LMFDB - Embedded newform 2925.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(1,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, 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("2925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.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:	$23.3562425912$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	2925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53919 q^{2} +0.369102 q^{4} +1.70928 q^{7} -2.51026 q^{8} +O(q^{10})$	$q+1.53919 q^{2} +0.369102 q^{4} +1.70928 q^{7} -2.51026 q^{8} +2.53919 q^{11} +1.00000 q^{13} +2.63090 q^{14} -4.60197 q^{16} +0.921622 q^{17} -0.539189 q^{19} +3.90829 q^{22} +2.82991 q^{23} +1.53919 q^{26} +0.630898 q^{28} +5.12783 q^{29} +0.879362 q^{31} -2.06278 q^{32} +1.41855 q^{34} +6.04945 q^{37} -0.829914 q^{38} -1.26180 q^{41} -6.43188 q^{43} +0.937221 q^{44} +4.35577 q^{46} +5.70928 q^{47} -4.07838 q^{49} +0.369102 q^{52} +8.49693 q^{53} -4.29072 q^{56} +7.89269 q^{58} +4.72261 q^{59} +8.04945 q^{61} +1.35350 q^{62} +6.02893 q^{64} -7.86603 q^{67} +0.340173 q^{68} +14.4813 q^{71} +1.95055 q^{73} +9.31124 q^{74} -0.199016 q^{76} +4.34017 q^{77} +0.496928 q^{79} -1.94214 q^{82} +8.63090 q^{83} -9.89988 q^{86} -6.37402 q^{88} -12.8371 q^{89} +1.70928 q^{91} +1.04453 q^{92} +8.78765 q^{94} -5.91548 q^{97} -6.27739 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 5 * q^4 - 2 * q^7 + 9 * q^8	$ 3 q + 3 q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8} + 6 q^{11} + 3 q^{13} + 4 q^{14} + 5 q^{16} + 6 q^{17} + 14 q^{22} + 14 q^{23} + 3 q^{26} - 2 q^{28} - 6 q^{29} - 10 q^{31} + 11 q^{32} - 10 q^{34} - 8 q^{38} + 4 q^{41} - 6 q^{43} + 20 q^{44} + 16 q^{46} + 10 q^{47} - 9 q^{49} + 5 q^{52} + 8 q^{53} - 20 q^{56} + 12 q^{58} + 8 q^{59} + 6 q^{61} - 6 q^{62} + 33 q^{64} - 10 q^{67} - 10 q^{68} + 12 q^{71} + 24 q^{73} + 2 q^{74} - 10 q^{76} + 2 q^{77} - 16 q^{79} + 24 q^{82} + 22 q^{83} + 16 q^{86} + 24 q^{88} - 10 q^{89} - 2 q^{91} + 32 q^{92} + 16 q^{94} + 14 q^{97} - 25 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 5 * q^4 - 2 * q^7 + 9 * q^8 + 6 * q^11 + 3 * q^13 + 4 * q^14 + 5 * q^16 + 6 * q^17 + 14 * q^22 + 14 * q^23 + 3 * q^26 - 2 * q^28 - 6 * q^29 - 10 * q^31 + 11 * q^32 - 10 * q^34 - 8 * q^38 + 4 * q^41 - 6 * q^43 + 20 * q^44 + 16 * q^46 + 10 * q^47 - 9 * q^49 + 5 * q^52 + 8 * q^53 - 20 * q^56 + 12 * q^58 + 8 * q^59 + 6 * q^61 - 6 * q^62 + 33 * q^64 - 10 * q^67 - 10 * q^68 + 12 * q^71 + 24 * q^73 + 2 * q^74 - 10 * q^76 + 2 * q^77 - 16 * q^79 + 24 * q^82 + 22 * q^83 + 16 * q^86 + 24 * q^88 - 10 * q^89 - 2 * q^91 + 32 * q^92 + 16 * q^94 + 14 * q^97 - 25 * 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 $

$2$	1.53919	1.08837	0.544185	−	0.838965i	$-0.316839\pi$
$2$	1.53919	1.08837	0.544185	+	0.838965i	$0.316839\pi$
$3$	0	0
$3$	0	0
$4$	0.369102	0.184551
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.70928	0.646045	0.323023	−	0.946391i	$-0.395301\pi$
$7$	1.70928	0.646045	0.323023	+	0.946391i	$0.395301\pi$
$8$	−2.51026	−0.887511
$9$	0	0
$10$	0	0
$11$	2.53919	0.765594	0.382797	−	0.923832i	$-0.374961\pi$
$11$	2.53919	0.765594	0.382797	+	0.923832i	$0.374961\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	2.63090	0.703137
$15$	0	0
$16$	−4.60197	−1.15049
$17$	0.921622	0.223526	0.111763	−	0.993735i	$-0.464350\pi$
$17$	0.921622	0.223526	0.111763	+	0.993735i	$0.464350\pi$
$18$	0	0
$19$	−0.539189	−0.123698	−0.0618492	−	0.998086i	$-0.519700\pi$
$19$	−0.539189	−0.123698	−0.0618492	+	0.998086i	$0.519700\pi$
$20$	0	0
$21$	0	0
$22$	3.90829	0.833250
$23$	2.82991	0.590078	0.295039	−	0.955485i	$-0.404667\pi$
$23$	2.82991	0.590078	0.295039	+	0.955485i	$0.404667\pi$
$24$	0	0
$25$	0	0
$26$	1.53919	0.301860
$27$	0	0
$28$	0.630898	0.119228
$29$	5.12783	0.952213	0.476107	−	0.879388i	$-0.342048\pi$
$29$	5.12783	0.952213	0.476107	+	0.879388i	$0.342048\pi$
$30$	0	0
$31$	0.879362	0.157938	0.0789690	−	0.996877i	$-0.474837\pi$
$31$	0.879362	0.157938	0.0789690	+	0.996877i	$0.474837\pi$
$32$	−2.06278	−0.364651
$33$	0	0
$34$	1.41855	0.243279
$35$	0	0
$36$	0	0
$37$	6.04945	0.994523	0.497262	−	0.867601i	$-0.334339\pi$
$37$	6.04945	0.994523	0.497262	+	0.867601i	$0.334339\pi$
$38$	−0.829914	−0.134630
$39$	0	0
$40$	0	0
$41$	−1.26180	−0.197059	−0.0985297	−	0.995134i	$-0.531414\pi$
$41$	−1.26180	−0.197059	−0.0985297	+	0.995134i	$0.531414\pi$
$42$	0	0
$43$	−6.43188	−0.980853	−0.490426	−	0.871483i	$-0.663159\pi$
$43$	−6.43188	−0.980853	−0.490426	+	0.871483i	$0.663159\pi$
$44$	0.937221	0.141291
$45$	0	0
$46$	4.35577	0.642223
$47$	5.70928	0.832783	0.416392	−	0.909185i	$-0.363295\pi$
$47$	5.70928	0.832783	0.416392	+	0.909185i	$0.363295\pi$
$48$	0	0
$49$	−4.07838	−0.582625
$50$	0	0
$51$	0	0
$52$	0.369102	0.0511853
$53$	8.49693	1.16714	0.583571	−	0.812062i	$-0.301655\pi$
$53$	8.49693	1.16714	0.583571	+	0.812062i	$0.301655\pi$
$54$	0	0
$55$	0	0
$56$	−4.29072	−0.573372
$57$	0	0
$58$	7.89269	1.03636
$59$	4.72261	0.614831	0.307415	−	0.951575i	$-0.400536\pi$
$59$	4.72261	0.614831	0.307415	+	0.951575i	$0.400536\pi$
$60$	0	0
$61$	8.04945	1.03063	0.515313	−	0.857002i	$-0.327676\pi$
$61$	8.04945	1.03063	0.515313	+	0.857002i	$0.327676\pi$
$62$	1.35350	0.171895
$63$	0	0
$64$	6.02893	0.753616
$65$	0	0
$66$	0	0
$67$	−7.86603	−0.960989	−0.480494	−	0.876998i	$-0.659543\pi$
$67$	−7.86603	−0.960989	−0.480494	+	0.876998i	$0.659543\pi$
$68$	0.340173	0.0412520
$69$	0	0
$70$	0	0
$71$	14.4813	1.71862	0.859309	−	0.511457i	$-0.170894\pi$
$71$	14.4813	1.71862	0.859309	+	0.511457i	$0.170894\pi$
$72$	0	0
$73$	1.95055	0.228295	0.114147	−	0.993464i	$-0.463586\pi$
$73$	1.95055	0.228295	0.114147	+	0.993464i	$0.463586\pi$
$74$	9.31124	1.08241
$75$	0	0
$76$	−0.199016	−0.0228287
$77$	4.34017	0.494609
$78$	0	0
$79$	0.496928	0.0559088	0.0279544	−	0.999609i	$-0.491101\pi$
$79$	0.496928	0.0559088	0.0279544	+	0.999609i	$0.491101\pi$
$80$	0	0
$81$	0	0
$82$	−1.94214	−0.214474
$83$	8.63090	0.947364	0.473682	−	0.880696i	$-0.342925\pi$
$83$	8.63090	0.947364	0.473682	+	0.880696i	$0.342925\pi$
$84$	0	0
$85$	0	0
$86$	−9.89988	−1.06753
$87$	0	0
$88$	−6.37402	−0.679473
$89$	−12.8371	−1.36073	−0.680365	−	0.732873i	$-0.738179\pi$
$89$	−12.8371	−1.36073	−0.680365	+	0.732873i	$0.738179\pi$
$90$	0	0
$91$	1.70928	0.179181
$92$	1.04453	0.108900
$93$	0	0
$94$	8.78765	0.906377
$95$	0	0
$96$	0	0
$97$	−5.91548	−0.600626	−0.300313	−	0.953841i	$-0.597091\pi$
$97$	−5.91548	−0.600626	−0.300313	+	0.953841i	$0.597091\pi$
$98$	−6.27739	−0.634113
$99$	0	0
$100$	0	0
$101$	16.4391	1.63575	0.817874	−	0.575397i	$-0.195152\pi$
$101$	16.4391	1.63575	0.817874	+	0.575397i	$0.195152\pi$
$102$	0	0
$103$	10.1906	1.00411	0.502055	−	0.864836i	$-0.332577\pi$
$103$	10.1906	1.00411	0.502055	+	0.864836i	$0.332577\pi$
$104$	−2.51026	−0.246151
$105$	0	0
$106$	13.0784	1.27028
$107$	9.75154	0.942717	0.471358	−	0.881942i	$-0.343764\pi$
$107$	9.75154	0.942717	0.471358	+	0.881942i	$0.343764\pi$
$108$	0	0
$109$	−16.8638	−1.61526	−0.807628	−	0.589693i	$-0.799249\pi$
$109$	−16.8638	−1.61526	−0.807628	+	0.589693i	$0.799249\pi$
$110$	0	0
$111$	0	0
$112$	−7.86603	−0.743270
$113$	−11.7587	−1.10617	−0.553084	−	0.833126i	$-0.686549\pi$
$113$	−11.7587	−1.10617	−0.553084	+	0.833126i	$0.686549\pi$
$114$	0	0
$115$	0	0
$116$	1.89269	0.175732
$117$	0	0
$118$	7.26898	0.669164
$119$	1.57531	0.144408
$120$	0	0
$121$	−4.55252	−0.413865
$122$	12.3896	1.12170
$123$	0	0
$124$	0.324575	0.0291477
$125$	0	0
$126$	0	0
$127$	18.0072	1.59788	0.798940	−	0.601411i	$-0.205395\pi$
$127$	18.0072	1.59788	0.798940	+	0.601411i	$0.205395\pi$
$128$	13.4052	1.18487
$129$	0	0
$130$	0	0
$131$	−14.2557	−1.24552	−0.622761	−	0.782412i	$-0.713989\pi$
$131$	−14.2557	−1.24552	−0.622761	+	0.782412i	$0.713989\pi$
$132$	0	0
$133$	−0.921622	−0.0799148
$134$	−12.1073	−1.04591
$135$	0	0
$136$	−2.31351	−0.198382
$137$	13.7854	1.17776	0.588882	−	0.808219i	$-0.299568\pi$
$137$	13.7854	1.17776	0.588882	+	0.808219i	$0.299568\pi$
$138$	0	0
$139$	−6.65368	−0.564358	−0.282179	−	0.959362i	$-0.591057\pi$
$139$	−6.65368	−0.564358	−0.282179	+	0.959362i	$0.591057\pi$
$140$	0	0
$141$	0	0
$142$	22.2895	1.87049
$143$	2.53919	0.212338
$144$	0	0
$145$	0	0
$146$	3.00227	0.248469
$147$	0	0
$148$	2.23287	0.183540
$149$	9.07838	0.743730	0.371865	−	0.928287i	$-0.378718\pi$
$149$	9.07838	0.743730	0.371865	+	0.928287i	$0.378718\pi$
$150$	0	0
$151$	3.27739	0.266711	0.133355	−	0.991068i	$-0.457425\pi$
$151$	3.27739	0.266711	0.133355	+	0.991068i	$0.457425\pi$
$152$	1.35350	0.109784
$153$	0	0
$154$	6.68035	0.538318
$155$	0	0
$156$	0	0
$157$	−12.8371	−1.02451	−0.512256	−	0.858833i	$-0.671190\pi$
$157$	−12.8371	−1.02451	−0.512256	+	0.858833i	$0.671190\pi$
$158$	0.764867	0.0608495
$159$	0	0
$160$	0	0
$161$	4.83710	0.381217
$162$	0	0
$163$	−12.0494	−0.943786	−0.471893	−	0.881656i	$-0.656429\pi$
$163$	−12.0494	−0.943786	−0.471893	+	0.881656i	$0.656429\pi$
$164$	−0.465732	−0.0363675
$165$	0	0
$166$	13.2846	1.03108
$167$	8.72979	0.675532	0.337766	−	0.941230i	$-0.390329\pi$
$167$	8.72979	0.675532	0.337766	+	0.941230i	$0.390329\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−2.37402	−0.181018
$173$	0.863763	0.0656707	0.0328354	−	0.999461i	$-0.489546\pi$
$173$	0.863763	0.0656707	0.0328354	+	0.999461i	$0.489546\pi$
$174$	0	0
$175$	0	0
$176$	−11.6853	−0.880810
$177$	0	0
$178$	−19.7587	−1.48098
$179$	−19.9155	−1.48855	−0.744276	−	0.667872i	$-0.767205\pi$
$179$	−19.9155	−1.48855	−0.744276	+	0.667872i	$0.767205\pi$
$180$	0	0
$181$	14.3896	1.06957	0.534786	−	0.844987i	$-0.320392\pi$
$181$	14.3896	1.06957	0.534786	+	0.844987i	$0.320392\pi$
$182$	2.63090	0.195015
$183$	0	0
$184$	−7.10382	−0.523700
$185$	0	0
$186$	0	0
$187$	2.34017	0.171130
$188$	2.10731	0.153691
$189$	0	0
$190$	0	0
$191$	−1.47641	−0.106829	−0.0534146	−	0.998572i	$-0.517010\pi$
$191$	−1.47641	−0.106829	−0.0534146	+	0.998572i	$0.517010\pi$
$192$	0	0
$193$	17.7321	1.27638	0.638191	−	0.769878i	$-0.279683\pi$
$193$	17.7321	1.27638	0.638191	+	0.769878i	$0.279683\pi$
$194$	−9.10504	−0.653704
$195$	0	0
$196$	−1.50534	−0.107524
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−5.39189	−0.382221	−0.191110	−	0.981569i	$-0.561209\pi$
$199$	−5.39189	−0.382221	−0.191110	+	0.981569i	$0.561209\pi$
$200$	0	0
$201$	0	0
$202$	25.3028	1.78030
$203$	8.76487	0.615173
$204$	0	0
$205$	0	0
$206$	15.6853	1.09284
$207$	0	0
$208$	−4.60197	−0.319089
$209$	−1.36910	−0.0947028
$210$	0	0
$211$	−22.7526	−1.56635	−0.783176	−	0.621800i	$-0.786402\pi$
$211$	−22.7526	−1.56635	−0.783176	+	0.621800i	$0.786402\pi$
$212$	3.13624	0.215398
$213$	0	0
$214$	15.0095	1.02603
$215$	0	0
$216$	0	0
$217$	1.50307	0.102035
$218$	−25.9565	−1.75800
$219$	0	0
$220$	0	0
$221$	0.921622	0.0619950
$222$	0	0
$223$	−8.76099	−0.586679	−0.293340	−	0.956008i	$-0.594767\pi$
$223$	−8.76099	−0.586679	−0.293340	+	0.956008i	$0.594767\pi$
$224$	−3.52586	−0.235581
$225$	0	0
$226$	−18.0989	−1.20392
$227$	−17.2267	−1.14338	−0.571689	−	0.820470i	$-0.693712\pi$
$227$	−17.2267	−1.14338	−0.571689	+	0.820470i	$0.693712\pi$
$228$	0	0
$229$	3.07838	0.203425	0.101712	−	0.994814i	$-0.467568\pi$
$229$	3.07838	0.203425	0.101712	+	0.994814i	$0.467568\pi$
$230$	0	0
$231$	0	0
$232$	−12.8722	−0.845100
$233$	18.9360	1.24054	0.620269	−	0.784389i	$-0.287023\pi$
$233$	18.9360	1.24054	0.620269	+	0.784389i	$0.287023\pi$
$234$	0	0
$235$	0	0
$236$	1.74313	0.113468
$237$	0	0
$238$	2.42469	0.157170
$239$	−6.63809	−0.429382	−0.214691	−	0.976682i	$-0.568874\pi$
$239$	−6.63809	−0.429382	−0.214691	+	0.976682i	$0.568874\pi$
$240$	0	0
$241$	−9.47641	−0.610429	−0.305215	−	0.952284i	$-0.598728\pi$
$241$	−9.47641	−0.610429	−0.305215	+	0.952284i	$0.598728\pi$
$242$	−7.00719	−0.450439
$243$	0	0
$244$	2.97107	0.190203
$245$	0	0
$246$	0	0
$247$	−0.539189	−0.0343078
$248$	−2.20743	−0.140172
$249$	0	0
$250$	0	0
$251$	−29.4596	−1.85947	−0.929736	−	0.368226i	$-0.879965\pi$
$251$	−29.4596	−1.85947	−0.929736	+	0.368226i	$0.879965\pi$
$252$	0	0
$253$	7.18568	0.451760
$254$	27.7165	1.73909
$255$	0	0
$256$	8.57531	0.535957
$257$	−20.4657	−1.27662	−0.638309	−	0.769781i	$-0.720366\pi$
$257$	−20.4657	−1.27662	−0.638309	+	0.769781i	$0.720366\pi$
$258$	0	0
$259$	10.3402	0.642507
$260$	0	0
$261$	0	0
$262$	−21.9421	−1.35559
$263$	9.14342	0.563808	0.281904	−	0.959443i	$-0.409034\pi$
$263$	9.14342	0.563808	0.281904	+	0.959443i	$0.409034\pi$
$264$	0	0
$265$	0	0
$266$	−1.41855	−0.0869769
$267$	0	0
$268$	−2.90337	−0.177352
$269$	−11.3919	−0.694576	−0.347288	−	0.937759i	$-0.612897\pi$
$269$	−11.3919	−0.694576	−0.347288	+	0.937759i	$0.612897\pi$
$270$	0	0
$271$	−21.1350	−1.28386	−0.641930	−	0.766763i	$-0.721866\pi$
$271$	−21.1350	−1.28386	−0.641930	+	0.766763i	$0.721866\pi$
$272$	−4.24128	−0.257165
$273$	0	0
$274$	21.2183	1.28185
$275$	0	0
$276$	0	0
$277$	13.0784	0.785804	0.392902	−	0.919580i	$-0.371471\pi$
$277$	13.0784	0.785804	0.392902	+	0.919580i	$0.371471\pi$
$278$	−10.2413	−0.614231
$279$	0	0
$280$	0	0
$281$	0.680346	0.0405860	0.0202930	−	0.999794i	$-0.493540\pi$
$281$	0.680346	0.0405860	0.0202930	+	0.999794i	$0.493540\pi$
$282$	0	0
$283$	19.2956	1.14701	0.573504	−	0.819203i	$-0.305584\pi$
$283$	19.2956	1.14701	0.573504	+	0.819203i	$0.305584\pi$
$284$	5.34509	0.317173
$285$	0	0
$286$	3.90829	0.231102
$287$	−2.15676	−0.127309
$288$	0	0
$289$	−16.1506	−0.950036
$290$	0	0
$291$	0	0
$292$	0.719953	0.0421321
$293$	−9.46800	−0.553126	−0.276563	−	0.960996i	$-0.589196\pi$
$293$	−9.46800	−0.553126	−0.276563	+	0.960996i	$0.589196\pi$
$294$	0	0
$295$	0	0
$296$	−15.1857	−0.882650
$297$	0	0
$298$	13.9733	0.809454
$299$	2.82991	0.163658
$300$	0	0
$301$	−10.9939	−0.633675
$302$	5.04453	0.290280
$303$	0	0
$304$	2.48133	0.142314
$305$	0	0
$306$	0	0
$307$	0.264063	0.0150709	0.00753543	−	0.999972i	$-0.497601\pi$
$307$	0.264063	0.0150709	0.00753543	+	0.999972i	$0.497601\pi$
$308$	1.60197	0.0912806
$309$	0	0
$310$	0	0
$311$	−13.0472	−0.739838	−0.369919	−	0.929064i	$-0.620615\pi$
$311$	−13.0472	−0.739838	−0.369919	+	0.929064i	$0.620615\pi$
$312$	0	0
$313$	33.7009	1.90489	0.952443	−	0.304718i	$-0.0985621\pi$
$313$	33.7009	1.90489	0.952443	+	0.304718i	$0.0985621\pi$
$314$	−19.7587	−1.11505
$315$	0	0
$316$	0.183417	0.0103180
$317$	−13.9506	−0.783541	−0.391771	−	0.920063i	$-0.628137\pi$
$317$	−13.9506	−0.783541	−0.391771	+	0.920063i	$0.628137\pi$
$318$	0	0
$319$	13.0205	0.729009
$320$	0	0
$321$	0	0
$322$	7.44521	0.414905
$323$	−0.496928	−0.0276498
$324$	0	0
$325$	0	0
$326$	−18.5464	−1.02719
$327$	0	0
$328$	3.16743	0.174892
$329$	9.75872	0.538016
$330$	0	0
$331$	−18.4547	−1.01436	−0.507180	−	0.861840i	$-0.669312\pi$
$331$	−18.4547	−1.01436	−0.507180	+	0.861840i	$0.669312\pi$
$332$	3.18568	0.174837
$333$	0	0
$334$	13.4368	0.735229
$335$	0	0
$336$	0	0
$337$	15.8576	0.863820	0.431910	−	0.901917i	$-0.357840\pi$
$337$	15.8576	0.863820	0.431910	+	0.901917i	$0.357840\pi$
$338$	1.53919	0.0837208
$339$	0	0
$340$	0	0
$341$	2.23287	0.120916
$342$	0	0
$343$	−18.9360	−1.02245
$344$	16.1457	0.870517
$345$	0	0
$346$	1.32950	0.0714741
$347$	9.72487	0.522059	0.261029	−	0.965331i	$-0.415938\pi$
$347$	9.72487	0.522059	0.261029	+	0.965331i	$0.415938\pi$
$348$	0	0
$349$	−30.9093	−1.65454	−0.827269	−	0.561805i	$-0.810107\pi$
$349$	−30.9093	−1.65454	−0.827269	+	0.561805i	$0.810107\pi$
$350$	0	0
$351$	0	0
$352$	−5.23779	−0.279175
$353$	5.95055	0.316716	0.158358	−	0.987382i	$-0.449380\pi$
$353$	5.95055	0.316716	0.158358	+	0.987382i	$0.449380\pi$
$354$	0	0
$355$	0	0
$356$	−4.73820	−0.251124
$357$	0	0
$358$	−30.6537	−1.62010
$359$	−10.9783	−0.579410	−0.289705	−	0.957116i	$-0.593557\pi$
$359$	−10.9783	−0.579410	−0.289705	+	0.957116i	$0.593557\pi$
$360$	0	0
$361$	−18.7093	−0.984699
$362$	22.1483	1.16409
$363$	0	0
$364$	0.630898	0.0330680
$365$	0	0
$366$	0	0
$367$	−10.3740	−0.541520	−0.270760	−	0.962647i	$-0.587275\pi$
$367$	−10.3740	−0.541520	−0.270760	+	0.962647i	$0.587275\pi$
$368$	−13.0232	−0.678880
$369$	0	0
$370$	0	0
$371$	14.5236	0.754027
$372$	0	0
$373$	−23.9877	−1.24204	−0.621018	−	0.783796i	$-0.713281\pi$
$373$	−23.9877	−1.24204	−0.621018	+	0.783796i	$0.713281\pi$
$374$	3.60197	0.186253
$375$	0	0
$376$	−14.3318	−0.739104
$377$	5.12783	0.264096
$378$	0	0
$379$	29.7575	1.52854	0.764270	−	0.644896i	$-0.223099\pi$
$379$	29.7575	1.52854	0.764270	+	0.644896i	$0.223099\pi$
$380$	0	0
$381$	0	0
$382$	−2.27247	−0.116270
$383$	12.4163	0.634442	0.317221	−	0.948352i	$-0.397250\pi$
$383$	12.4163	0.634442	0.317221	+	0.948352i	$0.397250\pi$
$384$	0	0
$385$	0	0
$386$	27.2930	1.38918
$387$	0	0
$388$	−2.18342	−0.110846
$389$	16.8371	0.853675	0.426837	−	0.904328i	$-0.359628\pi$
$389$	16.8371	0.853675	0.426837	+	0.904328i	$0.359628\pi$
$390$	0	0
$391$	2.60811	0.131898
$392$	10.2378	0.517086
$393$	0	0
$394$	3.07838	0.155086
$395$	0	0
$396$	0	0
$397$	3.89269	0.195369	0.0976843	−	0.995217i	$-0.468856\pi$
$397$	3.89269	0.195369	0.0976843	+	0.995217i	$0.468856\pi$
$398$	−8.29914	−0.415998
$399$	0	0
$400$	0	0
$401$	9.10504	0.454684	0.227342	−	0.973815i	$-0.426996\pi$
$401$	9.10504	0.454684	0.227342	+	0.973815i	$0.426996\pi$
$402$	0	0
$403$	0.879362	0.0438041
$404$	6.06770	0.301879
$405$	0	0
$406$	13.4908	0.669536
$407$	15.3607	0.761401
$408$	0	0
$409$	−19.4186	−0.960186	−0.480093	−	0.877218i	$-0.659397\pi$
$409$	−19.4186	−0.960186	−0.480093	+	0.877218i	$0.659397\pi$
$410$	0	0
$411$	0	0
$412$	3.76138	0.185310
$413$	8.07223	0.397209
$414$	0	0
$415$	0	0
$416$	−2.06278	−0.101136
$417$	0	0
$418$	−2.10731	−0.103072
$419$	16.7792	0.819720	0.409860	−	0.912149i	$-0.365578\pi$
$419$	16.7792	0.819720	0.409860	+	0.912149i	$0.365578\pi$
$420$	0	0
$421$	−19.0205	−0.927003	−0.463502	−	0.886096i	$-0.653407\pi$
$421$	−19.0205	−0.927003	−0.463502	+	0.886096i	$0.653407\pi$
$422$	−35.0205	−1.70477
$423$	0	0
$424$	−21.3295	−1.03585
$425$	0	0
$426$	0	0
$427$	13.7587	0.665831
$428$	3.59932	0.173979
$429$	0	0
$430$	0	0
$431$	−8.02997	−0.386790	−0.193395	−	0.981121i	$-0.561950\pi$
$431$	−8.02997	−0.386790	−0.193395	+	0.981121i	$0.561950\pi$
$432$	0	0
$433$	−13.0472	−0.627008	−0.313504	−	0.949587i	$-0.601503\pi$
$433$	−13.0472	−0.627008	−0.313504	+	0.949587i	$0.601503\pi$
$434$	2.31351	0.111052
$435$	0	0
$436$	−6.22446	−0.298097
$437$	−1.52586	−0.0729917
$438$	0	0
$439$	−7.70086	−0.367542	−0.183771	−	0.982969i	$-0.558831\pi$
$439$	−7.70086	−0.367542	−0.183771	+	0.982969i	$0.558831\pi$
$440$	0	0
$441$	0	0
$442$	1.41855	0.0674736
$443$	6.39084	0.303638	0.151819	−	0.988408i	$-0.451487\pi$
$443$	6.39084	0.303638	0.151819	+	0.988408i	$0.451487\pi$
$444$	0	0
$445$	0	0
$446$	−13.4848	−0.638525
$447$	0	0
$448$	10.3051	0.486870
$449$	−31.6163	−1.49207	−0.746034	−	0.665908i	$-0.768044\pi$
$449$	−31.6163	−1.49207	−0.746034	+	0.665908i	$0.768044\pi$
$450$	0	0
$451$	−3.20394	−0.150867
$452$	−4.34017	−0.204145
$453$	0	0
$454$	−26.5152	−1.24442
$455$	0	0
$456$	0	0
$457$	−35.6430	−1.66731	−0.833655	−	0.552286i	$-0.813756\pi$
$457$	−35.6430	−1.66731	−0.833655	+	0.552286i	$0.813756\pi$
$458$	4.73820	0.221402
$459$	0	0
$460$	0	0
$461$	14.9795	0.697664	0.348832	−	0.937185i	$-0.386578\pi$
$461$	14.9795	0.697664	0.348832	+	0.937185i	$0.386578\pi$
$462$	0	0
$463$	−9.09663	−0.422756	−0.211378	−	0.977404i	$-0.567795\pi$
$463$	−9.09663	−0.422756	−0.211378	+	0.977404i	$0.567795\pi$
$464$	−23.5981	−1.09551
$465$	0	0
$466$	29.1461	1.35017
$467$	−1.87709	−0.0868616	−0.0434308	−	0.999056i	$-0.513829\pi$
$467$	−1.87709	−0.0868616	−0.0434308	+	0.999056i	$0.513829\pi$
$468$	0	0
$469$	−13.4452	−0.620842
$470$	0	0
$471$	0	0
$472$	−11.8550	−0.545669
$473$	−16.3318	−0.750935
$474$	0	0
$475$	0	0
$476$	0.581449	0.0266507
$477$	0	0
$478$	−10.2173	−0.467327
$479$	15.7431	0.719322	0.359661	−	0.933083i	$-0.382892\pi$
$479$	15.7431	0.719322	0.359661	+	0.933083i	$0.382892\pi$
$480$	0	0
$481$	6.04945	0.275831
$482$	−14.5860	−0.664373
$483$	0	0
$484$	−1.68035	−0.0763794
$485$	0	0
$486$	0	0
$487$	4.94441	0.224053	0.112026	−	0.993705i	$-0.464266\pi$
$487$	4.94441	0.224053	0.112026	+	0.993705i	$0.464266\pi$
$488$	−20.2062	−0.914692
$489$	0	0
$490$	0	0
$491$	−39.4863	−1.78199	−0.890995	−	0.454014i	$-0.849992\pi$
$491$	−39.4863	−1.78199	−0.890995	+	0.454014i	$0.849992\pi$
$492$	0	0
$493$	4.72592	0.212845
$494$	−0.829914	−0.0373396
$495$	0	0
$496$	−4.04680	−0.181706
$497$	24.7526	1.11030
$498$	0	0
$499$	1.67089	0.0747993	0.0373997	−	0.999300i	$-0.488093\pi$
$499$	1.67089	0.0747993	0.0373997	+	0.999300i	$0.488093\pi$
$500$	0	0
$501$	0	0
$502$	−45.3439	−2.02380
$503$	−9.08557	−0.405105	−0.202553	−	0.979271i	$-0.564924\pi$
$503$	−9.08557	−0.405105	−0.202553	+	0.979271i	$0.564924\pi$
$504$	0	0
$505$	0	0
$506$	11.0601	0.491683
$507$	0	0
$508$	6.64650	0.294891
$509$	19.5441	0.866277	0.433139	−	0.901327i	$-0.357406\pi$
$509$	19.5441	0.866277	0.433139	+	0.901327i	$0.357406\pi$
$510$	0	0
$511$	3.33403	0.147489
$512$	−13.6114	−0.601546
$513$	0	0
$514$	−31.5006	−1.38943
$515$	0	0
$516$	0	0
$517$	14.4969	0.637574
$518$	15.9155	0.699286
$519$	0	0
$520$	0	0
$521$	−6.50534	−0.285004	−0.142502	−	0.989795i	$-0.545515\pi$
$521$	−6.50534	−0.285004	−0.142502	+	0.989795i	$0.545515\pi$
$522$	0	0
$523$	36.5452	1.59801	0.799004	−	0.601326i	$-0.205361\pi$
$523$	36.5452	1.59801	0.799004	+	0.601326i	$0.205361\pi$
$524$	−5.26180	−0.229863
$525$	0	0
$526$	14.0735	0.613632
$527$	0.810439	0.0353033
$528$	0	0
$529$	−14.9916	−0.651808
$530$	0	0
$531$	0	0
$532$	−0.340173	−0.0147484
$533$	−1.26180	−0.0546544
$534$	0	0
$535$	0	0
$536$	19.7458	0.852888
$537$	0	0
$538$	−17.5343	−0.755956
$539$	−10.3558	−0.446055
$540$	0	0
$541$	20.3402	0.874492	0.437246	−	0.899342i	$-0.355954\pi$
$541$	20.3402	0.874492	0.437246	+	0.899342i	$0.355954\pi$
$542$	−32.5308	−1.39732
$543$	0	0
$544$	−1.90110	−0.0815091
$545$	0	0
$546$	0	0
$547$	−11.5948	−0.495757	−0.247879	−	0.968791i	$-0.579733\pi$
$547$	−11.5948	−0.495757	−0.247879	+	0.968791i	$0.579733\pi$
$548$	5.08822	0.217358
$549$	0	0
$550$	0	0
$551$	−2.76487	−0.117787
$552$	0	0
$553$	0.849388	0.0361196
$554$	20.1301	0.855246
$555$	0	0
$556$	−2.45589	−0.104153
$557$	−10.7298	−0.454636	−0.227318	−	0.973821i	$-0.572996\pi$
$557$	−10.7298	−0.454636	−0.227318	+	0.973821i	$0.572996\pi$
$558$	0	0
$559$	−6.43188	−0.272040
$560$	0	0
$561$	0	0
$562$	1.04718	0.0441727
$563$	−10.2485	−0.431921	−0.215961	−	0.976402i	$-0.569288\pi$
$563$	−10.2485	−0.431921	−0.215961	+	0.976402i	$0.569288\pi$
$564$	0	0
$565$	0	0
$566$	29.6996	1.24837
$567$	0	0
$568$	−36.3519	−1.52529
$569$	8.84551	0.370823	0.185412	−	0.982661i	$-0.440638\pi$
$569$	8.84551	0.370823	0.185412	+	0.982661i	$0.440638\pi$
$570$	0	0
$571$	9.29299	0.388900	0.194450	−	0.980912i	$-0.437708\pi$
$571$	9.29299	0.388900	0.194450	+	0.980912i	$0.437708\pi$
$572$	0.937221	0.0391872
$573$	0	0
$574$	−3.31965	−0.138560
$575$	0	0
$576$	0	0
$577$	−19.5259	−0.812872	−0.406436	−	0.913679i	$-0.633229\pi$
$577$	−19.5259	−0.812872	−0.406436	+	0.913679i	$0.633229\pi$
$578$	−24.8588	−1.03399
$579$	0	0
$580$	0	0
$581$	14.7526	0.612040
$582$	0	0
$583$	21.5753	0.893558
$584$	−4.89639	−0.202614
$585$	0	0
$586$	−14.5730	−0.602007
$587$	−22.5029	−0.928794	−0.464397	−	0.885627i	$-0.653729\pi$
$587$	−22.5029	−0.928794	−0.464397	+	0.885627i	$0.653729\pi$
$588$	0	0
$589$	−0.474142	−0.0195367
$590$	0	0
$591$	0	0
$592$	−27.8394	−1.14419
$593$	−4.43907	−0.182291	−0.0911454	−	0.995838i	$-0.529053\pi$
$593$	−4.43907	−0.182291	−0.0911454	+	0.995838i	$0.529053\pi$
$594$	0	0
$595$	0	0
$596$	3.35085	0.137256
$597$	0	0
$598$	4.35577	0.178121
$599$	−33.3607	−1.36308	−0.681540	−	0.731780i	$-0.738690\pi$
$599$	−33.3607	−1.36308	−0.681540	+	0.731780i	$0.738690\pi$
$600$	0	0
$601$	13.3197	0.543320	0.271660	−	0.962393i	$-0.412427\pi$
$601$	13.3197	0.543320	0.271660	+	0.962393i	$0.412427\pi$
$602$	−16.9216	−0.689674
$603$	0	0
$604$	1.20969	0.0492217
$605$	0	0
$606$	0	0
$607$	14.1184	0.573047	0.286523	−	0.958073i	$-0.407500\pi$
$607$	14.1184	0.573047	0.286523	+	0.958073i	$0.407500\pi$
$608$	1.11223	0.0451068
$609$	0	0
$610$	0	0
$611$	5.70928	0.230973
$612$	0	0
$613$	−26.8104	−1.08286	−0.541432	−	0.840745i	$-0.682118\pi$
$613$	−26.8104	−1.08286	−0.541432	+	0.840745i	$0.682118\pi$
$614$	0.406442	0.0164027
$615$	0	0
$616$	−10.8950	−0.438970
$617$	−14.8950	−0.599649	−0.299824	−	0.953994i	$-0.596928\pi$
$617$	−14.8950	−0.599649	−0.299824	+	0.953994i	$0.596928\pi$
$618$	0	0
$619$	45.3184	1.82150	0.910751	−	0.412956i	$-0.135504\pi$
$619$	45.3184	1.82150	0.910751	+	0.412956i	$0.135504\pi$
$620$	0	0
$621$	0	0
$622$	−20.0821	−0.805218
$623$	−21.9421	−0.879093
$624$	0	0
$625$	0	0
$626$	51.8720	2.07322
$627$	0	0
$628$	−4.73820	−0.189075
$629$	5.57531	0.222302
$630$	0	0
$631$	37.8876	1.50828	0.754141	−	0.656713i	$-0.228054\pi$
$631$	37.8876	1.50828	0.754141	+	0.656713i	$0.228054\pi$
$632$	−1.24742	−0.0496197
$633$	0	0
$634$	−21.4725	−0.852783
$635$	0	0
$636$	0	0
$637$	−4.07838	−0.161591
$638$	20.0410	0.793432
$639$	0	0
$640$	0	0
$641$	8.47027	0.334555	0.167278	−	0.985910i	$-0.446502\pi$
$641$	8.47027	0.334555	0.167278	+	0.985910i	$0.446502\pi$
$642$	0	0
$643$	34.1750	1.34773	0.673865	−	0.738854i	$-0.264633\pi$
$643$	34.1750	1.34773	0.673865	+	0.738854i	$0.264633\pi$
$644$	1.78539	0.0703541
$645$	0	0
$646$	−0.764867	−0.0300933
$647$	13.8238	0.543468	0.271734	−	0.962372i	$-0.412403\pi$
$647$	13.8238	0.543468	0.271734	+	0.962372i	$0.412403\pi$
$648$	0	0
$649$	11.9916	0.470711
$650$	0	0
$651$	0	0
$652$	−4.44748	−0.174177
$653$	42.8781	1.67795	0.838976	−	0.544169i	$-0.183155\pi$
$653$	42.8781	1.67795	0.838976	+	0.544169i	$0.183155\pi$
$654$	0	0
$655$	0	0
$656$	5.80674	0.226715
$657$	0	0
$658$	15.0205	0.585561
$659$	23.2495	0.905672	0.452836	−	0.891594i	$-0.350412\pi$
$659$	23.2495	0.905672	0.452836	+	0.891594i	$0.350412\pi$
$660$	0	0
$661$	27.0661	1.05275	0.526374	−	0.850253i	$-0.323551\pi$
$661$	27.0661	1.05275	0.526374	+	0.850253i	$0.323551\pi$
$662$	−28.4052	−1.10400
$663$	0	0
$664$	−21.6658	−0.840796
$665$	0	0
$666$	0	0
$667$	14.5113	0.561880
$668$	3.22219	0.124670
$669$	0	0
$670$	0	0
$671$	20.4391	0.789042
$672$	0	0
$673$	16.1711	0.623351	0.311676	−	0.950189i	$-0.399110\pi$
$673$	16.1711	0.623351	0.311676	+	0.950189i	$0.399110\pi$
$674$	24.4079	0.940156
$675$	0	0
$676$	0.369102	0.0141962
$677$	−43.1194	−1.65721	−0.828607	−	0.559831i	$-0.810866\pi$
$677$	−43.1194	−1.65721	−0.828607	+	0.559831i	$0.810866\pi$
$678$	0	0
$679$	−10.1112	−0.388032
$680$	0	0
$681$	0	0
$682$	3.43680	0.131602
$683$	17.7093	0.677627	0.338813	−	0.940854i	$-0.389974\pi$
$683$	17.7093	0.677627	0.338813	+	0.940854i	$0.389974\pi$
$684$	0	0
$685$	0	0
$686$	−29.1461	−1.11280
$687$	0	0
$688$	29.5993	1.12846
$689$	8.49693	0.323707
$690$	0	0
$691$	−24.8794	−0.946456	−0.473228	−	0.880940i	$-0.656911\pi$
$691$	−24.8794	−0.946456	−0.473228	+	0.880940i	$0.656911\pi$
$692$	0.318817	0.0121196
$693$	0	0
$694$	14.9684	0.568193
$695$	0	0
$696$	0	0
$697$	−1.16290	−0.0440479
$698$	−47.5753	−1.80075
$699$	0	0
$700$	0	0
$701$	−33.0661	−1.24889	−0.624445	−	0.781069i	$-0.714675\pi$
$701$	−33.0661	−1.24889	−0.624445	+	0.781069i	$0.714675\pi$
$702$	0	0
$703$	−3.26180	−0.123021
$704$	15.3086	0.576964
$705$	0	0
$706$	9.15902	0.344704
$707$	28.0989	1.05677
$708$	0	0
$709$	2.18342	0.0820000	0.0410000	−	0.999159i	$-0.486946\pi$
$709$	2.18342	0.0820000	0.0410000	+	0.999159i	$0.486946\pi$
$710$	0	0
$711$	0	0
$712$	32.2245	1.20766
$713$	2.48852	0.0931957
$714$	0	0
$715$	0	0
$716$	−7.35085	−0.274714
$717$	0	0
$718$	−16.8976	−0.630613
$719$	−5.20847	−0.194243	−0.0971216	−	0.995273i	$-0.530964\pi$
$719$	−5.20847	−0.194243	−0.0971216	+	0.995273i	$0.530964\pi$
$720$	0	0
$721$	17.4186	0.648701
$722$	−28.7971	−1.07172
$723$	0	0
$724$	5.31124	0.197391
$725$	0	0
$726$	0	0
$727$	−3.52464	−0.130721	−0.0653607	−	0.997862i	$-0.520820\pi$
$727$	−3.52464	−0.130721	−0.0653607	+	0.997862i	$0.520820\pi$
$728$	−4.29072	−0.159025
$729$	0	0
$730$	0	0
$731$	−5.92777	−0.219246
$732$	0	0
$733$	−21.8310	−0.806345	−0.403172	−	0.915124i	$-0.632093\pi$
$733$	−21.8310	−0.806345	−0.403172	+	0.915124i	$0.632093\pi$
$734$	−15.9676	−0.589374
$735$	0	0
$736$	−5.83749	−0.215173
$737$	−19.9733	−0.735727
$738$	0	0
$739$	50.3533	1.85228	0.926139	−	0.377184i	$-0.123107\pi$
$739$	50.3533	1.85228	0.926139	+	0.377184i	$0.123107\pi$
$740$	0	0
$741$	0	0
$742$	22.3545	0.820661
$743$	−30.7877	−1.12949	−0.564745	−	0.825266i	$-0.691025\pi$
$743$	−30.7877	−1.12949	−0.564745	+	0.825266i	$0.691025\pi$
$744$	0	0
$745$	0	0
$746$	−36.9216	−1.35180
$747$	0	0
$748$	0.863763	0.0315823
$749$	16.6681	0.609038
$750$	0	0
$751$	−10.6225	−0.387620	−0.193810	−	0.981039i	$-0.562085\pi$
$751$	−10.6225	−0.387620	−0.193810	+	0.981039i	$0.562085\pi$
$752$	−26.2739	−0.958111
$753$	0	0
$754$	7.89269	0.287435
$755$	0	0
$756$	0	0
$757$	−7.98562	−0.290242	−0.145121	−	0.989414i	$-0.546357\pi$
$757$	−7.98562	−0.290242	−0.145121	+	0.989414i	$0.546357\pi$
$758$	45.8024	1.66362
$759$	0	0
$760$	0	0
$761$	48.9360	1.77393	0.886964	−	0.461838i	$-0.152810\pi$
$761$	48.9360	1.77393	0.886964	+	0.461838i	$0.152810\pi$
$762$	0	0
$763$	−28.8248	−1.04353
$764$	−0.544946	−0.0197155
$765$	0	0
$766$	19.1110	0.690509
$767$	4.72261	0.170523
$768$	0	0
$769$	−7.99547	−0.288324	−0.144162	−	0.989554i	$-0.546049\pi$
$769$	−7.99547	−0.288324	−0.144162	+	0.989554i	$0.546049\pi$
$770$	0	0
$771$	0	0
$772$	6.54495	0.235558
$773$	−26.6141	−0.957242	−0.478621	−	0.878022i	$-0.658863\pi$
$773$	−26.6141	−0.957242	−0.478621	+	0.878022i	$0.658863\pi$
$774$	0	0
$775$	0	0
$776$	14.8494	0.533062
$777$	0	0
$778$	25.9155	0.929115
$779$	0.680346	0.0243759
$780$	0	0
$781$	36.7708	1.31576
$782$	4.01438	0.143554
$783$	0	0
$784$	18.7686	0.670306
$785$	0	0
$786$	0	0
$787$	9.25792	0.330009	0.165005	−	0.986293i	$-0.447236\pi$
$787$	9.25792	0.330009	0.165005	+	0.986293i	$0.447236\pi$
$788$	0.738205	0.0262975
$789$	0	0
$790$	0	0
$791$	−20.0989	−0.714634
$792$	0	0
$793$	8.04945	0.285844
$794$	5.99159	0.212634
$795$	0	0
$796$	−1.99016	−0.0705393
$797$	−15.9421	−0.564700	−0.282350	−	0.959312i	$-0.591114\pi$
$797$	−15.9421	−0.564700	−0.282350	+	0.959312i	$0.591114\pi$
$798$	0	0
$799$	5.26180	0.186149
$800$	0	0
$801$	0	0
$802$	14.0144	0.494865
$803$	4.95282	0.174781
$804$	0	0
$805$	0	0
$806$	1.35350	0.0476751
$807$	0	0
$808$	−41.2663	−1.45174
$809$	−17.9239	−0.630170	−0.315085	−	0.949063i	$-0.602033\pi$
$809$	−17.9239	−0.630170	−0.315085	+	0.949063i	$0.602033\pi$
$810$	0	0
$811$	7.43415	0.261048	0.130524	−	0.991445i	$-0.458334\pi$
$811$	7.43415	0.261048	0.130524	+	0.991445i	$0.458334\pi$
$812$	3.23513	0.113531
$813$	0	0
$814$	23.6430	0.828687
$815$	0	0
$816$	0	0
$817$	3.46800	0.121330
$818$	−29.8888	−1.04504
$819$	0	0
$820$	0	0
$821$	−20.4801	−0.714761	−0.357380	−	0.933959i	$-0.616330\pi$
$821$	−20.4801	−0.714761	−0.357380	+	0.933959i	$0.616330\pi$
$822$	0	0
$823$	−3.75154	−0.130770	−0.0653852	−	0.997860i	$-0.520828\pi$
$823$	−3.75154	−0.130770	−0.0653852	+	0.997860i	$0.520828\pi$
$824$	−25.5811	−0.891159
$825$	0	0
$826$	12.4247	0.432310
$827$	48.1483	1.67428	0.837141	−	0.546987i	$-0.184225\pi$
$827$	48.1483	1.67428	0.837141	+	0.546987i	$0.184225\pi$
$828$	0	0
$829$	36.5608	1.26981	0.634904	−	0.772591i	$-0.281040\pi$
$829$	36.5608	1.26981	0.634904	+	0.772591i	$0.281040\pi$
$830$	0	0
$831$	0	0
$832$	6.02893	0.209016
$833$	−3.75872	−0.130232
$834$	0	0
$835$	0	0
$836$	−0.505339	−0.0174775
$837$	0	0
$838$	25.8264	0.892159
$839$	−45.2294	−1.56149	−0.780746	−	0.624849i	$-0.785161\pi$
$839$	−45.2294	−1.56149	−0.780746	+	0.624849i	$0.785161\pi$
$840$	0	0
$841$	−2.70540	−0.0932896
$842$	−29.2762	−1.00892
$843$	0	0
$844$	−8.39803	−0.289072
$845$	0	0
$846$	0	0
$847$	−7.78151	−0.267376
$848$	−39.1026	−1.34279
$849$	0	0
$850$	0	0
$851$	17.1194	0.586846
$852$	0	0
$853$	−37.2534	−1.27553	−0.637766	−	0.770230i	$-0.720141\pi$
$853$	−37.2534	−1.27553	−0.637766	+	0.770230i	$0.720141\pi$
$854$	21.1773	0.724671
$855$	0	0
$856$	−24.4789	−0.836671
$857$	−10.8371	−0.370188	−0.185094	−	0.982721i	$-0.559259\pi$
$857$	−10.8371	−0.370188	−0.185094	+	0.982721i	$0.559259\pi$
$858$	0	0
$859$	14.6081	0.498422	0.249211	−	0.968449i	$-0.419829\pi$
$859$	14.6081	0.498422	0.249211	+	0.968449i	$0.419829\pi$
$860$	0	0
$861$	0	0
$862$	−12.3596	−0.420971
$863$	10.3440	0.352116	0.176058	−	0.984380i	$-0.443665\pi$
$863$	10.3440	0.352116	0.176058	+	0.984380i	$0.443665\pi$
$864$	0	0
$865$	0	0
$866$	−20.0821	−0.682417
$867$	0	0
$868$	0.554787	0.0188307
$869$	1.26180	0.0428035
$870$	0	0
$871$	−7.86603	−0.266530
$872$	42.3324	1.43356
$873$	0	0
$874$	−2.34858	−0.0794420
$875$	0	0
$876$	0	0
$877$	38.0677	1.28545	0.642727	−	0.766095i	$-0.277803\pi$
$877$	38.0677	1.28545	0.642727	+	0.766095i	$0.277803\pi$
$878$	−11.8531	−0.400022
$879$	0	0
$880$	0	0
$881$	−12.0494	−0.405956	−0.202978	−	0.979183i	$-0.565062\pi$
$881$	−12.0494	−0.405956	−0.202978	+	0.979183i	$0.565062\pi$
$882$	0	0
$883$	−0.320699	−0.0107924	−0.00539619	−	0.999985i	$-0.501718\pi$
$883$	−0.320699	−0.0107924	−0.00539619	+	0.999985i	$0.501718\pi$
$884$	0.340173	0.0114413
$885$	0	0
$886$	9.83672	0.330471
$887$	−3.62144	−0.121596	−0.0607981	−	0.998150i	$-0.519365\pi$
$887$	−3.62144	−0.121596	−0.0607981	+	0.998150i	$0.519365\pi$
$888$	0	0
$889$	30.7792	1.03230
$890$	0	0
$891$	0	0
$892$	−3.23370	−0.108272
$893$	−3.07838	−0.103014
$894$	0	0
$895$	0	0
$896$	22.9132	0.765477
$897$	0	0
$898$	−48.6635	−1.62392
$899$	4.50921	0.150391
$900$	0	0
$901$	7.83096	0.260887
$902$	−4.93146	−0.164200
$903$	0	0
$904$	29.5174	0.981736
$905$	0	0
$906$	0	0
$907$	10.9333	0.363036	0.181518	−	0.983388i	$-0.441899\pi$
$907$	10.9333	0.363036	0.181518	+	0.983388i	$0.441899\pi$
$908$	−6.35842	−0.211012
$909$	0	0
$910$	0	0
$911$	37.5897	1.24540	0.622701	−	0.782460i	$-0.286035\pi$
$911$	37.5897	1.24540	0.622701	+	0.782460i	$0.286035\pi$
$912$	0	0
$913$	21.9155	0.725297
$914$	−54.8613	−1.81465
$915$	0	0
$916$	1.13624	0.0375423
$917$	−24.3668	−0.804664
$918$	0	0
$919$	33.6742	1.11081	0.555405	−	0.831580i	$-0.312563\pi$
$919$	33.6742	1.11081	0.555405	+	0.831580i	$0.312563\pi$
$920$	0	0
$921$	0	0
$922$	23.0563	0.759317
$923$	14.4813	0.476659
$924$	0	0
$925$	0	0
$926$	−14.0014	−0.460116
$927$	0	0
$928$	−10.5776	−0.347226
$929$	43.2039	1.41748	0.708738	−	0.705472i	$-0.249265\pi$
$929$	43.2039	1.41748	0.708738	+	0.705472i	$0.249265\pi$
$930$	0	0
$931$	2.19902	0.0720698
$932$	6.98932	0.228943
$933$	0	0
$934$	−2.88920	−0.0945376
$935$	0	0
$936$	0	0
$937$	−27.5630	−0.900445	−0.450222	−	0.892916i	$-0.648655\pi$
$937$	−27.5630	−0.900445	−0.450222	+	0.892916i	$0.648655\pi$
$938$	−20.6947	−0.675707
$939$	0	0
$940$	0	0
$941$	−58.1666	−1.89618	−0.948088	−	0.318007i	$-0.896987\pi$
$941$	−58.1666	−1.89618	−0.948088	+	0.318007i	$0.896987\pi$
$942$	0	0
$943$	−3.57077	−0.116280
$944$	−21.7333	−0.707358
$945$	0	0
$946$	−25.1377	−0.817296
$947$	48.5152	1.57653	0.788266	−	0.615335i	$-0.210979\pi$
$947$	48.5152	1.57653	0.788266	+	0.615335i	$0.210979\pi$
$948$	0	0
$949$	1.95055	0.0633176
$950$	0	0
$951$	0	0
$952$	−3.95443	−0.128164
$953$	−23.0349	−0.746173	−0.373087	−	0.927796i	$-0.621701\pi$
$953$	−23.0349	−0.746173	−0.373087	+	0.927796i	$0.621701\pi$
$954$	0	0
$955$	0	0
$956$	−2.45013	−0.0792430
$957$	0	0
$958$	24.2316	0.782889
$959$	23.5630	0.760890
$960$	0	0
$961$	−30.2267	−0.975056
$962$	9.31124	0.300207
$963$	0	0
$964$	−3.49777	−0.112655
$965$	0	0
$966$	0	0
$967$	−54.9998	−1.76868	−0.884338	−	0.466848i	$-0.845389\pi$
$967$	−54.9998	−1.76868	−0.884338	+	0.466848i	$0.845389\pi$
$968$	11.4280	0.367310
$969$	0	0
$970$	0	0
$971$	−9.70540	−0.311461	−0.155731	−	0.987800i	$-0.549773\pi$
$971$	−9.70540	−0.311461	−0.155731	+	0.987800i	$0.549773\pi$
$972$	0	0
$973$	−11.3730	−0.364601
$974$	7.61038	0.243852
$975$	0	0
$976$	−37.0433	−1.18573
$977$	32.2062	1.03037	0.515184	−	0.857080i	$-0.327724\pi$
$977$	32.2062	1.03037	0.515184	+	0.857080i	$0.327724\pi$
$978$	0	0
$979$	−32.5958	−1.04177
$980$	0	0
$981$	0	0
$982$	−60.7768	−1.93947
$983$	44.0782	1.40588	0.702938	−	0.711251i	$-0.251871\pi$
$983$	44.0782	1.40588	0.702938	+	0.711251i	$0.251871\pi$
$984$	0	0
$985$	0	0
$986$	7.27408	0.231654
$987$	0	0
$988$	−0.199016	−0.00633154
$989$	−18.2017	−0.578779
$990$	0	0
$991$	12.0677	0.383343	0.191672	−	0.981459i	$-0.438609\pi$
$991$	12.0677	0.383343	0.191672	+	0.981459i	$0.438609\pi$
$992$	−1.81393	−0.0575923
$993$	0	0
$994$	38.0989	1.20842
$995$	0	0
$996$	0	0
$997$	−45.7587	−1.44919	−0.724597	−	0.689173i	$-0.757974\pi$
$997$	−45.7587	−1.44919	−0.724597	+	0.689173i	$0.757974\pi$
$998$	2.57182	0.0814094
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.a.bj.1.2		3
3.2	odd	2		325.2.a.j.1.2		3
5.2	odd	4		585.2.c.b.469.5		6
5.3	odd	4		585.2.c.b.469.2		6
5.4	even	2		2925.2.a.bf.1.2		3
12.11	even	2		5200.2.a.cj.1.3		3
15.2	even	4		65.2.b.a.14.2	✓	6
15.8	even	4		65.2.b.a.14.5	yes	6
15.14	odd	2		325.2.a.k.1.2		3
39.38	odd	2		4225.2.a.bh.1.2		3
60.23	odd	4		1040.2.d.c.209.6		6
60.47	odd	4		1040.2.d.c.209.1		6
60.59	even	2		5200.2.a.cb.1.1		3
195.2	odd	12		845.2.l.e.654.3		12
195.8	odd	4		845.2.d.a.844.3		6
195.17	even	12		845.2.n.g.484.5		12
195.23	even	12		845.2.n.g.529.5		12
195.32	odd	12		845.2.l.e.699.4		12
195.38	even	4		845.2.b.c.339.2		6
195.47	odd	4		845.2.d.b.844.4		6
195.62	even	12		845.2.n.g.529.2		12
195.68	even	12		845.2.n.f.529.2		12
195.77	even	4		845.2.b.c.339.5		6
195.83	odd	4		845.2.d.b.844.3		6
195.98	odd	12		845.2.l.e.699.3		12
195.107	even	12		845.2.n.f.529.5		12
195.113	even	12		845.2.n.f.484.5		12
195.122	odd	4		845.2.d.a.844.4		6
195.128	odd	12		845.2.l.e.654.4		12
195.137	odd	12		845.2.l.d.699.4		12
195.152	even	12		845.2.n.f.484.2		12
195.158	odd	12		845.2.l.d.654.4		12
195.167	odd	12		845.2.l.d.654.3		12
195.173	even	12		845.2.n.g.484.2		12
195.188	odd	12		845.2.l.d.699.3		12
195.194	odd	2		4225.2.a.ba.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.b.a.14.2	✓	6	15.2	even	4
65.2.b.a.14.5	yes	6	15.8	even	4
325.2.a.j.1.2		3	3.2	odd	2
325.2.a.k.1.2		3	15.14	odd	2
585.2.c.b.469.2		6	5.3	odd	4
585.2.c.b.469.5		6	5.2	odd	4
845.2.b.c.339.2		6	195.38	even	4
845.2.b.c.339.5		6	195.77	even	4
845.2.d.a.844.3		6	195.8	odd	4
845.2.d.a.844.4		6	195.122	odd	4
845.2.d.b.844.3		6	195.83	odd	4
845.2.d.b.844.4		6	195.47	odd	4
845.2.l.d.654.3		12	195.167	odd	12
845.2.l.d.654.4		12	195.158	odd	12
845.2.l.d.699.3		12	195.188	odd	12
845.2.l.d.699.4		12	195.137	odd	12
845.2.l.e.654.3		12	195.2	odd	12
845.2.l.e.654.4		12	195.128	odd	12
845.2.l.e.699.3		12	195.98	odd	12
845.2.l.e.699.4		12	195.32	odd	12
845.2.n.f.484.2		12	195.152	even	12
845.2.n.f.484.5		12	195.113	even	12
845.2.n.f.529.2		12	195.68	even	12
845.2.n.f.529.5		12	195.107	even	12
845.2.n.g.484.2		12	195.173	even	12
845.2.n.g.484.5		12	195.17	even	12
845.2.n.g.529.2		12	195.62	even	12
845.2.n.g.529.5		12	195.23	even	12
1040.2.d.c.209.1		6	60.47	odd	4
1040.2.d.c.209.6		6	60.23	odd	4
2925.2.a.bf.1.2		3	5.4	even	2
2925.2.a.bj.1.2		3	1.1	even	1	trivial
4225.2.a.ba.1.2		3	195.194	odd	2
4225.2.a.bh.1.2		3	39.38	odd	2
5200.2.a.cb.1.1		3	60.59	even	2
5200.2.a.cj.1.3		3	12.11	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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53919 q^{2} +0.369102 q^{4} +1.70928 q^{7} -2.51026 q^{8} +O(q^{10})\)	\(q+1.53919 q^{2} +0.369102 q^{4} +1.70928 q^{7} -2.51026 q^{8} +2.53919 q^{11} +1.00000 q^{13} +2.63090 q^{14} -4.60197 q^{16} +0.921622 q^{17} -0.539189 q^{19} +3.90829 q^{22} +2.82991 q^{23} +1.53919 q^{26} +0.630898 q^{28} +5.12783 q^{29} +0.879362 q^{31} -2.06278 q^{32} +1.41855 q^{34} +6.04945 q^{37} -0.829914 q^{38} -1.26180 q^{41} -6.43188 q^{43} +0.937221 q^{44} +4.35577 q^{46} +5.70928 q^{47} -4.07838 q^{49} +0.369102 q^{52} +8.49693 q^{53} -4.29072 q^{56} +7.89269 q^{58} +4.72261 q^{59} +8.04945 q^{61} +1.35350 q^{62} +6.02893 q^{64} -7.86603 q^{67} +0.340173 q^{68} +14.4813 q^{71} +1.95055 q^{73} +9.31124 q^{74} -0.199016 q^{76} +4.34017 q^{77} +0.496928 q^{79} -1.94214 q^{82} +8.63090 q^{83} -9.89988 q^{86} -6.37402 q^{88} -12.8371 q^{89} +1.70928 q^{91} +1.04453 q^{92} +8.78765 q^{94} -5.91548 q^{97} -6.27739 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 5 * q^4 - 2 * q^7 + 9 * q^8	\( 3 q + 3 q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8} + 6 q^{11} + 3 q^{13} + 4 q^{14} + 5 q^{16} + 6 q^{17} + 14 q^{22} + 14 q^{23} + 3 q^{26} - 2 q^{28} - 6 q^{29} - 10 q^{31} + 11 q^{32} - 10 q^{34} - 8 q^{38} + 4 q^{41} - 6 q^{43} + 20 q^{44} + 16 q^{46} + 10 q^{47} - 9 q^{49} + 5 q^{52} + 8 q^{53} - 20 q^{56} + 12 q^{58} + 8 q^{59} + 6 q^{61} - 6 q^{62} + 33 q^{64} - 10 q^{67} - 10 q^{68} + 12 q^{71} + 24 q^{73} + 2 q^{74} - 10 q^{76} + 2 q^{77} - 16 q^{79} + 24 q^{82} + 22 q^{83} + 16 q^{86} + 24 q^{88} - 10 q^{89} - 2 q^{91} + 32 q^{92} + 16 q^{94} + 14 q^{97} - 25 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 5 * q^4 - 2 * q^7 + 9 * q^8 + 6 * q^11 + 3 * q^13 + 4 * q^14 + 5 * q^16 + 6 * q^17 + 14 * q^22 + 14 * q^23 + 3 * q^26 - 2 * q^28 - 6 * q^29 - 10 * q^31 + 11 * q^32 - 10 * q^34 - 8 * q^38 + 4 * q^41 - 6 * q^43 + 20 * q^44 + 16 * q^46 + 10 * q^47 - 9 * q^49 + 5 * q^52 + 8 * q^53 - 20 * q^56 + 12 * q^58 + 8 * q^59 + 6 * q^61 - 6 * q^62 + 33 * q^64 - 10 * q^67 - 10 * q^68 + 12 * q^71 + 24 * q^73 + 2 * q^74 - 10 * q^76 + 2 * q^77 - 16 * q^79 + 24 * q^82 + 22 * q^83 + 16 * q^86 + 24 * q^88 - 10 * q^89 - 2 * q^91 + 32 * q^92 + 16 * q^94 + 14 * q^97 - 25 * q^98

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