LMFDB - Embedded newform 77.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [77,4,Mod(1,77)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(77, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("77.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 77 = 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	77.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:	$4.54314707044$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	77.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{2} +4.00000 q^{3} +1.00000 q^{4} +12.0000 q^{5} +12.0000 q^{6} +7.00000 q^{7} -21.0000 q^{8} -11.0000 q^{9} +O(q^{10})$

\(q+3.00000 q^{2} +4.00000 q^{3} +1.00000 q^{4} +12.0000 q^{5} +12.0000 q^{6} +7.00000 q^{7} -21.0000 q^{8} -11.0000 q^{9} +36.0000 q^{10} +11.0000 q^{11} +4.00000 q^{12} +38.0000 q^{13} +21.0000 q^{14} +48.0000 q^{15} -71.0000 q^{16} -48.0000 q^{17} -33.0000 q^{18} -70.0000 q^{19} +12.0000 q^{20} +28.0000 q^{21} +33.0000 q^{22} +12.0000 q^{23} -84.0000 q^{24} +19.0000 q^{25} +114.000 q^{26} -152.000 q^{27} +7.00000 q^{28} +126.000 q^{29} +144.000 q^{30} -70.0000 q^{31} -45.0000 q^{32} +44.0000 q^{33} -144.000 q^{34} +84.0000 q^{35} -11.0000 q^{36} -358.000 q^{37} -210.000 q^{38} +152.000 q^{39} -252.000 q^{40} -216.000 q^{41} +84.0000 q^{42} +344.000 q^{43} +11.0000 q^{44} -132.000 q^{45} +36.0000 q^{46} +390.000 q^{47} -284.000 q^{48} +49.0000 q^{49} +57.0000 q^{50} -192.000 q^{51} +38.0000 q^{52} +438.000 q^{53} -456.000 q^{54} +132.000 q^{55} -147.000 q^{56} -280.000 q^{57} +378.000 q^{58} -552.000 q^{59} +48.0000 q^{60} +830.000 q^{61} -210.000 q^{62} -77.0000 q^{63} +433.000 q^{64} +456.000 q^{65} +132.000 q^{66} -196.000 q^{67} -48.0000 q^{68} +48.0000 q^{69} +252.000 q^{70} +648.000 q^{71} +231.000 q^{72} -16.0000 q^{73} -1074.00 q^{74} +76.0000 q^{75} -70.0000 q^{76} +77.0000 q^{77} +456.000 q^{78} +1352.00 q^{79} -852.000 q^{80} -311.000 q^{81} -648.000 q^{82} +90.0000 q^{83} +28.0000 q^{84} -576.000 q^{85} +1032.00 q^{86} +504.000 q^{87} -231.000 q^{88} +1146.00 q^{89} -396.000 q^{90} +266.000 q^{91} +12.0000 q^{92} -280.000 q^{93} +1170.00 q^{94} -840.000 q^{95} -180.000 q^{96} -70.0000 q^{97} +147.000 q^{98} -121.000 q^{99} +O(q^{100})\)

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 $

$2$	3.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	3.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	4.00000	0.769800	0.384900	−	0.922958i	$-0.374236\pi$
$3$	4.00000	0.769800	0.384900	+	0.922958i	$0.374236\pi$
$4$	1.00000	0.125000
$5$	12.0000	1.07331	0.536656	−	0.843801i	$-0.319687\pi$
$5$	12.0000	1.07331	0.536656	+	0.843801i	$0.319687\pi$
$6$	12.0000	0.816497
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−21.0000	−0.928078
$9$	−11.0000	−0.407407
$10$	36.0000	1.13842
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	4.00000	0.0962250
$13$	38.0000	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$13$	38.0000	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$14$	21.0000	0.400892
$15$	48.0000	0.826236
$16$	−71.0000	−1.10938
$17$	−48.0000	−0.684806	−0.342403	−	0.939553i	$-0.611241\pi$
$17$	−48.0000	−0.684806	−0.342403	+	0.939553i	$0.611241\pi$
$18$	−33.0000	−0.432121
$19$	−70.0000	−0.845216	−0.422608	−	0.906313i	$-0.638885\pi$
$19$	−70.0000	−0.845216	−0.422608	+	0.906313i	$0.638885\pi$
$20$	12.0000	0.134164
$21$	28.0000	0.290957
$22$	33.0000	0.319801
$23$	12.0000	0.108790	0.0543951	−	0.998519i	$-0.482677\pi$
$23$	12.0000	0.108790	0.0543951	+	0.998519i	$0.482677\pi$
$24$	−84.0000	−0.714435
$25$	19.0000	0.152000
$26$	114.000	0.859894
$27$	−152.000	−1.08342
$28$	7.00000	0.0472456
$29$	126.000	0.806814	0.403407	−	0.915021i	$-0.367826\pi$
$29$	126.000	0.806814	0.403407	+	0.915021i	$0.367826\pi$
$30$	144.000	0.876356
$31$	−70.0000	−0.405560	−0.202780	−	0.979224i	$-0.564998\pi$
$31$	−70.0000	−0.405560	−0.202780	+	0.979224i	$0.564998\pi$
$32$	−45.0000	−0.248592
$33$	44.0000	0.232104
$34$	−144.000	−0.726347
$35$	84.0000	0.405674
$36$	−11.0000	−0.0509259
$37$	−358.000	−1.59067	−0.795336	−	0.606169i	$-0.792705\pi$
$37$	−358.000	−1.59067	−0.795336	+	0.606169i	$0.792705\pi$
$38$	−210.000	−0.896487
$39$	152.000	0.624089
$40$	−252.000	−0.996117
$41$	−216.000	−0.822769	−0.411385	−	0.911462i	$-0.634955\pi$
$41$	−216.000	−0.822769	−0.411385	+	0.911462i	$0.634955\pi$
$42$	84.0000	0.308607
$43$	344.000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	344.000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	11.0000	0.0376889
$45$	−132.000	−0.437276
$46$	36.0000	0.115389
$47$	390.000	1.21037	0.605185	−	0.796085i	$-0.293099\pi$
$47$	390.000	1.21037	0.605185	+	0.796085i	$0.293099\pi$
$48$	−284.000	−0.853997
$49$	49.0000	0.142857
$50$	57.0000	0.161220
$51$	−192.000	−0.527164
$52$	38.0000	0.101339
$53$	438.000	1.13517	0.567584	−	0.823315i	$-0.307878\pi$
$53$	438.000	1.13517	0.567584	+	0.823315i	$0.307878\pi$
$54$	−456.000	−1.14914
$55$	132.000	0.323616
$56$	−147.000	−0.350780
$57$	−280.000	−0.650647
$58$	378.000	0.855756
$59$	−552.000	−1.21804	−0.609019	−	0.793155i	$-0.708437\pi$
$59$	−552.000	−1.21804	−0.609019	+	0.793155i	$0.708437\pi$
$60$	48.0000	0.103280
$61$	830.000	1.74214	0.871071	−	0.491158i	$-0.163426\pi$
$61$	830.000	1.74214	0.871071	+	0.491158i	$0.163426\pi$
$62$	−210.000	−0.430162
$63$	−77.0000	−0.153986
$64$	433.000	0.845703
$65$	456.000	0.870151
$66$	132.000	0.246183
$67$	−196.000	−0.357391	−0.178696	−	0.983904i	$-0.557188\pi$
$67$	−196.000	−0.357391	−0.178696	+	0.983904i	$0.557188\pi$
$68$	−48.0000	−0.0856008
$69$	48.0000	0.0837467
$70$	252.000	0.430282
$71$	648.000	1.08315	0.541574	−	0.840653i	$-0.317829\pi$
$71$	648.000	1.08315	0.541574	+	0.840653i	$0.317829\pi$
$72$	231.000	0.378106
$73$	−16.0000	−0.0256529	−0.0128264	−	0.999918i	$-0.504083\pi$
$73$	−16.0000	−0.0256529	−0.0128264	+	0.999918i	$0.504083\pi$
$74$	−1074.00	−1.68716
$75$	76.0000	0.117010
$76$	−70.0000	−0.105652
$77$	77.0000	0.113961
$78$	456.000	0.661947
$79$	1352.00	1.92547	0.962733	−	0.270452i	$-0.0871732\pi$
$79$	1352.00	1.92547	0.962733	+	0.270452i	$0.0871732\pi$
$80$	−852.000	−1.19071
$81$	−311.000	−0.426612
$82$	−648.000	−0.872678
$83$	90.0000	0.119021	0.0595107	−	0.998228i	$-0.481046\pi$
$83$	90.0000	0.119021	0.0595107	+	0.998228i	$0.481046\pi$
$84$	28.0000	0.0363696
$85$	−576.000	−0.735011
$86$	1032.00	1.29399
$87$	504.000	0.621086
$88$	−231.000	−0.279826
$89$	1146.00	1.36490	0.682448	−	0.730934i	$-0.260915\pi$
$89$	1146.00	1.36490	0.682448	+	0.730934i	$0.260915\pi$
$90$	−396.000	−0.463801
$91$	266.000	0.306422
$92$	12.0000	0.0135988
$93$	−280.000	−0.312201
$94$	1170.00	1.28379
$95$	−840.000	−0.907181
$96$	−180.000	−0.191366
$97$	−70.0000	−0.0732724	−0.0366362	−	0.999329i	$-0.511664\pi$
$97$	−70.0000	−0.0732724	−0.0366362	+	0.999329i	$0.511664\pi$
$98$	147.000	0.151523
$99$	−121.000	−0.122838
$100$	19.0000	0.0190000
$101$	−1254.00	−1.23542	−0.617711	−	0.786405i	$-0.711940\pi$
$101$	−1254.00	−1.23542	−0.617711	+	0.786405i	$0.711940\pi$
$102$	−576.000	−0.559142
$103$	−682.000	−0.652422	−0.326211	−	0.945297i	$-0.605772\pi$
$103$	−682.000	−0.652422	−0.326211	+	0.945297i	$0.605772\pi$
$104$	−798.000	−0.752407
$105$	336.000	0.312288
$106$	1314.00	1.20403
$107$	−384.000	−0.346941	−0.173470	−	0.984839i	$-0.555498\pi$
$107$	−384.000	−0.346941	−0.173470	+	0.984839i	$0.555498\pi$
$108$	−152.000	−0.135428
$109$	−646.000	−0.567666	−0.283833	−	0.958874i	$-0.591606\pi$
$109$	−646.000	−0.567666	−0.283833	+	0.958874i	$0.591606\pi$
$110$	396.000	0.343247
$111$	−1432.00	−1.22450
$112$	−497.000	−0.419304
$113$	−1314.00	−1.09390	−0.546950	−	0.837165i	$-0.684211\pi$
$113$	−1314.00	−1.09390	−0.546950	+	0.837165i	$0.684211\pi$
$114$	−840.000	−0.690116
$115$	144.000	0.116766
$116$	126.000	0.100852
$117$	−418.000	−0.330292
$118$	−1656.00	−1.29193
$119$	−336.000	−0.258833
$120$	−1008.00	−0.766812
$121$	121.000	0.0909091
$122$	2490.00	1.84782
$123$	−864.000	−0.633368
$124$	−70.0000	−0.0506950
$125$	−1272.00	−0.910169
$126$	−231.000	−0.163326
$127$	344.000	0.240355	0.120177	−	0.992752i	$-0.461654\pi$
$127$	344.000	0.240355	0.120177	+	0.992752i	$0.461654\pi$
$128$	1659.00	1.14560
$129$	1376.00	0.939148
$130$	1368.00	0.922935
$131$	−258.000	−0.172073	−0.0860365	−	0.996292i	$-0.527420\pi$
$131$	−258.000	−0.172073	−0.0860365	+	0.996292i	$0.527420\pi$
$132$	44.0000	0.0290129
$133$	−490.000	−0.319462
$134$	−588.000	−0.379071
$135$	−1824.00	−1.16285
$136$	1008.00	0.635554
$137$	−2730.00	−1.70248	−0.851240	−	0.524777i	$-0.824149\pi$
$137$	−2730.00	−1.70248	−0.851240	+	0.524777i	$0.824149\pi$
$138$	144.000	0.0888268
$139$	1838.00	1.12156	0.560781	−	0.827964i	$-0.310501\pi$
$139$	1838.00	1.12156	0.560781	+	0.827964i	$0.310501\pi$
$140$	84.0000	0.0507093
$141$	1560.00	0.931743
$142$	1944.00	1.14885
$143$	418.000	0.244440
$144$	781.000	0.451968
$145$	1512.00	0.865964
$146$	−48.0000	−0.0272090
$147$	196.000	0.109971
$148$	−358.000	−0.198834
$149$	−510.000	−0.280408	−0.140204	−	0.990123i	$-0.544776\pi$
$149$	−510.000	−0.280408	−0.140204	+	0.990123i	$0.544776\pi$
$150$	228.000	0.124107
$151$	2864.00	1.54350	0.771752	−	0.635924i	$-0.219381\pi$
$151$	2864.00	1.54350	0.771752	+	0.635924i	$0.219381\pi$
$152$	1470.00	0.784426
$153$	528.000	0.278995
$154$	231.000	0.120873
$155$	−840.000	−0.435293
$156$	152.000	0.0780112
$157$	−2968.00	−1.50874	−0.754370	−	0.656449i	$-0.772058\pi$
$157$	−2968.00	−1.50874	−0.754370	+	0.656449i	$0.772058\pi$
$158$	4056.00	2.04227
$159$	1752.00	0.873853
$160$	−540.000	−0.266817
$161$	84.0000	0.0411188
$162$	−933.000	−0.452490
$163$	1604.00	0.770767	0.385383	−	0.922757i	$-0.374069\pi$
$163$	1604.00	0.770767	0.385383	+	0.922757i	$0.374069\pi$
$164$	−216.000	−0.102846
$165$	528.000	0.249120
$166$	270.000	0.126241
$167$	180.000	0.0834061	0.0417030	−	0.999130i	$-0.486722\pi$
$167$	180.000	0.0834061	0.0417030	+	0.999130i	$0.486722\pi$
$168$	−588.000	−0.270031
$169$	−753.000	−0.342740
$170$	−1728.00	−0.779597
$171$	770.000	0.344347
$172$	344.000	0.152499
$173$	−1626.00	−0.714581	−0.357290	−	0.933993i	$-0.616299\pi$
$173$	−1626.00	−0.714581	−0.357290	+	0.933993i	$0.616299\pi$
$174$	1512.00	0.658761
$175$	133.000	0.0574506
$176$	−781.000	−0.334489
$177$	−2208.00	−0.937647
$178$	3438.00	1.44769
$179$	−3252.00	−1.35791	−0.678955	−	0.734180i	$-0.737567\pi$
$179$	−3252.00	−1.35791	−0.678955	+	0.734180i	$0.737567\pi$
$180$	−132.000	−0.0546594
$181$	1820.00	0.747401	0.373700	−	0.927549i	$-0.378089\pi$
$181$	1820.00	0.747401	0.373700	+	0.927549i	$0.378089\pi$
$182$	798.000	0.325009
$183$	3320.00	1.34110
$184$	−252.000	−0.100966
$185$	−4296.00	−1.70729
$186$	−840.000	−0.331139
$187$	−528.000	−0.206477
$188$	390.000	0.151296
$189$	−1064.00	−0.409495
$190$	−2520.00	−0.962211
$191$	−1212.00	−0.459148	−0.229574	−	0.973291i	$-0.573733\pi$
$191$	−1212.00	−0.459148	−0.229574	+	0.973291i	$0.573733\pi$
$192$	1732.00	0.651023
$193$	2522.00	0.940609	0.470304	−	0.882504i	$-0.344144\pi$
$193$	2522.00	0.940609	0.470304	+	0.882504i	$0.344144\pi$
$194$	−210.000	−0.0777171
$195$	1824.00	0.669843
$196$	49.0000	0.0178571
$197$	−3474.00	−1.25641	−0.628204	−	0.778049i	$-0.716210\pi$
$197$	−3474.00	−1.25641	−0.628204	+	0.778049i	$0.716210\pi$
$198$	−363.000	−0.130289
$199$	−2842.00	−1.01238	−0.506191	−	0.862421i	$-0.668947\pi$
$199$	−2842.00	−1.01238	−0.506191	+	0.862421i	$0.668947\pi$
$200$	−399.000	−0.141068
$201$	−784.000	−0.275120
$202$	−3762.00	−1.31036
$203$	882.000	0.304947
$204$	−192.000	−0.0658955
$205$	−2592.00	−0.883088
$206$	−2046.00	−0.691998
$207$	−132.000	−0.0443219
$208$	−2698.00	−0.899388
$209$	−770.000	−0.254842
$210$	1008.00	0.331231
$211$	5528.00	1.80362	0.901809	−	0.432136i	$-0.142240\pi$
$211$	5528.00	1.80362	0.901809	+	0.432136i	$0.142240\pi$
$212$	438.000	0.141896
$213$	2592.00	0.833807
$214$	−1152.00	−0.367986
$215$	4128.00	1.30943
$216$	3192.00	1.00550
$217$	−490.000	−0.153287
$218$	−1938.00	−0.602101
$219$	−64.0000	−0.0197476
$220$	132.000	0.0404520
$221$	−1824.00	−0.555183
$222$	−4296.00	−1.29878
$223$	4034.00	1.21137	0.605687	−	0.795703i	$-0.292898\pi$
$223$	4034.00	1.21137	0.605687	+	0.795703i	$0.292898\pi$
$224$	−315.000	−0.0939590
$225$	−209.000	−0.0619259
$226$	−3942.00	−1.16026
$227$	726.000	0.212275	0.106137	−	0.994351i	$-0.466152\pi$
$227$	726.000	0.212275	0.106137	+	0.994351i	$0.466152\pi$
$228$	−280.000	−0.0813309
$229$	−2788.00	−0.804525	−0.402263	−	0.915524i	$-0.631776\pi$
$229$	−2788.00	−0.804525	−0.402263	+	0.915524i	$0.631776\pi$
$230$	432.000	0.123849
$231$	308.000	0.0877269
$232$	−2646.00	−0.748786
$233$	2694.00	0.757467	0.378733	−	0.925506i	$-0.376360\pi$
$233$	2694.00	0.757467	0.378733	+	0.925506i	$0.376360\pi$
$234$	−1254.00	−0.350327
$235$	4680.00	1.29910
$236$	−552.000	−0.152255
$237$	5408.00	1.48223
$238$	−1008.00	−0.274533
$239$	6480.00	1.75379	0.876896	−	0.480680i	$-0.159610\pi$
$239$	6480.00	1.75379	0.876896	+	0.480680i	$0.159610\pi$
$240$	−3408.00	−0.916606
$241$	−2320.00	−0.620101	−0.310050	−	0.950720i	$-0.600346\pi$
$241$	−2320.00	−0.620101	−0.310050	+	0.950720i	$0.600346\pi$
$242$	363.000	0.0964237
$243$	2860.00	0.755017
$244$	830.000	0.217768
$245$	588.000	0.153330
$246$	−2592.00	−0.671788
$247$	−2660.00	−0.685230
$248$	1470.00	0.376392
$249$	360.000	0.0916228
$250$	−3816.00	−0.965380
$251$	2088.00	0.525073	0.262537	−	0.964922i	$-0.415441\pi$
$251$	2088.00	0.525073	0.262537	+	0.964922i	$0.415441\pi$
$252$	−77.0000	−0.0192482
$253$	132.000	0.0328015
$254$	1032.00	0.254935
$255$	−2304.00	−0.565812
$256$	1513.00	0.369385
$257$	−4182.00	−1.01504	−0.507521	−	0.861639i	$-0.669438\pi$
$257$	−4182.00	−1.01504	−0.507521	+	0.861639i	$0.669438\pi$
$258$	4128.00	0.996116
$259$	−2506.00	−0.601217
$260$	456.000	0.108769
$261$	−1386.00	−0.328702
$262$	−774.000	−0.182511
$263$	3696.00	0.866559	0.433280	−	0.901260i	$-0.357356\pi$
$263$	3696.00	0.866559	0.433280	+	0.901260i	$0.357356\pi$
$264$	−924.000	−0.215410
$265$	5256.00	1.21839
$266$	−1470.00	−0.338840
$267$	4584.00	1.05070
$268$	−196.000	−0.0446739
$269$	−6060.00	−1.37355	−0.686775	−	0.726870i	$-0.740974\pi$
$269$	−6060.00	−1.37355	−0.686775	+	0.726870i	$0.740974\pi$
$270$	−5472.00	−1.23339
$271$	−8764.00	−1.96448	−0.982242	−	0.187619i	$-0.939923\pi$
$271$	−8764.00	−1.96448	−0.982242	+	0.187619i	$0.939923\pi$
$272$	3408.00	0.759707
$273$	1064.00	0.235884
$274$	−8190.00	−1.80575
$275$	209.000	0.0458297
$276$	48.0000	0.0104683
$277$	5186.00	1.12490	0.562449	−	0.826832i	$-0.309859\pi$
$277$	5186.00	1.12490	0.562449	+	0.826832i	$0.309859\pi$
$278$	5514.00	1.18960
$279$	770.000	0.165228
$280$	−1764.00	−0.376497
$281$	3006.00	0.638160	0.319080	−	0.947728i	$-0.396626\pi$
$281$	3006.00	0.638160	0.319080	+	0.947728i	$0.396626\pi$
$282$	4680.00	0.988262
$283$	−3922.00	−0.823812	−0.411906	−	0.911226i	$-0.635137\pi$
$283$	−3922.00	−0.823812	−0.411906	+	0.911226i	$0.635137\pi$
$284$	648.000	0.135393
$285$	−3360.00	−0.698348
$286$	1254.00	0.259268
$287$	−1512.00	−0.310977
$288$	495.000	0.101278
$289$	−2609.00	−0.531040
$290$	4536.00	0.918493
$291$	−280.000	−0.0564051
$292$	−16.0000	−0.00320661
$293$	−5778.00	−1.15206	−0.576031	−	0.817428i	$-0.695399\pi$
$293$	−5778.00	−1.15206	−0.576031	+	0.817428i	$0.695399\pi$
$294$	588.000	0.116642
$295$	−6624.00	−1.30734
$296$	7518.00	1.47627
$297$	−1672.00	−0.326664
$298$	−1530.00	−0.297418
$299$	456.000	0.0881979
$300$	76.0000	0.0146262
$301$	2408.00	0.461112
$302$	8592.00	1.63713
$303$	−5016.00	−0.951029
$304$	4970.00	0.937661
$305$	9960.00	1.86986
$306$	1584.00	0.295919
$307$	−610.000	−0.113402	−0.0567012	−	0.998391i	$-0.518058\pi$
$307$	−610.000	−0.113402	−0.0567012	+	0.998391i	$0.518058\pi$
$308$	77.0000	0.0142451
$309$	−2728.00	−0.502235
$310$	−2520.00	−0.461698
$311$	6882.00	1.25480	0.627399	−	0.778698i	$-0.284119\pi$
$311$	6882.00	1.25480	0.627399	+	0.778698i	$0.284119\pi$
$312$	−3192.00	−0.579203
$313$	10334.0	1.86617	0.933087	−	0.359652i	$-0.117105\pi$
$313$	10334.0	1.86617	0.933087	+	0.359652i	$0.117105\pi$
$314$	−8904.00	−1.60026
$315$	−924.000	−0.165275
$316$	1352.00	0.240683
$317$	5934.00	1.05138	0.525689	−	0.850677i	$-0.323808\pi$
$317$	5934.00	1.05138	0.525689	+	0.850677i	$0.323808\pi$
$318$	5256.00	0.926861
$319$	1386.00	0.243264
$320$	5196.00	0.907704
$321$	−1536.00	−0.267075
$322$	252.000	0.0436131
$323$	3360.00	0.578809
$324$	−311.000	−0.0533265
$325$	722.000	0.123229
$326$	4812.00	0.817522
$327$	−2584.00	−0.436989
$328$	4536.00	0.763594
$329$	2730.00	0.457477
$330$	1584.00	0.264231
$331$	−3220.00	−0.534705	−0.267352	−	0.963599i	$-0.586149\pi$
$331$	−3220.00	−0.534705	−0.267352	+	0.963599i	$0.586149\pi$
$332$	90.0000	0.0148777
$333$	3938.00	0.648051
$334$	540.000	0.0884655
$335$	−2352.00	−0.383592
$336$	−1988.00	−0.322781
$337$	−6658.00	−1.07621	−0.538107	−	0.842876i	$-0.680861\pi$
$337$	−6658.00	−1.07621	−0.538107	+	0.842876i	$0.680861\pi$
$338$	−2259.00	−0.363531
$339$	−5256.00	−0.842085
$340$	−576.000	−0.0918764
$341$	−770.000	−0.122281
$342$	2310.00	0.365235
$343$	343.000	0.0539949
$344$	−7224.00	−1.13224
$345$	576.000	0.0898864
$346$	−4878.00	−0.757927
$347$	−6888.00	−1.06561	−0.532806	−	0.846238i	$-0.678862\pi$
$347$	−6888.00	−1.06561	−0.532806	+	0.846238i	$0.678862\pi$
$348$	504.000	0.0776357
$349$	−6190.00	−0.949407	−0.474704	−	0.880146i	$-0.657445\pi$
$349$	−6190.00	−0.949407	−0.474704	+	0.880146i	$0.657445\pi$
$350$	399.000	0.0609356
$351$	−5776.00	−0.878348
$352$	−495.000	−0.0749534
$353$	−3990.00	−0.601604	−0.300802	−	0.953687i	$-0.597254\pi$
$353$	−3990.00	−0.601604	−0.300802	+	0.953687i	$0.597254\pi$
$354$	−6624.00	−0.994524
$355$	7776.00	1.16256
$356$	1146.00	0.170612
$357$	−1344.00	−0.199249
$358$	−9756.00	−1.44028
$359$	−7656.00	−1.12554	−0.562769	−	0.826614i	$-0.690264\pi$
$359$	−7656.00	−1.12554	−0.562769	+	0.826614i	$0.690264\pi$
$360$	2772.00	0.405826
$361$	−1959.00	−0.285610
$362$	5460.00	0.792738
$363$	484.000	0.0699819
$364$	266.000	0.0383027
$365$	−192.000	−0.0275335
$366$	9960.00	1.42245
$367$	8426.00	1.19846	0.599228	−	0.800578i	$-0.295474\pi$
$367$	8426.00	1.19846	0.599228	+	0.800578i	$0.295474\pi$
$368$	−852.000	−0.120689
$369$	2376.00	0.335202
$370$	−12888.0	−1.81085
$371$	3066.00	0.429053
$372$	−280.000	−0.0390251
$373$	11918.0	1.65440	0.827199	−	0.561909i	$-0.189933\pi$
$373$	11918.0	1.65440	0.827199	+	0.561909i	$0.189933\pi$
$374$	−1584.00	−0.219002
$375$	−5088.00	−0.700649
$376$	−8190.00	−1.12332
$377$	4788.00	0.654097
$378$	−3192.00	−0.434335
$379$	3908.00	0.529658	0.264829	−	0.964295i	$-0.414684\pi$
$379$	3908.00	0.529658	0.264829	+	0.964295i	$0.414684\pi$
$380$	−840.000	−0.113398
$381$	1376.00	0.185025
$382$	−3636.00	−0.487000
$383$	−3246.00	−0.433062	−0.216531	−	0.976276i	$-0.569474\pi$
$383$	−3246.00	−0.433062	−0.216531	+	0.976276i	$0.569474\pi$
$384$	6636.00	0.881880
$385$	924.000	0.122315
$386$	7566.00	0.997666
$387$	−3784.00	−0.497032
$388$	−70.0000	−0.00915905
$389$	−8166.00	−1.06435	−0.532176	−	0.846634i	$-0.678625\pi$
$389$	−8166.00	−1.06435	−0.532176	+	0.846634i	$0.678625\pi$
$390$	5472.00	0.710476
$391$	−576.000	−0.0745002
$392$	−1029.00	−0.132583
$393$	−1032.00	−0.132462
$394$	−10422.0	−1.33262
$395$	16224.0	2.06663
$396$	−121.000	−0.0153547
$397$	−2824.00	−0.357009	−0.178504	−	0.983939i	$-0.557126\pi$
$397$	−2824.00	−0.357009	−0.178504	+	0.983939i	$0.557126\pi$
$398$	−8526.00	−1.07379
$399$	−1960.00	−0.245922
$400$	−1349.00	−0.168625
$401$	−10482.0	−1.30535	−0.652676	−	0.757637i	$-0.726354\pi$
$401$	−10482.0	−1.30535	−0.652676	+	0.757637i	$0.726354\pi$
$402$	−2352.00	−0.291809
$403$	−2660.00	−0.328794
$404$	−1254.00	−0.154428
$405$	−3732.00	−0.457888
$406$	2646.00	0.323445
$407$	−3938.00	−0.479605
$408$	4032.00	0.489249
$409$	8156.00	0.986035	0.493017	−	0.870019i	$-0.335894\pi$
$409$	8156.00	0.986035	0.493017	+	0.870019i	$0.335894\pi$
$410$	−7776.00	−0.936657
$411$	−10920.0	−1.31057
$412$	−682.000	−0.0815527
$413$	−3864.00	−0.460375
$414$	−396.000	−0.0470105
$415$	1080.00	0.127747
$416$	−1710.00	−0.201538
$417$	7352.00	0.863379
$418$	−2310.00	−0.270301
$419$	−11052.0	−1.28861	−0.644303	−	0.764771i	$-0.722852\pi$
$419$	−11052.0	−1.28861	−0.644303	+	0.764771i	$0.722852\pi$
$420$	336.000	0.0390360
$421$	5006.00	0.579519	0.289760	−	0.957099i	$-0.406425\pi$
$421$	5006.00	0.579519	0.289760	+	0.957099i	$0.406425\pi$
$422$	16584.0	1.91302
$423$	−4290.00	−0.493113
$424$	−9198.00	−1.05352
$425$	−912.000	−0.104091
$426$	7776.00	0.884386
$427$	5810.00	0.658467
$428$	−384.000	−0.0433676
$429$	1672.00	0.188170
$430$	12384.0	1.38886
$431$	−9480.00	−1.05948	−0.529740	−	0.848160i	$-0.677710\pi$
$431$	−9480.00	−1.05948	−0.529740	+	0.848160i	$0.677710\pi$
$432$	10792.0	1.20192
$433$	−1942.00	−0.215535	−0.107767	−	0.994176i	$-0.534370\pi$
$433$	−1942.00	−0.215535	−0.107767	+	0.994176i	$0.534370\pi$
$434$	−1470.00	−0.162586
$435$	6048.00	0.666619
$436$	−646.000	−0.0709582
$437$	−840.000	−0.0919511
$438$	−192.000	−0.0209455
$439$	−13660.0	−1.48509	−0.742547	−	0.669794i	$-0.766382\pi$
$439$	−13660.0	−1.48509	−0.742547	+	0.669794i	$0.766382\pi$
$440$	−2772.00	−0.300341
$441$	−539.000	−0.0582011
$442$	−5472.00	−0.588861
$443$	3828.00	0.410550	0.205275	−	0.978704i	$-0.434191\pi$
$443$	3828.00	0.410550	0.205275	+	0.978704i	$0.434191\pi$
$444$	−1432.00	−0.153062
$445$	13752.0	1.46496
$446$	12102.0	1.28486
$447$	−2040.00	−0.215858
$448$	3031.00	0.319646
$449$	18270.0	1.92030	0.960150	−	0.279486i	$-0.0901639\pi$
$449$	18270.0	1.92030	0.960150	+	0.279486i	$0.0901639\pi$
$450$	−627.000	−0.0656824
$451$	−2376.00	−0.248074
$452$	−1314.00	−0.136738
$453$	11456.0	1.18819
$454$	2178.00	0.225151
$455$	3192.00	0.328886
$456$	5880.00	0.603851
$457$	10154.0	1.03935	0.519676	−	0.854363i	$-0.326053\pi$
$457$	10154.0	1.03935	0.519676	+	0.854363i	$0.326053\pi$
$458$	−8364.00	−0.853328
$459$	7296.00	0.741935
$460$	144.000	0.0145957
$461$	−17190.0	−1.73670	−0.868349	−	0.495953i	$-0.834819\pi$
$461$	−17190.0	−1.73670	−0.868349	+	0.495953i	$0.834819\pi$
$462$	924.000	0.0930484
$463$	4448.00	0.446471	0.223236	−	0.974765i	$-0.428338\pi$
$463$	4448.00	0.446471	0.223236	+	0.974765i	$0.428338\pi$
$464$	−8946.00	−0.895060
$465$	−3360.00	−0.335089
$466$	8082.00	0.803415
$467$	11100.0	1.09989	0.549943	−	0.835202i	$-0.314649\pi$
$467$	11100.0	1.09989	0.549943	+	0.835202i	$0.314649\pi$
$468$	−418.000	−0.0412864
$469$	−1372.00	−0.135081
$470$	14040.0	1.37791
$471$	−11872.0	−1.16143
$472$	11592.0	1.13043
$473$	3784.00	0.367840
$474$	16224.0	1.57214
$475$	−1330.00	−0.128473
$476$	−336.000	−0.0323541
$477$	−4818.00	−0.462476
$478$	19440.0	1.86018
$479$	−15816.0	−1.50867	−0.754333	−	0.656491i	$-0.772040\pi$
$479$	−15816.0	−1.50867	−0.754333	+	0.656491i	$0.772040\pi$
$480$	−2160.00	−0.205396
$481$	−13604.0	−1.28958
$482$	−6960.00	−0.657716
$483$	336.000	0.0316533
$484$	121.000	0.0113636
$485$	−840.000	−0.0786442
$486$	8580.00	0.800816
$487$	−1924.00	−0.179024	−0.0895121	−	0.995986i	$-0.528531\pi$
$487$	−1924.00	−0.179024	−0.0895121	+	0.995986i	$0.528531\pi$
$488$	−17430.0	−1.61684
$489$	6416.00	0.593337
$490$	1764.00	0.162631
$491$	13068.0	1.20112	0.600561	−	0.799579i	$-0.294944\pi$
$491$	13068.0	1.20112	0.600561	+	0.799579i	$0.294944\pi$
$492$	−864.000	−0.0791710
$493$	−6048.00	−0.552512
$494$	−7980.00	−0.726796
$495$	−1452.00	−0.131844
$496$	4970.00	0.449919
$497$	4536.00	0.409391
$498$	1080.00	0.0971806
$499$	17876.0	1.60369	0.801843	−	0.597534i	$-0.203853\pi$
$499$	17876.0	1.60369	0.801843	+	0.597534i	$0.203853\pi$
$500$	−1272.00	−0.113771
$501$	720.000	0.0642060
$502$	6264.00	0.556924
$503$	−3852.00	−0.341456	−0.170728	−	0.985318i	$-0.554612\pi$
$503$	−3852.00	−0.341456	−0.170728	+	0.985318i	$0.554612\pi$
$504$	1617.00	0.142911
$505$	−15048.0	−1.32599
$506$	396.000	0.0347912
$507$	−3012.00	−0.263841
$508$	344.000	0.0300444
$509$	15132.0	1.31771	0.658855	−	0.752270i	$-0.271041\pi$
$509$	15132.0	1.31771	0.658855	+	0.752270i	$0.271041\pi$
$510$	−6912.00	−0.600134
$511$	−112.000	−0.00969587
$512$	−8733.00	−0.753804
$513$	10640.0	0.915726
$514$	−12546.0	−1.07662
$515$	−8184.00	−0.700253
$516$	1376.00	0.117393
$517$	4290.00	0.364940
$518$	−7518.00	−0.637687
$519$	−6504.00	−0.550085
$520$	−9576.00	−0.807568
$521$	−3054.00	−0.256810	−0.128405	−	0.991722i	$-0.540986\pi$
$521$	−3054.00	−0.256810	−0.128405	+	0.991722i	$0.540986\pi$
$522$	−4158.00	−0.348641
$523$	−11770.0	−0.984065	−0.492033	−	0.870577i	$-0.663746\pi$
$523$	−11770.0	−0.984065	−0.492033	+	0.870577i	$0.663746\pi$
$524$	−258.000	−0.0215091
$525$	532.000	0.0442255
$526$	11088.0	0.919125
$527$	3360.00	0.277730
$528$	−3124.00	−0.257490
$529$	−12023.0	−0.988165
$530$	15768.0	1.29230
$531$	6072.00	0.496238
$532$	−490.000	−0.0399327
$533$	−8208.00	−0.667032
$534$	13752.0	1.11443
$535$	−4608.00	−0.372376
$536$	4116.00	0.331687
$537$	−13008.0	−1.04532
$538$	−18180.0	−1.45687
$539$	539.000	0.0430730
$540$	−1824.00	−0.145356
$541$	10694.0	0.849854	0.424927	−	0.905228i	$-0.360300\pi$
$541$	10694.0	0.849854	0.424927	+	0.905228i	$0.360300\pi$
$542$	−26292.0	−2.08365
$543$	7280.00	0.575349
$544$	2160.00	0.170238
$545$	−7752.00	−0.609283
$546$	3192.00	0.250192
$547$	5636.00	0.440545	0.220272	−	0.975438i	$-0.429305\pi$
$547$	5636.00	0.440545	0.220272	+	0.975438i	$0.429305\pi$
$548$	−2730.00	−0.212810
$549$	−9130.00	−0.709761
$550$	627.000	0.0486098
$551$	−8820.00	−0.681932
$552$	−1008.00	−0.0777234
$553$	9464.00	0.727758
$554$	15558.0	1.19313
$555$	−17184.0	−1.31427
$556$	1838.00	0.140195
$557$	15126.0	1.15064	0.575322	−	0.817927i	$-0.304877\pi$
$557$	15126.0	1.15064	0.575322	+	0.817927i	$0.304877\pi$
$558$	2310.00	0.175251
$559$	13072.0	0.989064
$560$	−5964.00	−0.450045
$561$	−2112.00	−0.158946
$562$	9018.00	0.676871
$563$	3246.00	0.242989	0.121494	−	0.992592i	$-0.461231\pi$
$563$	3246.00	0.242989	0.121494	+	0.992592i	$0.461231\pi$
$564$	1560.00	0.116468
$565$	−15768.0	−1.17410
$566$	−11766.0	−0.873784
$567$	−2177.00	−0.161244
$568$	−13608.0	−1.00524
$569$	−1050.00	−0.0773608	−0.0386804	−	0.999252i	$-0.512315\pi$
$569$	−1050.00	−0.0773608	−0.0386804	+	0.999252i	$0.512315\pi$
$570$	−10080.0	−0.740710
$571$	6860.00	0.502771	0.251385	−	0.967887i	$-0.419114\pi$
$571$	6860.00	0.502771	0.251385	+	0.967887i	$0.419114\pi$
$572$	418.000	0.0305550
$573$	−4848.00	−0.353452
$574$	−4536.00	−0.329841
$575$	228.000	0.0165361
$576$	−4763.00	−0.344546
$577$	−12634.0	−0.911543	−0.455771	−	0.890097i	$-0.650637\pi$
$577$	−12634.0	−0.911543	−0.455771	+	0.890097i	$0.650637\pi$
$578$	−7827.00	−0.563253
$579$	10088.0	0.724081
$580$	1512.00	0.108245
$581$	630.000	0.0449859
$582$	−840.000	−0.0598267
$583$	4818.00	0.342266
$584$	336.000	0.0238078
$585$	−5016.00	−0.354506
$586$	−17334.0	−1.22195
$587$	18144.0	1.27578	0.637890	−	0.770127i	$-0.279807\pi$
$587$	18144.0	1.27578	0.637890	+	0.770127i	$0.279807\pi$
$588$	196.000	0.0137464
$589$	4900.00	0.342786
$590$	−19872.0	−1.38664
$591$	−13896.0	−0.967183
$592$	25418.0	1.76465
$593$	−10896.0	−0.754545	−0.377272	−	0.926102i	$-0.623138\pi$
$593$	−10896.0	−0.754545	−0.377272	+	0.926102i	$0.623138\pi$
$594$	−5016.00	−0.346480
$595$	−4032.00	−0.277808
$596$	−510.000	−0.0350510
$597$	−11368.0	−0.779332
$598$	1368.00	0.0935480
$599$	−20280.0	−1.38334	−0.691668	−	0.722216i	$-0.743124\pi$
$599$	−20280.0	−1.38334	−0.691668	+	0.722216i	$0.743124\pi$
$600$	−1596.00	−0.108594
$601$	12332.0	0.836993	0.418496	−	0.908218i	$-0.362557\pi$
$601$	12332.0	0.836993	0.418496	+	0.908218i	$0.362557\pi$
$602$	7224.00	0.489083
$603$	2156.00	0.145604
$604$	2864.00	0.192938
$605$	1452.00	0.0975739
$606$	−15048.0	−1.00872
$607$	21800.0	1.45772	0.728859	−	0.684664i	$-0.240051\pi$
$607$	21800.0	1.45772	0.728859	+	0.684664i	$0.240051\pi$
$608$	3150.00	0.210114
$609$	3528.00	0.234748
$610$	29880.0	1.98329
$611$	14820.0	0.981265
$612$	528.000	0.0348744
$613$	18542.0	1.22170	0.610852	−	0.791745i	$-0.290827\pi$
$613$	18542.0	1.22170	0.610852	+	0.791745i	$0.290827\pi$
$614$	−1830.00	−0.120281
$615$	−10368.0	−0.679802
$616$	−1617.00	−0.105764
$617$	10098.0	0.658882	0.329441	−	0.944176i	$-0.393140\pi$
$617$	10098.0	0.658882	0.329441	+	0.944176i	$0.393140\pi$
$618$	−8184.00	−0.532700
$619$	−124.000	−0.00805167	−0.00402583	−	0.999992i	$-0.501281\pi$
$619$	−124.000	−0.00805167	−0.00402583	+	0.999992i	$0.501281\pi$
$620$	−840.000	−0.0544116
$621$	−1824.00	−0.117866
$622$	20646.0	1.33092
$623$	8022.00	0.515882
$624$	−10792.0	−0.692349
$625$	−17639.0	−1.12890
$626$	31002.0	1.97938
$627$	−3080.00	−0.196178
$628$	−2968.00	−0.188593
$629$	17184.0	1.08930
$630$	−2772.00	−0.175300
$631$	−14308.0	−0.902682	−0.451341	−	0.892351i	$-0.649054\pi$
$631$	−14308.0	−0.902682	−0.451341	+	0.892351i	$0.649054\pi$
$632$	−28392.0	−1.78698
$633$	22112.0	1.38843
$634$	17802.0	1.11515
$635$	4128.00	0.257976
$636$	1752.00	0.109232
$637$	1862.00	0.115817
$638$	4158.00	0.258020
$639$	−7128.00	−0.441282
$640$	19908.0	1.22958
$641$	−678.000	−0.0417775	−0.0208888	−	0.999782i	$-0.506650\pi$
$641$	−678.000	−0.0417775	−0.0208888	+	0.999782i	$0.506650\pi$
$642$	−4608.00	−0.283276
$643$	17408.0	1.06766	0.533829	−	0.845592i	$-0.320752\pi$
$643$	17408.0	1.06766	0.533829	+	0.845592i	$0.320752\pi$
$644$	84.0000	0.00513985
$645$	16512.0	1.00800
$646$	10080.0	0.613920
$647$	−28686.0	−1.74306	−0.871532	−	0.490338i	$-0.836873\pi$
$647$	−28686.0	−1.74306	−0.871532	+	0.490338i	$0.836873\pi$
$648$	6531.00	0.395929
$649$	−6072.00	−0.367252
$650$	2166.00	0.130704
$651$	−1960.00	−0.118001
$652$	1604.00	0.0963458
$653$	9858.00	0.590771	0.295385	−	0.955378i	$-0.404552\pi$
$653$	9858.00	0.590771	0.295385	+	0.955378i	$0.404552\pi$
$654$	−7752.00	−0.463497
$655$	−3096.00	−0.184688
$656$	15336.0	0.912759
$657$	176.000	0.0104512
$658$	8190.00	0.485227
$659$	−22824.0	−1.34916	−0.674580	−	0.738201i	$-0.735675\pi$
$659$	−22824.0	−1.34916	−0.674580	+	0.738201i	$0.735675\pi$
$660$	528.000	0.0311400
$661$	24212.0	1.42472	0.712358	−	0.701816i	$-0.247627\pi$
$661$	24212.0	1.42472	0.712358	+	0.701816i	$0.247627\pi$
$662$	−9660.00	−0.567140
$663$	−7296.00	−0.427380
$664$	−1890.00	−0.110461
$665$	−5880.00	−0.342882
$666$	11814.0	0.687362
$667$	1512.00	0.0877734
$668$	180.000	0.0104258
$669$	16136.0	0.932517
$670$	−7056.00	−0.406861
$671$	9130.00	0.525275
$672$	−1260.00	−0.0723297
$673$	−17458.0	−0.999935	−0.499968	−	0.866044i	$-0.666655\pi$
$673$	−17458.0	−0.999935	−0.499968	+	0.866044i	$0.666655\pi$
$674$	−19974.0	−1.14150
$675$	−2888.00	−0.164680
$676$	−753.000	−0.0428425
$677$	14574.0	0.827362	0.413681	−	0.910422i	$-0.364243\pi$
$677$	14574.0	0.827362	0.413681	+	0.910422i	$0.364243\pi$
$678$	−15768.0	−0.893166
$679$	−490.000	−0.0276944
$680$	12096.0	0.682148
$681$	2904.00	0.163409
$682$	−2310.00	−0.129699
$683$	−27588.0	−1.54557	−0.772786	−	0.634667i	$-0.781137\pi$
$683$	−27588.0	−1.54557	−0.772786	+	0.634667i	$0.781137\pi$
$684$	770.000	0.0430434
$685$	−32760.0	−1.82729
$686$	1029.00	0.0572703
$687$	−11152.0	−0.619324
$688$	−24424.0	−1.35342
$689$	16644.0	0.920299
$690$	1728.00	0.0953389
$691$	10424.0	0.573875	0.286938	−	0.957949i	$-0.407363\pi$
$691$	10424.0	0.573875	0.286938	+	0.957949i	$0.407363\pi$
$692$	−1626.00	−0.0893226
$693$	−847.000	−0.0464284
$694$	−20664.0	−1.13025
$695$	22056.0	1.20379
$696$	−10584.0	−0.576416
$697$	10368.0	0.563438
$698$	−18570.0	−1.00700
$699$	10776.0	0.583098
$700$	133.000	0.00718132
$701$	3978.00	0.214332	0.107166	−	0.994241i	$-0.465822\pi$
$701$	3978.00	0.214332	0.107166	+	0.994241i	$0.465822\pi$
$702$	−17328.0	−0.931629
$703$	25060.0	1.34446
$704$	4763.00	0.254989
$705$	18720.0	1.00005
$706$	−11970.0	−0.638098
$707$	−8778.00	−0.466946
$708$	−2208.00	−0.117206
$709$	18794.0	0.995520	0.497760	−	0.867315i	$-0.334156\pi$
$709$	18794.0	0.995520	0.497760	+	0.867315i	$0.334156\pi$
$710$	23328.0	1.23308
$711$	−14872.0	−0.784449
$712$	−24066.0	−1.26673
$713$	−840.000	−0.0441210
$714$	−4032.00	−0.211336
$715$	5016.00	0.262361
$716$	−3252.00	−0.169739
$717$	25920.0	1.35007
$718$	−22968.0	−1.19381
$719$	33906.0	1.75867	0.879333	−	0.476208i	$-0.157989\pi$
$719$	33906.0	1.75867	0.879333	+	0.476208i	$0.157989\pi$
$720$	9372.00	0.485103
$721$	−4774.00	−0.246592
$722$	−5877.00	−0.302935
$723$	−9280.00	−0.477354
$724$	1820.00	0.0934251
$725$	2394.00	0.122636
$726$	1452.00	0.0742270
$727$	−2446.00	−0.124783	−0.0623914	−	0.998052i	$-0.519873\pi$
$727$	−2446.00	−0.124783	−0.0623914	+	0.998052i	$0.519873\pi$
$728$	−5586.00	−0.284383
$729$	19837.0	1.00782
$730$	−576.000	−0.0292037
$731$	−16512.0	−0.835456
$732$	3320.00	0.167638
$733$	−20410.0	−1.02846	−0.514230	−	0.857653i	$-0.671922\pi$
$733$	−20410.0	−1.02846	−0.514230	+	0.857653i	$0.671922\pi$
$734$	25278.0	1.27116
$735$	2352.00	0.118034
$736$	−540.000	−0.0270444
$737$	−2156.00	−0.107758
$738$	7128.00	0.355536
$739$	14564.0	0.724960	0.362480	−	0.931992i	$-0.381930\pi$
$739$	14564.0	0.724960	0.362480	+	0.931992i	$0.381930\pi$
$740$	−4296.00	−0.213411
$741$	−10640.0	−0.527490
$742$	9198.00	0.455080
$743$	−7416.00	−0.366173	−0.183087	−	0.983097i	$-0.558609\pi$
$743$	−7416.00	−0.366173	−0.183087	+	0.983097i	$0.558609\pi$
$744$	5880.00	0.289746
$745$	−6120.00	−0.300966
$746$	35754.0	1.75475
$747$	−990.000	−0.0484902
$748$	−528.000	−0.0258096
$749$	−2688.00	−0.131131
$750$	−15264.0	−0.743150
$751$	−17980.0	−0.873635	−0.436817	−	0.899550i	$-0.643894\pi$
$751$	−17980.0	−0.873635	−0.436817	+	0.899550i	$0.643894\pi$
$752$	−27690.0	−1.34275
$753$	8352.00	0.404202
$754$	14364.0	0.693775
$755$	34368.0	1.65666
$756$	−1064.00	−0.0511869
$757$	3170.00	0.152200	0.0761001	−	0.997100i	$-0.475753\pi$
$757$	3170.00	0.152200	0.0761001	+	0.997100i	$0.475753\pi$
$758$	11724.0	0.561787
$759$	528.000	0.0252506
$760$	17640.0	0.841934
$761$	−27492.0	−1.30957	−0.654786	−	0.755814i	$-0.727241\pi$
$761$	−27492.0	−1.30957	−0.654786	+	0.755814i	$0.727241\pi$
$762$	4128.00	0.196249
$763$	−4522.00	−0.214558
$764$	−1212.00	−0.0573935
$765$	6336.00	0.299449
$766$	−9738.00	−0.459332
$767$	−20976.0	−0.987483
$768$	6052.00	0.284353
$769$	2108.00	0.0988510	0.0494255	−	0.998778i	$-0.484261\pi$
$769$	2108.00	0.0988510	0.0494255	+	0.998778i	$0.484261\pi$
$770$	2772.00	0.129735
$771$	−16728.0	−0.781380
$772$	2522.00	0.117576
$773$	32280.0	1.50198	0.750990	−	0.660313i	$-0.229577\pi$
$773$	32280.0	1.50198	0.750990	+	0.660313i	$0.229577\pi$
$774$	−11352.0	−0.527182
$775$	−1330.00	−0.0616452
$776$	1470.00	0.0680025
$777$	−10024.0	−0.462817
$778$	−24498.0	−1.12891
$779$	15120.0	0.695417
$780$	1824.00	0.0837304
$781$	7128.00	0.326581
$782$	−1728.00	−0.0790194
$783$	−19152.0	−0.874121
$784$	−3479.00	−0.158482
$785$	−35616.0	−1.61935
$786$	−3096.00	−0.140497
$787$	9578.00	0.433823	0.216912	−	0.976191i	$-0.430402\pi$
$787$	9578.00	0.433823	0.216912	+	0.976191i	$0.430402\pi$
$788$	−3474.00	−0.157051
$789$	14784.0	0.667078
$790$	48672.0	2.19199
$791$	−9198.00	−0.413455
$792$	2541.00	0.114003
$793$	31540.0	1.41238
$794$	−8472.00	−0.378665
$795$	21024.0	0.937918
$796$	−2842.00	−0.126548
$797$	11952.0	0.531194	0.265597	−	0.964084i	$-0.414431\pi$
$797$	11952.0	0.531194	0.265597	+	0.964084i	$0.414431\pi$
$798$	−5880.00	−0.260839
$799$	−18720.0	−0.828869
$800$	−855.000	−0.0377860
$801$	−12606.0	−0.556069
$802$	−31446.0	−1.38453
$803$	−176.000	−0.00773463
$804$	−784.000	−0.0343900
$805$	1008.00	0.0441333
$806$	−7980.00	−0.348739
$807$	−24240.0	−1.05736
$808$	26334.0	1.14657
$809$	−9030.00	−0.392433	−0.196216	−	0.980561i	$-0.562865\pi$
$809$	−9030.00	−0.392433	−0.196216	+	0.980561i	$0.562865\pi$
$810$	−11196.0	−0.485663
$811$	−37762.0	−1.63502	−0.817511	−	0.575913i	$-0.804647\pi$
$811$	−37762.0	−1.63502	−0.817511	+	0.575913i	$0.804647\pi$
$812$	882.000	0.0381184
$813$	−35056.0	−1.51226
$814$	−11814.0	−0.508698
$815$	19248.0	0.827274
$816$	13632.0	0.584823
$817$	−24080.0	−1.03115
$818$	24468.0	1.04585
$819$	−2926.00	−0.124838
$820$	−2592.00	−0.110386
$821$	−14334.0	−0.609330	−0.304665	−	0.952460i	$-0.598545\pi$
$821$	−14334.0	−0.609330	−0.304665	+	0.952460i	$0.598545\pi$
$822$	−32760.0	−1.39007
$823$	13988.0	0.592456	0.296228	−	0.955117i	$-0.404271\pi$
$823$	13988.0	0.592456	0.296228	+	0.955117i	$0.404271\pi$
$824$	14322.0	0.605498
$825$	836.000	0.0352797
$826$	−11592.0	−0.488302
$827$	−22284.0	−0.936990	−0.468495	−	0.883466i	$-0.655204\pi$
$827$	−22284.0	−0.936990	−0.468495	+	0.883466i	$0.655204\pi$
$828$	−132.000	−0.00554024
$829$	−12868.0	−0.539112	−0.269556	−	0.962985i	$-0.586877\pi$
$829$	−12868.0	−0.539112	−0.269556	+	0.962985i	$0.586877\pi$
$830$	3240.00	0.135496
$831$	20744.0	0.865946
$832$	16454.0	0.685625
$833$	−2352.00	−0.0978295
$834$	22056.0	0.915752
$835$	2160.00	0.0895208
$836$	−770.000	−0.0318553
$837$	10640.0	0.439393
$838$	−33156.0	−1.36677
$839$	2826.00	0.116286	0.0581432	−	0.998308i	$-0.481482\pi$
$839$	2826.00	0.116286	0.0581432	+	0.998308i	$0.481482\pi$
$840$	−7056.00	−0.289828
$841$	−8513.00	−0.349051
$842$	15018.0	0.614673
$843$	12024.0	0.491256
$844$	5528.00	0.225452
$845$	−9036.00	−0.367867
$846$	−12870.0	−0.523026
$847$	847.000	0.0343604
$848$	−31098.0	−1.25933
$849$	−15688.0	−0.634171
$850$	−2736.00	−0.110405
$851$	−4296.00	−0.173049
$852$	2592.00	0.104226
$853$	−17962.0	−0.720993	−0.360497	−	0.932761i	$-0.617393\pi$
$853$	−17962.0	−0.720993	−0.360497	+	0.932761i	$0.617393\pi$
$854$	17430.0	0.698410
$855$	9240.00	0.369592
$856$	8064.00	0.321988
$857$	47148.0	1.87928	0.939641	−	0.342161i	$-0.111159\pi$
$857$	47148.0	1.87928	0.939641	+	0.342161i	$0.111159\pi$
$858$	5016.00	0.199584
$859$	34904.0	1.38639	0.693195	−	0.720750i	$-0.256202\pi$
$859$	34904.0	1.38639	0.693195	+	0.720750i	$0.256202\pi$
$860$	4128.00	0.163679
$861$	−6048.00	−0.239391
$862$	−28440.0	−1.12375
$863$	−44052.0	−1.73760	−0.868799	−	0.495164i	$-0.835108\pi$
$863$	−44052.0	−1.73760	−0.868799	+	0.495164i	$0.835108\pi$
$864$	6840.00	0.269330
$865$	−19512.0	−0.766969
$866$	−5826.00	−0.228609
$867$	−10436.0	−0.408795
$868$	−490.000	−0.0191609
$869$	14872.0	0.580550
$870$	18144.0	0.707057
$871$	−7448.00	−0.289743
$872$	13566.0	0.526838
$873$	770.000	0.0298517
$874$	−2520.00	−0.0975289
$875$	−8904.00	−0.344012
$876$	−64.0000	−0.00246845
$877$	−36214.0	−1.39437	−0.697184	−	0.716893i	$-0.745564\pi$
$877$	−36214.0	−1.39437	−0.697184	+	0.716893i	$0.745564\pi$
$878$	−40980.0	−1.57518
$879$	−23112.0	−0.886858
$880$	−9372.00	−0.359011
$881$	17046.0	0.651866	0.325933	−	0.945393i	$-0.394322\pi$
$881$	17046.0	0.651866	0.325933	+	0.945393i	$0.394322\pi$
$882$	−1617.00	−0.0617315
$883$	41276.0	1.57310	0.786550	−	0.617526i	$-0.211865\pi$
$883$	41276.0	1.57310	0.786550	+	0.617526i	$0.211865\pi$
$884$	−1824.00	−0.0693979
$885$	−26496.0	−1.00639
$886$	11484.0	0.435454
$887$	6456.00	0.244387	0.122193	−	0.992506i	$-0.461007\pi$
$887$	6456.00	0.244387	0.122193	+	0.992506i	$0.461007\pi$
$888$	30072.0	1.13643
$889$	2408.00	0.0908456
$890$	41256.0	1.55383
$891$	−3421.00	−0.128628
$892$	4034.00	0.151422
$893$	−27300.0	−1.02302
$894$	−6120.00	−0.228952
$895$	−39024.0	−1.45746
$896$	11613.0	0.432995
$897$	1824.00	0.0678947
$898$	54810.0	2.03679
$899$	−8820.00	−0.327212
$900$	−209.000	−0.00774074
$901$	−21024.0	−0.777371
$902$	−7128.00	−0.263122
$903$	9632.00	0.354964
$904$	27594.0	1.01522
$905$	21840.0	0.802195
$906$	34368.0	1.26027
$907$	−11500.0	−0.421005	−0.210502	−	0.977593i	$-0.567510\pi$
$907$	−11500.0	−0.421005	−0.210502	+	0.977593i	$0.567510\pi$
$908$	726.000	0.0265343
$909$	13794.0	0.503320
$910$	9576.00	0.348837
$911$	27396.0	0.996345	0.498172	−	0.867078i	$-0.334005\pi$
$911$	27396.0	0.996345	0.498172	+	0.867078i	$0.334005\pi$
$912$	19880.0	0.721812
$913$	990.000	0.0358863
$914$	30462.0	1.10240
$915$	39840.0	1.43942
$916$	−2788.00	−0.100566
$917$	−1806.00	−0.0650375
$918$	21888.0	0.786941
$919$	8840.00	0.317307	0.158653	−	0.987334i	$-0.449285\pi$
$919$	8840.00	0.317307	0.158653	+	0.987334i	$0.449285\pi$
$920$	−3024.00	−0.108368
$921$	−2440.00	−0.0872972
$922$	−51570.0	−1.84205
$923$	24624.0	0.878124
$924$	308.000	0.0109659
$925$	−6802.00	−0.241782
$926$	13344.0	0.473554
$927$	7502.00	0.265802
$928$	−5670.00	−0.200568
$929$	2874.00	0.101499	0.0507497	−	0.998711i	$-0.483839\pi$
$929$	2874.00	0.101499	0.0507497	+	0.998711i	$0.483839\pi$
$930$	−10080.0	−0.355415
$931$	−3430.00	−0.120745
$932$	2694.00	0.0946834
$933$	27528.0	0.965945
$934$	33300.0	1.16661
$935$	−6336.00	−0.221614
$936$	8778.00	0.306536
$937$	7832.00	0.273063	0.136532	−	0.990636i	$-0.456404\pi$
$937$	7832.00	0.273063	0.136532	+	0.990636i	$0.456404\pi$
$938$	−4116.00	−0.143275
$939$	41336.0	1.43658
$940$	4680.00	0.162388
$941$	16926.0	0.586368	0.293184	−	0.956056i	$-0.405285\pi$
$941$	16926.0	0.586368	0.293184	+	0.956056i	$0.405285\pi$
$942$	−35616.0	−1.23188
$943$	−2592.00	−0.0895092
$944$	39192.0	1.35126
$945$	−12768.0	−0.439516
$946$	11352.0	0.390154
$947$	−17988.0	−0.617245	−0.308623	−	0.951185i	$-0.599868\pi$
$947$	−17988.0	−0.617245	−0.308623	+	0.951185i	$0.599868\pi$
$948$	5408.00	0.185278
$949$	−608.000	−0.0207972
$950$	−3990.00	−0.136266
$951$	23736.0	0.809351
$952$	7056.00	0.240217
$953$	−29142.0	−0.990558	−0.495279	−	0.868734i	$-0.664934\pi$
$953$	−29142.0	−0.990558	−0.495279	+	0.868734i	$0.664934\pi$
$954$	−14454.0	−0.490530
$955$	−14544.0	−0.492809
$956$	6480.00	0.219224
$957$	5544.00	0.187264
$958$	−47448.0	−1.60018
$959$	−19110.0	−0.643477
$960$	20784.0	0.698751
$961$	−24891.0	−0.835521
$962$	−40812.0	−1.36781
$963$	4224.00	0.141346
$964$	−2320.00	−0.0775126
$965$	30264.0	1.00957
$966$	1008.00	0.0335734
$967$	31160.0	1.03623	0.518117	−	0.855310i	$-0.326633\pi$
$967$	31160.0	1.03623	0.518117	+	0.855310i	$0.326633\pi$
$968$	−2541.00	−0.0843707
$969$	13440.0	0.445568
$970$	−2520.00	−0.0834148
$971$	−33036.0	−1.09184	−0.545920	−	0.837838i	$-0.683820\pi$
$971$	−33036.0	−1.09184	−0.545920	+	0.837838i	$0.683820\pi$
$972$	2860.00	0.0943771
$973$	12866.0	0.423911
$974$	−5772.00	−0.189884
$975$	2888.00	0.0948616
$976$	−58930.0	−1.93269
$977$	12786.0	0.418690	0.209345	−	0.977842i	$-0.432867\pi$
$977$	12786.0	0.418690	0.209345	+	0.977842i	$0.432867\pi$
$978$	19248.0	0.629328
$979$	12606.0	0.411532
$980$	588.000	0.0191663
$981$	7106.00	0.231271
$982$	39204.0	1.27398
$983$	24918.0	0.808505	0.404253	−	0.914647i	$-0.367532\pi$
$983$	24918.0	0.808505	0.404253	+	0.914647i	$0.367532\pi$
$984$	18144.0	0.587815
$985$	−41688.0	−1.34852
$986$	−18144.0	−0.586027
$987$	10920.0	0.352166
$988$	−2660.00	−0.0856537
$989$	4128.00	0.132723
$990$	−4356.00	−0.139841
$991$	−7648.00	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$991$	−7648.00	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$992$	3150.00	0.100819
$993$	−12880.0	−0.411616
$994$	13608.0	0.434225
$995$	−34104.0	−1.08660
$996$	360.000	0.0114528
$997$	−31750.0	−1.00856	−0.504279	−	0.863541i	$-0.668242\pi$
$997$	−31750.0	−1.00856	−0.504279	+	0.863541i	$0.668242\pi$
$998$	53628.0	1.70097
$999$	54416.0	1.72337

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.a.a.1.1	✓	1
3.2	odd	2		693.4.a.b.1.1		1
4.3	odd	2		1232.4.a.d.1.1		1
5.4	even	2		1925.4.a.a.1.1		1
7.6	odd	2		539.4.a.a.1.1		1
11.10	odd	2		847.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.a.1.1	✓	1	1.1	even	1	trivial
539.4.a.a.1.1		1	7.6	odd	2
693.4.a.b.1.1		1	3.2	odd	2
847.4.a.a.1.1		1	11.10	odd	2
1232.4.a.d.1.1		1	4.3	odd	2
1925.4.a.a.1.1		1	5.4	even	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:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	77.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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