LMFDB - Embedded newform 384.9.b.a.319.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,9,Mod(319,384)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("384.319");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	384.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$156.433386263$
Analytic rank:	$0$
Dimension:	$8$
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} - 4x^{7} + 200x^{6} - 586x^{5} + 10949x^{4} - 20926x^{3} + 78946x^{2} - 68580x + 222900 $ x^8 - 4x^7 + 200x^6 - 586x^5 + 10949x^4 - 20926x^3 + 78946x^2 - 68580*x + 222900
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.3
Root			$1.36603 - 2.01178i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.9.b.a.319.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-46.7654 q^{3} +402.176i q^{5} -1935.65i q^{7} +2187.00 q^{9} +O(q^{10})$	$q-46.7654 q^{3} +402.176i q^{5} -1935.65i q^{7} +2187.00 q^{9} -20109.3 q^{11} -43789.3i q^{13} -18807.9i q^{15} +101325. q^{17} -13935.8 q^{19} +90521.3i q^{21} -433050. i q^{23} +228879. q^{25} -102276. q^{27} -467978. i q^{29} +1.48318e6i q^{31} +940421. q^{33} +778471. q^{35} -2.57554e6i q^{37} +2.04782e6i q^{39} -3.84335e6 q^{41} -1.88197e6 q^{43} +879559. i q^{45} +5.93142e6i q^{47} +2.01807e6 q^{49} -4.73851e6 q^{51} +8.07128e6i q^{53} -8.08750e6i q^{55} +651715. q^{57} +1.78704e7 q^{59} -2.11682e7i q^{61} -4.23326e6i q^{63} +1.76110e7 q^{65} -7.11867e6 q^{67} +2.02518e7i q^{69} +1.36430e7i q^{71} +3.97099e7 q^{73} -1.07036e7 q^{75} +3.89246e7i q^{77} +829701. i q^{79} +4.78297e6 q^{81} +1.45656e7 q^{83} +4.07506e7i q^{85} +2.18851e7i q^{87} +7.77943e7 q^{89} -8.47607e7 q^{91} -6.93615e7i q^{93} -5.60466e6i q^{95} -9.95872e7 q^{97} -4.39791e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 17496 q^{9}+O(q^{10}) $ 8 * q + 17496 * q^9	$ 8 q + 17496 q^{9} + 93040 q^{17} + 395912 q^{25} + 1065312 q^{33} + 2978576 q^{41} + 16144520 q^{49} + 11671776 q^{57} + 77025024 q^{65} + 191388656 q^{73} + 38263752 q^{81} + 422872432 q^{89} + 31368560 q^{97}+O(q^{100}) $ 8 * q + 17496 * q^9 + 93040 * q^17 + 395912 * q^25 + 1065312 * q^33 + 2978576 * q^41 + 16144520 * q^49 + 11671776 * q^57 + 77025024 * q^65 + 191388656 * q^73 + 38263752 * q^81 + 422872432 * q^89 + 31368560 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−46.7654	−0.577350
$3$	−46.7654	−0.577350
$4$	0	0
$5$	402.176i	0.643482i	0.946828	+	0.321741i	$0.104268\pi$
$5$	402.176i	0.643482i	−0.946828	+	0.321741i	$0.895732\pi$
$6$	0	0
$7$	− 1935.65i	− 0.806184i	−0.915159	−	0.403092i	$-0.867935\pi$
$7$	− 1935.65i	− 0.806184i	0.915159	−	0.403092i	$-0.132065\pi$
$8$	0	0
$9$	2187.00	0.333333
$10$	0	0
$11$	−20109.3	−1.37350	−0.686748	−	0.726896i	$-0.740962\pi$
$11$	−20109.3	−1.37350	−0.686748	+	0.726896i	$0.740962\pi$
$12$	0	0
$13$	− 43789.3i	− 1.53319i	−0.642134	−	0.766593i	$-0.721951\pi$
$13$	− 43789.3i	− 1.53319i	0.642134	−	0.766593i	$-0.278049\pi$
$14$	0	0
$15$	− 18807.9i	− 0.371514i
$16$	0	0
$17$	101325.	1.21317	0.606585	−	0.795018i	$-0.292539\pi$
$17$	101325.	1.21317	0.606585	+	0.795018i	$0.292539\pi$
$18$	0	0
$19$	−13935.8	−0.106935	−0.0534674	−	0.998570i	$-0.517027\pi$
$19$	−13935.8	−0.106935	−0.0534674	+	0.998570i	$0.517027\pi$
$20$	0	0
$21$	90521.3i	0.465451i
$22$	0	0
$23$	− 433050.i	− 1.54749i	−0.633499	−	0.773743i	$-0.718382\pi$
$23$	− 433050.i	− 1.54749i	0.633499	−	0.773743i	$-0.281618\pi$
$24$	0	0
$25$	228879.	0.585931
$26$	0	0
$27$	−102276.	−0.192450
$28$	0	0
$29$	− 467978.i	− 0.661657i	−0.943691	−	0.330829i	$-0.892672\pi$
$29$	− 467978.i	− 0.661657i	0.943691	−	0.330829i	$-0.107328\pi$
$30$	0	0
$31$	1.48318e6i	1.60601i	0.595975	+	0.803003i	$0.296766\pi$
$31$	1.48318e6i	1.60601i	−0.595975	+	0.803003i	$0.703234\pi$
$32$	0	0
$33$	940421.	0.792988
$34$	0	0
$35$	778471.	0.518765
$36$	0	0
$37$	− 2.57554e6i	− 1.37424i	−0.726545	−	0.687119i	$-0.758875\pi$
$37$	− 2.57554e6i	− 1.37424i	0.726545	−	0.687119i	$-0.241125\pi$
$38$	0	0
$39$	2.04782e6i	0.885185i
$40$	0	0
$41$	−3.84335e6	−1.36011	−0.680057	−	0.733160i	$-0.738045\pi$
$41$	−3.84335e6	−1.36011	−0.680057	+	0.733160i	$0.738045\pi$
$42$	0	0
$43$	−1.88197e6	−0.550475	−0.275238	−	0.961376i	$-0.588757\pi$
$43$	−1.88197e6	−0.550475	−0.275238	+	0.961376i	$0.588757\pi$
$44$	0	0
$45$	879559.i	0.214494i
$46$	0	0
$47$	5.93142e6i	1.21553i	0.794115	+	0.607767i	$0.207935\pi$
$47$	5.93142e6i	1.21553i	−0.794115	+	0.607767i	$0.792065\pi$
$48$	0	0
$49$	2.01807e6	0.350067
$50$	0	0
$51$	−4.73851e6	−0.700425
$52$	0	0
$53$	8.07128e6i	1.02291i	0.859309	+	0.511457i	$0.170894\pi$
$53$	8.07128e6i	1.02291i	−0.859309	+	0.511457i	$0.829106\pi$
$54$	0	0
$55$	− 8.08750e6i	− 0.883819i
$56$	0	0
$57$	651715.	0.0617388
$58$	0	0
$59$	1.78704e7	1.47477	0.737386	−	0.675471i	$-0.236060\pi$
$59$	1.78704e7	1.47477	0.737386	+	0.675471i	$0.236060\pi$
$60$	0	0
$61$	− 2.11682e7i	− 1.52885i	−0.644715	−	0.764423i	$-0.723024\pi$
$61$	− 2.11682e7i	− 1.52885i	0.644715	−	0.764423i	$-0.276976\pi$
$62$	0	0
$63$	− 4.23326e6i	− 0.268728i
$64$	0	0
$65$	1.76110e7	0.986576
$66$	0	0
$67$	−7.11867e6	−0.353264	−0.176632	−	0.984277i	$-0.556520\pi$
$67$	−7.11867e6	−0.353264	−0.176632	+	0.984277i	$0.556520\pi$
$68$	0	0
$69$	2.02518e7i	0.893442i
$70$	0	0
$71$	1.36430e7i	0.536879i	0.963297	+	0.268439i	$0.0865079\pi$
$71$	1.36430e7i	0.536879i	−0.963297	+	0.268439i	$0.913492\pi$
$72$	0	0
$73$	3.97099e7	1.39832	0.699162	−	0.714963i	$-0.253557\pi$
$73$	3.97099e7	1.39832	0.699162	+	0.714963i	$0.253557\pi$
$74$	0	0
$75$	−1.07036e7	−0.338288
$76$	0	0
$77$	3.89246e7i	1.10729i
$78$	0	0
$79$	829701.i	0.0213017i	0.999943	+	0.0106508i	$0.00339033\pi$
$79$	829701.i	0.0213017i	−0.999943	+	0.0106508i	$0.996610\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	1.45656e7	0.306914	0.153457	−	0.988155i	$-0.450959\pi$
$83$	1.45656e7	0.306914	0.153457	+	0.988155i	$0.450959\pi$
$84$	0	0
$85$	4.07506e7i	0.780653i
$86$	0	0
$87$	2.18851e7i	0.382008i
$88$	0	0
$89$	7.77943e7	1.23990	0.619952	−	0.784640i	$-0.287152\pi$
$89$	7.77943e7	1.23990	0.619952	+	0.784640i	$0.287152\pi$
$90$	0	0
$91$	−8.47607e7	−1.23603
$92$	0	0
$93$	− 6.93615e7i	− 0.927228i
$94$	0	0
$95$	− 5.60466e6i	− 0.0688105i
$96$	0	0
$97$	−9.95872e7	−1.12491	−0.562454	−	0.826829i	$-0.690143\pi$
$97$	−9.95872e7	−1.12491	−0.562454	+	0.826829i	$0.690143\pi$
$98$	0	0
$99$	−4.39791e7	−0.457832
$100$	0	0
$101$	− 9.82839e7i	− 0.944489i	−0.881468	−	0.472244i	$-0.843444\pi$
$101$	− 9.82839e7i	− 0.944489i	0.881468	−	0.472244i	$-0.156556\pi$
$102$	0	0
$103$	− 1.83029e8i	− 1.62619i	−0.582130	−	0.813096i	$-0.697780\pi$
$103$	− 1.83029e8i	− 1.62619i	0.582130	−	0.813096i	$-0.302220\pi$
$104$	0	0
$105$	−3.64055e7	−0.299509
$106$	0	0
$107$	−4.13285e7	−0.315293	−0.157647	−	0.987496i	$-0.550391\pi$
$107$	−4.13285e7	−0.315293	−0.157647	+	0.987496i	$0.550391\pi$
$108$	0	0
$109$	− 3.44974e7i	− 0.244388i	−0.992506	−	0.122194i	$-0.961007\pi$
$109$	− 3.44974e7i	− 0.244388i	0.992506	−	0.122194i	$-0.0389930\pi$
$110$	0	0
$111$	1.20446e8i	0.793416i
$112$	0	0
$113$	−1.89315e7	−0.116110	−0.0580552	−	0.998313i	$-0.518490\pi$
$113$	−1.89315e7	−0.116110	−0.0580552	+	0.998313i	$0.518490\pi$
$114$	0	0
$115$	1.74162e8	0.995779
$116$	0	0
$117$	− 9.57672e7i	− 0.511062i
$118$	0	0
$119$	− 1.96130e8i	− 0.978039i
$120$	0	0
$121$	1.90027e8	0.886490
$122$	0	0
$123$	1.79736e8	0.785262
$124$	0	0
$125$	2.49150e8i	1.02052i
$126$	0	0
$127$	− 2.21326e8i	− 0.850779i	−0.905010	−	0.425389i	$-0.860137\pi$
$127$	− 2.21326e8i	− 0.850779i	0.905010	−	0.425389i	$-0.139863\pi$
$128$	0	0
$129$	8.80108e7	0.317817
$130$	0	0
$131$	−5.49555e8	−1.86606	−0.933030	−	0.359799i	$-0.882845\pi$
$131$	−5.49555e8	−1.86606	−0.933030	+	0.359799i	$0.882845\pi$
$132$	0	0
$133$	2.69749e7i	0.0862091i
$134$	0	0
$135$	− 4.11329e7i	− 0.123838i
$136$	0	0
$137$	−1.68597e8	−0.478595	−0.239298	−	0.970946i	$-0.576917\pi$
$137$	−1.68597e8	−0.478595	−0.239298	+	0.970946i	$0.576917\pi$
$138$	0	0
$139$	−4.01732e8	−1.07616	−0.538080	−	0.842894i	$-0.680850\pi$
$139$	−4.01732e8	−1.07616	−0.538080	+	0.842894i	$0.680850\pi$
$140$	0	0
$141$	− 2.77385e8i	− 0.701789i
$142$	0	0
$143$	8.80574e8i	2.10582i
$144$	0	0
$145$	1.88209e8	0.425764
$146$	0	0
$147$	−9.43756e7	−0.202111
$148$	0	0
$149$	4.43104e7i	0.0899002i	0.998989	+	0.0449501i	$0.0143129\pi$
$149$	4.43104e7i	0.0899002i	−0.998989	+	0.0449501i	$0.985687\pi$
$150$	0	0
$151$	7.21307e8i	1.38743i	0.720247	+	0.693717i	$0.244028\pi$
$151$	7.21307e8i	1.38743i	−0.720247	+	0.693717i	$0.755972\pi$
$152$	0	0
$153$	2.21598e8	0.404390
$154$	0	0
$155$	−5.96499e8	−1.03343
$156$	0	0
$157$	2.39344e8i	0.393934i	0.980410	+	0.196967i	$0.0631091\pi$
$157$	2.39344e8i	0.393934i	−0.980410	+	0.196967i	$0.936891\pi$
$158$	0	0
$159$	− 3.77456e8i	− 0.590579i
$160$	0	0
$161$	−8.38233e8	−1.24756
$162$	0	0
$163$	−3.87688e7	−0.0549202	−0.0274601	−	0.999623i	$-0.508742\pi$
$163$	−3.87688e7	−0.0549202	−0.0274601	+	0.999623i	$0.508742\pi$
$164$	0	0
$165$	3.78215e8i	0.510273i
$166$	0	0
$167$	− 2.25595e7i	− 0.0290044i	−0.999895	−	0.0145022i	$-0.995384\pi$
$167$	− 2.25595e7i	− 0.0290044i	0.999895	−	0.0145022i	$-0.00461635\pi$
$168$	0	0
$169$	−1.10177e9	−1.35066
$170$	0	0
$171$	−3.04777e7	−0.0356449
$172$	0	0
$173$	1.04991e9i	1.17210i	0.810273	+	0.586052i	$0.199319\pi$
$173$	1.04991e9i	1.17210i	−0.810273	+	0.586052i	$0.800681\pi$
$174$	0	0
$175$	− 4.43030e8i	− 0.472369i
$176$	0	0
$177$	−8.35714e8	−0.851460
$178$	0	0
$179$	−7.39481e8	−0.720302	−0.360151	−	0.932894i	$-0.617275\pi$
$179$	−7.39481e8	−0.720302	−0.360151	+	0.932894i	$0.617275\pi$
$180$	0	0
$181$	− 1.75347e9i	− 1.63374i	−0.576821	−	0.816870i	$-0.695707\pi$
$181$	− 1.75347e9i	− 1.63374i	0.576821	−	0.816870i	$-0.304293\pi$
$182$	0	0
$183$	9.89937e8i	0.882680i
$184$	0	0
$185$	1.03582e9	0.884296
$186$	0	0
$187$	−2.03758e9	−1.66628
$188$	0	0
$189$	1.97970e8i	0.155150i
$190$	0	0
$191$	− 1.91581e9i	− 1.43953i	−0.694219	−	0.719764i	$-0.744250\pi$
$191$	− 1.91581e9i	− 1.43953i	0.694219	−	0.719764i	$-0.255750\pi$
$192$	0	0
$193$	−1.58868e9	−1.14501	−0.572504	−	0.819902i	$-0.694028\pi$
$193$	−1.58868e9	−1.14501	−0.572504	+	0.819902i	$0.694028\pi$
$194$	0	0
$195$	−8.23585e8	−0.569600
$196$	0	0
$197$	2.10608e9i	1.39833i	0.714961	+	0.699165i	$0.246445\pi$
$197$	2.10608e9i	1.39833i	−0.714961	+	0.699165i	$0.753555\pi$
$198$	0	0
$199$	1.88646e8i	0.120291i	0.998190	+	0.0601456i	$0.0191565\pi$
$199$	1.88646e8i	0.120291i	−0.998190	+	0.0601456i	$0.980843\pi$
$200$	0	0
$201$	3.32907e8	0.203957
$202$	0	0
$203$	−9.05840e8	−0.533418
$204$	0	0
$205$	− 1.54570e9i	− 0.875208i
$206$	0	0
$207$	− 9.47081e8i	− 0.515829i
$208$	0	0
$209$	2.80241e8	0.146874
$210$	0	0
$211$	−3.09009e9	−1.55898	−0.779490	−	0.626414i	$-0.784522\pi$
$211$	−3.09009e9	−1.55898	−0.779490	+	0.626414i	$0.784522\pi$
$212$	0	0
$213$	− 6.38019e8i	− 0.309967i
$214$	0	0
$215$	− 7.56881e8i	− 0.354221i
$216$	0	0
$217$	2.87092e9	1.29474
$218$	0	0
$219$	−1.85705e9	−0.807323
$220$	0	0
$221$	− 4.43696e9i	− 1.86002i
$222$	0	0
$223$	3.32988e9i	1.34651i	0.739411	+	0.673255i	$0.235104\pi$
$223$	3.32988e9i	1.34651i	−0.739411	+	0.673255i	$0.764896\pi$
$224$	0	0
$225$	5.00559e8	0.195310
$226$	0	0
$227$	2.70537e8	0.101888	0.0509439	−	0.998702i	$-0.483777\pi$
$227$	2.70537e8	0.101888	0.0509439	+	0.998702i	$0.483777\pi$
$228$	0	0
$229$	7.89958e8i	0.287251i	0.989632	+	0.143626i	$0.0458762\pi$
$229$	7.89958e8i	0.287251i	−0.989632	+	0.143626i	$0.954124\pi$
$230$	0	0
$231$	− 1.82033e9i	− 0.639295i
$232$	0	0
$233$	1.01558e9	0.344579	0.172289	−	0.985046i	$-0.444884\pi$
$233$	1.01558e9	0.344579	0.172289	+	0.985046i	$0.444884\pi$
$234$	0	0
$235$	−2.38547e9	−0.782174
$236$	0	0
$237$	− 3.88013e7i	− 0.0122985i
$238$	0	0
$239$	1.85103e9i	0.567313i	0.958926	+	0.283656i	$0.0915475\pi$
$239$	1.85103e9i	0.567313i	−0.958926	+	0.283656i	$0.908452\pi$
$240$	0	0
$241$	9.15721e8	0.271453	0.135727	−	0.990746i	$-0.456663\pi$
$241$	9.15721e8	0.271453	0.135727	+	0.990746i	$0.456663\pi$
$242$	0	0
$243$	−2.23677e8	−0.0641500
$244$	0	0
$245$	8.11617e8i	0.225261i
$246$	0	0
$247$	6.10241e8i	0.163951i
$248$	0	0
$249$	−6.81166e8	−0.177197
$250$	0	0
$251$	−6.39220e9	−1.61048	−0.805240	−	0.592949i	$-0.797964\pi$
$251$	−6.39220e9	−1.61048	−0.805240	+	0.592949i	$0.797964\pi$
$252$	0	0
$253$	8.70836e9i	2.12547i
$254$	0	0
$255$	− 1.90572e9i	− 0.450710i
$256$	0	0
$257$	6.28995e8	0.144183	0.0720916	−	0.997398i	$-0.477033\pi$
$257$	6.28995e8	0.144183	0.0720916	+	0.997398i	$0.477033\pi$
$258$	0	0
$259$	−4.98535e9	−1.10789
$260$	0	0
$261$	− 1.02347e9i	− 0.220552i
$262$	0	0
$263$	− 4.04520e9i	− 0.845507i	−0.906245	−	0.422753i	$-0.861064\pi$
$263$	− 4.04520e9i	− 0.845507i	0.906245	−	0.422753i	$-0.138936\pi$
$264$	0	0
$265$	−3.24607e9	−0.658226
$266$	0	0
$267$	−3.63808e9	−0.715859
$268$	0	0
$269$	− 5.27436e9i	− 1.00730i	−0.863907	−	0.503652i	$-0.831989\pi$
$269$	− 5.27436e9i	− 1.00730i	0.863907	−	0.503652i	$-0.168011\pi$
$270$	0	0
$271$	− 9.68764e9i	− 1.79614i	−0.439851	−	0.898071i	$-0.644969\pi$
$271$	− 9.68764e9i	− 1.79614i	0.439851	−	0.898071i	$-0.355031\pi$
$272$	0	0
$273$	3.96387e9	0.713622
$274$	0	0
$275$	−4.60262e9	−0.804774
$276$	0	0
$277$	− 1.41820e8i	− 0.0240889i	−0.999927	−	0.0120445i	$-0.996166\pi$
$277$	− 1.41820e8i	− 0.0240889i	0.999927	−	0.0120445i	$-0.00383397\pi$
$278$	0	0
$279$	3.24371e9i	0.535335i
$280$	0	0
$281$	−5.45914e9	−0.875587	−0.437793	−	0.899076i	$-0.644240\pi$
$281$	−5.45914e9	−0.875587	−0.437793	+	0.899076i	$0.644240\pi$
$282$	0	0
$283$	−8.18396e9	−1.27590	−0.637952	−	0.770077i	$-0.720218\pi$
$283$	−8.18396e9	−1.27590	−0.637952	+	0.770077i	$0.720218\pi$
$284$	0	0
$285$	2.62104e8i	0.0397278i
$286$	0	0
$287$	7.43938e9i	1.09650i
$288$	0	0
$289$	3.29105e9	0.471783
$290$	0	0
$291$	4.65723e9	0.649466
$292$	0	0
$293$	2.39451e9i	0.324897i	0.986717	+	0.162448i	$0.0519391\pi$
$293$	2.39451e9i	0.324897i	−0.986717	+	0.162448i	$0.948061\pi$
$294$	0	0
$295$	7.18703e9i	0.948989i
$296$	0	0
$297$	2.05670e9	0.264329
$298$	0	0
$299$	−1.89630e10	−2.37258
$300$	0	0
$301$	3.64282e9i	0.443785i
$302$	0	0
$303$	4.59628e9i	0.545301i
$304$	0	0
$305$	8.51333e9	0.983784
$306$	0	0
$307$	9.66896e8	0.108849	0.0544247	−	0.998518i	$-0.482668\pi$
$307$	9.66896e8	0.108849	0.0544247	+	0.998518i	$0.482668\pi$
$308$	0	0
$309$	8.55944e9i	0.938882i
$310$	0	0
$311$	2.31046e9i	0.246978i	0.992346	+	0.123489i	$0.0394083\pi$
$311$	2.31046e9i	0.246978i	−0.992346	+	0.123489i	$0.960592\pi$
$312$	0	0
$313$	5.76591e8	0.0600745	0.0300373	−	0.999549i	$-0.490437\pi$
$313$	5.76591e8	0.0600745	0.0300373	+	0.999549i	$0.490437\pi$
$314$	0	0
$315$	1.70252e9	0.172922
$316$	0	0
$317$	3.90592e9i	0.386800i	0.981120	+	0.193400i	$0.0619515\pi$
$317$	3.90592e9i	0.386800i	−0.981120	+	0.193400i	$0.938049\pi$
$318$	0	0
$319$	9.41072e9i	0.908783i
$320$	0	0
$321$	1.93274e9	0.182035
$322$	0	0
$323$	−1.41205e9	−0.129730
$324$	0	0
$325$	− 1.00225e10i	− 0.898341i
$326$	0	0
$327$	1.61328e9i	0.141098i
$328$	0	0
$329$	1.14811e10	0.979944
$330$	0	0
$331$	5.78587e9	0.482011	0.241005	−	0.970524i	$-0.422523\pi$
$331$	5.78587e9	0.482011	0.241005	+	0.970524i	$0.422523\pi$
$332$	0	0
$333$	− 5.63271e9i	− 0.458079i
$334$	0	0
$335$	− 2.86296e9i	− 0.227319i
$336$	0	0
$337$	−1.20243e10	−0.932267	−0.466134	−	0.884714i	$-0.654353\pi$
$337$	−1.20243e10	−0.932267	−0.466134	+	0.884714i	$0.654353\pi$
$338$	0	0
$339$	8.85339e8	0.0670364
$340$	0	0
$341$	− 2.98258e10i	− 2.20584i
$342$	0	0
$343$	− 1.50649e10i	− 1.08840i
$344$	0	0
$345$	−8.14477e9	−0.574913
$346$	0	0
$347$	−6.76753e9	−0.466780	−0.233390	−	0.972383i	$-0.574982\pi$
$347$	−6.76753e9	−0.466780	−0.233390	+	0.972383i	$0.574982\pi$
$348$	0	0
$349$	− 5.68848e9i	− 0.383438i	−0.981450	−	0.191719i	$-0.938594\pi$
$349$	− 5.68848e9i	− 0.383438i	0.981450	−	0.191719i	$-0.0614062\pi$
$350$	0	0
$351$	4.47859e9i	0.295062i
$352$	0	0
$353$	1.78794e9	0.115147	0.0575736	−	0.998341i	$-0.481664\pi$
$353$	1.78794e9	0.115147	0.0575736	+	0.998341i	$0.481664\pi$
$354$	0	0
$355$	−5.48688e9	−0.345471
$356$	0	0
$357$	9.17210e9i	0.564671i
$358$	0	0
$359$	− 5.27395e9i	− 0.317511i	−0.987318	−	0.158755i	$-0.949252\pi$
$359$	− 5.27395e9i	− 0.317511i	0.987318	−	0.158755i	$-0.0507481\pi$
$360$	0	0
$361$	−1.67894e10	−0.988565
$362$	0	0
$363$	−8.88668e9	−0.511815
$364$	0	0
$365$	1.59704e10i	0.899796i
$366$	0	0
$367$	− 6.40472e9i	− 0.353050i	−0.984296	−	0.176525i	$-0.943514\pi$
$367$	− 6.40472e9i	− 0.353050i	0.984296	−	0.176525i	$-0.0564856\pi$
$368$	0	0
$369$	−8.40542e9	−0.453371
$370$	0	0
$371$	1.56232e10	0.824657
$372$	0	0
$373$	− 8.98900e9i	− 0.464383i	−0.972670	−	0.232191i	$-0.925410\pi$
$373$	− 8.98900e9i	− 0.464383i	0.972670	−	0.232191i	$-0.0745895\pi$
$374$	0	0
$375$	− 1.16516e10i	− 0.589196i
$376$	0	0
$377$	−2.04924e10	−1.01444
$378$	0	0
$379$	−3.67818e10	−1.78269	−0.891346	−	0.453324i	$-0.850238\pi$
$379$	−3.67818e10	−1.78269	−0.891346	+	0.453324i	$0.850238\pi$
$380$	0	0
$381$	1.03504e10i	0.491197i
$382$	0	0
$383$	− 1.51464e9i	− 0.0703907i	−0.999380	−	0.0351953i	$-0.988795\pi$
$383$	− 1.51464e9i	− 0.0703907i	0.999380	−	0.0351953i	$-0.0112053\pi$
$384$	0	0
$385$	−1.56546e10	−0.712521
$386$	0	0
$387$	−4.11586e9	−0.183492
$388$	0	0
$389$	3.35930e10i	1.46707i	0.679652	+	0.733534i	$0.262131\pi$
$389$	3.35930e10i	1.46707i	−0.679652	+	0.733534i	$0.737869\pi$
$390$	0	0
$391$	− 4.38789e10i	− 1.87737i
$392$	0	0
$393$	2.57001e10	1.07737
$394$	0	0
$395$	−3.33686e8	−0.0137072
$396$	0	0
$397$	− 8.19714e9i	− 0.329990i	−0.986294	−	0.164995i	$-0.947239\pi$
$397$	− 8.19714e9i	− 0.329990i	0.986294	−	0.164995i	$-0.0527608\pi$
$398$	0	0
$399$	− 1.26149e9i	− 0.0497729i
$400$	0	0
$401$	−1.33618e10	−0.516758	−0.258379	−	0.966044i	$-0.583188\pi$
$401$	−1.33618e10	−0.516758	−0.258379	+	0.966044i	$0.583188\pi$
$402$	0	0
$403$	6.49474e10	2.46230
$404$	0	0
$405$	1.92360e9i	0.0714980i
$406$	0	0
$407$	5.17925e10i	1.88751i
$408$	0	0
$409$	−4.12559e10	−1.47432	−0.737162	−	0.675716i	$-0.763835\pi$
$409$	−4.12559e10	−1.47432	−0.737162	+	0.675716i	$0.763835\pi$
$410$	0	0
$411$	7.88451e9	0.276317
$412$	0	0
$413$	− 3.45907e10i	− 1.18894i
$414$	0	0
$415$	5.85794e9i	0.197493i
$416$	0	0
$417$	1.87871e10	0.621322
$418$	0	0
$419$	5.55204e10	1.80134	0.900671	−	0.434502i	$-0.143076\pi$
$419$	5.55204e10	1.80134	0.900671	+	0.434502i	$0.143076\pi$
$420$	0	0
$421$	3.81918e10i	1.21574i	0.794036	+	0.607871i	$0.207976\pi$
$421$	3.81918e10i	1.21574i	−0.794036	+	0.607871i	$0.792024\pi$
$422$	0	0
$423$	1.29720e10i	0.405178i
$424$	0	0
$425$	2.31913e10	0.710835
$426$	0	0
$427$	−4.09741e10	−1.23253
$428$	0	0
$429$	− 4.11804e10i	− 1.21580i
$430$	0	0
$431$	− 4.58425e10i	− 1.32849i	−0.747514	−	0.664246i	$-0.768753\pi$
$431$	− 4.58425e10i	− 1.32849i	0.747514	−	0.664246i	$-0.231247\pi$
$432$	0	0
$433$	5.43662e10	1.54660	0.773299	−	0.634042i	$-0.218605\pi$
$433$	5.43662e10	1.54660	0.773299	+	0.634042i	$0.218605\pi$
$434$	0	0
$435$	−8.80168e9	−0.245815
$436$	0	0
$437$	6.03492e9i	0.165480i
$438$	0	0
$439$	6.56595e10i	1.76783i	0.467651	+	0.883913i	$0.345100\pi$
$439$	6.56595e10i	1.76783i	−0.467651	+	0.883913i	$0.654900\pi$
$440$	0	0
$441$	4.41351e9	0.116689
$442$	0	0
$443$	3.38733e10	0.879514	0.439757	−	0.898117i	$-0.355064\pi$
$443$	3.38733e10	0.879514	0.439757	+	0.898117i	$0.355064\pi$
$444$	0	0
$445$	3.12870e10i	0.797855i
$446$	0	0
$447$	− 2.07219e9i	− 0.0519039i
$448$	0	0
$449$	1.81346e10	0.446193	0.223097	−	0.974796i	$-0.428383\pi$
$449$	1.81346e10	0.446193	0.223097	+	0.974796i	$0.428383\pi$
$450$	0	0
$451$	7.72874e10	1.86811
$452$	0	0
$453$	− 3.37322e10i	− 0.801036i
$454$	0	0
$455$	− 3.40887e10i	− 0.795362i
$456$	0	0
$457$	−2.66606e10	−0.611231	−0.305615	−	0.952155i	$-0.598862\pi$
$457$	−2.66606e10	−0.611231	−0.305615	+	0.952155i	$0.598862\pi$
$458$	0	0
$459$	−1.03631e10	−0.233475
$460$	0	0
$461$	7.04638e10i	1.56014i	0.625695	+	0.780068i	$0.284816\pi$
$461$	7.04638e10i	1.56014i	−0.625695	+	0.780068i	$0.715184\pi$
$462$	0	0
$463$	4.09104e10i	0.890245i	0.895470	+	0.445123i	$0.146840\pi$
$463$	4.09104e10i	0.890245i	−0.895470	+	0.445123i	$0.853160\pi$
$464$	0	0
$465$	2.78955e10	0.596654
$466$	0	0
$467$	5.39912e10	1.13515	0.567577	−	0.823320i	$-0.307881\pi$
$467$	5.39912e10	1.13515	0.567577	+	0.823320i	$0.307881\pi$
$468$	0	0
$469$	1.37792e10i	0.284796i
$470$	0	0
$471$	− 1.11930e10i	− 0.227438i
$472$	0	0
$473$	3.78451e10	0.756075
$474$	0	0
$475$	−3.18963e9	−0.0626564
$476$	0	0
$477$	1.76519e10i	0.340971i
$478$	0	0
$479$	− 4.17582e10i	− 0.793231i	−0.917985	−	0.396616i	$-0.870185\pi$
$479$	− 4.17582e10i	− 0.793231i	0.917985	−	0.396616i	$-0.129815\pi$
$480$	0	0
$481$	−1.12781e11	−2.10696
$482$	0	0
$483$	3.92003e10	0.720279
$484$	0	0
$485$	− 4.00516e10i	− 0.723857i
$486$	0	0
$487$	5.40468e10i	0.960847i	0.877037	+	0.480423i	$0.159517\pi$
$487$	5.40468e10i	0.960847i	−0.877037	+	0.480423i	$0.840483\pi$
$488$	0	0
$489$	1.81304e9	0.0317082
$490$	0	0
$491$	−3.85347e10	−0.663019	−0.331510	−	0.943452i	$-0.607558\pi$
$491$	−3.85347e10	−0.663019	−0.331510	+	0.943452i	$0.607558\pi$
$492$	0	0
$493$	− 4.74179e10i	− 0.802703i
$494$	0	0
$495$	− 1.76874e10i	− 0.294606i
$496$	0	0
$497$	2.64080e10	0.432823
$498$	0	0
$499$	5.84419e10	0.942588	0.471294	−	0.881976i	$-0.343787\pi$
$499$	5.84419e10	0.942588	0.471294	+	0.881976i	$0.343787\pi$
$500$	0	0
$501$	1.05500e9i	0.0167457i
$502$	0	0
$503$	1.70744e10i	0.266731i	0.991067	+	0.133365i	$0.0425783\pi$
$503$	1.70744e10i	0.266731i	−0.991067	+	0.133365i	$0.957422\pi$
$504$	0	0
$505$	3.95274e10	0.607761
$506$	0	0
$507$	5.15248e10	0.779802
$508$	0	0
$509$	− 1.01105e11i	− 1.50627i	−0.657867	−	0.753134i	$-0.728541\pi$
$509$	− 1.01105e11i	− 1.50627i	0.657867	−	0.753134i	$-0.271459\pi$
$510$	0	0
$511$	− 7.68645e10i	− 1.12731i
$512$	0	0
$513$	1.42530e9	0.0205796
$514$	0	0
$515$	7.36100e10	1.04642
$516$	0	0
$517$	− 1.19277e11i	− 1.66953i
$518$	0	0
$519$	− 4.90993e10i	− 0.676715i
$520$	0	0
$521$	6.85240e10	0.930019	0.465010	−	0.885306i	$-0.346051\pi$
$521$	6.85240e10	0.930019	0.465010	+	0.885306i	$0.346051\pi$
$522$	0	0
$523$	1.17769e11	1.57407	0.787036	−	0.616907i	$-0.211615\pi$
$523$	1.17769e11	1.57407	0.787036	+	0.616907i	$0.211615\pi$
$524$	0	0
$525$	2.07185e10i	0.272722i
$526$	0	0
$527$	1.50284e11i	1.94836i
$528$	0	0
$529$	−1.09221e11	−1.39471
$530$	0	0
$531$	3.90825e10	0.491591
$532$	0	0
$533$	1.68298e11i	2.08531i
$534$	0	0
$535$	− 1.66213e10i	− 0.202885i
$536$	0	0
$537$	3.45821e10	0.415867
$538$	0	0
$539$	−4.05820e10	−0.480815
$540$	0	0
$541$	1.06808e10i	0.124685i	0.998055	+	0.0623424i	$0.0198571\pi$
$541$	1.06808e10i	0.124685i	−0.998055	+	0.0623424i	$0.980143\pi$
$542$	0	0
$543$	8.20015e10i	0.943240i
$544$	0	0
$545$	1.38740e10	0.157259
$546$	0	0
$547$	1.66204e11	1.85649	0.928245	−	0.371971i	$-0.121318\pi$
$547$	1.66204e11	1.85649	0.928245	+	0.371971i	$0.121318\pi$
$548$	0	0
$549$	− 4.62948e10i	− 0.509615i
$550$	0	0
$551$	6.52166e9i	0.0707541i
$552$	0	0
$553$	1.60601e9	0.0171731
$554$	0	0
$555$	−4.84406e10	−0.510549
$556$	0	0
$557$	7.12167e10i	0.739880i	0.929056	+	0.369940i	$0.120622\pi$
$557$	7.12167e10i	0.739880i	−0.929056	+	0.369940i	$0.879378\pi$
$558$	0	0
$559$	8.24099e10i	0.843980i
$560$	0	0
$561$	9.52884e10	0.962030
$562$	0	0
$563$	5.48999e10	0.546435	0.273217	−	0.961952i	$-0.411912\pi$
$563$	5.48999e10	0.546435	0.273217	+	0.961952i	$0.411912\pi$
$564$	0	0
$565$	− 7.61380e9i	− 0.0747149i
$566$	0	0
$567$	− 9.25815e9i	− 0.0895760i
$568$	0	0
$569$	1.65851e11	1.58222	0.791112	−	0.611671i	$-0.209502\pi$
$569$	1.65851e11	1.58222	0.791112	+	0.611671i	$0.209502\pi$
$570$	0	0
$571$	−1.77889e11	−1.67342	−0.836709	−	0.547648i	$-0.815523\pi$
$571$	−1.77889e11	−1.67342	−0.836709	+	0.547648i	$0.815523\pi$
$572$	0	0
$573$	8.95938e10i	0.831112i
$574$	0	0
$575$	− 9.91163e10i	− 0.906721i
$576$	0	0
$577$	−1.39402e11	−1.25767	−0.628834	−	0.777540i	$-0.716468\pi$
$577$	−1.39402e11	−1.25767	−0.628834	+	0.777540i	$0.716468\pi$
$578$	0	0
$579$	7.42954e10	0.661070
$580$	0	0
$581$	− 2.81939e10i	− 0.247429i
$582$	0	0
$583$	− 1.62308e11i	− 1.40497i
$584$	0	0
$585$	3.85153e10	0.328859
$586$	0	0
$587$	−8.09518e10	−0.681827	−0.340914	−	0.940095i	$-0.610736\pi$
$587$	−8.09518e10	−0.681827	−0.340914	+	0.940095i	$0.610736\pi$
$588$	0	0
$589$	− 2.06694e10i	− 0.171738i
$590$	0	0
$591$	− 9.84915e10i	− 0.807326i
$592$	0	0
$593$	1.00987e11	0.816674	0.408337	−	0.912831i	$-0.366109\pi$
$593$	1.00987e11	0.816674	0.408337	+	0.912831i	$0.366109\pi$
$594$	0	0
$595$	7.88788e10	0.629350
$596$	0	0
$597$	− 8.82208e9i	− 0.0694502i
$598$	0	0
$599$	− 1.02923e11i	− 0.799474i	−0.916630	−	0.399737i	$-0.869102\pi$
$599$	− 1.02923e11i	− 0.799474i	0.916630	−	0.399737i	$-0.130898\pi$
$600$	0	0
$601$	−2.21864e11	−1.70055	−0.850273	−	0.526342i	$-0.823563\pi$
$601$	−2.21864e11	−1.70055	−0.850273	+	0.526342i	$0.823563\pi$
$602$	0	0
$603$	−1.55685e10	−0.117755
$604$	0	0
$605$	7.64243e10i	0.570440i
$606$	0	0
$607$	5.54725e10i	0.408623i	0.978906	+	0.204311i	$0.0654955\pi$
$607$	5.54725e10i	0.408623i	−0.978906	+	0.204311i	$0.934504\pi$
$608$	0	0
$609$	4.23619e10	0.307969
$610$	0	0
$611$	2.59733e11	1.86364
$612$	0	0
$613$	1.49374e11i	1.05787i	0.848662	+	0.528935i	$0.177409\pi$
$613$	1.49374e11i	1.05787i	−0.848662	+	0.528935i	$0.822591\pi$
$614$	0	0
$615$	7.22855e10i	0.505301i
$616$	0	0
$617$	−1.45992e11	−1.00737	−0.503686	−	0.863887i	$-0.668023\pi$
$617$	−1.45992e11	−1.00737	−0.503686	+	0.863887i	$0.668023\pi$
$618$	0	0
$619$	2.29296e11	1.56183	0.780915	−	0.624637i	$-0.214753\pi$
$619$	2.29296e11	1.56183	0.780915	+	0.624637i	$0.214753\pi$
$620$	0	0
$621$	4.42906e10i	0.297814i
$622$	0	0
$623$	− 1.50582e11i	− 0.999591i
$624$	0	0
$625$	−1.07960e10	−0.0707528
$626$	0	0
$627$	−1.31056e10	−0.0847980
$628$	0	0
$629$	− 2.60967e11i	− 1.66718i
$630$	0	0
$631$	− 4.64365e10i	− 0.292915i	−0.989217	−	0.146458i	$-0.953213\pi$
$631$	− 4.64365e10i	− 0.292915i	0.989217	−	0.146458i	$-0.0467872\pi$
$632$	0	0
$633$	1.44509e11	0.900078
$634$	0	0
$635$	8.90118e10	0.547460
$636$	0	0
$637$	− 8.83697e10i	− 0.536717i
$638$	0	0
$639$	2.98372e10i	0.178960i
$640$	0	0
$641$	−2.85844e11	−1.69316	−0.846578	−	0.532264i	$-0.821341\pi$
$641$	−2.85844e11	−1.69316	−0.846578	+	0.532264i	$0.821341\pi$
$642$	0	0
$643$	−2.95497e11	−1.72866	−0.864328	−	0.502928i	$-0.832256\pi$
$643$	−2.95497e11	−1.72866	−0.864328	+	0.502928i	$0.832256\pi$
$644$	0	0
$645$	3.53958e10i	0.204509i
$646$	0	0
$647$	8.99469e10i	0.513297i	0.966505	+	0.256649i	$0.0826183\pi$
$647$	8.99469e10i	0.513297i	−0.966505	+	0.256649i	$0.917382\pi$
$648$	0	0
$649$	−3.59361e11	−2.02559
$650$	0	0
$651$	−1.34259e11	−0.747517
$652$	0	0
$653$	− 2.81927e10i	− 0.155055i	−0.996990	−	0.0775273i	$-0.975298\pi$
$653$	− 2.81927e10i	− 0.155055i	0.996990	−	0.0775273i	$-0.0247025\pi$
$654$	0	0
$655$	− 2.21018e11i	− 1.20078i
$656$	0	0
$657$	8.68456e10	0.466108
$658$	0	0
$659$	1.19224e11	0.632152	0.316076	−	0.948734i	$-0.397635\pi$
$659$	1.19224e11	0.632152	0.316076	+	0.948734i	$0.397635\pi$
$660$	0	0
$661$	2.48610e11i	1.30230i	0.758947	+	0.651152i	$0.225714\pi$
$661$	2.48610e11i	1.30230i	−0.758947	+	0.651152i	$0.774286\pi$
$662$	0	0
$663$	2.07496e11i	1.07388i
$664$	0	0
$665$	−1.08487e10	−0.0554740
$666$	0	0
$667$	−2.02658e11	−1.02391
$668$	0	0
$669$	− 1.55723e11i	− 0.777408i
$670$	0	0
$671$	4.25678e11i	2.09986i
$672$	0	0
$673$	−6.89295e10	−0.336004	−0.168002	−	0.985787i	$-0.553732\pi$
$673$	−6.89295e10	−0.336004	−0.168002	+	0.985787i	$0.553732\pi$
$674$	0	0
$675$	−2.34088e10	−0.112763
$676$	0	0
$677$	− 1.65608e11i	− 0.788365i	−0.919032	−	0.394182i	$-0.871028\pi$
$677$	− 1.65608e11i	− 0.788365i	0.919032	−	0.394182i	$-0.128972\pi$
$678$	0	0
$679$	1.92766e11i	0.906883i
$680$	0	0
$681$	−1.26517e10	−0.0588250
$682$	0	0
$683$	−2.44222e11	−1.12228	−0.561141	−	0.827720i	$-0.689638\pi$
$683$	−2.44222e11	−1.12228	−0.561141	+	0.827720i	$0.689638\pi$
$684$	0	0
$685$	− 6.78058e10i	− 0.307967i
$686$	0	0
$687$	− 3.69427e10i	− 0.165845i
$688$	0	0
$689$	3.53436e11	1.56831
$690$	0	0
$691$	7.38454e10	0.323900	0.161950	−	0.986799i	$-0.448222\pi$
$691$	7.38454e10	0.323900	0.161950	+	0.986799i	$0.448222\pi$
$692$	0	0
$693$	8.51282e10i	0.369097i
$694$	0	0
$695$	− 1.61567e11i	− 0.692490i
$696$	0	0
$697$	−3.89429e11	−1.65005
$698$	0	0
$699$	−4.74937e10	−0.198943
$700$	0	0
$701$	− 2.12728e11i	− 0.880953i	−0.897764	−	0.440476i	$-0.854810\pi$
$701$	− 2.12728e11i	− 0.880953i	0.897764	−	0.440476i	$-0.145190\pi$
$702$	0	0
$703$	3.58923e10i	0.146954i
$704$	0	0
$705$	1.11558e11	0.451588
$706$	0	0
$707$	−1.90243e11	−0.761432
$708$	0	0
$709$	1.44756e11i	0.572863i	0.958101	+	0.286432i	$0.0924691\pi$
$709$	1.44756e11i	0.572863i	−0.958101	+	0.286432i	$0.907531\pi$
$710$	0	0
$711$	1.81456e9i	0.00710055i
$712$	0	0
$713$	6.42291e11	2.48527
$714$	0	0
$715$	−3.54146e11	−1.35506
$716$	0	0
$717$	− 8.65642e10i	− 0.327538i
$718$	0	0
$719$	4.70245e11i	1.75958i	0.475363	+	0.879790i	$0.342317\pi$
$719$	4.70245e11i	1.75958i	−0.475363	+	0.879790i	$0.657683\pi$
$720$	0	0
$721$	−3.54281e11	−1.31101
$722$	0	0
$723$	−4.28240e10	−0.156724
$724$	0	0
$725$	− 1.07110e11i	− 0.387686i
$726$	0	0
$727$	− 8.92240e10i	− 0.319407i	−0.987165	−	0.159703i	$-0.948946\pi$
$727$	− 8.92240e10i	− 0.319407i	0.987165	−	0.159703i	$-0.0510537\pi$
$728$	0	0
$729$	1.04604e10	0.0370370
$730$	0	0
$731$	−1.90691e11	−0.667820
$732$	0	0
$733$	2.06258e11i	0.714487i	0.934011	+	0.357244i	$0.116283\pi$
$733$	2.06258e11i	0.714487i	−0.934011	+	0.357244i	$0.883717\pi$
$734$	0	0
$735$	− 3.79556e10i	− 0.130055i
$736$	0	0
$737$	1.43152e11	0.485207
$738$	0	0
$739$	−3.46428e10	−0.116154	−0.0580772	−	0.998312i	$-0.518497\pi$
$739$	−3.46428e10	−0.116154	−0.0580772	+	0.998312i	$0.518497\pi$
$740$	0	0
$741$	− 2.85381e10i	− 0.0946570i
$742$	0	0
$743$	− 3.92994e9i	− 0.0128953i	−0.999979	−	0.00644764i	$-0.997948\pi$
$743$	− 3.92994e9i	− 0.0128953i	0.999979	−	0.00644764i	$-0.00205236\pi$
$744$	0	0
$745$	−1.78206e10	−0.0578491
$746$	0	0
$747$	3.18550e10	0.102305
$748$	0	0
$749$	7.99975e10i	0.254184i
$750$	0	0
$751$	− 3.31276e11i	− 1.04143i	−0.853730	−	0.520716i	$-0.825665\pi$
$751$	− 3.31276e11i	− 1.04143i	0.853730	−	0.520716i	$-0.174335\pi$
$752$	0	0
$753$	2.98934e11	0.929811
$754$	0	0
$755$	−2.90092e11	−0.892789
$756$	0	0
$757$	− 6.32528e10i	− 0.192618i	−0.995351	−	0.0963089i	$-0.969296\pi$
$757$	− 6.32528e10i	− 0.192618i	0.995351	−	0.0963089i	$-0.0307037\pi$
$758$	0	0
$759$	− 4.07250e11i	− 1.22714i
$760$	0	0
$761$	2.86012e11	0.852797	0.426399	−	0.904535i	$-0.359782\pi$
$761$	2.86012e11	0.852797	0.426399	+	0.904535i	$0.359782\pi$
$762$	0	0
$763$	−6.67748e10	−0.197022
$764$	0	0
$765$	8.91215e10i	0.260218i
$766$	0	0
$767$	− 7.82530e11i	− 2.26110i
$768$	0	0
$769$	−1.53604e11	−0.439235	−0.219617	−	0.975586i	$-0.570481\pi$
$769$	−1.53604e11	−0.439235	−0.219617	+	0.975586i	$0.570481\pi$
$770$	0	0
$771$	−2.94152e10	−0.0832443
$772$	0	0
$773$	2.38819e11i	0.668883i	0.942416	+	0.334442i	$0.108548\pi$
$773$	2.38819e11i	0.668883i	−0.942416	+	0.334442i	$0.891452\pi$
$774$	0	0
$775$	3.39469e11i	0.941009i
$776$	0	0
$777$	2.33142e11	0.639640
$778$	0	0
$779$	5.35604e10	0.145443
$780$	0	0
$781$	− 2.74352e11i	− 0.737400i
$782$	0	0
$783$	4.78628e10i	0.127336i
$784$	0	0
$785$	−9.62582e10	−0.253489
$786$	0	0
$787$	−2.90235e11	−0.756572	−0.378286	−	0.925689i	$-0.623486\pi$
$787$	−2.90235e11	−0.756572	−0.378286	+	0.925689i	$0.623486\pi$
$788$	0	0
$789$	1.89175e11i	0.488153i
$790$	0	0
$791$	3.66447e10i	0.0936064i
$792$	0	0
$793$	−9.26939e11	−2.34400
$794$	0	0
$795$	1.51804e11	0.380027
$796$	0	0
$797$	5.71943e11i	1.41749i	0.705465	+	0.708744i	$0.250738\pi$
$797$	5.71943e11i	1.41749i	−0.705465	+	0.708744i	$0.749262\pi$
$798$	0	0
$799$	6.01002e11i	1.47465i
$800$	0	0
$801$	1.70136e11	0.413301
$802$	0	0
$803$	−7.98541e11	−1.92059
$804$	0	0
$805$	− 3.37117e11i	− 0.802781i
$806$	0	0
$807$	2.46657e11i	0.581567i
$808$	0	0
$809$	−5.26381e10	−0.122887	−0.0614435	−	0.998111i	$-0.519570\pi$
$809$	−5.26381e10	−0.122887	−0.0614435	+	0.998111i	$0.519570\pi$
$810$	0	0
$811$	−1.17986e11	−0.272739	−0.136369	−	0.990658i	$-0.543543\pi$
$811$	−1.17986e11	−0.272739	−0.136369	+	0.990658i	$0.543543\pi$
$812$	0	0
$813$	4.53046e11i	1.03700i
$814$	0	0
$815$	− 1.55919e10i	− 0.0353402i
$816$	0	0
$817$	2.62268e10	0.0588649
$818$	0	0
$819$	−1.85372e11	−0.412010
$820$	0	0
$821$	1.94904e11i	0.428992i	0.976725	+	0.214496i	$0.0688109\pi$
$821$	1.94904e11i	0.428992i	−0.976725	+	0.214496i	$0.931189\pi$
$822$	0	0
$823$	1.25481e11i	0.273513i	0.990605	+	0.136756i	$0.0436678\pi$
$823$	1.25481e11i	0.273513i	−0.990605	+	0.136756i	$0.956332\pi$
$824$	0	0
$825$	2.15243e11	0.464637
$826$	0	0
$827$	−5.47785e11	−1.17108	−0.585542	−	0.810642i	$-0.699118\pi$
$827$	−5.47785e11	−1.17108	−0.585542	+	0.810642i	$0.699118\pi$
$828$	0	0
$829$	− 1.15721e11i	− 0.245015i	−0.992468	−	0.122507i	$-0.960906\pi$
$829$	− 1.15721e11i	− 0.245015i	0.992468	−	0.122507i	$-0.0390935\pi$
$830$	0	0
$831$	6.63225e9i	0.0139078i
$832$	0	0
$833$	2.04481e11	0.424691
$834$	0	0
$835$	9.07289e9	0.0186638
$836$	0	0
$837$	− 1.51694e11i	− 0.309076i
$838$	0	0
$839$	− 2.81036e11i	− 0.567172i	−0.958947	−	0.283586i	$-0.908476\pi$
$839$	− 2.81036e11i	− 0.567172i	0.958947	−	0.283586i	$-0.0915241\pi$
$840$	0	0
$841$	2.81243e11	0.562210
$842$	0	0
$843$	2.55299e11	0.505520
$844$	0	0
$845$	− 4.43106e11i	− 0.869123i
$846$	0	0
$847$	− 3.67826e11i	− 0.714674i
$848$	0	0
$849$	3.82726e11	0.736643
$850$	0	0
$851$	−1.11534e12	−2.12661
$852$	0	0
$853$	− 8.36511e11i	− 1.58007i	−0.613063	−	0.790034i	$-0.710063\pi$
$853$	− 8.36511e11i	− 1.58007i	0.613063	−	0.790034i	$-0.289937\pi$
$854$	0	0
$855$	− 1.22574e10i	− 0.0229368i
$856$	0	0
$857$	8.69801e9	0.0161249	0.00806244	−	0.999967i	$-0.497434\pi$
$857$	8.69801e9	0.0161249	0.00806244	+	0.999967i	$0.497434\pi$
$858$	0	0
$859$	−2.46238e11	−0.452254	−0.226127	−	0.974098i	$-0.572606\pi$
$859$	−2.46238e11	−0.452254	−0.226127	+	0.974098i	$0.572606\pi$
$860$	0	0
$861$	− 3.47906e11i	− 0.633066i
$862$	0	0
$863$	− 8.52034e11i	− 1.53608i	−0.640402	−	0.768040i	$-0.721232\pi$
$863$	− 8.52034e11i	− 1.53608i	0.640402	−	0.768040i	$-0.278768\pi$
$864$	0	0
$865$	−4.22247e11	−0.754228
$866$	0	0
$867$	−1.53907e11	−0.272384
$868$	0	0
$869$	− 1.66848e10i	− 0.0292577i
$870$	0	0
$871$	3.11722e11i	0.541620i
$872$	0	0
$873$	−2.17797e11	−0.374969
$874$	0	0
$875$	4.82267e11	0.822725
$876$	0	0
$877$	9.14656e11i	1.54618i	0.634298	+	0.773089i	$0.281289\pi$
$877$	9.14656e11i	1.54618i	−0.634298	+	0.773089i	$0.718711\pi$
$878$	0	0
$879$	− 1.11980e11i	− 0.187579i
$880$	0	0
$881$	6.17443e11	1.02493	0.512464	−	0.858709i	$-0.328733\pi$
$881$	6.17443e11	1.02493	0.512464	+	0.858709i	$0.328733\pi$
$882$	0	0
$883$	−6.59286e11	−1.08450	−0.542252	−	0.840216i	$-0.682428\pi$
$883$	−6.59286e11	−1.08450	−0.542252	+	0.840216i	$0.682428\pi$
$884$	0	0
$885$	− 3.36104e11i	− 0.547899i
$886$	0	0
$887$	9.49677e11i	1.53420i	0.641529	+	0.767099i	$0.278300\pi$
$887$	9.49677e11i	1.53420i	−0.641529	+	0.767099i	$0.721700\pi$
$888$	0	0
$889$	−4.28409e11	−0.685885
$890$	0	0
$891$	−9.61824e10	−0.152611
$892$	0	0
$893$	− 8.26593e10i	− 0.129983i
$894$	0	0
$895$	− 2.97401e11i	− 0.463501i
$896$	0	0
$897$	8.86810e11	1.36981
$898$	0	0
$899$	6.94095e11	1.06263
$900$	0	0
$901$	8.17824e11i	1.24097i
$902$	0	0
$903$	− 1.70358e11i	− 0.256219i
$904$	0	0
$905$	7.05202e11	1.05128
$906$	0	0
$907$	−3.18987e11	−0.471350	−0.235675	−	0.971832i	$-0.575730\pi$
$907$	−3.18987e11	−0.471350	−0.235675	+	0.971832i	$0.575730\pi$
$908$	0	0
$909$	− 2.14947e11i	− 0.314830i
$910$	0	0
$911$	1.05774e12i	1.53570i	0.640628	+	0.767852i	$0.278674\pi$
$911$	1.05774e12i	1.53570i	−0.640628	+	0.767852i	$0.721326\pi$
$912$	0	0
$913$	−2.92905e11	−0.421545
$914$	0	0
$915$	−3.98129e11	−0.567988
$916$	0	0
$917$	1.06374e12i	1.50439i
$918$	0	0
$919$	− 1.29009e12i	− 1.80867i	−0.426828	−	0.904333i	$-0.640369\pi$
$919$	− 1.29009e12i	− 1.80867i	0.426828	−	0.904333i	$-0.359631\pi$
$920$	0	0
$921$	−4.52173e10	−0.0628443
$922$	0	0
$923$	5.97417e11	0.823134
$924$	0	0
$925$	− 5.89489e11i	− 0.805209i
$926$	0	0
$927$	− 4.00285e11i	− 0.542064i
$928$	0	0
$929$	−5.16906e11	−0.693983	−0.346991	−	0.937868i	$-0.612797\pi$
$929$	−5.16906e11	−0.693983	−0.346991	+	0.937868i	$0.612797\pi$
$930$	0	0
$931$	−2.81234e10	−0.0374343
$932$	0	0
$933$	− 1.08050e11i	− 0.142593i
$934$	0	0
$935$	− 8.19468e11i	− 1.07222i
$936$	0	0
$937$	−7.96069e11	−1.03274	−0.516371	−	0.856365i	$-0.672718\pi$
$937$	−7.96069e11	−1.03274	−0.516371	+	0.856365i	$0.672718\pi$
$938$	0	0
$939$	−2.69645e10	−0.0346841
$940$	0	0
$941$	− 5.93891e11i	− 0.757440i	−0.925511	−	0.378720i	$-0.876364\pi$
$941$	− 5.93891e11i	− 0.757440i	0.925511	−	0.378720i	$-0.123636\pi$
$942$	0	0
$943$	1.66437e12i	2.10476i
$944$	0	0
$945$	−7.96188e10	−0.0998363
$946$	0	0
$947$	−7.57641e10	−0.0942028	−0.0471014	−	0.998890i	$-0.514998\pi$
$947$	−7.57641e10	−0.0942028	−0.0471014	+	0.998890i	$0.514998\pi$
$948$	0	0
$949$	− 1.73887e12i	− 2.14389i
$950$	0	0
$951$	− 1.82662e11i	− 0.223319i
$952$	0	0
$953$	6.44761e11	0.781676	0.390838	−	0.920459i	$-0.372185\pi$
$953$	6.44761e11	0.781676	0.390838	+	0.920459i	$0.372185\pi$
$954$	0	0
$955$	7.70495e11	0.926309
$956$	0	0
$957$	− 4.40096e11i	− 0.524686i
$958$	0	0
$959$	3.26345e11i	0.385836i
$960$	0	0
$961$	−1.34693e12	−1.57925
$962$	0	0
$963$	−9.03854e10	−0.105098
$964$	0	0
$965$	− 6.38931e11i	− 0.736791i
$966$	0	0
$967$	1.70360e11i	0.194833i	0.995244	+	0.0974166i	$0.0310579\pi$
$967$	1.70360e11i	0.194833i	−0.995244	+	0.0974166i	$0.968942\pi$
$968$	0	0
$969$	6.60352e10	0.0748997
$970$	0	0
$971$	4.23154e11	0.476016	0.238008	−	0.971263i	$-0.423505\pi$
$971$	4.23154e11	0.476016	0.238008	+	0.971263i	$0.423505\pi$
$972$	0	0
$973$	7.77612e11i	0.867584i
$974$	0	0
$975$	4.68705e11i	0.518658i
$976$	0	0
$977$	−1.56737e12	−1.72025	−0.860127	−	0.510080i	$-0.829616\pi$
$977$	−1.56737e12	−1.72025	−0.860127	+	0.510080i	$0.829616\pi$
$978$	0	0
$979$	−1.56439e12	−1.70300
$980$	0	0
$981$	− 7.54458e10i	− 0.0814627i
$982$	0	0
$983$	9.58136e10i	0.102616i	0.998683	+	0.0513078i	$0.0163389\pi$
$983$	9.58136e10i	0.102616i	−0.998683	+	0.0513078i	$0.983661\pi$
$984$	0	0
$985$	−8.47014e11	−0.899799
$986$	0	0
$987$	−5.36920e11	−0.565771
$988$	0	0
$989$	8.14985e11i	0.851853i
$990$	0	0
$991$	6.32891e11i	0.656198i	0.944643	+	0.328099i	$0.106408\pi$
$991$	6.32891e11i	0.656198i	−0.944643	+	0.328099i	$0.893592\pi$
$992$	0	0
$993$	−2.70578e11	−0.278289
$994$	0	0
$995$	−7.58687e10	−0.0774052
$996$	0	0
$997$	1.96393e12i	1.98767i	0.110851	+	0.993837i	$0.464642\pi$
$997$	1.96393e12i	1.98767i	−0.110851	+	0.993837i	$0.535358\pi$
$998$	0	0
$999$	2.63416e11i	0.264472i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.b.a.319.3	yes	8
4.3	odd	2	inner	384.9.b.a.319.7	yes	8
8.3	odd	2	inner	384.9.b.a.319.2	✓	8
8.5	even	2	inner	384.9.b.a.319.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.b.a.319.2	✓	8	8.3	odd	2	inner
384.9.b.a.319.3	yes	8	1.1	even	1	trivial
384.9.b.a.319.6	yes	8	8.5	even	2	inner
384.9.b.a.319.7	yes	8	4.3	odd	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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	384.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-46.7654 q^{3} +402.176i q^{5} -1935.65i q^{7} +2187.00 q^{9} +O(q^{10})\)	\(q-46.7654 q^{3} +402.176i q^{5} -1935.65i q^{7} +2187.00 q^{9} -20109.3 q^{11} -43789.3i q^{13} -18807.9i q^{15} +101325. q^{17} -13935.8 q^{19} +90521.3i q^{21} -433050. i q^{23} +228879. q^{25} -102276. q^{27} -467978. i q^{29} +1.48318e6i q^{31} +940421. q^{33} +778471. q^{35} -2.57554e6i q^{37} +2.04782e6i q^{39} -3.84335e6 q^{41} -1.88197e6 q^{43} +879559. i q^{45} +5.93142e6i q^{47} +2.01807e6 q^{49} -4.73851e6 q^{51} +8.07128e6i q^{53} -8.08750e6i q^{55} +651715. q^{57} +1.78704e7 q^{59} -2.11682e7i q^{61} -4.23326e6i q^{63} +1.76110e7 q^{65} -7.11867e6 q^{67} +2.02518e7i q^{69} +1.36430e7i q^{71} +3.97099e7 q^{73} -1.07036e7 q^{75} +3.89246e7i q^{77} +829701. i q^{79} +4.78297e6 q^{81} +1.45656e7 q^{83} +4.07506e7i q^{85} +2.18851e7i q^{87} +7.77943e7 q^{89} -8.47607e7 q^{91} -6.93615e7i q^{93} -5.60466e6i q^{95} -9.95872e7 q^{97} -4.39791e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 17496 q^{9}+O(q^{10}) \) 8 * q + 17496 * q^9	\( 8 q + 17496 q^{9} + 93040 q^{17} + 395912 q^{25} + 1065312 q^{33} + 2978576 q^{41} + 16144520 q^{49} + 11671776 q^{57} + 77025024 q^{65} + 191388656 q^{73} + 38263752 q^{81} + 422872432 q^{89} + 31368560 q^{97}+O(q^{100}) \) 8 * q + 17496 * q^9 + 93040 * q^17 + 395912 * q^25 + 1065312 * q^33 + 2978576 * q^41 + 16144520 * q^49 + 11671776 * q^57 + 77025024 * q^65 + 191388656 * q^73 + 38263752 * q^81 + 422872432 * q^89 + 31368560 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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