LMFDB - Embedded newform 8046.2.a.g.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 9x^{7} + 25x^{6} + 29x^{5} - 58x^{4} - 43x^{3} + 34x^{2} + 25x + 2 $ x^9 - 3x^8 - 9x^7 + 25x^6 + 29x^5 - 58x^4 - 43x^3 + 34x^2 + 25x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.58022$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.72526 q^{5} -2.79628 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.72526 q^{5} -2.79628 q^{7} -1.00000 q^{8} -2.72526 q^{10} -3.26105 q^{11} -2.91542 q^{13} +2.79628 q^{14} +1.00000 q^{16} -0.639779 q^{17} +5.70570 q^{19} +2.72526 q^{20} +3.26105 q^{22} +1.32811 q^{23} +2.42702 q^{25} +2.91542 q^{26} -2.79628 q^{28} +3.22092 q^{29} -4.53396 q^{31} -1.00000 q^{32} +0.639779 q^{34} -7.62059 q^{35} +3.32382 q^{37} -5.70570 q^{38} -2.72526 q^{40} -8.09895 q^{41} +11.8813 q^{43} -3.26105 q^{44} -1.32811 q^{46} -5.42908 q^{47} +0.819203 q^{49} -2.42702 q^{50} -2.91542 q^{52} +8.89160 q^{53} -8.88719 q^{55} +2.79628 q^{56} -3.22092 q^{58} -0.723299 q^{59} +3.58006 q^{61} +4.53396 q^{62} +1.00000 q^{64} -7.94526 q^{65} +11.3440 q^{67} -0.639779 q^{68} +7.62059 q^{70} +16.6286 q^{71} -14.5294 q^{73} -3.32382 q^{74} +5.70570 q^{76} +9.11882 q^{77} -14.8665 q^{79} +2.72526 q^{80} +8.09895 q^{82} -5.26159 q^{83} -1.74356 q^{85} -11.8813 q^{86} +3.26105 q^{88} +16.2962 q^{89} +8.15234 q^{91} +1.32811 q^{92} +5.42908 q^{94} +15.5495 q^{95} -16.0563 q^{97} -0.819203 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) $ 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8	$ 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8} - 4 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 9 q^{16} + q^{17} - 10 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 3 q^{25} + 8 q^{26} - 4 q^{28} + 4 q^{29} - 17 q^{31} - 9 q^{32} - q^{34} + 10 q^{35} - 11 q^{37} + 10 q^{38} - 4 q^{40} - 16 q^{43} + 4 q^{44} - 8 q^{46} + 7 q^{47} - 5 q^{49} + 3 q^{50} - 8 q^{52} + 12 q^{53} - 23 q^{55} + 4 q^{56} - 4 q^{58} + 6 q^{59} - 13 q^{61} + 17 q^{62} + 9 q^{64} - 24 q^{65} - 14 q^{67} + q^{68} - 10 q^{70} + 30 q^{71} - 12 q^{73} + 11 q^{74} - 10 q^{76} + 12 q^{77} - 35 q^{79} + 4 q^{80} + 5 q^{83} - 27 q^{85} + 16 q^{86} - 4 q^{88} + 23 q^{89} - 28 q^{91} + 8 q^{92} - 7 q^{94} + 32 q^{95} - 21 q^{97} + 5 q^{98}+O(q^{100}) $ 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8 - 4 * q^10 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 9 * q^16 + q^17 - 10 * q^19 + 4 * q^20 - 4 * q^22 + 8 * q^23 - 3 * q^25 + 8 * q^26 - 4 * q^28 + 4 * q^29 - 17 * q^31 - 9 * q^32 - q^34 + 10 * q^35 - 11 * q^37 + 10 * q^38 - 4 * q^40 - 16 * q^43 + 4 * q^44 - 8 * q^46 + 7 * q^47 - 5 * q^49 + 3 * q^50 - 8 * q^52 + 12 * q^53 - 23 * q^55 + 4 * q^56 - 4 * q^58 + 6 * q^59 - 13 * q^61 + 17 * q^62 + 9 * q^64 - 24 * q^65 - 14 * q^67 + q^68 - 10 * q^70 + 30 * q^71 - 12 * q^73 + 11 * q^74 - 10 * q^76 + 12 * q^77 - 35 * q^79 + 4 * q^80 + 5 * q^83 - 27 * q^85 + 16 * q^86 - 4 * q^88 + 23 * q^89 - 28 * q^91 + 8 * q^92 - 7 * q^94 + 32 * q^95 - 21 * q^97 + 5 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.72526	1.21877	0.609386	−	0.792874i	$-0.291416\pi$
$5$	2.72526	1.21877	0.609386	+	0.792874i	$0.291416\pi$
$6$	0	0
$7$	−2.79628	−1.05690	−0.528448	−	0.848966i	$-0.677226\pi$
$7$	−2.79628	−1.05690	−0.528448	+	0.848966i	$0.677226\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.72526	−0.861802
$11$	−3.26105	−0.983243	−0.491622	−	0.870809i	$-0.663596\pi$
$11$	−3.26105	−0.983243	−0.491622	+	0.870809i	$0.663596\pi$
$12$	0	0
$13$	−2.91542	−0.808591	−0.404296	−	0.914628i	$-0.632483\pi$
$13$	−2.91542	−0.808591	−0.404296	+	0.914628i	$0.632483\pi$
$14$	2.79628	0.747338
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.639779	−0.155169	−0.0775846	−	0.996986i	$-0.524721\pi$
$17$	−0.639779	−0.155169	−0.0775846	+	0.996986i	$0.524721\pi$
$18$	0	0
$19$	5.70570	1.30898	0.654489	−	0.756072i	$-0.272884\pi$
$19$	5.70570	1.30898	0.654489	+	0.756072i	$0.272884\pi$
$20$	2.72526	0.609386
$21$	0	0
$22$	3.26105	0.695258
$23$	1.32811	0.276930	0.138465	−	0.990367i	$-0.455783\pi$
$23$	1.32811	0.276930	0.138465	+	0.990367i	$0.455783\pi$
$24$	0	0
$25$	2.42702	0.485404
$26$	2.91542	0.571761
$27$	0	0
$28$	−2.79628	−0.528448
$29$	3.22092	0.598110	0.299055	−	0.954236i	$-0.403329\pi$
$29$	3.22092	0.598110	0.299055	+	0.954236i	$0.403329\pi$
$30$	0	0
$31$	−4.53396	−0.814323	−0.407162	−	0.913356i	$-0.633481\pi$
$31$	−4.53396	−0.814323	−0.407162	+	0.913356i	$0.633481\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.639779	0.109721
$35$	−7.62059	−1.28811
$36$	0	0
$37$	3.32382	0.546433	0.273217	−	0.961953i	$-0.411912\pi$
$37$	3.32382	0.546433	0.273217	+	0.961953i	$0.411912\pi$
$38$	−5.70570	−0.925587
$39$	0	0
$40$	−2.72526	−0.430901
$41$	−8.09895	−1.26484	−0.632422	−	0.774624i	$-0.717939\pi$
$41$	−8.09895	−1.26484	−0.632422	+	0.774624i	$0.717939\pi$
$42$	0	0
$43$	11.8813	1.81188	0.905940	−	0.423406i	$-0.139166\pi$
$43$	11.8813	1.81188	0.905940	+	0.423406i	$0.139166\pi$
$44$	−3.26105	−0.491622
$45$	0	0
$46$	−1.32811	−0.195819
$47$	−5.42908	−0.791913	−0.395957	−	0.918269i	$-0.629587\pi$
$47$	−5.42908	−0.791913	−0.395957	+	0.918269i	$0.629587\pi$
$48$	0	0
$49$	0.819203	0.117029
$50$	−2.42702	−0.343232
$51$	0	0
$52$	−2.91542	−0.404296
$53$	8.89160	1.22135	0.610677	−	0.791879i	$-0.290897\pi$
$53$	8.89160	1.22135	0.610677	+	0.791879i	$0.290897\pi$
$54$	0	0
$55$	−8.88719	−1.19835
$56$	2.79628	0.373669
$57$	0	0
$58$	−3.22092	−0.422927
$59$	−0.723299	−0.0941655	−0.0470827	−	0.998891i	$-0.514992\pi$
$59$	−0.723299	−0.0941655	−0.0470827	+	0.998891i	$0.514992\pi$
$60$	0	0
$61$	3.58006	0.458380	0.229190	−	0.973382i	$-0.426392\pi$
$61$	3.58006	0.458380	0.229190	+	0.973382i	$0.426392\pi$
$62$	4.53396	0.575813
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.94526	−0.985488
$66$	0	0
$67$	11.3440	1.38588	0.692942	−	0.720993i	$-0.256314\pi$
$67$	11.3440	1.38588	0.692942	+	0.720993i	$0.256314\pi$
$68$	−0.639779	−0.0775846
$69$	0	0
$70$	7.62059	0.910835
$71$	16.6286	1.97345	0.986727	−	0.162387i	$-0.0519194\pi$
$71$	16.6286	1.97345	0.986727	+	0.162387i	$0.0519194\pi$
$72$	0	0
$73$	−14.5294	−1.70054	−0.850270	−	0.526346i	$-0.823562\pi$
$73$	−14.5294	−1.70054	−0.850270	+	0.526346i	$0.823562\pi$
$74$	−3.32382	−0.386387
$75$	0	0
$76$	5.70570	0.654489
$77$	9.11882	1.03919
$78$	0	0
$79$	−14.8665	−1.67261	−0.836304	−	0.548266i	$-0.815288\pi$
$79$	−14.8665	−1.67261	−0.836304	+	0.548266i	$0.815288\pi$
$80$	2.72526	0.304693
$81$	0	0
$82$	8.09895	0.894379
$83$	−5.26159	−0.577534	−0.288767	−	0.957399i	$-0.593245\pi$
$83$	−5.26159	−0.577534	−0.288767	+	0.957399i	$0.593245\pi$
$84$	0	0
$85$	−1.74356	−0.189116
$86$	−11.8813	−1.28119
$87$	0	0
$88$	3.26105	0.347629
$89$	16.2962	1.72739	0.863697	−	0.504011i	$-0.168143\pi$
$89$	16.2962	1.72739	0.863697	+	0.504011i	$0.168143\pi$
$90$	0	0
$91$	8.15234	0.854597
$92$	1.32811	0.138465
$93$	0	0
$94$	5.42908	0.559967
$95$	15.5495	1.59534
$96$	0	0
$97$	−16.0563	−1.63027	−0.815137	−	0.579268i	$-0.803338\pi$
$97$	−16.0563	−1.63027	−0.815137	+	0.579268i	$0.803338\pi$
$98$	−0.819203	−0.0827520
$99$	0	0
$100$	2.42702	0.242702
$101$	−10.8277	−1.07739	−0.538697	−	0.842500i	$-0.681083\pi$
$101$	−10.8277	−1.07739	−0.538697	+	0.842500i	$0.681083\pi$
$102$	0	0
$103$	10.1784	1.00291	0.501454	−	0.865184i	$-0.332799\pi$
$103$	10.1784	1.00291	0.501454	+	0.865184i	$0.332799\pi$
$104$	2.91542	0.285880
$105$	0	0
$106$	−8.89160	−0.863628
$107$	4.43299	0.428553	0.214277	−	0.976773i	$-0.431261\pi$
$107$	4.43299	0.428553	0.214277	+	0.976773i	$0.431261\pi$
$108$	0	0
$109$	−19.8234	−1.89874	−0.949368	−	0.314166i	$-0.898275\pi$
$109$	−19.8234	−1.89874	−0.949368	+	0.314166i	$0.898275\pi$
$110$	8.88719	0.847360
$111$	0	0
$112$	−2.79628	−0.264224
$113$	11.3336	1.06618	0.533088	−	0.846060i	$-0.321031\pi$
$113$	11.3336	1.06618	0.533088	+	0.846060i	$0.321031\pi$
$114$	0	0
$115$	3.61944	0.337514
$116$	3.22092	0.299055
$117$	0	0
$118$	0.723299	0.0665851
$119$	1.78900	0.163998
$120$	0	0
$121$	−0.365561	−0.0332328
$122$	−3.58006	−0.324124
$123$	0	0
$124$	−4.53396	−0.407162
$125$	−7.01203	−0.627175
$126$	0	0
$127$	−3.89207	−0.345365	−0.172683	−	0.984978i	$-0.555243\pi$
$127$	−3.89207	−0.345365	−0.172683	+	0.984978i	$0.555243\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	7.94526	0.696845
$131$	−1.37955	−0.120532	−0.0602660	−	0.998182i	$-0.519195\pi$
$131$	−1.37955	−0.120532	−0.0602660	+	0.998182i	$0.519195\pi$
$132$	0	0
$133$	−15.9548	−1.38345
$134$	−11.3440	−0.979968
$135$	0	0
$136$	0.639779	0.0548606
$137$	−15.5769	−1.33082	−0.665411	−	0.746478i	$-0.731744\pi$
$137$	−15.5769	−1.33082	−0.665411	+	0.746478i	$0.731744\pi$
$138$	0	0
$139$	−3.71888	−0.315432	−0.157716	−	0.987485i	$-0.550413\pi$
$139$	−3.71888	−0.315432	−0.157716	+	0.987485i	$0.550413\pi$
$140$	−7.62059	−0.644057
$141$	0	0
$142$	−16.6286	−1.39544
$143$	9.50732	0.795042
$144$	0	0
$145$	8.77783	0.728959
$146$	14.5294	1.20246
$147$	0	0
$148$	3.32382	0.273217
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−3.55019	−0.288910	−0.144455	−	0.989511i	$-0.546143\pi$
$151$	−3.55019	−0.288910	−0.144455	+	0.989511i	$0.546143\pi$
$152$	−5.70570	−0.462794
$153$	0	0
$154$	−9.11882	−0.734815
$155$	−12.3562	−0.992474
$156$	0	0
$157$	−22.1681	−1.76921	−0.884603	−	0.466345i	$-0.845571\pi$
$157$	−22.1681	−1.76921	−0.884603	+	0.466345i	$0.845571\pi$
$158$	14.8665	1.18271
$159$	0	0
$160$	−2.72526	−0.215450
$161$	−3.71377	−0.292686
$162$	0	0
$163$	−16.1548	−1.26534	−0.632670	−	0.774422i	$-0.718041\pi$
$163$	−16.1548	−1.26534	−0.632670	+	0.774422i	$0.718041\pi$
$164$	−8.09895	−0.632422
$165$	0	0
$166$	5.26159	0.408379
$167$	16.0147	1.23926	0.619628	−	0.784896i	$-0.287284\pi$
$167$	16.0147	1.23926	0.619628	+	0.784896i	$0.287284\pi$
$168$	0	0
$169$	−4.50034	−0.346180
$170$	1.74356	0.133725
$171$	0	0
$172$	11.8813	0.905940
$173$	−25.3904	−1.93039	−0.965197	−	0.261523i	$-0.915775\pi$
$173$	−25.3904	−1.93039	−0.965197	+	0.261523i	$0.915775\pi$
$174$	0	0
$175$	−6.78663	−0.513021
$176$	−3.26105	−0.245811
$177$	0	0
$178$	−16.2962	−1.22145
$179$	−7.95101	−0.594286	−0.297143	−	0.954833i	$-0.596034\pi$
$179$	−7.95101	−0.594286	−0.297143	+	0.954833i	$0.596034\pi$
$180$	0	0
$181$	5.41497	0.402492	0.201246	−	0.979541i	$-0.435501\pi$
$181$	5.41497	0.402492	0.201246	+	0.979541i	$0.435501\pi$
$182$	−8.15234	−0.604291
$183$	0	0
$184$	−1.32811	−0.0979095
$185$	9.05827	0.665977
$186$	0	0
$187$	2.08635	0.152569
$188$	−5.42908	−0.395957
$189$	0	0
$190$	−15.5495	−1.12808
$191$	−8.24768	−0.596781	−0.298391	−	0.954444i	$-0.596450\pi$
$191$	−8.24768	−0.596781	−0.298391	+	0.954444i	$0.596450\pi$
$192$	0	0
$193$	3.95012	0.284336	0.142168	−	0.989843i	$-0.454593\pi$
$193$	3.95012	0.284336	0.142168	+	0.989843i	$0.454593\pi$
$194$	16.0563	1.15278
$195$	0	0
$196$	0.819203	0.0585145
$197$	−18.1572	−1.29364	−0.646822	−	0.762641i	$-0.723903\pi$
$197$	−18.1572	−1.29364	−0.646822	+	0.762641i	$0.723903\pi$
$198$	0	0
$199$	−8.03230	−0.569395	−0.284697	−	0.958617i	$-0.591893\pi$
$199$	−8.03230	−0.569395	−0.284697	+	0.958617i	$0.591893\pi$
$200$	−2.42702	−0.171616
$201$	0	0
$202$	10.8277	0.761833
$203$	−9.00661	−0.632140
$204$	0	0
$205$	−22.0717	−1.54155
$206$	−10.1784	−0.709163
$207$	0	0
$208$	−2.91542	−0.202148
$209$	−18.6066	−1.28704
$210$	0	0
$211$	−26.9038	−1.85213	−0.926065	−	0.377363i	$-0.876831\pi$
$211$	−26.9038	−1.85213	−0.926065	+	0.377363i	$0.876831\pi$
$212$	8.89160	0.610677
$213$	0	0
$214$	−4.43299	−0.303033
$215$	32.3796	2.20827
$216$	0	0
$217$	12.6782	0.860655
$218$	19.8234	1.34261
$219$	0	0
$220$	−8.88719	−0.599174
$221$	1.86522	0.125469
$222$	0	0
$223$	23.1746	1.55189	0.775945	−	0.630801i	$-0.217274\pi$
$223$	23.1746	1.55189	0.775945	+	0.630801i	$0.217274\pi$
$224$	2.79628	0.186835
$225$	0	0
$226$	−11.3336	−0.753901
$227$	−11.7427	−0.779388	−0.389694	−	0.920944i	$-0.627419\pi$
$227$	−11.7427	−0.779388	−0.389694	+	0.920944i	$0.627419\pi$
$228$	0	0
$229$	−13.1092	−0.866279	−0.433140	−	0.901327i	$-0.642594\pi$
$229$	−13.1092	−0.866279	−0.433140	+	0.901327i	$0.642594\pi$
$230$	−3.61944	−0.238659
$231$	0	0
$232$	−3.22092	−0.211464
$233$	4.00233	0.262201	0.131101	−	0.991369i	$-0.458149\pi$
$233$	4.00233	0.262201	0.131101	+	0.991369i	$0.458149\pi$
$234$	0	0
$235$	−14.7956	−0.965161
$236$	−0.723299	−0.0470827
$237$	0	0
$238$	−1.78900	−0.115964
$239$	−22.3878	−1.44815	−0.724074	−	0.689722i	$-0.757733\pi$
$239$	−22.3878	−1.44815	−0.724074	+	0.689722i	$0.757733\pi$
$240$	0	0
$241$	−22.8159	−1.46970	−0.734852	−	0.678228i	$-0.762748\pi$
$241$	−22.8159	−1.46970	−0.734852	+	0.678228i	$0.762748\pi$
$242$	0.365561	0.0234992
$243$	0	0
$244$	3.58006	0.229190
$245$	2.23254	0.142632
$246$	0	0
$247$	−16.6345	−1.05843
$248$	4.53396	0.287907
$249$	0	0
$250$	7.01203	0.443480
$251$	−12.2272	−0.771771	−0.385885	−	0.922547i	$-0.626104\pi$
$251$	−12.2272	−0.771771	−0.385885	+	0.922547i	$0.626104\pi$
$252$	0	0
$253$	−4.33103	−0.272289
$254$	3.89207	0.244210
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.64557	0.476918	0.238459	−	0.971153i	$-0.423358\pi$
$257$	7.64557	0.476918	0.238459	+	0.971153i	$0.423358\pi$
$258$	0	0
$259$	−9.29435	−0.577523
$260$	−7.94526	−0.492744
$261$	0	0
$262$	1.37955	0.0852290
$263$	23.0025	1.41839	0.709196	−	0.705011i	$-0.249058\pi$
$263$	23.0025	1.41839	0.709196	+	0.705011i	$0.249058\pi$
$264$	0	0
$265$	24.2319	1.48855
$266$	15.9548	0.978249
$267$	0	0
$268$	11.3440	0.692942
$269$	23.1234	1.40986	0.704929	−	0.709278i	$-0.250979\pi$
$269$	23.1234	1.40986	0.704929	+	0.709278i	$0.250979\pi$
$270$	0	0
$271$	−7.10401	−0.431538	−0.215769	−	0.976444i	$-0.569226\pi$
$271$	−7.10401	−0.431538	−0.215769	+	0.976444i	$0.569226\pi$
$272$	−0.639779	−0.0387923
$273$	0	0
$274$	15.5769	0.941033
$275$	−7.91463	−0.477270
$276$	0	0
$277$	−16.2288	−0.975096	−0.487548	−	0.873096i	$-0.662109\pi$
$277$	−16.2288	−0.975096	−0.487548	+	0.873096i	$0.662109\pi$
$278$	3.71888	0.223044
$279$	0	0
$280$	7.62059	0.455417
$281$	23.2687	1.38809	0.694047	−	0.719930i	$-0.255826\pi$
$281$	23.2687	1.38809	0.694047	+	0.719930i	$0.255826\pi$
$282$	0	0
$283$	4.13901	0.246038	0.123019	−	0.992404i	$-0.460742\pi$
$283$	4.13901	0.246038	0.123019	+	0.992404i	$0.460742\pi$
$284$	16.6286	0.986727
$285$	0	0
$286$	−9.50732	−0.562180
$287$	22.6470	1.33681
$288$	0	0
$289$	−16.5907	−0.975923
$290$	−8.77783	−0.515452
$291$	0	0
$292$	−14.5294	−0.850270
$293$	11.8774	0.693888	0.346944	−	0.937886i	$-0.387219\pi$
$293$	11.8774	0.693888	0.346944	+	0.937886i	$0.387219\pi$
$294$	0	0
$295$	−1.97117	−0.114766
$296$	−3.32382	−0.193193
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−3.87199	−0.223923
$300$	0	0
$301$	−33.2235	−1.91497
$302$	3.55019	0.204290
$303$	0	0
$304$	5.70570	0.327244
$305$	9.75659	0.558661
$306$	0	0
$307$	8.07771	0.461019	0.230510	−	0.973070i	$-0.425961\pi$
$307$	8.07771	0.461019	0.230510	+	0.973070i	$0.425961\pi$
$308$	9.11882	0.519593
$309$	0	0
$310$	12.3562	0.701785
$311$	−22.4413	−1.27253	−0.636266	−	0.771470i	$-0.719522\pi$
$311$	−22.4413	−1.27253	−0.636266	+	0.771470i	$0.719522\pi$
$312$	0	0
$313$	1.79049	0.101204	0.0506022	−	0.998719i	$-0.483886\pi$
$313$	1.79049	0.101204	0.0506022	+	0.998719i	$0.483886\pi$
$314$	22.1681	1.25102
$315$	0	0
$316$	−14.8665	−0.836304
$317$	11.1489	0.626185	0.313092	−	0.949723i	$-0.398635\pi$
$317$	11.1489	0.626185	0.313092	+	0.949723i	$0.398635\pi$
$318$	0	0
$319$	−10.5036	−0.588087
$320$	2.72526	0.152346
$321$	0	0
$322$	3.71377	0.206960
$323$	−3.65039	−0.203113
$324$	0	0
$325$	−7.07577	−0.392493
$326$	16.1548	0.894730
$327$	0	0
$328$	8.09895	0.447190
$329$	15.1813	0.836970
$330$	0	0
$331$	3.96274	0.217812	0.108906	−	0.994052i	$-0.465265\pi$
$331$	3.96274	0.217812	0.108906	+	0.994052i	$0.465265\pi$
$332$	−5.26159	−0.288767
$333$	0	0
$334$	−16.0147	−0.876286
$335$	30.9152	1.68908
$336$	0	0
$337$	−23.3207	−1.27036	−0.635179	−	0.772365i	$-0.719074\pi$
$337$	−23.3207	−1.27036	−0.635179	+	0.772365i	$0.719074\pi$
$338$	4.50034	0.244786
$339$	0	0
$340$	−1.74356	−0.0945579
$341$	14.7855	0.800678
$342$	0	0
$343$	17.2833	0.933208
$344$	−11.8813	−0.640596
$345$	0	0
$346$	25.3904	1.36500
$347$	−3.89827	−0.209270	−0.104635	−	0.994511i	$-0.533367\pi$
$347$	−3.89827	−0.209270	−0.104635	+	0.994511i	$0.533367\pi$
$348$	0	0
$349$	20.7995	1.11337	0.556686	−	0.830723i	$-0.312073\pi$
$349$	20.7995	1.11337	0.556686	+	0.830723i	$0.312073\pi$
$350$	6.78663	0.362761
$351$	0	0
$352$	3.26105	0.173814
$353$	−9.98456	−0.531424	−0.265712	−	0.964052i	$-0.585607\pi$
$353$	−9.98456	−0.531424	−0.265712	+	0.964052i	$0.585607\pi$
$354$	0	0
$355$	45.3172	2.40519
$356$	16.2962	0.863697
$357$	0	0
$358$	7.95101	0.420224
$359$	30.8076	1.62596	0.812982	−	0.582288i	$-0.197843\pi$
$359$	30.8076	1.62596	0.812982	+	0.582288i	$0.197843\pi$
$360$	0	0
$361$	13.5550	0.713423
$362$	−5.41497	−0.284605
$363$	0	0
$364$	8.15234	0.427299
$365$	−39.5964	−2.07257
$366$	0	0
$367$	−32.2179	−1.68176	−0.840880	−	0.541221i	$-0.817962\pi$
$367$	−32.2179	−1.68176	−0.840880	+	0.541221i	$0.817962\pi$
$368$	1.32811	0.0692325
$369$	0	0
$370$	−9.05827	−0.470917
$371$	−24.8634	−1.29084
$372$	0	0
$373$	−12.5090	−0.647691	−0.323845	−	0.946110i	$-0.604976\pi$
$373$	−12.5090	−0.647691	−0.323845	+	0.946110i	$0.604976\pi$
$374$	−2.08635	−0.107883
$375$	0	0
$376$	5.42908	0.279984
$377$	−9.39033	−0.483626
$378$	0	0
$379$	20.8180	1.06935	0.534675	−	0.845058i	$-0.320434\pi$
$379$	20.8180	1.06935	0.534675	+	0.845058i	$0.320434\pi$
$380$	15.5495	0.797672
$381$	0	0
$382$	8.24768	0.421988
$383$	38.2768	1.95585	0.977926	−	0.208952i	$-0.0670053\pi$
$383$	38.2768	1.95585	0.977926	+	0.208952i	$0.0670053\pi$
$384$	0	0
$385$	24.8511	1.26653
$386$	−3.95012	−0.201056
$387$	0	0
$388$	−16.0563	−0.815137
$389$	−20.5628	−1.04257	−0.521287	−	0.853382i	$-0.674548\pi$
$389$	−20.5628	−1.04257	−0.521287	+	0.853382i	$0.674548\pi$
$390$	0	0
$391$	−0.849696	−0.0429710
$392$	−0.819203	−0.0413760
$393$	0	0
$394$	18.1572	0.914745
$395$	−40.5149	−2.03853
$396$	0	0
$397$	−29.0661	−1.45879	−0.729394	−	0.684094i	$-0.760198\pi$
$397$	−29.0661	−1.45879	−0.729394	+	0.684094i	$0.760198\pi$
$398$	8.03230	0.402623
$399$	0	0
$400$	2.42702	0.121351
$401$	−3.25582	−0.162588	−0.0812940	−	0.996690i	$-0.525905\pi$
$401$	−3.25582	−0.162588	−0.0812940	+	0.996690i	$0.525905\pi$
$402$	0	0
$403$	13.2184	0.658455
$404$	−10.8277	−0.538697
$405$	0	0
$406$	9.00661	0.446990
$407$	−10.8391	−0.537277
$408$	0	0
$409$	39.5930	1.95775	0.978873	−	0.204471i	$-0.0655473\pi$
$409$	39.5930	1.95775	0.978873	+	0.204471i	$0.0655473\pi$
$410$	22.0717	1.09004
$411$	0	0
$412$	10.1784	0.501454
$413$	2.02255	0.0995231
$414$	0	0
$415$	−14.3392	−0.703882
$416$	2.91542	0.142940
$417$	0	0
$418$	18.6066	0.910077
$419$	−7.77544	−0.379855	−0.189927	−	0.981798i	$-0.560825\pi$
$419$	−7.77544	−0.379855	−0.189927	+	0.981798i	$0.560825\pi$
$420$	0	0
$421$	−25.7009	−1.25259	−0.626293	−	0.779587i	$-0.715429\pi$
$421$	−25.7009	−1.25259	−0.626293	+	0.779587i	$0.715429\pi$
$422$	26.9038	1.30965
$423$	0	0
$424$	−8.89160	−0.431814
$425$	−1.55276	−0.0753197
$426$	0	0
$427$	−10.0109	−0.484460
$428$	4.43299	0.214277
$429$	0	0
$430$	−32.3796	−1.56148
$431$	−19.0736	−0.918744	−0.459372	−	0.888244i	$-0.651925\pi$
$431$	−19.0736	−0.918744	−0.459372	+	0.888244i	$0.651925\pi$
$432$	0	0
$433$	18.1129	0.870451	0.435225	−	0.900321i	$-0.356669\pi$
$433$	18.1129	0.870451	0.435225	+	0.900321i	$0.356669\pi$
$434$	−12.6782	−0.608575
$435$	0	0
$436$	−19.8234	−0.949368
$437$	7.57779	0.362495
$438$	0	0
$439$	4.42989	0.211427	0.105714	−	0.994397i	$-0.466287\pi$
$439$	4.42989	0.211427	0.105714	+	0.994397i	$0.466287\pi$
$440$	8.88719	0.423680
$441$	0	0
$442$	−1.86522	−0.0887196
$443$	−22.6132	−1.07438	−0.537192	−	0.843460i	$-0.680515\pi$
$443$	−22.6132	−1.07438	−0.537192	+	0.843460i	$0.680515\pi$
$444$	0	0
$445$	44.4113	2.10530
$446$	−23.1746	−1.09735
$447$	0	0
$448$	−2.79628	−0.132112
$449$	23.4810	1.10814	0.554069	−	0.832470i	$-0.313074\pi$
$449$	23.4810	1.10814	0.554069	+	0.832470i	$0.313074\pi$
$450$	0	0
$451$	26.4111	1.24365
$452$	11.3336	0.533088
$453$	0	0
$454$	11.7427	0.551110
$455$	22.2172	1.04156
$456$	0	0
$457$	−10.3125	−0.482400	−0.241200	−	0.970475i	$-0.577541\pi$
$457$	−10.3125	−0.482400	−0.241200	+	0.970475i	$0.577541\pi$
$458$	13.1092	0.612552
$459$	0	0
$460$	3.61944	0.168757
$461$	28.1519	1.31116	0.655582	−	0.755124i	$-0.272424\pi$
$461$	28.1519	1.31116	0.655582	+	0.755124i	$0.272424\pi$
$462$	0	0
$463$	23.6846	1.10072	0.550358	−	0.834929i	$-0.314491\pi$
$463$	23.6846	1.10072	0.550358	+	0.834929i	$0.314491\pi$
$464$	3.22092	0.149527
$465$	0	0
$466$	−4.00233	−0.185404
$467$	17.9826	0.832133	0.416067	−	0.909334i	$-0.363408\pi$
$467$	17.9826	0.832133	0.416067	+	0.909334i	$0.363408\pi$
$468$	0	0
$469$	−31.7209	−1.46474
$470$	14.7956	0.682472
$471$	0	0
$472$	0.723299	0.0332925
$473$	−38.7455	−1.78152
$474$	0	0
$475$	13.8478	0.635383
$476$	1.78900	0.0819989
$477$	0	0
$478$	22.3878	1.02400
$479$	8.58630	0.392318	0.196159	−	0.980572i	$-0.437153\pi$
$479$	8.58630	0.392318	0.196159	+	0.980572i	$0.437153\pi$
$480$	0	0
$481$	−9.69033	−0.441841
$482$	22.8159	1.03924
$483$	0	0
$484$	−0.365561	−0.0166164
$485$	−43.7576	−1.98693
$486$	0	0
$487$	−4.72199	−0.213974	−0.106987	−	0.994260i	$-0.534120\pi$
$487$	−4.72199	−0.213974	−0.106987	+	0.994260i	$0.534120\pi$
$488$	−3.58006	−0.162062
$489$	0	0
$490$	−2.23254	−0.100856
$491$	2.83154	0.127786	0.0638929	−	0.997957i	$-0.479648\pi$
$491$	2.83154	0.127786	0.0638929	+	0.997957i	$0.479648\pi$
$492$	0	0
$493$	−2.06068	−0.0928082
$494$	16.6345	0.748422
$495$	0	0
$496$	−4.53396	−0.203581
$497$	−46.4983	−2.08574
$498$	0	0
$499$	−35.6782	−1.59718	−0.798588	−	0.601878i	$-0.794419\pi$
$499$	−35.6782	−1.59718	−0.798588	+	0.601878i	$0.794419\pi$
$500$	−7.01203	−0.313588
$501$	0	0
$502$	12.2272	0.545724
$503$	−27.2478	−1.21492	−0.607461	−	0.794350i	$-0.707812\pi$
$503$	−27.2478	−1.21492	−0.607461	+	0.794350i	$0.707812\pi$
$504$	0	0
$505$	−29.5082	−1.31310
$506$	4.33103	0.192538
$507$	0	0
$508$	−3.89207	−0.172683
$509$	1.94505	0.0862129	0.0431064	−	0.999070i	$-0.486275\pi$
$509$	1.94505	0.0862129	0.0431064	+	0.999070i	$0.486275\pi$
$510$	0	0
$511$	40.6284	1.79729
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−7.64557	−0.337232
$515$	27.7387	1.22232
$516$	0	0
$517$	17.7045	0.778643
$518$	9.29435	0.408370
$519$	0	0
$520$	7.94526	0.348423
$521$	−30.2679	−1.32606	−0.663031	−	0.748592i	$-0.730730\pi$
$521$	−30.2679	−1.32606	−0.663031	+	0.748592i	$0.730730\pi$
$522$	0	0
$523$	4.47550	0.195700	0.0978501	−	0.995201i	$-0.468803\pi$
$523$	4.47550	0.195700	0.0978501	+	0.995201i	$0.468803\pi$
$524$	−1.37955	−0.0602660
$525$	0	0
$526$	−23.0025	−1.00295
$527$	2.90073	0.126358
$528$	0	0
$529$	−21.2361	−0.923310
$530$	−24.2319	−1.05257
$531$	0	0
$532$	−15.9548	−0.691727
$533$	23.6118	1.02274
$534$	0	0
$535$	12.0810	0.522308
$536$	−11.3440	−0.489984
$537$	0	0
$538$	−23.1234	−0.996920
$539$	−2.67146	−0.115068
$540$	0	0
$541$	1.13187	0.0486630	0.0243315	−	0.999704i	$-0.492254\pi$
$541$	1.13187	0.0486630	0.0243315	+	0.999704i	$0.492254\pi$
$542$	7.10401	0.305143
$543$	0	0
$544$	0.639779	0.0274303
$545$	−54.0238	−2.31413
$546$	0	0
$547$	−20.4555	−0.874616	−0.437308	−	0.899312i	$-0.644068\pi$
$547$	−20.4555	−0.874616	−0.437308	+	0.899312i	$0.644068\pi$
$548$	−15.5769	−0.665411
$549$	0	0
$550$	7.91463	0.337481
$551$	18.3776	0.782913
$552$	0	0
$553$	41.5709	1.76777
$554$	16.2288	0.689497
$555$	0	0
$556$	−3.71888	−0.157716
$557$	−44.2761	−1.87604	−0.938020	−	0.346581i	$-0.887343\pi$
$557$	−44.2761	−1.87604	−0.938020	+	0.346581i	$0.887343\pi$
$558$	0	0
$559$	−34.6389	−1.46507
$560$	−7.62059	−0.322029
$561$	0	0
$562$	−23.2687	−0.981530
$563$	30.8672	1.30090	0.650449	−	0.759550i	$-0.274581\pi$
$563$	30.8672	1.30090	0.650449	+	0.759550i	$0.274581\pi$
$564$	0	0
$565$	30.8870	1.29943
$566$	−4.13901	−0.173975
$567$	0	0
$568$	−16.6286	−0.697721
$569$	−4.86745	−0.204054	−0.102027	−	0.994782i	$-0.532533\pi$
$569$	−4.86745	−0.204054	−0.102027	+	0.994782i	$0.532533\pi$
$570$	0	0
$571$	−27.5505	−1.15295	−0.576477	−	0.817113i	$-0.695573\pi$
$571$	−27.5505	−1.15295	−0.576477	+	0.817113i	$0.695573\pi$
$572$	9.50732	0.397521
$573$	0	0
$574$	−22.6470	−0.945266
$575$	3.22335	0.134423
$576$	0	0
$577$	27.4508	1.14279	0.571397	−	0.820674i	$-0.306402\pi$
$577$	27.4508	1.14279	0.571397	+	0.820674i	$0.306402\pi$
$578$	16.5907	0.690081
$579$	0	0
$580$	8.77783	0.364480
$581$	14.7129	0.610394
$582$	0	0
$583$	−28.9959	−1.20089
$584$	14.5294	0.601232
$585$	0	0
$586$	−11.8774	−0.490653
$587$	8.74701	0.361028	0.180514	−	0.983572i	$-0.442224\pi$
$587$	8.74701	0.361028	0.180514	+	0.983572i	$0.442224\pi$
$588$	0	0
$589$	−25.8694	−1.06593
$590$	1.97117	0.0811520
$591$	0	0
$592$	3.32382	0.136608
$593$	46.3617	1.90385	0.951924	−	0.306334i	$-0.0991023\pi$
$593$	46.3617	1.90385	0.951924	+	0.306334i	$0.0991023\pi$
$594$	0	0
$595$	4.87549	0.199876
$596$	1.00000	0.0409616
$597$	0	0
$598$	3.87199	0.158338
$599$	−6.12967	−0.250451	−0.125226	−	0.992128i	$-0.539965\pi$
$599$	−6.12967	−0.250451	−0.125226	+	0.992128i	$0.539965\pi$
$600$	0	0
$601$	1.43528	0.0585463	0.0292732	−	0.999571i	$-0.490681\pi$
$601$	1.43528	0.0585463	0.0292732	+	0.999571i	$0.490681\pi$
$602$	33.2235	1.35409
$603$	0	0
$604$	−3.55019	−0.144455
$605$	−0.996248	−0.0405032
$606$	0	0
$607$	20.8849	0.847693	0.423847	−	0.905734i	$-0.360680\pi$
$607$	20.8849	0.847693	0.423847	+	0.905734i	$0.360680\pi$
$608$	−5.70570	−0.231397
$609$	0	0
$610$	−9.75659	−0.395033
$611$	15.8280	0.640334
$612$	0	0
$613$	−37.5686	−1.51738	−0.758691	−	0.651451i	$-0.774161\pi$
$613$	−37.5686	−1.51738	−0.758691	+	0.651451i	$0.774161\pi$
$614$	−8.07771	−0.325990
$615$	0	0
$616$	−9.11882	−0.367408
$617$	4.84106	0.194894	0.0974468	−	0.995241i	$-0.468932\pi$
$617$	4.84106	0.194894	0.0974468	+	0.995241i	$0.468932\pi$
$618$	0	0
$619$	−7.70576	−0.309721	−0.154860	−	0.987936i	$-0.549493\pi$
$619$	−7.70576	−0.309721	−0.154860	+	0.987936i	$0.549493\pi$
$620$	−12.3562	−0.496237
$621$	0	0
$622$	22.4413	0.899815
$623$	−45.5688	−1.82568
$624$	0	0
$625$	−31.2447	−1.24979
$626$	−1.79049	−0.0715624
$627$	0	0
$628$	−22.1681	−0.884603
$629$	−2.12651	−0.0847896
$630$	0	0
$631$	−2.62695	−0.104577	−0.0522887	−	0.998632i	$-0.516652\pi$
$631$	−2.62695	−0.104577	−0.0522887	+	0.998632i	$0.516652\pi$
$632$	14.8665	0.591356
$633$	0	0
$634$	−11.1489	−0.442779
$635$	−10.6069	−0.420921
$636$	0	0
$637$	−2.38832	−0.0946287
$638$	10.5036	0.415841
$639$	0	0
$640$	−2.72526	−0.107725
$641$	−22.1313	−0.874132	−0.437066	−	0.899429i	$-0.643982\pi$
$641$	−22.1313	−0.874132	−0.437066	+	0.899429i	$0.643982\pi$
$642$	0	0
$643$	0.528582	0.0208452	0.0104226	−	0.999946i	$-0.496682\pi$
$643$	0.528582	0.0208452	0.0104226	+	0.999946i	$0.496682\pi$
$644$	−3.71377	−0.146343
$645$	0	0
$646$	3.65039	0.143623
$647$	1.05402	0.0414377	0.0207189	−	0.999785i	$-0.493405\pi$
$647$	1.05402	0.0414377	0.0207189	+	0.999785i	$0.493405\pi$
$648$	0	0
$649$	2.35871	0.0925876
$650$	7.07577	0.277535
$651$	0	0
$652$	−16.1548	−0.632670
$653$	12.8500	0.502860	0.251430	−	0.967876i	$-0.419099\pi$
$653$	12.8500	0.502860	0.251430	+	0.967876i	$0.419099\pi$
$654$	0	0
$655$	−3.75963	−0.146901
$656$	−8.09895	−0.316211
$657$	0	0
$658$	−15.1813	−0.591827
$659$	3.36625	0.131130	0.0655652	−	0.997848i	$-0.479115\pi$
$659$	3.36625	0.131130	0.0655652	+	0.997848i	$0.479115\pi$
$660$	0	0
$661$	−29.1141	−1.13241	−0.566203	−	0.824266i	$-0.691588\pi$
$661$	−29.1141	−1.13241	−0.566203	+	0.824266i	$0.691588\pi$
$662$	−3.96274	−0.154016
$663$	0	0
$664$	5.26159	0.204189
$665$	−43.4808	−1.68611
$666$	0	0
$667$	4.27773	0.165634
$668$	16.0147	0.619628
$669$	0	0
$670$	−30.9152	−1.19436
$671$	−11.6748	−0.450699
$672$	0	0
$673$	14.9029	0.574463	0.287232	−	0.957861i	$-0.407265\pi$
$673$	14.9029	0.574463	0.287232	+	0.957861i	$0.407265\pi$
$674$	23.3207	0.898279
$675$	0	0
$676$	−4.50034	−0.173090
$677$	−21.7641	−0.836462	−0.418231	−	0.908341i	$-0.637350\pi$
$677$	−21.7641	−0.836462	−0.418231	+	0.908341i	$0.637350\pi$
$678$	0	0
$679$	44.8981	1.72303
$680$	1.74356	0.0668625
$681$	0	0
$682$	−14.7855	−0.566165
$683$	22.9350	0.877582	0.438791	−	0.898589i	$-0.355407\pi$
$683$	22.9350	0.877582	0.438791	+	0.898589i	$0.355407\pi$
$684$	0	0
$685$	−42.4509	−1.62197
$686$	−17.2833	−0.659878
$687$	0	0
$688$	11.8813	0.452970
$689$	−25.9227	−0.987577
$690$	0	0
$691$	27.1459	1.03268	0.516339	−	0.856384i	$-0.327294\pi$
$691$	27.1459	1.03268	0.516339	+	0.856384i	$0.327294\pi$
$692$	−25.3904	−0.965197
$693$	0	0
$694$	3.89827	0.147976
$695$	−10.1349	−0.384439
$696$	0	0
$697$	5.18154	0.196265
$698$	−20.7995	−0.787273
$699$	0	0
$700$	−6.78663	−0.256511
$701$	11.2381	0.424459	0.212229	−	0.977220i	$-0.431928\pi$
$701$	11.2381	0.424459	0.212229	+	0.977220i	$0.431928\pi$
$702$	0	0
$703$	18.9647	0.715269
$704$	−3.26105	−0.122905
$705$	0	0
$706$	9.98456	0.375774
$707$	30.2773	1.13869
$708$	0	0
$709$	32.9160	1.23619	0.618094	−	0.786105i	$-0.287905\pi$
$709$	32.9160	1.23619	0.618094	+	0.786105i	$0.287905\pi$
$710$	−45.3172	−1.70073
$711$	0	0
$712$	−16.2962	−0.610726
$713$	−6.02159	−0.225510
$714$	0	0
$715$	25.9099	0.968975
$716$	−7.95101	−0.297143
$717$	0	0
$718$	−30.8076	−1.14973
$719$	−18.7416	−0.698942	−0.349471	−	0.936947i	$-0.613639\pi$
$719$	−18.7416	−0.698942	−0.349471	+	0.936947i	$0.613639\pi$
$720$	0	0
$721$	−28.4617	−1.05997
$722$	−13.5550	−0.504466
$723$	0	0
$724$	5.41497	0.201246
$725$	7.81723	0.290325
$726$	0	0
$727$	45.3511	1.68198	0.840990	−	0.541051i	$-0.181973\pi$
$727$	45.3511	1.68198	0.840990	+	0.541051i	$0.181973\pi$
$728$	−8.15234	−0.302146
$729$	0	0
$730$	39.5964	1.46553
$731$	−7.60140	−0.281148
$732$	0	0
$733$	52.6932	1.94627	0.973133	−	0.230243i	$-0.0739523\pi$
$733$	52.6932	1.94627	0.973133	+	0.230243i	$0.0739523\pi$
$734$	32.2179	1.18918
$735$	0	0
$736$	−1.32811	−0.0489547
$737$	−36.9932	−1.36266
$738$	0	0
$739$	−22.6834	−0.834421	−0.417211	−	0.908810i	$-0.636992\pi$
$739$	−22.6834	−0.834421	−0.417211	+	0.908810i	$0.636992\pi$
$740$	9.05827	0.332989
$741$	0	0
$742$	24.8634	0.912765
$743$	17.9688	0.659211	0.329606	−	0.944119i	$-0.393084\pi$
$743$	17.9688	0.659211	0.329606	+	0.944119i	$0.393084\pi$
$744$	0	0
$745$	2.72526	0.0998456
$746$	12.5090	0.457986
$747$	0	0
$748$	2.08635	0.0762845
$749$	−12.3959	−0.452936
$750$	0	0
$751$	0.566495	0.0206717	0.0103359	−	0.999947i	$-0.496710\pi$
$751$	0.566495	0.0206717	0.0103359	+	0.999947i	$0.496710\pi$
$752$	−5.42908	−0.197978
$753$	0	0
$754$	9.39033	0.341976
$755$	−9.67518	−0.352116
$756$	0	0
$757$	26.4143	0.960042	0.480021	−	0.877257i	$-0.340629\pi$
$757$	26.4143	0.960042	0.480021	+	0.877257i	$0.340629\pi$
$758$	−20.8180	−0.756144
$759$	0	0
$760$	−15.5495	−0.564040
$761$	2.85181	0.103378	0.0516890	−	0.998663i	$-0.483540\pi$
$761$	2.85181	0.103378	0.0516890	+	0.998663i	$0.483540\pi$
$762$	0	0
$763$	55.4318	2.00677
$764$	−8.24768	−0.298391
$765$	0	0
$766$	−38.2768	−1.38300
$767$	2.10872	0.0761414
$768$	0	0
$769$	10.3071	0.371683	0.185842	−	0.982580i	$-0.440499\pi$
$769$	10.3071	0.371683	0.185842	+	0.982580i	$0.440499\pi$
$770$	−24.8511	−0.895572
$771$	0	0
$772$	3.95012	0.142168
$773$	−7.81989	−0.281262	−0.140631	−	0.990062i	$-0.544913\pi$
$773$	−7.81989	−0.281262	−0.140631	+	0.990062i	$0.544913\pi$
$774$	0	0
$775$	−11.0040	−0.395276
$776$	16.0563	0.576389
$777$	0	0
$778$	20.5628	0.737210
$779$	−46.2102	−1.65565
$780$	0	0
$781$	−54.2267	−1.94039
$782$	0.849696	0.0303851
$783$	0	0
$784$	0.819203	0.0292573
$785$	−60.4137	−2.15626
$786$	0	0
$787$	2.48811	0.0886915	0.0443458	−	0.999016i	$-0.485880\pi$
$787$	2.48811	0.0886915	0.0443458	+	0.999016i	$0.485880\pi$
$788$	−18.1572	−0.646822
$789$	0	0
$790$	40.5149	1.44146
$791$	−31.6920	−1.12684
$792$	0	0
$793$	−10.4374	−0.370642
$794$	29.0661	1.03152
$795$	0	0
$796$	−8.03230	−0.284697
$797$	−2.20148	−0.0779805	−0.0389902	−	0.999240i	$-0.512414\pi$
$797$	−2.20148	−0.0779805	−0.0389902	+	0.999240i	$0.512414\pi$
$798$	0	0
$799$	3.47341	0.122881
$800$	−2.42702	−0.0858081
$801$	0	0
$802$	3.25582	0.114967
$803$	47.3812	1.67205
$804$	0	0
$805$	−10.1210	−0.356717
$806$	−13.2184	−0.465598
$807$	0	0
$808$	10.8277	0.380916
$809$	−31.9340	−1.12274	−0.561369	−	0.827565i	$-0.689725\pi$
$809$	−31.9340	−1.12274	−0.561369	+	0.827565i	$0.689725\pi$
$810$	0	0
$811$	−24.9375	−0.875675	−0.437837	−	0.899054i	$-0.644255\pi$
$811$	−24.9375	−0.875675	−0.437837	+	0.899054i	$0.644255\pi$
$812$	−9.00661	−0.316070
$813$	0	0
$814$	10.8391	0.379912
$815$	−44.0259	−1.54216
$816$	0	0
$817$	67.7911	2.37171
$818$	−39.5930	−1.38434
$819$	0	0
$820$	−22.0717	−0.770777
$821$	−53.9606	−1.88324	−0.941619	−	0.336681i	$-0.890696\pi$
$821$	−53.9606	−1.88324	−0.941619	+	0.336681i	$0.890696\pi$
$822$	0	0
$823$	−4.62304	−0.161149	−0.0805745	−	0.996749i	$-0.525675\pi$
$823$	−4.62304	−0.161149	−0.0805745	+	0.996749i	$0.525675\pi$
$824$	−10.1784	−0.354581
$825$	0	0
$826$	−2.02255	−0.0703735
$827$	23.5970	0.820549	0.410274	−	0.911962i	$-0.365433\pi$
$827$	23.5970	0.820549	0.410274	+	0.911962i	$0.365433\pi$
$828$	0	0
$829$	−41.9815	−1.45808	−0.729039	−	0.684472i	$-0.760033\pi$
$829$	−41.9815	−1.45808	−0.729039	+	0.684472i	$0.760033\pi$
$830$	14.3392	0.497720
$831$	0	0
$832$	−2.91542	−0.101074
$833$	−0.524109	−0.0181593
$834$	0	0
$835$	43.6442	1.51037
$836$	−18.6066	−0.643522
$837$	0	0
$838$	7.77544	0.268598
$839$	10.7147	0.369912	0.184956	−	0.982747i	$-0.440786\pi$
$839$	10.7147	0.369912	0.184956	+	0.982747i	$0.440786\pi$
$840$	0	0
$841$	−18.6257	−0.642265
$842$	25.7009	0.885713
$843$	0	0
$844$	−26.9038	−0.926065
$845$	−12.2646	−0.421914
$846$	0	0
$847$	1.02221	0.0351237
$848$	8.89160	0.305339
$849$	0	0
$850$	1.55276	0.0532591
$851$	4.41440	0.151324
$852$	0	0
$853$	33.7916	1.15700	0.578500	−	0.815682i	$-0.303638\pi$
$853$	33.7916	1.15700	0.578500	+	0.815682i	$0.303638\pi$
$854$	10.0109	0.342565
$855$	0	0
$856$	−4.43299	−0.151516
$857$	−30.3647	−1.03724	−0.518619	−	0.855005i	$-0.673554\pi$
$857$	−30.3647	−1.03724	−0.518619	+	0.855005i	$0.673554\pi$
$858$	0	0
$859$	−10.3030	−0.351533	−0.175767	−	0.984432i	$-0.556240\pi$
$859$	−10.3030	−0.351533	−0.175767	+	0.984432i	$0.556240\pi$
$860$	32.3796	1.10413
$861$	0	0
$862$	19.0736	0.649650
$863$	16.8673	0.574168	0.287084	−	0.957905i	$-0.407314\pi$
$863$	16.8673	0.574168	0.287084	+	0.957905i	$0.407314\pi$
$864$	0	0
$865$	−69.1953	−2.35271
$866$	−18.1129	−0.615502
$867$	0	0
$868$	12.6782	0.430327
$869$	48.4803	1.64458
$870$	0	0
$871$	−33.0724	−1.12061
$872$	19.8234	0.671305
$873$	0	0
$874$	−7.57779	−0.256323
$875$	19.6076	0.662859
$876$	0	0
$877$	27.9275	0.943044	0.471522	−	0.881854i	$-0.343705\pi$
$877$	27.9275	0.943044	0.471522	+	0.881854i	$0.343705\pi$
$878$	−4.42989	−0.149502
$879$	0	0
$880$	−8.88719	−0.299587
$881$	37.2411	1.25468	0.627342	−	0.778744i	$-0.284143\pi$
$881$	37.2411	1.25468	0.627342	+	0.778744i	$0.284143\pi$
$882$	0	0
$883$	19.0955	0.642616	0.321308	−	0.946975i	$-0.395878\pi$
$883$	19.0955	0.642616	0.321308	+	0.946975i	$0.395878\pi$
$884$	1.86522	0.0627343
$885$	0	0
$886$	22.6132	0.759705
$887$	−46.8091	−1.57169	−0.785847	−	0.618420i	$-0.787773\pi$
$887$	−46.8091	−1.57169	−0.785847	+	0.618420i	$0.787773\pi$
$888$	0	0
$889$	10.8833	0.365015
$890$	−44.4113	−1.48867
$891$	0	0
$892$	23.1746	0.775945
$893$	−30.9767	−1.03660
$894$	0	0
$895$	−21.6685	−0.724299
$896$	2.79628	0.0934173
$897$	0	0
$898$	−23.4810	−0.783573
$899$	−14.6035	−0.487055
$900$	0	0
$901$	−5.68866	−0.189517
$902$	−26.4111	−0.879392
$903$	0	0
$904$	−11.3336	−0.376950
$905$	14.7572	0.490545
$906$	0	0
$907$	−21.2858	−0.706784	−0.353392	−	0.935475i	$-0.614972\pi$
$907$	−21.2858	−0.706784	−0.353392	+	0.935475i	$0.614972\pi$
$908$	−11.7427	−0.389694
$909$	0	0
$910$	−22.2172	−0.736493
$911$	44.5040	1.47448	0.737241	−	0.675629i	$-0.236128\pi$
$911$	44.5040	1.47448	0.737241	+	0.675629i	$0.236128\pi$
$912$	0	0
$913$	17.1583	0.567857
$914$	10.3125	0.341108
$915$	0	0
$916$	−13.1092	−0.433140
$917$	3.85762	0.127390
$918$	0	0
$919$	−35.3524	−1.16617	−0.583083	−	0.812412i	$-0.698154\pi$
$919$	−35.3524	−1.16617	−0.583083	+	0.812412i	$0.698154\pi$
$920$	−3.61944	−0.119329
$921$	0	0
$922$	−28.1519	−0.927132
$923$	−48.4794	−1.59572
$924$	0	0
$925$	8.06698	0.265241
$926$	−23.6846	−0.778323
$927$	0	0
$928$	−3.22092	−0.105732
$929$	−44.0940	−1.44668	−0.723338	−	0.690494i	$-0.757393\pi$
$929$	−44.0940	−1.44668	−0.723338	+	0.690494i	$0.757393\pi$
$930$	0	0
$931$	4.67413	0.153188
$932$	4.00233	0.131101
$933$	0	0
$934$	−17.9826	−0.588407
$935$	5.68584	0.185947
$936$	0	0
$937$	5.16133	0.168613	0.0843066	−	0.996440i	$-0.473132\pi$
$937$	5.16133	0.168613	0.0843066	+	0.996440i	$0.473132\pi$
$938$	31.7209	1.03572
$939$	0	0
$940$	−14.7956	−0.482581
$941$	25.8702	0.843344	0.421672	−	0.906748i	$-0.361443\pi$
$941$	25.8702	0.843344	0.421672	+	0.906748i	$0.361443\pi$
$942$	0	0
$943$	−10.7563	−0.350273
$944$	−0.723299	−0.0235414
$945$	0	0
$946$	38.7455	1.25972
$947$	−14.1037	−0.458308	−0.229154	−	0.973390i	$-0.573596\pi$
$947$	−14.1037	−0.458308	−0.229154	+	0.973390i	$0.573596\pi$
$948$	0	0
$949$	42.3594	1.37504
$950$	−13.8478	−0.449284
$951$	0	0
$952$	−1.78900	−0.0579820
$953$	−12.3360	−0.399603	−0.199801	−	0.979836i	$-0.564030\pi$
$953$	−12.3360	−0.399603	−0.199801	+	0.979836i	$0.564030\pi$
$954$	0	0
$955$	−22.4770	−0.727340
$956$	−22.3878	−0.724074
$957$	0	0
$958$	−8.58630	−0.277411
$959$	43.5573	1.40654
$960$	0	0
$961$	−10.4432	−0.336878
$962$	9.69033	0.312429
$963$	0	0
$964$	−22.8159	−0.734852
$965$	10.7651	0.346541
$966$	0	0
$967$	−16.0689	−0.516740	−0.258370	−	0.966046i	$-0.583185\pi$
$967$	−16.0689	−0.516740	−0.258370	+	0.966046i	$0.583185\pi$
$968$	0.365561	0.0117496
$969$	0	0
$970$	43.7576	1.40497
$971$	−51.5433	−1.65410	−0.827051	−	0.562126i	$-0.809983\pi$
$971$	−51.5433	−1.65410	−0.827051	+	0.562126i	$0.809983\pi$
$972$	0	0
$973$	10.3991	0.333378
$974$	4.72199	0.151302
$975$	0	0
$976$	3.58006	0.114595
$977$	−47.6254	−1.52367	−0.761836	−	0.647770i	$-0.775702\pi$
$977$	−47.6254	−1.52367	−0.761836	+	0.647770i	$0.775702\pi$
$978$	0	0
$979$	−53.1427	−1.69845
$980$	2.23254	0.0713158
$981$	0	0
$982$	−2.83154	−0.0903582
$983$	24.2854	0.774585	0.387293	−	0.921957i	$-0.373410\pi$
$983$	24.2854	0.774585	0.387293	+	0.921957i	$0.373410\pi$
$984$	0	0
$985$	−49.4829	−1.57666
$986$	2.06068	0.0656253
$987$	0	0
$988$	−16.6345	−0.529214
$989$	15.7796	0.501764
$990$	0	0
$991$	16.8119	0.534048	0.267024	−	0.963690i	$-0.413960\pi$
$991$	16.8119	0.534048	0.267024	+	0.963690i	$0.413960\pi$
$992$	4.53396	0.143953
$993$	0	0
$994$	46.4983	1.47484
$995$	−21.8901	−0.693962
$996$	0	0
$997$	33.9833	1.07626	0.538132	−	0.842861i	$-0.319130\pi$
$997$	33.9833	1.07626	0.538132	+	0.842861i	$0.319130\pi$
$998$	35.6782	1.12937
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.g.1.8	✓	9
3.2	odd	2		8046.2.a.h.1.2	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.g.1.8	✓	9	1.1	even	1	trivial
8046.2.a.h.1.2	yes	9	3.2	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.72526 q^{5} -2.79628 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.72526 q^{5} -2.79628 q^{7} -1.00000 q^{8} -2.72526 q^{10} -3.26105 q^{11} -2.91542 q^{13} +2.79628 q^{14} +1.00000 q^{16} -0.639779 q^{17} +5.70570 q^{19} +2.72526 q^{20} +3.26105 q^{22} +1.32811 q^{23} +2.42702 q^{25} +2.91542 q^{26} -2.79628 q^{28} +3.22092 q^{29} -4.53396 q^{31} -1.00000 q^{32} +0.639779 q^{34} -7.62059 q^{35} +3.32382 q^{37} -5.70570 q^{38} -2.72526 q^{40} -8.09895 q^{41} +11.8813 q^{43} -3.26105 q^{44} -1.32811 q^{46} -5.42908 q^{47} +0.819203 q^{49} -2.42702 q^{50} -2.91542 q^{52} +8.89160 q^{53} -8.88719 q^{55} +2.79628 q^{56} -3.22092 q^{58} -0.723299 q^{59} +3.58006 q^{61} +4.53396 q^{62} +1.00000 q^{64} -7.94526 q^{65} +11.3440 q^{67} -0.639779 q^{68} +7.62059 q^{70} +16.6286 q^{71} -14.5294 q^{73} -3.32382 q^{74} +5.70570 q^{76} +9.11882 q^{77} -14.8665 q^{79} +2.72526 q^{80} +8.09895 q^{82} -5.26159 q^{83} -1.74356 q^{85} -11.8813 q^{86} +3.26105 q^{88} +16.2962 q^{89} +8.15234 q^{91} +1.32811 q^{92} +5.42908 q^{94} +15.5495 q^{95} -16.0563 q^{97} -0.819203 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) \) 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8	\( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8} - 4 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 9 q^{16} + q^{17} - 10 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 3 q^{25} + 8 q^{26} - 4 q^{28} + 4 q^{29} - 17 q^{31} - 9 q^{32} - q^{34} + 10 q^{35} - 11 q^{37} + 10 q^{38} - 4 q^{40} - 16 q^{43} + 4 q^{44} - 8 q^{46} + 7 q^{47} - 5 q^{49} + 3 q^{50} - 8 q^{52} + 12 q^{53} - 23 q^{55} + 4 q^{56} - 4 q^{58} + 6 q^{59} - 13 q^{61} + 17 q^{62} + 9 q^{64} - 24 q^{65} - 14 q^{67} + q^{68} - 10 q^{70} + 30 q^{71} - 12 q^{73} + 11 q^{74} - 10 q^{76} + 12 q^{77} - 35 q^{79} + 4 q^{80} + 5 q^{83} - 27 q^{85} + 16 q^{86} - 4 q^{88} + 23 q^{89} - 28 q^{91} + 8 q^{92} - 7 q^{94} + 32 q^{95} - 21 q^{97} + 5 q^{98}+O(q^{100}) \) 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8 - 4 * q^10 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 9 * q^16 + q^17 - 10 * q^19 + 4 * q^20 - 4 * q^22 + 8 * q^23 - 3 * q^25 + 8 * q^26 - 4 * q^28 + 4 * q^29 - 17 * q^31 - 9 * q^32 - q^34 + 10 * q^35 - 11 * q^37 + 10 * q^38 - 4 * q^40 - 16 * q^43 + 4 * q^44 - 8 * q^46 + 7 * q^47 - 5 * q^49 + 3 * q^50 - 8 * q^52 + 12 * q^53 - 23 * q^55 + 4 * q^56 - 4 * q^58 + 6 * q^59 - 13 * q^61 + 17 * q^62 + 9 * q^64 - 24 * q^65 - 14 * q^67 + q^68 - 10 * q^70 + 30 * q^71 - 12 * q^73 + 11 * q^74 - 10 * q^76 + 12 * q^77 - 35 * q^79 + 4 * q^80 + 5 * q^83 - 27 * q^85 + 16 * q^86 - 4 * q^88 + 23 * q^89 - 28 * q^91 + 8 * q^92 - 7 * q^94 + 32 * q^95 - 21 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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