LMFDB - Embedded newform 54.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,8,Mod(1,54)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("54.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	54.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:	$16.8687913761$
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	$\chi$	$=$	54.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{2} +64.0000 q^{4} +312.000 q^{5} +323.000 q^{7} -512.000 q^{8} +O(q^{10})$

\(q-8.00000 q^{2} +64.0000 q^{4} +312.000 q^{5} +323.000 q^{7} -512.000 q^{8} -2496.00 q^{10} +3720.00 q^{11} -14179.0 q^{13} -2584.00 q^{14} +4096.00 q^{16} +15912.0 q^{17} +22421.0 q^{19} +19968.0 q^{20} -29760.0 q^{22} +57768.0 q^{23} +19219.0 q^{25} +113432. q^{26} +20672.0 q^{28} +166656. q^{29} +94820.0 q^{31} -32768.0 q^{32} -127296. q^{34} +100776. q^{35} +453971. q^{37} -179368. q^{38} -159744. q^{40} +627072. q^{41} -42472.0 q^{43} +238080. q^{44} -462144. q^{46} -1.23526e6 q^{47} -719214. q^{49} -153752. q^{50} -907456. q^{52} +107280. q^{53} +1.16064e6 q^{55} -165376. q^{56} -1.33325e6 q^{58} -2.47922e6 q^{59} +2.87438e6 q^{61} -758560. q^{62} +262144. q^{64} -4.42385e6 q^{65} +1.50110e6 q^{67} +1.01837e6 q^{68} -806208. q^{70} +4.73314e6 q^{71} -85111.0 q^{73} -3.63177e6 q^{74} +1.43494e6 q^{76} +1.20156e6 q^{77} -1.18082e6 q^{79} +1.27795e6 q^{80} -5.01658e6 q^{82} -1.11653e6 q^{83} +4.96454e6 q^{85} +339776. q^{86} -1.90464e6 q^{88} +9.36814e6 q^{89} -4.57982e6 q^{91} +3.69715e6 q^{92} +9.88205e6 q^{94} +6.99535e6 q^{95} -2.03999e6 q^{97} +5.75371e6 q^{98} +O(q^{100})\)

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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	312.000	1.11625	0.558123	−	0.829759i	$-0.311522\pi$
$5$	312.000	1.11625	0.558123	+	0.829759i	$0.311522\pi$
$6$	0	0
$7$	323.000	0.355926	0.177963	−	0.984037i	$-0.443049\pi$
$7$	323.000	0.355926	0.177963	+	0.984037i	$0.443049\pi$
$8$	−512.000	−0.353553
$9$	0	0
$10$	−2496.00	−0.789305
$11$	3720.00	0.842691	0.421346	−	0.906900i	$-0.361558\pi$
$11$	3720.00	0.842691	0.421346	+	0.906900i	$0.361558\pi$
$12$	0	0
$13$	−14179.0	−1.78996	−0.894981	−	0.446104i	$-0.852811\pi$
$13$	−14179.0	−1.78996	−0.894981	+	0.446104i	$0.852811\pi$
$14$	−2584.00	−0.251677
$15$	0	0
$16$	4096.00	0.250000
$17$	15912.0	0.785513	0.392757	−	0.919642i	$-0.371521\pi$
$17$	15912.0	0.785513	0.392757	+	0.919642i	$0.371521\pi$
$18$	0	0
$19$	22421.0	0.749924	0.374962	−	0.927040i	$-0.377656\pi$
$19$	22421.0	0.749924	0.374962	+	0.927040i	$0.377656\pi$
$20$	19968.0	0.558123
$21$	0	0
$22$	−29760.0	−0.595873
$23$	57768.0	0.990011	0.495005	−	0.868890i	$-0.335166\pi$
$23$	57768.0	0.990011	0.495005	+	0.868890i	$0.335166\pi$
$24$	0	0
$25$	19219.0	0.246003
$26$	113432.	1.26569
$27$	0	0
$28$	20672.0	0.177963
$29$	166656.	1.26890	0.634451	−	0.772963i	$-0.281226\pi$
$29$	166656.	1.26890	0.634451	+	0.772963i	$0.281226\pi$
$30$	0	0
$31$	94820.0	0.571655	0.285828	−	0.958281i	$-0.407732\pi$
$31$	94820.0	0.571655	0.285828	+	0.958281i	$0.407732\pi$
$32$	−32768.0	−0.176777
$33$	0	0
$34$	−127296.	−0.555442
$35$	100776.	0.397300
$36$	0	0
$37$	453971.	1.47340	0.736702	−	0.676217i	$-0.236382\pi$
$37$	453971.	1.47340	0.736702	+	0.676217i	$0.236382\pi$
$38$	−179368.	−0.530276
$39$	0	0
$40$	−159744.	−0.394652
$41$	627072.	1.42093	0.710467	−	0.703731i	$-0.248484\pi$
$41$	627072.	1.42093	0.710467	+	0.703731i	$0.248484\pi$
$42$	0	0
$43$	−42472.0	−0.0814635	−0.0407318	−	0.999170i	$-0.512969\pi$
$43$	−42472.0	−0.0814635	−0.0407318	+	0.999170i	$0.512969\pi$
$44$	238080.	0.421346
$45$	0	0
$46$	−462144.	−0.700043
$47$	−1.23526e6	−1.73546	−0.867730	−	0.497036i	$-0.834422\pi$
$47$	−1.23526e6	−1.73546	−0.867730	+	0.497036i	$0.834422\pi$
$48$	0	0
$49$	−719214.	−0.873317
$50$	−153752.	−0.173951
$51$	0	0
$52$	−907456.	−0.894981
$53$	107280.	0.0989813	0.0494907	−	0.998775i	$-0.484240\pi$
$53$	107280.	0.0989813	0.0494907	+	0.998775i	$0.484240\pi$
$54$	0	0
$55$	1.16064e6	0.940650
$56$	−165376.	−0.125839
$57$	0	0
$58$	−1.33325e6	−0.897249
$59$	−2.47922e6	−1.57157	−0.785785	−	0.618500i	$-0.787741\pi$
$59$	−2.47922e6	−1.57157	−0.785785	+	0.618500i	$0.787741\pi$
$60$	0	0
$61$	2.87438e6	1.62140	0.810700	−	0.585462i	$-0.199087\pi$
$61$	2.87438e6	1.62140	0.810700	+	0.585462i	$0.199087\pi$
$62$	−758560.	−0.404221
$63$	0	0
$64$	262144.	0.125000
$65$	−4.42385e6	−1.99804
$66$	0	0
$67$	1.50110e6	0.609743	0.304872	−	0.952393i	$-0.401386\pi$
$67$	1.50110e6	0.609743	0.304872	+	0.952393i	$0.401386\pi$
$68$	1.01837e6	0.392757
$69$	0	0
$70$	−806208.	−0.280934
$71$	4.73314e6	1.56944	0.784720	−	0.619850i	$-0.212807\pi$
$71$	4.73314e6	1.56944	0.784720	+	0.619850i	$0.212807\pi$
$72$	0	0
$73$	−85111.0	−0.0256068	−0.0128034	−	0.999918i	$-0.504076\pi$
$73$	−85111.0	−0.0256068	−0.0128034	+	0.999918i	$0.504076\pi$
$74$	−3.63177e6	−1.04185
$75$	0	0
$76$	1.43494e6	0.374962
$77$	1.20156e6	0.299936
$78$	0	0
$79$	−1.18082e6	−0.269456	−0.134728	−	0.990883i	$-0.543016\pi$
$79$	−1.18082e6	−0.269456	−0.134728	+	0.990883i	$0.543016\pi$
$80$	1.27795e6	0.279061
$81$	0	0
$82$	−5.01658e6	−1.00475
$83$	−1.11653e6	−0.214337	−0.107168	−	0.994241i	$-0.534178\pi$
$83$	−1.11653e6	−0.214337	−0.107168	+	0.994241i	$0.534178\pi$
$84$	0	0
$85$	4.96454e6	0.876825
$86$	339776.	0.0576034
$87$	0	0
$88$	−1.90464e6	−0.297936
$89$	9.36814e6	1.40860	0.704301	−	0.709902i	$-0.251261\pi$
$89$	9.36814e6	1.40860	0.704301	+	0.709902i	$0.251261\pi$
$90$	0	0
$91$	−4.57982e6	−0.637094
$92$	3.69715e6	0.495005
$93$	0	0
$94$	9.88205e6	1.22716
$95$	6.99535e6	0.837099
$96$	0	0
$97$	−2.03999e6	−0.226949	−0.113474	−	0.993541i	$-0.536198\pi$
$97$	−2.03999e6	−0.226949	−0.113474	+	0.993541i	$0.536198\pi$
$98$	5.75371e6	0.617528
$99$	0	0
$100$	1.23002e6	0.123002
$101$	−1.52575e7	−1.47353	−0.736763	−	0.676151i	$-0.763647\pi$
$101$	−1.52575e7	−1.47353	−0.736763	+	0.676151i	$0.763647\pi$
$102$	0	0
$103$	−1.92433e7	−1.73520	−0.867601	−	0.497260i	$-0.834339\pi$
$103$	−1.92433e7	−1.73520	−0.867601	+	0.497260i	$0.834339\pi$
$104$	7.25965e6	0.632847
$105$	0	0
$106$	−858240.	−0.0699904
$107$	1.28571e7	1.01461	0.507306	−	0.861766i	$-0.330641\pi$
$107$	1.28571e7	1.01461	0.507306	+	0.861766i	$0.330641\pi$
$108$	0	0
$109$	−1.02835e7	−0.760589	−0.380294	−	0.924865i	$-0.624177\pi$
$109$	−1.02835e7	−0.760589	−0.380294	+	0.924865i	$0.624177\pi$
$110$	−9.28512e6	−0.665140
$111$	0	0
$112$	1.32301e6	0.0889814
$113$	1.86773e7	1.21769	0.608847	−	0.793287i	$-0.291632\pi$
$113$	1.86773e7	1.21769	0.608847	+	0.793287i	$0.291632\pi$
$114$	0	0
$115$	1.80236e7	1.10509
$116$	1.06660e7	0.634451
$117$	0	0
$118$	1.98338e7	1.11127
$119$	5.13958e6	0.279584
$120$	0	0
$121$	−5.64877e6	−0.289871
$122$	−2.29951e7	−1.14650
$123$	0	0
$124$	6.06848e6	0.285828
$125$	−1.83787e7	−0.841645
$126$	0	0
$127$	3.53659e6	0.153204	0.0766022	−	0.997062i	$-0.475593\pi$
$127$	3.53659e6	0.153204	0.0766022	+	0.997062i	$0.475593\pi$
$128$	−2.09715e6	−0.0883883
$129$	0	0
$130$	3.53908e7	1.41283
$131$	−3.01005e7	−1.16983	−0.584917	−	0.811093i	$-0.698873\pi$
$131$	−3.01005e7	−1.16983	−0.584917	+	0.811093i	$0.698873\pi$
$132$	0	0
$133$	7.24198e6	0.266917
$134$	−1.20088e7	−0.431154
$135$	0	0
$136$	−8.14694e6	−0.277721
$137$	−7.37326e6	−0.244984	−0.122492	−	0.992470i	$-0.539089\pi$
$137$	−7.37326e6	−0.244984	−0.122492	+	0.992470i	$0.539089\pi$
$138$	0	0
$139$	−5.92522e7	−1.87134	−0.935670	−	0.352877i	$-0.885204\pi$
$139$	−5.92522e7	−1.87134	−0.935670	+	0.352877i	$0.885204\pi$
$140$	6.44966e6	0.198650
$141$	0	0
$142$	−3.78651e7	−1.10976
$143$	−5.27459e7	−1.50839
$144$	0	0
$145$	5.19967e7	1.41641
$146$	680888.	0.0181068
$147$	0	0
$148$	2.90541e7	0.736702
$149$	−2.32632e7	−0.576127	−0.288064	−	0.957611i	$-0.593011\pi$
$149$	−2.32632e7	−0.576127	−0.288064	+	0.957611i	$0.593011\pi$
$150$	0	0
$151$	−5.82460e7	−1.37672	−0.688361	−	0.725368i	$-0.741670\pi$
$151$	−5.82460e7	−1.37672	−0.688361	+	0.725368i	$0.741670\pi$
$152$	−1.14796e7	−0.265138
$153$	0	0
$154$	−9.61248e6	−0.212086
$155$	2.95838e7	0.638107
$156$	0	0
$157$	−2.93576e7	−0.605441	−0.302721	−	0.953079i	$-0.597895\pi$
$157$	−2.93576e7	−0.605441	−0.302721	+	0.953079i	$0.597895\pi$
$158$	9.44655e6	0.190534
$159$	0	0
$160$	−1.02236e7	−0.197326
$161$	1.86591e7	0.352370
$162$	0	0
$163$	−3.50196e7	−0.633366	−0.316683	−	0.948531i	$-0.602569\pi$
$163$	−3.50196e7	−0.633366	−0.316683	+	0.948531i	$0.602569\pi$
$164$	4.01326e7	0.710467
$165$	0	0
$166$	8.93222e6	0.151559
$167$	−6.04548e7	−1.00444	−0.502219	−	0.864741i	$-0.667483\pi$
$167$	−6.04548e7	−1.00444	−0.502219	+	0.864741i	$0.667483\pi$
$168$	0	0
$169$	1.38296e8	2.20396
$170$	−3.97164e7	−0.620009
$171$	0	0
$172$	−2.71821e6	−0.0407318
$173$	6.43480e7	0.944874	0.472437	−	0.881364i	$-0.343374\pi$
$173$	6.43480e7	0.944874	0.472437	+	0.881364i	$0.343374\pi$
$174$	0	0
$175$	6.20774e6	0.0875589
$176$	1.52371e7	0.210673
$177$	0	0
$178$	−7.49451e7	−0.996032
$179$	1.05862e7	0.137960	0.0689799	−	0.997618i	$-0.478026\pi$
$179$	1.05862e7	0.137960	0.0689799	+	0.997618i	$0.478026\pi$
$180$	0	0
$181$	6.41578e7	0.804219	0.402109	−	0.915592i	$-0.368277\pi$
$181$	6.41578e7	0.804219	0.402109	+	0.915592i	$0.368277\pi$
$182$	3.66385e7	0.450493
$183$	0	0
$184$	−2.95772e7	−0.350022
$185$	1.41639e8	1.64468
$186$	0	0
$187$	5.91926e7	0.661945
$188$	−7.90564e7	−0.867730
$189$	0	0
$190$	−5.59628e7	−0.591919
$191$	1.56333e8	1.62343	0.811715	−	0.584054i	$-0.198534\pi$
$191$	1.56333e8	1.62343	0.811715	+	0.584054i	$0.198534\pi$
$192$	0	0
$193$	−3.53259e7	−0.353706	−0.176853	−	0.984237i	$-0.556592\pi$
$193$	−3.53259e7	−0.353706	−0.176853	+	0.984237i	$0.556592\pi$
$194$	1.63200e7	0.160477
$195$	0	0
$196$	−4.60297e7	−0.436658
$197$	−1.06306e8	−0.990666	−0.495333	−	0.868703i	$-0.664954\pi$
$197$	−1.06306e8	−0.990666	−0.495333	+	0.868703i	$0.664954\pi$
$198$	0	0
$199$	1.59628e7	0.143590	0.0717949	−	0.997419i	$-0.477127\pi$
$199$	1.59628e7	0.143590	0.0717949	+	0.997419i	$0.477127\pi$
$200$	−9.84013e6	−0.0869753
$201$	0	0
$202$	1.22060e8	1.04194
$203$	5.38299e7	0.451635
$204$	0	0
$205$	1.95646e8	1.58611
$206$	1.53947e8	1.22697
$207$	0	0
$208$	−5.80772e7	−0.447491
$209$	8.34061e7	0.631955
$210$	0	0
$211$	−2.65780e7	−0.194775	−0.0973876	−	0.995247i	$-0.531049\pi$
$211$	−2.65780e7	−0.194775	−0.0973876	+	0.995247i	$0.531049\pi$
$212$	6.86592e6	0.0494907
$213$	0	0
$214$	−1.02857e8	−0.717439
$215$	−1.32513e7	−0.0909332
$216$	0	0
$217$	3.06269e7	0.203467
$218$	8.22683e7	0.537818
$219$	0	0
$220$	7.42810e7	0.470325
$221$	−2.25616e8	−1.40604
$222$	0	0
$223$	−1.71223e8	−1.03394	−0.516969	−	0.856004i	$-0.672940\pi$
$223$	−1.71223e8	−1.03394	−0.516969	+	0.856004i	$0.672940\pi$
$224$	−1.05841e7	−0.0629194
$225$	0	0
$226$	−1.49418e8	−0.861040
$227$	−4.33897e7	−0.246205	−0.123102	−	0.992394i	$-0.539284\pi$
$227$	−4.33897e7	−0.246205	−0.123102	+	0.992394i	$0.539284\pi$
$228$	0	0
$229$	2.64398e8	1.45490	0.727451	−	0.686159i	$-0.240705\pi$
$229$	2.64398e8	1.45490	0.727451	+	0.686159i	$0.240705\pi$
$230$	−1.44189e8	−0.781420
$231$	0	0
$232$	−8.53279e7	−0.448624
$233$	−1.69022e8	−0.875380	−0.437690	−	0.899126i	$-0.644203\pi$
$233$	−1.69022e8	−0.875380	−0.437690	+	0.899126i	$0.644203\pi$
$234$	0	0
$235$	−3.85400e8	−1.93720
$236$	−1.58670e8	−0.785785
$237$	0	0
$238$	−4.11166e7	−0.197696
$239$	6.00906e7	0.284717	0.142359	−	0.989815i	$-0.454531\pi$
$239$	6.00906e7	0.284717	0.142359	+	0.989815i	$0.454531\pi$
$240$	0	0
$241$	−4.84056e7	−0.222759	−0.111380	−	0.993778i	$-0.535527\pi$
$241$	−4.84056e7	−0.222759	−0.111380	+	0.993778i	$0.535527\pi$
$242$	4.51902e7	0.204970
$243$	0	0
$244$	1.83961e8	0.810700
$245$	−2.24395e8	−0.974836
$246$	0	0
$247$	−3.17907e8	−1.34234
$248$	−4.85478e7	−0.202111
$249$	0	0
$250$	1.47029e8	0.595133
$251$	2.72664e7	0.108835	0.0544176	−	0.998518i	$-0.482670\pi$
$251$	2.72664e7	0.108835	0.0544176	+	0.998518i	$0.482670\pi$
$252$	0	0
$253$	2.14897e8	0.834273
$254$	−2.82927e7	−0.108332
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−2.68776e7	−0.0987701	−0.0493850	−	0.998780i	$-0.515726\pi$
$257$	−2.68776e7	−0.0987701	−0.0493850	+	0.998780i	$0.515726\pi$
$258$	0	0
$259$	1.46633e8	0.524423
$260$	−2.83126e8	−0.999018
$261$	0	0
$262$	2.40804e8	0.827198
$263$	−4.03708e8	−1.36843	−0.684214	−	0.729281i	$-0.739855\pi$
$263$	−4.03708e8	−1.36843	−0.684214	+	0.729281i	$0.739855\pi$
$264$	0	0
$265$	3.34714e7	0.110487
$266$	−5.79359e7	−0.188739
$267$	0	0
$268$	9.60702e7	0.304872
$269$	3.44677e8	1.07964	0.539820	−	0.841780i	$-0.318492\pi$
$269$	3.44677e8	1.07964	0.539820	+	0.841780i	$0.318492\pi$
$270$	0	0
$271$	3.41243e8	1.04153	0.520765	−	0.853700i	$-0.325647\pi$
$271$	3.41243e8	1.04153	0.520765	+	0.853700i	$0.325647\pi$
$272$	6.51756e7	0.196378
$273$	0	0
$274$	5.89860e7	0.173230
$275$	7.14947e7	0.207305
$276$	0	0
$277$	−5.41619e7	−0.153114	−0.0765570	−	0.997065i	$-0.524393\pi$
$277$	−5.41619e7	−0.153114	−0.0765570	+	0.997065i	$0.524393\pi$
$278$	4.74018e8	1.32324
$279$	0	0
$280$	−5.15973e7	−0.140467
$281$	−5.49715e6	−0.0147797	−0.00738985	−	0.999973i	$-0.502352\pi$
$281$	−5.49715e6	−0.0147797	−0.00738985	+	0.999973i	$0.502352\pi$
$282$	0	0
$283$	−5.58773e8	−1.46549	−0.732745	−	0.680503i	$-0.761761\pi$
$283$	−5.58773e8	−1.46549	−0.732745	+	0.680503i	$0.761761\pi$
$284$	3.02921e8	0.784720
$285$	0	0
$286$	4.21967e8	1.06659
$287$	2.02544e8	0.505747
$288$	0	0
$289$	−1.57147e8	−0.382969
$290$	−4.15973e8	−1.00155
$291$	0	0
$292$	−5.44710e6	−0.0128034
$293$	−1.24737e7	−0.0289706	−0.0144853	−	0.999895i	$-0.504611\pi$
$293$	−1.24737e7	−0.0289706	−0.0144853	+	0.999895i	$0.504611\pi$
$294$	0	0
$295$	−7.73518e8	−1.75426
$296$	−2.32433e8	−0.520927
$297$	0	0
$298$	1.86106e8	0.407383
$299$	−8.19092e8	−1.77208
$300$	0	0
$301$	−1.37185e7	−0.0289950
$302$	4.65968e8	0.973490
$303$	0	0
$304$	9.18364e7	0.187481
$305$	8.96807e8	1.80988
$306$	0	0
$307$	−1.80343e7	−0.0355725	−0.0177863	−	0.999842i	$-0.505662\pi$
$307$	−1.80343e7	−0.0355725	−0.0177863	+	0.999842i	$0.505662\pi$
$308$	7.68998e7	0.149968
$309$	0	0
$310$	−2.36671e8	−0.451210
$311$	5.94654e8	1.12099	0.560497	−	0.828157i	$-0.310610\pi$
$311$	5.94654e8	1.12099	0.560497	+	0.828157i	$0.310610\pi$
$312$	0	0
$313$	3.73871e8	0.689154	0.344577	−	0.938758i	$-0.388022\pi$
$313$	3.73871e8	0.689154	0.344577	+	0.938758i	$0.388022\pi$
$314$	2.34861e8	0.428112
$315$	0	0
$316$	−7.55724e7	−0.134728
$317$	5.89363e8	1.03914	0.519572	−	0.854427i	$-0.326091\pi$
$317$	5.89363e8	1.03914	0.519572	+	0.854427i	$0.326091\pi$
$318$	0	0
$319$	6.19960e8	1.06929
$320$	8.17889e7	0.139531
$321$	0	0
$322$	−1.49273e8	−0.249163
$323$	3.56763e8	0.589075
$324$	0	0
$325$	−2.72506e8	−0.440336
$326$	2.80157e8	0.447858
$327$	0	0
$328$	−3.21061e8	−0.502376
$329$	−3.98988e8	−0.617695
$330$	0	0
$331$	−1.15657e9	−1.75297	−0.876487	−	0.481425i	$-0.840119\pi$
$331$	−1.15657e9	−1.75297	−0.876487	+	0.481425i	$0.840119\pi$
$332$	−7.14578e7	−0.107168
$333$	0	0
$334$	4.83638e8	0.710245
$335$	4.68342e8	0.680623
$336$	0	0
$337$	5.78818e8	0.823830	0.411915	−	0.911222i	$-0.364860\pi$
$337$	5.78818e8	0.823830	0.411915	+	0.911222i	$0.364860\pi$
$338$	−1.10636e9	−1.55844
$339$	0	0
$340$	3.17731e8	0.438413
$341$	3.52730e8	0.481729
$342$	0	0
$343$	−4.98311e8	−0.666762
$344$	2.17457e7	0.0288017
$345$	0	0
$346$	−5.14784e8	−0.668127
$347$	3.25000e8	0.417571	0.208785	−	0.977961i	$-0.433049\pi$
$347$	3.25000e8	0.417571	0.208785	+	0.977961i	$0.433049\pi$
$348$	0	0
$349$	2.50765e8	0.315776	0.157888	−	0.987457i	$-0.449532\pi$
$349$	2.50765e8	0.315776	0.157888	+	0.987457i	$0.449532\pi$
$350$	−4.96619e7	−0.0619135
$351$	0	0
$352$	−1.21897e8	−0.148968
$353$	−1.04075e9	−1.25931	−0.629656	−	0.776874i	$-0.716804\pi$
$353$	−1.04075e9	−1.25931	−0.629656	+	0.776874i	$0.716804\pi$
$354$	0	0
$355$	1.47674e9	1.75188
$356$	5.99561e8	0.704301
$357$	0	0
$358$	−8.46893e7	−0.0975524
$359$	−1.02411e9	−1.16820	−0.584101	−	0.811681i	$-0.698553\pi$
$359$	−1.02411e9	−1.16820	−0.584101	+	0.811681i	$0.698553\pi$
$360$	0	0
$361$	−3.91170e8	−0.437614
$362$	−5.13262e8	−0.568668
$363$	0	0
$364$	−2.93108e8	−0.318547
$365$	−2.65546e7	−0.0285835
$366$	0	0
$367$	9.99933e8	1.05594	0.527971	−	0.849263i	$-0.322953\pi$
$367$	9.99933e8	1.05594	0.527971	+	0.849263i	$0.322953\pi$
$368$	2.36618e8	0.247503
$369$	0	0
$370$	−1.13311e9	−1.16297
$371$	3.46514e7	0.0352300
$372$	0	0
$373$	1.92581e8	0.192146	0.0960732	−	0.995374i	$-0.469372\pi$
$373$	1.92581e8	0.192146	0.0960732	+	0.995374i	$0.469372\pi$
$374$	−4.73541e8	−0.468066
$375$	0	0
$376$	6.32451e8	0.613578
$377$	−2.36302e9	−2.27129
$378$	0	0
$379$	1.91512e9	1.80700	0.903501	−	0.428585i	$-0.140988\pi$
$379$	1.91512e9	1.80700	0.903501	+	0.428585i	$0.140988\pi$
$380$	4.47703e8	0.418550
$381$	0	0
$382$	−1.25066e9	−1.14794
$383$	−1.51110e9	−1.37435	−0.687175	−	0.726492i	$-0.741149\pi$
$383$	−1.51110e9	−1.37435	−0.687175	+	0.726492i	$0.741149\pi$
$384$	0	0
$385$	3.74887e8	0.334802
$386$	2.82607e8	0.250108
$387$	0	0
$388$	−1.30560e8	−0.113474
$389$	7.37168e8	0.634955	0.317477	−	0.948266i	$-0.397164\pi$
$389$	7.37168e8	0.634955	0.317477	+	0.948266i	$0.397164\pi$
$390$	0	0
$391$	9.19204e8	0.777667
$392$	3.68238e8	0.308764
$393$	0	0
$394$	8.50451e8	0.700507
$395$	−3.68416e8	−0.300779
$396$	0	0
$397$	−8.55916e8	−0.686538	−0.343269	−	0.939237i	$-0.611534\pi$
$397$	−8.55916e8	−0.686538	−0.343269	+	0.939237i	$0.611534\pi$
$398$	−1.27702e8	−0.101533
$399$	0	0
$400$	7.87210e7	0.0615008
$401$	2.38953e8	0.185058	0.0925288	−	0.995710i	$-0.470505\pi$
$401$	2.38953e8	0.185058	0.0925288	+	0.995710i	$0.470505\pi$
$402$	0	0
$403$	−1.34445e9	−1.02324
$404$	−9.76478e8	−0.736763
$405$	0	0
$406$	−4.30639e8	−0.319354
$407$	1.68877e9	1.24163
$408$	0	0
$409$	−9.53539e8	−0.689139	−0.344570	−	0.938761i	$-0.611975\pi$
$409$	−9.53539e8	−0.689139	−0.344570	+	0.938761i	$0.611975\pi$
$410$	−1.56517e9	−1.12155
$411$	0	0
$412$	−1.23157e9	−0.867601
$413$	−8.00789e8	−0.559362
$414$	0	0
$415$	−3.48357e8	−0.239252
$416$	4.64617e8	0.316424
$417$	0	0
$418$	−6.67249e8	−0.446859
$419$	5.44668e8	0.361729	0.180864	−	0.983508i	$-0.442111\pi$
$419$	5.44668e8	0.361729	0.180864	+	0.983508i	$0.442111\pi$
$420$	0	0
$421$	−6.61270e8	−0.431909	−0.215954	−	0.976403i	$-0.569286\pi$
$421$	−6.61270e8	−0.431909	−0.215954	+	0.976403i	$0.569286\pi$
$422$	2.12624e8	0.137727
$423$	0	0
$424$	−5.49274e7	−0.0349952
$425$	3.05813e8	0.193239
$426$	0	0
$427$	9.28426e8	0.577098
$428$	8.22855e8	0.507306
$429$	0	0
$430$	1.06010e8	0.0642995
$431$	2.10789e8	0.126817	0.0634086	−	0.997988i	$-0.479803\pi$
$431$	2.10789e8	0.126817	0.0634086	+	0.997988i	$0.479803\pi$
$432$	0	0
$433$	−1.72888e9	−1.02343	−0.511714	−	0.859156i	$-0.670989\pi$
$433$	−1.72888e9	−1.02343	−0.511714	+	0.859156i	$0.670989\pi$
$434$	−2.45015e8	−0.143873
$435$	0	0
$436$	−6.58147e8	−0.380294
$437$	1.29522e9	0.742433
$438$	0	0
$439$	1.39429e8	0.0786554	0.0393277	−	0.999226i	$-0.487478\pi$
$439$	1.39429e8	0.0786554	0.0393277	+	0.999226i	$0.487478\pi$
$440$	−5.94248e8	−0.332570
$441$	0	0
$442$	1.80493e9	0.994220
$443$	1.40121e9	0.765754	0.382877	−	0.923799i	$-0.374933\pi$
$443$	1.40121e9	0.765754	0.382877	+	0.923799i	$0.374933\pi$
$444$	0	0
$445$	2.92286e9	1.57234
$446$	1.36978e9	0.731104
$447$	0	0
$448$	8.46725e7	0.0444907
$449$	−1.78420e7	−0.00930209	−0.00465104	−	0.999989i	$-0.501480\pi$
$449$	−1.78420e7	−0.00930209	−0.00465104	+	0.999989i	$0.501480\pi$
$450$	0	0
$451$	2.33271e9	1.19741
$452$	1.19534e9	0.608847
$453$	0	0
$454$	3.47118e8	0.174093
$455$	−1.42890e9	−0.711153
$456$	0	0
$457$	−1.74413e9	−0.854816	−0.427408	−	0.904059i	$-0.640573\pi$
$457$	−1.74413e9	−0.854816	−0.427408	+	0.904059i	$0.640573\pi$
$458$	−2.11518e9	−1.02877
$459$	0	0
$460$	1.15351e9	0.552547
$461$	1.35991e9	0.646483	0.323242	−	0.946316i	$-0.395227\pi$
$461$	1.35991e9	0.646483	0.323242	+	0.946316i	$0.395227\pi$
$462$	0	0
$463$	−1.97759e9	−0.925981	−0.462991	−	0.886363i	$-0.653224\pi$
$463$	−1.97759e9	−0.925981	−0.462991	+	0.886363i	$0.653224\pi$
$464$	6.82623e8	0.317225
$465$	0	0
$466$	1.35217e9	0.618987
$467$	3.46852e9	1.57592	0.787961	−	0.615726i	$-0.211137\pi$
$467$	3.46852e9	1.57592	0.787961	+	0.615726i	$0.211137\pi$
$468$	0	0
$469$	4.84854e8	0.217023
$470$	3.08320e9	1.36981
$471$	0	0
$472$	1.26936e9	0.555634
$473$	−1.57996e8	−0.0686486
$474$	0	0
$475$	4.30909e8	0.184484
$476$	3.28933e8	0.139792
$477$	0	0
$478$	−4.80725e8	−0.201325
$479$	2.18042e9	0.906494	0.453247	−	0.891385i	$-0.350265\pi$
$479$	2.18042e9	0.906494	0.453247	+	0.891385i	$0.350265\pi$
$480$	0	0
$481$	−6.43685e9	−2.63734
$482$	3.87245e8	0.157515
$483$	0	0
$484$	−3.61521e8	−0.144936
$485$	−6.36478e8	−0.253331
$486$	0	0
$487$	3.00745e9	1.17990	0.589952	−	0.807439i	$-0.299147\pi$
$487$	3.00745e9	1.17990	0.589952	+	0.807439i	$0.299147\pi$
$488$	−1.47168e9	−0.573252
$489$	0	0
$490$	1.79516e9	0.689313
$491$	−2.62970e9	−1.00258	−0.501292	−	0.865278i	$-0.667142\pi$
$491$	−2.62970e9	−1.00258	−0.501292	+	0.865278i	$0.667142\pi$
$492$	0	0
$493$	2.65183e9	0.996739
$494$	2.54326e9	0.949175
$495$	0	0
$496$	3.88383e8	0.142914
$497$	1.52880e9	0.558604
$498$	0	0
$499$	−2.76990e8	−0.0997957	−0.0498978	−	0.998754i	$-0.515890\pi$
$499$	−2.76990e8	−0.0997957	−0.0498978	+	0.998754i	$0.515890\pi$
$500$	−1.17624e9	−0.420823
$501$	0	0
$502$	−2.18131e8	−0.0769582
$503$	−4.59216e9	−1.60890	−0.804451	−	0.594019i	$-0.797540\pi$
$503$	−4.59216e9	−1.60890	−0.804451	+	0.594019i	$0.797540\pi$
$504$	0	0
$505$	−4.76033e9	−1.64482
$506$	−1.71918e9	−0.589920
$507$	0	0
$508$	2.26342e8	0.0766022
$509$	3.25466e9	1.09394	0.546970	−	0.837152i	$-0.315781\pi$
$509$	3.25466e9	1.09394	0.546970	+	0.837152i	$0.315781\pi$
$510$	0	0
$511$	−2.74909e7	−0.00911413
$512$	−1.34218e8	−0.0441942
$513$	0	0
$514$	2.15021e8	0.0698410
$515$	−6.00392e9	−1.93691
$516$	0	0
$517$	−4.59515e9	−1.46246
$518$	−1.17306e9	−0.370823
$519$	0	0
$520$	2.26501e9	0.706413
$521$	5.57306e9	1.72648	0.863241	−	0.504793i	$-0.168431\pi$
$521$	5.57306e9	1.72648	0.863241	+	0.504793i	$0.168431\pi$
$522$	0	0
$523$	2.33783e9	0.714590	0.357295	−	0.933992i	$-0.383699\pi$
$523$	2.33783e9	0.714590	0.357295	+	0.933992i	$0.383699\pi$
$524$	−1.92643e9	−0.584917
$525$	0	0
$526$	3.22966e9	0.967625
$527$	1.50878e9	0.449043
$528$	0	0
$529$	−6.76836e7	−0.0198787
$530$	−2.67771e8	−0.0781264
$531$	0	0
$532$	4.63487e8	0.133459
$533$	−8.89125e9	−2.54342
$534$	0	0
$535$	4.01142e9	1.13256
$536$	−7.68562e8	−0.215577
$537$	0	0
$538$	−2.75741e9	−0.763421
$539$	−2.67548e9	−0.735937
$540$	0	0
$541$	−5.60874e9	−1.52291	−0.761456	−	0.648217i	$-0.775515\pi$
$541$	−5.60874e9	−1.52291	−0.761456	+	0.648217i	$0.775515\pi$
$542$	−2.72995e9	−0.736472
$543$	0	0
$544$	−5.21404e8	−0.138860
$545$	−3.20847e9	−0.849004
$546$	0	0
$547$	−4.51033e9	−1.17829	−0.589146	−	0.808027i	$-0.700536\pi$
$547$	−4.51033e9	−1.17829	−0.589146	+	0.808027i	$0.700536\pi$
$548$	−4.71888e8	−0.122492
$549$	0	0
$550$	−5.71957e8	−0.146587
$551$	3.73659e9	0.951580
$552$	0	0
$553$	−3.81405e8	−0.0959065
$554$	4.33295e8	0.108268
$555$	0	0
$556$	−3.79214e9	−0.935670
$557$	3.30015e8	0.0809171	0.0404585	−	0.999181i	$-0.487118\pi$
$557$	3.30015e8	0.0809171	0.0404585	+	0.999181i	$0.487118\pi$
$558$	0	0
$559$	6.02210e8	0.145817
$560$	4.12778e8	0.0993251
$561$	0	0
$562$	4.39772e7	0.0104508
$563$	2.57572e9	0.608303	0.304151	−	0.952624i	$-0.401627\pi$
$563$	2.57572e9	0.608303	0.304151	+	0.952624i	$0.401627\pi$
$564$	0	0
$565$	5.82730e9	1.35925
$566$	4.47018e9	1.03626
$567$	0	0
$568$	−2.42337e9	−0.554881
$569$	−2.22891e9	−0.507225	−0.253612	−	0.967306i	$-0.581619\pi$
$569$	−2.22891e9	−0.507225	−0.253612	+	0.967306i	$0.581619\pi$
$570$	0	0
$571$	2.83433e9	0.637123	0.318562	−	0.947902i	$-0.396800\pi$
$571$	2.83433e9	0.637123	0.318562	+	0.947902i	$0.396800\pi$
$572$	−3.37574e9	−0.754193
$573$	0	0
$574$	−1.62035e9	−0.357617
$575$	1.11024e9	0.243546
$576$	0	0
$577$	1.24427e7	0.00269649	0.00134825	−	0.999999i	$-0.499571\pi$
$577$	1.24427e7	0.00269649	0.00134825	+	0.999999i	$0.499571\pi$
$578$	1.25718e9	0.270800
$579$	0	0
$580$	3.32779e9	0.708203
$581$	−3.60639e8	−0.0762879
$582$	0	0
$583$	3.99082e8	0.0834107
$584$	4.35768e7	0.00905338
$585$	0	0
$586$	9.97895e7	0.0204853
$587$	−8.61410e9	−1.75783	−0.878914	−	0.476980i	$-0.841731\pi$
$587$	−8.61410e9	−1.75783	−0.878914	+	0.476980i	$0.841731\pi$
$588$	0	0
$589$	2.12596e9	0.428698
$590$	6.18814e9	1.24045
$591$	0	0
$592$	1.85947e9	0.368351
$593$	−4.55490e9	−0.896989	−0.448495	−	0.893786i	$-0.648040\pi$
$593$	−4.55490e9	−0.896989	−0.448495	+	0.893786i	$0.648040\pi$
$594$	0	0
$595$	1.60355e9	0.312085
$596$	−1.48885e9	−0.288064
$597$	0	0
$598$	6.55274e9	1.25305
$599$	6.57167e8	0.124934	0.0624672	−	0.998047i	$-0.480103\pi$
$599$	6.57167e8	0.124934	0.0624672	+	0.998047i	$0.480103\pi$
$600$	0	0
$601$	8.39406e9	1.57729	0.788645	−	0.614849i	$-0.210783\pi$
$601$	8.39406e9	1.57729	0.788645	+	0.614849i	$0.210783\pi$
$602$	1.09748e8	0.0205025
$603$	0	0
$604$	−3.72774e9	−0.688361
$605$	−1.76242e9	−0.323567
$606$	0	0
$607$	−1.04953e9	−0.190473	−0.0952363	−	0.995455i	$-0.530361\pi$
$607$	−1.04953e9	−0.190473	−0.0952363	+	0.995455i	$0.530361\pi$
$608$	−7.34691e8	−0.132569
$609$	0	0
$610$	−7.17446e9	−1.27978
$611$	1.75147e10	3.10641
$612$	0	0
$613$	4.22848e9	0.741433	0.370717	−	0.928746i	$-0.379112\pi$
$613$	4.22848e9	0.741433	0.370717	+	0.928746i	$0.379112\pi$
$614$	1.44274e8	0.0251536
$615$	0	0
$616$	−6.15199e8	−0.106043
$617$	4.85460e9	0.832061	0.416031	−	0.909351i	$-0.363421\pi$
$617$	4.85460e9	0.832061	0.416031	+	0.909351i	$0.363421\pi$
$618$	0	0
$619$	2.32275e9	0.393627	0.196813	−	0.980441i	$-0.436941\pi$
$619$	2.32275e9	0.393627	0.196813	+	0.980441i	$0.436941\pi$
$620$	1.89337e9	0.319054
$621$	0	0
$622$	−4.75723e9	−0.792662
$623$	3.02591e9	0.501358
$624$	0	0
$625$	−7.23563e9	−1.18549
$626$	−2.99097e9	−0.487306
$627$	0	0
$628$	−1.87889e9	−0.302721
$629$	7.22359e9	1.15738
$630$	0	0
$631$	5.84987e9	0.926923	0.463461	−	0.886117i	$-0.346607\pi$
$631$	5.84987e9	0.926923	0.463461	+	0.886117i	$0.346607\pi$
$632$	6.04579e8	0.0952672
$633$	0	0
$634$	−4.71491e9	−0.734786
$635$	1.10342e9	0.171014
$636$	0	0
$637$	1.01977e10	1.56320
$638$	−4.95968e9	−0.756104
$639$	0	0
$640$	−6.54311e8	−0.0986631
$641$	9.81719e9	1.47226	0.736129	−	0.676841i	$-0.236652\pi$
$641$	9.81719e9	1.47226	0.736129	+	0.676841i	$0.236652\pi$
$642$	0	0
$643$	−4.73044e9	−0.701719	−0.350859	−	0.936428i	$-0.614110\pi$
$643$	−4.73044e9	−0.701719	−0.350859	+	0.936428i	$0.614110\pi$
$644$	1.19418e9	0.176185
$645$	0	0
$646$	−2.85410e9	−0.416539
$647$	−4.28999e9	−0.622718	−0.311359	−	0.950292i	$-0.600784\pi$
$647$	−4.28999e9	−0.622718	−0.311359	+	0.950292i	$0.600784\pi$
$648$	0	0
$649$	−9.22271e9	−1.32435
$650$	2.18005e9	0.311365
$651$	0	0
$652$	−2.24126e9	−0.316683
$653$	−1.32108e10	−1.85666	−0.928328	−	0.371762i	$-0.878754\pi$
$653$	−1.32108e10	−1.85666	−0.928328	+	0.371762i	$0.878754\pi$
$654$	0	0
$655$	−9.39136e9	−1.30582
$656$	2.56849e9	0.355234
$657$	0	0
$658$	3.19190e9	0.436776
$659$	5.60547e9	0.762980	0.381490	−	0.924373i	$-0.375411\pi$
$659$	5.60547e9	0.762980	0.381490	+	0.924373i	$0.375411\pi$
$660$	0	0
$661$	−1.77070e9	−0.238474	−0.119237	−	0.992866i	$-0.538045\pi$
$661$	−1.77070e9	−0.238474	−0.119237	+	0.992866i	$0.538045\pi$
$662$	9.25259e9	1.23954
$663$	0	0
$664$	5.71662e8	0.0757794
$665$	2.25950e9	0.297945
$666$	0	0
$667$	9.62738e9	1.25623
$668$	−3.86911e9	−0.502219
$669$	0	0
$670$	−3.74674e9	−0.481273
$671$	1.06927e10	1.36634
$672$	0	0
$673$	−7.36036e8	−0.0930778	−0.0465389	−	0.998916i	$-0.514819\pi$
$673$	−7.36036e8	−0.0930778	−0.0465389	+	0.998916i	$0.514819\pi$
$674$	−4.63055e9	−0.582536
$675$	0	0
$676$	8.85091e9	1.10198
$677$	6.22562e9	0.771120	0.385560	−	0.922683i	$-0.374008\pi$
$677$	6.22562e9	0.771120	0.385560	+	0.922683i	$0.374008\pi$
$678$	0	0
$679$	−6.58918e8	−0.0807769
$680$	−2.54185e9	−0.310005
$681$	0	0
$682$	−2.82184e9	−0.340634
$683$	−6.78088e9	−0.814355	−0.407178	−	0.913349i	$-0.633487\pi$
$683$	−6.78088e9	−0.814355	−0.407178	+	0.913349i	$0.633487\pi$
$684$	0	0
$685$	−2.30046e9	−0.273462
$686$	3.98648e9	0.471472
$687$	0	0
$688$	−1.73965e8	−0.0203659
$689$	−1.52112e9	−0.177173
$690$	0	0
$691$	−4.14997e9	−0.478489	−0.239244	−	0.970959i	$-0.576900\pi$
$691$	−4.14997e9	−0.478489	−0.239244	+	0.970959i	$0.576900\pi$
$692$	4.11827e9	0.472437
$693$	0	0
$694$	−2.60000e9	−0.295267
$695$	−1.84867e10	−2.08887
$696$	0	0
$697$	9.97797e9	1.11616
$698$	−2.00612e9	−0.223287
$699$	0	0
$700$	3.97295e8	0.0437794
$701$	−1.61210e10	−1.76758	−0.883791	−	0.467881i	$-0.845017\pi$
$701$	−1.61210e10	−1.76758	−0.883791	+	0.467881i	$0.845017\pi$
$702$	0	0
$703$	1.01785e10	1.10494
$704$	9.75176e8	0.105336
$705$	0	0
$706$	8.32596e9	0.890468
$707$	−4.92816e9	−0.524466
$708$	0	0
$709$	1.13288e10	1.19378	0.596889	−	0.802324i	$-0.296403\pi$
$709$	1.13288e10	1.19378	0.596889	+	0.802324i	$0.296403\pi$
$710$	−1.18139e10	−1.23877
$711$	0	0
$712$	−4.79649e9	−0.498016
$713$	5.47756e9	0.565945
$714$	0	0
$715$	−1.64567e10	−1.68373
$716$	6.77514e8	0.0689799
$717$	0	0
$718$	8.19292e9	0.826044
$719$	−8.97759e9	−0.900760	−0.450380	−	0.892837i	$-0.648711\pi$
$719$	−8.97759e9	−0.900760	−0.450380	+	0.892837i	$0.648711\pi$
$720$	0	0
$721$	−6.21560e9	−0.617603
$722$	3.12936e9	0.309440
$723$	0	0
$724$	4.10610e9	0.402109
$725$	3.20296e9	0.312154
$726$	0	0
$727$	−1.98049e10	−1.91163	−0.955813	−	0.293976i	$-0.905021\pi$
$727$	−1.98049e10	−1.91163	−0.955813	+	0.293976i	$0.905021\pi$
$728$	2.34487e9	0.225247
$729$	0	0
$730$	2.12437e8	0.0202116
$731$	−6.75814e8	−0.0639907
$732$	0	0
$733$	1.88605e10	1.76884	0.884420	−	0.466692i	$-0.154554\pi$
$733$	1.88605e10	1.76884	0.884420	+	0.466692i	$0.154554\pi$
$734$	−7.99946e9	−0.746663
$735$	0	0
$736$	−1.89294e9	−0.175011
$737$	5.58408e9	0.513825
$738$	0	0
$739$	−1.38397e10	−1.26145	−0.630725	−	0.776006i	$-0.717243\pi$
$739$	−1.38397e10	−1.26145	−0.630725	+	0.776006i	$0.717243\pi$
$740$	9.06489e9	0.822340
$741$	0	0
$742$	−2.77212e8	−0.0249114
$743$	−1.80884e10	−1.61785	−0.808926	−	0.587911i	$-0.799951\pi$
$743$	−1.80884e10	−1.61785	−0.808926	+	0.587911i	$0.799951\pi$
$744$	0	0
$745$	−7.25813e9	−0.643099
$746$	−1.54065e9	−0.135868
$747$	0	0
$748$	3.78833e9	0.330973
$749$	4.15285e9	0.361127
$750$	0	0
$751$	−2.22027e9	−0.191278	−0.0956390	−	0.995416i	$-0.530489\pi$
$751$	−2.22027e9	−0.191278	−0.0956390	+	0.995416i	$0.530489\pi$
$752$	−5.05961e9	−0.433865
$753$	0	0
$754$	1.89041e10	1.60604
$755$	−1.81727e10	−1.53676
$756$	0	0
$757$	2.21883e10	1.85904	0.929521	−	0.368770i	$-0.120221\pi$
$757$	2.21883e10	1.85904	0.929521	+	0.368770i	$0.120221\pi$
$758$	−1.53210e10	−1.27774
$759$	0	0
$760$	−3.58162e9	−0.295959
$761$	3.03821e9	0.249903	0.124952	−	0.992163i	$-0.460122\pi$
$761$	3.03821e9	0.249903	0.124952	+	0.992163i	$0.460122\pi$
$762$	0	0
$763$	−3.32158e9	−0.270713
$764$	1.00053e10	0.811715
$765$	0	0
$766$	1.20888e10	0.971812
$767$	3.51529e10	2.81305
$768$	0	0
$769$	−1.82409e10	−1.44645	−0.723226	−	0.690611i	$-0.757342\pi$
$769$	−1.82409e10	−1.44645	−0.723226	+	0.690611i	$0.757342\pi$
$770$	−2.99909e9	−0.236740
$771$	0	0
$772$	−2.26086e9	−0.176853
$773$	−1.20659e10	−0.939576	−0.469788	−	0.882779i	$-0.655670\pi$
$773$	−1.20659e10	−0.939576	−0.469788	+	0.882779i	$0.655670\pi$
$774$	0	0
$775$	1.82235e9	0.140629
$776$	1.04448e9	0.0802385
$777$	0	0
$778$	−5.89734e9	−0.448981
$779$	1.40596e10	1.06559
$780$	0	0
$781$	1.76073e10	1.32255
$782$	−7.35364e9	−0.549893
$783$	0	0
$784$	−2.94590e9	−0.218329
$785$	−9.15958e9	−0.675821
$786$	0	0
$787$	1.26799e10	0.927269	0.463634	−	0.886027i	$-0.346545\pi$
$787$	1.26799e10	0.927269	0.463634	+	0.886027i	$0.346545\pi$
$788$	−6.80361e9	−0.495333
$789$	0	0
$790$	2.94732e9	0.212683
$791$	6.03275e9	0.433409
$792$	0	0
$793$	−4.07559e10	−2.90225
$794$	6.84733e9	0.485456
$795$	0	0
$796$	1.02162e9	0.0717949
$797$	4.86451e9	0.340357	0.170179	−	0.985413i	$-0.445565\pi$
$797$	4.86451e9	0.340357	0.170179	+	0.985413i	$0.445565\pi$
$798$	0	0
$799$	−1.96554e10	−1.36323
$800$	−6.29768e8	−0.0434876
$801$	0	0
$802$	−1.91162e9	−0.130856
$803$	−3.16613e8	−0.0215786
$804$	0	0
$805$	5.82163e9	0.393332
$806$	1.07556e10	0.723541
$807$	0	0
$808$	7.81183e9	0.520970
$809$	−1.07879e10	−0.716338	−0.358169	−	0.933657i	$-0.616599\pi$
$809$	−1.07879e10	−0.716338	−0.358169	+	0.933657i	$0.616599\pi$
$810$	0	0
$811$	−2.41592e10	−1.59041	−0.795204	−	0.606341i	$-0.792636\pi$
$811$	−2.41592e10	−1.59041	−0.795204	+	0.606341i	$0.792636\pi$
$812$	3.44511e9	0.225817
$813$	0	0
$814$	−1.35102e10	−0.877962
$815$	−1.09261e10	−0.706992
$816$	0	0
$817$	−9.52265e8	−0.0610915
$818$	7.62831e9	0.487295
$819$	0	0
$820$	1.25214e10	0.793055
$821$	2.71882e10	1.71467	0.857333	−	0.514763i	$-0.172120\pi$
$821$	2.71882e10	1.71467	0.857333	+	0.514763i	$0.172120\pi$
$822$	0	0
$823$	7.45342e9	0.466075	0.233038	−	0.972468i	$-0.425133\pi$
$823$	7.45342e9	0.466075	0.233038	+	0.972468i	$0.425133\pi$
$824$	9.85259e9	0.613487
$825$	0	0
$826$	6.40631e9	0.395529
$827$	7.48871e9	0.460402	0.230201	−	0.973143i	$-0.426062\pi$
$827$	7.48871e9	0.460402	0.230201	+	0.973143i	$0.426062\pi$
$828$	0	0
$829$	5.23995e9	0.319437	0.159719	−	0.987163i	$-0.448941\pi$
$829$	5.23995e9	0.319437	0.159719	+	0.987163i	$0.448941\pi$
$830$	2.78685e9	0.169177
$831$	0	0
$832$	−3.71694e9	−0.223745
$833$	−1.14441e10	−0.686002
$834$	0	0
$835$	−1.88619e10	−1.12120
$836$	5.33799e9	0.315977
$837$	0	0
$838$	−4.35734e9	−0.255781
$839$	−4.92901e9	−0.288133	−0.144066	−	0.989568i	$-0.546018\pi$
$839$	−4.92901e9	−0.288133	−0.144066	+	0.989568i	$0.546018\pi$
$840$	0	0
$841$	1.05243e10	0.610111
$842$	5.29016e9	0.305405
$843$	0	0
$844$	−1.70099e9	−0.0973876
$845$	4.31482e10	2.46017
$846$	0	0
$847$	−1.82455e9	−0.103173
$848$	4.39419e8	0.0247453
$849$	0	0
$850$	−2.44650e9	−0.136640
$851$	2.62250e10	1.45869
$852$	0	0
$853$	3.17690e10	1.75260	0.876300	−	0.481767i	$-0.160005\pi$
$853$	3.17690e10	1.75260	0.876300	+	0.481767i	$0.160005\pi$
$854$	−7.42741e9	−0.408070
$855$	0	0
$856$	−6.58284e9	−0.358720
$857$	−2.40083e10	−1.30295	−0.651475	−	0.758670i	$-0.725850\pi$
$857$	−2.40083e10	−1.30295	−0.651475	+	0.758670i	$0.725850\pi$
$858$	0	0
$859$	2.89979e10	1.56096	0.780478	−	0.625183i	$-0.214976\pi$
$859$	2.89979e10	1.56096	0.780478	+	0.625183i	$0.214976\pi$
$860$	−8.48081e8	−0.0454666
$861$	0	0
$862$	−1.68631e9	−0.0896733
$863$	9.49509e9	0.502877	0.251438	−	0.967873i	$-0.419096\pi$
$863$	9.49509e9	0.502877	0.251438	+	0.967873i	$0.419096\pi$
$864$	0	0
$865$	2.00766e10	1.05471
$866$	1.38310e10	0.723673
$867$	0	0
$868$	1.96012e9	0.101733
$869$	−4.39265e9	−0.227069
$870$	0	0
$871$	−2.12841e10	−1.09142
$872$	5.26517e9	0.268909
$873$	0	0
$874$	−1.03617e10	−0.524979
$875$	−5.93631e9	−0.299563
$876$	0	0
$877$	1.79366e10	0.897930	0.448965	−	0.893549i	$-0.351793\pi$
$877$	1.79366e10	0.897930	0.448965	+	0.893549i	$0.351793\pi$
$878$	−1.11543e9	−0.0556177
$879$	0	0
$880$	4.75398e9	0.235163
$881$	5.83458e8	0.0287471	0.0143735	−	0.999897i	$-0.495425\pi$
$881$	5.83458e8	0.0287471	0.0143735	+	0.999897i	$0.495425\pi$
$882$	0	0
$883$	3.60031e10	1.75986	0.879929	−	0.475106i	$-0.157590\pi$
$883$	3.60031e10	1.75986	0.879929	+	0.475106i	$0.157590\pi$
$884$	−1.44394e10	−0.703020
$885$	0	0
$886$	−1.12096e10	−0.541469
$887$	1.65301e10	0.795320	0.397660	−	0.917533i	$-0.369822\pi$
$887$	1.65301e10	0.795320	0.397660	+	0.917533i	$0.369822\pi$
$888$	0	0
$889$	1.14232e9	0.0545294
$890$	−2.33829e10	−1.11182
$891$	0	0
$892$	−1.09583e10	−0.516969
$893$	−2.76957e10	−1.30146
$894$	0	0
$895$	3.30288e9	0.153997
$896$	−6.77380e8	−0.0314597
$897$	0	0
$898$	1.42736e8	0.00657757
$899$	1.58023e10	0.725374
$900$	0	0
$901$	1.70704e9	0.0777511
$902$	−1.86617e10	−0.846696
$903$	0	0
$904$	−9.56276e9	−0.430520
$905$	2.00172e10	0.897705
$906$	0	0
$907$	−3.20080e10	−1.42440	−0.712201	−	0.701975i	$-0.752302\pi$
$907$	−3.20080e10	−1.42440	−0.712201	+	0.701975i	$0.752302\pi$
$908$	−2.77694e9	−0.123102
$909$	0	0
$910$	1.14312e10	0.502861
$911$	−1.23494e10	−0.541168	−0.270584	−	0.962696i	$-0.587217\pi$
$911$	−1.23494e10	−0.541168	−0.270584	+	0.962696i	$0.587217\pi$
$912$	0	0
$913$	−4.15348e9	−0.180620
$914$	1.39531e10	0.604446
$915$	0	0
$916$	1.69215e10	0.727451
$917$	−9.72247e9	−0.416374
$918$	0	0
$919$	−1.97181e10	−0.838032	−0.419016	−	0.907979i	$-0.637625\pi$
$919$	−1.97181e10	−0.838032	−0.419016	+	0.907979i	$0.637625\pi$
$920$	−9.22809e9	−0.390710
$921$	0	0
$922$	−1.08793e10	−0.457133
$923$	−6.71111e10	−2.80924
$924$	0	0
$925$	8.72487e9	0.362462
$926$	1.58207e10	0.654768
$927$	0	0
$928$	−5.46098e9	−0.224312
$929$	−2.86505e10	−1.17240	−0.586201	−	0.810166i	$-0.699377\pi$
$929$	−2.86505e10	−1.17240	−0.586201	+	0.810166i	$0.699377\pi$
$930$	0	0
$931$	−1.61255e10	−0.654921
$932$	−1.08174e10	−0.437690
$933$	0	0
$934$	−2.77481e10	−1.11434
$935$	1.84681e10	0.738893
$936$	0	0
$937$	−4.85335e8	−0.0192732	−0.00963659	−	0.999954i	$-0.503067\pi$
$937$	−4.85335e8	−0.0192732	−0.00963659	+	0.999954i	$0.503067\pi$
$938$	−3.87883e9	−0.153459
$939$	0	0
$940$	−2.46656e10	−0.968599
$941$	2.90015e9	0.113464	0.0567319	−	0.998389i	$-0.481932\pi$
$941$	2.90015e9	0.113464	0.0567319	+	0.998389i	$0.481932\pi$
$942$	0	0
$943$	3.62247e10	1.40674
$944$	−1.01549e10	−0.392892
$945$	0	0
$946$	1.26397e9	0.0485419
$947$	−1.38658e9	−0.0530544	−0.0265272	−	0.999648i	$-0.508445\pi$
$947$	−1.38658e9	−0.0530544	−0.0265272	+	0.999648i	$0.508445\pi$
$948$	0	0
$949$	1.20679e9	0.0458352
$950$	−3.44727e9	−0.130450
$951$	0	0
$952$	−2.63146e9	−0.0988480
$953$	−1.45920e10	−0.546123	−0.273062	−	0.961997i	$-0.588036\pi$
$953$	−1.45920e10	−0.546123	−0.273062	+	0.961997i	$0.588036\pi$
$954$	0	0
$955$	4.87759e10	1.81215
$956$	3.84580e9	0.142359
$957$	0	0
$958$	−1.74433e10	−0.640988
$959$	−2.38156e9	−0.0871960
$960$	0	0
$961$	−1.85218e10	−0.673211
$962$	5.14948e10	1.86488
$963$	0	0
$964$	−3.09796e9	−0.111380
$965$	−1.10217e10	−0.394823
$966$	0	0
$967$	3.50961e9	0.124815	0.0624074	−	0.998051i	$-0.480122\pi$
$967$	3.50961e9	0.124815	0.0624074	+	0.998051i	$0.480122\pi$
$968$	2.89217e9	0.102485
$969$	0	0
$970$	5.09183e9	0.179132
$971$	−1.19076e10	−0.417403	−0.208701	−	0.977979i	$-0.566924\pi$
$971$	−1.19076e10	−0.417403	−0.208701	+	0.977979i	$0.566924\pi$
$972$	0	0
$973$	−1.91385e10	−0.666058
$974$	−2.40596e10	−0.834317
$975$	0	0
$976$	1.17735e10	0.405350
$977$	1.27872e10	0.438676	0.219338	−	0.975649i	$-0.429610\pi$
$977$	1.27872e10	0.438676	0.219338	+	0.975649i	$0.429610\pi$
$978$	0	0
$979$	3.48495e10	1.18702
$980$	−1.43613e10	−0.487418
$981$	0	0
$982$	2.10376e10	0.708934
$983$	−2.75522e10	−0.925163	−0.462582	−	0.886577i	$-0.653077\pi$
$983$	−2.75522e10	−0.925163	−0.462582	+	0.886577i	$0.653077\pi$
$984$	0	0
$985$	−3.31676e10	−1.10583
$986$	−2.12146e10	−0.704801
$987$	0	0
$988$	−2.03461e10	−0.671168
$989$	−2.45352e9	−0.0806497
$990$	0	0
$991$	1.90312e9	0.0621167	0.0310584	−	0.999518i	$-0.490112\pi$
$991$	1.90312e9	0.0621167	0.0310584	+	0.999518i	$0.490112\pi$
$992$	−3.10706e9	−0.101055
$993$	0	0
$994$	−1.22304e10	−0.394993
$995$	4.98040e9	0.160281
$996$	0	0
$997$	−1.52301e10	−0.486711	−0.243355	−	0.969937i	$-0.578248\pi$
$997$	−1.52301e10	−0.486711	−0.243355	+	0.969937i	$0.578248\pi$
$998$	2.21592e9	0.0705662
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.8.a.c.1.1	✓	1
3.2	odd	2		54.8.a.d.1.1	yes	1
4.3	odd	2		432.8.a.h.1.1		1
9.2	odd	6		162.8.c.f.109.1		2
9.4	even	3		162.8.c.g.55.1		2
9.5	odd	6		162.8.c.f.55.1		2
9.7	even	3		162.8.c.g.109.1		2
12.11	even	2		432.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.8.a.c.1.1	✓	1	1.1	even	1	trivial
54.8.a.d.1.1	yes	1	3.2	odd	2
162.8.c.f.55.1		2	9.5	odd	6
162.8.c.f.109.1		2	9.2	odd	6
162.8.c.g.55.1		2	9.4	even	3
162.8.c.g.109.1		2	9.7	even	3
432.8.a.a.1.1		1	12.11	even	2
432.8.a.h.1.1		1	4.3	odd	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:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	54.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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