LMFDB - Embedded newform 6525.2.a.bl.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.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:	$52.1023873189$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.71457$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.71457 q^{2} +0.939748 q^{4} -0.654317 q^{7} -1.81788 q^{8} -0.163559 q^{11} +2.65432 q^{13} -1.12187 q^{14} -4.99637 q^{16} +3.86328 q^{17} -3.47954 q^{19} -0.280433 q^{22} -7.69972 q^{23} +4.55101 q^{26} -0.614893 q^{28} -1.00000 q^{29} +5.05274 q^{31} -4.93087 q^{32} +6.62385 q^{34} +10.5904 q^{37} -5.96591 q^{38} +6.17417 q^{41} -10.5547 q^{43} -0.153704 q^{44} -13.2017 q^{46} -10.3036 q^{47} -6.57187 q^{49} +2.49439 q^{52} -4.04766 q^{53} +1.18947 q^{56} -1.71457 q^{58} +0.328734 q^{59} -5.72054 q^{61} +8.66328 q^{62} +1.53842 q^{64} +3.51985 q^{67} +3.63050 q^{68} -11.7457 q^{71} -1.12143 q^{73} +18.1580 q^{74} -3.26989 q^{76} +0.107019 q^{77} -12.4074 q^{79} +10.5860 q^{82} -7.89306 q^{83} -18.0968 q^{86} +0.297329 q^{88} -5.04702 q^{89} -1.73677 q^{91} -7.23579 q^{92} -17.6663 q^{94} +8.49076 q^{97} -11.2679 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{2} + 5 q^{4} + 8 q^{7} - 9 q^{8} - 12 q^{11} + 2 q^{13} - 6 q^{14} + q^{16} - 2 q^{19} + 14 q^{22} - 8 q^{23} - 6 q^{28} - 5 q^{29} + 2 q^{31} - q^{32} + 4 q^{34} + 16 q^{37} + 14 q^{38} + 14 q^{41}+ \cdots + 9 q^{98}+O(q^{100}) $ 5 * q - 3 * q^2 + 5 * q^4 + 8 * q^7 - 9 * q^8 - 12 * q^11 + 2 * q^13 - 6 * q^14 + q^16 - 2 * q^19 + 14 * q^22 - 8 * q^23 - 6 * q^28 - 5 * q^29 + 2 * q^31 - q^32 + 4 * q^34 + 16 * q^37 + 14 * q^38 + 14 * q^41 - 20 * q^44 - 6 * q^46 - 2 * q^47 - 7 * q^49 + 16 * q^52 - 26 * q^53 + 2 * q^56 + 3 * q^58 - 4 * q^59 - 12 * q^61 - 9 * q^64 + 12 * q^67 + 20 * q^68 - 30 * q^71 - 12 * q^73 - 2 * q^74 - 44 * q^76 - 18 * q^77 - 18 * q^79 + 10 * q^82 - 2 * q^83 - 30 * q^86 + 42 * q^88 + 22 * q^89 - 12 * q^91 - 20 * q^92 - 50 * q^94 + 20 * q^97 + 9 * q^98

Coefficient data

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

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

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

$2$	1.71457	1.21238	0.606192	−	0.795319i	$-0.292696\pi$
$2$	1.71457	1.21238	0.606192	+	0.795319i	$0.292696\pi$
$3$	0	0
$3$	0	0
$4$	0.939748	0.469874
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.654317	−0.247309	−0.123654	−	0.992325i	$-0.539461\pi$
$7$	−0.654317	−0.247309	−0.123654	+	0.992325i	$0.539461\pi$
$8$	−1.81788	−0.642716
$9$	0	0
$10$	0	0
$11$	−0.163559	−0.0493148	−0.0246574	−	0.999696i	$-0.507849\pi$
$11$	−0.163559	−0.0493148	−0.0246574	+	0.999696i	$0.507849\pi$
$12$	0	0
$13$	2.65432	0.736175	0.368088	−	0.929791i	$-0.380013\pi$
$13$	2.65432	0.736175	0.368088	+	0.929791i	$0.380013\pi$
$14$	−1.12187	−0.299833
$15$	0	0
$16$	−4.99637	−1.24909
$17$	3.86328	0.936982	0.468491	−	0.883468i	$-0.344798\pi$
$17$	3.86328	0.936982	0.468491	+	0.883468i	$0.344798\pi$
$18$	0	0
$19$	−3.47954	−0.798260	−0.399130	−	0.916894i	$-0.630688\pi$
$19$	−3.47954	−0.798260	−0.399130	+	0.916894i	$0.630688\pi$
$20$	0	0
$21$	0	0
$22$	−0.280433	−0.0597884
$23$	−7.69972	−1.60550	−0.802751	−	0.596314i	$-0.796631\pi$
$23$	−7.69972	−1.60550	−0.802751	+	0.596314i	$0.796631\pi$
$24$	0	0
$25$	0	0
$26$	4.55101	0.892527
$27$	0	0
$28$	−0.614893	−0.116204
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.05274	0.907499	0.453750	−	0.891129i	$-0.350086\pi$
$31$	5.05274	0.907499	0.453750	+	0.891129i	$0.350086\pi$
$32$	−4.93087	−0.871663
$33$	0	0
$34$	6.62385	1.13598
$35$	0	0
$36$	0	0
$37$	10.5904	1.74106	0.870528	−	0.492118i	$-0.163777\pi$
$37$	10.5904	1.74106	0.870528	+	0.492118i	$0.163777\pi$
$38$	−5.96591	−0.967798
$39$	0	0
$40$	0	0
$41$	6.17417	0.964244	0.482122	−	0.876104i	$-0.339866\pi$
$41$	6.17417	0.964244	0.482122	+	0.876104i	$0.339866\pi$
$42$	0	0
$43$	−10.5547	−1.60958	−0.804790	−	0.593559i	$-0.797722\pi$
$43$	−10.5547	−1.60958	−0.804790	+	0.593559i	$0.797722\pi$
$44$	−0.153704	−0.0231717
$45$	0	0
$46$	−13.2017	−1.94648
$47$	−10.3036	−1.50294	−0.751470	−	0.659767i	$-0.770655\pi$
$47$	−10.3036	−1.50294	−0.751470	+	0.659767i	$0.770655\pi$
$48$	0	0
$49$	−6.57187	−0.938838
$50$	0	0
$51$	0	0
$52$	2.49439	0.345909
$53$	−4.04766	−0.555989	−0.277995	−	0.960583i	$-0.589670\pi$
$53$	−4.04766	−0.555989	−0.277995	+	0.960583i	$0.589670\pi$
$54$	0	0
$55$	0	0
$56$	1.18947	0.158949
$57$	0	0
$58$	−1.71457	−0.225134
$59$	0.328734	0.0427975	0.0213988	−	0.999771i	$-0.493188\pi$
$59$	0.328734	0.0427975	0.0213988	+	0.999771i	$0.493188\pi$
$60$	0	0
$61$	−5.72054	−0.732441	−0.366220	−	0.930528i	$-0.619348\pi$
$61$	−5.72054	−0.732441	−0.366220	+	0.930528i	$0.619348\pi$
$62$	8.66328	1.10024
$63$	0	0
$64$	1.53842	0.192303
$65$	0	0
$66$	0	0
$67$	3.51985	0.430019	0.215009	−	0.976612i	$-0.431022\pi$
$67$	3.51985	0.430019	0.215009	+	0.976612i	$0.431022\pi$
$68$	3.63050	0.440263
$69$	0	0
$70$	0	0
$71$	−11.7457	−1.39396	−0.696981	−	0.717090i	$-0.745474\pi$
$71$	−11.7457	−1.39396	−0.696981	+	0.717090i	$0.745474\pi$
$72$	0	0
$73$	−1.12143	−0.131253	−0.0656267	−	0.997844i	$-0.520905\pi$
$73$	−1.12143	−0.131253	−0.0656267	+	0.997844i	$0.520905\pi$
$74$	18.1580	2.11083
$75$	0	0
$76$	−3.26989	−0.375082
$77$	0.107019	0.0121960
$78$	0	0
$79$	−12.4074	−1.39594	−0.697970	−	0.716127i	$-0.745913\pi$
$79$	−12.4074	−1.39594	−0.697970	+	0.716127i	$0.745913\pi$
$80$	0	0
$81$	0	0
$82$	10.5860	1.16903
$83$	−7.89306	−0.866376	−0.433188	−	0.901304i	$-0.642611\pi$
$83$	−7.89306	−0.866376	−0.433188	+	0.901304i	$0.642611\pi$
$84$	0	0
$85$	0	0
$86$	−18.0968	−1.95143
$87$	0	0
$88$	0.297329	0.0316954
$89$	−5.04702	−0.534983	−0.267491	−	0.963560i	$-0.586195\pi$
$89$	−5.04702	−0.534983	−0.267491	+	0.963560i	$0.586195\pi$
$90$	0	0
$91$	−1.73677	−0.182062
$92$	−7.23579	−0.754383
$93$	0	0
$94$	−17.6663	−1.82214
$95$	0	0
$96$	0	0
$97$	8.49076	0.862106	0.431053	−	0.902327i	$-0.358142\pi$
$97$	8.49076	0.862106	0.431053	+	0.902327i	$0.358142\pi$
$98$	−11.2679	−1.13823
$99$	0	0
$100$	0	0
$101$	−1.81126	−0.180227	−0.0901136	−	0.995931i	$-0.528723\pi$
$101$	−1.81126	−0.180227	−0.0901136	+	0.995931i	$0.528723\pi$
$102$	0	0
$103$	5.80807	0.572286	0.286143	−	0.958187i	$-0.407627\pi$
$103$	5.80807	0.572286	0.286143	+	0.958187i	$0.407627\pi$
$104$	−4.82522	−0.473152
$105$	0	0
$106$	−6.94000	−0.674072
$107$	−3.48568	−0.336973	−0.168487	−	0.985704i	$-0.553888\pi$
$107$	−3.48568	−0.336973	−0.168487	+	0.985704i	$0.553888\pi$
$108$	0	0
$109$	8.11029	0.776825	0.388412	−	0.921486i	$-0.373024\pi$
$109$	8.11029	0.776825	0.388412	+	0.921486i	$0.373024\pi$
$110$	0	0
$111$	0	0
$112$	3.26921	0.308911
$113$	−2.06715	−0.194461	−0.0972307	−	0.995262i	$-0.530998\pi$
$113$	−2.06715	−0.194461	−0.0972307	+	0.995262i	$0.530998\pi$
$114$	0	0
$115$	0	0
$116$	−0.939748	−0.0872534
$117$	0	0
$118$	0.563637	0.0518870
$119$	−2.52781	−0.231724
$120$	0	0
$121$	−10.9732	−0.997568
$122$	−9.80827	−0.887999
$123$	0	0
$124$	4.74830	0.426410
$125$	0	0
$126$	0	0
$127$	−12.9962	−1.15323	−0.576613	−	0.817017i	$-0.695626\pi$
$127$	−12.9962	−1.15323	−0.576613	+	0.817017i	$0.695626\pi$
$128$	12.4995	1.10481
$129$	0	0
$130$	0	0
$131$	−8.69003	−0.759251	−0.379626	−	0.925140i	$-0.623947\pi$
$131$	−8.69003	−0.759251	−0.379626	+	0.925140i	$0.623947\pi$
$132$	0	0
$133$	2.27672	0.197417
$134$	6.03504	0.521348
$135$	0	0
$136$	−7.02295	−0.602213
$137$	−14.4550	−1.23498	−0.617489	−	0.786580i	$-0.711850\pi$
$137$	−14.4550	−1.23498	−0.617489	+	0.786580i	$0.711850\pi$
$138$	0	0
$139$	−1.72208	−0.146065	−0.0730324	−	0.997330i	$-0.523268\pi$
$139$	−1.72208	−0.146065	−0.0730324	+	0.997330i	$0.523268\pi$
$140$	0	0
$141$	0	0
$142$	−20.1389	−1.69002
$143$	−0.434137	−0.0363043
$144$	0	0
$145$	0	0
$146$	−1.92277	−0.159129
$147$	0	0
$148$	9.95234	0.818077
$149$	−16.1461	−1.32274	−0.661369	−	0.750060i	$-0.730024\pi$
$149$	−16.1461	−1.32274	−0.661369	+	0.750060i	$0.730024\pi$
$150$	0	0
$151$	−0.733083	−0.0596574	−0.0298287	−	0.999555i	$-0.509496\pi$
$151$	−0.733083	−0.0596574	−0.0298287	+	0.999555i	$0.509496\pi$
$152$	6.32536	0.513055
$153$	0	0
$154$	0.183492	0.0147862
$155$	0	0
$156$	0	0
$157$	10.8270	0.864085	0.432043	−	0.901853i	$-0.357793\pi$
$157$	10.8270	0.864085	0.432043	+	0.901853i	$0.357793\pi$
$158$	−21.2733	−1.69241
$159$	0	0
$160$	0	0
$161$	5.03806	0.397054
$162$	0	0
$163$	−13.5528	−1.06154	−0.530768	−	0.847517i	$-0.678096\pi$
$163$	−13.5528	−1.06154	−0.530768	+	0.847517i	$0.678096\pi$
$164$	5.80216	0.453073
$165$	0	0
$166$	−13.5332	−1.05038
$167$	−18.4671	−1.42902	−0.714512	−	0.699623i	$-0.753351\pi$
$167$	−18.4671	−1.42902	−0.714512	+	0.699623i	$0.753351\pi$
$168$	0	0
$169$	−5.95460	−0.458046
$170$	0	0
$171$	0	0
$172$	−9.91878	−0.756300
$173$	8.92785	0.678772	0.339386	−	0.940647i	$-0.389781\pi$
$173$	8.92785	0.678772	0.339386	+	0.940647i	$0.389781\pi$
$174$	0	0
$175$	0	0
$176$	0.817200	0.0615987
$177$	0	0
$178$	−8.65346	−0.648604
$179$	1.86896	0.139693	0.0698465	−	0.997558i	$-0.477749\pi$
$179$	1.86896	0.139693	0.0698465	+	0.997558i	$0.477749\pi$
$180$	0	0
$181$	16.8852	1.25507	0.627533	−	0.778590i	$-0.284065\pi$
$181$	16.8852	1.25507	0.627533	+	0.778590i	$0.284065\pi$
$182$	−2.97780	−0.220730
$183$	0	0
$184$	13.9971	1.03188
$185$	0	0
$186$	0	0
$187$	−0.631872	−0.0462071
$188$	−9.68282	−0.706192
$189$	0	0
$190$	0	0
$191$	−7.58677	−0.548960	−0.274480	−	0.961593i	$-0.588506\pi$
$191$	−7.58677	−0.548960	−0.274480	+	0.961593i	$0.588506\pi$
$192$	0	0
$193$	27.2794	1.96362	0.981808	−	0.189876i	$-0.0608086\pi$
$193$	27.2794	1.96362	0.981808	+	0.189876i	$0.0608086\pi$
$194$	14.5580	1.04520
$195$	0	0
$196$	−6.17590	−0.441136
$197$	9.70626	0.691542	0.345771	−	0.938319i	$-0.387617\pi$
$197$	9.70626	0.691542	0.345771	+	0.938319i	$0.387617\pi$
$198$	0	0
$199$	22.9520	1.62703	0.813513	−	0.581547i	$-0.197552\pi$
$199$	22.9520	1.62703	0.813513	+	0.581547i	$0.197552\pi$
$200$	0	0
$201$	0	0
$202$	−3.10553	−0.218504
$203$	0.654317	0.0459241
$204$	0	0
$205$	0	0
$206$	9.95834	0.693831
$207$	0	0
$208$	−13.2619	−0.919551
$209$	0.569108	0.0393660
$210$	0	0
$211$	18.0020	1.23931	0.619655	−	0.784874i	$-0.287273\pi$
$211$	18.0020	1.23931	0.619655	+	0.784874i	$0.287273\pi$
$212$	−3.80378	−0.261245
$213$	0	0
$214$	−5.97644	−0.408541
$215$	0	0
$216$	0	0
$217$	−3.30610	−0.224432
$218$	13.9057	0.941810
$219$	0	0
$220$	0	0
$221$	10.2544	0.689783
$222$	0	0
$223$	−19.1383	−1.28160	−0.640799	−	0.767708i	$-0.721397\pi$
$223$	−19.1383	−1.28160	−0.640799	+	0.767708i	$0.721397\pi$
$224$	3.22635	0.215570
$225$	0	0
$226$	−3.54428	−0.235762
$227$	13.3621	0.886872	0.443436	−	0.896306i	$-0.353759\pi$
$227$	13.3621	0.886872	0.443436	+	0.896306i	$0.353759\pi$
$228$	0	0
$229$	7.45145	0.492405	0.246203	−	0.969218i	$-0.420817\pi$
$229$	7.45145	0.492405	0.246203	+	0.969218i	$0.420817\pi$
$230$	0	0
$231$	0	0
$232$	1.81788	0.119349
$233$	−21.5080	−1.40904	−0.704518	−	0.709687i	$-0.748837\pi$
$233$	−21.5080	−1.40904	−0.704518	+	0.709687i	$0.748837\pi$
$234$	0	0
$235$	0	0
$236$	0.308927	0.0201094
$237$	0	0
$238$	−4.33410	−0.280938
$239$	−24.1169	−1.55999	−0.779997	−	0.625784i	$-0.784779\pi$
$239$	−24.1169	−1.55999	−0.779997	+	0.625784i	$0.784779\pi$
$240$	0	0
$241$	−7.22805	−0.465600	−0.232800	−	0.972525i	$-0.574789\pi$
$241$	−7.22805	−0.465600	−0.232800	+	0.972525i	$0.574789\pi$
$242$	−18.8144	−1.20944
$243$	0	0
$244$	−5.37587	−0.344155
$245$	0	0
$246$	0	0
$247$	−9.23579	−0.587659
$248$	−9.18526	−0.583264
$249$	0	0
$250$	0	0
$251$	8.74458	0.551953	0.275977	−	0.961164i	$-0.410999\pi$
$251$	8.74458	0.551953	0.275977	+	0.961164i	$0.410999\pi$
$252$	0	0
$253$	1.25936	0.0791750
$254$	−22.2829	−1.39815
$255$	0	0
$256$	18.3544	1.14715
$257$	10.1877	0.635494	0.317747	−	0.948176i	$-0.397074\pi$
$257$	10.1877	0.635494	0.317747	+	0.948176i	$0.397074\pi$
$258$	0	0
$259$	−6.92950	−0.430578
$260$	0	0
$261$	0	0
$262$	−14.8997	−0.920504
$263$	−11.6794	−0.720184	−0.360092	−	0.932917i	$-0.617255\pi$
$263$	−11.6794	−0.720184	−0.360092	+	0.932917i	$0.617255\pi$
$264$	0	0
$265$	0	0
$266$	3.90359	0.239345
$267$	0	0
$268$	3.30778	0.202055
$269$	−23.0385	−1.40468	−0.702342	−	0.711839i	$-0.747862\pi$
$269$	−23.0385	−1.40468	−0.702342	+	0.711839i	$0.747862\pi$
$270$	0	0
$271$	−4.45505	−0.270625	−0.135312	−	0.990803i	$-0.543204\pi$
$271$	−4.45505	−0.270625	−0.135312	+	0.990803i	$0.543204\pi$
$272$	−19.3024	−1.17038
$273$	0	0
$274$	−24.7842	−1.49727
$275$	0	0
$276$	0	0
$277$	15.7000	0.943321	0.471661	−	0.881780i	$-0.343655\pi$
$277$	15.7000	0.943321	0.471661	+	0.881780i	$0.343655\pi$
$278$	−2.95262	−0.177087
$279$	0	0
$280$	0	0
$281$	−13.4721	−0.803677	−0.401838	−	0.915711i	$-0.631629\pi$
$281$	−13.4721	−0.803677	−0.401838	+	0.915711i	$0.631629\pi$
$282$	0	0
$283$	5.86481	0.348627	0.174313	−	0.984690i	$-0.444229\pi$
$283$	5.86481	0.348627	0.174313	+	0.984690i	$0.444229\pi$
$284$	−11.0380	−0.654986
$285$	0	0
$286$	−0.744357	−0.0440148
$287$	−4.03987	−0.238466
$288$	0	0
$289$	−2.07510	−0.122065
$290$	0	0
$291$	0	0
$292$	−1.05386	−0.0616726
$293$	10.3642	0.605485	0.302742	−	0.953072i	$-0.402098\pi$
$293$	10.3642	0.605485	0.302742	+	0.953072i	$0.402098\pi$
$294$	0	0
$295$	0	0
$296$	−19.2521	−1.11901
$297$	0	0
$298$	−27.6836	−1.60367
$299$	−20.4375	−1.18193
$300$	0	0
$301$	6.90614	0.398063
$302$	−1.25692	−0.0723277
$303$	0	0
$304$	17.3850	0.997101
$305$	0	0
$306$	0	0
$307$	−16.0760	−0.917504	−0.458752	−	0.888564i	$-0.651703\pi$
$307$	−16.0760	−0.917504	−0.458752	+	0.888564i	$0.651703\pi$
$308$	0.100571	0.00573057
$309$	0	0
$310$	0	0
$311$	−16.9976	−0.963846	−0.481923	−	0.876214i	$-0.660061\pi$
$311$	−16.9976	−0.963846	−0.481923	+	0.876214i	$0.660061\pi$
$312$	0	0
$313$	25.2391	1.42660	0.713300	−	0.700858i	$-0.247200\pi$
$313$	25.2391	1.42660	0.713300	+	0.700858i	$0.247200\pi$
$314$	18.5636	1.04760
$315$	0	0
$316$	−11.6598	−0.655916
$317$	−3.06731	−0.172277	−0.0861386	−	0.996283i	$-0.527453\pi$
$317$	−3.06731	−0.172277	−0.0861386	+	0.996283i	$0.527453\pi$
$318$	0	0
$319$	0.163559	0.00915753
$320$	0	0
$321$	0	0
$322$	8.63810	0.481382
$323$	−13.4424	−0.747955
$324$	0	0
$325$	0	0
$326$	−23.2372	−1.28699
$327$	0	0
$328$	−11.2239	−0.619735
$329$	6.74185	0.371690
$330$	0	0
$331$	−19.9971	−1.09914	−0.549571	−	0.835447i	$-0.685209\pi$
$331$	−19.9971	−1.09914	−0.549571	+	0.835447i	$0.685209\pi$
$332$	−7.41749	−0.407088
$333$	0	0
$334$	−31.6631	−1.73253
$335$	0	0
$336$	0	0
$337$	−6.45613	−0.351688	−0.175844	−	0.984418i	$-0.556265\pi$
$337$	−6.45613	−0.351688	−0.175844	+	0.984418i	$0.556265\pi$
$338$	−10.2096	−0.555328
$339$	0	0
$340$	0	0
$341$	−0.826420	−0.0447531
$342$	0	0
$343$	8.88031	0.479491
$344$	19.1872	1.03450
$345$	0	0
$346$	15.3074	0.822932
$347$	14.9415	0.802099	0.401050	−	0.916056i	$-0.368645\pi$
$347$	14.9415	0.802099	0.401050	+	0.916056i	$0.368645\pi$
$348$	0	0
$349$	−12.1055	−0.647992	−0.323996	−	0.946058i	$-0.605026\pi$
$349$	−12.1055	−0.647992	−0.323996	+	0.946058i	$0.605026\pi$
$350$	0	0
$351$	0	0
$352$	0.806487	0.0429859
$353$	18.1937	0.968355	0.484178	−	0.874970i	$-0.339119\pi$
$353$	18.1937	0.968355	0.484178	+	0.874970i	$0.339119\pi$
$354$	0	0
$355$	0	0
$356$	−4.74292	−0.251374
$357$	0	0
$358$	3.20447	0.169361
$359$	35.1010	1.85256	0.926279	−	0.376838i	$-0.122989\pi$
$359$	35.1010	1.85256	0.926279	+	0.376838i	$0.122989\pi$
$360$	0	0
$361$	−6.89283	−0.362780
$362$	28.9508	1.52162
$363$	0	0
$364$	−1.63212	−0.0855464
$365$	0	0
$366$	0	0
$367$	23.1943	1.21073	0.605367	−	0.795946i	$-0.293026\pi$
$367$	23.1943	1.21073	0.605367	+	0.795946i	$0.293026\pi$
$368$	38.4706	2.00542
$369$	0	0
$370$	0	0
$371$	2.64845	0.137501
$372$	0	0
$373$	−4.99469	−0.258615	−0.129308	−	0.991605i	$-0.541275\pi$
$373$	−4.99469	−0.258615	−0.129308	+	0.991605i	$0.541275\pi$
$374$	−1.08339	−0.0560207
$375$	0	0
$376$	18.7307	0.965964
$377$	−2.65432	−0.136704
$378$	0	0
$379$	−28.7738	−1.47801	−0.739006	−	0.673699i	$-0.764704\pi$
$379$	−28.7738	−1.47801	−0.739006	+	0.673699i	$0.764704\pi$
$380$	0	0
$381$	0	0
$382$	−13.0080	−0.665550
$383$	24.9482	1.27479	0.637396	−	0.770537i	$-0.280012\pi$
$383$	24.9482	1.27479	0.637396	+	0.770537i	$0.280012\pi$
$384$	0	0
$385$	0	0
$386$	46.7725	2.38066
$387$	0	0
$388$	7.97917	0.405081
$389$	−31.0949	−1.57657	−0.788286	−	0.615308i	$-0.789031\pi$
$389$	−31.0949	−1.57657	−0.788286	+	0.615308i	$0.789031\pi$
$390$	0	0
$391$	−29.7461	−1.50433
$392$	11.9468	0.603407
$393$	0	0
$394$	16.6421	0.838414
$395$	0	0
$396$	0	0
$397$	−12.1948	−0.612038	−0.306019	−	0.952025i	$-0.598997\pi$
$397$	−12.1948	−0.612038	−0.306019	+	0.952025i	$0.598997\pi$
$398$	39.3528	1.97258
$399$	0	0
$400$	0	0
$401$	−4.33039	−0.216249	−0.108125	−	0.994137i	$-0.534485\pi$
$401$	−4.33039	−0.216249	−0.108125	+	0.994137i	$0.534485\pi$
$402$	0	0
$403$	13.4116	0.668078
$404$	−1.70213	−0.0846841
$405$	0	0
$406$	1.12187	0.0556776
$407$	−1.73216	−0.0858599
$408$	0	0
$409$	5.12178	0.253256	0.126628	−	0.991950i	$-0.459585\pi$
$409$	5.12178	0.253256	0.126628	+	0.991950i	$0.459585\pi$
$410$	0	0
$411$	0	0
$412$	5.45812	0.268902
$413$	−0.215096	−0.0105842
$414$	0	0
$415$	0	0
$416$	−13.0881	−0.641697
$417$	0	0
$418$	0.975776	0.0477267
$419$	−14.1945	−0.693445	−0.346723	−	0.937968i	$-0.612705\pi$
$419$	−14.1945	−0.693445	−0.346723	+	0.937968i	$0.612705\pi$
$420$	0	0
$421$	4.83364	0.235577	0.117789	−	0.993039i	$-0.462419\pi$
$421$	4.83364	0.235577	0.117789	+	0.993039i	$0.462419\pi$
$422$	30.8657	1.50252
$423$	0	0
$424$	7.35815	0.357343
$425$	0	0
$426$	0	0
$427$	3.74305	0.181139
$428$	−3.27566	−0.158335
$429$	0	0
$430$	0	0
$431$	18.0910	0.871411	0.435706	−	0.900089i	$-0.356499\pi$
$431$	18.0910	0.871411	0.435706	+	0.900089i	$0.356499\pi$
$432$	0	0
$433$	−1.71005	−0.0821798	−0.0410899	−	0.999155i	$-0.513083\pi$
$433$	−1.71005	−0.0821798	−0.0410899	+	0.999155i	$0.513083\pi$
$434$	−5.66853	−0.272098
$435$	0	0
$436$	7.62163	0.365010
$437$	26.7914	1.28161
$438$	0	0
$439$	24.3533	1.16232	0.581160	−	0.813789i	$-0.302599\pi$
$439$	24.3533	1.16232	0.581160	+	0.813789i	$0.302599\pi$
$440$	0	0
$441$	0	0
$442$	17.5818	0.836281
$443$	−18.9066	−0.898280	−0.449140	−	0.893461i	$-0.648270\pi$
$443$	−18.9066	−0.898280	−0.449140	+	0.893461i	$0.648270\pi$
$444$	0	0
$445$	0	0
$446$	−32.8140	−1.55379
$447$	0	0
$448$	−1.00661	−0.0475581
$449$	−25.2808	−1.19307	−0.596537	−	0.802586i	$-0.703457\pi$
$449$	−25.2808	−1.19307	−0.596537	+	0.802586i	$0.703457\pi$
$450$	0	0
$451$	−1.00984	−0.0475515
$452$	−1.94260	−0.0913723
$453$	0	0
$454$	22.9102	1.07523
$455$	0	0
$456$	0	0
$457$	4.25835	0.199197	0.0995986	−	0.995028i	$-0.468244\pi$
$457$	4.25835	0.199197	0.0995986	+	0.995028i	$0.468244\pi$
$458$	12.7760	0.596984
$459$	0	0
$460$	0	0
$461$	−2.48353	−0.115670	−0.0578349	−	0.998326i	$-0.518420\pi$
$461$	−2.48353	−0.115670	−0.0578349	+	0.998326i	$0.518420\pi$
$462$	0	0
$463$	27.4876	1.27746	0.638728	−	0.769433i	$-0.279461\pi$
$463$	27.4876	1.27746	0.638728	+	0.769433i	$0.279461\pi$
$464$	4.99637	0.231951
$465$	0	0
$466$	−36.8769	−1.70829
$467$	6.24698	0.289076	0.144538	−	0.989499i	$-0.453830\pi$
$467$	6.24698	0.289076	0.144538	+	0.989499i	$0.453830\pi$
$468$	0	0
$469$	−2.30310	−0.106347
$470$	0	0
$471$	0	0
$472$	−0.597598	−0.0275067
$473$	1.72632	0.0793761
$474$	0	0
$475$	0	0
$476$	−2.37550	−0.108881
$477$	0	0
$478$	−41.3501	−1.89131
$479$	11.5736	0.528812	0.264406	−	0.964412i	$-0.414824\pi$
$479$	11.5736	0.528812	0.264406	+	0.964412i	$0.414824\pi$
$480$	0	0
$481$	28.1104	1.28172
$482$	−12.3930	−0.564485
$483$	0	0
$484$	−10.3121	−0.468731
$485$	0	0
$486$	0	0
$487$	−31.1947	−1.41357	−0.706783	−	0.707431i	$-0.749854\pi$
$487$	−31.1947	−1.41357	−0.706783	+	0.707431i	$0.749854\pi$
$488$	10.3992	0.470751
$489$	0	0
$490$	0	0
$491$	7.93512	0.358107	0.179053	−	0.983839i	$-0.442697\pi$
$491$	7.93512	0.358107	0.179053	+	0.983839i	$0.442697\pi$
$492$	0	0
$493$	−3.86328	−0.173993
$494$	−15.8354	−0.712469
$495$	0	0
$496$	−25.2454	−1.13355
$497$	7.68543	0.344739
$498$	0	0
$499$	−32.8984	−1.47274	−0.736368	−	0.676581i	$-0.763461\pi$
$499$	−32.8984	−1.47274	−0.736368	+	0.676581i	$0.763461\pi$
$500$	0	0
$501$	0	0
$502$	14.9932	0.669179
$503$	40.7038	1.81489	0.907446	−	0.420168i	$-0.138029\pi$
$503$	40.7038	1.81489	0.907446	+	0.420168i	$0.138029\pi$
$504$	0	0
$505$	0	0
$506$	2.15925	0.0959905
$507$	0	0
$508$	−12.2132	−0.541871
$509$	36.9407	1.63737	0.818684	−	0.574245i	$-0.194704\pi$
$509$	36.9407	1.63737	0.818684	+	0.574245i	$0.194704\pi$
$510$	0	0
$511$	0.733771	0.0324601
$512$	6.47089	0.285976
$513$	0	0
$514$	17.4676	0.770462
$515$	0	0
$516$	0	0
$517$	1.68525	0.0741172
$518$	−11.8811	−0.522026
$519$	0	0
$520$	0	0
$521$	33.2555	1.45695	0.728476	−	0.685072i	$-0.240229\pi$
$521$	33.2555	1.45695	0.728476	+	0.685072i	$0.240229\pi$
$522$	0	0
$523$	−0.341001	−0.0149109	−0.00745547	−	0.999972i	$-0.502373\pi$
$523$	−0.341001	−0.0149109	−0.00745547	+	0.999972i	$0.502373\pi$
$524$	−8.16644	−0.356752
$525$	0	0
$526$	−20.0252	−0.873139
$527$	19.5201	0.850310
$528$	0	0
$529$	36.2856	1.57764
$530$	0	0
$531$	0	0
$532$	2.13954	0.0927609
$533$	16.3882	0.709852
$534$	0	0
$535$	0	0
$536$	−6.39866	−0.276380
$537$	0	0
$538$	−39.5012	−1.70302
$539$	1.07489	0.0462986
$540$	0	0
$541$	−18.4196	−0.791919	−0.395960	−	0.918268i	$-0.629588\pi$
$541$	−18.4196	−0.791919	−0.395960	+	0.918268i	$0.629588\pi$
$542$	−7.63849	−0.328101
$543$	0	0
$544$	−19.0493	−0.816732
$545$	0	0
$546$	0	0
$547$	42.5936	1.82117	0.910586	−	0.413320i	$-0.135631\pi$
$547$	42.5936	1.82117	0.910586	+	0.413320i	$0.135631\pi$
$548$	−13.5841	−0.580284
$549$	0	0
$550$	0	0
$551$	3.47954	0.148233
$552$	0	0
$553$	8.11836	0.345228
$554$	26.9187	1.14367
$555$	0	0
$556$	−1.61832	−0.0686321
$557$	−17.5805	−0.744908	−0.372454	−	0.928051i	$-0.621484\pi$
$557$	−17.5805	−0.744908	−0.372454	+	0.928051i	$0.621484\pi$
$558$	0	0
$559$	−28.0156	−1.18493
$560$	0	0
$561$	0	0
$562$	−23.0988	−0.974365
$563$	−6.29621	−0.265354	−0.132677	−	0.991159i	$-0.542357\pi$
$563$	−6.29621	−0.265354	−0.132677	+	0.991159i	$0.542357\pi$
$564$	0	0
$565$	0	0
$566$	10.0556	0.422669
$567$	0	0
$568$	21.3523	0.895921
$569$	42.3981	1.77742	0.888710	−	0.458469i	$-0.151602\pi$
$569$	42.3981	1.77742	0.888710	+	0.458469i	$0.151602\pi$
$570$	0	0
$571$	−18.4882	−0.773708	−0.386854	−	0.922141i	$-0.626438\pi$
$571$	−18.4882	−0.773708	−0.386854	+	0.922141i	$0.626438\pi$
$572$	−0.407979	−0.0170585
$573$	0	0
$574$	−6.92663	−0.289112
$575$	0	0
$576$	0	0
$577$	−22.0727	−0.918897	−0.459448	−	0.888204i	$-0.651953\pi$
$577$	−22.0727	−0.918897	−0.459448	+	0.888204i	$0.651953\pi$
$578$	−3.55791	−0.147990
$579$	0	0
$580$	0	0
$581$	5.16457	0.214262
$582$	0	0
$583$	0.662030	0.0274185
$584$	2.03862	0.0843587
$585$	0	0
$586$	17.7702	0.734080
$587$	44.1932	1.82405	0.912024	−	0.410136i	$-0.134519\pi$
$587$	44.1932	1.82405	0.912024	+	0.410136i	$0.134519\pi$
$588$	0	0
$589$	−17.5812	−0.724421
$590$	0	0
$591$	0	0
$592$	−52.9137	−2.17474
$593$	−17.4159	−0.715184	−0.357592	−	0.933878i	$-0.616402\pi$
$593$	−17.4159	−0.715184	−0.357592	+	0.933878i	$0.616402\pi$
$594$	0	0
$595$	0	0
$596$	−15.1732	−0.621520
$597$	0	0
$598$	−35.0415	−1.43295
$599$	−7.70544	−0.314836	−0.157418	−	0.987532i	$-0.550317\pi$
$599$	−7.70544	−0.314836	−0.157418	+	0.987532i	$0.550317\pi$
$600$	0	0
$601$	−31.5500	−1.28695	−0.643476	−	0.765466i	$-0.722508\pi$
$601$	−31.5500	−1.28695	−0.643476	+	0.765466i	$0.722508\pi$
$602$	11.8410	0.482605
$603$	0	0
$604$	−0.688913	−0.0280315
$605$	0	0
$606$	0	0
$607$	−13.2516	−0.537864	−0.268932	−	0.963159i	$-0.586671\pi$
$607$	−13.2516	−0.537864	−0.268932	+	0.963159i	$0.586671\pi$
$608$	17.1571	0.695814
$609$	0	0
$610$	0	0
$611$	−27.3491	−1.10643
$612$	0	0
$613$	29.4609	1.18991	0.594957	−	0.803758i	$-0.297169\pi$
$613$	29.4609	1.18991	0.594957	+	0.803758i	$0.297169\pi$
$614$	−27.5634	−1.11237
$615$	0	0
$616$	−0.194548	−0.00783855
$617$	5.61068	0.225877	0.112939	−	0.993602i	$-0.463974\pi$
$617$	5.61068	0.225877	0.112939	+	0.993602i	$0.463974\pi$
$618$	0	0
$619$	40.5248	1.62883	0.814415	−	0.580282i	$-0.197058\pi$
$619$	40.5248	1.62883	0.814415	+	0.580282i	$0.197058\pi$
$620$	0	0
$621$	0	0
$622$	−29.1436	−1.16855
$623$	3.30235	0.132306
$624$	0	0
$625$	0	0
$626$	43.2743	1.72959
$627$	0	0
$628$	10.1746	0.406011
$629$	40.9138	1.63134
$630$	0	0
$631$	−20.2197	−0.804935	−0.402467	−	0.915434i	$-0.631847\pi$
$631$	−20.2197	−0.804935	−0.402467	+	0.915434i	$0.631847\pi$
$632$	22.5551	0.897193
$633$	0	0
$634$	−5.25911	−0.208866
$635$	0	0
$636$	0	0
$637$	−17.4438	−0.691149
$638$	0.280433	0.0111024
$639$	0	0
$640$	0	0
$641$	44.0804	1.74107	0.870535	−	0.492107i	$-0.163773\pi$
$641$	44.0804	1.74107	0.870535	+	0.492107i	$0.163773\pi$
$642$	0	0
$643$	3.56303	0.140512	0.0702561	−	0.997529i	$-0.477618\pi$
$643$	3.56303	0.140512	0.0702561	+	0.997529i	$0.477618\pi$
$644$	4.73450	0.186566
$645$	0	0
$646$	−23.0479	−0.906809
$647$	40.7523	1.60214	0.801069	−	0.598573i	$-0.204265\pi$
$647$	40.7523	1.60214	0.801069	+	0.598573i	$0.204265\pi$
$648$	0	0
$649$	−0.0537673	−0.00211055
$650$	0	0
$651$	0	0
$652$	−12.7362	−0.498788
$653$	11.6902	0.457472	0.228736	−	0.973489i	$-0.426541\pi$
$653$	11.6902	0.457472	0.228736	+	0.973489i	$0.426541\pi$
$654$	0	0
$655$	0	0
$656$	−30.8484	−1.20443
$657$	0	0
$658$	11.5594	0.450631
$659$	28.1795	1.09772	0.548858	−	0.835915i	$-0.315063\pi$
$659$	28.1795	1.09772	0.548858	+	0.835915i	$0.315063\pi$
$660$	0	0
$661$	−20.6736	−0.804108	−0.402054	−	0.915616i	$-0.631704\pi$
$661$	−20.6736	−0.804108	−0.402054	+	0.915616i	$0.631704\pi$
$662$	−34.2865	−1.33258
$663$	0	0
$664$	14.3486	0.556834
$665$	0	0
$666$	0	0
$667$	7.69972	0.298134
$668$	−17.3544	−0.671461
$669$	0	0
$670$	0	0
$671$	0.935645	0.0361202
$672$	0	0
$673$	44.7915	1.72659	0.863293	−	0.504703i	$-0.168398\pi$
$673$	44.7915	1.72659	0.863293	+	0.504703i	$0.168398\pi$
$674$	−11.0695	−0.426380
$675$	0	0
$676$	−5.59582	−0.215224
$677$	−16.8746	−0.648542	−0.324271	−	0.945964i	$-0.605119\pi$
$677$	−16.8746	−0.648542	−0.324271	+	0.945964i	$0.605119\pi$
$678$	0	0
$679$	−5.55565	−0.213206
$680$	0	0
$681$	0	0
$682$	−1.41695	−0.0542580
$683$	28.6207	1.09514	0.547571	−	0.836759i	$-0.315553\pi$
$683$	28.6207	1.09514	0.547571	+	0.836759i	$0.315553\pi$
$684$	0	0
$685$	0	0
$686$	15.2259	0.581328
$687$	0	0
$688$	52.7353	2.01051
$689$	−10.7438	−0.409305
$690$	0	0
$691$	−28.5928	−1.08772	−0.543861	−	0.839175i	$-0.683038\pi$
$691$	−28.5928	−1.08772	−0.543861	+	0.839175i	$0.683038\pi$
$692$	8.38993	0.318937
$693$	0	0
$694$	25.6182	0.972452
$695$	0	0
$696$	0	0
$697$	23.8525	0.903479
$698$	−20.7557	−0.785615
$699$	0	0
$700$	0	0
$701$	−49.6636	−1.87577	−0.937885	−	0.346946i	$-0.887219\pi$
$701$	−49.6636	−1.87577	−0.937885	+	0.346946i	$0.887219\pi$
$702$	0	0
$703$	−36.8498	−1.38982
$704$	−0.251622	−0.00948336
$705$	0	0
$706$	31.1944	1.17402
$707$	1.18514	0.0445717
$708$	0	0
$709$	−19.4755	−0.731419	−0.365710	−	0.930729i	$-0.619174\pi$
$709$	−19.4755	−0.731419	−0.365710	+	0.930729i	$0.619174\pi$
$710$	0	0
$711$	0	0
$712$	9.17485	0.343842
$713$	−38.9047	−1.45699
$714$	0	0
$715$	0	0
$716$	1.75636	0.0656381
$717$	0	0
$718$	60.1830	2.24601
$719$	34.3195	1.27990	0.639951	−	0.768415i	$-0.278954\pi$
$719$	34.3195	1.27990	0.639951	+	0.768415i	$0.278954\pi$
$720$	0	0
$721$	−3.80032	−0.141531
$722$	−11.8182	−0.439829
$723$	0	0
$724$	15.8678	0.589723
$725$	0	0
$726$	0	0
$727$	19.3206	0.716560	0.358280	−	0.933614i	$-0.383363\pi$
$727$	19.3206	0.716560	0.358280	+	0.933614i	$0.383363\pi$
$728$	3.15722	0.117014
$729$	0	0
$730$	0	0
$731$	−40.7758	−1.50815
$732$	0	0
$733$	3.10588	0.114718	0.0573591	−	0.998354i	$-0.481732\pi$
$733$	3.10588	0.114718	0.0573591	+	0.998354i	$0.481732\pi$
$734$	39.7683	1.46787
$735$	0	0
$736$	37.9663	1.39946
$737$	−0.575703	−0.0212063
$738$	0	0
$739$	38.7106	1.42399	0.711997	−	0.702183i	$-0.247791\pi$
$739$	38.7106	1.42399	0.711997	+	0.702183i	$0.247791\pi$
$740$	0	0
$741$	0	0
$742$	4.54096	0.166704
$743$	8.41130	0.308581	0.154290	−	0.988026i	$-0.450691\pi$
$743$	8.41130	0.308581	0.154290	+	0.988026i	$0.450691\pi$
$744$	0	0
$745$	0	0
$746$	−8.56373	−0.313541
$747$	0	0
$748$	−0.593801	−0.0217115
$749$	2.28074	0.0833364
$750$	0	0
$751$	−30.5930	−1.11635	−0.558177	−	0.829722i	$-0.688499\pi$
$751$	−30.5930	−1.11635	−0.558177	+	0.829722i	$0.688499\pi$
$752$	51.4808	1.87731
$753$	0	0
$754$	−4.55101	−0.165738
$755$	0	0
$756$	0	0
$757$	−41.6526	−1.51389	−0.756945	−	0.653479i	$-0.773309\pi$
$757$	−41.6526	−1.51389	−0.756945	+	0.653479i	$0.773309\pi$
$758$	−49.3347	−1.79192
$759$	0	0
$760$	0	0
$761$	28.3957	1.02934	0.514671	−	0.857388i	$-0.327914\pi$
$761$	28.3957	1.02934	0.514671	+	0.857388i	$0.327914\pi$
$762$	0	0
$763$	−5.30670	−0.192115
$764$	−7.12965	−0.257942
$765$	0	0
$766$	42.7753	1.54554
$767$	0.872565	0.0315065
$768$	0	0
$769$	−48.6782	−1.75538	−0.877690	−	0.479229i	$-0.840916\pi$
$769$	−48.6782	−1.75538	−0.877690	+	0.479229i	$0.840916\pi$
$770$	0	0
$771$	0	0
$772$	25.6358	0.922652
$773$	54.9974	1.97812	0.989060	−	0.147513i	$-0.0471267\pi$
$773$	54.9974	1.97812	0.989060	+	0.147513i	$0.0471267\pi$
$774$	0	0
$775$	0	0
$776$	−15.4351	−0.554089
$777$	0	0
$778$	−53.3143	−1.91141
$779$	−21.4833	−0.769717
$780$	0	0
$781$	1.92112	0.0687429
$782$	−51.0018	−1.82382
$783$	0	0
$784$	32.8355	1.17270
$785$	0	0
$786$	0	0
$787$	32.7472	1.16731	0.583655	−	0.812002i	$-0.301622\pi$
$787$	32.7472	1.16731	0.583655	+	0.812002i	$0.301622\pi$
$788$	9.12143	0.324938
$789$	0	0
$790$	0	0
$791$	1.35257	0.0480920
$792$	0	0
$793$	−15.1841	−0.539205
$794$	−20.9088	−0.742025
$795$	0	0
$796$	21.5691	0.764497
$797$	−6.54714	−0.231912	−0.115956	−	0.993254i	$-0.536993\pi$
$797$	−6.54714	−0.231912	−0.115956	+	0.993254i	$0.536993\pi$
$798$	0	0
$799$	−39.8058	−1.40823
$800$	0	0
$801$	0	0
$802$	−7.42475	−0.262177
$803$	0.183420	0.00647274
$804$	0	0
$805$	0	0
$806$	22.9951	0.809967
$807$	0	0
$808$	3.29265	0.115835
$809$	−40.5849	−1.42689	−0.713444	−	0.700712i	$-0.752866\pi$
$809$	−40.5849	−1.42689	−0.713444	+	0.700712i	$0.752866\pi$
$810$	0	0
$811$	15.4588	0.542832	0.271416	−	0.962462i	$-0.412508\pi$
$811$	15.4588	0.542832	0.271416	+	0.962462i	$0.412508\pi$
$812$	0.614893	0.0215785
$813$	0	0
$814$	−2.96990	−0.104095
$815$	0	0
$816$	0	0
$817$	36.7255	1.28486
$818$	8.78165	0.307043
$819$	0	0
$820$	0	0
$821$	22.4828	0.784656	0.392328	−	0.919825i	$-0.371670\pi$
$821$	22.4828	0.784656	0.392328	+	0.919825i	$0.371670\pi$
$822$	0	0
$823$	45.1129	1.57254	0.786268	−	0.617886i	$-0.212011\pi$
$823$	45.1129	1.57254	0.786268	+	0.617886i	$0.212011\pi$
$824$	−10.5584	−0.367818
$825$	0	0
$826$	−0.368798	−0.0128321
$827$	−9.44115	−0.328301	−0.164150	−	0.986435i	$-0.552488\pi$
$827$	−9.44115	−0.328301	−0.164150	+	0.986435i	$0.552488\pi$
$828$	0	0
$829$	−55.4844	−1.92705	−0.963527	−	0.267612i	$-0.913765\pi$
$829$	−55.4844	−1.92705	−0.963527	+	0.267612i	$0.913765\pi$
$830$	0	0
$831$	0	0
$832$	4.08346	0.141568
$833$	−25.3889	−0.879675
$834$	0	0
$835$	0	0
$836$	0.534818	0.0184971
$837$	0	0
$838$	−24.3374	−0.840721
$839$	−36.6797	−1.26632	−0.633162	−	0.774019i	$-0.718243\pi$
$839$	−36.6797	−1.26632	−0.633162	+	0.774019i	$0.718243\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	8.28762	0.285610
$843$	0	0
$844$	16.9174	0.582319
$845$	0	0
$846$	0	0
$847$	7.17998	0.246707
$848$	20.2236	0.694482
$849$	0	0
$850$	0	0
$851$	−81.5433	−2.79527
$852$	0	0
$853$	−16.2128	−0.555114	−0.277557	−	0.960709i	$-0.589525\pi$
$853$	−16.2128	−0.555114	−0.277557	+	0.960709i	$0.589525\pi$
$854$	6.41772	0.219610
$855$	0	0
$856$	6.33653	0.216578
$857$	−52.9150	−1.80754	−0.903772	−	0.428015i	$-0.859213\pi$
$857$	−52.9150	−1.80754	−0.903772	+	0.428015i	$0.859213\pi$
$858$	0	0
$859$	−13.7986	−0.470803	−0.235401	−	0.971898i	$-0.575641\pi$
$859$	−13.7986	−0.470803	−0.235401	+	0.971898i	$0.575641\pi$
$860$	0	0
$861$	0	0
$862$	31.0182	1.05648
$863$	4.25394	0.144806	0.0724029	−	0.997375i	$-0.476933\pi$
$863$	4.25394	0.144806	0.0724029	+	0.997375i	$0.476933\pi$
$864$	0	0
$865$	0	0
$866$	−2.93200	−0.0996334
$867$	0	0
$868$	−3.10690	−0.105455
$869$	2.02934	0.0688405
$870$	0	0
$871$	9.34281	0.316569
$872$	−14.7435	−0.499278
$873$	0	0
$874$	45.9358	1.55380
$875$	0	0
$876$	0	0
$877$	49.9185	1.68563	0.842814	−	0.538205i	$-0.180897\pi$
$877$	49.9185	1.68563	0.842814	+	0.538205i	$0.180897\pi$
$878$	41.7554	1.40918
$879$	0	0
$880$	0	0
$881$	11.5561	0.389334	0.194667	−	0.980869i	$-0.437637\pi$
$881$	11.5561	0.389334	0.194667	+	0.980869i	$0.437637\pi$
$882$	0	0
$883$	10.5007	0.353375	0.176688	−	0.984267i	$-0.443462\pi$
$883$	10.5007	0.353375	0.176688	+	0.984267i	$0.443462\pi$
$884$	9.63651	0.324111
$885$	0	0
$886$	−32.4167	−1.08906
$887$	13.1447	0.441356	0.220678	−	0.975347i	$-0.429173\pi$
$887$	13.1447	0.441356	0.220678	+	0.975347i	$0.429173\pi$
$888$	0	0
$889$	8.50364	0.285203
$890$	0	0
$891$	0	0
$892$	−17.9852	−0.602190
$893$	35.8519	1.19974
$894$	0	0
$895$	0	0
$896$	−8.17862	−0.273228
$897$	0	0
$898$	−43.3456	−1.44646
$899$	−5.05274	−0.168518
$900$	0	0
$901$	−15.6372	−0.520952
$902$	−1.73144	−0.0576506
$903$	0	0
$904$	3.75783	0.124983
$905$	0	0
$906$	0	0
$907$	−8.45271	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$907$	−8.45271	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$908$	12.5570	0.416718
$909$	0	0
$910$	0	0
$911$	23.3000	0.771965	0.385982	−	0.922506i	$-0.373863\pi$
$911$	23.3000	0.771965	0.385982	+	0.922506i	$0.373863\pi$
$912$	0	0
$913$	1.29098	0.0427252
$914$	7.30123	0.241503
$915$	0	0
$916$	7.00248	0.231368
$917$	5.68603	0.187769
$918$	0	0
$919$	−37.7491	−1.24523	−0.622614	−	0.782529i	$-0.713929\pi$
$919$	−37.7491	−1.24523	−0.622614	+	0.782529i	$0.713929\pi$
$920$	0	0
$921$	0	0
$922$	−4.25819	−0.140236
$923$	−31.1769	−1.02620
$924$	0	0
$925$	0	0
$926$	47.1294	1.54877
$927$	0	0
$928$	4.93087	0.161864
$929$	−34.3843	−1.12811	−0.564055	−	0.825737i	$-0.690760\pi$
$929$	−34.3843	−1.12811	−0.564055	+	0.825737i	$0.690760\pi$
$930$	0	0
$931$	22.8671	0.749437
$932$	−20.2121	−0.662069
$933$	0	0
$934$	10.7109	0.350471
$935$	0	0
$936$	0	0
$937$	−8.85110	−0.289153	−0.144576	−	0.989494i	$-0.546182\pi$
$937$	−8.85110	−0.289153	−0.144576	+	0.989494i	$0.546182\pi$
$938$	−3.94883	−0.128934
$939$	0	0
$940$	0	0
$941$	20.4890	0.667921	0.333960	−	0.942587i	$-0.391615\pi$
$941$	20.4890	0.667921	0.333960	+	0.942587i	$0.391615\pi$
$942$	0	0
$943$	−47.5394	−1.54810
$944$	−1.64248	−0.0534581
$945$	0	0
$946$	2.95989	0.0962343
$947$	38.8206	1.26150	0.630750	−	0.775986i	$-0.282747\pi$
$947$	38.8206	1.26150	0.630750	+	0.775986i	$0.282747\pi$
$948$	0	0
$949$	−2.97663	−0.0966255
$950$	0	0
$951$	0	0
$952$	4.59524	0.148933
$953$	−33.3755	−1.08114	−0.540570	−	0.841299i	$-0.681791\pi$
$953$	−33.3755	−1.08114	−0.540570	+	0.841299i	$0.681791\pi$
$954$	0	0
$955$	0	0
$956$	−22.6638	−0.733000
$957$	0	0
$958$	19.8437	0.641122
$959$	9.45818	0.305421
$960$	0	0
$961$	−5.46979	−0.176445
$962$	48.1972	1.55394
$963$	0	0
$964$	−6.79254	−0.218773
$965$	0	0
$966$	0	0
$967$	14.2745	0.459036	0.229518	−	0.973304i	$-0.426285\pi$
$967$	14.2745	0.459036	0.229518	+	0.973304i	$0.426285\pi$
$968$	19.9480	0.641153
$969$	0	0
$970$	0	0
$971$	−58.2228	−1.86846	−0.934229	−	0.356675i	$-0.883910\pi$
$971$	−58.2228	−1.86846	−0.934229	+	0.356675i	$0.883910\pi$
$972$	0	0
$973$	1.12679	0.0361231
$974$	−53.4854	−1.71378
$975$	0	0
$976$	28.5820	0.914886
$977$	−57.0456	−1.82505	−0.912526	−	0.409019i	$-0.865871\pi$
$977$	−57.0456	−1.82505	−0.912526	+	0.409019i	$0.865871\pi$
$978$	0	0
$979$	0.825483	0.0263826
$980$	0	0
$981$	0	0
$982$	13.6053	0.434163
$983$	−2.76691	−0.0882506	−0.0441253	−	0.999026i	$-0.514050\pi$
$983$	−2.76691	−0.0882506	−0.0441253	+	0.999026i	$0.514050\pi$
$984$	0	0
$985$	0	0
$986$	−6.62385	−0.210946
$987$	0	0
$988$	−8.67932	−0.276126
$989$	81.2684	2.58418
$990$	0	0
$991$	26.2090	0.832557	0.416278	−	0.909237i	$-0.363334\pi$
$991$	26.2090	0.832557	0.416278	+	0.909237i	$0.363334\pi$
$992$	−24.9144	−0.791034
$993$	0	0
$994$	13.1772	0.417955
$995$	0	0
$996$	0	0
$997$	53.0937	1.68149	0.840747	−	0.541428i	$-0.182116\pi$
$997$	53.0937	1.68149	0.840747	+	0.541428i	$0.182116\pi$
$998$	−56.4066	−1.78552
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bl.1.5		5
3.2	odd	2		2175.2.a.z.1.1		5
5.2	odd	4		1305.2.c.j.784.8		10
5.3	odd	4		1305.2.c.j.784.3		10
5.4	even	2		6525.2.a.bs.1.1		5
15.2	even	4		435.2.c.e.349.3	✓	10
15.8	even	4		435.2.c.e.349.8	yes	10
15.14	odd	2		2175.2.a.w.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.c.e.349.3	✓	10	15.2	even	4
435.2.c.e.349.8	yes	10	15.8	even	4
1305.2.c.j.784.3		10	5.3	odd	4
1305.2.c.j.784.8		10	5.2	odd	4
2175.2.a.w.1.5		5	15.14	odd	2
2175.2.a.z.1.1		5	3.2	odd	2
6525.2.a.bl.1.5		5	1.1	even	1	trivial
6525.2.a.bs.1.1		5	5.4	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:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q+1.71457 q^{2} +0.939748 q^{4} -0.654317 q^{7} -1.81788 q^{8} -0.163559 q^{11} +2.65432 q^{13} -1.12187 q^{14} -4.99637 q^{16} +3.86328 q^{17} -3.47954 q^{19} -0.280433 q^{22} -7.69972 q^{23} +4.55101 q^{26} -0.614893 q^{28} -1.00000 q^{29} +5.05274 q^{31} -4.93087 q^{32} +6.62385 q^{34} +10.5904 q^{37} -5.96591 q^{38} +6.17417 q^{41} -10.5547 q^{43} -0.153704 q^{44} -13.2017 q^{46} -10.3036 q^{47} -6.57187 q^{49} +2.49439 q^{52} -4.04766 q^{53} +1.18947 q^{56} -1.71457 q^{58} +0.328734 q^{59} -5.72054 q^{61} +8.66328 q^{62} +1.53842 q^{64} +3.51985 q^{67} +3.63050 q^{68} -11.7457 q^{71} -1.12143 q^{73} +18.1580 q^{74} -3.26989 q^{76} +0.107019 q^{77} -12.4074 q^{79} +10.5860 q^{82} -7.89306 q^{83} -18.0968 q^{86} +0.297329 q^{88} -5.04702 q^{89} -1.73677 q^{91} -7.23579 q^{92} -17.6663 q^{94} +8.49076 q^{97} -11.2679 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{2} + 5 q^{4} + 8 q^{7} - 9 q^{8} - 12 q^{11} + 2 q^{13} - 6 q^{14} + q^{16} - 2 q^{19} + 14 q^{22} - 8 q^{23} - 6 q^{28} - 5 q^{29} + 2 q^{31} - q^{32} + 4 q^{34} + 16 q^{37} + 14 q^{38} + 14 q^{41}+ \cdots + 9 q^{98}+O(q^{100}) \) 5 * q - 3 * q^2 + 5 * q^4 + 8 * q^7 - 9 * q^8 - 12 * q^11 + 2 * q^13 - 6 * q^14 + q^16 - 2 * q^19 + 14 * q^22 - 8 * q^23 - 6 * q^28 - 5 * q^29 + 2 * q^31 - q^32 + 4 * q^34 + 16 * q^37 + 14 * q^38 + 14 * q^41 - 20 * q^44 - 6 * q^46 - 2 * q^47 - 7 * q^49 + 16 * q^52 - 26 * q^53 + 2 * q^56 + 3 * q^58 - 4 * q^59 - 12 * q^61 - 9 * q^64 + 12 * q^67 + 20 * q^68 - 30 * q^71 - 12 * q^73 - 2 * q^74 - 44 * q^76 - 18 * q^77 - 18 * q^79 + 10 * q^82 - 2 * q^83 - 30 * q^86 + 42 * q^88 + 22 * q^89 - 12 * q^91 - 20 * q^92 - 50 * q^94 + 20 * q^97 + 9 * q^98

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