LMFDB - Embedded newform 6025.2.a.p.1.43

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, 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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$1$
Dimension:	$46$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.43
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44838 q^{2} -0.992209 q^{3} +3.99457 q^{4} -2.42931 q^{6} +0.744704 q^{7} +4.88345 q^{8} -2.01552 q^{9} +O(q^{10})$	\(q+2.44838 q^{2} -0.992209 q^{3} +3.99457 q^{4} -2.42931 q^{6} +0.744704 q^{7} +4.88345 q^{8} -2.01552 q^{9} -2.60457 q^{11} -3.96344 q^{12} -4.18428 q^{13} +1.82332 q^{14} +3.96742 q^{16} -0.484012 q^{17} -4.93476 q^{18} +3.30187 q^{19} -0.738902 q^{21} -6.37697 q^{22} +1.44005 q^{23} -4.84541 q^{24} -10.2447 q^{26} +4.97645 q^{27} +2.97477 q^{28} -0.129471 q^{29} +4.43651 q^{31} -0.0531507 q^{32} +2.58428 q^{33} -1.18504 q^{34} -8.05113 q^{36} -4.67172 q^{37} +8.08424 q^{38} +4.15168 q^{39} -9.21272 q^{41} -1.80911 q^{42} +7.65578 q^{43} -10.4041 q^{44} +3.52579 q^{46} +2.28201 q^{47} -3.93651 q^{48} -6.44542 q^{49} +0.480241 q^{51} -16.7144 q^{52} -12.5450 q^{53} +12.1842 q^{54} +3.63673 q^{56} -3.27615 q^{57} -0.316993 q^{58} -7.30045 q^{59} -5.28903 q^{61} +10.8623 q^{62} -1.50097 q^{63} -8.06498 q^{64} +6.32729 q^{66} -5.69391 q^{67} -1.93342 q^{68} -1.42883 q^{69} +2.08218 q^{71} -9.84271 q^{72} -15.5697 q^{73} -11.4381 q^{74} +13.1895 q^{76} -1.93963 q^{77} +10.1649 q^{78} -10.0023 q^{79} +1.10889 q^{81} -22.5562 q^{82} +2.71488 q^{83} -2.95159 q^{84} +18.7442 q^{86} +0.128462 q^{87} -12.7193 q^{88} -9.86499 q^{89} -3.11605 q^{91} +5.75237 q^{92} -4.40195 q^{93} +5.58723 q^{94} +0.0527366 q^{96} -6.26809 q^{97} -15.7808 q^{98} +5.24956 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

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 $

$2$	2.44838	1.73127	0.865633	−	0.500679i	$-0.166916\pi$
$2$	2.44838	1.73127	0.865633	+	0.500679i	$0.166916\pi$
$3$	−0.992209	−0.572852	−0.286426	−	0.958102i	$-0.592467\pi$
$3$	−0.992209	−0.572852	−0.286426	+	0.958102i	$0.592467\pi$
$4$	3.99457	1.99728
$5$	0	0
$5$	0	0
$6$	−2.42931	−0.991760
$7$	0.744704	0.281472	0.140736	−	0.990047i	$-0.455053\pi$
$7$	0.744704	0.281472	0.140736	+	0.990047i	$0.455053\pi$
$8$	4.88345	1.72656
$9$	−2.01552	−0.671840
$10$	0	0
$11$	−2.60457	−0.785307	−0.392654	−	0.919686i	$-0.628443\pi$
$11$	−2.60457	−0.785307	−0.392654	+	0.919686i	$0.628443\pi$
$12$	−3.96344	−1.14415
$13$	−4.18428	−1.16051	−0.580256	−	0.814434i	$-0.697047\pi$
$13$	−4.18428	−1.16051	−0.580256	+	0.814434i	$0.697047\pi$
$14$	1.82332	0.487302
$15$	0	0
$16$	3.96742	0.991856
$17$	−0.484012	−0.117390	−0.0586950	−	0.998276i	$-0.518694\pi$
$17$	−0.484012	−0.117390	−0.0586950	+	0.998276i	$0.518694\pi$
$18$	−4.93476	−1.16313
$19$	3.30187	0.757502	0.378751	−	0.925499i	$-0.376354\pi$
$19$	3.30187	0.757502	0.378751	+	0.925499i	$0.376354\pi$
$20$	0	0
$21$	−0.738902	−0.161242
$22$	−6.37697	−1.35958
$23$	1.44005	0.300271	0.150135	−	0.988665i	$-0.452029\pi$
$23$	1.44005	0.300271	0.150135	+	0.988665i	$0.452029\pi$
$24$	−4.84541	−0.989065
$25$	0	0
$26$	−10.2447	−2.00915
$27$	4.97645	0.957717
$28$	2.97477	0.562179
$29$	−0.129471	−0.0240421	−0.0120210	−	0.999928i	$-0.503827\pi$
$29$	−0.129471	−0.0240421	−0.0120210	+	0.999928i	$0.503827\pi$
$30$	0	0
$31$	4.43651	0.796821	0.398410	−	0.917207i	$-0.369562\pi$
$31$	4.43651	0.796821	0.398410	+	0.917207i	$0.369562\pi$
$32$	−0.0531507	−0.00939580
$33$	2.58428	0.449865
$34$	−1.18504	−0.203234
$35$	0	0
$36$	−8.05113	−1.34186
$37$	−4.67172	−0.768026	−0.384013	−	0.923328i	$-0.625458\pi$
$37$	−4.67172	−0.768026	−0.384013	+	0.923328i	$0.625458\pi$
$38$	8.08424	1.31144
$39$	4.15168	0.664802
$40$	0	0
$41$	−9.21272	−1.43878	−0.719392	−	0.694604i	$-0.755580\pi$
$41$	−9.21272	−1.43878	−0.719392	+	0.694604i	$0.755580\pi$
$42$	−1.80911	−0.279152
$43$	7.65578	1.16749	0.583747	−	0.811935i	$-0.301586\pi$
$43$	7.65578	1.16749	0.583747	+	0.811935i	$0.301586\pi$
$44$	−10.4041	−1.56848
$45$	0	0
$46$	3.52579	0.519849
$47$	2.28201	0.332865	0.166433	−	0.986053i	$-0.446775\pi$
$47$	2.28201	0.332865	0.166433	+	0.986053i	$0.446775\pi$
$48$	−3.93651	−0.568187
$49$	−6.44542	−0.920774
$50$	0	0
$51$	0.480241	0.0672472
$52$	−16.7144	−2.31787
$53$	−12.5450	−1.72319	−0.861596	−	0.507595i	$-0.830535\pi$
$53$	−12.5450	−1.72319	−0.861596	+	0.507595i	$0.830535\pi$
$54$	12.1842	1.65806
$55$	0	0
$56$	3.63673	0.485978
$57$	−3.27615	−0.433936
$58$	−0.316993	−0.0416232
$59$	−7.30045	−0.950438	−0.475219	−	0.879868i	$-0.657631\pi$
$59$	−7.30045	−0.950438	−0.475219	+	0.879868i	$0.657631\pi$
$60$	0	0
$61$	−5.28903	−0.677190	−0.338595	−	0.940932i	$-0.609952\pi$
$61$	−5.28903	−0.677190	−0.338595	+	0.940932i	$0.609952\pi$
$62$	10.8623	1.37951
$63$	−1.50097	−0.189104
$64$	−8.06498	−1.00812
$65$	0	0
$66$	6.32729	0.778836
$67$	−5.69391	−0.695621	−0.347811	−	0.937565i	$-0.613075\pi$
$67$	−5.69391	−0.695621	−0.347811	+	0.937565i	$0.613075\pi$
$68$	−1.93342	−0.234461
$69$	−1.42883	−0.172011
$70$	0	0
$71$	2.08218	0.247109	0.123555	−	0.992338i	$-0.460571\pi$
$71$	2.08218	0.247109	0.123555	+	0.992338i	$0.460571\pi$
$72$	−9.84271	−1.15997
$73$	−15.5697	−1.82230	−0.911149	−	0.412076i	$-0.864804\pi$
$73$	−15.5697	−1.82230	−0.911149	+	0.412076i	$0.864804\pi$
$74$	−11.4381	−1.32966
$75$	0	0
$76$	13.1895	1.51294
$77$	−1.93963	−0.221042
$78$	10.1649	1.15095
$79$	−10.0023	−1.12534	−0.562671	−	0.826681i	$-0.690226\pi$
$79$	−10.0023	−1.12534	−0.562671	+	0.826681i	$0.690226\pi$
$80$	0	0
$81$	1.10889	0.123210
$82$	−22.5562	−2.49092
$83$	2.71488	0.297997	0.148999	−	0.988837i	$-0.452395\pi$
$83$	2.71488	0.297997	0.148999	+	0.988837i	$0.452395\pi$
$84$	−2.95159	−0.322045
$85$	0	0
$86$	18.7442	2.02124
$87$	0.128462	0.0137726
$88$	−12.7193	−1.35588
$89$	−9.86499	−1.04569	−0.522843	−	0.852429i	$-0.675129\pi$
$89$	−9.86499	−1.04569	−0.522843	+	0.852429i	$0.675129\pi$
$90$	0	0
$91$	−3.11605	−0.326651
$92$	5.75237	0.599726
$93$	−4.40195	−0.456461
$94$	5.58723	0.576279
$95$	0	0
$96$	0.0527366	0.00538241
$97$	−6.26809	−0.636428	−0.318214	−	0.948019i	$-0.603083\pi$
$97$	−6.26809	−0.636428	−0.318214	+	0.948019i	$0.603083\pi$
$98$	−15.7808	−1.59410
$99$	5.24956	0.527601
$100$	0	0
$101$	4.57863	0.455591	0.227796	−	0.973709i	$-0.426848\pi$
$101$	4.57863	0.455591	0.227796	+	0.973709i	$0.426848\pi$
$102$	1.17581	0.116423
$103$	−5.95554	−0.586817	−0.293409	−	0.955987i	$-0.594790\pi$
$103$	−5.95554	−0.586817	−0.293409	+	0.955987i	$0.594790\pi$
$104$	−20.4338	−2.00369
$105$	0	0
$106$	−30.7150	−2.98330
$107$	−4.83097	−0.467027	−0.233514	−	0.972354i	$-0.575022\pi$
$107$	−4.83097	−0.467027	−0.233514	+	0.972354i	$0.575022\pi$
$108$	19.8787	1.91283
$109$	9.62741	0.922138	0.461069	−	0.887364i	$-0.347466\pi$
$109$	9.62741	0.922138	0.461069	+	0.887364i	$0.347466\pi$
$110$	0	0
$111$	4.63532	0.439965
$112$	2.95456	0.279179
$113$	−4.74825	−0.446677	−0.223339	−	0.974741i	$-0.571696\pi$
$113$	−4.74825	−0.446677	−0.223339	+	0.974741i	$0.571696\pi$
$114$	−8.02126	−0.751260
$115$	0	0
$116$	−0.517179	−0.0480188
$117$	8.43351	0.779678
$118$	−17.8743	−1.64546
$119$	−0.360446	−0.0330420
$120$	0	0
$121$	−4.21622	−0.383293
$122$	−12.9495	−1.17240
$123$	9.14094	0.824211
$124$	17.7219	1.59148
$125$	0	0
$126$	−3.67494	−0.327389
$127$	2.13691	0.189620	0.0948099	−	0.995495i	$-0.469776\pi$
$127$	2.13691	0.189620	0.0948099	+	0.995495i	$0.469776\pi$
$128$	−19.6398	−1.73593
$129$	−7.59613	−0.668802
$130$	0	0
$131$	−15.2770	−1.33476	−0.667378	−	0.744719i	$-0.732584\pi$
$131$	−15.2770	−1.33476	−0.667378	+	0.744719i	$0.732584\pi$
$132$	10.3231	0.898507
$133$	2.45892	0.213215
$134$	−13.9408	−1.20431
$135$	0	0
$136$	−2.36365	−0.202681
$137$	15.6904	1.34052	0.670259	−	0.742128i	$-0.266183\pi$
$137$	15.6904	1.34052	0.670259	+	0.742128i	$0.266183\pi$
$138$	−3.49832	−0.297797
$139$	13.8387	1.17378	0.586891	−	0.809666i	$-0.300352\pi$
$139$	13.8387	1.17378	0.586891	+	0.809666i	$0.300352\pi$
$140$	0	0
$141$	−2.26423	−0.190683
$142$	5.09797	0.427812
$143$	10.8983	0.911358
$144$	−7.99642	−0.666369
$145$	0	0
$146$	−38.1206	−3.15488
$147$	6.39520	0.527467
$148$	−18.6615	−1.53396
$149$	−4.70517	−0.385463	−0.192731	−	0.981252i	$-0.561735\pi$
$149$	−4.70517	−0.385463	−0.192731	+	0.981252i	$0.561735\pi$
$150$	0	0
$151$	15.7760	1.28383	0.641914	−	0.766776i	$-0.278140\pi$
$151$	15.7760	1.28383	0.641914	+	0.766776i	$0.278140\pi$
$152$	16.1245	1.30787
$153$	0.975536	0.0788674
$154$	−4.74896	−0.382682
$155$	0	0
$156$	16.5842	1.32780
$157$	6.26473	0.499980	0.249990	−	0.968248i	$-0.419573\pi$
$157$	6.26473	0.499980	0.249990	+	0.968248i	$0.419573\pi$
$158$	−24.4893	−1.94827
$159$	12.4473	0.987134
$160$	0	0
$161$	1.07241	0.0845178
$162$	2.71498	0.213309
$163$	22.7092	1.77872	0.889362	−	0.457204i	$-0.151149\pi$
$163$	22.7092	1.77872	0.889362	+	0.457204i	$0.151149\pi$
$164$	−36.8008	−2.87366
$165$	0	0
$166$	6.64707	0.515913
$167$	18.4720	1.42941	0.714703	−	0.699428i	$-0.246562\pi$
$167$	18.4720	1.42941	0.714703	+	0.699428i	$0.246562\pi$
$168$	−3.60840	−0.278394
$169$	4.50823	0.346787
$170$	0	0
$171$	−6.65500	−0.508920
$172$	30.5815	2.33182
$173$	−11.4775	−0.872615	−0.436307	−	0.899798i	$-0.643714\pi$
$173$	−11.4775	−0.872615	−0.436307	+	0.899798i	$0.643714\pi$
$174$	0.314523	0.0238440
$175$	0	0
$176$	−10.3334	−0.778911
$177$	7.24358	0.544460
$178$	−24.1532	−1.81036
$179$	−12.4461	−0.930266	−0.465133	−	0.885241i	$-0.653994\pi$
$179$	−12.4461	−0.930266	−0.465133	+	0.885241i	$0.653994\pi$
$180$	0	0
$181$	−11.1005	−0.825090	−0.412545	−	0.910937i	$-0.635360\pi$
$181$	−11.1005	−0.825090	−0.412545	+	0.910937i	$0.635360\pi$
$182$	−7.62928	−0.565520
$183$	5.24782	0.387930
$184$	7.03241	0.518436
$185$	0	0
$186$	−10.7776	−0.790255
$187$	1.26064	0.0921873
$188$	9.11564	0.664826
$189$	3.70598	0.269570
$190$	0	0
$191$	5.17318	0.374318	0.187159	−	0.982330i	$-0.440072\pi$
$191$	5.17318	0.374318	0.187159	+	0.982330i	$0.440072\pi$
$192$	8.00214	0.577505
$193$	14.2338	1.02457	0.512285	−	0.858816i	$-0.328799\pi$
$193$	14.2338	1.02457	0.512285	+	0.858816i	$0.328799\pi$
$194$	−15.3467	−1.10183
$195$	0	0
$196$	−25.7466	−1.83905
$197$	6.64124	0.473169	0.236585	−	0.971611i	$-0.423972\pi$
$197$	6.64124	0.473169	0.236585	+	0.971611i	$0.423972\pi$
$198$	12.8529	0.913418
$199$	9.13468	0.647541	0.323770	−	0.946136i	$-0.395050\pi$
$199$	9.13468	0.647541	0.323770	+	0.946136i	$0.395050\pi$
$200$	0	0
$201$	5.64954	0.398488
$202$	11.2102	0.788750
$203$	−0.0964172	−0.00676716
$204$	1.91835	0.134312
$205$	0	0
$206$	−14.5814	−1.01594
$207$	−2.90245	−0.201734
$208$	−16.6008	−1.15106
$209$	−8.59996	−0.594871
$210$	0	0
$211$	−19.5990	−1.34925	−0.674626	−	0.738160i	$-0.735695\pi$
$211$	−19.5990	−1.34925	−0.674626	+	0.738160i	$0.735695\pi$
$212$	−50.1119	−3.44170
$213$	−2.06596	−0.141557
$214$	−11.8280	−0.808548
$215$	0	0
$216$	24.3022	1.65356
$217$	3.30389	0.224283
$218$	23.5716	1.59647
$219$	15.4484	1.04391
$220$	0	0
$221$	2.02524	0.136233
$222$	11.3490	0.761697
$223$	4.78461	0.320401	0.160200	−	0.987085i	$-0.448786\pi$
$223$	4.78461	0.320401	0.160200	+	0.987085i	$0.448786\pi$
$224$	−0.0395815	−0.00264465
$225$	0	0
$226$	−11.6255	−0.773318
$227$	21.1630	1.40464	0.702319	−	0.711862i	$-0.252148\pi$
$227$	21.1630	1.40464	0.702319	+	0.711862i	$0.252148\pi$
$228$	−13.0868	−0.866694
$229$	22.4594	1.48416	0.742080	−	0.670311i	$-0.233839\pi$
$229$	22.4594	1.48416	0.742080	+	0.670311i	$0.233839\pi$
$230$	0	0
$231$	1.92452	0.126624
$232$	−0.632263	−0.0415101
$233$	11.0004	0.720658	0.360329	−	0.932825i	$-0.382664\pi$
$233$	11.0004	0.720658	0.360329	+	0.932825i	$0.382664\pi$
$234$	20.6484	1.34983
$235$	0	0
$236$	−29.1621	−1.89829
$237$	9.92434	0.644655
$238$	−0.882508	−0.0572045
$239$	−2.21903	−0.143537	−0.0717686	−	0.997421i	$-0.522864\pi$
$239$	−2.21903	−0.143537	−0.0717686	+	0.997421i	$0.522864\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−10.3229	−0.663582
$243$	−16.0296	−1.02830
$244$	−21.1274	−1.35254
$245$	0	0
$246$	22.3805	1.42693
$247$	−13.8160	−0.879089
$248$	21.6655	1.37576
$249$	−2.69373	−0.170708
$250$	0	0
$251$	11.1504	0.703807	0.351903	−	0.936036i	$-0.385535\pi$
$251$	11.1504	0.703807	0.351903	+	0.936036i	$0.385535\pi$
$252$	−5.99571	−0.377694
$253$	−3.75071	−0.235805
$254$	5.23196	0.328282
$255$	0	0
$256$	−31.9558	−1.99724
$257$	8.50879	0.530763	0.265382	−	0.964143i	$-0.414502\pi$
$257$	8.50879	0.530763	0.265382	+	0.964143i	$0.414502\pi$
$258$	−18.5982	−1.15787
$259$	−3.47905	−0.216177
$260$	0	0
$261$	0.260951	0.0161524
$262$	−37.4039	−2.31082
$263$	−10.5798	−0.652377	−0.326188	−	0.945305i	$-0.605764\pi$
$263$	−10.5798	−0.652377	−0.326188	+	0.945305i	$0.605764\pi$
$264$	12.6202	0.776720
$265$	0	0
$266$	6.02037	0.369132
$267$	9.78813	0.599024
$268$	−22.7447	−1.38935
$269$	6.26143	0.381766	0.190883	−	0.981613i	$-0.438865\pi$
$269$	6.26143	0.381766	0.190883	+	0.981613i	$0.438865\pi$
$270$	0	0
$271$	6.22770	0.378306	0.189153	−	0.981948i	$-0.439426\pi$
$271$	6.22770	0.378306	0.189153	+	0.981948i	$0.439426\pi$
$272$	−1.92028	−0.116434
$273$	3.09178	0.187123
$274$	38.4160	2.32079
$275$	0	0
$276$	−5.70755	−0.343554
$277$	0.195324	0.0117359	0.00586793	−	0.999983i	$-0.498132\pi$
$277$	0.195324	0.0117359	0.00586793	+	0.999983i	$0.498132\pi$
$278$	33.8824	2.03213
$279$	−8.94188	−0.535336
$280$	0	0
$281$	−21.0354	−1.25486	−0.627432	−	0.778671i	$-0.715894\pi$
$281$	−21.0354	−1.25486	−0.627432	+	0.778671i	$0.715894\pi$
$282$	−5.54370	−0.330123
$283$	12.8269	0.762478	0.381239	−	0.924477i	$-0.375497\pi$
$283$	12.8269	0.762478	0.381239	+	0.924477i	$0.375497\pi$
$284$	8.31740	0.493547
$285$	0	0
$286$	26.6831	1.57780
$287$	−6.86075	−0.404977
$288$	0.107126	0.00631248
$289$	−16.7657	−0.986220
$290$	0	0
$291$	6.21926	0.364579
$292$	−62.1943	−3.63965
$293$	−25.7912	−1.50674	−0.753369	−	0.657598i	$-0.771573\pi$
$293$	−25.7912	−1.50674	−0.753369	+	0.657598i	$0.771573\pi$
$294$	15.6579	0.913186
$295$	0	0
$296$	−22.8141	−1.32604
$297$	−12.9615	−0.752102
$298$	−11.5200	−0.667338
$299$	−6.02557	−0.348468
$300$	0	0
$301$	5.70129	0.328617
$302$	38.6255	2.22265
$303$	−4.54296	−0.260986
$304$	13.0999	0.751332
$305$	0	0
$306$	2.38848	0.136540
$307$	28.5348	1.62856	0.814282	−	0.580469i	$-0.197131\pi$
$307$	28.5348	1.62856	0.814282	+	0.580469i	$0.197131\pi$
$308$	−7.74799	−0.441483
$309$	5.90914	0.336160
$310$	0	0
$311$	4.04451	0.229343	0.114672	−	0.993403i	$-0.463418\pi$
$311$	4.04451	0.229343	0.114672	+	0.993403i	$0.463418\pi$
$312$	20.2746	1.14782
$313$	−10.8140	−0.611243	−0.305622	−	0.952153i	$-0.598864\pi$
$313$	−10.8140	−0.611243	−0.305622	+	0.952153i	$0.598864\pi$
$314$	15.3384	0.865598
$315$	0	0
$316$	−39.9547	−2.24763
$317$	2.62335	0.147342	0.0736711	−	0.997283i	$-0.476528\pi$
$317$	2.62335	0.147342	0.0736711	+	0.997283i	$0.476528\pi$
$318$	30.4757	1.70899
$319$	0.337215	0.0188804
$320$	0	0
$321$	4.79333	0.267538
$322$	2.62567	0.146323
$323$	−1.59815	−0.0889232
$324$	4.42953	0.246085
$325$	0	0
$326$	55.6008	3.07944
$327$	−9.55240	−0.528249
$328$	−44.9899	−2.48415
$329$	1.69942	0.0936922
$330$	0	0
$331$	−10.6123	−0.583307	−0.291653	−	0.956524i	$-0.594205\pi$
$331$	−10.6123	−0.583307	−0.291653	+	0.956524i	$0.594205\pi$
$332$	10.8448	0.595185
$333$	9.41594	0.515991
$334$	45.2264	2.47468
$335$	0	0
$336$	−2.93154	−0.159928
$337$	3.53987	0.192829	0.0964146	−	0.995341i	$-0.469263\pi$
$337$	3.53987	0.192829	0.0964146	+	0.995341i	$0.469263\pi$
$338$	11.0379	0.600381
$339$	4.71125	0.255880
$340$	0	0
$341$	−11.5552	−0.625749
$342$	−16.2940	−0.881076
$343$	−10.0129	−0.540643
$344$	37.3866	2.01575
$345$	0	0
$346$	−28.1012	−1.51073
$347$	−13.6131	−0.730791	−0.365396	−	0.930852i	$-0.619066\pi$
$347$	−13.6131	−0.730791	−0.365396	+	0.930852i	$0.619066\pi$
$348$	0.513149	0.0275077
$349$	3.51082	0.187930	0.0939649	−	0.995576i	$-0.470046\pi$
$349$	3.51082	0.187930	0.0939649	+	0.995576i	$0.470046\pi$
$350$	0	0
$351$	−20.8229	−1.11144
$352$	0.138435	0.00737859
$353$	10.7398	0.571621	0.285810	−	0.958286i	$-0.407737\pi$
$353$	10.7398	0.571621	0.285810	+	0.958286i	$0.407737\pi$
$354$	17.7350	0.942606
$355$	0	0
$356$	−39.4063	−2.08853
$357$	0.357637	0.0189282
$358$	−30.4728	−1.61054
$359$	12.0903	0.638099	0.319050	−	0.947738i	$-0.396636\pi$
$359$	12.0903	0.638099	0.319050	+	0.947738i	$0.396636\pi$
$360$	0	0
$361$	−8.09763	−0.426191
$362$	−27.1781	−1.42845
$363$	4.18337	0.219570
$364$	−12.4473	−0.652415
$365$	0	0
$366$	12.8487	0.671610
$367$	2.89274	0.151000	0.0754999	−	0.997146i	$-0.475945\pi$
$367$	2.89274	0.151000	0.0754999	+	0.997146i	$0.475945\pi$
$368$	5.71328	0.297825
$369$	18.5684	0.966634
$370$	0	0
$371$	−9.34233	−0.485030
$372$	−17.5839	−0.911681
$373$	37.8889	1.96181	0.980907	−	0.194479i	$-0.0623015\pi$
$373$	37.8889	1.96181	0.980907	+	0.194479i	$0.0623015\pi$
$374$	3.08653	0.159601
$375$	0	0
$376$	11.1441	0.574713
$377$	0.541741	0.0279011
$378$	9.07365	0.466698
$379$	−9.33140	−0.479322	−0.239661	−	0.970857i	$-0.577036\pi$
$379$	−9.33140	−0.479322	−0.239661	+	0.970857i	$0.577036\pi$
$380$	0	0
$381$	−2.12026	−0.108624
$382$	12.6659	0.648044
$383$	34.9438	1.78554	0.892771	−	0.450510i	$-0.148758\pi$
$383$	34.9438	1.78554	0.892771	+	0.450510i	$0.148758\pi$
$384$	19.4868	0.994433
$385$	0	0
$386$	34.8497	1.77380
$387$	−15.4304	−0.784370
$388$	−25.0383	−1.27113
$389$	−22.2652	−1.12889	−0.564445	−	0.825470i	$-0.690910\pi$
$389$	−22.2652	−1.12889	−0.564445	+	0.825470i	$0.690910\pi$
$390$	0	0
$391$	−0.697000	−0.0352488
$392$	−31.4759	−1.58977
$393$	15.1580	0.764618
$394$	16.2603	0.819182
$395$	0	0
$396$	20.9697	1.05377
$397$	18.6917	0.938110	0.469055	−	0.883169i	$-0.344595\pi$
$397$	18.6917	0.938110	0.469055	+	0.883169i	$0.344595\pi$
$398$	22.3652	1.12107
$399$	−2.43976	−0.122141
$400$	0	0
$401$	30.3420	1.51521	0.757604	−	0.652715i	$-0.226370\pi$
$401$	30.3420	1.51521	0.757604	+	0.652715i	$0.226370\pi$
$402$	13.8322	0.689889
$403$	−18.5636	−0.924720
$404$	18.2897	0.909944
$405$	0	0
$406$	−0.236066	−0.0117158
$407$	12.1678	0.603136
$408$	2.34523	0.116106
$409$	16.1799	0.800046	0.400023	−	0.916505i	$-0.369002\pi$
$409$	16.1799	0.800046	0.400023	+	0.916505i	$0.369002\pi$
$410$	0	0
$411$	−15.5681	−0.767918
$412$	−23.7898	−1.17204
$413$	−5.43668	−0.267521
$414$	−7.10630	−0.349255
$415$	0	0
$416$	0.222398	0.0109039
$417$	−13.7309	−0.672404
$418$	−21.0560	−1.02988
$419$	14.4615	0.706488	0.353244	−	0.935531i	$-0.385079\pi$
$419$	14.4615	0.706488	0.353244	+	0.935531i	$0.385079\pi$
$420$	0	0
$421$	14.6314	0.713093	0.356546	−	0.934278i	$-0.383954\pi$
$421$	14.6314	0.713093	0.356546	+	0.934278i	$0.383954\pi$
$422$	−47.9858	−2.33591
$423$	−4.59944	−0.223632
$424$	−61.2631	−2.97520
$425$	0	0
$426$	−5.05825	−0.245073
$427$	−3.93876	−0.190610
$428$	−19.2976	−0.932785
$429$	−10.8133	−0.522073
$430$	0	0
$431$	17.3814	0.837233	0.418616	−	0.908163i	$-0.362515\pi$
$431$	17.3814	0.837233	0.418616	+	0.908163i	$0.362515\pi$
$432$	19.7437	0.949917
$433$	−22.7286	−1.09227	−0.546133	−	0.837698i	$-0.683901\pi$
$433$	−22.7286	−1.09227	−0.546133	+	0.837698i	$0.683901\pi$
$434$	8.08917	0.388293
$435$	0	0
$436$	38.4573	1.84177
$437$	4.75486	0.227456
$438$	37.8236	1.80728
$439$	−36.3500	−1.73489	−0.867446	−	0.497531i	$-0.834240\pi$
$439$	−36.3500	−1.73489	−0.867446	+	0.497531i	$0.834240\pi$
$440$	0	0
$441$	12.9909	0.618613
$442$	4.95856	0.235855
$443$	−41.2429	−1.95951	−0.979755	−	0.200201i	$-0.935841\pi$
$443$	−41.2429	−1.95951	−0.979755	+	0.200201i	$0.935841\pi$
$444$	18.5161	0.878735
$445$	0	0
$446$	11.7145	0.554699
$447$	4.66851	0.220813
$448$	−6.00602	−0.283758
$449$	1.81576	0.0856909	0.0428454	−	0.999082i	$-0.486358\pi$
$449$	1.81576	0.0856909	0.0428454	+	0.999082i	$0.486358\pi$
$450$	0	0
$451$	23.9952	1.12989
$452$	−18.9672	−0.892141
$453$	−15.6530	−0.735444
$454$	51.8151	2.43180
$455$	0	0
$456$	−15.9989	−0.749218
$457$	−9.57207	−0.447763	−0.223881	−	0.974616i	$-0.571873\pi$
$457$	−9.57207	−0.447763	−0.223881	+	0.974616i	$0.571873\pi$
$458$	54.9892	2.56948
$459$	−2.40866	−0.112427
$460$	0	0
$461$	−7.96834	−0.371123	−0.185561	−	0.982633i	$-0.559410\pi$
$461$	−7.96834	−0.371123	−0.185561	+	0.982633i	$0.559410\pi$
$462$	4.71196	0.219220
$463$	−20.5525	−0.955157	−0.477579	−	0.878589i	$-0.658485\pi$
$463$	−20.5525	−0.955157	−0.477579	+	0.878589i	$0.658485\pi$
$464$	−0.513664	−0.0238463
$465$	0	0
$466$	26.9331	1.24765
$467$	15.8957	0.735567	0.367783	−	0.929912i	$-0.380117\pi$
$467$	15.8957	0.735567	0.367783	+	0.929912i	$0.380117\pi$
$468$	33.6882	1.55724
$469$	−4.24027	−0.195798
$470$	0	0
$471$	−6.21592	−0.286415
$472$	−35.6514	−1.64099
$473$	−19.9400	−0.916842
$474$	24.2985	1.11607
$475$	0	0
$476$	−1.43982	−0.0659942
$477$	25.2848	1.15771
$478$	−5.43303	−0.248501
$479$	−41.6793	−1.90437	−0.952187	−	0.305516i	$-0.901171\pi$
$479$	−41.6793	−1.90437	−0.952187	+	0.305516i	$0.901171\pi$
$480$	0	0
$481$	19.5478	0.891303
$482$	2.44838	0.111521
$483$	−1.06406	−0.0484162
$484$	−16.8420	−0.765544
$485$	0	0
$486$	−39.2465	−1.78026
$487$	7.56632	0.342863	0.171431	−	0.985196i	$-0.445161\pi$
$487$	7.56632	0.342863	0.171431	+	0.985196i	$0.445161\pi$
$488$	−25.8287	−1.16921
$489$	−22.5323	−1.01895
$490$	0	0
$491$	−26.7723	−1.20822	−0.604108	−	0.796903i	$-0.706470\pi$
$491$	−26.7723	−1.20822	−0.604108	+	0.796903i	$0.706470\pi$
$492$	36.5141	1.64618
$493$	0.0626653	0.00282230
$494$	−33.8268	−1.52194
$495$	0	0
$496$	17.6015	0.790331
$497$	1.55061	0.0695543
$498$	−6.59528	−0.295542
$499$	−27.0405	−1.21050	−0.605249	−	0.796036i	$-0.706926\pi$
$499$	−27.0405	−1.21050	−0.605249	+	0.796036i	$0.706926\pi$
$500$	0	0
$501$	−18.3281	−0.818838
$502$	27.3004	1.21848
$503$	−0.950190	−0.0423669	−0.0211834	−	0.999776i	$-0.506743\pi$
$503$	−0.950190	−0.0423669	−0.0211834	+	0.999776i	$0.506743\pi$
$504$	−7.32990	−0.326500
$505$	0	0
$506$	−9.18315	−0.408241
$507$	−4.47311	−0.198658
$508$	8.53601	0.378724
$509$	−32.8758	−1.45719	−0.728596	−	0.684944i	$-0.759827\pi$
$509$	−32.8758	−1.45719	−0.728596	+	0.684944i	$0.759827\pi$
$510$	0	0
$511$	−11.5948	−0.512926
$512$	−38.9603	−1.72182
$513$	16.4316	0.725473
$514$	20.8327	0.918893
$515$	0	0
$516$	−30.3432	−1.33579
$517$	−5.94365	−0.261402
$518$	−8.51803	−0.374261
$519$	11.3880	0.499879
$520$	0	0
$521$	−7.96960	−0.349155	−0.174577	−	0.984643i	$-0.555856\pi$
$521$	−7.96960	−0.349155	−0.174577	+	0.984643i	$0.555856\pi$
$522$	0.638906	0.0279642
$523$	16.8372	0.736240	0.368120	−	0.929778i	$-0.380001\pi$
$523$	16.8372	0.736240	0.368120	+	0.929778i	$0.380001\pi$
$524$	−61.0249	−2.66589
$525$	0	0
$526$	−25.9033	−1.12944
$527$	−2.14732	−0.0935389
$528$	10.2529	0.446201
$529$	−20.9263	−0.909837
$530$	0	0
$531$	14.7142	0.638543
$532$	9.82231	0.425851
$533$	38.5486	1.66973
$534$	23.9651	1.03707
$535$	0	0
$536$	−27.8059	−1.20103
$537$	12.3491	0.532905
$538$	15.3304	0.660939
$539$	16.7875	0.723090
$540$	0	0
$541$	22.9397	0.986254	0.493127	−	0.869957i	$-0.335854\pi$
$541$	22.9397	0.986254	0.493127	+	0.869957i	$0.335854\pi$
$542$	15.2478	0.654948
$543$	11.0140	0.472655
$544$	0.0257256	0.00110297
$545$	0	0
$546$	7.56984	0.323959
$547$	−8.56387	−0.366164	−0.183082	−	0.983098i	$-0.558607\pi$
$547$	−8.56387	−0.366164	−0.183082	+	0.983098i	$0.558607\pi$
$548$	62.6761	2.67739
$549$	10.6601	0.454964
$550$	0	0
$551$	−0.427495	−0.0182119
$552$	−6.97762	−0.296987
$553$	−7.44873	−0.316752
$554$	0.478227	0.0203179
$555$	0	0
$556$	55.2795	2.34437
$557$	6.79073	0.287732	0.143866	−	0.989597i	$-0.454046\pi$
$557$	6.79073	0.287732	0.143866	+	0.989597i	$0.454046\pi$
$558$	−21.8931	−0.926810
$559$	−32.0339	−1.35489
$560$	0	0
$561$	−1.25082	−0.0528097
$562$	−51.5025	−2.17250
$563$	20.2542	0.853612	0.426806	−	0.904343i	$-0.359639\pi$
$563$	20.2542	0.853612	0.426806	+	0.904343i	$0.359639\pi$
$564$	−9.04462	−0.380847
$565$	0	0
$566$	31.4050	1.32005
$567$	0.825794	0.0346801
$568$	10.1682	0.426649
$569$	−31.2543	−1.31025	−0.655125	−	0.755521i	$-0.727384\pi$
$569$	−31.2543	−1.31025	−0.655125	+	0.755521i	$0.727384\pi$
$570$	0	0
$571$	−31.1637	−1.30416	−0.652081	−	0.758149i	$-0.726104\pi$
$571$	−31.1637	−1.30416	−0.652081	+	0.758149i	$0.726104\pi$
$572$	43.5338	1.82024
$573$	−5.13287	−0.214429
$574$	−16.7977	−0.701123
$575$	0	0
$576$	16.2551	0.677297
$577$	30.7522	1.28023	0.640114	−	0.768280i	$-0.278887\pi$
$577$	30.7522	1.28023	0.640114	+	0.768280i	$0.278887\pi$
$578$	−41.0489	−1.70741
$579$	−14.1229	−0.586927
$580$	0	0
$581$	2.02179	0.0838778
$582$	15.2271	0.631184
$583$	32.6744	1.35323
$584$	−76.0341	−3.14631
$585$	0	0
$586$	−63.1467	−2.60857
$587$	17.7844	0.734042	0.367021	−	0.930213i	$-0.380378\pi$
$587$	17.7844	0.734042	0.367021	+	0.930213i	$0.380378\pi$
$588$	25.5460	1.05350
$589$	14.6488	0.603593
$590$	0	0
$591$	−6.58950	−0.271056
$592$	−18.5347	−0.761770
$593$	−1.96286	−0.0806051	−0.0403026	−	0.999188i	$-0.512832\pi$
$593$	−1.96286	−0.0806051	−0.0403026	+	0.999188i	$0.512832\pi$
$594$	−31.7347	−1.30209
$595$	0	0
$596$	−18.7951	−0.769878
$597$	−9.06352	−0.370945
$598$	−14.7529	−0.603291
$599$	−26.8177	−1.09574	−0.547871	−	0.836563i	$-0.684562\pi$
$599$	−26.8177	−1.09574	−0.547871	+	0.836563i	$0.684562\pi$
$600$	0	0
$601$	31.3885	1.28036	0.640182	−	0.768223i	$-0.278859\pi$
$601$	31.3885	1.28036	0.640182	+	0.768223i	$0.278859\pi$
$602$	13.9589	0.568923
$603$	11.4762	0.467346
$604$	63.0181	2.56417
$605$	0	0
$606$	−11.1229	−0.451837
$607$	−31.9764	−1.29788	−0.648941	−	0.760839i	$-0.724788\pi$
$607$	−31.9764	−1.29788	−0.648941	+	0.760839i	$0.724788\pi$
$608$	−0.175497	−0.00711733
$609$	0.0956661	0.00387658
$610$	0	0
$611$	−9.54858	−0.386294
$612$	3.89684	0.157521
$613$	−19.1393	−0.773028	−0.386514	−	0.922283i	$-0.626321\pi$
$613$	−19.1393	−0.773028	−0.386514	+	0.922283i	$0.626321\pi$
$614$	69.8639	2.81948
$615$	0	0
$616$	−9.47211	−0.381642
$617$	40.5121	1.63096	0.815478	−	0.578788i	$-0.196474\pi$
$617$	40.5121	1.63096	0.815478	+	0.578788i	$0.196474\pi$
$618$	14.4678	0.581982
$619$	−8.35556	−0.335838	−0.167919	−	0.985801i	$-0.553705\pi$
$619$	−8.35556	−0.335838	−0.167919	+	0.985801i	$0.553705\pi$
$620$	0	0
$621$	7.16632	0.287575
$622$	9.90251	0.397054
$623$	−7.34650	−0.294331
$624$	16.4715	0.659387
$625$	0	0
$626$	−26.4768	−1.05822
$627$	8.53295	0.340773
$628$	25.0249	0.998601
$629$	2.26117	0.0901586
$630$	0	0
$631$	18.3668	0.731169	0.365585	−	0.930778i	$-0.380869\pi$
$631$	18.3668	0.731169	0.365585	+	0.930778i	$0.380869\pi$
$632$	−48.8456	−1.94297
$633$	19.4463	0.772922
$634$	6.42297	0.255089
$635$	0	0
$636$	49.7215	1.97159
$637$	26.9694	1.06857
$638$	0.825630	0.0326870
$639$	−4.19668	−0.166018
$640$	0	0
$641$	−21.6917	−0.856772	−0.428386	−	0.903596i	$-0.640918\pi$
$641$	−21.6917	−0.856772	−0.428386	+	0.903596i	$0.640918\pi$
$642$	11.7359	0.463179
$643$	23.2362	0.916347	0.458174	−	0.888863i	$-0.348504\pi$
$643$	23.2362	0.916347	0.458174	+	0.888863i	$0.348504\pi$
$644$	4.28381	0.168806
$645$	0	0
$646$	−3.91287	−0.153950
$647$	−24.4246	−0.960232	−0.480116	−	0.877205i	$-0.659405\pi$
$647$	−24.4246	−0.960232	−0.480116	+	0.877205i	$0.659405\pi$
$648$	5.41521	0.212729
$649$	19.0145	0.746386
$650$	0	0
$651$	−3.27815	−0.128481
$652$	90.7135	3.55261
$653$	−13.0994	−0.512618	−0.256309	−	0.966595i	$-0.582506\pi$
$653$	−13.0994	−0.512618	−0.256309	+	0.966595i	$0.582506\pi$
$654$	−23.3879	−0.914540
$655$	0	0
$656$	−36.5507	−1.42707
$657$	31.3811	1.22429
$658$	4.16083	0.162206
$659$	−37.3844	−1.45629	−0.728145	−	0.685423i	$-0.759617\pi$
$659$	−37.3844	−1.45629	−0.728145	+	0.685423i	$0.759617\pi$
$660$	0	0
$661$	30.5258	1.18731	0.593657	−	0.804718i	$-0.297683\pi$
$661$	30.5258	1.18731	0.593657	+	0.804718i	$0.297683\pi$
$662$	−25.9830	−1.00986
$663$	−2.00946	−0.0780411
$664$	13.2580	0.514511
$665$	0	0
$666$	23.0538	0.893317
$667$	−0.186444	−0.00721913
$668$	73.7876	2.85493
$669$	−4.74733	−0.183542
$670$	0	0
$671$	13.7756	0.531802
$672$	0.0392732	0.00151499
$673$	−18.0019	−0.693923	−0.346962	−	0.937879i	$-0.612787\pi$
$673$	−18.0019	−0.693923	−0.346962	+	0.937879i	$0.612787\pi$
$674$	8.66696	0.333839
$675$	0	0
$676$	18.0084	0.692632
$677$	−1.78554	−0.0686239	−0.0343119	−	0.999411i	$-0.510924\pi$
$677$	−1.78554	−0.0686239	−0.0343119	+	0.999411i	$0.510924\pi$
$678$	11.5349	0.442997
$679$	−4.66787	−0.179137
$680$	0	0
$681$	−20.9981	−0.804650
$682$	−28.2915	−1.08334
$683$	16.9962	0.650343	0.325171	−	0.945655i	$-0.394578\pi$
$683$	16.9962	0.650343	0.325171	+	0.945655i	$0.394578\pi$
$684$	−26.5838	−1.01646
$685$	0	0
$686$	−24.5153	−0.935998
$687$	−22.2844	−0.850205
$688$	30.3737	1.15799
$689$	52.4919	1.99978
$690$	0	0
$691$	−9.73250	−0.370242	−0.185121	−	0.982716i	$-0.559268\pi$
$691$	−9.73250	−0.370242	−0.185121	+	0.982716i	$0.559268\pi$
$692$	−45.8475	−1.74286
$693$	3.90937	0.148505
$694$	−33.3301	−1.26519
$695$	0	0
$696$	0.627338	0.0237792
$697$	4.45906	0.168899
$698$	8.59582	0.325356
$699$	−10.9147	−0.412831
$700$	0	0
$701$	31.1728	1.17738	0.588690	−	0.808359i	$-0.299644\pi$
$701$	31.1728	1.17738	0.588690	+	0.808359i	$0.299644\pi$
$702$	−50.9823	−1.92420
$703$	−15.4254	−0.581781
$704$	21.0058	0.791686
$705$	0	0
$706$	26.2951	0.989627
$707$	3.40973	0.128236
$708$	28.9349	1.08744
$709$	−47.2147	−1.77318	−0.886592	−	0.462552i	$-0.846934\pi$
$709$	−47.2147	−1.77318	−0.886592	+	0.462552i	$0.846934\pi$
$710$	0	0
$711$	20.1598	0.756050
$712$	−48.1752	−1.80544
$713$	6.38879	0.239262
$714$	0.875632	0.0327697
$715$	0	0
$716$	−49.7168	−1.85801
$717$	2.20174	0.0822256
$718$	29.6015	1.10472
$719$	10.9829	0.409592	0.204796	−	0.978805i	$-0.434347\pi$
$719$	10.9829	0.409592	0.204796	+	0.978805i	$0.434347\pi$
$720$	0	0
$721$	−4.43512	−0.165172
$722$	−19.8261	−0.737851
$723$	−0.992209	−0.0369007
$724$	−44.3415	−1.64794
$725$	0	0
$726$	10.2425	0.380134
$727$	−4.07723	−0.151216	−0.0756080	−	0.997138i	$-0.524090\pi$
$727$	−4.07723	−0.151216	−0.0756080	+	0.997138i	$0.524090\pi$
$728$	−15.2171	−0.563983
$729$	12.5780	0.465853
$730$	0	0
$731$	−3.70549	−0.137052
$732$	20.9628	0.774806
$733$	13.3951	0.494759	0.247379	−	0.968919i	$-0.420431\pi$
$733$	13.3951	0.494759	0.247379	+	0.968919i	$0.420431\pi$
$734$	7.08253	0.261421
$735$	0	0
$736$	−0.0765396	−0.00282129
$737$	14.8302	0.546276
$738$	45.4626	1.67350
$739$	22.3405	0.821810	0.410905	−	0.911678i	$-0.365213\pi$
$739$	22.3405	0.821810	0.410905	+	0.911678i	$0.365213\pi$
$740$	0	0
$741$	13.7083	0.503588
$742$	−22.8736	−0.839716
$743$	36.1839	1.32746	0.663730	−	0.747973i	$-0.268973\pi$
$743$	36.1839	1.32746	0.663730	+	0.747973i	$0.268973\pi$
$744$	−21.4967	−0.788107
$745$	0	0
$746$	92.7665	3.39642
$747$	−5.47191	−0.200207
$748$	5.03572	0.184124
$749$	−3.59764	−0.131455
$750$	0	0
$751$	−10.1216	−0.369341	−0.184671	−	0.982800i	$-0.559122\pi$
$751$	−10.1216	−0.369341	−0.184671	+	0.982800i	$0.559122\pi$
$752$	9.05370	0.330155
$753$	−11.0635	−0.403177
$754$	1.32639	0.0483042
$755$	0	0
$756$	14.8038	0.538408
$757$	−32.6680	−1.18734	−0.593670	−	0.804709i	$-0.702321\pi$
$757$	−32.6680	−1.18734	−0.593670	+	0.804709i	$0.702321\pi$
$758$	−22.8468	−0.829834
$759$	3.72148	0.135081
$760$	0	0
$761$	24.2568	0.879309	0.439655	−	0.898167i	$-0.355101\pi$
$761$	24.2568	0.879309	0.439655	+	0.898167i	$0.355101\pi$
$762$	−5.19120	−0.188057
$763$	7.16957	0.259556
$764$	20.6646	0.747619
$765$	0	0
$766$	85.5556	3.09125
$767$	30.5472	1.10299
$768$	31.7069	1.14412
$769$	−16.5744	−0.597689	−0.298845	−	0.954302i	$-0.596601\pi$
$769$	−16.5744	−0.597689	−0.298845	+	0.954302i	$0.596601\pi$
$770$	0	0
$771$	−8.44249	−0.304049
$772$	56.8578	2.04636
$773$	37.9826	1.36614	0.683070	−	0.730353i	$-0.260644\pi$
$773$	37.9826	1.36614	0.683070	+	0.730353i	$0.260644\pi$
$774$	−37.7794	−1.35795
$775$	0	0
$776$	−30.6099	−1.09883
$777$	3.45194	0.123838
$778$	−54.5137	−1.95441
$779$	−30.4192	−1.08988
$780$	0	0
$781$	−5.42318	−0.194057
$782$	−1.70652	−0.0610251
$783$	−0.644303	−0.0230255
$784$	−25.5717	−0.913275
$785$	0	0
$786$	37.1125	1.32376
$787$	−1.38906	−0.0495146	−0.0247573	−	0.999693i	$-0.507881\pi$
$787$	−1.38906	−0.0495146	−0.0247573	+	0.999693i	$0.507881\pi$
$788$	26.5289	0.945052
$789$	10.4973	0.373715
$790$	0	0
$791$	−3.53604	−0.125727
$792$	25.6360	0.910936
$793$	22.1308	0.785887
$794$	45.7644	1.62412
$795$	0	0
$796$	36.4891	1.29332
$797$	−48.2065	−1.70756	−0.853781	−	0.520632i	$-0.825696\pi$
$797$	−48.2065	−1.70756	−0.853781	+	0.520632i	$0.825696\pi$
$798$	−5.97346	−0.211458
$799$	−1.10452	−0.0390751
$800$	0	0
$801$	19.8831	0.702534
$802$	74.2888	2.62323
$803$	40.5524	1.43106
$804$	22.5675	0.795893
$805$	0	0
$806$	−45.4508	−1.60094
$807$	−6.21265	−0.218695
$808$	22.3596	0.786606
$809$	14.6738	0.515902	0.257951	−	0.966158i	$-0.416953\pi$
$809$	14.6738	0.515902	0.257951	+	0.966158i	$0.416953\pi$
$810$	0	0
$811$	−19.7745	−0.694377	−0.347189	−	0.937795i	$-0.612864\pi$
$811$	−19.7745	−0.694377	−0.347189	+	0.937795i	$0.612864\pi$
$812$	−0.385145	−0.0135159
$813$	−6.17918	−0.216713
$814$	29.7914	1.04419
$815$	0	0
$816$	1.90532	0.0666995
$817$	25.2784	0.884379
$818$	39.6146	1.38509
$819$	6.28047	0.219457
$820$	0	0
$821$	−28.5257	−0.995554	−0.497777	−	0.867305i	$-0.665850\pi$
$821$	−28.5257	−0.995554	−0.497777	+	0.867305i	$0.665850\pi$
$822$	−38.1167	−1.32947
$823$	−20.1194	−0.701320	−0.350660	−	0.936503i	$-0.614043\pi$
$823$	−20.1194	−0.701320	−0.350660	+	0.936503i	$0.614043\pi$
$824$	−29.0836	−1.01318
$825$	0	0
$826$	−13.3111	−0.463151
$827$	25.3763	0.882421	0.441211	−	0.897404i	$-0.354549\pi$
$827$	25.3763	0.882421	0.441211	+	0.897404i	$0.354549\pi$
$828$	−11.5940	−0.402920
$829$	40.5872	1.40965	0.704826	−	0.709380i	$-0.251025\pi$
$829$	40.5872	1.40965	0.704826	+	0.709380i	$0.251025\pi$
$830$	0	0
$831$	−0.193802	−0.00672291
$832$	33.7462	1.16994
$833$	3.11966	0.108090
$834$	−33.6184	−1.16411
$835$	0	0
$836$	−34.3531	−1.18813
$837$	22.0781	0.763129
$838$	35.4071	1.22312
$839$	−16.9800	−0.586216	−0.293108	−	0.956079i	$-0.594689\pi$
$839$	−16.9800	−0.586216	−0.293108	+	0.956079i	$0.594689\pi$
$840$	0	0
$841$	−28.9832	−0.999422
$842$	35.8233	1.23455
$843$	20.8715	0.718852
$844$	−78.2895	−2.69484
$845$	0	0
$846$	−11.2612	−0.387167
$847$	−3.13984	−0.107886
$848$	−49.7714	−1.70916
$849$	−12.7269	−0.436787
$850$	0	0
$851$	−6.72750	−0.230616
$852$	−8.25260	−0.282729
$853$	3.23644	0.110814	0.0554069	−	0.998464i	$-0.482354\pi$
$853$	3.23644	0.110814	0.0554069	+	0.998464i	$0.482354\pi$
$854$	−9.64358	−0.329997
$855$	0	0
$856$	−23.5918	−0.806351
$857$	−1.40877	−0.0481228	−0.0240614	−	0.999710i	$-0.507660\pi$
$857$	−1.40877	−0.0481228	−0.0240614	+	0.999710i	$0.507660\pi$
$858$	−26.4752	−0.903848
$859$	43.3662	1.47963	0.739817	−	0.672808i	$-0.234912\pi$
$859$	43.3662	1.47963	0.739817	+	0.672808i	$0.234912\pi$
$860$	0	0
$861$	6.80730	0.231992
$862$	42.5563	1.44947
$863$	−36.3292	−1.23666	−0.618329	−	0.785919i	$-0.712190\pi$
$863$	−36.3292	−1.23666	−0.618329	+	0.785919i	$0.712190\pi$
$864$	−0.264501	−0.00899852
$865$	0	0
$866$	−55.6482	−1.89100
$867$	16.6351	0.564958
$868$	13.1976	0.447956
$869$	26.0516	0.883739
$870$	0	0
$871$	23.8249	0.807276
$872$	47.0150	1.59213
$873$	12.6335	0.427578
$874$	11.6417	0.393786
$875$	0	0
$876$	61.7097	2.08498
$877$	30.8667	1.04229	0.521147	−	0.853467i	$-0.325504\pi$
$877$	30.8667	1.04229	0.521147	+	0.853467i	$0.325504\pi$
$878$	−88.9987	−3.00356
$879$	25.5903	0.863139
$880$	0	0
$881$	−34.9920	−1.17891	−0.589456	−	0.807801i	$-0.700658\pi$
$881$	−34.9920	−1.17891	−0.589456	+	0.807801i	$0.700658\pi$
$882$	31.8066	1.07098
$883$	44.7518	1.50602	0.753008	−	0.658011i	$-0.228602\pi$
$883$	44.7518	1.50602	0.753008	+	0.658011i	$0.228602\pi$
$884$	8.08996	0.272095
$885$	0	0
$886$	−100.978	−3.39243
$887$	21.2497	0.713495	0.356747	−	0.934201i	$-0.383886\pi$
$887$	21.2497	0.713495	0.356747	+	0.934201i	$0.383886\pi$
$888$	22.6364	0.759627
$889$	1.59136	0.0533726
$890$	0	0
$891$	−2.88818	−0.0967576
$892$	19.1124	0.639931
$893$	7.53491	0.252146
$894$	11.4303	0.382286
$895$	0	0
$896$	−14.6259	−0.488616
$897$	5.97863	0.199621
$898$	4.44566	0.148354
$899$	−0.574397	−0.0191572
$900$	0	0
$901$	6.07194	0.202286
$902$	58.7493	1.95614
$903$	−5.65687	−0.188249
$904$	−23.1878	−0.771216
$905$	0	0
$906$	−38.3246	−1.27325
$907$	7.07983	0.235082	0.117541	−	0.993068i	$-0.462499\pi$
$907$	7.07983	0.235082	0.117541	+	0.993068i	$0.462499\pi$
$908$	84.5370	2.80546
$909$	−9.22833	−0.306085
$910$	0	0
$911$	−40.6767	−1.34768	−0.673840	−	0.738877i	$-0.735356\pi$
$911$	−40.6767	−1.34768	−0.673840	+	0.738877i	$0.735356\pi$
$912$	−12.9979	−0.430402
$913$	−7.07110	−0.234019
$914$	−23.4361	−0.775196
$915$	0	0
$916$	89.7156	2.96429
$917$	−11.3768	−0.375696
$918$	−5.89731	−0.194640
$919$	−11.3936	−0.375841	−0.187920	−	0.982184i	$-0.560175\pi$
$919$	−11.3936	−0.375841	−0.187920	+	0.982184i	$0.560175\pi$
$920$	0	0
$921$	−28.3124	−0.932927
$922$	−19.5095	−0.642512
$923$	−8.71243	−0.286773
$924$	7.68763	0.252904
$925$	0	0
$926$	−50.3204	−1.65363
$927$	12.0035	0.394247
$928$	0.00688145	0.000225895 0
$929$	−38.5646	−1.26526	−0.632631	−	0.774453i	$-0.718025\pi$
$929$	−38.5646	−1.26526	−0.632631	+	0.774453i	$0.718025\pi$
$930$	0	0
$931$	−21.2819	−0.697488
$932$	43.9417	1.43936
$933$	−4.01300	−0.131380
$934$	38.9188	1.27346
$935$	0	0
$936$	41.1847	1.34616
$937$	−35.6493	−1.16461	−0.582306	−	0.812970i	$-0.697849\pi$
$937$	−35.6493	−1.16461	−0.582306	+	0.812970i	$0.697849\pi$
$938$	−10.3818	−0.338978
$939$	10.7297	0.350152
$940$	0	0
$941$	−0.459712	−0.0149862	−0.00749308	−	0.999972i	$-0.502385\pi$
$941$	−0.459712	−0.0149862	−0.00749308	+	0.999972i	$0.502385\pi$
$942$	−15.2189	−0.495860
$943$	−13.2668	−0.432025
$944$	−28.9640	−0.942697
$945$	0	0
$946$	−48.8207	−1.58730
$947$	16.4312	0.533944	0.266972	−	0.963704i	$-0.413977\pi$
$947$	16.4312	0.533944	0.266972	+	0.963704i	$0.413977\pi$
$948$	39.6434	1.28756
$949$	65.1482	2.11480
$950$	0	0
$951$	−2.60291	−0.0844053
$952$	−1.76022	−0.0570490
$953$	−27.4225	−0.888302	−0.444151	−	0.895952i	$-0.646495\pi$
$953$	−27.4225	−0.888302	−0.444151	+	0.895952i	$0.646495\pi$
$954$	61.9067	2.00430
$955$	0	0
$956$	−8.86406	−0.286684
$957$	−0.334588	−0.0108157
$958$	−102.047	−3.29698
$959$	11.6847	0.377318
$960$	0	0
$961$	−11.3174	−0.365076
$962$	47.8604	1.54308
$963$	9.73692	0.313768
$964$	3.99457	0.128656
$965$	0	0
$966$	−2.60521	−0.0838213
$967$	−45.1755	−1.45275	−0.726373	−	0.687300i	$-0.758796\pi$
$967$	−45.1755	−1.45275	−0.726373	+	0.687300i	$0.758796\pi$
$968$	−20.5897	−0.661779
$969$	1.58569	0.0509398
$970$	0	0
$971$	50.1290	1.60872	0.804359	−	0.594144i	$-0.202509\pi$
$971$	50.1290	1.60872	0.804359	+	0.594144i	$0.202509\pi$
$972$	−64.0312	−2.05380
$973$	10.3057	0.330386
$974$	18.5252	0.593587
$975$	0	0
$976$	−20.9838	−0.671675
$977$	−2.05208	−0.0656518	−0.0328259	−	0.999461i	$-0.510451\pi$
$977$	−2.05208	−0.0656518	−0.0328259	+	0.999461i	$0.510451\pi$
$978$	−55.1676	−1.76407
$979$	25.6940	0.821185
$980$	0	0
$981$	−19.4042	−0.619530
$982$	−65.5487	−2.09174
$983$	48.6503	1.55170	0.775852	−	0.630915i	$-0.217320\pi$
$983$	48.6503	1.55170	0.775852	+	0.630915i	$0.217320\pi$
$984$	44.6394	1.42305
$985$	0	0
$986$	0.153428	0.00488616
$987$	−1.68618	−0.0536718
$988$	−55.1888	−1.75579
$989$	11.0247	0.350565
$990$	0	0
$991$	50.2126	1.59506	0.797528	−	0.603282i	$-0.206141\pi$
$991$	50.2126	1.59506	0.797528	+	0.603282i	$0.206141\pi$
$992$	−0.235804	−0.00748677
$993$	10.5297	0.334149
$994$	3.79648	0.120417
$995$	0	0
$996$	−10.7603	−0.340953
$997$	−21.8288	−0.691325	−0.345663	−	0.938359i	$-0.612346\pi$
$997$	−21.8288	−0.691325	−0.345663	+	0.938359i	$0.612346\pi$
$998$	−66.2053	−2.09569
$999$	−23.2485	−0.735551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.43		46
5.2	odd	4		1205.2.b.c.724.43	yes	46
5.3	odd	4		1205.2.b.c.724.4	✓	46
5.4	even	2	inner	6025.2.a.p.1.4		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.4	✓	46	5.3	odd	4
1205.2.b.c.724.43	yes	46	5.2	odd	4
6025.2.a.p.1.4		46	5.4	even	2	inner
6025.2.a.p.1.43		46	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:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44838 q^{2} -0.992209 q^{3} +3.99457 q^{4} -2.42931 q^{6} +0.744704 q^{7} +4.88345 q^{8} -2.01552 q^{9} +O(q^{10})\)	\(q+2.44838 q^{2} -0.992209 q^{3} +3.99457 q^{4} -2.42931 q^{6} +0.744704 q^{7} +4.88345 q^{8} -2.01552 q^{9} -2.60457 q^{11} -3.96344 q^{12} -4.18428 q^{13} +1.82332 q^{14} +3.96742 q^{16} -0.484012 q^{17} -4.93476 q^{18} +3.30187 q^{19} -0.738902 q^{21} -6.37697 q^{22} +1.44005 q^{23} -4.84541 q^{24} -10.2447 q^{26} +4.97645 q^{27} +2.97477 q^{28} -0.129471 q^{29} +4.43651 q^{31} -0.0531507 q^{32} +2.58428 q^{33} -1.18504 q^{34} -8.05113 q^{36} -4.67172 q^{37} +8.08424 q^{38} +4.15168 q^{39} -9.21272 q^{41} -1.80911 q^{42} +7.65578 q^{43} -10.4041 q^{44} +3.52579 q^{46} +2.28201 q^{47} -3.93651 q^{48} -6.44542 q^{49} +0.480241 q^{51} -16.7144 q^{52} -12.5450 q^{53} +12.1842 q^{54} +3.63673 q^{56} -3.27615 q^{57} -0.316993 q^{58} -7.30045 q^{59} -5.28903 q^{61} +10.8623 q^{62} -1.50097 q^{63} -8.06498 q^{64} +6.32729 q^{66} -5.69391 q^{67} -1.93342 q^{68} -1.42883 q^{69} +2.08218 q^{71} -9.84271 q^{72} -15.5697 q^{73} -11.4381 q^{74} +13.1895 q^{76} -1.93963 q^{77} +10.1649 q^{78} -10.0023 q^{79} +1.10889 q^{81} -22.5562 q^{82} +2.71488 q^{83} -2.95159 q^{84} +18.7442 q^{86} +0.128462 q^{87} -12.7193 q^{88} -9.86499 q^{89} -3.11605 q^{91} +5.75237 q^{92} -4.40195 q^{93} +5.58723 q^{94} +0.0527366 q^{96} -6.26809 q^{97} -15.7808 q^{98} +5.24956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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