LMFDB - Embedded newform 384.8.d.b.193.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,8,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, 8, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
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:	$119.955849786$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 277x^{4} + 19236x^{2} + 9216 $ x^6 + 277x^4 + 19236x^2 + 9216
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.2
Root			$11.9101i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.8.d.b.193.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000i q^{3} -96.4778i q^{5} -1147.84 q^{7} -729.000 q^{9} +O(q^{10})$	$q-27.0000i q^{3} -96.4778i q^{5} -1147.84 q^{7} -729.000 q^{9} +2575.55i q^{11} -8284.61i q^{13} -2604.90 q^{15} +22608.7 q^{17} -47112.8i q^{19} +30991.8i q^{21} -23449.5 q^{23} +68817.0 q^{25} +19683.0i q^{27} -201286. i q^{29} +181255. q^{31} +69540.0 q^{33} +110742. i q^{35} -361399. i q^{37} -223685. q^{39} -635548. q^{41} +11777.5i q^{43} +70332.3i q^{45} -978789. q^{47} +494003. q^{49} -610435. i q^{51} +595078. i q^{53} +248484. q^{55} -1.27204e6 q^{57} -800309. i q^{59} +47956.7i q^{61} +836778. q^{63} -799281. q^{65} +3.61335e6i q^{67} +633137. i q^{69} -153370. q^{71} -5.84223e6 q^{73} -1.85806e6i q^{75} -2.95633e6i q^{77} +2.20156e6 q^{79} +531441. q^{81} +2.26908e6i q^{83} -2.18124e6i q^{85} -5.43473e6 q^{87} +9.01888e6 q^{89} +9.50944e6i q^{91} -4.89387e6i q^{93} -4.54534e6 q^{95} +876482. q^{97} -1.87758e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 136 q^{7} - 4374 q^{9}+O(q^{10}) $ 6 * q + 136 * q^7 - 4374 * q^9	$ 6 q + 136 q^{7} - 4374 q^{9} - 6048 q^{15} - 25804 q^{17} + 12688 q^{23} - 18418 q^{25} + 220472 q^{31} - 264600 q^{33} - 591192 q^{39} - 619652 q^{41} - 326032 q^{47} - 581482 q^{49} - 4724352 q^{55} - 2429784 q^{57} - 99144 q^{63} - 9518720 q^{65} + 5841776 q^{71} - 11075196 q^{73} + 495800 q^{79} + 3188646 q^{81} + 965520 q^{87} + 9660740 q^{89} - 32750208 q^{95} - 9511564 q^{97}+O(q^{100}) $ 6 * q + 136 * q^7 - 4374 * q^9 - 6048 * q^15 - 25804 * q^17 + 12688 * q^23 - 18418 * q^25 + 220472 * q^31 - 264600 * q^33 - 591192 * q^39 - 619652 * q^41 - 326032 * q^47 - 581482 * q^49 - 4724352 * q^55 - 2429784 * q^57 - 99144 * q^63 - 9518720 * q^65 + 5841776 * q^71 - 11075196 * q^73 + 495800 * q^79 + 3188646 * q^81 + 965520 * q^87 + 9660740 * q^89 - 32750208 * q^95 - 9511564 * 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
$3$	− 27.0000i	− 0.577350i
$3$	− 27.0000i	− 0.577350i
$4$	0	0
$5$	− 96.4778i	− 0.345170i	−0.984995	−	0.172585i	$-0.944788\pi$
$5$	− 96.4778i	− 0.345170i	0.984995	−	0.172585i	$-0.0552119\pi$
$6$	0	0
$7$	−1147.84	−1.26485	−0.632426	−	0.774621i	$-0.717941\pi$
$7$	−1147.84	−1.26485	−0.632426	+	0.774621i	$0.717941\pi$
$8$	0	0
$9$	−729.000	−0.333333
$10$	0	0
$11$	2575.55i	0.583440i	0.956504	+	0.291720i	$0.0942275\pi$
$11$	2575.55i	0.583440i	−0.956504	+	0.291720i	$0.905772\pi$
$12$	0	0
$13$	− 8284.61i	− 1.04585i	−0.852378	−	0.522926i	$-0.824840\pi$
$13$	− 8284.61i	− 1.04585i	0.852378	−	0.522926i	$-0.175160\pi$
$14$	0	0
$15$	−2604.90	−0.199284
$16$	0	0
$17$	22608.7	1.11610	0.558052	−	0.829806i	$-0.311549\pi$
$17$	22608.7	1.11610	0.558052	+	0.829806i	$0.311549\pi$
$18$	0	0
$19$	− 47112.8i	− 1.57580i	−0.615804	−	0.787900i	$-0.711169\pi$
$19$	− 47112.8i	− 1.57580i	0.615804	−	0.787900i	$-0.288831\pi$
$20$	0	0
$21$	30991.8i	0.730263i
$22$	0	0
$23$	−23449.5	−0.401871	−0.200935	−	0.979605i	$-0.564398\pi$
$23$	−23449.5	−0.401871	−0.200935	+	0.979605i	$0.564398\pi$
$24$	0	0
$25$	68817.0	0.880858
$26$	0	0
$27$	19683.0i	0.192450i
$28$	0	0
$29$	− 201286.i	− 1.53257i	−0.642498	−	0.766287i	$-0.722102\pi$
$29$	− 201286.i	− 1.53257i	0.642498	−	0.766287i	$-0.277898\pi$
$30$	0	0
$31$	181255.	1.09276	0.546378	−	0.837539i	$-0.316006\pi$
$31$	181255.	1.09276	0.546378	+	0.837539i	$0.316006\pi$
$32$	0	0
$33$	69540.0	0.336849
$34$	0	0
$35$	110742.i	0.436589i
$36$	0	0
$37$	− 361399.i	− 1.17295i	−0.809966	−	0.586476i	$-0.800515\pi$
$37$	− 361399.i	− 1.17295i	0.809966	−	0.586476i	$-0.199485\pi$
$38$	0	0
$39$	−223685.	−0.603823
$40$	0	0
$41$	−635548.	−1.44014	−0.720071	−	0.693901i	$-0.755891\pi$
$41$	−635548.	−1.44014	−0.720071	+	0.693901i	$0.755891\pi$
$42$	0	0
$43$	11777.5i	0.0225899i	0.999936	+	0.0112949i	$0.00359537\pi$
$43$	11777.5i	0.0225899i	−0.999936	+	0.0112949i	$0.996405\pi$
$44$	0	0
$45$	70332.3i	0.115057i
$46$	0	0
$47$	−978789.	−1.37514	−0.687570	−	0.726119i	$-0.741322\pi$
$47$	−978789.	−1.37514	−0.687570	+	0.726119i	$0.741322\pi$
$48$	0	0
$49$	494003.	0.599851
$50$	0	0
$51$	− 610435.i	− 0.644383i
$52$	0	0
$53$	595078.i	0.549046i	0.961581	+	0.274523i	$0.0885199\pi$
$53$	595078.i	0.549046i	−0.961581	+	0.274523i	$0.911480\pi$
$54$	0	0
$55$	248484.	0.201386
$56$	0	0
$57$	−1.27204e6	−0.909788
$58$	0	0
$59$	− 800309.i	− 0.507312i	−0.967294	−	0.253656i	$-0.918367\pi$
$59$	− 800309.i	− 0.507312i	0.967294	−	0.253656i	$-0.0816332\pi$
$60$	0	0
$61$	47956.7i	0.0270517i	0.999909	+	0.0135259i	$0.00430554\pi$
$61$	47956.7i	0.0270517i	−0.999909	+	0.0135259i	$0.995694\pi$
$62$	0	0
$63$	836778.	0.421617
$64$	0	0
$65$	−799281.	−0.360996
$66$	0	0
$67$	3.61335e6i	1.46774i	0.679292	+	0.733868i	$0.262287\pi$
$67$	3.61335e6i	1.46774i	−0.679292	+	0.733868i	$0.737713\pi$
$68$	0	0
$69$	633137.i	0.232020i
$70$	0	0
$71$	−153370.	−0.0508554	−0.0254277	−	0.999677i	$-0.508095\pi$
$71$	−153370.	−0.0508554	−0.0254277	+	0.999677i	$0.508095\pi$
$72$	0	0
$73$	−5.84223e6	−1.75772	−0.878858	−	0.477084i	$-0.841694\pi$
$73$	−5.84223e6	−1.75772	−0.878858	+	0.477084i	$0.841694\pi$
$74$	0	0
$75$	− 1.85806e6i	− 0.508564i
$76$	0	0
$77$	− 2.95633e6i	− 0.737965i
$78$	0	0
$79$	2.20156e6	0.502383	0.251192	−	0.967937i	$-0.419178\pi$
$79$	2.20156e6	0.502383	0.251192	+	0.967937i	$0.419178\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	2.26908e6i	0.435589i	0.975995	+	0.217794i	$0.0698862\pi$
$83$	2.26908e6i	0.435589i	−0.975995	+	0.217794i	$0.930114\pi$
$84$	0	0
$85$	− 2.18124e6i	− 0.385245i
$86$	0	0
$87$	−5.43473e6	−0.884832
$88$	0	0
$89$	9.01888e6	1.35609	0.678043	−	0.735022i	$-0.262828\pi$
$89$	9.01888e6	1.35609	0.678043	+	0.735022i	$0.262828\pi$
$90$	0	0
$91$	9.50944e6i	1.32285i
$92$	0	0
$93$	− 4.89387e6i	− 0.630903i
$94$	0	0
$95$	−4.54534e6	−0.543918
$96$	0	0
$97$	876482.	0.0975083	0.0487542	−	0.998811i	$-0.484475\pi$
$97$	876482.	0.0975083	0.0487542	+	0.998811i	$0.484475\pi$
$98$	0	0
$99$	− 1.87758e6i	− 0.194480i
$100$	0	0
$101$	− 3.86197e6i	− 0.372979i	−0.982457	−	0.186490i	$-0.940289\pi$
$101$	− 3.86197e6i	− 0.372979i	0.982457	−	0.186490i	$-0.0597111\pi$
$102$	0	0
$103$	1.00385e7	0.905187	0.452594	−	0.891717i	$-0.350499\pi$
$103$	1.00385e7	0.905187	0.452594	+	0.891717i	$0.350499\pi$
$104$	0	0
$105$	2.99002e6	0.252065
$106$	0	0
$107$	− 6.31903e6i	− 0.498663i	−0.968418	−	0.249332i	$-0.919789\pi$
$107$	− 6.31903e6i	− 0.498663i	0.968418	−	0.249332i	$-0.0802109\pi$
$108$	0	0
$109$	108557.i	0.00802905i	0.999992	+	0.00401452i	$0.00127787\pi$
$109$	108557.i	0.00802905i	−0.999992	+	0.00401452i	$0.998722\pi$
$110$	0	0
$111$	−9.75776e6	−0.677204
$112$	0	0
$113$	−9.00680e6	−0.587213	−0.293607	−	0.955926i	$-0.594856\pi$
$113$	−9.00680e6	−0.587213	−0.293607	+	0.955926i	$0.594856\pi$
$114$	0	0
$115$	2.26236e6i	0.138714i
$116$	0	0
$117$	6.03948e6i	0.348617i
$118$	0	0
$119$	−2.59513e7	−1.41171
$120$	0	0
$121$	1.28537e7	0.659598
$122$	0	0
$123$	1.71598e7i	0.831466i
$124$	0	0
$125$	− 1.41766e7i	− 0.649215i
$126$	0	0
$127$	−2.54740e7	−1.10353	−0.551765	−	0.833999i	$-0.686046\pi$
$127$	−2.54740e7	−1.10353	−0.551765	+	0.833999i	$0.686046\pi$
$128$	0	0
$129$	317993.	0.0130423
$130$	0	0
$131$	3.55480e7i	1.38155i	0.723071	+	0.690773i	$0.242730\pi$
$131$	3.55480e7i	1.38155i	−0.723071	+	0.690773i	$0.757270\pi$
$132$	0	0
$133$	5.40781e7i	1.99315i
$134$	0	0
$135$	1.89897e6	0.0664279
$136$	0	0
$137$	1.54025e7	0.511762	0.255881	−	0.966708i	$-0.417634\pi$
$137$	1.54025e7	0.511762	0.255881	+	0.966708i	$0.417634\pi$
$138$	0	0
$139$	3.32599e7i	1.05044i	0.850968	+	0.525218i	$0.176016\pi$
$139$	3.32599e7i	1.05044i	−0.850968	+	0.525218i	$0.823984\pi$
$140$	0	0
$141$	2.64273e7i	0.793937i
$142$	0	0
$143$	2.13375e7	0.610192
$144$	0	0
$145$	−1.94197e7	−0.528998
$146$	0	0
$147$	− 1.33381e7i	− 0.346324i
$148$	0	0
$149$	3.77329e7i	0.934475i	0.884132	+	0.467238i	$0.154751\pi$
$149$	3.77329e7i	0.934475i	−0.884132	+	0.467238i	$0.845249\pi$
$150$	0	0
$151$	−7.06300e7	−1.66944	−0.834718	−	0.550678i	$-0.814369\pi$
$151$	−7.06300e7	−1.66944	−0.834718	+	0.550678i	$0.814369\pi$
$152$	0	0
$153$	−1.64817e7	−0.372034
$154$	0	0
$155$	− 1.74871e7i	− 0.377186i
$156$	0	0
$157$	− 5.66806e7i	− 1.16892i	−0.811421	−	0.584461i	$-0.801306\pi$
$157$	− 5.66806e7i	− 1.16892i	0.811421	−	0.584461i	$-0.198694\pi$
$158$	0	0
$159$	1.60671e7	0.316992
$160$	0	0
$161$	2.69164e7	0.508307
$162$	0	0
$163$	4.72913e6i	0.0855313i	0.999085	+	0.0427656i	$0.0136169\pi$
$163$	4.72913e6i	0.0855313i	−0.999085	+	0.0427656i	$0.986383\pi$
$164$	0	0
$165$	− 6.70906e6i	− 0.116270i
$166$	0	0
$167$	−7.14230e7	−1.18667	−0.593336	−	0.804955i	$-0.702189\pi$
$167$	−7.14230e7	−1.18667	−0.593336	+	0.804955i	$0.702189\pi$
$168$	0	0
$169$	−5.88627e6	−0.0938073
$170$	0	0
$171$	3.43452e7i	0.525266i
$172$	0	0
$173$	5.02253e7i	0.737498i	0.929529	+	0.368749i	$0.120214\pi$
$173$	5.02253e7i	0.737498i	−0.929529	+	0.368749i	$0.879786\pi$
$174$	0	0
$175$	−7.89912e7	−1.11416
$176$	0	0
$177$	−2.16083e7	−0.292897
$178$	0	0
$179$	1.51051e8i	1.96852i	0.176737	+	0.984258i	$0.443446\pi$
$179$	1.51051e8i	1.96852i	−0.176737	+	0.984258i	$0.556554\pi$
$180$	0	0
$181$	− 9.51914e7i	− 1.19323i	−0.802529	−	0.596613i	$-0.796513\pi$
$181$	− 9.51914e7i	− 1.19323i	0.802529	−	0.596613i	$-0.203487\pi$
$182$	0	0
$183$	1.29483e6	0.0156183
$184$	0	0
$185$	−3.48670e7	−0.404868
$186$	0	0
$187$	5.82299e7i	0.651179i
$188$	0	0
$189$	− 2.25930e7i	− 0.243421i
$190$	0	0
$191$	−8.95545e7	−0.929974	−0.464987	−	0.885318i	$-0.653941\pi$
$191$	−8.95545e7	−0.929974	−0.464987	+	0.885318i	$0.653941\pi$
$192$	0	0
$193$	1.32739e8	1.32907	0.664536	−	0.747257i	$-0.268629\pi$
$193$	1.32739e8	1.32907	0.664536	+	0.747257i	$0.268629\pi$
$194$	0	0
$195$	2.15806e7i	0.208421i
$196$	0	0
$197$	− 2.80550e7i	− 0.261443i	−0.991419	−	0.130722i	$-0.958271\pi$
$197$	− 2.80550e7i	− 0.261443i	0.991419	−	0.130722i	$-0.0417294\pi$
$198$	0	0
$199$	−1.60193e8	−1.44098	−0.720490	−	0.693465i	$-0.756083\pi$
$199$	−1.60193e8	−1.44098	−0.720490	+	0.693465i	$0.756083\pi$
$200$	0	0
$201$	9.75604e7	0.847398
$202$	0	0
$203$	2.31045e8i	1.93848i
$204$	0	0
$205$	6.13163e7i	0.497093i
$206$	0	0
$207$	1.70947e7	0.133957
$208$	0	0
$209$	1.21341e8	0.919384
$210$	0	0
$211$	− 5.79426e7i	− 0.424629i	−0.977201	−	0.212314i	$-0.931900\pi$
$211$	− 5.79426e7i	− 0.424629i	0.977201	−	0.212314i	$-0.0681001\pi$
$212$	0	0
$213$	4.14100e6i	0.0293614i
$214$	0	0
$215$	1.13627e6	0.00779734
$216$	0	0
$217$	−2.08052e8	−1.38217
$218$	0	0
$219$	1.57740e8i	1.01482i
$220$	0	0
$221$	− 1.87304e8i	− 1.16728i
$222$	0	0
$223$	1.31138e8	0.791881	0.395941	−	0.918276i	$-0.370419\pi$
$223$	1.31138e8	0.791881	0.395941	+	0.918276i	$0.370419\pi$
$224$	0	0
$225$	−5.01676e7	−0.293619
$226$	0	0
$227$	1.33419e8i	0.757052i	0.925591	+	0.378526i	$0.123569\pi$
$227$	1.33419e8i	0.757052i	−0.925591	+	0.378526i	$0.876431\pi$
$228$	0	0
$229$	5.75413e7i	0.316632i	0.987388	+	0.158316i	$0.0506065\pi$
$229$	5.75413e7i	0.316632i	−0.987388	+	0.158316i	$0.949393\pi$
$230$	0	0
$231$	−7.98210e7	−0.426065
$232$	0	0
$233$	−1.60630e8	−0.831921	−0.415960	−	0.909383i	$-0.636554\pi$
$233$	−1.60630e8	−0.831921	−0.415960	+	0.909383i	$0.636554\pi$
$234$	0	0
$235$	9.44315e7i	0.474656i
$236$	0	0
$237$	− 5.94420e7i	− 0.290051i
$238$	0	0
$239$	−1.95430e8	−0.925972	−0.462986	−	0.886366i	$-0.653222\pi$
$239$	−1.95430e8	−0.925972	−0.462986	+	0.886366i	$0.653222\pi$
$240$	0	0
$241$	−3.08523e8	−1.41980	−0.709902	−	0.704301i	$-0.751261\pi$
$241$	−3.08523e8	−1.41980	−0.709902	+	0.704301i	$0.751261\pi$
$242$	0	0
$243$	− 1.43489e7i	− 0.0641500i
$244$	0	0
$245$	− 4.76604e7i	− 0.207050i
$246$	0	0
$247$	−3.90311e8	−1.64805
$248$	0	0
$249$	6.12652e7	0.251487
$250$	0	0
$251$	− 3.77358e8i	− 1.50624i	−0.657881	−	0.753122i	$-0.728547\pi$
$251$	− 3.77358e8i	− 1.50624i	0.657881	−	0.753122i	$-0.271453\pi$
$252$	0	0
$253$	− 6.03955e7i	− 0.234467i
$254$	0	0
$255$	−5.88934e7	−0.222421
$256$	0	0
$257$	−3.78045e8	−1.38924	−0.694621	−	0.719376i	$-0.744428\pi$
$257$	−3.78045e8	−1.38924	−0.694621	+	0.719376i	$0.744428\pi$
$258$	0	0
$259$	4.14829e8i	1.48361i
$260$	0	0
$261$	1.46738e8i	0.510858i
$262$	0	0
$263$	−4.16146e8	−1.41059	−0.705295	−	0.708914i	$-0.749185\pi$
$263$	−4.16146e8	−1.41059	−0.705295	+	0.708914i	$0.749185\pi$
$264$	0	0
$265$	5.74119e7	0.189514
$266$	0	0
$267$	− 2.43510e8i	− 0.782937i
$268$	0	0
$269$	5.97907e8i	1.87284i	0.350880	+	0.936420i	$0.385882\pi$
$269$	5.97907e8i	1.87284i	−0.350880	+	0.936420i	$0.614118\pi$
$270$	0	0
$271$	1.04343e8	0.318473	0.159236	−	0.987240i	$-0.449097\pi$
$271$	1.04343e8	0.318473	0.159236	+	0.987240i	$0.449097\pi$
$272$	0	0
$273$	2.56755e8	0.763747
$274$	0	0
$275$	1.77242e8i	0.513928i
$276$	0	0
$277$	− 4.52462e8i	− 1.27910i	−0.768752	−	0.639548i	$-0.779122\pi$
$277$	− 4.52462e8i	− 1.27910i	0.768752	−	0.639548i	$-0.220878\pi$
$278$	0	0
$279$	−1.32135e8	−0.364252
$280$	0	0
$281$	3.75702e8	1.01012	0.505058	−	0.863085i	$-0.331471\pi$
$281$	3.75702e8	1.01012	0.505058	+	0.863085i	$0.331471\pi$
$282$	0	0
$283$	4.52192e8i	1.18596i	0.805217	+	0.592980i	$0.202049\pi$
$283$	4.52192e8i	1.18596i	−0.805217	+	0.592980i	$0.797951\pi$
$284$	0	0
$285$	1.22724e8i	0.314031i
$286$	0	0
$287$	7.29510e8	1.82157
$288$	0	0
$289$	1.00815e8	0.245687
$290$	0	0
$291$	− 2.36650e7i	− 0.0562965i
$292$	0	0
$293$	4.87094e7i	0.113130i	0.998399	+	0.0565648i	$0.0180147\pi$
$293$	4.87094e7i	0.113130i	−0.998399	+	0.0565648i	$0.981985\pi$
$294$	0	0
$295$	−7.72120e7	−0.175109
$296$	0	0
$297$	−5.06946e7	−0.112283
$298$	0	0
$299$	1.94270e8i	0.420297i
$300$	0	0
$301$	− 1.35188e7i	− 0.0285729i
$302$	0	0
$303$	−1.04273e8	−0.215340
$304$	0	0
$305$	4.62676e6	0.00933743
$306$	0	0
$307$	1.59137e8i	0.313896i	0.987607	+	0.156948i	$0.0501655\pi$
$307$	1.59137e8i	0.313896i	−0.987607	+	0.156948i	$0.949834\pi$
$308$	0	0
$309$	− 2.71039e8i	− 0.522610i
$310$	0	0
$311$	−6.72549e8	−1.26783	−0.633917	−	0.773401i	$-0.718554\pi$
$311$	−6.72549e8	−1.26783	−0.633917	+	0.773401i	$0.718554\pi$
$312$	0	0
$313$	1.61251e8	0.297232	0.148616	−	0.988895i	$-0.452518\pi$
$313$	1.61251e8	0.297232	0.148616	+	0.988895i	$0.452518\pi$
$314$	0	0
$315$	− 8.07306e7i	− 0.145530i
$316$	0	0
$317$	− 6.19133e7i	− 0.109163i	−0.998509	−	0.0545816i	$-0.982618\pi$
$317$	− 6.19133e7i	− 0.109163i	0.998509	−	0.0545816i	$-0.0173825\pi$
$318$	0	0
$319$	5.18424e8	0.894165
$320$	0	0
$321$	−1.70614e8	−0.287903
$322$	0	0
$323$	− 1.06516e9i	− 1.75875i
$324$	0	0
$325$	− 5.70122e8i	− 0.921247i
$326$	0	0
$327$	2.93103e6	0.00463557
$328$	0	0
$329$	1.12350e9	1.73935
$330$	0	0
$331$	− 6.96658e8i	− 1.05590i	−0.849276	−	0.527949i	$-0.822961\pi$
$331$	− 6.96658e8i	− 1.05590i	0.849276	−	0.527949i	$-0.177039\pi$
$332$	0	0
$333$	2.63460e8i	0.390984i
$334$	0	0
$335$	3.48608e8	0.506618
$336$	0	0
$337$	2.03714e8	0.289945	0.144973	−	0.989436i	$-0.453691\pi$
$337$	2.03714e8	0.289945	0.144973	+	0.989436i	$0.453691\pi$
$338$	0	0
$339$	2.43184e8i	0.339028i
$340$	0	0
$341$	4.66831e8i	0.637557i
$342$	0	0
$343$	3.78260e8	0.506129
$344$	0	0
$345$	6.10837e7	0.0800863
$346$	0	0
$347$	9.44086e8i	1.21299i	0.795086	+	0.606496i	$0.207426\pi$
$347$	9.44086e8i	1.21299i	−0.795086	+	0.606496i	$0.792574\pi$
$348$	0	0
$349$	− 4.69120e8i	− 0.590738i	−0.955383	−	0.295369i	$-0.904557\pi$
$349$	− 4.69120e8i	− 0.590738i	0.955383	−	0.295369i	$-0.0954426\pi$
$350$	0	0
$351$	1.63066e8	0.201274
$352$	0	0
$353$	5.39766e8	0.653122	0.326561	−	0.945176i	$-0.394110\pi$
$353$	5.39766e8	0.653122	0.326561	+	0.945176i	$0.394110\pi$
$354$	0	0
$355$	1.47968e7i	0.0175537i
$356$	0	0
$357$	7.00684e8i	0.815049i
$358$	0	0
$359$	−7.44443e8	−0.849182	−0.424591	−	0.905385i	$-0.639582\pi$
$359$	−7.44443e8	−0.849182	−0.424591	+	0.905385i	$0.639582\pi$
$360$	0	0
$361$	−1.32574e9	−1.48314
$362$	0	0
$363$	− 3.47050e8i	− 0.380819i
$364$	0	0
$365$	5.63646e8i	0.606710i
$366$	0	0
$367$	1.73007e9	1.82698	0.913488	−	0.406865i	$-0.133378\pi$
$367$	1.73007e9	1.82698	0.913488	+	0.406865i	$0.133378\pi$
$368$	0	0
$369$	4.63315e8	0.480047
$370$	0	0
$371$	− 6.83057e8i	− 0.694462i
$372$	0	0
$373$	1.10211e9i	1.09962i	0.835290	+	0.549810i	$0.185300\pi$
$373$	1.10211e9i	1.09962i	−0.835290	+	0.549810i	$0.814700\pi$
$374$	0	0
$375$	−3.82770e8	−0.374824
$376$	0	0
$377$	−1.66758e9	−1.60285
$378$	0	0
$379$	− 1.34814e9i	− 1.27203i	−0.771675	−	0.636017i	$-0.780581\pi$
$379$	− 1.34814e9i	− 1.27203i	0.771675	−	0.636017i	$-0.219419\pi$
$380$	0	0
$381$	6.87798e8i	0.637124i
$382$	0	0
$383$	1.94903e9	1.77265	0.886324	−	0.463066i	$-0.153251\pi$
$383$	1.94903e9	1.77265	0.886324	+	0.463066i	$0.153251\pi$
$384$	0	0
$385$	−2.85221e8	−0.254723
$386$	0	0
$387$	− 8.58581e6i	− 0.00752996i
$388$	0	0
$389$	− 1.03589e9i	− 0.892259i	−0.894968	−	0.446129i	$-0.852802\pi$
$389$	− 1.03589e9i	− 0.892259i	0.894968	−	0.446129i	$-0.147198\pi$
$390$	0	0
$391$	−5.30163e8	−0.448529
$392$	0	0
$393$	9.59796e8	0.797636
$394$	0	0
$395$	− 2.12401e8i	− 0.173407i
$396$	0	0
$397$	− 1.25741e9i	− 1.00858i	−0.863535	−	0.504289i	$-0.831755\pi$
$397$	− 1.25741e9i	− 1.00858i	0.863535	−	0.504289i	$-0.168245\pi$
$398$	0	0
$399$	1.46011e9	1.15075
$400$	0	0
$401$	3.06297e8	0.237212	0.118606	−	0.992941i	$-0.462157\pi$
$401$	3.06297e8	0.237212	0.118606	+	0.992941i	$0.462157\pi$
$402$	0	0
$403$	− 1.50162e9i	− 1.14286i
$404$	0	0
$405$	− 5.12723e7i	− 0.0383522i
$406$	0	0
$407$	9.30801e8	0.684347
$408$	0	0
$409$	−2.03391e9	−1.46994	−0.734971	−	0.678099i	$-0.762804\pi$
$409$	−2.03391e9	−1.46994	−0.734971	+	0.678099i	$0.762804\pi$
$410$	0	0
$411$	− 4.15866e8i	− 0.295466i
$412$	0	0
$413$	9.18629e8i	0.641675i
$414$	0	0
$415$	2.18916e8	0.150352
$416$	0	0
$417$	8.98019e8	0.606470
$418$	0	0
$419$	− 5.68845e8i	− 0.377785i	−0.981998	−	0.188892i	$-0.939510\pi$
$419$	− 5.68845e8i	− 0.377785i	0.981998	−	0.188892i	$-0.0604898\pi$
$420$	0	0
$421$	− 1.75106e8i	− 0.114370i	−0.998364	−	0.0571851i	$-0.981787\pi$
$421$	− 1.75106e8i	− 0.114370i	0.998364	−	0.0571851i	$-0.0182125\pi$
$422$	0	0
$423$	7.13537e8	0.458380
$424$	0	0
$425$	1.55586e9	0.983128
$426$	0	0
$427$	− 5.50468e7i	− 0.0342164i
$428$	0	0
$429$	− 5.76111e8i	− 0.352295i
$430$	0	0
$431$	1.51429e9	0.911042	0.455521	−	0.890225i	$-0.349453\pi$
$431$	1.51429e9	0.911042	0.455521	+	0.890225i	$0.349453\pi$
$432$	0	0
$433$	−3.06266e9	−1.81297	−0.906486	−	0.422236i	$-0.861245\pi$
$433$	−3.06266e9	−1.81297	−0.906486	+	0.422236i	$0.861245\pi$
$434$	0	0
$435$	5.24331e8i	0.305417i
$436$	0	0
$437$	1.10477e9i	0.633268i
$438$	0	0
$439$	−3.33643e9	−1.88216	−0.941080	−	0.338185i	$-0.890187\pi$
$439$	−3.33643e9	−1.88216	−0.941080	+	0.338185i	$0.890187\pi$
$440$	0	0
$441$	−3.60128e8	−0.199950
$442$	0	0
$443$	− 1.50629e9i	− 0.823181i	−0.911369	−	0.411591i	$-0.864973\pi$
$443$	− 1.50629e9i	− 0.823181i	0.911369	−	0.411591i	$-0.135027\pi$
$444$	0	0
$445$	− 8.70122e8i	− 0.468080i
$446$	0	0
$447$	1.01879e9	0.539520
$448$	0	0
$449$	2.53664e9	1.32250	0.661251	−	0.750165i	$-0.270026\pi$
$449$	2.53664e9	1.32250	0.661251	+	0.750165i	$0.270026\pi$
$450$	0	0
$451$	− 1.63689e9i	− 0.840236i
$452$	0	0
$453$	1.90701e9i	0.963849i
$454$	0	0
$455$	9.17450e8	0.456607
$456$	0	0
$457$	1.67303e9	0.819966	0.409983	−	0.912093i	$-0.365535\pi$
$457$	1.67303e9	0.819966	0.409983	+	0.912093i	$0.365535\pi$
$458$	0	0
$459$	4.45007e8i	0.214794i
$460$	0	0
$461$	− 3.23110e9i	− 1.53602i	−0.640438	−	0.768010i	$-0.721247\pi$
$461$	− 3.23110e9i	− 1.53602i	0.640438	−	0.768010i	$-0.278753\pi$
$462$	0	0
$463$	9.58036e8	0.448589	0.224294	−	0.974521i	$-0.427992\pi$
$463$	9.58036e8	0.448589	0.224294	+	0.974521i	$0.427992\pi$
$464$	0	0
$465$	−4.72150e8	−0.217768
$466$	0	0
$467$	− 1.22755e9i	− 0.557736i	−0.960329	−	0.278868i	$-0.910041\pi$
$467$	− 1.22755e9i	− 0.557736i	0.960329	−	0.278868i	$-0.0899592\pi$
$468$	0	0
$469$	− 4.14756e9i	− 1.85647i
$470$	0	0
$471$	−1.53038e9	−0.674878
$472$	0	0
$473$	−3.03336e7	−0.0131798
$474$	0	0
$475$	− 3.24216e9i	− 1.38806i
$476$	0	0
$477$	− 4.33812e8i	− 0.183015i
$478$	0	0
$479$	1.80880e9	0.751996	0.375998	−	0.926620i	$-0.377300\pi$
$479$	1.80880e9	0.751996	0.375998	+	0.926620i	$0.377300\pi$
$480$	0	0
$481$	−2.99405e9	−1.22674
$482$	0	0
$483$	− 7.26742e8i	− 0.293471i
$484$	0	0
$485$	− 8.45610e7i	− 0.0336569i
$486$	0	0
$487$	1.83297e9	0.719123	0.359562	−	0.933121i	$-0.382926\pi$
$487$	1.83297e9	0.719123	0.359562	+	0.933121i	$0.382926\pi$
$488$	0	0
$489$	1.27687e8	0.0493815
$490$	0	0
$491$	− 1.07315e9i	− 0.409143i	−0.978852	−	0.204571i	$-0.934420\pi$
$491$	− 1.07315e9i	− 0.409143i	0.978852	−	0.204571i	$-0.0655800\pi$
$492$	0	0
$493$	− 4.55082e9i	− 1.71051i
$494$	0	0
$495$	−1.81145e8	−0.0671286
$496$	0	0
$497$	1.76045e8	0.0643245
$498$	0	0
$499$	− 4.22943e8i	− 0.152381i	−0.997093	−	0.0761903i	$-0.975724\pi$
$499$	− 4.22943e8i	− 0.152381i	0.997093	−	0.0761903i	$-0.0242756\pi$
$500$	0	0
$501$	1.92842e9i	0.685125i
$502$	0	0
$503$	−7.49586e8	−0.262623	−0.131312	−	0.991341i	$-0.541919\pi$
$503$	−7.49586e8	−0.262623	−0.131312	+	0.991341i	$0.541919\pi$
$504$	0	0
$505$	−3.72595e8	−0.128741
$506$	0	0
$507$	1.58929e8i	0.0541597i
$508$	0	0
$509$	4.30275e9i	1.44622i	0.690734	+	0.723109i	$0.257287\pi$
$509$	4.30275e9i	1.44622i	−0.690734	+	0.723109i	$0.742713\pi$
$510$	0	0
$511$	6.70597e9	2.22325
$512$	0	0
$513$	9.27320e8	0.303263
$514$	0	0
$515$	− 9.68492e8i	− 0.312443i
$516$	0	0
$517$	− 2.52092e9i	− 0.802311i
$518$	0	0
$519$	1.35608e9	0.425795
$520$	0	0
$521$	1.34467e8	0.0416566	0.0208283	−	0.999783i	$-0.493370\pi$
$521$	1.34467e8	0.0416566	0.0208283	+	0.999783i	$0.493370\pi$
$522$	0	0
$523$	2.98774e9i	0.913243i	0.889661	+	0.456622i	$0.150941\pi$
$523$	2.98774e9i	0.913243i	−0.889661	+	0.456622i	$0.849059\pi$
$524$	0	0
$525$	2.13276e9i	0.643258i
$526$	0	0
$527$	4.09793e9	1.21963
$528$	0	0
$529$	−2.85495e9	−0.838500
$530$	0	0
$531$	5.83425e8i	0.169104i
$532$	0	0
$533$	5.26527e9i	1.50618i
$534$	0	0
$535$	−6.09646e8	−0.172123
$536$	0	0
$537$	4.07839e9	1.13652
$538$	0	0
$539$	1.27233e9i	0.349977i
$540$	0	0
$541$	1.21706e9i	0.330461i	0.986255	+	0.165231i	$0.0528369\pi$
$541$	1.21706e9i	0.330461i	−0.986255	+	0.165231i	$0.947163\pi$
$542$	0	0
$543$	−2.57017e9	−0.688910
$544$	0	0
$545$	1.04733e7	0.00277138
$546$	0	0
$547$	1.42972e9i	0.373505i	0.982407	+	0.186752i	$0.0597962\pi$
$547$	1.42972e9i	0.373505i	−0.982407	+	0.186752i	$0.940204\pi$
$548$	0	0
$549$	− 3.49604e7i	− 0.00901724i
$550$	0	0
$551$	−9.48316e9	−2.41503
$552$	0	0
$553$	−2.52704e9	−0.635440
$554$	0	0
$555$	9.41408e8i	0.233750i
$556$	0	0
$557$	6.32182e9i	1.55006i	0.631924	+	0.775031i	$0.282266\pi$
$557$	6.32182e9i	1.55006i	−0.631924	+	0.775031i	$0.717734\pi$
$558$	0	0
$559$	9.75721e7	0.0236257
$560$	0	0
$561$	1.57221e9	0.375959
$562$	0	0
$563$	4.82782e9i	1.14017i	0.821584	+	0.570087i	$0.193091\pi$
$563$	4.82782e9i	1.14017i	−0.821584	+	0.570087i	$0.806909\pi$
$564$	0	0
$565$	8.68957e8i	0.202688i
$566$	0	0
$567$	−6.10011e8	−0.140539
$568$	0	0
$569$	2.28899e9	0.520896	0.260448	−	0.965488i	$-0.416130\pi$
$569$	2.28899e9	0.520896	0.260448	+	0.965488i	$0.416130\pi$
$570$	0	0
$571$	1.26198e9i	0.283678i	0.989890	+	0.141839i	$0.0453015\pi$
$571$	1.26198e9i	0.283678i	−0.989890	+	0.141839i	$0.954699\pi$
$572$	0	0
$573$	2.41797e9i	0.536920i
$574$	0	0
$575$	−1.61373e9	−0.353991
$576$	0	0
$577$	−1.05685e9	−0.229033	−0.114516	−	0.993421i	$-0.536532\pi$
$577$	−1.05685e9	−0.229033	−0.114516	+	0.993421i	$0.536532\pi$
$578$	0	0
$579$	− 3.58396e9i	− 0.767340i
$580$	0	0
$581$	− 2.60455e9i	− 0.550955i
$582$	0	0
$583$	−1.53266e9	−0.320335
$584$	0	0
$585$	5.82676e8	0.120332
$586$	0	0
$587$	− 8.64797e9i	− 1.76474i	−0.470556	−	0.882370i	$-0.655947\pi$
$587$	− 8.64797e9i	− 1.76474i	0.470556	−	0.882370i	$-0.344053\pi$
$588$	0	0
$589$	− 8.53940e9i	− 1.72196i
$590$	0	0
$591$	−7.57484e8	−0.150944
$592$	0	0
$593$	−1.34549e9	−0.264965	−0.132482	−	0.991185i	$-0.542295\pi$
$593$	−1.34549e9	−0.264965	−0.132482	+	0.991185i	$0.542295\pi$
$594$	0	0
$595$	2.50372e9i	0.487278i
$596$	0	0
$597$	4.32522e9i	0.831951i
$598$	0	0
$599$	9.60516e9	1.82604	0.913021	−	0.407913i	$-0.133743\pi$
$599$	9.60516e9	1.82604	0.913021	+	0.407913i	$0.133743\pi$
$600$	0	0
$601$	−9.64387e9	−1.81213	−0.906067	−	0.423133i	$-0.860930\pi$
$601$	−9.64387e9	−1.81213	−0.906067	+	0.423133i	$0.860930\pi$
$602$	0	0
$603$	− 2.63413e9i	− 0.489246i
$604$	0	0
$605$	− 1.24010e9i	− 0.227673i
$606$	0	0
$607$	−6.45216e9	−1.17097	−0.585483	−	0.810684i	$-0.699095\pi$
$607$	−6.45216e9	−1.17097	−0.585483	+	0.810684i	$0.699095\pi$
$608$	0	0
$609$	6.23823e9	1.11918
$610$	0	0
$611$	8.10889e9i	1.43819i
$612$	0	0
$613$	3.13541e9i	0.549771i	0.961477	+	0.274886i	$0.0886400\pi$
$613$	3.13541e9i	0.549771i	−0.961477	+	0.274886i	$0.911360\pi$
$614$	0	0
$615$	1.65554e9	0.286997
$616$	0	0
$617$	3.65719e9	0.626829	0.313415	−	0.949616i	$-0.398527\pi$
$617$	3.65719e9	0.626829	0.313415	+	0.949616i	$0.398527\pi$
$618$	0	0
$619$	7.28006e9i	1.23372i	0.787072	+	0.616861i	$0.211596\pi$
$619$	7.28006e9i	1.23372i	−0.787072	+	0.616861i	$0.788404\pi$
$620$	0	0
$621$	− 4.61557e8i	− 0.0773401i
$622$	0	0
$623$	−1.03523e10	−1.71525
$624$	0	0
$625$	4.00860e9	0.656769
$626$	0	0
$627$	− 3.27622e9i	− 0.530807i
$628$	0	0
$629$	− 8.17075e9i	− 1.30914i
$630$	0	0
$631$	9.05952e8	0.143550	0.0717749	−	0.997421i	$-0.477134\pi$
$631$	9.05952e8	0.143550	0.0717749	+	0.997421i	$0.477134\pi$
$632$	0	0
$633$	−1.56445e9	−0.245159
$634$	0	0
$635$	2.45768e9i	0.380905i
$636$	0	0
$637$	− 4.09262e9i	− 0.627356i
$638$	0	0
$639$	1.11807e8	0.0169518
$640$	0	0
$641$	8.15790e9	1.22342	0.611710	−	0.791082i	$-0.290482\pi$
$641$	8.15790e9	1.22342	0.611710	+	0.791082i	$0.290482\pi$
$642$	0	0
$643$	5.54137e9i	0.822014i	0.911632	+	0.411007i	$0.134823\pi$
$643$	5.54137e9i	0.822014i	−0.911632	+	0.411007i	$0.865177\pi$
$644$	0	0
$645$	− 3.06793e7i	− 0.00450180i
$646$	0	0
$647$	−1.25642e10	−1.82377	−0.911885	−	0.410446i	$-0.865373\pi$
$647$	−1.25642e10	−1.82377	−0.911885	+	0.410446i	$0.865373\pi$
$648$	0	0
$649$	2.06124e9	0.295986
$650$	0	0
$651$	5.61740e9i	0.797999i
$652$	0	0
$653$	1.18999e10i	1.67242i	0.548408	+	0.836211i	$0.315234\pi$
$653$	1.18999e10i	1.67242i	−0.548408	+	0.836211i	$0.684766\pi$
$654$	0	0
$655$	3.42959e9	0.476868
$656$	0	0
$657$	4.25899e9	0.585905
$658$	0	0
$659$	− 2.01415e9i	− 0.274153i	−0.990560	−	0.137077i	$-0.956229\pi$
$659$	− 2.01415e9i	− 0.274153i	0.990560	−	0.137077i	$-0.0437707\pi$
$660$	0	0
$661$	− 8.73934e9i	− 1.17699i	−0.808500	−	0.588496i	$-0.799720\pi$
$661$	− 8.73934e9i	− 1.17699i	0.808500	−	0.588496i	$-0.200280\pi$
$662$	0	0
$663$	−5.05722e9	−0.673929
$664$	0	0
$665$	5.21734e9	0.687976
$666$	0	0
$667$	4.72007e9i	0.615897i
$668$	0	0
$669$	− 3.54071e9i	− 0.457193i
$670$	0	0
$671$	−1.23515e8	−0.0157830
$672$	0	0
$673$	−1.16058e10	−1.46764	−0.733822	−	0.679342i	$-0.762265\pi$
$673$	−1.16058e10	−1.46764	−0.733822	+	0.679342i	$0.762265\pi$
$674$	0	0
$675$	1.35453e9i	0.169521i
$676$	0	0
$677$	6.87044e9i	0.850990i	0.904961	+	0.425495i	$0.139900\pi$
$677$	6.87044e9i	0.850990i	−0.904961	+	0.425495i	$0.860100\pi$
$678$	0	0
$679$	−1.00606e9	−0.123334
$680$	0	0
$681$	3.60230e9	0.437084
$682$	0	0
$683$	9.51856e9i	1.14314i	0.820554	+	0.571569i	$0.193665\pi$
$683$	9.51856e9i	1.14314i	−0.820554	+	0.571569i	$0.806335\pi$
$684$	0	0
$685$	− 1.48600e9i	− 0.176645i
$686$	0	0
$687$	1.55361e9	0.182808
$688$	0	0
$689$	4.92999e9	0.574221
$690$	0	0
$691$	− 6.06326e9i	− 0.699090i	−0.936920	−	0.349545i	$-0.886336\pi$
$691$	− 6.06326e9i	− 0.699090i	0.936920	−	0.349545i	$-0.113664\pi$
$692$	0	0
$693$	2.15517e9i	0.245988i
$694$	0	0
$695$	3.20885e9	0.362579
$696$	0	0
$697$	−1.43689e10	−1.60735
$698$	0	0
$699$	4.33702e9i	0.480310i
$700$	0	0
$701$	1.44873e10i	1.58845i	0.607624	+	0.794225i	$0.292123\pi$
$701$	1.44873e10i	1.58845i	−0.607624	+	0.794225i	$0.707877\pi$
$702$	0	0
$703$	−1.70265e10	−1.84834
$704$	0	0
$705$	2.54965e9	0.274043
$706$	0	0
$707$	4.43294e9i	0.471764i
$708$	0	0
$709$	− 1.77425e10i	− 1.86962i	−0.355145	−	0.934811i	$-0.615569\pi$
$709$	− 1.77425e10i	− 1.86962i	0.355145	−	0.934811i	$-0.384431\pi$
$710$	0	0
$711$	−1.60493e9	−0.167461
$712$	0	0
$713$	−4.25033e9	−0.439146
$714$	0	0
$715$	− 2.05859e9i	− 0.210620i
$716$	0	0
$717$	5.27660e9i	0.534610i
$718$	0	0
$719$	−1.19275e10	−1.19673	−0.598366	−	0.801223i	$-0.704183\pi$
$719$	−1.19275e10	−1.19673	−0.598366	+	0.801223i	$0.704183\pi$
$720$	0	0
$721$	−1.15226e10	−1.14493
$722$	0	0
$723$	8.33013e9i	0.819724i
$724$	0	0
$725$	− 1.38519e10i	− 1.34998i
$726$	0	0
$727$	−1.19578e10	−1.15420	−0.577099	−	0.816674i	$-0.695815\pi$
$727$	−1.19578e10	−1.15420	−0.577099	+	0.816674i	$0.695815\pi$
$728$	0	0
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	2.66274e8i	0.0252127i
$732$	0	0
$733$	− 1.35782e10i	− 1.27344i	−0.771097	−	0.636718i	$-0.780292\pi$
$733$	− 1.35782e10i	− 1.27344i	0.771097	−	0.636718i	$-0.219708\pi$
$734$	0	0
$735$	−1.28683e9	−0.119541
$736$	0	0
$737$	−9.30638e9	−0.856336
$738$	0	0
$739$	− 9.90831e9i	− 0.903117i	−0.892241	−	0.451559i	$-0.850868\pi$
$739$	− 9.90831e9i	− 0.903117i	0.892241	−	0.451559i	$-0.149132\pi$
$740$	0	0
$741$	1.05384e10i	0.951504i
$742$	0	0
$743$	1.38901e10	1.24235	0.621176	−	0.783671i	$-0.286655\pi$
$743$	1.38901e10	1.24235	0.621176	+	0.783671i	$0.286655\pi$
$744$	0	0
$745$	3.64039e9	0.322553
$746$	0	0
$747$	− 1.65416e9i	− 0.145196i
$748$	0	0
$749$	7.25326e9i	0.630735i
$750$	0	0
$751$	−8.19650e9	−0.706136	−0.353068	−	0.935598i	$-0.614862\pi$
$751$	−8.19650e9	−0.706136	−0.353068	+	0.935598i	$0.614862\pi$
$752$	0	0
$753$	−1.01887e10	−0.869630
$754$	0	0
$755$	6.81423e9i	0.576239i
$756$	0	0
$757$	9.87499e9i	0.827373i	0.910419	+	0.413686i	$0.135759\pi$
$757$	9.87499e9i	0.827373i	−0.910419	+	0.413686i	$0.864241\pi$
$758$	0	0
$759$	−1.63068e9	−0.135370
$760$	0	0
$761$	−6.36810e9	−0.523798	−0.261899	−	0.965095i	$-0.584349\pi$
$761$	−6.36810e9	−0.523798	−0.261899	+	0.965095i	$0.584349\pi$
$762$	0	0
$763$	− 1.24606e8i	− 0.0101556i
$764$	0	0
$765$	1.59012e9i	0.128415i
$766$	0	0
$767$	−6.63025e9	−0.530574
$768$	0	0
$769$	−7.56351e8	−0.0599765	−0.0299882	−	0.999550i	$-0.509547\pi$
$769$	−7.56351e8	−0.0599765	−0.0299882	+	0.999550i	$0.509547\pi$
$770$	0	0
$771$	1.02072e10i	0.802079i
$772$	0	0
$773$	6.95651e8i	0.0541706i	0.999633	+	0.0270853i	$0.00862257\pi$
$773$	6.95651e8i	0.0541706i	−0.999633	+	0.0270853i	$0.991377\pi$
$774$	0	0
$775$	1.24734e10	0.962562
$776$	0	0
$777$	1.12004e10	0.856563
$778$	0	0
$779$	2.99424e10i	2.26937i
$780$	0	0
$781$	− 3.95013e8i	− 0.0296711i
$782$	0	0
$783$	3.96192e9	0.294944
$784$	0	0
$785$	−5.46842e9	−0.403477
$786$	0	0
$787$	2.56522e10i	1.87592i	0.346748	+	0.937958i	$0.387286\pi$
$787$	2.56522e10i	1.87592i	−0.346748	+	0.937958i	$0.612714\pi$
$788$	0	0
$789$	1.12359e10i	0.814404i
$790$	0	0
$791$	1.03384e10	0.742738
$792$	0	0
$793$	3.97302e8	0.0282921
$794$	0	0
$795$	− 1.55012e9i	− 0.109416i
$796$	0	0
$797$	− 1.80112e10i	− 1.26020i	−0.776516	−	0.630098i	$-0.783015\pi$
$797$	− 1.80112e10i	− 1.26020i	0.776516	−	0.630098i	$-0.216985\pi$
$798$	0	0
$799$	−2.21292e10	−1.53480
$800$	0	0
$801$	−6.57476e9	−0.452029
$802$	0	0
$803$	− 1.50470e10i	− 1.02552i
$804$	0	0
$805$	− 2.59683e9i	− 0.175452i
$806$	0	0
$807$	1.61435e10	1.08129
$808$	0	0
$809$	2.44159e10	1.62126	0.810631	−	0.585558i	$-0.199125\pi$
$809$	2.44159e10	1.62126	0.810631	+	0.585558i	$0.199125\pi$
$810$	0	0
$811$	1.32403e9i	0.0871613i	0.999050	+	0.0435807i	$0.0138766\pi$
$811$	1.32403e9i	0.0871613i	−0.999050	+	0.0435807i	$0.986123\pi$
$812$	0	0
$813$	− 2.81727e9i	− 0.183870i
$814$	0	0
$815$	4.56256e8	0.0295228
$816$	0	0
$817$	5.54871e8	0.0355971
$818$	0	0
$819$	− 6.93238e9i	− 0.440950i
$820$	0	0
$821$	− 5.93906e9i	− 0.374556i	−0.982307	−	0.187278i	$-0.940033\pi$
$821$	− 5.93906e9i	− 0.374556i	0.982307	−	0.187278i	$-0.0599666\pi$
$822$	0	0
$823$	5.96646e9	0.373093	0.186547	−	0.982446i	$-0.440270\pi$
$823$	5.96646e9	0.373093	0.186547	+	0.982446i	$0.440270\pi$
$824$	0	0
$825$	4.78553e9	0.296716
$826$	0	0
$827$	− 2.80401e10i	− 1.72389i	−0.506999	−	0.861946i	$-0.669245\pi$
$827$	− 2.80401e10i	− 1.72389i	0.506999	−	0.861946i	$-0.330755\pi$
$828$	0	0
$829$	− 1.39964e10i	− 0.853249i	−0.904429	−	0.426625i	$-0.859703\pi$
$829$	− 1.39964e10i	− 0.853249i	0.904429	−	0.426625i	$-0.140297\pi$
$830$	0	0
$831$	−1.22165e10	−0.738486
$832$	0	0
$833$	1.11688e10	0.669496
$834$	0	0
$835$	6.89074e9i	0.409603i
$836$	0	0
$837$	3.56763e9i	0.210301i
$838$	0	0
$839$	−1.20132e10	−0.702251	−0.351126	−	0.936328i	$-0.614201\pi$
$839$	−1.20132e10	−0.702251	−0.351126	+	0.936328i	$0.614201\pi$
$840$	0	0
$841$	−2.32663e10	−1.34878
$842$	0	0
$843$	− 1.01440e10i	− 0.583191i
$844$	0	0
$845$	5.67894e8i	0.0323794i
$846$	0	0
$847$	−1.47540e10	−0.834294
$848$	0	0
$849$	1.22092e10	0.684714
$850$	0	0
$851$	8.47462e9i	0.471375i
$852$	0	0
$853$	4.12903e9i	0.227786i	0.993493	+	0.113893i	$0.0363321\pi$
$853$	4.12903e9i	0.227786i	−0.993493	+	0.113893i	$0.963668\pi$
$854$	0	0
$855$	3.31355e9	0.181306
$856$	0	0
$857$	−1.29240e9	−0.0701398	−0.0350699	−	0.999385i	$-0.511165\pi$
$857$	−1.29240e9	−0.0701398	−0.0350699	+	0.999385i	$0.511165\pi$
$858$	0	0
$859$	− 2.70500e10i	− 1.45610i	−0.685525	−	0.728049i	$-0.740427\pi$
$859$	− 2.70500e10i	− 1.45610i	0.685525	−	0.728049i	$-0.259573\pi$
$860$	0	0
$861$	− 1.96968e10i	− 1.05168i
$862$	0	0
$863$	6.58584e9	0.348797	0.174399	−	0.984675i	$-0.444202\pi$
$863$	6.58584e9	0.348797	0.174399	+	0.984675i	$0.444202\pi$
$864$	0	0
$865$	4.84563e9	0.254562
$866$	0	0
$867$	− 2.72200e9i	− 0.141847i
$868$	0	0
$869$	5.67023e9i	0.293110i
$870$	0	0
$871$	2.99352e10	1.53504
$872$	0	0
$873$	−6.38955e8	−0.0325028
$874$	0	0
$875$	1.62726e10i	0.821161i
$876$	0	0
$877$	8.93451e9i	0.447272i	0.974673	+	0.223636i	$0.0717927\pi$
$877$	8.93451e9i	0.447272i	−0.974673	+	0.223636i	$0.928207\pi$
$878$	0	0
$879$	1.31515e9	0.0653154
$880$	0	0
$881$	2.43819e10	1.20130	0.600652	−	0.799511i	$-0.294908\pi$
$881$	2.43819e10	1.20130	0.600652	+	0.799511i	$0.294908\pi$
$882$	0	0
$883$	− 9.20596e9i	− 0.449994i	−0.974359	−	0.224997i	$-0.927763\pi$
$883$	− 9.20596e9i	− 0.449994i	0.974359	−	0.224997i	$-0.0722372\pi$
$884$	0	0
$885$	2.08473e9i	0.101099i
$886$	0	0
$887$	1.74880e10	0.841409	0.420705	−	0.907198i	$-0.361783\pi$
$887$	1.74880e10	0.841409	0.420705	+	0.907198i	$0.361783\pi$
$888$	0	0
$889$	2.92402e10	1.39580
$890$	0	0
$891$	1.36875e9i	0.0648267i
$892$	0	0
$893$	4.61134e10i	2.16694i
$894$	0	0
$895$	1.45731e10	0.679472
$896$	0	0
$897$	5.24529e9	0.242659
$898$	0	0
$899$	− 3.64841e10i	− 1.67473i
$900$	0	0
$901$	1.34540e10i	0.612792i
$902$	0	0
$903$	−3.65006e8	−0.0164966
$904$	0	0
$905$	−9.18387e9	−0.411866
$906$	0	0
$907$	2.40512e9i	0.107031i	0.998567	+	0.0535157i	$0.0170427\pi$
$907$	2.40512e9i	0.107031i	−0.998567	+	0.0535157i	$0.982957\pi$
$908$	0	0
$909$	2.81538e9i	0.124326i
$910$	0	0
$911$	−3.04433e10	−1.33406	−0.667032	−	0.745029i	$-0.732436\pi$
$911$	−3.04433e10	−1.33406	−0.667032	+	0.745029i	$0.732436\pi$
$912$	0	0
$913$	−5.84414e9	−0.254140
$914$	0	0
$915$	− 1.24922e8i	− 0.00539097i
$916$	0	0
$917$	− 4.08036e10i	− 1.74745i
$918$	0	0
$919$	1.68544e10	0.716325	0.358162	−	0.933659i	$-0.383403\pi$
$919$	1.68544e10	0.716325	0.358162	+	0.933659i	$0.383403\pi$
$920$	0	0
$921$	4.29669e9	0.181228
$922$	0	0
$923$	1.27061e9i	0.0531872i
$924$	0	0
$925$	− 2.48704e10i	− 1.03320i
$926$	0	0
$927$	−7.31806e9	−0.301729
$928$	0	0
$929$	3.60530e9	0.147532	0.0737660	−	0.997276i	$-0.476498\pi$
$929$	3.60530e9	0.147532	0.0737660	+	0.997276i	$0.476498\pi$
$930$	0	0
$931$	− 2.32738e10i	− 0.945245i
$932$	0	0
$933$	1.81588e10i	0.731985i
$934$	0	0
$935$	5.61790e9	0.224767
$936$	0	0
$937$	−3.91656e10	−1.55531	−0.777653	−	0.628694i	$-0.783590\pi$
$937$	−3.91656e10	−1.55531	−0.777653	+	0.628694i	$0.783590\pi$
$938$	0	0
$939$	− 4.35377e9i	− 0.171607i
$940$	0	0
$941$	3.42516e10i	1.34004i	0.742343	+	0.670020i	$0.233714\pi$
$941$	3.42516e10i	1.34004i	−0.742343	+	0.670020i	$0.766286\pi$
$942$	0	0
$943$	1.49033e10	0.578750
$944$	0	0
$945$	−2.17973e9	−0.0840215
$946$	0	0
$947$	− 1.43745e10i	− 0.550005i	−0.961443	−	0.275003i	$-0.911321\pi$
$947$	− 1.43745e10i	− 0.550005i	0.961443	−	0.275003i	$-0.0886787\pi$
$948$	0	0
$949$	4.84006e10i	1.83831i
$950$	0	0
$951$	−1.67166e9	−0.0630254
$952$	0	0
$953$	6.08466e8	0.0227725	0.0113863	−	0.999935i	$-0.496376\pi$
$953$	6.08466e8	0.0227725	0.0113863	+	0.999935i	$0.496376\pi$
$954$	0	0
$955$	8.64002e9i	0.320999i
$956$	0	0
$957$	− 1.39974e10i	− 0.516246i
$958$	0	0
$959$	−1.76796e10	−0.647303
$960$	0	0
$961$	5.34061e9	0.194115
$962$	0	0
$963$	4.60657e9i	0.166221i
$964$	0	0
$965$	− 1.28064e10i	− 0.458755i
$966$	0	0
$967$	3.23365e10	1.15001	0.575003	−	0.818151i	$-0.305001\pi$
$967$	3.23365e10	1.15001	0.575003	+	0.818151i	$0.305001\pi$
$968$	0	0
$969$	−2.87593e10	−1.01542
$970$	0	0
$971$	− 7.23682e9i	− 0.253677i	−0.991923	−	0.126838i	$-0.959517\pi$
$971$	− 7.23682e9i	− 0.253677i	0.991923	−	0.126838i	$-0.0404829\pi$
$972$	0	0
$973$	− 3.81772e10i	− 1.32865i
$974$	0	0
$975$	−1.53933e10	−0.531882
$976$	0	0
$977$	−4.52858e8	−0.0155357	−0.00776786	−	0.999970i	$-0.502473\pi$
$977$	−4.52858e8	−0.0155357	−0.00776786	+	0.999970i	$0.502473\pi$
$978$	0	0
$979$	2.32286e10i	0.791195i
$980$	0	0
$981$	− 7.91379e7i	− 0.00267635i
$982$	0	0
$983$	4.03652e10	1.35541	0.677704	−	0.735335i	$-0.262975\pi$
$983$	4.03652e10	1.35541	0.677704	+	0.735335i	$0.262975\pi$
$984$	0	0
$985$	−2.70668e9	−0.0902423
$986$	0	0
$987$	− 3.03344e10i	− 1.00421i
$988$	0	0
$989$	− 2.76177e8i	− 0.00907822i
$990$	0	0
$991$	4.32048e10	1.41018	0.705089	−	0.709119i	$-0.250907\pi$
$991$	4.32048e10	1.41018	0.705089	+	0.709119i	$0.250907\pi$
$992$	0	0
$993$	−1.88098e10	−0.609623
$994$	0	0
$995$	1.54551e10i	0.497383i
$996$	0	0
$997$	3.72293e10i	1.18974i	0.803822	+	0.594870i	$0.202796\pi$
$997$	3.72293e10i	1.18974i	−0.803822	+	0.594870i	$0.797204\pi$
$998$	0	0
$999$	7.11341e9	0.225735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.d.b.193.2	yes	6
4.3	odd	2		384.8.d.a.193.5	yes	6
8.3	odd	2		384.8.d.a.193.2	✓	6
8.5	even	2	inner	384.8.d.b.193.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.d.a.193.2	✓	6	8.3	odd	2
384.8.d.a.193.5	yes	6	4.3	odd	2
384.8.d.b.193.2	yes	6	1.1	even	1	trivial
384.8.d.b.193.5	yes	6	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-27.0000i q^{3} -96.4778i q^{5} -1147.84 q^{7} -729.000 q^{9} +O(q^{10})\)	\(q-27.0000i q^{3} -96.4778i q^{5} -1147.84 q^{7} -729.000 q^{9} +2575.55i q^{11} -8284.61i q^{13} -2604.90 q^{15} +22608.7 q^{17} -47112.8i q^{19} +30991.8i q^{21} -23449.5 q^{23} +68817.0 q^{25} +19683.0i q^{27} -201286. i q^{29} +181255. q^{31} +69540.0 q^{33} +110742. i q^{35} -361399. i q^{37} -223685. q^{39} -635548. q^{41} +11777.5i q^{43} +70332.3i q^{45} -978789. q^{47} +494003. q^{49} -610435. i q^{51} +595078. i q^{53} +248484. q^{55} -1.27204e6 q^{57} -800309. i q^{59} +47956.7i q^{61} +836778. q^{63} -799281. q^{65} +3.61335e6i q^{67} +633137. i q^{69} -153370. q^{71} -5.84223e6 q^{73} -1.85806e6i q^{75} -2.95633e6i q^{77} +2.20156e6 q^{79} +531441. q^{81} +2.26908e6i q^{83} -2.18124e6i q^{85} -5.43473e6 q^{87} +9.01888e6 q^{89} +9.50944e6i q^{91} -4.89387e6i q^{93} -4.54534e6 q^{95} +876482. q^{97} -1.87758e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 136 q^{7} - 4374 q^{9}+O(q^{10}) \) 6 * q + 136 * q^7 - 4374 * q^9	\( 6 q + 136 q^{7} - 4374 q^{9} - 6048 q^{15} - 25804 q^{17} + 12688 q^{23} - 18418 q^{25} + 220472 q^{31} - 264600 q^{33} - 591192 q^{39} - 619652 q^{41} - 326032 q^{47} - 581482 q^{49} - 4724352 q^{55} - 2429784 q^{57} - 99144 q^{63} - 9518720 q^{65} + 5841776 q^{71} - 11075196 q^{73} + 495800 q^{79} + 3188646 q^{81} + 965520 q^{87} + 9660740 q^{89} - 32750208 q^{95} - 9511564 q^{97}+O(q^{100}) \) 6 * q + 136 * q^7 - 4374 * q^9 - 6048 * q^15 - 25804 * q^17 + 12688 * q^23 - 18418 * q^25 + 220472 * q^31 - 264600 * q^33 - 591192 * q^39 - 619652 * q^41 - 326032 * q^47 - 581482 * q^49 - 4724352 * q^55 - 2429784 * q^57 - 99144 * q^63 - 9518720 * q^65 + 5841776 * q^71 - 11075196 * q^73 + 495800 * q^79 + 3188646 * q^81 + 965520 * q^87 + 9660740 * q^89 - 32750208 * q^95 - 9511564 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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