LMFDB - Embedded newform 8092.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8092,2,Mod(1,8092)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8092, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8092.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8092 = 2^{2} \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8092.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:	$64.6149453156$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 7x^{6} + 12x^{5} + 15x^{4} - 16x^{3} - 13x^{2} + 2x + 1 $ x^8 - 2x^7 - 7x^6 + 12x^5 + 15x^4 - 16x^3 - 13x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 476)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.05350$ of defining polynomial
Character	$\chi$	$=$	8092.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.98074 q^{3} +1.77152 q^{5} +1.00000 q^{7} +5.88484 q^{9} +O(q^{10})$	$q-2.98074 q^{3} +1.77152 q^{5} +1.00000 q^{7} +5.88484 q^{9} +2.99979 q^{11} +2.59917 q^{13} -5.28045 q^{15} +4.57015 q^{19} -2.98074 q^{21} -4.37363 q^{23} -1.86172 q^{25} -8.59896 q^{27} +2.40860 q^{29} -2.64427 q^{31} -8.94162 q^{33} +1.77152 q^{35} -0.526932 q^{37} -7.74745 q^{39} -12.0588 q^{41} -2.63796 q^{43} +10.4251 q^{45} -9.67463 q^{47} +1.00000 q^{49} +3.88549 q^{53} +5.31420 q^{55} -13.6225 q^{57} -2.63190 q^{59} -9.36096 q^{61} +5.88484 q^{63} +4.60448 q^{65} -14.9050 q^{67} +13.0367 q^{69} -14.2979 q^{71} +3.15166 q^{73} +5.54930 q^{75} +2.99979 q^{77} -17.3112 q^{79} +7.97679 q^{81} +12.6749 q^{83} -7.17941 q^{87} -2.05547 q^{89} +2.59917 q^{91} +7.88190 q^{93} +8.09612 q^{95} +7.19765 q^{97} +17.6533 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - 4 q^{5} + 8 q^{7}+O(q^{10}) $ 8 * q - 4 * q^3 - 4 * q^5 + 8 * q^7	$ 8 q - 4 q^{3} - 4 q^{5} + 8 q^{7} - 8 q^{11} + 4 q^{15} + 16 q^{19} - 4 q^{21} - 20 q^{23} + 16 q^{25} - 16 q^{27} - 16 q^{29} + 8 q^{33} - 4 q^{35} - 4 q^{37} + 8 q^{39} - 8 q^{41} - 16 q^{45} - 20 q^{47} + 8 q^{49} - 12 q^{53} - 4 q^{55} - 4 q^{57} - 4 q^{59} - 4 q^{61} + 36 q^{65} - 20 q^{67} - 12 q^{69} - 12 q^{71} + 24 q^{73} - 12 q^{75} - 8 q^{77} - 32 q^{79} - 4 q^{81} + 20 q^{83} + 32 q^{87} + 8 q^{89} - 8 q^{93} - 32 q^{95} - 24 q^{97} + 20 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 - 4 * q^5 + 8 * q^7 - 8 * q^11 + 4 * q^15 + 16 * q^19 - 4 * q^21 - 20 * q^23 + 16 * q^25 - 16 * q^27 - 16 * q^29 + 8 * q^33 - 4 * q^35 - 4 * q^37 + 8 * q^39 - 8 * q^41 - 16 * q^45 - 20 * q^47 + 8 * q^49 - 12 * q^53 - 4 * q^55 - 4 * q^57 - 4 * q^59 - 4 * q^61 + 36 * q^65 - 20 * q^67 - 12 * q^69 - 12 * q^71 + 24 * q^73 - 12 * q^75 - 8 * q^77 - 32 * q^79 - 4 * q^81 + 20 * q^83 + 32 * q^87 + 8 * q^89 - 8 * q^93 - 32 * q^95 - 24 * q^97 + 20 * 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
$3$	−2.98074	−1.72093	−0.860467	−	0.509507i	$-0.829828\pi$
$3$	−2.98074	−1.72093	−0.860467	+	0.509507i	$0.829828\pi$
$4$	0	0
$5$	1.77152	0.792248	0.396124	−	0.918197i	$-0.370355\pi$
$5$	1.77152	0.792248	0.396124	+	0.918197i	$0.370355\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	5.88484	1.96161
$10$	0	0
$11$	2.99979	0.904472	0.452236	−	0.891898i	$-0.350626\pi$
$11$	2.99979	0.904472	0.452236	+	0.891898i	$0.350626\pi$
$12$	0	0
$13$	2.59917	0.720879	0.360440	−	0.932783i	$-0.382627\pi$
$13$	2.59917	0.720879	0.360440	+	0.932783i	$0.382627\pi$
$14$	0	0
$15$	−5.28045	−1.36341
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	4.57015	1.04846	0.524232	−	0.851575i	$-0.324352\pi$
$19$	4.57015	1.04846	0.524232	+	0.851575i	$0.324352\pi$
$20$	0	0
$21$	−2.98074	−0.650452
$22$	0	0
$23$	−4.37363	−0.911964	−0.455982	−	0.889989i	$-0.650712\pi$
$23$	−4.37363	−0.911964	−0.455982	+	0.889989i	$0.650712\pi$
$24$	0	0
$25$	−1.86172	−0.372343
$26$	0	0
$27$	−8.59896	−1.65487
$28$	0	0
$29$	2.40860	0.447265	0.223633	−	0.974674i	$-0.428208\pi$
$29$	2.40860	0.447265	0.223633	+	0.974674i	$0.428208\pi$
$30$	0	0
$31$	−2.64427	−0.474925	−0.237463	−	0.971397i	$-0.576316\pi$
$31$	−2.64427	−0.474925	−0.237463	+	0.971397i	$0.576316\pi$
$32$	0	0
$33$	−8.94162	−1.55654
$34$	0	0
$35$	1.77152	0.299442
$36$	0	0
$37$	−0.526932	−0.0866270	−0.0433135	−	0.999062i	$-0.513791\pi$
$37$	−0.526932	−0.0866270	−0.0433135	+	0.999062i	$0.513791\pi$
$38$	0	0
$39$	−7.74745	−1.24059
$40$	0	0
$41$	−12.0588	−1.88326	−0.941631	−	0.336646i	$-0.890707\pi$
$41$	−12.0588	−1.88326	−0.941631	+	0.336646i	$0.890707\pi$
$42$	0	0
$43$	−2.63796	−0.402285	−0.201143	−	0.979562i	$-0.564465\pi$
$43$	−2.63796	−0.402285	−0.201143	+	0.979562i	$0.564465\pi$
$44$	0	0
$45$	10.4251	1.55408
$46$	0	0
$47$	−9.67463	−1.41119	−0.705595	−	0.708616i	$-0.749320\pi$
$47$	−9.67463	−1.41119	−0.705595	+	0.708616i	$0.749320\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.88549	0.533713	0.266857	−	0.963736i	$-0.414015\pi$
$53$	3.88549	0.533713	0.266857	+	0.963736i	$0.414015\pi$
$54$	0	0
$55$	5.31420	0.716566
$56$	0	0
$57$	−13.6225	−1.80434
$58$	0	0
$59$	−2.63190	−0.342645	−0.171322	−	0.985215i	$-0.554804\pi$
$59$	−2.63190	−0.342645	−0.171322	+	0.985215i	$0.554804\pi$
$60$	0	0
$61$	−9.36096	−1.19855	−0.599274	−	0.800544i	$-0.704544\pi$
$61$	−9.36096	−1.19855	−0.599274	+	0.800544i	$0.704544\pi$
$62$	0	0
$63$	5.88484	0.741420
$64$	0	0
$65$	4.60448	0.571115
$66$	0	0
$67$	−14.9050	−1.82093	−0.910467	−	0.413581i	$-0.864278\pi$
$67$	−14.9050	−1.82093	−0.910467	+	0.413581i	$0.864278\pi$
$68$	0	0
$69$	13.0367	1.56943
$70$	0	0
$71$	−14.2979	−1.69684	−0.848422	−	0.529321i	$-0.822447\pi$
$71$	−14.2979	−1.69684	−0.848422	+	0.529321i	$0.822447\pi$
$72$	0	0
$73$	3.15166	0.368874	0.184437	−	0.982844i	$-0.440954\pi$
$73$	3.15166	0.368874	0.184437	+	0.982844i	$0.440954\pi$
$74$	0	0
$75$	5.54930	0.640778
$76$	0	0
$77$	2.99979	0.341858
$78$	0	0
$79$	−17.3112	−1.94766	−0.973831	−	0.227273i	$-0.927019\pi$
$79$	−17.3112	−1.94766	−0.973831	+	0.227273i	$0.927019\pi$
$80$	0	0
$81$	7.97679	0.886310
$82$	0	0
$83$	12.6749	1.39125	0.695624	−	0.718406i	$-0.255128\pi$
$83$	12.6749	1.39125	0.695624	+	0.718406i	$0.255128\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.17941	−0.769713
$88$	0	0
$89$	−2.05547	−0.217879	−0.108940	−	0.994048i	$-0.534746\pi$
$89$	−2.05547	−0.217879	−0.108940	+	0.994048i	$0.534746\pi$
$90$	0	0
$91$	2.59917	0.272467
$92$	0	0
$93$	7.88190	0.817314
$94$	0	0
$95$	8.09612	0.830644
$96$	0	0
$97$	7.19765	0.730811	0.365405	−	0.930849i	$-0.380930\pi$
$97$	7.19765	0.730811	0.365405	+	0.930849i	$0.380930\pi$
$98$	0	0
$99$	17.6533	1.77422
$100$	0	0
$101$	−1.15980	−0.115404	−0.0577021	−	0.998334i	$-0.518377\pi$
$101$	−1.15980	−0.115404	−0.0577021	+	0.998334i	$0.518377\pi$
$102$	0	0
$103$	−8.73357	−0.860545	−0.430272	−	0.902699i	$-0.641582\pi$
$103$	−8.73357	−0.860545	−0.430272	+	0.902699i	$0.641582\pi$
$104$	0	0
$105$	−5.28045	−0.515319
$106$	0	0
$107$	−13.2202	−1.27804	−0.639022	−	0.769188i	$-0.720661\pi$
$107$	−13.2202	−1.27804	−0.639022	+	0.769188i	$0.720661\pi$
$108$	0	0
$109$	−4.64129	−0.444555	−0.222277	−	0.974983i	$-0.571349\pi$
$109$	−4.64129	−0.444555	−0.222277	+	0.974983i	$0.571349\pi$
$110$	0	0
$111$	1.57065	0.149079
$112$	0	0
$113$	−0.401281	−0.0377494	−0.0188747	−	0.999822i	$-0.506008\pi$
$113$	−0.401281	−0.0377494	−0.0188747	+	0.999822i	$0.506008\pi$
$114$	0	0
$115$	−7.74797	−0.722502
$116$	0	0
$117$	15.2957	1.41409
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.00124	−0.181931
$122$	0	0
$123$	35.9441	3.24097
$124$	0	0
$125$	−12.1557	−1.08724
$126$	0	0
$127$	0.350103	0.0310666	0.0155333	−	0.999879i	$-0.495055\pi$
$127$	0.350103	0.0310666	0.0155333	+	0.999879i	$0.495055\pi$
$128$	0	0
$129$	7.86309	0.692306
$130$	0	0
$131$	−7.88711	−0.689100	−0.344550	−	0.938768i	$-0.611969\pi$
$131$	−7.88711	−0.689100	−0.344550	+	0.938768i	$0.611969\pi$
$132$	0	0
$133$	4.57015	0.396282
$134$	0	0
$135$	−15.2332	−1.31107
$136$	0	0
$137$	9.64144	0.823724	0.411862	−	0.911246i	$-0.364879\pi$
$137$	9.64144	0.823724	0.411862	+	0.911246i	$0.364879\pi$
$138$	0	0
$139$	12.9086	1.09489	0.547446	−	0.836841i	$-0.315600\pi$
$139$	12.9086	1.09489	0.547446	+	0.836841i	$0.315600\pi$
$140$	0	0
$141$	28.8376	2.42856
$142$	0	0
$143$	7.79696	0.652015
$144$	0	0
$145$	4.26688	0.354345
$146$	0	0
$147$	−2.98074	−0.245848
$148$	0	0
$149$	3.94430	0.323129	0.161565	−	0.986862i	$-0.448346\pi$
$149$	3.94430	0.323129	0.161565	+	0.986862i	$0.448346\pi$
$150$	0	0
$151$	23.8865	1.94385	0.971927	−	0.235281i	$-0.0756012\pi$
$151$	23.8865	1.94385	0.971927	+	0.235281i	$0.0756012\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.68438	−0.376258
$156$	0	0
$157$	18.4697	1.47404	0.737020	−	0.675871i	$-0.236232\pi$
$157$	18.4697	1.47404	0.737020	+	0.675871i	$0.236232\pi$
$158$	0	0
$159$	−11.5817	−0.918485
$160$	0	0
$161$	−4.37363	−0.344690
$162$	0	0
$163$	−18.3996	−1.44117	−0.720586	−	0.693366i	$-0.756127\pi$
$163$	−18.3996	−1.44117	−0.720586	+	0.693366i	$0.756127\pi$
$164$	0	0
$165$	−15.8403	−1.23316
$166$	0	0
$167$	−23.8278	−1.84385	−0.921925	−	0.387369i	$-0.873384\pi$
$167$	−23.8278	−1.84385	−0.921925	+	0.387369i	$0.873384\pi$
$168$	0	0
$169$	−6.24433	−0.480333
$170$	0	0
$171$	26.8946	2.05668
$172$	0	0
$173$	−6.26854	−0.476588	−0.238294	−	0.971193i	$-0.576588\pi$
$173$	−6.26854	−0.476588	−0.238294	+	0.971193i	$0.576588\pi$
$174$	0	0
$175$	−1.86172	−0.140733
$176$	0	0
$177$	7.84503	0.589669
$178$	0	0
$179$	−12.8732	−0.962190	−0.481095	−	0.876668i	$-0.659761\pi$
$179$	−12.8732	−0.962190	−0.481095	+	0.876668i	$0.659761\pi$
$180$	0	0
$181$	21.3869	1.58968	0.794838	−	0.606822i	$-0.207556\pi$
$181$	21.3869	1.58968	0.794838	+	0.606822i	$0.207556\pi$
$182$	0	0
$183$	27.9026	2.06262
$184$	0	0
$185$	−0.933470	−0.0686301
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−8.59896	−0.625482
$190$	0	0
$191$	17.5198	1.26769	0.633846	−	0.773460i	$-0.281475\pi$
$191$	17.5198	1.26769	0.633846	+	0.773460i	$0.281475\pi$
$192$	0	0
$193$	−5.15622	−0.371153	−0.185576	−	0.982630i	$-0.559415\pi$
$193$	−5.15622	−0.371153	−0.185576	+	0.982630i	$0.559415\pi$
$194$	0	0
$195$	−13.7248	−0.982851
$196$	0	0
$197$	−9.16634	−0.653075	−0.326537	−	0.945184i	$-0.605882\pi$
$197$	−9.16634	−0.653075	−0.326537	+	0.945184i	$0.605882\pi$
$198$	0	0
$199$	11.6411	0.825214	0.412607	−	0.910909i	$-0.364618\pi$
$199$	11.6411	0.825214	0.412607	+	0.910909i	$0.364618\pi$
$200$	0	0
$201$	44.4280	3.13371
$202$	0	0
$203$	2.40860	0.169050
$204$	0	0
$205$	−21.3623	−1.49201
$206$	0	0
$207$	−25.7381	−1.78892
$208$	0	0
$209$	13.7095	0.948307
$210$	0	0
$211$	3.56648	0.245526	0.122763	−	0.992436i	$-0.460824\pi$
$211$	3.56648	0.245526	0.122763	+	0.992436i	$0.460824\pi$
$212$	0	0
$213$	42.6183	2.92015
$214$	0	0
$215$	−4.67320	−0.318710
$216$	0	0
$217$	−2.64427	−0.179505
$218$	0	0
$219$	−9.39430	−0.634808
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−21.4094	−1.43368	−0.716840	−	0.697238i	$-0.754412\pi$
$223$	−21.4094	−1.43368	−0.716840	+	0.697238i	$0.754412\pi$
$224$	0	0
$225$	−10.9559	−0.730393
$226$	0	0
$227$	−21.0931	−1.39999	−0.699997	−	0.714145i	$-0.746816\pi$
$227$	−21.0931	−1.39999	−0.699997	+	0.714145i	$0.746816\pi$
$228$	0	0
$229$	10.2882	0.679866	0.339933	−	0.940450i	$-0.389596\pi$
$229$	10.2882	0.679866	0.339933	+	0.940450i	$0.389596\pi$
$230$	0	0
$231$	−8.94162	−0.588315
$232$	0	0
$233$	11.4160	0.747888	0.373944	−	0.927451i	$-0.378005\pi$
$233$	11.4160	0.747888	0.373944	+	0.927451i	$0.378005\pi$
$234$	0	0
$235$	−17.1388	−1.11801
$236$	0	0
$237$	51.6003	3.35180
$238$	0	0
$239$	−0.00572125	−0.000370077 0	−0.000185039	−	1.00000i	$-0.500059\pi$
$239$	−0.00572125	−0.000370077 0	−0.000185039		1.00000i	$0.500059\pi$
$240$	0	0
$241$	25.6937	1.65507	0.827537	−	0.561410i	$-0.189741\pi$
$241$	25.6937	1.65507	0.827537	+	0.561410i	$0.189741\pi$
$242$	0	0
$243$	2.02010	0.129590
$244$	0	0
$245$	1.77152	0.113178
$246$	0	0
$247$	11.8786	0.755816
$248$	0	0
$249$	−37.7805	−2.39424
$250$	0	0
$251$	−8.56127	−0.540383	−0.270191	−	0.962807i	$-0.587087\pi$
$251$	−8.56127	−0.540383	−0.270191	+	0.962807i	$0.587087\pi$
$252$	0	0
$253$	−13.1200	−0.824846
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.3786	−0.959291	−0.479646	−	0.877462i	$-0.659235\pi$
$257$	−15.3786	−0.959291	−0.479646	+	0.877462i	$0.659235\pi$
$258$	0	0
$259$	−0.526932	−0.0327419
$260$	0	0
$261$	14.1742	0.877361
$262$	0	0
$263$	1.18787	0.0732471	0.0366236	−	0.999329i	$-0.488340\pi$
$263$	1.18787	0.0732471	0.0366236	+	0.999329i	$0.488340\pi$
$264$	0	0
$265$	6.88322	0.422833
$266$	0	0
$267$	6.12683	0.374956
$268$	0	0
$269$	4.48101	0.273212	0.136606	−	0.990625i	$-0.456381\pi$
$269$	4.48101	0.273212	0.136606	+	0.990625i	$0.456381\pi$
$270$	0	0
$271$	−15.5763	−0.946192	−0.473096	−	0.881011i	$-0.656864\pi$
$271$	−15.5763	−0.946192	−0.473096	+	0.881011i	$0.656864\pi$
$272$	0	0
$273$	−7.74745	−0.468897
$274$	0	0
$275$	−5.58477	−0.336774
$276$	0	0
$277$	−18.1491	−1.09047	−0.545237	−	0.838282i	$-0.683560\pi$
$277$	−18.1491	−1.09047	−0.545237	+	0.838282i	$0.683560\pi$
$278$	0	0
$279$	−15.5611	−0.931619
$280$	0	0
$281$	27.0114	1.61137	0.805683	−	0.592347i	$-0.201798\pi$
$281$	27.0114	1.61137	0.805683	+	0.592347i	$0.201798\pi$
$282$	0	0
$283$	6.42153	0.381720	0.190860	−	0.981617i	$-0.438872\pi$
$283$	6.42153	0.381720	0.190860	+	0.981617i	$0.438872\pi$
$284$	0	0
$285$	−24.1324	−1.42948
$286$	0	0
$287$	−12.0588	−0.711806
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−21.4544	−1.25768
$292$	0	0
$293$	−12.4808	−0.729134	−0.364567	−	0.931177i	$-0.618783\pi$
$293$	−12.4808	−0.729134	−0.364567	+	0.931177i	$0.618783\pi$
$294$	0	0
$295$	−4.66247	−0.271460
$296$	0	0
$297$	−25.7951	−1.49678
$298$	0	0
$299$	−11.3678	−0.657416
$300$	0	0
$301$	−2.63796	−0.152050
$302$	0	0
$303$	3.45706	0.198603
$304$	0	0
$305$	−16.5831	−0.949547
$306$	0	0
$307$	29.3966	1.67775	0.838877	−	0.544321i	$-0.183213\pi$
$307$	29.3966	1.67775	0.838877	+	0.544321i	$0.183213\pi$
$308$	0	0
$309$	26.0325	1.48094
$310$	0	0
$311$	15.8689	0.899844	0.449922	−	0.893068i	$-0.351452\pi$
$311$	15.8689	0.899844	0.449922	+	0.893068i	$0.351452\pi$
$312$	0	0
$313$	−32.8643	−1.85760	−0.928801	−	0.370579i	$-0.879159\pi$
$313$	−32.8643	−1.85760	−0.928801	+	0.370579i	$0.879159\pi$
$314$	0	0
$315$	10.4251	0.587388
$316$	0	0
$317$	−26.5628	−1.49191	−0.745957	−	0.665994i	$-0.768008\pi$
$317$	−26.5628	−1.49191	−0.745957	+	0.665994i	$0.768008\pi$
$318$	0	0
$319$	7.22529	0.404539
$320$	0	0
$321$	39.4060	2.19943
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.83891	−0.268414
$326$	0	0
$327$	13.8345	0.765049
$328$	0	0
$329$	−9.67463	−0.533379
$330$	0	0
$331$	−17.9872	−0.988666	−0.494333	−	0.869273i	$-0.664588\pi$
$331$	−17.9872	−0.988666	−0.494333	+	0.869273i	$0.664588\pi$
$332$	0	0
$333$	−3.10091	−0.169929
$334$	0	0
$335$	−26.4045	−1.44263
$336$	0	0
$337$	28.0534	1.52816	0.764082	−	0.645119i	$-0.223192\pi$
$337$	28.0534	1.52816	0.764082	+	0.645119i	$0.223192\pi$
$338$	0	0
$339$	1.19612	0.0649642
$340$	0	0
$341$	−7.93227	−0.429556
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	23.0947	1.24338
$346$	0	0
$347$	−19.1197	−1.02640	−0.513199	−	0.858270i	$-0.671540\pi$
$347$	−19.1197	−1.02640	−0.513199	+	0.858270i	$0.671540\pi$
$348$	0	0
$349$	12.4741	0.667725	0.333863	−	0.942622i	$-0.391648\pi$
$349$	12.4741	0.667725	0.333863	+	0.942622i	$0.391648\pi$
$350$	0	0
$351$	−22.3501	−1.19296
$352$	0	0
$353$	32.0136	1.70391	0.851956	−	0.523613i	$-0.175416\pi$
$353$	32.0136	1.70391	0.851956	+	0.523613i	$0.175416\pi$
$354$	0	0
$355$	−25.3289	−1.34432
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.5673	0.610501	0.305251	−	0.952272i	$-0.401260\pi$
$359$	11.5673	0.610501	0.305251	+	0.952272i	$0.401260\pi$
$360$	0	0
$361$	1.88628	0.0992781
$362$	0	0
$363$	5.96517	0.313090
$364$	0	0
$365$	5.58324	0.292240
$366$	0	0
$367$	−5.64863	−0.294856	−0.147428	−	0.989073i	$-0.547099\pi$
$367$	−5.64863	−0.294856	−0.147428	+	0.989073i	$0.547099\pi$
$368$	0	0
$369$	−70.9638	−3.69423
$370$	0	0
$371$	3.88549	0.201725
$372$	0	0
$373$	−18.0723	−0.935750	−0.467875	−	0.883795i	$-0.654980\pi$
$373$	−18.0723	−0.935750	−0.467875	+	0.883795i	$0.654980\pi$
$374$	0	0
$375$	36.2329	1.87106
$376$	0	0
$377$	6.26034	0.322424
$378$	0	0
$379$	−4.27736	−0.219713	−0.109857	−	0.993947i	$-0.535039\pi$
$379$	−4.27736	−0.219713	−0.109857	+	0.993947i	$0.535039\pi$
$380$	0	0
$381$	−1.04357	−0.0534636
$382$	0	0
$383$	2.28017	0.116511	0.0582557	−	0.998302i	$-0.481446\pi$
$383$	2.28017	0.116511	0.0582557	+	0.998302i	$0.481446\pi$
$384$	0	0
$385$	5.31420	0.270836
$386$	0	0
$387$	−15.5240	−0.789128
$388$	0	0
$389$	3.16537	0.160491	0.0802453	−	0.996775i	$-0.474430\pi$
$389$	3.16537	0.160491	0.0802453	+	0.996775i	$0.474430\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	23.5095	1.18590
$394$	0	0
$395$	−30.6671	−1.54303
$396$	0	0
$397$	1.47062	0.0738084	0.0369042	−	0.999319i	$-0.488250\pi$
$397$	1.47062	0.0738084	0.0369042	+	0.999319i	$0.488250\pi$
$398$	0	0
$399$	−13.6225	−0.681976
$400$	0	0
$401$	32.9273	1.64431	0.822155	−	0.569264i	$-0.192772\pi$
$401$	32.9273	1.64431	0.822155	+	0.569264i	$0.192772\pi$
$402$	0	0
$403$	−6.87290	−0.342364
$404$	0	0
$405$	14.1310	0.702177
$406$	0	0
$407$	−1.58069	−0.0783517
$408$	0	0
$409$	11.6136	0.574255	0.287128	−	0.957892i	$-0.407300\pi$
$409$	11.6136	0.574255	0.287128	+	0.957892i	$0.407300\pi$
$410$	0	0
$411$	−28.7387	−1.41757
$412$	0	0
$413$	−2.63190	−0.129508
$414$	0	0
$415$	22.4538	1.10221
$416$	0	0
$417$	−38.4772	−1.88424
$418$	0	0
$419$	34.0454	1.66322	0.831612	−	0.555357i	$-0.187418\pi$
$419$	34.0454	1.66322	0.831612	+	0.555357i	$0.187418\pi$
$420$	0	0
$421$	−16.4720	−0.802797	−0.401399	−	0.915903i	$-0.631476\pi$
$421$	−16.4720	−0.802797	−0.401399	+	0.915903i	$0.631476\pi$
$422$	0	0
$423$	−56.9336	−2.76821
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−9.36096	−0.453009
$428$	0	0
$429$	−23.2408	−1.12207
$430$	0	0
$431$	3.56754	0.171843	0.0859213	−	0.996302i	$-0.472617\pi$
$431$	3.56754	0.171843	0.0859213	+	0.996302i	$0.472617\pi$
$432$	0	0
$433$	−13.7012	−0.658436	−0.329218	−	0.944254i	$-0.606785\pi$
$433$	−13.7012	−0.658436	−0.329218	+	0.944254i	$0.606785\pi$
$434$	0	0
$435$	−12.7185	−0.609804
$436$	0	0
$437$	−19.9881	−0.956162
$438$	0	0
$439$	−36.1111	−1.72349	−0.861745	−	0.507341i	$-0.830628\pi$
$439$	−36.1111	−1.72349	−0.861745	+	0.507341i	$0.830628\pi$
$440$	0	0
$441$	5.88484	0.280230
$442$	0	0
$443$	6.17342	0.293308	0.146654	−	0.989188i	$-0.453150\pi$
$443$	6.17342	0.293308	0.146654	+	0.989188i	$0.453150\pi$
$444$	0	0
$445$	−3.64131	−0.172615
$446$	0	0
$447$	−11.7569	−0.556084
$448$	0	0
$449$	−20.1363	−0.950290	−0.475145	−	0.879907i	$-0.657604\pi$
$449$	−20.1363	−0.950290	−0.475145	+	0.879907i	$0.657604\pi$
$450$	0	0
$451$	−36.1738	−1.70336
$452$	0	0
$453$	−71.1995	−3.34524
$454$	0	0
$455$	4.60448	0.215861
$456$	0	0
$457$	−13.0583	−0.610842	−0.305421	−	0.952217i	$-0.598797\pi$
$457$	−13.0583	−0.610842	−0.305421	+	0.952217i	$0.598797\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15.0878	−0.702710	−0.351355	−	0.936242i	$-0.614279\pi$
$461$	−15.0878	−0.702710	−0.351355	+	0.936242i	$0.614279\pi$
$462$	0	0
$463$	−30.0923	−1.39851	−0.699255	−	0.714873i	$-0.746485\pi$
$463$	−30.0923	−1.39851	−0.699255	+	0.714873i	$0.746485\pi$
$464$	0	0
$465$	13.9629	0.647516
$466$	0	0
$467$	13.0220	0.602587	0.301294	−	0.953531i	$-0.402582\pi$
$467$	13.0220	0.602587	0.301294	+	0.953531i	$0.402582\pi$
$468$	0	0
$469$	−14.9050	−0.688249
$470$	0	0
$471$	−55.0533	−2.53672
$472$	0	0
$473$	−7.91334	−0.363856
$474$	0	0
$475$	−8.50833	−0.390389
$476$	0	0
$477$	22.8655	1.04694
$478$	0	0
$479$	37.5456	1.71550	0.857752	−	0.514064i	$-0.171861\pi$
$479$	37.5456	1.71550	0.857752	+	0.514064i	$0.171861\pi$
$480$	0	0
$481$	−1.36958	−0.0624476
$482$	0	0
$483$	13.0367	0.593189
$484$	0	0
$485$	12.7508	0.578983
$486$	0	0
$487$	−36.4782	−1.65298	−0.826492	−	0.562948i	$-0.809667\pi$
$487$	−36.4782	−1.65298	−0.826492	+	0.562948i	$0.809667\pi$
$488$	0	0
$489$	54.8446	2.48016
$490$	0	0
$491$	−21.2512	−0.959053	−0.479526	−	0.877527i	$-0.659192\pi$
$491$	−21.2512	−0.959053	−0.479526	+	0.877527i	$0.659192\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	31.2732	1.40562
$496$	0	0
$497$	−14.2979	−0.641347
$498$	0	0
$499$	23.0774	1.03309	0.516544	−	0.856261i	$-0.327218\pi$
$499$	23.0774	1.03309	0.516544	+	0.856261i	$0.327218\pi$
$500$	0	0
$501$	71.0245	3.17314
$502$	0	0
$503$	−0.650809	−0.0290181	−0.0145091	−	0.999895i	$-0.504619\pi$
$503$	−0.650809	−0.0290181	−0.0145091	+	0.999895i	$0.504619\pi$
$504$	0	0
$505$	−2.05460	−0.0914287
$506$	0	0
$507$	18.6128	0.826622
$508$	0	0
$509$	25.6793	1.13822	0.569108	−	0.822263i	$-0.307289\pi$
$509$	25.6793	1.13822	0.569108	+	0.822263i	$0.307289\pi$
$510$	0	0
$511$	3.15166	0.139421
$512$	0	0
$513$	−39.2986	−1.73507
$514$	0	0
$515$	−15.4717	−0.681765
$516$	0	0
$517$	−29.0219	−1.27638
$518$	0	0
$519$	18.6849	0.820177
$520$	0	0
$521$	8.53401	0.373882	0.186941	−	0.982371i	$-0.440143\pi$
$521$	8.53401	0.373882	0.186941	+	0.982371i	$0.440143\pi$
$522$	0	0
$523$	−18.7723	−0.820856	−0.410428	−	0.911893i	$-0.634621\pi$
$523$	−18.7723	−0.820856	−0.410428	+	0.911893i	$0.634621\pi$
$524$	0	0
$525$	5.54930	0.242191
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−3.87138	−0.168321
$530$	0	0
$531$	−15.4883	−0.672136
$532$	0	0
$533$	−31.3427	−1.35760
$534$	0	0
$535$	−23.4198	−1.01253
$536$	0	0
$537$	38.3718	1.65587
$538$	0	0
$539$	2.99979	0.129210
$540$	0	0
$541$	−13.8125	−0.593845	−0.296923	−	0.954902i	$-0.595960\pi$
$541$	−13.8125	−0.593845	−0.296923	+	0.954902i	$0.595960\pi$
$542$	0	0
$543$	−63.7489	−2.73573
$544$	0	0
$545$	−8.22213	−0.352197
$546$	0	0
$547$	10.0774	0.430877	0.215438	−	0.976517i	$-0.430882\pi$
$547$	10.0774	0.430877	0.215438	+	0.976517i	$0.430882\pi$
$548$	0	0
$549$	−55.0877	−2.35109
$550$	0	0
$551$	11.0076	0.468942
$552$	0	0
$553$	−17.3112	−0.736147
$554$	0	0
$555$	2.78244	0.118108
$556$	0	0
$557$	−36.0805	−1.52878	−0.764389	−	0.644756i	$-0.776959\pi$
$557$	−36.0805	−1.52878	−0.764389	+	0.644756i	$0.776959\pi$
$558$	0	0
$559$	−6.85650	−0.289999
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21.6232	−0.911309	−0.455655	−	0.890157i	$-0.650595\pi$
$563$	−21.6232	−0.911309	−0.455655	+	0.890157i	$0.650595\pi$
$564$	0	0
$565$	−0.710878	−0.0299069
$566$	0	0
$567$	7.97679	0.334994
$568$	0	0
$569$	8.92834	0.374296	0.187148	−	0.982332i	$-0.440076\pi$
$569$	8.92834	0.374296	0.187148	+	0.982332i	$0.440076\pi$
$570$	0	0
$571$	37.8466	1.58383	0.791916	−	0.610631i	$-0.209084\pi$
$571$	37.8466	1.58383	0.791916	+	0.610631i	$0.209084\pi$
$572$	0	0
$573$	−52.2222	−2.18161
$574$	0	0
$575$	8.14245	0.339564
$576$	0	0
$577$	32.3336	1.34607	0.673033	−	0.739612i	$-0.264991\pi$
$577$	32.3336	1.34607	0.673033	+	0.739612i	$0.264991\pi$
$578$	0	0
$579$	15.3694	0.638729
$580$	0	0
$581$	12.6749	0.525842
$582$	0	0
$583$	11.6557	0.482729
$584$	0	0
$585$	27.0966	1.12031
$586$	0	0
$587$	18.0489	0.744957	0.372478	−	0.928041i	$-0.378508\pi$
$587$	18.0489	0.744957	0.372478	+	0.928041i	$0.378508\pi$
$588$	0	0
$589$	−12.0847	−0.497942
$590$	0	0
$591$	27.3225	1.12390
$592$	0	0
$593$	−8.95732	−0.367833	−0.183917	−	0.982942i	$-0.558878\pi$
$593$	−8.95732	−0.367833	−0.183917	+	0.982942i	$0.558878\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−34.6991	−1.42014
$598$	0	0
$599$	1.29732	0.0530070	0.0265035	−	0.999649i	$-0.491563\pi$
$599$	1.29732	0.0530070	0.0265035	+	0.999649i	$0.491563\pi$
$600$	0	0
$601$	31.4326	1.28216	0.641082	−	0.767473i	$-0.278486\pi$
$601$	31.4326	1.28216	0.641082	+	0.767473i	$0.278486\pi$
$602$	0	0
$603$	−87.7134	−3.57197
$604$	0	0
$605$	−3.54523	−0.144134
$606$	0	0
$607$	1.05652	0.0428830	0.0214415	−	0.999770i	$-0.493174\pi$
$607$	1.05652	0.0428830	0.0214415	+	0.999770i	$0.493174\pi$
$608$	0	0
$609$	−7.17941	−0.290924
$610$	0	0
$611$	−25.1460	−1.01730
$612$	0	0
$613$	−25.7088	−1.03837	−0.519183	−	0.854663i	$-0.673764\pi$
$613$	−25.7088	−1.03837	−0.519183	+	0.854663i	$0.673764\pi$
$614$	0	0
$615$	63.6757	2.56765
$616$	0	0
$617$	−7.59660	−0.305828	−0.152914	−	0.988240i	$-0.548866\pi$
$617$	−7.59660	−0.305828	−0.152914	+	0.988240i	$0.548866\pi$
$618$	0	0
$619$	−17.6467	−0.709282	−0.354641	−	0.935002i	$-0.615397\pi$
$619$	−17.6467	−0.709282	−0.354641	+	0.935002i	$0.615397\pi$
$620$	0	0
$621$	37.6087	1.50918
$622$	0	0
$623$	−2.05547	−0.0823507
$624$	0	0
$625$	−12.2254	−0.489017
$626$	0	0
$627$	−40.8646	−1.63197
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−20.2378	−0.805653	−0.402827	−	0.915276i	$-0.631972\pi$
$631$	−20.2378	−0.805653	−0.402827	+	0.915276i	$0.631972\pi$
$632$	0	0
$633$	−10.6308	−0.422535
$634$	0	0
$635$	0.620215	0.0246125
$636$	0	0
$637$	2.59917	0.102983
$638$	0	0
$639$	−84.1406	−3.32855
$640$	0	0
$641$	−37.2967	−1.47313	−0.736565	−	0.676367i	$-0.763553\pi$
$641$	−37.2967	−1.47313	−0.736565	+	0.676367i	$0.763553\pi$
$642$	0	0
$643$	23.6793	0.933820	0.466910	−	0.884305i	$-0.345367\pi$
$643$	23.6793	0.933820	0.466910	+	0.884305i	$0.345367\pi$
$644$	0	0
$645$	13.9296	0.548478
$646$	0	0
$647$	−28.2713	−1.11146	−0.555730	−	0.831363i	$-0.687561\pi$
$647$	−28.2713	−1.11146	−0.555730	+	0.831363i	$0.687561\pi$
$648$	0	0
$649$	−7.89517	−0.309912
$650$	0	0
$651$	7.88190	0.308916
$652$	0	0
$653$	13.6863	0.535586	0.267793	−	0.963476i	$-0.413706\pi$
$653$	13.6863	0.535586	0.267793	+	0.963476i	$0.413706\pi$
$654$	0	0
$655$	−13.9722	−0.545938
$656$	0	0
$657$	18.5470	0.723588
$658$	0	0
$659$	22.7157	0.884878	0.442439	−	0.896799i	$-0.354113\pi$
$659$	22.7157	0.884878	0.442439	+	0.896799i	$0.354113\pi$
$660$	0	0
$661$	47.6821	1.85462	0.927310	−	0.374294i	$-0.122115\pi$
$661$	47.6821	1.85462	0.927310	+	0.374294i	$0.122115\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.09612	0.313954
$666$	0	0
$667$	−10.5343	−0.407890
$668$	0	0
$669$	63.8159	2.46727
$670$	0	0
$671$	−28.0809	−1.08405
$672$	0	0
$673$	47.7677	1.84131	0.920654	−	0.390380i	$-0.127656\pi$
$673$	47.7677	1.84131	0.920654	+	0.390380i	$0.127656\pi$
$674$	0	0
$675$	16.0088	0.616180
$676$	0	0
$677$	−23.8735	−0.917532	−0.458766	−	0.888557i	$-0.651708\pi$
$677$	−23.8735	−0.917532	−0.458766	+	0.888557i	$0.651708\pi$
$678$	0	0
$679$	7.19765	0.276220
$680$	0	0
$681$	62.8730	2.40930
$682$	0	0
$683$	−20.8404	−0.797434	−0.398717	−	0.917074i	$-0.630544\pi$
$683$	−20.8404	−0.797434	−0.398717	+	0.917074i	$0.630544\pi$
$684$	0	0
$685$	17.0800	0.652593
$686$	0	0
$687$	−30.6666	−1.17001
$688$	0	0
$689$	10.0990	0.384743
$690$	0	0
$691$	−11.1028	−0.422370	−0.211185	−	0.977446i	$-0.567732\pi$
$691$	−11.1028	−0.422370	−0.211185	+	0.977446i	$0.567732\pi$
$692$	0	0
$693$	17.6533	0.670593
$694$	0	0
$695$	22.8678	0.867427
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−34.0282	−1.28707
$700$	0	0
$701$	−19.3141	−0.729482	−0.364741	−	0.931109i	$-0.618842\pi$
$701$	−19.3141	−0.729482	−0.364741	+	0.931109i	$0.618842\pi$
$702$	0	0
$703$	−2.40816	−0.0908254
$704$	0	0
$705$	51.0864	1.92402
$706$	0	0
$707$	−1.15980	−0.0436187
$708$	0	0
$709$	6.85560	0.257468	0.128734	−	0.991679i	$-0.458909\pi$
$709$	6.85560	0.257468	0.128734	+	0.991679i	$0.458909\pi$
$710$	0	0
$711$	−101.874	−3.82056
$712$	0	0
$713$	11.5651	0.433115
$714$	0	0
$715$	13.8125	0.516557
$716$	0	0
$717$	0.0170536	0.000636878 0
$718$	0	0
$719$	16.1354	0.601747	0.300874	−	0.953664i	$-0.402722\pi$
$719$	16.1354	0.601747	0.300874	+	0.953664i	$0.402722\pi$
$720$	0	0
$721$	−8.73357	−0.325255
$722$	0	0
$723$	−76.5863	−2.84827
$724$	0	0
$725$	−4.48412	−0.166536
$726$	0	0
$727$	10.8301	0.401665	0.200832	−	0.979626i	$-0.435635\pi$
$727$	10.8301	0.401665	0.200832	+	0.979626i	$0.435635\pi$
$728$	0	0
$729$	−29.9518	−1.10933
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−27.4997	−1.01573	−0.507863	−	0.861438i	$-0.669564\pi$
$733$	−27.4997	−1.01573	−0.507863	+	0.861438i	$0.669564\pi$
$734$	0	0
$735$	−5.28045	−0.194772
$736$	0	0
$737$	−44.7119	−1.64698
$738$	0	0
$739$	−6.42719	−0.236428	−0.118214	−	0.992988i	$-0.537717\pi$
$739$	−6.42719	−0.236428	−0.118214	+	0.992988i	$0.537717\pi$
$740$	0	0
$741$	−35.4070	−1.30071
$742$	0	0
$743$	27.8081	1.02018	0.510090	−	0.860121i	$-0.329612\pi$
$743$	27.8081	1.02018	0.510090	+	0.860121i	$0.329612\pi$
$744$	0	0
$745$	6.98740	0.255999
$746$	0	0
$747$	74.5895	2.72909
$748$	0	0
$749$	−13.2202	−0.483055
$750$	0	0
$751$	17.9244	0.654071	0.327035	−	0.945012i	$-0.393950\pi$
$751$	17.9244	0.654071	0.327035	+	0.945012i	$0.393950\pi$
$752$	0	0
$753$	25.5190	0.929963
$754$	0	0
$755$	42.3154	1.54001
$756$	0	0
$757$	−9.96117	−0.362045	−0.181023	−	0.983479i	$-0.557941\pi$
$757$	−9.96117	−0.362045	−0.181023	+	0.983479i	$0.557941\pi$
$758$	0	0
$759$	39.1073	1.41951
$760$	0	0
$761$	0.279750	0.0101409	0.00507046	−	0.999987i	$-0.498386\pi$
$761$	0.279750	0.0101409	0.00507046	+	0.999987i	$0.498386\pi$
$762$	0	0
$763$	−4.64129	−0.168026
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.84076	−0.247005
$768$	0	0
$769$	−18.8011	−0.677986	−0.338993	−	0.940789i	$-0.610086\pi$
$769$	−18.8011	−0.677986	−0.338993	+	0.940789i	$0.610086\pi$
$770$	0	0
$771$	45.8397	1.65088
$772$	0	0
$773$	−14.9445	−0.537518	−0.268759	−	0.963207i	$-0.586613\pi$
$773$	−14.9445	−0.537518	−0.268759	+	0.963207i	$0.586613\pi$
$774$	0	0
$775$	4.92288	0.176835
$776$	0	0
$777$	1.57065	0.0563467
$778$	0	0
$779$	−55.1104	−1.97453
$780$	0	0
$781$	−42.8906	−1.53475
$782$	0	0
$783$	−20.7114	−0.740166
$784$	0	0
$785$	32.7194	1.16780
$786$	0	0
$787$	52.8472	1.88380	0.941900	−	0.335894i	$-0.109038\pi$
$787$	52.8472	1.88380	0.941900	+	0.335894i	$0.109038\pi$
$788$	0	0
$789$	−3.54073	−0.126053
$790$	0	0
$791$	−0.401281	−0.0142679
$792$	0	0
$793$	−24.3307	−0.864008
$794$	0	0
$795$	−20.5171	−0.727668
$796$	0	0
$797$	−21.3791	−0.757286	−0.378643	−	0.925543i	$-0.623609\pi$
$797$	−21.3791	−0.757286	−0.378643	+	0.925543i	$0.623609\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−12.0961	−0.427395
$802$	0	0
$803$	9.45434	0.333636
$804$	0	0
$805$	−7.74797	−0.273080
$806$	0	0
$807$	−13.3567	−0.470180
$808$	0	0
$809$	6.86433	0.241337	0.120668	−	0.992693i	$-0.461496\pi$
$809$	6.86433	0.241337	0.120668	+	0.992693i	$0.461496\pi$
$810$	0	0
$811$	31.3596	1.10118	0.550592	−	0.834775i	$-0.314402\pi$
$811$	31.3596	1.10118	0.550592	+	0.834775i	$0.314402\pi$
$812$	0	0
$813$	46.4289	1.62833
$814$	0	0
$815$	−32.5953	−1.14176
$816$	0	0
$817$	−12.0559	−0.421782
$818$	0	0
$819$	15.2957	0.534474
$820$	0	0
$821$	−25.9312	−0.905007	−0.452503	−	0.891763i	$-0.649469\pi$
$821$	−25.9312	−0.905007	−0.452503	+	0.891763i	$0.649469\pi$
$822$	0	0
$823$	38.0689	1.32700	0.663499	−	0.748177i	$-0.269071\pi$
$823$	38.0689	1.32700	0.663499	+	0.748177i	$0.269071\pi$
$824$	0	0
$825$	16.6468	0.579566
$826$	0	0
$827$	26.3710	0.917011	0.458505	−	0.888692i	$-0.348385\pi$
$827$	26.3710	0.917011	0.458505	+	0.888692i	$0.348385\pi$
$828$	0	0
$829$	11.7846	0.409298	0.204649	−	0.978835i	$-0.434395\pi$
$829$	11.7846	0.409298	0.204649	+	0.978835i	$0.434395\pi$
$830$	0	0
$831$	54.0979	1.87663
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−42.2114	−1.46079
$836$	0	0
$837$	22.7380	0.785940
$838$	0	0
$839$	−42.6807	−1.47350	−0.736750	−	0.676165i	$-0.763641\pi$
$839$	−42.6807	−1.47350	−0.736750	+	0.676165i	$0.763641\pi$
$840$	0	0
$841$	−23.1987	−0.799954
$842$	0	0
$843$	−80.5141	−2.77305
$844$	0	0
$845$	−11.0620	−0.380543
$846$	0	0
$847$	−2.00124	−0.0687633
$848$	0	0
$849$	−19.1409	−0.656915
$850$	0	0
$851$	2.30460	0.0790008
$852$	0	0
$853$	9.93285	0.340094	0.170047	−	0.985436i	$-0.445608\pi$
$853$	9.93285	0.340094	0.170047	+	0.985436i	$0.445608\pi$
$854$	0	0
$855$	47.6443	1.62940
$856$	0	0
$857$	29.1572	0.995990	0.497995	−	0.867180i	$-0.334070\pi$
$857$	29.1572	0.995990	0.497995	+	0.867180i	$0.334070\pi$
$858$	0	0
$859$	25.5644	0.872246	0.436123	−	0.899887i	$-0.356351\pi$
$859$	25.5644	0.872246	0.436123	+	0.899887i	$0.356351\pi$
$860$	0	0
$861$	35.9441	1.22497
$862$	0	0
$863$	4.40501	0.149948	0.0749741	−	0.997185i	$-0.476113\pi$
$863$	4.40501	0.149948	0.0749741	+	0.997185i	$0.476113\pi$
$864$	0	0
$865$	−11.1048	−0.377576
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−51.9300	−1.76161
$870$	0	0
$871$	−38.7406	−1.31267
$872$	0	0
$873$	42.3570	1.43357
$874$	0	0
$875$	−12.1557	−0.410937
$876$	0	0
$877$	25.5494	0.862744	0.431372	−	0.902174i	$-0.358030\pi$
$877$	25.5494	0.862744	0.431372	+	0.902174i	$0.358030\pi$
$878$	0	0
$879$	37.2020	1.25479
$880$	0	0
$881$	39.2495	1.32235	0.661174	−	0.750233i	$-0.270059\pi$
$881$	39.2495	1.32235	0.661174	+	0.750233i	$0.270059\pi$
$882$	0	0
$883$	33.1225	1.11466	0.557330	−	0.830291i	$-0.311826\pi$
$883$	33.1225	1.11466	0.557330	+	0.830291i	$0.311826\pi$
$884$	0	0
$885$	13.8976	0.467164
$886$	0	0
$887$	27.1328	0.911030	0.455515	−	0.890228i	$-0.349455\pi$
$887$	27.1328	0.911030	0.455515	+	0.890228i	$0.349455\pi$
$888$	0	0
$889$	0.350103	0.0117421
$890$	0	0
$891$	23.9287	0.801643
$892$	0	0
$893$	−44.2145	−1.47958
$894$	0	0
$895$	−22.8052	−0.762293
$896$	0	0
$897$	33.8845	1.13137
$898$	0	0
$899$	−6.36898	−0.212417
$900$	0	0
$901$	0	0
$902$	0	0
$903$	7.86309	0.261667
$904$	0	0
$905$	37.8873	1.25942
$906$	0	0
$907$	−56.2710	−1.86845	−0.934224	−	0.356687i	$-0.883906\pi$
$907$	−56.2710	−1.86845	−0.934224	+	0.356687i	$0.883906\pi$
$908$	0	0
$909$	−6.82522	−0.226378
$910$	0	0
$911$	45.3479	1.50244	0.751221	−	0.660051i	$-0.229465\pi$
$911$	45.3479	1.50244	0.751221	+	0.660051i	$0.229465\pi$
$912$	0	0
$913$	38.0220	1.25834
$914$	0	0
$915$	49.4301	1.63411
$916$	0	0
$917$	−7.88711	−0.260455
$918$	0	0
$919$	−30.5827	−1.00883	−0.504415	−	0.863461i	$-0.668292\pi$
$919$	−30.5827	−1.00883	−0.504415	+	0.863461i	$0.668292\pi$
$920$	0	0
$921$	−87.6238	−2.88730
$922$	0	0
$923$	−37.1625	−1.22322
$924$	0	0
$925$	0.980997	0.0322550
$926$	0	0
$927$	−51.3957	−1.68805
$928$	0	0
$929$	−27.4531	−0.900707	−0.450354	−	0.892850i	$-0.648702\pi$
$929$	−27.4531	−0.900707	−0.450354	+	0.892850i	$0.648702\pi$
$930$	0	0
$931$	4.57015	0.149781
$932$	0	0
$933$	−47.3012	−1.54857
$934$	0	0
$935$	0	0
$936$	0	0
$937$	18.3154	0.598337	0.299169	−	0.954200i	$-0.403291\pi$
$937$	18.3154	0.598337	0.299169	+	0.954200i	$0.403291\pi$
$938$	0	0
$939$	97.9602	3.19681
$940$	0	0
$941$	30.8433	1.00546	0.502731	−	0.864443i	$-0.332329\pi$
$941$	30.8433	1.00546	0.502731	+	0.864443i	$0.332329\pi$
$942$	0	0
$943$	52.7405	1.71747
$944$	0	0
$945$	−15.2332	−0.495537
$946$	0	0
$947$	−15.7368	−0.511379	−0.255689	−	0.966759i	$-0.582302\pi$
$947$	−15.7368	−0.511379	−0.255689	+	0.966759i	$0.582302\pi$
$948$	0	0
$949$	8.19170	0.265914
$950$	0	0
$951$	79.1768	2.56748
$952$	0	0
$953$	−51.5074	−1.66849	−0.834244	−	0.551395i	$-0.814096\pi$
$953$	−51.5074	−1.66849	−0.834244	+	0.551395i	$0.814096\pi$
$954$	0	0
$955$	31.0367	1.00433
$956$	0	0
$957$	−21.5367	−0.696184
$958$	0	0
$959$	9.64144	0.311338
$960$	0	0
$961$	−24.0078	−0.774446
$962$	0	0
$963$	−77.7987	−2.50703
$964$	0	0
$965$	−9.13434	−0.294045
$966$	0	0
$967$	21.6915	0.697552	0.348776	−	0.937206i	$-0.386597\pi$
$967$	21.6915	0.697552	0.348776	+	0.937206i	$0.386597\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14.7980	0.474889	0.237444	−	0.971401i	$-0.423690\pi$
$971$	14.7980	0.474889	0.237444	+	0.971401i	$0.423690\pi$
$972$	0	0
$973$	12.9086	0.413831
$974$	0	0
$975$	14.4236	0.461924
$976$	0	0
$977$	23.5724	0.754147	0.377074	−	0.926183i	$-0.376930\pi$
$977$	23.5724	0.754147	0.377074	+	0.926183i	$0.376930\pi$
$978$	0	0
$979$	−6.16599	−0.197066
$980$	0	0
$981$	−27.3132	−0.872044
$982$	0	0
$983$	−20.0283	−0.638803	−0.319401	−	0.947620i	$-0.603482\pi$
$983$	−20.0283	−0.638803	−0.319401	+	0.947620i	$0.603482\pi$
$984$	0	0
$985$	−16.2384	−0.517397
$986$	0	0
$987$	28.8376	0.917910
$988$	0	0
$989$	11.5375	0.366870
$990$	0	0
$991$	15.5094	0.492672	0.246336	−	0.969184i	$-0.420773\pi$
$991$	15.5094	0.492672	0.246336	+	0.969184i	$0.420773\pi$
$992$	0	0
$993$	53.6152	1.70143
$994$	0	0
$995$	20.6224	0.653774
$996$	0	0
$997$	−27.4828	−0.870388	−0.435194	−	0.900337i	$-0.643320\pi$
$997$	−27.4828	−0.870388	−0.435194	+	0.900337i	$0.643320\pi$
$998$	0	0
$999$	4.53107	0.143357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8092.2.a.s.1.1		8
17.8	even	8		476.2.l.a.421.8	yes	16
17.15	even	8		476.2.l.a.225.8	✓	16
17.16	even	2		8092.2.a.t.1.8		8
51.8	odd	8		4284.2.z.b.3277.4		16
51.32	odd	8		4284.2.z.b.4033.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.l.a.225.8	✓	16	17.15	even	8
476.2.l.a.421.8	yes	16	17.8	even	8
4284.2.z.b.3277.4		16	51.8	odd	8
4284.2.z.b.4033.4		16	51.32	odd	8
8092.2.a.s.1.1		8	1.1	even	1	trivial
8092.2.a.t.1.8		8	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8092 = 2^{2} \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8092.a (trivial)

\(f(q)\)	\(=\)	\(q-2.98074 q^{3} +1.77152 q^{5} +1.00000 q^{7} +5.88484 q^{9} +O(q^{10})\)	\(q-2.98074 q^{3} +1.77152 q^{5} +1.00000 q^{7} +5.88484 q^{9} +2.99979 q^{11} +2.59917 q^{13} -5.28045 q^{15} +4.57015 q^{19} -2.98074 q^{21} -4.37363 q^{23} -1.86172 q^{25} -8.59896 q^{27} +2.40860 q^{29} -2.64427 q^{31} -8.94162 q^{33} +1.77152 q^{35} -0.526932 q^{37} -7.74745 q^{39} -12.0588 q^{41} -2.63796 q^{43} +10.4251 q^{45} -9.67463 q^{47} +1.00000 q^{49} +3.88549 q^{53} +5.31420 q^{55} -13.6225 q^{57} -2.63190 q^{59} -9.36096 q^{61} +5.88484 q^{63} +4.60448 q^{65} -14.9050 q^{67} +13.0367 q^{69} -14.2979 q^{71} +3.15166 q^{73} +5.54930 q^{75} +2.99979 q^{77} -17.3112 q^{79} +7.97679 q^{81} +12.6749 q^{83} -7.17941 q^{87} -2.05547 q^{89} +2.59917 q^{91} +7.88190 q^{93} +8.09612 q^{95} +7.19765 q^{97} +17.6533 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 4 q^{5} + 8 q^{7}+O(q^{10}) \) 8 * q - 4 * q^3 - 4 * q^5 + 8 * q^7	\( 8 q - 4 q^{3} - 4 q^{5} + 8 q^{7} - 8 q^{11} + 4 q^{15} + 16 q^{19} - 4 q^{21} - 20 q^{23} + 16 q^{25} - 16 q^{27} - 16 q^{29} + 8 q^{33} - 4 q^{35} - 4 q^{37} + 8 q^{39} - 8 q^{41} - 16 q^{45} - 20 q^{47} + 8 q^{49} - 12 q^{53} - 4 q^{55} - 4 q^{57} - 4 q^{59} - 4 q^{61} + 36 q^{65} - 20 q^{67} - 12 q^{69} - 12 q^{71} + 24 q^{73} - 12 q^{75} - 8 q^{77} - 32 q^{79} - 4 q^{81} + 20 q^{83} + 32 q^{87} + 8 q^{89} - 8 q^{93} - 32 q^{95} - 24 q^{97} + 20 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 - 4 * q^5 + 8 * q^7 - 8 * q^11 + 4 * q^15 + 16 * q^19 - 4 * q^21 - 20 * q^23 + 16 * q^25 - 16 * q^27 - 16 * q^29 + 8 * q^33 - 4 * q^35 - 4 * q^37 + 8 * q^39 - 8 * q^41 - 16 * q^45 - 20 * q^47 + 8 * q^49 - 12 * q^53 - 4 * q^55 - 4 * q^57 - 4 * q^59 - 4 * q^61 + 36 * q^65 - 20 * q^67 - 12 * q^69 - 12 * q^71 + 24 * q^73 - 12 * q^75 - 8 * q^77 - 32 * q^79 - 4 * q^81 + 20 * q^83 + 32 * q^87 + 8 * q^89 - 8 * q^93 - 32 * q^95 - 24 * q^97 + 20 * q^99

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