LMFDB - Embedded newform 48.12.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,12,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("48.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	48.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:	$36.8804726669$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 12)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-243.000 q^{3} +2862.00 q^{5} -9128.00 q^{7} +59049.0 q^{9} +O(q^{10})$

$q-243.000 q^{3} +2862.00 q^{5} -9128.00 q^{7} +59049.0 q^{9} -668196. q^{11} +2.05295e6 q^{13} -695466. q^{15} +1.60418e6 q^{17} +230500. q^{19} +2.21810e6 q^{21} +4.30127e7 q^{23} -4.06371e7 q^{25} -1.43489e7 q^{27} -1.41745e8 q^{29} -2.33222e8 q^{31} +1.62372e8 q^{33} -2.61243e7 q^{35} +2.78270e8 q^{37} -4.98867e8 q^{39} -1.18158e9 q^{41} -8.56975e8 q^{43} +1.68998e8 q^{45} +1.66405e9 q^{47} -1.89401e9 q^{49} -3.89815e8 q^{51} -3.85118e9 q^{53} -1.91238e9 q^{55} -5.60115e7 q^{57} -1.03390e10 q^{59} +1.85948e8 q^{61} -5.38999e8 q^{63} +5.87554e9 q^{65} -2.91501e9 q^{67} -1.04521e10 q^{69} -1.26623e10 q^{71} -1.52013e10 q^{73} +9.87481e9 q^{75} +6.09929e9 q^{77} +3.66440e10 q^{79} +3.48678e9 q^{81} +9.21764e9 q^{83} +4.59116e9 q^{85} +3.44441e10 q^{87} +3.05738e10 q^{89} -1.87393e10 q^{91} +5.66729e10 q^{93} +6.59691e8 q^{95} +1.45702e11 q^{97} -3.94563e10 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−243.000	−0.577350
$3$	−243.000	−0.577350
$4$	0	0
$5$	2862.00	0.409576	0.204788	−	0.978806i	$-0.434349\pi$
$5$	2862.00	0.409576	0.204788	+	0.978806i	$0.434349\pi$
$6$	0	0
$7$	−9128.00	−0.205275	−0.102638	−	0.994719i	$-0.532728\pi$
$7$	−9128.00	−0.205275	−0.102638	+	0.994719i	$0.532728\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	−668196.	−1.25096	−0.625481	−	0.780239i	$-0.715097\pi$
$11$	−668196.	−1.25096	−0.625481	+	0.780239i	$0.715097\pi$
$12$	0	0
$13$	2.05295e6	1.53352	0.766761	−	0.641933i	$-0.221867\pi$
$13$	2.05295e6	1.53352	0.766761	+	0.641933i	$0.221867\pi$
$14$	0	0
$15$	−695466.	−0.236469
$16$	0	0
$17$	1.60418e6	0.274021	0.137010	−	0.990570i	$-0.456251\pi$
$17$	1.60418e6	0.274021	0.137010	+	0.990570i	$0.456251\pi$
$18$	0	0
$19$	230500.	0.0213563	0.0106782	−	0.999943i	$-0.496601\pi$
$19$	230500.	0.0213563	0.0106782	+	0.999943i	$0.496601\pi$
$20$	0	0
$21$	2.21810e6	0.118516
$22$	0	0
$23$	4.30127e7	1.39346	0.696729	−	0.717334i	$-0.254638\pi$
$23$	4.30127e7	1.39346	0.696729	+	0.717334i	$0.254638\pi$
$24$	0	0
$25$	−4.06371e7	−0.832247
$26$	0	0
$27$	−1.43489e7	−0.192450
$28$	0	0
$29$	−1.41745e8	−1.28327	−0.641637	−	0.767008i	$-0.721744\pi$
$29$	−1.41745e8	−1.28327	−0.641637	+	0.767008i	$0.721744\pi$
$30$	0	0
$31$	−2.33222e8	−1.46312	−0.731560	−	0.681777i	$-0.761207\pi$
$31$	−2.33222e8	−1.46312	−0.731560	+	0.681777i	$0.761207\pi$
$32$	0	0
$33$	1.62372e8	0.722243
$34$	0	0
$35$	−2.61243e7	−0.0840758
$36$	0	0
$37$	2.78270e8	0.659715	0.329858	−	0.944031i	$-0.392999\pi$
$37$	2.78270e8	0.659715	0.329858	+	0.944031i	$0.392999\pi$
$38$	0	0
$39$	−4.98867e8	−0.885379
$40$	0	0
$41$	−1.18158e9	−1.59276	−0.796381	−	0.604795i	$-0.793255\pi$
$41$	−1.18158e9	−1.59276	−0.796381	+	0.604795i	$0.793255\pi$
$42$	0	0
$43$	−8.56975e8	−0.888979	−0.444490	−	0.895784i	$-0.646615\pi$
$43$	−8.56975e8	−0.888979	−0.444490	+	0.895784i	$0.646615\pi$
$44$	0	0
$45$	1.68998e8	0.136525
$46$	0	0
$47$	1.66405e9	1.05835	0.529175	−	0.848513i	$-0.322501\pi$
$47$	1.66405e9	1.05835	0.529175	+	0.848513i	$0.322501\pi$
$48$	0	0
$49$	−1.89401e9	−0.957862
$50$	0	0
$51$	−3.89815e8	−0.158206
$52$	0	0
$53$	−3.85118e9	−1.26496	−0.632480	−	0.774577i	$-0.717963\pi$
$53$	−3.85118e9	−1.26496	−0.632480	+	0.774577i	$0.717963\pi$
$54$	0	0
$55$	−1.91238e9	−0.512364
$56$	0	0
$57$	−5.60115e7	−0.0123301
$58$	0	0
$59$	−1.03390e10	−1.88275	−0.941375	−	0.337363i	$-0.890465\pi$
$59$	−1.03390e10	−1.88275	−0.941375	+	0.337363i	$0.890465\pi$
$60$	0	0
$61$	1.85948e8	0.0281889	0.0140944	−	0.999901i	$-0.495513\pi$
$61$	1.85948e8	0.0281889	0.0140944	+	0.999901i	$0.495513\pi$
$62$	0	0
$63$	−5.38999e8	−0.0684251
$64$	0	0
$65$	5.87554e9	0.628094
$66$	0	0
$67$	−2.91501e9	−0.263772	−0.131886	−	0.991265i	$-0.542103\pi$
$67$	−2.91501e9	−0.263772	−0.131886	+	0.991265i	$0.542103\pi$
$68$	0	0
$69$	−1.04521e10	−0.804513
$70$	0	0
$71$	−1.26623e10	−0.832899	−0.416449	−	0.909159i	$-0.636726\pi$
$71$	−1.26623e10	−0.832899	−0.416449	+	0.909159i	$0.636726\pi$
$72$	0	0
$73$	−1.52013e10	−0.858231	−0.429115	−	0.903250i	$-0.641175\pi$
$73$	−1.52013e10	−0.858231	−0.429115	+	0.903250i	$0.641175\pi$
$74$	0	0
$75$	9.87481e9	0.480498
$76$	0	0
$77$	6.09929e9	0.256791
$78$	0	0
$79$	3.66440e10	1.33984	0.669922	−	0.742432i	$-0.266328\pi$
$79$	3.66440e10	1.33984	0.669922	+	0.742432i	$0.266328\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	9.21764e9	0.256856	0.128428	−	0.991719i	$-0.459007\pi$
$83$	9.21764e9	0.256856	0.128428	+	0.991719i	$0.459007\pi$
$84$	0	0
$85$	4.59116e9	0.112232
$86$	0	0
$87$	3.44441e10	0.740899
$88$	0	0
$89$	3.05738e10	0.580370	0.290185	−	0.956971i	$-0.406283\pi$
$89$	3.05738e10	0.580370	0.290185	+	0.956971i	$0.406283\pi$
$90$	0	0
$91$	−1.87393e10	−0.314794
$92$	0	0
$93$	5.66729e10	0.844733
$94$	0	0
$95$	6.59691e8	0.00874703
$96$	0	0
$97$	1.45702e11	1.72274	0.861371	−	0.507976i	$-0.169606\pi$
$97$	1.45702e11	1.72274	0.861371	+	0.507976i	$0.169606\pi$
$98$	0	0
$99$	−3.94563e10	−0.416987
$100$	0	0
$101$	−5.59904e10	−0.530085	−0.265043	−	0.964237i	$-0.585386\pi$
$101$	−5.59904e10	−0.530085	−0.265043	+	0.964237i	$0.585386\pi$
$102$	0	0
$103$	4.39677e10	0.373705	0.186852	−	0.982388i	$-0.440171\pi$
$103$	4.39677e10	0.373705	0.186852	+	0.982388i	$0.440171\pi$
$104$	0	0
$105$	6.34821e9	0.0485412
$106$	0	0
$107$	1.49244e11	1.02869	0.514347	−	0.857582i	$-0.328034\pi$
$107$	1.49244e11	1.02869	0.514347	+	0.857582i	$0.328034\pi$
$108$	0	0
$109$	1.20912e9	0.00752704	0.00376352	−	0.999993i	$-0.498802\pi$
$109$	1.20912e9	0.00752704	0.00376352	+	0.999993i	$0.498802\pi$
$110$	0	0
$111$	−6.76195e10	−0.380887
$112$	0	0
$113$	−2.75811e11	−1.40825	−0.704125	−	0.710077i	$-0.748660\pi$
$113$	−2.75811e11	−1.40825	−0.704125	+	0.710077i	$0.748660\pi$
$114$	0	0
$115$	1.23102e11	0.570727
$116$	0	0
$117$	1.21225e11	0.511174
$118$	0	0
$119$	−1.46429e10	−0.0562497
$120$	0	0
$121$	1.61174e11	0.564906
$122$	0	0
$123$	2.87123e11	0.919582
$124$	0	0
$125$	−2.56049e11	−0.750445
$126$	0	0
$127$	−5.48292e10	−0.147262	−0.0736312	−	0.997286i	$-0.523459\pi$
$127$	−5.48292e10	−0.147262	−0.0736312	+	0.997286i	$0.523459\pi$
$128$	0	0
$129$	2.08245e11	0.513253
$130$	0	0
$131$	−7.69885e11	−1.74355	−0.871774	−	0.489909i	$-0.837030\pi$
$131$	−7.69885e11	−1.74355	−0.871774	+	0.489909i	$0.837030\pi$
$132$	0	0
$133$	−2.10400e9	−0.00438392
$134$	0	0
$135$	−4.10666e10	−0.0788230
$136$	0	0
$137$	3.90519e11	0.691321	0.345660	−	0.938360i	$-0.387655\pi$
$137$	3.90519e11	0.691321	0.345660	+	0.938360i	$0.387655\pi$
$138$	0	0
$139$	−5.14102e10	−0.0840365	−0.0420183	−	0.999117i	$-0.513379\pi$
$139$	−5.14102e10	−0.0840365	−0.0420183	+	0.999117i	$0.513379\pi$
$140$	0	0
$141$	−4.04365e11	−0.611039
$142$	0	0
$143$	−1.37177e12	−1.91838
$144$	0	0
$145$	−4.05675e11	−0.525598
$146$	0	0
$147$	4.60244e11	0.553022
$148$	0	0
$149$	−1.39815e12	−1.55966	−0.779830	−	0.625991i	$-0.784695\pi$
$149$	−1.39815e12	−1.55966	−0.779830	+	0.625991i	$0.784695\pi$
$150$	0	0
$151$	4.21187e11	0.436618	0.218309	−	0.975880i	$-0.429946\pi$
$151$	4.21187e11	0.436618	0.218309	+	0.975880i	$0.429946\pi$
$152$	0	0
$153$	9.47251e10	0.0913403
$154$	0	0
$155$	−6.67481e11	−0.599259
$156$	0	0
$157$	−1.27939e12	−1.07042	−0.535210	−	0.844719i	$-0.679768\pi$
$157$	−1.27939e12	−1.07042	−0.535210	+	0.844719i	$0.679768\pi$
$158$	0	0
$159$	9.35837e11	0.730325
$160$	0	0
$161$	−3.92620e11	−0.286042
$162$	0	0
$163$	4.72791e11	0.321838	0.160919	−	0.986968i	$-0.448554\pi$
$163$	4.72791e11	0.321838	0.160919	+	0.986968i	$0.448554\pi$
$164$	0	0
$165$	4.64708e11	0.295814
$166$	0	0
$167$	−1.04628e12	−0.623313	−0.311656	−	0.950195i	$-0.600884\pi$
$167$	−1.04628e12	−0.623313	−0.311656	+	0.950195i	$0.600884\pi$
$168$	0	0
$169$	2.42244e12	1.35169
$170$	0	0
$171$	1.36108e10	0.00711877
$172$	0	0
$173$	−1.46150e12	−0.717044	−0.358522	−	0.933521i	$-0.616719\pi$
$173$	−1.46150e12	−0.717044	−0.358522	+	0.933521i	$0.616719\pi$
$174$	0	0
$175$	3.70935e11	0.170840
$176$	0	0
$177$	2.51238e12	1.08701
$178$	0	0
$179$	4.23853e11	0.172395	0.0861974	−	0.996278i	$-0.472528\pi$
$179$	4.23853e11	0.172395	0.0861974	+	0.996278i	$0.472528\pi$
$180$	0	0
$181$	1.83136e12	0.700717	0.350358	−	0.936616i	$-0.386060\pi$
$181$	1.83136e12	0.700717	0.350358	+	0.936616i	$0.386060\pi$
$182$	0	0
$183$	−4.51854e10	−0.0162749
$184$	0	0
$185$	7.96408e11	0.270204
$186$	0	0
$187$	−1.07191e12	−0.342790
$188$	0	0
$189$	1.30977e11	0.0395052
$190$	0	0
$191$	2.94498e12	0.838297	0.419149	−	0.907918i	$-0.362329\pi$
$191$	2.94498e12	0.838297	0.419149	+	0.907918i	$0.362329\pi$
$192$	0	0
$193$	6.25393e12	1.68108	0.840538	−	0.541752i	$-0.182239\pi$
$193$	6.25393e12	1.68108	0.840538	+	0.541752i	$0.182239\pi$
$194$	0	0
$195$	−1.42776e12	−0.362630
$196$	0	0
$197$	−6.66278e12	−1.59989	−0.799947	−	0.600071i	$-0.795139\pi$
$197$	−6.66278e12	−1.59989	−0.799947	+	0.600071i	$0.795139\pi$
$198$	0	0
$199$	7.06530e12	1.60487	0.802433	−	0.596742i	$-0.203539\pi$
$199$	7.06530e12	1.60487	0.802433	+	0.596742i	$0.203539\pi$
$200$	0	0
$201$	7.08348e11	0.152289
$202$	0	0
$203$	1.29385e12	0.263424
$204$	0	0
$205$	−3.38167e12	−0.652357
$206$	0	0
$207$	2.53986e12	0.464486
$208$	0	0
$209$	−1.54019e11	−0.0267159
$210$	0	0
$211$	3.11902e12	0.513411	0.256705	−	0.966490i	$-0.417363\pi$
$211$	3.11902e12	0.513411	0.256705	+	0.966490i	$0.417363\pi$
$212$	0	0
$213$	3.07694e12	0.480874
$214$	0	0
$215$	−2.45266e12	−0.364105
$216$	0	0
$217$	2.12885e12	0.300342
$218$	0	0
$219$	3.69391e12	0.495500
$220$	0	0
$221$	3.29330e12	0.420217
$222$	0	0
$223$	−1.34244e13	−1.63011	−0.815055	−	0.579384i	$-0.803293\pi$
$223$	−1.34244e13	−1.63011	−0.815055	+	0.579384i	$0.803293\pi$
$224$	0	0
$225$	−2.39958e12	−0.277416
$226$	0	0
$227$	−9.95920e12	−1.09669	−0.548343	−	0.836254i	$-0.684741\pi$
$227$	−9.95920e12	−1.09669	−0.548343	+	0.836254i	$0.684741\pi$
$228$	0	0
$229$	−1.19130e13	−1.25005	−0.625023	−	0.780607i	$-0.714910\pi$
$229$	−1.19130e13	−1.25005	−0.625023	+	0.780607i	$0.714910\pi$
$230$	0	0
$231$	−1.48213e12	−0.148259
$232$	0	0
$233$	1.96588e13	1.87542	0.937710	−	0.347420i	$-0.112942\pi$
$233$	1.96588e13	1.87542	0.937710	+	0.347420i	$0.112942\pi$
$234$	0	0
$235$	4.76253e12	0.433475
$236$	0	0
$237$	−8.90450e12	−0.773559
$238$	0	0
$239$	−3.00146e11	−0.0248968	−0.0124484	−	0.999923i	$-0.503963\pi$
$239$	−3.00146e11	−0.0248968	−0.0124484	+	0.999923i	$0.503963\pi$
$240$	0	0
$241$	8.82505e12	0.699235	0.349618	−	0.936893i	$-0.386312\pi$
$241$	8.82505e12	0.699235	0.349618	+	0.936893i	$0.386312\pi$
$242$	0	0
$243$	−8.47289e11	−0.0641500
$244$	0	0
$245$	−5.42065e12	−0.392317
$246$	0	0
$247$	4.73205e11	0.0327504
$248$	0	0
$249$	−2.23989e12	−0.148296
$250$	0	0
$251$	−1.54036e12	−0.0975928	−0.0487964	−	0.998809i	$-0.515539\pi$
$251$	−1.54036e12	−0.0975928	−0.0487964	+	0.998809i	$0.515539\pi$
$252$	0	0
$253$	−2.87409e13	−1.74316
$254$	0	0
$255$	−1.11565e12	−0.0647974
$256$	0	0
$257$	−2.36700e12	−0.131694	−0.0658470	−	0.997830i	$-0.520975\pi$
$257$	−2.36700e12	−0.131694	−0.0658470	+	0.997830i	$0.520975\pi$
$258$	0	0
$259$	−2.54005e12	−0.135423
$260$	0	0
$261$	−8.36991e12	−0.427758
$262$	0	0
$263$	−1.69269e12	−0.0829508	−0.0414754	−	0.999140i	$-0.513206\pi$
$263$	−1.69269e12	−0.0829508	−0.0414754	+	0.999140i	$0.513206\pi$
$264$	0	0
$265$	−1.10221e13	−0.518097
$266$	0	0
$267$	−7.42944e12	−0.335077
$268$	0	0
$269$	−3.24983e13	−1.40677	−0.703386	−	0.710808i	$-0.748329\pi$
$269$	−3.24983e13	−1.40677	−0.703386	+	0.710808i	$0.748329\pi$
$270$	0	0
$271$	−8.84500e12	−0.367593	−0.183796	−	0.982964i	$-0.558839\pi$
$271$	−8.84500e12	−0.367593	−0.183796	+	0.982964i	$0.558839\pi$
$272$	0	0
$273$	4.55366e12	0.181746
$274$	0	0
$275$	2.71535e13	1.04111
$276$	0	0
$277$	4.16847e13	1.53581	0.767906	−	0.640563i	$-0.221299\pi$
$277$	4.16847e13	1.53581	0.767906	+	0.640563i	$0.221299\pi$
$278$	0	0
$279$	−1.37715e13	−0.487707
$280$	0	0
$281$	3.13771e13	1.06838	0.534192	−	0.845363i	$-0.320616\pi$
$281$	3.13771e13	1.06838	0.534192	+	0.845363i	$0.320616\pi$
$282$	0	0
$283$	−3.97011e13	−1.30010	−0.650051	−	0.759891i	$-0.725252\pi$
$283$	−3.97011e13	−1.30010	−0.650051	+	0.759891i	$0.725252\pi$
$284$	0	0
$285$	−1.60305e11	−0.00505010
$286$	0	0
$287$	1.07854e13	0.326955
$288$	0	0
$289$	−3.16985e13	−0.924913
$290$	0	0
$291$	−3.54055e13	−0.994626
$292$	0	0
$293$	3.71473e13	1.00497	0.502487	−	0.864584i	$-0.332418\pi$
$293$	3.71473e13	1.00497	0.502487	+	0.864584i	$0.332418\pi$
$294$	0	0
$295$	−2.95902e13	−0.771129
$296$	0	0
$297$	9.58788e12	0.240748
$298$	0	0
$299$	8.83030e13	2.13690
$300$	0	0
$301$	7.82247e12	0.182485
$302$	0	0
$303$	1.36057e13	0.306045
$304$	0	0
$305$	5.32183e11	0.0115455
$306$	0	0
$307$	5.82939e13	1.22001	0.610003	−	0.792399i	$-0.291168\pi$
$307$	5.82939e13	1.22001	0.610003	+	0.792399i	$0.291168\pi$
$308$	0	0
$309$	−1.06841e13	−0.215759
$310$	0	0
$311$	3.38262e13	0.659281	0.329641	−	0.944106i	$-0.393072\pi$
$311$	3.38262e13	0.659281	0.329641	+	0.944106i	$0.393072\pi$
$312$	0	0
$313$	7.69062e13	1.44700	0.723498	−	0.690326i	$-0.242533\pi$
$313$	7.69062e13	1.44700	0.723498	+	0.690326i	$0.242533\pi$
$314$	0	0
$315$	−1.54262e12	−0.0280253
$316$	0	0
$317$	4.59943e13	0.807008	0.403504	−	0.914978i	$-0.367792\pi$
$317$	4.59943e13	0.807008	0.403504	+	0.914978i	$0.367792\pi$
$318$	0	0
$319$	9.47136e13	1.60533
$320$	0	0
$321$	−3.62663e13	−0.593916
$322$	0	0
$323$	3.69763e11	0.00585207
$324$	0	0
$325$	−8.34259e13	−1.27627
$326$	0	0
$327$	−2.93817e11	−0.00434574
$328$	0	0
$329$	−1.51895e13	−0.217253
$330$	0	0
$331$	−1.12182e14	−1.55192	−0.775960	−	0.630782i	$-0.782734\pi$
$331$	−1.12182e14	−1.55192	−0.775960	+	0.630782i	$0.782734\pi$
$332$	0	0
$333$	1.64315e13	0.219905
$334$	0	0
$335$	−8.34276e12	−0.108035
$336$	0	0
$337$	−6.29192e13	−0.788531	−0.394265	−	0.918997i	$-0.629001\pi$
$337$	−6.29192e13	−0.788531	−0.394265	+	0.918997i	$0.629001\pi$
$338$	0	0
$339$	6.70220e13	0.813053
$340$	0	0
$341$	1.55838e14	1.83031
$342$	0	0
$343$	3.53375e13	0.401900
$344$	0	0
$345$	−2.99139e13	−0.329509
$346$	0	0
$347$	−1.34320e14	−1.43327	−0.716635	−	0.697448i	$-0.754319\pi$
$347$	−1.34320e14	−1.43327	−0.716635	+	0.697448i	$0.754319\pi$
$348$	0	0
$349$	6.49874e13	0.671876	0.335938	−	0.941884i	$-0.390947\pi$
$349$	6.49874e13	0.671876	0.335938	+	0.941884i	$0.390947\pi$
$350$	0	0
$351$	−2.94576e13	−0.295126
$352$	0	0
$353$	6.06398e13	0.588839	0.294420	−	0.955676i	$-0.404874\pi$
$353$	6.06398e13	0.588839	0.294420	+	0.955676i	$0.404874\pi$
$354$	0	0
$355$	−3.62395e13	−0.341135
$356$	0	0
$357$	3.55823e12	0.0324758
$358$	0	0
$359$	−1.95954e13	−0.173435	−0.0867173	−	0.996233i	$-0.527638\pi$
$359$	−1.95954e13	−0.173435	−0.0867173	+	0.996233i	$0.527638\pi$
$360$	0	0
$361$	−1.16437e14	−0.999544
$362$	0	0
$363$	−3.91653e13	−0.326149
$364$	0	0
$365$	−4.35060e13	−0.351511
$366$	0	0
$367$	−1.02500e14	−0.803636	−0.401818	−	0.915719i	$-0.631622\pi$
$367$	−1.02500e14	−0.803636	−0.401818	+	0.915719i	$0.631622\pi$
$368$	0	0
$369$	−6.97710e13	−0.530921
$370$	0	0
$371$	3.51536e13	0.259665
$372$	0	0
$373$	8.91645e13	0.639431	0.319715	−	0.947514i	$-0.396413\pi$
$373$	8.91645e13	0.639431	0.319715	+	0.947514i	$0.396413\pi$
$374$	0	0
$375$	6.22200e13	0.433269
$376$	0	0
$377$	−2.90996e14	−1.96793
$378$	0	0
$379$	2.61504e14	1.71776	0.858879	−	0.512179i	$-0.171161\pi$
$379$	2.61504e14	1.71776	0.858879	+	0.512179i	$0.171161\pi$
$380$	0	0
$381$	1.33235e13	0.0850220
$382$	0	0
$383$	−1.38948e14	−0.861508	−0.430754	−	0.902469i	$-0.641752\pi$
$383$	−1.38948e14	−0.861508	−0.430754	+	0.902469i	$0.641752\pi$
$384$	0	0
$385$	1.74562e13	0.105176
$386$	0	0
$387$	−5.06035e13	−0.296326
$388$	0	0
$389$	−1.83753e14	−1.04595	−0.522976	−	0.852347i	$-0.675178\pi$
$389$	−1.83753e14	−1.04595	−0.522976	+	0.852347i	$0.675178\pi$
$390$	0	0
$391$	6.90001e13	0.381836
$392$	0	0
$393$	1.87082e14	1.00664
$394$	0	0
$395$	1.04875e14	0.548768
$396$	0	0
$397$	1.94932e13	0.0992053	0.0496026	−	0.998769i	$-0.484205\pi$
$397$	1.94932e13	0.0992053	0.0496026	+	0.998769i	$0.484205\pi$
$398$	0	0
$399$	5.11273e11	0.00253106
$400$	0	0
$401$	−2.33384e14	−1.12403	−0.562013	−	0.827128i	$-0.689973\pi$
$401$	−2.33384e14	−1.12403	−0.562013	+	0.827128i	$0.689973\pi$
$402$	0	0
$403$	−4.78793e14	−2.24373
$404$	0	0
$405$	9.97918e12	0.0455085
$406$	0	0
$407$	−1.85939e14	−0.825279
$408$	0	0
$409$	−1.58513e14	−0.684837	−0.342419	−	0.939548i	$-0.611246\pi$
$409$	−1.58513e14	−0.684837	−0.342419	+	0.939548i	$0.611246\pi$
$410$	0	0
$411$	−9.48962e13	−0.399134
$412$	0	0
$413$	9.43744e13	0.386482
$414$	0	0
$415$	2.63809e13	0.105202
$416$	0	0
$417$	1.24927e13	0.0485185
$418$	0	0
$419$	2.82880e14	1.07010	0.535052	−	0.844819i	$-0.320292\pi$
$419$	2.82880e14	1.07010	0.535052	+	0.844819i	$0.320292\pi$
$420$	0	0
$421$	4.96375e14	1.82919	0.914593	−	0.404375i	$-0.132511\pi$
$421$	4.96375e14	1.82919	0.914593	+	0.404375i	$0.132511\pi$
$422$	0	0
$423$	9.82608e13	0.352783
$424$	0	0
$425$	−6.51891e13	−0.228053
$426$	0	0
$427$	−1.69733e12	−0.00578647
$428$	0	0
$429$	3.33341e14	1.10758
$430$	0	0
$431$	−3.97677e14	−1.28797	−0.643985	−	0.765038i	$-0.722720\pi$
$431$	−3.97677e14	−1.28797	−0.643985	+	0.765038i	$0.722720\pi$
$432$	0	0
$433$	−1.49632e14	−0.472433	−0.236217	−	0.971700i	$-0.575908\pi$
$433$	−1.49632e14	−0.472433	−0.236217	+	0.971700i	$0.575908\pi$
$434$	0	0
$435$	9.85790e13	0.303454
$436$	0	0
$437$	9.91443e12	0.0297591
$438$	0	0
$439$	1.05728e14	0.309481	0.154741	−	0.987955i	$-0.450546\pi$
$439$	1.05728e14	0.309481	0.154741	+	0.987955i	$0.450546\pi$
$440$	0	0
$441$	−1.11839e14	−0.319287
$442$	0	0
$443$	−3.31088e13	−0.0921984	−0.0460992	−	0.998937i	$-0.514679\pi$
$443$	−3.31088e13	−0.0921984	−0.0460992	+	0.998937i	$0.514679\pi$
$444$	0	0
$445$	8.75023e13	0.237706
$446$	0	0
$447$	3.39751e14	0.900470
$448$	0	0
$449$	−1.86449e14	−0.482176	−0.241088	−	0.970503i	$-0.577504\pi$
$449$	−1.86449e14	−0.482176	−0.241088	+	0.970503i	$0.577504\pi$
$450$	0	0
$451$	7.89525e14	1.99249
$452$	0	0
$453$	−1.02348e14	−0.252082
$454$	0	0
$455$	−5.36320e13	−0.128932
$456$	0	0
$457$	5.52462e13	0.129647	0.0648236	−	0.997897i	$-0.479352\pi$
$457$	5.52462e13	0.129647	0.0648236	+	0.997897i	$0.479352\pi$
$458$	0	0
$459$	−2.30182e13	−0.0527353
$460$	0	0
$461$	7.95826e12	0.0178018	0.00890088	−	0.999960i	$-0.497167\pi$
$461$	7.95826e12	0.0178018	0.00890088	+	0.999960i	$0.497167\pi$
$462$	0	0
$463$	5.56637e14	1.21584	0.607920	−	0.793998i	$-0.292004\pi$
$463$	5.56637e14	1.21584	0.607920	+	0.793998i	$0.292004\pi$
$464$	0	0
$465$	1.62198e14	0.345982
$466$	0	0
$467$	−9.04316e13	−0.188398	−0.0941992	−	0.995553i	$-0.530029\pi$
$467$	−9.04316e13	−0.188398	−0.0941992	+	0.995553i	$0.530029\pi$
$468$	0	0
$469$	2.66082e13	0.0541458
$470$	0	0
$471$	3.10892e14	0.618008
$472$	0	0
$473$	5.72627e14	1.11208
$474$	0	0
$475$	−9.36685e12	−0.0177737
$476$	0	0
$477$	−2.27408e14	−0.421653
$478$	0	0
$479$	−4.91081e14	−0.889832	−0.444916	−	0.895572i	$-0.646766\pi$
$479$	−4.91081e14	−0.889832	−0.444916	+	0.895572i	$0.646766\pi$
$480$	0	0
$481$	5.71274e14	1.01169
$482$	0	0
$483$	9.54067e13	0.165147
$484$	0	0
$485$	4.16999e14	0.705594
$486$	0	0
$487$	−2.00849e14	−0.332246	−0.166123	−	0.986105i	$-0.553125\pi$
$487$	−2.00849e14	−0.332246	−0.166123	+	0.986105i	$0.553125\pi$
$488$	0	0
$489$	−1.14888e14	−0.185813
$490$	0	0
$491$	−7.45240e13	−0.117855	−0.0589275	−	0.998262i	$-0.518768\pi$
$491$	−7.45240e13	−0.117855	−0.0589275	+	0.998262i	$0.518768\pi$
$492$	0	0
$493$	−2.27385e14	−0.351644
$494$	0	0
$495$	−1.12924e14	−0.170788
$496$	0	0
$497$	1.15582e14	0.170973
$498$	0	0
$499$	−6.79790e14	−0.983608	−0.491804	−	0.870706i	$-0.663662\pi$
$499$	−6.79790e14	−0.983608	−0.491804	+	0.870706i	$0.663662\pi$
$500$	0	0
$501$	2.54245e14	0.359870
$502$	0	0
$503$	3.23785e14	0.448367	0.224183	−	0.974547i	$-0.428029\pi$
$503$	3.23785e14	0.448367	0.224183	+	0.974547i	$0.428029\pi$
$504$	0	0
$505$	−1.60244e14	−0.217110
$506$	0	0
$507$	−5.88654e14	−0.780398
$508$	0	0
$509$	−5.26272e13	−0.0682751	−0.0341376	−	0.999417i	$-0.510868\pi$
$509$	−5.26272e13	−0.0682751	−0.0341376	+	0.999417i	$0.510868\pi$
$510$	0	0
$511$	1.38757e14	0.176173
$512$	0	0
$513$	−3.30742e12	−0.00411002
$514$	0	0
$515$	1.25835e14	0.153061
$516$	0	0
$517$	−1.11191e15	−1.32396
$518$	0	0
$519$	3.55145e14	0.413986
$520$	0	0
$521$	−8.60352e14	−0.981904	−0.490952	−	0.871187i	$-0.663351\pi$
$521$	−8.60352e14	−0.981904	−0.490952	+	0.871187i	$0.663351\pi$
$522$	0	0
$523$	−8.88719e14	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$523$	−8.88719e14	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$524$	0	0
$525$	−9.01373e13	−0.0986344
$526$	0	0
$527$	−3.74129e14	−0.400925
$528$	0	0
$529$	8.97285e14	0.941725
$530$	0	0
$531$	−6.10508e14	−0.627583
$532$	0	0
$533$	−2.42572e15	−2.44254
$534$	0	0
$535$	4.27136e14	0.421328
$536$	0	0
$537$	−1.02996e14	−0.0995322
$538$	0	0
$539$	1.26557e15	1.19825
$540$	0	0
$541$	−4.09513e14	−0.379912	−0.189956	−	0.981793i	$-0.560835\pi$
$541$	−4.09513e14	−0.379912	−0.189956	+	0.981793i	$0.560835\pi$
$542$	0	0
$543$	−4.45022e14	−0.404559
$544$	0	0
$545$	3.46051e12	0.00308290
$546$	0	0
$547$	1.99316e14	0.174025	0.0870124	−	0.996207i	$-0.472268\pi$
$547$	1.99316e14	0.174025	0.0870124	+	0.996207i	$0.472268\pi$
$548$	0	0
$549$	1.09800e13	0.00939629
$550$	0	0
$551$	−3.26723e13	−0.0274060
$552$	0	0
$553$	−3.34487e14	−0.275037
$554$	0	0
$555$	−1.93527e14	−0.156002
$556$	0	0
$557$	1.34904e15	1.06615	0.533077	−	0.846067i	$-0.321035\pi$
$557$	1.34904e15	1.06615	0.533077	+	0.846067i	$0.321035\pi$
$558$	0	0
$559$	−1.75933e15	−1.36327
$560$	0	0
$561$	2.60473e14	0.197910
$562$	0	0
$563$	2.54316e15	1.89486	0.947432	−	0.319958i	$-0.103669\pi$
$563$	2.54316e15	1.89486	0.947432	+	0.319958i	$0.103669\pi$
$564$	0	0
$565$	−7.89370e14	−0.576785
$566$	0	0
$567$	−3.18274e13	−0.0228084
$568$	0	0
$569$	2.74174e15	1.92712	0.963561	−	0.267490i	$-0.0861943\pi$
$569$	2.74174e15	1.92712	0.963561	+	0.267490i	$0.0861943\pi$
$570$	0	0
$571$	−2.76777e15	−1.90823	−0.954117	−	0.299433i	$-0.903202\pi$
$571$	−2.76777e15	−1.90823	−0.954117	+	0.299433i	$0.903202\pi$
$572$	0	0
$573$	−7.15629e14	−0.483991
$574$	0	0
$575$	−1.74791e15	−1.15970
$576$	0	0
$577$	6.94653e14	0.452169	0.226085	−	0.974108i	$-0.427407\pi$
$577$	6.94653e14	0.452169	0.226085	+	0.974108i	$0.427407\pi$
$578$	0	0
$579$	−1.51970e15	−0.970570
$580$	0	0
$581$	−8.41386e13	−0.0527262
$582$	0	0
$583$	2.57334e15	1.58242
$584$	0	0
$585$	3.46945e14	0.209365
$586$	0	0
$587$	9.04959e14	0.535944	0.267972	−	0.963427i	$-0.413647\pi$
$587$	9.04959e14	0.535944	0.267972	+	0.963427i	$0.413647\pi$
$588$	0	0
$589$	−5.37576e13	−0.0312468
$590$	0	0
$591$	1.61906e15	0.923699
$592$	0	0
$593$	1.41540e15	0.792645	0.396323	−	0.918111i	$-0.370286\pi$
$593$	1.41540e15	0.792645	0.396323	+	0.918111i	$0.370286\pi$
$594$	0	0
$595$	−4.19081e13	−0.0230385
$596$	0	0
$597$	−1.71687e15	−0.926570
$598$	0	0
$599$	−2.22145e15	−1.17703	−0.588516	−	0.808486i	$-0.700288\pi$
$599$	−2.22145e15	−1.17703	−0.588516	+	0.808486i	$0.700288\pi$
$600$	0	0
$601$	−2.80969e15	−1.46167	−0.730833	−	0.682556i	$-0.760868\pi$
$601$	−2.80969e15	−1.46167	−0.730833	+	0.682556i	$0.760868\pi$
$602$	0	0
$603$	−1.72128e14	−0.0879240
$604$	0	0
$605$	4.61281e14	0.231372
$606$	0	0
$607$	7.67540e14	0.378062	0.189031	−	0.981971i	$-0.439465\pi$
$607$	7.67540e14	0.378062	0.189031	+	0.981971i	$0.439465\pi$
$608$	0	0
$609$	−3.14406e14	−0.152088
$610$	0	0
$611$	3.41622e15	1.62300
$612$	0	0
$613$	1.47031e15	0.686083	0.343042	−	0.939320i	$-0.388543\pi$
$613$	1.47031e15	0.686083	0.343042	+	0.939320i	$0.388543\pi$
$614$	0	0
$615$	8.21747e14	0.376639
$616$	0	0
$617$	1.22736e15	0.552592	0.276296	−	0.961073i	$-0.410893\pi$
$617$	1.22736e15	0.552592	0.276296	+	0.961073i	$0.410893\pi$
$618$	0	0
$619$	1.00809e15	0.445862	0.222931	−	0.974834i	$-0.428437\pi$
$619$	1.00809e15	0.445862	0.222931	+	0.974834i	$0.428437\pi$
$620$	0	0
$621$	−6.17186e14	−0.268171
$622$	0	0
$623$	−2.79078e14	−0.119136
$624$	0	0
$625$	1.25142e15	0.524883
$626$	0	0
$627$	3.74267e13	0.0154244
$628$	0	0
$629$	4.46394e14	0.180776
$630$	0	0
$631$	2.98051e15	1.18612	0.593061	−	0.805158i	$-0.297919\pi$
$631$	2.98051e15	1.18612	0.593061	+	0.805158i	$0.297919\pi$
$632$	0	0
$633$	−7.57922e14	−0.296418
$634$	0	0
$635$	−1.56921e14	−0.0603151
$636$	0	0
$637$	−3.88830e15	−1.46890
$638$	0	0
$639$	−7.47697e14	−0.277633
$640$	0	0
$641$	3.59154e15	1.31087	0.655437	−	0.755249i	$-0.272484\pi$
$641$	3.59154e15	1.31087	0.655437	+	0.755249i	$0.272484\pi$
$642$	0	0
$643$	1.78687e15	0.641110	0.320555	−	0.947230i	$-0.396131\pi$
$643$	1.78687e15	0.641110	0.320555	+	0.947230i	$0.396131\pi$
$644$	0	0
$645$	5.95997e14	0.210216
$646$	0	0
$647$	−7.66508e14	−0.265793	−0.132896	−	0.991130i	$-0.542428\pi$
$647$	−7.66508e14	−0.265793	−0.132896	+	0.991130i	$0.542428\pi$
$648$	0	0
$649$	6.90848e15	2.35525
$650$	0	0
$651$	−5.17310e14	−0.173403
$652$	0	0
$653$	−2.56438e14	−0.0845201	−0.0422600	−	0.999107i	$-0.513456\pi$
$653$	−2.56438e14	−0.0845201	−0.0422600	+	0.999107i	$0.513456\pi$
$654$	0	0
$655$	−2.20341e15	−0.714115
$656$	0	0
$657$	−8.97620e14	−0.286077
$658$	0	0
$659$	3.63270e15	1.13857	0.569286	−	0.822140i	$-0.307220\pi$
$659$	3.63270e15	1.13857	0.569286	+	0.822140i	$0.307220\pi$
$660$	0	0
$661$	5.24024e14	0.161526	0.0807632	−	0.996733i	$-0.474264\pi$
$661$	5.24024e14	0.161526	0.0807632	+	0.996733i	$0.474264\pi$
$662$	0	0
$663$	−8.00271e14	−0.242612
$664$	0	0
$665$	−6.02166e12	−0.00179555
$666$	0	0
$667$	−6.09685e15	−1.78819
$668$	0	0
$669$	3.26212e15	0.941144
$670$	0	0
$671$	−1.24250e14	−0.0352632
$672$	0	0
$673$	−3.22347e15	−0.899996	−0.449998	−	0.893029i	$-0.648575\pi$
$673$	−3.22347e15	−0.899996	−0.449998	+	0.893029i	$0.648575\pi$
$674$	0	0
$675$	5.83098e14	0.160166
$676$	0	0
$677$	6.08105e14	0.164339	0.0821695	−	0.996618i	$-0.473815\pi$
$677$	6.08105e14	0.164339	0.0821695	+	0.996618i	$0.473815\pi$
$678$	0	0
$679$	−1.32997e15	−0.353636
$680$	0	0
$681$	2.42009e15	0.633172
$682$	0	0
$683$	−1.63384e14	−0.0420626	−0.0210313	−	0.999779i	$-0.506695\pi$
$683$	−1.63384e14	−0.0420626	−0.0210313	+	0.999779i	$0.506695\pi$
$684$	0	0
$685$	1.11767e15	0.283148
$686$	0	0
$687$	2.89486e15	0.721714
$688$	0	0
$689$	−7.90628e15	−1.93984
$690$	0	0
$691$	2.57486e15	0.621763	0.310882	−	0.950449i	$-0.399376\pi$
$691$	2.57486e15	0.621763	0.310882	+	0.950449i	$0.399376\pi$
$692$	0	0
$693$	3.60157e14	0.0855971
$694$	0	0
$695$	−1.47136e14	−0.0344194
$696$	0	0
$697$	−1.89546e15	−0.436450
$698$	0	0
$699$	−4.77708e15	−1.08277
$700$	0	0
$701$	−6.95102e15	−1.55096	−0.775478	−	0.631374i	$-0.782491\pi$
$701$	−6.95102e15	−1.55096	−0.775478	+	0.631374i	$0.782491\pi$
$702$	0	0
$703$	6.41412e13	0.0140891
$704$	0	0
$705$	−1.15729e15	−0.250267
$706$	0	0
$707$	5.11080e14	0.108813
$708$	0	0
$709$	−1.64191e15	−0.344187	−0.172094	−	0.985081i	$-0.555053\pi$
$709$	−1.64191e15	−0.344187	−0.172094	+	0.985081i	$0.555053\pi$
$710$	0	0
$711$	2.16379e15	0.446615
$712$	0	0
$713$	−1.00315e16	−2.03880
$714$	0	0
$715$	−3.92601e15	−0.785722
$716$	0	0
$717$	7.29354e13	0.0143742
$718$	0	0
$719$	−7.03155e15	−1.36472	−0.682358	−	0.731018i	$-0.739045\pi$
$719$	−7.03155e15	−1.36472	−0.682358	+	0.731018i	$0.739045\pi$
$720$	0	0
$721$	−4.01337e14	−0.0767123
$722$	0	0
$723$	−2.14449e15	−0.403704
$724$	0	0
$725$	5.76011e15	1.06800
$726$	0	0
$727$	9.65145e15	1.76260	0.881300	−	0.472558i	$-0.156669\pi$
$727$	9.65145e15	1.76260	0.881300	+	0.472558i	$0.156669\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	−1.37474e15	−0.243599
$732$	0	0
$733$	5.38706e15	0.940329	0.470164	−	0.882579i	$-0.344195\pi$
$733$	5.38706e15	0.940329	0.470164	+	0.882579i	$0.344195\pi$
$734$	0	0
$735$	1.31722e15	0.226505
$736$	0	0
$737$	1.94780e15	0.329969
$738$	0	0
$739$	−6.23212e15	−1.04014	−0.520070	−	0.854124i	$-0.674094\pi$
$739$	−6.23212e15	−1.04014	−0.520070	+	0.854124i	$0.674094\pi$
$740$	0	0
$741$	−1.14989e14	−0.0189084
$742$	0	0
$743$	−3.05132e15	−0.494366	−0.247183	−	0.968969i	$-0.579505\pi$
$743$	−3.05132e15	−0.494366	−0.247183	+	0.968969i	$0.579505\pi$
$744$	0	0
$745$	−4.00151e15	−0.638800
$746$	0	0
$747$	5.44292e14	0.0856187
$748$	0	0
$749$	−1.36230e15	−0.211165
$750$	0	0
$751$	−3.92991e15	−0.600292	−0.300146	−	0.953893i	$-0.597035\pi$
$751$	−3.92991e15	−0.600292	−0.300146	+	0.953893i	$0.597035\pi$
$752$	0	0
$753$	3.74308e14	0.0563452
$754$	0	0
$755$	1.20544e15	0.178828
$756$	0	0
$757$	5.24662e15	0.767101	0.383551	−	0.923520i	$-0.374701\pi$
$757$	5.24662e15	0.767101	0.383551	+	0.923520i	$0.374701\pi$
$758$	0	0
$759$	6.98405e15	1.00642
$760$	0	0
$761$	4.83177e15	0.686264	0.343132	−	0.939287i	$-0.388512\pi$
$761$	4.83177e15	0.686264	0.343132	+	0.939287i	$0.388512\pi$
$762$	0	0
$763$	−1.10369e13	−0.00154512
$764$	0	0
$765$	2.71103e14	0.0374108
$766$	0	0
$767$	−2.12255e16	−2.88724
$768$	0	0
$769$	−6.55230e15	−0.878615	−0.439308	−	0.898337i	$-0.644776\pi$
$769$	−6.55230e15	−0.878615	−0.439308	+	0.898337i	$0.644776\pi$
$770$	0	0
$771$	5.75181e14	0.0760336
$772$	0	0
$773$	2.31322e15	0.301460	0.150730	−	0.988575i	$-0.451838\pi$
$773$	2.31322e15	0.301460	0.150730	+	0.988575i	$0.451838\pi$
$774$	0	0
$775$	9.47746e15	1.21768
$776$	0	0
$777$	6.17231e14	0.0781866
$778$	0	0
$779$	−2.72354e14	−0.0340155
$780$	0	0
$781$	8.46091e15	1.04192
$782$	0	0
$783$	2.03389e15	0.246966
$784$	0	0
$785$	−3.66161e15	−0.438419
$786$	0	0
$787$	1.47182e16	1.73778	0.868888	−	0.495008i	$-0.164835\pi$
$787$	1.47182e16	1.73778	0.868888	+	0.495008i	$0.164835\pi$
$788$	0	0
$789$	4.11324e14	0.0478917
$790$	0	0
$791$	2.51760e15	0.289079
$792$	0	0
$793$	3.81742e14	0.0432282
$794$	0	0
$795$	2.67837e15	0.299124
$796$	0	0
$797$	2.50458e15	0.275877	0.137938	−	0.990441i	$-0.455952\pi$
$797$	2.50458e15	0.275877	0.137938	+	0.990441i	$0.455952\pi$
$798$	0	0
$799$	2.66944e15	0.290010
$800$	0	0
$801$	1.80535e15	0.193457
$802$	0	0
$803$	1.01574e16	1.07361
$804$	0	0
$805$	−1.12368e15	−0.117156
$806$	0	0
$807$	7.89709e15	0.812200
$808$	0	0
$809$	−1.29400e16	−1.31286	−0.656431	−	0.754386i	$-0.727934\pi$
$809$	−1.29400e16	−1.31286	−0.656431	+	0.754386i	$0.727934\pi$
$810$	0	0
$811$	4.70410e15	0.470828	0.235414	−	0.971895i	$-0.424355\pi$
$811$	4.70410e15	0.470828	0.235414	+	0.971895i	$0.424355\pi$
$812$	0	0
$813$	2.14934e15	0.212230
$814$	0	0
$815$	1.35313e15	0.131817
$816$	0	0
$817$	−1.97533e14	−0.0189853
$818$	0	0
$819$	−1.10654e15	−0.104931
$820$	0	0
$821$	1.12296e16	1.05069	0.525346	−	0.850889i	$-0.323936\pi$
$821$	1.12296e16	1.05069	0.525346	+	0.850889i	$0.323936\pi$
$822$	0	0
$823$	−8.53732e15	−0.788175	−0.394087	−	0.919073i	$-0.628939\pi$
$823$	−8.53732e15	−0.788175	−0.394087	+	0.919073i	$0.628939\pi$
$824$	0	0
$825$	−6.59831e15	−0.601085
$826$	0	0
$827$	−1.02091e16	−0.917712	−0.458856	−	0.888511i	$-0.651741\pi$
$827$	−1.02091e16	−0.917712	−0.458856	+	0.888511i	$0.651741\pi$
$828$	0	0
$829$	2.25785e15	0.200284	0.100142	−	0.994973i	$-0.468070\pi$
$829$	2.25785e15	0.200284	0.100142	+	0.994973i	$0.468070\pi$
$830$	0	0
$831$	−1.01294e16	−0.886701
$832$	0	0
$833$	−3.03832e15	−0.262474
$834$	0	0
$835$	−2.99444e15	−0.255294
$836$	0	0
$837$	3.34648e15	0.281578
$838$	0	0
$839$	2.05099e16	1.70322	0.851612	−	0.524172i	$-0.175625\pi$
$839$	2.05099e16	1.70322	0.851612	+	0.524172i	$0.175625\pi$
$840$	0	0
$841$	7.89119e15	0.646792
$842$	0	0
$843$	−7.62463e15	−0.616832
$844$	0	0
$845$	6.93303e15	0.553619
$846$	0	0
$847$	−1.47120e15	−0.115961
$848$	0	0
$849$	9.64737e15	0.750614
$850$	0	0
$851$	1.19691e16	0.919285
$852$	0	0
$853$	1.36009e16	1.03121	0.515606	−	0.856826i	$-0.327567\pi$
$853$	1.36009e16	1.03121	0.515606	+	0.856826i	$0.327567\pi$
$854$	0	0
$855$	3.89541e13	0.00291568
$856$	0	0
$857$	1.03910e16	0.767828	0.383914	−	0.923369i	$-0.374576\pi$
$857$	1.03910e16	0.767828	0.383914	+	0.923369i	$0.374576\pi$
$858$	0	0
$859$	−1.33142e16	−0.971298	−0.485649	−	0.874154i	$-0.661417\pi$
$859$	−1.33142e16	−0.971298	−0.485649	+	0.874154i	$0.661417\pi$
$860$	0	0
$861$	−2.62086e15	−0.188767
$862$	0	0
$863$	−8.38331e15	−0.596151	−0.298076	−	0.954542i	$-0.596345\pi$
$863$	−8.38331e15	−0.596151	−0.298076	+	0.954542i	$0.596345\pi$
$864$	0	0
$865$	−4.18282e15	−0.293684
$866$	0	0
$867$	7.70274e15	0.533999
$868$	0	0
$869$	−2.44854e16	−1.67609
$870$	0	0
$871$	−5.98437e15	−0.404500
$872$	0	0
$873$	8.60355e15	0.574248
$874$	0	0
$875$	2.33722e15	0.154048
$876$	0	0
$877$	−1.51493e16	−0.986042	−0.493021	−	0.870017i	$-0.664107\pi$
$877$	−1.51493e16	−0.986042	−0.493021	+	0.870017i	$0.664107\pi$
$878$	0	0
$879$	−9.02679e15	−0.580223
$880$	0	0
$881$	−2.86242e15	−0.181705	−0.0908524	−	0.995864i	$-0.528959\pi$
$881$	−2.86242e15	−0.181705	−0.0908524	+	0.995864i	$0.528959\pi$
$882$	0	0
$883$	4.51715e15	0.283192	0.141596	−	0.989925i	$-0.454777\pi$
$883$	4.51715e15	0.283192	0.141596	+	0.989925i	$0.454777\pi$
$884$	0	0
$885$	7.19042e15	0.445212
$886$	0	0
$887$	−1.46802e15	−0.0897741	−0.0448871	−	0.998992i	$-0.514293\pi$
$887$	−1.46802e15	−0.0897741	−0.0448871	+	0.998992i	$0.514293\pi$
$888$	0	0
$889$	5.00481e14	0.0302293
$890$	0	0
$891$	−2.32986e15	−0.138996
$892$	0	0
$893$	3.83565e14	0.0226025
$894$	0	0
$895$	1.21307e15	0.0706088
$896$	0	0
$897$	−2.14576e16	−1.23374
$898$	0	0
$899$	3.30581e16	1.87758
$900$	0	0
$901$	−6.17798e15	−0.346625
$902$	0	0
$903$	−1.90086e15	−0.105358
$904$	0	0
$905$	5.24136e15	0.286997
$906$	0	0
$907$	−2.09558e16	−1.13361	−0.566805	−	0.823852i	$-0.691821\pi$
$907$	−2.09558e16	−1.13361	−0.566805	+	0.823852i	$0.691821\pi$
$908$	0	0
$909$	−3.30618e15	−0.176695
$910$	0	0
$911$	−1.93579e15	−0.102213	−0.0511065	−	0.998693i	$-0.516275\pi$
$911$	−1.93579e15	−0.102213	−0.0511065	+	0.998693i	$0.516275\pi$
$912$	0	0
$913$	−6.15919e15	−0.321317
$914$	0	0
$915$	−1.29321e14	−0.00666579
$916$	0	0
$917$	7.02751e15	0.357907
$918$	0	0
$919$	3.35035e16	1.68599	0.842994	−	0.537923i	$-0.180791\pi$
$919$	3.35035e16	1.68599	0.842994	+	0.537923i	$0.180791\pi$
$920$	0	0
$921$	−1.41654e16	−0.704370
$922$	0	0
$923$	−2.59951e16	−1.27727
$924$	0	0
$925$	−1.13081e16	−0.549046
$926$	0	0
$927$	2.59625e15	0.124568
$928$	0	0
$929$	−1.65748e16	−0.785892	−0.392946	−	0.919562i	$-0.628544\pi$
$929$	−1.65748e16	−0.785892	−0.392946	+	0.919562i	$0.628544\pi$
$930$	0	0
$931$	−4.36568e14	−0.0204564
$932$	0	0
$933$	−8.21976e15	−0.380636
$934$	0	0
$935$	−3.06779e15	−0.140398
$936$	0	0
$937$	−1.11596e16	−0.504755	−0.252378	−	0.967629i	$-0.581212\pi$
$937$	−1.11596e16	−0.504755	−0.252378	+	0.967629i	$0.581212\pi$
$938$	0	0
$939$	−1.86882e16	−0.835424
$940$	0	0
$941$	2.08676e16	0.921997	0.460999	−	0.887401i	$-0.347491\pi$
$941$	2.08676e16	0.921997	0.460999	+	0.887401i	$0.347491\pi$
$942$	0	0
$943$	−5.08229e16	−2.21945
$944$	0	0
$945$	3.74856e14	0.0161804
$946$	0	0
$947$	−3.24403e16	−1.38407	−0.692037	−	0.721862i	$-0.743287\pi$
$947$	−3.24403e16	−1.38407	−0.692037	+	0.721862i	$0.743287\pi$
$948$	0	0
$949$	−3.12074e16	−1.31612
$950$	0	0
$951$	−1.11766e16	−0.465926
$952$	0	0
$953$	3.61199e16	1.48846	0.744228	−	0.667926i	$-0.232818\pi$
$953$	3.61199e16	1.48846	0.744228	+	0.667926i	$0.232818\pi$
$954$	0	0
$955$	8.42852e15	0.343347
$956$	0	0
$957$	−2.30154e16	−0.926836
$958$	0	0
$959$	−3.56466e15	−0.141911
$960$	0	0
$961$	2.89840e16	1.14072
$962$	0	0
$963$	8.81271e15	0.342898
$964$	0	0
$965$	1.78987e16	0.688529
$966$	0	0
$967$	4.35846e14	0.0165763	0.00828815	−	0.999966i	$-0.497362\pi$
$967$	4.35846e14	0.0165763	0.00828815	+	0.999966i	$0.497362\pi$
$968$	0	0
$969$	−8.98524e13	−0.00337869
$970$	0	0
$971$	4.01243e15	0.149177	0.0745884	−	0.997214i	$-0.476236\pi$
$971$	4.01243e15	0.149177	0.0745884	+	0.997214i	$0.476236\pi$
$972$	0	0
$973$	4.69273e14	0.0172506
$974$	0	0
$975$	2.02725e16	0.736855
$976$	0	0
$977$	4.27191e16	1.53533	0.767665	−	0.640851i	$-0.221418\pi$
$977$	4.27191e16	1.53533	0.767665	+	0.640851i	$0.221418\pi$
$978$	0	0
$979$	−2.04293e16	−0.726021
$980$	0	0
$981$	7.13975e13	0.00250901
$982$	0	0
$983$	3.72875e16	1.29574	0.647871	−	0.761750i	$-0.275660\pi$
$983$	3.72875e16	1.29574	0.647871	+	0.761750i	$0.275660\pi$
$984$	0	0
$985$	−1.90689e16	−0.655278
$986$	0	0
$987$	3.69105e15	0.125431
$988$	0	0
$989$	−3.68608e16	−1.23876
$990$	0	0
$991$	4.47101e16	1.48594	0.742969	−	0.669326i	$-0.233417\pi$
$991$	4.47101e16	1.48594	0.742969	+	0.669326i	$0.233417\pi$
$992$	0	0
$993$	2.72602e16	0.896002
$994$	0	0
$995$	2.02209e16	0.657315
$996$	0	0
$997$	−1.16976e16	−0.376073	−0.188037	−	0.982162i	$-0.560212\pi$
$997$	−1.16976e16	−0.376073	−0.188037	+	0.982162i	$0.560212\pi$
$998$	0	0
$999$	−3.99287e15	−0.126962

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.12.a.c.1.1		1
3.2	odd	2		144.12.a.f.1.1		1
4.3	odd	2		12.12.a.b.1.1	✓	1
8.3	odd	2		192.12.a.d.1.1		1
8.5	even	2		192.12.a.n.1.1		1
12.11	even	2		36.12.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.12.a.b.1.1	✓	1	4.3	odd	2
36.12.a.b.1.1		1	12.11	even	2
48.12.a.c.1.1		1	1.1	even	1	trivial
144.12.a.f.1.1		1	3.2	odd	2
192.12.a.d.1.1		1	8.3	odd	2
192.12.a.n.1.1		1	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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