LMFDB - Embedded newform 2475.2.c.r.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.45161 + 1.45161i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.r.199.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.90321i q^{2} -1.62222 q^{4} -4.42864i q^{7} -0.719004i q^{8} -1.00000 q^{11} +0.622216i q^{13} -8.42864 q^{14} -4.61285 q^{16} +5.18421i q^{17} -7.05086 q^{19} +1.90321i q^{22} +8.85728i q^{23} +1.18421 q^{26} +7.18421i q^{28} -7.80642 q^{29} +2.75557 q^{31} +7.34122i q^{32} +9.86665 q^{34} -2.00000i q^{37} +13.4193i q^{38} +0.193576 q^{41} -5.67307i q^{43} +1.62222 q^{44} +16.8573 q^{46} +2.75557i q^{47} -12.6128 q^{49} -1.00937i q^{52} -10.8573i q^{53} -3.18421 q^{56} +14.8573i q^{58} -4.85728 q^{59} +6.85728 q^{61} -5.24443i q^{62} +4.74620 q^{64} -1.24443i q^{67} -8.40990i q^{68} -2.75557 q^{71} -4.23506i q^{73} -3.80642 q^{74} +11.4380 q^{76} +4.42864i q^{77} -8.56199 q^{79} -0.368416i q^{82} +0.133353i q^{83} -10.7971 q^{86} +0.719004i q^{88} +5.61285 q^{89} +2.75557 q^{91} -14.3684i q^{92} +5.24443 q^{94} +7.24443i q^{97} +24.0049i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{4} - 6 q^{11} - 24 q^{14} + 26 q^{16} - 16 q^{19} - 20 q^{26} - 20 q^{29} + 16 q^{31} + 60 q^{34} + 28 q^{41} + 10 q^{44} + 48 q^{46} - 22 q^{49} + 8 q^{56} + 24 q^{59} - 12 q^{61} - 26 q^{64}+ \cdots + 32 q^{94}+O(q^{100}) $ 6 * q - 10 * q^4 - 6 * q^11 - 24 * q^14 + 26 * q^16 - 16 * q^19 - 20 * q^26 - 20 * q^29 + 16 * q^31 + 60 * q^34 + 28 * q^41 + 10 * q^44 + 48 * q^46 - 22 * q^49 + 8 * q^56 + 24 * q^59 - 12 * q^61 - 26 * q^64 - 16 * q^71 + 4 * q^74 + 96 * q^76 - 24 * q^79 + 16 * q^86 - 20 * q^89 + 16 * q^91 + 32 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 1.90321i	− 1.34577i	−0.739745	−	0.672887i	$-0.765054\pi$
$2$	− 1.90321i	− 1.34577i	0.739745	−	0.672887i	$-0.234946\pi$
$3$	0	0
$3$	0	0
$4$	−1.62222	−0.811108
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.42864i	− 1.67387i	−0.547304	−	0.836934i	$-0.684346\pi$
$7$	− 4.42864i	− 1.67387i	0.547304	−	0.836934i	$-0.315654\pi$
$8$	− 0.719004i	− 0.254206i
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.622216i	0.172572i	0.996270	+	0.0862858i	$0.0274998\pi$
$13$	0.622216i	0.172572i	−0.996270	+	0.0862858i	$0.972500\pi$
$14$	−8.42864	−2.25265
$15$	0	0
$16$	−4.61285	−1.15321
$17$	5.18421i	1.25736i	0.777666	+	0.628678i	$0.216403\pi$
$17$	5.18421i	1.25736i	−0.777666	+	0.628678i	$0.783597\pi$
$18$	0	0
$19$	−7.05086	−1.61758	−0.808789	−	0.588100i	$-0.799876\pi$
$19$	−7.05086	−1.61758	−0.808789	+	0.588100i	$0.799876\pi$
$20$	0	0
$21$	0	0
$22$	1.90321i	0.405766i
$23$	8.85728i	1.84687i	0.383754	+	0.923435i	$0.374631\pi$
$23$	8.85728i	1.84687i	−0.383754	+	0.923435i	$0.625369\pi$
$24$	0	0
$25$	0	0
$26$	1.18421	0.232242
$27$	0	0
$28$	7.18421i	1.35769i
$29$	−7.80642	−1.44962	−0.724808	−	0.688951i	$-0.758072\pi$
$29$	−7.80642	−1.44962	−0.724808	+	0.688951i	$0.758072\pi$
$30$	0	0
$31$	2.75557	0.494915	0.247457	−	0.968899i	$-0.420405\pi$
$31$	2.75557	0.494915	0.247457	+	0.968899i	$0.420405\pi$
$32$	7.34122i	1.29776i
$33$	0	0
$34$	9.86665	1.69212
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	13.4193i	2.17689i
$39$	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
$43$	− 5.67307i	− 0.865135i	−0.901602	−	0.432568i	$-0.857608\pi$
$43$	− 5.67307i	− 0.865135i	0.901602	−	0.432568i	$-0.142392\pi$
$44$	1.62222	0.244558
$45$	0	0
$46$	16.8573	2.48547
$47$	2.75557i	0.401941i	0.979597	+	0.200971i	$0.0644095\pi$
$47$	2.75557i	0.401941i	−0.979597	+	0.200971i	$0.935590\pi$
$48$	0	0
$49$	−12.6128	−1.80184
$50$	0	0
$51$	0	0
$52$	− 1.00937i	− 0.139974i
$53$	− 10.8573i	− 1.49136i	−0.666303	−	0.745681i	$-0.732124\pi$
$53$	− 10.8573i	− 1.49136i	0.666303	−	0.745681i	$-0.267876\pi$
$54$	0	0
$55$	0	0
$56$	−3.18421	−0.425508
$57$	0	0
$58$	14.8573i	1.95086i
$59$	−4.85728	−0.632364	−0.316182	−	0.948699i	$-0.602401\pi$
$59$	−4.85728	−0.632364	−0.316182	+	0.948699i	$0.602401\pi$
$60$	0	0
$61$	6.85728	0.877985	0.438992	−	0.898491i	$-0.355336\pi$
$61$	6.85728	0.877985	0.438992	+	0.898491i	$0.355336\pi$
$62$	− 5.24443i	− 0.666043i
$63$	0	0
$64$	4.74620	0.593275
$65$	0	0
$66$	0	0
$67$	− 1.24443i	− 0.152031i	−0.997107	−	0.0760157i	$-0.975780\pi$
$67$	− 1.24443i	− 0.152031i	0.997107	−	0.0760157i	$-0.0242199\pi$
$68$	− 8.40990i	− 1.01985i
$69$	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
$73$	− 4.23506i	− 0.495677i	−0.968801	−	0.247838i	$-0.920280\pi$
$73$	− 4.23506i	− 0.495677i	0.968801	−	0.247838i	$-0.0797202\pi$
$74$	−3.80642	−0.442488
$75$	0	0
$76$	11.4380	1.31203
$77$	4.42864i	0.504690i
$78$	0	0
$79$	−8.56199	−0.963299	−0.481650	−	0.876364i	$-0.659962\pi$
$79$	−8.56199	−0.963299	−0.481650	+	0.876364i	$0.659962\pi$
$80$	0	0
$81$	0	0
$82$	− 0.368416i	− 0.0406848i
$83$	0.133353i	0.0146374i	0.999973	+	0.00731870i	$0.00232964\pi$
$83$	0.133353i	0.0146374i	−0.999973	+	0.00731870i	$0.997670\pi$
$84$	0	0
$85$	0	0
$86$	−10.7971	−1.16428
$87$	0	0
$88$	0.719004i	0.0766461i
$89$	5.61285	0.594961	0.297480	−	0.954728i	$-0.403854\pi$
$89$	5.61285	0.594961	0.297480	+	0.954728i	$0.403854\pi$
$90$	0	0
$91$	2.75557	0.288862
$92$	− 14.3684i	− 1.49801i
$93$	0	0
$94$	5.24443	0.540922
$95$	0	0
$96$	0	0
$97$	7.24443i	0.735561i	0.929913	+	0.367780i	$0.119882\pi$
$97$	7.24443i	0.735561i	−0.929913	+	0.367780i	$0.880118\pi$
$98$	24.0049i	2.42486i
$99$	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
$103$	11.6128i	1.14425i	0.820167	+	0.572124i	$0.193880\pi$
$103$	11.6128i	1.14425i	−0.820167	+	0.572124i	$0.806120\pi$
$104$	0.447375	0.0438688
$105$	0	0
$106$	−20.6637	−2.00704
$107$	− 2.62222i	− 0.253499i	−0.991935	−	0.126750i	$-0.959546\pi$
$107$	− 2.62222i	− 0.253499i	0.991935	−	0.126750i	$-0.0404545\pi$
$108$	0	0
$109$	19.7146	1.88831	0.944156	−	0.329499i	$-0.106880\pi$
$109$	19.7146	1.88831	0.944156	+	0.329499i	$0.106880\pi$
$110$	0	0
$111$	0	0
$112$	20.4286i	1.93032i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	12.6637	1.17580
$117$	0	0
$118$	9.24443i	0.851019i
$119$	22.9590	2.10465
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 13.0509i	− 1.18157i
$123$	0	0
$124$	−4.47013	−0.401429
$125$	0	0
$126$	0	0
$127$	− 15.1842i	− 1.34738i	−0.739014	−	0.673690i	$-0.764708\pi$
$127$	− 15.1842i	− 1.34738i	0.739014	−	0.673690i	$-0.235292\pi$
$128$	5.64941i	0.499342i
$129$	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
$133$	31.2257i	2.70761i
$134$	−2.36842	−0.204600
$135$	0	0
$136$	3.72746	0.319627
$137$	0.488863i	0.0417663i	0.999782	+	0.0208832i	$0.00664780\pi$
$137$	0.488863i	0.0417663i	−0.999782	+	0.0208832i	$0.993352\pi$
$138$	0	0
$139$	−17.8064	−1.51032	−0.755161	−	0.655540i	$-0.772441\pi$
$139$	−17.8064	−1.51032	−0.755161	+	0.655540i	$0.772441\pi$
$140$	0	0
$141$	0	0
$142$	5.24443i	0.440103i
$143$	− 0.622216i	− 0.0520323i
$144$	0	0
$145$	0	0
$146$	−8.06022	−0.667069
$147$	0	0
$148$	3.24443i	0.266691i
$149$	−1.43801	−0.117806	−0.0589031	−	0.998264i	$-0.518760\pi$
$149$	−1.43801	−0.117806	−0.0589031	+	0.998264i	$0.518760\pi$
$150$	0	0
$151$	−12.1748	−0.990774	−0.495387	−	0.868672i	$-0.664974\pi$
$151$	−12.1748	−0.990774	−0.495387	+	0.868672i	$0.664974\pi$
$152$	5.06959i	0.411198i
$153$	0	0
$154$	8.42864	0.679199
$155$	0	0
$156$	0	0
$157$	18.4701i	1.47408i	0.675851	+	0.737038i	$0.263776\pi$
$157$	18.4701i	1.47408i	−0.675851	+	0.737038i	$0.736224\pi$
$158$	16.2953i	1.29638i
$159$	0	0
$160$	0	0
$161$	39.2257	3.09142
$162$	0	0
$163$	10.1017i	0.791227i	0.918417	+	0.395614i	$0.129468\pi$
$163$	10.1017i	0.791227i	−0.918417	+	0.395614i	$0.870532\pi$
$164$	−0.314022	−0.0245210
$165$	0	0
$166$	0.253799	0.0196986
$167$	16.3368i	1.26418i	0.774896	+	0.632089i	$0.217802\pi$
$167$	16.3368i	1.26418i	−0.774896	+	0.632089i	$0.782198\pi$
$168$	0	0
$169$	12.6128	0.970219
$170$	0	0
$171$	0	0
$172$	9.20294i	0.701718i
$173$	− 9.18421i	− 0.698262i	−0.937074	−	0.349131i	$-0.886477\pi$
$173$	− 9.18421i	− 0.698262i	0.937074	−	0.349131i	$-0.113523\pi$
$174$	0	0
$175$	0	0
$176$	4.61285	0.347706
$177$	0	0
$178$	− 10.6824i	− 0.800683i
$179$	−25.3274	−1.89306	−0.946530	−	0.322617i	$-0.895437\pi$
$179$	−25.3274	−1.89306	−0.946530	+	0.322617i	$0.895437\pi$
$180$	0	0
$181$	−13.6128	−1.01184	−0.505918	−	0.862582i	$-0.668846\pi$
$181$	−13.6128	−1.01184	−0.505918	+	0.862582i	$0.668846\pi$
$182$	− 5.24443i	− 0.388743i
$183$	0	0
$184$	6.36842	0.469486
$185$	0	0
$186$	0	0
$187$	− 5.18421i	− 0.379107i
$188$	− 4.47013i	− 0.326017i
$189$	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
$193$	− 18.3368i	− 1.31991i	−0.751305	−	0.659955i	$-0.770575\pi$
$193$	− 18.3368i	− 1.31991i	0.751305	−	0.659955i	$-0.229425\pi$
$194$	13.7877	0.989898
$195$	0	0
$196$	20.4608	1.46148
$197$	6.69535i	0.477024i	0.971140	+	0.238512i	$0.0766596\pi$
$197$	6.69535i	0.477024i	−0.971140	+	0.238512i	$0.923340\pi$
$198$	0	0
$199$	−14.1017	−0.999644	−0.499822	−	0.866128i	$-0.666601\pi$
$199$	−14.1017	−0.999644	−0.499822	+	0.866128i	$0.666601\pi$
$200$	0	0
$201$	0	0
$202$	8.87601i	0.624514i
$203$	34.5718i	2.42647i
$204$	0	0
$205$	0	0
$206$	22.1017	1.53990
$207$	0	0
$208$	− 2.87019i	− 0.199012i
$209$	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	$0.619636\pi$
$212$	17.6128i	1.20966i
$213$	0	0
$214$	−4.99063	−0.341153
$215$	0	0
$216$	0	0
$217$	− 12.2034i	− 0.828422i
$218$	− 37.5210i	− 2.54124i
$219$	0	0
$220$	0	0
$221$	−3.22570	−0.216984
$222$	0	0
$223$	− 8.85728i	− 0.593127i	−0.955013	−	0.296564i	$-0.904159\pi$
$223$	− 8.85728i	− 0.593127i	0.955013	−	0.296564i	$-0.0958407\pi$
$224$	32.5116	2.17227
$225$	0	0
$226$	−11.4193	−0.759599
$227$	− 13.3778i	− 0.887915i	−0.896048	−	0.443957i	$-0.853574\pi$
$227$	− 13.3778i	− 0.887915i	0.896048	−	0.443957i	$-0.146426\pi$
$228$	0	0
$229$	−11.5111	−0.760677	−0.380339	−	0.924847i	$-0.624193\pi$
$229$	−11.5111	−0.760677	−0.380339	+	0.924847i	$0.624193\pi$
$230$	0	0
$231$	0	0
$232$	5.61285i	0.368502i
$233$	− 4.32693i	− 0.283467i	−0.989905	−	0.141733i	$-0.954732\pi$
$233$	− 4.32693i	− 0.283467i	0.989905	−	0.141733i	$-0.0452675\pi$
$234$	0	0
$235$	0	0
$236$	7.87955	0.512915
$237$	0	0
$238$	− 43.6958i	− 2.83238i
$239$	−3.34614	−0.216444	−0.108222	−	0.994127i	$-0.534516\pi$
$239$	−3.34614	−0.216444	−0.108222	+	0.994127i	$0.534516\pi$
$240$	0	0
$241$	−1.34614	−0.0867126	−0.0433563	−	0.999060i	$-0.513805\pi$
$241$	−1.34614	−0.0867126	−0.0433563	+	0.999060i	$0.513805\pi$
$242$	− 1.90321i	− 0.122343i
$243$	0	0
$244$	−11.1240	−0.712140
$245$	0	0
$246$	0	0
$247$	− 4.38715i	− 0.279148i
$248$	− 1.98126i	− 0.125810i
$249$	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
$253$	− 8.85728i	− 0.556852i
$254$	−28.8988	−1.81327
$255$	0	0
$256$	20.2444	1.26528
$257$	6.85728i	0.427745i	0.976862	+	0.213873i	$0.0686078\pi$
$257$	6.85728i	0.427745i	−0.976862	+	0.213873i	$0.931392\pi$
$258$	0	0
$259$	−8.85728	−0.550365
$260$	0	0
$261$	0	0
$262$	2.36842i	0.146321i
$263$	− 29.5812i	− 1.82406i	−0.410129	−	0.912028i	$-0.634516\pi$
$263$	− 29.5812i	− 1.82406i	0.410129	−	0.912028i	$-0.365484\pi$
$264$	0	0
$265$	0	0
$266$	59.4291	3.64383
$267$	0	0
$268$	2.01874i	0.123314i
$269$	8.48886	0.517575	0.258788	−	0.965934i	$-0.416677\pi$
$269$	8.48886	0.517575	0.258788	+	0.965934i	$0.416677\pi$
$270$	0	0
$271$	14.6637	0.890757	0.445378	−	0.895343i	$-0.353069\pi$
$271$	14.6637	0.890757	0.445378	+	0.895343i	$0.353069\pi$
$272$	− 23.9140i	− 1.45000i
$273$	0	0
$274$	0.930409	0.0562081
$275$	0	0
$276$	0	0
$277$	14.6035i	0.877438i	0.898624	+	0.438719i	$0.144568\pi$
$277$	14.6035i	0.877438i	−0.898624	+	0.438719i	$0.855432\pi$
$278$	33.8894i	2.03255i
$279$	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
$283$	− 27.1842i	− 1.61593i	−0.589228	−	0.807967i	$-0.700568\pi$
$283$	− 27.1842i	− 1.61593i	0.589228	−	0.807967i	$-0.299432\pi$
$284$	4.47013	0.265253
$285$	0	0
$286$	−1.18421	−0.0700237
$287$	− 0.857279i	− 0.0506036i
$288$	0	0
$289$	−9.87601	−0.580942
$290$	0	0
$291$	0	0
$292$	6.87019i	0.402047i
$293$	− 2.81579i	− 0.164500i	−0.996612	−	0.0822502i	$-0.973789\pi$
$293$	− 2.81579i	− 0.164500i	0.996612	−	0.0822502i	$-0.0262106\pi$
$294$	0	0
$295$	0	0
$296$	−1.43801	−0.0835825
$297$	0	0
$298$	2.73683i	0.158540i
$299$	−5.51114	−0.318717
$300$	0	0
$301$	−25.1240	−1.44812
$302$	23.1713i	1.33336i
$303$	0	0
$304$	32.5245	1.86541
$305$	0	0
$306$	0	0
$307$	− 24.4286i	− 1.39422i	−0.716966	−	0.697108i	$-0.754470\pi$
$307$	− 24.4286i	− 1.39422i	0.716966	−	0.697108i	$-0.245530\pi$
$308$	− 7.18421i	− 0.409358i
$309$	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
$313$	15.7146i	0.888239i	0.895967	+	0.444120i	$0.146483\pi$
$313$	15.7146i	0.888239i	−0.895967	+	0.444120i	$0.853517\pi$
$314$	35.1526	1.98377
$315$	0	0
$316$	13.8894	0.781340
$317$	− 16.4889i	− 0.926107i	−0.886330	−	0.463053i	$-0.846754\pi$
$317$	− 16.4889i	− 0.926107i	0.886330	−	0.463053i	$-0.153246\pi$
$318$	0	0
$319$	7.80642	0.437076
$320$	0	0
$321$	0	0
$322$	− 74.6548i	− 4.16035i
$323$	− 36.5531i	− 2.03387i
$324$	0	0
$325$	0	0
$326$	19.2257	1.06481
$327$	0	0
$328$	− 0.139182i	− 0.00768504i
$329$	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	$0.361416\pi$
$332$	− 0.216327i	− 0.0118725i
$333$	0	0
$334$	31.0923	1.70130
$335$	0	0
$336$	0	0
$337$	28.2351i	1.53806i	0.639212	+	0.769031i	$0.279261\pi$
$337$	28.2351i	1.53806i	−0.639212	+	0.769031i	$0.720739\pi$
$338$	− 24.0049i	− 1.30570i
$339$	0	0
$340$	0	0
$341$	−2.75557	−0.149222
$342$	0	0
$343$	24.8573i	1.34217i
$344$	−4.07896	−0.219923
$345$	0	0
$346$	−17.4795	−0.939703
$347$	− 2.62222i	− 0.140768i	−0.997520	−	0.0703840i	$-0.977578\pi$
$347$	− 2.62222i	− 0.140768i	0.997520	−	0.0703840i	$-0.0224224\pi$
$348$	0	0
$349$	−5.14272	−0.275284	−0.137642	−	0.990482i	$-0.543952\pi$
$349$	−5.14272	−0.275284	−0.137642	+	0.990482i	$0.543952\pi$
$350$	0	0
$351$	0	0
$352$	− 7.34122i	− 0.391288i
$353$	− 9.34614i	− 0.497445i	−0.968575	−	0.248722i	$-0.919989\pi$
$353$	− 9.34614i	− 0.497445i	0.968575	−	0.248722i	$-0.0800107\pi$
$354$	0	0
$355$	0	0
$356$	−9.10525	−0.482577
$357$	0	0
$358$	48.2034i	2.54763i
$359$	10.7556	0.567657	0.283829	−	0.958875i	$-0.408395\pi$
$359$	10.7556	0.567657	0.283829	+	0.958875i	$0.408395\pi$
$360$	0	0
$361$	30.7146	1.61656
$362$	25.9081i	1.36170i
$363$	0	0
$364$	−4.47013	−0.234298
$365$	0	0
$366$	0	0
$367$	33.7975i	1.76422i	0.471046	+	0.882108i	$0.343876\pi$
$367$	33.7975i	1.76422i	−0.471046	+	0.882108i	$0.656124\pi$
$368$	− 40.8573i	− 2.12983i
$369$	0	0
$370$	0	0
$371$	−48.0830	−2.49634
$372$	0	0
$373$	− 33.9496i	− 1.75784i	−0.476965	−	0.878922i	$-0.658263\pi$
$373$	− 33.9496i	− 1.75784i	0.476965	−	0.878922i	$-0.341737\pi$
$374$	−9.86665	−0.510192
$375$	0	0
$376$	1.98126	0.102176
$377$	− 4.85728i	− 0.250163i
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	11.6128i	0.594165i
$383$	− 14.6351i	− 0.747820i	−0.927465	−	0.373910i	$-0.878017\pi$
$383$	− 14.6351i	− 0.747820i	0.927465	−	0.373910i	$-0.121983\pi$
$384$	0	0
$385$	0	0
$386$	−34.8988	−1.77630
$387$	0	0
$388$	− 11.7520i	− 0.596619i
$389$	−5.61285	−0.284583	−0.142291	−	0.989825i	$-0.545447\pi$
$389$	−5.61285	−0.284583	−0.142291	+	0.989825i	$0.545447\pi$
$390$	0	0
$391$	−45.9180	−2.32217
$392$	9.06868i	0.458038i
$393$	0	0
$394$	12.7427	0.641966
$395$	0	0
$396$	0	0
$397$	− 12.7556i	− 0.640184i	−0.947387	−	0.320092i	$-0.896286\pi$
$397$	− 12.7556i	− 0.640184i	0.947387	−	0.320092i	$-0.103714\pi$
$398$	26.8385i	1.34529i
$399$	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
$403$	1.71456i	0.0854082i
$404$	7.56553	0.376399
$405$	0	0
$406$	65.7975	3.26548
$407$	2.00000i	0.0991363i
$408$	0	0
$409$	7.12399	0.352258	0.176129	−	0.984367i	$-0.443642\pi$
$409$	7.12399	0.352258	0.176129	+	0.984367i	$0.443642\pi$
$410$	0	0
$411$	0	0
$412$	− 18.8385i	− 0.928108i
$413$	21.5111i	1.05849i
$414$	0	0
$415$	0	0
$416$	−4.56782	−0.223956
$417$	0	0
$418$	− 13.4193i	− 0.656358i
$419$	15.6128	0.762738	0.381369	−	0.924423i	$-0.375453\pi$
$419$	15.6128	0.762738	0.381369	+	0.924423i	$0.375453\pi$
$420$	0	0
$421$	7.89829	0.384939	0.192470	−	0.981303i	$-0.438350\pi$
$421$	7.89829	0.384939	0.192470	+	0.981303i	$0.438350\pi$
$422$	20.2953i	0.987959i
$423$	0	0
$424$	−7.80642	−0.379113
$425$	0	0
$426$	0	0
$427$	− 30.3684i	− 1.46963i
$428$	4.25380i	0.205615i
$429$	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
$433$	− 14.4701i	− 0.695390i	−0.937608	−	0.347695i	$-0.886964\pi$
$433$	− 14.4701i	− 0.695390i	0.937608	−	0.347695i	$-0.113036\pi$
$434$	−23.2257	−1.11487
$435$	0	0
$436$	−31.9813	−1.53162
$437$	− 62.4514i	− 2.98746i
$438$	0	0
$439$	19.3176	0.921977	0.460988	−	0.887406i	$-0.347495\pi$
$439$	19.3176	0.921977	0.460988	+	0.887406i	$0.347495\pi$
$440$	0	0
$441$	0	0
$442$	6.13918i	0.292011i
$443$	− 13.1240i	− 0.623539i	−0.950158	−	0.311770i	$-0.899078\pi$
$443$	− 13.1240i	− 0.623539i	0.950158	−	0.311770i	$-0.100922\pi$
$444$	0	0
$445$	0	0
$446$	−16.8573	−0.798215
$447$	0	0
$448$	− 21.0192i	− 0.993064i
$449$	−32.3051	−1.52457	−0.762287	−	0.647240i	$-0.775923\pi$
$449$	−32.3051	−1.52457	−0.762287	+	0.647240i	$0.775923\pi$
$450$	0	0
$451$	−0.193576	−0.00911514
$452$	9.73329i	0.457816i
$453$	0	0
$454$	−25.4608	−1.19493
$455$	0	0
$456$	0	0
$457$	− 23.4608i	− 1.09745i	−0.836004	−	0.548724i	$-0.815114\pi$
$457$	− 23.4608i	− 1.09745i	0.836004	−	0.548724i	$-0.184886\pi$
$458$	21.9081i	1.02370i
$459$	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
$463$	19.3461i	0.899091i	0.893257	+	0.449546i	$0.148414\pi$
$463$	19.3461i	0.899091i	−0.893257	+	0.449546i	$0.851586\pi$
$464$	36.0098	1.67172
$465$	0	0
$466$	−8.23506	−0.381482
$467$	− 3.14272i	− 0.145428i	−0.997353	−	0.0727139i	$-0.976834\pi$
$467$	− 3.14272i	− 0.145428i	0.997353	−	0.0727139i	$-0.0231660\pi$
$468$	0	0
$469$	−5.51114	−0.254481
$470$	0	0
$471$	0	0
$472$	3.49240i	0.160751i
$473$	5.67307i	0.260848i
$474$	0	0
$475$	0	0
$476$	−37.2444	−1.70710
$477$	0	0
$478$	6.36842i	0.291285i
$479$	−24.8573	−1.13576	−0.567879	−	0.823112i	$-0.692236\pi$
$479$	−24.8573	−1.13576	−0.567879	+	0.823112i	$0.692236\pi$
$480$	0	0
$481$	1.24443	0.0567412
$482$	2.56199i	0.116696i
$483$	0	0
$484$	−1.62222	−0.0737371
$485$	0	0
$486$	0	0
$487$	− 11.5299i	− 0.522468i	−0.965275	−	0.261234i	$-0.915871\pi$
$487$	− 11.5299i	− 0.522468i	0.965275	−	0.261234i	$-0.0841295\pi$
$488$	− 4.93041i	− 0.223189i
$489$	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
$493$	− 40.4701i	− 1.82268i
$494$	−8.34968	−0.375670
$495$	0	0
$496$	−12.7110	−0.570742
$497$	12.2034i	0.547398i
$498$	0	0
$499$	25.3274	1.13381	0.566905	−	0.823783i	$-0.308141\pi$
$499$	25.3274	1.13381	0.566905	+	0.823783i	$0.308141\pi$
$500$	0	0
$501$	0	0
$502$	43.3087i	1.93296i
$503$	19.0923i	0.851285i	0.904891	+	0.425643i	$0.139952\pi$
$503$	19.0923i	0.851285i	−0.904891	+	0.425643i	$0.860048\pi$
$504$	0	0
$505$	0	0
$506$	−16.8573	−0.749397
$507$	0	0
$508$	24.6321i	1.09287i
$509$	−32.4514	−1.43838	−0.719191	−	0.694812i	$-0.755488\pi$
$509$	−32.4514	−1.43838	−0.719191	+	0.694812i	$0.755488\pi$
$510$	0	0
$511$	−18.7556	−0.829698
$512$	− 27.2306i	− 1.20343i
$513$	0	0
$514$	13.0509	0.575649
$515$	0	0
$516$	0	0
$517$	− 2.75557i	− 0.121190i
$518$	16.8573i	0.740666i
$519$	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
$523$	− 6.71408i	− 0.293586i	−0.989167	−	0.146793i	$-0.953105\pi$
$523$	− 6.71408i	− 0.293586i	0.989167	−	0.146793i	$-0.0468952\pi$
$524$	2.01874	0.0881889
$525$	0	0
$526$	−56.2993	−2.45477
$527$	14.2854i	0.622284i
$528$	0	0
$529$	−55.4514	−2.41093
$530$	0	0
$531$	0	0
$532$	− 50.6548i	− 2.19616i
$533$	0.120446i	0.00521710i
$534$	0	0
$535$	0	0
$536$	−0.894751	−0.0386473
$537$	0	0
$538$	− 16.1561i	− 0.696539i
$539$	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	$0.626468\pi$
$542$	− 27.9081i	− 1.19876i
$543$	0	0
$544$	−38.0584	−1.63174
$545$	0	0
$546$	0	0
$547$	41.3689i	1.76881i	0.466724	+	0.884403i	$0.345434\pi$
$547$	41.3689i	1.76881i	−0.466724	+	0.884403i	$0.654566\pi$
$548$	− 0.793040i	− 0.0338770i
$549$	0	0
$550$	0	0
$551$	55.0420	2.34487
$552$	0	0
$553$	37.9180i	1.61244i
$554$	27.7935	1.18083
$555$	0	0
$556$	28.8859	1.22503
$557$	20.7971i	0.881200i	0.897704	+	0.440600i	$0.145234\pi$
$557$	20.7971i	0.881200i	−0.897704	+	0.440600i	$0.854766\pi$
$558$	0	0
$559$	3.52987	0.149298
$560$	0	0
$561$	0	0
$562$	− 0.368416i	− 0.0155407i
$563$	37.7275i	1.59002i	0.606594	+	0.795012i	$0.292535\pi$
$563$	37.7275i	1.59002i	−0.606594	+	0.795012i	$0.707465\pi$
$564$	0	0
$565$	0	0
$566$	−51.7373	−2.17468
$567$	0	0
$568$	1.98126i	0.0831320i
$569$	−7.33630	−0.307554	−0.153777	−	0.988106i	$-0.549144\pi$
$569$	−7.33630	−0.307554	−0.153777	+	0.988106i	$0.549144\pi$
$570$	0	0
$571$	36.6450	1.53354	0.766772	−	0.641919i	$-0.221862\pi$
$571$	36.6450	1.53354	0.766772	+	0.641919i	$0.221862\pi$
$572$	1.00937i	0.0422038i
$573$	0	0
$574$	−1.63158	−0.0681010
$575$	0	0
$576$	0	0
$577$	4.22216i	0.175771i	0.996131	+	0.0878853i	$0.0280109\pi$
$577$	4.22216i	0.175771i	−0.996131	+	0.0878853i	$0.971989\pi$
$578$	18.7961i	0.781817i
$579$	0	0
$580$	0	0
$581$	0.590573	0.0245011
$582$	0	0
$583$	10.8573i	0.449663i
$584$	−3.04503	−0.126004
$585$	0	0
$586$	−5.35905	−0.221380
$587$	− 34.3684i	− 1.41854i	−0.704939	−	0.709268i	$-0.749026\pi$
$587$	− 34.3684i	− 1.41854i	0.704939	−	0.709268i	$-0.250974\pi$
$588$	0	0
$589$	−19.4291	−0.800563
$590$	0	0
$591$	0	0
$592$	9.22570i	0.379174i
$593$	27.9398i	1.14735i	0.819083	+	0.573675i	$0.194483\pi$
$593$	27.9398i	1.14735i	−0.819083	+	0.573675i	$0.805517\pi$
$594$	0	0
$595$	0	0
$596$	2.33276	0.0955535
$597$	0	0
$598$	10.4889i	0.428921i
$599$	−31.2257	−1.27585	−0.637924	−	0.770100i	$-0.720206\pi$
$599$	−31.2257	−1.27585	−0.637924	+	0.770100i	$0.720206\pi$
$600$	0	0
$601$	−8.75557	−0.357147	−0.178574	−	0.983927i	$-0.557148\pi$
$601$	−8.75557	−0.357147	−0.178574	+	0.983927i	$0.557148\pi$
$602$	47.8163i	1.94885i
$603$	0	0
$604$	19.7502	0.803625
$605$	0	0
$606$	0	0
$607$	− 15.1842i	− 0.616308i	−0.951336	−	0.308154i	$-0.900289\pi$
$607$	− 15.1842i	− 0.616308i	0.951336	−	0.308154i	$-0.0997112\pi$
$608$	− 51.7619i	− 2.09922i
$609$	0	0
$610$	0	0
$611$	−1.71456	−0.0693636
$612$	0	0
$613$	− 42.7239i	− 1.72560i	−0.505543	−	0.862802i	$-0.668708\pi$
$613$	− 42.7239i	− 1.72560i	0.505543	−	0.862802i	$-0.331292\pi$
$614$	−46.4929	−1.87630
$615$	0	0
$616$	3.18421	0.128295
$617$	3.51114i	0.141353i	0.997499	+	0.0706765i	$0.0225158\pi$
$617$	3.51114i	0.141353i	−0.997499	+	0.0706765i	$0.977484\pi$
$618$	0	0
$619$	−17.5941	−0.707167	−0.353584	−	0.935403i	$-0.615037\pi$
$619$	−17.5941	−0.707167	−0.353584	+	0.935403i	$0.615037\pi$
$620$	0	0
$621$	0	0
$622$	37.8350i	1.51705i
$623$	− 24.8573i	− 0.995886i
$624$	0	0
$625$	0	0
$626$	29.9081	1.19537
$627$	0	0
$628$	− 29.9625i	− 1.19564i
$629$	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	$0.398057\pi$
$632$	6.15610i	0.244877i
$633$	0	0
$634$	−31.3818	−1.24633
$635$	0	0
$636$	0	0
$637$	− 7.84791i	− 0.310946i
$638$	− 14.8573i	− 0.588205i
$639$	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
$643$	− 18.1017i	− 0.713862i	−0.934131	−	0.356931i	$-0.883823\pi$
$643$	− 18.1017i	− 0.713862i	0.934131	−	0.356931i	$-0.116177\pi$
$644$	−63.6325	−2.50747
$645$	0	0
$646$	−69.5683	−2.73713
$647$	47.0420i	1.84941i	0.380684	+	0.924705i	$0.375689\pi$
$647$	47.0420i	1.84941i	−0.380684	+	0.924705i	$0.624311\pi$
$648$	0	0
$649$	4.85728	0.190665
$650$	0	0
$651$	0	0
$652$	− 16.3872i	− 0.641770i
$653$	30.0830i	1.17724i	0.808411	+	0.588619i	$0.200328\pi$
$653$	30.0830i	1.17724i	−0.808411	+	0.588619i	$0.799672\pi$
$654$	0	0
$655$	0	0
$656$	−0.892937	−0.0348633
$657$	0	0
$658$	− 23.2257i	− 0.905432i
$659$	−10.2854	−0.400664	−0.200332	−	0.979728i	$-0.564202\pi$
$659$	−10.2854	−0.400664	−0.200332	+	0.979728i	$0.564202\pi$
$660$	0	0
$661$	−27.7146	−1.07797	−0.538986	−	0.842315i	$-0.681192\pi$
$661$	−27.7146	−1.07797	−0.538986	+	0.842315i	$0.681192\pi$
$662$	− 29.2070i	− 1.13516i
$663$	0	0
$664$	0.0958814	0.00372092
$665$	0	0
$666$	0	0
$667$	− 69.1437i	− 2.67725i
$668$	− 26.5018i	− 1.02538i
$669$	0	0
$670$	0	0
$671$	−6.85728	−0.264722
$672$	0	0
$673$	9.86665i	0.380331i	0.981752	+	0.190166i	$0.0609025\pi$
$673$	9.86665i	0.380331i	−0.981752	+	0.190166i	$0.939097\pi$
$674$	53.7373	2.06988
$675$	0	0
$676$	−20.4608	−0.786952
$677$	5.65433i	0.217314i	0.994079	+	0.108657i	$0.0346550\pi$
$677$	5.65433i	0.217314i	−0.994079	+	0.108657i	$0.965345\pi$
$678$	0	0
$679$	32.0830	1.23123
$680$	0	0
$681$	0	0
$682$	5.24443i	0.200820i
$683$	34.1847i	1.30804i	0.756477	+	0.654020i	$0.226919\pi$
$683$	34.1847i	1.30804i	−0.756477	+	0.654020i	$0.773081\pi$
$684$	0	0
$685$	0	0
$686$	47.3087	1.80625
$687$	0	0
$688$	26.1690i	0.997684i
$689$	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	$0.619167\pi$
$692$	14.8988i	0.566366i
$693$	0	0
$694$	−4.99063	−0.189442
$695$	0	0
$696$	0	0
$697$	1.00354i	0.0380118i
$698$	9.78769i	0.370469i
$699$	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
$703$	14.1017i	0.531856i
$704$	−4.74620	−0.178879
$705$	0	0
$706$	−17.7877	−0.669448
$707$	20.6539i	0.776768i
$708$	0	0
$709$	15.3274	0.575633	0.287816	−	0.957686i	$-0.407071\pi$
$709$	15.3274	0.575633	0.287816	+	0.957686i	$0.407071\pi$
$710$	0	0
$711$	0	0
$712$	− 4.03566i	− 0.151243i
$713$	24.4068i	0.914043i
$714$	0	0
$715$	0	0
$716$	41.0865	1.53548
$717$	0	0
$718$	− 20.4701i	− 0.763938i
$719$	23.8163	0.888197	0.444098	−	0.895978i	$-0.353524\pi$
$719$	23.8163	0.888197	0.444098	+	0.895978i	$0.353524\pi$
$720$	0	0
$721$	51.4291	1.91532
$722$	− 58.4563i	− 2.17552i
$723$	0	0
$724$	22.0830	0.820707
$725$	0	0
$726$	0	0
$727$	− 32.9403i	− 1.22169i	−0.791752	−	0.610843i	$-0.790831\pi$
$727$	− 32.9403i	− 1.22169i	0.791752	−	0.610843i	$-0.209169\pi$
$728$	− 1.98126i	− 0.0734305i
$729$	0	0
$730$	0	0
$731$	29.4104	1.08778
$732$	0	0
$733$	29.8666i	1.10315i	0.834125	+	0.551575i	$0.185973\pi$
$733$	29.8666i	1.10315i	−0.834125	+	0.551575i	$0.814027\pi$
$734$	64.3239	2.37424
$735$	0	0
$736$	−65.0232	−2.39679
$737$	1.24443i	0.0458392i
$738$	0	0
$739$	5.06959	0.186488	0.0932440	−	0.995643i	$-0.470276\pi$
$739$	5.06959	0.186488	0.0932440	+	0.995643i	$0.470276\pi$
$740$	0	0
$741$	0	0
$742$	91.5121i	3.35951i
$743$	22.4385i	0.823188i	0.911367	+	0.411594i	$0.135028\pi$
$743$	22.4385i	0.823188i	−0.911367	+	0.411594i	$0.864972\pi$
$744$	0	0
$745$	0	0
$746$	−64.6133	−2.36566
$747$	0	0
$748$	8.40990i	0.307497i
$749$	−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	$0.538629\pi$
$752$	− 12.7110i	− 0.463523i
$753$	0	0
$754$	−9.24443	−0.336662
$755$	0	0
$756$	0	0
$757$	8.75557i	0.318227i	0.987260	+	0.159113i	$0.0508635\pi$
$757$	8.75557i	0.318227i	−0.987260	+	0.159113i	$0.949136\pi$
$758$	− 38.0642i	− 1.38256i
$759$	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
$763$	− 87.3087i	− 3.16079i
$764$	9.89829	0.358108
$765$	0	0
$766$	−27.8537	−1.00640
$767$	− 3.02227i	− 0.109128i
$768$	0	0
$769$	28.9590	1.04429	0.522144	−	0.852857i	$-0.325132\pi$
$769$	28.9590	1.04429	0.522144	+	0.852857i	$0.325132\pi$
$770$	0	0
$771$	0	0
$772$	29.7462i	1.07059i
$773$	29.1427i	1.04819i	0.851660	+	0.524095i	$0.175596\pi$
$773$	29.1427i	1.04819i	−0.851660	+	0.524095i	$0.824404\pi$
$774$	0	0
$775$	0	0
$776$	5.20877	0.186984
$777$	0	0
$778$	10.6824i	0.382984i
$779$	−1.36488	−0.0489018
$780$	0	0
$781$	2.75557	0.0986020
$782$	87.3916i	3.12512i
$783$	0	0
$784$	58.1811	2.07790
$785$	0	0
$786$	0	0
$787$	− 11.2672i	− 0.401632i	−0.979629	−	0.200816i	$-0.935641\pi$
$787$	− 11.2672i	− 0.401632i	0.979629	−	0.200816i	$-0.0643593\pi$
$788$	− 10.8613i	− 0.386918i
$789$	0	0
$790$	0	0
$791$	−26.5718	−0.944786
$792$	0	0
$793$	4.26671i	0.151515i
$794$	−24.2766	−0.861543
$795$	0	0
$796$	22.8760	0.810819
$797$	− 41.9625i	− 1.48639i	−0.669075	−	0.743195i	$-0.733310\pi$
$797$	− 41.9625i	− 1.48639i	0.669075	−	0.743195i	$-0.266690\pi$
$798$	0	0
$799$	−14.2854	−0.505383
$800$	0	0
$801$	0	0
$802$	3.80642i	0.134409i
$803$	4.23506i	0.149452i
$804$	0	0
$805$	0	0
$806$	3.26317	0.114940
$807$	0	0
$808$	3.35322i	0.117966i
$809$	−27.8064	−0.977622	−0.488811	−	0.872390i	$-0.662569\pi$
$809$	−27.8064	−0.977622	−0.488811	+	0.872390i	$0.662569\pi$
$810$	0	0
$811$	6.78415	0.238224	0.119112	−	0.992881i	$-0.461995\pi$
$811$	6.78415	0.238224	0.119112	+	0.992881i	$0.461995\pi$
$812$	− 56.0830i	− 1.96813i
$813$	0	0
$814$	3.80642	0.133415
$815$	0	0
$816$	0	0
$817$	40.0000i	1.39942i
$818$	− 13.5585i	− 0.474060i
$819$	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
$823$	− 42.0642i	− 1.46627i	−0.680085	−	0.733134i	$-0.738057\pi$
$823$	− 42.0642i	− 1.46627i	0.680085	−	0.733134i	$-0.261943\pi$
$824$	8.34968	0.290875
$825$	0	0
$826$	40.9403	1.42449
$827$	− 30.8256i	− 1.07191i	−0.844246	−	0.535956i	$-0.819951\pi$
$827$	− 30.8256i	− 1.07191i	0.844246	−	0.535956i	$-0.180049\pi$
$828$	0	0
$829$	−7.12399	−0.247426	−0.123713	−	0.992318i	$-0.539480\pi$
$829$	−7.12399	−0.247426	−0.123713	+	0.992318i	$0.539480\pi$
$830$	0	0
$831$	0	0
$832$	2.95316i	0.102382i
$833$	− 65.3876i	− 2.26555i
$834$	0	0
$835$	0	0
$836$	−11.4380	−0.395592
$837$	0	0
$838$	− 29.7146i	− 1.02647i
$839$	3.34614	0.115522	0.0577608	−	0.998330i	$-0.481604\pi$
$839$	3.34614	0.115522	0.0577608	+	0.998330i	$0.481604\pi$
$840$	0	0
$841$	31.9403	1.10139
$842$	− 15.0321i	− 0.518041i
$843$	0	0
$844$	17.2988	0.595450
$845$	0	0
$846$	0	0
$847$	− 4.42864i	− 0.152170i
$848$	50.0830i	1.71986i
$849$	0	0
$850$	0	0
$851$	17.7146	0.607247
$852$	0	0
$853$	26.4197i	0.904595i	0.891867	+	0.452297i	$0.149395\pi$
$853$	26.4197i	0.904595i	−0.891867	+	0.452297i	$0.850605\pi$
$854$	−57.7975	−1.97779
$855$	0	0
$856$	−1.88538	−0.0644411
$857$	− 38.7783i	− 1.32464i	−0.749220	−	0.662321i	$-0.769571\pi$
$857$	− 38.7783i	− 1.32464i	0.749220	−	0.662321i	$-0.230429\pi$
$858$	0	0
$859$	27.3087	0.931760	0.465880	−	0.884848i	$-0.345738\pi$
$859$	27.3087	0.931760	0.465880	+	0.884848i	$0.345738\pi$
$860$	0	0
$861$	0	0
$862$	65.2899i	2.22378i
$863$	− 49.5308i	− 1.68605i	−0.537875	−	0.843024i	$-0.680773\pi$
$863$	− 49.5308i	− 1.68605i	0.537875	−	0.843024i	$-0.319227\pi$
$864$	0	0
$865$	0	0
$866$	−27.5397	−0.935838
$867$	0	0
$868$	19.7966i	0.671940i
$869$	8.56199	0.290446
$870$	0	0
$871$	0.774305	0.0262363
$872$	− 14.1748i	− 0.480021i
$873$	0	0
$874$	−118.858	−4.02044
$875$	0	0
$876$	0	0
$877$	− 4.50177i	− 0.152014i	−0.997107	−	0.0760070i	$-0.975783\pi$
$877$	− 4.50177i	− 0.152014i	0.997107	−	0.0760070i	$-0.0242171\pi$
$878$	− 36.7654i	− 1.24077i
$879$	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
$883$	30.2480i	1.01793i	0.860789	+	0.508963i	$0.169971\pi$
$883$	30.2480i	1.01793i	−0.860789	+	0.508963i	$0.830029\pi$
$884$	5.23277	0.175997
$885$	0	0
$886$	−24.9777	−0.839143
$887$	− 57.1941i	− 1.92039i	−0.279333	−	0.960194i	$-0.590113\pi$
$887$	− 57.1941i	− 1.92039i	0.279333	−	0.960194i	$-0.409887\pi$
$888$	0	0
$889$	−67.2454	−2.25534
$890$	0	0
$891$	0	0
$892$	14.3684i	0.481090i
$893$	− 19.4291i	− 0.650171i
$894$	0	0
$895$	0	0
$896$	25.0192	0.835833
$897$	0	0
$898$	61.4835i	2.05173i
$899$	−21.5111	−0.717437
$900$	0	0
$901$	56.2864	1.87517
$902$	0.368416i	0.0122669i
$903$	0	0
$904$	−4.31402	−0.143482
$905$	0	0
$906$	0	0
$907$	− 53.2641i	− 1.76861i	−0.466913	−	0.884303i	$-0.654634\pi$
$907$	− 53.2641i	− 1.76861i	0.466913	−	0.884303i	$-0.345366\pi$
$908$	21.7017i	0.720195i
$909$	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
$913$	− 0.133353i	− 0.00441334i
$914$	−44.6508	−1.47692
$915$	0	0
$916$	18.6735	0.616991
$917$	5.51114i	0.181994i
$918$	0	0
$919$	−55.8707	−1.84300	−0.921502	−	0.388375i	$-0.873037\pi$
$919$	−55.8707	−1.84300	−0.921502	+	0.388375i	$0.873037\pi$
$920$	0	0
$921$	0	0
$922$	− 54.9403i	− 1.80936i
$923$	− 1.71456i	− 0.0564354i
$924$	0	0
$925$	0	0
$926$	36.8198	1.20997
$927$	0	0
$928$	− 57.3087i	− 1.88125i
$929$	15.3274	0.502876	0.251438	−	0.967873i	$-0.419097\pi$
$929$	15.3274	0.502876	0.251438	+	0.967873i	$0.419097\pi$
$930$	0	0
$931$	88.9314	2.91461
$932$	7.01921i	0.229922i
$933$	0	0
$934$	−5.98126	−0.195713
$935$	0	0
$936$	0	0
$937$	− 27.8479i	− 0.909752i	−0.890555	−	0.454876i	$-0.849684\pi$
$937$	− 27.8479i	− 0.909752i	0.890555	−	0.454876i	$-0.150316\pi$
$938$	10.4889i	0.342474i
$939$	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
$943$	1.71456i	0.0558337i
$944$	22.4059	0.729250
$945$	0	0
$946$	10.7971	0.351043
$947$	− 8.47013i	− 0.275242i	−0.990485	−	0.137621i	$-0.956054\pi$
$947$	− 8.47013i	− 0.275242i	0.990485	−	0.137621i	$-0.0439456\pi$
$948$	0	0
$949$	2.63512	0.0855397
$950$	0	0
$951$	0	0
$952$	− 16.5076i	− 0.535014i
$953$	− 8.71408i	− 0.282277i	−0.989990	−	0.141138i	$-0.954924\pi$
$953$	− 8.71408i	− 0.282277i	0.989990	−	0.141138i	$-0.0450763\pi$
$954$	0	0
$955$	0	0
$956$	5.42816	0.175559
$957$	0	0
$958$	47.3087i	1.52847i
$959$	2.16500	0.0699114
$960$	0	0
$961$	−23.4068	−0.755059
$962$	− 2.36842i	− 0.0763608i
$963$	0	0
$964$	2.18373	0.0703333
$965$	0	0
$966$	0	0
$967$	44.2449i	1.42282i	0.702777	+	0.711410i	$0.251943\pi$
$967$	44.2449i	1.42282i	−0.702777	+	0.711410i	$0.748057\pi$
$968$	− 0.719004i	− 0.0231097i
$969$	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
$973$	78.8582i	2.52808i
$974$	−21.9438	−0.703124
$975$	0	0
$976$	−31.6316	−1.01250
$977$	16.2480i	0.519819i	0.965633	+	0.259909i	$0.0836927\pi$
$977$	16.2480i	0.519819i	−0.965633	+	0.259909i	$0.916307\pi$
$978$	0	0
$979$	−5.61285	−0.179387
$980$	0	0
$981$	0	0
$982$	31.1882i	0.995256i
$983$	1.12399i	0.0358496i	0.999839	+	0.0179248i	$0.00570594\pi$
$983$	1.12399i	0.0358496i	−0.999839	+	0.0179248i	$0.994294\pi$
$984$	0	0
$985$	0	0
$986$	−77.0232	−2.45292
$987$	0	0
$988$	7.11691i	0.226419i
$989$	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	$0.824698\pi$
$992$	20.2292i	0.642279i
$993$	0	0
$994$	23.2257	0.736674
$995$	0	0
$996$	0	0
$997$	− 35.7275i	− 1.13150i	−0.824577	−	0.565750i	$-0.808587\pi$
$997$	− 35.7275i	− 1.13150i	0.824577	−	0.565750i	$-0.191413\pi$
$998$	− 48.2034i	− 1.52585i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.c.1.3	✓	3	15.8	even	4
495.2.a.e.1.1		3	5.3	odd	4
825.2.a.k.1.1		3	15.2	even	4
825.2.c.g.199.2		6	15.14	odd	2
825.2.c.g.199.5		6	3.2	odd	2
1815.2.a.m.1.1		3	165.98	odd	4
2475.2.a.bb.1.3		3	5.2	odd	4
2475.2.c.r.199.2		6	1.1	even	1	trivial
2475.2.c.r.199.5		6	5.4	even	2	inner
2640.2.a.be.1.3		3	60.23	odd	4
5445.2.a.z.1.3		3	55.43	even	4
7920.2.a.cj.1.3		3	20.3	even	4
8085.2.a.bk.1.3		3	105.83	odd	4
9075.2.a.cf.1.3		3	165.32	odd	4

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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.90321i q^{2} -1.62222 q^{4} -4.42864i q^{7} -0.719004i q^{8} -1.00000 q^{11} +0.622216i q^{13} -8.42864 q^{14} -4.61285 q^{16} +5.18421i q^{17} -7.05086 q^{19} +1.90321i q^{22} +8.85728i q^{23} +1.18421 q^{26} +7.18421i q^{28} -7.80642 q^{29} +2.75557 q^{31} +7.34122i q^{32} +9.86665 q^{34} -2.00000i q^{37} +13.4193i q^{38} +0.193576 q^{41} -5.67307i q^{43} +1.62222 q^{44} +16.8573 q^{46} +2.75557i q^{47} -12.6128 q^{49} -1.00937i q^{52} -10.8573i q^{53} -3.18421 q^{56} +14.8573i q^{58} -4.85728 q^{59} +6.85728 q^{61} -5.24443i q^{62} +4.74620 q^{64} -1.24443i q^{67} -8.40990i q^{68} -2.75557 q^{71} -4.23506i q^{73} -3.80642 q^{74} +11.4380 q^{76} +4.42864i q^{77} -8.56199 q^{79} -0.368416i q^{82} +0.133353i q^{83} -10.7971 q^{86} +0.719004i q^{88} +5.61285 q^{89} +2.75557 q^{91} -14.3684i q^{92} +5.24443 q^{94} +7.24443i q^{97} +24.0049i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} - 6 q^{11} - 24 q^{14} + 26 q^{16} - 16 q^{19} - 20 q^{26} - 20 q^{29} + 16 q^{31} + 60 q^{34} + 28 q^{41} + 10 q^{44} + 48 q^{46} - 22 q^{49} + 8 q^{56} + 24 q^{59} - 12 q^{61} - 26 q^{64}+ \cdots + 32 q^{94}+O(q^{100}) \) 6 * q - 10 * q^4 - 6 * q^11 - 24 * q^14 + 26 * q^16 - 16 * q^19 - 20 * q^26 - 20 * q^29 + 16 * q^31 + 60 * q^34 + 28 * q^41 + 10 * q^44 + 48 * q^46 - 22 * q^49 + 8 * q^56 + 24 * q^59 - 12 * q^61 - 26 * q^64 - 16 * q^71 + 4 * q^74 + 96 * q^76 - 24 * q^79 + 16 * q^86 - 20 * q^89 + 16 * q^91 + 32 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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