LMFDB - Embedded newform 147.16.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,16,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("147.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	147.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:	$209.759452497$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	147.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-234.000 q^{2} +2187.00 q^{3} +21988.0 q^{4} -280710. q^{5} -511758. q^{6} +2.52252e6 q^{8} +4.78297e6 q^{9} +O(q^{10})$

\(q-234.000 q^{2} +2187.00 q^{3} +21988.0 q^{4} -280710. q^{5} -511758. q^{6} +2.52252e6 q^{8} +4.78297e6 q^{9} +6.56861e7 q^{10} +3.40311e7 q^{11} +4.80878e7 q^{12} -3.84022e8 q^{13} -6.13913e8 q^{15} -1.31077e9 q^{16} -1.25921e9 q^{17} -1.11921e9 q^{18} +2.49907e9 q^{19} -6.17225e9 q^{20} -7.96327e9 q^{22} +1.12848e10 q^{23} +5.51675e9 q^{24} +4.82805e10 q^{25} +8.98612e10 q^{26} +1.04604e10 q^{27} -4.84135e10 q^{29} +1.43656e11 q^{30} -1.30547e11 q^{31} +2.24063e11 q^{32} +7.44259e10 q^{33} +2.94655e11 q^{34} +1.05168e11 q^{36} -2.00223e11 q^{37} -5.84783e11 q^{38} -8.39857e11 q^{39} -7.08097e11 q^{40} -6.79142e11 q^{41} +2.79482e11 q^{43} +7.48275e11 q^{44} -1.34263e12 q^{45} -2.64065e12 q^{46} -1.52067e12 q^{47} -2.86666e12 q^{48} -1.12976e13 q^{50} -2.75389e12 q^{51} -8.44388e12 q^{52} +2.64605e12 q^{53} -2.44772e12 q^{54} -9.55286e12 q^{55} +5.46547e12 q^{57} +1.13287e13 q^{58} -7.39937e12 q^{59} -1.34987e13 q^{60} +4.26596e13 q^{61} +3.05481e13 q^{62} -9.47931e12 q^{64} +1.07799e14 q^{65} -1.74157e13 q^{66} -5.64080e13 q^{67} -2.76875e13 q^{68} +2.46799e13 q^{69} -1.33150e14 q^{71} +1.20651e13 q^{72} -1.05603e14 q^{73} +4.68523e13 q^{74} +1.05590e14 q^{75} +5.49496e13 q^{76} +1.96526e14 q^{78} -5.56657e13 q^{79} +3.67947e14 q^{80} +2.28768e13 q^{81} +1.58919e14 q^{82} -3.78077e14 q^{83} +3.53472e14 q^{85} -6.53988e13 q^{86} -1.05880e14 q^{87} +8.58440e13 q^{88} -2.19315e14 q^{89} +3.14175e14 q^{90} +2.48131e14 q^{92} -2.85507e14 q^{93} +3.55837e14 q^{94} -7.01514e14 q^{95} +4.90025e14 q^{96} -7.03323e14 q^{97} +1.62769e14 q^{99} +O(q^{100})\)

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 $

$2$	−234.000	−1.29268	−0.646340	−	0.763050i	$-0.723701\pi$
$2$	−234.000	−1.29268	−0.646340	+	0.763050i	$0.723701\pi$
$3$	2187.00	0.577350
$3$	2187.00	0.577350
$4$	21988.0	0.671021
$5$	−280710.	−1.60688	−0.803439	−	0.595387i	$-0.796999\pi$
$5$	−280710.	−1.60688	−0.803439	+	0.595387i	$0.796999\pi$
$6$	−511758.	−0.746329
$7$	0	0
$7$	0	0
$8$	2.52252e6	0.425265
$9$	4.78297e6	0.333333
$10$	6.56861e7	2.07718
$11$	3.40311e7	0.526539	0.263269	−	0.964722i	$-0.415199\pi$
$11$	3.40311e7	0.526539	0.263269	+	0.964722i	$0.415199\pi$
$12$	4.80878e7	0.387414
$13$	−3.84022e8	−1.69739	−0.848694	−	0.528884i	$-0.822611\pi$
$13$	−3.84022e8	−1.69739	−0.848694	+	0.528884i	$0.822611\pi$
$14$	0	0
$15$	−6.13913e8	−0.927731
$16$	−1.31077e9	−1.22075
$17$	−1.25921e9	−0.744270	−0.372135	−	0.928179i	$-0.621374\pi$
$17$	−1.25921e9	−0.744270	−0.372135	+	0.928179i	$0.621374\pi$
$18$	−1.11921e9	−0.430893
$19$	2.49907e9	0.641397	0.320698	−	0.947181i	$-0.396082\pi$
$19$	2.49907e9	0.641397	0.320698	+	0.947181i	$0.396082\pi$
$20$	−6.17225e9	−1.07825
$21$	0	0
$22$	−7.96327e9	−0.680646
$23$	1.12848e10	0.691093	0.345546	−	0.938402i	$-0.387694\pi$
$23$	1.12848e10	0.691093	0.345546	+	0.938402i	$0.387694\pi$
$24$	5.51675e9	0.245527
$25$	4.82805e10	1.58206
$26$	8.98612e10	2.19418
$27$	1.04604e10	0.192450
$28$	0	0
$29$	−4.84135e10	−0.521172	−0.260586	−	0.965451i	$-0.583916\pi$
$29$	−4.84135e10	−0.521172	−0.260586	+	0.965451i	$0.583916\pi$
$30$	1.43656e11	1.19926
$31$	−1.30547e11	−0.852227	−0.426113	−	0.904670i	$-0.640118\pi$
$31$	−1.30547e11	−0.852227	−0.426113	+	0.904670i	$0.640118\pi$
$32$	2.24063e11	1.15278
$33$	7.44259e10	0.303997
$34$	2.94655e11	0.962102
$35$	0	0
$36$	1.05168e11	0.223674
$37$	−2.00223e11	−0.346738	−0.173369	−	0.984857i	$-0.555465\pi$
$37$	−2.00223e11	−0.346738	−0.173369	+	0.984857i	$0.555465\pi$
$38$	−5.84783e11	−0.829120
$39$	−8.39857e11	−0.979988
$40$	−7.08097e11	−0.683349
$41$	−6.79142e11	−0.544605	−0.272302	−	0.962212i	$-0.587785\pi$
$41$	−6.79142e11	−0.544605	−0.272302	+	0.962212i	$0.587785\pi$
$42$	0	0
$43$	2.79482e11	0.156798	0.0783990	−	0.996922i	$-0.475019\pi$
$43$	2.79482e11	0.156798	0.0783990	+	0.996922i	$0.475019\pi$
$44$	7.48275e11	0.353318
$45$	−1.34263e12	−0.535626
$46$	−2.64065e12	−0.893362
$47$	−1.52067e12	−0.437826	−0.218913	−	0.975744i	$-0.570251\pi$
$47$	−1.52067e12	−0.437826	−0.218913	+	0.975744i	$0.570251\pi$
$48$	−2.86666e12	−0.704801
$49$	0	0
$50$	−1.12976e13	−2.04509
$51$	−2.75389e12	−0.429704
$52$	−8.44388e12	−1.13898
$53$	2.64605e12	0.309407	0.154703	−	0.987961i	$-0.450558\pi$
$53$	2.64605e12	0.309407	0.154703	+	0.987961i	$0.450558\pi$
$54$	−2.44772e12	−0.248776
$55$	−9.55286e12	−0.846083
$56$	0	0
$57$	5.46547e12	0.370310
$58$	1.13287e13	0.673708
$59$	−7.39937e12	−0.387084	−0.193542	−	0.981092i	$-0.561998\pi$
$59$	−7.39937e12	−0.387084	−0.193542	+	0.981092i	$0.561998\pi$
$60$	−1.34987e13	−0.622527
$61$	4.26596e13	1.73797	0.868987	−	0.494835i	$-0.164772\pi$
$61$	4.26596e13	1.73797	0.868987	+	0.494835i	$0.164772\pi$
$62$	3.05481e13	1.10166
$63$	0	0
$64$	−9.47931e12	−0.269418
$65$	1.07799e14	2.72750
$66$	−1.74157e13	−0.392971
$67$	−5.64080e13	−1.13705	−0.568525	−	0.822666i	$-0.692486\pi$
$67$	−5.64080e13	−1.13705	−0.568525	+	0.822666i	$0.692486\pi$
$68$	−2.76875e13	−0.499420
$69$	2.46799e13	0.399003
$70$	0	0
$71$	−1.33150e14	−1.73741	−0.868705	−	0.495329i	$-0.835047\pi$
$71$	−1.33150e14	−1.73741	−0.868705	+	0.495329i	$0.835047\pi$
$72$	1.20651e13	0.141755
$73$	−1.05603e14	−1.11881	−0.559405	−	0.828895i	$-0.688970\pi$
$73$	−1.05603e14	−1.11881	−0.559405	+	0.828895i	$0.688970\pi$
$74$	4.68523e13	0.448221
$75$	1.05590e14	0.913401
$76$	5.49496e13	0.430390
$77$	0	0
$78$	1.96526e14	1.26681
$79$	−5.56657e13	−0.326125	−0.163063	−	0.986616i	$-0.552137\pi$
$79$	−5.56657e13	−0.326125	−0.163063	+	0.986616i	$0.552137\pi$
$80$	3.67947e14	1.96160
$81$	2.28768e13	0.111111
$82$	1.58919e14	0.704000
$83$	−3.78077e14	−1.52931	−0.764654	−	0.644441i	$-0.777090\pi$
$83$	−3.78077e14	−1.52931	−0.764654	+	0.644441i	$0.777090\pi$
$84$	0	0
$85$	3.53472e14	1.19595
$86$	−6.53988e13	−0.202690
$87$	−1.05880e14	−0.300899
$88$	8.58440e13	0.223918
$89$	−2.19315e14	−0.525586	−0.262793	−	0.964852i	$-0.584644\pi$
$89$	−2.19315e14	−0.525586	−0.262793	+	0.964852i	$0.584644\pi$
$90$	3.14175e14	0.692393
$91$	0	0
$92$	2.48131e14	0.463737
$93$	−2.85507e14	−0.492033
$94$	3.55837e14	0.565969
$95$	−7.01514e14	−1.03065
$96$	4.90025e14	0.665556
$97$	−7.03323e14	−0.883826	−0.441913	−	0.897058i	$-0.645700\pi$
$97$	−7.03323e14	−0.883826	−0.441913	+	0.897058i	$0.645700\pi$
$98$	0	0
$99$	1.62769e14	0.175513
$100$	1.06159e15	1.06159
$101$	−5.60434e14	−0.520132	−0.260066	−	0.965591i	$-0.583744\pi$
$101$	−5.60434e14	−0.520132	−0.260066	+	0.965591i	$0.583744\pi$
$102$	6.44410e14	0.555470
$103$	2.24049e15	1.79499	0.897497	−	0.441021i	$-0.145384\pi$
$103$	2.24049e15	1.79499	0.897497	+	0.441021i	$0.145384\pi$
$104$	−9.68704e14	−0.721840
$105$	0	0
$106$	−6.19177e14	−0.399964
$107$	−1.23611e15	−0.744184	−0.372092	−	0.928196i	$-0.621359\pi$
$107$	−1.23611e15	−0.744184	−0.372092	+	0.928196i	$0.621359\pi$
$108$	2.30002e14	0.129138
$109$	9.13052e14	0.478406	0.239203	−	0.970970i	$-0.423114\pi$
$109$	9.13052e14	0.478406	0.239203	+	0.970970i	$0.423114\pi$
$110$	2.23537e15	1.09371
$111$	−4.37888e14	−0.200189
$112$	0	0
$113$	−1.99316e15	−0.796990	−0.398495	−	0.917170i	$-0.630467\pi$
$113$	−1.99316e15	−0.796990	−0.398495	+	0.917170i	$0.630467\pi$
$114$	−1.27892e15	−0.478693
$115$	−3.16777e15	−1.11050
$116$	−1.06452e15	−0.349717
$117$	−1.83677e15	−0.565796
$118$	1.73145e15	0.500375
$119$	0	0
$120$	−1.54861e15	−0.394532
$121$	−3.01914e15	−0.722757
$122$	−9.98235e15	−2.24664
$123$	−1.48528e15	−0.314428
$124$	−2.87047e15	−0.571862
$125$	−4.98624e15	−0.935293
$126$	0	0
$127$	4.90777e15	0.817252	0.408626	−	0.912702i	$-0.366008\pi$
$127$	4.90777e15	0.817252	0.408626	+	0.912702i	$0.366008\pi$
$128$	−5.12393e15	−0.804505
$129$	6.11228e14	0.0905274
$130$	−2.52249e16	−3.52578
$131$	3.51036e15	0.463252	0.231626	−	0.972805i	$-0.425595\pi$
$131$	3.51036e15	0.463252	0.231626	+	0.972805i	$0.425595\pi$
$132$	1.63648e15	0.203988
$133$	0	0
$134$	1.31995e16	1.46984
$135$	−2.93633e15	−0.309244
$136$	−3.17638e15	−0.316512
$137$	−7.53799e15	−0.710970	−0.355485	−	0.934682i	$-0.615684\pi$
$137$	−7.53799e15	−0.710970	−0.355485	+	0.934682i	$0.615684\pi$
$138$	−5.77510e15	−0.515783
$139$	−1.28475e16	−1.08695	−0.543474	−	0.839426i	$-0.682891\pi$
$139$	−1.28475e16	−1.08695	−0.543474	+	0.839426i	$0.682891\pi$
$140$	0	0
$141$	−3.32571e15	−0.252779
$142$	3.11570e16	2.24592
$143$	−1.30687e16	−0.893741
$144$	−6.26938e15	−0.406917
$145$	1.35901e16	0.837460
$146$	2.47112e16	1.44626
$147$	0	0
$148$	−4.40251e15	−0.232668
$149$	−1.88397e16	−0.946620	−0.473310	−	0.880896i	$-0.656941\pi$
$149$	−1.88397e16	−0.946620	−0.473310	+	0.880896i	$0.656941\pi$
$150$	−2.47079e16	−1.18073
$151$	−2.86423e16	−1.30221	−0.651105	−	0.758988i	$-0.725694\pi$
$151$	−2.86423e16	−1.30221	−0.651105	+	0.758988i	$0.725694\pi$
$152$	6.30396e15	0.272764
$153$	−6.02275e15	−0.248090
$154$	0	0
$155$	3.66459e16	1.36942
$156$	−1.84668e16	−0.657592
$157$	4.28834e16	1.45560	0.727799	−	0.685790i	$-0.240543\pi$
$157$	4.28834e16	1.45560	0.727799	+	0.685790i	$0.240543\pi$
$158$	1.30258e16	0.421575
$159$	5.78692e15	0.178636
$160$	−6.28967e16	−1.85237
$161$	0	0
$162$	−5.35317e15	−0.143631
$163$	1.55067e16	0.397294	0.198647	−	0.980071i	$-0.436345\pi$
$163$	1.55067e16	0.397294	0.198647	+	0.980071i	$0.436345\pi$
$164$	−1.49330e16	−0.365441
$165$	−2.08921e16	−0.488486
$166$	8.84701e16	1.97691
$167$	−8.14637e16	−1.74017	−0.870083	−	0.492906i	$-0.835935\pi$
$167$	−8.14637e16	−1.74017	−0.870083	+	0.492906i	$0.835935\pi$
$168$	0	0
$169$	9.62872e16	1.88113
$170$	−8.27125e16	−1.54598
$171$	1.19530e16	0.213799
$172$	6.14525e15	0.105215
$173$	−4.87723e16	−0.799517	−0.399758	−	0.916621i	$-0.630906\pi$
$173$	−4.87723e16	−0.799517	−0.399758	+	0.916621i	$0.630906\pi$
$174$	2.47760e16	0.388966
$175$	0	0
$176$	−4.46070e16	−0.642773
$177$	−1.61824e16	−0.223483
$178$	5.13197e16	0.679414
$179$	−7.76361e16	−0.985521	−0.492760	−	0.870165i	$-0.664012\pi$
$179$	−7.76361e16	−0.985521	−0.492760	+	0.870165i	$0.664012\pi$
$180$	−2.95217e16	−0.359416
$181$	1.26205e16	0.147397	0.0736985	−	0.997281i	$-0.476520\pi$
$181$	1.26205e16	0.147397	0.0736985	+	0.997281i	$0.476520\pi$
$182$	0	0
$183$	9.32966e16	1.00342
$184$	2.84662e16	0.293898
$185$	5.62047e16	0.557166
$186$	6.68086e16	0.636041
$187$	−4.28522e16	−0.391887
$188$	−3.34366e16	−0.293790
$189$	0	0
$190$	1.64154e17	1.33229
$191$	−6.26505e16	−0.488849	−0.244424	−	0.969668i	$-0.578599\pi$
$191$	−6.26505e16	−0.488849	−0.244424	+	0.969668i	$0.578599\pi$
$192$	−2.07312e16	−0.155549
$193$	−2.21474e16	−0.159824	−0.0799122	−	0.996802i	$-0.525464\pi$
$193$	−2.21474e16	−0.159824	−0.0799122	+	0.996802i	$0.525464\pi$
$194$	1.64578e17	1.14250
$195$	2.35756e17	1.57472
$196$	0	0
$197$	−1.63106e17	−1.00919	−0.504594	−	0.863357i	$-0.668358\pi$
$197$	−1.63106e17	−1.00919	−0.504594	+	0.863357i	$0.668358\pi$
$198$	−3.80881e16	−0.226882
$199$	−7.34283e16	−0.421178	−0.210589	−	0.977575i	$-0.567538\pi$
$199$	−7.34283e16	−0.421178	−0.210589	+	0.977575i	$0.567538\pi$
$200$	1.21789e17	0.672793
$201$	−1.23364e17	−0.656477
$202$	1.31141e17	0.672364
$203$	0	0
$204$	−6.05525e16	−0.288340
$205$	1.90642e17	0.875113
$206$	−5.24274e17	−2.32035
$207$	5.39750e16	0.230364
$208$	5.03366e17	2.07209
$209$	8.50460e16	0.337720
$210$	0	0
$211$	−1.22953e15	−0.00454589	−0.00227295	−	0.999997i	$-0.500724\pi$
$211$	−1.22953e15	−0.00454589	−0.00227295	+	0.999997i	$0.500724\pi$
$212$	5.81814e16	0.207618
$213$	−2.91198e17	−1.00309
$214$	2.89251e17	0.961991
$215$	−7.84534e16	−0.251955
$216$	2.63865e16	0.0818423
$217$	0	0
$218$	−2.13654e17	−0.618426
$219$	−2.30955e17	−0.645945
$220$	−2.10048e17	−0.567739
$221$	4.83564e17	1.26332
$222$	1.02466e17	0.258781
$223$	8.18320e17	1.99819	0.999095	−	0.0425326i	$-0.0135426\pi$
$223$	8.18320e17	1.99819	0.999095	+	0.0425326i	$0.0135426\pi$
$224$	0	0
$225$	2.30924e17	0.527352
$226$	4.66399e17	1.03025
$227$	5.79062e17	1.23746	0.618730	−	0.785604i	$-0.287647\pi$
$227$	5.79062e17	1.23746	0.618730	+	0.785604i	$0.287647\pi$
$228$	1.20175e17	0.248486
$229$	−9.00391e17	−1.80163	−0.900813	−	0.434207i	$-0.857029\pi$
$229$	−9.00391e17	−1.80163	−0.900813	+	0.434207i	$0.857029\pi$
$230$	7.41257e17	1.43552
$231$	0	0
$232$	−1.22124e17	−0.221636
$233$	−3.97437e17	−0.698391	−0.349196	−	0.937050i	$-0.613545\pi$
$233$	−3.97437e17	−0.698391	−0.349196	+	0.937050i	$0.613545\pi$
$234$	4.29803e17	0.731393
$235$	4.26868e17	0.703533
$236$	−1.62697e17	−0.259741
$237$	−1.21741e17	−0.188288
$238$	0	0
$239$	1.06770e18	1.55047	0.775236	−	0.631672i	$-0.217631\pi$
$239$	1.06770e18	1.55047	0.775236	+	0.631672i	$0.217631\pi$
$240$	8.04700e17	1.13253
$241$	2.98003e17	0.406530	0.203265	−	0.979124i	$-0.434845\pi$
$241$	2.98003e17	0.406530	0.203265	+	0.979124i	$0.434845\pi$
$242$	7.06478e17	0.934293
$243$	5.00315e16	0.0641500
$244$	9.38000e17	1.16622
$245$	0	0
$246$	3.47556e17	0.406454
$247$	−9.59699e17	−1.08870
$248$	−3.29308e17	−0.362422
$249$	−8.26855e17	−0.882947
$250$	1.16678e18	1.20903
$251$	−8.01026e17	−0.805552	−0.402776	−	0.915299i	$-0.631955\pi$
$251$	−8.01026e17	−0.805552	−0.402776	+	0.915299i	$0.631955\pi$
$252$	0	0
$253$	3.84035e17	0.363887
$254$	−1.14842e18	−1.05644
$255$	7.73044e17	0.690482
$256$	1.50962e18	1.30939
$257$	1.35952e18	1.14522	0.572608	−	0.819829i	$-0.305932\pi$
$257$	1.35952e18	1.14522	0.572608	+	0.819829i	$0.305932\pi$
$258$	−1.43027e17	−0.117023
$259$	0	0
$260$	2.37028e18	1.83021
$261$	−2.31560e17	−0.173724
$262$	−8.21424e17	−0.598836
$263$	2.74388e18	1.94401	0.972003	−	0.234967i	$-0.0754983\pi$
$263$	2.74388e18	1.94401	0.972003	+	0.234967i	$0.0754983\pi$
$264$	1.87741e17	0.129279
$265$	−7.42774e17	−0.497179
$266$	0	0
$267$	−4.79642e17	−0.303447
$268$	−1.24030e18	−0.762984
$269$	1.71195e18	1.02412	0.512058	−	0.858951i	$-0.328883\pi$
$269$	1.71195e18	1.02412	0.512058	+	0.858951i	$0.328883\pi$
$270$	6.87100e17	0.399753
$271$	1.67001e18	0.945036	0.472518	−	0.881321i	$-0.343345\pi$
$271$	1.67001e18	0.945036	0.472518	+	0.881321i	$0.343345\pi$
$272$	1.65053e18	0.908569
$273$	0	0
$274$	1.76389e18	0.919057
$275$	1.64304e18	0.833014
$276$	5.42662e17	0.267739
$277$	−6.50407e17	−0.312311	−0.156155	−	0.987733i	$-0.549910\pi$
$277$	−6.50407e17	−0.312311	−0.156155	+	0.987733i	$0.549910\pi$
$278$	3.00632e18	1.40507
$279$	−6.24404e17	−0.284076
$280$	0	0
$281$	1.43557e18	0.619052	0.309526	−	0.950891i	$-0.399830\pi$
$281$	1.43557e18	0.619052	0.309526	+	0.950891i	$0.399830\pi$
$282$	7.78216e17	0.326762
$283$	−2.60218e18	−1.06399	−0.531996	−	0.846747i	$-0.678558\pi$
$283$	−2.60218e18	−1.06399	−0.531996	+	0.846747i	$0.678558\pi$
$284$	−2.92770e18	−1.16584
$285$	−1.53421e18	−0.595044
$286$	3.05807e18	1.15532
$287$	0	0
$288$	1.07169e18	0.384259
$289$	−1.27682e18	−0.446062
$290$	−3.18009e18	−1.08257
$291$	−1.53817e18	−0.510277
$292$	−2.32201e18	−0.750744
$293$	1.14850e18	0.361931	0.180965	−	0.983489i	$-0.442078\pi$
$293$	1.14850e18	0.361931	0.180965	+	0.983489i	$0.442078\pi$
$294$	0	0
$295$	2.07708e18	0.621996
$296$	−5.05067e17	−0.147456
$297$	3.55977e17	0.101332
$298$	4.40848e18	1.22368
$299$	−4.33363e18	−1.17305
$300$	2.32170e18	0.612911
$301$	0	0
$302$	6.70230e18	1.68334
$303$	−1.22567e18	−0.300299
$304$	−3.27571e18	−0.782986
$305$	−1.19750e19	−2.79271
$306$	1.40932e18	0.320701
$307$	5.53368e18	1.22879	0.614393	−	0.789000i	$-0.289401\pi$
$307$	5.53368e18	1.22879	0.614393	+	0.789000i	$0.289401\pi$
$308$	0	0
$309$	4.89994e18	1.03634
$310$	−8.57515e18	−1.77023
$311$	−2.57410e18	−0.518708	−0.259354	−	0.965782i	$-0.583510\pi$
$311$	−2.57410e18	−0.518708	−0.259354	+	0.965782i	$0.583510\pi$
$312$	−2.11856e18	−0.416755
$313$	5.90030e18	1.13316	0.566581	−	0.824006i	$-0.308266\pi$
$313$	5.90030e18	1.13316	0.566581	+	0.824006i	$0.308266\pi$
$314$	−1.00347e19	−1.88162
$315$	0	0
$316$	−1.22398e18	−0.218837
$317$	5.15806e18	0.900621	0.450310	−	0.892872i	$-0.351313\pi$
$317$	5.15806e18	0.900621	0.450310	+	0.892872i	$0.351313\pi$
$318$	−1.35414e18	−0.230919
$319$	−1.64756e18	−0.274417
$320$	2.66094e18	0.432922
$321$	−2.70338e18	−0.429655
$322$	0	0
$323$	−3.14685e18	−0.477372
$324$	5.03015e17	0.0745578
$325$	−1.85408e19	−2.68536
$326$	−3.62857e18	−0.513574
$327$	1.99684e18	0.276208
$328$	−1.71315e18	−0.231601
$329$	0	0
$330$	4.88875e18	0.631456
$331$	4.14841e18	0.523807	0.261903	−	0.965094i	$-0.415650\pi$
$331$	4.14841e18	0.523807	0.261903	+	0.965094i	$0.415650\pi$
$332$	−8.31317e18	−1.02620
$333$	−9.57662e17	−0.115579
$334$	1.90625e19	2.24948
$335$	1.58343e19	1.82710
$336$	0	0
$337$	3.83002e18	0.422646	0.211323	−	0.977416i	$-0.432223\pi$
$337$	3.83002e18	0.422646	0.211323	+	0.977416i	$0.432223\pi$
$338$	−2.25312e19	−2.43170
$339$	−4.35903e18	−0.460143
$340$	7.77215e18	0.802507
$341$	−4.44266e18	−0.448730
$342$	−2.79700e18	−0.276373
$343$	0	0
$344$	7.04999e17	0.0666807
$345$	−6.92790e18	−0.641148
$346$	1.14127e19	1.03352
$347$	1.29300e19	1.14585	0.572926	−	0.819607i	$-0.305808\pi$
$347$	1.29300e19	1.14585	0.572926	+	0.819607i	$0.305808\pi$
$348$	−2.32809e18	−0.201909
$349$	−4.28556e18	−0.363762	−0.181881	−	0.983321i	$-0.558219\pi$
$349$	−4.28556e18	−0.363762	−0.181881	+	0.983321i	$0.558219\pi$
$350$	0	0
$351$	−4.01701e18	−0.326663
$352$	7.62509e18	0.606981
$353$	1.23957e19	0.965967	0.482984	−	0.875629i	$-0.339553\pi$
$353$	1.23957e19	0.965967	0.482984	+	0.875629i	$0.339553\pi$
$354$	3.78669e18	0.288892
$355$	3.73764e19	2.79181
$356$	−4.82230e18	−0.352679
$357$	0	0
$358$	1.81668e19	1.27396
$359$	1.32770e19	0.911784	0.455892	−	0.890035i	$-0.349320\pi$
$359$	1.32770e19	0.911784	0.455892	+	0.890035i	$0.349320\pi$
$360$	−3.38680e18	−0.227783
$361$	−8.93577e18	−0.588611
$362$	−2.95320e18	−0.190537
$363$	−6.60285e18	−0.417284
$364$	0	0
$365$	2.96439e19	1.79779
$366$	−2.18314e19	−1.29710
$367$	4.37416e18	0.254624	0.127312	−	0.991863i	$-0.459365\pi$
$367$	4.37416e18	0.254624	0.127312	+	0.991863i	$0.459365\pi$
$368$	−1.47918e19	−0.843653
$369$	−3.24831e18	−0.181535
$370$	−1.31519e19	−0.720237
$371$	0	0
$372$	−6.27773e18	−0.330164
$373$	2.25672e19	1.16322	0.581608	−	0.813469i	$-0.302424\pi$
$373$	2.25672e19	1.16322	0.581608	+	0.813469i	$0.302424\pi$
$374$	1.00274e19	0.506584
$375$	−1.09049e19	−0.539992
$376$	−3.83593e18	−0.186192
$377$	1.85918e19	0.884631
$378$	0	0
$379$	−2.49747e19	−1.14211	−0.571053	−	0.820913i	$-0.693465\pi$
$379$	−2.49747e19	−1.14211	−0.571053	+	0.820913i	$0.693465\pi$
$380$	−1.54249e19	−0.691584
$381$	1.07333e19	0.471841
$382$	1.46602e19	0.631925
$383$	−5.61575e18	−0.237365	−0.118683	−	0.992932i	$-0.537867\pi$
$383$	−5.61575e18	−0.237365	−0.118683	+	0.992932i	$0.537867\pi$
$384$	−1.12060e19	−0.464481
$385$	0	0
$386$	5.18250e18	0.206602
$387$	1.33675e18	0.0522660
$388$	−1.54647e19	−0.593065
$389$	2.11288e19	0.794792	0.397396	−	0.917647i	$-0.369914\pi$
$389$	2.11288e19	0.794792	0.397396	+	0.917647i	$0.369914\pi$
$390$	−5.51669e19	−2.03561
$391$	−1.42099e19	−0.514360
$392$	0	0
$393$	7.67716e18	0.267458
$394$	3.81667e19	1.30456
$395$	1.56259e19	0.524043
$396$	3.57898e18	0.117773
$397$	2.47105e19	0.797909	0.398955	−	0.916971i	$-0.369373\pi$
$397$	2.47105e19	0.797909	0.398955	+	0.916971i	$0.369373\pi$
$398$	1.71822e19	0.544448
$399$	0	0
$400$	−6.32848e19	−1.93130
$401$	1.37114e18	0.0410675	0.0205338	−	0.999789i	$-0.493463\pi$
$401$	1.37114e18	0.0410675	0.0205338	+	0.999789i	$0.493463\pi$
$402$	2.88673e19	0.848614
$403$	5.01331e19	1.44656
$404$	−1.23228e19	−0.349019
$405$	−6.42174e18	−0.178542
$406$	0	0
$407$	−6.81381e18	−0.182571
$408$	−6.94674e18	−0.182738
$409$	4.55723e19	1.17700	0.588500	−	0.808497i	$-0.299719\pi$
$409$	4.55723e19	1.17700	0.588500	+	0.808497i	$0.299719\pi$
$410$	−4.46102e19	−1.13124
$411$	−1.64856e19	−0.410479
$412$	4.92638e19	1.20448
$413$	0	0
$414$	−1.26302e19	−0.297787
$415$	1.06130e20	2.45741
$416$	−8.60451e19	−1.95671
$417$	−2.80975e19	−0.627549
$418$	−1.99008e19	−0.436564
$419$	5.38156e18	0.115959	0.0579793	−	0.998318i	$-0.481534\pi$
$419$	5.38156e18	0.115959	0.0579793	+	0.998318i	$0.481534\pi$
$420$	0	0
$421$	−2.32562e19	−0.483530	−0.241765	−	0.970335i	$-0.577726\pi$
$421$	−2.32562e19	−0.483530	−0.241765	+	0.970335i	$0.577726\pi$
$422$	2.87709e17	0.00587638
$423$	−7.27333e18	−0.145942
$424$	6.67472e18	0.131580
$425$	−6.07952e19	−1.17748
$426$	6.81404e19	1.29668
$427$	0	0
$428$	−2.71797e19	−0.499363
$429$	−2.85812e19	−0.516001
$430$	1.83581e19	0.325698
$431$	9.18432e18	0.160128	0.0800640	−	0.996790i	$-0.474488\pi$
$431$	9.18432e18	0.160128	0.0800640	+	0.996790i	$0.474488\pi$
$432$	−1.37111e19	−0.234934
$433$	4.10991e19	0.692107	0.346053	−	0.938215i	$-0.387522\pi$
$433$	4.10991e19	0.692107	0.346053	+	0.938215i	$0.387522\pi$
$434$	0	0
$435$	2.97216e19	0.483508
$436$	2.00762e19	0.321020
$437$	2.82016e19	0.443265
$438$	5.40434e19	0.835000
$439$	1.19914e20	1.82132	0.910661	−	0.413154i	$-0.135573\pi$
$439$	1.19914e20	1.82132	0.910661	+	0.413154i	$0.135573\pi$
$440$	−2.40973e19	−0.359810
$441$	0	0
$442$	−1.13154e20	−1.63306
$443$	−8.85273e19	−1.25617	−0.628086	−	0.778144i	$-0.716162\pi$
$443$	−8.85273e19	−1.25617	−0.628086	+	0.778144i	$0.716162\pi$
$444$	−9.62829e18	−0.134331
$445$	6.15639e19	0.844552
$446$	−1.91487e20	−2.58302
$447$	−4.12023e19	−0.546532
$448$	0	0
$449$	−1.33463e20	−1.71203	−0.856016	−	0.516949i	$-0.827068\pi$
$449$	−1.33463e20	−1.71203	−0.856016	+	0.516949i	$0.827068\pi$
$450$	−5.40363e19	−0.681697
$451$	−2.31119e19	−0.286755
$452$	−4.38255e19	−0.534797
$453$	−6.26407e19	−0.751831
$454$	−1.35501e20	−1.59964
$455$	0	0
$456$	1.37868e19	0.157480
$457$	−1.02907e19	−0.115631	−0.0578154	−	0.998327i	$-0.518413\pi$
$457$	−1.02907e19	−0.115631	−0.0578154	+	0.998327i	$0.518413\pi$
$458$	2.10691e20	2.32893
$459$	−1.31718e19	−0.143235
$460$	−6.96528e19	−0.745169
$461$	−1.62617e20	−1.71163	−0.855815	−	0.517282i	$-0.826944\pi$
$461$	−1.62617e20	−1.71163	−0.855815	+	0.517282i	$0.826944\pi$
$462$	0	0
$463$	7.89872e18	0.0804821	0.0402410	−	0.999190i	$-0.487187\pi$
$463$	7.89872e18	0.0804821	0.0402410	+	0.999190i	$0.487187\pi$
$464$	6.34590e19	0.636222
$465$	8.01446e19	0.790637
$466$	9.30002e19	0.902796
$467$	−1.94989e20	−1.86266	−0.931328	−	0.364182i	$-0.881349\pi$
$467$	−1.94989e20	−1.86266	−0.931328	+	0.364182i	$0.881349\pi$
$468$	−4.03868e19	−0.379661
$469$	0	0
$470$	−9.98871e19	−0.909443
$471$	9.37859e19	0.840390
$472$	−1.86651e19	−0.164613
$473$	9.51107e18	0.0825602
$474$	2.84874e19	0.243397
$475$	1.20656e20	1.01473
$476$	0	0
$477$	1.26560e19	0.103136
$478$	−2.49841e20	−2.00426
$479$	1.29233e20	1.02061	0.510303	−	0.859995i	$-0.329533\pi$
$479$	1.29233e20	1.02061	0.510303	+	0.859995i	$0.329533\pi$
$480$	−1.37555e20	−1.06947
$481$	7.68902e19	0.588549
$482$	−6.97326e19	−0.525513
$483$	0	0
$484$	−6.63848e19	−0.484985
$485$	1.97430e20	1.42020
$486$	−1.17074e19	−0.0829254
$487$	−1.65971e19	−0.115762	−0.0578808	−	0.998323i	$-0.518434\pi$
$487$	−1.65971e19	−0.115762	−0.0578808	+	0.998323i	$0.518434\pi$
$488$	1.07610e20	0.739100
$489$	3.39132e19	0.229378
$490$	0	0
$491$	−5.60231e19	−0.367499	−0.183749	−	0.982973i	$-0.558823\pi$
$491$	−5.60231e19	−0.367499	−0.183749	+	0.982973i	$0.558823\pi$
$492$	−3.26584e19	−0.210987
$493$	6.09626e19	0.387893
$494$	2.24570e20	1.40734
$495$	−4.56910e19	−0.282028
$496$	1.71118e20	1.04036
$497$	0	0
$498$	1.93484e20	1.14137
$499$	2.21529e20	1.28729	0.643646	−	0.765324i	$-0.277421\pi$
$499$	2.21529e20	1.28729	0.643646	+	0.765324i	$0.277421\pi$
$500$	−1.09637e20	−0.627601
$501$	−1.78161e20	−1.00469
$502$	1.87440e20	1.04132
$503$	−1.93755e20	−1.06046	−0.530229	−	0.847855i	$-0.677894\pi$
$503$	−1.93755e20	−1.06046	−0.530229	+	0.847855i	$0.677894\pi$
$504$	0	0
$505$	1.57319e20	0.835789
$506$	−8.98641e19	−0.470389
$507$	2.10580e20	1.08607
$508$	1.07912e20	0.548393
$509$	2.73407e20	1.36907	0.684536	−	0.728979i	$-0.260005\pi$
$509$	2.73407e20	1.36907	0.684536	+	0.728979i	$0.260005\pi$
$510$	−1.80892e20	−0.892573
$511$	0	0
$512$	−1.85350e20	−0.888111
$513$	2.61412e19	0.123437
$514$	−3.18128e20	−1.48040
$515$	−6.28927e20	−2.88434
$516$	1.34397e19	0.0607457
$517$	−5.17501e19	−0.230532
$518$	0	0
$519$	−1.06665e20	−0.461601
$520$	2.71925e20	1.15991
$521$	−3.07240e20	−1.29180	−0.645900	−	0.763422i	$-0.723518\pi$
$521$	−3.07240e20	−1.29180	−0.645900	+	0.763422i	$0.723518\pi$
$522$	5.41851e19	0.224569
$523$	−1.31068e20	−0.535470	−0.267735	−	0.963493i	$-0.586275\pi$
$523$	−1.31068e20	−0.535470	−0.267735	+	0.963493i	$0.586275\pi$
$524$	7.71858e19	0.310851
$525$	0	0
$526$	−6.42069e20	−2.51298
$527$	1.64386e20	0.634287
$528$	−9.75554e19	−0.371105
$529$	−1.39288e20	−0.522391
$530$	1.73809e20	0.642693
$531$	−3.53910e19	−0.129028
$532$	0	0
$533$	2.60806e20	0.924406
$534$	1.12236e20	0.392260
$535$	3.46990e20	1.19581
$536$	−1.42290e20	−0.483548
$537$	−1.69790e20	−0.568991
$538$	−4.00597e20	−1.32385
$539$	0	0
$540$	−6.45639e19	−0.207509
$541$	−2.52109e20	−0.799114	−0.399557	−	0.916708i	$-0.630836\pi$
$541$	−2.52109e20	−0.799114	−0.399557	+	0.916708i	$0.630836\pi$
$542$	−3.90781e20	−1.22163
$543$	2.76011e19	0.0850996
$544$	−2.82142e20	−0.857976
$545$	−2.56303e20	−0.768740
$546$	0	0
$547$	3.05687e20	0.892014	0.446007	−	0.895029i	$-0.352846\pi$
$547$	3.05687e20	0.892014	0.446007	+	0.895029i	$0.352846\pi$
$548$	−1.65745e20	−0.477076
$549$	2.04040e20	0.579325
$550$	−3.84471e20	−1.07682
$551$	−1.20989e20	−0.334278
$552$	6.22556e19	0.169682
$553$	0	0
$554$	1.52195e20	0.403718
$555$	1.22920e20	0.321680
$556$	−2.82491e20	−0.729364
$557$	−6.09048e20	−1.55145	−0.775725	−	0.631071i	$-0.782616\pi$
$557$	−6.09048e20	−1.55145	−0.775725	+	0.631071i	$0.782616\pi$
$558$	1.46110e20	0.367219
$559$	−1.07327e20	−0.266147
$560$	0	0
$561$	−9.37177e19	−0.226256
$562$	−3.35923e20	−0.800236
$563$	−8.90973e19	−0.209436	−0.104718	−	0.994502i	$-0.533394\pi$
$563$	−8.90973e19	−0.209436	−0.104718	+	0.994502i	$0.533394\pi$
$564$	−7.31257e19	−0.169620
$565$	5.59499e20	1.28067
$566$	6.08909e20	1.37540
$567$	0	0
$568$	−3.35873e20	−0.738860
$569$	5.02180e19	0.109023	0.0545114	−	0.998513i	$-0.482640\pi$
$569$	5.02180e19	0.109023	0.0545114	+	0.998513i	$0.482640\pi$
$570$	3.59006e20	0.769201
$571$	1.61396e20	0.341289	0.170645	−	0.985333i	$-0.445415\pi$
$571$	1.61396e20	0.341289	0.170645	+	0.985333i	$0.445415\pi$
$572$	−2.87354e20	−0.599718
$573$	−1.37017e20	−0.282237
$574$	0	0
$575$	5.44838e20	1.09335
$576$	−4.53392e19	−0.0898060
$577$	−8.65106e20	−1.69142	−0.845709	−	0.533644i	$-0.820822\pi$
$577$	−8.65106e20	−1.69142	−0.845709	+	0.533644i	$0.820822\pi$
$578$	2.98776e20	0.576616
$579$	−4.84364e19	−0.0922747
$580$	2.98820e20	0.561953
$581$	0	0
$582$	3.59931e20	0.659625
$583$	9.00480e19	0.162915
$584$	−2.66387e20	−0.475791
$585$	5.15599e20	0.909165
$586$	−2.68750e20	−0.467860
$587$	−2.88926e20	−0.496593	−0.248296	−	0.968684i	$-0.579871\pi$
$587$	−2.88926e20	−0.496593	−0.248296	+	0.968684i	$0.579871\pi$
$588$	0	0
$589$	−3.26247e20	−0.546615
$590$	−4.86036e20	−0.804042
$591$	−3.56712e20	−0.582655
$592$	2.62447e20	0.423281
$593$	8.44010e20	1.34412	0.672059	−	0.740498i	$-0.265410\pi$
$593$	8.44010e20	1.34412	0.672059	+	0.740498i	$0.265410\pi$
$594$	−8.32986e19	−0.130990
$595$	0	0
$596$	−4.14246e20	−0.635202
$597$	−1.60588e20	−0.243167
$598$	1.01407e21	1.51638
$599$	8.76702e19	0.129465	0.0647323	−	0.997903i	$-0.479381\pi$
$599$	8.76702e19	0.129465	0.0647323	+	0.997903i	$0.479381\pi$
$600$	2.66352e20	0.388437
$601$	−3.17659e20	−0.457512	−0.228756	−	0.973484i	$-0.573466\pi$
$601$	−3.17659e20	−0.457512	−0.228756	+	0.973484i	$0.573466\pi$
$602$	0	0
$603$	−2.69798e20	−0.379017
$604$	−6.29787e20	−0.873809
$605$	8.47502e20	1.16138
$606$	2.86806e20	0.388190
$607$	−4.16004e19	−0.0556138	−0.0278069	−	0.999613i	$-0.508852\pi$
$607$	−4.16004e19	−0.0556138	−0.0278069	+	0.999613i	$0.508852\pi$
$608$	5.59949e20	0.739387
$609$	0	0
$610$	2.80215e21	3.61008
$611$	5.83972e20	0.743161
$612$	−1.32428e20	−0.166473
$613$	1.14258e21	1.41883	0.709417	−	0.704789i	$-0.248958\pi$
$613$	1.14258e21	1.41883	0.709417	+	0.704789i	$0.248958\pi$
$614$	−1.29488e21	−1.58843
$615$	4.16934e20	0.505247
$616$	0	0
$617$	−1.25983e21	−1.48996	−0.744979	−	0.667088i	$-0.767541\pi$
$617$	−1.25983e21	−1.48996	−0.744979	+	0.667088i	$0.767541\pi$
$618$	−1.14659e21	−1.33966
$619$	1.25354e21	1.44696	0.723482	−	0.690343i	$-0.242541\pi$
$619$	1.25354e21	1.44696	0.723482	+	0.690343i	$0.242541\pi$
$620$	8.05771e20	0.918912
$621$	1.18043e20	0.133001
$622$	6.02340e20	0.670523
$623$	0	0
$624$	1.10086e21	1.19632
$625$	−7.37181e19	−0.0791542
$626$	−1.38067e21	−1.46481
$627$	1.85996e20	0.194983
$628$	9.42919e20	0.976736
$629$	2.52123e20	0.258067
$630$	0	0
$631$	1.47433e21	1.47359	0.736794	−	0.676118i	$-0.236339\pi$
$631$	1.47433e21	1.47359	0.736794	+	0.676118i	$0.236339\pi$
$632$	−1.40418e20	−0.138690
$633$	−2.68897e18	−0.00262457
$634$	−1.20699e21	−1.16421
$635$	−1.37766e21	−1.31322
$636$	1.27243e20	0.119868
$637$	0	0
$638$	3.85529e20	0.354734
$639$	−6.36851e20	−0.579137
$640$	1.43834e21	1.29274
$641$	1.46302e21	1.29962	0.649808	−	0.760098i	$-0.274849\pi$
$641$	1.46302e21	1.29962	0.649808	+	0.760098i	$0.274849\pi$
$642$	6.32591e20	0.555406
$643$	−4.28205e20	−0.371595	−0.185798	−	0.982588i	$-0.559487\pi$
$643$	−4.28205e20	−0.371595	−0.185798	+	0.982588i	$0.559487\pi$
$644$	0	0
$645$	−1.71578e20	−0.145466
$646$	7.36363e20	0.617089
$647$	−3.19295e20	−0.264491	−0.132245	−	0.991217i	$-0.542219\pi$
$647$	−3.19295e20	−0.264491	−0.132245	+	0.991217i	$0.542219\pi$
$648$	5.77072e19	0.0472517
$649$	−2.51808e20	−0.203814
$650$	4.33855e21	3.47132
$651$	0	0
$652$	3.40961e20	0.266593
$653$	2.05259e21	1.58655	0.793276	−	0.608863i	$-0.208374\pi$
$653$	2.05259e21	1.58655	0.793276	+	0.608863i	$0.208374\pi$
$654$	−4.67262e20	−0.357048
$655$	−9.85393e20	−0.744389
$656$	8.90200e20	0.664827
$657$	−5.05098e20	−0.372937
$658$	0	0
$659$	−6.54661e20	−0.472472	−0.236236	−	0.971696i	$-0.575914\pi$
$659$	−6.54661e20	−0.472472	−0.236236	+	0.971696i	$0.575914\pi$
$660$	−4.59375e20	−0.327784
$661$	1.05234e19	0.00742415	0.00371207	−	0.999993i	$-0.498818\pi$
$661$	1.05234e19	0.00742415	0.00371207	+	0.999993i	$0.498818\pi$
$662$	−9.70727e20	−0.677115
$663$	1.05755e21	0.729375
$664$	−9.53708e20	−0.650361
$665$	0	0
$666$	2.24093e20	0.149407
$667$	−5.46338e20	−0.360178
$668$	−1.79122e21	−1.16769
$669$	1.78967e21	1.15366
$670$	−3.70523e21	−2.36186
$671$	1.45175e21	0.915110
$672$	0	0
$673$	1.38475e21	0.853606	0.426803	−	0.904345i	$-0.359640\pi$
$673$	1.38475e21	0.853606	0.426803	+	0.904345i	$0.359640\pi$
$674$	−8.96225e20	−0.546346
$675$	5.05031e20	0.304467
$676$	2.11716e21	1.26228
$677$	2.38617e21	1.40697	0.703487	−	0.710708i	$-0.251625\pi$
$677$	2.38617e21	1.40697	0.703487	+	0.710708i	$0.251625\pi$
$678$	1.02001e21	0.594817
$679$	0	0
$680$	8.91641e20	0.508596
$681$	1.26641e21	0.714448
$682$	1.03958e21	0.580064
$683$	7.22891e20	0.398949	0.199475	−	0.979903i	$-0.436076\pi$
$683$	7.22891e20	0.398949	0.199475	+	0.979903i	$0.436076\pi$
$684$	2.62822e20	0.143463
$685$	2.11599e21	1.14244
$686$	0	0
$687$	−1.96915e21	−1.04017
$688$	−3.66338e20	−0.191412
$689$	−1.01614e21	−0.525183
$690$	1.62113e21	0.828800
$691$	3.35500e21	1.69671	0.848355	−	0.529428i	$-0.177593\pi$
$691$	3.35500e21	1.69671	0.848355	+	0.529428i	$0.177593\pi$
$692$	−1.07241e21	−0.536492
$693$	0	0
$694$	−3.02563e21	−1.48122
$695$	3.60643e21	1.74659
$696$	−2.67085e20	−0.127962
$697$	8.55180e20	0.405333
$698$	1.00282e21	0.470227
$699$	−8.69194e20	−0.403216
$700$	0	0
$701$	−2.23696e21	−1.01572	−0.507859	−	0.861440i	$-0.669563\pi$
$701$	−2.23696e21	−1.01572	−0.507859	+	0.861440i	$0.669563\pi$
$702$	9.39980e20	0.422270
$703$	−5.00372e20	−0.222397
$704$	−3.22591e20	−0.141859
$705$	9.33560e20	0.406185
$706$	−2.90060e21	−1.24869
$707$	0	0
$708$	−3.55819e20	−0.149962
$709$	9.51303e20	0.396709	0.198355	−	0.980130i	$-0.436440\pi$
$709$	9.51303e20	0.396709	0.198355	+	0.980130i	$0.436440\pi$
$710$	−8.74609e21	−3.60891
$711$	−2.66247e20	−0.108708
$712$	−5.53227e20	−0.223513
$713$	−1.47320e21	−0.588968
$714$	0	0
$715$	3.66851e21	1.43613
$716$	−1.70706e21	−0.661305
$717$	2.33505e21	0.895165
$718$	−3.10682e21	−1.17864
$719$	2.88492e21	1.08310	0.541549	−	0.840669i	$-0.317838\pi$
$719$	2.88492e21	1.08310	0.541549	+	0.840669i	$0.317838\pi$
$720$	1.75988e21	0.653866
$721$	0	0
$722$	2.09097e21	0.760885
$723$	6.51732e20	0.234710
$724$	2.77500e20	0.0989064
$725$	−2.33743e21	−0.824524
$726$	1.54507e21	0.539414
$727$	5.91622e20	0.204426	0.102213	−	0.994763i	$-0.467408\pi$
$727$	5.91622e20	0.204426	0.102213	+	0.994763i	$0.467408\pi$
$728$	0	0
$729$	1.09419e20	0.0370370
$730$	−6.93668e21	−2.32397
$731$	−3.51926e20	−0.116700
$732$	2.05141e21	0.673315
$733$	−2.12276e21	−0.689639	−0.344819	−	0.938669i	$-0.612060\pi$
$733$	−2.12276e21	−0.689639	−0.344819	+	0.938669i	$0.612060\pi$
$734$	−1.02355e21	−0.329147
$735$	0	0
$736$	2.52851e21	0.796675
$737$	−1.91962e21	−0.598701
$738$	7.60105e20	0.234667
$739$	1.37093e21	0.418970	0.209485	−	0.977812i	$-0.432821\pi$
$739$	1.37093e21	0.418970	0.209485	+	0.977812i	$0.432821\pi$
$740$	1.23583e21	0.373870
$741$	−2.09886e21	−0.628561
$742$	0	0
$743$	3.08860e21	0.906452	0.453226	−	0.891396i	$-0.350273\pi$
$743$	3.08860e21	0.906452	0.453226	+	0.891396i	$0.350273\pi$
$744$	−7.20197e20	−0.209245
$745$	5.28848e21	1.52110
$746$	−5.28071e21	−1.50367
$747$	−1.80833e21	−0.509769
$748$	−9.42233e20	−0.262964
$749$	0	0
$750$	2.55175e21	0.698036
$751$	−3.91840e21	−1.06123	−0.530615	−	0.847613i	$-0.678039\pi$
$751$	−3.91840e21	−1.06123	−0.530615	+	0.847613i	$0.678039\pi$
$752$	1.99326e21	0.534477
$753$	−1.75184e21	−0.465086
$754$	−4.35049e21	−1.14355
$755$	8.04018e21	2.09249
$756$	0	0
$757$	5.36424e21	1.36864	0.684319	−	0.729182i	$-0.260099\pi$
$757$	5.36424e21	1.36864	0.684319	+	0.729182i	$0.260099\pi$
$758$	5.84409e21	1.47638
$759$	8.39884e20	0.210090
$760$	−1.76958e21	−0.438298
$761$	−2.53746e21	−0.622320	−0.311160	−	0.950358i	$-0.600717\pi$
$761$	−2.53746e21	−0.622320	−0.311160	+	0.950358i	$0.600717\pi$
$762$	−2.51159e21	−0.609939
$763$	0	0
$764$	−1.37756e21	−0.328027
$765$	1.69065e21	0.398650
$766$	1.31409e21	0.306837
$767$	2.84152e21	0.657031
$768$	3.30153e21	0.755974
$769$	−2.55340e21	−0.578990	−0.289495	−	0.957180i	$-0.593487\pi$
$769$	−2.55340e21	−0.578990	−0.289495	+	0.957180i	$0.593487\pi$
$770$	0	0
$771$	2.97327e21	0.661191
$772$	−4.86977e20	−0.107246
$773$	−1.21187e21	−0.264307	−0.132154	−	0.991229i	$-0.542189\pi$
$773$	−1.21187e21	−0.264307	−0.132154	+	0.991229i	$0.542189\pi$
$774$	−3.12801e20	−0.0675632
$775$	−6.30289e21	−1.34827
$776$	−1.77415e21	−0.375860
$777$	0	0
$778$	−4.94415e21	−1.02741
$779$	−1.69722e21	−0.349308
$780$	5.18381e21	1.05667
$781$	−4.53122e21	−0.914814
$782$	3.32513e21	0.664902
$783$	−5.06422e20	−0.100300
$784$	0	0
$785$	−1.20378e22	−2.33897
$786$	−1.79646e21	−0.345738
$787$	−5.53051e21	−1.05428	−0.527139	−	0.849779i	$-0.676735\pi$
$787$	−5.53051e21	−1.05428	−0.527139	+	0.849779i	$0.676735\pi$
$788$	−3.58636e21	−0.677186
$789$	6.00088e21	1.12237
$790$	−3.65646e21	−0.677420
$791$	0	0
$792$	4.10589e20	0.0746395
$793$	−1.63822e22	−2.95002
$794$	−5.78227e21	−1.03144
$795$	−1.62445e21	−0.287046
$796$	−1.61454e21	−0.282619
$797$	−9.38793e21	−1.62792	−0.813959	−	0.580922i	$-0.802692\pi$
$797$	−9.38793e21	−1.62792	−0.813959	+	0.580922i	$0.802692\pi$
$798$	0	0
$799$	1.91484e21	0.325861
$800$	1.08179e22	1.82376
$801$	−1.04898e21	−0.175195
$802$	−3.20846e20	−0.0530871
$803$	−3.59379e21	−0.589097
$804$	−2.71254e21	−0.440509
$805$	0	0
$806$	−1.17311e22	−1.86994
$807$	3.74404e21	0.591274
$808$	−1.41370e21	−0.221194
$809$	6.04998e21	0.937864	0.468932	−	0.883234i	$-0.344639\pi$
$809$	6.04998e21	0.937864	0.468932	+	0.883234i	$0.344639\pi$
$810$	1.50269e21	0.230798
$811$	3.97224e21	0.604475	0.302238	−	0.953233i	$-0.402266\pi$
$811$	3.97224e21	0.604475	0.302238	+	0.953233i	$0.402266\pi$
$812$	0	0
$813$	3.65230e21	0.545617
$814$	1.59443e21	0.236006
$815$	−4.35289e21	−0.638404
$816$	3.60972e21	0.524562
$817$	6.98446e20	0.100570
$818$	−1.06639e22	−1.52148
$819$	0	0
$820$	4.19183e21	0.587219
$821$	−1.22139e21	−0.169543	−0.0847717	−	0.996400i	$-0.527016\pi$
$821$	−1.22139e21	−0.169543	−0.0847717	+	0.996400i	$0.527016\pi$
$822$	3.85763e21	0.530618
$823$	−7.21274e21	−0.983109	−0.491554	−	0.870847i	$-0.663571\pi$
$823$	−7.21274e21	−0.983109	−0.491554	+	0.870847i	$0.663571\pi$
$824$	5.65167e21	0.763348
$825$	3.59332e21	0.480941
$826$	0	0
$827$	−1.42305e22	−1.87038	−0.935189	−	0.354150i	$-0.884770\pi$
$827$	−1.42305e22	−1.87038	−0.935189	+	0.354150i	$0.884770\pi$
$828$	1.18680e21	0.154579
$829$	−3.26530e21	−0.421468	−0.210734	−	0.977543i	$-0.567585\pi$
$829$	−3.26530e21	−0.421468	−0.210734	+	0.977543i	$0.567585\pi$
$830$	−2.48344e22	−3.17665
$831$	−1.42244e21	−0.180313
$832$	3.64027e21	0.457307
$833$	0	0
$834$	6.57482e21	0.811220
$835$	2.28677e22	2.79623
$836$	1.86999e21	0.226617
$837$	−1.36557e21	−0.164011
$838$	−1.25928e21	−0.149897
$839$	6.61493e21	0.780388	0.390194	−	0.920733i	$-0.372408\pi$
$839$	6.61493e21	0.780388	0.390194	+	0.920733i	$0.372408\pi$
$840$	0	0
$841$	−6.28533e21	−0.728380
$842$	5.44195e21	0.625049
$843$	3.13959e21	0.357410
$844$	−2.70348e19	−0.00305039
$845$	−2.70288e22	−3.02274
$846$	1.70196e21	0.188656
$847$	0	0
$848$	−3.46837e21	−0.377709
$849$	−5.69096e21	−0.614296
$850$	1.42261e22	1.52210
$851$	−2.25949e21	−0.239628
$852$	−6.40287e21	−0.673097
$853$	−1.18003e22	−1.22963	−0.614815	−	0.788672i	$-0.710769\pi$
$853$	−1.18003e22	−1.22963	−0.614815	+	0.788672i	$0.710769\pi$
$854$	0	0
$855$	−3.35532e21	−0.343549
$856$	−3.11812e21	−0.316475
$857$	−2.98481e21	−0.300304	−0.150152	−	0.988663i	$-0.547976\pi$
$857$	−2.98481e21	−0.300304	−0.150152	+	0.988663i	$0.547976\pi$
$858$	6.68800e21	0.667024
$859$	−2.59672e21	−0.256730	−0.128365	−	0.991727i	$-0.540973\pi$
$859$	−2.59672e21	−0.256730	−0.128365	+	0.991727i	$0.540973\pi$
$860$	−1.72503e21	−0.169067
$861$	0	0
$862$	−2.14913e21	−0.206994
$863$	2.37732e21	0.226990	0.113495	−	0.993539i	$-0.463795\pi$
$863$	2.37732e21	0.226990	0.113495	+	0.993539i	$0.463795\pi$
$864$	2.34378e21	0.221852
$865$	1.36909e22	1.28473
$866$	−9.61719e21	−0.894672
$867$	−2.79240e21	−0.257534
$868$	0	0
$869$	−1.89436e21	−0.171717
$870$	−6.95486e21	−0.625020
$871$	2.16619e22	1.93002
$872$	2.30319e21	0.203449
$873$	−3.36397e21	−0.294609
$874$	−6.59917e21	−0.572999
$875$	0	0
$876$	−5.07823e21	−0.433442
$877$	−6.08094e21	−0.514605	−0.257302	−	0.966331i	$-0.582834\pi$
$877$	−6.08094e21	−0.514605	−0.257302	+	0.966331i	$0.582834\pi$
$878$	−2.80599e22	−2.35439
$879$	2.51178e21	0.208961
$880$	1.25216e22	1.03286
$881$	−4.02742e20	−0.0329387	−0.0164694	−	0.999864i	$-0.505243\pi$
$881$	−4.02742e20	−0.0329387	−0.0164694	+	0.999864i	$0.505243\pi$
$882$	0	0
$883$	−1.25847e22	−1.01190	−0.505951	−	0.862562i	$-0.668858\pi$
$883$	−1.25847e22	−1.01190	−0.505951	+	0.862562i	$0.668858\pi$
$884$	1.06326e22	0.847710
$885$	4.54257e21	0.359110
$886$	2.07154e22	1.62383
$887$	3.44552e21	0.267811	0.133905	−	0.990994i	$-0.457248\pi$
$887$	3.44552e21	0.267811	0.133905	+	0.990994i	$0.457248\pi$
$888$	−1.10458e21	−0.0851336
$889$	0	0
$890$	−1.44060e22	−1.09173
$891$	7.78521e20	0.0585043
$892$	1.79932e22	1.34083
$893$	−3.80027e21	−0.280820
$894$	9.64134e21	0.706490
$895$	2.17932e22	1.58361
$896$	0	0
$897$	−9.47764e21	−0.677262
$898$	3.12303e22	2.21311
$899$	6.32024e21	0.444157
$900$	5.07756e21	0.353864
$901$	−3.33193e21	−0.230282
$902$	5.40819e21	0.370683
$903$	0	0
$904$	−5.02778e21	−0.338932
$905$	−3.54271e21	−0.236849
$906$	1.46579e22	0.971876
$907$	2.23611e22	1.47041	0.735204	−	0.677846i	$-0.237086\pi$
$907$	2.23611e22	1.47041	0.735204	+	0.677846i	$0.237086\pi$
$908$	1.27324e22	0.830361
$909$	−2.68054e21	−0.173377
$910$	0	0
$911$	2.26193e22	1.43910	0.719550	−	0.694441i	$-0.244348\pi$
$911$	2.26193e22	1.43910	0.719550	+	0.694441i	$0.244348\pi$
$912$	−7.16399e21	−0.452057
$913$	−1.28664e22	−0.805240
$914$	2.40802e21	0.149474
$915$	−2.61893e22	−1.61237
$916$	−1.97978e22	−1.20893
$917$	0	0
$918$	3.08219e21	0.185157
$919$	−2.65500e21	−0.158197	−0.0790986	−	0.996867i	$-0.525204\pi$
$919$	−2.65500e21	−0.158197	−0.0790986	+	0.996867i	$0.525204\pi$
$920$	−7.99075e21	−0.472258
$921$	1.21022e22	0.709440
$922$	3.80525e22	2.21259
$923$	5.11324e22	2.94906
$924$	0	0
$925$	−9.66689e21	−0.548559
$926$	−1.84830e21	−0.104038
$927$	1.07162e22	0.598331
$928$	−1.08477e22	−0.600795
$929$	−1.28056e22	−0.703532	−0.351766	−	0.936088i	$-0.614419\pi$
$929$	−1.28056e22	−0.703532	−0.351766	+	0.936088i	$0.614419\pi$
$930$	−1.87538e22	−1.02204
$931$	0	0
$932$	−8.73884e21	−0.468635
$933$	−5.62956e21	−0.299476
$934$	4.56273e22	2.40782
$935$	1.20290e22	0.629714
$936$	−4.63328e21	−0.240613
$937$	6.63077e21	0.341599	0.170800	−	0.985306i	$-0.445365\pi$
$937$	6.63077e21	0.341599	0.170800	+	0.985306i	$0.445365\pi$
$938$	0	0
$939$	1.29040e22	0.654231
$940$	9.38598e21	0.472085
$941$	3.64056e21	0.181654	0.0908272	−	0.995867i	$-0.471049\pi$
$941$	3.64056e21	0.181654	0.0908272	+	0.995867i	$0.471049\pi$
$942$	−2.19459e22	−1.08635
$943$	−7.66400e21	−0.376372
$944$	9.69889e21	0.472533
$945$	0	0
$946$	−2.22559e21	−0.106724
$947$	3.77681e21	0.179680	0.0898401	−	0.995956i	$-0.471364\pi$
$947$	3.77681e21	0.179680	0.0898401	+	0.995956i	$0.471364\pi$
$948$	−2.67684e21	−0.126345
$949$	4.05540e22	1.89905
$950$	−2.82336e22	−1.31171
$951$	1.12807e22	0.519974
$952$	0	0
$953$	−2.57524e22	−1.16848	−0.584238	−	0.811582i	$-0.698607\pi$
$953$	−2.57524e22	−1.16848	−0.584238	+	0.811582i	$0.698607\pi$
$954$	−2.96150e21	−0.133321
$955$	1.75866e22	0.785520
$956$	2.34765e22	1.04040
$957$	−3.60322e21	−0.158435
$958$	−3.02406e22	−1.31932
$959$	0	0
$960$	5.81947e21	0.249948
$961$	−6.42267e21	−0.273710
$962$	−1.79923e22	−0.760806
$963$	−5.91230e21	−0.248061
$964$	6.55248e21	0.272790
$965$	6.21700e21	0.256818
$966$	0	0
$967$	2.79126e22	1.13528	0.567639	−	0.823277i	$-0.307857\pi$
$967$	2.79126e22	1.13528	0.567639	+	0.823277i	$0.307857\pi$
$968$	−7.61583e21	−0.307363
$969$	−6.88216e21	−0.275611
$970$	−4.61986e22	−1.83586
$971$	−2.84337e22	−1.12122	−0.560608	−	0.828081i	$-0.689432\pi$
$971$	−2.84337e22	−1.12122	−0.560608	+	0.828081i	$0.689432\pi$
$972$	1.10009e21	0.0430460
$973$	0	0
$974$	3.88372e21	0.149643
$975$	−4.05487e22	−1.55040
$976$	−5.59171e22	−2.12163
$977$	−2.39578e22	−0.902064	−0.451032	−	0.892508i	$-0.648944\pi$
$977$	−2.39578e22	−0.902064	−0.451032	+	0.892508i	$0.648944\pi$
$978$	−7.93568e21	−0.296512
$979$	−7.46352e21	−0.276741
$980$	0	0
$981$	4.36710e21	0.159469
$982$	1.31094e22	0.475058
$983$	−2.01688e22	−0.725319	−0.362660	−	0.931922i	$-0.618131\pi$
$983$	−2.01688e22	−0.725319	−0.362660	+	0.931922i	$0.618131\pi$
$984$	−3.74666e21	−0.133715
$985$	4.57853e22	1.62164
$986$	−1.42652e22	−0.501421
$987$	0	0
$988$	−2.11019e22	−0.730539
$989$	3.15391e21	0.108362
$990$	1.06917e22	0.364572
$991$	6.46390e21	0.218747	0.109374	−	0.994001i	$-0.465116\pi$
$991$	6.46390e21	0.218747	0.109374	+	0.994001i	$0.465116\pi$
$992$	−2.92508e22	−0.982426
$993$	9.07256e21	0.302420
$994$	0	0
$995$	2.06121e22	0.676781
$996$	−1.81809e22	−0.592475
$997$	−2.36491e22	−0.764895	−0.382447	−	0.923977i	$-0.624919\pi$
$997$	−2.36491e22	−0.764895	−0.382447	+	0.923977i	$0.624919\pi$
$998$	−5.18378e22	−1.66406
$999$	−2.09441e21	−0.0667298

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.16.a.a.1.1		1
7.6	odd	2		3.16.a.a.1.1	✓	1
21.20	even	2		9.16.a.d.1.1		1
28.27	even	2		48.16.a.g.1.1		1
35.13	even	4		75.16.b.a.49.2		2
35.27	even	4		75.16.b.a.49.1		2
35.34	odd	2		75.16.a.b.1.1		1
84.83	odd	2		144.16.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.16.a.a.1.1	✓	1	7.6	odd	2
9.16.a.d.1.1		1	21.20	even	2
48.16.a.g.1.1		1	28.27	even	2
75.16.a.b.1.1		1	35.34	odd	2
75.16.b.a.49.1		2	35.27	even	4
75.16.b.a.49.2		2	35.13	even	4
144.16.a.b.1.1		1	84.83	odd	2
147.16.a.a.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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