LMFDB - Embedded newform 384.8.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,8,Mod(1,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, 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("384.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	384.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:	$119.955849786$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 286x - 1680 $ x^3 - 286*x - 1680
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-12.1582$ of defining polynomial
Character	$\chi$	$=$	384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000 q^{3} +223.082 q^{5} +1526.43 q^{7} +729.000 q^{9} +O(q^{10})$	$q-27.0000 q^{3} +223.082 q^{5} +1526.43 q^{7} +729.000 q^{9} +1604.21 q^{11} -2959.89 q^{13} -6023.23 q^{15} -20142.8 q^{17} +7745.63 q^{19} -41213.7 q^{21} -71478.4 q^{23} -28359.2 q^{25} -19683.0 q^{27} -25181.4 q^{29} -270532. q^{31} -43313.6 q^{33} +340520. q^{35} -394033. q^{37} +79917.0 q^{39} +181640. q^{41} -555492. q^{43} +162627. q^{45} -669993. q^{47} +1.50646e6 q^{49} +543856. q^{51} +772991. q^{53} +357870. q^{55} -209132. q^{57} -1.30252e6 q^{59} +1.68711e6 q^{61} +1.11277e6 q^{63} -660299. q^{65} -1.08822e6 q^{67} +1.92992e6 q^{69} +859754. q^{71} +2.61965e6 q^{73} +765699. q^{75} +2.44871e6 q^{77} -6.33710e6 q^{79} +531441. q^{81} +8.43435e6 q^{83} -4.49351e6 q^{85} +679899. q^{87} -1.18695e7 q^{89} -4.51807e6 q^{91} +7.30435e6 q^{93} +1.72791e6 q^{95} +4.05179e6 q^{97} +1.16947e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 81 q^{3} + 308 q^{5} - 6 q^{7} + 2187 q^{9}+O(q^{10}) $ 3 * q - 81 * q^3 + 308 * q^5 - 6 * q^7 + 2187 * q^9	$ 3 q - 81 q^{3} + 308 q^{5} - 6 q^{7} + 2187 q^{9} - 1468 q^{11} + 3070 q^{13} - 8316 q^{15} - 5126 q^{17} - 14168 q^{19} + 162 q^{21} - 59980 q^{23} + 31529 q^{25} - 59049 q^{27} + 71984 q^{29} - 177470 q^{31} + 39636 q^{33} + 84760 q^{35} - 184014 q^{37} - 82890 q^{39} + 657250 q^{41} + 120544 q^{43} + 224532 q^{45} - 665132 q^{47} + 1204647 q^{49} + 138402 q^{51} - 111888 q^{53} - 1854224 q^{55} + 382536 q^{57} + 1033308 q^{59} + 3584306 q^{61} - 4374 q^{63} + 1230472 q^{65} + 2344900 q^{67} + 1619460 q^{69} - 6311524 q^{71} - 2112174 q^{73} - 851283 q^{75} + 6670456 q^{77} - 2018830 q^{79} + 1594323 q^{81} + 7430484 q^{83} + 9821848 q^{85} - 1943568 q^{87} - 13300410 q^{89} - 10605052 q^{91} + 4791690 q^{93} - 303008 q^{95} + 2960414 q^{97} - 1070172 q^{99}+O(q^{100}) $ 3 * q - 81 * q^3 + 308 * q^5 - 6 * q^7 + 2187 * q^9 - 1468 * q^11 + 3070 * q^13 - 8316 * q^15 - 5126 * q^17 - 14168 * q^19 + 162 * q^21 - 59980 * q^23 + 31529 * q^25 - 59049 * q^27 + 71984 * q^29 - 177470 * q^31 + 39636 * q^33 + 84760 * q^35 - 184014 * q^37 - 82890 * q^39 + 657250 * q^41 + 120544 * q^43 + 224532 * q^45 - 665132 * q^47 + 1204647 * q^49 + 138402 * q^51 - 111888 * q^53 - 1854224 * q^55 + 382536 * q^57 + 1033308 * q^59 + 3584306 * q^61 - 4374 * q^63 + 1230472 * q^65 + 2344900 * q^67 + 1619460 * q^69 - 6311524 * q^71 - 2112174 * q^73 - 851283 * q^75 + 6670456 * q^77 - 2018830 * q^79 + 1594323 * q^81 + 7430484 * q^83 + 9821848 * q^85 - 1943568 * q^87 - 13300410 * q^89 - 10605052 * q^91 + 4791690 * q^93 - 303008 * q^95 + 2960414 * q^97 - 1070172 * 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
$5$	223.082	0.798124	0.399062	−	0.916924i	$-0.369336\pi$
$5$	223.082	0.798124	0.399062	+	0.916924i	$0.369336\pi$
$6$	0	0
$7$	1526.43	1.68203	0.841017	−	0.541009i	$-0.181957\pi$
$7$	1526.43	1.68203	0.841017	+	0.541009i	$0.181957\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	1604.21	0.363401	0.181700	−	0.983354i	$-0.441840\pi$
$11$	1604.21	0.363401	0.181700	+	0.983354i	$0.441840\pi$
$12$	0	0
$13$	−2959.89	−0.373657	−0.186829	−	0.982393i	$-0.559821\pi$
$13$	−2959.89	−0.373657	−0.186829	+	0.982393i	$0.559821\pi$
$14$	0	0
$15$	−6023.23	−0.460797
$16$	0	0
$17$	−20142.8	−0.994372	−0.497186	−	0.867644i	$-0.665633\pi$
$17$	−20142.8	−0.994372	−0.497186	+	0.867644i	$0.665633\pi$
$18$	0	0
$19$	7745.63	0.259071	0.129536	−	0.991575i	$-0.458651\pi$
$19$	7745.63	0.259071	0.129536	+	0.991575i	$0.458651\pi$
$20$	0	0
$21$	−41213.7	−0.971123
$22$	0	0
$23$	−71478.4	−1.22498	−0.612488	−	0.790480i	$-0.709831\pi$
$23$	−71478.4	−1.22498	−0.612488	+	0.790480i	$0.709831\pi$
$24$	0	0
$25$	−28359.2	−0.362998
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−25181.4	−0.191729	−0.0958645	−	0.995394i	$-0.530562\pi$
$29$	−25181.4	−0.191729	−0.0958645	+	0.995394i	$0.530562\pi$
$30$	0	0
$31$	−270532.	−1.63099	−0.815496	−	0.578762i	$-0.803536\pi$
$31$	−270532.	−1.63099	−0.815496	+	0.578762i	$0.803536\pi$
$32$	0	0
$33$	−43313.6	−0.209810
$34$	0	0
$35$	340520.	1.34247
$36$	0	0
$37$	−394033.	−1.27887	−0.639435	−	0.768845i	$-0.720832\pi$
$37$	−394033.	−1.27887	−0.639435	+	0.768845i	$0.720832\pi$
$38$	0	0
$39$	79917.0	0.215731
$40$	0	0
$41$	181640.	0.411594	0.205797	−	0.978595i	$-0.434021\pi$
$41$	181640.	0.411594	0.205797	+	0.978595i	$0.434021\pi$
$42$	0	0
$43$	−555492.	−1.06546	−0.532731	−	0.846285i	$-0.678834\pi$
$43$	−555492.	−1.06546	−0.532731	+	0.846285i	$0.678834\pi$
$44$	0	0
$45$	162627.	0.266041
$46$	0	0
$47$	−669993.	−0.941299	−0.470649	−	0.882320i	$-0.655980\pi$
$47$	−669993.	−0.941299	−0.470649	+	0.882320i	$0.655980\pi$
$48$	0	0
$49$	1.50646e6	1.82924
$50$	0	0
$51$	543856.	0.574101
$52$	0	0
$53$	772991.	0.713196	0.356598	−	0.934258i	$-0.383937\pi$
$53$	772991.	0.713196	0.356598	+	0.934258i	$0.383937\pi$
$54$	0	0
$55$	357870.	0.290039
$56$	0	0
$57$	−209132.	−0.149575
$58$	0	0
$59$	−1.30252e6	−0.825659	−0.412830	−	0.910808i	$-0.635459\pi$
$59$	−1.30252e6	−0.825659	−0.412830	+	0.910808i	$0.635459\pi$
$60$	0	0
$61$	1.68711e6	0.951677	0.475838	−	0.879533i	$-0.342145\pi$
$61$	1.68711e6	0.951677	0.475838	+	0.879533i	$0.342145\pi$
$62$	0	0
$63$	1.11277e6	0.560678
$64$	0	0
$65$	−660299.	−0.298225
$66$	0	0
$67$	−1.08822e6	−0.442032	−0.221016	−	0.975270i	$-0.570937\pi$
$67$	−1.08822e6	−0.442032	−0.221016	+	0.975270i	$0.570937\pi$
$68$	0	0
$69$	1.92992e6	0.707240
$70$	0	0
$71$	859754.	0.285082	0.142541	−	0.989789i	$-0.454473\pi$
$71$	859754.	0.285082	0.142541	+	0.989789i	$0.454473\pi$
$72$	0	0
$73$	2.61965e6	0.788158	0.394079	−	0.919077i	$-0.371064\pi$
$73$	2.61965e6	0.788158	0.394079	+	0.919077i	$0.371064\pi$
$74$	0	0
$75$	765699.	0.209577
$76$	0	0
$77$	2.44871e6	0.611253
$78$	0	0
$79$	−6.33710e6	−1.44609	−0.723046	−	0.690800i	$-0.757259\pi$
$79$	−6.33710e6	−1.44609	−0.723046	+	0.690800i	$0.757259\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	8.43435e6	1.61912	0.809559	−	0.587038i	$-0.199706\pi$
$83$	8.43435e6	1.61912	0.809559	+	0.587038i	$0.199706\pi$
$84$	0	0
$85$	−4.49351e6	−0.793632
$86$	0	0
$87$	679899.	0.110695
$88$	0	0
$89$	−1.18695e7	−1.78471	−0.892354	−	0.451337i	$-0.850947\pi$
$89$	−1.18695e7	−1.78471	−0.892354	+	0.451337i	$0.850947\pi$
$90$	0	0
$91$	−4.51807e6	−0.628504
$92$	0	0
$93$	7.30435e6	0.941654
$94$	0	0
$95$	1.72791e6	0.206771
$96$	0	0
$97$	4.05179e6	0.450761	0.225380	−	0.974271i	$-0.427638\pi$
$97$	4.05179e6	0.450761	0.225380	+	0.974271i	$0.427638\pi$
$98$	0	0
$99$	1.16947e6	0.121134
$100$	0	0
$101$	4.68684e6	0.452643	0.226321	−	0.974053i	$-0.427330\pi$
$101$	4.68684e6	0.452643	0.226321	+	0.974053i	$0.427330\pi$
$102$	0	0
$103$	−1.57360e7	−1.41894	−0.709468	−	0.704737i	$-0.751065\pi$
$103$	−1.57360e7	−1.41894	−0.709468	+	0.704737i	$0.751065\pi$
$104$	0	0
$105$	−9.19405e6	−0.775076
$106$	0	0
$107$	1.17323e7	0.925849	0.462924	−	0.886398i	$-0.346800\pi$
$107$	1.17323e7	0.925849	0.462924	+	0.886398i	$0.346800\pi$
$108$	0	0
$109$	−1.64178e6	−0.121429	−0.0607144	−	0.998155i	$-0.519338\pi$
$109$	−1.64178e6	−0.121429	−0.0607144	+	0.998155i	$0.519338\pi$
$110$	0	0
$111$	1.06389e7	0.738356
$112$	0	0
$113$	−1.65935e7	−1.08184	−0.540921	−	0.841073i	$-0.681924\pi$
$113$	−1.65935e7	−1.08184	−0.540921	+	0.841073i	$0.681924\pi$
$114$	0	0
$115$	−1.59456e7	−0.977682
$116$	0	0
$117$	−2.15776e6	−0.124552
$118$	0	0
$119$	−3.07467e7	−1.67257
$120$	0	0
$121$	−1.69137e7	−0.867940
$122$	0	0
$123$	−4.90429e6	−0.237634
$124$	0	0
$125$	−2.37548e7	−1.08784
$126$	0	0
$127$	2.38888e7	1.03486	0.517431	−	0.855725i	$-0.326889\pi$
$127$	2.38888e7	1.03486	0.517431	+	0.855725i	$0.326889\pi$
$128$	0	0
$129$	1.49983e7	0.615145
$130$	0	0
$131$	−3.03484e7	−1.17947	−0.589735	−	0.807597i	$-0.700768\pi$
$131$	−3.03484e7	−1.17947	−0.589735	+	0.807597i	$0.700768\pi$
$132$	0	0
$133$	1.18232e7	0.435767
$134$	0	0
$135$	−4.39093e6	−0.153599
$136$	0	0
$137$	3.84774e7	1.27845	0.639225	−	0.769020i	$-0.279255\pi$
$137$	3.84774e7	1.27845	0.639225	+	0.769020i	$0.279255\pi$
$138$	0	0
$139$	−5.72666e7	−1.80863	−0.904315	−	0.426867i	$-0.859617\pi$
$139$	−5.72666e7	−1.80863	−0.904315	+	0.426867i	$0.859617\pi$
$140$	0	0
$141$	1.80898e7	0.543459
$142$	0	0
$143$	−4.74827e6	−0.135787
$144$	0	0
$145$	−5.61754e6	−0.153024
$146$	0	0
$147$	−4.06743e7	−1.05611
$148$	0	0
$149$	5.12193e7	1.26847	0.634237	−	0.773139i	$-0.281314\pi$
$149$	5.12193e7	1.26847	0.634237	+	0.773139i	$0.281314\pi$
$150$	0	0
$151$	−2.07825e6	−0.0491223	−0.0245612	−	0.999698i	$-0.507819\pi$
$151$	−2.07825e6	−0.0491223	−0.0245612	+	0.999698i	$0.507819\pi$
$152$	0	0
$153$	−1.46841e7	−0.331457
$154$	0	0
$155$	−6.03508e7	−1.30173
$156$	0	0
$157$	7.49433e7	1.54555	0.772777	−	0.634678i	$-0.218867\pi$
$157$	7.49433e7	1.54555	0.772777	+	0.634678i	$0.218867\pi$
$158$	0	0
$159$	−2.08707e7	−0.411764
$160$	0	0
$161$	−1.09107e8	−2.06045
$162$	0	0
$163$	2.32916e7	0.421252	0.210626	−	0.977567i	$-0.432450\pi$
$163$	2.32916e7	0.421252	0.210626	+	0.977567i	$0.432450\pi$
$164$	0	0
$165$	−9.66250e6	−0.167454
$166$	0	0
$167$	−4.24114e7	−0.704653	−0.352326	−	0.935877i	$-0.614609\pi$
$167$	−4.24114e7	−0.704653	−0.352326	+	0.935877i	$0.614609\pi$
$168$	0	0
$169$	−5.39876e7	−0.860380
$170$	0	0
$171$	5.64657e6	0.0863571
$172$	0	0
$173$	−7.97832e7	−1.17152	−0.585760	−	0.810484i	$-0.699204\pi$
$173$	−7.97832e7	−1.17152	−0.585760	+	0.810484i	$0.699204\pi$
$174$	0	0
$175$	−4.32885e7	−0.610575
$176$	0	0
$177$	3.51679e7	0.476694
$178$	0	0
$179$	2.19077e6	0.0285503	0.0142751	−	0.999898i	$-0.495456\pi$
$179$	2.19077e6	0.0285503	0.0142751	+	0.999898i	$0.495456\pi$
$180$	0	0
$181$	5.36384e7	0.672358	0.336179	−	0.941798i	$-0.390865\pi$
$181$	5.36384e7	0.672358	0.336179	+	0.941798i	$0.390865\pi$
$182$	0	0
$183$	−4.55520e7	−0.549451
$184$	0	0
$185$	−8.79018e7	−1.02070
$186$	0	0
$187$	−3.23132e7	−0.361356
$188$	0	0
$189$	−3.00448e7	−0.323708
$190$	0	0
$191$	−1.09646e8	−1.13861	−0.569305	−	0.822127i	$-0.692787\pi$
$191$	−1.09646e8	−1.13861	−0.569305	+	0.822127i	$0.692787\pi$
$192$	0	0
$193$	−1.15412e8	−1.15558	−0.577788	−	0.816187i	$-0.696084\pi$
$193$	−1.15412e8	−1.15558	−0.577788	+	0.816187i	$0.696084\pi$
$194$	0	0
$195$	1.78281e7	0.172180
$196$	0	0
$197$	1.78654e8	1.66487	0.832435	−	0.554122i	$-0.186946\pi$
$197$	1.78654e8	1.66487	0.832435	+	0.554122i	$0.186946\pi$
$198$	0	0
$199$	1.06694e8	0.959745	0.479873	−	0.877338i	$-0.340683\pi$
$199$	1.06694e8	0.959745	0.479873	+	0.877338i	$0.340683\pi$
$200$	0	0
$201$	2.93819e7	0.255208
$202$	0	0
$203$	−3.84378e7	−0.322495
$204$	0	0
$205$	4.05208e7	0.328503
$206$	0	0
$207$	−5.21078e7	−0.408325
$208$	0	0
$209$	1.24256e7	0.0941467
$210$	0	0
$211$	1.79426e8	1.31491	0.657456	−	0.753493i	$-0.271632\pi$
$211$	1.79426e8	1.31491	0.657456	+	0.753493i	$0.271632\pi$
$212$	0	0
$213$	−2.32133e7	−0.164592
$214$	0	0
$215$	−1.23920e8	−0.850371
$216$	0	0
$217$	−4.12948e8	−2.74338
$218$	0	0
$219$	−7.07305e7	−0.455043
$220$	0	0
$221$	5.96205e7	0.371554
$222$	0	0
$223$	−2.37823e8	−1.43611	−0.718054	−	0.695988i	$-0.754967\pi$
$223$	−2.37823e8	−1.43611	−0.718054	+	0.695988i	$0.754967\pi$
$224$	0	0
$225$	−2.06739e7	−0.120999
$226$	0	0
$227$	−2.34015e8	−1.32786	−0.663932	−	0.747793i	$-0.731114\pi$
$227$	−2.34015e8	−1.32786	−0.663932	+	0.747793i	$0.731114\pi$
$228$	0	0
$229$	2.47626e8	1.36261	0.681306	−	0.731998i	$-0.261412\pi$
$229$	2.47626e8	1.36261	0.681306	+	0.731998i	$0.261412\pi$
$230$	0	0
$231$	−6.61153e7	−0.352907
$232$	0	0
$233$	2.29473e8	1.18846	0.594231	−	0.804295i	$-0.297457\pi$
$233$	2.29473e8	1.18846	0.594231	+	0.804295i	$0.297457\pi$
$234$	0	0
$235$	−1.49464e8	−0.751273
$236$	0	0
$237$	1.71102e8	0.834902
$238$	0	0
$239$	2.21111e7	0.104766	0.0523828	−	0.998627i	$-0.483318\pi$
$239$	2.21111e7	0.104766	0.0523828	+	0.998627i	$0.483318\pi$
$240$	0	0
$241$	2.31724e8	1.06638	0.533189	−	0.845996i	$-0.320994\pi$
$241$	2.31724e8	1.06638	0.533189	+	0.845996i	$0.320994\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	3.36064e8	1.45996
$246$	0	0
$247$	−2.29262e7	−0.0968039
$248$	0	0
$249$	−2.27728e8	−0.934798
$250$	0	0
$251$	−6.22516e6	−0.0248481	−0.0124240	−	0.999923i	$-0.503955\pi$
$251$	−6.22516e6	−0.0248481	−0.0124240	+	0.999923i	$0.503955\pi$
$252$	0	0
$253$	−1.14666e8	−0.445157
$254$	0	0
$255$	1.21325e8	0.458204
$256$	0	0
$257$	2.02916e8	0.745675	0.372837	−	0.927897i	$-0.378385\pi$
$257$	2.02916e8	0.745675	0.372837	+	0.927897i	$0.378385\pi$
$258$	0	0
$259$	−6.01465e8	−2.15110
$260$	0	0
$261$	−1.83573e7	−0.0639097
$262$	0	0
$263$	−8.91567e7	−0.302210	−0.151105	−	0.988518i	$-0.548283\pi$
$263$	−8.91567e7	−0.302210	−0.151105	+	0.988518i	$0.548283\pi$
$264$	0	0
$265$	1.72441e8	0.569219
$266$	0	0
$267$	3.20476e8	1.03040
$268$	0	0
$269$	3.29913e8	1.03339	0.516697	−	0.856168i	$-0.327161\pi$
$269$	3.29913e8	1.03339	0.516697	+	0.856168i	$0.327161\pi$
$270$	0	0
$271$	−2.51563e8	−0.767810	−0.383905	−	0.923372i	$-0.625421\pi$
$271$	−2.51563e8	−0.767810	−0.383905	+	0.923372i	$0.625421\pi$
$272$	0	0
$273$	1.21988e8	0.362867
$274$	0	0
$275$	−4.54940e7	−0.131914
$276$	0	0
$277$	−3.19740e8	−0.903894	−0.451947	−	0.892045i	$-0.649270\pi$
$277$	−3.19740e8	−0.903894	−0.451947	+	0.892045i	$0.649270\pi$
$278$	0	0
$279$	−1.97218e8	−0.543664
$280$	0	0
$281$	−5.79730e8	−1.55867	−0.779334	−	0.626609i	$-0.784442\pi$
$281$	−5.79730e8	−1.55867	−0.779334	+	0.626609i	$0.784442\pi$
$282$	0	0
$283$	−1.52928e8	−0.401082	−0.200541	−	0.979685i	$-0.564270\pi$
$283$	−1.52928e8	−0.401082	−0.200541	+	0.979685i	$0.564270\pi$
$284$	0	0
$285$	−4.66537e7	−0.119379
$286$	0	0
$287$	2.77262e8	0.692315
$288$	0	0
$289$	−4.60590e6	−0.0112246
$290$	0	0
$291$	−1.09398e8	−0.260247
$292$	0	0
$293$	6.33108e7	0.147042	0.0735210	−	0.997294i	$-0.476576\pi$
$293$	6.33108e7	0.147042	0.0735210	+	0.997294i	$0.476576\pi$
$294$	0	0
$295$	−2.90568e8	−0.658978
$296$	0	0
$297$	−3.15756e7	−0.0699365
$298$	0	0
$299$	2.11568e8	0.457721
$300$	0	0
$301$	−8.47921e8	−1.79214
$302$	0	0
$303$	−1.26545e8	−0.261333
$304$	0	0
$305$	3.76365e8	0.759556
$306$	0	0
$307$	−1.53352e8	−0.302485	−0.151243	−	0.988497i	$-0.548327\pi$
$307$	−1.53352e8	−0.302485	−0.151243	+	0.988497i	$0.548327\pi$
$308$	0	0
$309$	4.24871e8	0.819224
$310$	0	0
$311$	2.53440e8	0.477764	0.238882	−	0.971049i	$-0.423219\pi$
$311$	2.53440e8	0.477764	0.238882	+	0.971049i	$0.423219\pi$
$312$	0	0
$313$	−4.34566e8	−0.801033	−0.400517	−	0.916290i	$-0.631169\pi$
$313$	−4.34566e8	−0.801033	−0.400517	+	0.916290i	$0.631169\pi$
$314$	0	0
$315$	2.48239e8	0.447491
$316$	0	0
$317$	−5.39819e8	−0.951789	−0.475895	−	0.879502i	$-0.657876\pi$
$317$	−5.39819e8	−0.951789	−0.475895	+	0.879502i	$0.657876\pi$
$318$	0	0
$319$	−4.03963e7	−0.0696745
$320$	0	0
$321$	−3.16772e8	−0.534539
$322$	0	0
$323$	−1.56019e8	−0.257613
$324$	0	0
$325$	8.39401e7	0.135637
$326$	0	0
$327$	4.43280e7	0.0701069
$328$	0	0
$329$	−1.02270e9	−1.58330
$330$	0	0
$331$	5.78667e7	0.0877062	0.0438531	−	0.999038i	$-0.486037\pi$
$331$	5.78667e7	0.0877062	0.0438531	+	0.999038i	$0.486037\pi$
$332$	0	0
$333$	−2.87250e8	−0.426290
$334$	0	0
$335$	−2.42762e8	−0.352797
$336$	0	0
$337$	1.27253e8	0.181118	0.0905592	−	0.995891i	$-0.471135\pi$
$337$	1.27253e8	0.181118	0.0905592	+	0.995891i	$0.471135\pi$
$338$	0	0
$339$	4.48025e8	0.624602
$340$	0	0
$341$	−4.33989e8	−0.592704
$342$	0	0
$343$	1.04242e9	1.39481
$344$	0	0
$345$	4.30531e8	0.564465
$346$	0	0
$347$	8.70640e8	1.11863	0.559314	−	0.828956i	$-0.311065\pi$
$347$	8.70640e8	1.11863	0.559314	+	0.828956i	$0.311065\pi$
$348$	0	0
$349$	−6.54583e8	−0.824282	−0.412141	−	0.911120i	$-0.635219\pi$
$349$	−6.54583e8	−0.824282	−0.412141	+	0.911120i	$0.635219\pi$
$350$	0	0
$351$	5.82595e7	0.0719104
$352$	0	0
$353$	1.18126e9	1.42933	0.714667	−	0.699464i	$-0.246578\pi$
$353$	1.18126e9	1.42933	0.714667	+	0.699464i	$0.246578\pi$
$354$	0	0
$355$	1.91796e8	0.227531
$356$	0	0
$357$	8.30160e8	0.965657
$358$	0	0
$359$	2.73290e8	0.311740	0.155870	−	0.987778i	$-0.450182\pi$
$359$	2.73290e8	0.311740	0.155870	+	0.987778i	$0.450182\pi$
$360$	0	0
$361$	−8.33877e8	−0.932882
$362$	0	0
$363$	4.56670e8	0.501105
$364$	0	0
$365$	5.84398e8	0.629048
$366$	0	0
$367$	1.33000e9	1.40449	0.702246	−	0.711935i	$-0.252181\pi$
$367$	1.33000e9	1.40449	0.702246	+	0.711935i	$0.252181\pi$
$368$	0	0
$369$	1.32416e8	0.137198
$370$	0	0
$371$	1.17992e9	1.19962
$372$	0	0
$373$	4.52311e8	0.451290	0.225645	−	0.974210i	$-0.427551\pi$
$373$	4.52311e8	0.451290	0.225645	+	0.974210i	$0.427551\pi$
$374$	0	0
$375$	6.41379e8	0.628066
$376$	0	0
$377$	7.45343e7	0.0716409
$378$	0	0
$379$	−1.64402e9	−1.55121	−0.775603	−	0.631221i	$-0.782554\pi$
$379$	−1.64402e9	−1.55121	−0.775603	+	0.631221i	$0.782554\pi$
$380$	0	0
$381$	−6.44999e8	−0.597477
$382$	0	0
$383$	−6.98969e7	−0.0635714	−0.0317857	−	0.999495i	$-0.510119\pi$
$383$	−6.98969e7	−0.0635714	−0.0317857	+	0.999495i	$0.510119\pi$
$384$	0	0
$385$	5.46265e8	0.487855
$386$	0	0
$387$	−4.04953e8	−0.355154
$388$	0	0
$389$	8.76666e8	0.755110	0.377555	−	0.925987i	$-0.376765\pi$
$389$	8.76666e8	0.755110	0.377555	+	0.925987i	$0.376765\pi$
$390$	0	0
$391$	1.43978e9	1.21808
$392$	0	0
$393$	8.19408e8	0.680967
$394$	0	0
$395$	−1.41370e9	−1.15416
$396$	0	0
$397$	−1.95321e9	−1.56669	−0.783344	−	0.621588i	$-0.786488\pi$
$397$	−1.95321e9	−1.56669	−0.783344	+	0.621588i	$0.786488\pi$
$398$	0	0
$399$	−3.19226e8	−0.251590
$400$	0	0
$401$	1.56622e9	1.21296	0.606481	−	0.795098i	$-0.292580\pi$
$401$	1.56622e9	1.21296	0.606481	+	0.795098i	$0.292580\pi$
$402$	0	0
$403$	8.00743e8	0.609432
$404$	0	0
$405$	1.18555e8	0.0886805
$406$	0	0
$407$	−6.32110e8	−0.464742
$408$	0	0
$409$	1.15083e9	0.831724	0.415862	−	0.909428i	$-0.363480\pi$
$409$	1.15083e9	0.831724	0.415862	+	0.909428i	$0.363480\pi$
$410$	0	0
$411$	−1.03889e9	−0.738113
$412$	0	0
$413$	−1.98820e9	−1.38879
$414$	0	0
$415$	1.88156e9	1.29226
$416$	0	0
$417$	1.54620e9	1.04421
$418$	0	0
$419$	1.67886e9	1.11498	0.557488	−	0.830185i	$-0.311765\pi$
$419$	1.67886e9	1.11498	0.557488	+	0.830185i	$0.311765\pi$
$420$	0	0
$421$	1.42077e9	0.927976	0.463988	−	0.885842i	$-0.346418\pi$
$421$	1.42077e9	0.927976	0.463988	+	0.885842i	$0.346418\pi$
$422$	0	0
$423$	−4.88425e8	−0.313766
$424$	0	0
$425$	5.71234e8	0.360955
$426$	0	0
$427$	2.57526e9	1.60075
$428$	0	0
$429$	1.28203e8	0.0783969
$430$	0	0
$431$	−5.76731e8	−0.346978	−0.173489	−	0.984836i	$-0.555504\pi$
$431$	−5.76731e8	−0.346978	−0.173489	+	0.984836i	$0.555504\pi$
$432$	0	0
$433$	−1.29171e9	−0.764642	−0.382321	−	0.924030i	$-0.624875\pi$
$433$	−1.29171e9	−0.764642	−0.382321	+	0.924030i	$0.624875\pi$
$434$	0	0
$435$	1.51674e8	0.0883482
$436$	0	0
$437$	−5.53645e8	−0.317356
$438$	0	0
$439$	1.60873e9	0.907523	0.453761	−	0.891123i	$-0.350082\pi$
$439$	1.60873e9	0.907523	0.453761	+	0.891123i	$0.350082\pi$
$440$	0	0
$441$	1.09821e9	0.609746
$442$	0	0
$443$	2.37709e9	1.29907	0.649534	−	0.760333i	$-0.274964\pi$
$443$	2.37709e9	1.29907	0.649534	+	0.760333i	$0.274964\pi$
$444$	0	0
$445$	−2.64787e9	−1.42442
$446$	0	0
$447$	−1.38292e9	−0.732353
$448$	0	0
$449$	3.25417e9	1.69659	0.848297	−	0.529520i	$-0.177628\pi$
$449$	3.25417e9	1.69659	0.848297	+	0.529520i	$0.177628\pi$
$450$	0	0
$451$	2.91389e8	0.149574
$452$	0	0
$453$	5.61128e7	0.0283608
$454$	0	0
$455$	−1.00790e9	−0.501624
$456$	0	0
$457$	−3.30664e9	−1.62062	−0.810309	−	0.586003i	$-0.800701\pi$
$457$	−3.30664e9	−1.62062	−0.810309	+	0.586003i	$0.800701\pi$
$458$	0	0
$459$	3.96471e8	0.191367
$460$	0	0
$461$	−4.00472e9	−1.90379	−0.951894	−	0.306428i	$-0.900866\pi$
$461$	−4.00472e9	−1.90379	−0.951894	+	0.306428i	$0.900866\pi$
$462$	0	0
$463$	1.35702e9	0.635407	0.317703	−	0.948190i	$-0.397088\pi$
$463$	1.35702e9	0.635407	0.317703	+	0.948190i	$0.397088\pi$
$464$	0	0
$465$	1.62947e9	0.751557
$466$	0	0
$467$	3.42473e9	1.55603	0.778014	−	0.628247i	$-0.216227\pi$
$467$	3.42473e9	1.55603	0.778014	+	0.628247i	$0.216227\pi$
$468$	0	0
$469$	−1.66109e9	−0.743513
$470$	0	0
$471$	−2.02347e9	−0.892326
$472$	0	0
$473$	−8.91124e8	−0.387190
$474$	0	0
$475$	−2.19660e8	−0.0940423
$476$	0	0
$477$	5.63510e8	0.237732
$478$	0	0
$479$	−1.35267e8	−0.0562366	−0.0281183	−	0.999605i	$-0.508952\pi$
$479$	−1.35267e8	−0.0562366	−0.0281183	+	0.999605i	$0.508952\pi$
$480$	0	0
$481$	1.16629e9	0.477859
$482$	0	0
$483$	2.94589e9	1.18960
$484$	0	0
$485$	9.03884e8	0.359763
$486$	0	0
$487$	−1.21620e9	−0.477148	−0.238574	−	0.971124i	$-0.576680\pi$
$487$	−1.21620e9	−0.477148	−0.238574	+	0.971124i	$0.576680\pi$
$488$	0	0
$489$	−6.28873e8	−0.243210
$490$	0	0
$491$	−1.04610e8	−0.0398831	−0.0199416	−	0.999801i	$-0.506348\pi$
$491$	−1.04610e8	−0.0398831	−0.0199416	+	0.999801i	$0.506348\pi$
$492$	0	0
$493$	5.07225e8	0.190650
$494$	0	0
$495$	2.60888e8	0.0966797
$496$	0	0
$497$	1.31236e9	0.479517
$498$	0	0
$499$	−4.71953e9	−1.70038	−0.850192	−	0.526473i	$-0.823514\pi$
$499$	−4.71953e9	−1.70038	−0.850192	+	0.526473i	$0.823514\pi$
$500$	0	0
$501$	1.14511e9	0.406831
$502$	0	0
$503$	−2.98374e9	−1.04538	−0.522689	−	0.852524i	$-0.675071\pi$
$503$	−2.98374e9	−1.04538	−0.522689	+	0.852524i	$0.675071\pi$
$504$	0	0
$505$	1.04555e9	0.361265
$506$	0	0
$507$	1.45766e9	0.496741
$508$	0	0
$509$	3.51373e9	1.18102	0.590509	−	0.807031i	$-0.298927\pi$
$509$	3.51373e9	1.18102	0.590509	+	0.807031i	$0.298927\pi$
$510$	0	0
$511$	3.99872e9	1.32571
$512$	0	0
$513$	−1.52457e8	−0.0498583
$514$	0	0
$515$	−3.51042e9	−1.13249
$516$	0	0
$517$	−1.07481e9	−0.342069
$518$	0	0
$519$	2.15415e9	0.676378
$520$	0	0
$521$	−1.74420e7	−0.00540335	−0.00270168	−	0.999996i	$-0.500860\pi$
$521$	−1.74420e7	−0.00540335	−0.00270168	+	0.999996i	$0.500860\pi$
$522$	0	0
$523$	2.15669e9	0.659222	0.329611	−	0.944117i	$-0.393082\pi$
$523$	2.15669e9	0.659222	0.329611	+	0.944117i	$0.393082\pi$
$524$	0	0
$525$	1.16879e9	0.352516
$526$	0	0
$527$	5.44927e9	1.62181
$528$	0	0
$529$	1.70434e9	0.500565
$530$	0	0
$531$	−9.49534e8	−0.275220
$532$	0	0
$533$	−5.37635e8	−0.153795
$534$	0	0
$535$	2.61727e9	0.738942
$536$	0	0
$537$	−5.91507e7	−0.0164835
$538$	0	0
$539$	2.41667e9	0.664746
$540$	0	0
$541$	−4.97673e9	−1.35131	−0.675653	−	0.737220i	$-0.736138\pi$
$541$	−4.97673e9	−1.35131	−0.675653	+	0.737220i	$0.736138\pi$
$542$	0	0
$543$	−1.44824e9	−0.388186
$544$	0	0
$545$	−3.66252e8	−0.0969152
$546$	0	0
$547$	−3.97379e9	−1.03812	−0.519062	−	0.854737i	$-0.673719\pi$
$547$	−3.97379e9	−1.03812	−0.519062	+	0.854737i	$0.673719\pi$
$548$	0	0
$549$	1.22990e9	0.317226
$550$	0	0
$551$	−1.95046e8	−0.0496715
$552$	0	0
$553$	−9.67316e9	−2.43238
$554$	0	0
$555$	2.37335e9	0.589299
$556$	0	0
$557$	3.21200e9	0.787558	0.393779	−	0.919205i	$-0.371167\pi$
$557$	3.21200e9	0.787558	0.393779	+	0.919205i	$0.371167\pi$
$558$	0	0
$559$	1.64419e9	0.398118
$560$	0	0
$561$	8.72457e8	0.208629
$562$	0	0
$563$	7.69409e9	1.81710	0.908548	−	0.417781i	$-0.137192\pi$
$563$	7.69409e9	1.81710	0.908548	+	0.417781i	$0.137192\pi$
$564$	0	0
$565$	−3.70172e9	−0.863444
$566$	0	0
$567$	8.11209e8	0.186893
$568$	0	0
$569$	7.21967e9	1.64295	0.821475	−	0.570244i	$-0.193152\pi$
$569$	7.21967e9	1.64295	0.821475	+	0.570244i	$0.193152\pi$
$570$	0	0
$571$	−6.66999e9	−1.49934	−0.749668	−	0.661814i	$-0.769787\pi$
$571$	−6.66999e9	−1.49934	−0.749668	+	0.661814i	$0.769787\pi$
$572$	0	0
$573$	2.96043e9	0.657376
$574$	0	0
$575$	2.02707e9	0.444664
$576$	0	0
$577$	1.52193e9	0.329822	0.164911	−	0.986308i	$-0.447266\pi$
$577$	1.52193e9	0.329822	0.164911	+	0.986308i	$0.447266\pi$
$578$	0	0
$579$	3.11611e9	0.667172
$580$	0	0
$581$	1.28745e10	2.72341
$582$	0	0
$583$	1.24004e9	0.259176
$584$	0	0
$585$	−4.81358e8	−0.0994083
$586$	0	0
$587$	−9.26742e9	−1.89115	−0.945574	−	0.325408i	$-0.894498\pi$
$587$	−9.26742e9	−1.89115	−0.945574	+	0.325408i	$0.894498\pi$
$588$	0	0
$589$	−2.09544e9	−0.422543
$590$	0	0
$591$	−4.82365e9	−0.961214
$592$	0	0
$593$	−3.62133e9	−0.713142	−0.356571	−	0.934268i	$-0.616054\pi$
$593$	−3.62133e9	−0.713142	−0.356571	+	0.934268i	$0.616054\pi$
$594$	0	0
$595$	−6.85904e9	−1.33492
$596$	0	0
$597$	−2.88075e9	−0.554109
$598$	0	0
$599$	6.87148e9	1.30634	0.653170	−	0.757211i	$-0.273439\pi$
$599$	6.87148e9	1.30634	0.653170	+	0.757211i	$0.273439\pi$
$600$	0	0
$601$	−4.37454e9	−0.822001	−0.411000	−	0.911635i	$-0.634820\pi$
$601$	−4.37454e9	−0.822001	−0.411000	+	0.911635i	$0.634820\pi$
$602$	0	0
$603$	−7.93311e8	−0.147344
$604$	0	0
$605$	−3.77315e9	−0.692724
$606$	0	0
$607$	−1.79016e9	−0.324887	−0.162443	−	0.986718i	$-0.551937\pi$
$607$	−1.79016e9	−0.324887	−0.162443	+	0.986718i	$0.551937\pi$
$608$	0	0
$609$	1.03782e9	0.186192
$610$	0	0
$611$	1.98310e9	0.351723
$612$	0	0
$613$	−7.18426e9	−1.25971	−0.629855	−	0.776713i	$-0.716886\pi$
$613$	−7.18426e9	−1.25971	−0.629855	+	0.776713i	$0.716886\pi$
$614$	0	0
$615$	−1.09406e9	−0.189661
$616$	0	0
$617$	−4.00365e9	−0.686212	−0.343106	−	0.939297i	$-0.611479\pi$
$617$	−4.00365e9	−0.686212	−0.343106	+	0.939297i	$0.611479\pi$
$618$	0	0
$619$	−8.14329e9	−1.38001	−0.690006	−	0.723804i	$-0.742392\pi$
$619$	−8.14329e9	−1.38001	−0.690006	+	0.723804i	$0.742392\pi$
$620$	0	0
$621$	1.40691e9	0.235747
$622$	0	0
$623$	−1.81180e10	−3.00194
$624$	0	0
$625$	−3.08371e9	−0.505235
$626$	0	0
$627$	−3.35491e8	−0.0543556
$628$	0	0
$629$	7.93693e9	1.27167
$630$	0	0
$631$	−4.92407e9	−0.780229	−0.390114	−	0.920766i	$-0.627565\pi$
$631$	−4.92407e9	−0.780229	−0.390114	+	0.920766i	$0.627565\pi$
$632$	0	0
$633$	−4.84450e9	−0.759165
$634$	0	0
$635$	5.32918e9	0.825948
$636$	0	0
$637$	−4.45894e9	−0.683508
$638$	0	0
$639$	6.26760e8	0.0950273
$640$	0	0
$641$	−4.25248e9	−0.637734	−0.318867	−	0.947800i	$-0.603302\pi$
$641$	−4.25248e9	−0.637734	−0.318867	+	0.947800i	$0.603302\pi$
$642$	0	0
$643$	1.99844e9	0.296452	0.148226	−	0.988954i	$-0.452644\pi$
$643$	1.99844e9	0.296452	0.148226	+	0.988954i	$0.452644\pi$
$644$	0	0
$645$	3.34585e9	0.490962
$646$	0	0
$647$	−4.98078e9	−0.722990	−0.361495	−	0.932374i	$-0.617733\pi$
$647$	−4.98078e9	−0.722990	−0.361495	+	0.932374i	$0.617733\pi$
$648$	0	0
$649$	−2.08950e9	−0.300045
$650$	0	0
$651$	1.11496e10	1.58389
$652$	0	0
$653$	9.24503e8	0.129931	0.0649654	−	0.997888i	$-0.479306\pi$
$653$	9.24503e8	0.129931	0.0649654	+	0.997888i	$0.479306\pi$
$654$	0	0
$655$	−6.77020e9	−0.941363
$656$	0	0
$657$	1.90972e9	0.262719
$658$	0	0
$659$	−9.40009e9	−1.27948	−0.639740	−	0.768592i	$-0.720958\pi$
$659$	−9.40009e9	−1.27948	−0.639740	+	0.768592i	$0.720958\pi$
$660$	0	0
$661$	−1.20065e10	−1.61700	−0.808501	−	0.588495i	$-0.799721\pi$
$661$	−1.20065e10	−1.61700	−0.808501	+	0.588495i	$0.799721\pi$
$662$	0	0
$663$	−1.60975e9	−0.214517
$664$	0	0
$665$	2.63755e9	0.347796
$666$	0	0
$667$	1.79993e9	0.234863
$668$	0	0
$669$	6.42122e9	0.829137
$670$	0	0
$671$	2.70648e9	0.345840
$672$	0	0
$673$	−1.23701e10	−1.56431	−0.782154	−	0.623086i	$-0.785879\pi$
$673$	−1.23701e10	−1.56431	−0.782154	+	0.623086i	$0.785879\pi$
$674$	0	0
$675$	5.58194e8	0.0698590
$676$	0	0
$677$	−4.27900e9	−0.530007	−0.265004	−	0.964247i	$-0.585373\pi$
$677$	−4.27900e9	−0.530007	−0.265004	+	0.964247i	$0.585373\pi$
$678$	0	0
$679$	6.18479e9	0.758195
$680$	0	0
$681$	6.31841e9	0.766643
$682$	0	0
$683$	−1.62834e10	−1.95557	−0.977786	−	0.209607i	$-0.932782\pi$
$683$	−1.62834e10	−1.95557	−0.977786	+	0.209607i	$0.932782\pi$
$684$	0	0
$685$	8.58363e9	1.02036
$686$	0	0
$687$	−6.68591e9	−0.786705
$688$	0	0
$689$	−2.28797e9	−0.266491
$690$	0	0
$691$	−1.32503e10	−1.52775	−0.763875	−	0.645364i	$-0.776706\pi$
$691$	−1.32503e10	−1.52775	−0.763875	+	0.645364i	$0.776706\pi$
$692$	0	0
$693$	1.78511e9	0.203751
$694$	0	0
$695$	−1.27752e10	−1.44351
$696$	0	0
$697$	−3.65875e9	−0.409277
$698$	0	0
$699$	−6.19576e9	−0.686159
$700$	0	0
$701$	4.06887e9	0.446130	0.223065	−	0.974804i	$-0.428394\pi$
$701$	4.06887e9	0.446130	0.223065	+	0.974804i	$0.428394\pi$
$702$	0	0
$703$	−3.05203e9	−0.331318
$704$	0	0
$705$	4.03552e9	0.433748
$706$	0	0
$707$	7.15415e9	0.761360
$708$	0	0
$709$	1.59855e10	1.68447	0.842236	−	0.539109i	$-0.181239\pi$
$709$	1.59855e10	1.68447	0.842236	+	0.539109i	$0.181239\pi$
$710$	0	0
$711$	−4.61975e9	−0.482031
$712$	0	0
$713$	1.93372e10	1.99793
$714$	0	0
$715$	−1.05926e9	−0.108375
$716$	0	0
$717$	−5.97001e8	−0.0604864
$718$	0	0
$719$	4.52111e9	0.453622	0.226811	−	0.973939i	$-0.427170\pi$
$719$	4.52111e9	0.453622	0.226811	+	0.973939i	$0.427170\pi$
$720$	0	0
$721$	−2.40199e10	−2.38670
$722$	0	0
$723$	−6.25654e9	−0.615673
$724$	0	0
$725$	7.14126e8	0.0695972
$726$	0	0
$727$	1.67194e10	1.61380	0.806899	−	0.590689i	$-0.201144\pi$
$727$	1.67194e10	1.61380	0.806899	+	0.590689i	$0.201144\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	1.11892e10	1.05947
$732$	0	0
$733$	9.84678e9	0.923486	0.461743	−	0.887014i	$-0.347224\pi$
$733$	9.84678e9	0.923486	0.461743	+	0.887014i	$0.347224\pi$
$734$	0	0
$735$	−9.07372e9	−0.842907
$736$	0	0
$737$	−1.74573e9	−0.160635
$738$	0	0
$739$	−4.02442e9	−0.366816	−0.183408	−	0.983037i	$-0.558713\pi$
$739$	−4.02442e9	−0.366816	−0.183408	+	0.983037i	$0.558713\pi$
$740$	0	0
$741$	6.19007e8	0.0558897
$742$	0	0
$743$	−1.10258e10	−0.986161	−0.493080	−	0.869984i	$-0.664129\pi$
$743$	−1.10258e10	−0.986161	−0.493080	+	0.869984i	$0.664129\pi$
$744$	0	0
$745$	1.14261e10	1.01240
$746$	0	0
$747$	6.14864e9	0.539706
$748$	0	0
$749$	1.79086e10	1.55731
$750$	0	0
$751$	6.87188e9	0.592019	0.296010	−	0.955185i	$-0.404344\pi$
$751$	6.87188e9	0.592019	0.296010	+	0.955185i	$0.404344\pi$
$752$	0	0
$753$	1.68079e8	0.0143460
$754$	0	0
$755$	−4.63622e8	−0.0392057
$756$	0	0
$757$	2.04576e10	1.71404	0.857018	−	0.515287i	$-0.172315\pi$
$757$	2.04576e10	1.71404	0.857018	+	0.515287i	$0.172315\pi$
$758$	0	0
$759$	3.09599e9	0.257012
$760$	0	0
$761$	−2.23380e10	−1.83737	−0.918686	−	0.394989i	$-0.870748\pi$
$761$	−2.23380e10	−1.83737	−0.918686	+	0.394989i	$0.870748\pi$
$762$	0	0
$763$	−2.50606e9	−0.204247
$764$	0	0
$765$	−3.27577e9	−0.264544
$766$	0	0
$767$	3.85530e9	0.308514
$768$	0	0
$769$	−4.68174e9	−0.371249	−0.185624	−	0.982621i	$-0.559431\pi$
$769$	−4.68174e9	−0.371249	−0.185624	+	0.982621i	$0.559431\pi$
$770$	0	0
$771$	−5.47872e9	−0.430515
$772$	0	0
$773$	−5.45602e9	−0.424862	−0.212431	−	0.977176i	$-0.568138\pi$
$773$	−5.45602e9	−0.424862	−0.212431	+	0.977176i	$0.568138\pi$
$774$	0	0
$775$	7.67206e9	0.592047
$776$	0	0
$777$	1.62395e10	1.24194
$778$	0	0
$779$	1.40692e9	0.106632
$780$	0	0
$781$	1.37922e9	0.103599
$782$	0	0
$783$	4.95646e8	0.0368983
$784$	0	0
$785$	1.67185e10	1.23354
$786$	0	0
$787$	−4.83754e9	−0.353763	−0.176882	−	0.984232i	$-0.556601\pi$
$787$	−4.83754e9	−0.353763	−0.176882	+	0.984232i	$0.556601\pi$
$788$	0	0
$789$	2.40723e9	0.174481
$790$	0	0
$791$	−2.53289e10	−1.81969
$792$	0	0
$793$	−4.99366e9	−0.355601
$794$	0	0
$795$	−4.65590e9	−0.328639
$796$	0	0
$797$	2.10631e10	1.47373	0.736865	−	0.676040i	$-0.236305\pi$
$797$	2.10631e10	1.47373	0.736865	+	0.676040i	$0.236305\pi$
$798$	0	0
$799$	1.34955e10	0.936001
$800$	0	0
$801$	−8.65286e9	−0.594902
$802$	0	0
$803$	4.20246e9	0.286417
$804$	0	0
$805$	−2.43399e10	−1.64449
$806$	0	0
$807$	−8.90764e9	−0.596630
$808$	0	0
$809$	1.32648e10	0.880810	0.440405	−	0.897799i	$-0.354835\pi$
$809$	1.32648e10	0.880810	0.440405	+	0.897799i	$0.354835\pi$
$810$	0	0
$811$	1.24665e10	0.820676	0.410338	−	0.911933i	$-0.365411\pi$
$811$	1.24665e10	0.820676	0.410338	+	0.911933i	$0.365411\pi$
$812$	0	0
$813$	6.79220e9	0.443296
$814$	0	0
$815$	5.19594e9	0.336212
$816$	0	0
$817$	−4.30263e9	−0.276031
$818$	0	0
$819$	−3.29367e9	−0.209501
$820$	0	0
$821$	−1.83660e10	−1.15828	−0.579141	−	0.815228i	$-0.696612\pi$
$821$	−1.83660e10	−1.15828	−0.579141	+	0.815228i	$0.696612\pi$
$822$	0	0
$823$	4.65247e9	0.290927	0.145463	−	0.989364i	$-0.453533\pi$
$823$	4.65247e9	0.290927	0.145463	+	0.989364i	$0.453533\pi$
$824$	0	0
$825$	1.22834e9	0.0761605
$826$	0	0
$827$	6.62501e9	0.407302	0.203651	−	0.979044i	$-0.434719\pi$
$827$	6.62501e9	0.407302	0.203651	+	0.979044i	$0.434719\pi$
$828$	0	0
$829$	−6.24995e9	−0.381009	−0.190505	−	0.981686i	$-0.561012\pi$
$829$	−6.24995e9	−0.381009	−0.190505	+	0.981686i	$0.561012\pi$
$830$	0	0
$831$	8.63297e9	0.521863
$832$	0	0
$833$	−3.03442e10	−1.81894
$834$	0	0
$835$	−9.46124e9	−0.562400
$836$	0	0
$837$	5.32487e9	0.313885
$838$	0	0
$839$	−2.30459e10	−1.34719	−0.673593	−	0.739103i	$-0.735250\pi$
$839$	−2.30459e10	−1.34719	−0.673593	+	0.739103i	$0.735250\pi$
$840$	0	0
$841$	−1.66158e10	−0.963240
$842$	0	0
$843$	1.56527e10	0.899897
$844$	0	0
$845$	−1.20437e10	−0.686690
$846$	0	0
$847$	−2.58176e10	−1.45990
$848$	0	0
$849$	4.12905e9	0.231565
$850$	0	0
$851$	2.81648e10	1.56658
$852$	0	0
$853$	−3.01575e10	−1.66369	−0.831847	−	0.555006i	$-0.812716\pi$
$853$	−3.01575e10	−1.66369	−0.831847	+	0.555006i	$0.812716\pi$
$854$	0	0
$855$	1.25965e9	0.0689237
$856$	0	0
$857$	−1.33465e10	−0.724329	−0.362165	−	0.932114i	$-0.617962\pi$
$857$	−1.33465e10	−0.724329	−0.362165	+	0.932114i	$0.617962\pi$
$858$	0	0
$859$	−1.93560e10	−1.04193	−0.520965	−	0.853578i	$-0.674428\pi$
$859$	−1.93560e10	−1.04193	−0.520965	+	0.853578i	$0.674428\pi$
$860$	0	0
$861$	−7.48607e9	−0.399708
$862$	0	0
$863$	3.05481e9	0.161788	0.0808940	−	0.996723i	$-0.474222\pi$
$863$	3.05481e9	0.161788	0.0808940	+	0.996723i	$0.474222\pi$
$864$	0	0
$865$	−1.77982e10	−0.935019
$866$	0	0
$867$	1.24359e8	0.00648055
$868$	0	0
$869$	−1.01660e10	−0.525511
$870$	0	0
$871$	3.22100e9	0.165169
$872$	0	0
$873$	2.95376e9	0.150254
$874$	0	0
$875$	−3.62601e10	−1.82979
$876$	0	0
$877$	1.65091e10	0.826465	0.413233	−	0.910626i	$-0.364400\pi$
$877$	1.65091e10	0.826465	0.413233	+	0.910626i	$0.364400\pi$
$878$	0	0
$879$	−1.70939e9	−0.0848947
$880$	0	0
$881$	5.79401e9	0.285472	0.142736	−	0.989761i	$-0.454410\pi$
$881$	5.79401e9	0.285472	0.142736	+	0.989761i	$0.454410\pi$
$882$	0	0
$883$	2.02986e10	0.992209	0.496104	−	0.868263i	$-0.334763\pi$
$883$	2.02986e10	0.992209	0.496104	+	0.868263i	$0.334763\pi$
$884$	0	0
$885$	7.84534e9	0.380461
$886$	0	0
$887$	1.71123e10	0.823334	0.411667	−	0.911334i	$-0.364947\pi$
$887$	1.71123e10	0.823334	0.411667	+	0.911334i	$0.364947\pi$
$888$	0	0
$889$	3.64647e10	1.74067
$890$	0	0
$891$	8.52541e8	0.0403779
$892$	0	0
$893$	−5.18952e9	−0.243863
$894$	0	0
$895$	4.88722e8	0.0227867
$896$	0	0
$897$	−5.71234e9	−0.264265
$898$	0	0
$899$	6.81238e9	0.312709
$900$	0	0
$901$	−1.55702e10	−0.709182
$902$	0	0
$903$	2.28939e10	1.03469
$904$	0	0
$905$	1.19658e10	0.536625
$906$	0	0
$907$	−6.34285e9	−0.282266	−0.141133	−	0.989991i	$-0.545075\pi$
$907$	−6.34285e9	−0.282266	−0.141133	+	0.989991i	$0.545075\pi$
$908$	0	0
$909$	3.41671e9	0.150881
$910$	0	0
$911$	1.93974e10	0.850018	0.425009	−	0.905189i	$-0.360271\pi$
$911$	1.93974e10	0.850018	0.425009	+	0.905189i	$0.360271\pi$
$912$	0	0
$913$	1.35305e10	0.588389
$914$	0	0
$915$	−1.01619e10	−0.438530
$916$	0	0
$917$	−4.63249e10	−1.98391
$918$	0	0
$919$	2.42437e10	1.03038	0.515188	−	0.857077i	$-0.327722\pi$
$919$	2.42437e10	1.03038	0.515188	+	0.857077i	$0.327722\pi$
$920$	0	0
$921$	4.14050e9	0.174640
$922$	0	0
$923$	−2.54477e9	−0.106523
$924$	0	0
$925$	1.11745e10	0.464227
$926$	0	0
$927$	−1.14715e10	−0.472979
$928$	0	0
$929$	2.27725e10	0.931872	0.465936	−	0.884818i	$-0.345718\pi$
$929$	2.27725e10	0.931872	0.465936	+	0.884818i	$0.345718\pi$
$930$	0	0
$931$	1.16684e10	0.473903
$932$	0	0
$933$	−6.84288e9	−0.275837
$934$	0	0
$935$	−7.20852e9	−0.288407
$936$	0	0
$937$	2.68538e10	1.06639	0.533196	−	0.845992i	$-0.320991\pi$
$937$	2.68538e10	1.06639	0.533196	+	0.845992i	$0.320991\pi$
$938$	0	0
$939$	1.17333e10	0.462477
$940$	0	0
$941$	3.92886e10	1.53710	0.768552	−	0.639788i	$-0.220978\pi$
$941$	3.92886e10	1.53710	0.768552	+	0.639788i	$0.220978\pi$
$942$	0	0
$943$	−1.29834e10	−0.504193
$944$	0	0
$945$	−6.70246e9	−0.258359
$946$	0	0
$947$	9.61412e9	0.367862	0.183931	−	0.982939i	$-0.441118\pi$
$947$	9.61412e9	0.367862	0.183931	+	0.982939i	$0.441118\pi$
$948$	0	0
$949$	−7.75387e9	−0.294501
$950$	0	0
$951$	1.45751e10	0.549516
$952$	0	0
$953$	6.03874e9	0.226007	0.113003	−	0.993595i	$-0.463953\pi$
$953$	6.03874e9	0.226007	0.113003	+	0.993595i	$0.463953\pi$
$954$	0	0
$955$	−2.44600e10	−0.908751
$956$	0	0
$957$	1.09070e9	0.0402266
$958$	0	0
$959$	5.87332e10	2.15040
$960$	0	0
$961$	4.56747e10	1.66014
$962$	0	0
$963$	8.55285e9	0.308616
$964$	0	0
$965$	−2.57463e10	−0.922293
$966$	0	0
$967$	−2.51151e9	−0.0893187	−0.0446593	−	0.999002i	$-0.514220\pi$
$967$	−2.51151e9	−0.0893187	−0.0446593	+	0.999002i	$0.514220\pi$
$968$	0	0
$969$	4.21251e9	0.148733
$970$	0	0
$971$	−8.52174e9	−0.298718	−0.149359	−	0.988783i	$-0.547721\pi$
$971$	−8.52174e9	−0.298718	−0.149359	+	0.988783i	$0.547721\pi$
$972$	0	0
$973$	−8.74136e10	−3.04217
$974$	0	0
$975$	−2.26638e9	−0.0783100
$976$	0	0
$977$	−1.00947e10	−0.346307	−0.173153	−	0.984895i	$-0.555396\pi$
$977$	−1.00947e10	−0.346307	−0.173153	+	0.984895i	$0.555396\pi$
$978$	0	0
$979$	−1.90411e10	−0.648564
$980$	0	0
$981$	−1.19686e9	−0.0404762
$982$	0	0
$983$	2.91948e10	0.980322	0.490161	−	0.871632i	$-0.336938\pi$
$983$	2.91948e10	0.980322	0.490161	+	0.871632i	$0.336938\pi$
$984$	0	0
$985$	3.98545e10	1.32877
$986$	0	0
$987$	2.76129e10	0.914117
$988$	0	0
$989$	3.97057e10	1.30516
$990$	0	0
$991$	1.49553e10	0.488132	0.244066	−	0.969759i	$-0.421519\pi$
$991$	1.49553e10	0.488132	0.244066	+	0.969759i	$0.421519\pi$
$992$	0	0
$993$	−1.56240e9	−0.0506372
$994$	0	0
$995$	2.38017e10	0.765996
$996$	0	0
$997$	5.03923e9	0.161039	0.0805196	−	0.996753i	$-0.474342\pi$
$997$	5.03923e9	0.161039	0.0805196	+	0.996753i	$0.474342\pi$
$998$	0	0
$999$	7.75575e9	0.246119

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.a.j.1.2	yes	3
4.3	odd	2		384.8.a.l.1.2	yes	3
8.3	odd	2		384.8.a.i.1.2	✓	3
8.5	even	2		384.8.a.k.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.a.i.1.2	✓	3	8.3	odd	2
384.8.a.j.1.2	yes	3	1.1	even	1	trivial
384.8.a.k.1.2	yes	3	8.5	even	2
384.8.a.l.1.2	yes	3	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} +223.082 q^{5} +1526.43 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q-27.0000 q^{3} +223.082 q^{5} +1526.43 q^{7} +729.000 q^{9} +1604.21 q^{11} -2959.89 q^{13} -6023.23 q^{15} -20142.8 q^{17} +7745.63 q^{19} -41213.7 q^{21} -71478.4 q^{23} -28359.2 q^{25} -19683.0 q^{27} -25181.4 q^{29} -270532. q^{31} -43313.6 q^{33} +340520. q^{35} -394033. q^{37} +79917.0 q^{39} +181640. q^{41} -555492. q^{43} +162627. q^{45} -669993. q^{47} +1.50646e6 q^{49} +543856. q^{51} +772991. q^{53} +357870. q^{55} -209132. q^{57} -1.30252e6 q^{59} +1.68711e6 q^{61} +1.11277e6 q^{63} -660299. q^{65} -1.08822e6 q^{67} +1.92992e6 q^{69} +859754. q^{71} +2.61965e6 q^{73} +765699. q^{75} +2.44871e6 q^{77} -6.33710e6 q^{79} +531441. q^{81} +8.43435e6 q^{83} -4.49351e6 q^{85} +679899. q^{87} -1.18695e7 q^{89} -4.51807e6 q^{91} +7.30435e6 q^{93} +1.72791e6 q^{95} +4.05179e6 q^{97} +1.16947e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 81 q^{3} + 308 q^{5} - 6 q^{7} + 2187 q^{9}+O(q^{10}) \) 3 * q - 81 * q^3 + 308 * q^5 - 6 * q^7 + 2187 * q^9	\( 3 q - 81 q^{3} + 308 q^{5} - 6 q^{7} + 2187 q^{9} - 1468 q^{11} + 3070 q^{13} - 8316 q^{15} - 5126 q^{17} - 14168 q^{19} + 162 q^{21} - 59980 q^{23} + 31529 q^{25} - 59049 q^{27} + 71984 q^{29} - 177470 q^{31} + 39636 q^{33} + 84760 q^{35} - 184014 q^{37} - 82890 q^{39} + 657250 q^{41} + 120544 q^{43} + 224532 q^{45} - 665132 q^{47} + 1204647 q^{49} + 138402 q^{51} - 111888 q^{53} - 1854224 q^{55} + 382536 q^{57} + 1033308 q^{59} + 3584306 q^{61} - 4374 q^{63} + 1230472 q^{65} + 2344900 q^{67} + 1619460 q^{69} - 6311524 q^{71} - 2112174 q^{73} - 851283 q^{75} + 6670456 q^{77} - 2018830 q^{79} + 1594323 q^{81} + 7430484 q^{83} + 9821848 q^{85} - 1943568 q^{87} - 13300410 q^{89} - 10605052 q^{91} + 4791690 q^{93} - 303008 q^{95} + 2960414 q^{97} - 1070172 q^{99}+O(q^{100}) \) 3 * q - 81 * q^3 + 308 * q^5 - 6 * q^7 + 2187 * q^9 - 1468 * q^11 + 3070 * q^13 - 8316 * q^15 - 5126 * q^17 - 14168 * q^19 + 162 * q^21 - 59980 * q^23 + 31529 * q^25 - 59049 * q^27 + 71984 * q^29 - 177470 * q^31 + 39636 * q^33 + 84760 * q^35 - 184014 * q^37 - 82890 * q^39 + 657250 * q^41 + 120544 * q^43 + 224532 * q^45 - 665132 * q^47 + 1204647 * q^49 + 138402 * q^51 - 111888 * q^53 - 1854224 * q^55 + 382536 * q^57 + 1033308 * q^59 + 3584306 * q^61 - 4374 * q^63 + 1230472 * q^65 + 2344900 * q^67 + 1619460 * q^69 - 6311524 * q^71 - 2112174 * q^73 - 851283 * q^75 + 6670456 * q^77 - 2018830 * q^79 + 1594323 * q^81 + 7430484 * q^83 + 9821848 * q^85 - 1943568 * q^87 - 13300410 * q^89 - 10605052 * q^91 + 4791690 * q^93 - 303008 * q^95 + 2960414 * q^97 - 1070172 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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