LMFDB - Embedded newform 3825.2.a.bt.1.7

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			1.7
Root			$-1.38157$ 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)

$f(q)$	$=$	$q+2.69353 q^{2} +5.25508 q^{4} -4.61905 q^{7} +8.76763 q^{8} +1.48856 q^{11} +6.10761 q^{13} -12.4415 q^{14} +13.1057 q^{16} -1.00000 q^{17} +0.180969 q^{19} +4.00947 q^{22} +6.74252 q^{23} +16.4510 q^{26} -24.2735 q^{28} +1.31185 q^{29} +3.02511 q^{31} +17.7652 q^{32} -2.69353 q^{34} -2.11789 q^{37} +0.487443 q^{38} -8.22550 q^{41} +2.32446 q^{43} +7.82248 q^{44} +18.1612 q^{46} +6.56155 q^{47} +14.3357 q^{49} +32.0960 q^{52} +1.17450 q^{53} -40.4982 q^{56} +3.53351 q^{58} -3.42975 q^{59} -10.1482 q^{61} +8.14822 q^{62} +21.6397 q^{64} +12.6679 q^{67} -5.25508 q^{68} +7.21290 q^{71} +9.33079 q^{73} -5.70459 q^{74} +0.951004 q^{76} -6.87572 q^{77} -8.76117 q^{79} -22.1556 q^{82} -13.0203 q^{83} +6.26099 q^{86} +13.0511 q^{88} -8.43207 q^{89} -28.2114 q^{91} +35.4325 q^{92} +17.6737 q^{94} +6.28431 q^{97} +38.6134 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.69353	1.90461	0.952305	−	0.305148i	$-0.0987059\pi$
$2$	2.69353	1.90461	0.952305	+	0.305148i	$0.0987059\pi$
$3$	0	0
$3$	0	0
$4$	5.25508	2.62754
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.61905	−1.74584	−0.872919	−	0.487865i	$-0.837776\pi$
$7$	−4.61905	−1.74584	−0.872919	+	0.487865i	$0.837776\pi$
$8$	8.76763	3.09983
$9$	0	0
$10$	0	0
$11$	1.48856	0.448817	0.224408	−	0.974495i	$-0.427955\pi$
$11$	1.48856	0.448817	0.224408	+	0.974495i	$0.427955\pi$
$12$	0	0
$13$	6.10761	1.69395	0.846973	−	0.531636i	$-0.178422\pi$
$13$	6.10761	1.69395	0.846973	+	0.531636i	$0.178422\pi$
$14$	−12.4415	−3.32514
$15$	0	0
$16$	13.1057	3.27642
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	0.180969	0.0415170	0.0207585	−	0.999785i	$-0.493392\pi$
$19$	0.180969	0.0415170	0.0207585	+	0.999785i	$0.493392\pi$
$20$	0	0
$21$	0	0
$22$	4.00947	0.854821
$23$	6.74252	1.40591	0.702956	−	0.711233i	$-0.251863\pi$
$23$	6.74252	1.40591	0.702956	+	0.711233i	$0.251863\pi$
$24$	0	0
$25$	0	0
$26$	16.4510	3.22631
$27$	0	0
$28$	−24.2735	−4.58726
$29$	1.31185	0.243605	0.121803	−	0.992554i	$-0.461132\pi$
$29$	1.31185	0.243605	0.121803	+	0.992554i	$0.461132\pi$
$30$	0	0
$31$	3.02511	0.543326	0.271663	−	0.962392i	$-0.412426\pi$
$31$	3.02511	0.543326	0.271663	+	0.962392i	$0.412426\pi$
$32$	17.7652	3.14048
$33$	0	0
$34$	−2.69353	−0.461936
$35$	0	0
$36$	0	0
$37$	−2.11789	−0.348179	−0.174090	−	0.984730i	$-0.555698\pi$
$37$	−2.11789	−0.348179	−0.174090	+	0.984730i	$0.555698\pi$
$38$	0.487443	0.0790738
$39$	0	0
$40$	0	0
$41$	−8.22550	−1.28461	−0.642304	−	0.766450i	$-0.722021\pi$
$41$	−8.22550	−1.28461	−0.642304	+	0.766450i	$0.722021\pi$
$42$	0	0
$43$	2.32446	0.354477	0.177238	−	0.984168i	$-0.443284\pi$
$43$	2.32446	0.354477	0.177238	+	0.984168i	$0.443284\pi$
$44$	7.82248	1.17928
$45$	0	0
$46$	18.1612	2.67772
$47$	6.56155	0.957101	0.478550	−	0.878060i	$-0.341162\pi$
$47$	6.56155	0.957101	0.478550	+	0.878060i	$0.341162\pi$
$48$	0	0
$49$	14.3357	2.04795
$50$	0	0
$51$	0	0
$52$	32.0960	4.45091
$53$	1.17450	0.161330	0.0806652	−	0.996741i	$-0.474296\pi$
$53$	1.17450	0.161330	0.0806652	+	0.996741i	$0.474296\pi$
$54$	0	0
$55$	0	0
$56$	−40.4982	−5.41180
$57$	0	0
$58$	3.53351	0.463973
$59$	−3.42975	−0.446515	−0.223257	−	0.974760i	$-0.571669\pi$
$59$	−3.42975	−0.446515	−0.223257	+	0.974760i	$0.571669\pi$
$60$	0	0
$61$	−10.1482	−1.29935	−0.649673	−	0.760214i	$-0.725094\pi$
$61$	−10.1482	−1.29935	−0.649673	+	0.760214i	$0.725094\pi$
$62$	8.14822	1.03482
$63$	0	0
$64$	21.6397	2.70497
$65$	0	0
$66$	0	0
$67$	12.6679	1.54762	0.773812	−	0.633415i	$-0.218347\pi$
$67$	12.6679	1.54762	0.773812	+	0.633415i	$0.218347\pi$
$68$	−5.25508	−0.637272
$69$	0	0
$70$	0	0
$71$	7.21290	0.856013	0.428007	−	0.903776i	$-0.359216\pi$
$71$	7.21290	0.856013	0.428007	+	0.903776i	$0.359216\pi$
$72$	0	0
$73$	9.33079	1.09209	0.546043	−	0.837757i	$-0.316134\pi$
$73$	9.33079	1.09209	0.546043	+	0.837757i	$0.316134\pi$
$74$	−5.70459	−0.663145
$75$	0	0
$76$	0.951004	0.109088
$77$	−6.87572	−0.783561
$78$	0	0
$79$	−8.76117	−0.985708	−0.492854	−	0.870112i	$-0.664046\pi$
$79$	−8.76117	−0.985708	−0.492854	+	0.870112i	$0.664046\pi$
$80$	0	0
$81$	0	0
$82$	−22.1556	−2.44668
$83$	−13.0203	−1.42916	−0.714582	−	0.699551i	$-0.753383\pi$
$83$	−13.0203	−1.42916	−0.714582	+	0.699551i	$0.753383\pi$
$84$	0	0
$85$	0	0
$86$	6.26099	0.675140
$87$	0	0
$88$	13.0511	1.39125
$89$	−8.43207	−0.893798	−0.446899	−	0.894584i	$-0.647472\pi$
$89$	−8.43207	−0.893798	−0.446899	+	0.894584i	$0.647472\pi$
$90$	0	0
$91$	−28.2114	−2.95736
$92$	35.4325	3.69409
$93$	0	0
$94$	17.6737	1.82290
$95$	0	0
$96$	0	0
$97$	6.28431	0.638075	0.319038	−	0.947742i	$-0.396640\pi$
$97$	6.28431	0.638075	0.319038	+	0.947742i	$0.396640\pi$
$98$	38.6134	3.90055
$99$	0	0
$100$	0	0
$101$	4.23578	0.421476	0.210738	−	0.977543i	$-0.432413\pi$
$101$	4.23578	0.421476	0.210738	+	0.977543i	$0.432413\pi$
$102$	0	0
$103$	6.91365	0.681222	0.340611	−	0.940204i	$-0.389366\pi$
$103$	6.91365	0.681222	0.340611	+	0.940204i	$0.389366\pi$
$104$	53.5493	5.25094
$105$	0	0
$106$	3.16355	0.307271
$107$	−0.116639	−0.0112760	−0.00563798	−	0.999984i	$-0.501795\pi$
$107$	−0.116639	−0.0112760	−0.00563798	+	0.999984i	$0.501795\pi$
$108$	0	0
$109$	−6.95715	−0.666374	−0.333187	−	0.942861i	$-0.608124\pi$
$109$	−6.95715	−0.666374	−0.333187	+	0.942861i	$0.608124\pi$
$110$	0	0
$111$	0	0
$112$	−60.5359	−5.72010
$113$	8.90847	0.838039	0.419019	−	0.907977i	$-0.362374\pi$
$113$	8.90847	0.838039	0.419019	+	0.907977i	$0.362374\pi$
$114$	0	0
$115$	0	0
$116$	6.89390	0.640083
$117$	0	0
$118$	−9.23811	−0.850437
$119$	4.61905	0.423428
$120$	0	0
$121$	−8.78420	−0.798563
$122$	−27.3345	−2.47475
$123$	0	0
$124$	15.8972	1.42761
$125$	0	0
$126$	0	0
$127$	−10.3434	−0.917828	−0.458914	−	0.888481i	$-0.651761\pi$
$127$	−10.3434	−0.917828	−0.458914	+	0.888481i	$0.651761\pi$
$128$	22.7567	2.01143
$129$	0	0
$130$	0	0
$131$	14.5397	1.27034	0.635169	−	0.772373i	$-0.280930\pi$
$131$	14.5397	1.27034	0.635169	+	0.772373i	$0.280930\pi$
$132$	0	0
$133$	−0.835903	−0.0724820
$134$	34.1212	2.94762
$135$	0	0
$136$	−8.76763	−0.751818
$137$	11.3907	0.973172	0.486586	−	0.873633i	$-0.338242\pi$
$137$	11.3907	0.973172	0.486586	+	0.873633i	$0.338242\pi$
$138$	0	0
$139$	−5.48504	−0.465235	−0.232618	−	0.972568i	$-0.574729\pi$
$139$	−5.48504	−0.465235	−0.232618	+	0.972568i	$0.574729\pi$
$140$	0	0
$141$	0	0
$142$	19.4281	1.63037
$143$	9.09153	0.760272
$144$	0	0
$145$	0	0
$146$	25.1327	2.08000
$147$	0	0
$148$	−11.1297	−0.914854
$149$	−20.2342	−1.65765	−0.828823	−	0.559511i	$-0.810989\pi$
$149$	−20.2342	−1.65765	−0.828823	+	0.559511i	$0.810989\pi$
$150$	0	0
$151$	7.23766	0.588993	0.294496	−	0.955653i	$-0.404848\pi$
$151$	7.23766	0.588993	0.294496	+	0.955653i	$0.404848\pi$
$152$	1.58667	0.128696
$153$	0	0
$154$	−18.5199	−1.49238
$155$	0	0
$156$	0	0
$157$	−7.51215	−0.599534	−0.299767	−	0.954012i	$-0.596909\pi$
$157$	−7.51215	−0.599534	−0.299767	+	0.954012i	$0.596909\pi$
$158$	−23.5984	−1.87739
$159$	0	0
$160$	0	0
$161$	−31.1441	−2.45450
$162$	0	0
$163$	−6.64426	−0.520419	−0.260209	−	0.965552i	$-0.583792\pi$
$163$	−6.64426	−0.520419	−0.260209	+	0.965552i	$0.583792\pi$
$164$	−43.2257	−3.37536
$165$	0	0
$166$	−35.0705	−2.72200
$167$	−3.61461	−0.279707	−0.139854	−	0.990172i	$-0.544663\pi$
$167$	−3.61461	−0.279707	−0.139854	+	0.990172i	$0.544663\pi$
$168$	0	0
$169$	24.3029	1.86945
$170$	0	0
$171$	0	0
$172$	12.2152	0.931402
$173$	−7.35067	−0.558861	−0.279431	−	0.960166i	$-0.590146\pi$
$173$	−7.35067	−0.558861	−0.279431	+	0.960166i	$0.590146\pi$
$174$	0	0
$175$	0	0
$176$	19.5086	1.47051
$177$	0	0
$178$	−22.7120	−1.70234
$179$	−5.18768	−0.387746	−0.193873	−	0.981027i	$-0.562105\pi$
$179$	−5.18768	−0.387746	−0.193873	+	0.981027i	$0.562105\pi$
$180$	0	0
$181$	0.693208	0.0515258	0.0257629	−	0.999668i	$-0.491799\pi$
$181$	0.693208	0.0515258	0.0257629	+	0.999668i	$0.491799\pi$
$182$	−75.9881	−5.63261
$183$	0	0
$184$	59.1160	4.35809
$185$	0	0
$186$	0	0
$187$	−1.48856	−0.108854
$188$	34.4815	2.51482
$189$	0	0
$190$	0	0
$191$	−3.42975	−0.248168	−0.124084	−	0.992272i	$-0.539599\pi$
$191$	−3.42975	−0.248168	−0.124084	+	0.992272i	$0.539599\pi$
$192$	0	0
$193$	−5.33241	−0.383835	−0.191918	−	0.981411i	$-0.561471\pi$
$193$	−5.33241	−0.383835	−0.191918	+	0.981411i	$0.561471\pi$
$194$	16.9270	1.21528
$195$	0	0
$196$	75.3350	5.38107
$197$	−14.0138	−0.998445	−0.499223	−	0.866474i	$-0.666381\pi$
$197$	−14.0138	−0.998445	−0.499223	+	0.866474i	$0.666381\pi$
$198$	0	0
$199$	−20.0932	−1.42437	−0.712184	−	0.701993i	$-0.752294\pi$
$199$	−20.0932	−1.42437	−0.712184	+	0.701993i	$0.752294\pi$
$200$	0	0
$201$	0	0
$202$	11.4092	0.802747
$203$	−6.05953	−0.425295
$204$	0	0
$205$	0	0
$206$	18.6221	1.29746
$207$	0	0
$208$	80.0444	5.55008
$209$	0.269382	0.0186335
$210$	0	0
$211$	4.39923	0.302856	0.151428	−	0.988468i	$-0.451613\pi$
$211$	4.39923	0.302856	0.151428	+	0.988468i	$0.451613\pi$
$212$	6.17210	0.423902
$213$	0	0
$214$	−0.314171	−0.0214763
$215$	0	0
$216$	0	0
$217$	−13.9732	−0.948560
$218$	−18.7393	−1.26918
$219$	0	0
$220$	0	0
$221$	−6.10761	−0.410842
$222$	0	0
$223$	−22.7045	−1.52040	−0.760202	−	0.649687i	$-0.774900\pi$
$223$	−22.7045	−1.52040	−0.760202	+	0.649687i	$0.774900\pi$
$224$	−82.0586	−5.48277
$225$	0	0
$226$	23.9952	1.59614
$227$	−20.6900	−1.37324	−0.686620	−	0.727016i	$-0.740906\pi$
$227$	−20.6900	−1.37324	−0.686620	+	0.727016i	$0.740906\pi$
$228$	0	0
$229$	−22.7050	−1.50039	−0.750193	−	0.661219i	$-0.770040\pi$
$229$	−22.7050	−1.50039	−0.750193	+	0.661219i	$0.770040\pi$
$230$	0	0
$231$	0	0
$232$	11.5019	0.755134
$233$	−12.0818	−0.791505	−0.395753	−	0.918357i	$-0.629516\pi$
$233$	−12.0818	−0.791505	−0.395753	+	0.918357i	$0.629516\pi$
$234$	0	0
$235$	0	0
$236$	−18.0236	−1.17324
$237$	0	0
$238$	12.4415	0.806465
$239$	24.1165	1.55997	0.779984	−	0.625799i	$-0.215227\pi$
$239$	24.1165	1.55997	0.779984	+	0.625799i	$0.215227\pi$
$240$	0	0
$241$	−18.5733	−1.19641	−0.598206	−	0.801342i	$-0.704120\pi$
$241$	−18.5733	−1.19641	−0.598206	+	0.801342i	$0.704120\pi$
$242$	−23.6605	−1.52095
$243$	0	0
$244$	−53.3297	−3.41408
$245$	0	0
$246$	0	0
$247$	1.10529	0.0703276
$248$	26.5231	1.68422
$249$	0	0
$250$	0	0
$251$	12.6679	0.799588	0.399794	−	0.916605i	$-0.369082\pi$
$251$	12.6679	0.799588	0.399794	+	0.916605i	$0.369082\pi$
$252$	0	0
$253$	10.0366	0.630997
$254$	−27.8602	−1.74810
$255$	0	0
$256$	18.0162	1.12602
$257$	16.0502	1.00119	0.500593	−	0.865683i	$-0.333115\pi$
$257$	16.0502	1.00119	0.500593	+	0.865683i	$0.333115\pi$
$258$	0	0
$259$	9.78265	0.607864
$260$	0	0
$261$	0	0
$262$	39.1630	2.41950
$263$	0.481294	0.0296779	0.0148389	−	0.999890i	$-0.495276\pi$
$263$	0.481294	0.0296779	0.0148389	+	0.999890i	$0.495276\pi$
$264$	0	0
$265$	0	0
$266$	−2.25153	−0.138050
$267$	0	0
$268$	66.5706	4.06644
$269$	−3.03782	−0.185219	−0.0926095	−	0.995703i	$-0.529521\pi$
$269$	−3.03782	−0.185219	−0.0926095	+	0.995703i	$0.529521\pi$
$270$	0	0
$271$	−15.2527	−0.926534	−0.463267	−	0.886219i	$-0.653323\pi$
$271$	−15.2527	−0.926534	−0.463267	+	0.886219i	$0.653323\pi$
$272$	−13.1057	−0.794649
$273$	0	0
$274$	30.6811	1.85351
$275$	0	0
$276$	0	0
$277$	−19.7582	−1.18716	−0.593578	−	0.804777i	$-0.702285\pi$
$277$	−19.7582	−1.18716	−0.593578	+	0.804777i	$0.702285\pi$
$278$	−14.7741	−0.886092
$279$	0	0
$280$	0	0
$281$	−26.4557	−1.57821	−0.789106	−	0.614257i	$-0.789456\pi$
$281$	−26.4557	−1.57821	−0.789106	+	0.614257i	$0.789456\pi$
$282$	0	0
$283$	1.28854	0.0765955	0.0382977	−	0.999266i	$-0.487806\pi$
$283$	1.28854	0.0765955	0.0382977	+	0.999266i	$0.487806\pi$
$284$	37.9043	2.24921
$285$	0	0
$286$	24.4883	1.44802
$287$	37.9940	2.24272
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	49.0340	2.86950
$293$	−6.86279	−0.400929	−0.200464	−	0.979701i	$-0.564245\pi$
$293$	−6.86279	−0.400929	−0.200464	+	0.979701i	$0.564245\pi$
$294$	0	0
$295$	0	0
$296$	−18.5689	−1.07929
$297$	0	0
$298$	−54.5012	−3.15717
$299$	41.1807	2.38154
$300$	0	0
$301$	−10.7368	−0.618859
$302$	19.4948	1.12180
$303$	0	0
$304$	2.37172	0.136027
$305$	0	0
$306$	0	0
$307$	−19.0747	−1.08865	−0.544326	−	0.838874i	$-0.683214\pi$
$307$	−19.0747	−1.08865	−0.544326	+	0.838874i	$0.683214\pi$
$308$	−36.1325	−2.05884
$309$	0	0
$310$	0	0
$311$	16.6884	0.946313	0.473156	−	0.880978i	$-0.343115\pi$
$311$	16.6884	0.946313	0.473156	+	0.880978i	$0.343115\pi$
$312$	0	0
$313$	−19.5208	−1.10338	−0.551690	−	0.834049i	$-0.686017\pi$
$313$	−19.5208	−1.10338	−0.551690	+	0.834049i	$0.686017\pi$
$314$	−20.2342	−1.14188
$315$	0	0
$316$	−46.0406	−2.58999
$317$	27.9952	1.57237	0.786184	−	0.617993i	$-0.212054\pi$
$317$	27.9952	1.57237	0.786184	+	0.617993i	$0.212054\pi$
$318$	0	0
$319$	1.95277	0.109334
$320$	0	0
$321$	0	0
$322$	−83.8873	−4.67486
$323$	−0.180969	−0.0100694
$324$	0	0
$325$	0	0
$326$	−17.8965	−0.991195
$327$	0	0
$328$	−72.1182	−3.98206
$329$	−30.3082	−1.67094
$330$	0	0
$331$	0.449715	0.0247185	0.0123593	−	0.999924i	$-0.496066\pi$
$331$	0.449715	0.0247185	0.0123593	+	0.999924i	$0.496066\pi$
$332$	−68.4228	−3.75519
$333$	0	0
$334$	−9.73606	−0.532733
$335$	0	0
$336$	0	0
$337$	13.9797	0.761523	0.380762	−	0.924673i	$-0.375662\pi$
$337$	13.9797	0.761523	0.380762	+	0.924673i	$0.375662\pi$
$338$	65.4605	3.56058
$339$	0	0
$340$	0	0
$341$	4.50305	0.243854
$342$	0	0
$343$	−33.8838	−1.82955
$344$	20.3800	1.09882
$345$	0	0
$346$	−19.7992	−1.06441
$347$	18.8291	1.01080	0.505400	−	0.862885i	$-0.331345\pi$
$347$	18.8291	1.01080	0.505400	+	0.862885i	$0.331345\pi$
$348$	0	0
$349$	1.04013	0.0556769	0.0278384	−	0.999612i	$-0.491138\pi$
$349$	1.04013	0.0556769	0.0278384	+	0.999612i	$0.491138\pi$
$350$	0	0
$351$	0	0
$352$	26.4446	1.40950
$353$	−21.1017	−1.12313	−0.561566	−	0.827432i	$-0.689801\pi$
$353$	−21.1017	−1.12313	−0.561566	+	0.827432i	$0.689801\pi$
$354$	0	0
$355$	0	0
$356$	−44.3112	−2.34849
$357$	0	0
$358$	−13.9732	−0.738504
$359$	−5.80836	−0.306554	−0.153277	−	0.988183i	$-0.548983\pi$
$359$	−5.80836	−0.306554	−0.153277	+	0.988183i	$0.548983\pi$
$360$	0	0
$361$	−18.9673	−0.998276
$362$	1.86717	0.0981365
$363$	0	0
$364$	−148.253	−7.77057
$365$	0	0
$366$	0	0
$367$	15.6845	0.818722	0.409361	−	0.912372i	$-0.365752\pi$
$367$	15.6845	0.818722	0.409361	+	0.912372i	$0.365752\pi$
$368$	88.3654	4.60636
$369$	0	0
$370$	0	0
$371$	−5.42509	−0.281657
$372$	0	0
$373$	28.6662	1.48428	0.742140	−	0.670244i	$-0.233811\pi$
$373$	28.6662	1.48428	0.742140	+	0.670244i	$0.233811\pi$
$374$	−4.00947	−0.207325
$375$	0	0
$376$	57.5293	2.96685
$377$	8.01230	0.412654
$378$	0	0
$379$	14.0803	0.723254	0.361627	−	0.932323i	$-0.382221\pi$
$379$	14.0803	0.723254	0.361627	+	0.932323i	$0.382221\pi$
$380$	0	0
$381$	0	0
$382$	−9.23811	−0.472663
$383$	16.5445	0.845382	0.422691	−	0.906274i	$-0.361086\pi$
$383$	16.5445	0.845382	0.422691	+	0.906274i	$0.361086\pi$
$384$	0	0
$385$	0	0
$386$	−14.3630	−0.731056
$387$	0	0
$388$	33.0246	1.67657
$389$	21.4533	1.08773	0.543863	−	0.839174i	$-0.316961\pi$
$389$	21.4533	1.08773	0.543863	+	0.839174i	$0.316961\pi$
$390$	0	0
$391$	−6.74252	−0.340984
$392$	125.690	6.34829
$393$	0	0
$394$	−37.7466	−1.90165
$395$	0	0
$396$	0	0
$397$	−32.8093	−1.64665	−0.823326	−	0.567568i	$-0.807884\pi$
$397$	−32.8093	−1.64665	−0.823326	+	0.567568i	$0.807884\pi$
$398$	−54.1215	−2.71287
$399$	0	0
$400$	0	0
$401$	−0.246062	−0.0122878	−0.00614388	−	0.999981i	$-0.501956\pi$
$401$	−0.246062	−0.0122878	−0.00614388	+	0.999981i	$0.501956\pi$
$402$	0	0
$403$	18.4762	0.920366
$404$	22.2594	1.10744
$405$	0	0
$406$	−16.3215	−0.810022
$407$	−3.15260	−0.156269
$408$	0	0
$409$	−3.77694	−0.186757	−0.0933787	−	0.995631i	$-0.529767\pi$
$409$	−3.77694	−0.186757	−0.0933787	+	0.995631i	$0.529767\pi$
$410$	0	0
$411$	0	0
$412$	36.3318	1.78994
$413$	15.8422	0.779543
$414$	0	0
$415$	0	0
$416$	108.503	5.31980
$417$	0	0
$418$	0.725587	0.0354896
$419$	18.4069	0.899234	0.449617	−	0.893222i	$-0.351561\pi$
$419$	18.4069	0.899234	0.449617	+	0.893222i	$0.351561\pi$
$420$	0	0
$421$	−14.5252	−0.707914	−0.353957	−	0.935262i	$-0.615164\pi$
$421$	−14.5252	−0.707914	−0.353957	+	0.935262i	$0.615164\pi$
$422$	11.8494	0.576822
$423$	0	0
$424$	10.2976	0.500096
$425$	0	0
$426$	0	0
$427$	46.8752	2.26845
$428$	−0.612950	−0.0296280
$429$	0	0
$430$	0	0
$431$	−37.0684	−1.78552	−0.892762	−	0.450529i	$-0.851235\pi$
$431$	−37.0684	−1.78552	−0.892762	+	0.450529i	$0.851235\pi$
$432$	0	0
$433$	16.7503	0.804966	0.402483	−	0.915428i	$-0.368147\pi$
$433$	16.7503	0.804966	0.402483	+	0.915428i	$0.368147\pi$
$434$	−37.6371	−1.80664
$435$	0	0
$436$	−36.5604	−1.75092
$437$	1.22018	0.0583693
$438$	0	0
$439$	−17.4072	−0.830802	−0.415401	−	0.909638i	$-0.636359\pi$
$439$	−17.4072	−0.830802	−0.415401	+	0.909638i	$0.636359\pi$
$440$	0	0
$441$	0	0
$442$	−16.4510	−0.782494
$443$	33.9402	1.61255	0.806273	−	0.591543i	$-0.201481\pi$
$443$	33.9402	1.61255	0.806273	+	0.591543i	$0.201481\pi$
$444$	0	0
$445$	0	0
$446$	−61.1551	−2.89578
$447$	0	0
$448$	−99.9551	−4.72243
$449$	−18.1461	−0.856369	−0.428184	−	0.903691i	$-0.640847\pi$
$449$	−18.1461	−0.856369	−0.428184	+	0.903691i	$0.640847\pi$
$450$	0	0
$451$	−12.2441	−0.576553
$452$	46.8147	2.20198
$453$	0	0
$454$	−55.7289	−2.61549
$455$	0	0
$456$	0	0
$457$	−17.9694	−0.840574	−0.420287	−	0.907391i	$-0.638071\pi$
$457$	−17.9694	−0.840574	−0.420287	+	0.907391i	$0.638071\pi$
$458$	−61.1564	−2.85765
$459$	0	0
$460$	0	0
$461$	−12.8532	−0.598634	−0.299317	−	0.954154i	$-0.596759\pi$
$461$	−12.8532	−0.598634	−0.299317	+	0.954154i	$0.596759\pi$
$462$	0	0
$463$	−7.39826	−0.343826	−0.171913	−	0.985112i	$-0.554995\pi$
$463$	−7.39826	−0.343826	−0.171913	+	0.985112i	$0.554995\pi$
$464$	17.1928	0.798154
$465$	0	0
$466$	−32.5426	−1.50751
$467$	23.4159	1.08356	0.541780	−	0.840521i	$-0.317751\pi$
$467$	23.4159	1.08356	0.541780	+	0.840521i	$0.317751\pi$
$468$	0	0
$469$	−58.5135	−2.70190
$470$	0	0
$471$	0	0
$472$	−30.0708	−1.38412
$473$	3.46009	0.159095
$474$	0	0
$475$	0	0
$476$	24.2735	1.11257
$477$	0	0
$478$	64.9585	2.97113
$479$	33.6761	1.53870	0.769349	−	0.638828i	$-0.220581\pi$
$479$	33.6761	1.53870	0.769349	+	0.638828i	$0.220581\pi$
$480$	0	0
$481$	−12.9353	−0.589797
$482$	−50.0277	−2.27870
$483$	0	0
$484$	−46.1616	−2.09826
$485$	0	0
$486$	0	0
$487$	33.2554	1.50695	0.753473	−	0.657479i	$-0.228377\pi$
$487$	33.2554	1.50695	0.753473	+	0.657479i	$0.228377\pi$
$488$	−88.9759	−4.02775
$489$	0	0
$490$	0	0
$491$	35.2806	1.59219	0.796096	−	0.605170i	$-0.206895\pi$
$491$	35.2806	1.59219	0.796096	+	0.605170i	$0.206895\pi$
$492$	0	0
$493$	−1.31185	−0.0590830
$494$	2.97711	0.133947
$495$	0	0
$496$	39.6462	1.78017
$497$	−33.3167	−1.49446
$498$	0	0
$499$	−13.7992	−0.617738	−0.308869	−	0.951105i	$-0.599950\pi$
$499$	−13.7992	−0.617738	−0.308869	+	0.951105i	$0.599950\pi$
$500$	0	0
$501$	0	0
$502$	34.1212	1.52290
$503$	8.73772	0.389596	0.194798	−	0.980843i	$-0.437595\pi$
$503$	8.73772	0.389596	0.194798	+	0.980843i	$0.437595\pi$
$504$	0	0
$505$	0	0
$506$	27.0339	1.20180
$507$	0	0
$508$	−54.3553	−2.41163
$509$	−5.64032	−0.250003	−0.125001	−	0.992157i	$-0.539894\pi$
$509$	−5.64032	−0.250003	−0.125001	+	0.992157i	$0.539894\pi$
$510$	0	0
$511$	−43.0994	−1.90661
$512$	3.01385	0.133194
$513$	0	0
$514$	43.2317	1.90687
$515$	0	0
$516$	0	0
$517$	9.76725	0.429563
$518$	26.3498	1.15774
$519$	0	0
$520$	0	0
$521$	24.8619	1.08922	0.544609	−	0.838690i	$-0.316678\pi$
$521$	24.8619	1.08922	0.544609	+	0.838690i	$0.316678\pi$
$522$	0	0
$523$	−21.4533	−0.938089	−0.469044	−	0.883175i	$-0.655402\pi$
$523$	−21.4533	−0.938089	−0.469044	+	0.883175i	$0.655402\pi$
$524$	76.4072	3.33786
$525$	0	0
$526$	1.29638	0.0565247
$527$	−3.02511	−0.131776
$528$	0	0
$529$	22.4616	0.976591
$530$	0	0
$531$	0	0
$532$	−4.39274	−0.190449
$533$	−50.2382	−2.17606
$534$	0	0
$535$	0	0
$536$	111.067	4.79737
$537$	0	0
$538$	−8.18244	−0.352770
$539$	21.3394	0.919155
$540$	0	0
$541$	−8.95779	−0.385125	−0.192563	−	0.981285i	$-0.561680\pi$
$541$	−8.95779	−0.385125	−0.192563	+	0.981285i	$0.561680\pi$
$542$	−41.0835	−1.76469
$543$	0	0
$544$	−17.7652	−0.761678
$545$	0	0
$546$	0	0
$547$	−11.2335	−0.480308	−0.240154	−	0.970735i	$-0.577198\pi$
$547$	−11.2335	−0.480308	−0.240154	+	0.970735i	$0.577198\pi$
$548$	59.8589	2.55705
$549$	0	0
$550$	0	0
$551$	0.237404	0.0101138
$552$	0	0
$553$	40.4683	1.72089
$554$	−53.2192	−2.26107
$555$	0	0
$556$	−28.8243	−1.22242
$557$	38.6187	1.63633	0.818164	−	0.574985i	$-0.194992\pi$
$557$	38.6187	1.63633	0.818164	+	0.574985i	$0.194992\pi$
$558$	0	0
$559$	14.1969	0.600465
$560$	0	0
$561$	0	0
$562$	−71.2590	−3.00588
$563$	−19.3186	−0.814180	−0.407090	−	0.913388i	$-0.633456\pi$
$563$	−19.3186	−0.814180	−0.407090	+	0.913388i	$0.633456\pi$
$564$	0	0
$565$	0	0
$566$	3.47070	0.145884
$567$	0	0
$568$	63.2400	2.65349
$569$	−21.0402	−0.882051	−0.441025	−	0.897495i	$-0.645385\pi$
$569$	−21.0402	−0.882051	−0.441025	+	0.897495i	$0.645385\pi$
$570$	0	0
$571$	15.9522	0.667580	0.333790	−	0.942647i	$-0.391672\pi$
$571$	15.9522	0.667580	0.333790	+	0.942647i	$0.391672\pi$
$572$	47.7767	1.99764
$573$	0	0
$574$	102.338	4.27150
$575$	0	0
$576$	0	0
$577$	−10.0761	−0.419474	−0.209737	−	0.977758i	$-0.567261\pi$
$577$	−10.0761	−0.419474	−0.209737	+	0.977758i	$0.567261\pi$
$578$	2.69353	0.112036
$579$	0	0
$580$	0	0
$581$	60.1415	2.49509
$582$	0	0
$583$	1.74831	0.0724078
$584$	81.8089	3.38528
$585$	0	0
$586$	−18.4851	−0.763613
$587$	29.6248	1.22275	0.611374	−	0.791342i	$-0.290617\pi$
$587$	29.6248	1.22275	0.611374	+	0.791342i	$0.290617\pi$
$588$	0	0
$589$	0.547450	0.0225573
$590$	0	0
$591$	0	0
$592$	−27.7564	−1.14078
$593$	18.6199	0.764628	0.382314	−	0.924032i	$-0.375127\pi$
$593$	18.6199	0.764628	0.382314	+	0.924032i	$0.375127\pi$
$594$	0	0
$595$	0	0
$596$	−106.332	−4.35553
$597$	0	0
$598$	110.921	4.53591
$599$	−44.2120	−1.80645	−0.903226	−	0.429165i	$-0.858808\pi$
$599$	−44.2120	−1.80645	−0.903226	+	0.429165i	$0.858808\pi$
$600$	0	0
$601$	31.0357	1.26597	0.632986	−	0.774163i	$-0.281829\pi$
$601$	31.0357	1.26597	0.632986	+	0.774163i	$0.281829\pi$
$602$	−28.9199	−1.17869
$603$	0	0
$604$	38.0345	1.54760
$605$	0	0
$606$	0	0
$607$	35.1424	1.42639	0.713194	−	0.700967i	$-0.247248\pi$
$607$	35.1424	1.42639	0.713194	+	0.700967i	$0.247248\pi$
$608$	3.21495	0.130383
$609$	0	0
$610$	0	0
$611$	40.0754	1.62128
$612$	0	0
$613$	45.8296	1.85104	0.925520	−	0.378698i	$-0.123628\pi$
$613$	45.8296	1.85104	0.925520	+	0.378698i	$0.123628\pi$
$614$	−51.3782	−2.07346
$615$	0	0
$616$	−60.2838	−2.42891
$617$	−19.3645	−0.779586	−0.389793	−	0.920902i	$-0.627453\pi$
$617$	−19.3645	−0.779586	−0.389793	+	0.920902i	$0.627453\pi$
$618$	0	0
$619$	−24.1232	−0.969594	−0.484797	−	0.874627i	$-0.661107\pi$
$619$	−24.1232	−0.969594	−0.484797	+	0.874627i	$0.661107\pi$
$620$	0	0
$621$	0	0
$622$	44.9506	1.80236
$623$	38.9482	1.56043
$624$	0	0
$625$	0	0
$626$	−52.5798	−2.10151
$627$	0	0
$628$	−39.4769	−1.57530
$629$	2.11789	0.0844458
$630$	0	0
$631$	42.5276	1.69300	0.846499	−	0.532390i	$-0.178706\pi$
$631$	42.5276	1.69300	0.846499	+	0.532390i	$0.178706\pi$
$632$	−76.8147	−3.05553
$633$	0	0
$634$	75.4058	2.99475
$635$	0	0
$636$	0	0
$637$	87.5566	3.46912
$638$	5.25984	0.208239
$639$	0	0
$640$	0	0
$641$	31.3506	1.23828	0.619138	−	0.785282i	$-0.287482\pi$
$641$	31.3506	1.23828	0.619138	+	0.785282i	$0.287482\pi$
$642$	0	0
$643$	−20.3035	−0.800692	−0.400346	−	0.916364i	$-0.631110\pi$
$643$	−20.3035	−0.800692	−0.400346	+	0.916364i	$0.631110\pi$
$644$	−163.664	−6.44928
$645$	0	0
$646$	−0.487443	−0.0191782
$647$	38.8373	1.52685	0.763425	−	0.645896i	$-0.223516\pi$
$647$	38.8373	1.52685	0.763425	+	0.645896i	$0.223516\pi$
$648$	0	0
$649$	−5.10537	−0.200403
$650$	0	0
$651$	0	0
$652$	−34.9161	−1.36742
$653$	34.5694	1.35281	0.676403	−	0.736532i	$-0.263538\pi$
$653$	34.5694	1.35281	0.676403	+	0.736532i	$0.263538\pi$
$654$	0	0
$655$	0	0
$656$	−107.801	−4.20892
$657$	0	0
$658$	−81.6358	−3.18249
$659$	17.8777	0.696417	0.348208	−	0.937417i	$-0.386790\pi$
$659$	17.8777	0.696417	0.348208	+	0.937417i	$0.386790\pi$
$660$	0	0
$661$	−10.7639	−0.418667	−0.209333	−	0.977844i	$-0.567129\pi$
$661$	−10.7639	−0.418667	−0.209333	+	0.977844i	$0.567129\pi$
$662$	1.21132	0.0470792
$663$	0	0
$664$	−114.157	−4.43016
$665$	0	0
$666$	0	0
$667$	8.84521	0.342488
$668$	−18.9951	−0.734942
$669$	0	0
$670$	0	0
$671$	−15.1062	−0.583168
$672$	0	0
$673$	7.18958	0.277138	0.138569	−	0.990353i	$-0.455750\pi$
$673$	7.18958	0.277138	0.138569	+	0.990353i	$0.455750\pi$
$674$	37.6547	1.45040
$675$	0	0
$676$	127.714	4.91206
$677$	38.0770	1.46342	0.731709	−	0.681617i	$-0.238723\pi$
$677$	38.0770	1.46342	0.731709	+	0.681617i	$0.238723\pi$
$678$	0	0
$679$	−29.0276	−1.11398
$680$	0	0
$681$	0	0
$682$	12.1291	0.464447
$683$	7.36273	0.281727	0.140864	−	0.990029i	$-0.455012\pi$
$683$	7.36273	0.281727	0.140864	+	0.990029i	$0.455012\pi$
$684$	0	0
$685$	0	0
$686$	−91.2668	−3.48458
$687$	0	0
$688$	30.4637	1.16142
$689$	7.17340	0.273285
$690$	0	0
$691$	40.5159	1.54130	0.770650	−	0.637259i	$-0.219932\pi$
$691$	40.5159	1.54130	0.770650	+	0.637259i	$0.219932\pi$
$692$	−38.6283	−1.46843
$693$	0	0
$694$	50.7167	1.92518
$695$	0	0
$696$	0	0
$697$	8.22550	0.311563
$698$	2.80161	0.106043
$699$	0	0
$700$	0	0
$701$	−16.5969	−0.626855	−0.313428	−	0.949612i	$-0.601477\pi$
$701$	−16.5969	−0.626855	−0.313428	+	0.949612i	$0.601477\pi$
$702$	0	0
$703$	−0.383272	−0.0144554
$704$	32.2120	1.21403
$705$	0	0
$706$	−56.8381	−2.13913
$707$	−19.5653	−0.735829
$708$	0	0
$709$	23.7142	0.890604	0.445302	−	0.895381i	$-0.353096\pi$
$709$	23.7142	0.890604	0.445302	+	0.895381i	$0.353096\pi$
$710$	0	0
$711$	0	0
$712$	−73.9293	−2.77062
$713$	20.3969	0.763870
$714$	0	0
$715$	0	0
$716$	−27.2617	−1.01882
$717$	0	0
$718$	−15.6450	−0.583865
$719$	−2.14873	−0.0801339	−0.0400670	−	0.999197i	$-0.512757\pi$
$719$	−2.14873	−0.0801339	−0.0400670	+	0.999197i	$0.512757\pi$
$720$	0	0
$721$	−31.9345	−1.18930
$722$	−51.0888	−1.90133
$723$	0	0
$724$	3.64286	0.135386
$725$	0	0
$726$	0	0
$727$	−21.5131	−0.797875	−0.398938	−	0.916978i	$-0.630621\pi$
$727$	−21.5131	−0.797875	−0.398938	+	0.916978i	$0.630621\pi$
$728$	−247.347	−9.16729
$729$	0	0
$730$	0	0
$731$	−2.32446	−0.0859733
$732$	0	0
$733$	−37.4517	−1.38331	−0.691655	−	0.722228i	$-0.743118\pi$
$733$	−37.4517	−1.38331	−0.691655	+	0.722228i	$0.743118\pi$
$734$	42.2465	1.55935
$735$	0	0
$736$	119.782	4.41524
$737$	18.8568	0.694600
$738$	0	0
$739$	−29.9540	−1.10188	−0.550938	−	0.834546i	$-0.685730\pi$
$739$	−29.9540	−1.10188	−0.550938	+	0.834546i	$0.685730\pi$
$740$	0	0
$741$	0	0
$742$	−14.6126	−0.536446
$743$	4.89399	0.179543	0.0897716	−	0.995962i	$-0.471386\pi$
$743$	4.89399	0.179543	0.0897716	+	0.995962i	$0.471386\pi$
$744$	0	0
$745$	0	0
$746$	77.2132	2.82698
$747$	0	0
$748$	−7.82248	−0.286018
$749$	0.538764	0.0196860
$750$	0	0
$751$	17.7060	0.646102	0.323051	−	0.946382i	$-0.395291\pi$
$751$	17.7060	0.646102	0.323051	+	0.946382i	$0.395291\pi$
$752$	85.9937	3.13587
$753$	0	0
$754$	21.5813	0.785946
$755$	0	0
$756$	0	0
$757$	−49.1135	−1.78506	−0.892530	−	0.450988i	$-0.851072\pi$
$757$	−49.1135	−1.78506	−0.892530	+	0.450988i	$0.851072\pi$
$758$	37.9255	1.37752
$759$	0	0
$760$	0	0
$761$	9.03057	0.327358	0.163679	−	0.986514i	$-0.447664\pi$
$761$	9.03057	0.327358	0.163679	+	0.986514i	$0.447664\pi$
$762$	0	0
$763$	32.1355	1.16338
$764$	−18.0236	−0.652070
$765$	0	0
$766$	44.5629	1.61012
$767$	−20.9476	−0.756372
$768$	0	0
$769$	−46.2896	−1.66924	−0.834622	−	0.550823i	$-0.814314\pi$
$769$	−46.2896	−1.66924	−0.834622	+	0.550823i	$0.814314\pi$
$770$	0	0
$771$	0	0
$772$	−28.0222	−1.00854
$773$	−0.350175	−0.0125949	−0.00629746	−	0.999980i	$-0.502005\pi$
$773$	−0.350175	−0.0125949	−0.00629746	+	0.999980i	$0.502005\pi$
$774$	0	0
$775$	0	0
$776$	55.0986	1.97792
$777$	0	0
$778$	57.7851	2.07170
$779$	−1.48856	−0.0533331
$780$	0	0
$781$	10.7368	0.384193
$782$	−18.1612	−0.649441
$783$	0	0
$784$	187.879	6.70995
$785$	0	0
$786$	0	0
$787$	−18.6914	−0.666278	−0.333139	−	0.942878i	$-0.608108\pi$
$787$	−18.6914	−0.666278	−0.333139	+	0.942878i	$0.608108\pi$
$788$	−73.6439	−2.62345
$789$	0	0
$790$	0	0
$791$	−41.1487	−1.46308
$792$	0	0
$793$	−61.9814	−2.20102
$794$	−88.3727	−3.13623
$795$	0	0
$796$	−105.591	−3.74258
$797$	−32.4669	−1.15004	−0.575018	−	0.818141i	$-0.695005\pi$
$797$	−32.4669	−1.15004	−0.575018	+	0.818141i	$0.695005\pi$
$798$	0	0
$799$	−6.56155	−0.232131
$800$	0	0
$801$	0	0
$802$	−0.662775	−0.0234034
$803$	13.8894	0.490147
$804$	0	0
$805$	0	0
$806$	49.7661	1.75294
$807$	0	0
$808$	37.1378	1.30650
$809$	−7.30347	−0.256776	−0.128388	−	0.991724i	$-0.540980\pi$
$809$	−7.30347	−0.256776	−0.128388	+	0.991724i	$0.540980\pi$
$810$	0	0
$811$	−3.42434	−0.120245	−0.0601225	−	0.998191i	$-0.519149\pi$
$811$	−3.42434	−0.120245	−0.0601225	+	0.998191i	$0.519149\pi$
$812$	−31.8433	−1.11748
$813$	0	0
$814$	−8.49161	−0.297631
$815$	0	0
$816$	0	0
$817$	0.420654	0.0147168
$818$	−10.1733	−0.355700
$819$	0	0
$820$	0	0
$821$	40.2521	1.40481	0.702404	−	0.711778i	$-0.252110\pi$
$821$	40.2521	1.40481	0.702404	+	0.711778i	$0.252110\pi$
$822$	0	0
$823$	−9.80674	−0.341841	−0.170921	−	0.985285i	$-0.554674\pi$
$823$	−9.80674	−0.341841	−0.170921	+	0.985285i	$0.554674\pi$
$824$	60.6163	2.11167
$825$	0	0
$826$	42.6713	1.48472
$827$	4.78549	0.166408	0.0832038	−	0.996533i	$-0.473485\pi$
$827$	4.78549	0.166408	0.0832038	+	0.996533i	$0.473485\pi$
$828$	0	0
$829$	38.5349	1.33837	0.669185	−	0.743096i	$-0.266643\pi$
$829$	38.5349	1.33837	0.669185	+	0.743096i	$0.266643\pi$
$830$	0	0
$831$	0	0
$832$	132.167	4.58207
$833$	−14.3357	−0.496701
$834$	0	0
$835$	0	0
$836$	1.41562	0.0489604
$837$	0	0
$838$	49.5793	1.71269
$839$	3.20700	0.110718	0.0553590	−	0.998467i	$-0.482370\pi$
$839$	3.20700	0.110718	0.0553590	+	0.998467i	$0.482370\pi$
$840$	0	0
$841$	−27.2790	−0.940656
$842$	−39.1239	−1.34830
$843$	0	0
$844$	23.1183	0.795765
$845$	0	0
$846$	0	0
$847$	40.5747	1.39416
$848$	15.3927	0.528586
$849$	0	0
$850$	0	0
$851$	−14.2799	−0.489509
$852$	0	0
$853$	−18.4995	−0.633412	−0.316706	−	0.948524i	$-0.602577\pi$
$853$	−18.4995	−0.633412	−0.316706	+	0.948524i	$0.602577\pi$
$854$	126.259	4.32051
$855$	0	0
$856$	−1.02265	−0.0349535
$857$	−7.84698	−0.268048	−0.134024	−	0.990978i	$-0.542790\pi$
$857$	−7.84698	−0.268048	−0.134024	+	0.990978i	$0.542790\pi$
$858$	0	0
$859$	28.9487	0.987718	0.493859	−	0.869542i	$-0.335586\pi$
$859$	28.9487	0.987718	0.493859	+	0.869542i	$0.335586\pi$
$860$	0	0
$861$	0	0
$862$	−99.8447	−3.40073
$863$	34.1900	1.16384	0.581921	−	0.813245i	$-0.302301\pi$
$863$	34.1900	1.16384	0.581921	+	0.813245i	$0.302301\pi$
$864$	0	0
$865$	0	0
$866$	45.1172	1.53315
$867$	0	0
$868$	−73.4300	−2.49238
$869$	−13.0415	−0.442403
$870$	0	0
$871$	77.3703	2.62159
$872$	−60.9978	−2.06565
$873$	0	0
$874$	3.28660	0.111171
$875$	0	0
$876$	0	0
$877$	2.93107	0.0989753	0.0494877	−	0.998775i	$-0.484241\pi$
$877$	2.93107	0.0989753	0.0494877	+	0.998775i	$0.484241\pi$
$878$	−46.8869	−1.58235
$879$	0	0
$880$	0	0
$881$	−56.2657	−1.89564	−0.947820	−	0.318807i	$-0.896718\pi$
$881$	−56.2657	−1.89564	−0.947820	+	0.318807i	$0.896718\pi$
$882$	0	0
$883$	0.966565	0.0325275	0.0162638	−	0.999868i	$-0.494823\pi$
$883$	0.966565	0.0325275	0.0162638	+	0.999868i	$0.494823\pi$
$884$	−32.0960	−1.07950
$885$	0	0
$886$	91.4187	3.07127
$887$	−41.0673	−1.37890	−0.689452	−	0.724331i	$-0.742149\pi$
$887$	−41.0673	−1.37890	−0.689452	+	0.724331i	$0.742149\pi$
$888$	0	0
$889$	47.7767	1.60238
$890$	0	0
$891$	0	0
$892$	−119.314	−3.99492
$893$	1.18743	0.0397360
$894$	0	0
$895$	0	0
$896$	−105.114	−3.51162
$897$	0	0
$898$	−48.8771	−1.63105
$899$	3.96851	0.132357
$900$	0	0
$901$	−1.17450	−0.0391283
$902$	−32.9799	−1.09811
$903$	0	0
$904$	78.1062	2.59778
$905$	0	0
$906$	0	0
$907$	−43.2302	−1.43543	−0.717717	−	0.696334i	$-0.754813\pi$
$907$	−43.2302	−1.43543	−0.717717	+	0.696334i	$0.754813\pi$
$908$	−108.727	−3.60824
$909$	0	0
$910$	0	0
$911$	44.9471	1.48916	0.744581	−	0.667532i	$-0.232649\pi$
$911$	44.9471	1.48916	0.744581	+	0.667532i	$0.232649\pi$
$912$	0	0
$913$	−19.3815	−0.641433
$914$	−48.4011	−1.60097
$915$	0	0
$916$	−119.316	−3.94232
$917$	−67.1596	−2.21780
$918$	0	0
$919$	1.36957	0.0451781	0.0225890	−	0.999745i	$-0.492809\pi$
$919$	1.36957	0.0451781	0.0225890	+	0.999745i	$0.492809\pi$
$920$	0	0
$921$	0	0
$922$	−34.6205	−1.14016
$923$	44.0536	1.45004
$924$	0	0
$925$	0	0
$926$	−19.9274	−0.654855
$927$	0	0
$928$	23.3054	0.765038
$929$	−2.67132	−0.0876431	−0.0438215	−	0.999039i	$-0.513953\pi$
$929$	−2.67132	−0.0876431	−0.0438215	+	0.999039i	$0.513953\pi$
$930$	0	0
$931$	2.59430	0.0850248
$932$	−63.4908	−2.07971
$933$	0	0
$934$	63.0714	2.06376
$935$	0	0
$936$	0	0
$937$	−26.6015	−0.869034	−0.434517	−	0.900664i	$-0.643081\pi$
$937$	−26.6015	−0.869034	−0.434517	+	0.900664i	$0.643081\pi$
$938$	−157.608	−5.14607
$939$	0	0
$940$	0	0
$941$	24.8072	0.808690	0.404345	−	0.914606i	$-0.367499\pi$
$941$	24.8072	0.808690	0.404345	+	0.914606i	$0.367499\pi$
$942$	0	0
$943$	−55.4606	−1.80605
$944$	−44.9492	−1.46297
$945$	0	0
$946$	9.31984	0.303014
$947$	39.6057	1.28701	0.643506	−	0.765441i	$-0.277479\pi$
$947$	39.6057	1.28701	0.643506	+	0.765441i	$0.277479\pi$
$948$	0	0
$949$	56.9888	1.84994
$950$	0	0
$951$	0	0
$952$	40.4982	1.31255
$953$	−43.2247	−1.40019	−0.700093	−	0.714052i	$-0.746858\pi$
$953$	−43.2247	−1.40019	−0.700093	+	0.714052i	$0.746858\pi$
$954$	0	0
$955$	0	0
$956$	126.734	4.09888
$957$	0	0
$958$	90.7073	2.93062
$959$	−52.6142	−1.69900
$960$	0	0
$961$	−21.8487	−0.704796
$962$	−34.8414	−1.12333
$963$	0	0
$964$	−97.6042	−3.14362
$965$	0	0
$966$	0	0
$967$	0.630476	0.0202748	0.0101374	−	0.999949i	$-0.496773\pi$
$967$	0.630476	0.0202748	0.0101374	+	0.999949i	$0.496773\pi$
$968$	−77.0166	−2.47541
$969$	0	0
$970$	0	0
$971$	−62.1272	−1.99376	−0.996879	−	0.0789458i	$-0.974845\pi$
$971$	−62.1272	−1.99376	−0.996879	+	0.0789458i	$0.974845\pi$
$972$	0	0
$973$	25.3357	0.812225
$974$	89.5743	2.87015
$975$	0	0
$976$	−132.999	−4.25721
$977$	43.0462	1.37717	0.688585	−	0.725156i	$-0.258232\pi$
$977$	43.0462	1.37717	0.688585	+	0.725156i	$0.258232\pi$
$978$	0	0
$979$	−12.5516	−0.401151
$980$	0	0
$981$	0	0
$982$	95.0292	3.03250
$983$	−26.5628	−0.847222	−0.423611	−	0.905844i	$-0.639238\pi$
$983$	−26.5628	−0.847222	−0.423611	+	0.905844i	$0.639238\pi$
$984$	0	0
$985$	0	0
$986$	−3.53351	−0.112530
$987$	0	0
$988$	5.80836	0.184789
$989$	15.6727	0.498364
$990$	0	0
$991$	−35.8067	−1.13744	−0.568719	−	0.822532i	$-0.692561\pi$
$991$	−35.8067	−1.13744	−0.568719	+	0.822532i	$0.692561\pi$
$992$	53.7418	1.70631
$993$	0	0
$994$	−89.7395	−2.84636
$995$	0	0
$996$	0	0
$997$	2.94417	0.0932428	0.0466214	−	0.998913i	$-0.485155\pi$
$997$	2.94417	0.0932428	0.0466214	+	0.998913i	$0.485155\pi$
$998$	−37.1685	−1.17655
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3825.2.a.bt.1.7		8
3.2	odd	2		3825.2.a.bs.1.1		8
5.2	odd	4		765.2.b.e.154.16	yes	16
5.3	odd	4		765.2.b.e.154.2	yes	16
5.4	even	2		3825.2.a.bs.1.2		8
15.2	even	4		765.2.b.e.154.1	✓	16
15.8	even	4		765.2.b.e.154.15	yes	16
15.14	odd	2	inner	3825.2.a.bt.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
765.2.b.e.154.1	✓	16	15.2	even	4
765.2.b.e.154.2	yes	16	5.3	odd	4
765.2.b.e.154.15	yes	16	15.8	even	4
765.2.b.e.154.16	yes	16	5.2	odd	4
3825.2.a.bs.1.1		8	3.2	odd	2
3825.2.a.bs.1.2		8	5.4	even	2
3825.2.a.bt.1.7		8	1.1	even	1	trivial
3825.2.a.bt.1.8		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)

\(f(q)\)	\(=\)	\(q+2.69353 q^{2} +5.25508 q^{4} -4.61905 q^{7} +8.76763 q^{8} +1.48856 q^{11} +6.10761 q^{13} -12.4415 q^{14} +13.1057 q^{16} -1.00000 q^{17} +0.180969 q^{19} +4.00947 q^{22} +6.74252 q^{23} +16.4510 q^{26} -24.2735 q^{28} +1.31185 q^{29} +3.02511 q^{31} +17.7652 q^{32} -2.69353 q^{34} -2.11789 q^{37} +0.487443 q^{38} -8.22550 q^{41} +2.32446 q^{43} +7.82248 q^{44} +18.1612 q^{46} +6.56155 q^{47} +14.3357 q^{49} +32.0960 q^{52} +1.17450 q^{53} -40.4982 q^{56} +3.53351 q^{58} -3.42975 q^{59} -10.1482 q^{61} +8.14822 q^{62} +21.6397 q^{64} +12.6679 q^{67} -5.25508 q^{68} +7.21290 q^{71} +9.33079 q^{73} -5.70459 q^{74} +0.951004 q^{76} -6.87572 q^{77} -8.76117 q^{79} -22.1556 q^{82} -13.0203 q^{83} +6.26099 q^{86} +13.0511 q^{88} -8.43207 q^{89} -28.2114 q^{91} +35.4325 q^{92} +17.6737 q^{94} +6.28431 q^{97} +38.6134 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 8 q^{4} + 18 q^{8} + 16 q^{16} - 8 q^{17} + 2 q^{19} + 38 q^{23} - 12 q^{31} + 40 q^{32} - 4 q^{34} + 22 q^{38} + 40 q^{46} + 36 q^{47} + 28 q^{49} + 28 q^{53} - 12 q^{61} - 4 q^{62} + 54 q^{64}+ \cdots + 102 q^{98}+O(q^{100}) \) 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

L-functions

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