LMFDB - Embedded newform 8512.2.a.ci.1.6

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 10x^{5} + 31x^{4} + 12x^{3} - 45x^{2} - 15x + 8 $ x^7 - 3x^6 - 10x^5 + 31x^4 + 12x^3 - 45x^2 - 15x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4256)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.33548$ 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)

$f(q)$	$=$	$q+2.33548 q^{3} +4.35508 q^{5} +1.00000 q^{7} +2.45447 q^{9} +O(q^{10})$	$q+2.33548 q^{3} +4.35508 q^{5} +1.00000 q^{7} +2.45447 q^{9} +0.727784 q^{11} +6.02561 q^{13} +10.1712 q^{15} -0.705165 q^{17} -1.00000 q^{19} +2.33548 q^{21} +2.70940 q^{23} +13.9667 q^{25} -1.27408 q^{27} -6.12351 q^{29} -9.48843 q^{31} +1.69973 q^{33} +4.35508 q^{35} +2.17514 q^{37} +14.0727 q^{39} +4.90950 q^{41} -2.88454 q^{43} +10.6894 q^{45} +3.05692 q^{47} +1.00000 q^{49} -1.64690 q^{51} -7.16153 q^{53} +3.16956 q^{55} -2.33548 q^{57} +5.08712 q^{59} -8.14526 q^{61} +2.45447 q^{63} +26.2420 q^{65} +7.65689 q^{67} +6.32775 q^{69} -13.8880 q^{71} +12.8979 q^{73} +32.6190 q^{75} +0.727784 q^{77} +7.93879 q^{79} -10.3390 q^{81} -14.4906 q^{83} -3.07105 q^{85} -14.3013 q^{87} +9.13798 q^{89} +6.02561 q^{91} -22.1600 q^{93} -4.35508 q^{95} -5.08230 q^{97} +1.78632 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} + 5 q^{5} + 7 q^{7} + 8 q^{9}+O(q^{10}) $ 7 * q + 3 * q^3 + 5 * q^5 + 7 * q^7 + 8 * q^9	$ 7 q + 3 q^{3} + 5 q^{5} + 7 q^{7} + 8 q^{9} + 3 q^{11} + 16 q^{13} - 8 q^{15} + 8 q^{17} - 7 q^{19} + 3 q^{21} - 10 q^{23} + 8 q^{25} + 6 q^{27} + 11 q^{29} - 14 q^{31} + 3 q^{33} + 5 q^{35} + 13 q^{37} + 6 q^{39} + 11 q^{41} + 11 q^{43} - 4 q^{45} - 7 q^{47} + 7 q^{49} + 24 q^{51} + 9 q^{53} + 8 q^{55} - 3 q^{57} + 23 q^{59} + 23 q^{61} + 8 q^{63} + 40 q^{65} + 16 q^{67} + 10 q^{69} - 3 q^{71} + 4 q^{73} + 48 q^{75} + 3 q^{77} - 29 q^{79} + 23 q^{81} + 6 q^{83} + 6 q^{85} + 6 q^{87} + 15 q^{89} + 16 q^{91} - 42 q^{93} - 5 q^{95} - 13 q^{97} + 47 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 + 5 * q^5 + 7 * q^7 + 8 * q^9 + 3 * q^11 + 16 * q^13 - 8 * q^15 + 8 * q^17 - 7 * q^19 + 3 * q^21 - 10 * q^23 + 8 * q^25 + 6 * q^27 + 11 * q^29 - 14 * q^31 + 3 * q^33 + 5 * q^35 + 13 * q^37 + 6 * q^39 + 11 * q^41 + 11 * q^43 - 4 * q^45 - 7 * q^47 + 7 * q^49 + 24 * q^51 + 9 * q^53 + 8 * q^55 - 3 * q^57 + 23 * q^59 + 23 * q^61 + 8 * q^63 + 40 * q^65 + 16 * q^67 + 10 * q^69 - 3 * q^71 + 4 * q^73 + 48 * q^75 + 3 * q^77 - 29 * q^79 + 23 * q^81 + 6 * q^83 + 6 * q^85 + 6 * q^87 + 15 * q^89 + 16 * q^91 - 42 * q^93 - 5 * q^95 - 13 * q^97 + 47 * 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
$3$	2.33548	1.34839	0.674195	−	0.738553i	$-0.264491\pi$
$3$	2.33548	1.34839	0.674195	+	0.738553i	$0.264491\pi$
$4$	0	0
$5$	4.35508	1.94765	0.973826	−	0.227295i	$-0.0729883\pi$
$5$	4.35508	1.94765	0.973826	+	0.227295i	$0.0729883\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.45447	0.818156
$10$	0	0
$11$	0.727784	0.219435	0.109718	−	0.993963i	$-0.465005\pi$
$11$	0.727784	0.219435	0.109718	+	0.993963i	$0.465005\pi$
$12$	0	0
$13$	6.02561	1.67120	0.835602	−	0.549335i	$-0.185119\pi$
$13$	6.02561	1.67120	0.835602	+	0.549335i	$0.185119\pi$
$14$	0	0
$15$	10.1712	2.62619
$16$	0	0
$17$	−0.705165	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$17$	−0.705165	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.33548	0.509644
$22$	0	0
$23$	2.70940	0.564949	0.282474	−	0.959275i	$-0.408845\pi$
$23$	2.70940	0.564949	0.282474	+	0.959275i	$0.408845\pi$
$24$	0	0
$25$	13.9667	2.79335
$26$	0	0
$27$	−1.27408	−0.245197
$28$	0	0
$29$	−6.12351	−1.13711	−0.568554	−	0.822646i	$-0.692497\pi$
$29$	−6.12351	−1.13711	−0.568554	+	0.822646i	$0.692497\pi$
$30$	0	0
$31$	−9.48843	−1.70417	−0.852086	−	0.523401i	$-0.824663\pi$
$31$	−9.48843	−1.70417	−0.852086	+	0.523401i	$0.824663\pi$
$32$	0	0
$33$	1.69973	0.295884
$34$	0	0
$35$	4.35508	0.736143
$36$	0	0
$37$	2.17514	0.357591	0.178795	−	0.983886i	$-0.442780\pi$
$37$	2.17514	0.357591	0.178795	+	0.983886i	$0.442780\pi$
$38$	0	0
$39$	14.0727	2.25344
$40$	0	0
$41$	4.90950	0.766735	0.383368	−	0.923596i	$-0.374764\pi$
$41$	4.90950	0.766735	0.383368	+	0.923596i	$0.374764\pi$
$42$	0	0
$43$	−2.88454	−0.439888	−0.219944	−	0.975512i	$-0.570588\pi$
$43$	−2.88454	−0.439888	−0.219944	+	0.975512i	$0.570588\pi$
$44$	0	0
$45$	10.6894	1.59348
$46$	0	0
$47$	3.05692	0.445897	0.222949	−	0.974830i	$-0.428432\pi$
$47$	3.05692	0.445897	0.222949	+	0.974830i	$0.428432\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.64690	−0.230612
$52$	0	0
$53$	−7.16153	−0.983712	−0.491856	−	0.870677i	$-0.663681\pi$
$53$	−7.16153	−0.983712	−0.491856	+	0.870677i	$0.663681\pi$
$54$	0	0
$55$	3.16956	0.427383
$56$	0	0
$57$	−2.33548	−0.309342
$58$	0	0
$59$	5.08712	0.662287	0.331144	−	0.943580i	$-0.392566\pi$
$59$	5.08712	0.662287	0.331144	+	0.943580i	$0.392566\pi$
$60$	0	0
$61$	−8.14526	−1.04289	−0.521447	−	0.853284i	$-0.674607\pi$
$61$	−8.14526	−1.04289	−0.521447	+	0.853284i	$0.674607\pi$
$62$	0	0
$63$	2.45447	0.309234
$64$	0	0
$65$	26.2420	3.25492
$66$	0	0
$67$	7.65689	0.935438	0.467719	−	0.883877i	$-0.345076\pi$
$67$	7.65689	0.935438	0.467719	+	0.883877i	$0.345076\pi$
$68$	0	0
$69$	6.32775	0.761772
$70$	0	0
$71$	−13.8880	−1.64820	−0.824099	−	0.566446i	$-0.808318\pi$
$71$	−13.8880	−1.64820	−0.824099	+	0.566446i	$0.808318\pi$
$72$	0	0
$73$	12.8979	1.50958	0.754792	−	0.655964i	$-0.227738\pi$
$73$	12.8979	1.50958	0.754792	+	0.655964i	$0.227738\pi$
$74$	0	0
$75$	32.6190	3.76652
$76$	0	0
$77$	0.727784	0.0829387
$78$	0	0
$79$	7.93879	0.893183	0.446592	−	0.894738i	$-0.352638\pi$
$79$	7.93879	0.893183	0.446592	+	0.894738i	$0.352638\pi$
$80$	0	0
$81$	−10.3390	−1.14878
$82$	0	0
$83$	−14.4906	−1.59055	−0.795276	−	0.606248i	$-0.792674\pi$
$83$	−14.4906	−1.59055	−0.795276	+	0.606248i	$0.792674\pi$
$84$	0	0
$85$	−3.07105	−0.333102
$86$	0	0
$87$	−14.3013	−1.53326
$88$	0	0
$89$	9.13798	0.968624	0.484312	−	0.874895i	$-0.339070\pi$
$89$	9.13798	0.968624	0.484312	+	0.874895i	$0.339070\pi$
$90$	0	0
$91$	6.02561	0.631656
$92$	0	0
$93$	−22.1600	−2.29789
$94$	0	0
$95$	−4.35508	−0.446822
$96$	0	0
$97$	−5.08230	−0.516029	−0.258015	−	0.966141i	$-0.583068\pi$
$97$	−5.08230	−0.516029	−0.258015	+	0.966141i	$0.583068\pi$
$98$	0	0
$99$	1.78632	0.179532
$100$	0	0
$101$	13.7530	1.36848	0.684239	−	0.729258i	$-0.260134\pi$
$101$	13.7530	1.36848	0.684239	+	0.729258i	$0.260134\pi$
$102$	0	0
$103$	−9.94273	−0.979686	−0.489843	−	0.871811i	$-0.662946\pi$
$103$	−9.94273	−0.979686	−0.489843	+	0.871811i	$0.662946\pi$
$104$	0	0
$105$	10.1712	0.992608
$106$	0	0
$107$	−4.71157	−0.455485	−0.227742	−	0.973721i	$-0.573134\pi$
$107$	−4.71157	−0.455485	−0.227742	+	0.973721i	$0.573134\pi$
$108$	0	0
$109$	−10.2915	−0.985749	−0.492875	−	0.870100i	$-0.664054\pi$
$109$	−10.2915	−0.985749	−0.492875	+	0.870100i	$0.664054\pi$
$110$	0	0
$111$	5.08000	0.482172
$112$	0	0
$113$	8.54159	0.803525	0.401763	−	0.915744i	$-0.368398\pi$
$113$	8.54159	0.803525	0.401763	+	0.915744i	$0.368398\pi$
$114$	0	0
$115$	11.7997	1.10032
$116$	0	0
$117$	14.7897	1.36731
$118$	0	0
$119$	−0.705165	−0.0646424
$120$	0	0
$121$	−10.4703	−0.951848
$122$	0	0
$123$	11.4660	1.03386
$124$	0	0
$125$	39.0509	3.49282
$126$	0	0
$127$	3.75196	0.332933	0.166466	−	0.986047i	$-0.446764\pi$
$127$	3.75196	0.332933	0.166466	+	0.986047i	$0.446764\pi$
$128$	0	0
$129$	−6.73679	−0.593141
$130$	0	0
$131$	−13.3308	−1.16472	−0.582358	−	0.812933i	$-0.697870\pi$
$131$	−13.3308	−1.16472	−0.582358	+	0.812933i	$0.697870\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−5.54872	−0.477558
$136$	0	0
$137$	−7.26783	−0.620932	−0.310466	−	0.950584i	$-0.600485\pi$
$137$	−7.26783	−0.620932	−0.310466	+	0.950584i	$0.600485\pi$
$138$	0	0
$139$	3.01908	0.256075	0.128038	−	0.991769i	$-0.459132\pi$
$139$	3.01908	0.256075	0.128038	+	0.991769i	$0.459132\pi$
$140$	0	0
$141$	7.13937	0.601243
$142$	0	0
$143$	4.38535	0.366721
$144$	0	0
$145$	−26.6684	−2.21469
$146$	0	0
$147$	2.33548	0.192627
$148$	0	0
$149$	−16.6216	−1.36169	−0.680845	−	0.732427i	$-0.738387\pi$
$149$	−16.6216	−1.36169	−0.680845	+	0.732427i	$0.738387\pi$
$150$	0	0
$151$	−8.15095	−0.663315	−0.331658	−	0.943400i	$-0.607608\pi$
$151$	−8.15095	−0.663315	−0.331658	+	0.943400i	$0.607608\pi$
$152$	0	0
$153$	−1.73080	−0.139927
$154$	0	0
$155$	−41.3229	−3.31913
$156$	0	0
$157$	−12.9731	−1.03537	−0.517685	−	0.855571i	$-0.673206\pi$
$157$	−12.9731	−1.03537	−0.517685	+	0.855571i	$0.673206\pi$
$158$	0	0
$159$	−16.7256	−1.32643
$160$	0	0
$161$	2.70940	0.213531
$162$	0	0
$163$	1.49471	0.117075	0.0585376	−	0.998285i	$-0.481356\pi$
$163$	1.49471	0.117075	0.0585376	+	0.998285i	$0.481356\pi$
$164$	0	0
$165$	7.40244	0.576280
$166$	0	0
$167$	17.9188	1.38660	0.693298	−	0.720651i	$-0.256157\pi$
$167$	17.9188	1.38660	0.693298	+	0.720651i	$0.256157\pi$
$168$	0	0
$169$	23.3080	1.79292
$170$	0	0
$171$	−2.45447	−0.187698
$172$	0	0
$173$	19.3923	1.47437	0.737184	−	0.675692i	$-0.236155\pi$
$173$	19.3923	1.47437	0.737184	+	0.675692i	$0.236155\pi$
$174$	0	0
$175$	13.9667	1.05579
$176$	0	0
$177$	11.8809	0.893021
$178$	0	0
$179$	20.9036	1.56241	0.781206	−	0.624273i	$-0.214605\pi$
$179$	20.9036	1.56241	0.781206	+	0.624273i	$0.214605\pi$
$180$	0	0
$181$	−7.48651	−0.556468	−0.278234	−	0.960513i	$-0.589749\pi$
$181$	−7.48651	−0.556468	−0.278234	+	0.960513i	$0.589749\pi$
$182$	0	0
$183$	−19.0231	−1.40623
$184$	0	0
$185$	9.47292	0.696463
$186$	0	0
$187$	−0.513208	−0.0375295
$188$	0	0
$189$	−1.27408	−0.0926757
$190$	0	0
$191$	−17.6406	−1.27643	−0.638214	−	0.769859i	$-0.720326\pi$
$191$	−17.6406	−1.27643	−0.638214	+	0.769859i	$0.720326\pi$
$192$	0	0
$193$	−23.7133	−1.70692	−0.853461	−	0.521156i	$-0.825501\pi$
$193$	−23.7133	−1.70692	−0.853461	+	0.521156i	$0.825501\pi$
$194$	0	0
$195$	61.2877	4.38891
$196$	0	0
$197$	2.66673	0.189996	0.0949982	−	0.995477i	$-0.469715\pi$
$197$	2.66673	0.189996	0.0949982	+	0.995477i	$0.469715\pi$
$198$	0	0
$199$	−23.7518	−1.68372	−0.841861	−	0.539695i	$-0.818540\pi$
$199$	−23.7518	−1.68372	−0.841861	+	0.539695i	$0.818540\pi$
$200$	0	0
$201$	17.8825	1.26133
$202$	0	0
$203$	−6.12351	−0.429786
$204$	0	0
$205$	21.3813	1.49333
$206$	0	0
$207$	6.65014	0.462216
$208$	0	0
$209$	−0.727784	−0.0503419
$210$	0	0
$211$	18.8030	1.29445	0.647226	−	0.762298i	$-0.275929\pi$
$211$	18.8030	1.29445	0.647226	+	0.762298i	$0.275929\pi$
$212$	0	0
$213$	−32.4351	−2.22241
$214$	0	0
$215$	−12.5624	−0.856749
$216$	0	0
$217$	−9.48843	−0.644117
$218$	0	0
$219$	30.1228	2.03551
$220$	0	0
$221$	−4.24905	−0.285822
$222$	0	0
$223$	−0.0710894	−0.00476050	−0.00238025	−	0.999997i	$-0.500758\pi$
$223$	−0.0710894	−0.00476050	−0.00238025	+	0.999997i	$0.500758\pi$
$224$	0	0
$225$	34.2809	2.28539
$226$	0	0
$227$	11.3837	0.755562	0.377781	−	0.925895i	$-0.376687\pi$
$227$	11.3837	0.755562	0.377781	+	0.925895i	$0.376687\pi$
$228$	0	0
$229$	6.24546	0.412711	0.206356	−	0.978477i	$-0.433840\pi$
$229$	6.24546	0.412711	0.206356	+	0.978477i	$0.433840\pi$
$230$	0	0
$231$	1.69973	0.111834
$232$	0	0
$233$	0.967947	0.0634123	0.0317062	−	0.999497i	$-0.489906\pi$
$233$	0.967947	0.0634123	0.0317062	+	0.999497i	$0.489906\pi$
$234$	0	0
$235$	13.3131	0.868452
$236$	0	0
$237$	18.5409	1.20436
$238$	0	0
$239$	−0.240940	−0.0155851	−0.00779255	−	0.999970i	$-0.502480\pi$
$239$	−0.240940	−0.0155851	−0.00779255	+	0.999970i	$0.502480\pi$
$240$	0	0
$241$	10.9047	0.702437	0.351218	−	0.936294i	$-0.385767\pi$
$241$	10.9047	0.702437	0.351218	+	0.936294i	$0.385767\pi$
$242$	0	0
$243$	−20.3243	−1.30380
$244$	0	0
$245$	4.35508	0.278236
$246$	0	0
$247$	−6.02561	−0.383401
$248$	0	0
$249$	−33.8425	−2.14468
$250$	0	0
$251$	−14.0160	−0.884680	−0.442340	−	0.896848i	$-0.645851\pi$
$251$	−14.0160	−0.884680	−0.442340	+	0.896848i	$0.645851\pi$
$252$	0	0
$253$	1.97186	0.123970
$254$	0	0
$255$	−7.17238	−0.449152
$256$	0	0
$257$	−7.33803	−0.457734	−0.228867	−	0.973458i	$-0.573502\pi$
$257$	−7.33803	−0.457734	−0.228867	+	0.973458i	$0.573502\pi$
$258$	0	0
$259$	2.17514	0.135157
$260$	0	0
$261$	−15.0300	−0.930331
$262$	0	0
$263$	−6.05915	−0.373623	−0.186812	−	0.982396i	$-0.559815\pi$
$263$	−6.05915	−0.373623	−0.186812	+	0.982396i	$0.559815\pi$
$264$	0	0
$265$	−31.1891	−1.91593
$266$	0	0
$267$	21.3416	1.30608
$268$	0	0
$269$	17.9340	1.09346	0.546729	−	0.837310i	$-0.315873\pi$
$269$	17.9340	1.09346	0.546729	+	0.837310i	$0.315873\pi$
$270$	0	0
$271$	−16.7234	−1.01588	−0.507938	−	0.861394i	$-0.669592\pi$
$271$	−16.7234	−1.01588	−0.507938	+	0.861394i	$0.669592\pi$
$272$	0	0
$273$	14.0727	0.851718
$274$	0	0
$275$	10.1648	0.612959
$276$	0	0
$277$	5.27084	0.316694	0.158347	−	0.987384i	$-0.449384\pi$
$277$	5.27084	0.316694	0.158347	+	0.987384i	$0.449384\pi$
$278$	0	0
$279$	−23.2890	−1.39428
$280$	0	0
$281$	11.0307	0.658039	0.329020	−	0.944323i	$-0.393282\pi$
$281$	11.0307	0.658039	0.329020	+	0.944323i	$0.393282\pi$
$282$	0	0
$283$	4.23169	0.251548	0.125774	−	0.992059i	$-0.459859\pi$
$283$	4.23169	0.251548	0.125774	+	0.992059i	$0.459859\pi$
$284$	0	0
$285$	−10.1712	−0.602490
$286$	0	0
$287$	4.90950	0.289799
$288$	0	0
$289$	−16.5027	−0.970750
$290$	0	0
$291$	−11.8696	−0.695809
$292$	0	0
$293$	−28.0755	−1.64019	−0.820094	−	0.572229i	$-0.806079\pi$
$293$	−28.0755	−1.64019	−0.820094	+	0.572229i	$0.806079\pi$
$294$	0	0
$295$	22.1548	1.28990
$296$	0	0
$297$	−0.927255	−0.0538048
$298$	0	0
$299$	16.3258	0.944145
$300$	0	0
$301$	−2.88454	−0.166262
$302$	0	0
$303$	32.1199	1.84524
$304$	0	0
$305$	−35.4733	−2.03119
$306$	0	0
$307$	2.30684	0.131658	0.0658292	−	0.997831i	$-0.479031\pi$
$307$	2.30684	0.131658	0.0658292	+	0.997831i	$0.479031\pi$
$308$	0	0
$309$	−23.2210	−1.32100
$310$	0	0
$311$	11.1467	0.632071	0.316035	−	0.948747i	$-0.397648\pi$
$311$	11.1467	0.632071	0.316035	+	0.948747i	$0.397648\pi$
$312$	0	0
$313$	−10.4900	−0.592930	−0.296465	−	0.955044i	$-0.595808\pi$
$313$	−10.4900	−0.592930	−0.296465	+	0.955044i	$0.595808\pi$
$314$	0	0
$315$	10.6894	0.602280
$316$	0	0
$317$	−1.44510	−0.0811650	−0.0405825	−	0.999176i	$-0.512921\pi$
$317$	−1.44510	−0.0811650	−0.0405825	+	0.999176i	$0.512921\pi$
$318$	0	0
$319$	−4.45659	−0.249521
$320$	0	0
$321$	−11.0038	−0.614171
$322$	0	0
$323$	0.705165	0.0392364
$324$	0	0
$325$	84.1581	4.66825
$326$	0	0
$327$	−24.0357	−1.32917
$328$	0	0
$329$	3.05692	0.168533
$330$	0	0
$331$	29.8426	1.64030	0.820149	−	0.572150i	$-0.193890\pi$
$331$	29.8426	1.64030	0.820149	+	0.572150i	$0.193890\pi$
$332$	0	0
$333$	5.33881	0.292565
$334$	0	0
$335$	33.3464	1.82191
$336$	0	0
$337$	28.3083	1.54205	0.771027	−	0.636803i	$-0.219743\pi$
$337$	28.3083	1.54205	0.771027	+	0.636803i	$0.219743\pi$
$338$	0	0
$339$	19.9487	1.08347
$340$	0	0
$341$	−6.90553	−0.373955
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	27.5579	1.48367
$346$	0	0
$347$	−5.16047	−0.277028	−0.138514	−	0.990360i	$-0.544233\pi$
$347$	−5.16047	−0.277028	−0.138514	+	0.990360i	$0.544233\pi$
$348$	0	0
$349$	24.9633	1.33626	0.668128	−	0.744046i	$-0.267096\pi$
$349$	24.9633	1.33626	0.668128	+	0.744046i	$0.267096\pi$
$350$	0	0
$351$	−7.67711	−0.409774
$352$	0	0
$353$	7.92056	0.421569	0.210784	−	0.977533i	$-0.432398\pi$
$353$	7.92056	0.421569	0.210784	+	0.977533i	$0.432398\pi$
$354$	0	0
$355$	−60.4832	−3.21012
$356$	0	0
$357$	−1.64690	−0.0871631
$358$	0	0
$359$	−20.4147	−1.07744	−0.538722	−	0.842483i	$-0.681093\pi$
$359$	−20.4147	−1.07744	−0.538722	+	0.842483i	$0.681093\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−24.4532	−1.28346
$364$	0	0
$365$	56.1714	2.94014
$366$	0	0
$367$	4.64481	0.242457	0.121229	−	0.992625i	$-0.461317\pi$
$367$	4.64481	0.242457	0.121229	+	0.992625i	$0.461317\pi$
$368$	0	0
$369$	12.0502	0.627309
$370$	0	0
$371$	−7.16153	−0.371808
$372$	0	0
$373$	11.6472	0.603069	0.301534	−	0.953455i	$-0.402501\pi$
$373$	11.6472	0.603069	0.301534	+	0.953455i	$0.402501\pi$
$374$	0	0
$375$	91.2025	4.70968
$376$	0	0
$377$	−36.8979	−1.90034
$378$	0	0
$379$	34.3117	1.76248	0.881238	−	0.472674i	$-0.156711\pi$
$379$	34.3117	1.76248	0.881238	+	0.472674i	$0.156711\pi$
$380$	0	0
$381$	8.76263	0.448923
$382$	0	0
$383$	9.63514	0.492333	0.246166	−	0.969228i	$-0.420829\pi$
$383$	9.63514	0.492333	0.246166	+	0.969228i	$0.420829\pi$
$384$	0	0
$385$	3.16956	0.161536
$386$	0	0
$387$	−7.08001	−0.359897
$388$	0	0
$389$	−1.87587	−0.0951103	−0.0475551	−	0.998869i	$-0.515143\pi$
$389$	−1.87587	−0.0951103	−0.0475551	+	0.998869i	$0.515143\pi$
$390$	0	0
$391$	−1.91057	−0.0966219
$392$	0	0
$393$	−31.1338	−1.57049
$394$	0	0
$395$	34.5741	1.73961
$396$	0	0
$397$	−19.7490	−0.991173	−0.495586	−	0.868559i	$-0.665047\pi$
$397$	−19.7490	−0.991173	−0.495586	+	0.868559i	$0.665047\pi$
$398$	0	0
$399$	−2.33548	−0.116920
$400$	0	0
$401$	37.9273	1.89400	0.946999	−	0.321237i	$-0.104099\pi$
$401$	37.9273	1.89400	0.946999	+	0.321237i	$0.104099\pi$
$402$	0	0
$403$	−57.1736	−2.84802
$404$	0	0
$405$	−45.0272	−2.23742
$406$	0	0
$407$	1.58303	0.0784680
$408$	0	0
$409$	−36.6539	−1.81242	−0.906209	−	0.422829i	$-0.861037\pi$
$409$	−36.6539	−1.81242	−0.906209	+	0.422829i	$0.861037\pi$
$410$	0	0
$411$	−16.9739	−0.837259
$412$	0	0
$413$	5.08712	0.250321
$414$	0	0
$415$	−63.1078	−3.09784
$416$	0	0
$417$	7.05101	0.345289
$418$	0	0
$419$	−31.1212	−1.52037	−0.760186	−	0.649706i	$-0.774892\pi$
$419$	−31.1212	−1.52037	−0.760186	+	0.649706i	$0.774892\pi$
$420$	0	0
$421$	−36.5199	−1.77987	−0.889935	−	0.456087i	$-0.849251\pi$
$421$	−36.5199	−1.77987	−0.889935	+	0.456087i	$0.849251\pi$
$422$	0	0
$423$	7.50310	0.364813
$424$	0	0
$425$	−9.84885	−0.477740
$426$	0	0
$427$	−8.14526	−0.394177
$428$	0	0
$429$	10.2419	0.494483
$430$	0	0
$431$	17.5674	0.846193	0.423096	−	0.906085i	$-0.360943\pi$
$431$	17.5674	0.846193	0.423096	+	0.906085i	$0.360943\pi$
$432$	0	0
$433$	33.1226	1.59177	0.795885	−	0.605447i	$-0.207006\pi$
$433$	33.1226	1.59177	0.795885	+	0.605447i	$0.207006\pi$
$434$	0	0
$435$	−62.2835	−2.98627
$436$	0	0
$437$	−2.70940	−0.129608
$438$	0	0
$439$	−20.6332	−0.984771	−0.492385	−	0.870377i	$-0.663875\pi$
$439$	−20.6332	−0.984771	−0.492385	+	0.870377i	$0.663875\pi$
$440$	0	0
$441$	2.45447	0.116879
$442$	0	0
$443$	−31.5569	−1.49931	−0.749657	−	0.661827i	$-0.769781\pi$
$443$	−31.5569	−1.49931	−0.749657	+	0.661827i	$0.769781\pi$
$444$	0	0
$445$	39.7966	1.88654
$446$	0	0
$447$	−38.8193	−1.83609
$448$	0	0
$449$	−18.0569	−0.852156	−0.426078	−	0.904687i	$-0.640105\pi$
$449$	−18.0569	−0.852156	−0.426078	+	0.904687i	$0.640105\pi$
$450$	0	0
$451$	3.57306	0.168249
$452$	0	0
$453$	−19.0364	−0.894408
$454$	0	0
$455$	26.2420	1.23025
$456$	0	0
$457$	19.2646	0.901160	0.450580	−	0.892736i	$-0.351217\pi$
$457$	19.2646	0.901160	0.450580	+	0.892736i	$0.351217\pi$
$458$	0	0
$459$	0.898436	0.0419354
$460$	0	0
$461$	−20.0307	−0.932925	−0.466462	−	0.884541i	$-0.654472\pi$
$461$	−20.0307	−0.932925	−0.466462	+	0.884541i	$0.654472\pi$
$462$	0	0
$463$	−0.817978	−0.0380147	−0.0190073	−	0.999819i	$-0.506051\pi$
$463$	−0.817978	−0.0380147	−0.0190073	+	0.999819i	$0.506051\pi$
$464$	0	0
$465$	−96.5088	−4.47549
$466$	0	0
$467$	−15.1650	−0.701752	−0.350876	−	0.936422i	$-0.614116\pi$
$467$	−15.1650	−0.701752	−0.350876	+	0.936422i	$0.614116\pi$
$468$	0	0
$469$	7.65689	0.353562
$470$	0	0
$471$	−30.2985	−1.39608
$472$	0	0
$473$	−2.09932	−0.0965270
$474$	0	0
$475$	−13.9667	−0.640838
$476$	0	0
$477$	−17.5778	−0.804830
$478$	0	0
$479$	36.8821	1.68518	0.842592	−	0.538552i	$-0.181029\pi$
$479$	36.8821	1.68518	0.842592	+	0.538552i	$0.181029\pi$
$480$	0	0
$481$	13.1066	0.597607
$482$	0	0
$483$	6.32775	0.287923
$484$	0	0
$485$	−22.1338	−1.00505
$486$	0	0
$487$	−21.3963	−0.969557	−0.484779	−	0.874637i	$-0.661100\pi$
$487$	−21.3963	−0.969557	−0.484779	+	0.874637i	$0.661100\pi$
$488$	0	0
$489$	3.49088	0.157863
$490$	0	0
$491$	−1.07482	−0.0485058	−0.0242529	−	0.999706i	$-0.507721\pi$
$491$	−1.07482	−0.0485058	−0.0242529	+	0.999706i	$0.507721\pi$
$492$	0	0
$493$	4.31809	0.194477
$494$	0	0
$495$	7.77958	0.349666
$496$	0	0
$497$	−13.8880	−0.622960
$498$	0	0
$499$	−24.0930	−1.07855	−0.539275	−	0.842130i	$-0.681302\pi$
$499$	−24.0930	−1.07855	−0.539275	+	0.842130i	$0.681302\pi$
$500$	0	0
$501$	41.8489	1.86967
$502$	0	0
$503$	−42.9938	−1.91700	−0.958499	−	0.285096i	$-0.907975\pi$
$503$	−42.9938	−1.91700	−0.958499	+	0.285096i	$0.907975\pi$
$504$	0	0
$505$	59.8956	2.66532
$506$	0	0
$507$	54.4354	2.41756
$508$	0	0
$509$	2.74829	0.121816	0.0609079	−	0.998143i	$-0.480600\pi$
$509$	2.74829	0.121816	0.0609079	+	0.998143i	$0.480600\pi$
$510$	0	0
$511$	12.8979	0.570569
$512$	0	0
$513$	1.27408	0.0562520
$514$	0	0
$515$	−43.3014	−1.90809
$516$	0	0
$517$	2.22478	0.0978455
$518$	0	0
$519$	45.2903	1.98802
$520$	0	0
$521$	25.2211	1.10496	0.552479	−	0.833527i	$-0.313682\pi$
$521$	25.2211	1.10496	0.552479	+	0.833527i	$0.313682\pi$
$522$	0	0
$523$	−10.2603	−0.448650	−0.224325	−	0.974514i	$-0.572018\pi$
$523$	−10.2603	−0.448650	−0.224325	+	0.974514i	$0.572018\pi$
$524$	0	0
$525$	32.6190	1.42361
$526$	0	0
$527$	6.69091	0.291461
$528$	0	0
$529$	−15.6592	−0.680833
$530$	0	0
$531$	12.4862	0.541854
$532$	0	0
$533$	29.5827	1.28137
$534$	0	0
$535$	−20.5193	−0.887125
$536$	0	0
$537$	48.8201	2.10674
$538$	0	0
$539$	0.727784	0.0313479
$540$	0	0
$541$	40.0848	1.72338	0.861689	−	0.507436i	$-0.169407\pi$
$541$	40.0848	1.72338	0.861689	+	0.507436i	$0.169407\pi$
$542$	0	0
$543$	−17.4846	−0.750336
$544$	0	0
$545$	−44.8204	−1.91990
$546$	0	0
$547$	−32.4627	−1.38800	−0.694002	−	0.719973i	$-0.744154\pi$
$547$	−32.4627	−1.38800	−0.694002	+	0.719973i	$0.744154\pi$
$548$	0	0
$549$	−19.9923	−0.853249
$550$	0	0
$551$	6.12351	0.260870
$552$	0	0
$553$	7.93879	0.337592
$554$	0	0
$555$	22.1238	0.939103
$556$	0	0
$557$	28.6447	1.21371	0.606857	−	0.794811i	$-0.292430\pi$
$557$	28.6447	1.21371	0.606857	+	0.794811i	$0.292430\pi$
$558$	0	0
$559$	−17.3811	−0.735143
$560$	0	0
$561$	−1.19859	−0.0506044
$562$	0	0
$563$	−18.7955	−0.792134	−0.396067	−	0.918222i	$-0.629625\pi$
$563$	−18.7955	−0.792134	−0.396067	+	0.918222i	$0.629625\pi$
$564$	0	0
$565$	37.1993	1.56499
$566$	0	0
$567$	−10.3390	−0.434197
$568$	0	0
$569$	36.8454	1.54464	0.772319	−	0.635235i	$-0.219097\pi$
$569$	36.8454	1.54464	0.772319	+	0.635235i	$0.219097\pi$
$570$	0	0
$571$	8.63628	0.361417	0.180708	−	0.983537i	$-0.442161\pi$
$571$	8.63628	0.361417	0.180708	+	0.983537i	$0.442161\pi$
$572$	0	0
$573$	−41.1992	−1.72112
$574$	0	0
$575$	37.8415	1.57810
$576$	0	0
$577$	4.91081	0.204440	0.102220	−	0.994762i	$-0.467405\pi$
$577$	4.91081	0.204440	0.102220	+	0.994762i	$0.467405\pi$
$578$	0	0
$579$	−55.3820	−2.30160
$580$	0	0
$581$	−14.4906	−0.601172
$582$	0	0
$583$	−5.21205	−0.215861
$584$	0	0
$585$	64.4102	2.66304
$586$	0	0
$587$	−46.2542	−1.90911	−0.954557	−	0.298030i	$-0.903671\pi$
$587$	−46.2542	−1.90911	−0.954557	+	0.298030i	$0.903671\pi$
$588$	0	0
$589$	9.48843	0.390964
$590$	0	0
$591$	6.22808	0.256189
$592$	0	0
$593$	15.1955	0.624004	0.312002	−	0.950082i	$-0.399001\pi$
$593$	15.1955	0.624004	0.312002	+	0.950082i	$0.399001\pi$
$594$	0	0
$595$	−3.07105	−0.125901
$596$	0	0
$597$	−55.4719	−2.27031
$598$	0	0
$599$	23.2599	0.950375	0.475187	−	0.879885i	$-0.342380\pi$
$599$	23.2599	0.950375	0.475187	+	0.879885i	$0.342380\pi$
$600$	0	0
$601$	−22.1860	−0.904987	−0.452493	−	0.891768i	$-0.649465\pi$
$601$	−22.1860	−0.904987	−0.452493	+	0.891768i	$0.649465\pi$
$602$	0	0
$603$	18.7936	0.765334
$604$	0	0
$605$	−45.5991	−1.85387
$606$	0	0
$607$	−7.31616	−0.296954	−0.148477	−	0.988916i	$-0.547437\pi$
$607$	−7.31616	−0.296954	−0.148477	+	0.988916i	$0.547437\pi$
$608$	0	0
$609$	−14.3013	−0.579519
$610$	0	0
$611$	18.4198	0.745185
$612$	0	0
$613$	−40.2784	−1.62683	−0.813414	−	0.581685i	$-0.802393\pi$
$613$	−40.2784	−1.62683	−0.813414	+	0.581685i	$0.802393\pi$
$614$	0	0
$615$	49.9355	2.01360
$616$	0	0
$617$	5.72230	0.230371	0.115186	−	0.993344i	$-0.463254\pi$
$617$	5.72230	0.230371	0.115186	+	0.993344i	$0.463254\pi$
$618$	0	0
$619$	11.2326	0.451476	0.225738	−	0.974188i	$-0.427521\pi$
$619$	11.2326	0.451476	0.225738	+	0.974188i	$0.427521\pi$
$620$	0	0
$621$	−3.45199	−0.138524
$622$	0	0
$623$	9.13798	0.366105
$624$	0	0
$625$	100.236	4.00944
$626$	0	0
$627$	−1.69973	−0.0678805
$628$	0	0
$629$	−1.53383	−0.0611579
$630$	0	0
$631$	−1.91192	−0.0761125	−0.0380562	−	0.999276i	$-0.512117\pi$
$631$	−1.91192	−0.0761125	−0.0380562	+	0.999276i	$0.512117\pi$
$632$	0	0
$633$	43.9141	1.74543
$634$	0	0
$635$	16.3401	0.648437
$636$	0	0
$637$	6.02561	0.238743
$638$	0	0
$639$	−34.0876	−1.34848
$640$	0	0
$641$	0.0599616	0.00236834	0.00118417	−	0.999999i	$-0.499623\pi$
$641$	0.0599616	0.00236834	0.00118417	+	0.999999i	$0.499623\pi$
$642$	0	0
$643$	40.4809	1.59641	0.798205	−	0.602385i	$-0.205783\pi$
$643$	40.4809	1.59641	0.798205	+	0.602385i	$0.205783\pi$
$644$	0	0
$645$	−29.3393	−1.15523
$646$	0	0
$647$	−2.71670	−0.106805	−0.0534023	−	0.998573i	$-0.517007\pi$
$647$	−2.71670	−0.106805	−0.0534023	+	0.998573i	$0.517007\pi$
$648$	0	0
$649$	3.70233	0.145329
$650$	0	0
$651$	−22.1600	−0.868521
$652$	0	0
$653$	29.0110	1.13529	0.567644	−	0.823274i	$-0.307855\pi$
$653$	29.0110	1.13529	0.567644	+	0.823274i	$0.307855\pi$
$654$	0	0
$655$	−58.0566	−2.26846
$656$	0	0
$657$	31.6575	1.23508
$658$	0	0
$659$	−7.89025	−0.307361	−0.153680	−	0.988121i	$-0.549113\pi$
$659$	−7.89025	−0.307361	−0.153680	+	0.988121i	$0.549113\pi$
$660$	0	0
$661$	2.55127	0.0992328	0.0496164	−	0.998768i	$-0.484200\pi$
$661$	2.55127	0.0992328	0.0496164	+	0.998768i	$0.484200\pi$
$662$	0	0
$663$	−9.92357	−0.385400
$664$	0	0
$665$	−4.35508	−0.168883
$666$	0	0
$667$	−16.5910	−0.642408
$668$	0	0
$669$	−0.166028	−0.00641901
$670$	0	0
$671$	−5.92799	−0.228848
$672$	0	0
$673$	10.2319	0.394409	0.197205	−	0.980362i	$-0.436814\pi$
$673$	10.2319	0.394409	0.197205	+	0.980362i	$0.436814\pi$
$674$	0	0
$675$	−17.7947	−0.684920
$676$	0	0
$677$	18.0321	0.693031	0.346516	−	0.938044i	$-0.387365\pi$
$677$	18.0321	0.693031	0.346516	+	0.938044i	$0.387365\pi$
$678$	0	0
$679$	−5.08230	−0.195041
$680$	0	0
$681$	26.5864	1.01879
$682$	0	0
$683$	9.47324	0.362483	0.181242	−	0.983439i	$-0.441988\pi$
$683$	9.47324	0.362483	0.181242	+	0.983439i	$0.441988\pi$
$684$	0	0
$685$	−31.6520	−1.20936
$686$	0	0
$687$	14.5861	0.556496
$688$	0	0
$689$	−43.1526	−1.64398
$690$	0	0
$691$	32.9066	1.25183	0.625913	−	0.779893i	$-0.284727\pi$
$691$	32.9066	1.25183	0.625913	+	0.779893i	$0.284727\pi$
$692$	0	0
$693$	1.78632	0.0678568
$694$	0	0
$695$	13.1484	0.498745
$696$	0	0
$697$	−3.46201	−0.131133
$698$	0	0
$699$	2.26062	0.0855046
$700$	0	0
$701$	42.2380	1.59531	0.797654	−	0.603115i	$-0.206074\pi$
$701$	42.2380	1.59531	0.797654	+	0.603115i	$0.206074\pi$
$702$	0	0
$703$	−2.17514	−0.0820370
$704$	0	0
$705$	31.0925	1.17101
$706$	0	0
$707$	13.7530	0.517236
$708$	0	0
$709$	−13.4419	−0.504820	−0.252410	−	0.967620i	$-0.581223\pi$
$709$	−13.4419	−0.504820	−0.252410	+	0.967620i	$0.581223\pi$
$710$	0	0
$711$	19.4855	0.730763
$712$	0	0
$713$	−25.7080	−0.962771
$714$	0	0
$715$	19.0985	0.714245
$716$	0	0
$717$	−0.562710	−0.0210148
$718$	0	0
$719$	−20.9111	−0.779852	−0.389926	−	0.920846i	$-0.627499\pi$
$719$	−20.9111	−0.779852	−0.389926	+	0.920846i	$0.627499\pi$
$720$	0	0
$721$	−9.94273	−0.370287
$722$	0	0
$723$	25.4678	0.947159
$724$	0	0
$725$	−85.5255	−3.17634
$726$	0	0
$727$	6.26971	0.232531	0.116265	−	0.993218i	$-0.462908\pi$
$727$	6.26971	0.232531	0.116265	+	0.993218i	$0.462908\pi$
$728$	0	0
$729$	−16.4500	−0.609258
$730$	0	0
$731$	2.03408	0.0752331
$732$	0	0
$733$	14.0440	0.518728	0.259364	−	0.965780i	$-0.416487\pi$
$733$	14.0440	0.518728	0.259364	+	0.965780i	$0.416487\pi$
$734$	0	0
$735$	10.1712	0.375171
$736$	0	0
$737$	5.57256	0.205268
$738$	0	0
$739$	45.3175	1.66703	0.833515	−	0.552497i	$-0.186325\pi$
$739$	45.3175	1.66703	0.833515	+	0.552497i	$0.186325\pi$
$740$	0	0
$741$	−14.0727	−0.516973
$742$	0	0
$743$	−39.9970	−1.46735	−0.733674	−	0.679501i	$-0.762196\pi$
$743$	−39.9970	−1.46735	−0.733674	+	0.679501i	$0.762196\pi$
$744$	0	0
$745$	−72.3882	−2.65210
$746$	0	0
$747$	−35.5667	−1.30132
$748$	0	0
$749$	−4.71157	−0.172157
$750$	0	0
$751$	4.73813	0.172897	0.0864484	−	0.996256i	$-0.472448\pi$
$751$	4.73813	0.172897	0.0864484	+	0.996256i	$0.472448\pi$
$752$	0	0
$753$	−32.7340	−1.19289
$754$	0	0
$755$	−35.4981	−1.29191
$756$	0	0
$757$	28.5302	1.03695	0.518473	−	0.855094i	$-0.326501\pi$
$757$	28.5302	1.03695	0.518473	+	0.855094i	$0.326501\pi$
$758$	0	0
$759$	4.60524	0.167160
$760$	0	0
$761$	8.01449	0.290525	0.145263	−	0.989393i	$-0.453597\pi$
$761$	8.01449	0.290525	0.145263	+	0.989393i	$0.453597\pi$
$762$	0	0
$763$	−10.2915	−0.372578
$764$	0	0
$765$	−7.53780	−0.272530
$766$	0	0
$767$	30.6530	1.10682
$768$	0	0
$769$	24.5847	0.886545	0.443273	−	0.896387i	$-0.353817\pi$
$769$	24.5847	0.886545	0.443273	+	0.896387i	$0.353817\pi$
$770$	0	0
$771$	−17.1378	−0.617204
$772$	0	0
$773$	3.01169	0.108323	0.0541615	−	0.998532i	$-0.482751\pi$
$773$	3.01169	0.108323	0.0541615	+	0.998532i	$0.482751\pi$
$774$	0	0
$775$	−132.522	−4.76035
$776$	0	0
$777$	5.08000	0.182244
$778$	0	0
$779$	−4.90950	−0.175901
$780$	0	0
$781$	−10.1074	−0.361673
$782$	0	0
$783$	7.80184	0.278815
$784$	0	0
$785$	−56.4991	−2.01654
$786$	0	0
$787$	11.8626	0.422855	0.211428	−	0.977394i	$-0.432189\pi$
$787$	11.8626	0.422855	0.211428	+	0.977394i	$0.432189\pi$
$788$	0	0
$789$	−14.1510	−0.503790
$790$	0	0
$791$	8.54159	0.303704
$792$	0	0
$793$	−49.0802	−1.74289
$794$	0	0
$795$	−72.8414	−2.58342
$796$	0	0
$797$	−9.14690	−0.324000	−0.162000	−	0.986791i	$-0.551794\pi$
$797$	−9.14690	−0.324000	−0.162000	+	0.986791i	$0.551794\pi$
$798$	0	0
$799$	−2.15563	−0.0762607
$800$	0	0
$801$	22.4289	0.792485
$802$	0	0
$803$	9.38688	0.331256
$804$	0	0
$805$	11.7997	0.415883
$806$	0	0
$807$	41.8846	1.47441
$808$	0	0
$809$	−14.4081	−0.506561	−0.253281	−	0.967393i	$-0.581510\pi$
$809$	−14.4081	−0.506561	−0.253281	+	0.967393i	$0.581510\pi$
$810$	0	0
$811$	40.5482	1.42384	0.711919	−	0.702261i	$-0.247826\pi$
$811$	40.5482	1.42384	0.711919	+	0.702261i	$0.247826\pi$
$812$	0	0
$813$	−39.0572	−1.36980
$814$	0	0
$815$	6.50961	0.228022
$816$	0	0
$817$	2.88454	0.100917
$818$	0	0
$819$	14.7897	0.516793
$820$	0	0
$821$	−23.2182	−0.810322	−0.405161	−	0.914245i	$-0.632785\pi$
$821$	−23.2182	−0.810322	−0.405161	+	0.914245i	$0.632785\pi$
$822$	0	0
$823$	−35.3636	−1.23270	−0.616349	−	0.787473i	$-0.711389\pi$
$823$	−35.3636	−1.23270	−0.616349	+	0.787473i	$0.711389\pi$
$824$	0	0
$825$	23.7396	0.826507
$826$	0	0
$827$	32.2907	1.12286	0.561430	−	0.827524i	$-0.310252\pi$
$827$	32.2907	1.12286	0.561430	+	0.827524i	$0.310252\pi$
$828$	0	0
$829$	−19.3124	−0.670746	−0.335373	−	0.942085i	$-0.608862\pi$
$829$	−19.3124	−0.670746	−0.335373	+	0.942085i	$0.608862\pi$
$830$	0	0
$831$	12.3099	0.427027
$832$	0	0
$833$	−0.705165	−0.0244325
$834$	0	0
$835$	78.0377	2.70061
$836$	0	0
$837$	12.0890	0.417858
$838$	0	0
$839$	−27.0719	−0.934627	−0.467313	−	0.884092i	$-0.654778\pi$
$839$	−27.0719	−0.934627	−0.467313	+	0.884092i	$0.654778\pi$
$840$	0	0
$841$	8.49739	0.293013
$842$	0	0
$843$	25.7621	0.887293
$844$	0	0
$845$	101.508	3.49199
$846$	0	0
$847$	−10.4703	−0.359765
$848$	0	0
$849$	9.88303	0.339185
$850$	0	0
$851$	5.89333	0.202021
$852$	0	0
$853$	22.6348	0.774999	0.387500	−	0.921870i	$-0.373339\pi$
$853$	22.6348	0.774999	0.387500	+	0.921870i	$0.373339\pi$
$854$	0	0
$855$	−10.6894	−0.365570
$856$	0	0
$857$	−52.1981	−1.78305	−0.891526	−	0.452969i	$-0.850365\pi$
$857$	−52.1981	−1.78305	−0.891526	+	0.452969i	$0.850365\pi$
$858$	0	0
$859$	−11.0541	−0.377161	−0.188581	−	0.982058i	$-0.560389\pi$
$859$	−11.0541	−0.377161	−0.188581	+	0.982058i	$0.560389\pi$
$860$	0	0
$861$	11.4660	0.390762
$862$	0	0
$863$	12.2924	0.418439	0.209220	−	0.977869i	$-0.432908\pi$
$863$	12.2924	0.418439	0.209220	+	0.977869i	$0.432908\pi$
$864$	0	0
$865$	84.4550	2.87156
$866$	0	0
$867$	−38.5418	−1.30895
$868$	0	0
$869$	5.77773	0.195996
$870$	0	0
$871$	46.1374	1.56331
$872$	0	0
$873$	−12.4743	−0.422193
$874$	0	0
$875$	39.0509	1.32016
$876$	0	0
$877$	32.4581	1.09603	0.548016	−	0.836468i	$-0.315383\pi$
$877$	32.4581	1.09603	0.548016	+	0.836468i	$0.315383\pi$
$878$	0	0
$879$	−65.5698	−2.21161
$880$	0	0
$881$	24.8052	0.835707	0.417853	−	0.908514i	$-0.362783\pi$
$881$	24.8052	0.835707	0.417853	+	0.908514i	$0.362783\pi$
$882$	0	0
$883$	−25.7870	−0.867801	−0.433900	−	0.900961i	$-0.642863\pi$
$883$	−25.7870	−0.867801	−0.433900	+	0.900961i	$0.642863\pi$
$884$	0	0
$885$	51.7422	1.73929
$886$	0	0
$887$	−21.5687	−0.724207	−0.362104	−	0.932138i	$-0.617941\pi$
$887$	−21.5687	−0.724207	−0.362104	+	0.932138i	$0.617941\pi$
$888$	0	0
$889$	3.75196	0.125837
$890$	0	0
$891$	−7.52455	−0.252082
$892$	0	0
$893$	−3.05692	−0.102296
$894$	0	0
$895$	91.0371	3.04303
$896$	0	0
$897$	38.1286	1.27308
$898$	0	0
$899$	58.1025	1.93783
$900$	0	0
$901$	5.05006	0.168242
$902$	0	0
$903$	−6.73679	−0.224186
$904$	0	0
$905$	−32.6044	−1.08381
$906$	0	0
$907$	42.3718	1.40693	0.703466	−	0.710729i	$-0.251635\pi$
$907$	42.3718	1.40693	0.703466	+	0.710729i	$0.251635\pi$
$908$	0	0
$909$	33.7564	1.11963
$910$	0	0
$911$	15.4661	0.512416	0.256208	−	0.966622i	$-0.417527\pi$
$911$	15.4661	0.512416	0.256208	+	0.966622i	$0.417527\pi$
$912$	0	0
$913$	−10.5460	−0.349023
$914$	0	0
$915$	−82.8471	−2.73884
$916$	0	0
$917$	−13.3308	−0.440221
$918$	0	0
$919$	20.1894	0.665988	0.332994	−	0.942929i	$-0.391941\pi$
$919$	20.1894	0.665988	0.332994	+	0.942929i	$0.391941\pi$
$920$	0	0
$921$	5.38758	0.177527
$922$	0	0
$923$	−83.6835	−2.75448
$924$	0	0
$925$	30.3796	0.998876
$926$	0	0
$927$	−24.4041	−0.801536
$928$	0	0
$929$	−28.0532	−0.920396	−0.460198	−	0.887816i	$-0.652222\pi$
$929$	−28.0532	−0.920396	−0.460198	+	0.887816i	$0.652222\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	26.0329	0.852278
$934$	0	0
$935$	−2.23506	−0.0730944
$936$	0	0
$937$	0.685694	0.0224006	0.0112003	−	0.999937i	$-0.496435\pi$
$937$	0.685694	0.0224006	0.0112003	+	0.999937i	$0.496435\pi$
$938$	0	0
$939$	−24.4992	−0.799502
$940$	0	0
$941$	15.9255	0.519156	0.259578	−	0.965722i	$-0.416416\pi$
$941$	15.9255	0.519156	0.259578	+	0.965722i	$0.416416\pi$
$942$	0	0
$943$	13.3018	0.433166
$944$	0	0
$945$	−5.54872	−0.180500
$946$	0	0
$947$	−60.7269	−1.97336	−0.986679	−	0.162679i	$-0.947987\pi$
$947$	−60.7269	−1.97336	−0.986679	+	0.162679i	$0.947987\pi$
$948$	0	0
$949$	77.7177	2.52282
$950$	0	0
$951$	−3.37501	−0.109442
$952$	0	0
$953$	45.2720	1.46650	0.733251	−	0.679958i	$-0.238002\pi$
$953$	45.2720	1.46650	0.733251	+	0.679958i	$0.238002\pi$
$954$	0	0
$955$	−76.8261	−2.48604
$956$	0	0
$957$	−10.4083	−0.336452
$958$	0	0
$959$	−7.26783	−0.234690
$960$	0	0
$961$	59.0303	1.90420
$962$	0	0
$963$	−11.5644	−0.372657
$964$	0	0
$965$	−103.273	−3.32449
$966$	0	0
$967$	−9.38225	−0.301713	−0.150856	−	0.988556i	$-0.548203\pi$
$967$	−9.38225	−0.301713	−0.150856	+	0.988556i	$0.548203\pi$
$968$	0	0
$969$	1.64690	0.0529060
$970$	0	0
$971$	26.4501	0.848824	0.424412	−	0.905469i	$-0.360481\pi$
$971$	26.4501	0.848824	0.424412	+	0.905469i	$0.360481\pi$
$972$	0	0
$973$	3.01908	0.0967874
$974$	0	0
$975$	196.550	6.29463
$976$	0	0
$977$	26.6771	0.853475	0.426737	−	0.904376i	$-0.359663\pi$
$977$	26.6771	0.853475	0.426737	+	0.904376i	$0.359663\pi$
$978$	0	0
$979$	6.65048	0.212550
$980$	0	0
$981$	−25.2602	−0.806497
$982$	0	0
$983$	5.00623	0.159674	0.0798370	−	0.996808i	$-0.474560\pi$
$983$	5.00623	0.159674	0.0798370	+	0.996808i	$0.474560\pi$
$984$	0	0
$985$	11.6138	0.370047
$986$	0	0
$987$	7.13937	0.227249
$988$	0	0
$989$	−7.81537	−0.248514
$990$	0	0
$991$	26.1163	0.829613	0.414806	−	0.909910i	$-0.363849\pi$
$991$	26.1163	0.829613	0.414806	+	0.909910i	$0.363849\pi$
$992$	0	0
$993$	69.6968	2.21176
$994$	0	0
$995$	−103.441	−3.27930
$996$	0	0
$997$	31.8849	1.00980	0.504902	−	0.863177i	$-0.331529\pi$
$997$	31.8849	1.00980	0.504902	+	0.863177i	$0.331529\pi$
$998$	0	0
$999$	−2.77130	−0.0876801

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.ci.1.6		7
4.3	odd	2		8512.2.a.ch.1.2		7
8.3	odd	2		4256.2.a.q.1.6	yes	7
8.5	even	2		4256.2.a.p.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4256.2.a.p.1.2	✓	7	8.5	even	2
4256.2.a.q.1.6	yes	7	8.3	odd	2
8512.2.a.ch.1.2		7	4.3	odd	2
8512.2.a.ci.1.6		7	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.33548 q^{3} +4.35508 q^{5} +1.00000 q^{7} +2.45447 q^{9} +O(q^{10})\)	\(q+2.33548 q^{3} +4.35508 q^{5} +1.00000 q^{7} +2.45447 q^{9} +0.727784 q^{11} +6.02561 q^{13} +10.1712 q^{15} -0.705165 q^{17} -1.00000 q^{19} +2.33548 q^{21} +2.70940 q^{23} +13.9667 q^{25} -1.27408 q^{27} -6.12351 q^{29} -9.48843 q^{31} +1.69973 q^{33} +4.35508 q^{35} +2.17514 q^{37} +14.0727 q^{39} +4.90950 q^{41} -2.88454 q^{43} +10.6894 q^{45} +3.05692 q^{47} +1.00000 q^{49} -1.64690 q^{51} -7.16153 q^{53} +3.16956 q^{55} -2.33548 q^{57} +5.08712 q^{59} -8.14526 q^{61} +2.45447 q^{63} +26.2420 q^{65} +7.65689 q^{67} +6.32775 q^{69} -13.8880 q^{71} +12.8979 q^{73} +32.6190 q^{75} +0.727784 q^{77} +7.93879 q^{79} -10.3390 q^{81} -14.4906 q^{83} -3.07105 q^{85} -14.3013 q^{87} +9.13798 q^{89} +6.02561 q^{91} -22.1600 q^{93} -4.35508 q^{95} -5.08230 q^{97} +1.78632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} + 5 q^{5} + 7 q^{7} + 8 q^{9}+O(q^{10}) \) 7 * q + 3 * q^3 + 5 * q^5 + 7 * q^7 + 8 * q^9	\( 7 q + 3 q^{3} + 5 q^{5} + 7 q^{7} + 8 q^{9} + 3 q^{11} + 16 q^{13} - 8 q^{15} + 8 q^{17} - 7 q^{19} + 3 q^{21} - 10 q^{23} + 8 q^{25} + 6 q^{27} + 11 q^{29} - 14 q^{31} + 3 q^{33} + 5 q^{35} + 13 q^{37} + 6 q^{39} + 11 q^{41} + 11 q^{43} - 4 q^{45} - 7 q^{47} + 7 q^{49} + 24 q^{51} + 9 q^{53} + 8 q^{55} - 3 q^{57} + 23 q^{59} + 23 q^{61} + 8 q^{63} + 40 q^{65} + 16 q^{67} + 10 q^{69} - 3 q^{71} + 4 q^{73} + 48 q^{75} + 3 q^{77} - 29 q^{79} + 23 q^{81} + 6 q^{83} + 6 q^{85} + 6 q^{87} + 15 q^{89} + 16 q^{91} - 42 q^{93} - 5 q^{95} - 13 q^{97} + 47 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 + 5 * q^5 + 7 * q^7 + 8 * q^9 + 3 * q^11 + 16 * q^13 - 8 * q^15 + 8 * q^17 - 7 * q^19 + 3 * q^21 - 10 * q^23 + 8 * q^25 + 6 * q^27 + 11 * q^29 - 14 * q^31 + 3 * q^33 + 5 * q^35 + 13 * q^37 + 6 * q^39 + 11 * q^41 + 11 * q^43 - 4 * q^45 - 7 * q^47 + 7 * q^49 + 24 * q^51 + 9 * q^53 + 8 * q^55 - 3 * q^57 + 23 * q^59 + 23 * q^61 + 8 * q^63 + 40 * q^65 + 16 * q^67 + 10 * q^69 - 3 * q^71 + 4 * q^73 + 48 * q^75 + 3 * q^77 - 29 * q^79 + 23 * q^81 + 6 * q^83 + 6 * q^85 + 6 * q^87 + 15 * q^89 + 16 * q^91 - 42 * q^93 - 5 * q^95 - 13 * q^97 + 47 * q^99

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