LMFDB - Embedded newform 384.10.d.b.193.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,10,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, 10, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
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:	$197.773761087$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225 $ x^8 + 13062x^6 + 45211107x^4 + 45928424926*x^2 + 852972309225
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.6
Root			$-4.34998i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.10.d.b.193.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+81.0000i q^{3} -1428.10i q^{5} -6382.13 q^{7} -6561.00 q^{9} +O(q^{10})$	$q+81.0000i q^{3} -1428.10i q^{5} -6382.13 q^{7} -6561.00 q^{9} -4073.61i q^{11} -154396. i q^{13} +115676. q^{15} +255714. q^{17} -1.01702e6i q^{19} -516952. i q^{21} +1.56457e6 q^{23} -86331.5 q^{25} -531441. i q^{27} +7.26136e6i q^{29} +611897. q^{31} +329962. q^{33} +9.11429e6i q^{35} -464986. i q^{37} +1.25061e7 q^{39} +2.04743e7 q^{41} -2.95237e7i q^{43} +9.36973e6i q^{45} -1.31748e6 q^{47} +377951. q^{49} +2.07129e7i q^{51} +5.76562e7i q^{53} -5.81750e6 q^{55} +8.23785e7 q^{57} -6.35883e7i q^{59} -5.01862e7i q^{61} +4.18731e7 q^{63} -2.20493e8 q^{65} -2.90029e8i q^{67} +1.26730e8i q^{69} +3.18003e8 q^{71} -3.49122e8 q^{73} -6.99285e6i q^{75} +2.59983e7i q^{77} +4.29993e8 q^{79} +4.30467e7 q^{81} -3.40898e8i q^{83} -3.65185e8i q^{85} -5.88170e8 q^{87} +5.24211e8 q^{89} +9.85376e8i q^{91} +4.95636e7i q^{93} -1.45240e9 q^{95} -9.61784e8 q^{97} +2.67270e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 13632 q^{7} - 52488 q^{9}+O(q^{10}) $ 8 * q + 13632 * q^7 - 52488 * q^9	$ 8 q + 13632 q^{7} - 52488 q^{9} + 90720 q^{15} + 8304 q^{17} + 4612608 q^{23} - 4754904 q^{25} + 7499328 q^{31} - 6213024 q^{33} + 23211360 q^{39} - 43518896 q^{41} - 49382016 q^{47} - 74106808 q^{49} - 19030656 q^{55} + 38141280 q^{57} - 89439552 q^{63} - 110270336 q^{65} + 741751296 q^{71} - 1507903440 q^{73} - 1008373440 q^{79} + 344373768 q^{81} - 423468000 q^{87} - 1337034448 q^{89} + 543950208 q^{95} - 904817936 q^{97}+O(q^{100}) $ 8 * q + 13632 * q^7 - 52488 * q^9 + 90720 * q^15 + 8304 * q^17 + 4612608 * q^23 - 4754904 * q^25 + 7499328 * q^31 - 6213024 * q^33 + 23211360 * q^39 - 43518896 * q^41 - 49382016 * q^47 - 74106808 * q^49 - 19030656 * q^55 + 38141280 * q^57 - 89439552 * q^63 - 110270336 * q^65 + 741751296 * q^71 - 1507903440 * q^73 - 1008373440 * q^79 + 344373768 * q^81 - 423468000 * q^87 - 1337034448 * q^89 + 543950208 * q^95 - 904817936 * 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
$3$	81.0000i	0.577350i
$3$	81.0000i	0.577350i
$4$	0	0
$5$	− 1428.10i	− 1.02186i	−0.859622	−	0.510931i	$-0.829301\pi$
$5$	− 1428.10i	− 1.02186i	0.859622	−	0.510931i	$-0.170699\pi$
$6$	0	0
$7$	−6382.13	−1.00467	−0.502336	−	0.864672i	$-0.667526\pi$
$7$	−6382.13	−1.00467	−0.502336	+	0.864672i	$0.667526\pi$
$8$	0	0
$9$	−6561.00	−0.333333
$10$	0	0
$11$	− 4073.61i	− 0.0838904i	−0.999120	−	0.0419452i	$-0.986645\pi$
$11$	− 4073.61i	− 0.0838904i	0.999120	−	0.0419452i	$-0.0133555\pi$
$12$	0	0
$13$	− 154396.i	− 1.49931i	−0.661829	−	0.749655i	$-0.730219\pi$
$13$	− 154396.i	− 1.49931i	0.661829	−	0.749655i	$-0.269781\pi$
$14$	0	0
$15$	115676.	0.589972
$16$	0	0
$17$	255714.	0.742566	0.371283	−	0.928520i	$-0.378918\pi$
$17$	255714.	0.742566	0.371283	+	0.928520i	$0.378918\pi$
$18$	0	0
$19$	− 1.01702e6i	− 1.79035i	−0.445716	−	0.895175i	$-0.647051\pi$
$19$	− 1.01702e6i	− 1.79035i	0.445716	−	0.895175i	$-0.352949\pi$
$20$	0	0
$21$	− 516952.i	− 0.580048i
$22$	0	0
$23$	1.56457e6	1.16579	0.582894	−	0.812548i	$-0.301920\pi$
$23$	1.56457e6	1.16579	0.582894	+	0.812548i	$0.301920\pi$
$24$	0	0
$25$	−86331.5	−0.0442017
$26$	0	0
$27$	− 531441.i	− 0.192450i
$28$	0	0
$29$	7.26136e6i	1.90646i	0.302253	+	0.953228i	$0.402261\pi$
$29$	7.26136e6i	1.90646i	−0.302253	+	0.953228i	$0.597739\pi$
$30$	0	0
$31$	611897.	0.119001	0.0595005	−	0.998228i	$-0.481049\pi$
$31$	611897.	0.119001	0.0595005	+	0.998228i	$0.481049\pi$
$32$	0	0
$33$	329962.	0.0484342
$34$	0	0
$35$	9.11429e6i	1.02664i
$36$	0	0
$37$	− 464986.i	− 0.0407880i	−0.999792	−	0.0203940i	$-0.993508\pi$
$37$	− 464986.i	− 0.0407880i	0.999792	−	0.0203940i	$-0.00649205\pi$
$38$	0	0
$39$	1.25061e7	0.865627
$40$	0	0
$41$	2.04743e7	1.13157	0.565785	−	0.824553i	$-0.308573\pi$
$41$	2.04743e7	1.13157	0.565785	+	0.824553i	$0.308573\pi$
$42$	0	0
$43$	− 2.95237e7i	− 1.31693i	−0.752611	−	0.658465i	$-0.771206\pi$
$43$	− 2.95237e7i	− 1.31693i	0.752611	−	0.658465i	$-0.228794\pi$
$44$	0	0
$45$	9.36973e6i	0.340621i
$46$	0	0
$47$	−1.31748e6	−0.0393824	−0.0196912	−	0.999806i	$-0.506268\pi$
$47$	−1.31748e6	−0.0393824	−0.0196912	+	0.999806i	$0.506268\pi$
$48$	0	0
$49$	377951.	0.00936598
$50$	0	0
$51$	2.07129e7i	0.428721i
$52$	0	0
$53$	5.76562e7i	1.00370i	0.864954	+	0.501851i	$0.167347\pi$
$53$	5.76562e7i	1.00370i	−0.864954	+	0.501851i	$0.832653\pi$
$54$	0	0
$55$	−5.81750e6	−0.0857244
$56$	0	0
$57$	8.23785e7	1.03366
$58$	0	0
$59$	− 6.35883e7i	− 0.683193i	−0.939847	−	0.341596i	$-0.889032\pi$
$59$	− 6.35883e7i	− 0.683193i	0.939847	−	0.341596i	$-0.110968\pi$
$60$	0	0
$61$	− 5.01862e7i	− 0.464088i	−0.972705	−	0.232044i	$-0.925459\pi$
$61$	− 5.01862e7i	− 0.464088i	0.972705	−	0.232044i	$-0.0745413\pi$
$62$	0	0
$63$	4.18731e7	0.334891
$64$	0	0
$65$	−2.20493e8	−1.53209
$66$	0	0
$67$	− 2.90029e8i	− 1.75835i	−0.476499	−	0.879175i	$-0.658094\pi$
$67$	− 2.90029e8i	− 1.75835i	0.476499	−	0.879175i	$-0.341906\pi$
$68$	0	0
$69$	1.26730e8i	0.673068i
$70$	0	0
$71$	3.18003e8	1.48514	0.742572	−	0.669766i	$-0.233606\pi$
$71$	3.18003e8	1.48514	0.742572	+	0.669766i	$0.233606\pi$
$72$	0	0
$73$	−3.49122e8	−1.43888	−0.719440	−	0.694554i	$-0.755602\pi$
$73$	−3.49122e8	−1.43888	−0.719440	+	0.694554i	$0.755602\pi$
$74$	0	0
$75$	− 6.99285e6i	− 0.0255199i
$76$	0	0
$77$	2.59983e7i	0.0842824i
$78$	0	0
$79$	4.29993e8	1.24205	0.621025	−	0.783791i	$-0.286716\pi$
$79$	4.29993e8	1.24205	0.621025	+	0.783791i	$0.286716\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	− 3.40898e8i	− 0.788447i	−0.919015	−	0.394223i	$-0.871014\pi$
$83$	− 3.40898e8i	− 0.788447i	0.919015	−	0.394223i	$-0.128986\pi$
$84$	0	0
$85$	− 3.65185e8i	− 0.758800i
$86$	0	0
$87$	−5.88170e8	−1.10069
$88$	0	0
$89$	5.24211e8	0.885628	0.442814	−	0.896614i	$-0.353980\pi$
$89$	5.24211e8	0.885628	0.442814	+	0.896614i	$0.353980\pi$
$90$	0	0
$91$	9.85376e8i	1.50632i
$92$	0	0
$93$	4.95636e7i	0.0687053i
$94$	0	0
$95$	−1.45240e9	−1.82949
$96$	0	0
$97$	−9.61784e8	−1.10307	−0.551537	−	0.834150i	$-0.685958\pi$
$97$	−9.61784e8	−1.10307	−0.551537	+	0.834150i	$0.685958\pi$
$98$	0	0
$99$	2.67270e7i	0.0279635i
$100$	0	0
$101$	2.61811e8i	0.250346i	0.992135	+	0.125173i	$0.0399486\pi$
$101$	2.61811e8i	0.250346i	−0.992135	+	0.125173i	$0.960051\pi$
$102$	0	0
$103$	4.87298e8	0.426606	0.213303	−	0.976986i	$-0.431578\pi$
$103$	4.87298e8	0.426606	0.213303	+	0.976986i	$0.431578\pi$
$104$	0	0
$105$	−7.38257e8	−0.592729
$106$	0	0
$107$	1.63558e9i	1.20627i	0.797640	+	0.603134i	$0.206082\pi$
$107$	1.63558e9i	1.20627i	−0.797640	+	0.603134i	$0.793918\pi$
$108$	0	0
$109$	1.81852e9i	1.23396i	0.786981	+	0.616978i	$0.211643\pi$
$109$	1.81852e9i	1.23396i	−0.786981	+	0.616978i	$0.788357\pi$
$110$	0	0
$111$	3.76639e7	0.0235489
$112$	0	0
$113$	−2.89653e9	−1.67119	−0.835594	−	0.549348i	$-0.814876\pi$
$113$	−2.89653e9	−1.67119	−0.835594	+	0.549348i	$0.814876\pi$
$114$	0	0
$115$	− 2.23435e9i	− 1.19127i
$116$	0	0
$117$	1.01299e9i	0.499770i
$118$	0	0
$119$	−1.63200e9	−0.746036
$120$	0	0
$121$	2.34135e9	0.992962
$122$	0	0
$123$	1.65842e9i	0.653312i
$124$	0	0
$125$	− 2.66596e9i	− 0.976694i
$126$	0	0
$127$	4.08260e8	0.139258	0.0696290	−	0.997573i	$-0.477818\pi$
$127$	4.08260e8	0.139258	0.0696290	+	0.997573i	$0.477818\pi$
$128$	0	0
$129$	2.39142e9	0.760330
$130$	0	0
$131$	1.67425e9i	0.496706i	0.968670	+	0.248353i	$0.0798893\pi$
$131$	1.67425e9i	0.496706i	−0.968670	+	0.248353i	$0.920111\pi$
$132$	0	0
$133$	6.49074e9i	1.79871i
$134$	0	0
$135$	−7.58948e8	−0.196657
$136$	0	0
$137$	−1.57362e9	−0.381642	−0.190821	−	0.981625i	$-0.561115\pi$
$137$	−1.57362e9	−0.381642	−0.190821	+	0.981625i	$0.561115\pi$
$138$	0	0
$139$	− 5.36040e9i	− 1.21795i	−0.793188	−	0.608977i	$-0.791580\pi$
$139$	− 5.36040e9i	− 1.21795i	0.793188	−	0.608977i	$-0.208420\pi$
$140$	0	0
$141$	− 1.06715e8i	− 0.0227374i
$142$	0	0
$143$	−6.28950e8	−0.125778
$144$	0	0
$145$	1.03699e10	1.94813
$146$	0	0
$147$	3.06140e7i	0.00540745i
$148$	0	0
$149$	− 1.83174e9i	− 0.304457i	−0.988345	−	0.152228i	$-0.951355\pi$
$149$	− 1.83174e9i	− 0.304457i	0.988345	−	0.152228i	$-0.0486449\pi$
$150$	0	0
$151$	−4.73373e9	−0.740981	−0.370490	−	0.928836i	$-0.620810\pi$
$151$	−4.73373e9	−0.740981	−0.370490	+	0.928836i	$0.620810\pi$
$152$	0	0
$153$	−1.67774e9	−0.247522
$154$	0	0
$155$	− 8.73847e8i	− 0.121603i
$156$	0	0
$157$	− 8.73612e8i	− 0.114755i	−0.998353	−	0.0573773i	$-0.981726\pi$
$157$	− 8.73612e8i	− 0.114755i	0.998353	−	0.0573773i	$-0.0182738\pi$
$158$	0	0
$159$	−4.67015e9	−0.579487
$160$	0	0
$161$	−9.98528e9	−1.17123
$162$	0	0
$163$	− 8.32642e9i	− 0.923877i	−0.886912	−	0.461939i	$-0.847154\pi$
$163$	− 8.32642e9i	− 0.923877i	0.886912	−	0.461939i	$-0.152846\pi$
$164$	0	0
$165$	− 4.71218e8i	− 0.0494930i
$166$	0	0
$167$	−1.58090e9	−0.157283	−0.0786414	−	0.996903i	$-0.525058\pi$
$167$	−1.58090e9	−0.157283	−0.0786414	+	0.996903i	$0.525058\pi$
$168$	0	0
$169$	−1.32337e10	−1.24793
$170$	0	0
$171$	6.67266e9i	0.596783i
$172$	0	0
$173$	− 1.19275e10i	− 1.01238i	−0.862423	−	0.506189i	$-0.831054\pi$
$173$	− 1.19275e10i	− 1.01238i	0.862423	−	0.506189i	$-0.168946\pi$
$174$	0	0
$175$	5.50979e8	0.0444082
$176$	0	0
$177$	5.15065e9	0.394441
$178$	0	0
$179$	− 1.27369e10i	− 0.927309i	−0.886016	−	0.463654i	$-0.846538\pi$
$179$	− 1.27369e10i	− 0.927309i	0.886016	−	0.463654i	$-0.153462\pi$
$180$	0	0
$181$	− 8.46716e9i	− 0.586387i	−0.956053	−	0.293193i	$-0.905282\pi$
$181$	− 8.46716e9i	− 0.586387i	0.956053	−	0.293193i	$-0.0947180\pi$
$182$	0	0
$183$	4.06508e9	0.267941
$184$	0	0
$185$	−6.64044e8	−0.0416797
$186$	0	0
$187$	− 1.04168e9i	− 0.0622942i
$188$	0	0
$189$	3.39172e9i	0.193349i
$190$	0	0
$191$	3.90757e9	0.212450	0.106225	−	0.994342i	$-0.466124\pi$
$191$	3.90757e9	0.212450	0.106225	+	0.994342i	$0.466124\pi$
$192$	0	0
$193$	1.82705e10	0.947858	0.473929	−	0.880563i	$-0.342835\pi$
$193$	1.82705e10	0.947858	0.473929	+	0.880563i	$0.342835\pi$
$194$	0	0
$195$	− 1.78599e10i	− 0.884552i
$196$	0	0
$197$	− 1.05198e10i	− 0.497633i	−0.968551	−	0.248816i	$-0.919958\pi$
$197$	− 1.05198e10i	− 0.497633i	0.968551	−	0.248816i	$-0.0800416\pi$
$198$	0	0
$199$	3.02525e10	1.36748	0.683741	−	0.729725i	$-0.260352\pi$
$199$	3.02525e10	1.36748	0.683741	+	0.729725i	$0.260352\pi$
$200$	0	0
$201$	2.34924e10	1.01518
$202$	0	0
$203$	− 4.63429e10i	− 1.91536i
$204$	0	0
$205$	− 2.92392e10i	− 1.15631i
$206$	0	0
$207$	−1.02651e10	−0.388596
$208$	0	0
$209$	−4.14294e9	−0.150193
$210$	0	0
$211$	− 1.53444e10i	− 0.532940i	−0.963843	−	0.266470i	$-0.914143\pi$
$211$	− 1.53444e10i	− 0.532940i	0.963843	−	0.266470i	$-0.0858573\pi$
$212$	0	0
$213$	2.57582e10i	0.857448i
$214$	0	0
$215$	−4.21626e10	−1.34572
$216$	0	0
$217$	−3.90520e9	−0.119557
$218$	0	0
$219$	− 2.82789e10i	− 0.830738i
$220$	0	0
$221$	− 3.94814e10i	− 1.11334i
$222$	0	0
$223$	−6.26805e10	−1.69731	−0.848654	−	0.528948i	$-0.822586\pi$
$223$	−6.26805e10	−1.69731	−0.848654	+	0.528948i	$0.822586\pi$
$224$	0	0
$225$	5.66421e8	0.0147339
$226$	0	0
$227$	4.85806e10i	1.21436i	0.794565	+	0.607179i	$0.207699\pi$
$227$	4.85806e10i	1.21436i	−0.794565	+	0.607179i	$0.792301\pi$
$228$	0	0
$229$	6.64074e10i	1.59572i	0.602843	+	0.797860i	$0.294035\pi$
$229$	6.64074e10i	1.59572i	−0.602843	+	0.797860i	$0.705965\pi$
$230$	0	0
$231$	−2.10586e9	−0.0486605
$232$	0	0
$233$	1.53115e10	0.340342	0.170171	−	0.985415i	$-0.445568\pi$
$233$	1.53115e10	0.340342	0.170171	+	0.985415i	$0.445568\pi$
$234$	0	0
$235$	1.88148e9i	0.0402434i
$236$	0	0
$237$	3.48294e10i	0.717098i
$238$	0	0
$239$	−5.49921e10	−1.09021	−0.545105	−	0.838368i	$-0.683510\pi$
$239$	−5.49921e10	−1.09021	−0.545105	+	0.838368i	$0.683510\pi$
$240$	0	0
$241$	−1.88950e10	−0.360802	−0.180401	−	0.983593i	$-0.557740\pi$
$241$	−1.88950e10	−0.360802	−0.180401	+	0.983593i	$0.557740\pi$
$242$	0	0
$243$	3.48678e9i	0.0641500i
$244$	0	0
$245$	− 5.39750e8i	− 0.00957074i
$246$	0	0
$247$	−1.57024e11	−2.68429
$248$	0	0
$249$	2.76127e10	0.455210
$250$	0	0
$251$	− 1.10372e11i	− 1.75520i	−0.479395	−	0.877599i	$-0.659144\pi$
$251$	− 1.10372e11i	− 1.75520i	0.479395	−	0.877599i	$-0.340856\pi$
$252$	0	0
$253$	− 6.37344e9i	− 0.0977984i
$254$	0	0
$255$	2.95800e10	0.438093
$256$	0	0
$257$	2.31328e10	0.330772	0.165386	−	0.986229i	$-0.447113\pi$
$257$	2.31328e10	0.330772	0.165386	+	0.986229i	$0.447113\pi$
$258$	0	0
$259$	2.96760e9i	0.0409785i
$260$	0	0
$261$	− 4.76418e10i	− 0.635485i
$262$	0	0
$263$	1.88727e10	0.243239	0.121619	−	0.992577i	$-0.461191\pi$
$263$	1.88727e10	0.243239	0.121619	+	0.992577i	$0.461191\pi$
$264$	0	0
$265$	8.23385e10	1.02564
$266$	0	0
$267$	4.24611e10i	0.511317i
$268$	0	0
$269$	6.86708e10i	0.799626i	0.916597	+	0.399813i	$0.130925\pi$
$269$	6.86708e10i	0.799626i	−0.916597	+	0.399813i	$0.869075\pi$
$270$	0	0
$271$	−1.15508e11	−1.30092	−0.650460	−	0.759540i	$-0.725424\pi$
$271$	−1.15508e11	−1.30092	−0.650460	+	0.759540i	$0.725424\pi$
$272$	0	0
$273$	−7.98155e10	−0.869672
$274$	0	0
$275$	3.51681e8i	0.00370810i
$276$	0	0
$277$	− 4.52642e10i	− 0.461950i	−0.972960	−	0.230975i	$-0.925808\pi$
$277$	− 4.52642e10i	− 0.461950i	0.972960	−	0.230975i	$-0.0741916\pi$
$278$	0	0
$279$	−4.01465e9	−0.0396670
$280$	0	0
$281$	−9.00026e10	−0.861146	−0.430573	−	0.902556i	$-0.641688\pi$
$281$	−9.00026e10	−0.861146	−0.430573	+	0.902556i	$0.641688\pi$
$282$	0	0
$283$	− 9.87406e10i	− 0.915076i	−0.889190	−	0.457538i	$-0.848731\pi$
$283$	− 9.87406e10i	− 0.915076i	0.889190	−	0.457538i	$-0.151269\pi$
$284$	0	0
$285$	− 1.17644e11i	− 1.05626i
$286$	0	0
$287$	−1.30670e11	−1.13686
$288$	0	0
$289$	−5.31980e10	−0.448595
$290$	0	0
$291$	− 7.79045e10i	− 0.636860i
$292$	0	0
$293$	8.54059e10i	0.676992i	0.940968	+	0.338496i	$0.109918\pi$
$293$	8.54059e10i	0.676992i	−0.940968	+	0.338496i	$0.890082\pi$
$294$	0	0
$295$	−9.08101e10	−0.698128
$296$	0	0
$297$	−2.16488e9	−0.0161447
$298$	0	0
$299$	− 2.41563e11i	− 1.74788i
$300$	0	0
$301$	1.88424e11i	1.32308i
$302$	0	0
$303$	−2.12067e10	−0.144537
$304$	0	0
$305$	−7.16707e10	−0.474234
$306$	0	0
$307$	− 1.83582e11i	− 1.17953i	−0.807576	−	0.589763i	$-0.799221\pi$
$307$	− 1.83582e11i	− 1.17953i	0.807576	−	0.589763i	$-0.200779\pi$
$308$	0	0
$309$	3.94711e10i	0.246301i
$310$	0	0
$311$	1.13932e11	0.690598	0.345299	−	0.938493i	$-0.387778\pi$
$311$	1.13932e11	0.690598	0.345299	+	0.938493i	$0.387778\pi$
$312$	0	0
$313$	−2.41679e11	−1.42328	−0.711640	−	0.702545i	$-0.752047\pi$
$313$	−2.41679e11	−1.42328	−0.711640	+	0.702545i	$0.752047\pi$
$314$	0	0
$315$	− 5.97988e10i	− 0.342212i
$316$	0	0
$317$	− 1.06227e11i	− 0.590840i	−0.955367	−	0.295420i	$-0.904540\pi$
$317$	− 1.06227e11i	− 0.590840i	0.955367	−	0.295420i	$-0.0954596\pi$
$318$	0	0
$319$	2.95799e10	0.159933
$320$	0	0
$321$	−1.32482e11	−0.696440
$322$	0	0
$323$	− 2.60066e11i	− 1.32945i
$324$	0	0
$325$	1.33293e10i	0.0662721i
$326$	0	0
$327$	−1.47300e11	−0.712424
$328$	0	0
$329$	8.40830e9	0.0395664
$330$	0	0
$331$	6.82287e10i	0.312422i	0.987724	+	0.156211i	$0.0499279\pi$
$331$	6.82287e10i	0.312422i	−0.987724	+	0.156211i	$0.950072\pi$
$332$	0	0
$333$	3.05077e9i	0.0135960i
$334$	0	0
$335$	−4.14190e11	−1.79679
$336$	0	0
$337$	3.87135e11	1.63504	0.817518	−	0.575903i	$-0.195349\pi$
$337$	3.87135e11	1.63504	0.817518	+	0.575903i	$0.195349\pi$
$338$	0	0
$339$	− 2.34619e11i	− 0.964861i
$340$	0	0
$341$	− 2.49263e9i	− 0.00998304i
$342$	0	0
$343$	2.55130e11	0.995262
$344$	0	0
$345$	1.80983e11	0.687782
$346$	0	0
$347$	2.81800e11i	1.04342i	0.853124	+	0.521709i	$0.174705\pi$
$347$	2.81800e11i	1.04342i	−0.853124	+	0.521709i	$0.825295\pi$
$348$	0	0
$349$	− 1.80174e11i	− 0.650096i	−0.945697	−	0.325048i	$-0.894619\pi$
$349$	− 1.80174e11i	− 0.650096i	0.945697	−	0.325048i	$-0.105381\pi$
$350$	0	0
$351$	−8.20525e10	−0.288542
$352$	0	0
$353$	1.18821e11	0.407294	0.203647	−	0.979044i	$-0.434721\pi$
$353$	1.18821e11	0.407294	0.203647	+	0.979044i	$0.434721\pi$
$354$	0	0
$355$	− 4.54139e11i	− 1.51761i
$356$	0	0
$357$	− 1.32192e11i	− 0.430724i
$358$	0	0
$359$	−4.95475e11	−1.57433	−0.787167	−	0.616740i	$-0.788453\pi$
$359$	−4.95475e11	−1.57433	−0.787167	+	0.616740i	$0.788453\pi$
$360$	0	0
$361$	−7.11639e11	−2.20535
$362$	0	0
$363$	1.89650e11i	0.573287i
$364$	0	0
$365$	4.98580e11i	1.47034i
$366$	0	0
$367$	−5.16234e11	−1.48542	−0.742709	−	0.669614i	$-0.766460\pi$
$367$	−5.16234e11	−1.48542	−0.742709	+	0.669614i	$0.766460\pi$
$368$	0	0
$369$	−1.34332e11	−0.377190
$370$	0	0
$371$	− 3.67969e11i	− 1.00839i
$372$	0	0
$373$	− 2.26879e11i	− 0.606881i	−0.952850	−	0.303441i	$-0.901865\pi$
$373$	− 2.26879e11i	− 0.606881i	0.952850	−	0.303441i	$-0.0981354\pi$
$374$	0	0
$375$	2.15943e11	0.563894
$376$	0	0
$377$	1.12113e12	2.85837
$378$	0	0
$379$	− 5.96753e11i	− 1.48566i	−0.669482	−	0.742828i	$-0.733484\pi$
$379$	− 5.96753e11i	− 1.48566i	0.669482	−	0.742828i	$-0.266516\pi$
$380$	0	0
$381$	3.30691e10i	0.0804007i
$382$	0	0
$383$	3.71694e11	0.882654	0.441327	−	0.897346i	$-0.354508\pi$
$383$	3.71694e11	0.882654	0.441327	+	0.897346i	$0.354508\pi$
$384$	0	0
$385$	3.71281e10	0.0861249
$386$	0	0
$387$	1.93705e11i	0.438977i
$388$	0	0
$389$	2.71241e11i	0.600596i	0.953845	+	0.300298i	$0.0970862\pi$
$389$	2.71241e11i	0.600596i	−0.953845	+	0.300298i	$0.902914\pi$
$390$	0	0
$391$	4.00083e11	0.865674
$392$	0	0
$393$	−1.35614e11	−0.286773
$394$	0	0
$395$	− 6.14071e11i	− 1.26920i
$396$	0	0
$397$	3.00982e11i	0.608112i	0.952654	+	0.304056i	$0.0983410\pi$
$397$	3.00982e11i	0.608112i	−0.952654	+	0.304056i	$0.901659\pi$
$398$	0	0
$399$	−5.25750e11	−1.03849
$400$	0	0
$401$	3.10723e11	0.600101	0.300050	−	0.953923i	$-0.402997\pi$
$401$	3.10723e11	0.600101	0.300050	+	0.953923i	$0.402997\pi$
$402$	0	0
$403$	− 9.44746e10i	− 0.178419i
$404$	0	0
$405$	− 6.14748e10i	− 0.113540i
$406$	0	0
$407$	−1.89417e9	−0.00342172
$408$	0	0
$409$	−4.46642e11	−0.789231	−0.394616	−	0.918846i	$-0.629122\pi$
$409$	−4.46642e11	−0.789231	−0.394616	+	0.918846i	$0.629122\pi$
$410$	0	0
$411$	− 1.27463e11i	− 0.220341i
$412$	0	0
$413$	4.05829e11i	0.686384i
$414$	0	0
$415$	−4.86834e11	−0.805684
$416$	0	0
$417$	4.34193e11	0.703186
$418$	0	0
$419$	8.23618e11i	1.30546i	0.757592	+	0.652728i	$0.226376\pi$
$419$	8.23618e11i	1.30546i	−0.757592	+	0.652728i	$0.773624\pi$
$420$	0	0
$421$	− 6.04524e11i	− 0.937872i	−0.883232	−	0.468936i	$-0.844637\pi$
$421$	− 6.04524e11i	− 0.937872i	0.883232	−	0.468936i	$-0.155363\pi$
$422$	0	0
$423$	8.64395e9	0.0131275
$424$	0	0
$425$	−2.20762e10	−0.0328227
$426$	0	0
$427$	3.20295e11i	0.466256i
$428$	0	0
$429$	− 5.09450e10i	− 0.0726179i
$430$	0	0
$431$	−1.20184e12	−1.67765	−0.838823	−	0.544404i	$-0.816756\pi$
$431$	−1.20184e12	−1.67765	−0.838823	+	0.544404i	$0.816756\pi$
$432$	0	0
$433$	1.53930e11	0.210440	0.105220	−	0.994449i	$-0.466445\pi$
$433$	1.53930e11	0.210440	0.105220	+	0.994449i	$0.466445\pi$
$434$	0	0
$435$	8.39963e11i	1.12476i
$436$	0	0
$437$	− 1.59120e12i	− 2.08717i
$438$	0	0
$439$	−4.56899e11	−0.587124	−0.293562	−	0.955940i	$-0.594841\pi$
$439$	−4.56899e11	−0.587124	−0.293562	+	0.955940i	$0.594841\pi$
$440$	0	0
$441$	−2.47974e9	−0.00312199
$442$	0	0
$443$	1.43507e12i	1.77034i	0.465271	+	0.885168i	$0.345957\pi$
$443$	1.43507e12i	1.77034i	−0.465271	+	0.885168i	$0.654043\pi$
$444$	0	0
$445$	− 7.48623e11i	− 0.904989i
$446$	0	0
$447$	1.48371e11	0.175778
$448$	0	0
$449$	−9.54602e11	−1.10844	−0.554222	−	0.832369i	$-0.686984\pi$
$449$	−9.54602e11	−1.10844	−0.554222	+	0.832369i	$0.686984\pi$
$450$	0	0
$451$	− 8.34043e10i	− 0.0949279i
$452$	0	0
$453$	− 3.83432e11i	− 0.427805i
$454$	0	0
$455$	1.40721e12	1.53925
$456$	0	0
$457$	4.57857e11	0.491029	0.245515	−	0.969393i	$-0.421043\pi$
$457$	4.57857e11	0.491029	0.245515	+	0.969393i	$0.421043\pi$
$458$	0	0
$459$	− 1.35897e11i	− 0.142907i
$460$	0	0
$461$	− 1.58756e12i	− 1.63710i	−0.574437	−	0.818549i	$-0.694779\pi$
$461$	− 1.58756e12i	− 1.63710i	0.574437	−	0.818549i	$-0.305221\pi$
$462$	0	0
$463$	−7.26819e11	−0.735041	−0.367521	−	0.930015i	$-0.619793\pi$
$463$	−7.26819e11	−0.735041	−0.367521	+	0.930015i	$0.619793\pi$
$464$	0	0
$465$	7.07816e10	0.0702073
$466$	0	0
$467$	− 8.73721e11i	− 0.850054i	−0.905181	−	0.425027i	$-0.860265\pi$
$467$	− 8.73721e11i	− 0.850054i	0.905181	−	0.425027i	$-0.139735\pi$
$468$	0	0
$469$	1.85100e12i	1.76657i
$470$	0	0
$471$	7.07626e10	0.0662536
$472$	0	0
$473$	−1.20268e11	−0.110478
$474$	0	0
$475$	8.78007e10i	0.0791365i
$476$	0	0
$477$	− 3.78282e11i	− 0.334567i
$478$	0	0
$479$	−6.89800e11	−0.598706	−0.299353	−	0.954142i	$-0.596771\pi$
$479$	−6.89800e11	−0.598706	−0.299353	+	0.954142i	$0.596771\pi$
$480$	0	0
$481$	−7.17921e10	−0.0611538
$482$	0	0
$483$	− 8.08807e11i	− 0.676212i
$484$	0	0
$485$	1.37352e12i	1.12719i
$486$	0	0
$487$	2.05185e12	1.65297	0.826485	−	0.562959i	$-0.190337\pi$
$487$	2.05185e12	1.65297	0.826485	+	0.562959i	$0.190337\pi$
$488$	0	0
$489$	6.74440e11	0.533401
$490$	0	0
$491$	− 5.51164e11i	− 0.427971i	−0.976837	−	0.213985i	$-0.931356\pi$
$491$	− 5.51164e11i	− 0.427971i	0.976837	−	0.213985i	$-0.0686445\pi$
$492$	0	0
$493$	1.85683e12i	1.41567i
$494$	0	0
$495$	3.81686e10	0.0285748
$496$	0	0
$497$	−2.02954e12	−1.49208
$498$	0	0
$499$	1.99167e12i	1.43802i	0.695000	+	0.719009i	$0.255404\pi$
$499$	1.99167e12i	1.43802i	−0.695000	+	0.719009i	$0.744596\pi$
$500$	0	0
$501$	− 1.28053e11i	− 0.0908073i
$502$	0	0
$503$	2.42693e11	0.169044	0.0845221	−	0.996422i	$-0.473064\pi$
$503$	2.42693e11	0.169044	0.0845221	+	0.996422i	$0.473064\pi$
$504$	0	0
$505$	3.73891e11	0.255819
$506$	0	0
$507$	− 1.07193e12i	− 0.720494i
$508$	0	0
$509$	7.46973e11i	0.493259i	0.969110	+	0.246629i	$0.0793230\pi$
$509$	7.46973e11i	0.493259i	−0.969110	+	0.246629i	$0.920677\pi$
$510$	0	0
$511$	2.22814e12	1.44560
$512$	0	0
$513$	−5.40485e11	−0.344553
$514$	0	0
$515$	− 6.95908e11i	− 0.435933i
$516$	0	0
$517$	5.36688e9i	0.00330381i
$518$	0	0
$519$	9.66128e11	0.584496
$520$	0	0
$521$	1.38178e12	0.821619	0.410809	−	0.911721i	$-0.365246\pi$
$521$	1.38178e12	0.821619	0.410809	+	0.911721i	$0.365246\pi$
$522$	0	0
$523$	− 7.83681e10i	− 0.0458017i	−0.999738	−	0.0229008i	$-0.992710\pi$
$523$	− 7.83681e10i	− 0.0458017i	0.999738	−	0.0229008i	$-0.00729020\pi$
$524$	0	0
$525$	4.46293e10i	0.0256391i
$526$	0	0
$527$	1.56471e11	0.0883661
$528$	0	0
$529$	6.46722e11	0.359060
$530$	0	0
$531$	4.17203e11i	0.227731i
$532$	0	0
$533$	− 3.16115e12i	− 1.69658i
$534$	0	0
$535$	2.33576e12	1.23264
$536$	0	0
$537$	1.03169e12	0.535382
$538$	0	0
$539$	− 1.53963e9i	0 0.000785716i
$540$	0	0
$541$	1.88658e12i	0.946865i	0.880830	+	0.473433i	$0.156985\pi$
$541$	1.88658e12i	0.946865i	−0.880830	+	0.473433i	$0.843015\pi$
$542$	0	0
$543$	6.85840e11	0.338551
$544$	0	0
$545$	2.59702e12	1.26093
$546$	0	0
$547$	3.29535e12i	1.57383i	0.617061	+	0.786915i	$0.288323\pi$
$547$	3.29535e12i	1.57383i	−0.617061	+	0.786915i	$0.711677\pi$
$548$	0	0
$549$	3.29272e11i	0.154696i
$550$	0	0
$551$	7.38493e12	3.41322
$552$	0	0
$553$	−2.74427e12	−1.24785
$554$	0	0
$555$	− 5.37876e10i	− 0.0240638i
$556$	0	0
$557$	− 1.26338e12i	− 0.556143i	−0.960560	−	0.278072i	$-0.910305\pi$
$557$	− 1.26338e12i	− 0.556143i	0.960560	−	0.278072i	$-0.0896952\pi$
$558$	0	0
$559$	−4.55835e12	−1.97449
$560$	0	0
$561$	8.43762e10	0.0359656
$562$	0	0
$563$	2.18128e12i	0.915004i	0.889209	+	0.457502i	$0.151256\pi$
$563$	2.18128e12i	0.915004i	−0.889209	+	0.457502i	$0.848744\pi$
$564$	0	0
$565$	4.13652e12i	1.70772i
$566$	0	0
$567$	−2.74730e11	−0.111630
$568$	0	0
$569$	−3.77066e12	−1.50804	−0.754020	−	0.656852i	$-0.771888\pi$
$569$	−3.77066e12	−1.50804	−0.754020	+	0.656852i	$0.771888\pi$
$570$	0	0
$571$	− 4.82244e10i	− 0.0189847i	−0.999955	−	0.00949236i	$-0.996978\pi$
$571$	− 4.82244e10i	− 0.0189847i	0.999955	−	0.00949236i	$-0.00302156\pi$
$572$	0	0
$573$	3.16513e11i	0.122658i
$574$	0	0
$575$	−1.35072e11	−0.0515298
$576$	0	0
$577$	−3.41311e12	−1.28192	−0.640958	−	0.767576i	$-0.721463\pi$
$577$	−3.41311e12	−1.28192	−0.640958	+	0.767576i	$0.721463\pi$
$578$	0	0
$579$	1.47991e12i	0.547246i
$580$	0	0
$581$	2.17565e12i	0.792131i
$582$	0	0
$583$	2.34869e11	0.0842009
$584$	0	0
$585$	1.44665e12	0.510696
$586$	0	0
$587$	5.13080e12i	1.78366i	0.452366	+	0.891832i	$0.350580\pi$
$587$	5.13080e12i	1.78366i	−0.452366	+	0.891832i	$0.649420\pi$
$588$	0	0
$589$	− 6.22310e11i	− 0.213053i
$590$	0	0
$591$	8.52103e11	0.287308
$592$	0	0
$593$	3.28338e12	1.09037	0.545186	−	0.838315i	$-0.316459\pi$
$593$	3.28338e12	1.09037	0.545186	+	0.838315i	$0.316459\pi$
$594$	0	0
$595$	2.33066e12i	0.762345i
$596$	0	0
$597$	2.45045e12i	0.789516i
$598$	0	0
$599$	2.83692e12	0.900382	0.450191	−	0.892932i	$-0.351356\pi$
$599$	2.83692e12	0.900382	0.450191	+	0.892932i	$0.351356\pi$
$600$	0	0
$601$	−2.83184e11	−0.0885389	−0.0442694	−	0.999020i	$-0.514096\pi$
$601$	−2.83184e11	−0.0885389	−0.0442694	+	0.999020i	$0.514096\pi$
$602$	0	0
$603$	1.90288e12i	0.586117i
$604$	0	0
$605$	− 3.34368e12i	− 1.01467i
$606$	0	0
$607$	−2.38906e12	−0.714294	−0.357147	−	0.934048i	$-0.616251\pi$
$607$	−2.38906e12	−0.714294	−0.357147	+	0.934048i	$0.616251\pi$
$608$	0	0
$609$	3.75378e12	1.10584
$610$	0	0
$611$	2.03413e11i	0.0590464i
$612$	0	0
$613$	− 6.27251e11i	− 0.179419i	−0.995968	−	0.0897097i	$-0.971406\pi$
$613$	− 6.27251e11i	− 0.179419i	0.995968	−	0.0897097i	$-0.0285939\pi$
$614$	0	0
$615$	2.36838e12	0.667595
$616$	0	0
$617$	−6.57062e12	−1.82525	−0.912626	−	0.408794i	$-0.865949\pi$
$617$	−6.57062e12	−1.82525	−0.912626	+	0.408794i	$0.865949\pi$
$618$	0	0
$619$	1.27976e12i	0.350366i	0.984536	+	0.175183i	$0.0560517\pi$
$619$	1.27976e12i	0.350366i	−0.984536	+	0.175183i	$0.943948\pi$
$620$	0	0
$621$	− 8.31476e11i	− 0.224356i
$622$	0	0
$623$	−3.34558e12	−0.889765
$624$	0	0
$625$	−3.97586e12	−1.04225
$626$	0	0
$627$	− 3.35578e11i	− 0.0867141i
$628$	0	0
$629$	− 1.18904e11i	− 0.0302878i
$630$	0	0
$631$	6.41367e12	1.61055	0.805276	−	0.592900i	$-0.202017\pi$
$631$	6.41367e12	1.61055	0.805276	+	0.592900i	$0.202017\pi$
$632$	0	0
$633$	1.24289e12	0.307693
$634$	0	0
$635$	− 5.83035e11i	− 0.142302i
$636$	0	0
$637$	− 5.83542e10i	− 0.0140425i
$638$	0	0
$639$	−2.08642e12	−0.495048
$640$	0	0
$641$	2.19384e11	0.0513266	0.0256633	−	0.999671i	$-0.491830\pi$
$641$	2.19384e11	0.0513266	0.0256633	+	0.999671i	$0.491830\pi$
$642$	0	0
$643$	− 1.43315e12i	− 0.330629i	−0.986241	−	0.165315i	$-0.947136\pi$
$643$	− 1.43315e12i	− 0.330629i	0.986241	−	0.165315i	$-0.0528639\pi$
$644$	0	0
$645$	− 3.41517e12i	− 0.776952i
$646$	0	0
$647$	2.24505e12	0.503682	0.251841	−	0.967769i	$-0.418964\pi$
$647$	2.24505e12	0.503682	0.251841	+	0.967769i	$0.418964\pi$
$648$	0	0
$649$	−2.59034e11	−0.0573133
$650$	0	0
$651$	− 3.16322e11i	− 0.0690262i
$652$	0	0
$653$	− 4.25852e12i	− 0.916535i	−0.888814	−	0.458267i	$-0.848470\pi$
$653$	− 4.25852e12i	− 0.916535i	0.888814	−	0.458267i	$-0.151530\pi$
$654$	0	0
$655$	2.39099e12	0.507565
$656$	0	0
$657$	2.29059e12	0.479627
$658$	0	0
$659$	2.98594e12i	0.616733i	0.951268	+	0.308367i	$0.0997823\pi$
$659$	2.98594e12i	0.616733i	−0.951268	+	0.308367i	$0.900218\pi$
$660$	0	0
$661$	3.60046e11i	0.0733587i	0.999327	+	0.0366794i	$0.0116780\pi$
$661$	3.60046e11i	0.0733587i	−0.999327	+	0.0366794i	$0.988322\pi$
$662$	0	0
$663$	3.19799e12	0.642786
$664$	0	0
$665$	9.26940e12	1.83804
$666$	0	0
$667$	1.13609e13i	2.22252i
$668$	0	0
$669$	− 5.07712e12i	− 0.979941i
$670$	0	0
$671$	−2.04439e11	−0.0389325
$672$	0	0
$673$	−9.00012e12	−1.69114	−0.845572	−	0.533862i	$-0.820740\pi$
$673$	−9.00012e12	−1.69114	−0.845572	+	0.533862i	$0.820740\pi$
$674$	0	0
$675$	4.58801e10i	0.00850662i
$676$	0	0
$677$	− 3.09265e12i	− 0.565825i	−0.959146	−	0.282913i	$-0.908699\pi$
$677$	− 3.09265e12i	− 0.565825i	0.959146	−	0.282913i	$-0.0913006\pi$
$678$	0	0
$679$	6.13823e12	1.10823
$680$	0	0
$681$	−3.93503e12	−0.701109
$682$	0	0
$683$	9.11567e12i	1.60286i	0.598090	+	0.801429i	$0.295927\pi$
$683$	9.11567e12i	1.60286i	−0.598090	+	0.801429i	$0.704073\pi$
$684$	0	0
$685$	2.24727e12i	0.389985i
$686$	0	0
$687$	−5.37900e12	−0.921289
$688$	0	0
$689$	8.90190e12	1.50486
$690$	0	0
$691$	− 8.66620e12i	− 1.44603i	−0.690832	−	0.723016i	$-0.742755\pi$
$691$	− 8.66620e12i	− 1.44603i	0.690832	−	0.723016i	$-0.257245\pi$
$692$	0	0
$693$	− 1.70575e11i	− 0.0280941i
$694$	0	0
$695$	−7.65517e12	−1.24458
$696$	0	0
$697$	5.23557e12	0.840266
$698$	0	0
$699$	1.24023e12i	0.196497i
$700$	0	0
$701$	1.21255e13i	1.89657i	0.317417	+	0.948286i	$0.397184\pi$
$701$	1.21255e13i	1.89657i	−0.317417	+	0.948286i	$0.602816\pi$
$702$	0	0
$703$	−4.72899e11	−0.0730247
$704$	0	0
$705$	−1.52400e11	−0.0232345
$706$	0	0
$707$	− 1.67091e12i	− 0.251516i
$708$	0	0
$709$	6.84961e12i	1.01802i	0.860759	+	0.509012i	$0.169989\pi$
$709$	6.84961e12i	1.01802i	−0.860759	+	0.509012i	$0.830011\pi$
$710$	0	0
$711$	−2.82118e12	−0.414017
$712$	0	0
$713$	9.57354e11	0.138730
$714$	0	0
$715$	8.98201e11i	0.128528i
$716$	0	0
$717$	− 4.45436e12i	− 0.629433i
$718$	0	0
$719$	3.22078e12	0.449450	0.224725	−	0.974422i	$-0.427852\pi$
$719$	3.22078e12	0.449450	0.224725	+	0.974422i	$0.427852\pi$
$720$	0	0
$721$	−3.11000e12	−0.428599
$722$	0	0
$723$	− 1.53049e12i	− 0.208309i
$724$	0	0
$725$	− 6.26884e11i	− 0.0842686i
$726$	0	0
$727$	1.26034e13	1.67334	0.836669	−	0.547708i	$-0.184500\pi$
$727$	1.26034e13	1.67334	0.836669	+	0.547708i	$0.184500\pi$
$728$	0	0
$729$	−2.82430e11	−0.0370370
$730$	0	0
$731$	− 7.54964e12i	− 0.977908i
$732$	0	0
$733$	7.29094e12i	0.932858i	0.884559	+	0.466429i	$0.154460\pi$
$733$	7.29094e12i	0.932858i	−0.884559	+	0.466429i	$0.845540\pi$
$734$	0	0
$735$	4.37198e10	0.00552567
$736$	0	0
$737$	−1.18147e12	−0.147509
$738$	0	0
$739$	− 3.39916e10i	− 0.00419248i	−0.999998	−	0.00209624i	$-0.999333\pi$
$739$	− 3.39916e10i	− 0.00419248i	0.999998	−	0.00209624i	$-0.000667255\pi$
$740$	0	0
$741$	− 1.27189e13i	− 1.54978i
$742$	0	0
$743$	9.08965e12	1.09420	0.547101	−	0.837067i	$-0.315731\pi$
$743$	9.08965e12	1.09420	0.547101	+	0.837067i	$0.315731\pi$
$744$	0	0
$745$	−2.61590e12	−0.311113
$746$	0	0
$747$	2.23663e12i	0.262816i
$748$	0	0
$749$	− 1.04385e13i	− 1.21190i
$750$	0	0
$751$	−7.25185e12	−0.831896	−0.415948	−	0.909388i	$-0.636550\pi$
$751$	−7.25185e12	−0.831896	−0.415948	+	0.909388i	$0.636550\pi$
$752$	0	0
$753$	8.94012e12	1.01336
$754$	0	0
$755$	6.76021e12i	0.757180i
$756$	0	0
$757$	1.12343e13i	1.24341i	0.783253	+	0.621703i	$0.213559\pi$
$757$	1.12343e13i	1.24341i	−0.783253	+	0.621703i	$0.786441\pi$
$758$	0	0
$759$	5.16249e11	0.0564639
$760$	0	0
$761$	1.63034e13	1.76217	0.881083	−	0.472962i	$-0.156815\pi$
$761$	1.63034e13	1.76217	0.881083	+	0.472962i	$0.156815\pi$
$762$	0	0
$763$	− 1.16060e13i	− 1.23972i
$764$	0	0
$765$	2.39598e12i	0.252933i
$766$	0	0
$767$	−9.81779e12	−1.02432
$768$	0	0
$769$	4.19893e12	0.432982	0.216491	−	0.976285i	$-0.430539\pi$
$769$	4.19893e12	0.432982	0.216491	+	0.976285i	$0.430539\pi$
$770$	0	0
$771$	1.87375e12i	0.190971i
$772$	0	0
$773$	− 1.11644e13i	− 1.12467i	−0.826908	−	0.562337i	$-0.809902\pi$
$773$	− 1.11644e13i	− 1.12467i	0.826908	−	0.562337i	$-0.190098\pi$
$774$	0	0
$775$	−5.28260e10	−0.00526005
$776$	0	0
$777$	−2.40376e11	−0.0236590
$778$	0	0
$779$	− 2.08227e13i	− 2.02591i
$780$	0	0
$781$	− 1.29542e12i	− 0.124589i
$782$	0	0
$783$	3.85898e12	0.366898
$784$	0	0
$785$	−1.24760e12	−0.117263
$786$	0	0
$787$	− 3.40070e12i	− 0.315996i	−0.987439	−	0.157998i	$-0.949496\pi$
$787$	− 3.40070e12i	− 0.315996i	0.987439	−	0.157998i	$-0.0505040\pi$
$788$	0	0
$789$	1.52869e12i	0.140434i
$790$	0	0
$791$	1.84860e13	1.67900
$792$	0	0
$793$	−7.74856e12	−0.695812
$794$	0	0
$795$	6.66942e12i	0.592156i
$796$	0	0
$797$	8.35775e11i	0.0733714i	0.999327	+	0.0366857i	$0.0116800\pi$
$797$	8.35775e11i	0.0733714i	−0.999327	+	0.0366857i	$0.988320\pi$
$798$	0	0
$799$	−3.36898e11	−0.0292440
$800$	0	0
$801$	−3.43935e12	−0.295209
$802$	0	0
$803$	1.42219e12i	0.120708i
$804$	0	0
$805$	1.42599e13i	1.19684i
$806$	0	0
$807$	−5.56234e12	−0.461664
$808$	0	0
$809$	1.21985e13	1.00124	0.500622	−	0.865666i	$-0.333105\pi$
$809$	1.21985e13	1.00124	0.500622	+	0.865666i	$0.333105\pi$
$810$	0	0
$811$	− 1.61845e13i	− 1.31373i	−0.754009	−	0.656864i	$-0.771883\pi$
$811$	− 1.61845e13i	− 1.31373i	0.754009	−	0.656864i	$-0.228117\pi$
$812$	0	0
$813$	− 9.35616e12i	− 0.751087i
$814$	0	0
$815$	−1.18909e13	−0.944075
$816$	0	0
$817$	−3.00261e13	−2.35776
$818$	0	0
$819$	− 6.46506e12i	− 0.502105i
$820$	0	0
$821$	− 3.85044e12i	− 0.295778i	−0.989004	−	0.147889i	$-0.952752\pi$
$821$	− 3.85044e12i	− 0.295778i	0.989004	−	0.147889i	$-0.0472479\pi$
$822$	0	0
$823$	−8.54699e12	−0.649403	−0.324701	−	0.945817i	$-0.605264\pi$
$823$	−8.54699e12	−0.649403	−0.324701	+	0.945817i	$0.605264\pi$
$824$	0	0
$825$	−2.84861e10	−0.00214087
$826$	0	0
$827$	− 4.46202e12i	− 0.331709i	−0.986150	−	0.165854i	$-0.946962\pi$
$827$	− 4.46202e12i	− 0.331709i	0.986150	−	0.165854i	$-0.0530382\pi$
$828$	0	0
$829$	5.99259e12i	0.440675i	0.975424	+	0.220338i	$0.0707159\pi$
$829$	5.99259e12i	0.440675i	−0.975424	+	0.220338i	$0.929284\pi$
$830$	0	0
$831$	3.66640e12	0.266707
$832$	0	0
$833$	9.66476e10	0.00695486
$834$	0	0
$835$	2.25768e12i	0.160721i
$836$	0	0
$837$	− 3.25187e11i	− 0.0229018i
$838$	0	0
$839$	−9.87709e12	−0.688177	−0.344088	−	0.938937i	$-0.611812\pi$
$839$	−9.87709e12	−0.688177	−0.344088	+	0.938937i	$0.611812\pi$
$840$	0	0
$841$	−3.82201e13	−2.63457
$842$	0	0
$843$	− 7.29021e12i	− 0.497183i
$844$	0	0
$845$	1.88990e13i	1.27521i
$846$	0	0
$847$	−1.49428e13	−0.997602
$848$	0	0
$849$	7.99799e12	0.528319
$850$	0	0
$851$	− 7.27502e11i	− 0.0475501i
$852$	0	0
$853$	− 2.87375e13i	− 1.85857i	−0.369365	−	0.929284i	$-0.620425\pi$
$853$	− 2.87375e13i	− 1.85857i	0.369365	−	0.929284i	$-0.379575\pi$
$854$	0	0
$855$	9.52919e12	0.609830
$856$	0	0
$857$	−4.90239e12	−0.310452	−0.155226	−	0.987879i	$-0.549611\pi$
$857$	−4.90239e12	−0.310452	−0.155226	+	0.987879i	$0.549611\pi$
$858$	0	0
$859$	− 4.08811e12i	− 0.256185i	−0.991762	−	0.128092i	$-0.959115\pi$
$859$	− 4.08811e12i	− 0.256185i	0.991762	−	0.128092i	$-0.0408854\pi$
$860$	0	0
$861$	− 1.05842e13i	− 0.656365i
$862$	0	0
$863$	−2.90112e13	−1.78040	−0.890200	−	0.455570i	$-0.849435\pi$
$863$	−2.90112e13	−1.78040	−0.890200	+	0.455570i	$0.849435\pi$
$864$	0	0
$865$	−1.70336e13	−1.03451
$866$	0	0
$867$	− 4.30904e12i	− 0.258997i
$868$	0	0
$869$	− 1.75162e12i	− 0.104196i
$870$	0	0
$871$	−4.47794e13	−2.63631
$872$	0	0
$873$	6.31026e12	0.367691
$874$	0	0
$875$	1.70145e13i	0.981257i
$876$	0	0
$877$	3.38408e13i	1.93171i	0.259082	+	0.965855i	$0.416580\pi$
$877$	3.38408e13i	1.93171i	−0.259082	+	0.965855i	$0.583420\pi$
$878$	0	0
$879$	−6.91788e12	−0.390862
$880$	0	0
$881$	2.28323e13	1.27690	0.638452	−	0.769662i	$-0.279575\pi$
$881$	2.28323e13	1.27690	0.638452	+	0.769662i	$0.279575\pi$
$882$	0	0
$883$	2.70857e12i	0.149940i	0.997186	+	0.0749699i	$0.0238861\pi$
$883$	2.70857e12i	0.149940i	−0.997186	+	0.0749699i	$0.976114\pi$
$884$	0	0
$885$	− 7.35562e12i	− 0.403065i
$886$	0	0
$887$	−1.71218e13	−0.928735	−0.464368	−	0.885643i	$-0.653718\pi$
$887$	−1.71218e13	−0.928735	−0.464368	+	0.885643i	$0.653718\pi$
$888$	0	0
$889$	−2.60557e12	−0.139909
$890$	0	0
$891$	− 1.75356e11i	− 0.00932116i
$892$	0	0
$893$	1.33990e12i	0.0705082i
$894$	0	0
$895$	−1.81895e13	−0.947582
$896$	0	0
$897$	1.95666e13	1.00914
$898$	0	0
$899$	4.44320e12i	0.226870i
$900$	0	0
$901$	1.47435e13i	0.745315i
$902$	0	0
$903$	−1.52623e13	−0.763882
$904$	0	0
$905$	−1.20919e13	−0.599206
$906$	0	0
$907$	2.45440e13i	1.20424i	0.798406	+	0.602119i	$0.205677\pi$
$907$	2.45440e13i	1.20424i	−0.798406	+	0.602119i	$0.794323\pi$
$908$	0	0
$909$	− 1.71774e12i	− 0.0834488i
$910$	0	0
$911$	1.04402e13	0.502198	0.251099	−	0.967961i	$-0.419208\pi$
$911$	1.04402e13	0.502198	0.251099	+	0.967961i	$0.419208\pi$
$912$	0	0
$913$	−1.38868e12	−0.0661432
$914$	0	0
$915$	− 5.80533e12i	− 0.273799i
$916$	0	0
$917$	− 1.06853e13i	− 0.499027i
$918$	0	0
$919$	1.14054e13	0.527464	0.263732	−	0.964596i	$-0.415047\pi$
$919$	1.14054e13	0.527464	0.263732	+	0.964596i	$0.415047\pi$
$920$	0	0
$921$	1.48702e13	0.681000
$922$	0	0
$923$	− 4.90985e13i	− 2.22669i
$924$	0	0
$925$	4.01429e10i	0.00180290i
$926$	0	0
$927$	−3.19716e12	−0.142202
$928$	0	0
$929$	1.77202e13	0.780544	0.390272	−	0.920700i	$-0.372381\pi$
$929$	1.77202e13	0.780544	0.390272	+	0.920700i	$0.372381\pi$
$930$	0	0
$931$	− 3.84383e11i	− 0.0167684i
$932$	0	0
$933$	9.22852e12i	0.398717i
$934$	0	0
$935$	−1.48762e12	−0.0636561
$936$	0	0
$937$	1.13938e13	0.482880	0.241440	−	0.970416i	$-0.422380\pi$
$937$	1.13938e13	0.482880	0.241440	+	0.970416i	$0.422380\pi$
$938$	0	0
$939$	− 1.95760e13i	− 0.821731i
$940$	0	0
$941$	− 3.57056e12i	− 0.148451i	−0.997241	−	0.0742255i	$-0.976352\pi$
$941$	− 3.57056e12i	− 0.148451i	0.997241	−	0.0742255i	$-0.0236484\pi$
$942$	0	0
$943$	3.20334e13	1.31917
$944$	0	0
$945$	4.84371e12	0.197576
$946$	0	0
$947$	− 3.48838e13i	− 1.40945i	−0.709482	−	0.704724i	$-0.751071\pi$
$947$	− 3.48838e13i	− 1.40945i	0.709482	−	0.704724i	$-0.248929\pi$
$948$	0	0
$949$	5.39032e13i	2.15733i
$950$	0	0
$951$	8.60442e12	0.341122
$952$	0	0
$953$	−2.48299e13	−0.975117	−0.487558	−	0.873090i	$-0.662112\pi$
$953$	−2.48299e13	−0.975117	−0.487558	+	0.873090i	$0.662112\pi$
$954$	0	0
$955$	− 5.58038e12i	− 0.217094i
$956$	0	0
$957$	2.39597e12i	0.0923376i
$958$	0	0
$959$	1.00430e13	0.383425
$960$	0	0
$961$	−2.60652e13	−0.985839
$962$	0	0
$963$	− 1.07310e13i	− 0.402090i
$964$	0	0
$965$	− 2.60921e13i	− 0.968580i
$966$	0	0
$967$	1.43858e13	0.529071	0.264535	−	0.964376i	$-0.414781\pi$
$967$	1.43858e13	0.529071	0.264535	+	0.964376i	$0.414781\pi$
$968$	0	0
$969$	2.10654e13	0.767560
$970$	0	0
$971$	− 9.63124e12i	− 0.347693i	−0.984773	−	0.173846i	$-0.944380\pi$
$971$	− 9.63124e12i	− 0.347693i	0.984773	−	0.173846i	$-0.0556196\pi$
$972$	0	0
$973$	3.42108e13i	1.22364i
$974$	0	0
$975$	−1.07967e12	−0.0382622
$976$	0	0
$977$	−9.87031e12	−0.346581	−0.173291	−	0.984871i	$-0.555440\pi$
$977$	−9.87031e12	−0.346581	−0.173291	+	0.984871i	$0.555440\pi$
$978$	0	0
$979$	− 2.13543e12i	− 0.0742957i
$980$	0	0
$981$	− 1.19313e13i	− 0.411318i
$982$	0	0
$983$	−5.82371e13	−1.98934	−0.994669	−	0.103116i	$-0.967119\pi$
$983$	−5.82371e13	−1.98934	−0.994669	+	0.103116i	$0.967119\pi$
$984$	0	0
$985$	−1.50233e13	−0.508512
$986$	0	0
$987$	6.81072e11i	0.0228437i
$988$	0	0
$989$	− 4.61918e13i	− 1.53526i
$990$	0	0
$991$	−6.04603e13	−1.99131	−0.995655	−	0.0931229i	$-0.970315\pi$
$991$	−6.04603e13	−1.99131	−0.995655	+	0.0931229i	$0.970315\pi$
$992$	0	0
$993$	−5.52652e12	−0.180377
$994$	0	0
$995$	− 4.32034e13i	− 1.39738i
$996$	0	0
$997$	4.77299e13i	1.52990i	0.644091	+	0.764949i	$0.277236\pi$
$997$	4.77299e13i	1.52990i	−0.644091	+	0.764949i	$0.722764\pi$
$998$	0	0
$999$	−2.47113e11	−0.00784965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.d.b.193.6	yes	8
4.3	odd	2		384.10.d.a.193.2	✓	8
8.3	odd	2		384.10.d.a.193.7	yes	8
8.5	even	2	inner	384.10.d.b.193.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.d.a.193.2	✓	8	4.3	odd	2
384.10.d.a.193.7	yes	8	8.3	odd	2
384.10.d.b.193.3	yes	8	8.5	even	2	inner
384.10.d.b.193.6	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+81.0000i q^{3} -1428.10i q^{5} -6382.13 q^{7} -6561.00 q^{9} +O(q^{10})\)	\(q+81.0000i q^{3} -1428.10i q^{5} -6382.13 q^{7} -6561.00 q^{9} -4073.61i q^{11} -154396. i q^{13} +115676. q^{15} +255714. q^{17} -1.01702e6i q^{19} -516952. i q^{21} +1.56457e6 q^{23} -86331.5 q^{25} -531441. i q^{27} +7.26136e6i q^{29} +611897. q^{31} +329962. q^{33} +9.11429e6i q^{35} -464986. i q^{37} +1.25061e7 q^{39} +2.04743e7 q^{41} -2.95237e7i q^{43} +9.36973e6i q^{45} -1.31748e6 q^{47} +377951. q^{49} +2.07129e7i q^{51} +5.76562e7i q^{53} -5.81750e6 q^{55} +8.23785e7 q^{57} -6.35883e7i q^{59} -5.01862e7i q^{61} +4.18731e7 q^{63} -2.20493e8 q^{65} -2.90029e8i q^{67} +1.26730e8i q^{69} +3.18003e8 q^{71} -3.49122e8 q^{73} -6.99285e6i q^{75} +2.59983e7i q^{77} +4.29993e8 q^{79} +4.30467e7 q^{81} -3.40898e8i q^{83} -3.65185e8i q^{85} -5.88170e8 q^{87} +5.24211e8 q^{89} +9.85376e8i q^{91} +4.95636e7i q^{93} -1.45240e9 q^{95} -9.61784e8 q^{97} +2.67270e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 13632 q^{7} - 52488 q^{9}+O(q^{10}) \) 8 * q + 13632 * q^7 - 52488 * q^9	\( 8 q + 13632 q^{7} - 52488 q^{9} + 90720 q^{15} + 8304 q^{17} + 4612608 q^{23} - 4754904 q^{25} + 7499328 q^{31} - 6213024 q^{33} + 23211360 q^{39} - 43518896 q^{41} - 49382016 q^{47} - 74106808 q^{49} - 19030656 q^{55} + 38141280 q^{57} - 89439552 q^{63} - 110270336 q^{65} + 741751296 q^{71} - 1507903440 q^{73} - 1008373440 q^{79} + 344373768 q^{81} - 423468000 q^{87} - 1337034448 q^{89} + 543950208 q^{95} - 904817936 q^{97}+O(q^{100}) \) 8 * q + 13632 * q^7 - 52488 * q^9 + 90720 * q^15 + 8304 * q^17 + 4612608 * q^23 - 4754904 * q^25 + 7499328 * q^31 - 6213024 * q^33 + 23211360 * q^39 - 43518896 * q^41 - 49382016 * q^47 - 74106808 * q^49 - 19030656 * q^55 + 38141280 * q^57 - 89439552 * q^63 - 110270336 * q^65 + 741751296 * q^71 - 1507903440 * q^73 - 1008373440 * q^79 + 344373768 * q^81 - 423468000 * q^87 - 1337034448 * q^89 + 543950208 * q^95 - 904817936 * q^97

Introduction

L-functions

Modular forms

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