LMFDB - Embedded newform 384.10.d.f.193.4

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 300700 x^{18} + 6140664 x^{17} + 35387063979 x^{16} - 1130222504088 x^{15} + \cdots + 12\!\cdots\!52 $ x^20 - 8x^19 - 300700x^18 + 6140664x^17 + 35387063979x^16 - 1130222504088x^15 - 2027709461599244x^14 + 84850591004671480x^13 + 57031067836759165823x^12 - 2761881829653049004184x^11 - 715774112122919877685260x^10 + 33804686703728196561023832x^9 + 4440127592352179578492608037x^8 - 186062650255909170075184345736x^7 - 13450754766452318411315895909412x^6 + 457952106473696272472169073091112x^5 + 15298236539262291502286172763910964x^4 - 382483105264444074951018570546548352x^3 + 3006361776222337276289989101322653304x^2 - 10187939951443559461324435287012809264*x + 12842550816412602186535854197192532452
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{175}\cdot 3^{32} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.4
Root			$-271.678 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.10.d.f.193.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-81.0000i q^{3} -573.512i q^{5} +2185.20 q^{7} -6561.00 q^{9} +O(q^{10})$	$q-81.0000i q^{3} -573.512i q^{5} +2185.20 q^{7} -6561.00 q^{9} +25190.5i q^{11} +170686. i q^{13} -46454.5 q^{15} +461173. q^{17} -165769. i q^{19} -177001. i q^{21} +953690. q^{23} +1.62421e6 q^{25} +531441. i q^{27} -1.12414e6i q^{29} -886392. q^{31} +2.04043e6 q^{33} -1.25324e6i q^{35} -1.04360e7i q^{37} +1.38256e7 q^{39} -2.74090e7 q^{41} +3.48442e7i q^{43} +3.76281e6i q^{45} -5.83629e7 q^{47} -3.55785e7 q^{49} -3.73550e7i q^{51} +4.24392e7i q^{53} +1.44471e7 q^{55} -1.34273e7 q^{57} -5.50758e7i q^{59} +3.22137e7i q^{61} -1.43371e7 q^{63} +9.78904e7 q^{65} +1.97628e8i q^{67} -7.72489e7i q^{69} -1.31495e8 q^{71} +1.88866e7 q^{73} -1.31561e8i q^{75} +5.50464e7i q^{77} -6.76946e7 q^{79} +4.30467e7 q^{81} -3.73246e8i q^{83} -2.64488e8i q^{85} -9.10553e7 q^{87} -2.50175e8 q^{89} +3.72983e8i q^{91} +7.17978e7i q^{93} -9.50706e7 q^{95} -7.73257e8 q^{97} -1.65275e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 131220 q^{9}+O(q^{10}) $ 20 * q - 131220 * q^9	$ 20 q - 131220 q^{9} - 905768 q^{17} - 9682620 q^{25} + 12054096 q^{33} + 74264008 q^{41} + 252775700 q^{49} - 5335632 q^{57} + 245588672 q^{65} - 895193896 q^{73} + 860934420 q^{81} + 882422136 q^{89} + 433683736 q^{97}+O(q^{100}) $ 20 * q - 131220 * q^9 - 905768 * q^17 - 9682620 * q^25 + 12054096 * q^33 + 74264008 * q^41 + 252775700 * q^49 - 5335632 * q^57 + 245588672 * q^65 - 895193896 * q^73 + 860934420 * q^81 + 882422136 * q^89 + 433683736 * 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
$5$	− 573.512i	− 0.410372i	−0.978723	−	0.205186i	$-0.934220\pi$
$5$	− 573.512i	− 0.410372i	0.978723	−	0.205186i	$-0.0657799\pi$
$6$	0	0
$7$	2185.20	0.343994	0.171997	−	0.985097i	$-0.444978\pi$
$7$	2185.20	0.343994	0.171997	+	0.985097i	$0.444978\pi$
$8$	0	0
$9$	−6561.00	−0.333333
$10$	0	0
$11$	25190.5i	0.518764i	0.965775	+	0.259382i	$0.0835189\pi$
$11$	25190.5i	0.518764i	−0.965775	+	0.259382i	$0.916481\pi$
$12$	0	0
$13$	170686.i	1.65750i	0.559622	+	0.828748i	$0.310946\pi$
$13$	170686.i	1.65750i	−0.559622	+	0.828748i	$0.689054\pi$
$14$	0	0
$15$	−46454.5	−0.236928
$16$	0	0
$17$	461173.	1.33919	0.669597	−	0.742725i	$-0.266467\pi$
$17$	461173.	1.33919	0.669597	+	0.742725i	$0.266467\pi$
$18$	0	0
$19$	− 165769.i	− 0.291818i	−0.989298	−	0.145909i	$-0.953389\pi$
$19$	− 165769.i	− 0.291818i	0.989298	−	0.145909i	$-0.0466107\pi$
$20$	0	0
$21$	− 177001.i	− 0.198605i
$22$	0	0
$23$	953690.	0.710611	0.355306	−	0.934750i	$-0.384377\pi$
$23$	953690.	0.710611	0.355306	+	0.934750i	$0.384377\pi$
$24$	0	0
$25$	1.62421e6	0.831595
$26$	0	0
$27$	531441.i	0.192450i
$28$	0	0
$29$	− 1.12414e6i	− 0.295141i	−0.989052	−	0.147570i	$-0.952855\pi$
$29$	− 1.12414e6i	− 0.295141i	0.989052	−	0.147570i	$-0.0471453\pi$
$30$	0	0
$31$	−886392.	−0.172385	−0.0861923	−	0.996279i	$-0.527470\pi$
$31$	−886392.	−0.172385	−0.0861923	+	0.996279i	$0.527470\pi$
$32$	0	0
$33$	2.04043e6	0.299509
$34$	0	0
$35$	− 1.25324e6i	− 0.141165i
$36$	0	0
$37$	− 1.04360e7i	− 0.915434i	−0.889098	−	0.457717i	$-0.848667\pi$
$37$	− 1.04360e7i	− 0.915434i	0.889098	−	0.457717i	$-0.151333\pi$
$38$	0	0
$39$	1.38256e7	0.956956
$40$	0	0
$41$	−2.74090e7	−1.51484	−0.757420	−	0.652928i	$-0.773540\pi$
$41$	−2.74090e7	−1.51484	−0.757420	+	0.652928i	$0.773540\pi$
$42$	0	0
$43$	3.48442e7i	1.55426i	0.629342	+	0.777128i	$0.283324\pi$
$43$	3.48442e7i	1.55426i	−0.629342	+	0.777128i	$0.716676\pi$
$44$	0	0
$45$	3.76281e6i	0.136791i
$46$	0	0
$47$	−5.83629e7	−1.74460	−0.872301	−	0.488970i	$-0.837373\pi$
$47$	−5.83629e7	−1.74460	−0.872301	+	0.488970i	$0.837373\pi$
$48$	0	0
$49$	−3.55785e7	−0.881668
$50$	0	0
$51$	− 3.73550e7i	− 0.773184i
$52$	0	0
$53$	4.24392e7i	0.738797i	0.929271	+	0.369399i	$0.120436\pi$
$53$	4.24392e7i	0.738797i	−0.929271	+	0.369399i	$0.879564\pi$
$54$	0	0
$55$	1.44471e7	0.212886
$56$	0	0
$57$	−1.34273e7	−0.168481
$58$	0	0
$59$	− 5.50758e7i	− 0.591734i	−0.955229	−	0.295867i	$-0.904391\pi$
$59$	− 5.50758e7i	− 0.591734i	0.955229	−	0.295867i	$-0.0956086\pi$
$60$	0	0
$61$	3.22137e7i	0.297890i	0.988845	+	0.148945i	$0.0475877\pi$
$61$	3.22137e7i	0.297890i	−0.988845	+	0.148945i	$0.952412\pi$
$62$	0	0
$63$	−1.43371e7	−0.114665
$64$	0	0
$65$	9.78904e7	0.680190
$66$	0	0
$67$	1.97628e8i	1.19815i	0.800692	+	0.599076i	$0.204465\pi$
$67$	1.97628e8i	1.19815i	−0.800692	+	0.599076i	$0.795535\pi$
$68$	0	0
$69$	− 7.72489e7i	− 0.410272i
$70$	0	0
$71$	−1.31495e8	−0.614111	−0.307056	−	0.951692i	$-0.599344\pi$
$71$	−1.31495e8	−0.614111	−0.307056	+	0.951692i	$0.599344\pi$
$72$	0	0
$73$	1.88866e7	0.0778398	0.0389199	−	0.999242i	$-0.487608\pi$
$73$	1.88866e7	0.0778398	0.0389199	+	0.999242i	$0.487608\pi$
$74$	0	0
$75$	− 1.31561e8i	− 0.480122i
$76$	0	0
$77$	5.50464e7i	0.178452i
$78$	0	0
$79$	−6.76946e7	−0.195538	−0.0977692	−	0.995209i	$-0.531171\pi$
$79$	−6.76946e7	−0.195538	−0.0977692	+	0.995209i	$0.531171\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	− 3.73246e8i	− 0.863264i	−0.902050	−	0.431632i	$-0.857938\pi$
$83$	− 3.73246e8i	− 0.863264i	0.902050	−	0.431632i	$-0.142062\pi$
$84$	0	0
$85$	− 2.64488e8i	− 0.549567i
$86$	0	0
$87$	−9.10553e7	−0.170400
$88$	0	0
$89$	−2.50175e8	−0.422658	−0.211329	−	0.977415i	$-0.567779\pi$
$89$	−2.50175e8	−0.422658	−0.211329	+	0.977415i	$0.567779\pi$
$90$	0	0
$91$	3.72983e8i	0.570168i
$92$	0	0
$93$	7.17978e7i	0.0995262i
$94$	0	0
$95$	−9.50706e7	−0.119754
$96$	0	0
$97$	−7.73257e8	−0.886852	−0.443426	−	0.896311i	$-0.646237\pi$
$97$	−7.73257e8	−0.886852	−0.443426	+	0.896311i	$0.646237\pi$
$98$	0	0
$99$	− 1.65275e8i	− 0.172921i
$100$	0	0
$101$	8.34597e8i	0.798051i	0.916940	+	0.399025i	$0.130651\pi$
$101$	8.34597e8i	0.798051i	−0.916940	+	0.399025i	$0.869349\pi$
$102$	0	0
$103$	−1.44701e9	−1.26678	−0.633392	−	0.773831i	$-0.718338\pi$
$103$	−1.44701e9	−1.26678	−0.633392	+	0.773831i	$0.718338\pi$
$104$	0	0
$105$	−1.01513e8	−0.0815019
$106$	0	0
$107$	− 2.20932e8i	− 0.162942i	−0.996676	−	0.0814708i	$-0.974038\pi$
$107$	− 2.20932e8i	− 0.162942i	0.996676	−	0.0814708i	$-0.0259617\pi$
$108$	0	0
$109$	8.53497e8i	0.579139i	0.957157	+	0.289570i	$0.0935122\pi$
$109$	8.53497e8i	0.579139i	−0.957157	+	0.289570i	$0.906488\pi$
$110$	0	0
$111$	−8.45318e8	−0.528526
$112$	0	0
$113$	1.25755e9	0.725560	0.362780	−	0.931875i	$-0.381828\pi$
$113$	1.25755e9	0.725560	0.362780	+	0.931875i	$0.381828\pi$
$114$	0	0
$115$	− 5.46953e8i	− 0.291615i
$116$	0	0
$117$	− 1.11987e9i	− 0.552499i
$118$	0	0
$119$	1.00776e9	0.460674
$120$	0	0
$121$	1.72339e9	0.730884
$122$	0	0
$123$	2.22013e9i	0.874593i
$124$	0	0
$125$	− 2.05164e9i	− 0.751635i
$126$	0	0
$127$	−2.98428e9	−1.01794	−0.508971	−	0.860783i	$-0.669974\pi$
$127$	−2.98428e9	−1.01794	−0.508971	+	0.860783i	$0.669974\pi$
$128$	0	0
$129$	2.82238e9	0.897351
$130$	0	0
$131$	2.24225e9i	0.665217i	0.943065	+	0.332609i	$0.107929\pi$
$131$	2.24225e9i	0.665217i	−0.943065	+	0.332609i	$0.892071\pi$
$132$	0	0
$133$	− 3.62239e8i	− 0.100384i
$134$	0	0
$135$	3.04788e8	0.0789761
$136$	0	0
$137$	−1.92870e8	−0.0467759	−0.0233879	−	0.999726i	$-0.507445\pi$
$137$	−1.92870e8	−0.0467759	−0.0233879	+	0.999726i	$0.507445\pi$
$138$	0	0
$139$	− 5.70374e9i	− 1.29596i	−0.761655	−	0.647982i	$-0.775613\pi$
$139$	− 5.70374e9i	− 1.29596i	0.761655	−	0.647982i	$-0.224387\pi$
$140$	0	0
$141$	4.72739e9i	1.00725i
$142$	0	0
$143$	−4.29966e9	−0.859849
$144$	0	0
$145$	−6.44708e8	−0.121117
$146$	0	0
$147$	2.88186e9i	0.509031i
$148$	0	0
$149$	8.87026e9i	1.47434i	0.675707	+	0.737171i	$0.263839\pi$
$149$	8.87026e9i	1.47434i	−0.675707	+	0.737171i	$0.736161\pi$
$150$	0	0
$151$	1.12257e10	1.75719	0.878595	−	0.477568i	$-0.158481\pi$
$151$	1.12257e10	1.75719	0.878595	+	0.477568i	$0.158481\pi$
$152$	0	0
$153$	−3.02575e9	−0.446398
$154$	0	0
$155$	5.08357e8i	0.0707418i
$156$	0	0
$157$	6.26221e9i	0.822582i	0.911504	+	0.411291i	$0.134922\pi$
$157$	6.26221e9i	0.822582i	−0.911504	+	0.411291i	$0.865078\pi$
$158$	0	0
$159$	3.43757e9	0.426545
$160$	0	0
$161$	2.08401e9	0.244446
$162$	0	0
$163$	4.06846e9i	0.451425i	0.974194	+	0.225712i	$0.0724710\pi$
$163$	4.06846e9i	0.451425i	−0.974194	+	0.225712i	$0.927529\pi$
$164$	0	0
$165$	− 1.17021e9i	− 0.122910i
$166$	0	0
$167$	−7.42771e9	−0.738976	−0.369488	−	0.929235i	$-0.620467\pi$
$167$	−7.42771e9	−0.738976	−0.369488	+	0.929235i	$0.620467\pi$
$168$	0	0
$169$	−1.85292e10	−1.74729
$170$	0	0
$171$	1.08761e9i	0.0972727i
$172$	0	0
$173$	− 9.15808e9i	− 0.777315i	−0.921382	−	0.388658i	$-0.872939\pi$
$173$	− 9.15808e9i	− 0.777315i	0.921382	−	0.388658i	$-0.127061\pi$
$174$	0	0
$175$	3.54923e9	0.286064
$176$	0	0
$177$	−4.46114e9	−0.341638
$178$	0	0
$179$	− 2.24462e10i	− 1.63420i	−0.576499	−	0.817098i	$-0.695582\pi$
$179$	− 2.24462e10i	− 1.63420i	0.576499	−	0.817098i	$-0.304418\pi$
$180$	0	0
$181$	1.01662e10i	0.704049i	0.935991	+	0.352025i	$0.114507\pi$
$181$	1.01662e10i	0.704049i	−0.935991	+	0.352025i	$0.885493\pi$
$182$	0	0
$183$	2.60931e9	0.171987
$184$	0	0
$185$	−5.98519e9	−0.375669
$186$	0	0
$187$	1.16172e10i	0.694726i
$188$	0	0
$189$	1.16131e9i	0.0662017i
$190$	0	0
$191$	2.71383e10	1.47548	0.737738	−	0.675088i	$-0.235894\pi$
$191$	2.71383e10	1.47548	0.737738	+	0.675088i	$0.235894\pi$
$192$	0	0
$193$	2.59970e10	1.34870	0.674350	−	0.738412i	$-0.264424\pi$
$193$	2.59970e10	1.34870	0.674350	+	0.738412i	$0.264424\pi$
$194$	0	0
$195$	− 7.92912e9i	− 0.392708i
$196$	0	0
$197$	1.11903e10i	0.529351i	0.964337	+	0.264676i	$0.0852649\pi$
$197$	1.11903e10i	0.529351i	−0.964337	+	0.264676i	$0.914735\pi$
$198$	0	0
$199$	−2.24980e10	−1.01696	−0.508482	−	0.861073i	$-0.669793\pi$
$199$	−2.24980e10	−1.01696	−0.508482	+	0.861073i	$0.669793\pi$
$200$	0	0
$201$	1.60079e10	0.691753
$202$	0	0
$203$	− 2.45647e9i	− 0.101527i
$204$	0	0
$205$	1.57194e10i	0.621648i
$206$	0	0
$207$	−6.25716e9	−0.236870
$208$	0	0
$209$	4.17581e9	0.151385
$210$	0	0
$211$	2.12635e10i	0.738521i	0.929326	+	0.369261i	$0.120389\pi$
$211$	2.12635e10i	0.738521i	−0.929326	+	0.369261i	$0.879611\pi$
$212$	0	0
$213$	1.06511e10i	0.354557i
$214$	0	0
$215$	1.99836e10	0.637823
$216$	0	0
$217$	−1.93695e9	−0.0592992
$218$	0	0
$219$	− 1.52982e9i	− 0.0449408i
$220$	0	0
$221$	7.87156e10i	2.21971i
$222$	0	0
$223$	2.72002e10	0.736546	0.368273	−	0.929718i	$-0.379949\pi$
$223$	2.72002e10	0.736546	0.368273	+	0.929718i	$0.379949\pi$
$224$	0	0
$225$	−1.06564e10	−0.277198
$226$	0	0
$227$	− 4.16425e9i	− 0.104093i	−0.998645	−	0.0520464i	$-0.983426\pi$
$227$	− 4.16425e9i	− 0.104093i	0.998645	−	0.0520464i	$-0.0165744\pi$
$228$	0	0
$229$	5.12064e10i	1.23045i	0.788351	+	0.615226i	$0.210935\pi$
$229$	5.12064e10i	1.23045i	−0.788351	+	0.615226i	$0.789065\pi$
$230$	0	0
$231$	4.45876e9	0.103029
$232$	0	0
$233$	−2.39479e10	−0.532311	−0.266155	−	0.963930i	$-0.585753\pi$
$233$	−2.39479e10	−0.532311	−0.266155	+	0.963930i	$0.585753\pi$
$234$	0	0
$235$	3.34718e10i	0.715935i
$236$	0	0
$237$	5.48326e9i	0.112894i
$238$	0	0
$239$	7.69723e9	0.152596	0.0762981	−	0.997085i	$-0.475690\pi$
$239$	7.69723e9	0.152596	0.0762981	+	0.997085i	$0.475690\pi$
$240$	0	0
$241$	−9.87829e8	−0.0188628	−0.00943138	−	0.999956i	$-0.503002\pi$
$241$	−9.87829e8	−0.0188628	−0.00943138	+	0.999956i	$0.503002\pi$
$242$	0	0
$243$	− 3.48678e9i	− 0.0641500i
$244$	0	0
$245$	2.04047e10i	0.361812i
$246$	0	0
$247$	2.82944e10	0.483687
$248$	0	0
$249$	−3.02329e10	−0.498406
$250$	0	0
$251$	− 3.00907e10i	− 0.478521i	−0.970955	−	0.239261i	$-0.923095\pi$
$251$	− 3.00907e10i	− 0.478521i	0.970955	−	0.239261i	$-0.0769050\pi$
$252$	0	0
$253$	2.40239e10i	0.368640i
$254$	0	0
$255$	−2.14235e10	−0.317293
$256$	0	0
$257$	1.68520e10	0.240963	0.120482	−	0.992716i	$-0.461556\pi$
$257$	1.68520e10	0.240963	0.120482	+	0.992716i	$0.461556\pi$
$258$	0	0
$259$	− 2.28048e10i	− 0.314904i
$260$	0	0
$261$	7.37548e9i	0.0983803i
$262$	0	0
$263$	−2.47935e10	−0.319548	−0.159774	−	0.987154i	$-0.551077\pi$
$263$	−2.47935e10	−0.319548	−0.159774	+	0.987154i	$0.551077\pi$
$264$	0	0
$265$	2.43394e10	0.303182
$266$	0	0
$267$	2.02642e10i	0.244022i
$268$	0	0
$269$	1.44586e11i	1.68361i	0.539785	+	0.841803i	$0.318506\pi$
$269$	1.44586e11i	1.68361i	−0.539785	+	0.841803i	$0.681494\pi$
$270$	0	0
$271$	−9.33109e10	−1.05092	−0.525461	−	0.850818i	$-0.676107\pi$
$271$	−9.33109e10	−1.05092	−0.525461	+	0.850818i	$0.676107\pi$
$272$	0	0
$273$	3.02116e10	0.329187
$274$	0	0
$275$	4.09146e10i	0.431402i
$276$	0	0
$277$	1.41736e11i	1.44651i	0.690583	+	0.723253i	$0.257354\pi$
$277$	1.41736e11i	1.44651i	−0.690583	+	0.723253i	$0.742646\pi$
$278$	0	0
$279$	5.81562e9	0.0574615
$280$	0	0
$281$	4.57492e10	0.437729	0.218864	−	0.975755i	$-0.429765\pi$
$281$	4.57492e10	0.437729	0.218864	+	0.975755i	$0.429765\pi$
$282$	0	0
$283$	− 8.57214e10i	− 0.794420i	−0.917728	−	0.397210i	$-0.869978\pi$
$283$	− 8.57214e10i	− 0.794420i	0.917728	−	0.397210i	$-0.130022\pi$
$284$	0	0
$285$	7.70072e9i	0.0691400i
$286$	0	0
$287$	−5.98944e10	−0.521096
$288$	0	0
$289$	9.40923e10	0.793440
$290$	0	0
$291$	6.26338e10i	0.512024i
$292$	0	0
$293$	7.92826e10i	0.628454i	0.949348	+	0.314227i	$0.101745\pi$
$293$	7.92826e10i	0.628454i	−0.949348	+	0.314227i	$0.898255\pi$
$294$	0	0
$295$	−3.15866e10	−0.242831
$296$	0	0
$297$	−1.33873e10	−0.0998362
$298$	0	0
$299$	1.62781e11i	1.17784i
$300$	0	0
$301$	7.61417e10i	0.534655i
$302$	0	0
$303$	6.76023e10	0.460755
$304$	0	0
$305$	1.84749e10	0.122246
$306$	0	0
$307$	− 2.70824e11i	− 1.74006i	−0.492996	−	0.870031i	$-0.664098\pi$
$307$	− 2.70824e11i	− 1.74006i	0.492996	−	0.870031i	$-0.335902\pi$
$308$	0	0
$309$	1.17207e11i	0.731378i
$310$	0	0
$311$	−2.53125e11	−1.53431	−0.767155	−	0.641462i	$-0.778328\pi$
$311$	−2.53125e11	−1.53431	−0.767155	+	0.641462i	$0.778328\pi$
$312$	0	0
$313$	−3.80905e10	−0.224320	−0.112160	−	0.993690i	$-0.535777\pi$
$313$	−3.80905e10	−0.224320	−0.112160	+	0.993690i	$0.535777\pi$
$314$	0	0
$315$	8.22251e9i	0.0470551i
$316$	0	0
$317$	2.66537e11i	1.48249i	0.671235	+	0.741245i	$0.265764\pi$
$317$	2.66537e11i	1.48249i	−0.671235	+	0.741245i	$0.734236\pi$
$318$	0	0
$319$	2.83176e10	0.153108
$320$	0	0
$321$	−1.78955e10	−0.0940744
$322$	0	0
$323$	− 7.64482e10i	− 0.390801i
$324$	0	0
$325$	2.77229e11i	1.37836i
$326$	0	0
$327$	6.91333e10	0.334366
$328$	0	0
$329$	−1.27535e11	−0.600132
$330$	0	0
$331$	1.32548e11i	0.606942i	0.952841	+	0.303471i	$0.0981456\pi$
$331$	1.32548e11i	0.606942i	−0.952841	+	0.303471i	$0.901854\pi$
$332$	0	0
$333$	6.84707e10i	0.305145i
$334$	0	0
$335$	1.13342e11	0.491688
$336$	0	0
$337$	−3.91693e11	−1.65429	−0.827145	−	0.561988i	$-0.810037\pi$
$337$	−3.91693e11	−1.65429	−0.827145	+	0.561988i	$0.810037\pi$
$338$	0	0
$339$	− 1.01862e11i	− 0.418903i
$340$	0	0
$341$	− 2.23287e10i	− 0.0894269i
$342$	0	0
$343$	−1.65927e11	−0.647282
$344$	0	0
$345$	−4.43032e10	−0.168364
$346$	0	0
$347$	5.17702e10i	0.191689i	0.995396	+	0.0958445i	$0.0305551\pi$
$347$	5.17702e10i	0.191689i	−0.995396	+	0.0958445i	$0.969445\pi$
$348$	0	0
$349$	2.61383e11i	0.943113i	0.881836	+	0.471556i	$0.156308\pi$
$349$	2.61383e11i	0.943113i	−0.881836	+	0.471556i	$0.843692\pi$
$350$	0	0
$351$	−9.07095e10	−0.318985
$352$	0	0
$353$	−2.46204e11	−0.843934	−0.421967	−	0.906611i	$-0.638660\pi$
$353$	−2.46204e11	−0.843934	−0.421967	+	0.906611i	$0.638660\pi$
$354$	0	0
$355$	7.54141e10i	0.252014i
$356$	0	0
$357$	− 8.16282e10i	− 0.265971i
$358$	0	0
$359$	4.40472e11	1.39956	0.699782	−	0.714356i	$-0.253280\pi$
$359$	4.40472e11	1.39956	0.699782	+	0.714356i	$0.253280\pi$
$360$	0	0
$361$	2.95208e11	0.914842
$362$	0	0
$363$	− 1.39594e11i	− 0.421976i
$364$	0	0
$365$	− 1.08317e10i	− 0.0319433i
$366$	0	0
$367$	−2.54837e11	−0.733271	−0.366636	−	0.930365i	$-0.619490\pi$
$367$	−2.54837e11	−0.733271	−0.366636	+	0.930365i	$0.619490\pi$
$368$	0	0
$369$	1.79831e11	0.504947
$370$	0	0
$371$	9.27382e10i	0.254142i
$372$	0	0
$373$	4.92490e11i	1.31737i	0.752419	+	0.658685i	$0.228887\pi$
$373$	4.92490e11i	1.31737i	−0.752419	+	0.658685i	$0.771113\pi$
$374$	0	0
$375$	−1.66183e11	−0.433957
$376$	0	0
$377$	1.91875e11	0.489195
$378$	0	0
$379$	6.04503e11i	1.50495i	0.658621	+	0.752475i	$0.271140\pi$
$379$	6.04503e11i	1.50495i	−0.658621	+	0.752475i	$0.728860\pi$
$380$	0	0
$381$	2.41727e11i	0.587710i
$382$	0	0
$383$	9.93960e10	0.236034	0.118017	−	0.993012i	$-0.462346\pi$
$383$	9.93960e10	0.236034	0.118017	+	0.993012i	$0.462346\pi$
$384$	0	0
$385$	3.15698e10	0.0732315
$386$	0	0
$387$	− 2.28613e11i	− 0.518086i
$388$	0	0
$389$	8.77921e11i	1.94394i	0.235109	+	0.971969i	$0.424455\pi$
$389$	8.77921e11i	1.94394i	−0.235109	+	0.971969i	$0.575545\pi$
$390$	0	0
$391$	4.39816e11	0.951646
$392$	0	0
$393$	1.81622e11	0.384063
$394$	0	0
$395$	3.88237e10i	0.0802435i
$396$	0	0
$397$	− 3.40922e9i	− 0.00688808i	−0.999994	−	0.00344404i	$-0.998904\pi$
$397$	− 3.40922e9i	− 0.00688808i	0.999994	−	0.00344404i	$-0.00109627\pi$
$398$	0	0
$399$	−2.93414e10	−0.0579565
$400$	0	0
$401$	4.36366e11	0.842755	0.421378	−	0.906885i	$-0.361547\pi$
$401$	4.36366e11	0.842755	0.421378	+	0.906885i	$0.361547\pi$
$402$	0	0
$403$	− 1.51295e11i	− 0.285727i
$404$	0	0
$405$	− 2.46878e10i	− 0.0455969i
$406$	0	0
$407$	2.62889e11	0.474894
$408$	0	0
$409$	1.53167e9	0.00270652	0.00135326	−	0.999999i	$-0.499569\pi$
$409$	1.53167e9	0.00270652	0.00135326	+	0.999999i	$0.499569\pi$
$410$	0	0
$411$	1.56225e10i	0.0270061i
$412$	0	0
$413$	− 1.20352e11i	− 0.203553i
$414$	0	0
$415$	−2.14061e11	−0.354259
$416$	0	0
$417$	−4.62003e11	−0.748226
$418$	0	0
$419$	− 7.95996e11i	− 1.26168i	−0.775915	−	0.630838i	$-0.782711\pi$
$419$	− 7.95996e11i	− 1.26168i	0.775915	−	0.630838i	$-0.217289\pi$
$420$	0	0
$421$	− 7.16602e11i	− 1.11175i	−0.831265	−	0.555877i	$-0.812383\pi$
$421$	− 7.16602e11i	− 1.11175i	0.831265	−	0.555877i	$-0.187617\pi$
$422$	0	0
$423$	3.82919e11	0.581534
$424$	0	0
$425$	7.49041e11	1.11367
$426$	0	0
$427$	7.03934e10i	0.102472i
$428$	0	0
$429$	3.48273e11i	0.496434i
$430$	0	0
$431$	−2.72042e11	−0.379741	−0.189871	−	0.981809i	$-0.560807\pi$
$431$	−2.72042e11	−0.379741	−0.189871	+	0.981809i	$0.560807\pi$
$432$	0	0
$433$	−2.93461e11	−0.401195	−0.200598	−	0.979674i	$-0.564288\pi$
$433$	−2.93461e11	−0.401195	−0.200598	+	0.979674i	$0.564288\pi$
$434$	0	0
$435$	5.22213e10i	0.0699272i
$436$	0	0
$437$	− 1.58092e11i	− 0.207369i
$438$	0	0
$439$	−2.95559e11	−0.379799	−0.189900	−	0.981804i	$-0.560816\pi$
$439$	−2.95559e11	−0.379799	−0.189900	+	0.981804i	$0.560816\pi$
$440$	0	0
$441$	2.33430e11	0.293889
$442$	0	0
$443$	1.41221e12i	1.74214i	0.491163	+	0.871068i	$0.336572\pi$
$443$	1.41221e12i	1.74214i	−0.491163	+	0.871068i	$0.663428\pi$
$444$	0	0
$445$	1.43479e11i	0.173447i
$446$	0	0
$447$	7.18491e11	0.851211
$448$	0	0
$449$	−8.27145e11	−0.960447	−0.480223	−	0.877146i	$-0.659444\pi$
$449$	−8.27145e11	−0.960447	−0.480223	+	0.877146i	$0.659444\pi$
$450$	0	0
$451$	− 6.90448e11i	− 0.785844i
$452$	0	0
$453$	− 9.09285e11i	− 1.01451i
$454$	0	0
$455$	2.13910e11	0.233981
$456$	0	0
$457$	−8.57958e11	−0.920117	−0.460058	−	0.887889i	$-0.652172\pi$
$457$	−8.57958e11	−0.920117	−0.460058	+	0.887889i	$0.652172\pi$
$458$	0	0
$459$	2.45086e11i	0.257728i
$460$	0	0
$461$	− 4.68707e11i	− 0.483334i	−0.970359	−	0.241667i	$-0.922306\pi$
$461$	− 4.68707e11i	− 0.483334i	0.970359	−	0.241667i	$-0.0776941\pi$
$462$	0	0
$463$	8.90051e11	0.900120	0.450060	−	0.892998i	$-0.351403\pi$
$463$	8.90051e11	0.900120	0.450060	+	0.892998i	$0.351403\pi$
$464$	0	0
$465$	4.11769e10	0.0408428
$466$	0	0
$467$	− 1.17567e12i	− 1.14383i	−0.820314	−	0.571913i	$-0.806201\pi$
$467$	− 1.17567e12i	− 1.14383i	0.820314	−	0.571913i	$-0.193799\pi$
$468$	0	0
$469$	4.31857e11i	0.412157i
$470$	0	0
$471$	5.07239e11	0.474918
$472$	0	0
$473$	−8.77744e11	−0.806293
$474$	0	0
$475$	− 2.69244e11i	− 0.242674i
$476$	0	0
$477$	− 2.78443e11i	− 0.246266i
$478$	0	0
$479$	7.61386e11	0.660838	0.330419	−	0.943834i	$-0.392810\pi$
$479$	7.61386e11	0.660838	0.330419	+	0.943834i	$0.392810\pi$
$480$	0	0
$481$	1.78128e12	1.51733
$482$	0	0
$483$	− 1.68805e11i	− 0.141131i
$484$	0	0
$485$	4.43473e11i	0.363939i
$486$	0	0
$487$	−2.08842e12	−1.68243	−0.841217	−	0.540698i	$-0.818160\pi$
$487$	−2.08842e12	−1.68243	−0.841217	+	0.540698i	$0.818160\pi$
$488$	0	0
$489$	3.29545e11	0.260630
$490$	0	0
$491$	5.35372e11i	0.415709i	0.978160	+	0.207854i	$0.0666480\pi$
$491$	5.35372e11i	0.415709i	−0.978160	+	0.207854i	$0.933352\pi$
$492$	0	0
$493$	− 5.18422e11i	− 0.395251i
$494$	0	0
$495$	−9.47872e10	−0.0709621
$496$	0	0
$497$	−2.87344e11	−0.211251
$498$	0	0
$499$	− 1.24709e12i	− 0.900422i	−0.892922	−	0.450211i	$-0.851349\pi$
$499$	− 1.24709e12i	− 0.900422i	0.892922	−	0.450211i	$-0.148651\pi$
$500$	0	0
$501$	6.01644e11i	0.426648i
$502$	0	0
$503$	−4.38151e11	−0.305188	−0.152594	−	0.988289i	$-0.548763\pi$
$503$	−4.38151e11	−0.305188	−0.152594	+	0.988289i	$0.548763\pi$
$504$	0	0
$505$	4.78651e11	0.327498
$506$	0	0
$507$	1.50086e12i	1.00880i
$508$	0	0
$509$	8.18520e11i	0.540505i	0.962790	+	0.270252i	$0.0871071\pi$
$509$	8.18520e11i	0.540505i	−0.962790	+	0.270252i	$0.912893\pi$
$510$	0	0
$511$	4.12712e10	0.0267764
$512$	0	0
$513$	8.80965e10	0.0561604
$514$	0	0
$515$	8.29875e11i	0.519853i
$516$	0	0
$517$	− 1.47019e12i	− 0.905037i
$518$	0	0
$519$	−7.41805e11	−0.448783
$520$	0	0
$521$	1.40738e12	0.836835	0.418418	−	0.908255i	$-0.362585\pi$
$521$	1.40738e12	0.836835	0.418418	+	0.908255i	$0.362585\pi$
$522$	0	0
$523$	1.18942e12i	0.695147i	0.937653	+	0.347574i	$0.112994\pi$
$523$	1.18942e12i	0.695147i	−0.937653	+	0.347574i	$0.887006\pi$
$524$	0	0
$525$	− 2.87487e11i	− 0.165159i
$526$	0	0
$527$	−4.08780e11	−0.230856
$528$	0	0
$529$	−8.91627e11	−0.495032
$530$	0	0
$531$	3.61352e11i	0.197245i
$532$	0	0
$533$	− 4.67834e12i	− 2.51084i
$534$	0	0
$535$	−1.26707e11	−0.0668667
$536$	0	0
$537$	−1.81814e12	−0.943503
$538$	0	0
$539$	− 8.96240e11i	− 0.457378i
$540$	0	0
$541$	− 2.99696e12i	− 1.50416i	−0.659072	−	0.752080i	$-0.729051\pi$
$541$	− 2.99696e12i	− 1.50416i	0.659072	−	0.752080i	$-0.270949\pi$
$542$	0	0
$543$	8.23459e11	0.406483
$544$	0	0
$545$	4.89491e11	0.237662
$546$	0	0
$547$	− 1.19878e12i	− 0.572527i	−0.958151	−	0.286264i	$-0.907587\pi$
$547$	− 1.19878e12i	− 0.572527i	0.958151	−	0.286264i	$-0.0924133\pi$
$548$	0	0
$549$	− 2.11354e11i	− 0.0992966i
$550$	0	0
$551$	−1.86348e11	−0.0861274
$552$	0	0
$553$	−1.47927e11	−0.0672640
$554$	0	0
$555$	4.84800e11i	0.216892i
$556$	0	0
$557$	− 2.94419e12i	− 1.29604i	−0.761625	−	0.648018i	$-0.775598\pi$
$557$	− 2.94419e12i	− 1.29604i	0.761625	−	0.648018i	$-0.224402\pi$
$558$	0	0
$559$	−5.94742e12	−2.57617
$560$	0	0
$561$	9.40991e11	0.401100
$562$	0	0
$563$	− 2.99061e12i	− 1.25450i	−0.778816	−	0.627252i	$-0.784180\pi$
$563$	− 2.99061e12i	− 1.25450i	0.778816	−	0.627252i	$-0.215820\pi$
$564$	0	0
$565$	− 7.21222e11i	− 0.297750i
$566$	0	0
$567$	9.40658e10	0.0382215
$568$	0	0
$569$	−3.33223e12	−1.33269	−0.666346	−	0.745643i	$-0.732142\pi$
$569$	−3.33223e12	−1.33269	−0.666346	+	0.745643i	$0.732142\pi$
$570$	0	0
$571$	3.69698e12i	1.45541i	0.685891	+	0.727704i	$0.259413\pi$
$571$	3.69698e12i	1.45541i	−0.685891	+	0.727704i	$0.740587\pi$
$572$	0	0
$573$	− 2.19820e12i	− 0.851866i
$574$	0	0
$575$	1.54899e12	0.590941
$576$	0	0
$577$	−1.68443e12	−0.632647	−0.316323	−	0.948651i	$-0.602448\pi$
$577$	−1.68443e12	−0.632647	−0.316323	+	0.948651i	$0.602448\pi$
$578$	0	0
$579$	− 2.10576e12i	− 0.778672i
$580$	0	0
$581$	− 8.15618e11i	− 0.296958i
$582$	0	0
$583$	−1.06906e12	−0.383262
$584$	0	0
$585$	−6.42259e11	−0.226730
$586$	0	0
$587$	5.18518e12i	1.80257i	0.433224	+	0.901286i	$0.357376\pi$
$587$	5.18518e12i	1.80257i	−0.433224	+	0.901286i	$0.642624\pi$
$588$	0	0
$589$	1.46936e11i	0.0503049i
$590$	0	0
$591$	9.06415e11	0.305621
$592$	0	0
$593$	5.75208e10	0.0191020	0.00955100	−	0.999954i	$-0.496960\pi$
$593$	5.75208e10	0.0191020	0.00955100	+	0.999954i	$0.496960\pi$
$594$	0	0
$595$	− 5.77960e11i	− 0.189048i
$596$	0	0
$597$	1.82234e12i	0.587144i
$598$	0	0
$599$	5.13376e12	1.62935	0.814677	−	0.579916i	$-0.196914\pi$
$599$	5.13376e12	1.62935	0.814677	+	0.579916i	$0.196914\pi$
$600$	0	0
$601$	1.69179e12	0.528947	0.264474	−	0.964393i	$-0.414802\pi$
$601$	1.69179e12	0.528947	0.264474	+	0.964393i	$0.414802\pi$
$602$	0	0
$603$	− 1.29664e12i	− 0.399384i
$604$	0	0
$605$	− 9.88383e11i	− 0.299934i
$606$	0	0
$607$	−1.63728e12	−0.489524	−0.244762	−	0.969583i	$-0.578710\pi$
$607$	−1.63728e12	−0.489524	−0.244762	+	0.969583i	$0.578710\pi$
$608$	0	0
$609$	−1.98974e11	−0.0586164
$610$	0	0
$611$	− 9.96171e12i	− 2.89167i
$612$	0	0
$613$	− 2.61028e12i	− 0.746646i	−0.927701	−	0.373323i	$-0.878218\pi$
$613$	− 2.61028e12i	− 0.746646i	0.927701	−	0.373323i	$-0.121782\pi$
$614$	0	0
$615$	1.27327e12	0.358908
$616$	0	0
$617$	6.80946e11	0.189160	0.0945800	−	0.995517i	$-0.469849\pi$
$617$	6.80946e11	0.189160	0.0945800	+	0.995517i	$0.469849\pi$
$618$	0	0
$619$	5.97226e12i	1.63505i	0.575894	+	0.817524i	$0.304654\pi$
$619$	5.97226e12i	1.63505i	−0.575894	+	0.817524i	$0.695346\pi$
$620$	0	0
$621$	5.06830e11i	0.136757i
$622$	0	0
$623$	−5.46684e11	−0.145392
$624$	0	0
$625$	1.99564e12	0.523145
$626$	0	0
$627$	− 3.38240e11i	− 0.0874020i
$628$	0	0
$629$	− 4.81281e12i	− 1.22594i
$630$	0	0
$631$	−5.38581e12	−1.35244	−0.676221	−	0.736698i	$-0.736384\pi$
$631$	−5.38581e12	−1.35244	−0.676221	+	0.736698i	$0.736384\pi$
$632$	0	0
$633$	1.72234e12	0.426386
$634$	0	0
$635$	1.71152e12i	0.417735i
$636$	0	0
$637$	− 6.07275e12i	− 1.46136i
$638$	0	0
$639$	8.62740e11	0.204704
$640$	0	0
$641$	6.98854e12	1.63503	0.817514	−	0.575909i	$-0.195352\pi$
$641$	6.98854e12	1.63503	0.817514	+	0.575909i	$0.195352\pi$
$642$	0	0
$643$	− 1.21879e12i	− 0.281176i	−0.990068	−	0.140588i	$-0.955101\pi$
$643$	− 1.21879e12i	− 0.281176i	0.990068	−	0.140588i	$-0.0448994\pi$
$644$	0	0
$645$	− 1.61867e12i	− 0.368247i
$646$	0	0
$647$	−8.11593e11	−0.182083	−0.0910414	−	0.995847i	$-0.529020\pi$
$647$	−8.11593e11	−0.182083	−0.0910414	+	0.995847i	$0.529020\pi$
$648$	0	0
$649$	1.38739e12	0.306970
$650$	0	0
$651$	1.56893e11i	0.0342364i
$652$	0	0
$653$	− 2.33692e12i	− 0.502962i	−0.967862	−	0.251481i	$-0.919082\pi$
$653$	− 2.33692e12i	− 0.502962i	0.967862	−	0.251481i	$-0.0809175\pi$
$654$	0	0
$655$	1.28596e12	0.272986
$656$	0	0
$657$	−1.23915e11	−0.0259466
$658$	0	0
$659$	5.06294e12i	1.04573i	0.852416	+	0.522863i	$0.175136\pi$
$659$	5.06294e12i	1.04573i	−0.852416	+	0.522863i	$0.824864\pi$
$660$	0	0
$661$	− 1.34073e12i	− 0.273170i	−0.990628	−	0.136585i	$-0.956387\pi$
$661$	− 1.34073e12i	− 0.273170i	0.990628	−	0.136585i	$-0.0436127\pi$
$662$	0	0
$663$	6.37597e12	1.28155
$664$	0	0
$665$	−2.07749e11	−0.0411946
$666$	0	0
$667$	− 1.07208e12i	− 0.209730i
$668$	0	0
$669$	− 2.20321e12i	− 0.425245i
$670$	0	0
$671$	−8.11478e11	−0.154535
$672$	0	0
$673$	4.40610e12	0.827916	0.413958	−	0.910296i	$-0.364146\pi$
$673$	4.40610e12	0.827916	0.413958	+	0.910296i	$0.364146\pi$
$674$	0	0
$675$	8.63171e11i	0.160041i
$676$	0	0
$677$	2.41655e12i	0.442126i	0.975259	+	0.221063i	$0.0709527\pi$
$677$	2.41655e12i	0.442126i	−0.975259	+	0.221063i	$0.929047\pi$
$678$	0	0
$679$	−1.68972e12	−0.305072
$680$	0	0
$681$	−3.37304e11	−0.0600980
$682$	0	0
$683$	− 3.19393e12i	− 0.561606i	−0.959765	−	0.280803i	$-0.909399\pi$
$683$	− 3.19393e12i	− 0.561606i	0.959765	−	0.280803i	$-0.0906008\pi$
$684$	0	0
$685$	1.10613e11i	0.0191955i
$686$	0	0
$687$	4.14772e12	0.710401
$688$	0	0
$689$	−7.24376e12	−1.22455
$690$	0	0
$691$	− 1.11051e13i	− 1.85298i	−0.376318	−	0.926491i	$-0.622810\pi$
$691$	− 1.11051e13i	− 1.85298i	0.376318	−	0.926491i	$-0.377190\pi$
$692$	0	0
$693$	− 3.61159e11i	− 0.0594839i
$694$	0	0
$695$	−3.27116e12	−0.531828
$696$	0	0
$697$	−1.26403e13	−2.02866
$698$	0	0
$699$	1.93978e12i	0.307330i
$700$	0	0
$701$	− 1.08592e12i	− 0.169850i	−0.996387	−	0.0849249i	$-0.972935\pi$
$701$	− 1.08592e12i	− 0.169850i	0.996387	−	0.0849249i	$-0.0270650\pi$
$702$	0	0
$703$	−1.72997e12	−0.267140
$704$	0	0
$705$	2.71122e12	0.413346
$706$	0	0
$707$	1.82376e12i	0.274525i
$708$	0	0
$709$	− 4.38234e12i	− 0.651325i	−0.945486	−	0.325662i	$-0.894413\pi$
$709$	− 4.38234e12i	− 0.651325i	0.945486	−	0.325662i	$-0.105587\pi$
$710$	0	0
$711$	4.44144e11	0.0651795
$712$	0	0
$713$	−8.45344e11	−0.122498
$714$	0	0
$715$	2.46591e12i	0.352858i
$716$	0	0
$717$	− 6.23475e11i	− 0.0881015i
$718$	0	0
$719$	1.08321e13	1.51158	0.755789	−	0.654815i	$-0.227253\pi$
$719$	1.08321e13	1.51158	0.755789	+	0.654815i	$0.227253\pi$
$720$	0	0
$721$	−3.16200e12	−0.435766
$722$	0	0
$723$	8.00142e10i	0.0108904i
$724$	0	0
$725$	− 1.82584e12i	− 0.245438i
$726$	0	0
$727$	−4.52246e12	−0.600440	−0.300220	−	0.953870i	$-0.597060\pi$
$727$	−4.52246e12	−0.600440	−0.300220	+	0.953870i	$0.597060\pi$
$728$	0	0
$729$	−2.82430e11	−0.0370370
$730$	0	0
$731$	1.60692e13i	2.08145i
$732$	0	0
$733$	1.16220e13i	1.48701i	0.668729	+	0.743506i	$0.266839\pi$
$733$	1.16220e13i	1.48701i	−0.668729	+	0.743506i	$0.733161\pi$
$734$	0	0
$735$	1.65278e12	0.208892
$736$	0	0
$737$	−4.97835e12	−0.621558
$738$	0	0
$739$	− 2.43315e12i	− 0.300102i	−0.988678	−	0.150051i	$-0.952056\pi$
$739$	− 2.43315e12i	− 0.300102i	0.988678	−	0.150051i	$-0.0479438\pi$
$740$	0	0
$741$	− 2.29185e12i	− 0.279257i
$742$	0	0
$743$	1.63463e13	1.96774	0.983872	−	0.178875i	$-0.0572457\pi$
$743$	1.63463e13	1.96774	0.983872	+	0.178875i	$0.0572457\pi$
$744$	0	0
$745$	5.08720e12	0.605028
$746$	0	0
$747$	2.44887e12i	0.287755i
$748$	0	0
$749$	− 4.82782e11i	− 0.0560509i
$750$	0	0
$751$	−1.35795e13	−1.55778	−0.778888	−	0.627164i	$-0.784216\pi$
$751$	−1.35795e13	−1.55778	−0.778888	+	0.627164i	$0.784216\pi$
$752$	0	0
$753$	−2.43735e12	−0.276274
$754$	0	0
$755$	− 6.43810e12i	− 0.721101i
$756$	0	0
$757$	− 5.13921e12i	− 0.568807i	−0.958705	−	0.284403i	$-0.908205\pi$
$757$	− 5.13921e12i	− 0.568807i	0.958705	−	0.284403i	$-0.0917955\pi$
$758$	0	0
$759$	1.94594e12	0.212834
$760$	0	0
$761$	1.01011e13	1.09178	0.545892	−	0.837855i	$-0.316191\pi$
$761$	1.01011e13	1.09178	0.545892	+	0.837855i	$0.316191\pi$
$762$	0	0
$763$	1.86507e12i	0.199220i
$764$	0	0
$765$	1.73531e12i	0.183189i
$766$	0	0
$767$	9.40066e12	0.980797
$768$	0	0
$769$	−2.55912e12	−0.263889	−0.131945	−	0.991257i	$-0.542122\pi$
$769$	−2.55912e12	−0.263889	−0.131945	+	0.991257i	$0.542122\pi$
$770$	0	0
$771$	− 1.36501e12i	− 0.139120i
$772$	0	0
$773$	1.50470e13i	1.51580i	0.652372	+	0.757899i	$0.273774\pi$
$773$	1.50470e13i	1.51580i	−0.652372	+	0.757899i	$0.726226\pi$
$774$	0	0
$775$	−1.43969e12	−0.143354
$776$	0	0
$777$	−1.84719e12	−0.181810
$778$	0	0
$779$	4.54357e12i	0.442058i
$780$	0	0
$781$	− 3.31243e12i	− 0.318579i
$782$	0	0
$783$	5.97414e11	0.0567999
$784$	0	0
$785$	3.59145e12	0.337564
$786$	0	0
$787$	− 1.23403e13i	− 1.14667i	−0.819320	−	0.573337i	$-0.805649\pi$
$787$	− 1.23403e13i	− 1.14667i	0.819320	−	0.573337i	$-0.194351\pi$
$788$	0	0
$789$	2.00827e12i	0.184491i
$790$	0	0
$791$	2.74801e12	0.249588
$792$	0	0
$793$	−5.49841e12	−0.493751
$794$	0	0
$795$	− 1.97149e12i	− 0.175042i
$796$	0	0
$797$	− 7.96003e12i	− 0.698799i	−0.936974	−	0.349399i	$-0.886386\pi$
$797$	− 7.96003e12i	− 0.698799i	0.936974	−	0.349399i	$-0.113614\pi$
$798$	0	0
$799$	−2.69154e13	−2.33636
$800$	0	0
$801$	1.64140e12	0.140886
$802$	0	0
$803$	4.75764e11i	0.0403805i
$804$	0	0
$805$	− 1.19520e12i	− 0.100314i
$806$	0	0
$807$	1.17115e13	0.972030
$808$	0	0
$809$	1.33262e13	1.09380	0.546902	−	0.837197i	$-0.315807\pi$
$809$	1.33262e13	1.09380	0.546902	+	0.837197i	$0.315807\pi$
$810$	0	0
$811$	3.23877e12i	0.262897i	0.991323	+	0.131449i	$0.0419628\pi$
$811$	3.23877e12i	0.262897i	−0.991323	+	0.131449i	$0.958037\pi$
$812$	0	0
$813$	7.55818e12i	0.606750i
$814$	0	0
$815$	2.33331e12	0.185252
$816$	0	0
$817$	5.77609e12	0.453560
$818$	0	0
$819$	− 2.44714e12i	− 0.190056i
$820$	0	0
$821$	1.53482e13i	1.17900i	0.807768	+	0.589501i	$0.200676\pi$
$821$	1.53482e13i	1.17900i	−0.807768	+	0.589501i	$0.799324\pi$
$822$	0	0
$823$	−2.18794e13	−1.66240	−0.831202	−	0.555971i	$-0.812347\pi$
$823$	−2.18794e13	−1.66240	−0.831202	+	0.555971i	$0.812347\pi$
$824$	0	0
$825$	3.31409e12	0.249070
$826$	0	0
$827$	− 1.40288e12i	− 0.104291i	−0.998640	−	0.0521454i	$-0.983394\pi$
$827$	− 1.40288e12i	− 0.104291i	0.998640	−	0.0521454i	$-0.0166059\pi$
$828$	0	0
$829$	1.59294e13i	1.17139i	0.810530	+	0.585697i	$0.199179\pi$
$829$	1.59294e13i	1.17139i	−0.810530	+	0.585697i	$0.800821\pi$
$830$	0	0
$831$	1.14806e13	0.835140
$832$	0	0
$833$	−1.64078e13	−1.18072
$834$	0	0
$835$	4.25988e12i	0.303255i
$836$	0	0
$837$	− 4.71065e11i	− 0.0331754i
$838$	0	0
$839$	1.81369e12	0.126367	0.0631836	−	0.998002i	$-0.479875\pi$
$839$	1.81369e12	0.126367	0.0631836	+	0.998002i	$0.479875\pi$
$840$	0	0
$841$	1.32435e13	0.912892
$842$	0	0
$843$	− 3.70569e12i	− 0.252723i
$844$	0	0
$845$	1.06267e13i	0.717040i
$846$	0	0
$847$	3.76595e12	0.251420
$848$	0	0
$849$	−6.94343e12	−0.458659
$850$	0	0
$851$	− 9.95273e12i	− 0.650518i
$852$	0	0
$853$	2.21558e13i	1.43290i	0.697637	+	0.716451i	$0.254235\pi$
$853$	2.21558e13i	1.43290i	−0.697637	+	0.716451i	$0.745765\pi$
$854$	0	0
$855$	6.23758e11	0.0399180
$856$	0	0
$857$	−1.33185e13	−0.843416	−0.421708	−	0.906732i	$-0.638569\pi$
$857$	−1.33185e13	−0.843416	−0.421708	+	0.906732i	$0.638569\pi$
$858$	0	0
$859$	4.69713e12i	0.294349i	0.989111	+	0.147175i	$0.0470179\pi$
$859$	4.69713e12i	0.294349i	−0.989111	+	0.147175i	$0.952982\pi$
$860$	0	0
$861$	4.85144e12i	0.300855i
$862$	0	0
$863$	−1.20688e12	−0.0740653	−0.0370326	−	0.999314i	$-0.511791\pi$
$863$	−1.20688e12	−0.0740653	−0.0370326	+	0.999314i	$0.511791\pi$
$864$	0	0
$865$	−5.25227e12	−0.318988
$866$	0	0
$867$	− 7.62148e12i	− 0.458093i
$868$	0	0
$869$	− 1.70526e12i	− 0.101438i
$870$	0	0
$871$	−3.37323e13	−1.98593
$872$	0	0
$873$	5.07334e12	0.295617
$874$	0	0
$875$	− 4.48326e12i	− 0.258558i
$876$	0	0
$877$	2.40216e13i	1.37121i	0.727975	+	0.685604i	$0.240462\pi$
$877$	2.40216e13i	1.37121i	−0.727975	+	0.685604i	$0.759538\pi$
$878$	0	0
$879$	6.42189e12	0.362838
$880$	0	0
$881$	−1.51231e13	−0.845765	−0.422883	−	0.906184i	$-0.638982\pi$
$881$	−1.51231e13	−0.845765	−0.422883	+	0.906184i	$0.638982\pi$
$882$	0	0
$883$	2.63940e13i	1.46111i	0.682854	+	0.730555i	$0.260738\pi$
$883$	2.63940e13i	1.46111i	−0.682854	+	0.730555i	$0.739262\pi$
$884$	0	0
$885$	2.55852e12i	0.140199i
$886$	0	0
$887$	−2.20527e13	−1.19621	−0.598103	−	0.801419i	$-0.704079\pi$
$887$	−2.20527e13	−1.19621	−0.598103	+	0.801419i	$0.704079\pi$
$888$	0	0
$889$	−6.52127e12	−0.350166
$890$	0	0
$891$	1.08437e12i	0.0576405i
$892$	0	0
$893$	9.67476e12i	0.509106i
$894$	0	0
$895$	−1.28732e13	−0.670628
$896$	0	0
$897$	1.31853e13	0.680023
$898$	0	0
$899$	9.96428e11i	0.0508777i
$900$	0	0
$901$	1.95718e13i	0.989393i
$902$	0	0
$903$	6.16748e12	0.308683
$904$	0	0
$905$	5.83041e12	0.288922
$906$	0	0
$907$	5.06988e12i	0.248751i	0.992235	+	0.124376i	$0.0396927\pi$
$907$	5.06988e12i	0.248751i	−0.992235	+	0.124376i	$0.960307\pi$
$908$	0	0
$909$	− 5.47579e12i	− 0.266017i
$910$	0	0
$911$	2.16345e13	1.04067	0.520337	−	0.853961i	$-0.325806\pi$
$911$	2.16345e13	1.04067	0.520337	+	0.853961i	$0.325806\pi$
$912$	0	0
$913$	9.40225e12	0.447830
$914$	0	0
$915$	− 1.49647e12i	− 0.0705785i
$916$	0	0
$917$	4.89978e12i	0.228831i
$918$	0	0
$919$	8.37636e12	0.387378	0.193689	−	0.981063i	$-0.437955\pi$
$919$	8.37636e12	0.387378	0.193689	+	0.981063i	$0.437955\pi$
$920$	0	0
$921$	−2.19368e13	−1.00463
$922$	0	0
$923$	− 2.24444e13i	− 1.01789i
$924$	0	0
$925$	− 1.69503e13i	− 0.761271i
$926$	0	0
$927$	9.49380e12	0.422261
$928$	0	0
$929$	3.81132e13	1.67882	0.839412	−	0.543496i	$-0.182900\pi$
$929$	3.81132e13	1.67882	0.839412	+	0.543496i	$0.182900\pi$
$930$	0	0
$931$	5.89781e12i	0.257287i
$932$	0	0
$933$	2.05031e13i	0.885834i
$934$	0	0
$935$	6.66259e12	0.285096
$936$	0	0
$937$	2.86158e13	1.21277	0.606383	−	0.795172i	$-0.292620\pi$
$937$	2.86158e13	1.21277	0.606383	+	0.795172i	$0.292620\pi$
$938$	0	0
$939$	3.08533e12i	0.129511i
$940$	0	0
$941$	3.31159e13i	1.37684i	0.725312	+	0.688421i	$0.241696\pi$
$941$	3.31159e13i	1.37684i	−0.725312	+	0.688421i	$0.758304\pi$
$942$	0	0
$943$	−2.61397e13	−1.07646
$944$	0	0
$945$	6.66024e11	0.0271673
$946$	0	0
$947$	6.14318e12i	0.248209i	0.992269	+	0.124105i	$0.0396059\pi$
$947$	6.14318e12i	0.248209i	−0.992269	+	0.124105i	$0.960394\pi$
$948$	0	0
$949$	3.22368e12i	0.129019i
$950$	0	0
$951$	2.15895e13	0.855915
$952$	0	0
$953$	1.53935e13	0.604531	0.302266	−	0.953224i	$-0.402257\pi$
$953$	1.53935e13	0.604531	0.302266	+	0.953224i	$0.402257\pi$
$954$	0	0
$955$	− 1.55641e13i	− 0.605494i
$956$	0	0
$957$	− 2.29373e12i	− 0.0883972i
$958$	0	0
$959$	−4.21460e11	−0.0160906
$960$	0	0
$961$	−2.56539e13	−0.970284
$962$	0	0
$963$	1.44954e12i	0.0543139i
$964$	0	0
$965$	− 1.49096e13i	− 0.553468i
$966$	0	0
$967$	−4.90791e13	−1.80500	−0.902500	−	0.430691i	$-0.858270\pi$
$967$	−4.90791e13	−1.80500	−0.902500	+	0.430691i	$0.858270\pi$
$968$	0	0
$969$	−6.19230e12	−0.225629
$970$	0	0
$971$	2.88291e13i	1.04075i	0.853939	+	0.520373i	$0.174207\pi$
$971$	2.88291e13i	1.04075i	−0.853939	+	0.520373i	$0.825793\pi$
$972$	0	0
$973$	− 1.24638e13i	− 0.445804i
$974$	0	0
$975$	2.24556e13	0.795799
$976$	0	0
$977$	−1.64010e13	−0.575899	−0.287949	−	0.957646i	$-0.592973\pi$
$977$	−1.64010e13	−0.575899	−0.287949	+	0.957646i	$0.592973\pi$
$978$	0	0
$979$	− 6.30204e12i	− 0.219260i
$980$	0	0
$981$	− 5.59980e12i	− 0.193046i
$982$	0	0
$983$	5.06192e13	1.72912	0.864559	−	0.502531i	$-0.167598\pi$
$983$	5.06192e13	1.72912	0.864559	+	0.502531i	$0.167598\pi$
$984$	0	0
$985$	6.41778e12	0.217231
$986$	0	0
$987$	1.03303e13i	0.346486i
$988$	0	0
$989$	3.32306e13i	1.10447i
$990$	0	0
$991$	−2.53288e13	−0.834225	−0.417113	−	0.908855i	$-0.636958\pi$
$991$	−2.53288e13	−0.834225	−0.417113	+	0.908855i	$0.636958\pi$
$992$	0	0
$993$	1.07364e13	0.350418
$994$	0	0
$995$	1.29029e13i	0.417333i
$996$	0	0
$997$	5.28104e13i	1.69275i	0.532591	+	0.846373i	$0.321218\pi$
$997$	5.28104e13i	1.69275i	−0.532591	+	0.846373i	$0.678782\pi$
$998$	0	0
$999$	5.54613e12	0.176175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.d.f.193.4	✓	20
4.3	odd	2	inner	384.10.d.f.193.14	yes	20
8.3	odd	2	inner	384.10.d.f.193.7	yes	20
8.5	even	2	inner	384.10.d.f.193.17	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.d.f.193.4	✓	20	1.1	even	1	trivial
384.10.d.f.193.7	yes	20	8.3	odd	2	inner
384.10.d.f.193.14	yes	20	4.3	odd	2	inner
384.10.d.f.193.17	yes	20	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-81.0000i q^{3} -573.512i q^{5} +2185.20 q^{7} -6561.00 q^{9} +O(q^{10})\)	\(q-81.0000i q^{3} -573.512i q^{5} +2185.20 q^{7} -6561.00 q^{9} +25190.5i q^{11} +170686. i q^{13} -46454.5 q^{15} +461173. q^{17} -165769. i q^{19} -177001. i q^{21} +953690. q^{23} +1.62421e6 q^{25} +531441. i q^{27} -1.12414e6i q^{29} -886392. q^{31} +2.04043e6 q^{33} -1.25324e6i q^{35} -1.04360e7i q^{37} +1.38256e7 q^{39} -2.74090e7 q^{41} +3.48442e7i q^{43} +3.76281e6i q^{45} -5.83629e7 q^{47} -3.55785e7 q^{49} -3.73550e7i q^{51} +4.24392e7i q^{53} +1.44471e7 q^{55} -1.34273e7 q^{57} -5.50758e7i q^{59} +3.22137e7i q^{61} -1.43371e7 q^{63} +9.78904e7 q^{65} +1.97628e8i q^{67} -7.72489e7i q^{69} -1.31495e8 q^{71} +1.88866e7 q^{73} -1.31561e8i q^{75} +5.50464e7i q^{77} -6.76946e7 q^{79} +4.30467e7 q^{81} -3.73246e8i q^{83} -2.64488e8i q^{85} -9.10553e7 q^{87} -2.50175e8 q^{89} +3.72983e8i q^{91} +7.17978e7i q^{93} -9.50706e7 q^{95} -7.73257e8 q^{97} -1.65275e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 131220 q^{9}+O(q^{10}) \) 20 * q - 131220 * q^9	\( 20 q - 131220 q^{9} - 905768 q^{17} - 9682620 q^{25} + 12054096 q^{33} + 74264008 q^{41} + 252775700 q^{49} - 5335632 q^{57} + 245588672 q^{65} - 895193896 q^{73} + 860934420 q^{81} + 882422136 q^{89} + 433683736 q^{97}+O(q^{100}) \) 20 * q - 131220 * q^9 - 905768 * q^17 - 9682620 * q^25 + 12054096 * q^33 + 74264008 * q^41 + 252775700 * q^49 - 5335632 * q^57 + 245588672 * q^65 - 895193896 * q^73 + 860934420 * q^81 + 882422136 * q^89 + 433683736 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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