LMFDB - Embedded newform 847.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,4,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, 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("847.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	847.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:	$49.9746177749$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

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	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	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
$11$	0	0
$11$	0	0
$12$	4.00000	0.0962250
$13$	−38.0000	−0.810716	−0.405358	−	0.914158i	$-0.632853\pi$
$13$	−38.0000	−0.810716	−0.405358	+	0.914158i	$0.632853\pi$
$14$	21.0000	0.400892
$15$	48.0000	0.826236
$16$	−71.0000	−1.10938
$17$	48.0000	0.684806	0.342403	−	0.939553i	$-0.388759\pi$
$17$	48.0000	0.684806	0.342403	+	0.939553i	$0.388759\pi$
$18$	33.0000	0.432121
$19$	70.0000	0.845216	0.422608	−	0.906313i	$-0.361115\pi$
$19$	70.0000	0.845216	0.422608	+	0.906313i	$0.361115\pi$
$20$	12.0000	0.134164
$21$	−28.0000	−0.290957
$22$	0	0
$23$	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	$-0.632174\pi$
$29$	−126.000	−0.806814	−0.403407	+	0.915021i	$0.632174\pi$
$30$	−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
$33$	0	0
$34$	−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	$-0.365045\pi$
$41$	216.000	0.822769	0.411385	+	0.911462i	$0.365045\pi$
$42$	84.0000	0.308607
$43$	−344.000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−344.000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	−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
$55$	0	0
$56$	−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	$-0.836574\pi$
$61$	−830.000	−1.74214	−0.871071	+	0.491158i	$0.836574\pi$
$62$	210.000	0.430162
$63$	77.0000	0.153986
$64$	433.000	0.845703
$65$	−456.000	−0.870151
$66$	0	0
$67$	−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	$-0.495917\pi$
$73$	16.0000	0.0256529	0.0128264	+	0.999918i	$0.495917\pi$
$74$	1074.00	1.68716
$75$	76.0000	0.117010
$76$	70.0000	0.105652
$77$	0	0
$78$	456.000	0.661947
$79$	−1352.00	−1.92547	−0.962733	−	0.270452i	$-0.912827\pi$
$79$	−1352.00	−1.92547	−0.962733	+	0.270452i	$0.912827\pi$
$80$	−852.000	−1.19071
$81$	−311.000	−0.426612
$82$	−648.000	−0.872678
$83$	−90.0000	−0.119021	−0.0595107	−	0.998228i	$-0.518954\pi$
$83$	−90.0000	−0.119021	−0.0595107	+	0.998228i	$0.518954\pi$
$84$	−28.0000	−0.0363696
$85$	576.000	0.735011
$86$	1032.00	1.29399
$87$	−504.000	−0.621086
$88$	0	0
$89$	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
$99$	0	0
$100$	19.0000	0.0190000
$101$	1254.00	1.23542	0.617711	−	0.786405i	$-0.288060\pi$
$101$	1254.00	1.23542	0.617711	+	0.786405i	$0.288060\pi$
$102$	−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	$-0.444502\pi$
$107$	384.000	0.346941	0.173470	+	0.984839i	$0.444502\pi$
$108$	−152.000	−0.135428
$109$	646.000	0.567666	0.283833	−	0.958874i	$-0.408394\pi$
$109$	646.000	0.567666	0.283833	+	0.958874i	$0.408394\pi$
$110$	0	0
$111$	−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
$121$	0	0
$122$	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	$-0.538346\pi$
$127$	−344.000	−0.240355	−0.120177	+	0.992752i	$0.538346\pi$
$128$	−1659.00	−1.14560
$129$	−1376.00	−0.939148
$130$	1368.00	0.922935
$131$	258.000	0.172073	0.0860365	−	0.996292i	$-0.472580\pi$
$131$	258.000	0.172073	0.0860365	+	0.996292i	$0.472580\pi$
$132$	0	0
$133$	−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	$-0.689499\pi$
$139$	−1838.00	−1.12156	−0.560781	+	0.827964i	$0.689499\pi$
$140$	−84.0000	−0.0507093
$141$	1560.00	0.931743
$142$	−1944.00	−1.14885
$143$	0	0
$144$	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	$-0.455224\pi$
$149$	510.000	0.280408	0.140204	+	0.990123i	$0.455224\pi$
$150$	−228.000	−0.124107
$151$	−2864.00	−1.54350	−0.771752	−	0.635924i	$-0.780619\pi$
$151$	−2864.00	−1.54350	−0.771752	+	0.635924i	$0.780619\pi$
$152$	1470.00	0.784426
$153$	−528.000	−0.278995
$154$	0	0
$155$	−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
$165$	0	0
$166$	270.000	0.126241
$167$	−180.000	−0.0834061	−0.0417030	−	0.999130i	$-0.513278\pi$
$167$	−180.000	−0.0834061	−0.0417030	+	0.999130i	$0.513278\pi$
$168$	−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	$-0.383701\pi$
$173$	1626.00	0.714581	0.357290	+	0.933993i	$0.383701\pi$
$174$	1512.00	0.658761
$175$	−133.000	−0.0574506
$176$	0	0
$177$	−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
$187$	0	0
$188$	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	$-0.655856\pi$
$193$	−2522.00	−0.940609	−0.470304	+	0.882504i	$0.655856\pi$
$194$	210.000	0.0777171
$195$	−1824.00	−0.669843
$196$	49.0000	0.0178571
$197$	3474.00	1.25641	0.628204	−	0.778049i	$-0.283790\pi$
$197$	3474.00	1.25641	0.628204	+	0.778049i	$0.283790\pi$
$198$	0	0
$199$	−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
$209$	0	0
$210$	1008.00	0.331231
$211$	−5528.00	−1.80362	−0.901809	−	0.432136i	$-0.857760\pi$
$211$	−5528.00	−1.80362	−0.901809	+	0.432136i	$0.857760\pi$
$212$	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
$220$	0	0
$221$	−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	$-0.533848\pi$
$227$	−726.000	−0.212275	−0.106137	+	0.994351i	$0.533848\pi$
$228$	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
$231$	0	0
$232$	−2646.00	−0.748786
$233$	−2694.00	−0.757467	−0.378733	−	0.925506i	$-0.623640\pi$
$233$	−2694.00	−0.757467	−0.378733	+	0.925506i	$0.623640\pi$
$234$	−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	$-0.840390\pi$
$239$	−6480.00	−1.75379	−0.876896	+	0.480680i	$0.840390\pi$
$240$	−3408.00	−0.916606
$241$	2320.00	0.620101	0.310050	−	0.950720i	$-0.399654\pi$
$241$	2320.00	0.620101	0.310050	+	0.950720i	$0.399654\pi$
$242$	0	0
$243$	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
$253$	0	0
$254$	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	$-0.642644\pi$
$263$	−3696.00	−0.866559	−0.433280	+	0.901260i	$0.642644\pi$
$264$	0	0
$265$	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	$-0.0600771\pi$
$271$	8764.00	1.96448	0.982242	+	0.187619i	$0.0600771\pi$
$272$	−3408.00	−0.759707
$273$	1064.00	0.235884
$274$	8190.00	1.80575
$275$	0	0
$276$	48.0000	0.0104683
$277$	−5186.00	−1.12490	−0.562449	−	0.826832i	$-0.690141\pi$
$277$	−5186.00	−1.12490	−0.562449	+	0.826832i	$0.690141\pi$
$278$	5514.00	1.18960
$279$	770.000	0.165228
$280$	−1764.00	−0.376497
$281$	−3006.00	−0.638160	−0.319080	−	0.947728i	$-0.603374\pi$
$281$	−3006.00	−0.638160	−0.319080	+	0.947728i	$0.603374\pi$
$282$	−4680.00	−0.988262
$283$	3922.00	0.823812	0.411906	−	0.911226i	$-0.364863\pi$
$283$	3922.00	0.823812	0.411906	+	0.911226i	$0.364863\pi$
$284$	648.000	0.135393
$285$	3360.00	0.698348
$286$	0	0
$287$	−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	$-0.304601\pi$
$293$	5778.00	1.15206	0.576031	+	0.817428i	$0.304601\pi$
$294$	−588.000	−0.116642
$295$	−6624.00	−1.30734
$296$	−7518.00	−1.47627
$297$	0	0
$298$	−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	$-0.481942\pi$
$307$	610.000	0.113402	0.0567012	+	0.998391i	$0.481942\pi$
$308$	0	0
$309$	−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
$319$	0	0
$320$	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
$330$	0	0
$331$	−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	$-0.319139\pi$
$337$	6658.00	1.07621	0.538107	+	0.842876i	$0.319139\pi$
$338$	2259.00	0.363531
$339$	−5256.00	−0.842085
$340$	576.000	0.0918764
$341$	0	0
$342$	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	$-0.321138\pi$
$347$	6888.00	1.06561	0.532806	+	0.846238i	$0.321138\pi$
$348$	−504.000	−0.0776357
$349$	6190.00	0.949407	0.474704	−	0.880146i	$-0.342555\pi$
$349$	6190.00	0.949407	0.474704	+	0.880146i	$0.342555\pi$
$350$	399.000	0.0609356
$351$	5776.00	0.878348
$352$	0	0
$353$	−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	$-0.309736\pi$
$359$	7656.00	1.12554	0.562769	+	0.826614i	$0.309736\pi$
$360$	−2772.00	−0.405826
$361$	−1959.00	−0.285610
$362$	−5460.00	−0.792738
$363$	0	0
$364$	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	$-0.810067\pi$
$373$	−11918.0	−1.65440	−0.827199	+	0.561909i	$0.810067\pi$
$374$	0	0
$375$	−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
$385$	0	0
$386$	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
$396$	0	0
$397$	−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
$407$	0	0
$408$	4032.00	0.489249
$409$	−8156.00	−0.986035	−0.493017	−	0.870019i	$-0.664106\pi$
$409$	−8156.00	−0.986035	−0.493017	+	0.870019i	$0.664106\pi$
$410$	−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
$418$	0	0
$419$	−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
$429$	0	0
$430$	12384.0	1.38886
$431$	9480.00	1.05948	0.529740	−	0.848160i	$-0.322290\pi$
$431$	9480.00	1.05948	0.529740	+	0.848160i	$0.322290\pi$
$432$	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	$-0.233618\pi$
$439$	13660.0	1.48509	0.742547	+	0.669794i	$0.233618\pi$
$440$	0	0
$441$	−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
$451$	0	0
$452$	−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	$-0.673947\pi$
$457$	−10154.0	−1.03935	−0.519676	+	0.854363i	$0.673947\pi$
$458$	8364.00	0.853328
$459$	−7296.00	−0.741935
$460$	144.000	0.0145957
$461$	17190.0	1.73670	0.868349	−	0.495953i	$-0.165181\pi$
$461$	17190.0	1.73670	0.868349	+	0.495953i	$0.165181\pi$
$462$	0	0
$463$	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
$473$	0	0
$474$	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	$-0.227960\pi$
$479$	15816.0	1.50867	0.754333	+	0.656491i	$0.227960\pi$
$480$	2160.00	0.205396
$481$	13604.0	1.28958
$482$	−6960.00	−0.657716
$483$	−336.000	−0.0316533
$484$	0	0
$485$	−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	$-0.705056\pi$
$491$	−13068.0	−1.20112	−0.600561	+	0.799579i	$0.705056\pi$
$492$	864.000	0.0791710
$493$	−6048.00	−0.552512
$494$	7980.00	0.726796
$495$	0	0
$496$	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	$-0.445388\pi$
$503$	3852.00	0.341456	0.170728	+	0.985318i	$0.445388\pi$
$504$	1617.00	0.142911
$505$	15048.0	1.32599
$506$	0	0
$507$	−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
$517$	0	0
$518$	−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	$-0.336254\pi$
$523$	11770.0	0.984065	0.492033	+	0.870577i	$0.336254\pi$
$524$	258.000	0.0215091
$525$	−532.000	−0.0442255
$526$	11088.0	0.919125
$527$	−3360.00	−0.277730
$528$	0	0
$529$	−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
$539$	0	0
$540$	−1824.00	−0.145356
$541$	−10694.0	−0.849854	−0.424927	−	0.905228i	$-0.639700\pi$
$541$	−10694.0	−0.849854	−0.424927	+	0.905228i	$0.639700\pi$
$542$	−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	$-0.570695\pi$
$547$	−5636.00	−0.440545	−0.220272	+	0.975438i	$0.570695\pi$
$548$	−2730.00	−0.212810
$549$	9130.00	0.709761
$550$	0	0
$551$	−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	$-0.695123\pi$
$557$	−15126.0	−1.15064	−0.575322	+	0.817927i	$0.695123\pi$
$558$	−2310.00	−0.175251
$559$	13072.0	0.989064
$560$	5964.00	0.450045
$561$	0	0
$562$	9018.00	0.676871
$563$	−3246.00	−0.242989	−0.121494	−	0.992592i	$-0.538769\pi$
$563$	−3246.00	−0.242989	−0.121494	+	0.992592i	$0.538769\pi$
$564$	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	$-0.487685\pi$
$569$	1050.00	0.0773608	0.0386804	+	0.999252i	$0.487685\pi$
$570$	−10080.0	−0.740710
$571$	−6860.00	−0.502771	−0.251385	−	0.967887i	$-0.580886\pi$
$571$	−6860.00	−0.502771	−0.251385	+	0.967887i	$0.580886\pi$
$572$	0	0
$573$	−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
$583$	0	0
$584$	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	$-0.376862\pi$
$593$	10896.0	0.754545	0.377272	+	0.926102i	$0.376862\pi$
$594$	0	0
$595$	−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	$-0.637443\pi$
$601$	−12332.0	−0.836993	−0.418496	+	0.908218i	$0.637443\pi$
$602$	−7224.00	−0.489083
$603$	2156.00	0.145604
$604$	−2864.00	−0.192938
$605$	0	0
$606$	−15048.0	−1.00872
$607$	−21800.0	−1.45772	−0.728859	−	0.684664i	$-0.759949\pi$
$607$	−21800.0	−1.45772	−0.728859	+	0.684664i	$0.759949\pi$
$608$	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	$-0.709173\pi$
$613$	−18542.0	−1.22170	−0.610852	+	0.791745i	$0.709173\pi$
$614$	−1830.00	−0.120281
$615$	10368.0	0.679802
$616$	0	0
$617$	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
$627$	0	0
$628$	−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
$638$	0	0
$639$	−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
$649$	0	0
$650$	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	$-0.264325\pi$
$659$	22824.0	1.34916	0.674580	+	0.738201i	$0.264325\pi$
$660$	0	0
$661$	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
$671$	0	0
$672$	−1260.00	−0.0723297
$673$	17458.0	0.999935	0.499968	−	0.866044i	$-0.333345\pi$
$673$	17458.0	0.999935	0.499968	+	0.866044i	$0.333345\pi$
$674$	−19974.0	−1.14150
$675$	−2888.00	−0.164680
$676$	−753.000	−0.0428425
$677$	−14574.0	−0.827362	−0.413681	−	0.910422i	$-0.635757\pi$
$677$	−14574.0	−0.827362	−0.413681	+	0.910422i	$0.635757\pi$
$678$	15768.0	0.893166
$679$	490.000	0.0276944
$680$	12096.0	0.682148
$681$	−2904.00	−0.163409
$682$	0	0
$683$	−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
$693$	0	0
$694$	−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	$-0.534178\pi$
$701$	−3978.00	−0.214332	−0.107166	+	0.994241i	$0.534178\pi$
$702$	−17328.0	−0.931629
$703$	−25060.0	−1.34446
$704$	0	0
$705$	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
$715$	0	0
$716$	−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
$726$	0	0
$727$	−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	$-0.328078\pi$
$733$	20410.0	1.02846	0.514230	+	0.857653i	$0.328078\pi$
$734$	−25278.0	−1.27116
$735$	2352.00	0.118034
$736$	540.000	0.0270444
$737$	0	0
$738$	7128.00	0.355536
$739$	−14564.0	−0.724960	−0.362480	−	0.931992i	$-0.618070\pi$
$739$	−14564.0	−0.724960	−0.362480	+	0.931992i	$0.618070\pi$
$740$	−4296.00	−0.213411
$741$	−10640.0	−0.527490
$742$	9198.00	0.455080
$743$	7416.00	0.366173	0.183087	−	0.983097i	$-0.441391\pi$
$743$	7416.00	0.366173	0.183087	+	0.983097i	$0.441391\pi$
$744$	−5880.00	−0.289746
$745$	6120.00	0.300966
$746$	35754.0	1.75475
$747$	990.000	0.0484902
$748$	0	0
$749$	−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
$759$	0	0
$760$	17640.0	0.841934
$761$	27492.0	1.30957	0.654786	−	0.755814i	$-0.272759\pi$
$761$	27492.0	1.30957	0.654786	+	0.755814i	$0.272759\pi$
$762$	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	$-0.515739\pi$
$769$	−2108.00	−0.0988510	−0.0494255	+	0.998778i	$0.515739\pi$
$770$	0	0
$771$	−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
$781$	0	0
$782$	−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	$-0.569598\pi$
$787$	−9578.00	−0.433823	−0.216912	+	0.976191i	$0.569598\pi$
$788$	3474.00	0.157051
$789$	−14784.0	−0.667078
$790$	48672.0	2.19199
$791$	9198.00	0.413455
$792$	0	0
$793$	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
$803$	0	0
$804$	−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	$-0.437135\pi$
$809$	9030.00	0.392433	0.196216	+	0.980561i	$0.437135\pi$
$810$	11196.0	0.485663
$811$	37762.0	1.63502	0.817511	−	0.575913i	$-0.195353\pi$
$811$	37762.0	1.63502	0.817511	+	0.575913i	$0.195353\pi$
$812$	882.000	0.0381184
$813$	35056.0	1.51226
$814$	0	0
$815$	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	$-0.401455\pi$
$821$	14334.0	0.609330	0.304665	+	0.952460i	$0.401455\pi$
$822$	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
$825$	0	0
$826$	−11592.0	−0.488302
$827$	22284.0	0.936990	0.468495	−	0.883466i	$-0.344796\pi$
$827$	22284.0	0.936990	0.468495	+	0.883466i	$0.344796\pi$
$828$	−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
$836$	0	0
$837$	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
$847$	0	0
$848$	−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	$-0.382607\pi$
$853$	17962.0	0.720993	0.360497	+	0.932761i	$0.382607\pi$
$854$	−17430.0	−0.698410
$855$	−9240.00	−0.369592
$856$	8064.00	0.321988
$857$	−47148.0	−1.87928	−0.939641	−	0.342161i	$-0.888841\pi$
$857$	−47148.0	−1.87928	−0.939641	+	0.342161i	$0.888841\pi$
$858$	0	0
$859$	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
$869$	0	0
$870$	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	$-0.254436\pi$
$877$	36214.0	1.39437	0.697184	+	0.716893i	$0.254436\pi$
$878$	−40980.0	−1.57518
$879$	23112.0	0.886858
$880$	0	0
$881$	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	$-0.538993\pi$
$887$	−6456.00	−0.244387	−0.122193	+	0.992506i	$0.538993\pi$
$888$	−30072.0	−1.13643
$889$	2408.00	0.0908456
$890$	−41256.0	−1.55383
$891$	0	0
$892$	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
$902$	0	0
$903$	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
$913$	0	0
$914$	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	$-0.550715\pi$
$919$	−8840.00	−0.317307	−0.158653	+	0.987334i	$0.550715\pi$
$920$	3024.00	0.108368
$921$	2440.00	0.0872972
$922$	−51570.0	−1.84205
$923$	−24624.0	−0.878124
$924$	0	0
$925$	−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
$935$	0	0
$936$	8778.00	0.306536
$937$	−7832.00	−0.273063	−0.136532	−	0.990636i	$-0.543596\pi$
$937$	−7832.00	−0.273063	−0.136532	+	0.990636i	$0.543596\pi$
$938$	−4116.00	−0.143275
$939$	41336.0	1.43658
$940$	4680.00	0.162388
$941$	−16926.0	−0.586368	−0.293184	−	0.956056i	$-0.594715\pi$
$941$	−16926.0	−0.586368	−0.293184	+	0.956056i	$0.594715\pi$
$942$	35616.0	1.23188
$943$	2592.00	0.0895092
$944$	39192.0	1.35126
$945$	12768.0	0.439516
$946$	0	0
$947$	−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	$-0.335066\pi$
$953$	29142.0	0.990558	0.495279	+	0.868734i	$0.335066\pi$
$954$	14454.0	0.490530
$955$	−14544.0	−0.492809
$956$	−6480.00	−0.219224
$957$	0	0
$958$	−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	$-0.673367\pi$
$967$	−31160.0	−1.03623	−0.518117	+	0.855310i	$0.673367\pi$
$968$	0	0
$969$	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
$979$	0	0
$980$	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
$990$	0	0
$991$	−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	$-0.331758\pi$
$997$	31750.0	1.00856	0.504279	+	0.863541i	$0.331758\pi$
$998$	−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	847.4.a.a.1.1		1
11.10	odd	2		77.4.a.a.1.1	✓	1
33.32	even	2		693.4.a.b.1.1		1
44.43	even	2		1232.4.a.d.1.1		1
55.54	odd	2		1925.4.a.a.1.1		1
77.76	even	2		539.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.a.1.1	✓	1	11.10	odd	2
539.4.a.a.1.1		1	77.76	even	2
693.4.a.b.1.1		1	33.32	even	2
847.4.a.a.1.1		1	1.1	even	1	trivial
1232.4.a.d.1.1		1	44.43	even	2
1925.4.a.a.1.1		1	55.54	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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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