$919$	−22.9184	−0.756009	−0.378004	+	0.925804i	$0.623390\pi$
$920$	0	0
$921$	−86.8035	−2.86027
$922$	0	0
$923$	56.7034	1.86641
$924$	0	0
$925$	0	0
$926$	0	0
$927$	111.538	3.66340
$928$	0	0
$929$	−4.45921	−0.146302	−0.0731509	−	0.997321i	$-0.523305\pi$
$929$	−4.45921	−0.146302	−0.0731509	+	0.997321i	$0.523305\pi$
$930$	0	0
$931$	6.66761	0.218522
$932$	0	0
$933$	63.4511	2.07730
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.15548	0.135754	0.0678768	−	0.997694i	$-0.478378\pi$
$937$	4.15548	0.135754	0.0678768	+	0.997694i	$0.478378\pi$
$938$	0	0
$939$	−52.4883	−1.71289
$940$	0	0
$941$	33.5079	1.09233	0.546163	−	0.837679i	$-0.316088\pi$
$941$	33.5079	1.09233	0.546163	+	0.837679i	$0.316088\pi$
$942$	0	0
$943$	−25.8984	−0.843368
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.0329	0.618487	0.309243	−	0.950983i	$-0.399924\pi$
$947$	19.0329	0.618487	0.309243	+	0.950983i	$0.399924\pi$
$948$	0	0
$949$	−46.8217	−1.51990
$950$	0	0
$951$	91.0963	2.95400
$952$	0	0
$953$	35.1979	1.14017	0.570086	−	0.821585i	$-0.306910\pi$
$953$	35.1979	1.14017	0.570086	+	0.821585i	$0.306910\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−8.14640	−0.263336
$958$	0	0
$959$	1.32573	0.0428099
$960$	0	0
$961$	−26.3643	−0.850461
$962$	0	0
$963$	119.893	3.86350
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−27.4855	−0.883873	−0.441936	−	0.897046i	$-0.645708\pi$
$967$	−27.4855	−0.883873	−0.441936	+	0.897046i	$0.645708\pi$
$968$	0	0
$969$	−21.8984	−0.703479
$970$	0	0
$971$	8.92749	0.286497	0.143248	−	0.989687i	$-0.454245\pi$
$971$	8.92749	0.286497	0.143248	+	0.989687i	$0.454245\pi$
$972$	0	0
$973$	−0.824521	−0.0264329
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24.7191	−0.790834	−0.395417	−	0.918502i	$-0.629400\pi$
$977$	−24.7191	−0.790834	−0.395417	+	0.918502i	$0.629400\pi$
$978$	0	0
$979$	11.4525	0.366025
$980$	0	0
$981$	−33.1426	−1.05816
$982$	0	0
$983$	−12.2017	−0.389175	−0.194587	−	0.980885i	$-0.562337\pi$
$983$	−12.2017	−0.389175	−0.194587	+	0.980885i	$0.562337\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	24.5369	0.781020
$988$	0	0
$989$	−21.5369	−0.684835
$990$	0	0
$991$	−25.9737	−0.825083	−0.412542	−	0.910939i	$-0.635359\pi$
$991$	−25.9737	−0.825083	−0.412542	+	0.910939i	$0.635359\pi$
$992$	0	0
$993$	−33.4063	−1.06012
$994$	0	0
$995$	0	0
$996$	0	0
$997$	62.2980	1.97300	0.986499	−	0.163765i	$-0.0523640\pi$
$997$	62.2980	1.97300	0.986499	+	0.163765i	$0.0523640\pi$
$998$	0	0
$999$	−57.2732	−1.81204

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bx.1.1		3
4.3	odd	2		2200.2.a.w.1.3	yes	3
5.2	odd	4		4400.2.b.bc.4049.6		6
5.3	odd	4		4400.2.b.bc.4049.1		6
5.4	even	2		4400.2.a.ca.1.3		3
20.3	even	4		2200.2.b.l.1849.6		6
20.7	even	4		2200.2.b.l.1849.1		6
20.19	odd	2		2200.2.a.t.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.t.1.1	✓	3	20.19	odd	2
2200.2.a.w.1.3	yes	3	4.3	odd	2
2200.2.b.l.1849.1		6	20.7	even	4
2200.2.b.l.1849.6		6	20.3	even	4
4400.2.a.bx.1.1		3	1.1	even	1	trivial
4400.2.a.ca.1.3		3	5.4	even	2
4400.2.b.bc.4049.1		6	5.3	odd	4
4400.2.b.bc.4049.6		6	5.2	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q-3.36147 q^{3} +0.576535 q^{7} +8.29947 q^{9} +1.00000 q^{11} +3.72294 q^{13} -6.51454 q^{17} -1.00000 q^{19} -1.93800 q^{21} -4.36147 q^{23} -17.8140 q^{27} +2.42347 q^{29} -2.15307 q^{31} -3.36147 q^{33} +3.21507 q^{37} -12.5145 q^{39} +5.93800 q^{41} +4.93800 q^{43} -12.6609 q^{47} -6.66761 q^{49} +21.8984 q^{51} +8.02241 q^{53} +3.36147 q^{57} +0.660941 q^{59} -10.4525 q^{61} +4.78493 q^{63} +5.87601 q^{67} +14.6609 q^{69} +15.2308 q^{71} -12.5765 q^{73} +0.576535 q^{77} -12.0844 q^{79} +34.9828 q^{81} +7.45254 q^{83} -8.14640 q^{87} +11.4525 q^{89} +2.14640 q^{91} +7.23748 q^{93} -14.6543 q^{97} +8.29947 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{7} + 6 q^{9} + 3 q^{11} - 3 q^{13} - 3 q^{17} - 3 q^{19} + 6 q^{21} - 6 q^{23} - 18 q^{27} + 12 q^{29} + 3 q^{31} - 3 q^{33} + 12 q^{37} - 21 q^{39} + 6 q^{41} + 3 q^{43} - 12 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^7 + 6 * q^9 + 3 * q^11 - 3 * q^13 - 3 * q^17 - 3 * q^19 + 6 * q^21 - 6 * q^23 - 18 * q^27 + 12 * q^29 + 3 * q^31 - 3 * q^33 + 12 * q^37 - 21 * q^39 + 6 * q^41 + 3 * q^43 - 12 * q^47 + 6 * q^49 + 9 * q^51 - 9 * q^53 + 3 * q^57 - 24 * q^59 - 3 * q^61 + 12 * q^63 - 6 * q^67 + 18 * q^69 + 15 * q^71 - 33 * q^73 - 3 * q^77 - 15 * q^79 + 27 * q^81 - 6 * q^83 - 15 * q^87 + 6 * q^89 - 3 * q^91 - 9 * q^93 - 18 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more