LMFDB - Embedded newform 48.22.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,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, 22, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
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:	$134.149125258$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
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)

$f(q)$

$=$

$q-59049.0 q^{3} +2.64446e7 q^{5} -1.66116e8 q^{7} +3.48678e9 q^{9} +O(q^{10})$

$q-59049.0 q^{3} +2.64446e7 q^{5} -1.66116e8 q^{7} +3.48678e9 q^{9} +1.04879e11 q^{11} +3.35591e11 q^{13} -1.56152e12 q^{15} +1.45961e13 q^{17} -3.56953e12 q^{19} +9.80898e12 q^{21} -2.22369e14 q^{23} +2.22477e14 q^{25} -2.05891e14 q^{27} +2.19411e15 q^{29} +8.72363e15 q^{31} -6.19299e15 q^{33} -4.39286e15 q^{35} +3.74709e16 q^{37} -1.98163e16 q^{39} +8.66167e16 q^{41} -1.31417e17 q^{43} +9.22064e16 q^{45} -3.39041e17 q^{47} -5.30951e17 q^{49} -8.61888e17 q^{51} -1.57149e18 q^{53} +2.77347e18 q^{55} +2.10777e17 q^{57} -5.23298e18 q^{59} -4.78838e18 q^{61} -5.79210e17 q^{63} +8.87456e18 q^{65} +1.54803e19 q^{67} +1.31307e19 q^{69} +1.29309e19 q^{71} -4.42572e19 q^{73} -1.31370e19 q^{75} -1.74220e19 q^{77} +1.48886e19 q^{79} +1.21577e19 q^{81} -3.70851e19 q^{83} +3.85988e20 q^{85} -1.29560e20 q^{87} -1.05572e20 q^{89} -5.57470e19 q^{91} -5.15121e20 q^{93} -9.43946e19 q^{95} +1.38109e21 q^{97} +3.65690e20 q^{99} +O(q^{100})$

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
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	0	0
$5$	2.64446e7	1.21102	0.605510	−	0.795838i	$-0.292969\pi$
$5$	2.64446e7	1.21102	0.605510	+	0.795838i	$0.292969\pi$
$6$	0	0
$7$	−1.66116e8	−0.222270	−0.111135	−	0.993805i	$-0.535449\pi$
$7$	−1.66116e8	−0.222270	−0.111135	+	0.993805i	$0.535449\pi$
$8$	0	0
$9$	3.48678e9	0.333333
$10$	0	0
$11$	1.04879e11	1.21917	0.609585	−	0.792721i	$-0.291336\pi$
$11$	1.04879e11	1.21917	0.609585	+	0.792721i	$0.291336\pi$
$12$	0	0
$13$	3.35591e11	0.675158	0.337579	−	0.941297i	$-0.390392\pi$
$13$	3.35591e11	0.675158	0.337579	+	0.941297i	$0.390392\pi$
$14$	0	0
$15$	−1.56152e12	−0.699182
$16$	0	0
$17$	1.45961e13	1.75600	0.878000	−	0.478661i	$-0.158878\pi$
$17$	1.45961e13	1.75600	0.878000	+	0.478661i	$0.158878\pi$
$18$	0	0
$19$	−3.56953e12	−0.133567	−0.0667834	−	0.997767i	$-0.521274\pi$
$19$	−3.56953e12	−0.133567	−0.0667834	+	0.997767i	$0.521274\pi$
$20$	0	0
$21$	9.80898e12	0.128328
$22$	0	0
$23$	−2.22369e14	−1.11926	−0.559632	−	0.828741i	$-0.689057\pi$
$23$	−2.22369e14	−1.11926	−0.559632	+	0.828741i	$0.689057\pi$
$24$	0	0
$25$	2.22477e14	0.466568
$26$	0	0
$27$	−2.05891e14	−0.192450
$28$	0	0
$29$	2.19411e15	0.968455	0.484227	−	0.874942i	$-0.339101\pi$
$29$	2.19411e15	0.968455	0.484227	+	0.874942i	$0.339101\pi$
$30$	0	0
$31$	8.72363e15	1.91161	0.955805	−	0.294001i	$-0.0949870\pi$
$31$	8.72363e15	1.91161	0.955805	+	0.294001i	$0.0949870\pi$
$32$	0	0
$33$	−6.19299e15	−0.703888
$34$	0	0
$35$	−4.39286e15	−0.269174
$36$	0	0
$37$	3.74709e16	1.28108	0.640540	−	0.767925i	$-0.278711\pi$
$37$	3.74709e16	1.28108	0.640540	+	0.767925i	$0.278711\pi$
$38$	0	0
$39$	−1.98163e16	−0.389803
$40$	0	0
$41$	8.66167e16	1.00779	0.503897	−	0.863764i	$-0.331899\pi$
$41$	8.66167e16	1.00779	0.503897	+	0.863764i	$0.331899\pi$
$42$	0	0
$43$	−1.31417e17	−0.927326	−0.463663	−	0.886012i	$-0.653465\pi$
$43$	−1.31417e17	−0.927326	−0.463663	+	0.886012i	$0.653465\pi$
$44$	0	0
$45$	9.22064e16	0.403673
$46$	0	0
$47$	−3.39041e17	−0.940210	−0.470105	−	0.882610i	$-0.655784\pi$
$47$	−3.39041e17	−0.940210	−0.470105	+	0.882610i	$0.655784\pi$
$48$	0	0
$49$	−5.30951e17	−0.950596
$50$	0	0
$51$	−8.61888e17	−1.01383
$52$	0	0
$53$	−1.57149e18	−1.23429	−0.617144	−	0.786850i	$-0.711710\pi$
$53$	−1.57149e18	−1.23429	−0.617144	+	0.786850i	$0.711710\pi$
$54$	0	0
$55$	2.77347e18	1.47644
$56$	0	0
$57$	2.10777e17	0.0771148
$58$	0	0
$59$	−5.23298e18	−1.33292	−0.666459	−	0.745542i	$-0.732191\pi$
$59$	−5.23298e18	−1.33292	−0.666459	+	0.745542i	$0.732191\pi$
$60$	0	0
$61$	−4.78838e18	−0.859460	−0.429730	−	0.902957i	$-0.641391\pi$
$61$	−4.78838e18	−0.859460	−0.429730	+	0.902957i	$0.641391\pi$
$62$	0	0
$63$	−5.79210e17	−0.0740901
$64$	0	0
$65$	8.87456e18	0.817630
$66$	0	0
$67$	1.54803e19	1.03752	0.518758	−	0.854921i	$-0.326395\pi$
$67$	1.54803e19	1.03752	0.518758	+	0.854921i	$0.326395\pi$
$68$	0	0
$69$	1.31307e19	0.646207
$70$	0	0
$71$	1.29309e19	0.471429	0.235715	−	0.971822i	$-0.424257\pi$
$71$	1.29309e19	0.471429	0.235715	+	0.971822i	$0.424257\pi$
$72$	0	0
$73$	−4.42572e19	−1.20530	−0.602648	−	0.798007i	$-0.705888\pi$
$73$	−4.42572e19	−1.20530	−0.602648	+	0.798007i	$0.705888\pi$
$74$	0	0
$75$	−1.31370e19	−0.269373
$76$	0	0
$77$	−1.74220e19	−0.270985
$78$	0	0
$79$	1.48886e19	0.176917	0.0884583	−	0.996080i	$-0.471806\pi$
$79$	1.48886e19	0.176917	0.0884583	+	0.996080i	$0.471806\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	0	0
$83$	−3.70851e19	−0.262349	−0.131174	−	0.991359i	$-0.541875\pi$
$83$	−3.70851e19	−0.262349	−0.131174	+	0.991359i	$0.541875\pi$
$84$	0	0
$85$	3.85988e20	2.12655
$86$	0	0
$87$	−1.29560e20	−0.559138
$88$	0	0
$89$	−1.05572e20	−0.358884	−0.179442	−	0.983769i	$-0.557429\pi$
$89$	−1.05572e20	−0.358884	−0.179442	+	0.983769i	$0.557429\pi$
$90$	0	0
$91$	−5.57470e19	−0.150068
$92$	0	0
$93$	−5.15121e20	−1.10367
$94$	0	0
$95$	−9.43946e19	−0.161752
$96$	0	0
$97$	1.38109e21	1.90160	0.950801	−	0.309803i	$-0.100263\pi$
$97$	1.38109e21	1.90160	0.950801	+	0.309803i	$0.100263\pi$
$98$	0	0
$99$	3.65690e20	0.406390
$100$	0	0
$101$	−1.49419e21	−1.34596	−0.672979	−	0.739662i	$-0.734986\pi$
$101$	−1.49419e21	−1.34596	−0.672979	+	0.739662i	$0.734986\pi$
$102$	0	0
$103$	1.72821e21	1.26708	0.633542	−	0.773708i	$-0.281600\pi$
$103$	1.72821e21	1.26708	0.633542	+	0.773708i	$0.281600\pi$
$104$	0	0
$105$	2.59394e20	0.155408
$106$	0	0
$107$	3.21208e21	1.57854	0.789272	−	0.614044i	$-0.210458\pi$
$107$	3.21208e21	1.57854	0.789272	+	0.614044i	$0.210458\pi$
$108$	0	0
$109$	−5.26985e20	−0.213216	−0.106608	−	0.994301i	$-0.533999\pi$
$109$	−5.26985e20	−0.213216	−0.106608	+	0.994301i	$0.533999\pi$
$110$	0	0
$111$	−2.21262e21	−0.739631
$112$	0	0
$113$	2.65376e20	0.0735425	0.0367713	−	0.999324i	$-0.488293\pi$
$113$	2.65376e20	0.0735425	0.0367713	+	0.999324i	$0.488293\pi$
$114$	0	0
$115$	−5.88045e21	−1.35545
$116$	0	0
$117$	1.17013e21	0.225053
$118$	0	0
$119$	−2.42465e21	−0.390307
$120$	0	0
$121$	3.59930e21	0.486376
$122$	0	0
$123$	−5.11463e21	−0.581850
$124$	0	0
$125$	−6.72644e21	−0.645996
$126$	0	0
$127$	1.95043e22	1.58559	0.792797	−	0.609485i	$-0.208624\pi$
$127$	1.95043e22	1.58559	0.792797	+	0.609485i	$0.208624\pi$
$128$	0	0
$129$	7.76004e21	0.535392
$130$	0	0
$131$	9.92728e21	0.582749	0.291375	−	0.956609i	$-0.405887\pi$
$131$	9.92728e21	0.582749	0.291375	+	0.956609i	$0.405887\pi$
$132$	0	0
$133$	5.92956e20	0.0296879
$134$	0	0
$135$	−5.44470e21	−0.233061
$136$	0	0
$137$	4.13768e22	1.51772	0.758858	−	0.651256i	$-0.225758\pi$
$137$	4.13768e22	1.51772	0.758858	+	0.651256i	$0.225758\pi$
$138$	0	0
$139$	1.86724e22	0.588228	0.294114	−	0.955770i	$-0.404975\pi$
$139$	1.86724e22	0.588228	0.294114	+	0.955770i	$0.404975\pi$
$140$	0	0
$141$	2.00200e22	0.542831
$142$	0	0
$143$	3.51964e22	0.823133
$144$	0	0
$145$	5.80222e22	1.17282
$146$	0	0
$147$	3.13521e22	0.548827
$148$	0	0
$149$	−4.48377e22	−0.681064	−0.340532	−	0.940233i	$-0.610607\pi$
$149$	−4.48377e22	−0.681064	−0.340532	+	0.940233i	$0.610607\pi$
$150$	0	0
$151$	−5.27798e22	−0.696963	−0.348482	−	0.937316i	$-0.613303\pi$
$151$	−5.27798e22	−0.696963	−0.348482	+	0.937316i	$0.613303\pi$
$152$	0	0
$153$	5.08936e22	0.585333
$154$	0	0
$155$	2.30692e23	2.31500
$156$	0	0
$157$	4.76649e22	0.418074	0.209037	−	0.977908i	$-0.432967\pi$
$157$	4.76649e22	0.418074	0.209037	+	0.977908i	$0.432967\pi$
$158$	0	0
$159$	9.27952e22	0.712616
$160$	0	0
$161$	3.69391e22	0.248779
$162$	0	0
$163$	−4.68583e22	−0.277215	−0.138607	−	0.990347i	$-0.544263\pi$
$163$	−4.68583e22	−0.277215	−0.138607	+	0.990347i	$0.544263\pi$
$164$	0	0
$165$	−1.63771e23	−0.852422
$166$	0	0
$167$	−4.21072e22	−0.193123	−0.0965613	−	0.995327i	$-0.530784\pi$
$167$	−4.21072e22	−0.193123	−0.0965613	+	0.995327i	$0.530784\pi$
$168$	0	0
$169$	−1.34443e23	−0.544161
$170$	0	0
$171$	−1.24462e22	−0.0445222
$172$	0	0
$173$	4.51089e23	1.42816	0.714082	−	0.700062i	$-0.246844\pi$
$173$	4.51089e23	1.42816	0.714082	+	0.700062i	$0.246844\pi$
$174$	0	0
$175$	−3.69570e22	−0.103704
$176$	0	0
$177$	3.09003e23	0.769560
$178$	0	0
$179$	−2.06595e23	−0.457259	−0.228629	−	0.973514i	$-0.573424\pi$
$179$	−2.06595e23	−0.457259	−0.228629	+	0.973514i	$0.573424\pi$
$180$	0	0
$181$	7.17280e23	1.41274	0.706372	−	0.707841i	$-0.250331\pi$
$181$	7.17280e23	1.41274	0.706372	+	0.707841i	$0.250331\pi$
$182$	0	0
$183$	2.82749e23	0.496210
$184$	0	0
$185$	9.90901e23	1.55141
$186$	0	0
$187$	1.53083e24	2.14086
$188$	0	0
$189$	3.42018e22	0.0427760
$190$	0	0
$191$	4.48187e23	0.501890	0.250945	−	0.968001i	$-0.419259\pi$
$191$	4.48187e23	0.501890	0.250945	+	0.968001i	$0.419259\pi$
$192$	0	0
$193$	−7.88064e23	−0.791061	−0.395530	−	0.918453i	$-0.629439\pi$
$193$	−7.88064e23	−0.791061	−0.395530	+	0.918453i	$0.629439\pi$
$194$	0	0
$195$	−5.24034e23	−0.472059
$196$	0	0
$197$	−5.38083e23	−0.435466	−0.217733	−	0.976008i	$-0.569866\pi$
$197$	−5.38083e23	−0.435466	−0.217733	+	0.976008i	$0.569866\pi$
$198$	0	0
$199$	1.82843e24	1.33083	0.665414	−	0.746475i	$-0.268255\pi$
$199$	1.82843e24	1.33083	0.665414	+	0.746475i	$0.268255\pi$
$200$	0	0
$201$	−9.14098e23	−0.599010
$202$	0	0
$203$	−3.64476e23	−0.215259
$204$	0	0
$205$	2.29054e24	1.22046
$206$	0	0
$207$	−7.75354e23	−0.373088
$208$	0	0
$209$	−3.74368e23	−0.162841
$210$	0	0
$211$	2.17230e24	0.854974	0.427487	−	0.904021i	$-0.359399\pi$
$211$	2.17230e24	0.854974	0.427487	+	0.904021i	$0.359399\pi$
$212$	0	0
$213$	−7.63557e23	−0.272180
$214$	0	0
$215$	−3.47526e24	−1.12301
$216$	0	0
$217$	−1.44913e24	−0.424894
$218$	0	0
$219$	2.61334e24	0.695878
$220$	0	0
$221$	4.89834e24	1.18558
$222$	0	0
$223$	−8.40788e24	−1.85134	−0.925668	−	0.378336i	$-0.876496\pi$
$223$	−8.40788e24	−1.85134	−0.925668	+	0.378336i	$0.876496\pi$
$224$	0	0
$225$	7.75730e23	0.155523
$226$	0	0
$227$	6.09701e24	1.11390	0.556949	−	0.830547i	$-0.311972\pi$
$227$	6.09701e24	1.11390	0.556949	+	0.830547i	$0.311972\pi$
$228$	0	0
$229$	−6.53459e23	−0.108879	−0.0544397	−	0.998517i	$-0.517337\pi$
$229$	−6.53459e23	−0.108879	−0.0544397	+	0.998517i	$0.517337\pi$
$230$	0	0
$231$	1.02875e24	0.156454
$232$	0	0
$233$	−1.09090e25	−1.51547	−0.757735	−	0.652562i	$-0.773694\pi$
$233$	−1.09090e25	−1.51547	−0.757735	+	0.652562i	$0.773694\pi$
$234$	0	0
$235$	−8.96579e24	−1.13861
$236$	0	0
$237$	−8.79156e23	−0.102143
$238$	0	0
$239$	−5.96515e21	−0.000634517 0	−0.000317258	−	1.00000i	$-0.500101\pi$
$239$	−5.96515e21	−0.000634517 0	−0.000317258		1.00000i	$0.500101\pi$
$240$	0	0
$241$	8.17929e24	0.797144	0.398572	−	0.917137i	$-0.369506\pi$
$241$	8.17929e24	0.797144	0.398572	+	0.917137i	$0.369506\pi$
$242$	0	0
$243$	−7.17898e23	−0.0641500
$244$	0	0
$245$	−1.40408e25	−1.15119
$246$	0	0
$247$	−1.19790e24	−0.0901787
$248$	0	0
$249$	2.18984e24	0.151467
$250$	0	0
$251$	−1.37352e25	−0.873499	−0.436749	−	0.899583i	$-0.643870\pi$
$251$	−1.37352e25	−0.873499	−0.436749	+	0.899583i	$0.643870\pi$
$252$	0	0
$253$	−2.33218e25	−1.36457
$254$	0	0
$255$	−2.27922e25	−1.22776
$256$	0	0
$257$	4.53006e24	0.224805	0.112403	−	0.993663i	$-0.464145\pi$
$257$	4.53006e24	0.224805	0.112403	+	0.993663i	$0.464145\pi$
$258$	0	0
$259$	−6.22451e24	−0.284746
$260$	0	0
$261$	7.65039e24	0.322818
$262$	0	0
$263$	5.23061e24	0.203712	0.101856	−	0.994799i	$-0.467522\pi$
$263$	5.23061e24	0.203712	0.101856	+	0.994799i	$0.467522\pi$
$264$	0	0
$265$	−4.15575e25	−1.49475
$266$	0	0
$267$	6.23392e24	0.207202
$268$	0	0
$269$	−1.15814e25	−0.355927	−0.177964	−	0.984037i	$-0.556951\pi$
$269$	−1.15814e25	−0.355927	−0.177964	+	0.984037i	$0.556951\pi$
$270$	0	0
$271$	−1.66790e25	−0.474234	−0.237117	−	0.971481i	$-0.576202\pi$
$271$	−1.66790e25	−0.474234	−0.237117	+	0.971481i	$0.576202\pi$
$272$	0	0
$273$	3.29181e24	0.0866416
$274$	0	0
$275$	2.33331e25	0.568826
$276$	0	0
$277$	−4.99177e25	−1.12776	−0.563880	−	0.825857i	$-0.690692\pi$
$277$	−4.99177e25	−1.12776	−0.563880	+	0.825857i	$0.690692\pi$
$278$	0	0
$279$	3.04174e25	0.637203
$280$	0	0
$281$	2.39858e25	0.466162	0.233081	−	0.972457i	$-0.425119\pi$
$281$	2.39858e25	0.466162	0.233081	+	0.972457i	$0.425119\pi$
$282$	0	0
$283$	2.68871e25	0.485050	0.242525	−	0.970145i	$-0.422024\pi$
$283$	2.68871e25	0.485050	0.242525	+	0.970145i	$0.422024\pi$
$284$	0	0
$285$	5.57391e24	0.0933875
$286$	0	0
$287$	−1.43884e25	−0.224003
$288$	0	0
$289$	1.43956e26	2.08354
$290$	0	0
$291$	−8.15521e25	−1.09789
$292$	0	0
$293$	1.29122e25	0.161767	0.0808835	−	0.996724i	$-0.474226\pi$
$293$	1.29122e25	0.161767	0.0808835	+	0.996724i	$0.474226\pi$
$294$	0	0
$295$	−1.38384e26	−1.61419
$296$	0	0
$297$	−2.15936e25	−0.234629
$298$	0	0
$299$	−7.46252e25	−0.755680
$300$	0	0
$301$	2.18304e25	0.206117
$302$	0	0
$303$	8.82304e25	0.777089
$304$	0	0
$305$	−1.26627e26	−1.04082
$306$	0	0
$307$	1.45907e26	1.11976	0.559878	−	0.828575i	$-0.310848\pi$
$307$	1.45907e26	1.11976	0.559878	+	0.828575i	$0.310848\pi$
$308$	0	0
$309$	−1.02049e26	−0.731552
$310$	0	0
$311$	1.00016e25	0.0670015	0.0335008	−	0.999439i	$-0.489334\pi$
$311$	1.00016e25	0.0670015	0.0335008	+	0.999439i	$0.489334\pi$
$312$	0	0
$313$	2.08348e25	0.130489	0.0652445	−	0.997869i	$-0.479217\pi$
$313$	2.08348e25	0.130489	0.0652445	+	0.997869i	$0.479217\pi$
$314$	0	0
$315$	−1.53170e25	−0.0897246
$316$	0	0
$317$	1.30949e26	0.717762	0.358881	−	0.933383i	$-0.383158\pi$
$317$	1.30949e26	0.717762	0.358881	+	0.933383i	$0.383158\pi$
$318$	0	0
$319$	2.30116e26	1.18071
$320$	0	0
$321$	−1.89670e26	−0.911373
$322$	0	0
$323$	−5.21014e25	−0.234543
$324$	0	0
$325$	7.46614e25	0.315007
$326$	0	0
$327$	3.11179e25	0.123100
$328$	0	0
$329$	5.63201e25	0.208981
$330$	0	0
$331$	−5.53748e24	−0.0192805	−0.00964026	−	0.999954i	$-0.503069\pi$
$331$	−5.53748e24	−0.0192805	−0.00964026	+	0.999954i	$0.503069\pi$
$332$	0	0
$333$	1.30653e26	0.427026
$334$	0	0
$335$	4.09370e26	1.25645
$336$	0	0
$337$	−2.35227e26	−0.678223	−0.339112	−	0.940746i	$-0.610126\pi$
$337$	−2.35227e26	−0.678223	−0.339112	+	0.940746i	$0.610126\pi$
$338$	0	0
$339$	−1.56702e25	−0.0424598
$340$	0	0
$341$	9.14923e26	2.33058
$342$	0	0
$343$	1.80983e26	0.433560
$344$	0	0
$345$	3.47235e26	0.782570
$346$	0	0
$347$	1.60491e26	0.340401	0.170200	−	0.985409i	$-0.445559\pi$
$347$	1.60491e26	0.340401	0.170200	+	0.985409i	$0.445559\pi$
$348$	0	0
$349$	3.71912e26	0.742631	0.371315	−	0.928507i	$-0.378907\pi$
$349$	3.71912e26	0.742631	0.371315	+	0.928507i	$0.378907\pi$
$350$	0	0
$351$	−6.90953e25	−0.129934
$352$	0	0
$353$	7.03252e26	1.24588	0.622941	−	0.782269i	$-0.285938\pi$
$353$	7.03252e26	1.24588	0.622941	+	0.782269i	$0.285938\pi$
$354$	0	0
$355$	3.41952e26	0.570910
$356$	0	0
$357$	1.43173e26	0.225344
$358$	0	0
$359$	5.01731e26	0.744696	0.372348	−	0.928093i	$-0.378553\pi$
$359$	5.01731e26	0.744696	0.372348	+	0.928093i	$0.378553\pi$
$360$	0	0
$361$	−7.01468e26	−0.982160
$362$	0	0
$363$	−2.12535e26	−0.280809
$364$	0	0
$365$	−1.17036e27	−1.45964
$366$	0	0
$367$	−5.69112e26	−0.670200	−0.335100	−	0.942183i	$-0.608770\pi$
$367$	−5.69112e26	−0.670200	−0.335100	+	0.942183i	$0.608770\pi$
$368$	0	0
$369$	3.02014e26	0.335931
$370$	0	0
$371$	2.61050e26	0.274346
$372$	0	0
$373$	2.59366e26	0.257615	0.128807	−	0.991670i	$-0.458885\pi$
$373$	2.59366e26	0.257615	0.128807	+	0.991670i	$0.458885\pi$
$374$	0	0
$375$	3.97189e26	0.372966
$376$	0	0
$377$	7.36324e26	0.653860
$378$	0	0
$379$	−1.10384e27	−0.927241	−0.463621	−	0.886034i	$-0.653450\pi$
$379$	−1.10384e27	−0.927241	−0.463621	+	0.886034i	$0.653450\pi$
$380$	0	0
$381$	−1.15171e27	−0.915444
$382$	0	0
$383$	6.37501e26	0.479616	0.239808	−	0.970820i	$-0.422916\pi$
$383$	6.37501e26	0.479616	0.239808	+	0.970820i	$0.422916\pi$
$384$	0	0
$385$	−4.60718e26	−0.328169
$386$	0	0
$387$	−4.58222e26	−0.309109
$388$	0	0
$389$	3.61461e25	0.0230989	0.0115494	−	0.999933i	$-0.496324\pi$
$389$	3.61461e25	0.0230989	0.0115494	+	0.999933i	$0.496324\pi$
$390$	0	0
$391$	−3.24573e27	−1.96543
$392$	0	0
$393$	−5.86196e26	−0.336450
$394$	0	0
$395$	3.93722e26	0.214250
$396$	0	0
$397$	6.60817e26	0.341021	0.170511	−	0.985356i	$-0.445458\pi$
$397$	6.60817e26	0.341021	0.170511	+	0.985356i	$0.445458\pi$
$398$	0	0
$399$	−3.50134e25	−0.0171403
$400$	0	0
$401$	−3.16192e27	−1.46871	−0.734354	−	0.678767i	$-0.762515\pi$
$401$	−3.16192e27	−1.46871	−0.734354	+	0.678767i	$0.762515\pi$
$402$	0	0
$403$	2.92757e27	1.29064
$404$	0	0
$405$	3.21504e26	0.134558
$406$	0	0
$407$	3.92990e27	1.56185
$408$	0	0
$409$	1.26849e27	0.478843	0.239422	−	0.970916i	$-0.423042\pi$
$409$	1.26849e27	0.478843	0.239422	+	0.970916i	$0.423042\pi$
$410$	0	0
$411$	−2.44326e27	−0.876254
$412$	0	0
$413$	8.69282e26	0.296268
$414$	0	0
$415$	−9.80698e26	−0.317710
$416$	0	0
$417$	−1.10259e27	−0.339614
$418$	0	0
$419$	−4.00666e27	−1.17364	−0.586821	−	0.809717i	$-0.699621\pi$
$419$	−4.00666e27	−1.17364	−0.586821	+	0.809717i	$0.699621\pi$
$420$	0	0
$421$	−5.33195e27	−1.48568	−0.742838	−	0.669471i	$-0.766521\pi$
$421$	−5.33195e27	−1.48568	−0.742838	+	0.669471i	$0.766521\pi$
$422$	0	0
$423$	−1.18216e27	−0.313403
$424$	0	0
$425$	3.24731e27	0.819294
$426$	0	0
$427$	7.95427e26	0.191033
$428$	0	0
$429$	−2.07831e27	−0.475236
$430$	0	0
$431$	−2.82207e27	−0.614548	−0.307274	−	0.951621i	$-0.599417\pi$
$431$	−2.82207e27	−0.614548	−0.307274	+	0.951621i	$0.599417\pi$
$432$	0	0
$433$	−1.58069e26	−0.0327886	−0.0163943	−	0.999866i	$-0.505219\pi$
$433$	−1.58069e26	−0.0327886	−0.0163943	+	0.999866i	$0.505219\pi$
$434$	0	0
$435$	−3.42616e27	−0.677127
$436$	0	0
$437$	7.93754e26	0.149496
$438$	0	0
$439$	−8.74620e27	−1.57015	−0.785077	−	0.619398i	$-0.787377\pi$
$439$	−8.74620e27	−1.57015	−0.785077	+	0.619398i	$0.787377\pi$
$440$	0	0
$441$	−1.85131e27	−0.316865
$442$	0	0
$443$	−1.81520e27	−0.296269	−0.148135	−	0.988967i	$-0.547327\pi$
$443$	−1.81520e27	−0.296269	−0.148135	+	0.988967i	$0.547327\pi$
$444$	0	0
$445$	−2.79180e27	−0.434615
$446$	0	0
$447$	2.64762e27	0.393212
$448$	0	0
$449$	1.06970e28	1.51592	0.757958	−	0.652304i	$-0.226197\pi$
$449$	1.06970e28	1.51592	0.757958	+	0.652304i	$0.226197\pi$
$450$	0	0
$451$	9.08426e27	1.22867
$452$	0	0
$453$	3.11659e27	0.402392
$454$	0	0
$455$	−1.47421e27	−0.181735
$456$	0	0
$457$	−1.59372e28	−1.87626	−0.938129	−	0.346285i	$-0.887443\pi$
$457$	−1.59372e28	−1.87626	−0.938129	+	0.346285i	$0.887443\pi$
$458$	0	0
$459$	−3.00522e27	−0.337942
$460$	0	0
$461$	5.33663e27	0.573334	0.286667	−	0.958030i	$-0.407453\pi$
$461$	5.33663e27	0.573334	0.286667	+	0.958030i	$0.407453\pi$
$462$	0	0
$463$	−3.38648e27	−0.347655	−0.173828	−	0.984776i	$-0.555614\pi$
$463$	−3.38648e27	−0.347655	−0.173828	+	0.984776i	$0.555614\pi$
$464$	0	0
$465$	−1.36222e28	−1.33656
$466$	0	0
$467$	1.57577e28	1.47797	0.738986	−	0.673721i	$-0.235305\pi$
$467$	1.57577e28	1.47797	0.738986	+	0.673721i	$0.235305\pi$
$468$	0	0
$469$	−2.57153e27	−0.230609
$470$	0	0
$471$	−2.81457e27	−0.241375
$472$	0	0
$473$	−1.37828e28	−1.13057
$474$	0	0
$475$	−7.94139e26	−0.0623180
$476$	0	0
$477$	−5.47946e27	−0.411429
$478$	0	0
$479$	2.12973e28	1.53039	0.765193	−	0.643801i	$-0.222644\pi$
$479$	2.12973e28	1.53039	0.765193	+	0.643801i	$0.222644\pi$
$480$	0	0
$481$	1.25749e28	0.864931
$482$	0	0
$483$	−2.18121e27	−0.143633
$484$	0	0
$485$	3.65224e28	2.30288
$486$	0	0
$487$	−1.68508e28	−1.01757	−0.508786	−	0.860893i	$-0.669906\pi$
$487$	−1.68508e28	−1.01757	−0.508786	+	0.860893i	$0.669906\pi$
$488$	0	0
$489$	2.76694e27	0.160050
$490$	0	0
$491$	1.57697e28	0.873910	0.436955	−	0.899483i	$-0.356057\pi$
$491$	1.57697e28	0.873910	0.436955	+	0.899483i	$0.356057\pi$
$492$	0	0
$493$	3.20255e28	1.70061
$494$	0	0
$495$	9.67050e27	0.492146
$496$	0	0
$497$	−2.14803e27	−0.104785
$498$	0	0
$499$	−2.57185e28	−1.20279	−0.601395	−	0.798952i	$-0.705388\pi$
$499$	−2.57185e28	−1.20279	−0.601395	+	0.798952i	$0.705388\pi$
$500$	0	0
$501$	2.48639e27	0.111499
$502$	0	0
$503$	−2.83510e28	−1.21928	−0.609642	−	0.792677i	$-0.708687\pi$
$503$	−2.83510e28	−1.21928	−0.609642	+	0.792677i	$0.708687\pi$
$504$	0	0
$505$	−3.95132e28	−1.62998
$506$	0	0
$507$	7.93872e27	0.314172
$508$	0	0
$509$	2.41622e28	0.917485	0.458743	−	0.888569i	$-0.348300\pi$
$509$	2.41622e28	0.917485	0.458743	+	0.888569i	$0.348300\pi$
$510$	0	0
$511$	7.35182e27	0.267902
$512$	0	0
$513$	7.34935e26	0.0257049
$514$	0	0
$515$	4.57017e28	1.53446
$516$	0	0
$517$	−3.55582e28	−1.14628
$518$	0	0
$519$	−2.66363e28	−0.824551
$520$	0	0
$521$	4.35634e28	1.29517	0.647584	−	0.761994i	$-0.275780\pi$
$521$	4.35634e28	1.29517	0.647584	+	0.761994i	$0.275780\pi$
$522$	0	0
$523$	−1.92361e28	−0.549351	−0.274676	−	0.961537i	$-0.588570\pi$
$523$	−1.92361e28	−0.549351	−0.274676	+	0.961537i	$0.588570\pi$
$524$	0	0
$525$	2.18227e27	0.0598737
$526$	0	0
$527$	1.27331e29	3.35679
$528$	0	0
$529$	9.97650e27	0.252751
$530$	0	0
$531$	−1.82463e28	−0.444306
$532$	0	0
$533$	2.90678e28	0.680420
$534$	0	0
$535$	8.49420e28	1.91165
$536$	0	0
$537$	1.21992e28	0.263998
$538$	0	0
$539$	−5.56855e28	−1.15894
$540$	0	0
$541$	7.52925e28	1.50723	0.753617	−	0.657314i	$-0.228307\pi$
$541$	7.52925e28	1.50723	0.753617	+	0.657314i	$0.228307\pi$
$542$	0	0
$543$	−4.23546e28	−0.815648
$544$	0	0
$545$	−1.39359e28	−0.258209
$546$	0	0
$547$	6.21957e28	1.10890	0.554451	−	0.832216i	$-0.312928\pi$
$547$	6.21957e28	1.10890	0.554451	+	0.832216i	$0.312928\pi$
$548$	0	0
$549$	−1.66961e28	−0.286487
$550$	0	0
$551$	−7.83194e27	−0.129353
$552$	0	0
$553$	−2.47323e27	−0.0393233
$554$	0	0
$555$	−5.85117e28	−0.895708
$556$	0	0
$557$	5.50617e27	0.0811652	0.0405826	−	0.999176i	$-0.487079\pi$
$557$	5.50617e27	0.0811652	0.0405826	+	0.999176i	$0.487079\pi$
$558$	0	0
$559$	−4.41024e28	−0.626092
$560$	0	0
$561$	−9.03937e28	−1.23603
$562$	0	0
$563$	1.21599e29	1.60174	0.800868	−	0.598840i	$-0.204372\pi$
$563$	1.21599e29	1.60174	0.800868	+	0.598840i	$0.204372\pi$
$564$	0	0
$565$	7.01776e27	0.0890614
$566$	0	0
$567$	−2.01958e27	−0.0246967
$568$	0	0
$569$	−3.49889e28	−0.412337	−0.206168	−	0.978517i	$-0.566099\pi$
$569$	−3.49889e28	−0.412337	−0.206168	+	0.978517i	$0.566099\pi$
$570$	0	0
$571$	−9.42449e28	−1.07048	−0.535240	−	0.844700i	$-0.679779\pi$
$571$	−9.42449e28	−1.07048	−0.535240	+	0.844700i	$0.679779\pi$
$572$	0	0
$573$	−2.64650e28	−0.289766
$574$	0	0
$575$	−4.94721e28	−0.522213
$576$	0	0
$577$	−1.55441e29	−1.58205	−0.791025	−	0.611784i	$-0.790452\pi$
$577$	−1.55441e29	−1.58205	−0.791025	+	0.611784i	$0.790452\pi$
$578$	0	0
$579$	4.65344e28	0.456719
$580$	0	0
$581$	6.16042e27	0.0583124
$582$	0	0
$583$	−1.64816e29	−1.50481
$584$	0	0
$585$	3.09437e28	0.272543
$586$	0	0
$587$	−1.17789e29	−1.00093	−0.500465	−	0.865757i	$-0.666838\pi$
$587$	−1.17789e29	−1.00093	−0.500465	+	0.865757i	$0.666838\pi$
$588$	0	0
$589$	−3.11392e28	−0.255327
$590$	0	0
$591$	3.17733e28	0.251416
$592$	0	0
$593$	1.16653e29	0.890884	0.445442	−	0.895311i	$-0.353047\pi$
$593$	1.16653e29	0.890884	0.445442	+	0.895311i	$0.353047\pi$
$594$	0	0
$595$	−6.41188e28	−0.472669
$596$	0	0
$597$	−1.07967e29	−0.768353
$598$	0	0
$599$	5.27398e28	0.362374	0.181187	−	0.983449i	$-0.442006\pi$
$599$	5.27398e28	0.362374	0.181187	+	0.983449i	$0.442006\pi$
$600$	0	0
$601$	−5.00756e28	−0.332234	−0.166117	−	0.986106i	$-0.553123\pi$
$601$	−5.00756e28	−0.332234	−0.166117	+	0.986106i	$0.553123\pi$
$602$	0	0
$603$	5.39766e28	0.345839
$604$	0	0
$605$	9.51820e28	0.589011
$606$	0	0
$607$	6.44426e28	0.385205	0.192603	−	0.981277i	$-0.438307\pi$
$607$	6.44426e28	0.385205	0.192603	+	0.981277i	$0.438307\pi$
$608$	0	0
$609$	2.15220e28	0.124280
$610$	0	0
$611$	−1.13779e29	−0.634791
$612$	0	0
$613$	1.46854e29	0.791680	0.395840	−	0.918319i	$-0.370453\pi$
$613$	1.46854e29	0.791680	0.395840	+	0.918319i	$0.370453\pi$
$614$	0	0
$615$	−1.35254e29	−0.704631
$616$	0	0
$617$	−2.87513e29	−1.44765	−0.723823	−	0.689985i	$-0.757617\pi$
$617$	−2.87513e29	−1.44765	−0.723823	+	0.689985i	$0.757617\pi$
$618$	0	0
$619$	−3.83570e29	−1.86678	−0.933388	−	0.358869i	$-0.883163\pi$
$619$	−3.83570e29	−1.86678	−0.933388	+	0.358869i	$0.883163\pi$
$620$	0	0
$621$	4.57839e28	0.215402
$622$	0	0
$623$	1.75372e28	0.0797692
$624$	0	0
$625$	−2.83963e29	−1.24888
$626$	0	0
$627$	2.21060e28	0.0940160
$628$	0	0
$629$	5.46931e29	2.24957
$630$	0	0
$631$	4.12646e29	1.64161	0.820803	−	0.571211i	$-0.193526\pi$
$631$	4.12646e29	1.64161	0.820803	+	0.571211i	$0.193526\pi$
$632$	0	0
$633$	−1.28272e29	−0.493620
$634$	0	0
$635$	5.15784e29	1.92019
$636$	0	0
$637$	−1.78183e29	−0.641803
$638$	0	0
$639$	4.50873e28	0.157143
$640$	0	0
$641$	−4.45897e29	−1.50392	−0.751960	−	0.659208i	$-0.770891\pi$
$641$	−4.45897e29	−1.50392	−0.751960	+	0.659208i	$0.770891\pi$
$642$	0	0
$643$	1.19395e29	0.389736	0.194868	−	0.980829i	$-0.437572\pi$
$643$	1.19395e29	0.389736	0.194868	+	0.980829i	$0.437572\pi$
$644$	0	0
$645$	2.05211e29	0.648370
$646$	0	0
$647$	−5.06908e29	−1.55037	−0.775183	−	0.631737i	$-0.782342\pi$
$647$	−5.06908e29	−1.55037	−0.775183	+	0.631737i	$0.782342\pi$
$648$	0	0
$649$	−5.48829e29	−1.62505
$650$	0	0
$651$	8.55698e28	0.245313
$652$	0	0
$653$	2.19335e29	0.608862	0.304431	−	0.952534i	$-0.401534\pi$
$653$	2.19335e29	0.608862	0.304431	+	0.952534i	$0.401534\pi$
$654$	0	0
$655$	2.62523e29	0.705721
$656$	0	0
$657$	−1.54315e29	−0.401765
$658$	0	0
$659$	−2.49016e29	−0.627957	−0.313979	−	0.949430i	$-0.601662\pi$
$659$	−2.49016e29	−0.627957	−0.313979	+	0.949430i	$0.601662\pi$
$660$	0	0
$661$	3.02171e29	0.738137	0.369069	−	0.929402i	$-0.379677\pi$
$661$	3.02171e29	0.738137	0.369069	+	0.929402i	$0.379677\pi$
$662$	0	0
$663$	−2.89242e29	−0.684494
$664$	0	0
$665$	1.56804e28	0.0359527
$666$	0	0
$667$	−4.87903e29	−1.08396
$668$	0	0
$669$	4.96477e29	1.06887
$670$	0	0
$671$	−5.02200e29	−1.04783
$672$	0	0
$673$	1.15417e29	0.233406	0.116703	−	0.993167i	$-0.462767\pi$
$673$	1.15417e29	0.233406	0.116703	+	0.993167i	$0.462767\pi$
$674$	0	0
$675$	−4.58061e28	−0.0897911
$676$	0	0
$677$	−2.04966e29	−0.389494	−0.194747	−	0.980853i	$-0.562389\pi$
$677$	−2.04966e29	−0.389494	−0.194747	+	0.980853i	$0.562389\pi$
$678$	0	0
$679$	−2.29421e29	−0.422670
$680$	0	0
$681$	−3.60022e29	−0.643109
$682$	0	0
$683$	7.56950e29	1.31114	0.655571	−	0.755134i	$-0.272428\pi$
$683$	7.56950e29	1.31114	0.655571	+	0.755134i	$0.272428\pi$
$684$	0	0
$685$	1.09419e30	1.83798
$686$	0	0
$687$	3.85861e28	0.0628616
$688$	0	0
$689$	−5.27380e29	−0.833339
$690$	0	0
$691$	4.98998e29	0.764855	0.382427	−	0.923986i	$-0.375088\pi$
$691$	4.98998e29	0.764855	0.382427	+	0.923986i	$0.375088\pi$
$692$	0	0
$693$	−6.07468e28	−0.0903285
$694$	0	0
$695$	4.93784e29	0.712356
$696$	0	0
$697$	1.26427e30	1.76968
$698$	0	0
$699$	6.44165e29	0.874957
$700$	0	0
$701$	4.27981e29	0.564137	0.282069	−	0.959394i	$-0.408979\pi$
$701$	4.27981e29	0.564137	0.282069	+	0.959394i	$0.408979\pi$
$702$	0	0
$703$	−1.33753e29	−0.171110
$704$	0	0
$705$	5.29421e29	0.657378
$706$	0	0
$707$	2.48209e29	0.299167
$708$	0	0
$709$	3.31006e29	0.387303	0.193651	−	0.981070i	$-0.437967\pi$
$709$	3.31006e29	0.387303	0.193651	+	0.981070i	$0.437967\pi$
$710$	0	0
$711$	5.19133e28	0.0589722
$712$	0	0
$713$	−1.93987e30	−2.13960
$714$	0	0
$715$	9.30753e29	0.996830
$716$	0	0
$717$	3.52236e26	0.000366338 0
$718$	0	0
$719$	1.36160e30	1.37530	0.687648	−	0.726044i	$-0.258643\pi$
$719$	1.36160e30	1.37530	0.687648	+	0.726044i	$0.258643\pi$
$720$	0	0
$721$	−2.87083e29	−0.281636
$722$	0	0
$723$	−4.82979e29	−0.460231
$724$	0	0
$725$	4.88139e29	0.451850
$726$	0	0
$727$	5.58942e29	0.502638	0.251319	−	0.967904i	$-0.419136\pi$
$727$	5.58942e29	0.502638	0.251319	+	0.967904i	$0.419136\pi$
$728$	0	0
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−1.91818e30	−1.62838
$732$	0	0
$733$	1.20616e30	0.994975	0.497488	−	0.867471i	$-0.334256\pi$
$733$	1.20616e30	0.994975	0.497488	+	0.867471i	$0.334256\pi$
$734$	0	0
$735$	8.29093e29	0.664640
$736$	0	0
$737$	1.62356e30	1.26491
$738$	0	0
$739$	1.99646e30	1.51180	0.755901	−	0.654686i	$-0.227199\pi$
$739$	1.99646e30	1.51180	0.755901	+	0.654686i	$0.227199\pi$
$740$	0	0
$741$	7.07350e28	0.0520647
$742$	0	0
$743$	−1.25058e30	−0.894803	−0.447402	−	0.894333i	$-0.647650\pi$
$743$	−1.25058e30	−0.894803	−0.447402	+	0.894333i	$0.647650\pi$
$744$	0	0
$745$	−1.18571e30	−0.824782
$746$	0	0
$747$	−1.29308e29	−0.0874496
$748$	0	0
$749$	−5.33577e29	−0.350864
$750$	0	0
$751$	−1.17402e30	−0.750685	−0.375343	−	0.926886i	$-0.622475\pi$
$751$	−1.17402e30	−0.750685	−0.375343	+	0.926886i	$0.622475\pi$
$752$	0	0
$753$	8.11053e29	0.504315
$754$	0	0
$755$	−1.39574e30	−0.844036
$756$	0	0
$757$	5.86936e29	0.345210	0.172605	−	0.984991i	$-0.444782\pi$
$757$	5.86936e29	0.345210	0.172605	+	0.984991i	$0.444782\pi$
$758$	0	0
$759$	1.37713e30	0.787837
$760$	0	0
$761$	−1.13068e30	−0.629217	−0.314609	−	0.949221i	$-0.601873\pi$
$761$	−1.13068e30	−0.629217	−0.314609	+	0.949221i	$0.601873\pi$
$762$	0	0
$763$	8.75405e28	0.0473916
$764$	0	0
$765$	1.34586e30	0.708850
$766$	0	0
$767$	−1.75614e30	−0.899930
$768$	0	0
$769$	−3.28732e30	−1.63914	−0.819569	−	0.572981i	$-0.805787\pi$
$769$	−3.28732e30	−1.63914	−0.819569	+	0.572981i	$0.805787\pi$
$770$	0	0
$771$	−2.67496e29	−0.129791
$772$	0	0
$773$	−2.96976e30	−1.40229	−0.701144	−	0.713020i	$-0.747327\pi$
$773$	−2.96976e30	−1.40229	−0.701144	+	0.713020i	$0.747327\pi$
$774$	0	0
$775$	1.94081e30	0.891897
$776$	0	0
$777$	3.67551e29	0.164398
$778$	0	0
$779$	−3.09181e29	−0.134608
$780$	0	0
$781$	1.35618e30	0.574752
$782$	0	0
$783$	−4.51748e29	−0.186379
$784$	0	0
$785$	1.26048e30	0.506295
$786$	0	0
$787$	−7.59025e29	−0.296839	−0.148420	−	0.988924i	$-0.547419\pi$
$787$	−7.59025e29	−0.296839	−0.148420	+	0.988924i	$0.547419\pi$
$788$	0	0
$789$	−3.08862e29	−0.117613
$790$	0	0
$791$	−4.40832e28	−0.0163463
$792$	0	0
$793$	−1.60694e30	−0.580272
$794$	0	0
$795$	2.45393e30	0.862992
$796$	0	0
$797$	4.13590e30	1.41664	0.708318	−	0.705894i	$-0.249454\pi$
$797$	4.13590e30	1.41664	0.708318	+	0.705894i	$0.249454\pi$
$798$	0	0
$799$	−4.94869e30	−1.65101
$800$	0	0
$801$	−3.68107e29	−0.119628
$802$	0	0
$803$	−4.64164e30	−1.46946
$804$	0	0
$805$	9.76837e29	0.301277
$806$	0	0
$807$	6.83868e29	0.205495
$808$	0	0
$809$	−6.22694e30	−1.82312	−0.911560	−	0.411166i	$-0.865122\pi$
$809$	−6.22694e30	−1.82312	−0.911560	+	0.411166i	$0.865122\pi$
$810$	0	0
$811$	−1.69279e30	−0.482930	−0.241465	−	0.970409i	$-0.577628\pi$
$811$	−1.69279e30	−0.482930	−0.241465	+	0.970409i	$0.577628\pi$
$812$	0	0
$813$	9.84878e29	0.273799
$814$	0	0
$815$	−1.23915e30	−0.335713
$816$	0	0
$817$	4.69097e29	0.123860
$818$	0	0
$819$	−1.94378e29	−0.0500226
$820$	0	0
$821$	−8.06422e29	−0.202283	−0.101141	−	0.994872i	$-0.532249\pi$
$821$	−8.06422e29	−0.202283	−0.101141	+	0.994872i	$0.532249\pi$
$822$	0	0
$823$	2.36474e30	0.578210	0.289105	−	0.957297i	$-0.406642\pi$
$823$	2.36474e30	0.578210	0.289105	+	0.957297i	$0.406642\pi$
$824$	0	0
$825$	−1.37780e30	−0.328412
$826$	0	0
$827$	1.05624e30	0.245445	0.122722	−	0.992441i	$-0.460838\pi$
$827$	1.05624e30	0.245445	0.122722	+	0.992441i	$0.460838\pi$
$828$	0	0
$829$	−3.64602e30	−0.826032	−0.413016	−	0.910724i	$-0.635525\pi$
$829$	−3.64602e30	−0.826032	−0.413016	+	0.910724i	$0.635525\pi$
$830$	0	0
$831$	2.94759e30	0.651113
$832$	0	0
$833$	−7.74984e30	−1.66925
$834$	0	0
$835$	−1.11351e30	−0.233875
$836$	0	0
$837$	−1.79612e30	−0.367890
$838$	0	0
$839$	6.80400e29	0.135914	0.0679569	−	0.997688i	$-0.478352\pi$
$839$	6.80400e29	0.135914	0.0679569	+	0.997688i	$0.478352\pi$
$840$	0	0
$841$	−3.18725e29	−0.0620953
$842$	0	0
$843$	−1.41634e30	−0.269139
$844$	0	0
$845$	−3.55528e30	−0.658990
$846$	0	0
$847$	−5.97902e29	−0.108107
$848$	0	0
$849$	−1.58766e30	−0.280044
$850$	0	0
$851$	−8.33237e30	−1.43387
$852$	0	0
$853$	9.57477e30	1.60755	0.803773	−	0.594936i	$-0.202823\pi$
$853$	9.57477e30	1.60755	0.803773	+	0.594936i	$0.202823\pi$
$854$	0	0
$855$	−3.29134e29	−0.0539173
$856$	0	0
$857$	−4.04955e30	−0.647304	−0.323652	−	0.946176i	$-0.604911\pi$
$857$	−4.04955e30	−0.647304	−0.323652	+	0.946176i	$0.604911\pi$
$858$	0	0
$859$	9.44318e30	1.47296	0.736479	−	0.676461i	$-0.236487\pi$
$859$	9.44318e30	1.47296	0.736479	+	0.676461i	$0.236487\pi$
$860$	0	0
$861$	8.49622e29	0.129328
$862$	0	0
$863$	5.03037e30	0.747285	0.373642	−	0.927573i	$-0.378109\pi$
$863$	5.03037e30	0.747285	0.373642	+	0.927573i	$0.378109\pi$
$864$	0	0
$865$	1.19288e31	1.72953
$866$	0	0
$867$	−8.50043e30	−1.20293
$868$	0	0
$869$	1.56150e30	0.215692
$870$	0	0
$871$	5.19506e30	0.700487
$872$	0	0
$873$	4.81557e30	0.633867
$874$	0	0
$875$	1.11737e30	0.143586
$876$	0	0
$877$	1.56535e30	0.196388	0.0981942	−	0.995167i	$-0.468693\pi$
$877$	1.56535e30	0.196388	0.0981942	+	0.995167i	$0.468693\pi$
$878$	0	0
$879$	−7.62452e29	−0.0933963
$880$	0	0
$881$	4.04777e30	0.484138	0.242069	−	0.970259i	$-0.422174\pi$
$881$	4.04777e30	0.484138	0.242069	+	0.970259i	$0.422174\pi$
$882$	0	0
$883$	−6.12581e29	−0.0715445	−0.0357723	−	0.999360i	$-0.511389\pi$
$883$	−6.12581e29	−0.0715445	−0.0357723	+	0.999360i	$0.511389\pi$
$884$	0	0
$885$	8.17143e30	0.931953
$886$	0	0
$887$	−9.40602e30	−1.04763	−0.523815	−	0.851832i	$-0.675492\pi$
$887$	−9.40602e30	−1.04763	−0.523815	+	0.851832i	$0.675492\pi$
$888$	0	0
$889$	−3.23998e30	−0.352431
$890$	0	0
$891$	1.27508e30	0.135463
$892$	0	0
$893$	1.21022e30	0.125581
$894$	0	0
$895$	−5.46330e30	−0.553749
$896$	0	0
$897$	4.40654e30	0.436292
$898$	0	0
$899$	1.91406e31	1.85131
$900$	0	0
$901$	−2.29378e31	−2.16741
$902$	0	0
$903$	−1.28907e30	−0.119002
$904$	0	0
$905$	1.89681e31	1.71086
$906$	0	0
$907$	2.95076e30	0.260051	0.130025	−	0.991511i	$-0.458494\pi$
$907$	2.95076e30	0.260051	0.130025	+	0.991511i	$0.458494\pi$
$908$	0	0
$909$	−5.20992e30	−0.448653
$910$	0	0
$911$	−5.31915e30	−0.447610	−0.223805	−	0.974634i	$-0.571848\pi$
$911$	−5.31915e30	−0.447610	−0.223805	+	0.974634i	$0.571848\pi$
$912$	0	0
$913$	−3.88944e30	−0.319848
$914$	0	0
$915$	7.47718e30	0.600920
$916$	0	0
$917$	−1.64908e30	−0.129528
$918$	0	0
$919$	−3.46199e30	−0.265774	−0.132887	−	0.991131i	$-0.542425\pi$
$919$	−3.46199e30	−0.265774	−0.132887	+	0.991131i	$0.542425\pi$
$920$	0	0
$921$	−8.61567e30	−0.646492
$922$	0	0
$923$	4.33950e30	0.318289
$924$	0	0
$925$	8.33641e30	0.597711
$926$	0	0
$927$	6.02590e30	0.422362
$928$	0	0
$929$	−8.04165e30	−0.551036	−0.275518	−	0.961296i	$-0.588849\pi$
$929$	−8.04165e30	−0.551036	−0.275518	+	0.961296i	$0.588849\pi$
$930$	0	0
$931$	1.89525e30	0.126968
$932$	0	0
$933$	−5.90584e29	−0.0386834
$934$	0	0
$935$	4.04820e31	2.59263
$936$	0	0
$937$	5.36884e30	0.336213	0.168106	−	0.985769i	$-0.446235\pi$
$937$	5.36884e30	0.336213	0.168106	+	0.985769i	$0.446235\pi$
$938$	0	0
$939$	−1.23028e30	−0.0753379
$940$	0	0
$941$	3.46189e30	0.207311	0.103655	−	0.994613i	$-0.466946\pi$
$941$	3.46189e30	0.207311	0.103655	+	0.994613i	$0.466946\pi$
$942$	0	0
$943$	−1.92609e31	−1.12799
$944$	0	0
$945$	9.04451e29	0.0518025
$946$	0	0
$947$	−1.53479e31	−0.859753	−0.429877	−	0.902888i	$-0.641443\pi$
$947$	−1.53479e31	−0.859753	−0.429877	+	0.902888i	$0.641443\pi$
$948$	0	0
$949$	−1.48523e31	−0.813766
$950$	0	0
$951$	−7.73242e30	−0.414400
$952$	0	0
$953$	3.39202e31	1.77821	0.889105	−	0.457703i	$-0.151328\pi$
$953$	3.39202e31	1.77821	0.889105	+	0.457703i	$0.151328\pi$
$954$	0	0
$955$	1.18521e31	0.607799
$956$	0	0
$957$	−1.35881e31	−0.681684
$958$	0	0
$959$	−6.87334e30	−0.337343
$960$	0	0
$961$	5.52762e31	2.65425
$962$	0	0
$963$	1.11998e31	0.526181
$964$	0	0
$965$	−2.08400e31	−0.957990
$966$	0	0
$967$	5.24173e30	0.235775	0.117887	−	0.993027i	$-0.462388\pi$
$967$	5.24173e30	0.235775	0.117887	+	0.993027i	$0.462388\pi$
$968$	0	0
$969$	3.07653e30	0.135414
$970$	0	0
$971$	1.66001e31	0.715004	0.357502	−	0.933912i	$-0.383629\pi$
$971$	1.66001e31	0.715004	0.357502	+	0.933912i	$0.383629\pi$
$972$	0	0
$973$	−3.10179e30	−0.130746
$974$	0	0
$975$	−4.40868e30	−0.181870
$976$	0	0
$977$	4.02046e31	1.62324	0.811621	−	0.584185i	$-0.198586\pi$
$977$	4.02046e31	1.62324	0.811621	+	0.584185i	$0.198586\pi$
$978$	0	0
$979$	−1.10723e31	−0.437540
$980$	0	0
$981$	−1.83748e30	−0.0710720
$982$	0	0
$983$	−1.31702e31	−0.498633	−0.249317	−	0.968422i	$-0.580206\pi$
$983$	−1.31702e31	−0.498633	−0.249317	+	0.968422i	$0.580206\pi$
$984$	0	0
$985$	−1.42294e31	−0.527357
$986$	0	0
$987$	−3.32565e30	−0.120655
$988$	0	0
$989$	2.92231e31	1.03792
$990$	0	0
$991$	4.21282e31	1.46487	0.732436	−	0.680836i	$-0.238384\pi$
$991$	4.21282e31	1.46487	0.732436	+	0.680836i	$0.238384\pi$
$992$	0	0
$993$	3.26983e29	0.0111316
$994$	0	0
$995$	4.83521e31	1.61166
$996$	0	0
$997$	−5.77509e31	−1.88477	−0.942387	−	0.334524i	$-0.891424\pi$
$997$	−5.77509e31	−1.88477	−0.942387	+	0.334524i	$0.891424\pi$
$998$	0	0
$999$	−7.71492e30	−0.246544

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.22.a.c.1.1		1
4.3	odd	2		6.22.a.a.1.1	✓	1
12.11	even	2		18.22.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.22.a.a.1.1	✓	1	4.3	odd	2
18.22.a.d.1.1		1	12.11	even	2
48.22.a.c.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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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