LMFDB - Embedded newform 2240.2.a.bh.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2240,2,Mod(1,2240)] 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("2240.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,-2,0,-2,0,3,0,-1,0,-5] 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:	$17.8864900528$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	2240.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.56155 q^{9} -2.56155 q^{11} -4.56155 q^{13} -2.56155 q^{15} -4.56155 q^{17} -1.12311 q^{19} -2.56155 q^{21} -5.12311 q^{23} +1.00000 q^{25} +1.43845 q^{27} +5.68466 q^{29} -6.56155 q^{33} +1.00000 q^{35} -6.00000 q^{37} -11.6847 q^{39} -3.12311 q^{41} -9.12311 q^{43} -3.56155 q^{45} +3.68466 q^{47} +1.00000 q^{49} -11.6847 q^{51} -3.12311 q^{53} +2.56155 q^{55} -2.87689 q^{57} +4.00000 q^{59} +9.36932 q^{61} -3.56155 q^{63} +4.56155 q^{65} +6.24621 q^{67} -13.1231 q^{69} +8.00000 q^{71} +4.24621 q^{73} +2.56155 q^{75} +2.56155 q^{77} -6.56155 q^{79} -7.00000 q^{81} -4.00000 q^{83} +4.56155 q^{85} +14.5616 q^{87} +7.12311 q^{89} +4.56155 q^{91} +1.12311 q^{95} -14.8078 q^{97} -9.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 2 q^{5} - 2 q^{7} + 3 q^{9} - q^{11} - 5 q^{13} - q^{15} - 5 q^{17} + 6 q^{19} - q^{21} - 2 q^{23} + 2 q^{25} + 7 q^{27} - q^{29} - 9 q^{33} + 2 q^{35} - 12 q^{37} - 11 q^{39} + 2 q^{41}+ \cdots - 10 q^{99}+O(q^{100}) $ 2 * q + q^3 - 2 * q^5 - 2 * q^7 + 3 * q^9 - q^11 - 5 * q^13 - q^15 - 5 * q^17 + 6 * q^19 - q^21 - 2 * q^23 + 2 * q^25 + 7 * q^27 - q^29 - 9 * q^33 + 2 * q^35 - 12 * q^37 - 11 * q^39 + 2 * q^41 - 10 * q^43 - 3 * q^45 - 5 * q^47 + 2 * q^49 - 11 * q^51 + 2 * q^53 + q^55 - 14 * q^57 + 8 * q^59 - 6 * q^61 - 3 * q^63 + 5 * q^65 - 4 * q^67 - 18 * q^69 + 16 * q^71 - 8 * q^73 + q^75 + q^77 - 9 * q^79 - 14 * q^81 - 8 * q^83 + 5 * q^85 + 25 * q^87 + 6 * q^89 + 5 * q^91 - 6 * q^95 - 9 * q^97 - 10 * 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
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	−2.56155	−0.772337	−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155	−0.772337	−0.386169	+	0.922428i	$0.626202\pi$
$12$	0	0
$13$	−4.56155	−1.26515	−0.632574	−	0.774500i	$-0.718001\pi$
$13$	−4.56155	−1.26515	−0.632574	+	0.774500i	$0.718001\pi$
$14$	0	0
$15$	−2.56155	−0.661390
$16$	0	0
$17$	−4.56155	−1.10634	−0.553170	−	0.833069i	$-0.686582\pi$
$17$	−4.56155	−1.10634	−0.553170	+	0.833069i	$0.686582\pi$
$18$	0	0
$19$	−1.12311	−0.257658	−0.128829	−	0.991667i	$-0.541122\pi$
$19$	−1.12311	−0.257658	−0.128829	+	0.991667i	$0.541122\pi$
$20$	0	0
$21$	−2.56155	−0.558977
$22$	0	0
$23$	−5.12311	−1.06824	−0.534121	−	0.845408i	$-0.679357\pi$
$23$	−5.12311	−1.06824	−0.534121	+	0.845408i	$0.679357\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−6.56155	−1.14222
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	−11.6847	−1.87104
$40$	0	0
$41$	−3.12311	−0.487747	−0.243874	−	0.969807i	$-0.578418\pi$
$41$	−3.12311	−0.487747	−0.243874	+	0.969807i	$0.578418\pi$
$42$	0	0
$43$	−9.12311	−1.39126	−0.695630	−	0.718400i	$-0.744875\pi$
$43$	−9.12311	−1.39126	−0.695630	+	0.718400i	$0.744875\pi$
$44$	0	0
$45$	−3.56155	−0.530925
$46$	0	0
$47$	3.68466	0.537463	0.268731	−	0.963215i	$-0.413396\pi$
$47$	3.68466	0.537463	0.268731	+	0.963215i	$0.413396\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−11.6847	−1.63618
$52$	0	0
$53$	−3.12311	−0.428992	−0.214496	−	0.976725i	$-0.568811\pi$
$53$	−3.12311	−0.428992	−0.214496	+	0.976725i	$0.568811\pi$
$54$	0	0
$55$	2.56155	0.345400
$56$	0	0
$57$	−2.87689	−0.381054
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	9.36932	1.19962	0.599809	−	0.800143i	$-0.295243\pi$
$61$	9.36932	1.19962	0.599809	+	0.800143i	$0.295243\pi$
$62$	0	0
$63$	−3.56155	−0.448713
$64$	0	0
$65$	4.56155	0.565791
$66$	0	0
$67$	6.24621	0.763096	0.381548	−	0.924349i	$-0.375391\pi$
$67$	6.24621	0.763096	0.381548	+	0.924349i	$0.375391\pi$
$68$	0	0
$69$	−13.1231	−1.57984
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	4.24621	0.496981	0.248491	−	0.968634i	$-0.420065\pi$
$73$	4.24621	0.496981	0.248491	+	0.968634i	$0.420065\pi$
$74$	0	0
$75$	2.56155	0.295783
$76$	0	0
$77$	2.56155	0.291916
$78$	0	0
$79$	−6.56155	−0.738232	−0.369116	−	0.929383i	$-0.620340\pi$
$79$	−6.56155	−0.738232	−0.369116	+	0.929383i	$0.620340\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	4.56155	0.494770
$86$	0	0
$87$	14.5616	1.56116
$88$	0	0
$89$	7.12311	0.755048	0.377524	−	0.926000i	$-0.376776\pi$
$89$	7.12311	0.755048	0.377524	+	0.926000i	$0.376776\pi$
$90$	0	0
$91$	4.56155	0.478181
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.12311	0.115228
$96$	0	0
$97$	−14.8078	−1.50350	−0.751750	−	0.659448i	$-0.770790\pi$
$97$	−14.8078	−1.50350	−0.751750	+	0.659448i	$0.770790\pi$
$98$	0	0
$99$	−9.12311	−0.916907
$100$	0	0
$101$	−0.246211	−0.0244989	−0.0122495	−	0.999925i	$-0.503899\pi$
$101$	−0.246211	−0.0244989	−0.0122495	+	0.999925i	$0.503899\pi$
$102$	0	0
$103$	1.43845	0.141734	0.0708672	−	0.997486i	$-0.477423\pi$
$103$	1.43845	0.141734	0.0708672	+	0.997486i	$0.477423\pi$
$104$	0	0
$105$	2.56155	0.249982
$106$	0	0
$107$	11.3693	1.09911	0.549557	−	0.835456i	$-0.314797\pi$
$107$	11.3693	1.09911	0.549557	+	0.835456i	$0.314797\pi$
$108$	0	0
$109$	−17.6847	−1.69388	−0.846942	−	0.531686i	$-0.821559\pi$
$109$	−17.6847	−1.69388	−0.846942	+	0.531686i	$0.821559\pi$
$110$	0	0
$111$	−15.3693	−1.45879
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	5.12311	0.477732
$116$	0	0
$117$	−16.2462	−1.50196
$118$	0	0
$119$	4.56155	0.418157
$120$	0	0
$121$	−4.43845	−0.403495
$122$	0	0
$123$	−8.00000	−0.721336
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.2462	0.909204	0.454602	−	0.890695i	$-0.349781\pi$
$127$	10.2462	0.909204	0.454602	+	0.890695i	$0.349781\pi$
$128$	0	0
$129$	−23.3693	−2.05755
$130$	0	0
$131$	9.12311	0.797089	0.398545	−	0.917149i	$-0.369515\pi$
$131$	9.12311	0.797089	0.398545	+	0.917149i	$0.369515\pi$
$132$	0	0
$133$	1.12311	0.0973856
$134$	0	0
$135$	−1.43845	−0.123802
$136$	0	0
$137$	−8.87689	−0.758404	−0.379202	−	0.925314i	$-0.623801\pi$
$137$	−8.87689	−0.758404	−0.379202	+	0.925314i	$0.623801\pi$
$138$	0	0
$139$	6.87689	0.583291	0.291645	−	0.956527i	$-0.405797\pi$
$139$	6.87689	0.583291	0.291645	+	0.956527i	$0.405797\pi$
$140$	0	0
$141$	9.43845	0.794861
$142$	0	0
$143$	11.6847	0.977120
$144$	0	0
$145$	−5.68466	−0.472085
$146$	0	0
$147$	2.56155	0.211273
$148$	0	0
$149$	4.24621	0.347863	0.173932	−	0.984758i	$-0.444353\pi$
$149$	4.24621	0.347863	0.173932	+	0.984758i	$0.444353\pi$
$150$	0	0
$151$	21.9309	1.78471	0.892354	−	0.451335i	$-0.149052\pi$
$151$	21.9309	1.78471	0.892354	+	0.451335i	$0.149052\pi$
$152$	0	0
$153$	−16.2462	−1.31343
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.75379	−0.299585	−0.149792	−	0.988717i	$-0.547861\pi$
$157$	−3.75379	−0.299585	−0.149792	+	0.988717i	$0.547861\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	5.12311	0.403757
$162$	0	0
$163$	−1.12311	−0.0879684	−0.0439842	−	0.999032i	$-0.514005\pi$
$163$	−1.12311	−0.0879684	−0.0439842	+	0.999032i	$0.514005\pi$
$164$	0	0
$165$	6.56155	0.510816
$166$	0	0
$167$	21.9309	1.69706	0.848531	−	0.529146i	$-0.177488\pi$
$167$	21.9309	1.69706	0.848531	+	0.529146i	$0.177488\pi$
$168$	0	0
$169$	7.80776	0.600597
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	8.56155	0.650923	0.325461	−	0.945555i	$-0.394480\pi$
$173$	8.56155	0.650923	0.325461	+	0.945555i	$0.394480\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	10.2462	0.770152
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−23.6155	−1.75533	−0.877664	−	0.479276i	$-0.840899\pi$
$181$	−23.6155	−1.75533	−0.877664	+	0.479276i	$0.840899\pi$
$182$	0	0
$183$	24.0000	1.77413
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	11.6847	0.854467
$188$	0	0
$189$	−1.43845	−0.104632
$190$	0	0
$191$	−9.43845	−0.682942	−0.341471	−	0.939892i	$-0.610925\pi$
$191$	−9.43845	−0.682942	−0.341471	+	0.939892i	$0.610925\pi$
$192$	0	0
$193$	−5.36932	−0.386492	−0.193246	−	0.981150i	$-0.561902\pi$
$193$	−5.36932	−0.386492	−0.193246	+	0.981150i	$0.561902\pi$
$194$	0	0
$195$	11.6847	0.836756
$196$	0	0
$197$	7.12311	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$197$	7.12311	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$198$	0	0
$199$	−18.2462	−1.29344	−0.646720	−	0.762728i	$-0.723860\pi$
$199$	−18.2462	−1.29344	−0.646720	+	0.762728i	$0.723860\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	−5.68466	−0.398985
$204$	0	0
$205$	3.12311	0.218127
$206$	0	0
$207$	−18.2462	−1.26820
$208$	0	0
$209$	2.87689	0.198999
$210$	0	0
$211$	23.0540	1.58710	0.793551	−	0.608504i	$-0.208230\pi$
$211$	23.0540	1.58710	0.793551	+	0.608504i	$0.208230\pi$
$212$	0	0
$213$	20.4924	1.40412
$214$	0	0
$215$	9.12311	0.622191
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.8769	0.734992
$220$	0	0
$221$	20.8078	1.39968
$222$	0	0
$223$	−6.56155	−0.439394	−0.219697	−	0.975568i	$-0.570507\pi$
$223$	−6.56155	−0.439394	−0.219697	+	0.975568i	$0.570507\pi$
$224$	0	0
$225$	3.56155	0.237437
$226$	0	0
$227$	−23.6847	−1.57201	−0.786003	−	0.618223i	$-0.787853\pi$
$227$	−23.6847	−1.57201	−0.786003	+	0.618223i	$0.787853\pi$
$228$	0	0
$229$	−19.1231	−1.26369	−0.631845	−	0.775095i	$-0.717702\pi$
$229$	−19.1231	−1.26369	−0.631845	+	0.775095i	$0.717702\pi$
$230$	0	0
$231$	6.56155	0.431718
$232$	0	0
$233$	−3.12311	−0.204601	−0.102301	−	0.994754i	$-0.532620\pi$
$233$	−3.12311	−0.204601	−0.102301	+	0.994754i	$0.532620\pi$
$234$	0	0
$235$	−3.68466	−0.240361
$236$	0	0
$237$	−16.8078	−1.09178
$238$	0	0
$239$	−0.807764	−0.0522499	−0.0261250	−	0.999659i	$-0.508317\pi$
$239$	−0.807764	−0.0522499	−0.0261250	+	0.999659i	$0.508317\pi$
$240$	0	0
$241$	12.2462	0.788848	0.394424	−	0.918929i	$-0.370944\pi$
$241$	12.2462	0.788848	0.394424	+	0.918929i	$0.370944\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	5.12311	0.325975
$248$	0	0
$249$	−10.2462	−0.649327
$250$	0	0
$251$	17.1231	1.08080	0.540400	−	0.841408i	$-0.318273\pi$
$251$	17.1231	1.08080	0.540400	+	0.841408i	$0.318273\pi$
$252$	0	0
$253$	13.1231	0.825043
$254$	0	0
$255$	11.6847	0.731722
$256$	0	0
$257$	22.4924	1.40304	0.701519	−	0.712650i	$-0.252505\pi$
$257$	22.4924	1.40304	0.701519	+	0.712650i	$0.252505\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	20.2462	1.25321
$262$	0	0
$263$	−21.1231	−1.30251	−0.651253	−	0.758860i	$-0.725756\pi$
$263$	−21.1231	−1.30251	−0.651253	+	0.758860i	$0.725756\pi$
$264$	0	0
$265$	3.12311	0.191851
$266$	0	0
$267$	18.2462	1.11665
$268$	0	0
$269$	−28.7386	−1.75223	−0.876113	−	0.482106i	$-0.839872\pi$
$269$	−28.7386	−1.75223	−0.876113	+	0.482106i	$0.839872\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	11.6847	0.707188
$274$	0	0
$275$	−2.56155	−0.154467
$276$	0	0
$277$	−16.2462	−0.976140	−0.488070	−	0.872804i	$-0.662299\pi$
$277$	−16.2462	−0.976140	−0.488070	+	0.872804i	$0.662299\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.5616	0.987979	0.493990	−	0.869468i	$-0.335538\pi$
$281$	16.5616	0.987979	0.493990	+	0.869468i	$0.335538\pi$
$282$	0	0
$283$	23.6847	1.40791	0.703953	−	0.710246i	$-0.251416\pi$
$283$	23.6847	1.40791	0.703953	+	0.710246i	$0.251416\pi$
$284$	0	0
$285$	2.87689	0.170413
$286$	0	0
$287$	3.12311	0.184351
$288$	0	0
$289$	3.80776	0.223986
$290$	0	0
$291$	−37.9309	−2.22355
$292$	0	0
$293$	−9.68466	−0.565784	−0.282892	−	0.959152i	$-0.591294\pi$
$293$	−9.68466	−0.565784	−0.282892	+	0.959152i	$0.591294\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	−3.68466	−0.213806
$298$	0	0
$299$	23.3693	1.35148
$300$	0	0
$301$	9.12311	0.525847
$302$	0	0
$303$	−0.630683	−0.0362318
$304$	0	0
$305$	−9.36932	−0.536486
$306$	0	0
$307$	31.6847	1.80834	0.904169	−	0.427174i	$-0.140491\pi$
$307$	31.6847	1.80834	0.904169	+	0.427174i	$0.140491\pi$
$308$	0	0
$309$	3.68466	0.209613
$310$	0	0
$311$	−9.61553	−0.545247	−0.272623	−	0.962121i	$-0.587891\pi$
$311$	−9.61553	−0.545247	−0.272623	+	0.962121i	$0.587891\pi$
$312$	0	0
$313$	31.3002	1.76919	0.884596	−	0.466359i	$-0.154434\pi$
$313$	31.3002	1.76919	0.884596	+	0.466359i	$0.154434\pi$
$314$	0	0
$315$	3.56155	0.200671
$316$	0	0
$317$	22.4924	1.26330	0.631650	−	0.775254i	$-0.282378\pi$
$317$	22.4924	1.26330	0.631650	+	0.775254i	$0.282378\pi$
$318$	0	0
$319$	−14.5616	−0.815290
$320$	0	0
$321$	29.1231	1.62549
$322$	0	0
$323$	5.12311	0.285057
$324$	0	0
$325$	−4.56155	−0.253029
$326$	0	0
$327$	−45.3002	−2.50511
$328$	0	0
$329$	−3.68466	−0.203142
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	−21.3693	−1.17103
$334$	0	0
$335$	−6.24621	−0.341267
$336$	0	0
$337$	−34.4924	−1.87892	−0.939461	−	0.342656i	$-0.888674\pi$
$337$	−34.4924	−1.87892	−0.939461	+	0.342656i	$0.888674\pi$
$338$	0	0
$339$	−35.8617	−1.94774
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	13.1231	0.706524
$346$	0	0
$347$	1.12311	0.0602915	0.0301457	−	0.999546i	$-0.490403\pi$
$347$	1.12311	0.0602915	0.0301457	+	0.999546i	$0.490403\pi$
$348$	0	0
$349$	22.4924	1.20399	0.601996	−	0.798499i	$-0.294372\pi$
$349$	22.4924	1.20399	0.601996	+	0.798499i	$0.294372\pi$
$350$	0	0
$351$	−6.56155	−0.350230
$352$	0	0
$353$	−14.8078	−0.788138	−0.394069	−	0.919081i	$-0.628933\pi$
$353$	−14.8078	−0.788138	−0.394069	+	0.919081i	$0.628933\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	11.6847	0.618418
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	0	0
$363$	−11.3693	−0.596734
$364$	0	0
$365$	−4.24621	−0.222257
$366$	0	0
$367$	3.68466	0.192338	0.0961688	−	0.995365i	$-0.469341\pi$
$367$	3.68466	0.192338	0.0961688	+	0.995365i	$0.469341\pi$
$368$	0	0
$369$	−11.1231	−0.579046
$370$	0	0
$371$	3.12311	0.162144
$372$	0	0
$373$	−29.3693	−1.52069	−0.760343	−	0.649522i	$-0.774969\pi$
$373$	−29.3693	−1.52069	−0.760343	+	0.649522i	$0.774969\pi$
$374$	0	0
$375$	−2.56155	−0.132278
$376$	0	0
$377$	−25.9309	−1.33551
$378$	0	0
$379$	−16.4924	−0.847159	−0.423579	−	0.905859i	$-0.639227\pi$
$379$	−16.4924	−0.847159	−0.423579	+	0.905859i	$0.639227\pi$
$380$	0	0
$381$	26.2462	1.34463
$382$	0	0
$383$	−10.2462	−0.523557	−0.261778	−	0.965128i	$-0.584309\pi$
$383$	−10.2462	−0.523557	−0.261778	+	0.965128i	$0.584309\pi$
$384$	0	0
$385$	−2.56155	−0.130549
$386$	0	0
$387$	−32.4924	−1.65168
$388$	0	0
$389$	−3.93087	−0.199303	−0.0996515	−	0.995022i	$-0.531773\pi$
$389$	−3.93087	−0.199303	−0.0996515	+	0.995022i	$0.531773\pi$
$390$	0	0
$391$	23.3693	1.18184
$392$	0	0
$393$	23.3693	1.17883
$394$	0	0
$395$	6.56155	0.330148
$396$	0	0
$397$	−23.4384	−1.17634	−0.588171	−	0.808737i	$-0.700152\pi$
$397$	−23.4384	−1.17634	−0.588171	+	0.808737i	$0.700152\pi$
$398$	0	0
$399$	2.87689	0.144025
$400$	0	0
$401$	27.4384	1.37021	0.685105	−	0.728444i	$-0.259756\pi$
$401$	27.4384	1.37021	0.685105	+	0.728444i	$0.259756\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	7.00000	0.347833
$406$	0	0
$407$	15.3693	0.761829
$408$	0	0
$409$	−26.4924	−1.30997	−0.654983	−	0.755644i	$-0.727324\pi$
$409$	−26.4924	−1.30997	−0.654983	+	0.755644i	$0.727324\pi$
$410$	0	0
$411$	−22.7386	−1.12161
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	17.6155	0.862636
$418$	0	0
$419$	−9.75379	−0.476504	−0.238252	−	0.971203i	$-0.576574\pi$
$419$	−9.75379	−0.476504	−0.238252	+	0.971203i	$0.576574\pi$
$420$	0	0
$421$	−9.68466	−0.472001	−0.236001	−	0.971753i	$-0.575837\pi$
$421$	−9.68466	−0.472001	−0.236001	+	0.971753i	$0.575837\pi$
$422$	0	0
$423$	13.1231	0.638067
$424$	0	0
$425$	−4.56155	−0.221268
$426$	0	0
$427$	−9.36932	−0.453413
$428$	0	0
$429$	29.9309	1.44508
$430$	0	0
$431$	0.807764	0.0389086	0.0194543	−	0.999811i	$-0.493807\pi$
$431$	0.807764	0.0389086	0.0194543	+	0.999811i	$0.493807\pi$
$432$	0	0
$433$	−8.24621	−0.396288	−0.198144	−	0.980173i	$-0.563491\pi$
$433$	−8.24621	−0.396288	−0.198144	+	0.980173i	$0.563491\pi$
$434$	0	0
$435$	−14.5616	−0.698173
$436$	0	0
$437$	5.75379	0.275241
$438$	0	0
$439$	−15.3693	−0.733537	−0.366769	−	0.930312i	$-0.619536\pi$
$439$	−15.3693	−0.733537	−0.366769	+	0.930312i	$0.619536\pi$
$440$	0	0
$441$	3.56155	0.169598
$442$	0	0
$443$	27.3693	1.30036	0.650178	−	0.759782i	$-0.274694\pi$
$443$	27.3693	1.30036	0.650178	+	0.759782i	$0.274694\pi$
$444$	0	0
$445$	−7.12311	−0.337668
$446$	0	0
$447$	10.8769	0.514459
$448$	0	0
$449$	18.8078	0.887593	0.443797	−	0.896128i	$-0.353631\pi$
$449$	18.8078	0.887593	0.443797	+	0.896128i	$0.353631\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	56.1771	2.63943
$454$	0	0
$455$	−4.56155	−0.213849
$456$	0	0
$457$	−8.87689	−0.415244	−0.207622	−	0.978209i	$-0.566572\pi$
$457$	−8.87689	−0.415244	−0.207622	+	0.978209i	$0.566572\pi$
$458$	0	0
$459$	−6.56155	−0.306267
$460$	0	0
$461$	4.87689	0.227140	0.113570	−	0.993530i	$-0.463771\pi$
$461$	4.87689	0.227140	0.113570	+	0.993530i	$0.463771\pi$
$462$	0	0
$463$	−20.4924	−0.952364	−0.476182	−	0.879347i	$-0.657980\pi$
$463$	−20.4924	−0.952364	−0.476182	+	0.879347i	$0.657980\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−26.5616	−1.22912	−0.614561	−	0.788869i	$-0.710667\pi$
$467$	−26.5616	−1.22912	−0.614561	+	0.788869i	$0.710667\pi$
$468$	0	0
$469$	−6.24621	−0.288423
$470$	0	0
$471$	−9.61553	−0.443060
$472$	0	0
$473$	23.3693	1.07452
$474$	0	0
$475$	−1.12311	−0.0515316
$476$	0	0
$477$	−11.1231	−0.509292
$478$	0	0
$479$	13.1231	0.599610	0.299805	−	0.954001i	$-0.403078\pi$
$479$	13.1231	0.599610	0.299805	+	0.954001i	$0.403078\pi$
$480$	0	0
$481$	27.3693	1.24793
$482$	0	0
$483$	13.1231	0.597122
$484$	0	0
$485$	14.8078	0.672386
$486$	0	0
$487$	5.12311	0.232150	0.116075	−	0.993240i	$-0.462969\pi$
$487$	5.12311	0.232150	0.116075	+	0.993240i	$0.462969\pi$
$488$	0	0
$489$	−2.87689	−0.130098
$490$	0	0
$491$	−4.17708	−0.188509	−0.0942545	−	0.995548i	$-0.530047\pi$
$491$	−4.17708	−0.188509	−0.0942545	+	0.995548i	$0.530047\pi$
$492$	0	0
$493$	−25.9309	−1.16787
$494$	0	0
$495$	9.12311	0.410053
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	4.17708	0.186992	0.0934959	−	0.995620i	$-0.470196\pi$
$499$	4.17708	0.186992	0.0934959	+	0.995620i	$0.470196\pi$
$500$	0	0
$501$	56.1771	2.50981
$502$	0	0
$503$	10.0691	0.448960	0.224480	−	0.974479i	$-0.427932\pi$
$503$	10.0691	0.448960	0.224480	+	0.974479i	$0.427932\pi$
$504$	0	0
$505$	0.246211	0.0109563
$506$	0	0
$507$	20.0000	0.888231
$508$	0	0
$509$	28.2462	1.25199	0.625996	−	0.779827i	$-0.284693\pi$
$509$	28.2462	1.25199	0.625996	+	0.779827i	$0.284693\pi$
$510$	0	0
$511$	−4.24621	−0.187841
$512$	0	0
$513$	−1.61553	−0.0713273
$514$	0	0
$515$	−1.43845	−0.0633856
$516$	0	0
$517$	−9.43845	−0.415102
$518$	0	0
$519$	21.9309	0.962658
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	−7.50758	−0.328283	−0.164142	−	0.986437i	$-0.552485\pi$
$523$	−7.50758	−0.328283	−0.164142	+	0.986437i	$0.552485\pi$
$524$	0	0
$525$	−2.56155	−0.111795
$526$	0	0
$527$	0	0
$528$	0	0
$529$	3.24621	0.141140
$530$	0	0
$531$	14.2462	0.618233
$532$	0	0
$533$	14.2462	0.617072
$534$	0	0
$535$	−11.3693	−0.491538
$536$	0	0
$537$	−51.2311	−2.21078
$538$	0	0
$539$	−2.56155	−0.110334
$540$	0	0
$541$	17.1922	0.739152	0.369576	−	0.929201i	$-0.379503\pi$
$541$	17.1922	0.739152	0.369576	+	0.929201i	$0.379503\pi$
$542$	0	0
$543$	−60.4924	−2.59598
$544$	0	0
$545$	17.6847	0.757528
$546$	0	0
$547$	−14.2462	−0.609124	−0.304562	−	0.952493i	$-0.598510\pi$
$547$	−14.2462	−0.609124	−0.304562	+	0.952493i	$0.598510\pi$
$548$	0	0
$549$	33.3693	1.42417
$550$	0	0
$551$	−6.38447	−0.271988
$552$	0	0
$553$	6.56155	0.279026
$554$	0	0
$555$	15.3693	0.652391
$556$	0	0
$557$	4.87689	0.206641	0.103320	−	0.994648i	$-0.467053\pi$
$557$	4.87689	0.206641	0.103320	+	0.994648i	$0.467053\pi$
$558$	0	0
$559$	41.6155	1.76015
$560$	0	0
$561$	29.9309	1.26368
$562$	0	0
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	0	0
$567$	7.00000	0.293972
$568$	0	0
$569$	34.9848	1.46664	0.733320	−	0.679883i	$-0.237969\pi$
$569$	34.9848	1.46664	0.733320	+	0.679883i	$0.237969\pi$
$570$	0	0
$571$	−7.50758	−0.314182	−0.157091	−	0.987584i	$-0.550212\pi$
$571$	−7.50758	−0.314182	−0.157091	+	0.987584i	$0.550212\pi$
$572$	0	0
$573$	−24.1771	−1.01001
$574$	0	0
$575$	−5.12311	−0.213648
$576$	0	0
$577$	13.0540	0.543444	0.271722	−	0.962376i	$-0.412407\pi$
$577$	13.0540	0.543444	0.271722	+	0.962376i	$0.412407\pi$
$578$	0	0
$579$	−13.7538	−0.571588
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	8.00000	0.331326
$584$	0	0
$585$	16.2462	0.671698
$586$	0	0
$587$	9.75379	0.402582	0.201291	−	0.979531i	$-0.435486\pi$
$587$	9.75379	0.402582	0.201291	+	0.979531i	$0.435486\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	18.2462	0.750549
$592$	0	0
$593$	−23.4384	−0.962502	−0.481251	−	0.876583i	$-0.659817\pi$
$593$	−23.4384	−0.962502	−0.481251	+	0.876583i	$0.659817\pi$
$594$	0	0
$595$	−4.56155	−0.187005
$596$	0	0
$597$	−46.7386	−1.91288
$598$	0	0
$599$	8.80776	0.359875	0.179938	−	0.983678i	$-0.442410\pi$
$599$	8.80776	0.359875	0.179938	+	0.983678i	$0.442410\pi$
$600$	0	0
$601$	−26.4924	−1.08065	−0.540324	−	0.841457i	$-0.681698\pi$
$601$	−26.4924	−1.08065	−0.540324	+	0.841457i	$0.681698\pi$
$602$	0	0
$603$	22.2462	0.905936
$604$	0	0
$605$	4.43845	0.180449
$606$	0	0
$607$	−4.94602	−0.200753	−0.100376	−	0.994950i	$-0.532005\pi$
$607$	−4.94602	−0.200753	−0.100376	+	0.994950i	$0.532005\pi$
$608$	0	0
$609$	−14.5616	−0.590064
$610$	0	0
$611$	−16.8078	−0.679969
$612$	0	0
$613$	8.73863	0.352950	0.176475	−	0.984305i	$-0.443531\pi$
$613$	8.73863	0.352950	0.176475	+	0.984305i	$0.443531\pi$
$614$	0	0
$615$	8.00000	0.322591
$616$	0	0
$617$	15.7538	0.634224	0.317112	−	0.948388i	$-0.397287\pi$
$617$	15.7538	0.634224	0.317112	+	0.948388i	$0.397287\pi$
$618$	0	0
$619$	42.1080	1.69246	0.846231	−	0.532817i	$-0.178866\pi$
$619$	42.1080	1.69246	0.846231	+	0.532817i	$0.178866\pi$
$620$	0	0
$621$	−7.36932	−0.295720
$622$	0	0
$623$	−7.12311	−0.285381
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.36932	0.294302
$628$	0	0
$629$	27.3693	1.09129
$630$	0	0
$631$	8.80776	0.350632	0.175316	−	0.984512i	$-0.443905\pi$
$631$	8.80776	0.350632	0.175316	+	0.984512i	$0.443905\pi$
$632$	0	0
$633$	59.0540	2.34718
$634$	0	0
$635$	−10.2462	−0.406608
$636$	0	0
$637$	−4.56155	−0.180735
$638$	0	0
$639$	28.4924	1.12714
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	2.56155	0.101018	0.0505089	−	0.998724i	$-0.483916\pi$
$643$	2.56155	0.101018	0.0505089	+	0.998724i	$0.483916\pi$
$644$	0	0
$645$	23.3693	0.920166
$646$	0	0
$647$	3.50758	0.137897	0.0689486	−	0.997620i	$-0.478036\pi$
$647$	3.50758	0.137897	0.0689486	+	0.997620i	$0.478036\pi$
$648$	0	0
$649$	−10.2462	−0.402199
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−49.2311	−1.92656	−0.963280	−	0.268499i	$-0.913473\pi$
$653$	−49.2311	−1.92656	−0.963280	+	0.268499i	$0.913473\pi$
$654$	0	0
$655$	−9.12311	−0.356469
$656$	0	0
$657$	15.1231	0.590009
$658$	0	0
$659$	36.1771	1.40926	0.704629	−	0.709575i	$-0.251113\pi$
$659$	36.1771	1.40926	0.704629	+	0.709575i	$0.251113\pi$
$660$	0	0
$661$	−3.12311	−0.121475	−0.0607374	−	0.998154i	$-0.519345\pi$
$661$	−3.12311	−0.121475	−0.0607374	+	0.998154i	$0.519345\pi$
$662$	0	0
$663$	53.3002	2.07001
$664$	0	0
$665$	−1.12311	−0.0435522
$666$	0	0
$667$	−29.1231	−1.12765
$668$	0	0
$669$	−16.8078	−0.649826
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	−25.8617	−0.996897	−0.498448	−	0.866919i	$-0.666097\pi$
$673$	−25.8617	−0.996897	−0.498448	+	0.866919i	$0.666097\pi$
$674$	0	0
$675$	1.43845	0.0553659
$676$	0	0
$677$	23.9309	0.919738	0.459869	−	0.887987i	$-0.347896\pi$
$677$	23.9309	0.919738	0.459869	+	0.887987i	$0.347896\pi$
$678$	0	0
$679$	14.8078	0.568270
$680$	0	0
$681$	−60.6695	−2.32486
$682$	0	0
$683$	−42.7386	−1.63535	−0.817674	−	0.575681i	$-0.804737\pi$
$683$	−42.7386	−1.63535	−0.817674	+	0.575681i	$0.804737\pi$
$684$	0	0
$685$	8.87689	0.339169
$686$	0	0
$687$	−48.9848	−1.86889
$688$	0	0
$689$	14.2462	0.542737
$690$	0	0
$691$	−8.49242	−0.323067	−0.161533	−	0.986867i	$-0.551644\pi$
$691$	−8.49242	−0.323067	−0.161533	+	0.986867i	$0.551644\pi$
$692$	0	0
$693$	9.12311	0.346558
$694$	0	0
$695$	−6.87689	−0.260855
$696$	0	0
$697$	14.2462	0.539614
$698$	0	0
$699$	−8.00000	−0.302588
$700$	0	0
$701$	−0.0691303	−0.00261102	−0.00130551	−	0.999999i	$-0.500416\pi$
$701$	−0.0691303	−0.00261102	−0.00130551	+	0.999999i	$0.500416\pi$
$702$	0	0
$703$	6.73863	0.254152
$704$	0	0
$705$	−9.43845	−0.355472
$706$	0	0
$707$	0.246211	0.00925973
$708$	0	0
$709$	18.1771	0.682655	0.341327	−	0.939945i	$-0.389124\pi$
$709$	18.1771	0.682655	0.341327	+	0.939945i	$0.389124\pi$
$710$	0	0
$711$	−23.3693	−0.876418
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−11.6847	−0.436981
$716$	0	0
$717$	−2.06913	−0.0772731
$718$	0	0
$719$	49.6155	1.85035	0.925173	−	0.379544i	$-0.123919\pi$
$719$	49.6155	1.85035	0.925173	+	0.379544i	$0.123919\pi$
$720$	0	0
$721$	−1.43845	−0.0535706
$722$	0	0
$723$	31.3693	1.16664
$724$	0	0
$725$	5.68466	0.211123
$726$	0	0
$727$	19.5076	0.723496	0.361748	−	0.932276i	$-0.382180\pi$
$727$	19.5076	0.723496	0.361748	+	0.932276i	$0.382180\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	41.6155	1.53921
$732$	0	0
$733$	5.68466	0.209968	0.104984	−	0.994474i	$-0.466521\pi$
$733$	5.68466	0.209968	0.104984	+	0.994474i	$0.466521\pi$
$734$	0	0
$735$	−2.56155	−0.0944843
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	−6.06913	−0.223257	−0.111628	−	0.993750i	$-0.535607\pi$
$739$	−6.06913	−0.223257	−0.111628	+	0.993750i	$0.535607\pi$
$740$	0	0
$741$	13.1231	0.482089
$742$	0	0
$743$	32.9848	1.21010	0.605048	−	0.796189i	$-0.293154\pi$
$743$	32.9848	1.21010	0.605048	+	0.796189i	$0.293154\pi$
$744$	0	0
$745$	−4.24621	−0.155569
$746$	0	0
$747$	−14.2462	−0.521242
$748$	0	0
$749$	−11.3693	−0.415426
$750$	0	0
$751$	45.9309	1.67604	0.838021	−	0.545639i	$-0.183713\pi$
$751$	45.9309	1.67604	0.838021	+	0.545639i	$0.183713\pi$
$752$	0	0
$753$	43.8617	1.59841
$754$	0	0
$755$	−21.9309	−0.798146
$756$	0	0
$757$	−14.6307	−0.531761	−0.265881	−	0.964006i	$-0.585663\pi$
$757$	−14.6307	−0.531761	−0.265881	+	0.964006i	$0.585663\pi$
$758$	0	0
$759$	33.6155	1.22017
$760$	0	0
$761$	31.7538	1.15107	0.575537	−	0.817776i	$-0.304793\pi$
$761$	31.7538	1.15107	0.575537	+	0.817776i	$0.304793\pi$
$762$	0	0
$763$	17.6847	0.640228
$764$	0	0
$765$	16.2462	0.587383
$766$	0	0
$767$	−18.2462	−0.658833
$768$	0	0
$769$	−9.50758	−0.342852	−0.171426	−	0.985197i	$-0.554837\pi$
$769$	−9.50758	−0.342852	−0.171426	+	0.985197i	$0.554837\pi$
$770$	0	0
$771$	57.6155	2.07497
$772$	0	0
$773$	−8.06913	−0.290226	−0.145113	−	0.989415i	$-0.546355\pi$
$773$	−8.06913	−0.290226	−0.145113	+	0.989415i	$0.546355\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	15.3693	0.551371
$778$	0	0
$779$	3.50758	0.125672
$780$	0	0
$781$	−20.4924	−0.733277
$782$	0	0
$783$	8.17708	0.292225
$784$	0	0
$785$	3.75379	0.133978
$786$	0	0
$787$	3.82292	0.136272	0.0681362	−	0.997676i	$-0.478295\pi$
$787$	3.82292	0.136272	0.0681362	+	0.997676i	$0.478295\pi$
$788$	0	0
$789$	−54.1080	−1.92629
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	−42.7386	−1.51769
$794$	0	0
$795$	8.00000	0.283731
$796$	0	0
$797$	13.0540	0.462396	0.231198	−	0.972907i	$-0.425736\pi$
$797$	13.0540	0.462396	0.231198	+	0.972907i	$0.425736\pi$
$798$	0	0
$799$	−16.8078	−0.594616
$800$	0	0
$801$	25.3693	0.896381
$802$	0	0
$803$	−10.8769	−0.383837
$804$	0	0
$805$	−5.12311	−0.180566
$806$	0	0
$807$	−73.6155	−2.59139
$808$	0	0
$809$	−53.5464	−1.88259	−0.941296	−	0.337584i	$-0.890390\pi$
$809$	−53.5464	−1.88259	−0.941296	+	0.337584i	$0.890390\pi$
$810$	0	0
$811$	21.6155	0.759024	0.379512	−	0.925187i	$-0.376092\pi$
$811$	21.6155	0.759024	0.379512	+	0.925187i	$0.376092\pi$
$812$	0	0
$813$	−40.9848	−1.43740
$814$	0	0
$815$	1.12311	0.0393407
$816$	0	0
$817$	10.2462	0.358470
$818$	0	0
$819$	16.2462	0.567689
$820$	0	0
$821$	−40.4233	−1.41078	−0.705391	−	0.708818i	$-0.749229\pi$
$821$	−40.4233	−1.41078	−0.705391	+	0.708818i	$0.749229\pi$
$822$	0	0
$823$	−3.50758	−0.122266	−0.0611332	−	0.998130i	$-0.519471\pi$
$823$	−3.50758	−0.122266	−0.0611332	+	0.998130i	$0.519471\pi$
$824$	0	0
$825$	−6.56155	−0.228444
$826$	0	0
$827$	−19.3693	−0.673537	−0.336769	−	0.941587i	$-0.609334\pi$
$827$	−19.3693	−0.673537	−0.336769	+	0.941587i	$0.609334\pi$
$828$	0	0
$829$	−43.1231	−1.49773	−0.748864	−	0.662724i	$-0.769400\pi$
$829$	−43.1231	−1.49773	−0.748864	+	0.662724i	$0.769400\pi$
$830$	0	0
$831$	−41.6155	−1.44363
$832$	0	0
$833$	−4.56155	−0.158048
$834$	0	0
$835$	−21.9309	−0.758949
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−37.1231	−1.28163	−0.640816	−	0.767695i	$-0.721404\pi$
$839$	−37.1231	−1.28163	−0.640816	+	0.767695i	$0.721404\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	0	0
$843$	42.4233	1.46114
$844$	0	0
$845$	−7.80776	−0.268595
$846$	0	0
$847$	4.43845	0.152507
$848$	0	0
$849$	60.6695	2.08217
$850$	0	0
$851$	30.7386	1.05371
$852$	0	0
$853$	56.7386	1.94269	0.971347	−	0.237666i	$-0.0763824\pi$
$853$	56.7386	1.94269	0.971347	+	0.237666i	$0.0763824\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	−32.2462	−1.10151	−0.550755	−	0.834667i	$-0.685660\pi$
$857$	−32.2462	−1.10151	−0.550755	+	0.834667i	$0.685660\pi$
$858$	0	0
$859$	−16.4924	−0.562714	−0.281357	−	0.959603i	$-0.590785\pi$
$859$	−16.4924	−0.562714	−0.281357	+	0.959603i	$0.590785\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	0	0
$863$	−42.2462	−1.43808	−0.719039	−	0.694970i	$-0.755418\pi$
$863$	−42.2462	−1.43808	−0.719039	+	0.694970i	$0.755418\pi$
$864$	0	0
$865$	−8.56155	−0.291102
$866$	0	0
$867$	9.75379	0.331256
$868$	0	0
$869$	16.8078	0.570164
$870$	0	0
$871$	−28.4924	−0.965429
$872$	0	0
$873$	−52.7386	−1.78493
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	23.7538	0.802108	0.401054	−	0.916054i	$-0.368644\pi$
$877$	23.7538	0.802108	0.401054	+	0.916054i	$0.368644\pi$
$878$	0	0
$879$	−24.8078	−0.836745
$880$	0	0
$881$	45.8617	1.54512	0.772561	−	0.634941i	$-0.218976\pi$
$881$	45.8617	1.54512	0.772561	+	0.634941i	$0.218976\pi$
$882$	0	0
$883$	−24.4924	−0.824236	−0.412118	−	0.911131i	$-0.635211\pi$
$883$	−24.4924	−0.824236	−0.412118	+	0.911131i	$0.635211\pi$
$884$	0	0
$885$	−10.2462	−0.344423
$886$	0	0
$887$	−12.4924	−0.419454	−0.209727	−	0.977760i	$-0.567258\pi$
$887$	−12.4924	−0.419454	−0.209727	+	0.977760i	$0.567258\pi$
$888$	0	0
$889$	−10.2462	−0.343647
$890$	0	0
$891$	17.9309	0.600707
$892$	0	0
$893$	−4.13826	−0.138482
$894$	0	0
$895$	20.0000	0.668526
$896$	0	0
$897$	59.8617	1.99873
$898$	0	0
$899$	0	0
$900$	0	0
$901$	14.2462	0.474610
$902$	0	0
$903$	23.3693	0.777682
$904$	0	0
$905$	23.6155	0.785007
$906$	0	0
$907$	−50.1080	−1.66381	−0.831904	−	0.554920i	$-0.812749\pi$
$907$	−50.1080	−1.66381	−0.831904	+	0.554920i	$0.812749\pi$
$908$	0	0
$909$	−0.876894	−0.0290848
$910$	0	0
$911$	4.49242	0.148841	0.0744203	−	0.997227i	$-0.476289\pi$
$911$	4.49242	0.148841	0.0744203	+	0.997227i	$0.476289\pi$
$912$	0	0
$913$	10.2462	0.339100
$914$	0	0
$915$	−24.0000	−0.793416
$916$	0	0
$917$	−9.12311	−0.301271
$918$	0	0
$919$	−13.3002	−0.438733	−0.219366	−	0.975643i	$-0.570399\pi$
$919$	−13.3002	−0.438733	−0.219366	+	0.975643i	$0.570399\pi$
$920$	0	0
$921$	81.1619	2.67438
$922$	0	0
$923$	−36.4924	−1.20116
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	5.12311	0.168265
$928$	0	0
$929$	−52.1080	−1.70961	−0.854803	−	0.518952i	$-0.826322\pi$
$929$	−52.1080	−1.70961	−0.854803	+	0.518952i	$0.826322\pi$
$930$	0	0
$931$	−1.12311	−0.0368083
$932$	0	0
$933$	−24.6307	−0.806372
$934$	0	0
$935$	−11.6847	−0.382129
$936$	0	0
$937$	22.6695	0.740580	0.370290	−	0.928916i	$-0.379258\pi$
$937$	22.6695	0.740580	0.370290	+	0.928916i	$0.379258\pi$
$938$	0	0
$939$	80.1771	2.61648
$940$	0	0
$941$	13.8617	0.451880	0.225940	−	0.974141i	$-0.427455\pi$
$941$	13.8617	0.451880	0.225940	+	0.974141i	$0.427455\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	0	0
$945$	1.43845	0.0467927
$946$	0	0
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	−19.3693	−0.628755
$950$	0	0
$951$	57.6155	1.86831
$952$	0	0
$953$	−24.8769	−0.805842	−0.402921	−	0.915235i	$-0.632005\pi$
$953$	−24.8769	−0.805842	−0.402921	+	0.915235i	$0.632005\pi$
$954$	0	0
$955$	9.43845	0.305421
$956$	0	0
$957$	−37.3002	−1.20574
$958$	0	0
$959$	8.87689	0.286650
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	40.4924	1.30485
$964$	0	0
$965$	5.36932	0.172844
$966$	0	0
$967$	−26.8769	−0.864303	−0.432151	−	0.901801i	$-0.642245\pi$
$967$	−26.8769	−0.864303	−0.432151	+	0.901801i	$0.642245\pi$
$968$	0	0
$969$	13.1231	0.421575
$970$	0	0
$971$	49.4773	1.58780	0.793901	−	0.608048i	$-0.208047\pi$
$971$	49.4773	1.58780	0.793901	+	0.608048i	$0.208047\pi$
$972$	0	0
$973$	−6.87689	−0.220463
$974$	0	0
$975$	−11.6847	−0.374209
$976$	0	0
$977$	−49.2311	−1.57504	−0.787521	−	0.616288i	$-0.788636\pi$
$977$	−49.2311	−1.57504	−0.787521	+	0.616288i	$0.788636\pi$
$978$	0	0
$979$	−18.2462	−0.583151
$980$	0	0
$981$	−62.9848	−2.01095
$982$	0	0
$983$	−10.4233	−0.332451	−0.166226	−	0.986088i	$-0.553158\pi$
$983$	−10.4233	−0.332451	−0.166226	+	0.986088i	$0.553158\pi$
$984$	0	0
$985$	−7.12311	−0.226961
$986$	0	0
$987$	−9.43845	−0.300429
$988$	0	0
$989$	46.7386	1.48620
$990$	0	0
$991$	−20.4924	−0.650963	−0.325482	−	0.945548i	$-0.605526\pi$
$991$	−20.4924	−0.650963	−0.325482	+	0.945548i	$0.605526\pi$
$992$	0	0
$993$	−30.7386	−0.975461
$994$	0	0
$995$	18.2462	0.578444
$996$	0	0
$997$	−9.68466	−0.306716	−0.153358	−	0.988171i	$-0.549009\pi$
$997$	−9.68466	−0.306716	−0.153358	+	0.988171i	$0.549009\pi$
$998$	0	0
$999$	−8.63068	−0.273063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.a.bh.1.2		2
4.3	odd	2		2240.2.a.bd.1.1		2
8.3	odd	2		560.2.a.i.1.2		2
8.5	even	2		35.2.a.b.1.2	✓	2
24.5	odd	2		315.2.a.e.1.1		2
24.11	even	2		5040.2.a.bt.1.2		2
40.3	even	4		2800.2.g.t.449.4		4
40.13	odd	4		175.2.b.b.99.2		4
40.19	odd	2		2800.2.a.bi.1.1		2
40.27	even	4		2800.2.g.t.449.1		4
40.29	even	2		175.2.a.f.1.1		2
40.37	odd	4		175.2.b.b.99.3		4
56.5	odd	6		245.2.e.h.116.1		4
56.13	odd	2		245.2.a.d.1.2		2
56.27	even	2		3920.2.a.bs.1.1		2
56.37	even	6		245.2.e.i.116.1		4
56.45	odd	6		245.2.e.h.226.1		4
56.53	even	6		245.2.e.i.226.1		4
88.21	odd	2		4235.2.a.m.1.1		2
104.77	even	2		5915.2.a.l.1.1		2
120.29	odd	2		1575.2.a.p.1.2		2
120.53	even	4		1575.2.d.e.1324.3		4
120.77	even	4		1575.2.d.e.1324.2		4
168.125	even	2		2205.2.a.x.1.1		2
280.13	even	4		1225.2.b.f.99.2		4
280.69	odd	2		1225.2.a.s.1.1		2
280.237	even	4		1225.2.b.f.99.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.a.b.1.2	✓	2	8.5	even	2
175.2.a.f.1.1		2	40.29	even	2
175.2.b.b.99.2		4	40.13	odd	4
175.2.b.b.99.3		4	40.37	odd	4
245.2.a.d.1.2		2	56.13	odd	2
245.2.e.h.116.1		4	56.5	odd	6
245.2.e.h.226.1		4	56.45	odd	6
245.2.e.i.116.1		4	56.37	even	6
245.2.e.i.226.1		4	56.53	even	6
315.2.a.e.1.1		2	24.5	odd	2
560.2.a.i.1.2		2	8.3	odd	2
1225.2.a.s.1.1		2	280.69	odd	2
1225.2.b.f.99.2		4	280.13	even	4
1225.2.b.f.99.3		4	280.237	even	4
1575.2.a.p.1.2		2	120.29	odd	2
1575.2.d.e.1324.2		4	120.77	even	4
1575.2.d.e.1324.3		4	120.53	even	4
2205.2.a.x.1.1		2	168.125	even	2
2240.2.a.bd.1.1		2	4.3	odd	2
2240.2.a.bh.1.2		2	1.1	even	1	trivial
2800.2.a.bi.1.1		2	40.19	odd	2
2800.2.g.t.449.1		4	40.27	even	4
2800.2.g.t.449.4		4	40.3	even	4
3920.2.a.bs.1.1		2	56.27	even	2
4235.2.a.m.1.1		2	88.21	odd	2
5040.2.a.bt.1.2		2	24.11	even	2
5915.2.a.l.1.1		2	104.77	even	2

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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.56155 q^{9} -2.56155 q^{11} -4.56155 q^{13} -2.56155 q^{15} -4.56155 q^{17} -1.12311 q^{19} -2.56155 q^{21} -5.12311 q^{23} +1.00000 q^{25} +1.43845 q^{27} +5.68466 q^{29} -6.56155 q^{33} +1.00000 q^{35} -6.00000 q^{37} -11.6847 q^{39} -3.12311 q^{41} -9.12311 q^{43} -3.56155 q^{45} +3.68466 q^{47} +1.00000 q^{49} -11.6847 q^{51} -3.12311 q^{53} +2.56155 q^{55} -2.87689 q^{57} +4.00000 q^{59} +9.36932 q^{61} -3.56155 q^{63} +4.56155 q^{65} +6.24621 q^{67} -13.1231 q^{69} +8.00000 q^{71} +4.24621 q^{73} +2.56155 q^{75} +2.56155 q^{77} -6.56155 q^{79} -7.00000 q^{81} -4.00000 q^{83} +4.56155 q^{85} +14.5616 q^{87} +7.12311 q^{89} +4.56155 q^{91} +1.12311 q^{95} -14.8078 q^{97} -9.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 2 q^{5} - 2 q^{7} + 3 q^{9} - q^{11} - 5 q^{13} - q^{15} - 5 q^{17} + 6 q^{19} - q^{21} - 2 q^{23} + 2 q^{25} + 7 q^{27} - q^{29} - 9 q^{33} + 2 q^{35} - 12 q^{37} - 11 q^{39} + 2 q^{41}+ \cdots - 10 q^{99}+O(q^{100}) \) 2 * q + q^3 - 2 * q^5 - 2 * q^7 + 3 * q^9 - q^11 - 5 * q^13 - q^15 - 5 * q^17 + 6 * q^19 - q^21 - 2 * q^23 + 2 * q^25 + 7 * q^27 - q^29 - 9 * q^33 + 2 * q^35 - 12 * q^37 - 11 * q^39 + 2 * q^41 - 10 * q^43 - 3 * q^45 - 5 * q^47 + 2 * q^49 - 11 * q^51 + 2 * q^53 + q^55 - 14 * q^57 + 8 * q^59 - 6 * q^61 - 3 * q^63 + 5 * q^65 - 4 * q^67 - 18 * q^69 + 16 * q^71 - 8 * q^73 + q^75 + q^77 - 9 * q^79 - 14 * q^81 - 8 * q^83 + 5 * q^85 + 25 * q^87 + 6 * q^89 + 5 * q^91 - 6 * q^95 - 9 * q^97 - 10 * q^99

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