LMFDB - Embedded newform 384.8.d.f.193.1

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			193.1
Root			$9.25013 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.8.d.f.193.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000i q^{3} -484.521i q^{5} -826.966 q^{7} -729.000 q^{9} +O(q^{10})$	$q-27.0000i q^{3} -484.521i q^{5} -826.966 q^{7} -729.000 q^{9} +4559.22i q^{11} +3071.92i q^{13} -13082.1 q^{15} +21664.0 q^{17} -58806.6i q^{19} +22328.1i q^{21} -18663.3 q^{23} -156636. q^{25} +19683.0i q^{27} +210373. i q^{29} -98739.9 q^{31} +123099. q^{33} +400682. i q^{35} +549811. i q^{37} +82941.9 q^{39} +559912. q^{41} +503917. i q^{43} +353216. i q^{45} -970883. q^{47} -139671. q^{49} -584927. i q^{51} +1.49503e6i q^{53} +2.20904e6 q^{55} -1.58778e6 q^{57} -415361. i q^{59} -2.20016e6i q^{61} +602858. q^{63} +1.48841e6 q^{65} +1.95346e6i q^{67} +503909. i q^{69} +1.29191e6 q^{71} +3.21010e6 q^{73} +4.22916e6i q^{75} -3.77032e6i q^{77} -1.91140e6 q^{79} +531441. q^{81} +2.28696e6i q^{83} -1.04966e7i q^{85} +5.68008e6 q^{87} -3.17696e6 q^{89} -2.54037e6i q^{91} +2.66598e6i q^{93} -2.84930e7 q^{95} +2.37459e6 q^{97} -3.32367e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 484.521i	− 1.73348i	−0.498765	−	0.866738i	$-0.666213\pi$
$5$	− 484.521i	− 1.73348i	0.498765	−	0.866738i	$-0.333787\pi$
$6$	0	0
$7$	−826.966	−0.911264	−0.455632	−	0.890168i	$-0.650587\pi$
$7$	−826.966	−0.911264	−0.455632	+	0.890168i	$0.650587\pi$
$8$	0	0
$9$	−729.000	−0.333333
$10$	0	0
$11$	4559.22i	1.03280i	0.856348	+	0.516400i	$0.172728\pi$
$11$	4559.22i	1.03280i	−0.856348	+	0.516400i	$0.827272\pi$
$12$	0	0
$13$	3071.92i	0.387801i	0.981021	+	0.193900i	$0.0621138\pi$
$13$	3071.92i	0.387801i	−0.981021	+	0.193900i	$0.937886\pi$
$14$	0	0
$15$	−13082.1	−1.00082
$16$	0	0
$17$	21664.0	1.06947	0.534733	−	0.845021i	$-0.320412\pi$
$17$	21664.0	1.06947	0.534733	+	0.845021i	$0.320412\pi$
$18$	0	0
$19$	− 58806.6i	− 1.96693i	−0.181106	−	0.983464i	$-0.557968\pi$
$19$	− 58806.6i	− 1.96693i	0.181106	−	0.983464i	$-0.442032\pi$
$20$	0	0
$21$	22328.1i	0.526119i
$22$	0	0
$23$	−18663.3	−0.319846	−0.159923	−	0.987130i	$-0.551125\pi$
$23$	−18663.3	−0.319846	−0.159923	+	0.987130i	$0.551125\pi$
$24$	0	0
$25$	−156636.	−2.00494
$26$	0	0
$27$	19683.0i	0.192450i
$28$	0	0
$29$	210373.i	1.60176i	0.598824	+	0.800880i	$0.295635\pi$
$29$	210373.i	1.60176i	−0.598824	+	0.800880i	$0.704365\pi$
$30$	0	0
$31$	−98739.9	−0.595288	−0.297644	−	0.954677i	$-0.596201\pi$
$31$	−98739.9	−0.595288	−0.297644	+	0.954677i	$0.596201\pi$
$32$	0	0
$33$	123099.	0.596287
$34$	0	0
$35$	400682.i	1.57965i
$36$	0	0
$37$	549811.i	1.78446i	0.451579	+	0.892231i	$0.350861\pi$
$37$	549811.i	1.78446i	−0.451579	+	0.892231i	$0.649139\pi$
$38$	0	0
$39$	82941.9	0.223897
$40$	0	0
$41$	559912.	1.26875	0.634375	−	0.773025i	$-0.281257\pi$
$41$	559912.	1.26875	0.634375	+	0.773025i	$0.281257\pi$
$42$	0	0
$43$	503917.i	0.966538i	0.875472	+	0.483269i	$0.160551\pi$
$43$	503917.i	0.966538i	−0.875472	+	0.483269i	$0.839449\pi$
$44$	0	0
$45$	353216.i	0.577825i
$46$	0	0
$47$	−970883.	−1.36403	−0.682016	−	0.731337i	$-0.738897\pi$
$47$	−970883.	−1.36403	−0.682016	+	0.731337i	$0.738897\pi$
$48$	0	0
$49$	−139671.	−0.169597
$50$	0	0
$51$	− 584927.i	− 0.617456i
$52$	0	0
$53$	1.49503e6i	1.37938i	0.724106	+	0.689689i	$0.242253\pi$
$53$	1.49503e6i	1.37938i	−0.724106	+	0.689689i	$0.757747\pi$
$54$	0	0
$55$	2.20904e6	1.79033
$56$	0	0
$57$	−1.58778e6	−1.13561
$58$	0	0
$59$	− 415361.i	− 0.263296i	−0.991297	−	0.131648i	$-0.957973\pi$
$59$	− 415361.i	− 0.263296i	0.991297	−	0.131648i	$-0.0420268\pi$
$60$	0	0
$61$	− 2.20016e6i	− 1.24108i	−0.784175	−	0.620540i	$-0.786913\pi$
$61$	− 2.20016e6i	− 1.24108i	0.784175	−	0.620540i	$-0.213087\pi$
$62$	0	0
$63$	602858.	0.303755
$64$	0	0
$65$	1.48841e6	0.672243
$66$	0	0
$67$	1.95346e6i	0.793492i	0.917928	+	0.396746i	$0.129861\pi$
$67$	1.95346e6i	0.793492i	−0.917928	+	0.396746i	$0.870139\pi$
$68$	0	0
$69$	503909.i	0.184663i
$70$	0	0
$71$	1.29191e6	0.428377	0.214189	−	0.976792i	$-0.431289\pi$
$71$	1.29191e6	0.428377	0.214189	+	0.976792i	$0.431289\pi$
$72$	0	0
$73$	3.21010e6	0.965803	0.482902	−	0.875675i	$-0.339583\pi$
$73$	3.21010e6	0.965803	0.482902	+	0.875675i	$0.339583\pi$
$74$	0	0
$75$	4.22916e6i	1.15755i
$76$	0	0
$77$	− 3.77032e6i	− 0.941153i
$78$	0	0
$79$	−1.91140e6	−0.436171	−0.218086	−	0.975930i	$-0.569981\pi$
$79$	−1.91140e6	−0.436171	−0.218086	+	0.975930i	$0.569981\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	2.28696e6i	0.439021i	0.975610	+	0.219511i	$0.0704461\pi$
$83$	2.28696e6i	0.439021i	−0.975610	+	0.219511i	$0.929554\pi$
$84$	0	0
$85$	− 1.04966e7i	− 1.85389i
$86$	0	0
$87$	5.68008e6	0.924777
$88$	0	0
$89$	−3.17696e6	−0.477691	−0.238845	−	0.971058i	$-0.576769\pi$
$89$	−3.17696e6	−0.477691	−0.238845	+	0.971058i	$0.576769\pi$
$90$	0	0
$91$	− 2.54037e6i	− 0.353389i
$92$	0	0
$93$	2.66598e6i	0.343690i
$94$	0	0
$95$	−2.84930e7	−3.40962
$96$	0	0
$97$	2.37459e6	0.264172	0.132086	−	0.991238i	$-0.457832\pi$
$97$	2.37459e6	0.264172	0.132086	+	0.991238i	$0.457832\pi$
$98$	0	0
$99$	− 3.32367e6i	− 0.344267i
$100$	0	0
$101$	− 5.11158e6i	− 0.493663i	−0.969058	−	0.246831i	$-0.920611\pi$
$101$	− 5.11158e6i	− 0.493663i	0.969058	−	0.246831i	$-0.0793894\pi$
$102$	0	0
$103$	1.05378e7	0.950214	0.475107	−	0.879928i	$-0.342409\pi$
$103$	1.05378e7	0.950214	0.475107	+	0.879928i	$0.342409\pi$
$104$	0	0
$105$	1.08184e7	0.912014
$106$	0	0
$107$	6.18570e6i	0.488142i	0.969757	+	0.244071i	$0.0784830\pi$
$107$	6.18570e6i	0.488142i	−0.969757	+	0.244071i	$0.921517\pi$
$108$	0	0
$109$	− 9.62679e6i	− 0.712014i	−0.934483	−	0.356007i	$-0.884138\pi$
$109$	− 9.62679e6i	− 0.712014i	0.934483	−	0.356007i	$-0.115862\pi$
$110$	0	0
$111$	1.48449e7	1.03026
$112$	0	0
$113$	2.35874e7	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$113$	2.35874e7	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$114$	0	0
$115$	9.04275e6i	0.554445i
$116$	0	0
$117$	− 2.23943e6i	− 0.129267i
$118$	0	0
$119$	−1.79154e7	−0.974566
$120$	0	0
$121$	−1.29931e6	−0.0666749
$122$	0	0
$123$	− 1.51176e7i	− 0.732513i
$124$	0	0
$125$	3.80400e7i	1.74203i
$126$	0	0
$127$	2.70889e7	1.17349	0.586744	−	0.809773i	$-0.300410\pi$
$127$	2.70889e7	1.17349	0.586744	+	0.809773i	$0.300410\pi$
$128$	0	0
$129$	1.36057e7	0.558031
$130$	0	0
$131$	− 3.97308e7i	− 1.54411i	−0.635557	−	0.772054i	$-0.719230\pi$
$131$	− 3.97308e7i	− 1.54411i	0.635557	−	0.772054i	$-0.280770\pi$
$132$	0	0
$133$	4.86310e7i	1.79239i
$134$	0	0
$135$	9.53683e6	0.333607
$136$	0	0
$137$	−739485.	−0.0245701	−0.0122851	−	0.999925i	$-0.503911\pi$
$137$	−739485.	−0.0245701	−0.0122851	+	0.999925i	$0.503911\pi$
$138$	0	0
$139$	− 3.11339e7i	− 0.983290i	−0.870796	−	0.491645i	$-0.836396\pi$
$139$	− 3.11339e7i	− 0.983290i	0.870796	−	0.491645i	$-0.163604\pi$
$140$	0	0
$141$	2.62139e7i	0.787524i
$142$	0	0
$143$	−1.40056e7	−0.400520
$144$	0	0
$145$	1.01930e8	2.77661
$146$	0	0
$147$	3.77111e6i	0.0979170i
$148$	0	0
$149$	3.52064e6i	0.0871907i	0.999049	+	0.0435954i	$0.0138812\pi$
$149$	3.52064e6i	0.0871907i	−0.999049	+	0.0435954i	$0.986119\pi$
$150$	0	0
$151$	−1.10851e7	−0.262012	−0.131006	−	0.991382i	$-0.541821\pi$
$151$	−1.10851e7	−0.262012	−0.131006	+	0.991382i	$0.541821\pi$
$152$	0	0
$153$	−1.57930e7	−0.356488
$154$	0	0
$155$	4.78416e7i	1.03192i
$156$	0	0
$157$	1.41227e6i	0.0291252i	0.999894	+	0.0145626i	$0.00463559\pi$
$157$	1.41227e6i	0.0291252i	−0.999894	+	0.0145626i	$0.995364\pi$
$158$	0	0
$159$	4.03657e7	0.796384
$160$	0	0
$161$	1.54339e7	0.291464
$162$	0	0
$163$	− 2.04722e7i	− 0.370262i	−0.982714	−	0.185131i	$-0.940729\pi$
$163$	− 2.04722e7i	− 0.370262i	0.982714	−	0.185131i	$-0.0592709\pi$
$164$	0	0
$165$	− 5.96440e7i	− 1.03365i
$166$	0	0
$167$	−6.31414e7	−1.04907	−0.524537	−	0.851388i	$-0.675762\pi$
$167$	−6.31414e7	−1.04907	−0.524537	+	0.851388i	$0.675762\pi$
$168$	0	0
$169$	5.33118e7	0.849611
$170$	0	0
$171$	4.28700e7i	0.655642i
$172$	0	0
$173$	− 4.19687e7i	− 0.616261i	−0.951344	−	0.308130i	$-0.900297\pi$
$173$	− 4.19687e7i	− 0.616261i	0.951344	−	0.308130i	$-0.0997033\pi$
$174$	0	0
$175$	1.29532e8	1.82703
$176$	0	0
$177$	−1.12147e7	−0.152014
$178$	0	0
$179$	− 5.08183e7i	− 0.662270i	−0.943583	−	0.331135i	$-0.892569\pi$
$179$	− 5.08183e7i	− 0.662270i	0.943583	−	0.331135i	$-0.107431\pi$
$180$	0	0
$181$	9.81793e7i	1.23068i	0.788262	+	0.615340i	$0.210981\pi$
$181$	9.81793e7i	1.23068i	−0.788262	+	0.615340i	$0.789019\pi$
$182$	0	0
$183$	−5.94043e7	−0.716538
$184$	0	0
$185$	2.66395e8	3.09332
$186$	0	0
$187$	9.87708e7i	1.10454i
$188$	0	0
$189$	− 1.62772e7i	− 0.175373i
$190$	0	0
$191$	−5.63430e7	−0.585090	−0.292545	−	0.956252i	$-0.594502\pi$
$191$	−5.63430e7	−0.585090	−0.292545	+	0.956252i	$0.594502\pi$
$192$	0	0
$193$	3.05504e7	0.305891	0.152945	−	0.988235i	$-0.451124\pi$
$193$	3.05504e7	0.305891	0.152945	+	0.988235i	$0.451124\pi$
$194$	0	0
$195$	− 4.01871e7i	− 0.388120i
$196$	0	0
$197$	− 1.98819e8i	− 1.85279i	−0.376549	−	0.926397i	$-0.622889\pi$
$197$	− 1.98819e8i	− 1.85279i	0.376549	−	0.926397i	$-0.377111\pi$
$198$	0	0
$199$	−1.71467e8	−1.54240	−0.771198	−	0.636595i	$-0.780342\pi$
$199$	−1.71467e8	−1.54240	−0.771198	+	0.636595i	$0.780342\pi$
$200$	0	0
$201$	5.27434e7	0.458123
$202$	0	0
$203$	− 1.73972e8i	− 1.45963i
$204$	0	0
$205$	− 2.71289e8i	− 2.19935i
$206$	0	0
$207$	1.36055e7	0.106615
$208$	0	0
$209$	2.68112e8	2.03144
$210$	0	0
$211$	− 2.89713e7i	− 0.212315i	−0.994349	−	0.106157i	$-0.966145\pi$
$211$	− 2.89713e7i	− 0.212315i	0.994349	−	0.106157i	$-0.0338547\pi$
$212$	0	0
$213$	− 3.48814e7i	− 0.247324i
$214$	0	0
$215$	2.44158e8	1.67547
$216$	0	0
$217$	8.16546e7	0.542465
$218$	0	0
$219$	− 8.66727e7i	− 0.557607i
$220$	0	0
$221$	6.65500e7i	0.414739i
$222$	0	0
$223$	−2.79957e8	−1.69054	−0.845269	−	0.534342i	$-0.820560\pi$
$223$	−2.79957e8	−1.69054	−0.845269	+	0.534342i	$0.820560\pi$
$224$	0	0
$225$	1.14187e8	0.668312
$226$	0	0
$227$	3.01491e8i	1.71074i	0.518016	+	0.855371i	$0.326671\pi$
$227$	3.01491e8i	1.71074i	−0.518016	+	0.855371i	$0.673329\pi$
$228$	0	0
$229$	2.34566e8i	1.29075i	0.763867	+	0.645374i	$0.223298\pi$
$229$	2.34566e8i	1.29075i	−0.763867	+	0.645374i	$0.776702\pi$
$230$	0	0
$231$	−1.01799e8	−0.543375
$232$	0	0
$233$	−1.88761e8	−0.977610	−0.488805	−	0.872393i	$-0.662567\pi$
$233$	−1.88761e8	−0.977610	−0.488805	+	0.872393i	$0.662567\pi$
$234$	0	0
$235$	4.70413e8i	2.36452i
$236$	0	0
$237$	5.16078e7i	0.251823i
$238$	0	0
$239$	3.96557e8	1.87894	0.939470	−	0.342630i	$-0.111318\pi$
$239$	3.96557e8	1.87894	0.939470	+	0.342630i	$0.111318\pi$
$240$	0	0
$241$	7.31086e7	0.336441	0.168221	−	0.985749i	$-0.446198\pi$
$241$	7.31086e7	0.336441	0.168221	+	0.985749i	$0.446198\pi$
$242$	0	0
$243$	− 1.43489e7i	− 0.0641500i
$244$	0	0
$245$	6.76733e7i	0.293993i
$246$	0	0
$247$	1.80649e8	0.762776
$248$	0	0
$249$	6.17480e7	0.253469
$250$	0	0
$251$	4.10096e8i	1.63692i	0.574563	+	0.818460i	$0.305172\pi$
$251$	4.10096e8i	1.63692i	−0.574563	+	0.818460i	$0.694828\pi$
$252$	0	0
$253$	− 8.50900e7i	− 0.330337i
$254$	0	0
$255$	−2.83409e8	−1.07034
$256$	0	0
$257$	1.73230e8	0.636588	0.318294	−	0.947992i	$-0.396890\pi$
$257$	1.73230e8	0.636588	0.318294	+	0.947992i	$0.396890\pi$
$258$	0	0
$259$	− 4.54675e8i	− 1.62612i
$260$	0	0
$261$	− 1.53362e8i	− 0.533920i
$262$	0	0
$263$	5.15032e8	1.74578	0.872890	−	0.487917i	$-0.162243\pi$
$263$	5.15032e8	1.74578	0.872890	+	0.487917i	$0.162243\pi$
$264$	0	0
$265$	7.24371e8	2.39112
$266$	0	0
$267$	8.57779e7i	0.275795i
$268$	0	0
$269$	5.15831e8i	1.61575i	0.589354	+	0.807875i	$0.299382\pi$
$269$	5.15831e8i	1.61575i	−0.589354	+	0.807875i	$0.700618\pi$
$270$	0	0
$271$	−7.18405e6	−0.0219269	−0.0109634	−	0.999940i	$-0.503490\pi$
$271$	−7.18405e6	−0.0219269	−0.0109634	+	0.999940i	$0.503490\pi$
$272$	0	0
$273$	−6.85901e7	−0.204029
$274$	0	0
$275$	− 7.14136e8i	− 2.07070i
$276$	0	0
$277$	3.72429e8i	1.05284i	0.850223	+	0.526422i	$0.176467\pi$
$277$	3.72429e8i	1.05284i	−0.850223	+	0.526422i	$0.823533\pi$
$278$	0	0
$279$	7.19814e7	0.198429
$280$	0	0
$281$	2.30771e8	0.620453	0.310227	−	0.950663i	$-0.399595\pi$
$281$	2.30771e8	0.620453	0.310227	+	0.950663i	$0.399595\pi$
$282$	0	0
$283$	6.05188e7i	0.158722i	0.996846	+	0.0793612i	$0.0252880\pi$
$283$	6.05188e7i	0.158722i	−0.996846	+	0.0793612i	$0.974712\pi$
$284$	0	0
$285$	7.69311e8i	1.96854i
$286$	0	0
$287$	−4.63028e8	−1.15617
$288$	0	0
$289$	5.89888e7	0.143756
$290$	0	0
$291$	− 6.41139e7i	− 0.152520i
$292$	0	0
$293$	− 8.55981e7i	− 0.198805i	−0.995047	−	0.0994025i	$-0.968307\pi$
$293$	− 8.55981e7i	− 0.198805i	0.995047	−	0.0994025i	$-0.0316931\pi$
$294$	0	0
$295$	−2.01251e8	−0.456417
$296$	0	0
$297$	−8.97391e7	−0.198762
$298$	0	0
$299$	− 5.73322e7i	− 0.124036i
$300$	0	0
$301$	− 4.16722e8i	− 0.880772i
$302$	0	0
$303$	−1.38013e8	−0.285016
$304$	0	0
$305$	−1.06602e9	−2.15138
$306$	0	0
$307$	7.47809e8i	1.47505i	0.675320	+	0.737525i	$0.264005\pi$
$307$	7.47809e8i	1.47505i	−0.675320	+	0.737525i	$0.735995\pi$
$308$	0	0
$309$	− 2.84522e8i	− 0.548606i
$310$	0	0
$311$	8.83221e7	0.166498	0.0832488	−	0.996529i	$-0.473470\pi$
$311$	8.83221e7	0.166498	0.0832488	+	0.996529i	$0.473470\pi$
$312$	0	0
$313$	−2.79088e8	−0.514441	−0.257220	−	0.966353i	$-0.582807\pi$
$313$	−2.79088e8	−0.514441	−0.257220	+	0.966353i	$0.582807\pi$
$314$	0	0
$315$	− 2.92097e8i	− 0.526551i
$316$	0	0
$317$	2.42763e8i	0.428030i	0.976830	+	0.214015i	$0.0686542\pi$
$317$	2.42763e8i	0.428030i	−0.976830	+	0.214015i	$0.931346\pi$
$318$	0	0
$319$	−9.59138e8	−1.65430
$320$	0	0
$321$	1.67014e8	0.281829
$322$	0	0
$323$	− 1.27398e9i	− 2.10356i
$324$	0	0
$325$	− 4.81172e8i	− 0.777515i
$326$	0	0
$327$	−2.59923e8	−0.411082
$328$	0	0
$329$	8.02887e8	1.24299
$330$	0	0
$331$	7.87662e8i	1.19383i	0.802305	+	0.596914i	$0.203607\pi$
$331$	7.87662e8i	1.19383i	−0.802305	+	0.596914i	$0.796393\pi$
$332$	0	0
$333$	− 4.00812e8i	− 0.594821i
$334$	0	0
$335$	9.46492e8	1.37550
$336$	0	0
$337$	2.60075e7	0.0370163	0.0185082	−	0.999829i	$-0.494108\pi$
$337$	2.60075e7	0.0370163	0.0185082	+	0.999829i	$0.494108\pi$
$338$	0	0
$339$	− 6.36860e8i	− 0.887862i
$340$	0	0
$341$	− 4.50177e8i	− 0.614813i
$342$	0	0
$343$	7.96545e8	1.06581
$344$	0	0
$345$	2.44154e8	0.320109
$346$	0	0
$347$	7.99682e8i	1.02746i	0.857952	+	0.513729i	$0.171736\pi$
$347$	7.99682e8i	1.02746i	−0.857952	+	0.513729i	$0.828264\pi$
$348$	0	0
$349$	− 1.02045e9i	− 1.28500i	−0.766286	−	0.642500i	$-0.777897\pi$
$349$	− 1.02045e9i	− 1.28500i	0.766286	−	0.642500i	$-0.222103\pi$
$350$	0	0
$351$	−6.04647e7	−0.0746323
$352$	0	0
$353$	1.29119e8	0.156236	0.0781178	−	0.996944i	$-0.475109\pi$
$353$	1.29119e8	0.156236	0.0781178	+	0.996944i	$0.475109\pi$
$354$	0	0
$355$	− 6.25955e8i	− 0.742581i
$356$	0	0
$357$	4.83715e8i	0.562666i
$358$	0	0
$359$	−1.61953e9	−1.84738	−0.923692	−	0.383136i	$-0.874844\pi$
$359$	−1.61953e9	−1.84738	−0.923692	+	0.383136i	$0.874844\pi$
$360$	0	0
$361$	−2.56434e9	−2.86880
$362$	0	0
$363$	3.50812e7i	0.0384948i
$364$	0	0
$365$	− 1.55536e9i	− 1.67420i
$366$	0	0
$367$	1.58446e9	1.67321	0.836606	−	0.547805i	$-0.184536\pi$
$367$	1.58446e9	1.67321	0.836606	+	0.547805i	$0.184536\pi$
$368$	0	0
$369$	−4.08176e8	−0.422917
$370$	0	0
$371$	− 1.23633e9i	− 1.25698i
$372$	0	0
$373$	− 2.59891e8i	− 0.259305i	−0.991559	−	0.129653i	$-0.958614\pi$
$373$	− 2.59891e8i	− 0.259305i	0.991559	−	0.129653i	$-0.0413862\pi$
$374$	0	0
$375$	1.02708e9	1.00576
$376$	0	0
$377$	−6.46250e8	−0.621164
$378$	0	0
$379$	1.18982e9i	1.12265i	0.827597	+	0.561323i	$0.189707\pi$
$379$	1.18982e9i	1.12265i	−0.827597	+	0.561323i	$0.810293\pi$
$380$	0	0
$381$	− 7.31401e8i	− 0.677514i
$382$	0	0
$383$	−6.10325e8	−0.555092	−0.277546	−	0.960712i	$-0.589521\pi$
$383$	−6.10325e8	−0.555092	−0.277546	+	0.960712i	$0.589521\pi$
$384$	0	0
$385$	−1.82680e9	−1.63147
$386$	0	0
$387$	− 3.67355e8i	− 0.322179i
$388$	0	0
$389$	6.28519e8i	0.541371i	0.962668	+	0.270685i	$0.0872503\pi$
$389$	6.28519e8i	0.541371i	−0.962668	+	0.270685i	$0.912750\pi$
$390$	0	0
$391$	−4.04321e8	−0.342064
$392$	0	0
$393$	−1.07273e9	−0.891491
$394$	0	0
$395$	9.26114e8i	0.756092i
$396$	0	0
$397$	1.99778e9i	1.60243i	0.598374	+	0.801217i	$0.295814\pi$
$397$	1.99778e9i	1.60243i	−0.598374	+	0.801217i	$0.704186\pi$
$398$	0	0
$399$	1.31304e9	1.03484
$400$	0	0
$401$	−6.23030e7	−0.0482507	−0.0241254	−	0.999709i	$-0.507680\pi$
$401$	−6.23030e7	−0.0482507	−0.0241254	+	0.999709i	$0.507680\pi$
$402$	0	0
$403$	− 3.03321e8i	− 0.230853i
$404$	0	0
$405$	− 2.57494e8i	− 0.192608i
$406$	0	0
$407$	−2.50671e9	−1.84299
$408$	0	0
$409$	1.65740e9	1.19783	0.598914	−	0.800813i	$-0.295599\pi$
$409$	1.65740e9	1.19783	0.598914	+	0.800813i	$0.295599\pi$
$410$	0	0
$411$	1.99661e7i	0.0141856i
$412$	0	0
$413$	3.43489e8i	0.239932i
$414$	0	0
$415$	1.10808e9	0.761032
$416$	0	0
$417$	−8.40615e8	−0.567703
$418$	0	0
$419$	5.44718e8i	0.361762i	0.983505	+	0.180881i	$0.0578949\pi$
$419$	5.44718e8i	0.361762i	−0.983505	+	0.180881i	$0.942105\pi$
$420$	0	0
$421$	1.75919e9i	1.14901i	0.818500	+	0.574507i	$0.194806\pi$
$421$	1.75919e9i	1.14901i	−0.818500	+	0.574507i	$0.805194\pi$
$422$	0	0
$423$	7.07774e8	0.454677
$424$	0	0
$425$	−3.39335e9	−2.14421
$426$	0	0
$427$	1.81946e9i	1.13095i
$428$	0	0
$429$	3.78150e8i	0.231241i
$430$	0	0
$431$	6.38357e8	0.384055	0.192027	−	0.981390i	$-0.438494\pi$
$431$	6.38357e8	0.384055	0.192027	+	0.981390i	$0.438494\pi$
$432$	0	0
$433$	1.18840e9	0.703486	0.351743	−	0.936097i	$-0.385589\pi$
$433$	1.18840e9	0.703486	0.351743	+	0.936097i	$0.385589\pi$
$434$	0	0
$435$	− 2.75212e9i	− 1.60308i
$436$	0	0
$437$	1.09752e9i	0.629113i
$438$	0	0
$439$	−7.11390e8	−0.401312	−0.200656	−	0.979662i	$-0.564307\pi$
$439$	−7.11390e8	−0.401312	−0.200656	+	0.979662i	$0.564307\pi$
$440$	0	0
$441$	1.01820e8	0.0565324
$442$	0	0
$443$	− 1.92825e9i	− 1.05378i	−0.849934	−	0.526890i	$-0.823358\pi$
$443$	− 1.92825e9i	− 1.05378i	0.849934	−	0.526890i	$-0.176642\pi$
$444$	0	0
$445$	1.53930e9i	0.828065i
$446$	0	0
$447$	9.50574e7	0.0503396
$448$	0	0
$449$	−2.20494e9	−1.14957	−0.574784	−	0.818305i	$-0.694914\pi$
$449$	−2.20494e9	−1.14957	−0.574784	+	0.818305i	$0.694914\pi$
$450$	0	0
$451$	2.55276e9i	1.31036i
$452$	0	0
$453$	2.99298e8i	0.151273i
$454$	0	0
$455$	−1.23086e9	−0.612591
$456$	0	0
$457$	−2.84830e9	−1.39598	−0.697989	−	0.716109i	$-0.745921\pi$
$457$	−2.84830e9	−1.39598	−0.697989	+	0.716109i	$0.745921\pi$
$458$	0	0
$459$	4.26412e8i	0.205819i
$460$	0	0
$461$	1.27737e9i	0.607242i	0.952793	+	0.303621i	$0.0981957\pi$
$461$	1.27737e9i	0.607242i	−0.952793	+	0.303621i	$0.901804\pi$
$462$	0	0
$463$	8.68144e7	0.0406498	0.0203249	−	0.999793i	$-0.493530\pi$
$463$	8.68144e7	0.0406498	0.0203249	+	0.999793i	$0.493530\pi$
$464$	0	0
$465$	1.29172e9	0.595777
$466$	0	0
$467$	− 2.98319e9i	− 1.35541i	−0.735332	−	0.677707i	$-0.762974\pi$
$467$	− 2.98319e9i	− 1.35541i	0.735332	−	0.677707i	$-0.237026\pi$
$468$	0	0
$469$	− 1.61544e9i	− 0.723081i
$470$	0	0
$471$	3.81313e7	0.0168155
$472$	0	0
$473$	−2.29747e9	−0.998240
$474$	0	0
$475$	9.21120e9i	3.94356i
$476$	0	0
$477$	− 1.08987e9i	− 0.459792i
$478$	0	0
$479$	−2.69279e9	−1.11951	−0.559756	−	0.828658i	$-0.689105\pi$
$479$	−2.69279e9	−1.11951	−0.559756	+	0.828658i	$0.689105\pi$
$480$	0	0
$481$	−1.68898e9	−0.692016
$482$	0	0
$483$	− 4.16715e8i	− 0.168277i
$484$	0	0
$485$	− 1.15054e9i	− 0.457936i
$486$	0	0
$487$	1.99374e9	0.782199	0.391100	−	0.920348i	$-0.372095\pi$
$487$	1.99374e9	0.782199	0.391100	+	0.920348i	$0.372095\pi$
$488$	0	0
$489$	−5.52751e8	−0.213771
$490$	0	0
$491$	2.20091e9i	0.839106i	0.907731	+	0.419553i	$0.137813\pi$
$491$	2.20091e9i	0.839106i	−0.907731	+	0.419553i	$0.862187\pi$
$492$	0	0
$493$	4.55752e9i	1.71303i
$494$	0	0
$495$	−1.61039e9	−0.596777
$496$	0	0
$497$	−1.06836e9	−0.390365
$498$	0	0
$499$	1.86309e9i	0.671248i	0.941996	+	0.335624i	$0.108947\pi$
$499$	1.86309e9i	0.671248i	−0.941996	+	0.335624i	$0.891053\pi$
$500$	0	0
$501$	1.70482e9i	0.605683i
$502$	0	0
$503$	1.27812e9	0.447800	0.223900	−	0.974612i	$-0.428121\pi$
$503$	1.27812e9	0.447800	0.223900	+	0.974612i	$0.428121\pi$
$504$	0	0
$505$	−2.47667e9	−0.855752
$506$	0	0
$507$	− 1.43942e9i	− 0.490523i
$508$	0	0
$509$	− 2.98340e9i	− 1.00277i	−0.865226	−	0.501383i	$-0.832825\pi$
$509$	− 2.98340e9i	− 1.00277i	0.865226	−	0.501383i	$-0.167175\pi$
$510$	0	0
$511$	−2.65464e9	−0.880102
$512$	0	0
$513$	1.15749e9	0.378535
$514$	0	0
$515$	− 5.10580e9i	− 1.64717i
$516$	0	0
$517$	− 4.42647e9i	− 1.40877i
$518$	0	0
$519$	−1.13316e9	−0.355798
$520$	0	0
$521$	4.31306e9	1.33614	0.668072	−	0.744097i	$-0.267120\pi$
$521$	4.31306e9	1.33614	0.668072	+	0.744097i	$0.267120\pi$
$522$	0	0
$523$	− 1.68806e9i	− 0.515979i	−0.966148	−	0.257989i	$-0.916940\pi$
$523$	− 1.68806e9i	− 0.515979i	0.966148	−	0.257989i	$-0.0830599\pi$
$524$	0	0
$525$	− 3.49737e9i	− 1.05483i
$526$	0	0
$527$	−2.13910e9	−0.636640
$528$	0	0
$529$	−3.05651e9	−0.897699
$530$	0	0
$531$	3.02798e8i	0.0877652i
$532$	0	0
$533$	1.72001e9i	0.492022i
$534$	0	0
$535$	2.99710e9	0.846182
$536$	0	0
$537$	−1.37210e9	−0.382362
$538$	0	0
$539$	− 6.36789e8i	− 0.175160i
$540$	0	0
$541$	− 4.61857e9i	− 1.25406i	−0.778997	−	0.627028i	$-0.784271\pi$
$541$	− 4.61857e9i	− 1.25406i	0.778997	−	0.627028i	$-0.215729\pi$
$542$	0	0
$543$	2.65084e9	0.710533
$544$	0	0
$545$	−4.66438e9	−1.23426
$546$	0	0
$547$	− 5.23901e9i	− 1.36865i	−0.729176	−	0.684327i	$-0.760096\pi$
$547$	− 5.23901e9i	− 1.36865i	0.729176	−	0.684327i	$-0.239904\pi$
$548$	0	0
$549$	1.60392e9i	0.413694i
$550$	0	0
$551$	1.23713e10	3.15055
$552$	0	0
$553$	1.58066e9	0.397467
$554$	0	0
$555$	− 7.19266e9i	− 1.78593i
$556$	0	0
$557$	− 2.07267e9i	− 0.508203i	−0.967178	−	0.254101i	$-0.918220\pi$
$557$	− 2.07267e9i	− 0.508203i	0.967178	−	0.254101i	$-0.0817797\pi$
$558$	0	0
$559$	−1.54799e9	−0.374824
$560$	0	0
$561$	2.66681e9	0.637708
$562$	0	0
$563$	− 1.56961e9i	− 0.370692i	−0.982673	−	0.185346i	$-0.940659\pi$
$563$	− 1.56961e9i	− 0.370692i	0.982673	−	0.185346i	$-0.0593405\pi$
$564$	0	0
$565$	− 1.14286e10i	− 2.66578i
$566$	0	0
$567$	−4.39484e8	−0.101252
$568$	0	0
$569$	4.78553e9	1.08902	0.544512	−	0.838753i	$-0.316715\pi$
$569$	4.78553e9	1.08902	0.544512	+	0.838753i	$0.316715\pi$
$570$	0	0
$571$	7.06103e9i	1.58724i	0.608416	+	0.793618i	$0.291805\pi$
$571$	7.06103e9i	1.58724i	−0.608416	+	0.793618i	$0.708195\pi$
$572$	0	0
$573$	1.52126e9i	0.337802i
$574$	0	0
$575$	2.92333e9	0.641270
$576$	0	0
$577$	7.94695e9	1.72221	0.861103	−	0.508431i	$-0.169774\pi$
$577$	7.94695e9	1.72221	0.861103	+	0.508431i	$0.169774\pi$
$578$	0	0
$579$	− 8.24862e8i	− 0.176606i
$580$	0	0
$581$	− 1.89124e9i	− 0.400064i
$582$	0	0
$583$	−6.81615e9	−1.42462
$584$	0	0
$585$	−1.08505e9	−0.224081
$586$	0	0
$587$	3.79237e8i	0.0773886i	0.999251	+	0.0386943i	$0.0123199\pi$
$587$	3.79237e8i	0.0773886i	−0.999251	+	0.0386943i	$0.987680\pi$
$588$	0	0
$589$	5.80656e9i	1.17089i
$590$	0	0
$591$	−5.36813e9	−1.06971
$592$	0	0
$593$	6.12403e9	1.20600	0.602998	−	0.797742i	$-0.293973\pi$
$593$	6.12403e9	1.20600	0.602998	+	0.797742i	$0.293973\pi$
$594$	0	0
$595$	8.68037e9i	1.68939i
$596$	0	0
$597$	4.62962e9i	0.890503i
$598$	0	0
$599$	1.63280e9	0.310412	0.155206	−	0.987882i	$-0.450396\pi$
$599$	1.63280e9	0.310412	0.155206	+	0.987882i	$0.450396\pi$
$600$	0	0
$601$	1.80391e8	0.0338965	0.0169482	−	0.999856i	$-0.494605\pi$
$601$	1.80391e8	0.0338965	0.0169482	+	0.999856i	$0.494605\pi$
$602$	0	0
$603$	− 1.42407e9i	− 0.264497i
$604$	0	0
$605$	6.29541e8i	0.115579i
$606$	0	0
$607$	3.34640e9	0.607320	0.303660	−	0.952780i	$-0.401791\pi$
$607$	3.34640e9	0.607320	0.303660	+	0.952780i	$0.401791\pi$
$608$	0	0
$609$	−4.69723e9	−0.842716
$610$	0	0
$611$	− 2.98248e9i	− 0.528973i
$612$	0	0
$613$	5.73850e9i	1.00621i	0.864227	+	0.503103i	$0.167808\pi$
$613$	5.73850e9i	1.00621i	−0.864227	+	0.503103i	$0.832192\pi$
$614$	0	0
$615$	−7.32481e9	−1.26979
$616$	0	0
$617$	−2.70937e9	−0.464377	−0.232188	−	0.972671i	$-0.574589\pi$
$617$	−2.70937e9	−0.464377	−0.232188	+	0.972671i	$0.574589\pi$
$618$	0	0
$619$	8.14429e9i	1.38018i	0.723723	+	0.690090i	$0.242429\pi$
$619$	8.14429e9i	1.38018i	−0.723723	+	0.690090i	$0.757571\pi$
$620$	0	0
$621$	− 3.67349e8i	− 0.0615543i
$622$	0	0
$623$	2.62724e9	0.435303
$624$	0	0
$625$	6.19404e9	1.01483
$626$	0	0
$627$	− 7.23902e9i	− 1.17285i
$628$	0	0
$629$	1.19111e10i	1.90842i
$630$	0	0
$631$	−2.96870e9	−0.470397	−0.235198	−	0.971947i	$-0.575574\pi$
$631$	−2.96870e9	−0.470397	−0.235198	+	0.971947i	$0.575574\pi$
$632$	0	0
$633$	−7.82226e8	−0.122580
$634$	0	0
$635$	− 1.31251e10i	− 2.03421i
$636$	0	0
$637$	− 4.29057e8i	− 0.0657699i
$638$	0	0
$639$	−9.41799e8	−0.142792
$640$	0	0
$641$	2.66582e9	0.399787	0.199893	−	0.979818i	$-0.435940\pi$
$641$	2.66582e9	0.399787	0.199893	+	0.979818i	$0.435940\pi$
$642$	0	0
$643$	1.27436e10i	1.89040i	0.326490	+	0.945201i	$0.394134\pi$
$643$	1.27436e10i	1.89040i	−0.326490	+	0.945201i	$0.605866\pi$
$644$	0	0
$645$	− 6.59227e9i	− 0.967333i
$646$	0	0
$647$	4.37698e9	0.635345	0.317672	−	0.948201i	$-0.397099\pi$
$647$	4.37698e9	0.635345	0.317672	+	0.948201i	$0.397099\pi$
$648$	0	0
$649$	1.89372e9	0.271932
$650$	0	0
$651$	− 2.20467e9i	− 0.313192i
$652$	0	0
$653$	− 3.10731e9i	− 0.436705i	−0.975870	−	0.218353i	$-0.929932\pi$
$653$	− 3.10731e9i	− 0.436705i	0.975870	−	0.218353i	$-0.0700683\pi$
$654$	0	0
$655$	−1.92504e10	−2.67667
$656$	0	0
$657$	−2.34016e9	−0.321934
$658$	0	0
$659$	− 1.99805e9i	− 0.271962i	−0.990711	−	0.135981i	$-0.956581\pi$
$659$	− 1.99805e9i	− 0.271962i	0.990711	−	0.135981i	$-0.0434186\pi$
$660$	0	0
$661$	6.24765e9i	0.841417i	0.907196	+	0.420709i	$0.138219\pi$
$661$	6.24765e9i	0.841417i	−0.907196	+	0.420709i	$0.861781\pi$
$662$	0	0
$663$	1.79685e9	0.239450
$664$	0	0
$665$	2.35628e10	3.10706
$666$	0	0
$667$	− 3.92626e9i	− 0.512316i
$668$	0	0
$669$	7.55885e9i	0.976032i
$670$	0	0
$671$	1.00310e10	1.28179
$672$	0	0
$673$	2.67358e9	0.338096	0.169048	−	0.985608i	$-0.445931\pi$
$673$	2.67358e9	0.338096	0.169048	+	0.985608i	$0.445931\pi$
$674$	0	0
$675$	− 3.08306e9i	− 0.385850i
$676$	0	0
$677$	9.95715e9i	1.23332i	0.787231	+	0.616658i	$0.211514\pi$
$677$	9.95715e9i	1.23332i	−0.787231	+	0.616658i	$0.788486\pi$
$678$	0	0
$679$	−1.96370e9	−0.240731
$680$	0	0
$681$	8.14027e9	0.987697
$682$	0	0
$683$	− 7.62355e9i	− 0.915555i	−0.889067	−	0.457778i	$-0.848646\pi$
$683$	− 7.62355e9i	− 0.915555i	0.889067	−	0.457778i	$-0.151354\pi$
$684$	0	0
$685$	3.58296e8i	0.0425917i
$686$	0	0
$687$	6.33329e9	0.745214
$688$	0	0
$689$	−4.59260e9	−0.534923
$690$	0	0
$691$	3.62210e9i	0.417625i	0.977956	+	0.208813i	$0.0669599\pi$
$691$	3.62210e9i	0.417625i	−0.977956	+	0.208813i	$0.933040\pi$
$692$	0	0
$693$	2.74856e9i	0.313718i
$694$	0	0
$695$	−1.50850e10	−1.70451
$696$	0	0
$697$	1.21299e10	1.35688
$698$	0	0
$699$	5.09654e9i	0.564424i
$700$	0	0
$701$	1.00062e10i	1.09712i	0.836111	+	0.548560i	$0.184824\pi$
$701$	1.00062e10i	1.09712i	−0.836111	+	0.548560i	$0.815176\pi$
$702$	0	0
$703$	3.23325e10	3.50991
$704$	0	0
$705$	1.27012e10	1.36515
$706$	0	0
$707$	4.22710e9i	0.449857i
$708$	0	0
$709$	1.02466e10i	1.07974i	0.841748	+	0.539871i	$0.181527\pi$
$709$	1.02466e10i	1.07974i	−0.841748	+	0.539871i	$0.818473\pi$
$710$	0	0
$711$	1.39341e9	0.145390
$712$	0	0
$713$	1.84281e9	0.190400
$714$	0	0
$715$	6.78599e9i	0.694292i
$716$	0	0
$717$	− 1.07070e10i	− 1.08481i
$718$	0	0
$719$	−9.71442e9	−0.974689	−0.487344	−	0.873210i	$-0.662034\pi$
$719$	−9.71442e9	−0.974689	−0.487344	+	0.873210i	$0.662034\pi$
$720$	0	0
$721$	−8.71443e9	−0.865896
$722$	0	0
$723$	− 1.97393e9i	− 0.194244i
$724$	0	0
$725$	− 3.29519e10i	− 3.21143i
$726$	0	0
$727$	−1.01862e10	−0.983201	−0.491600	−	0.870821i	$-0.663588\pi$
$727$	−1.01862e10	−0.983201	−0.491600	+	0.870821i	$0.663588\pi$
$728$	0	0
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	1.09168e10i	1.03368i
$732$	0	0
$733$	− 1.49661e10i	− 1.40360i	−0.712374	−	0.701800i	$-0.752380\pi$
$733$	− 1.49661e10i	− 1.40360i	0.712374	−	0.701800i	$-0.247620\pi$
$734$	0	0
$735$	1.82718e9	0.169737
$736$	0	0
$737$	−8.90625e9	−0.819518
$738$	0	0
$739$	1.66094e10i	1.51390i	0.653471	+	0.756952i	$0.273312\pi$
$739$	1.66094e10i	1.51390i	−0.653471	+	0.756952i	$0.726688\pi$
$740$	0	0
$741$	− 4.87753e9i	− 0.440389i
$742$	0	0
$743$	4.30569e9	0.385107	0.192554	−	0.981286i	$-0.438323\pi$
$743$	4.30569e9	0.385107	0.192554	+	0.981286i	$0.438323\pi$
$744$	0	0
$745$	1.70583e9	0.151143
$746$	0	0
$747$	− 1.66720e9i	− 0.146340i
$748$	0	0
$749$	− 5.11537e9i	− 0.444826i
$750$	0	0
$751$	9.44644e9	0.813820	0.406910	−	0.913468i	$-0.366606\pi$
$751$	9.44644e9	0.813820	0.406910	+	0.913468i	$0.366606\pi$
$752$	0	0
$753$	1.10726e10	0.945076
$754$	0	0
$755$	5.37097e9i	0.454191i
$756$	0	0
$757$	1.38829e9i	0.116318i	0.998307	+	0.0581589i	$0.0185230\pi$
$757$	1.38829e9i	0.116318i	−0.998307	+	0.0581589i	$0.981477\pi$
$758$	0	0
$759$	−2.29743e9	−0.190720
$760$	0	0
$761$	1.38138e10	1.13623	0.568117	−	0.822948i	$-0.307672\pi$
$761$	1.38138e10	1.13623	0.568117	+	0.822948i	$0.307672\pi$
$762$	0	0
$763$	7.96102e9i	0.648833i
$764$	0	0
$765$	7.65206e9i	0.617964i
$766$	0	0
$767$	1.27596e9	0.102106
$768$	0	0
$769$	1.94935e10	1.54578	0.772889	−	0.634541i	$-0.218811\pi$
$769$	1.94935e10	1.54578	0.772889	+	0.634541i	$0.218811\pi$
$770$	0	0
$771$	− 4.67722e9i	− 0.367534i
$772$	0	0
$773$	− 1.67350e10i	− 1.30316i	−0.758580	−	0.651580i	$-0.774106\pi$
$773$	− 1.67350e10i	− 1.30316i	0.758580	−	0.651580i	$-0.225894\pi$
$774$	0	0
$775$	1.54662e10	1.19351
$776$	0	0
$777$	−1.22762e10	−0.938839
$778$	0	0
$779$	− 3.29265e10i	− 2.49554i
$780$	0	0
$781$	5.89008e9i	0.442428i
$782$	0	0
$783$	−4.14078e9	−0.308259
$784$	0	0
$785$	6.84275e8	0.0504878
$786$	0	0
$787$	1.84916e10i	1.35227i	0.736780	+	0.676133i	$0.236345\pi$
$787$	1.84916e10i	1.35227i	−0.736780	+	0.676133i	$0.763655\pi$
$788$	0	0
$789$	− 1.39059e10i	− 1.00793i
$790$	0	0
$791$	−1.95060e10	−1.40136
$792$	0	0
$793$	6.75872e9	0.481292
$794$	0	0
$795$	− 1.95580e10i	− 1.38051i
$796$	0	0
$797$	1.24812e10i	0.873279i	0.899636	+	0.436640i	$0.143832\pi$
$797$	1.24812e10i	0.873279i	−0.899636	+	0.436640i	$0.856168\pi$
$798$	0	0
$799$	−2.10332e10	−1.45879
$800$	0	0
$801$	2.31600e9	0.159230
$802$	0	0
$803$	1.46356e10i	0.997481i
$804$	0	0
$805$	− 7.47805e9i	− 0.505246i
$806$	0	0
$807$	1.39274e10	0.932854
$808$	0	0
$809$	4.91309e9	0.326238	0.163119	−	0.986606i	$-0.447845\pi$
$809$	4.91309e9	0.326238	0.163119	+	0.986606i	$0.447845\pi$
$810$	0	0
$811$	− 1.33853e10i	− 0.881163i	−0.897713	−	0.440582i	$-0.854772\pi$
$811$	− 1.33853e10i	− 0.881163i	0.897713	−	0.440582i	$-0.145228\pi$
$812$	0	0
$813$	1.93969e8i	0.0126595i
$814$	0	0
$815$	−9.91923e9	−0.641840
$816$	0	0
$817$	2.96336e10	1.90111
$818$	0	0
$819$	1.85193e9i	0.117796i
$820$	0	0
$821$	1.59686e10i	1.00708i	0.863972	+	0.503541i	$0.167970\pi$
$821$	1.59686e10i	1.00708i	−0.863972	+	0.503541i	$0.832030\pi$
$822$	0	0
$823$	2.10577e10	1.31678	0.658388	−	0.752679i	$-0.271239\pi$
$823$	2.10577e10	1.31678	0.658388	+	0.752679i	$0.271239\pi$
$824$	0	0
$825$	−1.92817e10	−1.19552
$826$	0	0
$827$	1.86289e10i	1.14529i	0.819802	+	0.572647i	$0.194084\pi$
$827$	1.86289e10i	1.14529i	−0.819802	+	0.572647i	$0.805916\pi$
$828$	0	0
$829$	− 1.26711e10i	− 0.772458i	−0.922403	−	0.386229i	$-0.873777\pi$
$829$	− 1.26711e10i	− 0.772458i	0.922403	−	0.386229i	$-0.126223\pi$
$830$	0	0
$831$	1.00556e10	0.607860
$832$	0	0
$833$	−3.02582e9	−0.181378
$834$	0	0
$835$	3.05933e10i	1.81854i
$836$	0	0
$837$	− 1.94350e9i	− 0.114563i
$838$	0	0
$839$	−2.02505e10	−1.18377	−0.591887	−	0.806021i	$-0.701617\pi$
$839$	−2.02505e10	−1.18377	−0.591887	+	0.806021i	$0.701617\pi$
$840$	0	0
$841$	−2.70070e10	−1.56564
$842$	0	0
$843$	− 6.23082e9i	− 0.358219i
$844$	0	0
$845$	− 2.58307e10i	− 1.47278i
$846$	0	0
$847$	1.07448e9	0.0607585
$848$	0	0
$849$	1.63401e9	0.0916384
$850$	0	0
$851$	− 1.02613e10i	− 0.570753i
$852$	0	0
$853$	2.31306e10i	1.27605i	0.770018	+	0.638023i	$0.220247\pi$
$853$	2.31306e10i	1.27605i	−0.770018	+	0.638023i	$0.779753\pi$
$854$	0	0
$855$	2.07714e10	1.13654
$856$	0	0
$857$	−1.26288e10	−0.685375	−0.342687	−	0.939450i	$-0.611337\pi$
$857$	−1.26288e10	−0.685375	−0.342687	+	0.939450i	$0.611337\pi$
$858$	0	0
$859$	− 2.10559e10i	− 1.13344i	−0.823912	−	0.566718i	$-0.808213\pi$
$859$	− 2.10559e10i	− 1.13344i	0.823912	−	0.566718i	$-0.191787\pi$
$860$	0	0
$861$	1.25018e10i	0.667513i
$862$	0	0
$863$	−1.68819e10	−0.894094	−0.447047	−	0.894510i	$-0.647524\pi$
$863$	−1.68819e10	−0.894094	−0.447047	+	0.894510i	$0.647524\pi$
$864$	0	0
$865$	−2.03347e10	−1.06827
$866$	0	0
$867$	− 1.59270e9i	− 0.0829977i
$868$	0	0
$869$	− 8.71449e9i	− 0.450477i
$870$	0	0
$871$	−6.00087e9	−0.307717
$872$	0	0
$873$	−1.73108e9	−0.0880575
$874$	0	0
$875$	− 3.14578e10i	− 1.58745i
$876$	0	0
$877$	1.53077e10i	0.766321i	0.923682	+	0.383161i	$0.125164\pi$
$877$	1.53077e10i	0.766321i	−0.923682	+	0.383161i	$0.874836\pi$
$878$	0	0
$879$	−2.31115e9	−0.114780
$880$	0	0
$881$	−3.25753e10	−1.60499	−0.802497	−	0.596657i	$-0.796495\pi$
$881$	−3.25753e10	−1.60499	−0.802497	+	0.596657i	$0.796495\pi$
$882$	0	0
$883$	− 1.59810e10i	− 0.781164i	−0.920568	−	0.390582i	$-0.872274\pi$
$883$	− 1.59810e10i	− 0.781164i	0.920568	−	0.390582i	$-0.127726\pi$
$884$	0	0
$885$	5.43378e9i	0.263512i
$886$	0	0
$887$	−1.40697e10	−0.676944	−0.338472	−	0.940976i	$-0.609910\pi$
$887$	−1.40697e10	−0.676944	−0.338472	+	0.940976i	$0.609910\pi$
$888$	0	0
$889$	−2.24016e10	−1.06936
$890$	0	0
$891$	2.42296e9i	0.114756i
$892$	0	0
$893$	5.70943e10i	2.68295i
$894$	0	0
$895$	−2.46226e10	−1.14803
$896$	0	0
$897$	−1.54797e9	−0.0716124
$898$	0	0
$899$	− 2.07722e10i	− 0.953508i
$900$	0	0
$901$	3.23882e10i	1.47520i
$902$	0	0
$903$	−1.12515e10	−0.508514
$904$	0	0
$905$	4.75700e10	2.13335
$906$	0	0
$907$	− 2.65877e10i	− 1.18319i	−0.806235	−	0.591596i	$-0.798498\pi$
$907$	− 2.65877e10i	− 1.18319i	0.806235	−	0.591596i	$-0.201502\pi$
$908$	0	0
$909$	3.72634e9i	0.164554i
$910$	0	0
$911$	−6.29192e9	−0.275721	−0.137860	−	0.990452i	$-0.544022\pi$
$911$	−6.29192e9	−0.275721	−0.137860	+	0.990452i	$0.544022\pi$
$912$	0	0
$913$	−1.04268e10	−0.453421
$914$	0	0
$915$	2.87826e10i	1.24210i
$916$	0	0
$917$	3.28560e10i	1.40709i
$918$	0	0
$919$	4.48465e10	1.90601	0.953003	−	0.302961i	$-0.0979750\pi$
$919$	4.48465e10	1.90601	0.953003	+	0.302961i	$0.0979750\pi$
$920$	0	0
$921$	2.01908e10	0.851620
$922$	0	0
$923$	3.96863e9i	0.166125i
$924$	0	0
$925$	− 8.61200e10i	− 3.57773i
$926$	0	0
$927$	−7.68209e9	−0.316738
$928$	0	0
$929$	1.21912e10	0.498873	0.249436	−	0.968391i	$-0.419755\pi$
$929$	1.21912e10	0.498873	0.249436	+	0.968391i	$0.419755\pi$
$930$	0	0
$931$	8.21355e9i	0.333585i
$932$	0	0
$933$	− 2.38470e9i	− 0.0961275i
$934$	0	0
$935$	4.78565e10	1.91470
$936$	0	0
$937$	−4.15735e9	−0.165093	−0.0825464	−	0.996587i	$-0.526305\pi$
$937$	−4.15735e9	−0.165093	−0.0825464	+	0.996587i	$0.526305\pi$
$938$	0	0
$939$	7.53536e9i	0.297012i
$940$	0	0
$941$	2.56765e9i	0.100455i	0.998738	+	0.0502276i	$0.0159947\pi$
$941$	2.56765e9i	0.100455i	−0.998738	+	0.0502276i	$0.984005\pi$
$942$	0	0
$943$	−1.04498e10	−0.405804
$944$	0	0
$945$	−7.88663e9	−0.304005
$946$	0	0
$947$	2.08480e10i	0.797699i	0.917016	+	0.398850i	$0.130590\pi$
$947$	2.08480e10i	0.797699i	−0.917016	+	0.398850i	$0.869410\pi$
$948$	0	0
$949$	9.86118e9i	0.374539i
$950$	0	0
$951$	6.55459e9	0.247123
$952$	0	0
$953$	−3.94449e10	−1.47627	−0.738136	−	0.674652i	$-0.764294\pi$
$953$	−3.94449e10	−1.47627	−0.738136	+	0.674652i	$0.764294\pi$
$954$	0	0
$955$	2.72993e10i	1.01424i
$956$	0	0
$957$	2.58967e10i	0.955109i
$958$	0	0
$959$	6.11528e8	0.0223899
$960$	0	0
$961$	−1.77630e10	−0.645633
$962$	0	0
$963$	− 4.50938e9i	− 0.162714i
$964$	0	0
$965$	− 1.48023e10i	− 0.530254i
$966$	0	0
$967$	−1.48161e10	−0.526917	−0.263459	−	0.964671i	$-0.584863\pi$
$967$	−1.48161e10	−0.526917	−0.263459	+	0.964671i	$0.584863\pi$
$968$	0	0
$969$	−3.43976e10	−1.21449
$970$	0	0
$971$	− 2.49709e10i	− 0.875320i	−0.899141	−	0.437660i	$-0.855807\pi$
$971$	− 2.49709e10i	− 0.875320i	0.899141	−	0.437660i	$-0.144193\pi$
$972$	0	0
$973$	2.57467e10i	0.896037i
$974$	0	0
$975$	−1.29917e10	−0.448899
$976$	0	0
$977$	−2.62341e10	−0.899986	−0.449993	−	0.893032i	$-0.648573\pi$
$977$	−2.62341e10	−0.899986	−0.449993	+	0.893032i	$0.648573\pi$
$978$	0	0
$979$	− 1.44845e10i	− 0.493359i
$980$	0	0
$981$	7.01793e9i	0.237338i
$982$	0	0
$983$	3.57411e10	1.20014	0.600068	−	0.799949i	$-0.295140\pi$
$983$	3.57411e10	1.20014	0.600068	+	0.799949i	$0.295140\pi$
$984$	0	0
$985$	−9.63322e10	−3.21177
$986$	0	0
$987$	− 2.16780e10i	− 0.717643i
$988$	0	0
$989$	− 9.40474e9i	− 0.309143i
$990$	0	0
$991$	3.91765e10	1.27870	0.639349	−	0.768917i	$-0.279204\pi$
$991$	3.91765e10	1.27870	0.639349	+	0.768917i	$0.279204\pi$
$992$	0	0
$993$	2.12669e10	0.689257
$994$	0	0
$995$	8.30796e10i	2.67370i
$996$	0	0
$997$	3.56332e8i	0.0113873i	0.999984	+	0.00569366i	$0.00181236\pi$
$997$	3.56332e8i	0.0113873i	−0.999984	+	0.00569366i	$0.998188\pi$
$998$	0	0
$999$	−1.08219e10	−0.343420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.d.f.193.1	✓	16
4.3	odd	2	inner	384.8.d.f.193.9	yes	16
8.3	odd	2	inner	384.8.d.f.193.8	yes	16
8.5	even	2	inner	384.8.d.f.193.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.d.f.193.1	✓	16	1.1	even	1	trivial
384.8.d.f.193.8	yes	16	8.3	odd	2	inner
384.8.d.f.193.9	yes	16	4.3	odd	2	inner
384.8.d.f.193.16	yes	16	8.5	even	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)

\(f(q)\)	\(=\)	\(q-27.0000i q^{3} -484.521i q^{5} -826.966 q^{7} -729.000 q^{9} +O(q^{10})\)	\(q-27.0000i q^{3} -484.521i q^{5} -826.966 q^{7} -729.000 q^{9} +4559.22i q^{11} +3071.92i q^{13} -13082.1 q^{15} +21664.0 q^{17} -58806.6i q^{19} +22328.1i q^{21} -18663.3 q^{23} -156636. q^{25} +19683.0i q^{27} +210373. i q^{29} -98739.9 q^{31} +123099. q^{33} +400682. i q^{35} +549811. i q^{37} +82941.9 q^{39} +559912. q^{41} +503917. i q^{43} +353216. i q^{45} -970883. q^{47} -139671. q^{49} -584927. i q^{51} +1.49503e6i q^{53} +2.20904e6 q^{55} -1.58778e6 q^{57} -415361. i q^{59} -2.20016e6i q^{61} +602858. q^{63} +1.48841e6 q^{65} +1.95346e6i q^{67} +503909. i q^{69} +1.29191e6 q^{71} +3.21010e6 q^{73} +4.22916e6i q^{75} -3.77032e6i q^{77} -1.91140e6 q^{79} +531441. q^{81} +2.28696e6i q^{83} -1.04966e7i q^{85} +5.68008e6 q^{87} -3.17696e6 q^{89} -2.54037e6i q^{91} +2.66598e6i q^{93} -2.84930e7 q^{95} +2.37459e6 q^{97} -3.32367e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 11664 q^{9}+O(q^{10}) \) 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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