LMFDB - Embedded newform 2475.2.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.90321 q^{2} +1.62222 q^{4} +4.42864 q^{7} -0.719004 q^{8} -1.00000 q^{11} +0.622216 q^{13} +8.42864 q^{14} -4.61285 q^{16} -5.18421 q^{17} +7.05086 q^{19} -1.90321 q^{22} +8.85728 q^{23} +1.18421 q^{26} +7.18421 q^{28} +7.80642 q^{29} +2.75557 q^{31} -7.34122 q^{32} -9.86665 q^{34} +2.00000 q^{37} +13.4193 q^{38} +0.193576 q^{41} -5.67307 q^{43} -1.62222 q^{44} +16.8573 q^{46} -2.75557 q^{47} +12.6128 q^{49} +1.00937 q^{52} -10.8573 q^{53} -3.18421 q^{56} +14.8573 q^{58} +4.85728 q^{59} +6.85728 q^{61} +5.24443 q^{62} -4.74620 q^{64} +1.24443 q^{67} -8.40990 q^{68} -2.75557 q^{71} -4.23506 q^{73} +3.80642 q^{74} +11.4380 q^{76} -4.42864 q^{77} +8.56199 q^{79} +0.368416 q^{82} +0.133353 q^{83} -10.7971 q^{86} +0.719004 q^{88} -5.61285 q^{89} +2.75557 q^{91} +14.3684 q^{92} -5.24443 q^{94} -7.24443 q^{97} +24.0049 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 5 q^{4} - 9 q^{8} - 3 q^{11} + 2 q^{13} + 12 q^{14} + 13 q^{16} - 2 q^{17} + 8 q^{19} + q^{22} - 10 q^{26} + 8 q^{28} + 10 q^{29} + 8 q^{31} - 29 q^{32} - 30 q^{34} + 6 q^{37} + 14 q^{41}+ \cdots + 39 q^{98}+O(q^{100}) $ 3 * q - q^2 + 5 * q^4 - 9 * q^8 - 3 * q^11 + 2 * q^13 + 12 * q^14 + 13 * q^16 - 2 * q^17 + 8 * q^19 + q^22 - 10 * q^26 + 8 * q^28 + 10 * q^29 + 8 * q^31 - 29 * q^32 - 30 * q^34 + 6 * q^37 + 14 * q^41 - 4 * q^43 - 5 * q^44 + 24 * q^46 - 8 * q^47 + 11 * q^49 + 30 * q^52 - 6 * q^53 + 4 * q^56 + 18 * q^58 - 12 * q^59 - 6 * q^61 + 16 * q^62 + 13 * q^64 + 4 * q^67 + 42 * q^68 - 8 * q^71 + 14 * q^73 - 2 * q^74 + 48 * q^76 + 12 * q^79 - 26 * q^82 + 8 * q^86 + 9 * q^88 + 10 * q^89 + 8 * q^91 + 16 * q^92 - 16 * q^94 - 22 * q^97 + 39 * 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 $

$2$	1.90321	1.34577	0.672887	−	0.739745i	$-0.265054\pi$
$2$	1.90321	1.34577	0.672887	+	0.739745i	$0.265054\pi$
$3$	0	0
$3$	0	0
$4$	1.62222	0.811108
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.42864	1.67387	0.836934	−	0.547304i	$-0.184346\pi$
$7$	4.42864	1.67387	0.836934	+	0.547304i	$0.184346\pi$
$8$	−0.719004	−0.254206
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.622216	0.172572	0.0862858	−	0.996270i	$-0.472500\pi$
$13$	0.622216	0.172572	0.0862858	+	0.996270i	$0.472500\pi$
$14$	8.42864	2.25265
$15$	0	0
$16$	−4.61285	−1.15321
$17$	−5.18421	−1.25736	−0.628678	−	0.777666i	$-0.716403\pi$
$17$	−5.18421	−1.25736	−0.628678	+	0.777666i	$0.716403\pi$
$18$	0	0
$19$	7.05086	1.61758	0.808789	−	0.588100i	$-0.200124\pi$
$19$	7.05086	1.61758	0.808789	+	0.588100i	$0.200124\pi$
$20$	0	0
$21$	0	0
$22$	−1.90321	−0.405766
$23$	8.85728	1.84687	0.923435	−	0.383754i	$-0.125369\pi$
$23$	8.85728	1.84687	0.923435	+	0.383754i	$0.125369\pi$
$24$	0	0
$25$	0	0
$26$	1.18421	0.232242
$27$	0	0
$28$	7.18421	1.35769
$29$	7.80642	1.44962	0.724808	−	0.688951i	$-0.241928\pi$
$29$	7.80642	1.44962	0.724808	+	0.688951i	$0.241928\pi$
$30$	0	0
$31$	2.75557	0.494915	0.247457	−	0.968899i	$-0.420405\pi$
$31$	2.75557	0.494915	0.247457	+	0.968899i	$0.420405\pi$
$32$	−7.34122	−1.29776
$33$	0	0
$34$	−9.86665	−1.69212
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	13.4193	2.17689
$39$	0	0
$40$	0	0
$41$	0.193576	0.0302315	0.0151158	−	0.999886i	$-0.495188\pi$
$41$	0.193576	0.0302315	0.0151158	+	0.999886i	$0.495188\pi$
$42$	0	0
$43$	−5.67307	−0.865135	−0.432568	−	0.901602i	$-0.642392\pi$
$43$	−5.67307	−0.865135	−0.432568	+	0.901602i	$0.642392\pi$
$44$	−1.62222	−0.244558
$45$	0	0
$46$	16.8573	2.48547
$47$	−2.75557	−0.401941	−0.200971	−	0.979597i	$-0.564410\pi$
$47$	−2.75557	−0.401941	−0.200971	+	0.979597i	$0.564410\pi$
$48$	0	0
$49$	12.6128	1.80184
$50$	0	0
$51$	0	0
$52$	1.00937	0.139974
$53$	−10.8573	−1.49136	−0.745681	−	0.666303i	$-0.767876\pi$
$53$	−10.8573	−1.49136	−0.745681	+	0.666303i	$0.767876\pi$
$54$	0	0
$55$	0	0
$56$	−3.18421	−0.425508
$57$	0	0
$58$	14.8573	1.95086
$59$	4.85728	0.632364	0.316182	−	0.948699i	$-0.397599\pi$
$59$	4.85728	0.632364	0.316182	+	0.948699i	$0.397599\pi$
$60$	0	0
$61$	6.85728	0.877985	0.438992	−	0.898491i	$-0.355336\pi$
$61$	6.85728	0.877985	0.438992	+	0.898491i	$0.355336\pi$
$62$	5.24443	0.666043
$63$	0	0
$64$	−4.74620	−0.593275
$65$	0	0
$66$	0	0
$67$	1.24443	0.152031	0.0760157	−	0.997107i	$-0.475780\pi$
$67$	1.24443	0.152031	0.0760157	+	0.997107i	$0.475780\pi$
$68$	−8.40990	−1.01985
$69$	0	0
$70$	0	0
$71$	−2.75557	−0.327026	−0.163513	−	0.986541i	$-0.552283\pi$
$71$	−2.75557	−0.327026	−0.163513	+	0.986541i	$0.552283\pi$
$72$	0	0
$73$	−4.23506	−0.495677	−0.247838	−	0.968801i	$-0.579720\pi$
$73$	−4.23506	−0.495677	−0.247838	+	0.968801i	$0.579720\pi$
$74$	3.80642	0.442488
$75$	0	0
$76$	11.4380	1.31203
$77$	−4.42864	−0.504690
$78$	0	0
$79$	8.56199	0.963299	0.481650	−	0.876364i	$-0.340038\pi$
$79$	8.56199	0.963299	0.481650	+	0.876364i	$0.340038\pi$
$80$	0	0
$81$	0	0
$82$	0.368416	0.0406848
$83$	0.133353	0.0146374	0.00731870	−	0.999973i	$-0.497670\pi$
$83$	0.133353	0.0146374	0.00731870	+	0.999973i	$0.497670\pi$
$84$	0	0
$85$	0	0
$86$	−10.7971	−1.16428
$87$	0	0
$88$	0.719004	0.0766461
$89$	−5.61285	−0.594961	−0.297480	−	0.954728i	$-0.596146\pi$
$89$	−5.61285	−0.594961	−0.297480	+	0.954728i	$0.596146\pi$
$90$	0	0
$91$	2.75557	0.288862
$92$	14.3684	1.49801
$93$	0	0
$94$	−5.24443	−0.540922
$95$	0	0
$96$	0	0
$97$	−7.24443	−0.735561	−0.367780	−	0.929913i	$-0.619882\pi$
$97$	−7.24443	−0.735561	−0.367780	+	0.929913i	$0.619882\pi$
$98$	24.0049	2.42486
$99$	0	0
$100$	0	0
$101$	−4.66370	−0.464056	−0.232028	−	0.972709i	$-0.574536\pi$
$101$	−4.66370	−0.464056	−0.232028	+	0.972709i	$0.574536\pi$
$102$	0	0
$103$	11.6128	1.14425	0.572124	−	0.820167i	$-0.306120\pi$
$103$	11.6128	1.14425	0.572124	+	0.820167i	$0.306120\pi$
$104$	−0.447375	−0.0438688
$105$	0	0
$106$	−20.6637	−2.00704
$107$	2.62222	0.253499	0.126750	−	0.991935i	$-0.459546\pi$
$107$	2.62222	0.253499	0.126750	+	0.991935i	$0.459546\pi$
$108$	0	0
$109$	−19.7146	−1.88831	−0.944156	−	0.329499i	$-0.893120\pi$
$109$	−19.7146	−1.88831	−0.944156	+	0.329499i	$0.893120\pi$
$110$	0	0
$111$	0	0
$112$	−20.4286	−1.93032
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	12.6637	1.17580
$117$	0	0
$118$	9.24443	0.851019
$119$	−22.9590	−2.10465
$120$	0	0
$121$	1.00000	0.0909091
$122$	13.0509	1.18157
$123$	0	0
$124$	4.47013	0.401429
$125$	0	0
$126$	0	0
$127$	15.1842	1.34738	0.673690	−	0.739014i	$-0.264708\pi$
$127$	15.1842	1.34738	0.673690	+	0.739014i	$0.264708\pi$
$128$	5.64941	0.499342
$129$	0	0
$130$	0	0
$131$	−1.24443	−0.108726	−0.0543632	−	0.998521i	$-0.517313\pi$
$131$	−1.24443	−0.108726	−0.0543632	+	0.998521i	$0.517313\pi$
$132$	0	0
$133$	31.2257	2.70761
$134$	2.36842	0.204600
$135$	0	0
$136$	3.72746	0.319627
$137$	−0.488863	−0.0417663	−0.0208832	−	0.999782i	$-0.506648\pi$
$137$	−0.488863	−0.0417663	−0.0208832	+	0.999782i	$0.506648\pi$
$138$	0	0
$139$	17.8064	1.51032	0.755161	−	0.655540i	$-0.227559\pi$
$139$	17.8064	1.51032	0.755161	+	0.655540i	$0.227559\pi$
$140$	0	0
$141$	0	0
$142$	−5.24443	−0.440103
$143$	−0.622216	−0.0520323
$144$	0	0
$145$	0	0
$146$	−8.06022	−0.667069
$147$	0	0
$148$	3.24443	0.266691
$149$	1.43801	0.117806	0.0589031	−	0.998264i	$-0.481240\pi$
$149$	1.43801	0.117806	0.0589031	+	0.998264i	$0.481240\pi$
$150$	0	0
$151$	−12.1748	−0.990774	−0.495387	−	0.868672i	$-0.664974\pi$
$151$	−12.1748	−0.990774	−0.495387	+	0.868672i	$0.664974\pi$
$152$	−5.06959	−0.411198
$153$	0	0
$154$	−8.42864	−0.679199
$155$	0	0
$156$	0	0
$157$	−18.4701	−1.47408	−0.737038	−	0.675851i	$-0.763776\pi$
$157$	−18.4701	−1.47408	−0.737038	+	0.675851i	$0.763776\pi$
$158$	16.2953	1.29638
$159$	0	0
$160$	0	0
$161$	39.2257	3.09142
$162$	0	0
$163$	10.1017	0.791227	0.395614	−	0.918417i	$-0.370532\pi$
$163$	10.1017	0.791227	0.395614	+	0.918417i	$0.370532\pi$
$164$	0.314022	0.0245210
$165$	0	0
$166$	0.253799	0.0196986
$167$	−16.3368	−1.26418	−0.632089	−	0.774896i	$-0.717802\pi$
$167$	−16.3368	−1.26418	−0.632089	+	0.774896i	$0.717802\pi$
$168$	0	0
$169$	−12.6128	−0.970219
$170$	0	0
$171$	0	0
$172$	−9.20294	−0.701718
$173$	−9.18421	−0.698262	−0.349131	−	0.937074i	$-0.613523\pi$
$173$	−9.18421	−0.698262	−0.349131	+	0.937074i	$0.613523\pi$
$174$	0	0
$175$	0	0
$176$	4.61285	0.347706
$177$	0	0
$178$	−10.6824	−0.800683
$179$	25.3274	1.89306	0.946530	−	0.322617i	$-0.104563\pi$
$179$	25.3274	1.89306	0.946530	+	0.322617i	$0.104563\pi$
$180$	0	0
$181$	−13.6128	−1.01184	−0.505918	−	0.862582i	$-0.668846\pi$
$181$	−13.6128	−1.01184	−0.505918	+	0.862582i	$0.668846\pi$
$182$	5.24443	0.388743
$183$	0	0
$184$	−6.36842	−0.469486
$185$	0	0
$186$	0	0
$187$	5.18421	0.379107
$188$	−4.47013	−0.326017
$189$	0	0
$190$	0	0
$191$	−6.10171	−0.441504	−0.220752	−	0.975330i	$-0.570851\pi$
$191$	−6.10171	−0.441504	−0.220752	+	0.975330i	$0.570851\pi$
$192$	0	0
$193$	−18.3368	−1.31991	−0.659955	−	0.751305i	$-0.729425\pi$
$193$	−18.3368	−1.31991	−0.659955	+	0.751305i	$0.729425\pi$
$194$	−13.7877	−0.989898
$195$	0	0
$196$	20.4608	1.46148
$197$	−6.69535	−0.477024	−0.238512	−	0.971140i	$-0.576660\pi$
$197$	−6.69535	−0.477024	−0.238512	+	0.971140i	$0.576660\pi$
$198$	0	0
$199$	14.1017	0.999644	0.499822	−	0.866128i	$-0.333399\pi$
$199$	14.1017	0.999644	0.499822	+	0.866128i	$0.333399\pi$
$200$	0	0
$201$	0	0
$202$	−8.87601	−0.624514
$203$	34.5718	2.42647
$204$	0	0
$205$	0	0
$206$	22.1017	1.53990
$207$	0	0
$208$	−2.87019	−0.199012
$209$	−7.05086	−0.487718
$210$	0	0
$211$	−10.6637	−0.734120	−0.367060	−	0.930197i	$-0.619636\pi$
$211$	−10.6637	−0.734120	−0.367060	+	0.930197i	$0.619636\pi$
$212$	−17.6128	−1.20966
$213$	0	0
$214$	4.99063	0.341153
$215$	0	0
$216$	0	0
$217$	12.2034	0.828422
$218$	−37.5210	−2.54124
$219$	0	0
$220$	0	0
$221$	−3.22570	−0.216984
$222$	0	0
$223$	−8.85728	−0.593127	−0.296564	−	0.955013i	$-0.595841\pi$
$223$	−8.85728	−0.593127	−0.296564	+	0.955013i	$0.595841\pi$
$224$	−32.5116	−2.17227
$225$	0	0
$226$	−11.4193	−0.759599
$227$	13.3778	0.887915	0.443957	−	0.896048i	$-0.353574\pi$
$227$	13.3778	0.887915	0.443957	+	0.896048i	$0.353574\pi$
$228$	0	0
$229$	11.5111	0.760677	0.380339	−	0.924847i	$-0.375807\pi$
$229$	11.5111	0.760677	0.380339	+	0.924847i	$0.375807\pi$
$230$	0	0
$231$	0	0
$232$	−5.61285	−0.368502
$233$	−4.32693	−0.283467	−0.141733	−	0.989905i	$-0.545268\pi$
$233$	−4.32693	−0.283467	−0.141733	+	0.989905i	$0.545268\pi$
$234$	0	0
$235$	0	0
$236$	7.87955	0.512915
$237$	0	0
$238$	−43.6958	−2.83238
$239$	3.34614	0.216444	0.108222	−	0.994127i	$-0.465484\pi$
$239$	3.34614	0.216444	0.108222	+	0.994127i	$0.465484\pi$
$240$	0	0
$241$	−1.34614	−0.0867126	−0.0433563	−	0.999060i	$-0.513805\pi$
$241$	−1.34614	−0.0867126	−0.0433563	+	0.999060i	$0.513805\pi$
$242$	1.90321	0.122343
$243$	0	0
$244$	11.1240	0.712140
$245$	0	0
$246$	0	0
$247$	4.38715	0.279148
$248$	−1.98126	−0.125810
$249$	0	0
$250$	0	0
$251$	−22.7556	−1.43632	−0.718159	−	0.695879i	$-0.755015\pi$
$251$	−22.7556	−1.43632	−0.718159	+	0.695879i	$0.755015\pi$
$252$	0	0
$253$	−8.85728	−0.556852
$254$	28.8988	1.81327
$255$	0	0
$256$	20.2444	1.26528
$257$	−6.85728	−0.427745	−0.213873	−	0.976862i	$-0.568608\pi$
$257$	−6.85728	−0.427745	−0.213873	+	0.976862i	$0.568608\pi$
$258$	0	0
$259$	8.85728	0.550365
$260$	0	0
$261$	0	0
$262$	−2.36842	−0.146321
$263$	−29.5812	−1.82406	−0.912028	−	0.410129i	$-0.865484\pi$
$263$	−29.5812	−1.82406	−0.912028	+	0.410129i	$0.865484\pi$
$264$	0	0
$265$	0	0
$266$	59.4291	3.64383
$267$	0	0
$268$	2.01874	0.123314
$269$	−8.48886	−0.517575	−0.258788	−	0.965934i	$-0.583323\pi$
$269$	−8.48886	−0.517575	−0.258788	+	0.965934i	$0.583323\pi$
$270$	0	0
$271$	14.6637	0.890757	0.445378	−	0.895343i	$-0.353069\pi$
$271$	14.6637	0.890757	0.445378	+	0.895343i	$0.353069\pi$
$272$	23.9140	1.45000
$273$	0	0
$274$	−0.930409	−0.0562081
$275$	0	0
$276$	0	0
$277$	−14.6035	−0.877438	−0.438719	−	0.898624i	$-0.644568\pi$
$277$	−14.6035	−0.877438	−0.438719	+	0.898624i	$0.644568\pi$
$278$	33.8894	2.03255
$279$	0	0
$280$	0	0
$281$	0.193576	0.0115478	0.00577389	−	0.999983i	$-0.498162\pi$
$281$	0.193576	0.0115478	0.00577389	+	0.999983i	$0.498162\pi$
$282$	0	0
$283$	−27.1842	−1.61593	−0.807967	−	0.589228i	$-0.799432\pi$
$283$	−27.1842	−1.61593	−0.807967	+	0.589228i	$0.799432\pi$
$284$	−4.47013	−0.265253
$285$	0	0
$286$	−1.18421	−0.0700237
$287$	0.857279	0.0506036
$288$	0	0
$289$	9.87601	0.580942
$290$	0	0
$291$	0	0
$292$	−6.87019	−0.402047
$293$	−2.81579	−0.164500	−0.0822502	−	0.996612i	$-0.526211\pi$
$293$	−2.81579	−0.164500	−0.0822502	+	0.996612i	$0.526211\pi$
$294$	0	0
$295$	0	0
$296$	−1.43801	−0.0835825
$297$	0	0
$298$	2.73683	0.158540
$299$	5.51114	0.318717
$300$	0	0
$301$	−25.1240	−1.44812
$302$	−23.1713	−1.33336
$303$	0	0
$304$	−32.5245	−1.86541
$305$	0	0
$306$	0	0
$307$	24.4286	1.39422	0.697108	−	0.716966i	$-0.254470\pi$
$307$	24.4286	1.39422	0.697108	+	0.716966i	$0.254470\pi$
$308$	−7.18421	−0.409358
$309$	0	0
$310$	0	0
$311$	−19.8796	−1.12727	−0.563633	−	0.826025i	$-0.690597\pi$
$311$	−19.8796	−1.12727	−0.563633	+	0.826025i	$0.690597\pi$
$312$	0	0
$313$	15.7146	0.888239	0.444120	−	0.895967i	$-0.353517\pi$
$313$	15.7146	0.888239	0.444120	+	0.895967i	$0.353517\pi$
$314$	−35.1526	−1.98377
$315$	0	0
$316$	13.8894	0.781340
$317$	16.4889	0.926107	0.463053	−	0.886330i	$-0.346754\pi$
$317$	16.4889	0.926107	0.463053	+	0.886330i	$0.346754\pi$
$318$	0	0
$319$	−7.80642	−0.437076
$320$	0	0
$321$	0	0
$322$	74.6548	4.16035
$323$	−36.5531	−2.03387
$324$	0	0
$325$	0	0
$326$	19.2257	1.06481
$327$	0	0
$328$	−0.139182	−0.00768504
$329$	−12.2034	−0.672796
$330$	0	0
$331$	15.3461	0.843500	0.421750	−	0.906712i	$-0.361416\pi$
$331$	15.3461	0.843500	0.421750	+	0.906712i	$0.361416\pi$
$332$	0.216327	0.0118725
$333$	0	0
$334$	−31.0923	−1.70130
$335$	0	0
$336$	0	0
$337$	−28.2351	−1.53806	−0.769031	−	0.639212i	$-0.779261\pi$
$337$	−28.2351	−1.53806	−0.769031	+	0.639212i	$0.779261\pi$
$338$	−24.0049	−1.30570
$339$	0	0
$340$	0	0
$341$	−2.75557	−0.149222
$342$	0	0
$343$	24.8573	1.34217
$344$	4.07896	0.219923
$345$	0	0
$346$	−17.4795	−0.939703
$347$	2.62222	0.140768	0.0703840	−	0.997520i	$-0.477578\pi$
$347$	2.62222	0.140768	0.0703840	+	0.997520i	$0.477578\pi$
$348$	0	0
$349$	5.14272	0.275284	0.137642	−	0.990482i	$-0.456048\pi$
$349$	5.14272	0.275284	0.137642	+	0.990482i	$0.456048\pi$
$350$	0	0
$351$	0	0
$352$	7.34122	0.391288
$353$	−9.34614	−0.497445	−0.248722	−	0.968575i	$-0.580011\pi$
$353$	−9.34614	−0.497445	−0.248722	+	0.968575i	$0.580011\pi$
$354$	0	0
$355$	0	0
$356$	−9.10525	−0.482577
$357$	0	0
$358$	48.2034	2.54763
$359$	−10.7556	−0.567657	−0.283829	−	0.958875i	$-0.591605\pi$
$359$	−10.7556	−0.567657	−0.283829	+	0.958875i	$0.591605\pi$
$360$	0	0
$361$	30.7146	1.61656
$362$	−25.9081	−1.36170
$363$	0	0
$364$	4.47013	0.234298
$365$	0	0
$366$	0	0
$367$	−33.7975	−1.76422	−0.882108	−	0.471046i	$-0.843876\pi$
$367$	−33.7975	−1.76422	−0.882108	+	0.471046i	$0.843876\pi$
$368$	−40.8573	−2.12983
$369$	0	0
$370$	0	0
$371$	−48.0830	−2.49634
$372$	0	0
$373$	−33.9496	−1.75784	−0.878922	−	0.476965i	$-0.841737\pi$
$373$	−33.9496	−1.75784	−0.878922	+	0.476965i	$0.841737\pi$
$374$	9.86665	0.510192
$375$	0	0
$376$	1.98126	0.102176
$377$	4.85728	0.250163
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	−11.6128	−0.594165
$383$	−14.6351	−0.747820	−0.373910	−	0.927465i	$-0.621983\pi$
$383$	−14.6351	−0.747820	−0.373910	+	0.927465i	$0.621983\pi$
$384$	0	0
$385$	0	0
$386$	−34.8988	−1.77630
$387$	0	0
$388$	−11.7520	−0.596619
$389$	5.61285	0.284583	0.142291	−	0.989825i	$-0.454553\pi$
$389$	5.61285	0.284583	0.142291	+	0.989825i	$0.454553\pi$
$390$	0	0
$391$	−45.9180	−2.32217
$392$	−9.06868	−0.458038
$393$	0	0
$394$	−12.7427	−0.641966
$395$	0	0
$396$	0	0
$397$	12.7556	0.640184	0.320092	−	0.947387i	$-0.396286\pi$
$397$	12.7556	0.640184	0.320092	+	0.947387i	$0.396286\pi$
$398$	26.8385	1.34529
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	1.71456	0.0854082
$404$	−7.56553	−0.376399
$405$	0	0
$406$	65.7975	3.26548
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−7.12399	−0.352258	−0.176129	−	0.984367i	$-0.556358\pi$
$409$	−7.12399	−0.352258	−0.176129	+	0.984367i	$0.556358\pi$
$410$	0	0
$411$	0	0
$412$	18.8385	0.928108
$413$	21.5111	1.05849
$414$	0	0
$415$	0	0
$416$	−4.56782	−0.223956
$417$	0	0
$418$	−13.4193	−0.656358
$419$	−15.6128	−0.762738	−0.381369	−	0.924423i	$-0.624547\pi$
$419$	−15.6128	−0.762738	−0.381369	+	0.924423i	$0.624547\pi$
$420$	0	0
$421$	7.89829	0.384939	0.192470	−	0.981303i	$-0.438350\pi$
$421$	7.89829	0.384939	0.192470	+	0.981303i	$0.438350\pi$
$422$	−20.2953	−0.987959
$423$	0	0
$424$	7.80642	0.379113
$425$	0	0
$426$	0	0
$427$	30.3684	1.46963
$428$	4.25380	0.205615
$429$	0	0
$430$	0	0
$431$	−34.3051	−1.65242	−0.826210	−	0.563362i	$-0.809508\pi$
$431$	−34.3051	−1.65242	−0.826210	+	0.563362i	$0.809508\pi$
$432$	0	0
$433$	−14.4701	−0.695390	−0.347695	−	0.937608i	$-0.613036\pi$
$433$	−14.4701	−0.695390	−0.347695	+	0.937608i	$0.613036\pi$
$434$	23.2257	1.11487
$435$	0	0
$436$	−31.9813	−1.53162
$437$	62.4514	2.98746
$438$	0	0
$439$	−19.3176	−0.921977	−0.460988	−	0.887406i	$-0.652505\pi$
$439$	−19.3176	−0.921977	−0.460988	+	0.887406i	$0.652505\pi$
$440$	0	0
$441$	0	0
$442$	−6.13918	−0.292011
$443$	−13.1240	−0.623539	−0.311770	−	0.950158i	$-0.600922\pi$
$443$	−13.1240	−0.623539	−0.311770	+	0.950158i	$0.600922\pi$
$444$	0	0
$445$	0	0
$446$	−16.8573	−0.798215
$447$	0	0
$448$	−21.0192	−0.993064
$449$	32.3051	1.52457	0.762287	−	0.647240i	$-0.224077\pi$
$449$	32.3051	1.52457	0.762287	+	0.647240i	$0.224077\pi$
$450$	0	0
$451$	−0.193576	−0.00911514
$452$	−9.73329	−0.457816
$453$	0	0
$454$	25.4608	1.19493
$455$	0	0
$456$	0	0
$457$	23.4608	1.09745	0.548724	−	0.836004i	$-0.315114\pi$
$457$	23.4608	1.09745	0.548724	+	0.836004i	$0.315114\pi$
$458$	21.9081	1.02370
$459$	0	0
$460$	0	0
$461$	28.8671	1.34448	0.672238	−	0.740335i	$-0.265333\pi$
$461$	28.8671	1.34448	0.672238	+	0.740335i	$0.265333\pi$
$462$	0	0
$463$	19.3461	0.899091	0.449546	−	0.893257i	$-0.351586\pi$
$463$	19.3461	0.899091	0.449546	+	0.893257i	$0.351586\pi$
$464$	−36.0098	−1.67172
$465$	0	0
$466$	−8.23506	−0.381482
$467$	3.14272	0.145428	0.0727139	−	0.997353i	$-0.476834\pi$
$467$	3.14272	0.145428	0.0727139	+	0.997353i	$0.476834\pi$
$468$	0	0
$469$	5.51114	0.254481
$470$	0	0
$471$	0	0
$472$	−3.49240	−0.160751
$473$	5.67307	0.260848
$474$	0	0
$475$	0	0
$476$	−37.2444	−1.70710
$477$	0	0
$478$	6.36842	0.291285
$479$	24.8573	1.13576	0.567879	−	0.823112i	$-0.307764\pi$
$479$	24.8573	1.13576	0.567879	+	0.823112i	$0.307764\pi$
$480$	0	0
$481$	1.24443	0.0567412
$482$	−2.56199	−0.116696
$483$	0	0
$484$	1.62222	0.0737371
$485$	0	0
$486$	0	0
$487$	11.5299	0.522468	0.261234	−	0.965275i	$-0.415871\pi$
$487$	11.5299	0.522468	0.261234	+	0.965275i	$0.415871\pi$
$488$	−4.93041	−0.223189
$489$	0	0
$490$	0	0
$491$	−16.3872	−0.739542	−0.369771	−	0.929123i	$-0.620564\pi$
$491$	−16.3872	−0.739542	−0.369771	+	0.929123i	$0.620564\pi$
$492$	0	0
$493$	−40.4701	−1.82268
$494$	8.34968	0.375670
$495$	0	0
$496$	−12.7110	−0.570742
$497$	−12.2034	−0.547398
$498$	0	0
$499$	−25.3274	−1.13381	−0.566905	−	0.823783i	$-0.691859\pi$
$499$	−25.3274	−1.13381	−0.566905	+	0.823783i	$0.691859\pi$
$500$	0	0
$501$	0	0
$502$	−43.3087	−1.93296
$503$	19.0923	0.851285	0.425643	−	0.904891i	$-0.360048\pi$
$503$	19.0923	0.851285	0.425643	+	0.904891i	$0.360048\pi$
$504$	0	0
$505$	0	0
$506$	−16.8573	−0.749397
$507$	0	0
$508$	24.6321	1.09287
$509$	32.4514	1.43838	0.719191	−	0.694812i	$-0.244512\pi$
$509$	32.4514	1.43838	0.719191	+	0.694812i	$0.244512\pi$
$510$	0	0
$511$	−18.7556	−0.829698
$512$	27.2306	1.20343
$513$	0	0
$514$	−13.0509	−0.575649
$515$	0	0
$516$	0	0
$517$	2.75557	0.121190
$518$	16.8573	0.740666
$519$	0	0
$520$	0	0
$521$	29.2257	1.28040	0.640200	−	0.768208i	$-0.278851\pi$
$521$	29.2257	1.28040	0.640200	+	0.768208i	$0.278851\pi$
$522$	0	0
$523$	−6.71408	−0.293586	−0.146793	−	0.989167i	$-0.546895\pi$
$523$	−6.71408	−0.293586	−0.146793	+	0.989167i	$0.546895\pi$
$524$	−2.01874	−0.0881889
$525$	0	0
$526$	−56.2993	−2.45477
$527$	−14.2854	−0.622284
$528$	0	0
$529$	55.4514	2.41093
$530$	0	0
$531$	0	0
$532$	50.6548	2.19616
$533$	0.120446	0.00521710
$534$	0	0
$535$	0	0
$536$	−0.894751	−0.0386473
$537$	0	0
$538$	−16.1561	−0.696539
$539$	−12.6128	−0.543274
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	27.9081	1.19876
$543$	0	0
$544$	38.0584	1.63174
$545$	0	0
$546$	0	0
$547$	−41.3689	−1.76881	−0.884403	−	0.466724i	$-0.845434\pi$
$547$	−41.3689	−1.76881	−0.884403	+	0.466724i	$0.845434\pi$
$548$	−0.793040	−0.0338770
$549$	0	0
$550$	0	0
$551$	55.0420	2.34487
$552$	0	0
$553$	37.9180	1.61244
$554$	−27.7935	−1.18083
$555$	0	0
$556$	28.8859	1.22503
$557$	−20.7971	−0.881200	−0.440600	−	0.897704i	$-0.645234\pi$
$557$	−20.7971	−0.881200	−0.440600	+	0.897704i	$0.645234\pi$
$558$	0	0
$559$	−3.52987	−0.149298
$560$	0	0
$561$	0	0
$562$	0.368416	0.0155407
$563$	37.7275	1.59002	0.795012	−	0.606594i	$-0.207465\pi$
$563$	37.7275	1.59002	0.795012	+	0.606594i	$0.207465\pi$
$564$	0	0
$565$	0	0
$566$	−51.7373	−2.17468
$567$	0	0
$568$	1.98126	0.0831320
$569$	7.33630	0.307554	0.153777	−	0.988106i	$-0.450856\pi$
$569$	7.33630	0.307554	0.153777	+	0.988106i	$0.450856\pi$
$570$	0	0
$571$	36.6450	1.53354	0.766772	−	0.641919i	$-0.221862\pi$
$571$	36.6450	1.53354	0.766772	+	0.641919i	$0.221862\pi$
$572$	−1.00937	−0.0422038
$573$	0	0
$574$	1.63158	0.0681010
$575$	0	0
$576$	0	0
$577$	−4.22216	−0.175771	−0.0878853	−	0.996131i	$-0.528011\pi$
$577$	−4.22216	−0.175771	−0.0878853	+	0.996131i	$0.528011\pi$
$578$	18.7961	0.781817
$579$	0	0
$580$	0	0
$581$	0.590573	0.0245011
$582$	0	0
$583$	10.8573	0.449663
$584$	3.04503	0.126004
$585$	0	0
$586$	−5.35905	−0.221380
$587$	34.3684	1.41854	0.709268	−	0.704939i	$-0.249026\pi$
$587$	34.3684	1.41854	0.709268	+	0.704939i	$0.249026\pi$
$588$	0	0
$589$	19.4291	0.800563
$590$	0	0
$591$	0	0
$592$	−9.22570	−0.379174
$593$	27.9398	1.14735	0.573675	−	0.819083i	$-0.305517\pi$
$593$	27.9398	1.14735	0.573675	+	0.819083i	$0.305517\pi$
$594$	0	0
$595$	0	0
$596$	2.33276	0.0955535
$597$	0	0
$598$	10.4889	0.428921
$599$	31.2257	1.27585	0.637924	−	0.770100i	$-0.279794\pi$
$599$	31.2257	1.27585	0.637924	+	0.770100i	$0.279794\pi$
$600$	0	0
$601$	−8.75557	−0.357147	−0.178574	−	0.983927i	$-0.557148\pi$
$601$	−8.75557	−0.357147	−0.178574	+	0.983927i	$0.557148\pi$
$602$	−47.8163	−1.94885
$603$	0	0
$604$	−19.7502	−0.803625
$605$	0	0
$606$	0	0
$607$	15.1842	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$607$	15.1842	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$608$	−51.7619	−2.09922
$609$	0	0
$610$	0	0
$611$	−1.71456	−0.0693636
$612$	0	0
$613$	−42.7239	−1.72560	−0.862802	−	0.505543i	$-0.831292\pi$
$613$	−42.7239	−1.72560	−0.862802	+	0.505543i	$0.831292\pi$
$614$	46.4929	1.87630
$615$	0	0
$616$	3.18421	0.128295
$617$	−3.51114	−0.141353	−0.0706765	−	0.997499i	$-0.522516\pi$
$617$	−3.51114	−0.141353	−0.0706765	+	0.997499i	$0.522516\pi$
$618$	0	0
$619$	17.5941	0.707167	0.353584	−	0.935403i	$-0.384963\pi$
$619$	17.5941	0.707167	0.353584	+	0.935403i	$0.384963\pi$
$620$	0	0
$621$	0	0
$622$	−37.8350	−1.51705
$623$	−24.8573	−0.995886
$624$	0	0
$625$	0	0
$626$	29.9081	1.19537
$627$	0	0
$628$	−29.9625	−1.19564
$629$	−10.3684	−0.413416
$630$	0	0
$631$	15.8163	0.629636	0.314818	−	0.949152i	$-0.398057\pi$
$631$	15.8163	0.629636	0.314818	+	0.949152i	$0.398057\pi$
$632$	−6.15610	−0.244877
$633$	0	0
$634$	31.3818	1.24633
$635$	0	0
$636$	0	0
$637$	7.84791	0.310946
$638$	−14.8573	−0.588205
$639$	0	0
$640$	0	0
$641$	−25.8163	−1.01968	−0.509841	−	0.860269i	$-0.670296\pi$
$641$	−25.8163	−1.01968	−0.509841	+	0.860269i	$0.670296\pi$
$642$	0	0
$643$	−18.1017	−0.713862	−0.356931	−	0.934131i	$-0.616177\pi$
$643$	−18.1017	−0.713862	−0.356931	+	0.934131i	$0.616177\pi$
$644$	63.6325	2.50747
$645$	0	0
$646$	−69.5683	−2.73713
$647$	−47.0420	−1.84941	−0.924705	−	0.380684i	$-0.875689\pi$
$647$	−47.0420	−1.84941	−0.924705	+	0.380684i	$0.875689\pi$
$648$	0	0
$649$	−4.85728	−0.190665
$650$	0	0
$651$	0	0
$652$	16.3872	0.641770
$653$	30.0830	1.17724	0.588619	−	0.808411i	$-0.299672\pi$
$653$	30.0830	1.17724	0.588619	+	0.808411i	$0.299672\pi$
$654$	0	0
$655$	0	0
$656$	−0.892937	−0.0348633
$657$	0	0
$658$	−23.2257	−0.905432
$659$	10.2854	0.400664	0.200332	−	0.979728i	$-0.435798\pi$
$659$	10.2854	0.400664	0.200332	+	0.979728i	$0.435798\pi$
$660$	0	0
$661$	−27.7146	−1.07797	−0.538986	−	0.842315i	$-0.681192\pi$
$661$	−27.7146	−1.07797	−0.538986	+	0.842315i	$0.681192\pi$
$662$	29.2070	1.13516
$663$	0	0
$664$	−0.0958814	−0.00372092
$665$	0	0
$666$	0	0
$667$	69.1437	2.67725
$668$	−26.5018	−1.02538
$669$	0	0
$670$	0	0
$671$	−6.85728	−0.264722
$672$	0	0
$673$	9.86665	0.380331	0.190166	−	0.981752i	$-0.439097\pi$
$673$	9.86665	0.380331	0.190166	+	0.981752i	$0.439097\pi$
$674$	−53.7373	−2.06988
$675$	0	0
$676$	−20.4608	−0.786952
$677$	−5.65433	−0.217314	−0.108657	−	0.994079i	$-0.534655\pi$
$677$	−5.65433	−0.217314	−0.108657	+	0.994079i	$0.534655\pi$
$678$	0	0
$679$	−32.0830	−1.23123
$680$	0	0
$681$	0	0
$682$	−5.24443	−0.200820
$683$	34.1847	1.30804	0.654020	−	0.756477i	$-0.273081\pi$
$683$	34.1847	1.30804	0.654020	+	0.756477i	$0.273081\pi$
$684$	0	0
$685$	0	0
$686$	47.3087	1.80625
$687$	0	0
$688$	26.1690	0.997684
$689$	−6.75557	−0.257367
$690$	0	0
$691$	−19.2257	−0.731380	−0.365690	−	0.930737i	$-0.619167\pi$
$691$	−19.2257	−0.731380	−0.365690	+	0.930737i	$0.619167\pi$
$692$	−14.8988	−0.566366
$693$	0	0
$694$	4.99063	0.189442
$695$	0	0
$696$	0	0
$697$	−1.00354	−0.0380118
$698$	9.78769	0.370469
$699$	0	0
$700$	0	0
$701$	29.9081	1.12961	0.564807	−	0.825223i	$-0.308950\pi$
$701$	29.9081	1.12961	0.564807	+	0.825223i	$0.308950\pi$
$702$	0	0
$703$	14.1017	0.531856
$704$	4.74620	0.178879
$705$	0	0
$706$	−17.7877	−0.669448
$707$	−20.6539	−0.776768
$708$	0	0
$709$	−15.3274	−0.575633	−0.287816	−	0.957686i	$-0.592929\pi$
$709$	−15.3274	−0.575633	−0.287816	+	0.957686i	$0.592929\pi$
$710$	0	0
$711$	0	0
$712$	4.03566	0.151243
$713$	24.4068	0.914043
$714$	0	0
$715$	0	0
$716$	41.0865	1.53548
$717$	0	0
$718$	−20.4701	−0.763938
$719$	−23.8163	−0.888197	−0.444098	−	0.895978i	$-0.646476\pi$
$719$	−23.8163	−0.888197	−0.444098	+	0.895978i	$0.646476\pi$
$720$	0	0
$721$	51.4291	1.91532
$722$	58.4563	2.17552
$723$	0	0
$724$	−22.0830	−0.820707
$725$	0	0
$726$	0	0
$727$	32.9403	1.22169	0.610843	−	0.791752i	$-0.290831\pi$
$727$	32.9403	1.22169	0.610843	+	0.791752i	$0.290831\pi$
$728$	−1.98126	−0.0734305
$729$	0	0
$730$	0	0
$731$	29.4104	1.08778
$732$	0	0
$733$	29.8666	1.10315	0.551575	−	0.834125i	$-0.314027\pi$
$733$	29.8666	1.10315	0.551575	+	0.834125i	$0.314027\pi$
$734$	−64.3239	−2.37424
$735$	0	0
$736$	−65.0232	−2.39679
$737$	−1.24443	−0.0458392
$738$	0	0
$739$	−5.06959	−0.186488	−0.0932440	−	0.995643i	$-0.529724\pi$
$739$	−5.06959	−0.186488	−0.0932440	+	0.995643i	$0.529724\pi$
$740$	0	0
$741$	0	0
$742$	−91.5121	−3.35951
$743$	22.4385	0.823188	0.411594	−	0.911367i	$-0.364972\pi$
$743$	22.4385	0.823188	0.411594	+	0.911367i	$0.364972\pi$
$744$	0	0
$745$	0	0
$746$	−64.6133	−2.36566
$747$	0	0
$748$	8.40990	0.307497
$749$	11.6128	0.424324
$750$	0	0
$751$	−6.63512	−0.242119	−0.121060	−	0.992645i	$-0.538629\pi$
$751$	−6.63512	−0.242119	−0.121060	+	0.992645i	$0.538629\pi$
$752$	12.7110	0.463523
$753$	0	0
$754$	9.24443	0.336662
$755$	0	0
$756$	0	0
$757$	−8.75557	−0.318227	−0.159113	−	0.987260i	$-0.550864\pi$
$757$	−8.75557	−0.318227	−0.159113	+	0.987260i	$0.550864\pi$
$758$	−38.0642	−1.38256
$759$	0	0
$760$	0	0
$761$	−3.15257	−0.114280	−0.0571402	−	0.998366i	$-0.518198\pi$
$761$	−3.15257	−0.114280	−0.0571402	+	0.998366i	$0.518198\pi$
$762$	0	0
$763$	−87.3087	−3.16079
$764$	−9.89829	−0.358108
$765$	0	0
$766$	−27.8537	−1.00640
$767$	3.02227	0.109128
$768$	0	0
$769$	−28.9590	−1.04429	−0.522144	−	0.852857i	$-0.674868\pi$
$769$	−28.9590	−1.04429	−0.522144	+	0.852857i	$0.674868\pi$
$770$	0	0
$771$	0	0
$772$	−29.7462	−1.07059
$773$	29.1427	1.04819	0.524095	−	0.851660i	$-0.324404\pi$
$773$	29.1427	1.04819	0.524095	+	0.851660i	$0.324404\pi$
$774$	0	0
$775$	0	0
$776$	5.20877	0.186984
$777$	0	0
$778$	10.6824	0.382984
$779$	1.36488	0.0489018
$780$	0	0
$781$	2.75557	0.0986020
$782$	−87.3916	−3.12512
$783$	0	0
$784$	−58.1811	−2.07790
$785$	0	0
$786$	0	0
$787$	11.2672	0.401632	0.200816	−	0.979629i	$-0.435641\pi$
$787$	11.2672	0.401632	0.200816	+	0.979629i	$0.435641\pi$
$788$	−10.8613	−0.386918
$789$	0	0
$790$	0	0
$791$	−26.5718	−0.944786
$792$	0	0
$793$	4.26671	0.151515
$794$	24.2766	0.861543
$795$	0	0
$796$	22.8760	0.810819
$797$	41.9625	1.48639	0.743195	−	0.669075i	$-0.233310\pi$
$797$	41.9625	1.48639	0.743195	+	0.669075i	$0.233310\pi$
$798$	0	0
$799$	14.2854	0.505383
$800$	0	0
$801$	0	0
$802$	−3.80642	−0.134409
$803$	4.23506	0.149452
$804$	0	0
$805$	0	0
$806$	3.26317	0.114940
$807$	0	0
$808$	3.35322	0.117966
$809$	27.8064	0.977622	0.488811	−	0.872390i	$-0.337431\pi$
$809$	27.8064	0.977622	0.488811	+	0.872390i	$0.337431\pi$
$810$	0	0
$811$	6.78415	0.238224	0.119112	−	0.992881i	$-0.461995\pi$
$811$	6.78415	0.238224	0.119112	+	0.992881i	$0.461995\pi$
$812$	56.0830	1.96813
$813$	0	0
$814$	−3.80642	−0.133415
$815$	0	0
$816$	0	0
$817$	−40.0000	−1.39942
$818$	−13.5585	−0.474060
$819$	0	0
$820$	0	0
$821$	−3.62269	−0.126433	−0.0632164	−	0.998000i	$-0.520136\pi$
$821$	−3.62269	−0.126433	−0.0632164	+	0.998000i	$0.520136\pi$
$822$	0	0
$823$	−42.0642	−1.46627	−0.733134	−	0.680085i	$-0.761943\pi$
$823$	−42.0642	−1.46627	−0.733134	+	0.680085i	$0.761943\pi$
$824$	−8.34968	−0.290875
$825$	0	0
$826$	40.9403	1.42449
$827$	30.8256	1.07191	0.535956	−	0.844246i	$-0.319951\pi$
$827$	30.8256	1.07191	0.535956	+	0.844246i	$0.319951\pi$
$828$	0	0
$829$	7.12399	0.247426	0.123713	−	0.992318i	$-0.460520\pi$
$829$	7.12399	0.247426	0.123713	+	0.992318i	$0.460520\pi$
$830$	0	0
$831$	0	0
$832$	−2.95316	−0.102382
$833$	−65.3876	−2.26555
$834$	0	0
$835$	0	0
$836$	−11.4380	−0.395592
$837$	0	0
$838$	−29.7146	−1.02647
$839$	−3.34614	−0.115522	−0.0577608	−	0.998330i	$-0.518396\pi$
$839$	−3.34614	−0.115522	−0.0577608	+	0.998330i	$0.518396\pi$
$840$	0	0
$841$	31.9403	1.10139
$842$	15.0321	0.518041
$843$	0	0
$844$	−17.2988	−0.595450
$845$	0	0
$846$	0	0
$847$	4.42864	0.152170
$848$	50.0830	1.71986
$849$	0	0
$850$	0	0
$851$	17.7146	0.607247
$852$	0	0
$853$	26.4197	0.904595	0.452297	−	0.891867i	$-0.350605\pi$
$853$	26.4197	0.904595	0.452297	+	0.891867i	$0.350605\pi$
$854$	57.7975	1.97779
$855$	0	0
$856$	−1.88538	−0.0644411
$857$	38.7783	1.32464	0.662321	−	0.749220i	$-0.269571\pi$
$857$	38.7783	1.32464	0.662321	+	0.749220i	$0.269571\pi$
$858$	0	0
$859$	−27.3087	−0.931760	−0.465880	−	0.884848i	$-0.654262\pi$
$859$	−27.3087	−0.931760	−0.465880	+	0.884848i	$0.654262\pi$
$860$	0	0
$861$	0	0
$862$	−65.2899	−2.22378
$863$	−49.5308	−1.68605	−0.843024	−	0.537875i	$-0.819227\pi$
$863$	−49.5308	−1.68605	−0.843024	+	0.537875i	$0.819227\pi$
$864$	0	0
$865$	0	0
$866$	−27.5397	−0.935838
$867$	0	0
$868$	19.7966	0.671940
$869$	−8.56199	−0.290446
$870$	0	0
$871$	0.774305	0.0262363
$872$	14.1748	0.480021
$873$	0	0
$874$	118.858	4.02044
$875$	0	0
$876$	0	0
$877$	4.50177	0.152014	0.0760070	−	0.997107i	$-0.475783\pi$
$877$	4.50177	0.152014	0.0760070	+	0.997107i	$0.475783\pi$
$878$	−36.7654	−1.24077
$879$	0	0
$880$	0	0
$881$	15.1240	0.509540	0.254770	−	0.967002i	$-0.418000\pi$
$881$	15.1240	0.509540	0.254770	+	0.967002i	$0.418000\pi$
$882$	0	0
$883$	30.2480	1.01793	0.508963	−	0.860789i	$-0.330029\pi$
$883$	30.2480	1.01793	0.508963	+	0.860789i	$0.330029\pi$
$884$	−5.23277	−0.175997
$885$	0	0
$886$	−24.9777	−0.839143
$887$	57.1941	1.92039	0.960194	−	0.279333i	$-0.0901135\pi$
$887$	57.1941	1.92039	0.960194	+	0.279333i	$0.0901135\pi$
$888$	0	0
$889$	67.2454	2.25534
$890$	0	0
$891$	0	0
$892$	−14.3684	−0.481090
$893$	−19.4291	−0.650171
$894$	0	0
$895$	0	0
$896$	25.0192	0.835833
$897$	0	0
$898$	61.4835	2.05173
$899$	21.5111	0.717437
$900$	0	0
$901$	56.2864	1.87517
$902$	−0.368416	−0.0122669
$903$	0	0
$904$	4.31402	0.143482
$905$	0	0
$906$	0	0
$907$	53.2641	1.76861	0.884303	−	0.466913i	$-0.154634\pi$
$907$	53.2641	1.76861	0.884303	+	0.466913i	$0.154634\pi$
$908$	21.7017	0.720195
$909$	0	0
$910$	0	0
$911$	0.590573	0.0195665	0.00978327	−	0.999952i	$-0.496886\pi$
$911$	0.590573	0.0195665	0.00978327	+	0.999952i	$0.496886\pi$
$912$	0	0
$913$	−0.133353	−0.00441334
$914$	44.6508	1.47692
$915$	0	0
$916$	18.6735	0.616991
$917$	−5.51114	−0.181994
$918$	0	0
$919$	55.8707	1.84300	0.921502	−	0.388375i	$-0.126963\pi$
$919$	55.8707	1.84300	0.921502	+	0.388375i	$0.126963\pi$
$920$	0	0
$921$	0	0
$922$	54.9403	1.80936
$923$	−1.71456	−0.0564354
$924$	0	0
$925$	0	0
$926$	36.8198	1.20997
$927$	0	0
$928$	−57.3087	−1.88125
$929$	−15.3274	−0.502876	−0.251438	−	0.967873i	$-0.580903\pi$
$929$	−15.3274	−0.502876	−0.251438	+	0.967873i	$0.580903\pi$
$930$	0	0
$931$	88.9314	2.91461
$932$	−7.01921	−0.229922
$933$	0	0
$934$	5.98126	0.195713
$935$	0	0
$936$	0	0
$937$	27.8479	0.909752	0.454876	−	0.890555i	$-0.349684\pi$
$937$	27.8479	0.909752	0.454876	+	0.890555i	$0.349684\pi$
$938$	10.4889	0.342474
$939$	0	0
$940$	0	0
$941$	−10.4157	−0.339543	−0.169772	−	0.985483i	$-0.554303\pi$
$941$	−10.4157	−0.339543	−0.169772	+	0.985483i	$0.554303\pi$
$942$	0	0
$943$	1.71456	0.0558337
$944$	−22.4059	−0.729250
$945$	0	0
$946$	10.7971	0.351043
$947$	8.47013	0.275242	0.137621	−	0.990485i	$-0.456054\pi$
$947$	8.47013	0.275242	0.137621	+	0.990485i	$0.456054\pi$
$948$	0	0
$949$	−2.63512	−0.0855397
$950$	0	0
$951$	0	0
$952$	16.5076	0.535014
$953$	−8.71408	−0.282277	−0.141138	−	0.989990i	$-0.545076\pi$
$953$	−8.71408	−0.282277	−0.141138	+	0.989990i	$0.545076\pi$
$954$	0	0
$955$	0	0
$956$	5.42816	0.175559
$957$	0	0
$958$	47.3087	1.52847
$959$	−2.16500	−0.0699114
$960$	0	0
$961$	−23.4068	−0.755059
$962$	2.36842	0.0763608
$963$	0	0
$964$	−2.18373	−0.0703333
$965$	0	0
$966$	0	0
$967$	−44.2449	−1.42282	−0.711410	−	0.702777i	$-0.751943\pi$
$967$	−44.2449	−1.42282	−0.711410	+	0.702777i	$0.751943\pi$
$968$	−0.719004	−0.0231097
$969$	0	0
$970$	0	0
$971$	57.1437	1.83383	0.916914	−	0.399085i	$-0.130672\pi$
$971$	57.1437	1.83383	0.916914	+	0.399085i	$0.130672\pi$
$972$	0	0
$973$	78.8582	2.52808
$974$	21.9438	0.703124
$975$	0	0
$976$	−31.6316	−1.01250
$977$	−16.2480	−0.519819	−0.259909	−	0.965633i	$-0.583693\pi$
$977$	−16.2480	−0.519819	−0.259909	+	0.965633i	$0.583693\pi$
$978$	0	0
$979$	5.61285	0.179387
$980$	0	0
$981$	0	0
$982$	−31.1882	−0.995256
$983$	1.12399	0.0358496	0.0179248	−	0.999839i	$-0.494294\pi$
$983$	1.12399	0.0358496	0.0179248	+	0.999839i	$0.494294\pi$
$984$	0	0
$985$	0	0
$986$	−77.0232	−2.45292
$987$	0	0
$988$	7.11691	0.226419
$989$	−50.2480	−1.59779
$990$	0	0
$991$	−53.6513	−1.70429	−0.852144	−	0.523307i	$-0.824698\pi$
$991$	−53.6513	−1.70429	−0.852144	+	0.523307i	$0.824698\pi$
$992$	−20.2292	−0.642279
$993$	0	0
$994$	−23.2257	−0.736674
$995$	0	0
$996$	0	0
$997$	35.7275	1.13150	0.565750	−	0.824577i	$-0.308587\pi$
$997$	35.7275	1.13150	0.565750	+	0.824577i	$0.308587\pi$
$998$	−48.2034	−1.52585
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.bb.1.3		3
3.2	odd	2		825.2.a.k.1.1		3
5.2	odd	4		2475.2.c.r.199.5		6
5.3	odd	4		2475.2.c.r.199.2		6
5.4	even	2		495.2.a.e.1.1		3
15.2	even	4		825.2.c.g.199.2		6
15.8	even	4		825.2.c.g.199.5		6
15.14	odd	2		165.2.a.c.1.3	✓	3
20.19	odd	2		7920.2.a.cj.1.3		3
33.32	even	2		9075.2.a.cf.1.3		3
55.54	odd	2		5445.2.a.z.1.3		3
60.59	even	2		2640.2.a.be.1.3		3
105.104	even	2		8085.2.a.bk.1.3		3
165.164	even	2		1815.2.a.m.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.c.1.3	✓	3	15.14	odd	2
495.2.a.e.1.1		3	5.4	even	2
825.2.a.k.1.1		3	3.2	odd	2
825.2.c.g.199.2		6	15.2	even	4
825.2.c.g.199.5		6	15.8	even	4
1815.2.a.m.1.1		3	165.164	even	2
2475.2.a.bb.1.3		3	1.1	even	1	trivial
2475.2.c.r.199.2		6	5.3	odd	4
2475.2.c.r.199.5		6	5.2	odd	4
2640.2.a.be.1.3		3	60.59	even	2
5445.2.a.z.1.3		3	55.54	odd	2
7920.2.a.cj.1.3		3	20.19	odd	2
8085.2.a.bk.1.3		3	105.104	even	2
9075.2.a.cf.1.3		3	33.32	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+1.90321 q^{2} +1.62222 q^{4} +4.42864 q^{7} -0.719004 q^{8} -1.00000 q^{11} +0.622216 q^{13} +8.42864 q^{14} -4.61285 q^{16} -5.18421 q^{17} +7.05086 q^{19} -1.90321 q^{22} +8.85728 q^{23} +1.18421 q^{26} +7.18421 q^{28} +7.80642 q^{29} +2.75557 q^{31} -7.34122 q^{32} -9.86665 q^{34} +2.00000 q^{37} +13.4193 q^{38} +0.193576 q^{41} -5.67307 q^{43} -1.62222 q^{44} +16.8573 q^{46} -2.75557 q^{47} +12.6128 q^{49} +1.00937 q^{52} -10.8573 q^{53} -3.18421 q^{56} +14.8573 q^{58} +4.85728 q^{59} +6.85728 q^{61} +5.24443 q^{62} -4.74620 q^{64} +1.24443 q^{67} -8.40990 q^{68} -2.75557 q^{71} -4.23506 q^{73} +3.80642 q^{74} +11.4380 q^{76} -4.42864 q^{77} +8.56199 q^{79} +0.368416 q^{82} +0.133353 q^{83} -10.7971 q^{86} +0.719004 q^{88} -5.61285 q^{89} +2.75557 q^{91} +14.3684 q^{92} -5.24443 q^{94} -7.24443 q^{97} +24.0049 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 5 q^{4} - 9 q^{8} - 3 q^{11} + 2 q^{13} + 12 q^{14} + 13 q^{16} - 2 q^{17} + 8 q^{19} + q^{22} - 10 q^{26} + 8 q^{28} + 10 q^{29} + 8 q^{31} - 29 q^{32} - 30 q^{34} + 6 q^{37} + 14 q^{41}+ \cdots + 39 q^{98}+O(q^{100}) \) 3 * q - q^2 + 5 * q^4 - 9 * q^8 - 3 * q^11 + 2 * q^13 + 12 * q^14 + 13 * q^16 - 2 * q^17 + 8 * q^19 + q^22 - 10 * q^26 + 8 * q^28 + 10 * q^29 + 8 * q^31 - 29 * q^32 - 30 * q^34 + 6 * q^37 + 14 * q^41 - 4 * q^43 - 5 * q^44 + 24 * q^46 - 8 * q^47 + 11 * q^49 + 30 * q^52 - 6 * q^53 + 4 * q^56 + 18 * q^58 - 12 * q^59 - 6 * q^61 + 16 * q^62 + 13 * q^64 + 4 * q^67 + 42 * q^68 - 8 * q^71 + 14 * q^73 - 2 * q^74 + 48 * q^76 + 12 * q^79 - 26 * q^82 + 8 * q^86 + 9 * q^88 + 10 * q^89 + 8 * q^91 + 16 * q^92 - 16 * q^94 - 22 * q^97 + 39 * q^98

Introduction

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