LMFDB - Embedded newform 320.3.b.c.191.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,3,Mod(191,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, 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("320.191");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	320.b (of order $2$, degree $1$, not 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:	$8.71936845953$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.3
Root			$0.809017 + 0.587785i$ of defining polynomial
Character	$\chi$	$=$	320.191
Dual form			320.3.b.c.191.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.35114i q^{3} +2.23607 q^{5} -5.25731i q^{7} +3.47214 q^{9} +O(q^{10})$	$q+2.35114i q^{3} +2.23607 q^{5} -5.25731i q^{7} +3.47214 q^{9} -19.9192i q^{11} +8.47214 q^{13} +5.25731i q^{15} +11.8885 q^{17} +15.2169i q^{19} +12.3607 q^{21} -0.555029i q^{23} +5.00000 q^{25} +29.3238i q^{27} +10.9443 q^{29} -8.29451i q^{31} +46.8328 q^{33} -11.7557i q^{35} +18.3607 q^{37} +19.9192i q^{39} -14.5836 q^{41} -22.2703i q^{43} +7.76393 q^{45} +53.3902i q^{47} +21.3607 q^{49} +27.9516i q^{51} +66.3607 q^{53} -44.5407i q^{55} -35.7771 q^{57} -17.4370i q^{59} -90.1378 q^{61} -18.2541i q^{63} +18.9443 q^{65} -50.2220i q^{67} +1.30495 q^{69} -80.7868i q^{71} -5.55418 q^{73} +11.7557i q^{75} -104.721 q^{77} -13.8448i q^{79} -37.6950 q^{81} +76.2155i q^{83} +26.5836 q^{85} +25.7315i q^{87} -111.443 q^{89} -44.5407i q^{91} +19.5016 q^{93} +34.0260i q^{95} -92.8328 q^{97} -69.1621i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 16 q^{13} - 24 q^{17} - 40 q^{21} + 20 q^{25} + 8 q^{29} + 80 q^{33} - 16 q^{37} - 112 q^{41} + 40 q^{45} - 4 q^{49} + 176 q^{53} - 128 q^{61} + 40 q^{65} - 120 q^{69} + 264 q^{73} - 240 q^{77} - 276 q^{81} + 160 q^{85} - 88 q^{89} + 400 q^{93} - 264 q^{97}+O(q^{100}) $ 4 * q - 4 * q^9 + 16 * q^13 - 24 * q^17 - 40 * q^21 + 20 * q^25 + 8 * q^29 + 80 * q^33 - 16 * q^37 - 112 * q^41 + 40 * q^45 - 4 * q^49 + 176 * q^53 - 128 * q^61 + 40 * q^65 - 120 * q^69 + 264 * q^73 - 240 * q^77 - 276 * q^81 + 160 * q^85 - 88 * q^89 + 400 * q^93 - 264 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.35114i	0.783714i	0.920026	+	0.391857i	$0.128167\pi$
$3$	2.35114i	0.783714i	−0.920026	+	0.391857i	$0.871833\pi$
$4$	0	0
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	− 5.25731i	− 0.751044i	−0.926813	−	0.375522i	$-0.877463\pi$
$7$	− 5.25731i	− 0.751044i	0.926813	−	0.375522i	$-0.122537\pi$
$8$	0	0
$9$	3.47214	0.385793
$10$	0	0
$11$	− 19.9192i	− 1.81084i	−0.424522	−	0.905418i	$-0.639558\pi$
$11$	− 19.9192i	− 1.81084i	0.424522	−	0.905418i	$-0.360442\pi$
$12$	0	0
$13$	8.47214	0.651703	0.325851	−	0.945421i	$-0.394349\pi$
$13$	8.47214	0.651703	0.325851	+	0.945421i	$0.394349\pi$
$14$	0	0
$15$	5.25731i	0.350487i
$16$	0	0
$17$	11.8885	0.699326	0.349663	−	0.936876i	$-0.386296\pi$
$17$	11.8885	0.699326	0.349663	+	0.936876i	$0.386296\pi$
$18$	0	0
$19$	15.2169i	0.800890i	0.916321	+	0.400445i	$0.131144\pi$
$19$	15.2169i	0.800890i	−0.916321	+	0.400445i	$0.868856\pi$
$20$	0	0
$21$	12.3607	0.588604
$22$	0	0
$23$	− 0.555029i	− 0.0241317i	−0.999927	−	0.0120659i	$-0.996159\pi$
$23$	− 0.555029i	− 0.0241317i	0.999927	−	0.0120659i	$-0.00384077\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	29.3238i	1.08606i
$28$	0	0
$29$	10.9443	0.377389	0.188694	−	0.982036i	$-0.439574\pi$
$29$	10.9443	0.377389	0.188694	+	0.982036i	$0.439574\pi$
$30$	0	0
$31$	− 8.29451i	− 0.267565i	−0.991011	−	0.133782i	$-0.957288\pi$
$31$	− 8.29451i	− 0.267565i	0.991011	−	0.133782i	$-0.0427123\pi$
$32$	0	0
$33$	46.8328	1.41918
$34$	0	0
$35$	− 11.7557i	− 0.335877i
$36$	0	0
$37$	18.3607	0.496235	0.248117	−	0.968730i	$-0.420188\pi$
$37$	18.3607	0.496235	0.248117	+	0.968730i	$0.420188\pi$
$38$	0	0
$39$	19.9192i	0.510748i
$40$	0	0
$41$	−14.5836	−0.355697	−0.177849	−	0.984058i	$-0.556914\pi$
$41$	−14.5836	−0.355697	−0.177849	+	0.984058i	$0.556914\pi$
$42$	0	0
$43$	− 22.2703i	− 0.517915i	−0.965889	−	0.258957i	$-0.916621\pi$
$43$	− 22.2703i	− 0.517915i	0.965889	−	0.258957i	$-0.0833789\pi$
$44$	0	0
$45$	7.76393	0.172532
$46$	0	0
$47$	53.3902i	1.13596i	0.823042	+	0.567981i	$0.192275\pi$
$47$	53.3902i	1.13596i	−0.823042	+	0.567981i	$0.807725\pi$
$48$	0	0
$49$	21.3607	0.435932
$50$	0	0
$51$	27.9516i	0.548071i
$52$	0	0
$53$	66.3607	1.25209	0.626044	−	0.779788i	$-0.284673\pi$
$53$	66.3607	1.25209	0.626044	+	0.779788i	$0.284673\pi$
$54$	0	0
$55$	− 44.5407i	− 0.809830i
$56$	0	0
$57$	−35.7771	−0.627668
$58$	0	0
$59$	− 17.4370i	− 0.295543i	−0.989022	−	0.147771i	$-0.952790\pi$
$59$	− 17.4370i	− 0.295543i	0.989022	−	0.147771i	$-0.0472100\pi$
$60$	0	0
$61$	−90.1378	−1.47767	−0.738834	−	0.673887i	$-0.764623\pi$
$61$	−90.1378	−1.47767	−0.738834	+	0.673887i	$0.764623\pi$
$62$	0	0
$63$	− 18.2541i	− 0.289748i
$64$	0	0
$65$	18.9443	0.291450
$66$	0	0
$67$	− 50.2220i	− 0.749582i	−0.927109	−	0.374791i	$-0.877715\pi$
$67$	− 50.2220i	− 0.749582i	0.927109	−	0.374791i	$-0.122285\pi$
$68$	0	0
$69$	1.30495	0.0189123
$70$	0	0
$71$	− 80.7868i	− 1.13784i	−0.822392	−	0.568921i	$-0.807361\pi$
$71$	− 80.7868i	− 1.13784i	0.822392	−	0.568921i	$-0.192639\pi$
$72$	0	0
$73$	−5.55418	−0.0760846	−0.0380423	−	0.999276i	$-0.512112\pi$
$73$	−5.55418	−0.0760846	−0.0380423	+	0.999276i	$0.512112\pi$
$74$	0	0
$75$	11.7557i	0.156743i
$76$	0	0
$77$	−104.721	−1.36002
$78$	0	0
$79$	− 13.8448i	− 0.175251i	−0.996154	−	0.0876253i	$-0.972072\pi$
$79$	− 13.8448i	− 0.175251i	0.996154	−	0.0876253i	$-0.0279278\pi$
$80$	0	0
$81$	−37.6950	−0.465371
$82$	0	0
$83$	76.2155i	0.918260i	0.888369	+	0.459130i	$0.151839\pi$
$83$	76.2155i	0.918260i	−0.888369	+	0.459130i	$0.848161\pi$
$84$	0	0
$85$	26.5836	0.312748
$86$	0	0
$87$	25.7315i	0.295765i
$88$	0	0
$89$	−111.443	−1.25217	−0.626083	−	0.779757i	$-0.715343\pi$
$89$	−111.443	−1.25217	−0.626083	+	0.779757i	$0.715343\pi$
$90$	0	0
$91$	− 44.5407i	− 0.489458i
$92$	0	0
$93$	19.5016	0.209694
$94$	0	0
$95$	34.0260i	0.358169i
$96$	0	0
$97$	−92.8328	−0.957039	−0.478520	−	0.878077i	$-0.658826\pi$
$97$	−92.8328	−0.957039	−0.478520	+	0.878077i	$0.658826\pi$
$98$	0	0
$99$	− 69.1621i	− 0.698607i
$100$	0	0
$101$	−64.1115	−0.634767	−0.317383	−	0.948297i	$-0.602804\pi$
$101$	−64.1115	−0.634767	−0.317383	+	0.948297i	$0.602804\pi$
$102$	0	0
$103$	137.769i	1.33757i	0.743458	+	0.668783i	$0.233184\pi$
$103$	137.769i	1.33757i	−0.743458	+	0.668783i	$0.766816\pi$
$104$	0	0
$105$	27.6393	0.263232
$106$	0	0
$107$	51.3320i	0.479739i	0.970805	+	0.239869i	$0.0771046\pi$
$107$	51.3320i	0.479739i	−0.970805	+	0.239869i	$0.922895\pi$
$108$	0	0
$109$	−133.469	−1.22449	−0.612243	−	0.790669i	$-0.709733\pi$
$109$	−133.469	−1.22449	−0.612243	+	0.790669i	$0.709733\pi$
$110$	0	0
$111$	43.1685i	0.388906i
$112$	0	0
$113$	170.721	1.51081	0.755404	−	0.655259i	$-0.227441\pi$
$113$	170.721	1.51081	0.755404	+	0.655259i	$0.227441\pi$
$114$	0	0
$115$	− 1.24108i	− 0.0107920i
$116$	0	0
$117$	29.4164	0.251422
$118$	0	0
$119$	− 62.5018i	− 0.525225i
$120$	0	0
$121$	−275.774	−2.27912
$122$	0	0
$123$	− 34.2881i	− 0.278765i
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	− 198.637i	− 1.56407i	−0.623235	−	0.782035i	$-0.714182\pi$
$127$	− 198.637i	− 1.56407i	0.623235	−	0.782035i	$-0.285818\pi$
$128$	0	0
$129$	52.3607	0.405897
$130$	0	0
$131$	− 7.77041i	− 0.0593161i	−0.999560	−	0.0296580i	$-0.990558\pi$
$131$	− 7.77041i	− 0.0593161i	0.999560	−	0.0296580i	$-0.00944183\pi$
$132$	0	0
$133$	80.0000	0.601504
$134$	0	0
$135$	65.5699i	0.485703i
$136$	0	0
$137$	0.832816	0.00607895	0.00303947	−	0.999995i	$-0.499033\pi$
$137$	0.832816	0.00607895	0.00303947	+	0.999995i	$0.499033\pi$
$138$	0	0
$139$	237.658i	1.70977i	0.518817	+	0.854885i	$0.326373\pi$
$139$	237.658i	1.70977i	−0.518817	+	0.854885i	$0.673627\pi$
$140$	0	0
$141$	−125.528	−0.890269
$142$	0	0
$143$	− 168.758i	− 1.18013i
$144$	0	0
$145$	24.4721	0.168773
$146$	0	0
$147$	50.2220i	0.341646i
$148$	0	0
$149$	36.9706	0.248125	0.124062	−	0.992274i	$-0.460408\pi$
$149$	36.9706	0.248125	0.124062	+	0.992274i	$0.460408\pi$
$150$	0	0
$151$	282.723i	1.87234i	0.351552	+	0.936168i	$0.385654\pi$
$151$	282.723i	1.87234i	−0.351552	+	0.936168i	$0.614346\pi$
$152$	0	0
$153$	41.2786	0.269795
$154$	0	0
$155$	− 18.5471i	− 0.119659i
$156$	0	0
$157$	−204.748	−1.30413	−0.652063	−	0.758165i	$-0.726096\pi$
$157$	−204.748	−1.30413	−0.652063	+	0.758165i	$0.726096\pi$
$158$	0	0
$159$	156.023i	0.981279i
$160$	0	0
$161$	−2.91796	−0.0181240
$162$	0	0
$163$	− 107.235i	− 0.657885i	−0.944350	−	0.328943i	$-0.893308\pi$
$163$	− 107.235i	− 0.657885i	0.944350	−	0.328943i	$-0.106692\pi$
$164$	0	0
$165$	104.721	0.634675
$166$	0	0
$167$	33.2090i	0.198856i	0.995045	+	0.0994280i	$0.0317013\pi$
$167$	33.2090i	0.198856i	−0.995045	+	0.0994280i	$0.968299\pi$
$168$	0	0
$169$	−97.2229	−0.575284
$170$	0	0
$171$	52.8352i	0.308978i
$172$	0	0
$173$	226.361	1.30844	0.654222	−	0.756303i	$-0.272996\pi$
$173$	226.361	1.30844	0.654222	+	0.756303i	$0.272996\pi$
$174$	0	0
$175$	− 26.2866i	− 0.150209i
$176$	0	0
$177$	40.9969	0.231621
$178$	0	0
$179$	− 224.337i	− 1.25328i	−0.779308	−	0.626641i	$-0.784429\pi$
$179$	− 224.337i	− 1.25328i	0.779308	−	0.626641i	$-0.215571\pi$
$180$	0	0
$181$	−86.2229	−0.476370	−0.238185	−	0.971220i	$-0.576552\pi$
$181$	−86.2229	−0.476370	−0.238185	+	0.971220i	$0.576552\pi$
$182$	0	0
$183$	− 211.927i	− 1.15807i
$184$	0	0
$185$	41.0557	0.221923
$186$	0	0
$187$	− 236.810i	− 1.26636i
$188$	0	0
$189$	154.164	0.815683
$190$	0	0
$191$	− 31.0198i	− 0.162407i	−0.996698	−	0.0812036i	$-0.974124\pi$
$191$	− 31.0198i	− 0.162407i	0.996698	−	0.0812036i	$-0.0258764\pi$
$192$	0	0
$193$	110.223	0.571103	0.285552	−	0.958363i	$-0.407823\pi$
$193$	110.223	0.571103	0.285552	+	0.958363i	$0.407823\pi$
$194$	0	0
$195$	44.5407i	0.228414i
$196$	0	0
$197$	172.525	0.875760	0.437880	−	0.899033i	$-0.355729\pi$
$197$	172.525	0.875760	0.437880	+	0.899033i	$0.355729\pi$
$198$	0	0
$199$	− 272.208i	− 1.36788i	−0.729538	−	0.683940i	$-0.760265\pi$
$199$	− 272.208i	− 1.36788i	0.729538	−	0.683940i	$-0.239735\pi$
$200$	0	0
$201$	118.079	0.587457
$202$	0	0
$203$	− 57.5374i	− 0.283436i
$204$	0	0
$205$	−32.6099	−0.159073
$206$	0	0
$207$	− 1.92714i	− 0.00930984i
$208$	0	0
$209$	303.108	1.45028
$210$	0	0
$211$	205.266i	0.972826i	0.873729	+	0.486413i	$0.161695\pi$
$211$	205.266i	0.972826i	−0.873729	+	0.486413i	$0.838305\pi$
$212$	0	0
$213$	189.941	0.891743
$214$	0	0
$215$	− 49.7980i	− 0.231618i
$216$	0	0
$217$	−43.6068	−0.200953
$218$	0	0
$219$	− 13.0586i	− 0.0596285i
$220$	0	0
$221$	100.721	0.455753
$222$	0	0
$223$	235.731i	1.05709i	0.848905	+	0.528545i	$0.177262\pi$
$223$	235.731i	1.05709i	−0.848905	+	0.528545i	$0.822738\pi$
$224$	0	0
$225$	17.3607	0.0771586
$226$	0	0
$227$	58.5165i	0.257782i	0.991659	+	0.128891i	$0.0411417\pi$
$227$	58.5165i	0.257782i	−0.991659	+	0.128891i	$0.958858\pi$
$228$	0	0
$229$	−162.721	−0.710574	−0.355287	−	0.934757i	$-0.615617\pi$
$229$	−162.721	−0.710574	−0.355287	+	0.934757i	$0.615617\pi$
$230$	0	0
$231$	− 246.215i	− 1.06586i
$232$	0	0
$233$	319.050	1.36931	0.684656	−	0.728867i	$-0.259953\pi$
$233$	319.050	1.36931	0.684656	+	0.728867i	$0.259953\pi$
$234$	0	0
$235$	119.384i	0.508017i
$236$	0	0
$237$	32.5511	0.137346
$238$	0	0
$239$	236.810i	0.990837i	0.868654	+	0.495419i	$0.164985\pi$
$239$	236.810i	0.990837i	−0.868654	+	0.495419i	$0.835015\pi$
$240$	0	0
$241$	−0.917961	−0.00380897	−0.00190448	−	0.999998i	$-0.500606\pi$
$241$	−0.917961	−0.00380897	−0.00190448	+	0.999998i	$0.500606\pi$
$242$	0	0
$243$	175.287i	0.721347i
$244$	0	0
$245$	47.7639	0.194955
$246$	0	0
$247$	128.920i	0.521942i
$248$	0	0
$249$	−179.193	−0.719653
$250$	0	0
$251$	136.690i	0.544582i	0.962215	+	0.272291i	$0.0877813\pi$
$251$	136.690i	0.544582i	−0.962215	+	0.272291i	$0.912219\pi$
$252$	0	0
$253$	−11.0557	−0.0436985
$254$	0	0
$255$	62.5018i	0.245105i
$256$	0	0
$257$	−274.944	−1.06982	−0.534911	−	0.844908i	$-0.679655\pi$
$257$	−274.944	−1.06982	−0.534911	+	0.844908i	$0.679655\pi$
$258$	0	0
$259$	− 96.5278i	− 0.372694i
$260$	0	0
$261$	38.0000	0.145594
$262$	0	0
$263$	406.385i	1.54519i	0.634899	+	0.772596i	$0.281042\pi$
$263$	406.385i	1.54519i	−0.634899	+	0.772596i	$0.718958\pi$
$264$	0	0
$265$	148.387	0.559951
$266$	0	0
$267$	− 262.018i	− 0.981339i
$268$	0	0
$269$	348.525	1.29563	0.647816	−	0.761797i	$-0.275683\pi$
$269$	348.525	1.29563	0.647816	+	0.761797i	$0.275683\pi$
$270$	0	0
$271$	− 247.849i	− 0.914571i	−0.889320	−	0.457286i	$-0.848822\pi$
$271$	− 247.849i	− 0.914571i	0.889320	−	0.457286i	$-0.151178\pi$
$272$	0	0
$273$	104.721	0.383595
$274$	0	0
$275$	− 99.5959i	− 0.362167i
$276$	0	0
$277$	54.7539	0.197667	0.0988337	−	0.995104i	$-0.468489\pi$
$277$	54.7539	0.197667	0.0988337	+	0.995104i	$0.468489\pi$
$278$	0	0
$279$	− 28.7997i	− 0.103225i
$280$	0	0
$281$	−50.3607	−0.179220	−0.0896098	−	0.995977i	$-0.528562\pi$
$281$	−50.3607	−0.179220	−0.0896098	+	0.995977i	$0.528562\pi$
$282$	0	0
$283$	147.336i	0.520621i	0.965525	+	0.260310i	$0.0838249\pi$
$283$	147.336i	0.520621i	−0.965525	+	0.260310i	$0.916175\pi$
$284$	0	0
$285$	−80.0000	−0.280702
$286$	0	0
$287$	76.6705i	0.267145i
$288$	0	0
$289$	−147.663	−0.510943
$290$	0	0
$291$	− 218.263i	− 0.750045i
$292$	0	0
$293$	−178.859	−0.610441	−0.305220	−	0.952282i	$-0.598730\pi$
$293$	−178.859	−0.610441	−0.305220	+	0.952282i	$0.598730\pi$
$294$	0	0
$295$	− 38.9904i	− 0.132171i
$296$	0	0
$297$	584.105	1.96668
$298$	0	0
$299$	− 4.70228i	− 0.0157267i
$300$	0	0
$301$	−117.082	−0.388977
$302$	0	0
$303$	− 150.735i	− 0.497475i
$304$	0	0
$305$	−201.554	−0.660833
$306$	0	0
$307$	284.550i	0.926873i	0.886130	+	0.463436i	$0.153384\pi$
$307$	284.550i	0.926873i	−0.886130	+	0.463436i	$0.846616\pi$
$308$	0	0
$309$	−323.915	−1.04827
$310$	0	0
$311$	282.199i	0.907392i	0.891157	+	0.453696i	$0.149895\pi$
$311$	282.199i	0.907392i	−0.891157	+	0.453696i	$0.850105\pi$
$312$	0	0
$313$	−567.548	−1.81325	−0.906626	−	0.421935i	$-0.861351\pi$
$313$	−567.548	−1.81325	−0.906626	+	0.421935i	$0.861351\pi$
$314$	0	0
$315$	− 40.8174i	− 0.129579i
$316$	0	0
$317$	−161.141	−0.508331	−0.254165	−	0.967161i	$-0.581801\pi$
$317$	−161.141	−0.508331	−0.254165	+	0.967161i	$0.581801\pi$
$318$	0	0
$319$	− 218.001i	− 0.683389i
$320$	0	0
$321$	−120.689	−0.375978
$322$	0	0
$323$	180.907i	0.560083i
$324$	0	0
$325$	42.3607	0.130341
$326$	0	0
$327$	− 313.805i	− 0.959647i
$328$	0	0
$329$	280.689	0.853158
$330$	0	0
$331$	− 331.966i	− 1.00292i	−0.865181	−	0.501459i	$-0.832797\pi$
$331$	− 331.966i	− 1.00292i	0.865181	−	0.501459i	$-0.167203\pi$
$332$	0	0
$333$	63.7508	0.191444
$334$	0	0
$335$	− 112.300i	− 0.335223i
$336$	0	0
$337$	−269.108	−0.798541	−0.399271	−	0.916833i	$-0.630737\pi$
$337$	−269.108	−0.798541	−0.399271	+	0.916833i	$0.630737\pi$
$338$	0	0
$339$	401.390i	1.18404i
$340$	0	0
$341$	−165.220	−0.484516
$342$	0	0
$343$	− 369.908i	− 1.07845i
$344$	0	0
$345$	2.91796	0.00845786
$346$	0	0
$347$	503.075i	1.44978i	0.688863	+	0.724892i	$0.258110\pi$
$347$	503.075i	1.44978i	−0.688863	+	0.724892i	$0.741890\pi$
$348$	0	0
$349$	0.504658	0.00144601	0.000723006	−	1.00000i	$-0.499770\pi$
$349$	0.504658	0.00144601	0.000723006		1.00000i	$0.499770\pi$
$350$	0	0
$351$	248.435i	0.707791i
$352$	0	0
$353$	−335.994	−0.951824	−0.475912	−	0.879493i	$-0.657882\pi$
$353$	−335.994	−0.951824	−0.475912	+	0.879493i	$0.657882\pi$
$354$	0	0
$355$	− 180.645i	− 0.508859i
$356$	0	0
$357$	146.950	0.411626
$358$	0	0
$359$	98.4859i	0.274334i	0.990548	+	0.137167i	$0.0437997\pi$
$359$	98.4859i	0.274334i	−0.990548	+	0.137167i	$0.956200\pi$
$360$	0	0
$361$	129.446	0.358576
$362$	0	0
$363$	− 648.384i	− 1.78618i
$364$	0	0
$365$	−12.4195	−0.0340261
$366$	0	0
$367$	498.473i	1.35824i	0.734029	+	0.679118i	$0.237638\pi$
$367$	498.473i	1.35824i	−0.734029	+	0.679118i	$0.762362\pi$
$368$	0	0
$369$	−50.6362	−0.137226
$370$	0	0
$371$	− 348.879i	− 0.940374i
$372$	0	0
$373$	−600.354	−1.60953	−0.804765	−	0.593594i	$-0.797709\pi$
$373$	−600.354	−1.60953	−0.804765	+	0.593594i	$0.797709\pi$
$374$	0	0
$375$	26.2866i	0.0700975i
$376$	0	0
$377$	92.7214	0.245945
$378$	0	0
$379$	− 303.490i	− 0.800765i	−0.916348	−	0.400383i	$-0.868877\pi$
$379$	− 303.490i	− 0.800765i	0.916348	−	0.400383i	$-0.131123\pi$
$380$	0	0
$381$	467.023	1.22578
$382$	0	0
$383$	− 332.583i	− 0.868362i	−0.900826	−	0.434181i	$-0.857038\pi$
$383$	− 332.583i	− 0.868362i	0.900826	−	0.434181i	$-0.142962\pi$
$384$	0	0
$385$	−234.164	−0.608218
$386$	0	0
$387$	− 77.3256i	− 0.199808i
$388$	0	0
$389$	−392.354	−1.00862	−0.504312	−	0.863522i	$-0.668254\pi$
$389$	−392.354	−1.00862	−0.504312	+	0.863522i	$0.668254\pi$
$390$	0	0
$391$	− 6.59849i	− 0.0168759i
$392$	0	0
$393$	18.2693	0.0464868
$394$	0	0
$395$	− 30.9579i	− 0.0783744i
$396$	0	0
$397$	−334.190	−0.841789	−0.420895	−	0.907110i	$-0.638284\pi$
$397$	−334.190	−0.841789	−0.420895	+	0.907110i	$0.638284\pi$
$398$	0	0
$399$	188.091i	0.471407i
$400$	0	0
$401$	121.003	0.301753	0.150877	−	0.988553i	$-0.451790\pi$
$401$	121.003	0.301753	0.150877	+	0.988553i	$0.451790\pi$
$402$	0	0
$403$	− 70.2722i	− 0.174373i
$404$	0	0
$405$	−84.2887	−0.208120
$406$	0	0
$407$	− 365.730i	− 0.898599i
$408$	0	0
$409$	−607.410	−1.48511	−0.742555	−	0.669785i	$-0.766386\pi$
$409$	−607.410	−1.48511	−0.742555	+	0.669785i	$0.766386\pi$
$410$	0	0
$411$	1.95807i	0.00476415i
$412$	0	0
$413$	−91.6718	−0.221966
$414$	0	0
$415$	170.423i	0.410658i
$416$	0	0
$417$	−558.768	−1.33997
$418$	0	0
$419$	466.760i	1.11398i	0.830518	+	0.556992i	$0.188045\pi$
$419$	466.760i	1.11398i	−0.830518	+	0.556992i	$0.811955\pi$
$420$	0	0
$421$	73.0883	0.173606	0.0868031	−	0.996225i	$-0.472335\pi$
$421$	73.0883	0.173606	0.0868031	+	0.996225i	$0.472335\pi$
$422$	0	0
$423$	185.378i	0.438246i
$424$	0	0
$425$	59.4427	0.139865
$426$	0	0
$427$	473.882i	1.10979i
$428$	0	0
$429$	396.774	0.924881
$430$	0	0
$431$	− 463.630i	− 1.07571i	−0.843038	−	0.537853i	$-0.819235\pi$
$431$	− 463.630i	− 1.07571i	0.843038	−	0.537853i	$-0.180765\pi$
$432$	0	0
$433$	−99.8359	−0.230568	−0.115284	−	0.993333i	$-0.536778\pi$
$433$	−99.8359	−0.230568	−0.115284	+	0.993333i	$0.536778\pi$
$434$	0	0
$435$	57.5374i	0.132270i
$436$	0	0
$437$	8.44582	0.0193268
$438$	0	0
$439$	374.086i	0.852133i	0.904692	+	0.426066i	$0.140101\pi$
$439$	374.086i	0.852133i	−0.904692	+	0.426066i	$0.859899\pi$
$440$	0	0
$441$	74.1672	0.168180
$442$	0	0
$443$	290.100i	0.654854i	0.944877	+	0.327427i	$0.106182\pi$
$443$	290.100i	0.654854i	−0.944877	+	0.327427i	$0.893818\pi$
$444$	0	0
$445$	−249.193	−0.559985
$446$	0	0
$447$	86.9231i	0.194459i
$448$	0	0
$449$	299.921	0.667976	0.333988	−	0.942577i	$-0.391606\pi$
$449$	299.921	0.667976	0.333988	+	0.942577i	$0.391606\pi$
$450$	0	0
$451$	290.493i	0.644109i
$452$	0	0
$453$	−664.721	−1.46738
$454$	0	0
$455$	− 99.5959i	− 0.218892i
$456$	0	0
$457$	822.328	1.79941	0.899703	−	0.436503i	$-0.143783\pi$
$457$	822.328	1.79941	0.899703	+	0.436503i	$0.143783\pi$
$458$	0	0
$459$	348.617i	0.759513i
$460$	0	0
$461$	456.885	0.991075	0.495537	−	0.868587i	$-0.334971\pi$
$461$	456.885	0.991075	0.495537	+	0.868587i	$0.334971\pi$
$462$	0	0
$463$	400.249i	0.864469i	0.901761	+	0.432234i	$0.142275\pi$
$463$	400.249i	0.864469i	−0.901761	+	0.432234i	$0.857725\pi$
$464$	0	0
$465$	43.6068	0.0937781
$466$	0	0
$467$	− 913.145i	− 1.95534i	−0.210139	−	0.977672i	$-0.567392\pi$
$467$	− 913.145i	− 1.95534i	0.210139	−	0.977672i	$-0.432608\pi$
$468$	0	0
$469$	−264.033	−0.562969
$470$	0	0
$471$	− 481.391i	− 1.02206i
$472$	0	0
$473$	−443.607	−0.937858
$474$	0	0
$475$	76.0845i	0.160178i
$476$	0	0
$477$	230.413	0.483047
$478$	0	0
$479$	− 526.131i	− 1.09840i	−0.835692	−	0.549198i	$-0.814933\pi$
$479$	− 526.131i	− 1.09840i	0.835692	−	0.549198i	$-0.185067\pi$
$480$	0	0
$481$	155.554	0.323397
$482$	0	0
$483$	− 6.86054i	− 0.0142040i
$484$	0	0
$485$	−207.580	−0.428001
$486$	0	0
$487$	443.541i	0.910762i	0.890297	+	0.455381i	$0.150497\pi$
$487$	443.541i	0.910762i	−0.890297	+	0.455381i	$0.849503\pi$
$488$	0	0
$489$	252.125	0.515594
$490$	0	0
$491$	287.163i	0.584854i	0.956288	+	0.292427i	$0.0944628\pi$
$491$	287.163i	0.584854i	−0.956288	+	0.292427i	$0.905537\pi$
$492$	0	0
$493$	130.111	0.263918
$494$	0	0
$495$	− 154.651i	− 0.312427i
$496$	0	0
$497$	−424.721	−0.854570
$498$	0	0
$499$	− 810.936i	− 1.62512i	−0.582876	−	0.812561i	$-0.698073\pi$
$499$	− 810.936i	− 1.62512i	0.582876	−	0.812561i	$-0.301927\pi$
$500$	0	0
$501$	−78.0789	−0.155846
$502$	0	0
$503$	642.471i	1.27728i	0.769506	+	0.638639i	$0.220502\pi$
$503$	642.471i	1.27728i	−0.769506	+	0.638639i	$0.779498\pi$
$504$	0	0
$505$	−143.358	−0.283876
$506$	0	0
$507$	− 228.585i	− 0.450858i
$508$	0	0
$509$	915.050	1.79774	0.898870	−	0.438216i	$-0.144389\pi$
$509$	915.050	1.79774	0.898870	+	0.438216i	$0.144389\pi$
$510$	0	0
$511$	29.2000i	0.0571429i
$512$	0	0
$513$	−446.217	−0.869818
$514$	0	0
$515$	308.061i	0.598177i
$516$	0	0
$517$	1063.49	2.05704
$518$	0	0
$519$	532.206i	1.02544i
$520$	0	0
$521$	1006.98	1.93279	0.966396	−	0.257058i	$-0.0827533\pi$
$521$	1006.98	1.93279	0.966396	+	0.257058i	$0.0827533\pi$
$522$	0	0
$523$	774.173i	1.48025i	0.672467	+	0.740127i	$0.265235\pi$
$523$	774.173i	1.48025i	−0.672467	+	0.740127i	$0.734765\pi$
$524$	0	0
$525$	61.8034	0.117721
$526$	0	0
$527$	− 98.6096i	− 0.187115i
$528$	0	0
$529$	528.692	0.999418
$530$	0	0
$531$	− 60.5437i	− 0.114018i
$532$	0	0
$533$	−123.554	−0.231809
$534$	0	0
$535$	114.782i	0.214546i
$536$	0	0
$537$	527.449	0.982214
$538$	0	0
$539$	− 425.487i	− 0.789401i
$540$	0	0
$541$	259.115	0.478955	0.239477	−	0.970902i	$-0.423024\pi$
$541$	259.115	0.478955	0.239477	+	0.970902i	$0.423024\pi$
$542$	0	0
$543$	− 202.722i	− 0.373337i
$544$	0	0
$545$	−298.446	−0.547607
$546$	0	0
$547$	149.818i	0.273890i	0.990579	+	0.136945i	$0.0437284\pi$
$547$	149.818i	0.273890i	−0.990579	+	0.136945i	$0.956272\pi$
$548$	0	0
$549$	−312.971	−0.570074
$550$	0	0
$551$	166.538i	0.302247i
$552$	0	0
$553$	−72.7864	−0.131621
$554$	0	0
$555$	96.5278i	0.173924i
$556$	0	0
$557$	−511.698	−0.918668	−0.459334	−	0.888264i	$-0.651912\pi$
$557$	−511.698	−0.918668	−0.459334	+	0.888264i	$0.651912\pi$
$558$	0	0
$559$	− 188.677i	− 0.337526i
$560$	0	0
$561$	556.774	0.992467
$562$	0	0
$563$	− 490.726i	− 0.871627i	−0.900037	−	0.435814i	$-0.856461\pi$
$563$	− 490.726i	− 0.871627i	0.900037	−	0.435814i	$-0.143539\pi$
$564$	0	0
$565$	381.745	0.675654
$566$	0	0
$567$	198.175i	0.349514i
$568$	0	0
$569$	−232.748	−0.409047	−0.204523	−	0.978862i	$-0.565564\pi$
$569$	−232.748	−0.409047	−0.204523	+	0.978862i	$0.565564\pi$
$570$	0	0
$571$	210.755i	0.369098i	0.982823	+	0.184549i	$0.0590824\pi$
$571$	210.755i	0.369098i	−0.982823	+	0.184549i	$0.940918\pi$
$572$	0	0
$573$	72.9318	0.127281
$574$	0	0
$575$	− 2.77515i	− 0.00482634i
$576$	0	0
$577$	341.712	0.592222	0.296111	−	0.955154i	$-0.404310\pi$
$577$	341.712	0.592222	0.296111	+	0.955154i	$0.404310\pi$
$578$	0	0
$579$	259.150i	0.447581i
$580$	0	0
$581$	400.689	0.689654
$582$	0	0
$583$	− 1321.85i	− 2.26733i
$584$	0	0
$585$	65.7771	0.112439
$586$	0	0
$587$	− 618.412i	− 1.05351i	−0.850016	−	0.526756i	$-0.823408\pi$
$587$	− 618.412i	− 1.05351i	0.850016	−	0.526756i	$-0.176592\pi$
$588$	0	0
$589$	126.217	0.214290
$590$	0	0
$591$	405.630i	0.686345i
$592$	0	0
$593$	120.663	0.203478	0.101739	−	0.994811i	$-0.467559\pi$
$593$	120.663	0.203478	0.101739	+	0.994811i	$0.467559\pi$
$594$	0	0
$595$	− 139.758i	− 0.234888i
$596$	0	0
$597$	640.000	1.07203
$598$	0	0
$599$	− 849.927i	− 1.41891i	−0.704751	−	0.709455i	$-0.748941\pi$
$599$	− 849.927i	− 1.41891i	0.704751	−	0.709455i	$-0.251059\pi$
$600$	0	0
$601$	−11.3576	−0.0188978	−0.00944890	−	0.999955i	$-0.503008\pi$
$601$	−11.3576	−0.0188978	−0.00944890	+	0.999955i	$0.503008\pi$
$602$	0	0
$603$	− 174.378i	− 0.289183i
$604$	0	0
$605$	−616.649	−1.01926
$606$	0	0
$607$	− 1115.12i	− 1.83710i	−0.395305	−	0.918550i	$-0.629361\pi$
$607$	− 1115.12i	− 1.83710i	0.395305	−	0.918550i	$-0.370639\pi$
$608$	0	0
$609$	135.279	0.222132
$610$	0	0
$611$	452.329i	0.740309i
$612$	0	0
$613$	−499.475	−0.814805	−0.407402	−	0.913249i	$-0.633565\pi$
$613$	−499.475	−0.814805	−0.407402	+	0.913249i	$0.633565\pi$
$614$	0	0
$615$	− 76.6705i	− 0.124667i
$616$	0	0
$617$	545.935	0.884822	0.442411	−	0.896813i	$-0.354123\pi$
$617$	545.935	0.884822	0.442411	+	0.896813i	$0.354123\pi$
$618$	0	0
$619$	− 455.011i	− 0.735075i	−0.930009	−	0.367537i	$-0.880201\pi$
$619$	− 455.011i	− 0.735075i	0.930009	−	0.367537i	$-0.119799\pi$
$620$	0	0
$621$	16.2755	0.0262086
$622$	0	0
$623$	585.889i	0.940432i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	712.650i	1.13660i
$628$	0	0
$629$	218.282	0.347030
$630$	0	0
$631$	267.706i	0.424257i	0.977242	+	0.212128i	$0.0680395\pi$
$631$	267.706i	0.424257i	−0.977242	+	0.212128i	$0.931960\pi$
$632$	0	0
$633$	−482.610	−0.762417
$634$	0	0
$635$	− 444.165i	− 0.699473i
$636$	0	0
$637$	180.971	0.284098
$638$	0	0
$639$	− 280.503i	− 0.438971i
$640$	0	0
$641$	−418.571	−0.652997	−0.326499	−	0.945198i	$-0.605869\pi$
$641$	−418.571	−0.652997	−0.326499	+	0.945198i	$0.605869\pi$
$642$	0	0
$643$	− 439.339i	− 0.683265i	−0.939834	−	0.341633i	$-0.889020\pi$
$643$	− 439.339i	− 0.683265i	0.939834	−	0.341633i	$-0.110980\pi$
$644$	0	0
$645$	117.082	0.181523
$646$	0	0
$647$	419.644i	0.648600i	0.945954	+	0.324300i	$0.105129\pi$
$647$	419.644i	0.648600i	−0.945954	+	0.324300i	$0.894871\pi$
$648$	0	0
$649$	−347.331	−0.535179
$650$	0	0
$651$	− 102.526i	− 0.157490i
$652$	0	0
$653$	370.085	0.566746	0.283373	−	0.959010i	$-0.408547\pi$
$653$	370.085	0.566746	0.283373	+	0.959010i	$0.408547\pi$
$654$	0	0
$655$	− 17.3752i	− 0.0265270i
$656$	0	0
$657$	−19.2849	−0.0293529
$658$	0	0
$659$	322.823i	0.489868i	0.969540	+	0.244934i	$0.0787664\pi$
$659$	322.823i	0.489868i	−0.969540	+	0.244934i	$0.921234\pi$
$660$	0	0
$661$	812.735	1.22955	0.614777	−	0.788701i	$-0.289246\pi$
$661$	812.735	1.22955	0.614777	+	0.788701i	$0.289246\pi$
$662$	0	0
$663$	236.810i	0.357180i
$664$	0	0
$665$	178.885	0.269001
$666$	0	0
$667$	− 6.07439i	− 0.00910703i
$668$	0	0
$669$	−554.237	−0.828456
$670$	0	0
$671$	1795.47i	2.67581i
$672$	0	0
$673$	467.378	0.694469	0.347235	−	0.937778i	$-0.387121\pi$
$673$	467.378	0.694469	0.347235	+	0.937778i	$0.387121\pi$
$674$	0	0
$675$	146.619i	0.217213i
$676$	0	0
$677$	−548.237	−0.809803	−0.404902	−	0.914360i	$-0.632694\pi$
$677$	−548.237	−0.809803	−0.404902	+	0.914360i	$0.632694\pi$
$678$	0	0
$679$	488.051i	0.718779i
$680$	0	0
$681$	−137.580	−0.202027
$682$	0	0
$683$	− 23.9663i	− 0.0350898i	−0.999846	−	0.0175449i	$-0.994415\pi$
$683$	− 23.9663i	− 0.0350898i	0.999846	−	0.0175449i	$-0.00558500\pi$
$684$	0	0
$685$	1.86223	0.00271859
$686$	0	0
$687$	− 382.581i	− 0.556886i
$688$	0	0
$689$	562.217	0.815989
$690$	0	0
$691$	186.981i	0.270595i	0.990805	+	0.135298i	$0.0431990\pi$
$691$	186.981i	0.270595i	−0.990805	+	0.135298i	$0.956801\pi$
$692$	0	0
$693$	−363.607	−0.524685
$694$	0	0
$695$	531.420i	0.764633i
$696$	0	0
$697$	−173.378	−0.248748
$698$	0	0
$699$	750.130i	1.07315i
$700$	0	0
$701$	706.636	1.00804	0.504020	−	0.863692i	$-0.331854\pi$
$701$	706.636	1.00804	0.504020	+	0.863692i	$0.331854\pi$
$702$	0	0
$703$	279.393i	0.397429i
$704$	0	0
$705$	−280.689	−0.398140
$706$	0	0
$707$	337.054i	0.476738i
$708$	0	0
$709$	188.597	0.266005	0.133002	−	0.991116i	$-0.457538\pi$
$709$	188.597	0.266005	0.133002	+	0.991116i	$0.457538\pi$
$710$	0	0
$711$	− 48.0710i	− 0.0676104i
$712$	0	0
$713$	−4.60369	−0.00645679
$714$	0	0
$715$	− 377.354i	− 0.527769i
$716$	0	0
$717$	−556.774	−0.776533
$718$	0	0
$719$	156.085i	0.217086i	0.994092	+	0.108543i	$0.0346186\pi$
$719$	156.085i	0.217086i	−0.994092	+	0.108543i	$0.965381\pi$
$720$	0	0
$721$	724.296	1.00457
$722$	0	0
$723$	− 2.15825i	− 0.00298514i
$724$	0	0
$725$	54.7214	0.0754777
$726$	0	0
$727$	− 715.164i	− 0.983719i	−0.870675	−	0.491859i	$-0.836317\pi$
$727$	− 715.164i	− 0.983719i	0.870675	−	0.491859i	$-0.163683\pi$
$728$	0	0
$729$	−751.381	−1.03070
$730$	0	0
$731$	− 264.762i	− 0.362191i
$732$	0	0
$733$	−1233.29	−1.68252	−0.841259	−	0.540632i	$-0.818185\pi$
$733$	−1233.29	−1.68252	−0.841259	+	0.540632i	$0.818185\pi$
$734$	0	0
$735$	112.300i	0.152789i
$736$	0	0
$737$	−1000.38	−1.35737
$738$	0	0
$739$	− 8.55656i	− 0.0115786i	−0.999983	−	0.00578928i	$-0.998157\pi$
$739$	− 8.55656i	− 0.0115786i	0.999983	−	0.00578928i	$-0.00184280\pi$
$740$	0	0
$741$	−303.108	−0.409053
$742$	0	0
$743$	− 1010.56i	− 1.36011i	−0.733163	−	0.680053i	$-0.761957\pi$
$743$	− 1010.56i	− 1.36011i	0.733163	−	0.680053i	$-0.238043\pi$
$744$	0	0
$745$	82.6687	0.110965
$746$	0	0
$747$	264.631i	0.354258i
$748$	0	0
$749$	269.868	0.360305
$750$	0	0
$751$	− 1104.31i	− 1.47046i	−0.677820	−	0.735228i	$-0.737075\pi$
$751$	− 1104.31i	− 1.47046i	0.677820	−	0.735228i	$-0.262925\pi$
$752$	0	0
$753$	−321.378	−0.426796
$754$	0	0
$755$	632.188i	0.837335i
$756$	0	0
$757$	875.633	1.15671	0.578357	−	0.815783i	$-0.303694\pi$
$757$	875.633	1.15671	0.578357	+	0.815783i	$0.303694\pi$
$758$	0	0
$759$	− 25.9936i	− 0.0342471i
$760$	0	0
$761$	−647.207	−0.850470	−0.425235	−	0.905083i	$-0.639808\pi$
$761$	−647.207	−0.850470	−0.425235	+	0.905083i	$0.639808\pi$
$762$	0	0
$763$	701.688i	0.919644i
$764$	0	0
$765$	92.3018	0.120656
$766$	0	0
$767$	− 147.729i	− 0.192606i
$768$	0	0
$769$	631.430	0.821106	0.410553	−	0.911837i	$-0.365336\pi$
$769$	631.430	0.821106	0.410553	+	0.911837i	$0.365336\pi$
$770$	0	0
$771$	− 646.433i	− 0.838434i
$772$	0	0
$773$	421.522	0.545306	0.272653	−	0.962112i	$-0.412099\pi$
$773$	421.522	0.545306	0.272653	+	0.962112i	$0.412099\pi$
$774$	0	0
$775$	− 41.4725i	− 0.0535129i
$776$	0	0
$777$	226.950	0.292086
$778$	0	0
$779$	− 221.917i	− 0.284874i
$780$	0	0
$781$	−1609.21	−2.06044
$782$	0	0
$783$	320.927i	0.409869i
$784$	0	0
$785$	−457.830	−0.583223
$786$	0	0
$787$	− 838.633i	− 1.06561i	−0.846239	−	0.532804i	$-0.821138\pi$
$787$	− 838.633i	− 1.06561i	0.846239	−	0.532804i	$-0.178862\pi$
$788$	0	0
$789$	−955.469	−1.21099
$790$	0	0
$791$	− 897.535i	− 1.13468i
$792$	0	0
$793$	−763.659	−0.963001
$794$	0	0
$795$	348.879i	0.438841i
$796$	0	0
$797$	−1213.57	−1.52268	−0.761339	−	0.648354i	$-0.775458\pi$
$797$	−1213.57	−1.52268	−0.761339	+	0.648354i	$0.775458\pi$
$798$	0	0
$799$	634.732i	0.794408i
$800$	0	0
$801$	−386.944	−0.483076
$802$	0	0
$803$	110.635i	0.137777i
$804$	0	0
$805$	−6.52476	−0.00810529
$806$	0	0
$807$	819.431i	1.01540i
$808$	0	0
$809$	−229.214	−0.283330	−0.141665	−	0.989915i	$-0.545246\pi$
$809$	−229.214	−0.283330	−0.141665	+	0.989915i	$0.545246\pi$
$810$	0	0
$811$	− 454.225i	− 0.560080i	−0.959988	−	0.280040i	$-0.909652\pi$
$811$	− 454.225i	− 0.560080i	0.959988	−	0.280040i	$-0.0903478\pi$
$812$	0	0
$813$	582.728	0.716762
$814$	0	0
$815$	− 239.785i	− 0.294215i
$816$	0	0
$817$	338.885	0.414792
$818$	0	0
$819$	− 154.651i	− 0.188829i
$820$	0	0
$821$	−1130.90	−1.37747	−0.688733	−	0.725015i	$-0.741833\pi$
$821$	−1130.90	−1.37747	−0.688733	+	0.725015i	$0.741833\pi$
$822$	0	0
$823$	780.148i	0.947931i	0.880543	+	0.473966i	$0.157178\pi$
$823$	780.148i	0.947931i	−0.880543	+	0.473966i	$0.842822\pi$
$824$	0	0
$825$	234.164	0.283835
$826$	0	0
$827$	− 209.175i	− 0.252932i	−0.991971	−	0.126466i	$-0.959636\pi$
$827$	− 209.175i	− 0.252932i	0.991971	−	0.126466i	$-0.0403635\pi$
$828$	0	0
$829$	508.525	0.613419	0.306710	−	0.951803i	$-0.400772\pi$
$829$	508.525	0.613419	0.306710	+	0.951803i	$0.400772\pi$
$830$	0	0
$831$	128.734i	0.154915i
$832$	0	0
$833$	253.947	0.304859
$834$	0	0
$835$	74.2575i	0.0889311i
$836$	0	0
$837$	243.226	0.290593
$838$	0	0
$839$	− 274.028i	− 0.326613i	−0.986575	−	0.163306i	$-0.947784\pi$
$839$	− 274.028i	− 0.326613i	0.986575	−	0.163306i	$-0.0522159\pi$
$840$	0	0
$841$	−721.223	−0.857578
$842$	0	0
$843$	− 118.405i	− 0.140457i
$844$	0	0
$845$	−217.397	−0.257275
$846$	0	0
$847$	1449.83i	1.71172i
$848$	0	0
$849$	−346.407	−0.408018
$850$	0	0
$851$	− 10.1907i	− 0.0119750i
$852$	0	0
$853$	1583.28	1.85613	0.928066	−	0.372416i	$-0.121471\pi$
$853$	1583.28	1.85613	0.928066	+	0.372416i	$0.121471\pi$
$854$	0	0
$855$	118.143i	0.138179i
$856$	0	0
$857$	−1007.38	−1.17547	−0.587735	−	0.809054i	$-0.699980\pi$
$857$	−1007.38	−1.17547	−0.587735	+	0.809054i	$0.699980\pi$
$858$	0	0
$859$	76.6086i	0.0891835i	0.999005	+	0.0445917i	$0.0141987\pi$
$859$	76.6086i	0.0891835i	−0.999005	+	0.0445917i	$0.985801\pi$
$860$	0	0
$861$	−180.263	−0.209365
$862$	0	0
$863$	− 255.450i	− 0.296002i	−0.988987	−	0.148001i	$-0.952716\pi$
$863$	− 255.450i	− 0.296002i	0.988987	−	0.148001i	$-0.0472839\pi$
$864$	0	0
$865$	506.158	0.585154
$866$	0	0
$867$	− 347.175i	− 0.400433i
$868$	0	0
$869$	−275.777	−0.317350
$870$	0	0
$871$	− 425.487i	− 0.488504i
$872$	0	0
$873$	−322.328	−0.369219
$874$	0	0
$875$	− 58.7785i	− 0.0671755i
$876$	0	0
$877$	−601.522	−0.685886	−0.342943	−	0.939356i	$-0.611424\pi$
$877$	−601.522	−0.685886	−0.342943	+	0.939356i	$0.611424\pi$
$878$	0	0
$879$	− 420.523i	− 0.478411i
$880$	0	0
$881$	237.850	0.269977	0.134989	−	0.990847i	$-0.456900\pi$
$881$	237.850	0.269977	0.134989	+	0.990847i	$0.456900\pi$
$882$	0	0
$883$	1.30294i	0.00147559i	1.00000		0.000737794i	$0.000234847\pi$
$883$	1.30294i	0.00147559i	−1.00000		0.000737794i	$0.999765\pi$
$884$	0	0
$885$	91.6718	0.103584
$886$	0	0
$887$	− 536.353i	− 0.604682i	−0.953200	−	0.302341i	$-0.902232\pi$
$887$	− 536.353i	− 0.604682i	0.953200	−	0.302341i	$-0.0977682\pi$
$888$	0	0
$889$	−1044.30	−1.17469
$890$	0	0
$891$	750.855i	0.842710i
$892$	0	0
$893$	−812.433	−0.909780
$894$	0	0
$895$	− 501.634i	− 0.560485i
$896$	0	0
$897$	11.0557	0.0123252
$898$	0	0
$899$	− 90.7773i	− 0.100976i
$900$	0	0
$901$	788.932	0.875618
$902$	0	0
$903$	− 275.276i	− 0.304846i
$904$	0	0
$905$	−192.800	−0.213039
$906$	0	0
$907$	− 332.159i	− 0.366217i	−0.983093	−	0.183108i	$-0.941384\pi$
$907$	− 332.159i	− 0.366217i	0.983093	−	0.183108i	$-0.0586159\pi$
$908$	0	0
$909$	−222.604	−0.244889
$910$	0	0
$911$	− 1450.06i	− 1.59172i	−0.605478	−	0.795862i	$-0.707018\pi$
$911$	− 1450.06i	− 1.59172i	0.605478	−	0.795862i	$-0.292982\pi$
$912$	0	0
$913$	1518.15	1.66282
$914$	0	0
$915$	− 473.882i	− 0.517904i
$916$	0	0
$917$	−40.8514	−0.0445490
$918$	0	0
$919$	814.405i	0.886186i	0.896476	+	0.443093i	$0.146119\pi$
$919$	814.405i	0.886186i	−0.896476	+	0.443093i	$0.853881\pi$
$920$	0	0
$921$	−669.017	−0.726403
$922$	0	0
$923$	− 684.437i	− 0.741535i
$924$	0	0
$925$	91.8034	0.0992469
$926$	0	0
$927$	478.353i	0.516023i
$928$	0	0
$929$	400.039	0.430612	0.215306	−	0.976547i	$-0.430925\pi$
$929$	400.039	0.430612	0.215306	+	0.976547i	$0.430925\pi$
$930$	0	0
$931$	325.043i	0.349134i
$932$	0	0
$933$	−663.489	−0.711135
$934$	0	0
$935$	− 529.524i	− 0.566335i
$936$	0	0
$937$	249.279	0.266039	0.133020	−	0.991113i	$-0.457533\pi$
$937$	249.279	0.266039	0.133020	+	0.991113i	$0.457533\pi$
$938$	0	0
$939$	− 1334.39i	− 1.42107i
$940$	0	0
$941$	−724.229	−0.769638	−0.384819	−	0.922992i	$-0.625736\pi$
$941$	−724.229	−0.769638	−0.384819	+	0.922992i	$0.625736\pi$
$942$	0	0
$943$	8.09432i	0.00858358i
$944$	0	0
$945$	344.721	0.364785
$946$	0	0
$947$	1141.54i	1.20542i	0.797959	+	0.602712i	$0.205913\pi$
$947$	1141.54i	1.20542i	−0.797959	+	0.602712i	$0.794087\pi$
$948$	0	0
$949$	−47.0557	−0.0495845
$950$	0	0
$951$	− 378.865i	− 0.398386i
$952$	0	0
$953$	1295.33	1.35921	0.679604	−	0.733579i	$-0.262152\pi$
$953$	1295.33	1.35921	0.679604	+	0.733579i	$0.262152\pi$
$954$	0	0
$955$	− 69.3623i	− 0.0726307i
$956$	0	0
$957$	512.551	0.535581
$958$	0	0
$959$	− 4.37837i	− 0.00456556i
$960$	0	0
$961$	892.201	0.928409
$962$	0	0
$963$	178.232i	0.185080i
$964$	0	0
$965$	246.466	0.255405
$966$	0	0
$967$	− 398.477i	− 0.412075i	−0.978544	−	0.206037i	$-0.933943\pi$
$967$	− 398.477i	− 0.412075i	0.978544	−	0.206037i	$-0.0660569\pi$
$968$	0	0
$969$	−425.337	−0.438945
$970$	0	0
$971$	928.093i	0.955811i	0.878411	+	0.477906i	$0.158604\pi$
$971$	928.093i	0.955811i	−0.878411	+	0.477906i	$0.841396\pi$
$972$	0	0
$973$	1249.44	1.28411
$974$	0	0
$975$	99.5959i	0.102150i
$976$	0	0
$977$	−1378.05	−1.41049	−0.705247	−	0.708962i	$-0.749164\pi$
$977$	−1378.05	−1.41049	−0.705247	+	0.708962i	$0.749164\pi$
$978$	0	0
$979$	2219.85i	2.26747i
$980$	0	0
$981$	−463.423	−0.472398
$982$	0	0
$983$	311.291i	0.316675i	0.987385	+	0.158337i	$0.0506134\pi$
$983$	311.291i	0.316675i	−0.987385	+	0.158337i	$0.949387\pi$
$984$	0	0
$985$	385.777	0.391652
$986$	0	0
$987$	659.939i	0.668631i
$988$	0	0
$989$	−12.3607	−0.0124982
$990$	0	0
$991$	961.147i	0.969876i	0.874549	+	0.484938i	$0.161158\pi$
$991$	961.147i	0.969876i	−0.874549	+	0.484938i	$0.838842\pi$
$992$	0	0
$993$	780.498	0.786000
$994$	0	0
$995$	− 608.676i	− 0.611735i
$996$	0	0
$997$	1089.68	1.09296	0.546479	−	0.837473i	$-0.315968\pi$
$997$	1089.68	1.09296	0.546479	+	0.837473i	$0.315968\pi$
$998$	0	0
$999$	538.404i	0.538943i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.3.b.c.191.3		4
3.2	odd	2		2880.3.e.e.2431.1		4
4.3	odd	2	inner	320.3.b.c.191.2		4
5.2	odd	4		1600.3.h.n.1599.5		8
5.3	odd	4		1600.3.h.n.1599.3		8
5.4	even	2		1600.3.b.s.1151.2		4
8.3	odd	2		20.3.b.a.11.3	✓	4
8.5	even	2		20.3.b.a.11.4	yes	4
12.11	even	2		2880.3.e.e.2431.2		4
16.3	odd	4		1280.3.g.e.1151.5		8
16.5	even	4		1280.3.g.e.1151.6		8
16.11	odd	4		1280.3.g.e.1151.4		8
16.13	even	4		1280.3.g.e.1151.3		8
20.3	even	4		1600.3.h.n.1599.6		8
20.7	even	4		1600.3.h.n.1599.4		8
20.19	odd	2		1600.3.b.s.1151.3		4
24.5	odd	2		180.3.c.a.91.1		4
24.11	even	2		180.3.c.a.91.2		4
40.3	even	4		100.3.d.b.99.1		8
40.13	odd	4		100.3.d.b.99.7		8
40.19	odd	2		100.3.b.f.51.2		4
40.27	even	4		100.3.d.b.99.8		8
40.29	even	2		100.3.b.f.51.1		4
40.37	odd	4		100.3.d.b.99.2		8
120.29	odd	2		900.3.c.k.451.4		4
120.53	even	4		900.3.f.e.199.2		8
120.59	even	2		900.3.c.k.451.3		4
120.77	even	4		900.3.f.e.199.7		8
120.83	odd	4		900.3.f.e.199.8		8
120.107	odd	4		900.3.f.e.199.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.3.b.a.11.3	✓	4	8.3	odd	2
20.3.b.a.11.4	yes	4	8.5	even	2
100.3.b.f.51.1		4	40.29	even	2
100.3.b.f.51.2		4	40.19	odd	2
100.3.d.b.99.1		8	40.3	even	4
100.3.d.b.99.2		8	40.37	odd	4
100.3.d.b.99.7		8	40.13	odd	4
100.3.d.b.99.8		8	40.27	even	4
180.3.c.a.91.1		4	24.5	odd	2
180.3.c.a.91.2		4	24.11	even	2
320.3.b.c.191.2		4	4.3	odd	2	inner
320.3.b.c.191.3		4	1.1	even	1	trivial
900.3.c.k.451.3		4	120.59	even	2
900.3.c.k.451.4		4	120.29	odd	2
900.3.f.e.199.1		8	120.107	odd	4
900.3.f.e.199.2		8	120.53	even	4
900.3.f.e.199.7		8	120.77	even	4
900.3.f.e.199.8		8	120.83	odd	4
1280.3.g.e.1151.3		8	16.13	even	4
1280.3.g.e.1151.4		8	16.11	odd	4
1280.3.g.e.1151.5		8	16.3	odd	4
1280.3.g.e.1151.6		8	16.5	even	4
1600.3.b.s.1151.2		4	5.4	even	2
1600.3.b.s.1151.3		4	20.19	odd	2
1600.3.h.n.1599.3		8	5.3	odd	4
1600.3.h.n.1599.4		8	20.7	even	4
1600.3.h.n.1599.5		8	5.2	odd	4
1600.3.h.n.1599.6		8	20.3	even	4
2880.3.e.e.2431.1		4	3.2	odd	2
2880.3.e.e.2431.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	320.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.35114i q^{3} +2.23607 q^{5} -5.25731i q^{7} +3.47214 q^{9} +O(q^{10})\)	\(q+2.35114i q^{3} +2.23607 q^{5} -5.25731i q^{7} +3.47214 q^{9} -19.9192i q^{11} +8.47214 q^{13} +5.25731i q^{15} +11.8885 q^{17} +15.2169i q^{19} +12.3607 q^{21} -0.555029i q^{23} +5.00000 q^{25} +29.3238i q^{27} +10.9443 q^{29} -8.29451i q^{31} +46.8328 q^{33} -11.7557i q^{35} +18.3607 q^{37} +19.9192i q^{39} -14.5836 q^{41} -22.2703i q^{43} +7.76393 q^{45} +53.3902i q^{47} +21.3607 q^{49} +27.9516i q^{51} +66.3607 q^{53} -44.5407i q^{55} -35.7771 q^{57} -17.4370i q^{59} -90.1378 q^{61} -18.2541i q^{63} +18.9443 q^{65} -50.2220i q^{67} +1.30495 q^{69} -80.7868i q^{71} -5.55418 q^{73} +11.7557i q^{75} -104.721 q^{77} -13.8448i q^{79} -37.6950 q^{81} +76.2155i q^{83} +26.5836 q^{85} +25.7315i q^{87} -111.443 q^{89} -44.5407i q^{91} +19.5016 q^{93} +34.0260i q^{95} -92.8328 q^{97} -69.1621i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 16 q^{13} - 24 q^{17} - 40 q^{21} + 20 q^{25} + 8 q^{29} + 80 q^{33} - 16 q^{37} - 112 q^{41} + 40 q^{45} - 4 q^{49} + 176 q^{53} - 128 q^{61} + 40 q^{65} - 120 q^{69} + 264 q^{73} - 240 q^{77} - 276 q^{81} + 160 q^{85} - 88 q^{89} + 400 q^{93} - 264 q^{97}+O(q^{100}) \) 4 * q - 4 * q^9 + 16 * q^13 - 24 * q^17 - 40 * q^21 + 20 * q^25 + 8 * q^29 + 80 * q^33 - 16 * q^37 - 112 * q^41 + 40 * q^45 - 4 * q^49 + 176 * q^53 - 128 * q^61 + 40 * q^65 - 120 * q^69 + 264 * q^73 - 240 * q^77 - 276 * q^81 + 160 * q^85 - 88 * q^89 + 400 * q^93 - 264 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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