LMFDB - Embedded newform 2450.2.c.t.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(99,2450)]

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([1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2450.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$19.5633484952$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 490)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2450.99
Dual form			2450.2.c.t.99.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -0.585786i q^{3} -1.00000 q^{4} -0.585786 q^{6} +1.00000i q^{8} +2.65685 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -0.585786i q^{3} -1.00000 q^{4} -0.585786 q^{6} +1.00000i q^{8} +2.65685 q^{9} +4.82843 q^{11} +0.585786i q^{12} -0.828427i q^{13} +1.00000 q^{16} +5.41421i q^{17} -2.65685i q^{18} -3.41421 q^{19} -4.82843i q^{22} +6.82843i q^{23} +0.585786 q^{24} -0.828427 q^{26} -3.31371i q^{27} -0.828427 q^{29} +2.82843 q^{31} -1.00000i q^{32} -2.82843i q^{33} +5.41421 q^{34} -2.65685 q^{36} +3.65685i q^{37} +3.41421i q^{38} -0.485281 q^{39} +11.0711 q^{41} +3.17157i q^{43} -4.82843 q^{44} +6.82843 q^{46} +10.8284i q^{47} -0.585786i q^{48} +3.17157 q^{51} +0.828427i q^{52} +10.4853i q^{53} -3.31371 q^{54} +2.00000i q^{57} +0.828427i q^{58} -11.4142 q^{59} -13.3137 q^{61} -2.82843i q^{62} -1.00000 q^{64} -2.82843 q^{66} -9.65685i q^{67} -5.41421i q^{68} +4.00000 q^{69} +12.4853 q^{71} +2.65685i q^{72} +6.58579i q^{73} +3.65685 q^{74} +3.41421 q^{76} +0.485281i q^{78} +1.17157 q^{79} +6.02944 q^{81} -11.0711i q^{82} -6.24264i q^{83} +3.17157 q^{86} +0.485281i q^{87} +4.82843i q^{88} +12.7279 q^{89} -6.82843i q^{92} -1.65685i q^{93} +10.8284 q^{94} -0.585786 q^{96} -16.2426i q^{97} +12.8284 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 8 q^{6} - 12 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 - 8 * q^6 - 12 * q^9	$ 4 q - 4 q^{4} - 8 q^{6} - 12 q^{9} + 8 q^{11} + 4 q^{16} - 8 q^{19} + 8 q^{24} + 8 q^{26} + 8 q^{29} + 16 q^{34} + 12 q^{36} + 32 q^{39} + 16 q^{41} - 8 q^{44} + 16 q^{46} + 24 q^{51} + 32 q^{54} - 40 q^{59} - 8 q^{61} - 4 q^{64} + 16 q^{69} + 16 q^{71} - 8 q^{74} + 8 q^{76} + 16 q^{79} + 92 q^{81} + 24 q^{86} + 32 q^{94} - 8 q^{96} + 40 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 8 * q^6 - 12 * q^9 + 8 * q^11 + 4 * q^16 - 8 * q^19 + 8 * q^24 + 8 * q^26 + 8 * q^29 + 16 * q^34 + 12 * q^36 + 32 * q^39 + 16 * q^41 - 8 * q^44 + 16 * q^46 + 24 * q^51 + 32 * q^54 - 40 * q^59 - 8 * q^61 - 4 * q^64 + 16 * q^69 + 16 * q^71 - 8 * q^74 + 8 * q^76 + 16 * q^79 + 92 * q^81 + 24 * q^86 + 32 * q^94 - 8 * q^96 + 40 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	− 0.585786i	− 0.338204i	−0.985599	−	0.169102i	$-0.945913\pi$
$3$	− 0.585786i	− 0.338204i	0.985599	−	0.169102i	$-0.0540867\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−0.585786	−0.239146
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	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	$0.240491\pi$
$12$	0.585786i	0.169102i
$13$	− 0.828427i	− 0.229764i	−0.993379	−	0.114882i	$-0.963351\pi$
$13$	− 0.828427i	− 0.229764i	0.993379	−	0.114882i	$-0.0366490\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	5.41421i	1.31314i	0.754265	+	0.656570i	$0.227993\pi$
$17$	5.41421i	1.31314i	−0.754265	+	0.656570i	$0.772007\pi$
$18$	− 2.65685i	− 0.626227i
$19$	−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
$22$	− 4.82843i	− 1.02942i
$23$	6.82843i	1.42383i	0.702268	+	0.711913i	$0.252171\pi$
$23$	6.82843i	1.42383i	−0.702268	+	0.711913i	$0.747829\pi$
$24$	0.585786	0.119573
$25$	0	0
$26$	−0.828427	−0.162468
$27$	− 3.31371i	− 0.637723i
$28$	0	0
$29$	−0.828427	−0.153835	−0.0769175	−	0.997037i	$-0.524508\pi$
$29$	−0.828427	−0.153835	−0.0769175	+	0.997037i	$0.524508\pi$
$30$	0	0
$31$	2.82843	0.508001	0.254000	−	0.967204i	$-0.418254\pi$
$31$	2.82843	0.508001	0.254000	+	0.967204i	$0.418254\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 2.82843i	− 0.492366i
$34$	5.41421	0.928530
$35$	0	0
$36$	−2.65685	−0.442809
$37$	3.65685i	0.601183i	0.953753	+	0.300592i	$0.0971841\pi$
$37$	3.65685i	0.601183i	−0.953753	+	0.300592i	$0.902816\pi$
$38$	3.41421i	0.553859i
$39$	−0.485281	−0.0777072
$40$	0	0
$41$	11.0711	1.72901	0.864505	−	0.502624i	$-0.167632\pi$
$41$	11.0711	1.72901	0.864505	+	0.502624i	$0.167632\pi$
$42$	0	0
$43$	3.17157i	0.483660i	0.970319	+	0.241830i	$0.0777477\pi$
$43$	3.17157i	0.483660i	−0.970319	+	0.241830i	$0.922252\pi$
$44$	−4.82843	−0.727913
$45$	0	0
$46$	6.82843	1.00680
$47$	10.8284i	1.57949i	0.613436	+	0.789744i	$0.289787\pi$
$47$	10.8284i	1.57949i	−0.613436	+	0.789744i	$0.710213\pi$
$48$	− 0.585786i	− 0.0845510i
$49$	0	0
$50$	0	0
$51$	3.17157	0.444109
$52$	0.828427i	0.114882i
$53$	10.4853i	1.44026i	0.693837	+	0.720132i	$0.255919\pi$
$53$	10.4853i	1.44026i	−0.693837	+	0.720132i	$0.744081\pi$
$54$	−3.31371	−0.450939
$55$	0	0
$56$	0	0
$57$	2.00000i	0.264906i
$58$	0.828427i	0.108778i
$59$	−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	$-0.824807\pi$
$61$	−13.3137	−1.70465	−0.852323	+	0.523016i	$0.824807\pi$
$62$	− 2.82843i	− 0.359211i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−2.82843	−0.348155
$67$	− 9.65685i	− 1.17977i	−0.807486	−	0.589886i	$-0.799173\pi$
$67$	− 9.65685i	− 1.17977i	0.807486	−	0.589886i	$-0.200827\pi$
$68$	− 5.41421i	− 0.656570i
$69$	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	$0.234416\pi$
$72$	2.65685i	0.313113i
$73$	6.58579i	0.770808i	0.922748	+	0.385404i	$0.125938\pi$
$73$	6.58579i	0.770808i	−0.922748	+	0.385404i	$0.874062\pi$
$74$	3.65685	0.425101
$75$	0	0
$76$	3.41421	0.391637
$77$	0	0
$78$	0.485281i	0.0549473i
$79$	1.17157	0.131812	0.0659061	−	0.997826i	$-0.479006\pi$
$79$	1.17157	0.131812	0.0659061	+	0.997826i	$0.479006\pi$
$80$	0	0
$81$	6.02944	0.669937
$82$	− 11.0711i	− 1.22259i
$83$	− 6.24264i	− 0.685219i	−0.939478	−	0.342609i	$-0.888689\pi$
$83$	− 6.24264i	− 0.685219i	0.939478	−	0.342609i	$-0.111311\pi$
$84$	0	0
$85$	0	0
$86$	3.17157	0.341999
$87$	0.485281i	0.0520276i
$88$	4.82843i	0.514712i
$89$	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
$92$	− 6.82843i	− 0.711913i
$93$	− 1.65685i	− 0.171808i
$94$	10.8284	1.11687
$95$	0	0
$96$	−0.585786	−0.0597866
$97$	− 16.2426i	− 1.64919i	−0.565723	−	0.824595i	$-0.691403\pi$
$97$	− 16.2426i	− 1.64919i	0.565723	−	0.824595i	$-0.308597\pi$
$98$	0	0
$99$	12.8284	1.28931
$100$	0	0
$101$	−9.31371	−0.926749	−0.463374	−	0.886163i	$-0.653361\pi$
$101$	−9.31371	−0.926749	−0.463374	+	0.886163i	$0.653361\pi$
$102$	− 3.17157i	− 0.314033i
$103$	9.17157i	0.903702i	0.892093	+	0.451851i	$0.149236\pi$
$103$	9.17157i	0.903702i	−0.892093	+	0.451851i	$0.850764\pi$
$104$	0.828427	0.0812340
$105$	0	0
$106$	10.4853	1.01842
$107$	− 1.65685i	− 0.160174i	−0.996788	−	0.0800871i	$-0.974480\pi$
$107$	− 1.65685i	− 0.160174i	0.996788	−	0.0800871i	$-0.0255198\pi$
$108$	3.31371i	0.318862i
$109$	14.4853	1.38744	0.693719	−	0.720246i	$-0.255971\pi$
$109$	14.4853	1.38744	0.693719	+	0.720246i	$0.255971\pi$
$110$	0	0
$111$	2.14214	0.203323
$112$	0	0
$113$	− 7.31371i	− 0.688016i	−0.938967	−	0.344008i	$-0.888215\pi$
$113$	− 7.31371i	− 0.688016i	0.938967	−	0.344008i	$-0.111785\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	0.828427	0.0769175
$117$	− 2.20101i	− 0.203483i
$118$	11.4142i	1.05076i
$119$	0	0
$120$	0	0
$121$	12.3137	1.11943
$122$	13.3137i	1.20537i
$123$	− 6.48528i	− 0.584758i
$124$	−2.82843	−0.254000
$125$	0	0
$126$	0	0
$127$	− 2.82843i	− 0.250982i	−0.992095	−	0.125491i	$-0.959949\pi$
$127$	− 2.82843i	− 0.250982i	0.992095	−	0.125491i	$-0.0400507\pi$
$128$	1.00000i	0.0883883i
$129$	1.85786	0.163576
$130$	0	0
$131$	2.24264	0.195940	0.0979702	−	0.995189i	$-0.468765\pi$
$131$	2.24264	0.195940	0.0979702	+	0.995189i	$0.468765\pi$
$132$	2.82843i	0.246183i
$133$	0	0
$134$	−9.65685	−0.834225
$135$	0	0
$136$	−5.41421	−0.464265
$137$	− 16.0000i	− 1.36697i	−0.729964	−	0.683486i	$-0.760463\pi$
$137$	− 16.0000i	− 1.36697i	0.729964	−	0.683486i	$-0.239537\pi$
$138$	− 4.00000i	− 0.340503i
$139$	−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
$142$	− 12.4853i	− 1.04774i
$143$	− 4.00000i	− 0.334497i
$144$	2.65685	0.221405
$145$	0	0
$146$	6.58579	0.545044
$147$	0	0
$148$	− 3.65685i	− 0.300592i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	11.3137	0.920697	0.460348	−	0.887738i	$-0.347725\pi$
$151$	11.3137	0.920697	0.460348	+	0.887738i	$0.347725\pi$
$152$	− 3.41421i	− 0.276929i
$153$	14.3848i	1.16294i
$154$	0	0
$155$	0	0
$156$	0.485281	0.0388536
$157$	− 10.4853i	− 0.836817i	−0.908259	−	0.418408i	$-0.862588\pi$
$157$	− 10.4853i	− 0.836817i	0.908259	−	0.418408i	$-0.137412\pi$
$158$	− 1.17157i	− 0.0932053i
$159$	6.14214	0.487103
$160$	0	0
$161$	0	0
$162$	− 6.02944i	− 0.473717i
$163$	− 8.14214i	− 0.637741i	−0.947798	−	0.318871i	$-0.896696\pi$
$163$	− 8.14214i	− 0.637741i	0.947798	−	0.318871i	$-0.103304\pi$
$164$	−11.0711	−0.864505
$165$	0	0
$166$	−6.24264	−0.484523
$167$	− 23.7990i	− 1.84162i	−0.390010	−	0.920811i	$-0.627529\pi$
$167$	− 23.7990i	− 1.84162i	0.390010	−	0.920811i	$-0.372471\pi$
$168$	0	0
$169$	12.3137	0.947208
$170$	0	0
$171$	−9.07107	−0.693682
$172$	− 3.17157i	− 0.241830i
$173$	3.17157i	0.241130i	0.992705	+	0.120565i	$0.0384707\pi$
$173$	3.17157i	0.241130i	−0.992705	+	0.120565i	$0.961529\pi$
$174$	0.485281	0.0367891
$175$	0	0
$176$	4.82843	0.363956
$177$	6.68629i	0.502572i
$178$	− 12.7279i	− 0.953998i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	14.4853	1.07668	0.538341	−	0.842727i	$-0.319051\pi$
$181$	14.4853	1.07668	0.538341	+	0.842727i	$0.319051\pi$
$182$	0	0
$183$	7.79899i	0.576518i
$184$	−6.82843	−0.503398
$185$	0	0
$186$	−1.65685	−0.121486
$187$	26.1421i	1.91170i
$188$	− 10.8284i	− 0.789744i
$189$	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	$0.727904\pi$
$192$	0.585786i	0.0422755i
$193$	5.65685i	0.407189i	0.979055	+	0.203595i	$0.0652625\pi$
$193$	5.65685i	0.407189i	−0.979055	+	0.203595i	$0.934738\pi$
$194$	−16.2426	−1.16615
$195$	0	0
$196$	0	0
$197$	13.7990i	0.983137i	0.870839	+	0.491569i	$0.163576\pi$
$197$	13.7990i	0.983137i	−0.870839	+	0.491569i	$0.836424\pi$
$198$	− 12.8284i	− 0.911677i
$199$	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
$202$	9.31371i	0.655310i
$203$	0	0
$204$	−3.17157	−0.222055
$205$	0	0
$206$	9.17157	0.639014
$207$	18.1421i	1.26097i
$208$	− 0.828427i	− 0.0574411i
$209$	−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	$0.869041\pi$
$212$	− 10.4853i	− 0.720132i
$213$	− 7.31371i	− 0.501127i
$214$	−1.65685	−0.113260
$215$	0	0
$216$	3.31371	0.225469
$217$	0	0
$218$	− 14.4853i	− 0.981067i
$219$	3.85786	0.260690
$220$	0	0
$221$	4.48528	0.301713
$222$	− 2.14214i	− 0.143771i
$223$	− 15.3137i	− 1.02548i	−0.858543	−	0.512741i	$-0.828630\pi$
$223$	− 15.3137i	− 1.02548i	0.858543	−	0.512741i	$-0.171370\pi$
$224$	0	0
$225$	0	0
$226$	−7.31371	−0.486501
$227$	9.75736i	0.647619i	0.946122	+	0.323809i	$0.104964\pi$
$227$	9.75736i	0.647619i	−0.946122	+	0.323809i	$0.895036\pi$
$228$	− 2.00000i	− 0.132453i
$229$	−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
$232$	− 0.828427i	− 0.0543889i
$233$	− 0.686292i	− 0.0449605i	−0.999747	−	0.0224802i	$-0.992844\pi$
$233$	− 0.686292i	− 0.0449605i	0.999747	−	0.0224802i	$-0.00715628\pi$
$234$	−2.20101	−0.143885
$235$	0	0
$236$	11.4142	0.743002
$237$	− 0.686292i	− 0.0445794i
$238$	0	0
$239$	9.65685	0.624650	0.312325	−	0.949975i	$-0.398892\pi$
$239$	9.65685	0.624650	0.312325	+	0.949975i	$0.398892\pi$
$240$	0	0
$241$	−10.5858	−0.681890	−0.340945	−	0.940083i	$-0.610747\pi$
$241$	−10.5858	−0.681890	−0.340945	+	0.940083i	$0.610747\pi$
$242$	− 12.3137i	− 0.791555i
$243$	− 13.4731i	− 0.864299i
$244$	13.3137	0.852323
$245$	0	0
$246$	−6.48528	−0.413486
$247$	2.82843i	0.179969i
$248$	2.82843i	0.179605i
$249$	−3.65685	−0.231744
$250$	0	0
$251$	3.41421	0.215503	0.107752	−	0.994178i	$-0.465635\pi$
$251$	3.41421	0.215503	0.107752	+	0.994178i	$0.465635\pi$
$252$	0	0
$253$	32.9706i	2.07284i
$254$	−2.82843	−0.177471
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.89949i	0.617514i	0.951141	+	0.308757i	$0.0999129\pi$
$257$	9.89949i	0.617514i	−0.951141	+	0.308757i	$0.900087\pi$
$258$	− 1.85786i	− 0.115666i
$259$	0	0
$260$	0	0
$261$	−2.20101	−0.136239
$262$	− 2.24264i	− 0.138551i
$263$	− 28.0000i	− 1.72655i	−0.504730	−	0.863277i	$-0.668408\pi$
$263$	− 28.0000i	− 1.72655i	0.504730	−	0.863277i	$-0.331592\pi$
$264$	2.82843	0.174078
$265$	0	0
$266$	0	0
$267$	− 7.45584i	− 0.456290i
$268$	9.65685i	0.589886i
$269$	−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	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	5.41421i	0.328285i
$273$	0	0
$274$	−16.0000	−0.966595
$275$	0	0
$276$	−4.00000	−0.240772
$277$	20.1421i	1.21022i	0.796140	+	0.605112i	$0.206872\pi$
$277$	20.1421i	1.21022i	−0.796140	+	0.605112i	$0.793128\pi$
$278$	0.100505i	0.00602789i
$279$	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	$0.423305\pi$
$282$	− 6.34315i	− 0.377729i
$283$	− 6.24264i	− 0.371086i	−0.982636	−	0.185543i	$-0.940596\pi$
$283$	− 6.24264i	− 0.371086i	0.982636	−	0.185543i	$-0.0594045\pi$
$284$	−12.4853	−0.740865
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	− 2.65685i	− 0.156557i
$289$	−12.3137	−0.724336
$290$	0	0
$291$	−9.51472	−0.557763
$292$	− 6.58579i	− 0.385404i
$293$	19.6569i	1.14837i	0.818727	+	0.574183i	$0.194680\pi$
$293$	19.6569i	1.14837i	−0.818727	+	0.574183i	$0.805320\pi$
$294$	0	0
$295$	0	0
$296$	−3.65685	−0.212550
$297$	− 16.0000i	− 0.928414i
$298$	− 6.00000i	− 0.347571i
$299$	5.65685	0.327144
$300$	0	0
$301$	0	0
$302$	− 11.3137i	− 0.651031i
$303$	5.45584i	0.313430i
$304$	−3.41421	−0.195819
$305$	0	0
$306$	14.3848	0.822323
$307$	29.0711i	1.65917i	0.558378	+	0.829587i	$0.311424\pi$
$307$	29.0711i	1.65917i	−0.558378	+	0.829587i	$0.688576\pi$
$308$	0	0
$309$	5.37258	0.305636
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	− 0.485281i	− 0.0274736i
$313$	22.3848i	1.26526i	0.774453	+	0.632631i	$0.218025\pi$
$313$	22.3848i	1.26526i	−0.774453	+	0.632631i	$0.781975\pi$
$314$	−10.4853	−0.591719
$315$	0	0
$316$	−1.17157	−0.0659061
$317$	− 6.48528i	− 0.364250i	−0.983275	−	0.182125i	$-0.941702\pi$
$317$	− 6.48528i	− 0.364250i	0.983275	−	0.182125i	$-0.0582975\pi$
$318$	− 6.14214i	− 0.344434i
$319$	−4.00000	−0.223957
$320$	0	0
$321$	−0.970563	−0.0541715
$322$	0	0
$323$	− 18.4853i	− 1.02855i
$324$	−6.02944	−0.334969
$325$	0	0
$326$	−8.14214	−0.450951
$327$	− 8.48528i	− 0.469237i
$328$	11.0711i	0.611297i
$329$	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	$0.550946\pi$
$332$	6.24264i	0.342609i
$333$	9.71573i	0.532419i
$334$	−23.7990	−1.30222
$335$	0	0
$336$	0	0
$337$	6.00000i	0.326841i	0.986557	+	0.163420i	$0.0522527\pi$
$337$	6.00000i	0.326841i	−0.986557	+	0.163420i	$0.947747\pi$
$338$	− 12.3137i	− 0.669777i
$339$	−4.28427	−0.232690
$340$	0	0
$341$	13.6569	0.739560
$342$	9.07107i	0.490507i
$343$	0	0
$344$	−3.17157	−0.171000
$345$	0	0
$346$	3.17157	0.170505
$347$	− 8.82843i	− 0.473935i	−0.971518	−	0.236967i	$-0.923847\pi$
$347$	− 8.82843i	− 0.473935i	0.971518	−	0.236967i	$-0.0761535\pi$
$348$	− 0.485281i	− 0.0260138i
$349$	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
$352$	− 4.82843i	− 0.257356i
$353$	− 34.3848i	− 1.83012i	−0.403321	−	0.915058i	$-0.632144\pi$
$353$	− 34.3848i	− 1.83012i	0.403321	−	0.915058i	$-0.367856\pi$
$354$	6.68629	0.355372
$355$	0	0
$356$	−12.7279	−0.674579
$357$	0	0
$358$	4.00000i	0.211407i
$359$	28.2843	1.49279	0.746393	−	0.665505i	$-0.231784\pi$
$359$	28.2843	1.49279	0.746393	+	0.665505i	$0.231784\pi$
$360$	0	0
$361$	−7.34315	−0.386481
$362$	− 14.4853i	− 0.761329i
$363$	− 7.21320i	− 0.378595i
$364$	0	0
$365$	0	0
$366$	7.79899	0.407660
$367$	− 8.97056i	− 0.468260i	−0.972205	−	0.234130i	$-0.924776\pi$
$367$	− 8.97056i	− 0.468260i	0.972205	−	0.234130i	$-0.0752241\pi$
$368$	6.82843i	0.355956i
$369$	29.4142	1.53124
$370$	0	0
$371$	0	0
$372$	1.65685i	0.0859039i
$373$	13.5147i	0.699766i	0.936793	+	0.349883i	$0.113779\pi$
$373$	13.5147i	0.699766i	−0.936793	+	0.349883i	$0.886221\pi$
$374$	26.1421	1.35178
$375$	0	0
$376$	−10.8284	−0.558433
$377$	0.686292i	0.0353458i
$378$	0	0
$379$	−17.5147	−0.899671	−0.449835	−	0.893112i	$-0.648517\pi$
$379$	−17.5147	−0.899671	−0.449835	+	0.893112i	$0.648517\pi$
$380$	0	0
$381$	−1.65685	−0.0848832
$382$	18.1421i	0.928232i
$383$	15.5147i	0.792765i	0.918085	+	0.396383i	$0.129735\pi$
$383$	15.5147i	0.792765i	−0.918085	+	0.396383i	$0.870265\pi$
$384$	0.585786	0.0298933
$385$	0	0
$386$	5.65685	0.287926
$387$	8.42641i	0.428338i
$388$	16.2426i	0.824595i
$389$	0.142136	0.00720656	0.00360328	−	0.999994i	$-0.498853\pi$
$389$	0.142136	0.00720656	0.00360328	+	0.999994i	$0.498853\pi$
$390$	0	0
$391$	−36.9706	−1.86968
$392$	0	0
$393$	− 1.31371i	− 0.0662678i
$394$	13.7990	0.695183
$395$	0	0
$396$	−12.8284	−0.644653
$397$	− 5.79899i	− 0.291043i	−0.989355	−	0.145521i	$-0.953514\pi$
$397$	− 5.79899i	− 0.291043i	0.989355	−	0.145521i	$-0.0464860\pi$
$398$	− 0.485281i	− 0.0243250i
$399$	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	$0.547867\pi$
$402$	5.65685i	0.282138i
$403$	− 2.34315i	− 0.116720i
$404$	9.31371	0.463374
$405$	0	0
$406$	0	0
$407$	17.6569i	0.875218i
$408$	3.17157i	0.157016i
$409$	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
$412$	− 9.17157i	− 0.451851i
$413$	0	0
$414$	18.1421	0.891637
$415$	0	0
$416$	−0.828427	−0.0406170
$417$	0.0588745i	0.00288310i
$418$	16.4853i	0.806321i
$419$	−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	$0.541333\pi$
$422$	26.6274i	1.29620i
$423$	28.7696i	1.39882i
$424$	−10.4853	−0.509210
$425$	0	0
$426$	−7.31371	−0.354350
$427$	0	0
$428$	1.65685i	0.0800871i
$429$	−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	$0.800857\pi$
$432$	− 3.31371i	− 0.159431i
$433$	− 13.4142i	− 0.644646i	−0.946630	−	0.322323i	$-0.895536\pi$
$433$	− 13.4142i	− 0.644646i	0.946630	−	0.322323i	$-0.104464\pi$
$434$	0	0
$435$	0	0
$436$	−14.4853	−0.693719
$437$	− 23.3137i	− 1.11525i
$438$	− 3.85786i	− 0.184336i
$439$	−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
$442$	− 4.48528i	− 0.213343i
$443$	− 36.9706i	− 1.75652i	−0.478179	−	0.878262i	$-0.658703\pi$
$443$	− 36.9706i	− 1.75652i	0.478179	−	0.878262i	$-0.341297\pi$
$444$	−2.14214	−0.101661
$445$	0	0
$446$	−15.3137	−0.725125
$447$	− 3.51472i	− 0.166240i
$448$	0	0
$449$	−28.6274	−1.35101	−0.675506	−	0.737355i	$-0.736075\pi$
$449$	−28.6274	−1.35101	−0.675506	+	0.737355i	$0.736075\pi$
$450$	0	0
$451$	53.4558	2.51714
$452$	7.31371i	0.344008i
$453$	− 6.62742i	− 0.311383i
$454$	9.75736	0.457936
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	10.3431i	0.483832i	0.970297	+	0.241916i	$0.0777758\pi$
$457$	10.3431i	0.483832i	−0.970297	+	0.241916i	$0.922224\pi$
$458$	12.1421i	0.567365i
$459$	17.9411	0.837420
$460$	0	0
$461$	−7.17157	−0.334013	−0.167007	−	0.985956i	$-0.553410\pi$
$461$	−7.17157	−0.334013	−0.167007	+	0.985956i	$0.553410\pi$
$462$	0	0
$463$	− 16.9706i	− 0.788689i	−0.918963	−	0.394344i	$-0.870972\pi$
$463$	− 16.9706i	− 0.788689i	0.918963	−	0.394344i	$-0.129028\pi$
$464$	−0.828427	−0.0384588
$465$	0	0
$466$	−0.686292	−0.0317918
$467$	3.89949i	0.180447i	0.995922	+	0.0902236i	$0.0287582\pi$
$467$	3.89949i	0.180447i	−0.995922	+	0.0902236i	$0.971242\pi$
$468$	2.20101i	0.101742i
$469$	0	0
$470$	0	0
$471$	−6.14214	−0.283015
$472$	− 11.4142i	− 0.525382i
$473$	15.3137i	0.704125i
$474$	−0.686292	−0.0315224
$475$	0	0
$476$	0	0
$477$	27.8579i	1.27552i
$478$	− 9.65685i	− 0.441694i
$479$	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
$482$	10.5858i	0.482169i
$483$	0	0
$484$	−12.3137	−0.559714
$485$	0	0
$486$	−13.4731	−0.611152
$487$	− 7.79899i	− 0.353406i	−0.984264	−	0.176703i	$-0.943457\pi$
$487$	− 7.79899i	− 0.353406i	0.984264	−	0.176703i	$-0.0565432\pi$
$488$	− 13.3137i	− 0.602683i
$489$	−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	$0.684598\pi$
$492$	6.48528i	0.292379i
$493$	− 4.48528i	− 0.202007i
$494$	2.82843	0.127257
$495$	0	0
$496$	2.82843	0.127000
$497$	0	0
$498$	3.65685i	0.163868i
$499$	−41.6569	−1.86482	−0.932408	−	0.361406i	$-0.882297\pi$
$499$	−41.6569	−1.86482	−0.932408	+	0.361406i	$0.882297\pi$
$500$	0	0
$501$	−13.9411	−0.622844
$502$	− 3.41421i	− 0.152384i
$503$	− 6.34315i	− 0.282827i	−0.989951	−	0.141413i	$-0.954835\pi$
$503$	− 6.34315i	− 0.282827i	0.989951	−	0.141413i	$-0.0451647\pi$
$504$	0	0
$505$	0	0
$506$	32.9706	1.46572
$507$	− 7.21320i	− 0.320350i
$508$	2.82843i	0.125491i
$509$	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
$512$	− 1.00000i	− 0.0441942i
$513$	11.3137i	0.499512i
$514$	9.89949	0.436648
$515$	0	0
$516$	−1.85786	−0.0817879
$517$	52.2843i	2.29946i
$518$	0	0
$519$	1.85786	0.0815512
$520$	0	0
$521$	4.92893	0.215940	0.107970	−	0.994154i	$-0.465565\pi$
$521$	4.92893	0.215940	0.107970	+	0.994154i	$0.465565\pi$
$522$	2.20101i	0.0963356i
$523$	− 4.10051i	− 0.179303i	−0.995973	−	0.0896513i	$-0.971425\pi$
$523$	− 4.10051i	− 0.179303i	0.995973	−	0.0896513i	$-0.0285753\pi$
$524$	−2.24264	−0.0979702
$525$	0	0
$526$	−28.0000	−1.22086
$527$	15.3137i	0.667076i
$528$	− 2.82843i	− 0.123091i
$529$	−23.6274	−1.02728
$530$	0	0
$531$	−30.3259	−1.31603
$532$	0	0
$533$	− 9.17157i	− 0.397265i
$534$	−7.45584	−0.322646
$535$	0	0
$536$	9.65685	0.417113
$537$	2.34315i	0.101114i
$538$	1.51472i	0.0653042i
$539$	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	$0.366295\pi$
$542$	− 12.0000i	− 0.515444i
$543$	− 8.48528i	− 0.364138i
$544$	5.41421	0.232132
$545$	0	0
$546$	0	0
$547$	6.48528i	0.277291i	0.990342	+	0.138645i	$0.0442748\pi$
$547$	6.48528i	0.277291i	−0.990342	+	0.138645i	$0.955725\pi$
$548$	16.0000i	0.683486i
$549$	−35.3726	−1.50967
$550$	0	0
$551$	2.82843	0.120495
$552$	4.00000i	0.170251i
$553$	0	0
$554$	20.1421	0.855757
$555$	0	0
$556$	0.100505	0.00426236
$557$	20.8284i	0.882529i	0.897377	+	0.441264i	$0.145470\pi$
$557$	20.8284i	0.882529i	−0.897377	+	0.441264i	$0.854530\pi$
$558$	− 7.51472i	− 0.318123i
$559$	2.62742	0.111128
$560$	0	0
$561$	15.3137	0.646545
$562$	− 8.00000i	− 0.337460i
$563$	39.4142i	1.66111i	0.556936	+	0.830556i	$0.311977\pi$
$563$	39.4142i	1.66111i	−0.556936	+	0.830556i	$0.688023\pi$
$564$	−6.34315	−0.267095
$565$	0	0
$566$	−6.24264	−0.262398
$567$	0	0
$568$	12.4853i	0.523871i
$569$	6.68629	0.280304	0.140152	−	0.990130i	$-0.455241\pi$
$569$	6.68629	0.280304	0.140152	+	0.990130i	$0.455241\pi$
$570$	0	0
$571$	−41.7990	−1.74923	−0.874617	−	0.484815i	$-0.838887\pi$
$571$	−41.7990	−1.74923	−0.874617	+	0.484815i	$0.838887\pi$
$572$	4.00000i	0.167248i
$573$	10.6274i	0.443967i
$574$	0	0
$575$	0	0
$576$	−2.65685	−0.110702
$577$	25.8995i	1.07821i	0.842239	+	0.539105i	$0.181237\pi$
$577$	25.8995i	1.07821i	−0.842239	+	0.539105i	$0.818763\pi$
$578$	12.3137i	0.512183i
$579$	3.31371	0.137713
$580$	0	0
$581$	0	0
$582$	9.51472i	0.394398i
$583$	50.6274i	2.09677i
$584$	−6.58579	−0.272522
$585$	0	0
$586$	19.6569	0.812017
$587$	2.92893i	0.120890i	0.998172	+	0.0604450i	$0.0192520\pi$
$587$	2.92893i	0.120890i	−0.998172	+	0.0604450i	$0.980748\pi$
$588$	0	0
$589$	−9.65685	−0.397904
$590$	0	0
$591$	8.08326	0.332501
$592$	3.65685i	0.150296i
$593$	− 28.7279i	− 1.17971i	−0.807508	−	0.589857i	$-0.799184\pi$
$593$	− 28.7279i	− 1.17971i	0.807508	−	0.589857i	$-0.200816\pi$
$594$	−16.0000	−0.656488
$595$	0	0
$596$	−6.00000	−0.245770
$597$	− 0.284271i	− 0.0116344i
$598$	− 5.65685i	− 0.231326i
$599$	5.17157	0.211305	0.105652	−	0.994403i	$-0.466307\pi$
$599$	5.17157	0.211305	0.105652	+	0.994403i	$0.466307\pi$
$600$	0	0
$601$	9.41421	0.384014	0.192007	−	0.981394i	$-0.438500\pi$
$601$	9.41421	0.384014	0.192007	+	0.981394i	$0.438500\pi$
$602$	0	0
$603$	− 25.6569i	− 1.04483i
$604$	−11.3137	−0.460348
$605$	0	0
$606$	5.45584	0.221629
$607$	− 40.2843i	− 1.63509i	−0.575866	−	0.817544i	$-0.695335\pi$
$607$	− 40.2843i	− 1.63509i	0.575866	−	0.817544i	$-0.304665\pi$
$608$	3.41421i	0.138465i
$609$	0	0
$610$	0	0
$611$	8.97056	0.362910
$612$	− 14.3848i	− 0.581470i
$613$	23.6569i	0.955491i	0.878498	+	0.477746i	$0.158546\pi$
$613$	23.6569i	0.955491i	−0.878498	+	0.477746i	$0.841454\pi$
$614$	29.0711	1.17321
$615$	0	0
$616$	0	0
$617$	− 10.6863i	− 0.430214i	−0.976590	−	0.215107i	$-0.930990\pi$
$617$	− 10.6863i	− 0.430214i	0.976590	−	0.215107i	$-0.0690100\pi$
$618$	− 5.37258i	− 0.216117i
$619$	−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
$622$	− 4.00000i	− 0.160385i
$623$	0	0
$624$	−0.485281	−0.0194268
$625$	0	0
$626$	22.3848	0.894676
$627$	9.65685i	0.385658i
$628$	10.4853i	0.418408i
$629$	−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	$0.471544\pi$
$632$	1.17157i	0.0466027i
$633$	15.5980i	0.619964i
$634$	−6.48528	−0.257563
$635$	0	0
$636$	−6.14214	−0.243552
$637$	0	0
$638$	4.00000i	0.158362i
$639$	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	$0.366447\pi$
$642$	0.970563i	0.0383051i
$643$	− 47.2132i	− 1.86191i	−0.365138	−	0.930953i	$-0.618978\pi$
$643$	− 47.2132i	− 1.86191i	0.365138	−	0.930953i	$-0.381022\pi$
$644$	0	0
$645$	0	0
$646$	−18.4853	−0.727294
$647$	39.1127i	1.53768i	0.639442	+	0.768839i	$0.279165\pi$
$647$	39.1127i	1.53768i	−0.639442	+	0.768839i	$0.720835\pi$
$648$	6.02944i	0.236859i
$649$	−55.1127	−2.16336
$650$	0	0
$651$	0	0
$652$	8.14214i	0.318871i
$653$	− 15.6569i	− 0.612700i	−0.951919	−	0.306350i	$-0.900892\pi$
$653$	− 15.6569i	− 0.612700i	0.951919	−	0.306350i	$-0.0991078\pi$
$654$	−8.48528	−0.331801
$655$	0	0
$656$	11.0711	0.432253
$657$	17.4975i	0.682642i
$658$	0	0
$659$	32.8284	1.27881	0.639407	−	0.768868i	$-0.279180\pi$
$659$	32.8284	1.27881	0.639407	+	0.768868i	$0.279180\pi$
$660$	0	0
$661$	18.2843	0.711176	0.355588	−	0.934643i	$-0.384281\pi$
$661$	18.2843	0.711176	0.355588	+	0.934643i	$0.384281\pi$
$662$	5.79899i	0.225384i
$663$	− 2.62742i	− 0.102040i
$664$	6.24264	0.242261
$665$	0	0
$666$	9.71573	0.376477
$667$	− 5.65685i	− 0.219034i
$668$	23.7990i	0.920811i
$669$	−8.97056	−0.346822
$670$	0	0
$671$	−64.2843	−2.48167
$672$	0	0
$673$	48.0000i	1.85026i	0.379646	+	0.925132i	$0.376046\pi$
$673$	48.0000i	1.85026i	−0.379646	+	0.925132i	$0.623954\pi$
$674$	6.00000	0.231111
$675$	0	0
$676$	−12.3137	−0.473604
$677$	− 11.4558i	− 0.440284i	−0.975468	−	0.220142i	$-0.929348\pi$
$677$	− 11.4558i	− 0.440284i	0.975468	−	0.220142i	$-0.0706521\pi$
$678$	4.28427i	0.164536i
$679$	0	0
$680$	0	0
$681$	5.71573	0.219027
$682$	− 13.6569i	− 0.522948i
$683$	− 22.3431i	− 0.854937i	−0.904030	−	0.427468i	$-0.859406\pi$
$683$	− 22.3431i	− 0.854937i	0.904030	−	0.427468i	$-0.140594\pi$
$684$	9.07107	0.346841
$685$	0	0
$686$	0	0
$687$	7.11270i	0.271366i
$688$	3.17157i	0.120915i
$689$	8.68629	0.330921
$690$	0	0
$691$	10.2426	0.389648	0.194824	−	0.980838i	$-0.437586\pi$
$691$	10.2426	0.389648	0.194824	+	0.980838i	$0.437586\pi$
$692$	− 3.17157i	− 0.120565i
$693$	0	0
$694$	−8.82843	−0.335123
$695$	0	0
$696$	−0.485281	−0.0183945
$697$	59.9411i	2.27043i
$698$	− 14.4853i	− 0.548276i
$699$	−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	$0.588198\pi$
$702$	2.74517i	0.103610i
$703$	− 12.4853i	− 0.470891i
$704$	−4.82843	−0.181978
$705$	0	0
$706$	−34.3848	−1.29409
$707$	0	0
$708$	− 6.68629i	− 0.251286i
$709$	17.1127	0.642681	0.321340	−	0.946964i	$-0.395867\pi$
$709$	17.1127	0.642681	0.321340	+	0.946964i	$0.395867\pi$
$710$	0	0
$711$	3.11270	0.116735
$712$	12.7279i	0.476999i
$713$	19.3137i	0.723304i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	− 5.65685i	− 0.211259i
$718$	− 28.2843i	− 1.05556i
$719$	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
$722$	7.34315i	0.273284i
$723$	6.20101i	0.230618i
$724$	−14.4853	−0.538341
$725$	0	0
$726$	−7.21320	−0.267707
$727$	− 20.4853i	− 0.759757i	−0.925036	−	0.379879i	$-0.875966\pi$
$727$	− 20.4853i	− 0.759757i	0.925036	−	0.379879i	$-0.124034\pi$
$728$	0	0
$729$	10.1960	0.377628
$730$	0	0
$731$	−17.1716	−0.635114
$732$	− 7.79899i	− 0.288259i
$733$	− 34.0000i	− 1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$	− 34.0000i	− 1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$	−8.97056	−0.331110
$735$	0	0
$736$	6.82843	0.251699
$737$	− 46.6274i	− 1.71754i
$738$	− 29.4142i	− 1.08275i
$739$	−8.82843	−0.324759	−0.162379	−	0.986728i	$-0.551917\pi$
$739$	−8.82843	−0.324759	−0.162379	+	0.986728i	$0.551917\pi$
$740$	0	0
$741$	1.65685	0.0608661
$742$	0	0
$743$	− 12.2010i	− 0.447612i	−0.974634	−	0.223806i	$-0.928152\pi$
$743$	− 12.2010i	− 0.447612i	0.974634	−	0.223806i	$-0.0718481\pi$
$744$	1.65685	0.0607432
$745$	0	0
$746$	13.5147	0.494809
$747$	− 16.5858i	− 0.606842i
$748$	− 26.1421i	− 0.955851i
$749$	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	$0.598471\pi$
$752$	10.8284i	0.394872i
$753$	− 2.00000i	− 0.0728841i
$754$	0.686292	0.0249933
$755$	0	0
$756$	0	0
$757$	− 7.65685i	− 0.278293i	−0.990272	−	0.139147i	$-0.955564\pi$
$757$	− 7.65685i	− 0.278293i	0.990272	−	0.139147i	$-0.0444359\pi$
$758$	17.5147i	0.636163i
$759$	19.3137	0.701043
$760$	0	0
$761$	−14.3848	−0.521448	−0.260724	−	0.965413i	$-0.583961\pi$
$761$	−14.3848	−0.521448	−0.260724	+	0.965413i	$0.583961\pi$
$762$	1.65685i	0.0600215i
$763$	0	0
$764$	18.1421	0.656359
$765$	0	0
$766$	15.5147	0.560570
$767$	9.45584i	0.341431i
$768$	− 0.585786i	− 0.0211377i
$769$	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
$772$	− 5.65685i	− 0.203595i
$773$	2.00000i	0.0719350i	0.999353	+	0.0359675i	$0.0114513\pi$
$773$	2.00000i	0.0719350i	−0.999353	+	0.0359675i	$0.988549\pi$
$774$	8.42641	0.302881
$775$	0	0
$776$	16.2426	0.583077
$777$	0	0
$778$	− 0.142136i	− 0.00509581i
$779$	−37.7990	−1.35429
$780$	0	0
$781$	60.2843	2.15714
$782$	36.9706i	1.32206i
$783$	2.74517i	0.0981042i
$784$	0	0
$785$	0	0
$786$	−1.31371	−0.0468584
$787$	− 26.7279i	− 0.952748i	−0.879243	−	0.476374i	$-0.841951\pi$
$787$	− 26.7279i	− 0.952748i	0.879243	−	0.476374i	$-0.158049\pi$
$788$	− 13.7990i	− 0.491569i
$789$	−16.4020	−0.583927
$790$	0	0
$791$	0	0
$792$	12.8284i	0.455838i
$793$	11.0294i	0.391667i
$794$	−5.79899	−0.205798
$795$	0	0
$796$	−0.485281	−0.0172003
$797$	− 2.20101i	− 0.0779638i	−0.999240	−	0.0389819i	$-0.987589\pi$
$797$	− 2.20101i	− 0.0779638i	0.999240	−	0.0389819i	$-0.0124115\pi$
$798$	0	0
$799$	−58.6274	−2.07409
$800$	0	0
$801$	33.8162	1.19484
$802$	6.00000i	0.211867i
$803$	31.7990i	1.12216i
$804$	5.65685	0.199502
$805$	0	0
$806$	−2.34315	−0.0825338
$807$	0.887302i	0.0312345i
$808$	− 9.31371i	− 0.327655i
$809$	−36.9706	−1.29982	−0.649908	−	0.760013i	$-0.725193\pi$
$809$	−36.9706	−1.29982	−0.649908	+	0.760013i	$0.725193\pi$
$810$	0	0
$811$	35.4142	1.24356	0.621781	−	0.783191i	$-0.286410\pi$
$811$	35.4142	1.24356	0.621781	+	0.783191i	$0.286410\pi$
$812$	0	0
$813$	− 7.02944i	− 0.246533i
$814$	17.6569	0.618872
$815$	0	0
$816$	3.17157	0.111027
$817$	− 10.8284i	− 0.378839i
$818$	− 13.4142i	− 0.469017i
$819$	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	$0.529558\pi$
$822$	9.37258i	0.326906i
$823$	36.2843i	1.26479i	0.774646	+	0.632395i	$0.217928\pi$
$823$	36.2843i	1.26479i	−0.774646	+	0.632395i	$0.782072\pi$
$824$	−9.17157	−0.319507
$825$	0	0
$826$	0	0
$827$	− 50.6274i	− 1.76049i	−0.474522	−	0.880244i	$-0.657379\pi$
$827$	− 50.6274i	− 1.76049i	0.474522	−	0.880244i	$-0.342621\pi$
$828$	− 18.1421i	− 0.630483i
$829$	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
$832$	0.828427i	0.0287205i
$833$	0	0
$834$	0.0588745	0.00203866
$835$	0	0
$836$	16.4853	0.570155
$837$	− 9.37258i	− 0.323964i
$838$	32.8701i	1.13548i
$839$	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
$842$	5.31371i	0.183122i
$843$	− 4.68629i	− 0.161404i
$844$	26.6274	0.916553
$845$	0	0
$846$	28.7696	0.989118
$847$	0	0
$848$	10.4853i	0.360066i
$849$	−3.65685	−0.125503
$850$	0	0
$851$	−24.9706	−0.855980
$852$	7.31371i	0.250564i
$853$	48.8284i	1.67185i	0.548841	+	0.835927i	$0.315069\pi$
$853$	48.8284i	1.67185i	−0.548841	+	0.835927i	$0.684931\pi$
$854$	0	0
$855$	0	0
$856$	1.65685	0.0566301
$857$	− 19.0711i	− 0.651455i	−0.945464	−	0.325728i	$-0.894391\pi$
$857$	− 19.0711i	− 0.651455i	0.945464	−	0.325728i	$-0.105609\pi$
$858$	2.34315i	0.0799937i
$859$	−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
$862$	33.6569i	1.14636i
$863$	− 28.9706i	− 0.986169i	−0.869981	−	0.493085i	$-0.835869\pi$
$863$	− 28.9706i	− 0.986169i	0.869981	−	0.493085i	$-0.164131\pi$
$864$	−3.31371	−0.112735
$865$	0	0
$866$	−13.4142	−0.455834
$867$	7.21320i	0.244973i
$868$	0	0
$869$	5.65685	0.191896
$870$	0	0
$871$	−8.00000	−0.271070
$872$	14.4853i	0.490534i
$873$	− 43.1543i	− 1.46055i
$874$	−23.3137	−0.788598
$875$	0	0
$876$	−3.85786	−0.130345
$877$	26.2843i	0.887557i	0.896137	+	0.443778i	$0.146362\pi$
$877$	26.2843i	0.887557i	−0.896137	+	0.443778i	$0.853638\pi$
$878$	8.97056i	0.302742i
$879$	11.5147	0.388382
$880$	0	0
$881$	34.3848	1.15845	0.579226	−	0.815167i	$-0.303355\pi$
$881$	34.3848	1.15845	0.579226	+	0.815167i	$0.303355\pi$
$882$	0	0
$883$	30.3431i	1.02113i	0.859840	+	0.510564i	$0.170563\pi$
$883$	30.3431i	1.02113i	−0.859840	+	0.510564i	$0.829437\pi$
$884$	−4.48528	−0.150856
$885$	0	0
$886$	−36.9706	−1.24205
$887$	7.11270i	0.238821i	0.992845	+	0.119411i	$0.0381005\pi$
$887$	7.11270i	0.238821i	−0.992845	+	0.119411i	$0.961900\pi$
$888$	2.14214i	0.0718854i
$889$	0	0
$890$	0	0
$891$	29.1127	0.975312
$892$	15.3137i	0.512741i
$893$	− 36.9706i	− 1.23717i
$894$	−3.51472	−0.117550
$895$	0	0
$896$	0	0
$897$	− 3.31371i	− 0.110642i
$898$	28.6274i	0.955309i
$899$	−2.34315	−0.0781483
$900$	0	0
$901$	−56.7696	−1.89127
$902$	− 53.4558i	− 1.77988i
$903$	0	0
$904$	7.31371	0.243250
$905$	0	0
$906$	−6.62742	−0.220181
$907$	− 56.2843i	− 1.86889i	−0.356109	−	0.934444i	$-0.615897\pi$
$907$	− 56.2843i	− 1.86889i	0.356109	−	0.934444i	$-0.384103\pi$
$908$	− 9.75736i	− 0.323809i
$909$	−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	$0.609082\pi$
$912$	2.00000i	0.0662266i
$913$	− 30.1421i	− 0.997559i
$914$	10.3431	0.342121
$915$	0	0
$916$	12.1421	0.401187
$917$	0	0
$918$	− 17.9411i	− 0.592145i
$919$	−32.4853	−1.07159	−0.535795	−	0.844348i	$-0.679988\pi$
$919$	−32.4853	−1.07159	−0.535795	+	0.844348i	$0.679988\pi$
$920$	0	0
$921$	17.0294	0.561139
$922$	7.17157i	0.236183i
$923$	− 10.3431i	− 0.340449i
$924$	0	0
$925$	0	0
$926$	−16.9706	−0.557687
$927$	24.3675i	0.800335i
$928$	0.828427i	0.0271945i
$929$	−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
$932$	0.686292i	0.0224802i
$933$	− 2.34315i	− 0.0767111i
$934$	3.89949	0.127595
$935$	0	0
$936$	2.20101	0.0719423
$937$	11.7574i	0.384096i	0.981386	+	0.192048i	$0.0615130\pi$
$937$	11.7574i	0.384096i	−0.981386	+	0.192048i	$0.938487\pi$
$938$	0	0
$939$	13.1127	0.427917
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	6.14214i	0.200122i
$943$	75.5980i	2.46181i
$944$	−11.4142	−0.371501
$945$	0	0
$946$	15.3137	0.497892
$947$	0.828427i	0.0269203i	0.999909	+	0.0134601i	$0.00428462\pi$
$947$	0.828427i	0.0269203i	−0.999909	+	0.0134601i	$0.995715\pi$
$948$	0.686292i	0.0222897i
$949$	5.45584	0.177104
$950$	0	0
$951$	−3.79899	−0.123191
$952$	0	0
$953$	− 11.6569i	− 0.377603i	−0.982015	−	0.188801i	$-0.939540\pi$
$953$	− 11.6569i	− 0.377603i	0.982015	−	0.188801i	$-0.0604602\pi$
$954$	27.8579	0.901932
$955$	0	0
$956$	−9.65685	−0.312325
$957$	2.34315i	0.0757431i
$958$	− 22.8284i	− 0.737553i
$959$	0	0
$960$	0	0
$961$	−23.0000	−0.741935
$962$	− 3.02944i	− 0.0976730i
$963$	− 4.40202i	− 0.141853i
$964$	10.5858	0.340945
$965$	0	0
$966$	0	0
$967$	13.4558i	0.432711i	0.976315	+	0.216355i	$0.0694170\pi$
$967$	13.4558i	0.432711i	−0.976315	+	0.216355i	$0.930583\pi$
$968$	12.3137i	0.395778i
$969$	−10.8284	−0.347859
$970$	0	0
$971$	37.3553	1.19879	0.599395	−	0.800453i	$-0.295408\pi$
$971$	37.3553	1.19879	0.599395	+	0.800453i	$0.295408\pi$
$972$	13.4731i	0.432150i
$973$	0	0
$974$	−7.79899	−0.249896
$975$	0	0
$976$	−13.3137	−0.426161
$977$	− 35.3137i	− 1.12979i	−0.825164	−	0.564893i	$-0.808918\pi$
$977$	− 35.3137i	− 1.12979i	0.825164	−	0.564893i	$-0.191082\pi$
$978$	4.76955i	0.152513i
$979$	61.4558	1.96414
$980$	0	0
$981$	38.4853	1.22874
$982$	24.2843i	0.774942i
$983$	51.7990i	1.65213i	0.563574	+	0.826066i	$0.309426\pi$
$983$	51.7990i	1.65213i	−0.563574	+	0.826066i	$0.690574\pi$
$984$	6.48528	0.206743
$985$	0	0
$986$	−4.48528	−0.142840
$987$	0	0
$988$	− 2.82843i	− 0.0899843i
$989$	−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	$0.651057\pi$
$992$	− 2.82843i	− 0.0898027i
$993$	3.39697i	0.107800i
$994$	0	0
$995$	0	0
$996$	3.65685	0.115872
$997$	38.2843i	1.21248i	0.795284	+	0.606238i	$0.207322\pi$
$997$	38.2843i	1.21248i	−0.795284	+	0.606238i	$0.792678\pi$
$998$	41.6569i	1.31862i
$999$	12.1177	0.383389

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.c.t.99.2		4
5.2	odd	4		2450.2.a.bn.1.2		2
5.3	odd	4		490.2.a.m.1.1	yes	2
5.4	even	2	inner	2450.2.c.t.99.3		4
7.6	odd	2		2450.2.c.w.99.1		4
15.8	even	4		4410.2.a.bt.1.1		2
20.3	even	4		3920.2.a.bm.1.2		2
35.3	even	12		490.2.e.j.471.1		4
35.13	even	4		490.2.a.l.1.2	✓	2
35.18	odd	12		490.2.e.i.471.2		4
35.23	odd	12		490.2.e.i.361.2		4
35.27	even	4		2450.2.a.bs.1.1		2
35.33	even	12		490.2.e.j.361.1		4
35.34	odd	2		2450.2.c.w.99.4		4
105.83	odd	4		4410.2.a.by.1.1		2
140.83	odd	4		3920.2.a.ca.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.a.l.1.2	✓	2	35.13	even	4
490.2.a.m.1.1	yes	2	5.3	odd	4
490.2.e.i.361.2		4	35.23	odd	12
490.2.e.i.471.2		4	35.18	odd	12
490.2.e.j.361.1		4	35.33	even	12
490.2.e.j.471.1		4	35.3	even	12
2450.2.a.bn.1.2		2	5.2	odd	4
2450.2.a.bs.1.1		2	35.27	even	4
2450.2.c.t.99.2		4	1.1	even	1	trivial
2450.2.c.t.99.3		4	5.4	even	2	inner
2450.2.c.w.99.1		4	7.6	odd	2
2450.2.c.w.99.4		4	35.34	odd	2
3920.2.a.bm.1.2		2	20.3	even	4
3920.2.a.ca.1.1		2	140.83	odd	4
4410.2.a.bt.1.1		2	15.8	even	4
4410.2.a.by.1.1		2	105.83	odd	4

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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -0.585786i q^{3} -1.00000 q^{4} -0.585786 q^{6} +1.00000i q^{8} +2.65685 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -0.585786i q^{3} -1.00000 q^{4} -0.585786 q^{6} +1.00000i q^{8} +2.65685 q^{9} +4.82843 q^{11} +0.585786i q^{12} -0.828427i q^{13} +1.00000 q^{16} +5.41421i q^{17} -2.65685i q^{18} -3.41421 q^{19} -4.82843i q^{22} +6.82843i q^{23} +0.585786 q^{24} -0.828427 q^{26} -3.31371i q^{27} -0.828427 q^{29} +2.82843 q^{31} -1.00000i q^{32} -2.82843i q^{33} +5.41421 q^{34} -2.65685 q^{36} +3.65685i q^{37} +3.41421i q^{38} -0.485281 q^{39} +11.0711 q^{41} +3.17157i q^{43} -4.82843 q^{44} +6.82843 q^{46} +10.8284i q^{47} -0.585786i q^{48} +3.17157 q^{51} +0.828427i q^{52} +10.4853i q^{53} -3.31371 q^{54} +2.00000i q^{57} +0.828427i q^{58} -11.4142 q^{59} -13.3137 q^{61} -2.82843i q^{62} -1.00000 q^{64} -2.82843 q^{66} -9.65685i q^{67} -5.41421i q^{68} +4.00000 q^{69} +12.4853 q^{71} +2.65685i q^{72} +6.58579i q^{73} +3.65685 q^{74} +3.41421 q^{76} +0.485281i q^{78} +1.17157 q^{79} +6.02944 q^{81} -11.0711i q^{82} -6.24264i q^{83} +3.17157 q^{86} +0.485281i q^{87} +4.82843i q^{88} +12.7279 q^{89} -6.82843i q^{92} -1.65685i q^{93} +10.8284 q^{94} -0.585786 q^{96} -16.2426i q^{97} +12.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 8 q^{6} - 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 8 * q^6 - 12 * q^9	\( 4 q - 4 q^{4} - 8 q^{6} - 12 q^{9} + 8 q^{11} + 4 q^{16} - 8 q^{19} + 8 q^{24} + 8 q^{26} + 8 q^{29} + 16 q^{34} + 12 q^{36} + 32 q^{39} + 16 q^{41} - 8 q^{44} + 16 q^{46} + 24 q^{51} + 32 q^{54} - 40 q^{59} - 8 q^{61} - 4 q^{64} + 16 q^{69} + 16 q^{71} - 8 q^{74} + 8 q^{76} + 16 q^{79} + 92 q^{81} + 24 q^{86} + 32 q^{94} - 8 q^{96} + 40 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 8 * q^6 - 12 * q^9 + 8 * q^11 + 4 * q^16 - 8 * q^19 + 8 * q^24 + 8 * q^26 + 8 * q^29 + 16 * q^34 + 12 * q^36 + 32 * q^39 + 16 * q^41 - 8 * q^44 + 16 * q^46 + 24 * q^51 + 32 * q^54 - 40 * q^59 - 8 * q^61 - 4 * q^64 + 16 * q^69 + 16 * q^71 - 8 * q^74 + 8 * q^76 + 16 * q^79 + 92 * q^81 + 24 * q^86 + 32 * q^94 - 8 * q^96 + 40 * q^99

Introduction

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