LMFDB - Embedded newform 9025.2.a.cb.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9025,2,Mod(1,9025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9025.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,16,0,0,0,0,24,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.280944640000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 16x^{6} + 80x^{4} - 128x^{2} + 19 $ x^8 - 16x^6 + 80x^4 - 128*x^2 + 19
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.8
Root			$2.79913$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.79913 q^{2} -1.12228 q^{3} +5.83513 q^{4} -3.14142 q^{6} +10.7350 q^{8} -1.74048 q^{9} +2.92978 q^{11} -6.54867 q^{12} +3.08876 q^{13} +18.3785 q^{16} -4.87183 q^{18} +8.20083 q^{22} -12.0477 q^{24} +8.64583 q^{26} +5.32016 q^{27} +29.9737 q^{32} -3.28804 q^{33} -10.1559 q^{36} +9.15124 q^{37} -3.46646 q^{39} +17.0956 q^{44} -20.6259 q^{48} -7.00000 q^{49} +18.0233 q^{52} -13.6404 q^{53} +14.8918 q^{54} +1.11908 q^{61} +47.1432 q^{64} -9.20366 q^{66} -13.7060 q^{67} -18.6841 q^{72} +25.6155 q^{74} -9.70308 q^{78} -0.749299 q^{81} +31.4512 q^{88} -33.6390 q^{96} +15.8849 q^{97} -19.5939 q^{98} -5.09921 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} + 24 q^{9} + 32 q^{16} + 8 q^{24} + 24 q^{26} - 8 q^{36} + 72 q^{44} - 56 q^{49} + 88 q^{54} + 64 q^{64} + 104 q^{66} + 72 q^{81} - 120 q^{96} + 32 q^{99}+O(q^{100}) $ 8 * q + 16 * q^4 + 24 * q^9 + 32 * q^16 + 8 * q^24 + 24 * q^26 - 8 * q^36 + 72 * q^44 - 56 * q^49 + 88 * q^54 + 64 * q^64 + 104 * q^66 + 72 * q^81 - 120 * q^96 + 32 * 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 $

$2$	2.79913	1.97928	0.989642	−	0.143559i	$-0.0458545\pi$
$2$	2.79913	1.97928	0.989642	+	0.143559i	$0.0458545\pi$
$3$	−1.12228	−0.647951	−0.323976	−	0.946065i	$-0.605020\pi$
$3$	−1.12228	−0.647951	−0.323976	+	0.946065i	$0.605020\pi$
$4$	5.83513	2.91756
$5$	0	0
$5$	0	0
$6$	−3.14142	−1.28248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	10.7350	3.79540
$9$	−1.74048	−0.580159
$10$	0	0
$11$	2.92978	0.883361	0.441680	−	0.897172i	$-0.354382\pi$
$11$	2.92978	0.883361	0.441680	+	0.897172i	$0.354382\pi$
$12$	−6.54867	−1.89044
$13$	3.08876	0.856667	0.428333	−	0.903621i	$-0.359101\pi$
$13$	3.08876	0.856667	0.428333	+	0.903621i	$0.359101\pi$
$14$	0	0
$15$	0	0
$16$	18.3785	4.59461
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−4.87183	−1.14830
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0	0
$22$	8.20083	1.74842
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−12.0477	−2.45924
$25$	0	0
$26$	8.64583	1.69559
$27$	5.32016	1.02387
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	29.9737	5.29864
$33$	−3.28804	−0.572375
$34$	0	0
$35$	0	0
$36$	−10.1559	−1.69265
$37$	9.15124	1.50445	0.752227	−	0.658904i	$-0.228980\pi$
$37$	9.15124	1.50445	0.752227	+	0.658904i	$0.228980\pi$
$38$	0	0
$39$	−3.46646	−0.555078
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	17.0956	2.57726
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−20.6259	−2.97709
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	18.0233	2.49938
$53$	−13.6404	−1.87365	−0.936825	−	0.349799i	$-0.886250\pi$
$53$	−13.6404	−1.87365	−0.936825	+	0.349799i	$0.886250\pi$
$54$	14.8918	2.02652
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	1.11908	0.143283	0.0716414	−	0.997430i	$-0.477176\pi$
$61$	1.11908	0.143283	0.0716414	+	0.997430i	$0.477176\pi$
$62$	0	0
$63$	0	0
$64$	47.1432	5.89290
$65$	0	0
$66$	−9.20366	−1.13289
$67$	−13.7060	−1.67446	−0.837229	−	0.546853i	$-0.815826\pi$
$67$	−13.7060	−1.67446	−0.837229	+	0.546853i	$0.815826\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−18.6841	−2.20194
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	25.6155	2.97774
$75$	0	0
$76$	0	0
$77$	0	0
$78$	−9.70308	−1.09866
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−0.749299	−0.0832555
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	31.4512	3.35271
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−33.6390	−3.43326
$97$	15.8849	1.61287	0.806436	−	0.591322i	$-0.201394\pi$
$97$	15.8849	1.61287	0.806436	+	0.591322i	$0.201394\pi$
$98$	−19.5939	−1.97928
$99$	−5.09921	−0.512490
$100$	0	0
$101$	20.0810	1.99813	0.999067	−	0.0431977i	$-0.0137545\pi$
$101$	20.0810	1.99813	0.999067	+	0.0431977i	$0.0137545\pi$
$102$	0	0
$103$	19.8835	1.95918	0.979591	−	0.200999i	$-0.0644188\pi$
$103$	19.8835	1.95918	0.979591	+	0.200999i	$0.0644188\pi$
$104$	33.1579	3.25140
$105$	0	0
$106$	−38.1812	−3.70848
$107$	3.66801	0.354600	0.177300	−	0.984157i	$-0.443264\pi$
$107$	3.66801	0.354600	0.177300	+	0.984157i	$0.443264\pi$
$108$	31.0438	2.98719
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	−10.2703	−0.974813
$112$	0	0
$113$	−1.93025	−0.181583	−0.0907914	−	0.995870i	$-0.528940\pi$
$113$	−1.93025	−0.181583	−0.0907914	+	0.995870i	$0.528940\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.37591	−0.497003
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.41641	−0.219673
$122$	3.13244	0.283597
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−21.6693	−1.92284	−0.961421	−	0.275082i	$-0.911295\pi$
$127$	−21.6693	−1.92284	−0.961421	+	0.275082i	$0.911295\pi$
$128$	72.0127	6.36508
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−19.1861	−1.66994
$133$	0	0
$134$	−38.3649	−3.31423
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	21.8917	1.85683	0.928414	−	0.371546i	$-0.121172\pi$
$139$	21.8917	1.85683	0.928414	+	0.371546i	$0.121172\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.04937	0.756746
$144$	−31.9873	−2.66561
$145$	0	0
$146$	0	0
$147$	7.85599	0.647951
$148$	53.3986	4.38934
$149$	8.36188	0.685032	0.342516	−	0.939512i	$-0.388721\pi$
$149$	8.36188	0.685032	0.342516	+	0.939512i	$0.388721\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−20.2272	−1.61948
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	15.3084	1.21403
$160$	0	0
$161$	0	0
$162$	−2.09739	−0.164786
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.1802	1.32944	0.664721	−	0.747091i	$-0.268550\pi$
$167$	17.1802	1.32944	0.664721	+	0.747091i	$0.268550\pi$
$168$	0	0
$169$	−3.45959	−0.266122
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.1268	−0.998010	−0.499005	−	0.866599i	$-0.666301\pi$
$173$	−13.1268	−0.998010	−0.499005	+	0.866599i	$0.666301\pi$
$174$	0	0
$175$	0	0
$176$	53.8448	4.05870
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	−1.25592	−0.0928403
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.94427	−0.647185	−0.323592	−	0.946197i	$-0.604891\pi$
$191$	−8.94427	−0.647185	−0.323592	+	0.946197i	$0.604891\pi$
$192$	−52.9081	−3.81831
$193$	2.41753	0.174018	0.0870089	−	0.996208i	$-0.472269\pi$
$193$	2.41753	0.174018	0.0870089	+	0.996208i	$0.472269\pi$
$194$	44.4640	3.19233
$195$	0	0
$196$	−40.8459	−2.91756
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	−14.2734	−1.01436
$199$	−26.8328	−1.90213	−0.951064	−	0.308994i	$-0.900008\pi$
$199$	−26.8328	−1.90213	−0.951064	+	0.308994i	$0.900008\pi$
$200$	0	0
$201$	15.3821	1.08497
$202$	56.2093	3.95487
$203$	0	0
$204$	0	0
$205$	0	0
$206$	55.6566	3.87778
$207$	0	0
$208$	56.7666	3.93605
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−79.5933	−5.46649
$213$	0	0
$214$	10.2672	0.701854
$215$	0	0
$216$	57.1121	3.88598
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	−28.7479	−1.92943
$223$	−14.9356	−1.00016	−0.500082	−	0.865978i	$-0.666697\pi$
$223$	−14.9356	−1.00016	−0.500082	+	0.865978i	$0.666697\pi$
$224$	0	0
$225$	0	0
$226$	−5.40302	−0.359404
$227$	−18.7250	−1.24282	−0.621412	−	0.783484i	$-0.713441\pi$
$227$	−18.7250	−1.24282	−0.621412	+	0.783484i	$0.713441\pi$
$228$	0	0
$229$	29.5619	1.95351	0.976754	−	0.214362i	$-0.0687674\pi$
$229$	29.5619	1.95351	0.976754	+	0.214362i	$0.0687674\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	−15.0479	−0.983711
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−6.76384	−0.434796
$243$	−15.1196	−0.969920
$244$	6.52995	0.418037
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	17.8885	1.12911	0.564557	−	0.825394i	$-0.309047\pi$
$251$	17.8885	1.12911	0.564557	+	0.825394i	$0.309047\pi$
$252$	0	0
$253$	0	0
$254$	−60.6553	−3.80585
$255$	0	0
$256$	107.286	6.70540
$257$	−20.3741	−1.27090	−0.635450	−	0.772142i	$-0.719185\pi$
$257$	−20.3741	−1.27090	−0.635450	+	0.772142i	$0.719185\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	33.5896	2.07517
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−35.2972	−2.17239
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−79.9764	−4.88534
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−24.1298	−1.46578	−0.732892	−	0.680345i	$-0.761830\pi$
$271$	−24.1298	−1.46578	−0.732892	+	0.680345i	$0.761830\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	61.2777	3.67519
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	25.3304	1.49781
$287$	0	0
$288$	−52.1685	−3.07406
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−17.8274	−1.04506
$292$	0	0
$293$	−0.172964	−0.0101047	−0.00505234	−	0.999987i	$-0.501608\pi$
$293$	−0.172964	−0.0101047	−0.00505234	+	0.999987i	$0.501608\pi$
$294$	21.9899	1.28248
$295$	0	0
$296$	98.2387	5.71001
$297$	15.5869	0.904443
$298$	23.4060	1.35587
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−22.5366	−1.29469
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.35100	0.0771055	0.0385528	−	0.999257i	$-0.487725\pi$
$307$	1.35100	0.0771055	0.0385528	+	0.999257i	$0.487725\pi$
$308$	0	0
$309$	−22.3150	−1.26945
$310$	0	0
$311$	−31.3726	−1.77898	−0.889490	−	0.456955i	$-0.848940\pi$
$311$	−31.3726	−1.77898	−0.889490	+	0.456955i	$0.848940\pi$
$312$	−37.2125	−2.10675
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.6593	−1.77816	−0.889082	−	0.457748i	$-0.848656\pi$
$317$	−31.6593	−1.77816	−0.889082	+	0.457748i	$0.848656\pi$
$318$	42.8501	2.40292
$319$	0	0
$320$	0	0
$321$	−4.11655	−0.229763
$322$	0	0
$323$	0	0
$324$	−4.37226	−0.242903
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	−15.9275	−0.872823
$334$	48.0896	2.63134
$335$	0	0
$336$	0	0
$337$	−36.6783	−1.99800	−0.998998	−	0.0447653i	$-0.985746\pi$
$337$	−36.6783	−1.99800	−0.998998	+	0.0447653i	$0.985746\pi$
$338$	−9.68383	−0.526731
$339$	2.16629	0.117657
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−36.7435	−1.97534
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	13.4164	0.718164	0.359082	−	0.933306i	$-0.383090\pi$
$349$	13.4164	0.718164	0.359082	+	0.933306i	$0.383090\pi$
$350$	0	0
$351$	16.4327	0.877112
$352$	87.8161	4.68061
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.5131	1.34653	0.673265	−	0.739401i	$-0.264891\pi$
$359$	25.5131	1.34653	0.673265	+	0.739401i	$0.264891\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	2.71190	0.142338
$364$	0	0
$365$	0	0
$366$	−3.51548	−0.183757
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−22.6186	−1.17115	−0.585575	−	0.810619i	$-0.699131\pi$
$373$	−22.6186	−1.17115	−0.585575	+	0.810619i	$0.699131\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	24.3191	1.24591
$382$	−25.0362	−1.28096
$383$	−29.9215	−1.52892	−0.764460	−	0.644671i	$-0.776994\pi$
$383$	−29.9215	−1.52892	−0.764460	+	0.644671i	$0.776994\pi$
$384$	−80.8187	−4.12426
$385$	0	0
$386$	6.76699	0.344431
$387$	0	0
$388$	92.6907	4.70566
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	−75.1451	−3.79540
$393$	−13.4674	−0.679341
$394$	0	0
$395$	0	0
$396$	−29.7546	−1.49522
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−75.1085	−3.76485
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	43.0564	2.14746
$403$	0	0
$404$	117.175	5.82968
$405$	0	0
$406$	0	0
$407$	26.8111	1.32898
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	116.023	5.71604
$413$	0	0
$414$	0	0
$415$	0	0
$416$	92.5813	4.53917
$417$	−24.5687	−1.20313
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	0	0
$424$	−146.430	−7.11125
$425$	0	0
$426$	0	0
$427$	0	0
$428$	21.4033	1.03457
$429$	−10.1560	−0.490334
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	97.7764	4.70427
$433$	18.1458	0.872030	0.436015	−	0.899939i	$-0.356389\pi$
$433$	18.1458	0.872030	0.436015	+	0.899939i	$0.356389\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	12.1833	0.580159
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−59.9284	−2.84408
$445$	0	0
$446$	−41.8067	−1.97961
$447$	−9.38441	−0.443867
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	−11.2633	−0.529779
$453$	0	0
$454$	−52.4138	−2.45990
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	82.7477	3.86655
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	−31.3691	−1.45004
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	23.7408	1.08702
$478$	67.1791	3.07270
$479$	−43.0918	−1.96891	−0.984456	−	0.175630i	$-0.943804\pi$
$479$	−43.0918	−1.96891	−0.984456	+	0.175630i	$0.943804\pi$
$480$	0	0
$481$	28.2659	1.28882
$482$	0	0
$483$	0	0
$484$	−14.1000	−0.640911
$485$	0	0
$486$	−42.3216	−1.91975
$487$	32.2386	1.46087	0.730434	−	0.682983i	$-0.239318\pi$
$487$	32.2386	1.46087	0.730434	+	0.682983i	$0.239318\pi$
$488$	12.0133	0.543816
$489$	0	0
$490$	0	0
$491$	−35.7771	−1.61460	−0.807299	−	0.590143i	$-0.799071\pi$
$491$	−35.7771	−1.61460	−0.807299	+	0.590143i	$0.799071\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.31303	0.193078	0.0965389	−	0.995329i	$-0.469223\pi$
$499$	4.31303	0.193078	0.0965389	+	0.995329i	$0.469223\pi$
$500$	0	0
$501$	−19.2811	−0.861414
$502$	50.0724	2.23484
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.88264	0.172434
$508$	−126.443	−5.61001
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	156.283	6.90681
$513$	0	0
$514$	−57.0297	−2.51547
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	14.7320	0.646661
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−39.9718	−1.74784	−0.873922	−	0.486066i	$-0.838432\pi$
$523$	−39.9718	−1.74784	−0.873922	+	0.486066i	$0.838432\pi$
$524$	70.0215	3.05890
$525$	0	0
$526$	0	0
$527$	0	0
$528$	−60.4291	−2.62984
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−147.134	−6.35524
$537$	0	0
$538$	0	0
$539$	−20.5084	−0.883361
$540$	0	0
$541$	−27.3238	−1.17474	−0.587371	−	0.809318i	$-0.699837\pi$
$541$	−27.3238	−1.17474	−0.587371	+	0.809318i	$0.699837\pi$
$542$	−67.5426	−2.90120
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−46.7055	−1.99698	−0.998492	−	0.0549052i	$-0.982514\pi$
$547$	−46.7055	−1.99698	−0.998492	+	0.0549052i	$0.982514\pi$
$548$	0	0
$549$	−1.94773	−0.0831269
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	127.741	5.41742
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.7272	1.59001	0.795007	−	0.606601i	$-0.207467\pi$
$563$	37.7272	1.59001	0.795007	+	0.606601i	$0.207467\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−39.4704	−1.65178	−0.825891	−	0.563829i	$-0.809328\pi$
$571$	−39.4704	−1.65178	−0.825891	+	0.563829i	$0.809328\pi$
$572$	52.8042	2.20785
$573$	10.0380	0.419344
$574$	0	0
$575$	0	0
$576$	−82.0518	−3.41882
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	−47.5852	−1.97928
$579$	−2.71316	−0.112755
$580$	0	0
$581$	0	0
$582$	−49.9013	−2.06847
$583$	−39.9632	−1.65511
$584$	0	0
$585$	0	0
$586$	−0.484149	−0.0200000
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	45.8407	1.89044
$589$	0	0
$590$	0	0
$591$	0	0
$592$	168.186	6.91239
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	43.6297	1.79015
$595$	0	0
$596$	48.7926	1.99862
$597$	30.1140	1.23249
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	23.8550	0.971452
$604$	0	0
$605$	0	0
$606$	−63.0828	−2.56256
$607$	−47.2956	−1.91967	−0.959834	−	0.280568i	$-0.909477\pi$
$607$	−47.2956	−1.91967	−0.959834	+	0.280568i	$0.909477\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	3.78162	0.152614
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	−62.4625	−2.51261
$619$	−34.9941	−1.40653	−0.703265	−	0.710928i	$-0.748275\pi$
$619$	−34.9941	−1.40653	−0.703265	+	0.710928i	$0.748275\pi$
$620$	0	0
$621$	0	0
$622$	−87.8161	−3.52111
$623$	0	0
$624$	−63.7082	−2.55037
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	11.0275	0.438997	0.219499	−	0.975613i	$-0.429558\pi$
$631$	11.0275	0.438997	0.219499	+	0.975613i	$0.429558\pi$
$632$	0	0
$633$	0	0
$634$	−88.6185	−3.51949
$635$	0	0
$636$	89.3263	3.54202
$637$	−21.6213	−0.856667
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	−11.5228	−0.454767
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−8.04374	−0.315988
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−44.5832	−1.72756
$667$	0	0
$668$	100.249	3.87873
$669$	16.7620	0.648057
$670$	0	0
$671$	3.27864	0.126571
$672$	0	0
$673$	11.9683	0.461343	0.230671	−	0.973032i	$-0.425908\pi$
$673$	11.9683	0.461343	0.230671	+	0.973032i	$0.425908\pi$
$674$	−102.667	−3.95460
$675$	0	0
$676$	−20.1871	−0.776428
$677$	−27.1078	−1.04184	−0.520918	−	0.853607i	$-0.674410\pi$
$677$	−27.1078	−1.04184	−0.520918	+	0.853607i	$0.674410\pi$
$678$	6.06373	0.232876
$679$	0	0
$680$	0	0
$681$	21.0148	0.805289
$682$	0	0
$683$	51.1946	1.95891	0.979454	−	0.201667i	$-0.0646357\pi$
$683$	51.1946	1.95891	0.979454	+	0.201667i	$0.0646357\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−33.1769	−1.26578
$688$	0	0
$689$	−42.1318	−1.60509
$690$	0	0
$691$	52.5727	1.99996	0.999980	−	0.00630823i	$-0.00200798\pi$
$691$	52.5727	1.99996	0.999980	+	0.00630823i	$0.00200798\pi$
$692$	−76.5964	−2.91176
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	37.5543	1.42145
$699$	0	0
$700$	0	0
$701$	48.5239	1.83272	0.916360	−	0.400354i	$-0.131113\pi$
$701$	48.5239	1.83272	0.916360	+	0.400354i	$0.131113\pi$
$702$	45.9972	1.73605
$703$	0	0
$704$	138.119	5.20556
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−40.2492	−1.51159	−0.755796	−	0.654808i	$-0.772750\pi$
$709$	−40.2492	−1.51159	−0.755796	+	0.654808i	$0.772750\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−26.9348	−1.00590
$718$	71.4145	2.66516
$719$	−13.7940	−0.514429	−0.257214	−	0.966354i	$-0.582805\pi$
$719$	−13.7940	−0.514429	−0.257214	+	0.966354i	$0.582805\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	7.59095	0.281727
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	19.2163	0.711716
$730$	0	0
$731$	0	0
$732$	−7.32845	−0.270867
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−40.1556	−1.47915
$738$	0	0
$739$	53.6656	1.97412	0.987061	−	0.160345i	$-0.0512606\pi$
$739$	53.6656	1.97412	0.987061	+	0.160345i	$0.0512606\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	37.3813	1.37139	0.685693	−	0.727890i	$-0.259499\pi$
$743$	37.3813	1.37139	0.685693	+	0.727890i	$0.259499\pi$
$744$	0	0
$745$	0	0
$746$	−63.3125	−2.31804
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	−20.0760	−0.731611
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	54.3834	1.97140	0.985699	−	0.168518i	$-0.0538981\pi$
$761$	54.3834	1.97140	0.985699	+	0.168518i	$0.0538981\pi$
$762$	68.0724	2.46600
$763$	0	0
$764$	−52.1910	−1.88820
$765$	0	0
$766$	−83.7543	−3.02617
$767$	0	0
$768$	−120.406	−4.34477
$769$	47.1406	1.69993	0.849967	−	0.526836i	$-0.176622\pi$
$769$	47.1406	1.69993	0.849967	+	0.526836i	$0.176622\pi$
$770$	0	0
$771$	22.8655	0.823481
$772$	14.1066	0.507708
$773$	−54.0523	−1.94413	−0.972064	−	0.234717i	$-0.924584\pi$
$773$	−54.0523	−1.94413	−0.972064	+	0.234717i	$0.924584\pi$
$774$	0	0
$775$	0	0
$776$	170.525	6.12150
$777$	0	0
$778$	16.7948	0.602122
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−128.649	−4.59461
$785$	0	0
$786$	−37.6970	−1.34461
$787$	53.4392	1.90490	0.952451	−	0.304692i	$-0.0985534\pi$
$787$	53.4392	1.90490	0.952451	+	0.304692i	$0.0985534\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−54.7402	−1.94511
$793$	3.45655	0.122746
$794$	0	0
$795$	0	0
$796$	−156.573	−5.54958
$797$	−54.7345	−1.93879	−0.969397	−	0.245499i	$-0.921048\pi$
$797$	−54.7345	−1.93879	−0.969397	+	0.245499i	$0.921048\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	89.7562	3.16546
$805$	0	0
$806$	0	0
$807$	0	0
$808$	215.570	7.58372
$809$	−22.3607	−0.786160	−0.393080	−	0.919504i	$-0.628590\pi$
$809$	−22.3607	−0.786160	−0.393080	+	0.919504i	$0.628590\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	27.0805	0.949756
$814$	75.0477	2.63042
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	213.450	7.43589
$825$	0	0
$826$	0	0
$827$	−30.9935	−1.07775	−0.538875	−	0.842386i	$-0.681151\pi$
$827$	−30.9935	−1.07775	−0.538875	+	0.842386i	$0.681151\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	145.614	5.04825
$833$	0	0
$834$	−68.7710	−2.38134
$835$	0	0
$836$	0	0
$837$	0	0
$838$	100.769	3.48100
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−250.689	−8.60870
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	39.3761	1.34585
$857$	−48.0008	−1.63967	−0.819837	−	0.572597i	$-0.805936\pi$
$857$	−48.0008	−1.63967	−0.819837	+	0.572597i	$0.805936\pi$
$858$	−28.4278	−0.970511
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.71278	−0.126385	−0.0631923	−	0.998001i	$-0.520128\pi$
$863$	−3.71278	−0.126385	−0.0631923	+	0.998001i	$0.520128\pi$
$864$	159.465	5.42510
$865$	0	0
$866$	50.7924	1.72600
$867$	19.0788	0.647951
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−42.3346	−1.43445
$872$	0	0
$873$	−27.6474	−0.935723
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−59.2236	−1.99984	−0.999919	−	0.0127028i	$-0.995956\pi$
$877$	−59.2236	−1.99984	−0.999919	+	0.0127028i	$0.995956\pi$
$878$	0	0
$879$	0.194115	0.00654733
$880$	0	0
$881$	−9.21678	−0.310521	−0.155261	−	0.987874i	$-0.549622\pi$
$881$	−9.21678	−0.310521	−0.155261	+	0.987874i	$0.549622\pi$
$882$	34.1028	1.14830
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−52.3146	−1.75655	−0.878276	−	0.478154i	$-0.841306\pi$
$887$	−52.3146	−1.75655	−0.878276	+	0.478154i	$0.841306\pi$
$888$	−110.252	−3.69981
$889$	0	0
$890$	0	0
$891$	−2.19528	−0.0735446
$892$	−87.1513	−2.91804
$893$	0	0
$894$	−26.2682	−0.878539
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−20.7213	−0.689180
$905$	0	0
$906$	0	0
$907$	−28.7630	−0.955061	−0.477531	−	0.878615i	$-0.658468\pi$
$907$	−28.7630	−0.955061	−0.477531	+	0.878615i	$0.658468\pi$
$908$	−109.263	−3.62602
$909$	−34.9505	−1.15924
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	172.498	5.69949
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−1.51620	−0.0499606
$922$	50.3843	1.65932
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−34.6069	−1.13664
$928$	0	0
$929$	−31.3050	−1.02708	−0.513541	−	0.858065i	$-0.671667\pi$
$929$	−31.3050	−1.02708	−0.513541	+	0.858065i	$0.671667\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	35.2090	1.15269
$934$	0	0
$935$	0	0
$936$	−57.7105	−1.88633
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	35.5307	1.15216
$952$	0	0
$953$	−61.4682	−1.99115	−0.995575	−	0.0939754i	$-0.970043\pi$
$953$	−61.4682	−1.99115	−0.995575	+	0.0939754i	$0.970043\pi$
$954$	66.4535	2.15151
$955$	0	0
$956$	140.043	4.52932
$957$	0	0
$958$	−120.619	−3.89704
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	79.1200	2.55093
$963$	−6.38409	−0.205724
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−25.9402	−0.833749
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	−88.2246	−2.82980
$973$	0	0
$974$	90.2399	2.89147
$975$	0	0
$976$	20.5669	0.658330
$977$	16.6023	0.531154	0.265577	−	0.964090i	$-0.414438\pi$
$977$	16.6023	0.531154	0.265577	+	0.964090i	$0.414438\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−100.145	−3.19575
$983$	0.192493	0.00613957	0.00306979	−	0.999995i	$-0.499023\pi$
$983$	0.192493	0.00613957	0.00306979	+	0.999995i	$0.499023\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	12.0727	0.382156
$999$	48.6861	1.54036

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.cb.1.8		8
5.2	odd	4		1805.2.b.h.1084.8	yes	8
5.3	odd	4		1805.2.b.h.1084.1	✓	8
5.4	even	2	inner	9025.2.a.cb.1.1		8
19.18	odd	2	inner	9025.2.a.cb.1.1		8
95.18	even	4		1805.2.b.h.1084.8	yes	8
95.37	even	4		1805.2.b.h.1084.1	✓	8
95.94	odd	2	CM	9025.2.a.cb.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.h.1084.1	✓	8	5.3	odd	4
1805.2.b.h.1084.1	✓	8	95.37	even	4
1805.2.b.h.1084.8	yes	8	5.2	odd	4
1805.2.b.h.1084.8	yes	8	95.18	even	4
9025.2.a.cb.1.1		8	5.4	even	2	inner
9025.2.a.cb.1.1		8	19.18	odd	2	inner
9025.2.a.cb.1.8		8	1.1	even	1	trivial
9025.2.a.cb.1.8		8	95.94	odd	2	CM

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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.79913 q^{2} -1.12228 q^{3} +5.83513 q^{4} -3.14142 q^{6} +10.7350 q^{8} -1.74048 q^{9} +2.92978 q^{11} -6.54867 q^{12} +3.08876 q^{13} +18.3785 q^{16} -4.87183 q^{18} +8.20083 q^{22} -12.0477 q^{24} +8.64583 q^{26} +5.32016 q^{27} +29.9737 q^{32} -3.28804 q^{33} -10.1559 q^{36} +9.15124 q^{37} -3.46646 q^{39} +17.0956 q^{44} -20.6259 q^{48} -7.00000 q^{49} +18.0233 q^{52} -13.6404 q^{53} +14.8918 q^{54} +1.11908 q^{61} +47.1432 q^{64} -9.20366 q^{66} -13.7060 q^{67} -18.6841 q^{72} +25.6155 q^{74} -9.70308 q^{78} -0.749299 q^{81} +31.4512 q^{88} -33.6390 q^{96} +15.8849 q^{97} -19.5939 q^{98} -5.09921 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} + 24 q^{9} + 32 q^{16} + 8 q^{24} + 24 q^{26} - 8 q^{36} + 72 q^{44} - 56 q^{49} + 88 q^{54} + 64 q^{64} + 104 q^{66} + 72 q^{81} - 120 q^{96} + 32 q^{99}+O(q^{100}) \) 8 * q + 16 * q^4 + 24 * q^9 + 32 * q^16 + 8 * q^24 + 24 * q^26 - 8 * q^36 + 72 * q^44 - 56 * q^49 + 88 * q^54 + 64 * q^64 + 104 * q^66 + 72 * q^81 - 120 * q^96 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more