LMFDB - Embedded newform 384.9.g.b.127.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,9,Mod(127,384)]

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, 0, 0]))

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

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

chi := DirichletCharacter("384.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	384.g (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:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.15
Character	$\chi$	$=$	384.127
Dual form			384.9.g.b.127.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-46.7654i q^{3} +425.597 q^{5} -584.708i q^{7} -2187.00 q^{9} +O(q^{10})$	$q-46.7654i q^{3} +425.597 q^{5} -584.708i q^{7} -2187.00 q^{9} +24293.3i q^{11} -23313.7 q^{13} -19903.2i q^{15} -42504.3 q^{17} +122820. i q^{19} -27344.1 q^{21} -88938.3i q^{23} -209493. q^{25} +102276. i q^{27} -333559. q^{29} -522835. i q^{31} +1.13609e6 q^{33} -248850. i q^{35} -276058. q^{37} +1.09027e6i q^{39} +1.48281e6 q^{41} -5.21926e6i q^{43} -930780. q^{45} -4.82942e6i q^{47} +5.42292e6 q^{49} +1.98773e6i q^{51} +1.03784e7 q^{53} +1.03392e7i q^{55} +5.74371e6 q^{57} -1.42321e7i q^{59} -6.91381e6 q^{61} +1.27876e6i q^{63} -9.92223e6 q^{65} +3.84110e6i q^{67} -4.15923e6 q^{69} -3.13356e7i q^{71} -1.39061e7 q^{73} +9.79700e6i q^{75} +1.42045e7 q^{77} +3.00456e7i q^{79} +4.78297e6 q^{81} -3.11053e7i q^{83} -1.80897e7 q^{85} +1.55990e7i q^{87} +1.17019e7 q^{89} +1.36317e7i q^{91} -2.44506e7 q^{93} +5.22717e7i q^{95} +8.92489e7 q^{97} -5.31295e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 1344 q^{5} - 69984 q^{9}+O(q^{10}) $ 32 * q + 1344 * q^5 - 69984 * q^9	$ 32 q + 1344 q^{5} - 69984 q^{9} + 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} + 275520 q^{29} - 2421440 q^{37} - 4374720 q^{41} - 2939328 q^{45} - 14219104 q^{49} + 6224448 q^{53} + 3100032 q^{57} + 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} - 154517760 q^{77} + 153055008 q^{81} + 384830848 q^{85} - 182669760 q^{89} - 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) $ 32 * q + 1344 * q^5 - 69984 * q^9 + 114240 * q^13 - 154560 * q^17 + 1791712 * q^25 + 275520 * q^29 - 2421440 * q^37 - 4374720 * q^41 - 2939328 * q^45 - 14219104 * q^49 + 6224448 * q^53 + 3100032 * q^57 + 13005632 * q^61 + 75175296 * q^65 - 85710400 * q^73 - 154517760 * q^77 + 153055008 * q^81 + 384830848 * q^85 - 182669760 * q^89 - 149817600 * q^93 - 149408192 * 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
$3$	− 46.7654i	− 0.577350i
$3$	− 46.7654i	− 0.577350i
$4$	0	0
$5$	425.597	0.680955	0.340477	−	0.940253i	$-0.389411\pi$
$5$	425.597	0.680955	0.340477	+	0.940253i	$0.389411\pi$
$6$	0	0
$7$	− 584.708i	− 0.243527i	−0.992559	−	0.121763i	$-0.961145\pi$
$7$	− 584.708i	− 0.243527i	0.992559	−	0.121763i	$-0.0388549\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	24293.3i	1.65927i	0.558308	+	0.829634i	$0.311451\pi$
$11$	24293.3i	1.65927i	−0.558308	+	0.829634i	$0.688549\pi$
$12$	0	0
$13$	−23313.7	−0.816277	−0.408139	−	0.912920i	$-0.633822\pi$
$13$	−23313.7	−0.816277	−0.408139	+	0.912920i	$0.633822\pi$
$14$	0	0
$15$	− 19903.2i	− 0.393149i
$16$	0	0
$17$	−42504.3	−0.508905	−0.254453	−	0.967085i	$-0.581895\pi$
$17$	−42504.3	−0.508905	−0.254453	+	0.967085i	$0.581895\pi$
$18$	0	0
$19$	122820.i	0.942441i	0.882016	+	0.471220i	$0.156186\pi$
$19$	122820.i	0.942441i	−0.882016	+	0.471220i	$0.843814\pi$
$20$	0	0
$21$	−27344.1	−0.140600
$22$	0	0
$23$	− 88938.3i	− 0.317817i	−0.987293	−	0.158909i	$-0.949202\pi$
$23$	− 88938.3i	− 0.317817i	0.987293	−	0.158909i	$-0.0507976\pi$
$24$	0	0
$25$	−209493.	−0.536301
$26$	0	0
$27$	102276.i	0.192450i
$28$	0	0
$29$	−333559.	−0.471608	−0.235804	−	0.971801i	$-0.575772\pi$
$29$	−333559.	−0.471608	−0.235804	+	0.971801i	$0.575772\pi$
$30$	0	0
$31$	− 522835.i	− 0.566133i	−0.959100	−	0.283066i	$-0.908648\pi$
$31$	− 522835.i	− 0.566133i	0.959100	−	0.283066i	$-0.0913517\pi$
$32$	0	0
$33$	1.13609e6	0.957979
$34$	0	0
$35$	− 248850.i	− 0.165831i
$36$	0	0
$37$	−276058.	−0.147297	−0.0736484	−	0.997284i	$-0.523464\pi$
$37$	−276058.	−0.147297	−0.0736484	+	0.997284i	$0.523464\pi$
$38$	0	0
$39$	1.09027e6i	0.471278i
$40$	0	0
$41$	1.48281e6	0.524748	0.262374	−	0.964966i	$-0.415495\pi$
$41$	1.48281e6	0.524748	0.262374	+	0.964966i	$0.415495\pi$
$42$	0	0
$43$	− 5.21926e6i	− 1.52663i	−0.646024	−	0.763317i	$-0.723569\pi$
$43$	− 5.21926e6i	− 1.52663i	0.646024	−	0.763317i	$-0.276431\pi$
$44$	0	0
$45$	−930780.	−0.226985
$46$	0	0
$47$	− 4.82942e6i	− 0.989700i	−0.868978	−	0.494850i	$-0.835223\pi$
$47$	− 4.82942e6i	− 0.989700i	0.868978	−	0.494850i	$-0.164777\pi$
$48$	0	0
$49$	5.42292e6	0.940695
$50$	0	0
$51$	1.98773e6i	0.293817i
$52$	0	0
$53$	1.03784e7	1.31530	0.657651	−	0.753322i	$-0.271550\pi$
$53$	1.03784e7	1.31530	0.657651	+	0.753322i	$0.271550\pi$
$54$	0	0
$55$	1.03392e7i	1.12989i
$56$	0	0
$57$	5.74371e6	0.544118
$58$	0	0
$59$	− 1.42321e7i	− 1.17452i	−0.809397	−	0.587262i	$-0.800206\pi$
$59$	− 1.42321e7i	− 1.17452i	0.809397	−	0.587262i	$-0.199794\pi$
$60$	0	0
$61$	−6.91381e6	−0.499342	−0.249671	−	0.968331i	$-0.580322\pi$
$61$	−6.91381e6	−0.499342	−0.249671	+	0.968331i	$0.580322\pi$
$62$	0	0
$63$	1.27876e6i	0.0811756i
$64$	0	0
$65$	−9.92223e6	−0.555848
$66$	0	0
$67$	3.84110e6i	0.190615i	0.995448	+	0.0953074i	$0.0303834\pi$
$67$	3.84110e6i	0.190615i	−0.995448	+	0.0953074i	$0.969617\pi$
$68$	0	0
$69$	−4.15923e6	−0.183492
$70$	0	0
$71$	− 3.13356e7i	− 1.23312i	−0.787308	−	0.616559i	$-0.788526\pi$
$71$	− 3.13356e7i	− 1.23312i	0.787308	−	0.616559i	$-0.211474\pi$
$72$	0	0
$73$	−1.39061e7	−0.489681	−0.244841	−	0.969563i	$-0.578736\pi$
$73$	−1.39061e7	−0.489681	−0.244841	+	0.969563i	$0.578736\pi$
$74$	0	0
$75$	9.79700e6i	0.309633i
$76$	0	0
$77$	1.42045e7	0.404076
$78$	0	0
$79$	3.00456e7i	0.771387i	0.922627	+	0.385693i	$0.126038\pi$
$79$	3.00456e7i	0.771387i	−0.922627	+	0.385693i	$0.873962\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	− 3.11053e7i	− 0.655424i	−0.944778	−	0.327712i	$-0.893722\pi$
$83$	− 3.11053e7i	− 0.655424i	0.944778	−	0.327712i	$-0.106278\pi$
$84$	0	0
$85$	−1.80897e7	−0.346541
$86$	0	0
$87$	1.55990e7i	0.272283i
$88$	0	0
$89$	1.17019e7	0.186507	0.0932534	−	0.995642i	$-0.470273\pi$
$89$	1.17019e7	0.186507	0.0932534	+	0.995642i	$0.470273\pi$
$90$	0	0
$91$	1.36317e7i	0.198786i
$92$	0	0
$93$	−2.44506e7	−0.326857
$94$	0	0
$95$	5.22717e7i	0.641759i
$96$	0	0
$97$	8.92489e7	1.00813	0.504064	−	0.863666i	$-0.331837\pi$
$97$	8.92489e7	1.00813	0.504064	+	0.863666i	$0.331837\pi$
$98$	0	0
$99$	− 5.31295e7i	− 0.553089i
$100$	0	0
$101$	2.05738e8	1.97710	0.988551	−	0.150888i	$-0.0482132\pi$
$101$	2.05738e8	1.97710	0.988551	+	0.150888i	$0.0482132\pi$
$102$	0	0
$103$	6.65189e7i	0.591012i	0.955341	+	0.295506i	$0.0954883\pi$
$103$	6.65189e7i	0.591012i	−0.955341	+	0.295506i	$0.904512\pi$
$104$	0	0
$105$	−1.16376e7	−0.0957424
$106$	0	0
$107$	− 3.48590e7i	− 0.265937i	−0.991120	−	0.132969i	$-0.957549\pi$
$107$	− 3.48590e7i	− 0.265937i	0.991120	−	0.132969i	$-0.0424510\pi$
$108$	0	0
$109$	2.19465e8	1.55475	0.777373	−	0.629041i	$-0.216552\pi$
$109$	2.19465e8	1.55475	0.777373	+	0.629041i	$0.216552\pi$
$110$	0	0
$111$	1.29100e7i	0.0850419i
$112$	0	0
$113$	6.36984e7	0.390674	0.195337	−	0.980736i	$-0.437420\pi$
$113$	6.36984e7	0.390674	0.195337	+	0.980736i	$0.437420\pi$
$114$	0	0
$115$	− 3.78519e7i	− 0.216419i
$116$	0	0
$117$	5.09871e7	0.272092
$118$	0	0
$119$	2.48526e7i	0.123932i
$120$	0	0
$121$	−3.75808e8	−1.75317
$122$	0	0
$123$	− 6.93443e7i	− 0.302964i
$124$	0	0
$125$	−2.55408e8	−1.04615
$126$	0	0
$127$	− 3.48584e8i	− 1.33996i	−0.742377	−	0.669982i	$-0.766302\pi$
$127$	− 3.48584e8i	− 1.33996i	0.742377	−	0.669982i	$-0.233698\pi$
$128$	0	0
$129$	−2.44081e8	−0.881403
$130$	0	0
$131$	4.97297e8i	1.68862i	0.535858	+	0.844308i	$0.319988\pi$
$131$	4.97297e8i	1.68862i	−0.535858	+	0.844308i	$0.680012\pi$
$132$	0	0
$133$	7.18138e7	0.229510
$134$	0	0
$135$	4.35283e7i	0.131050i
$136$	0	0
$137$	1.23148e8	0.349579	0.174789	−	0.984606i	$-0.444076\pi$
$137$	1.23148e8	0.349579	0.174789	+	0.984606i	$0.444076\pi$
$138$	0	0
$139$	− 4.32112e8i	− 1.15754i	−0.815490	−	0.578772i	$-0.803532\pi$
$139$	− 4.32112e8i	− 1.15754i	0.815490	−	0.578772i	$-0.196468\pi$
$140$	0	0
$141$	−2.25850e8	−0.571404
$142$	0	0
$143$	− 5.66368e8i	− 1.35442i
$144$	0	0
$145$	−1.41962e8	−0.321144
$146$	0	0
$147$	− 2.53605e8i	− 0.543110i
$148$	0	0
$149$	−2.85179e8	−0.578593	−0.289296	−	0.957240i	$-0.593421\pi$
$149$	−2.85179e8	−0.578593	−0.289296	+	0.957240i	$0.593421\pi$
$150$	0	0
$151$	− 1.67567e8i	− 0.322316i	−0.986929	−	0.161158i	$-0.948477\pi$
$151$	− 1.67567e8i	− 0.322316i	0.986929	−	0.161158i	$-0.0515228\pi$
$152$	0	0
$153$	9.29569e7	0.169635
$154$	0	0
$155$	− 2.22517e8i	− 0.385511i
$156$	0	0
$157$	1.13988e8	0.187613	0.0938063	−	0.995590i	$-0.470097\pi$
$157$	1.13988e8	0.187613	0.0938063	+	0.995590i	$0.470097\pi$
$158$	0	0
$159$	− 4.85348e8i	− 0.759390i
$160$	0	0
$161$	−5.20030e7	−0.0773971
$162$	0	0
$163$	− 9.50214e7i	− 0.134608i	−0.997733	−	0.0673040i	$-0.978560\pi$
$163$	− 9.50214e7i	− 0.134608i	0.997733	−	0.0673040i	$-0.0214397\pi$
$164$	0	0
$165$	4.83515e8	0.652340
$166$	0	0
$167$	− 2.54182e8i	− 0.326798i	−0.986560	−	0.163399i	$-0.947754\pi$
$167$	− 2.54182e8i	− 0.326798i	0.986560	−	0.163399i	$-0.0522458\pi$
$168$	0	0
$169$	−2.72202e8	−0.333691
$170$	0	0
$171$	− 2.68607e8i	− 0.314147i
$172$	0	0
$173$	9.79172e8	1.09314	0.546569	−	0.837414i	$-0.315934\pi$
$173$	9.79172e8	1.09314	0.546569	+	0.837414i	$0.315934\pi$
$174$	0	0
$175$	1.22492e8i	0.130604i
$176$	0	0
$177$	−6.65571e8	−0.678112
$178$	0	0
$179$	− 6.36115e8i	− 0.619617i	−0.950799	−	0.309809i	$-0.899735\pi$
$179$	− 6.36115e8i	− 0.619617i	0.950799	−	0.309809i	$-0.100265\pi$
$180$	0	0
$181$	−1.09761e9	−1.02267	−0.511333	−	0.859383i	$-0.670848\pi$
$181$	−1.09761e9	−1.02267	−0.511333	+	0.859383i	$0.670848\pi$
$182$	0	0
$183$	3.23327e8i	0.288295i
$184$	0	0
$185$	−1.17489e8	−0.100302
$186$	0	0
$187$	− 1.03257e9i	− 0.844410i
$188$	0	0
$189$	5.98015e7	0.0468668
$190$	0	0
$191$	− 1.54416e9i	− 1.16027i	−0.814521	−	0.580134i	$-0.803000\pi$
$191$	− 1.54416e9i	− 1.16027i	0.814521	−	0.580134i	$-0.197000\pi$
$192$	0	0
$193$	−1.52519e9	−1.09924	−0.549622	−	0.835413i	$-0.685228\pi$
$193$	−1.52519e9	−1.09924	−0.549622	+	0.835413i	$0.685228\pi$
$194$	0	0
$195$	4.64017e8i	0.320919i
$196$	0	0
$197$	2.71469e9	1.80242	0.901209	−	0.433386i	$-0.142681\pi$
$197$	2.71469e9	1.80242	0.901209	+	0.433386i	$0.142681\pi$
$198$	0	0
$199$	− 1.01061e9i	− 0.644426i	−0.946667	−	0.322213i	$-0.895573\pi$
$199$	− 1.01061e9i	− 0.644426i	0.946667	−	0.322213i	$-0.104427\pi$
$200$	0	0
$201$	1.79630e8	0.110051
$202$	0	0
$203$	1.95035e8i	0.114849i
$204$	0	0
$205$	6.31080e8	0.357330
$206$	0	0
$207$	1.94508e8i	0.105939i
$208$	0	0
$209$	−2.98370e9	−1.56376
$210$	0	0
$211$	− 3.42535e8i	− 0.172812i	−0.996260	−	0.0864062i	$-0.972462\pi$
$211$	− 3.42535e8i	− 0.172812i	0.996260	−	0.0864062i	$-0.0275383\pi$
$212$	0	0
$213$	−1.46542e9	−0.711941
$214$	0	0
$215$	− 2.22130e9i	− 1.03957i
$216$	0	0
$217$	−3.05706e8	−0.137869
$218$	0	0
$219$	6.50324e8i	0.282718i
$220$	0	0
$221$	9.90932e8	0.415408
$222$	0	0
$223$	− 6.37197e8i	− 0.257664i	−0.991666	−	0.128832i	$-0.958877\pi$
$223$	− 6.37197e8i	− 0.257664i	0.991666	−	0.128832i	$-0.0411228\pi$
$224$	0	0
$225$	4.58160e8	0.178767
$226$	0	0
$227$	− 2.70397e9i	− 1.01835i	−0.860662	−	0.509176i	$-0.829950\pi$
$227$	− 2.70397e9i	− 1.01835i	0.860662	−	0.509176i	$-0.170050\pi$
$228$	0	0
$229$	4.13967e9	1.50530	0.752652	−	0.658419i	$-0.228774\pi$
$229$	4.13967e9	1.50530	0.752652	+	0.658419i	$0.228774\pi$
$230$	0	0
$231$	− 6.64279e8i	− 0.233294i
$232$	0	0
$233$	−3.40816e9	−1.15637	−0.578185	−	0.815906i	$-0.696239\pi$
$233$	−3.40816e9	−1.15637	−0.578185	+	0.815906i	$0.696239\pi$
$234$	0	0
$235$	− 2.05539e9i	− 0.673941i
$236$	0	0
$237$	1.40509e9	0.445360
$238$	0	0
$239$	− 2.63879e9i	− 0.808748i	−0.914594	−	0.404374i	$-0.867489\pi$
$239$	− 2.63879e9i	− 0.808748i	0.914594	−	0.404374i	$-0.132511\pi$
$240$	0	0
$241$	−1.91684e9	−0.568223	−0.284111	−	0.958791i	$-0.591699\pi$
$241$	−1.91684e9	−0.568223	−0.284111	+	0.958791i	$0.591699\pi$
$242$	0	0
$243$	− 2.23677e8i	− 0.0641500i
$244$	0	0
$245$	2.30798e9	0.640570
$246$	0	0
$247$	− 2.86338e9i	− 0.769293i
$248$	0	0
$249$	−1.45465e9	−0.378409
$250$	0	0
$251$	− 4.44444e9i	− 1.11975i	−0.828576	−	0.559877i	$-0.810848\pi$
$251$	− 4.44444e9i	− 1.11975i	0.828576	−	0.559877i	$-0.189152\pi$
$252$	0	0
$253$	2.16061e9	0.527344
$254$	0	0
$255$	8.45971e8i	0.200076i
$256$	0	0
$257$	−1.46470e9	−0.335749	−0.167875	−	0.985808i	$-0.553690\pi$
$257$	−1.46470e9	−0.335749	−0.167875	+	0.985808i	$0.553690\pi$
$258$	0	0
$259$	1.61413e8i	0.0358708i
$260$	0	0
$261$	7.29494e8	0.157203
$262$	0	0
$263$	5.24355e9i	1.09598i	0.836485	+	0.547990i	$0.184607\pi$
$263$	5.24355e9i	1.09598i	−0.836485	+	0.547990i	$0.815393\pi$
$264$	0	0
$265$	4.41700e9	0.895661
$266$	0	0
$267$	− 5.47242e8i	− 0.107680i
$268$	0	0
$269$	1.77476e9	0.338946	0.169473	−	0.985535i	$-0.445793\pi$
$269$	1.77476e9	0.338946	0.169473	+	0.985535i	$0.445793\pi$
$270$	0	0
$271$	1.01882e9i	0.188895i	0.995530	+	0.0944475i	$0.0301085\pi$
$271$	1.01882e9i	0.188895i	−0.995530	+	0.0944475i	$0.969892\pi$
$272$	0	0
$273$	6.37492e8	0.114769
$274$	0	0
$275$	− 5.08927e9i	− 0.889867i
$276$	0	0
$277$	1.10632e10	1.87914	0.939572	−	0.342350i	$-0.111223\pi$
$277$	1.10632e10	1.87914	0.939572	+	0.342350i	$0.111223\pi$
$278$	0	0
$279$	1.14344e9i	0.188711i
$280$	0	0
$281$	−1.73642e8	−0.0278503	−0.0139252	−	0.999903i	$-0.504433\pi$
$281$	−1.73642e8	−0.0278503	−0.0139252	+	0.999903i	$0.504433\pi$
$282$	0	0
$283$	− 1.19880e10i	− 1.86897i	−0.356007	−	0.934483i	$-0.615862\pi$
$283$	− 1.19880e10i	− 1.86897i	0.356007	−	0.934483i	$-0.384138\pi$
$284$	0	0
$285$	2.44451e9	0.370520
$286$	0	0
$287$	− 8.67013e8i	− 0.127790i
$288$	0	0
$289$	−5.16914e9	−0.741015
$290$	0	0
$291$	− 4.17376e9i	− 0.582043i
$292$	0	0
$293$	−2.84249e9	−0.385681	−0.192840	−	0.981230i	$-0.561770\pi$
$293$	−2.84249e9	−0.385681	−0.192840	+	0.981230i	$0.561770\pi$
$294$	0	0
$295$	− 6.05715e9i	− 0.799798i
$296$	0	0
$297$	−2.48462e9	−0.319326
$298$	0	0
$299$	2.07348e9i	0.259427i
$300$	0	0
$301$	−3.05174e9	−0.371777
$302$	0	0
$303$	− 9.62141e9i	− 1.14148i
$304$	0	0
$305$	−2.94249e9	−0.340029
$306$	0	0
$307$	− 2.31641e9i	− 0.260773i	−0.991463	−	0.130386i	$-0.958378\pi$
$307$	− 2.31641e9i	− 0.260773i	0.991463	−	0.130386i	$-0.0416218\pi$
$308$	0	0
$309$	3.11078e9	0.341221
$310$	0	0
$311$	− 5.91506e9i	− 0.632292i	−0.948711	−	0.316146i	$-0.897611\pi$
$311$	− 5.91506e9i	− 0.632292i	0.948711	−	0.316146i	$-0.102389\pi$
$312$	0	0
$313$	3.46268e9	0.360774	0.180387	−	0.983596i	$-0.442265\pi$
$313$	3.46268e9	0.360774	0.180387	+	0.983596i	$0.442265\pi$
$314$	0	0
$315$	5.44235e8i	0.0552769i
$316$	0	0
$317$	−1.25563e10	−1.24344	−0.621718	−	0.783241i	$-0.713565\pi$
$317$	−1.25563e10	−1.24344	−0.621718	+	0.783241i	$0.713565\pi$
$318$	0	0
$319$	− 8.10327e9i	− 0.782524i
$320$	0	0
$321$	−1.63019e9	−0.153539
$322$	0	0
$323$	− 5.22037e9i	− 0.479613i
$324$	0	0
$325$	4.88405e9	0.437770
$326$	0	0
$327$	− 1.02634e10i	− 0.897632i
$328$	0	0
$329$	−2.82380e9	−0.241019
$330$	0	0
$331$	8.31223e9i	0.692477i	0.938146	+	0.346239i	$0.112541\pi$
$331$	8.31223e9i	0.692477i	−0.938146	+	0.346239i	$0.887459\pi$
$332$	0	0
$333$	6.03739e8	0.0490990
$334$	0	0
$335$	1.63476e9i	0.129800i
$336$	0	0
$337$	−1.36914e10	−1.06152	−0.530758	−	0.847523i	$-0.678093\pi$
$337$	−1.36914e10	−1.06152	−0.530758	+	0.847523i	$0.678093\pi$
$338$	0	0
$339$	− 2.97888e9i	− 0.225556i
$340$	0	0
$341$	1.27014e10	0.939366
$342$	0	0
$343$	− 6.54155e9i	− 0.472611i
$344$	0	0
$345$	−1.77016e9	−0.124950
$346$	0	0
$347$	5.70122e9i	0.393233i	0.980480	+	0.196617i	$0.0629954\pi$
$347$	5.70122e9i	0.393233i	−0.980480	+	0.196617i	$0.937005\pi$
$348$	0	0
$349$	−7.21129e9	−0.486084	−0.243042	−	0.970016i	$-0.578145\pi$
$349$	−7.21129e9	−0.486084	−0.243042	+	0.970016i	$0.578145\pi$
$350$	0	0
$351$	− 2.38443e9i	− 0.157093i
$352$	0	0
$353$	−5.06540e9	−0.326223	−0.163112	−	0.986608i	$-0.552153\pi$
$353$	−5.06540e9	−0.326223	−0.163112	+	0.986608i	$0.552153\pi$
$354$	0	0
$355$	− 1.33363e10i	− 0.839698i
$356$	0	0
$357$	1.16224e9	0.0715523
$358$	0	0
$359$	− 1.28030e10i	− 0.770785i	−0.922753	−	0.385392i	$-0.874066\pi$
$359$	− 1.28030e10i	− 0.770785i	0.922753	−	0.385392i	$-0.125934\pi$
$360$	0	0
$361$	1.89886e9	0.111805
$362$	0	0
$363$	1.75748e10i	1.01219i
$364$	0	0
$365$	−5.91839e9	−0.333451
$366$	0	0
$367$	3.57832e10i	1.97249i	0.165290	+	0.986245i	$0.447144\pi$
$367$	3.57832e10i	1.97249i	−0.165290	+	0.986245i	$0.552856\pi$
$368$	0	0
$369$	−3.24291e9	−0.174916
$370$	0	0
$371$	− 6.06832e9i	− 0.320312i
$372$	0	0
$373$	3.68848e10	1.90552	0.952758	−	0.303731i	$-0.0982324\pi$
$373$	3.68848e10	1.90552	0.952758	+	0.303731i	$0.0982324\pi$
$374$	0	0
$375$	1.19442e10i	0.603996i
$376$	0	0
$377$	7.77650e9	0.384963
$378$	0	0
$379$	− 1.95039e9i	− 0.0945290i	−0.998882	−	0.0472645i	$-0.984950\pi$
$379$	− 1.95039e9i	− 0.0945290i	0.998882	−	0.0472645i	$-0.0150504\pi$
$380$	0	0
$381$	−1.63017e10	−0.773629
$382$	0	0
$383$	1.97660e10i	0.918592i	0.888283	+	0.459296i	$0.151898\pi$
$383$	1.97660e10i	0.918592i	−0.888283	+	0.459296i	$0.848102\pi$
$384$	0	0
$385$	6.04539e9	0.275158
$386$	0	0
$387$	1.14145e10i	0.508878i
$388$	0	0
$389$	−1.83144e8	−0.00799823	−0.00399911	−	0.999992i	$-0.501273\pi$
$389$	−1.83144e8	−0.00799823	−0.00399911	+	0.999992i	$0.501273\pi$
$390$	0	0
$391$	3.78026e9i	0.161739i
$392$	0	0
$393$	2.32563e10	0.974923
$394$	0	0
$395$	1.27873e10i	0.525279i
$396$	0	0
$397$	−7.20182e9	−0.289921	−0.144961	−	0.989437i	$-0.546306\pi$
$397$	−7.20182e9	−0.289921	−0.144961	+	0.989437i	$0.546306\pi$
$398$	0	0
$399$	− 3.35840e9i	− 0.132507i
$400$	0	0
$401$	−3.54914e10	−1.37260	−0.686302	−	0.727316i	$-0.740767\pi$
$401$	−3.54914e10	−1.37260	−0.686302	+	0.727316i	$0.740767\pi$
$402$	0	0
$403$	1.21892e10i	0.462121i
$404$	0	0
$405$	2.03562e9	0.0756616
$406$	0	0
$407$	− 6.70637e9i	− 0.244405i
$408$	0	0
$409$	4.66917e9	0.166858	0.0834289	−	0.996514i	$-0.473413\pi$
$409$	4.66917e9	0.166858	0.0834289	+	0.996514i	$0.473413\pi$
$410$	0	0
$411$	− 5.75906e9i	− 0.201829i
$412$	0	0
$413$	−8.32165e9	−0.286028
$414$	0	0
$415$	− 1.32383e10i	− 0.446314i
$416$	0	0
$417$	−2.02079e10	−0.668308
$418$	0	0
$419$	3.47311e10i	1.12684i	0.826170	+	0.563421i	$0.190515\pi$
$419$	3.47311e10i	1.12684i	−0.826170	+	0.563421i	$0.809485\pi$
$420$	0	0
$421$	−5.19134e10	−1.65254	−0.826269	−	0.563276i	$-0.809541\pi$
$421$	−5.19134e10	−1.65254	−0.826269	+	0.563276i	$0.809541\pi$
$422$	0	0
$423$	1.05619e10i	0.329900i
$424$	0	0
$425$	8.90433e9	0.272926
$426$	0	0
$427$	4.04256e9i	0.121603i
$428$	0	0
$429$	−2.64864e10	−0.781976
$430$	0	0
$431$	− 6.31069e10i	− 1.82881i	−0.404806	−	0.914403i	$-0.632661\pi$
$431$	− 6.31069e10i	− 1.82881i	0.404806	−	0.914403i	$-0.367339\pi$
$432$	0	0
$433$	2.71812e10	0.773245	0.386622	−	0.922238i	$-0.373642\pi$
$433$	2.71812e10	0.773245	0.386622	+	0.922238i	$0.373642\pi$
$434$	0	0
$435$	6.63889e9i	0.185412i
$436$	0	0
$437$	1.09234e10	0.299524
$438$	0	0
$439$	3.44182e10i	0.926681i	0.886180	+	0.463340i	$0.153349\pi$
$439$	3.44182e10i	0.926681i	−0.886180	+	0.463340i	$0.846651\pi$
$440$	0	0
$441$	−1.18599e10	−0.313565
$442$	0	0
$443$	− 2.24571e10i	− 0.583093i	−0.956557	−	0.291547i	$-0.905830\pi$
$443$	− 2.24571e10i	− 0.583093i	0.956557	−	0.291547i	$-0.0941699\pi$
$444$	0	0
$445$	4.98027e9	0.127003
$446$	0	0
$447$	1.33365e10i	0.334051i
$448$	0	0
$449$	−4.00449e10	−0.985284	−0.492642	−	0.870232i	$-0.663969\pi$
$449$	−4.00449e10	−0.985284	−0.492642	+	0.870232i	$0.663969\pi$
$450$	0	0
$451$	3.60225e10i	0.870698i
$452$	0	0
$453$	−7.83635e9	−0.186089
$454$	0	0
$455$	5.80161e9i	0.135364i
$456$	0	0
$457$	8.33836e10	1.91168	0.955841	−	0.293883i	$-0.0949477\pi$
$457$	8.33836e10	1.91168	0.955841	+	0.293883i	$0.0949477\pi$
$458$	0	0
$459$	− 4.34716e9i	− 0.0979389i
$460$	0	0
$461$	−5.16433e10	−1.14343	−0.571715	−	0.820452i	$-0.693722\pi$
$461$	−5.16433e10	−1.14343	−0.571715	+	0.820452i	$0.693722\pi$
$462$	0	0
$463$	4.77439e10i	1.03895i	0.854486	+	0.519475i	$0.173872\pi$
$463$	4.77439e10i	1.03895i	−0.854486	+	0.519475i	$0.826128\pi$
$464$	0	0
$465$	−1.04061e10	−0.222575
$466$	0	0
$467$	2.33765e10i	0.491488i	0.969335	+	0.245744i	$0.0790322\pi$
$467$	2.33765e10i	0.491488i	−0.969335	+	0.245744i	$0.920968\pi$
$468$	0	0
$469$	2.24592e9	0.0464198
$470$	0	0
$471$	− 5.33071e9i	− 0.108318i
$472$	0	0
$473$	1.26793e11	2.53310
$474$	0	0
$475$	− 2.57298e10i	− 0.505432i
$476$	0	0
$477$	−2.26975e10	−0.438434
$478$	0	0
$479$	5.12753e10i	0.974017i	0.873397	+	0.487008i	$0.161912\pi$
$479$	5.12753e10i	0.974017i	−0.873397	+	0.487008i	$0.838088\pi$
$480$	0	0
$481$	6.43594e9	0.120235
$482$	0	0
$483$	2.43194e9i	0.0446852i
$484$	0	0
$485$	3.79840e10	0.686490
$486$	0	0
$487$	8.40950e10i	1.49505i	0.664236	+	0.747523i	$0.268757\pi$
$487$	8.40950e10i	1.49505i	−0.664236	+	0.747523i	$0.731243\pi$
$488$	0	0
$489$	−4.44371e9	−0.0777160
$490$	0	0
$491$	6.45788e10i	1.11113i	0.831474	+	0.555564i	$0.187497\pi$
$491$	6.45788e10i	1.11113i	−0.831474	+	0.555564i	$0.812503\pi$
$492$	0	0
$493$	1.41777e10	0.240004
$494$	0	0
$495$	− 2.26118e10i	− 0.376629i
$496$	0	0
$497$	−1.83222e10	−0.300298
$498$	0	0
$499$	− 9.33890e10i	− 1.50624i	−0.657885	−	0.753119i	$-0.728549\pi$
$499$	− 9.33890e10i	− 1.50624i	0.657885	−	0.753119i	$-0.271451\pi$
$500$	0	0
$501$	−1.18869e10	−0.188677
$502$	0	0
$503$	7.08092e10i	1.10616i	0.833128	+	0.553080i	$0.186548\pi$
$503$	7.08092e10i	1.10616i	−0.833128	+	0.553080i	$0.813452\pi$
$504$	0	0
$505$	8.75614e10	1.34632
$506$	0	0
$507$	1.27296e10i	0.192657i
$508$	0	0
$509$	−9.13904e10	−1.36154	−0.680769	−	0.732498i	$-0.738354\pi$
$509$	−9.13904e10	−1.36154	−0.680769	+	0.732498i	$0.738354\pi$
$510$	0	0
$511$	8.13101e9i	0.119251i
$512$	0	0
$513$	−1.25615e10	−0.181373
$514$	0	0
$515$	2.83102e10i	0.402452i
$516$	0	0
$517$	1.17323e11	1.64218
$518$	0	0
$519$	− 4.57914e10i	− 0.631123i
$520$	0	0
$521$	1.08647e11	1.47458	0.737289	−	0.675577i	$-0.236105\pi$
$521$	1.08647e11	1.47458	0.737289	+	0.675577i	$0.236105\pi$
$522$	0	0
$523$	− 4.66215e10i	− 0.623131i	−0.950225	−	0.311566i	$-0.899147\pi$
$523$	− 4.66215e10i	− 0.623131i	0.950225	−	0.311566i	$-0.100853\pi$
$524$	0	0
$525$	5.72838e9	0.0754041
$526$	0	0
$527$	2.22227e10i	0.288108i
$528$	0	0
$529$	7.04010e10	0.898992
$530$	0	0
$531$	3.11257e10i	0.391508i
$532$	0	0
$533$	−3.45699e10	−0.428340
$534$	0	0
$535$	− 1.48359e10i	− 0.181091i
$536$	0	0
$537$	−2.97482e10	−0.357736
$538$	0	0
$539$	1.31741e11i	1.56086i
$540$	0	0
$541$	−1.67851e11	−1.95945	−0.979726	−	0.200340i	$-0.935795\pi$
$541$	−1.67851e11	−1.95945	−0.979726	+	0.200340i	$0.935795\pi$
$542$	0	0
$543$	5.13301e10i	0.590436i
$544$	0	0
$545$	9.34035e10	1.05871
$546$	0	0
$547$	3.61757e10i	0.404080i	0.979377	+	0.202040i	$0.0647571\pi$
$547$	3.61757e10i	0.404080i	−0.979377	+	0.202040i	$0.935243\pi$
$548$	0	0
$549$	1.51205e10	0.166447
$550$	0	0
$551$	− 4.09677e10i	− 0.444462i
$552$	0	0
$553$	1.75679e10	0.187853
$554$	0	0
$555$	5.49443e9i	0.0579097i
$556$	0	0
$557$	−5.93877e10	−0.616986	−0.308493	−	0.951227i	$-0.599825\pi$
$557$	−5.93877e10	−0.616986	−0.308493	+	0.951227i	$0.599825\pi$
$558$	0	0
$559$	1.21680e11i	1.24616i
$560$	0	0
$561$	−4.82886e10	−0.487521
$562$	0	0
$563$	5.66243e10i	0.563598i	0.959473	+	0.281799i	$0.0909311\pi$
$563$	5.66243e10i	0.563598i	−0.959473	+	0.281799i	$0.909069\pi$
$564$	0	0
$565$	2.71098e10	0.266031
$566$	0	0
$567$	− 2.79664e9i	− 0.0270585i
$568$	0	0
$569$	−4.58266e10	−0.437188	−0.218594	−	0.975816i	$-0.570147\pi$
$569$	−4.58266e10	−0.437188	−0.218594	+	0.975816i	$0.570147\pi$
$570$	0	0
$571$	− 5.45678e10i	− 0.513325i	−0.966501	−	0.256662i	$-0.917377\pi$
$571$	− 5.45678e10i	− 0.513325i	0.966501	−	0.256662i	$-0.0826228\pi$
$572$	0	0
$573$	−7.22132e10	−0.669881
$574$	0	0
$575$	1.86319e10i	0.170446i
$576$	0	0
$577$	−1.69500e11	−1.52921	−0.764604	−	0.644500i	$-0.777066\pi$
$577$	−1.69500e11	−1.52921	−0.764604	+	0.644500i	$0.777066\pi$
$578$	0	0
$579$	7.13260e10i	0.634649i
$580$	0	0
$581$	−1.81875e10	−0.159613
$582$	0	0
$583$	2.52125e11i	2.18244i
$584$	0	0
$585$	2.16999e10	0.185283
$586$	0	0
$587$	5.05989e10i	0.426175i	0.977033	+	0.213088i	$0.0683520\pi$
$587$	5.05989e10i	0.426175i	−0.977033	+	0.213088i	$0.931648\pi$
$588$	0	0
$589$	6.42145e10	0.533546
$590$	0	0
$591$	− 1.26953e11i	− 1.04063i
$592$	0	0
$593$	−3.12289e10	−0.252544	−0.126272	−	0.991996i	$-0.540301\pi$
$593$	−3.12289e10	−0.252544	−0.126272	+	0.991996i	$0.540301\pi$
$594$	0	0
$595$	1.05772e10i	0.0843922i
$596$	0	0
$597$	−4.72617e10	−0.372059
$598$	0	0
$599$	2.49376e10i	0.193708i	0.995299	+	0.0968538i	$0.0308779\pi$
$599$	2.49376e10i	0.193708i	−0.995299	+	0.0968538i	$0.969122\pi$
$600$	0	0
$601$	2.02149e11	1.54944	0.774719	−	0.632305i	$-0.217891\pi$
$601$	2.02149e11	1.54944	0.774719	+	0.632305i	$0.217891\pi$
$602$	0	0
$603$	− 8.40049e9i	− 0.0635382i
$604$	0	0
$605$	−1.59942e11	−1.19383
$606$	0	0
$607$	− 1.86515e11i	− 1.37391i	−0.726699	−	0.686956i	$-0.758946\pi$
$607$	− 1.86515e11i	− 1.37391i	0.726699	−	0.686956i	$-0.241054\pi$
$608$	0	0
$609$	9.12088e9	0.0663082
$610$	0	0
$611$	1.12592e11i	0.807870i
$612$	0	0
$613$	2.64658e11	1.87432	0.937159	−	0.348904i	$-0.113446\pi$
$613$	2.64658e11	1.87432	0.937159	+	0.348904i	$0.113446\pi$
$614$	0	0
$615$	− 2.95127e10i	− 0.206304i
$616$	0	0
$617$	−2.71229e11	−1.87152	−0.935761	−	0.352634i	$-0.885286\pi$
$617$	−2.71229e11	−1.87152	−0.935761	+	0.352634i	$0.885286\pi$
$618$	0	0
$619$	8.76820e10i	0.597239i	0.954372	+	0.298619i	$0.0965261\pi$
$619$	8.76820e10i	0.597239i	−0.954372	+	0.298619i	$0.903474\pi$
$620$	0	0
$621$	9.09625e9	0.0611640
$622$	0	0
$623$	− 6.84217e9i	− 0.0454194i
$624$	0	0
$625$	−2.68678e10	−0.176080
$626$	0	0
$627$	1.39534e11i	0.902838i
$628$	0	0
$629$	1.17337e10	0.0749602
$630$	0	0
$631$	1.73981e11i	1.09745i	0.836003	+	0.548724i	$0.184886\pi$
$631$	1.73981e11i	1.09745i	−0.836003	+	0.548724i	$0.815114\pi$
$632$	0	0
$633$	−1.60188e10	−0.0997733
$634$	0	0
$635$	− 1.48356e11i	− 0.912455i
$636$	0	0
$637$	−1.26428e11	−0.767868
$638$	0	0
$639$	6.85310e10i	0.411040i
$640$	0	0
$641$	−2.11172e11	−1.25085	−0.625424	−	0.780285i	$-0.715074\pi$
$641$	−2.11172e11	−1.25085	−0.625424	+	0.780285i	$0.715074\pi$
$642$	0	0
$643$	− 2.30983e11i	− 1.35125i	−0.737245	−	0.675626i	$-0.763873\pi$
$643$	− 2.30983e11i	− 1.35125i	0.737245	−	0.675626i	$-0.236127\pi$
$644$	0	0
$645$	−1.03880e11	−0.600195
$646$	0	0
$647$	− 2.25708e11i	− 1.28804i	−0.765007	−	0.644022i	$-0.777265\pi$
$647$	− 2.25708e11i	− 1.28804i	0.765007	−	0.644022i	$-0.222735\pi$
$648$	0	0
$649$	3.45746e11	1.94885
$650$	0	0
$651$	1.42965e10i	0.0795984i
$652$	0	0
$653$	1.78919e11	0.984019	0.492009	−	0.870590i	$-0.336263\pi$
$653$	1.78919e11	0.984019	0.492009	+	0.870590i	$0.336263\pi$
$654$	0	0
$655$	2.11648e11i	1.14987i
$656$	0	0
$657$	3.04126e10	0.163227
$658$	0	0
$659$	− 9.19117e10i	− 0.487337i	−0.969859	−	0.243668i	$-0.921649\pi$
$659$	− 9.19117e10i	− 0.487337i	0.969859	−	0.243668i	$-0.0783508\pi$
$660$	0	0
$661$	2.04482e11	1.07115	0.535573	−	0.844489i	$-0.320096\pi$
$661$	2.04482e11	1.07115	0.535573	+	0.844489i	$0.320096\pi$
$662$	0	0
$663$	− 4.63413e10i	− 0.239836i
$664$	0	0
$665$	3.05637e10	0.156286
$666$	0	0
$667$	2.96662e10i	0.149885i
$668$	0	0
$669$	−2.97988e10	−0.148763
$670$	0	0
$671$	− 1.67959e11i	− 0.828542i
$672$	0	0
$673$	3.26000e11	1.58912	0.794561	−	0.607184i	$-0.207701\pi$
$673$	3.26000e11	1.58912	0.794561	+	0.607184i	$0.207701\pi$
$674$	0	0
$675$	− 2.14260e10i	− 0.103211i
$676$	0	0
$677$	−1.18596e11	−0.564565	−0.282282	−	0.959331i	$-0.591092\pi$
$677$	−1.18596e11	−0.564565	−0.282282	+	0.959331i	$0.591092\pi$
$678$	0	0
$679$	− 5.21846e10i	− 0.245506i
$680$	0	0
$681$	−1.26452e11	−0.587946
$682$	0	0
$683$	− 1.72598e11i	− 0.793148i	−0.918003	−	0.396574i	$-0.870199\pi$
$683$	− 1.72598e11i	− 0.793148i	0.918003	−	0.396574i	$-0.129801\pi$
$684$	0	0
$685$	5.24114e10	0.238047
$686$	0	0
$687$	− 1.93593e11i	− 0.869087i
$688$	0	0
$689$	−2.41958e11	−1.07365
$690$	0	0
$691$	1.45326e11i	0.637428i	0.947851	+	0.318714i	$0.103251\pi$
$691$	1.45326e11i	0.637428i	−0.947851	+	0.318714i	$0.896749\pi$
$692$	0	0
$693$	−3.10653e10	−0.134692
$694$	0	0
$695$	− 1.83905e11i	− 0.788234i
$696$	0	0
$697$	−6.30259e10	−0.267047
$698$	0	0
$699$	1.59384e11i	0.667630i
$700$	0	0
$701$	−3.46244e11	−1.43387	−0.716935	−	0.697140i	$-0.754456\pi$
$701$	−3.46244e11	−1.43387	−0.716935	+	0.697140i	$0.754456\pi$
$702$	0	0
$703$	− 3.39054e10i	− 0.138819i
$704$	0	0
$705$	−9.61209e10	−0.389100
$706$	0	0
$707$	− 1.20297e11i	− 0.481478i
$708$	0	0
$709$	−8.96735e9	−0.0354878	−0.0177439	−	0.999843i	$-0.505648\pi$
$709$	−8.96735e9	−0.0354878	−0.0177439	+	0.999843i	$0.505648\pi$
$710$	0	0
$711$	− 6.57097e10i	− 0.257129i
$712$	0	0
$713$	−4.65001e10	−0.179927
$714$	0	0
$715$	− 2.41044e11i	− 0.922300i
$716$	0	0
$717$	−1.23404e11	−0.466931
$718$	0	0
$719$	9.39298e10i	0.351470i	0.984438	+	0.175735i	$0.0562301\pi$
$719$	9.39298e10i	0.351470i	−0.984438	+	0.175735i	$0.943770\pi$
$720$	0	0
$721$	3.88942e10	0.143927
$722$	0	0
$723$	8.96419e10i	0.328064i
$724$	0	0
$725$	6.98782e10	0.252924
$726$	0	0
$727$	− 2.29456e11i	− 0.821412i	−0.911768	−	0.410706i	$-0.865282\pi$
$727$	− 2.29456e11i	− 0.821412i	0.911768	−	0.410706i	$-0.134718\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	2.21841e11i	0.776912i
$732$	0	0
$733$	4.44865e11	1.54103	0.770516	−	0.637420i	$-0.219998\pi$
$733$	4.44865e11	1.54103	0.770516	+	0.637420i	$0.219998\pi$
$734$	0	0
$735$	− 1.07933e11i	− 0.369833i
$736$	0	0
$737$	−9.33132e10	−0.316281
$738$	0	0
$739$	− 5.10990e11i	− 1.71330i	−0.515895	−	0.856652i	$-0.672541\pi$
$739$	− 5.10990e11i	− 1.71330i	0.515895	−	0.856652i	$-0.327459\pi$
$740$	0	0
$741$	−1.33907e11	−0.444152
$742$	0	0
$743$	− 3.87724e11i	− 1.27224i	−0.771592	−	0.636118i	$-0.780539\pi$
$743$	− 3.87724e11i	− 1.27224i	0.771592	−	0.636118i	$-0.219461\pi$
$744$	0	0
$745$	−1.21371e11	−0.393995
$746$	0	0
$747$	6.80273e10i	0.218475i
$748$	0	0
$749$	−2.03823e10	−0.0647629
$750$	0	0
$751$	4.16206e8i	0.00130842i	1.00000		0.000654212i	$0.000208242\pi$
$751$	4.16206e8i	0.00130842i	−1.00000		0.000654212i	$0.999792\pi$
$752$	0	0
$753$	−2.07846e11	−0.646490
$754$	0	0
$755$	− 7.13161e10i	− 0.219482i
$756$	0	0
$757$	3.02639e11	0.921596	0.460798	−	0.887505i	$-0.347563\pi$
$757$	3.02639e11	0.921596	0.460798	+	0.887505i	$0.347563\pi$
$758$	0	0
$759$	− 1.01042e11i	− 0.304462i
$760$	0	0
$761$	−3.33821e11	−0.995348	−0.497674	−	0.867364i	$-0.665812\pi$
$761$	−3.33821e11	−0.995348	−0.497674	+	0.867364i	$0.665812\pi$
$762$	0	0
$763$	− 1.28323e11i	− 0.378622i
$764$	0	0
$765$	3.95621e10	0.115514
$766$	0	0
$767$	3.31804e11i	0.958738i
$768$	0	0
$769$	−4.15184e11	−1.18723	−0.593615	−	0.804749i	$-0.702300\pi$
$769$	−4.15184e11	−1.18723	−0.593615	+	0.804749i	$0.702300\pi$
$770$	0	0
$771$	6.84971e10i	0.193845i
$772$	0	0
$773$	6.71927e10	0.188193	0.0940967	−	0.995563i	$-0.470004\pi$
$773$	6.71927e10	0.188193	0.0940967	+	0.995563i	$0.470004\pi$
$774$	0	0
$775$	1.09530e11i	0.303617i
$776$	0	0
$777$	7.54856e9	0.0207100
$778$	0	0
$779$	1.82119e11i	0.494544i
$780$	0	0
$781$	7.61247e11	2.04607
$782$	0	0
$783$	− 3.41151e10i	− 0.0907610i
$784$	0	0
$785$	4.85130e10	0.127756
$786$	0	0
$787$	− 7.23571e11i	− 1.88618i	−0.332542	−	0.943088i	$-0.607906\pi$
$787$	− 7.23571e11i	− 1.88618i	0.332542	−	0.943088i	$-0.392094\pi$
$788$	0	0
$789$	2.45217e11	0.632765
$790$	0	0
$791$	− 3.72450e10i	− 0.0951397i
$792$	0	0
$793$	1.61186e11	0.407601
$794$	0	0
$795$	− 2.06563e11i	− 0.517110i
$796$	0	0
$797$	2.83695e11	0.703101	0.351550	−	0.936169i	$-0.385655\pi$
$797$	2.83695e11	0.703101	0.351550	+	0.936169i	$0.385655\pi$
$798$	0	0
$799$	2.05271e11i	0.503664i
$800$	0	0
$801$	−2.55920e10	−0.0621690
$802$	0	0
$803$	− 3.37825e11i	− 0.812513i
$804$	0	0
$805$	−2.21323e10	−0.0527039
$806$	0	0
$807$	− 8.29974e10i	− 0.195691i
$808$	0	0
$809$	4.24107e11	0.990105	0.495053	−	0.868863i	$-0.335149\pi$
$809$	4.24107e11	0.990105	0.495053	+	0.868863i	$0.335149\pi$
$810$	0	0
$811$	− 8.29372e11i	− 1.91719i	−0.284768	−	0.958597i	$-0.591916\pi$
$811$	− 8.29372e11i	− 1.91719i	0.284768	−	0.958597i	$-0.408084\pi$
$812$	0	0
$813$	4.76455e10	0.109059
$814$	0	0
$815$	− 4.04408e10i	− 0.0916619i
$816$	0	0
$817$	6.41028e11	1.43876
$818$	0	0
$819$	− 2.98126e10i	− 0.0662618i
$820$	0	0
$821$	3.59588e11	0.791466	0.395733	−	0.918366i	$-0.370491\pi$
$821$	3.59588e11	0.791466	0.395733	+	0.918366i	$0.370491\pi$
$822$	0	0
$823$	5.69282e11i	1.24088i	0.784256	+	0.620438i	$0.213045\pi$
$823$	5.69282e11i	1.24088i	−0.784256	+	0.620438i	$0.786955\pi$
$824$	0	0
$825$	−2.38002e11	−0.513765
$826$	0	0
$827$	4.99528e11i	1.06792i	0.845510	+	0.533959i	$0.179296\pi$
$827$	4.99528e11i	1.06792i	−0.845510	+	0.533959i	$0.820704\pi$
$828$	0	0
$829$	5.89395e11	1.24792	0.623962	−	0.781455i	$-0.285522\pi$
$829$	5.89395e11	1.24792	0.623962	+	0.781455i	$0.285522\pi$
$830$	0	0
$831$	− 5.17373e11i	− 1.08492i
$832$	0	0
$833$	−2.30497e11	−0.478725
$834$	0	0
$835$	− 1.08179e11i	− 0.222535i
$836$	0	0
$837$	5.34734e10	0.108952
$838$	0	0
$839$	− 2.66218e11i	− 0.537266i	−0.963243	−	0.268633i	$-0.913428\pi$
$839$	− 2.66218e11i	− 0.537266i	0.963243	−	0.268633i	$-0.0865719\pi$
$840$	0	0
$841$	−3.88985e11	−0.777586
$842$	0	0
$843$	8.12044e9i	0.0160794i
$844$	0	0
$845$	−1.15848e11	−0.227229
$846$	0	0
$847$	2.19738e11i	0.426944i
$848$	0	0
$849$	−5.60624e11	−1.07905
$850$	0	0
$851$	2.45522e10i	0.0468135i
$852$	0	0
$853$	4.44334e10	0.0839293	0.0419647	−	0.999119i	$-0.486638\pi$
$853$	4.44334e10	0.0839293	0.0419647	+	0.999119i	$0.486638\pi$
$854$	0	0
$855$	− 1.14318e11i	− 0.213920i
$856$	0	0
$857$	−3.92427e11	−0.727504	−0.363752	−	0.931496i	$-0.618504\pi$
$857$	−3.92427e11	−0.727504	−0.363752	+	0.931496i	$0.618504\pi$
$858$	0	0
$859$	− 5.05227e11i	− 0.927928i	−0.885854	−	0.463964i	$-0.846427\pi$
$859$	− 5.05227e11i	− 0.927928i	0.885854	−	0.463964i	$-0.153573\pi$
$860$	0	0
$861$	−4.05462e10	−0.0737798
$862$	0	0
$863$	− 4.89803e11i	− 0.883036i	−0.897252	−	0.441518i	$-0.854440\pi$
$863$	− 4.89803e11i	− 0.883036i	0.897252	−	0.441518i	$-0.145560\pi$
$864$	0	0
$865$	4.16732e11	0.744377
$866$	0	0
$867$	2.41737e11i	0.427825i
$868$	0	0
$869$	−7.29907e11	−1.27994
$870$	0	0
$871$	− 8.95503e10i	− 0.155594i
$872$	0	0
$873$	−1.95187e11	−0.336043
$874$	0	0
$875$	1.49339e11i	0.254766i
$876$	0	0
$877$	4.35062e11	0.735449	0.367725	−	0.929935i	$-0.380137\pi$
$877$	4.35062e11	0.735449	0.367725	+	0.929935i	$0.380137\pi$
$878$	0	0
$879$	1.32930e11i	0.222673i
$880$	0	0
$881$	−7.66566e11	−1.27247	−0.636233	−	0.771497i	$-0.719508\pi$
$881$	−7.66566e11	−1.27247	−0.636233	+	0.771497i	$0.719508\pi$
$882$	0	0
$883$	4.00425e11i	0.658686i	0.944210	+	0.329343i	$0.106827\pi$
$883$	4.00425e11i	0.658686i	−0.944210	+	0.329343i	$0.893173\pi$
$884$	0	0
$885$	−2.83265e11	−0.461763
$886$	0	0
$887$	− 2.09803e11i	− 0.338936i	−0.985536	−	0.169468i	$-0.945795\pi$
$887$	− 2.09803e11i	− 0.338936i	0.985536	−	0.169468i	$-0.0542049\pi$
$888$	0	0
$889$	−2.03820e11	−0.326317
$890$	0	0
$891$	1.16194e11i	0.184363i
$892$	0	0
$893$	5.93149e11	0.932734
$894$	0	0
$895$	− 2.70728e11i	− 0.421931i
$896$	0	0
$897$	9.69671e10	0.149780
$898$	0	0
$899$	1.74397e11i	0.266993i
$900$	0	0
$901$	−4.41125e11	−0.669365
$902$	0	0
$903$	1.42716e11i	0.214645i
$904$	0	0
$905$	−4.67139e11	−0.696389
$906$	0	0
$907$	− 6.37382e11i	− 0.941826i	−0.882180	−	0.470913i	$-0.843925\pi$
$907$	− 6.37382e11i	− 0.941826i	0.882180	−	0.470913i	$-0.156075\pi$
$908$	0	0
$909$	−4.49949e11	−0.659034
$910$	0	0
$911$	− 8.99658e11i	− 1.30618i	−0.757279	−	0.653091i	$-0.773472\pi$
$911$	− 8.99658e11i	− 1.30618i	0.757279	−	0.653091i	$-0.226528\pi$
$912$	0	0
$913$	7.55652e11	1.08752
$914$	0	0
$915$	1.37607e11i	0.196316i
$916$	0	0
$917$	2.90774e11	0.411223
$918$	0	0
$919$	5.06013e11i	0.709413i	0.934978	+	0.354707i	$0.115419\pi$
$919$	5.06013e11i	0.709413i	−0.934978	+	0.354707i	$0.884581\pi$
$920$	0	0
$921$	−1.08328e11	−0.150557
$922$	0	0
$923$	7.30549e11i	1.00657i
$924$	0	0
$925$	5.78321e10	0.0789955
$926$	0	0
$927$	− 1.45477e11i	− 0.197004i
$928$	0	0
$929$	1.05701e12	1.41912	0.709558	−	0.704647i	$-0.248895\pi$
$929$	1.05701e12	1.41912	0.709558	+	0.704647i	$0.248895\pi$
$930$	0	0
$931$	6.66042e11i	0.886549i
$932$	0	0
$933$	−2.76620e11	−0.365054
$934$	0	0
$935$	− 4.39459e11i	− 0.575005i
$936$	0	0
$937$	−7.81575e11	−1.01394	−0.506970	−	0.861964i	$-0.669235\pi$
$937$	−7.81575e11	−1.01394	−0.506970	+	0.861964i	$0.669235\pi$
$938$	0	0
$939$	− 1.61933e11i	− 0.208293i
$940$	0	0
$941$	8.91580e11	1.13711	0.568554	−	0.822646i	$-0.307503\pi$
$941$	8.91580e11	1.13711	0.568554	+	0.822646i	$0.307503\pi$
$942$	0	0
$943$	− 1.31879e11i	− 0.166774i
$944$	0	0
$945$	2.54513e10	0.0319141
$946$	0	0
$947$	− 9.28543e11i	− 1.15452i	−0.816560	−	0.577261i	$-0.804121\pi$
$947$	− 9.28543e11i	− 1.15452i	0.816560	−	0.577261i	$-0.195879\pi$
$948$	0	0
$949$	3.24202e11	0.399716
$950$	0	0
$951$	5.87199e11i	0.717898i
$952$	0	0
$953$	−1.14427e12	−1.38725	−0.693626	−	0.720335i	$-0.743988\pi$
$953$	−1.14427e12	−1.38725	−0.693626	+	0.720335i	$0.743988\pi$
$954$	0	0
$955$	− 6.57189e11i	− 0.790090i
$956$	0	0
$957$	−3.78952e11	−0.451790
$958$	0	0
$959$	− 7.20057e10i	− 0.0851319i
$960$	0	0
$961$	5.79534e11	0.679494
$962$	0	0
$963$	7.62365e10i	0.0886458i
$964$	0	0
$965$	−6.49115e11	−0.748536
$966$	0	0
$967$	− 2.31974e11i	− 0.265297i	−0.991163	−	0.132649i	$-0.957652\pi$
$967$	− 2.31974e11i	− 0.265297i	0.991163	−	0.132649i	$-0.0423482\pi$
$968$	0	0
$969$	−2.44132e11	−0.276905
$970$	0	0
$971$	− 6.93765e11i	− 0.780433i	−0.920723	−	0.390216i	$-0.872400\pi$
$971$	− 6.93765e11i	− 0.780433i	0.920723	−	0.390216i	$-0.127600\pi$
$972$	0	0
$973$	−2.52659e11	−0.281893
$974$	0	0
$975$	− 2.28404e11i	− 0.252747i
$976$	0	0
$977$	1.39487e12	1.53093	0.765465	−	0.643477i	$-0.222509\pi$
$977$	1.39487e12	1.53093	0.765465	+	0.643477i	$0.222509\pi$
$978$	0	0
$979$	2.84277e11i	0.309465i
$980$	0	0
$981$	−4.79970e11	−0.518248
$982$	0	0
$983$	− 1.19018e12i	− 1.27467i	−0.770588	−	0.637334i	$-0.780037\pi$
$983$	− 1.19018e12i	− 1.27467i	0.770588	−	0.637334i	$-0.219963\pi$
$984$	0	0
$985$	1.15536e12	1.22736
$986$	0	0
$987$	1.32056e11i	0.139152i
$988$	0	0
$989$	−4.64192e11	−0.485191
$990$	0	0
$991$	1.51631e12i	1.57215i	0.618131	+	0.786075i	$0.287890\pi$
$991$	1.51631e12i	1.57215i	−0.618131	+	0.786075i	$0.712110\pi$
$992$	0	0
$993$	3.88725e11	0.399802
$994$	0	0
$995$	− 4.30114e11i	− 0.438825i
$996$	0	0
$997$	8.43870e11	0.854073	0.427036	−	0.904234i	$-0.359558\pi$
$997$	8.43870e11	0.854073	0.427036	+	0.904234i	$0.359558\pi$
$998$	0	0
$999$	− 2.82341e10i	− 0.0283473i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.g.b.127.15	yes	32
4.3	odd	2	inner	384.9.g.b.127.16	yes	32
8.3	odd	2		384.9.g.a.127.17	✓	32
8.5	even	2		384.9.g.a.127.18	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.g.a.127.17	✓	32	8.3	odd	2
384.9.g.a.127.18	yes	32	8.5	even	2
384.9.g.b.127.15	yes	32	1.1	even	1	trivial
384.9.g.b.127.16	yes	32	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.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-46.7654i q^{3} +425.597 q^{5} -584.708i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q-46.7654i q^{3} +425.597 q^{5} -584.708i q^{7} -2187.00 q^{9} +24293.3i q^{11} -23313.7 q^{13} -19903.2i q^{15} -42504.3 q^{17} +122820. i q^{19} -27344.1 q^{21} -88938.3i q^{23} -209493. q^{25} +102276. i q^{27} -333559. q^{29} -522835. i q^{31} +1.13609e6 q^{33} -248850. i q^{35} -276058. q^{37} +1.09027e6i q^{39} +1.48281e6 q^{41} -5.21926e6i q^{43} -930780. q^{45} -4.82942e6i q^{47} +5.42292e6 q^{49} +1.98773e6i q^{51} +1.03784e7 q^{53} +1.03392e7i q^{55} +5.74371e6 q^{57} -1.42321e7i q^{59} -6.91381e6 q^{61} +1.27876e6i q^{63} -9.92223e6 q^{65} +3.84110e6i q^{67} -4.15923e6 q^{69} -3.13356e7i q^{71} -1.39061e7 q^{73} +9.79700e6i q^{75} +1.42045e7 q^{77} +3.00456e7i q^{79} +4.78297e6 q^{81} -3.11053e7i q^{83} -1.80897e7 q^{85} +1.55990e7i q^{87} +1.17019e7 q^{89} +1.36317e7i q^{91} -2.44506e7 q^{93} +5.22717e7i q^{95} +8.92489e7 q^{97} -5.31295e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 32 * q + 1344 * q^5 - 69984 * q^9	\( 32 q + 1344 q^{5} - 69984 q^{9} + 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} + 275520 q^{29} - 2421440 q^{37} - 4374720 q^{41} - 2939328 q^{45} - 14219104 q^{49} + 6224448 q^{53} + 3100032 q^{57} + 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} - 154517760 q^{77} + 153055008 q^{81} + 384830848 q^{85} - 182669760 q^{89} - 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) \) 32 * q + 1344 * q^5 - 69984 * q^9 + 114240 * q^13 - 154560 * q^17 + 1791712 * q^25 + 275520 * q^29 - 2421440 * q^37 - 4374720 * q^41 - 2939328 * q^45 - 14219104 * q^49 + 6224448 * q^53 + 3100032 * q^57 + 13005632 * q^61 + 75175296 * q^65 - 85710400 * q^73 - 154517760 * q^77 + 153055008 * q^81 + 384830848 * q^85 - 182669760 * q^89 - 149817600 * q^93 - 149408192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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