LMFDB - Embedded newform 384.8.a.n.1.3

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:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 430x^{2} - 2448x + 12138 $ x^4 - 430x^2 - 2448x + 12138
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$22.6791$ 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)

$f(q)$	$=$	$q-27.0000 q^{3} +76.3562 q^{5} +439.494 q^{7} +729.000 q^{9} +O(q^{10})$	$q-27.0000 q^{3} +76.3562 q^{5} +439.494 q^{7} +729.000 q^{9} -3312.93 q^{11} -11574.3 q^{13} -2061.62 q^{15} +17083.7 q^{17} -9140.80 q^{19} -11866.3 q^{21} +45538.9 q^{23} -72294.7 q^{25} -19683.0 q^{27} +185132. q^{29} +51220.8 q^{31} +89449.1 q^{33} +33558.1 q^{35} -159466. q^{37} +312506. q^{39} -54890.8 q^{41} +375528. q^{43} +55663.7 q^{45} -271249. q^{47} -630388. q^{49} -461261. q^{51} -362434. q^{53} -252963. q^{55} +246801. q^{57} -2.68482e6 q^{59} -1.65967e6 q^{61} +320391. q^{63} -883768. q^{65} +2.02598e6 q^{67} -1.22955e6 q^{69} +2.63088e6 q^{71} -1.87686e6 q^{73} +1.95196e6 q^{75} -1.45601e6 q^{77} +6.03702e6 q^{79} +531441. q^{81} +2.32458e6 q^{83} +1.30445e6 q^{85} -4.99856e6 q^{87} +2.23519e6 q^{89} -5.08683e6 q^{91} -1.38296e6 q^{93} -697956. q^{95} +1.18619e7 q^{97} -2.41513e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 108 q^{3} - 192 q^{5} + 680 q^{7} + 2916 q^{9}+O(q^{10}) $ 4 * q - 108 * q^3 - 192 * q^5 + 680 * q^7 + 2916 * q^9	$ 4 q - 108 q^{3} - 192 q^{5} + 680 q^{7} + 2916 q^{9} + 4496 q^{11} + 12840 q^{13} + 5184 q^{15} - 14952 q^{17} + 21504 q^{19} - 18360 q^{21} + 54992 q^{23} + 190732 q^{25} - 78732 q^{27} - 242384 q^{29} + 151432 q^{31} - 121392 q^{33} - 273984 q^{35} - 113288 q^{37} - 346680 q^{39} - 239176 q^{41} - 1495328 q^{43} - 139968 q^{45} + 772368 q^{47} - 1577100 q^{49} + 403704 q^{51} - 2389776 q^{53} + 2590080 q^{55} - 580608 q^{57} - 141232 q^{59} - 1231304 q^{61} + 495720 q^{63} - 1041024 q^{65} + 441392 q^{67} - 1484784 q^{69} + 1507504 q^{71} - 1516840 q^{73} - 5149764 q^{75} - 12340448 q^{77} + 9540936 q^{79} + 2125764 q^{81} + 4587600 q^{83} - 6382848 q^{85} + 6544368 q^{87} + 162376 q^{89} + 4681104 q^{91} - 4088664 q^{93} + 29221248 q^{95} + 2726760 q^{97} + 3277584 q^{99}+O(q^{100}) $ 4 * q - 108 * q^3 - 192 * q^5 + 680 * q^7 + 2916 * q^9 + 4496 * q^11 + 12840 * q^13 + 5184 * q^15 - 14952 * q^17 + 21504 * q^19 - 18360 * q^21 + 54992 * q^23 + 190732 * q^25 - 78732 * q^27 - 242384 * q^29 + 151432 * q^31 - 121392 * q^33 - 273984 * q^35 - 113288 * q^37 - 346680 * q^39 - 239176 * q^41 - 1495328 * q^43 - 139968 * q^45 + 772368 * q^47 - 1577100 * q^49 + 403704 * q^51 - 2389776 * q^53 + 2590080 * q^55 - 580608 * q^57 - 141232 * q^59 - 1231304 * q^61 + 495720 * q^63 - 1041024 * q^65 + 441392 * q^67 - 1484784 * q^69 + 1507504 * q^71 - 1516840 * q^73 - 5149764 * q^75 - 12340448 * q^77 + 9540936 * q^79 + 2125764 * q^81 + 4587600 * q^83 - 6382848 * q^85 + 6544368 * q^87 + 162376 * q^89 + 4681104 * q^91 - 4088664 * q^93 + 29221248 * q^95 + 2726760 * q^97 + 3277584 * 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
$5$	76.3562	0.273180	0.136590	−	0.990628i	$-0.456386\pi$
$5$	76.3562	0.273180	0.136590	+	0.990628i	$0.456386\pi$
$6$	0	0
$7$	439.494	0.484295	0.242147	−	0.970239i	$-0.422148\pi$
$7$	439.494	0.484295	0.242147	+	0.970239i	$0.422148\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−3312.93	−0.750478	−0.375239	−	0.926928i	$-0.622439\pi$
$11$	−3312.93	−0.750478	−0.375239	+	0.926928i	$0.622439\pi$
$12$	0	0
$13$	−11574.3	−1.46114	−0.730571	−	0.682837i	$-0.760746\pi$
$13$	−11574.3	−1.46114	−0.730571	+	0.682837i	$0.760746\pi$
$14$	0	0
$15$	−2061.62	−0.157721
$16$	0	0
$17$	17083.7	0.843358	0.421679	−	0.906745i	$-0.361441\pi$
$17$	17083.7	0.843358	0.421679	+	0.906745i	$0.361441\pi$
$18$	0	0
$19$	−9140.80	−0.305736	−0.152868	−	0.988247i	$-0.548851\pi$
$19$	−9140.80	−0.305736	−0.152868	+	0.988247i	$0.548851\pi$
$20$	0	0
$21$	−11866.3	−0.279608
$22$	0	0
$23$	45538.9	0.780433	0.390216	−	0.920723i	$-0.372400\pi$
$23$	45538.9	0.780433	0.390216	+	0.920723i	$0.372400\pi$
$24$	0	0
$25$	−72294.7	−0.925373
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	185132.	1.40958	0.704788	−	0.709418i	$-0.251042\pi$
$29$	185132.	1.40958	0.704788	+	0.709418i	$0.251042\pi$
$30$	0	0
$31$	51220.8	0.308802	0.154401	−	0.988008i	$-0.450655\pi$
$31$	51220.8	0.308802	0.154401	+	0.988008i	$0.450655\pi$
$32$	0	0
$33$	89449.1	0.433288
$34$	0	0
$35$	33558.1	0.132300
$36$	0	0
$37$	−159466.	−0.517561	−0.258781	−	0.965936i	$-0.583321\pi$
$37$	−159466.	−0.517561	−0.258781	+	0.965936i	$0.583321\pi$
$38$	0	0
$39$	312506.	0.843591
$40$	0	0
$41$	−54890.8	−0.124381	−0.0621907	−	0.998064i	$-0.519809\pi$
$41$	−54890.8	−0.124381	−0.0621907	+	0.998064i	$0.519809\pi$
$42$	0	0
$43$	375528.	0.720282	0.360141	−	0.932898i	$-0.382729\pi$
$43$	375528.	0.720282	0.360141	+	0.932898i	$0.382729\pi$
$44$	0	0
$45$	55663.7	0.0910601
$46$	0	0
$47$	−271249.	−0.381088	−0.190544	−	0.981679i	$-0.561025\pi$
$47$	−271249.	−0.381088	−0.190544	+	0.981679i	$0.561025\pi$
$48$	0	0
$49$	−630388.	−0.765459
$50$	0	0
$51$	−461261.	−0.486913
$52$	0	0
$53$	−362434.	−0.334398	−0.167199	−	0.985923i	$-0.553472\pi$
$53$	−362434.	−0.334398	−0.167199	+	0.985923i	$0.553472\pi$
$54$	0	0
$55$	−252963.	−0.205016
$56$	0	0
$57$	246801.	0.176517
$58$	0	0
$59$	−2.68482e6	−1.70190	−0.850949	−	0.525248i	$-0.823973\pi$
$59$	−2.68482e6	−1.70190	−0.850949	+	0.525248i	$0.823973\pi$
$60$	0	0
$61$	−1.65967e6	−0.936196	−0.468098	−	0.883677i	$-0.655060\pi$
$61$	−1.65967e6	−0.936196	−0.468098	+	0.883677i	$0.655060\pi$
$62$	0	0
$63$	320391.	0.161432
$64$	0	0
$65$	−883768.	−0.399155
$66$	0	0
$67$	2.02598e6	0.822950	0.411475	−	0.911421i	$-0.365014\pi$
$67$	2.02598e6	0.822950	0.411475	+	0.911421i	$0.365014\pi$
$68$	0	0
$69$	−1.22955e6	−0.450583
$70$	0	0
$71$	2.63088e6	0.872364	0.436182	−	0.899859i	$-0.356330\pi$
$71$	2.63088e6	0.872364	0.436182	+	0.899859i	$0.356330\pi$
$72$	0	0
$73$	−1.87686e6	−0.564680	−0.282340	−	0.959314i	$-0.591111\pi$
$73$	−1.87686e6	−0.564680	−0.282340	+	0.959314i	$0.591111\pi$
$74$	0	0
$75$	1.95196e6	0.534264
$76$	0	0
$77$	−1.45601e6	−0.363452
$78$	0	0
$79$	6.03702e6	1.37761	0.688807	−	0.724945i	$-0.258135\pi$
$79$	6.03702e6	1.37761	0.688807	+	0.724945i	$0.258135\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	2.32458e6	0.446242	0.223121	−	0.974791i	$-0.428376\pi$
$83$	2.32458e6	0.446242	0.223121	+	0.974791i	$0.428376\pi$
$84$	0	0
$85$	1.30445e6	0.230389
$86$	0	0
$87$	−4.99856e6	−0.813819
$88$	0	0
$89$	2.23519e6	0.336085	0.168043	−	0.985780i	$-0.446255\pi$
$89$	2.23519e6	0.336085	0.168043	+	0.985780i	$0.446255\pi$
$90$	0	0
$91$	−5.08683e6	−0.707624
$92$	0	0
$93$	−1.38296e6	−0.178287
$94$	0	0
$95$	−697956.	−0.0835210
$96$	0	0
$97$	1.18619e7	1.31963	0.659815	−	0.751428i	$-0.270635\pi$
$97$	1.18619e7	1.31963	0.659815	+	0.751428i	$0.270635\pi$
$98$	0	0
$99$	−2.41513e6	−0.250159
$100$	0	0
$101$	−6.80761e6	−0.657461	−0.328731	−	0.944424i	$-0.606621\pi$
$101$	−6.80761e6	−0.657461	−0.328731	+	0.944424i	$0.606621\pi$
$102$	0	0
$103$	246677.	0.0222432	0.0111216	−	0.999938i	$-0.496460\pi$
$103$	246677.	0.0222432	0.0111216	+	0.999938i	$0.496460\pi$
$104$	0	0
$105$	−906068.	−0.0763833
$106$	0	0
$107$	2.80685e6	0.221501	0.110751	−	0.993848i	$-0.464675\pi$
$107$	2.80685e6	0.221501	0.110751	+	0.993848i	$0.464675\pi$
$108$	0	0
$109$	5.49459e6	0.406390	0.203195	−	0.979138i	$-0.434867\pi$
$109$	5.49459e6	0.406390	0.203195	+	0.979138i	$0.434867\pi$
$110$	0	0
$111$	4.30558e6	0.298814
$112$	0	0
$113$	1.08462e7	0.707139	0.353569	−	0.935408i	$-0.384968\pi$
$113$	1.08462e7	0.707139	0.353569	+	0.935408i	$0.384968\pi$
$114$	0	0
$115$	3.47718e6	0.213199
$116$	0	0
$117$	−8.43765e6	−0.487047
$118$	0	0
$119$	7.50820e6	0.408434
$120$	0	0
$121$	−8.51167e6	−0.436783
$122$	0	0
$123$	1.48205e6	0.0718117
$124$	0	0
$125$	−1.14855e7	−0.525974
$126$	0	0
$127$	1.30044e7	0.563348	0.281674	−	0.959510i	$-0.409110\pi$
$127$	1.30044e7	0.563348	0.281674	+	0.959510i	$0.409110\pi$
$128$	0	0
$129$	−1.01393e7	−0.415855
$130$	0	0
$131$	3.07071e7	1.19341	0.596705	−	0.802460i	$-0.296476\pi$
$131$	3.07071e7	1.19341	0.596705	+	0.802460i	$0.296476\pi$
$132$	0	0
$133$	−4.01732e6	−0.148066
$134$	0	0
$135$	−1.50292e6	−0.0525736
$136$	0	0
$137$	4.23002e7	1.40547	0.702733	−	0.711454i	$-0.251963\pi$
$137$	4.23002e7	1.40547	0.702733	+	0.711454i	$0.251963\pi$
$138$	0	0
$139$	5.27426e7	1.66575	0.832874	−	0.553463i	$-0.186694\pi$
$139$	5.27426e7	1.66575	0.832874	+	0.553463i	$0.186694\pi$
$140$	0	0
$141$	7.32371e6	0.220021
$142$	0	0
$143$	3.83448e7	1.09655
$144$	0	0
$145$	1.41360e7	0.385068
$146$	0	0
$147$	1.70205e7	0.441938
$148$	0	0
$149$	2.72912e7	0.675881	0.337940	−	0.941167i	$-0.390270\pi$
$149$	2.72912e7	0.675881	0.337940	+	0.941167i	$0.390270\pi$
$150$	0	0
$151$	−1.16360e7	−0.275032	−0.137516	−	0.990500i	$-0.543912\pi$
$151$	−1.16360e7	−0.275032	−0.137516	+	0.990500i	$0.543912\pi$
$152$	0	0
$153$	1.24541e7	0.281119
$154$	0	0
$155$	3.91102e6	0.0843586
$156$	0	0
$157$	−7.99679e7	−1.64917	−0.824587	−	0.565735i	$-0.808593\pi$
$157$	−7.99679e7	−1.64917	−0.824587	+	0.565735i	$0.808593\pi$
$158$	0	0
$159$	9.78573e6	0.193065
$160$	0	0
$161$	2.00141e7	0.377960
$162$	0	0
$163$	6.78571e7	1.22727	0.613633	−	0.789592i	$-0.289708\pi$
$163$	6.78571e7	1.22727	0.613633	+	0.789592i	$0.289708\pi$
$164$	0	0
$165$	6.82999e6	0.118366
$166$	0	0
$167$	−1.10426e7	−0.183469	−0.0917345	−	0.995784i	$-0.529241\pi$
$167$	−1.10426e7	−0.183469	−0.0917345	+	0.995784i	$0.529241\pi$
$168$	0	0
$169$	7.12156e7	1.13494
$170$	0	0
$171$	−6.66364e6	−0.101912
$172$	0	0
$173$	−5.40382e7	−0.793487	−0.396743	−	0.917930i	$-0.629860\pi$
$173$	−5.40382e7	−0.793487	−0.396743	+	0.917930i	$0.629860\pi$
$174$	0	0
$175$	−3.17731e7	−0.448153
$176$	0	0
$177$	7.24902e7	0.982592
$178$	0	0
$179$	2.34615e7	0.305753	0.152876	−	0.988245i	$-0.451146\pi$
$179$	2.34615e7	0.305753	0.152876	+	0.988245i	$0.451146\pi$
$180$	0	0
$181$	−5.51310e7	−0.691069	−0.345534	−	0.938406i	$-0.612302\pi$
$181$	−5.51310e7	−0.691069	−0.345534	+	0.938406i	$0.612302\pi$
$182$	0	0
$183$	4.48110e7	0.540513
$184$	0	0
$185$	−1.21762e7	−0.141387
$186$	0	0
$187$	−5.65972e7	−0.632921
$188$	0	0
$189$	−8.65056e6	−0.0932026
$190$	0	0
$191$	1.52499e8	1.58362	0.791809	−	0.610768i	$-0.209139\pi$
$191$	1.52499e8	1.58362	0.791809	+	0.610768i	$0.209139\pi$
$192$	0	0
$193$	1.14829e8	1.14974	0.574870	−	0.818245i	$-0.305053\pi$
$193$	1.14829e8	1.14974	0.574870	+	0.818245i	$0.305053\pi$
$194$	0	0
$195$	2.38617e7	0.230452
$196$	0	0
$197$	−4.56776e7	−0.425668	−0.212834	−	0.977088i	$-0.568269\pi$
$197$	−4.56776e7	−0.425668	−0.212834	+	0.977088i	$0.568269\pi$
$198$	0	0
$199$	1.66969e8	1.50193	0.750966	−	0.660341i	$-0.229588\pi$
$199$	1.66969e8	1.50193	0.750966	+	0.660341i	$0.229588\pi$
$200$	0	0
$201$	−5.47015e7	−0.475131
$202$	0	0
$203$	8.13643e7	0.682650
$204$	0	0
$205$	−4.19125e6	−0.0339786
$206$	0	0
$207$	3.31979e7	0.260144
$208$	0	0
$209$	3.02828e7	0.229448
$210$	0	0
$211$	1.42139e8	1.04166	0.520829	−	0.853661i	$-0.325623\pi$
$211$	1.42139e8	1.04166	0.520829	+	0.853661i	$0.325623\pi$
$212$	0	0
$213$	−7.10339e7	−0.503659
$214$	0	0
$215$	2.86739e7	0.196767
$216$	0	0
$217$	2.25112e7	0.149551
$218$	0	0
$219$	5.06752e7	0.326018
$220$	0	0
$221$	−1.97732e8	−1.23227
$222$	0	0
$223$	7.94734e7	0.479905	0.239952	−	0.970785i	$-0.422868\pi$
$223$	7.94734e7	0.479905	0.239952	+	0.970785i	$0.422868\pi$
$224$	0	0
$225$	−5.27029e7	−0.308458
$226$	0	0
$227$	−1.11402e8	−0.632123	−0.316061	−	0.948739i	$-0.602361\pi$
$227$	−1.11402e8	−0.632123	−0.316061	+	0.948739i	$0.602361\pi$
$228$	0	0
$229$	−1.95312e7	−0.107475	−0.0537373	−	0.998555i	$-0.517113\pi$
$229$	−1.95312e7	−0.107475	−0.0537373	+	0.998555i	$0.517113\pi$
$230$	0	0
$231$	3.93123e7	0.209839
$232$	0	0
$233$	−978752.	−0.00506905	−0.00253453	−	0.999997i	$-0.500807\pi$
$233$	−978752.	−0.00506905	−0.00253453	+	0.999997i	$0.500807\pi$
$234$	0	0
$235$	−2.07115e7	−0.104106
$236$	0	0
$237$	−1.62999e8	−0.795366
$238$	0	0
$239$	−6.02161e7	−0.285312	−0.142656	−	0.989772i	$-0.545564\pi$
$239$	−6.02161e7	−0.285312	−0.142656	+	0.989772i	$0.545564\pi$
$240$	0	0
$241$	2.54709e8	1.17216	0.586078	−	0.810255i	$-0.300671\pi$
$241$	2.54709e8	1.17216	0.586078	+	0.810255i	$0.300671\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−4.81340e7	−0.209108
$246$	0	0
$247$	1.05798e8	0.446724
$248$	0	0
$249$	−6.27635e7	−0.257638
$250$	0	0
$251$	4.21816e8	1.68370	0.841850	−	0.539711i	$-0.181467\pi$
$251$	4.21816e8	1.68370	0.841850	+	0.539711i	$0.181467\pi$
$252$	0	0
$253$	−1.50867e8	−0.585697
$254$	0	0
$255$	−3.52201e7	−0.133015
$256$	0	0
$257$	3.29504e8	1.21086	0.605431	−	0.795898i	$-0.293001\pi$
$257$	3.29504e8	1.21086	0.605431	+	0.795898i	$0.293001\pi$
$258$	0	0
$259$	−7.00843e7	−0.250652
$260$	0	0
$261$	1.34961e8	0.469858
$262$	0	0
$263$	2.84255e8	0.963523	0.481762	−	0.876302i	$-0.339997\pi$
$263$	2.84255e8	0.963523	0.481762	+	0.876302i	$0.339997\pi$
$264$	0	0
$265$	−2.76741e7	−0.0913509
$266$	0	0
$267$	−6.03502e7	−0.194039
$268$	0	0
$269$	−5.10459e8	−1.59892	−0.799462	−	0.600717i	$-0.794882\pi$
$269$	−5.10459e8	−1.59892	−0.799462	+	0.600717i	$0.794882\pi$
$270$	0	0
$271$	1.49956e8	0.457691	0.228845	−	0.973463i	$-0.426505\pi$
$271$	1.49956e8	0.457691	0.228845	+	0.973463i	$0.426505\pi$
$272$	0	0
$273$	1.37344e8	0.408547
$274$	0	0
$275$	2.39507e8	0.694471
$276$	0	0
$277$	1.58913e8	0.449242	0.224621	−	0.974446i	$-0.427886\pi$
$277$	1.58913e8	0.449242	0.224621	+	0.974446i	$0.427886\pi$
$278$	0	0
$279$	3.73400e7	0.102934
$280$	0	0
$281$	−3.23969e8	−0.871026	−0.435513	−	0.900183i	$-0.643433\pi$
$281$	−3.23969e8	−0.871026	−0.435513	+	0.900183i	$0.643433\pi$
$282$	0	0
$283$	4.95798e8	1.30033	0.650163	−	0.759795i	$-0.274701\pi$
$283$	4.95798e8	1.30033	0.650163	+	0.759795i	$0.274701\pi$
$284$	0	0
$285$	1.88448e7	0.0482209
$286$	0	0
$287$	−2.41242e7	−0.0602373
$288$	0	0
$289$	−1.18484e8	−0.288748
$290$	0	0
$291$	−3.20271e8	−0.761889
$292$	0	0
$293$	2.37825e8	0.552359	0.276180	−	0.961106i	$-0.410932\pi$
$293$	2.37825e8	0.552359	0.276180	+	0.961106i	$0.410932\pi$
$294$	0	0
$295$	−2.05003e8	−0.464925
$296$	0	0
$297$	6.52084e7	0.144429
$298$	0	0
$299$	−5.27081e8	−1.14032
$300$	0	0
$301$	1.65042e8	0.348829
$302$	0	0
$303$	1.83806e8	0.379585
$304$	0	0
$305$	−1.26726e8	−0.255750
$306$	0	0
$307$	1.53353e8	0.302488	0.151244	−	0.988496i	$-0.451672\pi$
$307$	1.53353e8	0.302488	0.151244	+	0.988496i	$0.451672\pi$
$308$	0	0
$309$	−6.66027e6	−0.0128421
$310$	0	0
$311$	3.22614e8	0.608166	0.304083	−	0.952646i	$-0.401650\pi$
$311$	3.22614e8	0.608166	0.304083	+	0.952646i	$0.401650\pi$
$312$	0	0
$313$	−1.69267e8	−0.312008	−0.156004	−	0.987756i	$-0.549861\pi$
$313$	−1.69267e8	−0.312008	−0.156004	+	0.987756i	$0.549861\pi$
$314$	0	0
$315$	2.44638e7	0.0440999
$316$	0	0
$317$	2.94851e8	0.519871	0.259935	−	0.965626i	$-0.416299\pi$
$317$	2.94851e8	0.519871	0.259935	+	0.965626i	$0.416299\pi$
$318$	0	0
$319$	−6.13329e8	−1.05785
$320$	0	0
$321$	−7.57849e7	−0.127884
$322$	0	0
$323$	−1.56159e8	−0.257845
$324$	0	0
$325$	8.36760e8	1.35210
$326$	0	0
$327$	−1.48354e8	−0.234629
$328$	0	0
$329$	−1.19212e8	−0.184559
$330$	0	0
$331$	7.23744e8	1.09695	0.548475	−	0.836167i	$-0.315208\pi$
$331$	7.23744e8	1.09695	0.548475	+	0.836167i	$0.315208\pi$
$332$	0	0
$333$	−1.16251e8	−0.172520
$334$	0	0
$335$	1.54696e8	0.224814
$336$	0	0
$337$	1.28591e9	1.83023	0.915113	−	0.403197i	$-0.132101\pi$
$337$	1.28591e9	1.83023	0.915113	+	0.403197i	$0.132101\pi$
$338$	0	0
$339$	−2.92848e8	−0.408267
$340$	0	0
$341$	−1.69691e8	−0.231749
$342$	0	0
$343$	−6.38994e8	−0.855002
$344$	0	0
$345$	−9.38838e7	−0.123090
$346$	0	0
$347$	−9.53827e8	−1.22551	−0.612755	−	0.790273i	$-0.709939\pi$
$347$	−9.53827e8	−1.22551	−0.612755	+	0.790273i	$0.709939\pi$
$348$	0	0
$349$	−4.76597e8	−0.600153	−0.300077	−	0.953915i	$-0.597012\pi$
$349$	−4.76597e8	−0.600153	−0.300077	+	0.953915i	$0.597012\pi$
$350$	0	0
$351$	2.27817e8	0.281197
$352$	0	0
$353$	1.55707e9	1.88407	0.942034	−	0.335517i	$-0.108911\pi$
$353$	1.55707e9	1.88407	0.942034	+	0.335517i	$0.108911\pi$
$354$	0	0
$355$	2.00884e8	0.238313
$356$	0	0
$357$	−2.02721e8	−0.235809
$358$	0	0
$359$	−1.18215e9	−1.34847	−0.674234	−	0.738518i	$-0.735526\pi$
$359$	−1.18215e9	−1.34847	−0.674234	+	0.738518i	$0.735526\pi$
$360$	0	0
$361$	−8.10318e8	−0.906526
$362$	0	0
$363$	2.29815e8	0.252177
$364$	0	0
$365$	−1.43310e8	−0.154259
$366$	0	0
$367$	−1.34994e9	−1.42556	−0.712779	−	0.701389i	$-0.752564\pi$
$367$	−1.34994e9	−1.42556	−0.712779	+	0.701389i	$0.752564\pi$
$368$	0	0
$369$	−4.00154e7	−0.0414605
$370$	0	0
$371$	−1.59288e8	−0.161947
$372$	0	0
$373$	1.53688e9	1.53341	0.766706	−	0.641998i	$-0.221894\pi$
$373$	1.53688e9	1.53341	0.766706	+	0.641998i	$0.221894\pi$
$374$	0	0
$375$	3.10108e8	0.303671
$376$	0	0
$377$	−2.14277e9	−2.05959
$378$	0	0
$379$	−8.13409e8	−0.767488	−0.383744	−	0.923439i	$-0.625366\pi$
$379$	−8.13409e8	−0.767488	−0.383744	+	0.923439i	$0.625366\pi$
$380$	0	0
$381$	−3.51118e8	−0.325249
$382$	0	0
$383$	−9.88814e8	−0.899330	−0.449665	−	0.893197i	$-0.648457\pi$
$383$	−9.88814e8	−0.899330	−0.449665	+	0.893197i	$0.648457\pi$
$384$	0	0
$385$	−1.11176e8	−0.0992880
$386$	0	0
$387$	2.73760e8	0.240094
$388$	0	0
$389$	2.18585e9	1.88277	0.941383	−	0.337341i	$-0.109528\pi$
$389$	2.18585e9	1.88277	0.941383	+	0.337341i	$0.109528\pi$
$390$	0	0
$391$	7.77976e8	0.658184
$392$	0	0
$393$	−8.29093e8	−0.689016
$394$	0	0
$395$	4.60964e8	0.376337
$396$	0	0
$397$	−3.96669e8	−0.318172	−0.159086	−	0.987265i	$-0.550855\pi$
$397$	−3.96669e8	−0.318172	−0.159086	+	0.987265i	$0.550855\pi$
$398$	0	0
$399$	1.08468e8	0.0854861
$400$	0	0
$401$	−2.24188e9	−1.73623	−0.868115	−	0.496363i	$-0.834668\pi$
$401$	−2.24188e9	−1.73623	−0.868115	+	0.496363i	$0.834668\pi$
$402$	0	0
$403$	−5.92844e8	−0.451204
$404$	0	0
$405$	4.05788e7	0.0303534
$406$	0	0
$407$	5.28299e8	0.388418
$408$	0	0
$409$	5.60604e8	0.405158	0.202579	−	0.979266i	$-0.435068\pi$
$409$	5.60604e8	0.405158	0.202579	+	0.979266i	$0.435068\pi$
$410$	0	0
$411$	−1.14210e9	−0.811446
$412$	0	0
$413$	−1.17996e9	−0.824221
$414$	0	0
$415$	1.77496e8	0.121904
$416$	0	0
$417$	−1.42405e9	−0.961720
$418$	0	0
$419$	−1.01738e9	−0.675668	−0.337834	−	0.941206i	$-0.609694\pi$
$419$	−1.01738e9	−0.675668	−0.337834	+	0.941206i	$0.609694\pi$
$420$	0	0
$421$	3.03218e9	1.98047	0.990233	−	0.139425i	$-0.0445253\pi$
$421$	3.03218e9	1.98047	0.990233	+	0.139425i	$0.0445253\pi$
$422$	0	0
$423$	−1.97740e8	−0.127029
$424$	0	0
$425$	−1.23506e9	−0.780420
$426$	0	0
$427$	−7.29414e8	−0.453395
$428$	0	0
$429$	−1.03531e9	−0.633096
$430$	0	0
$431$	6.30825e8	0.379523	0.189762	−	0.981830i	$-0.439228\pi$
$431$	6.30825e8	0.379523	0.189762	+	0.981830i	$0.439228\pi$
$432$	0	0
$433$	1.71886e9	1.01750	0.508748	−	0.860916i	$-0.330109\pi$
$433$	1.71886e9	1.01750	0.508748	+	0.860916i	$0.330109\pi$
$434$	0	0
$435$	−3.81671e8	−0.222319
$436$	0	0
$437$	−4.16262e8	−0.238606
$438$	0	0
$439$	−3.44241e9	−1.94195	−0.970973	−	0.239189i	$-0.923118\pi$
$439$	−3.44241e9	−1.94195	−0.970973	+	0.239189i	$0.923118\pi$
$440$	0	0
$441$	−4.59553e8	−0.255153
$442$	0	0
$443$	−2.15018e9	−1.17506	−0.587531	−	0.809202i	$-0.699900\pi$
$443$	−2.15018e9	−1.17506	−0.587531	+	0.809202i	$0.699900\pi$
$444$	0	0
$445$	1.70671e8	0.0918119
$446$	0	0
$447$	−7.36862e8	−0.390220
$448$	0	0
$449$	2.07404e8	0.108132	0.0540662	−	0.998537i	$-0.482782\pi$
$449$	2.07404e8	0.108132	0.0540662	+	0.998537i	$0.482782\pi$
$450$	0	0
$451$	1.81849e8	0.0933455
$452$	0	0
$453$	3.14172e8	0.158790
$454$	0	0
$455$	−3.88411e8	−0.193309
$456$	0	0
$457$	−2.24254e9	−1.09909	−0.549544	−	0.835465i	$-0.685199\pi$
$457$	−2.24254e9	−1.09909	−0.549544	+	0.835465i	$0.685199\pi$
$458$	0	0
$459$	−3.36259e8	−0.162304
$460$	0	0
$461$	1.21004e9	0.575239	0.287619	−	0.957745i	$-0.407136\pi$
$461$	1.21004e9	0.575239	0.287619	+	0.957745i	$0.407136\pi$
$462$	0	0
$463$	−1.85560e9	−0.868861	−0.434431	−	0.900705i	$-0.643050\pi$
$463$	−1.85560e9	−0.868861	−0.434431	+	0.900705i	$0.643050\pi$
$464$	0	0
$465$	−1.05598e8	−0.0487045
$466$	0	0
$467$	−1.85352e9	−0.842149	−0.421075	−	0.907026i	$-0.638347\pi$
$467$	−1.85352e9	−0.842149	−0.421075	+	0.907026i	$0.638347\pi$
$468$	0	0
$469$	8.90407e8	0.398551
$470$	0	0
$471$	2.15913e9	0.952152
$472$	0	0
$473$	−1.24410e9	−0.540555
$474$	0	0
$475$	6.60831e8	0.282920
$476$	0	0
$477$	−2.64215e8	−0.111466
$478$	0	0
$479$	4.44100e9	1.84632	0.923159	−	0.384418i	$-0.125598\pi$
$479$	4.44100e9	1.84632	0.923159	+	0.384418i	$0.125598\pi$
$480$	0	0
$481$	1.84570e9	0.756231
$482$	0	0
$483$	−5.40380e8	−0.218215
$484$	0	0
$485$	9.05727e8	0.360497
$486$	0	0
$487$	−3.96987e9	−1.55749	−0.778744	−	0.627342i	$-0.784143\pi$
$487$	−3.96987e9	−1.55749	−0.778744	+	0.627342i	$0.784143\pi$
$488$	0	0
$489$	−1.83214e9	−0.708562
$490$	0	0
$491$	−2.81191e9	−1.07205	−0.536026	−	0.844202i	$-0.680075\pi$
$491$	−2.81191e9	−1.07205	−0.536026	+	0.844202i	$0.680075\pi$
$492$	0	0
$493$	3.16275e9	1.18878
$494$	0	0
$495$	−1.84410e8	−0.0683385
$496$	0	0
$497$	1.15626e9	0.422481
$498$	0	0
$499$	3.02168e9	1.08867	0.544335	−	0.838868i	$-0.316782\pi$
$499$	3.02168e9	1.08867	0.544335	+	0.838868i	$0.316782\pi$
$500$	0	0
$501$	2.98149e8	0.105926
$502$	0	0
$503$	−1.42962e9	−0.500879	−0.250439	−	0.968132i	$-0.580575\pi$
$503$	−1.42962e9	−0.500879	−0.250439	+	0.968132i	$0.580575\pi$
$504$	0	0
$505$	−5.19803e8	−0.179605
$506$	0	0
$507$	−1.92282e9	−0.655256
$508$	0	0
$509$	−4.70717e9	−1.58215	−0.791076	−	0.611718i	$-0.790479\pi$
$509$	−4.70717e9	−1.58215	−0.791076	+	0.611718i	$0.790479\pi$
$510$	0	0
$511$	−8.24869e8	−0.273471
$512$	0	0
$513$	1.79918e8	0.0588389
$514$	0	0
$515$	1.88353e7	0.00607641
$516$	0	0
$517$	8.98627e8	0.285998
$518$	0	0
$519$	1.45903e9	0.458120
$520$	0	0
$521$	−3.74833e9	−1.16120	−0.580598	−	0.814190i	$-0.697181\pi$
$521$	−3.74833e9	−1.16120	−0.580598	+	0.814190i	$0.697181\pi$
$522$	0	0
$523$	1.76656e9	0.539974	0.269987	−	0.962864i	$-0.412981\pi$
$523$	1.76656e9	0.539974	0.269987	+	0.962864i	$0.412981\pi$
$524$	0	0
$525$	8.57874e8	0.258741
$526$	0	0
$527$	8.75043e8	0.260431
$528$	0	0
$529$	−1.33103e9	−0.390925
$530$	0	0
$531$	−1.95724e9	−0.567300
$532$	0	0
$533$	6.35321e8	0.181739
$534$	0	0
$535$	2.14320e8	0.0605097
$536$	0	0
$537$	−6.33461e8	−0.176527
$538$	0	0
$539$	2.08843e9	0.574459
$540$	0	0
$541$	1.33269e9	0.361858	0.180929	−	0.983496i	$-0.442090\pi$
$541$	1.33269e9	0.361858	0.180929	+	0.983496i	$0.442090\pi$
$542$	0	0
$543$	1.48854e9	0.398989
$544$	0	0
$545$	4.19546e8	0.111018
$546$	0	0
$547$	5.11973e8	0.133749	0.0668746	−	0.997761i	$-0.478697\pi$
$547$	5.11973e8	0.133749	0.0668746	+	0.997761i	$0.478697\pi$
$548$	0	0
$549$	−1.20990e9	−0.312065
$550$	0	0
$551$	−1.69225e9	−0.430958
$552$	0	0
$553$	2.65323e9	0.667171
$554$	0	0
$555$	3.28758e8	0.0816301
$556$	0	0
$557$	−1.73985e9	−0.426597	−0.213299	−	0.976987i	$-0.568421\pi$
$557$	−1.73985e9	−0.426597	−0.213299	+	0.976987i	$0.568421\pi$
$558$	0	0
$559$	−4.34647e9	−1.05243
$560$	0	0
$561$	1.52813e9	0.365417
$562$	0	0
$563$	1.63894e9	0.387064	0.193532	−	0.981094i	$-0.438006\pi$
$563$	1.63894e9	0.387064	0.193532	+	0.981094i	$0.438006\pi$
$564$	0	0
$565$	8.28177e8	0.193176
$566$	0	0
$567$	2.33565e8	0.0538105
$568$	0	0
$569$	2.32749e8	0.0529657	0.0264828	−	0.999649i	$-0.491569\pi$
$569$	2.32749e8	0.0529657	0.0264828	+	0.999649i	$0.491569\pi$
$570$	0	0
$571$	4.78665e9	1.07598	0.537992	−	0.842950i	$-0.319183\pi$
$571$	4.78665e9	1.07598	0.537992	+	0.842950i	$0.319183\pi$
$572$	0	0
$573$	−4.11748e9	−0.914303
$574$	0	0
$575$	−3.29223e9	−0.722191
$576$	0	0
$577$	6.15565e9	1.33401	0.667004	−	0.745054i	$-0.267576\pi$
$577$	6.15565e9	1.33401	0.667004	+	0.745054i	$0.267576\pi$
$578$	0	0
$579$	−3.10037e9	−0.663802
$580$	0	0
$581$	1.02164e9	0.216113
$582$	0	0
$583$	1.20072e9	0.250958
$584$	0	0
$585$	−6.44267e8	−0.133052
$586$	0	0
$587$	−6.03516e9	−1.23156	−0.615779	−	0.787919i	$-0.711159\pi$
$587$	−6.03516e9	−1.23156	−0.615779	+	0.787919i	$0.711159\pi$
$588$	0	0
$589$	−4.68199e8	−0.0944119
$590$	0	0
$591$	1.23329e9	0.245760
$592$	0	0
$593$	2.94796e8	0.0580538	0.0290269	−	0.999579i	$-0.490759\pi$
$593$	2.94796e8	0.0580538	0.0290269	+	0.999579i	$0.490759\pi$
$594$	0	0
$595$	5.73298e8	0.111576
$596$	0	0
$597$	−4.50817e9	−0.867141
$598$	0	0
$599$	1.81433e9	0.344923	0.172462	−	0.985016i	$-0.444828\pi$
$599$	1.81433e9	0.344923	0.172462	+	0.985016i	$0.444828\pi$
$600$	0	0
$601$	7.25906e9	1.36402	0.682008	−	0.731344i	$-0.261107\pi$
$601$	7.25906e9	1.36402	0.682008	+	0.731344i	$0.261107\pi$
$602$	0	0
$603$	1.47694e9	0.274317
$604$	0	0
$605$	−6.49919e8	−0.119321
$606$	0	0
$607$	−1.88686e9	−0.342435	−0.171218	−	0.985233i	$-0.554770\pi$
$607$	−1.88686e9	−0.342435	−0.171218	+	0.985233i	$0.554770\pi$
$608$	0	0
$609$	−2.19684e9	−0.394128
$610$	0	0
$611$	3.13951e9	0.556823
$612$	0	0
$613$	1.05202e10	1.84464	0.922318	−	0.386431i	$-0.126292\pi$
$613$	1.05202e10	1.84464	0.922318	+	0.386431i	$0.126292\pi$
$614$	0	0
$615$	1.13164e8	0.0196175
$616$	0	0
$617$	2.72740e9	0.467467	0.233734	−	0.972301i	$-0.424906\pi$
$617$	2.72740e9	0.467467	0.233734	+	0.972301i	$0.424906\pi$
$618$	0	0
$619$	1.00353e10	1.70064	0.850320	−	0.526265i	$-0.176408\pi$
$619$	1.00353e10	1.70064	0.850320	+	0.526265i	$0.176408\pi$
$620$	0	0
$621$	−8.96343e8	−0.150194
$622$	0	0
$623$	9.82353e8	0.162764
$624$	0	0
$625$	4.77104e9	0.781687
$626$	0	0
$627$	−8.17636e8	−0.132472
$628$	0	0
$629$	−2.72427e9	−0.436489
$630$	0	0
$631$	9.38184e8	0.148657	0.0743285	−	0.997234i	$-0.476319\pi$
$631$	9.38184e8	0.148657	0.0743285	+	0.997234i	$0.476319\pi$
$632$	0	0
$633$	−3.83776e9	−0.601402
$634$	0	0
$635$	9.92965e8	0.153895
$636$	0	0
$637$	7.29629e9	1.11844
$638$	0	0
$639$	1.91792e9	0.290788
$640$	0	0
$641$	−7.41892e9	−1.11260	−0.556298	−	0.830983i	$-0.687779\pi$
$641$	−7.41892e9	−1.11260	−0.556298	+	0.830983i	$0.687779\pi$
$642$	0	0
$643$	5.34059e9	0.792229	0.396115	−	0.918201i	$-0.370358\pi$
$643$	5.34059e9	0.792229	0.396115	+	0.918201i	$0.370358\pi$
$644$	0	0
$645$	−7.74195e8	−0.113603
$646$	0	0
$647$	1.02197e10	1.48345	0.741725	−	0.670704i	$-0.234008\pi$
$647$	1.02197e10	1.48345	0.741725	+	0.670704i	$0.234008\pi$
$648$	0	0
$649$	8.89463e9	1.27724
$650$	0	0
$651$	−6.07803e8	−0.0863435
$652$	0	0
$653$	−5.43441e9	−0.763759	−0.381880	−	0.924212i	$-0.624723\pi$
$653$	−5.43441e9	−0.763759	−0.381880	+	0.924212i	$0.624723\pi$
$654$	0	0
$655$	2.34468e9	0.326016
$656$	0	0
$657$	−1.36823e9	−0.188227
$658$	0	0
$659$	8.02278e9	1.09201	0.546004	−	0.837782i	$-0.316148\pi$
$659$	8.02278e9	1.09201	0.546004	+	0.837782i	$0.316148\pi$
$660$	0	0
$661$	−7.05653e9	−0.950356	−0.475178	−	0.879890i	$-0.657616\pi$
$661$	−7.05653e9	−0.950356	−0.475178	+	0.879890i	$0.657616\pi$
$662$	0	0
$663$	5.33877e9	0.711449
$664$	0	0
$665$	−3.06748e8	−0.0404488
$666$	0	0
$667$	8.43071e9	1.10008
$668$	0	0
$669$	−2.14578e9	−0.277073
$670$	0	0
$671$	5.49836e9	0.702594
$672$	0	0
$673$	8.83514e9	1.11728	0.558638	−	0.829411i	$-0.311324\pi$
$673$	8.83514e9	1.11728	0.558638	+	0.829411i	$0.311324\pi$
$674$	0	0
$675$	1.42298e9	0.178088
$676$	0	0
$677$	−1.15304e10	−1.42819	−0.714093	−	0.700050i	$-0.753161\pi$
$677$	−1.15304e10	−1.42819	−0.714093	+	0.700050i	$0.753161\pi$
$678$	0	0
$679$	5.21322e9	0.639090
$680$	0	0
$681$	3.00785e9	0.364956
$682$	0	0
$683$	−7.10762e9	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$683$	−7.10762e9	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$684$	0	0
$685$	3.22988e9	0.383945
$686$	0	0
$687$	5.27343e8	0.0620505
$688$	0	0
$689$	4.19492e9	0.488603
$690$	0	0
$691$	−5.29302e9	−0.610281	−0.305141	−	0.952307i	$-0.598703\pi$
$691$	−5.29302e9	−0.610281	−0.305141	+	0.952307i	$0.598703\pi$
$692$	0	0
$693$	−1.06143e9	−0.121151
$694$	0	0
$695$	4.02722e9	0.455049
$696$	0	0
$697$	−9.37740e8	−0.104898
$698$	0	0
$699$	2.64263e7	0.00292662
$700$	0	0
$701$	−5.43289e9	−0.595687	−0.297844	−	0.954615i	$-0.596267\pi$
$701$	−5.43289e9	−0.595687	−0.297844	+	0.954615i	$0.596267\pi$
$702$	0	0
$703$	1.45765e9	0.158237
$704$	0	0
$705$	5.59211e8	0.0601054
$706$	0	0
$707$	−2.99191e9	−0.318405
$708$	0	0
$709$	−5.73137e9	−0.603944	−0.301972	−	0.953317i	$-0.597645\pi$
$709$	−5.73137e9	−0.603944	−0.301972	+	0.953317i	$0.597645\pi$
$710$	0	0
$711$	4.40098e9	0.459205
$712$	0	0
$713$	2.33254e9	0.240999
$714$	0	0
$715$	2.92786e9	0.299557
$716$	0	0
$717$	1.62584e9	0.164725
$718$	0	0
$719$	7.72918e9	0.775501	0.387750	−	0.921764i	$-0.373252\pi$
$719$	7.72918e9	0.775501	0.387750	+	0.921764i	$0.373252\pi$
$720$	0	0
$721$	1.08413e8	0.0107723
$722$	0	0
$723$	−6.87715e9	−0.676744
$724$	0	0
$725$	−1.33841e10	−1.30438
$726$	0	0
$727$	−9.33807e9	−0.901336	−0.450668	−	0.892692i	$-0.648814\pi$
$727$	−9.33807e9	−0.901336	−0.450668	+	0.892692i	$0.648814\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	6.41542e9	0.607455
$732$	0	0
$733$	−3.86170e9	−0.362172	−0.181086	−	0.983467i	$-0.557961\pi$
$733$	−3.86170e9	−0.362172	−0.181086	+	0.983467i	$0.557961\pi$
$734$	0	0
$735$	1.29962e9	0.120729
$736$	0	0
$737$	−6.71193e9	−0.617606
$738$	0	0
$739$	9.56400e9	0.871734	0.435867	−	0.900011i	$-0.356442\pi$
$739$	9.56400e9	0.871734	0.435867	+	0.900011i	$0.356442\pi$
$740$	0	0
$741$	−2.85655e9	−0.257916
$742$	0	0
$743$	−7.57986e9	−0.677954	−0.338977	−	0.940795i	$-0.610081\pi$
$743$	−7.57986e9	−0.677954	−0.338977	+	0.940795i	$0.610081\pi$
$744$	0	0
$745$	2.08385e9	0.184637
$746$	0	0
$747$	1.69462e9	0.148747
$748$	0	0
$749$	1.23359e9	0.107272
$750$	0	0
$751$	−2.11989e10	−1.82630	−0.913151	−	0.407621i	$-0.866359\pi$
$751$	−2.11989e10	−1.82630	−0.913151	+	0.407621i	$0.866359\pi$
$752$	0	0
$753$	−1.13890e10	−0.972085
$754$	0	0
$755$	−8.88480e8	−0.0751334
$756$	0	0
$757$	−4.59045e9	−0.384609	−0.192304	−	0.981335i	$-0.561596\pi$
$757$	−4.59045e9	−0.384609	−0.192304	+	0.981335i	$0.561596\pi$
$758$	0	0
$759$	4.07342e9	0.338152
$760$	0	0
$761$	1.86108e10	1.53080	0.765400	−	0.643555i	$-0.222541\pi$
$761$	1.86108e10	1.53080	0.765400	+	0.643555i	$0.222541\pi$
$762$	0	0
$763$	2.41484e9	0.196813
$764$	0	0
$765$	9.50944e8	0.0767962
$766$	0	0
$767$	3.10749e10	2.48672
$768$	0	0
$769$	−5.44062e9	−0.431426	−0.215713	−	0.976457i	$-0.569208\pi$
$769$	−5.44062e9	−0.431426	−0.215713	+	0.976457i	$0.569208\pi$
$770$	0	0
$771$	−8.89661e9	−0.699092
$772$	0	0
$773$	3.50685e9	0.273080	0.136540	−	0.990635i	$-0.456402\pi$
$773$	3.50685e9	0.273080	0.136540	+	0.990635i	$0.456402\pi$
$774$	0	0
$775$	−3.70299e9	−0.285757
$776$	0	0
$777$	1.89228e9	0.144714
$778$	0	0
$779$	5.01745e8	0.0380279
$780$	0	0
$781$	−8.71593e9	−0.654689
$782$	0	0
$783$	−3.64395e9	−0.271273
$784$	0	0
$785$	−6.10604e9	−0.450522
$786$	0	0
$787$	1.31231e10	0.959677	0.479839	−	0.877357i	$-0.340695\pi$
$787$	1.31231e10	0.959677	0.479839	+	0.877357i	$0.340695\pi$
$788$	0	0
$789$	−7.67487e9	−0.556291
$790$	0	0
$791$	4.76686e9	0.342464
$792$	0	0
$793$	1.92095e10	1.36791
$794$	0	0
$795$	7.47201e8	0.0527415
$796$	0	0
$797$	1.81380e10	1.26907	0.634534	−	0.772895i	$-0.281192\pi$
$797$	1.81380e10	1.26907	0.634534	+	0.772895i	$0.281192\pi$
$798$	0	0
$799$	−4.63394e9	−0.321393
$800$	0	0
$801$	1.62945e9	0.112028
$802$	0	0
$803$	6.21791e9	0.423779
$804$	0	0
$805$	1.52820e9	0.103251
$806$	0	0
$807$	1.37824e10	0.923139
$808$	0	0
$809$	7.98777e9	0.530403	0.265202	−	0.964193i	$-0.414562\pi$
$809$	7.98777e9	0.530403	0.265202	+	0.964193i	$0.414562\pi$
$810$	0	0
$811$	1.22955e10	0.809417	0.404709	−	0.914446i	$-0.367373\pi$
$811$	1.22955e10	0.809417	0.404709	+	0.914446i	$0.367373\pi$
$812$	0	0
$813$	−4.04882e9	−0.264248
$814$	0	0
$815$	5.18131e9	0.335265
$816$	0	0
$817$	−3.43262e9	−0.220216
$818$	0	0
$819$	−3.70830e9	−0.235875
$820$	0	0
$821$	−1.00831e10	−0.635906	−0.317953	−	0.948107i	$-0.602995\pi$
$821$	−1.00831e10	−0.635906	−0.317953	+	0.948107i	$0.602995\pi$
$822$	0	0
$823$	2.97302e10	1.85908	0.929541	−	0.368718i	$-0.120203\pi$
$823$	2.97302e10	1.85908	0.929541	+	0.368718i	$0.120203\pi$
$824$	0	0
$825$	−6.46670e9	−0.400953
$826$	0	0
$827$	−1.25578e10	−0.772045	−0.386023	−	0.922489i	$-0.626151\pi$
$827$	−1.25578e10	−0.772045	−0.386023	+	0.922489i	$0.626151\pi$
$828$	0	0
$829$	1.11503e10	0.679745	0.339873	−	0.940471i	$-0.389616\pi$
$829$	1.11503e10	0.679745	0.339873	+	0.940471i	$0.389616\pi$
$830$	0	0
$831$	−4.29065e9	−0.259370
$832$	0	0
$833$	−1.07694e10	−0.645555
$834$	0	0
$835$	−8.43169e8	−0.0501201
$836$	0	0
$837$	−1.00818e9	−0.0594290
$838$	0	0
$839$	−2.71777e10	−1.58871	−0.794357	−	0.607451i	$-0.792192\pi$
$839$	−2.71777e10	−1.58871	−0.794357	+	0.607451i	$0.792192\pi$
$840$	0	0
$841$	1.70239e10	0.986902
$842$	0	0
$843$	8.74716e9	0.502887
$844$	0	0
$845$	5.43775e9	0.310042
$846$	0	0
$847$	−3.74083e9	−0.211532
$848$	0	0
$849$	−1.33865e10	−0.750744
$850$	0	0
$851$	−7.26191e9	−0.403922
$852$	0	0
$853$	1.29803e9	0.0716082	0.0358041	−	0.999359i	$-0.488601\pi$
$853$	1.29803e9	0.0716082	0.0358041	+	0.999359i	$0.488601\pi$
$854$	0	0
$855$	−5.08810e8	−0.0278403
$856$	0	0
$857$	−2.25267e10	−1.22254	−0.611272	−	0.791421i	$-0.709342\pi$
$857$	−2.25267e10	−1.22254	−0.611272	+	0.791421i	$0.709342\pi$
$858$	0	0
$859$	−2.78894e10	−1.50129	−0.750643	−	0.660708i	$-0.770256\pi$
$859$	−2.78894e10	−1.50129	−0.750643	+	0.660708i	$0.770256\pi$
$860$	0	0
$861$	6.51352e8	0.0347780
$862$	0	0
$863$	1.10615e10	0.585838	0.292919	−	0.956137i	$-0.405373\pi$
$863$	1.10615e10	0.585838	0.292919	+	0.956137i	$0.405373\pi$
$864$	0	0
$865$	−4.12615e9	−0.216765
$866$	0	0
$867$	3.19908e9	0.166709
$868$	0	0
$869$	−2.00002e10	−1.03387
$870$	0	0
$871$	−2.34493e10	−1.20245
$872$	0	0
$873$	8.64730e9	0.439877
$874$	0	0
$875$	−5.04780e9	−0.254726
$876$	0	0
$877$	3.78522e10	1.89493	0.947463	−	0.319867i	$-0.103638\pi$
$877$	3.78522e10	1.89493	0.947463	+	0.319867i	$0.103638\pi$
$878$	0	0
$879$	−6.42129e9	−0.318905
$880$	0	0
$881$	2.38870e10	1.17692	0.588459	−	0.808527i	$-0.299735\pi$
$881$	2.38870e10	1.17692	0.588459	+	0.808527i	$0.299735\pi$
$882$	0	0
$883$	−1.59642e10	−0.780340	−0.390170	−	0.920743i	$-0.627584\pi$
$883$	−1.59642e10	−0.780340	−0.390170	+	0.920743i	$0.627584\pi$
$884$	0	0
$885$	5.53508e9	0.268425
$886$	0	0
$887$	−2.71062e10	−1.30417	−0.652087	−	0.758144i	$-0.726106\pi$
$887$	−2.71062e10	−1.30417	−0.652087	+	0.758144i	$0.726106\pi$
$888$	0	0
$889$	5.71535e9	0.272826
$890$	0	0
$891$	−1.76063e9	−0.0833864
$892$	0	0
$893$	2.47943e9	0.116512
$894$	0	0
$895$	1.79143e9	0.0835256
$896$	0	0
$897$	1.42312e10	0.658366
$898$	0	0
$899$	9.48260e9	0.435280
$900$	0	0
$901$	−6.19174e9	−0.282017
$902$	0	0
$903$	−4.45614e9	−0.201396
$904$	0	0
$905$	−4.20960e9	−0.188786
$906$	0	0
$907$	2.24099e10	0.997274	0.498637	−	0.866811i	$-0.333834\pi$
$907$	2.24099e10	0.997274	0.498637	+	0.866811i	$0.333834\pi$
$908$	0	0
$909$	−4.96275e9	−0.219154
$910$	0	0
$911$	−2.45581e10	−1.07617	−0.538084	−	0.842891i	$-0.680852\pi$
$911$	−2.45581e10	−1.07617	−0.538084	+	0.842891i	$0.680852\pi$
$912$	0	0
$913$	−7.70115e9	−0.334895
$914$	0	0
$915$	3.42160e9	0.147657
$916$	0	0
$917$	1.34956e10	0.577963
$918$	0	0
$919$	1.73862e10	0.738926	0.369463	−	0.929245i	$-0.379542\pi$
$919$	1.73862e10	0.738926	0.369463	+	0.929245i	$0.379542\pi$
$920$	0	0
$921$	−4.14053e9	−0.174641
$922$	0	0
$923$	−3.04506e10	−1.27465
$924$	0	0
$925$	1.15285e10	0.478937
$926$	0	0
$927$	1.79827e8	0.00741441
$928$	0	0
$929$	−3.31995e10	−1.35855	−0.679277	−	0.733882i	$-0.737706\pi$
$929$	−3.31995e10	−1.35855	−0.679277	+	0.733882i	$0.737706\pi$
$930$	0	0
$931$	5.76225e9	0.234028
$932$	0	0
$933$	−8.71058e9	−0.351125
$934$	0	0
$935$	−4.32155e9	−0.172902
$936$	0	0
$937$	−1.03678e10	−0.411718	−0.205859	−	0.978582i	$-0.565999\pi$
$937$	−1.03678e10	−0.411718	−0.205859	+	0.978582i	$0.565999\pi$
$938$	0	0
$939$	4.57020e9	0.180138
$940$	0	0
$941$	3.97122e10	1.55368	0.776838	−	0.629700i	$-0.216822\pi$
$941$	3.97122e10	1.55368	0.776838	+	0.629700i	$0.216822\pi$
$942$	0	0
$943$	−2.49967e9	−0.0970714
$944$	0	0
$945$	−6.60524e8	−0.0254611
$946$	0	0
$947$	3.71199e10	1.42030	0.710152	−	0.704048i	$-0.248626\pi$
$947$	3.71199e10	1.42030	0.710152	+	0.704048i	$0.248626\pi$
$948$	0	0
$949$	2.17233e10	0.825077
$950$	0	0
$951$	−7.96098e9	−0.300147
$952$	0	0
$953$	5.81376e8	0.0217587	0.0108793	−	0.999941i	$-0.496537\pi$
$953$	5.81376e8	0.0217587	0.0108793	+	0.999941i	$0.496537\pi$
$954$	0	0
$955$	1.16443e10	0.432613
$956$	0	0
$957$	1.65599e10	0.610753
$958$	0	0
$959$	1.85907e10	0.680659
$960$	0	0
$961$	−2.48890e10	−0.904641
$962$	0	0
$963$	2.04619e9	0.0738337
$964$	0	0
$965$	8.76787e9	0.314086
$966$	0	0
$967$	−2.62148e10	−0.932295	−0.466148	−	0.884707i	$-0.654358\pi$
$967$	−2.62148e10	−0.932295	−0.466148	+	0.884707i	$0.654358\pi$
$968$	0	0
$969$	4.21629e9	0.148867
$970$	0	0
$971$	−4.40945e10	−1.54567	−0.772835	−	0.634607i	$-0.781162\pi$
$971$	−4.40945e10	−1.54567	−0.772835	+	0.634607i	$0.781162\pi$
$972$	0	0
$973$	2.31800e10	0.806713
$974$	0	0
$975$	−2.25925e10	−0.780636
$976$	0	0
$977$	−2.13372e10	−0.731991	−0.365996	−	0.930617i	$-0.619271\pi$
$977$	−2.13372e10	−0.731991	−0.365996	+	0.930617i	$0.619271\pi$
$978$	0	0
$979$	−7.40503e9	−0.252225
$980$	0	0
$981$	4.00556e9	0.135463
$982$	0	0
$983$	−2.38085e10	−0.799457	−0.399728	−	0.916634i	$-0.630896\pi$
$983$	−2.38085e10	−0.799457	−0.399728	+	0.916634i	$0.630896\pi$
$984$	0	0
$985$	−3.48777e9	−0.116284
$986$	0	0
$987$	3.21873e9	0.106555
$988$	0	0
$989$	1.71011e10	0.562132
$990$	0	0
$991$	−1.81193e10	−0.591402	−0.295701	−	0.955281i	$-0.595553\pi$
$991$	−1.81193e10	−0.591402	−0.295701	+	0.955281i	$0.595553\pi$
$992$	0	0
$993$	−1.95411e10	−0.633325
$994$	0	0
$995$	1.27491e10	0.410298
$996$	0	0
$997$	−4.87009e10	−1.55634	−0.778169	−	0.628055i	$-0.783851\pi$
$997$	−4.87009e10	−1.55634	−0.778169	+	0.628055i	$0.783851\pi$
$998$	0	0
$999$	3.13877e9	0.0996047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.a.n.1.3	✓	4
4.3	odd	2		384.8.a.r.1.3	yes	4
8.3	odd	2		384.8.a.o.1.2	yes	4
8.5	even	2		384.8.a.s.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.a.n.1.3	✓	4	1.1	even	1	trivial
384.8.a.o.1.2	yes	4	8.3	odd	2
384.8.a.r.1.3	yes	4	4.3	odd	2
384.8.a.s.1.2	yes	4	8.5	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} +76.3562 q^{5} +439.494 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q-27.0000 q^{3} +76.3562 q^{5} +439.494 q^{7} +729.000 q^{9} -3312.93 q^{11} -11574.3 q^{13} -2061.62 q^{15} +17083.7 q^{17} -9140.80 q^{19} -11866.3 q^{21} +45538.9 q^{23} -72294.7 q^{25} -19683.0 q^{27} +185132. q^{29} +51220.8 q^{31} +89449.1 q^{33} +33558.1 q^{35} -159466. q^{37} +312506. q^{39} -54890.8 q^{41} +375528. q^{43} +55663.7 q^{45} -271249. q^{47} -630388. q^{49} -461261. q^{51} -362434. q^{53} -252963. q^{55} +246801. q^{57} -2.68482e6 q^{59} -1.65967e6 q^{61} +320391. q^{63} -883768. q^{65} +2.02598e6 q^{67} -1.22955e6 q^{69} +2.63088e6 q^{71} -1.87686e6 q^{73} +1.95196e6 q^{75} -1.45601e6 q^{77} +6.03702e6 q^{79} +531441. q^{81} +2.32458e6 q^{83} +1.30445e6 q^{85} -4.99856e6 q^{87} +2.23519e6 q^{89} -5.08683e6 q^{91} -1.38296e6 q^{93} -697956. q^{95} +1.18619e7 q^{97} -2.41513e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 108 q^{3} - 192 q^{5} + 680 q^{7} + 2916 q^{9}+O(q^{10}) \) 4 * q - 108 * q^3 - 192 * q^5 + 680 * q^7 + 2916 * q^9	\( 4 q - 108 q^{3} - 192 q^{5} + 680 q^{7} + 2916 q^{9} + 4496 q^{11} + 12840 q^{13} + 5184 q^{15} - 14952 q^{17} + 21504 q^{19} - 18360 q^{21} + 54992 q^{23} + 190732 q^{25} - 78732 q^{27} - 242384 q^{29} + 151432 q^{31} - 121392 q^{33} - 273984 q^{35} - 113288 q^{37} - 346680 q^{39} - 239176 q^{41} - 1495328 q^{43} - 139968 q^{45} + 772368 q^{47} - 1577100 q^{49} + 403704 q^{51} - 2389776 q^{53} + 2590080 q^{55} - 580608 q^{57} - 141232 q^{59} - 1231304 q^{61} + 495720 q^{63} - 1041024 q^{65} + 441392 q^{67} - 1484784 q^{69} + 1507504 q^{71} - 1516840 q^{73} - 5149764 q^{75} - 12340448 q^{77} + 9540936 q^{79} + 2125764 q^{81} + 4587600 q^{83} - 6382848 q^{85} + 6544368 q^{87} + 162376 q^{89} + 4681104 q^{91} - 4088664 q^{93} + 29221248 q^{95} + 2726760 q^{97} + 3277584 q^{99}+O(q^{100}) \) 4 * q - 108 * q^3 - 192 * q^5 + 680 * q^7 + 2916 * q^9 + 4496 * q^11 + 12840 * q^13 + 5184 * q^15 - 14952 * q^17 + 21504 * q^19 - 18360 * q^21 + 54992 * q^23 + 190732 * q^25 - 78732 * q^27 - 242384 * q^29 + 151432 * q^31 - 121392 * q^33 - 273984 * q^35 - 113288 * q^37 - 346680 * q^39 - 239176 * q^41 - 1495328 * q^43 - 139968 * q^45 + 772368 * q^47 - 1577100 * q^49 + 403704 * q^51 - 2389776 * q^53 + 2590080 * q^55 - 580608 * q^57 - 141232 * q^59 - 1231304 * q^61 + 495720 * q^63 - 1041024 * q^65 + 441392 * q^67 - 1484784 * q^69 + 1507504 * q^71 - 1516840 * q^73 - 5149764 * q^75 - 12340448 * q^77 + 9540936 * q^79 + 2125764 * q^81 + 4587600 * q^83 - 6382848 * q^85 + 6544368 * q^87 + 162376 * q^89 + 4681104 * q^91 - 4088664 * q^93 + 29221248 * q^95 + 2726760 * q^97 + 3277584 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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