LMFDB - Embedded newform 6025.2.a.p.1.29

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			1.29
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)

$f(q)$	$=$	$q+0.760721 q^{2} -1.12938 q^{3} -1.42130 q^{4} -0.859145 q^{6} -2.53853 q^{7} -2.60266 q^{8} -1.72450 q^{9} +O(q^{10})$	\(q+0.760721 q^{2} -1.12938 q^{3} -1.42130 q^{4} -0.859145 q^{6} -2.53853 q^{7} -2.60266 q^{8} -1.72450 q^{9} -2.06253 q^{11} +1.60519 q^{12} +4.79651 q^{13} -1.93111 q^{14} +0.862709 q^{16} +1.95237 q^{17} -1.31186 q^{18} -1.03825 q^{19} +2.86697 q^{21} -1.56901 q^{22} -2.68585 q^{23} +2.93940 q^{24} +3.64880 q^{26} +5.33576 q^{27} +3.60802 q^{28} +5.84308 q^{29} +7.76074 q^{31} +5.86160 q^{32} +2.32939 q^{33} +1.48521 q^{34} +2.45103 q^{36} -9.08041 q^{37} -0.789817 q^{38} -5.41709 q^{39} +9.18849 q^{41} +2.18096 q^{42} -3.23632 q^{43} +2.93148 q^{44} -2.04318 q^{46} +4.22721 q^{47} -0.974328 q^{48} -0.555881 q^{49} -2.20497 q^{51} -6.81729 q^{52} +6.32114 q^{53} +4.05903 q^{54} +6.60692 q^{56} +1.17258 q^{57} +4.44495 q^{58} +2.97769 q^{59} -11.0977 q^{61} +5.90376 q^{62} +4.37768 q^{63} +2.73362 q^{64} +1.77202 q^{66} +1.40319 q^{67} -2.77490 q^{68} +3.03335 q^{69} -6.51732 q^{71} +4.48827 q^{72} -1.89995 q^{73} -6.90766 q^{74} +1.47566 q^{76} +5.23580 q^{77} -4.12089 q^{78} -13.9020 q^{79} -0.852626 q^{81} +6.98988 q^{82} +6.49943 q^{83} -4.07483 q^{84} -2.46194 q^{86} -6.59906 q^{87} +5.36807 q^{88} -0.236437 q^{89} -12.1761 q^{91} +3.81741 q^{92} -8.76484 q^{93} +3.21573 q^{94} -6.61998 q^{96} +13.5524 q^{97} -0.422870 q^{98} +3.55683 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.760721	0.537911	0.268956	−	0.963153i	$-0.413322\pi$
$2$	0.760721	0.537911	0.268956	+	0.963153i	$0.413322\pi$
$3$	−1.12938	−0.652049	−0.326025	−	0.945361i	$-0.605709\pi$
$3$	−1.12938	−0.652049	−0.326025	+	0.945361i	$0.605709\pi$
$4$	−1.42130	−0.710652
$5$	0	0
$5$	0	0
$6$	−0.859145	−0.350745
$7$	−2.53853	−0.959473	−0.479737	−	0.877413i	$-0.659268\pi$
$7$	−2.53853	−0.959473	−0.479737	+	0.877413i	$0.659268\pi$
$8$	−2.60266	−0.920179
$9$	−1.72450	−0.574832
$10$	0	0
$11$	−2.06253	−0.621877	−0.310939	−	0.950430i	$-0.600643\pi$
$11$	−2.06253	−0.621877	−0.310939	+	0.950430i	$0.600643\pi$
$12$	1.60519	0.463380
$13$	4.79651	1.33031	0.665156	−	0.746705i	$-0.268365\pi$
$13$	4.79651	1.33031	0.665156	+	0.746705i	$0.268365\pi$
$14$	−1.93111	−0.516111
$15$	0	0
$16$	0.862709	0.215677
$17$	1.95237	0.473518	0.236759	−	0.971568i	$-0.423915\pi$
$17$	1.95237	0.473518	0.236759	+	0.971568i	$0.423915\pi$
$18$	−1.31186	−0.309209
$19$	−1.03825	−0.238190	−0.119095	−	0.992883i	$-0.537999\pi$
$19$	−1.03825	−0.238190	−0.119095	+	0.992883i	$0.537999\pi$
$20$	0	0
$21$	2.86697	0.625623
$22$	−1.56901	−0.334515
$23$	−2.68585	−0.560039	−0.280019	−	0.959994i	$-0.590341\pi$
$23$	−2.68585	−0.560039	−0.280019	+	0.959994i	$0.590341\pi$
$24$	2.93940	0.600002
$25$	0	0
$26$	3.64880	0.715589
$27$	5.33576	1.02687
$28$	3.60802	0.681851
$29$	5.84308	1.08503	0.542516	−	0.840046i	$-0.317472\pi$
$29$	5.84308	1.08503	0.542516	+	0.840046i	$0.317472\pi$
$30$	0	0
$31$	7.76074	1.39387	0.696935	−	0.717134i	$-0.254547\pi$
$31$	7.76074	1.39387	0.696935	+	0.717134i	$0.254547\pi$
$32$	5.86160	1.03619
$33$	2.32939	0.405494
$34$	1.48521	0.254711
$35$	0	0
$36$	2.45103	0.408505
$37$	−9.08041	−1.49281	−0.746405	−	0.665492i	$-0.768222\pi$
$37$	−9.08041	−1.49281	−0.746405	+	0.665492i	$0.768222\pi$
$38$	−0.789817	−0.128125
$39$	−5.41709	−0.867428
$40$	0	0
$41$	9.18849	1.43500	0.717501	−	0.696558i	$-0.245286\pi$
$41$	9.18849	1.43500	0.717501	+	0.696558i	$0.245286\pi$
$42$	2.18096	0.336530
$43$	−3.23632	−0.493534	−0.246767	−	0.969075i	$-0.579368\pi$
$43$	−3.23632	−0.493534	−0.246767	+	0.969075i	$0.579368\pi$
$44$	2.93148	0.441938
$45$	0	0
$46$	−2.04318	−0.301251
$47$	4.22721	0.616603	0.308301	−	0.951289i	$-0.400239\pi$
$47$	4.22721	0.616603	0.308301	+	0.951289i	$0.400239\pi$
$48$	−0.974328	−0.140632
$49$	−0.555881	−0.0794116
$50$	0	0
$51$	−2.20497	−0.308757
$52$	−6.81729	−0.945388
$53$	6.32114	0.868276	0.434138	−	0.900846i	$-0.357053\pi$
$53$	6.32114	0.868276	0.434138	+	0.900846i	$0.357053\pi$
$54$	4.05903	0.552364
$55$	0	0
$56$	6.60692	0.882887
$57$	1.17258	0.155312
$58$	4.44495	0.583651
$59$	2.97769	0.387662	0.193831	−	0.981035i	$-0.437909\pi$
$59$	2.97769	0.387662	0.193831	+	0.981035i	$0.437909\pi$
$60$	0	0
$61$	−11.0977	−1.42091	−0.710456	−	0.703741i	$-0.751511\pi$
$61$	−11.0977	−1.42091	−0.710456	+	0.703741i	$0.751511\pi$
$62$	5.90376	0.749778
$63$	4.37768	0.551536
$64$	2.73362	0.341703
$65$	0	0
$66$	1.77202	0.218120
$67$	1.40319	0.171427	0.0857134	−	0.996320i	$-0.472683\pi$
$67$	1.40319	0.171427	0.0857134	+	0.996320i	$0.472683\pi$
$68$	−2.77490	−0.336506
$69$	3.03335	0.365173
$70$	0	0
$71$	−6.51732	−0.773464	−0.386732	−	0.922192i	$-0.626396\pi$
$71$	−6.51732	−0.773464	−0.386732	+	0.922192i	$0.626396\pi$
$72$	4.48827	0.528948
$73$	−1.89995	−0.222373	−0.111186	−	0.993800i	$-0.535465\pi$
$73$	−1.89995	−0.222373	−0.111186	+	0.993800i	$0.535465\pi$
$74$	−6.90766	−0.802999
$75$	0	0
$76$	1.47566	0.169270
$77$	5.23580	0.596674
$78$	−4.12089	−0.466599
$79$	−13.9020	−1.56410	−0.782050	−	0.623215i	$-0.785826\pi$
$79$	−13.9020	−1.56410	−0.782050	+	0.623215i	$0.785826\pi$
$80$	0	0
$81$	−0.852626	−0.0947362
$82$	6.98988	0.771904
$83$	6.49943	0.713405	0.356703	−	0.934218i	$-0.383901\pi$
$83$	6.49943	0.713405	0.356703	+	0.934218i	$0.383901\pi$
$84$	−4.07483	−0.444600
$85$	0	0
$86$	−2.46194	−0.265477
$87$	−6.59906	−0.707494
$88$	5.36807	0.572238
$89$	−0.236437	−0.0250623	−0.0125311	−	0.999921i	$-0.503989\pi$
$89$	−0.236437	−0.0250623	−0.0125311	+	0.999921i	$0.503989\pi$
$90$	0	0
$91$	−12.1761	−1.27640
$92$	3.81741	0.397992
$93$	−8.76484	−0.908871
$94$	3.21573	0.331677
$95$	0	0
$96$	−6.61998	−0.675649
$97$	13.5524	1.37604	0.688020	−	0.725691i	$-0.258480\pi$
$97$	13.5524	1.37604	0.688020	+	0.725691i	$0.258480\pi$
$98$	−0.422870	−0.0427164
$99$	3.55683	0.357475
$100$	0	0
$101$	−12.0004	−1.19409	−0.597044	−	0.802209i	$-0.703658\pi$
$101$	−12.0004	−1.19409	−0.597044	+	0.802209i	$0.703658\pi$
$102$	−1.67737	−0.166084
$103$	−13.6216	−1.34218	−0.671088	−	0.741377i	$-0.734173\pi$
$103$	−13.6216	−1.34218	−0.671088	+	0.741377i	$0.734173\pi$
$104$	−12.4837	−1.22412
$105$	0	0
$106$	4.80863	0.467055
$107$	13.8286	1.33686	0.668432	−	0.743773i	$-0.266966\pi$
$107$	13.8286	1.33686	0.668432	+	0.743773i	$0.266966\pi$
$108$	−7.58373	−0.729745
$109$	13.7652	1.31846	0.659232	−	0.751940i	$-0.270882\pi$
$109$	13.7652	1.31846	0.659232	+	0.751940i	$0.270882\pi$
$110$	0	0
$111$	10.2553	0.973386
$112$	−2.19001	−0.206936
$113$	−6.68313	−0.628696	−0.314348	−	0.949308i	$-0.601786\pi$
$113$	−6.68313	−0.628696	−0.314348	+	0.949308i	$0.601786\pi$
$114$	0.892005	0.0835439
$115$	0	0
$116$	−8.30478	−0.771080
$117$	−8.27155	−0.764705
$118$	2.26519	0.208528
$119$	−4.95613	−0.454328
$120$	0	0
$121$	−6.74596	−0.613269
$122$	−8.44224	−0.764325
$123$	−10.3773	−0.935692
$124$	−11.0304	−0.990556
$125$	0	0
$126$	3.33019	0.296677
$127$	−4.63183	−0.411008	−0.205504	−	0.978656i	$-0.565883\pi$
$127$	−4.63183	−0.411008	−0.205504	+	0.978656i	$0.565883\pi$
$128$	−9.64367	−0.852388
$129$	3.65504	0.321808
$130$	0	0
$131$	−10.6049	−0.926558	−0.463279	−	0.886212i	$-0.653327\pi$
$131$	−10.6049	−0.926558	−0.463279	+	0.886212i	$0.653327\pi$
$132$	−3.31077	−0.288165
$133$	2.63562	0.228537
$134$	1.06744	0.0922124
$135$	0	0
$136$	−5.08134	−0.435721
$137$	18.3739	1.56979	0.784895	−	0.619629i	$-0.212717\pi$
$137$	18.3739	1.56979	0.784895	+	0.619629i	$0.212717\pi$
$138$	2.30754	0.196431
$139$	−5.98608	−0.507733	−0.253866	−	0.967239i	$-0.581702\pi$
$139$	−5.98608	−0.507733	−0.253866	+	0.967239i	$0.581702\pi$
$140$	0	0
$141$	−4.77414	−0.402055
$142$	−4.95787	−0.416055
$143$	−9.89295	−0.827290
$144$	−1.48774	−0.123978
$145$	0	0
$146$	−1.44534	−0.119617
$147$	0.627802	0.0517802
$148$	12.9060	1.06087
$149$	−6.99343	−0.572924	−0.286462	−	0.958092i	$-0.592479\pi$
$149$	−6.99343	−0.572924	−0.286462	+	0.958092i	$0.592479\pi$
$150$	0	0
$151$	−23.3270	−1.89833	−0.949164	−	0.314783i	$-0.898068\pi$
$151$	−23.3270	−1.89833	−0.949164	+	0.314783i	$0.898068\pi$
$152$	2.70220	0.219178
$153$	−3.36685	−0.272193
$154$	3.98298	0.320958
$155$	0	0
$156$	7.69932	0.616439
$157$	6.99961	0.558630	0.279315	−	0.960200i	$-0.409893\pi$
$157$	6.99961	0.558630	0.279315	+	0.960200i	$0.409893\pi$
$158$	−10.5756	−0.841347
$159$	−7.13899	−0.566158
$160$	0	0
$161$	6.81811	0.537342
$162$	−0.648610	−0.0509596
$163$	7.32590	0.573809	0.286904	−	0.957959i	$-0.407374\pi$
$163$	7.32590	0.573809	0.286904	+	0.957959i	$0.407374\pi$
$164$	−13.0596	−1.01979
$165$	0	0
$166$	4.94426	0.383749
$167$	−10.8149	−0.836882	−0.418441	−	0.908244i	$-0.637423\pi$
$167$	−10.8149	−0.836882	−0.418441	+	0.908244i	$0.637423\pi$
$168$	−7.46174	−0.575685
$169$	10.0065	0.769728
$170$	0	0
$171$	1.79045	0.136919
$172$	4.59979	0.350731
$173$	−8.58809	−0.652941	−0.326470	−	0.945207i	$-0.605859\pi$
$173$	−8.58809	−0.652941	−0.326470	+	0.945207i	$0.605859\pi$
$174$	−5.02005	−0.380569
$175$	0	0
$176$	−1.77936	−0.134125
$177$	−3.36295	−0.252775
$178$	−0.179863	−0.0134813
$179$	7.06513	0.528073	0.264036	−	0.964513i	$-0.414946\pi$
$179$	7.06513	0.528073	0.264036	+	0.964513i	$0.414946\pi$
$180$	0	0
$181$	−22.5251	−1.67427	−0.837137	−	0.546993i	$-0.815773\pi$
$181$	−22.5251	−1.67427	−0.837137	+	0.546993i	$0.815773\pi$
$182$	−9.26259	−0.686589
$183$	12.5335	0.926505
$184$	6.99035	0.515336
$185$	0	0
$186$	−6.66760	−0.488892
$187$	−4.02682	−0.294470
$188$	−6.00815	−0.438190
$189$	−13.5450	−0.985252
$190$	0	0
$191$	16.0033	1.15796	0.578981	−	0.815341i	$-0.303451\pi$
$191$	16.0033	1.15796	0.578981	+	0.815341i	$0.303451\pi$
$192$	−3.08731	−0.222807
$193$	15.9736	1.14981	0.574903	−	0.818221i	$-0.305040\pi$
$193$	15.9736	1.14981	0.574903	+	0.818221i	$0.305040\pi$
$194$	10.3096	0.740188
$195$	0	0
$196$	0.790075	0.0564339
$197$	−24.2137	−1.72516	−0.862579	−	0.505922i	$-0.831152\pi$
$197$	−24.2137	−1.72516	−0.862579	+	0.505922i	$0.831152\pi$
$198$	2.70576	0.192290
$199$	−12.3227	−0.873535	−0.436768	−	0.899574i	$-0.643877\pi$
$199$	−12.3227	−0.873535	−0.436768	+	0.899574i	$0.643877\pi$
$200$	0	0
$201$	−1.58474	−0.111779
$202$	−9.12899	−0.642313
$203$	−14.8328	−1.04106
$204$	3.13393	0.219419
$205$	0	0
$206$	−10.3622	−0.721972
$207$	4.63174	0.321928
$208$	4.13799	0.286918
$209$	2.14142	0.148125
$210$	0	0
$211$	26.5630	1.82867	0.914335	−	0.404959i	$-0.132714\pi$
$211$	26.5630	1.82867	0.914335	+	0.404959i	$0.132714\pi$
$212$	−8.98426	−0.617042
$213$	7.36055	0.504336
$214$	10.5197	0.719114
$215$	0	0
$216$	−13.8872	−0.944902
$217$	−19.7008	−1.33738
$218$	10.4715	0.709216
$219$	2.14577	0.144998
$220$	0	0
$221$	9.36453	0.629927
$222$	7.80139	0.523595
$223$	8.33195	0.557949	0.278974	−	0.960299i	$-0.410006\pi$
$223$	8.33195	0.557949	0.278974	+	0.960299i	$0.410006\pi$
$224$	−14.8798	−0.994200
$225$	0	0
$226$	−5.08400	−0.338183
$227$	−4.33870	−0.287970	−0.143985	−	0.989580i	$-0.545992\pi$
$227$	−4.33870	−0.287970	−0.143985	+	0.989580i	$0.545992\pi$
$228$	−1.66659	−0.110373
$229$	29.3533	1.93972	0.969859	−	0.243665i	$-0.0783499\pi$
$229$	29.3533	1.93972	0.969859	+	0.243665i	$0.0783499\pi$
$230$	0	0
$231$	−5.91321	−0.389061
$232$	−15.2075	−0.998423
$233$	13.0291	0.853563	0.426781	−	0.904355i	$-0.359647\pi$
$233$	13.0291	0.853563	0.426781	+	0.904355i	$0.359647\pi$
$234$	−6.29235	−0.411344
$235$	0	0
$236$	−4.23220	−0.275492
$237$	15.7007	1.01987
$238$	−3.77024	−0.244388
$239$	7.27224	0.470402	0.235201	−	0.971947i	$-0.424425\pi$
$239$	7.27224	0.470402	0.235201	+	0.971947i	$0.424425\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−5.13179	−0.329884
$243$	−15.0443	−0.965095
$244$	15.7732	1.00977
$245$	0	0
$246$	−7.89425	−0.503319
$247$	−4.97996	−0.316867
$248$	−20.1985	−1.28261
$249$	−7.34034	−0.465175
$250$	0	0
$251$	−27.8592	−1.75846	−0.879229	−	0.476400i	$-0.841941\pi$
$251$	−27.8592	−1.75846	−0.879229	+	0.476400i	$0.841941\pi$
$252$	−6.22201	−0.391950
$253$	5.53966	0.348275
$254$	−3.52353	−0.221086
$255$	0	0
$256$	−12.8034	−0.800212
$257$	15.9448	0.994607	0.497303	−	0.867577i	$-0.334324\pi$
$257$	15.9448	0.994607	0.497303	+	0.867577i	$0.334324\pi$
$258$	2.78047	0.173104
$259$	23.0509	1.43231
$260$	0	0
$261$	−10.0764	−0.623711
$262$	−8.06741	−0.498406
$263$	−0.603786	−0.0372310	−0.0186155	−	0.999827i	$-0.505926\pi$
$263$	−0.603786	−0.0372310	−0.0186155	+	0.999827i	$0.505926\pi$
$264$	−6.06260	−0.373127
$265$	0	0
$266$	2.00497	0.122933
$267$	0.267028	0.0163418
$268$	−1.99436	−0.121825
$269$	−23.5550	−1.43618	−0.718088	−	0.695952i	$-0.754982\pi$
$269$	−23.5550	−1.43618	−0.718088	+	0.695952i	$0.754982\pi$
$270$	0	0
$271$	−9.80993	−0.595911	−0.297955	−	0.954580i	$-0.596305\pi$
$271$	−9.80993	−0.595911	−0.297955	+	0.954580i	$0.596305\pi$
$272$	1.68432	0.102127
$273$	13.7514	0.832274
$274$	13.9774	0.844407
$275$	0	0
$276$	−4.31131	−0.259511
$277$	−7.76495	−0.466551	−0.233275	−	0.972411i	$-0.574944\pi$
$277$	−7.76495	−0.466551	−0.233275	+	0.972411i	$0.574944\pi$
$278$	−4.55374	−0.273115
$279$	−13.3834	−0.801241
$280$	0	0
$281$	−20.7632	−1.23863	−0.619314	−	0.785143i	$-0.712589\pi$
$281$	−20.7632	−1.23863	−0.619314	+	0.785143i	$0.712589\pi$
$282$	−3.63179	−0.216270
$283$	−8.45091	−0.502355	−0.251177	−	0.967941i	$-0.580818\pi$
$283$	−8.45091	−0.502355	−0.251177	+	0.967941i	$0.580818\pi$
$284$	9.26309	0.549663
$285$	0	0
$286$	−7.52578	−0.445009
$287$	−23.3252	−1.37685
$288$	−10.1083	−0.595637
$289$	−13.1883	−0.775781
$290$	0	0
$291$	−15.3059	−0.897246
$292$	2.70041	0.158030
$293$	13.4756	0.787253	0.393626	−	0.919271i	$-0.371220\pi$
$293$	13.4756	0.787253	0.393626	+	0.919271i	$0.371220\pi$
$294$	0.477582	0.0278532
$295$	0	0
$296$	23.6332	1.37365
$297$	−11.0052	−0.638586
$298$	−5.32005	−0.308182
$299$	−12.8827	−0.745026
$300$	0	0
$301$	8.21548	0.473533
$302$	−17.7454	−1.02113
$303$	13.5531	0.778604
$304$	−0.895705	−0.0513722
$305$	0	0
$306$	−2.56123	−0.146416
$307$	−9.67527	−0.552197	−0.276099	−	0.961129i	$-0.589042\pi$
$307$	−9.67527	−0.552197	−0.276099	+	0.961129i	$0.589042\pi$
$308$	−7.44165	−0.424028
$309$	15.3840	0.875165
$310$	0	0
$311$	−34.8734	−1.97749	−0.988746	−	0.149607i	$-0.952199\pi$
$311$	−34.8734	−1.97749	−0.988746	+	0.149607i	$0.952199\pi$
$312$	14.0988	0.798189
$313$	−27.2238	−1.53878	−0.769391	−	0.638779i	$-0.779440\pi$
$313$	−27.2238	−1.53878	−0.769391	+	0.638779i	$0.779440\pi$
$314$	5.32475	0.300493
$315$	0	0
$316$	19.7590	1.11153
$317$	−2.90326	−0.163063	−0.0815316	−	0.996671i	$-0.525981\pi$
$317$	−2.90326	−0.163063	−0.0815316	+	0.996671i	$0.525981\pi$
$318$	−5.43078	−0.304543
$319$	−12.0515	−0.674756
$320$	0	0
$321$	−15.6178	−0.871701
$322$	5.18668	0.289042
$323$	−2.02704	−0.112787
$324$	1.21184	0.0673244
$325$	0	0
$326$	5.57297	0.308658
$327$	−15.5461	−0.859703
$328$	−23.9145	−1.32046
$329$	−10.7309	−0.591614
$330$	0	0
$331$	4.82961	0.265459	0.132730	−	0.991152i	$-0.457626\pi$
$331$	4.82961	0.265459	0.132730	+	0.991152i	$0.457626\pi$
$332$	−9.23766	−0.506983
$333$	15.6591	0.858115
$334$	−8.22712	−0.450168
$335$	0	0
$336$	2.47336	0.134933
$337$	22.1454	1.20634	0.603168	−	0.797614i	$-0.293905\pi$
$337$	22.1454	1.20634	0.603168	+	0.797614i	$0.293905\pi$
$338$	7.61213	0.414045
$339$	7.54781	0.409941
$340$	0	0
$341$	−16.0068	−0.866816
$342$	1.36204	0.0736505
$343$	19.1808	1.03567
$344$	8.42303	0.454139
$345$	0	0
$346$	−6.53315	−0.351224
$347$	20.1874	1.08372	0.541859	−	0.840470i	$-0.317721\pi$
$347$	20.1874	1.08372	0.541859	+	0.840470i	$0.317721\pi$
$348$	9.37927	0.502782
$349$	4.22280	0.226041	0.113021	−	0.993593i	$-0.463947\pi$
$349$	4.22280	0.226041	0.113021	+	0.993593i	$0.463947\pi$
$350$	0	0
$351$	25.5930	1.36605
$352$	−12.0897	−0.644385
$353$	20.2277	1.07661	0.538305	−	0.842750i	$-0.319065\pi$
$353$	20.2277	1.07661	0.538305	+	0.842750i	$0.319065\pi$
$354$	−2.55826	−0.135970
$355$	0	0
$356$	0.336048	0.0178105
$357$	5.59737	0.296244
$358$	5.37460	0.284056
$359$	−23.0788	−1.21805	−0.609027	−	0.793149i	$-0.708440\pi$
$359$	−23.0788	−1.21805	−0.609027	+	0.793149i	$0.708440\pi$
$360$	0	0
$361$	−17.9220	−0.943265
$362$	−17.1353	−0.900611
$363$	7.61876	0.399881
$364$	17.3059	0.907074
$365$	0	0
$366$	9.53452	0.498377
$367$	−15.5656	−0.812516	−0.406258	−	0.913758i	$-0.633167\pi$
$367$	−15.5656	−0.812516	−0.406258	+	0.913758i	$0.633167\pi$
$368$	−2.31711	−0.120788
$369$	−15.8455	−0.824885
$370$	0	0
$371$	−16.0464	−0.833087
$372$	12.4575	0.645891
$373$	22.9059	1.18602	0.593011	−	0.805194i	$-0.297939\pi$
$373$	22.9059	1.18602	0.593011	+	0.805194i	$0.297939\pi$
$374$	−3.06329	−0.158399
$375$	0	0
$376$	−11.0020	−0.567385
$377$	28.0263	1.44343
$378$	−10.3040	−0.529978
$379$	−9.63127	−0.494725	−0.247363	−	0.968923i	$-0.579564\pi$
$379$	−9.63127	−0.494725	−0.247363	+	0.968923i	$0.579564\pi$
$380$	0	0
$381$	5.23110	0.267997
$382$	12.1741	0.622880
$383$	1.19113	0.0608640	0.0304320	−	0.999537i	$-0.490312\pi$
$383$	1.19113	0.0608640	0.0304320	+	0.999537i	$0.490312\pi$
$384$	10.8914	0.555799
$385$	0	0
$386$	12.1515	0.618494
$387$	5.58102	0.283699
$388$	−19.2621	−0.977885
$389$	−12.0151	−0.609188	−0.304594	−	0.952482i	$-0.598521\pi$
$389$	−12.0151	−0.609188	−0.304594	+	0.952482i	$0.598521\pi$
$390$	0	0
$391$	−5.24376	−0.265189
$392$	1.44677	0.0730728
$393$	11.9770	0.604161
$394$	−18.4199	−0.927982
$395$	0	0
$396$	−5.05533	−0.254040
$397$	−1.30174	−0.0653324	−0.0326662	−	0.999466i	$-0.510400\pi$
$397$	−1.30174	−0.0653324	−0.0326662	+	0.999466i	$0.510400\pi$
$398$	−9.37416	−0.469884
$399$	−2.97662	−0.149017
$400$	0	0
$401$	17.7958	0.888680	0.444340	−	0.895858i	$-0.353438\pi$
$401$	17.7958	0.888680	0.444340	+	0.895858i	$0.353438\pi$
$402$	−1.20554	−0.0601270
$403$	37.2244	1.85428
$404$	17.0563	0.848580
$405$	0	0
$406$	−11.2836	−0.559997
$407$	18.7286	0.928345
$408$	5.73877	0.284112
$409$	1.21607	0.0601305	0.0300653	−	0.999548i	$-0.490428\pi$
$409$	1.21607	0.0601305	0.0300653	+	0.999548i	$0.490428\pi$
$410$	0	0
$411$	−20.7512	−1.02358
$412$	19.3604	0.953820
$413$	−7.55894	−0.371951
$414$	3.52346	0.173169
$415$	0	0
$416$	28.1152	1.37846
$417$	6.76057	0.331067
$418$	1.62902	0.0796781
$419$	−23.5246	−1.14925	−0.574626	−	0.818416i	$-0.694853\pi$
$419$	−23.5246	−1.14925	−0.574626	+	0.818416i	$0.694853\pi$
$420$	0	0
$421$	24.1129	1.17519	0.587596	−	0.809154i	$-0.300074\pi$
$421$	24.1129	1.17519	0.587596	+	0.809154i	$0.300074\pi$
$422$	20.2070	0.983662
$423$	−7.28982	−0.354443
$424$	−16.4518	−0.798969
$425$	0	0
$426$	5.59933	0.271288
$427$	28.1718	1.36333
$428$	−19.6547	−0.950045
$429$	11.1729	0.539434
$430$	0	0
$431$	9.23057	0.444621	0.222310	−	0.974976i	$-0.428640\pi$
$431$	9.23057	0.444621	0.222310	+	0.974976i	$0.428640\pi$
$432$	4.60321	0.221472
$433$	40.5437	1.94841	0.974203	−	0.225673i	$-0.0724582\pi$
$433$	40.5437	1.94841	0.974203	+	0.225673i	$0.0724582\pi$
$434$	−14.9869	−0.719392
$435$	0	0
$436$	−19.5645	−0.936968
$437$	2.78858	0.133396
$438$	1.63234	0.0779960
$439$	19.5448	0.932823	0.466411	−	0.884568i	$-0.345547\pi$
$439$	19.5448	0.932823	0.466411	+	0.884568i	$0.345547\pi$
$440$	0	0
$441$	0.958614	0.0456483
$442$	7.12380	0.338845
$443$	−38.3533	−1.82222	−0.911109	−	0.412165i	$-0.864773\pi$
$443$	−38.3533	−1.82222	−0.911109	+	0.412165i	$0.864773\pi$
$444$	−14.5758	−0.691738
$445$	0	0
$446$	6.33829	0.300127
$447$	7.89825	0.373574
$448$	−6.93938	−0.327855
$449$	−7.15454	−0.337643	−0.168822	−	0.985647i	$-0.553996\pi$
$449$	−7.15454	−0.337643	−0.168822	+	0.985647i	$0.553996\pi$
$450$	0	0
$451$	−18.9516	−0.892395
$452$	9.49876	0.446784
$453$	26.3451	1.23780
$454$	−3.30054	−0.154902
$455$	0	0
$456$	−3.05182	−0.142915
$457$	13.2079	0.617838	0.308919	−	0.951088i	$-0.400033\pi$
$457$	13.2079	0.617838	0.308919	+	0.951088i	$0.400033\pi$
$458$	22.3297	1.04340
$459$	10.4174	0.486241
$460$	0	0
$461$	25.6149	1.19300	0.596502	−	0.802612i	$-0.296557\pi$
$461$	25.6149	1.19300	0.596502	+	0.802612i	$0.296557\pi$
$462$	−4.49831	−0.209280
$463$	−36.7408	−1.70749	−0.853745	−	0.520691i	$-0.825674\pi$
$463$	−36.7408	−1.70749	−0.853745	+	0.520691i	$0.825674\pi$
$464$	5.04087	0.234017
$465$	0	0
$466$	9.91149	0.459141
$467$	11.5404	0.534028	0.267014	−	0.963693i	$-0.413963\pi$
$467$	11.5404	0.534028	0.267014	+	0.963693i	$0.413963\pi$
$468$	11.7564	0.543439
$469$	−3.56203	−0.164479
$470$	0	0
$471$	−7.90524	−0.364254
$472$	−7.74990	−0.356718
$473$	6.67502	0.306918
$474$	11.9439	0.548600
$475$	0	0
$476$	7.04417	0.322869
$477$	−10.9008	−0.499113
$478$	5.53215	0.253035
$479$	29.9925	1.37039	0.685197	−	0.728358i	$-0.259716\pi$
$479$	29.9925	1.37039	0.685197	+	0.728358i	$0.259716\pi$
$480$	0	0
$481$	−43.5542	−1.98590
$482$	0.760721	0.0346499
$483$	−7.70025	−0.350373
$484$	9.58805	0.435820
$485$	0	0
$486$	−11.4446	−0.519135
$487$	−8.15043	−0.369331	−0.184666	−	0.982801i	$-0.559120\pi$
$487$	−8.15043	−0.369331	−0.184666	+	0.982801i	$0.559120\pi$
$488$	28.8835	1.30749
$489$	−8.27374	−0.374152
$490$	0	0
$491$	−7.89709	−0.356391	−0.178195	−	0.983995i	$-0.557026\pi$
$491$	−7.89709	−0.356391	−0.178195	+	0.983995i	$0.557026\pi$
$492$	14.7493	0.664951
$493$	11.4078	0.513782
$494$	−3.78836	−0.170446
$495$	0	0
$496$	6.69526	0.300626
$497$	16.5444	0.742118
$498$	−5.58396	−0.250223
$499$	−22.4351	−1.00433	−0.502166	−	0.864771i	$-0.667463\pi$
$499$	−22.4351	−1.00433	−0.502166	+	0.864771i	$0.667463\pi$
$500$	0	0
$501$	12.2142	0.545688
$502$	−21.1931	−0.945894
$503$	−7.14129	−0.318415	−0.159207	−	0.987245i	$-0.550894\pi$
$503$	−7.14129	−0.318415	−0.159207	+	0.987245i	$0.550894\pi$
$504$	−11.3936	−0.507511
$505$	0	0
$506$	4.21414	0.187341
$507$	−11.3011	−0.501900
$508$	6.58323	0.292084
$509$	−12.5554	−0.556506	−0.278253	−	0.960508i	$-0.589755\pi$
$509$	−12.5554	−0.556506	−0.278253	+	0.960508i	$0.589755\pi$
$510$	0	0
$511$	4.82308	0.213361
$512$	9.54752	0.421945
$513$	−5.53984	−0.244590
$514$	12.1295	0.535010
$515$	0	0
$516$	−5.19492	−0.228694
$517$	−8.71877	−0.383451
$518$	17.5353	0.770456
$519$	9.69924	0.425749
$520$	0	0
$521$	4.94447	0.216621	0.108310	−	0.994117i	$-0.465456\pi$
$521$	4.94447	0.216621	0.108310	+	0.994117i	$0.465456\pi$
$522$	−7.66530	−0.335501
$523$	1.07988	0.0472200	0.0236100	−	0.999721i	$-0.492484\pi$
$523$	1.07988	0.0472200	0.0236100	+	0.999721i	$0.492484\pi$
$524$	15.0728	0.658460
$525$	0	0
$526$	−0.459313	−0.0200270
$527$	15.1518	0.660023
$528$	2.00958	0.0874559
$529$	−15.7862	−0.686357
$530$	0	0
$531$	−5.13501	−0.222840
$532$	−3.74601	−0.162410
$533$	44.0727	1.90900
$534$	0.203134	0.00879045
$535$	0	0
$536$	−3.65202	−0.157743
$537$	−7.97923	−0.344329
$538$	−17.9188	−0.772535
$539$	1.14652	0.0493842
$540$	0	0
$541$	−15.7184	−0.675785	−0.337892	−	0.941185i	$-0.609714\pi$
$541$	−15.7184	−0.675785	−0.337892	+	0.941185i	$0.609714\pi$
$542$	−7.46262	−0.320547
$543$	25.4394	1.09171
$544$	11.4440	0.490657
$545$	0	0
$546$	10.4610	0.447689
$547$	−18.6002	−0.795286	−0.397643	−	0.917540i	$-0.630172\pi$
$547$	−18.6002	−0.795286	−0.397643	+	0.917540i	$0.630172\pi$
$548$	−26.1149	−1.11557
$549$	19.1379	0.816786
$550$	0	0
$551$	−6.06656	−0.258444
$552$	−7.89478	−0.336024
$553$	35.2907	1.50071
$554$	−5.90697	−0.250963
$555$	0	0
$556$	8.50803	0.360821
$557$	−1.59319	−0.0675055	−0.0337528	−	0.999430i	$-0.510746\pi$
$557$	−1.59319	−0.0675055	−0.0337528	+	0.999430i	$0.510746\pi$
$558$	−10.1810	−0.430996
$559$	−15.5230	−0.656554
$560$	0	0
$561$	4.54782	0.192009
$562$	−15.7950	−0.666272
$563$	−30.0999	−1.26856	−0.634280	−	0.773103i	$-0.718703\pi$
$563$	−30.0999	−1.26856	−0.634280	+	0.773103i	$0.718703\pi$
$564$	6.78550	0.285721
$565$	0	0
$566$	−6.42879	−0.270222
$567$	2.16441	0.0908968
$568$	16.9624	0.711725
$569$	−1.17755	−0.0493656	−0.0246828	−	0.999695i	$-0.507858\pi$
$569$	−1.17755	−0.0493656	−0.0246828	+	0.999695i	$0.507858\pi$
$570$	0	0
$571$	−9.39221	−0.393052	−0.196526	−	0.980499i	$-0.562966\pi$
$571$	−9.39221	−0.393052	−0.196526	+	0.980499i	$0.562966\pi$
$572$	14.0609	0.587915
$573$	−18.0739	−0.755048
$574$	−17.7440	−0.740621
$575$	0	0
$576$	−4.71413	−0.196422
$577$	−38.4047	−1.59881	−0.799405	−	0.600792i	$-0.794852\pi$
$577$	−38.4047	−1.59881	−0.799405	+	0.600792i	$0.794852\pi$
$578$	−10.0326	−0.417301
$579$	−18.0403	−0.749730
$580$	0	0
$581$	−16.4990	−0.684493
$582$	−11.6435	−0.482639
$583$	−13.0376	−0.539961
$584$	4.94493	0.204623
$585$	0	0
$586$	10.2512	0.423472
$587$	−32.9101	−1.35834	−0.679172	−	0.733979i	$-0.737661\pi$
$587$	−32.9101	−1.35834	−0.679172	+	0.733979i	$0.737661\pi$
$588$	−0.892297	−0.0367977
$589$	−8.05757	−0.332006
$590$	0	0
$591$	27.3466	1.12489
$592$	−7.83375	−0.321965
$593$	−25.5785	−1.05038	−0.525191	−	0.850984i	$-0.676006\pi$
$593$	−25.5785	−1.05038	−0.525191	+	0.850984i	$0.676006\pi$
$594$	−8.37188	−0.343502
$595$	0	0
$596$	9.93978	0.407149
$597$	13.9171	0.569588
$598$	−9.80015	−0.400758
$599$	−0.739459	−0.0302135	−0.0151067	−	0.999886i	$-0.504809\pi$
$599$	−0.739459	−0.0302135	−0.0151067	+	0.999886i	$0.504809\pi$
$600$	0	0
$601$	−12.2755	−0.500730	−0.250365	−	0.968152i	$-0.580551\pi$
$601$	−12.2755	−0.500730	−0.250365	+	0.968152i	$0.580551\pi$
$602$	6.24969	0.254718
$603$	−2.41979	−0.0985417
$604$	33.1548	1.34905
$605$	0	0
$606$	10.3101	0.418820
$607$	13.7567	0.558368	0.279184	−	0.960238i	$-0.409936\pi$
$607$	13.7567	0.558368	0.279184	+	0.960238i	$0.409936\pi$
$608$	−6.08579	−0.246811
$609$	16.7519	0.678821
$610$	0	0
$611$	20.2759	0.820273
$612$	4.78531	0.193435
$613$	−32.1681	−1.29926	−0.649629	−	0.760251i	$-0.725076\pi$
$613$	−32.1681	−1.29926	−0.649629	+	0.760251i	$0.725076\pi$
$614$	−7.36019	−0.297033
$615$	0	0
$616$	−13.6270	−0.549047
$617$	29.1316	1.17279	0.586397	−	0.810024i	$-0.300546\pi$
$617$	29.1316	1.17279	0.586397	+	0.810024i	$0.300546\pi$
$618$	11.7029	0.470761
$619$	4.25398	0.170982	0.0854909	−	0.996339i	$-0.472754\pi$
$619$	4.25398	0.170982	0.0854909	+	0.996339i	$0.472754\pi$
$620$	0	0
$621$	−14.3311	−0.575086
$622$	−26.5290	−1.06371
$623$	0.600201	0.0240466
$624$	−4.67337	−0.187084
$625$	0	0
$626$	−20.7097	−0.827728
$627$	−2.41848	−0.0965848
$628$	−9.94857	−0.396991
$629$	−17.7283	−0.706873
$630$	0	0
$631$	8.94679	0.356166	0.178083	−	0.984015i	$-0.443010\pi$
$631$	8.94679	0.356166	0.178083	+	0.984015i	$0.443010\pi$
$632$	36.1822	1.43925
$633$	−29.9997	−1.19238
$634$	−2.20857	−0.0877135
$635$	0	0
$636$	10.1467	0.402341
$637$	−2.66629	−0.105642
$638$	−9.16786	−0.362959
$639$	11.2391	0.444612
$640$	0	0
$641$	11.6413	0.459803	0.229902	−	0.973214i	$-0.426160\pi$
$641$	11.6413	0.459803	0.229902	+	0.973214i	$0.426160\pi$
$642$	−11.8808	−0.468898
$643$	−31.5371	−1.24370	−0.621851	−	0.783136i	$-0.713619\pi$
$643$	−31.5371	−1.24370	−0.621851	+	0.783136i	$0.713619\pi$
$644$	−9.69060	−0.381863
$645$	0	0
$646$	−1.54201	−0.0606696
$647$	7.63153	0.300027	0.150013	−	0.988684i	$-0.452068\pi$
$647$	7.63153	0.300027	0.150013	+	0.988684i	$0.452068\pi$
$648$	2.21909	0.0871742
$649$	−6.14158	−0.241078
$650$	0	0
$651$	22.2498	0.872038
$652$	−10.4123	−0.407778
$653$	25.3578	0.992328	0.496164	−	0.868229i	$-0.334741\pi$
$653$	25.3578	0.992328	0.496164	+	0.868229i	$0.334741\pi$
$654$	−11.8263	−0.462444
$655$	0	0
$656$	7.92699	0.309497
$657$	3.27646	0.127827
$658$	−8.16322	−0.318236
$659$	−39.0160	−1.51985	−0.759924	−	0.650011i	$-0.774764\pi$
$659$	−39.0160	−1.51985	−0.759924	+	0.650011i	$0.774764\pi$
$660$	0	0
$661$	48.6676	1.89295	0.946474	−	0.322779i	$-0.104617\pi$
$661$	48.6676	1.89295	0.946474	+	0.322779i	$0.104617\pi$
$662$	3.67399	0.142794
$663$	−10.5761	−0.410743
$664$	−16.9158	−0.656460
$665$	0	0
$666$	11.9122	0.461590
$667$	−15.6936	−0.607660
$668$	15.3713	0.594732
$669$	−9.40995	−0.363810
$670$	0	0
$671$	22.8893	0.883633
$672$	16.8050	0.648267
$673$	20.8488	0.803664	0.401832	−	0.915713i	$-0.368374\pi$
$673$	20.8488	0.803664	0.401832	+	0.915713i	$0.368374\pi$
$674$	16.8465	0.648902
$675$	0	0
$676$	−14.2222	−0.547008
$677$	−30.6907	−1.17954	−0.589769	−	0.807572i	$-0.700781\pi$
$677$	−30.6907	−1.17954	−0.589769	+	0.807572i	$0.700781\pi$
$678$	5.74178	0.220512
$679$	−34.4032	−1.32027
$680$	0	0
$681$	4.90005	0.187770
$682$	−12.1767	−0.466270
$683$	−22.3276	−0.854344	−0.427172	−	0.904170i	$-0.640490\pi$
$683$	−22.3276	−0.854344	−0.427172	+	0.904170i	$0.640490\pi$
$684$	−2.54478	−0.0973020
$685$	0	0
$686$	14.5912	0.557096
$687$	−33.1511	−1.26479
$688$	−2.79200	−0.106444
$689$	30.3194	1.15508
$690$	0	0
$691$	−27.0489	−1.02899	−0.514495	−	0.857493i	$-0.672021\pi$
$691$	−27.0489	−1.02899	−0.514495	+	0.857493i	$0.672021\pi$
$692$	12.2063	0.464013
$693$	−9.02911	−0.342987
$694$	15.3570	0.582944
$695$	0	0
$696$	17.1751	0.651021
$697$	17.9393	0.679499
$698$	3.21238	0.121590
$699$	−14.7148	−0.556565
$700$	0	0
$701$	−39.0100	−1.47339	−0.736693	−	0.676227i	$-0.763614\pi$
$701$	−39.0100	−1.47339	−0.736693	+	0.676227i	$0.763614\pi$
$702$	19.4691	0.734816
$703$	9.42771	0.355573
$704$	−5.63819	−0.212497
$705$	0	0
$706$	15.3876	0.579121
$707$	30.4634	1.14570
$708$	4.77977	0.179635
$709$	−12.6642	−0.475612	−0.237806	−	0.971313i	$-0.576428\pi$
$709$	−12.6642	−0.475612	−0.237806	+	0.971313i	$0.576428\pi$
$710$	0	0
$711$	23.9740	0.899095
$712$	0.615364	0.0230618
$713$	−20.8442	−0.780621
$714$	4.25804	0.159353
$715$	0	0
$716$	−10.0417	−0.375276
$717$	−8.21314	−0.306725
$718$	−17.5566	−0.655205
$719$	18.1316	0.676193	0.338096	−	0.941111i	$-0.390217\pi$
$719$	18.1316	0.676193	0.338096	+	0.941111i	$0.390217\pi$
$720$	0	0
$721$	34.5788	1.28778
$722$	−13.6337	−0.507393
$723$	−1.12938	−0.0420022
$724$	32.0149	1.18983
$725$	0	0
$726$	5.79576	0.215101
$727$	−21.4455	−0.795368	−0.397684	−	0.917522i	$-0.630186\pi$
$727$	−21.4455	−0.795368	−0.397684	+	0.917522i	$0.630186\pi$
$728$	31.6901	1.17451
$729$	19.5487	0.724026
$730$	0	0
$731$	−6.31848	−0.233697
$732$	−17.8139	−0.658422
$733$	0.379985	0.0140350	0.00701752	−	0.999975i	$-0.497766\pi$
$733$	0.379985	0.0140350	0.00701752	+	0.999975i	$0.497766\pi$
$734$	−11.8411	−0.437061
$735$	0	0
$736$	−15.7434	−0.580309
$737$	−2.89412	−0.106606
$738$	−12.0540	−0.443715
$739$	42.6584	1.56921	0.784607	−	0.619993i	$-0.212865\pi$
$739$	42.6584	1.56921	0.784607	+	0.619993i	$0.212865\pi$
$740$	0	0
$741$	5.62428	0.206613
$742$	−12.2068	−0.448127
$743$	−17.0856	−0.626810	−0.313405	−	0.949620i	$-0.601470\pi$
$743$	−17.0856	−0.626810	−0.313405	+	0.949620i	$0.601470\pi$
$744$	22.8119	0.836324
$745$	0	0
$746$	17.4250	0.637975
$747$	−11.2082	−0.410088
$748$	5.72333	0.209266
$749$	−35.1044	−1.28269
$750$	0	0
$751$	−33.7135	−1.23022	−0.615111	−	0.788441i	$-0.710889\pi$
$751$	−33.7135	−1.23022	−0.615111	+	0.788441i	$0.710889\pi$
$752$	3.64685	0.132987
$753$	31.4637	1.14660
$754$	21.3202	0.776437
$755$	0	0
$756$	19.2515	0.700171
$757$	−16.7814	−0.609932	−0.304966	−	0.952363i	$-0.598645\pi$
$757$	−16.7814	−0.609932	−0.304966	+	0.952363i	$0.598645\pi$
$758$	−7.32671	−0.266118
$759$	−6.25639	−0.227093
$760$	0	0
$761$	−29.8307	−1.08136	−0.540681	−	0.841227i	$-0.681834\pi$
$761$	−29.8307	−1.08136	−0.540681	+	0.841227i	$0.681834\pi$
$762$	3.97941	0.144159
$763$	−34.9432	−1.26503
$764$	−22.7456	−0.822907
$765$	0	0
$766$	0.906120	0.0327394
$767$	14.2825	0.515711
$768$	14.4599	0.521778
$769$	7.89994	0.284879	0.142440	−	0.989803i	$-0.454505\pi$
$769$	7.89994	0.284879	0.142440	+	0.989803i	$0.454505\pi$
$770$	0	0
$771$	−18.0077	−0.648532
$772$	−22.7034	−0.817112
$773$	25.4969	0.917061	0.458530	−	0.888679i	$-0.348376\pi$
$773$	25.4969	0.917061	0.458530	+	0.888679i	$0.348376\pi$
$774$	4.24560	0.152605
$775$	0	0
$776$	−35.2723	−1.26620
$777$	−26.0332	−0.933937
$778$	−9.14012	−0.327689
$779$	−9.53993	−0.341803
$780$	0	0
$781$	13.4422	0.480999
$782$	−3.98904	−0.142648
$783$	31.1773	1.11418
$784$	−0.479563	−0.0171273
$785$	0	0
$786$	9.11119	0.324985
$787$	49.0412	1.74813	0.874065	−	0.485809i	$-0.161475\pi$
$787$	49.0412	1.74813	0.874065	+	0.485809i	$0.161475\pi$
$788$	34.4151	1.22599
$789$	0.681905	0.0242764
$790$	0	0
$791$	16.9653	0.603217
$792$	−9.25721	−0.328941
$793$	−53.2301	−1.89026
$794$	−0.990260	−0.0351430
$795$	0	0
$796$	17.5143	0.620779
$797$	1.15733	0.0409948	0.0204974	−	0.999790i	$-0.493475\pi$
$797$	1.15733	0.0409948	0.0204974	+	0.999790i	$0.493475\pi$
$798$	−2.26438	−0.0801581
$799$	8.25307	0.291973
$800$	0	0
$801$	0.407734	0.0144066
$802$	13.5377	0.478031
$803$	3.91872	0.138289
$804$	2.25239	0.0794357
$805$	0	0
$806$	28.3174	0.997438
$807$	26.6026	0.936457
$808$	31.2330	1.09877
$809$	−31.6665	−1.11334	−0.556668	−	0.830735i	$-0.687920\pi$
$809$	−31.6665	−1.11334	−0.556668	+	0.830735i	$0.687920\pi$
$810$	0	0
$811$	36.0360	1.26540	0.632698	−	0.774398i	$-0.281947\pi$
$811$	36.0360	1.26540	0.632698	+	0.774398i	$0.281947\pi$
$812$	21.0819	0.739830
$813$	11.0792	0.388563
$814$	14.2473	0.499367
$815$	0	0
$816$	−1.90224	−0.0665918
$817$	3.36010	0.117555
$818$	0.925087	0.0323449
$819$	20.9976	0.733714
$820$	0	0
$821$	30.1203	1.05121	0.525603	−	0.850730i	$-0.323840\pi$
$821$	30.1203	1.05121	0.525603	+	0.850730i	$0.323840\pi$
$822$	−15.7859	−0.550595
$823$	−21.5975	−0.752843	−0.376421	−	0.926449i	$-0.622845\pi$
$823$	−21.5975	−0.752843	−0.376421	+	0.926449i	$0.622845\pi$
$824$	35.4524	1.23504
$825$	0	0
$826$	−5.75025	−0.200077
$827$	−2.37687	−0.0826519	−0.0413259	−	0.999146i	$-0.513158\pi$
$827$	−2.37687	−0.0826519	−0.0413259	+	0.999146i	$0.513158\pi$
$828$	−6.58311	−0.228779
$829$	−20.4204	−0.709230	−0.354615	−	0.935012i	$-0.615388\pi$
$829$	−20.4204	−0.709230	−0.354615	+	0.935012i	$0.615388\pi$
$830$	0	0
$831$	8.76960	0.304214
$832$	13.1118	0.454571
$833$	−1.08528	−0.0376028
$834$	5.14291	0.178084
$835$	0	0
$836$	−3.04361	−0.105265
$837$	41.4094	1.43132
$838$	−17.8957	−0.618196
$839$	2.84185	0.0981115	0.0490557	−	0.998796i	$-0.484379\pi$
$839$	2.84185	0.0981115	0.0490557	+	0.998796i	$0.484379\pi$
$840$	0	0
$841$	5.14153	0.177294
$842$	18.3432	0.632149
$843$	23.4496	0.807646
$844$	−37.7540	−1.29955
$845$	0	0
$846$	−5.54552	−0.190659
$847$	17.1248	0.588415
$848$	5.45330	0.187267
$849$	9.54431	0.327560
$850$	0	0
$851$	24.3886	0.836032
$852$	−10.4616	−0.358407
$853$	−13.8008	−0.472530	−0.236265	−	0.971689i	$-0.575923\pi$
$853$	−13.8008	−0.472530	−0.236265	+	0.971689i	$0.575923\pi$
$854$	21.4309	0.733349
$855$	0	0
$856$	−35.9912	−1.23015
$857$	2.06240	0.0704502	0.0352251	−	0.999379i	$-0.488785\pi$
$857$	2.06240	0.0704502	0.0352251	+	0.999379i	$0.488785\pi$
$858$	8.49948	0.290167
$859$	−22.7067	−0.774742	−0.387371	−	0.921924i	$-0.626617\pi$
$859$	−22.7067	−0.774742	−0.387371	+	0.921924i	$0.626617\pi$
$860$	0	0
$861$	26.3431	0.897771
$862$	7.02189	0.239166
$863$	25.2543	0.859667	0.429833	−	0.902908i	$-0.358572\pi$
$863$	25.2543	0.859667	0.429833	+	0.902908i	$0.358572\pi$
$864$	31.2761	1.06403
$865$	0	0
$866$	30.8425	1.04807
$867$	14.8946	0.505847
$868$	28.0009	0.950411
$869$	28.6734	0.972678
$870$	0	0
$871$	6.73041	0.228051
$872$	−35.8260	−1.21322
$873$	−23.3711	−0.790992
$874$	2.12133	0.0717551
$875$	0	0
$876$	−3.04980	−0.103043
$877$	23.5255	0.794401	0.397200	−	0.917732i	$-0.369982\pi$
$877$	23.5255	0.794401	0.397200	+	0.917732i	$0.369982\pi$
$878$	14.8681	0.501776
$879$	−15.2191	−0.513327
$880$	0	0
$881$	−23.8946	−0.805029	−0.402515	−	0.915414i	$-0.631864\pi$
$881$	−23.8946	−0.805029	−0.402515	+	0.915414i	$0.631864\pi$
$882$	0.729238	0.0245547
$883$	12.9066	0.434342	0.217171	−	0.976134i	$-0.430317\pi$
$883$	12.9066	0.434342	0.217171	+	0.976134i	$0.430317\pi$
$884$	−13.3098	−0.447658
$885$	0	0
$886$	−29.1761	−0.980192
$887$	−26.6396	−0.894471	−0.447235	−	0.894416i	$-0.647591\pi$
$887$	−26.6396	−0.894471	−0.447235	+	0.894416i	$0.647591\pi$
$888$	−26.6909	−0.895689
$889$	11.7580	0.394351
$890$	0	0
$891$	1.75857	0.0589143
$892$	−11.8422	−0.396507
$893$	−4.38889	−0.146869
$894$	6.00837	0.200950
$895$	0	0
$896$	24.4807	0.817843
$897$	14.5495	0.485793
$898$	−5.44261	−0.181622
$899$	45.3466	1.51239
$900$	0	0
$901$	12.3412	0.411144
$902$	−14.4169	−0.480029
$903$	−9.27842	−0.308766
$904$	17.3939	0.578513
$905$	0	0
$906$	20.0413	0.665828
$907$	21.2193	0.704574	0.352287	−	0.935892i	$-0.385404\pi$
$907$	21.2193	0.704574	0.352287	+	0.935892i	$0.385404\pi$
$908$	6.16661	0.204646
$909$	20.6947	0.686400
$910$	0	0
$911$	−6.50032	−0.215365	−0.107683	−	0.994185i	$-0.534343\pi$
$911$	−6.50032	−0.215365	−0.107683	+	0.994185i	$0.534343\pi$
$912$	1.01159	0.0334972
$913$	−13.4053	−0.443650
$914$	10.0475	0.332342
$915$	0	0
$916$	−41.7199	−1.37846
$917$	26.9209	0.889008
$918$	7.92471	0.261554
$919$	19.1428	0.631461	0.315731	−	0.948849i	$-0.397750\pi$
$919$	19.1428	0.631461	0.315731	+	0.948849i	$0.397750\pi$
$920$	0	0
$921$	10.9271	0.360060
$922$	19.4858	0.641730
$923$	−31.2604	−1.02895
$924$	8.40447	0.276487
$925$	0	0
$926$	−27.9495	−0.918478
$927$	23.4904	0.771526
$928$	34.2498	1.12430
$929$	−58.3414	−1.91412	−0.957059	−	0.289892i	$-0.906381\pi$
$929$	−58.3414	−1.91412	−0.957059	+	0.289892i	$0.906381\pi$
$930$	0	0
$931$	0.577142	0.0189151
$932$	−18.5183	−0.606586
$933$	39.3854	1.28942
$934$	8.77906	0.287260
$935$	0	0
$936$	21.5280	0.703666
$937$	−15.6711	−0.511952	−0.255976	−	0.966683i	$-0.582397\pi$
$937$	−15.6711	−0.511952	−0.255976	+	0.966683i	$0.582397\pi$
$938$	−2.70972	−0.0884753
$939$	30.7461	1.00336
$940$	0	0
$941$	12.8470	0.418799	0.209399	−	0.977830i	$-0.432849\pi$
$941$	12.8470	0.418799	0.209399	+	0.977830i	$0.432849\pi$
$942$	−6.01368	−0.195936
$943$	−24.6789	−0.803657
$944$	2.56888	0.0836098
$945$	0	0
$946$	5.07783	0.165094
$947$	−22.7709	−0.739954	−0.369977	−	0.929041i	$-0.620634\pi$
$947$	−22.7709	−0.739954	−0.369977	+	0.929041i	$0.620634\pi$
$948$	−22.3155	−0.724772
$949$	−9.11314	−0.295825
$950$	0	0
$951$	3.27889	0.106325
$952$	12.8991	0.418063
$953$	23.0127	0.745453	0.372727	−	0.927941i	$-0.378423\pi$
$953$	23.0127	0.745453	0.372727	+	0.927941i	$0.378423\pi$
$954$	−8.29246	−0.268478
$955$	0	0
$956$	−10.3361	−0.334292
$957$	13.6108	0.439974
$958$	22.8160	0.737150
$959$	−46.6427	−1.50617
$960$	0	0
$961$	29.2291	0.942873
$962$	−33.1326	−1.06824
$963$	−23.8474	−0.768472
$964$	−1.42130	−0.0457771
$965$	0	0
$966$	−5.85774	−0.188470
$967$	8.23926	0.264957	0.132478	−	0.991186i	$-0.457706\pi$
$967$	8.23926	0.264957	0.132478	+	0.991186i	$0.457706\pi$
$968$	17.5574	0.564317
$969$	2.28930	0.0735429
$970$	0	0
$971$	−42.6419	−1.36845	−0.684223	−	0.729273i	$-0.739858\pi$
$971$	−42.6419	−1.36845	−0.684223	+	0.729273i	$0.739858\pi$
$972$	21.3826	0.685846
$973$	15.1958	0.487156
$974$	−6.20021	−0.198667
$975$	0	0
$976$	−9.57407	−0.306458
$977$	−5.17109	−0.165438	−0.0827189	−	0.996573i	$-0.526360\pi$
$977$	−5.17109	−0.165438	−0.0827189	+	0.996573i	$0.526360\pi$
$978$	−6.29401	−0.201260
$979$	0.487659	0.0155856
$980$	0	0
$981$	−23.7380	−0.757895
$982$	−6.00749	−0.191707
$983$	−20.8634	−0.665438	−0.332719	−	0.943026i	$-0.607966\pi$
$983$	−20.8634	−0.665438	−0.332719	+	0.943026i	$0.607966\pi$
$984$	27.0086	0.861003
$985$	0	0
$986$	8.67817	0.276369
$987$	12.1193	0.385761
$988$	7.07803	0.225182
$989$	8.69227	0.276398
$990$	0	0
$991$	16.2535	0.516311	0.258155	−	0.966103i	$-0.416885\pi$
$991$	16.2535	0.516311	0.258155	+	0.966103i	$0.416885\pi$
$992$	45.4903	1.44432
$993$	−5.45448	−0.173093
$994$	12.5857	0.399193
$995$	0	0
$996$	10.4329	0.330578
$997$	−27.4154	−0.868254	−0.434127	−	0.900852i	$-0.642943\pi$
$997$	−27.4154	−0.868254	−0.434127	+	0.900852i	$0.642943\pi$
$998$	−17.0668	−0.540241
$999$	−48.4509	−1.53292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.29		46
5.2	odd	4		1205.2.b.c.724.29	yes	46
5.3	odd	4		1205.2.b.c.724.18	✓	46
5.4	even	2	inner	6025.2.a.p.1.18		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.18	✓	46	5.3	odd	4
1205.2.b.c.724.29	yes	46	5.2	odd	4
6025.2.a.p.1.18		46	5.4	even	2	inner
6025.2.a.p.1.29		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)

\(f(q)\)	\(=\)	\(q+0.760721 q^{2} -1.12938 q^{3} -1.42130 q^{4} -0.859145 q^{6} -2.53853 q^{7} -2.60266 q^{8} -1.72450 q^{9} +O(q^{10})\)	\(q+0.760721 q^{2} -1.12938 q^{3} -1.42130 q^{4} -0.859145 q^{6} -2.53853 q^{7} -2.60266 q^{8} -1.72450 q^{9} -2.06253 q^{11} +1.60519 q^{12} +4.79651 q^{13} -1.93111 q^{14} +0.862709 q^{16} +1.95237 q^{17} -1.31186 q^{18} -1.03825 q^{19} +2.86697 q^{21} -1.56901 q^{22} -2.68585 q^{23} +2.93940 q^{24} +3.64880 q^{26} +5.33576 q^{27} +3.60802 q^{28} +5.84308 q^{29} +7.76074 q^{31} +5.86160 q^{32} +2.32939 q^{33} +1.48521 q^{34} +2.45103 q^{36} -9.08041 q^{37} -0.789817 q^{38} -5.41709 q^{39} +9.18849 q^{41} +2.18096 q^{42} -3.23632 q^{43} +2.93148 q^{44} -2.04318 q^{46} +4.22721 q^{47} -0.974328 q^{48} -0.555881 q^{49} -2.20497 q^{51} -6.81729 q^{52} +6.32114 q^{53} +4.05903 q^{54} +6.60692 q^{56} +1.17258 q^{57} +4.44495 q^{58} +2.97769 q^{59} -11.0977 q^{61} +5.90376 q^{62} +4.37768 q^{63} +2.73362 q^{64} +1.77202 q^{66} +1.40319 q^{67} -2.77490 q^{68} +3.03335 q^{69} -6.51732 q^{71} +4.48827 q^{72} -1.89995 q^{73} -6.90766 q^{74} +1.47566 q^{76} +5.23580 q^{77} -4.12089 q^{78} -13.9020 q^{79} -0.852626 q^{81} +6.98988 q^{82} +6.49943 q^{83} -4.07483 q^{84} -2.46194 q^{86} -6.59906 q^{87} +5.36807 q^{88} -0.236437 q^{89} -12.1761 q^{91} +3.81741 q^{92} -8.76484 q^{93} +3.21573 q^{94} -6.61998 q^{96} +13.5524 q^{97} -0.422870 q^{98} +3.55683 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 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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