LMFDB - Embedded newform 192.8.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,8,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("192.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	192.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:	$59.9779248930$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{235}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 235 $ x^2 - 235
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 96)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-15.3297$ of defining polynomial
Character	$\chi$	$=$	192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.0000 q^{3} -335.275 q^{5} +710.377 q^{7} +729.000 q^{9} +O(q^{10})$	$q+27.0000 q^{3} -335.275 q^{5} +710.377 q^{7} +729.000 q^{9} +2891.65 q^{11} -10131.6 q^{13} -9052.43 q^{15} -14241.5 q^{17} +33189.3 q^{19} +19180.2 q^{21} +75372.9 q^{23} +34284.6 q^{25} +19683.0 q^{27} -150880. q^{29} -55120.5 q^{31} +78074.6 q^{33} -238172. q^{35} -510853. q^{37} -273552. q^{39} +604684. q^{41} -479815. q^{43} -244416. q^{45} -159323. q^{47} -318908. q^{49} -384520. q^{51} -319003. q^{53} -969500. q^{55} +896110. q^{57} +1.81794e6 q^{59} -3.00891e6 q^{61} +517865. q^{63} +3.39686e6 q^{65} -3.12562e6 q^{67} +2.03507e6 q^{69} -4.15152e6 q^{71} +1.69454e6 q^{73} +925683. q^{75} +2.05416e6 q^{77} +4.26998e6 q^{79} +531441. q^{81} -7.78317e6 q^{83} +4.77482e6 q^{85} -4.07375e6 q^{87} -9.58244e6 q^{89} -7.19723e6 q^{91} -1.48825e6 q^{93} -1.11275e7 q^{95} -1.01226e7 q^{97} +2.10801e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 54 q^{3} - 180 q^{5} - 1032 q^{7} + 1458 q^{9}+O(q^{10}) $ 2 * q + 54 * q^3 - 180 * q^5 - 1032 * q^7 + 1458 * q^9	$ 2 q + 54 q^{3} - 180 q^{5} - 1032 q^{7} + 1458 q^{9} + 2840 q^{11} + 340 q^{13} - 4860 q^{15} + 9780 q^{17} + 32040 q^{19} - 27864 q^{21} - 11136 q^{23} - 19730 q^{25} + 39366 q^{27} - 304212 q^{29} + 77640 q^{31} + 76680 q^{33} - 508720 q^{35} - 1015820 q^{37} + 9180 q^{39} + 704100 q^{41} - 395496 q^{43} - 131220 q^{45} - 1157488 q^{47} + 1893426 q^{49} + 264060 q^{51} - 1568580 q^{53} - 977520 q^{55} + 865080 q^{57} + 139240 q^{59} - 2603580 q^{61} - 752328 q^{63} + 5022840 q^{65} - 5289768 q^{67} - 300672 q^{69} - 5721760 q^{71} - 1190700 q^{73} - 532710 q^{75} + 2144160 q^{77} + 398280 q^{79} + 1062882 q^{81} - 6986616 q^{83} + 8504760 q^{85} - 8213724 q^{87} - 8166732 q^{89} - 25442640 q^{91} + 2096280 q^{93} - 11306000 q^{95} - 10361500 q^{97} + 2070360 q^{99}+O(q^{100}) $ 2 * q + 54 * q^3 - 180 * q^5 - 1032 * q^7 + 1458 * q^9 + 2840 * q^11 + 340 * q^13 - 4860 * q^15 + 9780 * q^17 + 32040 * q^19 - 27864 * q^21 - 11136 * q^23 - 19730 * q^25 + 39366 * q^27 - 304212 * q^29 + 77640 * q^31 + 76680 * q^33 - 508720 * q^35 - 1015820 * q^37 + 9180 * q^39 + 704100 * q^41 - 395496 * q^43 - 131220 * q^45 - 1157488 * q^47 + 1893426 * q^49 + 264060 * q^51 - 1568580 * q^53 - 977520 * q^55 + 865080 * q^57 + 139240 * q^59 - 2603580 * q^61 - 752328 * q^63 + 5022840 * q^65 - 5289768 * q^67 - 300672 * q^69 - 5721760 * q^71 - 1190700 * q^73 - 532710 * q^75 + 2144160 * q^77 + 398280 * q^79 + 1062882 * q^81 - 6986616 * q^83 + 8504760 * q^85 - 8213724 * q^87 - 8166732 * q^89 - 25442640 * q^91 + 2096280 * q^93 - 11306000 * q^95 - 10361500 * q^97 + 2070360 * q^99

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
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	0	0
$5$	−335.275	−1.19952	−0.599759	−	0.800181i	$-0.704737\pi$
$5$	−335.275	−1.19952	−0.599759	+	0.800181i	$0.704737\pi$
$6$	0	0
$7$	710.377	0.782791	0.391395	−	0.920223i	$-0.371993\pi$
$7$	710.377	0.782791	0.391395	+	0.920223i	$0.371993\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	2891.65	0.655046	0.327523	−	0.944843i	$-0.393786\pi$
$11$	2891.65	0.655046	0.327523	+	0.944843i	$0.393786\pi$
$12$	0	0
$13$	−10131.6	−1.27901	−0.639506	−	0.768786i	$-0.720861\pi$
$13$	−10131.6	−1.27901	−0.639506	+	0.768786i	$0.720861\pi$
$14$	0	0
$15$	−9052.43	−0.692542
$16$	0	0
$17$	−14241.5	−0.703046	−0.351523	−	0.936179i	$-0.614336\pi$
$17$	−14241.5	−0.703046	−0.351523	+	0.936179i	$0.614336\pi$
$18$	0	0
$19$	33189.3	1.11010	0.555048	−	0.831819i	$-0.312700\pi$
$19$	33189.3	1.11010	0.555048	+	0.831819i	$0.312700\pi$
$20$	0	0
$21$	19180.2	0.451944
$22$	0	0
$23$	75372.9	1.29172	0.645859	−	0.763457i	$-0.276499\pi$
$23$	75372.9	1.29172	0.645859	+	0.763457i	$0.276499\pi$
$24$	0	0
$25$	34284.6	0.438842
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−150880.	−1.14878	−0.574391	−	0.818581i	$-0.694761\pi$
$29$	−150880.	−1.14878	−0.574391	+	0.818581i	$0.694761\pi$
$30$	0	0
$31$	−55120.5	−0.332313	−0.166156	−	0.986099i	$-0.553136\pi$
$31$	−55120.5	−0.332313	−0.166156	+	0.986099i	$0.553136\pi$
$32$	0	0
$33$	78074.6	0.378191
$34$	0	0
$35$	−238172.	−0.938971
$36$	0	0
$37$	−510853.	−1.65802	−0.829011	−	0.559233i	$-0.811096\pi$
$37$	−510853.	−1.65802	−0.829011	+	0.559233i	$0.811096\pi$
$38$	0	0
$39$	−273552.	−0.738438
$40$	0	0
$41$	604684.	1.37020	0.685101	−	0.728448i	$-0.259758\pi$
$41$	604684.	1.37020	0.685101	+	0.728448i	$0.259758\pi$
$42$	0	0
$43$	−479815.	−0.920310	−0.460155	−	0.887839i	$-0.652206\pi$
$43$	−479815.	−0.920310	−0.460155	+	0.887839i	$0.652206\pi$
$44$	0	0
$45$	−244416.	−0.399839
$46$	0	0
$47$	−159323.	−0.223839	−0.111920	−	0.993717i	$-0.535700\pi$
$47$	−159323.	−0.223839	−0.111920	+	0.993717i	$0.535700\pi$
$48$	0	0
$49$	−318908.	−0.387239
$50$	0	0
$51$	−384520.	−0.405904
$52$	0	0
$53$	−319003.	−0.294326	−0.147163	−	0.989112i	$-0.547014\pi$
$53$	−319003.	−0.294326	−0.147163	+	0.989112i	$0.547014\pi$
$54$	0	0
$55$	−969500.	−0.785739
$56$	0	0
$57$	896110.	0.640914
$58$	0	0
$59$	1.81794e6	1.15239	0.576193	−	0.817313i	$-0.304538\pi$
$59$	1.81794e6	1.15239	0.576193	+	0.817313i	$0.304538\pi$
$60$	0	0
$61$	−3.00891e6	−1.69728	−0.848642	−	0.528968i	$-0.822579\pi$
$61$	−3.00891e6	−1.69728	−0.848642	+	0.528968i	$0.822579\pi$
$62$	0	0
$63$	517865.	0.260930
$64$	0	0
$65$	3.39686e6	1.53420
$66$	0	0
$67$	−3.12562e6	−1.26962	−0.634812	−	0.772667i	$-0.718922\pi$
$67$	−3.12562e6	−1.26962	−0.634812	+	0.772667i	$0.718922\pi$
$68$	0	0
$69$	2.03507e6	0.745774
$70$	0	0
$71$	−4.15152e6	−1.37658	−0.688292	−	0.725434i	$-0.741639\pi$
$71$	−4.15152e6	−1.37658	−0.688292	+	0.725434i	$0.741639\pi$
$72$	0	0
$73$	1.69454e6	0.509826	0.254913	−	0.966964i	$-0.417953\pi$
$73$	1.69454e6	0.509826	0.254913	+	0.966964i	$0.417953\pi$
$74$	0	0
$75$	925683.	0.253366
$76$	0	0
$77$	2.05416e6	0.512764
$78$	0	0
$79$	4.26998e6	0.974385	0.487192	−	0.873295i	$-0.338021\pi$
$79$	4.26998e6	0.974385	0.487192	+	0.873295i	$0.338021\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−7.78317e6	−1.49411	−0.747056	−	0.664761i	$-0.768534\pi$
$83$	−7.78317e6	−1.49411	−0.747056	+	0.664761i	$0.768534\pi$
$84$	0	0
$85$	4.77482e6	0.843316
$86$	0	0
$87$	−4.07375e6	−0.663250
$88$	0	0
$89$	−9.58244e6	−1.44082	−0.720412	−	0.693546i	$-0.756047\pi$
$89$	−9.58244e6	−1.44082	−0.720412	+	0.693546i	$0.756047\pi$
$90$	0	0
$91$	−7.19723e6	−1.00120
$92$	0	0
$93$	−1.48825e6	−0.191861
$94$	0	0
$95$	−1.11275e7	−1.33158
$96$	0	0
$97$	−1.01226e7	−1.12613	−0.563066	−	0.826412i	$-0.690378\pi$
$97$	−1.01226e7	−1.12613	−0.563066	+	0.826412i	$0.690378\pi$
$98$	0	0
$99$	2.10801e6	0.218349
$100$	0	0
$101$	4.00358e6	0.386655	0.193327	−	0.981134i	$-0.438072\pi$
$101$	4.00358e6	0.386655	0.193327	+	0.981134i	$0.438072\pi$
$102$	0	0
$103$	−1.25940e7	−1.13562	−0.567809	−	0.823160i	$-0.692209\pi$
$103$	−1.25940e7	−1.13562	−0.567809	+	0.823160i	$0.692209\pi$
$104$	0	0
$105$	−6.43064e6	−0.542115
$106$	0	0
$107$	−1.44326e7	−1.13894	−0.569470	−	0.822012i	$-0.692851\pi$
$107$	−1.44326e7	−1.13894	−0.569470	+	0.822012i	$0.692851\pi$
$108$	0	0
$109$	6.17092e6	0.456412	0.228206	−	0.973613i	$-0.426714\pi$
$109$	6.17092e6	0.456412	0.228206	+	0.973613i	$0.426714\pi$
$110$	0	0
$111$	−1.37930e7	−0.957259
$112$	0	0
$113$	4.83221e6	0.315044	0.157522	−	0.987515i	$-0.449649\pi$
$113$	4.83221e6	0.315044	0.157522	+	0.987515i	$0.449649\pi$
$114$	0	0
$115$	−2.52707e7	−1.54944
$116$	0	0
$117$	−7.38591e6	−0.426338
$118$	0	0
$119$	−1.01168e7	−0.550338
$120$	0	0
$121$	−1.11255e7	−0.570915
$122$	0	0
$123$	1.63265e7	0.791087
$124$	0	0
$125$	1.46986e7	0.673118
$126$	0	0
$127$	2.67682e7	1.15959	0.579796	−	0.814762i	$-0.303132\pi$
$127$	2.67682e7	1.15959	0.579796	+	0.814762i	$0.303132\pi$
$128$	0	0
$129$	−1.29550e7	−0.531341
$130$	0	0
$131$	3.86901e7	1.50366	0.751832	−	0.659355i	$-0.229171\pi$
$131$	3.86901e7	1.50366	0.751832	+	0.659355i	$0.229171\pi$
$132$	0	0
$133$	2.35769e7	0.868972
$134$	0	0
$135$	−6.59922e6	−0.230847
$136$	0	0
$137$	−1.95562e7	−0.649773	−0.324887	−	0.945753i	$-0.605326\pi$
$137$	−1.95562e7	−0.649773	−0.324887	+	0.945753i	$0.605326\pi$
$138$	0	0
$139$	−4.88761e7	−1.54363	−0.771817	−	0.635844i	$-0.780652\pi$
$139$	−4.88761e7	−1.54363	−0.771817	+	0.635844i	$0.780652\pi$
$140$	0	0
$141$	−4.30172e6	−0.129234
$142$	0	0
$143$	−2.92970e7	−0.837812
$144$	0	0
$145$	5.05862e7	1.37798
$146$	0	0
$147$	−8.61051e6	−0.223572
$148$	0	0
$149$	−1.88273e7	−0.466270	−0.233135	−	0.972444i	$-0.574898\pi$
$149$	−1.88273e7	−0.466270	−0.233135	+	0.972444i	$0.574898\pi$
$150$	0	0
$151$	5.29908e7	1.25251	0.626255	−	0.779618i	$-0.284587\pi$
$151$	5.29908e7	1.25251	0.626255	+	0.779618i	$0.284587\pi$
$152$	0	0
$153$	−1.03820e7	−0.234349
$154$	0	0
$155$	1.84805e7	0.398615
$156$	0	0
$157$	−6.61106e7	−1.36340	−0.681699	−	0.731633i	$-0.738759\pi$
$157$	−6.61106e7	−1.36340	−0.681699	+	0.731633i	$0.738759\pi$
$158$	0	0
$159$	−8.61307e6	−0.169929
$160$	0	0
$161$	5.35431e7	1.01114
$162$	0	0
$163$	−4.13135e7	−0.747197	−0.373599	−	0.927590i	$-0.621876\pi$
$163$	−4.13135e7	−0.747197	−0.373599	+	0.927590i	$0.621876\pi$
$164$	0	0
$165$	−2.61765e7	−0.453647
$166$	0	0
$167$	203790.	0.00338590	0.00169295	−	0.999999i	$-0.499461\pi$
$167$	203790.	0.00338590	0.00169295	+	0.999999i	$0.499461\pi$
$168$	0	0
$169$	3.99001e7	0.635873
$170$	0	0
$171$	2.41950e7	0.370032
$172$	0	0
$173$	−2.42803e7	−0.356528	−0.178264	−	0.983983i	$-0.557048\pi$
$173$	−2.42803e7	−0.356528	−0.178264	+	0.983983i	$0.557048\pi$
$174$	0	0
$175$	2.43550e7	0.343522
$176$	0	0
$177$	4.90845e7	0.665331
$178$	0	0
$179$	1.03121e8	1.34388	0.671941	−	0.740604i	$-0.265461\pi$
$179$	1.03121e8	1.34388	0.671941	+	0.740604i	$0.265461\pi$
$180$	0	0
$181$	6.74847e6	0.0845922	0.0422961	−	0.999105i	$-0.486533\pi$
$181$	6.74847e6	0.0845922	0.0422961	+	0.999105i	$0.486533\pi$
$182$	0	0
$183$	−8.12405e7	−0.979927
$184$	0	0
$185$	1.71277e8	1.98883
$186$	0	0
$187$	−4.11814e7	−0.460527
$188$	0	0
$189$	1.39823e7	0.150648
$190$	0	0
$191$	−1.92545e7	−0.199947	−0.0999735	−	0.994990i	$-0.531876\pi$
$191$	−1.92545e7	−0.199947	−0.0999735	+	0.994990i	$0.531876\pi$
$192$	0	0
$193$	−1.68684e8	−1.68897	−0.844486	−	0.535577i	$-0.820094\pi$
$193$	−1.68684e8	−1.68897	−0.844486	+	0.535577i	$0.820094\pi$
$194$	0	0
$195$	9.17153e7	0.885770
$196$	0	0
$197$	1.57329e8	1.46615	0.733073	−	0.680149i	$-0.238085\pi$
$197$	1.57329e8	1.46615	0.733073	+	0.680149i	$0.238085\pi$
$198$	0	0
$199$	1.33321e8	1.19926	0.599630	−	0.800277i	$-0.295314\pi$
$199$	1.33321e8	1.19926	0.599630	+	0.800277i	$0.295314\pi$
$200$	0	0
$201$	−8.43918e7	−0.733017
$202$	0	0
$203$	−1.07181e8	−0.899256
$204$	0	0
$205$	−2.02736e8	−1.64358
$206$	0	0
$207$	5.49468e7	0.430573
$208$	0	0
$209$	9.59718e7	0.727163
$210$	0	0
$211$	2.66491e8	1.95296	0.976480	−	0.215609i	$-0.0691736\pi$
$211$	2.66491e8	1.95296	0.976480	+	0.215609i	$0.0691736\pi$
$212$	0	0
$213$	−1.12091e8	−0.794771
$214$	0	0
$215$	1.60870e8	1.10393
$216$	0	0
$217$	−3.91563e7	−0.260131
$218$	0	0
$219$	4.57526e7	0.294348
$220$	0	0
$221$	1.44288e8	0.899205
$222$	0	0
$223$	2.59409e8	1.56645	0.783226	−	0.621737i	$-0.213573\pi$
$223$	2.59409e8	1.56645	0.783226	+	0.621737i	$0.213573\pi$
$224$	0	0
$225$	2.49934e7	0.146281
$226$	0	0
$227$	1.14021e8	0.646987	0.323494	−	0.946230i	$-0.395143\pi$
$227$	1.14021e8	0.646987	0.323494	+	0.946230i	$0.395143\pi$
$228$	0	0
$229$	−9.63317e7	−0.530084	−0.265042	−	0.964237i	$-0.585386\pi$
$229$	−9.63317e7	−0.530084	−0.265042	+	0.964237i	$0.585386\pi$
$230$	0	0
$231$	5.54624e7	0.296044
$232$	0	0
$233$	−1.31106e8	−0.679009	−0.339504	−	0.940604i	$-0.610259\pi$
$233$	−1.31106e8	−0.679009	−0.339504	+	0.940604i	$0.610259\pi$
$234$	0	0
$235$	5.34171e7	0.268499
$236$	0	0
$237$	1.15289e8	0.562561
$238$	0	0
$239$	1.92716e8	0.913116	0.456558	−	0.889694i	$-0.349082\pi$
$239$	1.92716e8	0.913116	0.456558	+	0.889694i	$0.349082\pi$
$240$	0	0
$241$	−3.36481e8	−1.54846	−0.774231	−	0.632904i	$-0.781863\pi$
$241$	−3.36481e8	−1.54846	−0.774231	+	0.632904i	$0.781863\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	1.06922e8	0.464500
$246$	0	0
$247$	−3.36259e8	−1.41983
$248$	0	0
$249$	−2.10146e8	−0.862626
$250$	0	0
$251$	2.10771e8	0.841306	0.420653	−	0.907222i	$-0.361801\pi$
$251$	2.10771e8	0.841306	0.420653	+	0.907222i	$0.361801\pi$
$252$	0	0
$253$	2.17952e8	0.846134
$254$	0	0
$255$	1.28920e8	0.486889
$256$	0	0
$257$	−1.23129e7	−0.0452476	−0.0226238	−	0.999744i	$-0.507202\pi$
$257$	−1.23129e7	−0.0452476	−0.0226238	+	0.999744i	$0.507202\pi$
$258$	0	0
$259$	−3.62898e8	−1.29788
$260$	0	0
$261$	−1.09991e8	−0.382927
$262$	0	0
$263$	−4.33617e8	−1.46981	−0.734905	−	0.678170i	$-0.762773\pi$
$263$	−4.33617e8	−1.46981	−0.734905	+	0.678170i	$0.762773\pi$
$264$	0	0
$265$	1.06954e8	0.353049
$266$	0	0
$267$	−2.58726e8	−0.831860
$268$	0	0
$269$	5.84288e8	1.83018	0.915090	−	0.403250i	$-0.132119\pi$
$269$	5.84288e8	1.83018	0.915090	+	0.403250i	$0.132119\pi$
$270$	0	0
$271$	2.27111e7	0.0693180	0.0346590	−	0.999399i	$-0.488965\pi$
$271$	2.27111e7	0.0693180	0.0346590	+	0.999399i	$0.488965\pi$
$272$	0	0
$273$	−1.94325e8	−0.578043
$274$	0	0
$275$	9.91390e7	0.287462
$276$	0	0
$277$	2.89498e8	0.818400	0.409200	−	0.912445i	$-0.365808\pi$
$277$	2.89498e8	0.818400	0.409200	+	0.912445i	$0.365808\pi$
$278$	0	0
$279$	−4.01828e7	−0.110771
$280$	0	0
$281$	3.03041e8	0.814758	0.407379	−	0.913259i	$-0.366443\pi$
$281$	3.03041e8	0.814758	0.407379	+	0.913259i	$0.366443\pi$
$282$	0	0
$283$	5.31646e8	1.39434	0.697172	−	0.716903i	$-0.254441\pi$
$283$	5.31646e8	1.39434	0.697172	+	0.716903i	$0.254441\pi$
$284$	0	0
$285$	−3.00444e8	−0.768787
$286$	0	0
$287$	4.29553e8	1.07258
$288$	0	0
$289$	−2.07519e8	−0.505726
$290$	0	0
$291$	−2.73309e8	−0.650172
$292$	0	0
$293$	−1.69254e8	−0.393099	−0.196550	−	0.980494i	$-0.562974\pi$
$293$	−1.69254e8	−0.393099	−0.196550	+	0.980494i	$0.562974\pi$
$294$	0	0
$295$	−6.09511e8	−1.38231
$296$	0	0
$297$	5.69164e7	0.126064
$298$	0	0
$299$	−7.63645e8	−1.65212
$300$	0	0
$301$	−3.40849e8	−0.720410
$302$	0	0
$303$	1.08097e8	0.223235
$304$	0	0
$305$	1.00881e9	2.03592
$306$	0	0
$307$	7.89572e8	1.55743	0.778713	−	0.627380i	$-0.215873\pi$
$307$	7.89572e8	1.55743	0.778713	+	0.627380i	$0.215873\pi$
$308$	0	0
$309$	−3.40037e8	−0.655650
$310$	0	0
$311$	−1.66047e8	−0.313018	−0.156509	−	0.987677i	$-0.550024\pi$
$311$	−1.66047e8	−0.313018	−0.156509	+	0.987677i	$0.550024\pi$
$312$	0	0
$313$	4.18218e8	0.770899	0.385450	−	0.922729i	$-0.374046\pi$
$313$	4.18218e8	0.770899	0.385450	+	0.922729i	$0.374046\pi$
$314$	0	0
$315$	−1.73627e8	−0.312990
$316$	0	0
$317$	5.75019e8	1.01385	0.506926	−	0.861990i	$-0.330782\pi$
$317$	5.75019e8	1.01385	0.506926	+	0.861990i	$0.330782\pi$
$318$	0	0
$319$	−4.36291e8	−0.752505
$320$	0	0
$321$	−3.89679e8	−0.657567
$322$	0	0
$323$	−4.72664e8	−0.780448
$324$	0	0
$325$	−3.47356e8	−0.561285
$326$	0	0
$327$	1.66615e8	0.263510
$328$	0	0
$329$	−1.13179e8	−0.175219
$330$	0	0
$331$	−1.14042e9	−1.72850	−0.864249	−	0.503065i	$-0.832206\pi$
$331$	−1.14042e9	−1.72850	−0.864249	+	0.503065i	$0.832206\pi$
$332$	0	0
$333$	−3.72412e8	−0.552674
$334$	0	0
$335$	1.04794e9	1.52294
$336$	0	0
$337$	1.16512e8	0.165832	0.0829159	−	0.996557i	$-0.473577\pi$
$337$	1.16512e8	0.165832	0.0829159	+	0.996557i	$0.473577\pi$
$338$	0	0
$339$	1.30470e8	0.181891
$340$	0	0
$341$	−1.59389e8	−0.217680
$342$	0	0
$343$	−8.11571e8	−1.08592
$344$	0	0
$345$	−6.82308e8	−0.894568
$346$	0	0
$347$	−6.08010e8	−0.781192	−0.390596	−	0.920562i	$-0.627731\pi$
$347$	−6.08010e8	−0.781192	−0.390596	+	0.920562i	$0.627731\pi$
$348$	0	0
$349$	−8.63094e8	−1.08685	−0.543424	−	0.839458i	$-0.682872\pi$
$349$	−8.63094e8	−1.08685	−0.543424	+	0.839458i	$0.682872\pi$
$350$	0	0
$351$	−1.99420e8	−0.246146
$352$	0	0
$353$	6.33693e8	0.766774	0.383387	−	0.923588i	$-0.374757\pi$
$353$	6.33693e8	0.766774	0.383387	+	0.923588i	$0.374757\pi$
$354$	0	0
$355$	1.39190e9	1.65124
$356$	0	0
$357$	−2.73154e8	−0.317738
$358$	0	0
$359$	−9.24215e8	−1.05425	−0.527123	−	0.849789i	$-0.676729\pi$
$359$	−9.24215e8	−1.05425	−0.527123	+	0.849789i	$0.676729\pi$
$360$	0	0
$361$	2.07656e8	0.232311
$362$	0	0
$363$	−3.00389e8	−0.329618
$364$	0	0
$365$	−5.68138e8	−0.611545
$366$	0	0
$367$	−1.07990e9	−1.14039	−0.570193	−	0.821511i	$-0.693132\pi$
$367$	−1.07990e9	−1.14039	−0.570193	+	0.821511i	$0.693132\pi$
$368$	0	0
$369$	4.40814e8	0.456734
$370$	0	0
$371$	−2.26612e8	−0.230396
$372$	0	0
$373$	−3.27025e8	−0.326287	−0.163143	−	0.986602i	$-0.552163\pi$
$373$	−3.27025e8	−0.326287	−0.163143	+	0.986602i	$0.552163\pi$
$374$	0	0
$375$	3.96863e8	0.388625
$376$	0	0
$377$	1.52865e9	1.46931
$378$	0	0
$379$	−9.75385e8	−0.920320	−0.460160	−	0.887836i	$-0.652208\pi$
$379$	−9.75385e8	−0.920320	−0.460160	+	0.887836i	$0.652208\pi$
$380$	0	0
$381$	7.22740e8	0.669491
$382$	0	0
$383$	−9.77181e8	−0.888750	−0.444375	−	0.895841i	$-0.646574\pi$
$383$	−9.77181e8	−0.888750	−0.444375	+	0.895841i	$0.646574\pi$
$384$	0	0
$385$	−6.88710e8	−0.615069
$386$	0	0
$387$	−3.49785e8	−0.306770
$388$	0	0
$389$	1.13465e9	0.977327	0.488663	−	0.872472i	$-0.337485\pi$
$389$	1.13465e9	0.977327	0.488663	+	0.872472i	$0.337485\pi$
$390$	0	0
$391$	−1.07342e9	−0.908137
$392$	0	0
$393$	1.04463e9	0.868141
$394$	0	0
$395$	−1.43162e9	−1.16879
$396$	0	0
$397$	−9.26214e8	−0.742924	−0.371462	−	0.928448i	$-0.621143\pi$
$397$	−9.26214e8	−0.742924	−0.371462	+	0.928448i	$0.621143\pi$
$398$	0	0
$399$	6.36576e8	0.501701
$400$	0	0
$401$	−1.30250e9	−1.00872	−0.504362	−	0.863492i	$-0.668272\pi$
$401$	−1.30250e9	−1.00872	−0.504362	+	0.863492i	$0.668272\pi$
$402$	0	0
$403$	5.58457e8	0.425032
$404$	0	0
$405$	−1.78179e8	−0.133280
$406$	0	0
$407$	−1.47721e9	−1.08608
$408$	0	0
$409$	−2.43668e9	−1.76103	−0.880515	−	0.474018i	$-0.842803\pi$
$409$	−2.43668e9	−1.76103	−0.880515	+	0.474018i	$0.842803\pi$
$410$	0	0
$411$	−5.28016e8	−0.375147
$412$	0	0
$413$	1.29142e9	0.902077
$414$	0	0
$415$	2.60951e9	1.79221
$416$	0	0
$417$	−1.31965e9	−0.891218
$418$	0	0
$419$	1.61835e9	1.07479	0.537393	−	0.843332i	$-0.319409\pi$
$419$	1.61835e9	1.07479	0.537393	+	0.843332i	$0.319409\pi$
$420$	0	0
$421$	2.08462e8	0.136157	0.0680786	−	0.997680i	$-0.478313\pi$
$421$	2.08462e8	0.136157	0.0680786	+	0.997680i	$0.478313\pi$
$422$	0	0
$423$	−1.16147e8	−0.0746131
$424$	0	0
$425$	−4.88263e8	−0.308526
$426$	0	0
$427$	−2.13746e9	−1.32862
$428$	0	0
$429$	−7.91018e8	−0.483711
$430$	0	0
$431$	9.30757e8	0.559971	0.279986	−	0.960004i	$-0.409670\pi$
$431$	9.30757e8	0.559971	0.279986	+	0.960004i	$0.409670\pi$
$432$	0	0
$433$	−1.38986e9	−0.822741	−0.411371	−	0.911468i	$-0.634950\pi$
$433$	−1.38986e9	−0.822741	−0.411371	+	0.911468i	$0.634950\pi$
$434$	0	0
$435$	1.36583e9	0.795580
$436$	0	0
$437$	2.50157e9	1.43393
$438$	0	0
$439$	7.74625e8	0.436984	0.218492	−	0.975839i	$-0.429886\pi$
$439$	7.74625e8	0.436984	0.218492	+	0.975839i	$0.429886\pi$
$440$	0	0
$441$	−2.32484e8	−0.129080
$442$	0	0
$443$	1.87372e9	1.02398	0.511991	−	0.858991i	$-0.328908\pi$
$443$	1.87372e9	1.02398	0.511991	+	0.858991i	$0.328908\pi$
$444$	0	0
$445$	3.21276e9	1.72829
$446$	0	0
$447$	−5.08338e8	−0.269201
$448$	0	0
$449$	2.43736e9	1.27074	0.635372	−	0.772207i	$-0.280847\pi$
$449$	2.43736e9	1.27074	0.635372	+	0.772207i	$0.280847\pi$
$450$	0	0
$451$	1.74853e9	0.897545
$452$	0	0
$453$	1.43075e9	0.723137
$454$	0	0
$455$	2.41305e9	1.20096
$456$	0	0
$457$	−4.41576e8	−0.216421	−0.108210	−	0.994128i	$-0.534512\pi$
$457$	−4.41576e8	−0.216421	−0.108210	+	0.994128i	$0.534512\pi$
$458$	0	0
$459$	−2.80315e8	−0.135301
$460$	0	0
$461$	−1.77394e9	−0.843309	−0.421654	−	0.906757i	$-0.638550\pi$
$461$	−1.77394e9	−0.843309	−0.421654	+	0.906757i	$0.638550\pi$
$462$	0	0
$463$	2.81182e9	1.31660	0.658300	−	0.752756i	$-0.271276\pi$
$463$	2.81182e9	1.31660	0.658300	+	0.752756i	$0.271276\pi$
$464$	0	0
$465$	4.98974e8	0.230140
$466$	0	0
$467$	1.67741e9	0.762131	0.381066	−	0.924548i	$-0.375557\pi$
$467$	1.67741e9	0.762131	0.381066	+	0.924548i	$0.375557\pi$
$468$	0	0
$469$	−2.22037e9	−0.993849
$470$	0	0
$471$	−1.78499e9	−0.787158
$472$	0	0
$473$	−1.38746e9	−0.602845
$474$	0	0
$475$	1.13788e9	0.487157
$476$	0	0
$477$	−2.32553e8	−0.0981087
$478$	0	0
$479$	2.38838e9	0.992953	0.496477	−	0.868050i	$-0.334627\pi$
$479$	2.38838e9	0.992953	0.496477	+	0.868050i	$0.334627\pi$
$480$	0	0
$481$	5.17574e9	2.12063
$482$	0	0
$483$	1.44566e9	0.583785
$484$	0	0
$485$	3.39384e9	1.35081
$486$	0	0
$487$	−6.54404e8	−0.256741	−0.128370	−	0.991726i	$-0.540975\pi$
$487$	−6.54404e8	−0.256741	−0.128370	+	0.991726i	$0.540975\pi$
$488$	0	0
$489$	−1.11546e9	−0.431395
$490$	0	0
$491$	−8.01260e8	−0.305484	−0.152742	−	0.988266i	$-0.548810\pi$
$491$	−8.01260e8	−0.305484	−0.152742	+	0.988266i	$0.548810\pi$
$492$	0	0
$493$	2.14875e9	0.807647
$494$	0	0
$495$	−7.06765e8	−0.261913
$496$	0	0
$497$	−2.94914e9	−1.07758
$498$	0	0
$499$	−2.30647e9	−0.830989	−0.415494	−	0.909596i	$-0.636391\pi$
$499$	−2.30647e9	−0.830989	−0.415494	+	0.909596i	$0.636391\pi$
$500$	0	0
$501$	5.50232e6	0.00195485
$502$	0	0
$503$	−1.82507e9	−0.639426	−0.319713	−	0.947514i	$-0.603587\pi$
$503$	−1.82507e9	−0.639426	−0.319713	+	0.947514i	$0.603587\pi$
$504$	0	0
$505$	−1.34230e9	−0.463799
$506$	0	0
$507$	1.07730e9	0.367121
$508$	0	0
$509$	−5.36457e9	−1.80311	−0.901556	−	0.432663i	$-0.857574\pi$
$509$	−5.36457e9	−1.80311	−0.901556	+	0.432663i	$0.857574\pi$
$510$	0	0
$511$	1.20376e9	0.399087
$512$	0	0
$513$	6.53264e8	0.213638
$514$	0	0
$515$	4.22245e9	1.36219
$516$	0	0
$517$	−4.60707e8	−0.146625
$518$	0	0
$519$	−6.55569e8	−0.205841
$520$	0	0
$521$	−1.11907e9	−0.346677	−0.173339	−	0.984862i	$-0.555455\pi$
$521$	−1.11907e9	−0.346677	−0.173339	+	0.984862i	$0.555455\pi$
$522$	0	0
$523$	−1.51072e8	−0.0461772	−0.0230886	−	0.999733i	$-0.507350\pi$
$523$	−1.51072e8	−0.0461772	−0.0230886	+	0.999733i	$0.507350\pi$
$524$	0	0
$525$	6.57584e8	0.198332
$526$	0	0
$527$	7.84997e8	0.233631
$528$	0	0
$529$	2.27624e9	0.668535
$530$	0	0
$531$	1.32528e9	0.384129
$532$	0	0
$533$	−6.12639e9	−1.75251
$534$	0	0
$535$	4.83889e9	1.36618
$536$	0	0
$537$	2.78427e9	0.775891
$538$	0	0
$539$	−9.22171e8	−0.253659
$540$	0	0
$541$	1.71582e9	0.465888	0.232944	−	0.972490i	$-0.425164\pi$
$541$	1.71582e9	0.465888	0.232944	+	0.972490i	$0.425164\pi$
$542$	0	0
$543$	1.82209e8	0.0488393
$544$	0	0
$545$	−2.06896e9	−0.547475
$546$	0	0
$547$	−3.72932e9	−0.974259	−0.487129	−	0.873330i	$-0.661956\pi$
$547$	−3.72932e9	−0.974259	−0.487129	+	0.873330i	$0.661956\pi$
$548$	0	0
$549$	−2.19349e9	−0.565761
$550$	0	0
$551$	−5.00759e9	−1.27526
$552$	0	0
$553$	3.03329e9	0.762739
$554$	0	0
$555$	4.62447e9	1.14825
$556$	0	0
$557$	−3.74622e9	−0.918544	−0.459272	−	0.888296i	$-0.651890\pi$
$557$	−3.74622e9	−0.918544	−0.459272	+	0.888296i	$0.651890\pi$
$558$	0	0
$559$	4.86127e9	1.17709
$560$	0	0
$561$	−1.11190e9	−0.265886
$562$	0	0
$563$	6.71873e8	0.158675	0.0793374	−	0.996848i	$-0.474720\pi$
$563$	6.71873e8	0.158675	0.0793374	+	0.996848i	$0.474720\pi$
$564$	0	0
$565$	−1.62012e9	−0.377901
$566$	0	0
$567$	3.77523e8	0.0869767
$568$	0	0
$569$	1.38230e9	0.314564	0.157282	−	0.987554i	$-0.449727\pi$
$569$	1.38230e9	0.314564	0.157282	+	0.987554i	$0.449727\pi$
$570$	0	0
$571$	−4.22076e9	−0.948778	−0.474389	−	0.880315i	$-0.657331\pi$
$571$	−4.22076e9	−0.948778	−0.474389	+	0.880315i	$0.657331\pi$
$572$	0	0
$573$	−5.19871e8	−0.115439
$574$	0	0
$575$	2.58413e9	0.566861
$576$	0	0
$577$	3.00405e9	0.651017	0.325508	−	0.945539i	$-0.394465\pi$
$577$	3.00405e9	0.651017	0.325508	+	0.945539i	$0.394465\pi$
$578$	0	0
$579$	−4.55446e9	−0.975129
$580$	0	0
$581$	−5.52899e9	−1.16958
$582$	0	0
$583$	−9.22445e8	−0.192797
$584$	0	0
$585$	2.47631e9	0.511399
$586$	0	0
$587$	4.90415e9	1.00076	0.500380	−	0.865806i	$-0.333194\pi$
$587$	4.90415e9	1.00076	0.500380	+	0.865806i	$0.333194\pi$
$588$	0	0
$589$	−1.82941e9	−0.368899
$590$	0	0
$591$	4.24789e9	0.846480
$592$	0	0
$593$	5.24962e9	1.03380	0.516900	−	0.856046i	$-0.327086\pi$
$593$	5.24962e9	1.03380	0.516900	+	0.856046i	$0.327086\pi$
$594$	0	0
$595$	3.39192e9	0.660140
$596$	0	0
$597$	3.59967e9	0.692393
$598$	0	0
$599$	−2.43405e9	−0.462739	−0.231369	−	0.972866i	$-0.574321\pi$
$599$	−2.43405e9	−0.462739	−0.231369	+	0.972866i	$0.574321\pi$
$600$	0	0
$601$	2.91489e9	0.547723	0.273862	−	0.961769i	$-0.411699\pi$
$601$	2.91489e9	0.547723	0.273862	+	0.961769i	$0.411699\pi$
$602$	0	0
$603$	−2.27858e9	−0.423208
$604$	0	0
$605$	3.73011e9	0.684823
$606$	0	0
$607$	3.86425e9	0.701302	0.350651	−	0.936506i	$-0.385960\pi$
$607$	3.86425e9	0.701302	0.350651	+	0.936506i	$0.385960\pi$
$608$	0	0
$609$	−2.89390e9	−0.519186
$610$	0	0
$611$	1.61419e9	0.286293
$612$	0	0
$613$	5.13751e9	0.900826	0.450413	−	0.892820i	$-0.351277\pi$
$613$	5.13751e9	0.900826	0.450413	+	0.892820i	$0.351277\pi$
$614$	0	0
$615$	−5.47386e9	−0.948922
$616$	0	0
$617$	−7.30389e9	−1.25186	−0.625930	−	0.779879i	$-0.715281\pi$
$617$	−7.30389e9	−1.25186	−0.625930	+	0.779879i	$0.715281\pi$
$618$	0	0
$619$	8.08687e9	1.37045	0.685225	−	0.728332i	$-0.259704\pi$
$619$	8.08687e9	1.37045	0.685225	+	0.728332i	$0.259704\pi$
$620$	0	0
$621$	1.48356e9	0.248591
$622$	0	0
$623$	−6.80714e9	−1.12786
$624$	0	0
$625$	−7.60657e9	−1.24626
$626$	0	0
$627$	2.59124e9	0.419828
$628$	0	0
$629$	7.27531e9	1.16567
$630$	0	0
$631$	−3.02687e9	−0.479613	−0.239807	−	0.970821i	$-0.577084\pi$
$631$	−3.02687e9	−0.479613	−0.239807	+	0.970821i	$0.577084\pi$
$632$	0	0
$633$	7.19525e9	1.12754
$634$	0	0
$635$	−8.97470e9	−1.39095
$636$	0	0
$637$	3.23104e9	0.495283
$638$	0	0
$639$	−3.02646e9	−0.458861
$640$	0	0
$641$	−8.84536e9	−1.32652	−0.663258	−	0.748391i	$-0.730827\pi$
$641$	−8.84536e9	−1.32652	−0.663258	+	0.748391i	$0.730827\pi$
$642$	0	0
$643$	−6.32368e9	−0.938062	−0.469031	−	0.883182i	$-0.655397\pi$
$643$	−6.32368e9	−0.938062	−0.469031	+	0.883182i	$0.655397\pi$
$644$	0	0
$645$	4.34349e9	0.637353
$646$	0	0
$647$	3.92152e9	0.569233	0.284616	−	0.958641i	$-0.408134\pi$
$647$	3.92152e9	0.569233	0.284616	+	0.958641i	$0.408134\pi$
$648$	0	0
$649$	5.25686e9	0.754866
$650$	0	0
$651$	−1.05722e9	−0.150187
$652$	0	0
$653$	−7.02434e9	−0.987210	−0.493605	−	0.869686i	$-0.664321\pi$
$653$	−7.02434e9	−0.987210	−0.493605	+	0.869686i	$0.664321\pi$
$654$	0	0
$655$	−1.29718e10	−1.80367
$656$	0	0
$657$	1.23532e9	0.169942
$658$	0	0
$659$	−9.72714e9	−1.32399	−0.661997	−	0.749506i	$-0.730291\pi$
$659$	−9.72714e9	−1.32399	−0.661997	+	0.749506i	$0.730291\pi$
$660$	0	0
$661$	−5.19552e9	−0.699719	−0.349860	−	0.936802i	$-0.613771\pi$
$661$	−5.19552e9	−0.699719	−0.349860	+	0.936802i	$0.613771\pi$
$662$	0	0
$663$	3.89579e9	0.519156
$664$	0	0
$665$	−7.90475e9	−1.04235
$666$	0	0
$667$	−1.13722e10	−1.48390
$668$	0	0
$669$	7.00403e9	0.904392
$670$	0	0
$671$	−8.70071e9	−1.11180
$672$	0	0
$673$	−2.24492e9	−0.283889	−0.141945	−	0.989875i	$-0.545335\pi$
$673$	−2.24492e9	−0.283889	−0.141945	+	0.989875i	$0.545335\pi$
$674$	0	0
$675$	6.74823e8	0.0844553
$676$	0	0
$677$	−1.59249e9	−0.197250	−0.0986251	−	0.995125i	$-0.531444\pi$
$677$	−1.59249e9	−0.197250	−0.0986251	+	0.995125i	$0.531444\pi$
$678$	0	0
$679$	−7.19083e9	−0.881525
$680$	0	0
$681$	3.07858e9	0.373538
$682$	0	0
$683$	−9.68643e9	−1.16330	−0.581649	−	0.813440i	$-0.697592\pi$
$683$	−9.68643e9	−1.16330	−0.581649	+	0.813440i	$0.697592\pi$
$684$	0	0
$685$	6.55670e9	0.779414
$686$	0	0
$687$	−2.60096e9	−0.306044
$688$	0	0
$689$	3.23200e9	0.376447
$690$	0	0
$691$	2.97399e9	0.342899	0.171449	−	0.985193i	$-0.445155\pi$
$691$	2.97399e9	0.342899	0.171449	+	0.985193i	$0.445155\pi$
$692$	0	0
$693$	1.49748e9	0.170921
$694$	0	0
$695$	1.63869e10	1.85162
$696$	0	0
$697$	−8.61159e9	−0.963315
$698$	0	0
$699$	−3.53985e9	−0.392026
$700$	0	0
$701$	1.90884e9	0.209294	0.104647	−	0.994509i	$-0.466629\pi$
$701$	1.90884e9	0.209294	0.104647	+	0.994509i	$0.466629\pi$
$702$	0	0
$703$	−1.69549e10	−1.84056
$704$	0	0
$705$	1.44226e9	0.155018
$706$	0	0
$707$	2.84405e9	0.302670
$708$	0	0
$709$	3.59009e9	0.378306	0.189153	−	0.981948i	$-0.439426\pi$
$709$	3.59009e9	0.378306	0.189153	+	0.981948i	$0.439426\pi$
$710$	0	0
$711$	3.11281e9	0.324795
$712$	0	0
$713$	−4.15459e9	−0.429254
$714$	0	0
$715$	9.82255e9	1.00497
$716$	0	0
$717$	5.20334e9	0.527188
$718$	0	0
$719$	2.83832e9	0.284780	0.142390	−	0.989811i	$-0.454521\pi$
$719$	2.83832e9	0.284780	0.142390	+	0.989811i	$0.454521\pi$
$720$	0	0
$721$	−8.94646e9	−0.888951
$722$	0	0
$723$	−9.08497e9	−0.894004
$724$	0	0
$725$	−5.17284e9	−0.504134
$726$	0	0
$727$	4.47195e6	0.000431645 0	0.000215822	−	1.00000i	$-0.499931\pi$
$727$	4.47195e6	0.000431645 0	0.000215822		1.00000i	$0.499931\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	6.83327e9	0.647020
$732$	0	0
$733$	8.23619e9	0.772436	0.386218	−	0.922408i	$-0.373781\pi$
$733$	8.23619e9	0.772436	0.386218	+	0.922408i	$0.373781\pi$
$734$	0	0
$735$	2.88689e9	0.268179
$736$	0	0
$737$	−9.03822e9	−0.831661
$738$	0	0
$739$	3.33578e9	0.304048	0.152024	−	0.988377i	$-0.451421\pi$
$739$	3.33578e9	0.304048	0.152024	+	0.988377i	$0.451421\pi$
$740$	0	0
$741$	−9.07900e9	−0.819737
$742$	0	0
$743$	5.44577e9	0.487078	0.243539	−	0.969891i	$-0.421692\pi$
$743$	5.44577e9	0.487078	0.243539	+	0.969891i	$0.421692\pi$
$744$	0	0
$745$	6.31234e9	0.559299
$746$	0	0
$747$	−5.67393e9	−0.498038
$748$	0	0
$749$	−1.02526e10	−0.891551
$750$	0	0
$751$	−7.09357e6	−0.000611118 0	−0.000305559	−	1.00000i	$-0.500097\pi$
$751$	−7.09357e6	−0.000611118 0	−0.000305559		1.00000i	$0.500097\pi$
$752$	0	0
$753$	5.69083e9	0.485728
$754$	0	0
$755$	−1.77665e10	−1.50241
$756$	0	0
$757$	4.93300e9	0.413310	0.206655	−	0.978414i	$-0.433742\pi$
$757$	4.93300e9	0.413310	0.206655	+	0.978414i	$0.433742\pi$
$758$	0	0
$759$	5.88471e9	0.488516
$760$	0	0
$761$	−1.28057e10	−1.05331	−0.526654	−	0.850079i	$-0.676554\pi$
$761$	−1.28057e10	−1.05331	−0.526654	+	0.850079i	$0.676554\pi$
$762$	0	0
$763$	4.38368e9	0.357275
$764$	0	0
$765$	3.48084e9	0.281105
$766$	0	0
$767$	−1.84186e10	−1.47392
$768$	0	0
$769$	1.36621e10	1.08336	0.541681	−	0.840584i	$-0.317788\pi$
$769$	1.36621e10	1.08336	0.541681	+	0.840584i	$0.317788\pi$
$770$	0	0
$771$	−3.32449e8	−0.0261237
$772$	0	0
$773$	3.39415e9	0.264304	0.132152	−	0.991229i	$-0.457811\pi$
$773$	3.39415e9	0.264304	0.132152	+	0.991229i	$0.457811\pi$
$774$	0	0
$775$	−1.88978e9	−0.145833
$776$	0	0
$777$	−9.79825e9	−0.749334
$778$	0	0
$779$	2.00690e10	1.52105
$780$	0	0
$781$	−1.20047e10	−0.901726
$782$	0	0
$783$	−2.96976e9	−0.221083
$784$	0	0
$785$	2.21653e10	1.63542
$786$	0	0
$787$	8.38934e9	0.613503	0.306751	−	0.951790i	$-0.400758\pi$
$787$	8.38934e9	0.613503	0.306751	+	0.951790i	$0.400758\pi$
$788$	0	0
$789$	−1.17077e10	−0.848595
$790$	0	0
$791$	3.43269e9	0.246613
$792$	0	0
$793$	3.04849e10	2.17085
$794$	0	0
$795$	2.88775e9	0.203833
$796$	0	0
$797$	−1.19751e10	−0.837867	−0.418934	−	0.908017i	$-0.637596\pi$
$797$	−1.19751e10	−0.837867	−0.418934	+	0.908017i	$0.637596\pi$
$798$	0	0
$799$	2.26900e9	0.157369
$800$	0	0
$801$	−6.98560e9	−0.480275
$802$	0	0
$803$	4.90002e9	0.333959
$804$	0	0
$805$	−1.79517e10	−1.21289
$806$	0	0
$807$	1.57758e10	1.05665
$808$	0	0
$809$	8.77076e9	0.582395	0.291197	−	0.956663i	$-0.405946\pi$
$809$	8.77076e9	0.582395	0.291197	+	0.956663i	$0.405946\pi$
$810$	0	0
$811$	1.19935e10	0.789535	0.394767	−	0.918781i	$-0.370825\pi$
$811$	1.19935e10	0.789535	0.394767	+	0.918781i	$0.370825\pi$
$812$	0	0
$813$	6.13200e8	0.0400207
$814$	0	0
$815$	1.38514e10	0.896276
$816$	0	0
$817$	−1.59247e10	−1.02163
$818$	0	0
$819$	−5.24678e9	−0.333733
$820$	0	0
$821$	−1.24857e9	−0.0787428	−0.0393714	−	0.999225i	$-0.512536\pi$
$821$	−1.24857e9	−0.0787428	−0.0393714	+	0.999225i	$0.512536\pi$
$822$	0	0
$823$	3.45738e9	0.216196	0.108098	−	0.994140i	$-0.465524\pi$
$823$	3.45738e9	0.216196	0.108098	+	0.994140i	$0.465524\pi$
$824$	0	0
$825$	2.67675e9	0.165966
$826$	0	0
$827$	−1.76043e9	−0.108231	−0.0541153	−	0.998535i	$-0.517234\pi$
$827$	−1.76043e9	−0.108231	−0.0541153	+	0.998535i	$0.517234\pi$
$828$	0	0
$829$	−8.84745e9	−0.539358	−0.269679	−	0.962950i	$-0.586918\pi$
$829$	−8.84745e9	−0.539358	−0.269679	+	0.962950i	$0.586918\pi$
$830$	0	0
$831$	7.81644e9	0.472504
$832$	0	0
$833$	4.54172e9	0.272247
$834$	0	0
$835$	−6.83257e7	−0.00406145
$836$	0	0
$837$	−1.08494e9	−0.0639536
$838$	0	0
$839$	−2.90818e10	−1.70002	−0.850010	−	0.526767i	$-0.823404\pi$
$839$	−2.90818e10	−1.70002	−0.850010	+	0.526767i	$0.823404\pi$
$840$	0	0
$841$	5.51478e9	0.319700
$842$	0	0
$843$	8.18209e9	0.470401
$844$	0	0
$845$	−1.33775e10	−0.762741
$846$	0	0
$847$	−7.90331e9	−0.446907
$848$	0	0
$849$	1.43544e10	0.805025
$850$	0	0
$851$	−3.85045e10	−2.14170
$852$	0	0
$853$	2.00633e10	1.10683	0.553414	−	0.832906i	$-0.313325\pi$
$853$	2.00633e10	1.10683	0.553414	+	0.832906i	$0.313325\pi$
$854$	0	0
$855$	−8.11198e9	−0.443860
$856$	0	0
$857$	−2.22093e10	−1.20532	−0.602661	−	0.797998i	$-0.705893\pi$
$857$	−2.22093e10	−1.20532	−0.602661	+	0.797998i	$0.705893\pi$
$858$	0	0
$859$	2.71682e10	1.46246	0.731232	−	0.682128i	$-0.238946\pi$
$859$	2.71682e10	1.46246	0.731232	+	0.682128i	$0.238946\pi$
$860$	0	0
$861$	1.15979e10	0.619255
$862$	0	0
$863$	3.23767e10	1.71473	0.857364	−	0.514710i	$-0.172100\pi$
$863$	3.23767e10	1.71473	0.857364	+	0.514710i	$0.172100\pi$
$864$	0	0
$865$	8.14060e9	0.427661
$866$	0	0
$867$	−5.60301e9	−0.291981
$868$	0	0
$869$	1.23473e10	0.638267
$870$	0	0
$871$	3.16675e10	1.62386
$872$	0	0
$873$	−7.37934e9	−0.375377
$874$	0	0
$875$	1.04416e10	0.526911
$876$	0	0
$877$	2.43044e10	1.21671	0.608354	−	0.793666i	$-0.291830\pi$
$877$	2.43044e10	1.21671	0.608354	+	0.793666i	$0.291830\pi$
$878$	0	0
$879$	−4.56986e9	−0.226956
$880$	0	0
$881$	2.63625e10	1.29889	0.649444	−	0.760409i	$-0.275002\pi$
$881$	2.63625e10	1.29889	0.649444	+	0.760409i	$0.275002\pi$
$882$	0	0
$883$	−1.59986e10	−0.782021	−0.391011	−	0.920386i	$-0.627874\pi$
$883$	−1.59986e10	−0.782021	−0.391011	+	0.920386i	$0.627874\pi$
$884$	0	0
$885$	−1.64568e10	−0.798076
$886$	0	0
$887$	3.86500e10	1.85959	0.929794	−	0.368081i	$-0.119985\pi$
$887$	3.86500e10	1.85959	0.929794	+	0.368081i	$0.119985\pi$
$888$	0	0
$889$	1.90155e10	0.907718
$890$	0	0
$891$	1.53674e9	0.0727829
$892$	0	0
$893$	−5.28782e9	−0.248483
$894$	0	0
$895$	−3.45739e10	−1.61201
$896$	0	0
$897$	−2.06184e10	−0.953854
$898$	0	0
$899$	8.31655e9	0.381755
$900$	0	0
$901$	4.54307e9	0.206925
$902$	0	0
$903$	−9.20293e9	−0.415929
$904$	0	0
$905$	−2.26259e9	−0.101470
$906$	0	0
$907$	5.99871e9	0.266951	0.133476	−	0.991052i	$-0.457386\pi$
$907$	5.99871e9	0.266951	0.133476	+	0.991052i	$0.457386\pi$
$908$	0	0
$909$	2.91861e9	0.128885
$910$	0	0
$911$	−1.01805e10	−0.446121	−0.223061	−	0.974805i	$-0.571605\pi$
$911$	−1.01805e10	−0.446121	−0.223061	+	0.974805i	$0.571605\pi$
$912$	0	0
$913$	−2.25062e10	−0.978712
$914$	0	0
$915$	2.72379e10	1.17544
$916$	0	0
$917$	2.74846e10	1.17705
$918$	0	0
$919$	3.51377e10	1.49338	0.746688	−	0.665175i	$-0.231643\pi$
$919$	3.51377e10	1.49338	0.746688	+	0.665175i	$0.231643\pi$
$920$	0	0
$921$	2.13184e10	0.899180
$922$	0	0
$923$	4.20614e10	1.76067
$924$	0	0
$925$	−1.75144e10	−0.727610
$926$	0	0
$927$	−9.18100e9	−0.378539
$928$	0	0
$929$	1.77161e10	0.724958	0.362479	−	0.931992i	$-0.381930\pi$
$929$	1.77161e10	0.724958	0.362479	+	0.931992i	$0.381930\pi$
$930$	0	0
$931$	−1.05843e10	−0.429872
$932$	0	0
$933$	−4.48327e9	−0.180721
$934$	0	0
$935$	1.38071e10	0.552411
$936$	0	0
$937$	−4.17094e10	−1.65633	−0.828163	−	0.560487i	$-0.810614\pi$
$937$	−4.17094e10	−1.65633	−0.828163	+	0.560487i	$0.810614\pi$
$938$	0	0
$939$	1.12919e10	0.445079
$940$	0	0
$941$	2.80091e10	1.09581	0.547905	−	0.836541i	$-0.315425\pi$
$941$	2.80091e10	1.09581	0.547905	+	0.836541i	$0.315425\pi$
$942$	0	0
$943$	4.55767e10	1.76991
$944$	0	0
$945$	−4.68794e9	−0.180705
$946$	0	0
$947$	4.94004e10	1.89019	0.945094	−	0.326798i	$-0.105970\pi$
$947$	4.94004e10	1.89019	0.945094	+	0.326798i	$0.105970\pi$
$948$	0	0
$949$	−1.71683e10	−0.652074
$950$	0	0
$951$	1.55255e10	0.585348
$952$	0	0
$953$	−8.75488e9	−0.327661	−0.163831	−	0.986488i	$-0.552385\pi$
$953$	−8.75488e9	−0.327661	−0.163831	+	0.986488i	$0.552385\pi$
$954$	0	0
$955$	6.45555e9	0.239840
$956$	0	0
$957$	−1.17799e10	−0.434459
$958$	0	0
$959$	−1.38922e10	−0.508636
$960$	0	0
$961$	−2.44743e10	−0.889568
$962$	0	0
$963$	−1.05213e10	−0.379646
$964$	0	0
$965$	5.65555e10	2.02595
$966$	0	0
$967$	−2.37991e10	−0.846385	−0.423192	−	0.906040i	$-0.639091\pi$
$967$	−2.37991e10	−0.846385	−0.423192	+	0.906040i	$0.639091\pi$
$968$	0	0
$969$	−1.27619e10	−0.450592
$970$	0	0
$971$	−6.16413e9	−0.216075	−0.108037	−	0.994147i	$-0.534457\pi$
$971$	−6.16413e9	−0.216075	−0.108037	+	0.994147i	$0.534457\pi$
$972$	0	0
$973$	−3.47204e10	−1.20834
$974$	0	0
$975$	−9.37862e9	−0.324058
$976$	0	0
$977$	1.77272e10	0.608147	0.304074	−	0.952649i	$-0.401653\pi$
$977$	1.77272e10	0.608147	0.304074	+	0.952649i	$0.401653\pi$
$978$	0	0
$979$	−2.77091e10	−0.943806
$980$	0	0
$981$	4.49860e9	0.152137
$982$	0	0
$983$	−1.61779e10	−0.543232	−0.271616	−	0.962406i	$-0.587558\pi$
$983$	−1.61779e10	−0.543232	−0.271616	+	0.962406i	$0.587558\pi$
$984$	0	0
$985$	−5.27486e10	−1.75867
$986$	0	0
$987$	−3.05585e9	−0.101163
$988$	0	0
$989$	−3.61650e10	−1.18878
$990$	0	0
$991$	−2.06432e9	−0.0673780	−0.0336890	−	0.999432i	$-0.510726\pi$
$991$	−2.06432e9	−0.0673780	−0.0336890	+	0.999432i	$0.510726\pi$
$992$	0	0
$993$	−3.07915e10	−0.997948
$994$	0	0
$995$	−4.46993e10	−1.43853
$996$	0	0
$997$	9.93049e9	0.317349	0.158675	−	0.987331i	$-0.449278\pi$
$997$	9.93049e9	0.317349	0.158675	+	0.987331i	$0.449278\pi$
$998$	0	0
$999$	−1.00551e10	−0.319086

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.8.a.u.1.1		2
3.2	odd	2		576.8.a.bl.1.2		2
4.3	odd	2		192.8.a.r.1.1		2
8.3	odd	2		96.8.a.g.1.2	yes	2
8.5	even	2		96.8.a.d.1.2	✓	2
12.11	even	2		576.8.a.bm.1.2		2
24.5	odd	2		288.8.a.i.1.1		2
24.11	even	2		288.8.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.8.a.d.1.2	✓	2	8.5	even	2
96.8.a.g.1.2	yes	2	8.3	odd	2
192.8.a.r.1.1		2	4.3	odd	2
192.8.a.u.1.1		2	1.1	even	1	trivial
288.8.a.i.1.1		2	24.5	odd	2
288.8.a.j.1.1		2	24.11	even	2
576.8.a.bl.1.2		2	3.2	odd	2
576.8.a.bm.1.2		2	12.11	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} -335.275 q^{5} +710.377 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+27.0000 q^{3} -335.275 q^{5} +710.377 q^{7} +729.000 q^{9} +2891.65 q^{11} -10131.6 q^{13} -9052.43 q^{15} -14241.5 q^{17} +33189.3 q^{19} +19180.2 q^{21} +75372.9 q^{23} +34284.6 q^{25} +19683.0 q^{27} -150880. q^{29} -55120.5 q^{31} +78074.6 q^{33} -238172. q^{35} -510853. q^{37} -273552. q^{39} +604684. q^{41} -479815. q^{43} -244416. q^{45} -159323. q^{47} -318908. q^{49} -384520. q^{51} -319003. q^{53} -969500. q^{55} +896110. q^{57} +1.81794e6 q^{59} -3.00891e6 q^{61} +517865. q^{63} +3.39686e6 q^{65} -3.12562e6 q^{67} +2.03507e6 q^{69} -4.15152e6 q^{71} +1.69454e6 q^{73} +925683. q^{75} +2.05416e6 q^{77} +4.26998e6 q^{79} +531441. q^{81} -7.78317e6 q^{83} +4.77482e6 q^{85} -4.07375e6 q^{87} -9.58244e6 q^{89} -7.19723e6 q^{91} -1.48825e6 q^{93} -1.11275e7 q^{95} -1.01226e7 q^{97} +2.10801e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 54 q^{3} - 180 q^{5} - 1032 q^{7} + 1458 q^{9}+O(q^{10}) \) 2 * q + 54 * q^3 - 180 * q^5 - 1032 * q^7 + 1458 * q^9	\( 2 q + 54 q^{3} - 180 q^{5} - 1032 q^{7} + 1458 q^{9} + 2840 q^{11} + 340 q^{13} - 4860 q^{15} + 9780 q^{17} + 32040 q^{19} - 27864 q^{21} - 11136 q^{23} - 19730 q^{25} + 39366 q^{27} - 304212 q^{29} + 77640 q^{31} + 76680 q^{33} - 508720 q^{35} - 1015820 q^{37} + 9180 q^{39} + 704100 q^{41} - 395496 q^{43} - 131220 q^{45} - 1157488 q^{47} + 1893426 q^{49} + 264060 q^{51} - 1568580 q^{53} - 977520 q^{55} + 865080 q^{57} + 139240 q^{59} - 2603580 q^{61} - 752328 q^{63} + 5022840 q^{65} - 5289768 q^{67} - 300672 q^{69} - 5721760 q^{71} - 1190700 q^{73} - 532710 q^{75} + 2144160 q^{77} + 398280 q^{79} + 1062882 q^{81} - 6986616 q^{83} + 8504760 q^{85} - 8213724 q^{87} - 8166732 q^{89} - 25442640 q^{91} + 2096280 q^{93} - 11306000 q^{95} - 10361500 q^{97} + 2070360 q^{99}+O(q^{100}) \) 2 * q + 54 * q^3 - 180 * q^5 - 1032 * q^7 + 1458 * q^9 + 2840 * q^11 + 340 * q^13 - 4860 * q^15 + 9780 * q^17 + 32040 * q^19 - 27864 * q^21 - 11136 * q^23 - 19730 * q^25 + 39366 * q^27 - 304212 * q^29 + 77640 * q^31 + 76680 * q^33 - 508720 * q^35 - 1015820 * q^37 + 9180 * q^39 + 704100 * q^41 - 395496 * q^43 - 131220 * q^45 - 1157488 * q^47 + 1893426 * q^49 + 264060 * q^51 - 1568580 * q^53 - 977520 * q^55 + 865080 * q^57 + 139240 * q^59 - 2603580 * q^61 - 752328 * q^63 + 5022840 * q^65 - 5289768 * q^67 - 300672 * q^69 - 5721760 * q^71 - 1190700 * q^73 - 532710 * q^75 + 2144160 * q^77 + 398280 * q^79 + 1062882 * q^81 - 6986616 * q^83 + 8504760 * q^85 - 8213724 * q^87 - 8166732 * q^89 - 25442640 * q^91 + 2096280 * q^93 - 11306000 * q^95 - 10361500 * q^97 + 2070360 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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