LMFDB - Embedded newform 7605.2.a.cn.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:	$5$
Coefficient field:	5.5.3352656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 6x - 6 $ x^5 - 2x^4 - 6x^3 + 10x^2 + 6x - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.946366$ 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)

$f(q)$	$=$	$q-0.946366 q^{2} -1.10439 q^{4} +1.00000 q^{5} +1.56306 q^{7} +2.93789 q^{8} +O(q^{10})$	$q-0.946366 q^{2} -1.10439 q^{4} +1.00000 q^{5} +1.56306 q^{7} +2.93789 q^{8} -0.946366 q^{10} -1.50942 q^{11} -1.47922 q^{14} -0.571534 q^{16} +3.26242 q^{17} +3.10439 q^{19} -1.10439 q^{20} +1.42847 q^{22} +2.93789 q^{23} +1.00000 q^{25} -1.72623 q^{28} +2.60534 q^{29} +2.65897 q^{31} -5.33490 q^{32} -3.08744 q^{34} +1.56306 q^{35} -0.571534 q^{37} -2.93789 q^{38} +2.93789 q^{40} -5.68801 q^{41} +1.26530 q^{43} +1.66700 q^{44} -2.78032 q^{46} +5.13459 q^{47} -4.55685 q^{49} -0.946366 q^{50} +2.91733 q^{53} -1.50942 q^{55} +4.59209 q^{56} -2.46560 q^{58} +12.8141 q^{59} +6.65897 q^{61} -2.51636 q^{62} +6.19183 q^{64} -9.49807 q^{67} -3.60299 q^{68} -1.47922 q^{70} +1.22014 q^{71} +12.8785 q^{73} +0.540880 q^{74} -3.42847 q^{76} -2.35932 q^{77} +7.76563 q^{79} -0.571534 q^{80} +5.38294 q^{82} -11.2312 q^{83} +3.26242 q^{85} -1.19743 q^{86} -4.43452 q^{88} +0.796815 q^{89} -3.24458 q^{92} -4.85920 q^{94} +3.10439 q^{95} +5.80042 q^{97} +4.31245 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8}+O(q^{10}) $ 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8	$ 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{11} + 4 q^{14} + 4 q^{16} + 4 q^{19} + 6 q^{20} + 14 q^{22} + 6 q^{23} + 5 q^{25} + 2 q^{28} - 16 q^{29} - 9 q^{31} + 14 q^{32} - q^{35} + 4 q^{37} - 6 q^{38} + 6 q^{40} + 6 q^{41} + 15 q^{43} + 14 q^{44} + 16 q^{46} + 10 q^{47} + 10 q^{49} + 2 q^{50} + 20 q^{53} + 8 q^{55} + 2 q^{56} + 4 q^{58} + 12 q^{59} + 11 q^{61} + 22 q^{62} + 4 q^{64} - 5 q^{67} - 50 q^{68} + 4 q^{70} + 10 q^{71} - q^{73} + 26 q^{74} - 24 q^{76} - 42 q^{77} - 17 q^{79} + 4 q^{80} + 16 q^{82} + 16 q^{83} + 44 q^{86} + 20 q^{88} + 4 q^{89} + 34 q^{92} - 16 q^{94} + 4 q^{95} + 11 q^{97} + 30 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 2 * q^10 + 8 * q^11 + 4 * q^14 + 4 * q^16 + 4 * q^19 + 6 * q^20 + 14 * q^22 + 6 * q^23 + 5 * q^25 + 2 * q^28 - 16 * q^29 - 9 * q^31 + 14 * q^32 - q^35 + 4 * q^37 - 6 * q^38 + 6 * q^40 + 6 * q^41 + 15 * q^43 + 14 * q^44 + 16 * q^46 + 10 * q^47 + 10 * q^49 + 2 * q^50 + 20 * q^53 + 8 * q^55 + 2 * q^56 + 4 * q^58 + 12 * q^59 + 11 * q^61 + 22 * q^62 + 4 * q^64 - 5 * q^67 - 50 * q^68 + 4 * q^70 + 10 * q^71 - q^73 + 26 * q^74 - 24 * q^76 - 42 * q^77 - 17 * q^79 + 4 * q^80 + 16 * q^82 + 16 * q^83 + 44 * q^86 + 20 * q^88 + 4 * q^89 + 34 * q^92 - 16 * q^94 + 4 * q^95 + 11 * q^97 + 30 * 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 $

$2$	−0.946366	−0.669182	−0.334591	−	0.942364i	$-0.608598\pi$
$2$	−0.946366	−0.669182	−0.334591	+	0.942364i	$0.608598\pi$
$3$	0	0
$3$	0	0
$4$	−1.10439	−0.552196
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.56306	0.590780	0.295390	−	0.955377i	$-0.404550\pi$
$7$	1.56306	0.590780	0.295390	+	0.955377i	$0.404550\pi$
$8$	2.93789	1.03870
$9$	0	0
$10$	−0.946366	−0.299267
$11$	−1.50942	−0.455108	−0.227554	−	0.973765i	$-0.573073\pi$
$11$	−1.50942	−0.455108	−0.227554	+	0.973765i	$0.573073\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−1.47922	−0.395339
$15$	0	0
$16$	−0.571534	−0.142883
$17$	3.26242	0.791253	0.395626	−	0.918412i	$-0.370528\pi$
$17$	3.26242	0.791253	0.395626	+	0.918412i	$0.370528\pi$
$18$	0	0
$19$	3.10439	0.712196	0.356098	−	0.934449i	$-0.384107\pi$
$19$	3.10439	0.712196	0.356098	+	0.934449i	$0.384107\pi$
$20$	−1.10439	−0.246950
$21$	0	0
$22$	1.42847	0.304550
$23$	2.93789	0.612592	0.306296	−	0.951936i	$-0.400910\pi$
$23$	2.93789	0.612592	0.306296	+	0.951936i	$0.400910\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−1.72623	−0.326227
$29$	2.60534	0.483799	0.241900	−	0.970301i	$-0.422229\pi$
$29$	2.60534	0.483799	0.241900	+	0.970301i	$0.422229\pi$
$30$	0	0
$31$	2.65897	0.477566	0.238783	−	0.971073i	$-0.423252\pi$
$31$	2.65897	0.477566	0.238783	+	0.971073i	$0.423252\pi$
$32$	−5.33490	−0.943086
$33$	0	0
$34$	−3.08744	−0.529492
$35$	1.56306	0.264205
$36$	0	0
$37$	−0.571534	−0.0939595	−0.0469798	−	0.998896i	$-0.514960\pi$
$37$	−0.571534	−0.0939595	−0.0469798	+	0.998896i	$0.514960\pi$
$38$	−2.93789	−0.476589
$39$	0	0
$40$	2.93789	0.464521
$41$	−5.68801	−0.888318	−0.444159	−	0.895948i	$-0.646497\pi$
$41$	−5.68801	−0.888318	−0.444159	+	0.895948i	$0.646497\pi$
$42$	0	0
$43$	1.26530	0.192956	0.0964779	−	0.995335i	$-0.469242\pi$
$43$	1.26530	0.192956	0.0964779	+	0.995335i	$0.469242\pi$
$44$	1.66700	0.251309
$45$	0	0
$46$	−2.78032	−0.409936
$47$	5.13459	0.748957	0.374479	−	0.927236i	$-0.377822\pi$
$47$	5.13459	0.748957	0.374479	+	0.927236i	$0.377822\pi$
$48$	0	0
$49$	−4.55685	−0.650979
$50$	−0.946366	−0.133836
$51$	0	0
$52$	0	0
$53$	2.91733	0.400726	0.200363	−	0.979722i	$-0.435788\pi$
$53$	2.91733	0.400726	0.200363	+	0.979722i	$0.435788\pi$
$54$	0	0
$55$	−1.50942	−0.203531
$56$	4.59209	0.613644
$57$	0	0
$58$	−2.46560	−0.323750
$59$	12.8141	1.66826	0.834128	−	0.551570i	$-0.185971\pi$
$59$	12.8141	1.66826	0.834128	+	0.551570i	$0.185971\pi$
$60$	0	0
$61$	6.65897	0.852594	0.426297	−	0.904583i	$-0.359818\pi$
$61$	6.65897	0.852594	0.426297	+	0.904583i	$0.359818\pi$
$62$	−2.51636	−0.319578
$63$	0	0
$64$	6.19183	0.773979
$65$	0	0
$66$	0	0
$67$	−9.49807	−1.16037	−0.580187	−	0.814483i	$-0.697021\pi$
$67$	−9.49807	−1.16037	−0.580187	+	0.814483i	$0.697021\pi$
$68$	−3.60299	−0.436927
$69$	0	0
$70$	−1.47922	−0.176801
$71$	1.22014	0.144804	0.0724018	−	0.997376i	$-0.476934\pi$
$71$	1.22014	0.144804	0.0724018	+	0.997376i	$0.476934\pi$
$72$	0	0
$73$	12.8785	1.50731	0.753657	−	0.657268i	$-0.228288\pi$
$73$	12.8785	1.50731	0.753657	+	0.657268i	$0.228288\pi$
$74$	0.540880	0.0628760
$75$	0	0
$76$	−3.42847	−0.393272
$77$	−2.35932	−0.268869
$78$	0	0
$79$	7.76563	0.873702	0.436851	−	0.899534i	$-0.356094\pi$
$79$	7.76563	0.873702	0.436851	+	0.899534i	$0.356094\pi$
$80$	−0.571534	−0.0638994
$81$	0	0
$82$	5.38294	0.594446
$83$	−11.2312	−1.23279	−0.616394	−	0.787438i	$-0.711407\pi$
$83$	−11.2312	−1.23279	−0.616394	+	0.787438i	$0.711407\pi$
$84$	0	0
$85$	3.26242	0.353859
$86$	−1.19743	−0.129122
$87$	0	0
$88$	−4.43452	−0.472721
$89$	0.796815	0.0844622	0.0422311	−	0.999108i	$-0.486553\pi$
$89$	0.796815	0.0844622	0.0422311	+	0.999108i	$0.486553\pi$
$90$	0	0
$91$	0	0
$92$	−3.24458	−0.338271
$93$	0	0
$94$	−4.85920	−0.501188
$95$	3.10439	0.318504
$96$	0	0
$97$	5.80042	0.588944	0.294472	−	0.955660i	$-0.404856\pi$
$97$	5.80042	0.588944	0.294472	+	0.955660i	$0.404856\pi$
$98$	4.31245	0.435623
$99$	0	0
$100$	−1.10439	−0.110439
$101$	6.04001	0.601004	0.300502	−	0.953781i	$-0.402846\pi$
$101$	6.04001	0.601004	0.300502	+	0.953781i	$0.402846\pi$
$102$	0	0
$103$	−9.34411	−0.920702	−0.460351	−	0.887737i	$-0.652276\pi$
$103$	−9.34411	−0.920702	−0.460351	+	0.887737i	$0.652276\pi$
$104$	0	0
$105$	0	0
$106$	−2.76086	−0.268159
$107$	−11.0104	−1.06441	−0.532206	−	0.846615i	$-0.678637\pi$
$107$	−11.0104	−1.06441	−0.532206	+	0.846615i	$0.678637\pi$
$108$	0	0
$109$	−18.5677	−1.77846	−0.889230	−	0.457460i	$-0.848759\pi$
$109$	−18.5677	−1.77846	−0.889230	+	0.457460i	$0.848759\pi$
$110$	1.42847	0.136199
$111$	0	0
$112$	−0.893340	−0.0844127
$113$	−13.0430	−1.22698	−0.613492	−	0.789701i	$-0.710236\pi$
$113$	−13.0430	−1.22698	−0.613492	+	0.789701i	$0.710236\pi$
$114$	0	0
$115$	2.93789	0.273960
$116$	−2.87732	−0.267152
$117$	0	0
$118$	−12.1268	−1.11637
$119$	5.09935	0.467457
$120$	0	0
$121$	−8.72164	−0.792876
$122$	−6.30182	−0.570540
$123$	0	0
$124$	−2.93655	−0.263710
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.60400	−0.674746	−0.337373	−	0.941371i	$-0.609538\pi$
$127$	−7.60400	−0.674746	−0.337373	+	0.941371i	$0.609538\pi$
$128$	4.81006	0.425153
$129$	0	0
$130$	0	0
$131$	19.0332	1.66294	0.831469	−	0.555571i	$-0.187500\pi$
$131$	19.0332	1.66294	0.831469	+	0.555571i	$0.187500\pi$
$132$	0	0
$133$	4.85234	0.420752
$134$	8.98865	0.776501
$135$	0	0
$136$	9.58463	0.821875
$137$	19.0684	1.62913	0.814563	−	0.580075i	$-0.196977\pi$
$137$	19.0684	1.62913	0.814563	+	0.580075i	$0.196977\pi$
$138$	0	0
$139$	−14.8291	−1.25779	−0.628893	−	0.777492i	$-0.716492\pi$
$139$	−14.8291	−1.25779	−0.628893	+	0.777492i	$0.716492\pi$
$140$	−1.72623	−0.145893
$141$	0	0
$142$	−1.15470	−0.0968999
$143$	0	0
$144$	0	0
$145$	2.60534	0.216362
$146$	−12.1878	−1.00867
$147$	0	0
$148$	0.631197	0.0518841
$149$	22.2798	1.82523	0.912617	−	0.408815i	$-0.134058\pi$
$149$	22.2798	1.82523	0.912617	+	0.408815i	$0.134058\pi$
$150$	0	0
$151$	16.7896	1.36632	0.683160	−	0.730269i	$-0.260605\pi$
$151$	16.7896	1.36632	0.683160	+	0.730269i	$0.260605\pi$
$152$	9.12036	0.739759
$153$	0	0
$154$	2.23278	0.179922
$155$	2.65897	0.213574
$156$	0	0
$157$	3.26279	0.260399	0.130200	−	0.991488i	$-0.458438\pi$
$157$	3.26279	0.260399	0.130200	+	0.991488i	$0.458438\pi$
$158$	−7.34913	−0.584665
$159$	0	0
$160$	−5.33490	−0.421761
$161$	4.59209	0.361908
$162$	0	0
$163$	−3.14549	−0.246374	−0.123187	−	0.992383i	$-0.539311\pi$
$163$	−3.14549	−0.246374	−0.123187	+	0.992383i	$0.539311\pi$
$164$	6.28179	0.490526
$165$	0	0
$166$	10.6289	0.824959
$167$	10.4279	0.806932	0.403466	−	0.914995i	$-0.367805\pi$
$167$	10.4279	0.806932	0.403466	+	0.914995i	$0.367805\pi$
$168$	0	0
$169$	0	0
$170$	−3.08744	−0.236796
$171$	0	0
$172$	−1.39738	−0.106549
$173$	12.4055	0.943172	0.471586	−	0.881820i	$-0.343682\pi$
$173$	12.4055	0.943172	0.471586	+	0.881820i	$0.343682\pi$
$174$	0	0
$175$	1.56306	0.118156
$176$	0.862686	0.0650274
$177$	0	0
$178$	−0.754078	−0.0565205
$179$	−9.16353	−0.684914	−0.342457	−	0.939533i	$-0.611259\pi$
$179$	−9.16353	−0.684914	−0.342457	+	0.939533i	$0.611259\pi$
$180$	0	0
$181$	−4.71081	−0.350152	−0.175076	−	0.984555i	$-0.556017\pi$
$181$	−4.71081	−0.350152	−0.175076	+	0.984555i	$0.556017\pi$
$182$	0	0
$183$	0	0
$184$	8.63120	0.636300
$185$	−0.571534	−0.0420200
$186$	0	0
$187$	−4.92437	−0.360106
$188$	−5.67060	−0.413571
$189$	0	0
$190$	−2.93789	−0.213137
$191$	−15.1506	−1.09626	−0.548128	−	0.836394i	$-0.684659\pi$
$191$	−15.1506	−1.09626	−0.548128	+	0.836394i	$0.684659\pi$
$192$	0	0
$193$	−10.2900	−0.740692	−0.370346	−	0.928894i	$-0.620761\pi$
$193$	−10.2900	−0.740692	−0.370346	+	0.928894i	$0.620761\pi$
$194$	−5.48932	−0.394110
$195$	0	0
$196$	5.03255	0.359468
$197$	−5.19012	−0.369781	−0.184890	−	0.982759i	$-0.559193\pi$
$197$	−5.19012	−0.369781	−0.184890	+	0.982759i	$0.559193\pi$
$198$	0	0
$199$	−12.6178	−0.894452	−0.447226	−	0.894421i	$-0.647588\pi$
$199$	−12.6178	−0.894452	−0.447226	+	0.894421i	$0.647588\pi$
$200$	2.93789	0.207740
$201$	0	0
$202$	−5.71606	−0.402181
$203$	4.07230	0.285819
$204$	0	0
$205$	−5.68801	−0.397268
$206$	8.84294	0.616117
$207$	0	0
$208$	0	0
$209$	−4.68584	−0.324127
$210$	0	0
$211$	−6.10916	−0.420572	−0.210286	−	0.977640i	$-0.567440\pi$
$211$	−6.10916	−0.420572	−0.210286	+	0.977640i	$0.567440\pi$
$212$	−3.22188	−0.221279
$213$	0	0
$214$	10.4198	0.712285
$215$	1.26530	0.0862924
$216$	0	0
$217$	4.15613	0.282137
$218$	17.5718	1.19011
$219$	0	0
$220$	1.66700	0.112389
$221$	0	0
$222$	0	0
$223$	6.37663	0.427010	0.213505	−	0.976942i	$-0.431512\pi$
$223$	6.37663	0.427010	0.213505	+	0.976942i	$0.431512\pi$
$224$	−8.33876	−0.557157
$225$	0	0
$226$	12.3435	0.821075
$227$	13.4506	0.892751	0.446375	−	0.894846i	$-0.352715\pi$
$227$	13.4506	0.892751	0.446375	+	0.894846i	$0.352715\pi$
$228$	0	0
$229$	9.14533	0.604341	0.302170	−	0.953254i	$-0.402289\pi$
$229$	9.14533	0.604341	0.302170	+	0.953254i	$0.402289\pi$
$230$	−2.78032	−0.183329
$231$	0	0
$232$	7.65420	0.502523
$233$	−22.1655	−1.45211	−0.726056	−	0.687636i	$-0.758649\pi$
$233$	−22.1655	−1.45211	−0.726056	+	0.687636i	$0.758649\pi$
$234$	0	0
$235$	5.13459	0.334944
$236$	−14.1518	−0.921205
$237$	0	0
$238$	−4.82585	−0.312813
$239$	17.5940	1.13806	0.569030	−	0.822317i	$-0.307319\pi$
$239$	17.5940	1.13806	0.569030	+	0.822317i	$0.307319\pi$
$240$	0	0
$241$	−28.1384	−1.81255	−0.906277	−	0.422684i	$-0.861088\pi$
$241$	−28.1384	−1.81255	−0.906277	+	0.422684i	$0.861088\pi$
$242$	8.25386	0.530578
$243$	0	0
$244$	−7.35412	−0.470799
$245$	−4.55685	−0.291126
$246$	0	0
$247$	0	0
$248$	7.81177	0.496048
$249$	0	0
$250$	−0.946366	−0.0598534
$251$	28.0822	1.77254	0.886268	−	0.463173i	$-0.153289\pi$
$251$	28.0822	1.77254	0.886268	+	0.463173i	$0.153289\pi$
$252$	0	0
$253$	−4.43452	−0.278796
$254$	7.19616	0.451528
$255$	0	0
$256$	−16.9357	−1.05848
$257$	30.2564	1.88734	0.943672	−	0.330883i	$-0.107347\pi$
$257$	30.2564	1.88734	0.943672	+	0.330883i	$0.107347\pi$
$258$	0	0
$259$	−0.893340	−0.0555094
$260$	0	0
$261$	0	0
$262$	−18.0124	−1.11281
$263$	−28.1241	−1.73420	−0.867102	−	0.498131i	$-0.834020\pi$
$263$	−28.1241	−1.73420	−0.867102	+	0.498131i	$0.834020\pi$
$264$	0	0
$265$	2.91733	0.179210
$266$	−4.59209	−0.281559
$267$	0	0
$268$	10.4896	0.640754
$269$	20.7242	1.26357	0.631787	−	0.775142i	$-0.282322\pi$
$269$	20.7242	1.26357	0.631787	+	0.775142i	$0.282322\pi$
$270$	0	0
$271$	7.94816	0.482816	0.241408	−	0.970424i	$-0.422391\pi$
$271$	7.94816	0.482816	0.241408	+	0.970424i	$0.422391\pi$
$272$	−1.86458	−0.113057
$273$	0	0
$274$	−18.0457	−1.09018
$275$	−1.50942	−0.0910217
$276$	0	0
$277$	−6.17238	−0.370862	−0.185431	−	0.982657i	$-0.559368\pi$
$277$	−6.17238	−0.370862	−0.185431	+	0.982657i	$0.559368\pi$
$278$	14.0337	0.841687
$279$	0	0
$280$	4.59209	0.274430
$281$	18.5723	1.10793	0.553967	−	0.832539i	$-0.313114\pi$
$281$	18.5723	1.10793	0.553967	+	0.832539i	$0.313114\pi$
$282$	0	0
$283$	−7.27912	−0.432699	−0.216349	−	0.976316i	$-0.569415\pi$
$283$	−7.27912	−0.432699	−0.216349	+	0.976316i	$0.569415\pi$
$284$	−1.34751	−0.0799600
$285$	0	0
$286$	0	0
$287$	−8.89069	−0.524801
$288$	0	0
$289$	−6.35662	−0.373919
$290$	−2.46560	−0.144785
$291$	0	0
$292$	−14.2229	−0.832333
$293$	6.44209	0.376351	0.188175	−	0.982135i	$-0.439743\pi$
$293$	6.44209	0.376351	0.188175	+	0.982135i	$0.439743\pi$
$294$	0	0
$295$	12.8141	0.746067
$296$	−1.67910	−0.0975959
$297$	0	0
$298$	−21.0849	−1.22141
$299$	0	0
$300$	0	0
$301$	1.97773	0.113994
$302$	−15.8891	−0.914316
$303$	0	0
$304$	−1.77426	−0.101761
$305$	6.65897	0.381292
$306$	0	0
$307$	−4.94599	−0.282283	−0.141141	−	0.989989i	$-0.545077\pi$
$307$	−4.94599	−0.282283	−0.141141	+	0.989989i	$0.545077\pi$
$308$	2.60561	0.148468
$309$	0	0
$310$	−2.51636	−0.142920
$311$	−30.0486	−1.70390	−0.851950	−	0.523623i	$-0.824580\pi$
$311$	−30.0486	−1.70390	−0.851950	+	0.523623i	$0.824580\pi$
$312$	0	0
$313$	−4.57620	−0.258662	−0.129331	−	0.991601i	$-0.541283\pi$
$313$	−4.57620	−0.258662	−0.129331	+	0.991601i	$0.541283\pi$
$314$	−3.08779	−0.174254
$315$	0	0
$316$	−8.57631	−0.482455
$317$	1.50960	0.0847875	0.0423937	−	0.999101i	$-0.486502\pi$
$317$	1.50960	0.0847875	0.0423937	+	0.999101i	$0.486502\pi$
$318$	0	0
$319$	−3.93256	−0.220181
$320$	6.19183	0.346134
$321$	0	0
$322$	−4.34580	−0.242182
$323$	10.1278	0.563527
$324$	0	0
$325$	0	0
$326$	2.97678	0.164869
$327$	0	0
$328$	−16.7107	−0.922696
$329$	8.02566	0.442469
$330$	0	0
$331$	−8.35889	−0.459446	−0.229723	−	0.973256i	$-0.573782\pi$
$331$	−8.35889	−0.459446	−0.229723	+	0.973256i	$0.573782\pi$
$332$	12.4037	0.680741
$333$	0	0
$334$	−9.86857	−0.539984
$335$	−9.49807	−0.518935
$336$	0	0
$337$	15.5359	0.846297	0.423148	−	0.906060i	$-0.360925\pi$
$337$	15.5359	0.846297	0.423148	+	0.906060i	$0.360925\pi$
$338$	0	0
$339$	0	0
$340$	−3.60299	−0.195400
$341$	−4.01352	−0.217344
$342$	0	0
$343$	−18.0640	−0.975366
$344$	3.71730	0.200423
$345$	0	0
$346$	−11.7401	−0.631153
$347$	−5.78093	−0.310336	−0.155168	−	0.987888i	$-0.549592\pi$
$347$	−5.78093	−0.310336	−0.155168	+	0.987888i	$0.549592\pi$
$348$	0	0
$349$	−5.45624	−0.292066	−0.146033	−	0.989280i	$-0.546651\pi$
$349$	−5.45624	−0.292066	−0.146033	+	0.989280i	$0.546651\pi$
$350$	−1.47922	−0.0790679
$351$	0	0
$352$	8.05262	0.429206
$353$	12.2625	0.652667	0.326334	−	0.945255i	$-0.394187\pi$
$353$	12.2625	0.652667	0.326334	+	0.945255i	$0.394187\pi$
$354$	0	0
$355$	1.22014	0.0647581
$356$	−0.879996	−0.0466397
$357$	0	0
$358$	8.67205	0.458332
$359$	27.4508	1.44880	0.724398	−	0.689382i	$-0.242118\pi$
$359$	27.4508	1.44880	0.724398	+	0.689382i	$0.242118\pi$
$360$	0	0
$361$	−9.36275	−0.492776
$362$	4.45815	0.234315
$363$	0	0
$364$	0	0
$365$	12.8785	0.674092
$366$	0	0
$367$	33.8221	1.76550	0.882749	−	0.469845i	$-0.155690\pi$
$367$	33.8221	1.76550	0.882749	+	0.469845i	$0.155690\pi$
$368$	−1.67910	−0.0875293
$369$	0	0
$370$	0.540880	0.0281190
$371$	4.55996	0.236741
$372$	0	0
$373$	−3.05192	−0.158023	−0.0790113	−	0.996874i	$-0.525176\pi$
$373$	−3.05192	−0.158023	−0.0790113	+	0.996874i	$0.525176\pi$
$374$	4.66026	0.240976
$375$	0	0
$376$	15.0849	0.777942
$377$	0	0
$378$	0	0
$379$	3.01218	0.154725	0.0773626	−	0.997003i	$-0.475350\pi$
$379$	3.01218	0.154725	0.0773626	+	0.997003i	$0.475350\pi$
$380$	−3.42847	−0.175877
$381$	0	0
$382$	14.3380	0.733594
$383$	−3.21916	−0.164491	−0.0822456	−	0.996612i	$-0.526209\pi$
$383$	−3.21916	−0.164491	−0.0822456	+	0.996612i	$0.526209\pi$
$384$	0	0
$385$	−2.35932	−0.120242
$386$	9.73812	0.495657
$387$	0	0
$388$	−6.40594	−0.325212
$389$	−12.8066	−0.649322	−0.324661	−	0.945830i	$-0.605250\pi$
$389$	−12.8066	−0.649322	−0.324661	+	0.945830i	$0.605250\pi$
$390$	0	0
$391$	9.58463	0.484715
$392$	−13.3875	−0.676172
$393$	0	0
$394$	4.91175	0.247451
$395$	7.76563	0.390731
$396$	0	0
$397$	34.2250	1.71771	0.858853	−	0.512223i	$-0.171178\pi$
$397$	34.2250	1.71771	0.858853	+	0.512223i	$0.171178\pi$
$398$	11.9410	0.598551
$399$	0	0
$400$	−0.571534	−0.0285767
$401$	−26.1906	−1.30790	−0.653948	−	0.756539i	$-0.726889\pi$
$401$	−26.1906	−1.30790	−0.653948	+	0.756539i	$0.726889\pi$
$402$	0	0
$403$	0	0
$404$	−6.67054	−0.331872
$405$	0	0
$406$	−3.85388	−0.191265
$407$	0.862686	0.0427618
$408$	0	0
$409$	−14.2937	−0.706779	−0.353390	−	0.935476i	$-0.614971\pi$
$409$	−14.2937	−0.706779	−0.353390	+	0.935476i	$0.614971\pi$
$410$	5.38294	0.265844
$411$	0	0
$412$	10.3196	0.508408
$413$	20.0292	0.985573
$414$	0	0
$415$	−11.2312	−0.551320
$416$	0	0
$417$	0	0
$418$	4.43452	0.216899
$419$	−23.4850	−1.14732	−0.573659	−	0.819094i	$-0.694477\pi$
$419$	−23.4850	−1.14732	−0.573659	+	0.819094i	$0.694477\pi$
$420$	0	0
$421$	12.7618	0.621971	0.310985	−	0.950415i	$-0.399341\pi$
$421$	12.7618	0.621971	0.310985	+	0.950415i	$0.399341\pi$
$422$	5.78150	0.281439
$423$	0	0
$424$	8.57080	0.416235
$425$	3.26242	0.158251
$426$	0	0
$427$	10.4084	0.503696
$428$	12.1598	0.587765
$429$	0	0
$430$	−1.19743	−0.0577453
$431$	14.6963	0.707895	0.353948	−	0.935265i	$-0.384839\pi$
$431$	14.6963	0.707895	0.353948	+	0.935265i	$0.384839\pi$
$432$	0	0
$433$	−19.5243	−0.938279	−0.469139	−	0.883124i	$-0.655436\pi$
$433$	−19.5243	−0.938279	−0.469139	+	0.883124i	$0.655436\pi$
$434$	−3.93322	−0.188801
$435$	0	0
$436$	20.5060	0.982059
$437$	9.12036	0.436286
$438$	0	0
$439$	30.4160	1.45168	0.725838	−	0.687866i	$-0.241452\pi$
$439$	30.4160	1.45168	0.725838	+	0.687866i	$0.241452\pi$
$440$	−4.43452	−0.211407
$441$	0	0
$442$	0	0
$443$	8.83602	0.419812	0.209906	−	0.977722i	$-0.432684\pi$
$443$	8.83602	0.419812	0.209906	+	0.977722i	$0.432684\pi$
$444$	0	0
$445$	0.796815	0.0377726
$446$	−6.03462	−0.285748
$447$	0	0
$448$	9.67819	0.457252
$449$	21.1204	0.996732	0.498366	−	0.866967i	$-0.333934\pi$
$449$	21.1204	0.996732	0.498366	+	0.866967i	$0.333934\pi$
$450$	0	0
$451$	8.58561	0.404281
$452$	14.4046	0.677535
$453$	0	0
$454$	−12.7292	−0.597412
$455$	0	0
$456$	0	0
$457$	−39.4184	−1.84391	−0.921957	−	0.387292i	$-0.873410\pi$
$457$	−39.4184	−1.84391	−0.921957	+	0.387292i	$0.873410\pi$
$458$	−8.65483	−0.404414
$459$	0	0
$460$	−3.24458	−0.151279
$461$	−0.424154	−0.0197548	−0.00987742	−	0.999951i	$-0.503144\pi$
$461$	−0.424154	−0.0197548	−0.00987742	+	0.999951i	$0.503144\pi$
$462$	0	0
$463$	−30.1509	−1.40123	−0.700616	−	0.713538i	$-0.747091\pi$
$463$	−30.1509	−1.40123	−0.700616	+	0.713538i	$0.747091\pi$
$464$	−1.48904	−0.0691269
$465$	0	0
$466$	20.9767	0.971726
$467$	8.07034	0.373451	0.186725	−	0.982412i	$-0.440213\pi$
$467$	8.07034	0.373451	0.186725	+	0.982412i	$0.440213\pi$
$468$	0	0
$469$	−14.8460	−0.685526
$470$	−4.85920	−0.224138
$471$	0	0
$472$	37.6465	1.73282
$473$	−1.90987	−0.0878158
$474$	0	0
$475$	3.10439	0.142439
$476$	−5.63168	−0.258128
$477$	0	0
$478$	−16.6503	−0.761569
$479$	37.2158	1.70044	0.850218	−	0.526431i	$-0.176470\pi$
$479$	37.2158	1.70044	0.850218	+	0.526431i	$0.176470\pi$
$480$	0	0
$481$	0	0
$482$	26.6292	1.21293
$483$	0	0
$484$	9.63211	0.437823
$485$	5.80042	0.263384
$486$	0	0
$487$	11.4572	0.519176	0.259588	−	0.965719i	$-0.416413\pi$
$487$	11.4572	0.519176	0.259588	+	0.965719i	$0.416413\pi$
$488$	19.5633	0.885591
$489$	0	0
$490$	4.31245	0.194816
$491$	−21.1039	−0.952405	−0.476202	−	0.879336i	$-0.657987\pi$
$491$	−21.1039	−0.952405	−0.476202	+	0.879336i	$0.657987\pi$
$492$	0	0
$493$	8.49971	0.382808
$494$	0	0
$495$	0	0
$496$	−1.51969	−0.0682362
$497$	1.90714	0.0855471
$498$	0	0
$499$	19.9900	0.894876	0.447438	−	0.894315i	$-0.352337\pi$
$499$	19.9900	0.894876	0.447438	+	0.894315i	$0.352337\pi$
$500$	−1.10439	−0.0493899
$501$	0	0
$502$	−26.5761	−1.18615
$503$	10.7419	0.478960	0.239480	−	0.970901i	$-0.423023\pi$
$503$	10.7419	0.478960	0.239480	+	0.970901i	$0.423023\pi$
$504$	0	0
$505$	6.04001	0.268777
$506$	4.19668	0.186565
$507$	0	0
$508$	8.39780	0.372592
$509$	−2.23185	−0.0989249	−0.0494624	−	0.998776i	$-0.515751\pi$
$509$	−2.23185	−0.0989249	−0.0494624	+	0.998776i	$0.515751\pi$
$510$	0	0
$511$	20.1298	0.890492
$512$	6.40728	0.283164
$513$	0	0
$514$	−28.6336	−1.26298
$515$	−9.34411	−0.411751
$516$	0	0
$517$	−7.75027	−0.340857
$518$	0.845426	0.0371459
$519$	0	0
$520$	0	0
$521$	33.9482	1.48730	0.743649	−	0.668570i	$-0.233093\pi$
$521$	33.9482	1.48730	0.743649	+	0.668570i	$0.233093\pi$
$522$	0	0
$523$	5.20347	0.227532	0.113766	−	0.993508i	$-0.463709\pi$
$523$	5.20347	0.227532	0.113766	+	0.993508i	$0.463709\pi$
$524$	−21.0201	−0.918268
$525$	0	0
$526$	26.6156	1.16050
$527$	8.67469	0.377875
$528$	0	0
$529$	−14.3688	−0.624731
$530$	−2.76086	−0.119924
$531$	0	0
$532$	−5.35889	−0.232337
$533$	0	0
$534$	0	0
$535$	−11.0104	−0.476020
$536$	−27.9043	−1.20528
$537$	0	0
$538$	−19.6126	−0.845561
$539$	6.87822	0.296266
$540$	0	0
$541$	5.87388	0.252538	0.126269	−	0.991996i	$-0.459700\pi$
$541$	5.87388	0.252538	0.126269	+	0.991996i	$0.459700\pi$
$542$	−7.52186	−0.323092
$543$	0	0
$544$	−17.4047	−0.746219
$545$	−18.5677	−0.795352
$546$	0	0
$547$	43.6009	1.86424	0.932119	−	0.362151i	$-0.117958\pi$
$547$	43.6009	1.86424	0.932119	+	0.362151i	$0.117958\pi$
$548$	−21.0590	−0.899597
$549$	0	0
$550$	1.42847	0.0609100
$551$	8.08800	0.344560
$552$	0	0
$553$	12.1381	0.516166
$554$	5.84133	0.248174
$555$	0	0
$556$	16.3771	0.694545
$557$	35.6499	1.51054	0.755268	−	0.655416i	$-0.227507\pi$
$557$	35.6499	1.51054	0.755268	+	0.655416i	$0.227507\pi$
$558$	0	0
$559$	0	0
$560$	−0.893340	−0.0377505
$561$	0	0
$562$	−17.5762	−0.741408
$563$	18.2191	0.767842	0.383921	−	0.923366i	$-0.374573\pi$
$563$	18.2191	0.767842	0.383921	+	0.923366i	$0.374573\pi$
$564$	0	0
$565$	−13.0430	−0.548724
$566$	6.88871	0.289554
$567$	0	0
$568$	3.58463	0.150408
$569$	26.5426	1.11272	0.556362	−	0.830940i	$-0.312197\pi$
$569$	26.5426	1.11272	0.556362	+	0.830940i	$0.312197\pi$
$570$	0	0
$571$	26.7099	1.11778	0.558888	−	0.829243i	$-0.311228\pi$
$571$	26.7099	1.11778	0.558888	+	0.829243i	$0.311228\pi$
$572$	0	0
$573$	0	0
$574$	8.41384	0.351187
$575$	2.93789	0.122518
$576$	0	0
$577$	12.0139	0.500144	0.250072	−	0.968227i	$-0.419546\pi$
$577$	12.0139	0.500144	0.250072	+	0.968227i	$0.419546\pi$
$578$	6.01569	0.250220
$579$	0	0
$580$	−2.87732	−0.119474
$581$	−17.5551	−0.728307
$582$	0	0
$583$	−4.40349	−0.182374
$584$	37.8356	1.56565
$585$	0	0
$586$	−6.09657	−0.251847
$587$	43.3120	1.78768	0.893840	−	0.448387i	$-0.148001\pi$
$587$	43.3120	1.78768	0.893840	+	0.448387i	$0.148001\pi$
$588$	0	0
$589$	8.25450	0.340121
$590$	−12.1268	−0.499254
$591$	0	0
$592$	0.326651	0.0134253
$593$	1.84854	0.0759104	0.0379552	−	0.999279i	$-0.487916\pi$
$593$	1.84854	0.0759104	0.0379552	+	0.999279i	$0.487916\pi$
$594$	0	0
$595$	5.09935	0.209053
$596$	−24.6057	−1.00789
$597$	0	0
$598$	0	0
$599$	25.2041	1.02981	0.514906	−	0.857247i	$-0.327827\pi$
$599$	25.2041	1.02981	0.514906	+	0.857247i	$0.327827\pi$
$600$	0	0
$601$	8.08666	0.329862	0.164931	−	0.986305i	$-0.447260\pi$
$601$	8.08666	0.329862	0.164931	+	0.986305i	$0.447260\pi$
$602$	−1.87166	−0.0762830
$603$	0	0
$604$	−18.5423	−0.754477
$605$	−8.72164	−0.354585
$606$	0	0
$607$	45.0393	1.82809	0.914045	−	0.405613i	$-0.132942\pi$
$607$	45.0393	1.82809	0.914045	+	0.405613i	$0.132942\pi$
$608$	−16.5616	−0.671662
$609$	0	0
$610$	−6.30182	−0.255153
$611$	0	0
$612$	0	0
$613$	41.2960	1.66793	0.833965	−	0.551817i	$-0.186065\pi$
$613$	41.2960	1.66793	0.833965	+	0.551817i	$0.186065\pi$
$614$	4.68072	0.188898
$615$	0	0
$616$	−6.93141	−0.279275
$617$	6.86361	0.276319	0.138159	−	0.990410i	$-0.455881\pi$
$617$	6.86361	0.276319	0.138159	+	0.990410i	$0.455881\pi$
$618$	0	0
$619$	37.3813	1.50248	0.751241	−	0.660028i	$-0.229456\pi$
$619$	37.3813	1.50248	0.751241	+	0.660028i	$0.229456\pi$
$620$	−2.93655	−0.117935
$621$	0	0
$622$	28.4370	1.14022
$623$	1.24547	0.0498986
$624$	0	0
$625$	1.00000	0.0400000
$626$	4.33076	0.173092
$627$	0	0
$628$	−3.60340	−0.143791
$629$	−1.86458	−0.0743457
$630$	0	0
$631$	−8.80986	−0.350715	−0.175358	−	0.984505i	$-0.556108\pi$
$631$	−8.80986	−0.350715	−0.175358	+	0.984505i	$0.556108\pi$
$632$	22.8146	0.907515
$633$	0	0
$634$	−1.42863	−0.0567382
$635$	−7.60400	−0.301756
$636$	0	0
$637$	0	0
$638$	3.72164	0.147341
$639$	0	0
$640$	4.81006	0.190134
$641$	8.17331	0.322826	0.161413	−	0.986887i	$-0.448395\pi$
$641$	8.17331	0.322826	0.161413	+	0.986887i	$0.448395\pi$
$642$	0	0
$643$	−25.3271	−0.998803	−0.499402	−	0.866371i	$-0.666447\pi$
$643$	−25.3271	−0.998803	−0.499402	+	0.866371i	$0.666447\pi$
$644$	−5.07147	−0.199844
$645$	0	0
$646$	−9.58463	−0.377102
$647$	1.14651	0.0450739	0.0225370	−	0.999746i	$-0.492826\pi$
$647$	1.14651	0.0450739	0.0225370	+	0.999746i	$0.492826\pi$
$648$	0	0
$649$	−19.3419	−0.759238
$650$	0	0
$651$	0	0
$652$	3.47385	0.136047
$653$	22.8429	0.893911	0.446956	−	0.894556i	$-0.352508\pi$
$653$	22.8429	0.893911	0.446956	+	0.894556i	$0.352508\pi$
$654$	0	0
$655$	19.0332	0.743689
$656$	3.25089	0.126926
$657$	0	0
$658$	−7.59521	−0.296092
$659$	−19.0819	−0.743326	−0.371663	−	0.928368i	$-0.621212\pi$
$659$	−19.0819	−0.743326	−0.371663	+	0.928368i	$0.621212\pi$
$660$	0	0
$661$	−11.5885	−0.450740	−0.225370	−	0.974273i	$-0.572359\pi$
$661$	−11.5885	−0.450740	−0.225370	+	0.974273i	$0.572359\pi$
$662$	7.91057	0.307453
$663$	0	0
$664$	−32.9961	−1.28050
$665$	4.85234	0.188166
$666$	0	0
$667$	7.65420	0.296372
$668$	−11.5164	−0.445585
$669$	0	0
$670$	8.98865	0.347262
$671$	−10.0512	−0.388023
$672$	0	0
$673$	35.4376	1.36602	0.683010	−	0.730409i	$-0.260671\pi$
$673$	35.4376	1.36602	0.683010	+	0.730409i	$0.260671\pi$
$674$	−14.7027	−0.566326
$675$	0	0
$676$	0	0
$677$	−30.5442	−1.17391	−0.586954	−	0.809620i	$-0.699673\pi$
$677$	−30.5442	−1.17391	−0.586954	+	0.809620i	$0.699673\pi$
$678$	0	0
$679$	9.06640	0.347936
$680$	9.58463	0.367554
$681$	0	0
$682$	3.79826	0.145443
$683$	19.5727	0.748929	0.374465	−	0.927241i	$-0.377827\pi$
$683$	19.5727	0.748929	0.374465	+	0.927241i	$0.377827\pi$
$684$	0	0
$685$	19.0684	0.728568
$686$	17.0952	0.652697
$687$	0	0
$688$	−0.723159	−0.0275702
$689$	0	0
$690$	0	0
$691$	16.7710	0.637997	0.318999	−	0.947755i	$-0.396653\pi$
$691$	16.7710	0.637997	0.318999	+	0.947755i	$0.396653\pi$
$692$	−13.7005	−0.520816
$693$	0	0
$694$	5.47087	0.207671
$695$	−14.8291	−0.562499
$696$	0	0
$697$	−18.5567	−0.702884
$698$	5.16360	0.195445
$699$	0	0
$700$	−1.72623	−0.0652453
$701$	−40.4052	−1.52608	−0.763041	−	0.646350i	$-0.776294\pi$
$701$	−40.4052	−1.52608	−0.763041	+	0.646350i	$0.776294\pi$
$702$	0	0
$703$	−1.77426	−0.0669176
$704$	−9.34610	−0.352244
$705$	0	0
$706$	−11.6048	−0.436753
$707$	9.44089	0.355061
$708$	0	0
$709$	−17.3812	−0.652763	−0.326382	−	0.945238i	$-0.605829\pi$
$709$	−17.3812	−0.652763	−0.326382	+	0.945238i	$0.605829\pi$
$710$	−1.15470	−0.0433349
$711$	0	0
$712$	2.34095	0.0877310
$713$	7.81177	0.292553
$714$	0	0
$715$	0	0
$716$	10.1201	0.378207
$717$	0	0
$718$	−25.9785	−0.969508
$719$	−2.46149	−0.0917981	−0.0458990	−	0.998946i	$-0.514615\pi$
$719$	−2.46149	−0.0917981	−0.0458990	+	0.998946i	$0.514615\pi$
$720$	0	0
$721$	−14.6054	−0.543933
$722$	8.86058	0.329757
$723$	0	0
$724$	5.20259	0.193353
$725$	2.60534	0.0967599
$726$	0	0
$727$	30.0547	1.11467	0.557333	−	0.830289i	$-0.311825\pi$
$727$	30.0547	1.11467	0.557333	+	0.830289i	$0.311825\pi$
$728$	0	0
$729$	0	0
$730$	−12.1878	−0.451090
$731$	4.12792	0.152677
$732$	0	0
$733$	−18.4533	−0.681587	−0.340794	−	0.940138i	$-0.610696\pi$
$733$	−18.4533	−0.681587	−0.340794	+	0.940138i	$0.610696\pi$
$734$	−32.0080	−1.18144
$735$	0	0
$736$	−15.6733	−0.577727
$737$	14.3366	0.528096
$738$	0	0
$739$	−35.7206	−1.31400	−0.657002	−	0.753889i	$-0.728176\pi$
$739$	−35.7206	−1.31400	−0.657002	+	0.753889i	$0.728176\pi$
$740$	0.631197	0.0232033
$741$	0	0
$742$	−4.31539	−0.158423
$743$	−11.1474	−0.408960	−0.204480	−	0.978871i	$-0.565550\pi$
$743$	−11.1474	−0.408960	−0.204480	+	0.978871i	$0.565550\pi$
$744$	0	0
$745$	22.2798	0.816270
$746$	2.88823	0.105746
$747$	0	0
$748$	5.43844	0.198849
$749$	−17.2098	−0.628834
$750$	0	0
$751$	12.6456	0.461443	0.230722	−	0.973020i	$-0.425891\pi$
$751$	12.6456	0.461443	0.230722	+	0.973020i	$0.425891\pi$
$752$	−2.93459	−0.107014
$753$	0	0
$754$	0	0
$755$	16.7896	0.611037
$756$	0	0
$757$	5.55281	0.201820	0.100910	−	0.994896i	$-0.467825\pi$
$757$	5.55281	0.201820	0.100910	+	0.994896i	$0.467825\pi$
$758$	−2.85062	−0.103539
$759$	0	0
$760$	9.12036	0.330830
$761$	−30.2645	−1.09709	−0.548544	−	0.836122i	$-0.684818\pi$
$761$	−30.2645	−1.09709	−0.548544	+	0.836122i	$0.684818\pi$
$762$	0	0
$763$	−29.0224	−1.05068
$764$	16.7322	0.605348
$765$	0	0
$766$	3.04650	0.110074
$767$	0	0
$768$	0	0
$769$	52.2489	1.88414	0.942072	−	0.335410i	$-0.108875\pi$
$769$	52.2489	1.88414	0.942072	+	0.335410i	$0.108875\pi$
$770$	2.23278	0.0804637
$771$	0	0
$772$	11.3642	0.409007
$773$	39.4777	1.41991	0.709957	−	0.704245i	$-0.248714\pi$
$773$	39.4777	1.41991	0.709957	+	0.704245i	$0.248714\pi$
$774$	0	0
$775$	2.65897	0.0955132
$776$	17.0410	0.611736
$777$	0	0
$778$	12.1198	0.434514
$779$	−17.6578	−0.632657
$780$	0	0
$781$	−1.84170	−0.0659013
$782$	−9.07056	−0.324363
$783$	0	0
$784$	2.60439	0.0930140
$785$	3.26279	0.116454
$786$	0	0
$787$	31.3300	1.11679	0.558396	−	0.829574i	$-0.311417\pi$
$787$	31.3300	1.11679	0.558396	+	0.829574i	$0.311417\pi$
$788$	5.73193	0.204192
$789$	0	0
$790$	−7.34913	−0.261470
$791$	−20.3870	−0.724878
$792$	0	0
$793$	0	0
$794$	−32.3894	−1.14946
$795$	0	0
$796$	13.9350	0.493913
$797$	14.9236	0.528623	0.264311	−	0.964437i	$-0.414855\pi$
$797$	14.9236	0.528623	0.264311	+	0.964437i	$0.414855\pi$
$798$	0	0
$799$	16.7512	0.592614
$800$	−5.33490	−0.188617
$801$	0	0
$802$	24.7859	0.875220
$803$	−19.4391	−0.685991
$804$	0	0
$805$	4.59209	0.161850
$806$	0	0
$807$	0	0
$808$	17.7449	0.624263
$809$	5.28116	0.185676	0.0928379	−	0.995681i	$-0.470406\pi$
$809$	5.28116	0.185676	0.0928379	+	0.995681i	$0.470406\pi$
$810$	0	0
$811$	−18.5871	−0.652682	−0.326341	−	0.945252i	$-0.605816\pi$
$811$	−18.5871	−0.652682	−0.326341	+	0.945252i	$0.605816\pi$
$812$	−4.49741	−0.157828
$813$	0	0
$814$	−0.816417	−0.0286154
$815$	−3.14549	−0.110182
$816$	0	0
$817$	3.92797	0.137422
$818$	13.5271	0.472963
$819$	0	0
$820$	6.28179	0.219370
$821$	−33.2271	−1.15963	−0.579817	−	0.814746i	$-0.696876\pi$
$821$	−33.2271	−1.15963	−0.579817	+	0.814746i	$0.696876\pi$
$822$	0	0
$823$	15.7718	0.549769	0.274884	−	0.961477i	$-0.411360\pi$
$823$	15.7718	0.549769	0.274884	+	0.961477i	$0.411360\pi$
$824$	−27.4520	−0.956334
$825$	0	0
$826$	−18.9550	−0.659528
$827$	1.81664	0.0631709	0.0315854	−	0.999501i	$-0.489944\pi$
$827$	1.81664	0.0631709	0.0315854	+	0.999501i	$0.489944\pi$
$828$	0	0
$829$	29.6050	1.02822	0.514112	−	0.857723i	$-0.328122\pi$
$829$	29.6050	1.02822	0.514112	+	0.857723i	$0.328122\pi$
$830$	10.6289	0.368933
$831$	0	0
$832$	0	0
$833$	−14.8664	−0.515089
$834$	0	0
$835$	10.4279	0.360871
$836$	5.17501	0.178981
$837$	0	0
$838$	22.2254	0.767764
$839$	6.93085	0.239280	0.119640	−	0.992817i	$-0.461826\pi$
$839$	6.93085	0.239280	0.119640	+	0.992817i	$0.461826\pi$
$840$	0	0
$841$	−22.2122	−0.765938
$842$	−12.0773	−0.416211
$843$	0	0
$844$	6.74691	0.232238
$845$	0	0
$846$	0	0
$847$	−13.6324	−0.468416
$848$	−1.66735	−0.0572571
$849$	0	0
$850$	−3.08744	−0.105898
$851$	−1.67910	−0.0575589
$852$	0	0
$853$	−6.90807	−0.236528	−0.118264	−	0.992982i	$-0.537733\pi$
$853$	−6.90807	−0.236528	−0.118264	+	0.992982i	$0.537733\pi$
$854$	−9.85012	−0.337064
$855$	0	0
$856$	−32.3473	−1.10561
$857$	−29.6052	−1.01129	−0.505646	−	0.862741i	$-0.668746\pi$
$857$	−29.6052	−1.01129	−0.505646	+	0.862741i	$0.668746\pi$
$858$	0	0
$859$	−51.0371	−1.74136	−0.870681	−	0.491848i	$-0.836322\pi$
$859$	−51.0371	−1.74136	−0.870681	+	0.491848i	$0.836322\pi$
$860$	−1.39738	−0.0476503
$861$	0	0
$862$	−13.9081	−0.473710
$863$	22.2757	0.758272	0.379136	−	0.925341i	$-0.376221\pi$
$863$	22.2757	0.758272	0.379136	+	0.925341i	$0.376221\pi$
$864$	0	0
$865$	12.4055	0.421799
$866$	18.4771	0.627879
$867$	0	0
$868$	−4.59000	−0.155795
$869$	−11.7216	−0.397629
$870$	0	0
$871$	0	0
$872$	−54.5498	−1.84729
$873$	0	0
$874$	−8.63120	−0.291955
$875$	1.56306	0.0528410
$876$	0	0
$877$	−13.9954	−0.472590	−0.236295	−	0.971681i	$-0.575933\pi$
$877$	−13.9954	−0.472590	−0.236295	+	0.971681i	$0.575933\pi$
$878$	−28.7846	−0.971434
$879$	0	0
$880$	0.862686	0.0290811
$881$	−55.6993	−1.87656	−0.938279	−	0.345879i	$-0.887581\pi$
$881$	−55.6993	−1.87656	−0.938279	+	0.345879i	$0.887581\pi$
$882$	0	0
$883$	35.1425	1.18264	0.591320	−	0.806437i	$-0.298607\pi$
$883$	35.1425	1.18264	0.591320	+	0.806437i	$0.298607\pi$
$884$	0	0
$885$	0	0
$886$	−8.36211	−0.280930
$887$	36.4172	1.22277	0.611386	−	0.791333i	$-0.290612\pi$
$887$	36.4172	1.22277	0.611386	+	0.791333i	$0.290612\pi$
$888$	0	0
$889$	−11.8855	−0.398627
$890$	−0.754078	−0.0252768
$891$	0	0
$892$	−7.04230	−0.235794
$893$	15.9398	0.533405
$894$	0	0
$895$	−9.16353	−0.306303
$896$	7.51841	0.251172
$897$	0	0
$898$	−19.9876	−0.666994
$899$	6.92753	0.231046
$900$	0	0
$901$	9.51756	0.317076
$902$	−8.12513	−0.270537
$903$	0	0
$904$	−38.3189	−1.27447
$905$	−4.71081	−0.156593
$906$	0	0
$907$	21.8530	0.725618	0.362809	−	0.931864i	$-0.381818\pi$
$907$	21.8530	0.725618	0.362809	+	0.931864i	$0.381818\pi$
$908$	−14.8548	−0.492973
$909$	0	0
$910$	0	0
$911$	−10.6483	−0.352794	−0.176397	−	0.984319i	$-0.556444\pi$
$911$	−10.6483	−0.352794	−0.176397	+	0.984319i	$0.556444\pi$
$912$	0	0
$913$	16.9527	0.561052
$914$	37.3042	1.23391
$915$	0	0
$916$	−10.1000	−0.333715
$917$	29.7500	0.982431
$918$	0	0
$919$	−47.9731	−1.58249	−0.791243	−	0.611502i	$-0.790566\pi$
$919$	−47.9731	−1.58249	−0.791243	+	0.611502i	$0.790566\pi$
$920$	8.63120	0.284562
$921$	0	0
$922$	0.401405	0.0132196
$923$	0	0
$924$	0	0
$925$	−0.571534	−0.0187919
$926$	28.5338	0.937679
$927$	0	0
$928$	−13.8992	−0.456265
$929$	−13.2246	−0.433884	−0.216942	−	0.976185i	$-0.569608\pi$
$929$	−13.2246	−0.433884	−0.216942	+	0.976185i	$0.569608\pi$
$930$	0	0
$931$	−14.1462	−0.463625
$932$	24.4794	0.801850
$933$	0	0
$934$	−7.63749	−0.249906
$935$	−4.92437	−0.161044
$936$	0	0
$937$	46.3689	1.51481	0.757403	−	0.652948i	$-0.226468\pi$
$937$	46.3689	1.51481	0.757403	+	0.652948i	$0.226468\pi$
$938$	14.0498	0.458741
$939$	0	0
$940$	−5.67060	−0.184955
$941$	13.3676	0.435772	0.217886	−	0.975974i	$-0.430084\pi$
$941$	13.3676	0.435772	0.217886	+	0.975974i	$0.430084\pi$
$942$	0	0
$943$	−16.7107	−0.544177
$944$	−7.32370	−0.238366
$945$	0	0
$946$	1.80743	0.0587647
$947$	40.7629	1.32461	0.662307	−	0.749232i	$-0.269577\pi$
$947$	40.7629	1.32461	0.662307	+	0.749232i	$0.269577\pi$
$948$	0	0
$949$	0	0
$950$	−2.93789	−0.0953177
$951$	0	0
$952$	14.9813	0.485548
$953$	−48.6303	−1.57529	−0.787645	−	0.616129i	$-0.788700\pi$
$953$	−48.6303	−1.57529	−0.787645	+	0.616129i	$0.788700\pi$
$954$	0	0
$955$	−15.1506	−0.490261
$956$	−19.4307	−0.628433
$957$	0	0
$958$	−35.2198	−1.13790
$959$	29.8051	0.962456
$960$	0	0
$961$	−23.9299	−0.771931
$962$	0	0
$963$	0	0
$964$	31.0758	1.00089
$965$	−10.2900	−0.331247
$966$	0	0
$967$	36.5694	1.17599	0.587996	−	0.808864i	$-0.299917\pi$
$967$	36.5694	1.17599	0.587996	+	0.808864i	$0.299917\pi$
$968$	−25.6232	−0.823561
$969$	0	0
$970$	−5.48932	−0.176251
$971$	−49.7370	−1.59614	−0.798068	−	0.602567i	$-0.794144\pi$
$971$	−49.7370	−1.59614	−0.798068	+	0.602567i	$0.794144\pi$
$972$	0	0
$973$	−23.1787	−0.743076
$974$	−10.8427	−0.347423
$975$	0	0
$976$	−3.80583	−0.121822
$977$	19.0019	0.607925	0.303962	−	0.952684i	$-0.401690\pi$
$977$	19.0019	0.607925	0.303962	+	0.952684i	$0.401690\pi$
$978$	0	0
$979$	−1.20273	−0.0384394
$980$	5.03255	0.160759
$981$	0	0
$982$	19.9720	0.637332
$983$	−3.16020	−0.100795	−0.0503974	−	0.998729i	$-0.516049\pi$
$983$	−3.16020	−0.100795	−0.0503974	+	0.998729i	$0.516049\pi$
$984$	0	0
$985$	−5.19012	−0.165371
$986$	−8.04383	−0.256168
$987$	0	0
$988$	0	0
$989$	3.71730	0.118203
$990$	0	0
$991$	−15.5149	−0.492848	−0.246424	−	0.969162i	$-0.579256\pi$
$991$	−15.5149	−0.492848	−0.246424	+	0.969162i	$0.579256\pi$
$992$	−14.1854	−0.450386
$993$	0	0
$994$	−1.80486	−0.0572465
$995$	−12.6178	−0.400011
$996$	0	0
$997$	16.8096	0.532365	0.266183	−	0.963923i	$-0.414238\pi$
$997$	16.8096	0.532365	0.266183	+	0.963923i	$0.414238\pi$
$998$	−18.9179	−0.598835
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.cn.1.2		5
3.2	odd	2		7605.2.a.cl.1.4		5
13.4	even	6		585.2.j.i.406.2	yes	10
13.10	even	6		585.2.j.i.451.2	yes	10
13.12	even	2		7605.2.a.cm.1.4		5
39.17	odd	6		585.2.j.h.406.4	✓	10
39.23	odd	6		585.2.j.h.451.4	yes	10
39.38	odd	2		7605.2.a.co.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.j.h.406.4	✓	10	39.17	odd	6
585.2.j.h.451.4	yes	10	39.23	odd	6
585.2.j.i.406.2	yes	10	13.4	even	6
585.2.j.i.451.2	yes	10	13.10	even	6
7605.2.a.cl.1.4		5	3.2	odd	2
7605.2.a.cm.1.4		5	13.12	even	2
7605.2.a.cn.1.2		5	1.1	even	1	trivial
7605.2.a.co.1.2		5	39.38	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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q-0.946366 q^{2} -1.10439 q^{4} +1.00000 q^{5} +1.56306 q^{7} +2.93789 q^{8} +O(q^{10})\)	\(q-0.946366 q^{2} -1.10439 q^{4} +1.00000 q^{5} +1.56306 q^{7} +2.93789 q^{8} -0.946366 q^{10} -1.50942 q^{11} -1.47922 q^{14} -0.571534 q^{16} +3.26242 q^{17} +3.10439 q^{19} -1.10439 q^{20} +1.42847 q^{22} +2.93789 q^{23} +1.00000 q^{25} -1.72623 q^{28} +2.60534 q^{29} +2.65897 q^{31} -5.33490 q^{32} -3.08744 q^{34} +1.56306 q^{35} -0.571534 q^{37} -2.93789 q^{38} +2.93789 q^{40} -5.68801 q^{41} +1.26530 q^{43} +1.66700 q^{44} -2.78032 q^{46} +5.13459 q^{47} -4.55685 q^{49} -0.946366 q^{50} +2.91733 q^{53} -1.50942 q^{55} +4.59209 q^{56} -2.46560 q^{58} +12.8141 q^{59} +6.65897 q^{61} -2.51636 q^{62} +6.19183 q^{64} -9.49807 q^{67} -3.60299 q^{68} -1.47922 q^{70} +1.22014 q^{71} +12.8785 q^{73} +0.540880 q^{74} -3.42847 q^{76} -2.35932 q^{77} +7.76563 q^{79} -0.571534 q^{80} +5.38294 q^{82} -11.2312 q^{83} +3.26242 q^{85} -1.19743 q^{86} -4.43452 q^{88} +0.796815 q^{89} -3.24458 q^{92} -4.85920 q^{94} +3.10439 q^{95} +5.80042 q^{97} +4.31245 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8}+O(q^{10}) \) 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8	\( 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{11} + 4 q^{14} + 4 q^{16} + 4 q^{19} + 6 q^{20} + 14 q^{22} + 6 q^{23} + 5 q^{25} + 2 q^{28} - 16 q^{29} - 9 q^{31} + 14 q^{32} - q^{35} + 4 q^{37} - 6 q^{38} + 6 q^{40} + 6 q^{41} + 15 q^{43} + 14 q^{44} + 16 q^{46} + 10 q^{47} + 10 q^{49} + 2 q^{50} + 20 q^{53} + 8 q^{55} + 2 q^{56} + 4 q^{58} + 12 q^{59} + 11 q^{61} + 22 q^{62} + 4 q^{64} - 5 q^{67} - 50 q^{68} + 4 q^{70} + 10 q^{71} - q^{73} + 26 q^{74} - 24 q^{76} - 42 q^{77} - 17 q^{79} + 4 q^{80} + 16 q^{82} + 16 q^{83} + 44 q^{86} + 20 q^{88} + 4 q^{89} + 34 q^{92} - 16 q^{94} + 4 q^{95} + 11 q^{97} + 30 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 2 * q^10 + 8 * q^11 + 4 * q^14 + 4 * q^16 + 4 * q^19 + 6 * q^20 + 14 * q^22 + 6 * q^23 + 5 * q^25 + 2 * q^28 - 16 * q^29 - 9 * q^31 + 14 * q^32 - q^35 + 4 * q^37 - 6 * q^38 + 6 * q^40 + 6 * q^41 + 15 * q^43 + 14 * q^44 + 16 * q^46 + 10 * q^47 + 10 * q^49 + 2 * q^50 + 20 * q^53 + 8 * q^55 + 2 * q^56 + 4 * q^58 + 12 * q^59 + 11 * q^61 + 22 * q^62 + 4 * q^64 - 5 * q^67 - 50 * q^68 + 4 * q^70 + 10 * q^71 - q^73 + 26 * q^74 - 24 * q^76 - 42 * q^77 - 17 * q^79 + 4 * q^80 + 16 * q^82 + 16 * q^83 + 44 * q^86 + 20 * q^88 + 4 * q^89 + 34 * q^92 - 16 * q^94 + 4 * q^95 + 11 * q^97 + 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more