LMFDB - Embedded newform 147.12.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,12,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, 12, names="a")

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

chi := DirichletCharacter("147.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 12 $
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:	$112.946447542$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
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)

$f(q)$

$=$

$q+8.00000 q^{2} -243.000 q^{3} -1984.00 q^{4} -4390.00 q^{5} -1944.00 q^{6} -32256.0 q^{8} +59049.0 q^{9} +O(q^{10})$

\(q+8.00000 q^{2} -243.000 q^{3} -1984.00 q^{4} -4390.00 q^{5} -1944.00 q^{6} -32256.0 q^{8} +59049.0 q^{9} -35120.0 q^{10} -804836. q^{11} +482112. q^{12} -358294. q^{13} +1.06677e6 q^{15} +3.80518e6 q^{16} +5.65786e6 q^{17} +472392. q^{18} +1.46020e7 q^{19} +8.70976e6 q^{20} -6.43869e6 q^{22} -3.67248e7 q^{23} +7.83821e6 q^{24} -2.95560e7 q^{25} -2.86635e6 q^{26} -1.43489e7 q^{27} +5.11270e7 q^{29} +8.53416e6 q^{30} +2.08102e8 q^{31} +9.65018e7 q^{32} +1.95575e8 q^{33} +4.52629e7 q^{34} -1.17153e8 q^{36} +6.52146e8 q^{37} +1.16816e8 q^{38} +8.70654e7 q^{39} +1.41604e8 q^{40} -9.51188e8 q^{41} +8.58608e8 q^{43} +1.59679e9 q^{44} -2.59225e8 q^{45} -2.93798e8 q^{46} +1.33655e9 q^{47} -9.24660e8 q^{48} -2.36448e8 q^{50} -1.37486e9 q^{51} +7.10855e8 q^{52} +1.49760e9 q^{53} -1.14791e8 q^{54} +3.53323e9 q^{55} -3.54829e9 q^{57} +4.09016e8 q^{58} -7.06794e9 q^{59} -2.11647e9 q^{60} +7.64393e9 q^{61} +1.66482e9 q^{62} -7.02100e9 q^{64} +1.57291e9 q^{65} +1.56460e9 q^{66} -5.08676e9 q^{67} -1.12252e10 q^{68} +8.92413e9 q^{69} +2.80141e9 q^{71} -1.90468e9 q^{72} +7.84428e9 q^{73} +5.21717e9 q^{74} +7.18211e9 q^{75} -2.89704e10 q^{76} +6.96524e8 q^{78} -2.11567e10 q^{79} -1.67048e10 q^{80} +3.48678e9 q^{81} -7.60951e9 q^{82} +1.08949e10 q^{83} -2.48380e10 q^{85} +6.86886e9 q^{86} -1.24239e10 q^{87} +2.59608e10 q^{88} -7.07888e10 q^{89} -2.07380e9 q^{90} +7.28620e10 q^{92} -5.05688e10 q^{93} +1.06924e10 q^{94} -6.41028e10 q^{95} -2.34499e10 q^{96} -8.22238e10 q^{97} -4.75248e10 q^{99} +O(q^{100})\)

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 $

$2$	8.00000	0.176777	0.0883883	−	0.996086i	$-0.471828\pi$
$2$	8.00000	0.176777	0.0883883	+	0.996086i	$0.471828\pi$
$3$	−243.000	−0.577350
$3$	−243.000	−0.577350
$4$	−1984.00	−0.968750
$5$	−4390.00	−0.628246	−0.314123	−	0.949382i	$-0.601710\pi$
$5$	−4390.00	−0.628246	−0.314123	+	0.949382i	$0.601710\pi$
$6$	−1944.00	−0.102062
$7$	0	0
$7$	0	0
$8$	−32256.0	−0.348029
$9$	59049.0	0.333333
$10$	−35120.0	−0.111059
$11$	−804836.	−1.50677	−0.753386	−	0.657578i	$-0.771581\pi$
$11$	−804836.	−1.50677	−0.753386	+	0.657578i	$0.771581\pi$
$12$	482112.	0.559308
$13$	−358294.	−0.267640	−0.133820	−	0.991006i	$-0.542724\pi$
$13$	−358294.	−0.267640	−0.133820	+	0.991006i	$0.542724\pi$
$14$	0	0
$15$	1.06677e6	0.362718
$16$	3.80518e6	0.907227
$17$	5.65786e6	0.966459	0.483229	−	0.875494i	$-0.339464\pi$
$17$	5.65786e6	0.966459	0.483229	+	0.875494i	$0.339464\pi$
$18$	472392.	0.0589256
$19$	1.46020e7	1.35291	0.676453	−	0.736486i	$-0.263516\pi$
$19$	1.46020e7	1.35291	0.676453	+	0.736486i	$0.263516\pi$
$20$	8.70976e6	0.608613
$21$	0	0
$22$	−6.43869e6	−0.266362
$23$	−3.67248e7	−1.18975	−0.594876	−	0.803818i	$-0.702799\pi$
$23$	−3.67248e7	−1.18975	−0.594876	+	0.803818i	$0.702799\pi$
$24$	7.83821e6	0.200935
$25$	−2.95560e7	−0.605307
$26$	−2.86635e6	−0.0473125
$27$	−1.43489e7	−0.192450
$28$	0	0
$29$	5.11270e7	0.462872	0.231436	−	0.972850i	$-0.425658\pi$
$29$	5.11270e7	0.462872	0.231436	+	0.972850i	$0.425658\pi$
$30$	8.53416e6	0.0641201
$31$	2.08102e8	1.30553	0.652765	−	0.757560i	$-0.273609\pi$
$31$	2.08102e8	1.30553	0.652765	+	0.757560i	$0.273609\pi$
$32$	9.65018e7	0.508406
$33$	1.95575e8	0.869935
$34$	4.52629e7	0.170847
$35$	0	0
$36$	−1.17153e8	−0.322917
$37$	6.52146e8	1.54609	0.773046	−	0.634350i	$-0.218732\pi$
$37$	6.52146e8	1.54609	0.773046	+	0.634350i	$0.218732\pi$
$38$	1.16816e8	0.239162
$39$	8.70654e7	0.154522
$40$	1.41604e8	0.218648
$41$	−9.51188e8	−1.28220	−0.641099	−	0.767458i	$-0.721521\pi$
$41$	−9.51188e8	−1.28220	−0.641099	+	0.767458i	$0.721521\pi$
$42$	0	0
$43$	8.58608e8	0.890673	0.445337	−	0.895363i	$-0.353084\pi$
$43$	8.58608e8	0.890673	0.445337	+	0.895363i	$0.353084\pi$
$44$	1.59679e9	1.45969
$45$	−2.59225e8	−0.209415
$46$	−2.93798e8	−0.210320
$47$	1.33655e9	0.850058	0.425029	−	0.905180i	$-0.360264\pi$
$47$	1.33655e9	0.850058	0.425029	+	0.905180i	$0.360264\pi$
$48$	−9.24660e8	−0.523788
$49$	0	0
$50$	−2.36448e8	−0.107004
$51$	−1.37486e9	−0.557985
$52$	7.10855e8	0.259276
$53$	1.49760e9	0.491901	0.245950	−	0.969282i	$-0.420900\pi$
$53$	1.49760e9	0.491901	0.245950	+	0.969282i	$0.420900\pi$
$54$	−1.14791e8	−0.0340207
$55$	3.53323e9	0.946623
$56$	0	0
$57$	−3.54829e9	−0.781101
$58$	4.09016e8	0.0818250
$59$	−7.06794e9	−1.28708	−0.643542	−	0.765411i	$-0.722536\pi$
$59$	−7.06794e9	−1.28708	−0.643542	+	0.765411i	$0.722536\pi$
$60$	−2.11647e9	−0.351383
$61$	7.64393e9	1.15878	0.579392	−	0.815049i	$-0.303290\pi$
$61$	7.64393e9	1.15878	0.579392	+	0.815049i	$0.303290\pi$
$62$	1.66482e9	0.230787
$63$	0	0
$64$	−7.02100e9	−0.817352
$65$	1.57291e9	0.168144
$66$	1.56460e9	0.153784
$67$	−5.08676e9	−0.460288	−0.230144	−	0.973157i	$-0.573920\pi$
$67$	−5.08676e9	−0.460288	−0.230144	+	0.973157i	$0.573920\pi$
$68$	−1.12252e10	−0.936257
$69$	8.92413e9	0.686903
$70$	0	0
$71$	2.80141e9	0.184271	0.0921353	−	0.995746i	$-0.470631\pi$
$71$	2.80141e9	0.184271	0.0921353	+	0.995746i	$0.470631\pi$
$72$	−1.90468e9	−0.116010
$73$	7.84428e9	0.442871	0.221436	−	0.975175i	$-0.428926\pi$
$73$	7.84428e9	0.442871	0.221436	+	0.975175i	$0.428926\pi$
$74$	5.21717e9	0.273313
$75$	7.18211e9	0.349474
$76$	−2.89704e10	−1.31063
$77$	0	0
$78$	6.96524e8	0.0273159
$79$	−2.11567e10	−0.773567	−0.386784	−	0.922170i	$-0.626414\pi$
$79$	−2.11567e10	−0.773567	−0.386784	+	0.922170i	$0.626414\pi$
$80$	−1.67048e10	−0.569961
$81$	3.48678e9	0.111111
$82$	−7.60951e9	−0.226663
$83$	1.08949e10	0.303596	0.151798	−	0.988412i	$-0.451494\pi$
$83$	1.08949e10	0.303596	0.151798	+	0.988412i	$0.451494\pi$
$84$	0	0
$85$	−2.48380e10	−0.607173
$86$	6.86886e9	0.157450
$87$	−1.24239e10	−0.267239
$88$	2.59608e10	0.524401
$89$	−7.07888e10	−1.34375	−0.671876	−	0.740663i	$-0.734511\pi$
$89$	−7.07888e10	−1.34375	−0.671876	+	0.740663i	$0.734511\pi$
$90$	−2.07380e9	−0.0370197
$91$	0	0
$92$	7.28620e10	1.15257
$93$	−5.05688e10	−0.753749
$94$	1.06924e10	0.150270
$95$	−6.41028e10	−0.849957
$96$	−2.34499e10	−0.293528
$97$	−8.22238e10	−0.972194	−0.486097	−	0.873905i	$-0.661580\pi$
$97$	−8.22238e10	−0.972194	−0.486097	+	0.873905i	$0.661580\pi$
$98$	0	0
$99$	−4.75248e10	−0.502257
$100$	5.86392e10	0.586392
$101$	1.38046e11	1.30694	0.653472	−	0.756950i	$-0.273312\pi$
$101$	1.38046e11	1.30694	0.653472	+	0.756950i	$0.273312\pi$
$102$	−1.09989e10	−0.0986388
$103$	6.01027e10	0.510845	0.255422	−	0.966830i	$-0.417785\pi$
$103$	6.01027e10	0.510845	0.255422	+	0.966830i	$0.417785\pi$
$104$	1.15571e10	0.0931465
$105$	0	0
$106$	1.19808e10	0.0869566
$107$	1.33576e11	0.920702	0.460351	−	0.887737i	$-0.347724\pi$
$107$	1.33576e11	0.920702	0.460351	+	0.887737i	$0.347724\pi$
$108$	2.84682e10	0.186436
$109$	−2.16913e11	−1.35033	−0.675164	−	0.737668i	$-0.735927\pi$
$109$	−2.16913e11	−1.35033	−0.675164	+	0.737668i	$0.735927\pi$
$110$	2.82658e10	0.167341
$111$	−1.58471e11	−0.892637
$112$	0	0
$113$	−3.52345e11	−1.79902	−0.899512	−	0.436896i	$-0.856078\pi$
$113$	−3.52345e11	−1.79902	−0.899512	+	0.436896i	$0.856078\pi$
$114$	−2.83863e10	−0.138080
$115$	1.61222e11	0.747456
$116$	−1.01436e11	−0.448408
$117$	−2.11569e10	−0.0892133
$118$	−5.65436e10	−0.227527
$119$	0	0
$120$	−3.44097e10	−0.126236
$121$	3.62449e11	1.27036
$122$	6.11514e10	0.204846
$123$	2.31139e11	0.740278
$124$	−4.12875e11	−1.26473
$125$	3.44106e11	1.00853
$126$	0	0
$127$	−3.23741e11	−0.869514	−0.434757	−	0.900548i	$-0.643166\pi$
$127$	−3.23741e11	−0.869514	−0.434757	+	0.900548i	$0.643166\pi$
$128$	−2.53804e11	−0.652894
$129$	−2.08642e11	−0.514230
$130$	1.25833e10	0.0297239
$131$	−2.68716e11	−0.608557	−0.304278	−	0.952583i	$-0.598415\pi$
$131$	−2.68716e11	−0.608557	−0.304278	+	0.952583i	$0.598415\pi$
$132$	−3.88021e11	−0.842750
$133$	0	0
$134$	−4.06941e10	−0.0813682
$135$	6.29917e10	0.120906
$136$	−1.82500e11	−0.336356
$137$	−3.76903e11	−0.667215	−0.333608	−	0.942712i	$-0.608266\pi$
$137$	−3.76903e11	−0.667215	−0.333608	+	0.942712i	$0.608266\pi$
$138$	7.13930e10	0.121429
$139$	−6.81630e11	−1.11421	−0.557105	−	0.830442i	$-0.688088\pi$
$139$	−6.81630e11	−1.11421	−0.557105	+	0.830442i	$0.688088\pi$
$140$	0	0
$141$	−3.24783e11	−0.490781
$142$	2.24113e10	0.0325747
$143$	2.88368e11	0.403273
$144$	2.24692e11	0.302409
$145$	−2.24447e11	−0.290797
$146$	6.27542e10	0.0782893
$147$	0	0
$148$	−1.29386e12	−1.49778
$149$	1.38806e12	1.54840	0.774201	−	0.632940i	$-0.218152\pi$
$149$	1.38806e12	1.54840	0.774201	+	0.632940i	$0.218152\pi$
$150$	5.74569e10	0.0617789
$151$	1.53008e12	1.58614	0.793068	−	0.609133i	$-0.208482\pi$
$151$	1.53008e12	1.58614	0.793068	+	0.609133i	$0.208482\pi$
$152$	−4.71002e11	−0.470851
$153$	3.34091e11	0.322153
$154$	0	0
$155$	−9.13568e11	−0.820194
$156$	−1.72738e11	−0.149693
$157$	8.76928e11	0.733696	0.366848	−	0.930281i	$-0.380437\pi$
$157$	8.76928e11	0.733696	0.366848	+	0.930281i	$0.380437\pi$
$158$	−1.69253e11	−0.136749
$159$	−3.63916e11	−0.283999
$160$	−4.23643e11	−0.319404
$161$	0	0
$162$	2.78943e10	0.0196419
$163$	−2.02497e12	−1.37844	−0.689220	−	0.724552i	$-0.742047\pi$
$163$	−2.02497e12	−1.37844	−0.689220	+	0.724552i	$0.742047\pi$
$164$	1.88716e12	1.24213
$165$	−8.58575e11	−0.546533
$166$	8.71596e10	0.0536687
$167$	−3.19750e11	−0.190489	−0.0952444	−	0.995454i	$-0.530363\pi$
$167$	−3.19750e11	−0.190489	−0.0952444	+	0.995454i	$0.530363\pi$
$168$	0	0
$169$	−1.66379e12	−0.928369
$170$	−1.98704e11	−0.107334
$171$	8.62234e11	0.450969
$172$	−1.70348e12	−0.862839
$173$	−1.53948e12	−0.755301	−0.377650	−	0.925948i	$-0.623268\pi$
$173$	−1.53948e12	−0.755301	−0.377650	+	0.925948i	$0.623268\pi$
$174$	−9.93909e10	−0.0472417
$175$	0	0
$176$	−3.06255e12	−1.36698
$177$	1.71751e12	0.743099
$178$	−5.66310e11	−0.237544
$179$	1.81701e10	0.00739036	0.00369518	−	0.999993i	$-0.498824\pi$
$179$	1.81701e10	0.00739036	0.00369518	+	0.999993i	$0.498824\pi$
$180$	5.14303e11	0.202871
$181$	−1.75588e12	−0.671834	−0.335917	−	0.941892i	$-0.609046\pi$
$181$	−1.75588e12	−0.671834	−0.335917	+	0.941892i	$0.609046\pi$
$182$	0	0
$183$	−1.85747e12	−0.669024
$184$	1.18460e12	0.414068
$185$	−2.86292e12	−0.971326
$186$	−4.04550e11	−0.133245
$187$	−4.55365e12	−1.45623
$188$	−2.65172e12	−0.823494
$189$	0	0
$190$	−5.12822e11	−0.150253
$191$	5.77511e12	1.64390	0.821952	−	0.569556i	$-0.192885\pi$
$191$	5.77511e12	1.64390	0.821952	+	0.569556i	$0.192885\pi$
$192$	1.70610e12	0.471899
$193$	−4.14036e12	−1.11294	−0.556471	−	0.830867i	$-0.687845\pi$
$193$	−4.14036e12	−1.11294	−0.556471	+	0.830867i	$0.687845\pi$
$194$	−6.57790e11	−0.171861
$195$	−3.82217e11	−0.0970778
$196$	0	0
$197$	6.91145e11	0.165961	0.0829803	−	0.996551i	$-0.473556\pi$
$197$	6.91145e11	0.165961	0.0829803	+	0.996551i	$0.473556\pi$
$198$	−3.80198e11	−0.0887874
$199$	7.42261e12	1.68603	0.843014	−	0.537891i	$-0.180779\pi$
$199$	7.42261e12	1.68603	0.843014	+	0.537891i	$0.180779\pi$
$200$	9.53359e11	0.210665
$201$	1.23608e12	0.265747
$202$	1.10437e12	0.231037
$203$	0	0
$204$	2.72772e12	0.540548
$205$	4.17572e12	0.805536
$206$	4.80821e11	0.0903055
$207$	−2.16856e12	−0.396584
$208$	−1.36337e12	−0.242810
$209$	−1.17522e13	−2.03852
$210$	0	0
$211$	−6.43036e11	−0.105848	−0.0529239	−	0.998599i	$-0.516854\pi$
$211$	−6.43036e11	−0.105848	−0.0529239	+	0.998599i	$0.516854\pi$
$212$	−2.97123e12	−0.476529
$213$	−6.80743e11	−0.106389
$214$	1.06861e12	0.162759
$215$	−3.76929e12	−0.559561
$216$	4.62838e11	0.0669782
$217$	0	0
$218$	−1.73530e12	−0.238706
$219$	−1.90616e12	−0.255692
$220$	−7.00993e12	−0.917041
$221$	−2.02718e12	−0.258663
$222$	−1.26777e12	−0.157797
$223$	1.22383e13	1.48609	0.743045	−	0.669242i	$-0.233381\pi$
$223$	1.22383e13	1.48609	0.743045	+	0.669242i	$0.233381\pi$
$224$	0	0
$225$	−1.74525e12	−0.201769
$226$	−2.81876e12	−0.318026
$227$	1.05402e12	0.116067	0.0580334	−	0.998315i	$-0.481517\pi$
$227$	1.05402e12	0.116067	0.0580334	+	0.998315i	$0.481517\pi$
$228$	7.03980e12	0.756691
$229$	−3.31679e12	−0.348035	−0.174018	−	0.984743i	$-0.555675\pi$
$229$	−3.31679e12	−0.348035	−0.174018	+	0.984743i	$0.555675\pi$
$230$	1.28977e12	0.132133
$231$	0	0
$232$	−1.64915e12	−0.161093
$233$	−1.63955e13	−1.56410	−0.782052	−	0.623213i	$-0.785827\pi$
$233$	−1.63955e13	−1.56410	−0.782052	+	0.623213i	$0.785827\pi$
$234$	−1.69255e11	−0.0157708
$235$	−5.86748e12	−0.534045
$236$	1.40228e13	1.24686
$237$	5.14107e12	0.446619
$238$	0	0
$239$	−1.28637e13	−1.06703	−0.533516	−	0.845790i	$-0.679130\pi$
$239$	−1.28637e13	−1.06703	−0.533516	+	0.845790i	$0.679130\pi$
$240$	4.05926e12	0.329067
$241$	5.00644e12	0.396675	0.198338	−	0.980134i	$-0.436446\pi$
$241$	5.00644e12	0.396675	0.198338	+	0.980134i	$0.436446\pi$
$242$	2.89959e12	0.224571
$243$	−8.47289e11	−0.0641500
$244$	−1.51656e13	−1.12257
$245$	0	0
$246$	1.84911e12	0.130864
$247$	−5.23181e12	−0.362092
$248$	−6.71254e12	−0.454363
$249$	−2.64747e12	−0.175281
$250$	2.75285e12	0.178284
$251$	1.74289e13	1.10424	0.552121	−	0.833764i	$-0.313818\pi$
$251$	1.74289e13	1.10424	0.552121	+	0.833764i	$0.313818\pi$
$252$	0	0
$253$	2.95574e13	1.79268
$254$	−2.58993e12	−0.153710
$255$	6.03564e12	0.350552
$256$	1.23486e13	0.701936
$257$	1.01846e13	0.566644	0.283322	−	0.959025i	$-0.408564\pi$
$257$	1.01846e13	0.566644	0.283322	+	0.959025i	$0.408564\pi$
$258$	−1.66913e12	−0.0909039
$259$	0	0
$260$	−3.12065e12	−0.162889
$261$	3.01900e12	0.154291
$262$	−2.14973e12	−0.107579
$263$	6.78972e12	0.332733	0.166366	−	0.986064i	$-0.446797\pi$
$263$	6.78972e12	0.332733	0.166366	+	0.986064i	$0.446797\pi$
$264$	−6.30847e12	−0.302763
$265$	−6.57445e12	−0.309034
$266$	0	0
$267$	1.72017e13	0.775816
$268$	1.00921e13	0.445904
$269$	−2.94099e13	−1.27308	−0.636540	−	0.771244i	$-0.719635\pi$
$269$	−2.94099e13	−1.27308	−0.636540	+	0.771244i	$0.719635\pi$
$270$	5.03934e11	0.0213734
$271$	3.10163e13	1.28902	0.644509	−	0.764596i	$-0.277062\pi$
$271$	3.10163e13	1.28902	0.644509	+	0.764596i	$0.277062\pi$
$272$	2.15292e13	0.876797
$273$	0	0
$274$	−3.01522e12	−0.117948
$275$	2.37878e13	0.912060
$276$	−1.77055e13	−0.665438
$277$	1.12341e13	0.413903	0.206952	−	0.978351i	$-0.433646\pi$
$277$	1.12341e13	0.413903	0.206952	+	0.978351i	$0.433646\pi$
$278$	−5.45304e12	−0.196966
$279$	1.22882e13	0.435177
$280$	0	0
$281$	2.65890e13	0.905351	0.452676	−	0.891675i	$-0.350470\pi$
$281$	2.65890e13	0.905351	0.452676	+	0.891675i	$0.350470\pi$
$282$	−2.59826e12	−0.0867587
$283$	−1.78795e13	−0.585503	−0.292751	−	0.956189i	$-0.594571\pi$
$283$	−1.78795e13	−0.585503	−0.292751	+	0.956189i	$0.594571\pi$
$284$	−5.55800e12	−0.178512
$285$	1.55770e13	0.490723
$286$	2.30694e12	0.0712892
$287$	0	0
$288$	5.69833e12	0.169469
$289$	−2.26049e12	−0.0659577
$290$	−1.79558e12	−0.0514062
$291$	1.99804e13	0.561297
$292$	−1.55631e13	−0.429031
$293$	1.90224e13	0.514627	0.257314	−	0.966328i	$-0.417163\pi$
$293$	1.90224e13	0.514627	0.257314	+	0.966328i	$0.417163\pi$
$294$	0	0
$295$	3.10283e13	0.808605
$296$	−2.10356e13	−0.538085
$297$	1.15485e13	0.289978
$298$	1.11045e13	0.273721
$299$	1.31583e13	0.318425
$300$	−1.42493e13	−0.338553
$301$	0	0
$302$	1.22406e13	0.280392
$303$	−3.35453e13	−0.754565
$304$	5.55633e13	1.22739
$305$	−3.35568e13	−0.728001
$306$	2.67273e12	0.0569491
$307$	2.84379e13	0.595164	0.297582	−	0.954696i	$-0.403820\pi$
$307$	2.84379e13	0.595164	0.297582	+	0.954696i	$0.403820\pi$
$308$	0	0
$309$	−1.46049e13	−0.294936
$310$	−7.30855e12	−0.144991
$311$	−2.74409e13	−0.534831	−0.267416	−	0.963581i	$-0.586170\pi$
$311$	−2.74409e13	−0.534831	−0.267416	+	0.963581i	$0.586170\pi$
$312$	−2.80838e12	−0.0537782
$313$	4.29785e13	0.808643	0.404322	−	0.914617i	$-0.367508\pi$
$313$	4.29785e13	0.808643	0.404322	+	0.914617i	$0.367508\pi$
$314$	7.01543e12	0.129700
$315$	0	0
$316$	4.19748e13	0.749393
$317$	6.59529e13	1.15720	0.578600	−	0.815612i	$-0.303599\pi$
$317$	6.59529e13	1.15720	0.578600	+	0.815612i	$0.303599\pi$
$318$	−2.91133e12	−0.0502044
$319$	−4.11488e13	−0.697443
$320$	3.08222e13	0.513498
$321$	−3.24591e13	−0.531568
$322$	0	0
$323$	8.26161e13	1.30753
$324$	−6.91778e12	−0.107639
$325$	1.05897e13	0.162004
$326$	−1.61998e13	−0.243676
$327$	5.27098e13	0.779612
$328$	3.06815e13	0.446242
$329$	0	0
$330$	−6.86860e12	−0.0966143
$331$	−1.19488e14	−1.65299	−0.826495	−	0.562944i	$-0.809669\pi$
$331$	−1.19488e14	−1.65299	−0.826495	+	0.562944i	$0.809669\pi$
$332$	−2.16156e13	−0.294108
$333$	3.85086e13	0.515364
$334$	−2.55800e12	−0.0336740
$335$	2.23309e13	0.289174
$336$	0	0
$337$	6.39053e13	0.800890	0.400445	−	0.916321i	$-0.368856\pi$
$337$	6.39053e13	0.800890	0.400445	+	0.916321i	$0.368856\pi$
$338$	−1.33103e13	−0.164114
$339$	8.56199e13	1.03867
$340$	4.92786e13	0.588199
$341$	−1.67488e14	−1.96714
$342$	6.89787e12	0.0797208
$343$	0	0
$344$	−2.76953e13	−0.309980
$345$	−3.91769e13	−0.431544
$346$	−1.23158e13	−0.133520
$347$	9.53921e13	1.01789	0.508944	−	0.860800i	$-0.330036\pi$
$347$	9.53921e13	1.01789	0.508944	+	0.860800i	$0.330036\pi$
$348$	2.46489e13	0.258888
$349$	−1.14672e13	−0.118554	−0.0592772	−	0.998242i	$-0.518880\pi$
$349$	−1.14672e13	−0.118554	−0.0592772	+	0.998242i	$0.518880\pi$
$350$	0	0
$351$	5.14113e12	0.0515074
$352$	−7.76681e13	−0.766052
$353$	−1.96672e14	−1.90977	−0.954886	−	0.296971i	$-0.904024\pi$
$353$	−1.96672e14	−1.90977	−0.954886	+	0.296971i	$0.904024\pi$
$354$	1.37401e13	0.131363
$355$	−1.22982e13	−0.115767
$356$	1.40445e14	1.30176
$357$	0	0
$358$	1.45361e11	0.00130644
$359$	3.53266e13	0.312668	0.156334	−	0.987704i	$-0.450032\pi$
$359$	3.53266e13	0.312668	0.156334	+	0.987704i	$0.450032\pi$
$360$	8.36157e12	0.0728826
$361$	9.67283e13	0.830355
$362$	−1.40470e13	−0.118765
$363$	−8.80752e13	−0.733444
$364$	0	0
$365$	−3.44364e13	−0.278232
$366$	−1.48598e13	−0.118268
$367$	−1.65633e14	−1.29862	−0.649311	−	0.760523i	$-0.724943\pi$
$367$	−1.65633e14	−1.29862	−0.649311	+	0.760523i	$0.724943\pi$
$368$	−1.39745e14	−1.07937
$369$	−5.61667e13	−0.427400
$370$	−2.29034e13	−0.171708
$371$	0	0
$372$	1.00329e14	0.730194
$373$	6.00067e13	0.430330	0.215165	−	0.976578i	$-0.430971\pi$
$373$	6.00067e13	0.430330	0.215165	+	0.976578i	$0.430971\pi$
$374$	−3.64292e13	−0.257428
$375$	−8.36179e13	−0.582274
$376$	−4.31119e13	−0.295845
$377$	−1.83185e13	−0.123883
$378$	0	0
$379$	−2.02889e14	−1.33273	−0.666367	−	0.745624i	$-0.732152\pi$
$379$	−2.02889e14	−1.33273	−0.666367	+	0.745624i	$0.732152\pi$
$380$	1.27180e14	0.823396
$381$	7.86690e13	0.502014
$382$	4.62009e13	0.290604
$383$	2.99017e14	1.85397	0.926985	−	0.375099i	$-0.122391\pi$
$383$	2.99017e14	1.85397	0.926985	+	0.375099i	$0.122391\pi$
$384$	6.16743e13	0.376949
$385$	0	0
$386$	−3.31229e13	−0.196742
$387$	5.06999e13	0.296891
$388$	1.63132e14	0.941813
$389$	−2.48539e14	−1.41472	−0.707362	−	0.706852i	$-0.750115\pi$
$389$	−2.48539e14	−1.41472	−0.707362	+	0.706852i	$0.750115\pi$
$390$	−3.05774e12	−0.0171611
$391$	−2.07784e14	−1.14985
$392$	0	0
$393$	6.52980e13	0.351350
$394$	5.52916e12	0.0293380
$395$	9.28777e13	0.485990
$396$	9.42891e13	0.486562
$397$	1.74060e14	0.885833	0.442917	−	0.896563i	$-0.353944\pi$
$397$	1.74060e14	0.885833	0.442917	+	0.896563i	$0.353944\pi$
$398$	5.93809e13	0.298051
$399$	0	0
$400$	−1.12466e14	−0.549151
$401$	2.15357e14	1.03720	0.518602	−	0.855016i	$-0.326453\pi$
$401$	2.15357e14	1.03720	0.518602	+	0.855016i	$0.326453\pi$
$402$	9.88866e12	0.0469779
$403$	−7.45617e13	−0.349412
$404$	−2.73884e14	−1.26610
$405$	−1.53070e13	−0.0698051
$406$	0	0
$407$	−5.24871e14	−2.32961
$408$	4.43475e13	0.194195
$409$	2.90910e14	1.25684	0.628422	−	0.777873i	$-0.283701\pi$
$409$	2.90910e14	1.25684	0.628422	+	0.777873i	$0.283701\pi$
$410$	3.34057e13	0.142400
$411$	9.15873e13	0.385217
$412$	−1.19244e14	−0.494881
$413$	0	0
$414$	−1.73485e13	−0.0701068
$415$	−4.78288e13	−0.190733
$416$	−3.45760e13	−0.136070
$417$	1.65636e14	0.643290
$418$	−9.40177e13	−0.360363
$419$	−1.64921e14	−0.623875	−0.311938	−	0.950103i	$-0.600978\pi$
$419$	−1.64921e14	−0.623875	−0.311938	+	0.950103i	$0.600978\pi$
$420$	0	0
$421$	2.67118e14	0.984353	0.492177	−	0.870495i	$-0.336201\pi$
$421$	2.67118e14	0.984353	0.492177	+	0.870495i	$0.336201\pi$
$422$	−5.14429e12	−0.0187114
$423$	7.89222e13	0.283353
$424$	−4.83065e13	−0.171196
$425$	−1.67224e14	−0.585005
$426$	−5.44594e12	−0.0188070
$427$	0	0
$428$	−2.65016e14	−0.891930
$429$	−7.00734e13	−0.232830
$430$	−3.01543e13	−0.0989174
$431$	−5.64399e14	−1.82794	−0.913968	−	0.405787i	$-0.866998\pi$
$431$	−5.64399e14	−1.82794	−0.913968	+	0.405787i	$0.866998\pi$
$432$	−5.46002e13	−0.174596
$433$	1.37266e14	0.433390	0.216695	−	0.976239i	$-0.430472\pi$
$433$	1.37266e14	0.433390	0.216695	+	0.976239i	$0.430472\pi$
$434$	0	0
$435$	5.45407e13	0.167892
$436$	4.30355e14	1.30813
$437$	−5.36256e14	−1.60962
$438$	−1.52493e13	−0.0452003
$439$	−8.97819e12	−0.0262805	−0.0131403	−	0.999914i	$-0.504183\pi$
$439$	−8.97819e12	−0.0262805	−0.0131403	+	0.999914i	$0.504183\pi$
$440$	−1.13968e14	−0.329452
$441$	0	0
$442$	−1.62174e13	−0.0457256
$443$	−4.04100e14	−1.12530	−0.562650	−	0.826696i	$-0.690218\pi$
$443$	−4.04100e14	−1.12530	−0.562650	+	0.826696i	$0.690218\pi$
$444$	3.14407e14	0.864742
$445$	3.10763e14	0.844207
$446$	9.79065e13	0.262706
$447$	−3.37299e14	−0.893971
$448$	0	0
$449$	−6.27613e14	−1.62307	−0.811534	−	0.584305i	$-0.801367\pi$
$449$	−6.27613e14	−1.62307	−0.811534	+	0.584305i	$0.801367\pi$
$450$	−1.39620e13	−0.0356681
$451$	7.65551e14	1.93198
$452$	6.99053e14	1.74280
$453$	−3.71809e14	−0.915756
$454$	8.43219e12	0.0205179
$455$	0	0
$456$	1.14454e14	0.271846
$457$	3.01238e14	0.706921	0.353460	−	0.935449i	$-0.385005\pi$
$457$	3.01238e14	0.706921	0.353460	+	0.935449i	$0.385005\pi$
$458$	−2.65343e13	−0.0615245
$459$	−8.11841e13	−0.185995
$460$	−3.19864e14	−0.724098
$461$	−3.50379e14	−0.783759	−0.391880	−	0.920017i	$-0.628175\pi$
$461$	−3.50379e14	−0.783759	−0.391880	+	0.920017i	$0.628175\pi$
$462$	0	0
$463$	4.90390e14	1.07114	0.535569	−	0.844491i	$-0.320097\pi$
$463$	4.90390e14	1.07114	0.535569	+	0.844491i	$0.320097\pi$
$464$	1.94548e14	0.419930
$465$	2.21997e14	0.473539
$466$	−1.31164e14	−0.276497
$467$	−6.80842e14	−1.41842	−0.709208	−	0.704999i	$-0.750947\pi$
$467$	−6.80842e14	−1.41842	−0.709208	+	0.704999i	$0.750947\pi$
$468$	4.19753e13	0.0864254
$469$	0	0
$470$	−4.69398e13	−0.0944067
$471$	−2.13094e14	−0.423600
$472$	2.27984e14	0.447943
$473$	−6.91038e14	−1.34204
$474$	4.11285e13	0.0789519
$475$	−4.31577e14	−0.818924
$476$	0	0
$477$	8.84315e13	0.163967
$478$	−1.02910e14	−0.188626
$479$	−7.91988e14	−1.43507	−0.717535	−	0.696523i	$-0.754730\pi$
$479$	−7.91988e14	−1.43507	−0.717535	+	0.696523i	$0.754730\pi$
$480$	1.02945e14	0.184408
$481$	−2.33660e14	−0.413796
$482$	4.00515e13	0.0701229
$483$	0	0
$484$	−7.19099e14	−1.23066
$485$	3.60962e14	0.610777
$486$	−6.77831e12	−0.0113402
$487$	−4.92313e14	−0.814388	−0.407194	−	0.913342i	$-0.633493\pi$
$487$	−4.92313e14	−0.814388	−0.407194	+	0.913342i	$0.633493\pi$
$488$	−2.46562e14	−0.403290
$489$	4.92069e14	0.795842
$490$	0	0
$491$	−1.07948e15	−1.70713	−0.853564	−	0.520989i	$-0.825563\pi$
$491$	−1.07948e15	−1.70713	−0.853564	+	0.520989i	$0.825563\pi$
$492$	−4.58579e14	−0.717144
$493$	2.89269e14	0.447347
$494$	−4.18545e13	−0.0640094
$495$	2.08634e14	0.315541
$496$	7.91867e14	1.18441
$497$	0	0
$498$	−2.11798e13	−0.0309856
$499$	−1.04552e15	−1.51280	−0.756400	−	0.654110i	$-0.773044\pi$
$499$	−1.04552e15	−1.51280	−0.756400	+	0.654110i	$0.773044\pi$
$500$	−6.82707e14	−0.977011
$501$	7.76992e13	0.109979
$502$	1.39431e14	0.195204
$503$	4.47866e14	0.620189	0.310094	−	0.950706i	$-0.399639\pi$
$503$	4.47866e14	0.620189	0.310094	+	0.950706i	$0.399639\pi$
$504$	0	0
$505$	−6.06023e14	−0.821082
$506$	2.36460e14	0.316905
$507$	4.04300e14	0.535994
$508$	6.42301e14	0.842342
$509$	−4.18511e14	−0.542949	−0.271475	−	0.962446i	$-0.587511\pi$
$509$	−4.18511e14	−0.542949	−0.271475	+	0.962446i	$0.587511\pi$
$510$	4.82851e13	0.0619694
$511$	0	0
$512$	6.18579e14	0.776980
$513$	−2.09523e14	−0.260367
$514$	8.14765e13	0.100169
$515$	−2.63851e14	−0.320936
$516$	4.13945e14	0.498161
$517$	−1.07571e15	−1.28084
$518$	0	0
$519$	3.74093e14	0.436073
$520$	−5.07358e13	−0.0585189
$521$	−7.11671e14	−0.812216	−0.406108	−	0.913825i	$-0.633114\pi$
$521$	−7.11671e14	−0.812216	−0.406108	+	0.913825i	$0.633114\pi$
$522$	2.41520e13	0.0272750
$523$	9.56769e14	1.06917	0.534586	−	0.845114i	$-0.320467\pi$
$523$	9.56769e14	1.06917	0.534586	+	0.845114i	$0.320467\pi$
$524$	5.33132e14	0.589539
$525$	0	0
$526$	5.43178e13	0.0588194
$527$	1.17741e15	1.26174
$528$	7.44199e14	0.789228
$529$	3.95901e14	0.415509
$530$	−5.25956e13	−0.0546301
$531$	−4.17355e14	−0.429028
$532$	0	0
$533$	3.40805e14	0.343168
$534$	1.37613e14	0.137146
$535$	−5.86401e14	−0.578427
$536$	1.64078e14	0.160194
$537$	−4.41533e12	−0.00426683
$538$	−2.35279e14	−0.225051
$539$	0	0
$540$	−1.24976e14	−0.117128
$541$	2.13753e14	0.198302	0.0991509	−	0.995072i	$-0.468387\pi$
$541$	2.13753e14	0.198302	0.0991509	+	0.995072i	$0.468387\pi$
$542$	2.48131e14	0.227868
$543$	4.26678e14	0.387884
$544$	5.45994e14	0.491353
$545$	9.52247e14	0.848338
$546$	0	0
$547$	1.98142e15	1.73000	0.865001	−	0.501771i	$-0.167318\pi$
$547$	1.98142e15	1.73000	0.865001	+	0.501771i	$0.167318\pi$
$548$	7.47775e14	0.646365
$549$	4.51366e14	0.386261
$550$	1.90302e14	0.161231
$551$	7.46556e14	0.626223
$552$	−2.87857e14	−0.239062
$553$	0	0
$554$	8.98726e13	0.0731684
$555$	6.95690e14	0.560795
$556$	1.35235e15	1.07939
$557$	−1.45121e15	−1.14691	−0.573453	−	0.819238i	$-0.694397\pi$
$557$	−1.45121e15	−1.14691	−0.573453	+	0.819238i	$0.694397\pi$
$558$	9.83058e13	0.0769291
$559$	−3.07634e14	−0.238380
$560$	0	0
$561$	1.10654e15	0.840757
$562$	2.12712e14	0.160045
$563$	−1.69873e15	−1.26569	−0.632846	−	0.774278i	$-0.718113\pi$
$563$	−1.69873e15	−1.26569	−0.632846	+	0.774278i	$0.718113\pi$
$564$	6.44369e14	0.475444
$565$	1.54680e15	1.13023
$566$	−1.43036e14	−0.103503
$567$	0	0
$568$	−9.03623e13	−0.0641315
$569$	−1.69686e14	−0.119270	−0.0596348	−	0.998220i	$-0.518994\pi$
$569$	−1.69686e14	−0.119270	−0.0596348	+	0.998220i	$0.518994\pi$
$570$	1.24616e14	0.0867484
$571$	1.03308e15	0.712254	0.356127	−	0.934438i	$-0.384097\pi$
$571$	1.03308e15	0.712254	0.356127	+	0.934438i	$0.384097\pi$
$572$	−5.72122e14	−0.390670
$573$	−1.40335e15	−0.949109
$574$	0	0
$575$	1.08544e15	0.720166
$576$	−4.14583e14	−0.272451
$577$	−1.45083e15	−0.944384	−0.472192	−	0.881496i	$-0.656537\pi$
$577$	−1.45083e15	−0.944384	−0.472192	+	0.881496i	$0.656537\pi$
$578$	−1.80840e13	−0.0116598
$579$	1.00611e15	0.642557
$580$	4.45304e14	0.281710
$581$	0	0
$582$	1.59843e14	0.0992241
$583$	−1.20532e15	−0.741182
$584$	−2.53025e14	−0.154132
$585$	9.28788e13	0.0560479
$586$	1.52179e14	0.0909741
$587$	1.76163e15	1.04329	0.521646	−	0.853162i	$-0.325318\pi$
$587$	1.76163e15	1.04329	0.521646	+	0.853162i	$0.325318\pi$
$588$	0	0
$589$	3.03871e15	1.76626
$590$	2.48226e14	0.142943
$591$	−1.67948e14	−0.0958174
$592$	2.48154e15	1.40266
$593$	2.51119e15	1.40630	0.703151	−	0.711041i	$-0.251776\pi$
$593$	2.51119e15	1.40630	0.703151	+	0.711041i	$0.251776\pi$
$594$	9.23881e13	0.0512614
$595$	0	0
$596$	−2.75391e15	−1.50001
$597$	−1.80370e15	−0.973429
$598$	1.05266e14	0.0562902
$599$	−2.36908e15	−1.25525	−0.627627	−	0.778514i	$-0.715974\pi$
$599$	−2.36908e15	−1.25525	−0.627627	+	0.778514i	$0.715974\pi$
$600$	−2.31666e14	−0.121627
$601$	2.51626e15	1.30902	0.654509	−	0.756054i	$-0.272875\pi$
$601$	2.51626e15	1.30902	0.654509	+	0.756054i	$0.272875\pi$
$602$	0	0
$603$	−3.00368e14	−0.153429
$604$	−3.03568e15	−1.53657
$605$	−1.59115e15	−0.798100
$606$	−2.68362e14	−0.133390
$607$	−3.45512e15	−1.70186	−0.850932	−	0.525276i	$-0.823962\pi$
$607$	−3.45512e15	−1.70186	−0.850932	+	0.525276i	$0.823962\pi$
$608$	1.40912e15	0.687825
$609$	0	0
$610$	−2.68455e14	−0.128694
$611$	−4.78880e14	−0.227510
$612$	−6.62837e14	−0.312086
$613$	−3.60688e15	−1.68306	−0.841529	−	0.540212i	$-0.818344\pi$
$613$	−3.60688e15	−1.68306	−0.841529	+	0.540212i	$0.818344\pi$
$614$	2.27503e14	0.105211
$615$	−1.01470e15	−0.465076
$616$	0	0
$617$	2.66149e15	1.19827	0.599137	−	0.800646i	$-0.295510\pi$
$617$	2.66149e15	1.19827	0.599137	+	0.800646i	$0.295510\pi$
$618$	−1.16840e14	−0.0521379
$619$	1.02858e14	0.0454925	0.0227463	−	0.999741i	$-0.492759\pi$
$619$	1.02858e14	0.0454925	0.0227463	+	0.999741i	$0.492759\pi$
$620$	1.81252e15	0.794563
$621$	5.26961e14	0.228968
$622$	−2.19527e14	−0.0945457
$623$	0	0
$624$	3.31300e14	0.140187
$625$	−6.74619e13	−0.0282956
$626$	3.43828e14	0.142949
$627$	2.85579e15	1.17694
$628$	−1.73983e15	−0.710768
$629$	3.68975e15	1.49423
$630$	0	0
$631$	2.00659e14	0.0798541	0.0399270	−	0.999203i	$-0.487287\pi$
$631$	2.00659e14	0.0798541	0.0399270	+	0.999203i	$0.487287\pi$
$632$	6.82429e14	0.269224
$633$	1.56258e14	0.0611112
$634$	5.27623e14	0.204566
$635$	1.42122e15	0.546269
$636$	7.22009e14	0.275124
$637$	0	0
$638$	−3.29191e14	−0.123292
$639$	1.65421e14	0.0614235
$640$	1.11420e15	0.410178
$641$	−2.24823e15	−0.820581	−0.410291	−	0.911955i	$-0.634573\pi$
$641$	−2.24823e15	−0.820581	−0.410291	+	0.911955i	$0.634573\pi$
$642$	−2.59673e14	−0.0939688
$643$	−2.81606e15	−1.01037	−0.505187	−	0.863010i	$-0.668576\pi$
$643$	−2.81606e15	−1.01037	−0.505187	+	0.863010i	$0.668576\pi$
$644$	0	0
$645$	9.15937e14	0.323063
$646$	6.60929e14	0.231140
$647$	−3.23571e15	−1.12201	−0.561004	−	0.827813i	$-0.689585\pi$
$647$	−3.23571e15	−1.12201	−0.561004	+	0.827813i	$0.689585\pi$
$648$	−1.12470e14	−0.0386699
$649$	5.68854e15	1.93934
$650$	8.47180e13	0.0286386
$651$	0	0
$652$	4.01755e15	1.33536
$653$	−3.14088e15	−1.03521	−0.517606	−	0.855619i	$-0.673177\pi$
$653$	−3.14088e15	−1.03521	−0.517606	+	0.855619i	$0.673177\pi$
$654$	4.21678e14	0.137817
$655$	1.17966e15	0.382323
$656$	−3.61945e15	−1.16324
$657$	4.63197e14	0.147624
$658$	0	0
$659$	8.48017e14	0.265788	0.132894	−	0.991130i	$-0.457573\pi$
$659$	8.48017e14	0.265788	0.132894	+	0.991130i	$0.457573\pi$
$660$	1.70341e15	0.529454
$661$	−2.31505e15	−0.713596	−0.356798	−	0.934182i	$-0.616132\pi$
$661$	−2.31505e15	−0.713596	−0.356798	+	0.934182i	$0.616132\pi$
$662$	−9.55904e14	−0.292210
$663$	4.92604e14	0.149339
$664$	−3.51427e14	−0.105660
$665$	0	0
$666$	3.08069e14	0.0911043
$667$	−1.87763e15	−0.550703
$668$	6.34384e14	0.184536
$669$	−2.97391e15	−0.857994
$670$	1.78647e14	0.0511192
$671$	−6.15211e15	−1.74602
$672$	0	0
$673$	−3.64280e15	−1.01707	−0.508536	−	0.861041i	$-0.669813\pi$
$673$	−3.64280e15	−1.01707	−0.508536	+	0.861041i	$0.669813\pi$
$674$	5.11243e14	0.141579
$675$	4.24097e14	0.116491
$676$	3.30095e15	0.899357
$677$	−4.64522e15	−1.25536	−0.627681	−	0.778471i	$-0.715996\pi$
$677$	−4.64522e15	−1.25536	−0.627681	+	0.778471i	$0.715996\pi$
$678$	6.84959e14	0.183612
$679$	0	0
$680$	8.01175e14	0.211314
$681$	−2.56128e14	−0.0670113
$682$	−1.33990e15	−0.347744
$683$	3.94390e15	1.01534	0.507671	−	0.861551i	$-0.330507\pi$
$683$	3.94390e15	1.01534	0.507671	+	0.861551i	$0.330507\pi$
$684$	−1.71067e15	−0.436876
$685$	1.65460e15	0.419175
$686$	0	0
$687$	8.05980e14	0.200938
$688$	3.26716e15	0.808042
$689$	−5.36580e14	−0.131652
$690$	−3.13415e14	−0.0762869
$691$	−1.73361e15	−0.418621	−0.209310	−	0.977849i	$-0.567122\pi$
$691$	−1.73361e15	−0.418621	−0.209310	+	0.977849i	$0.567122\pi$
$692$	3.05433e15	0.731698
$693$	0	0
$694$	7.63136e14	0.179939
$695$	2.99236e15	0.699998
$696$	4.00744e14	0.0930071
$697$	−5.38169e15	−1.23919
$698$	−9.17377e13	−0.0209577
$699$	3.98410e15	0.903036
$700$	0	0
$701$	−4.34481e15	−0.969441	−0.484720	−	0.874669i	$-0.661079\pi$
$701$	−4.34481e15	−0.969441	−0.484720	+	0.874669i	$0.661079\pi$
$702$	4.11290e13	0.00910530
$703$	9.52264e15	2.09172
$704$	5.65076e15	1.23156
$705$	1.42580e15	0.308331
$706$	−1.57338e15	−0.337603
$707$	0	0
$708$	−3.40754e15	−0.719877
$709$	3.46868e15	0.727126	0.363563	−	0.931570i	$-0.381560\pi$
$709$	3.46868e15	0.727126	0.363563	+	0.931570i	$0.381560\pi$
$710$	−9.83856e13	−0.0204649
$711$	−1.24928e15	−0.257856
$712$	2.28336e15	0.467665
$713$	−7.64251e15	−1.55326
$714$	0	0
$715$	−1.26594e15	−0.253354
$716$	−3.60495e13	−0.00715941
$717$	3.12588e15	0.616051
$718$	2.82613e14	0.0552723
$719$	8.98716e15	1.74427	0.872134	−	0.489266i	$-0.162735\pi$
$719$	8.98716e15	1.74427	0.872134	+	0.489266i	$0.162735\pi$
$720$	−9.86399e14	−0.189987
$721$	0	0
$722$	7.73826e14	0.146787
$723$	−1.21656e15	−0.229021
$724$	3.48366e15	0.650839
$725$	−1.51111e15	−0.280180
$726$	−7.04601e14	−0.129656
$727$	−2.99194e15	−0.546404	−0.273202	−	0.961957i	$-0.588083\pi$
$727$	−2.99194e15	−0.546404	−0.273202	+	0.961957i	$0.588083\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	−2.75491e14	−0.0491849
$731$	4.85788e15	0.860799
$732$	3.68523e15	0.648117
$733$	−4.15129e15	−0.724622	−0.362311	−	0.932057i	$-0.618012\pi$
$733$	−4.15129e15	−0.724622	−0.362311	+	0.932057i	$0.618012\pi$
$734$	−1.32506e15	−0.229566
$735$	0	0
$736$	−3.54401e15	−0.604876
$737$	4.09401e15	0.693549
$738$	−4.49334e14	−0.0755543
$739$	−7.42523e15	−1.23927	−0.619635	−	0.784890i	$-0.712719\pi$
$739$	−7.42523e15	−1.23927	−0.619635	+	0.784890i	$0.712719\pi$
$740$	5.68003e15	0.940972
$741$	1.27133e15	0.209054
$742$	0	0
$743$	6.33952e14	0.102711	0.0513556	−	0.998680i	$-0.483646\pi$
$743$	6.33952e14	0.102711	0.0513556	+	0.998680i	$0.483646\pi$
$744$	1.63115e15	0.262326
$745$	−6.09358e15	−0.972777
$746$	4.80054e14	0.0760723
$747$	6.43336e14	0.101199
$748$	9.03444e15	1.41073
$749$	0	0
$750$	−6.68943e14	−0.102932
$751$	3.55460e15	0.542964	0.271482	−	0.962444i	$-0.412486\pi$
$751$	3.55460e15	0.542964	0.271482	+	0.962444i	$0.412486\pi$
$752$	5.08584e15	0.771195
$753$	−4.23522e15	−0.637535
$754$	−1.46548e14	−0.0218997
$755$	−6.71704e15	−0.996483
$756$	0	0
$757$	−8.89494e15	−1.30052	−0.650258	−	0.759713i	$-0.725339\pi$
$757$	−8.89494e15	−1.30052	−0.650258	+	0.759713i	$0.725339\pi$
$758$	−1.62311e15	−0.235596
$759$	−7.18246e15	−1.03501
$760$	2.06770e15	0.295810
$761$	−1.04253e16	−1.48072	−0.740360	−	0.672210i	$-0.765345\pi$
$761$	−1.04253e16	−1.48072	−0.740360	+	0.672210i	$0.765345\pi$
$762$	6.29352e14	0.0887444
$763$	0	0
$764$	−1.14578e16	−1.59253
$765$	−1.46666e15	−0.202391
$766$	2.39214e15	0.327739
$767$	2.53240e15	0.344475
$768$	−3.00071e15	−0.405263
$769$	−9.07977e14	−0.121753	−0.0608765	−	0.998145i	$-0.519390\pi$
$769$	−9.07977e14	−0.121753	−0.0608765	+	0.998145i	$0.519390\pi$
$770$	0	0
$771$	−2.47485e15	−0.327152
$772$	8.21447e15	1.07816
$773$	−2.88244e15	−0.375641	−0.187820	−	0.982203i	$-0.560142\pi$
$773$	−2.88244e15	−0.375641	−0.187820	+	0.982203i	$0.560142\pi$
$774$	4.05599e14	0.0524834
$775$	−6.15067e15	−0.790247
$776$	2.65221e15	0.338352
$777$	0	0
$778$	−1.98831e15	−0.250090
$779$	−1.38893e16	−1.73469
$780$	7.58319e14	0.0940441
$781$	−2.25468e15	−0.277654
$782$	−1.66227e15	−0.203266
$783$	−7.33616e14	−0.0890798
$784$	0	0
$785$	−3.84972e15	−0.460941
$786$	5.22384e14	0.0621106
$787$	−9.61839e15	−1.13564	−0.567821	−	0.823152i	$-0.692213\pi$
$787$	−9.61839e15	−1.13564	−0.567821	+	0.823152i	$0.692213\pi$
$788$	−1.37123e15	−0.160774
$789$	−1.64990e15	−0.192103
$790$	7.43022e14	0.0859118
$791$	0	0
$792$	1.53296e15	0.174800
$793$	−2.73877e15	−0.310137
$794$	1.39248e15	0.156595
$795$	1.59759e15	0.178421
$796$	−1.47265e16	−1.63334
$797$	−1.07318e16	−1.18209	−0.591046	−	0.806638i	$-0.701285\pi$
$797$	−1.07318e16	−1.18209	−0.591046	+	0.806638i	$0.701285\pi$
$798$	0	0
$799$	7.56204e15	0.821546
$800$	−2.85221e15	−0.307742
$801$	−4.18001e15	−0.447918
$802$	1.72285e15	0.183353
$803$	−6.31336e15	−0.667306
$804$	−2.45239e15	−0.257443
$805$	0	0
$806$	−5.96494e14	−0.0617680
$807$	7.14660e15	0.735013
$808$	−4.45282e15	−0.454855
$809$	3.94794e15	0.400547	0.200273	−	0.979740i	$-0.435817\pi$
$809$	3.94794e15	0.400547	0.200273	+	0.979740i	$0.435817\pi$
$810$	−1.22456e14	−0.0123399
$811$	−7.55537e15	−0.756207	−0.378104	−	0.925763i	$-0.623424\pi$
$811$	−7.55537e15	−0.756207	−0.378104	+	0.925763i	$0.623424\pi$
$812$	0	0
$813$	−7.53697e15	−0.744215
$814$	−4.19896e15	−0.411821
$815$	8.88964e15	0.865999
$816$	−5.23160e15	−0.506219
$817$	1.25374e16	1.20500
$818$	2.32728e15	0.222181
$819$	0	0
$820$	−8.28462e15	−0.780363
$821$	9.48454e14	0.0887420	0.0443710	−	0.999015i	$-0.485872\pi$
$821$	9.48454e14	0.0887420	0.0443710	+	0.999015i	$0.485872\pi$
$822$	7.32699e14	0.0680974
$823$	−2.39832e15	−0.221416	−0.110708	−	0.993853i	$-0.535312\pi$
$823$	−2.39832e15	−0.221416	−0.110708	+	0.993853i	$0.535312\pi$
$824$	−1.93867e15	−0.177789
$825$	−5.78042e15	−0.526578
$826$	0	0
$827$	1.20594e16	1.08404	0.542019	−	0.840366i	$-0.317660\pi$
$827$	1.20594e16	1.08404	0.542019	+	0.840366i	$0.317660\pi$
$828$	4.30243e15	0.384191
$829$	3.00246e15	0.266334	0.133167	−	0.991094i	$-0.457485\pi$
$829$	3.00246e15	0.266334	0.133167	+	0.991094i	$0.457485\pi$
$830$	−3.82631e14	−0.0337171
$831$	−2.72988e15	−0.238967
$832$	2.51558e15	0.218756
$833$	0	0
$834$	1.32509e15	0.113719
$835$	1.40370e15	0.119674
$836$	2.33164e16	1.97482
$837$	−2.98604e15	−0.251250
$838$	−1.31936e15	−0.110287
$839$	7.55751e15	0.627607	0.313804	−	0.949488i	$-0.398397\pi$
$839$	7.55751e15	0.627607	0.313804	+	0.949488i	$0.398397\pi$
$840$	0	0
$841$	−9.58654e15	−0.785749
$842$	2.13694e15	0.174011
$843$	−6.46112e15	−0.522705
$844$	1.27578e15	0.102540
$845$	7.30402e15	0.583244
$846$	6.31378e14	0.0500901
$847$	0	0
$848$	5.69863e15	0.446265
$849$	4.34471e15	0.338040
$850$	−1.33779e15	−0.103415
$851$	−2.39499e16	−1.83947
$852$	1.35059e15	0.103064
$853$	−1.04461e16	−0.792015	−0.396008	−	0.918247i	$-0.629605\pi$
$853$	−1.04461e16	−0.792015	−0.396008	+	0.918247i	$0.629605\pi$
$854$	0	0
$855$	−3.78521e15	−0.283319
$856$	−4.30864e15	−0.320431
$857$	2.13498e16	1.57761	0.788804	−	0.614645i	$-0.210701\pi$
$857$	2.13498e16	1.57761	0.788804	+	0.614645i	$0.210701\pi$
$858$	−5.60587e14	−0.0411588
$859$	−1.22807e16	−0.895905	−0.447952	−	0.894057i	$-0.647847\pi$
$859$	−1.22807e16	−0.895905	−0.447952	+	0.894057i	$0.647847\pi$
$860$	7.47827e15	0.542075
$861$	0	0
$862$	−4.51519e15	−0.323136
$863$	1.68658e15	0.119935	0.0599676	−	0.998200i	$-0.480900\pi$
$863$	1.68658e15	0.119935	0.0599676	+	0.998200i	$0.480900\pi$
$864$	−1.38469e15	−0.0978427
$865$	6.75831e15	0.474515
$866$	1.09813e15	0.0766133
$867$	5.49300e14	0.0380807
$868$	0	0
$869$	1.70276e16	1.16559
$870$	4.36326e14	0.0296794
$871$	1.82255e15	0.123191
$872$	6.99674e15	0.469953
$873$	−4.85523e15	−0.324065
$874$	−4.29005e15	−0.284544
$875$	0	0
$876$	3.78182e15	0.247701
$877$	−1.36349e16	−0.887470	−0.443735	−	0.896158i	$-0.646347\pi$
$877$	−1.36349e16	−0.887470	−0.443735	+	0.896158i	$0.646347\pi$
$878$	−7.18255e13	−0.00464578
$879$	−4.62244e15	−0.297120
$880$	1.34446e16	0.858802
$881$	−4.78019e15	−0.303443	−0.151722	−	0.988423i	$-0.548482\pi$
$881$	−4.78019e15	−0.303443	−0.151722	+	0.988423i	$0.548482\pi$
$882$	0	0
$883$	2.35034e15	0.147349	0.0736743	−	0.997282i	$-0.476527\pi$
$883$	2.35034e15	0.147349	0.0736743	+	0.997282i	$0.476527\pi$
$884$	4.02192e15	0.250580
$885$	−7.53987e15	−0.466848
$886$	−3.23280e15	−0.198927
$887$	2.80378e16	1.71461	0.857304	−	0.514811i	$-0.172138\pi$
$887$	2.80378e16	1.71461	0.857304	+	0.514811i	$0.172138\pi$
$888$	5.11166e15	0.310664
$889$	0	0
$890$	2.48610e15	0.149236
$891$	−2.80629e15	−0.167419
$892$	−2.42808e16	−1.43965
$893$	1.95164e16	1.15005
$894$	−2.69839e15	−0.158033
$895$	−7.97667e13	−0.00464296
$896$	0	0
$897$	−3.19746e15	−0.183843
$898$	−5.02090e15	−0.286921
$899$	1.06396e16	0.604294
$900$	3.46258e15	0.195464
$901$	8.47319e15	0.475402
$902$	6.12441e15	0.341529
$903$	0	0
$904$	1.13652e16	0.626113
$905$	7.70830e15	0.422077
$906$	−2.97447e15	−0.161884
$907$	−7.92531e15	−0.428723	−0.214361	−	0.976754i	$-0.568767\pi$
$907$	−7.92531e15	−0.428723	−0.214361	+	0.976754i	$0.568767\pi$
$908$	−2.09118e15	−0.112440
$909$	8.15150e15	0.435648
$910$	0	0
$911$	1.57528e16	0.831777	0.415888	−	0.909416i	$-0.363471\pi$
$911$	1.57528e16	0.831777	0.415888	+	0.909416i	$0.363471\pi$
$912$	−1.35019e16	−0.708635
$913$	−8.76865e15	−0.457450
$914$	2.40990e15	0.124967
$915$	8.15431e15	0.420312
$916$	6.58051e15	0.337159
$917$	0	0
$918$	−6.49473e14	−0.0328796
$919$	2.85627e16	1.43735	0.718677	−	0.695344i	$-0.244748\pi$
$919$	2.85627e16	1.43735	0.718677	+	0.695344i	$0.244748\pi$
$920$	−5.20037e15	−0.260137
$921$	−6.91042e15	−0.343618
$922$	−2.80303e15	−0.138550
$923$	−1.00373e15	−0.0493182
$924$	0	0
$925$	−1.92748e16	−0.935861
$926$	3.92312e15	0.189352
$927$	3.54900e15	0.170282
$928$	4.93384e15	0.235327
$929$	−2.08352e16	−0.987897	−0.493949	−	0.869491i	$-0.664447\pi$
$929$	−2.08352e16	−0.987897	−0.493949	+	0.869491i	$0.664447\pi$
$930$	1.77598e15	0.0837107
$931$	0	0
$932$	3.25286e16	1.51523
$933$	6.66815e15	0.308785
$934$	−5.44674e15	−0.250743
$935$	1.99905e16	0.914872
$936$	6.82437e14	0.0310488
$937$	3.63422e16	1.64378	0.821889	−	0.569648i	$-0.192920\pi$
$937$	3.63422e16	1.64378	0.821889	+	0.569648i	$0.192920\pi$
$938$	0	0
$939$	−1.04438e16	−0.466870
$940$	1.16411e16	0.517356
$941$	−1.70015e16	−0.751180	−0.375590	−	0.926786i	$-0.622560\pi$
$941$	−1.70015e16	−0.751180	−0.375590	+	0.926786i	$0.622560\pi$
$942$	−1.70475e15	−0.0748825
$943$	3.49322e16	1.52550
$944$	−2.68948e16	−1.16768
$945$	0	0
$946$	−5.52831e15	−0.237242
$947$	−3.90585e16	−1.66644	−0.833222	−	0.552938i	$-0.813507\pi$
$947$	−3.90585e16	−1.66644	−0.833222	+	0.552938i	$0.813507\pi$
$948$	−1.01999e16	−0.432662
$949$	−2.81056e15	−0.118530
$950$	−3.45262e15	−0.144767
$951$	−1.60266e16	−0.668109
$952$	0	0
$953$	3.25047e16	1.33948	0.669739	−	0.742596i	$-0.266406\pi$
$953$	3.25047e16	1.33948	0.669739	+	0.742596i	$0.266406\pi$
$954$	7.07452e14	0.0289855
$955$	−2.53527e16	−1.03278
$956$	2.55216e16	1.03369
$957$	9.99917e15	0.402669
$958$	−6.33590e15	−0.253687
$959$	0	0
$960$	−7.48980e15	−0.296468
$961$	1.78980e16	0.704411
$962$	−1.86928e15	−0.0731495
$963$	7.88756e15	0.306901
$964$	−9.93277e15	−0.384279
$965$	1.81762e16	0.699201
$966$	0	0
$967$	2.41922e16	0.920090	0.460045	−	0.887896i	$-0.347833\pi$
$967$	2.41922e16	0.920090	0.460045	+	0.887896i	$0.347833\pi$
$968$	−1.16912e16	−0.442123
$969$	−2.00757e16	−0.754902
$970$	2.88770e15	0.107971
$971$	4.32072e15	0.160639	0.0803193	−	0.996769i	$-0.474406\pi$
$971$	4.32072e15	0.160639	0.0803193	+	0.996769i	$0.474406\pi$
$972$	1.68102e15	0.0621453
$973$	0	0
$974$	−3.93850e15	−0.143965
$975$	−2.57331e15	−0.0935333
$976$	2.90865e16	1.05128
$977$	2.18589e16	0.785614	0.392807	−	0.919621i	$-0.371504\pi$
$977$	2.18589e16	0.785614	0.392807	+	0.919621i	$0.371504\pi$
$978$	3.93655e15	0.140686
$979$	5.69734e16	2.02473
$980$	0	0
$981$	−1.28085e16	−0.450109
$982$	−8.63583e15	−0.301780
$983$	4.03990e16	1.40387	0.701934	−	0.712242i	$-0.252320\pi$
$983$	4.03990e16	1.40387	0.701934	+	0.712242i	$0.252320\pi$
$984$	−7.45561e15	−0.257638
$985$	−3.03413e15	−0.104264
$986$	2.31416e15	0.0790805
$987$	0	0
$988$	1.03799e16	0.350776
$989$	−3.15322e16	−1.05968
$990$	1.66907e15	0.0557803
$991$	9.82831e15	0.326643	0.163322	−	0.986573i	$-0.447779\pi$
$991$	9.82831e15	0.326643	0.163322	+	0.986573i	$0.447779\pi$
$992$	2.00822e16	0.663739
$993$	2.90356e16	0.954355
$994$	0	0
$995$	−3.25853e16	−1.05924
$996$	5.25259e15	0.169804
$997$	−4.40587e16	−1.41647	−0.708236	−	0.705976i	$-0.750509\pi$
$997$	−4.40587e16	−1.41647	−0.708236	+	0.705976i	$0.750509\pi$
$998$	−8.36420e15	−0.267428
$999$	−9.35758e15	−0.297546

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.12.a.b.1.1		1
7.6	odd	2		21.12.a.b.1.1	✓	1
21.20	even	2		63.12.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.12.a.b.1.1	✓	1	7.6	odd	2
63.12.a.a.1.1		1	21.20	even	2
147.12.a.b.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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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