LMFDB - Embedded newform 760.2.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [760,2,Mod(1,760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(760, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 760 = 2^{3} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	760.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$6.06863055362$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.76156$ of defining polynomial
Character	$\chi$	$=$	760.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.76156 q^{3} -1.00000 q^{5} -4.62620 q^{7} +0.103084 q^{9} -5.52311 q^{11} +5.49084 q^{13} -1.76156 q^{15} -6.62620 q^{17} -1.00000 q^{19} -8.14931 q^{21} +4.14931 q^{23} +1.00000 q^{25} -5.10308 q^{27} -7.87859 q^{29} +1.25240 q^{31} -9.72928 q^{33} +4.62620 q^{35} -0.387755 q^{37} +9.67243 q^{39} +6.77551 q^{41} -10.9817 q^{43} -0.103084 q^{45} +1.72928 q^{47} +14.4017 q^{49} -11.6724 q^{51} +1.49084 q^{53} +5.52311 q^{55} -1.76156 q^{57} +0.626198 q^{59} +15.0462 q^{61} -0.476886 q^{63} -5.49084 q^{65} +5.22012 q^{67} +7.30925 q^{69} -11.0462 q^{71} -4.83237 q^{73} +1.76156 q^{75} +25.5510 q^{77} -2.98168 q^{79} -9.29862 q^{81} -2.74760 q^{83} +6.62620 q^{85} -13.8786 q^{87} +4.27072 q^{89} -25.4017 q^{91} +2.20617 q^{93} +1.00000 q^{95} -13.7047 q^{97} -0.569343 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - 3 q^{5} - 5 q^{7} + 4 q^{9} - 4 q^{11} + 5 q^{13} + q^{15} - 11 q^{17} - 3 q^{19} - 3 q^{21} - 9 q^{23} + 3 q^{25} - 19 q^{27} + 3 q^{29} - 14 q^{31} - 24 q^{33} + 5 q^{35} + 14 q^{37} - 5 q^{39}+ \cdots + 36 q^{99}+O(q^{100}) $ 3 * q - q^3 - 3 * q^5 - 5 * q^7 + 4 * q^9 - 4 * q^11 + 5 * q^13 + q^15 - 11 * q^17 - 3 * q^19 - 3 * q^21 - 9 * q^23 + 3 * q^25 - 19 * q^27 + 3 * q^29 - 14 * q^31 - 24 * q^33 + 5 * q^35 + 14 * q^37 - 5 * q^39 - 10 * q^41 - 10 * q^43 - 4 * q^45 + 4 * q^49 - q^51 - 7 * q^53 + 4 * q^55 + q^57 - 7 * q^59 + 20 * q^61 - 14 * q^63 - 5 * q^65 - q^67 + 33 * q^69 - 8 * q^71 - 13 * q^73 - q^75 + 16 * q^77 + 14 * q^79 + 15 * q^81 - 26 * q^83 + 11 * q^85 - 15 * q^87 + 18 * q^89 - 37 * q^91 + 14 * q^93 + 3 * q^95 - 6 * q^97 + 36 * q^99

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
$3$	1.76156	1.01704	0.508518	−	0.861052i	$-0.330194\pi$
$3$	1.76156	1.01704	0.508518	+	0.861052i	$0.330194\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.62620	−1.74854	−0.874269	−	0.485441i	$-0.838659\pi$
$7$	−4.62620	−1.74854	−0.874269	+	0.485441i	$0.838659\pi$
$8$	0	0
$9$	0.103084	0.0343612
$10$	0	0
$11$	−5.52311	−1.66528	−0.832641	−	0.553813i	$-0.813172\pi$
$11$	−5.52311	−1.66528	−0.832641	+	0.553813i	$0.813172\pi$
$12$	0	0
$13$	5.49084	1.52288	0.761442	−	0.648233i	$-0.224492\pi$
$13$	5.49084	1.52288	0.761442	+	0.648233i	$0.224492\pi$
$14$	0	0
$15$	−1.76156	−0.454832
$16$	0	0
$17$	−6.62620	−1.60709	−0.803545	−	0.595245i	$-0.797055\pi$
$17$	−6.62620	−1.60709	−0.803545	+	0.595245i	$0.797055\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−8.14931	−1.77833
$22$	0	0
$23$	4.14931	0.865191	0.432596	−	0.901588i	$-0.357598\pi$
$23$	4.14931	0.865191	0.432596	+	0.901588i	$0.357598\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.10308	−0.982089
$28$	0	0
$29$	−7.87859	−1.46302	−0.731509	−	0.681832i	$-0.761184\pi$
$29$	−7.87859	−1.46302	−0.731509	+	0.681832i	$0.761184\pi$
$30$	0	0
$31$	1.25240	0.224937	0.112468	−	0.993655i	$-0.464124\pi$
$31$	1.25240	0.224937	0.112468	+	0.993655i	$0.464124\pi$
$32$	0	0
$33$	−9.72928	−1.69365
$34$	0	0
$35$	4.62620	0.781970
$36$	0	0
$37$	−0.387755	−0.0637466	−0.0318733	−	0.999492i	$-0.510147\pi$
$37$	−0.387755	−0.0637466	−0.0318733	+	0.999492i	$0.510147\pi$
$38$	0	0
$39$	9.67243	1.54883
$40$	0	0
$41$	6.77551	1.05816	0.529078	−	0.848573i	$-0.322538\pi$
$41$	6.77551	1.05816	0.529078	+	0.848573i	$0.322538\pi$
$42$	0	0
$43$	−10.9817	−1.67469	−0.837345	−	0.546675i	$-0.815893\pi$
$43$	−10.9817	−1.67469	−0.837345	+	0.546675i	$0.815893\pi$
$44$	0	0
$45$	−0.103084	−0.0153668
$46$	0	0
$47$	1.72928	0.252242	0.126121	−	0.992015i	$-0.459747\pi$
$47$	1.72928	0.252242	0.126121	+	0.992015i	$0.459747\pi$
$48$	0	0
$49$	14.4017	2.05739
$50$	0	0
$51$	−11.6724	−1.63447
$52$	0	0
$53$	1.49084	0.204782	0.102391	−	0.994744i	$-0.467351\pi$
$53$	1.49084	0.204782	0.102391	+	0.994744i	$0.467351\pi$
$54$	0	0
$55$	5.52311	0.744737
$56$	0	0
$57$	−1.76156	−0.233324
$58$	0	0
$59$	0.626198	0.0815240	0.0407620	−	0.999169i	$-0.487021\pi$
$59$	0.626198	0.0815240	0.0407620	+	0.999169i	$0.487021\pi$
$60$	0	0
$61$	15.0462	1.92647	0.963236	−	0.268656i	$-0.0865796\pi$
$61$	15.0462	1.92647	0.963236	+	0.268656i	$0.0865796\pi$
$62$	0	0
$63$	−0.476886	−0.0600819
$64$	0	0
$65$	−5.49084	−0.681055
$66$	0	0
$67$	5.22012	0.637739	0.318870	−	0.947799i	$-0.396697\pi$
$67$	5.22012	0.637739	0.318870	+	0.947799i	$0.396697\pi$
$68$	0	0
$69$	7.30925	0.879930
$70$	0	0
$71$	−11.0462	−1.31095	−0.655473	−	0.755219i	$-0.727531\pi$
$71$	−11.0462	−1.31095	−0.655473	+	0.755219i	$0.727531\pi$
$72$	0	0
$73$	−4.83237	−0.565586	−0.282793	−	0.959181i	$-0.591261\pi$
$73$	−4.83237	−0.565586	−0.282793	+	0.959181i	$0.591261\pi$
$74$	0	0
$75$	1.76156	0.203407
$76$	0	0
$77$	25.5510	2.91181
$78$	0	0
$79$	−2.98168	−0.335465	−0.167732	−	0.985833i	$-0.553644\pi$
$79$	−2.98168	−0.335465	−0.167732	+	0.985833i	$0.553644\pi$
$80$	0	0
$81$	−9.29862	−1.03318
$82$	0	0
$83$	−2.74760	−0.301589	−0.150794	−	0.988565i	$-0.548183\pi$
$83$	−2.74760	−0.301589	−0.150794	+	0.988565i	$0.548183\pi$
$84$	0	0
$85$	6.62620	0.718712
$86$	0	0
$87$	−13.8786	−1.48794
$88$	0	0
$89$	4.27072	0.452695	0.226348	−	0.974047i	$-0.427321\pi$
$89$	4.27072	0.452695	0.226348	+	0.974047i	$0.427321\pi$
$90$	0	0
$91$	−25.4017	−2.66282
$92$	0	0
$93$	2.20617	0.228769
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−13.7047	−1.39150	−0.695751	−	0.718283i	$-0.744928\pi$
$97$	−13.7047	−1.39150	−0.695751	+	0.718283i	$0.744928\pi$
$98$	0	0
$99$	−0.569343	−0.0572211
$100$	0	0
$101$	7.04623	0.701126	0.350563	−	0.936539i	$-0.385990\pi$
$101$	7.04623	0.701126	0.350563	+	0.936539i	$0.385990\pi$
$102$	0	0
$103$	0.658473	0.0648813	0.0324407	−	0.999474i	$-0.489672\pi$
$103$	0.658473	0.0648813	0.0324407	+	0.999474i	$0.489672\pi$
$104$	0	0
$105$	8.14931	0.795291
$106$	0	0
$107$	−5.22012	−0.504648	−0.252324	−	0.967643i	$-0.581195\pi$
$107$	−5.22012	−0.504648	−0.252324	+	0.967643i	$0.581195\pi$
$108$	0	0
$109$	−11.8140	−1.13158	−0.565790	−	0.824549i	$-0.691429\pi$
$109$	−11.8140	−1.13158	−0.565790	+	0.824549i	$0.691429\pi$
$110$	0	0
$111$	−0.683053	−0.0648325
$112$	0	0
$113$	−0.0891304	−0.00838468	−0.00419234	−	0.999991i	$-0.501334\pi$
$113$	−0.0891304	−0.00838468	−0.00419234	+	0.999991i	$0.501334\pi$
$114$	0	0
$115$	−4.14931	−0.386925
$116$	0	0
$117$	0.566016	0.0523282
$118$	0	0
$119$	30.6541	2.81006
$120$	0	0
$121$	19.5048	1.77316
$122$	0	0
$123$	11.9354	1.07618
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.3694	1.18635	0.593173	−	0.805075i	$-0.297875\pi$
$127$	13.3694	1.18635	0.593173	+	0.805075i	$0.297875\pi$
$128$	0	0
$129$	−19.3449	−1.70322
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	4.62620	0.401142
$134$	0	0
$135$	5.10308	0.439204
$136$	0	0
$137$	−10.6262	−0.907857	−0.453929	−	0.891038i	$-0.649978\pi$
$137$	−10.6262	−0.907857	−0.453929	+	0.891038i	$0.649978\pi$
$138$	0	0
$139$	−12.5693	−1.06612	−0.533059	−	0.846078i	$-0.678958\pi$
$139$	−12.5693	−1.06612	−0.533059	+	0.846078i	$0.678958\pi$
$140$	0	0
$141$	3.04623	0.256539
$142$	0	0
$143$	−30.3265	−2.53603
$144$	0	0
$145$	7.87859	0.654282
$146$	0	0
$147$	25.3694	2.09244
$148$	0	0
$149$	7.04623	0.577250	0.288625	−	0.957442i	$-0.406802\pi$
$149$	7.04623	0.577250	0.288625	+	0.957442i	$0.406802\pi$
$150$	0	0
$151$	4.47689	0.364324	0.182162	−	0.983269i	$-0.441691\pi$
$151$	4.47689	0.364324	0.182162	+	0.983269i	$0.441691\pi$
$152$	0	0
$153$	−0.683053	−0.0552216
$154$	0	0
$155$	−1.25240	−0.100595
$156$	0	0
$157$	8.27072	0.660075	0.330038	−	0.943968i	$-0.392939\pi$
$157$	8.27072	0.660075	0.330038	+	0.943968i	$0.392939\pi$
$158$	0	0
$159$	2.62620	0.208271
$160$	0	0
$161$	−19.1955	−1.51282
$162$	0	0
$163$	−18.5693	−1.45446	−0.727232	−	0.686392i	$-0.759193\pi$
$163$	−18.5693	−1.45446	−0.727232	+	0.686392i	$0.759193\pi$
$164$	0	0
$165$	9.72928	0.757424
$166$	0	0
$167$	−21.3694	−1.65362	−0.826808	−	0.562484i	$-0.809846\pi$
$167$	−21.3694	−1.65362	−0.826808	+	0.562484i	$0.809846\pi$
$168$	0	0
$169$	17.1493	1.31918
$170$	0	0
$171$	−0.103084	−0.00788301
$172$	0	0
$173$	−12.3878	−0.941824	−0.470912	−	0.882180i	$-0.656075\pi$
$173$	−12.3878	−0.941824	−0.470912	+	0.882180i	$0.656075\pi$
$174$	0	0
$175$	−4.62620	−0.349708
$176$	0	0
$177$	1.10308	0.0829128
$178$	0	0
$179$	12.7755	0.954886	0.477443	−	0.878663i	$-0.341564\pi$
$179$	12.7755	0.954886	0.477443	+	0.878663i	$0.341564\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	26.5048	1.95929
$184$	0	0
$185$	0.387755	0.0285083
$186$	0	0
$187$	36.5972	2.67626
$188$	0	0
$189$	23.6079	1.71722
$190$	0	0
$191$	−19.1955	−1.38894	−0.694470	−	0.719521i	$-0.744361\pi$
$191$	−19.1955	−1.38894	−0.694470	+	0.719521i	$0.744361\pi$
$192$	0	0
$193$	−13.7047	−0.986486	−0.493243	−	0.869892i	$-0.664189\pi$
$193$	−13.7047	−0.986486	−0.493243	+	0.869892i	$0.664189\pi$
$194$	0	0
$195$	−9.67243	−0.692657
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−0.561647	−0.0398141	−0.0199071	−	0.999802i	$-0.506337\pi$
$199$	−0.561647	−0.0398141	−0.0199071	+	0.999802i	$0.506337\pi$
$200$	0	0
$201$	9.19554	0.648603
$202$	0	0
$203$	36.4479	2.55814
$204$	0	0
$205$	−6.77551	−0.473222
$206$	0	0
$207$	0.427726	0.0297290
$208$	0	0
$209$	5.52311	0.382042
$210$	0	0
$211$	15.6724	1.07893	0.539467	−	0.842007i	$-0.318626\pi$
$211$	15.6724	1.07893	0.539467	+	0.842007i	$0.318626\pi$
$212$	0	0
$213$	−19.4586	−1.33328
$214$	0	0
$215$	10.9817	0.748944
$216$	0	0
$217$	−5.79383	−0.393311
$218$	0	0
$219$	−8.51249	−0.575221
$220$	0	0
$221$	−36.3834	−2.44741
$222$	0	0
$223$	12.0525	0.807094	0.403547	−	0.914959i	$-0.367777\pi$
$223$	12.0525	0.807094	0.403547	+	0.914959i	$0.367777\pi$
$224$	0	0
$225$	0.103084	0.00687225
$226$	0	0
$227$	3.07850	0.204327	0.102164	−	0.994768i	$-0.467423\pi$
$227$	3.07850	0.204327	0.102164	+	0.994768i	$0.467423\pi$
$228$	0	0
$229$	17.7938	1.17585	0.587925	−	0.808916i	$-0.299945\pi$
$229$	17.7938	1.17585	0.587925	+	0.808916i	$0.299945\pi$
$230$	0	0
$231$	45.0096	2.96141
$232$	0	0
$233$	8.50479	0.557167	0.278584	−	0.960412i	$-0.410135\pi$
$233$	8.50479	0.557167	0.278584	+	0.960412i	$0.410135\pi$
$234$	0	0
$235$	−1.72928	−0.112806
$236$	0	0
$237$	−5.25240	−0.341180
$238$	0	0
$239$	−23.6079	−1.52707	−0.763533	−	0.645768i	$-0.776537\pi$
$239$	−23.6079	−1.52707	−0.763533	+	0.645768i	$0.776537\pi$
$240$	0	0
$241$	9.11078	0.586877	0.293438	−	0.955978i	$-0.405200\pi$
$241$	9.11078	0.586877	0.293438	+	0.955978i	$0.405200\pi$
$242$	0	0
$243$	−1.07081	−0.0686924
$244$	0	0
$245$	−14.4017	−0.920091
$246$	0	0
$247$	−5.49084	−0.349374
$248$	0	0
$249$	−4.84006	−0.306726
$250$	0	0
$251$	13.5510	0.855333	0.427666	−	0.903937i	$-0.359336\pi$
$251$	13.5510	0.855333	0.427666	+	0.903937i	$0.359336\pi$
$252$	0	0
$253$	−22.9171	−1.44079
$254$	0	0
$255$	11.6724	0.730956
$256$	0	0
$257$	10.6585	0.664857	0.332429	−	0.943128i	$-0.392132\pi$
$257$	10.6585	0.664857	0.332429	+	0.943128i	$0.392132\pi$
$258$	0	0
$259$	1.79383	0.111463
$260$	0	0
$261$	−0.812155	−0.0502711
$262$	0	0
$263$	−17.9634	−1.10767	−0.553834	−	0.832627i	$-0.686836\pi$
$263$	−17.9634	−1.10767	−0.553834	+	0.832627i	$0.686836\pi$
$264$	0	0
$265$	−1.49084	−0.0915815
$266$	0	0
$267$	7.52311	0.460407
$268$	0	0
$269$	15.5510	0.948162	0.474081	−	0.880481i	$-0.342780\pi$
$269$	15.5510	0.948162	0.474081	+	0.880481i	$0.342780\pi$
$270$	0	0
$271$	8.29093	0.503638	0.251819	−	0.967774i	$-0.418971\pi$
$271$	8.29093	0.503638	0.251819	+	0.967774i	$0.418971\pi$
$272$	0	0
$273$	−44.7466	−2.70819
$274$	0	0
$275$	−5.52311	−0.333056
$276$	0	0
$277$	−28.5693	−1.71657	−0.858283	−	0.513177i	$-0.828468\pi$
$277$	−28.5693	−1.71657	−0.858283	+	0.513177i	$0.828468\pi$
$278$	0	0
$279$	0.129102	0.00772911
$280$	0	0
$281$	−19.3169	−1.15235	−0.576176	−	0.817325i	$-0.695456\pi$
$281$	−19.3169	−1.15235	−0.576176	+	0.817325i	$0.695456\pi$
$282$	0	0
$283$	16.2986	0.968853	0.484426	−	0.874832i	$-0.339028\pi$
$283$	16.2986	0.968853	0.484426	+	0.874832i	$0.339028\pi$
$284$	0	0
$285$	1.76156	0.104346
$286$	0	0
$287$	−31.3449	−1.85023
$288$	0	0
$289$	26.9065	1.58274
$290$	0	0
$291$	−24.1416	−1.41521
$292$	0	0
$293$	−26.3955	−1.54204	−0.771019	−	0.636812i	$-0.780253\pi$
$293$	−26.3955	−1.54204	−0.771019	+	0.636812i	$0.780253\pi$
$294$	0	0
$295$	−0.626198	−0.0364587
$296$	0	0
$297$	28.1849	1.63545
$298$	0	0
$299$	22.7832	1.31759
$300$	0	0
$301$	50.8034	2.92826
$302$	0	0
$303$	12.4123	0.713070
$304$	0	0
$305$	−15.0462	−0.861545
$306$	0	0
$307$	−30.2095	−1.72415	−0.862073	−	0.506783i	$-0.830834\pi$
$307$	−30.2095	−1.72415	−0.862073	+	0.506783i	$0.830834\pi$
$308$	0	0
$309$	1.15994	0.0659866
$310$	0	0
$311$	−4.12141	−0.233703	−0.116852	−	0.993149i	$-0.537280\pi$
$311$	−4.12141	−0.233703	−0.116852	+	0.993149i	$0.537280\pi$
$312$	0	0
$313$	−15.1031	−0.853677	−0.426838	−	0.904328i	$-0.640373\pi$
$313$	−15.1031	−0.853677	−0.426838	+	0.904328i	$0.640373\pi$
$314$	0	0
$315$	0.476886	0.0268695
$316$	0	0
$317$	−13.3126	−0.747709	−0.373854	−	0.927487i	$-0.621964\pi$
$317$	−13.3126	−0.747709	−0.373854	+	0.927487i	$0.621964\pi$
$318$	0	0
$319$	43.5144	2.43634
$320$	0	0
$321$	−9.19554	−0.513245
$322$	0	0
$323$	6.62620	0.368692
$324$	0	0
$325$	5.49084	0.304577
$326$	0	0
$327$	−20.8111	−1.15086
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	16.9248	0.930272	0.465136	−	0.885239i	$-0.346005\pi$
$331$	16.9248	0.930272	0.465136	+	0.885239i	$0.346005\pi$
$332$	0	0
$333$	−0.0399712	−0.00219041
$334$	0	0
$335$	−5.22012	−0.285206
$336$	0	0
$337$	1.16327	0.0633671	0.0316836	−	0.999498i	$-0.489913\pi$
$337$	1.16327	0.0633671	0.0316836	+	0.999498i	$0.489913\pi$
$338$	0	0
$339$	−0.157008	−0.00852752
$340$	0	0
$341$	−6.91713	−0.374583
$342$	0	0
$343$	−34.2418	−1.84888
$344$	0	0
$345$	−7.30925	−0.393517
$346$	0	0
$347$	3.70138	0.198700	0.0993501	−	0.995053i	$-0.468324\pi$
$347$	3.70138	0.198700	0.0993501	+	0.995053i	$0.468324\pi$
$348$	0	0
$349$	−2.54144	−0.136040	−0.0680200	−	0.997684i	$-0.521668\pi$
$349$	−2.54144	−0.136040	−0.0680200	+	0.997684i	$0.521668\pi$
$350$	0	0
$351$	−28.0202	−1.49561
$352$	0	0
$353$	−21.3738	−1.13761	−0.568806	−	0.822471i	$-0.692595\pi$
$353$	−21.3738	−1.13761	−0.568806	+	0.822471i	$0.692595\pi$
$354$	0	0
$355$	11.0462	0.586273
$356$	0	0
$357$	53.9990	2.85793
$358$	0	0
$359$	−4.42003	−0.233280	−0.116640	−	0.993174i	$-0.537212\pi$
$359$	−4.42003	−0.233280	−0.116640	+	0.993174i	$0.537212\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	34.3588	1.80337
$364$	0	0
$365$	4.83237	0.252938
$366$	0	0
$367$	3.28030	0.171230	0.0856152	−	0.996328i	$-0.472714\pi$
$367$	3.28030	0.171230	0.0856152	+	0.996328i	$0.472714\pi$
$368$	0	0
$369$	0.698445	0.0363596
$370$	0	0
$371$	−6.89692	−0.358070
$372$	0	0
$373$	−14.8078	−0.766718	−0.383359	−	0.923599i	$-0.625233\pi$
$373$	−14.8078	−0.766718	−0.383359	+	0.923599i	$0.625233\pi$
$374$	0	0
$375$	−1.76156	−0.0909664
$376$	0	0
$377$	−43.2601	−2.22801
$378$	0	0
$379$	−19.7370	−1.01382	−0.506910	−	0.861999i	$-0.669212\pi$
$379$	−19.7370	−1.01382	−0.506910	+	0.861999i	$0.669212\pi$
$380$	0	0
$381$	23.5510	1.20656
$382$	0	0
$383$	−10.2095	−0.521681	−0.260840	−	0.965382i	$-0.584000\pi$
$383$	−10.2095	−0.521681	−0.260840	+	0.965382i	$0.584000\pi$
$384$	0	0
$385$	−25.5510	−1.30220
$386$	0	0
$387$	−1.13203	−0.0575444
$388$	0	0
$389$	10.7110	0.543067	0.271534	−	0.962429i	$-0.412469\pi$
$389$	10.7110	0.543067	0.271534	+	0.962429i	$0.412469\pi$
$390$	0	0
$391$	−27.4942	−1.39044
$392$	0	0
$393$	−7.04623	−0.355435
$394$	0	0
$395$	2.98168	0.150024
$396$	0	0
$397$	20.5693	1.03235	0.516173	−	0.856484i	$-0.327356\pi$
$397$	20.5693	1.03235	0.516173	+	0.856484i	$0.327356\pi$
$398$	0	0
$399$	8.14931	0.407976
$400$	0	0
$401$	−23.9634	−1.19667	−0.598336	−	0.801245i	$-0.704171\pi$
$401$	−23.9634	−1.19667	−0.598336	+	0.801245i	$0.704171\pi$
$402$	0	0
$403$	6.87671	0.342553
$404$	0	0
$405$	9.29862	0.462052
$406$	0	0
$407$	2.14162	0.106156
$408$	0	0
$409$	−4.50479	−0.222748	−0.111374	−	0.993779i	$-0.535525\pi$
$409$	−4.50479	−0.222748	−0.111374	+	0.993779i	$0.535525\pi$
$410$	0	0
$411$	−18.7187	−0.923323
$412$	0	0
$413$	−2.89692	−0.142548
$414$	0	0
$415$	2.74760	0.134875
$416$	0	0
$417$	−22.1416	−1.08428
$418$	0	0
$419$	−31.7572	−1.55144	−0.775720	−	0.631077i	$-0.782613\pi$
$419$	−31.7572	−1.55144	−0.775720	+	0.631077i	$0.782613\pi$
$420$	0	0
$421$	−21.4296	−1.04442	−0.522208	−	0.852818i	$-0.674891\pi$
$421$	−21.4296	−1.04442	−0.522208	+	0.852818i	$0.674891\pi$
$422$	0	0
$423$	0.178261	0.00866734
$424$	0	0
$425$	−6.62620	−0.321418
$426$	0	0
$427$	−69.6068	−3.36851
$428$	0	0
$429$	−53.4219	−2.57923
$430$	0	0
$431$	−10.6831	−0.514585	−0.257292	−	0.966334i	$-0.582830\pi$
$431$	−10.6831	−0.514585	−0.257292	+	0.966334i	$0.582830\pi$
$432$	0	0
$433$	−24.6864	−1.18635	−0.593176	−	0.805073i	$-0.702126\pi$
$433$	−24.6864	−1.18635	−0.593176	+	0.805073i	$0.702126\pi$
$434$	0	0
$435$	13.8786	0.665428
$436$	0	0
$437$	−4.14931	−0.198489
$438$	0	0
$439$	−14.9817	−0.715036	−0.357518	−	0.933906i	$-0.616377\pi$
$439$	−14.9817	−0.715036	−0.357518	+	0.933906i	$0.616377\pi$
$440$	0	0
$441$	1.48458	0.0706944
$442$	0	0
$443$	−4.54144	−0.215770	−0.107885	−	0.994163i	$-0.534408\pi$
$443$	−4.54144	−0.215770	−0.107885	+	0.994163i	$0.534408\pi$
$444$	0	0
$445$	−4.27072	−0.202451
$446$	0	0
$447$	12.4123	0.587083
$448$	0	0
$449$	−16.0925	−0.759450	−0.379725	−	0.925099i	$-0.623981\pi$
$449$	−16.0925	−0.759450	−0.379725	+	0.925099i	$0.623981\pi$
$450$	0	0
$451$	−37.4219	−1.76213
$452$	0	0
$453$	7.88629	0.370530
$454$	0	0
$455$	25.4017	1.19085
$456$	0	0
$457$	−4.65410	−0.217710	−0.108855	−	0.994058i	$-0.534718\pi$
$457$	−4.65410	−0.217710	−0.108855	+	0.994058i	$0.534718\pi$
$458$	0	0
$459$	33.8140	1.57830
$460$	0	0
$461$	19.0096	0.885365	0.442682	−	0.896679i	$-0.354027\pi$
$461$	19.0096	0.885365	0.442682	+	0.896679i	$0.354027\pi$
$462$	0	0
$463$	−10.6339	−0.494199	−0.247099	−	0.968990i	$-0.579477\pi$
$463$	−10.6339	−0.494199	−0.247099	+	0.968990i	$0.579477\pi$
$464$	0	0
$465$	−2.20617	−0.102309
$466$	0	0
$467$	32.8805	1.52153	0.760764	−	0.649029i	$-0.224825\pi$
$467$	32.8805	1.52153	0.760764	+	0.649029i	$0.224825\pi$
$468$	0	0
$469$	−24.1493	−1.11511
$470$	0	0
$471$	14.5693	0.671320
$472$	0	0
$473$	60.6531	2.78883
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0.153681	0.00703658
$478$	0	0
$479$	12.4402	0.568409	0.284205	−	0.958764i	$-0.408271\pi$
$479$	12.4402	0.568409	0.284205	+	0.958764i	$0.408271\pi$
$480$	0	0
$481$	−2.12910	−0.0970787
$482$	0	0
$483$	−33.8140	−1.53859
$484$	0	0
$485$	13.7047	0.622298
$486$	0	0
$487$	40.9571	1.85594	0.927972	−	0.372651i	$-0.121551\pi$
$487$	40.9571	1.85594	0.927972	+	0.372651i	$0.121551\pi$
$488$	0	0
$489$	−32.7110	−1.47924
$490$	0	0
$491$	43.3449	1.95613	0.978063	−	0.208310i	$-0.0667962\pi$
$491$	43.3449	1.95613	0.978063	+	0.208310i	$0.0667962\pi$
$492$	0	0
$493$	52.2051	2.35120
$494$	0	0
$495$	0.569343	0.0255901
$496$	0	0
$497$	51.1020	2.29224
$498$	0	0
$499$	25.4094	1.13748	0.568741	−	0.822517i	$-0.307431\pi$
$499$	25.4094	1.13748	0.568741	+	0.822517i	$0.307431\pi$
$500$	0	0
$501$	−37.6435	−1.68179
$502$	0	0
$503$	34.1127	1.52101	0.760504	−	0.649333i	$-0.224952\pi$
$503$	34.1127	1.52101	0.760504	+	0.649333i	$0.224952\pi$
$504$	0	0
$505$	−7.04623	−0.313553
$506$	0	0
$507$	30.2095	1.34165
$508$	0	0
$509$	−22.3632	−0.991230	−0.495615	−	0.868542i	$-0.665057\pi$
$509$	−22.3632	−0.991230	−0.495615	+	0.868542i	$0.665057\pi$
$510$	0	0
$511$	22.3555	0.988948
$512$	0	0
$513$	5.10308	0.225307
$514$	0	0
$515$	−0.658473	−0.0290158
$516$	0	0
$517$	−9.55102	−0.420053
$518$	0	0
$519$	−21.8217	−0.957868
$520$	0	0
$521$	2.29862	0.100705	0.0503523	−	0.998732i	$-0.483966\pi$
$521$	2.29862	0.100705	0.0503523	+	0.998732i	$0.483966\pi$
$522$	0	0
$523$	27.3771	1.19712	0.598559	−	0.801079i	$-0.295740\pi$
$523$	27.3771	1.19712	0.598559	+	0.801079i	$0.295740\pi$
$524$	0	0
$525$	−8.14931	−0.355665
$526$	0	0
$527$	−8.29862	−0.361494
$528$	0	0
$529$	−5.78321	−0.251444
$530$	0	0
$531$	0.0645508	0.00280127
$532$	0	0
$533$	37.2032	1.61145
$534$	0	0
$535$	5.22012	0.225685
$536$	0	0
$537$	22.5048	0.971153
$538$	0	0
$539$	−79.5423	−3.42613
$540$	0	0
$541$	15.4586	0.664616	0.332308	−	0.943171i	$-0.392173\pi$
$541$	15.4586	0.664616	0.332308	+	0.943171i	$0.392173\pi$
$542$	0	0
$543$	38.7543	1.66310
$544$	0	0
$545$	11.8140	0.506058
$546$	0	0
$547$	−13.9754	−0.597546	−0.298773	−	0.954324i	$-0.596577\pi$
$547$	−13.9754	−0.597546	−0.298773	+	0.954324i	$0.596577\pi$
$548$	0	0
$549$	1.55102	0.0661960
$550$	0	0
$551$	7.87859	0.335639
$552$	0	0
$553$	13.7938	0.586573
$554$	0	0
$555$	0.683053	0.0289940
$556$	0	0
$557$	30.6618	1.29918	0.649591	−	0.760284i	$-0.274940\pi$
$557$	30.6618	1.29918	0.649591	+	0.760284i	$0.274940\pi$
$558$	0	0
$559$	−60.2986	−2.55036
$560$	0	0
$561$	64.4681	2.72185
$562$	0	0
$563$	−2.02458	−0.0853259	−0.0426629	−	0.999090i	$-0.513584\pi$
$563$	−2.02458	−0.0853259	−0.0426629	+	0.999090i	$0.513584\pi$
$564$	0	0
$565$	0.0891304	0.00374974
$566$	0	0
$567$	43.0173	1.80656
$568$	0	0
$569$	−28.2620	−1.18480	−0.592402	−	0.805643i	$-0.701820\pi$
$569$	−28.2620	−1.18480	−0.592402	+	0.805643i	$0.701820\pi$
$570$	0	0
$571$	10.6464	0.445538	0.222769	−	0.974871i	$-0.428490\pi$
$571$	10.6464	0.445538	0.222769	+	0.974871i	$0.428490\pi$
$572$	0	0
$573$	−33.8140	−1.41260
$574$	0	0
$575$	4.14931	0.173038
$576$	0	0
$577$	−9.30925	−0.387549	−0.193775	−	0.981046i	$-0.562073\pi$
$577$	−9.30925	−0.387549	−0.193775	+	0.981046i	$0.562073\pi$
$578$	0	0
$579$	−24.1416	−1.00329
$580$	0	0
$581$	12.7110	0.527339
$582$	0	0
$583$	−8.23407	−0.341021
$584$	0	0
$585$	−0.566016	−0.0234019
$586$	0	0
$587$	−42.3911	−1.74967	−0.874834	−	0.484424i	$-0.839029\pi$
$587$	−42.3911	−1.74967	−0.874834	+	0.484424i	$0.839029\pi$
$588$	0	0
$589$	−1.25240	−0.0516041
$590$	0	0
$591$	−3.52311	−0.144922
$592$	0	0
$593$	28.5048	1.17055	0.585276	−	0.810834i	$-0.300986\pi$
$593$	28.5048	1.17055	0.585276	+	0.810834i	$0.300986\pi$
$594$	0	0
$595$	−30.6541	−1.25670
$596$	0	0
$597$	−0.989374	−0.0404924
$598$	0	0
$599$	25.6156	1.04662	0.523312	−	0.852141i	$-0.324696\pi$
$599$	25.6156	1.04662	0.523312	+	0.852141i	$0.324696\pi$
$600$	0	0
$601$	−29.2803	−1.19437	−0.597184	−	0.802104i	$-0.703714\pi$
$601$	−29.2803	−1.19437	−0.597184	+	0.802104i	$0.703714\pi$
$602$	0	0
$603$	0.538109	0.0219135
$604$	0	0
$605$	−19.5048	−0.792983
$606$	0	0
$607$	−20.0525	−0.813905	−0.406953	−	0.913449i	$-0.633409\pi$
$607$	−20.0525	−0.813905	−0.406953	+	0.913449i	$0.633409\pi$
$608$	0	0
$609$	64.2051	2.60172
$610$	0	0
$611$	9.49521	0.384135
$612$	0	0
$613$	−7.89881	−0.319030	−0.159515	−	0.987196i	$-0.550993\pi$
$613$	−7.89881	−0.319030	−0.159515	+	0.987196i	$0.550993\pi$
$614$	0	0
$615$	−11.9354	−0.481284
$616$	0	0
$617$	−41.4094	−1.66708	−0.833540	−	0.552459i	$-0.813689\pi$
$617$	−41.4094	−1.66708	−0.833540	+	0.552459i	$0.813689\pi$
$618$	0	0
$619$	26.9450	1.08301	0.541506	−	0.840697i	$-0.317854\pi$
$619$	26.9450	1.08301	0.541506	+	0.840697i	$0.317854\pi$
$620$	0	0
$621$	−21.1743	−0.849695
$622$	0	0
$623$	−19.7572	−0.791555
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	9.72928	0.388550
$628$	0	0
$629$	2.56934	0.102446
$630$	0	0
$631$	−15.4865	−0.616507	−0.308253	−	0.951304i	$-0.599744\pi$
$631$	−15.4865	−0.616507	−0.308253	+	0.951304i	$0.599744\pi$
$632$	0	0
$633$	27.6079	1.09731
$634$	0	0
$635$	−13.3694	−0.530550
$636$	0	0
$637$	79.0775	3.13316
$638$	0	0
$639$	−1.13869	−0.0450457
$640$	0	0
$641$	9.93545	0.392427	0.196213	−	0.980561i	$-0.437135\pi$
$641$	9.93545	0.392427	0.196213	+	0.980561i	$0.437135\pi$
$642$	0	0
$643$	5.25240	0.207134	0.103567	−	0.994622i	$-0.466974\pi$
$643$	5.25240	0.207134	0.103567	+	0.994622i	$0.466974\pi$
$644$	0	0
$645$	19.3449	0.761703
$646$	0	0
$647$	1.64452	0.0646528	0.0323264	−	0.999477i	$-0.489708\pi$
$647$	1.64452	0.0646528	0.0323264	+	0.999477i	$0.489708\pi$
$648$	0	0
$649$	−3.45856	−0.135760
$650$	0	0
$651$	−10.2062	−0.400011
$652$	0	0
$653$	−30.9450	−1.21097	−0.605486	−	0.795856i	$-0.707021\pi$
$653$	−30.9450	−1.21097	−0.605486	+	0.795856i	$0.707021\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	−0.498138	−0.0194342
$658$	0	0
$659$	17.3372	0.675360	0.337680	−	0.941261i	$-0.390358\pi$
$659$	17.3372	0.675360	0.337680	+	0.941261i	$0.390358\pi$
$660$	0	0
$661$	8.83237	0.343539	0.171770	−	0.985137i	$-0.445052\pi$
$661$	8.83237	0.343539	0.171770	+	0.985137i	$0.445052\pi$
$662$	0	0
$663$	−64.0914	−2.48910
$664$	0	0
$665$	−4.62620	−0.179396
$666$	0	0
$667$	−32.6907	−1.26579
$668$	0	0
$669$	21.2311	0.820843
$670$	0	0
$671$	−83.1020	−3.20812
$672$	0	0
$673$	29.6402	1.14254	0.571272	−	0.820761i	$-0.306450\pi$
$673$	29.6402	1.14254	0.571272	+	0.820761i	$0.306450\pi$
$674$	0	0
$675$	−5.10308	−0.196418
$676$	0	0
$677$	23.2847	0.894903	0.447451	−	0.894308i	$-0.352332\pi$
$677$	23.2847	0.894903	0.447451	+	0.894308i	$0.352332\pi$
$678$	0	0
$679$	63.4007	2.43309
$680$	0	0
$681$	5.42296	0.207808
$682$	0	0
$683$	27.3415	1.04619	0.523097	−	0.852273i	$-0.324776\pi$
$683$	27.3415	1.04619	0.523097	+	0.852273i	$0.324776\pi$
$684$	0	0
$685$	10.6262	0.406006
$686$	0	0
$687$	31.3449	1.19588
$688$	0	0
$689$	8.18596	0.311860
$690$	0	0
$691$	38.8313	1.47721	0.738607	−	0.674137i	$-0.235484\pi$
$691$	38.8313	1.47721	0.738607	+	0.674137i	$0.235484\pi$
$692$	0	0
$693$	2.63389	0.100053
$694$	0	0
$695$	12.5693	0.476782
$696$	0	0
$697$	−44.8959	−1.70055
$698$	0	0
$699$	14.9817	0.566659
$700$	0	0
$701$	−19.5144	−0.737048	−0.368524	−	0.929618i	$-0.620137\pi$
$701$	−19.5144	−0.737048	−0.368524	+	0.929618i	$0.620137\pi$
$702$	0	0
$703$	0.387755	0.0146245
$704$	0	0
$705$	−3.04623	−0.114728
$706$	0	0
$707$	−32.5972	−1.22595
$708$	0	0
$709$	−11.1387	−0.418322	−0.209161	−	0.977881i	$-0.567073\pi$
$709$	−11.1387	−0.418322	−0.209161	+	0.977881i	$0.567073\pi$
$710$	0	0
$711$	−0.307362	−0.0115270
$712$	0	0
$713$	5.19658	0.194614
$714$	0	0
$715$	30.3265	1.13415
$716$	0	0
$717$	−41.5866	−1.55308
$718$	0	0
$719$	−16.1214	−0.601227	−0.300613	−	0.953746i	$-0.597191\pi$
$719$	−16.1214	−0.601227	−0.300613	+	0.953746i	$0.597191\pi$
$720$	0	0
$721$	−3.04623	−0.113447
$722$	0	0
$723$	16.0492	0.596875
$724$	0	0
$725$	−7.87859	−0.292604
$726$	0	0
$727$	−8.21386	−0.304635	−0.152318	−	0.988332i	$-0.548674\pi$
$727$	−8.21386	−0.304635	−0.152318	+	0.988332i	$0.548674\pi$
$728$	0	0
$729$	26.0096	0.963318
$730$	0	0
$731$	72.7668	2.69138
$732$	0	0
$733$	28.6743	1.05911	0.529555	−	0.848276i	$-0.322359\pi$
$733$	28.6743	1.05911	0.529555	+	0.848276i	$0.322359\pi$
$734$	0	0
$735$	−25.3694	−0.935766
$736$	0	0
$737$	−28.8313	−1.06202
$738$	0	0
$739$	−11.7014	−0.430442	−0.215221	−	0.976565i	$-0.569047\pi$
$739$	−11.7014	−0.430442	−0.215221	+	0.976565i	$0.569047\pi$
$740$	0	0
$741$	−9.67243	−0.355325
$742$	0	0
$743$	1.91087	0.0701030	0.0350515	−	0.999386i	$-0.488840\pi$
$743$	1.91087	0.0701030	0.0350515	+	0.999386i	$0.488840\pi$
$744$	0	0
$745$	−7.04623	−0.258154
$746$	0	0
$747$	−0.283233	−0.0103630
$748$	0	0
$749$	24.1493	0.882397
$750$	0	0
$751$	9.55976	0.348841	0.174420	−	0.984671i	$-0.444195\pi$
$751$	9.55976	0.348841	0.174420	+	0.984671i	$0.444195\pi$
$752$	0	0
$753$	23.8709	0.869904
$754$	0	0
$755$	−4.47689	−0.162931
$756$	0	0
$757$	−33.6435	−1.22279	−0.611397	−	0.791324i	$-0.709392\pi$
$757$	−33.6435	−1.22279	−0.611397	+	0.791324i	$0.709392\pi$
$758$	0	0
$759$	−40.3698	−1.46533
$760$	0	0
$761$	−22.4113	−0.812409	−0.406204	−	0.913782i	$-0.633148\pi$
$761$	−22.4113	−0.812409	−0.406204	+	0.913782i	$0.633148\pi$
$762$	0	0
$763$	54.6541	1.97861
$764$	0	0
$765$	0.683053	0.0246958
$766$	0	0
$767$	3.43835	0.124152
$768$	0	0
$769$	4.98937	0.179921	0.0899607	−	0.995945i	$-0.471326\pi$
$769$	4.98937	0.179921	0.0899607	+	0.995945i	$0.471326\pi$
$770$	0	0
$771$	18.7755	0.676183
$772$	0	0
$773$	16.6508	0.598887	0.299443	−	0.954114i	$-0.403199\pi$
$773$	16.6508	0.598887	0.299443	+	0.954114i	$0.403199\pi$
$774$	0	0
$775$	1.25240	0.0449874
$776$	0	0
$777$	3.15994	0.113362
$778$	0	0
$779$	−6.77551	−0.242758
$780$	0	0
$781$	61.0096	2.18309
$782$	0	0
$783$	40.2051	1.43681
$784$	0	0
$785$	−8.27072	−0.295195
$786$	0	0
$787$	−43.8453	−1.56292	−0.781458	−	0.623958i	$-0.785524\pi$
$787$	−43.8453	−1.56292	−0.781458	+	0.623958i	$0.785524\pi$
$788$	0	0
$789$	−31.6435	−1.12654
$790$	0	0
$791$	0.412335	0.0146609
$792$	0	0
$793$	82.6164	2.93380
$794$	0	0
$795$	−2.62620	−0.0931416
$796$	0	0
$797$	49.0052	1.73585	0.867927	−	0.496692i	$-0.165452\pi$
$797$	49.0052	1.73585	0.867927	+	0.496692i	$0.165452\pi$
$798$	0	0
$799$	−11.4586	−0.405375
$800$	0	0
$801$	0.440241	0.0155552
$802$	0	0
$803$	26.6897	0.941859
$804$	0	0
$805$	19.1955	0.676554
$806$	0	0
$807$	27.3940	0.964315
$808$	0	0
$809$	9.19554	0.323298	0.161649	−	0.986848i	$-0.448319\pi$
$809$	9.19554	0.323298	0.161649	+	0.986848i	$0.448319\pi$
$810$	0	0
$811$	−18.9527	−0.665520	−0.332760	−	0.943011i	$-0.607980\pi$
$811$	−18.9527	−0.665520	−0.332760	+	0.943011i	$0.607980\pi$
$812$	0	0
$813$	14.6049	0.512218
$814$	0	0
$815$	18.5693	0.650456
$816$	0	0
$817$	10.9817	0.384200
$818$	0	0
$819$	−2.61850	−0.0914979
$820$	0	0
$821$	−45.7572	−1.59694	−0.798468	−	0.602037i	$-0.794356\pi$
$821$	−45.7572	−1.59694	−0.798468	+	0.602037i	$0.794356\pi$
$822$	0	0
$823$	−11.3738	−0.396466	−0.198233	−	0.980155i	$-0.563520\pi$
$823$	−11.3738	−0.396466	−0.198233	+	0.980155i	$0.563520\pi$
$824$	0	0
$825$	−9.72928	−0.338730
$826$	0	0
$827$	−11.8386	−0.411669	−0.205835	−	0.978587i	$-0.565991\pi$
$827$	−11.8386	−0.411669	−0.205835	+	0.978587i	$0.565991\pi$
$828$	0	0
$829$	−26.9894	−0.937380	−0.468690	−	0.883363i	$-0.655274\pi$
$829$	−26.9894	−0.937380	−0.468690	+	0.883363i	$0.655274\pi$
$830$	0	0
$831$	−50.3265	−1.74581
$832$	0	0
$833$	−95.4286	−3.30640
$834$	0	0
$835$	21.3694	0.739520
$836$	0	0
$837$	−6.39108	−0.220908
$838$	0	0
$839$	2.81215	0.0970864	0.0485432	−	0.998821i	$-0.484542\pi$
$839$	2.81215	0.0970864	0.0485432	+	0.998821i	$0.484542\pi$
$840$	0	0
$841$	33.0722	1.14042
$842$	0	0
$843$	−34.0279	−1.17198
$844$	0	0
$845$	−17.1493	−0.589954
$846$	0	0
$847$	−90.2330	−3.10044
$848$	0	0
$849$	28.7110	0.985358
$850$	0	0
$851$	−1.60892	−0.0551530
$852$	0	0
$853$	−39.6068	−1.35611	−0.678056	−	0.735010i	$-0.737177\pi$
$853$	−39.6068	−1.35611	−0.678056	+	0.735010i	$0.737177\pi$
$854$	0	0
$855$	0.103084	0.00352539
$856$	0	0
$857$	17.8742	0.610572	0.305286	−	0.952261i	$-0.401248\pi$
$857$	17.8742	0.610572	0.305286	+	0.952261i	$0.401248\pi$
$858$	0	0
$859$	4.12910	0.140883	0.0704416	−	0.997516i	$-0.477559\pi$
$859$	4.12910	0.140883	0.0704416	+	0.997516i	$0.477559\pi$
$860$	0	0
$861$	−55.2158	−1.88175
$862$	0	0
$863$	14.7509	0.502128	0.251064	−	0.967971i	$-0.419220\pi$
$863$	14.7509	0.502128	0.251064	+	0.967971i	$0.419220\pi$
$864$	0	0
$865$	12.3878	0.421196
$866$	0	0
$867$	47.3973	1.60970
$868$	0	0
$869$	16.4681	0.558644
$870$	0	0
$871$	28.6628	0.971203
$872$	0	0
$873$	−1.41273	−0.0478137
$874$	0	0
$875$	4.62620	0.156394
$876$	0	0
$877$	16.8357	0.568501	0.284250	−	0.958750i	$-0.408255\pi$
$877$	16.8357	0.568501	0.284250	+	0.958750i	$0.408255\pi$
$878$	0	0
$879$	−46.4971	−1.56831
$880$	0	0
$881$	45.4373	1.53082	0.765411	−	0.643542i	$-0.222536\pi$
$881$	45.4373	1.53082	0.765411	+	0.643542i	$0.222536\pi$
$882$	0	0
$883$	38.0837	1.28162	0.640810	−	0.767700i	$-0.278599\pi$
$883$	38.0837	1.28162	0.640810	+	0.767700i	$0.278599\pi$
$884$	0	0
$885$	−1.10308	−0.0370798
$886$	0	0
$887$	21.3049	0.715348	0.357674	−	0.933847i	$-0.383570\pi$
$887$	21.3049	0.715348	0.357674	+	0.933847i	$0.383570\pi$
$888$	0	0
$889$	−61.8496	−2.07437
$890$	0	0
$891$	51.3574	1.72054
$892$	0	0
$893$	−1.72928	−0.0578682
$894$	0	0
$895$	−12.7755	−0.427038
$896$	0	0
$897$	40.1339	1.34003
$898$	0	0
$899$	−9.86712	−0.329087
$900$	0	0
$901$	−9.87859	−0.329104
$902$	0	0
$903$	89.4931	2.97814
$904$	0	0
$905$	−22.0000	−0.731305
$906$	0	0
$907$	12.4600	0.413728	0.206864	−	0.978370i	$-0.433674\pi$
$907$	12.4600	0.413728	0.206864	+	0.978370i	$0.433674\pi$
$908$	0	0
$909$	0.726351	0.0240916
$910$	0	0
$911$	35.2803	1.16889	0.584444	−	0.811434i	$-0.301313\pi$
$911$	35.2803	1.16889	0.584444	+	0.811434i	$0.301313\pi$
$912$	0	0
$913$	15.1753	0.502230
$914$	0	0
$915$	−26.5048	−0.876221
$916$	0	0
$917$	18.5048	0.611082
$918$	0	0
$919$	−0.392124	−0.0129350	−0.00646749	−	0.999979i	$-0.502059\pi$
$919$	−0.392124	−0.0129350	−0.00646749	+	0.999979i	$0.502059\pi$
$920$	0	0
$921$	−53.2158	−1.75352
$922$	0	0
$923$	−60.6531	−1.99642
$924$	0	0
$925$	−0.387755	−0.0127493
$926$	0	0
$927$	0.0678779	0.00222940
$928$	0	0
$929$	−13.4942	−0.442729	−0.221365	−	0.975191i	$-0.571051\pi$
$929$	−13.4942	−0.442729	−0.221365	+	0.975191i	$0.571051\pi$
$930$	0	0
$931$	−14.4017	−0.471997
$932$	0	0
$933$	−7.26009	−0.237685
$934$	0	0
$935$	−36.5972	−1.19686
$936$	0	0
$937$	−53.4296	−1.74547	−0.872735	−	0.488195i	$-0.837656\pi$
$937$	−53.4296	−1.74547	−0.872735	+	0.488195i	$0.837656\pi$
$938$	0	0
$939$	−26.6049	−0.868220
$940$	0	0
$941$	−0.0568550	−0.00185342	−0.000926710	−	1.00000i	$-0.500295\pi$
$941$	−0.0568550	−0.00185342	−0.000926710		1.00000i	$0.500295\pi$
$942$	0	0
$943$	28.1137	0.915508
$944$	0	0
$945$	−23.6079	−0.767964
$946$	0	0
$947$	2.03664	0.0661820	0.0330910	−	0.999452i	$-0.489465\pi$
$947$	2.03664	0.0661820	0.0330910	+	0.999452i	$0.489465\pi$
$948$	0	0
$949$	−26.5337	−0.861322
$950$	0	0
$951$	−23.4509	−0.760446
$952$	0	0
$953$	17.5352	0.568020	0.284010	−	0.958821i	$-0.408335\pi$
$953$	17.5352	0.568020	0.284010	+	0.958821i	$0.408335\pi$
$954$	0	0
$955$	19.1955	0.621153
$956$	0	0
$957$	76.6531	2.47784
$958$	0	0
$959$	49.1589	1.58742
$960$	0	0
$961$	−29.4315	−0.949403
$962$	0	0
$963$	−0.538109	−0.0173403
$964$	0	0
$965$	13.7047	0.441170
$966$	0	0
$967$	−14.9904	−0.482059	−0.241030	−	0.970518i	$-0.577485\pi$
$967$	−14.9904	−0.482059	−0.241030	+	0.970518i	$0.577485\pi$
$968$	0	0
$969$	11.6724	0.374972
$970$	0	0
$971$	−20.8805	−0.670087	−0.335043	−	0.942203i	$-0.608751\pi$
$971$	−20.8805	−0.670087	−0.335043	+	0.942203i	$0.608751\pi$
$972$	0	0
$973$	58.1483	1.86415
$974$	0	0
$975$	9.67243	0.309766
$976$	0	0
$977$	−24.0246	−0.768614	−0.384307	−	0.923205i	$-0.625560\pi$
$977$	−24.0246	−0.768614	−0.384307	+	0.923205i	$0.625560\pi$
$978$	0	0
$979$	−23.5877	−0.753865
$980$	0	0
$981$	−1.21784	−0.0388825
$982$	0	0
$983$	−30.9205	−0.986209	−0.493105	−	0.869970i	$-0.664138\pi$
$983$	−30.9205	−0.986209	−0.493105	+	0.869970i	$0.664138\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	−14.0925	−0.448568
$988$	0	0
$989$	−45.5664	−1.44893
$990$	0	0
$991$	2.56934	0.0816179	0.0408089	−	0.999167i	$-0.487007\pi$
$991$	2.56934	0.0816179	0.0408089	+	0.999167i	$0.487007\pi$
$992$	0	0
$993$	29.8140	0.946120
$994$	0	0
$995$	0.561647	0.0178054
$996$	0	0
$997$	35.1878	1.11441	0.557205	−	0.830375i	$-0.311874\pi$
$997$	35.1878	1.11441	0.557205	+	0.830375i	$0.311874\pi$
$998$	0	0
$999$	1.97875	0.0626048

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	760.2.a.h.1.3	✓	3
3.2	odd	2		6840.2.a.bj.1.1		3
4.3	odd	2		1520.2.a.r.1.1		3
5.2	odd	4		3800.2.d.k.3649.2		6
5.3	odd	4		3800.2.d.k.3649.5		6
5.4	even	2		3800.2.a.y.1.1		3
8.3	odd	2		6080.2.a.bs.1.3		3
8.5	even	2		6080.2.a.bw.1.1		3
20.19	odd	2		7600.2.a.bo.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.h.1.3	✓	3	1.1	even	1	trivial
1520.2.a.r.1.1		3	4.3	odd	2
3800.2.a.y.1.1		3	5.4	even	2
3800.2.d.k.3649.2		6	5.2	odd	4
3800.2.d.k.3649.5		6	5.3	odd	4
6080.2.a.bs.1.3		3	8.3	odd	2
6080.2.a.bw.1.1		3	8.5	even	2
6840.2.a.bj.1.1		3	3.2	odd	2
7600.2.a.bo.1.3		3	20.19	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:	\( N \)	\(=\)	\( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	760.a (trivial)

\(f(q)\)	\(=\)	\(q+1.76156 q^{3} -1.00000 q^{5} -4.62620 q^{7} +0.103084 q^{9} -5.52311 q^{11} +5.49084 q^{13} -1.76156 q^{15} -6.62620 q^{17} -1.00000 q^{19} -8.14931 q^{21} +4.14931 q^{23} +1.00000 q^{25} -5.10308 q^{27} -7.87859 q^{29} +1.25240 q^{31} -9.72928 q^{33} +4.62620 q^{35} -0.387755 q^{37} +9.67243 q^{39} +6.77551 q^{41} -10.9817 q^{43} -0.103084 q^{45} +1.72928 q^{47} +14.4017 q^{49} -11.6724 q^{51} +1.49084 q^{53} +5.52311 q^{55} -1.76156 q^{57} +0.626198 q^{59} +15.0462 q^{61} -0.476886 q^{63} -5.49084 q^{65} +5.22012 q^{67} +7.30925 q^{69} -11.0462 q^{71} -4.83237 q^{73} +1.76156 q^{75} +25.5510 q^{77} -2.98168 q^{79} -9.29862 q^{81} -2.74760 q^{83} +6.62620 q^{85} -13.8786 q^{87} +4.27072 q^{89} -25.4017 q^{91} +2.20617 q^{93} +1.00000 q^{95} -13.7047 q^{97} -0.569343 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 3 q^{5} - 5 q^{7} + 4 q^{9} - 4 q^{11} + 5 q^{13} + q^{15} - 11 q^{17} - 3 q^{19} - 3 q^{21} - 9 q^{23} + 3 q^{25} - 19 q^{27} + 3 q^{29} - 14 q^{31} - 24 q^{33} + 5 q^{35} + 14 q^{37} - 5 q^{39}+ \cdots + 36 q^{99}+O(q^{100}) \) 3 * q - q^3 - 3 * q^5 - 5 * q^7 + 4 * q^9 - 4 * q^11 + 5 * q^13 + q^15 - 11 * q^17 - 3 * q^19 - 3 * q^21 - 9 * q^23 + 3 * q^25 - 19 * q^27 + 3 * q^29 - 14 * q^31 - 24 * q^33 + 5 * q^35 + 14 * q^37 - 5 * q^39 - 10 * q^41 - 10 * q^43 - 4 * q^45 + 4 * q^49 - q^51 - 7 * q^53 + 4 * q^55 + q^57 - 7 * q^59 + 20 * q^61 - 14 * q^63 - 5 * q^65 - q^67 + 33 * q^69 - 8 * q^71 - 13 * q^73 - q^75 + 16 * q^77 + 14 * q^79 + 15 * q^81 - 26 * q^83 + 11 * q^85 - 15 * q^87 + 18 * q^89 - 37 * q^91 + 14 * q^93 + 3 * q^95 - 6 * q^97 + 36 * q^99

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