LMFDB - Embedded newform 2450.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, 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("2450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.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:	$19.5633484952$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 490)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.585786 q^{3} +1.00000 q^{4} +0.585786 q^{6} +1.00000 q^{8} -2.65685 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.585786 q^{3} +1.00000 q^{4} +0.585786 q^{6} +1.00000 q^{8} -2.65685 q^{9} +4.82843 q^{11} +0.585786 q^{12} +0.828427 q^{13} +1.00000 q^{16} +5.41421 q^{17} -2.65685 q^{18} -3.41421 q^{19} +4.82843 q^{22} +6.82843 q^{23} +0.585786 q^{24} +0.828427 q^{26} -3.31371 q^{27} +0.828427 q^{29} -2.82843 q^{31} +1.00000 q^{32} +2.82843 q^{33} +5.41421 q^{34} -2.65685 q^{36} -3.65685 q^{37} -3.41421 q^{38} +0.485281 q^{39} -11.0711 q^{41} +3.17157 q^{43} +4.82843 q^{44} +6.82843 q^{46} +10.8284 q^{47} +0.585786 q^{48} +3.17157 q^{51} +0.828427 q^{52} +10.4853 q^{53} -3.31371 q^{54} -2.00000 q^{57} +0.828427 q^{58} -11.4142 q^{59} +13.3137 q^{61} -2.82843 q^{62} +1.00000 q^{64} +2.82843 q^{66} +9.65685 q^{67} +5.41421 q^{68} +4.00000 q^{69} +12.4853 q^{71} -2.65685 q^{72} -6.58579 q^{73} -3.65685 q^{74} -3.41421 q^{76} +0.485281 q^{78} -1.17157 q^{79} +6.02944 q^{81} -11.0711 q^{82} +6.24264 q^{83} +3.17157 q^{86} +0.485281 q^{87} +4.82843 q^{88} +12.7279 q^{89} +6.82843 q^{92} -1.65685 q^{93} +10.8284 q^{94} +0.585786 q^{96} -16.2426 q^{97} -12.8284 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 2 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 4 * q^6 + 2 * q^8 + 6 * q^9	$ 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 2 q^{8} + 6 q^{9} + 4 q^{11} + 4 q^{12} - 4 q^{13} + 2 q^{16} + 8 q^{17} + 6 q^{18} - 4 q^{19} + 4 q^{22} + 8 q^{23} + 4 q^{24} - 4 q^{26} + 16 q^{27} - 4 q^{29} + 2 q^{32} + 8 q^{34} + 6 q^{36} + 4 q^{37} - 4 q^{38} - 16 q^{39} - 8 q^{41} + 12 q^{43} + 4 q^{44} + 8 q^{46} + 16 q^{47} + 4 q^{48} + 12 q^{51} - 4 q^{52} + 4 q^{53} + 16 q^{54} - 4 q^{57} - 4 q^{58} - 20 q^{59} + 4 q^{61} + 2 q^{64} + 8 q^{67} + 8 q^{68} + 8 q^{69} + 8 q^{71} + 6 q^{72} - 16 q^{73} + 4 q^{74} - 4 q^{76} - 16 q^{78} - 8 q^{79} + 46 q^{81} - 8 q^{82} + 4 q^{83} + 12 q^{86} - 16 q^{87} + 4 q^{88} + 8 q^{92} + 8 q^{93} + 16 q^{94} + 4 q^{96} - 24 q^{97} - 20 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 4 * q^6 + 2 * q^8 + 6 * q^9 + 4 * q^11 + 4 * q^12 - 4 * q^13 + 2 * q^16 + 8 * q^17 + 6 * q^18 - 4 * q^19 + 4 * q^22 + 8 * q^23 + 4 * q^24 - 4 * q^26 + 16 * q^27 - 4 * q^29 + 2 * q^32 + 8 * q^34 + 6 * q^36 + 4 * q^37 - 4 * q^38 - 16 * q^39 - 8 * q^41 + 12 * q^43 + 4 * q^44 + 8 * q^46 + 16 * q^47 + 4 * q^48 + 12 * q^51 - 4 * q^52 + 4 * q^53 + 16 * q^54 - 4 * q^57 - 4 * q^58 - 20 * q^59 + 4 * q^61 + 2 * q^64 + 8 * q^67 + 8 * q^68 + 8 * q^69 + 8 * q^71 + 6 * q^72 - 16 * q^73 + 4 * q^74 - 4 * q^76 - 16 * q^78 - 8 * q^79 + 46 * q^81 - 8 * q^82 + 4 * q^83 + 12 * q^86 - 16 * q^87 + 4 * q^88 + 8 * q^92 + 8 * q^93 + 16 * q^94 + 4 * q^96 - 24 * q^97 - 20 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.585786	0.338204	0.169102	−	0.985599i	$-0.445913\pi$
$3$	0.585786	0.338204	0.169102	+	0.985599i	$0.445913\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.585786	0.239146
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−2.65685	−0.885618
$10$	0	0
$11$	4.82843	1.45583	0.727913	−	0.685670i	$-0.240491\pi$
$11$	4.82843	1.45583	0.727913	+	0.685670i	$0.240491\pi$
$12$	0.585786	0.169102
$13$	0.828427	0.229764	0.114882	−	0.993379i	$-0.463351\pi$
$13$	0.828427	0.229764	0.114882	+	0.993379i	$0.463351\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	5.41421	1.31314	0.656570	−	0.754265i	$-0.272007\pi$
$17$	5.41421	1.31314	0.656570	+	0.754265i	$0.272007\pi$
$18$	−2.65685	−0.626227
$19$	−3.41421	−0.783274	−0.391637	−	0.920120i	$-0.628091\pi$
$19$	−3.41421	−0.783274	−0.391637	+	0.920120i	$0.628091\pi$
$20$	0	0
$21$	0	0
$22$	4.82843	1.02942
$23$	6.82843	1.42383	0.711913	−	0.702268i	$-0.247829\pi$
$23$	6.82843	1.42383	0.711913	+	0.702268i	$0.247829\pi$
$24$	0.585786	0.119573
$25$	0	0
$26$	0.828427	0.162468
$27$	−3.31371	−0.637723
$28$	0	0
$29$	0.828427	0.153835	0.0769175	−	0.997037i	$-0.475492\pi$
$29$	0.828427	0.153835	0.0769175	+	0.997037i	$0.475492\pi$
$30$	0	0
$31$	−2.82843	−0.508001	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843	−0.508001	−0.254000	+	0.967204i	$0.581746\pi$
$32$	1.00000	0.176777
$33$	2.82843	0.492366
$34$	5.41421	0.928530
$35$	0	0
$36$	−2.65685	−0.442809
$37$	−3.65685	−0.601183	−0.300592	−	0.953753i	$-0.597184\pi$
$37$	−3.65685	−0.601183	−0.300592	+	0.953753i	$0.597184\pi$
$38$	−3.41421	−0.553859
$39$	0.485281	0.0777072
$40$	0	0
$41$	−11.0711	−1.72901	−0.864505	−	0.502624i	$-0.832368\pi$
$41$	−11.0711	−1.72901	−0.864505	+	0.502624i	$0.832368\pi$
$42$	0	0
$43$	3.17157	0.483660	0.241830	−	0.970319i	$-0.422252\pi$
$43$	3.17157	0.483660	0.241830	+	0.970319i	$0.422252\pi$
$44$	4.82843	0.727913
$45$	0	0
$46$	6.82843	1.00680
$47$	10.8284	1.57949	0.789744	−	0.613436i	$-0.210213\pi$
$47$	10.8284	1.57949	0.789744	+	0.613436i	$0.210213\pi$
$48$	0.585786	0.0845510
$49$	0	0
$50$	0	0
$51$	3.17157	0.444109
$52$	0.828427	0.114882
$53$	10.4853	1.44026	0.720132	−	0.693837i	$-0.244081\pi$
$53$	10.4853	1.44026	0.720132	+	0.693837i	$0.244081\pi$
$54$	−3.31371	−0.450939
$55$	0	0
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0.828427	0.108778
$59$	−11.4142	−1.48600	−0.743002	−	0.669289i	$-0.766599\pi$
$59$	−11.4142	−1.48600	−0.743002	+	0.669289i	$0.766599\pi$
$60$	0	0
$61$	13.3137	1.70465	0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137	1.70465	0.852323	+	0.523016i	$0.175193\pi$
$62$	−2.82843	−0.359211
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	2.82843	0.348155
$67$	9.65685	1.17977	0.589886	−	0.807486i	$-0.299173\pi$
$67$	9.65685	1.17977	0.589886	+	0.807486i	$0.299173\pi$
$68$	5.41421	0.656570
$69$	4.00000	0.481543
$70$	0	0
$71$	12.4853	1.48173	0.740865	−	0.671654i	$-0.234416\pi$
$71$	12.4853	1.48173	0.740865	+	0.671654i	$0.234416\pi$
$72$	−2.65685	−0.313113
$73$	−6.58579	−0.770808	−0.385404	−	0.922748i	$-0.625938\pi$
$73$	−6.58579	−0.770808	−0.385404	+	0.922748i	$0.625938\pi$
$74$	−3.65685	−0.425101
$75$	0	0
$76$	−3.41421	−0.391637
$77$	0	0
$78$	0.485281	0.0549473
$79$	−1.17157	−0.131812	−0.0659061	−	0.997826i	$-0.520994\pi$
$79$	−1.17157	−0.131812	−0.0659061	+	0.997826i	$0.520994\pi$
$80$	0	0
$81$	6.02944	0.669937
$82$	−11.0711	−1.22259
$83$	6.24264	0.685219	0.342609	−	0.939478i	$-0.388689\pi$
$83$	6.24264	0.685219	0.342609	+	0.939478i	$0.388689\pi$
$84$	0	0
$85$	0	0
$86$	3.17157	0.341999
$87$	0.485281	0.0520276
$88$	4.82843	0.514712
$89$	12.7279	1.34916	0.674579	−	0.738203i	$-0.264325\pi$
$89$	12.7279	1.34916	0.674579	+	0.738203i	$0.264325\pi$
$90$	0	0
$91$	0	0
$92$	6.82843	0.711913
$93$	−1.65685	−0.171808
$94$	10.8284	1.11687
$95$	0	0
$96$	0.585786	0.0597866
$97$	−16.2426	−1.64919	−0.824595	−	0.565723i	$-0.808597\pi$
$97$	−16.2426	−1.64919	−0.824595	+	0.565723i	$0.808597\pi$
$98$	0	0
$99$	−12.8284	−1.28931
$100$	0	0
$101$	9.31371	0.926749	0.463374	−	0.886163i	$-0.346639\pi$
$101$	9.31371	0.926749	0.463374	+	0.886163i	$0.346639\pi$
$102$	3.17157	0.314033
$103$	−9.17157	−0.903702	−0.451851	−	0.892093i	$-0.649236\pi$
$103$	−9.17157	−0.903702	−0.451851	+	0.892093i	$0.649236\pi$
$104$	0.828427	0.0812340
$105$	0	0
$106$	10.4853	1.01842
$107$	1.65685	0.160174	0.0800871	−	0.996788i	$-0.474480\pi$
$107$	1.65685	0.160174	0.0800871	+	0.996788i	$0.474480\pi$
$108$	−3.31371	−0.318862
$109$	−14.4853	−1.38744	−0.693719	−	0.720246i	$-0.744029\pi$
$109$	−14.4853	−1.38744	−0.693719	+	0.720246i	$0.744029\pi$
$110$	0	0
$111$	−2.14214	−0.203323
$112$	0	0
$113$	−7.31371	−0.688016	−0.344008	−	0.938967i	$-0.611785\pi$
$113$	−7.31371	−0.688016	−0.344008	+	0.938967i	$0.611785\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	0.828427	0.0769175
$117$	−2.20101	−0.203483
$118$	−11.4142	−1.05076
$119$	0	0
$120$	0	0
$121$	12.3137	1.11943
$122$	13.3137	1.20537
$123$	−6.48528	−0.584758
$124$	−2.82843	−0.254000
$125$	0	0
$126$	0	0
$127$	2.82843	0.250982	0.125491	−	0.992095i	$-0.459949\pi$
$127$	2.82843	0.250982	0.125491	+	0.992095i	$0.459949\pi$
$128$	1.00000	0.0883883
$129$	1.85786	0.163576
$130$	0	0
$131$	−2.24264	−0.195940	−0.0979702	−	0.995189i	$-0.531235\pi$
$131$	−2.24264	−0.195940	−0.0979702	+	0.995189i	$0.531235\pi$
$132$	2.82843	0.246183
$133$	0	0
$134$	9.65685	0.834225
$135$	0	0
$136$	5.41421	0.464265
$137$	16.0000	1.36697	0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000	1.36697	0.683486	+	0.729964i	$0.260463\pi$
$138$	4.00000	0.340503
$139$	−0.100505	−0.00852473	−0.00426236	−	0.999991i	$-0.501357\pi$
$139$	−0.100505	−0.00852473	−0.00426236	+	0.999991i	$0.501357\pi$
$140$	0	0
$141$	6.34315	0.534189
$142$	12.4853	1.04774
$143$	4.00000	0.334497
$144$	−2.65685	−0.221405
$145$	0	0
$146$	−6.58579	−0.545044
$147$	0	0
$148$	−3.65685	−0.300592
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	11.3137	0.920697	0.460348	−	0.887738i	$-0.347725\pi$
$151$	11.3137	0.920697	0.460348	+	0.887738i	$0.347725\pi$
$152$	−3.41421	−0.276929
$153$	−14.3848	−1.16294
$154$	0	0
$155$	0	0
$156$	0.485281	0.0388536
$157$	−10.4853	−0.836817	−0.418408	−	0.908259i	$-0.637412\pi$
$157$	−10.4853	−0.836817	−0.418408	+	0.908259i	$0.637412\pi$
$158$	−1.17157	−0.0932053
$159$	6.14214	0.487103
$160$	0	0
$161$	0	0
$162$	6.02944	0.473717
$163$	−8.14214	−0.637741	−0.318871	−	0.947798i	$-0.603304\pi$
$163$	−8.14214	−0.637741	−0.318871	+	0.947798i	$0.603304\pi$
$164$	−11.0711	−0.864505
$165$	0	0
$166$	6.24264	0.484523
$167$	−23.7990	−1.84162	−0.920811	−	0.390010i	$-0.872471\pi$
$167$	−23.7990	−1.84162	−0.920811	+	0.390010i	$0.872471\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	0	0
$171$	9.07107	0.693682
$172$	3.17157	0.241830
$173$	−3.17157	−0.241130	−0.120565	−	0.992705i	$-0.538471\pi$
$173$	−3.17157	−0.241130	−0.120565	+	0.992705i	$0.538471\pi$
$174$	0.485281	0.0367891
$175$	0	0
$176$	4.82843	0.363956
$177$	−6.68629	−0.502572
$178$	12.7279	0.953998
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−14.4853	−1.07668	−0.538341	−	0.842727i	$-0.680949\pi$
$181$	−14.4853	−1.07668	−0.538341	+	0.842727i	$0.680949\pi$
$182$	0	0
$183$	7.79899	0.576518
$184$	6.82843	0.503398
$185$	0	0
$186$	−1.65685	−0.121486
$187$	26.1421	1.91170
$188$	10.8284	0.789744
$189$	0	0
$190$	0	0
$191$	−18.1421	−1.31272	−0.656359	−	0.754448i	$-0.727904\pi$
$191$	−18.1421	−1.31272	−0.656359	+	0.754448i	$0.727904\pi$
$192$	0.585786	0.0422755
$193$	5.65685	0.407189	0.203595	−	0.979055i	$-0.434738\pi$
$193$	5.65685	0.407189	0.203595	+	0.979055i	$0.434738\pi$
$194$	−16.2426	−1.16615
$195$	0	0
$196$	0	0
$197$	−13.7990	−0.983137	−0.491569	−	0.870839i	$-0.663576\pi$
$197$	−13.7990	−0.983137	−0.491569	+	0.870839i	$0.663576\pi$
$198$	−12.8284	−0.911677
$199$	0.485281	0.0344007	0.0172003	−	0.999852i	$-0.494525\pi$
$199$	0.485281	0.0344007	0.0172003	+	0.999852i	$0.494525\pi$
$200$	0	0
$201$	5.65685	0.399004
$202$	9.31371	0.655310
$203$	0	0
$204$	3.17157	0.222055
$205$	0	0
$206$	−9.17157	−0.639014
$207$	−18.1421	−1.26097
$208$	0.828427	0.0574411
$209$	−16.4853	−1.14031
$210$	0	0
$211$	−26.6274	−1.83311	−0.916553	−	0.399912i	$-0.869041\pi$
$211$	−26.6274	−1.83311	−0.916553	+	0.399912i	$0.869041\pi$
$212$	10.4853	0.720132
$213$	7.31371	0.501127
$214$	1.65685	0.113260
$215$	0	0
$216$	−3.31371	−0.225469
$217$	0	0
$218$	−14.4853	−0.981067
$219$	−3.85786	−0.260690
$220$	0	0
$221$	4.48528	0.301713
$222$	−2.14214	−0.143771
$223$	15.3137	1.02548	0.512741	−	0.858543i	$-0.328630\pi$
$223$	15.3137	1.02548	0.512741	+	0.858543i	$0.328630\pi$
$224$	0	0
$225$	0	0
$226$	−7.31371	−0.486501
$227$	9.75736	0.647619	0.323809	−	0.946122i	$-0.395036\pi$
$227$	9.75736	0.647619	0.323809	+	0.946122i	$0.395036\pi$
$228$	−2.00000	−0.132453
$229$	−12.1421	−0.802375	−0.401187	−	0.915996i	$-0.631402\pi$
$229$	−12.1421	−0.802375	−0.401187	+	0.915996i	$0.631402\pi$
$230$	0	0
$231$	0	0
$232$	0.828427	0.0543889
$233$	−0.686292	−0.0449605	−0.0224802	−	0.999747i	$-0.507156\pi$
$233$	−0.686292	−0.0449605	−0.0224802	+	0.999747i	$0.507156\pi$
$234$	−2.20101	−0.143885
$235$	0	0
$236$	−11.4142	−0.743002
$237$	−0.686292	−0.0445794
$238$	0	0
$239$	−9.65685	−0.624650	−0.312325	−	0.949975i	$-0.601108\pi$
$239$	−9.65685	−0.624650	−0.312325	+	0.949975i	$0.601108\pi$
$240$	0	0
$241$	10.5858	0.681890	0.340945	−	0.940083i	$-0.389253\pi$
$241$	10.5858	0.681890	0.340945	+	0.940083i	$0.389253\pi$
$242$	12.3137	0.791555
$243$	13.4731	0.864299
$244$	13.3137	0.852323
$245$	0	0
$246$	−6.48528	−0.413486
$247$	−2.82843	−0.179969
$248$	−2.82843	−0.179605
$249$	3.65685	0.231744
$250$	0	0
$251$	−3.41421	−0.215503	−0.107752	−	0.994178i	$-0.534365\pi$
$251$	−3.41421	−0.215503	−0.107752	+	0.994178i	$0.534365\pi$
$252$	0	0
$253$	32.9706	2.07284
$254$	2.82843	0.177471
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.89949	0.617514	0.308757	−	0.951141i	$-0.400087\pi$
$257$	9.89949	0.617514	0.308757	+	0.951141i	$0.400087\pi$
$258$	1.85786	0.115666
$259$	0	0
$260$	0	0
$261$	−2.20101	−0.136239
$262$	−2.24264	−0.138551
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	2.82843	0.174078
$265$	0	0
$266$	0	0
$267$	7.45584	0.456290
$268$	9.65685	0.589886
$269$	−1.51472	−0.0923540	−0.0461770	−	0.998933i	$-0.514704\pi$
$269$	−1.51472	−0.0923540	−0.0461770	+	0.998933i	$0.514704\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	5.41421	0.328285
$273$	0	0
$274$	16.0000	0.966595
$275$	0	0
$276$	4.00000	0.240772
$277$	−20.1421	−1.21022	−0.605112	−	0.796140i	$-0.706872\pi$
$277$	−20.1421	−1.21022	−0.605112	+	0.796140i	$0.706872\pi$
$278$	−0.100505	−0.00602789
$279$	7.51472	0.449894
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	6.34315	0.377729
$283$	6.24264	0.371086	0.185543	−	0.982636i	$-0.440596\pi$
$283$	6.24264	0.371086	0.185543	+	0.982636i	$0.440596\pi$
$284$	12.4853	0.740865
$285$	0	0
$286$	4.00000	0.236525
$287$	0	0
$288$	−2.65685	−0.156557
$289$	12.3137	0.724336
$290$	0	0
$291$	−9.51472	−0.557763
$292$	−6.58579	−0.385404
$293$	−19.6569	−1.14837	−0.574183	−	0.818727i	$-0.694680\pi$
$293$	−19.6569	−1.14837	−0.574183	+	0.818727i	$0.694680\pi$
$294$	0	0
$295$	0	0
$296$	−3.65685	−0.212550
$297$	−16.0000	−0.928414
$298$	−6.00000	−0.347571
$299$	5.65685	0.327144
$300$	0	0
$301$	0	0
$302$	11.3137	0.651031
$303$	5.45584	0.313430
$304$	−3.41421	−0.195819
$305$	0	0
$306$	−14.3848	−0.822323
$307$	29.0711	1.65917	0.829587	−	0.558378i	$-0.188576\pi$
$307$	29.0711	1.65917	0.829587	+	0.558378i	$0.188576\pi$
$308$	0	0
$309$	−5.37258	−0.305636
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0.485281	0.0274736
$313$	−22.3848	−1.26526	−0.632631	−	0.774453i	$-0.718025\pi$
$313$	−22.3848	−1.26526	−0.632631	+	0.774453i	$0.718025\pi$
$314$	−10.4853	−0.591719
$315$	0	0
$316$	−1.17157	−0.0659061
$317$	6.48528	0.364250	0.182125	−	0.983275i	$-0.441702\pi$
$317$	6.48528	0.364250	0.182125	+	0.983275i	$0.441702\pi$
$318$	6.14214	0.344434
$319$	4.00000	0.223957
$320$	0	0
$321$	0.970563	0.0541715
$322$	0	0
$323$	−18.4853	−1.02855
$324$	6.02944	0.334969
$325$	0	0
$326$	−8.14214	−0.450951
$327$	−8.48528	−0.469237
$328$	−11.0711	−0.611297
$329$	0	0
$330$	0	0
$331$	−5.79899	−0.318741	−0.159371	−	0.987219i	$-0.550946\pi$
$331$	−5.79899	−0.318741	−0.159371	+	0.987219i	$0.550946\pi$
$332$	6.24264	0.342609
$333$	9.71573	0.532419
$334$	−23.7990	−1.30222
$335$	0	0
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	−12.3137	−0.669777
$339$	−4.28427	−0.232690
$340$	0	0
$341$	−13.6569	−0.739560
$342$	9.07107	0.490507
$343$	0	0
$344$	3.17157	0.171000
$345$	0	0
$346$	−3.17157	−0.170505
$347$	8.82843	0.473935	0.236967	−	0.971518i	$-0.423847\pi$
$347$	8.82843	0.473935	0.236967	+	0.971518i	$0.423847\pi$
$348$	0.485281	0.0260138
$349$	14.4853	0.775379	0.387690	−	0.921790i	$-0.373273\pi$
$349$	14.4853	0.775379	0.387690	+	0.921790i	$0.373273\pi$
$350$	0	0
$351$	−2.74517	−0.146526
$352$	4.82843	0.257356
$353$	34.3848	1.83012	0.915058	−	0.403321i	$-0.132144\pi$
$353$	34.3848	1.83012	0.915058	+	0.403321i	$0.132144\pi$
$354$	−6.68629	−0.355372
$355$	0	0
$356$	12.7279	0.674579
$357$	0	0
$358$	4.00000	0.211407
$359$	−28.2843	−1.49279	−0.746393	−	0.665505i	$-0.768216\pi$
$359$	−28.2843	−1.49279	−0.746393	+	0.665505i	$0.768216\pi$
$360$	0	0
$361$	−7.34315	−0.386481
$362$	−14.4853	−0.761329
$363$	7.21320	0.378595
$364$	0	0
$365$	0	0
$366$	7.79899	0.407660
$367$	−8.97056	−0.468260	−0.234130	−	0.972205i	$-0.575224\pi$
$367$	−8.97056	−0.468260	−0.234130	+	0.972205i	$0.575224\pi$
$368$	6.82843	0.355956
$369$	29.4142	1.53124
$370$	0	0
$371$	0	0
$372$	−1.65685	−0.0859039
$373$	13.5147	0.699766	0.349883	−	0.936793i	$-0.386221\pi$
$373$	13.5147	0.699766	0.349883	+	0.936793i	$0.386221\pi$
$374$	26.1421	1.35178
$375$	0	0
$376$	10.8284	0.558433
$377$	0.686292	0.0353458
$378$	0	0
$379$	17.5147	0.899671	0.449835	−	0.893112i	$-0.351483\pi$
$379$	17.5147	0.899671	0.449835	+	0.893112i	$0.351483\pi$
$380$	0	0
$381$	1.65685	0.0848832
$382$	−18.1421	−0.928232
$383$	−15.5147	−0.792765	−0.396383	−	0.918085i	$-0.629735\pi$
$383$	−15.5147	−0.792765	−0.396383	+	0.918085i	$0.629735\pi$
$384$	0.585786	0.0298933
$385$	0	0
$386$	5.65685	0.287926
$387$	−8.42641	−0.428338
$388$	−16.2426	−0.824595
$389$	−0.142136	−0.00720656	−0.00360328	−	0.999994i	$-0.501147\pi$
$389$	−0.142136	−0.00720656	−0.00360328	+	0.999994i	$0.501147\pi$
$390$	0	0
$391$	36.9706	1.86968
$392$	0	0
$393$	−1.31371	−0.0662678
$394$	−13.7990	−0.695183
$395$	0	0
$396$	−12.8284	−0.644653
$397$	−5.79899	−0.291043	−0.145521	−	0.989355i	$-0.546486\pi$
$397$	−5.79899	−0.291043	−0.145521	+	0.989355i	$0.546486\pi$
$398$	0.485281	0.0243250
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	5.65685	0.282138
$403$	−2.34315	−0.116720
$404$	9.31371	0.463374
$405$	0	0
$406$	0	0
$407$	−17.6569	−0.875218
$408$	3.17157	0.157016
$409$	13.4142	0.663290	0.331645	−	0.943404i	$-0.392396\pi$
$409$	13.4142	0.663290	0.331645	+	0.943404i	$0.392396\pi$
$410$	0	0
$411$	9.37258	0.462315
$412$	−9.17157	−0.451851
$413$	0	0
$414$	−18.1421	−0.891637
$415$	0	0
$416$	0.828427	0.0406170
$417$	−0.0588745	−0.00288310
$418$	−16.4853	−0.806321
$419$	−32.8701	−1.60581	−0.802904	−	0.596109i	$-0.796713\pi$
$419$	−32.8701	−1.60581	−0.802904	+	0.596109i	$0.796713\pi$
$420$	0	0
$421$	−5.31371	−0.258974	−0.129487	−	0.991581i	$-0.541333\pi$
$421$	−5.31371	−0.258974	−0.129487	+	0.991581i	$0.541333\pi$
$422$	−26.6274	−1.29620
$423$	−28.7696	−1.39882
$424$	10.4853	0.509210
$425$	0	0
$426$	7.31371	0.354350
$427$	0	0
$428$	1.65685	0.0800871
$429$	2.34315	0.113128
$430$	0	0
$431$	−33.6569	−1.62119	−0.810597	−	0.585605i	$-0.800857\pi$
$431$	−33.6569	−1.62119	−0.810597	+	0.585605i	$0.800857\pi$
$432$	−3.31371	−0.159431
$433$	13.4142	0.644646	0.322323	−	0.946630i	$-0.395536\pi$
$433$	13.4142	0.644646	0.322323	+	0.946630i	$0.395536\pi$
$434$	0	0
$435$	0	0
$436$	−14.4853	−0.693719
$437$	−23.3137	−1.11525
$438$	−3.85786	−0.184336
$439$	−8.97056	−0.428142	−0.214071	−	0.976818i	$-0.568672\pi$
$439$	−8.97056	−0.428142	−0.214071	+	0.976818i	$0.568672\pi$
$440$	0	0
$441$	0	0
$442$	4.48528	0.213343
$443$	−36.9706	−1.75652	−0.878262	−	0.478179i	$-0.841297\pi$
$443$	−36.9706	−1.75652	−0.878262	+	0.478179i	$0.841297\pi$
$444$	−2.14214	−0.101661
$445$	0	0
$446$	15.3137	0.725125
$447$	−3.51472	−0.166240
$448$	0	0
$449$	28.6274	1.35101	0.675506	−	0.737355i	$-0.263925\pi$
$449$	28.6274	1.35101	0.675506	+	0.737355i	$0.263925\pi$
$450$	0	0
$451$	−53.4558	−2.51714
$452$	−7.31371	−0.344008
$453$	6.62742	0.311383
$454$	9.75736	0.457936
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	−10.3431	−0.483832	−0.241916	−	0.970297i	$-0.577776\pi$
$457$	−10.3431	−0.483832	−0.241916	+	0.970297i	$0.577776\pi$
$458$	−12.1421	−0.567365
$459$	−17.9411	−0.837420
$460$	0	0
$461$	7.17157	0.334013	0.167007	−	0.985956i	$-0.446590\pi$
$461$	7.17157	0.334013	0.167007	+	0.985956i	$0.446590\pi$
$462$	0	0
$463$	−16.9706	−0.788689	−0.394344	−	0.918963i	$-0.629028\pi$
$463$	−16.9706	−0.788689	−0.394344	+	0.918963i	$0.629028\pi$
$464$	0.828427	0.0384588
$465$	0	0
$466$	−0.686292	−0.0317918
$467$	3.89949	0.180447	0.0902236	−	0.995922i	$-0.471242\pi$
$467$	3.89949	0.180447	0.0902236	+	0.995922i	$0.471242\pi$
$468$	−2.20101	−0.101742
$469$	0	0
$470$	0	0
$471$	−6.14214	−0.283015
$472$	−11.4142	−0.525382
$473$	15.3137	0.704125
$474$	−0.686292	−0.0315224
$475$	0	0
$476$	0	0
$477$	−27.8579	−1.27552
$478$	−9.65685	−0.441694
$479$	22.8284	1.04306	0.521529	−	0.853234i	$-0.325362\pi$
$479$	22.8284	1.04306	0.521529	+	0.853234i	$0.325362\pi$
$480$	0	0
$481$	−3.02944	−0.138130
$482$	10.5858	0.482169
$483$	0	0
$484$	12.3137	0.559714
$485$	0	0
$486$	13.4731	0.611152
$487$	7.79899	0.353406	0.176703	−	0.984264i	$-0.443457\pi$
$487$	7.79899	0.353406	0.176703	+	0.984264i	$0.443457\pi$
$488$	13.3137	0.602683
$489$	−4.76955	−0.215687
$490$	0	0
$491$	−24.2843	−1.09593	−0.547967	−	0.836500i	$-0.684598\pi$
$491$	−24.2843	−1.09593	−0.547967	+	0.836500i	$0.684598\pi$
$492$	−6.48528	−0.292379
$493$	4.48528	0.202007
$494$	−2.82843	−0.127257
$495$	0	0
$496$	−2.82843	−0.127000
$497$	0	0
$498$	3.65685	0.163868
$499$	41.6569	1.86482	0.932408	−	0.361406i	$-0.117703\pi$
$499$	41.6569	1.86482	0.932408	+	0.361406i	$0.117703\pi$
$500$	0	0
$501$	−13.9411	−0.622844
$502$	−3.41421	−0.152384
$503$	6.34315	0.282827	0.141413	−	0.989951i	$-0.454835\pi$
$503$	6.34315	0.282827	0.141413	+	0.989951i	$0.454835\pi$
$504$	0	0
$505$	0	0
$506$	32.9706	1.46572
$507$	−7.21320	−0.320350
$508$	2.82843	0.125491
$509$	33.7990	1.49811	0.749057	−	0.662506i	$-0.230507\pi$
$509$	33.7990	1.49811	0.749057	+	0.662506i	$0.230507\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	11.3137	0.499512
$514$	9.89949	0.436648
$515$	0	0
$516$	1.85786	0.0817879
$517$	52.2843	2.29946
$518$	0	0
$519$	−1.85786	−0.0815512
$520$	0	0
$521$	−4.92893	−0.215940	−0.107970	−	0.994154i	$-0.534435\pi$
$521$	−4.92893	−0.215940	−0.107970	+	0.994154i	$0.534435\pi$
$522$	−2.20101	−0.0963356
$523$	4.10051	0.179303	0.0896513	−	0.995973i	$-0.471425\pi$
$523$	4.10051	0.179303	0.0896513	+	0.995973i	$0.471425\pi$
$524$	−2.24264	−0.0979702
$525$	0	0
$526$	−28.0000	−1.22086
$527$	−15.3137	−0.667076
$528$	2.82843	0.123091
$529$	23.6274	1.02728
$530$	0	0
$531$	30.3259	1.31603
$532$	0	0
$533$	−9.17157	−0.397265
$534$	7.45584	0.322646
$535$	0	0
$536$	9.65685	0.417113
$537$	2.34315	0.101114
$538$	−1.51472	−0.0653042
$539$	0	0
$540$	0	0
$541$	18.9706	0.815608	0.407804	−	0.913069i	$-0.366295\pi$
$541$	18.9706	0.815608	0.407804	+	0.913069i	$0.366295\pi$
$542$	−12.0000	−0.515444
$543$	−8.48528	−0.364138
$544$	5.41421	0.232132
$545$	0	0
$546$	0	0
$547$	−6.48528	−0.277291	−0.138645	−	0.990342i	$-0.544275\pi$
$547$	−6.48528	−0.277291	−0.138645	+	0.990342i	$0.544275\pi$
$548$	16.0000	0.683486
$549$	−35.3726	−1.50967
$550$	0	0
$551$	−2.82843	−0.120495
$552$	4.00000	0.170251
$553$	0	0
$554$	−20.1421	−0.855757
$555$	0	0
$556$	−0.100505	−0.00426236
$557$	−20.8284	−0.882529	−0.441264	−	0.897377i	$-0.645470\pi$
$557$	−20.8284	−0.882529	−0.441264	+	0.897377i	$0.645470\pi$
$558$	7.51472	0.318123
$559$	2.62742	0.111128
$560$	0	0
$561$	15.3137	0.646545
$562$	8.00000	0.337460
$563$	−39.4142	−1.66111	−0.830556	−	0.556936i	$-0.811977\pi$
$563$	−39.4142	−1.66111	−0.830556	+	0.556936i	$0.811977\pi$
$564$	6.34315	0.267095
$565$	0	0
$566$	6.24264	0.262398
$567$	0	0
$568$	12.4853	0.523871
$569$	−6.68629	−0.280304	−0.140152	−	0.990130i	$-0.544759\pi$
$569$	−6.68629	−0.280304	−0.140152	+	0.990130i	$0.544759\pi$
$570$	0	0
$571$	−41.7990	−1.74923	−0.874617	−	0.484815i	$-0.838887\pi$
$571$	−41.7990	−1.74923	−0.874617	+	0.484815i	$0.838887\pi$
$572$	4.00000	0.167248
$573$	−10.6274	−0.443967
$574$	0	0
$575$	0	0
$576$	−2.65685	−0.110702
$577$	25.8995	1.07821	0.539105	−	0.842239i	$-0.318763\pi$
$577$	25.8995	1.07821	0.539105	+	0.842239i	$0.318763\pi$
$578$	12.3137	0.512183
$579$	3.31371	0.137713
$580$	0	0
$581$	0	0
$582$	−9.51472	−0.394398
$583$	50.6274	2.09677
$584$	−6.58579	−0.272522
$585$	0	0
$586$	−19.6569	−0.812017
$587$	2.92893	0.120890	0.0604450	−	0.998172i	$-0.480748\pi$
$587$	2.92893	0.120890	0.0604450	+	0.998172i	$0.480748\pi$
$588$	0	0
$589$	9.65685	0.397904
$590$	0	0
$591$	−8.08326	−0.332501
$592$	−3.65685	−0.150296
$593$	28.7279	1.17971	0.589857	−	0.807508i	$-0.299184\pi$
$593$	28.7279	1.17971	0.589857	+	0.807508i	$0.299184\pi$
$594$	−16.0000	−0.656488
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0.284271	0.0116344
$598$	5.65685	0.231326
$599$	−5.17157	−0.211305	−0.105652	−	0.994403i	$-0.533693\pi$
$599$	−5.17157	−0.211305	−0.105652	+	0.994403i	$0.533693\pi$
$600$	0	0
$601$	−9.41421	−0.384014	−0.192007	−	0.981394i	$-0.561500\pi$
$601$	−9.41421	−0.384014	−0.192007	+	0.981394i	$0.561500\pi$
$602$	0	0
$603$	−25.6569	−1.04483
$604$	11.3137	0.460348
$605$	0	0
$606$	5.45584	0.221629
$607$	−40.2843	−1.63509	−0.817544	−	0.575866i	$-0.804665\pi$
$607$	−40.2843	−1.63509	−0.817544	+	0.575866i	$0.804665\pi$
$608$	−3.41421	−0.138465
$609$	0	0
$610$	0	0
$611$	8.97056	0.362910
$612$	−14.3848	−0.581470
$613$	23.6569	0.955491	0.477746	−	0.878498i	$-0.341454\pi$
$613$	23.6569	0.955491	0.477746	+	0.878498i	$0.341454\pi$
$614$	29.0711	1.17321
$615$	0	0
$616$	0	0
$617$	10.6863	0.430214	0.215107	−	0.976590i	$-0.430990\pi$
$617$	10.6863	0.430214	0.215107	+	0.976590i	$0.430990\pi$
$618$	−5.37258	−0.216117
$619$	−14.9289	−0.600044	−0.300022	−	0.953932i	$-0.596994\pi$
$619$	−14.9289	−0.600044	−0.300022	+	0.953932i	$0.596994\pi$
$620$	0	0
$621$	−22.6274	−0.908007
$622$	−4.00000	−0.160385
$623$	0	0
$624$	0.485281	0.0194268
$625$	0	0
$626$	−22.3848	−0.894676
$627$	−9.65685	−0.385658
$628$	−10.4853	−0.418408
$629$	−19.7990	−0.789437
$630$	0	0
$631$	4.48528	0.178556	0.0892781	−	0.996007i	$-0.471544\pi$
$631$	4.48528	0.178556	0.0892781	+	0.996007i	$0.471544\pi$
$632$	−1.17157	−0.0466027
$633$	−15.5980	−0.619964
$634$	6.48528	0.257563
$635$	0	0
$636$	6.14214	0.243552
$637$	0	0
$638$	4.00000	0.158362
$639$	−33.1716	−1.31225
$640$	0	0
$641$	20.6274	0.814734	0.407367	−	0.913265i	$-0.366447\pi$
$641$	20.6274	0.814734	0.407367	+	0.913265i	$0.366447\pi$
$642$	0.970563	0.0383051
$643$	47.2132	1.86191	0.930953	−	0.365138i	$-0.118978\pi$
$643$	47.2132	1.86191	0.930953	+	0.365138i	$0.118978\pi$
$644$	0	0
$645$	0	0
$646$	−18.4853	−0.727294
$647$	39.1127	1.53768	0.768839	−	0.639442i	$-0.220835\pi$
$647$	39.1127	1.53768	0.768839	+	0.639442i	$0.220835\pi$
$648$	6.02944	0.236859
$649$	−55.1127	−2.16336
$650$	0	0
$651$	0	0
$652$	−8.14214	−0.318871
$653$	−15.6569	−0.612700	−0.306350	−	0.951919i	$-0.599108\pi$
$653$	−15.6569	−0.612700	−0.306350	+	0.951919i	$0.599108\pi$
$654$	−8.48528	−0.331801
$655$	0	0
$656$	−11.0711	−0.432253
$657$	17.4975	0.682642
$658$	0	0
$659$	−32.8284	−1.27881	−0.639407	−	0.768868i	$-0.720820\pi$
$659$	−32.8284	−1.27881	−0.639407	+	0.768868i	$0.720820\pi$
$660$	0	0
$661$	−18.2843	−0.711176	−0.355588	−	0.934643i	$-0.615719\pi$
$661$	−18.2843	−0.711176	−0.355588	+	0.934643i	$0.615719\pi$
$662$	−5.79899	−0.225384
$663$	2.62742	0.102040
$664$	6.24264	0.242261
$665$	0	0
$666$	9.71573	0.376477
$667$	5.65685	0.219034
$668$	−23.7990	−0.920811
$669$	8.97056	0.346822
$670$	0	0
$671$	64.2843	2.48167
$672$	0	0
$673$	48.0000	1.85026	0.925132	−	0.379646i	$-0.123954\pi$
$673$	48.0000	1.85026	0.925132	+	0.379646i	$0.123954\pi$
$674$	−6.00000	−0.231111
$675$	0	0
$676$	−12.3137	−0.473604
$677$	−11.4558	−0.440284	−0.220142	−	0.975468i	$-0.570652\pi$
$677$	−11.4558	−0.440284	−0.220142	+	0.975468i	$0.570652\pi$
$678$	−4.28427	−0.164536
$679$	0	0
$680$	0	0
$681$	5.71573	0.219027
$682$	−13.6569	−0.522948
$683$	−22.3431	−0.854937	−0.427468	−	0.904030i	$-0.640594\pi$
$683$	−22.3431	−0.854937	−0.427468	+	0.904030i	$0.640594\pi$
$684$	9.07107	0.346841
$685$	0	0
$686$	0	0
$687$	−7.11270	−0.271366
$688$	3.17157	0.120915
$689$	8.68629	0.330921
$690$	0	0
$691$	−10.2426	−0.389648	−0.194824	−	0.980838i	$-0.562414\pi$
$691$	−10.2426	−0.389648	−0.194824	+	0.980838i	$0.562414\pi$
$692$	−3.17157	−0.120565
$693$	0	0
$694$	8.82843	0.335123
$695$	0	0
$696$	0.485281	0.0183945
$697$	−59.9411	−2.27043
$698$	14.4853	0.548276
$699$	−0.402020	−0.0152058
$700$	0	0
$701$	−14.4853	−0.547102	−0.273551	−	0.961858i	$-0.588198\pi$
$701$	−14.4853	−0.547102	−0.273551	+	0.961858i	$0.588198\pi$
$702$	−2.74517	−0.103610
$703$	12.4853	0.470891
$704$	4.82843	0.181978
$705$	0	0
$706$	34.3848	1.29409
$707$	0	0
$708$	−6.68629	−0.251286
$709$	−17.1127	−0.642681	−0.321340	−	0.946964i	$-0.604133\pi$
$709$	−17.1127	−0.642681	−0.321340	+	0.946964i	$0.604133\pi$
$710$	0	0
$711$	3.11270	0.116735
$712$	12.7279	0.476999
$713$	−19.3137	−0.723304
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	−5.65685	−0.211259
$718$	−28.2843	−1.05556
$719$	9.45584	0.352643	0.176322	−	0.984333i	$-0.443580\pi$
$719$	9.45584	0.352643	0.176322	+	0.984333i	$0.443580\pi$
$720$	0	0
$721$	0	0
$722$	−7.34315	−0.273284
$723$	6.20101	0.230618
$724$	−14.4853	−0.538341
$725$	0	0
$726$	7.21320	0.267707
$727$	−20.4853	−0.759757	−0.379879	−	0.925036i	$-0.624034\pi$
$727$	−20.4853	−0.759757	−0.379879	+	0.925036i	$0.624034\pi$
$728$	0	0
$729$	−10.1960	−0.377628
$730$	0	0
$731$	17.1716	0.635114
$732$	7.79899	0.288259
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−8.97056	−0.331110
$735$	0	0
$736$	6.82843	0.251699
$737$	46.6274	1.71754
$738$	29.4142	1.08275
$739$	8.82843	0.324759	0.162379	−	0.986728i	$-0.448083\pi$
$739$	8.82843	0.324759	0.162379	+	0.986728i	$0.448083\pi$
$740$	0	0
$741$	−1.65685	−0.0608661
$742$	0	0
$743$	−12.2010	−0.447612	−0.223806	−	0.974634i	$-0.571848\pi$
$743$	−12.2010	−0.447612	−0.223806	+	0.974634i	$0.571848\pi$
$744$	−1.65685	−0.0607432
$745$	0	0
$746$	13.5147	0.494809
$747$	−16.5858	−0.606842
$748$	26.1421	0.955851
$749$	0	0
$750$	0	0
$751$	−16.6863	−0.608891	−0.304446	−	0.952530i	$-0.598471\pi$
$751$	−16.6863	−0.608891	−0.304446	+	0.952530i	$0.598471\pi$
$752$	10.8284	0.394872
$753$	−2.00000	−0.0728841
$754$	0.686292	0.0249933
$755$	0	0
$756$	0	0
$757$	7.65685	0.278293	0.139147	−	0.990272i	$-0.455564\pi$
$757$	7.65685	0.278293	0.139147	+	0.990272i	$0.455564\pi$
$758$	17.5147	0.636163
$759$	19.3137	0.701043
$760$	0	0
$761$	14.3848	0.521448	0.260724	−	0.965413i	$-0.416039\pi$
$761$	14.3848	0.521448	0.260724	+	0.965413i	$0.416039\pi$
$762$	1.65685	0.0600215
$763$	0	0
$764$	−18.1421	−0.656359
$765$	0	0
$766$	−15.5147	−0.560570
$767$	−9.45584	−0.341431
$768$	0.585786	0.0211377
$769$	11.5563	0.416733	0.208366	−	0.978051i	$-0.433185\pi$
$769$	11.5563	0.416733	0.208366	+	0.978051i	$0.433185\pi$
$770$	0	0
$771$	5.79899	0.208846
$772$	5.65685	0.203595
$773$	−2.00000	−0.0719350	−0.0359675	−	0.999353i	$-0.511451\pi$
$773$	−2.00000	−0.0719350	−0.0359675	+	0.999353i	$0.511451\pi$
$774$	−8.42641	−0.302881
$775$	0	0
$776$	−16.2426	−0.583077
$777$	0	0
$778$	−0.142136	−0.00509581
$779$	37.7990	1.35429
$780$	0	0
$781$	60.2843	2.15714
$782$	36.9706	1.32206
$783$	−2.74517	−0.0981042
$784$	0	0
$785$	0	0
$786$	−1.31371	−0.0468584
$787$	−26.7279	−0.952748	−0.476374	−	0.879243i	$-0.658049\pi$
$787$	−26.7279	−0.952748	−0.476374	+	0.879243i	$0.658049\pi$
$788$	−13.7990	−0.491569
$789$	−16.4020	−0.583927
$790$	0	0
$791$	0	0
$792$	−12.8284	−0.455838
$793$	11.0294	0.391667
$794$	−5.79899	−0.205798
$795$	0	0
$796$	0.485281	0.0172003
$797$	−2.20101	−0.0779638	−0.0389819	−	0.999240i	$-0.512411\pi$
$797$	−2.20101	−0.0779638	−0.0389819	+	0.999240i	$0.512411\pi$
$798$	0	0
$799$	58.6274	2.07409
$800$	0	0
$801$	−33.8162	−1.19484
$802$	−6.00000	−0.211867
$803$	−31.7990	−1.12216
$804$	5.65685	0.199502
$805$	0	0
$806$	−2.34315	−0.0825338
$807$	−0.887302	−0.0312345
$808$	9.31371	0.327655
$809$	36.9706	1.29982	0.649908	−	0.760013i	$-0.274807\pi$
$809$	36.9706	1.29982	0.649908	+	0.760013i	$0.274807\pi$
$810$	0	0
$811$	−35.4142	−1.24356	−0.621781	−	0.783191i	$-0.713590\pi$
$811$	−35.4142	−1.24356	−0.621781	+	0.783191i	$0.713590\pi$
$812$	0	0
$813$	−7.02944	−0.246533
$814$	−17.6569	−0.618872
$815$	0	0
$816$	3.17157	0.111027
$817$	−10.8284	−0.378839
$818$	13.4142	0.469017
$819$	0	0
$820$	0	0
$821$	−5.31371	−0.185450	−0.0927249	−	0.995692i	$-0.529558\pi$
$821$	−5.31371	−0.185450	−0.0927249	+	0.995692i	$0.529558\pi$
$822$	9.37258	0.326906
$823$	36.2843	1.26479	0.632395	−	0.774646i	$-0.282072\pi$
$823$	36.2843	1.26479	0.632395	+	0.774646i	$0.282072\pi$
$824$	−9.17157	−0.319507
$825$	0	0
$826$	0	0
$827$	50.6274	1.76049	0.880244	−	0.474522i	$-0.157379\pi$
$827$	50.6274	1.76049	0.880244	+	0.474522i	$0.157379\pi$
$828$	−18.1421	−0.630483
$829$	38.9706	1.35350	0.676752	−	0.736211i	$-0.263387\pi$
$829$	38.9706	1.35350	0.676752	+	0.736211i	$0.263387\pi$
$830$	0	0
$831$	−11.7990	−0.409302
$832$	0.828427	0.0287205
$833$	0	0
$834$	−0.0588745	−0.00203866
$835$	0	0
$836$	−16.4853	−0.570155
$837$	9.37258	0.323964
$838$	−32.8701	−1.13548
$839$	13.8579	0.478427	0.239213	−	0.970967i	$-0.423110\pi$
$839$	13.8579	0.478427	0.239213	+	0.970967i	$0.423110\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	−5.31371	−0.183122
$843$	4.68629	0.161404
$844$	−26.6274	−0.916553
$845$	0	0
$846$	−28.7696	−0.989118
$847$	0	0
$848$	10.4853	0.360066
$849$	3.65685	0.125503
$850$	0	0
$851$	−24.9706	−0.855980
$852$	7.31371	0.250564
$853$	−48.8284	−1.67185	−0.835927	−	0.548841i	$-0.815069\pi$
$853$	−48.8284	−1.67185	−0.835927	+	0.548841i	$0.815069\pi$
$854$	0	0
$855$	0	0
$856$	1.65685	0.0566301
$857$	−19.0711	−0.651455	−0.325728	−	0.945464i	$-0.605609\pi$
$857$	−19.0711	−0.651455	−0.325728	+	0.945464i	$0.605609\pi$
$858$	2.34315	0.0799937
$859$	−35.2132	−1.20146	−0.600729	−	0.799452i	$-0.705123\pi$
$859$	−35.2132	−1.20146	−0.600729	+	0.799452i	$0.705123\pi$
$860$	0	0
$861$	0	0
$862$	−33.6569	−1.14636
$863$	−28.9706	−0.986169	−0.493085	−	0.869981i	$-0.664131\pi$
$863$	−28.9706	−0.986169	−0.493085	+	0.869981i	$0.664131\pi$
$864$	−3.31371	−0.112735
$865$	0	0
$866$	13.4142	0.455834
$867$	7.21320	0.244973
$868$	0	0
$869$	−5.65685	−0.191896
$870$	0	0
$871$	8.00000	0.271070
$872$	−14.4853	−0.490534
$873$	43.1543	1.46055
$874$	−23.3137	−0.788598
$875$	0	0
$876$	−3.85786	−0.130345
$877$	−26.2843	−0.887557	−0.443778	−	0.896137i	$-0.646362\pi$
$877$	−26.2843	−0.887557	−0.443778	+	0.896137i	$0.646362\pi$
$878$	−8.97056	−0.302742
$879$	−11.5147	−0.388382
$880$	0	0
$881$	−34.3848	−1.15845	−0.579226	−	0.815167i	$-0.696645\pi$
$881$	−34.3848	−1.15845	−0.579226	+	0.815167i	$0.696645\pi$
$882$	0	0
$883$	30.3431	1.02113	0.510564	−	0.859840i	$-0.329437\pi$
$883$	30.3431	1.02113	0.510564	+	0.859840i	$0.329437\pi$
$884$	4.48528	0.150856
$885$	0	0
$886$	−36.9706	−1.24205
$887$	7.11270	0.238821	0.119411	−	0.992845i	$-0.461900\pi$
$887$	7.11270	0.238821	0.119411	+	0.992845i	$0.461900\pi$
$888$	−2.14214	−0.0718854
$889$	0	0
$890$	0	0
$891$	29.1127	0.975312
$892$	15.3137	0.512741
$893$	−36.9706	−1.23717
$894$	−3.51472	−0.117550
$895$	0	0
$896$	0	0
$897$	3.31371	0.110642
$898$	28.6274	0.955309
$899$	−2.34315	−0.0781483
$900$	0	0
$901$	56.7696	1.89127
$902$	−53.4558	−1.77988
$903$	0	0
$904$	−7.31371	−0.243250
$905$	0	0
$906$	6.62742	0.220181
$907$	56.2843	1.86889	0.934444	−	0.356109i	$-0.115897\pi$
$907$	56.2843	1.86889	0.934444	+	0.356109i	$0.115897\pi$
$908$	9.75736	0.323809
$909$	−24.7452	−0.820745
$910$	0	0
$911$	−20.2843	−0.672048	−0.336024	−	0.941853i	$-0.609082\pi$
$911$	−20.2843	−0.672048	−0.336024	+	0.941853i	$0.609082\pi$
$912$	−2.00000	−0.0662266
$913$	30.1421	0.997559
$914$	−10.3431	−0.342121
$915$	0	0
$916$	−12.1421	−0.401187
$917$	0	0
$918$	−17.9411	−0.592145
$919$	32.4853	1.07159	0.535795	−	0.844348i	$-0.320012\pi$
$919$	32.4853	1.07159	0.535795	+	0.844348i	$0.320012\pi$
$920$	0	0
$921$	17.0294	0.561139
$922$	7.17157	0.236183
$923$	10.3431	0.340449
$924$	0	0
$925$	0	0
$926$	−16.9706	−0.557687
$927$	24.3675	0.800335
$928$	0.828427	0.0271945
$929$	−25.2132	−0.827218	−0.413609	−	0.910455i	$-0.635732\pi$
$929$	−25.2132	−0.827218	−0.413609	+	0.910455i	$0.635732\pi$
$930$	0	0
$931$	0	0
$932$	−0.686292	−0.0224802
$933$	−2.34315	−0.0767111
$934$	3.89949	0.127595
$935$	0	0
$936$	−2.20101	−0.0719423
$937$	11.7574	0.384096	0.192048	−	0.981386i	$-0.438487\pi$
$937$	11.7574	0.384096	0.192048	+	0.981386i	$0.438487\pi$
$938$	0	0
$939$	−13.1127	−0.427917
$940$	0	0
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	−6.14214	−0.200122
$943$	−75.5980	−2.46181
$944$	−11.4142	−0.371501
$945$	0	0
$946$	15.3137	0.497892
$947$	−0.828427	−0.0269203	−0.0134601	−	0.999909i	$-0.504285\pi$
$947$	−0.828427	−0.0269203	−0.0134601	+	0.999909i	$0.504285\pi$
$948$	−0.686292	−0.0222897
$949$	−5.45584	−0.177104
$950$	0	0
$951$	3.79899	0.123191
$952$	0	0
$953$	−11.6569	−0.377603	−0.188801	−	0.982015i	$-0.560460\pi$
$953$	−11.6569	−0.377603	−0.188801	+	0.982015i	$0.560460\pi$
$954$	−27.8579	−0.901932
$955$	0	0
$956$	−9.65685	−0.312325
$957$	2.34315	0.0757431
$958$	22.8284	0.737553
$959$	0	0
$960$	0	0
$961$	−23.0000	−0.741935
$962$	−3.02944	−0.0976730
$963$	−4.40202	−0.141853
$964$	10.5858	0.340945
$965$	0	0
$966$	0	0
$967$	−13.4558	−0.432711	−0.216355	−	0.976315i	$-0.569417\pi$
$967$	−13.4558	−0.432711	−0.216355	+	0.976315i	$0.569417\pi$
$968$	12.3137	0.395778
$969$	−10.8284	−0.347859
$970$	0	0
$971$	−37.3553	−1.19879	−0.599395	−	0.800453i	$-0.704592\pi$
$971$	−37.3553	−1.19879	−0.599395	+	0.800453i	$0.704592\pi$
$972$	13.4731	0.432150
$973$	0	0
$974$	7.79899	0.249896
$975$	0	0
$976$	13.3137	0.426161
$977$	35.3137	1.12979	0.564893	−	0.825164i	$-0.308918\pi$
$977$	35.3137	1.12979	0.564893	+	0.825164i	$0.308918\pi$
$978$	−4.76955	−0.152513
$979$	61.4558	1.96414
$980$	0	0
$981$	38.4853	1.22874
$982$	−24.2843	−0.774942
$983$	−51.7990	−1.65213	−0.826066	−	0.563574i	$-0.809426\pi$
$983$	−51.7990	−1.65213	−0.826066	+	0.563574i	$0.809426\pi$
$984$	−6.48528	−0.206743
$985$	0	0
$986$	4.48528	0.142840
$987$	0	0
$988$	−2.82843	−0.0899843
$989$	21.6569	0.688648
$990$	0	0
$991$	−28.7696	−0.913895	−0.456947	−	0.889494i	$-0.651057\pi$
$991$	−28.7696	−0.913895	−0.456947	+	0.889494i	$0.651057\pi$
$992$	−2.82843	−0.0898027
$993$	−3.39697	−0.107800
$994$	0	0
$995$	0	0
$996$	3.65685	0.115872
$997$	38.2843	1.21248	0.606238	−	0.795284i	$-0.292678\pi$
$997$	38.2843	1.21248	0.606238	+	0.795284i	$0.292678\pi$
$998$	41.6569	1.31862
$999$	12.1177	0.383389

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.a.bs.1.1		2
5.2	odd	4		2450.2.c.w.99.4		4
5.3	odd	4		2450.2.c.w.99.1		4
5.4	even	2		490.2.a.l.1.2	✓	2
7.6	odd	2		2450.2.a.bn.1.2		2
15.14	odd	2		4410.2.a.by.1.1		2
20.19	odd	2		3920.2.a.ca.1.1		2
35.4	even	6		490.2.e.j.471.1		4
35.9	even	6		490.2.e.j.361.1		4
35.13	even	4		2450.2.c.t.99.2		4
35.19	odd	6		490.2.e.i.361.2		4
35.24	odd	6		490.2.e.i.471.2		4
35.27	even	4		2450.2.c.t.99.3		4
35.34	odd	2		490.2.a.m.1.1	yes	2
105.104	even	2		4410.2.a.bt.1.1		2
140.139	even	2		3920.2.a.bm.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.a.l.1.2	✓	2	5.4	even	2
490.2.a.m.1.1	yes	2	35.34	odd	2
490.2.e.i.361.2		4	35.19	odd	6
490.2.e.i.471.2		4	35.24	odd	6
490.2.e.j.361.1		4	35.9	even	6
490.2.e.j.471.1		4	35.4	even	6
2450.2.a.bn.1.2		2	7.6	odd	2
2450.2.a.bs.1.1		2	1.1	even	1	trivial
2450.2.c.t.99.2		4	35.13	even	4
2450.2.c.t.99.3		4	35.27	even	4
2450.2.c.w.99.1		4	5.3	odd	4
2450.2.c.w.99.4		4	5.2	odd	4
3920.2.a.bm.1.2		2	140.139	even	2
3920.2.a.ca.1.1		2	20.19	odd	2
4410.2.a.bt.1.1		2	105.104	even	2
4410.2.a.by.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.585786 q^{3} +1.00000 q^{4} +0.585786 q^{6} +1.00000 q^{8} -2.65685 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.585786 q^{3} +1.00000 q^{4} +0.585786 q^{6} +1.00000 q^{8} -2.65685 q^{9} +4.82843 q^{11} +0.585786 q^{12} +0.828427 q^{13} +1.00000 q^{16} +5.41421 q^{17} -2.65685 q^{18} -3.41421 q^{19} +4.82843 q^{22} +6.82843 q^{23} +0.585786 q^{24} +0.828427 q^{26} -3.31371 q^{27} +0.828427 q^{29} -2.82843 q^{31} +1.00000 q^{32} +2.82843 q^{33} +5.41421 q^{34} -2.65685 q^{36} -3.65685 q^{37} -3.41421 q^{38} +0.485281 q^{39} -11.0711 q^{41} +3.17157 q^{43} +4.82843 q^{44} +6.82843 q^{46} +10.8284 q^{47} +0.585786 q^{48} +3.17157 q^{51} +0.828427 q^{52} +10.4853 q^{53} -3.31371 q^{54} -2.00000 q^{57} +0.828427 q^{58} -11.4142 q^{59} +13.3137 q^{61} -2.82843 q^{62} +1.00000 q^{64} +2.82843 q^{66} +9.65685 q^{67} +5.41421 q^{68} +4.00000 q^{69} +12.4853 q^{71} -2.65685 q^{72} -6.58579 q^{73} -3.65685 q^{74} -3.41421 q^{76} +0.485281 q^{78} -1.17157 q^{79} +6.02944 q^{81} -11.0711 q^{82} +6.24264 q^{83} +3.17157 q^{86} +0.485281 q^{87} +4.82843 q^{88} +12.7279 q^{89} +6.82843 q^{92} -1.65685 q^{93} +10.8284 q^{94} +0.585786 q^{96} -16.2426 q^{97} -12.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 2 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 4 * q^6 + 2 * q^8 + 6 * q^9	\( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 2 q^{8} + 6 q^{9} + 4 q^{11} + 4 q^{12} - 4 q^{13} + 2 q^{16} + 8 q^{17} + 6 q^{18} - 4 q^{19} + 4 q^{22} + 8 q^{23} + 4 q^{24} - 4 q^{26} + 16 q^{27} - 4 q^{29} + 2 q^{32} + 8 q^{34} + 6 q^{36} + 4 q^{37} - 4 q^{38} - 16 q^{39} - 8 q^{41} + 12 q^{43} + 4 q^{44} + 8 q^{46} + 16 q^{47} + 4 q^{48} + 12 q^{51} - 4 q^{52} + 4 q^{53} + 16 q^{54} - 4 q^{57} - 4 q^{58} - 20 q^{59} + 4 q^{61} + 2 q^{64} + 8 q^{67} + 8 q^{68} + 8 q^{69} + 8 q^{71} + 6 q^{72} - 16 q^{73} + 4 q^{74} - 4 q^{76} - 16 q^{78} - 8 q^{79} + 46 q^{81} - 8 q^{82} + 4 q^{83} + 12 q^{86} - 16 q^{87} + 4 q^{88} + 8 q^{92} + 8 q^{93} + 16 q^{94} + 4 q^{96} - 24 q^{97} - 20 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 4 * q^6 + 2 * q^8 + 6 * q^9 + 4 * q^11 + 4 * q^12 - 4 * q^13 + 2 * q^16 + 8 * q^17 + 6 * q^18 - 4 * q^19 + 4 * q^22 + 8 * q^23 + 4 * q^24 - 4 * q^26 + 16 * q^27 - 4 * q^29 + 2 * q^32 + 8 * q^34 + 6 * q^36 + 4 * q^37 - 4 * q^38 - 16 * q^39 - 8 * q^41 + 12 * q^43 + 4 * q^44 + 8 * q^46 + 16 * q^47 + 4 * q^48 + 12 * q^51 - 4 * q^52 + 4 * q^53 + 16 * q^54 - 4 * q^57 - 4 * q^58 - 20 * q^59 + 4 * q^61 + 2 * q^64 + 8 * q^67 + 8 * q^68 + 8 * q^69 + 8 * q^71 + 6 * q^72 - 16 * q^73 + 4 * q^74 - 4 * q^76 - 16 * q^78 - 8 * q^79 + 46 * q^81 - 8 * q^82 + 4 * q^83 + 12 * q^86 - 16 * q^87 + 4 * q^88 + 8 * q^92 + 8 * q^93 + 16 * q^94 + 4 * q^96 - 24 * q^97 - 20 * q^99

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