LMFDB - Embedded newform 7360.2.a.ce.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7360,2,Mod(1,7360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7360, 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("7360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7360 = 2^{6} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7360.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:	$58.7698958877$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.68740$ of defining polynomial
Character	$\chi$	$=$	7360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.68740 q^{3} +1.00000 q^{5} +4.59692 q^{7} +4.22212 q^{9} +O(q^{10})$	$q+2.68740 q^{3} +1.00000 q^{5} +4.59692 q^{7} +4.22212 q^{9} +5.13163 q^{11} +1.22212 q^{13} +2.68740 q^{15} -4.68740 q^{17} -4.59692 q^{19} +12.3537 q^{21} +1.00000 q^{23} +1.00000 q^{25} +3.28432 q^{27} -3.37480 q^{29} +0.777884 q^{31} +13.7907 q^{33} +4.59692 q^{35} -5.81903 q^{37} +3.28432 q^{39} -8.50643 q^{41} +8.00000 q^{43} +4.22212 q^{45} +6.44423 q^{47} +14.1316 q^{49} -12.5969 q^{51} +6.00000 q^{53} +5.13163 q^{55} -12.3537 q^{57} +9.37480 q^{59} -10.9507 q^{61} +19.4087 q^{63} +1.22212 q^{65} +15.6381 q^{67} +2.68740 q^{69} -1.31260 q^{71} -4.44423 q^{73} +2.68740 q^{75} +23.5897 q^{77} +4.88847 q^{79} -3.84008 q^{81} -3.81903 q^{83} -4.68740 q^{85} -9.06943 q^{87} +8.93057 q^{89} +5.61797 q^{91} +2.09048 q^{93} -4.59692 q^{95} -18.0622 q^{97} +21.6663 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 3 q^{5} - 3 q^{7} + 10 q^{9}+O(q^{10}) $ 3 * q + q^3 + 3 * q^5 - 3 * q^7 + 10 * q^9	$ 3 q + q^{3} + 3 q^{5} - 3 q^{7} + 10 q^{9} + 3 q^{11} + q^{13} + q^{15} - 7 q^{17} + 3 q^{19} + 22 q^{21} + 3 q^{23} + 3 q^{25} - 14 q^{27} + 4 q^{29} + 5 q^{31} - 9 q^{33} - 3 q^{35} + 2 q^{37} - 14 q^{39} + q^{41} + 24 q^{43} + 10 q^{45} + 14 q^{47} + 30 q^{49} - 21 q^{51} + 18 q^{53} + 3 q^{55} - 22 q^{57} + 14 q^{59} - q^{61} - 8 q^{63} + q^{65} + 8 q^{67} + q^{69} - 11 q^{71} - 8 q^{73} + q^{75} + 24 q^{77} + 4 q^{79} + 7 q^{81} + 8 q^{83} - 7 q^{85} - 36 q^{87} + 18 q^{89} + q^{91} + 16 q^{93} + 3 q^{95} - 33 q^{97} + 57 q^{99}+O(q^{100}) $ 3 * q + q^3 + 3 * q^5 - 3 * q^7 + 10 * q^9 + 3 * q^11 + q^13 + q^15 - 7 * q^17 + 3 * q^19 + 22 * q^21 + 3 * q^23 + 3 * q^25 - 14 * q^27 + 4 * q^29 + 5 * q^31 - 9 * q^33 - 3 * q^35 + 2 * q^37 - 14 * q^39 + q^41 + 24 * q^43 + 10 * q^45 + 14 * q^47 + 30 * q^49 - 21 * q^51 + 18 * q^53 + 3 * q^55 - 22 * q^57 + 14 * q^59 - q^61 - 8 * q^63 + q^65 + 8 * q^67 + q^69 - 11 * q^71 - 8 * q^73 + q^75 + 24 * q^77 + 4 * q^79 + 7 * q^81 + 8 * q^83 - 7 * q^85 - 36 * q^87 + 18 * q^89 + q^91 + 16 * q^93 + 3 * q^95 - 33 * q^97 + 57 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.68740	1.55157	0.775785	−	0.630997i	$-0.217354\pi$
$3$	2.68740	1.55157	0.775785	+	0.630997i	$0.217354\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.59692	1.73747	0.868735	−	0.495277i	$-0.164933\pi$
$7$	4.59692	1.73747	0.868735	+	0.495277i	$0.164933\pi$
$8$	0	0
$9$	4.22212	1.40737
$10$	0	0
$11$	5.13163	1.54725	0.773623	−	0.633647i	$-0.218443\pi$
$11$	5.13163	1.54725	0.773623	+	0.633647i	$0.218443\pi$
$12$	0	0
$13$	1.22212	0.338954	0.169477	−	0.985534i	$-0.445792\pi$
$13$	1.22212	0.338954	0.169477	+	0.985534i	$0.445792\pi$
$14$	0	0
$15$	2.68740	0.693884
$16$	0	0
$17$	−4.68740	−1.13686	−0.568431	−	0.822731i	$-0.692449\pi$
$17$	−4.68740	−1.13686	−0.568431	+	0.822731i	$0.692449\pi$
$18$	0	0
$19$	−4.59692	−1.05460	−0.527302	−	0.849678i	$-0.676796\pi$
$19$	−4.59692	−1.05460	−0.527302	+	0.849678i	$0.676796\pi$
$20$	0	0
$21$	12.3537	2.69581
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.28432	0.632067
$28$	0	0
$29$	−3.37480	−0.626684	−0.313342	−	0.949640i	$-0.601449\pi$
$29$	−3.37480	−0.626684	−0.313342	+	0.949640i	$0.601449\pi$
$30$	0	0
$31$	0.777884	0.139712	0.0698560	−	0.997557i	$-0.477746\pi$
$31$	0.777884	0.139712	0.0698560	+	0.997557i	$0.477746\pi$
$32$	0	0
$33$	13.7907	2.40066
$34$	0	0
$35$	4.59692	0.777021
$36$	0	0
$37$	−5.81903	−0.956643	−0.478321	−	0.878185i	$-0.658755\pi$
$37$	−5.81903	−0.956643	−0.478321	+	0.878185i	$0.658755\pi$
$38$	0	0
$39$	3.28432	0.525911
$40$	0	0
$41$	−8.50643	−1.32848	−0.664241	−	0.747519i	$-0.731245\pi$
$41$	−8.50643	−1.32848	−0.664241	+	0.747519i	$0.731245\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	4.22212	0.629396
$46$	0	0
$47$	6.44423	0.939988	0.469994	−	0.882670i	$-0.344256\pi$
$47$	6.44423	0.939988	0.469994	+	0.882670i	$0.344256\pi$
$48$	0	0
$49$	14.1316	2.01880
$50$	0	0
$51$	−12.5969	−1.76392
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	5.13163	0.691949
$56$	0	0
$57$	−12.3537	−1.63629
$58$	0	0
$59$	9.37480	1.22049	0.610247	−	0.792211i	$-0.291070\pi$
$59$	9.37480	1.22049	0.610247	+	0.792211i	$0.291070\pi$
$60$	0	0
$61$	−10.9507	−1.40209	−0.701044	−	0.713118i	$-0.747283\pi$
$61$	−10.9507	−1.40209	−0.701044	+	0.713118i	$0.747283\pi$
$62$	0	0
$63$	19.4087	2.44527
$64$	0	0
$65$	1.22212	0.151585
$66$	0	0
$67$	15.6381	1.91049	0.955247	−	0.295810i	$-0.0955895\pi$
$67$	15.6381	1.91049	0.955247	+	0.295810i	$0.0955895\pi$
$68$	0	0
$69$	2.68740	0.323525
$70$	0	0
$71$	−1.31260	−0.155777	−0.0778885	−	0.996962i	$-0.524818\pi$
$71$	−1.31260	−0.155777	−0.0778885	+	0.996962i	$0.524818\pi$
$72$	0	0
$73$	−4.44423	−0.520158	−0.260079	−	0.965587i	$-0.583749\pi$
$73$	−4.44423	−0.520158	−0.260079	+	0.965587i	$0.583749\pi$
$74$	0	0
$75$	2.68740	0.310314
$76$	0	0
$77$	23.5897	2.68829
$78$	0	0
$79$	4.88847	0.549995	0.274998	−	0.961445i	$-0.411323\pi$
$79$	4.88847	0.549995	0.274998	+	0.961445i	$0.411323\pi$
$80$	0	0
$81$	−3.84008	−0.426676
$82$	0	0
$83$	−3.81903	−0.419193	−0.209597	−	0.977788i	$-0.567215\pi$
$83$	−3.81903	−0.419193	−0.209597	+	0.977788i	$0.567215\pi$
$84$	0	0
$85$	−4.68740	−0.508420
$86$	0	0
$87$	−9.06943	−0.972345
$88$	0	0
$89$	8.93057	0.946638	0.473319	−	0.880891i	$-0.343056\pi$
$89$	8.93057	0.946638	0.473319	+	0.880891i	$0.343056\pi$
$90$	0	0
$91$	5.61797	0.588923
$92$	0	0
$93$	2.09048	0.216773
$94$	0	0
$95$	−4.59692	−0.471634
$96$	0	0
$97$	−18.0622	−1.83394	−0.916969	−	0.398958i	$-0.869372\pi$
$97$	−18.0622	−1.83394	−0.916969	+	0.398958i	$0.869372\pi$
$98$	0	0
$99$	21.6663	2.17755
$100$	0	0
$101$	−3.37480	−0.335805	−0.167903	−	0.985804i	$-0.553699\pi$
$101$	−3.37480	−0.335805	−0.167903	+	0.985804i	$0.553699\pi$
$102$	0	0
$103$	−13.1316	−1.29390	−0.646949	−	0.762533i	$-0.723955\pi$
$103$	−13.1316	−1.29390	−0.646949	+	0.762533i	$0.723955\pi$
$104$	0	0
$105$	12.3537	1.20560
$106$	0	0
$107$	12.3054	1.18960	0.594802	−	0.803872i	$-0.297230\pi$
$107$	12.3054	1.18960	0.594802	+	0.803872i	$0.297230\pi$
$108$	0	0
$109$	10.5969	1.01500	0.507500	−	0.861652i	$-0.330570\pi$
$109$	10.5969	1.01500	0.507500	+	0.861652i	$0.330570\pi$
$110$	0	0
$111$	−15.6381	−1.48430
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	5.15992	0.477035
$118$	0	0
$119$	−21.5476	−1.97526
$120$	0	0
$121$	15.3337	1.39397
$122$	0	0
$123$	−22.8602	−2.06123
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.26326	0.200832	0.100416	−	0.994946i	$-0.467983\pi$
$127$	2.26326	0.200832	0.100416	+	0.994946i	$0.467983\pi$
$128$	0	0
$129$	21.4992	1.89290
$130$	0	0
$131$	2.93057	0.256045	0.128022	−	0.991771i	$-0.459137\pi$
$131$	2.93057	0.256045	0.128022	+	0.991771i	$0.459137\pi$
$132$	0	0
$133$	−21.1316	−1.83234
$134$	0	0
$135$	3.28432	0.282669
$136$	0	0
$137$	−19.7907	−1.69084	−0.845419	−	0.534104i	$-0.820649\pi$
$137$	−19.7907	−1.69084	−0.845419	+	0.534104i	$0.820649\pi$
$138$	0	0
$139$	6.26326	0.531243	0.265622	−	0.964077i	$-0.414423\pi$
$139$	6.26326	0.531243	0.265622	+	0.964077i	$0.414423\pi$
$140$	0	0
$141$	17.3182	1.45846
$142$	0	0
$143$	6.27145	0.524445
$144$	0	0
$145$	−3.37480	−0.280262
$146$	0	0
$147$	37.9773	3.13232
$148$	0	0
$149$	19.7907	1.62132	0.810661	−	0.585516i	$-0.199108\pi$
$149$	19.7907	1.62132	0.810661	+	0.585516i	$0.199108\pi$
$150$	0	0
$151$	−4.06220	−0.330577	−0.165289	−	0.986245i	$-0.552856\pi$
$151$	−4.06220	−0.330577	−0.165289	+	0.986245i	$0.552856\pi$
$152$	0	0
$153$	−19.7907	−1.59999
$154$	0	0
$155$	0.777884	0.0624811
$156$	0	0
$157$	−4.62520	−0.369131	−0.184566	−	0.982820i	$-0.559088\pi$
$157$	−4.62520	−0.369131	−0.184566	+	0.982820i	$0.559088\pi$
$158$	0	0
$159$	16.1244	1.27875
$160$	0	0
$161$	4.59692	0.362288
$162$	0	0
$163$	12.4159	0.972492	0.486246	−	0.873822i	$-0.338366\pi$
$163$	12.4159	0.972492	0.486246	+	0.873822i	$0.338366\pi$
$164$	0	0
$165$	13.7907	1.07361
$166$	0	0
$167$	−12.8885	−0.997339	−0.498670	−	0.866792i	$-0.666178\pi$
$167$	−12.8885	−0.997339	−0.498670	+	0.866792i	$0.666178\pi$
$168$	0	0
$169$	−11.5064	−0.885110
$170$	0	0
$171$	−19.4087	−1.48422
$172$	0	0
$173$	10.2432	0.778774	0.389387	−	0.921074i	$-0.372687\pi$
$173$	10.2432	0.778774	0.389387	+	0.921074i	$0.372687\pi$
$174$	0	0
$175$	4.59692	0.347494
$176$	0	0
$177$	25.1938	1.89368
$178$	0	0
$179$	−13.1938	−0.986153	−0.493077	−	0.869986i	$-0.664128\pi$
$179$	−13.1938	−0.986153	−0.493077	+	0.869986i	$0.664128\pi$
$180$	0	0
$181$	−15.7907	−1.17372	−0.586858	−	0.809690i	$-0.699636\pi$
$181$	−15.7907	−1.17372	−0.586858	+	0.809690i	$0.699636\pi$
$182$	0	0
$183$	−29.4288	−2.17544
$184$	0	0
$185$	−5.81903	−0.427824
$186$	0	0
$187$	−24.0540	−1.75900
$188$	0	0
$189$	15.0977	1.09820
$190$	0	0
$191$	−16.1244	−1.16672	−0.583360	−	0.812214i	$-0.698262\pi$
$191$	−16.1244	−1.16672	−0.583360	+	0.812214i	$0.698262\pi$
$192$	0	0
$193$	−17.9434	−1.29160	−0.645798	−	0.763508i	$-0.723475\pi$
$193$	−17.9434	−1.29160	−0.645798	+	0.763508i	$0.723475\pi$
$194$	0	0
$195$	3.28432	0.235195
$196$	0	0
$197$	−5.88123	−0.419020	−0.209510	−	0.977806i	$-0.567187\pi$
$197$	−5.88123	−0.419020	−0.209510	+	0.977806i	$0.567187\pi$
$198$	0	0
$199$	−6.56863	−0.465638	−0.232819	−	0.972520i	$-0.574795\pi$
$199$	−6.56863	−0.465638	−0.232819	+	0.972520i	$0.574795\pi$
$200$	0	0
$201$	42.0257	2.96427
$202$	0	0
$203$	−15.5137	−1.08885
$204$	0	0
$205$	−8.50643	−0.594115
$206$	0	0
$207$	4.22212	0.293457
$208$	0	0
$209$	−23.5897	−1.63173
$210$	0	0
$211$	−15.4571	−1.06411	−0.532055	−	0.846710i	$-0.678580\pi$
$211$	−15.4571	−1.06411	−0.532055	+	0.846710i	$0.678580\pi$
$212$	0	0
$213$	−3.52748	−0.241699
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	3.57587	0.242746
$218$	0	0
$219$	−11.9434	−0.807062
$220$	0	0
$221$	−5.72855	−0.385344
$222$	0	0
$223$	10.7496	0.719846	0.359923	−	0.932982i	$-0.382803\pi$
$223$	10.7496	0.719846	0.359923	+	0.932982i	$0.382803\pi$
$224$	0	0
$225$	4.22212	0.281474
$226$	0	0
$227$	−12.3054	−0.816736	−0.408368	−	0.912817i	$-0.633902\pi$
$227$	−12.3054	−0.816736	−0.408368	+	0.912817i	$0.633902\pi$
$228$	0	0
$229$	−9.63806	−0.636901	−0.318451	−	0.947939i	$-0.603162\pi$
$229$	−9.63806	−0.636901	−0.318451	+	0.947939i	$0.603162\pi$
$230$	0	0
$231$	63.3949	4.17108
$232$	0	0
$233$	13.9434	0.913464	0.456732	−	0.889604i	$-0.349020\pi$
$233$	13.9434	0.913464	0.456732	+	0.889604i	$0.349020\pi$
$234$	0	0
$235$	6.44423	0.420375
$236$	0	0
$237$	13.1373	0.853357
$238$	0	0
$239$	26.3877	1.70688	0.853438	−	0.521194i	$-0.174513\pi$
$239$	26.3877	1.70688	0.853438	+	0.521194i	$0.174513\pi$
$240$	0	0
$241$	−7.06943	−0.455382	−0.227691	−	0.973733i	$-0.573118\pi$
$241$	−7.06943	−0.455382	−0.227691	+	0.973733i	$0.573118\pi$
$242$	0	0
$243$	−20.1728	−1.29408
$244$	0	0
$245$	14.1316	0.902837
$246$	0	0
$247$	−5.61797	−0.357463
$248$	0	0
$249$	−10.2633	−0.650408
$250$	0	0
$251$	−24.9023	−1.57182	−0.785909	−	0.618342i	$-0.787805\pi$
$251$	−24.9023	−1.57182	−0.785909	+	0.618342i	$0.787805\pi$
$252$	0	0
$253$	5.13163	0.322623
$254$	0	0
$255$	−12.5969	−0.788849
$256$	0	0
$257$	0.444233	0.0277105	0.0138552	−	0.999904i	$-0.495590\pi$
$257$	0.444233	0.0277105	0.0138552	+	0.999904i	$0.495590\pi$
$258$	0	0
$259$	−26.7496	−1.66214
$260$	0	0
$261$	−14.2488	−0.881978
$262$	0	0
$263$	23.8812	1.47258	0.736290	−	0.676666i	$-0.236576\pi$
$263$	23.8812	1.47258	0.736290	+	0.676666i	$0.236576\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	24.0000	1.46878
$268$	0	0
$269$	−16.2633	−0.991589	−0.495794	−	0.868440i	$-0.665123\pi$
$269$	−16.2633	−0.991589	−0.495794	+	0.868440i	$0.665123\pi$
$270$	0	0
$271$	−25.6098	−1.55568	−0.777842	−	0.628460i	$-0.783685\pi$
$271$	−25.6098	−1.55568	−0.777842	+	0.628460i	$0.783685\pi$
$272$	0	0
$273$	15.0977	0.913756
$274$	0	0
$275$	5.13163	0.309449
$276$	0	0
$277$	−2.88847	−0.173551	−0.0867755	−	0.996228i	$-0.527656\pi$
$277$	−2.88847	−0.173551	−0.0867755	+	0.996228i	$0.527656\pi$
$278$	0	0
$279$	3.28432	0.196627
$280$	0	0
$281$	4.26326	0.254325	0.127163	−	0.991882i	$-0.459413\pi$
$281$	4.26326	0.254325	0.127163	+	0.991882i	$0.459413\pi$
$282$	0	0
$283$	30.5686	1.81712	0.908558	−	0.417758i	$-0.137184\pi$
$283$	30.5686	1.81712	0.908558	+	0.417758i	$0.137184\pi$
$284$	0	0
$285$	−12.3537	−0.731773
$286$	0	0
$287$	−39.1033	−2.30820
$288$	0	0
$289$	4.97171	0.292454
$290$	0	0
$291$	−48.5403	−2.84549
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	9.37480	0.545822
$296$	0	0
$297$	16.8539	0.977962
$298$	0	0
$299$	1.22212	0.0706768
$300$	0	0
$301$	36.7753	2.11969
$302$	0	0
$303$	−9.06943	−0.521025
$304$	0	0
$305$	−10.9507	−0.627033
$306$	0	0
$307$	−8.54853	−0.487891	−0.243945	−	0.969789i	$-0.578442\pi$
$307$	−8.54853	−0.487891	−0.243945	+	0.969789i	$0.578442\pi$
$308$	0	0
$309$	−35.2899	−2.00757
$310$	0	0
$311$	7.63806	0.433115	0.216557	−	0.976270i	$-0.430517\pi$
$311$	7.63806	0.433115	0.216557	+	0.976270i	$0.430517\pi$
$312$	0	0
$313$	−18.2350	−1.03070	−0.515351	−	0.856979i	$-0.672338\pi$
$313$	−18.2350	−1.03070	−0.515351	+	0.856979i	$0.672338\pi$
$314$	0	0
$315$	19.4087	1.09356
$316$	0	0
$317$	16.8602	0.946962	0.473481	−	0.880804i	$-0.342997\pi$
$317$	16.8602	0.946962	0.473481	+	0.880804i	$0.342997\pi$
$318$	0	0
$319$	−17.3182	−0.969635
$320$	0	0
$321$	33.0694	1.84576
$322$	0	0
$323$	21.5476	1.19894
$324$	0	0
$325$	1.22212	0.0677908
$326$	0	0
$327$	28.4781	1.57485
$328$	0	0
$329$	29.6236	1.63320
$330$	0	0
$331$	19.1517	1.05267	0.526337	−	0.850276i	$-0.323565\pi$
$331$	19.1517	1.05267	0.526337	+	0.850276i	$0.323565\pi$
$332$	0	0
$333$	−24.5686	−1.34635
$334$	0	0
$335$	15.6381	0.854399
$336$	0	0
$337$	1.70845	0.0930652	0.0465326	−	0.998917i	$-0.485183\pi$
$337$	1.70845	0.0930652	0.0465326	+	0.998917i	$0.485183\pi$
$338$	0	0
$339$	−16.1244	−0.875757
$340$	0	0
$341$	3.99181	0.216169
$342$	0	0
$343$	32.7835	1.77014
$344$	0	0
$345$	2.68740	0.144685
$346$	0	0
$347$	10.6874	0.573730	0.286865	−	0.957971i	$-0.407387\pi$
$347$	10.6874	0.573730	0.286865	+	0.957971i	$0.407387\pi$
$348$	0	0
$349$	12.3877	0.663096	0.331548	−	0.943438i	$-0.392429\pi$
$349$	12.3877	0.663096	0.331548	+	0.943438i	$0.392429\pi$
$350$	0	0
$351$	4.01382	0.214242
$352$	0	0
$353$	5.45710	0.290452	0.145226	−	0.989399i	$-0.453609\pi$
$353$	5.45710	0.290452	0.145226	+	0.989399i	$0.453609\pi$
$354$	0	0
$355$	−1.31260	−0.0696656
$356$	0	0
$357$	−57.9070	−3.06476
$358$	0	0
$359$	−20.1810	−1.06511	−0.532555	−	0.846395i	$-0.678768\pi$
$359$	−20.1810	−1.06511	−0.532555	+	0.846395i	$0.678768\pi$
$360$	0	0
$361$	2.13163	0.112191
$362$	0	0
$363$	41.2076	2.16284
$364$	0	0
$365$	−4.44423	−0.232622
$366$	0	0
$367$	−2.74960	−0.143528	−0.0717639	−	0.997422i	$-0.522863\pi$
$367$	−2.74960	−0.143528	−0.0717639	+	0.997422i	$0.522863\pi$
$368$	0	0
$369$	−35.9151	−1.86967
$370$	0	0
$371$	27.5815	1.43196
$372$	0	0
$373$	12.0823	0.625598	0.312799	−	0.949819i	$-0.398733\pi$
$373$	12.0823	0.625598	0.312799	+	0.949819i	$0.398733\pi$
$374$	0	0
$375$	2.68740	0.138777
$376$	0	0
$377$	−4.12440	−0.212417
$378$	0	0
$379$	5.25603	0.269984	0.134992	−	0.990847i	$-0.456899\pi$
$379$	5.25603	0.269984	0.134992	+	0.990847i	$0.456899\pi$
$380$	0	0
$381$	6.08230	0.311605
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	23.5897	1.20224
$386$	0	0
$387$	33.7769	1.71698
$388$	0	0
$389$	−0.325463	−0.0165017	−0.00825083	−	0.999966i	$-0.502626\pi$
$389$	−0.325463	−0.0165017	−0.00825083	+	0.999966i	$0.502626\pi$
$390$	0	0
$391$	−4.68740	−0.237052
$392$	0	0
$393$	7.87560	0.397272
$394$	0	0
$395$	4.88847	0.245965
$396$	0	0
$397$	17.7568	0.891190	0.445595	−	0.895235i	$-0.352992\pi$
$397$	17.7568	0.891190	0.445595	+	0.895235i	$0.352992\pi$
$398$	0	0
$399$	−56.7891	−2.84301
$400$	0	0
$401$	3.91770	0.195641	0.0978204	−	0.995204i	$-0.468813\pi$
$401$	3.91770	0.195641	0.0978204	+	0.995204i	$0.468813\pi$
$402$	0	0
$403$	0.950664	0.0473560
$404$	0	0
$405$	−3.84008	−0.190815
$406$	0	0
$407$	−29.8611	−1.48016
$408$	0	0
$409$	9.58405	0.473901	0.236950	−	0.971522i	$-0.423852\pi$
$409$	9.58405	0.473901	0.236950	+	0.971522i	$0.423852\pi$
$410$	0	0
$411$	−53.1856	−2.62345
$412$	0	0
$413$	43.0952	2.12057
$414$	0	0
$415$	−3.81903	−0.187469
$416$	0	0
$417$	16.8319	0.824261
$418$	0	0
$419$	−20.5265	−1.00279	−0.501393	−	0.865219i	$-0.667179\pi$
$419$	−20.5265	−1.00279	−0.501393	+	0.865219i	$0.667179\pi$
$420$	0	0
$421$	−2.29155	−0.111683	−0.0558417	−	0.998440i	$-0.517784\pi$
$421$	−2.29155	−0.111683	−0.0558417	+	0.998440i	$0.517784\pi$
$422$	0	0
$423$	27.2083	1.32291
$424$	0	0
$425$	−4.68740	−0.227372
$426$	0	0
$427$	−50.3393	−2.43609
$428$	0	0
$429$	16.8539	0.813714
$430$	0	0
$431$	8.83189	0.425417	0.212709	−	0.977116i	$-0.431771\pi$
$431$	8.83189	0.425417	0.212709	+	0.977116i	$0.431771\pi$
$432$	0	0
$433$	33.3465	1.60253	0.801266	−	0.598309i	$-0.204160\pi$
$433$	33.3465	1.60253	0.801266	+	0.598309i	$0.204160\pi$
$434$	0	0
$435$	−9.06943	−0.434846
$436$	0	0
$437$	−4.59692	−0.219900
$438$	0	0
$439$	6.02829	0.287714	0.143857	−	0.989598i	$-0.454049\pi$
$439$	6.02829	0.287714	0.143857	+	0.989598i	$0.454049\pi$
$440$	0	0
$441$	59.6654	2.84121
$442$	0	0
$443$	−15.5275	−0.737733	−0.368866	−	0.929482i	$-0.620254\pi$
$443$	−15.5275	−0.737733	−0.368866	+	0.929482i	$0.620254\pi$
$444$	0	0
$445$	8.93057	0.423349
$446$	0	0
$447$	53.1856	2.51559
$448$	0	0
$449$	−18.3594	−0.866433	−0.433216	−	0.901290i	$-0.642621\pi$
$449$	−18.3594	−0.866433	−0.433216	+	0.901290i	$0.642621\pi$
$450$	0	0
$451$	−43.6519	−2.05549
$452$	0	0
$453$	−10.9168	−0.512914
$454$	0	0
$455$	5.61797	0.263374
$456$	0	0
$457$	−11.4992	−0.537910	−0.268955	−	0.963153i	$-0.586678\pi$
$457$	−11.4992	−0.537910	−0.268955	+	0.963153i	$0.586678\pi$
$458$	0	0
$459$	−15.3949	−0.718572
$460$	0	0
$461$	−1.33270	−0.0620700	−0.0310350	−	0.999518i	$-0.509880\pi$
$461$	−1.33270	−0.0620700	−0.0310350	+	0.999518i	$0.509880\pi$
$462$	0	0
$463$	−35.8190	−1.66465	−0.832326	−	0.554287i	$-0.812991\pi$
$463$	−35.8190	−1.66465	−0.832326	+	0.554287i	$0.812991\pi$
$464$	0	0
$465$	2.09048	0.0969439
$466$	0	0
$467$	−23.7625	−1.09960	−0.549798	−	0.835298i	$-0.685295\pi$
$467$	−23.7625	−1.09960	−0.549798	+	0.835298i	$0.685295\pi$
$468$	0	0
$469$	71.8869	3.31943
$470$	0	0
$471$	−12.4298	−0.572733
$472$	0	0
$473$	41.0531	1.88762
$474$	0	0
$475$	−4.59692	−0.210921
$476$	0	0
$477$	25.3327	1.15990
$478$	0	0
$479$	2.04210	0.0933060	0.0466530	−	0.998911i	$-0.485145\pi$
$479$	2.04210	0.0933060	0.0466530	+	0.998911i	$0.485145\pi$
$480$	0	0
$481$	−7.11153	−0.324258
$482$	0	0
$483$	12.3537	0.562115
$484$	0	0
$485$	−18.0622	−0.820162
$486$	0	0
$487$	18.6252	0.843988	0.421994	−	0.906599i	$-0.361330\pi$
$487$	18.6252	0.843988	0.421994	+	0.906599i	$0.361330\pi$
$488$	0	0
$489$	33.3666	1.50889
$490$	0	0
$491$	33.7204	1.52178	0.760889	−	0.648882i	$-0.224763\pi$
$491$	33.7204	1.52178	0.760889	+	0.648882i	$0.224763\pi$
$492$	0	0
$493$	15.8190	0.712453
$494$	0	0
$495$	21.6663	0.973830
$496$	0	0
$497$	−6.03391	−0.270658
$498$	0	0
$499$	−16.8885	−0.756032	−0.378016	−	0.925799i	$-0.623393\pi$
$499$	−16.8885	−0.756032	−0.378016	+	0.925799i	$0.623393\pi$
$500$	0	0
$501$	−34.6365	−1.54744
$502$	0	0
$503$	−11.4031	−0.508438	−0.254219	−	0.967147i	$-0.581818\pi$
$503$	−11.4031	−0.508438	−0.254219	+	0.967147i	$0.581818\pi$
$504$	0	0
$505$	−3.37480	−0.150177
$506$	0	0
$507$	−30.9224	−1.37331
$508$	0	0
$509$	1.87560	0.0831346	0.0415673	−	0.999136i	$-0.486765\pi$
$509$	1.87560	0.0831346	0.0415673	+	0.999136i	$0.486765\pi$
$510$	0	0
$511$	−20.4298	−0.903759
$512$	0	0
$513$	−15.0977	−0.666581
$514$	0	0
$515$	−13.1316	−0.578649
$516$	0	0
$517$	33.0694	1.45439
$518$	0	0
$519$	27.5275	1.20832
$520$	0	0
$521$	5.11153	0.223940	0.111970	−	0.993712i	$-0.464284\pi$
$521$	5.11153	0.223940	0.111970	+	0.993712i	$0.464284\pi$
$522$	0	0
$523$	19.4571	0.850799	0.425400	−	0.905006i	$-0.360134\pi$
$523$	19.4571	0.850799	0.425400	+	0.905006i	$0.360134\pi$
$524$	0	0
$525$	12.3537	0.539162
$526$	0	0
$527$	−3.64625	−0.158833
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	39.5815	1.71769
$532$	0	0
$533$	−10.3958	−0.450294
$534$	0	0
$535$	12.3054	0.532007
$536$	0	0
$537$	−35.4571	−1.53009
$538$	0	0
$539$	72.5183	3.12359
$540$	0	0
$541$	2.70750	0.116404	0.0582022	−	0.998305i	$-0.481463\pi$
$541$	2.70750	0.116404	0.0582022	+	0.998305i	$0.481463\pi$
$542$	0	0
$543$	−42.4360	−1.82111
$544$	0	0
$545$	10.5969	0.453922
$546$	0	0
$547$	−1.66635	−0.0712479	−0.0356240	−	0.999365i	$-0.511342\pi$
$547$	−1.66635	−0.0712479	−0.0356240	+	0.999365i	$0.511342\pi$
$548$	0	0
$549$	−46.2350	−1.97326
$550$	0	0
$551$	15.5137	0.660904
$552$	0	0
$553$	22.4719	0.955601
$554$	0	0
$555$	−15.6381	−0.663799
$556$	0	0
$557$	20.9306	0.886857	0.443428	−	0.896310i	$-0.353762\pi$
$557$	20.9306	0.886857	0.443428	+	0.896310i	$0.353762\pi$
$558$	0	0
$559$	9.77693	0.413520
$560$	0	0
$561$	−64.6427	−2.72922
$562$	0	0
$563$	−15.2761	−0.643812	−0.321906	−	0.946772i	$-0.604324\pi$
$563$	−15.2761	−0.643812	−0.321906	+	0.946772i	$0.604324\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	−17.6525	−0.741337
$568$	0	0
$569$	20.3877	0.854695	0.427348	−	0.904087i	$-0.359448\pi$
$569$	20.3877	0.854695	0.427348	+	0.904087i	$0.359448\pi$
$570$	0	0
$571$	5.49357	0.229899	0.114949	−	0.993371i	$-0.463329\pi$
$571$	5.49357	0.229899	0.114949	+	0.993371i	$0.463329\pi$
$572$	0	0
$573$	−43.3327	−1.81025
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	17.2761	0.719215	0.359607	−	0.933104i	$-0.382911\pi$
$577$	17.2761	0.719215	0.359607	+	0.933104i	$0.382911\pi$
$578$	0	0
$579$	−48.2212	−2.00400
$580$	0	0
$581$	−17.5558	−0.728336
$582$	0	0
$583$	30.7898	1.27518
$584$	0	0
$585$	5.15992	0.213336
$586$	0	0
$587$	31.8247	1.31354	0.656772	−	0.754089i	$-0.271921\pi$
$587$	31.8247	1.31354	0.656772	+	0.754089i	$0.271921\pi$
$588$	0	0
$589$	−3.57587	−0.147341
$590$	0	0
$591$	−15.8052	−0.650140
$592$	0	0
$593$	−12.4442	−0.511023	−0.255512	−	0.966806i	$-0.582244\pi$
$593$	−12.4442	−0.511023	−0.255512	+	0.966806i	$0.582244\pi$
$594$	0	0
$595$	−21.5476	−0.883365
$596$	0	0
$597$	−17.6525	−0.722470
$598$	0	0
$599$	−17.4370	−0.712456	−0.356228	−	0.934399i	$-0.615937\pi$
$599$	−17.4370	−0.712456	−0.356228	+	0.934399i	$0.615937\pi$
$600$	0	0
$601$	−0.916751	−0.0373951	−0.0186975	−	0.999825i	$-0.505952\pi$
$601$	−0.916751	−0.0373951	−0.0186975	+	0.999825i	$0.505952\pi$
$602$	0	0
$603$	66.0257	2.68878
$604$	0	0
$605$	15.3337	0.623402
$606$	0	0
$607$	−36.7753	−1.49266	−0.746332	−	0.665574i	$-0.768187\pi$
$607$	−36.7753	−1.49266	−0.746332	+	0.665574i	$0.768187\pi$
$608$	0	0
$609$	−41.6914	−1.68942
$610$	0	0
$611$	7.87560	0.318613
$612$	0	0
$613$	−4.38766	−0.177216	−0.0886080	−	0.996067i	$-0.528242\pi$
$613$	−4.38766	−0.177216	−0.0886080	+	0.996067i	$0.528242\pi$
$614$	0	0
$615$	−22.8602	−0.921811
$616$	0	0
$617$	40.4499	1.62845	0.814225	−	0.580549i	$-0.197162\pi$
$617$	40.4499	1.62845	0.814225	+	0.580549i	$0.197162\pi$
$618$	0	0
$619$	−39.8165	−1.60036	−0.800180	−	0.599760i	$-0.795263\pi$
$619$	−39.8165	−1.60036	−0.800180	+	0.599760i	$0.795263\pi$
$620$	0	0
$621$	3.28432	0.131795
$622$	0	0
$623$	41.0531	1.64476
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−63.3949	−2.53175
$628$	0	0
$629$	27.2761	1.08757
$630$	0	0
$631$	−25.9013	−1.03112	−0.515558	−	0.856855i	$-0.672415\pi$
$631$	−25.9013	−1.03112	−0.515558	+	0.856855i	$0.672415\pi$
$632$	0	0
$633$	−41.5394	−1.65104
$634$	0	0
$635$	2.26326	0.0898149
$636$	0	0
$637$	17.2705	0.684282
$638$	0	0
$639$	−5.54195	−0.219236
$640$	0	0
$641$	−43.8448	−1.73176	−0.865882	−	0.500248i	$-0.833242\pi$
$641$	−43.8448	−1.73176	−0.865882	+	0.500248i	$0.833242\pi$
$642$	0	0
$643$	3.94343	0.155514	0.0777568	−	0.996972i	$-0.475224\pi$
$643$	3.94343	0.155514	0.0777568	+	0.996972i	$0.475224\pi$
$644$	0	0
$645$	21.4992	0.846530
$646$	0	0
$647$	−24.3456	−0.957123	−0.478561	−	0.878054i	$-0.658842\pi$
$647$	−24.3456	−0.957123	−0.478561	+	0.878054i	$0.658842\pi$
$648$	0	0
$649$	48.1080	1.88841
$650$	0	0
$651$	9.60978	0.376637
$652$	0	0
$653$	−37.6921	−1.47500	−0.737502	−	0.675344i	$-0.763995\pi$
$653$	−37.6921	−1.47500	−0.737502	+	0.675344i	$0.763995\pi$
$654$	0	0
$655$	2.93057	0.114507
$656$	0	0
$657$	−18.7641	−0.732056
$658$	0	0
$659$	24.6107	0.958698	0.479349	−	0.877624i	$-0.340873\pi$
$659$	24.6107	0.958698	0.479349	+	0.877624i	$0.340873\pi$
$660$	0	0
$661$	−27.3126	−1.06234	−0.531169	−	0.847266i	$-0.678247\pi$
$661$	−27.3126	−1.06234	−0.531169	+	0.847266i	$0.678247\pi$
$662$	0	0
$663$	−15.3949	−0.597888
$664$	0	0
$665$	−21.1316	−0.819450
$666$	0	0
$667$	−3.37480	−0.130673
$668$	0	0
$669$	28.8885	1.11689
$670$	0	0
$671$	−56.1948	−2.16938
$672$	0	0
$673$	22.5265	0.868334	0.434167	−	0.900832i	$-0.357043\pi$
$673$	22.5265	0.868334	0.434167	+	0.900832i	$0.357043\pi$
$674$	0	0
$675$	3.28432	0.126413
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	−83.0304	−3.18641
$680$	0	0
$681$	−33.0694	−1.26722
$682$	0	0
$683$	−31.9974	−1.22435	−0.612174	−	0.790723i	$-0.709705\pi$
$683$	−31.9974	−1.22435	−0.612174	+	0.790723i	$0.709705\pi$
$684$	0	0
$685$	−19.7907	−0.756166
$686$	0	0
$687$	−25.9013	−0.988197
$688$	0	0
$689$	7.33270	0.279354
$690$	0	0
$691$	−2.80617	−0.106752	−0.0533758	−	0.998574i	$-0.516998\pi$
$691$	−2.80617	−0.106752	−0.0533758	+	0.998574i	$0.516998\pi$
$692$	0	0
$693$	99.5984	3.78343
$694$	0	0
$695$	6.26326	0.237579
$696$	0	0
$697$	39.8730	1.51030
$698$	0	0
$699$	37.4716	1.41730
$700$	0	0
$701$	−8.37385	−0.316276	−0.158138	−	0.987417i	$-0.550549\pi$
$701$	−8.37385	−0.316276	−0.158138	+	0.987417i	$0.550549\pi$
$702$	0	0
$703$	26.7496	1.00888
$704$	0	0
$705$	17.3182	0.652242
$706$	0	0
$707$	−15.5137	−0.583451
$708$	0	0
$709$	−21.2139	−0.796706	−0.398353	−	0.917232i	$-0.630418\pi$
$709$	−21.2139	−0.796706	−0.398353	+	0.917232i	$0.630418\pi$
$710$	0	0
$711$	20.6397	0.774048
$712$	0	0
$713$	0.777884	0.0291320
$714$	0	0
$715$	6.27145	0.234539
$716$	0	0
$717$	70.9142	2.64834
$718$	0	0
$719$	28.1106	1.04835	0.524174	−	0.851611i	$-0.324374\pi$
$719$	28.1106	1.04835	0.524174	+	0.851611i	$0.324374\pi$
$720$	0	0
$721$	−60.3650	−2.24811
$722$	0	0
$723$	−18.9984	−0.706558
$724$	0	0
$725$	−3.37480	−0.125337
$726$	0	0
$727$	39.0330	1.44765	0.723826	−	0.689982i	$-0.242382\pi$
$727$	39.0330	1.44765	0.723826	+	0.689982i	$0.242382\pi$
$728$	0	0
$729$	−42.6921	−1.58119
$730$	0	0
$731$	−37.4992	−1.38696
$732$	0	0
$733$	−47.1373	−1.74105	−0.870527	−	0.492120i	$-0.836222\pi$
$733$	−47.1373	−1.74105	−0.870527	+	0.492120i	$0.836222\pi$
$734$	0	0
$735$	37.9773	1.40082
$736$	0	0
$737$	80.2488	2.95600
$738$	0	0
$739$	−24.5265	−0.902223	−0.451111	−	0.892468i	$-0.648972\pi$
$739$	−24.5265	−0.902223	−0.451111	+	0.892468i	$0.648972\pi$
$740$	0	0
$741$	−15.0977	−0.554629
$742$	0	0
$743$	7.34651	0.269517	0.134759	−	0.990878i	$-0.456974\pi$
$743$	7.34651	0.269517	0.134759	+	0.990878i	$0.456974\pi$
$744$	0	0
$745$	19.7907	0.725077
$746$	0	0
$747$	−16.1244	−0.589961
$748$	0	0
$749$	56.5667	2.06690
$750$	0	0
$751$	−11.2359	−0.410005	−0.205002	−	0.978761i	$-0.565720\pi$
$751$	−11.2359	−0.410005	−0.205002	+	0.978761i	$0.565720\pi$
$752$	0	0
$753$	−66.9224	−2.43879
$754$	0	0
$755$	−4.06220	−0.147839
$756$	0	0
$757$	−49.1794	−1.78745	−0.893727	−	0.448611i	$-0.851919\pi$
$757$	−49.1794	−1.78745	−0.893727	+	0.448611i	$0.851919\pi$
$758$	0	0
$759$	13.7907	0.500572
$760$	0	0
$761$	43.2478	1.56773	0.783867	−	0.620929i	$-0.213245\pi$
$761$	43.2478	1.56773	0.783867	+	0.620929i	$0.213245\pi$
$762$	0	0
$763$	48.7131	1.76353
$764$	0	0
$765$	−19.7907	−0.715536
$766$	0	0
$767$	11.4571	0.413692
$768$	0	0
$769$	−39.6638	−1.43031	−0.715156	−	0.698964i	$-0.753645\pi$
$769$	−39.6638	−1.43031	−0.715156	+	0.698964i	$0.753645\pi$
$770$	0	0
$771$	1.19383	0.0429948
$772$	0	0
$773$	2.18097	0.0784440	0.0392220	−	0.999231i	$-0.487512\pi$
$773$	2.18097	0.0784440	0.0392220	+	0.999231i	$0.487512\pi$
$774$	0	0
$775$	0.777884	0.0279424
$776$	0	0
$777$	−71.8869	−2.57893
$778$	0	0
$779$	39.1033	1.40102
$780$	0	0
$781$	−6.73578	−0.241025
$782$	0	0
$783$	−11.0839	−0.396106
$784$	0	0
$785$	−4.62520	−0.165080
$786$	0	0
$787$	4.76407	0.169821	0.0849103	−	0.996389i	$-0.472940\pi$
$787$	4.76407	0.169821	0.0849103	+	0.996389i	$0.472940\pi$
$788$	0	0
$789$	64.1784	2.28481
$790$	0	0
$791$	−27.5815	−0.980685
$792$	0	0
$793$	−13.3830	−0.475244
$794$	0	0
$795$	16.1244	0.571873
$796$	0	0
$797$	−39.7204	−1.40697	−0.703484	−	0.710711i	$-0.748373\pi$
$797$	−39.7204	−1.40697	−0.703484	+	0.710711i	$0.748373\pi$
$798$	0	0
$799$	−30.2067	−1.06864
$800$	0	0
$801$	37.7059	1.33227
$802$	0	0
$803$	−22.8062	−0.804812
$804$	0	0
$805$	4.59692	0.162020
$806$	0	0
$807$	−43.7059	−1.53852
$808$	0	0
$809$	46.8941	1.64871	0.824354	−	0.566074i	$-0.191538\pi$
$809$	46.8941	1.64871	0.824354	+	0.566074i	$0.191538\pi$
$810$	0	0
$811$	−36.8319	−1.29334	−0.646671	−	0.762769i	$-0.723839\pi$
$811$	−36.8319	−1.29334	−0.646671	+	0.762769i	$0.723839\pi$
$812$	0	0
$813$	−68.8237	−2.41375
$814$	0	0
$815$	12.4159	0.434912
$816$	0	0
$817$	−36.7753	−1.28661
$818$	0	0
$819$	23.7197	0.828834
$820$	0	0
$821$	18.6107	0.649519	0.324759	−	0.945797i	$-0.394717\pi$
$821$	18.6107	0.649519	0.324759	+	0.945797i	$0.394717\pi$
$822$	0	0
$823$	9.31823	0.324813	0.162407	−	0.986724i	$-0.448074\pi$
$823$	9.31823	0.324813	0.162407	+	0.986724i	$0.448074\pi$
$824$	0	0
$825$	13.7907	0.480132
$826$	0	0
$827$	53.0129	1.84344	0.921719	−	0.387858i	$-0.126785\pi$
$827$	53.0129	1.84344	0.921719	+	0.387858i	$0.126785\pi$
$828$	0	0
$829$	25.2761	0.877876	0.438938	−	0.898517i	$-0.355355\pi$
$829$	25.2761	0.877876	0.438938	+	0.898517i	$0.355355\pi$
$830$	0	0
$831$	−7.76246	−0.269277
$832$	0	0
$833$	−66.2406	−2.29510
$834$	0	0
$835$	−12.8885	−0.446024
$836$	0	0
$837$	2.55481	0.0883073
$838$	0	0
$839$	−11.6946	−0.403744	−0.201872	−	0.979412i	$-0.564702\pi$
$839$	−11.6946	−0.403744	−0.201872	+	0.979412i	$0.564702\pi$
$840$	0	0
$841$	−17.6107	−0.607267
$842$	0	0
$843$	11.4571	0.394603
$844$	0	0
$845$	−11.5064	−0.395833
$846$	0	0
$847$	70.4875	2.42198
$848$	0	0
$849$	82.1501	2.81938
$850$	0	0
$851$	−5.81903	−0.199474
$852$	0	0
$853$	−38.6371	−1.32291	−0.661455	−	0.749985i	$-0.730061\pi$
$853$	−38.6371	−1.32291	−0.661455	+	0.749985i	$0.730061\pi$
$854$	0	0
$855$	−19.4087	−0.663764
$856$	0	0
$857$	26.2488	0.896642	0.448321	−	0.893873i	$-0.352022\pi$
$857$	26.2488	0.896642	0.448321	+	0.893873i	$0.352022\pi$
$858$	0	0
$859$	37.5558	1.28139	0.640693	−	0.767797i	$-0.278647\pi$
$859$	37.5558	1.28139	0.640693	+	0.767797i	$0.278647\pi$
$860$	0	0
$861$	−105.086	−3.58133
$862$	0	0
$863$	21.9855	0.748396	0.374198	−	0.927349i	$-0.377918\pi$
$863$	21.9855	0.748396	0.374198	+	0.927349i	$0.377918\pi$
$864$	0	0
$865$	10.2432	0.348278
$866$	0	0
$867$	13.3610	0.453763
$868$	0	0
$869$	25.0858	0.850978
$870$	0	0
$871$	19.1115	0.647570
$872$	0	0
$873$	−76.2607	−2.58103
$874$	0	0
$875$	4.59692	0.155404
$876$	0	0
$877$	36.5064	1.23273	0.616367	−	0.787459i	$-0.288604\pi$
$877$	36.5064	1.23273	0.616367	+	0.787459i	$0.288604\pi$
$878$	0	0
$879$	16.1244	0.543862
$880$	0	0
$881$	−17.4571	−0.588145	−0.294072	−	0.955783i	$-0.595011\pi$
$881$	−17.4571	−0.588145	−0.294072	+	0.955783i	$0.595011\pi$
$882$	0	0
$883$	22.9444	0.772140	0.386070	−	0.922470i	$-0.373832\pi$
$883$	22.9444	0.772140	0.386070	+	0.922470i	$0.373832\pi$
$884$	0	0
$885$	25.1938	0.846881
$886$	0	0
$887$	11.7223	0.393595	0.196798	−	0.980444i	$-0.436946\pi$
$887$	11.7223	0.393595	0.196798	+	0.980444i	$0.436946\pi$
$888$	0	0
$889$	10.4040	0.348940
$890$	0	0
$891$	−19.7059	−0.660172
$892$	0	0
$893$	−29.6236	−0.991316
$894$	0	0
$895$	−13.1938	−0.441021
$896$	0	0
$897$	3.28432	0.109660
$898$	0	0
$899$	−2.62520	−0.0875554
$900$	0	0
$901$	−28.1244	−0.936960
$902$	0	0
$903$	98.8300	3.28886
$904$	0	0
$905$	−15.7907	−0.524902
$906$	0	0
$907$	−1.91770	−0.0636763	−0.0318381	−	0.999493i	$-0.510136\pi$
$907$	−1.91770	−0.0636763	−0.0318381	+	0.999493i	$0.510136\pi$
$908$	0	0
$909$	−14.2488	−0.472603
$910$	0	0
$911$	56.8319	1.88292	0.941462	−	0.337118i	$-0.109452\pi$
$911$	56.8319	1.88292	0.941462	+	0.337118i	$0.109452\pi$
$912$	0	0
$913$	−19.5979	−0.648595
$914$	0	0
$915$	−29.4288	−0.972886
$916$	0	0
$917$	13.4716	0.444870
$918$	0	0
$919$	26.0257	0.858509	0.429255	−	0.903183i	$-0.358776\pi$
$919$	26.0257	0.858509	0.429255	+	0.903183i	$0.358776\pi$
$920$	0	0
$921$	−22.9733	−0.756997
$922$	0	0
$923$	−1.60415	−0.0528013
$924$	0	0
$925$	−5.81903	−0.191329
$926$	0	0
$927$	−55.4433	−1.82100
$928$	0	0
$929$	−17.1115	−0.561411	−0.280706	−	0.959794i	$-0.590568\pi$
$929$	−17.1115	−0.561411	−0.280706	+	0.959794i	$0.590568\pi$
$930$	0	0
$931$	−64.9619	−2.12904
$932$	0	0
$933$	20.5265	0.672008
$934$	0	0
$935$	−24.0540	−0.786650
$936$	0	0
$937$	16.3738	0.534910	0.267455	−	0.963570i	$-0.413817\pi$
$937$	16.3738	0.534910	0.267455	+	0.963570i	$0.413817\pi$
$938$	0	0
$939$	−49.0047	−1.59921
$940$	0	0
$941$	16.4097	0.534940	0.267470	−	0.963566i	$-0.413812\pi$
$941$	16.4097	0.534940	0.267470	+	0.963566i	$0.413812\pi$
$942$	0	0
$943$	−8.50643	−0.277008
$944$	0	0
$945$	15.0977	0.491129
$946$	0	0
$947$	−21.3886	−0.695037	−0.347518	−	0.937673i	$-0.612976\pi$
$947$	−21.3886	−0.695037	−0.347518	+	0.937673i	$0.612976\pi$
$948$	0	0
$949$	−5.43137	−0.176310
$950$	0	0
$951$	45.3100	1.46928
$952$	0	0
$953$	−16.7276	−0.541860	−0.270930	−	0.962599i	$-0.587331\pi$
$953$	−16.7276	−0.541860	−0.270930	+	0.962599i	$0.587331\pi$
$954$	0	0
$955$	−16.1244	−0.521773
$956$	0	0
$957$	−46.5410	−1.50446
$958$	0	0
$959$	−90.9764	−2.93778
$960$	0	0
$961$	−30.3949	−0.980481
$962$	0	0
$963$	51.9547	1.67422
$964$	0	0
$965$	−17.9434	−0.577619
$966$	0	0
$967$	−45.2617	−1.45552	−0.727758	−	0.685834i	$-0.759438\pi$
$967$	−45.2617	−1.45552	−0.727758	+	0.685834i	$0.759438\pi$
$968$	0	0
$969$	57.9070	1.86024
$970$	0	0
$971$	45.2560	1.45234	0.726168	−	0.687518i	$-0.241300\pi$
$971$	45.2560	1.45234	0.726168	+	0.687518i	$0.241300\pi$
$972$	0	0
$973$	28.7917	0.923019
$974$	0	0
$975$	3.28432	0.105182
$976$	0	0
$977$	−13.3465	−0.426993	−0.213496	−	0.976944i	$-0.568485\pi$
$977$	−13.3465	−0.426993	−0.213496	+	0.976944i	$0.568485\pi$
$978$	0	0
$979$	45.8284	1.46468
$980$	0	0
$981$	44.7414	1.42848
$982$	0	0
$983$	−5.13163	−0.163674	−0.0818368	−	0.996646i	$-0.526079\pi$
$983$	−5.13163	−0.163674	−0.0818368	+	0.996646i	$0.526079\pi$
$984$	0	0
$985$	−5.88123	−0.187392
$986$	0	0
$987$	79.6104	2.53403
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	−17.7989	−0.565402	−0.282701	−	0.959208i	$-0.591230\pi$
$991$	−17.7989	−0.565402	−0.282701	+	0.959208i	$0.591230\pi$
$992$	0	0
$993$	51.4684	1.63330
$994$	0	0
$995$	−6.56863	−0.208240
$996$	0	0
$997$	−40.3877	−1.27909	−0.639545	−	0.768754i	$-0.720877\pi$
$997$	−40.3877	−1.27909	−0.639545	+	0.768754i	$0.720877\pi$
$998$	0	0
$999$	−19.1115	−0.604662

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7360.2.a.ce.1.3		3
4.3	odd	2		7360.2.a.bz.1.1		3
8.3	odd	2		230.2.a.d.1.3	✓	3
8.5	even	2		1840.2.a.r.1.1		3
24.11	even	2		2070.2.a.z.1.1		3
40.3	even	4		1150.2.b.j.599.3		6
40.19	odd	2		1150.2.a.q.1.1		3
40.27	even	4		1150.2.b.j.599.4		6
40.29	even	2		9200.2.a.cf.1.3		3
184.91	even	2		5290.2.a.r.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.d.1.3	✓	3	8.3	odd	2
1150.2.a.q.1.1		3	40.19	odd	2
1150.2.b.j.599.3		6	40.3	even	4
1150.2.b.j.599.4		6	40.27	even	4
1840.2.a.r.1.1		3	8.5	even	2
2070.2.a.z.1.1		3	24.11	even	2
5290.2.a.r.1.3		3	184.91	even	2
7360.2.a.bz.1.1		3	4.3	odd	2
7360.2.a.ce.1.3		3	1.1	even	1	trivial
9200.2.a.cf.1.3		3	40.29	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:	\( N \)	\(=\)	\( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7360.a (trivial)

\(f(q)\)	\(=\)	\(q+2.68740 q^{3} +1.00000 q^{5} +4.59692 q^{7} +4.22212 q^{9} +O(q^{10})\)	\(q+2.68740 q^{3} +1.00000 q^{5} +4.59692 q^{7} +4.22212 q^{9} +5.13163 q^{11} +1.22212 q^{13} +2.68740 q^{15} -4.68740 q^{17} -4.59692 q^{19} +12.3537 q^{21} +1.00000 q^{23} +1.00000 q^{25} +3.28432 q^{27} -3.37480 q^{29} +0.777884 q^{31} +13.7907 q^{33} +4.59692 q^{35} -5.81903 q^{37} +3.28432 q^{39} -8.50643 q^{41} +8.00000 q^{43} +4.22212 q^{45} +6.44423 q^{47} +14.1316 q^{49} -12.5969 q^{51} +6.00000 q^{53} +5.13163 q^{55} -12.3537 q^{57} +9.37480 q^{59} -10.9507 q^{61} +19.4087 q^{63} +1.22212 q^{65} +15.6381 q^{67} +2.68740 q^{69} -1.31260 q^{71} -4.44423 q^{73} +2.68740 q^{75} +23.5897 q^{77} +4.88847 q^{79} -3.84008 q^{81} -3.81903 q^{83} -4.68740 q^{85} -9.06943 q^{87} +8.93057 q^{89} +5.61797 q^{91} +2.09048 q^{93} -4.59692 q^{95} -18.0622 q^{97} +21.6663 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 3 q^{5} - 3 q^{7} + 10 q^{9}+O(q^{10}) \) 3 * q + q^3 + 3 * q^5 - 3 * q^7 + 10 * q^9	\( 3 q + q^{3} + 3 q^{5} - 3 q^{7} + 10 q^{9} + 3 q^{11} + q^{13} + q^{15} - 7 q^{17} + 3 q^{19} + 22 q^{21} + 3 q^{23} + 3 q^{25} - 14 q^{27} + 4 q^{29} + 5 q^{31} - 9 q^{33} - 3 q^{35} + 2 q^{37} - 14 q^{39} + q^{41} + 24 q^{43} + 10 q^{45} + 14 q^{47} + 30 q^{49} - 21 q^{51} + 18 q^{53} + 3 q^{55} - 22 q^{57} + 14 q^{59} - q^{61} - 8 q^{63} + q^{65} + 8 q^{67} + q^{69} - 11 q^{71} - 8 q^{73} + q^{75} + 24 q^{77} + 4 q^{79} + 7 q^{81} + 8 q^{83} - 7 q^{85} - 36 q^{87} + 18 q^{89} + q^{91} + 16 q^{93} + 3 q^{95} - 33 q^{97} + 57 q^{99}+O(q^{100}) \) 3 * q + q^3 + 3 * q^5 - 3 * q^7 + 10 * q^9 + 3 * q^11 + q^13 + q^15 - 7 * q^17 + 3 * q^19 + 22 * q^21 + 3 * q^23 + 3 * q^25 - 14 * q^27 + 4 * q^29 + 5 * q^31 - 9 * q^33 - 3 * q^35 + 2 * q^37 - 14 * q^39 + q^41 + 24 * q^43 + 10 * q^45 + 14 * q^47 + 30 * q^49 - 21 * q^51 + 18 * q^53 + 3 * q^55 - 22 * q^57 + 14 * q^59 - q^61 - 8 * q^63 + q^65 + 8 * q^67 + q^69 - 11 * q^71 - 8 * q^73 + q^75 + 24 * q^77 + 4 * q^79 + 7 * q^81 + 8 * q^83 - 7 * q^85 - 36 * q^87 + 18 * q^89 + q^91 + 16 * q^93 + 3 * q^95 - 33 * q^97 + 57 * q^99

Introduction

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