LMFDB - Embedded newform 765.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [765,2,Mod(1,765)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(765, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("765.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 765 = 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	765.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:	$6.10855575463$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	765.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{7} -1.73205 q^{8} -1.73205 q^{10} -4.73205 q^{11} -4.00000 q^{13} -4.73205 q^{14} -5.00000 q^{16} +1.00000 q^{17} -1.46410 q^{19} -1.00000 q^{20} -8.19615 q^{22} +8.19615 q^{23} +1.00000 q^{25} -6.92820 q^{26} -2.73205 q^{28} +3.46410 q^{29} +3.26795 q^{31} -5.19615 q^{32} +1.73205 q^{34} +2.73205 q^{35} -0.535898 q^{37} -2.53590 q^{38} +1.73205 q^{40} +3.46410 q^{41} -0.535898 q^{43} -4.73205 q^{44} +14.1962 q^{46} -12.9282 q^{47} +0.464102 q^{49} +1.73205 q^{50} -4.00000 q^{52} -6.00000 q^{53} +4.73205 q^{55} +4.73205 q^{56} +6.00000 q^{58} -2.53590 q^{59} -4.92820 q^{61} +5.66025 q^{62} +1.00000 q^{64} +4.00000 q^{65} -10.0000 q^{67} +1.00000 q^{68} +4.73205 q^{70} -11.6603 q^{71} +6.39230 q^{73} -0.928203 q^{74} -1.46410 q^{76} +12.9282 q^{77} +14.5885 q^{79} +5.00000 q^{80} +6.00000 q^{82} -8.53590 q^{83} -1.00000 q^{85} -0.928203 q^{86} +8.19615 q^{88} -4.39230 q^{89} +10.9282 q^{91} +8.19615 q^{92} -22.3923 q^{94} +1.46410 q^{95} -4.92820 q^{97} +0.803848 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{11} - 8 q^{13} - 6 q^{14} - 10 q^{16} + 2 q^{17} + 4 q^{19} - 2 q^{20} - 6 q^{22} + 6 q^{23} + 2 q^{25} - 2 q^{28} + 10 q^{31} + 2 q^{35} - 8 q^{37} - 12 q^{38}+ \cdots + 12 q^{98}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^11 - 8 * q^13 - 6 * q^14 - 10 * q^16 + 2 * q^17 + 4 * q^19 - 2 * q^20 - 6 * q^22 + 6 * q^23 + 2 * q^25 - 2 * q^28 + 10 * q^31 + 2 * q^35 - 8 * q^37 - 12 * q^38 - 8 * q^43 - 6 * q^44 + 18 * q^46 - 12 * q^47 - 6 * q^49 - 8 * q^52 - 12 * q^53 + 6 * q^55 + 6 * q^56 + 12 * q^58 - 12 * q^59 + 4 * q^61 - 6 * q^62 + 2 * q^64 + 8 * q^65 - 20 * q^67 + 2 * q^68 + 6 * q^70 - 6 * q^71 - 8 * q^73 + 12 * q^74 + 4 * q^76 + 12 * q^77 - 2 * q^79 + 10 * q^80 + 12 * q^82 - 24 * q^83 - 2 * q^85 + 12 * q^86 + 6 * q^88 + 12 * q^89 + 8 * q^91 + 6 * q^92 - 24 * q^94 - 4 * q^95 + 4 * q^97 + 12 * q^98

Coefficient data

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

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

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

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.73205	−1.03262	−0.516309	−	0.856402i	$-0.672694\pi$
$7$	−2.73205	−1.03262	−0.516309	+	0.856402i	$0.672694\pi$
$8$	−1.73205	−0.612372
$9$	0	0
$10$	−1.73205	−0.547723
$11$	−4.73205	−1.42677	−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205	−1.42677	−0.713384	+	0.700774i	$0.752838\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−4.73205	−1.26469
$15$	0	0
$16$	−5.00000	−1.25000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−1.46410	−0.335888	−0.167944	−	0.985797i	$-0.553713\pi$
$19$	−1.46410	−0.335888	−0.167944	+	0.985797i	$0.553713\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−8.19615	−1.74743
$23$	8.19615	1.70902	0.854508	−	0.519438i	$-0.173859\pi$
$23$	8.19615	1.70902	0.854508	+	0.519438i	$0.173859\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.92820	−1.35873
$27$	0	0
$28$	−2.73205	−0.516309
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	3.26795	0.586941	0.293471	−	0.955968i	$-0.405190\pi$
$31$	3.26795	0.586941	0.293471	+	0.955968i	$0.405190\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	1.73205	0.297044
$35$	2.73205	0.461801
$36$	0	0
$37$	−0.535898	−0.0881012	−0.0440506	−	0.999029i	$-0.514026\pi$
$37$	−0.535898	−0.0881012	−0.0440506	+	0.999029i	$0.514026\pi$
$38$	−2.53590	−0.411377
$39$	0	0
$40$	1.73205	0.273861
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	−0.535898	−0.0817237	−0.0408619	−	0.999165i	$-0.513010\pi$
$43$	−0.535898	−0.0817237	−0.0408619	+	0.999165i	$0.513010\pi$
$44$	−4.73205	−0.713384
$45$	0	0
$46$	14.1962	2.09311
$47$	−12.9282	−1.88577	−0.942886	−	0.333115i	$-0.891900\pi$
$47$	−12.9282	−1.88577	−0.942886	+	0.333115i	$0.891900\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	1.73205	0.244949
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	4.73205	0.638070
$56$	4.73205	0.632347
$57$	0	0
$58$	6.00000	0.787839
$59$	−2.53590	−0.330146	−0.165073	−	0.986281i	$-0.552786\pi$
$59$	−2.53590	−0.330146	−0.165073	+	0.986281i	$0.552786\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	5.66025	0.718853
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	4.73205	0.565588
$71$	−11.6603	−1.38382	−0.691909	−	0.721985i	$-0.743230\pi$
$71$	−11.6603	−1.38382	−0.691909	+	0.721985i	$0.743230\pi$
$72$	0	0
$73$	6.39230	0.748163	0.374081	−	0.927396i	$-0.377958\pi$
$73$	6.39230	0.748163	0.374081	+	0.927396i	$0.377958\pi$
$74$	−0.928203	−0.107901
$75$	0	0
$76$	−1.46410	−0.167944
$77$	12.9282	1.47331
$78$	0	0
$79$	14.5885	1.64133	0.820665	−	0.571410i	$-0.193603\pi$
$79$	14.5885	1.64133	0.820665	+	0.571410i	$0.193603\pi$
$80$	5.00000	0.559017
$81$	0	0
$82$	6.00000	0.662589
$83$	−8.53590	−0.936937	−0.468468	−	0.883480i	$-0.655194\pi$
$83$	−8.53590	−0.936937	−0.468468	+	0.883480i	$0.655194\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	−0.928203	−0.100091
$87$	0	0
$88$	8.19615	0.873713
$89$	−4.39230	−0.465583	−0.232792	−	0.972527i	$-0.574786\pi$
$89$	−4.39230	−0.465583	−0.232792	+	0.972527i	$0.574786\pi$
$90$	0	0
$91$	10.9282	1.14559
$92$	8.19615	0.854508
$93$	0	0
$94$	−22.3923	−2.30959
$95$	1.46410	0.150214
$96$	0	0
$97$	−4.92820	−0.500383	−0.250192	−	0.968196i	$-0.580494\pi$
$97$	−4.92820	−0.500383	−0.250192	+	0.968196i	$0.580494\pi$
$98$	0.803848	0.0812009
$99$	0	0
$100$	1.00000	0.100000
$101$	9.46410	0.941713	0.470857	−	0.882210i	$-0.343945\pi$
$101$	9.46410	0.941713	0.470857	+	0.882210i	$0.343945\pi$
$102$	0	0
$103$	8.92820	0.879722	0.439861	−	0.898066i	$-0.355028\pi$
$103$	8.92820	0.879722	0.439861	+	0.898066i	$0.355028\pi$
$104$	6.92820	0.679366
$105$	0	0
$106$	−10.3923	−1.00939
$107$	−17.6603	−1.70728	−0.853641	−	0.520862i	$-0.825610\pi$
$107$	−17.6603	−1.70728	−0.853641	+	0.520862i	$0.825610\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	8.19615	0.781472
$111$	0	0
$112$	13.6603	1.29077
$113$	17.3205	1.62938	0.814688	−	0.579899i	$-0.196908\pi$
$113$	17.3205	1.62938	0.814688	+	0.579899i	$0.196908\pi$
$114$	0	0
$115$	−8.19615	−0.764295
$116$	3.46410	0.321634
$117$	0	0
$118$	−4.39230	−0.404344
$119$	−2.73205	−0.250447
$120$	0	0
$121$	11.3923	1.03566
$122$	−8.53590	−0.772804
$123$	0	0
$124$	3.26795	0.293471
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.3923	−1.27711	−0.638555	−	0.769576i	$-0.720468\pi$
$127$	−14.3923	−1.27711	−0.638555	+	0.769576i	$0.720468\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	6.92820	0.607644
$131$	2.19615	0.191879	0.0959394	−	0.995387i	$-0.469415\pi$
$131$	2.19615	0.191879	0.0959394	+	0.995387i	$0.469415\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−17.3205	−1.49626
$135$	0	0
$136$	−1.73205	−0.148522
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−3.66025	−0.310459	−0.155229	−	0.987878i	$-0.549612\pi$
$139$	−3.66025	−0.310459	−0.155229	+	0.987878i	$0.549612\pi$
$140$	2.73205	0.230900
$141$	0	0
$142$	−20.1962	−1.69482
$143$	18.9282	1.58286
$144$	0	0
$145$	−3.46410	−0.287678
$146$	11.0718	0.916308
$147$	0	0
$148$	−0.535898	−0.0440506
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−1.46410	−0.119147	−0.0595734	−	0.998224i	$-0.518974\pi$
$151$	−1.46410	−0.119147	−0.0595734	+	0.998224i	$0.518974\pi$
$152$	2.53590	0.205689
$153$	0	0
$154$	22.3923	1.80442
$155$	−3.26795	−0.262488
$156$	0	0
$157$	8.92820	0.712548	0.356274	−	0.934381i	$-0.384047\pi$
$157$	8.92820	0.712548	0.356274	+	0.934381i	$0.384047\pi$
$158$	25.2679	2.01021
$159$	0	0
$160$	5.19615	0.410792
$161$	−22.3923	−1.76476
$162$	0	0
$163$	−0.196152	−0.0153638	−0.00768192	−	0.999970i	$-0.502445\pi$
$163$	−0.196152	−0.0153638	−0.00768192	+	0.999970i	$0.502445\pi$
$164$	3.46410	0.270501
$165$	0	0
$166$	−14.7846	−1.14751
$167$	−12.5885	−0.974124	−0.487062	−	0.873367i	$-0.661931\pi$
$167$	−12.5885	−0.974124	−0.487062	+	0.873367i	$0.661931\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−1.73205	−0.132842
$171$	0	0
$172$	−0.535898	−0.0408619
$173$	−3.46410	−0.263371	−0.131685	−	0.991292i	$-0.542039\pi$
$173$	−3.46410	−0.263371	−0.131685	+	0.991292i	$0.542039\pi$
$174$	0	0
$175$	−2.73205	−0.206524
$176$	23.6603	1.78346
$177$	0	0
$178$	−7.60770	−0.570221
$179$	11.3205	0.846135	0.423067	−	0.906098i	$-0.360953\pi$
$179$	11.3205	0.846135	0.423067	+	0.906098i	$0.360953\pi$
$180$	0	0
$181$	−2.39230	−0.177819	−0.0889093	−	0.996040i	$-0.528338\pi$
$181$	−2.39230	−0.177819	−0.0889093	+	0.996040i	$0.528338\pi$
$182$	18.9282	1.40305
$183$	0	0
$184$	−14.1962	−1.04655
$185$	0.535898	0.0394000
$186$	0	0
$187$	−4.73205	−0.346042
$188$	−12.9282	−0.942886
$189$	0	0
$190$	2.53590	0.183973
$191$	−1.85641	−0.134325	−0.0671624	−	0.997742i	$-0.521395\pi$
$191$	−1.85641	−0.134325	−0.0671624	+	0.997742i	$0.521395\pi$
$192$	0	0
$193$	16.5359	1.19028	0.595140	−	0.803622i	$-0.297097\pi$
$193$	16.5359	1.19028	0.595140	+	0.803622i	$0.297097\pi$
$194$	−8.53590	−0.612842
$195$	0	0
$196$	0.464102	0.0331501
$197$	−17.3205	−1.23404	−0.617018	−	0.786949i	$-0.711659\pi$
$197$	−17.3205	−1.23404	−0.617018	+	0.786949i	$0.711659\pi$
$198$	0	0
$199$	10.1962	0.722786	0.361393	−	0.932414i	$-0.382301\pi$
$199$	10.1962	0.722786	0.361393	+	0.932414i	$0.382301\pi$
$200$	−1.73205	−0.122474
$201$	0	0
$202$	16.3923	1.15336
$203$	−9.46410	−0.664250
$204$	0	0
$205$	−3.46410	−0.241943
$206$	15.4641	1.07744
$207$	0	0
$208$	20.0000	1.38675
$209$	6.92820	0.479234
$210$	0	0
$211$	10.1962	0.701932	0.350966	−	0.936388i	$-0.385853\pi$
$211$	10.1962	0.701932	0.350966	+	0.936388i	$0.385853\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−30.5885	−2.09098
$215$	0.535898	0.0365480
$216$	0	0
$217$	−8.92820	−0.606086
$218$	−17.3205	−1.17309
$219$	0	0
$220$	4.73205	0.319035
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−26.3923	−1.76736	−0.883680	−	0.468092i	$-0.844942\pi$
$223$	−26.3923	−1.76736	−0.883680	+	0.468092i	$0.844942\pi$
$224$	14.1962	0.948520
$225$	0	0
$226$	30.0000	1.99557
$227$	−22.7321	−1.50878	−0.754390	−	0.656427i	$-0.772067\pi$
$227$	−22.7321	−1.50878	−0.754390	+	0.656427i	$0.772067\pi$
$228$	0	0
$229$	−8.39230	−0.554579	−0.277290	−	0.960786i	$-0.589436\pi$
$229$	−8.39230	−0.554579	−0.277290	+	0.960786i	$0.589436\pi$
$230$	−14.1962	−0.936067
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	12.9282	0.843343
$236$	−2.53590	−0.165073
$237$	0	0
$238$	−4.73205	−0.306733
$239$	20.7846	1.34444	0.672222	−	0.740349i	$-0.265340\pi$
$239$	20.7846	1.34444	0.672222	+	0.740349i	$0.265340\pi$
$240$	0	0
$241$	−5.60770	−0.361223	−0.180612	−	0.983554i	$-0.557808\pi$
$241$	−5.60770	−0.361223	−0.180612	+	0.983554i	$0.557808\pi$
$242$	19.7321	1.26842
$243$	0	0
$244$	−4.92820	−0.315496
$245$	−0.464102	−0.0296504
$246$	0	0
$247$	5.85641	0.372634
$248$	−5.66025	−0.359426
$249$	0	0
$250$	−1.73205	−0.109545
$251$	−6.92820	−0.437304	−0.218652	−	0.975803i	$-0.570166\pi$
$251$	−6.92820	−0.437304	−0.218652	+	0.975803i	$0.570166\pi$
$252$	0	0
$253$	−38.7846	−2.43837
$254$	−24.9282	−1.56413
$255$	0	0
$256$	19.0000	1.18750
$257$	6.92820	0.432169	0.216085	−	0.976375i	$-0.430671\pi$
$257$	6.92820	0.432169	0.216085	+	0.976375i	$0.430671\pi$
$258$	0	0
$259$	1.46410	0.0909748
$260$	4.00000	0.248069
$261$	0	0
$262$	3.80385	0.235002
$263$	1.60770	0.0991347	0.0495674	−	0.998771i	$-0.484216\pi$
$263$	1.60770	0.0991347	0.0495674	+	0.998771i	$0.484216\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	6.92820	0.424795
$267$	0	0
$268$	−10.0000	−0.610847
$269$	−0.928203	−0.0565935	−0.0282968	−	0.999600i	$-0.509008\pi$
$269$	−0.928203	−0.0565935	−0.0282968	+	0.999600i	$0.509008\pi$
$270$	0	0
$271$	2.92820	0.177876	0.0889378	−	0.996037i	$-0.471653\pi$
$271$	2.92820	0.177876	0.0889378	+	0.996037i	$0.471653\pi$
$272$	−5.00000	−0.303170
$273$	0	0
$274$	0	0
$275$	−4.73205	−0.285353
$276$	0	0
$277$	20.9282	1.25745	0.628727	−	0.777626i	$-0.283576\pi$
$277$	20.9282	1.25745	0.628727	+	0.777626i	$0.283576\pi$
$278$	−6.33975	−0.380233
$279$	0	0
$280$	−4.73205	−0.282794
$281$	12.9282	0.771232	0.385616	−	0.922659i	$-0.373989\pi$
$281$	12.9282	0.771232	0.385616	+	0.922659i	$0.373989\pi$
$282$	0	0
$283$	−5.26795	−0.313147	−0.156574	−	0.987666i	$-0.550045\pi$
$283$	−5.26795	−0.313147	−0.156574	+	0.987666i	$0.550045\pi$
$284$	−11.6603	−0.691909
$285$	0	0
$286$	32.7846	1.93859
$287$	−9.46410	−0.558648
$288$	0	0
$289$	1.00000	0.0588235
$290$	−6.00000	−0.352332
$291$	0	0
$292$	6.39230	0.374081
$293$	0.928203	0.0542262	0.0271131	−	0.999632i	$-0.491369\pi$
$293$	0.928203	0.0542262	0.0271131	+	0.999632i	$0.491369\pi$
$294$	0	0
$295$	2.53590	0.147646
$296$	0.928203	0.0539507
$297$	0	0
$298$	10.3923	0.602010
$299$	−32.7846	−1.89598
$300$	0	0
$301$	1.46410	0.0843894
$302$	−2.53590	−0.145925
$303$	0	0
$304$	7.32051	0.419860
$305$	4.92820	0.282188
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	12.9282	0.736653
$309$	0	0
$310$	−5.66025	−0.321481
$311$	16.0526	0.910257	0.455129	−	0.890426i	$-0.349593\pi$
$311$	16.0526	0.910257	0.455129	+	0.890426i	$0.349593\pi$
$312$	0	0
$313$	−26.3923	−1.49178	−0.745891	−	0.666068i	$-0.767976\pi$
$313$	−26.3923	−1.49178	−0.745891	+	0.666068i	$0.767976\pi$
$314$	15.4641	0.872690
$315$	0	0
$316$	14.5885	0.820665
$317$	24.9282	1.40011	0.700054	−	0.714090i	$-0.253159\pi$
$317$	24.9282	1.40011	0.700054	+	0.714090i	$0.253159\pi$
$318$	0	0
$319$	−16.3923	−0.917793
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−38.7846	−2.16138
$323$	−1.46410	−0.0814648
$324$	0	0
$325$	−4.00000	−0.221880
$326$	−0.339746	−0.0188168
$327$	0	0
$328$	−6.00000	−0.331295
$329$	35.3205	1.94728
$330$	0	0
$331$	−6.53590	−0.359245	−0.179623	−	0.983736i	$-0.557488\pi$
$331$	−6.53590	−0.359245	−0.179623	+	0.983736i	$0.557488\pi$
$332$	−8.53590	−0.468468
$333$	0	0
$334$	−21.8038	−1.19305
$335$	10.0000	0.546358
$336$	0	0
$337$	−6.78461	−0.369581	−0.184791	−	0.982778i	$-0.559161\pi$
$337$	−6.78461	−0.369581	−0.184791	+	0.982778i	$0.559161\pi$
$338$	5.19615	0.282633
$339$	0	0
$340$	−1.00000	−0.0542326
$341$	−15.4641	−0.837428
$342$	0	0
$343$	17.8564	0.964155
$344$	0.928203	0.0500454
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−3.80385	−0.204201	−0.102101	−	0.994774i	$-0.532556\pi$
$347$	−3.80385	−0.204201	−0.102101	+	0.994774i	$0.532556\pi$
$348$	0	0
$349$	10.7846	0.577287	0.288643	−	0.957437i	$-0.406796\pi$
$349$	10.7846	0.577287	0.288643	+	0.957437i	$0.406796\pi$
$350$	−4.73205	−0.252939
$351$	0	0
$352$	24.5885	1.31057
$353$	−26.7846	−1.42560	−0.712800	−	0.701367i	$-0.752573\pi$
$353$	−26.7846	−1.42560	−0.712800	+	0.701367i	$0.752573\pi$
$354$	0	0
$355$	11.6603	0.618862
$356$	−4.39230	−0.232792
$357$	0	0
$358$	19.6077	1.03630
$359$	21.4641	1.13283	0.566416	−	0.824119i	$-0.308330\pi$
$359$	21.4641	1.13283	0.566416	+	0.824119i	$0.308330\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	−4.14359	−0.217782
$363$	0	0
$364$	10.9282	0.572793
$365$	−6.39230	−0.334589
$366$	0	0
$367$	−7.80385	−0.407358	−0.203679	−	0.979038i	$-0.565290\pi$
$367$	−7.80385	−0.407358	−0.203679	+	0.979038i	$0.565290\pi$
$368$	−40.9808	−2.13627
$369$	0	0
$370$	0.928203	0.0482550
$371$	16.3923	0.851046
$372$	0	0
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	−8.19615	−0.423813
$375$	0	0
$376$	22.3923	1.15479
$377$	−13.8564	−0.713641
$378$	0	0
$379$	17.8038	0.914522	0.457261	−	0.889332i	$-0.348830\pi$
$379$	17.8038	0.914522	0.457261	+	0.889332i	$0.348830\pi$
$380$	1.46410	0.0751068
$381$	0	0
$382$	−3.21539	−0.164514
$383$	−15.4641	−0.790179	−0.395089	−	0.918643i	$-0.629286\pi$
$383$	−15.4641	−0.790179	−0.395089	+	0.918643i	$0.629286\pi$
$384$	0	0
$385$	−12.9282	−0.658882
$386$	28.6410	1.45779
$387$	0	0
$388$	−4.92820	−0.250192
$389$	4.39230	0.222699	0.111349	−	0.993781i	$-0.464483\pi$
$389$	4.39230	0.222699	0.111349	+	0.993781i	$0.464483\pi$
$390$	0	0
$391$	8.19615	0.414497
$392$	−0.803848	−0.0406004
$393$	0	0
$394$	−30.0000	−1.51138
$395$	−14.5885	−0.734025
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	17.6603	0.885229
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−12.9282	−0.645604	−0.322802	−	0.946467i	$-0.604625\pi$
$401$	−12.9282	−0.645604	−0.322802	+	0.946467i	$0.604625\pi$
$402$	0	0
$403$	−13.0718	−0.651153
$404$	9.46410	0.470857
$405$	0	0
$406$	−16.3923	−0.813536
$407$	2.53590	0.125700
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	8.92820	0.439861
$413$	6.92820	0.340915
$414$	0	0
$415$	8.53590	0.419011
$416$	20.7846	1.01905
$417$	0	0
$418$	12.0000	0.586939
$419$	−38.1962	−1.86600	−0.933002	−	0.359871i	$-0.882821\pi$
$419$	−38.1962	−1.86600	−0.933002	+	0.359871i	$0.882821\pi$
$420$	0	0
$421$	5.46410	0.266304	0.133152	−	0.991096i	$-0.457490\pi$
$421$	5.46410	0.266304	0.133152	+	0.991096i	$0.457490\pi$
$422$	17.6603	0.859688
$423$	0	0
$424$	10.3923	0.504695
$425$	1.00000	0.0485071
$426$	0	0
$427$	13.4641	0.651574
$428$	−17.6603	−0.853641
$429$	0	0
$430$	0.928203	0.0447619
$431$	−9.80385	−0.472235	−0.236117	−	0.971725i	$-0.575875\pi$
$431$	−9.80385	−0.472235	−0.236117	+	0.971725i	$0.575875\pi$
$432$	0	0
$433$	−17.8564	−0.858124	−0.429062	−	0.903275i	$-0.641156\pi$
$433$	−17.8564	−0.858124	−0.429062	+	0.903275i	$0.641156\pi$
$434$	−15.4641	−0.742301
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−12.0000	−0.574038
$438$	0	0
$439$	−20.0526	−0.957056	−0.478528	−	0.878072i	$-0.658830\pi$
$439$	−20.0526	−0.957056	−0.478528	+	0.878072i	$0.658830\pi$
$440$	−8.19615	−0.390736
$441$	0	0
$442$	−6.92820	−0.329541
$443$	12.9282	0.614237	0.307119	−	0.951671i	$-0.400635\pi$
$443$	12.9282	0.614237	0.307119	+	0.951671i	$0.400635\pi$
$444$	0	0
$445$	4.39230	0.208215
$446$	−45.7128	−2.16456
$447$	0	0
$448$	−2.73205	−0.129077
$449$	34.3923	1.62307	0.811537	−	0.584302i	$-0.198631\pi$
$449$	34.3923	1.62307	0.811537	+	0.584302i	$0.198631\pi$
$450$	0	0
$451$	−16.3923	−0.771883
$452$	17.3205	0.814688
$453$	0	0
$454$	−39.3731	−1.84787
$455$	−10.9282	−0.512322
$456$	0	0
$457$	−36.7846	−1.72071	−0.860356	−	0.509694i	$-0.829759\pi$
$457$	−36.7846	−1.72071	−0.860356	+	0.509694i	$0.829759\pi$
$458$	−14.5359	−0.679218
$459$	0	0
$460$	−8.19615	−0.382148
$461$	24.9282	1.16102	0.580511	−	0.814252i	$-0.302853\pi$
$461$	24.9282	1.16102	0.580511	+	0.814252i	$0.302853\pi$
$462$	0	0
$463$	−23.8564	−1.10870	−0.554351	−	0.832283i	$-0.687033\pi$
$463$	−23.8564	−1.10870	−0.554351	+	0.832283i	$0.687033\pi$
$464$	−17.3205	−0.804084
$465$	0	0
$466$	−10.3923	−0.481414
$467$	−1.60770	−0.0743953	−0.0371976	−	0.999308i	$-0.511843\pi$
$467$	−1.60770	−0.0743953	−0.0371976	+	0.999308i	$0.511843\pi$
$468$	0	0
$469$	27.3205	1.26154
$470$	22.3923	1.03288
$471$	0	0
$472$	4.39230	0.202172
$473$	2.53590	0.116601
$474$	0	0
$475$	−1.46410	−0.0671776
$476$	−2.73205	−0.125223
$477$	0	0
$478$	36.0000	1.64660
$479$	11.6603	0.532771	0.266385	−	0.963867i	$-0.414171\pi$
$479$	11.6603	0.532771	0.266385	+	0.963867i	$0.414171\pi$
$480$	0	0
$481$	2.14359	0.0977395
$482$	−9.71281	−0.442407
$483$	0	0
$484$	11.3923	0.517832
$485$	4.92820	0.223778
$486$	0	0
$487$	24.9808	1.13199	0.565993	−	0.824410i	$-0.308493\pi$
$487$	24.9808	1.13199	0.565993	+	0.824410i	$0.308493\pi$
$488$	8.53590	0.386402
$489$	0	0
$490$	−0.803848	−0.0363141
$491$	19.6077	0.884883	0.442441	−	0.896797i	$-0.354112\pi$
$491$	19.6077	0.884883	0.442441	+	0.896797i	$0.354112\pi$
$492$	0	0
$493$	3.46410	0.156015
$494$	10.1436	0.456382
$495$	0	0
$496$	−16.3397	−0.733676
$497$	31.8564	1.42896
$498$	0	0
$499$	−15.6603	−0.701049	−0.350525	−	0.936554i	$-0.613997\pi$
$499$	−15.6603	−0.701049	−0.350525	+	0.936554i	$0.613997\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−12.0000	−0.535586
$503$	−15.1244	−0.674362	−0.337181	−	0.941440i	$-0.609473\pi$
$503$	−15.1244	−0.674362	−0.337181	+	0.941440i	$0.609473\pi$
$504$	0	0
$505$	−9.46410	−0.421147
$506$	−67.1769	−2.98638
$507$	0	0
$508$	−14.3923	−0.638555
$509$	−19.8564	−0.880120	−0.440060	−	0.897968i	$-0.645043\pi$
$509$	−19.8564	−0.880120	−0.440060	+	0.897968i	$0.645043\pi$
$510$	0	0
$511$	−17.4641	−0.772566
$512$	8.66025	0.382733
$513$	0	0
$514$	12.0000	0.529297
$515$	−8.92820	−0.393424
$516$	0	0
$517$	61.1769	2.69056
$518$	2.53590	0.111421
$519$	0	0
$520$	−6.92820	−0.303822
$521$	−4.14359	−0.181534	−0.0907671	−	0.995872i	$-0.528932\pi$
$521$	−4.14359	−0.181534	−0.0907671	+	0.995872i	$0.528932\pi$
$522$	0	0
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	2.19615	0.0959394
$525$	0	0
$526$	2.78461	0.121415
$527$	3.26795	0.142354
$528$	0	0
$529$	44.1769	1.92074
$530$	10.3923	0.451413
$531$	0	0
$532$	4.00000	0.173422
$533$	−13.8564	−0.600188
$534$	0	0
$535$	17.6603	0.763519
$536$	17.3205	0.748132
$537$	0	0
$538$	−1.60770	−0.0693127
$539$	−2.19615	−0.0945950
$540$	0	0
$541$	39.1769	1.68435	0.842174	−	0.539207i	$-0.181276\pi$
$541$	39.1769	1.68435	0.842174	+	0.539207i	$0.181276\pi$
$542$	5.07180	0.217852
$543$	0	0
$544$	−5.19615	−0.222783
$545$	10.0000	0.428353
$546$	0	0
$547$	−39.9090	−1.70638	−0.853192	−	0.521597i	$-0.825337\pi$
$547$	−39.9090	−1.70638	−0.853192	+	0.521597i	$0.825337\pi$
$548$	0	0
$549$	0	0
$550$	−8.19615	−0.349485
$551$	−5.07180	−0.216066
$552$	0	0
$553$	−39.8564	−1.69487
$554$	36.2487	1.54006
$555$	0	0
$556$	−3.66025	−0.155229
$557$	6.92820	0.293557	0.146779	−	0.989169i	$-0.453109\pi$
$557$	6.92820	0.293557	0.146779	+	0.989169i	$0.453109\pi$
$558$	0	0
$559$	2.14359	0.0906643
$560$	−13.6603	−0.577251
$561$	0	0
$562$	22.3923	0.944562
$563$	−27.4641	−1.15747	−0.578737	−	0.815514i	$-0.696454\pi$
$563$	−27.4641	−1.15747	−0.578737	+	0.815514i	$0.696454\pi$
$564$	0	0
$565$	−17.3205	−0.728679
$566$	−9.12436	−0.383525
$567$	0	0
$568$	20.1962	0.847412
$569$	−40.6410	−1.70376	−0.851880	−	0.523737i	$-0.824537\pi$
$569$	−40.6410	−1.70376	−0.851880	+	0.523737i	$0.824537\pi$
$570$	0	0
$571$	−36.4449	−1.52517	−0.762585	−	0.646888i	$-0.776070\pi$
$571$	−36.4449	−1.52517	−0.762585	+	0.646888i	$0.776070\pi$
$572$	18.9282	0.791428
$573$	0	0
$574$	−16.3923	−0.684202
$575$	8.19615	0.341803
$576$	0	0
$577$	−38.6410	−1.60865	−0.804323	−	0.594192i	$-0.797472\pi$
$577$	−38.6410	−1.60865	−0.804323	+	0.594192i	$0.797472\pi$
$578$	1.73205	0.0720438
$579$	0	0
$580$	−3.46410	−0.143839
$581$	23.3205	0.967498
$582$	0	0
$583$	28.3923	1.17589
$584$	−11.0718	−0.458154
$585$	0	0
$586$	1.60770	0.0664133
$587$	−46.3923	−1.91482	−0.957408	−	0.288740i	$-0.906764\pi$
$587$	−46.3923	−1.91482	−0.957408	+	0.288740i	$0.906764\pi$
$588$	0	0
$589$	−4.78461	−0.197146
$590$	4.39230	0.180828
$591$	0	0
$592$	2.67949	0.110126
$593$	−19.8564	−0.815405	−0.407702	−	0.913115i	$-0.633670\pi$
$593$	−19.8564	−0.815405	−0.407702	+	0.913115i	$0.633670\pi$
$594$	0	0
$595$	2.73205	0.112003
$596$	6.00000	0.245770
$597$	0	0
$598$	−56.7846	−2.32210
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	8.24871	0.336472	0.168236	−	0.985747i	$-0.446193\pi$
$601$	8.24871	0.336472	0.168236	+	0.985747i	$0.446193\pi$
$602$	2.53590	0.103356
$603$	0	0
$604$	−1.46410	−0.0595734
$605$	−11.3923	−0.463163
$606$	0	0
$607$	−21.6603	−0.879163	−0.439581	−	0.898203i	$-0.644873\pi$
$607$	−21.6603	−0.879163	−0.439581	+	0.898203i	$0.644873\pi$
$608$	7.60770	0.308533
$609$	0	0
$610$	8.53590	0.345608
$611$	51.7128	2.09208
$612$	0	0
$613$	15.8564	0.640434	0.320217	−	0.947344i	$-0.396244\pi$
$613$	15.8564	0.640434	0.320217	+	0.947344i	$0.396244\pi$
$614$	−17.3205	−0.698999
$615$	0	0
$616$	−22.3923	−0.902212
$617$	27.4641	1.10566	0.552832	−	0.833293i	$-0.313547\pi$
$617$	27.4641	1.10566	0.552832	+	0.833293i	$0.313547\pi$
$618$	0	0
$619$	38.5885	1.55100	0.775501	−	0.631347i	$-0.217498\pi$
$619$	38.5885	1.55100	0.775501	+	0.631347i	$0.217498\pi$
$620$	−3.26795	−0.131244
$621$	0	0
$622$	27.8038	1.11483
$623$	12.0000	0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	−45.7128	−1.82705
$627$	0	0
$628$	8.92820	0.356274
$629$	−0.535898	−0.0213677
$630$	0	0
$631$	−32.3923	−1.28952	−0.644759	−	0.764386i	$-0.723042\pi$
$631$	−32.3923	−1.28952	−0.644759	+	0.764386i	$0.723042\pi$
$632$	−25.2679	−1.00511
$633$	0	0
$634$	43.1769	1.71477
$635$	14.3923	0.571141
$636$	0	0
$637$	−1.85641	−0.0735535
$638$	−28.3923	−1.12406
$639$	0	0
$640$	−12.1244	−0.479257
$641$	−31.1769	−1.23141	−0.615707	−	0.787975i	$-0.711130\pi$
$641$	−31.1769	−1.23141	−0.615707	+	0.787975i	$0.711130\pi$
$642$	0	0
$643$	−24.1962	−0.954203	−0.477102	−	0.878848i	$-0.658313\pi$
$643$	−24.1962	−0.954203	−0.477102	+	0.878848i	$0.658313\pi$
$644$	−22.3923	−0.882380
$645$	0	0
$646$	−2.53590	−0.0997736
$647$	38.7846	1.52478	0.762390	−	0.647118i	$-0.224026\pi$
$647$	38.7846	1.52478	0.762390	+	0.647118i	$0.224026\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	−6.92820	−0.271746
$651$	0	0
$652$	−0.196152	−0.00768192
$653$	25.6077	1.00211	0.501053	−	0.865416i	$-0.332946\pi$
$653$	25.6077	1.00211	0.501053	+	0.865416i	$0.332946\pi$
$654$	0	0
$655$	−2.19615	−0.0858108
$656$	−17.3205	−0.676252
$657$	0	0
$658$	61.1769	2.38492
$659$	32.7846	1.27711	0.638554	−	0.769577i	$-0.279533\pi$
$659$	32.7846	1.27711	0.638554	+	0.769577i	$0.279533\pi$
$660$	0	0
$661$	−8.14359	−0.316749	−0.158375	−	0.987379i	$-0.550625\pi$
$661$	−8.14359	−0.316749	−0.158375	+	0.987379i	$0.550625\pi$
$662$	−11.3205	−0.439984
$663$	0	0
$664$	14.7846	0.573754
$665$	−4.00000	−0.155113
$666$	0	0
$667$	28.3923	1.09935
$668$	−12.5885	−0.487062
$669$	0	0
$670$	17.3205	0.669150
$671$	23.3205	0.900278
$672$	0	0
$673$	23.4641	0.904475	0.452237	−	0.891898i	$-0.350626\pi$
$673$	23.4641	0.904475	0.452237	+	0.891898i	$0.350626\pi$
$674$	−11.7513	−0.452643
$675$	0	0
$676$	3.00000	0.115385
$677$	−2.78461	−0.107021	−0.0535106	−	0.998567i	$-0.517041\pi$
$677$	−2.78461	−0.107021	−0.0535106	+	0.998567i	$0.517041\pi$
$678$	0	0
$679$	13.4641	0.516705
$680$	1.73205	0.0664211
$681$	0	0
$682$	−26.7846	−1.02564
$683$	30.8372	1.17995	0.589976	−	0.807421i	$-0.299137\pi$
$683$	30.8372	1.17995	0.589976	+	0.807421i	$0.299137\pi$
$684$	0	0
$685$	0	0
$686$	30.9282	1.18084
$687$	0	0
$688$	2.67949	0.102155
$689$	24.0000	0.914327
$690$	0	0
$691$	−38.9808	−1.48290	−0.741449	−	0.671009i	$-0.765861\pi$
$691$	−38.9808	−1.48290	−0.741449	+	0.671009i	$0.765861\pi$
$692$	−3.46410	−0.131685
$693$	0	0
$694$	−6.58846	−0.250094
$695$	3.66025	0.138841
$696$	0	0
$697$	3.46410	0.131212
$698$	18.6795	0.707029
$699$	0	0
$700$	−2.73205	−0.103262
$701$	−11.3205	−0.427570	−0.213785	−	0.976881i	$-0.568579\pi$
$701$	−11.3205	−0.427570	−0.213785	+	0.976881i	$0.568579\pi$
$702$	0	0
$703$	0.784610	0.0295921
$704$	−4.73205	−0.178346
$705$	0	0
$706$	−46.3923	−1.74600
$707$	−25.8564	−0.972430
$708$	0	0
$709$	4.53590	0.170349	0.0851746	−	0.996366i	$-0.472855\pi$
$709$	4.53590	0.170349	0.0851746	+	0.996366i	$0.472855\pi$
$710$	20.1962	0.757948
$711$	0	0
$712$	7.60770	0.285110
$713$	26.7846	1.00309
$714$	0	0
$715$	−18.9282	−0.707875
$716$	11.3205	0.423067
$717$	0	0
$718$	37.1769	1.38743
$719$	5.41154	0.201816	0.100908	−	0.994896i	$-0.467825\pi$
$719$	5.41154	0.201816	0.100908	+	0.994896i	$0.467825\pi$
$720$	0	0
$721$	−24.3923	−0.908417
$722$	−29.1962	−1.08657
$723$	0	0
$724$	−2.39230	−0.0889093
$725$	3.46410	0.128654
$726$	0	0
$727$	0.143594	0.00532559	0.00266279	−	0.999996i	$-0.499152\pi$
$727$	0.143594	0.00532559	0.00266279	+	0.999996i	$0.499152\pi$
$728$	−18.9282	−0.701526
$729$	0	0
$730$	−11.0718	−0.409786
$731$	−0.535898	−0.0198209
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−13.5167	−0.498909
$735$	0	0
$736$	−42.5885	−1.56983
$737$	47.3205	1.74307
$738$	0	0
$739$	−11.6077	−0.426996	−0.213498	−	0.976944i	$-0.568486\pi$
$739$	−11.6077	−0.426996	−0.213498	+	0.976944i	$0.568486\pi$
$740$	0.535898	0.0197000
$741$	0	0
$742$	28.3923	1.04231
$743$	22.7321	0.833958	0.416979	−	0.908916i	$-0.363089\pi$
$743$	22.7321	0.833958	0.416979	+	0.908916i	$0.363089\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	34.6410	1.26830
$747$	0	0
$748$	−4.73205	−0.173021
$749$	48.2487	1.76297
$750$	0	0
$751$	43.6603	1.59319	0.796593	−	0.604516i	$-0.206634\pi$
$751$	43.6603	1.59319	0.796593	+	0.604516i	$0.206634\pi$
$752$	64.6410	2.35722
$753$	0	0
$754$	−24.0000	−0.874028
$755$	1.46410	0.0532841
$756$	0	0
$757$	30.6410	1.11367	0.556833	−	0.830624i	$-0.312016\pi$
$757$	30.6410	1.11367	0.556833	+	0.830624i	$0.312016\pi$
$758$	30.8372	1.12006
$759$	0	0
$760$	−2.53590	−0.0919867
$761$	28.3923	1.02922	0.514610	−	0.857424i	$-0.327937\pi$
$761$	28.3923	1.02922	0.514610	+	0.857424i	$0.327937\pi$
$762$	0	0
$763$	27.3205	0.989069
$764$	−1.85641	−0.0671624
$765$	0	0
$766$	−26.7846	−0.967767
$767$	10.1436	0.366264
$768$	0	0
$769$	36.3923	1.31234	0.656170	−	0.754613i	$-0.272175\pi$
$769$	36.3923	1.31234	0.656170	+	0.754613i	$0.272175\pi$
$770$	−22.3923	−0.806963
$771$	0	0
$772$	16.5359	0.595140
$773$	46.6410	1.67756	0.838780	−	0.544470i	$-0.183269\pi$
$773$	46.6410	1.67756	0.838780	+	0.544470i	$0.183269\pi$
$774$	0	0
$775$	3.26795	0.117388
$776$	8.53590	0.306421
$777$	0	0
$778$	7.60770	0.272749
$779$	−5.07180	−0.181716
$780$	0	0
$781$	55.1769	1.97439
$782$	14.1962	0.507653
$783$	0	0
$784$	−2.32051	−0.0828753
$785$	−8.92820	−0.318661
$786$	0	0
$787$	43.9090	1.56519	0.782593	−	0.622534i	$-0.213897\pi$
$787$	43.9090	1.56519	0.782593	+	0.622534i	$0.213897\pi$
$788$	−17.3205	−0.617018
$789$	0	0
$790$	−25.2679	−0.898993
$791$	−47.3205	−1.68252
$792$	0	0
$793$	19.7128	0.700023
$794$	24.2487	0.860555
$795$	0	0
$796$	10.1962	0.361393
$797$	24.9282	0.883002	0.441501	−	0.897261i	$-0.354446\pi$
$797$	24.9282	0.883002	0.441501	+	0.897261i	$0.354446\pi$
$798$	0	0
$799$	−12.9282	−0.457367
$800$	−5.19615	−0.183712
$801$	0	0
$802$	−22.3923	−0.790700
$803$	−30.2487	−1.06745
$804$	0	0
$805$	22.3923	0.789225
$806$	−22.6410	−0.797496
$807$	0	0
$808$	−16.3923	−0.576679
$809$	55.8564	1.96381	0.981903	−	0.189383i	$-0.0606487\pi$
$809$	55.8564	1.96381	0.981903	+	0.189383i	$0.0606487\pi$
$810$	0	0
$811$	44.8372	1.57445	0.787223	−	0.616668i	$-0.211518\pi$
$811$	44.8372	1.57445	0.787223	+	0.616668i	$0.211518\pi$
$812$	−9.46410	−0.332125
$813$	0	0
$814$	4.39230	0.153950
$815$	0.196152	0.00687092
$816$	0	0
$817$	0.784610	0.0274500
$818$	45.0333	1.57455
$819$	0	0
$820$	−3.46410	−0.120972
$821$	24.9282	0.870000	0.435000	−	0.900430i	$-0.356748\pi$
$821$	24.9282	0.870000	0.435000	+	0.900430i	$0.356748\pi$
$822$	0	0
$823$	−8.98076	−0.313050	−0.156525	−	0.987674i	$-0.550029\pi$
$823$	−8.98076	−0.313050	−0.156525	+	0.987674i	$0.550029\pi$
$824$	−15.4641	−0.538718
$825$	0	0
$826$	12.0000	0.417533
$827$	15.8038	0.549554	0.274777	−	0.961508i	$-0.411396\pi$
$827$	15.8038	0.549554	0.274777	+	0.961508i	$0.411396\pi$
$828$	0	0
$829$	17.7128	0.615191	0.307596	−	0.951517i	$-0.400476\pi$
$829$	17.7128	0.615191	0.307596	+	0.951517i	$0.400476\pi$
$830$	14.7846	0.513181
$831$	0	0
$832$	−4.00000	−0.138675
$833$	0.464102	0.0160802
$834$	0	0
$835$	12.5885	0.435642
$836$	6.92820	0.239617
$837$	0	0
$838$	−66.1577	−2.28538
$839$	−19.2679	−0.665203	−0.332602	−	0.943067i	$-0.607926\pi$
$839$	−19.2679	−0.665203	−0.332602	+	0.943067i	$0.607926\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	9.46410	0.326154
$843$	0	0
$844$	10.1962	0.350966
$845$	−3.00000	−0.103203
$846$	0	0
$847$	−31.1244	−1.06945
$848$	30.0000	1.03020
$849$	0	0
$850$	1.73205	0.0594089
$851$	−4.39230	−0.150566
$852$	0	0
$853$	−23.1769	−0.793562	−0.396781	−	0.917913i	$-0.629873\pi$
$853$	−23.1769	−0.793562	−0.396781	+	0.917913i	$0.629873\pi$
$854$	23.3205	0.798011
$855$	0	0
$856$	30.5885	1.04549
$857$	−31.1769	−1.06498	−0.532492	−	0.846435i	$-0.678744\pi$
$857$	−31.1769	−1.06498	−0.532492	+	0.846435i	$0.678744\pi$
$858$	0	0
$859$	−25.4641	−0.868824	−0.434412	−	0.900714i	$-0.643044\pi$
$859$	−25.4641	−0.868824	−0.434412	+	0.900714i	$0.643044\pi$
$860$	0.535898	0.0182740
$861$	0	0
$862$	−16.9808	−0.578367
$863$	−23.0718	−0.785373	−0.392687	−	0.919672i	$-0.628454\pi$
$863$	−23.0718	−0.785373	−0.392687	+	0.919672i	$0.628454\pi$
$864$	0	0
$865$	3.46410	0.117783
$866$	−30.9282	−1.05098
$867$	0	0
$868$	−8.92820	−0.303043
$869$	−69.0333	−2.34180
$870$	0	0
$871$	40.0000	1.35535
$872$	17.3205	0.586546
$873$	0	0
$874$	−20.7846	−0.703050
$875$	2.73205	0.0923602
$876$	0	0
$877$	−1.21539	−0.0410408	−0.0205204	−	0.999789i	$-0.506532\pi$
$877$	−1.21539	−0.0410408	−0.0205204	+	0.999789i	$0.506532\pi$
$878$	−34.7321	−1.17215
$879$	0	0
$880$	−23.6603	−0.797587
$881$	41.3205	1.39212	0.696062	−	0.717982i	$-0.254934\pi$
$881$	41.3205	1.39212	0.696062	+	0.717982i	$0.254934\pi$
$882$	0	0
$883$	−10.0000	−0.336527	−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000	−0.336527	−0.168263	+	0.985742i	$0.553816\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	22.3923	0.752284
$887$	−3.12436	−0.104906	−0.0524528	−	0.998623i	$-0.516704\pi$
$887$	−3.12436	−0.104906	−0.0524528	+	0.998623i	$0.516704\pi$
$888$	0	0
$889$	39.3205	1.31877
$890$	7.60770	0.255011
$891$	0	0
$892$	−26.3923	−0.883680
$893$	18.9282	0.633408
$894$	0	0
$895$	−11.3205	−0.378403
$896$	−33.1244	−1.10661
$897$	0	0
$898$	59.5692	1.98785
$899$	11.3205	0.377560
$900$	0	0
$901$	−6.00000	−0.199889
$902$	−28.3923	−0.945360
$903$	0	0
$904$	−30.0000	−0.997785
$905$	2.39230	0.0795229
$906$	0	0
$907$	30.7321	1.02044	0.510220	−	0.860044i	$-0.329564\pi$
$907$	30.7321	1.02044	0.510220	+	0.860044i	$0.329564\pi$
$908$	−22.7321	−0.754390
$909$	0	0
$910$	−18.9282	−0.627464
$911$	−36.3397	−1.20399	−0.601995	−	0.798500i	$-0.705627\pi$
$911$	−36.3397	−1.20399	−0.601995	+	0.798500i	$0.705627\pi$
$912$	0	0
$913$	40.3923	1.33679
$914$	−63.7128	−2.10743
$915$	0	0
$916$	−8.39230	−0.277290
$917$	−6.00000	−0.198137
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	14.1962	0.468033
$921$	0	0
$922$	43.1769	1.42196
$923$	46.6410	1.53521
$924$	0	0
$925$	−0.535898	−0.0176202
$926$	−41.3205	−1.35788
$927$	0	0
$928$	−18.0000	−0.590879
$929$	3.46410	0.113653	0.0568267	−	0.998384i	$-0.481902\pi$
$929$	3.46410	0.113653	0.0568267	+	0.998384i	$0.481902\pi$
$930$	0	0
$931$	−0.679492	−0.0222694
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−2.78461	−0.0911152
$935$	4.73205	0.154755
$936$	0	0
$937$	−32.6410	−1.06634	−0.533168	−	0.846010i	$-0.678999\pi$
$937$	−32.6410	−1.06634	−0.533168	+	0.846010i	$0.678999\pi$
$938$	47.3205	1.54507
$939$	0	0
$940$	12.9282	0.421671
$941$	−48.2487	−1.57286	−0.786432	−	0.617677i	$-0.788074\pi$
$941$	−48.2487	−1.57286	−0.786432	+	0.617677i	$0.788074\pi$
$942$	0	0
$943$	28.3923	0.924581
$944$	12.6795	0.412682
$945$	0	0
$946$	4.39230	0.142806
$947$	−8.19615	−0.266339	−0.133170	−	0.991093i	$-0.542515\pi$
$947$	−8.19615	−0.266339	−0.133170	+	0.991093i	$0.542515\pi$
$948$	0	0
$949$	−25.5692	−0.830012
$950$	−2.53590	−0.0822754
$951$	0	0
$952$	4.73205	0.153367
$953$	8.78461	0.284561	0.142281	−	0.989826i	$-0.454556\pi$
$953$	8.78461	0.284561	0.142281	+	0.989826i	$0.454556\pi$
$954$	0	0
$955$	1.85641	0.0600719
$956$	20.7846	0.672222
$957$	0	0
$958$	20.1962	0.652508
$959$	0	0
$960$	0	0
$961$	−20.3205	−0.655500
$962$	3.71281	0.119706
$963$	0	0
$964$	−5.60770	−0.180612
$965$	−16.5359	−0.532309
$966$	0	0
$967$	39.1769	1.25984	0.629922	−	0.776658i	$-0.283087\pi$
$967$	39.1769	1.25984	0.629922	+	0.776658i	$0.283087\pi$
$968$	−19.7321	−0.634212
$969$	0	0
$970$	8.53590	0.274071
$971$	−54.2487	−1.74092	−0.870462	−	0.492236i	$-0.836180\pi$
$971$	−54.2487	−1.74092	−0.870462	+	0.492236i	$0.836180\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	43.2679	1.38639
$975$	0	0
$976$	24.6410	0.788740
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	20.7846	0.664279
$980$	−0.464102	−0.0148252
$981$	0	0
$982$	33.9615	1.08376
$983$	−58.7321	−1.87326	−0.936631	−	0.350318i	$-0.886073\pi$
$983$	−58.7321	−1.87326	−0.936631	+	0.350318i	$0.886073\pi$
$984$	0	0
$985$	17.3205	0.551877
$986$	6.00000	0.191079
$987$	0	0
$988$	5.85641	0.186317
$989$	−4.39230	−0.139667
$990$	0	0
$991$	−2.98076	−0.0946870	−0.0473435	−	0.998879i	$-0.515076\pi$
$991$	−2.98076	−0.0946870	−0.0473435	+	0.998879i	$0.515076\pi$
$992$	−16.9808	−0.539140
$993$	0	0
$994$	55.1769	1.75011
$995$	−10.1962	−0.323240
$996$	0	0
$997$	−2.39230	−0.0757651	−0.0378825	−	0.999282i	$-0.512061\pi$
$997$	−2.39230	−0.0757651	−0.0378825	+	0.999282i	$0.512061\pi$
$998$	−27.1244	−0.858606
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	765.2.a.g.1.2		2
3.2	odd	2		85.2.a.c.1.1	✓	2
5.4	even	2		3825.2.a.v.1.1		2
12.11	even	2		1360.2.a.k.1.1		2
15.2	even	4		425.2.b.d.324.1		4
15.8	even	4		425.2.b.d.324.4		4
15.14	odd	2		425.2.a.e.1.2		2
21.20	even	2		4165.2.a.t.1.1		2
24.5	odd	2		5440.2.a.bb.1.1		2
24.11	even	2		5440.2.a.bl.1.2		2
51.38	odd	4		1445.2.d.e.866.3		4
51.47	odd	4		1445.2.d.e.866.4		4
51.50	odd	2		1445.2.a.g.1.1		2
60.59	even	2		6800.2.a.bg.1.2		2
255.254	odd	2		7225.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.c.1.1	✓	2	3.2	odd	2
425.2.a.e.1.2		2	15.14	odd	2
425.2.b.d.324.1		4	15.2	even	4
425.2.b.d.324.4		4	15.8	even	4
765.2.a.g.1.2		2	1.1	even	1	trivial
1360.2.a.k.1.1		2	12.11	even	2
1445.2.a.g.1.1		2	51.50	odd	2
1445.2.d.e.866.3		4	51.38	odd	4
1445.2.d.e.866.4		4	51.47	odd	4
3825.2.a.v.1.1		2	5.4	even	2
4165.2.a.t.1.1		2	21.20	even	2
5440.2.a.bb.1.1		2	24.5	odd	2
5440.2.a.bl.1.2		2	24.11	even	2
6800.2.a.bg.1.2		2	60.59	even	2
7225.2.a.l.1.2		2	255.254	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:	\( N \)	\(=\)	\( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	765.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{7} -1.73205 q^{8} -1.73205 q^{10} -4.73205 q^{11} -4.00000 q^{13} -4.73205 q^{14} -5.00000 q^{16} +1.00000 q^{17} -1.46410 q^{19} -1.00000 q^{20} -8.19615 q^{22} +8.19615 q^{23} +1.00000 q^{25} -6.92820 q^{26} -2.73205 q^{28} +3.46410 q^{29} +3.26795 q^{31} -5.19615 q^{32} +1.73205 q^{34} +2.73205 q^{35} -0.535898 q^{37} -2.53590 q^{38} +1.73205 q^{40} +3.46410 q^{41} -0.535898 q^{43} -4.73205 q^{44} +14.1962 q^{46} -12.9282 q^{47} +0.464102 q^{49} +1.73205 q^{50} -4.00000 q^{52} -6.00000 q^{53} +4.73205 q^{55} +4.73205 q^{56} +6.00000 q^{58} -2.53590 q^{59} -4.92820 q^{61} +5.66025 q^{62} +1.00000 q^{64} +4.00000 q^{65} -10.0000 q^{67} +1.00000 q^{68} +4.73205 q^{70} -11.6603 q^{71} +6.39230 q^{73} -0.928203 q^{74} -1.46410 q^{76} +12.9282 q^{77} +14.5885 q^{79} +5.00000 q^{80} +6.00000 q^{82} -8.53590 q^{83} -1.00000 q^{85} -0.928203 q^{86} +8.19615 q^{88} -4.39230 q^{89} +10.9282 q^{91} +8.19615 q^{92} -22.3923 q^{94} +1.46410 q^{95} -4.92820 q^{97} +0.803848 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{11} - 8 q^{13} - 6 q^{14} - 10 q^{16} + 2 q^{17} + 4 q^{19} - 2 q^{20} - 6 q^{22} + 6 q^{23} + 2 q^{25} - 2 q^{28} + 10 q^{31} + 2 q^{35} - 8 q^{37} - 12 q^{38}+ \cdots + 12 q^{98}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^11 - 8 * q^13 - 6 * q^14 - 10 * q^16 + 2 * q^17 + 4 * q^19 - 2 * q^20 - 6 * q^22 + 6 * q^23 + 2 * q^25 - 2 * q^28 + 10 * q^31 + 2 * q^35 - 8 * q^37 - 12 * q^38 - 8 * q^43 - 6 * q^44 + 18 * q^46 - 12 * q^47 - 6 * q^49 - 8 * q^52 - 12 * q^53 + 6 * q^55 + 6 * q^56 + 12 * q^58 - 12 * q^59 + 4 * q^61 - 6 * q^62 + 2 * q^64 + 8 * q^65 - 20 * q^67 + 2 * q^68 + 6 * q^70 - 6 * q^71 - 8 * q^73 + 12 * q^74 + 4 * q^76 + 12 * q^77 - 2 * q^79 + 10 * q^80 + 12 * q^82 - 24 * q^83 - 2 * q^85 + 12 * q^86 + 6 * q^88 + 12 * q^89 + 8 * q^91 + 6 * q^92 - 24 * q^94 - 4 * q^95 + 4 * q^97 + 12 * q^98

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