LMFDB - Embedded newform 756.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,2,Mod(1,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("756.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	756.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:	$6.03669039281$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	756.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.60555 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+3.60555 q^{5} -1.00000 q^{7} -3.60555 q^{11} +6.00000 q^{13} +7.21110 q^{17} -1.00000 q^{19} -3.60555 q^{23} +8.00000 q^{25} -7.21110 q^{29} +9.00000 q^{31} -3.60555 q^{35} -1.00000 q^{37} -10.8167 q^{41} +8.00000 q^{43} +1.00000 q^{49} -13.0000 q^{55} +14.4222 q^{59} +21.6333 q^{65} -2.00000 q^{67} -3.60555 q^{71} +4.00000 q^{73} +3.60555 q^{77} +7.21110 q^{83} +26.0000 q^{85} -10.8167 q^{89} -6.00000 q^{91} -3.60555 q^{95} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^7	$ 2 q - 2 q^{7} + 12 q^{13} - 2 q^{19} + 16 q^{25} + 18 q^{31} - 2 q^{37} + 16 q^{43} + 2 q^{49} - 26 q^{55} - 4 q^{67} + 8 q^{73} + 52 q^{85} - 12 q^{91} - 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 + 12 * q^13 - 2 * q^19 + 16 * q^25 + 18 * q^31 - 2 * q^37 + 16 * q^43 + 2 * q^49 - 26 * q^55 - 4 * q^67 + 8 * q^73 + 52 * q^85 - 12 * q^91 - 16 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.60555	1.61245	0.806226	−	0.591608i	$-0.201507\pi$
$5$	3.60555	1.61245	0.806226	+	0.591608i	$0.201507\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.60555	−1.08711	−0.543557	−	0.839372i	$-0.682923\pi$
$11$	−3.60555	−1.08711	−0.543557	+	0.839372i	$0.682923\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.21110	1.74895	0.874475	−	0.485071i	$-0.161206\pi$
$17$	7.21110	1.74895	0.874475	+	0.485071i	$0.161206\pi$
$18$	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
$21$	0	0
$22$	0	0
$23$	−3.60555	−0.751809	−0.375905	−	0.926658i	$-0.622668\pi$
$23$	−3.60555	−0.751809	−0.375905	+	0.926658i	$0.622668\pi$
$24$	0	0
$25$	8.00000	1.60000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.21110	−1.33907	−0.669534	−	0.742781i	$-0.733506\pi$
$29$	−7.21110	−1.33907	−0.669534	+	0.742781i	$0.733506\pi$
$30$	0	0
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.60555	−0.609449
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.8167	−1.68928	−0.844639	−	0.535337i	$-0.820185\pi$
$41$	−10.8167	−1.68928	−0.844639	+	0.535337i	$0.820185\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−13.0000	−1.75292
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.4222	1.87761	0.938806	−	0.344447i	$-0.111934\pi$
$59$	14.4222	1.87761	0.938806	+	0.344447i	$0.111934\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	21.6333	2.68328
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.60555	−0.427900	−0.213950	−	0.976845i	$-0.568633\pi$
$71$	−3.60555	−0.427900	−0.213950	+	0.976845i	$0.568633\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.60555	0.410891
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.21110	0.791521	0.395761	−	0.918354i	$-0.370481\pi$
$83$	7.21110	0.791521	0.395761	+	0.918354i	$0.370481\pi$
$84$	0	0
$85$	26.0000	2.82010
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.8167	−1.14656	−0.573282	−	0.819358i	$-0.694330\pi$
$89$	−10.8167	−1.14656	−0.573282	+	0.819358i	$0.694330\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.60555	−0.369922
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.21110	−0.717532	−0.358766	−	0.933428i	$-0.616802\pi$
$101$	−7.21110	−0.717532	−0.358766	+	0.933428i	$0.616802\pi$
$102$	0	0
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.21110	−0.678363	−0.339182	−	0.940721i	$-0.610150\pi$
$113$	−7.21110	−0.678363	−0.339182	+	0.940721i	$0.610150\pi$
$114$	0	0
$115$	−13.0000	−1.21226
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.21110	−0.661041
$120$	0	0
$121$	2.00000	0.181818
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.8167	0.967471
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.4222	1.26007	0.630037	−	0.776565i	$-0.283040\pi$
$131$	14.4222	1.26007	0.630037	+	0.776565i	$0.283040\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.21110	−0.616086	−0.308043	−	0.951372i	$-0.599674\pi$
$137$	−7.21110	−0.616086	−0.308043	+	0.951372i	$0.599674\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−21.6333	−1.80907
$144$	0	0
$145$	−26.0000	−2.15918
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−21.6333	−1.77227	−0.886135	−	0.463428i	$-0.846619\pi$
$149$	−21.6333	−1.77227	−0.886135	+	0.463428i	$0.846619\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	32.4500	2.60644
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.60555	0.284157
$162$	0	0
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.21110	−0.558012	−0.279006	−	0.960289i	$-0.590005\pi$
$167$	−7.21110	−0.558012	−0.279006	+	0.960289i	$0.590005\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.0278	−1.37062	−0.685312	−	0.728249i	$-0.740334\pi$
$173$	−18.0278	−1.37062	−0.685312	+	0.728249i	$0.740334\pi$
$174$	0	0
$175$	−8.00000	−0.604743
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.60555	−0.265085
$186$	0	0
$187$	−26.0000	−1.90131
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.8167	−0.782666	−0.391333	−	0.920249i	$-0.627986\pi$
$191$	−10.8167	−0.782666	−0.391333	+	0.920249i	$0.627986\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.21110	0.513770	0.256885	−	0.966442i	$-0.417304\pi$
$197$	7.21110	0.513770	0.256885	+	0.966442i	$0.417304\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.21110	0.506120
$204$	0	0
$205$	−39.0000	−2.72388
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.60555	0.249401
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	28.8444	1.96717
$216$	0	0
$217$	−9.00000	−0.610960
$218$	0	0
$219$	0	0
$220$	0	0
$221$	43.2666	2.91043
$222$	0	0
$223$	11.0000	0.736614	0.368307	−	0.929704i	$-0.379937\pi$
$223$	11.0000	0.736614	0.368307	+	0.929704i	$0.379937\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.6333	−1.43585	−0.717927	−	0.696119i	$-0.754909\pi$
$227$	−21.6333	−1.43585	−0.717927	+	0.696119i	$0.754909\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.21110	0.472415	0.236208	−	0.971703i	$-0.424095\pi$
$233$	7.21110	0.472415	0.236208	+	0.971703i	$0.424095\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.60555	0.230350
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	13.0000	0.817303
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.8167	−0.674724	−0.337362	−	0.941375i	$-0.609535\pi$
$257$	−10.8167	−0.674724	−0.337362	+	0.941375i	$0.609535\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	0	0
$261$	0	0
$262$	0	0
$263$	10.8167	0.666983	0.333492	−	0.942753i	$-0.391773\pi$
$263$	10.8167	0.666983	0.333492	+	0.942753i	$0.391773\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.60555	0.219834	0.109917	−	0.993941i	$-0.464941\pi$
$269$	3.60555	0.219834	0.109917	+	0.993941i	$0.464941\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−28.8444	−1.73938
$276$	0	0
$277$	−13.0000	−0.781094	−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000	−0.781094	−0.390547	+	0.920583i	$0.627714\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−14.4222	−0.860357	−0.430178	−	0.902744i	$-0.641549\pi$
$281$	−14.4222	−0.860357	−0.430178	+	0.902744i	$0.641549\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.8167	0.638487
$288$	0	0
$289$	35.0000	2.05882
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.6333	1.26383	0.631916	−	0.775037i	$-0.282269\pi$
$293$	21.6333	1.26383	0.631916	+	0.775037i	$0.282269\pi$
$294$	0	0
$295$	52.0000	3.02756
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−21.6333	−1.25109
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−29.0000	−1.65512	−0.827559	−	0.561379i	$-0.810271\pi$
$307$	−29.0000	−1.65512	−0.827559	+	0.561379i	$0.810271\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.21110	−0.408904	−0.204452	−	0.978877i	$-0.565541\pi$
$311$	−7.21110	−0.408904	−0.204452	+	0.978877i	$0.565541\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	26.0000	1.45572
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7.21110	−0.401236
$324$	0	0
$325$	48.0000	2.66256
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.21110	−0.393985
$336$	0	0
$337$	3.00000	0.163420	0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000	0.163420	0.0817102	+	0.996656i	$0.473962\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−32.4500	−1.75726
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−32.4500	−1.74201	−0.871003	−	0.491278i	$-0.836530\pi$
$347$	−32.4500	−1.74201	−0.871003	+	0.491278i	$0.836530\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.8167	0.575712	0.287856	−	0.957674i	$-0.407057\pi$
$353$	10.8167	0.575712	0.287856	+	0.957674i	$0.407057\pi$
$354$	0	0
$355$	−13.0000	−0.689968
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.4222	0.761175	0.380587	−	0.924745i	$-0.375722\pi$
$359$	14.4222	0.761175	0.380587	+	0.924745i	$0.375722\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.4222	0.754893
$366$	0	0
$367$	−31.0000	−1.61819	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	−31.0000	−1.61819	−0.809093	+	0.587680i	$0.800041\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.00000	0.0517780	0.0258890	−	0.999665i	$-0.491758\pi$
$373$	1.00000	0.0517780	0.0258890	+	0.999665i	$0.491758\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−43.2666	−2.22834
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.21110	0.368470	0.184235	−	0.982882i	$-0.441019\pi$
$383$	7.21110	0.368470	0.184235	+	0.982882i	$0.441019\pi$
$384$	0	0
$385$	13.0000	0.662541
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14.4222	0.731235	0.365617	−	0.930765i	$-0.380858\pi$
$389$	14.4222	0.731235	0.365617	+	0.930765i	$0.380858\pi$
$390$	0	0
$391$	−26.0000	−1.31488
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.6333	1.08032	0.540158	−	0.841564i	$-0.318365\pi$
$401$	21.6333	1.08032	0.540158	+	0.841564i	$0.318365\pi$
$402$	0	0
$403$	54.0000	2.68993
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.60555	0.178721
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.4222	−0.709670
$414$	0	0
$415$	26.0000	1.27629
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.6333	1.05686	0.528428	−	0.848978i	$-0.322782\pi$
$419$	21.6333	1.05686	0.528428	+	0.848978i	$0.322782\pi$
$420$	0	0
$421$	15.0000	0.731055	0.365528	−	0.930800i	$-0.380889\pi$
$421$	15.0000	0.731055	0.365528	+	0.930800i	$0.380889\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	57.6888	2.79832
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.8167	−0.521020	−0.260510	−	0.965471i	$-0.583891\pi$
$431$	−10.8167	−0.521020	−0.260510	+	0.965471i	$0.583891\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.60555	0.172477
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.0278	0.856525	0.428262	−	0.903654i	$-0.359126\pi$
$443$	18.0278	0.856525	0.428262	+	0.903654i	$0.359126\pi$
$444$	0	0
$445$	−39.0000	−1.84878
$446$	0	0
$447$	0	0
$448$	0	0
$449$	36.0555	1.70156	0.850782	−	0.525518i	$-0.176128\pi$
$449$	36.0555	1.70156	0.850782	+	0.525518i	$0.176128\pi$
$450$	0	0
$451$	39.0000	1.83644
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−21.6333	−1.01419
$456$	0	0
$457$	−23.0000	−1.07589	−0.537947	−	0.842978i	$-0.680800\pi$
$457$	−23.0000	−1.07589	−0.537947	+	0.842978i	$0.680800\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	25.2389	1.17549	0.587745	−	0.809046i	$-0.300016\pi$
$461$	25.2389	1.17549	0.587745	+	0.809046i	$0.300016\pi$
$462$	0	0
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.4222	0.667380	0.333690	−	0.942683i	$-0.391706\pi$
$467$	14.4222	0.667380	0.333690	+	0.942683i	$0.391706\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−28.8444	−1.32627
$474$	0	0
$475$	−8.00000	−0.367065
$476$	0	0
$477$	0	0
$478$	0	0
$479$	28.8444	1.31793	0.658967	−	0.752172i	$-0.270994\pi$
$479$	28.8444	1.31793	0.658967	+	0.752172i	$0.270994\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−28.8444	−1.30976
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−32.4500	−1.46445	−0.732223	−	0.681065i	$-0.761517\pi$
$491$	−32.4500	−1.46445	−0.732223	+	0.681065i	$0.761517\pi$
$492$	0	0
$493$	−52.0000	−2.34196
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.60555	0.161731
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	21.6333	0.964582	0.482291	−	0.876011i	$-0.339805\pi$
$503$	21.6333	0.964582	0.482291	+	0.876011i	$0.339805\pi$
$504$	0	0
$505$	−26.0000	−1.15698
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−21.6333	−0.958880	−0.479440	−	0.877575i	$-0.659160\pi$
$509$	−21.6333	−0.958880	−0.479440	+	0.877575i	$0.659160\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−25.2389	−1.11216
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.8167	0.473886	0.236943	−	0.971524i	$-0.423855\pi$
$521$	10.8167	0.473886	0.236943	+	0.971524i	$0.423855\pi$
$522$	0	0
$523$	35.0000	1.53044	0.765222	−	0.643767i	$-0.222629\pi$
$523$	35.0000	1.53044	0.765222	+	0.643767i	$0.222629\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	64.8999	2.82709
$528$	0	0
$529$	−10.0000	−0.434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−64.8999	−2.81113
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.60555	−0.155302
$540$	0	0
$541$	−9.00000	−0.386940	−0.193470	−	0.981106i	$-0.561974\pi$
$541$	−9.00000	−0.386940	−0.193470	+	0.981106i	$0.561974\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−39.6611	−1.69889
$546$	0	0
$547$	−24.0000	−1.02617	−0.513083	−	0.858339i	$-0.671497\pi$
$547$	−24.0000	−1.02617	−0.513083	+	0.858339i	$0.671497\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.21110	0.307203
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.21110	−0.305544	−0.152772	−	0.988261i	$-0.548820\pi$
$557$	−7.21110	−0.305544	−0.152772	+	0.988261i	$0.548820\pi$
$558$	0	0
$559$	48.0000	2.03018
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.21110	0.303912	0.151956	−	0.988387i	$-0.451443\pi$
$563$	7.21110	0.303912	0.151956	+	0.988387i	$0.451443\pi$
$564$	0	0
$565$	−26.0000	−1.09383
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.6333	−0.906915	−0.453458	−	0.891278i	$-0.649810\pi$
$569$	−21.6333	−0.906915	−0.453458	+	0.891278i	$0.649810\pi$
$570$	0	0
$571$	−18.0000	−0.753277	−0.376638	−	0.926360i	$-0.622920\pi$
$571$	−18.0000	−0.753277	−0.376638	+	0.926360i	$0.622920\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−28.8444	−1.20290
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7.21110	−0.299167
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−36.0555	−1.48817	−0.744085	−	0.668085i	$-0.767114\pi$
$587$	−36.0555	−1.48817	−0.744085	+	0.668085i	$0.767114\pi$
$588$	0	0
$589$	−9.00000	−0.370839
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10.8167	0.444187	0.222093	−	0.975025i	$-0.428711\pi$
$593$	10.8167	0.444187	0.222093	+	0.975025i	$0.428711\pi$
$594$	0	0
$595$	−26.0000	−1.06590
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.4500	1.32587	0.662935	−	0.748677i	$-0.269311\pi$
$599$	32.4500	1.32587	0.662935	+	0.748677i	$0.269311\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.21110	0.293173
$606$	0	0
$607$	−36.0000	−1.46119	−0.730597	−	0.682808i	$-0.760758\pi$
$607$	−36.0000	−1.46119	−0.730597	+	0.682808i	$0.760758\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	15.0000	0.605844	0.302922	−	0.953015i	$-0.402038\pi$
$613$	15.0000	0.605844	0.302922	+	0.953015i	$0.402038\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	28.8444	1.16123	0.580616	−	0.814177i	$-0.302812\pi$
$617$	28.8444	1.16123	0.580616	+	0.814177i	$0.302812\pi$
$618$	0	0
$619$	−23.0000	−0.924448	−0.462224	−	0.886763i	$-0.652948\pi$
$619$	−23.0000	−0.924448	−0.462224	+	0.886763i	$0.652948\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.8167	0.433360
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.21110	−0.287525
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−21.6333	−0.858492
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−36.0555	−1.42411	−0.712054	−	0.702125i	$-0.752235\pi$
$641$	−36.0555	−1.42411	−0.712054	+	0.702125i	$0.752235\pi$
$642$	0	0
$643$	−1.00000	−0.0394362	−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000	−0.0394362	−0.0197181	+	0.999806i	$0.506277\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	43.2666	1.70099	0.850493	−	0.525986i	$-0.176304\pi$
$647$	43.2666	1.70099	0.850493	+	0.525986i	$0.176304\pi$
$648$	0	0
$649$	−52.0000	−2.04118
$650$	0	0
$651$	0	0
$652$	0	0
$653$	43.2666	1.69315	0.846577	−	0.532267i	$-0.178660\pi$
$653$	43.2666	1.69315	0.846577	+	0.532267i	$0.178660\pi$
$654$	0	0
$655$	52.0000	2.03181
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3.60555	−0.140452	−0.0702262	−	0.997531i	$-0.522372\pi$
$659$	−3.60555	−0.140452	−0.0702262	+	0.997531i	$0.522372\pi$
$660$	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
$663$	0	0
$664$	0	0
$665$	3.60555	0.139817
$666$	0	0
$667$	26.0000	1.00672
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.8167	0.415718	0.207859	−	0.978159i	$-0.433351\pi$
$677$	10.8167	0.415718	0.207859	+	0.978159i	$0.433351\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.4500	1.24166	0.620832	−	0.783944i	$-0.286795\pi$
$683$	32.4500	1.24166	0.620832	+	0.783944i	$0.286795\pi$
$684$	0	0
$685$	−26.0000	−0.993409
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−72.1110	−2.73533
$696$	0	0
$697$	−78.0000	−2.95446
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−7.21110	−0.272360	−0.136180	−	0.990684i	$-0.543482\pi$
$701$	−7.21110	−0.272360	−0.136180	+	0.990684i	$0.543482\pi$
$702$	0	0
$703$	1.00000	0.0377157
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.21110	0.271201
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−32.4500	−1.21526
$714$	0	0
$715$	−78.0000	−2.91703
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−43.2666	−1.61357	−0.806786	−	0.590843i	$-0.798795\pi$
$719$	−43.2666	−1.61357	−0.806786	+	0.590843i	$0.798795\pi$
$720$	0	0
$721$	7.00000	0.260694
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−57.6888	−2.14251
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	57.6888	2.13370
$732$	0	0
$733$	24.0000	0.886460	0.443230	−	0.896408i	$-0.353832\pi$
$733$	24.0000	0.886460	0.443230	+	0.896408i	$0.353832\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.21110	0.265624
$738$	0	0
$739$	−42.0000	−1.54499	−0.772497	−	0.635018i	$-0.780993\pi$
$739$	−42.0000	−1.54499	−0.772497	+	0.635018i	$0.780993\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.8167	0.396825	0.198412	−	0.980119i	$-0.436422\pi$
$743$	10.8167	0.396825	0.198412	+	0.980119i	$0.436422\pi$
$744$	0	0
$745$	−78.0000	−2.85770
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	36.0555	1.31219
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.21110	0.261402	0.130701	−	0.991422i	$-0.458277\pi$
$761$	7.21110	0.261402	0.130701	+	0.991422i	$0.458277\pi$
$762$	0	0
$763$	11.0000	0.398227
$764$	0	0
$765$	0	0
$766$	0	0
$767$	86.5332	3.12453
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	25.2389	0.907779	0.453889	−	0.891058i	$-0.350036\pi$
$773$	25.2389	0.907779	0.453889	+	0.891058i	$0.350036\pi$
$774$	0	0
$775$	72.0000	2.58632
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.8167	0.387547
$780$	0	0
$781$	13.0000	0.465177
$782$	0	0
$783$	0	0
$784$	0	0
$785$	50.4777	1.80163
$786$	0	0
$787$	−12.0000	−0.427754	−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000	−0.427754	−0.213877	+	0.976861i	$0.568609\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.21110	0.256397
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	54.0833	1.91573	0.957864	−	0.287223i	$-0.0927320\pi$
$797$	54.0833	1.91573	0.957864	+	0.287223i	$0.0927320\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−14.4222	−0.508949
$804$	0	0
$805$	13.0000	0.458190
$806$	0	0
$807$	0	0
$808$	0	0
$809$	36.0555	1.26764	0.633822	−	0.773479i	$-0.281485\pi$
$809$	36.0555	1.26764	0.633822	+	0.773479i	$0.281485\pi$
$810$	0	0
$811$	−35.0000	−1.22902	−0.614508	−	0.788911i	$-0.710645\pi$
$811$	−35.0000	−1.22902	−0.614508	+	0.788911i	$0.710645\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−79.3221	−2.77853
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14.4222	−0.503338	−0.251669	−	0.967813i	$-0.580979\pi$
$821$	−14.4222	−0.503338	−0.251669	+	0.967813i	$0.580979\pi$
$822$	0	0
$823$	34.0000	1.18517	0.592583	−	0.805510i	$-0.298108\pi$
$823$	34.0000	1.18517	0.592583	+	0.805510i	$0.298108\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−54.0833	−1.88066	−0.940330	−	0.340264i	$-0.889483\pi$
$827$	−54.0833	−1.88066	−0.940330	+	0.340264i	$0.889483\pi$
$828$	0	0
$829$	40.0000	1.38926	0.694629	−	0.719368i	$-0.255569\pi$
$829$	40.0000	1.38926	0.694629	+	0.719368i	$0.255569\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.21110	0.249850
$834$	0	0
$835$	−26.0000	−0.899767
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−36.0555	−1.24477	−0.622387	−	0.782709i	$-0.713837\pi$
$839$	−36.0555	−1.24477	−0.622387	+	0.782709i	$0.713837\pi$
$840$	0	0
$841$	23.0000	0.793103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	82.9277	2.85280
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.60555	0.123597
$852$	0	0
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−54.0833	−1.84745	−0.923725	−	0.383057i	$-0.874871\pi$
$857$	−54.0833	−1.84745	−0.923725	+	0.383057i	$0.874871\pi$
$858$	0	0
$859$	35.0000	1.19418	0.597092	−	0.802173i	$-0.296323\pi$
$859$	35.0000	1.19418	0.597092	+	0.802173i	$0.296323\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−28.8444	−0.981875	−0.490938	−	0.871195i	$-0.663346\pi$
$863$	−28.8444	−0.981875	−0.490938	+	0.871195i	$0.663346\pi$
$864$	0	0
$865$	−65.0000	−2.21007
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−10.8167	−0.365670
$876$	0	0
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10.8167	0.364422	0.182211	−	0.983259i	$-0.441675\pi$
$881$	10.8167	0.364422	0.182211	+	0.983259i	$0.441675\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−21.6333	−0.726375	−0.363188	−	0.931716i	$-0.618312\pi$
$887$	−21.6333	−0.726375	−0.363188	+	0.931716i	$0.618312\pi$
$888$	0	0
$889$	6.00000	0.201234
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−64.8999	−2.16453
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21.6333	−0.719115
$906$	0	0
$907$	−6.00000	−0.199227	−0.0996134	−	0.995026i	$-0.531761\pi$
$907$	−6.00000	−0.199227	−0.0996134	+	0.995026i	$0.531761\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	43.2666	1.43349	0.716743	−	0.697337i	$-0.245632\pi$
$911$	43.2666	1.43349	0.716743	+	0.697337i	$0.245632\pi$
$912$	0	0
$913$	−26.0000	−0.860474
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14.4222	−0.476263
$918$	0	0
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−21.6333	−0.712069
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21.6333	0.709766	0.354883	−	0.934911i	$-0.384521\pi$
$929$	21.6333	0.709766	0.354883	+	0.934911i	$0.384521\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−93.7443	−3.06577
$936$	0	0
$937$	−4.00000	−0.130674	−0.0653372	−	0.997863i	$-0.520812\pi$
$937$	−4.00000	−0.130674	−0.0653372	+	0.997863i	$0.520812\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−39.6611	−1.29291	−0.646457	−	0.762951i	$-0.723750\pi$
$941$	−39.6611	−1.29291	−0.646457	+	0.762951i	$0.723750\pi$
$942$	0	0
$943$	39.0000	1.27001
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.60555	0.117165	0.0585823	−	0.998283i	$-0.481342\pi$
$947$	3.60555	0.117165	0.0585823	+	0.998283i	$0.481342\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	0	0
$952$	0	0
$953$	7.21110	0.233591	0.116795	−	0.993156i	$-0.462738\pi$
$953$	7.21110	0.233591	0.116795	+	0.993156i	$0.462738\pi$
$954$	0	0
$955$	−39.0000	−1.26201
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.21110	0.232859
$960$	0	0
$961$	50.0000	1.61290
$962$	0	0
$963$	0	0
$964$	0	0
$965$	50.4777	1.62494
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−21.6333	−0.694246	−0.347123	−	0.937820i	$-0.612841\pi$
$971$	−21.6333	−0.694246	−0.347123	+	0.937820i	$0.612841\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	39.0000	1.24645
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	26.0000	0.828429
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−28.8444	−0.917199
$990$	0	0
$991$	−22.0000	−0.698853	−0.349427	−	0.936964i	$-0.613624\pi$
$991$	−22.0000	−0.698853	−0.349427	+	0.936964i	$0.613624\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	61.2944	1.94316
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.a.g.1.2	yes	2
3.2	odd	2	inner	756.2.a.g.1.1	✓	2
4.3	odd	2		3024.2.a.bj.1.2		2
7.6	odd	2		5292.2.a.o.1.1		2
9.2	odd	6		2268.2.j.q.757.2		4
9.4	even	3		2268.2.j.q.1513.1		4
9.5	odd	6		2268.2.j.q.1513.2		4
9.7	even	3		2268.2.j.q.757.1		4
12.11	even	2		3024.2.a.bj.1.1		2
21.20	even	2		5292.2.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.a.g.1.1	✓	2	3.2	odd	2	inner
756.2.a.g.1.2	yes	2	1.1	even	1	trivial
2268.2.j.q.757.1		4	9.7	even	3
2268.2.j.q.757.2		4	9.2	odd	6
2268.2.j.q.1513.1		4	9.4	even	3
2268.2.j.q.1513.2		4	9.5	odd	6
3024.2.a.bj.1.1		2	12.11	even	2
3024.2.a.bj.1.2		2	4.3	odd	2
5292.2.a.o.1.1		2	7.6	odd	2
5292.2.a.o.1.2		2	21.20	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.a (trivial)

\(f(q)\)	\(=\)	\(q+3.60555 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+3.60555 q^{5} -1.00000 q^{7} -3.60555 q^{11} +6.00000 q^{13} +7.21110 q^{17} -1.00000 q^{19} -3.60555 q^{23} +8.00000 q^{25} -7.21110 q^{29} +9.00000 q^{31} -3.60555 q^{35} -1.00000 q^{37} -10.8167 q^{41} +8.00000 q^{43} +1.00000 q^{49} -13.0000 q^{55} +14.4222 q^{59} +21.6333 q^{65} -2.00000 q^{67} -3.60555 q^{71} +4.00000 q^{73} +3.60555 q^{77} +7.21110 q^{83} +26.0000 q^{85} -10.8167 q^{89} -6.00000 q^{91} -3.60555 q^{95} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^7	\( 2 q - 2 q^{7} + 12 q^{13} - 2 q^{19} + 16 q^{25} + 18 q^{31} - 2 q^{37} + 16 q^{43} + 2 q^{49} - 26 q^{55} - 4 q^{67} + 8 q^{73} + 52 q^{85} - 12 q^{91} - 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 + 12 * q^13 - 2 * q^19 + 16 * q^25 + 18 * q^31 - 2 * q^37 + 16 * q^43 + 2 * q^49 - 26 * q^55 - 4 * q^67 + 8 * q^73 + 52 * q^85 - 12 * q^91 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more