LMFDB - Embedded newform 8512.2.a.ch.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			$-0.838827$ 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)

$f(q)$	$=$	$q+0.838827 q^{3} +2.50991 q^{5} -1.00000 q^{7} -2.29637 q^{9} +O(q^{10})$	$q+0.838827 q^{3} +2.50991 q^{5} -1.00000 q^{7} -2.29637 q^{9} +3.46016 q^{11} +6.36062 q^{13} +2.10538 q^{15} +6.34077 q^{17} +1.00000 q^{19} -0.838827 q^{21} -5.29877 q^{23} +1.29966 q^{25} -4.44274 q^{27} +10.1298 q^{29} -9.39225 q^{31} +2.90248 q^{33} -2.50991 q^{35} -2.14446 q^{37} +5.33546 q^{39} -7.43345 q^{41} +1.15432 q^{43} -5.76368 q^{45} +5.96586 q^{47} +1.00000 q^{49} +5.31881 q^{51} +6.50168 q^{53} +8.68469 q^{55} +0.838827 q^{57} +9.26281 q^{59} +7.12100 q^{61} +2.29637 q^{63} +15.9646 q^{65} +7.54487 q^{67} -4.44476 q^{69} +1.10596 q^{71} +0.0584070 q^{73} +1.09019 q^{75} -3.46016 q^{77} +16.5650 q^{79} +3.16241 q^{81} -5.59617 q^{83} +15.9148 q^{85} +8.49719 q^{87} -14.8293 q^{89} -6.36062 q^{91} -7.87848 q^{93} +2.50991 q^{95} -1.89047 q^{97} -7.94580 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.838827	0.484297	0.242149	−	0.970239i	$-0.422148\pi$
$3$	0.838827	0.484297	0.242149	+	0.970239i	$0.422148\pi$
$4$	0	0
$5$	2.50991	1.12247	0.561233	−	0.827658i	$-0.310327\pi$
$5$	2.50991	1.12247	0.561233	+	0.827658i	$0.310327\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.29637	−0.765456
$10$	0	0
$11$	3.46016	1.04328	0.521639	−	0.853167i	$-0.325321\pi$
$11$	3.46016	1.04328	0.521639	+	0.853167i	$0.325321\pi$
$12$	0	0
$13$	6.36062	1.76412	0.882059	−	0.471139i	$-0.156157\pi$
$13$	6.36062	1.76412	0.882059	+	0.471139i	$0.156157\pi$
$14$	0	0
$15$	2.10538	0.543608
$16$	0	0
$17$	6.34077	1.53786	0.768931	−	0.639332i	$-0.220789\pi$
$17$	6.34077	1.53786	0.768931	+	0.639332i	$0.220789\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.838827	−0.183047
$22$	0	0
$23$	−5.29877	−1.10487	−0.552435	−	0.833556i	$-0.686301\pi$
$23$	−5.29877	−1.10487	−0.552435	+	0.833556i	$0.686301\pi$
$24$	0	0
$25$	1.29966	0.259932
$26$	0	0
$27$	−4.44274	−0.855006
$28$	0	0
$29$	10.1298	1.88106	0.940532	−	0.339704i	$-0.110327\pi$
$29$	10.1298	1.88106	0.940532	+	0.339704i	$0.110327\pi$
$30$	0	0
$31$	−9.39225	−1.68690	−0.843449	−	0.537209i	$-0.819479\pi$
$31$	−9.39225	−1.68690	−0.843449	+	0.537209i	$0.819479\pi$
$32$	0	0
$33$	2.90248	0.505256
$34$	0	0
$35$	−2.50991	−0.424253
$36$	0	0
$37$	−2.14446	−0.352546	−0.176273	−	0.984341i	$-0.556404\pi$
$37$	−2.14446	−0.352546	−0.176273	+	0.984341i	$0.556404\pi$
$38$	0	0
$39$	5.33546	0.854357
$40$	0	0
$41$	−7.43345	−1.16091	−0.580455	−	0.814292i	$-0.697125\pi$
$41$	−7.43345	−1.16091	−0.580455	+	0.814292i	$0.697125\pi$
$42$	0	0
$43$	1.15432	0.176032	0.0880159	−	0.996119i	$-0.471947\pi$
$43$	1.15432	0.176032	0.0880159	+	0.996119i	$0.471947\pi$
$44$	0	0
$45$	−5.76368	−0.859199
$46$	0	0
$47$	5.96586	0.870211	0.435105	−	0.900380i	$-0.356711\pi$
$47$	5.96586	0.870211	0.435105	+	0.900380i	$0.356711\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.31881	0.744782
$52$	0	0
$53$	6.50168	0.893074	0.446537	−	0.894765i	$-0.352657\pi$
$53$	6.50168	0.893074	0.446537	+	0.894765i	$0.352657\pi$
$54$	0	0
$55$	8.68469	1.17104
$56$	0	0
$57$	0.838827	0.111105
$58$	0	0
$59$	9.26281	1.20591	0.602957	−	0.797773i	$-0.293989\pi$
$59$	9.26281	1.20591	0.602957	+	0.797773i	$0.293989\pi$
$60$	0	0
$61$	7.12100	0.911751	0.455875	−	0.890044i	$-0.349326\pi$
$61$	7.12100	0.911751	0.455875	+	0.890044i	$0.349326\pi$
$62$	0	0
$63$	2.29637	0.289315
$64$	0	0
$65$	15.9646	1.98016
$66$	0	0
$67$	7.54487	0.921753	0.460876	−	0.887464i	$-0.347535\pi$
$67$	7.54487	0.921753	0.460876	+	0.887464i	$0.347535\pi$
$68$	0	0
$69$	−4.44476	−0.535086
$70$	0	0
$71$	1.10596	0.131254	0.0656269	−	0.997844i	$-0.479095\pi$
$71$	1.10596	0.131254	0.0656269	+	0.997844i	$0.479095\pi$
$72$	0	0
$73$	0.0584070	0.00683603	0.00341801	−	0.999994i	$-0.498912\pi$
$73$	0.0584070	0.00683603	0.00341801	+	0.999994i	$0.498912\pi$
$74$	0	0
$75$	1.09019	0.125884
$76$	0	0
$77$	−3.46016	−0.394322
$78$	0	0
$79$	16.5650	1.86371	0.931857	−	0.362827i	$-0.118188\pi$
$79$	16.5650	1.86371	0.931857	+	0.362827i	$0.118188\pi$
$80$	0	0
$81$	3.16241	0.351379
$82$	0	0
$83$	−5.59617	−0.614260	−0.307130	−	0.951668i	$-0.599369\pi$
$83$	−5.59617	−0.614260	−0.307130	+	0.951668i	$0.599369\pi$
$84$	0	0
$85$	15.9148	1.72620
$86$	0	0
$87$	8.49719	0.910994
$88$	0	0
$89$	−14.8293	−1.57191	−0.785953	−	0.618286i	$-0.787827\pi$
$89$	−14.8293	−1.57191	−0.785953	+	0.618286i	$0.787827\pi$
$90$	0	0
$91$	−6.36062	−0.666774
$92$	0	0
$93$	−7.87848	−0.816960
$94$	0	0
$95$	2.50991	0.257512
$96$	0	0
$97$	−1.89047	−0.191948	−0.0959741	−	0.995384i	$-0.530597\pi$
$97$	−1.89047	−0.191948	−0.0959741	+	0.995384i	$0.530597\pi$
$98$	0	0
$99$	−7.94580	−0.798583
$100$	0	0
$101$	3.59919	0.358132	0.179066	−	0.983837i	$-0.442692\pi$
$101$	3.59919	0.358132	0.179066	+	0.983837i	$0.442692\pi$
$102$	0	0
$103$	9.29050	0.915420	0.457710	−	0.889101i	$-0.348670\pi$
$103$	9.29050	0.915420	0.457710	+	0.889101i	$0.348670\pi$
$104$	0	0
$105$	−2.10538	−0.205464
$106$	0	0
$107$	−19.8956	−1.92338	−0.961690	−	0.274138i	$-0.911608\pi$
$107$	−19.8956	−1.92338	−0.961690	+	0.274138i	$0.911608\pi$
$108$	0	0
$109$	−0.787338	−0.0754133	−0.0377067	−	0.999289i	$-0.512005\pi$
$109$	−0.787338	−0.0754133	−0.0377067	+	0.999289i	$0.512005\pi$
$110$	0	0
$111$	−1.79883	−0.170737
$112$	0	0
$113$	−10.5592	−0.993324	−0.496662	−	0.867944i	$-0.665441\pi$
$113$	−10.5592	−0.993324	−0.496662	+	0.867944i	$0.665441\pi$
$114$	0	0
$115$	−13.2995	−1.24018
$116$	0	0
$117$	−14.6063	−1.35035
$118$	0	0
$119$	−6.34077	−0.581257
$120$	0	0
$121$	0.972699	0.0884272
$122$	0	0
$123$	−6.23539	−0.562226
$124$	0	0
$125$	−9.28753	−0.830702
$126$	0	0
$127$	−1.27500	−0.113138	−0.0565688	−	0.998399i	$-0.518016\pi$
$127$	−1.27500	−0.113138	−0.0565688	+	0.998399i	$0.518016\pi$
$128$	0	0
$129$	0.968273	0.0852517
$130$	0	0
$131$	8.07361	0.705394	0.352697	−	0.935738i	$-0.385265\pi$
$131$	8.07361	0.705394	0.352697	+	0.935738i	$0.385265\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−11.1509	−0.959715
$136$	0	0
$137$	10.8874	0.930172	0.465086	−	0.885265i	$-0.346023\pi$
$137$	10.8874	0.930172	0.465086	+	0.885265i	$0.346023\pi$
$138$	0	0
$139$	−12.1523	−1.03075	−0.515374	−	0.856966i	$-0.672347\pi$
$139$	−12.1523	−1.03075	−0.515374	+	0.856966i	$0.672347\pi$
$140$	0	0
$141$	5.00433	0.421441
$142$	0	0
$143$	22.0087	1.84046
$144$	0	0
$145$	25.4250	2.11143
$146$	0	0
$147$	0.838827	0.0691853
$148$	0	0
$149$	−13.5985	−1.11403	−0.557014	−	0.830503i	$-0.688053\pi$
$149$	−13.5985	−1.11403	−0.557014	+	0.830503i	$0.688053\pi$
$150$	0	0
$151$	−16.4025	−1.33482	−0.667409	−	0.744692i	$-0.732597\pi$
$151$	−16.4025	−1.33482	−0.667409	+	0.744692i	$0.732597\pi$
$152$	0	0
$153$	−14.5607	−1.17717
$154$	0	0
$155$	−23.5737	−1.89349
$156$	0	0
$157$	17.2552	1.37711	0.688557	−	0.725183i	$-0.258245\pi$
$157$	17.2552	1.37711	0.688557	+	0.725183i	$0.258245\pi$
$158$	0	0
$159$	5.45379	0.432514
$160$	0	0
$161$	5.29877	0.417602
$162$	0	0
$163$	−14.9989	−1.17480	−0.587401	−	0.809296i	$-0.699849\pi$
$163$	−14.9989	−1.17480	−0.587401	+	0.809296i	$0.699849\pi$
$164$	0	0
$165$	7.28496	0.567133
$166$	0	0
$167$	4.30131	0.332845	0.166423	−	0.986055i	$-0.446778\pi$
$167$	4.30131	0.332845	0.166423	+	0.986055i	$0.446778\pi$
$168$	0	0
$169$	27.4575	2.11211
$170$	0	0
$171$	−2.29637	−0.175608
$172$	0	0
$173$	24.4226	1.85682	0.928410	−	0.371558i	$-0.121176\pi$
$173$	24.4226	1.85682	0.928410	+	0.371558i	$0.121176\pi$
$174$	0	0
$175$	−1.29966	−0.0982450
$176$	0	0
$177$	7.76990	0.584021
$178$	0	0
$179$	−8.51414	−0.636377	−0.318188	−	0.948028i	$-0.603074\pi$
$179$	−8.51414	−0.636377	−0.318188	+	0.948028i	$0.603074\pi$
$180$	0	0
$181$	16.5481	1.23001	0.615004	−	0.788524i	$-0.289154\pi$
$181$	16.5481	1.23001	0.615004	+	0.788524i	$0.289154\pi$
$182$	0	0
$183$	5.97329	0.441558
$184$	0	0
$185$	−5.38240	−0.395722
$186$	0	0
$187$	21.9401	1.60442
$188$	0	0
$189$	4.44274	0.323162
$190$	0	0
$191$	−15.6296	−1.13092	−0.565459	−	0.824776i	$-0.691301\pi$
$191$	−15.6296	−1.13092	−0.565459	+	0.824776i	$0.691301\pi$
$192$	0	0
$193$	−13.4908	−0.971089	−0.485545	−	0.874212i	$-0.661379\pi$
$193$	−13.4908	−0.971089	−0.485545	+	0.874212i	$0.661379\pi$
$194$	0	0
$195$	13.3915	0.958988
$196$	0	0
$197$	−13.3172	−0.948811	−0.474405	−	0.880306i	$-0.657337\pi$
$197$	−13.3172	−0.948811	−0.474405	+	0.880306i	$0.657337\pi$
$198$	0	0
$199$	8.56699	0.607298	0.303649	−	0.952784i	$-0.401795\pi$
$199$	8.56699	0.607298	0.303649	+	0.952784i	$0.401795\pi$
$200$	0	0
$201$	6.32885	0.446402
$202$	0	0
$203$	−10.1298	−0.710976
$204$	0	0
$205$	−18.6573	−1.30308
$206$	0	0
$207$	12.1679	0.845730
$208$	0	0
$209$	3.46016	0.239344
$210$	0	0
$211$	−9.06901	−0.624337	−0.312168	−	0.950027i	$-0.601055\pi$
$211$	−9.06901	−0.624337	−0.312168	+	0.950027i	$0.601055\pi$
$212$	0	0
$213$	0.927714	0.0635659
$214$	0	0
$215$	2.89723	0.197590
$216$	0	0
$217$	9.39225	0.637588
$218$	0	0
$219$	0.0489934	0.00331067
$220$	0	0
$221$	40.3312	2.71297
$222$	0	0
$223$	−2.44936	−0.164021	−0.0820105	−	0.996631i	$-0.526134\pi$
$223$	−2.44936	−0.164021	−0.0820105	+	0.996631i	$0.526134\pi$
$224$	0	0
$225$	−2.98450	−0.198966
$226$	0	0
$227$	11.5126	0.764117	0.382058	−	0.924138i	$-0.375215\pi$
$227$	11.5126	0.764117	0.382058	+	0.924138i	$0.375215\pi$
$228$	0	0
$229$	8.76061	0.578918	0.289459	−	0.957190i	$-0.406525\pi$
$229$	8.76061	0.578918	0.289459	+	0.957190i	$0.406525\pi$
$230$	0	0
$231$	−2.90248	−0.190969
$232$	0	0
$233$	10.1559	0.665333	0.332666	−	0.943045i	$-0.392052\pi$
$233$	10.1559	0.665333	0.332666	+	0.943045i	$0.392052\pi$
$234$	0	0
$235$	14.9738	0.976783
$236$	0	0
$237$	13.8952	0.902591
$238$	0	0
$239$	−16.4419	−1.06354	−0.531770	−	0.846889i	$-0.678473\pi$
$239$	−16.4419	−1.06354	−0.531770	+	0.846889i	$0.678473\pi$
$240$	0	0
$241$	−5.10981	−0.329152	−0.164576	−	0.986364i	$-0.552626\pi$
$241$	−5.10981	−0.329152	−0.164576	+	0.986364i	$0.552626\pi$
$242$	0	0
$243$	15.9809	1.02518
$244$	0	0
$245$	2.50991	0.160352
$246$	0	0
$247$	6.36062	0.404716
$248$	0	0
$249$	−4.69422	−0.297484
$250$	0	0
$251$	−6.82819	−0.430992	−0.215496	−	0.976505i	$-0.569137\pi$
$251$	−6.82819	−0.430992	−0.215496	+	0.976505i	$0.569137\pi$
$252$	0	0
$253$	−18.3346	−1.15269
$254$	0	0
$255$	13.3497	0.835994
$256$	0	0
$257$	−13.2450	−0.826202	−0.413101	−	0.910685i	$-0.635554\pi$
$257$	−13.2450	−0.826202	−0.413101	+	0.910685i	$0.635554\pi$
$258$	0	0
$259$	2.14446	0.133250
$260$	0	0
$261$	−23.2619	−1.43987
$262$	0	0
$263$	1.19868	0.0739139	0.0369569	−	0.999317i	$-0.488234\pi$
$263$	1.19868	0.0739139	0.0369569	+	0.999317i	$0.488234\pi$
$264$	0	0
$265$	16.3186	1.00245
$266$	0	0
$267$	−12.4393	−0.761270
$268$	0	0
$269$	29.0920	1.77377	0.886885	−	0.461989i	$-0.152864\pi$
$269$	29.0920	1.77377	0.886885	+	0.461989i	$0.152864\pi$
$270$	0	0
$271$	12.7017	0.771572	0.385786	−	0.922588i	$-0.373930\pi$
$271$	12.7017	0.771572	0.385786	+	0.922588i	$0.373930\pi$
$272$	0	0
$273$	−5.33546	−0.322917
$274$	0	0
$275$	4.49703	0.271181
$276$	0	0
$277$	−7.98143	−0.479558	−0.239779	−	0.970828i	$-0.577075\pi$
$277$	−7.98143	−0.479558	−0.239779	+	0.970828i	$0.577075\pi$
$278$	0	0
$279$	21.5681	1.29125
$280$	0	0
$281$	20.4089	1.21749	0.608745	−	0.793366i	$-0.291673\pi$
$281$	20.4089	1.21749	0.608745	+	0.793366i	$0.291673\pi$
$282$	0	0
$283$	18.9925	1.12899	0.564493	−	0.825438i	$-0.309072\pi$
$283$	18.9925	1.12899	0.564493	+	0.825438i	$0.309072\pi$
$284$	0	0
$285$	2.10538	0.124712
$286$	0	0
$287$	7.43345	0.438783
$288$	0	0
$289$	23.2053	1.36502
$290$	0	0
$291$	−1.58578	−0.0929600
$292$	0	0
$293$	−19.2730	−1.12594	−0.562971	−	0.826476i	$-0.690342\pi$
$293$	−19.2730	−1.12594	−0.562971	+	0.826476i	$0.690342\pi$
$294$	0	0
$295$	23.2488	1.35360
$296$	0	0
$297$	−15.3726	−0.892008
$298$	0	0
$299$	−33.7035	−1.94912
$300$	0	0
$301$	−1.15432	−0.0665337
$302$	0	0
$303$	3.01910	0.173443
$304$	0	0
$305$	17.8731	1.02341
$306$	0	0
$307$	−8.77122	−0.500600	−0.250300	−	0.968168i	$-0.580529\pi$
$307$	−8.77122	−0.500600	−0.250300	+	0.968168i	$0.580529\pi$
$308$	0	0
$309$	7.79313	0.443336
$310$	0	0
$311$	5.49554	0.311623	0.155812	−	0.987787i	$-0.450201\pi$
$311$	5.49554	0.311623	0.155812	+	0.987787i	$0.450201\pi$
$312$	0	0
$313$	−25.2854	−1.42922	−0.714608	−	0.699526i	$-0.753395\pi$
$313$	−25.2854	−1.42922	−0.714608	+	0.699526i	$0.753395\pi$
$314$	0	0
$315$	5.76368	0.324747
$316$	0	0
$317$	6.49512	0.364802	0.182401	−	0.983224i	$-0.441613\pi$
$317$	6.49512	0.364802	0.182401	+	0.983224i	$0.441613\pi$
$318$	0	0
$319$	35.0509	1.96247
$320$	0	0
$321$	−16.6890	−0.931488
$322$	0	0
$323$	6.34077	0.352810
$324$	0	0
$325$	8.26663	0.458550
$326$	0	0
$327$	−0.660441	−0.0365225
$328$	0	0
$329$	−5.96586	−0.328909
$330$	0	0
$331$	12.1880	0.669914	0.334957	−	0.942233i	$-0.391278\pi$
$331$	12.1880	0.669914	0.334957	+	0.942233i	$0.391278\pi$
$332$	0	0
$333$	4.92446	0.269859
$334$	0	0
$335$	18.9370	1.03464
$336$	0	0
$337$	9.73440	0.530267	0.265133	−	0.964212i	$-0.414584\pi$
$337$	9.73440	0.530267	0.265133	+	0.964212i	$0.414584\pi$
$338$	0	0
$339$	−8.85733	−0.481064
$340$	0	0
$341$	−32.4987	−1.75990
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−11.1559	−0.600616
$346$	0	0
$347$	−3.90135	−0.209435	−0.104718	−	0.994502i	$-0.533394\pi$
$347$	−3.90135	−0.209435	−0.104718	+	0.994502i	$0.533394\pi$
$348$	0	0
$349$	3.55550	0.190322	0.0951608	−	0.995462i	$-0.469663\pi$
$349$	3.55550	0.190322	0.0951608	+	0.995462i	$0.469663\pi$
$350$	0	0
$351$	−28.2586	−1.50833
$352$	0	0
$353$	2.90189	0.154452	0.0772261	−	0.997014i	$-0.475394\pi$
$353$	2.90189	0.154452	0.0772261	+	0.997014i	$0.475394\pi$
$354$	0	0
$355$	2.77587	0.147328
$356$	0	0
$357$	−5.31881	−0.281501
$358$	0	0
$359$	1.40056	0.0739185	0.0369592	−	0.999317i	$-0.488233\pi$
$359$	1.40056	0.0739185	0.0369592	+	0.999317i	$0.488233\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.815926	0.0428250
$364$	0	0
$365$	0.146597	0.00767321
$366$	0	0
$367$	10.1574	0.530213	0.265106	−	0.964219i	$-0.414593\pi$
$367$	10.1574	0.530213	0.265106	+	0.964219i	$0.414593\pi$
$368$	0	0
$369$	17.0700	0.888626
$370$	0	0
$371$	−6.50168	−0.337550
$372$	0	0
$373$	1.10281	0.0571012	0.0285506	−	0.999592i	$-0.490911\pi$
$373$	1.10281	0.0571012	0.0285506	+	0.999592i	$0.490911\pi$
$374$	0	0
$375$	−7.79064	−0.402307
$376$	0	0
$377$	64.4321	3.31842
$378$	0	0
$379$	−18.6517	−0.958071	−0.479036	−	0.877795i	$-0.659014\pi$
$379$	−18.6517	−0.958071	−0.479036	+	0.877795i	$0.659014\pi$
$380$	0	0
$381$	−1.06950	−0.0547923
$382$	0	0
$383$	−3.13654	−0.160270	−0.0801348	−	0.996784i	$-0.525535\pi$
$383$	−3.13654	−0.160270	−0.0801348	+	0.996784i	$0.525535\pi$
$384$	0	0
$385$	−8.68469	−0.442613
$386$	0	0
$387$	−2.65074	−0.134745
$388$	0	0
$389$	33.4216	1.69454	0.847270	−	0.531162i	$-0.178244\pi$
$389$	33.4216	1.69454	0.847270	+	0.531162i	$0.178244\pi$
$390$	0	0
$391$	−33.5983	−1.69914
$392$	0	0
$393$	6.77236	0.341620
$394$	0	0
$395$	41.5768	2.09196
$396$	0	0
$397$	10.9045	0.547283	0.273642	−	0.961832i	$-0.411772\pi$
$397$	10.9045	0.547283	0.273642	+	0.961832i	$0.411772\pi$
$398$	0	0
$399$	−0.838827	−0.0419939
$400$	0	0
$401$	5.52313	0.275812	0.137906	−	0.990445i	$-0.455963\pi$
$401$	5.52313	0.275812	0.137906	+	0.990445i	$0.455963\pi$
$402$	0	0
$403$	−59.7405	−2.97589
$404$	0	0
$405$	7.93738	0.394412
$406$	0	0
$407$	−7.42016	−0.367804
$408$	0	0
$409$	14.7459	0.729137	0.364568	−	0.931177i	$-0.381217\pi$
$409$	14.7459	0.729137	0.364568	+	0.931177i	$0.381217\pi$
$410$	0	0
$411$	9.13264	0.450480
$412$	0	0
$413$	−9.26281	−0.455793
$414$	0	0
$415$	−14.0459	−0.689486
$416$	0	0
$417$	−10.1937	−0.499188
$418$	0	0
$419$	16.1788	0.790386	0.395193	−	0.918598i	$-0.370678\pi$
$419$	16.1788	0.790386	0.395193	+	0.918598i	$0.370678\pi$
$420$	0	0
$421$	−7.24074	−0.352892	−0.176446	−	0.984310i	$-0.556460\pi$
$421$	−7.24074	−0.352892	−0.176446	+	0.984310i	$0.556460\pi$
$422$	0	0
$423$	−13.6998	−0.666108
$424$	0	0
$425$	8.24084	0.399739
$426$	0	0
$427$	−7.12100	−0.344609
$428$	0	0
$429$	18.4615	0.891332
$430$	0	0
$431$	−11.6634	−0.561807	−0.280903	−	0.959736i	$-0.590634\pi$
$431$	−11.6634	−0.561807	−0.280903	+	0.959736i	$0.590634\pi$
$432$	0	0
$433$	5.52522	0.265525	0.132762	−	0.991148i	$-0.457615\pi$
$433$	5.52522	0.265525	0.132762	+	0.991148i	$0.457615\pi$
$434$	0	0
$435$	21.3272	1.02256
$436$	0	0
$437$	−5.29877	−0.253475
$438$	0	0
$439$	34.9712	1.66908	0.834542	−	0.550945i	$-0.185733\pi$
$439$	34.9712	1.66908	0.834542	+	0.550945i	$0.185733\pi$
$440$	0	0
$441$	−2.29637	−0.109351
$442$	0	0
$443$	31.2216	1.48338	0.741691	−	0.670742i	$-0.234024\pi$
$443$	31.2216	1.48338	0.741691	+	0.670742i	$0.234024\pi$
$444$	0	0
$445$	−37.2203	−1.76441
$446$	0	0
$447$	−11.4068	−0.539521
$448$	0	0
$449$	−10.5208	−0.496508	−0.248254	−	0.968695i	$-0.579857\pi$
$449$	−10.5208	−0.496508	−0.248254	+	0.968695i	$0.579857\pi$
$450$	0	0
$451$	−25.7209	−1.21115
$452$	0	0
$453$	−13.7589	−0.646448
$454$	0	0
$455$	−15.9646	−0.748432
$456$	0	0
$457$	16.0833	0.752346	0.376173	−	0.926550i	$-0.377240\pi$
$457$	16.0833	0.752346	0.376173	+	0.926550i	$0.377240\pi$
$458$	0	0
$459$	−28.1704	−1.31488
$460$	0	0
$461$	28.7877	1.34078	0.670388	−	0.742011i	$-0.266128\pi$
$461$	28.7877	1.34078	0.670388	+	0.742011i	$0.266128\pi$
$462$	0	0
$463$	17.7307	0.824017	0.412009	−	0.911180i	$-0.364827\pi$
$463$	17.7307	0.824017	0.412009	+	0.911180i	$0.364827\pi$
$464$	0	0
$465$	−19.7743	−0.917011
$466$	0	0
$467$	28.9193	1.33823	0.669114	−	0.743160i	$-0.266674\pi$
$467$	28.9193	1.33823	0.669114	+	0.743160i	$0.266674\pi$
$468$	0	0
$469$	−7.54487	−0.348390
$470$	0	0
$471$	14.4741	0.666932
$472$	0	0
$473$	3.99412	0.183650
$474$	0	0
$475$	1.29966	0.0596325
$476$	0	0
$477$	−14.9303	−0.683609
$478$	0	0
$479$	−39.4956	−1.80460	−0.902300	−	0.431109i	$-0.858122\pi$
$479$	−39.4956	−1.80460	−0.902300	+	0.431109i	$0.858122\pi$
$480$	0	0
$481$	−13.6401	−0.621934
$482$	0	0
$483$	4.44476	0.202243
$484$	0	0
$485$	−4.74491	−0.215455
$486$	0	0
$487$	2.43774	0.110464	0.0552322	−	0.998474i	$-0.482410\pi$
$487$	2.43774	0.110464	0.0552322	+	0.998474i	$0.482410\pi$
$488$	0	0
$489$	−12.5815	−0.568954
$490$	0	0
$491$	28.5143	1.28683	0.643416	−	0.765516i	$-0.277516\pi$
$491$	28.5143	1.28683	0.643416	+	0.765516i	$0.277516\pi$
$492$	0	0
$493$	64.2310	2.89282
$494$	0	0
$495$	−19.9433	−0.896383
$496$	0	0
$497$	−1.10596	−0.0496093
$498$	0	0
$499$	−24.7694	−1.10883	−0.554416	−	0.832240i	$-0.687058\pi$
$499$	−24.7694	−1.10883	−0.554416	+	0.832240i	$0.687058\pi$
$500$	0	0
$501$	3.60805	0.161196
$502$	0	0
$503$	−25.3299	−1.12940	−0.564702	−	0.825295i	$-0.691009\pi$
$503$	−25.3299	−1.12940	−0.564702	+	0.825295i	$0.691009\pi$
$504$	0	0
$505$	9.03364	0.401992
$506$	0	0
$507$	23.0321	1.02289
$508$	0	0
$509$	−27.6943	−1.22753	−0.613764	−	0.789490i	$-0.710345\pi$
$509$	−27.6943	−1.22753	−0.613764	+	0.789490i	$0.710345\pi$
$510$	0	0
$511$	−0.0584070	−0.00258377
$512$	0	0
$513$	−4.44274	−0.196152
$514$	0	0
$515$	23.3183	1.02753
$516$	0	0
$517$	20.6428	0.907871
$518$	0	0
$519$	20.4864	0.899253
$520$	0	0
$521$	13.6577	0.598353	0.299177	−	0.954198i	$-0.403288\pi$
$521$	13.6577	0.598353	0.299177	+	0.954198i	$0.403288\pi$
$522$	0	0
$523$	26.2806	1.14917	0.574586	−	0.818444i	$-0.305163\pi$
$523$	26.2806	1.14917	0.574586	+	0.818444i	$0.305163\pi$
$524$	0	0
$525$	−1.09019	−0.0475798
$526$	0	0
$527$	−59.5541	−2.59422
$528$	0	0
$529$	5.07700	0.220739
$530$	0	0
$531$	−21.2708	−0.923075
$532$	0	0
$533$	−47.2814	−2.04798
$534$	0	0
$535$	−49.9362	−2.15893
$536$	0	0
$537$	−7.14189	−0.308195
$538$	0	0
$539$	3.46016	0.149040
$540$	0	0
$541$	−3.56729	−0.153370	−0.0766849	−	0.997055i	$-0.524434\pi$
$541$	−3.56729	−0.153370	−0.0766849	+	0.997055i	$0.524434\pi$
$542$	0	0
$543$	13.8810	0.595690
$544$	0	0
$545$	−1.97615	−0.0846490
$546$	0	0
$547$	−7.75085	−0.331402	−0.165701	−	0.986176i	$-0.552989\pi$
$547$	−7.75085	−0.331402	−0.165701	+	0.986176i	$0.552989\pi$
$548$	0	0
$549$	−16.3524	−0.697905
$550$	0	0
$551$	10.1298	0.431546
$552$	0	0
$553$	−16.5650	−0.704417
$554$	0	0
$555$	−4.51490	−0.191647
$556$	0	0
$557$	−0.922909	−0.0391049	−0.0195524	−	0.999809i	$-0.506224\pi$
$557$	−0.922909	−0.0391049	−0.0195524	+	0.999809i	$0.506224\pi$
$558$	0	0
$559$	7.34217	0.310541
$560$	0	0
$561$	18.4039	0.777014
$562$	0	0
$563$	−30.6656	−1.29240	−0.646201	−	0.763168i	$-0.723643\pi$
$563$	−30.6656	−1.29240	−0.646201	+	0.763168i	$0.723643\pi$
$564$	0	0
$565$	−26.5026	−1.11497
$566$	0	0
$567$	−3.16241	−0.132809
$568$	0	0
$569$	−11.5203	−0.482955	−0.241477	−	0.970406i	$-0.577632\pi$
$569$	−11.5203	−0.482955	−0.241477	+	0.970406i	$0.577632\pi$
$570$	0	0
$571$	15.4530	0.646689	0.323344	−	0.946281i	$-0.395193\pi$
$571$	15.4530	0.646689	0.323344	+	0.946281i	$0.395193\pi$
$572$	0	0
$573$	−13.1105	−0.547701
$574$	0	0
$575$	−6.88660	−0.287191
$576$	0	0
$577$	−39.0625	−1.62619	−0.813097	−	0.582128i	$-0.802220\pi$
$577$	−39.0625	−1.62619	−0.813097	+	0.582128i	$0.802220\pi$
$578$	0	0
$579$	−11.3165	−0.470296
$580$	0	0
$581$	5.59617	0.232168
$582$	0	0
$583$	22.4968	0.931724
$584$	0	0
$585$	−36.6606	−1.51573
$586$	0	0
$587$	23.2106	0.958003	0.479001	−	0.877814i	$-0.340999\pi$
$587$	23.2106	0.958003	0.479001	+	0.877814i	$0.340999\pi$
$588$	0	0
$589$	−9.39225	−0.387001
$590$	0	0
$591$	−11.1708	−0.459507
$592$	0	0
$593$	−25.4167	−1.04374	−0.521870	−	0.853025i	$-0.674765\pi$
$593$	−25.4167	−1.04374	−0.521870	+	0.853025i	$0.674765\pi$
$594$	0	0
$595$	−15.9148	−0.652442
$596$	0	0
$597$	7.18623	0.294113
$598$	0	0
$599$	−47.0201	−1.92119	−0.960594	−	0.277955i	$-0.910343\pi$
$599$	−47.0201	−1.92119	−0.960594	+	0.277955i	$0.910343\pi$
$600$	0	0
$601$	−5.46252	−0.222821	−0.111410	−	0.993774i	$-0.535537\pi$
$601$	−5.46252	−0.222821	−0.111410	+	0.993774i	$0.535537\pi$
$602$	0	0
$603$	−17.3258	−0.705562
$604$	0	0
$605$	2.44139	0.0992565
$606$	0	0
$607$	12.7810	0.518766	0.259383	−	0.965775i	$-0.416481\pi$
$607$	12.7810	0.518766	0.259383	+	0.965775i	$0.416481\pi$
$608$	0	0
$609$	−8.49719	−0.344324
$610$	0	0
$611$	37.9466	1.53515
$612$	0	0
$613$	10.8772	0.439328	0.219664	−	0.975576i	$-0.429504\pi$
$613$	10.8772	0.439328	0.219664	+	0.975576i	$0.429504\pi$
$614$	0	0
$615$	−15.6503	−0.631080
$616$	0	0
$617$	−38.7821	−1.56131	−0.780655	−	0.624962i	$-0.785114\pi$
$617$	−38.7821	−1.56131	−0.780655	+	0.624962i	$0.785114\pi$
$618$	0	0
$619$	−24.9373	−1.00232	−0.501158	−	0.865356i	$-0.667092\pi$
$619$	−24.9373	−1.00232	−0.501158	+	0.865356i	$0.667092\pi$
$620$	0	0
$621$	23.5411	0.944671
$622$	0	0
$623$	14.8293	0.594125
$624$	0	0
$625$	−29.8092	−1.19237
$626$	0	0
$627$	2.90248	0.115914
$628$	0	0
$629$	−13.5975	−0.542168
$630$	0	0
$631$	45.7661	1.82192	0.910959	−	0.412496i	$-0.135343\pi$
$631$	45.7661	1.82192	0.910959	+	0.412496i	$0.135343\pi$
$632$	0	0
$633$	−7.60734	−0.302364
$634$	0	0
$635$	−3.20013	−0.126993
$636$	0	0
$637$	6.36062	0.252017
$638$	0	0
$639$	−2.53970	−0.100469
$640$	0	0
$641$	34.7497	1.37253	0.686265	−	0.727351i	$-0.259249\pi$
$641$	34.7497	1.37253	0.686265	+	0.727351i	$0.259249\pi$
$642$	0	0
$643$	−12.0973	−0.477070	−0.238535	−	0.971134i	$-0.576667\pi$
$643$	−12.0973	−0.477070	−0.238535	+	0.971134i	$0.576667\pi$
$644$	0	0
$645$	2.43028	0.0956922
$646$	0	0
$647$	−40.1375	−1.57797	−0.788985	−	0.614413i	$-0.789393\pi$
$647$	−40.1375	−1.57797	−0.788985	+	0.614413i	$0.789393\pi$
$648$	0	0
$649$	32.0508	1.25810
$650$	0	0
$651$	7.87848	0.308782
$652$	0	0
$653$	26.1580	1.02364	0.511820	−	0.859093i	$-0.328971\pi$
$653$	26.1580	1.02364	0.511820	+	0.859093i	$0.328971\pi$
$654$	0	0
$655$	20.2640	0.791782
$656$	0	0
$657$	−0.134124	−0.00523268
$658$	0	0
$659$	32.5535	1.26810	0.634052	−	0.773290i	$-0.281390\pi$
$659$	32.5535	1.26810	0.634052	+	0.773290i	$0.281390\pi$
$660$	0	0
$661$	12.6399	0.491634	0.245817	−	0.969316i	$-0.420944\pi$
$661$	12.6399	0.491634	0.245817	+	0.969316i	$0.420944\pi$
$662$	0	0
$663$	33.8309	1.31388
$664$	0	0
$665$	−2.50991	−0.0973302
$666$	0	0
$667$	−53.6757	−2.07833
$668$	0	0
$669$	−2.05459	−0.0794350
$670$	0	0
$671$	24.6398	0.951209
$672$	0	0
$673$	12.2360	0.471665	0.235832	−	0.971794i	$-0.424218\pi$
$673$	12.2360	0.471665	0.235832	+	0.971794i	$0.424218\pi$
$674$	0	0
$675$	−5.77405	−0.222243
$676$	0	0
$677$	33.7535	1.29725	0.648626	−	0.761107i	$-0.275344\pi$
$677$	33.7535	1.29725	0.648626	+	0.761107i	$0.275344\pi$
$678$	0	0
$679$	1.89047	0.0725496
$680$	0	0
$681$	9.65707	0.370060
$682$	0	0
$683$	−13.4124	−0.513212	−0.256606	−	0.966516i	$-0.582604\pi$
$683$	−13.4124	−0.513212	−0.256606	+	0.966516i	$0.582604\pi$
$684$	0	0
$685$	27.3264	1.04409
$686$	0	0
$687$	7.34864	0.280368
$688$	0	0
$689$	41.3547	1.57549
$690$	0	0
$691$	−12.6257	−0.480304	−0.240152	−	0.970735i	$-0.577197\pi$
$691$	−12.6257	−0.480304	−0.240152	+	0.970735i	$0.577197\pi$
$692$	0	0
$693$	7.94580	0.301836
$694$	0	0
$695$	−30.5013	−1.15698
$696$	0	0
$697$	−47.1338	−1.78532
$698$	0	0
$699$	8.51902	0.322219
$700$	0	0
$701$	−0.500105	−0.0188887	−0.00944435	−	0.999955i	$-0.503006\pi$
$701$	−0.500105	−0.0188887	−0.00944435	+	0.999955i	$0.503006\pi$
$702$	0	0
$703$	−2.14446	−0.0808797
$704$	0	0
$705$	12.5604	0.473053
$706$	0	0
$707$	−3.59919	−0.135361
$708$	0	0
$709$	3.59561	0.135036	0.0675180	−	0.997718i	$-0.478492\pi$
$709$	3.59561	0.135036	0.0675180	+	0.997718i	$0.478492\pi$
$710$	0	0
$711$	−38.0394	−1.42659
$712$	0	0
$713$	49.7674	1.86380
$714$	0	0
$715$	55.2400	2.06586
$716$	0	0
$717$	−13.7919	−0.515069
$718$	0	0
$719$	16.2473	0.605923	0.302961	−	0.953003i	$-0.402025\pi$
$719$	16.2473	0.605923	0.302961	+	0.953003i	$0.402025\pi$
$720$	0	0
$721$	−9.29050	−0.345996
$722$	0	0
$723$	−4.28625	−0.159407
$724$	0	0
$725$	13.1653	0.488949
$726$	0	0
$727$	−13.4012	−0.497022	−0.248511	−	0.968629i	$-0.579941\pi$
$727$	−13.4012	−0.497022	−0.248511	+	0.968629i	$0.579941\pi$
$728$	0	0
$729$	3.91801	0.145111
$730$	0	0
$731$	7.31926	0.270713
$732$	0	0
$733$	−9.33243	−0.344701	−0.172351	−	0.985036i	$-0.555136\pi$
$733$	−9.33243	−0.344701	−0.172351	+	0.985036i	$0.555136\pi$
$734$	0	0
$735$	2.10538	0.0776582
$736$	0	0
$737$	26.1065	0.961644
$738$	0	0
$739$	−10.7733	−0.396303	−0.198151	−	0.980171i	$-0.563494\pi$
$739$	−10.7733	−0.396303	−0.198151	+	0.980171i	$0.563494\pi$
$740$	0	0
$741$	5.33546	0.196003
$742$	0	0
$743$	−51.2919	−1.88172	−0.940859	−	0.338797i	$-0.889980\pi$
$743$	−51.2919	−1.88172	−0.940859	+	0.338797i	$0.889980\pi$
$744$	0	0
$745$	−34.1309	−1.25046
$746$	0	0
$747$	12.8509	0.470189
$748$	0	0
$749$	19.8956	0.726970
$750$	0	0
$751$	1.58651	0.0578926	0.0289463	−	0.999581i	$-0.490785\pi$
$751$	1.58651	0.0578926	0.0289463	+	0.999581i	$0.490785\pi$
$752$	0	0
$753$	−5.72767	−0.208728
$754$	0	0
$755$	−41.1689	−1.49829
$756$	0	0
$757$	−3.36632	−0.122351	−0.0611755	−	0.998127i	$-0.519485\pi$
$757$	−3.36632	−0.122351	−0.0611755	+	0.998127i	$0.519485\pi$
$758$	0	0
$759$	−15.3796	−0.558243
$760$	0	0
$761$	−10.5896	−0.383874	−0.191937	−	0.981407i	$-0.561477\pi$
$761$	−10.5896	−0.383874	−0.191937	+	0.981407i	$0.561477\pi$
$762$	0	0
$763$	0.787338	0.0285036
$764$	0	0
$765$	−36.5462	−1.32133
$766$	0	0
$767$	58.9172	2.12738
$768$	0	0
$769$	25.9476	0.935695	0.467847	−	0.883809i	$-0.345030\pi$
$769$	25.9476	0.935695	0.467847	+	0.883809i	$0.345030\pi$
$770$	0	0
$771$	−11.1103	−0.400127
$772$	0	0
$773$	−5.86293	−0.210875	−0.105438	−	0.994426i	$-0.533624\pi$
$773$	−5.86293	−0.210875	−0.105438	+	0.994426i	$0.533624\pi$
$774$	0	0
$775$	−12.2067	−0.438479
$776$	0	0
$777$	1.79883	0.0645326
$778$	0	0
$779$	−7.43345	−0.266331
$780$	0	0
$781$	3.82681	0.136934
$782$	0	0
$783$	−45.0042	−1.60832
$784$	0	0
$785$	43.3090	1.54576
$786$	0	0
$787$	26.0191	0.927480	0.463740	−	0.885971i	$-0.346507\pi$
$787$	26.0191	0.927480	0.463740	+	0.885971i	$0.346507\pi$
$788$	0	0
$789$	1.00549	0.0357963
$790$	0	0
$791$	10.5592	0.375441
$792$	0	0
$793$	45.2940	1.60844
$794$	0	0
$795$	13.6885	0.485482
$796$	0	0
$797$	−4.96085	−0.175722	−0.0878611	−	0.996133i	$-0.528003\pi$
$797$	−4.96085	−0.175722	−0.0878611	+	0.996133i	$0.528003\pi$
$798$	0	0
$799$	37.8282	1.33826
$800$	0	0
$801$	34.0536	1.20323
$802$	0	0
$803$	0.202098	0.00713187
$804$	0	0
$805$	13.2995	0.468744
$806$	0	0
$807$	24.4032	0.859032
$808$	0	0
$809$	−36.8839	−1.29677	−0.648385	−	0.761313i	$-0.724555\pi$
$809$	−36.8839	−1.29677	−0.648385	+	0.761313i	$0.724555\pi$
$810$	0	0
$811$	−46.2723	−1.62484	−0.812421	−	0.583072i	$-0.801851\pi$
$811$	−46.2723	−1.62484	−0.812421	+	0.583072i	$0.801851\pi$
$812$	0	0
$813$	10.6545	0.373670
$814$	0	0
$815$	−37.6459	−1.31868
$816$	0	0
$817$	1.15432	0.0403844
$818$	0	0
$819$	14.6063	0.510386
$820$	0	0
$821$	−35.3927	−1.23521	−0.617607	−	0.786487i	$-0.711898\pi$
$821$	−35.3927	−1.23521	−0.617607	+	0.786487i	$0.711898\pi$
$822$	0	0
$823$	−11.4699	−0.399815	−0.199907	−	0.979815i	$-0.564064\pi$
$823$	−11.4699	−0.399815	−0.199907	+	0.979815i	$0.564064\pi$
$824$	0	0
$825$	3.77223	0.131332
$826$	0	0
$827$	−22.4282	−0.779905	−0.389953	−	0.920835i	$-0.627509\pi$
$827$	−22.4282	−0.779905	−0.389953	+	0.920835i	$0.627509\pi$
$828$	0	0
$829$	53.8294	1.86957	0.934786	−	0.355211i	$-0.115591\pi$
$829$	53.8294	1.86957	0.934786	+	0.355211i	$0.115591\pi$
$830$	0	0
$831$	−6.69504	−0.232248
$832$	0	0
$833$	6.34077	0.219695
$834$	0	0
$835$	10.7959	0.373608
$836$	0	0
$837$	41.7273	1.44231
$838$	0	0
$839$	1.60165	0.0552952	0.0276476	−	0.999618i	$-0.491198\pi$
$839$	1.60165	0.0552952	0.0276476	+	0.999618i	$0.491198\pi$
$840$	0	0
$841$	73.6137	2.53840
$842$	0	0
$843$	17.1195	0.589627
$844$	0	0
$845$	68.9158	2.37078
$846$	0	0
$847$	−0.972699	−0.0334223
$848$	0	0
$849$	15.9314	0.546765
$850$	0	0
$851$	11.3630	0.389518
$852$	0	0
$853$	−37.7120	−1.29123	−0.645617	−	0.763662i	$-0.723400\pi$
$853$	−37.7120	−1.29123	−0.645617	+	0.763662i	$0.723400\pi$
$854$	0	0
$855$	−5.76368	−0.197114
$856$	0	0
$857$	28.3095	0.967035	0.483517	−	0.875335i	$-0.339359\pi$
$857$	28.3095	0.967035	0.483517	+	0.875335i	$0.339359\pi$
$858$	0	0
$859$	3.40350	0.116126	0.0580629	−	0.998313i	$-0.481508\pi$
$859$	3.40350	0.116126	0.0580629	+	0.998313i	$0.481508\pi$
$860$	0	0
$861$	6.23539	0.212501
$862$	0	0
$863$	14.3494	0.488458	0.244229	−	0.969718i	$-0.421465\pi$
$863$	14.3494	0.488458	0.244229	+	0.969718i	$0.421465\pi$
$864$	0	0
$865$	61.2987	2.08422
$866$	0	0
$867$	19.4653	0.661076
$868$	0	0
$869$	57.3177	1.94437
$870$	0	0
$871$	47.9901	1.62608
$872$	0	0
$873$	4.34122	0.146928
$874$	0	0
$875$	9.28753	0.313976
$876$	0	0
$877$	−41.1281	−1.38880	−0.694398	−	0.719591i	$-0.744329\pi$
$877$	−41.1281	−1.38880	−0.694398	+	0.719591i	$0.744329\pi$
$878$	0	0
$879$	−16.1667	−0.545291
$880$	0	0
$881$	−14.2298	−0.479416	−0.239708	−	0.970845i	$-0.577052\pi$
$881$	−14.2298	−0.479416	−0.239708	+	0.970845i	$0.577052\pi$
$882$	0	0
$883$	35.2550	1.18643	0.593213	−	0.805046i	$-0.297859\pi$
$883$	35.2550	1.18643	0.593213	+	0.805046i	$0.297859\pi$
$884$	0	0
$885$	19.5018	0.655545
$886$	0	0
$887$	−30.0916	−1.01038	−0.505189	−	0.863009i	$-0.668577\pi$
$887$	−30.0916	−1.01038	−0.505189	+	0.863009i	$0.668577\pi$
$888$	0	0
$889$	1.27500	0.0427620
$890$	0	0
$891$	10.9425	0.366586
$892$	0	0
$893$	5.96586	0.199640
$894$	0	0
$895$	−21.3697	−0.714312
$896$	0	0
$897$	−28.2714	−0.943954
$898$	0	0
$899$	−95.1421	−3.17317
$900$	0	0
$901$	41.2257	1.37343
$902$	0	0
$903$	−0.968273	−0.0322221
$904$	0	0
$905$	41.5342	1.38064
$906$	0	0
$907$	49.2512	1.63536	0.817680	−	0.575674i	$-0.195260\pi$
$907$	49.2512	1.63536	0.817680	+	0.575674i	$0.195260\pi$
$908$	0	0
$909$	−8.26506	−0.274135
$910$	0	0
$911$	−40.2177	−1.33247	−0.666236	−	0.745741i	$-0.732096\pi$
$911$	−40.2177	−1.33247	−0.666236	+	0.745741i	$0.732096\pi$
$912$	0	0
$913$	−19.3637	−0.640843
$914$	0	0
$915$	14.9924	0.495635
$916$	0	0
$917$	−8.07361	−0.266614
$918$	0	0
$919$	−35.3387	−1.16572	−0.582859	−	0.812573i	$-0.698066\pi$
$919$	−35.3387	−1.16572	−0.582859	+	0.812573i	$0.698066\pi$
$920$	0	0
$921$	−7.35754	−0.242439
$922$	0	0
$923$	7.03462	0.231547
$924$	0	0
$925$	−2.78706	−0.0916380
$926$	0	0
$927$	−21.3344	−0.700714
$928$	0	0
$929$	−37.5422	−1.23172	−0.615860	−	0.787856i	$-0.711191\pi$
$929$	−37.5422	−1.23172	−0.615860	+	0.787856i	$0.711191\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	4.60981	0.150918
$934$	0	0
$935$	55.0676	1.80090
$936$	0	0
$937$	−33.3138	−1.08831	−0.544157	−	0.838983i	$-0.683150\pi$
$937$	−33.3138	−1.08831	−0.544157	+	0.838983i	$0.683150\pi$
$938$	0	0
$939$	−21.2101	−0.692165
$940$	0	0
$941$	−51.7592	−1.68730	−0.843650	−	0.536893i	$-0.819598\pi$
$941$	−51.7592	−1.68730	−0.843650	+	0.536893i	$0.819598\pi$
$942$	0	0
$943$	39.3882	1.28266
$944$	0	0
$945$	11.1509	0.362738
$946$	0	0
$947$	−33.8775	−1.10087	−0.550435	−	0.834878i	$-0.685538\pi$
$947$	−33.8775	−1.10087	−0.550435	+	0.834878i	$0.685538\pi$
$948$	0	0
$949$	0.371505	0.0120596
$950$	0	0
$951$	5.44828	0.176673
$952$	0	0
$953$	−55.7038	−1.80442	−0.902211	−	0.431295i	$-0.858057\pi$
$953$	−55.7038	−1.80442	−0.902211	+	0.431295i	$0.858057\pi$
$954$	0	0
$955$	−39.2289	−1.26942
$956$	0	0
$957$	29.4016	0.950420
$958$	0	0
$959$	−10.8874	−0.351572
$960$	0	0
$961$	57.2144	1.84563
$962$	0	0
$963$	45.6876	1.47226
$964$	0	0
$965$	−33.8607	−1.09002
$966$	0	0
$967$	−2.96518	−0.0953536	−0.0476768	−	0.998863i	$-0.515182\pi$
$967$	−2.96518	−0.0953536	−0.0476768	+	0.998863i	$0.515182\pi$
$968$	0	0
$969$	5.31881	0.170865
$970$	0	0
$971$	−1.53029	−0.0491093	−0.0245547	−	0.999698i	$-0.507817\pi$
$971$	−1.53029	−0.0491093	−0.0245547	+	0.999698i	$0.507817\pi$
$972$	0	0
$973$	12.1523	0.389586
$974$	0	0
$975$	6.93428	0.222075
$976$	0	0
$977$	−23.1611	−0.740988	−0.370494	−	0.928835i	$-0.620812\pi$
$977$	−23.1611	−0.740988	−0.370494	+	0.928835i	$0.620812\pi$
$978$	0	0
$979$	−51.3119	−1.63993
$980$	0	0
$981$	1.80802	0.0577256
$982$	0	0
$983$	−53.1318	−1.69464	−0.847320	−	0.531082i	$-0.821786\pi$
$983$	−53.1318	−1.69464	−0.847320	+	0.531082i	$0.821786\pi$
$984$	0	0
$985$	−33.4250	−1.06501
$986$	0	0
$987$	−5.00433	−0.159290
$988$	0	0
$989$	−6.11646	−0.194492
$990$	0	0
$991$	35.2726	1.12047	0.560236	−	0.828333i	$-0.310711\pi$
$991$	35.2726	1.12047	0.560236	+	0.828333i	$0.310711\pi$
$992$	0	0
$993$	10.2236	0.324438
$994$	0	0
$995$	21.5024	0.681672
$996$	0	0
$997$	−39.2627	−1.24346	−0.621731	−	0.783230i	$-0.713570\pi$
$997$	−39.2627	−1.24346	−0.621731	+	0.783230i	$0.713570\pi$
$998$	0	0
$999$	9.52726	0.301429

Twists

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

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

\(f(q)\)	\(=\)	\(q+0.838827 q^{3} +2.50991 q^{5} -1.00000 q^{7} -2.29637 q^{9} +O(q^{10})\)	\(q+0.838827 q^{3} +2.50991 q^{5} -1.00000 q^{7} -2.29637 q^{9} +3.46016 q^{11} +6.36062 q^{13} +2.10538 q^{15} +6.34077 q^{17} +1.00000 q^{19} -0.838827 q^{21} -5.29877 q^{23} +1.29966 q^{25} -4.44274 q^{27} +10.1298 q^{29} -9.39225 q^{31} +2.90248 q^{33} -2.50991 q^{35} -2.14446 q^{37} +5.33546 q^{39} -7.43345 q^{41} +1.15432 q^{43} -5.76368 q^{45} +5.96586 q^{47} +1.00000 q^{49} +5.31881 q^{51} +6.50168 q^{53} +8.68469 q^{55} +0.838827 q^{57} +9.26281 q^{59} +7.12100 q^{61} +2.29637 q^{63} +15.9646 q^{65} +7.54487 q^{67} -4.44476 q^{69} +1.10596 q^{71} +0.0584070 q^{73} +1.09019 q^{75} -3.46016 q^{77} +16.5650 q^{79} +3.16241 q^{81} -5.59617 q^{83} +15.9148 q^{85} +8.49719 q^{87} -14.8293 q^{89} -6.36062 q^{91} -7.87848 q^{93} +2.50991 q^{95} -1.89047 q^{97} -7.94580 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} + 5 q^{5} - 7 q^{7} + 8 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more