LMFDB - Embedded newform 160.3.b.a.31.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,3,Mod(31,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("160.31");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	160.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$4.35968422976$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.2
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	160.31
Dual form			160.3.b.a.31.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.763932i q^{3} -2.23607 q^{5} +12.1803i q^{7} +8.41641 q^{9} +O(q^{10})$	$q-0.763932i q^{3} -2.23607 q^{5} +12.1803i q^{7} +8.41641 q^{9} +5.52786i q^{11} +20.4721 q^{13} +1.70820i q^{15} -1.05573 q^{17} -12.0000i q^{19} +9.30495 q^{21} +31.5967i q^{23} +5.00000 q^{25} -13.3050i q^{27} -44.8328 q^{29} +27.4164i q^{31} +4.22291 q^{33} -27.2361i q^{35} +23.5279 q^{37} -15.6393i q^{39} +8.69505 q^{41} -26.6525i q^{43} -18.8197 q^{45} -44.5410i q^{47} -99.3607 q^{49} +0.806504i q^{51} +0.695048 q^{53} -12.3607i q^{55} -9.16718 q^{57} -94.6099i q^{59} -33.1935 q^{61} +102.515i q^{63} -45.7771 q^{65} +82.2067i q^{67} +24.1378 q^{69} -122.249i q^{71} +132.164 q^{73} -3.81966i q^{75} -67.3313 q^{77} -134.833i q^{79} +65.5836 q^{81} +19.4590i q^{83} +2.36068 q^{85} +34.2492i q^{87} -30.0000 q^{89} +249.358i q^{91} +20.9443 q^{93} +26.8328i q^{95} +12.3344 q^{97} +46.5248i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{9}+O(q^{10}) $ 4 * q - 20 * q^9	$ 4 q - 20 q^{9} + 64 q^{13} - 40 q^{17} - 88 q^{21} + 20 q^{25} - 72 q^{29} + 160 q^{33} + 112 q^{37} + 160 q^{41} - 120 q^{45} - 308 q^{49} + 128 q^{53} - 144 q^{57} + 64 q^{61} - 40 q^{65} - 136 q^{69} - 8 q^{73} + 160 q^{77} + 316 q^{81} - 80 q^{85} - 120 q^{89} + 48 q^{93} + 264 q^{97}+O(q^{100}) $ 4 * q - 20 * q^9 + 64 * q^13 - 40 * q^17 - 88 * q^21 + 20 * q^25 - 72 * q^29 + 160 * q^33 + 112 * q^37 + 160 * q^41 - 120 * q^45 - 308 * q^49 + 128 * q^53 - 144 * q^57 + 64 * q^61 - 40 * q^65 - 136 * q^69 - 8 * q^73 + 160 * q^77 + 316 * q^81 - 80 * q^85 - 120 * q^89 + 48 * q^93 + 264 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 0.763932i	− 0.254644i	−0.991861	−	0.127322i	$-0.959362\pi$
$3$	− 0.763932i	− 0.254644i	0.991861	−	0.127322i	$-0.0406382\pi$
$4$	0	0
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	12.1803i	1.74005i	0.493009	+	0.870024i	$0.335897\pi$
$7$	12.1803i	1.74005i	−0.493009	+	0.870024i	$0.664103\pi$
$8$	0	0
$9$	8.41641	0.935156
$10$	0	0
$11$	5.52786i	0.502533i	0.967918	+	0.251267i	$0.0808471\pi$
$11$	5.52786i	0.502533i	−0.967918	+	0.251267i	$0.919153\pi$
$12$	0	0
$13$	20.4721	1.57478	0.787390	−	0.616455i	$-0.211432\pi$
$13$	20.4721	1.57478	0.787390	+	0.616455i	$0.211432\pi$
$14$	0	0
$15$	1.70820i	0.113880i
$16$	0	0
$17$	−1.05573	−0.0621017	−0.0310508	−	0.999518i	$-0.509885\pi$
$17$	−1.05573	−0.0621017	−0.0310508	+	0.999518i	$0.509885\pi$
$18$	0	0
$19$	− 12.0000i	− 0.631579i	−0.948829	−	0.315789i	$-0.897731\pi$
$19$	− 12.0000i	− 0.631579i	0.948829	−	0.315789i	$-0.102269\pi$
$20$	0	0
$21$	9.30495	0.443093
$22$	0	0
$23$	31.5967i	1.37377i	0.726765	+	0.686886i	$0.241023\pi$
$23$	31.5967i	1.37377i	−0.726765	+	0.686886i	$0.758977\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 13.3050i	− 0.492776i
$28$	0	0
$29$	−44.8328	−1.54596	−0.772980	−	0.634431i	$-0.781235\pi$
$29$	−44.8328	−1.54596	−0.772980	+	0.634431i	$0.781235\pi$
$30$	0	0
$31$	27.4164i	0.884400i	0.896916	+	0.442200i	$0.145802\pi$
$31$	27.4164i	0.884400i	−0.896916	+	0.442200i	$0.854198\pi$
$32$	0	0
$33$	4.22291	0.127967
$34$	0	0
$35$	− 27.2361i	− 0.778173i
$36$	0	0
$37$	23.5279	0.635888	0.317944	−	0.948109i	$-0.397008\pi$
$37$	23.5279	0.635888	0.317944	+	0.948109i	$0.397008\pi$
$38$	0	0
$39$	− 15.6393i	− 0.401008i
$40$	0	0
$41$	8.69505	0.212074	0.106037	−	0.994362i	$-0.466184\pi$
$41$	8.69505	0.212074	0.106037	+	0.994362i	$0.466184\pi$
$42$	0	0
$43$	− 26.6525i	− 0.619825i	−0.950765	−	0.309913i	$-0.899700\pi$
$43$	− 26.6525i	− 0.619825i	0.950765	−	0.309913i	$-0.100300\pi$
$44$	0	0
$45$	−18.8197	−0.418215
$46$	0	0
$47$	− 44.5410i	− 0.947681i	−0.880611	−	0.473841i	$-0.842867\pi$
$47$	− 44.5410i	− 0.947681i	0.880611	−	0.473841i	$-0.157133\pi$
$48$	0	0
$49$	−99.3607	−2.02777
$50$	0	0
$51$	0.806504i	0.0158138i
$52$	0	0
$53$	0.695048	0.0131141	0.00655706	−	0.999979i	$-0.497913\pi$
$53$	0.695048	0.0131141	0.00655706	+	0.999979i	$0.497913\pi$
$54$	0	0
$55$	− 12.3607i	− 0.224740i
$56$	0	0
$57$	−9.16718	−0.160828
$58$	0	0
$59$	− 94.6099i	− 1.60356i	−0.597621	−	0.801779i	$-0.703887\pi$
$59$	− 94.6099i	− 1.60356i	0.597621	−	0.801779i	$-0.296113\pi$
$60$	0	0
$61$	−33.1935	−0.544156	−0.272078	−	0.962275i	$-0.587711\pi$
$61$	−33.1935	−0.544156	−0.272078	+	0.962275i	$0.587711\pi$
$62$	0	0
$63$	102.515i	1.62722i
$64$	0	0
$65$	−45.7771	−0.704263
$66$	0	0
$67$	82.2067i	1.22696i	0.789708	+	0.613482i	$0.210232\pi$
$67$	82.2067i	1.22696i	−0.789708	+	0.613482i	$0.789768\pi$
$68$	0	0
$69$	24.1378	0.349823
$70$	0	0
$71$	− 122.249i	− 1.72182i	−0.508757	−	0.860910i	$-0.669895\pi$
$71$	− 122.249i	− 1.72182i	0.508757	−	0.860910i	$-0.330105\pi$
$72$	0	0
$73$	132.164	1.81047	0.905233	−	0.424915i	$-0.139696\pi$
$73$	132.164	1.81047	0.905233	+	0.424915i	$0.139696\pi$
$74$	0	0
$75$	− 3.81966i	− 0.0509288i
$76$	0	0
$77$	−67.3313	−0.874432
$78$	0	0
$79$	− 134.833i	− 1.70674i	−0.521302	−	0.853372i	$-0.674553\pi$
$79$	− 134.833i	− 1.70674i	0.521302	−	0.853372i	$-0.325447\pi$
$80$	0	0
$81$	65.5836	0.809674
$82$	0	0
$83$	19.4590i	0.234446i	0.993106	+	0.117223i	$0.0373992\pi$
$83$	19.4590i	0.234446i	−0.993106	+	0.117223i	$0.962601\pi$
$84$	0	0
$85$	2.36068	0.0277727
$86$	0	0
$87$	34.2492i	0.393669i
$88$	0	0
$89$	−30.0000	−0.337079	−0.168539	−	0.985695i	$-0.553905\pi$
$89$	−30.0000	−0.337079	−0.168539	+	0.985695i	$0.553905\pi$
$90$	0	0
$91$	249.358i	2.74019i
$92$	0	0
$93$	20.9443	0.225207
$94$	0	0
$95$	26.8328i	0.282451i
$96$	0	0
$97$	12.3344	0.127158	0.0635792	−	0.997977i	$-0.479748\pi$
$97$	12.3344	0.127158	0.0635792	+	0.997977i	$0.479748\pi$
$98$	0	0
$99$	46.5248i	0.469947i
$100$	0	0
$101$	105.554	1.04509	0.522545	−	0.852611i	$-0.324983\pi$
$101$	105.554	1.04509	0.522545	+	0.852611i	$0.324983\pi$
$102$	0	0
$103$	100.987i	0.980455i	0.871595	+	0.490227i	$0.163086\pi$
$103$	100.987i	0.980455i	−0.871595	+	0.490227i	$0.836914\pi$
$104$	0	0
$105$	−20.8065	−0.198157
$106$	0	0
$107$	22.8754i	0.213789i	0.994270	+	0.106894i	$0.0340907\pi$
$107$	22.8754i	0.213789i	−0.994270	+	0.106894i	$0.965909\pi$
$108$	0	0
$109$	−35.5279	−0.325944	−0.162972	−	0.986631i	$-0.552108\pi$
$109$	−35.5279	−0.325944	−0.162972	+	0.986631i	$0.552108\pi$
$110$	0	0
$111$	− 17.9737i	− 0.161925i
$112$	0	0
$113$	−70.7214	−0.625853	−0.312926	−	0.949777i	$-0.601309\pi$
$113$	−70.7214	−0.625853	−0.312926	+	0.949777i	$0.601309\pi$
$114$	0	0
$115$	− 70.6525i	− 0.614369i
$116$	0	0
$117$	172.302	1.47267
$118$	0	0
$119$	− 12.8591i	− 0.108060i
$120$	0	0
$121$	90.4427	0.747460
$122$	0	0
$123$	− 6.64243i	− 0.0540035i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	− 148.541i	− 1.16961i	−0.811172	−	0.584807i	$-0.801170\pi$
$127$	− 148.541i	− 1.16961i	0.811172	−	0.584807i	$-0.198830\pi$
$128$	0	0
$129$	−20.3607	−0.157835
$130$	0	0
$131$	34.9180i	0.266549i	0.991079	+	0.133275i	$0.0425492\pi$
$131$	34.9180i	0.266549i	−0.991079	+	0.133275i	$0.957451\pi$
$132$	0	0
$133$	146.164	1.09898
$134$	0	0
$135$	29.7508i	0.220376i
$136$	0	0
$137$	100.610	0.734379	0.367189	−	0.930146i	$-0.380320\pi$
$137$	100.610	0.734379	0.367189	+	0.930146i	$0.380320\pi$
$138$	0	0
$139$	198.610i	1.42885i	0.699713	+	0.714424i	$0.253311\pi$
$139$	198.610i	1.42885i	−0.699713	+	0.714424i	$0.746689\pi$
$140$	0	0
$141$	−34.0263	−0.241321
$142$	0	0
$143$	113.167i	0.791379i
$144$	0	0
$145$	100.249	0.691374
$146$	0	0
$147$	75.9048i	0.516359i
$148$	0	0
$149$	−203.082	−1.36297	−0.681483	−	0.731834i	$-0.738665\pi$
$149$	−203.082	−1.36297	−0.681483	+	0.731834i	$0.738665\pi$
$150$	0	0
$151$	− 113.803i	− 0.753665i	−0.926281	−	0.376832i	$-0.877013\pi$
$151$	− 113.803i	− 0.753665i	0.926281	−	0.376832i	$-0.122987\pi$
$152$	0	0
$153$	−8.88544	−0.0580748
$154$	0	0
$155$	− 61.3050i	− 0.395516i
$156$	0	0
$157$	−144.918	−0.923044	−0.461522	−	0.887129i	$-0.652697\pi$
$157$	−144.918	−0.923044	−0.461522	+	0.887129i	$0.652697\pi$
$158$	0	0
$159$	− 0.530970i	− 0.00333943i
$160$	0	0
$161$	−384.859	−2.39043
$162$	0	0
$163$	− 203.374i	− 1.24769i	−0.781547	−	0.623846i	$-0.785569\pi$
$163$	− 203.374i	− 1.24769i	0.781547	−	0.623846i	$-0.214431\pi$
$164$	0	0
$165$	−9.44272	−0.0572286
$166$	0	0
$167$	− 88.7639i	− 0.531521i	−0.964039	−	0.265760i	$-0.914377\pi$
$167$	− 88.7639i	− 0.531521i	0.964039	−	0.265760i	$-0.0856230\pi$
$168$	0	0
$169$	250.108	1.47993
$170$	0	0
$171$	− 100.997i	− 0.590625i
$172$	0	0
$173$	−135.580	−0.783702	−0.391851	−	0.920029i	$-0.628165\pi$
$173$	−135.580	−0.783702	−0.391851	+	0.920029i	$0.628165\pi$
$174$	0	0
$175$	60.9017i	0.348010i
$176$	0	0
$177$	−72.2755	−0.408336
$178$	0	0
$179$	− 203.830i	− 1.13871i	−0.822091	−	0.569357i	$-0.807192\pi$
$179$	− 203.830i	− 1.13871i	0.822091	−	0.569357i	$-0.192808\pi$
$180$	0	0
$181$	22.6687	0.125242	0.0626208	−	0.998037i	$-0.480054\pi$
$181$	22.6687	0.125242	0.0626208	+	0.998037i	$0.480054\pi$
$182$	0	0
$183$	25.3576i	0.138566i
$184$	0	0
$185$	−52.6099	−0.284378
$186$	0	0
$187$	− 5.83592i	− 0.0312081i
$188$	0	0
$189$	162.059	0.857454
$190$	0	0
$191$	− 150.748i	− 0.789255i	−0.918841	−	0.394627i	$-0.870874\pi$
$191$	− 150.748i	− 0.789255i	0.918841	−	0.394627i	$-0.129126\pi$
$192$	0	0
$193$	−210.498	−1.09067	−0.545333	−	0.838220i	$-0.683597\pi$
$193$	−210.498	−1.09067	−0.545333	+	0.838220i	$0.683597\pi$
$194$	0	0
$195$	34.9706i	0.179336i
$196$	0	0
$197$	−97.0232	−0.492504	−0.246252	−	0.969206i	$-0.579199\pi$
$197$	−97.0232	−0.492504	−0.246252	+	0.969206i	$0.579199\pi$
$198$	0	0
$199$	104.892i	0.527094i	0.964647	+	0.263547i	$0.0848924\pi$
$199$	104.892i	0.527094i	−0.964647	+	0.263547i	$0.915108\pi$
$200$	0	0
$201$	62.8003	0.312439
$202$	0	0
$203$	− 546.079i	− 2.69004i
$204$	0	0
$205$	−19.4427	−0.0948425
$206$	0	0
$207$	265.931i	1.28469i
$208$	0	0
$209$	66.3344	0.317389
$210$	0	0
$211$	240.584i	1.14021i	0.821573	+	0.570103i	$0.193097\pi$
$211$	240.584i	1.14021i	−0.821573	+	0.570103i	$0.806903\pi$
$212$	0	0
$213$	−93.3901	−0.438451
$214$	0	0
$215$	59.5967i	0.277194i
$216$	0	0
$217$	−333.941	−1.53890
$218$	0	0
$219$	− 100.964i	− 0.461025i
$220$	0	0
$221$	−21.6130	−0.0977964
$222$	0	0
$223$	− 18.2918i	− 0.0820260i	−0.999159	−	0.0410130i	$-0.986941\pi$
$223$	− 18.2918i	− 0.0820260i	0.999159	−	0.0410130i	$-0.0130585\pi$
$224$	0	0
$225$	42.0820	0.187031
$226$	0	0
$227$	− 92.4559i	− 0.407295i	−0.979044	−	0.203647i	$-0.934720\pi$
$227$	− 92.4559i	− 0.407295i	0.979044	−	0.203647i	$-0.0652796\pi$
$228$	0	0
$229$	165.161	0.721227	0.360613	−	0.932715i	$-0.382567\pi$
$229$	165.161	0.721227	0.360613	+	0.932715i	$0.382567\pi$
$230$	0	0
$231$	51.4365i	0.222669i
$232$	0	0
$233$	37.3313	0.160220	0.0801100	−	0.996786i	$-0.474473\pi$
$233$	37.3313	0.160220	0.0801100	+	0.996786i	$0.474473\pi$
$234$	0	0
$235$	99.5967i	0.423816i
$236$	0	0
$237$	−103.003	−0.434612
$238$	0	0
$239$	397.495i	1.66316i	0.555405	+	0.831580i	$0.312563\pi$
$239$	397.495i	1.66316i	−0.555405	+	0.831580i	$0.687437\pi$
$240$	0	0
$241$	237.915	0.987199	0.493599	−	0.869689i	$-0.335681\pi$
$241$	237.915	0.987199	0.493599	+	0.869689i	$0.335681\pi$
$242$	0	0
$243$	− 169.846i	− 0.698955i
$244$	0	0
$245$	222.177	0.906846
$246$	0	0
$247$	− 245.666i	− 0.994598i
$248$	0	0
$249$	14.8653	0.0597002
$250$	0	0
$251$	− 277.803i	− 1.10679i	−0.832920	−	0.553393i	$-0.813333\pi$
$251$	− 277.803i	− 1.10679i	0.832920	−	0.553393i	$-0.186667\pi$
$252$	0	0
$253$	−174.663	−0.690366
$254$	0	0
$255$	− 1.80340i	− 0.00707215i
$256$	0	0
$257$	446.498	1.73735	0.868674	−	0.495384i	$-0.164973\pi$
$257$	446.498	1.73735	0.868674	+	0.495384i	$0.164973\pi$
$258$	0	0
$259$	286.577i	1.10648i
$260$	0	0
$261$	−377.331	−1.44571
$262$	0	0
$263$	105.039i	0.399390i	0.979858	+	0.199695i	$0.0639951\pi$
$263$	105.039i	0.399390i	−0.979858	+	0.199695i	$0.936005\pi$
$264$	0	0
$265$	−1.55418	−0.00586481
$266$	0	0
$267$	22.9180i	0.0858351i
$268$	0	0
$269$	504.354	1.87492	0.937462	−	0.348088i	$-0.113169\pi$
$269$	504.354	1.87492	0.937462	+	0.348088i	$0.113169\pi$
$270$	0	0
$271$	− 164.413i	− 0.606691i	−0.952881	−	0.303346i	$-0.901896\pi$
$271$	− 164.413i	− 0.606691i	0.952881	−	0.303346i	$-0.0981037\pi$
$272$	0	0
$273$	190.492	0.697774
$274$	0	0
$275$	27.6393i	0.100507i
$276$	0	0
$277$	315.580	1.13928	0.569640	−	0.821894i	$-0.307083\pi$
$277$	315.580	1.13928	0.569640	+	0.821894i	$0.307083\pi$
$278$	0	0
$279$	230.748i	0.827053i
$280$	0	0
$281$	−207.410	−0.738115	−0.369057	−	0.929407i	$-0.620319\pi$
$281$	−207.410	−0.738115	−0.369057	+	0.929407i	$0.620319\pi$
$282$	0	0
$283$	438.895i	1.55087i	0.631429	+	0.775434i	$0.282469\pi$
$283$	438.895i	1.55087i	−0.631429	+	0.775434i	$0.717531\pi$
$284$	0	0
$285$	20.4984	0.0719244
$286$	0	0
$287$	105.909i	0.369020i
$288$	0	0
$289$	−287.885	−0.996143
$290$	0	0
$291$	− 9.42262i	− 0.0323801i
$292$	0	0
$293$	−272.918	−0.931461	−0.465730	−	0.884927i	$-0.654208\pi$
$293$	−272.918	−0.931461	−0.465730	+	0.884927i	$0.654208\pi$
$294$	0	0
$295$	211.554i	0.717133i
$296$	0	0
$297$	73.5480	0.247636
$298$	0	0
$299$	646.853i	2.16339i
$300$	0	0
$301$	324.636	1.07853
$302$	0	0
$303$	− 80.6362i	− 0.266126i
$304$	0	0
$305$	74.2229	0.243354
$306$	0	0
$307$	− 251.459i	− 0.819085i	−0.912291	−	0.409542i	$-0.865688\pi$
$307$	− 251.459i	− 0.819085i	0.912291	−	0.409542i	$-0.134312\pi$
$308$	0	0
$309$	77.1471	0.249667
$310$	0	0
$311$	352.912i	1.13476i	0.823455	+	0.567382i	$0.192044\pi$
$311$	352.912i	1.13476i	−0.823455	+	0.567382i	$0.807956\pi$
$312$	0	0
$313$	−435.220	−1.39048	−0.695239	−	0.718778i	$-0.744702\pi$
$313$	−435.220	−1.39048	−0.695239	+	0.718778i	$0.744702\pi$
$314$	0	0
$315$	− 229.230i	− 0.727714i
$316$	0	0
$317$	41.4164	0.130651	0.0653256	−	0.997864i	$-0.479191\pi$
$317$	41.4164	0.130651	0.0653256	+	0.997864i	$0.479191\pi$
$318$	0	0
$319$	− 247.830i	− 0.776896i
$320$	0	0
$321$	17.4752	0.0544400
$322$	0	0
$323$	12.6687i	0.0392221i
$324$	0	0
$325$	102.361	0.314956
$326$	0	0
$327$	27.1409i	0.0829996i
$328$	0	0
$329$	542.525	1.64901
$330$	0	0
$331$	− 103.416i	− 0.312436i	−0.987723	−	0.156218i	$-0.950070\pi$
$331$	− 103.416i	− 0.312436i	0.987723	−	0.156218i	$-0.0499303\pi$
$332$	0	0
$333$	198.020	0.594655
$334$	0	0
$335$	− 183.820i	− 0.548715i
$336$	0	0
$337$	106.997	0.317498	0.158749	−	0.987319i	$-0.449254\pi$
$337$	106.997	0.317498	0.158749	+	0.987319i	$0.449254\pi$
$338$	0	0
$339$	54.0263i	0.159370i
$340$	0	0
$341$	−151.554	−0.444440
$342$	0	0
$343$	− 613.410i	− 1.78837i
$344$	0	0
$345$	−53.9737	−0.156445
$346$	0	0
$347$	511.426i	1.47385i	0.675974	+	0.736926i	$0.263723\pi$
$347$	511.426i	1.47385i	−0.675974	+	0.736926i	$0.736277\pi$
$348$	0	0
$349$	−63.1672	−0.180995	−0.0904974	−	0.995897i	$-0.528846\pi$
$349$	−63.1672	−0.180995	−0.0904974	+	0.995897i	$0.528846\pi$
$350$	0	0
$351$	− 272.381i	− 0.776014i
$352$	0	0
$353$	62.7740	0.177830	0.0889150	−	0.996039i	$-0.471660\pi$
$353$	62.7740	0.177830	0.0889150	+	0.996039i	$0.471660\pi$
$354$	0	0
$355$	273.358i	0.770021i
$356$	0	0
$357$	−9.82350	−0.0275168
$358$	0	0
$359$	− 176.105i	− 0.490544i	−0.969454	−	0.245272i	$-0.921123\pi$
$359$	− 176.105i	− 0.490544i	0.969454	−	0.245272i	$-0.0788772\pi$
$360$	0	0
$361$	217.000	0.601108
$362$	0	0
$363$	− 69.0921i	− 0.190336i
$364$	0	0
$365$	−295.528	−0.809665
$366$	0	0
$367$	− 190.259i	− 0.518418i	−0.965821	−	0.259209i	$-0.916538\pi$
$367$	− 190.259i	− 0.518418i	0.965821	−	0.259209i	$-0.0834618\pi$
$368$	0	0
$369$	73.1811	0.198323
$370$	0	0
$371$	8.46592i	0.0228192i
$372$	0	0
$373$	−525.076	−1.40771	−0.703855	−	0.710344i	$-0.748540\pi$
$373$	−525.076	−1.40771	−0.703855	+	0.710344i	$0.748540\pi$
$374$	0	0
$375$	8.54102i	0.0227761i
$376$	0	0
$377$	−917.823	−2.43455
$378$	0	0
$379$	− 530.217i	− 1.39899i	−0.714638	−	0.699494i	$-0.753409\pi$
$379$	− 530.217i	− 1.39899i	0.714638	−	0.699494i	$-0.246591\pi$
$380$	0	0
$381$	−113.475	−0.297835
$382$	0	0
$383$	− 165.177i	− 0.431272i	−0.976474	−	0.215636i	$-0.930818\pi$
$383$	− 165.177i	− 0.431272i	0.976474	−	0.215636i	$-0.0691825\pi$
$384$	0	0
$385$	150.557	0.391058
$386$	0	0
$387$	− 224.318i	− 0.579633i
$388$	0	0
$389$	−532.079	−1.36781	−0.683906	−	0.729570i	$-0.739720\pi$
$389$	−532.079	−1.36781	−0.683906	+	0.729570i	$0.739720\pi$
$390$	0	0
$391$	− 33.3576i	− 0.0853135i
$392$	0	0
$393$	26.6749	0.0678752
$394$	0	0
$395$	301.495i	0.763279i
$396$	0	0
$397$	−512.407	−1.29070	−0.645349	−	0.763888i	$-0.723288\pi$
$397$	−512.407	−1.29070	−0.645349	+	0.763888i	$0.723288\pi$
$398$	0	0
$399$	− 111.659i	− 0.279848i
$400$	0	0
$401$	22.3282	0.0556812	0.0278406	−	0.999612i	$-0.491137\pi$
$401$	22.3282	0.0556812	0.0278406	+	0.999612i	$0.491137\pi$
$402$	0	0
$403$	561.272i	1.39274i
$404$	0	0
$405$	−146.649	−0.362097
$406$	0	0
$407$	130.059i	0.319555i
$408$	0	0
$409$	−42.4659	−0.103829	−0.0519143	−	0.998652i	$-0.516532\pi$
$409$	−42.4659	−0.103829	−0.0519143	+	0.998652i	$0.516532\pi$
$410$	0	0
$411$	− 76.8591i	− 0.187005i
$412$	0	0
$413$	1152.38	2.79027
$414$	0	0
$415$	− 43.5116i	− 0.104847i
$416$	0	0
$417$	151.724	0.363848
$418$	0	0
$419$	− 284.381i	− 0.678713i	−0.940658	−	0.339357i	$-0.889791\pi$
$419$	− 284.381i	− 0.678713i	0.940658	−	0.339357i	$-0.110209\pi$
$420$	0	0
$421$	201.246	0.478019	0.239010	−	0.971017i	$-0.423177\pi$
$421$	201.246	0.478019	0.239010	+	0.971017i	$0.423177\pi$
$422$	0	0
$423$	− 374.875i	− 0.886230i
$424$	0	0
$425$	−5.27864	−0.0124203
$426$	0	0
$427$	− 404.308i	− 0.946857i
$428$	0	0
$429$	86.4520	0.201520
$430$	0	0
$431$	81.9086i	0.190043i	0.995475	+	0.0950216i	$0.0302920\pi$
$431$	81.9086i	0.190043i	−0.995475	+	0.0950216i	$0.969708\pi$
$432$	0	0
$433$	−195.325	−0.451097	−0.225549	−	0.974232i	$-0.572417\pi$
$433$	−195.325	−0.451097	−0.225549	+	0.974232i	$0.572417\pi$
$434$	0	0
$435$	− 76.5836i	− 0.176054i
$436$	0	0
$437$	379.161	0.867645
$438$	0	0
$439$	− 466.780i	− 1.06328i	−0.846970	−	0.531640i	$-0.821576\pi$
$439$	− 466.780i	− 1.06328i	0.846970	−	0.531640i	$-0.178424\pi$
$440$	0	0
$441$	−836.260	−1.89628
$442$	0	0
$443$	376.508i	0.849906i	0.905215	+	0.424953i	$0.139709\pi$
$443$	376.508i	0.849906i	−0.905215	+	0.424953i	$0.860291\pi$
$444$	0	0
$445$	67.0820	0.150746
$446$	0	0
$447$	155.141i	0.347071i
$448$	0	0
$449$	−247.029	−0.550177	−0.275088	−	0.961419i	$-0.588707\pi$
$449$	−247.029	−0.550177	−0.275088	+	0.961419i	$0.588707\pi$
$450$	0	0
$451$	48.0650i	0.106574i
$452$	0	0
$453$	−86.9381	−0.191916
$454$	0	0
$455$	− 557.580i	− 1.22545i
$456$	0	0
$457$	−229.003	−0.501101	−0.250550	−	0.968104i	$-0.580612\pi$
$457$	−229.003	−0.501101	−0.250550	+	0.968104i	$0.580612\pi$
$458$	0	0
$459$	14.0464i	0.0306022i
$460$	0	0
$461$	62.7740	0.136169	0.0680846	−	0.997680i	$-0.478311\pi$
$461$	62.7740	0.136169	0.0680846	+	0.997680i	$0.478311\pi$
$462$	0	0
$463$	− 204.075i	− 0.440767i	−0.975413	−	0.220383i	$-0.929269\pi$
$463$	− 204.075i	− 0.440767i	0.975413	−	0.220383i	$-0.0707309\pi$
$464$	0	0
$465$	−46.8328	−0.100716
$466$	0	0
$467$	412.371i	0.883021i	0.897256	+	0.441510i	$0.145557\pi$
$467$	412.371i	0.883021i	−0.897256	+	0.441510i	$0.854443\pi$
$468$	0	0
$469$	−1001.30	−2.13498
$470$	0	0
$471$	110.707i	0.235048i
$472$	0	0
$473$	147.331	0.311483
$474$	0	0
$475$	− 60.0000i	− 0.126316i
$476$	0	0
$477$	5.84981	0.0122638
$478$	0	0
$479$	638.597i	1.33319i	0.745421	+	0.666594i	$0.232249\pi$
$479$	638.597i	1.33319i	−0.745421	+	0.666594i	$0.767751\pi$
$480$	0	0
$481$	481.666	1.00138
$482$	0	0
$483$	294.006i	0.608709i
$484$	0	0
$485$	−27.5805	−0.0568670
$486$	0	0
$487$	− 24.8692i	− 0.0510661i	−0.999674	−	0.0255330i	$-0.991872\pi$
$487$	− 24.8692i	− 0.0510661i	0.999674	−	0.0255330i	$-0.00812830\pi$
$488$	0	0
$489$	−155.364	−0.317717
$490$	0	0
$491$	− 320.584i	− 0.652920i	−0.945211	−	0.326460i	$-0.894144\pi$
$491$	− 320.584i	− 0.652920i	0.945211	−	0.326460i	$-0.105856\pi$
$492$	0	0
$493$	47.3313	0.0960066
$494$	0	0
$495$	− 104.033i	− 0.210167i
$496$	0	0
$497$	1489.04	2.99605
$498$	0	0
$499$	− 168.774i	− 0.338224i	−0.985597	−	0.169112i	$-0.945910\pi$
$499$	− 168.774i	− 0.338224i	0.985597	−	0.169112i	$-0.0540900\pi$
$500$	0	0
$501$	−67.8096	−0.135349
$502$	0	0
$503$	662.089i	1.31628i	0.752895	+	0.658140i	$0.228657\pi$
$503$	662.089i	1.31628i	−0.752895	+	0.658140i	$0.771343\pi$
$504$	0	0
$505$	−236.026	−0.467379
$506$	0	0
$507$	− 191.066i	− 0.376856i
$508$	0	0
$509$	651.823	1.28060	0.640298	−	0.768126i	$-0.278811\pi$
$509$	651.823	1.28060	0.640298	+	0.768126i	$0.278811\pi$
$510$	0	0
$511$	1609.80i	3.15030i
$512$	0	0
$513$	−159.659	−0.311227
$514$	0	0
$515$	− 225.813i	− 0.438473i
$516$	0	0
$517$	246.217	0.476241
$518$	0	0
$519$	103.574i	0.199565i
$520$	0	0
$521$	750.984	1.44143	0.720714	−	0.693232i	$-0.243814\pi$
$521$	750.984	1.44143	0.720714	+	0.693232i	$0.243814\pi$
$522$	0	0
$523$	328.128i	0.627395i	0.949523	+	0.313698i	$0.101568\pi$
$523$	328.128i	0.627395i	−0.949523	+	0.313698i	$0.898432\pi$
$524$	0	0
$525$	46.5248	0.0886186
$526$	0	0
$527$	− 28.9443i	− 0.0549227i
$528$	0	0
$529$	−469.354	−0.887249
$530$	0	0
$531$	− 796.276i	− 1.49958i
$532$	0	0
$533$	178.006	0.333970
$534$	0	0
$535$	− 51.1509i	− 0.0956092i
$536$	0	0
$537$	−155.712	−0.289967
$538$	0	0
$539$	− 549.252i	− 1.01902i
$540$	0	0
$541$	238.563	0.440968	0.220484	−	0.975391i	$-0.429236\pi$
$541$	238.563	0.440968	0.220484	+	0.975391i	$0.429236\pi$
$542$	0	0
$543$	− 17.3174i	− 0.0318920i
$544$	0	0
$545$	79.4427	0.145766
$546$	0	0
$547$	− 341.453i	− 0.624228i	−0.950045	−	0.312114i	$-0.898963\pi$
$547$	− 341.453i	− 0.624228i	0.950045	−	0.312114i	$-0.101037\pi$
$548$	0	0
$549$	−279.370	−0.508871
$550$	0	0
$551$	537.994i	0.976395i
$552$	0	0
$553$	1642.31	2.96982
$554$	0	0
$555$	40.1904i	0.0724151i
$556$	0	0
$557$	349.574	0.627602	0.313801	−	0.949489i	$-0.398398\pi$
$557$	349.574	0.627602	0.313801	+	0.949489i	$0.398398\pi$
$558$	0	0
$559$	− 545.633i	− 0.976088i
$560$	0	0
$561$	−4.45825	−0.00794696
$562$	0	0
$563$	746.778i	1.32643i	0.748431	+	0.663213i	$0.230808\pi$
$563$	746.778i	1.32643i	−0.748431	+	0.663213i	$0.769192\pi$
$564$	0	0
$565$	158.138	0.279890
$566$	0	0
$567$	798.830i	1.40887i
$568$	0	0
$569$	−660.906	−1.16152	−0.580761	−	0.814074i	$-0.697245\pi$
$569$	−660.906	−1.16152	−0.580761	+	0.814074i	$0.697245\pi$
$570$	0	0
$571$	529.240i	0.926865i	0.886132	+	0.463432i	$0.153382\pi$
$571$	529.240i	0.926865i	−0.886132	+	0.463432i	$0.846618\pi$
$572$	0	0
$573$	−115.161	−0.200979
$574$	0	0
$575$	157.984i	0.274754i
$576$	0	0
$577$	−677.830	−1.17475	−0.587374	−	0.809316i	$-0.699838\pi$
$577$	−677.830	−1.17475	−0.587374	+	0.809316i	$0.699838\pi$
$578$	0	0
$579$	160.807i	0.277731i
$580$	0	0
$581$	−237.017	−0.407947
$582$	0	0
$583$	3.84213i	0.00659028i
$584$	0	0
$585$	−385.279	−0.658596
$586$	0	0
$587$	− 80.6137i	− 0.137332i	−0.997640	−	0.0686659i	$-0.978126\pi$
$587$	− 80.6137i	− 0.137332i	0.997640	−	0.0686659i	$-0.0218742\pi$
$588$	0	0
$589$	328.997	0.558569
$590$	0	0
$591$	74.1191i	0.125413i
$592$	0	0
$593$	347.548	0.586084	0.293042	−	0.956100i	$-0.405332\pi$
$593$	347.548	0.586084	0.293042	+	0.956100i	$0.405332\pi$
$594$	0	0
$595$	28.7539i	0.0483259i
$596$	0	0
$597$	80.1301	0.134221
$598$	0	0
$599$	78.2817i	0.130687i	0.997863	+	0.0653437i	$0.0208144\pi$
$599$	78.2817i	0.130687i	−0.997863	+	0.0653437i	$0.979186\pi$
$600$	0	0
$601$	−39.3050	−0.0653993	−0.0326996	−	0.999465i	$-0.510410\pi$
$601$	−39.3050	−0.0653993	−0.0326996	+	0.999465i	$0.510410\pi$
$602$	0	0
$603$	691.885i	1.14740i
$604$	0	0
$605$	−202.236	−0.334274
$606$	0	0
$607$	432.443i	0.712427i	0.934405	+	0.356214i	$0.115932\pi$
$607$	432.443i	0.712427i	−0.934405	+	0.356214i	$0.884068\pi$
$608$	0	0
$609$	−417.167	−0.685004
$610$	0	0
$611$	− 911.850i	− 1.49239i
$612$	0	0
$613$	498.413	0.813072	0.406536	−	0.913635i	$-0.366737\pi$
$613$	498.413	0.813072	0.406536	+	0.913635i	$0.366737\pi$
$614$	0	0
$615$	14.8529i	0.0241511i
$616$	0	0
$617$	−611.325	−0.990802	−0.495401	−	0.868664i	$-0.664979\pi$
$617$	−611.325	−0.990802	−0.495401	+	0.868664i	$0.664979\pi$
$618$	0	0
$619$	− 490.663i	− 0.792670i	−0.918106	−	0.396335i	$-0.870282\pi$
$619$	− 490.663i	− 0.792670i	0.918106	−	0.396335i	$-0.129718\pi$
$620$	0	0
$621$	420.393	0.676962
$622$	0	0
$623$	− 365.410i	− 0.586533i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 50.6749i	− 0.0808213i
$628$	0	0
$629$	−24.8390	−0.0394897
$630$	0	0
$631$	104.636i	0.165826i	0.996557	+	0.0829130i	$0.0264224\pi$
$631$	104.636i	0.165826i	−0.996557	+	0.0829130i	$0.973578\pi$
$632$	0	0
$633$	183.790	0.290347
$634$	0	0
$635$	332.148i	0.523067i
$636$	0	0
$637$	−2034.13	−3.19329
$638$	0	0
$639$	− 1028.90i	− 1.61017i
$640$	0	0
$641$	150.020	0.234041	0.117020	−	0.993130i	$-0.462666\pi$
$641$	150.020	0.234041	0.117020	+	0.993130i	$0.462666\pi$
$642$	0	0
$643$	− 646.810i	− 1.00593i	−0.864308	−	0.502963i	$-0.832243\pi$
$643$	− 646.810i	− 1.00593i	0.864308	−	0.502963i	$-0.167757\pi$
$644$	0	0
$645$	45.5279	0.0705858
$646$	0	0
$647$	694.450i	1.07334i	0.843793	+	0.536669i	$0.180318\pi$
$647$	694.450i	1.07334i	−0.843793	+	0.536669i	$0.819682\pi$
$648$	0	0
$649$	522.991	0.805841
$650$	0	0
$651$	255.108i	0.391872i
$652$	0	0
$653$	−698.361	−1.06947	−0.534733	−	0.845021i	$-0.679588\pi$
$653$	−698.361	−1.06947	−0.534733	+	0.845021i	$0.679588\pi$
$654$	0	0
$655$	− 78.0789i	− 0.119204i
$656$	0	0
$657$	1112.35	1.69307
$658$	0	0
$659$	− 834.282i	− 1.26598i	−0.774159	−	0.632991i	$-0.781827\pi$
$659$	− 834.282i	− 1.26598i	0.774159	−	0.632991i	$-0.218173\pi$
$660$	0	0
$661$	978.584	1.48046	0.740230	−	0.672354i	$-0.234717\pi$
$661$	978.584	1.48046	0.740230	+	0.672354i	$0.234717\pi$
$662$	0	0
$663$	16.5109i	0.0249033i
$664$	0	0
$665$	−326.833	−0.491478
$666$	0	0
$667$	− 1416.57i	− 2.12379i
$668$	0	0
$669$	−13.9737	−0.0208874
$670$	0	0
$671$	− 183.489i	− 0.273456i
$672$	0	0
$673$	−682.669	−1.01437	−0.507183	−	0.861838i	$-0.669313\pi$
$673$	−682.669	−1.01437	−0.507183	+	0.861838i	$0.669313\pi$
$674$	0	0
$675$	− 66.5248i	− 0.0985552i
$676$	0	0
$677$	948.067	1.40039	0.700197	−	0.713950i	$-0.253096\pi$
$677$	948.067	1.40039	0.700197	+	0.713950i	$0.253096\pi$
$678$	0	0
$679$	150.237i	0.221262i
$680$	0	0
$681$	−70.6300	−0.103715
$682$	0	0
$683$	341.688i	0.500275i	0.968210	+	0.250138i	$0.0804759\pi$
$683$	341.688i	0.500275i	−0.968210	+	0.250138i	$0.919524\pi$
$684$	0	0
$685$	−224.971	−0.328424
$686$	0	0
$687$	− 126.172i	− 0.183656i
$688$	0	0
$689$	14.2291	0.0206518
$690$	0	0
$691$	215.416i	0.311746i	0.987777	+	0.155873i	$0.0498190\pi$
$691$	215.416i	0.311746i	−0.987777	+	0.155873i	$0.950181\pi$
$692$	0	0
$693$	−566.687	−0.817731
$694$	0	0
$695$	− 444.105i	− 0.639000i
$696$	0	0
$697$	−9.17961	−0.0131702
$698$	0	0
$699$	− 28.5185i	− 0.0407991i
$700$	0	0
$701$	−1137.40	−1.62254	−0.811272	−	0.584668i	$-0.801225\pi$
$701$	−1137.40	−1.62254	−0.811272	+	0.584668i	$0.801225\pi$
$702$	0	0
$703$	− 282.334i	− 0.401614i
$704$	0	0
$705$	76.0851	0.107922
$706$	0	0
$707$	1285.69i	1.81851i
$708$	0	0
$709$	−853.149	−1.20331	−0.601656	−	0.798755i	$-0.705492\pi$
$709$	−853.149	−1.20331	−0.601656	+	0.798755i	$0.705492\pi$
$710$	0	0
$711$	− 1134.81i	− 1.59607i
$712$	0	0
$713$	−866.269	−1.21496
$714$	0	0
$715$	− 253.050i	− 0.353915i
$716$	0	0
$717$	303.659	0.423514
$718$	0	0
$719$	696.616i	0.968868i	0.874828	+	0.484434i	$0.160974\pi$
$719$	696.616i	0.968868i	−0.874828	+	0.484434i	$0.839026\pi$
$720$	0	0
$721$	−1230.05	−1.70604
$722$	0	0
$723$	− 181.751i	− 0.251384i
$724$	0	0
$725$	−224.164	−0.309192
$726$	0	0
$727$	− 588.265i	− 0.809168i	−0.914501	−	0.404584i	$-0.867416\pi$
$727$	− 588.265i	− 0.809168i	0.914501	−	0.404584i	$-0.132584\pi$
$728$	0	0
$729$	460.502	0.631689
$730$	0	0
$731$	28.1378i	0.0384922i
$732$	0	0
$733$	1106.86	1.51004	0.755020	−	0.655702i	$-0.227627\pi$
$733$	1106.86	1.51004	0.755020	+	0.655702i	$0.227627\pi$
$734$	0	0
$735$	− 169.728i	− 0.230923i
$736$	0	0
$737$	−454.427	−0.616590
$738$	0	0
$739$	720.053i	0.974361i	0.873301	+	0.487180i	$0.161975\pi$
$739$	720.053i	0.974361i	−0.873301	+	0.487180i	$0.838025\pi$
$740$	0	0
$741$	−187.672	−0.253268
$742$	0	0
$743$	− 317.793i	− 0.427716i	−0.976865	−	0.213858i	$-0.931397\pi$
$743$	− 317.793i	− 0.427716i	0.976865	−	0.213858i	$-0.0686031\pi$
$744$	0	0
$745$	454.105	0.609537
$746$	0	0
$747$	163.775i	0.219243i
$748$	0	0
$749$	−278.630	−0.372003
$750$	0	0
$751$	606.512i	0.807606i	0.914846	+	0.403803i	$0.132312\pi$
$751$	606.512i	0.807606i	−0.914846	+	0.403803i	$0.867688\pi$
$752$	0	0
$753$	−212.223	−0.281837
$754$	0	0
$755$	254.472i	0.337049i
$756$	0	0
$757$	359.463	0.474852	0.237426	−	0.971406i	$-0.423696\pi$
$757$	359.463	0.474852	0.237426	+	0.971406i	$0.423696\pi$
$758$	0	0
$759$	133.430i	0.175797i
$760$	0	0
$761$	239.876	0.315212	0.157606	−	0.987502i	$-0.449622\pi$
$761$	239.876	0.315212	0.157606	+	0.987502i	$0.449622\pi$
$762$	0	0
$763$	− 432.741i	− 0.567158i
$764$	0	0
$765$	19.8684	0.0259718
$766$	0	0
$767$	− 1936.87i	− 2.52525i
$768$	0	0
$769$	−768.885	−0.999851	−0.499926	−	0.866068i	$-0.666639\pi$
$769$	−768.885	−0.999851	−0.499926	+	0.866068i	$0.666639\pi$
$770$	0	0
$771$	− 341.094i	− 0.442405i
$772$	0	0
$773$	−33.5743	−0.0434337	−0.0217169	−	0.999764i	$-0.506913\pi$
$773$	−33.5743	−0.0434337	−0.0217169	+	0.999764i	$0.506913\pi$
$774$	0	0
$775$	137.082i	0.176880i
$776$	0	0
$777$	218.926	0.281758
$778$	0	0
$779$	− 104.341i	− 0.133942i
$780$	0	0
$781$	675.777	0.865272
$782$	0	0
$783$	596.498i	0.761812i
$784$	0	0
$785$	324.046	0.412798
$786$	0	0
$787$	− 426.738i	− 0.542233i	−0.962546	−	0.271117i	$-0.912607\pi$
$787$	− 426.738i	− 0.542233i	0.962546	−	0.271117i	$-0.0873929\pi$
$788$	0	0
$789$	80.2430	0.101702
$790$	0	0
$791$	− 861.410i	− 1.08901i
$792$	0	0
$793$	−679.542	−0.856925
$794$	0	0
$795$	1.18728i	0.00149344i
$796$	0	0
$797$	194.203	0.243667	0.121834	−	0.992551i	$-0.461123\pi$
$797$	194.203	0.243667	0.121834	+	0.992551i	$0.461123\pi$
$798$	0	0
$799$	47.0232i	0.0588526i
$800$	0	0
$801$	−252.492	−0.315221
$802$	0	0
$803$	730.585i	0.909819i
$804$	0	0
$805$	860.571	1.06903
$806$	0	0
$807$	− 385.293i	− 0.477438i
$808$	0	0
$809$	1446.33	1.78780	0.893899	−	0.448269i	$-0.147959\pi$
$809$	1446.33	1.78780	0.893899	+	0.448269i	$0.147959\pi$
$810$	0	0
$811$	23.3112i	0.0287437i	0.999897	+	0.0143719i	$0.00457486\pi$
$811$	23.3112i	0.0287437i	−0.999897	+	0.0143719i	$0.995425\pi$
$812$	0	0
$813$	−125.601	−0.154490
$814$	0	0
$815$	454.758i	0.557985i
$816$	0	0
$817$	−319.830	−0.391468
$818$	0	0
$819$	2098.70i	2.56251i
$820$	0	0
$821$	−701.416	−0.854344	−0.427172	−	0.904170i	$-0.640490\pi$
$821$	−701.416	−0.854344	−0.427172	+	0.904170i	$0.640490\pi$
$822$	0	0
$823$	− 270.961i	− 0.329235i	−0.986357	−	0.164618i	$-0.947361\pi$
$823$	− 270.961i	− 0.329235i	0.986357	−	0.164618i	$-0.0526390\pi$
$824$	0	0
$825$	21.1146	0.0255934
$826$	0	0
$827$	1035.10i	1.25163i	0.779971	+	0.625815i	$0.215234\pi$
$827$	1035.10i	1.25163i	−0.779971	+	0.625815i	$0.784766\pi$
$828$	0	0
$829$	−332.735	−0.401369	−0.200685	−	0.979656i	$-0.564317\pi$
$829$	−332.735	−0.401369	−0.200685	+	0.979656i	$0.564317\pi$
$830$	0	0
$831$	− 241.082i	− 0.290111i
$832$	0	0
$833$	104.898	0.125928
$834$	0	0
$835$	198.482i	0.237703i
$836$	0	0
$837$	364.774	0.435811
$838$	0	0
$839$	102.177i	0.121784i	0.998144	+	0.0608918i	$0.0193945\pi$
$839$	102.177i	0.121784i	−0.998144	+	0.0608918i	$0.980606\pi$
$840$	0	0
$841$	1168.98	1.38999
$842$	0	0
$843$	158.447i	0.187956i
$844$	0	0
$845$	−559.259	−0.661845
$846$	0	0
$847$	1101.62i	1.30062i
$848$	0	0
$849$	335.286	0.394919
$850$	0	0
$851$	743.404i	0.873565i
$852$	0	0
$853$	358.401	0.420165	0.210083	−	0.977684i	$-0.432627\pi$
$853$	358.401	0.420165	0.210083	+	0.977684i	$0.432627\pi$
$854$	0	0
$855$	225.836i	0.264136i
$856$	0	0
$857$	33.3839	0.0389544	0.0194772	−	0.999810i	$-0.493800\pi$
$857$	33.3839	0.0389544	0.0194772	+	0.999810i	$0.493800\pi$
$858$	0	0
$859$	989.483i	1.15190i	0.817485	+	0.575950i	$0.195368\pi$
$859$	989.483i	1.15190i	−0.817485	+	0.575950i	$0.804632\pi$
$860$	0	0
$861$	80.9070	0.0939686
$862$	0	0
$863$	− 767.341i	− 0.889156i	−0.895740	−	0.444578i	$-0.853354\pi$
$863$	− 767.341i	− 0.889156i	0.895740	−	0.444578i	$-0.146646\pi$
$864$	0	0
$865$	303.167	0.350482
$866$	0	0
$867$	219.925i	0.253662i
$868$	0	0
$869$	745.337	0.857696
$870$	0	0
$871$	1682.95i	1.93220i
$872$	0	0
$873$	103.811	0.118913
$874$	0	0
$875$	− 136.180i	− 0.155635i
$876$	0	0
$877$	−1013.80	−1.15598	−0.577992	−	0.816043i	$-0.696163\pi$
$877$	−1013.80	−1.15598	−0.577992	+	0.816043i	$0.696163\pi$
$878$	0	0
$879$	208.491i	0.237191i
$880$	0	0
$881$	−280.748	−0.318669	−0.159335	−	0.987225i	$-0.550935\pi$
$881$	−280.748	−0.318669	−0.159335	+	0.987225i	$0.550935\pi$
$882$	0	0
$883$	− 1096.87i	− 1.24221i	−0.783728	−	0.621104i	$-0.786685\pi$
$883$	− 1096.87i	− 1.24221i	0.783728	−	0.621104i	$-0.213315\pi$
$884$	0	0
$885$	161.613	0.182614
$886$	0	0
$887$	− 1164.82i	− 1.31321i	−0.754235	−	0.656605i	$-0.771992\pi$
$887$	− 1164.82i	− 1.31321i	0.754235	−	0.656605i	$-0.228008\pi$
$888$	0	0
$889$	1809.28	2.03519
$890$	0	0
$891$	362.537i	0.406888i
$892$	0	0
$893$	−534.492	−0.598536
$894$	0	0
$895$	455.777i	0.509248i
$896$	0	0
$897$	494.152	0.550894
$898$	0	0
$899$	− 1229.15i	− 1.36725i
$900$	0	0
$901$	−0.733782	−0.000814408 0
$902$	0	0
$903$	− 248.000i	− 0.274640i
$904$	0	0
$905$	−50.6888	−0.0560098
$906$	0	0
$907$	1132.82i	1.24897i	0.781036	+	0.624485i	$0.214691\pi$
$907$	1132.82i	1.24897i	−0.781036	+	0.624485i	$0.785309\pi$
$908$	0	0
$909$	888.387	0.977323
$910$	0	0
$911$	− 1793.23i	− 1.96842i	−0.177014	−	0.984208i	$-0.556644\pi$
$911$	− 1793.23i	− 1.96842i	0.177014	−	0.984208i	$-0.443356\pi$
$912$	0	0
$913$	−107.567	−0.117817
$914$	0	0
$915$	− 56.7013i	− 0.0619686i
$916$	0	0
$917$	−425.313	−0.463809
$918$	0	0
$919$	− 205.390i	− 0.223493i	−0.993737	−	0.111747i	$-0.964356\pi$
$919$	− 205.390i	− 0.223493i	0.993737	−	0.111747i	$-0.0356445\pi$
$920$	0	0
$921$	−192.098	−0.208575
$922$	0	0
$923$	− 2502.70i	− 2.71149i
$924$	0	0
$925$	117.639	0.127178
$926$	0	0
$927$	849.946i	0.916879i
$928$	0	0
$929$	−736.237	−0.792505	−0.396252	−	0.918142i	$-0.629689\pi$
$929$	−736.237	−0.792505	−0.396252	+	0.918142i	$0.629689\pi$
$930$	0	0
$931$	1192.33i	1.28070i
$932$	0	0
$933$	269.601	0.288961
$934$	0	0
$935$	13.0495i	0.0139567i
$936$	0	0
$937$	−40.3344	−0.0430463	−0.0215231	−	0.999768i	$-0.506852\pi$
$937$	−40.3344	−0.0430463	−0.0215231	+	0.999768i	$0.506852\pi$
$938$	0	0
$939$	332.478i	0.354077i
$940$	0	0
$941$	447.115	0.475148	0.237574	−	0.971369i	$-0.423648\pi$
$941$	447.115	0.475148	0.237574	+	0.971369i	$0.423648\pi$
$942$	0	0
$943$	274.735i	0.291342i
$944$	0	0
$945$	−362.375	−0.383465
$946$	0	0
$947$	144.553i	0.152644i	0.997083	+	0.0763218i	$0.0243176\pi$
$947$	144.553i	0.152644i	−0.997083	+	0.0763218i	$0.975682\pi$
$948$	0	0
$949$	2705.68	2.85109
$950$	0	0
$951$	− 31.6393i	− 0.0332695i
$952$	0	0
$953$	432.006	0.453312	0.226656	−	0.973975i	$-0.427221\pi$
$953$	432.006	0.453312	0.226656	+	0.973975i	$0.427221\pi$
$954$	0	0
$955$	337.082i	0.352965i
$956$	0	0
$957$	−189.325	−0.197832
$958$	0	0
$959$	1225.46i	1.27785i
$960$	0	0
$961$	209.341	0.217836
$962$	0	0
$963$	192.529i	0.199926i
$964$	0	0
$965$	470.689	0.487760
$966$	0	0
$967$	1472.95i	1.52322i	0.648035	+	0.761610i	$0.275591\pi$
$967$	1472.95i	1.52322i	−0.648035	+	0.761610i	$0.724409\pi$
$968$	0	0
$969$	9.67805	0.00998767
$970$	0	0
$971$	− 308.190i	− 0.317395i	−0.987327	−	0.158697i	$-0.949271\pi$
$971$	− 308.190i	− 0.317395i	0.987327	−	0.158697i	$-0.0507294\pi$
$972$	0	0
$973$	−2419.14	−2.48627
$974$	0	0
$975$	− 78.1966i	− 0.0802016i
$976$	0	0
$977$	−1376.78	−1.40919	−0.704596	−	0.709609i	$-0.748872\pi$
$977$	−1376.78	−1.40919	−0.704596	+	0.709609i	$0.748872\pi$
$978$	0	0
$979$	− 165.836i	− 0.169393i
$980$	0	0
$981$	−299.017	−0.304808
$982$	0	0
$983$	1250.38i	1.27200i	0.771689	+	0.636000i	$0.219412\pi$
$983$	1250.38i	1.27200i	−0.771689	+	0.636000i	$0.780588\pi$
$984$	0	0
$985$	216.950	0.220254
$986$	0	0
$987$	− 414.452i	− 0.419911i
$988$	0	0
$989$	842.132	0.851498
$990$	0	0
$991$	− 1727.44i	− 1.74313i	−0.490278	−	0.871566i	$-0.663105\pi$
$991$	− 1727.44i	− 1.74313i	0.490278	−	0.871566i	$-0.336895\pi$
$992$	0	0
$993$	−79.0031	−0.0795600
$994$	0	0
$995$	− 234.545i	− 0.235723i
$996$	0	0
$997$	1131.90	1.13530	0.567651	−	0.823269i	$-0.307852\pi$
$997$	1131.90	1.13530	0.567651	+	0.823269i	$0.307852\pi$
$998$	0	0
$999$	− 313.037i	− 0.313350i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.b.a.31.2	✓	4
3.2	odd	2		1440.3.e.b.991.4		4
4.3	odd	2	inner	160.3.b.a.31.3	yes	4
5.2	odd	4		800.3.h.c.799.4		4
5.3	odd	4		800.3.h.j.799.2		4
5.4	even	2		800.3.b.d.351.3		4
8.3	odd	2		320.3.b.b.191.2		4
8.5	even	2		320.3.b.b.191.3		4
12.11	even	2		1440.3.e.b.991.3		4
16.3	odd	4		1280.3.g.a.1151.4		4
16.5	even	4		1280.3.g.a.1151.3		4
16.11	odd	4		1280.3.g.d.1151.1		4
16.13	even	4		1280.3.g.d.1151.2		4
20.3	even	4		800.3.h.c.799.3		4
20.7	even	4		800.3.h.j.799.1		4
20.19	odd	2		800.3.b.d.351.2		4
24.5	odd	2		2880.3.e.a.2431.2		4
24.11	even	2		2880.3.e.a.2431.1		4
40.3	even	4		1600.3.h.m.1599.2		4
40.13	odd	4		1600.3.h.d.1599.3		4
40.19	odd	2		1600.3.b.n.1151.3		4
40.27	even	4		1600.3.h.d.1599.4		4
40.29	even	2		1600.3.b.n.1151.2		4
40.37	odd	4		1600.3.h.m.1599.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.b.a.31.2	✓	4	1.1	even	1	trivial
160.3.b.a.31.3	yes	4	4.3	odd	2	inner
320.3.b.b.191.2		4	8.3	odd	2
320.3.b.b.191.3		4	8.5	even	2
800.3.b.d.351.2		4	20.19	odd	2
800.3.b.d.351.3		4	5.4	even	2
800.3.h.c.799.3		4	20.3	even	4
800.3.h.c.799.4		4	5.2	odd	4
800.3.h.j.799.1		4	20.7	even	4
800.3.h.j.799.2		4	5.3	odd	4
1280.3.g.a.1151.3		4	16.5	even	4
1280.3.g.a.1151.4		4	16.3	odd	4
1280.3.g.d.1151.1		4	16.11	odd	4
1280.3.g.d.1151.2		4	16.13	even	4
1440.3.e.b.991.3		4	12.11	even	2
1440.3.e.b.991.4		4	3.2	odd	2
1600.3.b.n.1151.2		4	40.29	even	2
1600.3.b.n.1151.3		4	40.19	odd	2
1600.3.h.d.1599.3		4	40.13	odd	4
1600.3.h.d.1599.4		4	40.27	even	4
1600.3.h.m.1599.1		4	40.37	odd	4
1600.3.h.m.1599.2		4	40.3	even	4
2880.3.e.a.2431.1		4	24.11	even	2
2880.3.e.a.2431.2		4	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	160.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.763932i q^{3} -2.23607 q^{5} +12.1803i q^{7} +8.41641 q^{9} +O(q^{10})\)	\(q-0.763932i q^{3} -2.23607 q^{5} +12.1803i q^{7} +8.41641 q^{9} +5.52786i q^{11} +20.4721 q^{13} +1.70820i q^{15} -1.05573 q^{17} -12.0000i q^{19} +9.30495 q^{21} +31.5967i q^{23} +5.00000 q^{25} -13.3050i q^{27} -44.8328 q^{29} +27.4164i q^{31} +4.22291 q^{33} -27.2361i q^{35} +23.5279 q^{37} -15.6393i q^{39} +8.69505 q^{41} -26.6525i q^{43} -18.8197 q^{45} -44.5410i q^{47} -99.3607 q^{49} +0.806504i q^{51} +0.695048 q^{53} -12.3607i q^{55} -9.16718 q^{57} -94.6099i q^{59} -33.1935 q^{61} +102.515i q^{63} -45.7771 q^{65} +82.2067i q^{67} +24.1378 q^{69} -122.249i q^{71} +132.164 q^{73} -3.81966i q^{75} -67.3313 q^{77} -134.833i q^{79} +65.5836 q^{81} +19.4590i q^{83} +2.36068 q^{85} +34.2492i q^{87} -30.0000 q^{89} +249.358i q^{91} +20.9443 q^{93} +26.8328i q^{95} +12.3344 q^{97} +46.5248i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{9}+O(q^{10}) \) 4 * q - 20 * q^9	\( 4 q - 20 q^{9} + 64 q^{13} - 40 q^{17} - 88 q^{21} + 20 q^{25} - 72 q^{29} + 160 q^{33} + 112 q^{37} + 160 q^{41} - 120 q^{45} - 308 q^{49} + 128 q^{53} - 144 q^{57} + 64 q^{61} - 40 q^{65} - 136 q^{69} - 8 q^{73} + 160 q^{77} + 316 q^{81} - 80 q^{85} - 120 q^{89} + 48 q^{93} + 264 q^{97}+O(q^{100}) \) 4 * q - 20 * q^9 + 64 * q^13 - 40 * q^17 - 88 * q^21 + 20 * q^25 - 72 * q^29 + 160 * q^33 + 112 * q^37 + 160 * q^41 - 120 * q^45 - 308 * q^49 + 128 * q^53 - 144 * q^57 + 64 * q^61 - 40 * q^65 - 136 * q^69 - 8 * q^73 + 160 * q^77 + 316 * q^81 - 80 * q^85 - 120 * q^89 + 48 * q^93 + 264 * q^97

Introduction

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