LMFDB - Embedded newform 5610.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5610.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:	$44.7960755339$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	5610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} -1.00000 q^{22} +5.65685 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -7.65685 q^{29} +1.00000 q^{30} -5.65685 q^{31} +1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} +11.6569 q^{37} +2.00000 q^{39} -1.00000 q^{40} -3.65685 q^{41} -5.65685 q^{43} -1.00000 q^{44} -1.00000 q^{45} +5.65685 q^{46} -4.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +1.00000 q^{55} -7.65685 q^{58} +11.3137 q^{59} +1.00000 q^{60} +7.65685 q^{61} -5.65685 q^{62} +1.00000 q^{64} +2.00000 q^{65} +1.00000 q^{66} -4.00000 q^{67} +1.00000 q^{68} -5.65685 q^{69} +5.65685 q^{71} +1.00000 q^{72} +9.31371 q^{73} +11.6569 q^{74} -1.00000 q^{75} +2.00000 q^{78} -11.3137 q^{79} -1.00000 q^{80} +1.00000 q^{81} -3.65685 q^{82} +4.00000 q^{83} -1.00000 q^{85} -5.65685 q^{86} +7.65685 q^{87} -1.00000 q^{88} -14.0000 q^{89} -1.00000 q^{90} +5.65685 q^{92} +5.65685 q^{93} -4.00000 q^{94} -1.00000 q^{96} -0.343146 q^{97} -7.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 2 q^{20} - 2 q^{22} - 2 q^{24} + 2 q^{25} - 4 q^{26} - 2 q^{27} - 4 q^{29} + 2 q^{30} + 2 q^{32} + 2 q^{33} + 2 q^{34} + 2 q^{36} + 12 q^{37} + 4 q^{39} - 2 q^{40} + 4 q^{41} - 2 q^{44} - 2 q^{45} - 8 q^{47} - 2 q^{48} - 14 q^{49} + 2 q^{50} - 2 q^{51} - 4 q^{52} - 20 q^{53} - 2 q^{54} + 2 q^{55} - 4 q^{58} + 2 q^{60} + 4 q^{61} + 2 q^{64} + 4 q^{65} + 2 q^{66} - 8 q^{67} + 2 q^{68} + 2 q^{72} - 4 q^{73} + 12 q^{74} - 2 q^{75} + 4 q^{78} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} - 2 q^{85} + 4 q^{87} - 2 q^{88} - 28 q^{89} - 2 q^{90} - 8 q^{94} - 2 q^{96} - 12 q^{97} - 14 q^{98} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^15 + 2 * q^16 + 2 * q^17 + 2 * q^18 - 2 * q^20 - 2 * q^22 - 2 * q^24 + 2 * q^25 - 4 * q^26 - 2 * q^27 - 4 * q^29 + 2 * q^30 + 2 * q^32 + 2 * q^33 + 2 * q^34 + 2 * q^36 + 12 * q^37 + 4 * q^39 - 2 * q^40 + 4 * q^41 - 2 * q^44 - 2 * q^45 - 8 * q^47 - 2 * q^48 - 14 * q^49 + 2 * q^50 - 2 * q^51 - 4 * q^52 - 20 * q^53 - 2 * q^54 + 2 * q^55 - 4 * q^58 + 2 * q^60 + 4 * q^61 + 2 * q^64 + 4 * q^65 + 2 * q^66 - 8 * q^67 + 2 * q^68 + 2 * q^72 - 4 * q^73 + 12 * q^74 - 2 * q^75 + 4 * q^78 - 2 * q^80 + 2 * q^81 + 4 * q^82 + 8 * q^83 - 2 * q^85 + 4 * q^87 - 2 * q^88 - 28 * q^89 - 2 * q^90 - 8 * q^94 - 2 * q^96 - 12 * q^97 - 14 * q^98 - 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	5.65685	1.17954	0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685	1.17954	0.589768	+	0.807573i	$0.299219\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.65685	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	1.00000	0.182574
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	11.6569	1.91638	0.958188	−	0.286141i	$-0.0923726\pi$
$37$	11.6569	1.91638	0.958188	+	0.286141i	$0.0923726\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	−1.00000	−0.158114
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	0	0
$43$	−5.65685	−0.862662	−0.431331	−	0.902194i	$-0.641956\pi$
$43$	−5.65685	−0.862662	−0.431331	+	0.902194i	$0.641956\pi$
$44$	−1.00000	−0.150756
$45$	−1.00000	−0.149071
$46$	5.65685	0.834058
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	−1.00000	−0.140028
$52$	−2.00000	−0.277350
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	−1.00000	−0.136083
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	−7.65685	−1.00539
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	1.00000	0.129099
$61$	7.65685	0.980360	0.490180	−	0.871621i	$-0.336931\pi$
$61$	7.65685	0.980360	0.490180	+	0.871621i	$0.336931\pi$
$62$	−5.65685	−0.718421
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	1.00000	0.123091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	1.00000	0.121268
$69$	−5.65685	−0.681005
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	1.00000	0.117851
$73$	9.31371	1.09009	0.545044	−	0.838408i	$-0.316513\pi$
$73$	9.31371	1.09009	0.545044	+	0.838408i	$0.316513\pi$
$74$	11.6569	1.35508
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	2.00000	0.226455
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−3.65685	−0.403832
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	−5.65685	−0.609994
$87$	7.65685	0.820901
$88$	−1.00000	−0.106600
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	5.65685	0.589768
$93$	5.65685	0.586588
$94$	−4.00000	−0.412568
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−0.343146	−0.0348412	−0.0174206	−	0.999848i	$-0.505545\pi$
$97$	−0.343146	−0.0348412	−0.0174206	+	0.999848i	$0.505545\pi$
$98$	−7.00000	−0.707107
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−1.00000	−0.0990148
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−10.0000	−0.971286
$107$	−17.6569	−1.70695	−0.853476	−	0.521132i	$-0.825510\pi$
$107$	−17.6569	−1.70695	−0.853476	+	0.521132i	$0.825510\pi$
$108$	−1.00000	−0.0962250
$109$	−8.34315	−0.799128	−0.399564	−	0.916705i	$-0.630839\pi$
$109$	−8.34315	−0.799128	−0.399564	+	0.916705i	$0.630839\pi$
$110$	1.00000	0.0953463
$111$	−11.6569	−1.10642
$112$	0	0
$113$	−7.65685	−0.720296	−0.360148	−	0.932895i	$-0.617274\pi$
$113$	−7.65685	−0.720296	−0.360148	+	0.932895i	$0.617274\pi$
$114$	0	0
$115$	−5.65685	−0.527504
$116$	−7.65685	−0.710921
$117$	−2.00000	−0.184900
$118$	11.3137	1.04151
$119$	0	0
$120$	1.00000	0.0912871
$121$	1.00000	0.0909091
$122$	7.65685	0.693219
$123$	3.65685	0.329727
$124$	−5.65685	−0.508001
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.3137	−1.35887	−0.679436	−	0.733735i	$-0.737775\pi$
$127$	−15.3137	−1.35887	−0.679436	+	0.733735i	$0.737775\pi$
$128$	1.00000	0.0883883
$129$	5.65685	0.498058
$130$	2.00000	0.175412
$131$	−7.31371	−0.639002	−0.319501	−	0.947586i	$-0.603515\pi$
$131$	−7.31371	−0.639002	−0.319501	+	0.947586i	$0.603515\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−4.00000	−0.345547
$135$	1.00000	0.0860663
$136$	1.00000	0.0857493
$137$	−8.34315	−0.712803	−0.356402	−	0.934333i	$-0.615996\pi$
$137$	−8.34315	−0.712803	−0.356402	+	0.934333i	$0.615996\pi$
$138$	−5.65685	−0.481543
$139$	−12.9706	−1.10015	−0.550074	−	0.835116i	$-0.685401\pi$
$139$	−12.9706	−1.10015	−0.550074	+	0.835116i	$0.685401\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	5.65685	0.474713
$143$	2.00000	0.167248
$144$	1.00000	0.0833333
$145$	7.65685	0.635867
$146$	9.31371	0.770808
$147$	7.00000	0.577350
$148$	11.6569	0.958188
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	−17.6569	−1.43689	−0.718447	−	0.695581i	$-0.755147\pi$
$151$	−17.6569	−1.43689	−0.718447	+	0.695581i	$0.755147\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	5.65685	0.454369
$156$	2.00000	0.160128
$157$	−5.31371	−0.424080	−0.212040	−	0.977261i	$-0.568011\pi$
$157$	−5.31371	−0.424080	−0.212040	+	0.977261i	$0.568011\pi$
$158$	−11.3137	−0.900070
$159$	10.0000	0.793052
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	15.3137	1.19946	0.599731	−	0.800202i	$-0.295274\pi$
$163$	15.3137	1.19946	0.599731	+	0.800202i	$0.295274\pi$
$164$	−3.65685	−0.285552
$165$	−1.00000	−0.0778499
$166$	4.00000	0.310460
$167$	5.65685	0.437741	0.218870	−	0.975754i	$-0.429763\pi$
$167$	5.65685	0.437741	0.218870	+	0.975754i	$0.429763\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−1.00000	−0.0766965
$171$	0	0
$172$	−5.65685	−0.431331
$173$	−21.3137	−1.62045	−0.810226	−	0.586118i	$-0.800655\pi$
$173$	−21.3137	−1.62045	−0.810226	+	0.586118i	$0.800655\pi$
$174$	7.65685	0.580465
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−11.3137	−0.850390
$178$	−14.0000	−1.04934
$179$	11.3137	0.845626	0.422813	−	0.906217i	$-0.361043\pi$
$179$	11.3137	0.845626	0.422813	+	0.906217i	$0.361043\pi$
$180$	−1.00000	−0.0745356
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	−7.65685	−0.566011
$184$	5.65685	0.417029
$185$	−11.6569	−0.857029
$186$	5.65685	0.414781
$187$	−1.00000	−0.0731272
$188$	−4.00000	−0.291730
$189$	0	0
$190$	0	0
$191$	−17.6569	−1.27761	−0.638803	−	0.769371i	$-0.720570\pi$
$191$	−17.6569	−1.27761	−0.638803	+	0.769371i	$0.720570\pi$
$192$	−1.00000	−0.0721688
$193$	−21.3137	−1.53419	−0.767097	−	0.641531i	$-0.778300\pi$
$193$	−21.3137	−1.53419	−0.767097	+	0.641531i	$0.778300\pi$
$194$	−0.343146	−0.0246364
$195$	−2.00000	−0.143223
$196$	−7.00000	−0.500000
$197$	−13.3137	−0.948562	−0.474281	−	0.880373i	$-0.657292\pi$
$197$	−13.3137	−0.948562	−0.474281	+	0.880373i	$0.657292\pi$
$198$	−1.00000	−0.0710669
$199$	16.9706	1.20301	0.601506	−	0.798869i	$-0.294568\pi$
$199$	16.9706	1.20301	0.601506	+	0.798869i	$0.294568\pi$
$200$	1.00000	0.0707107
$201$	4.00000	0.282138
$202$	6.00000	0.422159
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	3.65685	0.255406
$206$	0	0
$207$	5.65685	0.393179
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	20.9706	1.44367	0.721837	−	0.692064i	$-0.243298\pi$
$211$	20.9706	1.44367	0.721837	+	0.692064i	$0.243298\pi$
$212$	−10.0000	−0.686803
$213$	−5.65685	−0.387601
$214$	−17.6569	−1.20700
$215$	5.65685	0.385794
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−8.34315	−0.565069
$219$	−9.31371	−0.629362
$220$	1.00000	0.0674200
$221$	−2.00000	−0.134535
$222$	−11.6569	−0.782357
$223$	3.31371	0.221902	0.110951	−	0.993826i	$-0.464610\pi$
$223$	3.31371	0.221902	0.110951	+	0.993826i	$0.464610\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−7.65685	−0.509326
$227$	−6.34315	−0.421009	−0.210505	−	0.977593i	$-0.567511\pi$
$227$	−6.34315	−0.421009	−0.210505	+	0.977593i	$0.567511\pi$
$228$	0	0
$229$	1.31371	0.0868123	0.0434062	−	0.999058i	$-0.486179\pi$
$229$	1.31371	0.0868123	0.0434062	+	0.999058i	$0.486179\pi$
$230$	−5.65685	−0.373002
$231$	0	0
$232$	−7.65685	−0.502697
$233$	−25.3137	−1.65836	−0.829178	−	0.558985i	$-0.811191\pi$
$233$	−25.3137	−1.65836	−0.829178	+	0.558985i	$0.811191\pi$
$234$	−2.00000	−0.130744
$235$	4.00000	0.260931
$236$	11.3137	0.736460
$237$	11.3137	0.734904
$238$	0	0
$239$	−4.68629	−0.303131	−0.151565	−	0.988447i	$-0.548431\pi$
$239$	−4.68629	−0.303131	−0.151565	+	0.988447i	$0.548431\pi$
$240$	1.00000	0.0645497
$241$	17.3137	1.11527	0.557637	−	0.830085i	$-0.311708\pi$
$241$	17.3137	1.11527	0.557637	+	0.830085i	$0.311708\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	7.65685	0.490180
$245$	7.00000	0.447214
$246$	3.65685	0.233153
$247$	0	0
$248$	−5.65685	−0.359211
$249$	−4.00000	−0.253490
$250$	−1.00000	−0.0632456
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	−5.65685	−0.355643
$254$	−15.3137	−0.960868
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	−30.9706	−1.93189	−0.965945	−	0.258746i	$-0.916691\pi$
$257$	−30.9706	−1.93189	−0.965945	+	0.258746i	$0.916691\pi$
$258$	5.65685	0.352180
$259$	0	0
$260$	2.00000	0.124035
$261$	−7.65685	−0.473947
$262$	−7.31371	−0.451842
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	1.00000	0.0615457
$265$	10.0000	0.614295
$266$	0	0
$267$	14.0000	0.856786
$268$	−4.00000	−0.244339
$269$	29.3137	1.78729	0.893644	−	0.448776i	$-0.148140\pi$
$269$	29.3137	1.78729	0.893644	+	0.448776i	$0.148140\pi$
$270$	1.00000	0.0608581
$271$	−9.65685	−0.586612	−0.293306	−	0.956019i	$-0.594755\pi$
$271$	−9.65685	−0.586612	−0.293306	+	0.956019i	$0.594755\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−8.34315	−0.504028
$275$	−1.00000	−0.0603023
$276$	−5.65685	−0.340503
$277$	10.9706	0.659157	0.329579	−	0.944128i	$-0.393093\pi$
$277$	10.9706	0.659157	0.329579	+	0.944128i	$0.393093\pi$
$278$	−12.9706	−0.777923
$279$	−5.65685	−0.338667
$280$	0	0
$281$	24.6274	1.46915	0.734574	−	0.678528i	$-0.237382\pi$
$281$	24.6274	1.46915	0.734574	+	0.678528i	$0.237382\pi$
$282$	4.00000	0.238197
$283$	−1.65685	−0.0984898	−0.0492449	−	0.998787i	$-0.515681\pi$
$283$	−1.65685	−0.0984898	−0.0492449	+	0.998787i	$0.515681\pi$
$284$	5.65685	0.335673
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	7.65685	0.449626
$291$	0.343146	0.0201156
$292$	9.31371	0.545044
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	7.00000	0.408248
$295$	−11.3137	−0.658710
$296$	11.6569	0.677541
$297$	1.00000	0.0580259
$298$	6.00000	0.347571
$299$	−11.3137	−0.654289
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	−17.6569	−1.01604
$303$	−6.00000	−0.344691
$304$	0	0
$305$	−7.65685	−0.438430
$306$	1.00000	0.0571662
$307$	13.6569	0.779438	0.389719	−	0.920934i	$-0.372572\pi$
$307$	13.6569	0.779438	0.389719	+	0.920934i	$0.372572\pi$
$308$	0	0
$309$	0	0
$310$	5.65685	0.321288
$311$	16.9706	0.962312	0.481156	−	0.876635i	$-0.340217\pi$
$311$	16.9706	0.962312	0.481156	+	0.876635i	$0.340217\pi$
$312$	2.00000	0.113228
$313$	−14.9706	−0.846186	−0.423093	−	0.906086i	$-0.639056\pi$
$313$	−14.9706	−0.846186	−0.423093	+	0.906086i	$0.639056\pi$
$314$	−5.31371	−0.299870
$315$	0	0
$316$	−11.3137	−0.636446
$317$	−2.68629	−0.150877	−0.0754386	−	0.997150i	$-0.524036\pi$
$317$	−2.68629	−0.150877	−0.0754386	+	0.997150i	$0.524036\pi$
$318$	10.0000	0.560772
$319$	7.65685	0.428702
$320$	−1.00000	−0.0559017
$321$	17.6569	0.985510
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	15.3137	0.848148
$327$	8.34315	0.461377
$328$	−3.65685	−0.201916
$329$	0	0
$330$	−1.00000	−0.0550482
$331$	−7.31371	−0.401998	−0.200999	−	0.979591i	$-0.564419\pi$
$331$	−7.31371	−0.401998	−0.200999	+	0.979591i	$0.564419\pi$
$332$	4.00000	0.219529
$333$	11.6569	0.638792
$334$	5.65685	0.309529
$335$	4.00000	0.218543
$336$	0	0
$337$	17.3137	0.943138	0.471569	−	0.881829i	$-0.343688\pi$
$337$	17.3137	0.943138	0.471569	+	0.881829i	$0.343688\pi$
$338$	−9.00000	−0.489535
$339$	7.65685	0.415863
$340$	−1.00000	−0.0542326
$341$	5.65685	0.306336
$342$	0	0
$343$	0	0
$344$	−5.65685	−0.304997
$345$	5.65685	0.304555
$346$	−21.3137	−1.14583
$347$	9.65685	0.518407	0.259204	−	0.965823i	$-0.416540\pi$
$347$	9.65685	0.518407	0.259204	+	0.965823i	$0.416540\pi$
$348$	7.65685	0.410450
$349$	8.34315	0.446598	0.223299	−	0.974750i	$-0.428317\pi$
$349$	8.34315	0.446598	0.223299	+	0.974750i	$0.428317\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	−1.00000	−0.0533002
$353$	18.9706	1.00970	0.504851	−	0.863207i	$-0.331548\pi$
$353$	18.9706	1.00970	0.504851	+	0.863207i	$0.331548\pi$
$354$	−11.3137	−0.601317
$355$	−5.65685	−0.300235
$356$	−14.0000	−0.741999
$357$	0	0
$358$	11.3137	0.597948
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	−1.00000	−0.0527046
$361$	−19.0000	−1.00000
$362$	14.0000	0.735824
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−9.31371	−0.487502
$366$	−7.65685	−0.400230
$367$	−3.31371	−0.172974	−0.0864871	−	0.996253i	$-0.527564\pi$
$367$	−3.31371	−0.172974	−0.0864871	+	0.996253i	$0.527564\pi$
$368$	5.65685	0.294884
$369$	−3.65685	−0.190368
$370$	−11.6569	−0.606011
$371$	0	0
$372$	5.65685	0.293294
$373$	−32.6274	−1.68938	−0.844692	−	0.535253i	$-0.820216\pi$
$373$	−32.6274	−1.68938	−0.844692	+	0.535253i	$0.820216\pi$
$374$	−1.00000	−0.0517088
$375$	1.00000	0.0516398
$376$	−4.00000	−0.206284
$377$	15.3137	0.788696
$378$	0	0
$379$	−1.65685	−0.0851069	−0.0425534	−	0.999094i	$-0.513549\pi$
$379$	−1.65685	−0.0851069	−0.0425534	+	0.999094i	$0.513549\pi$
$380$	0	0
$381$	15.3137	0.784545
$382$	−17.6569	−0.903403
$383$	−31.3137	−1.60006	−0.800028	−	0.599963i	$-0.795182\pi$
$383$	−31.3137	−1.60006	−0.800028	+	0.599963i	$0.795182\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−21.3137	−1.08484
$387$	−5.65685	−0.287554
$388$	−0.343146	−0.0174206
$389$	−15.6569	−0.793834	−0.396917	−	0.917855i	$-0.629920\pi$
$389$	−15.6569	−0.793834	−0.396917	+	0.917855i	$0.629920\pi$
$390$	−2.00000	−0.101274
$391$	5.65685	0.286079
$392$	−7.00000	−0.353553
$393$	7.31371	0.368928
$394$	−13.3137	−0.670735
$395$	11.3137	0.569254
$396$	−1.00000	−0.0502519
$397$	5.02944	0.252420	0.126210	−	0.992004i	$-0.459719\pi$
$397$	5.02944	0.252420	0.126210	+	0.992004i	$0.459719\pi$
$398$	16.9706	0.850657
$399$	0	0
$400$	1.00000	0.0500000
$401$	−18.9706	−0.947345	−0.473672	−	0.880701i	$-0.657072\pi$
$401$	−18.9706	−0.947345	−0.473672	+	0.880701i	$0.657072\pi$
$402$	4.00000	0.199502
$403$	11.3137	0.563576
$404$	6.00000	0.298511
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−11.6569	−0.577809
$408$	−1.00000	−0.0495074
$409$	−17.3137	−0.856108	−0.428054	−	0.903753i	$-0.640801\pi$
$409$	−17.3137	−0.856108	−0.428054	+	0.903753i	$0.640801\pi$
$410$	3.65685	0.180599
$411$	8.34315	0.411537
$412$	0	0
$413$	0	0
$414$	5.65685	0.278019
$415$	−4.00000	−0.196352
$416$	−2.00000	−0.0980581
$417$	12.9706	0.635171
$418$	0	0
$419$	−18.6274	−0.910009	−0.455004	−	0.890489i	$-0.650362\pi$
$419$	−18.6274	−0.910009	−0.455004	+	0.890489i	$0.650362\pi$
$420$	0	0
$421$	28.6274	1.39521	0.697607	−	0.716480i	$-0.254248\pi$
$421$	28.6274	1.39521	0.697607	+	0.716480i	$0.254248\pi$
$422$	20.9706	1.02083
$423$	−4.00000	−0.194487
$424$	−10.0000	−0.485643
$425$	1.00000	0.0485071
$426$	−5.65685	−0.274075
$427$	0	0
$428$	−17.6569	−0.853476
$429$	−2.00000	−0.0965609
$430$	5.65685	0.272798
$431$	−14.6274	−0.704578	−0.352289	−	0.935891i	$-0.614597\pi$
$431$	−14.6274	−0.704578	−0.352289	+	0.935891i	$0.614597\pi$
$432$	−1.00000	−0.0481125
$433$	−9.31371	−0.447588	−0.223794	−	0.974636i	$-0.571844\pi$
$433$	−9.31371	−0.447588	−0.223794	+	0.974636i	$0.571844\pi$
$434$	0	0
$435$	−7.65685	−0.367118
$436$	−8.34315	−0.399564
$437$	0	0
$438$	−9.31371	−0.445026
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	1.00000	0.0476731
$441$	−7.00000	−0.333333
$442$	−2.00000	−0.0951303
$443$	−36.2843	−1.72392	−0.861959	−	0.506978i	$-0.830762\pi$
$443$	−36.2843	−1.72392	−0.861959	+	0.506978i	$0.830762\pi$
$444$	−11.6569	−0.553210
$445$	14.0000	0.663664
$446$	3.31371	0.156909
$447$	−6.00000	−0.283790
$448$	0	0
$449$	−26.9706	−1.27282	−0.636410	−	0.771351i	$-0.719581\pi$
$449$	−26.9706	−1.27282	−0.636410	+	0.771351i	$0.719581\pi$
$450$	1.00000	0.0471405
$451$	3.65685	0.172195
$452$	−7.65685	−0.360148
$453$	17.6569	0.829591
$454$	−6.34315	−0.297699
$455$	0	0
$456$	0	0
$457$	−14.9706	−0.700293	−0.350147	−	0.936695i	$-0.613868\pi$
$457$	−14.9706	−0.700293	−0.350147	+	0.936695i	$0.613868\pi$
$458$	1.31371	0.0613856
$459$	−1.00000	−0.0466760
$460$	−5.65685	−0.263752
$461$	−6.68629	−0.311412	−0.155706	−	0.987803i	$-0.549765\pi$
$461$	−6.68629	−0.311412	−0.155706	+	0.987803i	$0.549765\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−7.65685	−0.355461
$465$	−5.65685	−0.262330
$466$	−25.3137	−1.17263
$467$	24.9706	1.15550	0.577750	−	0.816214i	$-0.303931\pi$
$467$	24.9706	1.15550	0.577750	+	0.816214i	$0.303931\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	4.00000	0.184506
$471$	5.31371	0.244843
$472$	11.3137	0.520756
$473$	5.65685	0.260102
$474$	11.3137	0.519656
$475$	0	0
$476$	0	0
$477$	−10.0000	−0.457869
$478$	−4.68629	−0.214346
$479$	27.3137	1.24800	0.623998	−	0.781426i	$-0.285507\pi$
$479$	27.3137	1.24800	0.623998	+	0.781426i	$0.285507\pi$
$480$	1.00000	0.0456435
$481$	−23.3137	−1.06301
$482$	17.3137	0.788618
$483$	0	0
$484$	1.00000	0.0454545
$485$	0.343146	0.0155814
$486$	−1.00000	−0.0453609
$487$	30.6274	1.38786	0.693930	−	0.720042i	$-0.255878\pi$
$487$	30.6274	1.38786	0.693930	+	0.720042i	$0.255878\pi$
$488$	7.65685	0.346610
$489$	−15.3137	−0.692510
$490$	7.00000	0.316228
$491$	0.686292	0.0309719	0.0154860	−	0.999880i	$-0.495070\pi$
$491$	0.686292	0.0309719	0.0154860	+	0.999880i	$0.495070\pi$
$492$	3.65685	0.164864
$493$	−7.65685	−0.344847
$494$	0	0
$495$	1.00000	0.0449467
$496$	−5.65685	−0.254000
$497$	0	0
$498$	−4.00000	−0.179244
$499$	32.2843	1.44524	0.722621	−	0.691244i	$-0.242937\pi$
$499$	32.2843	1.44524	0.722621	+	0.691244i	$0.242937\pi$
$500$	−1.00000	−0.0447214
$501$	−5.65685	−0.252730
$502$	8.00000	0.357057
$503$	20.2843	0.904431	0.452215	−	0.891909i	$-0.350634\pi$
$503$	20.2843	0.904431	0.452215	+	0.891909i	$0.350634\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	−5.65685	−0.251478
$507$	9.00000	0.399704
$508$	−15.3137	−0.679436
$509$	−28.3431	−1.25629	−0.628144	−	0.778097i	$-0.716185\pi$
$509$	−28.3431	−1.25629	−0.628144	+	0.778097i	$0.716185\pi$
$510$	1.00000	0.0442807
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−30.9706	−1.36605
$515$	0	0
$516$	5.65685	0.249029
$517$	4.00000	0.175920
$518$	0	0
$519$	21.3137	0.935568
$520$	2.00000	0.0877058
$521$	5.02944	0.220344	0.110172	−	0.993913i	$-0.464860\pi$
$521$	5.02944	0.220344	0.110172	+	0.993913i	$0.464860\pi$
$522$	−7.65685	−0.335131
$523$	24.9706	1.09189	0.545943	−	0.837822i	$-0.316171\pi$
$523$	24.9706	1.09189	0.545943	+	0.837822i	$0.316171\pi$
$524$	−7.31371	−0.319501
$525$	0	0
$526$	−16.0000	−0.697633
$527$	−5.65685	−0.246416
$528$	1.00000	0.0435194
$529$	9.00000	0.391304
$530$	10.0000	0.434372
$531$	11.3137	0.490973
$532$	0	0
$533$	7.31371	0.316792
$534$	14.0000	0.605839
$535$	17.6569	0.763372
$536$	−4.00000	−0.172774
$537$	−11.3137	−0.488223
$538$	29.3137	1.26380
$539$	7.00000	0.301511
$540$	1.00000	0.0430331
$541$	9.02944	0.388206	0.194103	−	0.980981i	$-0.437820\pi$
$541$	9.02944	0.388206	0.194103	+	0.980981i	$0.437820\pi$
$542$	−9.65685	−0.414797
$543$	−14.0000	−0.600798
$544$	1.00000	0.0428746
$545$	8.34315	0.357381
$546$	0	0
$547$	17.6569	0.754953	0.377476	−	0.926019i	$-0.376792\pi$
$547$	17.6569	0.754953	0.377476	+	0.926019i	$0.376792\pi$
$548$	−8.34315	−0.356402
$549$	7.65685	0.326787
$550$	−1.00000	−0.0426401
$551$	0	0
$552$	−5.65685	−0.240772
$553$	0	0
$554$	10.9706	0.466095
$555$	11.6569	0.494806
$556$	−12.9706	−0.550074
$557$	−40.6274	−1.72144	−0.860719	−	0.509080i	$-0.829986\pi$
$557$	−40.6274	−1.72144	−0.860719	+	0.509080i	$0.829986\pi$
$558$	−5.65685	−0.239474
$559$	11.3137	0.478519
$560$	0	0
$561$	1.00000	0.0422200
$562$	24.6274	1.03884
$563$	23.3137	0.982556	0.491278	−	0.871003i	$-0.336530\pi$
$563$	23.3137	0.982556	0.491278	+	0.871003i	$0.336530\pi$
$564$	4.00000	0.168430
$565$	7.65685	0.322126
$566$	−1.65685	−0.0696428
$567$	0	0
$568$	5.65685	0.237356
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	30.3431	1.26982	0.634911	−	0.772586i	$-0.281037\pi$
$571$	30.3431	1.26982	0.634911	+	0.772586i	$0.281037\pi$
$572$	2.00000	0.0836242
$573$	17.6569	0.737626
$574$	0	0
$575$	5.65685	0.235907
$576$	1.00000	0.0416667
$577$	40.6274	1.69134	0.845671	−	0.533705i	$-0.179201\pi$
$577$	40.6274	1.69134	0.845671	+	0.533705i	$0.179201\pi$
$578$	1.00000	0.0415945
$579$	21.3137	0.885767
$580$	7.65685	0.317934
$581$	0	0
$582$	0.343146	0.0142238
$583$	10.0000	0.414158
$584$	9.31371	0.385404
$585$	2.00000	0.0826898
$586$	−26.0000	−1.07405
$587$	−13.6569	−0.563679	−0.281839	−	0.959462i	$-0.590945\pi$
$587$	−13.6569	−0.563679	−0.281839	+	0.959462i	$0.590945\pi$
$588$	7.00000	0.288675
$589$	0	0
$590$	−11.3137	−0.465778
$591$	13.3137	0.547653
$592$	11.6569	0.479094
$593$	35.9411	1.47593	0.737963	−	0.674842i	$-0.235788\pi$
$593$	35.9411	1.47593	0.737963	+	0.674842i	$0.235788\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	6.00000	0.245770
$597$	−16.9706	−0.694559
$598$	−11.3137	−0.462652
$599$	3.02944	0.123779	0.0618897	−	0.998083i	$-0.480287\pi$
$599$	3.02944	0.123779	0.0618897	+	0.998083i	$0.480287\pi$
$600$	−1.00000	−0.0408248
$601$	−45.3137	−1.84838	−0.924192	−	0.381927i	$-0.875260\pi$
$601$	−45.3137	−1.84838	−0.924192	+	0.381927i	$0.875260\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	−17.6569	−0.718447
$605$	−1.00000	−0.0406558
$606$	−6.00000	−0.243733
$607$	−6.62742	−0.268999	−0.134499	−	0.990914i	$-0.542943\pi$
$607$	−6.62742	−0.268999	−0.134499	+	0.990914i	$0.542943\pi$
$608$	0	0
$609$	0	0
$610$	−7.65685	−0.310017
$611$	8.00000	0.323645
$612$	1.00000	0.0404226
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	13.6569	0.551146
$615$	−3.65685	−0.147459
$616$	0	0
$617$	37.5980	1.51364	0.756819	−	0.653625i	$-0.226752\pi$
$617$	37.5980	1.51364	0.756819	+	0.653625i	$0.226752\pi$
$618$	0	0
$619$	−1.65685	−0.0665946	−0.0332973	−	0.999445i	$-0.510601\pi$
$619$	−1.65685	−0.0665946	−0.0332973	+	0.999445i	$0.510601\pi$
$620$	5.65685	0.227185
$621$	−5.65685	−0.227002
$622$	16.9706	0.680458
$623$	0	0
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	−14.9706	−0.598344
$627$	0	0
$628$	−5.31371	−0.212040
$629$	11.6569	0.464789
$630$	0	0
$631$	−6.62742	−0.263833	−0.131917	−	0.991261i	$-0.542113\pi$
$631$	−6.62742	−0.263833	−0.131917	+	0.991261i	$0.542113\pi$
$632$	−11.3137	−0.450035
$633$	−20.9706	−0.833505
$634$	−2.68629	−0.106686
$635$	15.3137	0.607706
$636$	10.0000	0.396526
$637$	14.0000	0.554700
$638$	7.65685	0.303138
$639$	5.65685	0.223782
$640$	−1.00000	−0.0395285
$641$	−30.2843	−1.19616	−0.598078	−	0.801438i	$-0.704069\pi$
$641$	−30.2843	−1.19616	−0.598078	+	0.801438i	$0.704069\pi$
$642$	17.6569	0.696860
$643$	8.68629	0.342554	0.171277	−	0.985223i	$-0.445211\pi$
$643$	8.68629	0.342554	0.171277	+	0.985223i	$0.445211\pi$
$644$	0	0
$645$	−5.65685	−0.222738
$646$	0	0
$647$	−37.9411	−1.49162	−0.745810	−	0.666159i	$-0.767937\pi$
$647$	−37.9411	−1.49162	−0.745810	+	0.666159i	$0.767937\pi$
$648$	1.00000	0.0392837
$649$	−11.3137	−0.444102
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	15.3137	0.599731
$653$	16.6274	0.650681	0.325341	−	0.945597i	$-0.394521\pi$
$653$	16.6274	0.650681	0.325341	+	0.945597i	$0.394521\pi$
$654$	8.34315	0.326243
$655$	7.31371	0.285770
$656$	−3.65685	−0.142776
$657$	9.31371	0.363362
$658$	0	0
$659$	15.3137	0.596537	0.298269	−	0.954482i	$-0.403591\pi$
$659$	15.3137	0.596537	0.298269	+	0.954482i	$0.403591\pi$
$660$	−1.00000	−0.0389249
$661$	39.9411	1.55353	0.776765	−	0.629791i	$-0.216859\pi$
$661$	39.9411	1.55353	0.776765	+	0.629791i	$0.216859\pi$
$662$	−7.31371	−0.284255
$663$	2.00000	0.0776736
$664$	4.00000	0.155230
$665$	0	0
$666$	11.6569	0.451694
$667$	−43.3137	−1.67711
$668$	5.65685	0.218870
$669$	−3.31371	−0.128115
$670$	4.00000	0.154533
$671$	−7.65685	−0.295590
$672$	0	0
$673$	−14.6863	−0.566115	−0.283057	−	0.959103i	$-0.591349\pi$
$673$	−14.6863	−0.566115	−0.283057	+	0.959103i	$0.591349\pi$
$674$	17.3137	0.666899
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	−24.6274	−0.946509	−0.473254	−	0.880926i	$-0.656921\pi$
$677$	−24.6274	−0.946509	−0.473254	+	0.880926i	$0.656921\pi$
$678$	7.65685	0.294060
$679$	0	0
$680$	−1.00000	−0.0383482
$681$	6.34315	0.243070
$682$	5.65685	0.216612
$683$	−37.9411	−1.45178	−0.725888	−	0.687812i	$-0.758571\pi$
$683$	−37.9411	−1.45178	−0.725888	+	0.687812i	$0.758571\pi$
$684$	0	0
$685$	8.34315	0.318775
$686$	0	0
$687$	−1.31371	−0.0501211
$688$	−5.65685	−0.215666
$689$	20.0000	0.761939
$690$	5.65685	0.215353
$691$	32.2843	1.22815	0.614076	−	0.789247i	$-0.289529\pi$
$691$	32.2843	1.22815	0.614076	+	0.789247i	$0.289529\pi$
$692$	−21.3137	−0.810226
$693$	0	0
$694$	9.65685	0.366569
$695$	12.9706	0.492001
$696$	7.65685	0.290232
$697$	−3.65685	−0.138513
$698$	8.34315	0.315793
$699$	25.3137	0.957452
$700$	0	0
$701$	47.9411	1.81071	0.905356	−	0.424654i	$-0.139604\pi$
$701$	47.9411	1.81071	0.905356	+	0.424654i	$0.139604\pi$
$702$	2.00000	0.0754851
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	−4.00000	−0.150649
$706$	18.9706	0.713967
$707$	0	0
$708$	−11.3137	−0.425195
$709$	41.3137	1.55157	0.775784	−	0.630998i	$-0.217354\pi$
$709$	41.3137	1.55157	0.775784	+	0.630998i	$0.217354\pi$
$710$	−5.65685	−0.212298
$711$	−11.3137	−0.424297
$712$	−14.0000	−0.524672
$713$	−32.0000	−1.19841
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	11.3137	0.422813
$717$	4.68629	0.175013
$718$	−8.00000	−0.298557
$719$	26.3431	0.982434	0.491217	−	0.871037i	$-0.336552\pi$
$719$	26.3431	0.982434	0.491217	+	0.871037i	$0.336552\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−19.0000	−0.707107
$723$	−17.3137	−0.643904
$724$	14.0000	0.520306
$725$	−7.65685	−0.284368
$726$	−1.00000	−0.0371135
$727$	48.0000	1.78022	0.890111	−	0.455744i	$-0.150627\pi$
$727$	48.0000	1.78022	0.890111	+	0.455744i	$0.150627\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−9.31371	−0.344716
$731$	−5.65685	−0.209226
$732$	−7.65685	−0.283005
$733$	−22.6863	−0.837937	−0.418969	−	0.908001i	$-0.637608\pi$
$733$	−22.6863	−0.837937	−0.418969	+	0.908001i	$0.637608\pi$
$734$	−3.31371	−0.122311
$735$	−7.00000	−0.258199
$736$	5.65685	0.208514
$737$	4.00000	0.147342
$738$	−3.65685	−0.134611
$739$	3.31371	0.121897	0.0609484	−	0.998141i	$-0.480588\pi$
$739$	3.31371	0.121897	0.0609484	+	0.998141i	$0.480588\pi$
$740$	−11.6569	−0.428514
$741$	0	0
$742$	0	0
$743$	24.9706	0.916081	0.458041	−	0.888931i	$-0.348551\pi$
$743$	24.9706	0.916081	0.458041	+	0.888931i	$0.348551\pi$
$744$	5.65685	0.207390
$745$	−6.00000	−0.219823
$746$	−32.6274	−1.19457
$747$	4.00000	0.146352
$748$	−1.00000	−0.0365636
$749$	0	0
$750$	1.00000	0.0365148
$751$	−13.6569	−0.498346	−0.249173	−	0.968459i	$-0.580159\pi$
$751$	−13.6569	−0.498346	−0.249173	+	0.968459i	$0.580159\pi$
$752$	−4.00000	−0.145865
$753$	−8.00000	−0.291536
$754$	15.3137	0.557692
$755$	17.6569	0.642599
$756$	0	0
$757$	−19.9411	−0.724773	−0.362386	−	0.932028i	$-0.618038\pi$
$757$	−19.9411	−0.724773	−0.362386	+	0.932028i	$0.618038\pi$
$758$	−1.65685	−0.0601797
$759$	5.65685	0.205331
$760$	0	0
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	15.3137	0.554757
$763$	0	0
$764$	−17.6569	−0.638803
$765$	−1.00000	−0.0361551
$766$	−31.3137	−1.13141
$767$	−22.6274	−0.817029
$768$	−1.00000	−0.0360844
$769$	−12.6274	−0.455356	−0.227678	−	0.973736i	$-0.573113\pi$
$769$	−12.6274	−0.455356	−0.227678	+	0.973736i	$0.573113\pi$
$770$	0	0
$771$	30.9706	1.11538
$772$	−21.3137	−0.767097
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	−5.65685	−0.203331
$775$	−5.65685	−0.203200
$776$	−0.343146	−0.0123182
$777$	0	0
$778$	−15.6569	−0.561325
$779$	0	0
$780$	−2.00000	−0.0716115
$781$	−5.65685	−0.202418
$782$	5.65685	0.202289
$783$	7.65685	0.273634
$784$	−7.00000	−0.250000
$785$	5.31371	0.189654
$786$	7.31371	0.260871
$787$	19.0294	0.678326	0.339163	−	0.940728i	$-0.389856\pi$
$787$	19.0294	0.678326	0.339163	+	0.940728i	$0.389856\pi$
$788$	−13.3137	−0.474281
$789$	16.0000	0.569615
$790$	11.3137	0.402524
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	−15.3137	−0.543806
$794$	5.02944	0.178488
$795$	−10.0000	−0.354663
$796$	16.9706	0.601506
$797$	46.0000	1.62940	0.814702	−	0.579880i	$-0.196901\pi$
$797$	46.0000	1.62940	0.814702	+	0.579880i	$0.196901\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	1.00000	0.0353553
$801$	−14.0000	−0.494666
$802$	−18.9706	−0.669874
$803$	−9.31371	−0.328674
$804$	4.00000	0.141069
$805$	0	0
$806$	11.3137	0.398508
$807$	−29.3137	−1.03189
$808$	6.00000	0.211079
$809$	−34.2843	−1.20537	−0.602685	−	0.797979i	$-0.705903\pi$
$809$	−34.2843	−1.20537	−0.602685	+	0.797979i	$0.705903\pi$
$810$	−1.00000	−0.0351364
$811$	48.2843	1.69549	0.847745	−	0.530404i	$-0.177960\pi$
$811$	48.2843	1.69549	0.847745	+	0.530404i	$0.177960\pi$
$812$	0	0
$813$	9.65685	0.338681
$814$	−11.6569	−0.408573
$815$	−15.3137	−0.536416
$816$	−1.00000	−0.0350070
$817$	0	0
$818$	−17.3137	−0.605360
$819$	0	0
$820$	3.65685	0.127703
$821$	−42.9706	−1.49968	−0.749841	−	0.661618i	$-0.769870\pi$
$821$	−42.9706	−1.49968	−0.749841	+	0.661618i	$0.769870\pi$
$822$	8.34315	0.291001
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	38.3431	1.33332	0.666661	−	0.745361i	$-0.267723\pi$
$827$	38.3431	1.33332	0.666661	+	0.745361i	$0.267723\pi$
$828$	5.65685	0.196589
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	−4.00000	−0.138842
$831$	−10.9706	−0.380565
$832$	−2.00000	−0.0693375
$833$	−7.00000	−0.242536
$834$	12.9706	0.449134
$835$	−5.65685	−0.195764
$836$	0	0
$837$	5.65685	0.195529
$838$	−18.6274	−0.643473
$839$	−29.6569	−1.02387	−0.511934	−	0.859025i	$-0.671071\pi$
$839$	−29.6569	−1.02387	−0.511934	+	0.859025i	$0.671071\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	28.6274	0.986566
$843$	−24.6274	−0.848213
$844$	20.9706	0.721837
$845$	9.00000	0.309609
$846$	−4.00000	−0.137523
$847$	0	0
$848$	−10.0000	−0.343401
$849$	1.65685	0.0568631
$850$	1.00000	0.0342997
$851$	65.9411	2.26043
$852$	−5.65685	−0.193801
$853$	15.6569	0.536080	0.268040	−	0.963408i	$-0.413624\pi$
$853$	15.6569	0.536080	0.268040	+	0.963408i	$0.413624\pi$
$854$	0	0
$855$	0	0
$856$	−17.6569	−0.603499
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	−2.00000	−0.0682789
$859$	24.6863	0.842285	0.421143	−	0.906994i	$-0.361629\pi$
$859$	24.6863	0.842285	0.421143	+	0.906994i	$0.361629\pi$
$860$	5.65685	0.192897
$861$	0	0
$862$	−14.6274	−0.498212
$863$	31.3137	1.06593	0.532966	−	0.846137i	$-0.321078\pi$
$863$	31.3137	1.06593	0.532966	+	0.846137i	$0.321078\pi$
$864$	−1.00000	−0.0340207
$865$	21.3137	0.724688
$866$	−9.31371	−0.316493
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	11.3137	0.383791
$870$	−7.65685	−0.259592
$871$	8.00000	0.271070
$872$	−8.34315	−0.282535
$873$	−0.343146	−0.0116137
$874$	0	0
$875$	0	0
$876$	−9.31371	−0.314681
$877$	38.2843	1.29277	0.646384	−	0.763012i	$-0.276280\pi$
$877$	38.2843	1.29277	0.646384	+	0.763012i	$0.276280\pi$
$878$	8.00000	0.269987
$879$	26.0000	0.876958
$880$	1.00000	0.0337100
$881$	−23.6569	−0.797020	−0.398510	−	0.917164i	$-0.630473\pi$
$881$	−23.6569	−0.797020	−0.398510	+	0.917164i	$0.630473\pi$
$882$	−7.00000	−0.235702
$883$	−23.3137	−0.784569	−0.392284	−	0.919844i	$-0.628315\pi$
$883$	−23.3137	−0.784569	−0.392284	+	0.919844i	$0.628315\pi$
$884$	−2.00000	−0.0672673
$885$	11.3137	0.380306
$886$	−36.2843	−1.21899
$887$	0.970563	0.0325883	0.0162942	−	0.999867i	$-0.494813\pi$
$887$	0.970563	0.0325883	0.0162942	+	0.999867i	$0.494813\pi$
$888$	−11.6569	−0.391178
$889$	0	0
$890$	14.0000	0.469281
$891$	−1.00000	−0.0335013
$892$	3.31371	0.110951
$893$	0	0
$894$	−6.00000	−0.200670
$895$	−11.3137	−0.378176
$896$	0	0
$897$	11.3137	0.377754
$898$	−26.9706	−0.900019
$899$	43.3137	1.44459
$900$	1.00000	0.0333333
$901$	−10.0000	−0.333148
$902$	3.65685	0.121760
$903$	0	0
$904$	−7.65685	−0.254663
$905$	−14.0000	−0.465376
$906$	17.6569	0.586610
$907$	45.9411	1.52545	0.762725	−	0.646723i	$-0.223861\pi$
$907$	45.9411	1.52545	0.762725	+	0.646723i	$0.223861\pi$
$908$	−6.34315	−0.210505
$909$	6.00000	0.199007
$910$	0	0
$911$	−7.02944	−0.232896	−0.116448	−	0.993197i	$-0.537151\pi$
$911$	−7.02944	−0.232896	−0.116448	+	0.993197i	$0.537151\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	−14.9706	−0.495182
$915$	7.65685	0.253128
$916$	1.31371	0.0434062
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	49.6569	1.63803	0.819014	−	0.573773i	$-0.194521\pi$
$919$	49.6569	1.63803	0.819014	+	0.573773i	$0.194521\pi$
$920$	−5.65685	−0.186501
$921$	−13.6569	−0.450009
$922$	−6.68629	−0.220201
$923$	−11.3137	−0.372395
$924$	0	0
$925$	11.6569	0.383275
$926$	−16.0000	−0.525793
$927$	0	0
$928$	−7.65685	−0.251349
$929$	45.5980	1.49602	0.748011	−	0.663687i	$-0.231009\pi$
$929$	45.5980	1.49602	0.748011	+	0.663687i	$0.231009\pi$
$930$	−5.65685	−0.185496
$931$	0	0
$932$	−25.3137	−0.829178
$933$	−16.9706	−0.555591
$934$	24.9706	0.817062
$935$	1.00000	0.0327035
$936$	−2.00000	−0.0653720
$937$	36.3431	1.18728	0.593639	−	0.804731i	$-0.297691\pi$
$937$	36.3431	1.18728	0.593639	+	0.804731i	$0.297691\pi$
$938$	0	0
$939$	14.9706	0.488546
$940$	4.00000	0.130466
$941$	10.2843	0.335258	0.167629	−	0.985850i	$-0.446389\pi$
$941$	10.2843	0.335258	0.167629	+	0.985850i	$0.446389\pi$
$942$	5.31371	0.173130
$943$	−20.6863	−0.673638
$944$	11.3137	0.368230
$945$	0	0
$946$	5.65685	0.183920
$947$	53.9411	1.75285	0.876426	−	0.481537i	$-0.159921\pi$
$947$	53.9411	1.75285	0.876426	+	0.481537i	$0.159921\pi$
$948$	11.3137	0.367452
$949$	−18.6274	−0.604672
$950$	0	0
$951$	2.68629	0.0871090
$952$	0	0
$953$	55.2548	1.78988	0.894940	−	0.446187i	$-0.147218\pi$
$953$	55.2548	1.78988	0.894940	+	0.446187i	$0.147218\pi$
$954$	−10.0000	−0.323762
$955$	17.6569	0.571362
$956$	−4.68629	−0.151565
$957$	−7.65685	−0.247511
$958$	27.3137	0.882466
$959$	0	0
$960$	1.00000	0.0322749
$961$	1.00000	0.0322581
$962$	−23.3137	−0.751664
$963$	−17.6569	−0.568984
$964$	17.3137	0.557637
$965$	21.3137	0.686113
$966$	0	0
$967$	49.2548	1.58393	0.791964	−	0.610567i	$-0.209059\pi$
$967$	49.2548	1.58393	0.791964	+	0.610567i	$0.209059\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	0.343146	0.0110177
$971$	−11.3137	−0.363074	−0.181537	−	0.983384i	$-0.558107\pi$
$971$	−11.3137	−0.363074	−0.181537	+	0.983384i	$0.558107\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	30.6274	0.981366
$975$	2.00000	0.0640513
$976$	7.65685	0.245090
$977$	49.5980	1.58678	0.793390	−	0.608714i	$-0.208314\pi$
$977$	49.5980	1.58678	0.793390	+	0.608714i	$0.208314\pi$
$978$	−15.3137	−0.489678
$979$	14.0000	0.447442
$980$	7.00000	0.223607
$981$	−8.34315	−0.266376
$982$	0.686292	0.0219004
$983$	−42.3431	−1.35054	−0.675268	−	0.737572i	$-0.735972\pi$
$983$	−42.3431	−1.35054	−0.675268	+	0.737572i	$0.735972\pi$
$984$	3.65685	0.116576
$985$	13.3137	0.424210
$986$	−7.65685	−0.243844
$987$	0	0
$988$	0	0
$989$	−32.0000	−1.01754
$990$	1.00000	0.0317821
$991$	−31.5980	−1.00374	−0.501871	−	0.864942i	$-0.667355\pi$
$991$	−31.5980	−1.00374	−0.501871	+	0.864942i	$0.667355\pi$
$992$	−5.65685	−0.179605
$993$	7.31371	0.232094
$994$	0	0
$995$	−16.9706	−0.538003
$996$	−4.00000	−0.126745
$997$	−21.0294	−0.666009	−0.333004	−	0.942925i	$-0.608062\pi$
$997$	−21.0294	−0.666009	−0.333004	+	0.942925i	$0.608062\pi$
$998$	32.2843	1.02194
$999$	−11.6569	−0.368807

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5610.2.a.bt.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.bt.1.2	✓	2	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:	\( N \)	\(=\)	\( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5610.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} -1.00000 q^{22} +5.65685 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -7.65685 q^{29} +1.00000 q^{30} -5.65685 q^{31} +1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} +11.6569 q^{37} +2.00000 q^{39} -1.00000 q^{40} -3.65685 q^{41} -5.65685 q^{43} -1.00000 q^{44} -1.00000 q^{45} +5.65685 q^{46} -4.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +1.00000 q^{55} -7.65685 q^{58} +11.3137 q^{59} +1.00000 q^{60} +7.65685 q^{61} -5.65685 q^{62} +1.00000 q^{64} +2.00000 q^{65} +1.00000 q^{66} -4.00000 q^{67} +1.00000 q^{68} -5.65685 q^{69} +5.65685 q^{71} +1.00000 q^{72} +9.31371 q^{73} +11.6569 q^{74} -1.00000 q^{75} +2.00000 q^{78} -11.3137 q^{79} -1.00000 q^{80} +1.00000 q^{81} -3.65685 q^{82} +4.00000 q^{83} -1.00000 q^{85} -5.65685 q^{86} +7.65685 q^{87} -1.00000 q^{88} -14.0000 q^{89} -1.00000 q^{90} +5.65685 q^{92} +5.65685 q^{93} -4.00000 q^{94} -1.00000 q^{96} -0.343146 q^{97} -7.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 2 q^{20} - 2 q^{22} - 2 q^{24} + 2 q^{25} - 4 q^{26} - 2 q^{27} - 4 q^{29} + 2 q^{30} + 2 q^{32} + 2 q^{33} + 2 q^{34} + 2 q^{36} + 12 q^{37} + 4 q^{39} - 2 q^{40} + 4 q^{41} - 2 q^{44} - 2 q^{45} - 8 q^{47} - 2 q^{48} - 14 q^{49} + 2 q^{50} - 2 q^{51} - 4 q^{52} - 20 q^{53} - 2 q^{54} + 2 q^{55} - 4 q^{58} + 2 q^{60} + 4 q^{61} + 2 q^{64} + 4 q^{65} + 2 q^{66} - 8 q^{67} + 2 q^{68} + 2 q^{72} - 4 q^{73} + 12 q^{74} - 2 q^{75} + 4 q^{78} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} - 2 q^{85} + 4 q^{87} - 2 q^{88} - 28 q^{89} - 2 q^{90} - 8 q^{94} - 2 q^{96} - 12 q^{97} - 14 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^15 + 2 * q^16 + 2 * q^17 + 2 * q^18 - 2 * q^20 - 2 * q^22 - 2 * q^24 + 2 * q^25 - 4 * q^26 - 2 * q^27 - 4 * q^29 + 2 * q^30 + 2 * q^32 + 2 * q^33 + 2 * q^34 + 2 * q^36 + 12 * q^37 + 4 * q^39 - 2 * q^40 + 4 * q^41 - 2 * q^44 - 2 * q^45 - 8 * q^47 - 2 * q^48 - 14 * q^49 + 2 * q^50 - 2 * q^51 - 4 * q^52 - 20 * q^53 - 2 * q^54 + 2 * q^55 - 4 * q^58 + 2 * q^60 + 4 * q^61 + 2 * q^64 + 4 * q^65 + 2 * q^66 - 8 * q^67 + 2 * q^68 + 2 * q^72 - 4 * q^73 + 12 * q^74 - 2 * q^75 + 4 * q^78 - 2 * q^80 + 2 * q^81 + 4 * q^82 + 8 * q^83 - 2 * q^85 + 4 * q^87 - 2 * q^88 - 28 * q^89 - 2 * q^90 - 8 * q^94 - 2 * q^96 - 12 * q^97 - 14 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more