LMFDB - Embedded newform 384.8.d.f.193.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,8,Mod(193,384)]

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, 1, 0]))

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

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	384.d (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$119.955849786$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} - 1020 x^{14} + 7280 x^{13} + 388150 x^{12} - 2423904 x^{11} - 70542796 x^{10} + \cdots + 694045717832241 $ x^16 - 8x^15 - 1020x^14 + 7280x^13 + 388150x^12 - 2423904x^11 - 70542796x^10 + 375107560x^9 + 6894477923x^8 - 29855519152x^7 - 372292484316x^6 + 1222974198808x^5 + 10528455140834x^4 - 23130795757272x^3 - 133992602927280x^2 + 145766920335192*x + 694045717832241
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{108}\cdot 3^{28} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.7
Root			$-8.47155 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.8.d.f.193.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000i q^{3} +295.990i q^{5} -1208.55 q^{7} -729.000 q^{9} +O(q^{10})$	$q-27.0000i q^{3} +295.990i q^{5} -1208.55 q^{7} -729.000 q^{9} +8341.26i q^{11} +9842.29i q^{13} +7991.72 q^{15} +9080.24 q^{17} +23035.9i q^{19} +32630.9i q^{21} +92905.1 q^{23} -9484.89 q^{25} +19683.0i q^{27} -38163.5i q^{29} +78442.9 q^{31} +225214. q^{33} -357719. i q^{35} +539121. i q^{37} +265742. q^{39} -872572. q^{41} -548781. i q^{43} -215776. i q^{45} -59537.2 q^{47} +637052. q^{49} -245167. i q^{51} +1.43192e6i q^{53} -2.46893e6 q^{55} +621968. q^{57} +404986. i q^{59} +1.46875e6i q^{61} +881033. q^{63} -2.91322e6 q^{65} +586820. i q^{67} -2.50844e6i q^{69} -3.70658e6 q^{71} +1.57329e6 q^{73} +256092. i q^{75} -1.00808e7i q^{77} +7.58995e6 q^{79} +531441. q^{81} -5.50373e6i q^{83} +2.68766e6i q^{85} -1.03041e6 q^{87} -2.90039e6 q^{89} -1.18949e7i q^{91} -2.11796e6i q^{93} -6.81838e6 q^{95} -9.32490e6 q^{97} -6.08078e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 11664 q^{9}+O(q^{10}) $ 16 * q - 11664 * q^9	$ 16 q - 11664 q^{9} + 77280 q^{17} - 236464 q^{25} + 544320 q^{33} - 930912 q^{41} + 1163024 q^{49} - 3032640 q^{57} + 6283008 q^{65} - 2727200 q^{73} + 8503056 q^{81} - 43093152 q^{89} + 35537120 q^{97}+O(q^{100}) $ 16 * q - 11664 * q^9 + 77280 * q^17 - 236464 * q^25 + 544320 * q^33 - 930912 * q^41 + 1163024 * q^49 - 3032640 * q^57 + 6283008 * q^65 - 2727200 * q^73 + 8503056 * q^81 - 43093152 * q^89 + 35537120 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 27.0000i	− 0.577350i
$3$	− 27.0000i	− 0.577350i
$4$	0	0
$5$	295.990i	1.05896i	0.848321	+	0.529482i	$0.177614\pi$
$5$	295.990i	1.05896i	−0.848321	+	0.529482i	$0.822386\pi$
$6$	0	0
$7$	−1208.55	−1.33175	−0.665873	−	0.746065i	$-0.731941\pi$
$7$	−1208.55	−1.33175	−0.665873	+	0.746065i	$0.731941\pi$
$8$	0	0
$9$	−729.000	−0.333333
$10$	0	0
$11$	8341.26i	1.88955i	0.327727	+	0.944773i	$0.393717\pi$
$11$	8341.26i	1.88955i	−0.327727	+	0.944773i	$0.606283\pi$
$12$	0	0
$13$	9842.29i	1.24249i	0.783615	+	0.621247i	$0.213374\pi$
$13$	9842.29i	1.24249i	−0.783615	+	0.621247i	$0.786626\pi$
$14$	0	0
$15$	7991.72	0.611394
$16$	0	0
$17$	9080.24	0.448256	0.224128	−	0.974560i	$-0.428047\pi$
$17$	9080.24	0.448256	0.224128	+	0.974560i	$0.428047\pi$
$18$	0	0
$19$	23035.9i	0.770490i	0.922814	+	0.385245i	$0.125883\pi$
$19$	23035.9i	0.770490i	−0.922814	+	0.385245i	$0.874117\pi$
$20$	0	0
$21$	32630.9i	0.768884i
$22$	0	0
$23$	92905.1	1.59218	0.796090	−	0.605178i	$-0.206898\pi$
$23$	92905.1	1.59218	0.796090	+	0.605178i	$0.206898\pi$
$24$	0	0
$25$	−9484.89	−0.121407
$26$	0	0
$27$	19683.0i	0.192450i
$28$	0	0
$29$	− 38163.5i	− 0.290573i	−0.989390	−	0.145287i	$-0.953590\pi$
$29$	− 38163.5i	− 0.290573i	0.989390	−	0.145287i	$-0.0464104\pi$
$30$	0	0
$31$	78442.9	0.472920	0.236460	−	0.971641i	$-0.424013\pi$
$31$	78442.9	0.472920	0.236460	+	0.971641i	$0.424013\pi$
$32$	0	0
$33$	225214.	1.09093
$34$	0	0
$35$	− 357719.i	− 1.41027i
$36$	0	0
$37$	539121.i	1.74977i	0.484334	+	0.874883i	$0.339062\pi$
$37$	539121.i	1.74977i	−0.484334	+	0.874883i	$0.660938\pi$
$38$	0	0
$39$	265742.	0.717355
$40$	0	0
$41$	−872572.	−1.97723	−0.988616	−	0.150460i	$-0.951924\pi$
$41$	−872572.	−1.97723	−0.988616	+	0.150460i	$0.951924\pi$
$42$	0	0
$43$	− 548781.i	− 1.05259i	−0.850302	−	0.526296i	$-0.823581\pi$
$43$	− 548781.i	− 1.05259i	0.850302	−	0.526296i	$-0.176419\pi$
$44$	0	0
$45$	− 215776.i	− 0.352988i
$46$	0	0
$47$	−59537.2	−0.0836462	−0.0418231	−	0.999125i	$-0.513317\pi$
$47$	−59537.2	−0.0836462	−0.0418231	+	0.999125i	$0.513317\pi$
$48$	0	0
$49$	637052.	0.773550
$50$	0	0
$51$	− 245167.i	− 0.258801i
$52$	0	0
$53$	1.43192e6i	1.32115i	0.750758	+	0.660577i	$0.229688\pi$
$53$	1.43192e6i	1.32115i	−0.750758	+	0.660577i	$0.770312\pi$
$54$	0	0
$55$	−2.46893e6	−2.00096
$56$	0	0
$57$	621968.	0.444842
$58$	0	0
$59$	404986.i	0.256719i	0.991728	+	0.128359i	$0.0409711\pi$
$59$	404986.i	0.256719i	−0.991728	+	0.128359i	$0.959029\pi$
$60$	0	0
$61$	1.46875e6i	0.828501i	0.910163	+	0.414251i	$0.135956\pi$
$61$	1.46875e6i	0.828501i	−0.910163	+	0.414251i	$0.864044\pi$
$62$	0	0
$63$	881033.	0.443916
$64$	0	0
$65$	−2.91322e6	−1.31576
$66$	0	0
$67$	586820.i	0.238365i	0.992872	+	0.119183i	$0.0380274\pi$
$67$	586820.i	0.238365i	−0.992872	+	0.119183i	$0.961973\pi$
$68$	0	0
$69$	− 2.50844e6i	− 0.919245i
$70$	0	0
$71$	−3.70658e6	−1.22905	−0.614525	−	0.788897i	$-0.710652\pi$
$71$	−3.70658e6	−1.22905	−0.614525	+	0.788897i	$0.710652\pi$
$72$	0	0
$73$	1.57329e6	0.473346	0.236673	−	0.971589i	$-0.423943\pi$
$73$	1.57329e6	0.473346	0.236673	+	0.971589i	$0.423943\pi$
$74$	0	0
$75$	256092.i	0.0700941i
$76$	0	0
$77$	− 1.00808e7i	− 2.51640i
$78$	0	0
$79$	7.58995e6	1.73198	0.865992	−	0.500058i	$-0.166688\pi$
$79$	7.58995e6	1.73198	0.865992	+	0.500058i	$0.166688\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	− 5.50373e6i	− 1.05654i	−0.849078	−	0.528268i	$-0.822842\pi$
$83$	− 5.50373e6i	− 1.05654i	0.849078	−	0.528268i	$-0.177158\pi$
$84$	0	0
$85$	2.68766e6i	0.474687i
$86$	0	0
$87$	−1.03041e6	−0.167762
$88$	0	0
$89$	−2.90039e6	−0.436106	−0.218053	−	0.975937i	$-0.569971\pi$
$89$	−2.90039e6	−0.436106	−0.218053	+	0.975937i	$0.569971\pi$
$90$	0	0
$91$	− 1.18949e7i	− 1.65469i
$92$	0	0
$93$	− 2.11796e6i	− 0.273041i
$94$	0	0
$95$	−6.81838e6	−0.815922
$96$	0	0
$97$	−9.32490e6	−1.03739	−0.518696	−	0.854959i	$-0.673582\pi$
$97$	−9.32490e6	−1.03739	−0.518696	+	0.854959i	$0.673582\pi$
$98$	0	0
$99$	− 6.08078e6i	− 0.629848i
$100$	0	0
$101$	− 6.16540e6i	− 0.595437i	−0.954654	−	0.297719i	$-0.903774\pi$
$101$	− 6.16540e6i	− 0.595437i	0.954654	−	0.297719i	$-0.0962257\pi$
$102$	0	0
$103$	1.55521e7	1.40236	0.701179	−	0.712986i	$-0.252658\pi$
$103$	1.55521e7	1.40236	0.701179	+	0.712986i	$0.252658\pi$
$104$	0	0
$105$	−9.65840e6	−0.814222
$106$	0	0
$107$	8.89410e6i	0.701874i	0.936399	+	0.350937i	$0.114137\pi$
$107$	8.89410e6i	0.701874i	−0.936399	+	0.350937i	$0.885863\pi$
$108$	0	0
$109$	2.19219e7i	1.62139i	0.585472	+	0.810693i	$0.300909\pi$
$109$	2.19219e7i	1.62139i	−0.585472	+	0.810693i	$0.699091\pi$
$110$	0	0
$111$	1.45563e7	1.01023
$112$	0	0
$113$	2.22685e7	1.45183	0.725915	−	0.687784i	$-0.241416\pi$
$113$	2.22685e7	1.45183	0.725915	+	0.687784i	$0.241416\pi$
$114$	0	0
$115$	2.74989e7i	1.68606i
$116$	0	0
$117$	− 7.17503e6i	− 0.414165i
$118$	0	0
$119$	−1.09739e7	−0.596964
$120$	0	0
$121$	−5.00895e7	−2.57038
$122$	0	0
$123$	2.35594e7i	1.14156i
$124$	0	0
$125$	2.03168e7i	0.930400i
$126$	0	0
$127$	6.31332e6	0.273492	0.136746	−	0.990606i	$-0.456336\pi$
$127$	6.31332e6	0.273492	0.136746	+	0.990606i	$0.456336\pi$
$128$	0	0
$129$	−1.48171e7	−0.607714
$130$	0	0
$131$	1.73606e6i	0.0674707i	0.999431	+	0.0337354i	$0.0107403\pi$
$131$	1.73606e6i	0.0674707i	−0.999431	+	0.0337354i	$0.989260\pi$
$132$	0	0
$133$	− 2.78400e7i	− 1.02610i
$134$	0	0
$135$	−5.82596e6	−0.203798
$136$	0	0
$137$	−3.01382e7	−1.00137	−0.500686	−	0.865629i	$-0.666919\pi$
$137$	−3.01382e7	−1.00137	−0.500686	+	0.865629i	$0.666919\pi$
$138$	0	0
$139$	2.65011e6i	0.0836975i	0.999124	+	0.0418488i	$0.0133248\pi$
$139$	2.65011e6i	0.0836975i	−0.999124	+	0.0418488i	$0.986675\pi$
$140$	0	0
$141$	1.60751e6i	0.0482932i
$142$	0	0
$143$	−8.20971e7	−2.34775
$144$	0	0
$145$	1.12960e7	0.307707
$146$	0	0
$147$	− 1.72004e7i	− 0.446609i
$148$	0	0
$149$	4.54803e7i	1.12634i	0.826340	+	0.563172i	$0.190419\pi$
$149$	4.54803e7i	1.12634i	−0.826340	+	0.563172i	$0.809581\pi$
$150$	0	0
$151$	−5.25623e7	−1.24238	−0.621191	−	0.783659i	$-0.713351\pi$
$151$	−5.25623e7	−1.24238	−0.621191	+	0.783659i	$0.713351\pi$
$152$	0	0
$153$	−6.61950e6	−0.149419
$154$	0	0
$155$	2.32183e7i	0.500806i
$156$	0	0
$157$	1.91709e7i	0.395360i	0.980267	+	0.197680i	$0.0633407\pi$
$157$	1.91709e7i	0.395360i	−0.980267	+	0.197680i	$0.936659\pi$
$158$	0	0
$159$	3.86619e7	0.762769
$160$	0	0
$161$	−1.12281e8	−2.12038
$162$	0	0
$163$	− 4.93967e7i	− 0.893390i	−0.894686	−	0.446695i	$-0.852601\pi$
$163$	− 4.93967e7i	− 0.893390i	0.894686	−	0.446695i	$-0.147399\pi$
$164$	0	0
$165$	6.66610e7i	1.15526i
$166$	0	0
$167$	1.03440e8	1.71862	0.859311	−	0.511453i	$-0.170893\pi$
$167$	1.03440e8	1.71862	0.859311	+	0.511453i	$0.170893\pi$
$168$	0	0
$169$	−3.41222e7	−0.543793
$170$	0	0
$171$	− 1.67931e7i	− 0.256830i
$172$	0	0
$173$	− 4.80500e7i	− 0.705557i	−0.935707	−	0.352778i	$-0.885237\pi$
$173$	− 4.80500e7i	− 0.705557i	0.935707	−	0.352778i	$-0.114763\pi$
$174$	0	0
$175$	1.14630e7	0.161683
$176$	0	0
$177$	1.09346e7	0.148217
$178$	0	0
$179$	1.17664e6i	0.0153341i	0.999971	+	0.00766705i	$0.00244052\pi$
$179$	1.17664e6i	0.0153341i	−0.999971	+	0.00766705i	$0.997559\pi$
$180$	0	0
$181$	− 6.72784e7i	− 0.843336i	−0.906750	−	0.421668i	$-0.861445\pi$
$181$	− 6.72784e7i	− 0.843336i	0.906750	−	0.421668i	$-0.138555\pi$
$182$	0	0
$183$	3.96562e7	0.478336
$184$	0	0
$185$	−1.59574e8	−1.85294
$186$	0	0
$187$	7.57407e7i	0.847000i
$188$	0	0
$189$	− 2.37879e7i	− 0.256295i
$190$	0	0
$191$	−4.22053e7	−0.438279	−0.219139	−	0.975694i	$-0.570325\pi$
$191$	−4.22053e7	−0.438279	−0.219139	+	0.975694i	$0.570325\pi$
$192$	0	0
$193$	−1.72792e8	−1.73011	−0.865056	−	0.501675i	$-0.832717\pi$
$193$	−1.72792e8	−1.73011	−0.865056	+	0.501675i	$0.832717\pi$
$194$	0	0
$195$	7.86568e7i	0.759653i
$196$	0	0
$197$	− 4.37361e7i	− 0.407575i	−0.979015	−	0.203788i	$-0.934675\pi$
$197$	− 4.37361e7i	− 0.407575i	0.979015	−	0.203788i	$-0.0653252\pi$
$198$	0	0
$199$	7.18697e7	0.646487	0.323243	−	0.946316i	$-0.395227\pi$
$199$	7.18697e7	0.646487	0.323243	+	0.946316i	$0.395227\pi$
$200$	0	0
$201$	1.58441e7	0.137620
$202$	0	0
$203$	4.61225e7i	0.386970i
$204$	0	0
$205$	− 2.58272e8i	− 2.09382i
$206$	0	0
$207$	−6.77278e7	−0.530727
$208$	0	0
$209$	−1.92148e8	−1.45588
$210$	0	0
$211$	− 1.42645e8i	− 1.04537i	−0.852527	−	0.522684i	$-0.824931\pi$
$211$	− 1.42645e8i	− 1.04537i	0.852527	−	0.522684i	$-0.175069\pi$
$212$	0	0
$213$	1.00078e8i	0.709593i
$214$	0	0
$215$	1.62434e8	1.11466
$216$	0	0
$217$	−9.48022e7	−0.629810
$218$	0	0
$219$	− 4.24788e7i	− 0.273286i
$220$	0	0
$221$	8.93704e7i	0.556956i
$222$	0	0
$223$	8.29773e7	0.501063	0.250531	−	0.968108i	$-0.419395\pi$
$223$	8.29773e7	0.501063	0.250531	+	0.968108i	$0.419395\pi$
$224$	0	0
$225$	6.91448e6	0.0404689
$226$	0	0
$227$	− 1.22220e8i	− 0.693507i	−0.937956	−	0.346753i	$-0.887284\pi$
$227$	− 1.22220e8i	− 0.693507i	0.937956	−	0.346753i	$-0.112716\pi$
$228$	0	0
$229$	− 1.21931e8i	− 0.670948i	−0.942049	−	0.335474i	$-0.891104\pi$
$229$	− 1.21931e8i	− 0.670948i	0.942049	−	0.335474i	$-0.108896\pi$
$230$	0	0
$231$	−2.72183e8	−1.45284
$232$	0	0
$233$	−1.74651e8	−0.904536	−0.452268	−	0.891882i	$-0.649385\pi$
$233$	−1.74651e8	−0.904536	−0.452268	+	0.891882i	$0.649385\pi$
$234$	0	0
$235$	− 1.76224e7i	− 0.0885784i
$236$	0	0
$237$	− 2.04929e8i	− 0.999962i
$238$	0	0
$239$	−3.65760e7	−0.173302	−0.0866510	−	0.996239i	$-0.527616\pi$
$239$	−3.65760e7	−0.173302	−0.0866510	+	0.996239i	$0.527616\pi$
$240$	0	0
$241$	−1.08643e8	−0.499970	−0.249985	−	0.968250i	$-0.580426\pi$
$241$	−1.08643e8	−0.499970	−0.249985	+	0.968250i	$0.580426\pi$
$242$	0	0
$243$	− 1.43489e7i	− 0.0641500i
$244$	0	0
$245$	1.88561e8i	0.819162i
$246$	0	0
$247$	−2.26726e8	−0.957329
$248$	0	0
$249$	−1.48601e8	−0.609991
$250$	0	0
$251$	− 1.03642e8i	− 0.413691i	−0.978374	−	0.206846i	$-0.933680\pi$
$251$	− 1.03642e8i	− 0.413691i	0.978374	−	0.206846i	$-0.0663198\pi$
$252$	0	0
$253$	7.74946e8i	3.00850i
$254$	0	0
$255$	7.25668e7	0.274061
$256$	0	0
$257$	1.53253e8	0.563175	0.281587	−	0.959536i	$-0.409139\pi$
$257$	1.53253e8	0.563175	0.281587	+	0.959536i	$0.409139\pi$
$258$	0	0
$259$	− 6.51555e8i	− 2.33025i
$260$	0	0
$261$	2.78212e7i	0.0968577i
$262$	0	0
$263$	−1.84479e8	−0.625320	−0.312660	−	0.949865i	$-0.601220\pi$
$263$	−1.84479e8	−0.625320	−0.312660	+	0.949865i	$0.601220\pi$
$264$	0	0
$265$	−4.23834e8	−1.39906
$266$	0	0
$267$	7.83106e7i	0.251786i
$268$	0	0
$269$	− 1.76958e8i	− 0.554291i	−0.960828	−	0.277145i	$-0.910612\pi$
$269$	− 1.76958e8i	− 0.554291i	0.960828	−	0.277145i	$-0.0893884\pi$
$270$	0	0
$271$	−2.88497e8	−0.880540	−0.440270	−	0.897865i	$-0.645117\pi$
$271$	−2.88497e8	−0.880540	−0.440270	+	0.897865i	$0.645117\pi$
$272$	0	0
$273$	−3.21163e8	−0.955335
$274$	0	0
$275$	− 7.91159e7i	− 0.229403i
$276$	0	0
$277$	− 5.63159e7i	− 0.159203i	−0.996827	−	0.0796016i	$-0.974635\pi$
$277$	− 5.63159e7i	− 0.159203i	0.996827	−	0.0796016i	$-0.0253648\pi$
$278$	0	0
$279$	−5.71849e7	−0.157640
$280$	0	0
$281$	5.01509e8	1.34836	0.674181	−	0.738566i	$-0.264497\pi$
$281$	5.01509e8	1.34836	0.674181	+	0.738566i	$0.264497\pi$
$282$	0	0
$283$	6.60409e8i	1.73205i	0.500001	+	0.866025i	$0.333333\pi$
$283$	6.60409e8i	1.73205i	−0.500001	+	0.866025i	$0.666667\pi$
$284$	0	0
$285$	1.84096e8i	0.471073i
$286$	0	0
$287$	1.05455e9	2.63317
$288$	0	0
$289$	−3.27888e8	−0.799066
$290$	0	0
$291$	2.51772e8i	0.598939i
$292$	0	0
$293$	− 2.05344e8i	− 0.476920i	−0.971152	−	0.238460i	$-0.923357\pi$
$293$	− 2.05344e8i	− 0.476920i	0.971152	−	0.238460i	$-0.0766426\pi$
$294$	0	0
$295$	−1.19872e8	−0.271856
$296$	0	0
$297$	−1.64181e8	−0.363643
$298$	0	0
$299$	9.14399e8i	1.97827i
$300$	0	0
$301$	6.63230e8i	1.40179i
$302$	0	0
$303$	−1.66466e8	−0.343776
$304$	0	0
$305$	−4.34735e8	−0.877354
$306$	0	0
$307$	− 2.35013e8i	− 0.463562i	−0.972768	−	0.231781i	$-0.925545\pi$
$307$	− 2.35013e8i	− 0.463562i	0.972768	−	0.231781i	$-0.0744552\pi$
$308$	0	0
$309$	− 4.19907e8i	− 0.809651i
$310$	0	0
$311$	−8.80142e7	−0.165917	−0.0829586	−	0.996553i	$-0.526437\pi$
$311$	−8.80142e7	−0.165917	−0.0829586	+	0.996553i	$0.526437\pi$
$312$	0	0
$313$	8.77867e8	1.61817	0.809084	−	0.587693i	$-0.199964\pi$
$313$	8.77867e8	1.61817	0.809084	+	0.587693i	$0.199964\pi$
$314$	0	0
$315$	2.60777e8i	0.470091i
$316$	0	0
$317$	− 7.52195e8i	− 1.32624i	−0.748512	−	0.663121i	$-0.769231\pi$
$317$	− 7.52195e8i	− 1.32624i	0.748512	−	0.663121i	$-0.230769\pi$
$318$	0	0
$319$	3.18332e8	0.549051
$320$	0	0
$321$	2.40141e8	0.405227
$322$	0	0
$323$	2.09171e8i	0.345377i
$324$	0	0
$325$	− 9.33530e7i	− 0.150847i
$326$	0	0
$327$	5.91893e8	0.936107
$328$	0	0
$329$	7.19538e7	0.111396
$330$	0	0
$331$	− 5.61554e8i	− 0.851125i	−0.904929	−	0.425562i	$-0.860076\pi$
$331$	− 5.61554e8i	− 0.851125i	0.904929	−	0.425562i	$-0.139924\pi$
$332$	0	0
$333$	− 3.93019e8i	− 0.583256i
$334$	0	0
$335$	−1.73693e8	−0.252420
$336$	0	0
$337$	−2.50757e8	−0.356902	−0.178451	−	0.983949i	$-0.557109\pi$
$337$	−2.50757e8	−0.356902	−0.178451	+	0.983949i	$0.557109\pi$
$338$	0	0
$339$	− 6.01249e8i	− 0.838215i
$340$	0	0
$341$	6.54313e8i	0.893604i
$342$	0	0
$343$	2.25384e8	0.301574
$344$	0	0
$345$	7.42472e8	0.973449
$346$	0	0
$347$	7.58059e8i	0.973979i	0.873408	+	0.486990i	$0.161905\pi$
$347$	7.58059e8i	0.973979i	−0.873408	+	0.486990i	$0.838095\pi$
$348$	0	0
$349$	− 4.46508e8i	− 0.562264i	−0.959669	−	0.281132i	$-0.909290\pi$
$349$	− 4.46508e8i	− 0.562264i	0.959669	−	0.281132i	$-0.0907098\pi$
$350$	0	0
$351$	−1.93726e8	−0.239118
$352$	0	0
$353$	−1.29035e8	−0.156134	−0.0780668	−	0.996948i	$-0.524875\pi$
$353$	−1.29035e8	−0.156134	−0.0780668	+	0.996948i	$0.524875\pi$
$354$	0	0
$355$	− 1.09711e9i	− 1.30152i
$356$	0	0
$357$	2.96296e8i	0.344657i
$358$	0	0
$359$	4.14531e8	0.472854	0.236427	−	0.971649i	$-0.424024\pi$
$359$	4.14531e8	0.472854	0.236427	+	0.971649i	$0.424024\pi$
$360$	0	0
$361$	3.63221e8	0.406346
$362$	0	0
$363$	1.35242e9i	1.48401i
$364$	0	0
$365$	4.65678e8i	0.501257i
$366$	0	0
$367$	−1.12231e9	−1.18517	−0.592584	−	0.805508i	$-0.701892\pi$
$367$	−1.12231e9	−1.18517	−0.592584	+	0.805508i	$0.701892\pi$
$368$	0	0
$369$	6.36105e8	0.659077
$370$	0	0
$371$	− 1.73055e9i	− 1.75944i
$372$	0	0
$373$	− 6.16853e8i	− 0.615461i	−0.951474	−	0.307731i	$-0.900430\pi$
$373$	− 6.16853e8i	− 0.615461i	0.951474	−	0.307731i	$-0.0995696\pi$
$374$	0	0
$375$	5.48553e8	0.537166
$376$	0	0
$377$	3.75616e8	0.361035
$378$	0	0
$379$	− 3.45320e8i	− 0.325825i	−0.986640	−	0.162913i	$-0.947911\pi$
$379$	− 3.45320e8i	− 0.325825i	0.986640	−	0.162913i	$-0.0520889\pi$
$380$	0	0
$381$	− 1.70460e8i	− 0.157901i
$382$	0	0
$383$	−1.34038e9	−1.21908	−0.609541	−	0.792755i	$-0.708646\pi$
$383$	−1.34038e9	−1.21908	−0.609541	+	0.792755i	$0.708646\pi$
$384$	0	0
$385$	2.98382e9	2.66477
$386$	0	0
$387$	4.00062e8i	0.350864i
$388$	0	0
$389$	− 8.09565e8i	− 0.697314i	−0.937250	−	0.348657i	$-0.886638\pi$
$389$	− 8.09565e8i	− 0.697314i	0.937250	−	0.348657i	$-0.113362\pi$
$390$	0	0
$391$	8.43601e8	0.713704
$392$	0	0
$393$	4.68736e7	0.0389542
$394$	0	0
$395$	2.24655e9i	1.83411i
$396$	0	0
$397$	− 1.77379e9i	− 1.42277i	−0.702801	−	0.711387i	$-0.748067\pi$
$397$	− 1.77379e9i	− 1.42277i	0.702801	−	0.711387i	$-0.251933\pi$
$398$	0	0
$399$	−7.51680e8	−0.592418
$400$	0	0
$401$	1.24387e9	0.963315	0.481658	−	0.876360i	$-0.340035\pi$
$401$	1.24387e9	0.963315	0.481658	+	0.876360i	$0.340035\pi$
$402$	0	0
$403$	7.72058e8i	0.587601i
$404$	0	0
$405$	1.57301e8i	0.117663i
$406$	0	0
$407$	−4.49695e9	−3.30626
$408$	0	0
$409$	−6.54473e8	−0.472999	−0.236499	−	0.971632i	$-0.576000\pi$
$409$	−6.54473e8	−0.472999	−0.236499	+	0.971632i	$0.576000\pi$
$410$	0	0
$411$	8.13731e8i	0.578142i
$412$	0	0
$413$	− 4.89446e8i	− 0.341884i
$414$	0	0
$415$	1.62905e9	1.11883
$416$	0	0
$417$	7.15531e7	0.0483228
$418$	0	0
$419$	1.61210e8i	0.107064i	0.998566	+	0.0535320i	$0.0170479\pi$
$419$	1.61210e8i	0.107064i	−0.998566	+	0.0535320i	$0.982952\pi$
$420$	0	0
$421$	1.92687e9i	1.25853i	0.777190	+	0.629266i	$0.216644\pi$
$421$	1.92687e9i	1.25853i	−0.777190	+	0.629266i	$0.783356\pi$
$422$	0	0
$423$	4.34026e7	0.0278821
$424$	0	0
$425$	−8.61251e7	−0.0544212
$426$	0	0
$427$	− 1.77506e9i	− 1.10335i
$428$	0	0
$429$	2.21662e9i	1.35547i
$430$	0	0
$431$	5.61070e8	0.337557	0.168778	−	0.985654i	$-0.446018\pi$
$431$	5.61070e8	0.337557	0.168778	+	0.985654i	$0.446018\pi$
$432$	0	0
$433$	2.63369e9	1.55904	0.779519	−	0.626378i	$-0.215463\pi$
$433$	2.63369e9	1.55904	0.779519	+	0.626378i	$0.215463\pi$
$434$	0	0
$435$	− 3.04992e8i	− 0.177655i
$436$	0	0
$437$	2.14015e9i	1.22676i
$438$	0	0
$439$	3.02903e9	1.70874	0.854372	−	0.519662i	$-0.173942\pi$
$439$	3.02903e9	1.70874	0.854372	+	0.519662i	$0.173942\pi$
$440$	0	0
$441$	−4.64411e8	−0.257850
$442$	0	0
$443$	2.23887e9i	1.22353i	0.791038	+	0.611767i	$0.209541\pi$
$443$	2.23887e9i	1.22353i	−0.791038	+	0.611767i	$0.790459\pi$
$444$	0	0
$445$	− 8.58487e8i	− 0.461821i
$446$	0	0
$447$	1.22797e9	0.650295
$448$	0	0
$449$	1.86974e9	0.974807	0.487403	−	0.873177i	$-0.337944\pi$
$449$	1.86974e9	0.974807	0.487403	+	0.873177i	$0.337944\pi$
$450$	0	0
$451$	− 7.27835e9i	− 3.73607i
$452$	0	0
$453$	1.41918e9i	0.717290i
$454$	0	0
$455$	3.52077e9	1.75226
$456$	0	0
$457$	2.45091e9	1.20121	0.600607	−	0.799545i	$-0.294926\pi$
$457$	2.45091e9	1.20121	0.600607	+	0.799545i	$0.294926\pi$
$458$	0	0
$459$	1.78726e8i	0.0862669i
$460$	0	0
$461$	3.52974e9i	1.67799i	0.544139	+	0.838995i	$0.316856\pi$
$461$	3.52974e9i	1.67799i	−0.544139	+	0.838995i	$0.683144\pi$
$462$	0	0
$463$	−1.30367e9	−0.610427	−0.305214	−	0.952284i	$-0.598728\pi$
$463$	−1.30367e9	−0.610427	−0.305214	+	0.952284i	$0.598728\pi$
$464$	0	0
$465$	6.26894e8	0.289140
$466$	0	0
$467$	− 3.11104e9i	− 1.41350i	−0.707463	−	0.706751i	$-0.750160\pi$
$467$	− 3.11104e9i	− 1.41350i	0.707463	−	0.706751i	$-0.249840\pi$
$468$	0	0
$469$	− 7.09201e8i	− 0.317442i
$470$	0	0
$471$	5.17613e8	0.228261
$472$	0	0
$473$	4.57753e9	1.98892
$474$	0	0
$475$	− 2.18493e8i	− 0.0935425i
$476$	0	0
$477$	− 1.04387e9i	− 0.440385i
$478$	0	0
$479$	−1.91637e8	−0.0796718	−0.0398359	−	0.999206i	$-0.512684\pi$
$479$	−1.91637e8	−0.0796718	−0.0398359	+	0.999206i	$0.512684\pi$
$480$	0	0
$481$	−5.30618e9	−2.17408
$482$	0	0
$483$	3.03157e9i	1.22420i
$484$	0	0
$485$	− 2.76007e9i	− 1.09856i
$486$	0	0
$487$	−2.25790e9	−0.885837	−0.442918	−	0.896562i	$-0.646057\pi$
$487$	−2.25790e9	−0.885837	−0.442918	+	0.896562i	$0.646057\pi$
$488$	0	0
$489$	−1.33371e9	−0.515799
$490$	0	0
$491$	− 1.90733e8i	− 0.0727176i	−0.999339	−	0.0363588i	$-0.988424\pi$
$491$	− 1.90733e8i	− 0.0727176i	0.999339	−	0.0363588i	$-0.0115759\pi$
$492$	0	0
$493$	− 3.46534e8i	− 0.130251i
$494$	0	0
$495$	1.79985e9	0.666987
$496$	0	0
$497$	4.47960e9	1.63678
$498$	0	0
$499$	− 3.86435e9i	− 1.39227i	−0.717909	−	0.696137i	$-0.754900\pi$
$499$	− 3.86435e9i	− 1.39227i	0.717909	−	0.696137i	$-0.245100\pi$
$500$	0	0
$501$	− 2.79288e9i	− 0.992247i
$502$	0	0
$503$	−8.43474e8	−0.295518	−0.147759	−	0.989023i	$-0.547206\pi$
$503$	−8.43474e8	−0.295518	−0.147759	+	0.989023i	$0.547206\pi$
$504$	0	0
$505$	1.82489e9	0.630547
$506$	0	0
$507$	9.21299e8i	0.313959i
$508$	0	0
$509$	3.07188e9i	1.03250i	0.856436	+	0.516252i	$0.172673\pi$
$509$	3.07188e9i	1.03250i	−0.856436	+	0.516252i	$0.827327\pi$
$510$	0	0
$511$	−1.90140e9	−0.630377
$512$	0	0
$513$	−4.53415e8	−0.148281
$514$	0	0
$515$	4.60326e9i	1.48505i
$516$	0	0
$517$	− 4.96616e8i	− 0.158053i
$518$	0	0
$519$	−1.29735e9	−0.407353
$520$	0	0
$521$	3.10240e9	0.961092	0.480546	−	0.876969i	$-0.340438\pi$
$521$	3.10240e9	0.961092	0.480546	+	0.876969i	$0.340438\pi$
$522$	0	0
$523$	2.05170e9i	0.627131i	0.949567	+	0.313566i	$0.101524\pi$
$523$	2.05170e9i	0.627131i	−0.949567	+	0.313566i	$0.898476\pi$
$524$	0	0
$525$	− 3.09500e8i	− 0.0933476i
$526$	0	0
$527$	7.12281e8	0.211989
$528$	0	0
$529$	5.22653e9	1.53504
$530$	0	0
$531$	− 2.95235e8i	− 0.0855729i
$532$	0	0
$533$	− 8.58811e9i	− 2.45670i
$534$	0	0
$535$	−2.63256e9	−0.743260
$536$	0	0
$537$	3.17693e7	0.00885315
$538$	0	0
$539$	5.31381e9i	1.46166i
$540$	0	0
$541$	− 2.25941e9i	− 0.613487i	−0.951792	−	0.306743i	$-0.900761\pi$
$541$	− 2.25941e9i	− 0.613487i	0.951792	−	0.306743i	$-0.0992394\pi$
$542$	0	0
$543$	−1.81652e9	−0.486900
$544$	0	0
$545$	−6.48867e9	−1.71699
$546$	0	0
$547$	− 7.29294e9i	− 1.90523i	−0.304180	−	0.952615i	$-0.598382\pi$
$547$	− 7.29294e9i	− 1.90523i	0.304180	−	0.952615i	$-0.401618\pi$
$548$	0	0
$549$	− 1.07072e9i	− 0.276167i
$550$	0	0
$551$	8.79129e8	0.223884
$552$	0	0
$553$	−9.17284e9	−2.30656
$554$	0	0
$555$	4.30850e9i	1.06980i
$556$	0	0
$557$	− 1.49041e9i	− 0.365436i	−0.983165	−	0.182718i	$-0.941510\pi$
$557$	− 1.49041e9i	− 0.365436i	0.983165	−	0.182718i	$-0.0584895\pi$
$558$	0	0
$559$	5.40127e9	1.30784
$560$	0	0
$561$	2.04500e9	0.489016
$562$	0	0
$563$	4.98977e9i	1.17842i	0.807979	+	0.589212i	$0.200562\pi$
$563$	4.98977e9i	1.17842i	−0.807979	+	0.589212i	$0.799438\pi$
$564$	0	0
$565$	6.59124e9i	1.53744i
$566$	0	0
$567$	−6.42273e8	−0.147972
$568$	0	0
$569$	−5.02061e9	−1.14252	−0.571260	−	0.820769i	$-0.693545\pi$
$569$	−5.02061e9	−1.14252	−0.571260	+	0.820769i	$0.693545\pi$
$570$	0	0
$571$	− 2.27701e8i	− 0.0511846i	−0.999672	−	0.0255923i	$-0.991853\pi$
$571$	− 2.27701e8i	− 0.0511846i	0.999672	−	0.0255923i	$-0.00814717\pi$
$572$	0	0
$573$	1.13954e9i	0.253040i
$574$	0	0
$575$	−8.81194e8	−0.193301
$576$	0	0
$577$	4.73674e9	1.02651	0.513256	−	0.858235i	$-0.328439\pi$
$577$	4.73674e9	1.02651	0.513256	+	0.858235i	$0.328439\pi$
$578$	0	0
$579$	4.66540e9i	0.998881i
$580$	0	0
$581$	6.65154e9i	1.40704i
$582$	0	0
$583$	−1.19440e10	−2.49638
$584$	0	0
$585$	2.12373e9	0.438586
$586$	0	0
$587$	− 2.02047e9i	− 0.412306i	−0.978520	−	0.206153i	$-0.933906\pi$
$587$	− 2.02047e9i	− 0.412306i	0.978520	−	0.206153i	$-0.0660944\pi$
$588$	0	0
$589$	1.80700e9i	0.364380i
$590$	0	0
$591$	−1.18087e9	−0.235314
$592$	0	0
$593$	−6.44514e9	−1.26923	−0.634616	−	0.772828i	$-0.718842\pi$
$593$	−6.44514e9	−1.26923	−0.634616	+	0.772828i	$0.718842\pi$
$594$	0	0
$595$	− 3.24817e9i	− 0.632164i
$596$	0	0
$597$	− 1.94048e9i	− 0.373249i
$598$	0	0
$599$	5.73444e9	1.09018	0.545088	−	0.838379i	$-0.316496\pi$
$599$	5.73444e9	1.09018	0.545088	+	0.838379i	$0.316496\pi$
$600$	0	0
$601$	4.98830e9	0.937328	0.468664	−	0.883377i	$-0.344736\pi$
$601$	4.98830e9	0.937328	0.468664	+	0.883377i	$0.344736\pi$
$602$	0	0
$603$	− 4.27792e8i	− 0.0794551i
$604$	0	0
$605$	− 1.48260e10i	− 2.72194i
$606$	0	0
$607$	4.01757e9	0.729127	0.364563	−	0.931179i	$-0.381218\pi$
$607$	4.01757e9	0.729127	0.364563	+	0.931179i	$0.381218\pi$
$608$	0	0
$609$	1.24531e9	0.223417
$610$	0	0
$611$	− 5.85983e8i	− 0.103930i
$612$	0	0
$613$	5.59669e9i	0.981341i	0.871345	+	0.490670i	$0.163248\pi$
$613$	5.59669e9i	0.981341i	−0.871345	+	0.490670i	$0.836752\pi$
$614$	0	0
$615$	−6.97335e9	−1.20887
$616$	0	0
$617$	−7.49818e9	−1.28516	−0.642581	−	0.766218i	$-0.722136\pi$
$617$	−7.49818e9	−1.28516	−0.642581	+	0.766218i	$0.722136\pi$
$618$	0	0
$619$	− 1.06598e10i	− 1.80647i	−0.429148	−	0.903234i	$-0.641186\pi$
$619$	− 1.06598e10i	− 1.80647i	0.429148	−	0.903234i	$-0.358814\pi$
$620$	0	0
$621$	1.82865e9i	0.306415i
$622$	0	0
$623$	3.50527e9	0.580783
$624$	0	0
$625$	−6.75456e9	−1.10667
$626$	0	0
$627$	5.18800e9i	0.840550i
$628$	0	0
$629$	4.89535e9i	0.784344i
$630$	0	0
$631$	8.72314e9	1.38220	0.691099	−	0.722761i	$-0.257127\pi$
$631$	8.72314e9	1.38220	0.691099	+	0.722761i	$0.257127\pi$
$632$	0	0
$633$	−3.85142e9	−0.603543
$634$	0	0
$635$	1.86868e9i	0.289619i
$636$	0	0
$637$	6.27005e9i	0.961132i
$638$	0	0
$639$	2.70210e9	0.409684
$640$	0	0
$641$	−7.26338e9	−1.08927	−0.544635	−	0.838673i	$-0.683332\pi$
$641$	−7.26338e9	−1.08927	−0.544635	+	0.838673i	$0.683332\pi$
$642$	0	0
$643$	− 4.78860e9i	− 0.710347i	−0.934800	−	0.355173i	$-0.884422\pi$
$643$	− 4.78860e9i	− 0.710347i	0.934800	−	0.355173i	$-0.115578\pi$
$644$	0	0
$645$	− 4.38571e9i	− 0.643548i
$646$	0	0
$647$	1.10498e10	1.60394	0.801971	−	0.597363i	$-0.203785\pi$
$647$	1.10498e10	1.60394	0.801971	+	0.597363i	$0.203785\pi$
$648$	0	0
$649$	−3.37809e9	−0.485082
$650$	0	0
$651$	2.55966e9i	0.363621i
$652$	0	0
$653$	8.16878e9i	1.14805i	0.818837	+	0.574026i	$0.194619\pi$
$653$	8.16878e9i	1.14805i	−0.818837	+	0.574026i	$0.805381\pi$
$654$	0	0
$655$	−5.13856e8	−0.0714491
$656$	0	0
$657$	−1.14693e9	−0.157782
$658$	0	0
$659$	6.84404e9i	0.931566i	0.884899	+	0.465783i	$0.154227\pi$
$659$	6.84404e9i	0.931566i	−0.884899	+	0.465783i	$0.845773\pi$
$660$	0	0
$661$	− 1.19963e9i	− 0.161563i	−0.996732	−	0.0807816i	$-0.974258\pi$
$661$	− 1.19963e9i	− 0.161563i	0.996732	−	0.0807816i	$-0.0257416\pi$
$662$	0	0
$663$	2.41300e9	0.321559
$664$	0	0
$665$	8.24035e9	1.08660
$666$	0	0
$667$	− 3.54558e9i	− 0.462645i
$668$	0	0
$669$	− 2.24039e9i	− 0.289289i
$670$	0	0
$671$	−1.22512e10	−1.56549
$672$	0	0
$673$	−6.16862e9	−0.780072	−0.390036	−	0.920800i	$-0.627538\pi$
$673$	−6.16862e9	−0.780072	−0.390036	+	0.920800i	$0.627538\pi$
$674$	0	0
$675$	− 1.86691e8i	− 0.0233647i
$676$	0	0
$677$	6.01999e9i	0.745651i	0.927901	+	0.372826i	$0.121611\pi$
$677$	6.01999e9i	0.745651i	−0.927901	+	0.372826i	$0.878389\pi$
$678$	0	0
$679$	1.12696e10	1.38154
$680$	0	0
$681$	−3.29993e9	−0.400396
$682$	0	0
$683$	1.27889e10i	1.53590i	0.640511	+	0.767949i	$0.278722\pi$
$683$	1.27889e10i	1.53590i	−0.640511	+	0.767949i	$0.721278\pi$
$684$	0	0
$685$	− 8.92059e9i	− 1.06042i
$686$	0	0
$687$	−3.29213e9	−0.387372
$688$	0	0
$689$	−1.40934e10	−1.64153
$690$	0	0
$691$	− 2.77755e9i	− 0.320250i	−0.987097	−	0.160125i	$-0.948810\pi$
$691$	− 2.77755e9i	− 0.320250i	0.987097	−	0.160125i	$-0.0511898\pi$
$692$	0	0
$693$	7.34893e9i	0.838799i
$694$	0	0
$695$	−7.84406e8	−0.0886327
$696$	0	0
$697$	−7.92316e9	−0.886306
$698$	0	0
$699$	4.71558e9i	0.522234i
$700$	0	0
$701$	7.94638e9i	0.871277i	0.900122	+	0.435639i	$0.143477\pi$
$701$	7.94638e9i	0.871277i	−0.900122	+	0.435639i	$0.856523\pi$
$702$	0	0
$703$	−1.24191e10	−1.34818
$704$	0	0
$705$	−4.75805e8	−0.0511407
$706$	0	0
$707$	7.45119e9i	0.792972i
$708$	0	0
$709$	− 1.68125e10i	− 1.77162i	−0.464049	−	0.885810i	$-0.653604\pi$
$709$	− 1.68125e10i	− 1.77162i	0.464049	−	0.885810i	$-0.346396\pi$
$710$	0	0
$711$	−5.53307e9	−0.577328
$712$	0	0
$713$	7.28775e9	0.752974
$714$	0	0
$715$	− 2.42999e10i	− 2.48618i
$716$	0	0
$717$	9.87552e8i	0.100056i
$718$	0	0
$719$	5.38427e9	0.540227	0.270113	−	0.962829i	$-0.412939\pi$
$719$	5.38427e9	0.540227	0.270113	+	0.962829i	$0.412939\pi$
$720$	0	0
$721$	−1.87955e10	−1.86758
$722$	0	0
$723$	2.93337e9i	0.288658i
$724$	0	0
$725$	3.61977e8i	0.0352775i
$726$	0	0
$727$	−7.83636e9	−0.756387	−0.378193	−	0.925727i	$-0.623455\pi$
$727$	−7.83636e9	−0.756387	−0.378193	+	0.925727i	$0.623455\pi$
$728$	0	0
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	− 4.98307e9i	− 0.471830i
$732$	0	0
$733$	1.28847e10i	1.20840i	0.796833	+	0.604200i	$0.206507\pi$
$733$	1.28847e10i	1.20840i	−0.796833	+	0.604200i	$0.793493\pi$
$734$	0	0
$735$	5.09114e9	0.472943
$736$	0	0
$737$	−4.89482e9	−0.450402
$738$	0	0
$739$	1.03527e10i	0.943624i	0.881699	+	0.471812i	$0.156400\pi$
$739$	1.03527e10i	0.943624i	−0.881699	+	0.471812i	$0.843600\pi$
$740$	0	0
$741$	6.12159e9i	0.552714i
$742$	0	0
$743$	−9.03056e9	−0.807707	−0.403854	−	0.914824i	$-0.632330\pi$
$743$	−9.03056e9	−0.807707	−0.403854	+	0.914824i	$0.632330\pi$
$744$	0	0
$745$	−1.34617e10	−1.19276
$746$	0	0
$747$	4.01222e9i	0.352179i
$748$	0	0
$749$	− 1.07490e10i	− 0.934718i
$750$	0	0
$751$	−1.27926e10	−1.10209	−0.551046	−	0.834475i	$-0.685771\pi$
$751$	−1.27926e10	−1.10209	−0.551046	+	0.834475i	$0.685771\pi$
$752$	0	0
$753$	−2.79832e9	−0.238845
$754$	0	0
$755$	− 1.55579e10i	− 1.31564i
$756$	0	0
$757$	1.64139e9i	0.137524i	0.997633	+	0.0687618i	$0.0219048\pi$
$757$	1.64139e9i	0.137524i	−0.997633	+	0.0687618i	$0.978095\pi$
$758$	0	0
$759$	2.09235e10	1.73696
$760$	0	0
$761$	1.25169e10	1.02956	0.514778	−	0.857323i	$-0.327874\pi$
$761$	1.25169e10	1.02956	0.514778	+	0.857323i	$0.327874\pi$
$762$	0	0
$763$	− 2.64938e10i	− 2.15928i
$764$	0	0
$765$	− 1.95930e9i	− 0.158229i
$766$	0	0
$767$	−3.98599e9	−0.318972
$768$	0	0
$769$	1.05617e10	0.837511	0.418756	−	0.908099i	$-0.362466\pi$
$769$	1.05617e10	0.837511	0.418756	+	0.908099i	$0.362466\pi$
$770$	0	0
$771$	− 4.13783e9i	− 0.325149i
$772$	0	0
$773$	− 1.85897e10i	− 1.44759i	−0.690016	−	0.723794i	$-0.742397\pi$
$773$	− 1.85897e10i	− 1.44759i	0.690016	−	0.723794i	$-0.257603\pi$
$774$	0	0
$775$	−7.44022e8	−0.0574156
$776$	0	0
$777$	−1.75920e10	−1.34537
$778$	0	0
$779$	− 2.01004e10i	− 1.52344i
$780$	0	0
$781$	− 3.09176e10i	− 2.32235i
$782$	0	0
$783$	7.51172e8	0.0559208
$784$	0	0
$785$	−5.67438e9	−0.418672
$786$	0	0
$787$	− 4.81123e9i	− 0.351840i	−0.984404	−	0.175920i	$-0.943710\pi$
$787$	− 4.81123e9i	− 0.351840i	0.984404	−	0.175920i	$-0.0562899\pi$
$788$	0	0
$789$	4.98094e9i	0.361029i
$790$	0	0
$791$	−2.69126e10	−1.93347
$792$	0	0
$793$	−1.44559e10	−1.02941
$794$	0	0
$795$	1.14435e10i	0.807746i
$796$	0	0
$797$	1.86151e10i	1.30245i	0.758883	+	0.651226i	$0.225745\pi$
$797$	1.86151e10i	1.30245i	−0.758883	+	0.651226i	$0.774255\pi$
$798$	0	0
$799$	−5.40613e8	−0.0374949
$800$	0	0
$801$	2.11439e9	0.145369
$802$	0	0
$803$	1.31232e10i	0.894409i
$804$	0	0
$805$	− 3.32339e10i	− 2.24541i
$806$	0	0
$807$	−4.77787e9	−0.320020
$808$	0	0
$809$	−1.02158e10	−0.678345	−0.339172	−	0.940724i	$-0.610147\pi$
$809$	−1.02158e10	−0.678345	−0.339172	+	0.940724i	$0.610147\pi$
$810$	0	0
$811$	− 2.00546e10i	− 1.32020i	−0.751177	−	0.660100i	$-0.770514\pi$
$811$	− 2.00546e10i	− 1.32020i	0.751177	−	0.660100i	$-0.229486\pi$
$812$	0	0
$813$	7.78942e9i	0.508380i
$814$	0	0
$815$	1.46209e10	0.946069
$816$	0	0
$817$	1.26417e10	0.811011
$818$	0	0
$819$	8.67139e9i	0.551563i
$820$	0	0
$821$	− 1.79346e10i	− 1.13107i	−0.824724	−	0.565536i	$-0.808669\pi$
$821$	− 1.79346e10i	− 1.13107i	0.824724	−	0.565536i	$-0.191331\pi$
$822$	0	0
$823$	1.24280e10	0.777144	0.388572	−	0.921418i	$-0.372968\pi$
$823$	1.24280e10	0.777144	0.388572	+	0.921418i	$0.372968\pi$
$824$	0	0
$825$	−2.13613e9	−0.132446
$826$	0	0
$827$	2.13534e10i	1.31280i	0.754414	+	0.656399i	$0.227921\pi$
$827$	2.13534e10i	1.31280i	−0.754414	+	0.656399i	$0.772079\pi$
$828$	0	0
$829$	3.18974e10i	1.94453i	0.233881	+	0.972265i	$0.424857\pi$
$829$	3.18974e10i	1.94453i	−0.233881	+	0.972265i	$0.575143\pi$
$830$	0	0
$831$	−1.52053e9	−0.0919160
$832$	0	0
$833$	5.78458e9	0.346748
$834$	0	0
$835$	3.06171e10i	1.81996i
$836$	0	0
$837$	1.54399e9i	0.0910135i
$838$	0	0
$839$	1.89333e10	1.10677	0.553386	−	0.832925i	$-0.313335\pi$
$839$	1.89333e10	1.10677	0.553386	+	0.832925i	$0.313335\pi$
$840$	0	0
$841$	1.57934e10	0.915567
$842$	0	0
$843$	− 1.35407e10i	− 0.778477i
$844$	0	0
$845$	− 1.00998e10i	− 0.575857i
$846$	0	0
$847$	6.05356e10	3.42310
$848$	0	0
$849$	1.78310e10	0.999999
$850$	0	0
$851$	5.00871e10i	2.78594i
$852$	0	0
$853$	1.46867e10i	0.810220i	0.914268	+	0.405110i	$0.132767\pi$
$853$	1.46867e10i	0.810220i	−0.914268	+	0.405110i	$0.867233\pi$
$854$	0	0
$855$	4.97060e9	0.271974
$856$	0	0
$857$	−6.66249e9	−0.361579	−0.180790	−	0.983522i	$-0.557865\pi$
$857$	−6.66249e9	−0.361579	−0.180790	+	0.983522i	$0.557865\pi$
$858$	0	0
$859$	1.11084e10i	0.597966i	0.954258	+	0.298983i	$0.0966474\pi$
$859$	1.11084e10i	0.597966i	−0.954258	+	0.298983i	$0.903353\pi$
$860$	0	0
$861$	− 2.84728e10i	− 1.52026i
$862$	0	0
$863$	−1.57309e10	−0.833134	−0.416567	−	0.909105i	$-0.636767\pi$
$863$	−1.57309e10	−0.833134	−0.416567	+	0.909105i	$0.636767\pi$
$864$	0	0
$865$	1.42223e10	0.747160
$866$	0	0
$867$	8.85297e9i	0.461341i
$868$	0	0
$869$	6.33097e10i	3.27266i
$870$	0	0
$871$	−5.77565e9	−0.296167
$872$	0	0
$873$	6.79785e9	0.345797
$874$	0	0
$875$	− 2.45538e10i	− 1.23906i
$876$	0	0
$877$	2.22672e10i	1.11472i	0.830270	+	0.557362i	$0.188186\pi$
$877$	2.22672e10i	1.11472i	−0.830270	+	0.557362i	$0.811814\pi$
$878$	0	0
$879$	−5.54429e9	−0.275350
$880$	0	0
$881$	−2.20896e10	−1.08836	−0.544180	−	0.838968i	$-0.683159\pi$
$881$	−2.20896e10	−1.08836	−0.544180	+	0.838968i	$0.683159\pi$
$882$	0	0
$883$	− 2.65630e10i	− 1.29842i	−0.760610	−	0.649210i	$-0.775100\pi$
$883$	− 2.65630e10i	− 1.29842i	0.760610	−	0.649210i	$-0.224900\pi$
$884$	0	0
$885$	3.23653e9i	0.156956i
$886$	0	0
$887$	−4.12167e10	−1.98308	−0.991540	−	0.129801i	$-0.958566\pi$
$887$	−4.12167e10	−1.98308	−0.991540	+	0.129801i	$0.958566\pi$
$888$	0	0
$889$	−7.62997e9	−0.364222
$890$	0	0
$891$	4.43289e9i	0.209949i
$892$	0	0
$893$	− 1.37149e9i	− 0.0644485i
$894$	0	0
$895$	−3.48274e8	−0.0162383
$896$	0	0
$897$	2.46888e10	1.14216
$898$	0	0
$899$	− 2.99366e9i	− 0.137418i
$900$	0	0
$901$	1.30022e10i	0.592216i
$902$	0	0
$903$	1.79072e10	0.809321
$904$	0	0
$905$	1.99137e10	0.893063
$906$	0	0
$907$	2.16876e10i	0.965132i	0.875860	+	0.482566i	$0.160295\pi$
$907$	2.16876e10i	0.965132i	−0.875860	+	0.482566i	$0.839705\pi$
$908$	0	0
$909$	4.49457e9i	0.198479i
$910$	0	0
$911$	1.46518e10	0.642062	0.321031	−	0.947069i	$-0.395971\pi$
$911$	1.46518e10	0.642062	0.321031	+	0.947069i	$0.395971\pi$
$912$	0	0
$913$	4.59081e10	1.99637
$914$	0	0
$915$	1.17378e10i	0.506541i
$916$	0	0
$917$	− 2.09812e9i	− 0.0898539i
$918$	0	0
$919$	9.27067e9	0.394010	0.197005	−	0.980403i	$-0.436879\pi$
$919$	9.27067e9	0.394010	0.197005	+	0.980403i	$0.436879\pi$
$920$	0	0
$921$	−6.34535e9	−0.267638
$922$	0	0
$923$	− 3.64813e10i	− 1.52709i
$924$	0	0
$925$	− 5.11350e9i	− 0.212433i
$926$	0	0
$927$	−1.13375e10	−0.467452
$928$	0	0
$929$	−3.16010e10	−1.29314	−0.646570	−	0.762855i	$-0.723797\pi$
$929$	−3.16010e10	−1.29314	−0.646570	+	0.762855i	$0.723797\pi$
$930$	0	0
$931$	1.46750e10i	0.596012i
$932$	0	0
$933$	2.37638e9i	0.0957923i
$934$	0	0
$935$	−2.24185e10	−0.896943
$936$	0	0
$937$	6.72794e9	0.267174	0.133587	−	0.991037i	$-0.457351\pi$
$937$	6.72794e9	0.267174	0.133587	+	0.991037i	$0.457351\pi$
$938$	0	0
$939$	− 2.37024e10i	− 0.934250i
$940$	0	0
$941$	− 7.67548e9i	− 0.300291i	−0.988664	−	0.150145i	$-0.952026\pi$
$941$	− 7.67548e9i	− 0.300291i	0.988664	−	0.150145i	$-0.0479742\pi$
$942$	0	0
$943$	−8.10664e10	−3.14811
$944$	0	0
$945$	7.04097e9	0.271407
$946$	0	0
$947$	2.29327e10i	0.877464i	0.898618	+	0.438732i	$0.144572\pi$
$947$	2.29327e10i	0.877464i	−0.898618	+	0.438732i	$0.855428\pi$
$948$	0	0
$949$	1.54848e10i	0.588130i
$950$	0	0
$951$	−2.03093e10	−0.765706
$952$	0	0
$953$	4.11486e10	1.54003	0.770016	−	0.638025i	$-0.220248\pi$
$953$	4.11486e10	1.54003	0.770016	+	0.638025i	$0.220248\pi$
$954$	0	0
$955$	− 1.24923e10i	− 0.464122i
$956$	0	0
$957$	− 8.59496e9i	− 0.316995i
$958$	0	0
$959$	3.64235e10	1.33357
$960$	0	0
$961$	−2.13593e10	−0.776346
$962$	0	0
$963$	− 6.48380e9i	− 0.233958i
$964$	0	0
$965$	− 5.11448e10i	− 1.83213i
$966$	0	0
$967$	2.35579e9	0.0837807	0.0418903	−	0.999122i	$-0.486662\pi$
$967$	2.35579e9	0.0837807	0.0418903	+	0.999122i	$0.486662\pi$
$968$	0	0
$969$	5.64762e9	0.199403
$970$	0	0
$971$	− 6.53180e9i	− 0.228963i	−0.993425	−	0.114482i	$-0.963479\pi$
$971$	− 6.53180e9i	− 0.228963i	0.993425	−	0.114482i	$-0.0365207\pi$
$972$	0	0
$973$	− 3.20280e9i	− 0.111464i
$974$	0	0
$975$	−2.52053e9	−0.0870915
$976$	0	0
$977$	2.56047e10	0.878394	0.439197	−	0.898391i	$-0.355263\pi$
$977$	2.56047e10	0.878394	0.439197	+	0.898391i	$0.355263\pi$
$978$	0	0
$979$	− 2.41929e10i	− 0.824042i
$980$	0	0
$981$	− 1.59811e10i	− 0.540462i
$982$	0	0
$983$	2.64231e10	0.887252	0.443626	−	0.896212i	$-0.353692\pi$
$983$	2.64231e10	0.887252	0.443626	+	0.896212i	$0.353692\pi$
$984$	0	0
$985$	1.29454e10	0.431608
$986$	0	0
$987$	− 1.94275e9i	− 0.0643143i
$988$	0	0
$989$	− 5.09846e10i	− 1.67591i
$990$	0	0
$991$	4.41414e10	1.44075	0.720375	−	0.693585i	$-0.243970\pi$
$991$	4.41414e10	1.44075	0.720375	+	0.693585i	$0.243970\pi$
$992$	0	0
$993$	−1.51619e10	−0.491397
$994$	0	0
$995$	2.12727e10i	0.684607i
$996$	0	0
$997$	3.85869e10i	1.23312i	0.787306	+	0.616562i	$0.211475\pi$
$997$	3.85869e10i	1.23312i	−0.787306	+	0.616562i	$0.788525\pi$
$998$	0	0
$999$	−1.06115e10	−0.336743

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.d.f.193.7	yes	16
4.3	odd	2	inner	384.8.d.f.193.15	yes	16
8.3	odd	2	inner	384.8.d.f.193.2	✓	16
8.5	even	2	inner	384.8.d.f.193.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.d.f.193.2	✓	16	8.3	odd	2	inner
384.8.d.f.193.7	yes	16	1.1	even	1	trivial
384.8.d.f.193.10	yes	16	8.5	even	2	inner
384.8.d.f.193.15	yes	16	4.3	odd	2	inner

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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-27.0000i q^{3} +295.990i q^{5} -1208.55 q^{7} -729.000 q^{9} +O(q^{10})\)	\(q-27.0000i q^{3} +295.990i q^{5} -1208.55 q^{7} -729.000 q^{9} +8341.26i q^{11} +9842.29i q^{13} +7991.72 q^{15} +9080.24 q^{17} +23035.9i q^{19} +32630.9i q^{21} +92905.1 q^{23} -9484.89 q^{25} +19683.0i q^{27} -38163.5i q^{29} +78442.9 q^{31} +225214. q^{33} -357719. i q^{35} +539121. i q^{37} +265742. q^{39} -872572. q^{41} -548781. i q^{43} -215776. i q^{45} -59537.2 q^{47} +637052. q^{49} -245167. i q^{51} +1.43192e6i q^{53} -2.46893e6 q^{55} +621968. q^{57} +404986. i q^{59} +1.46875e6i q^{61} +881033. q^{63} -2.91322e6 q^{65} +586820. i q^{67} -2.50844e6i q^{69} -3.70658e6 q^{71} +1.57329e6 q^{73} +256092. i q^{75} -1.00808e7i q^{77} +7.58995e6 q^{79} +531441. q^{81} -5.50373e6i q^{83} +2.68766e6i q^{85} -1.03041e6 q^{87} -2.90039e6 q^{89} -1.18949e7i q^{91} -2.11796e6i q^{93} -6.81838e6 q^{95} -9.32490e6 q^{97} -6.08078e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 11664 q^{9}+O(q^{10}) \) 16 * q - 11664 * q^9	\( 16 q - 11664 q^{9} + 77280 q^{17} - 236464 q^{25} + 544320 q^{33} - 930912 q^{41} + 1163024 q^{49} - 3032640 q^{57} + 6283008 q^{65} - 2727200 q^{73} + 8503056 q^{81} - 43093152 q^{89} + 35537120 q^{97}+O(q^{100}) \) 16 * q - 11664 * q^9 + 77280 * q^17 - 236464 * q^25 + 544320 * q^33 - 930912 * q^41 + 1163024 * q^49 - 3032640 * q^57 + 6283008 * q^65 - 2727200 * q^73 + 8503056 * q^81 - 43093152 * q^89 + 35537120 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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