LMFDB - Embedded newform 384.8.d.f.193.3

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} - 1020 x^{14} + 7280 x^{13} + 388150 x^{12} - 2423904 x^{11} - 70542796 x^{10} + \cdots + 694045717832241 $ x^16 - 8x^15 - 1020x^14 + 7280x^13 + 388150x^12 - 2423904x^11 - 70542796x^10 + 375107560x^9 + 6894477923x^8 - 29855519152x^7 - 372292484316x^6 + 1222974198808x^5 + 10528455140834x^4 - 23130795757272x^3 - 133992602927280x^2 + 145766920335192*x + 694045717832241
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{108}\cdot 3^{28} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.3
Root			$19.4308 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.8.d.f.193.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000i q^{3} -183.972i q^{5} +284.564 q^{7} -729.000 q^{9} +O(q^{10})$	$q-27.0000i q^{3} -183.972i q^{5} +284.564 q^{7} -729.000 q^{9} -5594.99i q^{11} +14239.5i q^{13} -4967.25 q^{15} +6689.16 q^{17} -25535.9i q^{19} -7683.23i q^{21} +38266.6 q^{23} +44279.3 q^{25} +19683.0i q^{27} -159373. i q^{29} -315025. q^{31} -151065. q^{33} -52351.8i q^{35} -445274. i q^{37} +384467. q^{39} +298003. q^{41} +112165. i q^{43} +134116. i q^{45} +444602. q^{47} -742566. q^{49} -180607. i q^{51} +677212. i q^{53} -1.02932e6 q^{55} -689469. q^{57} -1.80472e6i q^{59} +513377. i q^{61} -207447. q^{63} +2.61967e6 q^{65} -3.66168e6i q^{67} -1.03320e6i q^{69} +3.21898e6 q^{71} -3.10349e6 q^{73} -1.19554e6i q^{75} -1.59213e6i q^{77} -520868. q^{79} +531441. q^{81} +2.29441e6i q^{83} -1.23062e6i q^{85} -4.30308e6 q^{87} -7.41532e6 q^{89} +4.05205e6i q^{91} +8.50569e6i q^{93} -4.69789e6 q^{95} +3.06744e6 q^{97} +4.07874e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 11664 q^{9}+O(q^{10}) $ 16 * q - 11664 * q^9	$ 16 q - 11664 q^{9} + 77280 q^{17} - 236464 q^{25} + 544320 q^{33} - 930912 q^{41} + 1163024 q^{49} - 3032640 q^{57} + 6283008 q^{65} - 2727200 q^{73} + 8503056 q^{81} - 43093152 q^{89} + 35537120 q^{97}+O(q^{100}) $ 16 * q - 11664 * q^9 + 77280 * q^17 - 236464 * q^25 + 544320 * q^33 - 930912 * q^41 + 1163024 * q^49 - 3032640 * q^57 + 6283008 * q^65 - 2727200 * q^73 + 8503056 * q^81 - 43093152 * q^89 + 35537120 * 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
$5$	− 183.972i	− 0.658199i	−0.944295	−	0.329099i	$-0.893255\pi$
$5$	− 183.972i	− 0.658199i	0.944295	−	0.329099i	$-0.106745\pi$
$6$	0	0
$7$	284.564	0.313572	0.156786	−	0.987633i	$-0.449887\pi$
$7$	284.564	0.313572	0.156786	+	0.987633i	$0.449887\pi$
$8$	0	0
$9$	−729.000	−0.333333
$10$	0	0
$11$	− 5594.99i	− 1.26743i	−0.773566	−	0.633716i	$-0.781529\pi$
$11$	− 5594.99i	− 1.26743i	0.773566	−	0.633716i	$-0.218471\pi$
$12$	0	0
$13$	14239.5i	1.79760i	0.438358	+	0.898800i	$0.355560\pi$
$13$	14239.5i	1.79760i	−0.438358	+	0.898800i	$0.644440\pi$
$14$	0	0
$15$	−4967.25	−0.380011
$16$	0	0
$17$	6689.16	0.330218	0.165109	−	0.986275i	$-0.447202\pi$
$17$	6689.16	0.330218	0.165109	+	0.986275i	$0.447202\pi$
$18$	0	0
$19$	− 25535.9i	− 0.854109i	−0.904226	−	0.427054i	$-0.859551\pi$
$19$	− 25535.9i	− 0.854109i	0.904226	−	0.427054i	$-0.140449\pi$
$20$	0	0
$21$	− 7683.23i	− 0.181041i
$22$	0	0
$23$	38266.6	0.655802	0.327901	−	0.944712i	$-0.393659\pi$
$23$	38266.6	0.655802	0.327901	+	0.944712i	$0.393659\pi$
$24$	0	0
$25$	44279.3	0.566775
$26$	0	0
$27$	19683.0i	0.192450i
$28$	0	0
$29$	− 159373.i	− 1.21345i	−0.794911	−	0.606726i	$-0.792482\pi$
$29$	− 159373.i	− 1.21345i	0.794911	−	0.606726i	$-0.207518\pi$
$30$	0	0
$31$	−315025.	−1.89924	−0.949620	−	0.313405i	$-0.898530\pi$
$31$	−315025.	−1.89924	−0.949620	+	0.313405i	$0.898530\pi$
$32$	0	0
$33$	−151065.	−0.731752
$34$	0	0
$35$	− 52351.8i	− 0.206392i
$36$	0	0
$37$	− 445274.i	− 1.44518i	−0.691278	−	0.722589i	$-0.742952\pi$
$37$	− 445274.i	− 1.44518i	0.691278	−	0.722589i	$-0.257048\pi$
$38$	0	0
$39$	384467.	1.03785
$40$	0	0
$41$	298003.	0.675268	0.337634	−	0.941277i	$-0.390373\pi$
$41$	298003.	0.675268	0.337634	+	0.941277i	$0.390373\pi$
$42$	0	0
$43$	112165.i	0.215138i	0.994198	+	0.107569i	$0.0343066\pi$
$43$	112165.i	0.215138i	−0.994198	+	0.107569i	$0.965693\pi$
$44$	0	0
$45$	134116.i	0.219400i
$46$	0	0
$47$	444602.	0.624639	0.312320	−	0.949977i	$-0.398894\pi$
$47$	444602.	0.624639	0.312320	+	0.949977i	$0.398894\pi$
$48$	0	0
$49$	−742566.	−0.901673
$50$	0	0
$51$	− 180607.i	− 0.190651i
$52$	0	0
$53$	677212.i	0.624826i	0.949946	+	0.312413i	$0.101137\pi$
$53$	677212.i	0.624826i	−0.949946	+	0.312413i	$0.898863\pi$
$54$	0	0
$55$	−1.02932e6	−0.834222
$56$	0	0
$57$	−689469.	−0.493120
$58$	0	0
$59$	− 1.80472e6i	− 1.14401i	−0.820251	−	0.572004i	$-0.806166\pi$
$59$	− 1.80472e6i	− 1.14401i	0.820251	−	0.572004i	$-0.193834\pi$
$60$	0	0
$61$	513377.i	0.289589i	0.989462	+	0.144795i	$0.0462521\pi$
$61$	513377.i	0.289589i	−0.989462	+	0.144795i	$0.953748\pi$
$62$	0	0
$63$	−207447.	−0.104524
$64$	0	0
$65$	2.61967e6	1.18318
$66$	0	0
$67$	− 3.66168e6i	− 1.48737i	−0.668530	−	0.743685i	$-0.733076\pi$
$67$	− 3.66168e6i	− 1.48737i	0.668530	−	0.743685i	$-0.266924\pi$
$68$	0	0
$69$	− 1.03320e6i	− 0.378627i
$70$	0	0
$71$	3.21898e6	1.06737	0.533684	−	0.845684i	$-0.320807\pi$
$71$	3.21898e6	1.06737	0.533684	+	0.845684i	$0.320807\pi$
$72$	0	0
$73$	−3.10349e6	−0.933727	−0.466864	−	0.884329i	$-0.654616\pi$
$73$	−3.10349e6	−0.933727	−0.466864	+	0.884329i	$0.654616\pi$
$74$	0	0
$75$	− 1.19554e6i	− 0.327227i
$76$	0	0
$77$	− 1.59213e6i	− 0.397431i
$78$	0	0
$79$	−520868.	−0.118859	−0.0594296	−	0.998232i	$-0.518928\pi$
$79$	−520868.	−0.118859	−0.0594296	+	0.998232i	$0.518928\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	2.29441e6i	0.440451i	0.975449	+	0.220226i	$0.0706793\pi$
$83$	2.29441e6i	0.440451i	−0.975449	+	0.220226i	$0.929321\pi$
$84$	0	0
$85$	− 1.23062e6i	− 0.217349i
$86$	0	0
$87$	−4.30308e6	−0.700587
$88$	0	0
$89$	−7.41532e6	−1.11497	−0.557487	−	0.830186i	$-0.688234\pi$
$89$	−7.41532e6	−1.11497	−0.557487	+	0.830186i	$0.688234\pi$
$90$	0	0
$91$	4.05205e6i	0.563676i
$92$	0	0
$93$	8.50569e6i	1.09653i
$94$	0	0
$95$	−4.69789e6	−0.562173
$96$	0	0
$97$	3.06744e6	0.341252	0.170626	−	0.985336i	$-0.445421\pi$
$97$	3.06744e6	0.341252	0.170626	+	0.985336i	$0.445421\pi$
$98$	0	0
$99$	4.07874e6i	0.422477i
$100$	0	0
$101$	1.40466e7i	1.35659i	0.734791	+	0.678294i	$0.237280\pi$
$101$	1.40466e7i	1.35659i	−0.734791	+	0.678294i	$0.762720\pi$
$102$	0	0
$103$	1.49402e7	1.34719	0.673593	−	0.739103i	$-0.264750\pi$
$103$	1.49402e7	1.34719	0.673593	+	0.739103i	$0.264750\pi$
$104$	0	0
$105$	−1.41350e6	−0.119161
$106$	0	0
$107$	− 3.93955e6i	− 0.310888i	−0.987845	−	0.155444i	$-0.950319\pi$
$107$	− 3.93955e6i	− 0.310888i	0.987845	−	0.155444i	$-0.0496808\pi$
$108$	0	0
$109$	− 42469.4i	− 0.00314111i	−0.999999	−	0.00157056i	$-0.999500\pi$
$109$	− 42469.4i	− 0.00314111i	0.999999	−	0.00157056i	$-0.000499924\pi$
$110$	0	0
$111$	−1.20224e7	−0.834374
$112$	0	0
$113$	−2.55574e7	−1.66626	−0.833129	−	0.553078i	$-0.813453\pi$
$113$	−2.55574e7	−1.66626	−0.833129	+	0.553078i	$0.813453\pi$
$114$	0	0
$115$	− 7.03999e6i	− 0.431648i
$116$	0	0
$117$	− 1.03806e7i	− 0.599200i
$118$	0	0
$119$	1.90349e6	0.103547
$120$	0	0
$121$	−1.18167e7	−0.606383
$122$	0	0
$123$	− 8.04607e6i	− 0.389866i
$124$	0	0
$125$	− 2.25190e7i	− 1.03125i
$126$	0	0
$127$	−1.68496e7	−0.729924	−0.364962	−	0.931023i	$-0.618918\pi$
$127$	−1.68496e7	−0.729924	−0.364962	+	0.931023i	$0.618918\pi$
$128$	0	0
$129$	3.02845e6	0.124210
$130$	0	0
$131$	421999.i	0.0164007i	0.999966	+	0.00820035i	$0.00261028\pi$
$131$	421999.i	0.0164007i	−0.999966	+	0.00820035i	$0.997390\pi$
$132$	0	0
$133$	− 7.26659e6i	− 0.267824i
$134$	0	0
$135$	3.62112e6	0.126670
$136$	0	0
$137$	−4.61817e7	−1.53443	−0.767217	−	0.641388i	$-0.778359\pi$
$137$	−4.61817e7	−1.53443	−0.767217	+	0.641388i	$0.778359\pi$
$138$	0	0
$139$	3.34792e7i	1.05736i	0.848821	+	0.528680i	$0.177313\pi$
$139$	3.34792e7i	1.05736i	−0.848821	+	0.528680i	$0.822687\pi$
$140$	0	0
$141$	− 1.20043e7i	− 0.360636i
$142$	0	0
$143$	7.96698e7	2.27834
$144$	0	0
$145$	−2.93203e7	−0.798693
$146$	0	0
$147$	2.00493e7i	0.520581i
$148$	0	0
$149$	− 2.65368e7i	− 0.657198i	−0.944470	−	0.328599i	$-0.893424\pi$
$149$	− 2.65368e7i	− 0.657198i	0.944470	−	0.328599i	$-0.106576\pi$
$150$	0	0
$151$	−6.94638e7	−1.64187	−0.820936	−	0.571021i	$-0.806548\pi$
$151$	−6.94638e7	−1.64187	−0.820936	+	0.571021i	$0.806548\pi$
$152$	0	0
$153$	−4.87640e6	−0.110073
$154$	0	0
$155$	5.79559e7i	1.25008i
$156$	0	0
$157$	− 2.41307e7i	− 0.497646i	−0.968549	−	0.248823i	$-0.919956\pi$
$157$	− 2.41307e7i	− 0.497646i	0.968549	−	0.248823i	$-0.0800438\pi$
$158$	0	0
$159$	1.82847e7	0.360744
$160$	0	0
$161$	1.08893e7	0.205641
$162$	0	0
$163$	− 1.36103e7i	− 0.246157i	−0.992397	−	0.123078i	$-0.960723\pi$
$163$	− 1.36103e7i	− 0.246157i	0.992397	−	0.123078i	$-0.0392766\pi$
$164$	0	0
$165$	2.77917e7i	0.481638i
$166$	0	0
$167$	−1.38336e7	−0.229841	−0.114921	−	0.993375i	$-0.536661\pi$
$167$	−1.38336e7	−0.229841	−0.114921	+	0.993375i	$0.536661\pi$
$168$	0	0
$169$	−1.40015e8	−2.23137
$170$	0	0
$171$	1.86157e7i	0.284703i
$172$	0	0
$173$	1.04444e8i	1.53364i	0.641862	+	0.766820i	$0.278162\pi$
$173$	1.04444e8i	1.53364i	−0.641862	+	0.766820i	$0.721838\pi$
$174$	0	0
$175$	1.26003e7	0.177724
$176$	0	0
$177$	−4.87275e7	−0.660493
$178$	0	0
$179$	− 7.08133e6i	− 0.0922846i	−0.998935	−	0.0461423i	$-0.985307\pi$
$179$	− 7.08133e6i	− 0.0922846i	0.998935	−	0.0461423i	$-0.0146928\pi$
$180$	0	0
$181$	1.08452e8i	1.35945i	0.733465	+	0.679727i	$0.237902\pi$
$181$	1.08452e8i	1.35945i	−0.733465	+	0.679727i	$0.762098\pi$
$182$	0	0
$183$	1.38612e7	0.167194
$184$	0	0
$185$	−8.19180e7	−0.951214
$186$	0	0
$187$	− 3.74258e7i	− 0.418528i
$188$	0	0
$189$	5.60107e6i	0.0603469i
$190$	0	0
$191$	−9.38858e7	−0.974951	−0.487476	−	0.873137i	$-0.662082\pi$
$191$	−9.38858e7	−0.974951	−0.487476	+	0.873137i	$0.662082\pi$
$192$	0	0
$193$	−1.03956e8	−1.04088	−0.520439	−	0.853899i	$-0.674232\pi$
$193$	−1.03956e8	−1.04088	−0.520439	+	0.853899i	$0.674232\pi$
$194$	0	0
$195$	− 7.07311e7i	− 0.683108i
$196$	0	0
$197$	− 1.56759e8i	− 1.46083i	−0.683002	−	0.730416i	$-0.739326\pi$
$197$	− 1.56759e8i	− 1.46083i	0.683002	−	0.730416i	$-0.260674\pi$
$198$	0	0
$199$	3.30545e7	0.297334	0.148667	−	0.988887i	$-0.452502\pi$
$199$	3.30545e7	0.297334	0.148667	+	0.988887i	$0.452502\pi$
$200$	0	0
$201$	−9.88655e7	−0.858734
$202$	0	0
$203$	− 4.53519e7i	− 0.380504i
$204$	0	0
$205$	− 5.48242e7i	− 0.444461i
$206$	0	0
$207$	−2.78964e7	−0.218601
$208$	0	0
$209$	−1.42873e8	−1.08252
$210$	0	0
$211$	− 7.58975e7i	− 0.556210i	−0.960551	−	0.278105i	$-0.910294\pi$
$211$	− 7.58975e7i	− 0.556210i	0.960551	−	0.278105i	$-0.0897064\pi$
$212$	0	0
$213$	− 8.69125e7i	− 0.616246i
$214$	0	0
$215$	2.06352e7	0.141603
$216$	0	0
$217$	−8.96449e7	−0.595547
$218$	0	0
$219$	8.37941e7i	0.539088i
$220$	0	0
$221$	9.52503e7i	0.593600i
$222$	0	0
$223$	7.24762e7	0.437652	0.218826	−	0.975764i	$-0.429777\pi$
$223$	7.24762e7	0.437652	0.218826	+	0.975764i	$0.429777\pi$
$224$	0	0
$225$	−3.22796e7	−0.188925
$226$	0	0
$227$	− 1.54924e8i	− 0.879077i	−0.898224	−	0.439539i	$-0.855142\pi$
$227$	− 1.54924e8i	− 0.879077i	0.898224	−	0.439539i	$-0.144858\pi$
$228$	0	0
$229$	1.71007e8i	0.940998i	0.882400	+	0.470499i	$0.155926\pi$
$229$	1.71007e8i	0.940998i	−0.882400	+	0.470499i	$0.844074\pi$
$230$	0	0
$231$	−4.29875e7	−0.229457
$232$	0	0
$233$	−8.41003e7	−0.435564	−0.217782	−	0.975997i	$-0.569882\pi$
$233$	−8.41003e7	−0.435564	−0.217782	+	0.975997i	$0.569882\pi$
$234$	0	0
$235$	− 8.17944e7i	− 0.411137i
$236$	0	0
$237$	1.40634e7i	0.0686234i
$238$	0	0
$239$	−6.01887e7	−0.285182	−0.142591	−	0.989782i	$-0.545543\pi$
$239$	−6.01887e7	−0.285182	−0.142591	+	0.989782i	$0.545543\pi$
$240$	0	0
$241$	−2.16773e8	−0.997575	−0.498788	−	0.866724i	$-0.666221\pi$
$241$	−2.16773e8	−0.997575	−0.498788	+	0.866724i	$0.666221\pi$
$242$	0	0
$243$	− 1.43489e7i	− 0.0641500i
$244$	0	0
$245$	1.36612e8i	0.593480i
$246$	0	0
$247$	3.63618e8	1.53535
$248$	0	0
$249$	6.19491e7	0.254295
$250$	0	0
$251$	2.90911e8i	1.16119i	0.814193	+	0.580594i	$0.197180\pi$
$251$	2.90911e8i	1.16119i	−0.814193	+	0.580594i	$0.802820\pi$
$252$	0	0
$253$	− 2.14101e8i	− 0.831184i
$254$	0	0
$255$	−3.32267e7	−0.125486
$256$	0	0
$257$	−4.97350e8	−1.82766	−0.913831	−	0.406095i	$-0.866890\pi$
$257$	−4.97350e8	−1.82766	−0.913831	+	0.406095i	$0.866890\pi$
$258$	0	0
$259$	− 1.26709e8i	− 0.453167i
$260$	0	0
$261$	1.16183e8i	0.404484i
$262$	0	0
$263$	−4.98800e8	−1.69076	−0.845378	−	0.534168i	$-0.820625\pi$
$263$	−4.98800e8	−1.69076	−0.845378	+	0.534168i	$0.820625\pi$
$264$	0	0
$265$	1.24588e8	0.411260
$266$	0	0
$267$	2.00214e8i	0.643730i
$268$	0	0
$269$	− 9.27422e7i	− 0.290499i	−0.989395	−	0.145249i	$-0.953602\pi$
$269$	− 9.27422e7i	− 0.290499i	0.989395	−	0.145249i	$-0.0463984\pi$
$270$	0	0
$271$	5.16293e7	0.157581	0.0787905	−	0.996891i	$-0.474894\pi$
$271$	5.16293e7	0.157581	0.0787905	+	0.996891i	$0.474894\pi$
$272$	0	0
$273$	1.09405e8	0.325439
$274$	0	0
$275$	− 2.47742e8i	− 0.718348i
$276$	0	0
$277$	− 3.92885e8i	− 1.11067i	−0.831625	−	0.555337i	$-0.812589\pi$
$277$	− 3.92885e8i	− 1.11067i	0.831625	−	0.555337i	$-0.187411\pi$
$278$	0	0
$279$	2.29654e8	0.633080
$280$	0	0
$281$	−1.37674e8	−0.370152	−0.185076	−	0.982724i	$-0.559253\pi$
$281$	−1.37674e8	−0.370152	−0.185076	+	0.982724i	$0.559253\pi$
$282$	0	0
$283$	− 6.74922e7i	− 0.177011i	−0.996076	−	0.0885057i	$-0.971791\pi$
$283$	− 6.74922e7i	− 0.177011i	0.996076	−	0.0885057i	$-0.0282091\pi$
$284$	0	0
$285$	1.26843e8i	0.324571i
$286$	0	0
$287$	8.48008e7	0.211745
$288$	0	0
$289$	−3.65594e8	−0.890956
$290$	0	0
$291$	− 8.28208e7i	− 0.197022i
$292$	0	0
$293$	− 3.45006e8i	− 0.801289i	−0.916233	−	0.400645i	$-0.868786\pi$
$293$	− 3.45006e8i	− 0.801289i	0.916233	−	0.400645i	$-0.131214\pi$
$294$	0	0
$295$	−3.32019e8	−0.752984
$296$	0	0
$297$	1.10126e8	0.243917
$298$	0	0
$299$	5.44898e8i	1.17887i
$300$	0	0
$301$	3.19180e7i	0.0674611i
$302$	0	0
$303$	3.79260e8	0.783226
$304$	0	0
$305$	9.44471e7	0.190607
$306$	0	0
$307$	− 9.09040e8i	− 1.79308i	−0.442968	−	0.896538i	$-0.646074\pi$
$307$	− 9.09040e8i	− 1.79308i	0.442968	−	0.896538i	$-0.353926\pi$
$308$	0	0
$309$	− 4.03387e8i	− 0.777798i
$310$	0	0
$311$	7.17772e8	1.35308	0.676542	−	0.736404i	$-0.263478\pi$
$311$	7.17772e8	1.35308	0.676542	+	0.736404i	$0.263478\pi$
$312$	0	0
$313$	−8.85927e7	−0.163303	−0.0816513	−	0.996661i	$-0.526019\pi$
$313$	−8.85927e7	−0.163303	−0.0816513	+	0.996661i	$0.526019\pi$
$314$	0	0
$315$	3.81645e7i	0.0687975i
$316$	0	0
$317$	− 4.89237e8i	− 0.862605i	−0.902207	−	0.431302i	$-0.858054\pi$
$317$	− 4.89237e8i	− 0.862605i	0.902207	−	0.431302i	$-0.141946\pi$
$318$	0	0
$319$	−8.91692e8	−1.53797
$320$	0	0
$321$	−1.06368e8	−0.179491
$322$	0	0
$323$	− 1.70814e8i	− 0.282042i
$324$	0	0
$325$	6.30515e8i	1.01883i
$326$	0	0
$327$	−1.14667e6	−0.00181352
$328$	0	0
$329$	1.26518e8	0.195869
$330$	0	0
$331$	9.88413e8i	1.49810i	0.662514	+	0.749050i	$0.269489\pi$
$331$	9.88413e8i	1.49810i	−0.662514	+	0.749050i	$0.730511\pi$
$332$	0	0
$333$	3.24605e8i	0.481726i
$334$	0	0
$335$	−6.73648e8	−0.978985
$336$	0	0
$337$	1.18765e9	1.69038	0.845188	−	0.534469i	$-0.179488\pi$
$337$	1.18765e9	1.69038	0.845188	+	0.534469i	$0.179488\pi$
$338$	0	0
$339$	6.90050e8i	0.962015i
$340$	0	0
$341$	1.76256e9i	2.40716i
$342$	0	0
$343$	−4.45658e8	−0.596311
$344$	0	0
$345$	−1.90080e8	−0.249212
$346$	0	0
$347$	1.37733e9i	1.76964i	0.465930	+	0.884821i	$0.345720\pi$
$347$	1.37733e9i	1.76964i	−0.465930	+	0.884821i	$0.654280\pi$
$348$	0	0
$349$	3.14137e8i	0.395576i	0.980245	+	0.197788i	$0.0633757\pi$
$349$	3.14137e8i	0.395576i	−0.980245	+	0.197788i	$0.936624\pi$
$350$	0	0
$351$	−2.80276e8	−0.345948
$352$	0	0
$353$	6.67291e8	0.807428	0.403714	−	0.914885i	$-0.367719\pi$
$353$	6.67291e8	0.807428	0.403714	+	0.914885i	$0.367719\pi$
$354$	0	0
$355$	− 5.92203e8i	− 0.702541i
$356$	0	0
$357$	− 5.13943e7i	− 0.0597828i
$358$	0	0
$359$	1.01929e9	1.16269	0.581347	−	0.813656i	$-0.302526\pi$
$359$	1.01929e9	1.16269	0.581347	+	0.813656i	$0.302526\pi$
$360$	0	0
$361$	2.41791e8	0.270498
$362$	0	0
$363$	3.19051e8i	0.350096i
$364$	0	0
$365$	5.70955e8i	0.614578i
$366$	0	0
$367$	1.63115e9	1.72252	0.861259	−	0.508167i	$-0.169677\pi$
$367$	1.63115e9	1.72252	0.861259	+	0.508167i	$0.169677\pi$
$368$	0	0
$369$	−2.17244e8	−0.225089
$370$	0	0
$371$	1.92710e8i	0.195928i
$372$	0	0
$373$	− 9.79337e8i	− 0.977128i	−0.872528	−	0.488564i	$-0.837521\pi$
$373$	− 9.79337e8i	− 0.977128i	0.872528	−	0.488564i	$-0.162479\pi$
$374$	0	0
$375$	−6.08012e8	−0.595392
$376$	0	0
$377$	2.26940e9	2.18130
$378$	0	0
$379$	− 7.09025e7i	− 0.0668997i	−0.999440	−	0.0334498i	$-0.989351\pi$
$379$	− 7.09025e7i	− 0.0668997i	0.999440	−	0.0334498i	$-0.0106494\pi$
$380$	0	0
$381$	4.54940e8i	0.421422i
$382$	0	0
$383$	2.89421e8	0.263230	0.131615	−	0.991301i	$-0.457984\pi$
$383$	2.89421e8	0.263230	0.131615	+	0.991301i	$0.457984\pi$
$384$	0	0
$385$	−2.92908e8	−0.261588
$386$	0	0
$387$	− 8.17680e7i	− 0.0717126i
$388$	0	0
$389$	− 1.78931e9i	− 1.54121i	−0.637314	−	0.770604i	$-0.719955\pi$
$389$	− 1.78931e9i	− 1.54121i	0.637314	−	0.770604i	$-0.280045\pi$
$390$	0	0
$391$	2.55972e8	0.216557
$392$	0	0
$393$	1.13940e7	0.00946894
$394$	0	0
$395$	9.58252e7i	0.0782330i
$396$	0	0
$397$	− 4.16405e8i	− 0.334002i	−0.985957	−	0.167001i	$-0.946592\pi$
$397$	− 4.16405e8i	− 0.334002i	0.985957	−	0.167001i	$-0.0534083\pi$
$398$	0	0
$399$	−1.96198e8	−0.154628
$400$	0	0
$401$	1.22663e9	0.949965	0.474983	−	0.879995i	$-0.342454\pi$
$401$	1.22663e9	0.949965	0.474983	+	0.879995i	$0.342454\pi$
$402$	0	0
$403$	− 4.48581e9i	− 3.41407i
$404$	0	0
$405$	− 9.77703e7i	− 0.0731332i
$406$	0	0
$407$	−2.49130e9	−1.83166
$408$	0	0
$409$	1.90034e9	1.37341	0.686704	−	0.726937i	$-0.259057\pi$
$409$	1.90034e9	1.37341	0.686704	+	0.726937i	$0.259057\pi$
$410$	0	0
$411$	1.24691e9i	0.885905i
$412$	0	0
$413$	− 5.13559e8i	− 0.358728i
$414$	0	0
$415$	4.22107e8	0.289904
$416$	0	0
$417$	9.03938e8	0.610467
$418$	0	0
$419$	− 9.70717e8i	− 0.644679i	−0.946624	−	0.322339i	$-0.895531\pi$
$419$	− 9.70717e8i	− 0.644679i	0.946624	−	0.322339i	$-0.104469\pi$
$420$	0	0
$421$	1.33738e9i	0.873509i	0.899581	+	0.436755i	$0.143872\pi$
$421$	1.33738e9i	0.873509i	−0.899581	+	0.436755i	$0.856128\pi$
$422$	0	0
$423$	−3.24115e8	−0.208213
$424$	0	0
$425$	2.96191e8	0.187159
$426$	0	0
$427$	1.46089e8i	0.0908069i
$428$	0	0
$429$	− 2.15109e9i	− 1.31540i
$430$	0	0
$431$	3.12208e9	1.87834	0.939169	−	0.343455i	$-0.111597\pi$
$431$	3.12208e9	1.87834	0.939169	+	0.343455i	$0.111597\pi$
$432$	0	0
$433$	1.31888e9	0.780724	0.390362	−	0.920661i	$-0.372350\pi$
$433$	1.31888e9	0.780724	0.390362	+	0.920661i	$0.372350\pi$
$434$	0	0
$435$	7.91647e8i	0.461125i
$436$	0	0
$437$	− 9.77172e8i	− 0.560126i
$438$	0	0
$439$	1.36568e8	0.0770410	0.0385205	−	0.999258i	$-0.487736\pi$
$439$	1.36568e8	0.0770410	0.0385205	+	0.999258i	$0.487736\pi$
$440$	0	0
$441$	5.41331e8	0.300558
$442$	0	0
$443$	1.12150e9i	0.612894i	0.951888	+	0.306447i	$0.0991402\pi$
$443$	1.12150e9i	0.612894i	−0.951888	+	0.306447i	$0.900860\pi$
$444$	0	0
$445$	1.36421e9i	0.733874i
$446$	0	0
$447$	−7.16493e8	−0.379433
$448$	0	0
$449$	3.24802e8	0.169339	0.0846693	−	0.996409i	$-0.473017\pi$
$449$	3.24802e8	0.169339	0.0846693	+	0.996409i	$0.473017\pi$
$450$	0	0
$451$	− 1.66732e9i	− 0.855857i
$452$	0	0
$453$	1.87552e9i	0.947935i
$454$	0	0
$455$	7.45464e8	0.371011
$456$	0	0
$457$	−1.60194e9	−0.785126	−0.392563	−	0.919725i	$-0.628411\pi$
$457$	−1.60194e9	−0.785126	−0.392563	+	0.919725i	$0.628411\pi$
$458$	0	0
$459$	1.31663e8i	0.0635504i
$460$	0	0
$461$	1.41415e9i	0.672268i	0.941814	+	0.336134i	$0.109120\pi$
$461$	1.41415e9i	0.672268i	−0.941814	+	0.336134i	$0.890880\pi$
$462$	0	0
$463$	−2.32759e9	−1.08986	−0.544932	−	0.838480i	$-0.683445\pi$
$463$	−2.32759e9	−1.08986	−0.544932	+	0.838480i	$0.683445\pi$
$464$	0	0
$465$	1.56481e9	0.721732
$466$	0	0
$467$	6.69027e8i	0.303973i	0.988383	+	0.151986i	$0.0485670\pi$
$467$	6.69027e8i	0.303973i	−0.988383	+	0.151986i	$0.951433\pi$
$468$	0	0
$469$	− 1.04198e9i	− 0.466397i
$470$	0	0
$471$	−6.51529e8	−0.287316
$472$	0	0
$473$	6.27560e8	0.272672
$474$	0	0
$475$	− 1.13071e9i	− 0.484087i
$476$	0	0
$477$	− 4.93688e8i	− 0.208275i
$478$	0	0
$479$	−2.55573e9	−1.06253	−0.531265	−	0.847206i	$-0.678283\pi$
$479$	−2.55573e9	−1.06253	−0.531265	+	0.847206i	$0.678283\pi$
$480$	0	0
$481$	6.34048e9	2.59785
$482$	0	0
$483$	− 2.94011e8i	− 0.118727i
$484$	0	0
$485$	− 5.64323e8i	− 0.224611i
$486$	0	0
$487$	−8.14252e8	−0.319453	−0.159727	−	0.987161i	$-0.551061\pi$
$487$	−8.14252e8	−0.319453	−0.159727	+	0.987161i	$0.551061\pi$
$488$	0	0
$489$	−3.67478e8	−0.142119
$490$	0	0
$491$	4.42000e9i	1.68514i	0.538584	+	0.842572i	$0.318960\pi$
$491$	4.42000e9i	1.68514i	−0.538584	+	0.842572i	$0.681040\pi$
$492$	0	0
$493$	− 1.06607e9i	− 0.400703i
$494$	0	0
$495$	7.50375e8	0.278074
$496$	0	0
$497$	9.16006e8	0.334696
$498$	0	0
$499$	− 5.22005e9i	− 1.88071i	−0.340188	−	0.940357i	$-0.610491\pi$
$499$	− 5.22005e9i	− 1.88071i	0.340188	−	0.940357i	$-0.389509\pi$
$500$	0	0
$501$	3.73508e8i	0.132699i
$502$	0	0
$503$	5.31577e9	1.86242	0.931212	−	0.364479i	$-0.118753\pi$
$503$	5.31577e9	1.86242	0.931212	+	0.364479i	$0.118753\pi$
$504$	0	0
$505$	2.58419e9	0.892904
$506$	0	0
$507$	3.78041e9i	1.28828i
$508$	0	0
$509$	− 2.25207e7i	− 0.00756954i	−0.999993	−	0.00378477i	$-0.998795\pi$
$509$	− 2.25207e7i	− 0.00756954i	0.999993	−	0.00378477i	$-0.00120473\pi$
$510$	0	0
$511$	−8.83140e8	−0.292790
$512$	0	0
$513$	5.02623e8	0.164373
$514$	0	0
$515$	− 2.74859e9i	− 0.886716i
$516$	0	0
$517$	− 2.48754e9i	− 0.791688i
$518$	0	0
$519$	2.82000e9	0.885448
$520$	0	0
$521$	−2.69906e9	−0.836143	−0.418071	−	0.908414i	$-0.637294\pi$
$521$	−2.69906e9	−0.836143	−0.418071	+	0.908414i	$0.637294\pi$
$522$	0	0
$523$	− 5.30973e9i	− 1.62299i	−0.584357	−	0.811497i	$-0.698653\pi$
$523$	− 5.30973e9i	− 1.62299i	0.584357	−	0.811497i	$-0.301347\pi$
$524$	0	0
$525$	− 3.40208e8i	− 0.102609i
$526$	0	0
$527$	−2.10726e9	−0.627162
$528$	0	0
$529$	−1.94049e9	−0.569924
$530$	0	0
$531$	1.31564e9i	0.381336i
$532$	0	0
$533$	4.24341e9i	1.21386i
$534$	0	0
$535$	−7.24767e8	−0.204626
$536$	0	0
$537$	−1.91196e8	−0.0532805
$538$	0	0
$539$	4.15465e9i	1.14281i
$540$	0	0
$541$	2.01180e7i	0.00546255i	0.999996	+	0.00273127i	$0.000869393\pi$
$541$	2.01180e7i	0.00546255i	−0.999996	+	0.00273127i	$0.999131\pi$
$542$	0	0
$543$	2.92822e9	0.784881
$544$	0	0
$545$	−7.81319e6	−0.00206748
$546$	0	0
$547$	− 6.92319e8i	− 0.180863i	−0.995903	−	0.0904317i	$-0.971175\pi$
$547$	− 6.92319e8i	− 0.180863i	0.995903	−	0.0904317i	$-0.0288247\pi$
$548$	0	0
$549$	− 3.74252e8i	− 0.0965297i
$550$	0	0
$551$	−4.06974e9	−1.03642
$552$	0	0
$553$	−1.48220e8	−0.0372709
$554$	0	0
$555$	2.21179e9i	0.549184i
$556$	0	0
$557$	− 2.79188e9i	− 0.684549i	−0.939600	−	0.342274i	$-0.888803\pi$
$557$	− 2.79188e9i	− 0.684549i	0.939600	−	0.342274i	$-0.111197\pi$
$558$	0	0
$559$	−1.59717e9	−0.386732
$560$	0	0
$561$	−1.01050e9	−0.241638
$562$	0	0
$563$	6.93722e8i	0.163835i	0.996639	+	0.0819173i	$0.0261043\pi$
$563$	6.93722e8i	0.163835i	−0.996639	+	0.0819173i	$0.973896\pi$
$564$	0	0
$565$	4.70185e9i	1.09673i
$566$	0	0
$567$	1.51229e8	0.0348413
$568$	0	0
$569$	4.92736e9	1.12130	0.560649	−	0.828053i	$-0.310552\pi$
$569$	4.92736e9	1.12130	0.560649	+	0.828053i	$0.310552\pi$
$570$	0	0
$571$	− 3.97256e9i	− 0.892984i	−0.894788	−	0.446492i	$-0.852673\pi$
$571$	− 3.97256e9i	− 0.892984i	0.894788	−	0.446492i	$-0.147327\pi$
$572$	0	0
$573$	2.53492e9i	0.562889i
$574$	0	0
$575$	1.69442e9	0.371692
$576$	0	0
$577$	−5.68322e9	−1.23163	−0.615814	−	0.787892i	$-0.711173\pi$
$577$	−5.68322e9	−1.23163	−0.615814	+	0.787892i	$0.711173\pi$
$578$	0	0
$579$	2.80682e9i	0.600951i
$580$	0	0
$581$	6.52906e8i	0.138113i
$582$	0	0
$583$	3.78899e9	0.791925
$584$	0	0
$585$	−1.90974e9	−0.394393
$586$	0	0
$587$	4.92999e9i	1.00603i	0.864277	+	0.503017i	$0.167777\pi$
$587$	4.92999e9i	1.00603i	−0.864277	+	0.503017i	$0.832223\pi$
$588$	0	0
$589$	8.04445e9i	1.62216i
$590$	0	0
$591$	−4.23249e9	−0.843412
$592$	0	0
$593$	−5.08495e9	−1.00137	−0.500685	−	0.865629i	$-0.666919\pi$
$593$	−5.08495e9	−1.00137	−0.500685	+	0.865629i	$0.666919\pi$
$594$	0	0
$595$	− 3.50190e8i	− 0.0681544i
$596$	0	0
$597$	− 8.92470e8i	− 0.171666i
$598$	0	0
$599$	−3.33152e9	−0.633357	−0.316679	−	0.948533i	$-0.602568\pi$
$599$	−3.33152e9	−0.633357	−0.316679	+	0.948533i	$0.602568\pi$
$600$	0	0
$601$	−5.84837e9	−1.09894	−0.549470	−	0.835513i	$-0.685170\pi$
$601$	−5.84837e9	−1.09894	−0.549470	+	0.835513i	$0.685170\pi$
$602$	0	0
$603$	2.66937e9i	0.495790i
$604$	0	0
$605$	2.17394e9i	0.399121i
$606$	0	0
$607$	9.27070e9	1.68249	0.841244	−	0.540655i	$-0.181824\pi$
$607$	9.27070e9	1.68249	0.841244	+	0.540655i	$0.181824\pi$
$608$	0	0
$609$	−1.22450e9	−0.219684
$610$	0	0
$611$	6.33092e9i	1.12285i
$612$	0	0
$613$	− 6.35417e9i	− 1.11416i	−0.830459	−	0.557080i	$-0.811922\pi$
$613$	− 6.35417e9i	− 1.11416i	0.830459	−	0.557080i	$-0.188078\pi$
$614$	0	0
$615$	−1.48025e9	−0.256610
$616$	0	0
$617$	6.20833e9	1.06409	0.532043	−	0.846718i	$-0.321425\pi$
$617$	6.20833e9	1.06409	0.532043	+	0.846718i	$0.321425\pi$
$618$	0	0
$619$	− 4.14645e9i	− 0.702682i	−0.936248	−	0.351341i	$-0.885726\pi$
$619$	− 4.14645e9i	− 0.702682i	0.936248	−	0.351341i	$-0.114274\pi$
$620$	0	0
$621$	7.53202e8i	0.126209i
$622$	0	0
$623$	−2.11013e9	−0.349624
$624$	0	0
$625$	−6.83545e8	−0.111992
$626$	0	0
$627$	3.85757e9i	0.624996i
$628$	0	0
$629$	− 2.97851e9i	− 0.477223i
$630$	0	0
$631$	2.67069e9	0.423176	0.211588	−	0.977359i	$-0.432136\pi$
$631$	2.67069e9	0.423176	0.211588	+	0.977359i	$0.432136\pi$
$632$	0	0
$633$	−2.04923e9	−0.321128
$634$	0	0
$635$	3.09986e9i	0.480435i
$636$	0	0
$637$	− 1.05738e10i	− 1.62085i
$638$	0	0
$639$	−2.34664e9	−0.355790
$640$	0	0
$641$	−9.26474e9	−1.38941	−0.694705	−	0.719295i	$-0.744465\pi$
$641$	−9.26474e9	−1.38941	−0.694705	+	0.719295i	$0.744465\pi$
$642$	0	0
$643$	− 6.02904e9i	− 0.894354i	−0.894445	−	0.447177i	$-0.852429\pi$
$643$	− 6.02904e9i	− 0.894354i	0.894445	−	0.447177i	$-0.147571\pi$
$644$	0	0
$645$	− 5.57150e8i	− 0.0817547i
$646$	0	0
$647$	−9.87910e9	−1.43401	−0.717005	−	0.697068i	$-0.754488\pi$
$647$	−9.87910e9	−1.43401	−0.717005	+	0.697068i	$0.754488\pi$
$648$	0	0
$649$	−1.00974e10	−1.44995
$650$	0	0
$651$	2.42041e9i	0.343839i
$652$	0	0
$653$	− 1.01069e10i	− 1.42044i	−0.703980	−	0.710220i	$-0.748595\pi$
$653$	− 1.01069e10i	− 1.42044i	0.703980	−	0.710220i	$-0.251405\pi$
$654$	0	0
$655$	7.76361e7	0.0107949
$656$	0	0
$657$	2.26244e9	0.311242
$658$	0	0
$659$	7.18052e9i	0.977366i	0.872461	+	0.488683i	$0.162523\pi$
$659$	7.18052e9i	0.977366i	−0.872461	+	0.488683i	$0.837477\pi$
$660$	0	0
$661$	− 1.28840e10i	− 1.73518i	−0.497277	−	0.867592i	$-0.665667\pi$
$661$	− 1.28840e10i	− 1.73518i	0.497277	−	0.867592i	$-0.334333\pi$
$662$	0	0
$663$	2.57176e9	0.342715
$664$	0	0
$665$	−1.33685e9	−0.176282
$666$	0	0
$667$	− 6.09868e9i	− 0.795784i
$668$	0	0
$669$	− 1.95686e9i	− 0.252678i
$670$	0	0
$671$	2.87234e9	0.367035
$672$	0	0
$673$	−4.98882e9	−0.630877	−0.315439	−	0.948946i	$-0.602152\pi$
$673$	−4.98882e9	−0.630877	−0.315439	+	0.948946i	$0.602152\pi$
$674$	0	0
$675$	8.71549e8i	0.109076i
$676$	0	0
$677$	1.13046e10i	1.40021i	0.714040	+	0.700105i	$0.246863\pi$
$677$	1.13046e10i	1.40021i	−0.714040	+	0.700105i	$0.753137\pi$
$678$	0	0
$679$	8.72882e8	0.107007
$680$	0	0
$681$	−4.18294e9	−0.507535
$682$	0	0
$683$	− 1.06414e10i	− 1.27798i	−0.769214	−	0.638991i	$-0.779352\pi$
$683$	− 1.06414e10i	− 1.27798i	0.769214	−	0.638991i	$-0.220648\pi$
$684$	0	0
$685$	8.49615e9i	1.00996i
$686$	0	0
$687$	4.61718e9	0.543286
$688$	0	0
$689$	−9.64317e9	−1.12319
$690$	0	0
$691$	− 1.57441e10i	− 1.81529i	−0.419743	−	0.907643i	$-0.637880\pi$
$691$	− 1.57441e10i	− 1.81529i	0.419743	−	0.907643i	$-0.362120\pi$
$692$	0	0
$693$	1.16066e9i	0.132477i
$694$	0	0
$695$	6.15924e9	0.695953
$696$	0	0
$697$	1.99339e9	0.222986
$698$	0	0
$699$	2.27071e9i	0.251473i
$700$	0	0
$701$	4.43950e9i	0.486767i	0.969930	+	0.243384i	$0.0782574\pi$
$701$	4.43950e9i	0.486767i	−0.969930	+	0.243384i	$0.921743\pi$
$702$	0	0
$703$	−1.13705e10	−1.23434
$704$	0	0
$705$	−2.20845e9	−0.237370
$706$	0	0
$707$	3.99717e9i	0.425387i
$708$	0	0
$709$	1.36199e10i	1.43520i	0.696457	+	0.717599i	$0.254759\pi$
$709$	1.36199e10i	1.43520i	−0.696457	+	0.717599i	$0.745241\pi$
$710$	0	0
$711$	3.79713e8	0.0396198
$712$	0	0
$713$	−1.20550e10	−1.24552
$714$	0	0
$715$	− 1.46570e10i	− 1.49960i
$716$	0	0
$717$	1.62510e9i	0.164650i
$718$	0	0
$719$	−9.21125e9	−0.924204	−0.462102	−	0.886827i	$-0.652905\pi$
$719$	−9.21125e9	−0.924204	−0.462102	+	0.886827i	$0.652905\pi$
$720$	0	0
$721$	4.25146e9	0.422439
$722$	0	0
$723$	5.85287e9i	0.575950i
$724$	0	0
$725$	− 7.05693e9i	− 0.687754i
$726$	0	0
$727$	−3.19112e9	−0.308015	−0.154008	−	0.988070i	$-0.549218\pi$
$727$	−3.19112e9	−0.308015	−0.154008	+	0.988070i	$0.549218\pi$
$728$	0	0
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	7.50287e8i	0.0710423i
$732$	0	0
$733$	6.29300e9i	0.590193i	0.955467	+	0.295097i	$0.0953518\pi$
$733$	6.29300e9i	0.590193i	−0.955467	+	0.295097i	$0.904648\pi$
$734$	0	0
$735$	3.68851e9	0.342646
$736$	0	0
$737$	−2.04871e10	−1.88514
$738$	0	0
$739$	− 7.41064e9i	− 0.675461i	−0.941243	−	0.337730i	$-0.890341\pi$
$739$	− 7.41064e9i	− 0.675461i	0.941243	−	0.337730i	$-0.109659\pi$
$740$	0	0
$741$	− 9.81769e9i	− 0.886433i
$742$	0	0
$743$	1.34972e10	1.20721	0.603605	−	0.797283i	$-0.293730\pi$
$743$	1.34972e10	1.20721	0.603605	+	0.797283i	$0.293730\pi$
$744$	0	0
$745$	−4.88202e9	−0.432567
$746$	0	0
$747$	− 1.67262e9i	− 0.146817i
$748$	0	0
$749$	− 1.12105e9i	− 0.0974855i
$750$	0	0
$751$	3.51489e9	0.302811	0.151406	−	0.988472i	$-0.451620\pi$
$751$	3.51489e9	0.302811	0.151406	+	0.988472i	$0.451620\pi$
$752$	0	0
$753$	7.85461e9	0.670413
$754$	0	0
$755$	1.27794e10i	1.08068i
$756$	0	0
$757$	1.76869e10i	1.48189i	0.671567	+	0.740944i	$0.265622\pi$
$757$	1.76869e10i	1.48189i	−0.671567	+	0.740944i	$0.734378\pi$
$758$	0	0
$759$	−5.78073e9	−0.479884
$760$	0	0
$761$	−5.31006e9	−0.436770	−0.218385	−	0.975863i	$-0.570079\pi$
$761$	−5.31006e9	−0.436770	−0.218385	+	0.975863i	$0.570079\pi$
$762$	0	0
$763$	− 1.20853e7i	0 0.000984964i
$764$	0	0
$765$	8.97121e8i	0.0724496i
$766$	0	0
$767$	2.56984e10	2.05647
$768$	0	0
$769$	−1.07584e10	−0.853114	−0.426557	−	0.904461i	$-0.640274\pi$
$769$	−1.07584e10	−0.853114	−0.426557	+	0.904461i	$0.640274\pi$
$770$	0	0
$771$	1.34284e10i	1.05520i
$772$	0	0
$773$	− 4.62681e9i	− 0.360291i	−0.983640	−	0.180146i	$-0.942343\pi$
$773$	− 4.62681e9i	− 0.360291i	0.983640	−	0.180146i	$-0.0576569\pi$
$774$	0	0
$775$	−1.39491e10	−1.07644
$776$	0	0
$777$	−3.42114e9	−0.261636
$778$	0	0
$779$	− 7.60976e9i	− 0.576753i
$780$	0	0
$781$	− 1.80102e10i	− 1.35282i
$782$	0	0
$783$	3.13695e9	0.233529
$784$	0	0
$785$	−4.43937e9	−0.327550
$786$	0	0
$787$	7.11869e9i	0.520581i	0.965530	+	0.260291i	$0.0838184\pi$
$787$	7.11869e9i	0.520581i	−0.965530	+	0.260291i	$0.916182\pi$
$788$	0	0
$789$	1.34676e10i	0.976159i
$790$	0	0
$791$	−7.27272e9	−0.522491
$792$	0	0
$793$	−7.31024e9	−0.520566
$794$	0	0
$795$	− 3.36388e9i	− 0.237441i
$796$	0	0
$797$	1.53742e10i	1.07569i	0.843043	+	0.537846i	$0.180762\pi$
$797$	1.53742e10i	1.07569i	−0.843043	+	0.537846i	$0.819238\pi$
$798$	0	0
$799$	2.97402e9	0.206267
$800$	0	0
$801$	5.40577e9	0.371658
$802$	0	0
$803$	1.73640e10i	1.18344i
$804$	0	0
$805$	− 2.00333e9i	− 0.135353i
$806$	0	0
$807$	−2.50404e9	−0.167720
$808$	0	0
$809$	2.11744e10	1.40602	0.703008	−	0.711182i	$-0.251840\pi$
$809$	2.11744e10	1.40602	0.703008	+	0.711182i	$0.251840\pi$
$810$	0	0
$811$	− 1.97139e10i	− 1.29778i	−0.760883	−	0.648889i	$-0.775234\pi$
$811$	− 1.97139e10i	− 1.29778i	0.760883	−	0.648889i	$-0.224766\pi$
$812$	0	0
$813$	− 1.39399e9i	− 0.0909794i
$814$	0	0
$815$	−2.50392e9	−0.162020
$816$	0	0
$817$	2.86422e9	0.183751
$818$	0	0
$819$	− 2.95394e9i	− 0.187892i
$820$	0	0
$821$	− 2.39351e10i	− 1.50951i	−0.656009	−	0.754753i	$-0.727757\pi$
$821$	− 2.39351e10i	− 1.50951i	0.656009	−	0.754753i	$-0.272243\pi$
$822$	0	0
$823$	−2.42443e10	−1.51604	−0.758020	−	0.652231i	$-0.773833\pi$
$823$	−2.42443e10	−1.51604	−0.758020	+	0.652231i	$0.773833\pi$
$824$	0	0
$825$	−6.68903e9	−0.414738
$826$	0	0
$827$	− 1.22177e10i	− 0.751137i	−0.926795	−	0.375568i	$-0.877448\pi$
$827$	− 1.22177e10i	− 0.751137i	0.926795	−	0.375568i	$-0.122552\pi$
$828$	0	0
$829$	− 2.53509e10i	− 1.54544i	−0.634747	−	0.772720i	$-0.718895\pi$
$829$	− 2.53509e10i	− 1.54544i	0.634747	−	0.772720i	$-0.281105\pi$
$830$	0	0
$831$	−1.06079e10	−0.641248
$832$	0	0
$833$	−4.96715e9	−0.297748
$834$	0	0
$835$	2.54500e9i	0.151281i
$836$	0	0
$837$	− 6.20065e9i	− 0.365509i
$838$	0	0
$839$	1.80130e10	1.05298	0.526488	−	0.850182i	$-0.323508\pi$
$839$	1.80130e10	1.05298	0.526488	+	0.850182i	$0.323508\pi$
$840$	0	0
$841$	−8.14999e9	−0.472467
$842$	0	0
$843$	3.71720e9i	0.213707i
$844$	0	0
$845$	2.57589e10i	1.46868i
$846$	0	0
$847$	−3.36261e9	−0.190145
$848$	0	0
$849$	−1.82229e9	−0.102198
$850$	0	0
$851$	− 1.70391e10i	− 0.947750i
$852$	0	0
$853$	1.29463e10i	0.714207i	0.934065	+	0.357104i	$0.116236\pi$
$853$	1.29463e10i	0.714207i	−0.934065	+	0.357104i	$0.883764\pi$
$854$	0	0
$855$	3.42476e9	0.187391
$856$	0	0
$857$	2.35856e10	1.28001	0.640006	−	0.768370i	$-0.278932\pi$
$857$	2.35856e10	1.28001	0.640006	+	0.768370i	$0.278932\pi$
$858$	0	0
$859$	− 1.37171e10i	− 0.738391i	−0.929352	−	0.369196i	$-0.879633\pi$
$859$	− 1.37171e10i	− 0.738391i	0.929352	−	0.369196i	$-0.120367\pi$
$860$	0	0
$861$	− 2.28962e9i	− 0.122251i
$862$	0	0
$863$	−2.29995e10	−1.21809	−0.609046	−	0.793135i	$-0.708448\pi$
$863$	−2.29995e10	−1.21809	−0.609046	+	0.793135i	$0.708448\pi$
$864$	0	0
$865$	1.92148e10	1.00944
$866$	0	0
$867$	9.87103e9i	0.514394i
$868$	0	0
$869$	2.91425e9i	0.150646i
$870$	0	0
$871$	5.21406e10	2.67370
$872$	0	0
$873$	−2.23616e9	−0.113751
$874$	0	0
$875$	− 6.40809e9i	− 0.323370i
$876$	0	0
$877$	2.42484e10i	1.21390i	0.794739	+	0.606952i	$0.207608\pi$
$877$	2.42484e10i	1.21390i	−0.794739	+	0.606952i	$0.792392\pi$
$878$	0	0
$879$	−9.31515e9	−0.462625
$880$	0	0
$881$	4.91329e9	0.242079	0.121039	−	0.992648i	$-0.461377\pi$
$881$	4.91329e9	0.242079	0.121039	+	0.992648i	$0.461377\pi$
$882$	0	0
$883$	2.82033e10i	1.37860i	0.724478	+	0.689298i	$0.242081\pi$
$883$	2.82033e10i	1.37860i	−0.724478	+	0.689298i	$0.757919\pi$
$884$	0	0
$885$	8.96451e9i	0.434735i
$886$	0	0
$887$	−1.26085e10	−0.606640	−0.303320	−	0.952889i	$-0.598095\pi$
$887$	−1.26085e10	−0.606640	−0.303320	+	0.952889i	$0.598095\pi$
$888$	0	0
$889$	−4.79480e9	−0.228883
$890$	0	0
$891$	− 2.97340e9i	− 0.140826i
$892$	0	0
$893$	− 1.13533e10i	− 0.533510i
$894$	0	0
$895$	−1.30277e9	−0.0607416
$896$	0	0
$897$	1.47122e10	0.680621
$898$	0	0
$899$	5.02067e10i	2.30464i
$900$	0	0
$901$	4.52998e9i	0.206329i
$902$	0	0
$903$	8.61786e8	0.0389487
$904$	0	0
$905$	1.99522e10	0.894791
$906$	0	0
$907$	− 2.92782e10i	− 1.30292i	−0.758681	−	0.651462i	$-0.774156\pi$
$907$	− 2.92782e10i	− 1.30292i	0.758681	−	0.651462i	$-0.225844\pi$
$908$	0	0
$909$	− 1.02400e10i	− 0.452196i
$910$	0	0
$911$	2.95238e9	0.129377	0.0646887	−	0.997905i	$-0.479395\pi$
$911$	2.95238e9	0.129377	0.0646887	+	0.997905i	$0.479395\pi$
$912$	0	0
$913$	1.28372e10	0.558242
$914$	0	0
$915$	− 2.55007e9i	− 0.110047i
$916$	0	0
$917$	1.20086e8i	0.00514279i
$918$	0	0
$919$	1.90924e10	0.811441	0.405720	−	0.913997i	$-0.367021\pi$
$919$	1.90924e10	0.811441	0.405720	+	0.913997i	$0.367021\pi$
$920$	0	0
$921$	−2.45441e10	−1.03523
$922$	0	0
$923$	4.58367e10i	1.91870i
$924$	0	0
$925$	− 1.97164e10i	− 0.819090i
$926$	0	0
$927$	−1.08914e10	−0.449062
$928$	0	0
$929$	1.96279e10	0.803192	0.401596	−	0.915817i	$-0.368456\pi$
$929$	1.96279e10	0.803192	0.401596	+	0.915817i	$0.368456\pi$
$930$	0	0
$931$	1.89621e10i	0.770127i
$932$	0	0
$933$	− 1.93798e10i	− 0.781204i
$934$	0	0
$935$	−6.88530e9	−0.275475
$936$	0	0
$937$	−1.31934e8	−0.00523924	−0.00261962	−	0.999997i	$-0.500834\pi$
$937$	−1.31934e8	−0.00523924	−0.00261962	+	0.999997i	$0.500834\pi$
$938$	0	0
$939$	2.39200e9i	0.0942828i
$940$	0	0
$941$	− 3.52276e10i	− 1.37822i	−0.724655	−	0.689112i	$-0.758001\pi$
$941$	− 3.52276e10i	− 1.37822i	0.724655	−	0.689112i	$-0.241999\pi$
$942$	0	0
$943$	1.14036e10	0.442842
$944$	0	0
$945$	1.03044e9	0.0397202
$946$	0	0
$947$	− 1.17267e10i	− 0.448696i	−0.974509	−	0.224348i	$-0.927975\pi$
$947$	− 1.17267e10i	− 0.448696i	0.974509	−	0.224348i	$-0.0720253\pi$
$948$	0	0
$949$	− 4.41921e10i	− 1.67847i
$950$	0	0
$951$	−1.32094e10	−0.498025
$952$	0	0
$953$	2.67093e10	0.999627	0.499813	−	0.866133i	$-0.333402\pi$
$953$	2.67093e10	0.999627	0.499813	+	0.866133i	$0.333402\pi$
$954$	0	0
$955$	1.72724e10i	0.641712i
$956$	0	0
$957$	2.40757e10i	0.887946i
$958$	0	0
$959$	−1.31416e10	−0.481155
$960$	0	0
$961$	7.17284e10	2.60711
$962$	0	0
$963$	2.87193e9i	0.103629i
$964$	0	0
$965$	1.91250e10i	0.685104i
$966$	0	0
$967$	−3.16153e10	−1.12436	−0.562179	−	0.827015i	$-0.690037\pi$
$967$	−3.16153e10	−1.12436	−0.562179	+	0.827015i	$0.690037\pi$
$968$	0	0
$969$	−4.61197e9	−0.162837
$970$	0	0
$971$	− 5.58741e10i	− 1.95859i	−0.202444	−	0.979294i	$-0.564888\pi$
$971$	− 5.58741e10i	− 1.95859i	0.202444	−	0.979294i	$-0.435112\pi$
$972$	0	0
$973$	9.52696e9i	0.331558i
$974$	0	0
$975$	1.70239e10	0.588224
$976$	0	0
$977$	−2.53125e10	−0.868369	−0.434184	−	0.900824i	$-0.642963\pi$
$977$	−2.53125e10	−0.868369	−0.434184	+	0.900824i	$0.642963\pi$
$978$	0	0
$979$	4.14886e10i	1.41315i
$980$	0	0
$981$	3.09602e7i	0.00104704i
$982$	0	0
$983$	5.54874e10	1.86319	0.931595	−	0.363497i	$-0.118417\pi$
$983$	5.54874e10	1.86319	0.931595	+	0.363497i	$0.118417\pi$
$984$	0	0
$985$	−2.88393e10	−0.961518
$986$	0	0
$987$	− 3.41598e9i	− 0.113085i
$988$	0	0
$989$	4.29216e9i	0.141088i
$990$	0	0
$991$	−1.49201e10	−0.486982	−0.243491	−	0.969903i	$-0.578293\pi$
$991$	−1.49201e10	−0.486982	−0.243491	+	0.969903i	$0.578293\pi$
$992$	0	0
$993$	2.66872e10	0.864928
$994$	0	0
$995$	− 6.08110e9i	− 0.195705i
$996$	0	0
$997$	− 4.99025e9i	− 0.159474i	−0.996816	−	0.0797369i	$-0.974592\pi$
$997$	− 4.99025e9i	− 0.159474i	0.996816	−	0.0797369i	$-0.0254080\pi$
$998$	0	0
$999$	8.76433e9	0.278125

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.d.f.193.3	✓	16
4.3	odd	2	inner	384.8.d.f.193.11	yes	16
8.3	odd	2	inner	384.8.d.f.193.6	yes	16
8.5	even	2	inner	384.8.d.f.193.14	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.d.f.193.3	✓	16	1.1	even	1	trivial
384.8.d.f.193.6	yes	16	8.3	odd	2	inner
384.8.d.f.193.11	yes	16	4.3	odd	2	inner
384.8.d.f.193.14	yes	16	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)

\(f(q)\)	\(=\)	\(q-27.0000i q^{3} -183.972i q^{5} +284.564 q^{7} -729.000 q^{9} +O(q^{10})\)	\(q-27.0000i q^{3} -183.972i q^{5} +284.564 q^{7} -729.000 q^{9} -5594.99i q^{11} +14239.5i q^{13} -4967.25 q^{15} +6689.16 q^{17} -25535.9i q^{19} -7683.23i q^{21} +38266.6 q^{23} +44279.3 q^{25} +19683.0i q^{27} -159373. i q^{29} -315025. q^{31} -151065. q^{33} -52351.8i q^{35} -445274. i q^{37} +384467. q^{39} +298003. q^{41} +112165. i q^{43} +134116. i q^{45} +444602. q^{47} -742566. q^{49} -180607. i q^{51} +677212. i q^{53} -1.02932e6 q^{55} -689469. q^{57} -1.80472e6i q^{59} +513377. i q^{61} -207447. q^{63} +2.61967e6 q^{65} -3.66168e6i q^{67} -1.03320e6i q^{69} +3.21898e6 q^{71} -3.10349e6 q^{73} -1.19554e6i q^{75} -1.59213e6i q^{77} -520868. q^{79} +531441. q^{81} +2.29441e6i q^{83} -1.23062e6i q^{85} -4.30308e6 q^{87} -7.41532e6 q^{89} +4.05205e6i q^{91} +8.50569e6i q^{93} -4.69789e6 q^{95} +3.06744e6 q^{97} +4.07874e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 11664 q^{9}+O(q^{10}) \) 16 * q - 11664 * q^9	\( 16 q - 11664 q^{9} + 77280 q^{17} - 236464 q^{25} + 544320 q^{33} - 930912 q^{41} + 1163024 q^{49} - 3032640 q^{57} + 6283008 q^{65} - 2727200 q^{73} + 8503056 q^{81} - 43093152 q^{89} + 35537120 q^{97}+O(q^{100}) \) 16 * q - 11664 * q^9 + 77280 * q^17 - 236464 * q^25 + 544320 * q^33 - 930912 * q^41 + 1163024 * q^49 - 3032640 * q^57 + 6283008 * q^65 - 2727200 * q^73 + 8503056 * q^81 - 43093152 * q^89 + 35537120 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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