LMFDB - Embedded newform 348.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [348,2,Mod(1,348)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(348, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("348.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 348 = 2^{2} \cdot 3 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	348.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:	$2.77879399034$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	348.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} +5.00000 q^{13} +2.00000 q^{15} -1.00000 q^{17} +6.00000 q^{19} -1.00000 q^{21} +4.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{29} -3.00000 q^{33} -2.00000 q^{35} +8.00000 q^{37} -5.00000 q^{39} -10.0000 q^{41} +4.00000 q^{43} -2.00000 q^{45} +7.00000 q^{47} -6.00000 q^{49} +1.00000 q^{51} -2.00000 q^{53} -6.00000 q^{55} -6.00000 q^{57} +6.00000 q^{59} -8.00000 q^{61} +1.00000 q^{63} -10.0000 q^{65} +3.00000 q^{67} -4.00000 q^{69} -4.00000 q^{71} +4.00000 q^{73} +1.00000 q^{75} +3.00000 q^{77} +6.00000 q^{79} +1.00000 q^{81} -14.0000 q^{83} +2.00000 q^{85} +1.00000 q^{87} -7.00000 q^{89} +5.00000 q^{91} -12.0000 q^{95} -2.00000 q^{97} +3.00000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	−5.00000	−0.800641
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−10.0000	−1.24035
$66$	0	0
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−7.00000	−0.741999	−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000	−0.741999	−0.370999	+	0.928633i	$0.620985\pi$
$90$	0	0
$91$	5.00000	0.524142
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−12.0000	−1.23117
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	0	0
$101$	7.00000	0.696526	0.348263	−	0.937397i	$-0.386772\pi$
$101$	7.00000	0.696526	0.348263	+	0.937397i	$0.386772\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	−21.0000	−1.97551	−0.987757	−	0.156001i	$-0.950140\pi$
$113$	−21.0000	−1.97551	−0.987757	+	0.156001i	$0.950140\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	5.00000	0.462250
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	10.0000	0.901670
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	−7.00000	−0.589506
$142$	0	0
$143$	15.0000	1.25436
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	0	0
$165$	6.00000	0.467099
$166$	0	0
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	0	0
$181$	−23.0000	−1.70958	−0.854788	−	0.518977i	$-0.826313\pi$
$181$	−23.0000	−1.70958	−0.854788	+	0.518977i	$0.826313\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	−16.0000	−1.17634
$186$	0	0
$187$	−3.00000	−0.219382
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	10.0000	0.716115
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	0	0
$201$	−3.00000	−0.211604
$202$	0	0
$203$	−1.00000	−0.0701862
$204$	0	0
$205$	20.0000	1.39686
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	18.0000	1.24509
$210$	0	0
$211$	−6.00000	−0.413057	−0.206529	−	0.978441i	$-0.566217\pi$
$211$	−6.00000	−0.413057	−0.206529	+	0.978441i	$0.566217\pi$
$212$	0	0
$213$	4.00000	0.274075
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−5.00000	−0.336336
$222$	0	0
$223$	−5.00000	−0.334825	−0.167412	−	0.985887i	$-0.553541\pi$
$223$	−5.00000	−0.334825	−0.167412	+	0.985887i	$0.553541\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	−14.0000	−0.913259
$236$	0	0
$237$	−6.00000	−0.389742
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−13.0000	−0.837404	−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000	−0.837404	−0.418702	+	0.908124i	$0.637515\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	12.0000	0.766652
$246$	0	0
$247$	30.0000	1.90885
$248$	0	0
$249$	14.0000	0.887214
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	−2.00000	−0.125245
$256$	0	0
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	7.00000	0.428393
$268$	0	0
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	0	0
$273$	−5.00000	−0.302614
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−5.00000	−0.300421	−0.150210	−	0.988654i	$-0.547995\pi$
$277$	−5.00000	−0.300421	−0.150210	+	0.988654i	$0.547995\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	0	0
$285$	12.0000	0.710819
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	−7.00000	−0.408944	−0.204472	−	0.978872i	$-0.565548\pi$
$293$	−7.00000	−0.408944	−0.204472	+	0.978872i	$0.565548\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	−3.00000	−0.174078
$298$	0	0
$299$	20.0000	1.15663
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	−7.00000	−0.402139
$304$	0	0
$305$	16.0000	0.916157
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	−29.0000	−1.64444	−0.822220	−	0.569170i	$-0.807264\pi$
$311$	−29.0000	−1.64444	−0.822220	+	0.569170i	$0.807264\pi$
$312$	0	0
$313$	−27.0000	−1.52613	−0.763065	−	0.646322i	$-0.776306\pi$
$313$	−27.0000	−1.52613	−0.763065	+	0.646322i	$0.776306\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	9.00000	0.505490	0.252745	−	0.967533i	$-0.418667\pi$
$317$	9.00000	0.505490	0.252745	+	0.967533i	$0.418667\pi$
$318$	0	0
$319$	−3.00000	−0.167968
$320$	0	0
$321$	−10.0000	−0.558146
$322$	0	0
$323$	−6.00000	−0.333849
$324$	0	0
$325$	−5.00000	−0.277350
$326$	0	0
$327$	−11.0000	−0.608301
$328$	0	0
$329$	7.00000	0.385922
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	0	0
$333$	8.00000	0.438397
$334$	0	0
$335$	−6.00000	−0.327815
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	21.0000	1.14056
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	0	0
$357$	1.00000	0.0529256
$358$	0	0
$359$	−28.0000	−1.47778	−0.738892	−	0.673824i	$-0.764651\pi$
$359$	−28.0000	−1.47778	−0.738892	+	0.673824i	$0.764651\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	−8.00000	−0.418739
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	−5.00000	−0.257513
$378$	0	0
$379$	−22.0000	−1.13006	−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000	−1.13006	−0.565032	+	0.825069i	$0.691136\pi$
$380$	0	0
$381$	−14.0000	−0.717242
$382$	0	0
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	−3.00000	−0.151330
$394$	0	0
$395$	−12.0000	−0.603786
$396$	0	0
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	0	0
$413$	6.00000	0.295241
$414$	0	0
$415$	28.0000	1.37447
$416$	0	0
$417$	11.0000	0.538672
$418$	0	0
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	0	0
$423$	7.00000	0.340352
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	−8.00000	−0.387147
$428$	0	0
$429$	−15.0000	−0.724207
$430$	0	0
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	0	0
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	0	0
$437$	24.0000	1.14808
$438$	0	0
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	37.0000	1.75792	0.878962	−	0.476893i	$-0.158237\pi$
$443$	37.0000	1.75792	0.878962	+	0.476893i	$0.158237\pi$
$444$	0	0
$445$	14.0000	0.663664
$446$	0	0
$447$	0	0
$448$	0	0
$449$	23.0000	1.08544	0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000	1.08544	0.542719	+	0.839915i	$0.317395\pi$
$450$	0	0
$451$	−30.0000	−1.41264
$452$	0	0
$453$	4.00000	0.187936
$454$	0	0
$455$	−10.0000	−0.468807
$456$	0	0
$457$	−35.0000	−1.63723	−0.818615	−	0.574342i	$-0.805258\pi$
$457$	−35.0000	−1.63723	−0.818615	+	0.574342i	$0.805258\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	0	0
$463$	−5.00000	−0.232370	−0.116185	−	0.993228i	$-0.537067\pi$
$463$	−5.00000	−0.232370	−0.116185	+	0.993228i	$0.537067\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	3.00000	0.138527
$470$	0	0
$471$	−6.00000	−0.276465
$472$	0	0
$473$	12.0000	0.551761
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	40.0000	1.82384
$482$	0	0
$483$	−4.00000	−0.182006
$484$	0	0
$485$	4.00000	0.181631
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	18.0000	0.813988
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	1.00000	0.0450377
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	13.0000	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$500$	0	0
$501$	−4.00000	−0.178707
$502$	0	0
$503$	19.0000	0.847168	0.423584	−	0.905857i	$-0.360772\pi$
$503$	19.0000	0.847168	0.423584	+	0.905857i	$0.360772\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	−6.00000	−0.264906
$514$	0	0
$515$	32.0000	1.41009
$516$	0	0
$517$	21.0000	0.923579
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−40.0000	−1.75243	−0.876216	−	0.481919i	$-0.839940\pi$
$521$	−40.0000	−1.75243	−0.876216	+	0.481919i	$0.839940\pi$
$522$	0	0
$523$	21.0000	0.918266	0.459133	−	0.888368i	$-0.348160\pi$
$523$	21.0000	0.918266	0.459133	+	0.888368i	$0.348160\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	−50.0000	−2.16574
$534$	0	0
$535$	−20.0000	−0.864675
$536$	0	0
$537$	−8.00000	−0.345225
$538$	0	0
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	23.0000	0.987024
$544$	0	0
$545$	−22.0000	−0.942376
$546$	0	0
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	0	0
$549$	−8.00000	−0.341432
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	6.00000	0.255146
$554$	0	0
$555$	16.0000	0.679162
$556$	0	0
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	0	0
$559$	20.0000	0.845910
$560$	0	0
$561$	3.00000	0.126660
$562$	0	0
$563$	−11.0000	−0.463595	−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000	−0.463595	−0.231797	+	0.972764i	$0.574461\pi$
$564$	0	0
$565$	42.0000	1.76695
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−3.00000	−0.125767	−0.0628833	−	0.998021i	$-0.520030\pi$
$569$	−3.00000	−0.125767	−0.0628833	+	0.998021i	$0.520030\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	−16.0000	−0.668410
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−4.00000	−0.166522	−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000	−0.166522	−0.0832611	+	0.996528i	$0.526534\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	−14.0000	−0.580818
$582$	0	0
$583$	−6.00000	−0.248495
$584$	0	0
$585$	−10.0000	−0.413449
$586$	0	0
$587$	−26.0000	−1.07313	−0.536567	−	0.843857i	$-0.680279\pi$
$587$	−26.0000	−1.07313	−0.536567	+	0.843857i	$0.680279\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	0	0
$597$	25.0000	1.02318
$598$	0	0
$599$	−17.0000	−0.694601	−0.347301	−	0.937754i	$-0.612902\pi$
$599$	−17.0000	−0.694601	−0.347301	+	0.937754i	$0.612902\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	3.00000	0.122169
$604$	0	0
$605$	4.00000	0.162623
$606$	0	0
$607$	30.0000	1.21766	0.608831	−	0.793300i	$-0.291639\pi$
$607$	30.0000	1.21766	0.608831	+	0.793300i	$0.291639\pi$
$608$	0	0
$609$	1.00000	0.0405220
$610$	0	0
$611$	35.0000	1.41595
$612$	0	0
$613$	−19.0000	−0.767403	−0.383701	−	0.923457i	$-0.625351\pi$
$613$	−19.0000	−0.767403	−0.383701	+	0.923457i	$0.625351\pi$
$614$	0	0
$615$	−20.0000	−0.806478
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	22.0000	0.884255	0.442127	−	0.896952i	$-0.354224\pi$
$619$	22.0000	0.884255	0.442127	+	0.896952i	$0.354224\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−7.00000	−0.280449
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−18.0000	−0.718851
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	43.0000	1.71180	0.855901	−	0.517139i	$-0.173003\pi$
$631$	43.0000	1.71180	0.855901	+	0.517139i	$0.173003\pi$
$632$	0	0
$633$	6.00000	0.238479
$634$	0	0
$635$	−28.0000	−1.11115
$636$	0	0
$637$	−30.0000	−1.18864
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	13.0000	0.513469	0.256735	−	0.966482i	$-0.417353\pi$
$641$	13.0000	0.513469	0.256735	+	0.966482i	$0.417353\pi$
$642$	0	0
$643$	−5.00000	−0.197181	−0.0985904	−	0.995128i	$-0.531433\pi$
$643$	−5.00000	−0.197181	−0.0985904	+	0.995128i	$0.531433\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−23.0000	−0.900060	−0.450030	−	0.893014i	$-0.648587\pi$
$653$	−23.0000	−0.900060	−0.450030	+	0.893014i	$0.648587\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	0	0
$661$	−39.0000	−1.51692	−0.758462	−	0.651717i	$-0.774049\pi$
$661$	−39.0000	−1.51692	−0.758462	+	0.651717i	$0.774049\pi$
$662$	0	0
$663$	5.00000	0.194184
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	−4.00000	−0.154881
$668$	0	0
$669$	5.00000	0.193311
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	−27.0000	−1.04077	−0.520387	−	0.853931i	$-0.674212\pi$
$673$	−27.0000	−1.04077	−0.520387	+	0.853931i	$0.674212\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−1.00000	−0.0384331	−0.0192166	−	0.999815i	$-0.506117\pi$
$677$	−1.00000	−0.0384331	−0.0192166	+	0.999815i	$0.506117\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	−6.00000	−0.229920
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	−36.0000	−1.37549
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	23.0000	0.874961	0.437481	−	0.899228i	$-0.355871\pi$
$691$	23.0000	0.874961	0.437481	+	0.899228i	$0.355871\pi$
$692$	0	0
$693$	3.00000	0.113961
$694$	0	0
$695$	22.0000	0.834508
$696$	0	0
$697$	10.0000	0.378777
$698$	0	0
$699$	−26.0000	−0.983410
$700$	0	0
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	0	0
$703$	48.0000	1.81035
$704$	0	0
$705$	14.0000	0.527271
$706$	0	0
$707$	7.00000	0.263262
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	6.00000	0.225018
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−30.0000	−1.12194
$716$	0	0
$717$	6.00000	0.224074
$718$	0	0
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	13.0000	0.483475
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.00000	−0.147945
$732$	0	0
$733$	−48.0000	−1.77292	−0.886460	−	0.462805i	$-0.846843\pi$
$733$	−48.0000	−1.77292	−0.886460	+	0.462805i	$0.846843\pi$
$734$	0	0
$735$	−12.0000	−0.442627
$736$	0	0
$737$	9.00000	0.331519
$738$	0	0
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	0	0
$741$	−30.0000	−1.10208
$742$	0	0
$743$	−37.0000	−1.35740	−0.678699	−	0.734416i	$-0.737456\pi$
$743$	−37.0000	−1.35740	−0.678699	+	0.734416i	$0.737456\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−14.0000	−0.512233
$748$	0	0
$749$	10.0000	0.365392
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	9.00000	0.327978
$754$	0	0
$755$	8.00000	0.291150
$756$	0	0
$757$	−40.0000	−1.45382	−0.726912	−	0.686730i	$-0.759045\pi$
$757$	−40.0000	−1.45382	−0.726912	+	0.686730i	$0.759045\pi$
$758$	0	0
$759$	−12.0000	−0.435572
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	11.0000	0.398227
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	30.0000	1.08324
$768$	0	0
$769$	4.00000	0.144244	0.0721218	−	0.997396i	$-0.477023\pi$
$769$	4.00000	0.144244	0.0721218	+	0.997396i	$0.477023\pi$
$770$	0	0
$771$	8.00000	0.288113
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	−60.0000	−2.14972
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−12.0000	−0.428298
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−21.0000	−0.746674
$792$	0	0
$793$	−40.0000	−1.42044
$794$	0	0
$795$	−4.00000	−0.141865
$796$	0	0
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	−7.00000	−0.247642
$800$	0	0
$801$	−7.00000	−0.247333
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	−8.00000	−0.281963
$806$	0	0
$807$	9.00000	0.316815
$808$	0	0
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	0	0
$811$	−21.0000	−0.737410	−0.368705	−	0.929547i	$-0.620199\pi$
$811$	−21.0000	−0.737410	−0.368705	+	0.929547i	$0.620199\pi$
$812$	0	0
$813$	−14.0000	−0.491001
$814$	0	0
$815$	36.0000	1.26102
$816$	0	0
$817$	24.0000	0.839654
$818$	0	0
$819$	5.00000	0.174714
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	3.00000	0.104447
$826$	0	0
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	0	0
$829$	40.0000	1.38926	0.694629	−	0.719368i	$-0.255569\pi$
$829$	40.0000	1.38926	0.694629	+	0.719368i	$0.255569\pi$
$830$	0	0
$831$	5.00000	0.173448
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	30.0000	1.03325
$844$	0	0
$845$	−24.0000	−0.825625
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	32.0000	1.09824
$850$	0	0
$851$	32.0000	1.09695
$852$	0	0
$853$	−40.0000	−1.36957	−0.684787	−	0.728743i	$-0.740105\pi$
$853$	−40.0000	−1.36957	−0.684787	+	0.728743i	$0.740105\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	0	0
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	−10.0000	−0.341196	−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000	−0.341196	−0.170598	+	0.985341i	$0.554570\pi$
$860$	0	0
$861$	10.0000	0.340799
$862$	0	0
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	18.0000	0.610608
$870$	0	0
$871$	15.0000	0.508256
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	7.00000	0.236104
$880$	0	0
$881$	33.0000	1.11180	0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000	1.11180	0.555899	+	0.831250i	$0.312374\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	0	0
$887$	−39.0000	−1.30949	−0.654746	−	0.755849i	$-0.727224\pi$
$887$	−39.0000	−1.30949	−0.654746	+	0.755849i	$0.727224\pi$
$888$	0	0
$889$	14.0000	0.469545
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	42.0000	1.40548
$894$	0	0
$895$	−16.0000	−0.534821
$896$	0	0
$897$	−20.0000	−0.667781
$898$	0	0
$899$	0	0
$900$	0	0
$901$	2.00000	0.0666297
$902$	0	0
$903$	−4.00000	−0.133112
$904$	0	0
$905$	46.0000	1.52909
$906$	0	0
$907$	22.0000	0.730498	0.365249	−	0.930910i	$-0.380984\pi$
$907$	22.0000	0.730498	0.365249	+	0.930910i	$0.380984\pi$
$908$	0	0
$909$	7.00000	0.232175
$910$	0	0
$911$	−49.0000	−1.62344	−0.811721	−	0.584045i	$-0.801469\pi$
$911$	−49.0000	−1.62344	−0.811721	+	0.584045i	$0.801469\pi$
$912$	0	0
$913$	−42.0000	−1.39000
$914$	0	0
$915$	−16.0000	−0.528944
$916$	0	0
$917$	3.00000	0.0990687
$918$	0	0
$919$	3.00000	0.0989609	0.0494804	−	0.998775i	$-0.484243\pi$
$919$	3.00000	0.0989609	0.0494804	+	0.998775i	$0.484243\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	−20.0000	−0.658308
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	0	0
$933$	29.0000	0.949417
$934$	0	0
$935$	6.00000	0.196221
$936$	0	0
$937$	43.0000	1.40475	0.702374	−	0.711808i	$-0.252123\pi$
$937$	43.0000	1.40475	0.702374	+	0.711808i	$0.252123\pi$
$938$	0	0
$939$	27.0000	0.881112
$940$	0	0
$941$	4.00000	0.130396	0.0651981	−	0.997872i	$-0.479232\pi$
$941$	4.00000	0.130396	0.0651981	+	0.997872i	$0.479232\pi$
$942$	0	0
$943$	−40.0000	−1.30258
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	−45.0000	−1.46230	−0.731152	−	0.682215i	$-0.761017\pi$
$947$	−45.0000	−1.46230	−0.731152	+	0.682215i	$0.761017\pi$
$948$	0	0
$949$	20.0000	0.649227
$950$	0	0
$951$	−9.00000	−0.291845
$952$	0	0
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	0	0
$955$	−32.0000	−1.03550
$956$	0	0
$957$	3.00000	0.0969762
$958$	0	0
$959$	18.0000	0.581250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	10.0000	0.322245
$964$	0	0
$965$	−28.0000	−0.901352
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	−11.0000	−0.352644
$974$	0	0
$975$	5.00000	0.160128
$976$	0	0
$977$	−32.0000	−1.02377	−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000	−1.02377	−0.511885	+	0.859054i	$0.671053\pi$
$978$	0	0
$979$	−21.0000	−0.671163
$980$	0	0
$981$	11.0000	0.351203
$982$	0	0
$983$	−20.0000	−0.637901	−0.318950	−	0.947771i	$-0.603330\pi$
$983$	−20.0000	−0.637901	−0.318950	+	0.947771i	$0.603330\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	−7.00000	−0.222812
$988$	0	0
$989$	16.0000	0.508770
$990$	0	0
$991$	25.0000	0.794151	0.397076	−	0.917786i	$-0.370025\pi$
$991$	25.0000	0.794151	0.397076	+	0.917786i	$0.370025\pi$
$992$	0	0
$993$	26.0000	0.825085
$994$	0	0
$995$	50.0000	1.58511
$996$	0	0
$997$	−52.0000	−1.64686	−0.823428	−	0.567420i	$-0.807941\pi$
$997$	−52.0000	−1.64686	−0.823428	+	0.567420i	$0.807941\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	348.2.a.a.1.1	✓	1
3.2	odd	2		1044.2.a.h.1.1		1
4.3	odd	2		1392.2.a.j.1.1		1
5.2	odd	4		8700.2.g.j.349.2		2
5.3	odd	4		8700.2.g.j.349.1		2
5.4	even	2		8700.2.a.p.1.1		1
8.3	odd	2		5568.2.a.n.1.1		1
8.5	even	2		5568.2.a.be.1.1		1
12.11	even	2		4176.2.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
348.2.a.a.1.1	✓	1	1.1	even	1	trivial
1044.2.a.h.1.1		1	3.2	odd	2
1392.2.a.j.1.1		1	4.3	odd	2
4176.2.a.ba.1.1		1	12.11	even	2
5568.2.a.n.1.1		1	8.3	odd	2
5568.2.a.be.1.1		1	8.5	even	2
8700.2.a.p.1.1		1	5.4	even	2
8700.2.g.j.349.1		2	5.3	odd	4
8700.2.g.j.349.2		2	5.2	odd	4

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:	\( N \)	\(=\)	\( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	348.a (trivial)

Introduction

L-functions

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Fields

Representations

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Database

Properties

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