LMFDB - Embedded newform 384.9.g.a.127.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,9,Mod(127,384)]

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([1, 0, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("384.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	384.g (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:	$156.433386263$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.11
Character	$\chi$	$=$	384.127
Dual form			384.9.g.a.127.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-46.7654i q^{3} -1118.17 q^{5} +1267.11i q^{7} -2187.00 q^{9} +O(q^{10})$	$q-46.7654i q^{3} -1118.17 q^{5} +1267.11i q^{7} -2187.00 q^{9} -21758.4i q^{11} -39559.4 q^{13} +52291.6i q^{15} +135213. q^{17} -116904. i q^{19} +59256.9 q^{21} -68145.5i q^{23} +859680. q^{25} +102276. i q^{27} +440003. q^{29} +42048.8i q^{31} -1.01754e6 q^{33} -1.41685e6i q^{35} +23739.5 q^{37} +1.85001e6i q^{39} +3.67813e6 q^{41} +1.32272e6i q^{43} +2.44544e6 q^{45} +1.47124e6i q^{47} +4.15923e6 q^{49} -6.32331e6i q^{51} +3.57675e6 q^{53} +2.43296e7i q^{55} -5.46705e6 q^{57} -1.45148e7i q^{59} +7.17147e6 q^{61} -2.77117e6i q^{63} +4.42341e7 q^{65} -1.79084e7i q^{67} -3.18685e6 q^{69} -4.38306e7i q^{71} -9.56930e6 q^{73} -4.02032e7i q^{75} +2.75703e7 q^{77} -6.26180e7i q^{79} +4.78297e6 q^{81} +7.23294e7i q^{83} -1.51192e8 q^{85} -2.05769e7i q^{87} -8.15953e7 q^{89} -5.01261e7i q^{91} +1.96643e6 q^{93} +1.30718e8i q^{95} -7.87759e7 q^{97} +4.75856e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) $ 32 * q - 1344 * q^5 - 69984 * q^9	$ 32 q - 1344 q^{5} - 69984 q^{9} - 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} - 275520 q^{29} + 2421440 q^{37} - 4374720 q^{41} + 2939328 q^{45} - 14219104 q^{49} - 6224448 q^{53} + 3100032 q^{57} - 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} + 154517760 q^{77} + 153055008 q^{81} - 384830848 q^{85} - 182669760 q^{89} + 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) $ 32 * q - 1344 * q^5 - 69984 * q^9 - 114240 * q^13 - 154560 * q^17 + 1791712 * q^25 - 275520 * q^29 + 2421440 * q^37 - 4374720 * q^41 + 2939328 * q^45 - 14219104 * q^49 - 6224448 * q^53 + 3100032 * q^57 - 13005632 * q^61 + 75175296 * q^65 - 85710400 * q^73 + 154517760 * q^77 + 153055008 * q^81 - 384830848 * q^85 - 182669760 * q^89 + 149817600 * q^93 - 149408192 * 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
$3$	− 46.7654i	− 0.577350i
$3$	− 46.7654i	− 0.577350i
$4$	0	0
$5$	−1118.17	−1.78907	−0.894536	−	0.446996i	$-0.852494\pi$
$5$	−1118.17	−1.78907	−0.894536	+	0.446996i	$0.852494\pi$
$6$	0	0
$7$	1267.11i	0.527743i	0.964558	+	0.263871i	$0.0849995\pi$
$7$	1267.11i	0.527743i	−0.964558	+	0.263871i	$0.915001\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	− 21758.4i	− 1.48613i	−0.669220	−	0.743064i	$-0.733372\pi$
$11$	− 21758.4i	− 1.48613i	0.669220	−	0.743064i	$-0.266628\pi$
$12$	0	0
$13$	−39559.4	−1.38508	−0.692542	−	0.721378i	$-0.743509\pi$
$13$	−39559.4	−1.38508	−0.692542	+	0.721378i	$0.743509\pi$
$14$	0	0
$15$	52291.6i	1.03292i
$16$	0	0
$17$	135213.	1.61892	0.809458	−	0.587178i	$-0.199761\pi$
$17$	135213.	1.61892	0.809458	+	0.587178i	$0.199761\pi$
$18$	0	0
$19$	− 116904.i	− 0.897044i	−0.893772	−	0.448522i	$-0.851951\pi$
$19$	− 116904.i	− 0.897044i	0.893772	−	0.448522i	$-0.148049\pi$
$20$	0	0
$21$	59256.9	0.304693
$22$	0	0
$23$	− 68145.5i	− 0.243515i	−0.992560	−	0.121758i	$-0.961147\pi$
$23$	− 68145.5i	− 0.243515i	0.992560	−	0.121758i	$-0.0388530\pi$
$24$	0	0
$25$	859680.	2.20078
$26$	0	0
$27$	102276.i	0.192450i
$28$	0	0
$29$	440003.	0.622105	0.311053	−	0.950393i	$-0.399318\pi$
$29$	440003.	0.622105	0.311053	+	0.950393i	$0.399318\pi$
$30$	0	0
$31$	42048.8i	0.0455309i	0.999741	+	0.0227655i	$0.00724710\pi$
$31$	42048.8i	0.0455309i	−0.999741	+	0.0227655i	$0.992753\pi$
$32$	0	0
$33$	−1.01754e6	−0.858016
$34$	0	0
$35$	− 1.41685e6i	− 0.944170i
$36$	0	0
$37$	23739.5	0.0126667	0.00633337	−	0.999980i	$-0.497984\pi$
$37$	23739.5	0.0126667	0.00633337	+	0.999980i	$0.497984\pi$
$38$	0	0
$39$	1.85001e6i	0.799678i
$40$	0	0
$41$	3.67813e6	1.30164	0.650822	−	0.759230i	$-0.274424\pi$
$41$	3.67813e6	1.30164	0.650822	+	0.759230i	$0.274424\pi$
$42$	0	0
$43$	1.32272e6i	0.386897i	0.981110	+	0.193448i	$0.0619672\pi$
$43$	1.32272e6i	0.386897i	−0.981110	+	0.193448i	$0.938033\pi$
$44$	0	0
$45$	2.44544e6	0.596357
$46$	0	0
$47$	1.47124e6i	0.301504i	0.988572	+	0.150752i	$0.0481694\pi$
$47$	1.47124e6i	0.301504i	−0.988572	+	0.150752i	$0.951831\pi$
$48$	0	0
$49$	4.15923e6	0.721487
$50$	0	0
$51$	− 6.32331e6i	− 0.934681i
$52$	0	0
$53$	3.57675e6	0.453299	0.226650	−	0.973976i	$-0.427223\pi$
$53$	3.57675e6	0.453299	0.226650	+	0.973976i	$0.427223\pi$
$54$	0	0
$55$	2.43296e7i	2.65879i
$56$	0	0
$57$	−5.46705e6	−0.517909
$58$	0	0
$59$	− 1.45148e7i	− 1.19785i	−0.800806	−	0.598924i	$-0.795595\pi$
$59$	− 1.45148e7i	− 1.19785i	0.800806	−	0.598924i	$-0.204405\pi$
$60$	0	0
$61$	7.17147e6	0.517951	0.258976	−	0.965884i	$-0.416615\pi$
$61$	7.17147e6	0.517951	0.258976	+	0.965884i	$0.416615\pi$
$62$	0	0
$63$	− 2.77117e6i	− 0.175914i
$64$	0	0
$65$	4.42341e7	2.47801
$66$	0	0
$67$	− 1.79084e7i	− 0.888705i	−0.895852	−	0.444352i	$-0.853434\pi$
$67$	− 1.79084e7i	− 0.888705i	0.895852	−	0.444352i	$-0.146566\pi$
$68$	0	0
$69$	−3.18685e6	−0.140593
$70$	0	0
$71$	− 4.38306e7i	− 1.72482i	−0.506211	−	0.862410i	$-0.668954\pi$
$71$	− 4.38306e7i	− 1.72482i	0.506211	−	0.862410i	$-0.331046\pi$
$72$	0	0
$73$	−9.56930e6	−0.336968	−0.168484	−	0.985704i	$-0.553887\pi$
$73$	−9.56930e6	−0.336968	−0.168484	+	0.985704i	$0.553887\pi$
$74$	0	0
$75$	− 4.02032e7i	− 1.27062i
$76$	0	0
$77$	2.75703e7	0.784294
$78$	0	0
$79$	− 6.26180e7i	− 1.60765i	−0.594868	−	0.803823i	$-0.702796\pi$
$79$	− 6.26180e7i	− 1.60765i	0.594868	−	0.803823i	$-0.297204\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	7.23294e7i	1.52406i	0.647541	+	0.762030i	$0.275797\pi$
$83$	7.23294e7i	1.52406i	−0.647541	+	0.762030i	$0.724203\pi$
$84$	0	0
$85$	−1.51192e8	−2.89636
$86$	0	0
$87$	− 2.05769e7i	− 0.359173i
$88$	0	0
$89$	−8.15953e7	−1.30048	−0.650242	−	0.759727i	$-0.725333\pi$
$89$	−8.15953e7	−1.30048	−0.650242	+	0.759727i	$0.725333\pi$
$90$	0	0
$91$	− 5.01261e7i	− 0.730968i
$92$	0	0
$93$	1.96643e6	0.0262873
$94$	0	0
$95$	1.30718e8i	1.60488i
$96$	0	0
$97$	−7.87759e7	−0.889828	−0.444914	−	0.895573i	$-0.646766\pi$
$97$	−7.87759e7	−0.889828	−0.444914	+	0.895573i	$0.646766\pi$
$98$	0	0
$99$	4.75856e7i	0.495376i
$100$	0	0
$101$	−1.66849e8	−1.60338	−0.801691	−	0.597739i	$-0.796066\pi$
$101$	−1.66849e8	−1.60338	−0.801691	+	0.597739i	$0.796066\pi$
$102$	0	0
$103$	− 1.61269e8i	− 1.43286i	−0.697661	−	0.716428i	$-0.745776\pi$
$103$	− 1.61269e8i	− 1.43286i	0.697661	−	0.716428i	$-0.254224\pi$
$104$	0	0
$105$	−6.62593e7	−0.545117
$106$	0	0
$107$	2.16933e8i	1.65497i	0.561487	+	0.827485i	$0.310229\pi$
$107$	2.16933e8i	1.65497i	−0.561487	+	0.827485i	$0.689771\pi$
$108$	0	0
$109$	1.52744e8	1.08207	0.541037	−	0.840999i	$-0.318032\pi$
$109$	1.52744e8	1.08207	0.541037	+	0.840999i	$0.318032\pi$
$110$	0	0
$111$	− 1.11019e6i	− 0.00731314i
$112$	0	0
$113$	2.60152e8	1.59556	0.797781	−	0.602948i	$-0.206007\pi$
$113$	2.60152e8	1.59556	0.797781	+	0.602948i	$0.206007\pi$
$114$	0	0
$115$	7.61983e7i	0.435666i
$116$	0	0
$117$	8.65163e7	0.461694
$118$	0	0
$119$	1.71330e8i	0.854371i
$120$	0	0
$121$	−2.59069e8	−1.20858
$122$	0	0
$123$	− 1.72009e8i	− 0.751504i
$124$	0	0
$125$	−5.24483e8	−2.14828
$126$	0	0
$127$	− 2.04882e8i	− 0.787570i	−0.919202	−	0.393785i	$-0.871165\pi$
$127$	− 2.04882e8i	− 0.787570i	0.919202	−	0.393785i	$-0.128835\pi$
$128$	0	0
$129$	6.18576e7	0.223375
$130$	0	0
$131$	− 4.34627e8i	− 1.47581i	−0.674903	−	0.737906i	$-0.735815\pi$
$131$	− 4.34627e8i	− 1.47581i	0.674903	−	0.737906i	$-0.264185\pi$
$132$	0	0
$133$	1.48130e8	0.473409
$134$	0	0
$135$	− 1.14362e8i	− 0.344307i
$136$	0	0
$137$	1.18719e8	0.337007	0.168504	−	0.985701i	$-0.446107\pi$
$137$	1.18719e8	0.337007	0.168504	+	0.985701i	$0.446107\pi$
$138$	0	0
$139$	− 3.96718e8i	− 1.06273i	−0.847143	−	0.531364i	$-0.821679\pi$
$139$	− 3.96718e8i	− 1.06273i	0.847143	−	0.531364i	$-0.178321\pi$
$140$	0	0
$141$	6.88031e7	0.174073
$142$	0	0
$143$	8.60748e8i	2.05841i
$144$	0	0
$145$	−4.91998e8	−1.11299
$146$	0	0
$147$	− 1.94508e8i	− 0.416551i
$148$	0	0
$149$	−7.88759e8	−1.60029	−0.800146	−	0.599805i	$-0.795245\pi$
$149$	−7.88759e8	−1.60029	−0.800146	+	0.599805i	$0.795245\pi$
$150$	0	0
$151$	4.33457e8i	0.833755i	0.908963	+	0.416878i	$0.136876\pi$
$151$	4.33457e8i	0.833755i	−0.908963	+	0.416878i	$0.863124\pi$
$152$	0	0
$153$	−2.95712e8	−0.539639
$154$	0	0
$155$	− 4.70177e7i	− 0.0814581i
$156$	0	0
$157$	7.68635e8	1.26509	0.632545	−	0.774524i	$-0.282010\pi$
$157$	7.68635e8	1.26509	0.632545	+	0.774524i	$0.282010\pi$
$158$	0	0
$159$	− 1.67268e8i	− 0.261712i
$160$	0	0
$161$	8.63479e7	0.128513
$162$	0	0
$163$	1.05341e8i	0.149227i	0.997213	+	0.0746135i	$0.0237723\pi$
$163$	1.05341e8i	0.149227i	−0.997213	+	0.0746135i	$0.976228\pi$
$164$	0	0
$165$	1.13778e9	1.53505
$166$	0	0
$167$	8.03903e8i	1.03357i	0.856117	+	0.516783i	$0.172870\pi$
$167$	8.03903e8i	1.03357i	−0.856117	+	0.516783i	$0.827130\pi$
$168$	0	0
$169$	7.49213e8	0.918456
$170$	0	0
$171$	2.55668e8i	0.299015i
$172$	0	0
$173$	6.80734e8	0.759964	0.379982	−	0.924994i	$-0.375930\pi$
$173$	6.80734e8	0.759964	0.379982	+	0.924994i	$0.375930\pi$
$174$	0	0
$175$	1.08931e9i	1.16145i
$176$	0	0
$177$	−6.78788e8	−0.691578
$178$	0	0
$179$	7.98811e8i	0.778094i	0.921218	+	0.389047i	$0.127196\pi$
$179$	7.98811e8i	0.778094i	−0.921218	+	0.389047i	$0.872804\pi$
$180$	0	0
$181$	−5.18227e8	−0.482843	−0.241421	−	0.970420i	$-0.577614\pi$
$181$	−5.18227e8	−0.482843	−0.241421	+	0.970420i	$0.577614\pi$
$182$	0	0
$183$	− 3.35377e8i	− 0.299039i
$184$	0	0
$185$	−2.65448e7	−0.0226617
$186$	0	0
$187$	− 2.94203e9i	− 2.40592i
$188$	0	0
$189$	−1.29595e8	−0.101564
$190$	0	0
$191$	2.60139e9i	1.95466i	0.211717	+	0.977331i	$0.432095\pi$
$191$	2.60139e9i	1.95466i	−0.211717	+	0.977331i	$0.567905\pi$
$192$	0	0
$193$	−2.10211e9	−1.51505	−0.757524	−	0.652808i	$-0.773591\pi$
$193$	−2.10211e9	−1.51505	−0.757524	+	0.652808i	$0.773591\pi$
$194$	0	0
$195$	− 2.06862e9i	− 1.43068i
$196$	0	0
$197$	5.64262e8	0.374642	0.187321	−	0.982299i	$-0.440020\pi$
$197$	5.64262e8	0.374642	0.187321	+	0.982299i	$0.440020\pi$
$198$	0	0
$199$	5.55443e8i	0.354182i	0.984194	+	0.177091i	$0.0566688\pi$
$199$	5.55443e8i	0.354182i	−0.984194	+	0.177091i	$0.943331\pi$
$200$	0	0
$201$	−8.37493e8	−0.513094
$202$	0	0
$203$	5.57533e8i	0.328312i
$204$	0	0
$205$	−4.11278e9	−2.32874
$206$	0	0
$207$	1.49034e8i	0.0811717i
$208$	0	0
$209$	−2.54364e9	−1.33312
$210$	0	0
$211$	7.13300e8i	0.359867i	0.983679	+	0.179934i	$0.0575883\pi$
$211$	7.13300e8i	0.359867i	−0.983679	+	0.179934i	$0.942412\pi$
$212$	0	0
$213$	−2.04975e9	−0.995825
$214$	0	0
$215$	− 1.47903e9i	− 0.692186i
$216$	0	0
$217$	−5.32804e7	−0.0240286
$218$	0	0
$219$	4.47512e8i	0.194549i
$220$	0	0
$221$	−5.34896e9	−2.24233
$222$	0	0
$223$	− 9.82818e7i	− 0.0397423i	−0.999803	−	0.0198712i	$-0.993674\pi$
$223$	− 9.82818e7i	− 0.0397423i	0.999803	−	0.0198712i	$-0.00632561\pi$
$224$	0	0
$225$	−1.88012e9	−0.733593
$226$	0	0
$227$	2.42754e9i	0.914244i	0.889404	+	0.457122i	$0.151120\pi$
$227$	2.42754e9i	0.914244i	−0.889404	+	0.457122i	$0.848880\pi$
$228$	0	0
$229$	−1.14656e8	−0.0416923	−0.0208462	−	0.999783i	$-0.506636\pi$
$229$	−1.14656e8	−0.0416923	−0.0208462	+	0.999783i	$0.506636\pi$
$230$	0	0
$231$	− 1.28934e9i	− 0.452812i
$232$	0	0
$233$	−2.45565e8	−0.0833189	−0.0416594	−	0.999132i	$-0.513264\pi$
$233$	−2.45565e8	−0.0833189	−0.0416594	+	0.999132i	$0.513264\pi$
$234$	0	0
$235$	− 1.64510e9i	− 0.539412i
$236$	0	0
$237$	−2.92835e9	−0.928175
$238$	0	0
$239$	− 4.51412e9i	− 1.38351i	−0.722133	−	0.691754i	$-0.756838\pi$
$239$	− 4.51412e9i	− 1.38351i	0.722133	−	0.691754i	$-0.243162\pi$
$240$	0	0
$241$	−4.00102e9	−1.18605	−0.593024	−	0.805185i	$-0.702066\pi$
$241$	−4.00102e9	−1.18605	−0.593024	+	0.805185i	$0.702066\pi$
$242$	0	0
$243$	− 2.23677e8i	− 0.0641500i
$244$	0	0
$245$	−4.65073e9	−1.29079
$246$	0	0
$247$	4.62464e9i	1.24248i
$248$	0	0
$249$	3.38251e9	0.879917
$250$	0	0
$251$	− 5.61278e9i	− 1.41411i	−0.707158	−	0.707055i	$-0.750023\pi$
$251$	− 5.61278e9i	− 1.41411i	0.707158	−	0.707055i	$-0.249977\pi$
$252$	0	0
$253$	−1.48274e9	−0.361895
$254$	0	0
$255$	7.07053e9i	1.67221i
$256$	0	0
$257$	−1.97676e9	−0.453129	−0.226565	−	0.973996i	$-0.572749\pi$
$257$	−1.97676e9	−0.453129	−0.226565	+	0.973996i	$0.572749\pi$
$258$	0	0
$259$	3.00806e7i	0.00668478i
$260$	0	0
$261$	−9.62287e8	−0.207368
$262$	0	0
$263$	4.00767e9i	0.837662i	0.908064	+	0.418831i	$0.137560\pi$
$263$	4.00767e9i	0.837662i	−0.908064	+	0.418831i	$0.862440\pi$
$264$	0	0
$265$	−3.99941e9	−0.810985
$266$	0	0
$267$	3.81583e9i	0.750835i
$268$	0	0
$269$	8.14052e9	1.55469	0.777344	−	0.629076i	$-0.216567\pi$
$269$	8.14052e9	1.55469	0.777344	+	0.629076i	$0.216567\pi$
$270$	0	0
$271$	− 5.83113e9i	− 1.08112i	−0.841304	−	0.540562i	$-0.818212\pi$
$271$	− 5.83113e9i	− 1.08112i	0.841304	−	0.540562i	$-0.181788\pi$
$272$	0	0
$273$	−2.34417e9	−0.422025
$274$	0	0
$275$	− 1.87053e10i	− 3.27064i
$276$	0	0
$277$	−8.67294e8	−0.147315	−0.0736576	−	0.997284i	$-0.523467\pi$
$277$	−8.67294e8	−0.147315	−0.0736576	+	0.997284i	$0.523467\pi$
$278$	0	0
$279$	− 9.19606e7i	− 0.0151770i
$280$	0	0
$281$	−7.70559e9	−1.23589	−0.617946	−	0.786220i	$-0.712035\pi$
$281$	−7.70559e9	−1.23589	−0.617946	+	0.786220i	$0.712035\pi$
$282$	0	0
$283$	− 1.09214e10i	− 1.70268i	−0.524610	−	0.851342i	$-0.675789\pi$
$283$	− 1.09214e10i	− 1.70268i	0.524610	−	0.851342i	$-0.324211\pi$
$284$	0	0
$285$	6.11309e9	0.926576
$286$	0	0
$287$	4.66060e9i	0.686933i
$288$	0	0
$289$	1.13069e10	1.62089
$290$	0	0
$291$	3.68398e9i	0.513743i
$292$	0	0
$293$	−7.75593e9	−1.05236	−0.526179	−	0.850374i	$-0.676376\pi$
$293$	−7.75593e9	−1.05236	−0.526179	+	0.850374i	$0.676376\pi$
$294$	0	0
$295$	1.62300e10i	2.14304i
$296$	0	0
$297$	2.22536e9	0.286005
$298$	0	0
$299$	2.69579e9i	0.337289i
$300$	0	0
$301$	−1.67604e9	−0.204182
$302$	0	0
$303$	7.80274e9i	0.925713i
$304$	0	0
$305$	−8.01893e9	−0.926653
$306$	0	0
$307$	− 1.05748e10i	− 1.19047i	−0.803552	−	0.595234i	$-0.797059\pi$
$307$	− 1.05748e10i	− 1.19047i	0.803552	−	0.595234i	$-0.202941\pi$
$308$	0	0
$309$	−7.54181e9	−0.827259
$310$	0	0
$311$	− 3.80297e9i	− 0.406519i	−0.979125	−	0.203260i	$-0.934846\pi$
$311$	− 3.80297e9i	− 0.406519i	0.979125	−	0.203260i	$-0.0651536\pi$
$312$	0	0
$313$	3.27745e9	0.341475	0.170737	−	0.985317i	$-0.445385\pi$
$313$	3.27745e9	0.341475	0.170737	+	0.985317i	$0.445385\pi$
$314$	0	0
$315$	3.09864e9i	0.314723i
$316$	0	0
$317$	1.47128e10	1.45700	0.728498	−	0.685048i	$-0.240219\pi$
$317$	1.47128e10	1.45700	0.728498	+	0.685048i	$0.240219\pi$
$318$	0	0
$319$	− 9.57377e9i	− 0.924528i
$320$	0	0
$321$	1.01449e10	0.955498
$322$	0	0
$323$	− 1.58070e10i	− 1.45224i
$324$	0	0
$325$	−3.40084e10	−3.04826
$326$	0	0
$327$	− 7.14311e9i	− 0.624736i
$328$	0	0
$329$	−1.86423e9	−0.159116
$330$	0	0
$331$	8.01925e9i	0.668070i	0.942561	+	0.334035i	$0.108410\pi$
$331$	8.01925e9i	0.668070i	−0.942561	+	0.334035i	$0.891590\pi$
$332$	0	0
$333$	−5.19183e7	−0.00422224
$334$	0	0
$335$	2.00246e10i	1.58996i
$336$	0	0
$337$	2.21506e9	0.171738	0.0858690	−	0.996306i	$-0.472633\pi$
$337$	2.21506e9	0.171738	0.0858690	+	0.996306i	$0.472633\pi$
$338$	0	0
$339$	− 1.21661e10i	− 0.921198i
$340$	0	0
$341$	9.14913e8	0.0676648
$342$	0	0
$343$	1.25748e10i	0.908503i
$344$	0	0
$345$	3.56344e9	0.251532
$346$	0	0
$347$	− 1.29354e10i	− 0.892201i	−0.894983	−	0.446100i	$-0.852812\pi$
$347$	− 1.29354e10i	− 0.892201i	0.894983	−	0.446100i	$-0.147188\pi$
$348$	0	0
$349$	8.30381e9	0.559726	0.279863	−	0.960040i	$-0.409711\pi$
$349$	8.30381e9	0.559726	0.279863	+	0.960040i	$0.409711\pi$
$350$	0	0
$351$	− 4.04597e9i	− 0.266559i
$352$	0	0
$353$	−2.37013e10	−1.52642	−0.763209	−	0.646152i	$-0.776377\pi$
$353$	−2.37013e10	−1.52642	−0.763209	+	0.646152i	$0.776377\pi$
$354$	0	0
$355$	4.90100e10i	3.08583i
$356$	0	0
$357$	8.01233e9	0.493272
$358$	0	0
$359$	− 1.97017e10i	− 1.18611i	−0.805162	−	0.593055i	$-0.797922\pi$
$359$	− 1.97017e10i	− 1.18611i	0.805162	−	0.593055i	$-0.202078\pi$
$360$	0	0
$361$	3.31709e9	0.195312
$362$	0	0
$363$	1.21155e10i	0.697772i
$364$	0	0
$365$	1.07001e10	0.602860
$366$	0	0
$367$	1.03987e10i	0.573210i	0.958049	+	0.286605i	$0.0925268\pi$
$367$	1.03987e10i	0.573210i	−0.958049	+	0.286605i	$0.907473\pi$
$368$	0	0
$369$	−8.04408e9	−0.433881
$370$	0	0
$371$	4.53214e9i	0.239225i
$372$	0	0
$373$	1.09119e10	0.563721	0.281860	−	0.959455i	$-0.409049\pi$
$373$	1.09119e10	0.563721	0.281860	+	0.959455i	$0.409049\pi$
$374$	0	0
$375$	2.45276e10i	1.24031i
$376$	0	0
$377$	−1.74062e10	−0.861668
$378$	0	0
$379$	− 3.01464e10i	− 1.46109i	−0.682862	−	0.730547i	$-0.739265\pi$
$379$	− 3.01464e10i	− 1.46109i	0.682862	−	0.730547i	$-0.260735\pi$
$380$	0	0
$381$	−9.58139e9	−0.454704
$382$	0	0
$383$	2.09774e9i	0.0974892i	0.998811	+	0.0487446i	$0.0155220\pi$
$383$	2.09774e9i	0.0974892i	−0.998811	+	0.0487446i	$0.984478\pi$
$384$	0	0
$385$	−3.08283e10	−1.40316
$386$	0	0
$387$	− 2.89280e9i	− 0.128966i
$388$	0	0
$389$	−6.03957e8	−0.0263759	−0.0131880	−	0.999913i	$-0.504198\pi$
$389$	−6.03957e8	−0.0263759	−0.0131880	+	0.999913i	$0.504198\pi$
$390$	0	0
$391$	− 9.21419e9i	− 0.394230i
$392$	0	0
$393$	−2.03255e10	−0.852060
$394$	0	0
$395$	7.00175e10i	2.87620i
$396$	0	0
$397$	−1.62856e10	−0.655603	−0.327801	−	0.944747i	$-0.606308\pi$
$397$	−1.62856e10	−0.655603	−0.327801	+	0.944747i	$0.606308\pi$
$398$	0	0
$399$	− 6.92735e9i	− 0.273323i
$400$	0	0
$401$	−1.32909e10	−0.514016	−0.257008	−	0.966409i	$-0.582737\pi$
$401$	−1.32909e10	−0.514016	−0.257008	+	0.966409i	$0.582737\pi$
$402$	0	0
$403$	− 1.66342e9i	− 0.0630641i
$404$	0	0
$405$	−5.34817e9	−0.198786
$406$	0	0
$407$	− 5.16533e8i	− 0.0188244i
$408$	0	0
$409$	4.03255e10	1.44107	0.720537	−	0.693417i	$-0.243896\pi$
$409$	4.03255e10	1.44107	0.720537	+	0.693417i	$0.243896\pi$
$410$	0	0
$411$	− 5.55195e9i	− 0.194571i
$412$	0	0
$413$	1.83918e10	0.632156
$414$	0	0
$415$	− 8.08765e10i	− 2.72665i
$416$	0	0
$417$	−1.85527e10	−0.613567
$418$	0	0
$419$	3.42893e10i	1.11251i	0.831012	+	0.556254i	$0.187762\pi$
$419$	3.42893e10i	1.11251i	−0.831012	+	0.556254i	$0.812238\pi$
$420$	0	0
$421$	−4.80086e10	−1.52824	−0.764119	−	0.645075i	$-0.776826\pi$
$421$	−4.80086e10	−1.52824	−0.764119	+	0.645075i	$0.776826\pi$
$422$	0	0
$423$	− 3.21760e9i	− 0.100501i
$424$	0	0
$425$	1.16240e11	3.56288
$426$	0	0
$427$	9.08705e9i	0.273345i
$428$	0	0
$429$	4.02532e10	1.18842
$430$	0	0
$431$	− 6.06838e10i	− 1.75859i	−0.476281	−	0.879293i	$-0.658015\pi$
$431$	− 6.06838e10i	− 1.75859i	0.476281	−	0.879293i	$-0.341985\pi$
$432$	0	0
$433$	−3.62460e10	−1.03112	−0.515559	−	0.856854i	$-0.672416\pi$
$433$	−3.62460e10	−1.03112	−0.515559	+	0.856854i	$0.672416\pi$
$434$	0	0
$435$	2.30085e10i	0.642586i
$436$	0	0
$437$	−7.96646e9	−0.218444
$438$	0	0
$439$	− 3.78921e10i	− 1.02021i	−0.860112	−	0.510106i	$-0.829606\pi$
$439$	− 3.78921e10i	− 1.02021i	0.860112	−	0.510106i	$-0.170394\pi$
$440$	0	0
$441$	−9.09624e9	−0.240496
$442$	0	0
$443$	1.15468e10i	0.299811i	0.988700	+	0.149905i	$0.0478969\pi$
$443$	1.15468e10i	0.299811i	−0.988700	+	0.149905i	$0.952103\pi$
$444$	0	0
$445$	9.12374e10	2.32666
$446$	0	0
$447$	3.68866e10i	0.923929i
$448$	0	0
$449$	1.60943e10	0.395992	0.197996	−	0.980203i	$-0.436557\pi$
$449$	1.60943e10	0.395992	0.197996	+	0.980203i	$0.436557\pi$
$450$	0	0
$451$	− 8.00303e10i	− 1.93441i
$452$	0	0
$453$	2.02708e10	0.481369
$454$	0	0
$455$	5.60495e10i	1.30775i
$456$	0	0
$457$	4.21553e10	0.966469	0.483234	−	0.875491i	$-0.339462\pi$
$457$	4.21553e10	0.966469	0.483234	+	0.875491i	$0.339462\pi$
$458$	0	0
$459$	1.38291e10i	0.311560i
$460$	0	0
$461$	−7.82467e10	−1.73246	−0.866229	−	0.499648i	$-0.833463\pi$
$461$	−7.82467e10	−1.73246	−0.866229	+	0.499648i	$0.833463\pi$
$462$	0	0
$463$	4.24716e10i	0.924219i	0.886823	+	0.462110i	$0.152907\pi$
$463$	4.24716e10i	0.924219i	−0.886823	+	0.462110i	$0.847093\pi$
$464$	0	0
$465$	−2.19880e9	−0.0470298
$466$	0	0
$467$	− 2.15518e10i	− 0.453122i	−0.973997	−	0.226561i	$-0.927252\pi$
$467$	− 2.15518e10i	− 0.453122i	0.973997	−	0.226561i	$-0.0727483\pi$
$468$	0	0
$469$	2.26919e10	0.469008
$470$	0	0
$471$	− 3.59455e10i	− 0.730400i
$472$	0	0
$473$	2.87803e10	0.574978
$474$	0	0
$475$	− 1.00500e11i	− 1.97420i
$476$	0	0
$477$	−7.82235e9	−0.151100
$478$	0	0
$479$	7.02539e10i	1.33453i	0.744821	+	0.667265i	$0.232535\pi$
$479$	7.02539e10i	1.33453i	−0.744821	+	0.667265i	$0.767465\pi$
$480$	0	0
$481$	−9.39119e8	−0.0175445
$482$	0	0
$483$	− 4.03809e9i	− 0.0741972i
$484$	0	0
$485$	8.80848e10	1.59197
$486$	0	0
$487$	− 3.47879e10i	− 0.618462i	−0.950987	−	0.309231i	$-0.899928\pi$
$487$	− 3.47879e10i	− 0.618462i	0.950987	−	0.309231i	$-0.100072\pi$
$488$	0	0
$489$	4.92632e9	0.0861563
$490$	0	0
$491$	4.77077e10i	0.820847i	0.911895	+	0.410424i	$0.134619\pi$
$491$	4.77077e10i	0.820847i	−0.911895	+	0.410424i	$0.865381\pi$
$492$	0	0
$493$	5.94944e10	1.00714
$494$	0	0
$495$	− 5.32088e10i	− 0.886263i
$496$	0	0
$497$	5.55382e10	0.910261
$498$	0	0
$499$	3.69486e10i	0.595932i	0.954577	+	0.297966i	$0.0963081\pi$
$499$	3.69486e10i	0.595932i	−0.954577	+	0.297966i	$0.903692\pi$
$500$	0	0
$501$	3.75948e10	0.596729
$502$	0	0
$503$	5.22919e10i	0.816888i	0.912783	+	0.408444i	$0.133928\pi$
$503$	5.22919e10i	0.816888i	−0.912783	+	0.408444i	$0.866072\pi$
$504$	0	0
$505$	1.86565e11	2.86857
$506$	0	0
$507$	− 3.50372e10i	− 0.530271i
$508$	0	0
$509$	1.01697e11	1.51508	0.757539	−	0.652790i	$-0.226402\pi$
$509$	1.01697e11	1.51508	0.757539	+	0.652790i	$0.226402\pi$
$510$	0	0
$511$	− 1.21254e10i	− 0.177833i
$512$	0	0
$513$	1.19564e10	0.172636
$514$	0	0
$515$	1.80326e11i	2.56348i
$516$	0	0
$517$	3.20119e10	0.448073
$518$	0	0
$519$	− 3.18348e10i	− 0.438765i
$520$	0	0
$521$	5.82969e10	0.791214	0.395607	−	0.918420i	$-0.370534\pi$
$521$	5.82969e10	0.791214	0.395607	+	0.918420i	$0.370534\pi$
$522$	0	0
$523$	1.09455e11i	1.46295i	0.681867	+	0.731476i	$0.261168\pi$
$523$	1.09455e11i	1.46295i	−0.681867	+	0.731476i	$0.738832\pi$
$524$	0	0
$525$	5.09420e10	0.670561
$526$	0	0
$527$	5.68556e9i	0.0737107i
$528$	0	0
$529$	7.36672e10	0.940700
$530$	0	0
$531$	3.17438e10i	0.399282i
$532$	0	0
$533$	−1.45505e11	−1.80289
$534$	0	0
$535$	− 2.42568e11i	− 2.96086i
$536$	0	0
$537$	3.73567e10	0.449233
$538$	0	0
$539$	− 9.04982e10i	− 1.07222i
$540$	0	0
$541$	−2.46413e10	−0.287657	−0.143829	−	0.989603i	$-0.545941\pi$
$541$	−2.46413e10	−0.287657	−0.143829	+	0.989603i	$0.545941\pi$
$542$	0	0
$543$	2.42351e10i	0.278769i
$544$	0	0
$545$	−1.70793e11	−1.93591
$546$	0	0
$547$	− 8.79139e10i	− 0.981993i	−0.871162	−	0.490996i	$-0.836633\pi$
$547$	− 8.79139e10i	− 0.981993i	0.871162	−	0.490996i	$-0.163367\pi$
$548$	0	0
$549$	−1.56840e10	−0.172650
$550$	0	0
$551$	− 5.14380e10i	− 0.558056i
$552$	0	0
$553$	7.93439e10	0.848424
$554$	0	0
$555$	1.24138e9i	0.0130837i
$556$	0	0
$557$	6.59656e10	0.685325	0.342662	−	0.939459i	$-0.388671\pi$
$557$	6.59656e10	0.685325	0.342662	+	0.939459i	$0.388671\pi$
$558$	0	0
$559$	− 5.23261e10i	− 0.535884i
$560$	0	0
$561$	−1.37585e11	−1.38906
$562$	0	0
$563$	1.55327e11i	1.54601i	0.634400	+	0.773005i	$0.281247\pi$
$563$	1.55327e11i	1.54601i	−0.634400	+	0.773005i	$0.718753\pi$
$564$	0	0
$565$	−2.90894e11	−2.85457
$566$	0	0
$567$	6.06055e9i	0.0586381i
$568$	0	0
$569$	1.27518e10	0.121652	0.0608262	−	0.998148i	$-0.480626\pi$
$569$	1.27518e10	0.121652	0.0608262	+	0.998148i	$0.480626\pi$
$570$	0	0
$571$	1.11266e11i	1.04669i	0.852120	+	0.523347i	$0.175317\pi$
$571$	1.11266e11i	1.04669i	−0.852120	+	0.523347i	$0.824683\pi$
$572$	0	0
$573$	1.21655e11	1.12852
$574$	0	0
$575$	− 5.85833e10i	− 0.535923i
$576$	0	0
$577$	6.06159e10	0.546869	0.273434	−	0.961891i	$-0.411840\pi$
$577$	6.06159e10	0.546869	0.273434	+	0.961891i	$0.411840\pi$
$578$	0	0
$579$	9.83060e10i	0.874713i
$580$	0	0
$581$	−9.16493e10	−0.804312
$582$	0	0
$583$	− 7.78243e10i	− 0.673660i
$584$	0	0
$585$	−9.67400e10	−0.826005
$586$	0	0
$587$	− 8.89189e10i	− 0.748931i	−0.927241	−	0.374465i	$-0.877826\pi$
$587$	− 8.89189e10i	− 0.748931i	0.927241	−	0.374465i	$-0.122174\pi$
$588$	0	0
$589$	4.91565e9	0.0408432
$590$	0	0
$591$	− 2.63879e10i	− 0.216299i
$592$	0	0
$593$	−1.24694e11	−1.00839	−0.504195	−	0.863590i	$-0.668211\pi$
$593$	−1.24694e11	−1.00839	−0.504195	+	0.863590i	$0.668211\pi$
$594$	0	0
$595$	− 1.91577e11i	− 1.52853i
$596$	0	0
$597$	2.59755e10	0.204487
$598$	0	0
$599$	− 4.14040e10i	− 0.321614i	−0.986986	−	0.160807i	$-0.948590\pi$
$599$	− 4.14040e10i	− 0.321614i	0.986986	−	0.160807i	$-0.0514096\pi$
$600$	0	0
$601$	−5.99811e10	−0.459744	−0.229872	−	0.973221i	$-0.573831\pi$
$601$	−5.99811e10	−0.459744	−0.229872	+	0.973221i	$0.573831\pi$
$602$	0	0
$603$	3.91657e10i	0.296235i
$604$	0	0
$605$	2.89683e11	2.16223
$606$	0	0
$607$	− 1.31345e11i	− 0.967514i	−0.875202	−	0.483757i	$-0.839272\pi$
$607$	− 1.31345e11i	− 0.967514i	0.875202	−	0.483757i	$-0.160728\pi$
$608$	0	0
$609$	2.60732e10	0.189551
$610$	0	0
$611$	− 5.82014e10i	− 0.417608i
$612$	0	0
$613$	7.62677e10	0.540130	0.270065	−	0.962842i	$-0.412955\pi$
$613$	7.62677e10	0.540130	0.270065	+	0.962842i	$0.412955\pi$
$614$	0	0
$615$	1.92336e11i	1.34450i
$616$	0	0
$617$	−7.06605e10	−0.487569	−0.243785	−	0.969829i	$-0.578389\pi$
$617$	−7.06605e10	−0.487569	−0.243785	+	0.969829i	$0.578389\pi$
$618$	0	0
$619$	1.06019e11i	0.722139i	0.932539	+	0.361070i	$0.117588\pi$
$619$	1.06019e11i	0.722139i	−0.932539	+	0.361070i	$0.882412\pi$
$620$	0	0
$621$	6.96964e9	0.0468645
$622$	0	0
$623$	− 1.03390e11i	− 0.686321i
$624$	0	0
$625$	2.50649e11	1.64265
$626$	0	0
$627$	1.18954e11i	0.769679i
$628$	0	0
$629$	3.20990e9	0.0205064
$630$	0	0
$631$	− 7.84803e10i	− 0.495043i	−0.968882	−	0.247522i	$-0.920384\pi$
$631$	− 7.84803e10i	− 0.495043i	0.968882	−	0.247522i	$-0.0796162\pi$
$632$	0	0
$633$	3.33577e10	0.207770
$634$	0	0
$635$	2.29093e11i	1.40902i
$636$	0	0
$637$	−1.64537e11	−0.999320
$638$	0	0
$639$	9.58575e10i	0.574940i
$640$	0	0
$641$	1.72460e11	1.02155	0.510773	−	0.859716i	$-0.329359\pi$
$641$	1.72460e11	1.02155	0.510773	+	0.859716i	$0.329359\pi$
$642$	0	0
$643$	8.96566e10i	0.524491i	0.965001	+	0.262246i	$0.0844630\pi$
$643$	8.96566e10i	0.524491i	−0.965001	+	0.262246i	$0.915537\pi$
$644$	0	0
$645$	−6.91674e10	−0.399634
$646$	0	0
$647$	4.36971e10i	0.249365i	0.992197	+	0.124682i	$0.0397912\pi$
$647$	4.36971e10i	0.249365i	−0.992197	+	0.124682i	$0.960209\pi$
$648$	0	0
$649$	−3.15818e11	−1.78015
$650$	0	0
$651$	2.49168e9i	0.0138729i
$652$	0	0
$653$	3.54464e11	1.94948	0.974741	−	0.223338i	$-0.0716953\pi$
$653$	3.54464e11	1.94948	0.974741	+	0.223338i	$0.0716953\pi$
$654$	0	0
$655$	4.85986e11i	2.64033i
$656$	0	0
$657$	2.09281e10	0.112323
$658$	0	0
$659$	− 1.26353e11i	− 0.669954i	−0.942226	−	0.334977i	$-0.891272\pi$
$659$	− 1.26353e11i	− 0.669954i	0.942226	−	0.334977i	$-0.108728\pi$
$660$	0	0
$661$	−1.65621e11	−0.867581	−0.433791	−	0.901014i	$-0.642824\pi$
$661$	−1.65621e11	−0.867581	−0.433791	+	0.901014i	$0.642824\pi$
$662$	0	0
$663$	2.50146e11i	1.29461i
$664$	0	0
$665$	−1.65635e11	−0.846963
$666$	0	0
$667$	− 2.99842e10i	− 0.151492i
$668$	0	0
$669$	−4.59618e9	−0.0229453
$670$	0	0
$671$	− 1.56040e11i	− 0.769742i
$672$	0	0
$673$	−2.70034e11	−1.31631	−0.658154	−	0.752883i	$-0.728662\pi$
$673$	−2.70034e11	−1.31631	−0.658154	+	0.752883i	$0.728662\pi$
$674$	0	0
$675$	8.79245e10i	0.423540i
$676$	0	0
$677$	9.29389e10	0.442428	0.221214	−	0.975225i	$-0.428998\pi$
$677$	9.29389e10	0.442428	0.221214	+	0.975225i	$0.428998\pi$
$678$	0	0
$679$	− 9.98177e10i	− 0.469601i
$680$	0	0
$681$	1.13525e11	0.527839
$682$	0	0
$683$	3.19850e10i	0.146982i	0.997296	+	0.0734910i	$0.0234140\pi$
$683$	3.19850e10i	0.146982i	−0.997296	+	0.0734910i	$0.976586\pi$
$684$	0	0
$685$	−1.32748e11	−0.602930
$686$	0	0
$687$	5.36195e9i	0.0240711i
$688$	0	0
$689$	−1.41494e11	−0.627857
$690$	0	0
$691$	− 1.28544e11i	− 0.563820i	−0.959441	−	0.281910i	$-0.909032\pi$
$691$	− 1.28544e11i	− 0.563820i	0.959441	−	0.281910i	$-0.0909680\pi$
$692$	0	0
$693$	−6.02963e10	−0.261431
$694$	0	0
$695$	4.43598e11i	1.90130i
$696$	0	0
$697$	4.97333e11	2.10725
$698$	0	0
$699$	1.14840e10i	0.0481042i
$700$	0	0
$701$	1.25719e11	0.520628	0.260314	−	0.965524i	$-0.416174\pi$
$701$	1.25719e11	0.520628	0.260314	+	0.965524i	$0.416174\pi$
$702$	0	0
$703$	− 2.77523e9i	− 0.0113626i
$704$	0	0
$705$	−7.69336e10	−0.311430
$706$	0	0
$707$	− 2.11416e11i	− 0.846174i
$708$	0	0
$709$	−1.89275e11	−0.749044	−0.374522	−	0.927218i	$-0.622193\pi$
$709$	−1.89275e11	−0.749044	−0.374522	+	0.927218i	$0.622193\pi$
$710$	0	0
$711$	1.36945e11i	0.535882i
$712$	0	0
$713$	2.86543e9	0.0110875
$714$	0	0
$715$	− 9.62463e11i	− 3.68265i
$716$	0	0
$717$	−2.11105e11	−0.798768
$718$	0	0
$719$	− 5.59792e10i	− 0.209465i	−0.994500	−	0.104732i	$-0.966601\pi$
$719$	− 5.59792e10i	− 0.209465i	0.994500	−	0.104732i	$-0.0333986\pi$
$720$	0	0
$721$	2.04346e11	0.756179
$722$	0	0
$723$	1.87109e11i	0.684765i
$724$	0	0
$725$	3.78262e11	1.36912
$726$	0	0
$727$	− 1.70674e11i	− 0.610984i	−0.952195	−	0.305492i	$-0.901179\pi$
$727$	− 1.70674e11i	− 0.610984i	0.952195	−	0.305492i	$-0.0988210\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	1.78850e11i	0.626353i
$732$	0	0
$733$	−6.36088e10	−0.220344	−0.110172	−	0.993913i	$-0.535140\pi$
$733$	−6.36088e10	−0.220344	−0.110172	+	0.993913i	$0.535140\pi$
$734$	0	0
$735$	2.17493e11i	0.745240i
$736$	0	0
$737$	−3.89658e11	−1.32073
$738$	0	0
$739$	− 3.87743e11i	− 1.30007i	−0.759905	−	0.650034i	$-0.774755\pi$
$739$	− 3.87743e11i	− 1.30007i	0.759905	−	0.650034i	$-0.225245\pi$
$740$	0	0
$741$	2.16273e11	0.717347
$742$	0	0
$743$	1.56732e11i	0.514284i	0.966374	+	0.257142i	$0.0827809\pi$
$743$	1.56732e11i	0.514284i	−0.966374	+	0.257142i	$0.917219\pi$
$744$	0	0
$745$	8.81967e11	2.86304
$746$	0	0
$747$	− 1.58184e11i	− 0.508020i
$748$	0	0
$749$	−2.74878e11	−0.873399
$750$	0	0
$751$	− 5.52528e11i	− 1.73698i	−0.495707	−	0.868490i	$-0.665091\pi$
$751$	− 5.52528e11i	− 1.73698i	0.495707	−	0.868490i	$-0.334909\pi$
$752$	0	0
$753$	−2.62484e11	−0.816437
$754$	0	0
$755$	− 4.84679e11i	− 1.49165i
$756$	0	0
$757$	−2.78318e11	−0.847535	−0.423767	−	0.905771i	$-0.639293\pi$
$757$	−2.78318e11	−0.847535	−0.423767	+	0.905771i	$0.639293\pi$
$758$	0	0
$759$	6.93408e10i	0.208940i
$760$	0	0
$761$	−1.56451e11	−0.466488	−0.233244	−	0.972418i	$-0.574934\pi$
$761$	−1.56451e11	−0.466488	−0.233244	+	0.972418i	$0.574934\pi$
$762$	0	0
$763$	1.93543e11i	0.571057i
$764$	0	0
$765$	3.30656e11	0.965452
$766$	0	0
$767$	5.74194e11i	1.65912i
$768$	0	0
$769$	−1.10347e11	−0.315541	−0.157770	−	0.987476i	$-0.550431\pi$
$769$	−1.10347e11	−0.315541	−0.157770	+	0.987476i	$0.550431\pi$
$770$	0	0
$771$	9.24441e10i	0.261614i
$772$	0	0
$773$	−3.11984e10	−0.0873804	−0.0436902	−	0.999045i	$-0.513911\pi$
$773$	−3.11984e10	−0.0873804	−0.0436902	+	0.999045i	$0.513911\pi$
$774$	0	0
$775$	3.61485e10i	0.100204i
$776$	0	0
$777$	1.40673e9	0.00385946
$778$	0	0
$779$	− 4.29988e11i	− 1.16763i
$780$	0	0
$781$	−9.53683e11	−2.56330
$782$	0	0
$783$	4.50017e10i	0.119724i
$784$	0	0
$785$	−8.59464e11	−2.26334
$786$	0	0
$787$	2.03839e11i	0.531359i	0.964061	+	0.265679i	$0.0855963\pi$
$787$	2.03839e11i	0.531359i	−0.964061	+	0.265679i	$0.914404\pi$
$788$	0	0
$789$	1.87420e11	0.483624
$790$	0	0
$791$	3.29642e11i	0.842046i
$792$	0	0
$793$	−2.83699e11	−0.717406
$794$	0	0
$795$	1.87034e11i	0.468222i
$796$	0	0
$797$	−7.25099e11	−1.79707	−0.898533	−	0.438907i	$-0.855366\pi$
$797$	−7.25099e11	−1.79707	−0.898533	+	0.438907i	$0.855366\pi$
$798$	0	0
$799$	1.98932e11i	0.488109i
$800$	0	0
$801$	1.78449e11	0.433495
$802$	0	0
$803$	2.08213e11i	0.500778i
$804$	0	0
$805$	−9.65517e10	−0.229920
$806$	0	0
$807$	− 3.80695e11i	− 0.897599i
$808$	0	0
$809$	−7.92894e11	−1.85106	−0.925531	−	0.378673i	$-0.876381\pi$
$809$	−7.92894e11	−1.85106	−0.925531	+	0.378673i	$0.876381\pi$
$810$	0	0
$811$	− 6.06750e10i	− 0.140258i	−0.997538	−	0.0701288i	$-0.977659\pi$
$811$	− 6.06750e10i	− 0.140258i	0.997538	−	0.0701288i	$-0.0223410\pi$
$812$	0	0
$813$	−2.72695e11	−0.624187
$814$	0	0
$815$	− 1.17789e11i	− 0.266978i
$816$	0	0
$817$	1.54631e11	0.347064
$818$	0	0
$819$	1.09626e11i	0.243656i
$820$	0	0
$821$	2.37908e11	0.523645	0.261822	−	0.965116i	$-0.415677\pi$
$821$	2.37908e11	0.523645	0.261822	+	0.965116i	$0.415677\pi$
$822$	0	0
$823$	− 7.54074e10i	− 0.164367i	−0.996617	−	0.0821835i	$-0.973811\pi$
$823$	− 7.54074e10i	− 0.164367i	0.996617	−	0.0821835i	$-0.0261894\pi$
$824$	0	0
$825$	−8.74758e11	−1.88831
$826$	0	0
$827$	6.39123e11i	1.36635i	0.730254	+	0.683175i	$0.239402\pi$
$827$	6.39123e11i	1.36635i	−0.730254	+	0.683175i	$0.760598\pi$
$828$	0	0
$829$	1.76013e11	0.372673	0.186336	−	0.982486i	$-0.440339\pi$
$829$	1.76013e11	0.372673	0.186336	+	0.982486i	$0.440339\pi$
$830$	0	0
$831$	4.05593e10i	0.0850525i
$832$	0	0
$833$	5.62384e11	1.16803
$834$	0	0
$835$	− 8.98901e11i	− 1.84912i
$836$	0	0
$837$	−4.30057e9	−0.00876243
$838$	0	0
$839$	− 2.85446e11i	− 0.576072i	−0.957620	−	0.288036i	$-0.906998\pi$
$839$	− 2.85446e11i	− 0.576072i	0.957620	−	0.288036i	$-0.0930022\pi$
$840$	0	0
$841$	−3.06644e11	−0.612985
$842$	0	0
$843$	3.60355e11i	0.713543i
$844$	0	0
$845$	−8.37747e11	−1.64318
$846$	0	0
$847$	− 3.28269e11i	− 0.637818i
$848$	0	0
$849$	−5.10745e11	−0.983046
$850$	0	0
$851$	− 1.61774e9i	− 0.00308454i
$852$	0	0
$853$	−7.25416e11	−1.37022	−0.685111	−	0.728439i	$-0.740246\pi$
$853$	−7.25416e11	−1.37022	−0.685111	+	0.728439i	$0.740246\pi$
$854$	0	0
$855$	− 2.85881e11i	− 0.534959i
$856$	0	0
$857$	−2.75236e11	−0.510248	−0.255124	−	0.966908i	$-0.582116\pi$
$857$	−2.75236e11	−0.510248	−0.255124	+	0.966908i	$0.582116\pi$
$858$	0	0
$859$	8.84849e10i	0.162516i	0.996693	+	0.0812581i	$0.0258938\pi$
$859$	8.84849e10i	0.162516i	−0.996693	+	0.0812581i	$0.974106\pi$
$860$	0	0
$861$	2.17955e11	0.396601
$862$	0	0
$863$	− 7.42243e10i	− 0.133814i	−0.997759	−	0.0669072i	$-0.978687\pi$
$863$	− 7.42243e10i	− 0.133814i	0.997759	−	0.0669072i	$-0.0213131\pi$
$864$	0	0
$865$	−7.61176e11	−1.35963
$866$	0	0
$867$	− 5.28773e11i	− 0.935820i
$868$	0	0
$869$	−1.36247e12	−2.38917
$870$	0	0
$871$	7.08445e11i	1.23093i
$872$	0	0
$873$	1.72283e11	0.296609
$874$	0	0
$875$	− 6.64578e11i	− 1.13374i
$876$	0	0
$877$	−4.28236e11	−0.723910	−0.361955	−	0.932196i	$-0.617891\pi$
$877$	−4.28236e11	−0.723910	−0.361955	+	0.932196i	$0.617891\pi$
$878$	0	0
$879$	3.62709e11i	0.607579i
$880$	0	0
$881$	8.73821e11	1.45050	0.725252	−	0.688484i	$-0.241723\pi$
$881$	8.73821e11	1.45050	0.725252	+	0.688484i	$0.241723\pi$
$882$	0	0
$883$	− 2.92688e11i	− 0.481462i	−0.970592	−	0.240731i	$-0.922613\pi$
$883$	− 2.92688e11i	− 0.481462i	0.970592	−	0.240731i	$-0.0773871\pi$
$884$	0	0
$885$	7.59000e11	1.23728
$886$	0	0
$887$	− 2.65792e11i	− 0.429385i	−0.976682	−	0.214692i	$-0.931125\pi$
$887$	− 2.65792e11i	− 0.429385i	0.976682	−	0.214692i	$-0.0688749\pi$
$888$	0	0
$889$	2.59609e11	0.415635
$890$	0	0
$891$	− 1.04070e11i	− 0.165125i
$892$	0	0
$893$	1.71994e11	0.270462
$894$	0	0
$895$	− 8.93207e11i	− 1.39207i
$896$	0	0
$897$	1.26070e11	0.194734
$898$	0	0
$899$	1.85016e10i	0.0283250i
$900$	0	0
$901$	4.83625e11	0.733853
$902$	0	0
$903$	7.83805e10i	0.117885i
$904$	0	0
$905$	5.79466e11	0.863841
$906$	0	0
$907$	− 8.44675e11i	− 1.24813i	−0.781372	−	0.624066i	$-0.785480\pi$
$907$	− 8.44675e11i	− 1.24813i	0.781372	−	0.624066i	$-0.214520\pi$
$908$	0	0
$909$	3.64898e11	0.534461
$910$	0	0
$911$	9.01110e11i	1.30829i	0.756369	+	0.654146i	$0.226972\pi$
$911$	9.01110e11i	1.30829i	−0.756369	+	0.654146i	$0.773028\pi$
$912$	0	0
$913$	1.57377e12	2.26495
$914$	0	0
$915$	3.75008e11i	0.535003i
$916$	0	0
$917$	5.50720e11	0.778849
$918$	0	0
$919$	− 5.27757e11i	− 0.739898i	−0.929052	−	0.369949i	$-0.879375\pi$
$919$	− 5.27757e11i	− 0.739898i	0.929052	−	0.369949i	$-0.120625\pi$
$920$	0	0
$921$	−4.94534e11	−0.687317
$922$	0	0
$923$	1.73391e12i	2.38902i
$924$	0	0
$925$	2.04084e10	0.0278767
$926$	0	0
$927$	3.52696e11i	0.477618i
$928$	0	0
$929$	8.40665e11	1.12865	0.564326	−	0.825552i	$-0.309136\pi$
$929$	8.40665e11	1.12865	0.564326	+	0.825552i	$0.309136\pi$
$930$	0	0
$931$	− 4.86230e11i	− 0.647206i
$932$	0	0
$933$	−1.77847e11	−0.234704
$934$	0	0
$935$	3.28969e12i	4.30436i
$936$	0	0
$937$	−1.07526e12	−1.39494	−0.697471	−	0.716613i	$-0.745691\pi$
$937$	−1.07526e12	−1.39494	−0.697471	+	0.716613i	$0.745691\pi$
$938$	0	0
$939$	− 1.53271e11i	− 0.197150i
$940$	0	0
$941$	3.12494e11	0.398550	0.199275	−	0.979944i	$-0.436141\pi$
$941$	3.12494e11	0.398550	0.199275	+	0.979944i	$0.436141\pi$
$942$	0	0
$943$	− 2.50648e11i	− 0.316970i
$944$	0	0
$945$	1.44909e11	0.181706
$946$	0	0
$947$	1.00927e12i	1.25489i	0.778661	+	0.627445i	$0.215899\pi$
$947$	1.00927e12i	1.25489i	−0.778661	+	0.627445i	$0.784101\pi$
$948$	0	0
$949$	3.78555e11	0.466729
$950$	0	0
$951$	− 6.88050e11i	− 0.841197i
$952$	0	0
$953$	2.62051e11	0.317697	0.158849	−	0.987303i	$-0.449222\pi$
$953$	2.62051e11	0.317697	0.158849	+	0.987303i	$0.449222\pi$
$954$	0	0
$955$	− 2.90879e12i	− 3.49703i
$956$	0	0
$957$	−4.47721e11	−0.533776
$958$	0	0
$959$	1.50431e11i	0.177853i
$960$	0	0
$961$	8.51123e11	0.997927
$962$	0	0
$963$	− 4.74432e11i	− 0.551657i
$964$	0	0
$965$	2.35052e12	2.71053
$966$	0	0
$967$	− 4.10467e11i	− 0.469431i	−0.972064	−	0.234716i	$-0.924584\pi$
$967$	− 4.10467e11i	− 0.469431i	0.972064	−	0.234716i	$-0.0754159\pi$
$968$	0	0
$969$	−7.39218e11	−0.838451
$970$	0	0
$971$	1.68403e12i	1.89440i	0.320641	+	0.947201i	$0.396102\pi$
$971$	1.68403e12i	1.89440i	−0.320641	+	0.947201i	$0.603898\pi$
$972$	0	0
$973$	5.02685e11	0.560848
$974$	0	0
$975$	1.59041e12i	1.75992i
$976$	0	0
$977$	−1.80830e12	−1.98469	−0.992343	−	0.123513i	$-0.960584\pi$
$977$	−1.80830e12	−1.98469	−0.992343	+	0.123513i	$0.960584\pi$
$978$	0	0
$979$	1.77538e12i	1.93269i
$980$	0	0
$981$	−3.34050e11	−0.360691
$982$	0	0
$983$	− 3.71916e11i	− 0.398319i	−0.979967	−	0.199159i	$-0.936179\pi$
$983$	− 3.71916e11i	− 0.398319i	0.979967	−	0.199159i	$-0.0638211\pi$
$984$	0	0
$985$	−6.30941e11	−0.670261
$986$	0	0
$987$	8.71812e10i	0.0918659i
$988$	0	0
$989$	9.01376e10	0.0942152
$990$	0	0
$991$	7.68294e11i	0.796586i	0.917258	+	0.398293i	$0.130397\pi$
$991$	7.68294e11i	0.796586i	−0.917258	+	0.398293i	$0.869603\pi$
$992$	0	0
$993$	3.75023e11	0.385710
$994$	0	0
$995$	− 6.21079e11i	− 0.633658i
$996$	0	0
$997$	−6.17551e11	−0.625018	−0.312509	−	0.949915i	$-0.601169\pi$
$997$	−6.17551e11	−0.625018	−0.312509	+	0.949915i	$0.601169\pi$
$998$	0	0
$999$	2.42798e9i	0.00243771i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.g.a.127.11	✓	32
4.3	odd	2	inner	384.9.g.a.127.12	yes	32
8.3	odd	2		384.9.g.b.127.21	yes	32
8.5	even	2		384.9.g.b.127.22	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.g.a.127.11	✓	32	1.1	even	1	trivial
384.9.g.a.127.12	yes	32	4.3	odd	2	inner
384.9.g.b.127.21	yes	32	8.3	odd	2
384.9.g.b.127.22	yes	32	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	384.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-46.7654i q^{3} -1118.17 q^{5} +1267.11i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q-46.7654i q^{3} -1118.17 q^{5} +1267.11i q^{7} -2187.00 q^{9} -21758.4i q^{11} -39559.4 q^{13} +52291.6i q^{15} +135213. q^{17} -116904. i q^{19} +59256.9 q^{21} -68145.5i q^{23} +859680. q^{25} +102276. i q^{27} +440003. q^{29} +42048.8i q^{31} -1.01754e6 q^{33} -1.41685e6i q^{35} +23739.5 q^{37} +1.85001e6i q^{39} +3.67813e6 q^{41} +1.32272e6i q^{43} +2.44544e6 q^{45} +1.47124e6i q^{47} +4.15923e6 q^{49} -6.32331e6i q^{51} +3.57675e6 q^{53} +2.43296e7i q^{55} -5.46705e6 q^{57} -1.45148e7i q^{59} +7.17147e6 q^{61} -2.77117e6i q^{63} +4.42341e7 q^{65} -1.79084e7i q^{67} -3.18685e6 q^{69} -4.38306e7i q^{71} -9.56930e6 q^{73} -4.02032e7i q^{75} +2.75703e7 q^{77} -6.26180e7i q^{79} +4.78297e6 q^{81} +7.23294e7i q^{83} -1.51192e8 q^{85} -2.05769e7i q^{87} -8.15953e7 q^{89} -5.01261e7i q^{91} +1.96643e6 q^{93} +1.30718e8i q^{95} -7.87759e7 q^{97} +4.75856e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 32 * q - 1344 * q^5 - 69984 * q^9	\( 32 q - 1344 q^{5} - 69984 q^{9} - 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} - 275520 q^{29} + 2421440 q^{37} - 4374720 q^{41} + 2939328 q^{45} - 14219104 q^{49} - 6224448 q^{53} + 3100032 q^{57} - 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} + 154517760 q^{77} + 153055008 q^{81} - 384830848 q^{85} - 182669760 q^{89} + 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) \) 32 * q - 1344 * q^5 - 69984 * q^9 - 114240 * q^13 - 154560 * q^17 + 1791712 * q^25 - 275520 * q^29 + 2421440 * q^37 - 4374720 * q^41 + 2939328 * q^45 - 14219104 * q^49 - 6224448 * q^53 + 3100032 * q^57 - 13005632 * q^61 + 75175296 * q^65 - 85710400 * q^73 + 154517760 * q^77 + 153055008 * q^81 - 384830848 * q^85 - 182669760 * q^89 + 149817600 * q^93 - 149408192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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