LMFDB - Embedded newform 5610.2.a.bt.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} -1.00000 q^{22} -5.65685 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +3.65685 q^{29} +1.00000 q^{30} +5.65685 q^{31} +1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} +0.343146 q^{37} +2.00000 q^{39} -1.00000 q^{40} +7.65685 q^{41} +5.65685 q^{43} -1.00000 q^{44} -1.00000 q^{45} -5.65685 q^{46} -4.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +1.00000 q^{55} +3.65685 q^{58} -11.3137 q^{59} +1.00000 q^{60} -3.65685 q^{61} +5.65685 q^{62} +1.00000 q^{64} +2.00000 q^{65} +1.00000 q^{66} -4.00000 q^{67} +1.00000 q^{68} +5.65685 q^{69} -5.65685 q^{71} +1.00000 q^{72} -13.3137 q^{73} +0.343146 q^{74} -1.00000 q^{75} +2.00000 q^{78} +11.3137 q^{79} -1.00000 q^{80} +1.00000 q^{81} +7.65685 q^{82} +4.00000 q^{83} -1.00000 q^{85} +5.65685 q^{86} -3.65685 q^{87} -1.00000 q^{88} -14.0000 q^{89} -1.00000 q^{90} -5.65685 q^{92} -5.65685 q^{93} -4.00000 q^{94} -1.00000 q^{96} -11.6569 q^{97} -7.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.65685	0.679061	0.339530	−	0.940595i	$-0.389732\pi$
$29$	3.65685	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$30$	1.00000	0.182574
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	0.343146	0.0564128	0.0282064	−	0.999602i	$-0.491020\pi$
$37$	0.343146	0.0564128	0.0282064	+	0.999602i	$0.491020\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	−1.00000	−0.158114
$41$	7.65685	1.19580	0.597900	−	0.801571i	$-0.296002\pi$
$41$	7.65685	1.19580	0.597900	+	0.801571i	$0.296002\pi$
$42$	0	0
$43$	5.65685	0.862662	0.431331	−	0.902194i	$-0.358044\pi$
$43$	5.65685	0.862662	0.431331	+	0.902194i	$0.358044\pi$
$44$	−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
$58$	3.65685	0.480168
$59$	−11.3137	−1.47292	−0.736460	−	0.676481i	$-0.763504\pi$
$59$	−11.3137	−1.47292	−0.736460	+	0.676481i	$0.763504\pi$
$60$	1.00000	0.129099
$61$	−3.65685	−0.468212	−0.234106	−	0.972211i	$-0.575216\pi$
$61$	−3.65685	−0.468212	−0.234106	+	0.972211i	$0.575216\pi$
$62$	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	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	1.00000	0.117851
$73$	−13.3137	−1.55825	−0.779126	−	0.626868i	$-0.784337\pi$
$73$	−13.3137	−1.55825	−0.779126	+	0.626868i	$0.784337\pi$
$74$	0.343146	0.0398899
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	2.00000	0.226455
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	7.65685	0.845558
$83$	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
$87$	−3.65685	−0.392056
$88$	−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
$97$	−11.6569	−1.18357	−0.591787	−	0.806094i	$-0.701577\pi$
$97$	−11.6569	−1.18357	−0.591787	+	0.806094i	$0.701577\pi$
$98$	−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
$107$	−6.34315	−0.613215	−0.306608	−	0.951836i	$-0.599194\pi$
$107$	−6.34315	−0.613215	−0.306608	+	0.951836i	$0.599194\pi$
$108$	−1.00000	−0.0962250
$109$	−19.6569	−1.88279	−0.941393	−	0.337313i	$-0.890482\pi$
$109$	−19.6569	−1.88279	−0.941393	+	0.337313i	$0.890482\pi$
$110$	1.00000	0.0953463
$111$	−0.343146	−0.0325700
$112$	0	0
$113$	3.65685	0.344008	0.172004	−	0.985096i	$-0.444976\pi$
$113$	3.65685	0.344008	0.172004	+	0.985096i	$0.444976\pi$
$114$	0	0
$115$	5.65685	0.527504
$116$	3.65685	0.339530
$117$	−2.00000	−0.184900
$118$	−11.3137	−1.04151
$119$	0	0
$120$	1.00000	0.0912871
$121$	1.00000	0.0909091
$122$	−3.65685	−0.331076
$123$	−7.65685	−0.690395
$124$	5.65685	0.508001
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	7.31371	0.648987	0.324493	−	0.945888i	$-0.394806\pi$
$127$	7.31371	0.648987	0.324493	+	0.945888i	$0.394806\pi$
$128$	1.00000	0.0883883
$129$	−5.65685	−0.498058
$130$	2.00000	0.175412
$131$	15.3137	1.33796	0.668982	−	0.743278i	$-0.266730\pi$
$131$	15.3137	1.33796	0.668982	+	0.743278i	$0.266730\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−4.00000	−0.345547
$135$	1.00000	0.0860663
$136$	1.00000	0.0857493
$137$	−19.6569	−1.67940	−0.839699	−	0.543052i	$-0.817269\pi$
$137$	−19.6569	−1.67940	−0.839699	+	0.543052i	$0.817269\pi$
$138$	5.65685	0.481543
$139$	20.9706	1.77870	0.889350	−	0.457227i	$-0.151157\pi$
$139$	20.9706	1.77870	0.889350	+	0.457227i	$0.151157\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	−5.65685	−0.474713
$143$	2.00000	0.167248
$144$	1.00000	0.0833333
$145$	−3.65685	−0.303685
$146$	−13.3137	−1.10185
$147$	7.00000	0.577350
$148$	0.343146	0.0282064
$149$	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
$151$	−6.34315	−0.516198	−0.258099	−	0.966118i	$-0.583096\pi$
$151$	−6.34315	−0.516198	−0.258099	+	0.966118i	$0.583096\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	−5.65685	−0.454369
$156$	2.00000	0.160128
$157$	17.3137	1.38178	0.690892	−	0.722958i	$-0.257218\pi$
$157$	17.3137	1.38178	0.690892	+	0.722958i	$0.257218\pi$
$158$	11.3137	0.900070
$159$	10.0000	0.793052
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	−7.31371	−0.572854	−0.286427	−	0.958102i	$-0.592468\pi$
$163$	−7.31371	−0.572854	−0.286427	+	0.958102i	$0.592468\pi$
$164$	7.65685	0.597900
$165$	−1.00000	−0.0778499
$166$	4.00000	0.310460
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−1.00000	−0.0766965
$171$	0	0
$172$	5.65685	0.431331
$173$	1.31371	0.0998794	0.0499397	−	0.998752i	$-0.484097\pi$
$173$	1.31371	0.0998794	0.0499397	+	0.998752i	$0.484097\pi$
$174$	−3.65685	−0.277225
$175$	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	$-0.638957\pi$
$179$	−11.3137	−0.845626	−0.422813	+	0.906217i	$0.638957\pi$
$180$	−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
$183$	3.65685	0.270322
$184$	−5.65685	−0.417029
$185$	−0.343146	−0.0252286
$186$	−5.65685	−0.414781
$187$	−1.00000	−0.0731272
$188$	−4.00000	−0.291730
$189$	0	0
$190$	0	0
$191$	−6.34315	−0.458974	−0.229487	−	0.973312i	$-0.573705\pi$
$191$	−6.34315	−0.458974	−0.229487	+	0.973312i	$0.573705\pi$
$192$	−1.00000	−0.0721688
$193$	1.31371	0.0945628	0.0472814	−	0.998882i	$-0.484944\pi$
$193$	1.31371	0.0945628	0.0472814	+	0.998882i	$0.484944\pi$
$194$	−11.6569	−0.836913
$195$	−2.00000	−0.143223
$196$	−7.00000	−0.500000
$197$	9.31371	0.663574	0.331787	−	0.943354i	$-0.392348\pi$
$197$	9.31371	0.663574	0.331787	+	0.943354i	$0.392348\pi$
$198$	−1.00000	−0.0710669
$199$	−16.9706	−1.20301	−0.601506	−	0.798869i	$-0.705432\pi$
$199$	−16.9706	−1.20301	−0.601506	+	0.798869i	$0.705432\pi$
$200$	1.00000	0.0707107
$201$	4.00000	0.282138
$202$	6.00000	0.422159
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	−7.65685	−0.534778
$206$	0	0
$207$	−5.65685	−0.393179
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	−12.9706	−0.892930	−0.446465	−	0.894801i	$-0.647317\pi$
$211$	−12.9706	−0.892930	−0.446465	+	0.894801i	$0.647317\pi$
$212$	−10.0000	−0.686803
$213$	5.65685	0.387601
$214$	−6.34315	−0.433609
$215$	−5.65685	−0.385794
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−19.6569	−1.33133
$219$	13.3137	0.899657
$220$	1.00000	0.0674200
$221$	−2.00000	−0.134535
$222$	−0.343146	−0.0230304
$223$	−19.3137	−1.29334	−0.646671	−	0.762769i	$-0.723839\pi$
$223$	−19.3137	−1.29334	−0.646671	+	0.762769i	$0.723839\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	3.65685	0.243250
$227$	−17.6569	−1.17193	−0.585963	−	0.810338i	$-0.699284\pi$
$227$	−17.6569	−1.17193	−0.585963	+	0.810338i	$0.699284\pi$
$228$	0	0
$229$	−21.3137	−1.40845	−0.704225	−	0.709977i	$-0.748705\pi$
$229$	−21.3137	−1.40845	−0.704225	+	0.709977i	$0.748705\pi$
$230$	5.65685	0.373002
$231$	0	0
$232$	3.65685	0.240084
$233$	−2.68629	−0.175985	−0.0879924	−	0.996121i	$-0.528045\pi$
$233$	−2.68629	−0.175985	−0.0879924	+	0.996121i	$0.528045\pi$
$234$	−2.00000	−0.130744
$235$	4.00000	0.260931
$236$	−11.3137	−0.736460
$237$	−11.3137	−0.734904
$238$	0	0
$239$	−27.3137	−1.76678	−0.883388	−	0.468641i	$-0.844744\pi$
$239$	−27.3137	−1.76678	−0.883388	+	0.468641i	$0.844744\pi$
$240$	1.00000	0.0645497
$241$	−5.31371	−0.342286	−0.171143	−	0.985246i	$-0.554746\pi$
$241$	−5.31371	−0.342286	−0.171143	+	0.985246i	$0.554746\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	−3.65685	−0.234106
$245$	7.00000	0.447214
$246$	−7.65685	−0.488183
$247$	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
$254$	7.31371	0.458903
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	2.97056	0.185299	0.0926493	−	0.995699i	$-0.470466\pi$
$257$	2.97056	0.185299	0.0926493	+	0.995699i	$0.470466\pi$
$258$	−5.65685	−0.352180
$259$	0	0
$260$	2.00000	0.124035
$261$	3.65685	0.226354
$262$	15.3137	0.946084
$263$	−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
$269$	6.68629	0.407670	0.203835	−	0.979005i	$-0.434659\pi$
$269$	6.68629	0.407670	0.203835	+	0.979005i	$0.434659\pi$
$270$	1.00000	0.0608581
$271$	1.65685	0.100647	0.0503234	−	0.998733i	$-0.483975\pi$
$271$	1.65685	0.100647	0.0503234	+	0.998733i	$0.483975\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−19.6569	−1.18751
$275$	−1.00000	−0.0603023
$276$	5.65685	0.340503
$277$	−22.9706	−1.38017	−0.690084	−	0.723730i	$-0.742426\pi$
$277$	−22.9706	−1.38017	−0.690084	+	0.723730i	$0.742426\pi$
$278$	20.9706	1.25773
$279$	5.65685	0.338667
$280$	0	0
$281$	−20.6274	−1.23053	−0.615264	−	0.788321i	$-0.710951\pi$
$281$	−20.6274	−1.23053	−0.615264	+	0.788321i	$0.710951\pi$
$282$	4.00000	0.238197
$283$	9.65685	0.574040	0.287020	−	0.957925i	$-0.407335\pi$
$283$	9.65685	0.574040	0.287020	+	0.957925i	$0.407335\pi$
$284$	−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
$290$	−3.65685	−0.214738
$291$	11.6569	0.683337
$292$	−13.3137	−0.779126
$293$	−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
$296$	0.343146	0.0199449
$297$	1.00000	0.0580259
$298$	6.00000	0.347571
$299$	11.3137	0.654289
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	−6.34315	−0.365007
$303$	−6.00000	−0.344691
$304$	0	0
$305$	3.65685	0.209391
$306$	1.00000	0.0571662
$307$	2.34315	0.133730	0.0668652	−	0.997762i	$-0.478700\pi$
$307$	2.34315	0.133730	0.0668652	+	0.997762i	$0.478700\pi$
$308$	0	0
$309$	0	0
$310$	−5.65685	−0.321288
$311$	−16.9706	−0.962312	−0.481156	−	0.876635i	$-0.659783\pi$
$311$	−16.9706	−0.962312	−0.481156	+	0.876635i	$0.659783\pi$
$312$	2.00000	0.113228
$313$	18.9706	1.07228	0.536140	−	0.844129i	$-0.319882\pi$
$313$	18.9706	1.07228	0.536140	+	0.844129i	$0.319882\pi$
$314$	17.3137	0.977069
$315$	0	0
$316$	11.3137	0.636446
$317$	−25.3137	−1.42176	−0.710880	−	0.703314i	$-0.751703\pi$
$317$	−25.3137	−1.42176	−0.710880	+	0.703314i	$0.751703\pi$
$318$	10.0000	0.560772
$319$	−3.65685	−0.204745
$320$	−1.00000	−0.0559017
$321$	6.34315	0.354040
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	−7.31371	−0.405069
$327$	19.6569	1.08703
$328$	7.65685	0.422779
$329$	0	0
$330$	−1.00000	−0.0550482
$331$	15.3137	0.841718	0.420859	−	0.907126i	$-0.361729\pi$
$331$	15.3137	0.841718	0.420859	+	0.907126i	$0.361729\pi$
$332$	4.00000	0.219529
$333$	0.343146	0.0188043
$334$	−5.65685	−0.309529
$335$	4.00000	0.218543
$336$	0	0
$337$	−5.31371	−0.289456	−0.144728	−	0.989471i	$-0.546231\pi$
$337$	−5.31371	−0.289456	−0.144728	+	0.989471i	$0.546231\pi$
$338$	−9.00000	−0.489535
$339$	−3.65685	−0.198613
$340$	−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
$346$	1.31371	0.0706254
$347$	−1.65685	−0.0889446	−0.0444723	−	0.999011i	$-0.514161\pi$
$347$	−1.65685	−0.0889446	−0.0444723	+	0.999011i	$0.514161\pi$
$348$	−3.65685	−0.196028
$349$	19.6569	1.05221	0.526104	−	0.850420i	$-0.323652\pi$
$349$	19.6569	1.05221	0.526104	+	0.850420i	$0.323652\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	−1.00000	−0.0533002
$353$	−14.9706	−0.796803	−0.398401	−	0.917211i	$-0.630435\pi$
$353$	−14.9706	−0.796803	−0.398401	+	0.917211i	$0.630435\pi$
$354$	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
$365$	13.3137	0.696871
$366$	3.65685	0.191147
$367$	19.3137	1.00817	0.504084	−	0.863655i	$-0.331830\pi$
$367$	19.3137	1.00817	0.504084	+	0.863655i	$0.331830\pi$
$368$	−5.65685	−0.294884
$369$	7.65685	0.398600
$370$	−0.343146	−0.0178393
$371$	0	0
$372$	−5.65685	−0.293294
$373$	12.6274	0.653823	0.326911	−	0.945055i	$-0.393992\pi$
$373$	12.6274	0.653823	0.326911	+	0.945055i	$0.393992\pi$
$374$	−1.00000	−0.0517088
$375$	1.00000	0.0516398
$376$	−4.00000	−0.206284
$377$	−7.31371	−0.376675
$378$	0	0
$379$	9.65685	0.496039	0.248020	−	0.968755i	$-0.420220\pi$
$379$	9.65685	0.496039	0.248020	+	0.968755i	$0.420220\pi$
$380$	0	0
$381$	−7.31371	−0.374693
$382$	−6.34315	−0.324544
$383$	−8.68629	−0.443849	−0.221924	−	0.975064i	$-0.571234\pi$
$383$	−8.68629	−0.443849	−0.221924	+	0.975064i	$0.571234\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	1.31371	0.0668660
$387$	5.65685	0.287554
$388$	−11.6569	−0.591787
$389$	−4.34315	−0.220206	−0.110103	−	0.993920i	$-0.535118\pi$
$389$	−4.34315	−0.220206	−0.110103	+	0.993920i	$0.535118\pi$
$390$	−2.00000	−0.101274
$391$	−5.65685	−0.286079
$392$	−7.00000	−0.353553
$393$	−15.3137	−0.772474
$394$	9.31371	0.469218
$395$	−11.3137	−0.569254
$396$	−1.00000	−0.0502519
$397$	38.9706	1.95588	0.977938	−	0.208894i	$-0.0669864\pi$
$397$	38.9706	1.95588	0.977938	+	0.208894i	$0.0669864\pi$
$398$	−16.9706	−0.850657
$399$	0	0
$400$	1.00000	0.0500000
$401$	14.9706	0.747594	0.373797	−	0.927510i	$-0.378056\pi$
$401$	14.9706	0.747594	0.373797	+	0.927510i	$0.378056\pi$
$402$	4.00000	0.199502
$403$	−11.3137	−0.563576
$404$	6.00000	0.298511
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−0.343146	−0.0170091
$408$	−1.00000	−0.0495074
$409$	5.31371	0.262746	0.131373	−	0.991333i	$-0.458061\pi$
$409$	5.31371	0.262746	0.131373	+	0.991333i	$0.458061\pi$
$410$	−7.65685	−0.378145
$411$	19.6569	0.969601
$412$	0	0
$413$	0	0
$414$	−5.65685	−0.278019
$415$	−4.00000	−0.196352
$416$	−2.00000	−0.0980581
$417$	−20.9706	−1.02693
$418$	0	0
$419$	26.6274	1.30083	0.650417	−	0.759577i	$-0.274594\pi$
$419$	26.6274	1.30083	0.650417	+	0.759577i	$0.274594\pi$
$420$	0	0
$421$	−16.6274	−0.810371	−0.405185	−	0.914235i	$-0.632793\pi$
$421$	−16.6274	−0.810371	−0.405185	+	0.914235i	$0.632793\pi$
$422$	−12.9706	−0.631397
$423$	−4.00000	−0.194487
$424$	−10.0000	−0.485643
$425$	1.00000	0.0485071
$426$	5.65685	0.274075
$427$	0	0
$428$	−6.34315	−0.306608
$429$	−2.00000	−0.0965609
$430$	−5.65685	−0.272798
$431$	30.6274	1.47527	0.737635	−	0.675199i	$-0.235942\pi$
$431$	30.6274	1.47527	0.737635	+	0.675199i	$0.235942\pi$
$432$	−1.00000	−0.0481125
$433$	13.3137	0.639816	0.319908	−	0.947449i	$-0.396348\pi$
$433$	13.3137	0.639816	0.319908	+	0.947449i	$0.396348\pi$
$434$	0	0
$435$	3.65685	0.175333
$436$	−19.6569	−0.941393
$437$	0	0
$438$	13.3137	0.636154
$439$	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
$443$	20.2843	0.963735	0.481867	−	0.876244i	$-0.339959\pi$
$443$	20.2843	0.963735	0.481867	+	0.876244i	$0.339959\pi$
$444$	−0.343146	−0.0162850
$445$	14.0000	0.663664
$446$	−19.3137	−0.914531
$447$	−6.00000	−0.283790
$448$	0	0
$449$	6.97056	0.328961	0.164481	−	0.986380i	$-0.447405\pi$
$449$	6.97056	0.328961	0.164481	+	0.986380i	$0.447405\pi$
$450$	1.00000	0.0471405
$451$	−7.65685	−0.360547
$452$	3.65685	0.172004
$453$	6.34315	0.298027
$454$	−17.6569	−0.828677
$455$	0	0
$456$	0	0
$457$	18.9706	0.887405	0.443703	−	0.896174i	$-0.353665\pi$
$457$	18.9706	0.887405	0.443703	+	0.896174i	$0.353665\pi$
$458$	−21.3137	−0.995924
$459$	−1.00000	−0.0466760
$460$	5.65685	0.263752
$461$	−29.3137	−1.36528	−0.682638	−	0.730757i	$-0.739167\pi$
$461$	−29.3137	−1.36528	−0.682638	+	0.730757i	$0.739167\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	3.65685	0.169765
$465$	5.65685	0.262330
$466$	−2.68629	−0.124440
$467$	−8.97056	−0.415108	−0.207554	−	0.978224i	$-0.566550\pi$
$467$	−8.97056	−0.415108	−0.207554	+	0.978224i	$0.566550\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	4.00000	0.184506
$471$	−17.3137	−0.797774
$472$	−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
$478$	−27.3137	−1.24930
$479$	4.68629	0.214122	0.107061	−	0.994252i	$-0.465856\pi$
$479$	4.68629	0.214122	0.107061	+	0.994252i	$0.465856\pi$
$480$	1.00000	0.0456435
$481$	−0.686292	−0.0312922
$482$	−5.31371	−0.242033
$483$	0	0
$484$	1.00000	0.0454545
$485$	11.6569	0.529310
$486$	−1.00000	−0.0453609
$487$	−14.6274	−0.662832	−0.331416	−	0.943485i	$-0.607526\pi$
$487$	−14.6274	−0.662832	−0.331416	+	0.943485i	$0.607526\pi$
$488$	−3.65685	−0.165538
$489$	7.31371	0.330737
$490$	7.00000	0.316228
$491$	23.3137	1.05213	0.526066	−	0.850443i	$-0.323666\pi$
$491$	23.3137	1.05213	0.526066	+	0.850443i	$0.323666\pi$
$492$	−7.65685	−0.345198
$493$	3.65685	0.164696
$494$	0	0
$495$	1.00000	0.0449467
$496$	5.65685	0.254000
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−24.2843	−1.08711	−0.543557	−	0.839372i	$-0.682923\pi$
$499$	−24.2843	−1.08711	−0.543557	+	0.839372i	$0.682923\pi$
$500$	−1.00000	−0.0447214
$501$	5.65685	0.252730
$502$	8.00000	0.357057
$503$	−36.2843	−1.61784	−0.808918	−	0.587922i	$-0.799946\pi$
$503$	−36.2843	−1.61784	−0.808918	+	0.587922i	$0.799946\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	5.65685	0.251478
$507$	9.00000	0.399704
$508$	7.31371	0.324493
$509$	−39.6569	−1.75776	−0.878880	−	0.477044i	$-0.841708\pi$
$509$	−39.6569	−1.75776	−0.878880	+	0.477044i	$0.841708\pi$
$510$	1.00000	0.0442807
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.97056	0.131026
$515$	0	0
$516$	−5.65685	−0.249029
$517$	4.00000	0.175920
$518$	0	0
$519$	−1.31371	−0.0576654
$520$	2.00000	0.0877058
$521$	38.9706	1.70733	0.853666	−	0.520821i	$-0.174374\pi$
$521$	38.9706	1.70733	0.853666	+	0.520821i	$0.174374\pi$
$522$	3.65685	0.160056
$523$	−8.97056	−0.392255	−0.196128	−	0.980578i	$-0.562837\pi$
$523$	−8.97056	−0.392255	−0.196128	+	0.980578i	$0.562837\pi$
$524$	15.3137	0.668982
$525$	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
$533$	−15.3137	−0.663310
$534$	14.0000	0.605839
$535$	6.34315	0.274238
$536$	−4.00000	−0.172774
$537$	11.3137	0.488223
$538$	6.68629	0.288266
$539$	7.00000	0.301511
$540$	1.00000	0.0430331
$541$	42.9706	1.84745	0.923724	−	0.383058i	$-0.125129\pi$
$541$	42.9706	1.84745	0.923724	+	0.383058i	$0.125129\pi$
$542$	1.65685	0.0711680
$543$	−14.0000	−0.600798
$544$	1.00000	0.0428746
$545$	19.6569	0.842007
$546$	0	0
$547$	6.34315	0.271213	0.135607	−	0.990763i	$-0.456702\pi$
$547$	6.34315	0.271213	0.135607	+	0.990763i	$0.456702\pi$
$548$	−19.6569	−0.839699
$549$	−3.65685	−0.156071
$550$	−1.00000	−0.0426401
$551$	0	0
$552$	5.65685	0.240772
$553$	0	0
$554$	−22.9706	−0.975926
$555$	0.343146	0.0145657
$556$	20.9706	0.889350
$557$	4.62742	0.196070	0.0980350	−	0.995183i	$-0.468744\pi$
$557$	4.62742	0.196070	0.0980350	+	0.995183i	$0.468744\pi$
$558$	5.65685	0.239474
$559$	−11.3137	−0.478519
$560$	0	0
$561$	1.00000	0.0422200
$562$	−20.6274	−0.870115
$563$	0.686292	0.0289237	0.0144619	−	0.999895i	$-0.495396\pi$
$563$	0.686292	0.0289237	0.0144619	+	0.999895i	$0.495396\pi$
$564$	4.00000	0.168430
$565$	−3.65685	−0.153845
$566$	9.65685	0.405908
$567$	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
$571$	41.6569	1.74329	0.871643	−	0.490142i	$-0.163055\pi$
$571$	41.6569	1.74329	0.871643	+	0.490142i	$0.163055\pi$
$572$	2.00000	0.0836242
$573$	6.34315	0.264989
$574$	0	0
$575$	−5.65685	−0.235907
$576$	1.00000	0.0416667
$577$	−4.62742	−0.192642	−0.0963209	−	0.995350i	$-0.530708\pi$
$577$	−4.62742	−0.192642	−0.0963209	+	0.995350i	$0.530708\pi$
$578$	1.00000	0.0415945
$579$	−1.31371	−0.0545959
$580$	−3.65685	−0.151843
$581$	0	0
$582$	11.6569	0.483192
$583$	10.0000	0.414158
$584$	−13.3137	−0.550925
$585$	2.00000	0.0826898
$586$	−26.0000	−1.07405
$587$	−2.34315	−0.0967120	−0.0483560	−	0.998830i	$-0.515398\pi$
$587$	−2.34315	−0.0967120	−0.0483560	+	0.998830i	$0.515398\pi$
$588$	7.00000	0.288675
$589$	0	0
$590$	11.3137	0.465778
$591$	−9.31371	−0.383115
$592$	0.343146	0.0141032
$593$	−31.9411	−1.31166	−0.655832	−	0.754907i	$-0.727682\pi$
$593$	−31.9411	−1.31166	−0.655832	+	0.754907i	$0.727682\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	6.00000	0.245770
$597$	16.9706	0.694559
$598$	11.3137	0.462652
$599$	36.9706	1.51058	0.755288	−	0.655393i	$-0.227497\pi$
$599$	36.9706	1.51058	0.755288	+	0.655393i	$0.227497\pi$
$600$	−1.00000	−0.0408248
$601$	−22.6863	−0.925393	−0.462697	−	0.886517i	$-0.653118\pi$
$601$	−22.6863	−0.925393	−0.462697	+	0.886517i	$0.653118\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	−6.34315	−0.258099
$605$	−1.00000	−0.0406558
$606$	−6.00000	−0.243733
$607$	38.6274	1.56784	0.783919	−	0.620863i	$-0.213218\pi$
$607$	38.6274	1.56784	0.783919	+	0.620863i	$0.213218\pi$
$608$	0	0
$609$	0	0
$610$	3.65685	0.148062
$611$	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	$0.618421\pi$
$614$	2.34315	0.0945617
$615$	7.65685	0.308754
$616$	0	0
$617$	−41.5980	−1.67467	−0.837336	−	0.546689i	$-0.815888\pi$
$617$	−41.5980	−1.67467	−0.837336	+	0.546689i	$0.815888\pi$
$618$	0	0
$619$	9.65685	0.388142	0.194071	−	0.980988i	$-0.437831\pi$
$619$	9.65685	0.388142	0.194071	+	0.980988i	$0.437831\pi$
$620$	−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
$626$	18.9706	0.758216
$627$	0	0
$628$	17.3137	0.690892
$629$	0.343146	0.0136821
$630$	0	0
$631$	38.6274	1.53773	0.768867	−	0.639409i	$-0.220821\pi$
$631$	38.6274	1.53773	0.768867	+	0.639409i	$0.220821\pi$
$632$	11.3137	0.450035
$633$	12.9706	0.515534
$634$	−25.3137	−1.00534
$635$	−7.31371	−0.290236
$636$	10.0000	0.396526
$637$	14.0000	0.554700
$638$	−3.65685	−0.144776
$639$	−5.65685	−0.223782
$640$	−1.00000	−0.0395285
$641$	26.2843	1.03817	0.519083	−	0.854724i	$-0.326273\pi$
$641$	26.2843	1.03817	0.519083	+	0.854724i	$0.326273\pi$
$642$	6.34315	0.250344
$643$	31.3137	1.23489	0.617446	−	0.786613i	$-0.288167\pi$
$643$	31.3137	1.23489	0.617446	+	0.786613i	$0.288167\pi$
$644$	0	0
$645$	5.65685	0.222738
$646$	0	0
$647$	29.9411	1.17711	0.588554	−	0.808458i	$-0.299698\pi$
$647$	29.9411	1.17711	0.588554	+	0.808458i	$0.299698\pi$
$648$	1.00000	0.0392837
$649$	11.3137	0.444102
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−7.31371	−0.286427
$653$	−28.6274	−1.12028	−0.560139	−	0.828399i	$-0.689252\pi$
$653$	−28.6274	−1.12028	−0.560139	+	0.828399i	$0.689252\pi$
$654$	19.6569	0.768644
$655$	−15.3137	−0.598356
$656$	7.65685	0.298950
$657$	−13.3137	−0.519417
$658$	0	0
$659$	−7.31371	−0.284902	−0.142451	−	0.989802i	$-0.545498\pi$
$659$	−7.31371	−0.284902	−0.142451	+	0.989802i	$0.545498\pi$
$660$	−1.00000	−0.0389249
$661$	−27.9411	−1.08678	−0.543392	−	0.839479i	$-0.682860\pi$
$661$	−27.9411	−1.08678	−0.543392	+	0.839479i	$0.682860\pi$
$662$	15.3137	0.595184
$663$	2.00000	0.0776736
$664$	4.00000	0.155230
$665$	0	0
$666$	0.343146	0.0132966
$667$	−20.6863	−0.800976
$668$	−5.65685	−0.218870
$669$	19.3137	0.746711
$670$	4.00000	0.154533
$671$	3.65685	0.141171
$672$	0	0
$673$	−37.3137	−1.43834	−0.719169	−	0.694835i	$-0.755477\pi$
$673$	−37.3137	−1.43834	−0.719169	+	0.694835i	$0.755477\pi$
$674$	−5.31371	−0.204676
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	20.6274	0.792776	0.396388	−	0.918083i	$-0.370264\pi$
$677$	20.6274	0.792776	0.396388	+	0.918083i	$0.370264\pi$
$678$	−3.65685	−0.140441
$679$	0	0
$680$	−1.00000	−0.0383482
$681$	17.6569	0.676612
$682$	−5.65685	−0.216612
$683$	29.9411	1.14567	0.572833	−	0.819672i	$-0.305844\pi$
$683$	29.9411	1.14567	0.572833	+	0.819672i	$0.305844\pi$
$684$	0	0
$685$	19.6569	0.751050
$686$	0	0
$687$	21.3137	0.813169
$688$	5.65685	0.215666
$689$	20.0000	0.761939
$690$	−5.65685	−0.215353
$691$	−24.2843	−0.923817	−0.461909	−	0.886928i	$-0.652835\pi$
$691$	−24.2843	−0.923817	−0.461909	+	0.886928i	$0.652835\pi$
$692$	1.31371	0.0499397
$693$	0	0
$694$	−1.65685	−0.0628933
$695$	−20.9706	−0.795459
$696$	−3.65685	−0.138613
$697$	7.65685	0.290024
$698$	19.6569	0.744023
$699$	2.68629	0.101605
$700$	0	0
$701$	−19.9411	−0.753166	−0.376583	−	0.926383i	$-0.622901\pi$
$701$	−19.9411	−0.753166	−0.376583	+	0.926383i	$0.622901\pi$
$702$	2.00000	0.0754851
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	−4.00000	−0.150649
$706$	−14.9706	−0.563425
$707$	0	0
$708$	11.3137	0.425195
$709$	18.6863	0.701778	0.350889	−	0.936417i	$-0.385879\pi$
$709$	18.6863	0.701778	0.350889	+	0.936417i	$0.385879\pi$
$710$	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
$717$	27.3137	1.02005
$718$	−8.00000	−0.298557
$719$	37.6569	1.40436	0.702182	−	0.711998i	$-0.252209\pi$
$719$	37.6569	1.40436	0.702182	+	0.711998i	$0.252209\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−19.0000	−0.707107
$723$	5.31371	0.197619
$724$	14.0000	0.520306
$725$	3.65685	0.135812
$726$	−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
$730$	13.3137	0.492762
$731$	5.65685	0.209226
$732$	3.65685	0.135161
$733$	−45.3137	−1.67370	−0.836850	−	0.547432i	$-0.815605\pi$
$733$	−45.3137	−1.67370	−0.836850	+	0.547432i	$0.815605\pi$
$734$	19.3137	0.712882
$735$	−7.00000	−0.258199
$736$	−5.65685	−0.208514
$737$	4.00000	0.147342
$738$	7.65685	0.281853
$739$	−19.3137	−0.710466	−0.355233	−	0.934778i	$-0.615599\pi$
$739$	−19.3137	−0.710466	−0.355233	+	0.934778i	$0.615599\pi$
$740$	−0.343146	−0.0126143
$741$	0	0
$742$	0	0
$743$	−8.97056	−0.329098	−0.164549	−	0.986369i	$-0.552617\pi$
$743$	−8.97056	−0.329098	−0.164549	+	0.986369i	$0.552617\pi$
$744$	−5.65685	−0.207390
$745$	−6.00000	−0.219823
$746$	12.6274	0.462323
$747$	4.00000	0.146352
$748$	−1.00000	−0.0365636
$749$	0	0
$750$	1.00000	0.0365148
$751$	−2.34315	−0.0855026	−0.0427513	−	0.999086i	$-0.513612\pi$
$751$	−2.34315	−0.0855026	−0.0427513	+	0.999086i	$0.513612\pi$
$752$	−4.00000	−0.145865
$753$	−8.00000	−0.291536
$754$	−7.31371	−0.266350
$755$	6.34315	0.230851
$756$	0	0
$757$	47.9411	1.74245	0.871225	−	0.490884i	$-0.163326\pi$
$757$	47.9411	1.74245	0.871225	+	0.490884i	$0.163326\pi$
$758$	9.65685	0.350753
$759$	−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	$0.288651\pi$
$762$	−7.31371	−0.264948
$763$	0	0
$764$	−6.34315	−0.229487
$765$	−1.00000	−0.0361551
$766$	−8.68629	−0.313848
$767$	22.6274	0.817029
$768$	−1.00000	−0.0360844
$769$	32.6274	1.17657	0.588287	−	0.808652i	$-0.299802\pi$
$769$	32.6274	1.17657	0.588287	+	0.808652i	$0.299802\pi$
$770$	0	0
$771$	−2.97056	−0.106982
$772$	1.31371	0.0472814
$773$	−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
$776$	−11.6569	−0.418457
$777$	0	0
$778$	−4.34315	−0.155709
$779$	0	0
$780$	−2.00000	−0.0716115
$781$	5.65685	0.202418
$782$	−5.65685	−0.202289
$783$	−3.65685	−0.130685
$784$	−7.00000	−0.250000
$785$	−17.3137	−0.617953
$786$	−15.3137	−0.546222
$787$	52.9706	1.88820	0.944098	−	0.329664i	$-0.106935\pi$
$787$	52.9706	1.88820	0.944098	+	0.329664i	$0.106935\pi$
$788$	9.31371	0.331787
$789$	16.0000	0.569615
$790$	−11.3137	−0.402524
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	7.31371	0.259717
$794$	38.9706	1.38301
$795$	−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
$802$	14.9706	0.528629
$803$	13.3137	0.469831
$804$	4.00000	0.141069
$805$	0	0
$806$	−11.3137	−0.398508
$807$	−6.68629	−0.235369
$808$	6.00000	0.211079
$809$	22.2843	0.783473	0.391737	−	0.920077i	$-0.371874\pi$
$809$	22.2843	0.783473	0.391737	+	0.920077i	$0.371874\pi$
$810$	−1.00000	−0.0351364
$811$	−8.28427	−0.290900	−0.145450	−	0.989366i	$-0.546463\pi$
$811$	−8.28427	−0.290900	−0.145450	+	0.989366i	$0.546463\pi$
$812$	0	0
$813$	−1.65685	−0.0581084
$814$	−0.343146	−0.0120273
$815$	7.31371	0.256188
$816$	−1.00000	−0.0350070
$817$	0	0
$818$	5.31371	0.185789
$819$	0	0
$820$	−7.65685	−0.267389
$821$	−9.02944	−0.315130	−0.157565	−	0.987509i	$-0.550364\pi$
$821$	−9.02944	−0.315130	−0.157565	+	0.987509i	$0.550364\pi$
$822$	19.6569	0.685612
$823$	−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
$827$	49.6569	1.72674	0.863369	−	0.504573i	$-0.168350\pi$
$827$	49.6569	1.72674	0.863369	+	0.504573i	$0.168350\pi$
$828$	−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
$831$	22.9706	0.796840
$832$	−2.00000	−0.0693375
$833$	−7.00000	−0.242536
$834$	−20.9706	−0.726151
$835$	5.65685	0.195764
$836$	0	0
$837$	−5.65685	−0.195529
$838$	26.6274	0.919829
$839$	−18.3431	−0.633276	−0.316638	−	0.948547i	$-0.602554\pi$
$839$	−18.3431	−0.633276	−0.316638	+	0.948547i	$0.602554\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	−16.6274	−0.573019
$843$	20.6274	0.710446
$844$	−12.9706	−0.446465
$845$	9.00000	0.309609
$846$	−4.00000	−0.137523
$847$	0	0
$848$	−10.0000	−0.343401
$849$	−9.65685	−0.331422
$850$	1.00000	0.0342997
$851$	−1.94113	−0.0665409
$852$	5.65685	0.193801
$853$	4.34315	0.148706	0.0743532	−	0.997232i	$-0.476311\pi$
$853$	4.34315	0.148706	0.0743532	+	0.997232i	$0.476311\pi$
$854$	0	0
$855$	0	0
$856$	−6.34315	−0.216804
$857$	−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
$859$	47.3137	1.61432	0.807161	−	0.590331i	$-0.201003\pi$
$859$	47.3137	1.61432	0.807161	+	0.590331i	$0.201003\pi$
$860$	−5.65685	−0.192897
$861$	0	0
$862$	30.6274	1.04317
$863$	8.68629	0.295685	0.147842	−	0.989011i	$-0.452767\pi$
$863$	8.68629	0.295685	0.147842	+	0.989011i	$0.452767\pi$
$864$	−1.00000	−0.0340207
$865$	−1.31371	−0.0446674
$866$	13.3137	0.452418
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−11.3137	−0.383791
$870$	3.65685	0.123979
$871$	8.00000	0.271070
$872$	−19.6569	−0.665665
$873$	−11.6569	−0.394525
$874$	0	0
$875$	0	0
$876$	13.3137	0.449829
$877$	−18.2843	−0.617416	−0.308708	−	0.951157i	$-0.599897\pi$
$877$	−18.2843	−0.617416	−0.308708	+	0.951157i	$0.599897\pi$
$878$	8.00000	0.269987
$879$	26.0000	0.876958
$880$	1.00000	0.0337100
$881$	−12.3431	−0.415851	−0.207926	−	0.978145i	$-0.566671\pi$
$881$	−12.3431	−0.415851	−0.207926	+	0.978145i	$0.566671\pi$
$882$	−7.00000	−0.235702
$883$	−0.686292	−0.0230955	−0.0115478	−	0.999933i	$-0.503676\pi$
$883$	−0.686292	−0.0230955	−0.0115478	+	0.999933i	$0.503676\pi$
$884$	−2.00000	−0.0672673
$885$	−11.3137	−0.380306
$886$	20.2843	0.681463
$887$	−32.9706	−1.10704	−0.553522	−	0.832835i	$-0.686716\pi$
$887$	−32.9706	−1.10704	−0.553522	+	0.832835i	$0.686716\pi$
$888$	−0.343146	−0.0115152
$889$	0	0
$890$	14.0000	0.469281
$891$	−1.00000	−0.0335013
$892$	−19.3137	−0.646671
$893$	0	0
$894$	−6.00000	−0.200670
$895$	11.3137	0.378176
$896$	0	0
$897$	−11.3137	−0.377754
$898$	6.97056	0.232611
$899$	20.6863	0.689926
$900$	1.00000	0.0333333
$901$	−10.0000	−0.333148
$902$	−7.65685	−0.254945
$903$	0	0
$904$	3.65685	0.121625
$905$	−14.0000	−0.465376
$906$	6.34315	0.210737
$907$	−21.9411	−0.728543	−0.364272	−	0.931293i	$-0.618682\pi$
$907$	−21.9411	−0.728543	−0.364272	+	0.931293i	$0.618682\pi$
$908$	−17.6569	−0.585963
$909$	6.00000	0.199007
$910$	0	0
$911$	−40.9706	−1.35742	−0.678708	−	0.734409i	$-0.737460\pi$
$911$	−40.9706	−1.35742	−0.678708	+	0.734409i	$0.737460\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	18.9706	0.627490
$915$	−3.65685	−0.120892
$916$	−21.3137	−0.704225
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	38.3431	1.26482	0.632412	−	0.774632i	$-0.282065\pi$
$919$	38.3431	1.26482	0.632412	+	0.774632i	$0.282065\pi$
$920$	5.65685	0.186501
$921$	−2.34315	−0.0772093
$922$	−29.3137	−0.965396
$923$	11.3137	0.372395
$924$	0	0
$925$	0.343146	0.0112826
$926$	−16.0000	−0.525793
$927$	0	0
$928$	3.65685	0.120042
$929$	−33.5980	−1.10231	−0.551157	−	0.834402i	$-0.685813\pi$
$929$	−33.5980	−1.10231	−0.551157	+	0.834402i	$0.685813\pi$
$930$	5.65685	0.185496
$931$	0	0
$932$	−2.68629	−0.0879924
$933$	16.9706	0.555591
$934$	−8.97056	−0.293526
$935$	1.00000	0.0327035
$936$	−2.00000	−0.0653720
$937$	47.6569	1.55688	0.778441	−	0.627718i	$-0.216011\pi$
$937$	47.6569	1.55688	0.778441	+	0.627718i	$0.216011\pi$
$938$	0	0
$939$	−18.9706	−0.619081
$940$	4.00000	0.130466
$941$	−46.2843	−1.50882	−0.754412	−	0.656401i	$-0.772078\pi$
$941$	−46.2843	−1.50882	−0.754412	+	0.656401i	$0.772078\pi$
$942$	−17.3137	−0.564111
$943$	−43.3137	−1.41049
$944$	−11.3137	−0.368230
$945$	0	0
$946$	−5.65685	−0.183920
$947$	−13.9411	−0.453026	−0.226513	−	0.974008i	$-0.572733\pi$
$947$	−13.9411	−0.453026	−0.226513	+	0.974008i	$0.572733\pi$
$948$	−11.3137	−0.367452
$949$	26.6274	0.864363
$950$	0	0
$951$	25.3137	0.820853
$952$	0	0
$953$	−35.2548	−1.14202	−0.571008	−	0.820944i	$-0.693447\pi$
$953$	−35.2548	−1.14202	−0.571008	+	0.820944i	$0.693447\pi$
$954$	−10.0000	−0.323762
$955$	6.34315	0.205259
$956$	−27.3137	−0.883388
$957$	3.65685	0.118209
$958$	4.68629	0.151407
$959$	0	0
$960$	1.00000	0.0322749
$961$	1.00000	0.0322581
$962$	−0.686292	−0.0221269
$963$	−6.34315	−0.204405
$964$	−5.31371	−0.171143
$965$	−1.31371	−0.0422898
$966$	0	0
$967$	−41.2548	−1.32667	−0.663333	−	0.748324i	$-0.730859\pi$
$967$	−41.2548	−1.32667	−0.663333	+	0.748324i	$0.730859\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	11.6569	0.374279
$971$	11.3137	0.363074	0.181537	−	0.983384i	$-0.441893\pi$
$971$	11.3137	0.363074	0.181537	+	0.983384i	$0.441893\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−14.6274	−0.468693
$975$	2.00000	0.0640513
$976$	−3.65685	−0.117053
$977$	−29.5980	−0.946923	−0.473462	−	0.880814i	$-0.656996\pi$
$977$	−29.5980	−0.946923	−0.473462	+	0.880814i	$0.656996\pi$
$978$	7.31371	0.233867
$979$	14.0000	0.447442
$980$	7.00000	0.223607
$981$	−19.6569	−0.627595
$982$	23.3137	0.743970
$983$	−53.6569	−1.71139	−0.855694	−	0.517482i	$-0.826869\pi$
$983$	−53.6569	−1.71139	−0.855694	+	0.517482i	$0.826869\pi$
$984$	−7.65685	−0.244092
$985$	−9.31371	−0.296759
$986$	3.65685	0.116458
$987$	0	0
$988$	0	0
$989$	−32.0000	−1.01754
$990$	1.00000	0.0317821
$991$	47.5980	1.51200	0.756000	−	0.654572i	$-0.227151\pi$
$991$	47.5980	1.51200	0.756000	+	0.654572i	$0.227151\pi$
$992$	5.65685	0.179605
$993$	−15.3137	−0.485966
$994$	0	0
$995$	16.9706	0.538003
$996$	−4.00000	−0.126745
$997$	−54.9706	−1.74094	−0.870468	−	0.492226i	$-0.836183\pi$
$997$	−54.9706	−1.74094	−0.870468	+	0.492226i	$0.836183\pi$
$998$	−24.2843	−0.768705
$999$	−0.343146	−0.0108567

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.bt.1.1	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} -1.00000 q^{22} -5.65685 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +3.65685 q^{29} +1.00000 q^{30} +5.65685 q^{31} +1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} +0.343146 q^{37} +2.00000 q^{39} -1.00000 q^{40} +7.65685 q^{41} +5.65685 q^{43} -1.00000 q^{44} -1.00000 q^{45} -5.65685 q^{46} -4.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +1.00000 q^{55} +3.65685 q^{58} -11.3137 q^{59} +1.00000 q^{60} -3.65685 q^{61} +5.65685 q^{62} +1.00000 q^{64} +2.00000 q^{65} +1.00000 q^{66} -4.00000 q^{67} +1.00000 q^{68} +5.65685 q^{69} -5.65685 q^{71} +1.00000 q^{72} -13.3137 q^{73} +0.343146 q^{74} -1.00000 q^{75} +2.00000 q^{78} +11.3137 q^{79} -1.00000 q^{80} +1.00000 q^{81} +7.65685 q^{82} +4.00000 q^{83} -1.00000 q^{85} +5.65685 q^{86} -3.65685 q^{87} -1.00000 q^{88} -14.0000 q^{89} -1.00000 q^{90} -5.65685 q^{92} -5.65685 q^{93} -4.00000 q^{94} -1.00000 q^{96} -11.6569 q^{97} -7.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 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

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