LMFDB - Embedded newform 6512.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6512,2,Mod(1,6512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6512, 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("6512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6512 = 2^{4} \cdot 11 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6512.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:	$51.9985817963$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 16 x^{9} + 32 x^{8} + 89 x^{7} - 179 x^{6} - 201 x^{5} + 407 x^{4} + 168 x^{3} + \cdots + 75 $ x^11 - 2x^10 - 16x^9 + 32x^8 + 89x^7 - 179x^6 - 201x^5 + 407x^4 + 168x^3 - 333x^2 - 51x + 75
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 407)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.12601$ of defining polynomial
Character	$\chi$	$=$	6512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.05580 q^{3} -1.92378 q^{5} +2.08676 q^{7} +6.33790 q^{9} +O(q^{10})$	$q-3.05580 q^{3} -1.92378 q^{5} +2.08676 q^{7} +6.33790 q^{9} +1.00000 q^{11} -3.35798 q^{13} +5.87867 q^{15} +6.78748 q^{17} -0.267136 q^{19} -6.37671 q^{21} +4.66206 q^{23} -1.29909 q^{25} -10.1999 q^{27} +9.44474 q^{29} +3.22663 q^{31} -3.05580 q^{33} -4.01445 q^{35} +1.00000 q^{37} +10.2613 q^{39} +0.209019 q^{41} -3.75750 q^{43} -12.1927 q^{45} +11.2166 q^{47} -2.64544 q^{49} -20.7412 q^{51} +2.74980 q^{53} -1.92378 q^{55} +0.816313 q^{57} +0.914017 q^{59} +13.1842 q^{61} +13.2257 q^{63} +6.46000 q^{65} -3.42780 q^{67} -14.2463 q^{69} +0.854668 q^{71} +9.96667 q^{73} +3.96976 q^{75} +2.08676 q^{77} -13.6528 q^{79} +12.1553 q^{81} -6.20180 q^{83} -13.0576 q^{85} -28.8612 q^{87} -9.11340 q^{89} -7.00730 q^{91} -9.85993 q^{93} +0.513909 q^{95} +11.7443 q^{97} +6.33790 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + q^{5} - 9 q^{7} + 15 q^{9}+O(q^{10}) $ 11 * q + q^5 - 9 * q^7 + 15 * q^9	$ 11 q + q^{5} - 9 q^{7} + 15 q^{9} + 11 q^{11} + 22 q^{13} + 10 q^{15} + 19 q^{17} - 14 q^{19} - 13 q^{21} + 14 q^{23} + 38 q^{25} + 9 q^{27} + 13 q^{29} - 12 q^{31} - 12 q^{35} + 11 q^{37} + 8 q^{39} + 8 q^{41} - 21 q^{43} - 8 q^{45} + 14 q^{47} + 20 q^{49} + 2 q^{51} - 2 q^{53} + q^{55} - 3 q^{57} + 30 q^{59} + 20 q^{61} - 31 q^{63} - 7 q^{67} + 9 q^{69} + 15 q^{71} + 47 q^{73} + 40 q^{75} - 9 q^{77} - 2 q^{79} - 17 q^{81} - 24 q^{83} - 25 q^{85} - 21 q^{87} + 4 q^{89} - 21 q^{91} - 37 q^{93} + 36 q^{95} + 25 q^{97} + 15 q^{99}+O(q^{100}) $ 11 * q + q^5 - 9 * q^7 + 15 * q^9 + 11 * q^11 + 22 * q^13 + 10 * q^15 + 19 * q^17 - 14 * q^19 - 13 * q^21 + 14 * q^23 + 38 * q^25 + 9 * q^27 + 13 * q^29 - 12 * q^31 - 12 * q^35 + 11 * q^37 + 8 * q^39 + 8 * q^41 - 21 * q^43 - 8 * q^45 + 14 * q^47 + 20 * q^49 + 2 * q^51 - 2 * q^53 + q^55 - 3 * q^57 + 30 * q^59 + 20 * q^61 - 31 * q^63 - 7 * q^67 + 9 * q^69 + 15 * q^71 + 47 * q^73 + 40 * q^75 - 9 * q^77 - 2 * q^79 - 17 * q^81 - 24 * q^83 - 25 * q^85 - 21 * q^87 + 4 * q^89 - 21 * q^91 - 37 * q^93 + 36 * q^95 + 25 * q^97 + 15 * 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
$3$	−3.05580	−1.76427	−0.882133	−	0.471001i	$-0.843893\pi$
$3$	−3.05580	−1.76427	−0.882133	+	0.471001i	$0.843893\pi$
$4$	0	0
$5$	−1.92378	−0.860338	−0.430169	−	0.902748i	$-0.641546\pi$
$5$	−1.92378	−0.860338	−0.430169	+	0.902748i	$0.641546\pi$
$6$	0	0
$7$	2.08676	0.788720	0.394360	−	0.918956i	$-0.370966\pi$
$7$	2.08676	0.788720	0.394360	+	0.918956i	$0.370966\pi$
$8$	0	0
$9$	6.33790	2.11263
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.35798	−0.931337	−0.465668	−	0.884959i	$-0.654186\pi$
$13$	−3.35798	−0.931337	−0.465668	+	0.884959i	$0.654186\pi$
$14$	0	0
$15$	5.87867	1.51787
$16$	0	0
$17$	6.78748	1.64620	0.823102	−	0.567893i	$-0.192241\pi$
$17$	6.78748	1.64620	0.823102	+	0.567893i	$0.192241\pi$
$18$	0	0
$19$	−0.267136	−0.0612852	−0.0306426	−	0.999530i	$-0.509755\pi$
$19$	−0.267136	−0.0612852	−0.0306426	+	0.999530i	$0.509755\pi$
$20$	0	0
$21$	−6.37671	−1.39151
$22$	0	0
$23$	4.66206	0.972107	0.486053	−	0.873929i	$-0.338436\pi$
$23$	4.66206	0.972107	0.486053	+	0.873929i	$0.338436\pi$
$24$	0	0
$25$	−1.29909	−0.259818
$26$	0	0
$27$	−10.1999	−1.96298
$28$	0	0
$29$	9.44474	1.75384	0.876922	−	0.480633i	$-0.159593\pi$
$29$	9.44474	1.75384	0.876922	+	0.480633i	$0.159593\pi$
$30$	0	0
$31$	3.22663	0.579520	0.289760	−	0.957099i	$-0.406424\pi$
$31$	3.22663	0.579520	0.289760	+	0.957099i	$0.406424\pi$
$32$	0	0
$33$	−3.05580	−0.531946
$34$	0	0
$35$	−4.01445	−0.678566
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	10.2613	1.64313
$40$	0	0
$41$	0.209019	0.0326432	0.0163216	−	0.999867i	$-0.494804\pi$
$41$	0.209019	0.0326432	0.0163216	+	0.999867i	$0.494804\pi$
$42$	0	0
$43$	−3.75750	−0.573013	−0.286507	−	0.958078i	$-0.592494\pi$
$43$	−3.75750	−0.573013	−0.286507	+	0.958078i	$0.592494\pi$
$44$	0	0
$45$	−12.1927	−1.81758
$46$	0	0
$47$	11.2166	1.63610	0.818052	−	0.575144i	$-0.195054\pi$
$47$	11.2166	1.63610	0.818052	+	0.575144i	$0.195054\pi$
$48$	0	0
$49$	−2.64544	−0.377920
$50$	0	0
$51$	−20.7412	−2.90434
$52$	0	0
$53$	2.74980	0.377714	0.188857	−	0.982005i	$-0.439522\pi$
$53$	2.74980	0.377714	0.188857	+	0.982005i	$0.439522\pi$
$54$	0	0
$55$	−1.92378	−0.259402
$56$	0	0
$57$	0.816313	0.108123
$58$	0	0
$59$	0.914017	0.118995	0.0594974	−	0.998228i	$-0.481050\pi$
$59$	0.914017	0.118995	0.0594974	+	0.998228i	$0.481050\pi$
$60$	0	0
$61$	13.1842	1.68807	0.844033	−	0.536291i	$-0.180175\pi$
$61$	13.1842	1.68807	0.844033	+	0.536291i	$0.180175\pi$
$62$	0	0
$63$	13.2257	1.66628
$64$	0	0
$65$	6.46000	0.801265
$66$	0	0
$67$	−3.42780	−0.418772	−0.209386	−	0.977833i	$-0.567147\pi$
$67$	−3.42780	−0.418772	−0.209386	+	0.977833i	$0.567147\pi$
$68$	0	0
$69$	−14.2463	−1.71505
$70$	0	0
$71$	0.854668	0.101430	0.0507152	−	0.998713i	$-0.483850\pi$
$71$	0.854668	0.101430	0.0507152	+	0.998713i	$0.483850\pi$
$72$	0	0
$73$	9.96667	1.16651	0.583255	−	0.812289i	$-0.301779\pi$
$73$	9.96667	1.16651	0.583255	+	0.812289i	$0.301779\pi$
$74$	0	0
$75$	3.96976	0.458388
$76$	0	0
$77$	2.08676	0.237808
$78$	0	0
$79$	−13.6528	−1.53606	−0.768032	−	0.640411i	$-0.778764\pi$
$79$	−13.6528	−1.53606	−0.768032	+	0.640411i	$0.778764\pi$
$80$	0	0
$81$	12.1553	1.35059
$82$	0	0
$83$	−6.20180	−0.680736	−0.340368	−	0.940292i	$-0.610552\pi$
$83$	−6.20180	−0.680736	−0.340368	+	0.940292i	$0.610552\pi$
$84$	0	0
$85$	−13.0576	−1.41629
$86$	0	0
$87$	−28.8612	−3.09425
$88$	0	0
$89$	−9.11340	−0.966019	−0.483009	−	0.875615i	$-0.660456\pi$
$89$	−9.11340	−0.966019	−0.483009	+	0.875615i	$0.660456\pi$
$90$	0	0
$91$	−7.00730	−0.734564
$92$	0	0
$93$	−9.85993	−1.02243
$94$	0	0
$95$	0.513909	0.0527260
$96$	0	0
$97$	11.7443	1.19245	0.596225	−	0.802817i	$-0.296666\pi$
$97$	11.7443	1.19245	0.596225	+	0.802817i	$0.296666\pi$
$98$	0	0
$99$	6.33790	0.636983
$100$	0	0
$101$	−6.43172	−0.639980	−0.319990	−	0.947421i	$-0.603680\pi$
$101$	−6.43172	−0.639980	−0.319990	+	0.947421i	$0.603680\pi$
$102$	0	0
$103$	−17.7730	−1.75122	−0.875612	−	0.483016i	$-0.839541\pi$
$103$	−17.7730	−1.75122	−0.875612	+	0.483016i	$0.839541\pi$
$104$	0	0
$105$	12.2674	1.19717
$106$	0	0
$107$	−0.604301	−0.0584200	−0.0292100	−	0.999573i	$-0.509299\pi$
$107$	−0.604301	−0.0584200	−0.0292100	+	0.999573i	$0.509299\pi$
$108$	0	0
$109$	−17.6594	−1.69146	−0.845732	−	0.533608i	$-0.820836\pi$
$109$	−17.6594	−1.69146	−0.845732	+	0.533608i	$0.820836\pi$
$110$	0	0
$111$	−3.05580	−0.290043
$112$	0	0
$113$	−10.6523	−1.00208	−0.501040	−	0.865424i	$-0.667049\pi$
$113$	−10.6523	−1.00208	−0.501040	+	0.865424i	$0.667049\pi$
$114$	0	0
$115$	−8.96875	−0.836341
$116$	0	0
$117$	−21.2826	−1.96757
$118$	0	0
$119$	14.1638	1.29840
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.638719	−0.0575913
$124$	0	0
$125$	12.1180	1.08387
$126$	0	0
$127$	12.5416	1.11289	0.556443	−	0.830886i	$-0.312166\pi$
$127$	12.5416	1.11289	0.556443	+	0.830886i	$0.312166\pi$
$128$	0	0
$129$	11.4822	1.01095
$130$	0	0
$131$	7.04400	0.615437	0.307719	−	0.951477i	$-0.400434\pi$
$131$	7.04400	0.615437	0.307719	+	0.951477i	$0.400434\pi$
$132$	0	0
$133$	−0.557448	−0.0483369
$134$	0	0
$135$	19.6224	1.68883
$136$	0	0
$137$	−14.5382	−1.24208	−0.621042	−	0.783777i	$-0.713291\pi$
$137$	−14.5382	−1.24208	−0.621042	+	0.783777i	$0.713291\pi$
$138$	0	0
$139$	18.6991	1.58604	0.793019	−	0.609197i	$-0.208508\pi$
$139$	18.6991	1.58604	0.793019	+	0.609197i	$0.208508\pi$
$140$	0	0
$141$	−34.2756	−2.88652
$142$	0	0
$143$	−3.35798	−0.280809
$144$	0	0
$145$	−18.1696	−1.50890
$146$	0	0
$147$	8.08393	0.666751
$148$	0	0
$149$	−8.76200	−0.717811	−0.358905	−	0.933374i	$-0.616850\pi$
$149$	−8.76200	−0.717811	−0.358905	+	0.933374i	$0.616850\pi$
$150$	0	0
$151$	−9.03624	−0.735359	−0.367680	−	0.929953i	$-0.619848\pi$
$151$	−9.03624	−0.735359	−0.367680	+	0.929953i	$0.619848\pi$
$152$	0	0
$153$	43.0183	3.47783
$154$	0	0
$155$	−6.20731	−0.498583
$156$	0	0
$157$	11.7809	0.940220	0.470110	−	0.882608i	$-0.344214\pi$
$157$	11.7809	0.940220	0.470110	+	0.882608i	$0.344214\pi$
$158$	0	0
$159$	−8.40284	−0.666388
$160$	0	0
$161$	9.72859	0.766720
$162$	0	0
$163$	0.518576	0.0406180	0.0203090	−	0.999794i	$-0.493535\pi$
$163$	0.518576	0.0406180	0.0203090	+	0.999794i	$0.493535\pi$
$164$	0	0
$165$	5.87867	0.457654
$166$	0	0
$167$	4.12508	0.319208	0.159604	−	0.987181i	$-0.448978\pi$
$167$	4.12508	0.319208	0.159604	+	0.987181i	$0.448978\pi$
$168$	0	0
$169$	−1.72396	−0.132612
$170$	0	0
$171$	−1.69308	−0.129473
$172$	0	0
$173$	0.362357	0.0275495	0.0137748	−	0.999905i	$-0.495615\pi$
$173$	0.362357	0.0275495	0.0137748	+	0.999905i	$0.495615\pi$
$174$	0	0
$175$	−2.71089	−0.204924
$176$	0	0
$177$	−2.79305	−0.209938
$178$	0	0
$179$	5.97272	0.446422	0.223211	−	0.974770i	$-0.428346\pi$
$179$	5.97272	0.446422	0.223211	+	0.974770i	$0.428346\pi$
$180$	0	0
$181$	4.95609	0.368383	0.184192	−	0.982890i	$-0.441033\pi$
$181$	4.95609	0.368383	0.184192	+	0.982890i	$0.441033\pi$
$182$	0	0
$183$	−40.2883	−2.97820
$184$	0	0
$185$	−1.92378	−0.141439
$186$	0	0
$187$	6.78748	0.496349
$188$	0	0
$189$	−21.2848	−1.54824
$190$	0	0
$191$	−20.1437	−1.45755	−0.728774	−	0.684754i	$-0.759910\pi$
$191$	−20.1437	−1.45755	−0.728774	+	0.684754i	$0.759910\pi$
$192$	0	0
$193$	22.4900	1.61886	0.809431	−	0.587214i	$-0.199775\pi$
$193$	22.4900	1.61886	0.809431	+	0.587214i	$0.199775\pi$
$194$	0	0
$195$	−19.7405	−1.41364
$196$	0	0
$197$	−19.9181	−1.41911	−0.709554	−	0.704651i	$-0.751104\pi$
$197$	−19.9181	−1.41911	−0.709554	+	0.704651i	$0.751104\pi$
$198$	0	0
$199$	27.7838	1.96954	0.984772	−	0.173849i	$-0.0556205\pi$
$199$	27.7838	1.96954	0.984772	+	0.173849i	$0.0556205\pi$
$200$	0	0
$201$	10.4747	0.738825
$202$	0	0
$203$	19.7089	1.38329
$204$	0	0
$205$	−0.402105	−0.0280842
$206$	0	0
$207$	29.5477	2.05370
$208$	0	0
$209$	−0.267136	−0.0184782
$210$	0	0
$211$	12.9636	0.892450	0.446225	−	0.894921i	$-0.352768\pi$
$211$	12.9636	0.892450	0.446225	+	0.894921i	$0.352768\pi$
$212$	0	0
$213$	−2.61169	−0.178950
$214$	0	0
$215$	7.22858	0.492985
$216$	0	0
$217$	6.73320	0.457079
$218$	0	0
$219$	−30.4561	−2.05804
$220$	0	0
$221$	−22.7922	−1.53317
$222$	0	0
$223$	−4.83430	−0.323729	−0.161864	−	0.986813i	$-0.551751\pi$
$223$	−4.83430	−0.323729	−0.161864	+	0.986813i	$0.551751\pi$
$224$	0	0
$225$	−8.23350	−0.548900
$226$	0	0
$227$	−8.12378	−0.539194	−0.269597	−	0.962973i	$-0.586891\pi$
$227$	−8.12378	−0.539194	−0.269597	+	0.962973i	$0.586891\pi$
$228$	0	0
$229$	−2.65582	−0.175502	−0.0877509	−	0.996142i	$-0.527968\pi$
$229$	−2.65582	−0.175502	−0.0877509	+	0.996142i	$0.527968\pi$
$230$	0	0
$231$	−6.37671	−0.419557
$232$	0	0
$233$	7.79662	0.510774	0.255387	−	0.966839i	$-0.417797\pi$
$233$	7.79662	0.510774	0.255387	+	0.966839i	$0.417797\pi$
$234$	0	0
$235$	−21.5782	−1.40760
$236$	0	0
$237$	41.7203	2.71003
$238$	0	0
$239$	−6.04134	−0.390782	−0.195391	−	0.980725i	$-0.562598\pi$
$239$	−6.04134	−0.390782	−0.195391	+	0.980725i	$0.562598\pi$
$240$	0	0
$241$	−15.9812	−1.02944	−0.514720	−	0.857358i	$-0.672104\pi$
$241$	−15.9812	−1.02944	−0.514720	+	0.857358i	$0.672104\pi$
$242$	0	0
$243$	−6.54423	−0.419813
$244$	0	0
$245$	5.08923	0.325139
$246$	0	0
$247$	0.897038	0.0570771
$248$	0	0
$249$	18.9514	1.20100
$250$	0	0
$251$	−3.10149	−0.195764	−0.0978822	−	0.995198i	$-0.531207\pi$
$251$	−3.10149	−0.195764	−0.0978822	+	0.995198i	$0.531207\pi$
$252$	0	0
$253$	4.66206	0.293101
$254$	0	0
$255$	39.9013	2.49872
$256$	0	0
$257$	23.3276	1.45513	0.727566	−	0.686037i	$-0.240651\pi$
$257$	23.3276	1.45513	0.727566	+	0.686037i	$0.240651\pi$
$258$	0	0
$259$	2.08676	0.129665
$260$	0	0
$261$	59.8598	3.70523
$262$	0	0
$263$	21.4858	1.32487	0.662435	−	0.749120i	$-0.269523\pi$
$263$	21.4858	1.32487	0.662435	+	0.749120i	$0.269523\pi$
$264$	0	0
$265$	−5.29000	−0.324962
$266$	0	0
$267$	27.8487	1.70431
$268$	0	0
$269$	25.8012	1.57313	0.786564	−	0.617509i	$-0.211858\pi$
$269$	25.8012	1.57313	0.786564	+	0.617509i	$0.211858\pi$
$270$	0	0
$271$	−22.7410	−1.38142	−0.690709	−	0.723132i	$-0.742702\pi$
$271$	−22.7410	−1.38142	−0.690709	+	0.723132i	$0.742702\pi$
$272$	0	0
$273$	21.4129	1.29597
$274$	0	0
$275$	−1.29909	−0.0783381
$276$	0	0
$277$	−2.80145	−0.168323	−0.0841615	−	0.996452i	$-0.526821\pi$
$277$	−2.80145	−0.168323	−0.0841615	+	0.996452i	$0.526821\pi$
$278$	0	0
$279$	20.4501	1.22431
$280$	0	0
$281$	−14.7748	−0.881388	−0.440694	−	0.897657i	$-0.645268\pi$
$281$	−14.7748	−0.881388	−0.440694	+	0.897657i	$0.645268\pi$
$282$	0	0
$283$	15.4996	0.921357	0.460679	−	0.887567i	$-0.347606\pi$
$283$	15.4996	0.921357	0.460679	+	0.887567i	$0.347606\pi$
$284$	0	0
$285$	−1.57040	−0.0930227
$286$	0	0
$287$	0.436171	0.0257464
$288$	0	0
$289$	29.0698	1.70999
$290$	0	0
$291$	−35.8881	−2.10380
$292$	0	0
$293$	−0.910769	−0.0532077	−0.0266038	−	0.999646i	$-0.508469\pi$
$293$	−0.910769	−0.0532077	−0.0266038	+	0.999646i	$0.508469\pi$
$294$	0	0
$295$	−1.75836	−0.102376
$296$	0	0
$297$	−10.1999	−0.591861
$298$	0	0
$299$	−15.6551	−0.905358
$300$	0	0
$301$	−7.84099	−0.451947
$302$	0	0
$303$	19.6540	1.12909
$304$	0	0
$305$	−25.3635	−1.45231
$306$	0	0
$307$	−2.62103	−0.149590	−0.0747950	−	0.997199i	$-0.523830\pi$
$307$	−2.62103	−0.149590	−0.0747950	+	0.997199i	$0.523830\pi$
$308$	0	0
$309$	54.3106	3.08962
$310$	0	0
$311$	21.5713	1.22320	0.611599	−	0.791168i	$-0.290527\pi$
$311$	21.5713	1.22320	0.611599	+	0.791168i	$0.290527\pi$
$312$	0	0
$313$	15.7157	0.888302	0.444151	−	0.895952i	$-0.353505\pi$
$313$	15.7157	0.888302	0.444151	+	0.895952i	$0.353505\pi$
$314$	0	0
$315$	−25.4432	−1.43356
$316$	0	0
$317$	1.05813	0.0594307	0.0297154	−	0.999558i	$-0.490540\pi$
$317$	1.05813	0.0594307	0.0297154	+	0.999558i	$0.490540\pi$
$318$	0	0
$319$	9.44474	0.528804
$320$	0	0
$321$	1.84662	0.103068
$322$	0	0
$323$	−1.81318	−0.100888
$324$	0	0
$325$	4.36232	0.241978
$326$	0	0
$327$	53.9635	2.98419
$328$	0	0
$329$	23.4063	1.29043
$330$	0	0
$331$	−24.7129	−1.35835	−0.679173	−	0.733978i	$-0.737661\pi$
$331$	−24.7129	−1.35835	−0.679173	+	0.733978i	$0.737661\pi$
$332$	0	0
$333$	6.33790	0.347315
$334$	0	0
$335$	6.59431	0.360286
$336$	0	0
$337$	−16.7192	−0.910750	−0.455375	−	0.890300i	$-0.650495\pi$
$337$	−16.7192	−0.910750	−0.455375	+	0.890300i	$0.650495\pi$
$338$	0	0
$339$	32.5512	1.76794
$340$	0	0
$341$	3.22663	0.174732
$342$	0	0
$343$	−20.1277	−1.08679
$344$	0	0
$345$	27.4067	1.47553
$346$	0	0
$347$	23.3650	1.25430	0.627150	−	0.778899i	$-0.284221\pi$
$347$	23.3650	1.25430	0.627150	+	0.778899i	$0.284221\pi$
$348$	0	0
$349$	29.6422	1.58671	0.793354	−	0.608760i	$-0.208333\pi$
$349$	29.6422	1.58671	0.793354	+	0.608760i	$0.208333\pi$
$350$	0	0
$351$	34.2512	1.82820
$352$	0	0
$353$	−11.2952	−0.601182	−0.300591	−	0.953753i	$-0.597184\pi$
$353$	−11.2952	−0.601182	−0.300591	+	0.953753i	$0.597184\pi$
$354$	0	0
$355$	−1.64419	−0.0872645
$356$	0	0
$357$	−43.2818	−2.29071
$358$	0	0
$359$	14.1843	0.748621	0.374310	−	0.927303i	$-0.377879\pi$
$359$	14.1843	0.748621	0.374310	+	0.927303i	$0.377879\pi$
$360$	0	0
$361$	−18.9286	−0.996244
$362$	0	0
$363$	−3.05580	−0.160388
$364$	0	0
$365$	−19.1736	−1.00359
$366$	0	0
$367$	13.9885	0.730194	0.365097	−	0.930969i	$-0.381036\pi$
$367$	13.9885	0.730194	0.365097	+	0.930969i	$0.381036\pi$
$368$	0	0
$369$	1.32474	0.0689632
$370$	0	0
$371$	5.73817	0.297911
$372$	0	0
$373$	20.9389	1.08418	0.542089	−	0.840321i	$-0.317634\pi$
$373$	20.9389	1.08418	0.542089	+	0.840321i	$0.317634\pi$
$374$	0	0
$375$	−37.0303	−1.91223
$376$	0	0
$377$	−31.7153	−1.63342
$378$	0	0
$379$	29.7250	1.52687	0.763436	−	0.645884i	$-0.223511\pi$
$379$	29.7250	1.52687	0.763436	+	0.645884i	$0.223511\pi$
$380$	0	0
$381$	−38.3246	−1.96343
$382$	0	0
$383$	−7.92443	−0.404919	−0.202460	−	0.979291i	$-0.564894\pi$
$383$	−7.92443	−0.404919	−0.202460	+	0.979291i	$0.564894\pi$
$384$	0	0
$385$	−4.01445	−0.204595
$386$	0	0
$387$	−23.8147	−1.21057
$388$	0	0
$389$	−27.9035	−1.41476	−0.707382	−	0.706831i	$-0.750124\pi$
$389$	−27.9035	−1.41476	−0.707382	+	0.706831i	$0.750124\pi$
$390$	0	0
$391$	31.6436	1.60029
$392$	0	0
$393$	−21.5250	−1.08579
$394$	0	0
$395$	26.2650	1.32153
$396$	0	0
$397$	−23.4461	−1.17673	−0.588364	−	0.808596i	$-0.700228\pi$
$397$	−23.4461	−1.17673	−0.588364	+	0.808596i	$0.700228\pi$
$398$	0	0
$399$	1.70345	0.0852791
$400$	0	0
$401$	−11.6855	−0.583544	−0.291772	−	0.956488i	$-0.594245\pi$
$401$	−11.6855	−0.583544	−0.291772	+	0.956488i	$0.594245\pi$
$402$	0	0
$403$	−10.8350	−0.539728
$404$	0	0
$405$	−23.3840	−1.16196
$406$	0	0
$407$	1.00000	0.0495682
$408$	0	0
$409$	31.8911	1.57691	0.788457	−	0.615090i	$-0.210880\pi$
$409$	31.8911	1.57691	0.788457	+	0.615090i	$0.210880\pi$
$410$	0	0
$411$	44.4259	2.19137
$412$	0	0
$413$	1.90733	0.0938536
$414$	0	0
$415$	11.9309	0.585663
$416$	0	0
$417$	−57.1407	−2.79819
$418$	0	0
$419$	−5.60124	−0.273638	−0.136819	−	0.990596i	$-0.543688\pi$
$419$	−5.60124	−0.273638	−0.136819	+	0.990596i	$0.543688\pi$
$420$	0	0
$421$	26.9004	1.31104	0.655522	−	0.755176i	$-0.272449\pi$
$421$	26.9004	1.31104	0.655522	+	0.755176i	$0.272449\pi$
$422$	0	0
$423$	71.0895	3.45649
$424$	0	0
$425$	−8.81754	−0.427713
$426$	0	0
$427$	27.5123	1.33141
$428$	0	0
$429$	10.2613	0.495421
$430$	0	0
$431$	−37.6161	−1.81190	−0.905952	−	0.423380i	$-0.860844\pi$
$431$	−37.6161	−1.81190	−0.905952	+	0.423380i	$0.860844\pi$
$432$	0	0
$433$	31.0239	1.49091	0.745457	−	0.666554i	$-0.232231\pi$
$433$	31.0239	1.49091	0.745457	+	0.666554i	$0.232231\pi$
$434$	0	0
$435$	55.5225	2.66210
$436$	0	0
$437$	−1.24540	−0.0595757
$438$	0	0
$439$	4.01368	0.191563	0.0957813	−	0.995402i	$-0.469465\pi$
$439$	4.01368	0.191563	0.0957813	+	0.995402i	$0.469465\pi$
$440$	0	0
$441$	−16.7665	−0.798407
$442$	0	0
$443$	23.6652	1.12437	0.562184	−	0.827012i	$-0.309961\pi$
$443$	23.6652	1.12437	0.562184	+	0.827012i	$0.309961\pi$
$444$	0	0
$445$	17.5321	0.831103
$446$	0	0
$447$	26.7749	1.26641
$448$	0	0
$449$	−10.1536	−0.479178	−0.239589	−	0.970874i	$-0.577013\pi$
$449$	−10.1536	−0.479178	−0.239589	+	0.970874i	$0.577013\pi$
$450$	0	0
$451$	0.209019	0.00984231
$452$	0	0
$453$	27.6129	1.29737
$454$	0	0
$455$	13.4805	0.631974
$456$	0	0
$457$	34.8210	1.62886	0.814428	−	0.580265i	$-0.197051\pi$
$457$	34.8210	1.62886	0.814428	+	0.580265i	$0.197051\pi$
$458$	0	0
$459$	−69.2319	−3.23147
$460$	0	0
$461$	−19.3500	−0.901218	−0.450609	−	0.892721i	$-0.648793\pi$
$461$	−19.3500	−0.901218	−0.450609	+	0.892721i	$0.648793\pi$
$462$	0	0
$463$	5.80193	0.269639	0.134819	−	0.990870i	$-0.456955\pi$
$463$	5.80193	0.269639	0.134819	+	0.990870i	$0.456955\pi$
$464$	0	0
$465$	18.9683	0.879634
$466$	0	0
$467$	37.2343	1.72300	0.861498	−	0.507760i	$-0.169527\pi$
$467$	37.2343	1.72300	0.861498	+	0.507760i	$0.169527\pi$
$468$	0	0
$469$	−7.15298	−0.330294
$470$	0	0
$471$	−36.0001	−1.65880
$472$	0	0
$473$	−3.75750	−0.172770
$474$	0	0
$475$	0.347034	0.0159230
$476$	0	0
$477$	17.4280	0.797972
$478$	0	0
$479$	1.60037	0.0731227	0.0365614	−	0.999331i	$-0.488360\pi$
$479$	1.60037	0.0731227	0.0365614	+	0.999331i	$0.488360\pi$
$480$	0	0
$481$	−3.35798	−0.153111
$482$	0	0
$483$	−29.7286	−1.35270
$484$	0	0
$485$	−22.5933	−1.02591
$486$	0	0
$487$	−25.1312	−1.13880	−0.569401	−	0.822060i	$-0.692825\pi$
$487$	−25.1312	−1.13880	−0.569401	+	0.822060i	$0.692825\pi$
$488$	0	0
$489$	−1.58466	−0.0716609
$490$	0	0
$491$	21.4265	0.966964	0.483482	−	0.875354i	$-0.339372\pi$
$491$	21.4265	0.966964	0.483482	+	0.875354i	$0.339372\pi$
$492$	0	0
$493$	64.1059	2.88719
$494$	0	0
$495$	−12.1927	−0.548021
$496$	0	0
$497$	1.78349	0.0800003
$498$	0	0
$499$	38.7484	1.73462	0.867308	−	0.497772i	$-0.165848\pi$
$499$	38.7484	1.73462	0.867308	+	0.497772i	$0.165848\pi$
$500$	0	0
$501$	−12.6054	−0.563168
$502$	0	0
$503$	−9.68221	−0.431708	−0.215854	−	0.976426i	$-0.569254\pi$
$503$	−9.68221	−0.431708	−0.215854	+	0.976426i	$0.569254\pi$
$504$	0	0
$505$	12.3732	0.550599
$506$	0	0
$507$	5.26807	0.233963
$508$	0	0
$509$	6.38700	0.283099	0.141549	−	0.989931i	$-0.454792\pi$
$509$	6.38700	0.283099	0.141549	+	0.989931i	$0.454792\pi$
$510$	0	0
$511$	20.7980	0.920051
$512$	0	0
$513$	2.72477	0.120302
$514$	0	0
$515$	34.1912	1.50664
$516$	0	0
$517$	11.2166	0.493304
$518$	0	0
$519$	−1.10729	−0.0486046
$520$	0	0
$521$	1.72590	0.0756129	0.0378065	−	0.999285i	$-0.487963\pi$
$521$	1.72590	0.0756129	0.0378065	+	0.999285i	$0.487963\pi$
$522$	0	0
$523$	−1.96843	−0.0860736	−0.0430368	−	0.999073i	$-0.513703\pi$
$523$	−1.96843	−0.0860736	−0.0430368	+	0.999073i	$0.513703\pi$
$524$	0	0
$525$	8.28392	0.361540
$526$	0	0
$527$	21.9007	0.954009
$528$	0	0
$529$	−1.26520	−0.0550088
$530$	0	0
$531$	5.79295	0.251392
$532$	0	0
$533$	−0.701881	−0.0304018
$534$	0	0
$535$	1.16254	0.0502610
$536$	0	0
$537$	−18.2514	−0.787607
$538$	0	0
$539$	−2.64544	−0.113947
$540$	0	0
$541$	6.73387	0.289512	0.144756	−	0.989467i	$-0.453760\pi$
$541$	6.73387	0.289512	0.144756	+	0.989467i	$0.453760\pi$
$542$	0	0
$543$	−15.1448	−0.649926
$544$	0	0
$545$	33.9727	1.45523
$546$	0	0
$547$	19.7548	0.844656	0.422328	−	0.906443i	$-0.361213\pi$
$547$	19.7548	0.844656	0.422328	+	0.906443i	$0.361213\pi$
$548$	0	0
$549$	83.5603	3.56626
$550$	0	0
$551$	−2.52303	−0.107485
$552$	0	0
$553$	−28.4902	−1.21153
$554$	0	0
$555$	5.87867	0.249536
$556$	0	0
$557$	−37.2839	−1.57977	−0.789885	−	0.613255i	$-0.789860\pi$
$557$	−37.2839	−1.57977	−0.789885	+	0.613255i	$0.789860\pi$
$558$	0	0
$559$	12.6176	0.533668
$560$	0	0
$561$	−20.7412	−0.875692
$562$	0	0
$563$	−29.9539	−1.26241	−0.631204	−	0.775617i	$-0.717439\pi$
$563$	−29.9539	−1.26241	−0.631204	+	0.775617i	$0.717439\pi$
$564$	0	0
$565$	20.4926	0.862128
$566$	0	0
$567$	25.3651	1.06524
$568$	0	0
$569$	38.5030	1.61413	0.807065	−	0.590462i	$-0.201054\pi$
$569$	38.5030	1.61413	0.807065	+	0.590462i	$0.201054\pi$
$570$	0	0
$571$	12.3146	0.515352	0.257676	−	0.966231i	$-0.417043\pi$
$571$	12.3146	0.515352	0.257676	+	0.966231i	$0.417043\pi$
$572$	0	0
$573$	61.5551	2.57150
$574$	0	0
$575$	−6.05643	−0.252571
$576$	0	0
$577$	28.5802	1.18981	0.594905	−	0.803796i	$-0.297190\pi$
$577$	28.5802	1.18981	0.594905	+	0.803796i	$0.297190\pi$
$578$	0	0
$579$	−68.7248	−2.85610
$580$	0	0
$581$	−12.9417	−0.536910
$582$	0	0
$583$	2.74980	0.113885
$584$	0	0
$585$	40.9428	1.69278
$586$	0	0
$587$	−33.2549	−1.37258	−0.686289	−	0.727329i	$-0.740762\pi$
$587$	−33.2549	−1.37258	−0.686289	+	0.727329i	$0.740762\pi$
$588$	0	0
$589$	−0.861949	−0.0355160
$590$	0	0
$591$	60.8658	2.50368
$592$	0	0
$593$	23.9716	0.984396	0.492198	−	0.870483i	$-0.336194\pi$
$593$	23.9716	0.984396	0.492198	+	0.870483i	$0.336194\pi$
$594$	0	0
$595$	−27.2480	−1.11706
$596$	0	0
$597$	−84.9018	−3.47480
$598$	0	0
$599$	12.2965	0.502419	0.251210	−	0.967933i	$-0.419172\pi$
$599$	12.2965	0.502419	0.251210	+	0.967933i	$0.419172\pi$
$600$	0	0
$601$	−2.67070	−0.108940	−0.0544701	−	0.998515i	$-0.517347\pi$
$601$	−2.67070	−0.108940	−0.0544701	+	0.998515i	$0.517347\pi$
$602$	0	0
$603$	−21.7250	−0.884712
$604$	0	0
$605$	−1.92378	−0.0782126
$606$	0	0
$607$	15.4519	0.627173	0.313586	−	0.949560i	$-0.398469\pi$
$607$	15.4519	0.627173	0.313586	+	0.949560i	$0.398469\pi$
$608$	0	0
$609$	−60.2264	−2.44050
$610$	0	0
$611$	−37.6650	−1.52376
$612$	0	0
$613$	−15.9922	−0.645921	−0.322960	−	0.946412i	$-0.604678\pi$
$613$	−15.9922	−0.645921	−0.322960	+	0.946412i	$0.604678\pi$
$614$	0	0
$615$	1.22875	0.0495480
$616$	0	0
$617$	17.2120	0.692931	0.346466	−	0.938063i	$-0.387382\pi$
$617$	17.2120	0.692931	0.346466	+	0.938063i	$0.387382\pi$
$618$	0	0
$619$	−26.1156	−1.04967	−0.524836	−	0.851203i	$-0.675874\pi$
$619$	−26.1156	−1.04967	−0.524836	+	0.851203i	$0.675874\pi$
$620$	0	0
$621$	−47.5528	−1.90823
$622$	0	0
$623$	−19.0175	−0.761919
$624$	0	0
$625$	−16.8169	−0.672677
$626$	0	0
$627$	0.816313	0.0326004
$628$	0	0
$629$	6.78748	0.270634
$630$	0	0
$631$	−41.1080	−1.63648	−0.818241	−	0.574875i	$-0.805051\pi$
$631$	−41.1080	−1.63648	−0.818241	+	0.574875i	$0.805051\pi$
$632$	0	0
$633$	−39.6141	−1.57452
$634$	0	0
$635$	−24.1272	−0.957459
$636$	0	0
$637$	8.88334	0.351971
$638$	0	0
$639$	5.41680	0.214285
$640$	0	0
$641$	−34.8883	−1.37801	−0.689003	−	0.724759i	$-0.741951\pi$
$641$	−34.8883	−1.37801	−0.689003	+	0.724759i	$0.741951\pi$
$642$	0	0
$643$	10.3654	0.408773	0.204387	−	0.978890i	$-0.434480\pi$
$643$	10.3654	0.408773	0.204387	+	0.978890i	$0.434480\pi$
$644$	0	0
$645$	−22.0891	−0.869757
$646$	0	0
$647$	−44.0522	−1.73187	−0.865936	−	0.500155i	$-0.833276\pi$
$647$	−44.0522	−1.73187	−0.865936	+	0.500155i	$0.833276\pi$
$648$	0	0
$649$	0.914017	0.0358783
$650$	0	0
$651$	−20.5753	−0.806409
$652$	0	0
$653$	20.0366	0.784093	0.392046	−	0.919945i	$-0.371767\pi$
$653$	20.0366	0.784093	0.392046	+	0.919945i	$0.371767\pi$
$654$	0	0
$655$	−13.5511	−0.529484
$656$	0	0
$657$	63.1678	2.46441
$658$	0	0
$659$	−25.6086	−0.997570	−0.498785	−	0.866726i	$-0.666220\pi$
$659$	−25.6086	−0.997570	−0.498785	+	0.866726i	$0.666220\pi$
$660$	0	0
$661$	−10.9122	−0.424434	−0.212217	−	0.977223i	$-0.568068\pi$
$661$	−10.9122	−0.424434	−0.212217	+	0.977223i	$0.568068\pi$
$662$	0	0
$663$	69.6484	2.70492
$664$	0	0
$665$	1.07240	0.0415861
$666$	0	0
$667$	44.0319	1.70492
$668$	0	0
$669$	14.7726	0.571144
$670$	0	0
$671$	13.1842	0.508971
$672$	0	0
$673$	−9.42551	−0.363327	−0.181663	−	0.983361i	$-0.558148\pi$
$673$	−9.42551	−0.363327	−0.181663	+	0.983361i	$0.558148\pi$
$674$	0	0
$675$	13.2506	0.510018
$676$	0	0
$677$	19.9013	0.764870	0.382435	−	0.923982i	$-0.375086\pi$
$677$	19.9013	0.764870	0.382435	+	0.923982i	$0.375086\pi$
$678$	0	0
$679$	24.5075	0.940510
$680$	0	0
$681$	24.8246	0.951282
$682$	0	0
$683$	42.0283	1.60817	0.804084	−	0.594515i	$-0.202656\pi$
$683$	42.0283	1.60817	0.804084	+	0.594515i	$0.202656\pi$
$684$	0	0
$685$	27.9683	1.06861
$686$	0	0
$687$	8.11566	0.309632
$688$	0	0
$689$	−9.23378	−0.351779
$690$	0	0
$691$	−28.7121	−1.09226	−0.546130	−	0.837700i	$-0.683900\pi$
$691$	−28.7121	−1.09226	−0.546130	+	0.837700i	$0.683900\pi$
$692$	0	0
$693$	13.2257	0.502401
$694$	0	0
$695$	−35.9729	−1.36453
$696$	0	0
$697$	1.41871	0.0537374
$698$	0	0
$699$	−23.8249	−0.901141
$700$	0	0
$701$	3.74884	0.141592	0.0707959	−	0.997491i	$-0.477446\pi$
$701$	3.74884	0.141592	0.0707959	+	0.997491i	$0.477446\pi$
$702$	0	0
$703$	−0.267136	−0.0100752
$704$	0	0
$705$	65.9385	2.48339
$706$	0	0
$707$	−13.4214	−0.504765
$708$	0	0
$709$	−36.0268	−1.35302	−0.676508	−	0.736436i	$-0.736507\pi$
$709$	−36.0268	−1.35302	−0.676508	+	0.736436i	$0.736507\pi$
$710$	0	0
$711$	−86.5303	−3.24514
$712$	0	0
$713$	15.0427	0.563355
$714$	0	0
$715$	6.46000	0.241590
$716$	0	0
$717$	18.4611	0.689443
$718$	0	0
$719$	−20.3249	−0.757992	−0.378996	−	0.925398i	$-0.623731\pi$
$719$	−20.3249	−0.757992	−0.378996	+	0.925398i	$0.623731\pi$
$720$	0	0
$721$	−37.0879	−1.38123
$722$	0	0
$723$	48.8354	1.81621
$724$	0	0
$725$	−12.2696	−0.455680
$726$	0	0
$727$	33.0084	1.22422	0.612108	−	0.790774i	$-0.290322\pi$
$727$	33.0084	1.22422	0.612108	+	0.790774i	$0.290322\pi$
$728$	0	0
$729$	−16.4680	−0.609926
$730$	0	0
$731$	−25.5039	−0.943297
$732$	0	0
$733$	20.6847	0.764008	0.382004	−	0.924161i	$-0.375234\pi$
$733$	20.6847	0.764008	0.382004	+	0.924161i	$0.375234\pi$
$734$	0	0
$735$	−15.5517	−0.573632
$736$	0	0
$737$	−3.42780	−0.126264
$738$	0	0
$739$	26.4355	0.972446	0.486223	−	0.873835i	$-0.338374\pi$
$739$	26.4355	0.972446	0.486223	+	0.873835i	$0.338374\pi$
$740$	0	0
$741$	−2.74117	−0.100699
$742$	0	0
$743$	4.74696	0.174149	0.0870745	−	0.996202i	$-0.472248\pi$
$743$	4.74696	0.174149	0.0870745	+	0.996202i	$0.472248\pi$
$744$	0	0
$745$	16.8561	0.617560
$746$	0	0
$747$	−39.3064	−1.43814
$748$	0	0
$749$	−1.26103	−0.0460770
$750$	0	0
$751$	−7.84002	−0.286087	−0.143043	−	0.989716i	$-0.545689\pi$
$751$	−7.84002	−0.286087	−0.143043	+	0.989716i	$0.545689\pi$
$752$	0	0
$753$	9.47753	0.345380
$754$	0	0
$755$	17.3837	0.632658
$756$	0	0
$757$	2.28648	0.0831035	0.0415517	−	0.999136i	$-0.486770\pi$
$757$	2.28648	0.0831035	0.0415517	+	0.999136i	$0.486770\pi$
$758$	0	0
$759$	−14.2463	−0.517108
$760$	0	0
$761$	36.3530	1.31779	0.658897	−	0.752233i	$-0.271023\pi$
$761$	36.3530	1.31779	0.658897	+	0.752233i	$0.271023\pi$
$762$	0	0
$763$	−36.8509	−1.33409
$764$	0	0
$765$	−82.7576	−2.99211
$766$	0	0
$767$	−3.06925	−0.110824
$768$	0	0
$769$	26.8095	0.966775	0.483388	−	0.875406i	$-0.339406\pi$
$769$	26.8095	0.966775	0.483388	+	0.875406i	$0.339406\pi$
$770$	0	0
$771$	−71.2843	−2.56724
$772$	0	0
$773$	−36.4425	−1.31074	−0.655372	−	0.755306i	$-0.727488\pi$
$773$	−36.4425	−1.31074	−0.655372	+	0.755306i	$0.727488\pi$
$774$	0	0
$775$	−4.19168	−0.150570
$776$	0	0
$777$	−6.37671	−0.228763
$778$	0	0
$779$	−0.0558364	−0.00200055
$780$	0	0
$781$	0.854668	0.0305824
$782$	0	0
$783$	−96.3359	−3.44276
$784$	0	0
$785$	−22.6638	−0.808907
$786$	0	0
$787$	−0.944598	−0.0336713	−0.0168356	−	0.999858i	$-0.505359\pi$
$787$	−0.944598	−0.0336713	−0.0168356	+	0.999858i	$0.505359\pi$
$788$	0	0
$789$	−65.6562	−2.33742
$790$	0	0
$791$	−22.2287	−0.790362
$792$	0	0
$793$	−44.2724	−1.57216
$794$	0	0
$795$	16.1652	0.573319
$796$	0	0
$797$	−7.04884	−0.249683	−0.124841	−	0.992177i	$-0.539842\pi$
$797$	−7.04884	−0.249683	−0.124841	+	0.992177i	$0.539842\pi$
$798$	0	0
$799$	76.1322	2.69336
$800$	0	0
$801$	−57.7598	−2.04084
$802$	0	0
$803$	9.96667	0.351716
$804$	0	0
$805$	−18.7156	−0.659639
$806$	0	0
$807$	−78.8433	−2.77541
$808$	0	0
$809$	3.60812	0.126855	0.0634274	−	0.997986i	$-0.479797\pi$
$809$	3.60812	0.126855	0.0634274	+	0.997986i	$0.479797\pi$
$810$	0	0
$811$	−27.7320	−0.973802	−0.486901	−	0.873457i	$-0.661873\pi$
$811$	−27.7320	−0.973802	−0.486901	+	0.873457i	$0.661873\pi$
$812$	0	0
$813$	69.4920	2.43719
$814$	0	0
$815$	−0.997623	−0.0349452
$816$	0	0
$817$	1.00376	0.0351172
$818$	0	0
$819$	−44.4115	−1.55186
$820$	0	0
$821$	−6.23573	−0.217628	−0.108814	−	0.994062i	$-0.534705\pi$
$821$	−6.23573	−0.217628	−0.108814	+	0.994062i	$0.534705\pi$
$822$	0	0
$823$	51.8846	1.80858	0.904292	−	0.426914i	$-0.140399\pi$
$823$	51.8846	1.80858	0.904292	+	0.426914i	$0.140399\pi$
$824$	0	0
$825$	3.96976	0.138209
$826$	0	0
$827$	31.1839	1.08437	0.542185	−	0.840259i	$-0.317597\pi$
$827$	31.1839	1.08437	0.542185	+	0.840259i	$0.317597\pi$
$828$	0	0
$829$	−11.3360	−0.393715	−0.196858	−	0.980432i	$-0.563074\pi$
$829$	−11.3360	−0.393715	−0.196858	+	0.980432i	$0.563074\pi$
$830$	0	0
$831$	8.56068	0.296967
$832$	0	0
$833$	−17.9559	−0.622134
$834$	0	0
$835$	−7.93572	−0.274627
$836$	0	0
$837$	−32.9115	−1.13759
$838$	0	0
$839$	16.6723	0.575593	0.287796	−	0.957692i	$-0.407077\pi$
$839$	16.6723	0.575593	0.287796	+	0.957692i	$0.407077\pi$
$840$	0	0
$841$	60.2031	2.07597
$842$	0	0
$843$	45.1487	1.55500
$844$	0	0
$845$	3.31651	0.114091
$846$	0	0
$847$	2.08676	0.0717019
$848$	0	0
$849$	−47.3637	−1.62552
$850$	0	0
$851$	4.66206	0.159813
$852$	0	0
$853$	−18.3944	−0.629814	−0.314907	−	0.949123i	$-0.601973\pi$
$853$	−18.3944	−0.629814	−0.314907	+	0.949123i	$0.601973\pi$
$854$	0	0
$855$	3.25711	0.111391
$856$	0	0
$857$	−3.58503	−0.122462	−0.0612312	−	0.998124i	$-0.519503\pi$
$857$	−3.58503	−0.122462	−0.0612312	+	0.998124i	$0.519503\pi$
$858$	0	0
$859$	−20.5747	−0.702001	−0.351001	−	0.936375i	$-0.614158\pi$
$859$	−20.5747	−0.702001	−0.351001	+	0.936375i	$0.614158\pi$
$860$	0	0
$861$	−1.33285	−0.0454235
$862$	0	0
$863$	−30.2180	−1.02863	−0.514316	−	0.857601i	$-0.671954\pi$
$863$	−30.2180	−1.02863	−0.514316	+	0.857601i	$0.671954\pi$
$864$	0	0
$865$	−0.697094	−0.0237019
$866$	0	0
$867$	−88.8315	−3.01688
$868$	0	0
$869$	−13.6528	−0.463141
$870$	0	0
$871$	11.5105	0.390018
$872$	0	0
$873$	74.4341	2.51921
$874$	0	0
$875$	25.2874	0.854870
$876$	0	0
$877$	11.6760	0.394271	0.197136	−	0.980376i	$-0.436836\pi$
$877$	11.6760	0.394271	0.197136	+	0.980376i	$0.436836\pi$
$878$	0	0
$879$	2.78313	0.0938725
$880$	0	0
$881$	−15.7736	−0.531427	−0.265714	−	0.964052i	$-0.585608\pi$
$881$	−15.7736	−0.531427	−0.265714	+	0.964052i	$0.585608\pi$
$882$	0	0
$883$	−43.8740	−1.47648	−0.738238	−	0.674540i	$-0.764342\pi$
$883$	−43.8740	−1.47648	−0.738238	+	0.674540i	$0.764342\pi$
$884$	0	0
$885$	5.37320	0.180618
$886$	0	0
$887$	−19.4896	−0.654398	−0.327199	−	0.944955i	$-0.606105\pi$
$887$	−19.4896	−0.654398	−0.327199	+	0.944955i	$0.606105\pi$
$888$	0	0
$889$	26.1713	0.877756
$890$	0	0
$891$	12.1553	0.407217
$892$	0	0
$893$	−2.99635	−0.100269
$894$	0	0
$895$	−11.4902	−0.384074
$896$	0	0
$897$	47.8389	1.59729
$898$	0	0
$899$	30.4747	1.01639
$900$	0	0
$901$	18.6642	0.621795
$902$	0	0
$903$	23.9605	0.797355
$904$	0	0
$905$	−9.53440	−0.316934
$906$	0	0
$907$	21.6115	0.717599	0.358799	−	0.933415i	$-0.383186\pi$
$907$	21.6115	0.717599	0.358799	+	0.933415i	$0.383186\pi$
$908$	0	0
$909$	−40.7636	−1.35204
$910$	0	0
$911$	−13.6899	−0.453567	−0.226784	−	0.973945i	$-0.572821\pi$
$911$	−13.6899	−0.453567	−0.226784	+	0.973945i	$0.572821\pi$
$912$	0	0
$913$	−6.20180	−0.205250
$914$	0	0
$915$	77.5056	2.56226
$916$	0	0
$917$	14.6991	0.485408
$918$	0	0
$919$	21.4138	0.706375	0.353187	−	0.935553i	$-0.385098\pi$
$919$	21.4138	0.706375	0.353187	+	0.935553i	$0.385098\pi$
$920$	0	0
$921$	8.00934	0.263917
$922$	0	0
$923$	−2.86996	−0.0944659
$924$	0	0
$925$	−1.29909	−0.0427138
$926$	0	0
$927$	−112.643	−3.69969
$928$	0	0
$929$	−30.3444	−0.995567	−0.497784	−	0.867301i	$-0.665853\pi$
$929$	−30.3444	−0.995567	−0.497784	+	0.867301i	$0.665853\pi$
$930$	0	0
$931$	0.706692	0.0231609
$932$	0	0
$933$	−65.9177	−2.15805
$934$	0	0
$935$	−13.0576	−0.427028
$936$	0	0
$937$	8.51073	0.278034	0.139017	−	0.990290i	$-0.455606\pi$
$937$	8.51073	0.278034	0.139017	+	0.990290i	$0.455606\pi$
$938$	0	0
$939$	−48.0239	−1.56720
$940$	0	0
$941$	28.8295	0.939816	0.469908	−	0.882715i	$-0.344287\pi$
$941$	28.8295	0.939816	0.469908	+	0.882715i	$0.344287\pi$
$942$	0	0
$943$	0.974458	0.0317327
$944$	0	0
$945$	40.9472	1.33201
$946$	0	0
$947$	−13.0404	−0.423756	−0.211878	−	0.977296i	$-0.567958\pi$
$947$	−13.0404	−0.423756	−0.211878	+	0.977296i	$0.567958\pi$
$948$	0	0
$949$	−33.4679	−1.08641
$950$	0	0
$951$	−3.23344	−0.104852
$952$	0	0
$953$	35.9090	1.16321	0.581603	−	0.813473i	$-0.302426\pi$
$953$	35.9090	1.16321	0.581603	+	0.813473i	$0.302426\pi$
$954$	0	0
$955$	38.7520	1.25399
$956$	0	0
$957$	−28.8612	−0.932951
$958$	0	0
$959$	−30.3378	−0.979658
$960$	0	0
$961$	−20.5888	−0.664156
$962$	0	0
$963$	−3.83000	−0.123420
$964$	0	0
$965$	−43.2656	−1.39277
$966$	0	0
$967$	−0.668271	−0.0214902	−0.0107451	−	0.999942i	$-0.503420\pi$
$967$	−0.668271	−0.0214902	−0.0107451	+	0.999942i	$0.503420\pi$
$968$	0	0
$969$	5.54071	0.177993
$970$	0	0
$971$	−14.8373	−0.476150	−0.238075	−	0.971247i	$-0.576516\pi$
$971$	−14.8373	−0.476150	−0.238075	+	0.971247i	$0.576516\pi$
$972$	0	0
$973$	39.0205	1.25094
$974$	0	0
$975$	−13.3304	−0.426913
$976$	0	0
$977$	−16.0924	−0.514840	−0.257420	−	0.966300i	$-0.582872\pi$
$977$	−16.0924	−0.514840	−0.257420	+	0.966300i	$0.582872\pi$
$978$	0	0
$979$	−9.11340	−0.291266
$980$	0	0
$981$	−111.924	−3.57344
$982$	0	0
$983$	12.6587	0.403749	0.201875	−	0.979411i	$-0.435297\pi$
$983$	12.6587	0.403749	0.201875	+	0.979411i	$0.435297\pi$
$984$	0	0
$985$	38.3180	1.22091
$986$	0	0
$987$	−71.5248	−2.27666
$988$	0	0
$989$	−17.5177	−0.557030
$990$	0	0
$991$	−10.3785	−0.329685	−0.164843	−	0.986320i	$-0.552712\pi$
$991$	−10.3785	−0.329685	−0.164843	+	0.986320i	$0.552712\pi$
$992$	0	0
$993$	75.5177	2.39648
$994$	0	0
$995$	−53.4499	−1.69447
$996$	0	0
$997$	14.3188	0.453482	0.226741	−	0.973955i	$-0.427193\pi$
$997$	14.3188	0.453482	0.226741	+	0.973955i	$0.427193\pi$
$998$	0	0
$999$	−10.1999	−0.322712

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6512.2.a.bb.1.1		11
4.3	odd	2		407.2.a.c.1.10	✓	11
12.11	even	2		3663.2.a.u.1.2		11
44.43	even	2		4477.2.a.k.1.2		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
407.2.a.c.1.10	✓	11	4.3	odd	2
3663.2.a.u.1.2		11	12.11	even	2
4477.2.a.k.1.2		11	44.43	even	2
6512.2.a.bb.1.1		11	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:	\( N \)	\(=\)	\( 6512 = 2^{4} \cdot 11 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6512.a (trivial)

\(f(q)\)	\(=\)	\(q-3.05580 q^{3} -1.92378 q^{5} +2.08676 q^{7} +6.33790 q^{9} +O(q^{10})\)	\(q-3.05580 q^{3} -1.92378 q^{5} +2.08676 q^{7} +6.33790 q^{9} +1.00000 q^{11} -3.35798 q^{13} +5.87867 q^{15} +6.78748 q^{17} -0.267136 q^{19} -6.37671 q^{21} +4.66206 q^{23} -1.29909 q^{25} -10.1999 q^{27} +9.44474 q^{29} +3.22663 q^{31} -3.05580 q^{33} -4.01445 q^{35} +1.00000 q^{37} +10.2613 q^{39} +0.209019 q^{41} -3.75750 q^{43} -12.1927 q^{45} +11.2166 q^{47} -2.64544 q^{49} -20.7412 q^{51} +2.74980 q^{53} -1.92378 q^{55} +0.816313 q^{57} +0.914017 q^{59} +13.1842 q^{61} +13.2257 q^{63} +6.46000 q^{65} -3.42780 q^{67} -14.2463 q^{69} +0.854668 q^{71} +9.96667 q^{73} +3.96976 q^{75} +2.08676 q^{77} -13.6528 q^{79} +12.1553 q^{81} -6.20180 q^{83} -13.0576 q^{85} -28.8612 q^{87} -9.11340 q^{89} -7.00730 q^{91} -9.85993 q^{93} +0.513909 q^{95} +11.7443 q^{97} +6.33790 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + q^{5} - 9 q^{7} + 15 q^{9}+O(q^{10}) \) 11 * q + q^5 - 9 * q^7 + 15 * q^9	\( 11 q + q^{5} - 9 q^{7} + 15 q^{9} + 11 q^{11} + 22 q^{13} + 10 q^{15} + 19 q^{17} - 14 q^{19} - 13 q^{21} + 14 q^{23} + 38 q^{25} + 9 q^{27} + 13 q^{29} - 12 q^{31} - 12 q^{35} + 11 q^{37} + 8 q^{39} + 8 q^{41} - 21 q^{43} - 8 q^{45} + 14 q^{47} + 20 q^{49} + 2 q^{51} - 2 q^{53} + q^{55} - 3 q^{57} + 30 q^{59} + 20 q^{61} - 31 q^{63} - 7 q^{67} + 9 q^{69} + 15 q^{71} + 47 q^{73} + 40 q^{75} - 9 q^{77} - 2 q^{79} - 17 q^{81} - 24 q^{83} - 25 q^{85} - 21 q^{87} + 4 q^{89} - 21 q^{91} - 37 q^{93} + 36 q^{95} + 25 q^{97} + 15 q^{99}+O(q^{100}) \) 11 * q + q^5 - 9 * q^7 + 15 * q^9 + 11 * q^11 + 22 * q^13 + 10 * q^15 + 19 * q^17 - 14 * q^19 - 13 * q^21 + 14 * q^23 + 38 * q^25 + 9 * q^27 + 13 * q^29 - 12 * q^31 - 12 * q^35 + 11 * q^37 + 8 * q^39 + 8 * q^41 - 21 * q^43 - 8 * q^45 + 14 * q^47 + 20 * q^49 + 2 * q^51 - 2 * q^53 + q^55 - 3 * q^57 + 30 * q^59 + 20 * q^61 - 31 * q^63 - 7 * q^67 + 9 * q^69 + 15 * q^71 + 47 * q^73 + 40 * q^75 - 9 * q^77 - 2 * q^79 - 17 * q^81 - 24 * q^83 - 25 * q^85 - 21 * q^87 + 4 * q^89 - 21 * q^91 - 37 * q^93 + 36 * q^95 + 25 * q^97 + 15 * q^99

Introduction

L-functions

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