LMFDB - Embedded newform 7605.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, 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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.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:	$60.7262307372$
Analytic rank:	$0$
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 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205 q^{2} +5.46410 q^{4} +1.00000 q^{5} -2.46410 q^{7} +9.46410 q^{8} +O(q^{10})$	$q+2.73205 q^{2} +5.46410 q^{4} +1.00000 q^{5} -2.46410 q^{7} +9.46410 q^{8} +2.73205 q^{10} -3.46410 q^{11} -6.73205 q^{14} +14.9282 q^{16} +3.26795 q^{17} +1.46410 q^{19} +5.46410 q^{20} -9.46410 q^{22} +7.46410 q^{23} +1.00000 q^{25} -13.4641 q^{28} -0.732051 q^{29} +7.19615 q^{31} +21.8564 q^{32} +8.92820 q^{34} -2.46410 q^{35} +4.00000 q^{37} +4.00000 q^{38} +9.46410 q^{40} +8.73205 q^{41} -3.73205 q^{43} -18.9282 q^{44} +20.3923 q^{46} -10.1962 q^{47} -0.928203 q^{49} +2.73205 q^{50} -6.92820 q^{53} -3.46410 q^{55} -23.3205 q^{56} -2.00000 q^{58} +10.7321 q^{59} -2.46410 q^{61} +19.6603 q^{62} +29.8564 q^{64} +5.53590 q^{67} +17.8564 q^{68} -6.73205 q^{70} +9.26795 q^{71} +5.39230 q^{73} +10.9282 q^{74} +8.00000 q^{76} +8.53590 q^{77} -11.9282 q^{79} +14.9282 q^{80} +23.8564 q^{82} +9.46410 q^{83} +3.26795 q^{85} -10.1962 q^{86} -32.7846 q^{88} +4.73205 q^{89} +40.7846 q^{92} -27.8564 q^{94} +1.46410 q^{95} -9.53590 q^{97} -2.53590 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 + 12 * q^8	$ 2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} + 12 q^{8} + 2 q^{10} - 10 q^{14} + 16 q^{16} + 10 q^{17} - 4 q^{19} + 4 q^{20} - 12 q^{22} + 8 q^{23} + 2 q^{25} - 20 q^{28} + 2 q^{29} + 4 q^{31} + 16 q^{32} + 4 q^{34} + 2 q^{35} + 8 q^{37} + 8 q^{38} + 12 q^{40} + 14 q^{41} - 4 q^{43} - 24 q^{44} + 20 q^{46} - 10 q^{47} + 12 q^{49} + 2 q^{50} - 12 q^{56} - 4 q^{58} + 18 q^{59} + 2 q^{61} + 22 q^{62} + 32 q^{64} + 18 q^{67} + 8 q^{68} - 10 q^{70} + 22 q^{71} - 10 q^{73} + 8 q^{74} + 16 q^{76} + 24 q^{77} - 10 q^{79} + 16 q^{80} + 20 q^{82} + 12 q^{83} + 10 q^{85} - 10 q^{86} - 24 q^{88} + 6 q^{89} + 40 q^{92} - 28 q^{94} - 4 q^{95} - 26 q^{97} - 12 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 + 12 * q^8 + 2 * q^10 - 10 * q^14 + 16 * q^16 + 10 * q^17 - 4 * q^19 + 4 * q^20 - 12 * q^22 + 8 * q^23 + 2 * q^25 - 20 * q^28 + 2 * q^29 + 4 * q^31 + 16 * q^32 + 4 * q^34 + 2 * q^35 + 8 * q^37 + 8 * q^38 + 12 * q^40 + 14 * q^41 - 4 * q^43 - 24 * q^44 + 20 * q^46 - 10 * q^47 + 12 * q^49 + 2 * q^50 - 12 * q^56 - 4 * q^58 + 18 * q^59 + 2 * q^61 + 22 * q^62 + 32 * q^64 + 18 * q^67 + 8 * q^68 - 10 * q^70 + 22 * q^71 - 10 * q^73 + 8 * q^74 + 16 * q^76 + 24 * q^77 - 10 * q^79 + 16 * q^80 + 20 * q^82 + 12 * q^83 + 10 * q^85 - 10 * q^86 - 24 * q^88 + 6 * q^89 + 40 * q^92 - 28 * q^94 - 4 * q^95 - 26 * q^97 - 12 * q^98

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 $

$2$	2.73205	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$2$	2.73205	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$3$	0	0
$3$	0	0
$4$	5.46410	2.73205
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.46410	−0.931343	−0.465671	−	0.884958i	$-0.654187\pi$
$7$	−2.46410	−0.931343	−0.465671	+	0.884958i	$0.654187\pi$
$8$	9.46410	3.34607
$9$	0	0
$10$	2.73205	0.863950
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−6.73205	−1.79922
$15$	0	0
$16$	14.9282	3.73205
$17$	3.26795	0.792594	0.396297	−	0.918122i	$-0.370295\pi$
$17$	3.26795	0.792594	0.396297	+	0.918122i	$0.370295\pi$
$18$	0	0
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	5.46410	1.22181
$21$	0	0
$22$	−9.46410	−2.01775
$23$	7.46410	1.55637	0.778186	−	0.628033i	$-0.216140\pi$
$23$	7.46410	1.55637	0.778186	+	0.628033i	$0.216140\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−13.4641	−2.54448
$29$	−0.732051	−0.135938	−0.0679692	−	0.997687i	$-0.521652\pi$
$29$	−0.732051	−0.135938	−0.0679692	+	0.997687i	$0.521652\pi$
$30$	0	0
$31$	7.19615	1.29247	0.646234	−	0.763140i	$-0.276343\pi$
$31$	7.19615	1.29247	0.646234	+	0.763140i	$0.276343\pi$
$32$	21.8564	3.86370
$33$	0	0
$34$	8.92820	1.53117
$35$	−2.46410	−0.416509
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	9.46410	1.49641
$41$	8.73205	1.36372	0.681859	−	0.731484i	$-0.261172\pi$
$41$	8.73205	1.36372	0.681859	+	0.731484i	$0.261172\pi$
$42$	0	0
$43$	−3.73205	−0.569132	−0.284566	−	0.958656i	$-0.591850\pi$
$43$	−3.73205	−0.569132	−0.284566	+	0.958656i	$0.591850\pi$
$44$	−18.9282	−2.85353
$45$	0	0
$46$	20.3923	3.00668
$47$	−10.1962	−1.48726	−0.743631	−	0.668590i	$-0.766898\pi$
$47$	−10.1962	−1.48726	−0.743631	+	0.668590i	$0.766898\pi$
$48$	0	0
$49$	−0.928203	−0.132600
$50$	2.73205	0.386370
$51$	0	0
$52$	0	0
$53$	−6.92820	−0.951662	−0.475831	−	0.879537i	$-0.657853\pi$
$53$	−6.92820	−0.951662	−0.475831	+	0.879537i	$0.657853\pi$
$54$	0	0
$55$	−3.46410	−0.467099
$56$	−23.3205	−3.11633
$57$	0	0
$58$	−2.00000	−0.262613
$59$	10.7321	1.39719	0.698597	−	0.715515i	$-0.253808\pi$
$59$	10.7321	1.39719	0.698597	+	0.715515i	$0.253808\pi$
$60$	0	0
$61$	−2.46410	−0.315496	−0.157748	−	0.987479i	$-0.550423\pi$
$61$	−2.46410	−0.315496	−0.157748	+	0.987479i	$0.550423\pi$
$62$	19.6603	2.49685
$63$	0	0
$64$	29.8564	3.73205
$65$	0	0
$66$	0	0
$67$	5.53590	0.676318	0.338159	−	0.941089i	$-0.390196\pi$
$67$	5.53590	0.676318	0.338159	+	0.941089i	$0.390196\pi$
$68$	17.8564	2.16541
$69$	0	0
$70$	−6.73205	−0.804634
$71$	9.26795	1.09990	0.549952	−	0.835197i	$-0.314646\pi$
$71$	9.26795	1.09990	0.549952	+	0.835197i	$0.314646\pi$
$72$	0	0
$73$	5.39230	0.631122	0.315561	−	0.948905i	$-0.397807\pi$
$73$	5.39230	0.631122	0.315561	+	0.948905i	$0.397807\pi$
$74$	10.9282	1.27038
$75$	0	0
$76$	8.00000	0.917663
$77$	8.53590	0.972756
$78$	0	0
$79$	−11.9282	−1.34203	−0.671014	−	0.741445i	$-0.734141\pi$
$79$	−11.9282	−1.34203	−0.671014	+	0.741445i	$0.734141\pi$
$80$	14.9282	1.66902
$81$	0	0
$82$	23.8564	2.63450
$83$	9.46410	1.03882	0.519410	−	0.854525i	$-0.326152\pi$
$83$	9.46410	1.03882	0.519410	+	0.854525i	$0.326152\pi$
$84$	0	0
$85$	3.26795	0.354459
$86$	−10.1962	−1.09948
$87$	0	0
$88$	−32.7846	−3.49485
$89$	4.73205	0.501596	0.250798	−	0.968039i	$-0.419307\pi$
$89$	4.73205	0.501596	0.250798	+	0.968039i	$0.419307\pi$
$90$	0	0
$91$	0	0
$92$	40.7846	4.25209
$93$	0	0
$94$	−27.8564	−2.87317
$95$	1.46410	0.150214
$96$	0	0
$97$	−9.53590	−0.968224	−0.484112	−	0.875006i	$-0.660857\pi$
$97$	−9.53590	−0.968224	−0.484112	+	0.875006i	$0.660857\pi$
$98$	−2.53590	−0.256164
$99$	0	0
$100$	5.46410	0.546410
$101$	−8.92820	−0.888389	−0.444195	−	0.895930i	$-0.646510\pi$
$101$	−8.92820	−0.888389	−0.444195	+	0.895930i	$0.646510\pi$
$102$	0	0
$103$	7.19615	0.709058	0.354529	−	0.935045i	$-0.384641\pi$
$103$	7.19615	0.709058	0.354529	+	0.935045i	$0.384641\pi$
$104$	0	0
$105$	0	0
$106$	−18.9282	−1.83847
$107$	7.12436	0.688737	0.344369	−	0.938834i	$-0.388093\pi$
$107$	7.12436	0.688737	0.344369	+	0.938834i	$0.388093\pi$
$108$	0	0
$109$	−11.7321	−1.12373	−0.561863	−	0.827230i	$-0.689915\pi$
$109$	−11.7321	−1.12373	−0.561863	+	0.827230i	$0.689915\pi$
$110$	−9.46410	−0.902367
$111$	0	0
$112$	−36.7846	−3.47582
$113$	−12.5359	−1.17928	−0.589639	−	0.807667i	$-0.700730\pi$
$113$	−12.5359	−1.17928	−0.589639	+	0.807667i	$0.700730\pi$
$114$	0	0
$115$	7.46410	0.696031
$116$	−4.00000	−0.371391
$117$	0	0
$118$	29.3205	2.69917
$119$	−8.05256	−0.738177
$120$	0	0
$121$	1.00000	0.0909091
$122$	−6.73205	−0.609491
$123$	0	0
$124$	39.3205	3.53109
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.66025	−0.236059	−0.118030	−	0.993010i	$-0.537658\pi$
$127$	−2.66025	−0.236059	−0.118030	+	0.993010i	$0.537658\pi$
$128$	37.8564	3.34607
$129$	0	0
$130$	0	0
$131$	−4.73205	−0.413441	−0.206721	−	0.978400i	$-0.566279\pi$
$131$	−4.73205	−0.413441	−0.206721	+	0.978400i	$0.566279\pi$
$132$	0	0
$133$	−3.60770	−0.312827
$134$	15.1244	1.30655
$135$	0	0
$136$	30.9282	2.65207
$137$	15.1244	1.29216	0.646080	−	0.763269i	$-0.276407\pi$
$137$	15.1244	1.29216	0.646080	+	0.763269i	$0.276407\pi$
$138$	0	0
$139$	−4.07180	−0.345365	−0.172683	−	0.984978i	$-0.555244\pi$
$139$	−4.07180	−0.345365	−0.172683	+	0.984978i	$0.555244\pi$
$140$	−13.4641	−1.13792
$141$	0	0
$142$	25.3205	2.12485
$143$	0	0
$144$	0	0
$145$	−0.732051	−0.0607935
$146$	14.7321	1.21923
$147$	0	0
$148$	21.8564	1.79659
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	9.85641	0.802103	0.401051	−	0.916056i	$-0.368645\pi$
$151$	9.85641	0.802103	0.401051	+	0.916056i	$0.368645\pi$
$152$	13.8564	1.12390
$153$	0	0
$154$	23.3205	1.87922
$155$	7.19615	0.578009
$156$	0	0
$157$	6.26795	0.500237	0.250118	−	0.968215i	$-0.419530\pi$
$157$	6.26795	0.500237	0.250118	+	0.968215i	$0.419530\pi$
$158$	−32.5885	−2.59260
$159$	0	0
$160$	21.8564	1.72790
$161$	−18.3923	−1.44952
$162$	0	0
$163$	−12.4641	−0.976264	−0.488132	−	0.872770i	$-0.662322\pi$
$163$	−12.4641	−0.976264	−0.488132	+	0.872770i	$0.662322\pi$
$164$	47.7128	3.72574
$165$	0	0
$166$	25.8564	2.00685
$167$	10.5359	0.815292	0.407646	−	0.913140i	$-0.366350\pi$
$167$	10.5359	0.815292	0.407646	+	0.913140i	$0.366350\pi$
$168$	0	0
$169$	0	0
$170$	8.92820	0.684762
$171$	0	0
$172$	−20.3923	−1.55490
$173$	−6.73205	−0.511828	−0.255914	−	0.966700i	$-0.582376\pi$
$173$	−6.73205	−0.511828	−0.255914	+	0.966700i	$0.582376\pi$
$174$	0	0
$175$	−2.46410	−0.186269
$176$	−51.7128	−3.89800
$177$	0	0
$178$	12.9282	0.969010
$179$	15.1244	1.13045	0.565224	−	0.824938i	$-0.308790\pi$
$179$	15.1244	1.13045	0.565224	+	0.824938i	$0.308790\pi$
$180$	0	0
$181$	−21.4641	−1.59541	−0.797707	−	0.603045i	$-0.793954\pi$
$181$	−21.4641	−1.59541	−0.797707	+	0.603045i	$0.793954\pi$
$182$	0	0
$183$	0	0
$184$	70.6410	5.20772
$185$	4.00000	0.294086
$186$	0	0
$187$	−11.3205	−0.827838
$188$	−55.7128	−4.06327
$189$	0	0
$190$	4.00000	0.290191
$191$	−17.5167	−1.26746	−0.633731	−	0.773554i	$-0.718477\pi$
$191$	−17.5167	−1.26746	−0.633731	+	0.773554i	$0.718477\pi$
$192$	0	0
$193$	19.9282	1.43446	0.717232	−	0.696835i	$-0.245409\pi$
$193$	19.9282	1.43446	0.717232	+	0.696835i	$0.245409\pi$
$194$	−26.0526	−1.87046
$195$	0	0
$196$	−5.07180	−0.362271
$197$	10.9282	0.778602	0.389301	−	0.921111i	$-0.372717\pi$
$197$	10.9282	0.778602	0.389301	+	0.921111i	$0.372717\pi$
$198$	0	0
$199$	−15.0000	−1.06332	−0.531661	−	0.846957i	$-0.678432\pi$
$199$	−15.0000	−1.06332	−0.531661	+	0.846957i	$0.678432\pi$
$200$	9.46410	0.669213
$201$	0	0
$202$	−24.3923	−1.71624
$203$	1.80385	0.126605
$204$	0	0
$205$	8.73205	0.609873
$206$	19.6603	1.36979
$207$	0	0
$208$	0	0
$209$	−5.07180	−0.350824
$210$	0	0
$211$	18.4641	1.27112	0.635561	−	0.772051i	$-0.280769\pi$
$211$	18.4641	1.27112	0.635561	+	0.772051i	$0.280769\pi$
$212$	−37.8564	−2.59999
$213$	0	0
$214$	19.4641	1.33054
$215$	−3.73205	−0.254524
$216$	0	0
$217$	−17.7321	−1.20373
$218$	−32.0526	−2.17087
$219$	0	0
$220$	−18.9282	−1.27614
$221$	0	0
$222$	0	0
$223$	−15.8564	−1.06182	−0.530912	−	0.847427i	$-0.678150\pi$
$223$	−15.8564	−1.06182	−0.530912	+	0.847427i	$0.678150\pi$
$224$	−53.8564	−3.59843
$225$	0	0
$226$	−34.2487	−2.27819
$227$	5.66025	0.375684	0.187842	−	0.982199i	$-0.439851\pi$
$227$	5.66025	0.375684	0.187842	+	0.982199i	$0.439851\pi$
$228$	0	0
$229$	−2.39230	−0.158088	−0.0790440	−	0.996871i	$-0.525187\pi$
$229$	−2.39230	−0.158088	−0.0790440	+	0.996871i	$0.525187\pi$
$230$	20.3923	1.34463
$231$	0	0
$232$	−6.92820	−0.454859
$233$	3.85641	0.252642	0.126321	−	0.991989i	$-0.459683\pi$
$233$	3.85641	0.252642	0.126321	+	0.991989i	$0.459683\pi$
$234$	0	0
$235$	−10.1962	−0.665124
$236$	58.6410	3.81721
$237$	0	0
$238$	−22.0000	−1.42605
$239$	−1.46410	−0.0947049	−0.0473524	−	0.998878i	$-0.515078\pi$
$239$	−1.46410	−0.0947049	−0.0473524	+	0.998878i	$0.515078\pi$
$240$	0	0
$241$	−22.3923	−1.44242	−0.721208	−	0.692719i	$-0.756413\pi$
$241$	−22.3923	−1.44242	−0.721208	+	0.692719i	$0.756413\pi$
$242$	2.73205	0.175623
$243$	0	0
$244$	−13.4641	−0.861951
$245$	−0.928203	−0.0593007
$246$	0	0
$247$	0	0
$248$	68.1051	4.32468
$249$	0	0
$250$	2.73205	0.172790
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−25.8564	−1.62558
$254$	−7.26795	−0.456032
$255$	0	0
$256$	43.7128	2.73205
$257$	−7.26795	−0.453362	−0.226681	−	0.973969i	$-0.572788\pi$
$257$	−7.26795	−0.453362	−0.226681	+	0.973969i	$0.572788\pi$
$258$	0	0
$259$	−9.85641	−0.612447
$260$	0	0
$261$	0	0
$262$	−12.9282	−0.798707
$263$	23.6603	1.45895	0.729477	−	0.684005i	$-0.239764\pi$
$263$	23.6603	1.45895	0.729477	+	0.684005i	$0.239764\pi$
$264$	0	0
$265$	−6.92820	−0.425596
$266$	−9.85641	−0.604335
$267$	0	0
$268$	30.2487	1.84773
$269$	1.66025	0.101227	0.0506137	−	0.998718i	$-0.483882\pi$
$269$	1.66025	0.101227	0.0506137	+	0.998718i	$0.483882\pi$
$270$	0	0
$271$	−0.267949	−0.0162768	−0.00813838	−	0.999967i	$-0.502591\pi$
$271$	−0.267949	−0.0162768	−0.00813838	+	0.999967i	$0.502591\pi$
$272$	48.7846	2.95800
$273$	0	0
$274$	41.3205	2.49626
$275$	−3.46410	−0.208893
$276$	0	0
$277$	−28.2487	−1.69730	−0.848650	−	0.528954i	$-0.822584\pi$
$277$	−28.2487	−1.69730	−0.848650	+	0.528954i	$0.822584\pi$
$278$	−11.1244	−0.667195
$279$	0	0
$280$	−23.3205	−1.39367
$281$	−3.12436	−0.186383	−0.0931917	−	0.995648i	$-0.529707\pi$
$281$	−3.12436	−0.186383	−0.0931917	+	0.995648i	$0.529707\pi$
$282$	0	0
$283$	−3.73205	−0.221847	−0.110924	−	0.993829i	$-0.535381\pi$
$283$	−3.73205	−0.221847	−0.110924	+	0.993829i	$0.535381\pi$
$284$	50.6410	3.00499
$285$	0	0
$286$	0	0
$287$	−21.5167	−1.27009
$288$	0	0
$289$	−6.32051	−0.371795
$290$	−2.00000	−0.117444
$291$	0	0
$292$	29.4641	1.72426
$293$	−6.58846	−0.384902	−0.192451	−	0.981307i	$-0.561644\pi$
$293$	−6.58846	−0.384902	−0.192451	+	0.981307i	$0.561644\pi$
$294$	0	0
$295$	10.7321	0.624844
$296$	37.8564	2.20036
$297$	0	0
$298$	10.9282	0.633054
$299$	0	0
$300$	0	0
$301$	9.19615	0.530057
$302$	26.9282	1.54954
$303$	0	0
$304$	21.8564	1.25355
$305$	−2.46410	−0.141094
$306$	0	0
$307$	−25.9282	−1.47980	−0.739900	−	0.672716i	$-0.765127\pi$
$307$	−25.9282	−1.47980	−0.739900	+	0.672716i	$0.765127\pi$
$308$	46.6410	2.65762
$309$	0	0
$310$	19.6603	1.11663
$311$	0.196152	0.0111228	0.00556139	−	0.999985i	$-0.498230\pi$
$311$	0.196152	0.0111228	0.00556139	+	0.999985i	$0.498230\pi$
$312$	0	0
$313$	−21.1962	−1.19808	−0.599039	−	0.800720i	$-0.704450\pi$
$313$	−21.1962	−1.19808	−0.599039	+	0.800720i	$0.704450\pi$
$314$	17.1244	0.966383
$315$	0	0
$316$	−65.1769	−3.66649
$317$	−10.5359	−0.591755	−0.295878	−	0.955226i	$-0.595612\pi$
$317$	−10.5359	−0.591755	−0.295878	+	0.955226i	$0.595612\pi$
$318$	0	0
$319$	2.53590	0.141983
$320$	29.8564	1.66902
$321$	0	0
$322$	−50.2487	−2.80025
$323$	4.78461	0.266223
$324$	0	0
$325$	0	0
$326$	−34.0526	−1.88600
$327$	0	0
$328$	82.6410	4.56309
$329$	25.1244	1.38515
$330$	0	0
$331$	−16.1244	−0.886275	−0.443137	−	0.896454i	$-0.646135\pi$
$331$	−16.1244	−0.886275	−0.443137	+	0.896454i	$0.646135\pi$
$332$	51.7128	2.83811
$333$	0	0
$334$	28.7846	1.57502
$335$	5.53590	0.302458
$336$	0	0
$337$	−7.05256	−0.384177	−0.192089	−	0.981378i	$-0.561526\pi$
$337$	−7.05256	−0.384177	−0.192089	+	0.981378i	$0.561526\pi$
$338$	0	0
$339$	0	0
$340$	17.8564	0.968400
$341$	−24.9282	−1.34994
$342$	0	0
$343$	19.5359	1.05484
$344$	−35.3205	−1.90435
$345$	0	0
$346$	−18.3923	−0.988776
$347$	−13.4641	−0.722791	−0.361395	−	0.932413i	$-0.617700\pi$
$347$	−13.4641	−0.722791	−0.361395	+	0.932413i	$0.617700\pi$
$348$	0	0
$349$	−21.9808	−1.17660	−0.588302	−	0.808642i	$-0.700203\pi$
$349$	−21.9808	−1.17660	−0.588302	+	0.808642i	$0.700203\pi$
$350$	−6.73205	−0.359843
$351$	0	0
$352$	−75.7128	−4.03551
$353$	−23.0718	−1.22799	−0.613994	−	0.789311i	$-0.710438\pi$
$353$	−23.0718	−1.22799	−0.613994	+	0.789311i	$0.710438\pi$
$354$	0	0
$355$	9.26795	0.491892
$356$	25.8564	1.37039
$357$	0	0
$358$	41.3205	2.18386
$359$	−16.3397	−0.862379	−0.431189	−	0.902261i	$-0.641906\pi$
$359$	−16.3397	−0.862379	−0.431189	+	0.902261i	$0.641906\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	−58.6410	−3.08210
$363$	0	0
$364$	0	0
$365$	5.39230	0.282246
$366$	0	0
$367$	6.12436	0.319689	0.159844	−	0.987142i	$-0.448901\pi$
$367$	6.12436	0.319689	0.159844	+	0.987142i	$0.448901\pi$
$368$	111.426	5.80846
$369$	0	0
$370$	10.9282	0.568130
$371$	17.0718	0.886324
$372$	0	0
$373$	−26.1244	−1.35267	−0.676334	−	0.736595i	$-0.736432\pi$
$373$	−26.1244	−1.35267	−0.676334	+	0.736595i	$0.736432\pi$
$374$	−30.9282	−1.59926
$375$	0	0
$376$	−96.4974	−4.97647
$377$	0	0
$378$	0	0
$379$	16.2679	0.835628	0.417814	−	0.908532i	$-0.362796\pi$
$379$	16.2679	0.835628	0.417814	+	0.908532i	$0.362796\pi$
$380$	8.00000	0.410391
$381$	0	0
$382$	−47.8564	−2.44855
$383$	−25.6603	−1.31118	−0.655589	−	0.755118i	$-0.727580\pi$
$383$	−25.6603	−1.31118	−0.655589	+	0.755118i	$0.727580\pi$
$384$	0	0
$385$	8.53590	0.435030
$386$	54.4449	2.77117
$387$	0	0
$388$	−52.1051	−2.64524
$389$	−10.5359	−0.534191	−0.267096	−	0.963670i	$-0.586064\pi$
$389$	−10.5359	−0.534191	−0.267096	+	0.963670i	$0.586064\pi$
$390$	0	0
$391$	24.3923	1.23357
$392$	−8.78461	−0.443690
$393$	0	0
$394$	29.8564	1.50414
$395$	−11.9282	−0.600173
$396$	0	0
$397$	10.3205	0.517971	0.258986	−	0.965881i	$-0.416612\pi$
$397$	10.3205	0.517971	0.258986	+	0.965881i	$0.416612\pi$
$398$	−40.9808	−2.05418
$399$	0	0
$400$	14.9282	0.746410
$401$	−19.3205	−0.964820	−0.482410	−	0.875946i	$-0.660238\pi$
$401$	−19.3205	−0.964820	−0.482410	+	0.875946i	$0.660238\pi$
$402$	0	0
$403$	0	0
$404$	−48.7846	−2.42713
$405$	0	0
$406$	4.92820	0.244583
$407$	−13.8564	−0.686837
$408$	0	0
$409$	−14.6603	−0.724903	−0.362451	−	0.932003i	$-0.618060\pi$
$409$	−14.6603	−0.724903	−0.362451	+	0.932003i	$0.618060\pi$
$410$	23.8564	1.17818
$411$	0	0
$412$	39.3205	1.93718
$413$	−26.4449	−1.30127
$414$	0	0
$415$	9.46410	0.464574
$416$	0	0
$417$	0	0
$418$	−13.8564	−0.677739
$419$	38.4449	1.87815	0.939077	−	0.343706i	$-0.111682\pi$
$419$	38.4449	1.87815	0.939077	+	0.343706i	$0.111682\pi$
$420$	0	0
$421$	7.58846	0.369839	0.184919	−	0.982754i	$-0.440798\pi$
$421$	7.58846	0.369839	0.184919	+	0.982754i	$0.440798\pi$
$422$	50.4449	2.45562
$423$	0	0
$424$	−65.5692	−3.18432
$425$	3.26795	0.158519
$426$	0	0
$427$	6.07180	0.293835
$428$	38.9282	1.88167
$429$	0	0
$430$	−10.1962	−0.491702
$431$	11.3205	0.545290	0.272645	−	0.962115i	$-0.412102\pi$
$431$	11.3205	0.545290	0.272645	+	0.962115i	$0.412102\pi$
$432$	0	0
$433$	15.1962	0.730280	0.365140	−	0.930953i	$-0.381021\pi$
$433$	15.1962	0.730280	0.365140	+	0.930953i	$0.381021\pi$
$434$	−48.4449	−2.32543
$435$	0	0
$436$	−64.1051	−3.07008
$437$	10.9282	0.522767
$438$	0	0
$439$	−5.39230	−0.257361	−0.128680	−	0.991686i	$-0.541074\pi$
$439$	−5.39230	−0.257361	−0.128680	+	0.991686i	$0.541074\pi$
$440$	−32.7846	−1.56294
$441$	0	0
$442$	0	0
$443$	21.1244	1.00365	0.501824	−	0.864970i	$-0.332662\pi$
$443$	21.1244	1.00365	0.501824	+	0.864970i	$0.332662\pi$
$444$	0	0
$445$	4.73205	0.224321
$446$	−43.3205	−2.05129
$447$	0	0
$448$	−73.5692	−3.47582
$449$	24.9282	1.17643	0.588217	−	0.808703i	$-0.299830\pi$
$449$	24.9282	1.17643	0.588217	+	0.808703i	$0.299830\pi$
$450$	0	0
$451$	−30.2487	−1.42436
$452$	−68.4974	−3.22185
$453$	0	0
$454$	15.4641	0.725766
$455$	0	0
$456$	0	0
$457$	−15.3923	−0.720022	−0.360011	−	0.932948i	$-0.617227\pi$
$457$	−15.3923	−0.720022	−0.360011	+	0.932948i	$0.617227\pi$
$458$	−6.53590	−0.305402
$459$	0	0
$460$	40.7846	1.90159
$461$	33.5167	1.56103	0.780513	−	0.625139i	$-0.214958\pi$
$461$	33.5167	1.56103	0.780513	+	0.625139i	$0.214958\pi$
$462$	0	0
$463$	−39.7846	−1.84895	−0.924474	−	0.381246i	$-0.875495\pi$
$463$	−39.7846	−1.84895	−0.924474	+	0.381246i	$0.875495\pi$
$464$	−10.9282	−0.507329
$465$	0	0
$466$	10.5359	0.488066
$467$	6.33975	0.293368	0.146684	−	0.989183i	$-0.453140\pi$
$467$	6.33975	0.293368	0.146684	+	0.989183i	$0.453140\pi$
$468$	0	0
$469$	−13.6410	−0.629884
$470$	−27.8564	−1.28492
$471$	0	0
$472$	101.569	4.67510
$473$	12.9282	0.594439
$474$	0	0
$475$	1.46410	0.0671776
$476$	−44.0000	−2.01674
$477$	0	0
$478$	−4.00000	−0.182956
$479$	−29.9090	−1.36658	−0.683288	−	0.730149i	$-0.739451\pi$
$479$	−29.9090	−1.36658	−0.683288	+	0.730149i	$0.739451\pi$
$480$	0	0
$481$	0	0
$482$	−61.1769	−2.78653
$483$	0	0
$484$	5.46410	0.248368
$485$	−9.53590	−0.433003
$486$	0	0
$487$	−20.7846	−0.941841	−0.470920	−	0.882176i	$-0.656078\pi$
$487$	−20.7846	−0.941841	−0.470920	+	0.882176i	$0.656078\pi$
$488$	−23.3205	−1.05567
$489$	0	0
$490$	−2.53590	−0.114560
$491$	17.1244	0.772811	0.386406	−	0.922329i	$-0.373716\pi$
$491$	17.1244	0.772811	0.386406	+	0.922329i	$0.373716\pi$
$492$	0	0
$493$	−2.39230	−0.107744
$494$	0	0
$495$	0	0
$496$	107.426	4.82355
$497$	−22.8372	−1.02439
$498$	0	0
$499$	20.3923	0.912885	0.456442	−	0.889753i	$-0.349123\pi$
$499$	20.3923	0.912885	0.456442	+	0.889753i	$0.349123\pi$
$500$	5.46410	0.244362
$501$	0	0
$502$	0	0
$503$	−10.7846	−0.480862	−0.240431	−	0.970666i	$-0.577289\pi$
$503$	−10.7846	−0.480862	−0.240431	+	0.970666i	$0.577289\pi$
$504$	0	0
$505$	−8.92820	−0.397300
$506$	−70.6410	−3.14038
$507$	0	0
$508$	−14.5359	−0.644926
$509$	−0.928203	−0.0411419	−0.0205709	−	0.999788i	$-0.506548\pi$
$509$	−0.928203	−0.0411419	−0.0205709	+	0.999788i	$0.506548\pi$
$510$	0	0
$511$	−13.2872	−0.587791
$512$	43.7128	1.93185
$513$	0	0
$514$	−19.8564	−0.875829
$515$	7.19615	0.317100
$516$	0	0
$517$	35.3205	1.55339
$518$	−26.9282	−1.18316
$519$	0	0
$520$	0	0
$521$	9.26795	0.406036	0.203018	−	0.979175i	$-0.434925\pi$
$521$	9.26795	0.406036	0.203018	+	0.979175i	$0.434925\pi$
$522$	0	0
$523$	−30.5359	−1.33524	−0.667621	−	0.744501i	$-0.732687\pi$
$523$	−30.5359	−1.33524	−0.667621	+	0.744501i	$0.732687\pi$
$524$	−25.8564	−1.12954
$525$	0	0
$526$	64.6410	2.81848
$527$	23.5167	1.02440
$528$	0	0
$529$	32.7128	1.42230
$530$	−18.9282	−0.822189
$531$	0	0
$532$	−19.7128	−0.854659
$533$	0	0
$534$	0	0
$535$	7.12436	0.308013
$536$	52.3923	2.26300
$537$	0	0
$538$	4.53590	0.195556
$539$	3.21539	0.138497
$540$	0	0
$541$	13.5885	0.584213	0.292107	−	0.956386i	$-0.405644\pi$
$541$	13.5885	0.584213	0.292107	+	0.956386i	$0.405644\pi$
$542$	−0.732051	−0.0314443
$543$	0	0
$544$	71.4256	3.06235
$545$	−11.7321	−0.502546
$546$	0	0
$547$	−7.33975	−0.313825	−0.156912	−	0.987613i	$-0.550154\pi$
$547$	−7.33975	−0.313825	−0.156912	+	0.987613i	$0.550154\pi$
$548$	82.6410	3.53025
$549$	0	0
$550$	−9.46410	−0.403551
$551$	−1.07180	−0.0456601
$552$	0	0
$553$	29.3923	1.24989
$554$	−77.1769	−3.27893
$555$	0	0
$556$	−22.2487	−0.943556
$557$	14.5359	0.615906	0.307953	−	0.951402i	$-0.400356\pi$
$557$	14.5359	0.615906	0.307953	+	0.951402i	$0.400356\pi$
$558$	0	0
$559$	0	0
$560$	−36.7846	−1.55443
$561$	0	0
$562$	−8.53590	−0.360065
$563$	−3.46410	−0.145994	−0.0729972	−	0.997332i	$-0.523256\pi$
$563$	−3.46410	−0.145994	−0.0729972	+	0.997332i	$0.523256\pi$
$564$	0	0
$565$	−12.5359	−0.527389
$566$	−10.1962	−0.428576
$567$	0	0
$568$	87.7128	3.68035
$569$	33.8038	1.41713	0.708566	−	0.705645i	$-0.249343\pi$
$569$	33.8038	1.41713	0.708566	+	0.705645i	$0.249343\pi$
$570$	0	0
$571$	23.8564	0.998360	0.499180	−	0.866498i	$-0.333635\pi$
$571$	23.8564	0.998360	0.499180	+	0.866498i	$0.333635\pi$
$572$	0	0
$573$	0	0
$574$	−58.7846	−2.45362
$575$	7.46410	0.311275
$576$	0	0
$577$	−43.7128	−1.81979	−0.909894	−	0.414841i	$-0.863837\pi$
$577$	−43.7128	−1.81979	−0.909894	+	0.414841i	$0.863837\pi$
$578$	−17.2679	−0.718252
$579$	0	0
$580$	−4.00000	−0.166091
$581$	−23.3205	−0.967498
$582$	0	0
$583$	24.0000	0.993978
$584$	51.0333	2.11177
$585$	0	0
$586$	−18.0000	−0.743573
$587$	−8.33975	−0.344218	−0.172109	−	0.985078i	$-0.555058\pi$
$587$	−8.33975	−0.344218	−0.172109	+	0.985078i	$0.555058\pi$
$588$	0	0
$589$	10.5359	0.434124
$590$	29.3205	1.20711
$591$	0	0
$592$	59.7128	2.45418
$593$	−13.3205	−0.547008	−0.273504	−	0.961871i	$-0.588183\pi$
$593$	−13.3205	−0.547008	−0.273504	+	0.961871i	$0.588183\pi$
$594$	0	0
$595$	−8.05256	−0.330123
$596$	21.8564	0.895273
$597$	0	0
$598$	0	0
$599$	−1.12436	−0.0459399	−0.0229700	−	0.999736i	$-0.507312\pi$
$599$	−1.12436	−0.0459399	−0.0229700	+	0.999736i	$0.507312\pi$
$600$	0	0
$601$	26.2487	1.07071	0.535354	−	0.844628i	$-0.320178\pi$
$601$	26.2487	1.07071	0.535354	+	0.844628i	$0.320178\pi$
$602$	25.1244	1.02399
$603$	0	0
$604$	53.8564	2.19139
$605$	1.00000	0.0406558
$606$	0	0
$607$	38.3923	1.55830	0.779148	−	0.626840i	$-0.215652\pi$
$607$	38.3923	1.55830	0.779148	+	0.626840i	$0.215652\pi$
$608$	32.0000	1.29777
$609$	0	0
$610$	−6.73205	−0.272573
$611$	0	0
$612$	0	0
$613$	43.7846	1.76844	0.884222	−	0.467067i	$-0.154689\pi$
$613$	43.7846	1.76844	0.884222	+	0.467067i	$0.154689\pi$
$614$	−70.8372	−2.85876
$615$	0	0
$616$	80.7846	3.25490
$617$	−44.6410	−1.79718	−0.898590	−	0.438790i	$-0.855407\pi$
$617$	−44.6410	−1.79718	−0.898590	+	0.438790i	$0.855407\pi$
$618$	0	0
$619$	−19.5885	−0.787327	−0.393663	−	0.919255i	$-0.628792\pi$
$619$	−19.5885	−0.787327	−0.393663	+	0.919255i	$0.628792\pi$
$620$	39.3205	1.57915
$621$	0	0
$622$	0.535898	0.0214876
$623$	−11.6603	−0.467158
$624$	0	0
$625$	1.00000	0.0400000
$626$	−57.9090	−2.31451
$627$	0	0
$628$	34.2487	1.36667
$629$	13.0718	0.521207
$630$	0	0
$631$	41.5885	1.65561	0.827805	−	0.561016i	$-0.189589\pi$
$631$	41.5885	1.65561	0.827805	+	0.561016i	$0.189589\pi$
$632$	−112.890	−4.49051
$633$	0	0
$634$	−28.7846	−1.14318
$635$	−2.66025	−0.105569
$636$	0	0
$637$	0	0
$638$	6.92820	0.274290
$639$	0	0
$640$	37.8564	1.49641
$641$	2.78461	0.109985	0.0549927	−	0.998487i	$-0.482486\pi$
$641$	2.78461	0.109985	0.0549927	+	0.998487i	$0.482486\pi$
$642$	0	0
$643$	37.7846	1.49008	0.745040	−	0.667020i	$-0.232430\pi$
$643$	37.7846	1.49008	0.745040	+	0.667020i	$0.232430\pi$
$644$	−100.497	−3.96015
$645$	0	0
$646$	13.0718	0.514303
$647$	−2.00000	−0.0786281	−0.0393141	−	0.999227i	$-0.512517\pi$
$647$	−2.00000	−0.0786281	−0.0393141	+	0.999227i	$0.512517\pi$
$648$	0	0
$649$	−37.1769	−1.45932
$650$	0	0
$651$	0	0
$652$	−68.1051	−2.66720
$653$	45.5167	1.78120	0.890602	−	0.454783i	$-0.150283\pi$
$653$	45.5167	1.78120	0.890602	+	0.454783i	$0.150283\pi$
$654$	0	0
$655$	−4.73205	−0.184897
$656$	130.354	5.08946
$657$	0	0
$658$	68.6410	2.67591
$659$	−25.1769	−0.980753	−0.490377	−	0.871511i	$-0.663141\pi$
$659$	−25.1769	−0.980753	−0.490377	+	0.871511i	$0.663141\pi$
$660$	0	0
$661$	−17.0526	−0.663268	−0.331634	−	0.943408i	$-0.607600\pi$
$661$	−17.0526	−0.663268	−0.331634	+	0.943408i	$0.607600\pi$
$662$	−44.0526	−1.71215
$663$	0	0
$664$	89.5692	3.47596
$665$	−3.60770	−0.139900
$666$	0	0
$667$	−5.46410	−0.211571
$668$	57.5692	2.22742
$669$	0	0
$670$	15.1244	0.584305
$671$	8.53590	0.329525
$672$	0	0
$673$	−1.33975	−0.0516434	−0.0258217	−	0.999667i	$-0.508220\pi$
$673$	−1.33975	−0.0516434	−0.0258217	+	0.999667i	$0.508220\pi$
$674$	−19.2679	−0.742174
$675$	0	0
$676$	0	0
$677$	8.78461	0.337620	0.168810	−	0.985649i	$-0.446008\pi$
$677$	8.78461	0.337620	0.168810	+	0.985649i	$0.446008\pi$
$678$	0	0
$679$	23.4974	0.901748
$680$	30.9282	1.18604
$681$	0	0
$682$	−68.1051	−2.60788
$683$	44.7321	1.71162	0.855812	−	0.517287i	$-0.173058\pi$
$683$	44.7321	1.71162	0.855812	+	0.517287i	$0.173058\pi$
$684$	0	0
$685$	15.1244	0.577872
$686$	53.3731	2.03779
$687$	0	0
$688$	−55.7128	−2.12403
$689$	0	0
$690$	0	0
$691$	12.4115	0.472157	0.236079	−	0.971734i	$-0.424138\pi$
$691$	12.4115	0.472157	0.236079	+	0.971734i	$0.424138\pi$
$692$	−36.7846	−1.39834
$693$	0	0
$694$	−36.7846	−1.39632
$695$	−4.07180	−0.154452
$696$	0	0
$697$	28.5359	1.08087
$698$	−60.0526	−2.27302
$699$	0	0
$700$	−13.4641	−0.508895
$701$	−32.7846	−1.23826	−0.619129	−	0.785289i	$-0.712514\pi$
$701$	−32.7846	−1.23826	−0.619129	+	0.785289i	$0.712514\pi$
$702$	0	0
$703$	5.85641	0.220879
$704$	−103.426	−3.89800
$705$	0	0
$706$	−63.0333	−2.37229
$707$	22.0000	0.827395
$708$	0	0
$709$	13.3397	0.500985	0.250492	−	0.968119i	$-0.419408\pi$
$709$	13.3397	0.500985	0.250492	+	0.968119i	$0.419408\pi$
$710$	25.3205	0.950262
$711$	0	0
$712$	44.7846	1.67837
$713$	53.7128	2.01156
$714$	0	0
$715$	0	0
$716$	82.6410	3.08844
$717$	0	0
$718$	−44.6410	−1.66599
$719$	−31.2679	−1.16610	−0.583049	−	0.812437i	$-0.698140\pi$
$719$	−31.2679	−1.16610	−0.583049	+	0.812437i	$0.698140\pi$
$720$	0	0
$721$	−17.7321	−0.660376
$722$	−46.0526	−1.71390
$723$	0	0
$724$	−117.282	−4.35875
$725$	−0.732051	−0.0271877
$726$	0	0
$727$	−9.73205	−0.360942	−0.180471	−	0.983580i	$-0.557762\pi$
$727$	−9.73205	−0.360942	−0.180471	+	0.983580i	$0.557762\pi$
$728$	0	0
$729$	0	0
$730$	14.7321	0.545258
$731$	−12.1962	−0.451091
$732$	0	0
$733$	−32.3205	−1.19379	−0.596893	−	0.802321i	$-0.703598\pi$
$733$	−32.3205	−1.19379	−0.596893	+	0.802321i	$0.703598\pi$
$734$	16.7321	0.617591
$735$	0	0
$736$	163.138	6.01336
$737$	−19.1769	−0.706391
$738$	0	0
$739$	−10.3923	−0.382287	−0.191144	−	0.981562i	$-0.561220\pi$
$739$	−10.3923	−0.382287	−0.191144	+	0.981562i	$0.561220\pi$
$740$	21.8564	0.803457
$741$	0	0
$742$	46.6410	1.71225
$743$	−25.9090	−0.950508	−0.475254	−	0.879849i	$-0.657644\pi$
$743$	−25.9090	−0.950508	−0.475254	+	0.879849i	$0.657644\pi$
$744$	0	0
$745$	4.00000	0.146549
$746$	−71.3731	−2.61315
$747$	0	0
$748$	−61.8564	−2.26169
$749$	−17.5551	−0.641451
$750$	0	0
$751$	13.3205	0.486072	0.243036	−	0.970017i	$-0.421857\pi$
$751$	13.3205	0.486072	0.243036	+	0.970017i	$0.421857\pi$
$752$	−152.210	−5.55054
$753$	0	0
$754$	0	0
$755$	9.85641	0.358711
$756$	0	0
$757$	13.0718	0.475103	0.237551	−	0.971375i	$-0.423655\pi$
$757$	13.0718	0.475103	0.237551	+	0.971375i	$0.423655\pi$
$758$	44.4449	1.61431
$759$	0	0
$760$	13.8564	0.502625
$761$	21.8564	0.792294	0.396147	−	0.918187i	$-0.370347\pi$
$761$	21.8564	0.792294	0.396147	+	0.918187i	$0.370347\pi$
$762$	0	0
$763$	28.9090	1.04657
$764$	−95.7128	−3.46277
$765$	0	0
$766$	−70.1051	−2.53300
$767$	0	0
$768$	0	0
$769$	−29.6077	−1.06768	−0.533840	−	0.845585i	$-0.679252\pi$
$769$	−29.6077	−1.06768	−0.533840	+	0.845585i	$0.679252\pi$
$770$	23.3205	0.840413
$771$	0	0
$772$	108.890	3.91903
$773$	19.8564	0.714185	0.357093	−	0.934069i	$-0.383768\pi$
$773$	19.8564	0.714185	0.357093	+	0.934069i	$0.383768\pi$
$774$	0	0
$775$	7.19615	0.258493
$776$	−90.2487	−3.23974
$777$	0	0
$778$	−28.7846	−1.03198
$779$	12.7846	0.458056
$780$	0	0
$781$	−32.1051	−1.14881
$782$	66.6410	2.38308
$783$	0	0
$784$	−13.8564	−0.494872
$785$	6.26795	0.223713
$786$	0	0
$787$	28.4641	1.01464	0.507318	−	0.861759i	$-0.330637\pi$
$787$	28.4641	1.01464	0.507318	+	0.861759i	$0.330637\pi$
$788$	59.7128	2.12718
$789$	0	0
$790$	−32.5885	−1.15945
$791$	30.8897	1.09831
$792$	0	0
$793$	0	0
$794$	28.1962	1.00064
$795$	0	0
$796$	−81.9615	−2.90505
$797$	−28.4449	−1.00757	−0.503784	−	0.863829i	$-0.668059\pi$
$797$	−28.4449	−1.00757	−0.503784	+	0.863829i	$0.668059\pi$
$798$	0	0
$799$	−33.3205	−1.17879
$800$	21.8564	0.772741
$801$	0	0
$802$	−52.7846	−1.86389
$803$	−18.6795	−0.659185
$804$	0	0
$805$	−18.3923	−0.648244
$806$	0	0
$807$	0	0
$808$	−84.4974	−2.97261
$809$	−36.0000	−1.26569	−0.632846	−	0.774277i	$-0.718114\pi$
$809$	−36.0000	−1.26569	−0.632846	+	0.774277i	$0.718114\pi$
$810$	0	0
$811$	39.0526	1.37132	0.685660	−	0.727922i	$-0.259514\pi$
$811$	39.0526	1.37132	0.685660	+	0.727922i	$0.259514\pi$
$812$	9.85641	0.345892
$813$	0	0
$814$	−37.8564	−1.32687
$815$	−12.4641	−0.436598
$816$	0	0
$817$	−5.46410	−0.191165
$818$	−40.0526	−1.40040
$819$	0	0
$820$	47.7128	1.66620
$821$	12.9282	0.451197	0.225599	−	0.974220i	$-0.427566\pi$
$821$	12.9282	0.451197	0.225599	+	0.974220i	$0.427566\pi$
$822$	0	0
$823$	49.1769	1.71420	0.857100	−	0.515150i	$-0.172264\pi$
$823$	49.1769	1.71420	0.857100	+	0.515150i	$0.172264\pi$
$824$	68.1051	2.37255
$825$	0	0
$826$	−72.2487	−2.51385
$827$	28.5885	0.994118	0.497059	−	0.867717i	$-0.334413\pi$
$827$	28.5885	0.994118	0.497059	+	0.867717i	$0.334413\pi$
$828$	0	0
$829$	24.1769	0.839699	0.419849	−	0.907594i	$-0.362083\pi$
$829$	24.1769	0.839699	0.419849	+	0.907594i	$0.362083\pi$
$830$	25.8564	0.897489
$831$	0	0
$832$	0	0
$833$	−3.03332	−0.105098
$834$	0	0
$835$	10.5359	0.364610
$836$	−27.7128	−0.958468
$837$	0	0
$838$	105.033	3.62832
$839$	−27.4641	−0.948166	−0.474083	−	0.880480i	$-0.657220\pi$
$839$	−27.4641	−0.948166	−0.474083	+	0.880480i	$0.657220\pi$
$840$	0	0
$841$	−28.4641	−0.981521
$842$	20.7321	0.714474
$843$	0	0
$844$	100.890	3.47277
$845$	0	0
$846$	0	0
$847$	−2.46410	−0.0846675
$848$	−103.426	−3.55165
$849$	0	0
$850$	8.92820	0.306235
$851$	29.8564	1.02346
$852$	0	0
$853$	37.3923	1.28029	0.640144	−	0.768255i	$-0.278875\pi$
$853$	37.3923	1.28029	0.640144	+	0.768255i	$0.278875\pi$
$854$	16.5885	0.567645
$855$	0	0
$856$	67.4256	2.30456
$857$	−7.12436	−0.243363	−0.121682	−	0.992569i	$-0.538829\pi$
$857$	−7.12436	−0.243363	−0.121682	+	0.992569i	$0.538829\pi$
$858$	0	0
$859$	−33.7846	−1.15272	−0.576358	−	0.817197i	$-0.695527\pi$
$859$	−33.7846	−1.15272	−0.576358	+	0.817197i	$0.695527\pi$
$860$	−20.3923	−0.695372
$861$	0	0
$862$	30.9282	1.05342
$863$	13.6077	0.463211	0.231606	−	0.972810i	$-0.425602\pi$
$863$	13.6077	0.463211	0.231606	+	0.972810i	$0.425602\pi$
$864$	0	0
$865$	−6.73205	−0.228897
$866$	41.5167	1.41079
$867$	0	0
$868$	−96.8897	−3.28865
$869$	41.3205	1.40170
$870$	0	0
$871$	0	0
$872$	−111.033	−3.76006
$873$	0	0
$874$	29.8564	1.00991
$875$	−2.46410	−0.0833018
$876$	0	0
$877$	50.6410	1.71003	0.855013	−	0.518607i	$-0.173549\pi$
$877$	50.6410	1.71003	0.855013	+	0.518607i	$0.173549\pi$
$878$	−14.7321	−0.497183
$879$	0	0
$880$	−51.7128	−1.74324
$881$	−37.1769	−1.25252	−0.626261	−	0.779613i	$-0.715416\pi$
$881$	−37.1769	−1.25252	−0.626261	+	0.779613i	$0.715416\pi$
$882$	0	0
$883$	16.6603	0.560662	0.280331	−	0.959903i	$-0.409556\pi$
$883$	16.6603	0.560662	0.280331	+	0.959903i	$0.409556\pi$
$884$	0	0
$885$	0	0
$886$	57.7128	1.93890
$887$	−48.9808	−1.64461	−0.822307	−	0.569045i	$-0.807313\pi$
$887$	−48.9808	−1.64461	−0.822307	+	0.569045i	$0.807313\pi$
$888$	0	0
$889$	6.55514	0.219852
$890$	12.9282	0.433354
$891$	0	0
$892$	−86.6410	−2.90096
$893$	−14.9282	−0.499553
$894$	0	0
$895$	15.1244	0.505551
$896$	−93.2820	−3.11633
$897$	0	0
$898$	68.1051	2.27270
$899$	−5.26795	−0.175696
$900$	0	0
$901$	−22.6410	−0.754282
$902$	−82.6410	−2.75164
$903$	0	0
$904$	−118.641	−3.94594
$905$	−21.4641	−0.713491
$906$	0	0
$907$	−31.4641	−1.04475	−0.522374	−	0.852716i	$-0.674954\pi$
$907$	−31.4641	−1.04475	−0.522374	+	0.852716i	$0.674954\pi$
$908$	30.9282	1.02639
$909$	0	0
$910$	0	0
$911$	−47.7128	−1.58080	−0.790398	−	0.612594i	$-0.790126\pi$
$911$	−47.7128	−1.58080	−0.790398	+	0.612594i	$0.790126\pi$
$912$	0	0
$913$	−32.7846	−1.08501
$914$	−42.0526	−1.39098
$915$	0	0
$916$	−13.0718	−0.431904
$917$	11.6603	0.385056
$918$	0	0
$919$	−32.9282	−1.08620	−0.543101	−	0.839668i	$-0.682750\pi$
$919$	−32.9282	−1.08620	−0.543101	+	0.839668i	$0.682750\pi$
$920$	70.6410	2.32897
$921$	0	0
$922$	91.5692	3.01567
$923$	0	0
$924$	0	0
$925$	4.00000	0.131519
$926$	−108.694	−3.57189
$927$	0	0
$928$	−16.0000	−0.525226
$929$	−19.3205	−0.633885	−0.316943	−	0.948445i	$-0.602656\pi$
$929$	−19.3205	−0.633885	−0.316943	+	0.948445i	$0.602656\pi$
$930$	0	0
$931$	−1.35898	−0.0445389
$932$	21.0718	0.690230
$933$	0	0
$934$	17.3205	0.566744
$935$	−11.3205	−0.370220
$936$	0	0
$937$	−16.2487	−0.530822	−0.265411	−	0.964135i	$-0.585508\pi$
$937$	−16.2487	−0.530822	−0.265411	+	0.964135i	$0.585508\pi$
$938$	−37.2679	−1.21684
$939$	0	0
$940$	−55.7128	−1.81715
$941$	−40.5885	−1.32315	−0.661573	−	0.749881i	$-0.730111\pi$
$941$	−40.5885	−1.32315	−0.661573	+	0.749881i	$0.730111\pi$
$942$	0	0
$943$	65.1769	2.12245
$944$	160.210	5.21440
$945$	0	0
$946$	35.3205	1.14837
$947$	36.5885	1.18897	0.594483	−	0.804109i	$-0.297357\pi$
$947$	36.5885	1.18897	0.594483	+	0.804109i	$0.297357\pi$
$948$	0	0
$949$	0	0
$950$	4.00000	0.129777
$951$	0	0
$952$	−76.2102	−2.46999
$953$	24.2487	0.785493	0.392746	−	0.919647i	$-0.371525\pi$
$953$	24.2487	0.785493	0.392746	+	0.919647i	$0.371525\pi$
$954$	0	0
$955$	−17.5167	−0.566826
$956$	−8.00000	−0.258738
$957$	0	0
$958$	−81.7128	−2.64002
$959$	−37.2679	−1.20344
$960$	0	0
$961$	20.7846	0.670471
$962$	0	0
$963$	0	0
$964$	−122.354	−3.94075
$965$	19.9282	0.641512
$966$	0	0
$967$	32.2487	1.03705	0.518524	−	0.855063i	$-0.326482\pi$
$967$	32.2487	1.03705	0.518524	+	0.855063i	$0.326482\pi$
$968$	9.46410	0.304188
$969$	0	0
$970$	−26.0526	−0.836497
$971$	1.17691	0.0377690	0.0188845	−	0.999822i	$-0.493989\pi$
$971$	1.17691	0.0377690	0.0188845	+	0.999822i	$0.493989\pi$
$972$	0	0
$973$	10.0333	0.321654
$974$	−56.7846	−1.81950
$975$	0	0
$976$	−36.7846	−1.17745
$977$	−1.07180	−0.0342898	−0.0171449	−	0.999853i	$-0.505458\pi$
$977$	−1.07180	−0.0342898	−0.0171449	+	0.999853i	$0.505458\pi$
$978$	0	0
$979$	−16.3923	−0.523900
$980$	−5.07180	−0.162013
$981$	0	0
$982$	46.7846	1.49296
$983$	23.8564	0.760901	0.380451	−	0.924801i	$-0.375769\pi$
$983$	23.8564	0.760901	0.380451	+	0.924801i	$0.375769\pi$
$984$	0	0
$985$	10.9282	0.348202
$986$	−6.53590	−0.208145
$987$	0	0
$988$	0	0
$989$	−27.8564	−0.885782
$990$	0	0
$991$	28.9282	0.918935	0.459467	−	0.888195i	$-0.348040\pi$
$991$	28.9282	0.918935	0.459467	+	0.888195i	$0.348040\pi$
$992$	157.282	4.99371
$993$	0	0
$994$	−62.3923	−1.97896
$995$	−15.0000	−0.475532
$996$	0	0
$997$	−57.5885	−1.82384	−0.911922	−	0.410363i	$-0.865402\pi$
$997$	−57.5885	−1.82384	−0.911922	+	0.410363i	$0.865402\pi$
$998$	55.7128	1.76356
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.bk.1.2		2
3.2	odd	2		2535.2.a.n.1.1		2
13.2	odd	12		585.2.bu.a.316.2		4
13.7	odd	12		585.2.bu.a.361.2		4
13.12	even	2		7605.2.a.y.1.1		2
39.2	even	12		195.2.bb.a.121.1	✓	4
39.20	even	12		195.2.bb.a.166.1	yes	4
39.38	odd	2		2535.2.a.s.1.2		2
195.2	odd	12		975.2.w.f.199.2		4
195.59	even	12		975.2.bc.h.751.2		4
195.98	odd	12		975.2.w.f.49.2		4
195.119	even	12		975.2.bc.h.901.2		4
195.137	odd	12		975.2.w.a.49.1		4
195.158	odd	12		975.2.w.a.199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.bb.a.121.1	✓	4	39.2	even	12
195.2.bb.a.166.1	yes	4	39.20	even	12
585.2.bu.a.316.2		4	13.2	odd	12
585.2.bu.a.361.2		4	13.7	odd	12
975.2.w.a.49.1		4	195.137	odd	12
975.2.w.a.199.1		4	195.158	odd	12
975.2.w.f.49.2		4	195.98	odd	12
975.2.w.f.199.2		4	195.2	odd	12
975.2.bc.h.751.2		4	195.59	even	12
975.2.bc.h.901.2		4	195.119	even	12
2535.2.a.n.1.1		2	3.2	odd	2
2535.2.a.s.1.2		2	39.38	odd	2
7605.2.a.y.1.1		2	13.12	even	2
7605.2.a.bk.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q+2.73205 q^{2} +5.46410 q^{4} +1.00000 q^{5} -2.46410 q^{7} +9.46410 q^{8} +O(q^{10})\)	\(q+2.73205 q^{2} +5.46410 q^{4} +1.00000 q^{5} -2.46410 q^{7} +9.46410 q^{8} +2.73205 q^{10} -3.46410 q^{11} -6.73205 q^{14} +14.9282 q^{16} +3.26795 q^{17} +1.46410 q^{19} +5.46410 q^{20} -9.46410 q^{22} +7.46410 q^{23} +1.00000 q^{25} -13.4641 q^{28} -0.732051 q^{29} +7.19615 q^{31} +21.8564 q^{32} +8.92820 q^{34} -2.46410 q^{35} +4.00000 q^{37} +4.00000 q^{38} +9.46410 q^{40} +8.73205 q^{41} -3.73205 q^{43} -18.9282 q^{44} +20.3923 q^{46} -10.1962 q^{47} -0.928203 q^{49} +2.73205 q^{50} -6.92820 q^{53} -3.46410 q^{55} -23.3205 q^{56} -2.00000 q^{58} +10.7321 q^{59} -2.46410 q^{61} +19.6603 q^{62} +29.8564 q^{64} +5.53590 q^{67} +17.8564 q^{68} -6.73205 q^{70} +9.26795 q^{71} +5.39230 q^{73} +10.9282 q^{74} +8.00000 q^{76} +8.53590 q^{77} -11.9282 q^{79} +14.9282 q^{80} +23.8564 q^{82} +9.46410 q^{83} +3.26795 q^{85} -10.1962 q^{86} -32.7846 q^{88} +4.73205 q^{89} +40.7846 q^{92} -27.8564 q^{94} +1.46410 q^{95} -9.53590 q^{97} -2.53590 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 + 12 * q^8	\( 2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} + 12 q^{8} + 2 q^{10} - 10 q^{14} + 16 q^{16} + 10 q^{17} - 4 q^{19} + 4 q^{20} - 12 q^{22} + 8 q^{23} + 2 q^{25} - 20 q^{28} + 2 q^{29} + 4 q^{31} + 16 q^{32} + 4 q^{34} + 2 q^{35} + 8 q^{37} + 8 q^{38} + 12 q^{40} + 14 q^{41} - 4 q^{43} - 24 q^{44} + 20 q^{46} - 10 q^{47} + 12 q^{49} + 2 q^{50} - 12 q^{56} - 4 q^{58} + 18 q^{59} + 2 q^{61} + 22 q^{62} + 32 q^{64} + 18 q^{67} + 8 q^{68} - 10 q^{70} + 22 q^{71} - 10 q^{73} + 8 q^{74} + 16 q^{76} + 24 q^{77} - 10 q^{79} + 16 q^{80} + 20 q^{82} + 12 q^{83} + 10 q^{85} - 10 q^{86} - 24 q^{88} + 6 q^{89} + 40 q^{92} - 28 q^{94} - 4 q^{95} - 26 q^{97} - 12 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 + 12 * q^8 + 2 * q^10 - 10 * q^14 + 16 * q^16 + 10 * q^17 - 4 * q^19 + 4 * q^20 - 12 * q^22 + 8 * q^23 + 2 * q^25 - 20 * q^28 + 2 * q^29 + 4 * q^31 + 16 * q^32 + 4 * q^34 + 2 * q^35 + 8 * q^37 + 8 * q^38 + 12 * q^40 + 14 * q^41 - 4 * q^43 - 24 * q^44 + 20 * q^46 - 10 * q^47 + 12 * q^49 + 2 * q^50 - 12 * q^56 - 4 * q^58 + 18 * q^59 + 2 * q^61 + 22 * q^62 + 32 * q^64 + 18 * q^67 + 8 * q^68 - 10 * q^70 + 22 * q^71 - 10 * q^73 + 8 * q^74 + 16 * q^76 + 24 * q^77 - 10 * q^79 + 16 * q^80 + 20 * q^82 + 12 * q^83 + 10 * q^85 - 10 * q^86 - 24 * q^88 + 6 * q^89 + 40 * q^92 - 28 * q^94 - 4 * q^95 - 26 * q^97 - 12 * q^98

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