LMFDB - Embedded newform 7865.2.a.u.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7865,2,Mod(1,7865)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7865, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7865.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7865 = 5 \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7865.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,1,-5,7,-8,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$62.8023411897$
Analytic rank:	$0$
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} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 13x^{3} - 35x^{2} - 3x + 5 $ x^8 - x^7 - 11x^6 + 9x^5 + 34x^4 - 13x^3 - 35x^2 - 3x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.336128$ of defining polynomial
Character	$\chi$	$=$	7865.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.336128 q^{2} -2.12071 q^{3} -1.88702 q^{4} -1.00000 q^{5} -0.712829 q^{6} -1.25721 q^{7} -1.30653 q^{8} +1.49741 q^{9} -0.336128 q^{10} +4.00182 q^{12} -1.00000 q^{13} -0.422583 q^{14} +2.12071 q^{15} +3.33487 q^{16} +1.59876 q^{17} +0.503321 q^{18} +6.34435 q^{19} +1.88702 q^{20} +2.66618 q^{21} +2.01720 q^{23} +2.77078 q^{24} +1.00000 q^{25} -0.336128 q^{26} +3.18655 q^{27} +2.37238 q^{28} -3.54131 q^{29} +0.712829 q^{30} -4.93400 q^{31} +3.73401 q^{32} +0.537387 q^{34} +1.25721 q^{35} -2.82564 q^{36} -2.92059 q^{37} +2.13251 q^{38} +2.12071 q^{39} +1.30653 q^{40} +0.127199 q^{41} +0.896177 q^{42} -6.69458 q^{43} -1.49741 q^{45} +0.678035 q^{46} -6.75748 q^{47} -7.07230 q^{48} -5.41942 q^{49} +0.336128 q^{50} -3.39050 q^{51} +1.88702 q^{52} +4.99469 q^{53} +1.07109 q^{54} +1.64259 q^{56} -13.4545 q^{57} -1.19033 q^{58} -12.9532 q^{59} -4.00182 q^{60} +2.78047 q^{61} -1.65845 q^{62} -1.88256 q^{63} -5.41464 q^{64} +1.00000 q^{65} -0.638510 q^{67} -3.01689 q^{68} -4.27789 q^{69} +0.422583 q^{70} +6.26909 q^{71} -1.95642 q^{72} -5.31914 q^{73} -0.981692 q^{74} -2.12071 q^{75} -11.9719 q^{76} +0.712829 q^{78} -14.3733 q^{79} -3.33487 q^{80} -11.2500 q^{81} +0.0427550 q^{82} -9.60590 q^{83} -5.03113 q^{84} -1.59876 q^{85} -2.25023 q^{86} +7.51010 q^{87} +7.67233 q^{89} -0.503321 q^{90} +1.25721 q^{91} -3.80648 q^{92} +10.4636 q^{93} -2.27138 q^{94} -6.34435 q^{95} -7.91876 q^{96} +9.78708 q^{97} -1.82162 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} - 5 q^{3} + 7 q^{4} - 8 q^{5} - 7 q^{6} + 9 q^{7} + 3 q^{8} + 3 q^{9} - q^{10} - 15 q^{12} - 8 q^{13} - 7 q^{14} + 5 q^{15} + 9 q^{16} + 4 q^{17} + 8 q^{18} + 16 q^{19} - 7 q^{20} - 23 q^{21}+ \cdots + 41 q^{98}+O(q^{100}) $ 8 * q + q^2 - 5 * q^3 + 7 * q^4 - 8 * q^5 - 7 * q^6 + 9 * q^7 + 3 * q^8 + 3 * q^9 - q^10 - 15 * q^12 - 8 * q^13 - 7 * q^14 + 5 * q^15 + 9 * q^16 + 4 * q^17 + 8 * q^18 + 16 * q^19 - 7 * q^20 - 23 * q^21 - 9 * q^23 + 5 * q^24 + 8 * q^25 - q^26 - 11 * q^27 + 11 * q^28 - 13 * q^29 + 7 * q^30 - 28 * q^32 + 13 * q^34 - 9 * q^35 + 12 * q^36 - 5 * q^37 + 55 * q^38 + 5 * q^39 - 3 * q^40 + 13 * q^41 - 5 * q^42 + 5 * q^43 - 3 * q^45 + 27 * q^46 + 10 * q^47 - 36 * q^48 + 23 * q^49 + q^50 - 37 * q^51 - 7 * q^52 + 14 * q^53 - 45 * q^54 - 18 * q^56 - 21 * q^57 + 3 * q^58 - 20 * q^59 + 15 * q^60 + 6 * q^61 + 25 * q^62 + 26 * q^63 - 3 * q^64 + 8 * q^65 + 4 * q^67 - 9 * q^69 + 7 * q^70 - 9 * q^71 + 35 * q^72 + 10 * q^73 + 42 * q^74 - 5 * q^75 + 19 * q^76 + 7 * q^78 + 13 * q^79 - 9 * q^80 + 16 * q^82 + 26 * q^83 - 58 * q^84 - 4 * q^85 + 26 * q^86 + 11 * q^87 - 6 * q^89 - 8 * q^90 - 9 * q^91 - 16 * q^92 + 3 * q^93 + 23 * q^94 - 16 * q^95 + 21 * q^96 + 26 * q^97 + 41 * q^98

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 $

$2$	0.336128	0.237678	0.118839	−	0.992914i	$-0.462083\pi$
$2$	0.336128	0.237678	0.118839	+	0.992914i	$0.462083\pi$
$3$	−2.12071	−1.22439	−0.612196	−	0.790706i	$-0.709714\pi$
$3$	−2.12071	−1.22439	−0.612196	+	0.790706i	$0.709714\pi$
$4$	−1.88702	−0.943509
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.712829	−0.291011
$7$	−1.25721	−0.475181	−0.237591	−	0.971365i	$-0.576358\pi$
$7$	−1.25721	−0.475181	−0.237591	+	0.971365i	$0.576358\pi$
$8$	−1.30653	−0.461930
$9$	1.49741	0.499137
$10$	−0.336128	−0.106293
$11$	0	0
$11$	0	0
$12$	4.00182	1.15523
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.422583	−0.112940
$15$	2.12071	0.547565
$16$	3.33487	0.833719
$17$	1.59876	0.387756	0.193878	−	0.981026i	$-0.437893\pi$
$17$	1.59876	0.387756	0.193878	+	0.981026i	$0.437893\pi$
$18$	0.503321	0.118634
$19$	6.34435	1.45549	0.727746	−	0.685846i	$-0.240568\pi$
$19$	6.34435	1.45549	0.727746	+	0.685846i	$0.240568\pi$
$20$	1.88702	0.421950
$21$	2.66618	0.581808
$22$	0	0
$23$	2.01720	0.420614	0.210307	−	0.977635i	$-0.432554\pi$
$23$	2.01720	0.420614	0.210307	+	0.977635i	$0.432554\pi$
$24$	2.77078	0.565583
$25$	1.00000	0.200000
$26$	−0.336128	−0.0659200
$27$	3.18655	0.613253
$28$	2.37238	0.448338
$29$	−3.54131	−0.657606	−0.328803	−	0.944399i	$-0.606645\pi$
$29$	−3.54131	−0.657606	−0.328803	+	0.944399i	$0.606645\pi$
$30$	0.712829	0.130144
$31$	−4.93400	−0.886172	−0.443086	−	0.896479i	$-0.646116\pi$
$31$	−4.93400	−0.886172	−0.443086	+	0.896479i	$0.646116\pi$
$32$	3.73401	0.660086
$33$	0	0
$34$	0.537387	0.0921611
$35$	1.25721	0.212508
$36$	−2.82564	−0.470941
$37$	−2.92059	−0.480143	−0.240071	−	0.970755i	$-0.577171\pi$
$37$	−2.92059	−0.480143	−0.240071	+	0.970755i	$0.577171\pi$
$38$	2.13251	0.345939
$39$	2.12071	0.339585
$40$	1.30653	0.206581
$41$	0.127199	0.0198651	0.00993256	−	0.999951i	$-0.496838\pi$
$41$	0.127199	0.0198651	0.00993256	+	0.999951i	$0.496838\pi$
$42$	0.896177	0.138283
$43$	−6.69458	−1.02091	−0.510457	−	0.859903i	$-0.670524\pi$
$43$	−6.69458	−1.02091	−0.510457	+	0.859903i	$0.670524\pi$
$44$	0	0
$45$	−1.49741	−0.223221
$46$	0.678035	0.0999708
$47$	−6.75748	−0.985680	−0.492840	−	0.870120i	$-0.664041\pi$
$47$	−6.75748	−0.985680	−0.492840	+	0.870120i	$0.664041\pi$
$48$	−7.07230	−1.02080
$49$	−5.41942	−0.774203
$50$	0.336128	0.0475356
$51$	−3.39050	−0.474766
$52$	1.88702	0.261682
$53$	4.99469	0.686073	0.343037	−	0.939322i	$-0.388544\pi$
$53$	4.99469	0.686073	0.343037	+	0.939322i	$0.388544\pi$
$54$	1.07109	0.145757
$55$	0	0
$56$	1.64259	0.219500
$57$	−13.4545	−1.78209
$58$	−1.19033	−0.156298
$59$	−12.9532	−1.68637	−0.843185	−	0.537624i	$-0.819322\pi$
$59$	−12.9532	−1.68637	−0.843185	+	0.537624i	$0.819322\pi$
$60$	−4.00182	−0.516633
$61$	2.78047	0.356003	0.178002	−	0.984030i	$-0.443037\pi$
$61$	2.78047	0.356003	0.178002	+	0.984030i	$0.443037\pi$
$62$	−1.65845	−0.210624
$63$	−1.88256	−0.237181
$64$	−5.41464	−0.676831
$65$	1.00000	0.124035
$66$	0	0
$67$	−0.638510	−0.0780064	−0.0390032	−	0.999239i	$-0.512418\pi$
$67$	−0.638510	−0.0780064	−0.0390032	+	0.999239i	$0.512418\pi$
$68$	−3.01689	−0.365851
$69$	−4.27789	−0.514997
$70$	0.422583	0.0505084
$71$	6.26909	0.744004	0.372002	−	0.928232i	$-0.378671\pi$
$71$	6.26909	0.744004	0.372002	+	0.928232i	$0.378671\pi$
$72$	−1.95642	−0.230566
$73$	−5.31914	−0.622558	−0.311279	−	0.950319i	$-0.600757\pi$
$73$	−5.31914	−0.622558	−0.311279	+	0.950319i	$0.600757\pi$
$74$	−0.981692	−0.114119
$75$	−2.12071	−0.244879
$76$	−11.9719	−1.37327
$77$	0	0
$78$	0.712829	0.0807120
$79$	−14.3733	−1.61712	−0.808559	−	0.588415i	$-0.799752\pi$
$79$	−14.3733	−1.61712	−0.808559	+	0.588415i	$0.799752\pi$
$80$	−3.33487	−0.372850
$81$	−11.2500	−1.25000
$82$	0.0427550	0.00472150
$83$	−9.60590	−1.05438	−0.527192	−	0.849746i	$-0.676755\pi$
$83$	−9.60590	−1.05438	−0.527192	+	0.849746i	$0.676755\pi$
$84$	−5.03113	−0.548942
$85$	−1.59876	−0.173410
$86$	−2.25023	−0.242649
$87$	7.51010	0.805167
$88$	0	0
$89$	7.67233	0.813265	0.406633	−	0.913592i	$-0.366703\pi$
$89$	7.67233	0.813265	0.406633	+	0.913592i	$0.366703\pi$
$90$	−0.503321	−0.0530547
$91$	1.25721	0.131792
$92$	−3.80648	−0.396853
$93$	10.4636	1.08502
$94$	−2.27138	−0.234275
$95$	−6.34435	−0.650916
$96$	−7.91876	−0.808205
$97$	9.78708	0.993727	0.496863	−	0.867829i	$-0.334485\pi$
$97$	9.78708	0.993727	0.496863	+	0.867829i	$0.334485\pi$
$98$	−1.82162	−0.184011
$99$	0	0
$100$	−1.88702	−0.188702
$101$	−6.04138	−0.601139	−0.300570	−	0.953760i	$-0.597177\pi$
$101$	−6.04138	−0.601139	−0.300570	+	0.953760i	$0.597177\pi$
$102$	−1.13964	−0.112841
$103$	15.5344	1.53065	0.765326	−	0.643643i	$-0.222578\pi$
$103$	15.5344	1.53065	0.765326	+	0.643643i	$0.222578\pi$
$104$	1.30653	0.128116
$105$	−2.66618	−0.260193
$106$	1.67885	0.163065
$107$	4.24575	0.410452	0.205226	−	0.978715i	$-0.434207\pi$
$107$	4.24575	0.410452	0.205226	+	0.978715i	$0.434207\pi$
$108$	−6.01309	−0.578609
$109$	1.55214	0.148668	0.0743340	−	0.997233i	$-0.476317\pi$
$109$	1.55214	0.148668	0.0743340	+	0.997233i	$0.476317\pi$
$110$	0	0
$111$	6.19373	0.587883
$112$	−4.19264	−0.396167
$113$	−12.2725	−1.15450	−0.577250	−	0.816567i	$-0.695874\pi$
$113$	−12.2725	−1.15450	−0.577250	+	0.816567i	$0.695874\pi$
$114$	−4.52243	−0.423565
$115$	−2.01720	−0.188104
$116$	6.68253	0.620457
$117$	−1.49741	−0.138436
$118$	−4.35394	−0.400813
$119$	−2.00998	−0.184254
$120$	−2.77078	−0.252936
$121$	0	0
$122$	0.934594	0.0846142
$123$	−0.269752	−0.0243227
$124$	9.31054	0.836111
$125$	−1.00000	−0.0894427
$126$	−0.632781	−0.0563726
$127$	17.7780	1.57755	0.788773	−	0.614684i	$-0.210717\pi$
$127$	17.7780	1.57755	0.788773	+	0.614684i	$0.210717\pi$
$128$	−9.28803	−0.820954
$129$	14.1973	1.25000
$130$	0.336128	0.0294803
$131$	2.45505	0.214498	0.107249	−	0.994232i	$-0.465796\pi$
$131$	2.45505	0.214498	0.107249	+	0.994232i	$0.465796\pi$
$132$	0	0
$133$	−7.97618	−0.691623
$134$	−0.214621	−0.0185404
$135$	−3.18655	−0.274255
$136$	−2.08883	−0.179116
$137$	5.45301	0.465882	0.232941	−	0.972491i	$-0.425165\pi$
$137$	5.45301	0.465882	0.232941	+	0.972491i	$0.425165\pi$
$138$	−1.43792	−0.122404
$139$	−22.7508	−1.92970	−0.964850	−	0.262802i	$-0.915353\pi$
$139$	−22.7508	−1.92970	−0.964850	+	0.262802i	$0.915353\pi$
$140$	−2.37238	−0.200503
$141$	14.3307	1.20686
$142$	2.10721	0.176834
$143$	0	0
$144$	4.99368	0.416140
$145$	3.54131	0.294090
$146$	−1.78791	−0.147968
$147$	11.4930	0.947928
$148$	5.51121	0.453019
$149$	−10.1385	−0.830580	−0.415290	−	0.909689i	$-0.636320\pi$
$149$	−10.1385	−0.830580	−0.415290	+	0.909689i	$0.636320\pi$
$150$	−0.712829	−0.0582023
$151$	−3.60329	−0.293232	−0.146616	−	0.989194i	$-0.546838\pi$
$151$	−3.60329	−0.293232	−0.146616	+	0.989194i	$0.546838\pi$
$152$	−8.28910	−0.672335
$153$	2.39400	0.193543
$154$	0	0
$155$	4.93400	0.396308
$156$	−4.00182	−0.320402
$157$	−13.6875	−1.09238	−0.546190	−	0.837661i	$-0.683922\pi$
$157$	−13.6875	−1.09238	−0.546190	+	0.837661i	$0.683922\pi$
$158$	−4.83125	−0.384353
$159$	−10.5923	−0.840023
$160$	−3.73401	−0.295200
$161$	−2.53604	−0.199868
$162$	−3.78143	−0.297097
$163$	9.94504	0.778956	0.389478	−	0.921036i	$-0.372655\pi$
$163$	9.94504	0.778956	0.389478	+	0.921036i	$0.372655\pi$
$164$	−0.240027	−0.0187429
$165$	0	0
$166$	−3.22881	−0.250604
$167$	15.3760	1.18983	0.594914	−	0.803790i	$-0.297186\pi$
$167$	15.3760	1.18983	0.594914	+	0.803790i	$0.297186\pi$
$168$	−3.48346	−0.268754
$169$	1.00000	0.0769231
$170$	−0.537387	−0.0412157
$171$	9.50010	0.726491
$172$	12.6328	0.963242
$173$	11.3713	0.864542	0.432271	−	0.901744i	$-0.357712\pi$
$173$	11.3713	0.864542	0.432271	+	0.901744i	$0.357712\pi$
$174$	2.52435	0.191371
$175$	−1.25721	−0.0950362
$176$	0	0
$177$	27.4701	2.06478
$178$	2.57888	0.193295
$179$	14.4377	1.07913	0.539563	−	0.841945i	$-0.318590\pi$
$179$	14.4377	1.07913	0.539563	+	0.841945i	$0.318590\pi$
$180$	2.82564	0.210611
$181$	3.78746	0.281520	0.140760	−	0.990044i	$-0.455045\pi$
$181$	3.78746	0.281520	0.140760	+	0.990044i	$0.455045\pi$
$182$	0.422583	0.0313240
$183$	−5.89658	−0.435888
$184$	−2.63553	−0.194294
$185$	2.92059	0.214726
$186$	3.51710	0.257886
$187$	0	0
$188$	12.7515	0.929998
$189$	−4.00617	−0.291406
$190$	−2.13251	−0.154708
$191$	8.56733	0.619910	0.309955	−	0.950751i	$-0.399686\pi$
$191$	8.56733	0.619910	0.309955	+	0.950751i	$0.399686\pi$
$192$	11.4829	0.828706
$193$	−22.0513	−1.58728	−0.793642	−	0.608385i	$-0.791818\pi$
$193$	−22.0513	−1.58728	−0.793642	+	0.608385i	$0.791818\pi$
$194$	3.28971	0.236187
$195$	−2.12071	−0.151867
$196$	10.2265	0.730467
$197$	−16.6673	−1.18750	−0.593748	−	0.804651i	$-0.702352\pi$
$197$	−16.6673	−1.18750	−0.593748	+	0.804651i	$0.702352\pi$
$198$	0	0
$199$	−13.6275	−0.966027	−0.483014	−	0.875613i	$-0.660458\pi$
$199$	−13.6275	−0.966027	−0.483014	+	0.875613i	$0.660458\pi$
$200$	−1.30653	−0.0923859
$201$	1.35410	0.0955105
$202$	−2.03067	−0.142878
$203$	4.45218	0.312482
$204$	6.39794	0.447946
$205$	−0.127199	−0.00888395
$206$	5.22155	0.363802
$207$	3.02057	0.209944
$208$	−3.33487	−0.231232
$209$	0	0
$210$	−0.896177	−0.0618421
$211$	9.82022	0.676052	0.338026	−	0.941137i	$-0.390241\pi$
$211$	9.82022	0.676052	0.338026	+	0.941137i	$0.390241\pi$
$212$	−9.42507	−0.647316
$213$	−13.2949	−0.910953
$214$	1.42711	0.0975554
$215$	6.69458	0.456567
$216$	−4.16334	−0.283279
$217$	6.20308	0.421092
$218$	0.521717	0.0353351
$219$	11.2804	0.762255
$220$	0	0
$221$	−1.59876	−0.107544
$222$	2.08188	0.139727
$223$	−6.00036	−0.401814	−0.200907	−	0.979610i	$-0.564389\pi$
$223$	−6.00036	−0.401814	−0.200907	+	0.979610i	$0.564389\pi$
$224$	−4.69444	−0.313661
$225$	1.49741	0.0998275
$226$	−4.12513	−0.274399
$227$	8.42756	0.559357	0.279678	−	0.960094i	$-0.409772\pi$
$227$	8.42756	0.559357	0.279678	+	0.960094i	$0.409772\pi$
$228$	25.3889	1.68142
$229$	24.0748	1.59091	0.795453	−	0.606015i	$-0.207233\pi$
$229$	24.0748	1.59091	0.795453	+	0.606015i	$0.207233\pi$
$230$	−0.678035	−0.0447083
$231$	0	0
$232$	4.62685	0.303767
$233$	−0.903711	−0.0592041	−0.0296020	−	0.999562i	$-0.509424\pi$
$233$	−0.903711	−0.0592041	−0.0296020	+	0.999562i	$0.509424\pi$
$234$	−0.503321	−0.0329031
$235$	6.75748	0.440810
$236$	24.4430	1.59110
$237$	30.4815	1.97999
$238$	−0.675609	−0.0437932
$239$	16.0679	1.03934	0.519672	−	0.854366i	$-0.326054\pi$
$239$	16.0679	1.03934	0.519672	+	0.854366i	$0.326054\pi$
$240$	7.07230	0.456515
$241$	−4.20389	−0.270797	−0.135398	−	0.990791i	$-0.543231\pi$
$241$	−4.20389	−0.270797	−0.135398	+	0.990791i	$0.543231\pi$
$242$	0	0
$243$	14.2983	0.917237
$244$	−5.24681	−0.335892
$245$	5.41942	0.346234
$246$	−0.0906710	−0.00578097
$247$	−6.34435	−0.403681
$248$	6.44644	0.409349
$249$	20.3713	1.29098
$250$	−0.336128	−0.0212586
$251$	−16.3960	−1.03491	−0.517454	−	0.855711i	$-0.673120\pi$
$251$	−16.3960	−1.03491	−0.517454	+	0.855711i	$0.673120\pi$
$252$	3.55243	0.223782
$253$	0	0
$254$	5.97569	0.374948
$255$	3.39050	0.212322
$256$	7.70733	0.481708
$257$	−1.15296	−0.0719197	−0.0359599	−	0.999353i	$-0.511449\pi$
$257$	−1.15296	−0.0719197	−0.0359599	+	0.999353i	$0.511449\pi$
$258$	4.77209	0.297097
$259$	3.67180	0.228155
$260$	−1.88702	−0.117028
$261$	−5.30281	−0.328235
$262$	0.825208	0.0509815
$263$	−15.8898	−0.979806	−0.489903	−	0.871777i	$-0.662968\pi$
$263$	−15.8898	−0.979806	−0.489903	+	0.871777i	$0.662968\pi$
$264$	0	0
$265$	−4.99469	−0.306821
$266$	−2.68102	−0.164384
$267$	−16.2708	−0.995756
$268$	1.20488	0.0735998
$269$	0.825752	0.0503470	0.0251735	−	0.999683i	$-0.491986\pi$
$269$	0.825752	0.0503470	0.0251735	+	0.999683i	$0.491986\pi$
$270$	−1.07109	−0.0651844
$271$	−9.36641	−0.568969	−0.284484	−	0.958681i	$-0.591822\pi$
$271$	−9.36641	−0.568969	−0.284484	+	0.958681i	$0.591822\pi$
$272$	5.33166	0.323279
$273$	−2.66618	−0.161365
$274$	1.83291	0.110730
$275$	0	0
$276$	8.07245	0.485904
$277$	14.4310	0.867076	0.433538	−	0.901135i	$-0.357265\pi$
$277$	14.4310	0.867076	0.433538	+	0.901135i	$0.357265\pi$
$278$	−7.64718	−0.458647
$279$	−7.38823	−0.442322
$280$	−1.64259	−0.0981635
$281$	−11.1323	−0.664100	−0.332050	−	0.943262i	$-0.607740\pi$
$281$	−11.1323	−0.664100	−0.332050	+	0.943262i	$0.607740\pi$
$282$	4.81693	0.286844
$283$	12.8896	0.766206	0.383103	−	0.923706i	$-0.374855\pi$
$283$	12.8896	0.766206	0.383103	+	0.923706i	$0.374855\pi$
$284$	−11.8299	−0.701975
$285$	13.4545	0.796977
$286$	0	0
$287$	−0.159916	−0.00943953
$288$	5.59135	0.329474
$289$	−14.4440	−0.849645
$290$	1.19033	0.0698988
$291$	−20.7556	−1.21671
$292$	10.0373	0.587389
$293$	−21.9596	−1.28289	−0.641447	−	0.767167i	$-0.721666\pi$
$293$	−21.9596	−1.28289	−0.641447	+	0.767167i	$0.721666\pi$
$294$	3.86312	0.225302
$295$	12.9532	0.754167
$296$	3.81586	0.221792
$297$	0	0
$298$	−3.40784	−0.197411
$299$	−2.01720	−0.116657
$300$	4.00182	0.231045
$301$	8.41650	0.485119
$302$	−1.21117	−0.0696947
$303$	12.8120	0.736031
$304$	21.1576	1.21347
$305$	−2.78047	−0.159210
$306$	0.804690	0.0460010
$307$	14.4840	0.826644	0.413322	−	0.910585i	$-0.364368\pi$
$307$	14.4840	0.826644	0.413322	+	0.910585i	$0.364368\pi$
$308$	0	0
$309$	−32.9440	−1.87412
$310$	1.65845	0.0941938
$311$	−32.2312	−1.82767	−0.913833	−	0.406090i	$-0.866892\pi$
$311$	−32.2312	−1.82767	−0.913833	+	0.406090i	$0.866892\pi$
$312$	−2.77078	−0.156865
$313$	18.6044	1.05158	0.525792	−	0.850613i	$-0.323769\pi$
$313$	18.6044	1.05158	0.525792	+	0.850613i	$0.323769\pi$
$314$	−4.60074	−0.259635
$315$	1.88256	0.106070
$316$	27.1226	1.52577
$317$	24.7501	1.39010	0.695051	−	0.718960i	$-0.255382\pi$
$317$	24.7501	1.39010	0.695051	+	0.718960i	$0.255382\pi$
$318$	−3.56036	−0.199655
$319$	0	0
$320$	5.41464	0.302688
$321$	−9.00400	−0.502554
$322$	−0.852433	−0.0475042
$323$	10.1431	0.564376
$324$	21.2289	1.17939
$325$	−1.00000	−0.0554700
$326$	3.34280	0.185141
$327$	−3.29164	−0.182028
$328$	−0.166190	−0.00917629
$329$	8.49558	0.468377
$330$	0	0
$331$	−17.0382	−0.936503	−0.468252	−	0.883595i	$-0.655116\pi$
$331$	−17.0382	−0.936503	−0.468252	+	0.883595i	$0.655116\pi$
$332$	18.1265	0.994822
$333$	−4.37333	−0.239657
$334$	5.16828	0.282796
$335$	0.638510	0.0348855
$336$	8.89138	0.485064
$337$	21.4192	1.16678	0.583389	−	0.812193i	$-0.301726\pi$
$337$	21.4192	1.16678	0.583389	+	0.812193i	$0.301726\pi$
$338$	0.336128	0.0182829
$339$	26.0264	1.41356
$340$	3.01689	0.163614
$341$	0	0
$342$	3.19324	0.172671
$343$	15.6138	0.843068
$344$	8.74670	0.471590
$345$	4.27789	0.230314
$346$	3.82220	0.205483
$347$	3.47333	0.186458	0.0932290	−	0.995645i	$-0.470281\pi$
$347$	3.47333	0.186458	0.0932290	+	0.995645i	$0.470281\pi$
$348$	−14.1717	−0.759683
$349$	2.97477	0.159236	0.0796178	−	0.996825i	$-0.474630\pi$
$349$	2.97477	0.159236	0.0796178	+	0.996825i	$0.474630\pi$
$350$	−0.422583	−0.0225880
$351$	−3.18655	−0.170086
$352$	0	0
$353$	−10.3433	−0.550521	−0.275260	−	0.961370i	$-0.588764\pi$
$353$	−10.3433	−0.550521	−0.275260	+	0.961370i	$0.588764\pi$
$354$	9.23345	0.490752
$355$	−6.26909	−0.332729
$356$	−14.4778	−0.767323
$357$	4.26258	0.225600
$358$	4.85291	0.256484
$359$	2.56651	0.135455	0.0677276	−	0.997704i	$-0.478425\pi$
$359$	2.56651	0.135455	0.0677276	+	0.997704i	$0.478425\pi$
$360$	1.95642	0.103112
$361$	21.2507	1.11846
$362$	1.27307	0.0669111
$363$	0	0
$364$	−2.37238	−0.124347
$365$	5.31914	0.278416
$366$	−1.98200	−0.103601
$367$	−5.04531	−0.263363	−0.131682	−	0.991292i	$-0.542038\pi$
$367$	−5.04531	−0.263363	−0.131682	+	0.991292i	$0.542038\pi$
$368$	6.72709	0.350674
$369$	0.190469	0.00991542
$370$	0.981692	0.0510357
$371$	−6.27938	−0.326009
$372$	−19.7450	−1.02373
$373$	−5.04186	−0.261058	−0.130529	−	0.991445i	$-0.541668\pi$
$373$	−5.04186	−0.261058	−0.130529	+	0.991445i	$0.541668\pi$
$374$	0	0
$375$	2.12071	0.109513
$376$	8.82888	0.455315
$377$	3.54131	0.182387
$378$	−1.34658	−0.0692609
$379$	19.6349	1.00858	0.504289	−	0.863535i	$-0.331755\pi$
$379$	19.6349	1.00858	0.504289	+	0.863535i	$0.331755\pi$
$380$	11.9719	0.614145
$381$	−37.7021	−1.93154
$382$	2.87972	0.147339
$383$	−15.9095	−0.812940	−0.406470	−	0.913664i	$-0.633240\pi$
$383$	−15.9095	−0.812940	−0.406470	+	0.913664i	$0.633240\pi$
$384$	19.6972	1.00517
$385$	0	0
$386$	−7.41204	−0.377263
$387$	−10.0245	−0.509576
$388$	−18.4684	−0.937590
$389$	−22.1659	−1.12385	−0.561927	−	0.827187i	$-0.689940\pi$
$389$	−22.1659	−1.12385	−0.561927	+	0.827187i	$0.689940\pi$
$390$	−0.712829	−0.0360955
$391$	3.22501	0.163096
$392$	7.08066	0.357627
$393$	−5.20644	−0.262630
$394$	−5.60234	−0.282242
$395$	14.3733	0.723197
$396$	0	0
$397$	−17.5209	−0.879347	−0.439673	−	0.898158i	$-0.644906\pi$
$397$	−17.5209	−0.879347	−0.439673	+	0.898158i	$0.644906\pi$
$398$	−4.58057	−0.229603
$399$	16.9152	0.846818
$400$	3.33487	0.166744
$401$	−0.853210	−0.0426073	−0.0213036	−	0.999773i	$-0.506782\pi$
$401$	−0.853210	−0.0426073	−0.0213036	+	0.999773i	$0.506782\pi$
$402$	0.455149	0.0227008
$403$	4.93400	0.245780
$404$	11.4002	0.567181
$405$	11.2500	0.559017
$406$	1.49650	0.0742701
$407$	0	0
$408$	4.42981	0.219308
$409$	32.1662	1.59051	0.795257	−	0.606272i	$-0.207336\pi$
$409$	32.1662	1.59051	0.795257	+	0.606272i	$0.207336\pi$
$410$	−0.0427550	−0.00211152
$411$	−11.5642	−0.570422
$412$	−29.3137	−1.44418
$413$	16.2850	0.801331
$414$	1.01530	0.0498992
$415$	9.60590	0.471535
$416$	−3.73401	−0.183075
$417$	48.2479	2.36271
$418$	0	0
$419$	−27.4922	−1.34308	−0.671542	−	0.740966i	$-0.734368\pi$
$419$	−27.4922	−1.34308	−0.671542	+	0.740966i	$0.734368\pi$
$420$	5.03113	0.245494
$421$	−4.50415	−0.219519	−0.109760	−	0.993958i	$-0.535008\pi$
$421$	−4.50415	−0.219519	−0.109760	+	0.993958i	$0.535008\pi$
$422$	3.30085	0.160683
$423$	−10.1187	−0.491990
$424$	−6.52573	−0.316918
$425$	1.59876	0.0775512
$426$	−4.46879	−0.216514
$427$	−3.49564	−0.169166
$428$	−8.01180	−0.387265
$429$	0	0
$430$	2.25023	0.108516
$431$	10.9158	0.525795	0.262897	−	0.964824i	$-0.415322\pi$
$431$	10.9158	0.525795	0.262897	+	0.964824i	$0.415322\pi$
$432$	10.6268	0.511280
$433$	−27.7153	−1.33191	−0.665955	−	0.745992i	$-0.731976\pi$
$433$	−27.7153	−1.33191	−0.665955	+	0.745992i	$0.731976\pi$
$434$	2.08503	0.100084
$435$	−7.51010	−0.360082
$436$	−2.92892	−0.140270
$437$	12.7978	0.612201
$438$	3.79164	0.181171
$439$	12.8592	0.613734	0.306867	−	0.951752i	$-0.400719\pi$
$439$	12.8592	0.613734	0.306867	+	0.951752i	$0.400719\pi$
$440$	0	0
$441$	−8.11510	−0.386433
$442$	−0.537387	−0.0255609
$443$	16.3529	0.776950	0.388475	−	0.921459i	$-0.373002\pi$
$443$	16.3529	0.776950	0.388475	+	0.921459i	$0.373002\pi$
$444$	−11.6877	−0.554673
$445$	−7.67233	−0.363703
$446$	−2.01689	−0.0955024
$447$	21.5009	1.01696
$448$	6.80735	0.321617
$449$	0.809366	0.0381963	0.0190982	−	0.999818i	$-0.493920\pi$
$449$	0.809366	0.0381963	0.0190982	+	0.999818i	$0.493920\pi$
$450$	0.503321	0.0237268
$451$	0	0
$452$	23.1584	1.08928
$453$	7.64153	0.359031
$454$	2.83273	0.132947
$455$	−1.25721	−0.0589390
$456$	17.5788	0.823202
$457$	11.0545	0.517108	0.258554	−	0.965997i	$-0.416754\pi$
$457$	11.0545	0.517108	0.258554	+	0.965997i	$0.416754\pi$
$458$	8.09220	0.378124
$459$	5.09453	0.237792
$460$	3.80648	0.177478
$461$	17.6411	0.821629	0.410815	−	0.911719i	$-0.365244\pi$
$461$	17.6411	0.821629	0.410815	+	0.911719i	$0.365244\pi$
$462$	0	0
$463$	9.67331	0.449557	0.224778	−	0.974410i	$-0.427834\pi$
$463$	9.67331	0.449557	0.224778	+	0.974410i	$0.427834\pi$
$464$	−11.8098	−0.548258
$465$	−10.4636	−0.485237
$466$	−0.303762	−0.0140715
$467$	7.21255	0.333757	0.166879	−	0.985977i	$-0.446631\pi$
$467$	7.21255	0.333757	0.166879	+	0.985977i	$0.446631\pi$
$468$	2.82564	0.130615
$469$	0.802742	0.0370672
$470$	2.27138	0.104771
$471$	29.0272	1.33750
$472$	16.9239	0.778984
$473$	0	0
$474$	10.2457	0.470600
$475$	6.34435	0.291099
$476$	3.79286	0.173846
$477$	7.47911	0.342445
$478$	5.40086	0.247029
$479$	−34.7347	−1.58707	−0.793535	−	0.608524i	$-0.791762\pi$
$479$	−34.7347	−1.58707	−0.793535	+	0.608524i	$0.791762\pi$
$480$	7.91876	0.361440
$481$	2.92059	0.133168
$482$	−1.41304	−0.0643624
$483$	5.37821	0.244717
$484$	0	0
$485$	−9.78708	−0.444408
$486$	4.80606	0.218007
$487$	−14.8116	−0.671179	−0.335589	−	0.942008i	$-0.608935\pi$
$487$	−14.8116	−0.671179	−0.335589	+	0.942008i	$0.608935\pi$
$488$	−3.63279	−0.164448
$489$	−21.0905	−0.953748
$490$	1.82162	0.0822922
$491$	−26.7581	−1.20757	−0.603787	−	0.797145i	$-0.706342\pi$
$491$	−26.7581	−1.20757	−0.603787	+	0.797145i	$0.706342\pi$
$492$	0.509027	0.0229487
$493$	−5.66171	−0.254991
$494$	−2.13251	−0.0959461
$495$	0	0
$496$	−16.4543	−0.738818
$497$	−7.88157	−0.353537
$498$	6.84737	0.306838
$499$	−28.9178	−1.29454	−0.647269	−	0.762262i	$-0.724089\pi$
$499$	−28.9178	−1.29454	−0.647269	+	0.762262i	$0.724089\pi$
$500$	1.88702	0.0843900
$501$	−32.6080	−1.45682
$502$	−5.51116	−0.245975
$503$	38.2908	1.70730	0.853650	−	0.520846i	$-0.174384\pi$
$503$	38.2908	1.70730	0.853650	+	0.520846i	$0.174384\pi$
$504$	2.45963	0.109561
$505$	6.04138	0.268838
$506$	0	0
$507$	−2.12071	−0.0941840
$508$	−33.5475	−1.48843
$509$	39.5880	1.75471	0.877353	−	0.479846i	$-0.159308\pi$
$509$	39.5880	1.75471	0.877353	+	0.479846i	$0.159308\pi$
$510$	1.13964	0.0504642
$511$	6.68728	0.295828
$512$	21.1667	0.935445
$513$	20.2166	0.892585
$514$	−0.387542	−0.0170937
$515$	−15.5344	−0.684528
$516$	−26.7905	−1.17939
$517$	0	0
$518$	1.23419	0.0542274
$519$	−24.1152	−1.05854
$520$	−1.30653	−0.0572953
$521$	26.2445	1.14979	0.574897	−	0.818226i	$-0.305042\pi$
$521$	26.2445	1.14979	0.574897	+	0.818226i	$0.305042\pi$
$522$	−1.78242	−0.0780144
$523$	7.57119	0.331065	0.165533	−	0.986204i	$-0.447066\pi$
$523$	7.57119	0.331065	0.165533	+	0.986204i	$0.447066\pi$
$524$	−4.63271	−0.202381
$525$	2.66618	0.116362
$526$	−5.34100	−0.232879
$527$	−7.88827	−0.343619
$528$	0	0
$529$	−18.9309	−0.823084
$530$	−1.67885	−0.0729247
$531$	−19.3963	−0.841730
$532$	15.0512	0.652552
$533$	−0.127199	−0.00550959
$534$	−5.46906	−0.236669
$535$	−4.24575	−0.183560
$536$	0.834235	0.0360335
$537$	−30.6182	−1.32127
$538$	0.277558	0.0119664
$539$	0	0
$540$	6.01309	0.258762
$541$	14.3784	0.618176	0.309088	−	0.951033i	$-0.399976\pi$
$541$	14.3784	0.618176	0.309088	+	0.951033i	$0.399976\pi$
$542$	−3.14831	−0.135231
$543$	−8.03211	−0.344691
$544$	5.96978	0.255952
$545$	−1.55214	−0.0664863
$546$	−0.896177	−0.0383528
$547$	−1.14789	−0.0490801	−0.0245400	−	0.999699i	$-0.507812\pi$
$547$	−1.14789	−0.0490801	−0.0245400	+	0.999699i	$0.507812\pi$
$548$	−10.2899	−0.439564
$549$	4.16352	0.177695
$550$	0	0
$551$	−22.4673	−0.957140
$552$	5.58920	0.237892
$553$	18.0702	0.768424
$554$	4.85066	0.206085
$555$	−6.19373	−0.262909
$556$	42.9312	1.82069
$557$	19.1833	0.812823	0.406412	−	0.913690i	$-0.366780\pi$
$557$	19.1833	0.812823	0.406412	+	0.913690i	$0.366780\pi$
$558$	−2.48339	−0.105130
$559$	6.69458	0.283151
$560$	4.19264	0.177171
$561$	0	0
$562$	−3.74189	−0.157842
$563$	34.8962	1.47070	0.735351	−	0.677687i	$-0.237018\pi$
$563$	34.8962	1.47070	0.735351	+	0.677687i	$0.237018\pi$
$564$	−27.0422	−1.13868
$565$	12.2725	0.516308
$566$	4.33254	0.182110
$567$	14.1436	0.593976
$568$	−8.19078	−0.343678
$569$	13.9688	0.585602	0.292801	−	0.956173i	$-0.405413\pi$
$569$	13.9688	0.585602	0.292801	+	0.956173i	$0.405413\pi$
$570$	4.52243	0.189424
$571$	3.90306	0.163338	0.0816690	−	0.996660i	$-0.473975\pi$
$571$	3.90306	0.163338	0.0816690	+	0.996660i	$0.473975\pi$
$572$	0	0
$573$	−18.1688	−0.759014
$574$	−0.0537521	−0.00224357
$575$	2.01720	0.0841229
$576$	−8.10795	−0.337831
$577$	−24.7826	−1.03171	−0.515856	−	0.856675i	$-0.672526\pi$
$577$	−24.7826	−1.03171	−0.515856	+	0.856675i	$0.672526\pi$
$578$	−4.85502	−0.201942
$579$	46.7643	1.94346
$580$	−6.68253	−0.277477
$581$	12.0767	0.501024
$582$	−6.97651	−0.289186
$583$	0	0
$584$	6.94963	0.287578
$585$	1.49741	0.0619104
$586$	−7.38123	−0.304916
$587$	20.3244	0.838877	0.419439	−	0.907784i	$-0.362227\pi$
$587$	20.3244	0.838877	0.419439	+	0.907784i	$0.362227\pi$
$588$	−21.6875	−0.894379
$589$	−31.3030	−1.28982
$590$	4.35394	0.179249
$591$	35.3465	1.45396
$592$	−9.73981	−0.400304
$593$	33.3920	1.37124	0.685622	−	0.727958i	$-0.259530\pi$
$593$	33.3920	1.37124	0.685622	+	0.727958i	$0.259530\pi$
$594$	0	0
$595$	2.00998	0.0824011
$596$	19.1316	0.783660
$597$	28.9000	1.18280
$598$	−0.678035	−0.0277269
$599$	22.2780	0.910253	0.455126	−	0.890427i	$-0.349594\pi$
$599$	22.2780	0.910253	0.455126	+	0.890427i	$0.349594\pi$
$600$	2.77078	0.113117
$601$	1.07057	0.0436696	0.0218348	−	0.999762i	$-0.493049\pi$
$601$	1.07057	0.0436696	0.0218348	+	0.999762i	$0.493049\pi$
$602$	2.82902	0.115302
$603$	−0.956113	−0.0389359
$604$	6.79947	0.276667
$605$	0	0
$606$	4.30647	0.174938
$607$	18.0347	0.732006	0.366003	−	0.930614i	$-0.380726\pi$
$607$	18.0347	0.732006	0.366003	+	0.930614i	$0.380726\pi$
$608$	23.6899	0.960751
$609$	−9.44179	−0.382600
$610$	−0.934594	−0.0378406
$611$	6.75748	0.273378
$612$	−4.51752	−0.182610
$613$	−11.5137	−0.465034	−0.232517	−	0.972592i	$-0.574696\pi$
$613$	−11.5137	−0.465034	−0.232517	+	0.972592i	$0.574696\pi$
$614$	4.86846	0.196475
$615$	0.269752	0.0108774
$616$	0	0
$617$	23.6167	0.950772	0.475386	−	0.879777i	$-0.342308\pi$
$617$	23.6167	0.950772	0.475386	+	0.879777i	$0.342308\pi$
$618$	−11.0734	−0.445437
$619$	43.9312	1.76574	0.882871	−	0.469615i	$-0.155607\pi$
$619$	43.9312	1.76574	0.882871	+	0.469615i	$0.155607\pi$
$620$	−9.31054	−0.373920
$621$	6.42790	0.257943
$622$	−10.8338	−0.434396
$623$	−9.64574	−0.386448
$624$	7.07230	0.283119
$625$	1.00000	0.0400000
$626$	6.25345	0.249938
$627$	0	0
$628$	25.8285	1.03067
$629$	−4.66933	−0.186178
$630$	0.632781	0.0252106
$631$	27.9264	1.11173	0.555867	−	0.831271i	$-0.312386\pi$
$631$	27.9264	1.11173	0.555867	+	0.831271i	$0.312386\pi$
$632$	18.7791	0.746994
$633$	−20.8258	−0.827753
$634$	8.31918	0.330397
$635$	−17.7780	−0.705500
$636$	19.9878	0.792569
$637$	5.41942	0.214725
$638$	0	0
$639$	9.38741	0.371360
$640$	9.28803	0.367142
$641$	10.5862	0.418130	0.209065	−	0.977902i	$-0.432958\pi$
$641$	10.5862	0.418130	0.209065	+	0.977902i	$0.432958\pi$
$642$	−3.02649	−0.119446
$643$	−33.8548	−1.33510	−0.667551	−	0.744564i	$-0.732657\pi$
$643$	−33.8548	−1.33510	−0.667551	+	0.744564i	$0.732657\pi$
$644$	4.78556	0.188577
$645$	−14.1973	−0.559017
$646$	3.40937	0.134140
$647$	10.3375	0.406408	0.203204	−	0.979136i	$-0.434865\pi$
$647$	10.3375	0.406408	0.203204	+	0.979136i	$0.434865\pi$
$648$	14.6985	0.577412
$649$	0	0
$650$	−0.336128	−0.0131840
$651$	−13.1549	−0.515582
$652$	−18.7665	−0.734952
$653$	−37.3541	−1.46178	−0.730890	−	0.682495i	$-0.760895\pi$
$653$	−37.3541	−1.46178	−0.730890	+	0.682495i	$0.760895\pi$
$654$	−1.10641	−0.0432641
$655$	−2.45505	−0.0959265
$656$	0.424192	0.0165619
$657$	−7.96494	−0.310742
$658$	2.85560	0.111323
$659$	−2.84526	−0.110836	−0.0554178	−	0.998463i	$-0.517649\pi$
$659$	−2.84526	−0.110836	−0.0554178	+	0.998463i	$0.517649\pi$
$660$	0	0
$661$	−29.5252	−1.14840	−0.574199	−	0.818716i	$-0.694687\pi$
$661$	−29.5252	−1.14840	−0.574199	+	0.818716i	$0.694687\pi$
$662$	−5.72700	−0.222586
$663$	3.39050	0.131676
$664$	12.5504	0.487052
$665$	7.97618	0.309303
$666$	−1.47000	−0.0569612
$667$	−7.14352	−0.276598
$668$	−29.0147	−1.12261
$669$	12.7250	0.491978
$670$	0.214621	0.00829153
$671$	0	0
$672$	9.95555	0.384044
$673$	27.7232	1.06865	0.534325	−	0.845279i	$-0.320566\pi$
$673$	27.7232	1.06865	0.534325	+	0.845279i	$0.320566\pi$
$674$	7.19959	0.277318
$675$	3.18655	0.122651
$676$	−1.88702	−0.0725776
$677$	−0.520642	−0.0200099	−0.0100050	−	0.999950i	$-0.503185\pi$
$677$	−0.520642	−0.0200099	−0.0100050	+	0.999950i	$0.503185\pi$
$678$	8.74820	0.335973
$679$	−12.3044	−0.472200
$680$	2.08883	0.0801031
$681$	−17.8724	−0.684872
$682$	0	0
$683$	−2.60913	−0.0998355	−0.0499177	−	0.998753i	$-0.515896\pi$
$683$	−2.60913	−0.0998355	−0.0499177	+	0.998753i	$0.515896\pi$
$684$	−17.9269	−0.685451
$685$	−5.45301	−0.208349
$686$	5.24824	0.200379
$687$	−51.0557	−1.94789
$688$	−22.3256	−0.851155
$689$	−4.99469	−0.190283
$690$	1.43792	0.0547405
$691$	43.4438	1.65268	0.826339	−	0.563173i	$-0.190419\pi$
$691$	43.4438	1.65268	0.826339	+	0.563173i	$0.190419\pi$
$692$	−21.4578	−0.815703
$693$	0	0
$694$	1.16748	0.0443170
$695$	22.7508	0.862988
$696$	−9.81220	−0.371931
$697$	0.203360	0.00770282
$698$	0.999902	0.0378468
$699$	1.91651	0.0724890
$700$	2.37238	0.0896676
$701$	−4.76785	−0.180079	−0.0900395	−	0.995938i	$-0.528699\pi$
$701$	−4.76785	−0.180079	−0.0900395	+	0.995938i	$0.528699\pi$
$702$	−1.07109	−0.0404256
$703$	−18.5293	−0.698844
$704$	0	0
$705$	−14.3307	−0.539724
$706$	−3.47668	−0.130847
$707$	7.59529	0.285650
$708$	−51.8365	−1.94814
$709$	2.48416	0.0932945	0.0466473	−	0.998911i	$-0.485146\pi$
$709$	2.48416	0.0932945	0.0466473	+	0.998911i	$0.485146\pi$
$710$	−2.10721	−0.0790823
$711$	−21.5227	−0.807164
$712$	−10.0242	−0.375671
$713$	−9.95284	−0.372737
$714$	1.43277	0.0536201
$715$	0	0
$716$	−27.2442	−1.01816
$717$	−34.0753	−1.27257
$718$	0.862675	0.0321947
$719$	−4.07962	−0.152144	−0.0760720	−	0.997102i	$-0.524238\pi$
$719$	−4.07962	−0.152144	−0.0760720	+	0.997102i	$0.524238\pi$
$720$	−4.99368	−0.186103
$721$	−19.5300	−0.727337
$722$	7.14295	0.265833
$723$	8.91524	0.331561
$724$	−7.14701	−0.265617
$725$	−3.54131	−0.131521
$726$	0	0
$727$	6.62965	0.245880	0.122940	−	0.992414i	$-0.460768\pi$
$727$	6.62965	0.245880	0.122940	+	0.992414i	$0.460768\pi$
$728$	−1.64259	−0.0608784
$729$	3.42740	0.126941
$730$	1.78791	0.0661735
$731$	−10.7030	−0.395865
$732$	11.1270	0.411264
$733$	−1.44392	−0.0533325	−0.0266663	−	0.999644i	$-0.508489\pi$
$733$	−1.44392	−0.0533325	−0.0266663	+	0.999644i	$0.508489\pi$
$734$	−1.69587	−0.0625956
$735$	−11.4930	−0.423926
$736$	7.53223	0.277642
$737$	0	0
$738$	0.0640219	0.00235668
$739$	50.7703	1.86762	0.933808	−	0.357774i	$-0.116464\pi$
$739$	50.7703	1.86762	0.933808	+	0.357774i	$0.116464\pi$
$740$	−5.51121	−0.202596
$741$	13.4545	0.494264
$742$	−2.11067	−0.0774852
$743$	5.66758	0.207923	0.103962	−	0.994581i	$-0.466848\pi$
$743$	5.66758	0.207923	0.103962	+	0.994581i	$0.466848\pi$
$744$	−13.6710	−0.501204
$745$	10.1385	0.371447
$746$	−1.69471	−0.0620477
$747$	−14.3840	−0.526283
$748$	0	0
$749$	−5.33780	−0.195039
$750$	0.712829	0.0260288
$751$	11.8202	0.431327	0.215663	−	0.976468i	$-0.430809\pi$
$751$	11.8202	0.431327	0.215663	+	0.976468i	$0.430809\pi$
$752$	−22.5354	−0.821780
$753$	34.7712	1.26713
$754$	1.19033	0.0433494
$755$	3.60329	0.131137
$756$	7.55972	0.274944
$757$	7.17784	0.260883	0.130441	−	0.991456i	$-0.458361\pi$
$757$	7.17784	0.260883	0.130441	+	0.991456i	$0.458361\pi$
$758$	6.59984	0.239717
$759$	0	0
$760$	8.28910	0.300677
$761$	12.1408	0.440104	0.220052	−	0.975488i	$-0.429377\pi$
$761$	12.1408	0.440104	0.220052	+	0.975488i	$0.429377\pi$
$762$	−12.6727	−0.459084
$763$	−1.95137	−0.0706442
$764$	−16.1667	−0.584891
$765$	−2.39400	−0.0865553
$766$	−5.34764	−0.193218
$767$	12.9532	0.467715
$768$	−16.3450	−0.589799
$769$	−19.7789	−0.713246	−0.356623	−	0.934248i	$-0.616072\pi$
$769$	−19.7789	−0.713246	−0.356623	+	0.934248i	$0.616072\pi$
$770$	0	0
$771$	2.44510	0.0880580
$772$	41.6111	1.49762
$773$	−16.0190	−0.576162	−0.288081	−	0.957606i	$-0.593017\pi$
$773$	−16.0190	−0.576162	−0.288081	+	0.957606i	$0.593017\pi$
$774$	−3.36953	−0.121115
$775$	−4.93400	−0.177234
$776$	−12.7871	−0.459032
$777$	−7.78683	−0.279351
$778$	−7.45056	−0.267115
$779$	0.806993	0.0289135
$780$	4.00182	0.143288
$781$	0	0
$782$	1.08401	0.0387643
$783$	−11.2846	−0.403278
$784$	−18.0731	−0.645467
$785$	13.6875	0.488527
$786$	−1.75003	−0.0624214
$787$	29.3542	1.04636	0.523182	−	0.852221i	$-0.324745\pi$
$787$	29.3542	1.04636	0.523182	+	0.852221i	$0.324745\pi$
$788$	31.4515	1.12041
$789$	33.6976	1.19967
$790$	4.83125	0.171888
$791$	15.4291	0.548597
$792$	0	0
$793$	−2.78047	−0.0987376
$794$	−5.88925	−0.209002
$795$	10.5923	0.375670
$796$	25.7153	0.911455
$797$	−29.8671	−1.05795	−0.528973	−	0.848639i	$-0.677423\pi$
$797$	−29.8671	−1.05795	−0.528973	+	0.848639i	$0.677423\pi$
$798$	5.68566	0.201270
$799$	−10.8036	−0.382203
$800$	3.73401	0.132017
$801$	11.4886	0.405931
$802$	−0.286787	−0.0101268
$803$	0	0
$804$	−2.55520	−0.0901150
$805$	2.53604	0.0893837
$806$	1.65845	0.0584165
$807$	−1.75118	−0.0616445
$808$	7.89326	0.277684
$809$	−34.7607	−1.22212	−0.611061	−	0.791583i	$-0.709257\pi$
$809$	−34.7607	−1.22212	−0.611061	+	0.791583i	$0.709257\pi$
$810$	3.78143	0.132866
$811$	−37.4762	−1.31597	−0.657984	−	0.753032i	$-0.728591\pi$
$811$	−37.4762	−1.31597	−0.657984	+	0.753032i	$0.728591\pi$
$812$	−8.40135	−0.294829
$813$	19.8634	0.696641
$814$	0	0
$815$	−9.94504	−0.348360
$816$	−11.3069	−0.395821
$817$	−42.4727	−1.48593
$818$	10.8119	0.378031
$819$	1.88256	0.0657821
$820$	0.240027	0.00838209
$821$	−16.4552	−0.574289	−0.287144	−	0.957887i	$-0.592706\pi$
$821$	−16.4552	−0.574289	−0.287144	+	0.957887i	$0.592706\pi$
$822$	−3.88706	−0.135577
$823$	−50.9494	−1.77598	−0.887992	−	0.459860i	$-0.847900\pi$
$823$	−50.9494	−1.77598	−0.887992	+	0.459860i	$0.847900\pi$
$824$	−20.2962	−0.707053
$825$	0	0
$826$	5.47383	0.190459
$827$	−4.11991	−0.143263	−0.0716317	−	0.997431i	$-0.522821\pi$
$827$	−4.11991	−0.143263	−0.0716317	+	0.997431i	$0.522821\pi$
$828$	−5.69987	−0.198084
$829$	20.1973	0.701481	0.350741	−	0.936473i	$-0.385930\pi$
$829$	20.1973	0.701481	0.350741	+	0.936473i	$0.385930\pi$
$830$	3.22881	0.112074
$831$	−30.6040	−1.06164
$832$	5.41464	0.187719
$833$	−8.66435	−0.300202
$834$	16.2175	0.561564
$835$	−15.3760	−0.532107
$836$	0	0
$837$	−15.7225	−0.543447
$838$	−9.24090	−0.319222
$839$	43.8429	1.51362	0.756812	−	0.653632i	$-0.226756\pi$
$839$	43.8429	1.51362	0.756812	+	0.653632i	$0.226756\pi$
$840$	3.48346	0.120191
$841$	−16.4591	−0.567555
$842$	−1.51397	−0.0521749
$843$	23.6085	0.813119
$844$	−18.5309	−0.637861
$845$	−1.00000	−0.0344010
$846$	−3.40119	−0.116935
$847$	0	0
$848$	16.6567	0.571992
$849$	−27.3351	−0.938137
$850$	0.537387	0.0184322
$851$	−5.89141	−0.201955
$852$	25.0878	0.859493
$853$	−8.74666	−0.299480	−0.149740	−	0.988725i	$-0.547844\pi$
$853$	−8.74666	−0.299480	−0.149740	+	0.988725i	$0.547844\pi$
$854$	−1.17498	−0.0402071
$855$	−9.50010	−0.324897
$856$	−5.54721	−0.189600
$857$	44.7039	1.52706	0.763528	−	0.645775i	$-0.223466\pi$
$857$	44.7039	1.52706	0.763528	+	0.645775i	$0.223466\pi$
$858$	0	0
$859$	4.21902	0.143951	0.0719755	−	0.997406i	$-0.477070\pi$
$859$	4.21902	0.143951	0.0719755	+	0.997406i	$0.477070\pi$
$860$	−12.6328	−0.430775
$861$	0.339135	0.0115577
$862$	3.66909	0.124970
$863$	33.2729	1.13262	0.566311	−	0.824192i	$-0.308370\pi$
$863$	33.2729	1.13262	0.566311	+	0.824192i	$0.308370\pi$
$864$	11.8986	0.404800
$865$	−11.3713	−0.386635
$866$	−9.31586	−0.316566
$867$	30.6315	1.04030
$868$	−11.7053	−0.397304
$869$	0	0
$870$	−2.52435	−0.0855836
$871$	0.638510	0.0216351
$872$	−2.02792	−0.0686741
$873$	14.6553	0.496006
$874$	4.30169	0.145507
$875$	1.25721	0.0425015
$876$	−21.2862	−0.719195
$877$	23.4583	0.792131	0.396066	−	0.918222i	$-0.370375\pi$
$877$	23.4583	0.792131	0.396066	+	0.918222i	$0.370375\pi$
$878$	4.32232	0.145871
$879$	46.5700	1.57077
$880$	0	0
$881$	−50.7212	−1.70884	−0.854420	−	0.519583i	$-0.826087\pi$
$881$	−50.7212	−1.70884	−0.854420	+	0.519583i	$0.826087\pi$
$882$	−2.72771	−0.0918468
$883$	−17.8801	−0.601713	−0.300856	−	0.953669i	$-0.597272\pi$
$883$	−17.8801	−0.601713	−0.300856	+	0.953669i	$0.597272\pi$
$884$	3.01689	0.101469
$885$	−27.4701	−0.923397
$886$	5.49667	0.184664
$887$	31.4391	1.05562	0.527810	−	0.849362i	$-0.323013\pi$
$887$	31.4391	1.05562	0.527810	+	0.849362i	$0.323013\pi$
$888$	−8.09232	−0.271561
$889$	−22.3508	−0.749620
$890$	−2.57888	−0.0864443
$891$	0	0
$892$	11.3228	0.379115
$893$	−42.8718	−1.43465
$894$	7.22703	0.241708
$895$	−14.4377	−0.482600
$896$	11.6770	0.390102
$897$	4.27789	0.142834
$898$	0.272050	0.00907843
$899$	17.4728	0.582752
$900$	−2.82564	−0.0941881
$901$	7.98530	0.266029
$902$	0	0
$903$	−17.8490	−0.593976
$904$	16.0344	0.533298
$905$	−3.78746	−0.125900
$906$	2.56853	0.0853337
$907$	5.81037	0.192930	0.0964651	−	0.995336i	$-0.469246\pi$
$907$	5.81037	0.192930	0.0964651	+	0.995336i	$0.469246\pi$
$908$	−15.9030	−0.527758
$909$	−9.04643	−0.300051
$910$	−0.422583	−0.0140085
$911$	44.5831	1.47710	0.738551	−	0.674197i	$-0.235510\pi$
$911$	44.5831	1.47710	0.738551	+	0.674197i	$0.235510\pi$
$912$	−44.8691	−1.48577
$913$	0	0
$914$	3.71573	0.122905
$915$	5.89658	0.194935
$916$	−45.4296	−1.50104
$917$	−3.08651	−0.101926
$918$	1.71241	0.0565180
$919$	36.6414	1.20869	0.604343	−	0.796724i	$-0.293436\pi$
$919$	36.6414	1.20869	0.604343	+	0.796724i	$0.293436\pi$
$920$	2.63553	0.0868910
$921$	−30.7163	−1.01214
$922$	5.92967	0.195283
$923$	−6.26909	−0.206350
$924$	0	0
$925$	−2.92059	−0.0960285
$926$	3.25146	0.106850
$927$	23.2614	0.764005
$928$	−13.2233	−0.434076
$929$	−12.7328	−0.417751	−0.208876	−	0.977942i	$-0.566980\pi$
$929$	−12.7328	−0.417751	−0.208876	+	0.977942i	$0.566980\pi$
$930$	−3.51710	−0.115330
$931$	−34.3827	−1.12685
$932$	1.70532	0.0558596
$933$	68.3531	2.23778
$934$	2.42434	0.0793268
$935$	0	0
$936$	1.95642	0.0639476
$937$	24.5637	0.802462	0.401231	−	0.915977i	$-0.368583\pi$
$937$	24.5637	0.802462	0.401231	+	0.915977i	$0.368583\pi$
$938$	0.269824	0.00881006
$939$	−39.4545	−1.28755
$940$	−12.7515	−0.415908
$941$	14.3627	0.468210	0.234105	−	0.972211i	$-0.424784\pi$
$941$	14.3627	0.468210	0.234105	+	0.972211i	$0.424784\pi$
$942$	9.75683	0.317895
$943$	0.256585	0.00835555
$944$	−43.1974	−1.40596
$945$	4.00617	0.130321
$946$	0	0
$947$	56.7586	1.84441	0.922204	−	0.386705i	$-0.126387\pi$
$947$	56.7586	1.84441	0.922204	+	0.386705i	$0.126387\pi$
$948$	−57.5192	−1.86814
$949$	5.31914	0.172667
$950$	2.13251	0.0691877
$951$	−52.4877	−1.70203
$952$	2.62610	0.0851125
$953$	−46.7552	−1.51455	−0.757275	−	0.653097i	$-0.773470\pi$
$953$	−46.7552	−1.51455	−0.757275	+	0.653097i	$0.773470\pi$
$954$	2.51393	0.0813916
$955$	−8.56733	−0.277232
$956$	−30.3204	−0.980631
$957$	0	0
$958$	−11.6753	−0.377212
$959$	−6.85558	−0.221378
$960$	−11.4829	−0.370609
$961$	−6.65567	−0.214699
$962$	0.981692	0.0316510
$963$	6.35763	0.204872
$964$	7.93282	0.255499
$965$	22.0513	0.709855
$966$	1.80776	0.0581639
$967$	53.1619	1.70957	0.854786	−	0.518981i	$-0.173688\pi$
$967$	53.1619	1.70957	0.854786	+	0.518981i	$0.173688\pi$
$968$	0	0
$969$	−21.5105	−0.691018
$970$	−3.28971	−0.105626
$971$	−43.0786	−1.38246	−0.691229	−	0.722636i	$-0.742930\pi$
$971$	−43.0786	−1.38246	−0.691229	+	0.722636i	$0.742930\pi$
$972$	−26.9812	−0.865422
$973$	28.6026	0.916957
$974$	−4.97859	−0.159524
$975$	2.12071	0.0679171
$976$	9.27253	0.296807
$977$	27.4889	0.879449	0.439725	−	0.898133i	$-0.355076\pi$
$977$	27.4889	0.879449	0.439725	+	0.898133i	$0.355076\pi$
$978$	−7.08911	−0.226685
$979$	0	0
$980$	−10.2265	−0.326675
$981$	2.32419	0.0742057
$982$	−8.99412	−0.287014
$983$	−5.56017	−0.177342	−0.0886709	−	0.996061i	$-0.528262\pi$
$983$	−5.56017	−0.177342	−0.0886709	+	0.996061i	$0.528262\pi$
$984$	0.352440	0.0112354
$985$	16.6673	0.531064
$986$	−1.90306	−0.0606057
$987$	−18.0167	−0.573477
$988$	11.9719	0.380877
$989$	−13.5043	−0.429411
$990$	0	0
$991$	19.1507	0.608341	0.304171	−	0.952618i	$-0.401621\pi$
$991$	19.1507	0.608341	0.304171	+	0.952618i	$0.401621\pi$
$992$	−18.4236	−0.584950
$993$	36.1331	1.14665
$994$	−2.64921	−0.0840280
$995$	13.6275	0.432020
$996$	−38.4411	−1.21805
$997$	−17.6921	−0.560316	−0.280158	−	0.959954i	$-0.590387\pi$
$997$	−17.6921	−0.560316	−0.280158	+	0.959954i	$0.590387\pi$
$998$	−9.72007	−0.307683
$999$	−9.30663	−0.294449

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7865.2.a.u.1.5	yes	8
11.10	odd	2		7865.2.a.t.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7865.2.a.t.1.4	✓	8	11.10	odd	2
7865.2.a.u.1.5	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7865 = 5 \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7865.a (trivial)

\(f(q)\)	\(=\)	\(q+0.336128 q^{2} -2.12071 q^{3} -1.88702 q^{4} -1.00000 q^{5} -0.712829 q^{6} -1.25721 q^{7} -1.30653 q^{8} +1.49741 q^{9} -0.336128 q^{10} +4.00182 q^{12} -1.00000 q^{13} -0.422583 q^{14} +2.12071 q^{15} +3.33487 q^{16} +1.59876 q^{17} +0.503321 q^{18} +6.34435 q^{19} +1.88702 q^{20} +2.66618 q^{21} +2.01720 q^{23} +2.77078 q^{24} +1.00000 q^{25} -0.336128 q^{26} +3.18655 q^{27} +2.37238 q^{28} -3.54131 q^{29} +0.712829 q^{30} -4.93400 q^{31} +3.73401 q^{32} +0.537387 q^{34} +1.25721 q^{35} -2.82564 q^{36} -2.92059 q^{37} +2.13251 q^{38} +2.12071 q^{39} +1.30653 q^{40} +0.127199 q^{41} +0.896177 q^{42} -6.69458 q^{43} -1.49741 q^{45} +0.678035 q^{46} -6.75748 q^{47} -7.07230 q^{48} -5.41942 q^{49} +0.336128 q^{50} -3.39050 q^{51} +1.88702 q^{52} +4.99469 q^{53} +1.07109 q^{54} +1.64259 q^{56} -13.4545 q^{57} -1.19033 q^{58} -12.9532 q^{59} -4.00182 q^{60} +2.78047 q^{61} -1.65845 q^{62} -1.88256 q^{63} -5.41464 q^{64} +1.00000 q^{65} -0.638510 q^{67} -3.01689 q^{68} -4.27789 q^{69} +0.422583 q^{70} +6.26909 q^{71} -1.95642 q^{72} -5.31914 q^{73} -0.981692 q^{74} -2.12071 q^{75} -11.9719 q^{76} +0.712829 q^{78} -14.3733 q^{79} -3.33487 q^{80} -11.2500 q^{81} +0.0427550 q^{82} -9.60590 q^{83} -5.03113 q^{84} -1.59876 q^{85} -2.25023 q^{86} +7.51010 q^{87} +7.67233 q^{89} -0.503321 q^{90} +1.25721 q^{91} -3.80648 q^{92} +10.4636 q^{93} -2.27138 q^{94} -6.34435 q^{95} -7.91876 q^{96} +9.78708 q^{97} -1.82162 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} - 5 q^{3} + 7 q^{4} - 8 q^{5} - 7 q^{6} + 9 q^{7} + 3 q^{8} + 3 q^{9} - q^{10} - 15 q^{12} - 8 q^{13} - 7 q^{14} + 5 q^{15} + 9 q^{16} + 4 q^{17} + 8 q^{18} + 16 q^{19} - 7 q^{20} - 23 q^{21}+ \cdots + 41 q^{98}+O(q^{100}) \) 8 * q + q^2 - 5 * q^3 + 7 * q^4 - 8 * q^5 - 7 * q^6 + 9 * q^7 + 3 * q^8 + 3 * q^9 - q^10 - 15 * q^12 - 8 * q^13 - 7 * q^14 + 5 * q^15 + 9 * q^16 + 4 * q^17 + 8 * q^18 + 16 * q^19 - 7 * q^20 - 23 * q^21 - 9 * q^23 + 5 * q^24 + 8 * q^25 - q^26 - 11 * q^27 + 11 * q^28 - 13 * q^29 + 7 * q^30 - 28 * q^32 + 13 * q^34 - 9 * q^35 + 12 * q^36 - 5 * q^37 + 55 * q^38 + 5 * q^39 - 3 * q^40 + 13 * q^41 - 5 * q^42 + 5 * q^43 - 3 * q^45 + 27 * q^46 + 10 * q^47 - 36 * q^48 + 23 * q^49 + q^50 - 37 * q^51 - 7 * q^52 + 14 * q^53 - 45 * q^54 - 18 * q^56 - 21 * q^57 + 3 * q^58 - 20 * q^59 + 15 * q^60 + 6 * q^61 + 25 * q^62 + 26 * q^63 - 3 * q^64 + 8 * q^65 + 4 * q^67 - 9 * q^69 + 7 * q^70 - 9 * q^71 + 35 * q^72 + 10 * q^73 + 42 * q^74 - 5 * q^75 + 19 * q^76 + 7 * q^78 + 13 * q^79 - 9 * q^80 + 16 * q^82 + 26 * q^83 - 58 * q^84 - 4 * q^85 + 26 * q^86 + 11 * q^87 - 6 * q^89 - 8 * q^90 - 9 * q^91 - 16 * q^92 + 3 * q^93 + 23 * q^94 - 16 * q^95 + 21 * q^96 + 26 * q^97 + 41 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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