LMFDB - Embedded newform 384.10.d.a.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			$-53.8781i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.10.d.a.193.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+81.0000i q^{3} -473.533i q^{5} -574.939 q^{7} -6561.00 q^{9} +O(q^{10})$	$q+81.0000i q^{3} -473.533i q^{5} -574.939 q^{7} -6561.00 q^{9} -72774.7i q^{11} +14829.2i q^{13} +38356.2 q^{15} +109932. q^{17} +497573. i q^{19} -46570.1i q^{21} -104764. q^{23} +1.72889e6 q^{25} -531441. i q^{27} +7.06390e6i q^{29} -2.30578e6 q^{31} +5.89475e6 q^{33} +272253. i q^{35} -5.20960e6i q^{37} -1.20116e6 q^{39} -1.08234e7 q^{41} +2.04081e7i q^{43} +3.10685e6i q^{45} +3.47476e7 q^{47} -4.00231e7 q^{49} +8.90445e6i q^{51} -9.34226e7i q^{53} -3.44612e7 q^{55} -4.03034e7 q^{57} -9.04786e7i q^{59} +2.00269e7i q^{61} +3.77218e6 q^{63} +7.02210e6 q^{65} +1.09151e8i q^{67} -8.48588e6i q^{69} +2.65931e8 q^{71} +5.05525e7 q^{73} +1.40040e8i q^{75} +4.18410e7i q^{77} -1.24238e7 q^{79} +4.30467e7 q^{81} -2.72920e8i q^{83} -5.20562e7i q^{85} -5.72176e8 q^{87} -2.70637e8 q^{89} -8.52588e6i q^{91} -1.86768e8i q^{93} +2.35617e8 q^{95} -4.06136e8 q^{97} +4.77475e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 473.533i	− 0.338833i	−0.985545	−	0.169416i	$-0.945812\pi$
$5$	− 473.533i	− 0.338833i	0.985545	−	0.169416i	$-0.0541882\pi$
$6$	0	0
$7$	−574.939	−0.0905067	−0.0452534	−	0.998976i	$-0.514410\pi$
$7$	−574.939	−0.0905067	−0.0452534	+	0.998976i	$0.514410\pi$
$8$	0	0
$9$	−6561.00	−0.333333
$10$	0	0
$11$	− 72774.7i	− 1.49869i	−0.662177	−	0.749347i	$-0.730367\pi$
$11$	− 72774.7i	− 1.49869i	0.662177	−	0.749347i	$-0.269633\pi$
$12$	0	0
$13$	14829.2i	0.144003i	0.997405	+	0.0720016i	$0.0229387\pi$
$13$	14829.2i	0.144003i	−0.997405	+	0.0720016i	$0.977061\pi$
$14$	0	0
$15$	38356.2	0.195625
$16$	0	0
$17$	109932.	0.319229	0.159614	−	0.987179i	$-0.448975\pi$
$17$	109932.	0.319229	0.159614	+	0.987179i	$0.448975\pi$
$18$	0	0
$19$	497573.i	0.875922i	0.898994	+	0.437961i	$0.144299\pi$
$19$	497573.i	0.875922i	−0.898994	+	0.437961i	$0.855701\pi$
$20$	0	0
$21$	− 46570.1i	− 0.0522541i
$22$	0	0
$23$	−104764.	−0.0780614	−0.0390307	−	0.999238i	$-0.512427\pi$
$23$	−104764.	−0.0780614	−0.0390307	+	0.999238i	$0.512427\pi$
$24$	0	0
$25$	1.72889e6	0.885192
$26$	0	0
$27$	− 531441.i	− 0.192450i
$28$	0	0
$29$	7.06390e6i	1.85461i	0.374301	+	0.927307i	$0.377883\pi$
$29$	7.06390e6i	1.85461i	−0.374301	+	0.927307i	$0.622117\pi$
$30$	0	0
$31$	−2.30578e6	−0.448426	−0.224213	−	0.974540i	$-0.571981\pi$
$31$	−2.30578e6	−0.448426	−0.224213	+	0.974540i	$0.571981\pi$
$32$	0	0
$33$	5.89475e6	0.865272
$34$	0	0
$35$	272253.i	0.0306666i
$36$	0	0
$37$	− 5.20960e6i	− 0.456979i	−0.973546	−	0.228490i	$-0.926621\pi$
$37$	− 5.20960e6i	− 0.456979i	0.973546	−	0.228490i	$-0.0733787\pi$
$38$	0	0
$39$	−1.20116e6	−0.0831402
$40$	0	0
$41$	−1.08234e7	−0.598185	−0.299092	−	0.954224i	$-0.596684\pi$
$41$	−1.08234e7	−0.598185	−0.299092	+	0.954224i	$0.596684\pi$
$42$	0	0
$43$	2.04081e7i	0.910319i	0.890410	+	0.455160i	$0.150418\pi$
$43$	2.04081e7i	0.910319i	−0.890410	+	0.455160i	$0.849582\pi$
$44$	0	0
$45$	3.10685e6i	0.112944i
$46$	0	0
$47$	3.47476e7	1.03869	0.519343	−	0.854566i	$-0.326177\pi$
$47$	3.47476e7	1.03869	0.519343	+	0.854566i	$0.326177\pi$
$48$	0	0
$49$	−4.00231e7	−0.991809
$50$	0	0
$51$	8.90445e6i	0.184307i
$52$	0	0
$53$	− 9.34226e7i	− 1.62634i	−0.582028	−	0.813169i	$-0.697741\pi$
$53$	− 9.34226e7i	− 1.62634i	0.582028	−	0.813169i	$-0.302259\pi$
$54$	0	0
$55$	−3.44612e7	−0.507807
$56$	0	0
$57$	−4.03034e7	−0.505714
$58$	0	0
$59$	− 9.04786e7i	− 0.972102i	−0.873930	−	0.486051i	$-0.838437\pi$
$59$	− 9.04786e7i	− 0.972102i	0.873930	−	0.486051i	$-0.161563\pi$
$60$	0	0
$61$	2.00269e7i	0.185195i	0.995704	+	0.0925976i	$0.0295170\pi$
$61$	2.00269e7i	0.185195i	−0.995704	+	0.0925976i	$0.970483\pi$
$62$	0	0
$63$	3.77218e6	0.0301689
$64$	0	0
$65$	7.02210e6	0.0487930
$66$	0	0
$67$	1.09151e8i	0.661745i	0.943676	+	0.330872i	$0.107343\pi$
$67$	1.09151e8i	0.661745i	−0.943676	+	0.330872i	$0.892657\pi$
$68$	0	0
$69$	− 8.48588e6i	− 0.0450688i
$70$	0	0
$71$	2.65931e8	1.24196	0.620978	−	0.783828i	$-0.286736\pi$
$71$	2.65931e8	1.24196	0.620978	+	0.783828i	$0.286736\pi$
$72$	0	0
$73$	5.05525e7	0.208348	0.104174	−	0.994559i	$-0.466780\pi$
$73$	5.05525e7	0.208348	0.104174	+	0.994559i	$0.466780\pi$
$74$	0	0
$75$	1.40040e8i	0.511066i
$76$	0	0
$77$	4.18410e7i	0.135642i
$78$	0	0
$79$	−1.24238e7	−0.0358865	−0.0179433	−	0.999839i	$-0.505712\pi$
$79$	−1.24238e7	−0.0358865	−0.0179433	+	0.999839i	$0.505712\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	− 2.72920e8i	− 0.631225i	−0.948888	−	0.315613i	$-0.897790\pi$
$83$	− 2.72920e8i	− 0.631225i	0.948888	−	0.315613i	$-0.102210\pi$
$84$	0	0
$85$	− 5.20562e7i	− 0.108165i
$86$	0	0
$87$	−5.72176e8	−1.07076
$88$	0	0
$89$	−2.70637e8	−0.457227	−0.228613	−	0.973517i	$-0.573419\pi$
$89$	−2.70637e8	−0.457227	−0.228613	+	0.973517i	$0.573419\pi$
$90$	0	0
$91$	− 8.52588e6i	− 0.0130333i
$92$	0	0
$93$	− 1.86768e8i	− 0.258899i
$94$	0	0
$95$	2.35617e8	0.296791
$96$	0	0
$97$	−4.06136e8	−0.465799	−0.232899	−	0.972501i	$-0.574821\pi$
$97$	−4.06136e8	−0.465799	−0.232899	+	0.972501i	$0.574821\pi$
$98$	0	0
$99$	4.77475e8i	0.499565i
$100$	0	0
$101$	− 3.58458e8i	− 0.342762i	−0.985205	−	0.171381i	$-0.945177\pi$
$101$	− 3.58458e8i	− 0.342762i	0.985205	−	0.171381i	$-0.0548229\pi$
$102$	0	0
$103$	3.12727e8	0.273777	0.136889	−	0.990586i	$-0.456290\pi$
$103$	3.12727e8	0.273777	0.136889	+	0.990586i	$0.456290\pi$
$104$	0	0
$105$	−2.20525e7	−0.0177054
$106$	0	0
$107$	− 9.28960e8i	− 0.685125i	−0.939495	−	0.342562i	$-0.888705\pi$
$107$	− 9.28960e8i	− 0.685125i	0.939495	−	0.342562i	$-0.111295\pi$
$108$	0	0
$109$	− 1.54301e9i	− 1.04701i	−0.852024	−	0.523503i	$-0.824625\pi$
$109$	− 1.54301e9i	− 1.04701i	0.852024	−	0.523503i	$-0.175375\pi$
$110$	0	0
$111$	4.21977e8	0.263837
$112$	0	0
$113$	−5.98385e8	−0.345245	−0.172623	−	0.984988i	$-0.555224\pi$
$113$	−5.98385e8	−0.345245	−0.172623	+	0.984988i	$0.555224\pi$
$114$	0	0
$115$	4.96092e7i	0.0264498i
$116$	0	0
$117$	− 9.72942e7i	− 0.0480010i
$118$	0	0
$119$	−6.32039e7	−0.0288924
$120$	0	0
$121$	−2.93820e9	−1.24609
$122$	0	0
$123$	− 8.76693e8i	− 0.345362i
$124$	0	0
$125$	− 1.74356e9i	− 0.638765i
$126$	0	0
$127$	−2.52986e9	−0.862940	−0.431470	−	0.902127i	$-0.642005\pi$
$127$	−2.52986e9	−0.862940	−0.431470	+	0.902127i	$0.642005\pi$
$128$	0	0
$129$	−1.65305e9	−0.525573
$130$	0	0
$131$	1.70203e8i	0.0504948i	0.999681	+	0.0252474i	$0.00803734\pi$
$131$	1.70203e8i	0.0504948i	−0.999681	+	0.0252474i	$0.991963\pi$
$132$	0	0
$133$	− 2.86074e8i	− 0.0792768i
$134$	0	0
$135$	−2.51655e8	−0.0652084
$136$	0	0
$137$	2.59035e9	0.628226	0.314113	−	0.949386i	$-0.398293\pi$
$137$	2.59035e9	0.628226	0.314113	+	0.949386i	$0.398293\pi$
$138$	0	0
$139$	− 2.75040e9i	− 0.624928i	−0.949930	−	0.312464i	$-0.898846\pi$
$139$	− 2.75040e9i	− 0.624928i	0.949930	−	0.312464i	$-0.101154\pi$
$140$	0	0
$141$	2.81456e9i	0.599686i
$142$	0	0
$143$	1.07919e9	0.215817
$144$	0	0
$145$	3.34499e9	0.628404
$146$	0	0
$147$	− 3.24187e9i	− 0.572621i
$148$	0	0
$149$	− 2.97566e9i	− 0.494590i	−0.968940	−	0.247295i	$-0.920458\pi$
$149$	− 2.97566e9i	− 0.494590i	0.968940	−	0.247295i	$-0.0795416\pi$
$150$	0	0
$151$	1.10799e10	1.73435	0.867177	−	0.498000i	$-0.165932\pi$
$151$	1.10799e10	1.73435	0.867177	+	0.498000i	$0.165932\pi$
$152$	0	0
$153$	−7.21261e8	−0.106410
$154$	0	0
$155$	1.09186e9i	0.151941i
$156$	0	0
$157$	1.20320e10i	1.58048i	0.612797	+	0.790240i	$0.290044\pi$
$157$	1.20320e10i	1.58048i	−0.612797	+	0.790240i	$0.709956\pi$
$158$	0	0
$159$	7.56723e9	0.938966
$160$	0	0
$161$	6.02329e7	0.00706508
$162$	0	0
$163$	− 1.21499e9i	− 0.134812i	−0.997726	−	0.0674062i	$-0.978528\pi$
$163$	− 1.21499e9i	− 0.134812i	0.997726	−	0.0674062i	$-0.0214724\pi$
$164$	0	0
$165$	− 2.79136e9i	− 0.293182i
$166$	0	0
$167$	−1.01298e10	−1.00780	−0.503902	−	0.863761i	$-0.668103\pi$
$167$	−1.01298e10	−1.00780	−0.503902	+	0.863761i	$0.668103\pi$
$168$	0	0
$169$	1.03846e10	0.979263
$170$	0	0
$171$	− 3.26458e9i	− 0.291974i
$172$	0	0
$173$	1.12366e10i	0.953738i	0.878974	+	0.476869i	$0.158228\pi$
$173$	1.12366e10i	0.953738i	−0.878974	+	0.476869i	$0.841772\pi$
$174$	0	0
$175$	−9.94008e8	−0.0801159
$176$	0	0
$177$	7.32877e9	0.561244
$178$	0	0
$179$	3.72939e9i	0.271518i	0.990742	+	0.135759i	$0.0433473\pi$
$179$	3.72939e9i	0.271518i	−0.990742	+	0.135759i	$0.956653\pi$
$180$	0	0
$181$	− 1.47052e10i	− 1.01840i	−0.860649	−	0.509198i	$-0.829942\pi$
$181$	− 1.47052e10i	− 1.01840i	0.860649	−	0.509198i	$-0.170058\pi$
$182$	0	0
$183$	−1.62218e9	−0.106922
$184$	0	0
$185$	−2.46692e9	−0.154839
$186$	0	0
$187$	− 8.00023e9i	− 0.478426i
$188$	0	0
$189$	3.05546e8i	0.0174180i
$190$	0	0
$191$	2.16788e10	1.17865	0.589325	−	0.807896i	$-0.299394\pi$
$191$	2.16788e10	1.17865	0.589325	+	0.807896i	$0.299394\pi$
$192$	0	0
$193$	−2.45321e10	−1.27270	−0.636350	−	0.771400i	$-0.719557\pi$
$193$	−2.45321e10	−1.27270	−0.636350	+	0.771400i	$0.719557\pi$
$194$	0	0
$195$	5.68790e8i	0.0281706i
$196$	0	0
$197$	− 1.44518e10i	− 0.683632i	−0.939767	−	0.341816i	$-0.888958\pi$
$197$	− 1.44518e10i	− 0.683632i	0.939767	−	0.341816i	$-0.111042\pi$
$198$	0	0
$199$	2.71655e10	1.22794	0.613972	−	0.789328i	$-0.289571\pi$
$199$	2.71655e10	1.22794	0.613972	+	0.789328i	$0.289571\pi$
$200$	0	0
$201$	−8.84122e9	−0.382058
$202$	0	0
$203$	− 4.06132e9i	− 0.167855i
$204$	0	0
$205$	5.12522e9i	0.202684i
$206$	0	0
$207$	6.87356e8	0.0260205
$208$	0	0
$209$	3.62107e10	1.31274
$210$	0	0
$211$	− 3.52449e10i	− 1.22412i	−0.790810	−	0.612062i	$-0.790340\pi$
$211$	− 3.52449e10i	− 1.22412i	0.790810	−	0.612062i	$-0.209660\pi$
$212$	0	0
$213$	2.15404e10i	0.717044i
$214$	0	0
$215$	9.66389e9	0.308446
$216$	0	0
$217$	1.32568e9	0.0405855
$218$	0	0
$219$	4.09475e9i	0.120290i
$220$	0	0
$221$	1.63019e9i	0.0459699i
$222$	0	0
$223$	1.45878e10	0.395019	0.197510	−	0.980301i	$-0.436715\pi$
$223$	1.45878e10	0.395019	0.197510	+	0.980301i	$0.436715\pi$
$224$	0	0
$225$	−1.13433e10	−0.295064
$226$	0	0
$227$	− 6.30500e10i	− 1.57605i	−0.615646	−	0.788023i	$-0.711105\pi$
$227$	− 6.30500e10i	− 1.57605i	0.615646	−	0.788023i	$-0.288895\pi$
$228$	0	0
$229$	4.91705e10i	1.18153i	0.806843	+	0.590765i	$0.201174\pi$
$229$	4.91705e10i	1.18153i	−0.806843	+	0.590765i	$0.798826\pi$
$230$	0	0
$231$	−3.38912e9	−0.0783129
$232$	0	0
$233$	3.30380e10	0.734366	0.367183	−	0.930149i	$-0.380322\pi$
$233$	3.30380e10	0.734366	0.367183	+	0.930149i	$0.380322\pi$
$234$	0	0
$235$	− 1.64541e10i	− 0.351941i
$236$	0	0
$237$	− 1.00633e9i	− 0.0207191i
$238$	0	0
$239$	7.33801e10	1.45475	0.727374	−	0.686241i	$-0.240740\pi$
$239$	7.33801e10	1.45475	0.727374	+	0.686241i	$0.240740\pi$
$240$	0	0
$241$	5.53816e10	1.05752	0.528760	−	0.848771i	$-0.322657\pi$
$241$	5.53816e10	1.05752	0.528760	+	0.848771i	$0.322657\pi$
$242$	0	0
$243$	3.48678e9i	0.0641500i
$244$	0	0
$245$	1.89522e10i	0.336057i
$246$	0	0
$247$	−7.37860e9	−0.126136
$248$	0	0
$249$	2.21065e10	0.364438
$250$	0	0
$251$	− 9.51298e10i	− 1.51281i	−0.654103	−	0.756406i	$-0.726954\pi$
$251$	− 9.51298e10i	− 1.51281i	0.654103	−	0.756406i	$-0.273046\pi$
$252$	0	0
$253$	7.62416e9i	0.116990i
$254$	0	0
$255$	4.21655e9	0.0624492
$256$	0	0
$257$	−5.82111e10	−0.832351	−0.416176	−	0.909284i	$-0.636630\pi$
$257$	−5.82111e10	−0.832351	−0.416176	+	0.909284i	$0.636630\pi$
$258$	0	0
$259$	2.99520e9i	0.0413597i
$260$	0	0
$261$	− 4.63463e10i	− 0.618205i
$262$	0	0
$263$	1.26131e11	1.62563	0.812814	−	0.582524i	$-0.197935\pi$
$263$	1.26131e11	1.62563	0.812814	+	0.582524i	$0.197935\pi$
$264$	0	0
$265$	−4.42387e10	−0.551056
$266$	0	0
$267$	− 2.19216e10i	− 0.263980i
$268$	0	0
$269$	1.23726e10i	0.144071i	0.997402	+	0.0720356i	$0.0229495\pi$
$269$	1.23726e10i	0.144071i	−0.997402	+	0.0720356i	$0.977050\pi$
$270$	0	0
$271$	6.40678e10	0.721569	0.360785	−	0.932649i	$-0.382509\pi$
$271$	6.40678e10	0.721569	0.360785	+	0.932649i	$0.382509\pi$
$272$	0	0
$273$	6.90596e8	0.00752475
$274$	0	0
$275$	− 1.25820e11i	− 1.32663i
$276$	0	0
$277$	− 1.56675e11i	− 1.59897i	−0.600688	−	0.799484i	$-0.705107\pi$
$277$	− 1.56675e11i	− 1.59897i	0.600688	−	0.799484i	$-0.294893\pi$
$278$	0	0
$279$	1.51282e10	0.149475
$280$	0	0
$281$	−9.55568e10	−0.914288	−0.457144	−	0.889393i	$-0.651128\pi$
$281$	−9.55568e10	−0.914288	−0.457144	+	0.889393i	$0.651128\pi$
$282$	0	0
$283$	− 7.99388e10i	− 0.740830i	−0.928866	−	0.370415i	$-0.879215\pi$
$283$	− 7.99388e10i	− 0.740830i	0.928866	−	0.370415i	$-0.120785\pi$
$284$	0	0
$285$	1.90850e10i	0.171352i
$286$	0	0
$287$	6.22278e9	0.0541397
$288$	0	0
$289$	−1.06503e11	−0.898093
$290$	0	0
$291$	− 3.28970e10i	− 0.268929i
$292$	0	0
$293$	− 9.87325e10i	− 0.782629i	−0.920257	−	0.391315i	$-0.872020\pi$
$293$	− 9.87325e10i	− 0.782629i	0.920257	−	0.391315i	$-0.127980\pi$
$294$	0	0
$295$	−4.28446e10	−0.329380
$296$	0	0
$297$	−3.86754e10	−0.288424
$298$	0	0
$299$	− 1.55356e9i	− 0.0112411i
$300$	0	0
$301$	− 1.17334e10i	− 0.0823900i
$302$	0	0
$303$	2.90351e10	0.197894
$304$	0	0
$305$	9.48340e9	0.0627502
$306$	0	0
$307$	− 2.01628e11i	− 1.29547i	−0.761864	−	0.647737i	$-0.775716\pi$
$307$	− 2.01628e11i	− 1.29547i	0.761864	−	0.647737i	$-0.224284\pi$
$308$	0	0
$309$	2.53309e10i	0.158065i
$310$	0	0
$311$	1.25577e11	0.761182	0.380591	−	0.924744i	$-0.375721\pi$
$311$	1.25577e11	0.761182	0.380591	+	0.924744i	$0.375721\pi$
$312$	0	0
$313$	9.72056e10	0.572455	0.286228	−	0.958162i	$-0.407599\pi$
$313$	9.72056e10	0.572455	0.286228	+	0.958162i	$0.407599\pi$
$314$	0	0
$315$	− 1.78625e9i	− 0.0102222i
$316$	0	0
$317$	− 3.24299e10i	− 0.180376i	−0.995925	−	0.0901880i	$-0.971253\pi$
$317$	− 3.24299e10i	− 0.180376i	0.995925	−	0.0901880i	$-0.0287468\pi$
$318$	0	0
$319$	5.14073e11	2.77950
$320$	0	0
$321$	7.52457e10	0.395557
$322$	0	0
$323$	5.46989e10i	0.279620i
$324$	0	0
$325$	2.56380e10i	0.127470i
$326$	0	0
$327$	1.24984e11	0.604490
$328$	0	0
$329$	−1.99778e10	−0.0940081
$330$	0	0
$331$	− 2.13971e11i	− 0.979781i	−0.871784	−	0.489890i	$-0.837037\pi$
$331$	− 2.13971e11i	− 0.979781i	0.871784	−	0.489890i	$-0.162963\pi$
$332$	0	0
$333$	3.41802e10i	0.152326i
$334$	0	0
$335$	5.16865e10	0.224221
$336$	0	0
$337$	−1.67278e11	−0.706489	−0.353244	−	0.935531i	$-0.614922\pi$
$337$	−1.67278e11	−0.706489	−0.353244	+	0.935531i	$0.614922\pi$
$338$	0	0
$339$	− 4.84692e10i	− 0.199328i
$340$	0	0
$341$	1.67802e11i	0.672053i
$342$	0	0
$343$	4.62117e10	0.180272
$344$	0	0
$345$	−4.01834e9	−0.0152708
$346$	0	0
$347$	− 1.93452e11i	− 0.716293i	−0.933665	−	0.358146i	$-0.883409\pi$
$347$	− 1.93452e11i	− 0.716293i	0.933665	−	0.358146i	$-0.116591\pi$
$348$	0	0
$349$	− 1.79870e11i	− 0.649000i	−0.945886	−	0.324500i	$-0.894804\pi$
$349$	− 1.79870e11i	− 0.649000i	0.945886	−	0.324500i	$-0.105196\pi$
$350$	0	0
$351$	7.88083e9	0.0277134
$352$	0	0
$353$	2.13154e11	0.730646	0.365323	−	0.930881i	$-0.380959\pi$
$353$	2.13154e11	0.730646	0.365323	+	0.930881i	$0.380959\pi$
$354$	0	0
$355$	− 1.25927e11i	− 0.420815i
$356$	0	0
$357$	− 5.11952e9i	− 0.0166810i
$358$	0	0
$359$	−1.85269e11	−0.588677	−0.294339	−	0.955701i	$-0.595099\pi$
$359$	−1.85269e11	−0.588677	−0.294339	+	0.955701i	$0.595099\pi$
$360$	0	0
$361$	7.51089e10	0.232760
$362$	0	0
$363$	− 2.37995e11i	− 0.719428i
$364$	0	0
$365$	− 2.39383e10i	− 0.0705952i
$366$	0	0
$367$	3.99652e11	1.14997	0.574983	−	0.818166i	$-0.305009\pi$
$367$	3.99652e11	1.14997	0.574983	+	0.818166i	$0.305009\pi$
$368$	0	0
$369$	7.10122e10	0.199395
$370$	0	0
$371$	5.37123e10i	0.147194i
$372$	0	0
$373$	3.17540e11i	0.849394i	0.905335	+	0.424697i	$0.139619\pi$
$373$	3.17540e11i	0.849394i	−0.905335	+	0.424697i	$0.860381\pi$
$374$	0	0
$375$	1.41228e11	0.368791
$376$	0	0
$377$	−1.04752e11	−0.267070
$378$	0	0
$379$	− 2.10387e11i	− 0.523771i	−0.965099	−	0.261886i	$-0.915656\pi$
$379$	− 2.10387e11i	− 0.523771i	0.965099	−	0.261886i	$-0.0843443\pi$
$380$	0	0
$381$	− 2.04919e11i	− 0.498219i
$382$	0	0
$383$	6.26458e11	1.48764	0.743819	−	0.668381i	$-0.233012\pi$
$383$	6.26458e11	1.48764	0.743819	+	0.668381i	$0.233012\pi$
$384$	0	0
$385$	1.98131e10	0.0459599
$386$	0	0
$387$	− 1.33897e11i	− 0.303440i
$388$	0	0
$389$	4.71840e11i	1.04477i	0.852709	+	0.522386i	$0.174958\pi$
$389$	4.71840e11i	1.04477i	−0.852709	+	0.522386i	$0.825042\pi$
$390$	0	0
$391$	−1.15169e10	−0.0249194
$392$	0	0
$393$	−1.37864e10	−0.0291532
$394$	0	0
$395$	5.88307e9i	0.0121595i
$396$	0	0
$397$	− 4.08614e11i	− 0.825573i	−0.910828	−	0.412786i	$-0.864556\pi$
$397$	− 4.08614e11i	− 0.825573i	0.910828	−	0.412786i	$-0.135444\pi$
$398$	0	0
$399$	2.31720e10	0.0457705
$400$	0	0
$401$	−4.75227e11	−0.917807	−0.458903	−	0.888486i	$-0.651758\pi$
$401$	−4.75227e11	−0.917807	−0.458903	+	0.888486i	$0.651758\pi$
$402$	0	0
$403$	− 3.41928e10i	− 0.0645747i
$404$	0	0
$405$	− 2.03840e10i	− 0.0376481i
$406$	0	0
$407$	−3.79127e11	−0.684872
$408$	0	0
$409$	1.47969e11	0.261466	0.130733	−	0.991418i	$-0.458267\pi$
$409$	1.47969e11	0.261466	0.130733	+	0.991418i	$0.458267\pi$
$410$	0	0
$411$	2.09818e11i	0.362706i
$412$	0	0
$413$	5.20197e10i	0.0879818i
$414$	0	0
$415$	−1.29237e11	−0.213880
$416$	0	0
$417$	2.22783e11	0.360802
$418$	0	0
$419$	− 8.99050e11i	− 1.42502i	−0.701663	−	0.712509i	$-0.747559\pi$
$419$	− 8.99050e11i	− 1.42502i	0.701663	−	0.712509i	$-0.252441\pi$
$420$	0	0
$421$	− 9.67201e10i	− 0.150054i	−0.997182	−	0.0750269i	$-0.976096\pi$
$421$	− 9.67201e10i	− 0.150054i	0.997182	−	0.0750269i	$-0.0239043\pi$
$422$	0	0
$423$	−2.27979e11	−0.346229
$424$	0	0
$425$	1.90060e11	0.282579
$426$	0	0
$427$	− 1.15143e10i	− 0.0167614i
$428$	0	0
$429$	8.74143e10i	0.124602i
$430$	0	0
$431$	4.91630e11	0.686263	0.343131	−	0.939287i	$-0.388512\pi$
$431$	4.91630e11	0.686263	0.343131	+	0.939287i	$0.388512\pi$
$432$	0	0
$433$	−2.39033e10	−0.0326786	−0.0163393	−	0.999867i	$-0.505201\pi$
$433$	−2.39033e10	−0.0326786	−0.0163393	+	0.999867i	$0.505201\pi$
$434$	0	0
$435$	2.70944e11i	0.362809i
$436$	0	0
$437$	− 5.21277e10i	− 0.0683757i
$438$	0	0
$439$	−8.29279e11	−1.06564	−0.532820	−	0.846229i	$-0.678868\pi$
$439$	−8.29279e11	−1.06564	−0.532820	+	0.846229i	$0.678868\pi$
$440$	0	0
$441$	2.62591e11	0.330603
$442$	0	0
$443$	− 5.71843e11i	− 0.705440i	−0.935729	−	0.352720i	$-0.885257\pi$
$443$	− 5.71843e11i	− 0.705440i	0.935729	−	0.352720i	$-0.114743\pi$
$444$	0	0
$445$	1.28155e11i	0.154923i
$446$	0	0
$447$	2.41028e11	0.285551
$448$	0	0
$449$	1.02914e12	1.19500	0.597500	−	0.801869i	$-0.296161\pi$
$449$	1.02914e12	1.19500	0.597500	+	0.801869i	$0.296161\pi$
$450$	0	0
$451$	7.87668e11i	0.896496i
$452$	0	0
$453$	8.97468e11i	1.00133i
$454$	0	0
$455$	−4.03728e9	−0.00441609
$456$	0	0
$457$	−7.44956e11	−0.798928	−0.399464	−	0.916749i	$-0.630804\pi$
$457$	−7.44956e11	−0.798928	−0.399464	+	0.916749i	$0.630804\pi$
$458$	0	0
$459$	− 5.84221e10i	− 0.0614356i
$460$	0	0
$461$	1.35785e11i	0.140023i	0.997546	+	0.0700115i	$0.0223036\pi$
$461$	1.35785e11i	0.140023i	−0.997546	+	0.0700115i	$0.977696\pi$
$462$	0	0
$463$	6.39280e10	0.0646512	0.0323256	−	0.999477i	$-0.489709\pi$
$463$	6.39280e10	0.0646512	0.0323256	+	0.999477i	$0.489709\pi$
$464$	0	0
$465$	−8.84409e10	−0.0877233
$466$	0	0
$467$	8.08293e11i	0.786399i	0.919453	+	0.393199i	$0.128632\pi$
$467$	8.08293e11i	0.786399i	−0.919453	+	0.393199i	$0.871368\pi$
$468$	0	0
$469$	− 6.27551e10i	− 0.0598923i
$470$	0	0
$471$	−9.74591e11	−0.912491
$472$	0	0
$473$	1.48519e12	1.36429
$474$	0	0
$475$	8.60250e11i	0.775360i
$476$	0	0
$477$	6.12946e11i	0.542112i
$478$	0	0
$479$	1.38621e12	1.20315	0.601573	−	0.798818i	$-0.294541\pi$
$479$	1.38621e12	1.20315	0.601573	+	0.798818i	$0.294541\pi$
$480$	0	0
$481$	7.72540e10	0.0658064
$482$	0	0
$483$	4.87886e9i	0.00407903i
$484$	0	0
$485$	1.92319e11i	0.157828i
$486$	0	0
$487$	−1.20417e11	−0.0970078	−0.0485039	−	0.998823i	$-0.515445\pi$
$487$	−1.20417e11	−0.0970078	−0.0485039	+	0.998823i	$0.515445\pi$
$488$	0	0
$489$	9.84146e10	0.0778340
$490$	0	0
$491$	1.43283e11i	0.111257i	0.998452	+	0.0556287i	$0.0177163\pi$
$491$	1.43283e11i	0.111257i	−0.998452	+	0.0556287i	$0.982284\pi$
$492$	0	0
$493$	7.76545e11i	0.592046i
$494$	0	0
$495$	2.26100e11	0.169269
$496$	0	0
$497$	−1.52894e11	−0.112405
$498$	0	0
$499$	− 1.93465e12i	− 1.39685i	−0.715683	−	0.698426i	$-0.753884\pi$
$499$	− 1.93465e12i	− 1.39685i	0.715683	−	0.698426i	$-0.246116\pi$
$500$	0	0
$501$	− 8.20513e11i	− 0.581856i
$502$	0	0
$503$	−2.14780e12	−1.49602	−0.748010	−	0.663687i	$-0.768991\pi$
$503$	−2.14780e12	−1.49602	−0.748010	+	0.663687i	$0.768991\pi$
$504$	0	0
$505$	−1.69742e11	−0.116139
$506$	0	0
$507$	8.41152e11i	0.565378i
$508$	0	0
$509$	4.93770e10i	0.0326058i	0.999867	+	0.0163029i	$0.00518961\pi$
$509$	4.93770e10i	0.0326058i	−0.999867	+	0.0163029i	$0.994810\pi$
$510$	0	0
$511$	−2.90646e10	−0.0188569
$512$	0	0
$513$	2.64431e11	0.168571
$514$	0	0
$515$	− 1.48086e11i	− 0.0927646i
$516$	0	0
$517$	− 2.52875e12i	− 1.55667i
$518$	0	0
$519$	−9.10168e11	−0.550641
$520$	0	0
$521$	−3.06642e12	−1.82331	−0.911657	−	0.410951i	$-0.865197\pi$
$521$	−3.06642e12	−1.82331	−0.911657	+	0.410951i	$0.865197\pi$
$522$	0	0
$523$	1.36512e12i	0.797836i	0.916987	+	0.398918i	$0.130614\pi$
$523$	1.36512e12i	0.797836i	−0.916987	+	0.398918i	$0.869386\pi$
$524$	0	0
$525$	− 8.05146e10i	− 0.0462549i
$526$	0	0
$527$	−2.53478e11	−0.143150
$528$	0	0
$529$	−1.79018e12	−0.993906
$530$	0	0
$531$	5.93630e11i	0.324034i
$532$	0	0
$533$	− 1.60502e11i	− 0.0861405i
$534$	0	0
$535$	−4.39893e11	−0.232143
$536$	0	0
$537$	−3.02081e11	−0.156761
$538$	0	0
$539$	2.91266e12i	1.48642i
$540$	0	0
$541$	− 2.34837e12i	− 1.17863i	−0.807902	−	0.589317i	$-0.799397\pi$
$541$	− 2.34837e12i	− 1.17863i	0.807902	−	0.589317i	$-0.200603\pi$
$542$	0	0
$543$	1.19112e12	0.587971
$544$	0	0
$545$	−7.30666e11	−0.354760
$546$	0	0
$547$	3.90038e12i	1.86279i	0.364007	+	0.931396i	$0.381408\pi$
$547$	3.90038e12i	1.86279i	−0.364007	+	0.931396i	$0.618592\pi$
$548$	0	0
$549$	− 1.31397e11i	− 0.0617317i
$550$	0	0
$551$	−3.51481e12	−1.62450
$552$	0	0
$553$	7.14292e9	0.00324797
$554$	0	0
$555$	− 1.99820e11i	− 0.0893966i
$556$	0	0
$557$	− 9.85631e11i	− 0.433876i	−0.976185	−	0.216938i	$-0.930393\pi$
$557$	− 9.85631e11i	− 0.433876i	0.976185	−	0.216938i	$-0.0696070\pi$
$558$	0	0
$559$	−3.02635e11	−0.131089
$560$	0	0
$561$	6.48019e11	0.276220
$562$	0	0
$563$	3.71992e12i	1.56044i	0.625508	+	0.780218i	$0.284892\pi$
$563$	3.71992e12i	1.56044i	−0.625508	+	0.780218i	$0.715108\pi$
$564$	0	0
$565$	2.83355e11i	0.116980i
$566$	0	0
$567$	−2.47493e10	−0.0100563
$568$	0	0
$569$	2.34608e12	0.938293	0.469146	−	0.883120i	$-0.344562\pi$
$569$	2.34608e12	0.938293	0.469146	+	0.883120i	$0.344562\pi$
$570$	0	0
$571$	− 1.58941e12i	− 0.625710i	−0.949801	−	0.312855i	$-0.898715\pi$
$571$	− 1.58941e12i	− 0.625710i	0.949801	−	0.312855i	$-0.101285\pi$
$572$	0	0
$573$	1.75598e12i	0.680493i
$574$	0	0
$575$	−1.81125e11	−0.0690994
$576$	0	0
$577$	−4.20772e12	−1.58036	−0.790179	−	0.612876i	$-0.790013\pi$
$577$	−4.20772e12	−1.58036	−0.790179	+	0.612876i	$0.790013\pi$
$578$	0	0
$579$	− 1.98710e12i	− 0.734794i
$580$	0	0
$581$	1.56913e11i	0.0571302i
$582$	0	0
$583$	−6.79880e12	−2.43738
$584$	0	0
$585$	−4.60720e10	−0.0162643
$586$	0	0
$587$	1.63748e12i	0.569252i	0.958639	+	0.284626i	$0.0918694\pi$
$587$	1.63748e12i	0.569252i	−0.958639	+	0.284626i	$0.908131\pi$
$588$	0	0
$589$	− 1.14729e12i	− 0.392786i
$590$	0	0
$591$	1.17059e12	0.394695
$592$	0	0
$593$	5.53148e12	1.83694	0.918471	−	0.395489i	$-0.129425\pi$
$593$	5.53148e12	1.83694	0.918471	+	0.395489i	$0.129425\pi$
$594$	0	0
$595$	2.99292e10i	0.00978967i
$596$	0	0
$597$	2.20040e12i	0.708954i
$598$	0	0
$599$	−4.16661e12	−1.32240	−0.661199	−	0.750211i	$-0.729952\pi$
$599$	−4.16661e12	−1.32240	−0.661199	+	0.750211i	$0.729952\pi$
$600$	0	0
$601$	−2.90464e12	−0.908149	−0.454075	−	0.890964i	$-0.650030\pi$
$601$	−2.90464e12	−0.908149	−0.454075	+	0.890964i	$0.650030\pi$
$602$	0	0
$603$	− 7.16139e11i	− 0.220582i
$604$	0	0
$605$	1.39134e12i	0.422214i
$606$	0	0
$607$	4.66728e12	1.39545	0.697726	−	0.716365i	$-0.254195\pi$
$607$	4.66728e12	1.39545	0.697726	+	0.716365i	$0.254195\pi$
$608$	0	0
$609$	3.28967e11	0.0969112
$610$	0	0
$611$	5.15278e11i	0.149574i
$612$	0	0
$613$	3.12221e11i	0.0893078i	0.999003	+	0.0446539i	$0.0142185\pi$
$613$	3.12221e11i	0.0893078i	−0.999003	+	0.0446539i	$0.985781\pi$
$614$	0	0
$615$	−4.15143e11	−0.117020
$616$	0	0
$617$	3.59628e12	0.999011	0.499505	−	0.866311i	$-0.333515\pi$
$617$	3.59628e12	0.999011	0.499505	+	0.866311i	$0.333515\pi$
$618$	0	0
$619$	4.85670e12i	1.32964i	0.747004	+	0.664819i	$0.231491\pi$
$619$	4.85670e12i	1.32964i	−0.747004	+	0.664819i	$0.768509\pi$
$620$	0	0
$621$	5.56758e10i	0.0150229i
$622$	0	0
$623$	1.55600e11	0.0413821
$624$	0	0
$625$	2.55111e12	0.668758
$626$	0	0
$627$	2.93307e12i	0.757911i
$628$	0	0
$629$	− 5.72699e11i	− 0.145881i
$630$	0	0
$631$	−6.19329e12	−1.55521	−0.777606	−	0.628752i	$-0.783566\pi$
$631$	−6.19329e12	−1.55521	−0.777606	+	0.628752i	$0.783566\pi$
$632$	0	0
$633$	2.85484e12	0.706748
$634$	0	0
$635$	1.19797e12i	0.292392i
$636$	0	0
$637$	− 5.93509e11i	− 0.142824i
$638$	0	0
$639$	−1.74477e12	−0.413985
$640$	0	0
$641$	−5.96307e12	−1.39511	−0.697555	−	0.716531i	$-0.745729\pi$
$641$	−5.96307e12	−1.39511	−0.697555	+	0.716531i	$0.745729\pi$
$642$	0	0
$643$	7.48485e11i	0.172677i	0.996266	+	0.0863383i	$0.0275166\pi$
$643$	7.48485e11i	0.172677i	−0.996266	+	0.0863383i	$0.972483\pi$
$644$	0	0
$645$	7.82775e11i	0.178081i
$646$	0	0
$647$	−5.21646e12	−1.17032	−0.585162	−	0.810916i	$-0.698969\pi$
$647$	−5.21646e12	−1.17032	−0.585162	+	0.810916i	$0.698969\pi$
$648$	0	0
$649$	−6.58455e12	−1.45688
$650$	0	0
$651$	1.07380e11i	0.0234321i
$652$	0	0
$653$	5.05968e12i	1.08896i	0.838772	+	0.544482i	$0.183274\pi$
$653$	5.05968e12i	1.08896i	−0.838772	+	0.544482i	$0.816726\pi$
$654$	0	0
$655$	8.05967e10	0.0171093
$656$	0	0
$657$	−3.31675e11	−0.0694494
$658$	0	0
$659$	1.64097e12i	0.338936i	0.985536	+	0.169468i	$0.0542049\pi$
$659$	1.64097e12i	0.338936i	−0.985536	+	0.169468i	$0.945795\pi$
$660$	0	0
$661$	− 9.39337e12i	− 1.91388i	−0.290284	−	0.956941i	$-0.593750\pi$
$661$	− 9.39337e12i	− 1.91388i	0.290284	−	0.956941i	$-0.406250\pi$
$662$	0	0
$663$	−1.32046e11	−0.0265408
$664$	0	0
$665$	−1.35466e11	−0.0268616
$666$	0	0
$667$	− 7.40042e11i	− 0.144774i
$668$	0	0
$669$	1.18161e12i	0.228065i
$670$	0	0
$671$	1.45745e12	0.277551
$672$	0	0
$673$	6.84794e11	0.128674	0.0643372	−	0.997928i	$-0.479507\pi$
$673$	6.84794e11	0.128674	0.0643372	+	0.997928i	$0.479507\pi$
$674$	0	0
$675$	− 9.18804e11i	− 0.170355i
$676$	0	0
$677$	− 2.15344e12i	− 0.393988i	−0.980405	−	0.196994i	$-0.936882\pi$
$677$	− 2.15344e12i	− 0.393988i	0.980405	−	0.196994i	$-0.0631180\pi$
$678$	0	0
$679$	2.33503e11	0.0421579
$680$	0	0
$681$	5.10705e12	0.909930
$682$	0	0
$683$	− 3.32483e12i	− 0.584623i	−0.956323	−	0.292311i	$-0.905576\pi$
$683$	− 3.32483e12i	− 0.584623i	0.956323	−	0.292311i	$-0.0944244\pi$
$684$	0	0
$685$	− 1.22662e12i	− 0.212863i
$686$	0	0
$687$	−3.98281e12	−0.682157
$688$	0	0
$689$	1.38538e12	0.234198
$690$	0	0
$691$	1.50122e12i	0.250491i	0.992126	+	0.125246i	$0.0399719\pi$
$691$	1.50122e12i	0.250491i	−0.992126	+	0.125246i	$0.960028\pi$
$692$	0	0
$693$	− 2.74519e11i	− 0.0452140i
$694$	0	0
$695$	−1.30241e12	−0.211746
$696$	0	0
$697$	−1.18983e12	−0.190958
$698$	0	0
$699$	2.67608e12i	0.423987i
$700$	0	0
$701$	3.02156e12i	0.472607i	0.971679	+	0.236304i	$0.0759360\pi$
$701$	3.02156e12i	0.472607i	−0.971679	+	0.236304i	$0.924064\pi$
$702$	0	0
$703$	2.59215e12	0.400278
$704$	0	0
$705$	1.33279e12	0.203193
$706$	0	0
$707$	2.06092e11i	0.0310223i
$708$	0	0
$709$	− 3.67440e12i	− 0.546108i	−0.961999	−	0.273054i	$-0.911966\pi$
$709$	− 3.67440e12i	− 0.546108i	0.961999	−	0.273054i	$-0.0880337\pi$
$710$	0	0
$711$	8.15124e10	0.0119622
$712$	0	0
$713$	2.41563e11	0.0350047
$714$	0	0
$715$	− 5.11031e11i	− 0.0731257i
$716$	0	0
$717$	5.94379e12i	0.839899i
$718$	0	0
$719$	9.45558e12	1.31950	0.659748	−	0.751487i	$-0.270663\pi$
$719$	9.45558e12	1.31950	0.659748	+	0.751487i	$0.270663\pi$
$720$	0	0
$721$	−1.79799e11	−0.0247787
$722$	0	0
$723$	4.48591e12i	0.610560i
$724$	0	0
$725$	1.22127e13i	1.64169i
$726$	0	0
$727$	−1.44431e13	−1.91760	−0.958798	−	0.284090i	$-0.908309\pi$
$727$	−1.44431e13	−1.91760	−0.958798	+	0.284090i	$0.908309\pi$
$728$	0	0
$729$	−2.82430e11	−0.0370370
$730$	0	0
$731$	2.24349e12i	0.290600i
$732$	0	0
$733$	4.09211e12i	0.523576i	0.965125	+	0.261788i	$0.0843121\pi$
$733$	4.09211e12i	0.523576i	−0.965125	+	0.261788i	$0.915688\pi$
$734$	0	0
$735$	−1.53513e12	−0.194023
$736$	0	0
$737$	7.94342e12	0.991753
$738$	0	0
$739$	9.48921e12i	1.17039i	0.810893	+	0.585195i	$0.198982\pi$
$739$	9.48921e12i	1.17039i	−0.810893	+	0.585195i	$0.801018\pi$
$740$	0	0
$741$	− 5.97666e11i	− 0.0728244i
$742$	0	0
$743$	2.23279e12	0.268781	0.134390	−	0.990928i	$-0.457092\pi$
$743$	2.23279e12	0.268781	0.134390	+	0.990928i	$0.457092\pi$
$744$	0	0
$745$	−1.40907e12	−0.167583
$746$	0	0
$747$	1.79063e12i	0.210408i
$748$	0	0
$749$	5.34095e11i	0.0620084i
$750$	0	0
$751$	1.33252e13	1.52860	0.764298	−	0.644863i	$-0.223086\pi$
$751$	1.33252e13	1.52860	0.764298	+	0.644863i	$0.223086\pi$
$752$	0	0
$753$	7.70552e12	0.873422
$754$	0	0
$755$	− 5.24667e12i	− 0.587656i
$756$	0	0
$757$	− 2.46238e12i	− 0.272536i	−0.990672	−	0.136268i	$-0.956489\pi$
$757$	− 2.46238e12i	− 0.272536i	0.990672	−	0.136268i	$-0.0435108\pi$
$758$	0	0
$759$	−6.17557e11	−0.0675443
$760$	0	0
$761$	7.93783e12	0.857968	0.428984	−	0.903312i	$-0.358872\pi$
$761$	7.93783e12	0.857968	0.428984	+	0.903312i	$0.358872\pi$
$762$	0	0
$763$	8.87137e11i	0.0947611i
$764$	0	0
$765$	3.41541e11i	0.0360550i
$766$	0	0
$767$	1.34172e12	0.139986
$768$	0	0
$769$	−1.58369e13	−1.63305	−0.816527	−	0.577307i	$-0.804104\pi$
$769$	−1.58369e13	−1.63305	−0.816527	+	0.577307i	$0.804104\pi$
$770$	0	0
$771$	− 4.71510e12i	− 0.480558i
$772$	0	0
$773$	1.21012e13i	1.21904i	0.792769	+	0.609522i	$0.208639\pi$
$773$	1.21012e13i	1.21904i	−0.792769	+	0.609522i	$0.791361\pi$
$774$	0	0
$775$	−3.98645e12	−0.396943
$776$	0	0
$777$	−2.42611e11	−0.0238790
$778$	0	0
$779$	− 5.38542e12i	− 0.523963i
$780$	0	0
$781$	− 1.93530e13i	− 1.86131i
$782$	0	0
$783$	3.75405e12	0.356921
$784$	0	0
$785$	5.69754e12	0.535518
$786$	0	0
$787$	− 1.64934e13i	− 1.53258i	−0.642495	−	0.766290i	$-0.722101\pi$
$787$	− 1.64934e13i	− 1.53258i	0.642495	−	0.766290i	$-0.277899\pi$
$788$	0	0
$789$	1.02166e13i	0.938557i
$790$	0	0
$791$	3.44035e11	0.0312470
$792$	0	0
$793$	−2.96983e11	−0.0266687
$794$	0	0
$795$	− 3.58333e12i	− 0.318152i
$796$	0	0
$797$	− 9.14539e12i	− 0.802860i	−0.915890	−	0.401430i	$-0.868513\pi$
$797$	− 9.14539e12i	− 0.802860i	0.915890	−	0.401430i	$-0.131487\pi$
$798$	0	0
$799$	3.81986e12	0.331579
$800$	0	0
$801$	1.77565e12	0.152409
$802$	0	0
$803$	− 3.67894e12i	− 0.312250i
$804$	0	0
$805$	− 2.85223e10i	− 0.00239388i
$806$	0	0
$807$	−1.00218e12	−0.0831796
$808$	0	0
$809$	1.10523e12	0.0907160	0.0453580	−	0.998971i	$-0.485557\pi$
$809$	1.10523e12	0.0907160	0.0453580	+	0.998971i	$0.485557\pi$
$810$	0	0
$811$	5.17595e12i	0.420142i	0.977686	+	0.210071i	$0.0673695\pi$
$811$	5.17595e12i	0.420142i	−0.977686	+	0.210071i	$0.932631\pi$
$812$	0	0
$813$	5.18949e12i	0.416598i
$814$	0	0
$815$	−5.75340e11	−0.0456789
$816$	0	0
$817$	−1.01545e13	−0.797369
$818$	0	0
$819$	5.59383e10i	0.00434442i
$820$	0	0
$821$	1.06367e13i	0.817076i	0.912741	+	0.408538i	$0.133961\pi$
$821$	1.06367e13i	0.817076i	−0.912741	+	0.408538i	$0.866039\pi$
$822$	0	0
$823$	1.89127e13	1.43699	0.718495	−	0.695532i	$-0.244831\pi$
$823$	1.89127e13	1.43699	0.718495	+	0.695532i	$0.244831\pi$
$824$	0	0
$825$	1.01914e13	0.765932
$826$	0	0
$827$	− 6.62866e11i	− 0.0492777i	−0.999696	−	0.0246389i	$-0.992156\pi$
$827$	− 6.62866e11i	− 0.0492777i	0.999696	−	0.0246389i	$-0.00784359\pi$
$828$	0	0
$829$	− 2.27273e13i	− 1.67129i	−0.549267	−	0.835647i	$-0.685093\pi$
$829$	− 2.27273e13i	− 1.67129i	0.549267	−	0.835647i	$-0.314907\pi$
$830$	0	0
$831$	1.26906e13	0.923164
$832$	0	0
$833$	−4.39979e12	−0.316614
$834$	0	0
$835$	4.79679e12i	0.341477i
$836$	0	0
$837$	1.22539e12i	0.0862996i
$838$	0	0
$839$	2.12551e12	0.148093	0.0740463	−	0.997255i	$-0.476409\pi$
$839$	2.12551e12	0.148093	0.0740463	+	0.997255i	$0.476409\pi$
$840$	0	0
$841$	−3.53916e13	−2.43960
$842$	0	0
$843$	− 7.74010e12i	− 0.527865i
$844$	0	0
$845$	− 4.91745e12i	− 0.331806i
$846$	0	0
$847$	1.68929e12	0.112779
$848$	0	0
$849$	6.47504e12	0.427718
$850$	0	0
$851$	5.45778e11i	0.0356724i
$852$	0	0
$853$	− 2.52949e13i	− 1.63592i	−0.575276	−	0.817960i	$-0.695105\pi$
$853$	− 2.52949e13i	− 1.63592i	0.575276	−	0.817960i	$-0.304895\pi$
$854$	0	0
$855$	−1.54588e12	−0.0989303
$856$	0	0
$857$	2.42903e12	0.153822	0.0769112	−	0.997038i	$-0.475494\pi$
$857$	2.42903e12	0.153822	0.0769112	+	0.997038i	$0.475494\pi$
$858$	0	0
$859$	− 2.92636e13i	− 1.83383i	−0.399083	−	0.916915i	$-0.630671\pi$
$859$	− 2.92636e13i	− 1.83383i	0.399083	−	0.916915i	$-0.369329\pi$
$860$	0	0
$861$	5.04046e11i	0.0312576i
$862$	0	0
$863$	1.25262e13	0.768724	0.384362	−	0.923182i	$-0.374421\pi$
$863$	1.25262e13	0.768724	0.384362	+	0.923182i	$0.374421\pi$
$864$	0	0
$865$	5.32092e12	0.323157
$866$	0	0
$867$	− 8.62674e12i	− 0.518514i
$868$	0	0
$869$	9.04136e11i	0.0537830i
$870$	0	0
$871$	−1.61862e12	−0.0952933
$872$	0	0
$873$	2.66466e12	0.155266
$874$	0	0
$875$	1.00244e12i	0.0578125i
$876$	0	0
$877$	6.98911e11i	0.0398955i	0.999801	+	0.0199477i	$0.00634998\pi$
$877$	6.98911e11i	0.0398955i	−0.999801	+	0.0199477i	$0.993650\pi$
$878$	0	0
$879$	7.99733e12	0.451851
$880$	0	0
$881$	−1.80581e13	−1.00991	−0.504953	−	0.863147i	$-0.668490\pi$
$881$	−1.80581e13	−1.00991	−0.504953	+	0.863147i	$0.668490\pi$
$882$	0	0
$883$	− 2.54103e13i	− 1.40665i	−0.710867	−	0.703326i	$-0.751697\pi$
$883$	− 2.54103e13i	− 1.40665i	0.710867	−	0.703326i	$-0.248303\pi$
$884$	0	0
$885$	− 3.47041e12i	− 0.190168i
$886$	0	0
$887$	−2.87381e12	−0.155884	−0.0779421	−	0.996958i	$-0.524835\pi$
$887$	−2.87381e12	−0.155884	−0.0779421	+	0.996958i	$0.524835\pi$
$888$	0	0
$889$	1.45452e12	0.0781019
$890$	0	0
$891$	− 3.13271e12i	− 0.166522i
$892$	0	0
$893$	1.72895e13i	0.909809i
$894$	0	0
$895$	1.76599e12	0.0919993
$896$	0	0
$897$	1.25839e11	0.00649004
$898$	0	0
$899$	− 1.62878e13i	− 0.831657i
$900$	0	0
$901$	− 1.02701e13i	− 0.519174i
$902$	0	0
$903$	9.50405e11	0.0475679
$904$	0	0
$905$	−6.96339e12	−0.345066
$906$	0	0
$907$	− 9.81693e10i	− 0.00481662i	−0.999997	−	0.00240831i	$-0.999233\pi$
$907$	− 9.81693e10i	− 0.00481662i	0.999997	−	0.00240831i	$-0.000766590\pi$
$908$	0	0
$909$	2.35185e12i	0.114254i
$910$	0	0
$911$	3.70352e12	0.178148	0.0890741	−	0.996025i	$-0.471609\pi$
$911$	3.70352e12	0.178148	0.0890741	+	0.996025i	$0.471609\pi$
$912$	0	0
$913$	−1.98617e13	−0.946014
$914$	0	0
$915$	7.68156e11i	0.0362288i
$916$	0	0
$917$	− 9.78564e10i	− 0.00457012i
$918$	0	0
$919$	3.79299e13	1.75413	0.877066	−	0.480370i	$-0.159498\pi$
$919$	3.79299e13	1.75413	0.877066	+	0.480370i	$0.159498\pi$
$920$	0	0
$921$	1.63319e13	0.747942
$922$	0	0
$923$	3.94354e12i	0.178846i
$924$	0	0
$925$	− 9.00683e12i	− 0.404515i
$926$	0	0
$927$	−2.05180e12	−0.0912591
$928$	0	0
$929$	3.49219e13	1.53825	0.769124	−	0.639099i	$-0.220693\pi$
$929$	3.49219e13	1.53825	0.769124	+	0.639099i	$0.220693\pi$
$930$	0	0
$931$	− 1.99144e13i	− 0.868747i
$932$	0	0
$933$	1.01717e13i	0.439468i
$934$	0	0
$935$	−3.78837e12	−0.162106
$936$	0	0
$937$	−2.39866e13	−1.01658	−0.508288	−	0.861187i	$-0.669722\pi$
$937$	−2.39866e13	−1.01658	−0.508288	+	0.861187i	$0.669722\pi$
$938$	0	0
$939$	7.87365e12i	0.330507i
$940$	0	0
$941$	− 8.10164e12i	− 0.336837i	−0.985716	−	0.168418i	$-0.946134\pi$
$941$	− 8.10164e12i	− 0.336837i	0.985716	−	0.168418i	$-0.0538660\pi$
$942$	0	0
$943$	1.13390e12	0.0466951
$944$	0	0
$945$	1.44686e11	0.00590180
$946$	0	0
$947$	− 1.66583e12i	− 0.0673061i	−0.999434	−	0.0336531i	$-0.989286\pi$
$947$	− 1.66583e12i	− 0.0673061i	0.999434	−	0.0336531i	$-0.0107141\pi$
$948$	0	0
$949$	7.49652e11i	0.0300028i
$950$	0	0
$951$	2.62682e12	0.104140
$952$	0	0
$953$	3.98680e13	1.56569	0.782846	−	0.622215i	$-0.213767\pi$
$953$	3.98680e13	1.56569	0.782846	+	0.622215i	$0.213767\pi$
$954$	0	0
$955$	− 1.02656e13i	− 0.399365i
$956$	0	0
$957$	4.16399e13i	1.60475i
$958$	0	0
$959$	−1.48929e12	−0.0568587
$960$	0	0
$961$	−2.11230e13	−0.798914
$962$	0	0
$963$	6.09490e12i	0.228375i
$964$	0	0
$965$	1.16167e13i	0.431232i
$966$	0	0
$967$	3.53153e13	1.29880	0.649402	−	0.760446i	$-0.275019\pi$
$967$	3.53153e13	1.29880	0.649402	+	0.760446i	$0.275019\pi$
$968$	0	0
$969$	−4.43061e12	−0.161438
$970$	0	0
$971$	− 2.43725e13i	− 0.879862i	−0.898032	−	0.439931i	$-0.855003\pi$
$971$	− 2.43725e13i	− 0.879862i	0.898032	−	0.439931i	$-0.144997\pi$
$972$	0	0
$973$	1.58132e12i	0.0565602i
$974$	0	0
$975$	−2.07668e12	−0.0735951
$976$	0	0
$977$	3.11200e13	1.09273	0.546366	−	0.837546i	$-0.316011\pi$
$977$	3.11200e13	1.09273	0.546366	+	0.837546i	$0.316011\pi$
$978$	0	0
$979$	1.96955e13i	0.685243i
$980$	0	0
$981$	1.01237e13i	0.349002i
$982$	0	0
$983$	2.69352e13	0.920088	0.460044	−	0.887896i	$-0.347834\pi$
$983$	2.69352e13	0.920088	0.460044	+	0.887896i	$0.347834\pi$
$984$	0	0
$985$	−6.84338e12	−0.231637
$986$	0	0
$987$	− 1.61820e12i	− 0.0542756i
$988$	0	0
$989$	− 2.13803e12i	− 0.0710608i
$990$	0	0
$991$	−3.56596e13	−1.17448	−0.587240	−	0.809413i	$-0.699785\pi$
$991$	−3.56596e13	−1.17448	−0.587240	+	0.809413i	$0.699785\pi$
$992$	0	0
$993$	1.73317e13	0.565677
$994$	0	0
$995$	− 1.28638e13i	− 0.416068i
$996$	0	0
$997$	− 3.95731e13i	− 1.26844i	−0.773151	−	0.634222i	$-0.781320\pi$
$997$	− 3.95731e13i	− 1.26844i	0.773151	−	0.634222i	$-0.218680\pi$
$998$	0	0
$999$	−2.76859e12	−0.0879457

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.d.a.193.3	✓	8	8.5	even	2	inner
384.10.d.a.193.6	yes	8	1.1	even	1	trivial
384.10.d.b.193.2	yes	8	4.3	odd	2
384.10.d.b.193.7	yes	8	8.3	odd	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+81.0000i q^{3} -473.533i q^{5} -574.939 q^{7} -6561.00 q^{9} +O(q^{10})\)	\(q+81.0000i q^{3} -473.533i q^{5} -574.939 q^{7} -6561.00 q^{9} -72774.7i q^{11} +14829.2i q^{13} +38356.2 q^{15} +109932. q^{17} +497573. i q^{19} -46570.1i q^{21} -104764. q^{23} +1.72889e6 q^{25} -531441. i q^{27} +7.06390e6i q^{29} -2.30578e6 q^{31} +5.89475e6 q^{33} +272253. i q^{35} -5.20960e6i q^{37} -1.20116e6 q^{39} -1.08234e7 q^{41} +2.04081e7i q^{43} +3.10685e6i q^{45} +3.47476e7 q^{47} -4.00231e7 q^{49} +8.90445e6i q^{51} -9.34226e7i q^{53} -3.44612e7 q^{55} -4.03034e7 q^{57} -9.04786e7i q^{59} +2.00269e7i q^{61} +3.77218e6 q^{63} +7.02210e6 q^{65} +1.09151e8i q^{67} -8.48588e6i q^{69} +2.65931e8 q^{71} +5.05525e7 q^{73} +1.40040e8i q^{75} +4.18410e7i q^{77} -1.24238e7 q^{79} +4.30467e7 q^{81} -2.72920e8i q^{83} -5.20562e7i q^{85} -5.72176e8 q^{87} -2.70637e8 q^{89} -8.52588e6i q^{91} -1.86768e8i q^{93} +2.35617e8 q^{95} -4.06136e8 q^{97} +4.77475e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 13632 q^{7} - 52488 q^{9}+O(q^{10}) \) 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

Varieties

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Database

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