LMFDB - Embedded newform 4650.2.a.ce.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.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:	$37.1304369399$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	4650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.56155 q^{11} +1.00000 q^{12} -2.00000 q^{13} +2.56155 q^{14} +1.00000 q^{16} +3.12311 q^{17} -1.00000 q^{18} -7.68466 q^{19} -2.56155 q^{21} -2.56155 q^{22} -1.43845 q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} -2.56155 q^{28} +7.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} +2.56155 q^{33} -3.12311 q^{34} +1.00000 q^{36} +3.12311 q^{37} +7.68466 q^{38} -2.00000 q^{39} +7.12311 q^{41} +2.56155 q^{42} -12.8078 q^{43} +2.56155 q^{44} +1.43845 q^{46} -5.12311 q^{47} +1.00000 q^{48} -0.438447 q^{49} +3.12311 q^{51} -2.00000 q^{52} -7.43845 q^{53} -1.00000 q^{54} +2.56155 q^{56} -7.68466 q^{57} -7.12311 q^{58} -13.1231 q^{59} +6.00000 q^{61} -1.00000 q^{62} -2.56155 q^{63} +1.00000 q^{64} -2.56155 q^{66} +15.3693 q^{67} +3.12311 q^{68} -1.43845 q^{69} -7.68466 q^{71} -1.00000 q^{72} +10.8078 q^{73} -3.12311 q^{74} -7.68466 q^{76} -6.56155 q^{77} +2.00000 q^{78} -4.31534 q^{79} +1.00000 q^{81} -7.12311 q^{82} -14.2462 q^{83} -2.56155 q^{84} +12.8078 q^{86} +7.12311 q^{87} -2.56155 q^{88} -13.6847 q^{89} +5.12311 q^{91} -1.43845 q^{92} +1.00000 q^{93} +5.12311 q^{94} -1.00000 q^{96} +6.00000 q^{97} +0.438447 q^{98} +2.56155 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + q^{11} + 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 3 q^{19} - q^{21} - q^{22} - 7 q^{23} - 2 q^{24} + 4 q^{26} + 2 q^{27} - q^{28} + 6 q^{29} + 2 q^{31} - 2 q^{32} + q^{33} + 2 q^{34} + 2 q^{36} - 2 q^{37} + 3 q^{38} - 4 q^{39} + 6 q^{41} + q^{42} - 5 q^{43} + q^{44} + 7 q^{46} - 2 q^{47} + 2 q^{48} - 5 q^{49} - 2 q^{51} - 4 q^{52} - 19 q^{53} - 2 q^{54} + q^{56} - 3 q^{57} - 6 q^{58} - 18 q^{59} + 12 q^{61} - 2 q^{62} - q^{63} + 2 q^{64} - q^{66} + 6 q^{67} - 2 q^{68} - 7 q^{69} - 3 q^{71} - 2 q^{72} + q^{73} + 2 q^{74} - 3 q^{76} - 9 q^{77} + 4 q^{78} - 21 q^{79} + 2 q^{81} - 6 q^{82} - 12 q^{83} - q^{84} + 5 q^{86} + 6 q^{87} - q^{88} - 15 q^{89} + 2 q^{91} - 7 q^{92} + 2 q^{93} + 2 q^{94} - 2 q^{96} + 12 q^{97} + 5 q^{98} + q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 + q^11 + 2 * q^12 - 4 * q^13 + q^14 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 3 * q^19 - q^21 - q^22 - 7 * q^23 - 2 * q^24 + 4 * q^26 + 2 * q^27 - q^28 + 6 * q^29 + 2 * q^31 - 2 * q^32 + q^33 + 2 * q^34 + 2 * q^36 - 2 * q^37 + 3 * q^38 - 4 * q^39 + 6 * q^41 + q^42 - 5 * q^43 + q^44 + 7 * q^46 - 2 * q^47 + 2 * q^48 - 5 * q^49 - 2 * q^51 - 4 * q^52 - 19 * q^53 - 2 * q^54 + q^56 - 3 * q^57 - 6 * q^58 - 18 * q^59 + 12 * q^61 - 2 * q^62 - q^63 + 2 * q^64 - q^66 + 6 * q^67 - 2 * q^68 - 7 * q^69 - 3 * q^71 - 2 * q^72 + q^73 + 2 * q^74 - 3 * q^76 - 9 * q^77 + 4 * q^78 - 21 * q^79 + 2 * q^81 - 6 * q^82 - 12 * q^83 - q^84 + 5 * q^86 + 6 * q^87 - q^88 - 15 * q^89 + 2 * q^91 - 7 * q^92 + 2 * q^93 + 2 * q^94 - 2 * q^96 + 12 * q^97 + 5 * q^98 + 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
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−2.56155	−0.968176	−0.484088	−	0.875019i	$-0.660849\pi$
$7$	−2.56155	−0.968176	−0.484088	+	0.875019i	$0.660849\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	2.56155	0.772337	0.386169	−	0.922428i	$-0.373798\pi$
$11$	2.56155	0.772337	0.386169	+	0.922428i	$0.373798\pi$
$12$	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	$0.589456\pi$
$14$	2.56155	0.684604
$15$	0	0
$16$	1.00000	0.250000
$17$	3.12311	0.757464	0.378732	−	0.925506i	$-0.376360\pi$
$17$	3.12311	0.757464	0.378732	+	0.925506i	$0.376360\pi$
$18$	−1.00000	−0.235702
$19$	−7.68466	−1.76298	−0.881491	−	0.472201i	$-0.843460\pi$
$19$	−7.68466	−1.76298	−0.881491	+	0.472201i	$0.843460\pi$
$20$	0	0
$21$	−2.56155	−0.558977
$22$	−2.56155	−0.546125
$23$	−1.43845	−0.299937	−0.149968	−	0.988691i	$-0.547917\pi$
$23$	−1.43845	−0.299937	−0.149968	+	0.988691i	$0.547917\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	−2.56155	−0.484088
$29$	7.12311	1.32273	0.661364	−	0.750065i	$-0.269978\pi$
$29$	7.12311	1.32273	0.661364	+	0.750065i	$0.269978\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	2.56155	0.445909
$34$	−3.12311	−0.535608
$35$	0	0
$36$	1.00000	0.166667
$37$	3.12311	0.513435	0.256718	−	0.966486i	$-0.417359\pi$
$37$	3.12311	0.513435	0.256718	+	0.966486i	$0.417359\pi$
$38$	7.68466	1.24662
$39$	−2.00000	−0.320256
$40$	0	0
$41$	7.12311	1.11244	0.556221	−	0.831034i	$-0.312251\pi$
$41$	7.12311	1.11244	0.556221	+	0.831034i	$0.312251\pi$
$42$	2.56155	0.395256
$43$	−12.8078	−1.95317	−0.976583	−	0.215142i	$-0.930979\pi$
$43$	−12.8078	−1.95317	−0.976583	+	0.215142i	$0.930979\pi$
$44$	2.56155	0.386169
$45$	0	0
$46$	1.43845	0.212087
$47$	−5.12311	−0.747282	−0.373641	−	0.927573i	$-0.621891\pi$
$47$	−5.12311	−0.747282	−0.373641	+	0.927573i	$0.621891\pi$
$48$	1.00000	0.144338
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	3.12311	0.437322
$52$	−2.00000	−0.277350
$53$	−7.43845	−1.02175	−0.510875	−	0.859655i	$-0.670678\pi$
$53$	−7.43845	−1.02175	−0.510875	+	0.859655i	$0.670678\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	2.56155	0.342302
$57$	−7.68466	−1.01786
$58$	−7.12311	−0.935310
$59$	−13.1231	−1.70848	−0.854241	−	0.519877i	$-0.825978\pi$
$59$	−13.1231	−1.70848	−0.854241	+	0.519877i	$0.825978\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−1.00000	−0.127000
$63$	−2.56155	−0.322725
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.56155	−0.315305
$67$	15.3693	1.87766	0.938830	−	0.344380i	$-0.111911\pi$
$67$	15.3693	1.87766	0.938830	+	0.344380i	$0.111911\pi$
$68$	3.12311	0.378732
$69$	−1.43845	−0.173169
$70$	0	0
$71$	−7.68466	−0.912001	−0.456001	−	0.889979i	$-0.650719\pi$
$71$	−7.68466	−0.912001	−0.456001	+	0.889979i	$0.650719\pi$
$72$	−1.00000	−0.117851
$73$	10.8078	1.26495	0.632477	−	0.774580i	$-0.282038\pi$
$73$	10.8078	1.26495	0.632477	+	0.774580i	$0.282038\pi$
$74$	−3.12311	−0.363054
$75$	0	0
$76$	−7.68466	−0.881491
$77$	−6.56155	−0.747758
$78$	2.00000	0.226455
$79$	−4.31534	−0.485514	−0.242757	−	0.970087i	$-0.578052\pi$
$79$	−4.31534	−0.485514	−0.242757	+	0.970087i	$0.578052\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−7.12311	−0.786615
$83$	−14.2462	−1.56372	−0.781862	−	0.623451i	$-0.785730\pi$
$83$	−14.2462	−1.56372	−0.781862	+	0.623451i	$0.785730\pi$
$84$	−2.56155	−0.279488
$85$	0	0
$86$	12.8078	1.38110
$87$	7.12311	0.763677
$88$	−2.56155	−0.273062
$89$	−13.6847	−1.45057	−0.725285	−	0.688448i	$-0.758292\pi$
$89$	−13.6847	−1.45057	−0.725285	+	0.688448i	$0.758292\pi$
$90$	0	0
$91$	5.12311	0.537047
$92$	−1.43845	−0.149968
$93$	1.00000	0.103695
$94$	5.12311	0.528408
$95$	0	0
$96$	−1.00000	−0.102062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0.438447	0.0442899
$99$	2.56155	0.257446
$100$	0	0
$101$	−0.561553	−0.0558766	−0.0279383	−	0.999610i	$-0.508894\pi$
$101$	−0.561553	−0.0558766	−0.0279383	+	0.999610i	$0.508894\pi$
$102$	−3.12311	−0.309234
$103$	1.75379	0.172806	0.0864030	−	0.996260i	$-0.472463\pi$
$103$	1.75379	0.172806	0.0864030	+	0.996260i	$0.472463\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	7.43845	0.722486
$107$	2.56155	0.247635	0.123817	−	0.992305i	$-0.460486\pi$
$107$	2.56155	0.247635	0.123817	+	0.992305i	$0.460486\pi$
$108$	1.00000	0.0962250
$109$	5.36932	0.514287	0.257144	−	0.966373i	$-0.417219\pi$
$109$	5.36932	0.514287	0.257144	+	0.966373i	$0.417219\pi$
$110$	0	0
$111$	3.12311	0.296432
$112$	−2.56155	−0.242044
$113$	1.68466	0.158479	0.0792397	−	0.996856i	$-0.474751\pi$
$113$	1.68466	0.158479	0.0792397	+	0.996856i	$0.474751\pi$
$114$	7.68466	0.719734
$115$	0	0
$116$	7.12311	0.661364
$117$	−2.00000	−0.184900
$118$	13.1231	1.20808
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−4.43845	−0.403495
$122$	−6.00000	−0.543214
$123$	7.12311	0.642269
$124$	1.00000	0.0898027
$125$	0	0
$126$	2.56155	0.228201
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	−12.8078	−1.12766
$130$	0	0
$131$	−15.3693	−1.34282	−0.671412	−	0.741085i	$-0.734312\pi$
$131$	−15.3693	−1.34282	−0.671412	+	0.741085i	$0.734312\pi$
$132$	2.56155	0.222955
$133$	19.6847	1.70688
$134$	−15.3693	−1.32771
$135$	0	0
$136$	−3.12311	−0.267804
$137$	−20.2462	−1.72975	−0.864875	−	0.501987i	$-0.832603\pi$
$137$	−20.2462	−1.72975	−0.864875	+	0.501987i	$0.832603\pi$
$138$	1.43845	0.122449
$139$	−17.1231	−1.45236	−0.726181	−	0.687503i	$-0.758707\pi$
$139$	−17.1231	−1.45236	−0.726181	+	0.687503i	$0.758707\pi$
$140$	0	0
$141$	−5.12311	−0.431443
$142$	7.68466	0.644882
$143$	−5.12311	−0.428416
$144$	1.00000	0.0833333
$145$	0	0
$146$	−10.8078	−0.894457
$147$	−0.438447	−0.0361625
$148$	3.12311	0.256718
$149$	17.0540	1.39712	0.698558	−	0.715553i	$-0.253825\pi$
$149$	17.0540	1.39712	0.698558	+	0.715553i	$0.253825\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	7.68466	0.623308
$153$	3.12311	0.252488
$154$	6.56155	0.528745
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−17.0540	−1.36106	−0.680528	−	0.732722i	$-0.738249\pi$
$157$	−17.0540	−1.36106	−0.680528	+	0.732722i	$0.738249\pi$
$158$	4.31534	0.343310
$159$	−7.43845	−0.589907
$160$	0	0
$161$	3.68466	0.290392
$162$	−1.00000	−0.0785674
$163$	15.3693	1.20382	0.601909	−	0.798565i	$-0.294407\pi$
$163$	15.3693	1.20382	0.601909	+	0.798565i	$0.294407\pi$
$164$	7.12311	0.556221
$165$	0	0
$166$	14.2462	1.10572
$167$	6.56155	0.507748	0.253874	−	0.967237i	$-0.418295\pi$
$167$	6.56155	0.507748	0.253874	+	0.967237i	$0.418295\pi$
$168$	2.56155	0.197628
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−7.68466	−0.587661
$172$	−12.8078	−0.976583
$173$	−8.24621	−0.626948	−0.313474	−	0.949597i	$-0.601493\pi$
$173$	−8.24621	−0.626948	−0.313474	+	0.949597i	$0.601493\pi$
$174$	−7.12311	−0.540001
$175$	0	0
$176$	2.56155	0.193084
$177$	−13.1231	−0.986393
$178$	13.6847	1.02571
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−13.0540	−0.970294	−0.485147	−	0.874433i	$-0.661234\pi$
$181$	−13.0540	−0.970294	−0.485147	+	0.874433i	$0.661234\pi$
$182$	−5.12311	−0.379750
$183$	6.00000	0.443533
$184$	1.43845	0.106044
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	8.00000	0.585018
$188$	−5.12311	−0.373641
$189$	−2.56155	−0.186326
$190$	0	0
$191$	−14.2462	−1.03082	−0.515410	−	0.856944i	$-0.672360\pi$
$191$	−14.2462	−1.03082	−0.515410	+	0.856944i	$0.672360\pi$
$192$	1.00000	0.0721688
$193$	−14.4924	−1.04319	−0.521594	−	0.853194i	$-0.674662\pi$
$193$	−14.4924	−1.04319	−0.521594	+	0.853194i	$0.674662\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	−0.438447	−0.0313177
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−2.56155	−0.182042
$199$	−16.8078	−1.19147	−0.595735	−	0.803181i	$-0.703139\pi$
$199$	−16.8078	−1.19147	−0.595735	+	0.803181i	$0.703139\pi$
$200$	0	0
$201$	15.3693	1.08407
$202$	0.561553	0.0395107
$203$	−18.2462	−1.28063
$204$	3.12311	0.218661
$205$	0	0
$206$	−1.75379	−0.122192
$207$	−1.43845	−0.0999790
$208$	−2.00000	−0.138675
$209$	−19.6847	−1.36162
$210$	0	0
$211$	−23.6847	−1.63052	−0.815260	−	0.579096i	$-0.803406\pi$
$211$	−23.6847	−1.63052	−0.815260	+	0.579096i	$0.803406\pi$
$212$	−7.43845	−0.510875
$213$	−7.68466	−0.526544
$214$	−2.56155	−0.175104
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−2.56155	−0.173890
$218$	−5.36932	−0.363656
$219$	10.8078	0.730321
$220$	0	0
$221$	−6.24621	−0.420166
$222$	−3.12311	−0.209609
$223$	21.1231	1.41451	0.707254	−	0.706960i	$-0.249934\pi$
$223$	21.1231	1.41451	0.707254	+	0.706960i	$0.249934\pi$
$224$	2.56155	0.171151
$225$	0	0
$226$	−1.68466	−0.112062
$227$	−7.68466	−0.510049	−0.255024	−	0.966935i	$-0.582083\pi$
$227$	−7.68466	−0.510049	−0.255024	+	0.966935i	$0.582083\pi$
$228$	−7.68466	−0.508929
$229$	−16.5616	−1.09442	−0.547209	−	0.836996i	$-0.684310\pi$
$229$	−16.5616	−1.09442	−0.547209	+	0.836996i	$0.684310\pi$
$230$	0	0
$231$	−6.56155	−0.431718
$232$	−7.12311	−0.467655
$233$	17.0540	1.11724	0.558622	−	0.829423i	$-0.311330\pi$
$233$	17.0540	1.11724	0.558622	+	0.829423i	$0.311330\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−13.1231	−0.854241
$237$	−4.31534	−0.280312
$238$	8.00000	0.518563
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	12.2462	0.788848	0.394424	−	0.918929i	$-0.370944\pi$
$241$	12.2462	0.788848	0.394424	+	0.918929i	$0.370944\pi$
$242$	4.43845	0.285314
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	0	0
$246$	−7.12311	−0.454153
$247$	15.3693	0.977926
$248$	−1.00000	−0.0635001
$249$	−14.2462	−0.902817
$250$	0	0
$251$	16.4924	1.04099	0.520496	−	0.853864i	$-0.325747\pi$
$251$	16.4924	1.04099	0.520496	+	0.853864i	$0.325747\pi$
$252$	−2.56155	−0.161363
$253$	−3.68466	−0.231652
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.68466	0.105086	0.0525431	−	0.998619i	$-0.483267\pi$
$257$	1.68466	0.105086	0.0525431	+	0.998619i	$0.483267\pi$
$258$	12.8078	0.797377
$259$	−8.00000	−0.497096
$260$	0	0
$261$	7.12311	0.440909
$262$	15.3693	0.949520
$263$	20.4924	1.26362	0.631808	−	0.775125i	$-0.282313\pi$
$263$	20.4924	1.26362	0.631808	+	0.775125i	$0.282313\pi$
$264$	−2.56155	−0.157653
$265$	0	0
$266$	−19.6847	−1.20694
$267$	−13.6847	−0.837487
$268$	15.3693	0.938830
$269$	−0.246211	−0.0150118	−0.00750588	−	0.999972i	$-0.502389\pi$
$269$	−0.246211	−0.0150118	−0.00750588	+	0.999972i	$0.502389\pi$
$270$	0	0
$271$	27.0540	1.64341	0.821706	−	0.569912i	$-0.193023\pi$
$271$	27.0540	1.64341	0.821706	+	0.569912i	$0.193023\pi$
$272$	3.12311	0.189366
$273$	5.12311	0.310064
$274$	20.2462	1.22312
$275$	0	0
$276$	−1.43845	−0.0865843
$277$	5.36932	0.322611	0.161305	−	0.986905i	$-0.448430\pi$
$277$	5.36932	0.322611	0.161305	+	0.986905i	$0.448430\pi$
$278$	17.1231	1.02698
$279$	1.00000	0.0598684
$280$	0	0
$281$	4.87689	0.290931	0.145466	−	0.989363i	$-0.453532\pi$
$281$	4.87689	0.290931	0.145466	+	0.989363i	$0.453532\pi$
$282$	5.12311	0.305077
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−7.68466	−0.456001
$285$	0	0
$286$	5.12311	0.302936
$287$	−18.2462	−1.07704
$288$	−1.00000	−0.0589256
$289$	−7.24621	−0.426248
$290$	0	0
$291$	6.00000	0.351726
$292$	10.8078	0.632477
$293$	−26.4924	−1.54770	−0.773852	−	0.633367i	$-0.781673\pi$
$293$	−26.4924	−1.54770	−0.773852	+	0.633367i	$0.781673\pi$
$294$	0.438447	0.0255708
$295$	0	0
$296$	−3.12311	−0.181527
$297$	2.56155	0.148636
$298$	−17.0540	−0.987910
$299$	2.87689	0.166375
$300$	0	0
$301$	32.8078	1.89101
$302$	0	0
$303$	−0.561553	−0.0322604
$304$	−7.68466	−0.440745
$305$	0	0
$306$	−3.12311	−0.178536
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	−6.56155	−0.373879
$309$	1.75379	0.0997696
$310$	0	0
$311$	−1.75379	−0.0994482	−0.0497241	−	0.998763i	$-0.515834\pi$
$311$	−1.75379	−0.0994482	−0.0497241	+	0.998763i	$0.515834\pi$
$312$	2.00000	0.113228
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	17.0540	0.962412
$315$	0	0
$316$	−4.31534	−0.242757
$317$	−16.8769	−0.947901	−0.473950	−	0.880552i	$-0.657172\pi$
$317$	−16.8769	−0.947901	−0.473950	+	0.880552i	$0.657172\pi$
$318$	7.43845	0.417127
$319$	18.2462	1.02159
$320$	0	0
$321$	2.56155	0.142972
$322$	−3.68466	−0.205338
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	0	0
$326$	−15.3693	−0.851228
$327$	5.36932	0.296924
$328$	−7.12311	−0.393308
$329$	13.1231	0.723500
$330$	0	0
$331$	−6.24621	−0.343323	−0.171661	−	0.985156i	$-0.554914\pi$
$331$	−6.24621	−0.343323	−0.171661	+	0.985156i	$0.554914\pi$
$332$	−14.2462	−0.781862
$333$	3.12311	0.171145
$334$	−6.56155	−0.359032
$335$	0	0
$336$	−2.56155	−0.139744
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	9.00000	0.489535
$339$	1.68466	0.0914981
$340$	0	0
$341$	2.56155	0.138716
$342$	7.68466	0.415539
$343$	19.0540	1.02882
$344$	12.8078	0.690548
$345$	0	0
$346$	8.24621	0.443319
$347$	−14.2462	−0.764777	−0.382388	−	0.924002i	$-0.624898\pi$
$347$	−14.2462	−0.764777	−0.382388	+	0.924002i	$0.624898\pi$
$348$	7.12311	0.381839
$349$	5.36932	0.287413	0.143706	−	0.989620i	$-0.454098\pi$
$349$	5.36932	0.287413	0.143706	+	0.989620i	$0.454098\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−2.56155	−0.136531
$353$	0.246211	0.0131045	0.00655225	−	0.999979i	$-0.497914\pi$
$353$	0.246211	0.0131045	0.00655225	+	0.999979i	$0.497914\pi$
$354$	13.1231	0.697485
$355$	0	0
$356$	−13.6847	−0.725285
$357$	−8.00000	−0.423405
$358$	12.0000	0.634220
$359$	−31.6847	−1.67225	−0.836126	−	0.548537i	$-0.815185\pi$
$359$	−31.6847	−1.67225	−0.836126	+	0.548537i	$0.815185\pi$
$360$	0	0
$361$	40.0540	2.10810
$362$	13.0540	0.686102
$363$	−4.43845	−0.232958
$364$	5.12311	0.268524
$365$	0	0
$366$	−6.00000	−0.313625
$367$	−15.3693	−0.802272	−0.401136	−	0.916019i	$-0.631384\pi$
$367$	−15.3693	−0.802272	−0.401136	+	0.916019i	$0.631384\pi$
$368$	−1.43845	−0.0749842
$369$	7.12311	0.370814
$370$	0	0
$371$	19.0540	0.989233
$372$	1.00000	0.0518476
$373$	5.68466	0.294340	0.147170	−	0.989111i	$-0.452983\pi$
$373$	5.68466	0.294340	0.147170	+	0.989111i	$0.452983\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	5.12311	0.264204
$377$	−14.2462	−0.733717
$378$	2.56155	0.131752
$379$	−7.05398	−0.362338	−0.181169	−	0.983452i	$-0.557988\pi$
$379$	−7.05398	−0.362338	−0.181169	+	0.983452i	$0.557988\pi$
$380$	0	0
$381$	0	0
$382$	14.2462	0.728900
$383$	10.2462	0.523557	0.261778	−	0.965128i	$-0.415691\pi$
$383$	10.2462	0.523557	0.261778	+	0.965128i	$0.415691\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	14.4924	0.737645
$387$	−12.8078	−0.651055
$388$	6.00000	0.304604
$389$	7.75379	0.393133	0.196566	−	0.980491i	$-0.437021\pi$
$389$	7.75379	0.393133	0.196566	+	0.980491i	$0.437021\pi$
$390$	0	0
$391$	−4.49242	−0.227192
$392$	0.438447	0.0221449
$393$	−15.3693	−0.775279
$394$	6.00000	0.302276
$395$	0	0
$396$	2.56155	0.128723
$397$	−9.05398	−0.454406	−0.227203	−	0.973847i	$-0.572958\pi$
$397$	−9.05398	−0.454406	−0.227203	+	0.973847i	$0.572958\pi$
$398$	16.8078	0.842497
$399$	19.6847	0.985466
$400$	0	0
$401$	−13.6847	−0.683379	−0.341690	−	0.939813i	$-0.610999\pi$
$401$	−13.6847	−0.683379	−0.341690	+	0.939813i	$0.610999\pi$
$402$	−15.3693	−0.766552
$403$	−2.00000	−0.0996271
$404$	−0.561553	−0.0279383
$405$	0	0
$406$	18.2462	0.905544
$407$	8.00000	0.396545
$408$	−3.12311	−0.154617
$409$	−37.3693	−1.84779	−0.923897	−	0.382642i	$-0.875014\pi$
$409$	−37.3693	−1.84779	−0.923897	+	0.382642i	$0.875014\pi$
$410$	0	0
$411$	−20.2462	−0.998672
$412$	1.75379	0.0864030
$413$	33.6155	1.65411
$414$	1.43845	0.0706958
$415$	0	0
$416$	2.00000	0.0980581
$417$	−17.1231	−0.838522
$418$	19.6847	0.962808
$419$	5.75379	0.281091	0.140545	−	0.990074i	$-0.455114\pi$
$419$	5.75379	0.281091	0.140545	+	0.990074i	$0.455114\pi$
$420$	0	0
$421$	−14.4924	−0.706317	−0.353159	−	0.935563i	$-0.614892\pi$
$421$	−14.4924	−0.706317	−0.353159	+	0.935563i	$0.614892\pi$
$422$	23.6847	1.15295
$423$	−5.12311	−0.249094
$424$	7.43845	0.361243
$425$	0	0
$426$	7.68466	0.372323
$427$	−15.3693	−0.743773
$428$	2.56155	0.123817
$429$	−5.12311	−0.247346
$430$	0	0
$431$	−30.2462	−1.45691	−0.728454	−	0.685094i	$-0.759761\pi$
$431$	−30.2462	−1.45691	−0.728454	+	0.685094i	$0.759761\pi$
$432$	1.00000	0.0481125
$433$	−35.3002	−1.69642	−0.848209	−	0.529661i	$-0.822319\pi$
$433$	−35.3002	−1.69642	−0.848209	+	0.529661i	$0.822319\pi$
$434$	2.56155	0.122958
$435$	0	0
$436$	5.36932	0.257144
$437$	11.0540	0.528783
$438$	−10.8078	−0.516415
$439$	9.61553	0.458924	0.229462	−	0.973318i	$-0.426303\pi$
$439$	9.61553	0.458924	0.229462	+	0.973318i	$0.426303\pi$
$440$	0	0
$441$	−0.438447	−0.0208784
$442$	6.24621	0.297102
$443$	−23.0540	−1.09533	−0.547664	−	0.836699i	$-0.684483\pi$
$443$	−23.0540	−1.09533	−0.547664	+	0.836699i	$0.684483\pi$
$444$	3.12311	0.148216
$445$	0	0
$446$	−21.1231	−1.00021
$447$	17.0540	0.806625
$448$	−2.56155	−0.121022
$449$	10.4924	0.495168	0.247584	−	0.968866i	$-0.420363\pi$
$449$	10.4924	0.495168	0.247584	+	0.968866i	$0.420363\pi$
$450$	0	0
$451$	18.2462	0.859181
$452$	1.68466	0.0792397
$453$	0	0
$454$	7.68466	0.360659
$455$	0	0
$456$	7.68466	0.359867
$457$	24.7386	1.15722	0.578612	−	0.815603i	$-0.303594\pi$
$457$	24.7386	1.15722	0.578612	+	0.815603i	$0.303594\pi$
$458$	16.5616	0.773871
$459$	3.12311	0.145774
$460$	0	0
$461$	9.36932	0.436373	0.218186	−	0.975907i	$-0.429986\pi$
$461$	9.36932	0.436373	0.218186	+	0.975907i	$0.429986\pi$
$462$	6.56155	0.305271
$463$	−30.7386	−1.42855	−0.714273	−	0.699867i	$-0.753242\pi$
$463$	−30.7386	−1.42855	−0.714273	+	0.699867i	$0.753242\pi$
$464$	7.12311	0.330682
$465$	0	0
$466$	−17.0540	−0.790010
$467$	9.75379	0.451352	0.225676	−	0.974202i	$-0.427541\pi$
$467$	9.75379	0.451352	0.225676	+	0.974202i	$0.427541\pi$
$468$	−2.00000	−0.0924500
$469$	−39.3693	−1.81791
$470$	0	0
$471$	−17.0540	−0.785806
$472$	13.1231	0.604040
$473$	−32.8078	−1.50850
$474$	4.31534	0.198210
$475$	0	0
$476$	−8.00000	−0.366679
$477$	−7.43845	−0.340583
$478$	−24.0000	−1.09773
$479$	10.5616	0.482570	0.241285	−	0.970454i	$-0.422431\pi$
$479$	10.5616	0.482570	0.241285	+	0.970454i	$0.422431\pi$
$480$	0	0
$481$	−6.24621	−0.284803
$482$	−12.2462	−0.557800
$483$	3.68466	0.167658
$484$	−4.43845	−0.201748
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	−6.00000	−0.271607
$489$	15.3693	0.695025
$490$	0	0
$491$	−25.9309	−1.17024	−0.585122	−	0.810945i	$-0.698953\pi$
$491$	−25.9309	−1.17024	−0.585122	+	0.810945i	$0.698953\pi$
$492$	7.12311	0.321134
$493$	22.2462	1.00192
$494$	−15.3693	−0.691498
$495$	0	0
$496$	1.00000	0.0449013
$497$	19.6847	0.882978
$498$	14.2462	0.638388
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	6.56155	0.293149
$502$	−16.4924	−0.736093
$503$	−26.2462	−1.17026	−0.585130	−	0.810939i	$-0.698957\pi$
$503$	−26.2462	−1.17026	−0.585130	+	0.810939i	$0.698957\pi$
$504$	2.56155	0.114101
$505$	0	0
$506$	3.68466	0.163803
$507$	−9.00000	−0.399704
$508$	0	0
$509$	19.6155	0.869443	0.434721	−	0.900565i	$-0.356847\pi$
$509$	19.6155	0.869443	0.434721	+	0.900565i	$0.356847\pi$
$510$	0	0
$511$	−27.6847	−1.22470
$512$	−1.00000	−0.0441942
$513$	−7.68466	−0.339286
$514$	−1.68466	−0.0743071
$515$	0	0
$516$	−12.8078	−0.563830
$517$	−13.1231	−0.577154
$518$	8.00000	0.351500
$519$	−8.24621	−0.361968
$520$	0	0
$521$	−32.2462	−1.41273	−0.706366	−	0.707847i	$-0.749667\pi$
$521$	−32.2462	−1.41273	−0.706366	+	0.707847i	$0.749667\pi$
$522$	−7.12311	−0.311770
$523$	22.4233	0.980502	0.490251	−	0.871581i	$-0.336905\pi$
$523$	22.4233	0.980502	0.490251	+	0.871581i	$0.336905\pi$
$524$	−15.3693	−0.671412
$525$	0	0
$526$	−20.4924	−0.893512
$527$	3.12311	0.136045
$528$	2.56155	0.111477
$529$	−20.9309	−0.910038
$530$	0	0
$531$	−13.1231	−0.569494
$532$	19.6847	0.853438
$533$	−14.2462	−0.617072
$534$	13.6847	0.592193
$535$	0	0
$536$	−15.3693	−0.663853
$537$	−12.0000	−0.517838
$538$	0.246211	0.0106149
$539$	−1.12311	−0.0483756
$540$	0	0
$541$	19.7538	0.849282	0.424641	−	0.905362i	$-0.360400\pi$
$541$	19.7538	0.849282	0.424641	+	0.905362i	$0.360400\pi$
$542$	−27.0540	−1.16207
$543$	−13.0540	−0.560200
$544$	−3.12311	−0.133902
$545$	0	0
$546$	−5.12311	−0.219249
$547$	−26.8769	−1.14917	−0.574587	−	0.818444i	$-0.694837\pi$
$547$	−26.8769	−1.14917	−0.574587	+	0.818444i	$0.694837\pi$
$548$	−20.2462	−0.864875
$549$	6.00000	0.256074
$550$	0	0
$551$	−54.7386	−2.33194
$552$	1.43845	0.0612244
$553$	11.0540	0.470063
$554$	−5.36932	−0.228120
$555$	0	0
$556$	−17.1231	−0.726181
$557$	26.8078	1.13588	0.567941	−	0.823069i	$-0.307740\pi$
$557$	26.8078	1.13588	0.567941	+	0.823069i	$0.307740\pi$
$558$	−1.00000	−0.0423334
$559$	25.6155	1.08342
$560$	0	0
$561$	8.00000	0.337760
$562$	−4.87689	−0.205719
$563$	−16.4924	−0.695073	−0.347536	−	0.937667i	$-0.612982\pi$
$563$	−16.4924	−0.695073	−0.347536	+	0.937667i	$0.612982\pi$
$564$	−5.12311	−0.215722
$565$	0	0
$566$	0	0
$567$	−2.56155	−0.107575
$568$	7.68466	0.322441
$569$	30.8078	1.29153	0.645764	−	0.763537i	$-0.276539\pi$
$569$	30.8078	1.29153	0.645764	+	0.763537i	$0.276539\pi$
$570$	0	0
$571$	41.1231	1.72095	0.860474	−	0.509494i	$-0.170167\pi$
$571$	41.1231	1.72095	0.860474	+	0.509494i	$0.170167\pi$
$572$	−5.12311	−0.214208
$573$	−14.2462	−0.595144
$574$	18.2462	0.761582
$575$	0	0
$576$	1.00000	0.0416667
$577$	−15.1231	−0.629583	−0.314792	−	0.949161i	$-0.601935\pi$
$577$	−15.1231	−0.629583	−0.314792	+	0.949161i	$0.601935\pi$
$578$	7.24621	0.301403
$579$	−14.4924	−0.602285
$580$	0	0
$581$	36.4924	1.51396
$582$	−6.00000	−0.248708
$583$	−19.0540	−0.789135
$584$	−10.8078	−0.447228
$585$	0	0
$586$	26.4924	1.09439
$587$	0.492423	0.0203245	0.0101622	−	0.999948i	$-0.496765\pi$
$587$	0.492423	0.0203245	0.0101622	+	0.999948i	$0.496765\pi$
$588$	−0.438447	−0.0180813
$589$	−7.68466	−0.316641
$590$	0	0
$591$	−6.00000	−0.246807
$592$	3.12311	0.128359
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	−2.56155	−0.105102
$595$	0	0
$596$	17.0540	0.698558
$597$	−16.8078	−0.687896
$598$	−2.87689	−0.117645
$599$	38.4233	1.56993	0.784967	−	0.619538i	$-0.212680\pi$
$599$	38.4233	1.56993	0.784967	+	0.619538i	$0.212680\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−32.8078	−1.33714
$603$	15.3693	0.625887
$604$	0	0
$605$	0	0
$606$	0.561553	0.0228115
$607$	−7.05398	−0.286312	−0.143156	−	0.989700i	$-0.545725\pi$
$607$	−7.05398	−0.286312	−0.143156	+	0.989700i	$0.545725\pi$
$608$	7.68466	0.311654
$609$	−18.2462	−0.739374
$610$	0	0
$611$	10.2462	0.414517
$612$	3.12311	0.126244
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0	0
$615$	0	0
$616$	6.56155	0.264372
$617$	15.4384	0.621528	0.310764	−	0.950487i	$-0.399415\pi$
$617$	15.4384	0.621528	0.310764	+	0.950487i	$0.399415\pi$
$618$	−1.75379	−0.0705477
$619$	11.3693	0.456971	0.228486	−	0.973547i	$-0.426623\pi$
$619$	11.3693	0.456971	0.228486	+	0.973547i	$0.426623\pi$
$620$	0	0
$621$	−1.43845	−0.0577229
$622$	1.75379	0.0703205
$623$	35.0540	1.40441
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−10.0000	−0.399680
$627$	−19.6847	−0.786130
$628$	−17.0540	−0.680528
$629$	9.75379	0.388909
$630$	0	0
$631$	29.3002	1.16642	0.583211	−	0.812321i	$-0.301796\pi$
$631$	29.3002	1.16642	0.583211	+	0.812321i	$0.301796\pi$
$632$	4.31534	0.171655
$633$	−23.6847	−0.941381
$634$	16.8769	0.670267
$635$	0	0
$636$	−7.43845	−0.294954
$637$	0.876894	0.0347438
$638$	−18.2462	−0.722374
$639$	−7.68466	−0.304000
$640$	0	0
$641$	−5.50758	−0.217536	−0.108768	−	0.994067i	$-0.534691\pi$
$641$	−5.50758	−0.217536	−0.108768	+	0.994067i	$0.534691\pi$
$642$	−2.56155	−0.101096
$643$	7.05398	0.278182	0.139091	−	0.990280i	$-0.455582\pi$
$643$	7.05398	0.278182	0.139091	+	0.990280i	$0.455582\pi$
$644$	3.68466	0.145196
$645$	0	0
$646$	24.0000	0.944267
$647$	45.9309	1.80573	0.902864	−	0.429925i	$-0.141460\pi$
$647$	45.9309	1.80573	0.902864	+	0.429925i	$0.141460\pi$
$648$	−1.00000	−0.0392837
$649$	−33.6155	−1.31952
$650$	0	0
$651$	−2.56155	−0.100395
$652$	15.3693	0.601909
$653$	−40.2462	−1.57496	−0.787478	−	0.616343i	$-0.788614\pi$
$653$	−40.2462	−1.57496	−0.787478	+	0.616343i	$0.788614\pi$
$654$	−5.36932	−0.209957
$655$	0	0
$656$	7.12311	0.278111
$657$	10.8078	0.421651
$658$	−13.1231	−0.511592
$659$	−30.7386	−1.19741	−0.598704	−	0.800971i	$-0.704317\pi$
$659$	−30.7386	−1.19741	−0.598704	+	0.800971i	$0.704317\pi$
$660$	0	0
$661$	26.4924	1.03044	0.515218	−	0.857059i	$-0.327711\pi$
$661$	26.4924	1.03044	0.515218	+	0.857059i	$0.327711\pi$
$662$	6.24621	0.242766
$663$	−6.24621	−0.242583
$664$	14.2462	0.552860
$665$	0	0
$666$	−3.12311	−0.121018
$667$	−10.2462	−0.396735
$668$	6.56155	0.253874
$669$	21.1231	0.816666
$670$	0	0
$671$	15.3693	0.593326
$672$	2.56155	0.0988140
$673$	−11.7538	−0.453075	−0.226538	−	0.974002i	$-0.572741\pi$
$673$	−11.7538	−0.453075	−0.226538	+	0.974002i	$0.572741\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−32.4233	−1.24613	−0.623064	−	0.782171i	$-0.714112\pi$
$677$	−32.4233	−1.24613	−0.623064	+	0.782171i	$0.714112\pi$
$678$	−1.68466	−0.0646989
$679$	−15.3693	−0.589820
$680$	0	0
$681$	−7.68466	−0.294477
$682$	−2.56155	−0.0980869
$683$	−31.6847	−1.21238	−0.606190	−	0.795320i	$-0.707303\pi$
$683$	−31.6847	−1.21238	−0.606190	+	0.795320i	$0.707303\pi$
$684$	−7.68466	−0.293830
$685$	0	0
$686$	−19.0540	−0.727484
$687$	−16.5616	−0.631863
$688$	−12.8078	−0.488291
$689$	14.8769	0.566765
$690$	0	0
$691$	15.0540	0.572680	0.286340	−	0.958128i	$-0.407561\pi$
$691$	15.0540	0.572680	0.286340	+	0.958128i	$0.407561\pi$
$692$	−8.24621	−0.313474
$693$	−6.56155	−0.249253
$694$	14.2462	0.540779
$695$	0	0
$696$	−7.12311	−0.270001
$697$	22.2462	0.842635
$698$	−5.36932	−0.203232
$699$	17.0540	0.645041
$700$	0	0
$701$	5.19224	0.196108	0.0980540	−	0.995181i	$-0.468738\pi$
$701$	5.19224	0.196108	0.0980540	+	0.995181i	$0.468738\pi$
$702$	2.00000	0.0754851
$703$	−24.0000	−0.905177
$704$	2.56155	0.0965422
$705$	0	0
$706$	−0.246211	−0.00926628
$707$	1.43845	0.0540984
$708$	−13.1231	−0.493197
$709$	17.0540	0.640475	0.320238	−	0.947337i	$-0.396237\pi$
$709$	17.0540	0.640475	0.320238	+	0.947337i	$0.396237\pi$
$710$	0	0
$711$	−4.31534	−0.161838
$712$	13.6847	0.512854
$713$	−1.43845	−0.0538703
$714$	8.00000	0.299392
$715$	0	0
$716$	−12.0000	−0.448461
$717$	24.0000	0.896296
$718$	31.6847	1.18246
$719$	−34.8769	−1.30069	−0.650344	−	0.759640i	$-0.725375\pi$
$719$	−34.8769	−1.30069	−0.650344	+	0.759640i	$0.725375\pi$
$720$	0	0
$721$	−4.49242	−0.167307
$722$	−40.0540	−1.49065
$723$	12.2462	0.455441
$724$	−13.0540	−0.485147
$725$	0	0
$726$	4.43845	0.164726
$727$	−23.0540	−0.855025	−0.427512	−	0.904010i	$-0.640610\pi$
$727$	−23.0540	−0.855025	−0.427512	+	0.904010i	$0.640610\pi$
$728$	−5.12311	−0.189875
$729$	1.00000	0.0370370
$730$	0	0
$731$	−40.0000	−1.47945
$732$	6.00000	0.221766
$733$	6.49242	0.239803	0.119902	−	0.992786i	$-0.461742\pi$
$733$	6.49242	0.239803	0.119902	+	0.992786i	$0.461742\pi$
$734$	15.3693	0.567292
$735$	0	0
$736$	1.43845	0.0530219
$737$	39.3693	1.45019
$738$	−7.12311	−0.262205
$739$	52.9848	1.94908	0.974540	−	0.224216i	$-0.0719821\pi$
$739$	52.9848	1.94908	0.974540	+	0.224216i	$0.0719821\pi$
$740$	0	0
$741$	15.3693	0.564606
$742$	−19.0540	−0.699493
$743$	50.4233	1.84985	0.924926	−	0.380148i	$-0.124127\pi$
$743$	50.4233	1.84985	0.924926	+	0.380148i	$0.124127\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−5.68466	−0.208130
$747$	−14.2462	−0.521242
$748$	8.00000	0.292509
$749$	−6.56155	−0.239754
$750$	0	0
$751$	45.1231	1.64657	0.823283	−	0.567631i	$-0.192140\pi$
$751$	45.1231	1.64657	0.823283	+	0.567631i	$0.192140\pi$
$752$	−5.12311	−0.186820
$753$	16.4924	0.601017
$754$	14.2462	0.518816
$755$	0	0
$756$	−2.56155	−0.0931628
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	7.05398	0.256212
$759$	−3.68466	−0.133745
$760$	0	0
$761$	−20.4233	−0.740344	−0.370172	−	0.928963i	$-0.620701\pi$
$761$	−20.4233	−0.740344	−0.370172	+	0.928963i	$0.620701\pi$
$762$	0	0
$763$	−13.7538	−0.497921
$764$	−14.2462	−0.515410
$765$	0	0
$766$	−10.2462	−0.370211
$767$	26.2462	0.947696
$768$	1.00000	0.0360844
$769$	−20.5616	−0.741469	−0.370734	−	0.928739i	$-0.620894\pi$
$769$	−20.5616	−0.741469	−0.370734	+	0.928739i	$0.620894\pi$
$770$	0	0
$771$	1.68466	0.0606715
$772$	−14.4924	−0.521594
$773$	11.4384	0.411412	0.205706	−	0.978614i	$-0.434051\pi$
$773$	11.4384	0.411412	0.205706	+	0.978614i	$0.434051\pi$
$774$	12.8078	0.460366
$775$	0	0
$776$	−6.00000	−0.215387
$777$	−8.00000	−0.286998
$778$	−7.75379	−0.277987
$779$	−54.7386	−1.96122
$780$	0	0
$781$	−19.6847	−0.704372
$782$	4.49242	0.160649
$783$	7.12311	0.254559
$784$	−0.438447	−0.0156588
$785$	0	0
$786$	15.3693	0.548205
$787$	17.9309	0.639166	0.319583	−	0.947558i	$-0.396457\pi$
$787$	17.9309	0.639166	0.319583	+	0.947558i	$0.396457\pi$
$788$	−6.00000	−0.213741
$789$	20.4924	0.729550
$790$	0	0
$791$	−4.31534	−0.153436
$792$	−2.56155	−0.0910208
$793$	−12.0000	−0.426132
$794$	9.05398	0.321314
$795$	0	0
$796$	−16.8078	−0.595735
$797$	26.9848	0.955852	0.477926	−	0.878400i	$-0.341389\pi$
$797$	26.9848	0.955852	0.477926	+	0.878400i	$0.341389\pi$
$798$	−19.6847	−0.696829
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−13.6847	−0.483524
$802$	13.6847	0.483222
$803$	27.6847	0.976970
$804$	15.3693	0.542034
$805$	0	0
$806$	2.00000	0.0704470
$807$	−0.246211	−0.00866705
$808$	0.561553	0.0197554
$809$	−37.6847	−1.32492	−0.662461	−	0.749096i	$-0.730488\pi$
$809$	−37.6847	−1.32492	−0.662461	+	0.749096i	$0.730488\pi$
$810$	0	0
$811$	−24.6695	−0.866263	−0.433132	−	0.901331i	$-0.642592\pi$
$811$	−24.6695	−0.866263	−0.433132	+	0.901331i	$0.642592\pi$
$812$	−18.2462	−0.640316
$813$	27.0540	0.948824
$814$	−8.00000	−0.280400
$815$	0	0
$816$	3.12311	0.109331
$817$	98.4233	3.44340
$818$	37.3693	1.30659
$819$	5.12311	0.179016
$820$	0	0
$821$	19.6155	0.684587	0.342293	−	0.939593i	$-0.388796\pi$
$821$	19.6155	0.684587	0.342293	+	0.939593i	$0.388796\pi$
$822$	20.2462	0.706168
$823$	25.6155	0.892901	0.446451	−	0.894808i	$-0.352688\pi$
$823$	25.6155	0.892901	0.446451	+	0.894808i	$0.352688\pi$
$824$	−1.75379	−0.0610961
$825$	0	0
$826$	−33.6155	−1.16963
$827$	−1.75379	−0.0609852	−0.0304926	−	0.999535i	$-0.509708\pi$
$827$	−1.75379	−0.0609852	−0.0304926	+	0.999535i	$0.509708\pi$
$828$	−1.43845	−0.0499895
$829$	24.4233	0.848256	0.424128	−	0.905602i	$-0.360581\pi$
$829$	24.4233	0.848256	0.424128	+	0.905602i	$0.360581\pi$
$830$	0	0
$831$	5.36932	0.186260
$832$	−2.00000	−0.0693375
$833$	−1.36932	−0.0474440
$834$	17.1231	0.592925
$835$	0	0
$836$	−19.6847	−0.680808
$837$	1.00000	0.0345651
$838$	−5.75379	−0.198761
$839$	6.06913	0.209530	0.104765	−	0.994497i	$-0.466591\pi$
$839$	6.06913	0.209530	0.104765	+	0.994497i	$0.466591\pi$
$840$	0	0
$841$	21.7386	0.749608
$842$	14.4924	0.499442
$843$	4.87689	0.167969
$844$	−23.6847	−0.815260
$845$	0	0
$846$	5.12311	0.176136
$847$	11.3693	0.390654
$848$	−7.43845	−0.255437
$849$	0	0
$850$	0	0
$851$	−4.49242	−0.153998
$852$	−7.68466	−0.263272
$853$	−47.7926	−1.63639	−0.818194	−	0.574942i	$-0.805024\pi$
$853$	−47.7926	−1.63639	−0.818194	+	0.574942i	$0.805024\pi$
$854$	15.3693	0.525927
$855$	0	0
$856$	−2.56155	−0.0875521
$857$	−29.2311	−0.998514	−0.499257	−	0.866454i	$-0.666394\pi$
$857$	−29.2311	−0.998514	−0.499257	+	0.866454i	$0.666394\pi$
$858$	5.12311	0.174900
$859$	26.7386	0.912310	0.456155	−	0.889900i	$-0.349226\pi$
$859$	26.7386	0.912310	0.456155	+	0.889900i	$0.349226\pi$
$860$	0	0
$861$	−18.2462	−0.621829
$862$	30.2462	1.03019
$863$	−39.5464	−1.34618	−0.673088	−	0.739563i	$-0.735032\pi$
$863$	−39.5464	−1.34618	−0.673088	+	0.739563i	$0.735032\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	35.3002	1.19955
$867$	−7.24621	−0.246094
$868$	−2.56155	−0.0869448
$869$	−11.0540	−0.374980
$870$	0	0
$871$	−30.7386	−1.04154
$872$	−5.36932	−0.181828
$873$	6.00000	0.203069
$874$	−11.0540	−0.373906
$875$	0	0
$876$	10.8078	0.365161
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	−9.61553	−0.324508
$879$	−26.4924	−0.893567
$880$	0	0
$881$	9.50758	0.320318	0.160159	−	0.987091i	$-0.448799\pi$
$881$	9.50758	0.320318	0.160159	+	0.987091i	$0.448799\pi$
$882$	0.438447	0.0147633
$883$	−30.4233	−1.02383	−0.511913	−	0.859038i	$-0.671063\pi$
$883$	−30.4233	−1.02383	−0.511913	+	0.859038i	$0.671063\pi$
$884$	−6.24621	−0.210083
$885$	0	0
$886$	23.0540	0.774513
$887$	32.6307	1.09563	0.547816	−	0.836599i	$-0.315460\pi$
$887$	32.6307	1.09563	0.547816	+	0.836599i	$0.315460\pi$
$888$	−3.12311	−0.104805
$889$	0	0
$890$	0	0
$891$	2.56155	0.0858152
$892$	21.1231	0.707254
$893$	39.3693	1.31744
$894$	−17.0540	−0.570370
$895$	0	0
$896$	2.56155	0.0855755
$897$	2.87689	0.0960567
$898$	−10.4924	−0.350137
$899$	7.12311	0.237569
$900$	0	0
$901$	−23.2311	−0.773939
$902$	−18.2462	−0.607532
$903$	32.8078	1.09177
$904$	−1.68466	−0.0560309
$905$	0	0
$906$	0	0
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	−7.68466	−0.255024
$909$	−0.561553	−0.0186255
$910$	0	0
$911$	−11.5076	−0.381263	−0.190632	−	0.981662i	$-0.561054\pi$
$911$	−11.5076	−0.381263	−0.190632	+	0.981662i	$0.561054\pi$
$912$	−7.68466	−0.254464
$913$	−36.4924	−1.20772
$914$	−24.7386	−0.818281
$915$	0	0
$916$	−16.5616	−0.547209
$917$	39.3693	1.30009
$918$	−3.12311	−0.103078
$919$	−1.61553	−0.0532914	−0.0266457	−	0.999645i	$-0.508483\pi$
$919$	−1.61553	−0.0532914	−0.0266457	+	0.999645i	$0.508483\pi$
$920$	0	0
$921$	0	0
$922$	−9.36932	−0.308562
$923$	15.3693	0.505887
$924$	−6.56155	−0.215859
$925$	0	0
$926$	30.7386	1.01013
$927$	1.75379	0.0576020
$928$	−7.12311	−0.233827
$929$	7.43845	0.244048	0.122024	−	0.992527i	$-0.461062\pi$
$929$	7.43845	0.244048	0.122024	+	0.992527i	$0.461062\pi$
$930$	0	0
$931$	3.36932	0.110425
$932$	17.0540	0.558622
$933$	−1.75379	−0.0574165
$934$	−9.75379	−0.319154
$935$	0	0
$936$	2.00000	0.0653720
$937$	−41.3693	−1.35148	−0.675738	−	0.737142i	$-0.736175\pi$
$937$	−41.3693	−1.35148	−0.675738	+	0.737142i	$0.736175\pi$
$938$	39.3693	1.28545
$939$	10.0000	0.326338
$940$	0	0
$941$	2.63068	0.0857578	0.0428789	−	0.999080i	$-0.486347\pi$
$941$	2.63068	0.0857578	0.0428789	+	0.999080i	$0.486347\pi$
$942$	17.0540	0.555649
$943$	−10.2462	−0.333663
$944$	−13.1231	−0.427121
$945$	0	0
$946$	32.8078	1.06667
$947$	9.12311	0.296461	0.148231	−	0.988953i	$-0.452642\pi$
$947$	9.12311	0.296461	0.148231	+	0.988953i	$0.452642\pi$
$948$	−4.31534	−0.140156
$949$	−21.6155	−0.701670
$950$	0	0
$951$	−16.8769	−0.547271
$952$	8.00000	0.259281
$953$	−16.1080	−0.521788	−0.260894	−	0.965367i	$-0.584017\pi$
$953$	−16.1080	−0.521788	−0.260894	+	0.965367i	$0.584017\pi$
$954$	7.43845	0.240829
$955$	0	0
$956$	24.0000	0.776215
$957$	18.2462	0.589816
$958$	−10.5616	−0.341228
$959$	51.8617	1.67470
$960$	0	0
$961$	1.00000	0.0322581
$962$	6.24621	0.201386
$963$	2.56155	0.0825449
$964$	12.2462	0.394424
$965$	0	0
$966$	−3.68466	−0.118552
$967$	42.2462	1.35855	0.679273	−	0.733885i	$-0.262295\pi$
$967$	42.2462	1.35855	0.679273	+	0.733885i	$0.262295\pi$
$968$	4.43845	0.142657
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−37.1231	−1.19134	−0.595669	−	0.803230i	$-0.703113\pi$
$971$	−37.1231	−1.19134	−0.595669	+	0.803230i	$0.703113\pi$
$972$	1.00000	0.0320750
$973$	43.8617	1.40614
$974$	24.0000	0.769010
$975$	0	0
$976$	6.00000	0.192055
$977$	−42.9848	−1.37521	−0.687604	−	0.726086i	$-0.741337\pi$
$977$	−42.9848	−1.37521	−0.687604	+	0.726086i	$0.741337\pi$
$978$	−15.3693	−0.491457
$979$	−35.0540	−1.12033
$980$	0	0
$981$	5.36932	0.171429
$982$	25.9309	0.827487
$983$	−40.9848	−1.30721	−0.653607	−	0.756834i	$-0.726745\pi$
$983$	−40.9848	−1.30721	−0.653607	+	0.756834i	$0.726745\pi$
$984$	−7.12311	−0.227076
$985$	0	0
$986$	−22.2462	−0.708464
$987$	13.1231	0.417713
$988$	15.3693	0.488963
$989$	18.4233	0.585827
$990$	0	0
$991$	35.6847	1.13356	0.566780	−	0.823869i	$-0.308189\pi$
$991$	35.6847	1.13356	0.566780	+	0.823869i	$0.308189\pi$
$992$	−1.00000	−0.0317500
$993$	−6.24621	−0.198218
$994$	−19.6847	−0.624359
$995$	0	0
$996$	−14.2462	−0.451408
$997$	29.5076	0.934514	0.467257	−	0.884121i	$-0.345242\pi$
$997$	29.5076	0.934514	0.467257	+	0.884121i	$0.345242\pi$
$998$	−20.0000	−0.633089
$999$	3.12311	0.0988107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.ce.1.1		2
5.2	odd	4		4650.2.d.bd.3349.1		4
5.3	odd	4		4650.2.d.bd.3349.4		4
5.4	even	2		930.2.a.p.1.2	✓	2
15.14	odd	2		2790.2.a.be.1.2		2
20.19	odd	2		7440.2.a.bl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.p.1.2	✓	2	5.4	even	2
2790.2.a.be.1.2		2	15.14	odd	2
4650.2.a.ce.1.1		2	1.1	even	1	trivial
4650.2.d.bd.3349.1		4	5.2	odd	4
4650.2.d.bd.3349.4		4	5.3	odd	4
7440.2.a.bl.1.1		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.56155 q^{11} +1.00000 q^{12} -2.00000 q^{13} +2.56155 q^{14} +1.00000 q^{16} +3.12311 q^{17} -1.00000 q^{18} -7.68466 q^{19} -2.56155 q^{21} -2.56155 q^{22} -1.43845 q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} -2.56155 q^{28} +7.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} +2.56155 q^{33} -3.12311 q^{34} +1.00000 q^{36} +3.12311 q^{37} +7.68466 q^{38} -2.00000 q^{39} +7.12311 q^{41} +2.56155 q^{42} -12.8078 q^{43} +2.56155 q^{44} +1.43845 q^{46} -5.12311 q^{47} +1.00000 q^{48} -0.438447 q^{49} +3.12311 q^{51} -2.00000 q^{52} -7.43845 q^{53} -1.00000 q^{54} +2.56155 q^{56} -7.68466 q^{57} -7.12311 q^{58} -13.1231 q^{59} +6.00000 q^{61} -1.00000 q^{62} -2.56155 q^{63} +1.00000 q^{64} -2.56155 q^{66} +15.3693 q^{67} +3.12311 q^{68} -1.43845 q^{69} -7.68466 q^{71} -1.00000 q^{72} +10.8078 q^{73} -3.12311 q^{74} -7.68466 q^{76} -6.56155 q^{77} +2.00000 q^{78} -4.31534 q^{79} +1.00000 q^{81} -7.12311 q^{82} -14.2462 q^{83} -2.56155 q^{84} +12.8078 q^{86} +7.12311 q^{87} -2.56155 q^{88} -13.6847 q^{89} +5.12311 q^{91} -1.43845 q^{92} +1.00000 q^{93} +5.12311 q^{94} -1.00000 q^{96} +6.00000 q^{97} +0.438447 q^{98} +2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + q^{11} + 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 3 q^{19} - q^{21} - q^{22} - 7 q^{23} - 2 q^{24} + 4 q^{26} + 2 q^{27} - q^{28} + 6 q^{29} + 2 q^{31} - 2 q^{32} + q^{33} + 2 q^{34} + 2 q^{36} - 2 q^{37} + 3 q^{38} - 4 q^{39} + 6 q^{41} + q^{42} - 5 q^{43} + q^{44} + 7 q^{46} - 2 q^{47} + 2 q^{48} - 5 q^{49} - 2 q^{51} - 4 q^{52} - 19 q^{53} - 2 q^{54} + q^{56} - 3 q^{57} - 6 q^{58} - 18 q^{59} + 12 q^{61} - 2 q^{62} - q^{63} + 2 q^{64} - q^{66} + 6 q^{67} - 2 q^{68} - 7 q^{69} - 3 q^{71} - 2 q^{72} + q^{73} + 2 q^{74} - 3 q^{76} - 9 q^{77} + 4 q^{78} - 21 q^{79} + 2 q^{81} - 6 q^{82} - 12 q^{83} - q^{84} + 5 q^{86} + 6 q^{87} - q^{88} - 15 q^{89} + 2 q^{91} - 7 q^{92} + 2 q^{93} + 2 q^{94} - 2 q^{96} + 12 q^{97} + 5 q^{98} + q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 + q^11 + 2 * q^12 - 4 * q^13 + q^14 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 3 * q^19 - q^21 - q^22 - 7 * q^23 - 2 * q^24 + 4 * q^26 + 2 * q^27 - q^28 + 6 * q^29 + 2 * q^31 - 2 * q^32 + q^33 + 2 * q^34 + 2 * q^36 - 2 * q^37 + 3 * q^38 - 4 * q^39 + 6 * q^41 + q^42 - 5 * q^43 + q^44 + 7 * q^46 - 2 * q^47 + 2 * q^48 - 5 * q^49 - 2 * q^51 - 4 * q^52 - 19 * q^53 - 2 * q^54 + q^56 - 3 * q^57 - 6 * q^58 - 18 * q^59 + 12 * q^61 - 2 * q^62 - q^63 + 2 * q^64 - q^66 + 6 * q^67 - 2 * q^68 - 7 * q^69 - 3 * q^71 - 2 * q^72 + q^73 + 2 * q^74 - 3 * q^76 - 9 * q^77 + 4 * q^78 - 21 * q^79 + 2 * q^81 - 6 * q^82 - 12 * q^83 - q^84 + 5 * q^86 + 6 * q^87 - q^88 - 15 * q^89 + 2 * q^91 - 7 * q^92 + 2 * q^93 + 2 * q^94 - 2 * q^96 + 12 * q^97 + 5 * q^98 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more