LMFDB - Embedded newform 770.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [770,2,Mod(1,770)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(770, 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("770.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 770 = 2 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	770.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:	$6.14848095564$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 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	$\chi$	$=$	770.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -2.82843 q^{12} +2.00000 q^{13} -1.00000 q^{14} +2.82843 q^{15} +1.00000 q^{16} -3.65685 q^{17} +5.00000 q^{18} +2.82843 q^{19} -1.00000 q^{20} +2.82843 q^{21} -1.00000 q^{22} +2.82843 q^{23} -2.82843 q^{24} +1.00000 q^{25} +2.00000 q^{26} -5.65685 q^{27} -1.00000 q^{28} +8.82843 q^{29} +2.82843 q^{30} +5.65685 q^{31} +1.00000 q^{32} +2.82843 q^{33} -3.65685 q^{34} +1.00000 q^{35} +5.00000 q^{36} +8.82843 q^{37} +2.82843 q^{38} -5.65685 q^{39} -1.00000 q^{40} +4.82843 q^{41} +2.82843 q^{42} +1.65685 q^{43} -1.00000 q^{44} -5.00000 q^{45} +2.82843 q^{46} -8.00000 q^{47} -2.82843 q^{48} +1.00000 q^{49} +1.00000 q^{50} +10.3431 q^{51} +2.00000 q^{52} +6.48528 q^{53} -5.65685 q^{54} +1.00000 q^{55} -1.00000 q^{56} -8.00000 q^{57} +8.82843 q^{58} -8.00000 q^{59} +2.82843 q^{60} +13.3137 q^{61} +5.65685 q^{62} -5.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +2.82843 q^{66} -7.31371 q^{67} -3.65685 q^{68} -8.00000 q^{69} +1.00000 q^{70} +5.65685 q^{71} +5.00000 q^{72} -3.65685 q^{73} +8.82843 q^{74} -2.82843 q^{75} +2.82843 q^{76} +1.00000 q^{77} -5.65685 q^{78} -10.8284 q^{79} -1.00000 q^{80} +1.00000 q^{81} +4.82843 q^{82} -13.6569 q^{83} +2.82843 q^{84} +3.65685 q^{85} +1.65685 q^{86} -24.9706 q^{87} -1.00000 q^{88} -0.343146 q^{89} -5.00000 q^{90} -2.00000 q^{91} +2.82843 q^{92} -16.0000 q^{93} -8.00000 q^{94} -2.82843 q^{95} -2.82843 q^{96} +4.82843 q^{97} +1.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} + 10 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 + 10 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} + 10 q^{9} - 2 q^{10} - 2 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} + 4 q^{17} + 10 q^{18} - 2 q^{20} - 2 q^{22} + 2 q^{25} + 4 q^{26} - 2 q^{28} + 12 q^{29} + 2 q^{32} + 4 q^{34} + 2 q^{35} + 10 q^{36} + 12 q^{37} - 2 q^{40} + 4 q^{41} - 8 q^{43} - 2 q^{44} - 10 q^{45} - 16 q^{47} + 2 q^{49} + 2 q^{50} + 32 q^{51} + 4 q^{52} - 4 q^{53} + 2 q^{55} - 2 q^{56} - 16 q^{57} + 12 q^{58} - 16 q^{59} + 4 q^{61} - 10 q^{63} + 2 q^{64} - 4 q^{65} + 8 q^{67} + 4 q^{68} - 16 q^{69} + 2 q^{70} + 10 q^{72} + 4 q^{73} + 12 q^{74} + 2 q^{77} - 16 q^{79} - 2 q^{80} + 2 q^{81} + 4 q^{82} - 16 q^{83} - 4 q^{85} - 8 q^{86} - 16 q^{87} - 2 q^{88} - 12 q^{89} - 10 q^{90} - 4 q^{91} - 32 q^{93} - 16 q^{94} + 4 q^{97} + 2 q^{98} - 10 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 + 10 * q^9 - 2 * q^10 - 2 * q^11 + 4 * q^13 - 2 * q^14 + 2 * q^16 + 4 * q^17 + 10 * q^18 - 2 * q^20 - 2 * q^22 + 2 * q^25 + 4 * q^26 - 2 * q^28 + 12 * q^29 + 2 * q^32 + 4 * q^34 + 2 * q^35 + 10 * q^36 + 12 * q^37 - 2 * q^40 + 4 * q^41 - 8 * q^43 - 2 * q^44 - 10 * q^45 - 16 * q^47 + 2 * q^49 + 2 * q^50 + 32 * q^51 + 4 * q^52 - 4 * q^53 + 2 * q^55 - 2 * q^56 - 16 * q^57 + 12 * q^58 - 16 * q^59 + 4 * q^61 - 10 * q^63 + 2 * q^64 - 4 * q^65 + 8 * q^67 + 4 * q^68 - 16 * q^69 + 2 * q^70 + 10 * q^72 + 4 * q^73 + 12 * q^74 + 2 * q^77 - 16 * q^79 - 2 * q^80 + 2 * q^81 + 4 * q^82 - 16 * q^83 - 4 * q^85 - 8 * q^86 - 16 * q^87 - 2 * q^88 - 12 * q^89 - 10 * q^90 - 4 * q^91 - 32 * q^93 - 16 * q^94 + 4 * q^97 + 2 * q^98 - 10 * 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
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.82843	−1.15470
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	5.00000	1.66667
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.82843	−0.816497
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	2.82843	0.730297
$16$	1.00000	0.250000
$17$	−3.65685	−0.886917	−0.443459	−	0.896295i	$-0.646249\pi$
$17$	−3.65685	−0.886917	−0.443459	+	0.896295i	$0.646249\pi$
$18$	5.00000	1.17851
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	−1.00000	−0.223607
$21$	2.82843	0.617213
$22$	−1.00000	−0.213201
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	−2.82843	−0.577350
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−5.65685	−1.08866
$28$	−1.00000	−0.188982
$29$	8.82843	1.63940	0.819699	−	0.572795i	$-0.194141\pi$
$29$	8.82843	1.63940	0.819699	+	0.572795i	$0.194141\pi$
$30$	2.82843	0.516398
$31$	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
$33$	2.82843	0.492366
$34$	−3.65685	−0.627145
$35$	1.00000	0.169031
$36$	5.00000	0.833333
$37$	8.82843	1.45138	0.725692	−	0.688019i	$-0.241520\pi$
$37$	8.82843	1.45138	0.725692	+	0.688019i	$0.241520\pi$
$38$	2.82843	0.458831
$39$	−5.65685	−0.905822
$40$	−1.00000	−0.158114
$41$	4.82843	0.754074	0.377037	−	0.926198i	$-0.376943\pi$
$41$	4.82843	0.754074	0.377037	+	0.926198i	$0.376943\pi$
$42$	2.82843	0.436436
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	−1.00000	−0.150756
$45$	−5.00000	−0.745356
$46$	2.82843	0.417029
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−2.82843	−0.408248
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	10.3431	1.44833
$52$	2.00000	0.277350
$53$	6.48528	0.890822	0.445411	−	0.895326i	$-0.353058\pi$
$53$	6.48528	0.890822	0.445411	+	0.895326i	$0.353058\pi$
$54$	−5.65685	−0.769800
$55$	1.00000	0.134840
$56$	−1.00000	−0.133631
$57$	−8.00000	−1.05963
$58$	8.82843	1.15923
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	2.82843	0.365148
$61$	13.3137	1.70465	0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137	1.70465	0.852323	+	0.523016i	$0.175193\pi$
$62$	5.65685	0.718421
$63$	−5.00000	−0.629941
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	2.82843	0.348155
$67$	−7.31371	−0.893512	−0.446756	−	0.894656i	$-0.647421\pi$
$67$	−7.31371	−0.893512	−0.446756	+	0.894656i	$0.647421\pi$
$68$	−3.65685	−0.443459
$69$	−8.00000	−0.963087
$70$	1.00000	0.119523
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	5.00000	0.589256
$73$	−3.65685	−0.428002	−0.214001	−	0.976833i	$-0.568650\pi$
$73$	−3.65685	−0.428002	−0.214001	+	0.976833i	$0.568650\pi$
$74$	8.82843	1.02628
$75$	−2.82843	−0.326599
$76$	2.82843	0.324443
$77$	1.00000	0.113961
$78$	−5.65685	−0.640513
$79$	−10.8284	−1.21829	−0.609147	−	0.793058i	$-0.708488\pi$
$79$	−10.8284	−1.21829	−0.609147	+	0.793058i	$0.708488\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	4.82843	0.533211
$83$	−13.6569	−1.49903	−0.749517	−	0.661985i	$-0.769714\pi$
$83$	−13.6569	−1.49903	−0.749517	+	0.661985i	$0.769714\pi$
$84$	2.82843	0.308607
$85$	3.65685	0.396642
$86$	1.65685	0.178663
$87$	−24.9706	−2.67713
$88$	−1.00000	−0.106600
$89$	−0.343146	−0.0363734	−0.0181867	−	0.999835i	$-0.505789\pi$
$89$	−0.343146	−0.0363734	−0.0181867	+	0.999835i	$0.505789\pi$
$90$	−5.00000	−0.527046
$91$	−2.00000	−0.209657
$92$	2.82843	0.294884
$93$	−16.0000	−1.65912
$94$	−8.00000	−0.825137
$95$	−2.82843	−0.290191
$96$	−2.82843	−0.288675
$97$	4.82843	0.490252	0.245126	−	0.969491i	$-0.421171\pi$
$97$	4.82843	0.490252	0.245126	+	0.969491i	$0.421171\pi$
$98$	1.00000	0.101015
$99$	−5.00000	−0.502519
$100$	1.00000	0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	10.3431	1.02412
$103$	5.65685	0.557386	0.278693	−	0.960380i	$-0.410099\pi$
$103$	5.65685	0.557386	0.278693	+	0.960380i	$0.410099\pi$
$104$	2.00000	0.196116
$105$	−2.82843	−0.276026
$106$	6.48528	0.629906
$107$	9.65685	0.933563	0.466782	−	0.884373i	$-0.345413\pi$
$107$	9.65685	0.933563	0.466782	+	0.884373i	$0.345413\pi$
$108$	−5.65685	−0.544331
$109$	−2.48528	−0.238047	−0.119023	−	0.992891i	$-0.537976\pi$
$109$	−2.48528	−0.238047	−0.119023	+	0.992891i	$0.537976\pi$
$110$	1.00000	0.0953463
$111$	−24.9706	−2.37010
$112$	−1.00000	−0.0944911
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−8.00000	−0.749269
$115$	−2.82843	−0.263752
$116$	8.82843	0.819699
$117$	10.0000	0.924500
$118$	−8.00000	−0.736460
$119$	3.65685	0.335223
$120$	2.82843	0.258199
$121$	1.00000	0.0909091
$122$	13.3137	1.20537
$123$	−13.6569	−1.23140
$124$	5.65685	0.508001
$125$	−1.00000	−0.0894427
$126$	−5.00000	−0.445435
$127$	−3.31371	−0.294044	−0.147022	−	0.989133i	$-0.546969\pi$
$127$	−3.31371	−0.294044	−0.147022	+	0.989133i	$0.546969\pi$
$128$	1.00000	0.0883883
$129$	−4.68629	−0.412605
$130$	−2.00000	−0.175412
$131$	13.1716	1.15081	0.575403	−	0.817870i	$-0.304845\pi$
$131$	13.1716	1.15081	0.575403	+	0.817870i	$0.304845\pi$
$132$	2.82843	0.246183
$133$	−2.82843	−0.245256
$134$	−7.31371	−0.631808
$135$	5.65685	0.486864
$136$	−3.65685	−0.313573
$137$	−17.3137	−1.47921	−0.739605	−	0.673041i	$-0.764988\pi$
$137$	−17.3137	−1.47921	−0.739605	+	0.673041i	$0.764988\pi$
$138$	−8.00000	−0.681005
$139$	13.1716	1.11720	0.558599	−	0.829438i	$-0.311339\pi$
$139$	13.1716	1.11720	0.558599	+	0.829438i	$0.311339\pi$
$140$	1.00000	0.0845154
$141$	22.6274	1.90557
$142$	5.65685	0.474713
$143$	−2.00000	−0.167248
$144$	5.00000	0.416667
$145$	−8.82843	−0.733161
$146$	−3.65685	−0.302643
$147$	−2.82843	−0.233285
$148$	8.82843	0.725692
$149$	−18.4853	−1.51437	−0.757187	−	0.653199i	$-0.773427\pi$
$149$	−18.4853	−1.51437	−0.757187	+	0.653199i	$0.773427\pi$
$150$	−2.82843	−0.230940
$151$	5.17157	0.420857	0.210428	−	0.977609i	$-0.432514\pi$
$151$	5.17157	0.420857	0.210428	+	0.977609i	$0.432514\pi$
$152$	2.82843	0.229416
$153$	−18.2843	−1.47820
$154$	1.00000	0.0805823
$155$	−5.65685	−0.454369
$156$	−5.65685	−0.452911
$157$	4.34315	0.346621	0.173310	−	0.984867i	$-0.444554\pi$
$157$	4.34315	0.346621	0.173310	+	0.984867i	$0.444554\pi$
$158$	−10.8284	−0.861463
$159$	−18.3431	−1.45471
$160$	−1.00000	−0.0790569
$161$	−2.82843	−0.222911
$162$	1.00000	0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	4.82843	0.377037
$165$	−2.82843	−0.220193
$166$	−13.6569	−1.05998
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	2.82843	0.218218
$169$	−9.00000	−0.692308
$170$	3.65685	0.280468
$171$	14.1421	1.08148
$172$	1.65685	0.126334
$173$	21.3137	1.62045	0.810226	−	0.586118i	$-0.199345\pi$
$173$	21.3137	1.62045	0.810226	+	0.586118i	$0.199345\pi$
$174$	−24.9706	−1.89301
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	22.6274	1.70078
$178$	−0.343146	−0.0257199
$179$	17.6569	1.31974	0.659868	−	0.751382i	$-0.270612\pi$
$179$	17.6569	1.31974	0.659868	+	0.751382i	$0.270612\pi$
$180$	−5.00000	−0.372678
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	−2.00000	−0.148250
$183$	−37.6569	−2.78367
$184$	2.82843	0.208514
$185$	−8.82843	−0.649079
$186$	−16.0000	−1.17318
$187$	3.65685	0.267416
$188$	−8.00000	−0.583460
$189$	5.65685	0.411476
$190$	−2.82843	−0.205196
$191$	−3.31371	−0.239772	−0.119886	−	0.992788i	$-0.538253\pi$
$191$	−3.31371	−0.239772	−0.119886	+	0.992788i	$0.538253\pi$
$192$	−2.82843	−0.204124
$193$	−1.31371	−0.0945628	−0.0472814	−	0.998882i	$-0.515056\pi$
$193$	−1.31371	−0.0945628	−0.0472814	+	0.998882i	$0.515056\pi$
$194$	4.82843	0.346661
$195$	5.65685	0.405096
$196$	1.00000	0.0714286
$197$	−23.6569	−1.68548	−0.842741	−	0.538320i	$-0.819059\pi$
$197$	−23.6569	−1.68548	−0.842741	+	0.538320i	$0.819059\pi$
$198$	−5.00000	−0.355335
$199$	11.3137	0.802008	0.401004	−	0.916076i	$-0.368661\pi$
$199$	11.3137	0.802008	0.401004	+	0.916076i	$0.368661\pi$
$200$	1.00000	0.0707107
$201$	20.6863	1.45910
$202$	10.0000	0.703598
$203$	−8.82843	−0.619634
$204$	10.3431	0.724165
$205$	−4.82843	−0.337232
$206$	5.65685	0.394132
$207$	14.1421	0.982946
$208$	2.00000	0.138675
$209$	−2.82843	−0.195646
$210$	−2.82843	−0.195180
$211$	−7.31371	−0.503496	−0.251748	−	0.967793i	$-0.581005\pi$
$211$	−7.31371	−0.503496	−0.251748	+	0.967793i	$0.581005\pi$
$212$	6.48528	0.445411
$213$	−16.0000	−1.09630
$214$	9.65685	0.660129
$215$	−1.65685	−0.112997
$216$	−5.65685	−0.384900
$217$	−5.65685	−0.384012
$218$	−2.48528	−0.168324
$219$	10.3431	0.698925
$220$	1.00000	0.0674200
$221$	−7.31371	−0.491973
$222$	−24.9706	−1.67591
$223$	28.2843	1.89405	0.947027	−	0.321153i	$-0.104070\pi$
$223$	28.2843	1.89405	0.947027	+	0.321153i	$0.104070\pi$
$224$	−1.00000	−0.0668153
$225$	5.00000	0.333333
$226$	2.00000	0.133038
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−8.00000	−0.529813
$229$	21.3137	1.40845	0.704225	−	0.709977i	$-0.251295\pi$
$229$	21.3137	1.40845	0.704225	+	0.709977i	$0.251295\pi$
$230$	−2.82843	−0.186501
$231$	−2.82843	−0.186097
$232$	8.82843	0.579615
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	10.0000	0.653720
$235$	8.00000	0.521862
$236$	−8.00000	−0.520756
$237$	30.6274	1.98946
$238$	3.65685	0.237039
$239$	−24.4853	−1.58382	−0.791911	−	0.610637i	$-0.790913\pi$
$239$	−24.4853	−1.58382	−0.791911	+	0.610637i	$0.790913\pi$
$240$	2.82843	0.182574
$241$	−3.17157	−0.204299	−0.102149	−	0.994769i	$-0.532572\pi$
$241$	−3.17157	−0.204299	−0.102149	+	0.994769i	$0.532572\pi$
$242$	1.00000	0.0642824
$243$	14.1421	0.907218
$244$	13.3137	0.852323
$245$	−1.00000	−0.0638877
$246$	−13.6569	−0.870729
$247$	5.65685	0.359937
$248$	5.65685	0.359211
$249$	38.6274	2.44791
$250$	−1.00000	−0.0632456
$251$	11.3137	0.714115	0.357057	−	0.934082i	$-0.383780\pi$
$251$	11.3137	0.714115	0.357057	+	0.934082i	$0.383780\pi$
$252$	−5.00000	−0.314970
$253$	−2.82843	−0.177822
$254$	−3.31371	−0.207921
$255$	−10.3431	−0.647713
$256$	1.00000	0.0625000
$257$	28.8284	1.79827	0.899134	−	0.437674i	$-0.144197\pi$
$257$	28.8284	1.79827	0.899134	+	0.437674i	$0.144197\pi$
$258$	−4.68629	−0.291756
$259$	−8.82843	−0.548572
$260$	−2.00000	−0.124035
$261$	44.1421	2.73233
$262$	13.1716	0.813742
$263$	30.6274	1.88857	0.944284	−	0.329133i	$-0.106756\pi$
$263$	30.6274	1.88857	0.944284	+	0.329133i	$0.106756\pi$
$264$	2.82843	0.174078
$265$	−6.48528	−0.398388
$266$	−2.82843	−0.173422
$267$	0.970563	0.0593975
$268$	−7.31371	−0.446756
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	5.65685	0.344265
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	−3.65685	−0.221729
$273$	5.65685	0.342368
$274$	−17.3137	−1.04596
$275$	−1.00000	−0.0603023
$276$	−8.00000	−0.481543
$277$	9.31371	0.559607	0.279803	−	0.960057i	$-0.409731\pi$
$277$	9.31371	0.559607	0.279803	+	0.960057i	$0.409731\pi$
$278$	13.1716	0.789978
$279$	28.2843	1.69334
$280$	1.00000	0.0597614
$281$	26.9706	1.60893	0.804464	−	0.594001i	$-0.202452\pi$
$281$	26.9706	1.60893	0.804464	+	0.594001i	$0.202452\pi$
$282$	22.6274	1.34744
$283$	22.6274	1.34506	0.672530	−	0.740070i	$-0.265208\pi$
$283$	22.6274	1.34506	0.672530	+	0.740070i	$0.265208\pi$
$284$	5.65685	0.335673
$285$	8.00000	0.473879
$286$	−2.00000	−0.118262
$287$	−4.82843	−0.285013
$288$	5.00000	0.294628
$289$	−3.62742	−0.213377
$290$	−8.82843	−0.518423
$291$	−13.6569	−0.800579
$292$	−3.65685	−0.214001
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	−2.82843	−0.164957
$295$	8.00000	0.465778
$296$	8.82843	0.513142
$297$	5.65685	0.328244
$298$	−18.4853	−1.07082
$299$	5.65685	0.327144
$300$	−2.82843	−0.163299
$301$	−1.65685	−0.0954995
$302$	5.17157	0.297591
$303$	−28.2843	−1.62489
$304$	2.82843	0.162221
$305$	−13.3137	−0.762341
$306$	−18.2843	−1.04524
$307$	24.9706	1.42515	0.712573	−	0.701598i	$-0.247530\pi$
$307$	24.9706	1.42515	0.712573	+	0.701598i	$0.247530\pi$
$308$	1.00000	0.0569803
$309$	−16.0000	−0.910208
$310$	−5.65685	−0.321288
$311$	−32.9706	−1.86959	−0.934795	−	0.355189i	$-0.884417\pi$
$311$	−32.9706	−1.86959	−0.934795	+	0.355189i	$0.884417\pi$
$312$	−5.65685	−0.320256
$313$	−3.17157	−0.179268	−0.0896339	−	0.995975i	$-0.528570\pi$
$313$	−3.17157	−0.179268	−0.0896339	+	0.995975i	$0.528570\pi$
$314$	4.34315	0.245098
$315$	5.00000	0.281718
$316$	−10.8284	−0.609147
$317$	8.82843	0.495854	0.247927	−	0.968779i	$-0.420251\pi$
$317$	8.82843	0.495854	0.247927	+	0.968779i	$0.420251\pi$
$318$	−18.3431	−1.02863
$319$	−8.82843	−0.494297
$320$	−1.00000	−0.0559017
$321$	−27.3137	−1.52450
$322$	−2.82843	−0.157622
$323$	−10.3431	−0.575508
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	12.0000	0.664619
$327$	7.02944	0.388729
$328$	4.82843	0.266605
$329$	8.00000	0.441054
$330$	−2.82843	−0.155700
$331$	−4.97056	−0.273207	−0.136603	−	0.990626i	$-0.543619\pi$
$331$	−4.97056	−0.273207	−0.136603	+	0.990626i	$0.543619\pi$
$332$	−13.6569	−0.749517
$333$	44.1421	2.41897
$334$	−16.0000	−0.875481
$335$	7.31371	0.399591
$336$	2.82843	0.154303
$337$	6.68629	0.364226	0.182113	−	0.983278i	$-0.441706\pi$
$337$	6.68629	0.364226	0.182113	+	0.983278i	$0.441706\pi$
$338$	−9.00000	−0.489535
$339$	−5.65685	−0.307238
$340$	3.65685	0.198321
$341$	−5.65685	−0.306336
$342$	14.1421	0.764719
$343$	−1.00000	−0.0539949
$344$	1.65685	0.0893316
$345$	8.00000	0.430706
$346$	21.3137	1.14583
$347$	−4.97056	−0.266834	−0.133417	−	0.991060i	$-0.542595\pi$
$347$	−4.97056	−0.266834	−0.133417	+	0.991060i	$0.542595\pi$
$348$	−24.9706	−1.33856
$349$	−6.97056	−0.373126	−0.186563	−	0.982443i	$-0.559735\pi$
$349$	−6.97056	−0.373126	−0.186563	+	0.982443i	$0.559735\pi$
$350$	−1.00000	−0.0534522
$351$	−11.3137	−0.603881
$352$	−1.00000	−0.0533002
$353$	−19.1716	−1.02040	−0.510200	−	0.860056i	$-0.670429\pi$
$353$	−19.1716	−1.02040	−0.510200	+	0.860056i	$0.670429\pi$
$354$	22.6274	1.20263
$355$	−5.65685	−0.300235
$356$	−0.343146	−0.0181867
$357$	−10.3431	−0.547417
$358$	17.6569	0.933194
$359$	−17.4558	−0.921284	−0.460642	−	0.887586i	$-0.652381\pi$
$359$	−17.4558	−0.921284	−0.460642	+	0.887586i	$0.652381\pi$
$360$	−5.00000	−0.263523
$361$	−11.0000	−0.578947
$362$	−14.0000	−0.735824
$363$	−2.82843	−0.148454
$364$	−2.00000	−0.104828
$365$	3.65685	0.191408
$366$	−37.6569	−1.96836
$367$	−28.2843	−1.47643	−0.738213	−	0.674567i	$-0.764330\pi$
$367$	−28.2843	−1.47643	−0.738213	+	0.674567i	$0.764330\pi$
$368$	2.82843	0.147442
$369$	24.1421	1.25679
$370$	−8.82843	−0.458968
$371$	−6.48528	−0.336699
$372$	−16.0000	−0.829561
$373$	−7.65685	−0.396457	−0.198228	−	0.980156i	$-0.563519\pi$
$373$	−7.65685	−0.396457	−0.198228	+	0.980156i	$0.563519\pi$
$374$	3.65685	0.189091
$375$	2.82843	0.146059
$376$	−8.00000	−0.412568
$377$	17.6569	0.909374
$378$	5.65685	0.290957
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−2.82843	−0.145095
$381$	9.37258	0.480172
$382$	−3.31371	−0.169544
$383$	−18.3431	−0.937291	−0.468645	−	0.883386i	$-0.655258\pi$
$383$	−18.3431	−0.937291	−0.468645	+	0.883386i	$0.655258\pi$
$384$	−2.82843	−0.144338
$385$	−1.00000	−0.0509647
$386$	−1.31371	−0.0668660
$387$	8.28427	0.421113
$388$	4.82843	0.245126
$389$	−2.97056	−0.150614	−0.0753068	−	0.997160i	$-0.523994\pi$
$389$	−2.97056	−0.150614	−0.0753068	+	0.997160i	$0.523994\pi$
$390$	5.65685	0.286446
$391$	−10.3431	−0.523075
$392$	1.00000	0.0505076
$393$	−37.2548	−1.87926
$394$	−23.6569	−1.19182
$395$	10.8284	0.544837
$396$	−5.00000	−0.251259
$397$	10.9706	0.550597	0.275298	−	0.961359i	$-0.411223\pi$
$397$	10.9706	0.550597	0.275298	+	0.961359i	$0.411223\pi$
$398$	11.3137	0.567105
$399$	8.00000	0.400501
$400$	1.00000	0.0500000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	20.6863	1.03174
$403$	11.3137	0.563576
$404$	10.0000	0.497519
$405$	−1.00000	−0.0496904
$406$	−8.82843	−0.438147
$407$	−8.82843	−0.437609
$408$	10.3431	0.512062
$409$	16.1421	0.798177	0.399089	−	0.916912i	$-0.369327\pi$
$409$	16.1421	0.798177	0.399089	+	0.916912i	$0.369327\pi$
$410$	−4.82843	−0.238459
$411$	48.9706	2.41554
$412$	5.65685	0.278693
$413$	8.00000	0.393654
$414$	14.1421	0.695048
$415$	13.6569	0.670389
$416$	2.00000	0.0980581
$417$	−37.2548	−1.82438
$418$	−2.82843	−0.138343
$419$	14.6274	0.714596	0.357298	−	0.933990i	$-0.383698\pi$
$419$	14.6274	0.714596	0.357298	+	0.933990i	$0.383698\pi$
$420$	−2.82843	−0.138013
$421$	−16.6274	−0.810371	−0.405185	−	0.914235i	$-0.632793\pi$
$421$	−16.6274	−0.810371	−0.405185	+	0.914235i	$0.632793\pi$
$422$	−7.31371	−0.356026
$423$	−40.0000	−1.94487
$424$	6.48528	0.314953
$425$	−3.65685	−0.177383
$426$	−16.0000	−0.775203
$427$	−13.3137	−0.644296
$428$	9.65685	0.466782
$429$	5.65685	0.273115
$430$	−1.65685	−0.0799006
$431$	19.7990	0.953684	0.476842	−	0.878989i	$-0.341781\pi$
$431$	19.7990	0.953684	0.476842	+	0.878989i	$0.341781\pi$
$432$	−5.65685	−0.272166
$433$	15.1716	0.729099	0.364550	−	0.931184i	$-0.381223\pi$
$433$	15.1716	0.729099	0.364550	+	0.931184i	$0.381223\pi$
$434$	−5.65685	−0.271538
$435$	24.9706	1.19725
$436$	−2.48528	−0.119023
$437$	8.00000	0.382692
$438$	10.3431	0.494215
$439$	−8.97056	−0.428142	−0.214071	−	0.976818i	$-0.568672\pi$
$439$	−8.97056	−0.428142	−0.214071	+	0.976818i	$0.568672\pi$
$440$	1.00000	0.0476731
$441$	5.00000	0.238095
$442$	−7.31371	−0.347878
$443$	−31.3137	−1.48776	−0.743880	−	0.668314i	$-0.767016\pi$
$443$	−31.3137	−1.48776	−0.743880	+	0.668314i	$0.767016\pi$
$444$	−24.9706	−1.18505
$445$	0.343146	0.0162667
$446$	28.2843	1.33930
$447$	52.2843	2.47296
$448$	−1.00000	−0.0472456
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	5.00000	0.235702
$451$	−4.82843	−0.227362
$452$	2.00000	0.0940721
$453$	−14.6274	−0.687256
$454$	−24.0000	−1.12638
$455$	2.00000	0.0937614
$456$	−8.00000	−0.374634
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	21.3137	0.995924
$459$	20.6863	0.965553
$460$	−2.82843	−0.131876
$461$	−0.343146	−0.0159819	−0.00799095	−	0.999968i	$-0.502544\pi$
$461$	−0.343146	−0.0159819	−0.00799095	+	0.999968i	$0.502544\pi$
$462$	−2.82843	−0.131590
$463$	41.4558	1.92662	0.963308	−	0.268398i	$-0.0864941\pi$
$463$	41.4558	1.92662	0.963308	+	0.268398i	$0.0864941\pi$
$464$	8.82843	0.409849
$465$	16.0000	0.741982
$466$	−6.00000	−0.277945
$467$	−27.7990	−1.28638	−0.643192	−	0.765705i	$-0.722390\pi$
$467$	−27.7990	−1.28638	−0.643192	+	0.765705i	$0.722390\pi$
$468$	10.0000	0.462250
$469$	7.31371	0.337716
$470$	8.00000	0.369012
$471$	−12.2843	−0.566029
$472$	−8.00000	−0.368230
$473$	−1.65685	−0.0761822
$474$	30.6274	1.40676
$475$	2.82843	0.129777
$476$	3.65685	0.167612
$477$	32.4264	1.48470
$478$	−24.4853	−1.11993
$479$	12.6863	0.579651	0.289826	−	0.957079i	$-0.406403\pi$
$479$	12.6863	0.579651	0.289826	+	0.957079i	$0.406403\pi$
$480$	2.82843	0.129099
$481$	17.6569	0.805083
$482$	−3.17157	−0.144461
$483$	8.00000	0.364013
$484$	1.00000	0.0454545
$485$	−4.82843	−0.219248
$486$	14.1421	0.641500
$487$	36.7696	1.66619	0.833094	−	0.553132i	$-0.186567\pi$
$487$	36.7696	1.66619	0.833094	+	0.553132i	$0.186567\pi$
$488$	13.3137	0.602683
$489$	−33.9411	−1.53487
$490$	−1.00000	−0.0451754
$491$	−31.3137	−1.41317	−0.706584	−	0.707629i	$-0.749765\pi$
$491$	−31.3137	−1.41317	−0.706584	+	0.707629i	$0.749765\pi$
$492$	−13.6569	−0.615699
$493$	−32.2843	−1.45401
$494$	5.65685	0.254514
$495$	5.00000	0.224733
$496$	5.65685	0.254000
$497$	−5.65685	−0.253745
$498$	38.6274	1.73094
$499$	−4.97056	−0.222513	−0.111256	−	0.993792i	$-0.535488\pi$
$499$	−4.97056	−0.222513	−0.111256	+	0.993792i	$0.535488\pi$
$500$	−1.00000	−0.0447214
$501$	45.2548	2.02184
$502$	11.3137	0.504956
$503$	19.3137	0.861156	0.430578	−	0.902553i	$-0.358310\pi$
$503$	19.3137	0.861156	0.430578	+	0.902553i	$0.358310\pi$
$504$	−5.00000	−0.222718
$505$	−10.0000	−0.444994
$506$	−2.82843	−0.125739
$507$	25.4558	1.13053
$508$	−3.31371	−0.147022
$509$	−33.3137	−1.47660	−0.738302	−	0.674470i	$-0.764372\pi$
$509$	−33.3137	−1.47660	−0.738302	+	0.674470i	$0.764372\pi$
$510$	−10.3431	−0.458002
$511$	3.65685	0.161770
$512$	1.00000	0.0441942
$513$	−16.0000	−0.706417
$514$	28.8284	1.27157
$515$	−5.65685	−0.249271
$516$	−4.68629	−0.206302
$517$	8.00000	0.351840
$518$	−8.82843	−0.387899
$519$	−60.2843	−2.64619
$520$	−2.00000	−0.0877058
$521$	−37.5980	−1.64720	−0.823599	−	0.567173i	$-0.808037\pi$
$521$	−37.5980	−1.64720	−0.823599	+	0.567173i	$0.808037\pi$
$522$	44.1421	1.93205
$523$	−21.6569	−0.946988	−0.473494	−	0.880797i	$-0.657007\pi$
$523$	−21.6569	−0.946988	−0.473494	+	0.880797i	$0.657007\pi$
$524$	13.1716	0.575403
$525$	2.82843	0.123443
$526$	30.6274	1.33542
$527$	−20.6863	−0.901109
$528$	2.82843	0.123091
$529$	−15.0000	−0.652174
$530$	−6.48528	−0.281703
$531$	−40.0000	−1.73585
$532$	−2.82843	−0.122628
$533$	9.65685	0.418285
$534$	0.970563	0.0420004
$535$	−9.65685	−0.417502
$536$	−7.31371	−0.315904
$537$	−49.9411	−2.15512
$538$	2.00000	0.0862261
$539$	−1.00000	−0.0430730
$540$	5.65685	0.243432
$541$	29.1127	1.25165	0.625826	−	0.779962i	$-0.284762\pi$
$541$	29.1127	1.25165	0.625826	+	0.779962i	$0.284762\pi$
$542$	−16.9706	−0.728948
$543$	39.5980	1.69931
$544$	−3.65685	−0.156786
$545$	2.48528	0.106458
$546$	5.65685	0.242091
$547$	0.686292	0.0293437	0.0146719	−	0.999892i	$-0.495330\pi$
$547$	0.686292	0.0293437	0.0146719	+	0.999892i	$0.495330\pi$
$548$	−17.3137	−0.739605
$549$	66.5685	2.84108
$550$	−1.00000	−0.0426401
$551$	24.9706	1.06378
$552$	−8.00000	−0.340503
$553$	10.8284	0.460472
$554$	9.31371	0.395702
$555$	24.9706	1.05994
$556$	13.1716	0.558599
$557$	6.97056	0.295352	0.147676	−	0.989036i	$-0.452821\pi$
$557$	6.97056	0.295352	0.147676	+	0.989036i	$0.452821\pi$
$558$	28.2843	1.19737
$559$	3.31371	0.140155
$560$	1.00000	0.0422577
$561$	−10.3431	−0.436688
$562$	26.9706	1.13768
$563$	22.6274	0.953632	0.476816	−	0.879003i	$-0.341791\pi$
$563$	22.6274	0.953632	0.476816	+	0.879003i	$0.341791\pi$
$564$	22.6274	0.952786
$565$	−2.00000	−0.0841406
$566$	22.6274	0.951101
$567$	−1.00000	−0.0419961
$568$	5.65685	0.237356
$569$	−28.6274	−1.20012	−0.600062	−	0.799954i	$-0.704857\pi$
$569$	−28.6274	−1.20012	−0.600062	+	0.799954i	$0.704857\pi$
$570$	8.00000	0.335083
$571$	43.5980	1.82452	0.912259	−	0.409613i	$-0.134336\pi$
$571$	43.5980	1.82452	0.912259	+	0.409613i	$0.134336\pi$
$572$	−2.00000	−0.0836242
$573$	9.37258	0.391545
$574$	−4.82843	−0.201535
$575$	2.82843	0.117954
$576$	5.00000	0.208333
$577$	−37.1127	−1.54502	−0.772511	−	0.635001i	$-0.780999\pi$
$577$	−37.1127	−1.54502	−0.772511	+	0.635001i	$0.780999\pi$
$578$	−3.62742	−0.150881
$579$	3.71573	0.154420
$580$	−8.82843	−0.366580
$581$	13.6569	0.566582
$582$	−13.6569	−0.566095
$583$	−6.48528	−0.268593
$584$	−3.65685	−0.151322
$585$	−10.0000	−0.413449
$586$	2.00000	0.0826192
$587$	−9.85786	−0.406878	−0.203439	−	0.979088i	$-0.565212\pi$
$587$	−9.85786	−0.406878	−0.203439	+	0.979088i	$0.565212\pi$
$588$	−2.82843	−0.116642
$589$	16.0000	0.659269
$590$	8.00000	0.329355
$591$	66.9117	2.75238
$592$	8.82843	0.362846
$593$	31.6569	1.29999	0.649996	−	0.759938i	$-0.274771\pi$
$593$	31.6569	1.29999	0.649996	+	0.759938i	$0.274771\pi$
$594$	5.65685	0.232104
$595$	−3.65685	−0.149916
$596$	−18.4853	−0.757187
$597$	−32.0000	−1.30967
$598$	5.65685	0.231326
$599$	10.3431	0.422609	0.211305	−	0.977420i	$-0.432229\pi$
$599$	10.3431	0.422609	0.211305	+	0.977420i	$0.432229\pi$
$600$	−2.82843	−0.115470
$601$	7.17157	0.292535	0.146267	−	0.989245i	$-0.453274\pi$
$601$	7.17157	0.292535	0.146267	+	0.989245i	$0.453274\pi$
$602$	−1.65685	−0.0675283
$603$	−36.5685	−1.48919
$604$	5.17157	0.210428
$605$	−1.00000	−0.0406558
$606$	−28.2843	−1.14897
$607$	18.3431	0.744525	0.372263	−	0.928127i	$-0.378582\pi$
$607$	18.3431	0.744525	0.372263	+	0.928127i	$0.378582\pi$
$608$	2.82843	0.114708
$609$	24.9706	1.01186
$610$	−13.3137	−0.539056
$611$	−16.0000	−0.647291
$612$	−18.2843	−0.739098
$613$	−10.9706	−0.443097	−0.221548	−	0.975149i	$-0.571111\pi$
$613$	−10.9706	−0.443097	−0.221548	+	0.975149i	$0.571111\pi$
$614$	24.9706	1.00773
$615$	13.6569	0.550698
$616$	1.00000	0.0402911
$617$	−43.6569	−1.75756	−0.878779	−	0.477228i	$-0.841642\pi$
$617$	−43.6569	−1.75756	−0.878779	+	0.477228i	$0.841642\pi$
$618$	−16.0000	−0.643614
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	−5.65685	−0.227185
$621$	−16.0000	−0.642058
$622$	−32.9706	−1.32200
$623$	0.343146	0.0137478
$624$	−5.65685	−0.226455
$625$	1.00000	0.0400000
$626$	−3.17157	−0.126762
$627$	8.00000	0.319489
$628$	4.34315	0.173310
$629$	−32.2843	−1.28726
$630$	5.00000	0.199205
$631$	22.6274	0.900783	0.450392	−	0.892831i	$-0.351284\pi$
$631$	22.6274	0.900783	0.450392	+	0.892831i	$0.351284\pi$
$632$	−10.8284	−0.430732
$633$	20.6863	0.822206
$634$	8.82843	0.350622
$635$	3.31371	0.131501
$636$	−18.3431	−0.727353
$637$	2.00000	0.0792429
$638$	−8.82843	−0.349521
$639$	28.2843	1.11891
$640$	−1.00000	−0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−27.3137	−1.07799
$643$	−14.1421	−0.557711	−0.278856	−	0.960333i	$-0.589955\pi$
$643$	−14.1421	−0.557711	−0.278856	+	0.960333i	$0.589955\pi$
$644$	−2.82843	−0.111456
$645$	4.68629	0.184523
$646$	−10.3431	−0.406946
$647$	−44.2843	−1.74099	−0.870497	−	0.492173i	$-0.836203\pi$
$647$	−44.2843	−1.74099	−0.870497	+	0.492173i	$0.836203\pi$
$648$	1.00000	0.0392837
$649$	8.00000	0.314027
$650$	2.00000	0.0784465
$651$	16.0000	0.627089
$652$	12.0000	0.469956
$653$	−43.4558	−1.70056	−0.850279	−	0.526332i	$-0.823567\pi$
$653$	−43.4558	−1.70056	−0.850279	+	0.526332i	$0.823567\pi$
$654$	7.02944	0.274873
$655$	−13.1716	−0.514656
$656$	4.82843	0.188518
$657$	−18.2843	−0.713337
$658$	8.00000	0.311872
$659$	1.65685	0.0645419	0.0322709	−	0.999479i	$-0.489726\pi$
$659$	1.65685	0.0645419	0.0322709	+	0.999479i	$0.489726\pi$
$660$	−2.82843	−0.110096
$661$	−6.00000	−0.233373	−0.116686	−	0.993169i	$-0.537227\pi$
$661$	−6.00000	−0.233373	−0.116686	+	0.993169i	$0.537227\pi$
$662$	−4.97056	−0.193186
$663$	20.6863	0.803389
$664$	−13.6569	−0.529989
$665$	2.82843	0.109682
$666$	44.1421	1.71047
$667$	24.9706	0.966864
$668$	−16.0000	−0.619059
$669$	−80.0000	−3.09298
$670$	7.31371	0.282553
$671$	−13.3137	−0.513970
$672$	2.82843	0.109109
$673$	8.62742	0.332562	0.166281	−	0.986078i	$-0.446824\pi$
$673$	8.62742	0.332562	0.166281	+	0.986078i	$0.446824\pi$
$674$	6.68629	0.257546
$675$	−5.65685	−0.217732
$676$	−9.00000	−0.346154
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	−5.65685	−0.217250
$679$	−4.82843	−0.185298
$680$	3.65685	0.140234
$681$	67.8823	2.60125
$682$	−5.65685	−0.216612
$683$	33.6569	1.28784	0.643922	−	0.765091i	$-0.277306\pi$
$683$	33.6569	1.28784	0.643922	+	0.765091i	$0.277306\pi$
$684$	14.1421	0.540738
$685$	17.3137	0.661523
$686$	−1.00000	−0.0381802
$687$	−60.2843	−2.29999
$688$	1.65685	0.0631670
$689$	12.9706	0.494139
$690$	8.00000	0.304555
$691$	−32.9706	−1.25426	−0.627130	−	0.778915i	$-0.715770\pi$
$691$	−32.9706	−1.25426	−0.627130	+	0.778915i	$0.715770\pi$
$692$	21.3137	0.810226
$693$	5.00000	0.189934
$694$	−4.97056	−0.188680
$695$	−13.1716	−0.499626
$696$	−24.9706	−0.946507
$697$	−17.6569	−0.668801
$698$	−6.97056	−0.263840
$699$	16.9706	0.641886
$700$	−1.00000	−0.0377964
$701$	35.1716	1.32841	0.664206	−	0.747550i	$-0.268770\pi$
$701$	35.1716	1.32841	0.664206	+	0.747550i	$0.268770\pi$
$702$	−11.3137	−0.427008
$703$	24.9706	0.941783
$704$	−1.00000	−0.0376889
$705$	−22.6274	−0.852198
$706$	−19.1716	−0.721532
$707$	−10.0000	−0.376089
$708$	22.6274	0.850390
$709$	−30.2843	−1.13735	−0.568675	−	0.822562i	$-0.692544\pi$
$709$	−30.2843	−1.13735	−0.568675	+	0.822562i	$0.692544\pi$
$710$	−5.65685	−0.212298
$711$	−54.1421	−2.03049
$712$	−0.343146	−0.0128599
$713$	16.0000	0.599205
$714$	−10.3431	−0.387083
$715$	2.00000	0.0747958
$716$	17.6569	0.659868
$717$	69.2548	2.58637
$718$	−17.4558	−0.651446
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	−5.00000	−0.186339
$721$	−5.65685	−0.210672
$722$	−11.0000	−0.409378
$723$	8.97056	0.333619
$724$	−14.0000	−0.520306
$725$	8.82843	0.327880
$726$	−2.82843	−0.104973
$727$	27.3137	1.01301	0.506505	−	0.862237i	$-0.330937\pi$
$727$	27.3137	1.01301	0.506505	+	0.862237i	$0.330937\pi$
$728$	−2.00000	−0.0741249
$729$	−43.0000	−1.59259
$730$	3.65685	0.135346
$731$	−6.05887	−0.224096
$732$	−37.6569	−1.39184
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	−28.2843	−1.04399
$735$	2.82843	0.104328
$736$	2.82843	0.104257
$737$	7.31371	0.269404
$738$	24.1421	0.888684
$739$	−49.2548	−1.81187	−0.905934	−	0.423419i	$-0.860830\pi$
$739$	−49.2548	−1.81187	−0.905934	+	0.423419i	$0.860830\pi$
$740$	−8.82843	−0.324539
$741$	−16.0000	−0.587775
$742$	−6.48528	−0.238082
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	−16.0000	−0.586588
$745$	18.4853	0.677248
$746$	−7.65685	−0.280337
$747$	−68.2843	−2.49839
$748$	3.65685	0.133708
$749$	−9.65685	−0.352854
$750$	2.82843	0.103280
$751$	−39.5980	−1.44495	−0.722475	−	0.691397i	$-0.756996\pi$
$751$	−39.5980	−1.44495	−0.722475	+	0.691397i	$0.756996\pi$
$752$	−8.00000	−0.291730
$753$	−32.0000	−1.16614
$754$	17.6569	0.643025
$755$	−5.17157	−0.188213
$756$	5.65685	0.205738
$757$	37.1127	1.34888	0.674442	−	0.738328i	$-0.264384\pi$
$757$	37.1127	1.34888	0.674442	+	0.738328i	$0.264384\pi$
$758$	4.00000	0.145287
$759$	8.00000	0.290382
$760$	−2.82843	−0.102598
$761$	43.4558	1.57527	0.787637	−	0.616140i	$-0.211305\pi$
$761$	43.4558	1.57527	0.787637	+	0.616140i	$0.211305\pi$
$762$	9.37258	0.339533
$763$	2.48528	0.0899732
$764$	−3.31371	−0.119886
$765$	18.2843	0.661069
$766$	−18.3431	−0.662765
$767$	−16.0000	−0.577727
$768$	−2.82843	−0.102062
$769$	2.48528	0.0896215	0.0448108	−	0.998995i	$-0.485732\pi$
$769$	2.48528	0.0896215	0.0448108	+	0.998995i	$0.485732\pi$
$770$	−1.00000	−0.0360375
$771$	−81.5391	−2.93656
$772$	−1.31371	−0.0472814
$773$	−33.3137	−1.19821	−0.599105	−	0.800670i	$-0.704477\pi$
$773$	−33.3137	−1.19821	−0.599105	+	0.800670i	$0.704477\pi$
$774$	8.28427	0.297772
$775$	5.65685	0.203200
$776$	4.82843	0.173330
$777$	24.9706	0.895814
$778$	−2.97056	−0.106500
$779$	13.6569	0.489308
$780$	5.65685	0.202548
$781$	−5.65685	−0.202418
$782$	−10.3431	−0.369870
$783$	−49.9411	−1.78475
$784$	1.00000	0.0357143
$785$	−4.34315	−0.155014
$786$	−37.2548	−1.32884
$787$	−14.6274	−0.521411	−0.260706	−	0.965418i	$-0.583955\pi$
$787$	−14.6274	−0.521411	−0.260706	+	0.965418i	$0.583955\pi$
$788$	−23.6569	−0.842741
$789$	−86.6274	−3.08402
$790$	10.8284	0.385258
$791$	−2.00000	−0.0711118
$792$	−5.00000	−0.177667
$793$	26.6274	0.945567
$794$	10.9706	0.389331
$795$	18.3431	0.650564
$796$	11.3137	0.401004
$797$	−50.2843	−1.78116	−0.890580	−	0.454826i	$-0.849701\pi$
$797$	−50.2843	−1.78116	−0.890580	+	0.454826i	$0.849701\pi$
$798$	8.00000	0.283197
$799$	29.2548	1.03496
$800$	1.00000	0.0353553
$801$	−1.71573	−0.0606223
$802$	10.0000	0.353112
$803$	3.65685	0.129048
$804$	20.6863	0.729549
$805$	2.82843	0.0996890
$806$	11.3137	0.398508
$807$	−5.65685	−0.199131
$808$	10.0000	0.351799
$809$	−34.2843	−1.20537	−0.602685	−	0.797979i	$-0.705903\pi$
$809$	−34.2843	−1.20537	−0.602685	+	0.797979i	$0.705903\pi$
$810$	−1.00000	−0.0351364
$811$	−6.14214	−0.215680	−0.107840	−	0.994168i	$-0.534393\pi$
$811$	−6.14214	−0.215680	−0.107840	+	0.994168i	$0.534393\pi$
$812$	−8.82843	−0.309817
$813$	48.0000	1.68343
$814$	−8.82843	−0.309436
$815$	−12.0000	−0.420342
$816$	10.3431	0.362083
$817$	4.68629	0.163953
$818$	16.1421	0.564397
$819$	−10.0000	−0.349428
$820$	−4.82843	−0.168616
$821$	9.79899	0.341987	0.170994	−	0.985272i	$-0.445302\pi$
$821$	9.79899	0.341987	0.170994	+	0.985272i	$0.445302\pi$
$822$	48.9706	1.70804
$823$	26.8284	0.935180	0.467590	−	0.883945i	$-0.345122\pi$
$823$	26.8284	0.935180	0.467590	+	0.883945i	$0.345122\pi$
$824$	5.65685	0.197066
$825$	2.82843	0.0984732
$826$	8.00000	0.278356
$827$	41.2548	1.43457	0.717286	−	0.696779i	$-0.245384\pi$
$827$	41.2548	1.43457	0.717286	+	0.696779i	$0.245384\pi$
$828$	14.1421	0.491473
$829$	−51.2548	−1.78015	−0.890077	−	0.455810i	$-0.849350\pi$
$829$	−51.2548	−1.78015	−0.890077	+	0.455810i	$0.849350\pi$
$830$	13.6569	0.474036
$831$	−26.3431	−0.913834
$832$	2.00000	0.0693375
$833$	−3.65685	−0.126702
$834$	−37.2548	−1.29003
$835$	16.0000	0.553703
$836$	−2.82843	−0.0978232
$837$	−32.0000	−1.10608
$838$	14.6274	0.505296
$839$	−45.6569	−1.57625	−0.788125	−	0.615515i	$-0.788948\pi$
$839$	−45.6569	−1.57625	−0.788125	+	0.615515i	$0.788948\pi$
$840$	−2.82843	−0.0975900
$841$	48.9411	1.68763
$842$	−16.6274	−0.573019
$843$	−76.2843	−2.62737
$844$	−7.31371	−0.251748
$845$	9.00000	0.309609
$846$	−40.0000	−1.37523
$847$	−1.00000	−0.0343604
$848$	6.48528	0.222705
$849$	−64.0000	−2.19647
$850$	−3.65685	−0.125429
$851$	24.9706	0.855980
$852$	−16.0000	−0.548151
$853$	−2.68629	−0.0919769	−0.0459884	−	0.998942i	$-0.514644\pi$
$853$	−2.68629	−0.0919769	−0.0459884	+	0.998942i	$0.514644\pi$
$854$	−13.3137	−0.455586
$855$	−14.1421	−0.483651
$856$	9.65685	0.330064
$857$	26.9706	0.921297	0.460648	−	0.887583i	$-0.347617\pi$
$857$	26.9706	0.921297	0.460648	+	0.887583i	$0.347617\pi$
$858$	5.65685	0.193122
$859$	27.3137	0.931932	0.465966	−	0.884803i	$-0.345707\pi$
$859$	27.3137	0.931932	0.465966	+	0.884803i	$0.345707\pi$
$860$	−1.65685	−0.0564983
$861$	13.6569	0.465424
$862$	19.7990	0.674356
$863$	36.7696	1.25165	0.625825	−	0.779963i	$-0.284762\pi$
$863$	36.7696	1.25165	0.625825	+	0.779963i	$0.284762\pi$
$864$	−5.65685	−0.192450
$865$	−21.3137	−0.724688
$866$	15.1716	0.515551
$867$	10.2599	0.348444
$868$	−5.65685	−0.192006
$869$	10.8284	0.367329
$870$	24.9706	0.846581
$871$	−14.6274	−0.495631
$872$	−2.48528	−0.0841622
$873$	24.1421	0.817087
$874$	8.00000	0.270604
$875$	1.00000	0.0338062
$876$	10.3431	0.349463
$877$	30.9706	1.04580	0.522901	−	0.852394i	$-0.324850\pi$
$877$	30.9706	1.04580	0.522901	+	0.852394i	$0.324850\pi$
$878$	−8.97056	−0.302742
$879$	−5.65685	−0.190801
$880$	1.00000	0.0337100
$881$	−25.3137	−0.852841	−0.426420	−	0.904525i	$-0.640226\pi$
$881$	−25.3137	−0.852841	−0.426420	+	0.904525i	$0.640226\pi$
$882$	5.00000	0.168359
$883$	30.3431	1.02113	0.510564	−	0.859840i	$-0.329437\pi$
$883$	30.3431	1.02113	0.510564	+	0.859840i	$0.329437\pi$
$884$	−7.31371	−0.245987
$885$	−22.6274	−0.760612
$886$	−31.3137	−1.05200
$887$	22.6274	0.759754	0.379877	−	0.925037i	$-0.375966\pi$
$887$	22.6274	0.759754	0.379877	+	0.925037i	$0.375966\pi$
$888$	−24.9706	−0.837957
$889$	3.31371	0.111138
$890$	0.343146	0.0115023
$891$	−1.00000	−0.0335013
$892$	28.2843	0.947027
$893$	−22.6274	−0.757198
$894$	52.2843	1.74865
$895$	−17.6569	−0.590204
$896$	−1.00000	−0.0334077
$897$	−16.0000	−0.534224
$898$	−6.00000	−0.200223
$899$	49.9411	1.66563
$900$	5.00000	0.166667
$901$	−23.7157	−0.790085
$902$	−4.82843	−0.160769
$903$	4.68629	0.155950
$904$	2.00000	0.0665190
$905$	14.0000	0.465376
$906$	−14.6274	−0.485963
$907$	6.34315	0.210621	0.105310	−	0.994439i	$-0.466416\pi$
$907$	6.34315	0.210621	0.105310	+	0.994439i	$0.466416\pi$
$908$	−24.0000	−0.796468
$909$	50.0000	1.65840
$910$	2.00000	0.0662994
$911$	−10.3431	−0.342684	−0.171342	−	0.985212i	$-0.554810\pi$
$911$	−10.3431	−0.342684	−0.171342	+	0.985212i	$0.554810\pi$
$912$	−8.00000	−0.264906
$913$	13.6569	0.451976
$914$	−22.0000	−0.727695
$915$	37.6569	1.24490
$916$	21.3137	0.704225
$917$	−13.1716	−0.434964
$918$	20.6863	0.682749
$919$	−52.7696	−1.74071	−0.870353	−	0.492428i	$-0.836110\pi$
$919$	−52.7696	−1.74071	−0.870353	+	0.492428i	$0.836110\pi$
$920$	−2.82843	−0.0932505
$921$	−70.6274	−2.32725
$922$	−0.343146	−0.0113009
$923$	11.3137	0.372395
$924$	−2.82843	−0.0930484
$925$	8.82843	0.290277
$926$	41.4558	1.36232
$927$	28.2843	0.928977
$928$	8.82843	0.289807
$929$	−5.02944	−0.165010	−0.0825052	−	0.996591i	$-0.526292\pi$
$929$	−5.02944	−0.165010	−0.0825052	+	0.996591i	$0.526292\pi$
$930$	16.0000	0.524661
$931$	2.82843	0.0926980
$932$	−6.00000	−0.196537
$933$	93.2548	3.05303
$934$	−27.7990	−0.909611
$935$	−3.65685	−0.119592
$936$	10.0000	0.326860
$937$	−20.6274	−0.673868	−0.336934	−	0.941528i	$-0.609390\pi$
$937$	−20.6274	−0.673868	−0.336934	+	0.941528i	$0.609390\pi$
$938$	7.31371	0.238801
$939$	8.97056	0.292743
$940$	8.00000	0.260931
$941$	−9.31371	−0.303618	−0.151809	−	0.988410i	$-0.548510\pi$
$941$	−9.31371	−0.303618	−0.151809	+	0.988410i	$0.548510\pi$
$942$	−12.2843	−0.400243
$943$	13.6569	0.444728
$944$	−8.00000	−0.260378
$945$	−5.65685	−0.184017
$946$	−1.65685	−0.0538690
$947$	−43.5980	−1.41674	−0.708372	−	0.705839i	$-0.750570\pi$
$947$	−43.5980	−1.41674	−0.708372	+	0.705839i	$0.750570\pi$
$948$	30.6274	0.994732
$949$	−7.31371	−0.237413
$950$	2.82843	0.0917663
$951$	−24.9706	−0.809726
$952$	3.65685	0.118519
$953$	47.2548	1.53073	0.765367	−	0.643594i	$-0.222557\pi$
$953$	47.2548	1.53073	0.765367	+	0.643594i	$0.222557\pi$
$954$	32.4264	1.04984
$955$	3.31371	0.107229
$956$	−24.4853	−0.791911
$957$	24.9706	0.807184
$958$	12.6863	0.409875
$959$	17.3137	0.559089
$960$	2.82843	0.0912871
$961$	1.00000	0.0322581
$962$	17.6569	0.569280
$963$	48.2843	1.55594
$964$	−3.17157	−0.102149
$965$	1.31371	0.0422898
$966$	8.00000	0.257396
$967$	36.6863	1.17975	0.589876	−	0.807494i	$-0.299177\pi$
$967$	36.6863	1.17975	0.589876	+	0.807494i	$0.299177\pi$
$968$	1.00000	0.0321412
$969$	29.2548	0.939801
$970$	−4.82843	−0.155031
$971$	25.9411	0.832490	0.416245	−	0.909252i	$-0.363346\pi$
$971$	25.9411	0.832490	0.416245	+	0.909252i	$0.363346\pi$
$972$	14.1421	0.453609
$973$	−13.1716	−0.422261
$974$	36.7696	1.17817
$975$	−5.65685	−0.181164
$976$	13.3137	0.426161
$977$	24.6274	0.787901	0.393950	−	0.919132i	$-0.371108\pi$
$977$	24.6274	0.787901	0.393950	+	0.919132i	$0.371108\pi$
$978$	−33.9411	−1.08532
$979$	0.343146	0.0109670
$980$	−1.00000	−0.0319438
$981$	−12.4264	−0.396745
$982$	−31.3137	−0.999261
$983$	−39.5980	−1.26298	−0.631490	−	0.775384i	$-0.717556\pi$
$983$	−39.5980	−1.26298	−0.631490	+	0.775384i	$0.717556\pi$
$984$	−13.6569	−0.435365
$985$	23.6569	0.753770
$986$	−32.2843	−1.02814
$987$	−22.6274	−0.720239
$988$	5.65685	0.179969
$989$	4.68629	0.149015
$990$	5.00000	0.158910
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	5.65685	0.179605
$993$	14.0589	0.446145
$994$	−5.65685	−0.179425
$995$	−11.3137	−0.358669
$996$	38.6274	1.22396
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	−4.97056	−0.157340
$999$	−49.9411	−1.58007

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	770.2.a.i.1.1	✓	2
3.2	odd	2		6930.2.a.br.1.2		2
4.3	odd	2		6160.2.a.ba.1.2		2
5.2	odd	4		3850.2.c.u.1849.4		4
5.3	odd	4		3850.2.c.u.1849.1		4
5.4	even	2		3850.2.a.bi.1.2		2
7.6	odd	2		5390.2.a.bt.1.2		2
11.10	odd	2		8470.2.a.bo.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.i.1.1	✓	2	1.1	even	1	trivial
3850.2.a.bi.1.2		2	5.4	even	2
3850.2.c.u.1849.1		4	5.3	odd	4
3850.2.c.u.1849.4		4	5.2	odd	4
5390.2.a.bt.1.2		2	7.6	odd	2
6160.2.a.ba.1.2		2	4.3	odd	2
6930.2.a.br.1.2		2	3.2	odd	2
8470.2.a.bo.1.1		2	11.10	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:	\( N \)	\(=\)	\( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	770.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -2.82843 q^{12} +2.00000 q^{13} -1.00000 q^{14} +2.82843 q^{15} +1.00000 q^{16} -3.65685 q^{17} +5.00000 q^{18} +2.82843 q^{19} -1.00000 q^{20} +2.82843 q^{21} -1.00000 q^{22} +2.82843 q^{23} -2.82843 q^{24} +1.00000 q^{25} +2.00000 q^{26} -5.65685 q^{27} -1.00000 q^{28} +8.82843 q^{29} +2.82843 q^{30} +5.65685 q^{31} +1.00000 q^{32} +2.82843 q^{33} -3.65685 q^{34} +1.00000 q^{35} +5.00000 q^{36} +8.82843 q^{37} +2.82843 q^{38} -5.65685 q^{39} -1.00000 q^{40} +4.82843 q^{41} +2.82843 q^{42} +1.65685 q^{43} -1.00000 q^{44} -5.00000 q^{45} +2.82843 q^{46} -8.00000 q^{47} -2.82843 q^{48} +1.00000 q^{49} +1.00000 q^{50} +10.3431 q^{51} +2.00000 q^{52} +6.48528 q^{53} -5.65685 q^{54} +1.00000 q^{55} -1.00000 q^{56} -8.00000 q^{57} +8.82843 q^{58} -8.00000 q^{59} +2.82843 q^{60} +13.3137 q^{61} +5.65685 q^{62} -5.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +2.82843 q^{66} -7.31371 q^{67} -3.65685 q^{68} -8.00000 q^{69} +1.00000 q^{70} +5.65685 q^{71} +5.00000 q^{72} -3.65685 q^{73} +8.82843 q^{74} -2.82843 q^{75} +2.82843 q^{76} +1.00000 q^{77} -5.65685 q^{78} -10.8284 q^{79} -1.00000 q^{80} +1.00000 q^{81} +4.82843 q^{82} -13.6569 q^{83} +2.82843 q^{84} +3.65685 q^{85} +1.65685 q^{86} -24.9706 q^{87} -1.00000 q^{88} -0.343146 q^{89} -5.00000 q^{90} -2.00000 q^{91} +2.82843 q^{92} -16.0000 q^{93} -8.00000 q^{94} -2.82843 q^{95} -2.82843 q^{96} +4.82843 q^{97} +1.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} + 10 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 + 10 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} + 10 q^{9} - 2 q^{10} - 2 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} + 4 q^{17} + 10 q^{18} - 2 q^{20} - 2 q^{22} + 2 q^{25} + 4 q^{26} - 2 q^{28} + 12 q^{29} + 2 q^{32} + 4 q^{34} + 2 q^{35} + 10 q^{36} + 12 q^{37} - 2 q^{40} + 4 q^{41} - 8 q^{43} - 2 q^{44} - 10 q^{45} - 16 q^{47} + 2 q^{49} + 2 q^{50} + 32 q^{51} + 4 q^{52} - 4 q^{53} + 2 q^{55} - 2 q^{56} - 16 q^{57} + 12 q^{58} - 16 q^{59} + 4 q^{61} - 10 q^{63} + 2 q^{64} - 4 q^{65} + 8 q^{67} + 4 q^{68} - 16 q^{69} + 2 q^{70} + 10 q^{72} + 4 q^{73} + 12 q^{74} + 2 q^{77} - 16 q^{79} - 2 q^{80} + 2 q^{81} + 4 q^{82} - 16 q^{83} - 4 q^{85} - 8 q^{86} - 16 q^{87} - 2 q^{88} - 12 q^{89} - 10 q^{90} - 4 q^{91} - 32 q^{93} - 16 q^{94} + 4 q^{97} + 2 q^{98} - 10 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 + 10 * q^9 - 2 * q^10 - 2 * q^11 + 4 * q^13 - 2 * q^14 + 2 * q^16 + 4 * q^17 + 10 * q^18 - 2 * q^20 - 2 * q^22 + 2 * q^25 + 4 * q^26 - 2 * q^28 + 12 * q^29 + 2 * q^32 + 4 * q^34 + 2 * q^35 + 10 * q^36 + 12 * q^37 - 2 * q^40 + 4 * q^41 - 8 * q^43 - 2 * q^44 - 10 * q^45 - 16 * q^47 + 2 * q^49 + 2 * q^50 + 32 * q^51 + 4 * q^52 - 4 * q^53 + 2 * q^55 - 2 * q^56 - 16 * q^57 + 12 * q^58 - 16 * q^59 + 4 * q^61 - 10 * q^63 + 2 * q^64 - 4 * q^65 + 8 * q^67 + 4 * q^68 - 16 * q^69 + 2 * q^70 + 10 * q^72 + 4 * q^73 + 12 * q^74 + 2 * q^77 - 16 * q^79 - 2 * q^80 + 2 * q^81 + 4 * q^82 - 16 * q^83 - 4 * q^85 - 8 * q^86 - 16 * q^87 - 2 * q^88 - 12 * q^89 - 10 * q^90 - 4 * q^91 - 32 * q^93 - 16 * q^94 + 4 * q^97 + 2 * q^98 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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