LMFDB - Embedded newform 8512.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.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:	$67.9686622005$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.68740$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68740 q^{3} -3.90952 q^{5} -1.00000 q^{7} +4.22212 q^{9} +O(q^{10})$	$q-2.68740 q^{3} -3.90952 q^{5} -1.00000 q^{7} +4.22212 q^{9} +3.22212 q^{11} -1.46528 q^{13} +10.5064 q^{15} +1.22212 q^{17} -1.00000 q^{19} +2.68740 q^{21} +2.09048 q^{23} +10.2843 q^{25} -3.28432 q^{27} +1.22212 q^{29} -0.777884 q^{31} -8.65911 q^{33} +3.90952 q^{35} +11.7285 q^{37} +3.93780 q^{39} +4.15268 q^{41} -7.81903 q^{43} -16.5064 q^{45} +3.28432 q^{47} +1.00000 q^{49} -3.28432 q^{51} -12.6874 q^{53} -12.5969 q^{55} +2.68740 q^{57} +4.53472 q^{59} -11.9095 q^{61} -4.22212 q^{63} +5.72855 q^{65} +9.97171 q^{67} -5.61797 q^{69} -5.72855 q^{71} +8.68740 q^{73} -27.6381 q^{75} -3.22212 q^{77} -7.81903 q^{79} -3.84008 q^{81} +15.7003 q^{83} -4.77788 q^{85} -3.28432 q^{87} -12.2633 q^{89} +1.46528 q^{91} +2.09048 q^{93} +3.90952 q^{95} -17.1033 q^{97} +13.6042 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - 2 q^{5} - 3 q^{7} + 10 q^{9}+O(q^{10}) $ 3 * q - q^3 - 2 * q^5 - 3 * q^7 + 10 * q^9	$ 3 q - q^{3} - 2 q^{5} - 3 q^{7} + 10 q^{9} + 7 q^{11} + 5 q^{15} + q^{17} - 3 q^{19} + q^{21} + 16 q^{23} + 7 q^{25} + 14 q^{27} + q^{29} - 5 q^{31} + 12 q^{33} + 2 q^{35} + 6 q^{37} + 33 q^{39} + q^{41} - 4 q^{43} - 23 q^{45} - 14 q^{47} + 3 q^{49} + 14 q^{51} - 31 q^{53} - 21 q^{55} + q^{57} + 18 q^{59} - 26 q^{61} - 10 q^{63} - 12 q^{65} - q^{67} - q^{69} + 12 q^{71} + 19 q^{73} - 44 q^{75} - 7 q^{77} - 4 q^{79} + 7 q^{81} - 13 q^{83} - 17 q^{85} + 14 q^{87} - 12 q^{89} + 16 q^{93} + 2 q^{95} - 8 q^{97} + 54 q^{99}+O(q^{100}) $ 3 * q - q^3 - 2 * q^5 - 3 * q^7 + 10 * q^9 + 7 * q^11 + 5 * q^15 + q^17 - 3 * q^19 + q^21 + 16 * q^23 + 7 * q^25 + 14 * q^27 + q^29 - 5 * q^31 + 12 * q^33 + 2 * q^35 + 6 * q^37 + 33 * q^39 + q^41 - 4 * q^43 - 23 * q^45 - 14 * q^47 + 3 * q^49 + 14 * q^51 - 31 * q^53 - 21 * q^55 + q^57 + 18 * q^59 - 26 * q^61 - 10 * q^63 - 12 * q^65 - q^67 - q^69 + 12 * q^71 + 19 * q^73 - 44 * q^75 - 7 * q^77 - 4 * q^79 + 7 * q^81 - 13 * q^83 - 17 * q^85 + 14 * q^87 - 12 * q^89 + 16 * q^93 + 2 * q^95 - 8 * q^97 + 54 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.68740	−1.55157	−0.775785	−	0.630997i	$-0.782646\pi$
$3$	−2.68740	−1.55157	−0.775785	+	0.630997i	$0.782646\pi$
$4$	0	0
$5$	−3.90952	−1.74839	−0.874194	−	0.485576i	$-0.838610\pi$
$5$	−3.90952	−1.74839	−0.874194	+	0.485576i	$0.838610\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.22212	1.40737
$10$	0	0
$11$	3.22212	0.971505	0.485752	−	0.874096i	$-0.338546\pi$
$11$	3.22212	0.971505	0.485752	+	0.874096i	$0.338546\pi$
$12$	0	0
$13$	−1.46528	−0.406396	−0.203198	−	0.979138i	$-0.565134\pi$
$13$	−1.46528	−0.406396	−0.203198	+	0.979138i	$0.565134\pi$
$14$	0	0
$15$	10.5064	2.71275
$16$	0	0
$17$	1.22212	0.296407	0.148203	−	0.988957i	$-0.452651\pi$
$17$	1.22212	0.296407	0.148203	+	0.988957i	$0.452651\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.68740	0.586439
$22$	0	0
$23$	2.09048	0.435896	0.217948	−	0.975960i	$-0.430064\pi$
$23$	2.09048	0.435896	0.217948	+	0.975960i	$0.430064\pi$
$24$	0	0
$25$	10.2843	2.05686
$26$	0	0
$27$	−3.28432	−0.632067
$28$	0	0
$29$	1.22212	0.226941	0.113471	−	0.993541i	$-0.463803\pi$
$29$	1.22212	0.226941	0.113471	+	0.993541i	$0.463803\pi$
$30$	0	0
$31$	−0.777884	−0.139712	−0.0698560	−	0.997557i	$-0.522254\pi$
$31$	−0.777884	−0.139712	−0.0698560	+	0.997557i	$0.522254\pi$
$32$	0	0
$33$	−8.65911	−1.50736
$34$	0	0
$35$	3.90952	0.660829
$36$	0	0
$37$	11.7285	1.92816	0.964081	−	0.265610i	$-0.0855732\pi$
$37$	11.7285	1.92816	0.964081	+	0.265610i	$0.0855732\pi$
$38$	0	0
$39$	3.93780	0.630553
$40$	0	0
$41$	4.15268	0.648540	0.324270	−	0.945965i	$-0.394881\pi$
$41$	4.15268	0.648540	0.324270	+	0.945965i	$0.394881\pi$
$42$	0	0
$43$	−7.81903	−1.19239	−0.596196	−	0.802839i	$-0.703322\pi$
$43$	−7.81903	−1.19239	−0.596196	+	0.802839i	$0.703322\pi$
$44$	0	0
$45$	−16.5064	−2.46063
$46$	0	0
$47$	3.28432	0.479067	0.239533	−	0.970888i	$-0.423006\pi$
$47$	3.28432	0.479067	0.239533	+	0.970888i	$0.423006\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.28432	−0.459896
$52$	0	0
$53$	−12.6874	−1.74275	−0.871374	−	0.490619i	$-0.836771\pi$
$53$	−12.6874	−1.74275	−0.871374	+	0.490619i	$0.836771\pi$
$54$	0	0
$55$	−12.5969	−1.69857
$56$	0	0
$57$	2.68740	0.355955
$58$	0	0
$59$	4.53472	0.590370	0.295185	−	0.955440i	$-0.404619\pi$
$59$	4.53472	0.590370	0.295185	+	0.955440i	$0.404619\pi$
$60$	0	0
$61$	−11.9095	−1.52486	−0.762429	−	0.647072i	$-0.775993\pi$
$61$	−11.9095	−1.52486	−0.762429	+	0.647072i	$0.775993\pi$
$62$	0	0
$63$	−4.22212	−0.531937
$64$	0	0
$65$	5.72855	0.710539
$66$	0	0
$67$	9.97171	1.21824	0.609119	−	0.793079i	$-0.291523\pi$
$67$	9.97171	1.21824	0.609119	+	0.793079i	$0.291523\pi$
$68$	0	0
$69$	−5.61797	−0.676324
$70$	0	0
$71$	−5.72855	−0.679854	−0.339927	−	0.940452i	$-0.610402\pi$
$71$	−5.72855	−0.679854	−0.339927	+	0.940452i	$0.610402\pi$
$72$	0	0
$73$	8.68740	1.01678	0.508392	−	0.861126i	$-0.330240\pi$
$73$	8.68740	1.01678	0.508392	+	0.861126i	$0.330240\pi$
$74$	0	0
$75$	−27.6381	−3.19137
$76$	0	0
$77$	−3.22212	−0.367194
$78$	0	0
$79$	−7.81903	−0.879710	−0.439855	−	0.898069i	$-0.644970\pi$
$79$	−7.81903	−0.879710	−0.439855	+	0.898069i	$0.644970\pi$
$80$	0	0
$81$	−3.84008	−0.426676
$82$	0	0
$83$	15.7003	1.72333	0.861664	−	0.507479i	$-0.169423\pi$
$83$	15.7003	1.72333	0.861664	+	0.507479i	$0.169423\pi$
$84$	0	0
$85$	−4.77788	−0.518234
$86$	0	0
$87$	−3.28432	−0.352116
$88$	0	0
$89$	−12.2633	−1.29990	−0.649952	−	0.759975i	$-0.725211\pi$
$89$	−12.2633	−1.29990	−0.649952	+	0.759975i	$0.725211\pi$
$90$	0	0
$91$	1.46528	0.153603
$92$	0	0
$93$	2.09048	0.216773
$94$	0	0
$95$	3.90952	0.401108
$96$	0	0
$97$	−17.1033	−1.73658	−0.868291	−	0.496055i	$-0.834781\pi$
$97$	−17.1033	−1.73658	−0.868291	+	0.496055i	$0.834781\pi$
$98$	0	0
$99$	13.6042	1.36727
$100$	0	0
$101$	−13.4653	−1.33985	−0.669923	−	0.742431i	$-0.733673\pi$
$101$	−13.4653	−1.33985	−0.669923	+	0.742431i	$0.733673\pi$
$102$	0	0
$103$	11.2843	1.11188	0.555938	−	0.831224i	$-0.312359\pi$
$103$	11.2843	1.11188	0.555938	+	0.831224i	$0.312359\pi$
$104$	0	0
$105$	−10.5064	−1.02532
$106$	0	0
$107$	1.02105	0.0987087	0.0493543	−	0.998781i	$-0.484284\pi$
$107$	1.02105	0.0987087	0.0493543	+	0.998781i	$0.484284\pi$
$108$	0	0
$109$	9.46528	0.906610	0.453305	−	0.891356i	$-0.350245\pi$
$109$	9.46528	0.906610	0.453305	+	0.891356i	$0.350245\pi$
$110$	0	0
$111$	−31.5193	−2.99168
$112$	0	0
$113$	−12.3255	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$113$	−12.3255	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$114$	0	0
$115$	−8.17278	−0.762116
$116$	0	0
$117$	−6.18660	−0.571951
$118$	0	0
$119$	−1.22212	−0.112031
$120$	0	0
$121$	−0.617966	−0.0561787
$122$	0	0
$123$	−11.1599	−1.00626
$124$	0	0
$125$	−20.6591	−1.84781
$126$	0	0
$127$	2.80617	0.249007	0.124504	−	0.992219i	$-0.460266\pi$
$127$	2.80617	0.249007	0.124504	+	0.992219i	$0.460266\pi$
$128$	0	0
$129$	21.0129	1.85008
$130$	0	0
$131$	2.15268	0.188081	0.0940404	−	0.995568i	$-0.470022\pi$
$131$	2.15268	0.188081	0.0940404	+	0.995568i	$0.470022\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	12.8401	1.10510
$136$	0	0
$137$	−16.7496	−1.43101	−0.715507	−	0.698605i	$-0.753804\pi$
$137$	−16.7496	−1.43101	−0.715507	+	0.698605i	$0.753804\pi$
$138$	0	0
$139$	0.715685	0.0607036	0.0303518	−	0.999539i	$-0.490337\pi$
$139$	0.715685	0.0607036	0.0303518	+	0.999539i	$0.490337\pi$
$140$	0	0
$141$	−8.82627	−0.743306
$142$	0	0
$143$	−4.72131	−0.394816
$144$	0	0
$145$	−4.77788	−0.396782
$146$	0	0
$147$	−2.68740	−0.221653
$148$	0	0
$149$	17.1033	1.40116	0.700580	−	0.713573i	$-0.252925\pi$
$149$	17.1033	1.40116	0.700580	+	0.713573i	$0.252925\pi$
$150$	0	0
$151$	5.97171	0.485971	0.242986	−	0.970030i	$-0.421873\pi$
$151$	5.97171	0.485971	0.242986	+	0.970030i	$0.421873\pi$
$152$	0	0
$153$	5.15992	0.417155
$154$	0	0
$155$	3.04115	0.244271
$156$	0	0
$157$	−7.48538	−0.597398	−0.298699	−	0.954347i	$-0.596553\pi$
$157$	−7.48538	−0.597398	−0.298699	+	0.954347i	$0.596553\pi$
$158$	0	0
$159$	34.0961	2.70400
$160$	0	0
$161$	−2.09048	−0.164753
$162$	0	0
$163$	21.4370	1.67908	0.839538	−	0.543302i	$-0.182826\pi$
$163$	21.4370	1.67908	0.839538	+	0.543302i	$0.182826\pi$
$164$	0	0
$165$	33.8529	2.63545
$166$	0	0
$167$	−4.84008	−0.374537	−0.187268	−	0.982309i	$-0.559963\pi$
$167$	−4.84008	−0.374537	−0.187268	+	0.982309i	$0.559963\pi$
$168$	0	0
$169$	−10.8529	−0.834842
$170$	0	0
$171$	−4.22212	−0.322873
$172$	0	0
$173$	−12.2633	−0.932359	−0.466179	−	0.884690i	$-0.654370\pi$
$173$	−12.2633	−0.932359	−0.466179	+	0.884690i	$0.654370\pi$
$174$	0	0
$175$	−10.2843	−0.777421
$176$	0	0
$177$	−12.1866	−0.916001
$178$	0	0
$179$	22.9846	1.71795	0.858974	−	0.512019i	$-0.171102\pi$
$179$	22.9846	1.71795	0.858974	+	0.512019i	$0.171102\pi$
$180$	0	0
$181$	−18.4159	−1.36885	−0.684423	−	0.729085i	$-0.739946\pi$
$181$	−18.4159	−1.36885	−0.684423	+	0.729085i	$0.739946\pi$
$182$	0	0
$183$	32.0056	2.36592
$184$	0	0
$185$	−45.8529	−3.37118
$186$	0	0
$187$	3.93780	0.287961
$188$	0	0
$189$	3.28432	0.238899
$190$	0	0
$191$	16.9507	1.22651	0.613254	−	0.789886i	$-0.289860\pi$
$191$	16.9507	1.22651	0.613254	+	0.789886i	$0.289860\pi$
$192$	0	0
$193$	3.97171	0.285890	0.142945	−	0.989731i	$-0.454343\pi$
$193$	3.97171	0.285890	0.142945	+	0.989731i	$0.454343\pi$
$194$	0	0
$195$	−15.3949	−1.10245
$196$	0	0
$197$	14.0622	1.00189	0.500945	−	0.865479i	$-0.332986\pi$
$197$	14.0622	1.00189	0.500945	+	0.865479i	$0.332986\pi$
$198$	0	0
$199$	−15.4653	−1.09630	−0.548152	−	0.836378i	$-0.684669\pi$
$199$	−15.4653	−1.09630	−0.548152	+	0.836378i	$0.684669\pi$
$200$	0	0
$201$	−26.7980	−1.89018
$202$	0	0
$203$	−1.22212	−0.0857758
$204$	0	0
$205$	−16.2350	−1.13390
$206$	0	0
$207$	8.82627	0.613468
$208$	0	0
$209$	−3.22212	−0.222878
$210$	0	0
$211$	−18.5064	−1.27404	−0.637018	−	0.770849i	$-0.719832\pi$
$211$	−18.5064	−1.27404	−0.637018	+	0.770849i	$0.719832\pi$
$212$	0	0
$213$	15.3949	1.05484
$214$	0	0
$215$	30.5686	2.08476
$216$	0	0
$217$	0.777884	0.0528062
$218$	0	0
$219$	−23.3465	−1.57761
$220$	0	0
$221$	−1.79075	−0.120459
$222$	0	0
$223$	12.4781	0.835598	0.417799	−	0.908539i	$-0.362802\pi$
$223$	12.4781	0.835598	0.417799	+	0.908539i	$0.362802\pi$
$224$	0	0
$225$	43.4216	2.89477
$226$	0	0
$227$	−21.6098	−1.43429	−0.717146	−	0.696923i	$-0.754552\pi$
$227$	−21.6098	−1.43429	−0.717146	+	0.696923i	$0.754552\pi$
$228$	0	0
$229$	−16.3877	−1.08293	−0.541464	−	0.840724i	$-0.682130\pi$
$229$	−16.3877	−1.08293	−0.541464	+	0.840724i	$0.682130\pi$
$230$	0	0
$231$	8.65911	0.569728
$232$	0	0
$233$	−0.333651	−0.0218582	−0.0109291	−	0.999940i	$-0.503479\pi$
$233$	−0.333651	−0.0218582	−0.0109291	+	0.999940i	$0.503479\pi$
$234$	0	0
$235$	−12.8401	−0.837595
$236$	0	0
$237$	21.0129	1.36493
$238$	0	0
$239$	−6.97895	−0.451431	−0.225715	−	0.974193i	$-0.572472\pi$
$239$	−6.97895	−0.451431	−0.225715	+	0.974193i	$0.572472\pi$
$240$	0	0
$241$	23.1938	1.49405	0.747023	−	0.664798i	$-0.231482\pi$
$241$	23.1938	1.49405	0.747023	+	0.664798i	$0.231482\pi$
$242$	0	0
$243$	20.1728	1.29408
$244$	0	0
$245$	−3.90952	−0.249770
$246$	0	0
$247$	1.46528	0.0932337
$248$	0	0
$249$	−42.1929	−2.67387
$250$	0	0
$251$	−26.4982	−1.67255	−0.836277	−	0.548307i	$-0.815273\pi$
$251$	−26.4982	−1.67255	−0.836277	+	0.548307i	$0.815273\pi$
$252$	0	0
$253$	6.73578	0.423475
$254$	0	0
$255$	12.8401	0.804077
$256$	0	0
$257$	4.68740	0.292392	0.146196	−	0.989256i	$-0.453297\pi$
$257$	4.68740	0.292392	0.146196	+	0.989256i	$0.453297\pi$
$258$	0	0
$259$	−11.7285	−0.728777
$260$	0	0
$261$	5.15992	0.319391
$262$	0	0
$263$	3.93780	0.242815	0.121408	−	0.992603i	$-0.461259\pi$
$263$	3.93780	0.242815	0.121408	+	0.992603i	$0.461259\pi$
$264$	0	0
$265$	49.6016	3.04700
$266$	0	0
$267$	32.9563	2.01689
$268$	0	0
$269$	13.5193	0.824286	0.412143	−	0.911119i	$-0.364780\pi$
$269$	13.5193	0.824286	0.412143	+	0.911119i	$0.364780\pi$
$270$	0	0
$271$	−29.4288	−1.78767	−0.893836	−	0.448393i	$-0.851996\pi$
$271$	−29.4288	−1.78767	−0.893836	+	0.448393i	$0.851996\pi$
$272$	0	0
$273$	−3.93780	−0.238327
$274$	0	0
$275$	33.1373	1.99825
$276$	0	0
$277$	16.0823	0.966292	0.483146	−	0.875540i	$-0.339494\pi$
$277$	16.0823	0.966292	0.483146	+	0.875540i	$0.339494\pi$
$278$	0	0
$279$	−3.28432	−0.196627
$280$	0	0
$281$	16.3958	0.978094	0.489047	−	0.872257i	$-0.337345\pi$
$281$	16.3958	0.978094	0.489047	+	0.872257i	$0.337345\pi$
$282$	0	0
$283$	−5.31260	−0.315801	−0.157901	−	0.987455i	$-0.550473\pi$
$283$	−5.31260	−0.315801	−0.157901	+	0.987455i	$0.550473\pi$
$284$	0	0
$285$	−10.5064	−0.622347
$286$	0	0
$287$	−4.15268	−0.245125
$288$	0	0
$289$	−15.5064	−0.912143
$290$	0	0
$291$	45.9635	2.69443
$292$	0	0
$293$	19.9095	1.16313	0.581563	−	0.813501i	$-0.302441\pi$
$293$	19.9095	1.16313	0.581563	+	0.813501i	$0.302441\pi$
$294$	0	0
$295$	−17.7285	−1.03220
$296$	0	0
$297$	−10.5824	−0.614056
$298$	0	0
$299$	−3.06315	−0.177147
$300$	0	0
$301$	7.81903	0.450682
$302$	0	0
$303$	36.1866	2.07887
$304$	0	0
$305$	46.5604	2.66604
$306$	0	0
$307$	0.472518	0.0269680	0.0134840	−	0.999909i	$-0.495708\pi$
$307$	0.472518	0.0269680	0.0134840	+	0.999909i	$0.495708\pi$
$308$	0	0
$309$	−30.3255	−1.72516
$310$	0	0
$311$	−2.33365	−0.132329	−0.0661646	−	0.997809i	$-0.521076\pi$
$311$	−2.33365	−0.132329	−0.0661646	+	0.997809i	$0.521076\pi$
$312$	0	0
$313$	3.19383	0.180526	0.0902630	−	0.995918i	$-0.471229\pi$
$313$	3.19383	0.180526	0.0902630	+	0.995918i	$0.471229\pi$
$314$	0	0
$315$	16.5064	0.930032
$316$	0	0
$317$	12.7496	0.716089	0.358044	−	0.933705i	$-0.383444\pi$
$317$	12.7496	0.716089	0.358044	+	0.933705i	$0.383444\pi$
$318$	0	0
$319$	3.93780	0.220475
$320$	0	0
$321$	−2.74397	−0.153153
$322$	0	0
$323$	−1.22212	−0.0680004
$324$	0	0
$325$	−15.0694	−0.835902
$326$	0	0
$327$	−25.4370	−1.40667
$328$	0	0
$329$	−3.28432	−0.181070
$330$	0	0
$331$	29.2560	1.60806	0.804028	−	0.594591i	$-0.202686\pi$
$331$	29.2560	1.60806	0.804028	+	0.594591i	$0.202686\pi$
$332$	0	0
$333$	49.5193	2.71364
$334$	0	0
$335$	−38.9846	−2.12996
$336$	0	0
$337$	−25.8730	−1.40939	−0.704697	−	0.709508i	$-0.748917\pi$
$337$	−25.8730	−1.40939	−0.704697	+	0.709508i	$0.748917\pi$
$338$	0	0
$339$	33.1234	1.79902
$340$	0	0
$341$	−2.50643	−0.135731
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	21.9635	1.18248
$346$	0	0
$347$	−28.0056	−1.50342	−0.751710	−	0.659493i	$-0.770771\pi$
$347$	−28.0056	−1.50342	−0.751710	+	0.659493i	$0.770771\pi$
$348$	0	0
$349$	−18.2834	−0.978686	−0.489343	−	0.872091i	$-0.662763\pi$
$349$	−18.2834	−0.978686	−0.489343	+	0.872091i	$0.662763\pi$
$350$	0	0
$351$	4.81245	0.256870
$352$	0	0
$353$	7.79075	0.414660	0.207330	−	0.978271i	$-0.433523\pi$
$353$	7.79075	0.414660	0.207330	+	0.978271i	$0.433523\pi$
$354$	0	0
$355$	22.3958	1.18865
$356$	0	0
$357$	3.28432	0.173824
$358$	0	0
$359$	36.9425	1.94975	0.974875	−	0.222754i	$-0.0715047\pi$
$359$	36.9425	1.94975	0.974875	+	0.222754i	$0.0715047\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.66072	0.0871653
$364$	0	0
$365$	−33.9635	−1.77773
$366$	0	0
$367$	14.9789	0.781895	0.390947	−	0.920413i	$-0.372147\pi$
$367$	14.9789	0.781895	0.390947	+	0.920413i	$0.372147\pi$
$368$	0	0
$369$	17.5331	0.912737
$370$	0	0
$371$	12.6874	0.658697
$372$	0	0
$373$	10.4643	0.541823	0.270911	−	0.962604i	$-0.412675\pi$
$373$	10.4643	0.541823	0.270911	+	0.962604i	$0.412675\pi$
$374$	0	0
$375$	55.5193	2.86700
$376$	0	0
$377$	−1.79075	−0.0922281
$378$	0	0
$379$	−20.9645	−1.07687	−0.538436	−	0.842666i	$-0.680985\pi$
$379$	−20.9645	−1.07687	−0.538436	+	0.842666i	$0.680985\pi$
$380$	0	0
$381$	−7.54130	−0.386352
$382$	0	0
$383$	26.2149	1.33952	0.669759	−	0.742579i	$-0.266397\pi$
$383$	26.2149	1.33952	0.669759	+	0.742579i	$0.266397\pi$
$384$	0	0
$385$	12.5969	0.641998
$386$	0	0
$387$	−33.0129	−1.67814
$388$	0	0
$389$	11.9717	0.606990	0.303495	−	0.952833i	$-0.401846\pi$
$389$	11.9717	0.606990	0.303495	+	0.952833i	$0.401846\pi$
$390$	0	0
$391$	2.55481	0.129203
$392$	0	0
$393$	−5.78512	−0.291821
$394$	0	0
$395$	30.5686	1.53807
$396$	0	0
$397$	18.1244	0.909637	0.454819	−	0.890584i	$-0.349704\pi$
$397$	18.1244	0.909637	0.454819	+	0.890584i	$0.349704\pi$
$398$	0	0
$399$	−2.68740	−0.134538
$400$	0	0
$401$	20.3337	1.01541	0.507707	−	0.861530i	$-0.330493\pi$
$401$	20.3337	1.01541	0.507707	+	0.861530i	$0.330493\pi$
$402$	0	0
$403$	1.13982	0.0567785
$404$	0	0
$405$	15.0129	0.745995
$406$	0	0
$407$	37.7907	1.87322
$408$	0	0
$409$	14.9507	0.739263	0.369631	−	0.929178i	$-0.379484\pi$
$409$	14.9507	0.739263	0.369631	+	0.929178i	$0.379484\pi$
$410$	0	0
$411$	45.0129	2.22032
$412$	0	0
$413$	−4.53472	−0.223139
$414$	0	0
$415$	−61.3804	−3.01305
$416$	0	0
$417$	−1.92333	−0.0941859
$418$	0	0
$419$	1.42318	0.0695270	0.0347635	−	0.999396i	$-0.488932\pi$
$419$	1.42318	0.0695270	0.0347635	+	0.999396i	$0.488932\pi$
$420$	0	0
$421$	−17.7706	−0.866088	−0.433044	−	0.901373i	$-0.642561\pi$
$421$	−17.7706	−0.866088	−0.433044	+	0.901373i	$0.642561\pi$
$422$	0	0
$423$	13.8668	0.674225
$424$	0	0
$425$	12.5686	0.609668
$426$	0	0
$427$	11.9095	0.576342
$428$	0	0
$429$	12.6881	0.612585
$430$	0	0
$431$	−2.93057	−0.141160	−0.0705802	−	0.997506i	$-0.522485\pi$
$431$	−2.93057	−0.141160	−0.0705802	+	0.997506i	$0.522485\pi$
$432$	0	0
$433$	17.1599	0.824653	0.412327	−	0.911036i	$-0.364716\pi$
$433$	17.1599	0.824653	0.412327	+	0.911036i	$0.364716\pi$
$434$	0	0
$435$	12.8401	0.615635
$436$	0	0
$437$	−2.09048	−0.100001
$438$	0	0
$439$	13.4992	0.644281	0.322141	−	0.946692i	$-0.395598\pi$
$439$	13.4992	0.644281	0.322141	+	0.946692i	$0.395598\pi$
$440$	0	0
$441$	4.22212	0.201053
$442$	0	0
$443$	4.95066	0.235213	0.117607	−	0.993060i	$-0.462478\pi$
$443$	4.95066	0.235213	0.117607	+	0.993060i	$0.462478\pi$
$444$	0	0
$445$	47.9434	2.27274
$446$	0	0
$447$	−45.9635	−2.17400
$448$	0	0
$449$	8.68740	0.409984	0.204992	−	0.978764i	$-0.434283\pi$
$449$	8.68740	0.409984	0.204992	+	0.978764i	$0.434283\pi$
$450$	0	0
$451$	13.3804	0.630060
$452$	0	0
$453$	−16.0484	−0.754019
$454$	0	0
$455$	−5.72855	−0.268558
$456$	0	0
$457$	−22.9507	−1.07359	−0.536793	−	0.843714i	$-0.680365\pi$
$457$	−22.9507	−1.07359	−0.536793	+	0.843714i	$0.680365\pi$
$458$	0	0
$459$	−4.01382	−0.187349
$460$	0	0
$461$	19.0751	0.888414	0.444207	−	0.895924i	$-0.353485\pi$
$461$	19.0751	0.888414	0.444207	+	0.895924i	$0.353485\pi$
$462$	0	0
$463$	31.1033	1.44550	0.722748	−	0.691112i	$-0.242879\pi$
$463$	31.1033	1.44550	0.722748	+	0.691112i	$0.242879\pi$
$464$	0	0
$465$	−8.17278	−0.379004
$466$	0	0
$467$	−8.06220	−0.373074	−0.186537	−	0.982448i	$-0.559726\pi$
$467$	−8.06220	−0.373074	−0.186537	+	0.982448i	$0.559726\pi$
$468$	0	0
$469$	−9.97171	−0.460451
$470$	0	0
$471$	20.1162	0.926906
$472$	0	0
$473$	−25.1938	−1.15841
$474$	0	0
$475$	−10.2843	−0.471877
$476$	0	0
$477$	−53.5677	−2.45270
$478$	0	0
$479$	23.5677	1.07683	0.538417	−	0.842678i	$-0.319022\pi$
$479$	23.5677	1.07683	0.538417	+	0.842678i	$0.319022\pi$
$480$	0	0
$481$	−17.1856	−0.783598
$482$	0	0
$483$	5.61797	0.255626
$484$	0	0
$485$	66.8658	3.03622
$486$	0	0
$487$	−23.5055	−1.06513	−0.532567	−	0.846388i	$-0.678773\pi$
$487$	−23.5055	−1.06513	−0.532567	+	0.846388i	$0.678773\pi$
$488$	0	0
$489$	−57.6098	−2.60520
$490$	0	0
$491$	−8.52653	−0.384797	−0.192398	−	0.981317i	$-0.561627\pi$
$491$	−8.52653	−0.384797	−0.192398	+	0.981317i	$0.561627\pi$
$492$	0	0
$493$	1.49357	0.0672669
$494$	0	0
$495$	−53.1856	−2.39052
$496$	0	0
$497$	5.72855	0.256960
$498$	0	0
$499$	20.2432	0.906209	0.453104	−	0.891458i	$-0.350317\pi$
$499$	20.2432	0.906209	0.453104	+	0.891458i	$0.350317\pi$
$500$	0	0
$501$	13.0072	0.581120
$502$	0	0
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	0	0
$505$	52.6427	2.34257
$506$	0	0
$507$	29.1662	1.29532
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−8.68740	−0.384308
$512$	0	0
$513$	3.28432	0.145006
$514$	0	0
$515$	−44.1162	−1.94399
$516$	0	0
$517$	10.5824	0.465415
$518$	0	0
$519$	32.9563	1.44662
$520$	0	0
$521$	−38.1162	−1.66990	−0.834951	−	0.550325i	$-0.814504\pi$
$521$	−38.1162	−1.66990	−0.834951	+	0.550325i	$0.814504\pi$
$522$	0	0
$523$	15.7706	0.689602	0.344801	−	0.938676i	$-0.387946\pi$
$523$	15.7706	0.689602	0.344801	+	0.938676i	$0.387946\pi$
$524$	0	0
$525$	27.6381	1.20622
$526$	0	0
$527$	−0.950664	−0.0414116
$528$	0	0
$529$	−18.6299	−0.809995
$530$	0	0
$531$	19.1461	0.830870
$532$	0	0
$533$	−6.08486	−0.263564
$534$	0	0
$535$	−3.99181	−0.172581
$536$	0	0
$537$	−61.7687	−2.66552
$538$	0	0
$539$	3.22212	0.138786
$540$	0	0
$541$	−3.25040	−0.139746	−0.0698728	−	0.997556i	$-0.522259\pi$
$541$	−3.25040	−0.139746	−0.0698728	+	0.997556i	$0.522259\pi$
$542$	0	0
$543$	49.4910	2.12386
$544$	0	0
$545$	−37.0047	−1.58511
$546$	0	0
$547$	26.9928	1.15413	0.577064	−	0.816699i	$-0.304198\pi$
$547$	26.9928	1.15413	0.577064	+	0.816699i	$0.304198\pi$
$548$	0	0
$549$	−50.2834	−2.14604
$550$	0	0
$551$	−1.22212	−0.0520639
$552$	0	0
$553$	7.81903	0.332499
$554$	0	0
$555$	123.225	5.23062
$556$	0	0
$557$	−24.6792	−1.04569	−0.522846	−	0.852427i	$-0.675130\pi$
$557$	−24.6792	−1.04569	−0.522846	+	0.852427i	$0.675130\pi$
$558$	0	0
$559$	11.4571	0.484584
$560$	0	0
$561$	−10.5824	−0.446791
$562$	0	0
$563$	3.82722	0.161298	0.0806490	−	0.996743i	$-0.474301\pi$
$563$	3.82722	0.161298	0.0806490	+	0.996743i	$0.474301\pi$
$564$	0	0
$565$	48.1866	2.02723
$566$	0	0
$567$	3.84008	0.161268
$568$	0	0
$569$	−8.79798	−0.368831	−0.184415	−	0.982848i	$-0.559039\pi$
$569$	−8.79798	−0.368831	−0.184415	+	0.982848i	$0.559039\pi$
$570$	0	0
$571$	−13.8529	−0.579728	−0.289864	−	0.957068i	$-0.593610\pi$
$571$	−13.8529	−0.579728	−0.289864	+	0.957068i	$0.593610\pi$
$572$	0	0
$573$	−45.5532	−1.90301
$574$	0	0
$575$	21.4992	0.896578
$576$	0	0
$577$	20.5064	0.853694	0.426847	−	0.904324i	$-0.359624\pi$
$577$	20.5064	0.853694	0.426847	+	0.904324i	$0.359624\pi$
$578$	0	0
$579$	−10.6736	−0.443579
$580$	0	0
$581$	−15.7003	−0.651357
$582$	0	0
$583$	−40.8803	−1.69309
$584$	0	0
$585$	24.1866	0.999993
$586$	0	0
$587$	1.95790	0.0808112	0.0404056	−	0.999183i	$-0.487135\pi$
$587$	1.95790	0.0808112	0.0404056	+	0.999183i	$0.487135\pi$
$588$	0	0
$589$	0.777884	0.0320521
$590$	0	0
$591$	−37.7907	−1.55450
$592$	0	0
$593$	−13.4653	−0.552953	−0.276476	−	0.961021i	$-0.589167\pi$
$593$	−13.4653	−0.552953	−0.276476	+	0.961021i	$0.589167\pi$
$594$	0	0
$595$	4.77788	0.195874
$596$	0	0
$597$	41.5614	1.70099
$598$	0	0
$599$	−43.1574	−1.76336	−0.881681	−	0.471846i	$-0.843588\pi$
$599$	−43.1574	−1.76336	−0.881681	+	0.471846i	$0.843588\pi$
$600$	0	0
$601$	−3.89570	−0.158909	−0.0794545	−	0.996838i	$-0.525318\pi$
$601$	−3.89570	−0.158909	−0.0794545	+	0.996838i	$0.525318\pi$
$602$	0	0
$603$	42.1017	1.71452
$604$	0	0
$605$	2.41595	0.0982222
$606$	0	0
$607$	6.74141	0.273625	0.136813	−	0.990597i	$-0.456314\pi$
$607$	6.74141	0.273625	0.136813	+	0.990597i	$0.456314\pi$
$608$	0	0
$609$	3.28432	0.133087
$610$	0	0
$611$	−4.81245	−0.194691
$612$	0	0
$613$	−27.4772	−1.10979	−0.554897	−	0.831919i	$-0.687242\pi$
$613$	−27.4772	−1.10979	−0.554897	+	0.831919i	$0.687242\pi$
$614$	0	0
$615$	43.6299	1.75933
$616$	0	0
$617$	32.9846	1.32791	0.663955	−	0.747773i	$-0.268877\pi$
$617$	32.9846	1.32791	0.663955	+	0.747773i	$0.268877\pi$
$618$	0	0
$619$	−14.5064	−0.583063	−0.291531	−	0.956561i	$-0.594165\pi$
$619$	−14.5064	−0.583063	−0.291531	+	0.956561i	$0.594165\pi$
$620$	0	0
$621$	−6.86581	−0.275515
$622$	0	0
$623$	12.2633	0.491317
$624$	0	0
$625$	29.3456	1.17382
$626$	0	0
$627$	8.65911	0.345812
$628$	0	0
$629$	14.3337	0.571520
$630$	0	0
$631$	−39.8108	−1.58484	−0.792422	−	0.609973i	$-0.791180\pi$
$631$	−39.8108	−1.58484	−0.792422	+	0.609973i	$0.791180\pi$
$632$	0	0
$633$	49.7342	1.97676
$634$	0	0
$635$	−10.9708	−0.435361
$636$	0	0
$637$	−1.46528	−0.0580566
$638$	0	0
$639$	−24.1866	−0.956807
$640$	0	0
$641$	39.3044	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$641$	39.3044	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$642$	0	0
$643$	12.6591	0.499227	0.249613	−	0.968346i	$-0.419696\pi$
$643$	12.6591	0.499227	0.249613	+	0.968346i	$0.419696\pi$
$644$	0	0
$645$	−82.1501	−3.23466
$646$	0	0
$647$	35.2843	1.38717	0.693585	−	0.720375i	$-0.256030\pi$
$647$	35.2843	1.38717	0.693585	+	0.720375i	$0.256030\pi$
$648$	0	0
$649$	14.6114	0.573547
$650$	0	0
$651$	−2.09048	−0.0819325
$652$	0	0
$653$	45.0468	1.76282	0.881408	−	0.472355i	$-0.156596\pi$
$653$	45.0468	1.76282	0.881408	+	0.472355i	$0.156596\pi$
$654$	0	0
$655$	−8.41595	−0.328838
$656$	0	0
$657$	36.6792	1.43099
$658$	0	0
$659$	−0.194783	−0.00758768	−0.00379384	−	0.999993i	$-0.501208\pi$
$659$	−0.194783	−0.00758768	−0.00379384	+	0.999993i	$0.501208\pi$
$660$	0	0
$661$	26.7753	1.04144	0.520720	−	0.853728i	$-0.325664\pi$
$661$	26.7753	1.04144	0.520720	+	0.853728i	$0.325664\pi$
$662$	0	0
$663$	4.81245	0.186900
$664$	0	0
$665$	−3.90952	−0.151605
$666$	0	0
$667$	2.55481	0.0989228
$668$	0	0
$669$	−33.5338	−1.29649
$670$	0	0
$671$	−38.3738	−1.48141
$672$	0	0
$673$	−50.1866	−1.93455	−0.967276	−	0.253728i	$-0.918343\pi$
$673$	−50.1866	−1.93455	−0.967276	+	0.253728i	$0.918343\pi$
$674$	0	0
$675$	−33.7769	−1.30007
$676$	0	0
$677$	19.3804	0.744850	0.372425	−	0.928062i	$-0.378526\pi$
$677$	19.3804	0.744850	0.372425	+	0.928062i	$0.378526\pi$
$678$	0	0
$679$	17.1033	0.656366
$680$	0	0
$681$	58.0741	2.22540
$682$	0	0
$683$	−26.7980	−1.02540	−0.512698	−	0.858569i	$-0.671354\pi$
$683$	−26.7980	−1.02540	−0.512698	+	0.858569i	$0.671354\pi$
$684$	0	0
$685$	65.4828	2.50197
$686$	0	0
$687$	44.0402	1.68024
$688$	0	0
$689$	18.5906	0.708247
$690$	0	0
$691$	9.73674	0.370403	0.185201	−	0.982701i	$-0.440706\pi$
$691$	9.73674	0.370403	0.185201	+	0.982701i	$0.440706\pi$
$692$	0	0
$693$	−13.6042	−0.516779
$694$	0	0
$695$	−2.79798	−0.106133
$696$	0	0
$697$	5.07506	0.192232
$698$	0	0
$699$	0.896653	0.0339145
$700$	0	0
$701$	−22.9563	−0.867047	−0.433524	−	0.901142i	$-0.642730\pi$
$701$	−22.9563	−0.867047	−0.433524	+	0.901142i	$0.642730\pi$
$702$	0	0
$703$	−11.7285	−0.442351
$704$	0	0
$705$	34.5064	1.29959
$706$	0	0
$707$	13.4653	0.506414
$708$	0	0
$709$	−15.7285	−0.590698	−0.295349	−	0.955389i	$-0.595436\pi$
$709$	−15.7285	−0.590698	−0.295349	+	0.955389i	$0.595436\pi$
$710$	0	0
$711$	−33.0129	−1.23808
$712$	0	0
$713$	−1.62615	−0.0608999
$714$	0	0
$715$	18.4580	0.690292
$716$	0	0
$717$	18.7552	0.700427
$718$	0	0
$719$	31.6381	1.17990	0.589950	−	0.807440i	$-0.299147\pi$
$719$	31.6381	1.17990	0.589950	+	0.807440i	$0.299147\pi$
$720$	0	0
$721$	−11.2843	−0.420250
$722$	0	0
$723$	−62.3311	−2.31812
$724$	0	0
$725$	12.5686	0.466787
$726$	0	0
$727$	4.47815	0.166085	0.0830426	−	0.996546i	$-0.473536\pi$
$727$	4.47815	0.166085	0.0830426	+	0.996546i	$0.473536\pi$
$728$	0	0
$729$	−42.6921	−1.58119
$730$	0	0
$731$	−9.55577	−0.353433
$732$	0	0
$733$	32.2067	1.18958	0.594791	−	0.803881i	$-0.297235\pi$
$733$	32.2067	1.18958	0.594791	+	0.803881i	$0.297235\pi$
$734$	0	0
$735$	10.5064	0.387536
$736$	0	0
$737$	32.1300	1.18352
$738$	0	0
$739$	−38.6848	−1.42304	−0.711522	−	0.702663i	$-0.751994\pi$
$739$	−38.6848	−1.42304	−0.711522	+	0.702663i	$0.751994\pi$
$740$	0	0
$741$	−3.93780	−0.144659
$742$	0	0
$743$	−8.47815	−0.311033	−0.155517	−	0.987833i	$-0.549704\pi$
$743$	−8.47815	−0.311033	−0.155517	+	0.987833i	$0.549704\pi$
$744$	0	0
$745$	−66.8658	−2.44977
$746$	0	0
$747$	66.2883	2.42536
$748$	0	0
$749$	−1.02105	−0.0373084
$750$	0	0
$751$	−30.9928	−1.13094	−0.565471	−	0.824768i	$-0.691306\pi$
$751$	−30.9928	−1.13094	−0.565471	+	0.824768i	$0.691306\pi$
$752$	0	0
$753$	71.2114	2.59509
$754$	0	0
$755$	−23.3465	−0.849667
$756$	0	0
$757$	−1.47347	−0.0535542	−0.0267771	−	0.999641i	$-0.508524\pi$
$757$	−1.47347	−0.0535542	−0.0267771	+	0.999641i	$0.508524\pi$
$758$	0	0
$759$	−18.1017	−0.657052
$760$	0	0
$761$	−39.0047	−1.41392	−0.706959	−	0.707254i	$-0.749934\pi$
$761$	−39.0047	−1.41392	−0.706959	+	0.707254i	$0.749934\pi$
$762$	0	0
$763$	−9.46528	−0.342666
$764$	0	0
$765$	−20.1728	−0.729348
$766$	0	0
$767$	−6.64464	−0.239924
$768$	0	0
$769$	−28.9306	−1.04326	−0.521631	−	0.853171i	$-0.674676\pi$
$769$	−28.9306	−1.04326	−0.521631	+	0.853171i	$0.674676\pi$
$770$	0	0
$771$	−12.5969	−0.453667
$772$	0	0
$773$	−18.2212	−0.655370	−0.327685	−	0.944787i	$-0.606268\pi$
$773$	−18.2212	−0.655370	−0.327685	+	0.944787i	$0.606268\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	31.5193	1.13075
$778$	0	0
$779$	−4.15268	−0.148785
$780$	0	0
$781$	−18.4580	−0.660481
$782$	0	0
$783$	−4.01382	−0.143442
$784$	0	0
$785$	29.2642	1.04448
$786$	0	0
$787$	48.5265	1.72978	0.864892	−	0.501958i	$-0.167387\pi$
$787$	48.5265	1.72978	0.864892	+	0.501958i	$0.167387\pi$
$788$	0	0
$789$	−10.5824	−0.376745
$790$	0	0
$791$	12.3255	0.438243
$792$	0	0
$793$	17.4508	0.619697
$794$	0	0
$795$	−133.299	−4.72764
$796$	0	0
$797$	−31.2076	−1.10543	−0.552716	−	0.833370i	$-0.686408\pi$
$797$	−31.2076	−1.10543	−0.552716	+	0.833370i	$0.686408\pi$
$798$	0	0
$799$	4.01382	0.141999
$800$	0	0
$801$	−51.7769	−1.82945
$802$	0	0
$803$	27.9918	0.987810
$804$	0	0
$805$	8.17278	0.288053
$806$	0	0
$807$	−36.3317	−1.27894
$808$	0	0
$809$	43.3666	1.52469	0.762344	−	0.647172i	$-0.224048\pi$
$809$	43.3666	1.52469	0.762344	+	0.647172i	$0.224048\pi$
$810$	0	0
$811$	−49.8448	−1.75029	−0.875143	−	0.483864i	$-0.839233\pi$
$811$	−49.8448	−1.75029	−0.875143	+	0.483864i	$0.839233\pi$
$812$	0	0
$813$	79.0870	2.77370
$814$	0	0
$815$	−83.8083	−2.93568
$816$	0	0
$817$	7.81903	0.273553
$818$	0	0
$819$	6.18660	0.216177
$820$	0	0
$821$	−26.1810	−0.913722	−0.456861	−	0.889538i	$-0.651026\pi$
$821$	−26.1810	−0.913722	−0.456861	+	0.889538i	$0.651026\pi$
$822$	0	0
$823$	−31.7625	−1.10717	−0.553585	−	0.832793i	$-0.686741\pi$
$823$	−31.7625	−1.10717	−0.553585	+	0.832793i	$0.686741\pi$
$824$	0	0
$825$	−89.0531	−3.10043
$826$	0	0
$827$	−21.5476	−0.749283	−0.374641	−	0.927170i	$-0.622234\pi$
$827$	−21.5476	−0.749283	−0.374641	+	0.927170i	$0.622234\pi$
$828$	0	0
$829$	5.29250	0.183816	0.0919081	−	0.995767i	$-0.470703\pi$
$829$	5.29250	0.183816	0.0919081	+	0.995767i	$0.470703\pi$
$830$	0	0
$831$	−43.2196	−1.49927
$832$	0	0
$833$	1.22212	0.0423438
$834$	0	0
$835$	18.9224	0.654836
$836$	0	0
$837$	2.55481	0.0883073
$838$	0	0
$839$	41.6638	1.43839	0.719197	−	0.694806i	$-0.244510\pi$
$839$	41.6638	1.43839	0.719197	+	0.694806i	$0.244510\pi$
$840$	0	0
$841$	−27.5064	−0.948498
$842$	0	0
$843$	−44.0622	−1.51758
$844$	0	0
$845$	42.4298	1.45963
$846$	0	0
$847$	0.617966	0.0212336
$848$	0	0
$849$	14.2771	0.489988
$850$	0	0
$851$	24.5183	0.840478
$852$	0	0
$853$	13.5759	0.464829	0.232414	−	0.972617i	$-0.425337\pi$
$853$	13.5759	0.464829	0.232414	+	0.972617i	$0.425337\pi$
$854$	0	0
$855$	16.5064	0.564508
$856$	0	0
$857$	−14.4725	−0.494372	−0.247186	−	0.968968i	$-0.579506\pi$
$857$	−14.4725	−0.494372	−0.247186	+	0.968968i	$0.579506\pi$
$858$	0	0
$859$	−8.59692	−0.293323	−0.146661	−	0.989187i	$-0.546853\pi$
$859$	−8.59692	−0.293323	−0.146661	+	0.989187i	$0.546853\pi$
$860$	0	0
$861$	11.1599	0.380329
$862$	0	0
$863$	17.6180	0.599723	0.299861	−	0.953983i	$-0.403060\pi$
$863$	17.6180	0.599723	0.299861	+	0.953983i	$0.403060\pi$
$864$	0	0
$865$	47.9434	1.63013
$866$	0	0
$867$	41.6720	1.41525
$868$	0	0
$869$	−25.1938	−0.854642
$870$	0	0
$871$	−14.6114	−0.495088
$872$	0	0
$873$	−72.2123	−2.44402
$874$	0	0
$875$	20.6591	0.698406
$876$	0	0
$877$	7.37480	0.249029	0.124515	−	0.992218i	$-0.460263\pi$
$877$	7.37480	0.249029	0.124515	+	0.992218i	$0.460263\pi$
$878$	0	0
$879$	−53.5048	−1.80467
$880$	0	0
$881$	−12.1043	−0.407804	−0.203902	−	0.978991i	$-0.565362\pi$
$881$	−12.1043	−0.407804	−0.203902	+	0.978991i	$0.565362\pi$
$882$	0	0
$883$	−28.6591	−0.964456	−0.482228	−	0.876046i	$-0.660172\pi$
$883$	−28.6591	−0.964456	−0.482228	+	0.876046i	$0.660172\pi$
$884$	0	0
$885$	47.6437	1.60153
$886$	0	0
$887$	34.4360	1.15625	0.578125	−	0.815948i	$-0.303785\pi$
$887$	34.4360	1.15625	0.578125	+	0.815948i	$0.303785\pi$
$888$	0	0
$889$	−2.80617	−0.0941159
$890$	0	0
$891$	−12.3732	−0.414518
$892$	0	0
$893$	−3.28432	−0.109905
$894$	0	0
$895$	−89.8586	−3.00364
$896$	0	0
$897$	8.23191	0.274856
$898$	0	0
$899$	−0.950664	−0.0317064
$900$	0	0
$901$	−15.5055	−0.516562
$902$	0	0
$903$	−21.0129	−0.699264
$904$	0	0
$905$	71.9974	2.39328
$906$	0	0
$907$	2.92238	0.0970360	0.0485180	−	0.998822i	$-0.484550\pi$
$907$	2.92238	0.0970360	0.0485180	+	0.998822i	$0.484550\pi$
$908$	0	0
$909$	−56.8520	−1.88566
$910$	0	0
$911$	23.8190	0.789160	0.394580	−	0.918862i	$-0.370890\pi$
$911$	23.8190	0.789160	0.394580	+	0.918862i	$0.370890\pi$
$912$	0	0
$913$	50.5881	1.67422
$914$	0	0
$915$	−125.127	−4.13655
$916$	0	0
$917$	−2.15268	−0.0710878
$918$	0	0
$919$	34.2149	1.12864	0.564322	−	0.825554i	$-0.309137\pi$
$919$	34.2149	1.12864	0.564322	+	0.825554i	$0.309137\pi$
$920$	0	0
$921$	−1.26985	−0.0418428
$922$	0	0
$923$	8.39394	0.276290
$924$	0	0
$925$	120.620	3.96596
$926$	0	0
$927$	47.6437	1.56482
$928$	0	0
$929$	−13.8812	−0.455428	−0.227714	−	0.973728i	$-0.573125\pi$
$929$	−13.8812	−0.455428	−0.227714	+	0.973728i	$0.573125\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	6.27145	0.205318
$934$	0	0
$935$	−15.3949	−0.503467
$936$	0	0
$937$	41.4709	1.35480	0.677398	−	0.735617i	$-0.263108\pi$
$937$	41.4709	1.35480	0.677398	+	0.735617i	$0.263108\pi$
$938$	0	0
$939$	−8.58310	−0.280099
$940$	0	0
$941$	34.1646	1.11373	0.556867	−	0.830602i	$-0.312003\pi$
$941$	34.1646	1.11373	0.556867	+	0.830602i	$0.312003\pi$
$942$	0	0
$943$	8.68112	0.282696
$944$	0	0
$945$	−12.8401	−0.417688
$946$	0	0
$947$	−0.520901	−0.0169270	−0.00846351	−	0.999964i	$-0.502694\pi$
$947$	−0.520901	−0.0169270	−0.00846351	+	0.999964i	$0.502694\pi$
$948$	0	0
$949$	−12.7295	−0.413217
$950$	0	0
$951$	−34.2633	−1.11106
$952$	0	0
$953$	21.1655	0.685619	0.342810	−	0.939405i	$-0.388621\pi$
$953$	21.1655	0.685619	0.342810	+	0.939405i	$0.388621\pi$
$954$	0	0
$955$	−66.2689	−2.14441
$956$	0	0
$957$	−10.5824	−0.342082
$958$	0	0
$959$	16.7496	0.540873
$960$	0	0
$961$	−30.3949	−0.980481
$962$	0	0
$963$	4.31099	0.138920
$964$	0	0
$965$	−15.5275	−0.499847
$966$	0	0
$967$	−30.2607	−0.973119	−0.486559	−	0.873648i	$-0.661748\pi$
$967$	−30.2607	−0.973119	−0.486559	+	0.873648i	$0.661748\pi$
$968$	0	0
$969$	3.28432	0.105507
$970$	0	0
$971$	5.60415	0.179846	0.0899229	−	0.995949i	$-0.471338\pi$
$971$	5.60415	0.179846	0.0899229	+	0.995949i	$0.471338\pi$
$972$	0	0
$973$	−0.715685	−0.0229438
$974$	0	0
$975$	40.4976	1.29696
$976$	0	0
$977$	−13.0550	−0.417665	−0.208833	−	0.977951i	$-0.566966\pi$
$977$	−13.0550	−0.417665	−0.208833	+	0.977951i	$0.566966\pi$
$978$	0	0
$979$	−39.5137	−1.26286
$980$	0	0
$981$	39.9635	1.27594
$982$	0	0
$983$	48.7835	1.55595	0.777976	−	0.628294i	$-0.216246\pi$
$983$	48.7835	1.55595	0.777976	+	0.628294i	$0.216246\pi$
$984$	0	0
$985$	−54.9764	−1.75169
$986$	0	0
$987$	8.82627	0.280943
$988$	0	0
$989$	−16.3456	−0.519759
$990$	0	0
$991$	−5.90133	−0.187462	−0.0937309	−	0.995598i	$-0.529879\pi$
$991$	−5.90133	−0.187462	−0.0937309	+	0.995598i	$0.529879\pi$
$992$	0	0
$993$	−78.6226	−2.49501
$994$	0	0
$995$	60.4618	1.91677
$996$	0	0
$997$	−1.08953	−0.0345058	−0.0172529	−	0.999851i	$-0.505492\pi$
$997$	−1.08953	−0.0345058	−0.0172529	+	0.999851i	$0.505492\pi$
$998$	0	0
$999$	−38.5202	−1.21873

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.bk.1.1		3
4.3	odd	2		8512.2.a.bo.1.3		3
8.3	odd	2		1064.2.a.f.1.1	✓	3
8.5	even	2		2128.2.a.q.1.3		3
24.11	even	2		9576.2.a.ca.1.1		3
56.27	even	2		7448.2.a.bi.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.f.1.1	✓	3	8.3	odd	2
2128.2.a.q.1.3		3	8.5	even	2
7448.2.a.bi.1.3		3	56.27	even	2
8512.2.a.bk.1.1		3	1.1	even	1	trivial
8512.2.a.bo.1.3		3	4.3	odd	2
9576.2.a.ca.1.1		3	24.11	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:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q-2.68740 q^{3} -3.90952 q^{5} -1.00000 q^{7} +4.22212 q^{9} +O(q^{10})\)	\(q-2.68740 q^{3} -3.90952 q^{5} -1.00000 q^{7} +4.22212 q^{9} +3.22212 q^{11} -1.46528 q^{13} +10.5064 q^{15} +1.22212 q^{17} -1.00000 q^{19} +2.68740 q^{21} +2.09048 q^{23} +10.2843 q^{25} -3.28432 q^{27} +1.22212 q^{29} -0.777884 q^{31} -8.65911 q^{33} +3.90952 q^{35} +11.7285 q^{37} +3.93780 q^{39} +4.15268 q^{41} -7.81903 q^{43} -16.5064 q^{45} +3.28432 q^{47} +1.00000 q^{49} -3.28432 q^{51} -12.6874 q^{53} -12.5969 q^{55} +2.68740 q^{57} +4.53472 q^{59} -11.9095 q^{61} -4.22212 q^{63} +5.72855 q^{65} +9.97171 q^{67} -5.61797 q^{69} -5.72855 q^{71} +8.68740 q^{73} -27.6381 q^{75} -3.22212 q^{77} -7.81903 q^{79} -3.84008 q^{81} +15.7003 q^{83} -4.77788 q^{85} -3.28432 q^{87} -12.2633 q^{89} +1.46528 q^{91} +2.09048 q^{93} +3.90952 q^{95} -17.1033 q^{97} +13.6042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 2 q^{5} - 3 q^{7} + 10 q^{9}+O(q^{10}) \) 3 * q - q^3 - 2 * q^5 - 3 * q^7 + 10 * q^9	\( 3 q - q^{3} - 2 q^{5} - 3 q^{7} + 10 q^{9} + 7 q^{11} + 5 q^{15} + q^{17} - 3 q^{19} + q^{21} + 16 q^{23} + 7 q^{25} + 14 q^{27} + q^{29} - 5 q^{31} + 12 q^{33} + 2 q^{35} + 6 q^{37} + 33 q^{39} + q^{41} - 4 q^{43} - 23 q^{45} - 14 q^{47} + 3 q^{49} + 14 q^{51} - 31 q^{53} - 21 q^{55} + q^{57} + 18 q^{59} - 26 q^{61} - 10 q^{63} - 12 q^{65} - q^{67} - q^{69} + 12 q^{71} + 19 q^{73} - 44 q^{75} - 7 q^{77} - 4 q^{79} + 7 q^{81} - 13 q^{83} - 17 q^{85} + 14 q^{87} - 12 q^{89} + 16 q^{93} + 2 q^{95} - 8 q^{97} + 54 q^{99}+O(q^{100}) \) 3 * q - q^3 - 2 * q^5 - 3 * q^7 + 10 * q^9 + 7 * q^11 + 5 * q^15 + q^17 - 3 * q^19 + q^21 + 16 * q^23 + 7 * q^25 + 14 * q^27 + q^29 - 5 * q^31 + 12 * q^33 + 2 * q^35 + 6 * q^37 + 33 * q^39 + q^41 - 4 * q^43 - 23 * q^45 - 14 * q^47 + 3 * q^49 + 14 * q^51 - 31 * q^53 - 21 * q^55 + q^57 + 18 * q^59 - 26 * q^61 - 10 * q^63 - 12 * q^65 - q^67 - q^69 + 12 * q^71 + 19 * q^73 - 44 * q^75 - 7 * q^77 - 4 * q^79 + 7 * q^81 - 13 * q^83 - 17 * q^85 + 14 * q^87 - 12 * q^89 + 16 * q^93 + 2 * q^95 - 8 * q^97 + 54 * q^99

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