LMFDB - Embedded newform 2925.2.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(1,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, 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("2925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	2925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.77846 q^{2} +5.71982 q^{4} +2.71982 q^{7} +10.3354 q^{8} +O(q^{10})$	$q+2.77846 q^{2} +5.71982 q^{4} +2.71982 q^{7} +10.3354 q^{8} +2.71982 q^{11} -1.00000 q^{13} +7.55691 q^{14} +17.2767 q^{16} -2.83709 q^{17} -3.55691 q^{19} +7.55691 q^{22} -4.83709 q^{23} -2.77846 q^{26} +15.5569 q^{28} -6.00000 q^{29} +7.55691 q^{31} +27.3319 q^{32} -7.88273 q^{34} +4.27674 q^{37} -9.88273 q^{38} -2.83709 q^{41} -11.1138 q^{43} +15.5569 q^{44} -13.4396 q^{46} -11.5569 q^{47} +0.397442 q^{49} -5.71982 q^{52} +1.16291 q^{53} +28.1104 q^{56} -16.6707 q^{58} +2.11727 q^{59} +6.60256 q^{61} +20.9966 q^{62} +41.3871 q^{64} -1.88273 q^{67} -16.2277 q^{68} +6.71982 q^{71} -9.11383 q^{73} +11.8827 q^{74} -20.3449 q^{76} +7.39744 q^{77} +10.2767 q^{79} -7.88273 q^{82} +2.11727 q^{83} -30.8793 q^{86} +28.1104 q^{88} -1.16291 q^{89} -2.71982 q^{91} -27.6673 q^{92} -32.1104 q^{94} +10.8371 q^{97} +1.10428 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 8 q^{4} - q^{7} + 6 q^{8}+O(q^{10}) $ 3 * q + 8 * q^4 - q^7 + 6 * q^8	$ 3 q + 8 q^{4} - q^{7} + 6 q^{8} - q^{11} - 3 q^{13} + 6 q^{14} + 26 q^{16} - q^{17} + 6 q^{19} + 6 q^{22} - 7 q^{23} + 30 q^{28} - 18 q^{29} + 6 q^{31} + 22 q^{32} - 22 q^{34} - 13 q^{37} - 28 q^{38} - q^{41} + 30 q^{44} - 22 q^{46} - 18 q^{47} + 12 q^{49} - 8 q^{52} + 11 q^{53} + 16 q^{56} + 8 q^{59} + 9 q^{61} + 28 q^{62} + 30 q^{64} - 4 q^{67} + 18 q^{68} + 11 q^{71} + 6 q^{73} + 34 q^{74} + 4 q^{76} + 33 q^{77} + 5 q^{79} - 22 q^{82} + 8 q^{83} - 56 q^{86} + 16 q^{88} - 11 q^{89} + q^{91} + 2 q^{92} - 28 q^{94} + 25 q^{97} + 10 q^{98}+O(q^{100}) $ 3 * q + 8 * q^4 - q^7 + 6 * q^8 - q^11 - 3 * q^13 + 6 * q^14 + 26 * q^16 - q^17 + 6 * q^19 + 6 * q^22 - 7 * q^23 + 30 * q^28 - 18 * q^29 + 6 * q^31 + 22 * q^32 - 22 * q^34 - 13 * q^37 - 28 * q^38 - q^41 + 30 * q^44 - 22 * q^46 - 18 * q^47 + 12 * q^49 - 8 * q^52 + 11 * q^53 + 16 * q^56 + 8 * q^59 + 9 * q^61 + 28 * q^62 + 30 * q^64 - 4 * q^67 + 18 * q^68 + 11 * q^71 + 6 * q^73 + 34 * q^74 + 4 * q^76 + 33 * q^77 + 5 * q^79 - 22 * q^82 + 8 * q^83 - 56 * q^86 + 16 * q^88 - 11 * q^89 + q^91 + 2 * q^92 - 28 * q^94 + 25 * q^97 + 10 * 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 $

$2$	2.77846	1.96467	0.982333	−	0.187142i	$-0.0599223\pi$
$2$	2.77846	1.96467	0.982333	+	0.187142i	$0.0599223\pi$
$3$	0	0
$3$	0	0
$4$	5.71982	2.85991
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.71982	1.02800	0.513998	−	0.857791i	$-0.328164\pi$
$7$	2.71982	1.02800	0.513998	+	0.857791i	$0.328164\pi$
$8$	10.3354	3.65411
$9$	0	0
$10$	0	0
$11$	2.71982	0.820058	0.410029	−	0.912073i	$-0.365519\pi$
$11$	2.71982	0.820058	0.410029	+	0.912073i	$0.365519\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	7.55691	2.01967
$15$	0	0
$16$	17.2767	4.31918
$17$	−2.83709	−0.688095	−0.344048	−	0.938952i	$-0.611798\pi$
$17$	−2.83709	−0.688095	−0.344048	+	0.938952i	$0.611798\pi$
$18$	0	0
$19$	−3.55691	−0.816012	−0.408006	−	0.912979i	$-0.633776\pi$
$19$	−3.55691	−0.816012	−0.408006	+	0.912979i	$0.633776\pi$
$20$	0	0
$21$	0	0
$22$	7.55691	1.61114
$23$	−4.83709	−1.00860	−0.504302	−	0.863528i	$-0.668250\pi$
$23$	−4.83709	−1.00860	−0.504302	+	0.863528i	$0.668250\pi$
$24$	0	0
$25$	0	0
$26$	−2.77846	−0.544900
$27$	0	0
$28$	15.5569	2.93998
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	7.55691	1.35726	0.678631	−	0.734479i	$-0.262574\pi$
$31$	7.55691	1.35726	0.678631	+	0.734479i	$0.262574\pi$
$32$	27.3319	4.83165
$33$	0	0
$34$	−7.88273	−1.35188
$35$	0	0
$36$	0	0
$37$	4.27674	0.703091	0.351546	−	0.936171i	$-0.385656\pi$
$37$	4.27674	0.703091	0.351546	+	0.936171i	$0.385656\pi$
$38$	−9.88273	−1.60319
$39$	0	0
$40$	0	0
$41$	−2.83709	−0.443079	−0.221540	−	0.975151i	$-0.571108\pi$
$41$	−2.83709	−0.443079	−0.221540	+	0.975151i	$0.571108\pi$
$42$	0	0
$43$	−11.1138	−1.69484	−0.847421	−	0.530921i	$-0.821846\pi$
$43$	−11.1138	−1.69484	−0.847421	+	0.530921i	$0.821846\pi$
$44$	15.5569	2.34529
$45$	0	0
$46$	−13.4396	−1.98157
$47$	−11.5569	−1.68575	−0.842875	−	0.538110i	$-0.819138\pi$
$47$	−11.5569	−1.68575	−0.842875	+	0.538110i	$0.819138\pi$
$48$	0	0
$49$	0.397442	0.0567775
$50$	0	0
$51$	0	0
$52$	−5.71982	−0.793197
$53$	1.16291	0.159738	0.0798690	−	0.996805i	$-0.474550\pi$
$53$	1.16291	0.159738	0.0798690	+	0.996805i	$0.474550\pi$
$54$	0	0
$55$	0	0
$56$	28.1104	3.75641
$57$	0	0
$58$	−16.6707	−2.18898
$59$	2.11727	0.275645	0.137822	−	0.990457i	$-0.455990\pi$
$59$	2.11727	0.275645	0.137822	+	0.990457i	$0.455990\pi$
$60$	0	0
$61$	6.60256	0.845371	0.422685	−	0.906276i	$-0.361088\pi$
$61$	6.60256	0.845371	0.422685	+	0.906276i	$0.361088\pi$
$62$	20.9966	2.66657
$63$	0	0
$64$	41.3871	5.17339
$65$	0	0
$66$	0	0
$67$	−1.88273	−0.230013	−0.115006	−	0.993365i	$-0.536689\pi$
$67$	−1.88273	−0.230013	−0.115006	+	0.993365i	$0.536689\pi$
$68$	−16.2277	−1.96789
$69$	0	0
$70$	0	0
$71$	6.71982	0.797496	0.398748	−	0.917060i	$-0.369445\pi$
$71$	6.71982	0.797496	0.398748	+	0.917060i	$0.369445\pi$
$72$	0	0
$73$	−9.11383	−1.06669	−0.533346	−	0.845897i	$-0.679066\pi$
$73$	−9.11383	−1.06669	−0.533346	+	0.845897i	$0.679066\pi$
$74$	11.8827	1.38134
$75$	0	0
$76$	−20.3449	−2.33372
$77$	7.39744	0.843017
$78$	0	0
$79$	10.2767	1.15622	0.578112	−	0.815958i	$-0.303790\pi$
$79$	10.2767	1.15622	0.578112	+	0.815958i	$0.303790\pi$
$80$	0	0
$81$	0	0
$82$	−7.88273	−0.870502
$83$	2.11727	0.232400	0.116200	−	0.993226i	$-0.462929\pi$
$83$	2.11727	0.232400	0.116200	+	0.993226i	$0.462929\pi$
$84$	0	0
$85$	0	0
$86$	−30.8793	−3.32980
$87$	0	0
$88$	28.1104	2.99658
$89$	−1.16291	−0.123268	−0.0616341	−	0.998099i	$-0.519631\pi$
$89$	−1.16291	−0.123268	−0.0616341	+	0.998099i	$0.519631\pi$
$90$	0	0
$91$	−2.71982	−0.285115
$92$	−27.6673	−2.88452
$93$	0	0
$94$	−32.1104	−3.31193
$95$	0	0
$96$	0	0
$97$	10.8371	1.10034	0.550170	−	0.835053i	$-0.314563\pi$
$97$	10.8371	1.10034	0.550170	+	0.835053i	$0.314563\pi$
$98$	1.10428	0.111549
$99$	0	0
$100$	0	0
$101$	−7.67418	−0.763610	−0.381805	−	0.924243i	$-0.624697\pi$
$101$	−7.67418	−0.763610	−0.381805	+	0.924243i	$0.624697\pi$
$102$	0	0
$103$	−3.76547	−0.371023	−0.185511	−	0.982642i	$-0.559394\pi$
$103$	−3.76547	−0.371023	−0.185511	+	0.982642i	$0.559394\pi$
$104$	−10.3354	−1.01347
$105$	0	0
$106$	3.23109	0.313832
$107$	−12.6026	−1.21834	−0.609168	−	0.793041i	$-0.708496\pi$
$107$	−12.6026	−1.21834	−0.609168	+	0.793041i	$0.708496\pi$
$108$	0	0
$109$	11.4396	1.09572	0.547860	−	0.836570i	$-0.315443\pi$
$109$	11.4396	1.09572	0.547860	+	0.836570i	$0.315443\pi$
$110$	0	0
$111$	0	0
$112$	46.9897	4.44011
$113$	−13.1138	−1.23365	−0.616823	−	0.787102i	$-0.711580\pi$
$113$	−13.1138	−1.23365	−0.616823	+	0.787102i	$0.711580\pi$
$114$	0	0
$115$	0	0
$116$	−34.3189	−3.18643
$117$	0	0
$118$	5.88273	0.541550
$119$	−7.71639	−0.707360
$120$	0	0
$121$	−3.60256	−0.327505
$122$	18.3449	1.66087
$123$	0	0
$124$	43.2242	3.88165
$125$	0	0
$126$	0	0
$127$	13.4396	1.19258	0.596288	−	0.802771i	$-0.296642\pi$
$127$	13.4396	1.19258	0.596288	+	0.802771i	$0.296642\pi$
$128$	60.3285	5.33234
$129$	0	0
$130$	0	0
$131$	9.43965	0.824746	0.412373	−	0.911015i	$-0.364700\pi$
$131$	9.43965	0.824746	0.412373	+	0.911015i	$0.364700\pi$
$132$	0	0
$133$	−9.67418	−0.838858
$134$	−5.23109	−0.451898
$135$	0	0
$136$	−29.3224	−2.51437
$137$	1.76547	0.150834	0.0754170	−	0.997152i	$-0.475971\pi$
$137$	1.76547	0.150834	0.0754170	+	0.997152i	$0.475971\pi$
$138$	0	0
$139$	−6.27674	−0.532386	−0.266193	−	0.963920i	$-0.585766\pi$
$139$	−6.27674	−0.532386	−0.266193	+	0.963920i	$0.585766\pi$
$140$	0	0
$141$	0	0
$142$	18.6707	1.56681
$143$	−2.71982	−0.227443
$144$	0	0
$145$	0	0
$146$	−25.3224	−2.09570
$147$	0	0
$148$	24.4622	2.01078
$149$	20.8302	1.70648	0.853239	−	0.521520i	$-0.174635\pi$
$149$	20.8302	1.70648	0.853239	+	0.521520i	$0.174635\pi$
$150$	0	0
$151$	4.99656	0.406614	0.203307	−	0.979115i	$-0.434831\pi$
$151$	4.99656	0.406614	0.203307	+	0.979115i	$0.434831\pi$
$152$	−36.7620	−2.98179
$153$	0	0
$154$	20.5535	1.65625
$155$	0	0
$156$	0	0
$157$	−8.87930	−0.708645	−0.354322	−	0.935123i	$-0.615288\pi$
$157$	−8.87930	−0.708645	−0.354322	+	0.935123i	$0.615288\pi$
$158$	28.5535	2.27159
$159$	0	0
$160$	0	0
$161$	−13.1560	−1.03684
$162$	0	0
$163$	13.8337	1.08354	0.541768	−	0.840528i	$-0.317755\pi$
$163$	13.8337	1.08354	0.541768	+	0.840528i	$0.317755\pi$
$164$	−16.2277	−1.26717
$165$	0	0
$166$	5.88273	0.456589
$167$	−9.88273	−0.764749	−0.382374	−	0.924007i	$-0.624894\pi$
$167$	−9.88273	−0.764749	−0.382374	+	0.924007i	$0.624894\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−63.5691	−4.84710
$173$	13.1138	0.997026	0.498513	−	0.866882i	$-0.333880\pi$
$173$	13.1138	0.997026	0.498513	+	0.866882i	$0.333880\pi$
$174$	0	0
$175$	0	0
$176$	46.9897	3.54198
$177$	0	0
$178$	−3.23109	−0.242181
$179$	−8.55348	−0.639317	−0.319658	−	0.947533i	$-0.603568\pi$
$179$	−8.55348	−0.639317	−0.319658	+	0.947533i	$0.603568\pi$
$180$	0	0
$181$	3.72326	0.276748	0.138374	−	0.990380i	$-0.455812\pi$
$181$	3.72326	0.276748	0.138374	+	0.990380i	$0.455812\pi$
$182$	−7.55691	−0.560156
$183$	0	0
$184$	−49.9931	−3.68554
$185$	0	0
$186$	0	0
$187$	−7.71639	−0.564278
$188$	−66.1035	−4.82109
$189$	0	0
$190$	0	0
$191$	4.23453	0.306400	0.153200	−	0.988195i	$-0.451042\pi$
$191$	4.23453	0.306400	0.153200	+	0.988195i	$0.451042\pi$
$192$	0	0
$193$	23.3906	1.68369	0.841845	−	0.539719i	$-0.181470\pi$
$193$	23.3906	1.68369	0.841845	+	0.539719i	$0.181470\pi$
$194$	30.1104	2.16180
$195$	0	0
$196$	2.27330	0.162379
$197$	14.5535	1.03689	0.518446	−	0.855110i	$-0.326511\pi$
$197$	14.5535	1.03689	0.518446	+	0.855110i	$0.326511\pi$
$198$	0	0
$199$	−15.1138	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$199$	−15.1138	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$200$	0	0
$201$	0	0
$202$	−21.3224	−1.50024
$203$	−16.3189	−1.14537
$204$	0	0
$205$	0	0
$206$	−10.4622	−0.728935
$207$	0	0
$208$	−17.2767	−1.19793
$209$	−9.67418	−0.669177
$210$	0	0
$211$	−18.2277	−1.25484	−0.627422	−	0.778680i	$-0.715890\pi$
$211$	−18.2277	−1.25484	−0.627422	+	0.778680i	$0.715890\pi$
$212$	6.65164	0.456836
$213$	0	0
$214$	−35.0157	−2.39362
$215$	0	0
$216$	0	0
$217$	20.5535	1.39526
$218$	31.7846	2.15272
$219$	0	0
$220$	0	0
$221$	2.83709	0.190843
$222$	0	0
$223$	−10.1173	−0.677502	−0.338751	−	0.940876i	$-0.610004\pi$
$223$	−10.1173	−0.677502	−0.338751	+	0.940876i	$0.610004\pi$
$224$	74.3380	4.96692
$225$	0	0
$226$	−36.4362	−2.42370
$227$	11.3224	0.751493	0.375746	−	0.926723i	$-0.377386\pi$
$227$	11.3224	0.751493	0.375746	+	0.926723i	$0.377386\pi$
$228$	0	0
$229$	−6.23453	−0.411990	−0.205995	−	0.978553i	$-0.566043\pi$
$229$	−6.23453	−0.411990	−0.205995	+	0.978553i	$0.566043\pi$
$230$	0	0
$231$	0	0
$232$	−62.0122	−4.07130
$233$	6.83709	0.447913	0.223956	−	0.974599i	$-0.428103\pi$
$233$	6.83709	0.447913	0.223956	+	0.974599i	$0.428103\pi$
$234$	0	0
$235$	0	0
$236$	12.1104	0.788319
$237$	0	0
$238$	−21.4396	−1.38973
$239$	−1.28018	−0.0828077	−0.0414039	−	0.999142i	$-0.513183\pi$
$239$	−1.28018	−0.0828077	−0.0414039	+	0.999142i	$0.513183\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	−10.0096	−0.643438
$243$	0	0
$244$	37.7655	2.41769
$245$	0	0
$246$	0	0
$247$	3.55691	0.226321
$248$	78.1035	4.95958
$249$	0	0
$250$	0	0
$251$	−18.2277	−1.15052	−0.575260	−	0.817971i	$-0.695099\pi$
$251$	−18.2277	−1.15052	−0.575260	+	0.817971i	$0.695099\pi$
$252$	0	0
$253$	−13.1560	−0.827113
$254$	37.3415	2.34301
$255$	0	0
$256$	84.8459	5.30287
$257$	1.11383	0.0694787	0.0347394	−	0.999396i	$-0.488940\pi$
$257$	1.11383	0.0694787	0.0347394	+	0.999396i	$0.488940\pi$
$258$	0	0
$259$	11.6320	0.722776
$260$	0	0
$261$	0	0
$262$	26.2277	1.62035
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	0	0
$266$	−26.8793	−1.64808
$267$	0	0
$268$	−10.7689	−0.657816
$269$	−15.6742	−0.955672	−0.477836	−	0.878449i	$-0.658579\pi$
$269$	−15.6742	−0.955672	−0.477836	+	0.878449i	$0.658579\pi$
$270$	0	0
$271$	0.443086	0.0269155	0.0134578	−	0.999909i	$-0.495716\pi$
$271$	0.443086	0.0269155	0.0134578	+	0.999909i	$0.495716\pi$
$272$	−49.0157	−2.97201
$273$	0	0
$274$	4.90528	0.296339
$275$	0	0
$276$	0	0
$277$	4.87930	0.293168	0.146584	−	0.989198i	$-0.453172\pi$
$277$	4.87930	0.293168	0.146584	+	0.989198i	$0.453172\pi$
$278$	−17.4396	−1.04596
$279$	0	0
$280$	0	0
$281$	9.11383	0.543685	0.271843	−	0.962342i	$-0.412367\pi$
$281$	9.11383	0.543685	0.271843	+	0.962342i	$0.412367\pi$
$282$	0	0
$283$	−33.3415	−1.98195	−0.990973	−	0.134063i	$-0.957198\pi$
$283$	−33.3415	−1.98195	−0.990973	+	0.134063i	$0.957198\pi$
$284$	38.4362	2.28077
$285$	0	0
$286$	−7.55691	−0.446850
$287$	−7.71639	−0.455484
$288$	0	0
$289$	−8.95092	−0.526525
$290$	0	0
$291$	0	0
$292$	−52.1295	−3.05065
$293$	−29.4328	−1.71948	−0.859740	−	0.510731i	$-0.829375\pi$
$293$	−29.4328	−1.71948	−0.859740	+	0.510731i	$0.829375\pi$
$294$	0	0
$295$	0	0
$296$	44.2017	2.56917
$297$	0	0
$298$	57.8759	3.35266
$299$	4.83709	0.279736
$300$	0	0
$301$	−30.2277	−1.74229
$302$	13.8827	0.798862
$303$	0	0
$304$	−61.4519	−3.52451
$305$	0	0
$306$	0	0
$307$	21.8337	1.24611	0.623056	−	0.782177i	$-0.285891\pi$
$307$	21.8337	1.24611	0.623056	+	0.782177i	$0.285891\pi$
$308$	42.3121	2.41095
$309$	0	0
$310$	0	0
$311$	−25.1070	−1.42368	−0.711842	−	0.702339i	$-0.752139\pi$
$311$	−25.1070	−1.42368	−0.711842	+	0.702339i	$0.752139\pi$
$312$	0	0
$313$	−8.22766	−0.465055	−0.232527	−	0.972590i	$-0.574700\pi$
$313$	−8.22766	−0.465055	−0.232527	+	0.972590i	$0.574700\pi$
$314$	−24.6707	−1.39225
$315$	0	0
$316$	58.7811	3.30670
$317$	27.6742	1.55434	0.777168	−	0.629293i	$-0.216655\pi$
$317$	27.6742	1.55434	0.777168	+	0.629293i	$0.216655\pi$
$318$	0	0
$319$	−16.3189	−0.913685
$320$	0	0
$321$	0	0
$322$	−36.5535	−2.03705
$323$	10.0913	0.561494
$324$	0	0
$325$	0	0
$326$	38.4362	2.12878
$327$	0	0
$328$	−29.3224	−1.61906
$329$	−31.4328	−1.73294
$330$	0	0
$331$	−13.2311	−0.727247	−0.363623	−	0.931546i	$-0.618460\pi$
$331$	−13.2311	−0.727247	−0.363623	+	0.931546i	$0.618460\pi$
$332$	12.1104	0.664644
$333$	0	0
$334$	−27.4588	−1.50248
$335$	0	0
$336$	0	0
$337$	4.32582	0.235642	0.117821	−	0.993035i	$-0.462409\pi$
$337$	4.32582	0.235642	0.117821	+	0.993035i	$0.462409\pi$
$338$	2.77846	0.151128
$339$	0	0
$340$	0	0
$341$	20.5535	1.11303
$342$	0	0
$343$	−17.9578	−0.969630
$344$	−114.866	−6.19314
$345$	0	0
$346$	36.4362	1.95882
$347$	6.27674	0.336953	0.168476	−	0.985706i	$-0.446115\pi$
$347$	6.27674	0.336953	0.168476	+	0.985706i	$0.446115\pi$
$348$	0	0
$349$	17.6673	0.945709	0.472855	−	0.881140i	$-0.343224\pi$
$349$	17.6673	0.945709	0.472855	+	0.881140i	$0.343224\pi$
$350$	0	0
$351$	0	0
$352$	74.3380	3.96223
$353$	−13.7655	−0.732662	−0.366331	−	0.930485i	$-0.619386\pi$
$353$	−13.7655	−0.732662	−0.366331	+	0.930485i	$0.619386\pi$
$354$	0	0
$355$	0	0
$356$	−6.65164	−0.352536
$357$	0	0
$358$	−23.7655	−1.25604
$359$	−0.996562	−0.0525965	−0.0262983	−	0.999654i	$-0.508372\pi$
$359$	−0.996562	−0.0525965	−0.0262983	+	0.999654i	$0.508372\pi$
$360$	0	0
$361$	−6.34836	−0.334124
$362$	10.3449	0.543717
$363$	0	0
$364$	−15.5569	−0.815404
$365$	0	0
$366$	0	0
$367$	−14.2277	−0.742678	−0.371339	−	0.928497i	$-0.621101\pi$
$367$	−14.2277	−0.742678	−0.371339	+	0.928497i	$0.621101\pi$
$368$	−83.5691	−4.35634
$369$	0	0
$370$	0	0
$371$	3.16291	0.164210
$372$	0	0
$373$	−15.6742	−0.811578	−0.405789	−	0.913967i	$-0.633003\pi$
$373$	−15.6742	−0.811578	−0.405789	+	0.913967i	$0.633003\pi$
$374$	−21.4396	−1.10862
$375$	0	0
$376$	−119.445	−6.15991
$377$	6.00000	0.309016
$378$	0	0
$379$	26.2017	1.34589	0.672945	−	0.739693i	$-0.265029\pi$
$379$	26.2017	1.34589	0.672945	+	0.739693i	$0.265029\pi$
$380$	0	0
$381$	0	0
$382$	11.7655	0.601974
$383$	22.4362	1.14644	0.573218	−	0.819403i	$-0.305695\pi$
$383$	22.4362	1.14644	0.573218	+	0.819403i	$0.305695\pi$
$384$	0	0
$385$	0	0
$386$	64.9897	3.30789
$387$	0	0
$388$	61.9862	3.14687
$389$	−31.6742	−1.60594	−0.802972	−	0.596016i	$-0.796749\pi$
$389$	−31.6742	−1.60594	−0.802972	+	0.596016i	$0.796749\pi$
$390$	0	0
$391$	13.7233	0.694015
$392$	4.10771	0.207471
$393$	0	0
$394$	40.4362	2.03715
$395$	0	0
$396$	0	0
$397$	−17.9509	−0.900931	−0.450465	−	0.892794i	$-0.648742\pi$
$397$	−17.9509	−0.900931	−0.450465	+	0.892794i	$0.648742\pi$
$398$	−41.9931	−2.10493
$399$	0	0
$400$	0	0
$401$	−13.5829	−0.678297	−0.339149	−	0.940733i	$-0.610139\pi$
$401$	−13.5829	−0.678297	−0.339149	+	0.940733i	$0.610139\pi$
$402$	0	0
$403$	−7.55691	−0.376437
$404$	−43.8950	−2.18386
$405$	0	0
$406$	−45.3415	−2.25026
$407$	11.6320	0.576576
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	0	0
$412$	−21.5378	−1.06109
$413$	5.75859	0.283362
$414$	0	0
$415$	0	0
$416$	−27.3319	−1.34006
$417$	0	0
$418$	−26.8793	−1.31471
$419$	−12.3189	−0.601820	−0.300910	−	0.953653i	$-0.597290\pi$
$419$	−12.3189	−0.601820	−0.300910	+	0.953653i	$0.597290\pi$
$420$	0	0
$421$	22.7880	1.11062	0.555310	−	0.831644i	$-0.312600\pi$
$421$	22.7880	1.11062	0.555310	+	0.831644i	$0.312600\pi$
$422$	−50.6448	−2.46535
$423$	0	0
$424$	12.0191	0.583699
$425$	0	0
$426$	0	0
$427$	17.9578	0.869039
$428$	−72.0844	−3.48433
$429$	0	0
$430$	0	0
$431$	8.99656	0.433349	0.216675	−	0.976244i	$-0.430479\pi$
$431$	8.99656	0.433349	0.216675	+	0.976244i	$0.430479\pi$
$432$	0	0
$433$	20.3258	0.976797	0.488398	−	0.872621i	$-0.337581\pi$
$433$	20.3258	0.976797	0.488398	+	0.872621i	$0.337581\pi$
$434$	57.1070	2.74122
$435$	0	0
$436$	65.4328	3.13366
$437$	17.2051	0.823032
$438$	0	0
$439$	−25.3906	−1.21183	−0.605913	−	0.795531i	$-0.707192\pi$
$439$	−25.3906	−1.21183	−0.605913	+	0.795531i	$0.707192\pi$
$440$	0	0
$441$	0	0
$442$	7.88273	0.374943
$443$	10.9284	0.519223	0.259611	−	0.965713i	$-0.416406\pi$
$443$	10.9284	0.519223	0.259611	+	0.965713i	$0.416406\pi$
$444$	0	0
$445$	0	0
$446$	−28.1104	−1.33107
$447$	0	0
$448$	112.566	5.31823
$449$	−2.83709	−0.133891	−0.0669453	−	0.997757i	$-0.521325\pi$
$449$	−2.83709	−0.133891	−0.0669453	+	0.997757i	$0.521325\pi$
$450$	0	0
$451$	−7.71639	−0.363350
$452$	−75.0088	−3.52812
$453$	0	0
$454$	31.4588	1.47643
$455$	0	0
$456$	0	0
$457$	13.7164	0.641625	0.320813	−	0.947143i	$-0.396044\pi$
$457$	13.7164	0.641625	0.320813	+	0.947143i	$0.396044\pi$
$458$	−17.3224	−0.809422
$459$	0	0
$460$	0	0
$461$	19.6251	0.914032	0.457016	−	0.889458i	$-0.348918\pi$
$461$	19.6251	0.914032	0.457016	+	0.889458i	$0.348918\pi$
$462$	0	0
$463$	−27.0388	−1.25660	−0.628299	−	0.777972i	$-0.716249\pi$
$463$	−27.0388	−1.25660	−0.628299	+	0.777972i	$0.716249\pi$
$464$	−103.660	−4.81231
$465$	0	0
$466$	18.9966	0.879999
$467$	28.9215	1.33833	0.669164	−	0.743115i	$-0.266652\pi$
$467$	28.9215	1.33833	0.669164	+	0.743115i	$0.266652\pi$
$468$	0	0
$469$	−5.12070	−0.236452
$470$	0	0
$471$	0	0
$472$	21.8827	1.00723
$473$	−30.2277	−1.38987
$474$	0	0
$475$	0	0
$476$	−44.1364	−2.02299
$477$	0	0
$478$	−3.55691	−0.162689
$479$	12.1595	0.555580	0.277790	−	0.960642i	$-0.410398\pi$
$479$	12.1595	0.555580	0.277790	+	0.960642i	$0.410398\pi$
$480$	0	0
$481$	−4.27674	−0.195002
$482$	−16.6707	−0.759332
$483$	0	0
$484$	−20.6060	−0.936636
$485$	0	0
$486$	0	0
$487$	0.159472	0.00722636	0.00361318	−	0.999993i	$-0.498850\pi$
$487$	0.159472	0.00722636	0.00361318	+	0.999993i	$0.498850\pi$
$488$	68.2399	3.08907
$489$	0	0
$490$	0	0
$491$	42.2277	1.90571	0.952854	−	0.303430i	$-0.0981318\pi$
$491$	42.2277	1.90571	0.952854	+	0.303430i	$0.0981318\pi$
$492$	0	0
$493$	17.0225	0.766657
$494$	9.88273	0.444645
$495$	0	0
$496$	130.559	5.86226
$497$	18.2767	0.819824
$498$	0	0
$499$	7.79145	0.348793	0.174397	−	0.984676i	$-0.444203\pi$
$499$	7.79145	0.348793	0.174397	+	0.984676i	$0.444203\pi$
$500$	0	0
$501$	0	0
$502$	−50.6448	−2.26039
$503$	27.3484	1.21940	0.609702	−	0.792631i	$-0.291289\pi$
$503$	27.3484	1.21940	0.609702	+	0.792631i	$0.291289\pi$
$504$	0	0
$505$	0	0
$506$	−36.5535	−1.62500
$507$	0	0
$508$	76.8724	3.41066
$509$	33.4819	1.48406	0.742029	−	0.670368i	$-0.233864\pi$
$509$	33.4819	1.48406	0.742029	+	0.670368i	$0.233864\pi$
$510$	0	0
$511$	−24.7880	−1.09656
$512$	115.084	5.08603
$513$	0	0
$514$	3.09472	0.136502
$515$	0	0
$516$	0	0
$517$	−31.4328	−1.38241
$518$	32.3189	1.42001
$519$	0	0
$520$	0	0
$521$	17.3484	0.760045	0.380023	−	0.924977i	$-0.375916\pi$
$521$	17.3484	0.760045	0.380023	+	0.924977i	$0.375916\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	53.9931	2.35870
$525$	0	0
$526$	22.2277	0.969172
$527$	−21.4396	−0.933926
$528$	0	0
$529$	0.397442	0.0172801
$530$	0	0
$531$	0	0
$532$	−55.3346	−2.39906
$533$	2.83709	0.122888
$534$	0	0
$535$	0	0
$536$	−19.4588	−0.840490
$537$	0	0
$538$	−43.5500	−1.87758
$539$	1.08097	0.0465608
$540$	0	0
$541$	−32.6448	−1.40351	−0.701754	−	0.712419i	$-0.747599\pi$
$541$	−32.6448	−1.40351	−0.701754	+	0.712419i	$0.747599\pi$
$542$	1.23109	0.0528800
$543$	0	0
$544$	−77.5432	−3.32464
$545$	0	0
$546$	0	0
$547$	−34.2277	−1.46347	−0.731734	−	0.681590i	$-0.761289\pi$
$547$	−34.2277	−1.46347	−0.731734	+	0.681590i	$0.761289\pi$
$548$	10.0982	0.431372
$549$	0	0
$550$	0	0
$551$	21.3415	0.909178
$552$	0	0
$553$	27.9509	1.18859
$554$	13.5569	0.575978
$555$	0	0
$556$	−35.9018	−1.52258
$557$	6.65164	0.281839	0.140919	−	0.990021i	$-0.454994\pi$
$557$	6.65164	0.281839	0.140919	+	0.990021i	$0.454994\pi$
$558$	0	0
$559$	11.1138	0.470065
$560$	0	0
$561$	0	0
$562$	25.3224	1.06816
$563$	40.2699	1.69717	0.848586	−	0.529057i	$-0.177454\pi$
$563$	40.2699	1.69717	0.848586	+	0.529057i	$0.177454\pi$
$564$	0	0
$565$	0	0
$566$	−92.6379	−3.89386
$567$	0	0
$568$	69.4519	2.91414
$569$	13.4328	0.563131	0.281566	−	0.959542i	$-0.409146\pi$
$569$	13.4328	0.563131	0.281566	+	0.959542i	$0.409146\pi$
$570$	0	0
$571$	−35.7164	−1.49468	−0.747342	−	0.664439i	$-0.768670\pi$
$571$	−35.7164	−1.49468	−0.747342	+	0.664439i	$0.768670\pi$
$572$	−15.5569	−0.650467
$573$	0	0
$574$	−21.4396	−0.894874
$575$	0	0
$576$	0	0
$577$	13.7164	0.571021	0.285510	−	0.958376i	$-0.407837\pi$
$577$	13.7164	0.571021	0.285510	+	0.958376i	$0.407837\pi$
$578$	−24.8697	−1.03444
$579$	0	0
$580$	0	0
$581$	5.75859	0.238907
$582$	0	0
$583$	3.16291	0.130994
$584$	−94.1948	−3.89781
$585$	0	0
$586$	−81.7777	−3.37821
$587$	30.6707	1.26592	0.632959	−	0.774186i	$-0.281840\pi$
$587$	30.6707	1.26592	0.632959	+	0.774186i	$0.281840\pi$
$588$	0	0
$589$	−26.8793	−1.10754
$590$	0	0
$591$	0	0
$592$	73.8881	3.03678
$593$	−45.6673	−1.87533	−0.937666	−	0.347538i	$-0.887018\pi$
$593$	−45.6673	−1.87533	−0.937666	+	0.347538i	$0.887018\pi$
$594$	0	0
$595$	0	0
$596$	119.145	4.88038
$597$	0	0
$598$	13.4396	0.549588
$599$	40.2208	1.64338	0.821688	−	0.569937i	$-0.193032\pi$
$599$	40.2208	1.64338	0.821688	+	0.569937i	$0.193032\pi$
$600$	0	0
$601$	17.3974	0.709656	0.354828	−	0.934932i	$-0.384539\pi$
$601$	17.3974	0.709656	0.354828	+	0.934932i	$0.384539\pi$
$602$	−83.9862	−3.42302
$603$	0	0
$604$	28.5795	1.16288
$605$	0	0
$606$	0	0
$607$	14.2277	0.577483	0.288741	−	0.957407i	$-0.406763\pi$
$607$	14.2277	0.577483	0.288741	+	0.957407i	$0.406763\pi$
$608$	−97.2173	−3.94268
$609$	0	0
$610$	0	0
$611$	11.5569	0.467543
$612$	0	0
$613$	40.8302	1.64912	0.824558	−	0.565777i	$-0.191424\pi$
$613$	40.8302	1.64912	0.824558	+	0.565777i	$0.191424\pi$
$614$	60.6639	2.44819
$615$	0	0
$616$	76.4553	3.08047
$617$	5.11383	0.205875	0.102937	−	0.994688i	$-0.467176\pi$
$617$	5.11383	0.205875	0.102937	+	0.994688i	$0.467176\pi$
$618$	0	0
$619$	11.5569	0.464512	0.232256	−	0.972655i	$-0.425389\pi$
$619$	11.5569	0.464512	0.232256	+	0.972655i	$0.425389\pi$
$620$	0	0
$621$	0	0
$622$	−69.7586	−2.79706
$623$	−3.16291	−0.126719
$624$	0	0
$625$	0	0
$626$	−22.8602	−0.913677
$627$	0	0
$628$	−50.7880	−2.02666
$629$	−12.1335	−0.483794
$630$	0	0
$631$	35.2242	1.40225	0.701127	−	0.713036i	$-0.252681\pi$
$631$	35.2242	1.40225	0.701127	+	0.713036i	$0.252681\pi$
$632$	106.214	4.22496
$633$	0	0
$634$	76.8915	3.05375
$635$	0	0
$636$	0	0
$637$	−0.397442	−0.0157472
$638$	−45.3415	−1.79509
$639$	0	0
$640$	0	0
$641$	21.9018	0.865071	0.432535	−	0.901617i	$-0.357619\pi$
$641$	21.9018	0.865071	0.432535	+	0.901617i	$0.357619\pi$
$642$	0	0
$643$	−7.50783	−0.296080	−0.148040	−	0.988981i	$-0.547296\pi$
$643$	−7.50783	−0.296080	−0.148040	+	0.988981i	$0.547296\pi$
$644$	−75.2502	−2.96527
$645$	0	0
$646$	28.0382	1.10315
$647$	−14.0422	−0.552056	−0.276028	−	0.961150i	$-0.589018\pi$
$647$	−14.0422	−0.552056	−0.276028	+	0.961150i	$0.589018\pi$
$648$	0	0
$649$	5.75859	0.226044
$650$	0	0
$651$	0	0
$652$	79.1261	3.09882
$653$	7.99312	0.312795	0.156398	−	0.987694i	$-0.450012\pi$
$653$	7.99312	0.312795	0.156398	+	0.987694i	$0.450012\pi$
$654$	0	0
$655$	0	0
$656$	−49.0157	−1.91374
$657$	0	0
$658$	−87.3346	−3.40466
$659$	25.3415	0.987164	0.493582	−	0.869699i	$-0.335687\pi$
$659$	25.3415	0.987164	0.493582	+	0.869699i	$0.335687\pi$
$660$	0	0
$661$	27.4396	1.06728	0.533639	−	0.845712i	$-0.320824\pi$
$661$	27.4396	1.06728	0.533639	+	0.845712i	$0.320824\pi$
$662$	−36.7620	−1.42880
$663$	0	0
$664$	21.8827	0.849215
$665$	0	0
$666$	0	0
$667$	29.0225	1.12376
$668$	−56.5275	−2.18711
$669$	0	0
$670$	0	0
$671$	17.9578	0.693253
$672$	0	0
$673$	−27.1070	−1.04490	−0.522448	−	0.852671i	$-0.674981\pi$
$673$	−27.1070	−1.04490	−0.522448	+	0.852671i	$0.674981\pi$
$674$	12.0191	0.462959
$675$	0	0
$676$	5.71982	0.219993
$677$	−36.5957	−1.40649	−0.703243	−	0.710949i	$-0.748265\pi$
$677$	−36.5957	−1.40649	−0.703243	+	0.710949i	$0.748265\pi$
$678$	0	0
$679$	29.4750	1.13115
$680$	0	0
$681$	0	0
$682$	57.1070	2.18674
$683$	−13.4656	−0.515248	−0.257624	−	0.966245i	$-0.582940\pi$
$683$	−13.4656	−0.515248	−0.257624	+	0.966245i	$0.582940\pi$
$684$	0	0
$685$	0	0
$686$	−49.8950	−1.90500
$687$	0	0
$688$	−192.011	−7.32034
$689$	−1.16291	−0.0443033
$690$	0	0
$691$	29.5500	1.12414	0.562068	−	0.827091i	$-0.310006\pi$
$691$	29.5500	1.12414	0.562068	+	0.827091i	$0.310006\pi$
$692$	75.0088	2.85141
$693$	0	0
$694$	17.4396	0.662000
$695$	0	0
$696$	0	0
$697$	8.04908	0.304881
$698$	49.0878	1.85800
$699$	0	0
$700$	0	0
$701$	−43.6604	−1.64903	−0.824516	−	0.565839i	$-0.808552\pi$
$701$	−43.6604	−1.64903	−0.824516	+	0.565839i	$0.808552\pi$
$702$	0	0
$703$	−15.2120	−0.573731
$704$	112.566	4.24248
$705$	0	0
$706$	−38.2468	−1.43944
$707$	−20.8724	−0.784988
$708$	0	0
$709$	−26.7880	−1.00604	−0.503022	−	0.864273i	$-0.667779\pi$
$709$	−26.7880	−1.00604	−0.503022	+	0.864273i	$0.667779\pi$
$710$	0	0
$711$	0	0
$712$	−12.0191	−0.450435
$713$	−36.5535	−1.36894
$714$	0	0
$715$	0	0
$716$	−48.9244	−1.82839
$717$	0	0
$718$	−2.76891	−0.103335
$719$	34.8793	1.30078	0.650389	−	0.759601i	$-0.274606\pi$
$719$	34.8793	1.30078	0.650389	+	0.759601i	$0.274606\pi$
$720$	0	0
$721$	−10.2414	−0.381410
$722$	−17.6386	−0.656443
$723$	0	0
$724$	21.2964	0.791475
$725$	0	0
$726$	0	0
$727$	−37.4396	−1.38856	−0.694280	−	0.719705i	$-0.744277\pi$
$727$	−37.4396	−1.38856	−0.694280	+	0.719705i	$0.744277\pi$
$728$	−28.1104	−1.04184
$729$	0	0
$730$	0	0
$731$	31.5309	1.16621
$732$	0	0
$733$	47.1560	1.74175	0.870874	−	0.491506i	$-0.163554\pi$
$733$	47.1560	1.74175	0.870874	+	0.491506i	$0.163554\pi$
$734$	−39.5309	−1.45911
$735$	0	0
$736$	−132.207	−4.87322
$737$	−5.12070	−0.188624
$738$	0	0
$739$	−31.7914	−1.16947	−0.584734	−	0.811225i	$-0.698801\pi$
$739$	−31.7914	−1.16947	−0.584734	+	0.811225i	$0.698801\pi$
$740$	0	0
$741$	0	0
$742$	8.78801	0.322618
$743$	16.6776	0.611842	0.305921	−	0.952057i	$-0.401036\pi$
$743$	16.6776	0.611842	0.305921	+	0.952057i	$0.401036\pi$
$744$	0	0
$745$	0	0
$746$	−43.5500	−1.59448
$747$	0	0
$748$	−44.1364	−1.61379
$749$	−34.2767	−1.25244
$750$	0	0
$751$	16.1855	0.590616	0.295308	−	0.955402i	$-0.404578\pi$
$751$	16.1855	0.590616	0.295308	+	0.955402i	$0.404578\pi$
$752$	−199.666	−7.28106
$753$	0	0
$754$	16.6707	0.607113
$755$	0	0
$756$	0	0
$757$	−12.3258	−0.447990	−0.223995	−	0.974590i	$-0.571910\pi$
$757$	−12.3258	−0.447990	−0.223995	+	0.974590i	$0.571910\pi$
$758$	72.8002	2.64422
$759$	0	0
$760$	0	0
$761$	0.00687569	0.000249244 0	0.000124622	−	1.00000i	$-0.499960\pi$
$761$	0.00687569	0.000249244 0	0.000124622		1.00000i	$0.499960\pi$
$762$	0	0
$763$	31.1138	1.12640
$764$	24.2208	0.876277
$765$	0	0
$766$	62.3380	2.25237
$767$	−2.11727	−0.0764501
$768$	0	0
$769$	−20.3258	−0.732968	−0.366484	−	0.930424i	$-0.619439\pi$
$769$	−20.3258	−0.732968	−0.366484	+	0.930424i	$0.619439\pi$
$770$	0	0
$771$	0	0
$772$	133.790	4.81520
$773$	9.90184	0.356144	0.178072	−	0.984017i	$-0.443014\pi$
$773$	9.90184	0.356144	0.178072	+	0.984017i	$0.443014\pi$
$774$	0	0
$775$	0	0
$776$	112.005	4.02076
$777$	0	0
$778$	−88.0054	−3.15514
$779$	10.0913	0.361558
$780$	0	0
$781$	18.2767	0.653993
$782$	38.1295	1.36351
$783$	0	0
$784$	6.86651	0.245232
$785$	0	0
$786$	0	0
$787$	−36.3449	−1.29556	−0.647778	−	0.761829i	$-0.724302\pi$
$787$	−36.3449	−1.29556	−0.647778	+	0.761829i	$0.724302\pi$
$788$	83.2433	2.96542
$789$	0	0
$790$	0	0
$791$	−35.6673	−1.26818
$792$	0	0
$793$	−6.60256	−0.234464
$794$	−49.8759	−1.77003
$795$	0	0
$796$	−86.4484	−3.06408
$797$	18.8371	0.667244	0.333622	−	0.942707i	$-0.391729\pi$
$797$	18.8371	0.667244	0.333622	+	0.942707i	$0.391729\pi$
$798$	0	0
$799$	32.7880	1.15996
$800$	0	0
$801$	0	0
$802$	−37.7395	−1.33263
$803$	−24.7880	−0.874750
$804$	0	0
$805$	0	0
$806$	−20.9966	−0.739572
$807$	0	0
$808$	−79.3155	−2.79031
$809$	−32.2277	−1.13306	−0.566532	−	0.824040i	$-0.691715\pi$
$809$	−32.2277	−1.13306	−0.566532	+	0.824040i	$0.691715\pi$
$810$	0	0
$811$	−23.0034	−0.807760	−0.403880	−	0.914812i	$-0.632339\pi$
$811$	−23.0034	−0.807760	−0.403880	+	0.914812i	$0.632339\pi$
$812$	−93.3415	−3.27564
$813$	0	0
$814$	32.3189	1.13278
$815$	0	0
$816$	0	0
$817$	39.5309	1.38301
$818$	−38.8984	−1.36005
$819$	0	0
$820$	0	0
$821$	−49.9372	−1.74282	−0.871410	−	0.490556i	$-0.836794\pi$
$821$	−49.9372	−1.74282	−0.871410	+	0.490556i	$0.836794\pi$
$822$	0	0
$823$	−28.2345	−0.984194	−0.492097	−	0.870540i	$-0.663769\pi$
$823$	−28.2345	−0.984194	−0.492097	+	0.870540i	$0.663769\pi$
$824$	−38.9175	−1.35576
$825$	0	0
$826$	16.0000	0.556711
$827$	−9.55004	−0.332087	−0.166044	−	0.986118i	$-0.553099\pi$
$827$	−9.55004	−0.332087	−0.166044	+	0.986118i	$0.553099\pi$
$828$	0	0
$829$	−37.9862	−1.31932	−0.659658	−	0.751565i	$-0.729299\pi$
$829$	−37.9862	−1.31932	−0.659658	+	0.751565i	$0.729299\pi$
$830$	0	0
$831$	0	0
$832$	−41.3871	−1.43484
$833$	−1.12758	−0.0390683
$834$	0	0
$835$	0	0
$836$	−55.3346	−1.91379
$837$	0	0
$838$	−34.2277	−1.18237
$839$	4.72670	0.163184	0.0815919	−	0.996666i	$-0.474000\pi$
$839$	4.72670	0.163184	0.0815919	+	0.996666i	$0.474000\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	63.3155	2.18200
$843$	0	0
$844$	−104.259	−3.58874
$845$	0	0
$846$	0	0
$847$	−9.79832	−0.336674
$848$	20.0913	0.689938
$849$	0	0
$850$	0	0
$851$	−20.6870	−0.709140
$852$	0	0
$853$	48.3611	1.65585	0.827927	−	0.560836i	$-0.189520\pi$
$853$	48.3611	1.65585	0.827927	+	0.560836i	$0.189520\pi$
$854$	49.8950	1.70737
$855$	0	0
$856$	−130.252	−4.45193
$857$	6.83709	0.233551	0.116775	−	0.993158i	$-0.462744\pi$
$857$	6.83709	0.233551	0.116775	+	0.993158i	$0.462744\pi$
$858$	0	0
$859$	12.6026	0.429994	0.214997	−	0.976615i	$-0.431026\pi$
$859$	12.6026	0.429994	0.214997	+	0.976615i	$0.431026\pi$
$860$	0	0
$861$	0	0
$862$	24.9966	0.851386
$863$	8.20855	0.279422	0.139711	−	0.990192i	$-0.455383\pi$
$863$	8.20855	0.279422	0.139711	+	0.990192i	$0.455383\pi$
$864$	0	0
$865$	0	0
$866$	56.4744	1.91908
$867$	0	0
$868$	117.562	3.99032
$869$	27.9509	0.948170
$870$	0	0
$871$	1.88273	0.0637940
$872$	118.233	4.00387
$873$	0	0
$874$	47.8037	1.61698
$875$	0	0
$876$	0	0
$877$	−13.5309	−0.456907	−0.228454	−	0.973555i	$-0.573367\pi$
$877$	−13.5309	−0.456907	−0.228454	+	0.973555i	$0.573367\pi$
$878$	−70.5466	−2.38083
$879$	0	0
$880$	0	0
$881$	9.34836	0.314954	0.157477	−	0.987523i	$-0.449664\pi$
$881$	9.34836	0.314954	0.157477	+	0.987523i	$0.449664\pi$
$882$	0	0
$883$	55.1001	1.85427	0.927133	−	0.374733i	$-0.122266\pi$
$883$	55.1001	1.85427	0.927133	+	0.374733i	$0.122266\pi$
$884$	16.2277	0.545795
$885$	0	0
$886$	30.3640	1.02010
$887$	0.133492	0.00448223	0.00224112	−	0.999997i	$-0.499287\pi$
$887$	0.133492	0.00448223	0.00224112	+	0.999997i	$0.499287\pi$
$888$	0	0
$889$	36.5535	1.22596
$890$	0	0
$891$	0	0
$892$	−57.8690	−1.93760
$893$	41.1070	1.37559
$894$	0	0
$895$	0	0
$896$	164.083	5.48162
$897$	0	0
$898$	−7.88273	−0.263050
$899$	−45.3415	−1.51222
$900$	0	0
$901$	−3.29928	−0.109915
$902$	−21.4396	−0.713862
$903$	0	0
$904$	−135.536	−4.50787
$905$	0	0
$906$	0	0
$907$	58.5466	1.94401	0.972004	−	0.234964i	$-0.0754973\pi$
$907$	58.5466	1.94401	0.972004	+	0.234964i	$0.0754973\pi$
$908$	64.7620	2.14920
$909$	0	0
$910$	0	0
$911$	−50.4622	−1.67189	−0.835943	−	0.548816i	$-0.815079\pi$
$911$	−50.4622	−1.67189	−0.835943	+	0.548816i	$0.815079\pi$
$912$	0	0
$913$	5.75859	0.190582
$914$	38.1104	1.26058
$915$	0	0
$916$	−35.6604	−1.17825
$917$	25.6742	0.847836
$918$	0	0
$919$	56.9735	1.87938	0.939691	−	0.342026i	$-0.111113\pi$
$919$	56.9735	1.87938	0.939691	+	0.342026i	$0.111113\pi$
$920$	0	0
$921$	0	0
$922$	54.5275	1.79577
$923$	−6.71982	−0.221186
$924$	0	0
$925$	0	0
$926$	−75.1261	−2.46880
$927$	0	0
$928$	−163.992	−5.38329
$929$	36.5957	1.20067	0.600333	−	0.799750i	$-0.295035\pi$
$929$	36.5957	1.20067	0.600333	+	0.799750i	$0.295035\pi$
$930$	0	0
$931$	−1.41367	−0.0463311
$932$	39.1070	1.28099
$933$	0	0
$934$	80.3572	2.62937
$935$	0	0
$936$	0	0
$937$	47.1070	1.53892	0.769459	−	0.638697i	$-0.220526\pi$
$937$	47.1070	1.53892	0.769459	+	0.638697i	$0.220526\pi$
$938$	−14.2277	−0.464549
$939$	0	0
$940$	0	0
$941$	36.3611	1.18534	0.592670	−	0.805446i	$-0.298074\pi$
$941$	36.3611	1.18534	0.592670	+	0.805446i	$0.298074\pi$
$942$	0	0
$943$	13.7233	0.446891
$944$	36.5795	1.19056
$945$	0	0
$946$	−83.9862	−2.73063
$947$	44.5795	1.44864	0.724319	−	0.689465i	$-0.242154\pi$
$947$	44.5795	1.44864	0.724319	+	0.689465i	$0.242154\pi$
$948$	0	0
$949$	9.11383	0.295847
$950$	0	0
$951$	0	0
$952$	−79.7517	−2.58477
$953$	19.8596	0.643317	0.321658	−	0.946856i	$-0.395760\pi$
$953$	19.8596	0.643317	0.321658	+	0.946856i	$0.395760\pi$
$954$	0	0
$955$	0	0
$956$	−7.32238	−0.236823
$957$	0	0
$958$	33.7846	1.09153
$959$	4.80176	0.155057
$960$	0	0
$961$	26.1070	0.842160
$962$	−11.8827	−0.383115
$963$	0	0
$964$	−34.3189	−1.10534
$965$	0	0
$966$	0	0
$967$	−47.4068	−1.52450	−0.762250	−	0.647283i	$-0.775905\pi$
$967$	−47.4068	−1.52450	−0.762250	+	0.647283i	$0.775905\pi$
$968$	−37.2338	−1.19674
$969$	0	0
$970$	0	0
$971$	10.6448	0.341607	0.170803	−	0.985305i	$-0.445364\pi$
$971$	10.6448	0.341607	0.170803	+	0.985305i	$0.445364\pi$
$972$	0	0
$973$	−17.0716	−0.547291
$974$	0.443086	0.0141974
$975$	0	0
$976$	114.071	3.65131
$977$	−1.21199	−0.0387750	−0.0193875	−	0.999812i	$-0.506172\pi$
$977$	−1.21199	−0.0387750	−0.0193875	+	0.999812i	$0.506172\pi$
$978$	0	0
$979$	−3.16291	−0.101087
$980$	0	0
$981$	0	0
$982$	117.328	3.74408
$983$	51.8759	1.65458	0.827291	−	0.561773i	$-0.189881\pi$
$983$	51.8759	1.65458	0.827291	+	0.561773i	$0.189881\pi$
$984$	0	0
$985$	0	0
$986$	47.2964	1.50622
$987$	0	0
$988$	20.3449	0.647258
$989$	53.7586	1.70942
$990$	0	0
$991$	−21.6251	−0.686944	−0.343472	−	0.939163i	$-0.611603\pi$
$991$	−21.6251	−0.686944	−0.343472	+	0.939163i	$0.611603\pi$
$992$	206.545	6.55781
$993$	0	0
$994$	50.7811	1.61068
$995$	0	0
$996$	0	0
$997$	−23.2051	−0.734913	−0.367457	−	0.930041i	$-0.619771\pi$
$997$	−23.2051	−0.734913	−0.367457	+	0.930041i	$0.619771\pi$
$998$	21.6482	0.685262
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.a.bh.1.3		3
3.2	odd	2		975.2.a.o.1.1		3
5.2	odd	4		2925.2.c.w.2224.6		6
5.3	odd	4		2925.2.c.w.2224.1		6
5.4	even	2		585.2.a.n.1.1		3
15.2	even	4		975.2.c.i.274.1		6
15.8	even	4		975.2.c.i.274.6		6
15.14	odd	2		195.2.a.e.1.3	✓	3
20.19	odd	2		9360.2.a.dd.1.3		3
60.59	even	2		3120.2.a.bj.1.3		3
65.64	even	2		7605.2.a.bx.1.3		3
105.104	even	2		9555.2.a.bq.1.3		3
195.194	odd	2		2535.2.a.bc.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.e.1.3	✓	3	15.14	odd	2
585.2.a.n.1.1		3	5.4	even	2
975.2.a.o.1.1		3	3.2	odd	2
975.2.c.i.274.1		6	15.2	even	4
975.2.c.i.274.6		6	15.8	even	4
2535.2.a.bc.1.1		3	195.194	odd	2
2925.2.a.bh.1.3		3	1.1	even	1	trivial
2925.2.c.w.2224.1		6	5.3	odd	4
2925.2.c.w.2224.6		6	5.2	odd	4
3120.2.a.bj.1.3		3	60.59	even	2
7605.2.a.bx.1.3		3	65.64	even	2
9360.2.a.dd.1.3		3	20.19	odd	2
9555.2.a.bq.1.3		3	105.104	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.a (trivial)

\(f(q)\)	\(=\)	\(q+2.77846 q^{2} +5.71982 q^{4} +2.71982 q^{7} +10.3354 q^{8} +O(q^{10})\)	\(q+2.77846 q^{2} +5.71982 q^{4} +2.71982 q^{7} +10.3354 q^{8} +2.71982 q^{11} -1.00000 q^{13} +7.55691 q^{14} +17.2767 q^{16} -2.83709 q^{17} -3.55691 q^{19} +7.55691 q^{22} -4.83709 q^{23} -2.77846 q^{26} +15.5569 q^{28} -6.00000 q^{29} +7.55691 q^{31} +27.3319 q^{32} -7.88273 q^{34} +4.27674 q^{37} -9.88273 q^{38} -2.83709 q^{41} -11.1138 q^{43} +15.5569 q^{44} -13.4396 q^{46} -11.5569 q^{47} +0.397442 q^{49} -5.71982 q^{52} +1.16291 q^{53} +28.1104 q^{56} -16.6707 q^{58} +2.11727 q^{59} +6.60256 q^{61} +20.9966 q^{62} +41.3871 q^{64} -1.88273 q^{67} -16.2277 q^{68} +6.71982 q^{71} -9.11383 q^{73} +11.8827 q^{74} -20.3449 q^{76} +7.39744 q^{77} +10.2767 q^{79} -7.88273 q^{82} +2.11727 q^{83} -30.8793 q^{86} +28.1104 q^{88} -1.16291 q^{89} -2.71982 q^{91} -27.6673 q^{92} -32.1104 q^{94} +10.8371 q^{97} +1.10428 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 8 q^{4} - q^{7} + 6 q^{8}+O(q^{10}) \) 3 * q + 8 * q^4 - q^7 + 6 * q^8	\( 3 q + 8 q^{4} - q^{7} + 6 q^{8} - q^{11} - 3 q^{13} + 6 q^{14} + 26 q^{16} - q^{17} + 6 q^{19} + 6 q^{22} - 7 q^{23} + 30 q^{28} - 18 q^{29} + 6 q^{31} + 22 q^{32} - 22 q^{34} - 13 q^{37} - 28 q^{38} - q^{41} + 30 q^{44} - 22 q^{46} - 18 q^{47} + 12 q^{49} - 8 q^{52} + 11 q^{53} + 16 q^{56} + 8 q^{59} + 9 q^{61} + 28 q^{62} + 30 q^{64} - 4 q^{67} + 18 q^{68} + 11 q^{71} + 6 q^{73} + 34 q^{74} + 4 q^{76} + 33 q^{77} + 5 q^{79} - 22 q^{82} + 8 q^{83} - 56 q^{86} + 16 q^{88} - 11 q^{89} + q^{91} + 2 q^{92} - 28 q^{94} + 25 q^{97} + 10 q^{98}+O(q^{100}) \) 3 * q + 8 * q^4 - q^7 + 6 * q^8 - q^11 - 3 * q^13 + 6 * q^14 + 26 * q^16 - q^17 + 6 * q^19 + 6 * q^22 - 7 * q^23 + 30 * q^28 - 18 * q^29 + 6 * q^31 + 22 * q^32 - 22 * q^34 - 13 * q^37 - 28 * q^38 - q^41 + 30 * q^44 - 22 * q^46 - 18 * q^47 + 12 * q^49 - 8 * q^52 + 11 * q^53 + 16 * q^56 + 8 * q^59 + 9 * q^61 + 28 * q^62 + 30 * q^64 - 4 * q^67 + 18 * q^68 + 11 * q^71 + 6 * q^73 + 34 * q^74 + 4 * q^76 + 33 * q^77 + 5 * q^79 - 22 * q^82 + 8 * q^83 - 56 * q^86 + 16 * q^88 - 11 * q^89 + q^91 + 2 * q^92 - 28 * q^94 + 25 * q^97 + 10 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more