LMFDB - Embedded newform 384.9.g.a.127.21

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			127.21
Character	$\chi$	$=$	384.127
Dual form			384.9.g.a.127.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-46.7654i q^{3} +26.4806 q^{5} +1828.22i q^{7} -2187.00 q^{9} +O(q^{10})$	$q-46.7654i q^{3} +26.4806 q^{5} +1828.22i q^{7} -2187.00 q^{9} +877.801i q^{11} +13304.9 q^{13} -1238.37i q^{15} +142183. q^{17} -47907.0i q^{19} +85497.5 q^{21} -95162.9i q^{23} -389924. q^{25} +102276. i q^{27} -1.23844e6 q^{29} -841704. i q^{31} +41050.7 q^{33} +48412.4i q^{35} +347879. q^{37} -622207. i q^{39} +3.07799e6 q^{41} -431324. i q^{43} -57913.0 q^{45} +6.04692e6i q^{47} +2.42240e6 q^{49} -6.64925e6i q^{51} -2.11471e6 q^{53} +23244.7i q^{55} -2.24039e6 q^{57} +1.08465e7i q^{59} +3.50215e6 q^{61} -3.99832e6i q^{63} +352320. q^{65} +2.35736e7i q^{67} -4.45033e6 q^{69} +514773. i q^{71} +6.92754e6 q^{73} +1.82349e7i q^{75} -1.60482e6 q^{77} +4.59396e7i q^{79} +4.78297e6 q^{81} -4.83697e7i q^{83} +3.76509e6 q^{85} +5.79162e7i q^{87} -8.30165e6 q^{89} +2.43243e7i q^{91} -3.93626e7 q^{93} -1.26860e6i q^{95} +9.38015e7 q^{97} -1.91975e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	26.4806	0.0423689	0.0211844	−	0.999776i	$-0.493256\pi$
$5$	26.4806	0.0423689	0.0211844	+	0.999776i	$0.493256\pi$
$6$	0	0
$7$	1828.22i	0.761442i	0.924690	+	0.380721i	$0.124324\pi$
$7$	1828.22i	0.761442i	−0.924690	+	0.380721i	$0.875676\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	877.801i	0.0599550i	0.999551	+	0.0299775i	$0.00954357\pi$
$11$	877.801i	0.0599550i	−0.999551	+	0.0299775i	$0.990456\pi$
$12$	0	0
$13$	13304.9	0.465840	0.232920	−	0.972496i	$-0.425172\pi$
$13$	13304.9	0.465840	0.232920	+	0.972496i	$0.425172\pi$
$14$	0	0
$15$	− 1238.37i	− 0.0244617i
$16$	0	0
$17$	142183.	1.70236	0.851182	−	0.524871i	$-0.175886\pi$
$17$	142183.	1.70236	0.851182	+	0.524871i	$0.175886\pi$
$18$	0	0
$19$	− 47907.0i	− 0.367608i	−0.982963	−	0.183804i	$-0.941159\pi$
$19$	− 47907.0i	− 0.367608i	0.982963	−	0.183804i	$-0.0588411\pi$
$20$	0	0
$21$	85497.5	0.439619
$22$	0	0
$23$	− 95162.9i	− 0.340061i	−0.985439	−	0.170030i	$-0.945613\pi$
$23$	− 95162.9i	− 0.340061i	0.985439	−	0.170030i	$-0.0543866\pi$
$24$	0	0
$25$	−389924.	−0.998205
$26$	0	0
$27$	102276.i	0.192450i
$28$	0	0
$29$	−1.23844e6	−1.75099	−0.875494	−	0.483228i	$-0.839464\pi$
$29$	−1.23844e6	−1.75099	−0.875494	+	0.483228i	$0.839464\pi$
$30$	0	0
$31$	− 841704.i	− 0.911408i	−0.890131	−	0.455704i	$-0.849388\pi$
$31$	− 841704.i	− 0.911408i	0.890131	−	0.455704i	$-0.150612\pi$
$32$	0	0
$33$	41050.7	0.0346150
$34$	0	0
$35$	48412.4i	0.0322615i
$36$	0	0
$37$	347879.	0.185619	0.0928093	−	0.995684i	$-0.470415\pi$
$37$	347879.	0.185619	0.0928093	+	0.995684i	$0.470415\pi$
$38$	0	0
$39$	− 622207.i	− 0.268953i
$40$	0	0
$41$	3.07799e6	1.08926	0.544630	−	0.838677i	$-0.316670\pi$
$41$	3.07799e6	1.08926	0.544630	+	0.838677i	$0.316670\pi$
$42$	0	0
$43$	− 431324.i	− 0.126162i	−0.998008	−	0.0630812i	$-0.979907\pi$
$43$	− 431324.i	− 0.126162i	0.998008	−	0.0630812i	$-0.0200927\pi$
$44$	0	0
$45$	−57913.0	−0.0141230
$46$	0	0
$47$	6.04692e6i	1.23920i	0.784916	+	0.619602i	$0.212706\pi$
$47$	6.04692e6i	1.23920i	−0.784916	+	0.619602i	$0.787294\pi$
$48$	0	0
$49$	2.42240e6	0.420206
$50$	0	0
$51$	− 6.64925e6i	− 0.982860i
$52$	0	0
$53$	−2.11471e6	−0.268008	−0.134004	−	0.990981i	$-0.542784\pi$
$53$	−2.11471e6	−0.268008	−0.134004	+	0.990981i	$0.542784\pi$
$54$	0	0
$55$	23244.7i	0.00254023i
$56$	0	0
$57$	−2.24039e6	−0.212238
$58$	0	0
$59$	1.08465e7i	0.895123i	0.894253	+	0.447561i	$0.147707\pi$
$59$	1.08465e7i	0.895123i	−0.894253	+	0.447561i	$0.852293\pi$
$60$	0	0
$61$	3.50215e6	0.252939	0.126470	−	0.991970i	$-0.459635\pi$
$61$	3.50215e6	0.252939	0.126470	+	0.991970i	$0.459635\pi$
$62$	0	0
$63$	− 3.99832e6i	− 0.253814i
$64$	0	0
$65$	352320.	0.0197371
$66$	0	0
$67$	2.35736e7i	1.16984i	0.811091	+	0.584920i	$0.198874\pi$
$67$	2.35736e7i	1.16984i	−0.811091	+	0.584920i	$0.801126\pi$
$68$	0	0
$69$	−4.45033e6	−0.196334
$70$	0	0
$71$	514773.i	0.0202574i	0.999949	+	0.0101287i	$0.00322411\pi$
$71$	514773.i	0.0202574i	−0.999949	+	0.0101287i	$0.996776\pi$
$72$	0	0
$73$	6.92754e6	0.243943	0.121971	−	0.992534i	$-0.461078\pi$
$73$	6.92754e6	0.243943	0.121971	+	0.992534i	$0.461078\pi$
$74$	0	0
$75$	1.82349e7i	0.576314i
$76$	0	0
$77$	−1.60482e6	−0.0456523
$78$	0	0
$79$	4.59396e7i	1.17945i	0.807605	+	0.589724i	$0.200763\pi$
$79$	4.59396e7i	1.17945i	−0.807605	+	0.589724i	$0.799237\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	− 4.83697e7i	− 1.01920i	−0.860411	−	0.509602i	$-0.829793\pi$
$83$	− 4.83697e7i	− 1.01920i	0.860411	−	0.509602i	$-0.170207\pi$
$84$	0	0
$85$	3.76509e6	0.0721273
$86$	0	0
$87$	5.79162e7i	1.01093i
$88$	0	0
$89$	−8.30165e6	−0.132314	−0.0661568	−	0.997809i	$-0.521074\pi$
$89$	−8.30165e6	−0.132314	−0.0661568	+	0.997809i	$0.521074\pi$
$90$	0	0
$91$	2.43243e7i	0.354710i
$92$	0	0
$93$	−3.93626e7	−0.526201
$94$	0	0
$95$	− 1.26860e6i	− 0.0155751i
$96$	0	0
$97$	9.38015e7	1.05955	0.529777	−	0.848137i	$-0.322276\pi$
$97$	9.38015e7	1.05955	0.529777	+	0.848137i	$0.322276\pi$
$98$	0	0
$99$	− 1.91975e6i	− 0.0199850i
$100$	0	0
$101$	−1.58287e7	−0.152111	−0.0760555	−	0.997104i	$-0.524233\pi$
$101$	−1.58287e7	−0.152111	−0.0760555	+	0.997104i	$0.524233\pi$
$102$	0	0
$103$	− 1.01403e8i	− 0.900954i	−0.892788	−	0.450477i	$-0.851254\pi$
$103$	− 1.01403e8i	− 0.900954i	0.892788	−	0.450477i	$-0.148746\pi$
$104$	0	0
$105$	2.26402e6	0.0186262
$106$	0	0
$107$	4.03894e7i	0.308129i	0.988061	+	0.154064i	$0.0492363\pi$
$107$	4.03894e7i	0.308129i	−0.988061	+	0.154064i	$0.950764\pi$
$108$	0	0
$109$	2.06949e7	0.146608	0.0733039	−	0.997310i	$-0.476646\pi$
$109$	2.06949e7	0.146608	0.0733039	+	0.997310i	$0.476646\pi$
$110$	0	0
$111$	− 1.62687e7i	− 0.107167i
$112$	0	0
$113$	−2.37541e7	−0.145688	−0.0728441	−	0.997343i	$-0.523208\pi$
$113$	−2.37541e7	−0.145688	−0.0728441	+	0.997343i	$0.523208\pi$
$114$	0	0
$115$	− 2.51997e6i	− 0.0144080i
$116$	0	0
$117$	−2.90977e7	−0.155280
$118$	0	0
$119$	2.59942e8i	1.29625i
$120$	0	0
$121$	2.13588e8	0.996405
$122$	0	0
$123$	− 1.43943e8i	− 0.628884i
$124$	0	0
$125$	−2.06694e7	−0.0846617
$126$	0	0
$127$	− 1.62464e8i	− 0.624512i	−0.949998	−	0.312256i	$-0.898915\pi$
$127$	− 1.62464e8i	− 0.624512i	0.949998	−	0.312256i	$-0.101085\pi$
$128$	0	0
$129$	−2.01710e7	−0.0728399
$130$	0	0
$131$	− 1.78391e8i	− 0.605743i	−0.953031	−	0.302872i	$-0.902055\pi$
$131$	− 1.78391e8i	− 0.605743i	0.953031	−	0.302872i	$-0.0979453\pi$
$132$	0	0
$133$	8.75846e7	0.279912
$134$	0	0
$135$	2.70832e6i	0.00815390i
$136$	0	0
$137$	2.52292e8	0.716177	0.358089	−	0.933688i	$-0.383429\pi$
$137$	2.52292e8	0.716177	0.358089	+	0.933688i	$0.383429\pi$
$138$	0	0
$139$	− 1.78123e8i	− 0.477158i	−0.971123	−	0.238579i	$-0.923318\pi$
$139$	− 1.78123e8i	− 0.477158i	0.971123	−	0.238579i	$-0.0766816\pi$
$140$	0	0
$141$	2.82786e8	0.715455
$142$	0	0
$143$	1.16790e7i	0.0279295i
$144$	0	0
$145$	−3.27946e7	−0.0741875
$146$	0	0
$147$	− 1.13285e8i	− 0.242606i
$148$	0	0
$149$	4.06093e8	0.823911	0.411955	−	0.911204i	$-0.364846\pi$
$149$	4.06093e8	0.823911	0.411955	+	0.911204i	$0.364846\pi$
$150$	0	0
$151$	7.83590e8i	1.50723i	0.657313	+	0.753617i	$0.271693\pi$
$151$	7.83590e8i	1.50723i	−0.657313	+	0.753617i	$0.728307\pi$
$152$	0	0
$153$	−3.10954e8	−0.567455
$154$	0	0
$155$	− 2.22888e7i	− 0.0386153i
$156$	0	0
$157$	6.40743e8	1.05459	0.527297	−	0.849681i	$-0.323206\pi$
$157$	6.40743e8	1.05459	0.527297	+	0.849681i	$0.323206\pi$
$158$	0	0
$159$	9.88953e7i	0.154735i
$160$	0	0
$161$	1.73979e8	0.258936
$162$	0	0
$163$	− 4.95615e8i	− 0.702092i	−0.936358	−	0.351046i	$-0.885826\pi$
$163$	− 4.95615e8i	− 0.702092i	0.936358	−	0.351046i	$-0.114174\pi$
$164$	0	0
$165$	1.08705e6	0.00146660
$166$	0	0
$167$	1.35012e7i	0.0173583i	0.999962	+	0.00867916i	$0.00276270\pi$
$167$	1.35012e7i	0.0173583i	−0.999962	+	0.00867916i	$0.997237\pi$
$168$	0	0
$169$	−6.38711e8	−0.782993
$170$	0	0
$171$	1.04773e8i	0.122536i
$172$	0	0
$173$	−1.53328e8	−0.171174	−0.0855869	−	0.996331i	$-0.527277\pi$
$173$	−1.53328e8	−0.171174	−0.0855869	+	0.996331i	$0.527277\pi$
$174$	0	0
$175$	− 7.12867e8i	− 0.760075i
$176$	0	0
$177$	5.07242e8	0.516799
$178$	0	0
$179$	− 5.33994e8i	− 0.520145i	−0.965589	−	0.260072i	$-0.916254\pi$
$179$	− 5.33994e8i	− 0.520145i	0.965589	−	0.260072i	$-0.0837464\pi$
$180$	0	0
$181$	8.72675e8	0.813090	0.406545	−	0.913631i	$-0.366733\pi$
$181$	8.72675e8	0.813090	0.406545	+	0.913631i	$0.366733\pi$
$182$	0	0
$183$	− 1.63780e8i	− 0.146034i
$184$	0	0
$185$	9.21203e6	0.00786445
$186$	0	0
$187$	1.24809e8i	0.102065i
$188$	0	0
$189$	−1.86983e8	−0.146540
$190$	0	0
$191$	6.38894e8i	0.480060i	0.970765	+	0.240030i	$0.0771573\pi$
$191$	6.38894e8i	0.480060i	−0.970765	+	0.240030i	$0.922843\pi$
$192$	0	0
$193$	1.71319e9	1.23474	0.617371	−	0.786672i	$-0.288198\pi$
$193$	1.71319e9	1.23474	0.617371	+	0.786672i	$0.288198\pi$
$194$	0	0
$195$	− 1.64764e7i	− 0.0113952i
$196$	0	0
$197$	2.76646e9	1.83679	0.918396	−	0.395663i	$-0.129485\pi$
$197$	2.76646e9	1.83679	0.918396	+	0.395663i	$0.129485\pi$
$198$	0	0
$199$	2.04333e9i	1.30295i	0.758671	+	0.651474i	$0.225849\pi$
$199$	2.04333e9i	1.30295i	−0.758671	+	0.651474i	$0.774151\pi$
$200$	0	0
$201$	1.10243e9	0.675408
$202$	0	0
$203$	− 2.26415e9i	− 1.33328i
$204$	0	0
$205$	8.15068e7	0.0461507
$206$	0	0
$207$	2.08121e8i	0.113354i
$208$	0	0
$209$	4.20528e7	0.0220399
$210$	0	0
$211$	− 3.76713e9i	− 1.90056i	−0.311403	−	0.950278i	$-0.600799\pi$
$211$	− 3.76713e9i	− 1.90056i	0.311403	−	0.950278i	$-0.399201\pi$
$212$	0	0
$213$	2.40736e7	0.0116956
$214$	0	0
$215$	− 1.14217e7i	− 0.00534536i
$216$	0	0
$217$	1.53882e9	0.693984
$218$	0	0
$219$	− 3.23969e8i	− 0.140840i
$220$	0	0
$221$	1.89173e9	0.793030
$222$	0	0
$223$	3.77139e9i	1.52504i	0.646963	+	0.762521i	$0.276039\pi$
$223$	3.77139e9i	1.52504i	−0.646963	+	0.762521i	$0.723961\pi$
$224$	0	0
$225$	8.52763e8	0.332735
$226$	0	0
$227$	− 1.71723e9i	− 0.646732i	−0.946274	−	0.323366i	$-0.895186\pi$
$227$	− 1.71723e9i	− 0.646732i	0.946274	−	0.323366i	$-0.104814\pi$
$228$	0	0
$229$	−1.78050e9	−0.647439	−0.323720	−	0.946153i	$-0.604933\pi$
$229$	−1.78050e9	−0.647439	−0.323720	+	0.946153i	$0.604933\pi$
$230$	0	0
$231$	7.50498e7i	0.0263573i
$232$	0	0
$233$	4.32638e9	1.46792	0.733958	−	0.679195i	$-0.237671\pi$
$233$	4.32638e9	1.46792	0.733958	+	0.679195i	$0.237671\pi$
$234$	0	0
$235$	1.60126e8i	0.0525037i
$236$	0	0
$237$	2.14838e9	0.680955
$238$	0	0
$239$	1.42232e8i	0.0435919i	0.999762	+	0.0217960i	$0.00693842\pi$
$239$	1.42232e8i	0.0435919i	−0.999762	+	0.0217960i	$0.993062\pi$
$240$	0	0
$241$	4.84030e8	0.143484	0.0717421	−	0.997423i	$-0.477144\pi$
$241$	4.84030e8	0.143484	0.0717421	+	0.997423i	$0.477144\pi$
$242$	0	0
$243$	− 2.23677e8i	− 0.0641500i
$244$	0	0
$245$	6.41466e7	0.0178037
$246$	0	0
$247$	− 6.37396e8i	− 0.171246i
$248$	0	0
$249$	−2.26203e9	−0.588437
$250$	0	0
$251$	1.70342e9i	0.429167i	0.976706	+	0.214584i	$0.0688394\pi$
$251$	1.70342e9i	0.429167i	−0.976706	+	0.214584i	$0.931161\pi$
$252$	0	0
$253$	8.35341e7	0.0203883
$254$	0	0
$255$	− 1.76076e8i	− 0.0416427i
$256$	0	0
$257$	5.07794e9	1.16401	0.582003	−	0.813186i	$-0.302269\pi$
$257$	5.07794e9	1.16401	0.582003	+	0.813186i	$0.302269\pi$
$258$	0	0
$259$	6.36000e8i	0.141338i
$260$	0	0
$261$	2.70847e9	0.583663
$262$	0	0
$263$	8.70857e9i	1.82022i	0.414367	+	0.910110i	$0.364003\pi$
$263$	8.70857e9i	1.82022i	−0.414367	+	0.910110i	$0.635997\pi$
$264$	0	0
$265$	−5.59988e7	−0.0113552
$266$	0	0
$267$	3.88230e8i	0.0763913i
$268$	0	0
$269$	−3.47687e9	−0.664016	−0.332008	−	0.943276i	$-0.607726\pi$
$269$	−3.47687e9	−0.664016	−0.332008	+	0.943276i	$0.607726\pi$
$270$	0	0
$271$	6.85777e9i	1.27147i	0.771907	+	0.635735i	$0.219303\pi$
$271$	6.85777e9i	1.27147i	−0.771907	+	0.635735i	$0.780697\pi$
$272$	0	0
$273$	1.13753e9	0.204792
$274$	0	0
$275$	− 3.42276e8i	− 0.0598474i
$276$	0	0
$277$	1.01767e10	1.72858	0.864289	−	0.502995i	$-0.167769\pi$
$277$	1.01767e10	1.72858	0.864289	+	0.502995i	$0.167769\pi$
$278$	0	0
$279$	1.84081e9i	0.303803i
$280$	0	0
$281$	1.81840e9	0.291651	0.145826	−	0.989310i	$-0.453416\pi$
$281$	1.81840e9	0.291651	0.145826	+	0.989310i	$0.453416\pi$
$282$	0	0
$283$	− 9.97098e9i	− 1.55450i	−0.629189	−	0.777252i	$-0.716613\pi$
$283$	− 9.97098e9i	− 1.55450i	0.629189	−	0.777252i	$-0.283387\pi$
$284$	0	0
$285$	−5.93267e7	−0.00899230
$286$	0	0
$287$	5.62725e9i	0.829408i
$288$	0	0
$289$	1.32403e10	1.89804
$290$	0	0
$291$	− 4.38666e9i	− 0.611734i
$292$	0	0
$293$	1.10990e10	1.50596	0.752980	−	0.658044i	$-0.228616\pi$
$293$	1.10990e10	1.50596	0.752980	+	0.658044i	$0.228616\pi$
$294$	0	0
$295$	2.87222e8i	0.0379254i
$296$	0	0
$297$	−8.97779e7	−0.0115383
$298$	0	0
$299$	− 1.26613e9i	− 0.158414i
$300$	0	0
$301$	7.88556e8	0.0960653
$302$	0	0
$303$	7.40237e8i	0.0878213i
$304$	0	0
$305$	9.27390e7	0.0107168
$306$	0	0
$307$	1.09602e10i	1.23386i	0.787017	+	0.616931i	$0.211624\pi$
$307$	1.09602e10i	1.23386i	−0.787017	+	0.616931i	$0.788376\pi$
$308$	0	0
$309$	−4.74216e9	−0.520166
$310$	0	0
$311$	− 1.38108e9i	− 0.147631i	−0.997272	−	0.0738153i	$-0.976482\pi$
$311$	− 1.38108e9i	− 0.147631i	0.997272	−	0.0738153i	$-0.0235175\pi$
$312$	0	0
$313$	−6.01991e8	−0.0627210	−0.0313605	−	0.999508i	$-0.509984\pi$
$313$	−6.01991e8	−0.0627210	−0.0313605	+	0.999508i	$0.509984\pi$
$314$	0	0
$315$	− 1.05878e8i	− 0.0107538i
$316$	0	0
$317$	1.02712e9	0.101715	0.0508576	−	0.998706i	$-0.483805\pi$
$317$	1.02712e9	0.101715	0.0508576	+	0.998706i	$0.483805\pi$
$318$	0	0
$319$	− 1.08711e9i	− 0.104981i
$320$	0	0
$321$	1.88883e9	0.177898
$322$	0	0
$323$	− 6.81156e9i	− 0.625802i
$324$	0	0
$325$	−5.18788e9	−0.465004
$326$	0	0
$327$	− 9.67804e8i	− 0.0846440i
$328$	0	0
$329$	−1.10551e10	−0.943582
$330$	0	0
$331$	− 5.29749e9i	− 0.441324i	−0.975350	−	0.220662i	$-0.929178\pi$
$331$	− 5.29749e9i	− 0.441324i	0.975350	−	0.220662i	$-0.0708218\pi$
$332$	0	0
$333$	−7.60811e8	−0.0618728
$334$	0	0
$335$	6.24242e8i	0.0495649i
$336$	0	0
$337$	7.15344e9	0.554620	0.277310	−	0.960781i	$-0.410557\pi$
$337$	7.15344e9	0.554620	0.277310	+	0.960781i	$0.410557\pi$
$338$	0	0
$339$	1.11087e9i	0.0841131i
$340$	0	0
$341$	7.38849e8	0.0546434
$342$	0	0
$343$	1.49680e10i	1.08140i
$344$	0	0
$345$	−1.17847e8	−0.00831846
$346$	0	0
$347$	− 2.65445e10i	− 1.83086i	−0.402474	−	0.915432i	$-0.631849\pi$
$347$	− 2.65445e10i	− 1.83086i	0.402474	−	0.915432i	$-0.368151\pi$
$348$	0	0
$349$	−1.04508e9	−0.0704446	−0.0352223	−	0.999380i	$-0.511214\pi$
$349$	−1.04508e9	−0.0704446	−0.0352223	+	0.999380i	$0.511214\pi$
$350$	0	0
$351$	1.36077e9i	0.0896510i
$352$	0	0
$353$	1.33804e10	0.861725	0.430863	−	0.902418i	$-0.358209\pi$
$353$	1.33804e10	0.861725	0.430863	+	0.902418i	$0.358209\pi$
$354$	0	0
$355$	1.36315e7i	0 0.000858282i
$356$	0	0
$357$	1.21563e10	0.748391
$358$	0	0
$359$	− 2.40693e10i	− 1.44906i	−0.689245	−	0.724528i	$-0.742058\pi$
$359$	− 2.40693e10i	− 1.44906i	0.689245	−	0.724528i	$-0.257942\pi$
$360$	0	0
$361$	1.46885e10	0.864865
$362$	0	0
$363$	− 9.98854e9i	− 0.575275i
$364$	0	0
$365$	1.83445e8	0.0103356
$366$	0	0
$367$	− 3.68982e9i	− 0.203396i	−0.994815	−	0.101698i	$-0.967573\pi$
$367$	− 3.68982e9i	− 0.203396i	0.994815	−	0.101698i	$-0.0324275\pi$
$368$	0	0
$369$	−6.73156e9	−0.363087
$370$	0	0
$371$	− 3.86617e9i	− 0.204073i
$372$	0	0
$373$	3.00266e10	1.55121	0.775604	−	0.631220i	$-0.217445\pi$
$373$	3.00266e10	1.55121	0.775604	+	0.631220i	$0.217445\pi$
$374$	0	0
$375$	9.66611e8i	0.0488795i
$376$	0	0
$377$	−1.64773e10	−0.815681
$378$	0	0
$379$	3.33248e10i	1.61514i	0.589769	+	0.807572i	$0.299219\pi$
$379$	3.33248e10i	1.61514i	−0.589769	+	0.807572i	$0.700781\pi$
$380$	0	0
$381$	−7.59767e9	−0.360562
$382$	0	0
$383$	4.79513e8i	0.0222846i	0.999938	+	0.0111423i	$0.00354678\pi$
$383$	4.79513e8i	0.0222846i	−0.999938	+	0.0111423i	$0.996453\pi$
$384$	0	0
$385$	−4.24964e7	−0.00193424
$386$	0	0
$387$	9.43306e8i	0.0420541i
$388$	0	0
$389$	1.80399e10	0.787837	0.393919	−	0.919145i	$-0.371119\pi$
$389$	1.80399e10	0.787837	0.393919	+	0.919145i	$0.371119\pi$
$390$	0	0
$391$	− 1.35306e10i	− 0.578907i
$392$	0	0
$393$	−8.34253e9	−0.349726
$394$	0	0
$395$	1.21651e9i	0.0499719i
$396$	0	0
$397$	−4.45109e10	−1.79186	−0.895931	−	0.444193i	$-0.853491\pi$
$397$	−4.45109e10	−1.79186	−0.895931	+	0.444193i	$0.853491\pi$
$398$	0	0
$399$	− 4.09593e9i	− 0.161607i
$400$	0	0
$401$	6.71792e9	0.259811	0.129905	−	0.991526i	$-0.458533\pi$
$401$	6.71792e9	0.259811	0.129905	+	0.991526i	$0.458533\pi$
$402$	0	0
$403$	− 1.11988e10i	− 0.424570i
$404$	0	0
$405$	1.26656e8	0.00470766
$406$	0	0
$407$	3.05369e8i	0.0111288i
$408$	0	0
$409$	−5.03675e10	−1.79994	−0.899968	−	0.435956i	$-0.856410\pi$
$409$	−5.03675e10	−1.79994	−0.899968	+	0.435956i	$0.856410\pi$
$410$	0	0
$411$	− 1.17985e10i	− 0.413485i
$412$	0	0
$413$	−1.98299e10	−0.681584
$414$	0	0
$415$	− 1.28086e9i	− 0.0431825i
$416$	0	0
$417$	−8.33001e9	−0.275487
$418$	0	0
$419$	3.92080e10i	1.27209i	0.771651	+	0.636046i	$0.219431\pi$
$419$	3.92080e10i	1.27209i	−0.771651	+	0.636046i	$0.780569\pi$
$420$	0	0
$421$	−6.15806e9	−0.196027	−0.0980134	−	0.995185i	$-0.531249\pi$
$421$	−6.15806e9	−0.196027	−0.0980134	+	0.995185i	$0.531249\pi$
$422$	0	0
$423$	− 1.32246e10i	− 0.413068i
$424$	0	0
$425$	−5.54406e10	−1.69931
$426$	0	0
$427$	6.40272e9i	0.192599i
$428$	0	0
$429$	5.46174e8	0.0161251
$430$	0	0
$431$	2.00289e10i	0.580427i	0.956962	+	0.290214i	$0.0937264\pi$
$431$	2.00289e10i	0.580427i	−0.956962	+	0.290214i	$0.906274\pi$
$432$	0	0
$433$	2.85478e10	0.812120	0.406060	−	0.913846i	$-0.366902\pi$
$433$	2.85478e10	0.812120	0.406060	+	0.913846i	$0.366902\pi$
$434$	0	0
$435$	1.53365e9i	0.0428322i
$436$	0	0
$437$	−4.55897e9	−0.125009
$438$	0	0
$439$	− 1.19037e10i	− 0.320498i	−0.987077	−	0.160249i	$-0.948770\pi$
$439$	− 1.19037e10i	− 0.320498i	0.987077	−	0.160249i	$-0.0512297\pi$
$440$	0	0
$441$	−5.29780e9	−0.140069
$442$	0	0
$443$	4.78181e10i	1.24159i	0.783974	+	0.620794i	$0.213190\pi$
$443$	4.78181e10i	1.24159i	−0.783974	+	0.620794i	$0.786810\pi$
$444$	0	0
$445$	−2.19832e8	−0.00560598
$446$	0	0
$447$	− 1.89911e10i	− 0.475685i
$448$	0	0
$449$	1.66994e10	0.410882	0.205441	−	0.978670i	$-0.434137\pi$
$449$	1.66994e10	0.410882	0.205441	+	0.978670i	$0.434137\pi$
$450$	0	0
$451$	2.70186e9i	0.0653066i
$452$	0	0
$453$	3.66449e10	0.870203
$454$	0	0
$455$	6.44120e8i	0.0150287i
$456$	0	0
$457$	1.43480e9	0.0328949	0.0164474	−	0.999865i	$-0.494764\pi$
$457$	1.43480e9	0.0328949	0.0164474	+	0.999865i	$0.494764\pi$
$458$	0	0
$459$	1.45419e10i	0.327620i
$460$	0	0
$461$	−1.39386e10	−0.308613	−0.154306	−	0.988023i	$-0.549314\pi$
$461$	−1.39386e10	−0.308613	−0.154306	+	0.988023i	$0.549314\pi$
$462$	0	0
$463$	1.14947e10i	0.250135i	0.992148	+	0.125068i	$0.0399148\pi$
$463$	1.14947e10i	0.250135i	−0.992148	+	0.125068i	$0.960085\pi$
$464$	0	0
$465$	−1.04234e9	−0.0222946
$466$	0	0
$467$	− 3.36849e10i	− 0.708219i	−0.935204	−	0.354110i	$-0.884784\pi$
$467$	− 3.36849e10i	− 0.708219i	0.935204	−	0.354110i	$-0.115216\pi$
$468$	0	0
$469$	−4.30978e10	−0.890766
$470$	0	0
$471$	− 2.99646e10i	− 0.608870i
$472$	0	0
$473$	3.78617e8	0.00756407
$474$	0	0
$475$	1.86801e10i	0.366948i
$476$	0	0
$477$	4.62488e9	0.0893360
$478$	0	0
$479$	− 7.66650e10i	− 1.45631i	−0.685411	−	0.728157i	$-0.740377\pi$
$479$	− 7.66650e10i	− 1.45631i	0.685411	−	0.728157i	$-0.259623\pi$
$480$	0	0
$481$	4.62848e9	0.0864686
$482$	0	0
$483$	− 8.13619e9i	− 0.149497i
$484$	0	0
$485$	2.48392e9	0.0448921
$486$	0	0
$487$	6.77383e10i	1.20425i	0.798400	+	0.602127i	$0.205680\pi$
$487$	6.77383e10i	1.20425i	−0.798400	+	0.602127i	$0.794320\pi$
$488$	0	0
$489$	−2.31776e10	−0.405353
$490$	0	0
$491$	8.56926e9i	0.147441i	0.997279	+	0.0737203i	$0.0234872\pi$
$491$	8.56926e9i	0.147441i	−0.997279	+	0.0737203i	$0.976513\pi$
$492$	0	0
$493$	−1.76085e11	−2.98082
$494$	0	0
$495$	− 5.08361e7i	0 0.000846742i
$496$	0	0
$497$	−9.41120e8	−0.0154248
$498$	0	0
$499$	1.02766e11i	1.65747i	0.559640	+	0.828736i	$0.310939\pi$
$499$	1.02766e11i	1.65747i	−0.559640	+	0.828736i	$0.689061\pi$
$500$	0	0
$501$	6.31391e8	0.0100218
$502$	0	0
$503$	− 9.07272e10i	− 1.41731i	−0.705554	−	0.708656i	$-0.749302\pi$
$503$	− 9.07272e10i	− 1.41731i	0.705554	−	0.708656i	$-0.250698\pi$
$504$	0	0
$505$	−4.19154e8	−0.00644478
$506$	0	0
$507$	2.98696e10i	0.452061i
$508$	0	0
$509$	−3.89720e10	−0.580606	−0.290303	−	0.956935i	$-0.593756\pi$
$509$	−3.89720e10	−0.580606	−0.290303	+	0.956935i	$0.593756\pi$
$510$	0	0
$511$	1.26651e10i	0.185748i
$512$	0	0
$513$	4.89973e9	0.0707461
$514$	0	0
$515$	− 2.68521e9i	− 0.0381724i
$516$	0	0
$517$	−5.30799e9	−0.0742965
$518$	0	0
$519$	7.17045e9i	0.0988273i
$520$	0	0
$521$	−1.24381e11	−1.68811	−0.844057	−	0.536254i	$-0.819839\pi$
$521$	−1.24381e11	−1.68811	−0.844057	+	0.536254i	$0.819839\pi$
$522$	0	0
$523$	8.99303e10i	1.20199i	0.799255	+	0.600993i	$0.205228\pi$
$523$	8.99303e10i	1.20199i	−0.799255	+	0.600993i	$0.794772\pi$
$524$	0	0
$525$	−3.33375e10	−0.438830
$526$	0	0
$527$	− 1.19676e11i	− 1.55155i
$528$	0	0
$529$	6.92550e10	0.884359
$530$	0	0
$531$	− 2.37213e10i	− 0.298374i
$532$	0	0
$533$	4.09522e10	0.507421
$534$	0	0
$535$	1.06953e9i	0.0130551i
$536$	0	0
$537$	−2.49724e10	−0.300306
$538$	0	0
$539$	2.12639e9i	0.0251934i
$540$	0	0
$541$	−1.14157e11	−1.33264	−0.666319	−	0.745667i	$-0.732131\pi$
$541$	−1.14157e11	−1.33264	−0.666319	+	0.745667i	$0.732131\pi$
$542$	0	0
$543$	− 4.08110e10i	− 0.469438i
$544$	0	0
$545$	5.48012e8	0.00621161
$546$	0	0
$547$	7.65502e10i	0.855060i	0.904001	+	0.427530i	$0.140616\pi$
$547$	7.65502e10i	0.855060i	−0.904001	+	0.427530i	$0.859384\pi$
$548$	0	0
$549$	−7.65921e9	−0.0843130
$550$	0	0
$551$	5.93300e10i	0.643677i
$552$	0	0
$553$	−8.39878e10	−0.898081
$554$	0	0
$555$	− 4.30804e8i	− 0.00454054i
$556$	0	0
$557$	−7.76398e10	−0.806610	−0.403305	−	0.915066i	$-0.632139\pi$
$557$	−7.76398e10	−0.806610	−0.403305	+	0.915066i	$0.632139\pi$
$558$	0	0
$559$	− 5.73871e9i	− 0.0587715i
$560$	0	0
$561$	5.83672e9	0.0589274
$562$	0	0
$563$	7.00466e10i	0.697194i	0.937273	+	0.348597i	$0.113342\pi$
$563$	7.00466e10i	0.697194i	−0.937273	+	0.348597i	$0.886658\pi$
$564$	0	0
$565$	−6.29021e8	−0.00617265
$566$	0	0
$567$	8.74433e9i	0.0846047i
$568$	0	0
$569$	−1.43044e11	−1.36465	−0.682323	−	0.731051i	$-0.739030\pi$
$569$	−1.43044e11	−1.36465	−0.682323	+	0.731051i	$0.739030\pi$
$570$	0	0
$571$	1.49896e10i	0.141008i	0.997511	+	0.0705041i	$0.0224608\pi$
$571$	1.49896e10i	0.141008i	−0.997511	+	0.0705041i	$0.977539\pi$
$572$	0	0
$573$	2.98781e10	0.277163
$574$	0	0
$575$	3.71063e10i	0.339450i
$576$	0	0
$577$	3.25732e10	0.293872	0.146936	−	0.989146i	$-0.453059\pi$
$577$	3.25732e10	0.293872	0.146936	+	0.989146i	$0.453059\pi$
$578$	0	0
$579$	− 8.01180e10i	− 0.712879i
$580$	0	0
$581$	8.84305e10	0.776064
$582$	0	0
$583$	− 1.85630e9i	− 0.0160684i
$584$	0	0
$585$	−7.70524e8	−0.00657905
$586$	0	0
$587$	− 6.00083e10i	− 0.505427i	−0.967541	−	0.252714i	$-0.918677\pi$
$587$	− 6.00083e10i	− 0.505427i	0.967541	−	0.252714i	$-0.0813231\pi$
$588$	0	0
$589$	−4.03235e10	−0.335040
$590$	0	0
$591$	− 1.29375e11i	− 1.06047i
$592$	0	0
$593$	6.75490e10	0.546261	0.273130	−	0.961977i	$-0.411941\pi$
$593$	6.75490e10	0.546261	0.273130	+	0.961977i	$0.411941\pi$
$594$	0	0
$595$	6.88342e9i	0.0549207i
$596$	0	0
$597$	9.55573e10	0.752257
$598$	0	0
$599$	9.01397e9i	0.0700178i	0.999387	+	0.0350089i	$0.0111460\pi$
$599$	9.01397e9i	0.0700178i	−0.999387	+	0.0350089i	$0.988854\pi$
$600$	0	0
$601$	−1.29949e10	−0.0996034	−0.0498017	−	0.998759i	$-0.515859\pi$
$601$	−1.29949e10	−0.0996034	−0.0498017	+	0.998759i	$0.515859\pi$
$602$	0	0
$603$	− 5.15555e10i	− 0.389947i
$604$	0	0
$605$	5.65594e9	0.0422166
$606$	0	0
$607$	6.89317e10i	0.507767i	0.967235	+	0.253883i	$0.0817079\pi$
$607$	6.89317e10i	0.507767i	−0.967235	+	0.253883i	$0.918292\pi$
$608$	0	0
$609$	−1.05884e11	−0.769768
$610$	0	0
$611$	8.04535e10i	0.577271i
$612$	0	0
$613$	5.16800e10	0.366000	0.183000	−	0.983113i	$-0.441419\pi$
$613$	5.16800e10	0.366000	0.183000	+	0.983113i	$0.441419\pi$
$614$	0	0
$615$	− 3.81170e9i	− 0.0266451i
$616$	0	0
$617$	2.45740e10	0.169564	0.0847821	−	0.996400i	$-0.472981\pi$
$617$	2.45740e10	0.169564	0.0847821	+	0.996400i	$0.472981\pi$
$618$	0	0
$619$	2.06900e11i	1.40928i	0.709564	+	0.704641i	$0.248892\pi$
$619$	2.06900e11i	1.40928i	−0.709564	+	0.704641i	$0.751108\pi$
$620$	0	0
$621$	9.73287e9	0.0654447
$622$	0	0
$623$	− 1.51773e10i	− 0.100749i
$624$	0	0
$625$	1.51767e11	0.994618
$626$	0	0
$627$	− 1.96662e9i	− 0.0127247i
$628$	0	0
$629$	4.94625e10	0.315990
$630$	0	0
$631$	3.13779e10i	0.197928i	0.995091	+	0.0989639i	$0.0315528\pi$
$631$	3.13779e10i	0.197928i	−0.995091	+	0.0989639i	$0.968447\pi$
$632$	0	0
$633$	−1.76171e11	−1.09729
$634$	0	0
$635$	− 4.30212e9i	− 0.0264599i
$636$	0	0
$637$	3.22297e10	0.195749
$638$	0	0
$639$	− 1.12581e9i	− 0.00675245i
$640$	0	0
$641$	−1.86516e11	−1.10480	−0.552400	−	0.833579i	$-0.686288\pi$
$641$	−1.86516e11	−1.10480	−0.552400	+	0.833579i	$0.686288\pi$
$642$	0	0
$643$	− 2.52102e10i	− 0.147480i	−0.997278	−	0.0737399i	$-0.976507\pi$
$643$	− 2.52102e10i	− 0.147480i	0.997278	−	0.0737399i	$-0.0234935\pi$
$644$	0	0
$645$	−5.34140e8	−0.00308615
$646$	0	0
$647$	− 2.67970e11i	− 1.52922i	−0.644495	−	0.764608i	$-0.722932\pi$
$647$	− 2.67970e11i	− 1.52922i	0.644495	−	0.764608i	$-0.277068\pi$
$648$	0	0
$649$	−9.52109e9	−0.0536671
$650$	0	0
$651$	− 7.19636e10i	− 0.400672i
$652$	0	0
$653$	−1.85734e11	−1.02150	−0.510750	−	0.859729i	$-0.670632\pi$
$653$	−1.85734e11	−1.02150	−0.510750	+	0.859729i	$0.670632\pi$
$654$	0	0
$655$	− 4.72390e9i	− 0.0256647i
$656$	0	0
$657$	−1.51505e10	−0.0813142
$658$	0	0
$659$	− 3.12092e11i	− 1.65478i	−0.561626	−	0.827392i	$-0.689824\pi$
$659$	− 3.12092e11i	− 1.65478i	0.561626	−	0.827392i	$-0.310176\pi$
$660$	0	0
$661$	−2.70437e11	−1.41664	−0.708322	−	0.705889i	$-0.750548\pi$
$661$	−2.70437e11	−1.41664	−0.708322	+	0.705889i	$0.750548\pi$
$662$	0	0
$663$	− 8.84673e10i	− 0.457856i
$664$	0	0
$665$	2.31929e9	0.0118596
$666$	0	0
$667$	1.17854e11i	0.595442i
$668$	0	0
$669$	1.76370e11	0.880484
$670$	0	0
$671$	3.07420e9i	0.0151650i
$672$	0	0
$673$	2.73104e11	1.33128	0.665638	−	0.746275i	$-0.268160\pi$
$673$	2.73104e11	1.33128	0.665638	+	0.746275i	$0.268160\pi$
$674$	0	0
$675$	− 3.98798e10i	− 0.192105i
$676$	0	0
$677$	1.06110e11	0.505130	0.252565	−	0.967580i	$-0.418726\pi$
$677$	1.06110e11	0.505130	0.252565	+	0.967580i	$0.418726\pi$
$678$	0	0
$679$	1.71490e11i	0.806789i
$680$	0	0
$681$	−8.03067e10	−0.373391
$682$	0	0
$683$	− 3.30574e11i	− 1.51910i	−0.650449	−	0.759550i	$-0.725419\pi$
$683$	− 3.30574e11i	− 1.51910i	0.650449	−	0.759550i	$-0.274581\pi$
$684$	0	0
$685$	6.68082e9	0.0303436
$686$	0	0
$687$	8.32656e10i	0.373799i
$688$	0	0
$689$	−2.81360e10	−0.124849
$690$	0	0
$691$	9.33882e10i	0.409619i	0.978802	+	0.204809i	$0.0656574\pi$
$691$	9.33882e10i	0.409619i	−0.978802	+	0.204809i	$0.934343\pi$
$692$	0	0
$693$	3.50973e9	0.0152174
$694$	0	0
$695$	− 4.71681e9i	− 0.0202166i
$696$	0	0
$697$	4.37638e11	1.85432
$698$	0	0
$699$	− 2.02325e11i	− 0.847502i
$700$	0	0
$701$	3.05818e11	1.26646	0.633229	−	0.773964i	$-0.281729\pi$
$701$	3.05818e11	1.26646	0.633229	+	0.773964i	$0.281729\pi$
$702$	0	0
$703$	− 1.66658e10i	− 0.0682348i
$704$	0	0
$705$	7.48834e9	0.0303130
$706$	0	0
$707$	− 2.89384e10i	− 0.115824i
$708$	0	0
$709$	2.07512e11	0.821218	0.410609	−	0.911812i	$-0.365316\pi$
$709$	2.07512e11	0.821218	0.410609	+	0.911812i	$0.365316\pi$
$710$	0	0
$711$	− 1.00470e11i	− 0.393149i
$712$	0	0
$713$	−8.00990e10	−0.309934
$714$	0	0
$715$	3.09267e8i	0.00118334i
$716$	0	0
$717$	6.65154e9	0.0251678
$718$	0	0
$719$	3.01067e11i	1.12654i	0.826272	+	0.563272i	$0.190458\pi$
$719$	3.01067e11i	1.12654i	−0.826272	+	0.563272i	$0.809542\pi$
$720$	0	0
$721$	1.85388e11	0.686024
$722$	0	0
$723$	− 2.26359e10i	− 0.0828407i
$724$	0	0
$725$	4.82898e11	1.74785
$726$	0	0
$727$	3.22118e11i	1.15313i	0.817052	+	0.576563i	$0.195607\pi$
$727$	3.22118e11i	1.15313i	−0.817052	+	0.576563i	$0.804393\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	− 6.13270e10i	− 0.214774i
$732$	0	0
$733$	−4.34260e11	−1.50430	−0.752149	−	0.658993i	$-0.770983\pi$
$733$	−4.34260e11	−1.50430	−0.752149	+	0.658993i	$0.770983\pi$
$734$	0	0
$735$	− 2.99984e9i	− 0.0102789i
$736$	0	0
$737$	−2.06929e10	−0.0701378
$738$	0	0
$739$	− 4.88710e10i	− 0.163860i	−0.996638	−	0.0819301i	$-0.973892\pi$
$739$	− 4.88710e10i	− 0.163860i	0.996638	−	0.0819301i	$-0.0261084\pi$
$740$	0	0
$741$	−2.98081e10	−0.0988692
$742$	0	0
$743$	− 3.41638e11i	− 1.12101i	−0.828150	−	0.560506i	$-0.810607\pi$
$743$	− 3.41638e11i	− 1.12101i	0.828150	−	0.560506i	$-0.189393\pi$
$744$	0	0
$745$	1.07536e10	0.0349082
$746$	0	0
$747$	1.05784e11i	0.339734i
$748$	0	0
$749$	−7.38408e10	−0.234622
$750$	0	0
$751$	2.94217e11i	0.924927i	0.886638	+	0.462464i	$0.153034\pi$
$751$	2.94217e11i	0.924927i	−0.886638	+	0.462464i	$0.846966\pi$
$752$	0	0
$753$	7.96610e10	0.247780
$754$	0	0
$755$	2.07499e10i	0.0638599i
$756$	0	0
$757$	−2.12679e11	−0.647650	−0.323825	−	0.946117i	$-0.604969\pi$
$757$	−2.12679e11	−0.647650	−0.323825	+	0.946117i	$0.604969\pi$
$758$	0	0
$759$	− 3.90650e9i	− 0.0117712i
$760$	0	0
$761$	4.65349e10	0.138752	0.0693762	−	0.997591i	$-0.477899\pi$
$761$	4.65349e10	0.138752	0.0693762	+	0.997591i	$0.477899\pi$
$762$	0	0
$763$	3.78349e10i	0.111633i
$764$	0	0
$765$	−8.23425e9	−0.0240424
$766$	0	0
$767$	1.44312e11i	0.416984i
$768$	0	0
$769$	3.22102e11	0.921061	0.460531	−	0.887644i	$-0.347659\pi$
$769$	3.22102e11	0.921061	0.460531	+	0.887644i	$0.347659\pi$
$770$	0	0
$771$	− 2.37472e11i	− 0.672039i
$772$	0	0
$773$	1.61926e11	0.453522	0.226761	−	0.973950i	$-0.427186\pi$
$773$	1.61926e11	0.453522	0.226761	+	0.973950i	$0.427186\pi$
$774$	0	0
$775$	3.28200e11i	0.909771i
$776$	0	0
$777$	2.97428e10	0.0816014
$778$	0	0
$779$	− 1.47457e11i	− 0.400420i
$780$	0	0
$781$	−4.51869e8	−0.00121453
$782$	0	0
$783$	− 1.26663e11i	− 0.336978i
$784$	0	0
$785$	1.69672e10	0.0446820
$786$	0	0
$787$	4.64076e11i	1.20973i	0.796326	+	0.604867i	$0.206774\pi$
$787$	4.64076e11i	1.20973i	−0.796326	+	0.604867i	$0.793226\pi$
$788$	0	0
$789$	4.07259e11	1.05090
$790$	0	0
$791$	− 4.34277e10i	− 0.110933i
$792$	0	0
$793$	4.65957e10	0.117829
$794$	0	0
$795$	2.61880e9i	0.00655593i
$796$	0	0
$797$	6.66254e11	1.65123	0.825613	−	0.564237i	$-0.190830\pi$
$797$	6.66254e11	1.65123	0.825613	+	0.564237i	$0.190830\pi$
$798$	0	0
$799$	8.59770e11i	2.10958i
$800$	0	0
$801$	1.81557e10	0.0441045
$802$	0	0
$803$	6.08100e9i	0.0146256i
$804$	0	0
$805$	4.60706e9	0.0109709
$806$	0	0
$807$	1.62597e11i	0.383370i
$808$	0	0
$809$	4.57989e11	1.06921	0.534603	−	0.845104i	$-0.320461\pi$
$809$	4.57989e11	1.06921	0.534603	+	0.845104i	$0.320461\pi$
$810$	0	0
$811$	− 1.72588e11i	− 0.398958i	−0.979902	−	0.199479i	$-0.936075\pi$
$811$	− 1.72588e11i	− 0.398958i	0.979902	−	0.199479i	$-0.0639250\pi$
$812$	0	0
$813$	3.20706e11	0.734083
$814$	0	0
$815$	− 1.31242e10i	− 0.0297468i
$816$	0	0
$817$	−2.06634e10	−0.0463782
$818$	0	0
$819$	− 5.31971e10i	− 0.118237i
$820$	0	0
$821$	−4.72775e11	−1.04059	−0.520297	−	0.853985i	$-0.674179\pi$
$821$	−4.72775e11	−1.04059	−0.520297	+	0.853985i	$0.674179\pi$
$822$	0	0
$823$	4.00386e11i	0.872730i	0.899770	+	0.436365i	$0.143734\pi$
$823$	4.00386e11i	0.872730i	−0.899770	+	0.436365i	$0.856266\pi$
$824$	0	0
$825$	−1.60066e10	−0.0345529
$826$	0	0
$827$	− 8.51319e11i	− 1.82000i	−0.414613	−	0.909998i	$-0.636083\pi$
$827$	− 8.51319e11i	− 1.82000i	0.414613	−	0.909998i	$-0.363917\pi$
$828$	0	0
$829$	6.21161e11	1.31518	0.657592	−	0.753375i	$-0.271575\pi$
$829$	6.21161e11	1.31518	0.657592	+	0.753375i	$0.271575\pi$
$830$	0	0
$831$	− 4.75918e11i	− 0.997995i
$832$	0	0
$833$	3.44425e11	0.715343
$834$	0	0
$835$	3.57520e8i	0 0.000735453i
$836$	0	0
$837$	8.60860e10	0.175400
$838$	0	0
$839$	8.77761e11i	1.77145i	0.464212	+	0.885724i	$0.346338\pi$
$839$	8.77761e11i	1.77145i	−0.464212	+	0.885724i	$0.653662\pi$
$840$	0	0
$841$	1.03349e12	2.06596
$842$	0	0
$843$	− 8.50380e10i	− 0.168385i
$844$	0	0
$845$	−1.69134e10	−0.0331745
$846$	0	0
$847$	3.90487e11i	0.758705i
$848$	0	0
$849$	−4.66296e11	−0.897494
$850$	0	0
$851$	− 3.31052e10i	− 0.0631215i
$852$	0	0
$853$	−6.95270e11	−1.31328	−0.656640	−	0.754204i	$-0.728023\pi$
$853$	−6.95270e11	−1.31328	−0.656640	+	0.754204i	$0.728023\pi$
$854$	0	0
$855$	2.77444e9i	0.00519171i
$856$	0	0
$857$	1.89044e11	0.350461	0.175231	−	0.984527i	$-0.443933\pi$
$857$	1.89044e11	0.350461	0.175231	+	0.984527i	$0.443933\pi$
$858$	0	0
$859$	1.99782e11i	0.366930i	0.983026	+	0.183465i	$0.0587314\pi$
$859$	1.99782e11i	0.366930i	−0.983026	+	0.183465i	$0.941269\pi$
$860$	0	0
$861$	2.63160e11	0.478859
$862$	0	0
$863$	1.91816e11i	0.345813i	0.984938	+	0.172907i	$0.0553159\pi$
$863$	1.91816e11i	0.345813i	−0.984938	+	0.172907i	$0.944684\pi$
$864$	0	0
$865$	−4.06021e9	−0.00725245
$866$	0	0
$867$	− 6.19186e11i	− 1.09583i
$868$	0	0
$869$	−4.03258e10	−0.0707138
$870$	0	0
$871$	3.13644e11i	0.544959i
$872$	0	0
$873$	−2.05144e11	−0.353185
$874$	0	0
$875$	− 3.77882e10i	− 0.0644650i
$876$	0	0
$877$	−5.21303e11	−0.881236	−0.440618	−	0.897695i	$-0.645241\pi$
$877$	−5.21303e11	−0.881236	−0.440618	+	0.897695i	$0.645241\pi$
$878$	0	0
$879$	− 5.19049e11i	− 0.869466i
$880$	0	0
$881$	−3.82242e11	−0.634505	−0.317252	−	0.948341i	$-0.602760\pi$
$881$	−3.82242e11	−0.634505	−0.317252	+	0.948341i	$0.602760\pi$
$882$	0	0
$883$	− 3.60372e11i	− 0.592801i	−0.955064	−	0.296400i	$-0.904214\pi$
$883$	− 3.60372e11i	− 0.592801i	0.955064	−	0.296400i	$-0.0957863\pi$
$884$	0	0
$885$	1.34320e10	0.0218962
$886$	0	0
$887$	− 1.00334e12i	− 1.62089i	−0.585816	−	0.810444i	$-0.699226\pi$
$887$	− 1.00334e12i	− 1.62089i	0.585816	−	0.810444i	$-0.300774\pi$
$888$	0	0
$889$	2.97019e11	0.475530
$890$	0	0
$891$	4.19850e9i	0.00666167i
$892$	0	0
$893$	2.89690e11	0.455541
$894$	0	0
$895$	− 1.41405e10i	− 0.0220380i
$896$	0	0
$897$	−5.92110e10	−0.0914603
$898$	0	0
$899$	1.04240e12i	1.59586i
$900$	0	0
$901$	−3.00676e11	−0.456247
$902$	0	0
$903$	− 3.68771e10i	− 0.0554634i
$904$	0	0
$905$	2.31089e10	0.0344497
$906$	0	0
$907$	2.42968e11i	0.359022i	0.983756	+	0.179511i	$0.0574515\pi$
$907$	2.42968e11i	0.359022i	−0.983756	+	0.179511i	$0.942549\pi$
$908$	0	0
$909$	3.46174e10	0.0507037
$910$	0	0
$911$	− 4.24906e11i	− 0.616906i	−0.951240	−	0.308453i	$-0.900189\pi$
$911$	− 4.24906e11i	− 0.616906i	0.951240	−	0.308453i	$-0.0998113\pi$
$912$	0	0
$913$	4.24590e10	0.0611063
$914$	0	0
$915$	− 4.33697e9i	− 0.00618732i
$916$	0	0
$917$	3.26139e11	0.461238
$918$	0	0
$919$	− 4.64713e11i	− 0.651512i	−0.945454	−	0.325756i	$-0.894381\pi$
$919$	− 4.64713e11i	− 0.651512i	0.945454	−	0.325756i	$-0.105619\pi$
$920$	0	0
$921$	5.12560e11	0.712371
$922$	0	0
$923$	6.84899e9i	0.00943669i
$924$	0	0
$925$	−1.35646e11	−0.185285
$926$	0	0
$927$	2.21769e11i	0.300318i
$928$	0	0
$929$	−1.68867e11	−0.226716	−0.113358	−	0.993554i	$-0.536161\pi$
$929$	−1.68867e11	−0.226716	−0.113358	+	0.993554i	$0.536161\pi$
$930$	0	0
$931$	− 1.16050e11i	− 0.154471i
$932$	0	0
$933$	−6.45866e10	−0.0852346
$934$	0	0
$935$	3.30500e9i	0.00432439i
$936$	0	0
$937$	5.08337e11	0.659467	0.329734	−	0.944074i	$-0.393041\pi$
$937$	5.08337e11	0.659467	0.329734	+	0.944074i	$0.393041\pi$
$938$	0	0
$939$	2.81523e10i	0.0362120i
$940$	0	0
$941$	1.41917e12	1.80998	0.904992	−	0.425429i	$-0.139877\pi$
$941$	1.41917e12	1.80998	0.904992	+	0.425429i	$0.139877\pi$
$942$	0	0
$943$	− 2.92910e11i	− 0.370414i
$944$	0	0
$945$	−4.95142e9	−0.00620872
$946$	0	0
$947$	− 1.48729e12i	− 1.84925i	−0.380882	−	0.924624i	$-0.624380\pi$
$947$	− 1.48729e12i	− 1.84925i	0.380882	−	0.924624i	$-0.375620\pi$
$948$	0	0
$949$	9.21700e10	0.113638
$950$	0	0
$951$	− 4.80338e10i	− 0.0587253i
$952$	0	0
$953$	−5.60965e11	−0.680087	−0.340043	−	0.940410i	$-0.610442\pi$
$953$	−5.60965e11	−0.680087	−0.340043	+	0.940410i	$0.610442\pi$
$954$	0	0
$955$	1.69183e10i	0.0203396i
$956$	0	0
$957$	−5.08389e10	−0.0606105
$958$	0	0
$959$	4.61245e11i	0.545327i
$960$	0	0
$961$	1.44425e11	0.169336
$962$	0	0
$963$	− 8.83316e10i	− 0.102710i
$964$	0	0
$965$	4.53662e10	0.0523147
$966$	0	0
$967$	1.50321e12i	1.71915i	0.511013	+	0.859573i	$0.329270\pi$
$967$	1.50321e12i	1.71915i	−0.511013	+	0.859573i	$0.670730\pi$
$968$	0	0
$969$	−3.18545e11	−0.361307
$970$	0	0
$971$	− 1.10455e12i	− 1.24254i	−0.783598	−	0.621269i	$-0.786618\pi$
$971$	− 1.10455e12i	− 1.24254i	0.783598	−	0.621269i	$-0.213382\pi$
$972$	0	0
$973$	3.25649e11	0.363328
$974$	0	0
$975$	2.42613e11i	0.268470i
$976$	0	0
$977$	−8.31513e11	−0.912622	−0.456311	−	0.889820i	$-0.650830\pi$
$977$	−8.31513e11	−0.912622	−0.456311	+	0.889820i	$0.650830\pi$
$978$	0	0
$979$	− 7.28720e9i	− 0.00793286i
$980$	0	0
$981$	−4.52597e10	−0.0488693
$982$	0	0
$983$	− 7.08257e11i	− 0.758537i	−0.925287	−	0.379269i	$-0.876176\pi$
$983$	− 7.08257e11i	− 0.758537i	0.925287	−	0.379269i	$-0.123824\pi$
$984$	0	0
$985$	7.32575e10	0.0778228
$986$	0	0
$987$	5.16997e11i	0.544777i
$988$	0	0
$989$	−4.10460e10	−0.0429028
$990$	0	0
$991$	− 6.31212e11i	− 0.654457i	−0.944945	−	0.327228i	$-0.893885\pi$
$991$	− 6.31212e11i	− 0.654457i	0.944945	−	0.327228i	$-0.106115\pi$
$992$	0	0
$993$	−2.47739e11	−0.254799
$994$	0	0
$995$	5.41086e10i	0.0552045i
$996$	0	0
$997$	−1.31050e12	−1.32634	−0.663171	−	0.748468i	$-0.730790\pi$
$997$	−1.31050e12	−1.32634	−0.663171	+	0.748468i	$0.730790\pi$
$998$	0	0
$999$	3.55796e10i	0.0357223i

Twists

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

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

\(f(q)\)	\(=\)	\(q-46.7654i q^{3} +26.4806 q^{5} +1828.22i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q-46.7654i q^{3} +26.4806 q^{5} +1828.22i q^{7} -2187.00 q^{9} +877.801i q^{11} +13304.9 q^{13} -1238.37i q^{15} +142183. q^{17} -47907.0i q^{19} +85497.5 q^{21} -95162.9i q^{23} -389924. q^{25} +102276. i q^{27} -1.23844e6 q^{29} -841704. i q^{31} +41050.7 q^{33} +48412.4i q^{35} +347879. q^{37} -622207. i q^{39} +3.07799e6 q^{41} -431324. i q^{43} -57913.0 q^{45} +6.04692e6i q^{47} +2.42240e6 q^{49} -6.64925e6i q^{51} -2.11471e6 q^{53} +23244.7i q^{55} -2.24039e6 q^{57} +1.08465e7i q^{59} +3.50215e6 q^{61} -3.99832e6i q^{63} +352320. q^{65} +2.35736e7i q^{67} -4.45033e6 q^{69} +514773. i q^{71} +6.92754e6 q^{73} +1.82349e7i q^{75} -1.60482e6 q^{77} +4.59396e7i q^{79} +4.78297e6 q^{81} -4.83697e7i q^{83} +3.76509e6 q^{85} +5.79162e7i q^{87} -8.30165e6 q^{89} +2.43243e7i q^{91} -3.93626e7 q^{93} -1.26860e6i q^{95} +9.38015e7 q^{97} -1.91975e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 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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