LMFDB - Embedded newform 765.2.a.g.1.1

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			1.1
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)

$f(q)$	$=$	\(q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{7} +1.73205 q^{8} +1.73205 q^{10} -1.26795 q^{11} -4.00000 q^{13} -1.26795 q^{14} -5.00000 q^{16} +1.00000 q^{17} +5.46410 q^{19} -1.00000 q^{20} +2.19615 q^{22} -2.19615 q^{23} +1.00000 q^{25} +6.92820 q^{26} +0.732051 q^{28} -3.46410 q^{29} +6.73205 q^{31} +5.19615 q^{32} -1.73205 q^{34} -0.732051 q^{35} -7.46410 q^{37} -9.46410 q^{38} -1.73205 q^{40} -3.46410 q^{41} -7.46410 q^{43} -1.26795 q^{44} +3.80385 q^{46} +0.928203 q^{47} -6.46410 q^{49} -1.73205 q^{50} -4.00000 q^{52} -6.00000 q^{53} +1.26795 q^{55} +1.26795 q^{56} +6.00000 q^{58} -9.46410 q^{59} +8.92820 q^{61} -11.6603 q^{62} +1.00000 q^{64} +4.00000 q^{65} -10.0000 q^{67} +1.00000 q^{68} +1.26795 q^{70} +5.66025 q^{71} -14.3923 q^{73} +12.9282 q^{74} +5.46410 q^{76} -0.928203 q^{77} -16.5885 q^{79} +5.00000 q^{80} +6.00000 q^{82} -15.4641 q^{83} -1.00000 q^{85} +12.9282 q^{86} -2.19615 q^{88} +16.3923 q^{89} -2.92820 q^{91} -2.19615 q^{92} -1.60770 q^{94} -5.46410 q^{95} +8.92820 q^{97} +11.1962 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.732051	0.276689	0.138345	−	0.990384i	$-0.455822\pi$
$7$	0.732051	0.276689	0.138345	+	0.990384i	$0.455822\pi$
$8$	1.73205	0.612372
$9$	0	0
$10$	1.73205	0.547723
$11$	−1.26795	−0.382301	−0.191151	−	0.981561i	$-0.561222\pi$
$11$	−1.26795	−0.382301	−0.191151	+	0.981561i	$0.561222\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−1.26795	−0.338874
$15$	0	0
$16$	−5.00000	−1.25000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	5.46410	1.25355	0.626775	−	0.779200i	$-0.284374\pi$
$19$	5.46410	1.25355	0.626775	+	0.779200i	$0.284374\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.19615	0.468221
$23$	−2.19615	−0.457929	−0.228965	−	0.973435i	$-0.573534\pi$
$23$	−2.19615	−0.457929	−0.228965	+	0.973435i	$0.573534\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.92820	1.35873
$27$	0	0
$28$	0.732051	0.138345
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	6.73205	1.20911	0.604556	−	0.796563i	$-0.293351\pi$
$31$	6.73205	1.20911	0.604556	+	0.796563i	$0.293351\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	−1.73205	−0.297044
$35$	−0.732051	−0.123739
$36$	0	0
$37$	−7.46410	−1.22709	−0.613545	−	0.789659i	$-0.710257\pi$
$37$	−7.46410	−1.22709	−0.613545	+	0.789659i	$0.710257\pi$
$38$	−9.46410	−1.53528
$39$	0	0
$40$	−1.73205	−0.273861
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	−7.46410	−1.13826	−0.569132	−	0.822246i	$-0.692721\pi$
$43$	−7.46410	−1.13826	−0.569132	+	0.822246i	$0.692721\pi$
$44$	−1.26795	−0.191151
$45$	0	0
$46$	3.80385	0.560847
$47$	0.928203	0.135392	0.0676962	−	0.997706i	$-0.478435\pi$
$47$	0.928203	0.135392	0.0676962	+	0.997706i	$0.478435\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	−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
$55$	1.26795	0.170970
$56$	1.26795	0.169437
$57$	0	0
$58$	6.00000	0.787839
$59$	−9.46410	−1.23212	−0.616061	−	0.787699i	$-0.711272\pi$
$59$	−9.46410	−1.23212	−0.616061	+	0.787699i	$0.711272\pi$
$60$	0	0
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	−11.6603	−1.48085
$63$	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
$70$	1.26795	0.151549
$71$	5.66025	0.671749	0.335874	−	0.941907i	$-0.390968\pi$
$71$	5.66025	0.671749	0.335874	+	0.941907i	$0.390968\pi$
$72$	0	0
$73$	−14.3923	−1.68449	−0.842246	−	0.539093i	$-0.818767\pi$
$73$	−14.3923	−1.68449	−0.842246	+	0.539093i	$0.818767\pi$
$74$	12.9282	1.50287
$75$	0	0
$76$	5.46410	0.626775
$77$	−0.928203	−0.105779
$78$	0	0
$79$	−16.5885	−1.86635	−0.933174	−	0.359426i	$-0.882973\pi$
$79$	−16.5885	−1.86635	−0.933174	+	0.359426i	$0.882973\pi$
$80$	5.00000	0.559017
$81$	0	0
$82$	6.00000	0.662589
$83$	−15.4641	−1.69741	−0.848703	−	0.528870i	$-0.822616\pi$
$83$	−15.4641	−1.69741	−0.848703	+	0.528870i	$0.822616\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	12.9282	1.39408
$87$	0	0
$88$	−2.19615	−0.234111
$89$	16.3923	1.73758	0.868790	−	0.495180i	$-0.164898\pi$
$89$	16.3923	1.73758	0.868790	+	0.495180i	$0.164898\pi$
$90$	0	0
$91$	−2.92820	−0.306959
$92$	−2.19615	−0.228965
$93$	0	0
$94$	−1.60770	−0.165821
$95$	−5.46410	−0.560605
$96$	0	0
$97$	8.92820	0.906522	0.453261	−	0.891378i	$-0.350261\pi$
$97$	8.92820	0.906522	0.453261	+	0.891378i	$0.350261\pi$
$98$	11.1962	1.13098
$99$	0	0
$100$	1.00000	0.100000
$101$	2.53590	0.252331	0.126166	−	0.992009i	$-0.459733\pi$
$101$	2.53590	0.252331	0.126166	+	0.992009i	$0.459733\pi$
$102$	0	0
$103$	−4.92820	−0.485590	−0.242795	−	0.970078i	$-0.578064\pi$
$103$	−4.92820	−0.485590	−0.242795	+	0.970078i	$0.578064\pi$
$104$	−6.92820	−0.679366
$105$	0	0
$106$	10.3923	1.00939
$107$	−0.339746	−0.0328445	−0.0164222	−	0.999865i	$-0.505228\pi$
$107$	−0.339746	−0.0328445	−0.0164222	+	0.999865i	$0.505228\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−2.19615	−0.209395
$111$	0	0
$112$	−3.66025	−0.345861
$113$	−17.3205	−1.62938	−0.814688	−	0.579899i	$-0.803092\pi$
$113$	−17.3205	−1.62938	−0.814688	+	0.579899i	$0.803092\pi$
$114$	0	0
$115$	2.19615	0.204792
$116$	−3.46410	−0.321634
$117$	0	0
$118$	16.3923	1.50903
$119$	0.732051	0.0671070
$120$	0	0
$121$	−9.39230	−0.853846
$122$	−15.4641	−1.40005
$123$	0	0
$124$	6.73205	0.604556
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.39230	0.567225	0.283613	−	0.958939i	$-0.408467\pi$
$127$	6.39230	0.567225	0.283613	+	0.958939i	$0.408467\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	−6.92820	−0.607644
$131$	−8.19615	−0.716101	−0.358051	−	0.933702i	$-0.616558\pi$
$131$	−8.19615	−0.716101	−0.358051	+	0.933702i	$0.616558\pi$
$132$	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
$139$	13.6603	1.15865	0.579324	−	0.815097i	$-0.303317\pi$
$139$	13.6603	1.15865	0.579324	+	0.815097i	$0.303317\pi$
$140$	−0.732051	−0.0618696
$141$	0	0
$142$	−9.80385	−0.822721
$143$	5.07180	0.424125
$144$	0	0
$145$	3.46410	0.287678
$146$	24.9282	2.06307
$147$	0	0
$148$	−7.46410	−0.613545
$149$	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
$151$	5.46410	0.444662	0.222331	−	0.974971i	$-0.428633\pi$
$151$	5.46410	0.444662	0.222331	+	0.974971i	$0.428633\pi$
$152$	9.46410	0.767640
$153$	0	0
$154$	1.60770	0.129552
$155$	−6.73205	−0.540731
$156$	0	0
$157$	−4.92820	−0.393313	−0.196657	−	0.980472i	$-0.563008\pi$
$157$	−4.92820	−0.393313	−0.196657	+	0.980472i	$0.563008\pi$
$158$	28.7321	2.28580
$159$	0	0
$160$	−5.19615	−0.410792
$161$	−1.60770	−0.126704
$162$	0	0
$163$	10.1962	0.798624	0.399312	−	0.916815i	$-0.369249\pi$
$163$	10.1962	0.798624	0.399312	+	0.916815i	$0.369249\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	26.7846	2.07889
$167$	18.5885	1.43842	0.719209	−	0.694794i	$-0.244504\pi$
$167$	18.5885	1.43842	0.719209	+	0.694794i	$0.244504\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	1.73205	0.132842
$171$	0	0
$172$	−7.46410	−0.569132
$173$	3.46410	0.263371	0.131685	−	0.991292i	$-0.457961\pi$
$173$	3.46410	0.263371	0.131685	+	0.991292i	$0.457961\pi$
$174$	0	0
$175$	0.732051	0.0553378
$176$	6.33975	0.477876
$177$	0	0
$178$	−28.3923	−2.12809
$179$	−23.3205	−1.74306	−0.871528	−	0.490345i	$-0.836871\pi$
$179$	−23.3205	−1.74306	−0.871528	+	0.490345i	$0.836871\pi$
$180$	0	0
$181$	18.3923	1.36709	0.683545	−	0.729909i	$-0.260437\pi$
$181$	18.3923	1.36709	0.683545	+	0.729909i	$0.260437\pi$
$182$	5.07180	0.375947
$183$	0	0
$184$	−3.80385	−0.280423
$185$	7.46410	0.548772
$186$	0	0
$187$	−1.26795	−0.0927216
$188$	0.928203	0.0676962
$189$	0	0
$190$	9.46410	0.686598
$191$	25.8564	1.87090	0.935452	−	0.353454i	$-0.114993\pi$
$191$	25.8564	1.87090	0.935452	+	0.353454i	$0.114993\pi$
$192$	0	0
$193$	23.4641	1.68898	0.844491	−	0.535569i	$-0.179903\pi$
$193$	23.4641	1.68898	0.844491	+	0.535569i	$0.179903\pi$
$194$	−15.4641	−1.11026
$195$	0	0
$196$	−6.46410	−0.461722
$197$	17.3205	1.23404	0.617018	−	0.786949i	$-0.288341\pi$
$197$	17.3205	1.23404	0.617018	+	0.786949i	$0.288341\pi$
$198$	0	0
$199$	−0.196152	−0.0139049	−0.00695244	−	0.999976i	$-0.502213\pi$
$199$	−0.196152	−0.0139049	−0.00695244	+	0.999976i	$0.502213\pi$
$200$	1.73205	0.122474
$201$	0	0
$202$	−4.39230	−0.309041
$203$	−2.53590	−0.177985
$204$	0	0
$205$	3.46410	0.241943
$206$	8.53590	0.594724
$207$	0	0
$208$	20.0000	1.38675
$209$	−6.92820	−0.479234
$210$	0	0
$211$	−0.196152	−0.0135037	−0.00675184	−	0.999977i	$-0.502149\pi$
$211$	−0.196152	−0.0135037	−0.00675184	+	0.999977i	$0.502149\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	0.588457	0.0402261
$215$	7.46410	0.509048
$216$	0	0
$217$	4.92820	0.334548
$218$	17.3205	1.17309
$219$	0	0
$220$	1.26795	0.0854851
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−5.60770	−0.375519	−0.187760	−	0.982215i	$-0.560123\pi$
$223$	−5.60770	−0.375519	−0.187760	+	0.982215i	$0.560123\pi$
$224$	3.80385	0.254155
$225$	0	0
$226$	30.0000	1.99557
$227$	−19.2679	−1.27886	−0.639429	−	0.768850i	$-0.720829\pi$
$227$	−19.2679	−1.27886	−0.639429	+	0.768850i	$0.720829\pi$
$228$	0	0
$229$	12.3923	0.818907	0.409453	−	0.912331i	$-0.365719\pi$
$229$	12.3923	0.818907	0.409453	+	0.912331i	$0.365719\pi$
$230$	−3.80385	−0.250818
$231$	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
$235$	−0.928203	−0.0605493
$236$	−9.46410	−0.616061
$237$	0	0
$238$	−1.26795	−0.0821889
$239$	−20.7846	−1.34444	−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846	−1.34444	−0.672222	+	0.740349i	$0.734660\pi$
$240$	0	0
$241$	−26.3923	−1.70008	−0.850039	−	0.526720i	$-0.823422\pi$
$241$	−26.3923	−1.70008	−0.850039	+	0.526720i	$0.823422\pi$
$242$	16.2679	1.04574
$243$	0	0
$244$	8.92820	0.571570
$245$	6.46410	0.412976
$246$	0	0
$247$	−21.8564	−1.39069
$248$	11.6603	0.740427
$249$	0	0
$250$	1.73205	0.109545
$251$	6.92820	0.437304	0.218652	−	0.975803i	$-0.429834\pi$
$251$	6.92820	0.437304	0.218652	+	0.975803i	$0.429834\pi$
$252$	0	0
$253$	2.78461	0.175067
$254$	−11.0718	−0.694706
$255$	0	0
$256$	19.0000	1.18750
$257$	−6.92820	−0.432169	−0.216085	−	0.976375i	$-0.569329\pi$
$257$	−6.92820	−0.432169	−0.216085	+	0.976375i	$0.569329\pi$
$258$	0	0
$259$	−5.46410	−0.339523
$260$	4.00000	0.248069
$261$	0	0
$262$	14.1962	0.877041
$263$	22.3923	1.38077	0.690384	−	0.723443i	$-0.257441\pi$
$263$	22.3923	1.38077	0.690384	+	0.723443i	$0.257441\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	−6.92820	−0.424795
$267$	0	0
$268$	−10.0000	−0.610847
$269$	12.9282	0.788246	0.394123	−	0.919058i	$-0.371048\pi$
$269$	12.9282	0.788246	0.394123	+	0.919058i	$0.371048\pi$
$270$	0	0
$271$	−10.9282	−0.663841	−0.331921	−	0.943307i	$-0.607697\pi$
$271$	−10.9282	−0.663841	−0.331921	+	0.943307i	$0.607697\pi$
$272$	−5.00000	−0.303170
$273$	0	0
$274$	0	0
$275$	−1.26795	−0.0764602
$276$	0	0
$277$	7.07180	0.424903	0.212452	−	0.977172i	$-0.431855\pi$
$277$	7.07180	0.424903	0.212452	+	0.977172i	$0.431855\pi$
$278$	−23.6603	−1.41905
$279$	0	0
$280$	−1.26795	−0.0757745
$281$	−0.928203	−0.0553720	−0.0276860	−	0.999617i	$-0.508814\pi$
$281$	−0.928203	−0.0553720	−0.0276860	+	0.999617i	$0.508814\pi$
$282$	0	0
$283$	−8.73205	−0.519067	−0.259533	−	0.965734i	$-0.583569\pi$
$283$	−8.73205	−0.519067	−0.259533	+	0.965734i	$0.583569\pi$
$284$	5.66025	0.335874
$285$	0	0
$286$	−8.78461	−0.519445
$287$	−2.53590	−0.149689
$288$	0	0
$289$	1.00000	0.0588235
$290$	−6.00000	−0.352332
$291$	0	0
$292$	−14.3923	−0.842246
$293$	−12.9282	−0.755274	−0.377637	−	0.925954i	$-0.623263\pi$
$293$	−12.9282	−0.755274	−0.377637	+	0.925954i	$0.623263\pi$
$294$	0	0
$295$	9.46410	0.551021
$296$	−12.9282	−0.751437
$297$	0	0
$298$	−10.3923	−0.602010
$299$	8.78461	0.508027
$300$	0	0
$301$	−5.46410	−0.314946
$302$	−9.46410	−0.544598
$303$	0	0
$304$	−27.3205	−1.56694
$305$	−8.92820	−0.511227
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	−0.928203	−0.0528893
$309$	0	0
$310$	11.6603	0.662258
$311$	−22.0526	−1.25049	−0.625243	−	0.780430i	$-0.715000\pi$
$311$	−22.0526	−1.25049	−0.625243	+	0.780430i	$0.715000\pi$
$312$	0	0
$313$	−5.60770	−0.316966	−0.158483	−	0.987362i	$-0.550660\pi$
$313$	−5.60770	−0.316966	−0.158483	+	0.987362i	$0.550660\pi$
$314$	8.53590	0.481709
$315$	0	0
$316$	−16.5885	−0.933174
$317$	11.0718	0.621854	0.310927	−	0.950434i	$-0.399361\pi$
$317$	11.0718	0.621854	0.310927	+	0.950434i	$0.399361\pi$
$318$	0	0
$319$	4.39230	0.245922
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	2.78461	0.155180
$323$	5.46410	0.304031
$324$	0	0
$325$	−4.00000	−0.221880
$326$	−17.6603	−0.978111
$327$	0	0
$328$	−6.00000	−0.331295
$329$	0.679492	0.0374616
$330$	0	0
$331$	−13.4641	−0.740054	−0.370027	−	0.929021i	$-0.620652\pi$
$331$	−13.4641	−0.740054	−0.370027	+	0.929021i	$0.620652\pi$
$332$	−15.4641	−0.848703
$333$	0	0
$334$	−32.1962	−1.76170
$335$	10.0000	0.546358
$336$	0	0
$337$	34.7846	1.89484	0.947419	−	0.319995i	$-0.103681\pi$
$337$	34.7846	1.89484	0.947419	+	0.319995i	$0.103681\pi$
$338$	−5.19615	−0.282633
$339$	0	0
$340$	−1.00000	−0.0542326
$341$	−8.53590	−0.462245
$342$	0	0
$343$	−9.85641	−0.532196
$344$	−12.9282	−0.697042
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−14.1962	−0.762089	−0.381045	−	0.924557i	$-0.624436\pi$
$347$	−14.1962	−0.762089	−0.381045	+	0.924557i	$0.624436\pi$
$348$	0	0
$349$	−30.7846	−1.64786	−0.823931	−	0.566690i	$-0.808224\pi$
$349$	−30.7846	−1.64786	−0.823931	+	0.566690i	$0.808224\pi$
$350$	−1.26795	−0.0677747
$351$	0	0
$352$	−6.58846	−0.351166
$353$	14.7846	0.786905	0.393453	−	0.919345i	$-0.371281\pi$
$353$	14.7846	0.786905	0.393453	+	0.919345i	$0.371281\pi$
$354$	0	0
$355$	−5.66025	−0.300415
$356$	16.3923	0.868790
$357$	0	0
$358$	40.3923	2.13480
$359$	14.5359	0.767175	0.383588	−	0.923504i	$-0.374688\pi$
$359$	14.5359	0.767175	0.383588	+	0.923504i	$0.374688\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	−31.8564	−1.67434
$363$	0	0
$364$	−2.92820	−0.153480
$365$	14.3923	0.753328
$366$	0	0
$367$	−18.1962	−0.949831	−0.474916	−	0.880031i	$-0.657521\pi$
$367$	−18.1962	−0.949831	−0.474916	+	0.880031i	$0.657521\pi$
$368$	10.9808	0.572412
$369$	0	0
$370$	−12.9282	−0.672105
$371$	−4.39230	−0.228037
$372$	0	0
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	2.19615	0.113560
$375$	0	0
$376$	1.60770	0.0829105
$377$	13.8564	0.713641
$378$	0	0
$379$	28.1962	1.44834	0.724170	−	0.689622i	$-0.242223\pi$
$379$	28.1962	1.44834	0.724170	+	0.689622i	$0.242223\pi$
$380$	−5.46410	−0.280302
$381$	0	0
$382$	−44.7846	−2.29138
$383$	−8.53590	−0.436164	−0.218082	−	0.975930i	$-0.569980\pi$
$383$	−8.53590	−0.436164	−0.218082	+	0.975930i	$0.569980\pi$
$384$	0	0
$385$	0.928203	0.0473056
$386$	−40.6410	−2.06857
$387$	0	0
$388$	8.92820	0.453261
$389$	−16.3923	−0.831123	−0.415561	−	0.909565i	$-0.636415\pi$
$389$	−16.3923	−0.831123	−0.415561	+	0.909565i	$0.636415\pi$
$390$	0	0
$391$	−2.19615	−0.111064
$392$	−11.1962	−0.565491
$393$	0	0
$394$	−30.0000	−1.51138
$395$	16.5885	0.834656
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	0.339746	0.0170299
$399$	0	0
$400$	−5.00000	−0.250000
$401$	0.928203	0.0463523	0.0231761	−	0.999731i	$-0.492622\pi$
$401$	0.928203	0.0463523	0.0231761	+	0.999731i	$0.492622\pi$
$402$	0	0
$403$	−26.9282	−1.34139
$404$	2.53590	0.126166
$405$	0	0
$406$	4.39230	0.217986
$407$	9.46410	0.469118
$408$	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
$412$	−4.92820	−0.242795
$413$	−6.92820	−0.340915
$414$	0	0
$415$	15.4641	0.759103
$416$	−20.7846	−1.01905
$417$	0	0
$418$	12.0000	0.586939
$419$	−27.8038	−1.35831	−0.679153	−	0.733996i	$-0.737653\pi$
$419$	−27.8038	−1.35831	−0.679153	+	0.733996i	$0.737653\pi$
$420$	0	0
$421$	−1.46410	−0.0713559	−0.0356780	−	0.999363i	$-0.511359\pi$
$421$	−1.46410	−0.0713559	−0.0356780	+	0.999363i	$0.511359\pi$
$422$	0.339746	0.0165386
$423$	0	0
$424$	−10.3923	−0.504695
$425$	1.00000	0.0485071
$426$	0	0
$427$	6.53590	0.316294
$428$	−0.339746	−0.0164222
$429$	0	0
$430$	−12.9282	−0.623453
$431$	−20.1962	−0.972814	−0.486407	−	0.873732i	$-0.661693\pi$
$431$	−20.1962	−0.972814	−0.486407	+	0.873732i	$0.661693\pi$
$432$	0	0
$433$	9.85641	0.473669	0.236834	−	0.971550i	$-0.423890\pi$
$433$	9.85641	0.473669	0.236834	+	0.971550i	$0.423890\pi$
$434$	−8.53590	−0.409736
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−12.0000	−0.574038
$438$	0	0
$439$	18.0526	0.861602	0.430801	−	0.902447i	$-0.358231\pi$
$439$	18.0526	0.861602	0.430801	+	0.902447i	$0.358231\pi$
$440$	2.19615	0.104697
$441$	0	0
$442$	6.92820	0.329541
$443$	−0.928203	−0.0441003	−0.0220501	−	0.999757i	$-0.507019\pi$
$443$	−0.928203	−0.0441003	−0.0220501	+	0.999757i	$0.507019\pi$
$444$	0	0
$445$	−16.3923	−0.777070
$446$	9.71281	0.459915
$447$	0	0
$448$	0.732051	0.0345861
$449$	13.6077	0.642187	0.321093	−	0.947048i	$-0.395950\pi$
$449$	13.6077	0.642187	0.321093	+	0.947048i	$0.395950\pi$
$450$	0	0
$451$	4.39230	0.206826
$452$	−17.3205	−0.814688
$453$	0	0
$454$	33.3731	1.56628
$455$	2.92820	0.137276
$456$	0	0
$457$	4.78461	0.223815	0.111907	−	0.993719i	$-0.464304\pi$
$457$	4.78461	0.223815	0.111907	+	0.993719i	$0.464304\pi$
$458$	−21.4641	−1.00295
$459$	0	0
$460$	2.19615	0.102396
$461$	11.0718	0.515665	0.257832	−	0.966190i	$-0.416992\pi$
$461$	11.0718	0.515665	0.257832	+	0.966190i	$0.416992\pi$
$462$	0	0
$463$	3.85641	0.179222	0.0896112	−	0.995977i	$-0.471438\pi$
$463$	3.85641	0.179222	0.0896112	+	0.995977i	$0.471438\pi$
$464$	17.3205	0.804084
$465$	0	0
$466$	10.3923	0.481414
$467$	−22.3923	−1.03619	−0.518096	−	0.855322i	$-0.673359\pi$
$467$	−22.3923	−1.03619	−0.518096	+	0.855322i	$0.673359\pi$
$468$	0	0
$469$	−7.32051	−0.338030
$470$	1.60770	0.0741574
$471$	0	0
$472$	−16.3923	−0.754517
$473$	9.46410	0.435160
$474$	0	0
$475$	5.46410	0.250710
$476$	0.732051	0.0335535
$477$	0	0
$478$	36.0000	1.64660
$479$	−5.66025	−0.258624	−0.129312	−	0.991604i	$-0.541277\pi$
$479$	−5.66025	−0.258624	−0.129312	+	0.991604i	$0.541277\pi$
$480$	0	0
$481$	29.8564	1.36133
$482$	45.7128	2.08216
$483$	0	0
$484$	−9.39230	−0.426923
$485$	−8.92820	−0.405409
$486$	0	0
$487$	−26.9808	−1.22262	−0.611308	−	0.791393i	$-0.709356\pi$
$487$	−26.9808	−1.22262	−0.611308	+	0.791393i	$0.709356\pi$
$488$	15.4641	0.700027
$489$	0	0
$490$	−11.1962	−0.505791
$491$	40.3923	1.82288	0.911440	−	0.411434i	$-0.134972\pi$
$491$	40.3923	1.82288	0.911440	+	0.411434i	$0.134972\pi$
$492$	0	0
$493$	−3.46410	−0.156015
$494$	37.8564	1.70324
$495$	0	0
$496$	−33.6603	−1.51139
$497$	4.14359	0.185866
$498$	0	0
$499$	1.66025	0.0743232	0.0371616	−	0.999309i	$-0.488168\pi$
$499$	1.66025	0.0743232	0.0371616	+	0.999309i	$0.488168\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−12.0000	−0.535586
$503$	9.12436	0.406835	0.203417	−	0.979092i	$-0.434795\pi$
$503$	9.12436	0.406835	0.203417	+	0.979092i	$0.434795\pi$
$504$	0	0
$505$	−2.53590	−0.112846
$506$	−4.82309	−0.214412
$507$	0	0
$508$	6.39230	0.283613
$509$	7.85641	0.348229	0.174115	−	0.984725i	$-0.444294\pi$
$509$	7.85641	0.348229	0.174115	+	0.984725i	$0.444294\pi$
$510$	0	0
$511$	−10.5359	−0.466081
$512$	−8.66025	−0.382733
$513$	0	0
$514$	12.0000	0.529297
$515$	4.92820	0.217163
$516$	0	0
$517$	−1.17691	−0.0517606
$518$	9.46410	0.415829
$519$	0	0
$520$	6.92820	0.303822
$521$	−31.8564	−1.39565	−0.697827	−	0.716266i	$-0.745850\pi$
$521$	−31.8564	−1.39565	−0.697827	+	0.716266i	$0.745850\pi$
$522$	0	0
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	−8.19615	−0.358051
$525$	0	0
$526$	−38.7846	−1.69109
$527$	6.73205	0.293253
$528$	0	0
$529$	−18.1769	−0.790301
$530$	−10.3923	−0.451413
$531$	0	0
$532$	4.00000	0.173422
$533$	13.8564	0.600188
$534$	0	0
$535$	0.339746	0.0146885
$536$	−17.3205	−0.748132
$537$	0	0
$538$	−22.3923	−0.965401
$539$	8.19615	0.353033
$540$	0	0
$541$	−23.1769	−0.996453	−0.498227	−	0.867047i	$-0.666015\pi$
$541$	−23.1769	−0.996453	−0.498227	+	0.867047i	$0.666015\pi$
$542$	18.9282	0.813036
$543$	0	0
$544$	5.19615	0.222783
$545$	10.0000	0.428353
$546$	0	0
$547$	25.9090	1.10779	0.553894	−	0.832587i	$-0.313141\pi$
$547$	25.9090	1.10779	0.553894	+	0.832587i	$0.313141\pi$
$548$	0	0
$549$	0	0
$550$	2.19615	0.0936443
$551$	−18.9282	−0.806369
$552$	0	0
$553$	−12.1436	−0.516398
$554$	−12.2487	−0.520398
$555$	0	0
$556$	13.6603	0.579324
$557$	−6.92820	−0.293557	−0.146779	−	0.989169i	$-0.546891\pi$
$557$	−6.92820	−0.293557	−0.146779	+	0.989169i	$0.546891\pi$
$558$	0	0
$559$	29.8564	1.26279
$560$	3.66025	0.154674
$561$	0	0
$562$	1.60770	0.0678165
$563$	−20.5359	−0.865485	−0.432742	−	0.901518i	$-0.642454\pi$
$563$	−20.5359	−0.865485	−0.432742	+	0.901518i	$0.642454\pi$
$564$	0	0
$565$	17.3205	0.728679
$566$	15.1244	0.635724
$567$	0	0
$568$	9.80385	0.411360
$569$	28.6410	1.20069	0.600347	−	0.799740i	$-0.295029\pi$
$569$	28.6410	1.20069	0.600347	+	0.799740i	$0.295029\pi$
$570$	0	0
$571$	22.4449	0.939288	0.469644	−	0.882856i	$-0.344382\pi$
$571$	22.4449	0.939288	0.469644	+	0.882856i	$0.344382\pi$
$572$	5.07180	0.212062
$573$	0	0
$574$	4.39230	0.183331
$575$	−2.19615	−0.0915859
$576$	0	0
$577$	30.6410	1.27560	0.637801	−	0.770201i	$-0.279844\pi$
$577$	30.6410	1.27560	0.637801	+	0.770201i	$0.279844\pi$
$578$	−1.73205	−0.0720438
$579$	0	0
$580$	3.46410	0.143839
$581$	−11.3205	−0.469654
$582$	0	0
$583$	7.60770	0.315079
$584$	−24.9282	−1.03154
$585$	0	0
$586$	22.3923	0.925018
$587$	−25.6077	−1.05694	−0.528471	−	0.848951i	$-0.677235\pi$
$587$	−25.6077	−1.05694	−0.528471	+	0.848951i	$0.677235\pi$
$588$	0	0
$589$	36.7846	1.51568
$590$	−16.3923	−0.674861
$591$	0	0
$592$	37.3205	1.53386
$593$	7.85641	0.322624	0.161312	−	0.986903i	$-0.448427\pi$
$593$	7.85641	0.322624	0.161312	+	0.986903i	$0.448427\pi$
$594$	0	0
$595$	−0.732051	−0.0300112
$596$	6.00000	0.245770
$597$	0	0
$598$	−15.2154	−0.622204
$599$	−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
$601$	−40.2487	−1.64178	−0.820890	−	0.571087i	$-0.806522\pi$
$601$	−40.2487	−1.64178	−0.820890	+	0.571087i	$0.806522\pi$
$602$	9.46410	0.385728
$603$	0	0
$604$	5.46410	0.222331
$605$	9.39230	0.381851
$606$	0	0
$607$	−4.33975	−0.176145	−0.0880724	−	0.996114i	$-0.528071\pi$
$607$	−4.33975	−0.176145	−0.0880724	+	0.996114i	$0.528071\pi$
$608$	28.3923	1.15146
$609$	0	0
$610$	15.4641	0.626123
$611$	−3.71281	−0.150204
$612$	0	0
$613$	−11.8564	−0.478876	−0.239438	−	0.970912i	$-0.576963\pi$
$613$	−11.8564	−0.478876	−0.239438	+	0.970912i	$0.576963\pi$
$614$	17.3205	0.698999
$615$	0	0
$616$	−1.60770	−0.0647759
$617$	20.5359	0.826744	0.413372	−	0.910562i	$-0.364351\pi$
$617$	20.5359	0.826744	0.413372	+	0.910562i	$0.364351\pi$
$618$	0	0
$619$	7.41154	0.297895	0.148948	−	0.988845i	$-0.452411\pi$
$619$	7.41154	0.297895	0.148948	+	0.988845i	$0.452411\pi$
$620$	−6.73205	−0.270366
$621$	0	0
$622$	38.1962	1.53153
$623$	12.0000	0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	9.71281	0.388202
$627$	0	0
$628$	−4.92820	−0.196657
$629$	−7.46410	−0.297613
$630$	0	0
$631$	−11.6077	−0.462095	−0.231048	−	0.972942i	$-0.574215\pi$
$631$	−11.6077	−0.462095	−0.231048	+	0.972942i	$0.574215\pi$
$632$	−28.7321	−1.14290
$633$	0	0
$634$	−19.1769	−0.761613
$635$	−6.39230	−0.253671
$636$	0	0
$637$	25.8564	1.02447
$638$	−7.60770	−0.301192
$639$	0	0
$640$	12.1244	0.479257
$641$	31.1769	1.23141	0.615707	−	0.787975i	$-0.288870\pi$
$641$	31.1769	1.23141	0.615707	+	0.787975i	$0.288870\pi$
$642$	0	0
$643$	−13.8038	−0.544371	−0.272185	−	0.962245i	$-0.587746\pi$
$643$	−13.8038	−0.544371	−0.272185	+	0.962245i	$0.587746\pi$
$644$	−1.60770	−0.0633521
$645$	0	0
$646$	−9.46410	−0.372360
$647$	−2.78461	−0.109474	−0.0547372	−	0.998501i	$-0.517432\pi$
$647$	−2.78461	−0.109474	−0.0547372	+	0.998501i	$0.517432\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	6.92820	0.271746
$651$	0	0
$652$	10.1962	0.399312
$653$	46.3923	1.81547	0.907736	−	0.419543i	$-0.137810\pi$
$653$	46.3923	1.81547	0.907736	+	0.419543i	$0.137810\pi$
$654$	0	0
$655$	8.19615	0.320250
$656$	17.3205	0.676252
$657$	0	0
$658$	−1.17691	−0.0458809
$659$	−8.78461	−0.342200	−0.171100	−	0.985254i	$-0.554732\pi$
$659$	−8.78461	−0.342200	−0.171100	+	0.985254i	$0.554732\pi$
$660$	0	0
$661$	−35.8564	−1.39465	−0.697326	−	0.716754i	$-0.745627\pi$
$661$	−35.8564	−1.39465	−0.697326	+	0.716754i	$0.745627\pi$
$662$	23.3205	0.906377
$663$	0	0
$664$	−26.7846	−1.03944
$665$	−4.00000	−0.155113
$666$	0	0
$667$	7.60770	0.294571
$668$	18.5885	0.719209
$669$	0	0
$670$	−17.3205	−0.669150
$671$	−11.3205	−0.437023
$672$	0	0
$673$	16.5359	0.637412	0.318706	−	0.947854i	$-0.396752\pi$
$673$	16.5359	0.637412	0.318706	+	0.947854i	$0.396752\pi$
$674$	−60.2487	−2.32069
$675$	0	0
$676$	3.00000	0.115385
$677$	38.7846	1.49061	0.745307	−	0.666722i	$-0.232303\pi$
$677$	38.7846	1.49061	0.745307	+	0.666722i	$0.232303\pi$
$678$	0	0
$679$	6.53590	0.250825
$680$	−1.73205	−0.0664211
$681$	0	0
$682$	14.7846	0.566132
$683$	−48.8372	−1.86870	−0.934351	−	0.356354i	$-0.884020\pi$
$683$	−48.8372	−1.86870	−0.934351	+	0.356354i	$0.884020\pi$
$684$	0	0
$685$	0	0
$686$	17.0718	0.651804
$687$	0	0
$688$	37.3205	1.42283
$689$	24.0000	0.914327
$690$	0	0
$691$	12.9808	0.493811	0.246906	−	0.969040i	$-0.420586\pi$
$691$	12.9808	0.493811	0.246906	+	0.969040i	$0.420586\pi$
$692$	3.46410	0.131685
$693$	0	0
$694$	24.5885	0.933365
$695$	−13.6603	−0.518163
$696$	0	0
$697$	−3.46410	−0.131212
$698$	53.3205	2.01821
$699$	0	0
$700$	0.732051	0.0276689
$701$	23.3205	0.880803	0.440402	−	0.897801i	$-0.354836\pi$
$701$	23.3205	0.880803	0.440402	+	0.897801i	$0.354836\pi$
$702$	0	0
$703$	−40.7846	−1.53822
$704$	−1.26795	−0.0477876
$705$	0	0
$706$	−25.6077	−0.963758
$707$	1.85641	0.0698174
$708$	0	0
$709$	11.4641	0.430543	0.215272	−	0.976554i	$-0.430936\pi$
$709$	11.4641	0.430543	0.215272	+	0.976554i	$0.430936\pi$
$710$	9.80385	0.367932
$711$	0	0
$712$	28.3923	1.06405
$713$	−14.7846	−0.553688
$714$	0	0
$715$	−5.07180	−0.189674
$716$	−23.3205	−0.871528
$717$	0	0
$718$	−25.1769	−0.939594
$719$	36.5885	1.36452	0.682260	−	0.731110i	$-0.260997\pi$
$719$	36.5885	1.36452	0.682260	+	0.731110i	$0.260997\pi$
$720$	0	0
$721$	−3.60770	−0.134358
$722$	−18.8038	−0.699807
$723$	0	0
$724$	18.3923	0.683545
$725$	−3.46410	−0.128654
$726$	0	0
$727$	27.8564	1.03314	0.516568	−	0.856246i	$-0.327209\pi$
$727$	27.8564	1.03314	0.516568	+	0.856246i	$0.327209\pi$
$728$	−5.07180	−0.187973
$729$	0	0
$730$	−24.9282	−0.922634
$731$	−7.46410	−0.276070
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	31.5167	1.16330
$735$	0	0
$736$	−11.4115	−0.420635
$737$	12.6795	0.467055
$738$	0	0
$739$	−32.3923	−1.19157	−0.595785	−	0.803144i	$-0.703159\pi$
$739$	−32.3923	−1.19157	−0.595785	+	0.803144i	$0.703159\pi$
$740$	7.46410	0.274386
$741$	0	0
$742$	7.60770	0.279287
$743$	19.2679	0.706872	0.353436	−	0.935459i	$-0.385013\pi$
$743$	19.2679	0.706872	0.353436	+	0.935459i	$0.385013\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	−34.6410	−1.26830
$747$	0	0
$748$	−1.26795	−0.0463608
$749$	−0.248711	−0.00908771
$750$	0	0
$751$	26.3397	0.961151	0.480575	−	0.876953i	$-0.340428\pi$
$751$	26.3397	0.961151	0.480575	+	0.876953i	$0.340428\pi$
$752$	−4.64102	−0.169240
$753$	0	0
$754$	−24.0000	−0.874028
$755$	−5.46410	−0.198859
$756$	0	0
$757$	−38.6410	−1.40443	−0.702216	−	0.711964i	$-0.747806\pi$
$757$	−38.6410	−1.40443	−0.702216	+	0.711964i	$0.747806\pi$
$758$	−48.8372	−1.77385
$759$	0	0
$760$	−9.46410	−0.343299
$761$	7.60770	0.275779	0.137889	−	0.990448i	$-0.455968\pi$
$761$	7.60770	0.275779	0.137889	+	0.990448i	$0.455968\pi$
$762$	0	0
$763$	−7.32051	−0.265020
$764$	25.8564	0.935452
$765$	0	0
$766$	14.7846	0.534190
$767$	37.8564	1.36692
$768$	0	0
$769$	15.6077	0.562828	0.281414	−	0.959586i	$-0.409197\pi$
$769$	15.6077	0.562828	0.281414	+	0.959586i	$0.409197\pi$
$770$	−1.60770	−0.0579373
$771$	0	0
$772$	23.4641	0.844491
$773$	−22.6410	−0.814341	−0.407170	−	0.913352i	$-0.633484\pi$
$773$	−22.6410	−0.814341	−0.407170	+	0.913352i	$0.633484\pi$
$774$	0	0
$775$	6.73205	0.241822
$776$	15.4641	0.555129
$777$	0	0
$778$	28.3923	1.01791
$779$	−18.9282	−0.678173
$780$	0	0
$781$	−7.17691	−0.256810
$782$	3.80385	0.136025
$783$	0	0
$784$	32.3205	1.15430
$785$	4.92820	0.175895
$786$	0	0
$787$	−21.9090	−0.780970	−0.390485	−	0.920609i	$-0.627693\pi$
$787$	−21.9090	−0.780970	−0.390485	+	0.920609i	$0.627693\pi$
$788$	17.3205	0.617018
$789$	0	0
$790$	−28.7321	−1.02224
$791$	−12.6795	−0.450831
$792$	0	0
$793$	−35.7128	−1.26820
$794$	−24.2487	−0.860555
$795$	0	0
$796$	−0.196152	−0.00695244
$797$	11.0718	0.392183	0.196092	−	0.980586i	$-0.437175\pi$
$797$	11.0718	0.392183	0.196092	+	0.980586i	$0.437175\pi$
$798$	0	0
$799$	0.928203	0.0328375
$800$	5.19615	0.183712
$801$	0	0
$802$	−1.60770	−0.0567697
$803$	18.2487	0.643983
$804$	0	0
$805$	1.60770	0.0566638
$806$	46.6410	1.64286
$807$	0	0
$808$	4.39230	0.154521
$809$	28.1436	0.989476	0.494738	−	0.869042i	$-0.335264\pi$
$809$	28.1436	0.989476	0.494738	+	0.869042i	$0.335264\pi$
$810$	0	0
$811$	−34.8372	−1.22330	−0.611649	−	0.791129i	$-0.709494\pi$
$811$	−34.8372	−1.22330	−0.611649	+	0.791129i	$0.709494\pi$
$812$	−2.53590	−0.0889926
$813$	0	0
$814$	−16.3923	−0.574550
$815$	−10.1962	−0.357156
$816$	0	0
$817$	−40.7846	−1.42687
$818$	−45.0333	−1.57455
$819$	0	0
$820$	3.46410	0.120972
$821$	11.0718	0.386408	0.193204	−	0.981159i	$-0.438112\pi$
$821$	11.0718	0.386408	0.193204	+	0.981159i	$0.438112\pi$
$822$	0	0
$823$	42.9808	1.49822	0.749108	−	0.662448i	$-0.230483\pi$
$823$	42.9808	1.49822	0.749108	+	0.662448i	$0.230483\pi$
$824$	−8.53590	−0.297362
$825$	0	0
$826$	12.0000	0.417533
$827$	26.1962	0.910929	0.455465	−	0.890254i	$-0.349473\pi$
$827$	26.1962	0.910929	0.455465	+	0.890254i	$0.349473\pi$
$828$	0	0
$829$	−37.7128	−1.30982	−0.654910	−	0.755707i	$-0.727294\pi$
$829$	−37.7128	−1.30982	−0.654910	+	0.755707i	$0.727294\pi$
$830$	−26.7846	−0.929707
$831$	0	0
$832$	−4.00000	−0.138675
$833$	−6.46410	−0.223968
$834$	0	0
$835$	−18.5885	−0.643280
$836$	−6.92820	−0.239617
$837$	0	0
$838$	48.1577	1.66358
$839$	−22.7321	−0.784798	−0.392399	−	0.919795i	$-0.628355\pi$
$839$	−22.7321	−0.784798	−0.392399	+	0.919795i	$0.628355\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	2.53590	0.0873928
$843$	0	0
$844$	−0.196152	−0.00675184
$845$	−3.00000	−0.103203
$846$	0	0
$847$	−6.87564	−0.236250
$848$	30.0000	1.03020
$849$	0	0
$850$	−1.73205	−0.0594089
$851$	16.3923	0.561921
$852$	0	0
$853$	39.1769	1.34139	0.670696	−	0.741732i	$-0.265996\pi$
$853$	39.1769	1.34139	0.670696	+	0.741732i	$0.265996\pi$
$854$	−11.3205	−0.387380
$855$	0	0
$856$	−0.588457	−0.0201131
$857$	31.1769	1.06498	0.532492	−	0.846435i	$-0.321256\pi$
$857$	31.1769	1.06498	0.532492	+	0.846435i	$0.321256\pi$
$858$	0	0
$859$	−18.5359	−0.632437	−0.316218	−	0.948686i	$-0.602413\pi$
$859$	−18.5359	−0.632437	−0.316218	+	0.948686i	$0.602413\pi$
$860$	7.46410	0.254524
$861$	0	0
$862$	34.9808	1.19145
$863$	−36.9282	−1.25705	−0.628525	−	0.777789i	$-0.716341\pi$
$863$	−36.9282	−1.25705	−0.628525	+	0.777789i	$0.716341\pi$
$864$	0	0
$865$	−3.46410	−0.117783
$866$	−17.0718	−0.580123
$867$	0	0
$868$	4.92820	0.167274
$869$	21.0333	0.713507
$870$	0	0
$871$	40.0000	1.35535
$872$	−17.3205	−0.586546
$873$	0	0
$874$	20.7846	0.703050
$875$	−0.732051	−0.0247478
$876$	0	0
$877$	−42.7846	−1.44473	−0.722367	−	0.691510i	$-0.756946\pi$
$877$	−42.7846	−1.44473	−0.722367	+	0.691510i	$0.756946\pi$
$878$	−31.2679	−1.05524
$879$	0	0
$880$	−6.33975	−0.213713
$881$	6.67949	0.225038	0.112519	−	0.993650i	$-0.464108\pi$
$881$	6.67949	0.225038	0.112519	+	0.993650i	$0.464108\pi$
$882$	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
$886$	1.60770	0.0540116
$887$	21.1244	0.709286	0.354643	−	0.935002i	$-0.384602\pi$
$887$	21.1244	0.709286	0.354643	+	0.935002i	$0.384602\pi$
$888$	0	0
$889$	4.67949	0.156945
$890$	28.3923	0.951712
$891$	0	0
$892$	−5.60770	−0.187760
$893$	5.07180	0.169721
$894$	0	0
$895$	23.3205	0.779519
$896$	−8.87564	−0.296514
$897$	0	0
$898$	−23.5692	−0.786515
$899$	−23.3205	−0.777782
$900$	0	0
$901$	−6.00000	−0.199889
$902$	−7.60770	−0.253309
$903$	0	0
$904$	−30.0000	−0.997785
$905$	−18.3923	−0.611381
$906$	0	0
$907$	27.2679	0.905417	0.452709	−	0.891658i	$-0.350458\pi$
$907$	27.2679	0.905417	0.452709	+	0.891658i	$0.350458\pi$
$908$	−19.2679	−0.639429
$909$	0	0
$910$	−5.07180	−0.168128
$911$	−53.6603	−1.77784	−0.888922	−	0.458059i	$-0.848545\pi$
$911$	−53.6603	−1.77784	−0.888922	+	0.458059i	$0.848545\pi$
$912$	0	0
$913$	19.6077	0.648920
$914$	−8.28719	−0.274116
$915$	0	0
$916$	12.3923	0.409453
$917$	−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	$0.729333\pi$
$920$	3.80385	0.125409
$921$	0	0
$922$	−19.1769	−0.631558
$923$	−22.6410	−0.745238
$924$	0	0
$925$	−7.46410	−0.245418
$926$	−6.67949	−0.219502
$927$	0	0
$928$	−18.0000	−0.590879
$929$	−3.46410	−0.113653	−0.0568267	−	0.998384i	$-0.518098\pi$
$929$	−3.46410	−0.113653	−0.0568267	+	0.998384i	$0.518098\pi$
$930$	0	0
$931$	−35.3205	−1.15758
$932$	−6.00000	−0.196537
$933$	0	0
$934$	38.7846	1.26907
$935$	1.26795	0.0414664
$936$	0	0
$937$	36.6410	1.19701	0.598505	−	0.801119i	$-0.295762\pi$
$937$	36.6410	1.19701	0.598505	+	0.801119i	$0.295762\pi$
$938$	12.6795	0.414000
$939$	0	0
$940$	−0.928203	−0.0302747
$941$	0.248711	0.00810776	0.00405388	−	0.999992i	$-0.498710\pi$
$941$	0.248711	0.00810776	0.00405388	+	0.999992i	$0.498710\pi$
$942$	0	0
$943$	7.60770	0.247741
$944$	47.3205	1.54015
$945$	0	0
$946$	−16.3923	−0.532960
$947$	2.19615	0.0713654	0.0356827	−	0.999363i	$-0.488639\pi$
$947$	2.19615	0.0713654	0.0356827	+	0.999363i	$0.488639\pi$
$948$	0	0
$949$	57.5692	1.86878
$950$	−9.46410	−0.307056
$951$	0	0
$952$	1.26795	0.0410945
$953$	−32.7846	−1.06200	−0.530999	−	0.847373i	$-0.678183\pi$
$953$	−32.7846	−1.06200	−0.530999	+	0.847373i	$0.678183\pi$
$954$	0	0
$955$	−25.8564	−0.836694
$956$	−20.7846	−0.672222
$957$	0	0
$958$	9.80385	0.316748
$959$	0	0
$960$	0	0
$961$	14.3205	0.461952
$962$	−51.7128	−1.66729
$963$	0	0
$964$	−26.3923	−0.850039
$965$	−23.4641	−0.755336
$966$	0	0
$967$	−23.1769	−0.745319	−0.372660	−	0.927968i	$-0.621554\pi$
$967$	−23.1769	−0.745319	−0.372660	+	0.927968i	$0.621554\pi$
$968$	−16.2679	−0.522872
$969$	0	0
$970$	15.4641	0.496522
$971$	−5.75129	−0.184568	−0.0922838	−	0.995733i	$-0.529417\pi$
$971$	−5.75129	−0.184568	−0.0922838	+	0.995733i	$0.529417\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	46.7321	1.49739
$975$	0	0
$976$	−44.6410	−1.42892
$977$	−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
$980$	6.46410	0.206488
$981$	0	0
$982$	−69.9615	−2.23256
$983$	−55.2679	−1.76277	−0.881387	−	0.472395i	$-0.843390\pi$
$983$	−55.2679	−1.76277	−0.881387	+	0.472395i	$0.843390\pi$
$984$	0	0
$985$	−17.3205	−0.551877
$986$	6.00000	0.191079
$987$	0	0
$988$	−21.8564	−0.695345
$989$	16.3923	0.521245
$990$	0	0
$991$	48.9808	1.55593	0.777963	−	0.628311i	$-0.216253\pi$
$991$	48.9808	1.55593	0.777963	+	0.628311i	$0.216253\pi$
$992$	34.9808	1.11064
$993$	0	0
$994$	−7.17691	−0.227638
$995$	0.196152	0.00621845
$996$	0	0
$997$	18.3923	0.582490	0.291245	−	0.956648i	$-0.405930\pi$
$997$	18.3923	0.582490	0.291245	+	0.956648i	$0.405930\pi$
$998$	−2.87564	−0.0910269
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	765.2.a.g.1.1		2
3.2	odd	2		85.2.a.c.1.2	✓	2
5.4	even	2		3825.2.a.v.1.2		2
12.11	even	2		1360.2.a.k.1.2		2
15.2	even	4		425.2.b.d.324.3		4
15.8	even	4		425.2.b.d.324.2		4
15.14	odd	2		425.2.a.e.1.1		2
21.20	even	2		4165.2.a.t.1.2		2
24.5	odd	2		5440.2.a.bb.1.2		2
24.11	even	2		5440.2.a.bl.1.1		2
51.38	odd	4		1445.2.d.e.866.2		4
51.47	odd	4		1445.2.d.e.866.1		4
51.50	odd	2		1445.2.a.g.1.2		2
60.59	even	2		6800.2.a.bg.1.1		2
255.254	odd	2		7225.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.c.1.2	✓	2	3.2	odd	2
425.2.a.e.1.1		2	15.14	odd	2
425.2.b.d.324.2		4	15.8	even	4
425.2.b.d.324.3		4	15.2	even	4
765.2.a.g.1.1		2	1.1	even	1	trivial
1360.2.a.k.1.2		2	12.11	even	2
1445.2.a.g.1.2		2	51.50	odd	2
1445.2.d.e.866.1		4	51.47	odd	4
1445.2.d.e.866.2		4	51.38	odd	4
3825.2.a.v.1.2		2	5.4	even	2
4165.2.a.t.1.2		2	21.20	even	2
5440.2.a.bb.1.2		2	24.5	odd	2
5440.2.a.bl.1.1		2	24.11	even	2
6800.2.a.bg.1.1		2	60.59	even	2
7225.2.a.l.1.1		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)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{7} +1.73205 q^{8} +1.73205 q^{10} -1.26795 q^{11} -4.00000 q^{13} -1.26795 q^{14} -5.00000 q^{16} +1.00000 q^{17} +5.46410 q^{19} -1.00000 q^{20} +2.19615 q^{22} -2.19615 q^{23} +1.00000 q^{25} +6.92820 q^{26} +0.732051 q^{28} -3.46410 q^{29} +6.73205 q^{31} +5.19615 q^{32} -1.73205 q^{34} -0.732051 q^{35} -7.46410 q^{37} -9.46410 q^{38} -1.73205 q^{40} -3.46410 q^{41} -7.46410 q^{43} -1.26795 q^{44} +3.80385 q^{46} +0.928203 q^{47} -6.46410 q^{49} -1.73205 q^{50} -4.00000 q^{52} -6.00000 q^{53} +1.26795 q^{55} +1.26795 q^{56} +6.00000 q^{58} -9.46410 q^{59} +8.92820 q^{61} -11.6603 q^{62} +1.00000 q^{64} +4.00000 q^{65} -10.0000 q^{67} +1.00000 q^{68} +1.26795 q^{70} +5.66025 q^{71} -14.3923 q^{73} +12.9282 q^{74} +5.46410 q^{76} -0.928203 q^{77} -16.5885 q^{79} +5.00000 q^{80} +6.00000 q^{82} -15.4641 q^{83} -1.00000 q^{85} +12.9282 q^{86} -2.19615 q^{88} +16.3923 q^{89} -2.92820 q^{91} -2.19615 q^{92} -1.60770 q^{94} -5.46410 q^{95} +8.92820 q^{97} +11.1962 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{11} - 8 q^{13} - 6 q^{14} - 10 q^{16} + 2 q^{17} + 4 q^{19} - 2 q^{20} - 6 q^{22} + 6 q^{23} + 2 q^{25} - 2 q^{28} + 10 q^{31} + 2 q^{35} - 8 q^{37} - 12 q^{38}+ \cdots + 12 q^{98}+O(q^{100}) \) 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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