LMFDB - Embedded newform 384.8.a.s.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			$3.20119$ 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)

$f(q)$	$=$	$q+27.0000 q^{3} +170.323 q^{5} -803.427 q^{7} +729.000 q^{9} +O(q^{10})$	$q+27.0000 q^{3} +170.323 q^{5} -803.427 q^{7} +729.000 q^{9} -8558.72 q^{11} -1465.98 q^{13} +4598.72 q^{15} +30783.7 q^{17} -11397.6 q^{19} -21692.5 q^{21} -116211. q^{23} -49115.0 q^{25} +19683.0 q^{27} +184311. q^{29} +61312.9 q^{31} -231086. q^{33} -136842. q^{35} +357060. q^{37} -39581.6 q^{39} +389342. q^{41} +939452. q^{43} +124166. q^{45} +1.13965e6 q^{47} -178048. q^{49} +831160. q^{51} +324394. q^{53} -1.45775e6 q^{55} -307734. q^{57} -1.86418e6 q^{59} -1.14154e6 q^{61} -585698. q^{63} -249691. q^{65} +1.58844e6 q^{67} -3.13771e6 q^{69} +3.98069e6 q^{71} +3.61658e6 q^{73} -1.32611e6 q^{75} +6.87631e6 q^{77} +4.26185e6 q^{79} +531441. q^{81} -161844. q^{83} +5.24318e6 q^{85} +4.97639e6 q^{87} -9.25081e6 q^{89} +1.17781e6 q^{91} +1.65545e6 q^{93} -1.94127e6 q^{95} -8.61098e6 q^{97} -6.23931e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	170.323	0.609367	0.304683	−	0.952454i	$-0.401449\pi$
$5$	170.323	0.609367	0.304683	+	0.952454i	$0.401449\pi$
$6$	0	0
$7$	−803.427	−0.885326	−0.442663	−	0.896688i	$-0.645966\pi$
$7$	−803.427	−0.885326	−0.442663	+	0.896688i	$0.645966\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−8558.72	−1.93881	−0.969404	−	0.245472i	$-0.921057\pi$
$11$	−8558.72	−1.93881	−0.969404	+	0.245472i	$0.921057\pi$
$12$	0	0
$13$	−1465.98	−0.185066	−0.0925332	−	0.995710i	$-0.529496\pi$
$13$	−1465.98	−0.185066	−0.0925332	+	0.995710i	$0.529496\pi$
$14$	0	0
$15$	4598.72	0.351818
$16$	0	0
$17$	30783.7	1.51967	0.759836	−	0.650115i	$-0.225279\pi$
$17$	30783.7	1.51967	0.759836	+	0.650115i	$0.225279\pi$
$18$	0	0
$19$	−11397.6	−0.381219	−0.190610	−	0.981666i	$-0.561046\pi$
$19$	−11397.6	−0.381219	−0.190610	+	0.981666i	$0.561046\pi$
$20$	0	0
$21$	−21692.5	−0.511143
$22$	0	0
$23$	−116211.	−1.99160	−0.995799	−	0.0915692i	$-0.970812\pi$
$23$	−116211.	−1.99160	−0.995799	+	0.0915692i	$0.970812\pi$
$24$	0	0
$25$	−49115.0	−0.628672
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	184311.	1.40332	0.701661	−	0.712511i	$-0.252442\pi$
$29$	184311.	1.40332	0.701661	+	0.712511i	$0.252442\pi$
$30$	0	0
$31$	61312.9	0.369646	0.184823	−	0.982772i	$-0.440829\pi$
$31$	61312.9	0.369646	0.184823	+	0.982772i	$0.440829\pi$
$32$	0	0
$33$	−231086.	−1.11937
$34$	0	0
$35$	−136842.	−0.539488
$36$	0	0
$37$	357060.	1.15887	0.579436	−	0.815018i	$-0.303273\pi$
$37$	357060.	1.15887	0.579436	+	0.815018i	$0.303273\pi$
$38$	0	0
$39$	−39581.6	−0.106848
$40$	0	0
$41$	389342.	0.882243	0.441121	−	0.897447i	$-0.354581\pi$
$41$	389342.	0.882243	0.441121	+	0.897447i	$0.354581\pi$
$42$	0	0
$43$	939452.	1.80192	0.900959	−	0.433904i	$-0.142864\pi$
$43$	939452.	1.80192	0.900959	+	0.433904i	$0.142864\pi$
$44$	0	0
$45$	124166.	0.203122
$46$	0	0
$47$	1.13965e6	1.60113	0.800567	−	0.599243i	$-0.204532\pi$
$47$	1.13965e6	1.60113	0.800567	+	0.599243i	$0.204532\pi$
$48$	0	0
$49$	−178048.	−0.216198
$50$	0	0
$51$	831160.	0.877383
$52$	0	0
$53$	324394.	0.299300	0.149650	−	0.988739i	$-0.452185\pi$
$53$	324394.	0.299300	0.149650	+	0.988739i	$0.452185\pi$
$54$	0	0
$55$	−1.45775e6	−1.18144
$56$	0	0
$57$	−307734.	−0.220097
$58$	0	0
$59$	−1.86418e6	−1.18169	−0.590847	−	0.806784i	$-0.701206\pi$
$59$	−1.86418e6	−1.18169	−0.590847	+	0.806784i	$0.701206\pi$
$60$	0	0
$61$	−1.14154e6	−0.643929	−0.321965	−	0.946752i	$-0.604343\pi$
$61$	−1.14154e6	−0.643929	−0.321965	+	0.946752i	$0.604343\pi$
$62$	0	0
$63$	−585698.	−0.295109
$64$	0	0
$65$	−249691.	−0.112773
$66$	0	0
$67$	1.58844e6	0.645223	0.322612	−	0.946531i	$-0.395439\pi$
$67$	1.58844e6	0.645223	0.322612	+	0.946531i	$0.395439\pi$
$68$	0	0
$69$	−3.13771e6	−1.14985
$70$	0	0
$71$	3.98069e6	1.31994	0.659970	−	0.751292i	$-0.270569\pi$
$71$	3.98069e6	1.31994	0.659970	+	0.751292i	$0.270569\pi$
$72$	0	0
$73$	3.61658e6	1.08810	0.544050	−	0.839053i	$-0.316890\pi$
$73$	3.61658e6	1.08810	0.544050	+	0.839053i	$0.316890\pi$
$74$	0	0
$75$	−1.32611e6	−0.362964
$76$	0	0
$77$	6.87631e6	1.71648
$78$	0	0
$79$	4.26185e6	0.972530	0.486265	−	0.873811i	$-0.338359\pi$
$79$	4.26185e6	0.972530	0.486265	+	0.873811i	$0.338359\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−161844.	−0.0310686	−0.0155343	−	0.999879i	$-0.504945\pi$
$83$	−161844.	−0.0310686	−0.0155343	+	0.999879i	$0.504945\pi$
$84$	0	0
$85$	5.24318e6	0.926037
$86$	0	0
$87$	4.97639e6	0.810209
$88$	0	0
$89$	−9.25081e6	−1.39096	−0.695480	−	0.718545i	$-0.744808\pi$
$89$	−9.25081e6	−1.39096	−0.695480	+	0.718545i	$0.744808\pi$
$90$	0	0
$91$	1.17781e6	0.163844
$92$	0	0
$93$	1.65545e6	0.213415
$94$	0	0
$95$	−1.94127e6	−0.232302
$96$	0	0
$97$	−8.61098e6	−0.957969	−0.478984	−	0.877823i	$-0.658995\pi$
$97$	−8.61098e6	−0.957969	−0.478984	+	0.877823i	$0.658995\pi$
$98$	0	0
$99$	−6.23931e6	−0.646269
$100$	0	0
$101$	−4.60481e6	−0.444720	−0.222360	−	0.974965i	$-0.571376\pi$
$101$	−4.60481e6	−0.444720	−0.222360	+	0.974965i	$0.571376\pi$
$102$	0	0
$103$	−1.95825e6	−0.176579	−0.0882893	−	0.996095i	$-0.528140\pi$
$103$	−1.95825e6	−0.176579	−0.0882893	+	0.996095i	$0.528140\pi$
$104$	0	0
$105$	−3.69474e6	−0.311474
$106$	0	0
$107$	290533.	0.0229273	0.0114636	−	0.999934i	$-0.496351\pi$
$107$	290533.	0.0229273	0.0114636	+	0.999934i	$0.496351\pi$
$108$	0	0
$109$	1.20086e7	0.888178	0.444089	−	0.895983i	$-0.353527\pi$
$109$	1.20086e7	0.888178	0.444089	+	0.895983i	$0.353527\pi$
$110$	0	0
$111$	9.64063e6	0.669075
$112$	0	0
$113$	8.40975e6	0.548288	0.274144	−	0.961689i	$-0.411606\pi$
$113$	8.40975e6	0.548288	0.274144	+	0.961689i	$0.411606\pi$
$114$	0	0
$115$	−1.97935e7	−1.21361
$116$	0	0
$117$	−1.06870e6	−0.0616888
$118$	0	0
$119$	−2.47325e7	−1.34541
$120$	0	0
$121$	5.37646e7	2.75897
$122$	0	0
$123$	1.05122e7	0.509363
$124$	0	0
$125$	−2.16719e7	−0.992458
$126$	0	0
$127$	1.16293e7	0.503779	0.251890	−	0.967756i	$-0.418948\pi$
$127$	1.16293e7	0.503779	0.251890	+	0.967756i	$0.418948\pi$
$128$	0	0
$129$	2.53652e7	1.04034
$130$	0	0
$131$	−4.71549e7	−1.83264	−0.916320	−	0.400448i	$-0.868855\pi$
$131$	−4.71549e7	−1.83264	−0.916320	+	0.400448i	$0.868855\pi$
$132$	0	0
$133$	9.15711e6	0.337503
$134$	0	0
$135$	3.35247e6	0.117273
$136$	0	0
$137$	−1.51260e7	−0.502576	−0.251288	−	0.967912i	$-0.580854\pi$
$137$	−1.51260e7	−0.502576	−0.251288	+	0.967912i	$0.580854\pi$
$138$	0	0
$139$	5.10238e6	0.161146	0.0805732	−	0.996749i	$-0.474325\pi$
$139$	5.10238e6	0.161146	0.0805732	+	0.996749i	$0.474325\pi$
$140$	0	0
$141$	3.07705e7	0.924415
$142$	0	0
$143$	1.25470e7	0.358808
$144$	0	0
$145$	3.13924e7	0.855138
$146$	0	0
$147$	−4.80731e6	−0.124822
$148$	0	0
$149$	5.54568e7	1.37342	0.686709	−	0.726932i	$-0.259055\pi$
$149$	5.54568e7	1.37342	0.686709	+	0.726932i	$0.259055\pi$
$150$	0	0
$151$	4.11499e7	0.972633	0.486317	−	0.873783i	$-0.338340\pi$
$151$	4.11499e7	0.972633	0.486317	+	0.873783i	$0.338340\pi$
$152$	0	0
$153$	2.24413e7	0.506557
$154$	0	0
$155$	1.04430e7	0.225250
$156$	0	0
$157$	−2.41239e7	−0.497507	−0.248754	−	0.968567i	$-0.580021\pi$
$157$	−2.41239e7	−0.497507	−0.248754	+	0.968567i	$0.580021\pi$
$158$	0	0
$159$	8.75863e6	0.172801
$160$	0	0
$161$	9.33674e7	1.76321
$162$	0	0
$163$	2.99050e6	0.0540862	0.0270431	−	0.999634i	$-0.491391\pi$
$163$	2.99050e6	0.0540862	0.0270431	+	0.999634i	$0.491391\pi$
$164$	0	0
$165$	−3.93592e7	−0.682107
$166$	0	0
$167$	5.25574e7	0.873225	0.436613	−	0.899650i	$-0.356178\pi$
$167$	5.25574e7	0.873225	0.436613	+	0.899650i	$0.356178\pi$
$168$	0	0
$169$	−6.05994e7	−0.965750
$170$	0	0
$171$	−8.30883e6	−0.127073
$172$	0	0
$173$	1.30414e8	1.91498	0.957489	−	0.288470i	$-0.0931466\pi$
$173$	1.30414e8	1.91498	0.957489	+	0.288470i	$0.0931466\pi$
$174$	0	0
$175$	3.94603e7	0.556580
$176$	0	0
$177$	−5.03327e7	−0.682251
$178$	0	0
$179$	−3.42930e7	−0.446910	−0.223455	−	0.974714i	$-0.571734\pi$
$179$	−3.42930e7	−0.446910	−0.223455	+	0.974714i	$0.571734\pi$
$180$	0	0
$181$	1.04120e6	0.0130515	0.00652574	−	0.999979i	$-0.497923\pi$
$181$	1.04120e6	0.0130515	0.00652574	+	0.999979i	$0.497923\pi$
$182$	0	0
$183$	−3.08217e7	−0.371773
$184$	0	0
$185$	6.08156e7	0.706178
$186$	0	0
$187$	−2.63469e8	−2.94635
$188$	0	0
$189$	−1.58138e7	−0.170381
$190$	0	0
$191$	1.07300e8	1.11426	0.557128	−	0.830427i	$-0.311903\pi$
$191$	1.07300e8	1.11426	0.557128	+	0.830427i	$0.311903\pi$
$192$	0	0
$193$	8.41745e7	0.842810	0.421405	−	0.906873i	$-0.361537\pi$
$193$	8.41745e7	0.842810	0.421405	+	0.906873i	$0.361537\pi$
$194$	0	0
$195$	−6.74166e6	−0.0651097
$196$	0	0
$197$	1.24519e8	1.16039	0.580193	−	0.814479i	$-0.302977\pi$
$197$	1.24519e8	1.16039	0.580193	+	0.814479i	$0.302977\pi$
$198$	0	0
$199$	−5.14773e7	−0.463053	−0.231526	−	0.972829i	$-0.574372\pi$
$199$	−5.14773e7	−0.463053	−0.231526	+	0.972829i	$0.574372\pi$
$200$	0	0
$201$	4.28880e7	0.372520
$202$	0	0
$203$	−1.48080e8	−1.24240
$204$	0	0
$205$	6.63140e7	0.537609
$206$	0	0
$207$	−8.47182e7	−0.663866
$208$	0	0
$209$	9.75487e7	0.739110
$210$	0	0
$211$	−2.50525e7	−0.183596	−0.0917978	−	0.995778i	$-0.529261\pi$
$211$	−2.50525e7	−0.183596	−0.0917978	+	0.995778i	$0.529261\pi$
$212$	0	0
$213$	1.07479e8	0.762068
$214$	0	0
$215$	1.60010e8	1.09803
$216$	0	0
$217$	−4.92604e7	−0.327257
$218$	0	0
$219$	9.76478e7	0.628215
$220$	0	0
$221$	−4.51284e7	−0.281240
$222$	0	0
$223$	2.69254e8	1.62590	0.812952	−	0.582331i	$-0.197859\pi$
$223$	2.69254e8	1.62590	0.812952	+	0.582331i	$0.197859\pi$
$224$	0	0
$225$	−3.58049e7	−0.209557
$226$	0	0
$227$	−9.89081e7	−0.561231	−0.280615	−	0.959820i	$-0.590539\pi$
$227$	−9.89081e7	−0.561231	−0.280615	+	0.959820i	$0.590539\pi$
$228$	0	0
$229$	−1.76761e8	−0.972665	−0.486333	−	0.873774i	$-0.661666\pi$
$229$	−1.76761e8	−0.972665	−0.486333	+	0.873774i	$0.661666\pi$
$230$	0	0
$231$	1.85660e8	0.991008
$232$	0	0
$233$	−8.16179e7	−0.422707	−0.211354	−	0.977410i	$-0.567787\pi$
$233$	−8.16179e7	−0.422707	−0.211354	+	0.977410i	$0.567787\pi$
$234$	0	0
$235$	1.94108e8	0.975678
$236$	0	0
$237$	1.15070e8	0.561491
$238$	0	0
$239$	4.87021e7	0.230757	0.115379	−	0.993322i	$-0.463192\pi$
$239$	4.87021e7	0.230757	0.115379	+	0.993322i	$0.463192\pi$
$240$	0	0
$241$	2.21691e8	1.02021	0.510104	−	0.860113i	$-0.329607\pi$
$241$	2.21691e8	1.02021	0.510104	+	0.860113i	$0.329607\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−3.03258e7	−0.131744
$246$	0	0
$247$	1.67087e7	0.0705508
$248$	0	0
$249$	−4.36977e6	−0.0179375
$250$	0	0
$251$	1.90667e8	0.761059	0.380529	−	0.924769i	$-0.375742\pi$
$251$	1.90667e8	0.761059	0.380529	+	0.924769i	$0.375742\pi$
$252$	0	0
$253$	9.94622e8	3.86132
$254$	0	0
$255$	1.41566e8	0.534648
$256$	0	0
$257$	1.82788e8	0.671711	0.335855	−	0.941914i	$-0.390975\pi$
$257$	1.82788e8	0.671711	0.335855	+	0.941914i	$0.390975\pi$
$258$	0	0
$259$	−2.86872e8	−1.02598
$260$	0	0
$261$	1.34362e8	0.467774
$262$	0	0
$263$	2.64394e8	0.896204	0.448102	−	0.893982i	$-0.352100\pi$
$263$	2.64394e8	0.896204	0.448102	+	0.893982i	$0.352100\pi$
$264$	0	0
$265$	5.52518e7	0.182384
$266$	0	0
$267$	−2.49772e8	−0.803071
$268$	0	0
$269$	5.76393e6	0.0180545	0.00902726	−	0.999959i	$-0.497126\pi$
$269$	5.76393e6	0.0180545	0.00902726	+	0.999959i	$0.497126\pi$
$270$	0	0
$271$	4.54148e7	0.138613	0.0693066	−	0.997595i	$-0.477921\pi$
$271$	4.54148e7	0.138613	0.0693066	+	0.997595i	$0.477921\pi$
$272$	0	0
$273$	3.18009e7	0.0945954
$274$	0	0
$275$	4.20362e8	1.21887
$276$	0	0
$277$	−2.55561e8	−0.722463	−0.361231	−	0.932476i	$-0.617644\pi$
$277$	−2.55561e8	−0.722463	−0.361231	+	0.932476i	$0.617644\pi$
$278$	0	0
$279$	4.46971e7	0.123215
$280$	0	0
$281$	1.19922e8	0.322423	0.161211	−	0.986920i	$-0.448460\pi$
$281$	1.19922e8	0.322423	0.161211	+	0.986920i	$0.448460\pi$
$282$	0	0
$283$	−3.77072e8	−0.988943	−0.494472	−	0.869194i	$-0.664638\pi$
$283$	−3.77072e8	−0.988943	−0.494472	+	0.869194i	$0.664638\pi$
$284$	0	0
$285$	−5.24143e7	−0.134120
$286$	0	0
$287$	−3.12808e8	−0.781073
$288$	0	0
$289$	5.37299e8	1.30940
$290$	0	0
$291$	−2.32496e8	−0.553084
$292$	0	0
$293$	−4.64702e8	−1.07929	−0.539645	−	0.841893i	$-0.681441\pi$
$293$	−4.64702e8	−1.07929	−0.539645	+	0.841893i	$0.681441\pi$
$294$	0	0
$295$	−3.17512e8	−0.720084
$296$	0	0
$297$	−1.68461e8	−0.373124
$298$	0	0
$299$	1.70364e8	0.368578
$300$	0	0
$301$	−7.54781e8	−1.59528
$302$	0	0
$303$	−1.24330e8	−0.256759
$304$	0	0
$305$	−1.94431e8	−0.392389
$306$	0	0
$307$	2.92572e8	0.577096	0.288548	−	0.957465i	$-0.406828\pi$
$307$	2.92572e8	0.577096	0.288548	+	0.957465i	$0.406828\pi$
$308$	0	0
$309$	−5.28728e7	−0.101948
$310$	0	0
$311$	−8.46441e7	−0.159564	−0.0797821	−	0.996812i	$-0.525422\pi$
$311$	−8.46441e7	−0.159564	−0.0797821	+	0.996812i	$0.525422\pi$
$312$	0	0
$313$	2.45028e8	0.451660	0.225830	−	0.974167i	$-0.427491\pi$
$313$	2.45028e8	0.451660	0.225830	+	0.974167i	$0.427491\pi$
$314$	0	0
$315$	−9.97579e7	−0.179829
$316$	0	0
$317$	9.16827e8	1.61652	0.808258	−	0.588828i	$-0.200411\pi$
$317$	9.16827e8	1.61652	0.808258	+	0.588828i	$0.200411\pi$
$318$	0	0
$319$	−1.57746e9	−2.72077
$320$	0	0
$321$	7.84439e6	0.0132371
$322$	0	0
$323$	−3.50860e8	−0.579328
$324$	0	0
$325$	7.20019e7	0.116346
$326$	0	0
$327$	3.24233e8	0.512790
$328$	0	0
$329$	−9.15623e8	−1.41753
$330$	0	0
$331$	−8.63201e8	−1.30832	−0.654160	−	0.756356i	$-0.726978\pi$
$331$	−8.63201e8	−1.30832	−0.654160	+	0.756356i	$0.726978\pi$
$332$	0	0
$333$	2.60297e8	0.386291
$334$	0	0
$335$	2.70549e8	0.393177
$336$	0	0
$337$	1.99753e8	0.284307	0.142154	−	0.989845i	$-0.454597\pi$
$337$	1.99753e8	0.284307	0.142154	+	0.989845i	$0.454597\pi$
$338$	0	0
$339$	2.27063e8	0.316554
$340$	0	0
$341$	−5.24760e8	−0.716672
$342$	0	0
$343$	8.04705e8	1.07673
$344$	0	0
$345$	−5.34425e8	−0.700680
$346$	0	0
$347$	−4.63988e8	−0.596147	−0.298073	−	0.954543i	$-0.596344\pi$
$347$	−4.63988e8	−0.596147	−0.298073	+	0.954543i	$0.596344\pi$
$348$	0	0
$349$	−2.00082e8	−0.251953	−0.125976	−	0.992033i	$-0.540206\pi$
$349$	−2.00082e8	−0.251953	−0.125976	+	0.992033i	$0.540206\pi$
$350$	0	0
$351$	−2.88550e7	−0.0356160
$352$	0	0
$353$	−1.87624e8	−0.227026	−0.113513	−	0.993537i	$-0.536210\pi$
$353$	−1.87624e8	−0.227026	−0.113513	+	0.993537i	$0.536210\pi$
$354$	0	0
$355$	6.78004e8	0.804328
$356$	0	0
$357$	−6.67777e8	−0.776770
$358$	0	0
$359$	5.52437e8	0.630162	0.315081	−	0.949065i	$-0.397968\pi$
$359$	5.52437e8	0.630162	0.315081	+	0.949065i	$0.397968\pi$
$360$	0	0
$361$	−7.63967e8	−0.854672
$362$	0	0
$363$	1.45164e9	1.59289
$364$	0	0
$365$	6.15988e8	0.663052
$366$	0	0
$367$	4.84351e8	0.511480	0.255740	−	0.966746i	$-0.417681\pi$
$367$	4.84351e8	0.511480	0.255740	+	0.966746i	$0.417681\pi$
$368$	0	0
$369$	2.83831e8	0.294081
$370$	0	0
$371$	−2.60627e8	−0.264978
$372$	0	0
$373$	−1.23031e9	−1.22753	−0.613766	−	0.789488i	$-0.710346\pi$
$373$	−1.23031e9	−1.22753	−0.613766	+	0.789488i	$0.710346\pi$
$374$	0	0
$375$	−5.85142e8	−0.572996
$376$	0	0
$377$	−2.70196e8	−0.259708
$378$	0	0
$379$	3.92482e8	0.370325	0.185162	−	0.982708i	$-0.440719\pi$
$379$	3.92482e8	0.370325	0.185162	+	0.982708i	$0.440719\pi$
$380$	0	0
$381$	3.13991e8	0.290857
$382$	0	0
$383$	1.19343e9	1.08543	0.542713	−	0.839918i	$-0.317397\pi$
$383$	1.19343e9	1.08543	0.542713	+	0.839918i	$0.317397\pi$
$384$	0	0
$385$	1.17119e9	1.04596
$386$	0	0
$387$	6.84860e8	0.600639
$388$	0	0
$389$	−2.02141e9	−1.74113	−0.870566	−	0.492052i	$-0.836247\pi$
$389$	−2.02141e9	−1.74113	−0.870566	+	0.492052i	$0.836247\pi$
$390$	0	0
$391$	−3.57742e9	−3.02658
$392$	0	0
$393$	−1.27318e9	−1.05807
$394$	0	0
$395$	7.25891e8	0.592627
$396$	0	0
$397$	−2.00840e9	−1.61095	−0.805477	−	0.592627i	$-0.798091\pi$
$397$	−2.00840e9	−1.61095	−0.805477	+	0.592627i	$0.798091\pi$
$398$	0	0
$399$	2.47242e8	0.194858
$400$	0	0
$401$	1.97549e9	1.52992	0.764961	−	0.644077i	$-0.222758\pi$
$401$	1.97549e9	1.52992	0.764961	+	0.644077i	$0.222758\pi$
$402$	0	0
$403$	−8.98837e7	−0.0684090
$404$	0	0
$405$	9.05167e7	0.0677074
$406$	0	0
$407$	−3.05598e9	−2.24683
$408$	0	0
$409$	−2.30321e9	−1.66457	−0.832285	−	0.554348i	$-0.812968\pi$
$409$	−2.30321e9	−1.66457	−0.832285	+	0.554348i	$0.812968\pi$
$410$	0	0
$411$	−4.08402e8	−0.290162
$412$	0	0
$413$	1.49773e9	1.04618
$414$	0	0
$415$	−2.75657e7	−0.0189322
$416$	0	0
$417$	1.37764e8	0.0930379
$418$	0	0
$419$	8.11767e8	0.539116	0.269558	−	0.962984i	$-0.413122\pi$
$419$	8.11767e8	0.539116	0.269558	+	0.962984i	$0.413122\pi$
$420$	0	0
$421$	−1.79307e9	−1.17114	−0.585572	−	0.810621i	$-0.699130\pi$
$421$	−1.79307e9	−1.17114	−0.585572	+	0.810621i	$0.699130\pi$
$422$	0	0
$423$	8.30802e8	0.533711
$424$	0	0
$425$	−1.51194e9	−0.955376
$426$	0	0
$427$	9.17147e8	0.570087
$428$	0	0
$429$	3.38768e8	0.207158
$430$	0	0
$431$	−2.25406e9	−1.35611	−0.678056	−	0.735010i	$-0.737177\pi$
$431$	−2.25406e9	−1.35611	−0.678056	+	0.735010i	$0.737177\pi$
$432$	0	0
$433$	−5.79170e8	−0.342846	−0.171423	−	0.985198i	$-0.554836\pi$
$433$	−5.79170e8	−0.342846	−0.171423	+	0.985198i	$0.554836\pi$
$434$	0	0
$435$	8.47594e8	0.493714
$436$	0	0
$437$	1.32453e9	0.759235
$438$	0	0
$439$	2.10809e9	1.18922	0.594612	−	0.804013i	$-0.297306\pi$
$439$	2.10809e9	1.18922	0.594612	+	0.804013i	$0.297306\pi$
$440$	0	0
$441$	−1.29797e8	−0.0720660
$442$	0	0
$443$	3.30366e9	1.80544	0.902719	−	0.430231i	$-0.141568\pi$
$443$	3.30366e9	1.80544	0.902719	+	0.430231i	$0.141568\pi$
$444$	0	0
$445$	−1.57563e9	−0.847605
$446$	0	0
$447$	1.49733e9	0.792943
$448$	0	0
$449$	−1.97620e9	−1.03031	−0.515156	−	0.857096i	$-0.672266\pi$
$449$	−1.97620e9	−1.03031	−0.515156	+	0.857096i	$0.672266\pi$
$450$	0	0
$451$	−3.33227e9	−1.71050
$452$	0	0
$453$	1.11105e9	0.561550
$454$	0	0
$455$	2.00608e8	0.0998411
$456$	0	0
$457$	3.00615e9	1.47334	0.736672	−	0.676251i	$-0.236396\pi$
$457$	3.00615e9	1.47334	0.736672	+	0.676251i	$0.236396\pi$
$458$	0	0
$459$	6.05916e8	0.292461
$460$	0	0
$461$	2.10649e9	1.00140	0.500698	−	0.865622i	$-0.333077\pi$
$461$	2.10649e9	1.00140	0.500698	+	0.865622i	$0.333077\pi$
$462$	0	0
$463$	−2.63797e9	−1.23520	−0.617599	−	0.786493i	$-0.711894\pi$
$463$	−2.63797e9	−1.23520	−0.617599	+	0.786493i	$0.711894\pi$
$464$	0	0
$465$	2.81961e8	0.130048
$466$	0	0
$467$	3.52190e9	1.60018	0.800088	−	0.599882i	$-0.204786\pi$
$467$	3.52190e9	1.60018	0.800088	+	0.599882i	$0.204786\pi$
$468$	0	0
$469$	−1.27620e9	−0.571233
$470$	0	0
$471$	−6.51347e8	−0.287236
$472$	0	0
$473$	−8.04051e9	−3.49357
$474$	0	0
$475$	5.59792e8	0.239662
$476$	0	0
$477$	2.36483e8	0.0997667
$478$	0	0
$479$	7.26788e8	0.302158	0.151079	−	0.988522i	$-0.451725\pi$
$479$	7.26788e8	0.302158	0.151079	+	0.988522i	$0.451725\pi$
$480$	0	0
$481$	−5.23445e8	−0.214468
$482$	0	0
$483$	2.52092e9	1.01799
$484$	0	0
$485$	−1.46665e9	−0.583754
$486$	0	0
$487$	−2.74369e9	−1.07643	−0.538213	−	0.842809i	$-0.680900\pi$
$487$	−2.74369e9	−1.07643	−0.538213	+	0.842809i	$0.680900\pi$
$488$	0	0
$489$	8.07434e7	0.0312267
$490$	0	0
$491$	7.06210e8	0.269245	0.134623	−	0.990897i	$-0.457018\pi$
$491$	7.06210e8	0.269245	0.134623	+	0.990897i	$0.457018\pi$
$492$	0	0
$493$	5.67377e9	2.13259
$494$	0	0
$495$	−1.06270e9	−0.393815
$496$	0	0
$497$	−3.19819e9	−1.16858
$498$	0	0
$499$	−5.00895e8	−0.180466	−0.0902329	−	0.995921i	$-0.528761\pi$
$499$	−5.00895e8	−0.180466	−0.0902329	+	0.995921i	$0.528761\pi$
$500$	0	0
$501$	1.41905e9	0.504157
$502$	0	0
$503$	1.62878e9	0.570658	0.285329	−	0.958430i	$-0.407897\pi$
$503$	1.62878e9	0.570658	0.285329	+	0.958430i	$0.407897\pi$
$504$	0	0
$505$	−7.84305e8	−0.270997
$506$	0	0
$507$	−1.63618e9	−0.557576
$508$	0	0
$509$	−3.07202e9	−1.03255	−0.516277	−	0.856422i	$-0.672682\pi$
$509$	−3.07202e9	−1.03255	−0.516277	+	0.856422i	$0.672682\pi$
$510$	0	0
$511$	−2.90566e9	−0.963323
$512$	0	0
$513$	−2.24338e8	−0.0733656
$514$	0	0
$515$	−3.33535e8	−0.107601
$516$	0	0
$517$	−9.75392e9	−3.10429
$518$	0	0
$519$	3.52119e9	1.10561
$520$	0	0
$521$	2.94213e9	0.911444	0.455722	−	0.890122i	$-0.349381\pi$
$521$	2.94213e9	0.911444	0.455722	+	0.890122i	$0.349381\pi$
$522$	0	0
$523$	−1.96275e9	−0.599942	−0.299971	−	0.953948i	$-0.596977\pi$
$523$	−1.96275e9	−0.599942	−0.299971	+	0.953948i	$0.596977\pi$
$524$	0	0
$525$	1.06543e9	0.321342
$526$	0	0
$527$	1.88744e9	0.561741
$528$	0	0
$529$	1.01003e10	2.96646
$530$	0	0
$531$	−1.35898e9	−0.393898
$532$	0	0
$533$	−5.70770e8	−0.163273
$534$	0	0
$535$	4.94845e7	0.0139711
$536$	0	0
$537$	−9.25911e8	−0.258023
$538$	0	0
$539$	1.52387e9	0.419166
$540$	0	0
$541$	−2.58836e9	−0.702804	−0.351402	−	0.936225i	$-0.614295\pi$
$541$	−2.58836e9	−0.702804	−0.351402	+	0.936225i	$0.614295\pi$
$542$	0	0
$543$	2.81124e7	0.00753527
$544$	0	0
$545$	2.04535e9	0.541226
$546$	0	0
$547$	−2.21608e9	−0.578936	−0.289468	−	0.957188i	$-0.593478\pi$
$547$	−2.21608e9	−0.578936	−0.289468	+	0.957188i	$0.593478\pi$
$548$	0	0
$549$	−8.32186e8	−0.214643
$550$	0	0
$551$	−2.10069e9	−0.534973
$552$	0	0
$553$	−3.42408e9	−0.861006
$554$	0	0
$555$	1.64202e9	0.407712
$556$	0	0
$557$	−2.23234e9	−0.547353	−0.273677	−	0.961822i	$-0.588240\pi$
$557$	−2.23234e9	−0.547353	−0.273677	+	0.961822i	$0.588240\pi$
$558$	0	0
$559$	−1.37722e9	−0.333474
$560$	0	0
$561$	−7.11367e9	−1.70108
$562$	0	0
$563$	−4.05647e9	−0.958007	−0.479004	−	0.877813i	$-0.659002\pi$
$563$	−4.05647e9	−0.958007	−0.479004	+	0.877813i	$0.659002\pi$
$564$	0	0
$565$	1.43237e9	0.334108
$566$	0	0
$567$	−4.26974e8	−0.0983695
$568$	0	0
$569$	7.36385e9	1.67576	0.837881	−	0.545854i	$-0.183795\pi$
$569$	7.36385e9	1.67576	0.837881	+	0.545854i	$0.183795\pi$
$570$	0	0
$571$	−1.33947e9	−0.301096	−0.150548	−	0.988603i	$-0.548104\pi$
$571$	−1.33947e9	−0.301096	−0.150548	+	0.988603i	$0.548104\pi$
$572$	0	0
$573$	2.89711e9	0.643316
$574$	0	0
$575$	5.70773e9	1.25206
$576$	0	0
$577$	2.99535e9	0.649131	0.324565	−	0.945863i	$-0.394782\pi$
$577$	2.99535e9	0.649131	0.324565	+	0.945863i	$0.394782\pi$
$578$	0	0
$579$	2.27271e9	0.486597
$580$	0	0
$581$	1.30029e8	0.0275059
$582$	0	0
$583$	−2.77640e9	−0.580285
$584$	0	0
$585$	−1.82025e8	−0.0375911
$586$	0	0
$587$	−1.17790e8	−0.0240366	−0.0120183	−	0.999928i	$-0.503826\pi$
$587$	−1.17790e8	−0.0240366	−0.0120183	+	0.999928i	$0.503826\pi$
$588$	0	0
$589$	−6.98818e8	−0.140916
$590$	0	0
$591$	3.36200e9	0.669949
$592$	0	0
$593$	9.97185e8	0.196374	0.0981871	−	0.995168i	$-0.468696\pi$
$593$	9.97185e8	0.196374	0.0981871	+	0.995168i	$0.468696\pi$
$594$	0	0
$595$	−4.21251e9	−0.819845
$596$	0	0
$597$	−1.38989e9	−0.267344
$598$	0	0
$599$	1.00115e9	0.190328	0.0951641	−	0.995462i	$-0.469662\pi$
$599$	1.00115e9	0.190328	0.0951641	+	0.995462i	$0.469662\pi$
$600$	0	0
$601$	2.09863e9	0.394343	0.197172	−	0.980369i	$-0.436824\pi$
$601$	2.09863e9	0.394343	0.197172	+	0.980369i	$0.436824\pi$
$602$	0	0
$603$	1.15798e9	0.215074
$604$	0	0
$605$	9.15735e9	1.68123
$606$	0	0
$607$	−6.56157e9	−1.19082	−0.595411	−	0.803421i	$-0.703011\pi$
$607$	−6.56157e9	−1.19082	−0.595411	+	0.803421i	$0.703011\pi$
$608$	0	0
$609$	−3.99816e9	−0.717299
$610$	0	0
$611$	−1.67070e9	−0.296316
$612$	0	0
$613$	−2.41894e9	−0.424144	−0.212072	−	0.977254i	$-0.568021\pi$
$613$	−2.41894e9	−0.424144	−0.212072	+	0.977254i	$0.568021\pi$
$614$	0	0
$615$	1.79048e9	0.310389
$616$	0	0
$617$	−7.05422e9	−1.20907	−0.604534	−	0.796579i	$-0.706641\pi$
$617$	−7.05422e9	−1.20907	−0.604534	+	0.796579i	$0.706641\pi$
$618$	0	0
$619$	−5.23254e9	−0.886738	−0.443369	−	0.896339i	$-0.646217\pi$
$619$	−5.23254e9	−0.886738	−0.443369	+	0.896339i	$0.646217\pi$
$620$	0	0
$621$	−2.28739e9	−0.383283
$622$	0	0
$623$	7.43235e9	1.23145
$624$	0	0
$625$	1.45882e8	0.0239014
$626$	0	0
$627$	2.63381e9	0.426726
$628$	0	0
$629$	1.09916e10	1.76111
$630$	0	0
$631$	−4.89794e9	−0.776088	−0.388044	−	0.921641i	$-0.626849\pi$
$631$	−4.89794e9	−0.776088	−0.388044	+	0.921641i	$0.626849\pi$
$632$	0	0
$633$	−6.76417e8	−0.105999
$634$	0	0
$635$	1.98074e9	0.306986
$636$	0	0
$637$	2.61016e8	0.0400110
$638$	0	0
$639$	2.90192e9	0.439980
$640$	0	0
$641$	8.62626e9	1.29366	0.646829	−	0.762635i	$-0.276095\pi$
$641$	8.62626e9	1.29366	0.646829	+	0.762635i	$0.276095\pi$
$642$	0	0
$643$	8.94373e9	1.32672	0.663362	−	0.748299i	$-0.269129\pi$
$643$	8.94373e9	1.32672	0.663362	+	0.748299i	$0.269129\pi$
$644$	0	0
$645$	4.32028e9	0.633947
$646$	0	0
$647$	6.57546e9	0.954468	0.477234	−	0.878776i	$-0.341639\pi$
$647$	6.57546e9	0.954468	0.477234	+	0.878776i	$0.341639\pi$
$648$	0	0
$649$	1.59550e10	2.29108
$650$	0	0
$651$	−1.33003e9	−0.188942
$652$	0	0
$653$	−2.14306e9	−0.301189	−0.150594	−	0.988596i	$-0.548119\pi$
$653$	−2.14306e9	−0.301189	−0.150594	+	0.988596i	$0.548119\pi$
$654$	0	0
$655$	−8.03156e9	−1.11675
$656$	0	0
$657$	2.63649e9	0.362700
$658$	0	0
$659$	7.97182e8	0.108507	0.0542536	−	0.998527i	$-0.482722\pi$
$659$	7.97182e8	0.108507	0.0542536	+	0.998527i	$0.482722\pi$
$660$	0	0
$661$	3.11232e9	0.419159	0.209579	−	0.977792i	$-0.432791\pi$
$661$	3.11232e9	0.419159	0.209579	+	0.977792i	$0.432791\pi$
$662$	0	0
$663$	−1.21847e9	−0.162374
$664$	0	0
$665$	1.55967e9	0.205663
$666$	0	0
$667$	−2.14190e10	−2.79485
$668$	0	0
$669$	7.26985e9	0.938716
$670$	0	0
$671$	9.77016e9	1.24845
$672$	0	0
$673$	2.27827e9	0.288107	0.144053	−	0.989570i	$-0.453986\pi$
$673$	2.27827e9	0.288107	0.144053	+	0.989570i	$0.453986\pi$
$674$	0	0
$675$	−9.66731e8	−0.120988
$676$	0	0
$677$	9.55138e9	1.18306	0.591529	−	0.806284i	$-0.298525\pi$
$677$	9.55138e9	1.18306	0.591529	+	0.806284i	$0.298525\pi$
$678$	0	0
$679$	6.91829e9	0.848115
$680$	0	0
$681$	−2.67052e9	−0.324027
$682$	0	0
$683$	−1.27351e10	−1.52943	−0.764715	−	0.644369i	$-0.777120\pi$
$683$	−1.27351e10	−1.52943	−0.764715	+	0.644369i	$0.777120\pi$
$684$	0	0
$685$	−2.57631e9	−0.306253
$686$	0	0
$687$	−4.77256e9	−0.561569
$688$	0	0
$689$	−4.75556e8	−0.0553904
$690$	0	0
$691$	−1.48449e10	−1.71161	−0.855804	−	0.517300i	$-0.826937\pi$
$691$	−1.48449e10	−1.71161	−0.855804	+	0.517300i	$0.826937\pi$
$692$	0	0
$693$	5.01283e9	0.572159
$694$	0	0
$695$	8.69053e8	0.0981972
$696$	0	0
$697$	1.19854e10	1.34072
$698$	0	0
$699$	−2.20368e9	−0.244050
$700$	0	0
$701$	−6.56312e8	−0.0719610	−0.0359805	−	0.999352i	$-0.511455\pi$
$701$	−6.56312e8	−0.0719610	−0.0359805	+	0.999352i	$0.511455\pi$
$702$	0	0
$703$	−4.06962e9	−0.441784
$704$	0	0
$705$	5.24092e9	0.563308
$706$	0	0
$707$	3.69963e9	0.393722
$708$	0	0
$709$	3.03120e9	0.319413	0.159706	−	0.987165i	$-0.448945\pi$
$709$	3.03120e9	0.319413	0.159706	+	0.987165i	$0.448945\pi$
$710$	0	0
$711$	3.10689e9	0.324177
$712$	0	0
$713$	−7.12526e9	−0.736186
$714$	0	0
$715$	2.13704e9	0.218646
$716$	0	0
$717$	1.31496e9	0.133228
$718$	0	0
$719$	−5.92132e9	−0.594111	−0.297055	−	0.954860i	$-0.596005\pi$
$719$	−5.92132e9	−0.594111	−0.297055	+	0.954860i	$0.596005\pi$
$720$	0	0
$721$	1.57331e9	0.156330
$722$	0	0
$723$	5.98566e9	0.589018
$724$	0	0
$725$	−9.05242e9	−0.882230
$726$	0	0
$727$	1.77984e10	1.71795	0.858975	−	0.512017i	$-0.171102\pi$
$727$	1.77984e10	1.71795	0.858975	+	0.512017i	$0.171102\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	2.89198e10	2.73832
$732$	0	0
$733$	−5.50247e9	−0.516053	−0.258026	−	0.966138i	$-0.583072\pi$
$733$	−5.50247e9	−0.516053	−0.258026	+	0.966138i	$0.583072\pi$
$734$	0	0
$735$	−8.18795e8	−0.0760623
$736$	0	0
$737$	−1.35950e10	−1.25096
$738$	0	0
$739$	−2.95798e9	−0.269613	−0.134806	−	0.990872i	$-0.543041\pi$
$739$	−2.95798e9	−0.269613	−0.134806	+	0.990872i	$0.543041\pi$
$740$	0	0
$741$	4.51134e8	0.0407325
$742$	0	0
$743$	4.33970e9	0.388150	0.194075	−	0.980987i	$-0.437830\pi$
$743$	4.33970e9	0.388150	0.194075	+	0.980987i	$0.437830\pi$
$744$	0	0
$745$	9.44557e9	0.836915
$746$	0	0
$747$	−1.17984e8	−0.0103562
$748$	0	0
$749$	−2.33422e8	−0.0202981
$750$	0	0
$751$	1.15573e10	0.995675	0.497838	−	0.867270i	$-0.334128\pi$
$751$	1.15573e10	0.995675	0.497838	+	0.867270i	$0.334128\pi$
$752$	0	0
$753$	5.14802e9	0.439397
$754$	0	0
$755$	7.00877e9	0.592690
$756$	0	0
$757$	1.11105e10	0.930886	0.465443	−	0.885078i	$-0.345895\pi$
$757$	1.11105e10	0.930886	0.465443	+	0.885078i	$0.345895\pi$
$758$	0	0
$759$	2.68548e10	2.22934
$760$	0	0
$761$	4.48974e9	0.369297	0.184648	−	0.982805i	$-0.440885\pi$
$761$	4.48974e9	0.369297	0.184648	+	0.982805i	$0.440885\pi$
$762$	0	0
$763$	−9.64804e9	−0.786327
$764$	0	0
$765$	3.82228e9	0.308679
$766$	0	0
$767$	2.73285e9	0.218692
$768$	0	0
$769$	−1.08922e10	−0.863720	−0.431860	−	0.901941i	$-0.642143\pi$
$769$	−1.08922e10	−0.863720	−0.431860	+	0.901941i	$0.642143\pi$
$770$	0	0
$771$	4.93528e9	0.387813
$772$	0	0
$773$	2.05575e10	1.60081	0.800407	−	0.599456i	$-0.204617\pi$
$773$	2.05575e10	1.60081	0.800407	+	0.599456i	$0.204617\pi$
$774$	0	0
$775$	−3.01139e9	−0.232386
$776$	0	0
$777$	−7.74554e9	−0.592350
$778$	0	0
$779$	−4.43756e9	−0.336328
$780$	0	0
$781$	−3.40696e10	−2.55911
$782$	0	0
$783$	3.62779e9	0.270070
$784$	0	0
$785$	−4.10887e9	−0.303164
$786$	0	0
$787$	−7.87413e9	−0.575826	−0.287913	−	0.957657i	$-0.592961\pi$
$787$	−7.87413e9	−0.575826	−0.287913	+	0.957657i	$0.592961\pi$
$788$	0	0
$789$	7.13865e9	0.517424
$790$	0	0
$791$	−6.75662e9	−0.485413
$792$	0	0
$793$	1.67349e9	0.119170
$794$	0	0
$795$	1.49180e9	0.105299
$796$	0	0
$797$	1.81725e10	1.27148	0.635742	−	0.771901i	$-0.280694\pi$
$797$	1.81725e10	1.27148	0.635742	+	0.771901i	$0.280694\pi$
$798$	0	0
$799$	3.50826e10	2.43320
$800$	0	0
$801$	−6.74384e9	−0.463653
$802$	0	0
$803$	−3.09534e10	−2.10962
$804$	0	0
$805$	1.59026e10	1.07444
$806$	0	0
$807$	1.55626e8	0.0104238
$808$	0	0
$809$	1.56562e10	1.03960	0.519801	−	0.854288i	$-0.326006\pi$
$809$	1.56562e10	1.03960	0.519801	+	0.854288i	$0.326006\pi$
$810$	0	0
$811$	−7.03230e8	−0.0462940	−0.0231470	−	0.999732i	$-0.507369\pi$
$811$	−7.03230e8	−0.0462940	−0.0231470	+	0.999732i	$0.507369\pi$
$812$	0	0
$813$	1.22620e9	0.0800284
$814$	0	0
$815$	5.09351e8	0.0329583
$816$	0	0
$817$	−1.07075e10	−0.686925
$818$	0	0
$819$	8.58624e8	0.0546147
$820$	0	0
$821$	1.08592e10	0.684850	0.342425	−	0.939545i	$-0.388752\pi$
$821$	1.08592e10	0.684850	0.342425	+	0.939545i	$0.388752\pi$
$822$	0	0
$823$	−3.05577e10	−1.91083	−0.955414	−	0.295271i	$-0.904590\pi$
$823$	−3.05577e10	−1.91083	−0.955414	+	0.295271i	$0.904590\pi$
$824$	0	0
$825$	1.13498e10	0.703718
$826$	0	0
$827$	1.12107e10	0.689229	0.344615	−	0.938744i	$-0.388010\pi$
$827$	1.12107e10	0.689229	0.344615	+	0.938744i	$0.388010\pi$
$828$	0	0
$829$	−6.11945e9	−0.373054	−0.186527	−	0.982450i	$-0.559723\pi$
$829$	−6.11945e9	−0.373054	−0.186527	+	0.982450i	$0.559723\pi$
$830$	0	0
$831$	−6.90015e9	−0.417114
$832$	0	0
$833$	−5.48099e9	−0.328550
$834$	0	0
$835$	8.95174e9	0.532114
$836$	0	0
$837$	1.20682e9	0.0711384
$838$	0	0
$839$	−2.99122e10	−1.74856	−0.874282	−	0.485419i	$-0.838667\pi$
$839$	−2.99122e10	−1.74856	−0.874282	+	0.485419i	$0.838667\pi$
$840$	0	0
$841$	1.67205e10	0.969314
$842$	0	0
$843$	3.23789e9	0.186151
$844$	0	0
$845$	−1.03215e10	−0.588496
$846$	0	0
$847$	−4.31959e10	−2.44259
$848$	0	0
$849$	−1.01809e10	−0.570967
$850$	0	0
$851$	−4.14945e10	−2.30801
$852$	0	0
$853$	−2.07207e10	−1.14310	−0.571548	−	0.820569i	$-0.693657\pi$
$853$	−2.07207e10	−1.14310	−0.571548	+	0.820569i	$0.693657\pi$
$854$	0	0
$855$	−1.41519e9	−0.0774341
$856$	0	0
$857$	1.36122e10	0.738744	0.369372	−	0.929282i	$-0.379573\pi$
$857$	1.36122e10	0.738744	0.369372	+	0.929282i	$0.379573\pi$
$858$	0	0
$859$	−1.64286e10	−0.884351	−0.442175	−	0.896929i	$-0.645793\pi$
$859$	−1.64286e10	−0.884351	−0.442175	+	0.896929i	$0.645793\pi$
$860$	0	0
$861$	−8.44582e9	−0.450952
$862$	0	0
$863$	1.78780e10	0.946849	0.473425	−	0.880834i	$-0.343018\pi$
$863$	1.78780e10	0.946849	0.473425	+	0.880834i	$0.343018\pi$
$864$	0	0
$865$	2.22126e10	1.16692
$866$	0	0
$867$	1.45071e10	0.755984
$868$	0	0
$869$	−3.64760e10	−1.88555
$870$	0	0
$871$	−2.32863e9	−0.119409
$872$	0	0
$873$	−6.27740e9	−0.319323
$874$	0	0
$875$	1.74118e10	0.878649
$876$	0	0
$877$	9.42965e9	0.472060	0.236030	−	0.971746i	$-0.424154\pi$
$877$	9.42965e9	0.472060	0.236030	+	0.971746i	$0.424154\pi$
$878$	0	0
$879$	−1.25470e10	−0.623128
$880$	0	0
$881$	−3.21077e10	−1.58195	−0.790977	−	0.611845i	$-0.790428\pi$
$881$	−3.21077e10	−1.58195	−0.790977	+	0.611845i	$0.790428\pi$
$882$	0	0
$883$	1.75025e10	0.855533	0.427766	−	0.903889i	$-0.359301\pi$
$883$	1.75025e10	0.855533	0.427766	+	0.903889i	$0.359301\pi$
$884$	0	0
$885$	−8.57283e9	−0.415741
$886$	0	0
$887$	3.00901e10	1.44774	0.723870	−	0.689937i	$-0.242362\pi$
$887$	3.00901e10	1.44774	0.723870	+	0.689937i	$0.242362\pi$
$888$	0	0
$889$	−9.34328e9	−0.446009
$890$	0	0
$891$	−4.54846e9	−0.215423
$892$	0	0
$893$	−1.29892e10	−0.610383
$894$	0	0
$895$	−5.84089e9	−0.272332
$896$	0	0
$897$	4.59983e9	0.212798
$898$	0	0
$899$	1.13006e10	0.518733
$900$	0	0
$901$	9.98605e9	0.454838
$902$	0	0
$903$	−2.03791e10	−0.921038
$904$	0	0
$905$	1.77341e8	0.00795313
$906$	0	0
$907$	1.55723e10	0.692990	0.346495	−	0.938052i	$-0.387372\pi$
$907$	1.55723e10	0.692990	0.346495	+	0.938052i	$0.387372\pi$
$908$	0	0
$909$	−3.35690e9	−0.148240
$910$	0	0
$911$	1.98356e10	0.869221	0.434610	−	0.900619i	$-0.356886\pi$
$911$	1.98356e10	0.869221	0.434610	+	0.900619i	$0.356886\pi$
$912$	0	0
$913$	1.38517e9	0.0602361
$914$	0	0
$915$	−5.24965e9	−0.226546
$916$	0	0
$917$	3.78855e10	1.62248
$918$	0	0
$919$	−1.00319e10	−0.426360	−0.213180	−	0.977013i	$-0.568382\pi$
$919$	−1.00319e10	−0.426360	−0.213180	+	0.977013i	$0.568382\pi$
$920$	0	0
$921$	7.89944e9	0.333186
$922$	0	0
$923$	−5.83563e9	−0.244277
$924$	0	0
$925$	−1.75370e10	−0.728551
$926$	0	0
$927$	−1.42756e9	−0.0588595
$928$	0	0
$929$	−9.95372e9	−0.407315	−0.203657	−	0.979042i	$-0.565283\pi$
$929$	−9.95372e9	−0.407315	−0.203657	+	0.979042i	$0.565283\pi$
$930$	0	0
$931$	2.02932e9	0.0824188
$932$	0	0
$933$	−2.28539e9	−0.0921244
$934$	0	0
$935$	−4.48749e10	−1.79541
$936$	0	0
$937$	−2.47369e10	−0.982327	−0.491163	−	0.871068i	$-0.663428\pi$
$937$	−2.47369e10	−0.982327	−0.491163	+	0.871068i	$0.663428\pi$
$938$	0	0
$939$	6.61577e9	0.260766
$940$	0	0
$941$	3.39100e10	1.32667	0.663337	−	0.748321i	$-0.269140\pi$
$941$	3.39100e10	1.32667	0.663337	+	0.748321i	$0.269140\pi$
$942$	0	0
$943$	−4.52460e10	−1.75707
$944$	0	0
$945$	−2.69346e9	−0.103825
$946$	0	0
$947$	3.03467e10	1.16115	0.580573	−	0.814208i	$-0.302828\pi$
$947$	3.03467e10	1.16115	0.580573	+	0.814208i	$0.302828\pi$
$948$	0	0
$949$	−5.30186e9	−0.201371
$950$	0	0
$951$	2.47543e10	0.933296
$952$	0	0
$953$	4.43307e10	1.65913	0.829564	−	0.558412i	$-0.188589\pi$
$953$	4.43307e10	1.65913	0.829564	+	0.558412i	$0.188589\pi$
$954$	0	0
$955$	1.82758e10	0.678990
$956$	0	0
$957$	−4.25915e10	−1.57084
$958$	0	0
$959$	1.21526e10	0.444944
$960$	0	0
$961$	−2.37533e10	−0.863362
$962$	0	0
$963$	2.11799e8	0.00764242
$964$	0	0
$965$	1.43369e10	0.513580
$966$	0	0
$967$	3.65095e10	1.29841	0.649207	−	0.760612i	$-0.275101\pi$
$967$	3.65095e10	1.29841	0.649207	+	0.760612i	$0.275101\pi$
$968$	0	0
$969$	−9.47321e9	−0.334475
$970$	0	0
$971$	2.35243e10	0.824612	0.412306	−	0.911046i	$-0.364724\pi$
$971$	2.35243e10	0.824612	0.412306	+	0.911046i	$0.364724\pi$
$972$	0	0
$973$	−4.09939e9	−0.142667
$974$	0	0
$975$	1.94405e9	0.0671725
$976$	0	0
$977$	−2.90818e10	−0.997679	−0.498840	−	0.866694i	$-0.666240\pi$
$977$	−2.90818e10	−0.997679	−0.498840	+	0.866694i	$0.666240\pi$
$978$	0	0
$979$	7.91751e10	2.69680
$980$	0	0
$981$	8.75428e9	0.296059
$982$	0	0
$983$	−2.81209e10	−0.944259	−0.472130	−	0.881529i	$-0.656515\pi$
$983$	−2.81209e10	−0.944259	−0.472130	+	0.881529i	$0.656515\pi$
$984$	0	0
$985$	2.12084e10	0.707100
$986$	0	0
$987$	−2.47218e10	−0.818409
$988$	0	0
$989$	−1.09175e11	−3.58869
$990$	0	0
$991$	1.76726e10	0.576822	0.288411	−	0.957507i	$-0.406873\pi$
$991$	1.76726e10	0.576822	0.288411	+	0.957507i	$0.406873\pi$
$992$	0	0
$993$	−2.33064e10	−0.755359
$994$	0	0
$995$	−8.76778e9	−0.282169
$996$	0	0
$997$	−3.23958e10	−1.03528	−0.517638	−	0.855600i	$-0.673189\pi$
$997$	−3.23958e10	−1.03528	−0.517638	+	0.855600i	$0.673189\pi$
$998$	0	0
$999$	7.02802e9	0.223025

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} +170.323 q^{5} -803.427 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+27.0000 q^{3} +170.323 q^{5} -803.427 q^{7} +729.000 q^{9} -8558.72 q^{11} -1465.98 q^{13} +4598.72 q^{15} +30783.7 q^{17} -11397.6 q^{19} -21692.5 q^{21} -116211. q^{23} -49115.0 q^{25} +19683.0 q^{27} +184311. q^{29} +61312.9 q^{31} -231086. q^{33} -136842. q^{35} +357060. q^{37} -39581.6 q^{39} +389342. q^{41} +939452. q^{43} +124166. q^{45} +1.13965e6 q^{47} -178048. q^{49} +831160. q^{51} +324394. q^{53} -1.45775e6 q^{55} -307734. q^{57} -1.86418e6 q^{59} -1.14154e6 q^{61} -585698. q^{63} -249691. q^{65} +1.58844e6 q^{67} -3.13771e6 q^{69} +3.98069e6 q^{71} +3.61658e6 q^{73} -1.32611e6 q^{75} +6.87631e6 q^{77} +4.26185e6 q^{79} +531441. q^{81} -161844. q^{83} +5.24318e6 q^{85} +4.97639e6 q^{87} -9.25081e6 q^{89} +1.17781e6 q^{91} +1.65545e6 q^{93} -1.94127e6 q^{95} -8.61098e6 q^{97} -6.23931e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{3} + 192 q^{5} + 680 q^{7} + 2916 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more