LMFDB - Embedded newform 3825.2.a.bt.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3825,2,Mod(1,3825)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3825.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3825 = 3^{2} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3825.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,4,0,8,0,0,0,18,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$30.5427787731$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.17830397164.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 24x^{5} + 16x^{4} - 51x^{3} - 11x^{2} + 30x + 4 $ x^8 - 3x^7 - 8x^6 + 24x^5 + 16x^4 - 51x^3 - 11x^2 + 30*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 765)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.63205$ of defining polynomial
Character	$\chi$	$=$	3825.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.32973 q^{2} -0.231826 q^{4} -3.42890 q^{7} -2.96772 q^{8} -4.87695 q^{11} -1.44805 q^{13} -4.55950 q^{14} -3.48261 q^{16} -1.00000 q^{17} +5.06562 q^{19} -6.48501 q^{22} +7.50407 q^{23} -1.92551 q^{26} +0.794907 q^{28} +4.67201 q^{29} -9.47179 q^{31} +1.30452 q^{32} -1.32973 q^{34} -0.485294 q^{37} +6.73589 q^{38} +0.962754 q^{41} +12.4926 q^{43} +1.13060 q^{44} +9.97836 q^{46} +2.43845 q^{47} +4.75736 q^{49} +0.335695 q^{52} -0.221007 q^{53} +10.1760 q^{56} +6.21249 q^{58} -5.15730 q^{59} +10.5949 q^{61} -12.5949 q^{62} +8.69987 q^{64} +12.0151 q^{67} +0.231826 q^{68} -8.78331 q^{71} -8.29802 q^{73} -0.645308 q^{74} -1.17434 q^{76} +16.7226 q^{77} +9.25435 q^{79} +1.28020 q^{82} +8.92730 q^{83} +16.6117 q^{86} +14.4734 q^{88} -11.0445 q^{89} +4.96521 q^{91} -1.73964 q^{92} +3.24247 q^{94} -10.9970 q^{97} +6.32599 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 8 q^{4} + 18 q^{8} + 16 q^{16} - 8 q^{17} + 2 q^{19} + 38 q^{23} - 12 q^{31} + 40 q^{32} - 4 q^{34} + 22 q^{38} + 40 q^{46} + 36 q^{47} + 28 q^{49} + 28 q^{53} - 12 q^{61} - 4 q^{62} + 54 q^{64}+ \cdots + 102 q^{98}+O(q^{100}) $ 8 * q + 4 * q^2 + 8 * q^4 + 18 * q^8 + 16 * q^16 - 8 * q^17 + 2 * q^19 + 38 * q^23 - 12 * q^31 + 40 * q^32 - 4 * q^34 + 22 * q^38 + 40 * q^46 + 36 * q^47 + 28 * q^49 + 28 * q^53 - 12 * q^61 - 4 * q^62 + 54 * q^64 - 8 * q^68 + 6 * q^76 + 36 * q^77 - 36 * q^79 + 32 * q^83 - 48 * q^91 + 76 * q^92 + 18 * q^94 + 102 * 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 $

$2$	1.32973	0.940259	0.470130	−	0.882597i	$-0.344207\pi$
$2$	1.32973	0.940259	0.470130	+	0.882597i	$0.344207\pi$
$3$	0	0
$3$	0	0
$4$	−0.231826	−0.115913
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.42890	−1.29600	−0.648001	−	0.761639i	$-0.724395\pi$
$7$	−3.42890	−1.29600	−0.648001	+	0.761639i	$0.724395\pi$
$8$	−2.96772	−1.04925
$9$	0	0
$10$	0	0
$11$	−4.87695	−1.47046	−0.735228	−	0.677820i	$-0.762925\pi$
$11$	−4.87695	−1.47046	−0.735228	+	0.677820i	$0.762925\pi$
$12$	0	0
$13$	−1.44805	−0.401616	−0.200808	−	0.979631i	$-0.564357\pi$
$13$	−1.44805	−0.401616	−0.200808	+	0.979631i	$0.564357\pi$
$14$	−4.55950	−1.21858
$15$	0	0
$16$	−3.48261	−0.870651
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.06562	1.16213	0.581067	−	0.813856i	$-0.302636\pi$
$19$	5.06562	1.16213	0.581067	+	0.813856i	$0.302636\pi$
$20$	0	0
$21$	0	0
$22$	−6.48501	−1.38261
$23$	7.50407	1.56471	0.782353	−	0.622835i	$-0.214019\pi$
$23$	7.50407	1.56471	0.782353	+	0.622835i	$0.214019\pi$
$24$	0	0
$25$	0	0
$26$	−1.92551	−0.377623
$27$	0	0
$28$	0.794907	0.150223
$29$	4.67201	0.867570	0.433785	−	0.901017i	$-0.357178\pi$
$29$	4.67201	0.867570	0.433785	+	0.901017i	$0.357178\pi$
$30$	0	0
$31$	−9.47179	−1.70118	−0.850592	−	0.525827i	$-0.823756\pi$
$31$	−9.47179	−1.70118	−0.850592	+	0.525827i	$0.823756\pi$
$32$	1.30452	0.230609
$33$	0	0
$34$	−1.32973	−0.228046
$35$	0	0
$36$	0	0
$37$	−0.485294	−0.0797818	−0.0398909	−	0.999204i	$-0.512701\pi$
$37$	−0.485294	−0.0797818	−0.0398909	+	0.999204i	$0.512701\pi$
$38$	6.73589	1.09271
$39$	0	0
$40$	0	0
$41$	0.962754	0.150357	0.0751785	−	0.997170i	$-0.476047\pi$
$41$	0.962754	0.150357	0.0751785	+	0.997170i	$0.476047\pi$
$42$	0	0
$43$	12.4926	1.90510	0.952549	−	0.304386i	$-0.0984513\pi$
$43$	12.4926	1.90510	0.952549	+	0.304386i	$0.0984513\pi$
$44$	1.13060	0.170445
$45$	0	0
$46$	9.97836	1.47123
$47$	2.43845	0.355684	0.177842	−	0.984059i	$-0.443088\pi$
$47$	2.43845	0.355684	0.177842	+	0.984059i	$0.443088\pi$
$48$	0	0
$49$	4.75736	0.679622
$50$	0	0
$51$	0	0
$52$	0.335695	0.0465525
$53$	−0.221007	−0.0303577	−0.0151788	−	0.999885i	$-0.504832\pi$
$53$	−0.221007	−0.0303577	−0.0151788	+	0.999885i	$0.504832\pi$
$54$	0	0
$55$	0	0
$56$	10.1760	1.35983
$57$	0	0
$58$	6.21249	0.815740
$59$	−5.15730	−0.671423	−0.335712	−	0.941965i	$-0.608977\pi$
$59$	−5.15730	−0.671423	−0.335712	+	0.941965i	$0.608977\pi$
$60$	0	0
$61$	10.5949	1.35654	0.678269	−	0.734814i	$-0.262731\pi$
$61$	10.5949	1.35654	0.678269	+	0.734814i	$0.262731\pi$
$62$	−12.5949	−1.59955
$63$	0	0
$64$	8.69987	1.08748
$65$	0	0
$66$	0	0
$67$	12.0151	1.46788	0.733939	−	0.679215i	$-0.237680\pi$
$67$	12.0151	1.46788	0.733939	+	0.679215i	$0.237680\pi$
$68$	0.231826	0.0281130
$69$	0	0
$70$	0	0
$71$	−8.78331	−1.04239	−0.521193	−	0.853439i	$-0.674513\pi$
$71$	−8.78331	−1.04239	−0.521193	+	0.853439i	$0.674513\pi$
$72$	0	0
$73$	−8.29802	−0.971209	−0.485605	−	0.874179i	$-0.661401\pi$
$73$	−8.29802	−0.971209	−0.485605	+	0.874179i	$0.661401\pi$
$74$	−0.645308	−0.0750155
$75$	0	0
$76$	−1.17434	−0.134706
$77$	16.7226	1.90571
$78$	0	0
$79$	9.25435	1.04120	0.520598	−	0.853802i	$-0.325709\pi$
$79$	9.25435	1.04120	0.520598	+	0.853802i	$0.325709\pi$
$80$	0	0
$81$	0	0
$82$	1.28020	0.141374
$83$	8.92730	0.979899	0.489949	−	0.871751i	$-0.337015\pi$
$83$	8.92730	0.979899	0.489949	+	0.871751i	$0.337015\pi$
$84$	0	0
$85$	0	0
$86$	16.6117	1.79129
$87$	0	0
$88$	14.4734	1.54287
$89$	−11.0445	−1.17072	−0.585358	−	0.810775i	$-0.699046\pi$
$89$	−11.0445	−1.17072	−0.585358	+	0.810775i	$0.699046\pi$
$90$	0	0
$91$	4.96521	0.520496
$92$	−1.73964	−0.181370
$93$	0	0
$94$	3.24247	0.334435
$95$	0	0
$96$	0	0
$97$	−10.9970	−1.11658	−0.558288	−	0.829647i	$-0.688542\pi$
$97$	−10.9970	−1.11658	−0.558288	+	0.829647i	$0.688542\pi$
$98$	6.32599	0.639021
$99$	0	0
$100$	0	0
$101$	0.970587	0.0965770	0.0482885	−	0.998833i	$-0.484623\pi$
$101$	0.970587	0.0965770	0.0482885	+	0.998833i	$0.484623\pi$
$102$	0	0
$103$	−5.63476	−0.555209	−0.277605	−	0.960695i	$-0.589541\pi$
$103$	−5.63476	−0.555209	−0.277605	+	0.960695i	$0.589541\pi$
$104$	4.29740	0.421395
$105$	0	0
$106$	−0.293879	−0.0285441
$107$	14.4097	1.39304	0.696521	−	0.717537i	$-0.254730\pi$
$107$	14.4097	1.39304	0.696521	+	0.717537i	$0.254730\pi$
$108$	0	0
$109$	2.55698	0.244915	0.122457	−	0.992474i	$-0.460923\pi$
$109$	2.55698	0.244915	0.122457	+	0.992474i	$0.460923\pi$
$110$	0	0
$111$	0	0
$112$	11.9415	1.12837
$113$	10.9379	1.02896	0.514478	−	0.857504i	$-0.327986\pi$
$113$	10.9379	1.02896	0.514478	+	0.857504i	$0.327986\pi$
$114$	0	0
$115$	0	0
$116$	−1.08309	−0.100562
$117$	0	0
$118$	−6.85780	−0.631312
$119$	3.42890	0.314327
$120$	0	0
$121$	12.7846	1.16224
$122$	14.0883	1.27550
$123$	0	0
$124$	2.19580	0.197189
$125$	0	0
$126$	0	0
$127$	0.477461	0.0423678	0.0211839	−	0.999776i	$-0.493256\pi$
$127$	0.477461	0.0423678	0.0211839	+	0.999776i	$0.493256\pi$
$128$	8.95941	0.791907
$129$	0	0
$130$	0	0
$131$	9.59646	0.838447	0.419223	−	0.907883i	$-0.362302\pi$
$131$	9.59646	0.838447	0.419223	+	0.907883i	$0.362302\pi$
$132$	0	0
$133$	−17.3695	−1.50613
$134$	15.9768	1.38019
$135$	0	0
$136$	2.96772	0.254480
$137$	−13.7307	−1.17309	−0.586547	−	0.809916i	$-0.699513\pi$
$137$	−13.7307	−1.17309	−0.586547	+	0.809916i	$0.699513\pi$
$138$	0	0
$139$	−7.00814	−0.594422	−0.297211	−	0.954812i	$-0.596057\pi$
$139$	−7.00814	−0.594422	−0.297211	+	0.954812i	$0.596057\pi$
$140$	0	0
$141$	0	0
$142$	−11.6794	−0.980114
$143$	7.06205	0.590559
$144$	0	0
$145$	0	0
$146$	−11.0341	−0.913188
$147$	0	0
$148$	0.112503	0.00924773
$149$	15.8661	1.29980	0.649901	−	0.760019i	$-0.274810\pi$
$149$	15.8661	1.29980	0.649901	+	0.760019i	$0.274810\pi$
$150$	0	0
$151$	−6.59133	−0.536394	−0.268197	−	0.963364i	$-0.586428\pi$
$151$	−6.59133	−0.536394	−0.268197	+	0.963364i	$0.586428\pi$
$152$	−15.0333	−1.21936
$153$	0	0
$154$	22.2365	1.79186
$155$	0	0
$156$	0	0
$157$	11.9319	0.952266	0.476133	−	0.879373i	$-0.342038\pi$
$157$	11.9319	0.952266	0.476133	+	0.879373i	$0.342038\pi$
$158$	12.3058	0.978993
$159$	0	0
$160$	0	0
$161$	−25.7307	−2.02786
$162$	0	0
$163$	−19.0700	−1.49368	−0.746839	−	0.665005i	$-0.768430\pi$
$163$	−19.0700	−1.49368	−0.746839	+	0.665005i	$0.768430\pi$
$164$	−0.223191	−0.0174283
$165$	0	0
$166$	11.8709	0.921359
$167$	−3.17166	−0.245430	−0.122715	−	0.992442i	$-0.539160\pi$
$167$	−3.17166	−0.245430	−0.122715	+	0.992442i	$0.539160\pi$
$168$	0	0
$169$	−10.9032	−0.838704
$170$	0	0
$171$	0	0
$172$	−2.89610	−0.220825
$173$	−1.38910	−0.105611	−0.0528056	−	0.998605i	$-0.516816\pi$
$173$	−1.38910	−0.105611	−0.0528056	+	0.998605i	$0.516816\pi$
$174$	0	0
$175$	0	0
$176$	16.9845	1.28025
$177$	0	0
$178$	−14.6862	−1.10078
$179$	24.4244	1.82557	0.912783	−	0.408444i	$-0.133929\pi$
$179$	24.4244	1.82557	0.912783	+	0.408444i	$0.133929\pi$
$180$	0	0
$181$	−14.3395	−1.06585	−0.532925	−	0.846163i	$-0.678907\pi$
$181$	−14.3395	−1.06585	−0.532925	+	0.846163i	$0.678907\pi$
$182$	6.60238	0.489401
$183$	0	0
$184$	−22.2700	−1.64176
$185$	0	0
$186$	0	0
$187$	4.87695	0.356638
$188$	−0.565295	−0.0412283
$189$	0	0
$190$	0	0
$191$	−5.15730	−0.373169	−0.186585	−	0.982439i	$-0.559742\pi$
$191$	−5.15730	−0.373169	−0.186585	+	0.982439i	$0.559742\pi$
$192$	0	0
$193$	−14.3980	−1.03639	−0.518195	−	0.855262i	$-0.673396\pi$
$193$	−14.3980	−1.03639	−0.518195	+	0.855262i	$0.673396\pi$
$194$	−14.6230	−1.04987
$195$	0	0
$196$	−1.10288	−0.0787769
$197$	14.2139	1.01270	0.506350	−	0.862328i	$-0.330994\pi$
$197$	14.2139	1.01270	0.506350	+	0.862328i	$0.330994\pi$
$198$	0	0
$199$	−14.8932	−1.05575	−0.527874	−	0.849323i	$-0.677011\pi$
$199$	−14.8932	−1.05575	−0.527874	+	0.849323i	$0.677011\pi$
$200$	0	0
$201$	0	0
$202$	1.29062	0.0908074
$203$	−16.0198	−1.12437
$204$	0	0
$205$	0	0
$206$	−7.49269	−0.522041
$207$	0	0
$208$	5.04298	0.349668
$209$	−24.7048	−1.70886
$210$	0	0
$211$	−23.3856	−1.60993	−0.804965	−	0.593322i	$-0.797816\pi$
$211$	−23.3856	−1.60993	−0.804965	+	0.593322i	$0.797816\pi$
$212$	0.0512351	0.00351884
$213$	0	0
$214$	19.1610	1.30982
$215$	0	0
$216$	0	0
$217$	32.4778	2.20474
$218$	3.40009	0.230283
$219$	0	0
$220$	0	0
$221$	1.44805	0.0974062
$222$	0	0
$223$	24.5818	1.64612	0.823061	−	0.567953i	$-0.192264\pi$
$223$	24.5818	1.64612	0.823061	+	0.567953i	$0.192264\pi$
$224$	−4.47308	−0.298870
$225$	0	0
$226$	14.5445	0.967485
$227$	17.2437	1.14450	0.572252	−	0.820078i	$-0.306070\pi$
$227$	17.2437	1.14450	0.572252	+	0.820078i	$0.306070\pi$
$228$	0	0
$229$	11.6120	0.767340	0.383670	−	0.923470i	$-0.374660\pi$
$229$	11.6120	0.767340	0.383670	+	0.923470i	$0.374660\pi$
$230$	0	0
$231$	0	0
$232$	−13.8652	−0.910295
$233$	19.1287	1.25316	0.626582	−	0.779355i	$-0.284453\pi$
$233$	19.1287	1.25316	0.626582	+	0.779355i	$0.284453\pi$
$234$	0	0
$235$	0	0
$236$	1.19559	0.0778265
$237$	0	0
$238$	4.55950	0.295549
$239$	4.20238	0.271829	0.135915	−	0.990721i	$-0.456603\pi$
$239$	4.20238	0.271829	0.135915	+	0.990721i	$0.456603\pi$
$240$	0	0
$241$	4.83397	0.311383	0.155692	−	0.987806i	$-0.450239\pi$
$241$	4.83397	0.311383	0.155692	+	0.987806i	$0.450239\pi$
$242$	17.0001	1.09281
$243$	0	0
$244$	−2.45617	−0.157240
$245$	0	0
$246$	0	0
$247$	−7.33526	−0.466731
$248$	28.1096	1.78496
$249$	0	0
$250$	0	0
$251$	12.0151	0.758386	0.379193	−	0.925318i	$-0.376202\pi$
$251$	12.0151	0.758386	0.379193	+	0.925318i	$0.376202\pi$
$252$	0	0
$253$	−36.5969	−2.30083
$254$	0.634892	0.0398367
$255$	0	0
$256$	−5.48617	−0.342886
$257$	−8.94357	−0.557885	−0.278942	−	0.960308i	$-0.589984\pi$
$257$	−8.94357	−0.557885	−0.278942	+	0.960308i	$0.589984\pi$
$258$	0	0
$259$	1.66402	0.103397
$260$	0	0
$261$	0	0
$262$	12.7607	0.788357
$263$	14.1185	0.870586	0.435293	−	0.900289i	$-0.356645\pi$
$263$	14.1185	0.870586	0.435293	+	0.900289i	$0.356645\pi$
$264$	0	0
$265$	0	0
$266$	−23.0967	−1.41615
$267$	0	0
$268$	−2.78541	−0.170146
$269$	−23.4617	−1.43048	−0.715241	−	0.698877i	$-0.753683\pi$
$269$	−23.4617	−1.43048	−0.715241	+	0.698877i	$0.753683\pi$
$270$	0	0
$271$	−5.04042	−0.306184	−0.153092	−	0.988212i	$-0.548923\pi$
$271$	−5.04042	−0.306184	−0.153092	+	0.988212i	$0.548923\pi$
$272$	3.48261	0.211164
$273$	0	0
$274$	−18.2581	−1.10301
$275$	0	0
$276$	0	0
$277$	3.16861	0.190384	0.0951918	−	0.995459i	$-0.469654\pi$
$277$	3.16861	0.190384	0.0951918	+	0.995459i	$0.469654\pi$
$278$	−9.31891	−0.558911
$279$	0	0
$280$	0	0
$281$	−9.84892	−0.587537	−0.293769	−	0.955877i	$-0.594910\pi$
$281$	−9.84892	−0.587537	−0.293769	+	0.955877i	$0.594910\pi$
$282$	0	0
$283$	32.2807	1.91889	0.959444	−	0.281898i	$-0.0909641\pi$
$283$	32.2807	1.91889	0.959444	+	0.281898i	$0.0909641\pi$
$284$	2.03620	0.120826
$285$	0	0
$286$	9.39060	0.555278
$287$	−3.30119	−0.194863
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	1.92369	0.112576
$293$	29.2958	1.71148	0.855740	−	0.517406i	$-0.173102\pi$
$293$	29.2958	1.71148	0.855740	+	0.517406i	$0.173102\pi$
$294$	0	0
$295$	0	0
$296$	1.44021	0.0837108
$297$	0	0
$298$	21.0976	1.22215
$299$	−10.8662	−0.628411
$300$	0	0
$301$	−42.8357	−2.46901
$302$	−8.76466	−0.504350
$303$	0	0
$304$	−17.6416	−1.01181
$305$	0	0
$306$	0	0
$307$	−7.41850	−0.423396	−0.211698	−	0.977335i	$-0.567899\pi$
$307$	−7.41850	−0.423396	−0.211698	+	0.977335i	$0.567899\pi$
$308$	−3.87672	−0.220897
$309$	0	0
$310$	0	0
$311$	21.7411	1.23282	0.616412	−	0.787424i	$-0.288585\pi$
$311$	21.7411	1.23282	0.616412	+	0.787424i	$0.288585\pi$
$312$	0	0
$313$	26.8352	1.51682	0.758408	−	0.651780i	$-0.225977\pi$
$313$	26.8352	1.51682	0.758408	+	0.651780i	$0.225977\pi$
$314$	15.8661	0.895377
$315$	0	0
$316$	−2.14539	−0.120688
$317$	18.5445	1.04156	0.520781	−	0.853690i	$-0.325641\pi$
$317$	18.5445	1.04156	0.520781	+	0.853690i	$0.325641\pi$
$318$	0	0
$319$	−22.7851	−1.27572
$320$	0	0
$321$	0	0
$322$	−34.2148	−1.90672
$323$	−5.06562	−0.281859
$324$	0	0
$325$	0	0
$326$	−25.3579	−1.40444
$327$	0	0
$328$	−2.85718	−0.157762
$329$	−8.36119	−0.460967
$330$	0	0
$331$	20.3037	1.11599	0.557996	−	0.829844i	$-0.311570\pi$
$331$	20.3037	1.11599	0.557996	+	0.829844i	$0.311570\pi$
$332$	−2.06958	−0.113583
$333$	0	0
$334$	−4.21744	−0.230768
$335$	0	0
$336$	0	0
$337$	16.6871	0.909004	0.454502	−	0.890746i	$-0.349817\pi$
$337$	16.6871	0.909004	0.454502	+	0.890746i	$0.349817\pi$
$338$	−14.4982	−0.788599
$339$	0	0
$340$	0	0
$341$	46.1934	2.50151
$342$	0	0
$343$	7.68980	0.415210
$344$	−37.0744	−1.99892
$345$	0	0
$346$	−1.84712	−0.0993019
$347$	−2.16915	−0.116446	−0.0582230	−	0.998304i	$-0.518543\pi$
$347$	−2.16915	−0.116446	−0.0582230	+	0.998304i	$0.518543\pi$
$348$	0	0
$349$	−7.84004	−0.419668	−0.209834	−	0.977737i	$-0.567292\pi$
$349$	−7.84004	−0.419668	−0.209834	+	0.977737i	$0.567292\pi$
$350$	0	0
$351$	0	0
$352$	−6.36209	−0.339101
$353$	−13.2383	−0.704603	−0.352301	−	0.935887i	$-0.614601\pi$
$353$	−13.2383	−0.704603	−0.352301	+	0.935887i	$0.614601\pi$
$354$	0	0
$355$	0	0
$356$	2.56040	0.135701
$357$	0	0
$358$	32.4778	1.71651
$359$	−1.70050	−0.0897490	−0.0448745	−	0.998993i	$-0.514289\pi$
$359$	−1.70050	−0.0897490	−0.0448745	+	0.998993i	$0.514289\pi$
$360$	0	0
$361$	6.66051	0.350553
$362$	−19.0677	−1.00218
$363$	0	0
$364$	−1.15106	−0.0603321
$365$	0	0
$366$	0	0
$367$	−6.84213	−0.357157	−0.178578	−	0.983926i	$-0.557150\pi$
$367$	−6.84213	−0.357157	−0.178578	+	0.983926i	$0.557150\pi$
$368$	−26.1337	−1.36231
$369$	0	0
$370$	0	0
$371$	0.757811	0.0393436
$372$	0	0
$373$	−4.82160	−0.249653	−0.124827	−	0.992179i	$-0.539837\pi$
$373$	−4.82160	−0.249653	−0.124827	+	0.992179i	$0.539837\pi$
$374$	6.48501	0.335332
$375$	0	0
$376$	−7.23663	−0.373201
$377$	−6.76529	−0.348430
$378$	0	0
$379$	−3.68009	−0.189034	−0.0945168	−	0.995523i	$-0.530131\pi$
$379$	−3.68009	−0.189034	−0.0945168	+	0.995523i	$0.530131\pi$
$380$	0	0
$381$	0	0
$382$	−6.85780	−0.350876
$383$	17.7482	0.906892	0.453446	−	0.891284i	$-0.350195\pi$
$383$	17.7482	0.906892	0.453446	+	0.891284i	$0.350195\pi$
$384$	0	0
$385$	0	0
$386$	−19.1454	−0.974476
$387$	0	0
$388$	2.54939	0.129426
$389$	3.96170	0.200866	0.100433	−	0.994944i	$-0.467977\pi$
$389$	3.96170	0.200866	0.100433	+	0.994944i	$0.467977\pi$
$390$	0	0
$391$	−7.50407	−0.379497
$392$	−14.1185	−0.713092
$393$	0	0
$394$	18.9006	0.952201
$395$	0	0
$396$	0	0
$397$	−11.3048	−0.567372	−0.283686	−	0.958917i	$-0.591557\pi$
$397$	−11.3048	−0.567372	−0.283686	+	0.958917i	$0.591557\pi$
$398$	−19.8038	−0.992677
$399$	0	0
$400$	0	0
$401$	−2.90393	−0.145015	−0.0725076	−	0.997368i	$-0.523100\pi$
$401$	−2.90393	−0.145015	−0.0725076	+	0.997368i	$0.523100\pi$
$402$	0	0
$403$	13.7156	0.683223
$404$	−0.225007	−0.0111945
$405$	0	0
$406$	−21.3020	−1.05720
$407$	2.36675	0.117316
$408$	0	0
$409$	−26.9957	−1.33485	−0.667426	−	0.744677i	$-0.732604\pi$
$409$	−26.9957	−1.33485	−0.667426	+	0.744677i	$0.732604\pi$
$410$	0	0
$411$	0	0
$412$	1.30628	0.0643559
$413$	17.6839	0.870166
$414$	0	0
$415$	0	0
$416$	−1.88901	−0.0926164
$417$	0	0
$418$	−32.8506	−1.60678
$419$	1.26272	0.0616879	0.0308439	−	0.999524i	$-0.490181\pi$
$419$	1.26272	0.0616879	0.0308439	+	0.999524i	$0.490181\pi$
$420$	0	0
$421$	−7.16809	−0.349351	−0.174676	−	0.984626i	$-0.555888\pi$
$421$	−7.16809	−0.349351	−0.174676	+	0.984626i	$0.555888\pi$
$422$	−31.0965	−1.51375
$423$	0	0
$424$	0.655887	0.0318527
$425$	0	0
$426$	0	0
$427$	−36.3288	−1.75808
$428$	−3.34055	−0.161471
$429$	0	0
$430$	0	0
$431$	15.3333	0.738580	0.369290	−	0.929314i	$-0.379601\pi$
$431$	15.3333	0.738580	0.369290	+	0.929314i	$0.379601\pi$
$432$	0	0
$433$	−5.07406	−0.243844	−0.121922	−	0.992540i	$-0.538906\pi$
$433$	−5.07406	−0.243844	−0.121922	+	0.992540i	$0.538906\pi$
$434$	43.1866	2.07302
$435$	0	0
$436$	−0.592774	−0.0283887
$437$	38.0128	1.81840
$438$	0	0
$439$	17.5476	0.837501	0.418751	−	0.908101i	$-0.362468\pi$
$439$	17.5476	0.837501	0.418751	+	0.908101i	$0.362468\pi$
$440$	0	0
$441$	0	0
$442$	1.92551	0.0915871
$443$	40.0325	1.90200	0.951002	−	0.309185i	$-0.100056\pi$
$443$	40.0325	1.90200	0.951002	+	0.309185i	$0.100056\pi$
$444$	0	0
$445$	0	0
$446$	32.6872	1.54778
$447$	0	0
$448$	−29.8310	−1.40938
$449$	−5.20481	−0.245630	−0.122815	−	0.992430i	$-0.539192\pi$
$449$	−5.20481	−0.245630	−0.122815	+	0.992430i	$0.539192\pi$
$450$	0	0
$451$	−4.69530	−0.221093
$452$	−2.53570	−0.119269
$453$	0	0
$454$	22.9294	1.07613
$455$	0	0
$456$	0	0
$457$	−14.7538	−0.690152	−0.345076	−	0.938575i	$-0.612147\pi$
$457$	−14.7538	−0.690152	−0.345076	+	0.938575i	$0.612147\pi$
$458$	15.4407	0.721498
$459$	0	0
$460$	0	0
$461$	18.2965	0.852154	0.426077	−	0.904687i	$-0.359895\pi$
$461$	18.2965	0.852154	0.426077	+	0.904687i	$0.359895\pi$
$462$	0	0
$463$	39.0949	1.81690	0.908448	−	0.417998i	$-0.137268\pi$
$463$	39.0949	1.81690	0.908448	+	0.417998i	$0.137268\pi$
$464$	−16.2708	−0.755351
$465$	0	0
$466$	25.4360	1.17830
$467$	−3.92273	−0.181522	−0.0907612	−	0.995873i	$-0.528930\pi$
$467$	−3.92273	−0.181522	−0.0907612	+	0.995873i	$0.528930\pi$
$468$	0	0
$469$	−41.1986	−1.90237
$470$	0	0
$471$	0	0
$472$	15.3054	0.704489
$473$	−60.9256	−2.80136
$474$	0	0
$475$	0	0
$476$	−0.794907	−0.0364345
$477$	0	0
$478$	5.58802	0.255590
$479$	40.3336	1.84289	0.921446	−	0.388507i	$-0.127009\pi$
$479$	40.3336	1.84289	0.921446	+	0.388507i	$0.127009\pi$
$480$	0	0
$481$	0.702728	0.0320416
$482$	6.42786	0.292781
$483$	0	0
$484$	−2.96380	−0.134718
$485$	0	0
$486$	0	0
$487$	−22.9489	−1.03991	−0.519957	−	0.854192i	$-0.674052\pi$
$487$	−22.9489	−1.03991	−0.519957	+	0.854192i	$0.674052\pi$
$488$	−31.4427	−1.42334
$489$	0	0
$490$	0	0
$491$	−7.30781	−0.329797	−0.164899	−	0.986311i	$-0.552730\pi$
$491$	−7.30781	−0.329797	−0.164899	+	0.986311i	$0.552730\pi$
$492$	0	0
$493$	−4.67201	−0.210417
$494$	−9.75390	−0.438849
$495$	0	0
$496$	32.9865	1.48114
$497$	30.1171	1.35094
$498$	0	0
$499$	4.15288	0.185908	0.0929542	−	0.995670i	$-0.470369\pi$
$499$	4.15288	0.185908	0.0929542	+	0.995670i	$0.470369\pi$
$500$	0	0
$501$	0	0
$502$	15.9768	0.713080
$503$	0.0485525	0.00216485	0.00108242	−	0.999999i	$-0.499655\pi$
$503$	0.0485525	0.00216485	0.00108242	+	0.999999i	$0.499655\pi$
$504$	0	0
$505$	0	0
$506$	−48.6640	−2.16338
$507$	0	0
$508$	−0.110688	−0.00491097
$509$	9.51322	0.421666	0.210833	−	0.977522i	$-0.432382\pi$
$509$	9.51322	0.421666	0.210833	+	0.977522i	$0.432382\pi$
$510$	0	0
$511$	28.4531	1.25869
$512$	−25.2139	−1.11431
$513$	0	0
$514$	−11.8925	−0.524556
$515$	0	0
$516$	0	0
$517$	−11.8922	−0.523017
$518$	2.21270	0.0972203
$519$	0	0
$520$	0	0
$521$	−33.1999	−1.45451	−0.727257	−	0.686365i	$-0.759205\pi$
$521$	−33.1999	−1.45451	−0.727257	+	0.686365i	$0.759205\pi$
$522$	0	0
$523$	−3.96170	−0.173233	−0.0866166	−	0.996242i	$-0.527606\pi$
$523$	−3.96170	−0.173233	−0.0866166	+	0.996242i	$0.527606\pi$
$524$	−2.22471	−0.0971867
$525$	0	0
$526$	18.7738	0.818577
$527$	9.47179	0.412598
$528$	0	0
$529$	33.3110	1.44831
$530$	0	0
$531$	0	0
$532$	4.02670	0.174579
$533$	−1.39411	−0.0603858
$534$	0	0
$535$	0	0
$536$	−35.6574	−1.54017
$537$	0	0
$538$	−31.1976	−1.34502
$539$	−23.2014	−0.999354
$540$	0	0
$541$	−26.7816	−1.15143	−0.575715	−	0.817651i	$-0.695276\pi$
$541$	−26.7816	−1.15143	−0.575715	+	0.817651i	$0.695276\pi$
$542$	−6.70238	−0.287892
$543$	0	0
$544$	−1.30452	−0.0559310
$545$	0	0
$546$	0	0
$547$	−0.942690	−0.0403065	−0.0201533	−	0.999797i	$-0.506415\pi$
$547$	−0.942690	−0.0403065	−0.0201533	+	0.999797i	$0.506415\pi$
$548$	3.18313	0.135977
$549$	0	0
$550$	0	0
$551$	23.6666	1.00823
$552$	0	0
$553$	−31.7322	−1.34939
$554$	4.21339	0.179010
$555$	0	0
$556$	1.62467	0.0689012
$557$	−19.2331	−0.814931	−0.407465	−	0.913221i	$-0.633587\pi$
$557$	−19.2331	−0.814931	−0.407465	+	0.913221i	$0.633587\pi$
$558$	0	0
$559$	−18.0898	−0.765118
$560$	0	0
$561$	0	0
$562$	−13.0964	−0.552437
$563$	−15.0671	−0.635004	−0.317502	−	0.948258i	$-0.602844\pi$
$563$	−15.0671	−0.635004	−0.317502	+	0.948258i	$0.602844\pi$
$564$	0	0
$565$	0	0
$566$	42.9245	1.80425
$567$	0	0
$568$	26.0664	1.09372
$569$	20.0528	0.840658	0.420329	−	0.907372i	$-0.361915\pi$
$569$	20.0528	0.840658	0.420329	+	0.907372i	$0.361915\pi$
$570$	0	0
$571$	−13.2923	−0.556264	−0.278132	−	0.960543i	$-0.589715\pi$
$571$	−13.2923	−0.556264	−0.278132	+	0.960543i	$0.589715\pi$
$572$	−1.63717	−0.0684533
$573$	0	0
$574$	−4.38968	−0.183222
$575$	0	0
$576$	0	0
$577$	45.7003	1.90253	0.951264	−	0.308378i	$-0.0997862\pi$
$577$	45.7003	1.90253	0.951264	+	0.308378i	$0.0997862\pi$
$578$	1.32973	0.0553094
$579$	0	0
$580$	0	0
$581$	−30.6108	−1.26995
$582$	0	0
$583$	1.07784	0.0446396
$584$	24.6262	1.01904
$585$	0	0
$586$	38.9555	1.60923
$587$	23.3479	0.963670	0.481835	−	0.876262i	$-0.339970\pi$
$587$	23.3479	0.963670	0.481835	+	0.876262i	$0.339970\pi$
$588$	0	0
$589$	−47.9805	−1.97700
$590$	0	0
$591$	0	0
$592$	1.69009	0.0694621
$593$	−7.38739	−0.303364	−0.151682	−	0.988429i	$-0.548469\pi$
$593$	−7.38739	−0.303364	−0.151682	+	0.988429i	$0.548469\pi$
$594$	0	0
$595$	0	0
$596$	−3.67817	−0.150664
$597$	0	0
$598$	−14.4491	−0.590870
$599$	36.5695	1.49419	0.747095	−	0.664718i	$-0.231448\pi$
$599$	36.5695	1.49419	0.747095	+	0.664718i	$0.231448\pi$
$600$	0	0
$601$	−32.5899	−1.32937	−0.664684	−	0.747124i	$-0.731434\pi$
$601$	−32.5899	−1.32937	−0.664684	+	0.747124i	$0.731434\pi$
$602$	−56.9598	−2.32151
$603$	0	0
$604$	1.52804	0.0621750
$605$	0	0
$606$	0	0
$607$	3.03468	0.123174	0.0615869	−	0.998102i	$-0.480384\pi$
$607$	3.03468	0.123174	0.0615869	+	0.998102i	$0.480384\pi$
$608$	6.60822	0.267999
$609$	0	0
$610$	0	0
$611$	−3.53099	−0.142848
$612$	0	0
$613$	14.1189	0.570256	0.285128	−	0.958489i	$-0.407964\pi$
$613$	14.1189	0.570256	0.285128	+	0.958489i	$0.407964\pi$
$614$	−9.86458	−0.398102
$615$	0	0
$616$	−49.6279	−1.99956
$617$	14.8248	0.596825	0.298413	−	0.954437i	$-0.403543\pi$
$617$	14.8248	0.596825	0.298413	+	0.954437i	$0.403543\pi$
$618$	0	0
$619$	−26.1567	−1.05132	−0.525662	−	0.850693i	$-0.676182\pi$
$619$	−26.1567	−1.05132	−0.525662	+	0.850693i	$0.676182\pi$
$620$	0	0
$621$	0	0
$622$	28.9097	1.15917
$623$	37.8705	1.51725
$624$	0	0
$625$	0	0
$626$	35.6835	1.42620
$627$	0	0
$628$	−2.76611	−0.110380
$629$	0.485294	0.0193499
$630$	0	0
$631$	−19.0677	−0.759075	−0.379537	−	0.925176i	$-0.623917\pi$
$631$	−19.0677	−0.759075	−0.379537	+	0.925176i	$0.623917\pi$
$632$	−27.4643	−1.09247
$633$	0	0
$634$	24.6591	0.979338
$635$	0	0
$636$	0	0
$637$	−6.88888	−0.272947
$638$	−30.2980	−1.19951
$639$	0	0
$640$	0	0
$641$	37.7380	1.49056	0.745280	−	0.666752i	$-0.232316\pi$
$641$	37.7380	1.49056	0.745280	+	0.666752i	$0.232316\pi$
$642$	0	0
$643$	3.41323	0.134605	0.0673024	−	0.997733i	$-0.478561\pi$
$643$	3.41323	0.134605	0.0673024	+	0.997733i	$0.478561\pi$
$644$	5.96504	0.235055
$645$	0	0
$646$	−6.73589	−0.265020
$647$	20.9486	0.823574	0.411787	−	0.911280i	$-0.364905\pi$
$647$	20.9486	0.823574	0.411787	+	0.911280i	$0.364905\pi$
$648$	0	0
$649$	25.1519	0.987298
$650$	0	0
$651$	0	0
$652$	4.42092	0.173136
$653$	−39.0767	−1.52919	−0.764594	−	0.644512i	$-0.777061\pi$
$653$	−39.0767	−1.52919	−0.764594	+	0.644512i	$0.777061\pi$
$654$	0	0
$655$	0	0
$656$	−3.35289	−0.130908
$657$	0	0
$658$	−11.1181	−0.433429
$659$	20.0127	0.779584	0.389792	−	0.920903i	$-0.372547\pi$
$659$	20.0127	0.779584	0.389792	+	0.920903i	$0.372547\pi$
$660$	0	0
$661$	−11.1427	−0.433400	−0.216700	−	0.976238i	$-0.569529\pi$
$661$	−11.1427	−0.433400	−0.216700	+	0.976238i	$0.569529\pi$
$662$	26.9984	1.04932
$663$	0	0
$664$	−26.4937	−1.02816
$665$	0	0
$666$	0	0
$667$	35.0590	1.35749
$668$	0.735272	0.0284485
$669$	0	0
$670$	0	0
$671$	−51.6707	−1.99473
$672$	0	0
$673$	18.8254	0.725665	0.362833	−	0.931854i	$-0.381810\pi$
$673$	18.8254	0.725665	0.362833	+	0.931854i	$0.381810\pi$
$674$	22.1893	0.854700
$675$	0	0
$676$	2.52763	0.0972166
$677$	−2.58425	−0.0993208	−0.0496604	−	0.998766i	$-0.515814\pi$
$677$	−2.58425	−0.0993208	−0.0496604	+	0.998766i	$0.515814\pi$
$678$	0	0
$679$	37.7076	1.44709
$680$	0	0
$681$	0	0
$682$	61.4246	2.35207
$683$	−33.9357	−1.29851	−0.649257	−	0.760569i	$-0.724920\pi$
$683$	−33.9357	−1.29851	−0.649257	+	0.760569i	$0.724920\pi$
$684$	0	0
$685$	0	0
$686$	10.2253	0.390405
$687$	0	0
$688$	−43.5067	−1.65868
$689$	0.320029	0.0121921
$690$	0	0
$691$	−31.0373	−1.18071	−0.590356	−	0.807143i	$-0.701013\pi$
$691$	−31.0373	−1.18071	−0.590356	+	0.807143i	$0.701013\pi$
$692$	0.322029	0.0122417
$693$	0	0
$694$	−2.88438	−0.109489
$695$	0	0
$696$	0	0
$697$	−0.962754	−0.0364669
$698$	−10.4251	−0.394597
$699$	0	0
$700$	0	0
$701$	23.1338	0.873752	0.436876	−	0.899522i	$-0.356085\pi$
$701$	23.1338	0.873752	0.436876	+	0.899522i	$0.356085\pi$
$702$	0	0
$703$	−2.45831	−0.0927170
$704$	−42.4288	−1.59910
$705$	0	0
$706$	−17.6033	−0.662509
$707$	−3.32805	−0.125164
$708$	0	0
$709$	14.0717	0.528474	0.264237	−	0.964458i	$-0.414880\pi$
$709$	14.0717	0.528474	0.264237	+	0.964458i	$0.414880\pi$
$710$	0	0
$711$	0	0
$712$	32.7770	1.22837
$713$	−71.0769	−2.66185
$714$	0	0
$715$	0	0
$716$	−5.66221	−0.211607
$717$	0	0
$718$	−2.26120	−0.0843874
$719$	−12.1446	−0.452918	−0.226459	−	0.974021i	$-0.572715\pi$
$719$	−12.1446	−0.452918	−0.226459	+	0.974021i	$0.572715\pi$
$720$	0	0
$721$	19.3210	0.719553
$722$	8.85667	0.329611
$723$	0	0
$724$	3.32427	0.123546
$725$	0	0
$726$	0	0
$727$	−47.0741	−1.74588	−0.872942	−	0.487825i	$-0.837791\pi$
$727$	−47.0741	−1.74588	−0.872942	+	0.487825i	$0.837791\pi$
$728$	−14.7354	−0.546129
$729$	0	0
$730$	0	0
$731$	−12.4926	−0.462054
$732$	0	0
$733$	12.8750	0.475549	0.237774	−	0.971320i	$-0.423582\pi$
$733$	12.8750	0.475549	0.237774	+	0.971320i	$0.423582\pi$
$734$	−9.09817	−0.335820
$735$	0	0
$736$	9.78923	0.360836
$737$	−58.5970	−2.15845
$738$	0	0
$739$	−7.81861	−0.287612	−0.143806	−	0.989606i	$-0.545934\pi$
$739$	−7.81861	−0.287612	−0.143806	+	0.989606i	$0.545934\pi$
$740$	0	0
$741$	0	0
$742$	1.00768	0.0369932
$743$	7.81333	0.286643	0.143322	−	0.989676i	$-0.454222\pi$
$743$	7.81333	0.286643	0.143322	+	0.989676i	$0.454222\pi$
$744$	0	0
$745$	0	0
$746$	−6.41142	−0.234739
$747$	0	0
$748$	−1.13060	−0.0413389
$749$	−49.4095	−1.80539
$750$	0	0
$751$	4.95396	0.180772	0.0903862	−	0.995907i	$-0.471190\pi$
$751$	4.95396	0.180772	0.0903862	+	0.995907i	$0.471190\pi$
$752$	−8.49215	−0.309677
$753$	0	0
$754$	−8.99599	−0.327614
$755$	0	0
$756$	0	0
$757$	−40.4845	−1.47143	−0.735716	−	0.677290i	$-0.763154\pi$
$757$	−40.4845	−1.47143	−0.735716	+	0.677290i	$0.763154\pi$
$758$	−4.89352	−0.177740
$759$	0	0
$760$	0	0
$761$	4.74741	0.172094	0.0860468	−	0.996291i	$-0.472577\pi$
$761$	4.74741	0.172094	0.0860468	+	0.996291i	$0.472577\pi$
$762$	0	0
$763$	−8.76764	−0.317410
$764$	1.19559	0.0432551
$765$	0	0
$766$	23.6003	0.852713
$767$	7.46801	0.269654
$768$	0	0
$769$	−4.29621	−0.154925	−0.0774627	−	0.996995i	$-0.524682\pi$
$769$	−4.29621	−0.154925	−0.0774627	+	0.996995i	$0.524682\pi$
$770$	0	0
$771$	0	0
$772$	3.33783	0.120131
$773$	−29.4037	−1.05758	−0.528788	−	0.848754i	$-0.677353\pi$
$773$	−29.4037	−1.05758	−0.528788	+	0.848754i	$0.677353\pi$
$774$	0	0
$775$	0	0
$776$	32.6360	1.17156
$777$	0	0
$778$	5.26799	0.188866
$779$	4.87695	0.174735
$780$	0	0
$781$	42.8357	1.53278
$782$	−9.97836	−0.356826
$783$	0	0
$784$	−16.5680	−0.591714
$785$	0	0
$786$	0	0
$787$	−4.96019	−0.176812	−0.0884058	−	0.996085i	$-0.528177\pi$
$787$	−4.96019	−0.176812	−0.0884058	+	0.996085i	$0.528177\pi$
$788$	−3.29515	−0.117385
$789$	0	0
$790$	0	0
$791$	−37.5051	−1.33353
$792$	0	0
$793$	−15.3419	−0.544807
$794$	−15.0323	−0.533476
$795$	0	0
$796$	3.45262	0.122375
$797$	−17.7520	−0.628808	−0.314404	−	0.949289i	$-0.601805\pi$
$797$	−17.7520	−0.628808	−0.314404	+	0.949289i	$0.601805\pi$
$798$	0	0
$799$	−2.43845	−0.0862661
$800$	0	0
$801$	0	0
$802$	−3.86143	−0.136352
$803$	40.4690	1.42812
$804$	0	0
$805$	0	0
$806$	18.2380	0.642406
$807$	0	0
$808$	−2.88043	−0.101333
$809$	−45.9885	−1.61687	−0.808434	−	0.588587i	$-0.799685\pi$
$809$	−45.9885	−1.61687	−0.808434	+	0.588587i	$0.799685\pi$
$810$	0	0
$811$	36.8574	1.29424	0.647119	−	0.762389i	$-0.275974\pi$
$811$	36.8574	1.29424	0.647119	+	0.762389i	$0.275974\pi$
$812$	3.71381	0.130329
$813$	0	0
$814$	3.14713	0.110307
$815$	0	0
$816$	0	0
$817$	63.2826	2.21398
$818$	−35.8969	−1.25511
$819$	0	0
$820$	0	0
$821$	−13.0799	−0.456493	−0.228247	−	0.973603i	$-0.573299\pi$
$821$	−13.0799	−0.456493	−0.228247	+	0.973603i	$0.573299\pi$
$822$	0	0
$823$	20.9955	0.731858	0.365929	−	0.930643i	$-0.380751\pi$
$823$	20.9955	0.731858	0.365929	+	0.930643i	$0.380751\pi$
$824$	16.7224	0.582552
$825$	0	0
$826$	23.5147	0.818181
$827$	25.3408	0.881186	0.440593	−	0.897707i	$-0.354768\pi$
$827$	25.3408	0.881186	0.440593	+	0.897707i	$0.354768\pi$
$828$	0	0
$829$	20.8372	0.723705	0.361853	−	0.932235i	$-0.382144\pi$
$829$	20.8372	0.723705	0.361853	+	0.932235i	$0.382144\pi$
$830$	0	0
$831$	0	0
$832$	−12.5978	−0.436751
$833$	−4.75736	−0.164833
$834$	0	0
$835$	0	0
$836$	5.72720	0.198079
$837$	0	0
$838$	1.67907	0.0580026
$839$	−25.3553	−0.875363	−0.437682	−	0.899130i	$-0.644200\pi$
$839$	−25.3553	−0.875363	−0.437682	+	0.899130i	$0.644200\pi$
$840$	0	0
$841$	−7.17237	−0.247323
$842$	−9.53161	−0.328481
$843$	0	0
$844$	5.42138	0.186612
$845$	0	0
$846$	0	0
$847$	−43.8372	−1.50626
$848$	0.769681	0.0264309
$849$	0	0
$850$	0	0
$851$	−3.64168	−0.124835
$852$	0	0
$853$	13.8931	0.475691	0.237845	−	0.971303i	$-0.423559\pi$
$853$	13.8931	0.475691	0.237845	+	0.971303i	$0.423559\pi$
$854$	−48.3074	−1.65305
$855$	0	0
$856$	−42.7640	−1.46164
$857$	−19.1394	−0.653789	−0.326894	−	0.945061i	$-0.606002\pi$
$857$	−19.1394	−0.653789	−0.326894	+	0.945061i	$0.606002\pi$
$858$	0	0
$859$	32.3777	1.10471	0.552356	−	0.833609i	$-0.313729\pi$
$859$	32.3777	1.10471	0.552356	+	0.833609i	$0.313729\pi$
$860$	0	0
$861$	0	0
$862$	20.3891	0.694456
$863$	1.39617	0.0475263	0.0237632	−	0.999718i	$-0.492435\pi$
$863$	1.39617	0.0475263	0.0237632	+	0.999718i	$0.492435\pi$
$864$	0	0
$865$	0	0
$866$	−6.74711	−0.229276
$867$	0	0
$868$	−7.52919	−0.255557
$869$	−45.1330	−1.53103
$870$	0	0
$871$	−17.3984	−0.589524
$872$	−7.58841	−0.256976
$873$	0	0
$874$	50.5466	1.70976
$875$	0	0
$876$	0	0
$877$	−49.8155	−1.68215	−0.841075	−	0.540919i	$-0.818076\pi$
$877$	−49.8155	−1.68215	−0.841075	+	0.540919i	$0.818076\pi$
$878$	23.3335	0.787468
$879$	0	0
$880$	0	0
$881$	1.51441	0.0510216	0.0255108	−	0.999675i	$-0.491879\pi$
$881$	1.51441	0.0510216	0.0255108	+	0.999675i	$0.491879\pi$
$882$	0	0
$883$	−32.2410	−1.08500	−0.542498	−	0.840057i	$-0.682522\pi$
$883$	−32.2410	−1.08500	−0.542498	+	0.840057i	$0.682522\pi$
$884$	−0.335695	−0.0112906
$885$	0	0
$886$	53.2324	1.78838
$887$	28.7750	0.966171	0.483086	−	0.875573i	$-0.339516\pi$
$887$	28.7750	0.966171	0.483086	+	0.875573i	$0.339516\pi$
$888$	0	0
$889$	−1.63717	−0.0549088
$890$	0	0
$891$	0	0
$892$	−5.69870	−0.190807
$893$	12.3522	0.413352
$894$	0	0
$895$	0	0
$896$	−30.7209	−1.02631
$897$	0	0
$898$	−6.92098	−0.230956
$899$	−44.2522	−1.47589
$900$	0	0
$901$	0.221007	0.00736282
$902$	−6.24347	−0.207885
$903$	0	0
$904$	−32.4608	−1.07963
$905$	0	0
$906$	0	0
$907$	32.7307	1.08681	0.543403	−	0.839472i	$-0.317136\pi$
$907$	32.7307	1.08681	0.543403	+	0.839472i	$0.317136\pi$
$908$	−3.99753	−0.132663
$909$	0	0
$910$	0	0
$911$	51.9668	1.72174	0.860868	−	0.508829i	$-0.169921\pi$
$911$	51.9668	1.72174	0.860868	+	0.508829i	$0.169921\pi$
$912$	0	0
$913$	−43.5380	−1.44090
$914$	−19.6185	−0.648922
$915$	0	0
$916$	−2.69195	−0.0889445
$917$	−32.9053	−1.08663
$918$	0	0
$919$	49.2635	1.62505	0.812527	−	0.582923i	$-0.198091\pi$
$919$	49.2635	1.62505	0.812527	+	0.582923i	$0.198091\pi$
$920$	0	0
$921$	0	0
$922$	24.3294	0.801246
$923$	12.7187	0.418639
$924$	0	0
$925$	0	0
$926$	51.9856	1.70835
$927$	0	0
$928$	6.09474	0.200070
$929$	55.7703	1.82976	0.914881	−	0.403723i	$-0.132284\pi$
$929$	55.7703	1.82976	0.914881	+	0.403723i	$0.132284\pi$
$930$	0	0
$931$	24.0990	0.789812
$932$	−4.43453	−0.145258
$933$	0	0
$934$	−5.21617	−0.170678
$935$	0	0
$936$	0	0
$937$	11.3594	0.371095	0.185547	−	0.982635i	$-0.440594\pi$
$937$	11.3594	0.371095	0.185547	+	0.982635i	$0.440594\pi$
$938$	−54.7829	−1.78872
$939$	0	0
$940$	0	0
$941$	−52.4988	−1.71141	−0.855706	−	0.517462i	$-0.826877\pi$
$941$	−52.4988	−1.71141	−0.855706	+	0.517462i	$0.826877\pi$
$942$	0	0
$943$	7.22457	0.235264
$944$	17.9608	0.584575
$945$	0	0
$946$	−81.0144	−2.63400
$947$	−41.0861	−1.33512	−0.667559	−	0.744557i	$-0.732661\pi$
$947$	−41.0861	−1.33512	−0.667559	+	0.744557i	$0.732661\pi$
$948$	0	0
$949$	12.0159	0.390053
$950$	0	0
$951$	0	0
$952$	−10.1760	−0.329806
$953$	−16.8354	−0.545353	−0.272676	−	0.962106i	$-0.587909\pi$
$953$	−16.8354	−0.545353	−0.272676	+	0.962106i	$0.587909\pi$
$954$	0	0
$955$	0	0
$956$	−0.974219	−0.0315085
$957$	0	0
$958$	53.6327	1.73280
$959$	47.0812	1.52033
$960$	0	0
$961$	58.7148	1.89402
$962$	0.934437	0.0301275
$963$	0	0
$964$	−1.12064	−0.0360933
$965$	0	0
$966$	0	0
$967$	−54.6685	−1.75802	−0.879010	−	0.476804i	$-0.841795\pi$
$967$	−54.6685	−1.75802	−0.879010	+	0.476804i	$0.841795\pi$
$968$	−37.9412	−1.21948
$969$	0	0
$970$	0	0
$971$	5.86638	0.188261	0.0941306	−	0.995560i	$-0.469993\pi$
$971$	5.86638	0.188261	0.0941306	+	0.995560i	$0.469993\pi$
$972$	0	0
$973$	24.0302	0.770373
$974$	−30.5158	−0.977790
$975$	0	0
$976$	−36.8978	−1.18107
$977$	46.2192	1.47868	0.739342	−	0.673330i	$-0.235137\pi$
$977$	46.2192	1.47868	0.739342	+	0.673330i	$0.235137\pi$
$978$	0	0
$979$	53.8635	1.72149
$980$	0	0
$981$	0	0
$982$	−9.71740	−0.310095
$983$	−25.0422	−0.798722	−0.399361	−	0.916794i	$-0.630768\pi$
$983$	−25.0422	−0.798722	−0.399361	+	0.916794i	$0.630768\pi$
$984$	0	0
$985$	0	0
$986$	−6.21249	−0.197846
$987$	0	0
$988$	1.70050	0.0541002
$989$	93.7450	2.98092
$990$	0	0
$991$	6.37368	0.202467	0.101233	−	0.994863i	$-0.467721\pi$
$991$	6.37368	0.202467	0.101233	+	0.994863i	$0.467721\pi$
$992$	−12.3562	−0.392309
$993$	0	0
$994$	40.0475	1.27023
$995$	0	0
$996$	0	0
$997$	48.5144	1.53647	0.768233	−	0.640171i	$-0.221136\pi$
$997$	48.5144	1.53647	0.768233	+	0.640171i	$0.221136\pi$
$998$	5.52220	0.174802
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3825.2.a.bt.1.5		8
3.2	odd	2		3825.2.a.bs.1.3		8
5.2	odd	4		765.2.b.e.154.11	yes	16
5.3	odd	4		765.2.b.e.154.5	✓	16
5.4	even	2		3825.2.a.bs.1.4		8
15.2	even	4		765.2.b.e.154.6	yes	16
15.8	even	4		765.2.b.e.154.12	yes	16
15.14	odd	2	inner	3825.2.a.bt.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
765.2.b.e.154.5	✓	16	5.3	odd	4
765.2.b.e.154.6	yes	16	15.2	even	4
765.2.b.e.154.11	yes	16	5.2	odd	4
765.2.b.e.154.12	yes	16	15.8	even	4
3825.2.a.bs.1.3		8	3.2	odd	2
3825.2.a.bs.1.4		8	5.4	even	2
3825.2.a.bt.1.5		8	1.1	even	1	trivial
3825.2.a.bt.1.6		8	15.14	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3825 = 3^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3825.a (trivial)

\(f(q)\)	\(=\)	\(q+1.32973 q^{2} -0.231826 q^{4} -3.42890 q^{7} -2.96772 q^{8} -4.87695 q^{11} -1.44805 q^{13} -4.55950 q^{14} -3.48261 q^{16} -1.00000 q^{17} +5.06562 q^{19} -6.48501 q^{22} +7.50407 q^{23} -1.92551 q^{26} +0.794907 q^{28} +4.67201 q^{29} -9.47179 q^{31} +1.30452 q^{32} -1.32973 q^{34} -0.485294 q^{37} +6.73589 q^{38} +0.962754 q^{41} +12.4926 q^{43} +1.13060 q^{44} +9.97836 q^{46} +2.43845 q^{47} +4.75736 q^{49} +0.335695 q^{52} -0.221007 q^{53} +10.1760 q^{56} +6.21249 q^{58} -5.15730 q^{59} +10.5949 q^{61} -12.5949 q^{62} +8.69987 q^{64} +12.0151 q^{67} +0.231826 q^{68} -8.78331 q^{71} -8.29802 q^{73} -0.645308 q^{74} -1.17434 q^{76} +16.7226 q^{77} +9.25435 q^{79} +1.28020 q^{82} +8.92730 q^{83} +16.6117 q^{86} +14.4734 q^{88} -11.0445 q^{89} +4.96521 q^{91} -1.73964 q^{92} +3.24247 q^{94} -10.9970 q^{97} +6.32599 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 8 q^{4} + 18 q^{8} + 16 q^{16} - 8 q^{17} + 2 q^{19} + 38 q^{23} - 12 q^{31} + 40 q^{32} - 4 q^{34} + 22 q^{38} + 40 q^{46} + 36 q^{47} + 28 q^{49} + 28 q^{53} - 12 q^{61} - 4 q^{62} + 54 q^{64}+ \cdots + 102 q^{98}+O(q^{100}) \) 8 * q + 4 * q^2 + 8 * q^4 + 18 * q^8 + 16 * q^16 - 8 * q^17 + 2 * q^19 + 38 * q^23 - 12 * q^31 + 40 * q^32 - 4 * q^34 + 22 * q^38 + 40 * q^46 + 36 * q^47 + 28 * q^49 + 28 * q^53 - 12 * q^61 - 4 * q^62 + 54 * q^64 - 8 * q^68 + 6 * q^76 + 36 * q^77 - 36 * q^79 + 32 * q^83 - 48 * q^91 + 76 * q^92 + 18 * q^94 + 102 * q^98

Introduction

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