LMFDB - Embedded newform 6025.2.a.p.1.4

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			1.4
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	$-0.833084\pi$
$2$	−2.44838	−1.73127	−0.865633	+	0.500679i	$0.833084\pi$
$3$	0.992209	0.572852	0.286426	−	0.958102i	$-0.407533\pi$
$3$	0.992209	0.572852	0.286426	+	0.958102i	$0.407533\pi$
$4$	3.99457	1.99728
$5$	0	0
$5$	0	0
$6$	−2.42931	−0.991760
$7$	−0.744704	−0.281472	−0.140736	−	0.990047i	$-0.544947\pi$
$7$	−0.744704	−0.281472	−0.140736	+	0.990047i	$0.544947\pi$
$8$	−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	$-0.302953\pi$
$13$	4.18428	1.16051	0.580256	+	0.814434i	$0.302953\pi$
$14$	1.82332	0.487302
$15$	0	0
$16$	3.96742	0.991856
$17$	0.484012	0.117390	0.0586950	−	0.998276i	$-0.481306\pi$
$17$	0.484012	0.117390	0.0586950	+	0.998276i	$0.481306\pi$
$18$	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	$-0.547971\pi$
$23$	−1.44005	−0.300271	−0.150135	+	0.988665i	$0.547971\pi$
$24$	−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	$-0.374542\pi$
$37$	4.67172	0.768026	0.384013	+	0.923328i	$0.374542\pi$
$38$	−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	$-0.698414\pi$
$43$	−7.65578	−1.16749	−0.583747	+	0.811935i	$0.698414\pi$
$44$	−10.4041	−1.56848
$45$	0	0
$46$	3.52579	0.519849
$47$	−2.28201	−0.332865	−0.166433	−	0.986053i	$-0.553225\pi$
$47$	−2.28201	−0.332865	−0.166433	+	0.986053i	$0.553225\pi$
$48$	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	$-0.169465\pi$
$53$	12.5450	1.72319	0.861596	+	0.507595i	$0.169465\pi$
$54$	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	$-0.386925\pi$
$67$	5.69391	0.695621	0.347811	+	0.937565i	$0.386925\pi$
$68$	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	$-0.135196\pi$
$73$	15.5697	1.82230	0.911149	+	0.412076i	$0.135196\pi$
$74$	−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	$-0.547605\pi$
$83$	−2.71488	−0.297997	−0.148999	+	0.988837i	$0.547605\pi$
$84$	−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	$-0.396917\pi$
$97$	6.26809	0.636428	0.318214	+	0.948019i	$0.396917\pi$
$98$	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	$-0.405210\pi$
$103$	5.95554	0.586817	0.293409	+	0.955987i	$0.405210\pi$
$104$	−20.4338	−2.00369
$105$	0	0
$106$	−30.7150	−2.98330
$107$	4.83097	0.467027	0.233514	−	0.972354i	$-0.424978\pi$
$107$	4.83097	0.467027	0.233514	+	0.972354i	$0.424978\pi$
$108$	−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	$-0.428304\pi$
$113$	4.74825	0.446677	0.223339	+	0.974741i	$0.428304\pi$
$114$	−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	$-0.530224\pi$
$127$	−2.13691	−0.189620	−0.0948099	+	0.995495i	$0.530224\pi$
$128$	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	$-0.733817\pi$
$137$	−15.6904	−1.34052	−0.670259	+	0.742128i	$0.733817\pi$
$138$	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	$-0.580427\pi$
$157$	−6.26473	−0.499980	−0.249990	+	0.968248i	$0.580427\pi$
$158$	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	$-0.848851\pi$
$163$	−22.7092	−1.77872	−0.889362	+	0.457204i	$0.848851\pi$
$164$	−36.8008	−2.87366
$165$	0	0
$166$	6.64707	0.515913
$167$	−18.4720	−1.42941	−0.714703	−	0.699428i	$-0.753438\pi$
$167$	−18.4720	−1.42941	−0.714703	+	0.699428i	$0.753438\pi$
$168$	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	$-0.356286\pi$
$173$	11.4775	0.872615	0.436307	+	0.899798i	$0.356286\pi$
$174$	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	$-0.671201\pi$
$193$	−14.2338	−1.02457	−0.512285	+	0.858816i	$0.671201\pi$
$194$	−15.3467	−1.10183
$195$	0	0
$196$	−25.7466	−1.83905
$197$	−6.64124	−0.473169	−0.236585	−	0.971611i	$-0.576028\pi$
$197$	−6.64124	−0.473169	−0.236585	+	0.971611i	$0.576028\pi$
$198$	−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	$-0.551214\pi$
$223$	−4.78461	−0.320401	−0.160200	+	0.987085i	$0.551214\pi$
$224$	−0.0395815	−0.00264465
$225$	0	0
$226$	−11.6255	−0.773318
$227$	−21.1630	−1.40464	−0.702319	−	0.711862i	$-0.747852\pi$
$227$	−21.1630	−1.40464	−0.702319	+	0.711862i	$0.747852\pi$
$228$	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	$-0.617336\pi$
$233$	−11.0004	−0.720658	−0.360329	+	0.932825i	$0.617336\pi$
$234$	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	$-0.585498\pi$
$257$	−8.50879	−0.530763	−0.265382	+	0.964143i	$0.585498\pi$
$258$	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	$-0.394236\pi$
$263$	10.5798	0.652377	0.326188	+	0.945305i	$0.394236\pi$
$264$	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	$-0.501868\pi$
$277$	−0.195324	−0.0117359	−0.00586793	+	0.999983i	$0.501868\pi$
$278$	−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	$-0.624503\pi$
$283$	−12.8269	−0.762478	−0.381239	+	0.924477i	$0.624503\pi$
$284$	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	$-0.228427\pi$
$293$	25.7912	1.50674	0.753369	+	0.657598i	$0.228427\pi$
$294$	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	$-0.802869\pi$
$307$	−28.5348	−1.62856	−0.814282	+	0.580469i	$0.802869\pi$
$308$	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	$-0.401136\pi$
$313$	10.8140	0.611243	0.305622	+	0.952153i	$0.401136\pi$
$314$	15.3384	0.865598
$315$	0	0
$316$	−39.9547	−2.24763
$317$	−2.62335	−0.147342	−0.0736711	−	0.997283i	$-0.523472\pi$
$317$	−2.62335	−0.147342	−0.0736711	+	0.997283i	$0.523472\pi$
$318$	−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	$-0.530737\pi$
$337$	−3.53987	−0.192829	−0.0964146	+	0.995341i	$0.530737\pi$
$338$	−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	$-0.380934\pi$
$347$	13.6131	0.730791	0.365396	+	0.930852i	$0.380934\pi$
$348$	−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	$-0.592263\pi$
$353$	−10.7398	−0.571621	−0.285810	+	0.958286i	$0.592263\pi$
$354$	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	$-0.524055\pi$
$367$	−2.89274	−0.151000	−0.0754999	+	0.997146i	$0.524055\pi$
$368$	−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	$-0.937698\pi$
$373$	−37.8889	−1.96181	−0.980907	+	0.194479i	$0.937698\pi$
$374$	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	$-0.851242\pi$
$383$	−34.9438	−1.78554	−0.892771	+	0.450510i	$0.851242\pi$
$384$	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	$-0.655405\pi$
$397$	−18.6917	−0.938110	−0.469055	+	0.883169i	$0.655405\pi$
$398$	−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	$-0.316099\pi$
$433$	22.7286	1.09227	0.546133	+	0.837698i	$0.316099\pi$
$434$	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	$-0.0641594\pi$
$443$	41.2429	1.95951	0.979755	+	0.200201i	$0.0641594\pi$
$444$	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	$-0.428127\pi$
$457$	9.57207	0.447763	0.223881	+	0.974616i	$0.428127\pi$
$458$	−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	$-0.341515\pi$
$463$	20.5525	0.955157	0.477579	+	0.878589i	$0.341515\pi$
$464$	−0.513664	−0.0238463
$465$	0	0
$466$	26.9331	1.24765
$467$	−15.8957	−0.735567	−0.367783	−	0.929912i	$-0.619883\pi$
$467$	−15.8957	−0.735567	−0.367783	+	0.929912i	$0.619883\pi$
$468$	−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	$-0.554839\pi$
$487$	−7.56632	−0.342863	−0.171431	+	0.985196i	$0.554839\pi$
$488$	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	$-0.493257\pi$
$503$	0.950190	0.0423669	0.0211834	+	0.999776i	$0.493257\pi$
$504$	−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	$-0.619999\pi$
$523$	−16.8372	−0.736240	−0.368120	+	0.929778i	$0.619999\pi$
$524$	−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	$-0.441393\pi$
$547$	8.56387	0.366164	0.183082	+	0.983098i	$0.441393\pi$
$548$	−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	$-0.545954\pi$
$557$	−6.79073	−0.287732	−0.143866	+	0.989597i	$0.545954\pi$
$558$	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	$-0.640361\pi$
$563$	−20.2542	−0.853612	−0.426806	+	0.904343i	$0.640361\pi$
$564$	−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	$-0.721113\pi$
$577$	−30.7522	−1.28023	−0.640114	+	0.768280i	$0.721113\pi$
$578$	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	$-0.619622\pi$
$587$	−17.7844	−0.734042	−0.367021	+	0.930213i	$0.619622\pi$
$588$	−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	$-0.487168\pi$
$593$	1.96286	0.0806051	0.0403026	+	0.999188i	$0.487168\pi$
$594$	−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	$-0.275212\pi$
$607$	31.9764	1.29788	0.648941	+	0.760839i	$0.275212\pi$
$608$	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	$-0.373679\pi$
$613$	19.1393	0.773028	0.386514	+	0.922283i	$0.373679\pi$
$614$	69.8639	2.81948
$615$	0	0
$616$	−9.47211	−0.381642
$617$	−40.5121	−1.63096	−0.815478	−	0.578788i	$-0.803526\pi$
$617$	−40.5121	−1.63096	−0.815478	+	0.578788i	$0.803526\pi$
$618$	−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	$-0.651496\pi$
$643$	−23.2362	−0.916347	−0.458174	+	0.888863i	$0.651496\pi$
$644$	4.28381	0.168806
$645$	0	0
$646$	−3.91287	−0.153950
$647$	24.4246	0.960232	0.480116	−	0.877205i	$-0.340595\pi$
$647$	24.4246	0.960232	0.480116	+	0.877205i	$0.340595\pi$
$648$	−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	$-0.417494\pi$
$653$	13.0994	0.512618	0.256309	+	0.966595i	$0.417494\pi$
$654$	−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	$-0.387213\pi$
$673$	18.0019	0.693923	0.346962	+	0.937879i	$0.387213\pi$
$674$	8.66696	0.333839
$675$	0	0
$676$	18.0084	0.692632
$677$	1.78554	0.0686239	0.0343119	−	0.999411i	$-0.489076\pi$
$677$	1.78554	0.0686239	0.0343119	+	0.999411i	$0.489076\pi$
$678$	−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	$-0.605422\pi$
$683$	−16.9962	−0.650343	−0.325171	+	0.945655i	$0.605422\pi$
$684$	−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	$-0.475910\pi$
$727$	4.07723	0.151216	0.0756080	+	0.997138i	$0.475910\pi$
$728$	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	$-0.579569\pi$
$733$	−13.3951	−0.494759	−0.247379	+	0.968919i	$0.579569\pi$
$734$	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	$-0.731027\pi$
$743$	−36.1839	−1.32746	−0.663730	+	0.747973i	$0.731027\pi$
$744$	−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	$-0.297679\pi$
$757$	32.6680	1.18734	0.593670	+	0.804709i	$0.297679\pi$
$758$	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	$-0.739356\pi$
$773$	−37.9826	−1.36614	−0.683070	+	0.730353i	$0.739356\pi$
$774$	−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	$-0.492119\pi$
$787$	1.38906	0.0495146	0.0247573	+	0.999693i	$0.492119\pi$
$788$	−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	$-0.174304\pi$
$797$	48.2065	1.70756	0.853781	+	0.520632i	$0.174304\pi$
$798$	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	$-0.385957\pi$
$823$	20.1194	0.701320	0.350660	+	0.936503i	$0.385957\pi$
$824$	−29.0836	−1.01318
$825$	0	0
$826$	−13.3111	−0.463151
$827$	−25.3763	−0.882421	−0.441211	−	0.897404i	$-0.645451\pi$
$827$	−25.3763	−0.882421	−0.441211	+	0.897404i	$0.645451\pi$
$828$	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	$-0.517646\pi$
$853$	−3.23644	−0.110814	−0.0554069	+	0.998464i	$0.517646\pi$
$854$	−9.64358	−0.329997
$855$	0	0
$856$	−23.5918	−0.806351
$857$	1.40877	0.0481228	0.0240614	−	0.999710i	$-0.492340\pi$
$857$	1.40877	0.0481228	0.0240614	+	0.999710i	$0.492340\pi$
$858$	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	$-0.287810\pi$
$863$	36.3292	1.23666	0.618329	+	0.785919i	$0.287810\pi$
$864$	−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	$-0.674496\pi$
$877$	−30.8667	−1.04229	−0.521147	+	0.853467i	$0.674496\pi$
$878$	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	$-0.771398\pi$
$883$	−44.7518	−1.50602	−0.753008	+	0.658011i	$0.771398\pi$
$884$	8.08996	0.272095
$885$	0	0
$886$	−100.978	−3.39243
$887$	−21.2497	−0.713495	−0.356747	−	0.934201i	$-0.616114\pi$
$887$	−21.2497	−0.713495	−0.356747	+	0.934201i	$0.616114\pi$
$888$	−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	$-0.537501\pi$
$907$	−7.07983	−0.235082	−0.117541	+	0.993068i	$0.537501\pi$
$908$	−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	$-0.302151\pi$
$937$	35.6493	1.16461	0.582306	+	0.812970i	$0.302151\pi$
$938$	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	$-0.586023\pi$
$947$	−16.4312	−0.533944	−0.266972	+	0.963704i	$0.586023\pi$
$948$	−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	$-0.353505\pi$
$953$	27.4225	0.888302	0.444151	+	0.895952i	$0.353505\pi$
$954$	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	$-0.241204\pi$
$967$	45.1755	1.45275	0.726373	+	0.687300i	$0.241204\pi$
$968$	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	$-0.489549\pi$
$977$	2.05208	0.0656518	0.0328259	+	0.999461i	$0.489549\pi$
$978$	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	$-0.782680\pi$
$983$	−48.6503	−1.55170	−0.775852	+	0.630915i	$0.782680\pi$
$984$	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	$-0.387654\pi$
$997$	21.8288	0.691325	0.345663	+	0.938359i	$0.387654\pi$
$998$	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.1.4		46
5.2	odd	4		1205.2.b.c.724.4	✓	46
5.3	odd	4		1205.2.b.c.724.43	yes	46
5.4	even	2	inner	6025.2.a.p.1.43		46

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

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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