LMFDB - Embedded newform 1225.4.a.br.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.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:	$72.2773397570$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 83 x^{10} + 130 x^{9} + 2358 x^{8} - 2866 x^{7} - 28527 x^{6} + 24618 x^{5} + \cdots - 11012 $ x^12 - 2x^11 - 83x^10 + 130x^9 + 2358x^8 - 2866x^7 - 28527x^6 + 24618x^5 + 146563x^4 - 82640x^3 - 243048x^2 + 109224*x - 11012
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.73876$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.32455 q^{2} -6.52145 q^{3} +10.7017 q^{4} +28.2023 q^{6} -11.6837 q^{8} +15.5293 q^{9} +60.2876 q^{11} -69.7907 q^{12} +51.2386 q^{13} -35.0869 q^{16} -106.231 q^{17} -67.1571 q^{18} +87.0593 q^{19} -260.716 q^{22} -143.140 q^{23} +76.1949 q^{24} -221.584 q^{26} +74.8057 q^{27} -97.2962 q^{29} +111.587 q^{31} +245.205 q^{32} -393.162 q^{33} +459.403 q^{34} +166.190 q^{36} -345.001 q^{37} -376.492 q^{38} -334.150 q^{39} -237.933 q^{41} +162.845 q^{43} +645.181 q^{44} +619.017 q^{46} +125.041 q^{47} +228.817 q^{48} +692.783 q^{51} +548.341 q^{52} -436.168 q^{53} -323.501 q^{54} -567.753 q^{57} +420.762 q^{58} +512.605 q^{59} -597.469 q^{61} -482.566 q^{62} -779.705 q^{64} +1700.25 q^{66} -551.516 q^{67} -1136.86 q^{68} +933.481 q^{69} +736.134 q^{71} -181.440 q^{72} +578.126 q^{73} +1491.97 q^{74} +931.685 q^{76} +1445.05 q^{78} -412.915 q^{79} -907.132 q^{81} +1028.95 q^{82} +961.838 q^{83} -704.232 q^{86} +634.512 q^{87} -704.384 q^{88} -895.439 q^{89} -1531.85 q^{92} -727.712 q^{93} -540.748 q^{94} -1599.09 q^{96} +474.428 q^{97} +936.222 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 10 q^{2} + 58 q^{4} - 180 q^{8} + 140 q^{9} + 28 q^{11} + 386 q^{16} - 678 q^{18} - 118 q^{22} - 492 q^{23} - 488 q^{29} - 1390 q^{32} + 1810 q^{36} - 304 q^{37} - 80 q^{39} - 644 q^{43} + 610 q^{44}+ \cdots - 1048 q^{99}+O(q^{100}) $ 12 * q - 10 * q^2 + 58 * q^4 - 180 * q^8 + 140 * q^9 + 28 * q^11 + 386 * q^16 - 678 * q^18 - 118 * q^22 - 492 * q^23 - 488 * q^29 - 1390 * q^32 + 1810 * q^36 - 304 * q^37 - 80 * q^39 - 644 * q^43 + 610 * q^44 + 754 * q^46 - 1812 * q^51 - 2360 * q^53 - 2068 * q^57 + 314 * q^58 + 2792 * q^64 - 4168 * q^67 - 988 * q^71 - 11336 * q^72 - 514 * q^74 + 2572 * q^78 + 3092 * q^79 - 2936 * q^81 - 3190 * q^86 + 1764 * q^88 - 4270 * q^92 - 640 * q^93 - 1048 * q^99

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 $

$2$	−4.32455	−1.52896	−0.764479	−	0.644648i	$-0.777004\pi$
$2$	−4.32455	−1.52896	−0.764479	+	0.644648i	$0.777004\pi$
$3$	−6.52145	−1.25505	−0.627527	−	0.778595i	$-0.715933\pi$
$3$	−6.52145	−1.25505	−0.627527	+	0.778595i	$0.715933\pi$
$4$	10.7017	1.33772
$5$	0	0
$5$	0	0
$6$	28.2023	1.91892
$7$	0	0
$7$	0	0
$8$	−11.6837	−0.516353
$9$	15.5293	0.575159
$10$	0	0
$11$	60.2876	1.65249	0.826245	−	0.563312i	$-0.190473\pi$
$11$	60.2876	1.65249	0.826245	+	0.563312i	$0.190473\pi$
$12$	−69.7907	−1.67890
$13$	51.2386	1.09316	0.546578	−	0.837408i	$-0.315930\pi$
$13$	51.2386	1.09316	0.546578	+	0.837408i	$0.315930\pi$
$14$	0	0
$15$	0	0
$16$	−35.0869	−0.548233
$17$	−106.231	−1.51558	−0.757792	−	0.652497i	$-0.773722\pi$
$17$	−106.231	−1.51558	−0.757792	+	0.652497i	$0.773722\pi$
$18$	−67.1571	−0.879394
$19$	87.0593	1.05120	0.525600	−	0.850732i	$-0.323841\pi$
$19$	87.0593	1.05120	0.525600	+	0.850732i	$0.323841\pi$
$20$	0	0
$21$	0	0
$22$	−260.716	−2.52659
$23$	−143.140	−1.29769	−0.648843	−	0.760922i	$-0.724747\pi$
$23$	−143.140	−1.29769	−0.648843	+	0.760922i	$0.724747\pi$
$24$	76.1949	0.648051
$25$	0	0
$26$	−221.584	−1.67139
$27$	74.8057	0.533199
$28$	0	0
$29$	−97.2962	−0.623016	−0.311508	−	0.950244i	$-0.600834\pi$
$29$	−97.2962	−0.623016	−0.311508	+	0.950244i	$0.600834\pi$
$30$	0	0
$31$	111.587	0.646507	0.323253	−	0.946312i	$-0.395223\pi$
$31$	111.587	0.646507	0.323253	+	0.946312i	$0.395223\pi$
$32$	245.205	1.35458
$33$	−393.162	−2.07396
$34$	459.403	2.31726
$35$	0	0
$36$	166.190	0.769399
$37$	−345.001	−1.53291	−0.766457	−	0.642295i	$-0.777982\pi$
$37$	−345.001	−1.53291	−0.766457	+	0.642295i	$0.777982\pi$
$38$	−376.492	−1.60724
$39$	−334.150	−1.37197
$40$	0	0
$41$	−237.933	−0.906315	−0.453157	−	0.891431i	$-0.649702\pi$
$41$	−237.933	−0.906315	−0.453157	+	0.891431i	$0.649702\pi$
$42$	0	0
$43$	162.845	0.577527	0.288764	−	0.957400i	$-0.406756\pi$
$43$	162.845	0.577527	0.288764	+	0.957400i	$0.406756\pi$
$44$	645.181	2.21056
$45$	0	0
$46$	619.017	1.98411
$47$	125.041	0.388067	0.194034	−	0.980995i	$-0.437843\pi$
$47$	125.041	0.388067	0.194034	+	0.980995i	$0.437843\pi$
$48$	228.817	0.688061
$49$	0	0
$50$	0	0
$51$	692.783	1.90214
$52$	548.341	1.46233
$53$	−436.168	−1.13042	−0.565211	−	0.824947i	$-0.691205\pi$
$53$	−436.168	−1.13042	−0.565211	+	0.824947i	$0.691205\pi$
$54$	−323.501	−0.815239
$55$	0	0
$56$	0	0
$57$	−567.753	−1.31931
$58$	420.762	0.952565
$59$	512.605	1.13111	0.565555	−	0.824711i	$-0.308662\pi$
$59$	512.605	1.13111	0.565555	+	0.824711i	$0.308662\pi$
$60$	0	0
$61$	−597.469	−1.25407	−0.627034	−	0.778992i	$-0.715731\pi$
$61$	−597.469	−1.25407	−0.627034	+	0.778992i	$0.715731\pi$
$62$	−482.566	−0.988482
$63$	0	0
$64$	−779.705	−1.52286
$65$	0	0
$66$	1700.25	3.17100
$67$	−551.516	−1.00565	−0.502824	−	0.864389i	$-0.667706\pi$
$67$	−551.516	−1.00565	−0.502824	+	0.864389i	$0.667706\pi$
$68$	−1136.86	−2.02742
$69$	933.481	1.62867
$70$	0	0
$71$	736.134	1.23047	0.615233	−	0.788345i	$-0.289062\pi$
$71$	736.134	1.23047	0.615233	+	0.788345i	$0.289062\pi$
$72$	−181.440	−0.296985
$73$	578.126	0.926912	0.463456	−	0.886120i	$-0.346609\pi$
$73$	578.126	0.926912	0.463456	+	0.886120i	$0.346609\pi$
$74$	1491.97	2.34376
$75$	0	0
$76$	931.685	1.40621
$77$	0	0
$78$	1445.05	2.09768
$79$	−412.915	−0.588058	−0.294029	−	0.955796i	$-0.594996\pi$
$79$	−412.915	−0.588058	−0.294029	+	0.955796i	$0.594996\pi$
$80$	0	0
$81$	−907.132	−1.24435
$82$	1028.95	1.38572
$83$	961.838	1.27199	0.635996	−	0.771692i	$-0.280589\pi$
$83$	961.838	1.27199	0.635996	+	0.771692i	$0.280589\pi$
$84$	0	0
$85$	0	0
$86$	−704.232	−0.883015
$87$	634.512	0.781918
$88$	−704.384	−0.853268
$89$	−895.439	−1.06648	−0.533238	−	0.845965i	$-0.679025\pi$
$89$	−895.439	−1.06648	−0.533238	+	0.845965i	$0.679025\pi$
$90$	0	0
$91$	0	0
$92$	−1531.85	−1.73594
$93$	−727.712	−0.811400
$94$	−540.748	−0.593339
$95$	0	0
$96$	−1599.09	−1.70007
$97$	474.428	0.496607	0.248304	−	0.968682i	$-0.420127\pi$
$97$	474.428	0.496607	0.248304	+	0.968682i	$0.420127\pi$
$98$	0	0
$99$	936.222	0.950443
$100$	0	0
$101$	376.277	0.370703	0.185351	−	0.982672i	$-0.440658\pi$
$101$	376.277	0.370703	0.185351	+	0.982672i	$0.440658\pi$
$102$	−2995.97	−2.90829
$103$	1293.76	1.23765	0.618823	−	0.785530i	$-0.287610\pi$
$103$	1293.76	1.23765	0.618823	+	0.785530i	$0.287610\pi$
$104$	−598.658	−0.564454
$105$	0	0
$106$	1886.23	1.72837
$107$	−1159.17	−1.04730	−0.523650	−	0.851933i	$-0.675430\pi$
$107$	−1159.17	−1.04730	−0.523650	+	0.851933i	$0.675430\pi$
$108$	800.550	0.713268
$109$	1632.06	1.43416	0.717079	−	0.696992i	$-0.245479\pi$
$109$	1632.06	1.43416	0.717079	+	0.696992i	$0.245479\pi$
$110$	0	0
$111$	2249.91	1.92389
$112$	0	0
$113$	1493.26	1.24314	0.621568	−	0.783360i	$-0.286496\pi$
$113$	1493.26	1.24314	0.621568	+	0.783360i	$0.286496\pi$
$114$	2455.28	2.01717
$115$	0	0
$116$	−1041.24	−0.833418
$117$	795.698	0.628738
$118$	−2216.79	−1.72942
$119$	0	0
$120$	0	0
$121$	2303.59	1.73072
$122$	2583.79	1.91742
$123$	1551.67	1.13747
$124$	1194.18	0.864842
$125$	0	0
$126$	0	0
$127$	−54.8987	−0.0383580	−0.0191790	−	0.999816i	$-0.506105\pi$
$127$	−54.8987	−0.0383580	−0.0191790	+	0.999816i	$0.506105\pi$
$128$	1410.24	0.973815
$129$	−1061.99	−0.724827
$130$	0	0
$131$	−1849.60	−1.23359	−0.616795	−	0.787124i	$-0.711569\pi$
$131$	−1849.60	−1.23359	−0.616795	+	0.787124i	$0.711569\pi$
$132$	−4207.51	−2.77437
$133$	0	0
$134$	2385.06	1.53759
$135$	0	0
$136$	1241.18	0.782576
$137$	1758.35	1.09654	0.548269	−	0.836302i	$-0.315287\pi$
$137$	1758.35	1.09654	0.548269	+	0.836302i	$0.315287\pi$
$138$	−4036.89	−2.49016
$139$	283.853	0.173209	0.0866047	−	0.996243i	$-0.472398\pi$
$139$	283.853	0.173209	0.0866047	+	0.996243i	$0.472398\pi$
$140$	0	0
$141$	−815.451	−0.487045
$142$	−3183.45	−1.88133
$143$	3089.05	1.80643
$144$	−544.874	−0.315321
$145$	0	0
$146$	−2500.13	−1.41721
$147$	0	0
$148$	−3692.11	−2.05060
$149$	653.719	0.359428	0.179714	−	0.983719i	$-0.442483\pi$
$149$	653.719	0.359428	0.179714	+	0.983719i	$0.442483\pi$
$150$	0	0
$151$	−1550.48	−0.835607	−0.417803	−	0.908537i	$-0.637200\pi$
$151$	−1550.48	−0.835607	−0.417803	+	0.908537i	$0.637200\pi$
$152$	−1017.18	−0.542790
$153$	−1649.70	−0.871701
$154$	0	0
$155$	0	0
$156$	−3575.98	−1.83530
$157$	−2059.46	−1.04690	−0.523449	−	0.852057i	$-0.675355\pi$
$157$	−2059.46	−1.04690	−0.523449	+	0.852057i	$0.675355\pi$
$158$	1785.67	0.899117
$159$	2844.45	1.41874
$160$	0	0
$161$	0	0
$162$	3922.94	1.90256
$163$	−396.631	−0.190592	−0.0952962	−	0.995449i	$-0.530380\pi$
$163$	−396.631	−0.190592	−0.0952962	+	0.995449i	$0.530380\pi$
$164$	−2546.29	−1.21239
$165$	0	0
$166$	−4159.51	−1.94483
$167$	−1479.26	−0.685438	−0.342719	−	0.939438i	$-0.611348\pi$
$167$	−1479.26	−0.685438	−0.342719	+	0.939438i	$0.611348\pi$
$168$	0	0
$169$	428.390	0.194988
$170$	0	0
$171$	1351.97	0.604606
$172$	1742.72	0.772567
$173$	3375.25	1.48333	0.741664	−	0.670771i	$-0.234037\pi$
$173$	3375.25	1.48333	0.741664	+	0.670771i	$0.234037\pi$
$174$	−2743.98	−1.19552
$175$	0	0
$176$	−2115.30	−0.905949
$177$	−3342.93	−1.41960
$178$	3872.37	1.63060
$179$	−4072.82	−1.70065	−0.850326	−	0.526256i	$-0.823595\pi$
$179$	−4072.82	−1.70065	−0.850326	+	0.526256i	$0.823595\pi$
$180$	0	0
$181$	−1909.31	−0.784079	−0.392039	−	0.919948i	$-0.628230\pi$
$181$	−1909.31	−0.784079	−0.392039	+	0.919948i	$0.628230\pi$
$182$	0	0
$183$	3896.36	1.57392
$184$	1672.41	0.670064
$185$	0	0
$186$	3147.03	1.24060
$187$	−6404.44	−2.50449
$188$	1338.16	0.519124
$189$	0	0
$190$	0	0
$191$	−3892.04	−1.47444	−0.737220	−	0.675652i	$-0.763862\pi$
$191$	−3892.04	−1.47444	−0.737220	+	0.675652i	$0.763862\pi$
$192$	5084.81	1.91127
$193$	2257.27	0.841877	0.420938	−	0.907089i	$-0.361701\pi$
$193$	2257.27	0.841877	0.420938	+	0.907089i	$0.361701\pi$
$194$	−2051.69	−0.759292
$195$	0	0
$196$	0	0
$197$	−4042.45	−1.46199	−0.730997	−	0.682380i	$-0.760945\pi$
$197$	−4042.45	−1.46199	−0.730997	+	0.682380i	$0.760945\pi$
$198$	−4048.74	−1.45319
$199$	−3783.70	−1.34784	−0.673918	−	0.738806i	$-0.735390\pi$
$199$	−3783.70	−1.34784	−0.673918	+	0.738806i	$0.735390\pi$
$200$	0	0
$201$	3596.68	1.26214
$202$	−1627.23	−0.566790
$203$	0	0
$204$	7413.97	2.54452
$205$	0	0
$206$	−5594.91	−1.89231
$207$	−2222.86	−0.746376
$208$	−1797.80	−0.599304
$209$	5248.59	1.73710
$210$	0	0
$211$	1912.30	0.623924	0.311962	−	0.950095i	$-0.399014\pi$
$211$	1912.30	0.623924	0.311962	+	0.950095i	$0.399014\pi$
$212$	−4667.75	−1.51218
$213$	−4800.66	−1.54430
$214$	5012.89	1.60128
$215$	0	0
$216$	−874.010	−0.275319
$217$	0	0
$218$	−7057.93	−2.19277
$219$	−3770.22	−1.16332
$220$	0	0
$221$	−5443.15	−1.65677
$222$	−9729.83	−2.94155
$223$	1450.22	0.435489	0.217744	−	0.976006i	$-0.430130\pi$
$223$	1450.22	0.435489	0.217744	+	0.976006i	$0.430130\pi$
$224$	0	0
$225$	0	0
$226$	−6457.69	−1.90070
$227$	−2182.90	−0.638256	−0.319128	−	0.947712i	$-0.603390\pi$
$227$	−2182.90	−0.638256	−0.319128	+	0.947712i	$0.603390\pi$
$228$	−6075.94	−1.76486
$229$	3599.67	1.03875	0.519373	−	0.854548i	$-0.326165\pi$
$229$	3599.67	1.03875	0.519373	+	0.854548i	$0.326165\pi$
$230$	0	0
$231$	0	0
$232$	1136.78	0.321696
$233$	1722.93	0.484433	0.242217	−	0.970222i	$-0.422125\pi$
$233$	1722.93	0.484433	0.242217	+	0.970222i	$0.422125\pi$
$234$	−3441.03	−0.961314
$235$	0	0
$236$	5485.76	1.51310
$237$	2692.81	0.738045
$238$	0	0
$239$	−5594.30	−1.51408	−0.757040	−	0.653369i	$-0.773355\pi$
$239$	−5594.30	−1.51408	−0.757040	+	0.653369i	$0.773355\pi$
$240$	0	0
$241$	2374.82	0.634753	0.317376	−	0.948300i	$-0.397198\pi$
$241$	2374.82	0.634753	0.317376	+	0.948300i	$0.397198\pi$
$242$	−9961.98	−2.64620
$243$	3896.06	1.02853
$244$	−6393.95	−1.67759
$245$	0	0
$246$	−6710.26	−1.73915
$247$	4460.79	1.14912
$248$	−1303.76	−0.333826
$249$	−6272.58	−1.59642
$250$	0	0
$251$	−3420.57	−0.860178	−0.430089	−	0.902787i	$-0.641518\pi$
$251$	−3420.57	−0.860178	−0.430089	+	0.902787i	$0.641518\pi$
$252$	0	0
$253$	−8629.57	−2.14441
$254$	237.412	0.0586478
$255$	0	0
$256$	139.013	0.0339387
$257$	4847.27	1.17652	0.588258	−	0.808673i	$-0.299814\pi$
$257$	4847.27	1.17652	0.588258	+	0.808673i	$0.299814\pi$
$258$	4592.61	1.10823
$259$	0	0
$260$	0	0
$261$	−1510.94	−0.358333
$262$	7998.68	1.88611
$263$	−6046.80	−1.41773	−0.708863	−	0.705347i	$-0.750791\pi$
$263$	−6046.80	−1.41773	−0.708863	+	0.705347i	$0.750791\pi$
$264$	4593.60	1.07090
$265$	0	0
$266$	0	0
$267$	5839.56	1.33848
$268$	−5902.17	−1.34527
$269$	46.0036	0.0104271	0.00521354	−	0.999986i	$-0.498340\pi$
$269$	46.0036	0.0104271	0.00521354	+	0.999986i	$0.498340\pi$
$270$	0	0
$271$	4806.85	1.07747	0.538736	−	0.842474i	$-0.318902\pi$
$271$	4806.85	1.07747	0.538736	+	0.842474i	$0.318902\pi$
$272$	3727.33	0.830893
$273$	0	0
$274$	−7604.06	−1.67656
$275$	0	0
$276$	9989.86	2.17869
$277$	−2568.38	−0.557108	−0.278554	−	0.960421i	$-0.589855\pi$
$277$	−2568.38	−0.557108	−0.278554	+	0.960421i	$0.589855\pi$
$278$	−1227.54	−0.264830
$279$	1732.87	0.371844
$280$	0	0
$281$	−3056.49	−0.648879	−0.324440	−	0.945906i	$-0.605176\pi$
$281$	−3056.49	−0.648879	−0.324440	+	0.945906i	$0.605176\pi$
$282$	3526.46	0.744672
$283$	3732.87	0.784085	0.392042	−	0.919947i	$-0.371769\pi$
$283$	3732.87	0.784085	0.392042	+	0.919947i	$0.371769\pi$
$284$	7877.91	1.64601
$285$	0	0
$286$	−13358.7	−2.76195
$287$	0	0
$288$	3807.86	0.779097
$289$	6372.13	1.29699
$290$	0	0
$291$	−3093.96	−0.623268
$292$	6186.95	1.23994
$293$	−9110.55	−1.81653	−0.908266	−	0.418393i	$-0.862594\pi$
$293$	−9110.55	−1.81653	−0.908266	+	0.418393i	$0.862594\pi$
$294$	0	0
$295$	0	0
$296$	4030.90	0.791525
$297$	4509.85	0.881105
$298$	−2827.04	−0.549550
$299$	−7334.30	−1.41857
$300$	0	0
$301$	0	0
$302$	6705.15	1.27761
$303$	−2453.87	−0.465252
$304$	−3054.64	−0.576302
$305$	0	0
$306$	7134.20	1.33279
$307$	8496.89	1.57962	0.789810	−	0.613352i	$-0.210179\pi$
$307$	8496.89	1.57962	0.789810	+	0.613352i	$0.210179\pi$
$308$	0	0
$309$	−8437.16	−1.55331
$310$	0	0
$311$	−179.361	−0.0327029	−0.0163515	−	0.999866i	$-0.505205\pi$
$311$	−179.361	−0.0327029	−0.0163515	+	0.999866i	$0.505205\pi$
$312$	3904.12	0.708420
$313$	4622.25	0.834712	0.417356	−	0.908743i	$-0.362957\pi$
$313$	4622.25	0.834712	0.417356	+	0.908743i	$0.362957\pi$
$314$	8906.24	1.60066
$315$	0	0
$316$	−4418.91	−0.786655
$317$	−4251.64	−0.753300	−0.376650	−	0.926356i	$-0.622924\pi$
$317$	−4251.64	−0.753300	−0.376650	+	0.926356i	$0.622924\pi$
$318$	−12301.0	−2.16919
$319$	−5865.75	−1.02953
$320$	0	0
$321$	7559.46	1.31442
$322$	0	0
$323$	−9248.44	−1.59318
$324$	−9707.88	−1.66459
$325$	0	0
$326$	1715.25	0.291408
$327$	−10643.4	−1.79994
$328$	2779.95	0.467978
$329$	0	0
$330$	0	0
$331$	11024.5	1.83069	0.915347	−	0.402667i	$-0.131917\pi$
$331$	11024.5	1.83069	0.915347	+	0.402667i	$0.131917\pi$
$332$	10293.3	1.70156
$333$	−5357.62	−0.881669
$334$	6397.11	1.04801
$335$	0	0
$336$	0	0
$337$	−2758.01	−0.445812	−0.222906	−	0.974840i	$-0.571554\pi$
$337$	−2758.01	−0.445812	−0.222906	+	0.974840i	$0.571554\pi$
$338$	−1852.59	−0.298129
$339$	−9738.24	−1.56020
$340$	0	0
$341$	6727.34	1.06835
$342$	−5846.66	−0.924418
$343$	0	0
$344$	−1902.64	−0.298208
$345$	0	0
$346$	−14596.5	−2.26795
$347$	1450.14	0.224345	0.112172	−	0.993689i	$-0.464219\pi$
$347$	1450.14	0.224345	0.112172	+	0.993689i	$0.464219\pi$
$348$	6790.37	1.04598
$349$	7518.18	1.15312	0.576560	−	0.817055i	$-0.304395\pi$
$349$	7518.18	1.15312	0.576560	+	0.817055i	$0.304395\pi$
$350$	0	0
$351$	3832.94	0.582869
$352$	14782.8	2.23843
$353$	−453.117	−0.0683201	−0.0341600	−	0.999416i	$-0.510876\pi$
$353$	−453.117	−0.0683201	−0.0341600	+	0.999416i	$0.510876\pi$
$354$	14456.7	2.17052
$355$	0	0
$356$	−9582.74	−1.42664
$357$	0	0
$358$	17613.1	2.60023
$359$	2575.76	0.378673	0.189336	−	0.981912i	$-0.439366\pi$
$359$	2575.76	0.378673	0.189336	+	0.981912i	$0.439366\pi$
$360$	0	0
$361$	720.329	0.105020
$362$	8256.93	1.19882
$363$	−15022.7	−2.17215
$364$	0	0
$365$	0	0
$366$	−16850.0	−2.40646
$367$	−3489.74	−0.496356	−0.248178	−	0.968714i	$-0.579832\pi$
$367$	−3489.74	−0.496356	−0.248178	+	0.968714i	$0.579832\pi$
$368$	5022.35	0.711434
$369$	−3694.93	−0.521275
$370$	0	0
$371$	0	0
$372$	−7787.77	−1.08542
$373$	6276.24	0.871237	0.435619	−	0.900131i	$-0.356530\pi$
$373$	6276.24	0.871237	0.435619	+	0.900131i	$0.356530\pi$
$374$	27696.3	3.82925
$375$	0	0
$376$	−1460.95	−0.200380
$377$	−4985.32	−0.681053
$378$	0	0
$379$	−6156.67	−0.834425	−0.417212	−	0.908809i	$-0.636993\pi$
$379$	−6156.67	−0.834425	−0.417212	+	0.908809i	$0.636993\pi$
$380$	0	0
$381$	358.019	0.0481413
$382$	16831.3	2.25436
$383$	1191.17	0.158918	0.0794592	−	0.996838i	$-0.474681\pi$
$383$	1191.17	0.158918	0.0794592	+	0.996838i	$0.474681\pi$
$384$	−9196.77	−1.22219
$385$	0	0
$386$	−9761.70	−1.28719
$387$	2528.87	0.332170
$388$	5077.20	0.664319
$389$	−5766.53	−0.751606	−0.375803	−	0.926700i	$-0.622633\pi$
$389$	−5766.53	−0.751606	−0.375803	+	0.926700i	$0.622633\pi$
$390$	0	0
$391$	15206.0	1.96675
$392$	0	0
$393$	12062.1	1.54822
$394$	17481.8	2.23533
$395$	0	0
$396$	10019.2	1.27142
$397$	−7299.37	−0.922783	−0.461392	−	0.887197i	$-0.652650\pi$
$397$	−7299.37	−0.922783	−0.461392	+	0.887197i	$0.652650\pi$
$398$	16362.8	2.06078
$399$	0	0
$400$	0	0
$401$	−12087.8	−1.50533	−0.752664	−	0.658404i	$-0.771232\pi$
$401$	−12087.8	−1.50533	−0.752664	+	0.658404i	$0.771232\pi$
$402$	−15554.0	−1.92976
$403$	5717.58	0.706732
$404$	4026.82	0.495895
$405$	0	0
$406$	0	0
$407$	−20799.3	−2.53313
$408$	−8094.29	−0.982175
$409$	3828.88	0.462900	0.231450	−	0.972847i	$-0.425653\pi$
$409$	3828.88	0.462900	0.231450	+	0.972847i	$0.425653\pi$
$410$	0	0
$411$	−11467.0	−1.37621
$412$	13845.4	1.65562
$413$	0	0
$414$	9612.88	1.14118
$415$	0	0
$416$	12563.9	1.48076
$417$	−1851.13	−0.217387
$418$	−22697.8	−2.65595
$419$	−16111.9	−1.87857	−0.939283	−	0.343142i	$-0.888509\pi$
$419$	−16111.9	−1.87857	−0.939283	+	0.343142i	$0.888509\pi$
$420$	0	0
$421$	−3002.01	−0.347528	−0.173764	−	0.984787i	$-0.555593\pi$
$421$	−3002.01	−0.347528	−0.173764	+	0.984787i	$0.555593\pi$
$422$	−8269.82	−0.953954
$423$	1941.80	0.223200
$424$	5096.08	0.583697
$425$	0	0
$426$	20760.7	2.36117
$427$	0	0
$428$	−12405.1	−1.40099
$429$	−20145.1	−2.26716
$430$	0	0
$431$	−593.283	−0.0663050	−0.0331525	−	0.999450i	$-0.510555\pi$
$431$	−593.283	−0.0663050	−0.0331525	+	0.999450i	$0.510555\pi$
$432$	−2624.70	−0.292317
$433$	−15946.6	−1.76985	−0.884924	−	0.465736i	$-0.845790\pi$
$433$	−15946.6	−1.76985	−0.884924	+	0.465736i	$0.845790\pi$
$434$	0	0
$435$	0	0
$436$	17465.9	1.91849
$437$	−12461.7	−1.36413
$438$	16304.5	1.77867
$439$	11007.6	1.19673	0.598363	−	0.801225i	$-0.295818\pi$
$439$	11007.6	1.19673	0.598363	+	0.801225i	$0.295818\pi$
$440$	0	0
$441$	0	0
$442$	23539.2	2.53313
$443$	1208.79	0.129641	0.0648207	−	0.997897i	$-0.479352\pi$
$443$	1208.79	0.129641	0.0648207	+	0.997897i	$0.479352\pi$
$444$	24077.9	2.57362
$445$	0	0
$446$	−6271.55	−0.665844
$447$	−4263.19	−0.451101
$448$	0	0
$449$	4508.55	0.473878	0.236939	−	0.971524i	$-0.423856\pi$
$449$	4508.55	0.473878	0.236939	+	0.971524i	$0.423856\pi$
$450$	0	0
$451$	−14344.4	−1.49768
$452$	15980.5	1.66296
$453$	10111.4	1.04873
$454$	9440.05	0.975867
$455$	0	0
$456$	6633.48	0.681230
$457$	−5260.69	−0.538478	−0.269239	−	0.963073i	$-0.586772\pi$
$457$	−5260.69	−0.538478	−0.269239	+	0.963073i	$0.586772\pi$
$458$	−15567.0	−1.58820
$459$	−7946.72	−0.808107
$460$	0	0
$461$	2852.94	0.288231	0.144116	−	0.989561i	$-0.453966\pi$
$461$	2852.94	0.288231	0.144116	+	0.989561i	$0.453966\pi$
$462$	0	0
$463$	3579.03	0.359248	0.179624	−	0.983735i	$-0.442512\pi$
$463$	3579.03	0.359248	0.179624	+	0.983735i	$0.442512\pi$
$464$	3413.82	0.341558
$465$	0	0
$466$	−7450.90	−0.740679
$467$	9949.10	0.985845	0.492922	−	0.870073i	$-0.335929\pi$
$467$	9949.10	0.985845	0.492922	+	0.870073i	$0.335929\pi$
$468$	8515.34	0.841072
$469$	0	0
$470$	0	0
$471$	13430.7	1.31391
$472$	−5989.14	−0.584052
$473$	9817.54	0.954357
$474$	−11645.2	−1.12844
$475$	0	0
$476$	0	0
$477$	−6773.38	−0.650171
$478$	24192.8	2.31497
$479$	7005.69	0.668263	0.334132	−	0.942526i	$-0.391557\pi$
$479$	7005.69	0.668263	0.334132	+	0.942526i	$0.391557\pi$
$480$	0	0
$481$	−17677.4	−1.67571
$482$	−10270.0	−0.970511
$483$	0	0
$484$	24652.4	2.31521
$485$	0	0
$486$	−16848.7	−1.57258
$487$	−4772.63	−0.444083	−0.222041	−	0.975037i	$-0.571272\pi$
$487$	−4772.63	−0.444083	−0.222041	+	0.975037i	$0.571272\pi$
$488$	6980.67	0.647541
$489$	2586.61	0.239204
$490$	0	0
$491$	4640.45	0.426519	0.213259	−	0.976996i	$-0.431592\pi$
$491$	4640.45	0.426519	0.213259	+	0.976996i	$0.431592\pi$
$492$	16605.5	1.52162
$493$	10335.9	0.944232
$494$	−19290.9	−1.75696
$495$	0	0
$496$	−3915.26	−0.354436
$497$	0	0
$498$	27126.1	2.44086
$499$	15389.5	1.38062	0.690308	−	0.723515i	$-0.257475\pi$
$499$	15389.5	1.38062	0.690308	+	0.723515i	$0.257475\pi$
$500$	0	0
$501$	9646.89	0.860262
$502$	14792.4	1.31518
$503$	−3679.31	−0.326148	−0.163074	−	0.986614i	$-0.552141\pi$
$503$	−3679.31	−0.326148	−0.163074	+	0.986614i	$0.552141\pi$
$504$	0	0
$505$	0	0
$506$	37319.0	3.27872
$507$	−2793.72	−0.244721
$508$	−587.510	−0.0513121
$509$	−5349.69	−0.465856	−0.232928	−	0.972494i	$-0.574831\pi$
$509$	−5349.69	−0.465856	−0.232928	+	0.972494i	$0.574831\pi$
$510$	0	0
$511$	0	0
$512$	−11883.0	−1.02571
$513$	6512.53	0.560498
$514$	−20962.3	−1.79884
$515$	0	0
$516$	−11365.1	−0.969613
$517$	7538.44	0.641277
$518$	0	0
$519$	−22011.5	−1.86166
$520$	0	0
$521$	4032.59	0.339100	0.169550	−	0.985522i	$-0.445769\pi$
$521$	4032.59	0.339100	0.169550	+	0.985522i	$0.445769\pi$
$522$	6534.14	0.547876
$523$	−10125.6	−0.846580	−0.423290	−	0.905994i	$-0.639125\pi$
$523$	−10125.6	−0.846580	−0.423290	+	0.905994i	$0.639125\pi$
$524$	−19793.9	−1.65019
$525$	0	0
$526$	26149.7	2.16764
$527$	−11854.1	−0.979835
$528$	13794.8	1.13701
$529$	8322.11	0.683990
$530$	0	0
$531$	7960.39	0.650568
$532$	0	0
$533$	−12191.3	−0.990743
$534$	−25253.5	−2.04649
$535$	0	0
$536$	6443.77	0.519269
$537$	26560.7	2.13441
$538$	−198.945	−0.0159426
$539$	0	0
$540$	0	0
$541$	−5016.75	−0.398682	−0.199341	−	0.979930i	$-0.563880\pi$
$541$	−5016.75	−0.398682	−0.199341	+	0.979930i	$0.563880\pi$
$542$	−20787.4	−1.64741
$543$	12451.5	0.984061
$544$	−26048.5	−2.05298
$545$	0	0
$546$	0	0
$547$	7622.97	0.595858	0.297929	−	0.954588i	$-0.403704\pi$
$547$	7622.97	0.595858	0.297929	+	0.954588i	$0.403704\pi$
$548$	18817.3	1.46686
$549$	−9278.27	−0.721288
$550$	0	0
$551$	−8470.54	−0.654914
$552$	−10906.5	−0.840967
$553$	0	0
$554$	11107.1	0.851795
$555$	0	0
$556$	3037.72	0.231705
$557$	−6853.84	−0.521376	−0.260688	−	0.965423i	$-0.583949\pi$
$557$	−6853.84	−0.521376	−0.260688	+	0.965423i	$0.583949\pi$
$558$	−7493.90	−0.568534
$559$	8343.96	0.631327
$560$	0	0
$561$	41766.2	3.14326
$562$	13217.9	0.992110
$563$	−12704.7	−0.951050	−0.475525	−	0.879702i	$-0.657742\pi$
$563$	−12704.7	−0.951050	−0.475525	+	0.879702i	$0.657742\pi$
$564$	−8726.73	−0.651528
$565$	0	0
$566$	−16143.0	−1.19883
$567$	0	0
$568$	−8600.80	−0.635355
$569$	−21069.9	−1.55236	−0.776182	−	0.630509i	$-0.782846\pi$
$569$	−21069.9	−1.55236	−0.776182	+	0.630509i	$0.782846\pi$
$570$	0	0
$571$	−6627.96	−0.485764	−0.242882	−	0.970056i	$-0.578093\pi$
$571$	−6627.96	−0.485764	−0.242882	+	0.970056i	$0.578093\pi$
$572$	33058.1	2.41649
$573$	25381.7	1.85050
$574$	0	0
$575$	0	0
$576$	−12108.3	−0.875887
$577$	−18977.9	−1.36926	−0.684629	−	0.728892i	$-0.740036\pi$
$577$	−18977.9	−1.36926	−0.684629	+	0.728892i	$0.740036\pi$
$578$	−27556.6	−1.98305
$579$	−14720.7	−1.05660
$580$	0	0
$581$	0	0
$582$	13380.0	0.952952
$583$	−26295.5	−1.86801
$584$	−6754.67	−0.478614
$585$	0	0
$586$	39399.0	2.77740
$587$	−1824.82	−0.128311	−0.0641554	−	0.997940i	$-0.520435\pi$
$587$	−1824.82	−0.128311	−0.0641554	+	0.997940i	$0.520435\pi$
$588$	0	0
$589$	9714.73	0.679607
$590$	0	0
$591$	26362.7	1.83488
$592$	12105.0	0.840394
$593$	8897.81	0.616171	0.308085	−	0.951359i	$-0.400312\pi$
$593$	8897.81	0.616171	0.308085	+	0.951359i	$0.400312\pi$
$594$	−19503.1	−1.34717
$595$	0	0
$596$	6995.92	0.480812
$597$	24675.2	1.69160
$598$	31717.5	2.16894
$599$	−4699.67	−0.320573	−0.160286	−	0.987071i	$-0.551242\pi$
$599$	−4699.67	−0.320573	−0.160286	+	0.987071i	$0.551242\pi$
$600$	0	0
$601$	24866.7	1.68774	0.843871	−	0.536547i	$-0.180272\pi$
$601$	24866.7	1.68774	0.843871	+	0.536547i	$0.180272\pi$
$602$	0	0
$603$	−8564.65	−0.578407
$604$	−16592.9	−1.11780
$605$	0	0
$606$	10611.9	0.711351
$607$	17790.9	1.18964	0.594819	−	0.803860i	$-0.297224\pi$
$607$	17790.9	1.18964	0.594819	+	0.803860i	$0.297224\pi$
$608$	21347.4	1.42393
$609$	0	0
$610$	0	0
$611$	6406.94	0.424218
$612$	−17654.6	−1.16609
$613$	−14474.5	−0.953700	−0.476850	−	0.878985i	$-0.658222\pi$
$613$	−14474.5	−0.953700	−0.476850	+	0.878985i	$0.658222\pi$
$614$	−36745.2	−2.41517
$615$	0	0
$616$	0	0
$617$	13157.5	0.858509	0.429255	−	0.903184i	$-0.358776\pi$
$617$	13157.5	0.858509	0.429255	+	0.903184i	$0.358776\pi$
$618$	36486.9	2.37495
$619$	−17436.0	−1.13217	−0.566084	−	0.824348i	$-0.691542\pi$
$619$	−17436.0	−1.13217	−0.566084	+	0.824348i	$0.691542\pi$
$620$	0	0
$621$	−10707.7	−0.691925
$622$	775.654	0.0500015
$623$	0	0
$624$	11724.3	0.752158
$625$	0	0
$626$	−19989.1	−1.27624
$627$	−34228.4	−2.18015
$628$	−22039.8	−1.40045
$629$	36650.0	2.32326
$630$	0	0
$631$	−17388.4	−1.09702	−0.548512	−	0.836143i	$-0.684806\pi$
$631$	−17388.4	−1.09702	−0.548512	+	0.836143i	$0.684806\pi$
$632$	4824.39	0.303646
$633$	−12470.9	−0.783058
$634$	18386.4	1.15177
$635$	0	0
$636$	30440.5	1.89787
$637$	0	0
$638$	25366.7	1.57410
$639$	11431.6	0.707713
$640$	0	0
$641$	10838.2	0.667839	0.333919	−	0.942602i	$-0.391629\pi$
$641$	10838.2	0.667839	0.333919	+	0.942602i	$0.391629\pi$
$642$	−32691.3	−2.00969
$643$	21318.7	1.30751	0.653754	−	0.756707i	$-0.273193\pi$
$643$	21318.7	1.30751	0.653754	+	0.756707i	$0.273193\pi$
$644$	0	0
$645$	0	0
$646$	39995.3	2.43591
$647$	−2324.42	−0.141240	−0.0706200	−	0.997503i	$-0.522498\pi$
$647$	−2324.42	−0.141240	−0.0706200	+	0.997503i	$0.522498\pi$
$648$	10598.7	0.642525
$649$	30903.7	1.86915
$650$	0	0
$651$	0	0
$652$	−4244.64	−0.254958
$653$	−4990.89	−0.299094	−0.149547	−	0.988755i	$-0.547782\pi$
$653$	−4990.89	−0.299094	−0.149547	+	0.988755i	$0.547782\pi$
$654$	46027.9	2.75204
$655$	0	0
$656$	8348.33	0.496871
$657$	8977.88	0.533121
$658$	0	0
$659$	5720.85	0.338168	0.169084	−	0.985602i	$-0.445919\pi$
$659$	5720.85	0.338168	0.169084	+	0.985602i	$0.445919\pi$
$660$	0	0
$661$	−3357.64	−0.197575	−0.0987876	−	0.995109i	$-0.531496\pi$
$661$	−3357.64	−0.197575	−0.0987876	+	0.995109i	$0.531496\pi$
$662$	−47675.8	−2.79905
$663$	35497.2	2.07933
$664$	−11237.9	−0.656797
$665$	0	0
$666$	23169.3	1.34804
$667$	13927.0	0.808479
$668$	−15830.6	−0.916922
$669$	−9457.54	−0.546561
$670$	0	0
$671$	−36020.0	−2.07233
$672$	0	0
$673$	−16048.7	−0.919213	−0.459606	−	0.888123i	$-0.652010\pi$
$673$	−16048.7	−0.919213	−0.459606	+	0.888123i	$0.652010\pi$
$674$	11927.2	0.681628
$675$	0	0
$676$	4584.51	0.260839
$677$	−15438.9	−0.876463	−0.438232	−	0.898862i	$-0.644395\pi$
$677$	−15438.9	−0.876463	−0.438232	+	0.898862i	$0.644395\pi$
$678$	42113.5	2.38549
$679$	0	0
$680$	0	0
$681$	14235.7	0.801045
$682$	−29092.7	−1.63346
$683$	12079.0	0.676708	0.338354	−	0.941019i	$-0.390130\pi$
$683$	12079.0	0.676708	0.338354	+	0.941019i	$0.390130\pi$
$684$	14468.4	0.808791
$685$	0	0
$686$	0	0
$687$	−23475.1	−1.30368
$688$	−5713.74	−0.316619
$689$	−22348.6	−1.23573
$690$	0	0
$691$	−30561.1	−1.68249	−0.841243	−	0.540658i	$-0.818175\pi$
$691$	−30561.1	−1.68249	−0.841243	+	0.540658i	$0.818175\pi$
$692$	36121.0	1.98427
$693$	0	0
$694$	−6271.21	−0.343014
$695$	0	0
$696$	−7413.47	−0.403746
$697$	25276.0	1.37360
$698$	−32512.7	−1.76307
$699$	−11236.0	−0.607990
$700$	0	0
$701$	502.506	0.0270747	0.0135374	−	0.999908i	$-0.495691\pi$
$701$	502.506	0.0270747	0.0135374	+	0.999908i	$0.495691\pi$
$702$	−16575.7	−0.891183
$703$	−30035.6	−1.61140
$704$	−47006.5	−2.51651
$705$	0	0
$706$	1959.53	0.104459
$707$	0	0
$708$	−35775.1	−1.89903
$709$	−23820.1	−1.26175	−0.630876	−	0.775884i	$-0.717304\pi$
$709$	−23820.1	−1.26175	−0.630876	+	0.775884i	$0.717304\pi$
$710$	0	0
$711$	−6412.28	−0.338227
$712$	10462.1	0.550678
$713$	−15972.7	−0.838963
$714$	0	0
$715$	0	0
$716$	−43586.2	−2.27499
$717$	36482.9	1.90025
$718$	−11139.0	−0.578975
$719$	−28061.7	−1.45553	−0.727764	−	0.685828i	$-0.759440\pi$
$719$	−28061.7	−1.45553	−0.727764	+	0.685828i	$0.759440\pi$
$720$	0	0
$721$	0	0
$722$	−3115.10	−0.160571
$723$	−15487.2	−0.796648
$724$	−20433.0	−1.04887
$725$	0	0
$726$	64966.5	3.32112
$727$	13296.5	0.678324	0.339162	−	0.940728i	$-0.389857\pi$
$727$	13296.5	0.678324	0.339162	+	0.940728i	$0.389857\pi$
$728$	0	0
$729$	−915.390	−0.0465066
$730$	0	0
$731$	−17299.3	−0.875291
$732$	41697.8	2.10546
$733$	−18117.3	−0.912929	−0.456464	−	0.889742i	$-0.650884\pi$
$733$	−18117.3	−0.912929	−0.456464	+	0.889742i	$0.650884\pi$
$734$	15091.5	0.758909
$735$	0	0
$736$	−35098.7	−1.75782
$737$	−33249.6	−1.66182
$738$	15978.9	0.797007
$739$	5254.97	0.261579	0.130790	−	0.991410i	$-0.458249\pi$
$739$	5254.97	0.261579	0.130790	+	0.991410i	$0.458249\pi$
$740$	0	0
$741$	−29090.8	−1.44221
$742$	0	0
$743$	11949.0	0.589993	0.294997	−	0.955498i	$-0.404681\pi$
$743$	11949.0	0.589993	0.294997	+	0.955498i	$0.404681\pi$
$744$	8502.39	0.418969
$745$	0	0
$746$	−27141.9	−1.33209
$747$	14936.7	0.731598
$748$	−68538.5	−3.35029
$749$	0	0
$750$	0	0
$751$	2096.54	0.101869	0.0509346	−	0.998702i	$-0.483780\pi$
$751$	2096.54	0.101869	0.0509346	+	0.998702i	$0.483780\pi$
$752$	−4387.32	−0.212751
$753$	22307.1	1.07957
$754$	21559.3	1.04130
$755$	0	0
$756$	0	0
$757$	27081.7	1.30026	0.650132	−	0.759821i	$-0.274713\pi$
$757$	27081.7	1.30026	0.650132	+	0.759821i	$0.274713\pi$
$758$	26624.8	1.27580
$759$	56277.3	2.69135
$760$	0	0
$761$	−23196.2	−1.10494	−0.552472	−	0.833531i	$-0.686315\pi$
$761$	−23196.2	−1.10494	−0.552472	+	0.833531i	$0.686315\pi$
$762$	−1548.27	−0.0736061
$763$	0	0
$764$	−41651.5	−1.97238
$765$	0	0
$766$	−5151.26	−0.242980
$767$	26265.1	1.23648
$768$	−906.566	−0.0425949
$769$	−36309.1	−1.70265	−0.851327	−	0.524636i	$-0.824202\pi$
$769$	−36309.1	−1.70265	−0.851327	+	0.524636i	$0.824202\pi$
$770$	0	0
$771$	−31611.2	−1.47659
$772$	24156.7	1.12619
$773$	−28040.3	−1.30471	−0.652355	−	0.757914i	$-0.726219\pi$
$773$	−28040.3	−1.30471	−0.652355	+	0.757914i	$0.726219\pi$
$774$	−10936.2	−0.507874
$775$	0	0
$776$	−5543.09	−0.256425
$777$	0	0
$778$	24937.7	1.14917
$779$	−20714.3	−0.952717
$780$	0	0
$781$	44379.7	2.03333
$782$	−65759.1	−3.00708
$783$	−7278.31	−0.332191
$784$	0	0
$785$	0	0
$786$	−52163.0	−2.36716
$787$	−24756.0	−1.12129	−0.560647	−	0.828055i	$-0.689447\pi$
$787$	−24756.0	−1.12129	−0.560647	+	0.828055i	$0.689447\pi$
$788$	−43261.2	−1.95573
$789$	39433.9	1.77932
$790$	0	0
$791$	0	0
$792$	−10938.6	−0.490764
$793$	−30613.5	−1.37089
$794$	31566.5	1.41090
$795$	0	0
$796$	−40492.1	−1.80302
$797$	−13483.6	−0.599264	−0.299632	−	0.954055i	$-0.596864\pi$
$797$	−13483.6	−0.599264	−0.299632	+	0.954055i	$0.596864\pi$
$798$	0	0
$799$	−13283.3	−0.588148
$800$	0	0
$801$	−13905.5	−0.613393
$802$	52274.4	2.30159
$803$	34853.8	1.53171
$804$	38490.7	1.68839
$805$	0	0
$806$	−24726.0	−1.08056
$807$	−300.010	−0.0130866
$808$	−4396.33	−0.191414
$809$	11458.7	0.497980	0.248990	−	0.968506i	$-0.419901\pi$
$809$	11458.7	0.497980	0.248990	+	0.968506i	$0.419901\pi$
$810$	0	0
$811$	−15792.8	−0.683799	−0.341900	−	0.939737i	$-0.611070\pi$
$811$	−15792.8	−0.683799	−0.341900	+	0.939737i	$0.611070\pi$
$812$	0	0
$813$	−31347.6	−1.35229
$814$	89947.5	3.87304
$815$	0	0
$816$	−24307.6	−1.04281
$817$	14177.2	0.607096
$818$	−16558.2	−0.707755
$819$	0	0
$820$	0	0
$821$	20959.5	0.890977	0.445488	−	0.895288i	$-0.353030\pi$
$821$	20959.5	0.890977	0.445488	+	0.895288i	$0.353030\pi$
$822$	49589.5	2.10417
$823$	−20675.3	−0.875693	−0.437846	−	0.899050i	$-0.644259\pi$
$823$	−20675.3	−0.875693	−0.437846	+	0.899050i	$0.644259\pi$
$824$	−15115.9	−0.639062
$825$	0	0
$826$	0	0
$827$	−22034.0	−0.926480	−0.463240	−	0.886233i	$-0.653313\pi$
$827$	−22034.0	−0.926480	−0.463240	+	0.886233i	$0.653313\pi$
$828$	−23788.5	−0.998438
$829$	−38175.7	−1.59939	−0.799696	−	0.600405i	$-0.795006\pi$
$829$	−38175.7	−1.59939	−0.799696	+	0.600405i	$0.795006\pi$
$830$	0	0
$831$	16749.5	0.699200
$832$	−39951.0	−1.66472
$833$	0	0
$834$	8005.32	0.332376
$835$	0	0
$836$	56169.0	2.32374
$837$	8347.38	0.344716
$838$	69676.9	2.87225
$839$	−35043.1	−1.44198	−0.720991	−	0.692944i	$-0.756313\pi$
$839$	−35043.1	−1.44198	−0.720991	+	0.692944i	$0.756313\pi$
$840$	0	0
$841$	−14922.4	−0.611851
$842$	12982.3	0.531355
$843$	19932.8	0.814378
$844$	20464.9	0.834633
$845$	0	0
$846$	−8397.42	−0.341264
$847$	0	0
$848$	15303.8	0.619734
$849$	−24343.7	−0.984068
$850$	0	0
$851$	49383.5	1.98924
$852$	−51375.4	−2.06583
$853$	−39514.9	−1.58612	−0.793062	−	0.609140i	$-0.791515\pi$
$853$	−39514.9	−1.58612	−0.793062	+	0.609140i	$0.791515\pi$
$854$	0	0
$855$	0	0
$856$	13543.4	0.540777
$857$	43288.3	1.72544	0.862719	−	0.505683i	$-0.168759\pi$
$857$	43288.3	1.72544	0.862719	+	0.505683i	$0.168759\pi$
$858$	87118.3	3.46640
$859$	34716.8	1.37896	0.689478	−	0.724307i	$-0.257840\pi$
$859$	34716.8	1.37896	0.689478	+	0.724307i	$0.257840\pi$
$860$	0	0
$861$	0	0
$862$	2565.68	0.101378
$863$	−11893.4	−0.469124	−0.234562	−	0.972101i	$-0.575366\pi$
$863$	−11893.4	−0.469124	−0.234562	+	0.972101i	$0.575366\pi$
$864$	18342.7	0.722259
$865$	0	0
$866$	68961.8	2.70602
$867$	−41555.5	−1.62780
$868$	0	0
$869$	−24893.7	−0.971760
$870$	0	0
$871$	−28258.9	−1.09933
$872$	−19068.6	−0.740532
$873$	7367.53	0.285628
$874$	53891.2	2.08569
$875$	0	0
$876$	−40347.8	−1.55620
$877$	−21641.8	−0.833286	−0.416643	−	0.909070i	$-0.636793\pi$
$877$	−21641.8	−0.833286	−0.416643	+	0.909070i	$0.636793\pi$
$878$	−47602.7	−1.82974
$879$	59414.0	2.27985
$880$	0	0
$881$	20072.1	0.767591	0.383795	−	0.923418i	$-0.374617\pi$
$881$	20072.1	0.767591	0.383795	+	0.923418i	$0.374617\pi$
$882$	0	0
$883$	−13244.6	−0.504774	−0.252387	−	0.967626i	$-0.581216\pi$
$883$	−13244.6	−0.504774	−0.252387	+	0.967626i	$0.581216\pi$
$884$	−58251.1	−2.21628
$885$	0	0
$886$	−5227.45	−0.198216
$887$	−17615.9	−0.666837	−0.333419	−	0.942779i	$-0.608202\pi$
$887$	−17615.9	−0.666837	−0.333419	+	0.942779i	$0.608202\pi$
$888$	−26287.3	−0.993406
$889$	0	0
$890$	0	0
$891$	−54688.8	−2.05628
$892$	15519.9	0.582560
$893$	10886.0	0.407936
$894$	18436.4	0.689715
$895$	0	0
$896$	0	0
$897$	47830.2	1.78038
$898$	−19497.4	−0.724541
$899$	−10857.0	−0.402784
$900$	0	0
$901$	46334.8	1.71325
$902$	62033.1	2.28988
$903$	0	0
$904$	−17446.9	−0.641897
$905$	0	0
$906$	−43727.3	−1.60347
$907$	−7469.99	−0.273470	−0.136735	−	0.990608i	$-0.543661\pi$
$907$	−7469.99	−0.273470	−0.136735	+	0.990608i	$0.543661\pi$
$908$	−23360.8	−0.853805
$909$	5843.32	0.213213
$910$	0	0
$911$	−11906.6	−0.433023	−0.216511	−	0.976280i	$-0.569468\pi$
$911$	−11906.6	−0.433023	−0.216511	+	0.976280i	$0.569468\pi$
$912$	19920.7	0.723289
$913$	57986.8	2.10195
$914$	22750.1	0.823311
$915$	0	0
$916$	38522.7	1.38955
$917$	0	0
$918$	34366.0	1.23556
$919$	52532.1	1.88561	0.942804	−	0.333348i	$-0.108179\pi$
$919$	52532.1	1.88561	0.942804	+	0.333348i	$0.108179\pi$
$920$	0	0
$921$	−55412.0	−1.98251
$922$	−12337.7	−0.440694
$923$	37718.5	1.34509
$924$	0	0
$925$	0	0
$926$	−15477.7	−0.549275
$927$	20091.1	0.711843
$928$	−23857.5	−0.843924
$929$	35727.3	1.26176	0.630880	−	0.775880i	$-0.282694\pi$
$929$	35727.3	1.26176	0.630880	+	0.775880i	$0.282694\pi$
$930$	0	0
$931$	0	0
$932$	18438.3	0.648034
$933$	1169.69	0.0410439
$934$	−43025.4	−1.50732
$935$	0	0
$936$	−9296.73	−0.324651
$937$	26877.8	0.937096	0.468548	−	0.883438i	$-0.344777\pi$
$937$	26877.8	0.937096	0.468548	+	0.883438i	$0.344777\pi$
$938$	0	0
$939$	−30143.7	−1.04761
$940$	0	0
$941$	27316.9	0.946341	0.473170	−	0.880971i	$-0.343110\pi$
$941$	27316.9	0.946341	0.473170	+	0.880971i	$0.343110\pi$
$942$	−58081.6	−2.00892
$943$	34057.8	1.17611
$944$	−17985.7	−0.620112
$945$	0	0
$946$	−42456.4	−1.45917
$947$	23707.0	0.813489	0.406744	−	0.913542i	$-0.366664\pi$
$947$	23707.0	0.813489	0.406744	+	0.913542i	$0.366664\pi$
$948$	28817.7	0.987294
$949$	29622.3	1.01326
$950$	0	0
$951$	27726.9	0.945432
$952$	0	0
$953$	−7942.04	−0.269956	−0.134978	−	0.990849i	$-0.543096\pi$
$953$	−7942.04	−0.269956	−0.134978	+	0.990849i	$0.543096\pi$
$954$	29291.8	0.994086
$955$	0	0
$956$	−59868.6	−2.02541
$957$	38253.2	1.29211
$958$	−30296.4	−1.02175
$959$	0	0
$960$	0	0
$961$	−17339.2	−0.582029
$962$	76446.6	2.56210
$963$	−18001.1	−0.602364
$964$	25414.6	0.849118
$965$	0	0
$966$	0	0
$967$	−29259.7	−0.973039	−0.486519	−	0.873670i	$-0.661734\pi$
$967$	−29259.7	−0.973039	−0.486519	+	0.873670i	$0.661734\pi$
$968$	−26914.5	−0.893663
$969$	60313.2	1.99953
$970$	0	0
$971$	−21397.3	−0.707179	−0.353590	−	0.935401i	$-0.615039\pi$
$971$	−21397.3	−0.707179	−0.353590	+	0.935401i	$0.615039\pi$
$972$	41694.6	1.37588
$973$	0	0
$974$	20639.5	0.678984
$975$	0	0
$976$	20963.3	0.687521
$977$	4992.24	0.163476	0.0817380	−	0.996654i	$-0.473953\pi$
$977$	4992.24	0.163476	0.0817380	+	0.996654i	$0.473953\pi$
$978$	−11185.9	−0.365732
$979$	−53983.8	−1.76234
$980$	0	0
$981$	25344.7	0.824868
$982$	−20067.9	−0.652129
$983$	−22123.4	−0.717830	−0.358915	−	0.933370i	$-0.616853\pi$
$983$	−22123.4	−0.717830	−0.358915	+	0.933370i	$0.616853\pi$
$984$	−18129.3	−0.587338
$985$	0	0
$986$	−44698.2	−1.44369
$987$	0	0
$988$	47738.2	1.53720
$989$	−23309.7	−0.749449
$990$	0	0
$991$	−49240.9	−1.57839	−0.789197	−	0.614140i	$-0.789503\pi$
$991$	−49240.9	−1.57839	−0.789197	+	0.614140i	$0.789503\pi$
$992$	27361.8	0.875744
$993$	−71895.5	−2.29762
$994$	0	0
$995$	0	0
$996$	−67127.4	−2.13555
$997$	−13190.2	−0.418994	−0.209497	−	0.977809i	$-0.567183\pi$
$997$	−13190.2	−0.418994	−0.209497	+	0.977809i	$0.567183\pi$
$998$	−66552.6	−2.11091
$999$	−25808.1	−0.817348

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.br.1.3	✓	12
5.4	even	2		1225.4.a.bt.1.10	yes	12
7.6	odd	2	inner	1225.4.a.br.1.4	yes	12
35.34	odd	2		1225.4.a.bt.1.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1225.4.a.br.1.3	✓	12	1.1	even	1	trivial
1225.4.a.br.1.4	yes	12	7.6	odd	2	inner
1225.4.a.bt.1.9	yes	12	35.34	odd	2
1225.4.a.bt.1.10	yes	12	5.4	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q-4.32455 q^{2} -6.52145 q^{3} +10.7017 q^{4} +28.2023 q^{6} -11.6837 q^{8} +15.5293 q^{9} +60.2876 q^{11} -69.7907 q^{12} +51.2386 q^{13} -35.0869 q^{16} -106.231 q^{17} -67.1571 q^{18} +87.0593 q^{19} -260.716 q^{22} -143.140 q^{23} +76.1949 q^{24} -221.584 q^{26} +74.8057 q^{27} -97.2962 q^{29} +111.587 q^{31} +245.205 q^{32} -393.162 q^{33} +459.403 q^{34} +166.190 q^{36} -345.001 q^{37} -376.492 q^{38} -334.150 q^{39} -237.933 q^{41} +162.845 q^{43} +645.181 q^{44} +619.017 q^{46} +125.041 q^{47} +228.817 q^{48} +692.783 q^{51} +548.341 q^{52} -436.168 q^{53} -323.501 q^{54} -567.753 q^{57} +420.762 q^{58} +512.605 q^{59} -597.469 q^{61} -482.566 q^{62} -779.705 q^{64} +1700.25 q^{66} -551.516 q^{67} -1136.86 q^{68} +933.481 q^{69} +736.134 q^{71} -181.440 q^{72} +578.126 q^{73} +1491.97 q^{74} +931.685 q^{76} +1445.05 q^{78} -412.915 q^{79} -907.132 q^{81} +1028.95 q^{82} +961.838 q^{83} -704.232 q^{86} +634.512 q^{87} -704.384 q^{88} -895.439 q^{89} -1531.85 q^{92} -727.712 q^{93} -540.748 q^{94} -1599.09 q^{96} +474.428 q^{97} +936.222 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 10 q^{2} + 58 q^{4} - 180 q^{8} + 140 q^{9} + 28 q^{11} + 386 q^{16} - 678 q^{18} - 118 q^{22} - 492 q^{23} - 488 q^{29} - 1390 q^{32} + 1810 q^{36} - 304 q^{37} - 80 q^{39} - 644 q^{43} + 610 q^{44}+ \cdots - 1048 q^{99}+O(q^{100}) \) 12 * q - 10 * q^2 + 58 * q^4 - 180 * q^8 + 140 * q^9 + 28 * q^11 + 386 * q^16 - 678 * q^18 - 118 * q^22 - 492 * q^23 - 488 * q^29 - 1390 * q^32 + 1810 * q^36 - 304 * q^37 - 80 * q^39 - 644 * q^43 + 610 * q^44 + 754 * q^46 - 1812 * q^51 - 2360 * q^53 - 2068 * q^57 + 314 * q^58 + 2792 * q^64 - 4168 * q^67 - 988 * q^71 - 11336 * q^72 - 514 * q^74 + 2572 * q^78 + 3092 * q^79 - 2936 * q^81 - 3190 * q^86 + 1764 * q^88 - 4270 * q^92 - 640 * q^93 - 1048 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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