LMFDB - Embedded newform 5625.2.a.r.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.46840000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 $ x^6 - x^5 - 11x^4 + 8x^3 + 31x^2 - 15x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1875)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.13324$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.13324 q^{2} +2.55073 q^{4} +2.16876 q^{7} +1.17484 q^{8} +O(q^{10})$	$q+2.13324 q^{2} +2.55073 q^{4} +2.16876 q^{7} +1.17484 q^{8} +2.50913 q^{11} -4.33379 q^{13} +4.62650 q^{14} -2.59524 q^{16} +6.77562 q^{17} +6.83602 q^{19} +5.35259 q^{22} +1.67843 q^{23} -9.24504 q^{26} +5.53193 q^{28} -4.44927 q^{29} +6.56295 q^{31} -7.88596 q^{32} +14.4541 q^{34} +7.97720 q^{37} +14.5829 q^{38} -11.2249 q^{41} -4.25487 q^{43} +6.40012 q^{44} +3.58050 q^{46} -4.98652 q^{47} -2.29646 q^{49} -11.0543 q^{52} +8.21338 q^{53} +2.54795 q^{56} -9.49138 q^{58} -3.67416 q^{59} +4.93960 q^{61} +14.0004 q^{62} -11.6322 q^{64} +11.5812 q^{67} +17.2828 q^{68} +2.30251 q^{71} -1.11599 q^{73} +17.0173 q^{74} +17.4368 q^{76} +5.44172 q^{77} +7.78306 q^{79} -23.9454 q^{82} -9.87708 q^{83} -9.07667 q^{86} +2.94783 q^{88} +1.24025 q^{89} -9.39897 q^{91} +4.28122 q^{92} -10.6375 q^{94} +15.0488 q^{97} -4.89892 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} + 11 q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + q^2 + 11 * q^4 + 2 * q^7 + 6 * q^8	$ 6 q + q^{2} + 11 q^{4} + 2 q^{7} + 6 q^{8} + 4 q^{14} + 17 q^{16} + 2 q^{17} - 2 q^{19} - 9 q^{22} - q^{23} - 37 q^{26} + 44 q^{28} - 31 q^{29} - 2 q^{31} + 33 q^{32} + 37 q^{34} + 22 q^{37} + 27 q^{38} - 33 q^{41} + 3 q^{43} + 11 q^{44} - 12 q^{46} - 6 q^{47} + 4 q^{49} + 33 q^{52} + 14 q^{53} + 30 q^{56} + q^{58} + 8 q^{59} + 34 q^{61} + 31 q^{62} + 12 q^{64} - 2 q^{67} + 27 q^{68} + 3 q^{71} + 36 q^{73} - 36 q^{74} + 27 q^{76} + 16 q^{77} + 25 q^{79} - 36 q^{82} - 12 q^{83} + 30 q^{86} - 56 q^{88} - 18 q^{89} + 28 q^{91} + 3 q^{92} - 50 q^{94} - 7 q^{97} + 15 q^{98}+O(q^{100}) $ 6 * q + q^2 + 11 * q^4 + 2 * q^7 + 6 * q^8 + 4 * q^14 + 17 * q^16 + 2 * q^17 - 2 * q^19 - 9 * q^22 - q^23 - 37 * q^26 + 44 * q^28 - 31 * q^29 - 2 * q^31 + 33 * q^32 + 37 * q^34 + 22 * q^37 + 27 * q^38 - 33 * q^41 + 3 * q^43 + 11 * q^44 - 12 * q^46 - 6 * q^47 + 4 * q^49 + 33 * q^52 + 14 * q^53 + 30 * q^56 + q^58 + 8 * q^59 + 34 * q^61 + 31 * q^62 + 12 * q^64 - 2 * q^67 + 27 * q^68 + 3 * q^71 + 36 * q^73 - 36 * q^74 + 27 * q^76 + 16 * q^77 + 25 * q^79 - 36 * q^82 - 12 * q^83 + 30 * q^86 - 56 * q^88 - 18 * q^89 + 28 * q^91 + 3 * q^92 - 50 * q^94 - 7 * q^97 + 15 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	2.13324	1.50843	0.754216	−	0.656627i	$-0.228017\pi$
$2$	2.13324	1.50843	0.754216	+	0.656627i	$0.228017\pi$
$3$	0	0
$3$	0	0
$4$	2.55073	1.27536
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.16876	0.819716	0.409858	−	0.912149i	$-0.365578\pi$
$7$	2.16876	0.819716	0.409858	+	0.912149i	$0.365578\pi$
$8$	1.17484	0.415369
$9$	0	0
$10$	0	0
$11$	2.50913	0.756532	0.378266	−	0.925697i	$-0.376520\pi$
$11$	2.50913	0.756532	0.378266	+	0.925697i	$0.376520\pi$
$12$	0	0
$13$	−4.33379	−1.20198	−0.600989	−	0.799257i	$-0.705226\pi$
$13$	−4.33379	−1.20198	−0.600989	+	0.799257i	$0.705226\pi$
$14$	4.62650	1.23648
$15$	0	0
$16$	−2.59524	−0.648810
$17$	6.77562	1.64333	0.821665	−	0.569971i	$-0.193046\pi$
$17$	6.77562	1.64333	0.821665	+	0.569971i	$0.193046\pi$
$18$	0	0
$19$	6.83602	1.56829	0.784145	−	0.620578i	$-0.213102\pi$
$19$	6.83602	1.56829	0.784145	+	0.620578i	$0.213102\pi$
$20$	0	0
$21$	0	0
$22$	5.35259	1.14118
$23$	1.67843	0.349977	0.174989	−	0.984570i	$-0.444011\pi$
$23$	1.67843	0.349977	0.174989	+	0.984570i	$0.444011\pi$
$24$	0	0
$25$	0	0
$26$	−9.24504	−1.81310
$27$	0	0
$28$	5.53193	1.04544
$29$	−4.44927	−0.826209	−0.413104	−	0.910684i	$-0.635556\pi$
$29$	−4.44927	−0.826209	−0.413104	+	0.910684i	$0.635556\pi$
$30$	0	0
$31$	6.56295	1.17874	0.589371	−	0.807863i	$-0.299376\pi$
$31$	6.56295	1.17874	0.589371	+	0.807863i	$0.299376\pi$
$32$	−7.88596	−1.39405
$33$	0	0
$34$	14.4541	2.47885
$35$	0	0
$36$	0	0
$37$	7.97720	1.31144	0.655722	−	0.755002i	$-0.272364\pi$
$37$	7.97720	1.31144	0.655722	+	0.755002i	$0.272364\pi$
$38$	14.5829	2.36566
$39$	0	0
$40$	0	0
$41$	−11.2249	−1.75303	−0.876517	−	0.481371i	$-0.840139\pi$
$41$	−11.2249	−1.75303	−0.876517	+	0.481371i	$0.840139\pi$
$42$	0	0
$43$	−4.25487	−0.648861	−0.324431	−	0.945909i	$-0.605173\pi$
$43$	−4.25487	−0.648861	−0.324431	+	0.945909i	$0.605173\pi$
$44$	6.40012	0.964854
$45$	0	0
$46$	3.58050	0.527916
$47$	−4.98652	−0.727358	−0.363679	−	0.931524i	$-0.618479\pi$
$47$	−4.98652	−0.727358	−0.363679	+	0.931524i	$0.618479\pi$
$48$	0	0
$49$	−2.29646	−0.328066
$50$	0	0
$51$	0	0
$52$	−11.0543	−1.53296
$53$	8.21338	1.12820	0.564098	−	0.825708i	$-0.309224\pi$
$53$	8.21338	1.12820	0.564098	+	0.825708i	$0.309224\pi$
$54$	0	0
$55$	0	0
$56$	2.54795	0.340484
$57$	0	0
$58$	−9.49138	−1.24628
$59$	−3.67416	−0.478335	−0.239168	−	0.970978i	$-0.576875\pi$
$59$	−3.67416	−0.478335	−0.239168	+	0.970978i	$0.576875\pi$
$60$	0	0
$61$	4.93960	0.632451	0.316226	−	0.948684i	$-0.397584\pi$
$61$	4.93960	0.632451	0.316226	+	0.948684i	$0.397584\pi$
$62$	14.0004	1.77805
$63$	0	0
$64$	−11.6322	−1.45402
$65$	0	0
$66$	0	0
$67$	11.5812	1.41487	0.707436	−	0.706778i	$-0.249852\pi$
$67$	11.5812	1.41487	0.707436	+	0.706778i	$0.249852\pi$
$68$	17.2828	2.09584
$69$	0	0
$70$	0	0
$71$	2.30251	0.273257	0.136629	−	0.990622i	$-0.456373\pi$
$71$	2.30251	0.273257	0.136629	+	0.990622i	$0.456373\pi$
$72$	0	0
$73$	−1.11599	−0.130617	−0.0653084	−	0.997865i	$-0.520803\pi$
$73$	−1.11599	−0.130617	−0.0653084	+	0.997865i	$0.520803\pi$
$74$	17.0173	1.97822
$75$	0	0
$76$	17.4368	2.00014
$77$	5.44172	0.620141
$78$	0	0
$79$	7.78306	0.875663	0.437831	−	0.899057i	$-0.355747\pi$
$79$	7.78306	0.875663	0.437831	+	0.899057i	$0.355747\pi$
$80$	0	0
$81$	0	0
$82$	−23.9454	−2.64433
$83$	−9.87708	−1.08415	−0.542075	−	0.840330i	$-0.682361\pi$
$83$	−9.87708	−1.08415	−0.542075	+	0.840330i	$0.682361\pi$
$84$	0	0
$85$	0	0
$86$	−9.07667	−0.978763
$87$	0	0
$88$	2.94783	0.314240
$89$	1.24025	0.131466	0.0657329	−	0.997837i	$-0.479061\pi$
$89$	1.24025	0.131466	0.0657329	+	0.997837i	$0.479061\pi$
$90$	0	0
$91$	−9.39897	−0.985280
$92$	4.28122	0.446348
$93$	0	0
$94$	−10.6375	−1.09717
$95$	0	0
$96$	0	0
$97$	15.0488	1.52797	0.763985	−	0.645234i	$-0.223240\pi$
$97$	15.0488	1.52797	0.763985	+	0.645234i	$0.223240\pi$
$98$	−4.89892	−0.494866
$99$	0	0
$100$	0	0
$101$	17.0211	1.69366	0.846830	−	0.531864i	$-0.178508\pi$
$101$	17.0211	1.69366	0.846830	+	0.531864i	$0.178508\pi$
$102$	0	0
$103$	10.0859	0.993793	0.496897	−	0.867810i	$-0.334473\pi$
$103$	10.0859	0.993793	0.496897	+	0.867810i	$0.334473\pi$
$104$	−5.09151	−0.499264
$105$	0	0
$106$	17.5212	1.70180
$107$	−5.67218	−0.548351	−0.274175	−	0.961680i	$-0.588405\pi$
$107$	−5.67218	−0.548351	−0.274175	+	0.961680i	$0.588405\pi$
$108$	0	0
$109$	8.34557	0.799361	0.399680	−	0.916655i	$-0.369121\pi$
$109$	8.34557	0.799361	0.399680	+	0.916655i	$0.369121\pi$
$110$	0	0
$111$	0	0
$112$	−5.62846	−0.531839
$113$	14.8224	1.39438	0.697188	−	0.716888i	$-0.254434\pi$
$113$	14.8224	1.39438	0.697188	+	0.716888i	$0.254434\pi$
$114$	0	0
$115$	0	0
$116$	−11.3489	−1.05372
$117$	0	0
$118$	−7.83788	−0.721536
$119$	14.6947	1.34706
$120$	0	0
$121$	−4.70425	−0.427659
$122$	10.5374	0.954009
$123$	0	0
$124$	16.7403	1.50332
$125$	0	0
$126$	0	0
$127$	−0.502113	−0.0445553	−0.0222777	−	0.999752i	$-0.507092\pi$
$127$	−0.502113	−0.0445553	−0.0222777	+	0.999752i	$0.507092\pi$
$128$	−9.04239	−0.799242
$129$	0	0
$130$	0	0
$131$	−6.05788	−0.529280	−0.264640	−	0.964347i	$-0.585253\pi$
$131$	−6.05788	−0.529280	−0.264640	+	0.964347i	$0.585253\pi$
$132$	0	0
$133$	14.8257	1.28555
$134$	24.7056	2.13424
$135$	0	0
$136$	7.96027	0.682588
$137$	1.94014	0.165757	0.0828786	−	0.996560i	$-0.473589\pi$
$137$	1.94014	0.165757	0.0828786	+	0.996560i	$0.473589\pi$
$138$	0	0
$139$	19.2023	1.62872	0.814358	−	0.580363i	$-0.197089\pi$
$139$	19.2023	1.62872	0.814358	+	0.580363i	$0.197089\pi$
$140$	0	0
$141$	0	0
$142$	4.91181	0.412190
$143$	−10.8741	−0.909335
$144$	0	0
$145$	0	0
$146$	−2.38068	−0.197026
$147$	0	0
$148$	20.3477	1.67257
$149$	−9.49138	−0.777564	−0.388782	−	0.921330i	$-0.627104\pi$
$149$	−9.49138	−0.777564	−0.388782	+	0.921330i	$0.627104\pi$
$150$	0	0
$151$	−12.5747	−1.02332	−0.511659	−	0.859189i	$-0.670969\pi$
$151$	−12.5747	−1.02332	−0.511659	+	0.859189i	$0.670969\pi$
$152$	8.03123	0.651419
$153$	0	0
$154$	11.6085	0.935440
$155$	0	0
$156$	0	0
$157$	−18.7991	−1.50033	−0.750165	−	0.661250i	$-0.770026\pi$
$157$	−18.7991	−1.50033	−0.750165	+	0.661250i	$0.770026\pi$
$158$	16.6032	1.32088
$159$	0	0
$160$	0	0
$161$	3.64012	0.286882
$162$	0	0
$163$	−7.29949	−0.571740	−0.285870	−	0.958268i	$-0.592283\pi$
$163$	−7.29949	−0.571740	−0.285870	+	0.958268i	$0.592283\pi$
$164$	−28.6317	−2.23576
$165$	0	0
$166$	−21.0702	−1.63537
$167$	−14.7991	−1.14519	−0.572594	−	0.819839i	$-0.694063\pi$
$167$	−14.7991	−1.14519	−0.572594	+	0.819839i	$0.694063\pi$
$168$	0	0
$169$	5.78176	0.444750
$170$	0	0
$171$	0	0
$172$	−10.8530	−0.827535
$173$	−2.29460	−0.174455	−0.0872276	−	0.996188i	$-0.527801\pi$
$173$	−2.29460	−0.174455	−0.0872276	+	0.996188i	$0.527801\pi$
$174$	0	0
$175$	0	0
$176$	−6.51180	−0.490845
$177$	0	0
$178$	2.64575	0.198307
$179$	−13.6637	−1.02127	−0.510636	−	0.859797i	$-0.670590\pi$
$179$	−13.6637	−1.02127	−0.510636	+	0.859797i	$0.670590\pi$
$180$	0	0
$181$	8.71878	0.648062	0.324031	−	0.946046i	$-0.394962\pi$
$181$	8.71878	0.648062	0.324031	+	0.946046i	$0.394962\pi$
$182$	−20.0503	−1.48623
$183$	0	0
$184$	1.97189	0.145370
$185$	0	0
$186$	0	0
$187$	17.0009	1.24323
$188$	−12.7193	−0.927647
$189$	0	0
$190$	0	0
$191$	−11.6080	−0.839925	−0.419963	−	0.907541i	$-0.637957\pi$
$191$	−11.6080	−0.839925	−0.419963	+	0.907541i	$0.637957\pi$
$192$	0	0
$193$	14.8653	1.07002	0.535012	−	0.844844i	$-0.320307\pi$
$193$	14.8653	1.07002	0.535012	+	0.844844i	$0.320307\pi$
$194$	32.1027	2.30484
$195$	0	0
$196$	−5.85766	−0.418404
$197$	22.8292	1.62651	0.813256	−	0.581906i	$-0.197693\pi$
$197$	22.8292	1.62651	0.813256	+	0.581906i	$0.197693\pi$
$198$	0	0
$199$	5.87697	0.416607	0.208304	−	0.978064i	$-0.433206\pi$
$199$	5.87697	0.416607	0.208304	+	0.978064i	$0.433206\pi$
$200$	0	0
$201$	0	0
$202$	36.3101	2.55477
$203$	−9.64942	−0.677256
$204$	0	0
$205$	0	0
$206$	21.5157	1.49907
$207$	0	0
$208$	11.2472	0.779855
$209$	17.1525	1.18646
$210$	0	0
$211$	−24.6884	−1.69962	−0.849810	−	0.527090i	$-0.823283\pi$
$211$	−24.6884	−1.69962	−0.849810	+	0.527090i	$0.823283\pi$
$212$	20.9501	1.43886
$213$	0	0
$214$	−12.1002	−0.827149
$215$	0	0
$216$	0	0
$217$	14.2335	0.966232
$218$	17.8031	1.20578
$219$	0	0
$220$	0	0
$221$	−29.3641	−1.97525
$222$	0	0
$223$	3.31060	0.221694	0.110847	−	0.993837i	$-0.464644\pi$
$223$	3.31060	0.221694	0.110847	+	0.993837i	$0.464644\pi$
$224$	−17.1028	−1.14273
$225$	0	0
$226$	31.6198	2.10332
$227$	−2.70185	−0.179328	−0.0896641	−	0.995972i	$-0.528579\pi$
$227$	−2.70185	−0.179328	−0.0896641	+	0.995972i	$0.528579\pi$
$228$	0	0
$229$	−9.33473	−0.616856	−0.308428	−	0.951248i	$-0.599803\pi$
$229$	−9.33473	−0.616856	−0.308428	+	0.951248i	$0.599803\pi$
$230$	0	0
$231$	0	0
$232$	−5.22718	−0.343181
$233$	9.52491	0.623998	0.311999	−	0.950083i	$-0.399002\pi$
$233$	9.52491	0.623998	0.311999	+	0.950083i	$0.399002\pi$
$234$	0	0
$235$	0	0
$236$	−9.37179	−0.610052
$237$	0	0
$238$	31.3474	2.03195
$239$	−7.84648	−0.507546	−0.253773	−	0.967264i	$-0.581672\pi$
$239$	−7.84648	−0.507546	−0.253773	+	0.967264i	$0.581672\pi$
$240$	0	0
$241$	1.42783	0.0919747	0.0459874	−	0.998942i	$-0.485357\pi$
$241$	1.42783	0.0919747	0.0459874	+	0.998942i	$0.485357\pi$
$242$	−10.0353	−0.645095
$243$	0	0
$244$	12.5996	0.806606
$245$	0	0
$246$	0	0
$247$	−29.6259	−1.88505
$248$	7.71042	0.489612
$249$	0	0
$250$	0	0
$251$	17.5762	1.10940	0.554701	−	0.832050i	$-0.312833\pi$
$251$	17.5762	1.10940	0.554701	+	0.832050i	$0.312833\pi$
$252$	0	0
$253$	4.21141	0.264769
$254$	−1.07113	−0.0672087
$255$	0	0
$256$	3.97476	0.248423
$257$	−19.2890	−1.20322	−0.601608	−	0.798791i	$-0.705473\pi$
$257$	−19.2890	−1.20322	−0.601608	+	0.798791i	$0.705473\pi$
$258$	0	0
$259$	17.3007	1.07501
$260$	0	0
$261$	0	0
$262$	−12.9229	−0.798382
$263$	−22.2432	−1.37157	−0.685786	−	0.727803i	$-0.740541\pi$
$263$	−22.2432	−1.37157	−0.685786	+	0.727803i	$0.740541\pi$
$264$	0	0
$265$	0	0
$266$	31.6268	1.93917
$267$	0	0
$268$	29.5406	1.80448
$269$	−25.1785	−1.53516	−0.767581	−	0.640951i	$-0.778540\pi$
$269$	−25.1785	−1.53516	−0.767581	+	0.640951i	$0.778540\pi$
$270$	0	0
$271$	−5.70091	−0.346306	−0.173153	−	0.984895i	$-0.555395\pi$
$271$	−5.70091	−0.346306	−0.173153	+	0.984895i	$0.555395\pi$
$272$	−17.5843	−1.06621
$273$	0	0
$274$	4.13879	0.250033
$275$	0	0
$276$	0	0
$277$	−19.1611	−1.15128	−0.575638	−	0.817704i	$-0.695246\pi$
$277$	−19.1611	−1.15128	−0.575638	+	0.817704i	$0.695246\pi$
$278$	40.9631	2.45681
$279$	0	0
$280$	0	0
$281$	−4.36015	−0.260105	−0.130052	−	0.991507i	$-0.541515\pi$
$281$	−4.36015	−0.260105	−0.130052	+	0.991507i	$0.541515\pi$
$282$	0	0
$283$	24.9081	1.48063	0.740317	−	0.672258i	$-0.234676\pi$
$283$	24.9081	1.48063	0.740317	+	0.672258i	$0.234676\pi$
$284$	5.87307	0.348503
$285$	0	0
$286$	−23.1970	−1.37167
$287$	−24.3441	−1.43699
$288$	0	0
$289$	28.9090	1.70053
$290$	0	0
$291$	0	0
$292$	−2.84659	−0.166584
$293$	−13.5651	−0.792484	−0.396242	−	0.918146i	$-0.629686\pi$
$293$	−13.5651	−0.792484	−0.396242	+	0.918146i	$0.629686\pi$
$294$	0	0
$295$	0	0
$296$	9.37194	0.544733
$297$	0	0
$298$	−20.2474	−1.17290
$299$	−7.27397	−0.420665
$300$	0	0
$301$	−9.22780	−0.531882
$302$	−26.8250	−1.54360
$303$	0	0
$304$	−17.7411	−1.01752
$305$	0	0
$306$	0	0
$307$	11.9672	0.683007	0.341504	−	0.939880i	$-0.389064\pi$
$307$	11.9672	0.683007	0.341504	+	0.939880i	$0.389064\pi$
$308$	13.8803	0.790906
$309$	0	0
$310$	0	0
$311$	−7.66452	−0.434615	−0.217308	−	0.976103i	$-0.569727\pi$
$311$	−7.66452	−0.434615	−0.217308	+	0.976103i	$0.569727\pi$
$312$	0	0
$313$	−26.3840	−1.49131	−0.745655	−	0.666332i	$-0.767863\pi$
$313$	−26.3840	−1.49131	−0.745655	+	0.666332i	$0.767863\pi$
$314$	−40.1030	−2.26315
$315$	0	0
$316$	19.8525	1.11679
$317$	−10.8880	−0.611529	−0.305765	−	0.952107i	$-0.598912\pi$
$317$	−10.8880	−0.611529	−0.305765	+	0.952107i	$0.598912\pi$
$318$	0	0
$319$	−11.1638	−0.625053
$320$	0	0
$321$	0	0
$322$	7.76526	0.432741
$323$	46.3183	2.57722
$324$	0	0
$325$	0	0
$326$	−15.5716	−0.862431
$327$	0	0
$328$	−13.1875	−0.728155
$329$	−10.8146	−0.596227
$330$	0	0
$331$	−32.2137	−1.77062	−0.885312	−	0.464998i	$-0.846055\pi$
$331$	−32.2137	−1.77062	−0.885312	+	0.464998i	$0.846055\pi$
$332$	−25.1938	−1.38269
$333$	0	0
$334$	−31.5701	−1.72744
$335$	0	0
$336$	0	0
$337$	3.47169	0.189115	0.0945576	−	0.995519i	$-0.469856\pi$
$337$	3.47169	0.189115	0.0945576	+	0.995519i	$0.469856\pi$
$338$	12.3339	0.670875
$339$	0	0
$340$	0	0
$341$	16.4673	0.891755
$342$	0	0
$343$	−20.1618	−1.08864
$344$	−4.99879	−0.269517
$345$	0	0
$346$	−4.89494	−0.263154
$347$	−10.9805	−0.589465	−0.294733	−	0.955580i	$-0.595231\pi$
$347$	−10.9805	−0.589465	−0.294733	+	0.955580i	$0.595231\pi$
$348$	0	0
$349$	−26.3158	−1.40865	−0.704325	−	0.709878i	$-0.748750\pi$
$349$	−26.3158	−1.40865	−0.704325	+	0.709878i	$0.748750\pi$
$350$	0	0
$351$	0	0
$352$	−19.7869	−1.05465
$353$	−16.3668	−0.871119	−0.435559	−	0.900160i	$-0.643449\pi$
$353$	−16.3668	−0.871119	−0.435559	+	0.900160i	$0.643449\pi$
$354$	0	0
$355$	0	0
$356$	3.16353	0.167667
$357$	0	0
$358$	−29.1480	−1.54052
$359$	17.4871	0.922934	0.461467	−	0.887157i	$-0.347323\pi$
$359$	17.4871	0.922934	0.461467	+	0.887157i	$0.347323\pi$
$360$	0	0
$361$	27.7311	1.45953
$362$	18.5993	0.977557
$363$	0	0
$364$	−23.9742	−1.25659
$365$	0	0
$366$	0	0
$367$	−8.29594	−0.433044	−0.216522	−	0.976278i	$-0.569471\pi$
$367$	−8.29594	−0.433044	−0.216522	+	0.976278i	$0.569471\pi$
$368$	−4.35593	−0.227068
$369$	0	0
$370$	0	0
$371$	17.8129	0.924799
$372$	0	0
$373$	19.1295	0.990486	0.495243	−	0.868755i	$-0.335079\pi$
$373$	19.1295	0.990486	0.495243	+	0.868755i	$0.335079\pi$
$374$	36.2671	1.87533
$375$	0	0
$376$	−5.85836	−0.302122
$377$	19.2822	0.993085
$378$	0	0
$379$	−27.0952	−1.39179	−0.695894	−	0.718145i	$-0.744992\pi$
$379$	−27.0952	−1.39179	−0.695894	+	0.718145i	$0.744992\pi$
$380$	0	0
$381$	0	0
$382$	−24.7627	−1.26697
$383$	19.8026	1.01187	0.505933	−	0.862573i	$-0.331148\pi$
$383$	19.8026	1.01187	0.505933	+	0.862573i	$0.331148\pi$
$384$	0	0
$385$	0	0
$386$	31.7112	1.61406
$387$	0	0
$388$	38.3853	1.94872
$389$	23.0130	1.16680	0.583402	−	0.812183i	$-0.301721\pi$
$389$	23.0130	1.16680	0.583402	+	0.812183i	$0.301721\pi$
$390$	0	0
$391$	11.3724	0.575128
$392$	−2.69798	−0.136269
$393$	0	0
$394$	48.7002	2.45348
$395$	0	0
$396$	0	0
$397$	20.5745	1.03260	0.516301	−	0.856407i	$-0.327309\pi$
$397$	20.5745	1.03260	0.516301	+	0.856407i	$0.327309\pi$
$398$	12.5370	0.628423
$399$	0	0
$400$	0	0
$401$	13.1542	0.656889	0.328444	−	0.944523i	$-0.393476\pi$
$401$	13.1542	0.656889	0.328444	+	0.944523i	$0.393476\pi$
$402$	0	0
$403$	−28.4425	−1.41682
$404$	43.4161	2.16003
$405$	0	0
$406$	−20.5846	−1.02159
$407$	20.0159	0.992150
$408$	0	0
$409$	2.76531	0.136736	0.0683679	−	0.997660i	$-0.478221\pi$
$409$	2.76531	0.136736	0.0683679	+	0.997660i	$0.478221\pi$
$410$	0	0
$411$	0	0
$412$	25.7264	1.26745
$413$	−7.96839	−0.392099
$414$	0	0
$415$	0	0
$416$	34.1761	1.67562
$417$	0	0
$418$	36.5904	1.78970
$419$	9.70852	0.474292	0.237146	−	0.971474i	$-0.423788\pi$
$419$	9.70852	0.474292	0.237146	+	0.971474i	$0.423788\pi$
$420$	0	0
$421$	−4.21745	−0.205546	−0.102773	−	0.994705i	$-0.532772\pi$
$421$	−4.21745	−0.205546	−0.102773	+	0.994705i	$0.532772\pi$
$422$	−52.6664	−2.56376
$423$	0	0
$424$	9.64942	0.468617
$425$	0	0
$426$	0	0
$427$	10.7128	0.518430
$428$	−14.4682	−0.699347
$429$	0	0
$430$	0	0
$431$	−18.9785	−0.914164	−0.457082	−	0.889425i	$-0.651105\pi$
$431$	−18.9785	−0.914164	−0.457082	+	0.889425i	$0.651105\pi$
$432$	0	0
$433$	−13.7054	−0.658640	−0.329320	−	0.944218i	$-0.606819\pi$
$433$	−13.7054	−0.658640	−0.329320	+	0.944218i	$0.606819\pi$
$434$	30.3635	1.45750
$435$	0	0
$436$	21.2873	1.01948
$437$	11.4738	0.548866
$438$	0	0
$439$	−33.3843	−1.59335	−0.796673	−	0.604410i	$-0.793409\pi$
$439$	−33.3843	−1.59335	−0.796673	+	0.604410i	$0.793409\pi$
$440$	0	0
$441$	0	0
$442$	−62.6409	−2.97952
$443$	11.6290	0.552512	0.276256	−	0.961084i	$-0.410906\pi$
$443$	11.6290	0.552512	0.276256	+	0.961084i	$0.410906\pi$
$444$	0	0
$445$	0	0
$446$	7.06231	0.334410
$447$	0	0
$448$	−25.2275	−1.19189
$449$	13.3509	0.630069	0.315034	−	0.949080i	$-0.397984\pi$
$449$	13.3509	0.630069	0.315034	+	0.949080i	$0.397984\pi$
$450$	0	0
$451$	−28.1647	−1.32623
$452$	37.8080	1.77834
$453$	0	0
$454$	−5.76371	−0.270504
$455$	0	0
$456$	0	0
$457$	23.3042	1.09013	0.545063	−	0.838395i	$-0.316506\pi$
$457$	23.3042	1.09013	0.545063	+	0.838395i	$0.316506\pi$
$458$	−19.9132	−0.930485
$459$	0	0
$460$	0	0
$461$	−5.03269	−0.234396	−0.117198	−	0.993109i	$-0.537391\pi$
$461$	−5.03269	−0.234396	−0.117198	+	0.993109i	$0.537391\pi$
$462$	0	0
$463$	12.7759	0.593746	0.296873	−	0.954917i	$-0.404056\pi$
$463$	12.7759	0.593746	0.296873	+	0.954917i	$0.404056\pi$
$464$	11.5469	0.536052
$465$	0	0
$466$	20.3190	0.941257
$467$	4.86284	0.225025	0.112513	−	0.993650i	$-0.464110\pi$
$467$	4.86284	0.225025	0.112513	+	0.993650i	$0.464110\pi$
$468$	0	0
$469$	25.1169	1.15979
$470$	0	0
$471$	0	0
$472$	−4.31655	−0.198685
$473$	−10.6760	−0.490884
$474$	0	0
$475$	0	0
$476$	37.4823	1.71800
$477$	0	0
$478$	−16.7385	−0.765599
$479$	−4.79202	−0.218953	−0.109476	−	0.993989i	$-0.534917\pi$
$479$	−4.79202	−0.218953	−0.109476	+	0.993989i	$0.534917\pi$
$480$	0	0
$481$	−34.5715	−1.57633
$482$	3.04591	0.138738
$483$	0	0
$484$	−11.9993	−0.545421
$485$	0	0
$486$	0	0
$487$	−9.16949	−0.415509	−0.207755	−	0.978181i	$-0.566616\pi$
$487$	−9.16949	−0.415509	−0.207755	+	0.978181i	$0.566616\pi$
$488$	5.80325	0.262701
$489$	0	0
$490$	0	0
$491$	−31.9054	−1.43987	−0.719935	−	0.694042i	$-0.755828\pi$
$491$	−31.9054	−1.43987	−0.719935	+	0.694042i	$0.755828\pi$
$492$	0	0
$493$	−30.1466	−1.35773
$494$	−63.1992	−2.84347
$495$	0	0
$496$	−17.0324	−0.764778
$497$	4.99359	0.223993
$498$	0	0
$499$	9.07297	0.406162	0.203081	−	0.979162i	$-0.434905\pi$
$499$	9.07297	0.406162	0.203081	+	0.979162i	$0.434905\pi$
$500$	0	0
$501$	0	0
$502$	37.4944	1.67346
$503$	38.7485	1.72771	0.863856	−	0.503739i	$-0.168043\pi$
$503$	38.7485	1.72771	0.863856	+	0.503739i	$0.168043\pi$
$504$	0	0
$505$	0	0
$506$	8.98396	0.399386
$507$	0	0
$508$	−1.28075	−0.0568243
$509$	−32.9367	−1.45989	−0.729947	−	0.683503i	$-0.760455\pi$
$509$	−32.9367	−1.45989	−0.729947	+	0.683503i	$0.760455\pi$
$510$	0	0
$511$	−2.42032	−0.107069
$512$	26.5639	1.17397
$513$	0	0
$514$	−41.1482	−1.81497
$515$	0	0
$516$	0	0
$517$	−12.5118	−0.550270
$518$	36.9065	1.62158
$519$	0	0
$520$	0	0
$521$	17.7521	0.777735	0.388867	−	0.921294i	$-0.372866\pi$
$521$	17.7521	0.777735	0.388867	+	0.921294i	$0.372866\pi$
$522$	0	0
$523$	4.98678	0.218056	0.109028	−	0.994039i	$-0.465226\pi$
$523$	4.98678	0.218056	0.109028	+	0.994039i	$0.465226\pi$
$524$	−15.4520	−0.675025
$525$	0	0
$526$	−47.4501	−2.06892
$527$	44.4681	1.93706
$528$	0	0
$529$	−20.1829	−0.877516
$530$	0	0
$531$	0	0
$532$	37.8164	1.63955
$533$	48.6463	2.10711
$534$	0	0
$535$	0	0
$536$	13.6061	0.587693
$537$	0	0
$538$	−53.7120	−2.31569
$539$	−5.76214	−0.248193
$540$	0	0
$541$	−3.49960	−0.150460	−0.0752298	−	0.997166i	$-0.523969\pi$
$541$	−3.49960	−0.150460	−0.0752298	+	0.997166i	$0.523969\pi$
$542$	−12.1614	−0.522378
$543$	0	0
$544$	−53.4322	−2.29089
$545$	0	0
$546$	0	0
$547$	17.4640	0.746709	0.373354	−	0.927689i	$-0.378208\pi$
$547$	17.4640	0.746709	0.373354	+	0.927689i	$0.378208\pi$
$548$	4.94877	0.211401
$549$	0	0
$550$	0	0
$551$	−30.4153	−1.29574
$552$	0	0
$553$	16.8796	0.717795
$554$	−40.8752	−1.73662
$555$	0	0
$556$	48.9798	2.07721
$557$	6.02774	0.255403	0.127702	−	0.991813i	$-0.459240\pi$
$557$	6.02774	0.255403	0.127702	+	0.991813i	$0.459240\pi$
$558$	0	0
$559$	18.4397	0.779917
$560$	0	0
$561$	0	0
$562$	−9.30126	−0.392350
$563$	9.19823	0.387659	0.193830	−	0.981035i	$-0.437909\pi$
$563$	9.19823	0.387659	0.193830	+	0.981035i	$0.437909\pi$
$564$	0	0
$565$	0	0
$566$	53.1351	2.23343
$567$	0	0
$568$	2.70508	0.113503
$569$	−9.06517	−0.380032	−0.190016	−	0.981781i	$-0.560854\pi$
$569$	−9.06517	−0.380032	−0.190016	+	0.981781i	$0.560854\pi$
$570$	0	0
$571$	−6.34688	−0.265609	−0.132804	−	0.991142i	$-0.542398\pi$
$571$	−6.34688	−0.265609	−0.132804	+	0.991142i	$0.542398\pi$
$572$	−27.7368	−1.15973
$573$	0	0
$574$	−51.9320	−2.16760
$575$	0	0
$576$	0	0
$577$	−2.41418	−0.100504	−0.0502519	−	0.998737i	$-0.516002\pi$
$577$	−2.41418	−0.100504	−0.0502519	+	0.998737i	$0.516002\pi$
$578$	61.6700	2.56513
$579$	0	0
$580$	0	0
$581$	−21.4211	−0.888695
$582$	0	0
$583$	20.6085	0.853516
$584$	−1.31111	−0.0542541
$585$	0	0
$586$	−28.9378	−1.19541
$587$	4.50976	0.186138	0.0930689	−	0.995660i	$-0.470332\pi$
$587$	4.50976	0.186138	0.0930689	+	0.995660i	$0.470332\pi$
$588$	0	0
$589$	44.8645	1.84861
$590$	0	0
$591$	0	0
$592$	−20.7027	−0.850877
$593$	−33.2824	−1.36674	−0.683371	−	0.730071i	$-0.739487\pi$
$593$	−33.2824	−1.36674	−0.683371	+	0.730071i	$0.739487\pi$
$594$	0	0
$595$	0	0
$596$	−24.2099	−0.991678
$597$	0	0
$598$	−15.5172	−0.634544
$599$	−39.1061	−1.59783	−0.798915	−	0.601444i	$-0.794592\pi$
$599$	−39.1061	−1.59783	−0.798915	+	0.601444i	$0.794592\pi$
$600$	0	0
$601$	−28.8076	−1.17509	−0.587543	−	0.809193i	$-0.699905\pi$
$601$	−28.8076	−1.17509	−0.587543	+	0.809193i	$0.699905\pi$
$602$	−19.6852	−0.802307
$603$	0	0
$604$	−32.0747	−1.30510
$605$	0	0
$606$	0	0
$607$	23.0933	0.937327	0.468663	−	0.883377i	$-0.344736\pi$
$607$	23.0933	0.937327	0.468663	+	0.883377i	$0.344736\pi$
$608$	−53.9085	−2.18628
$609$	0	0
$610$	0	0
$611$	21.6105	0.874268
$612$	0	0
$613$	18.2228	0.736013	0.368007	−	0.929823i	$-0.380040\pi$
$613$	18.2228	0.736013	0.368007	+	0.929823i	$0.380040\pi$
$614$	25.5291	1.03027
$615$	0	0
$616$	6.39315	0.257587
$617$	15.4484	0.621930	0.310965	−	0.950421i	$-0.399348\pi$
$617$	15.4484	0.621930	0.310965	+	0.950421i	$0.399348\pi$
$618$	0	0
$619$	16.5042	0.663359	0.331680	−	0.943392i	$-0.392385\pi$
$619$	16.5042	0.663359	0.331680	+	0.943392i	$0.392385\pi$
$620$	0	0
$621$	0	0
$622$	−16.3503	−0.655587
$623$	2.68980	0.107765
$624$	0	0
$625$	0	0
$626$	−56.2835	−2.24954
$627$	0	0
$628$	−47.9514	−1.91347
$629$	54.0505	2.15513
$630$	0	0
$631$	−38.1237	−1.51768	−0.758841	−	0.651276i	$-0.774234\pi$
$631$	−38.1237	−1.51768	−0.758841	+	0.651276i	$0.774234\pi$
$632$	9.14386	0.363723
$633$	0	0
$634$	−23.2267	−0.922450
$635$	0	0
$636$	0	0
$637$	9.95240	0.394329
$638$	−23.8151	−0.942850
$639$	0	0
$640$	0	0
$641$	−38.9299	−1.53764	−0.768819	−	0.639466i	$-0.779155\pi$
$641$	−38.9299	−1.53764	−0.768819	+	0.639466i	$0.779155\pi$
$642$	0	0
$643$	−40.5346	−1.59853	−0.799265	−	0.600979i	$-0.794778\pi$
$643$	−40.5346	−1.59853	−0.799265	+	0.600979i	$0.794778\pi$
$644$	9.28496	0.365879
$645$	0	0
$646$	98.8082	3.88755
$647$	31.8752	1.25315	0.626573	−	0.779363i	$-0.284457\pi$
$647$	31.8752	1.25315	0.626573	+	0.779363i	$0.284457\pi$
$648$	0	0
$649$	−9.21896	−0.361876
$650$	0	0
$651$	0	0
$652$	−18.6190	−0.729177
$653$	−19.5285	−0.764210	−0.382105	−	0.924119i	$-0.624801\pi$
$653$	−19.5285	−0.764210	−0.382105	+	0.924119i	$0.624801\pi$
$654$	0	0
$655$	0	0
$656$	29.1313	1.13738
$657$	0	0
$658$	−23.0701	−0.899367
$659$	−47.6562	−1.85642	−0.928211	−	0.372055i	$-0.878653\pi$
$659$	−47.6562	−1.85642	−0.928211	+	0.372055i	$0.878653\pi$
$660$	0	0
$661$	3.91738	0.152368	0.0761842	−	0.997094i	$-0.475726\pi$
$661$	3.91738	0.152368	0.0761842	+	0.997094i	$0.475726\pi$
$662$	−68.7196	−2.67086
$663$	0	0
$664$	−11.6040	−0.450322
$665$	0	0
$666$	0	0
$667$	−7.46779	−0.289154
$668$	−37.7485	−1.46053
$669$	0	0
$670$	0	0
$671$	12.3941	0.478470
$672$	0	0
$673$	−28.1866	−1.08651	−0.543257	−	0.839566i	$-0.682809\pi$
$673$	−28.1866	−1.08651	−0.543257	+	0.839566i	$0.682809\pi$
$674$	7.40597	0.285267
$675$	0	0
$676$	14.7477	0.567219
$677$	−9.50611	−0.365350	−0.182675	−	0.983173i	$-0.558476\pi$
$677$	−9.50611	−0.365350	−0.182675	+	0.983173i	$0.558476\pi$
$678$	0	0
$679$	32.6372	1.25250
$680$	0	0
$681$	0	0
$682$	35.1288	1.34515
$683$	7.49565	0.286813	0.143407	−	0.989664i	$-0.454194\pi$
$683$	7.49565	0.286813	0.143407	+	0.989664i	$0.454194\pi$
$684$	0	0
$685$	0	0
$686$	−43.0101	−1.64213
$687$	0	0
$688$	11.0424	0.420987
$689$	−35.5951	−1.35607
$690$	0	0
$691$	−0.745188	−0.0283483	−0.0141741	−	0.999900i	$-0.504512\pi$
$691$	−0.745188	−0.0283483	−0.0141741	+	0.999900i	$0.504512\pi$
$692$	−5.85290	−0.222494
$693$	0	0
$694$	−23.4241	−0.889167
$695$	0	0
$696$	0	0
$697$	−76.0556	−2.88081
$698$	−56.1379	−2.12485
$699$	0	0
$700$	0	0
$701$	−31.6488	−1.19536	−0.597679	−	0.801735i	$-0.703910\pi$
$701$	−31.6488	−1.19536	−0.597679	+	0.801735i	$0.703910\pi$
$702$	0	0
$703$	54.5323	2.05672
$704$	−29.1867	−1.10002
$705$	0	0
$706$	−34.9145	−1.31402
$707$	36.9147	1.38832
$708$	0	0
$709$	−16.0181	−0.601571	−0.300785	−	0.953692i	$-0.597249\pi$
$709$	−16.0181	−0.601571	−0.300785	+	0.953692i	$0.597249\pi$
$710$	0	0
$711$	0	0
$712$	1.45709	0.0546068
$713$	11.0155	0.412532
$714$	0	0
$715$	0	0
$716$	−34.8524	−1.30250
$717$	0	0
$718$	37.3043	1.39218
$719$	−14.3316	−0.534478	−0.267239	−	0.963630i	$-0.586111\pi$
$719$	−14.3316	−0.534478	−0.267239	+	0.963630i	$0.586111\pi$
$720$	0	0
$721$	21.8739	0.814628
$722$	59.1573	2.20161
$723$	0	0
$724$	22.2393	0.826515
$725$	0	0
$726$	0	0
$727$	18.3551	0.680755	0.340377	−	0.940289i	$-0.389445\pi$
$727$	18.3551	0.680755	0.340377	+	0.940289i	$0.389445\pi$
$728$	−11.0423	−0.409254
$729$	0	0
$730$	0	0
$731$	−28.8294	−1.06629
$732$	0	0
$733$	0.665145	0.0245677	0.0122838	−	0.999925i	$-0.496090\pi$
$733$	0.665145	0.0245677	0.0122838	+	0.999925i	$0.496090\pi$
$734$	−17.6973	−0.653218
$735$	0	0
$736$	−13.2360	−0.487887
$737$	29.0588	1.07040
$738$	0	0
$739$	−32.8332	−1.20779	−0.603895	−	0.797064i	$-0.706385\pi$
$739$	−32.8332	−1.20779	−0.603895	+	0.797064i	$0.706385\pi$
$740$	0	0
$741$	0	0
$742$	37.9992	1.39500
$743$	29.6004	1.08593	0.542967	−	0.839754i	$-0.317301\pi$
$743$	29.6004	1.08593	0.542967	+	0.839754i	$0.317301\pi$
$744$	0	0
$745$	0	0
$746$	40.8078	1.49408
$747$	0	0
$748$	43.3648	1.58557
$749$	−12.3016	−0.449492
$750$	0	0
$751$	24.9709	0.911199	0.455600	−	0.890185i	$-0.349425\pi$
$751$	24.9709	0.911199	0.455600	+	0.890185i	$0.349425\pi$
$752$	12.9412	0.471917
$753$	0	0
$754$	41.1337	1.49800
$755$	0	0
$756$	0	0
$757$	−11.2173	−0.407700	−0.203850	−	0.979002i	$-0.565345\pi$
$757$	−11.2173	−0.407700	−0.203850	+	0.979002i	$0.565345\pi$
$758$	−57.8007	−2.09942
$759$	0	0
$760$	0	0
$761$	−19.3105	−0.700004	−0.350002	−	0.936749i	$-0.613819\pi$
$761$	−19.3105	−0.700004	−0.350002	+	0.936749i	$0.613819\pi$
$762$	0	0
$763$	18.0996	0.655249
$764$	−29.6089	−1.07121
$765$	0	0
$766$	42.2438	1.52633
$767$	15.9231	0.574948
$768$	0	0
$769$	−19.3372	−0.697318	−0.348659	−	0.937250i	$-0.613363\pi$
$769$	−19.3372	−0.697318	−0.348659	+	0.937250i	$0.613363\pi$
$770$	0	0
$771$	0	0
$772$	37.9172	1.36467
$773$	−7.79217	−0.280265	−0.140132	−	0.990133i	$-0.544753\pi$
$773$	−7.79217	−0.280265	−0.140132	+	0.990133i	$0.544753\pi$
$774$	0	0
$775$	0	0
$776$	17.6799	0.634671
$777$	0	0
$778$	49.0923	1.76004
$779$	−76.7336	−2.74926
$780$	0	0
$781$	5.77730	0.206728
$782$	24.2601	0.867540
$783$	0	0
$784$	5.95987	0.212853
$785$	0	0
$786$	0	0
$787$	−9.07666	−0.323548	−0.161774	−	0.986828i	$-0.551722\pi$
$787$	−9.07666	−0.323548	−0.161774	+	0.986828i	$0.551722\pi$
$788$	58.2311	2.07440
$789$	0	0
$790$	0	0
$791$	32.1463	1.14299
$792$	0	0
$793$	−21.4072	−0.760192
$794$	43.8903	1.55761
$795$	0	0
$796$	14.9906	0.531326
$797$	22.9954	0.814541	0.407270	−	0.913308i	$-0.366481\pi$
$797$	22.9954	0.814541	0.407270	+	0.913308i	$0.366481\pi$
$798$	0	0
$799$	−33.7867	−1.19529
$800$	0	0
$801$	0	0
$802$	28.0611	0.990872
$803$	−2.80017	−0.0988158
$804$	0	0
$805$	0	0
$806$	−60.6747	−2.13718
$807$	0	0
$808$	19.9970	0.703493
$809$	50.6773	1.78172	0.890860	−	0.454278i	$-0.150103\pi$
$809$	50.6773	1.78172	0.890860	+	0.454278i	$0.150103\pi$
$810$	0	0
$811$	26.7614	0.939720	0.469860	−	0.882741i	$-0.344304\pi$
$811$	26.7614	0.939720	0.469860	+	0.882741i	$0.344304\pi$
$812$	−24.6130	−0.863749
$813$	0	0
$814$	42.6987	1.49659
$815$	0	0
$816$	0	0
$817$	−29.0864	−1.01760
$818$	5.89908	0.206256
$819$	0	0
$820$	0	0
$821$	0.832947	0.0290700	0.0145350	−	0.999894i	$-0.495373\pi$
$821$	0.832947	0.0290700	0.0145350	+	0.999894i	$0.495373\pi$
$822$	0	0
$823$	−6.67034	−0.232514	−0.116257	−	0.993219i	$-0.537090\pi$
$823$	−6.67034	−0.232514	−0.116257	+	0.993219i	$0.537090\pi$
$824$	11.8493	0.412791
$825$	0	0
$826$	−16.9985	−0.591454
$827$	−1.34470	−0.0467599	−0.0233799	−	0.999727i	$-0.507443\pi$
$827$	−1.34470	−0.0467599	−0.0233799	+	0.999727i	$0.507443\pi$
$828$	0	0
$829$	−24.7770	−0.860541	−0.430270	−	0.902700i	$-0.641582\pi$
$829$	−24.7770	−0.860541	−0.430270	+	0.902700i	$0.641582\pi$
$830$	0	0
$831$	0	0
$832$	50.4115	1.74770
$833$	−15.5600	−0.539121
$834$	0	0
$835$	0	0
$836$	43.7513	1.51317
$837$	0	0
$838$	20.7106	0.715437
$839$	6.72534	0.232184	0.116092	−	0.993238i	$-0.462963\pi$
$839$	6.72534	0.232184	0.116092	+	0.993238i	$0.462963\pi$
$840$	0	0
$841$	−9.20399	−0.317379
$842$	−8.99685	−0.310052
$843$	0	0
$844$	−62.9734	−2.16763
$845$	0	0
$846$	0	0
$847$	−10.2024	−0.350559
$848$	−21.3157	−0.731984
$849$	0	0
$850$	0	0
$851$	13.3892	0.458975
$852$	0	0
$853$	16.7244	0.572632	0.286316	−	0.958135i	$-0.407569\pi$
$853$	16.7244	0.572632	0.286316	+	0.958135i	$0.407569\pi$
$854$	22.8531	0.782016
$855$	0	0
$856$	−6.66391	−0.227768
$857$	12.9930	0.443831	0.221915	−	0.975066i	$-0.428769\pi$
$857$	12.9930	0.443831	0.221915	+	0.975066i	$0.428769\pi$
$858$	0	0
$859$	−36.5682	−1.24769	−0.623845	−	0.781548i	$-0.714430\pi$
$859$	−36.5682	−1.24769	−0.623845	+	0.781548i	$0.714430\pi$
$860$	0	0
$861$	0	0
$862$	−40.4859	−1.37895
$863$	−46.3384	−1.57738	−0.788688	−	0.614794i	$-0.789239\pi$
$863$	−46.3384	−1.57738	−0.788688	+	0.614794i	$0.789239\pi$
$864$	0	0
$865$	0	0
$866$	−29.2370	−0.993513
$867$	0	0
$868$	36.3058	1.23230
$869$	19.5287	0.662467
$870$	0	0
$871$	−50.1906	−1.70064
$872$	9.80472	0.332030
$873$	0	0
$874$	24.4764	0.827926
$875$	0	0
$876$	0	0
$877$	44.2548	1.49438	0.747190	−	0.664611i	$-0.231403\pi$
$877$	44.2548	1.49438	0.747190	+	0.664611i	$0.231403\pi$
$878$	−71.2169	−2.40345
$879$	0	0
$880$	0	0
$881$	42.2495	1.42342	0.711712	−	0.702472i	$-0.247920\pi$
$881$	42.2495	1.42342	0.711712	+	0.702472i	$0.247920\pi$
$882$	0	0
$883$	−3.49289	−0.117545	−0.0587726	−	0.998271i	$-0.518719\pi$
$883$	−3.49289	−0.117545	−0.0587726	+	0.998271i	$0.518719\pi$
$884$	−74.9000	−2.51916
$885$	0	0
$886$	24.8076	0.833427
$887$	14.7275	0.494501	0.247250	−	0.968952i	$-0.420473\pi$
$887$	14.7275	0.494501	0.247250	+	0.968952i	$0.420473\pi$
$888$	0	0
$889$	−1.08896	−0.0365227
$890$	0	0
$891$	0	0
$892$	8.44444	0.282741
$893$	−34.0879	−1.14071
$894$	0	0
$895$	0	0
$896$	−19.6108	−0.655151
$897$	0	0
$898$	28.4808	0.950416
$899$	−29.2004	−0.973886
$900$	0	0
$901$	55.6508	1.85400
$902$	−60.0823	−2.00052
$903$	0	0
$904$	17.4140	0.579180
$905$	0	0
$906$	0	0
$907$	−7.33785	−0.243649	−0.121825	−	0.992552i	$-0.538875\pi$
$907$	−7.33785	−0.243649	−0.121825	+	0.992552i	$0.538875\pi$
$908$	−6.89169	−0.228709
$909$	0	0
$910$	0	0
$911$	53.5731	1.77496	0.887479	−	0.460849i	$-0.152455\pi$
$911$	53.5731	1.77496	0.887479	+	0.460849i	$0.152455\pi$
$912$	0	0
$913$	−24.7829	−0.820195
$914$	49.7136	1.64438
$915$	0	0
$916$	−23.8104	−0.786717
$917$	−13.1381	−0.433859
$918$	0	0
$919$	−43.2016	−1.42509	−0.712545	−	0.701627i	$-0.752458\pi$
$919$	−43.2016	−1.42509	−0.712545	+	0.701627i	$0.752458\pi$
$920$	0	0
$921$	0	0
$922$	−10.7359	−0.353570
$923$	−9.97859	−0.328449
$924$	0	0
$925$	0	0
$926$	27.2541	0.895625
$927$	0	0
$928$	35.0868	1.15178
$929$	14.8420	0.486951	0.243476	−	0.969907i	$-0.421712\pi$
$929$	14.8420	0.486951	0.243476	+	0.969907i	$0.421712\pi$
$930$	0	0
$931$	−15.6987	−0.514503
$932$	24.2955	0.795824
$933$	0	0
$934$	10.3736	0.339435
$935$	0	0
$936$	0	0
$937$	3.75611	0.122707	0.0613534	−	0.998116i	$-0.480458\pi$
$937$	3.75611	0.122707	0.0613534	+	0.998116i	$0.480458\pi$
$938$	53.5805	1.74947
$939$	0	0
$940$	0	0
$941$	−59.9208	−1.95336	−0.976682	−	0.214692i	$-0.931125\pi$
$941$	−59.9208	−1.95336	−0.976682	+	0.214692i	$0.931125\pi$
$942$	0	0
$943$	−18.8402	−0.613521
$944$	9.53532	0.310348
$945$	0	0
$946$	−22.7746	−0.740465
$947$	−24.4212	−0.793581	−0.396791	−	0.917909i	$-0.629876\pi$
$947$	−24.4212	−0.793581	−0.396791	+	0.917909i	$0.629876\pi$
$948$	0	0
$949$	4.83647	0.156998
$950$	0	0
$951$	0	0
$952$	17.2639	0.559528
$953$	−36.6466	−1.18710	−0.593550	−	0.804797i	$-0.702274\pi$
$953$	−36.6466	−1.18710	−0.593550	+	0.804797i	$0.702274\pi$
$954$	0	0
$955$	0	0
$956$	−20.0142	−0.647307
$957$	0	0
$958$	−10.2225	−0.330275
$959$	4.20770	0.135874
$960$	0	0
$961$	12.0723	0.389431
$962$	−73.7495	−2.37778
$963$	0	0
$964$	3.64201	0.117301
$965$	0	0
$966$	0	0
$967$	47.4395	1.52555	0.762776	−	0.646663i	$-0.223836\pi$
$967$	47.4395	1.52555	0.762776	+	0.646663i	$0.223836\pi$
$968$	−5.52674	−0.177636
$969$	0	0
$970$	0	0
$971$	5.68782	0.182531	0.0912655	−	0.995827i	$-0.470909\pi$
$971$	5.68782	0.182531	0.0912655	+	0.995827i	$0.470909\pi$
$972$	0	0
$973$	41.6452	1.33508
$974$	−19.5608	−0.626767
$975$	0	0
$976$	−12.8194	−0.410340
$977$	2.29901	0.0735518	0.0367759	−	0.999324i	$-0.488291\pi$
$977$	2.29901	0.0735518	0.0367759	+	0.999324i	$0.488291\pi$
$978$	0	0
$979$	3.11194	0.0994581
$980$	0	0
$981$	0	0
$982$	−68.0620	−2.17194
$983$	2.35879	0.0752336	0.0376168	−	0.999292i	$-0.488023\pi$
$983$	2.35879	0.0752336	0.0376168	+	0.999292i	$0.488023\pi$
$984$	0	0
$985$	0	0
$986$	−64.3100	−2.04805
$987$	0	0
$988$	−75.5676	−2.40413
$989$	−7.14150	−0.227087
$990$	0	0
$991$	−7.76677	−0.246720	−0.123360	−	0.992362i	$-0.539367\pi$
$991$	−7.76677	−0.246720	−0.123360	+	0.992362i	$0.539367\pi$
$992$	−51.7552	−1.64323
$993$	0	0
$994$	10.6526	0.337879
$995$	0	0
$996$	0	0
$997$	8.35947	0.264747	0.132374	−	0.991200i	$-0.457740\pi$
$997$	8.35947	0.264747	0.132374	+	0.991200i	$0.457740\pi$
$998$	19.3549	0.612668
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.r.1.5		6
3.2	odd	2		1875.2.a.i.1.2	✓	6
5.4	even	2		5625.2.a.o.1.2		6
15.2	even	4		1875.2.b.e.1249.3		12
15.8	even	4		1875.2.b.e.1249.10		12
15.14	odd	2		1875.2.a.l.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.i.1.2	✓	6	3.2	odd	2
1875.2.a.l.1.5	yes	6	15.14	odd	2
1875.2.b.e.1249.3		12	15.2	even	4
1875.2.b.e.1249.10		12	15.8	even	4
5625.2.a.o.1.2		6	5.4	even	2
5625.2.a.r.1.5		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q+2.13324 q^{2} +2.55073 q^{4} +2.16876 q^{7} +1.17484 q^{8} +O(q^{10})\)	\(q+2.13324 q^{2} +2.55073 q^{4} +2.16876 q^{7} +1.17484 q^{8} +2.50913 q^{11} -4.33379 q^{13} +4.62650 q^{14} -2.59524 q^{16} +6.77562 q^{17} +6.83602 q^{19} +5.35259 q^{22} +1.67843 q^{23} -9.24504 q^{26} +5.53193 q^{28} -4.44927 q^{29} +6.56295 q^{31} -7.88596 q^{32} +14.4541 q^{34} +7.97720 q^{37} +14.5829 q^{38} -11.2249 q^{41} -4.25487 q^{43} +6.40012 q^{44} +3.58050 q^{46} -4.98652 q^{47} -2.29646 q^{49} -11.0543 q^{52} +8.21338 q^{53} +2.54795 q^{56} -9.49138 q^{58} -3.67416 q^{59} +4.93960 q^{61} +14.0004 q^{62} -11.6322 q^{64} +11.5812 q^{67} +17.2828 q^{68} +2.30251 q^{71} -1.11599 q^{73} +17.0173 q^{74} +17.4368 q^{76} +5.44172 q^{77} +7.78306 q^{79} -23.9454 q^{82} -9.87708 q^{83} -9.07667 q^{86} +2.94783 q^{88} +1.24025 q^{89} -9.39897 q^{91} +4.28122 q^{92} -10.6375 q^{94} +15.0488 q^{97} -4.89892 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 11 q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + q^2 + 11 * q^4 + 2 * q^7 + 6 * q^8	\( 6 q + q^{2} + 11 q^{4} + 2 q^{7} + 6 q^{8} + 4 q^{14} + 17 q^{16} + 2 q^{17} - 2 q^{19} - 9 q^{22} - q^{23} - 37 q^{26} + 44 q^{28} - 31 q^{29} - 2 q^{31} + 33 q^{32} + 37 q^{34} + 22 q^{37} + 27 q^{38} - 33 q^{41} + 3 q^{43} + 11 q^{44} - 12 q^{46} - 6 q^{47} + 4 q^{49} + 33 q^{52} + 14 q^{53} + 30 q^{56} + q^{58} + 8 q^{59} + 34 q^{61} + 31 q^{62} + 12 q^{64} - 2 q^{67} + 27 q^{68} + 3 q^{71} + 36 q^{73} - 36 q^{74} + 27 q^{76} + 16 q^{77} + 25 q^{79} - 36 q^{82} - 12 q^{83} + 30 q^{86} - 56 q^{88} - 18 q^{89} + 28 q^{91} + 3 q^{92} - 50 q^{94} - 7 q^{97} + 15 q^{98}+O(q^{100}) \) 6 * q + q^2 + 11 * q^4 + 2 * q^7 + 6 * q^8 + 4 * q^14 + 17 * q^16 + 2 * q^17 - 2 * q^19 - 9 * q^22 - q^23 - 37 * q^26 + 44 * q^28 - 31 * q^29 - 2 * q^31 + 33 * q^32 + 37 * q^34 + 22 * q^37 + 27 * q^38 - 33 * q^41 + 3 * q^43 + 11 * q^44 - 12 * q^46 - 6 * q^47 + 4 * q^49 + 33 * q^52 + 14 * q^53 + 30 * q^56 + q^58 + 8 * q^59 + 34 * q^61 + 31 * q^62 + 12 * q^64 - 2 * q^67 + 27 * q^68 + 3 * q^71 + 36 * q^73 - 36 * q^74 + 27 * q^76 + 16 * q^77 + 25 * q^79 - 36 * q^82 - 12 * q^83 + 30 * q^86 - 56 * q^88 - 18 * q^89 + 28 * q^91 + 3 * q^92 - 50 * q^94 - 7 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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