LMFDB - Embedded newform 147.12.a.a.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)

$f(q)$

$=$

$q-62.0000 q^{2} -243.000 q^{3} +1796.00 q^{4} +3310.00 q^{5} +15066.0 q^{6} +15624.0 q^{8} +59049.0 q^{9} +O(q^{10})$

\(q-62.0000 q^{2} -243.000 q^{3} +1796.00 q^{4} +3310.00 q^{5} +15066.0 q^{6} +15624.0 q^{8} +59049.0 q^{9} -205220. q^{10} +628904. q^{11} -436428. q^{12} -176854. q^{13} -804330. q^{15} -4.64690e6 q^{16} -566958. q^{17} -3.66104e6 q^{18} +1.29161e7 q^{19} +5.94476e6 q^{20} -3.89920e7 q^{22} -2.56641e7 q^{23} -3.79663e6 q^{24} -3.78720e7 q^{25} +1.09649e7 q^{26} -1.43489e7 q^{27} -4.74115e7 q^{29} +4.98685e7 q^{30} -1.39427e7 q^{31} +2.56110e8 q^{32} -1.52824e8 q^{33} +3.51514e7 q^{34} +1.06052e8 q^{36} -6.41657e8 q^{37} -8.00800e8 q^{38} +4.29755e7 q^{39} +5.17154e7 q^{40} +6.00859e8 q^{41} -1.41775e9 q^{43} +1.12951e9 q^{44} +1.95452e8 q^{45} +1.59117e9 q^{46} -8.60414e8 q^{47} +1.12920e9 q^{48} +2.34807e9 q^{50} +1.37771e8 q^{51} -3.17630e8 q^{52} +3.22142e9 q^{53} +8.89632e8 q^{54} +2.08167e9 q^{55} -3.13862e9 q^{57} +2.93951e9 q^{58} +6.08296e9 q^{59} -1.44458e9 q^{60} +8.64141e8 q^{61} +8.64446e8 q^{62} -6.36195e9 q^{64} -5.85387e8 q^{65} +9.47507e9 q^{66} +1.18977e10 q^{67} -1.01826e9 q^{68} +6.23638e9 q^{69} -1.40778e10 q^{71} +9.22582e8 q^{72} +1.88142e10 q^{73} +3.97828e10 q^{74} +9.20290e9 q^{75} +2.31974e10 q^{76} -2.66448e9 q^{78} +1.70214e10 q^{79} -1.53812e10 q^{80} +3.48678e9 q^{81} -3.72533e10 q^{82} -4.76131e10 q^{83} -1.87663e9 q^{85} +8.79007e10 q^{86} +1.15210e10 q^{87} +9.82600e9 q^{88} -6.15621e10 q^{89} -1.21180e10 q^{90} -4.60927e10 q^{92} +3.38807e9 q^{93} +5.33457e10 q^{94} +4.27524e10 q^{95} -6.22346e10 q^{96} +1.66480e11 q^{97} +3.71362e10 q^{99} +O(q^{100})\)

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 $

$2$	−62.0000	−1.37002	−0.685010	−	0.728534i	$-0.740202\pi$
$2$	−62.0000	−1.37002	−0.685010	+	0.728534i	$0.740202\pi$
$3$	−243.000	−0.577350
$3$	−243.000	−0.577350
$4$	1796.00	0.876953
$5$	3310.00	0.473689	0.236844	−	0.971548i	$-0.423887\pi$
$5$	3310.00	0.473689	0.236844	+	0.971548i	$0.423887\pi$
$6$	15066.0	0.790981
$7$	0	0
$7$	0	0
$8$	15624.0	0.168577
$9$	59049.0	0.333333
$10$	−205220.	−0.648963
$11$	628904.	1.17740	0.588701	−	0.808351i	$-0.299640\pi$
$11$	628904.	1.17740	0.588701	+	0.808351i	$0.299640\pi$
$12$	−436428.	−0.506309
$13$	−176854.	−0.132107	−0.0660536	−	0.997816i	$-0.521041\pi$
$13$	−176854.	−0.132107	−0.0660536	+	0.997816i	$0.521041\pi$
$14$	0	0
$15$	−804330.	−0.273484
$16$	−4.64690e6	−1.10791
$17$	−566958.	−0.0968460	−0.0484230	−	0.998827i	$-0.515420\pi$
$17$	−566958.	−0.0968460	−0.0484230	+	0.998827i	$0.515420\pi$
$18$	−3.66104e6	−0.456673
$19$	1.29161e7	1.19671	0.598353	−	0.801233i	$-0.295822\pi$
$19$	1.29161e7	1.19671	0.598353	+	0.801233i	$0.295822\pi$
$20$	5.94476e6	0.415403
$21$	0	0
$22$	−3.89920e7	−1.61306
$23$	−2.56641e7	−0.831425	−0.415712	−	0.909496i	$-0.636468\pi$
$23$	−2.56641e7	−0.831425	−0.415712	+	0.909496i	$0.636468\pi$
$24$	−3.79663e6	−0.0973277
$25$	−3.78720e7	−0.775619
$26$	1.09649e7	0.180989
$27$	−1.43489e7	−0.192450
$28$	0	0
$29$	−4.74115e7	−0.429234	−0.214617	−	0.976698i	$-0.568850\pi$
$29$	−4.74115e7	−0.429234	−0.214617	+	0.976698i	$0.568850\pi$
$30$	4.98685e7	0.374679
$31$	−1.39427e7	−0.0874696	−0.0437348	−	0.999043i	$-0.513926\pi$
$31$	−1.39427e7	−0.0874696	−0.0437348	+	0.999043i	$0.513926\pi$
$32$	2.56110e8	1.34928
$33$	−1.52824e8	−0.679773
$34$	3.51514e7	0.132681
$35$	0	0
$36$	1.06052e8	0.292318
$37$	−6.41657e8	−1.52123	−0.760613	−	0.649206i	$-0.775101\pi$
$37$	−6.41657e8	−1.52123	−0.760613	+	0.649206i	$0.775101\pi$
$38$	−8.00800e8	−1.63951
$39$	4.29755e7	0.0762721
$40$	5.17154e7	0.0798528
$41$	6.00859e8	0.809956	0.404978	−	0.914326i	$-0.367279\pi$
$41$	6.00859e8	0.809956	0.404978	+	0.914326i	$0.367279\pi$
$42$	0	0
$43$	−1.41775e9	−1.47070	−0.735350	−	0.677687i	$-0.762982\pi$
$43$	−1.41775e9	−1.47070	−0.735350	+	0.677687i	$0.762982\pi$
$44$	1.12951e9	1.03253
$45$	1.95452e8	0.157896
$46$	1.59117e9	1.13907
$47$	−8.60414e8	−0.547229	−0.273615	−	0.961839i	$-0.588219\pi$
$47$	−8.60414e8	−0.547229	−0.273615	+	0.961839i	$0.588219\pi$
$48$	1.12920e9	0.639650
$49$	0	0
$50$	2.34807e9	1.06261
$51$	1.37771e8	0.0559141
$52$	−3.17630e8	−0.115852
$53$	3.22142e9	1.05811	0.529054	−	0.848588i	$-0.322547\pi$
$53$	3.22142e9	1.05811	0.529054	+	0.848588i	$0.322547\pi$
$54$	8.89632e8	0.263660
$55$	2.08167e9	0.557722
$56$	0	0
$57$	−3.13862e9	−0.690918
$58$	2.93951e9	0.588059
$59$	6.08296e9	1.10772	0.553859	−	0.832611i	$-0.313155\pi$
$59$	6.08296e9	1.10772	0.553859	+	0.832611i	$0.313155\pi$
$60$	−1.44458e9	−0.239833
$61$	8.64141e8	0.131000	0.0654999	−	0.997853i	$-0.479136\pi$
$61$	8.64141e8	0.131000	0.0654999	+	0.997853i	$0.479136\pi$
$62$	8.64446e8	0.119835
$63$	0	0
$64$	−6.36195e9	−0.740629
$65$	−5.85387e8	−0.0625777
$66$	9.47507e9	0.931302
$67$	1.18977e10	1.07659	0.538295	−	0.842756i	$-0.319069\pi$
$67$	1.18977e10	1.07659	0.538295	+	0.842756i	$0.319069\pi$
$68$	−1.01826e9	−0.0849294
$69$	6.23638e9	0.480023
$70$	0	0
$71$	−1.40778e10	−0.926006	−0.463003	−	0.886357i	$-0.653228\pi$
$71$	−1.40778e10	−0.926006	−0.463003	+	0.886357i	$0.653228\pi$
$72$	9.22582e8	0.0561922
$73$	1.88142e10	1.06221	0.531103	−	0.847307i	$-0.321778\pi$
$73$	1.88142e10	1.06221	0.531103	+	0.847307i	$0.321778\pi$
$74$	3.97828e10	2.08411
$75$	9.20290e9	0.447804
$76$	2.31974e10	1.04945
$77$	0	0
$78$	−2.66448e9	−0.104494
$79$	1.70214e10	0.622365	0.311183	−	0.950350i	$-0.399275\pi$
$79$	1.70214e10	0.622365	0.311183	+	0.950350i	$0.399275\pi$
$80$	−1.53812e10	−0.524803
$81$	3.48678e9	0.111111
$82$	−3.72533e10	−1.10966
$83$	−4.76131e10	−1.32677	−0.663387	−	0.748276i	$-0.730882\pi$
$83$	−4.76131e10	−1.32677	−0.663387	+	0.748276i	$0.730882\pi$
$84$	0	0
$85$	−1.87663e9	−0.0458749
$86$	8.79007e10	2.01489
$87$	1.15210e10	0.247818
$88$	9.82600e9	0.198482
$89$	−6.15621e10	−1.16861	−0.584303	−	0.811536i	$-0.698632\pi$
$89$	−6.15621e10	−1.16861	−0.584303	+	0.811536i	$0.698632\pi$
$90$	−1.21180e10	−0.216321
$91$	0	0
$92$	−4.60927e10	−0.729121
$93$	3.38807e9	0.0505006
$94$	5.33457e10	0.749715
$95$	4.27524e10	0.566866
$96$	−6.22346e10	−0.779005
$97$	1.66480e11	1.96841	0.984207	−	0.177024i	$-0.0566470\pi$
$97$	1.66480e11	1.96841	0.984207	+	0.177024i	$0.0566470\pi$
$98$	0	0
$99$	3.71362e10	0.392467
$100$	−6.80182e10	−0.680182
$101$	1.61844e11	1.53225	0.766123	−	0.642694i	$-0.222183\pi$
$101$	1.61844e11	1.53225	0.766123	+	0.642694i	$0.222183\pi$
$102$	−8.54179e9	−0.0766034
$103$	−9.71731e10	−0.825926	−0.412963	−	0.910748i	$-0.635506\pi$
$103$	−9.71731e10	−0.825926	−0.412963	+	0.910748i	$0.635506\pi$
$104$	−2.76317e9	−0.0222702
$105$	0	0
$106$	−1.99728e11	−1.44963
$107$	−2.57884e11	−1.77752	−0.888759	−	0.458375i	$-0.848432\pi$
$107$	−2.57884e11	−1.77752	−0.888759	+	0.458375i	$0.848432\pi$
$108$	−2.57706e10	−0.168770
$109$	7.81504e10	0.486503	0.243251	−	0.969963i	$-0.421786\pi$
$109$	7.81504e10	0.486503	0.243251	+	0.969963i	$0.421786\pi$
$110$	−1.29064e11	−0.764090
$111$	1.55923e11	0.878280
$112$	0	0
$113$	−1.75764e11	−0.897425	−0.448713	−	0.893676i	$-0.648117\pi$
$113$	−1.75764e11	−0.897425	−0.448713	+	0.893676i	$0.648117\pi$
$114$	1.94594e11	0.946572
$115$	−8.49482e10	−0.393836
$116$	−8.51510e10	−0.376418
$117$	−1.04431e10	−0.0440357
$118$	−3.77143e11	−1.51759
$119$	0	0
$120$	−1.25669e10	−0.0461030
$121$	1.10209e11	0.386274
$122$	−5.35767e10	−0.179472
$123$	−1.46009e11	−0.467628
$124$	−2.50411e10	−0.0767067
$125$	−2.86977e11	−0.841091
$126$	0	0
$127$	1.56005e11	0.419003	0.209501	−	0.977808i	$-0.432816\pi$
$127$	1.56005e11	0.419003	0.209501	+	0.977808i	$0.432816\pi$
$128$	−1.30071e11	−0.334601
$129$	3.44514e11	0.849109
$130$	3.62940e10	0.0857326
$131$	4.72989e11	1.07117	0.535586	−	0.844481i	$-0.320091\pi$
$131$	4.72989e11	1.07117	0.535586	+	0.844481i	$0.320091\pi$
$132$	−2.74471e11	−0.596129
$133$	0	0
$134$	−7.37655e11	−1.47495
$135$	−4.74949e10	−0.0911614
$136$	−8.85815e9	−0.0163260
$137$	−6.72271e11	−1.19009	−0.595047	−	0.803691i	$-0.702866\pi$
$137$	−6.72271e11	−1.19009	−0.595047	+	0.803691i	$0.702866\pi$
$138$	−3.86655e11	−0.657641
$139$	−6.65757e11	−1.08826	−0.544132	−	0.839000i	$-0.683141\pi$
$139$	−6.65757e11	−1.08826	−0.544132	+	0.839000i	$0.683141\pi$
$140$	0	0
$141$	2.09081e11	0.315943
$142$	8.72824e11	1.26865
$143$	−1.11224e11	−0.155543
$144$	−2.74395e11	−0.369302
$145$	−1.56932e11	−0.203323
$146$	−1.16648e12	−1.45524
$147$	0	0
$148$	−1.15242e12	−1.33404
$149$	3.53074e11	0.393860	0.196930	−	0.980418i	$-0.436903\pi$
$149$	3.53074e11	0.393860	0.196930	+	0.980418i	$0.436903\pi$
$150$	−5.70580e11	−0.613500
$151$	−1.10135e12	−1.14170	−0.570850	−	0.821054i	$-0.693386\pi$
$151$	−1.10135e12	−1.14170	−0.570850	+	0.821054i	$0.693386\pi$
$152$	2.01802e11	0.201737
$153$	−3.34783e10	−0.0322820
$154$	0	0
$155$	−4.61503e10	−0.0414333
$156$	7.71840e10	0.0668871
$157$	−6.25950e11	−0.523711	−0.261855	−	0.965107i	$-0.584334\pi$
$157$	−6.25950e11	−0.523711	−0.261855	+	0.965107i	$0.584334\pi$
$158$	−1.05532e12	−0.852652
$159$	−7.82805e11	−0.610899
$160$	8.47723e11	0.639137
$161$	0	0
$162$	−2.16181e11	−0.152224
$163$	1.83554e12	1.24949	0.624743	−	0.780830i	$-0.285204\pi$
$163$	1.83554e12	1.24949	0.624743	+	0.780830i	$0.285204\pi$
$164$	1.07914e12	0.710294
$165$	−5.05846e11	−0.322001
$166$	2.95201e12	1.81771
$167$	−2.10696e12	−1.25521	−0.627605	−	0.778532i	$-0.715965\pi$
$167$	−2.10696e12	−1.25521	−0.627605	+	0.778532i	$0.715965\pi$
$168$	0	0
$169$	−1.76088e12	−0.982548
$170$	1.16351e11	0.0628495
$171$	7.62684e11	0.398902
$172$	−2.54629e12	−1.28974
$173$	−3.15816e12	−1.54946	−0.774729	−	0.632293i	$-0.782114\pi$
$173$	−3.15816e12	−1.54946	−0.774729	+	0.632293i	$0.782114\pi$
$174$	−7.14301e11	−0.339516
$175$	0	0
$176$	−2.92245e12	−1.30445
$177$	−1.47816e12	−0.639541
$178$	3.81685e12	1.60101
$179$	−8.39250e10	−0.0341350	−0.0170675	−	0.999854i	$-0.505433\pi$
$179$	−8.39250e10	−0.0341350	−0.0170675	+	0.999854i	$0.505433\pi$
$180$	3.51032e11	0.138468
$181$	7.68001e11	0.293853	0.146926	−	0.989147i	$-0.453062\pi$
$181$	7.68001e11	0.293853	0.146926	+	0.989147i	$0.453062\pi$
$182$	0	0
$183$	−2.09986e11	−0.0756328
$184$	−4.00976e11	−0.140159
$185$	−2.12389e12	−0.720587
$186$	−2.10060e11	−0.0691868
$187$	−3.56562e11	−0.114027
$188$	−1.54530e12	−0.479894
$189$	0	0
$190$	−2.65065e12	−0.776617
$191$	−2.02526e12	−0.576497	−0.288249	−	0.957556i	$-0.593073\pi$
$191$	−2.02526e12	−0.576497	−0.288249	+	0.957556i	$0.593073\pi$
$192$	1.54595e12	0.427602
$193$	6.42414e12	1.72683	0.863416	−	0.504493i	$-0.168320\pi$
$193$	6.42414e12	1.72683	0.863416	+	0.504493i	$0.168320\pi$
$194$	−1.03217e13	−2.69676
$195$	1.42249e11	0.0361292
$196$	0	0
$197$	3.07067e12	0.737343	0.368671	−	0.929560i	$-0.379813\pi$
$197$	3.07067e12	0.737343	0.368671	+	0.929560i	$0.379813\pi$
$198$	−2.30244e12	−0.537688
$199$	−4.89049e12	−1.11086	−0.555431	−	0.831562i	$-0.687447\pi$
$199$	−4.89049e12	−1.11086	−0.555431	+	0.831562i	$0.687447\pi$
$200$	−5.91713e11	−0.130751
$201$	−2.89113e12	−0.621570
$202$	−1.00343e13	−2.09921
$203$	0	0
$204$	2.47436e11	0.0490340
$205$	1.98884e12	0.383667
$206$	6.02473e12	1.13153
$207$	−1.51544e12	−0.277142
$208$	8.21822e11	0.146362
$209$	8.12300e12	1.40900
$210$	0	0
$211$	−1.04782e13	−1.72478	−0.862392	−	0.506240i	$-0.831035\pi$
$211$	−1.04782e13	−1.72478	−0.862392	+	0.506240i	$0.831035\pi$
$212$	5.78567e12	0.927911
$213$	3.42091e12	0.534630
$214$	1.59888e13	2.43523
$215$	−4.69276e12	−0.696654
$216$	−2.24187e11	−0.0324426
$217$	0	0
$218$	−4.84532e12	−0.666518
$219$	−4.57184e12	−0.613265
$220$	3.73868e12	0.489096
$221$	1.00269e11	0.0127941
$222$	−9.66721e12	−1.20326
$223$	1.47335e13	1.78908	0.894539	−	0.446989i	$-0.147504\pi$
$223$	1.47335e13	1.78908	0.894539	+	0.446989i	$0.147504\pi$
$224$	0	0
$225$	−2.23631e12	−0.258540
$226$	1.08974e13	1.22949
$227$	7.42051e12	0.817131	0.408566	−	0.912729i	$-0.366029\pi$
$227$	7.42051e12	0.817131	0.408566	+	0.912729i	$0.366029\pi$
$228$	−5.63696e12	−0.605903
$229$	1.26763e13	1.33014	0.665069	−	0.746782i	$-0.268402\pi$
$229$	1.26763e13	1.33014	0.665069	+	0.746782i	$0.268402\pi$
$230$	5.26679e12	0.539564
$231$	0	0
$232$	−7.40757e11	−0.0723588
$233$	6.60440e12	0.630051	0.315026	−	0.949083i	$-0.397987\pi$
$233$	6.60440e12	0.630051	0.315026	+	0.949083i	$0.397987\pi$
$234$	6.47469e11	0.0603298
$235$	−2.84797e12	−0.259216
$236$	1.09250e13	0.971416
$237$	−4.13619e12	−0.359323
$238$	0	0
$239$	1.40120e13	1.16228	0.581141	−	0.813803i	$-0.302607\pi$
$239$	1.40120e13	1.16228	0.581141	+	0.813803i	$0.302607\pi$
$240$	3.73764e12	0.302995
$241$	4.01812e12	0.318368	0.159184	−	0.987249i	$-0.449114\pi$
$241$	4.01812e12	0.318368	0.159184	+	0.987249i	$0.449114\pi$
$242$	−6.83293e12	−0.529203
$243$	−8.47289e11	−0.0641500
$244$	1.55200e12	0.114881
$245$	0	0
$246$	9.05255e12	0.640660
$247$	−2.28427e12	−0.158093
$248$	−2.17840e11	−0.0147453
$249$	1.15700e13	0.766014
$250$	1.77926e13	1.15231
$251$	1.31534e13	0.833363	0.416681	−	0.909053i	$-0.363193\pi$
$251$	1.31534e13	0.833363	0.416681	+	0.909053i	$0.363193\pi$
$252$	0	0
$253$	−1.61403e13	−0.978921
$254$	−9.67229e12	−0.574042
$255$	4.56021e11	0.0264859
$256$	2.10937e13	1.19904
$257$	−3.38720e13	−1.88456	−0.942278	−	0.334833i	$-0.891320\pi$
$257$	−3.38720e13	−1.88456	−0.942278	+	0.334833i	$0.891320\pi$
$258$	−2.13599e13	−1.16330
$259$	0	0
$260$	−1.05135e12	−0.0548777
$261$	−2.79960e12	−0.143078
$262$	−2.93253e13	−1.46753
$263$	−1.39562e13	−0.683930	−0.341965	−	0.939713i	$-0.611092\pi$
$263$	−1.39562e13	−0.683930	−0.341965	+	0.939713i	$0.611092\pi$
$264$	−2.38772e12	−0.114594
$265$	1.06629e13	0.501214
$266$	0	0
$267$	1.49596e13	0.674695
$268$	2.13682e13	0.944119
$269$	−2.69859e13	−1.16815	−0.584077	−	0.811699i	$-0.698543\pi$
$269$	−2.69859e13	−1.16815	−0.584077	+	0.811699i	$0.698543\pi$
$270$	2.94468e12	0.124893
$271$	4.60650e13	1.91443	0.957215	−	0.289377i	$-0.0934483\pi$
$271$	4.60650e13	1.91443	0.957215	+	0.289377i	$0.0934483\pi$
$272$	2.63459e12	0.107296
$273$	0	0
$274$	4.16808e13	1.63045
$275$	−2.38179e13	−0.913215
$276$	1.12005e13	0.420958
$277$	3.39724e13	1.25166	0.625831	−	0.779959i	$-0.284760\pi$
$277$	3.39724e13	1.25166	0.625831	+	0.779959i	$0.284760\pi$
$278$	4.12769e13	1.49094
$279$	−8.23301e11	−0.0291565
$280$	0	0
$281$	2.78504e11	0.00948301	0.00474150	−	0.999989i	$-0.498491\pi$
$281$	2.78504e11	0.00948301	0.00474150	+	0.999989i	$0.498491\pi$
$282$	−1.29630e13	−0.432848
$283$	−1.75578e13	−0.574971	−0.287485	−	0.957785i	$-0.592819\pi$
$283$	−1.75578e13	−0.574971	−0.287485	+	0.957785i	$0.592819\pi$
$284$	−2.52837e13	−0.812064
$285$	−1.03888e13	−0.327280
$286$	6.89590e12	0.213097
$287$	0	0
$288$	1.51230e13	0.449759
$289$	−3.39505e13	−0.990621
$290$	9.72978e12	0.278557
$291$	−4.04545e13	−1.13646
$292$	3.37902e13	0.931505
$293$	−5.72209e13	−1.54804	−0.774021	−	0.633161i	$-0.781757\pi$
$293$	−5.72209e13	−1.54804	−0.774021	+	0.633161i	$0.781757\pi$
$294$	0	0
$295$	2.01346e13	0.524713
$296$	−1.00253e13	−0.256443
$297$	−9.02409e12	−0.226591
$298$	−2.18906e13	−0.539596
$299$	4.53880e12	0.109837
$300$	1.65284e13	0.392703
$301$	0	0
$302$	6.82837e13	1.56415
$303$	−3.93281e13	−0.884643
$304$	−6.00199e13	−1.32584
$305$	2.86031e12	0.0620531
$306$	2.07565e12	0.0442270
$307$	1.19584e13	0.250271	0.125136	−	0.992140i	$-0.460063\pi$
$307$	1.19584e13	0.250271	0.125136	+	0.992140i	$0.460063\pi$
$308$	0	0
$309$	2.36131e13	0.476849
$310$	2.86132e12	0.0567645
$311$	−7.81662e13	−1.52348	−0.761741	−	0.647882i	$-0.775655\pi$
$311$	−7.81662e13	−1.52348	−0.761741	+	0.647882i	$0.775655\pi$
$312$	6.71450e11	0.0128577
$313$	−1.06068e13	−0.199567	−0.0997837	−	0.995009i	$-0.531815\pi$
$313$	−1.06068e13	−0.199567	−0.0997837	+	0.995009i	$0.531815\pi$
$314$	3.88089e13	0.717494
$315$	0	0
$316$	3.05704e13	0.545785
$317$	−5.14215e13	−0.902234	−0.451117	−	0.892465i	$-0.648974\pi$
$317$	−5.14215e13	−0.902234	−0.451117	+	0.892465i	$0.648974\pi$
$318$	4.85339e13	0.836944
$319$	−2.98173e13	−0.505381
$320$	−2.10581e13	−0.350827
$321$	6.26658e13	1.02625
$322$	0	0
$323$	−7.32290e12	−0.115896
$324$	6.26226e12	0.0974392
$325$	6.69782e12	0.102465
$326$	−1.13803e14	−1.71182
$327$	−1.89905e13	−0.280883
$328$	9.38783e12	0.136540
$329$	0	0
$330$	3.13625e13	0.441147
$331$	−9.14328e13	−1.26488	−0.632438	−	0.774611i	$-0.717946\pi$
$331$	−9.14328e13	−1.26488	−0.632438	+	0.774611i	$0.717946\pi$
$332$	−8.55132e13	−1.16352
$333$	−3.78892e13	−0.507075
$334$	1.30632e14	1.71966
$335$	3.93813e13	0.509969
$336$	0	0
$337$	4.14566e13	0.519552	0.259776	−	0.965669i	$-0.416351\pi$
$337$	4.14566e13	0.519552	0.259776	+	0.965669i	$0.416351\pi$
$338$	1.09175e14	1.34611
$339$	4.27106e13	0.518129
$340$	−3.37043e12	−0.0402301
$341$	−8.76861e12	−0.102987
$342$	−4.72864e13	−0.546503
$343$	0	0
$344$	−2.21510e13	−0.247926
$345$	2.06424e13	0.227382
$346$	1.95806e14	2.12279
$347$	1.30695e14	1.39459	0.697295	−	0.716784i	$-0.254387\pi$
$347$	1.30695e14	1.39459	0.697295	+	0.716784i	$0.254387\pi$
$348$	2.06917e13	0.217325
$349$	−5.43941e13	−0.562356	−0.281178	−	0.959656i	$-0.590725\pi$
$349$	−5.43941e13	−0.562356	−0.281178	+	0.959656i	$0.590725\pi$
$350$	0	0
$351$	2.53766e12	0.0254240
$352$	1.61068e14	1.58864
$353$	−1.38354e14	−1.34348	−0.671738	−	0.740789i	$-0.734452\pi$
$353$	−1.38354e14	−1.34348	−0.671738	+	0.740789i	$0.734452\pi$
$354$	9.16459e13	0.876183
$355$	−4.65975e13	−0.438639
$356$	−1.10565e14	−1.02481
$357$	0	0
$358$	5.20335e12	0.0467656
$359$	−2.59947e13	−0.230073	−0.115037	−	0.993361i	$-0.536699\pi$
$359$	−2.59947e13	−0.230073	−0.115037	+	0.993361i	$0.536699\pi$
$360$	3.05375e12	0.0266176
$361$	5.03360e13	0.432105
$362$	−4.76161e13	−0.402584
$363$	−2.67807e13	−0.223016
$364$	0	0
$365$	6.22748e13	0.503155
$366$	1.30192e13	0.103618
$367$	1.54019e13	0.120756	0.0603782	−	0.998176i	$-0.480769\pi$
$367$	1.54019e13	0.120756	0.0603782	+	0.998176i	$0.480769\pi$
$368$	1.19258e14	0.921141
$369$	3.54801e13	0.269985
$370$	1.31681e14	0.987219
$371$	0	0
$372$	6.08498e12	0.0442866
$373$	5.42555e13	0.389086	0.194543	−	0.980894i	$-0.437678\pi$
$373$	5.42555e13	0.389086	0.194543	+	0.980894i	$0.437678\pi$
$374$	2.21069e13	0.156219
$375$	6.97355e13	0.485604
$376$	−1.34431e13	−0.0922500
$377$	8.38491e12	0.0567049
$378$	0	0
$379$	−6.54226e13	−0.429747	−0.214873	−	0.976642i	$-0.568934\pi$
$379$	−6.54226e13	−0.429747	−0.214873	+	0.976642i	$0.568934\pi$
$380$	7.67833e13	0.497115
$381$	−3.79091e13	−0.241911
$382$	1.25566e14	0.789812
$383$	−7.57373e13	−0.469587	−0.234794	−	0.972045i	$-0.575441\pi$
$383$	−7.57373e13	−0.469587	−0.234794	+	0.972045i	$0.575441\pi$
$384$	3.16074e13	0.193182
$385$	0	0
$386$	−3.98297e14	−2.36579
$387$	−8.37169e13	−0.490234
$388$	2.98997e14	1.72621
$389$	−2.09880e13	−0.119467	−0.0597335	−	0.998214i	$-0.519025\pi$
$389$	−2.09880e13	−0.119467	−0.0597335	+	0.998214i	$0.519025\pi$
$390$	−8.81944e12	−0.0494978
$391$	1.45505e13	0.0805202
$392$	0	0
$393$	−1.14936e14	−0.618441
$394$	−1.90382e14	−1.01017
$395$	5.63407e13	0.294807
$396$	6.66965e13	0.344175
$397$	−3.40304e14	−1.73189	−0.865944	−	0.500142i	$-0.833281\pi$
$397$	−3.40304e14	−1.73189	−0.865944	+	0.500142i	$0.833281\pi$
$398$	3.03210e14	1.52190
$399$	0	0
$400$	1.75987e14	0.859313
$401$	−4.37400e13	−0.210662	−0.105331	−	0.994437i	$-0.533590\pi$
$401$	−4.37400e13	−0.210662	−0.105331	+	0.994437i	$0.533590\pi$
$402$	1.79250e14	0.851562
$403$	2.46582e12	0.0115554
$404$	2.90672e14	1.34371
$405$	1.15413e13	0.0526321
$406$	0	0
$407$	−4.03541e14	−1.79109
$408$	2.15253e12	0.00942581
$409$	−1.41421e14	−0.610992	−0.305496	−	0.952193i	$-0.598822\pi$
$409$	−1.41421e14	−0.610992	−0.305496	+	0.952193i	$0.598822\pi$
$410$	−1.23308e14	−0.525631
$411$	1.63362e14	0.687101
$412$	−1.74523e14	−0.724298
$413$	0	0
$414$	9.39572e13	0.379689
$415$	−1.57599e14	−0.628478
$416$	−4.52940e13	−0.178249
$417$	1.61779e14	0.628310
$418$	−5.03626e14	−1.93036
$419$	4.70763e14	1.78084	0.890420	−	0.455139i	$-0.150411\pi$
$419$	4.70763e14	1.78084	0.890420	+	0.455139i	$0.150411\pi$
$420$	0	0
$421$	−5.22853e12	−0.0192676	−0.00963381	−	0.999954i	$-0.503067\pi$
$421$	−5.22853e12	−0.0192676	−0.00963381	+	0.999954i	$0.503067\pi$
$422$	6.49651e14	2.36299
$423$	−5.08066e13	−0.182410
$424$	5.03315e13	0.178372
$425$	2.14718e13	0.0751156
$426$	−2.12096e14	−0.732454
$427$	0	0
$428$	−4.63160e14	−1.55880
$429$	2.70275e13	0.0898029
$430$	2.90951e14	0.954430
$431$	−2.67746e14	−0.867159	−0.433579	−	0.901115i	$-0.642750\pi$
$431$	−2.67746e14	−0.867159	−0.433579	+	0.901115i	$0.642750\pi$
$432$	6.66779e13	0.213217
$433$	−4.81955e14	−1.52168	−0.760840	−	0.648940i	$-0.775213\pi$
$433$	−4.81955e14	−1.52168	−0.760840	+	0.648940i	$0.775213\pi$
$434$	0	0
$435$	3.81345e13	0.117389
$436$	1.40358e14	0.426640
$437$	−3.31481e14	−0.994971
$438$	2.83454e14	0.840185
$439$	−1.38232e14	−0.404625	−0.202313	−	0.979321i	$-0.564846\pi$
$439$	−1.38232e14	−0.404625	−0.202313	+	0.979321i	$0.564846\pi$
$440$	3.25240e13	0.0940188
$441$	0	0
$442$	−6.21666e12	−0.0175281
$443$	1.16434e14	0.324235	0.162117	−	0.986771i	$-0.448168\pi$
$443$	1.16434e14	0.324235	0.162117	+	0.986771i	$0.448168\pi$
$444$	2.80037e14	0.770210
$445$	−2.03770e14	−0.553555
$446$	−9.13477e14	−2.45107
$447$	−8.57971e13	−0.227395
$448$	0	0
$449$	−1.46736e14	−0.379474	−0.189737	−	0.981835i	$-0.560763\pi$
$449$	−1.46736e14	−0.379474	−0.189737	+	0.981835i	$0.560763\pi$
$450$	1.38651e14	0.354204
$451$	3.77883e14	0.953644
$452$	−3.15672e14	−0.787000
$453$	2.67628e14	0.659161
$454$	−4.60072e14	−1.11949
$455$	0	0
$456$	−4.90378e13	−0.116473
$457$	−6.68239e14	−1.56817	−0.784084	−	0.620654i	$-0.786867\pi$
$457$	−6.68239e14	−1.56817	−0.784084	+	0.620654i	$0.786867\pi$
$458$	−7.85930e14	−1.82232
$459$	8.13523e12	0.0186380
$460$	−1.52567e14	−0.345376
$461$	−4.70733e14	−1.05298	−0.526489	−	0.850182i	$-0.676492\pi$
$461$	−4.70733e14	−1.05298	−0.526489	+	0.850182i	$0.676492\pi$
$462$	0	0
$463$	−3.13454e14	−0.684666	−0.342333	−	0.939579i	$-0.611217\pi$
$463$	−3.13454e14	−0.684666	−0.342333	+	0.939579i	$0.611217\pi$
$464$	2.20316e14	0.475551
$465$	1.12145e13	0.0239215
$466$	−4.09473e14	−0.863183
$467$	3.55296e14	0.740197	0.370099	−	0.928992i	$-0.379324\pi$
$467$	3.55296e14	0.740197	0.370099	+	0.928992i	$0.379324\pi$
$468$	−1.87557e13	−0.0386173
$469$	0	0
$470$	1.76574e14	0.355131
$471$	1.52106e14	0.302365
$472$	9.50402e13	0.186735
$473$	−8.91631e14	−1.73161
$474$	2.56444e14	0.492279
$475$	−4.89160e14	−0.928188
$476$	0	0
$477$	1.90222e14	0.352703
$478$	−8.68743e14	−1.59235
$479$	1.71350e14	0.310483	0.155242	−	0.987877i	$-0.450384\pi$
$479$	1.71350e14	0.310483	0.155242	+	0.987877i	$0.450384\pi$
$480$	−2.05997e14	−0.369006
$481$	1.13480e14	0.200965
$482$	−2.49124e14	−0.436171
$483$	0	0
$484$	1.97935e14	0.338744
$485$	5.51047e14	0.932415
$486$	5.25319e13	0.0878868
$487$	−1.08143e15	−1.78890	−0.894452	−	0.447164i	$-0.852434\pi$
$487$	−1.08143e15	−1.78890	−0.894452	+	0.447164i	$0.852434\pi$
$488$	1.35013e13	0.0220835
$489$	−4.46036e14	−0.721391
$490$	0	0
$491$	3.20214e14	0.506398	0.253199	−	0.967414i	$-0.418517\pi$
$491$	3.20214e14	0.506398	0.253199	+	0.967414i	$0.418517\pi$
$492$	−2.62232e14	−0.410088
$493$	2.68803e13	0.0415696
$494$	1.41625e14	0.216591
$495$	1.22921e14	0.185907
$496$	6.47902e13	0.0969081
$497$	0	0
$498$	−7.17340e14	−1.04945
$499$	5.50970e13	0.0797214	0.0398607	−	0.999205i	$-0.487309\pi$
$499$	5.50970e13	0.0797214	0.0398607	+	0.999205i	$0.487309\pi$
$500$	−5.15412e14	−0.737597
$501$	5.11992e14	0.724696
$502$	−8.15514e14	−1.14172
$503$	6.15656e13	0.0852538	0.0426269	−	0.999091i	$-0.486427\pi$
$503$	6.15656e13	0.0852538	0.0426269	+	0.999091i	$0.486427\pi$
$504$	0	0
$505$	5.35703e14	0.725808
$506$	1.00070e15	1.34114
$507$	4.27895e14	0.567274
$508$	2.80184e14	0.367446
$509$	1.09624e15	1.42219	0.711093	−	0.703098i	$-0.248200\pi$
$509$	1.09624e15	1.42219	0.711093	+	0.703098i	$0.248200\pi$
$510$	−2.82733e13	−0.0362861
$511$	0	0
$512$	−1.04142e15	−1.30810
$513$	−1.85332e14	−0.230306
$514$	2.10006e15	2.58188
$515$	−3.21643e14	−0.391232
$516$	6.18747e14	0.744629
$517$	−5.41118e14	−0.644308
$518$	0	0
$519$	7.67432e14	0.894580
$520$	−9.14608e12	−0.0105491
$521$	−1.27254e15	−1.45232	−0.726161	−	0.687525i	$-0.758697\pi$
$521$	−1.27254e15	−1.45232	−0.726161	+	0.687525i	$0.758697\pi$
$522$	1.73575e14	0.196020
$523$	5.87195e14	0.656180	0.328090	−	0.944646i	$-0.393595\pi$
$523$	5.87195e14	0.656180	0.328090	+	0.944646i	$0.393595\pi$
$524$	8.49489e14	0.939367
$525$	0	0
$526$	8.65286e14	0.936997
$527$	7.90491e12	0.00847108
$528$	7.10156e14	0.753125
$529$	−2.94164e14	−0.308733
$530$	−6.61100e14	−0.686673
$531$	3.59193e14	0.369239
$532$	0	0
$533$	−1.06264e14	−0.107001
$534$	−9.27494e14	−0.924345
$535$	−8.53597e14	−0.841990
$536$	1.85889e14	0.181488
$537$	2.03938e13	0.0197078
$538$	1.67313e15	1.60039
$539$	0	0
$540$	−8.53008e13	−0.0799443
$541$	1.16872e14	0.108424	0.0542119	−	0.998529i	$-0.482735\pi$
$541$	1.16872e14	0.108424	0.0542119	+	0.998529i	$0.482735\pi$
$542$	−2.85603e15	−2.62281
$543$	−1.86624e14	−0.169656
$544$	−1.45203e14	−0.130672
$545$	2.58678e14	0.230451
$546$	0	0
$547$	1.31348e15	1.14681	0.573407	−	0.819271i	$-0.305621\pi$
$547$	1.31348e15	1.14681	0.573407	+	0.819271i	$0.305621\pi$
$548$	−1.20740e15	−1.04366
$549$	5.10267e13	0.0436666
$550$	1.47671e15	1.25112
$551$	−6.12372e14	−0.513667
$552$	9.74371e13	0.0809207
$553$	0	0
$554$	−2.10629e15	−1.71480
$555$	5.16104e14	0.416031
$556$	−1.19570e15	−0.954357
$557$	4.85099e14	0.383378	0.191689	−	0.981456i	$-0.438604\pi$
$557$	4.85099e14	0.383378	0.191689	+	0.981456i	$0.438604\pi$
$558$	5.10447e13	0.0399450
$559$	2.50735e14	0.194290
$560$	0	0
$561$	8.66446e13	0.0658333
$562$	−1.72672e13	−0.0129919
$563$	3.92854e14	0.292708	0.146354	−	0.989232i	$-0.453246\pi$
$563$	3.92854e14	0.292708	0.146354	+	0.989232i	$0.453246\pi$
$564$	3.75509e14	0.277067
$565$	−5.81779e14	−0.425100
$566$	1.08859e15	0.787721
$567$	0	0
$568$	−2.19952e14	−0.156103
$569$	1.82569e14	0.128324	0.0641622	−	0.997939i	$-0.479562\pi$
$569$	1.82569e14	0.128324	0.0641622	+	0.997939i	$0.479562\pi$
$570$	6.44107e14	0.448380
$571$	1.97498e15	1.36165	0.680824	−	0.732447i	$-0.261622\pi$
$571$	1.97498e15	1.36165	0.680824	+	0.732447i	$0.261622\pi$
$572$	−1.99759e14	−0.136404
$573$	4.92138e14	0.332841
$574$	0	0
$575$	9.71951e14	0.644869
$576$	−3.75667e14	−0.246876
$577$	1.26863e15	0.825787	0.412894	−	0.910779i	$-0.364518\pi$
$577$	1.26863e15	0.825787	0.412894	+	0.910779i	$0.364518\pi$
$578$	2.10493e15	1.35717
$579$	−1.56107e15	−0.996987
$580$	−2.81850e14	−0.178305
$581$	0	0
$582$	2.50818e15	1.55698
$583$	2.02596e15	1.24582
$584$	2.93952e14	0.179063
$585$	−3.45665e13	−0.0208592
$586$	3.54769e15	2.12085
$587$	−1.44418e15	−0.855288	−0.427644	−	0.903947i	$-0.640656\pi$
$587$	−1.44418e15	−0.855288	−0.427644	+	0.903947i	$0.640656\pi$
$588$	0	0
$589$	−1.80085e14	−0.104675
$590$	−1.24834e15	−0.718867
$591$	−7.46174e14	−0.425705
$592$	2.98171e15	1.68538
$593$	7.42951e14	0.416063	0.208032	−	0.978122i	$-0.433294\pi$
$593$	7.42951e14	0.416063	0.208032	+	0.978122i	$0.433294\pi$
$594$	5.59493e14	0.310434
$595$	0	0
$596$	6.34122e14	0.345397
$597$	1.18839e15	0.641357
$598$	−2.81406e14	−0.150479
$599$	−2.56220e15	−1.35758	−0.678790	−	0.734332i	$-0.737496\pi$
$599$	−2.56220e15	−1.35758	−0.678790	+	0.734332i	$0.737496\pi$
$600$	1.43786e14	0.0754893
$601$	4.73424e14	0.246286	0.123143	−	0.992389i	$-0.460703\pi$
$601$	4.73424e14	0.246286	0.123143	+	0.992389i	$0.460703\pi$
$602$	0	0
$603$	7.02545e14	0.358863
$604$	−1.97802e15	−1.00122
$605$	3.64790e14	0.182974
$606$	2.43834e15	1.21198
$607$	2.76784e15	1.36333	0.681667	−	0.731662i	$-0.261255\pi$
$607$	2.76784e15	1.36333	0.681667	+	0.731662i	$0.261255\pi$
$608$	3.30794e15	1.61469
$609$	0	0
$610$	−1.77339e14	−0.0850140
$611$	1.52168e14	0.0722929
$612$	−6.01270e13	−0.0283098
$613$	−2.34884e15	−1.09602	−0.548012	−	0.836470i	$-0.684615\pi$
$613$	−2.34884e15	−1.09602	−0.548012	+	0.836470i	$0.684615\pi$
$614$	−7.41418e14	−0.342876
$615$	−4.83289e14	−0.221510
$616$	0	0
$617$	−1.39888e15	−0.629814	−0.314907	−	0.949123i	$-0.601973\pi$
$617$	−1.39888e15	−0.629814	−0.314907	+	0.949123i	$0.601973\pi$
$618$	−1.46401e15	−0.653292
$619$	2.10728e15	0.932018	0.466009	−	0.884780i	$-0.345692\pi$
$619$	2.10728e15	0.932018	0.466009	+	0.884780i	$0.345692\pi$
$620$	−8.28859e13	−0.0363351
$621$	3.68252e14	0.160008
$622$	4.84631e15	2.08720
$623$	0	0
$624$	−1.99703e14	−0.0845024
$625$	8.99324e14	0.377204
$626$	6.57621e14	0.273411
$627$	−1.97389e15	−0.813488
$628$	−1.12421e15	−0.459270
$629$	3.63793e14	0.147325
$630$	0	0
$631$	7.81150e14	0.310866	0.155433	−	0.987846i	$-0.450323\pi$
$631$	7.81150e14	0.310866	0.155433	+	0.987846i	$0.450323\pi$
$632$	2.65942e14	0.104916
$633$	2.54621e15	0.995805
$634$	3.18814e15	1.23608
$635$	5.16375e14	0.198477
$636$	−1.40592e15	−0.535730
$637$	0	0
$638$	1.84867e15	0.692382
$639$	−8.31280e14	−0.308669
$640$	−4.30536e14	−0.158497
$641$	−3.84617e15	−1.40381	−0.701906	−	0.712269i	$-0.747667\pi$
$641$	−3.84617e15	−1.40381	−0.701906	+	0.712269i	$0.747667\pi$
$642$	−3.88528e15	−1.40598
$643$	−3.69999e15	−1.32752	−0.663759	−	0.747947i	$-0.731040\pi$
$643$	−3.69999e15	−1.32752	−0.663759	+	0.747947i	$0.731040\pi$
$644$	0	0
$645$	1.14034e15	0.402213
$646$	4.54020e14	0.158780
$647$	−1.47283e15	−0.510715	−0.255358	−	0.966847i	$-0.582193\pi$
$647$	−1.47283e15	−0.510715	−0.255358	+	0.966847i	$0.582193\pi$
$648$	5.44775e13	0.0187307
$649$	3.82560e15	1.30423
$650$	−4.15265e14	−0.140379
$651$	0	0
$652$	3.29662e15	1.09574
$653$	9.03049e14	0.297638	0.148819	−	0.988864i	$-0.452453\pi$
$653$	9.03049e14	0.297638	0.148819	+	0.988864i	$0.452453\pi$
$654$	1.17741e15	0.384815
$655$	1.56559e15	0.507402
$656$	−2.79213e15	−0.897356
$657$	1.11096e15	0.354069
$658$	0	0
$659$	3.92318e15	1.22961	0.614806	−	0.788678i	$-0.289234\pi$
$659$	3.92318e15	1.22961	0.614806	+	0.788678i	$0.289234\pi$
$660$	−9.08500e14	−0.282380
$661$	−4.11719e15	−1.26909	−0.634546	−	0.772885i	$-0.718813\pi$
$661$	−4.11719e15	−1.26909	−0.634546	+	0.772885i	$0.718813\pi$
$662$	5.66883e15	1.73291
$663$	−2.43653e13	−0.00738665
$664$	−7.43908e14	−0.223663
$665$	0	0
$666$	2.34913e15	0.694703
$667$	1.21677e15	0.356876
$668$	−3.78411e15	−1.10076
$669$	−3.58024e15	−1.03292
$670$	−2.44164e15	−0.698667
$671$	5.43462e14	0.154239
$672$	0	0
$673$	6.42241e14	0.179314	0.0896571	−	0.995973i	$-0.471423\pi$
$673$	6.42241e14	0.179314	0.0896571	+	0.995973i	$0.471423\pi$
$674$	−2.57031e15	−0.711796
$675$	5.43422e14	0.149268
$676$	−3.16255e15	−0.861648
$677$	−9.39738e14	−0.253962	−0.126981	−	0.991905i	$-0.540529\pi$
$677$	−9.39738e14	−0.253962	−0.126981	+	0.991905i	$0.540529\pi$
$678$	−2.64806e15	−0.709846
$679$	0	0
$680$	−2.93205e13	−0.00773343
$681$	−1.80318e15	−0.471771
$682$	5.43654e14	0.141094
$683$	−7.18610e15	−1.85003	−0.925016	−	0.379927i	$-0.875949\pi$
$683$	−7.18610e15	−1.85003	−0.925016	+	0.379927i	$0.875949\pi$
$684$	1.36978e15	0.349818
$685$	−2.22522e15	−0.563734
$686$	0	0
$687$	−3.08034e15	−0.767956
$688$	6.58815e15	1.62940
$689$	−5.69721e14	−0.139784
$690$	−1.27983e15	−0.311517
$691$	7.03052e15	1.69769	0.848845	−	0.528642i	$-0.177298\pi$
$691$	7.03052e15	1.69769	0.848845	+	0.528642i	$0.177298\pi$
$692$	−5.67205e15	−1.35880
$693$	0	0
$694$	−8.10309e15	−1.91062
$695$	−2.20366e15	−0.515498
$696$	1.80004e14	0.0417764
$697$	−3.40662e14	−0.0784410
$698$	3.37243e15	0.770439
$699$	−1.60487e15	−0.363760
$700$	0	0
$701$	2.18167e15	0.486787	0.243394	−	0.969928i	$-0.421739\pi$
$701$	2.18167e15	0.486787	0.243394	+	0.969928i	$0.421739\pi$
$702$	−1.57335e14	−0.0348314
$703$	−8.28773e15	−1.82046
$704$	−4.00106e15	−0.872017
$705$	6.92057e14	0.149659
$706$	8.57793e15	1.84059
$707$	0	0
$708$	−2.65477e15	−0.560847
$709$	−7.20135e15	−1.50959	−0.754796	−	0.655960i	$-0.772264\pi$
$709$	−7.20135e15	−1.50959	−0.754796	+	0.655960i	$0.772264\pi$
$710$	2.88905e15	0.600944
$711$	1.00509e15	0.207455
$712$	−9.61846e14	−0.197000
$713$	3.57826e14	0.0727244
$714$	0	0
$715$	−3.68152e14	−0.0736790
$716$	−1.50729e14	−0.0299348
$717$	−3.40491e15	−0.671043
$718$	1.61167e15	0.315205
$719$	−5.70783e15	−1.10780	−0.553901	−	0.832582i	$-0.686861\pi$
$719$	−5.70783e15	−1.10780	−0.553901	+	0.832582i	$0.686861\pi$
$720$	−9.08246e14	−0.174934
$721$	0	0
$722$	−3.12083e15	−0.591992
$723$	−9.76404e14	−0.183810
$724$	1.37933e15	0.257695
$725$	1.79557e15	0.332922
$726$	1.66040e15	0.305536
$727$	−7.29260e15	−1.33181	−0.665907	−	0.746035i	$-0.731955\pi$
$727$	−7.29260e15	−1.33181	−0.665907	+	0.746035i	$0.731955\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	−3.86104e15	−0.689332
$731$	8.03807e14	0.142432
$732$	−3.77135e14	−0.0663264
$733$	6.02275e15	1.05129	0.525646	−	0.850704i	$-0.323824\pi$
$733$	6.02275e15	1.05129	0.525646	+	0.850704i	$0.323824\pi$
$734$	−9.54916e14	−0.165439
$735$	0	0
$736$	−6.57282e15	−1.12182
$737$	7.48249e15	1.26758
$738$	−2.19977e15	−0.369885
$739$	−2.79105e14	−0.0465825	−0.0232913	−	0.999729i	$-0.507415\pi$
$739$	−2.79105e14	−0.0465825	−0.0232913	+	0.999729i	$0.507415\pi$
$740$	−3.81450e15	−0.631921
$741$	5.55077e14	0.0912753
$742$	0	0
$743$	2.77138e15	0.449011	0.224505	−	0.974473i	$-0.427923\pi$
$743$	2.77138e15	0.449011	0.224505	+	0.974473i	$0.427923\pi$
$744$	5.29352e13	0.00851322
$745$	1.16868e15	0.186567
$746$	−3.36384e15	−0.533055
$747$	−2.81151e15	−0.442258
$748$	−6.40386e14	−0.0999960
$749$	0	0
$750$	−4.32360e15	−0.665287
$751$	−4.76878e15	−0.728430	−0.364215	−	0.931315i	$-0.618663\pi$
$751$	−4.76878e15	−0.728430	−0.364215	+	0.931315i	$0.618663\pi$
$752$	3.99825e15	0.606279
$753$	−3.19629e15	−0.481142
$754$	−5.19864e14	−0.0776868
$755$	−3.64547e15	−0.540810
$756$	0	0
$757$	−9.90830e15	−1.44868	−0.724339	−	0.689444i	$-0.757855\pi$
$757$	−9.90830e15	−1.44868	−0.724339	+	0.689444i	$0.757855\pi$
$758$	4.05620e15	0.588761
$759$	3.92208e15	0.565180
$760$	6.67963e14	0.0955603
$761$	−4.09383e15	−0.581452	−0.290726	−	0.956806i	$-0.593897\pi$
$761$	−4.09383e15	−0.581452	−0.290726	+	0.956806i	$0.593897\pi$
$762$	2.35037e15	0.331423
$763$	0	0
$764$	−3.63737e15	−0.505561
$765$	−1.10813e14	−0.0152916
$766$	4.69571e15	0.643344
$767$	−1.07580e15	−0.146337
$768$	−5.12577e15	−0.692265
$769$	−5.37473e15	−0.720711	−0.360356	−	0.932815i	$-0.617345\pi$
$769$	−5.37473e15	−0.720711	−0.360356	+	0.932815i	$0.617345\pi$
$770$	0	0
$771$	8.23090e15	1.08805
$772$	1.15378e16	1.51435
$773$	2.37959e15	0.310109	0.155055	−	0.987906i	$-0.450445\pi$
$773$	2.37959e15	0.310109	0.155055	+	0.987906i	$0.450445\pi$
$774$	5.19045e15	0.671629
$775$	5.28038e14	0.0678431
$776$	2.60108e15	0.331828
$777$	0	0
$778$	1.30126e15	0.163672
$779$	7.76077e15	0.969279
$780$	2.55479e14	0.0316836
$781$	−8.85359e15	−1.09028
$782$	−9.02129e14	−0.110314
$783$	6.80303e14	0.0826062
$784$	0	0
$785$	−2.07190e15	−0.248076
$786$	7.12606e15	0.847277
$787$	2.44024e15	0.288118	0.144059	−	0.989569i	$-0.453984\pi$
$787$	2.44024e15	0.288118	0.144059	+	0.989569i	$0.453984\pi$
$788$	5.51493e15	0.646615
$789$	3.39136e15	0.394867
$790$	−3.49312e15	−0.403892
$791$	0	0
$792$	5.80215e14	0.0661608
$793$	−1.52827e14	−0.0173060
$794$	2.10989e16	2.37272
$795$	−2.59109e15	−0.289376
$796$	−8.78332e15	−0.974174
$797$	−3.47038e15	−0.382257	−0.191129	−	0.981565i	$-0.561215\pi$
$797$	−3.47038e15	−0.382257	−0.191129	+	0.981565i	$0.561215\pi$
$798$	0	0
$799$	4.87819e14	0.0529970
$800$	−9.69939e15	−1.04652
$801$	−3.63518e15	−0.389535
$802$	2.71188e15	0.288610
$803$	1.18323e16	1.25064
$804$	−5.19248e15	−0.545087
$805$	0	0
$806$	−1.52881e14	−0.0158311
$807$	6.55758e15	0.674433
$808$	2.52865e15	0.258301
$809$	1.04613e16	1.06138	0.530689	−	0.847567i	$-0.321933\pi$
$809$	1.04613e16	1.06138	0.530689	+	0.847567i	$0.321933\pi$
$810$	−7.15558e14	−0.0721070
$811$	4.10267e15	0.410631	0.205315	−	0.978696i	$-0.434178\pi$
$811$	4.10267e15	0.410631	0.205315	+	0.978696i	$0.434178\pi$
$812$	0	0
$813$	−1.11938e16	−1.10530
$814$	2.50195e16	2.45383
$815$	6.07563e15	0.591867
$816$	−6.40207e14	−0.0619476
$817$	−1.83119e16	−1.76000
$818$	8.76811e15	0.837072
$819$	0	0
$820$	3.57196e15	0.336458
$821$	−1.10577e16	−1.03462	−0.517308	−	0.855799i	$-0.673066\pi$
$821$	−1.10577e16	−1.03462	−0.517308	+	0.855799i	$0.673066\pi$
$822$	−1.01284e16	−0.941341
$823$	1.66587e16	1.53795	0.768973	−	0.639282i	$-0.220768\pi$
$823$	1.66587e16	1.53795	0.768973	+	0.639282i	$0.220768\pi$
$824$	−1.51823e15	−0.139232
$825$	5.78774e15	0.527245
$826$	0	0
$827$	−5.55836e15	−0.499650	−0.249825	−	0.968291i	$-0.580373\pi$
$827$	−5.55836e15	−0.499650	−0.249825	+	0.968291i	$0.580373\pi$
$828$	−2.72173e15	−0.243040
$829$	−3.80539e15	−0.337559	−0.168779	−	0.985654i	$-0.553983\pi$
$829$	−3.80539e15	−0.337559	−0.168779	+	0.985654i	$0.553983\pi$
$830$	9.77117e15	0.861027
$831$	−8.25528e15	−0.722647
$832$	1.12514e15	0.0978424
$833$	0	0
$834$	−1.00303e16	−0.860796
$835$	−6.97405e15	−0.594579
$836$	1.45889e16	1.23563
$837$	2.00062e14	0.0168335
$838$	−2.91873e16	−2.43979
$839$	−8.22572e14	−0.0683098	−0.0341549	−	0.999417i	$-0.510874\pi$
$839$	−8.22572e14	−0.0683098	−0.0341549	+	0.999417i	$0.510874\pi$
$840$	0	0
$841$	−9.95266e15	−0.815758
$842$	3.24169e14	0.0263970
$843$	−6.76764e13	−0.00547502
$844$	−1.88189e16	−1.51256
$845$	−5.82852e15	−0.465422
$846$	3.15001e15	0.249905
$847$	0	0
$848$	−1.49696e16	−1.17228
$849$	4.26655e15	0.331959
$850$	−1.33125e15	−0.102910
$851$	1.64676e16	1.26478
$852$	6.14395e15	0.468845
$853$	−6.20545e15	−0.470493	−0.235247	−	0.971936i	$-0.575590\pi$
$853$	−6.20545e15	−0.470493	−0.235247	+	0.971936i	$0.575590\pi$
$854$	0	0
$855$	2.52448e15	0.188955
$856$	−4.02918e15	−0.299648
$857$	−1.02852e16	−0.760010	−0.380005	−	0.924984i	$-0.624078\pi$
$857$	−1.02852e16	−0.760010	−0.380005	+	0.924984i	$0.624078\pi$
$858$	−1.67570e15	−0.123032
$859$	−1.24588e14	−0.00908899	−0.00454449	−	0.999990i	$-0.501447\pi$
$859$	−1.24588e14	−0.00908899	−0.00454449	+	0.999990i	$0.501447\pi$
$860$	−8.42820e15	−0.610933
$861$	0	0
$862$	1.66003e16	1.18802
$863$	−2.54402e16	−1.80910	−0.904549	−	0.426370i	$-0.859792\pi$
$863$	−2.54402e16	−1.80910	−0.904549	+	0.426370i	$0.859792\pi$
$864$	−3.67489e15	−0.259668
$865$	−1.04535e16	−0.733961
$866$	2.98812e16	2.08473
$867$	8.24996e15	0.571935
$868$	0	0
$869$	1.07048e16	0.732774
$870$	−2.36434e15	−0.160825
$871$	−2.10415e15	−0.142225
$872$	1.22102e15	0.0820130
$873$	9.83045e15	0.656138
$874$	2.05518e16	1.36313
$875$	0	0
$876$	−8.21102e15	−0.537805
$877$	−1.84724e16	−1.20234	−0.601169	−	0.799122i	$-0.705298\pi$
$877$	−1.84724e16	−1.20234	−0.601169	+	0.799122i	$0.705298\pi$
$878$	8.57037e15	0.554344
$879$	1.39047e16	0.893762
$880$	−9.67331e15	−0.617903
$881$	−5.44811e14	−0.0345842	−0.0172921	−	0.999850i	$-0.505505\pi$
$881$	−5.44811e14	−0.0345842	−0.0172921	+	0.999850i	$0.505505\pi$
$882$	0	0
$883$	−7.32941e15	−0.459499	−0.229750	−	0.973250i	$-0.573791\pi$
$883$	−7.32941e15	−0.459499	−0.229750	+	0.973250i	$0.573791\pi$
$884$	1.80083e14	0.0112198
$885$	−4.89271e15	−0.302943
$886$	−7.21891e15	−0.444208
$887$	−8.44266e15	−0.516297	−0.258148	−	0.966105i	$-0.583112\pi$
$887$	−8.44266e15	−0.516297	−0.258148	+	0.966105i	$0.583112\pi$
$888$	2.43614e15	0.148057
$889$	0	0
$890$	1.26338e16	0.758382
$891$	2.19285e15	0.130822
$892$	2.64614e16	1.56894
$893$	−1.11132e16	−0.654872
$894$	5.31942e15	0.311536
$895$	−2.77792e14	−0.0161694
$896$	0	0
$897$	−1.10293e15	−0.0634145
$898$	9.09763e15	0.519886
$899$	6.61043e14	0.0375449
$900$	−4.01640e15	−0.226727
$901$	−1.82641e15	−0.102474
$902$	−2.34287e16	−1.30651
$903$	0	0
$904$	−2.74614e15	−0.151285
$905$	2.54208e15	0.139195
$906$	−1.65929e16	−0.903063
$907$	−1.76079e16	−0.952504	−0.476252	−	0.879309i	$-0.658005\pi$
$907$	−1.76079e16	−0.952504	−0.476252	+	0.879309i	$0.658005\pi$
$908$	1.33272e16	0.716586
$909$	9.55672e15	0.510749
$910$	0	0
$911$	1.44504e16	0.763009	0.381505	−	0.924367i	$-0.375406\pi$
$911$	1.44504e16	0.763009	0.381505	+	0.924367i	$0.375406\pi$
$912$	1.45848e16	0.765473
$913$	−2.99441e16	−1.56215
$914$	4.14308e16	2.14842
$915$	−6.95055e14	−0.0358264
$916$	2.27666e16	1.16647
$917$	0	0
$918$	−5.04384e14	−0.0255345
$919$	−1.00654e16	−0.506519	−0.253259	−	0.967398i	$-0.581503\pi$
$919$	−1.00654e16	−0.506519	−0.253259	+	0.967398i	$0.581503\pi$
$920$	−1.32723e15	−0.0663916
$921$	−2.90588e15	−0.144494
$922$	2.91854e16	1.44260
$923$	2.48972e15	0.122332
$924$	0	0
$925$	2.43009e16	1.17989
$926$	1.94342e16	0.938005
$927$	−5.73797e15	−0.275309
$928$	−1.21425e16	−0.579156
$929$	2.66109e15	0.126175	0.0630874	−	0.998008i	$-0.479905\pi$
$929$	2.66109e15	0.126175	0.0630874	+	0.998008i	$0.479905\pi$
$930$	−6.95300e14	−0.0327730
$931$	0	0
$932$	1.18615e16	0.552526
$933$	1.89944e16	0.879582
$934$	−2.20284e16	−1.01408
$935$	−1.18022e15	−0.0540131
$936$	−1.63162e14	−0.00742339
$937$	−5.61498e15	−0.253969	−0.126984	−	0.991905i	$-0.540530\pi$
$937$	−5.61498e15	−0.253969	−0.126984	+	0.991905i	$0.540530\pi$
$938$	0	0
$939$	2.57745e15	0.115220
$940$	−5.11495e15	−0.227321
$941$	−2.37104e16	−1.04760	−0.523801	−	0.851840i	$-0.675487\pi$
$941$	−2.37104e16	−1.04760	−0.523801	+	0.851840i	$0.675487\pi$
$942$	−9.43057e15	−0.414245
$943$	−1.54205e16	−0.673418
$944$	−2.82669e16	−1.22725
$945$	0	0
$946$	5.52811e16	2.37233
$947$	−2.39148e16	−1.02033	−0.510167	−	0.860075i	$-0.670417\pi$
$947$	−2.39148e16	−1.02033	−0.510167	+	0.860075i	$0.670417\pi$
$948$	−7.42860e15	−0.315109
$949$	−3.32736e15	−0.140325
$950$	3.03279e16	1.27164
$951$	1.24954e16	0.520905
$952$	0	0
$953$	1.57974e16	0.650989	0.325494	−	0.945544i	$-0.394469\pi$
$953$	1.57974e16	0.650989	0.325494	+	0.945544i	$0.394469\pi$
$954$	−1.17937e16	−0.483210
$955$	−6.70361e15	−0.273080
$956$	2.51655e16	1.01927
$957$	7.24559e15	0.291782
$958$	−1.06237e16	−0.425368
$959$	0	0
$960$	5.11711e15	0.202550
$961$	−2.52141e16	−0.992349
$962$	−7.03574e15	−0.275326
$963$	−1.52278e16	−0.592506
$964$	7.21655e15	0.279194
$965$	2.12639e16	0.817981
$966$	0	0
$967$	−3.40244e16	−1.29403	−0.647016	−	0.762476i	$-0.723983\pi$
$967$	−3.40244e16	−1.29403	−0.647016	+	0.762476i	$0.723983\pi$
$968$	1.72190e15	0.0651168
$969$	1.77946e15	0.0669127
$970$	−3.41649e16	−1.27743
$971$	4.67428e16	1.73784	0.868918	−	0.494955i	$-0.164816\pi$
$971$	4.67428e16	1.73784	0.868918	+	0.494955i	$0.164816\pi$
$972$	−1.52173e15	−0.0562566
$973$	0	0
$974$	6.70484e16	2.45083
$975$	−1.62757e15	−0.0591581
$976$	−4.01557e15	−0.145135
$977$	−2.13162e16	−0.766106	−0.383053	−	0.923726i	$-0.625127\pi$
$977$	−2.13162e16	−0.766106	−0.383053	+	0.923726i	$0.625127\pi$
$978$	2.76542e16	0.988320
$979$	−3.87166e16	−1.37592
$980$	0	0
$981$	4.61470e15	0.162168
$982$	−1.98533e16	−0.693775
$983$	5.11191e16	1.77639	0.888195	−	0.459466i	$-0.151959\pi$
$983$	5.11191e16	1.77639	0.888195	+	0.459466i	$0.151959\pi$
$984$	−2.28124e15	−0.0788312
$985$	1.01639e16	0.349271
$986$	−1.66658e15	−0.0569512
$987$	0	0
$988$	−4.10255e15	−0.138641
$989$	3.63854e16	1.22278
$990$	−7.62108e15	−0.254697
$991$	1.74754e16	0.580795	0.290398	−	0.956906i	$-0.406212\pi$
$991$	1.74754e16	0.580795	0.290398	+	0.956906i	$0.406212\pi$
$992$	−3.57085e15	−0.118021
$993$	2.22182e16	0.730277
$994$	0	0
$995$	−1.61875e16	−0.526203
$996$	2.07797e16	0.671758
$997$	2.24429e16	0.721532	0.360766	−	0.932656i	$-0.382515\pi$
$997$	2.24429e16	0.721532	0.360766	+	0.932656i	$0.382515\pi$
$998$	−3.41601e15	−0.109220
$999$	9.20708e15	0.292760

Twists

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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