LMFDB - Embedded newform 384.9.b.b.319.2

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} + 4826x^{6} + 8748877x^{4} + 7060845096x^{2} + 2140819627716 $ x^8 + 4826x^6 + 8748877x^4 + 7060845096*x^2 + 2140819627716
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.2
Root			$1.73205 - 35.9608i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.9.b.b.319.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-46.7654 q^{3} -197.353i q^{5} +3794.06i q^{7} +2187.00 q^{9} +O(q^{10})$	$q-46.7654 q^{3} -197.353i q^{5} +3794.06i q^{7} +2187.00 q^{9} +16480.4 q^{11} -11110.6i q^{13} +9229.26i q^{15} +66738.9 q^{17} +20481.8 q^{19} -177431. i q^{21} +44505.2i q^{23} +351677. q^{25} -102276. q^{27} -725037. i q^{29} +44205.0i q^{31} -770713. q^{33} +748768. q^{35} -2.85851e6i q^{37} +519592. i q^{39} -10702.9 q^{41} -374259. q^{43} -431610. i q^{45} +2.11469e6i q^{47} -8.63010e6 q^{49} -3.12107e6 q^{51} -5.99442e6i q^{53} -3.25245e6i q^{55} -957837. q^{57} +1.11289e7 q^{59} +2.30232e7i q^{61} +8.29761e6i q^{63} -2.19271e6 q^{65} -1.77776e7 q^{67} -2.08130e6i q^{69} +2.44380e7i q^{71} +3.70522e6 q^{73} -1.64463e7 q^{75} +6.25277e7i q^{77} -5.98403e7i q^{79} +4.78297e6 q^{81} +5.83719e7 q^{83} -1.31711e7i q^{85} +3.39066e7i q^{87} -8.20248e7 q^{89} +4.21544e7 q^{91} -2.06726e6i q^{93} -4.04213e6i q^{95} -5.81980e7 q^{97} +3.60427e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 17496 q^{9}+O(q^{10}) $ 8 * q + 17496 * q^9	$ 8 q + 17496 q^{9} + 276848 q^{17} - 2841976 q^{25} + 775008 q^{33} + 5826832 q^{41} - 16599928 q^{49} - 42366240 q^{57} - 132963072 q^{65} - 57759760 q^{73} + 38263752 q^{81} - 258778768 q^{89} - 771489424 q^{97}+O(q^{100}) $ 8 * q + 17496 * q^9 + 276848 * q^17 - 2841976 * q^25 + 775008 * q^33 + 5826832 * q^41 - 16599928 * q^49 - 42366240 * q^57 - 132963072 * q^65 - 57759760 * q^73 + 38263752 * q^81 - 258778768 * q^89 - 771489424 * 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
$5$	− 197.353i	− 0.315764i	−0.987458	−	0.157882i	$-0.949533\pi$
$5$	− 197.353i	− 0.315764i	0.987458	−	0.157882i	$-0.0504666\pi$
$6$	0	0
$7$	3794.06i	1.58020i	0.612978	+	0.790100i	$0.289972\pi$
$7$	3794.06i	1.58020i	−0.612978	+	0.790100i	$0.710028\pi$
$8$	0	0
$9$	2187.00	0.333333
$10$	0	0
$11$	16480.4	1.12563	0.562817	−	0.826581i	$-0.309718\pi$
$11$	16480.4	1.12563	0.562817	+	0.826581i	$0.309718\pi$
$12$	0	0
$13$	− 11110.6i	− 0.389014i	−0.980901	−	0.194507i	$-0.937689\pi$
$13$	− 11110.6i	− 0.389014i	0.980901	−	0.194507i	$-0.0623106\pi$
$14$	0	0
$15$	9229.26i	0.182306i
$16$	0	0
$17$	66738.9	0.799067	0.399534	−	0.916719i	$-0.369172\pi$
$17$	66738.9	0.799067	0.399534	+	0.916719i	$0.369172\pi$
$18$	0	0
$19$	20481.8	0.157164	0.0785820	−	0.996908i	$-0.474961\pi$
$19$	20481.8	0.157164	0.0785820	+	0.996908i	$0.474961\pi$
$20$	0	0
$21$	− 177431.i	− 0.912329i
$22$	0	0
$23$	44505.2i	0.159037i	0.996833	+	0.0795187i	$0.0253383\pi$
$23$	44505.2i	0.159037i	−0.996833	+	0.0795187i	$0.974662\pi$
$24$	0	0
$25$	351677.	0.900293
$26$	0	0
$27$	−102276.	−0.192450
$28$	0	0
$29$	− 725037.i	− 1.02510i	−0.858656	−	0.512552i	$-0.828700\pi$
$29$	− 725037.i	− 1.02510i	0.858656	−	0.512552i	$-0.171300\pi$
$30$	0	0
$31$	44205.0i	0.0478657i	0.999714	+	0.0239329i	$0.00761880\pi$
$31$	44205.0i	0.0478657i	−0.999714	+	0.0239329i	$0.992381\pi$
$32$	0	0
$33$	−770713.	−0.649885
$34$	0	0
$35$	748768.	0.498971
$36$	0	0
$37$	− 2.85851e6i	− 1.52522i	−0.646857	−	0.762611i	$-0.723917\pi$
$37$	− 2.85851e6i	− 1.52522i	0.646857	−	0.762611i	$-0.276083\pi$
$38$	0	0
$39$	519592.i	0.224597i
$40$	0	0
$41$	−10702.9	−0.00378761	−0.00189381	−	0.999998i	$-0.500603\pi$
$41$	−10702.9	−0.00378761	−0.00189381	+	0.999998i	$0.500603\pi$
$42$	0	0
$43$	−374259.	−0.109471	−0.0547354	−	0.998501i	$-0.517432\pi$
$43$	−374259.	−0.109471	−0.0547354	+	0.998501i	$0.517432\pi$
$44$	0	0
$45$	− 431610.i	− 0.105255i
$46$	0	0
$47$	2.11469e6i	0.433365i	0.976242	+	0.216683i	$0.0695237\pi$
$47$	2.11469e6i	0.433365i	−0.976242	+	0.216683i	$0.930476\pi$
$48$	0	0
$49$	−8.63010e6	−1.49703
$50$	0	0
$51$	−3.12107e6	−0.461342
$52$	0	0
$53$	− 5.99442e6i	− 0.759703i	−0.925047	−	0.379852i	$-0.875975\pi$
$53$	− 5.99442e6i	− 0.759703i	0.925047	−	0.379852i	$-0.124025\pi$
$54$	0	0
$55$	− 3.25245e6i	− 0.355435i
$56$	0	0
$57$	−957837.	−0.0907387
$58$	0	0
$59$	1.11289e7	0.918424	0.459212	−	0.888327i	$-0.348132\pi$
$59$	1.11289e7	0.918424	0.459212	+	0.888327i	$0.348132\pi$
$60$	0	0
$61$	2.30232e7i	1.66282i	0.555657	+	0.831411i	$0.312467\pi$
$61$	2.30232e7i	1.66282i	−0.555657	+	0.831411i	$0.687533\pi$
$62$	0	0
$63$	8.29761e6i	0.526734i
$64$	0	0
$65$	−2.19271e6	−0.122837
$66$	0	0
$67$	−1.77776e7	−0.882212	−0.441106	−	0.897455i	$-0.645414\pi$
$67$	−1.77776e7	−0.882212	−0.441106	+	0.897455i	$0.645414\pi$
$68$	0	0
$69$	− 2.08130e6i	− 0.0918203i
$70$	0	0
$71$	2.44380e7i	0.961682i	0.876808	+	0.480841i	$0.159669\pi$
$71$	2.44380e7i	0.961682i	−0.876808	+	0.480841i	$0.840331\pi$
$72$	0	0
$73$	3.70522e6	0.130474	0.0652368	−	0.997870i	$-0.479220\pi$
$73$	3.70522e6	0.130474	0.0652368	+	0.997870i	$0.479220\pi$
$74$	0	0
$75$	−1.64463e7	−0.519784
$76$	0	0
$77$	6.25277e7i	1.77873i
$78$	0	0
$79$	− 5.98403e7i	− 1.53633i	−0.640251	−	0.768166i	$-0.721169\pi$
$79$	− 5.98403e7i	− 1.53633i	0.640251	−	0.768166i	$-0.278831\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	5.83719e7	1.22996	0.614981	−	0.788542i	$-0.289164\pi$
$83$	5.83719e7	1.22996	0.614981	+	0.788542i	$0.289164\pi$
$84$	0	0
$85$	− 1.31711e7i	− 0.252317i
$86$	0	0
$87$	3.39066e7i	0.591844i
$88$	0	0
$89$	−8.20248e7	−1.30733	−0.653665	−	0.756784i	$-0.726769\pi$
$89$	−8.20248e7	−1.30733	−0.653665	+	0.756784i	$0.726769\pi$
$90$	0	0
$91$	4.21544e7	0.614719
$92$	0	0
$93$	− 2.06726e6i	− 0.0276353i
$94$	0	0
$95$	− 4.04213e6i	− 0.0496267i
$96$	0	0
$97$	−5.81980e7	−0.657387	−0.328694	−	0.944437i	$-0.606608\pi$
$97$	−5.81980e7	−0.657387	−0.328694	+	0.944437i	$0.606608\pi$
$98$	0	0
$99$	3.60427e7	0.375211
$100$	0	0
$101$	− 7.64434e7i	− 0.734606i	−0.930101	−	0.367303i	$-0.880281\pi$
$101$	− 7.64434e7i	− 0.734606i	0.930101	−	0.367303i	$-0.119719\pi$
$102$	0	0
$103$	− 1.91556e8i	− 1.70195i	−0.525207	−	0.850974i	$-0.676012\pi$
$103$	− 1.91556e8i	− 1.70195i	0.525207	−	0.850974i	$-0.323988\pi$
$104$	0	0
$105$	−3.50164e7	−0.288081
$106$	0	0
$107$	−5.44812e7	−0.415635	−0.207817	−	0.978168i	$-0.566636\pi$
$107$	−5.44812e7	−0.415635	−0.207817	+	0.978168i	$0.566636\pi$
$108$	0	0
$109$	− 1.72940e8i	− 1.22515i	−0.790413	−	0.612574i	$-0.790134\pi$
$109$	− 1.72940e8i	− 1.22515i	0.790413	−	0.612574i	$-0.209866\pi$
$110$	0	0
$111$	1.33679e8i	0.880588i
$112$	0	0
$113$	−1.07735e8	−0.660761	−0.330380	−	0.943848i	$-0.607177\pi$
$113$	−1.07735e8	−0.660761	−0.330380	+	0.943848i	$0.607177\pi$
$114$	0	0
$115$	8.78321e6	0.0502183
$116$	0	0
$117$	− 2.42989e7i	− 0.129671i
$118$	0	0
$119$	2.53212e8i	1.26269i
$120$	0	0
$121$	5.72450e7	0.267052
$122$	0	0
$123$	500524.	0.00218678
$124$	0	0
$125$	− 1.46495e8i	− 0.600044i
$126$	0	0
$127$	− 2.05573e8i	− 0.790225i	−0.918633	−	0.395113i	$-0.870706\pi$
$127$	− 2.05573e8i	− 0.790225i	0.918633	−	0.395113i	$-0.129294\pi$
$128$	0	0
$129$	1.75024e7	0.0632030
$130$	0	0
$131$	3.21055e7	0.109017	0.0545084	−	0.998513i	$-0.482641\pi$
$131$	3.21055e7	0.109017	0.0545084	+	0.998513i	$0.482641\pi$
$132$	0	0
$133$	7.77091e7i	0.248351i
$134$	0	0
$135$	2.01844e7i	0.0607688i
$136$	0	0
$137$	6.85467e8	1.94583	0.972913	−	0.231171i	$-0.0742558\pi$
$137$	6.85467e8	1.94583	0.972913	+	0.231171i	$0.0742558\pi$
$138$	0	0
$139$	5.81121e8	1.55671	0.778354	−	0.627826i	$-0.216055\pi$
$139$	5.81121e8	1.55671	0.778354	+	0.627826i	$0.216055\pi$
$140$	0	0
$141$	− 9.88940e7i	− 0.250204i
$142$	0	0
$143$	− 1.83108e8i	− 0.437887i
$144$	0	0
$145$	−1.43088e8	−0.323691
$146$	0	0
$147$	4.03590e8	0.864313
$148$	0	0
$149$	8.86338e8i	1.79827i	0.437673	+	0.899134i	$0.355803\pi$
$149$	8.86338e8i	1.79827i	−0.437673	+	0.899134i	$0.644197\pi$
$150$	0	0
$151$	− 2.60061e8i	− 0.500227i	−0.968217	−	0.250114i	$-0.919532\pi$
$151$	− 2.60061e8i	− 0.500227i	0.968217	−	0.250114i	$-0.0804680\pi$
$152$	0	0
$153$	1.45958e8	0.266356
$154$	0	0
$155$	8.72397e6	0.0151143
$156$	0	0
$157$	− 8.08233e7i	− 0.133026i	−0.997786	−	0.0665132i	$-0.978813\pi$
$157$	− 8.08233e7i	− 0.133026i	0.997786	−	0.0665132i	$-0.0211875\pi$
$158$	0	0
$159$	2.80331e8i	0.438615i
$160$	0	0
$161$	−1.68855e8	−0.251311
$162$	0	0
$163$	4.11086e8	0.582348	0.291174	−	0.956670i	$-0.405954\pi$
$163$	4.11086e8	0.582348	0.291174	+	0.956670i	$0.405954\pi$
$164$	0	0
$165$	1.52102e8i	0.205210i
$166$	0	0
$167$	7.36188e8i	0.946504i	0.880927	+	0.473252i	$0.156920\pi$
$167$	7.36188e8i	0.946504i	−0.880927	+	0.473252i	$0.843080\pi$
$168$	0	0
$169$	6.92285e8	0.848668
$170$	0	0
$171$	4.47936e7	0.0523880
$172$	0	0
$173$	6.33738e8i	0.707498i	0.935340	+	0.353749i	$0.115093\pi$
$173$	6.33738e8i	0.707498i	−0.935340	+	0.353749i	$0.884907\pi$
$174$	0	0
$175$	1.33428e9i	1.42264i
$176$	0	0
$177$	−5.20446e8	−0.530252
$178$	0	0
$179$	1.50585e9	1.46679	0.733396	−	0.679802i	$-0.237934\pi$
$179$	1.50585e9	1.46679	0.733396	+	0.679802i	$0.237934\pi$
$180$	0	0
$181$	8.25523e8i	0.769157i	0.923092	+	0.384579i	$0.125653\pi$
$181$	8.25523e8i	0.769157i	−0.923092	+	0.384579i	$0.874347\pi$
$182$	0	0
$183$	− 1.07669e9i	− 0.960031i
$184$	0	0
$185$	−5.64135e8	−0.481611
$186$	0	0
$187$	1.09988e9	0.899458
$188$	0	0
$189$	− 3.88041e8i	− 0.304110i
$190$	0	0
$191$	− 1.94421e9i	− 1.46086i	−0.682986	−	0.730431i	$-0.739319\pi$
$191$	− 1.94421e9i	− 1.46086i	0.682986	−	0.730431i	$-0.260681\pi$
$192$	0	0
$193$	9.05725e8	0.652781	0.326390	−	0.945235i	$-0.394168\pi$
$193$	9.05725e8	0.652781	0.326390	+	0.945235i	$0.394168\pi$
$194$	0	0
$195$	1.02543e8	0.0709197
$196$	0	0
$197$	− 2.17129e9i	− 1.44163i	−0.693130	−	0.720813i	$-0.743769\pi$
$197$	− 2.17129e9i	− 1.44163i	0.693130	−	0.720813i	$-0.256231\pi$
$198$	0	0
$199$	5.33204e8i	0.340002i	0.985444	+	0.170001i	$0.0543771\pi$
$199$	5.33204e8i	0.340002i	−0.985444	+	0.170001i	$0.945623\pi$
$200$	0	0
$201$	8.31374e8	0.509345
$202$	0	0
$203$	2.75083e9	1.61987
$204$	0	0
$205$	2.11224e6i	0.00119599i
$206$	0	0
$207$	9.73328e7i	0.0530125i
$208$	0	0
$209$	3.37548e8	0.176909
$210$	0	0
$211$	1.80735e9	0.911828	0.455914	−	0.890024i	$-0.349312\pi$
$211$	1.80735e9	0.911828	0.455914	+	0.890024i	$0.349312\pi$
$212$	0	0
$213$	− 1.14285e9i	− 0.555227i
$214$	0	0
$215$	7.38610e7i	0.0345670i
$216$	0	0
$217$	−1.67717e8	−0.0756375
$218$	0	0
$219$	−1.73276e8	−0.0753289
$220$	0	0
$221$	− 7.41510e8i	− 0.310848i
$222$	0	0
$223$	3.30606e9i	1.33688i	0.743767	+	0.668439i	$0.233037\pi$
$223$	3.30606e9i	1.33688i	−0.743767	+	0.668439i	$0.766963\pi$
$224$	0	0
$225$	7.69118e8	0.300098
$226$	0	0
$227$	4.39061e9	1.65357	0.826783	−	0.562521i	$-0.190169\pi$
$227$	4.39061e9	1.65357	0.826783	+	0.562521i	$0.190169\pi$
$228$	0	0
$229$	− 4.51374e9i	− 1.64133i	−0.571413	−	0.820663i	$-0.693605\pi$
$229$	− 4.51374e9i	− 1.64133i	0.571413	−	0.820663i	$-0.306395\pi$
$230$	0	0
$231$	− 2.92413e9i	− 1.02695i
$232$	0	0
$233$	1.03865e9	0.352408	0.176204	−	0.984354i	$-0.443618\pi$
$233$	1.03865e9	0.352408	0.176204	+	0.984354i	$0.443618\pi$
$234$	0	0
$235$	4.17338e8	0.136841
$236$	0	0
$237$	2.79845e9i	0.887002i
$238$	0	0
$239$	− 4.05969e9i	− 1.24423i	−0.782925	−	0.622116i	$-0.786273\pi$
$239$	− 4.05969e9i	− 1.24423i	0.782925	−	0.622116i	$-0.213727\pi$
$240$	0	0
$241$	−1.83737e9	−0.544663	−0.272331	−	0.962204i	$-0.587795\pi$
$241$	−1.83737e9	−0.544663	−0.272331	+	0.962204i	$0.587795\pi$
$242$	0	0
$243$	−2.23677e8	−0.0641500
$244$	0	0
$245$	1.70317e9i	0.472710i
$246$	0	0
$247$	− 2.27565e8i	− 0.0611389i
$248$	0	0
$249$	−2.72979e9	−0.710119
$250$	0	0
$251$	2.99843e9	0.755437	0.377719	−	0.925920i	$-0.376709\pi$
$251$	2.99843e9	0.755437	0.377719	+	0.925920i	$0.376709\pi$
$252$	0	0
$253$	7.33464e8i	0.179018i
$254$	0	0
$255$	6.15951e8i	0.145675i
$256$	0	0
$257$	−4.06684e9	−0.932233	−0.466117	−	0.884723i	$-0.654347\pi$
$257$	−4.06684e9	−0.932233	−0.466117	+	0.884723i	$0.654347\pi$
$258$	0	0
$259$	1.08454e10	2.41016
$260$	0	0
$261$	− 1.58566e9i	− 0.341701i
$262$	0	0
$263$	3.67554e9i	0.768243i	0.923283	+	0.384122i	$0.125496\pi$
$263$	3.67554e9i	0.768243i	−0.923283	+	0.384122i	$0.874504\pi$
$264$	0	0
$265$	−1.18301e9	−0.239887
$266$	0	0
$267$	3.83592e9	0.754787
$268$	0	0
$269$	4.57397e9i	0.873542i	0.899573	+	0.436771i	$0.143878\pi$
$269$	4.57397e9i	0.873542i	−0.899573	+	0.436771i	$0.856122\pi$
$270$	0	0
$271$	1.80061e9i	0.333843i	0.985970	+	0.166922i	$0.0533827\pi$
$271$	1.80061e9i	0.333843i	−0.985970	+	0.166922i	$0.946617\pi$
$272$	0	0
$273$	−1.97136e9	−0.354908
$274$	0	0
$275$	5.79578e9	1.01340
$276$	0	0
$277$	1.08581e10i	1.84431i	0.386823	+	0.922154i	$0.373572\pi$
$277$	1.08581e10i	1.84431i	−0.386823	+	0.922154i	$0.626428\pi$
$278$	0	0
$279$	9.66764e7i	0.0159552i
$280$	0	0
$281$	−2.87260e9	−0.460734	−0.230367	−	0.973104i	$-0.573993\pi$
$281$	−2.87260e9	−0.460734	−0.230367	+	0.973104i	$0.573993\pi$
$282$	0	0
$283$	5.08258e8	0.0792389	0.0396195	−	0.999215i	$-0.487385\pi$
$283$	5.08258e8	0.0792389	0.0396195	+	0.999215i	$0.487385\pi$
$284$	0	0
$285$	1.89032e8i	0.0286520i
$286$	0	0
$287$	− 4.06074e7i	− 0.00598519i
$288$	0	0
$289$	−2.52168e9	−0.361491
$290$	0	0
$291$	2.72165e9	0.379543
$292$	0	0
$293$	− 2.13811e9i	− 0.290108i	−0.989424	−	0.145054i	$-0.953664\pi$
$293$	− 2.13811e9i	− 0.290108i	0.989424	−	0.145054i	$-0.0463356\pi$
$294$	0	0
$295$	− 2.19631e9i	− 0.290005i
$296$	0	0
$297$	−1.68555e9	−0.216628
$298$	0	0
$299$	4.94480e8	0.0618677
$300$	0	0
$301$	− 1.41996e9i	− 0.172986i
$302$	0	0
$303$	3.57490e9i	0.424125i
$304$	0	0
$305$	4.54368e9	0.525060
$306$	0	0
$307$	1.39905e10	1.57499	0.787497	−	0.616318i	$-0.211377\pi$
$307$	1.39905e10	1.57499	0.787497	+	0.616318i	$0.211377\pi$
$308$	0	0
$309$	8.95818e9i	0.982621i
$310$	0	0
$311$	5.84680e9i	0.624995i	0.949919	+	0.312497i	$0.101166\pi$
$311$	5.84680e9i	0.624995i	−0.949919	+	0.312497i	$0.898834\pi$
$312$	0	0
$313$	8.27511e9	0.862177	0.431088	−	0.902310i	$-0.358130\pi$
$313$	8.27511e9	0.862177	0.431088	+	0.902310i	$0.358130\pi$
$314$	0	0
$315$	1.63756e9	0.166324
$316$	0	0
$317$	− 1.37723e10i	− 1.36386i	−0.731419	−	0.681928i	$-0.761142\pi$
$317$	− 1.37723e10i	− 1.36386i	0.731419	−	0.681928i	$-0.238858\pi$
$318$	0	0
$319$	− 1.19489e10i	− 1.15389i
$320$	0	0
$321$	2.54784e9	0.239967
$322$	0	0
$323$	1.36693e9	0.125585
$324$	0	0
$325$	− 3.90735e9i	− 0.350226i
$326$	0	0
$327$	8.08759e9i	0.707340i
$328$	0	0
$329$	−8.02325e9	−0.684804
$330$	0	0
$331$	−2.73215e9	−0.227611	−0.113806	−	0.993503i	$-0.536304\pi$
$331$	−2.73215e9	−0.227611	−0.113806	+	0.993503i	$0.536304\pi$
$332$	0	0
$333$	− 6.25157e9i	− 0.508408i
$334$	0	0
$335$	3.50845e9i	0.278571i
$336$	0	0
$337$	2.19153e10	1.69913	0.849567	−	0.527481i	$-0.176863\pi$
$337$	2.19153e10	1.69913	0.849567	+	0.527481i	$0.176863\pi$
$338$	0	0
$339$	5.03828e9	0.381490
$340$	0	0
$341$	7.28517e8i	0.0538793i
$342$	0	0
$343$	− 1.08711e10i	− 0.785414i
$344$	0	0
$345$	−4.10750e8	−0.0289935
$346$	0	0
$347$	−9.17879e9	−0.633093	−0.316547	−	0.948577i	$-0.602523\pi$
$347$	−9.17879e9	−0.633093	−0.316547	+	0.948577i	$0.602523\pi$
$348$	0	0
$349$	− 1.80632e10i	− 1.21757i	−0.793335	−	0.608785i	$-0.791657\pi$
$349$	− 1.80632e10i	− 1.21757i	0.793335	−	0.608785i	$-0.208343\pi$
$350$	0	0
$351$	1.13635e9i	0.0748657i
$352$	0	0
$353$	4.40105e9	0.283438	0.141719	−	0.989907i	$-0.454737\pi$
$353$	4.40105e9	0.283438	0.141719	+	0.989907i	$0.454737\pi$
$354$	0	0
$355$	4.82289e9	0.303665
$356$	0	0
$357$	− 1.18415e10i	− 0.729013i
$358$	0	0
$359$	6.13107e8i	0.0369113i	0.999830	+	0.0184556i	$0.00587494\pi$
$359$	6.13107e8i	0.0369113i	−0.999830	+	0.0184556i	$0.994125\pi$
$360$	0	0
$361$	−1.65641e10	−0.975299
$362$	0	0
$363$	−2.67709e9	−0.154183
$364$	0	0
$365$	− 7.31234e8i	− 0.0411989i
$366$	0	0
$367$	− 1.46445e10i	− 0.807253i	−0.914924	−	0.403626i	$-0.867750\pi$
$367$	− 1.46445e10i	− 0.807253i	0.914924	−	0.403626i	$-0.132250\pi$
$368$	0	0
$369$	−2.34072e7	−0.00126254
$370$	0	0
$371$	2.27432e10	1.20048
$372$	0	0
$373$	1.37672e10i	0.711230i	0.934633	+	0.355615i	$0.115728\pi$
$373$	1.37672e10i	0.711230i	−0.934633	+	0.355615i	$0.884272\pi$
$374$	0	0
$375$	6.85090e9i	0.346436i
$376$	0	0
$377$	−8.05561e9	−0.398779
$378$	0	0
$379$	−4.02771e8	−0.0195210	−0.00976048	−	0.999952i	$-0.503107\pi$
$379$	−4.02771e8	−0.0195210	−0.00976048	+	0.999952i	$0.503107\pi$
$380$	0	0
$381$	9.61369e9i	0.456237i
$382$	0	0
$383$	2.88378e10i	1.34019i	0.742274	+	0.670097i	$0.233747\pi$
$383$	2.88378e10i	1.34019i	−0.742274	+	0.670097i	$0.766253\pi$
$384$	0	0
$385$	1.23400e10	0.561658
$386$	0	0
$387$	−8.18505e8	−0.0364903
$388$	0	0
$389$	− 1.15555e10i	− 0.504649i	−0.967643	−	0.252325i	$-0.918805\pi$
$389$	− 1.15555e10i	− 0.504649i	0.967643	−	0.252325i	$-0.0811951\pi$
$390$	0	0
$391$	2.97023e9i	0.127082i
$392$	0	0
$393$	−1.50142e9	−0.0629409
$394$	0	0
$395$	−1.18096e10	−0.485118
$396$	0	0
$397$	2.13939e10i	0.861247i	0.902532	+	0.430623i	$0.141706\pi$
$397$	2.13939e10i	0.861247i	−0.902532	+	0.430623i	$0.858294\pi$
$398$	0	0
$399$	− 3.63409e9i	− 0.143385i
$400$	0	0
$401$	2.83492e9	0.109639	0.0548194	−	0.998496i	$-0.482542\pi$
$401$	2.83492e9	0.109639	0.0548194	+	0.998496i	$0.482542\pi$
$402$	0	0
$403$	4.91145e8	0.0186204
$404$	0	0
$405$	− 9.43931e8i	− 0.0350849i
$406$	0	0
$407$	− 4.71095e10i	− 1.71684i
$408$	0	0
$409$	3.93438e10	1.40599	0.702997	−	0.711193i	$-0.251845\pi$
$409$	3.93438e10	1.40599	0.702997	+	0.711193i	$0.251845\pi$
$410$	0	0
$411$	−3.20561e10	−1.12342
$412$	0	0
$413$	4.22236e10i	1.45129i
$414$	0	0
$415$	− 1.15199e10i	− 0.388378i
$416$	0	0
$417$	−2.71763e10	−0.898766
$418$	0	0
$419$	3.17809e10	1.03112	0.515561	−	0.856853i	$-0.327583\pi$
$419$	3.17809e10	1.03112	0.515561	+	0.856853i	$0.327583\pi$
$420$	0	0
$421$	− 2.53716e9i	− 0.0807643i	−0.999184	−	0.0403821i	$-0.987142\pi$
$421$	− 2.53716e9i	− 0.0807643i	0.999184	−	0.0403821i	$-0.0128575\pi$
$422$	0	0
$423$	4.62482e9i	0.144455i
$424$	0	0
$425$	2.34705e10	0.719395
$426$	0	0
$427$	−8.73514e10	−2.62759
$428$	0	0
$429$	8.56309e9i	0.252814i
$430$	0	0
$431$	5.95521e9i	0.172579i	0.996270	+	0.0862894i	$0.0275010\pi$
$431$	5.95521e9i	0.172579i	−0.996270	+	0.0862894i	$0.972499\pi$
$432$	0	0
$433$	3.61421e10	1.02816	0.514081	−	0.857741i	$-0.328133\pi$
$433$	3.61421e10	1.02816	0.514081	+	0.857741i	$0.328133\pi$
$434$	0	0
$435$	6.69156e9	0.186883
$436$	0	0
$437$	9.11545e8i	0.0249949i
$438$	0	0
$439$	3.40899e10i	0.917840i	0.888478	+	0.458920i	$0.151764\pi$
$439$	3.40899e10i	0.917840i	−0.888478	+	0.458920i	$0.848236\pi$
$440$	0	0
$441$	−1.88740e10	−0.499011
$442$	0	0
$443$	−7.28042e10	−1.89035	−0.945174	−	0.326567i	$-0.894108\pi$
$443$	−7.28042e10	−1.89035	−0.945174	+	0.326567i	$0.894108\pi$
$444$	0	0
$445$	1.61878e10i	0.412808i
$446$	0	0
$447$	− 4.14499e10i	− 1.03823i
$448$	0	0
$449$	−3.63688e10	−0.894836	−0.447418	−	0.894325i	$-0.647656\pi$
$449$	−3.63688e10	−0.894836	−0.447418	+	0.894325i	$0.647656\pi$
$450$	0	0
$451$	−1.76388e8	−0.00426346
$452$	0	0
$453$	1.21618e10i	0.288806i
$454$	0	0
$455$	− 8.31927e9i	− 0.194106i
$456$	0	0
$457$	2.39306e10	0.548643	0.274321	−	0.961638i	$-0.411547\pi$
$457$	2.39306e10	0.548643	0.274321	+	0.961638i	$0.411547\pi$
$458$	0	0
$459$	−6.82578e9	−0.153781
$460$	0	0
$461$	1.37304e10i	0.304004i	0.988380	+	0.152002i	$0.0485720\pi$
$461$	1.37304e10i	0.304004i	−0.988380	+	0.152002i	$0.951428\pi$
$462$	0	0
$463$	5.34930e10i	1.16405i	0.813170	+	0.582026i	$0.197740\pi$
$463$	5.34930e10i	1.16405i	−0.813170	+	0.582026i	$0.802260\pi$
$464$	0	0
$465$	−4.07980e8	−0.00872623
$466$	0	0
$467$	−2.41485e8	−0.00507718	−0.00253859	−	0.999997i	$-0.500808\pi$
$467$	−2.41485e8	−0.00507718	−0.00253859	+	0.999997i	$0.500808\pi$
$468$	0	0
$469$	− 6.74492e10i	− 1.39407i
$470$	0	0
$471$	3.77973e9i	0.0768029i
$472$	0	0
$473$	−6.16794e9	−0.123224
$474$	0	0
$475$	7.20297e9	0.141494
$476$	0	0
$477$	− 1.31098e10i	− 0.253234i
$478$	0	0
$479$	1.58734e10i	0.301528i	0.988570	+	0.150764i	$0.0481734\pi$
$479$	1.58734e10i	0.301528i	−0.988570	+	0.150764i	$0.951827\pi$
$480$	0	0
$481$	−3.17598e10	−0.593332
$482$	0	0
$483$	7.89659e9	0.145094
$484$	0	0
$485$	1.14855e10i	0.207579i
$486$	0	0
$487$	6.42808e10i	1.14279i	0.820676	+	0.571393i	$0.193597\pi$
$487$	6.42808e10i	1.14279i	−0.820676	+	0.571393i	$0.806403\pi$
$488$	0	0
$489$	−1.92246e10	−0.336219
$490$	0	0
$491$	−1.06597e11	−1.83408	−0.917040	−	0.398795i	$-0.869429\pi$
$491$	−1.06597e11	−1.83408	−0.917040	+	0.398795i	$0.869429\pi$
$492$	0	0
$493$	− 4.83882e10i	− 0.819127i
$494$	0	0
$495$	− 7.11311e9i	− 0.118478i
$496$	0	0
$497$	−9.27191e10	−1.51965
$498$	0	0
$499$	−7.59553e10	−1.22506	−0.612528	−	0.790449i	$-0.709847\pi$
$499$	−7.59553e10	−1.22506	−0.612528	+	0.790449i	$0.709847\pi$
$500$	0	0
$501$	− 3.44281e10i	− 0.546465i
$502$	0	0
$503$	− 6.63220e10i	− 1.03606i	−0.855362	−	0.518031i	$-0.826665\pi$
$503$	− 6.63220e10i	− 1.03606i	0.855362	−	0.518031i	$-0.173335\pi$
$504$	0	0
$505$	−1.50863e10	−0.231962
$506$	0	0
$507$	−3.23750e10	−0.489979
$508$	0	0
$509$	− 6.87347e10i	− 1.02401i	−0.858982	−	0.512006i	$-0.828903\pi$
$509$	− 6.87347e10i	− 1.02401i	0.858982	−	0.512006i	$-0.171097\pi$
$510$	0	0
$511$	1.40578e10i	0.206174i
$512$	0	0
$513$	−2.09479e9	−0.0302462
$514$	0	0
$515$	−3.78040e10	−0.537414
$516$	0	0
$517$	3.48509e10i	0.487811i
$518$	0	0
$519$	− 2.96370e10i	− 0.408474i
$520$	0	0
$521$	−7.24827e10	−0.983747	−0.491874	−	0.870667i	$-0.663688\pi$
$521$	−7.24827e10	−0.983747	−0.491874	+	0.870667i	$0.663688\pi$
$522$	0	0
$523$	−1.21199e11	−1.61991	−0.809957	−	0.586489i	$-0.800510\pi$
$523$	−1.21199e11	−1.61991	−0.809957	+	0.586489i	$0.800510\pi$
$524$	0	0
$525$	− 6.23983e10i	− 0.821364i
$526$	0	0
$527$	2.95019e9i	0.0382479i
$528$	0	0
$529$	7.63303e10	0.974707
$530$	0	0
$531$	2.43388e10	0.306141
$532$	0	0
$533$	1.18916e8i	0.00147343i
$534$	0	0
$535$	1.07520e10i	0.131243i
$536$	0	0
$537$	−7.04215e10	−0.846853
$538$	0	0
$539$	−1.42228e11	−1.68511
$540$	0	0
$541$	3.08763e10i	0.360442i	0.983626	+	0.180221i	$0.0576813\pi$
$541$	3.08763e10i	0.360442i	−0.983626	+	0.180221i	$0.942319\pi$
$542$	0	0
$543$	− 3.86059e10i	− 0.444073i
$544$	0	0
$545$	−3.41301e10	−0.386858
$546$	0	0
$547$	1.59776e11	1.78468	0.892341	−	0.451361i	$-0.149061\pi$
$547$	1.59776e11	1.78468	0.892341	+	0.451361i	$0.149061\pi$
$548$	0	0
$549$	5.03517e10i	0.554274i
$550$	0	0
$551$	− 1.48500e10i	− 0.161109i
$552$	0	0
$553$	2.27038e11	2.42771
$554$	0	0
$555$	2.63820e10	0.278058
$556$	0	0
$557$	− 4.27428e10i	− 0.444061i	−0.975040	−	0.222030i	$-0.928732\pi$
$557$	− 4.27428e10i	− 0.444061i	0.975040	−	0.222030i	$-0.0712684\pi$
$558$	0	0
$559$	4.15825e9i	0.0425856i
$560$	0	0
$561$	−5.14365e10	−0.519302
$562$	0	0
$563$	−2.01627e10	−0.200685	−0.100343	−	0.994953i	$-0.531994\pi$
$563$	−2.01627e10	−0.200685	−0.100343	+	0.994953i	$0.531994\pi$
$564$	0	0
$565$	2.12618e10i	0.208644i
$566$	0	0
$567$	1.81469e10i	0.175578i
$568$	0	0
$569$	−6.07200e10	−0.579272	−0.289636	−	0.957137i	$-0.593534\pi$
$569$	−6.07200e10	−0.579272	−0.289636	+	0.957137i	$0.593534\pi$
$570$	0	0
$571$	−1.73024e11	−1.62765	−0.813825	−	0.581110i	$-0.802619\pi$
$571$	−1.73024e11	−1.62765	−0.813825	+	0.581110i	$0.802619\pi$
$572$	0	0
$573$	9.09216e10i	0.843429i
$574$	0	0
$575$	1.56514e10i	0.143180i
$576$	0	0
$577$	8.41388e10	0.759089	0.379545	−	0.925173i	$-0.376081\pi$
$577$	8.41388e10	0.759089	0.379545	+	0.925173i	$0.376081\pi$
$578$	0	0
$579$	−4.23566e10	−0.376883
$580$	0	0
$581$	2.21467e11i	1.94359i
$582$	0	0
$583$	− 9.87905e10i	− 0.855148i
$584$	0	0
$585$	−4.79545e9	−0.0409455
$586$	0	0
$587$	−2.93365e10	−0.247091	−0.123545	−	0.992339i	$-0.539426\pi$
$587$	−2.93365e10	−0.247091	−0.123545	+	0.992339i	$0.539426\pi$
$588$	0	0
$589$	9.05397e8i	0.00752277i
$590$	0	0
$591$	1.01541e11i	0.832323i
$592$	0	0
$593$	6.15955e10	0.498116	0.249058	−	0.968489i	$-0.419879\pi$
$593$	6.15955e10	0.498116	0.249058	+	0.968489i	$0.419879\pi$
$594$	0	0
$595$	4.99719e10	0.398711
$596$	0	0
$597$	− 2.49355e10i	− 0.196300i
$598$	0	0
$599$	1.86673e11i	1.45002i	0.688740	+	0.725008i	$0.258164\pi$
$599$	1.86673e11i	1.45002i	−0.688740	+	0.725008i	$0.741836\pi$
$600$	0	0
$601$	7.79316e10	0.597332	0.298666	−	0.954358i	$-0.403458\pi$
$601$	7.79316e10	0.597332	0.298666	+	0.954358i	$0.403458\pi$
$602$	0	0
$603$	−3.88795e10	−0.294071
$604$	0	0
$605$	− 1.12975e10i	− 0.0843255i
$606$	0	0
$607$	− 1.32801e11i	− 0.978242i	−0.872216	−	0.489121i	$-0.837318\pi$
$607$	− 1.32801e11i	− 0.978242i	0.872216	−	0.489121i	$-0.162682\pi$
$608$	0	0
$609$	−1.28644e11	−0.935233
$610$	0	0
$611$	2.34955e10	0.168585
$612$	0	0
$613$	1.41461e11i	1.00183i	0.865496	+	0.500916i	$0.167003\pi$
$613$	1.41461e11i	1.00183i	−0.865496	+	0.500916i	$0.832997\pi$
$614$	0	0
$615$	− 9.87797e7i	0 0.000690506i
$616$	0	0
$617$	1.08559e11	0.749076	0.374538	−	0.927212i	$-0.377801\pi$
$617$	1.08559e11	0.749076	0.374538	+	0.927212i	$0.377801\pi$
$618$	0	0
$619$	−1.65925e11	−1.13019	−0.565093	−	0.825027i	$-0.691160\pi$
$619$	−1.65925e11	−1.13019	−0.565093	+	0.825027i	$0.691160\pi$
$620$	0	0
$621$	− 4.55181e9i	− 0.0306068i
$622$	0	0
$623$	− 3.11207e11i	− 2.06584i
$624$	0	0
$625$	1.08463e11	0.710821
$626$	0	0
$627$	−1.57856e10	−0.102139
$628$	0	0
$629$	− 1.90774e11i	− 1.21876i
$630$	0	0
$631$	− 8.74514e10i	− 0.551632i	−0.961210	−	0.275816i	$-0.911052\pi$
$631$	− 8.74514e10i	− 0.551632i	0.961210	−	0.275816i	$-0.0889480\pi$
$632$	0	0
$633$	−8.45215e10	−0.526444
$634$	0	0
$635$	−4.05703e10	−0.249525
$636$	0	0
$637$	9.58858e10i	0.582367i
$638$	0	0
$639$	5.34458e10i	0.320561i
$640$	0	0
$641$	−2.77089e11	−1.64129	−0.820647	−	0.571435i	$-0.806387\pi$
$641$	−2.77089e11	−1.64129	−0.820647	+	0.571435i	$0.806387\pi$
$642$	0	0
$643$	−1.47094e10	−0.0860498	−0.0430249	−	0.999074i	$-0.513699\pi$
$643$	−1.47094e10	−0.0860498	−0.0430249	+	0.999074i	$0.513699\pi$
$644$	0	0
$645$	− 3.45414e9i	− 0.0199572i
$646$	0	0
$647$	2.14667e11i	1.22503i	0.790458	+	0.612517i	$0.209843\pi$
$647$	2.14667e11i	1.22503i	−0.790458	+	0.612517i	$0.790157\pi$
$648$	0	0
$649$	1.83408e11	1.03381
$650$	0	0
$651$	7.84333e9	0.0436693
$652$	0	0
$653$	8.93518e10i	0.491418i	0.969344	+	0.245709i	$0.0790207\pi$
$653$	8.93518e10i	0.491418i	−0.969344	+	0.245709i	$0.920979\pi$
$654$	0	0
$655$	− 6.33609e9i	− 0.0344236i
$656$	0	0
$657$	8.10331e9	0.0434912
$658$	0	0
$659$	1.96460e11	1.04167	0.520837	−	0.853656i	$-0.325620\pi$
$659$	1.96460e11	1.04167	0.520837	+	0.853656i	$0.325620\pi$
$660$	0	0
$661$	− 2.55340e11i	− 1.33756i	−0.743460	−	0.668780i	$-0.766817\pi$
$661$	− 2.55340e11i	− 1.33756i	0.743460	−	0.668780i	$-0.233183\pi$
$662$	0	0
$663$	3.46770e10i	0.179468i
$664$	0	0
$665$	1.53361e10	0.0784202
$666$	0	0
$667$	3.22679e10	0.163030
$668$	0	0
$669$	− 1.54609e11i	− 0.771847i
$670$	0	0
$671$	3.79431e11i	1.87173i
$672$	0	0
$673$	3.16723e11	1.54390	0.771950	−	0.635683i	$-0.219282\pi$
$673$	3.16723e11	1.54390	0.771950	+	0.635683i	$0.219282\pi$
$674$	0	0
$675$	−3.59681e10	−0.173261
$676$	0	0
$677$	− 2.24064e11i	− 1.06664i	−0.845914	−	0.533319i	$-0.820945\pi$
$677$	− 2.24064e11i	− 1.06664i	0.845914	−	0.533319i	$-0.179055\pi$
$678$	0	0
$679$	− 2.20807e11i	− 1.03880i
$680$	0	0
$681$	−2.05328e11	−0.954686
$682$	0	0
$683$	2.78839e11	1.28136	0.640679	−	0.767809i	$-0.278653\pi$
$683$	2.78839e11	1.28136	0.640679	+	0.767809i	$0.278653\pi$
$684$	0	0
$685$	− 1.35279e11i	− 0.614422i
$686$	0	0
$687$	2.11087e11i	0.947620i
$688$	0	0
$689$	−6.66017e10	−0.295535
$690$	0	0
$691$	2.74979e11	1.20611	0.603056	−	0.797699i	$-0.293949\pi$
$691$	2.74979e11	1.20611	0.603056	+	0.797699i	$0.293949\pi$
$692$	0	0
$693$	1.36748e11i	0.592909i
$694$	0	0
$695$	− 1.14686e11i	− 0.491552i
$696$	0	0
$697$	−7.14299e8	−0.00302656
$698$	0	0
$699$	−4.85728e10	−0.203463
$700$	0	0
$701$	1.21009e11i	0.501123i	0.968101	+	0.250562i	$0.0806152\pi$
$701$	1.21009e11i	0.501123i	−0.968101	+	0.250562i	$0.919385\pi$
$702$	0	0
$703$	− 5.85474e10i	− 0.239710i
$704$	0	0
$705$	−1.95170e10	−0.0790053
$706$	0	0
$707$	2.90031e11	1.16082
$708$	0	0
$709$	− 3.24092e11i	− 1.28258i	−0.767301	−	0.641288i	$-0.778401\pi$
$709$	− 3.24092e11i	− 1.28258i	0.767301	−	0.641288i	$-0.221599\pi$
$710$	0	0
$711$	− 1.30871e11i	− 0.512111i
$712$	0	0
$713$	−1.96735e9	−0.00761244
$714$	0	0
$715$	−3.61367e10	−0.138269
$716$	0	0
$717$	1.89853e11i	0.718358i
$718$	0	0
$719$	− 4.10633e11i	− 1.53652i	−0.640138	−	0.768260i	$-0.721123\pi$
$719$	− 4.10633e11i	− 1.53652i	0.640138	−	0.768260i	$-0.278877\pi$
$720$	0	0
$721$	7.26775e11	2.68942
$722$	0	0
$723$	8.59252e10	0.314461
$724$	0	0
$725$	− 2.54979e11i	− 0.922894i
$726$	0	0
$727$	1.06772e10i	0.0382225i	0.999817	+	0.0191112i	$0.00608367\pi$
$727$	1.06772e10i	0.0382225i	−0.999817	+	0.0191112i	$0.993916\pi$
$728$	0	0
$729$	1.04604e10	0.0370370
$730$	0	0
$731$	−2.49776e10	−0.0874746
$732$	0	0
$733$	2.08286e11i	0.721515i	0.932660	+	0.360757i	$0.117482\pi$
$733$	2.08286e11i	0.721515i	−0.932660	+	0.360757i	$0.882518\pi$
$734$	0	0
$735$	− 7.96495e10i	− 0.272919i
$736$	0	0
$737$	−2.92981e11	−0.993048
$738$	0	0
$739$	1.82789e11	0.612875	0.306437	−	0.951891i	$-0.400863\pi$
$739$	1.82789e11	0.612875	0.306437	+	0.951891i	$0.400863\pi$
$740$	0	0
$741$	1.06422e10i	0.0352986i
$742$	0	0
$743$	5.51013e11i	1.80803i	0.427498	+	0.904016i	$0.359395\pi$
$743$	5.51013e11i	1.80803i	−0.427498	+	0.904016i	$0.640605\pi$
$744$	0	0
$745$	1.74921e11	0.567829
$746$	0	0
$747$	1.27659e11	0.409987
$748$	0	0
$749$	− 2.06705e11i	− 0.656786i
$750$	0	0
$751$	4.25385e11i	1.33728i	0.743586	+	0.668640i	$0.233123\pi$
$751$	4.25385e11i	1.33728i	−0.743586	+	0.668640i	$0.766877\pi$
$752$	0	0
$753$	−1.40223e11	−0.436152
$754$	0	0
$755$	−5.13237e10	−0.157954
$756$	0	0
$757$	4.55433e11i	1.38689i	0.720512	+	0.693443i	$0.243907\pi$
$757$	4.55433e11i	1.38689i	−0.720512	+	0.693443i	$0.756093\pi$
$758$	0	0
$759$	− 3.43007e10i	− 0.103356i
$760$	0	0
$761$	−5.60243e11	−1.67047	−0.835233	−	0.549897i	$-0.814667\pi$
$761$	−5.60243e11	−1.67047	−0.835233	+	0.549897i	$0.814667\pi$
$762$	0	0
$763$	6.56144e11	1.93598
$764$	0	0
$765$	− 2.88052e10i	− 0.0841056i
$766$	0	0
$767$	− 1.23649e11i	− 0.357279i
$768$	0	0
$769$	2.49195e11	0.712581	0.356290	−	0.934375i	$-0.384041\pi$
$769$	2.49195e11	0.712581	0.356290	+	0.934375i	$0.384041\pi$
$770$	0	0
$771$	1.90187e11	0.538225
$772$	0	0
$773$	5.13512e11i	1.43824i	0.694884	+	0.719121i	$0.255456\pi$
$773$	5.13512e11i	1.43824i	−0.694884	+	0.719121i	$0.744544\pi$
$774$	0	0
$775$	1.55459e10i	0.0430932i
$776$	0	0
$777$	−5.07188e11	−1.39151
$778$	0	0
$779$	−2.19214e8	−0.000595276 0
$780$	0	0
$781$	4.02747e11i	1.08250i
$782$	0	0
$783$	7.41538e10i	0.197281i
$784$	0	0
$785$	−1.59507e10	−0.0420050
$786$	0	0
$787$	−1.88847e11	−0.492279	−0.246139	−	0.969234i	$-0.579162\pi$
$787$	−1.88847e11	−0.492279	−0.246139	+	0.969234i	$0.579162\pi$
$788$	0	0
$789$	− 1.71888e11i	− 0.443545i
$790$	0	0
$791$	− 4.08754e11i	− 1.04413i
$792$	0	0
$793$	2.55802e11	0.646861
$794$	0	0
$795$	5.53241e10	0.138499
$796$	0	0
$797$	− 4.98231e8i	− 0.00123480i	−1.00000		0.000617401i	$-0.999803\pi$
$797$	− 4.98231e8i	− 0.00123480i	1.00000		0.000617401i	$-0.000196525\pi$
$798$	0	0
$799$	1.41132e11i	0.346288i
$800$	0	0
$801$	−1.79388e11	−0.435777
$802$	0	0
$803$	6.10635e10	0.146865
$804$	0	0
$805$	3.33240e10i	0.0793550i
$806$	0	0
$807$	− 2.13903e11i	− 0.504340i
$808$	0	0
$809$	−4.76114e11	−1.11152	−0.555760	−	0.831343i	$-0.687573\pi$
$809$	−4.76114e11	−1.11152	−0.555760	+	0.831343i	$0.687573\pi$
$810$	0	0
$811$	7.86403e11	1.81787	0.908933	−	0.416942i	$-0.136898\pi$
$811$	7.86403e11	1.81787	0.908933	+	0.416942i	$0.136898\pi$
$812$	0	0
$813$	− 8.42063e10i	− 0.192745i
$814$	0	0
$815$	− 8.11289e10i	− 0.183885i
$816$	0	0
$817$	−7.66549e9	−0.0172049
$818$	0	0
$819$	9.21916e10	0.204906
$820$	0	0
$821$	7.14924e11i	1.57357i	0.617224	+	0.786787i	$0.288257\pi$
$821$	7.14924e11i	1.57357i	−0.617224	+	0.786787i	$0.711743\pi$
$822$	0	0
$823$	3.90745e11i	0.851715i	0.904790	+	0.425858i	$0.140028\pi$
$823$	3.90745e11i	0.851715i	−0.904790	+	0.425858i	$0.859972\pi$
$824$	0	0
$825$	−2.71042e11	−0.585087
$826$	0	0
$827$	4.83361e11	1.03336	0.516678	−	0.856180i	$-0.327169\pi$
$827$	4.83361e11	1.03336	0.516678	+	0.856180i	$0.327169\pi$
$828$	0	0
$829$	− 6.59993e11i	− 1.39740i	−0.715414	−	0.698701i	$-0.753762\pi$
$829$	− 6.59993e11i	− 1.39740i	0.715414	−	0.698701i	$-0.246238\pi$
$830$	0	0
$831$	− 5.07782e11i	− 1.06481i
$832$	0	0
$833$	−5.75964e11	−1.19623
$834$	0	0
$835$	1.45288e11	0.298872
$836$	0	0
$837$	− 4.52111e9i	− 0.00921176i
$838$	0	0
$839$	− 8.61080e11i	− 1.73778i	−0.495003	−	0.868891i	$-0.664833\pi$
$839$	− 8.61080e11i	− 1.73778i	0.495003	−	0.868891i	$-0.335167\pi$
$840$	0	0
$841$	−2.54319e10	−0.0508387
$842$	0	0
$843$	1.34338e11	0.266005
$844$	0	0
$845$	− 1.36624e11i	− 0.267979i
$846$	0	0
$847$	2.17191e11i	0.421996i
$848$	0	0
$849$	−2.37689e10	−0.0457486
$850$	0	0
$851$	1.27219e11	0.242567
$852$	0	0
$853$	5.20403e11i	0.982977i	0.870884	+	0.491489i	$0.163547\pi$
$853$	5.20403e11i	0.982977i	−0.870884	+	0.491489i	$0.836453\pi$
$854$	0	0
$855$	− 8.84014e9i	− 0.0165422i
$856$	0	0
$857$	8.53509e11	1.58229	0.791143	−	0.611632i	$-0.209486\pi$
$857$	8.53509e11	1.58229	0.791143	+	0.611632i	$0.209486\pi$
$858$	0	0
$859$	2.12057e11	0.389476	0.194738	−	0.980855i	$-0.437614\pi$
$859$	2.12057e11	0.389476	0.194738	+	0.980855i	$0.437614\pi$
$860$	0	0
$861$	1.89902e9i	0.00345555i
$862$	0	0
$863$	4.67543e11i	0.842905i	0.906851	+	0.421452i	$0.138480\pi$
$863$	4.67543e11i	0.842905i	−0.906851	+	0.421452i	$0.861520\pi$
$864$	0	0
$865$	1.25070e11	0.223403
$866$	0	0
$867$	1.17927e11	0.208707
$868$	0	0
$869$	− 9.86192e11i	− 1.72935i
$870$	0	0
$871$	1.97520e11i	0.343192i
$872$	0	0
$873$	−1.27279e11	−0.219129
$874$	0	0
$875$	5.55812e11	0.948190
$876$	0	0
$877$	− 5.09851e11i	− 0.861876i	−0.902382	−	0.430938i	$-0.858183\pi$
$877$	− 5.09851e11i	− 0.861876i	0.902382	−	0.430938i	$-0.141817\pi$
$878$	0	0
$879$	9.99895e10i	0.167494i
$880$	0	0
$881$	−7.67923e10	−0.127472	−0.0637359	−	0.997967i	$-0.520302\pi$
$881$	−7.67923e10	−0.127472	−0.0637359	+	0.997967i	$0.520302\pi$
$882$	0	0
$883$	−2.22619e11	−0.366201	−0.183101	−	0.983094i	$-0.558613\pi$
$883$	−2.22619e11	−0.366201	−0.183101	+	0.983094i	$0.558613\pi$
$884$	0	0
$885$	1.02711e11i	0.167435i
$886$	0	0
$887$	5.12545e11i	0.828014i	0.910274	+	0.414007i	$0.135871\pi$
$887$	5.12545e11i	0.828014i	−0.910274	+	0.414007i	$0.864129\pi$
$888$	0	0
$889$	7.79956e11	1.24871
$890$	0	0
$891$	7.88253e10	0.125070
$892$	0	0
$893$	4.33125e10i	0.0681094i
$894$	0	0
$895$	− 2.97183e11i	− 0.463160i
$896$	0	0
$897$	−2.31245e10	−0.0357193
$898$	0	0
$899$	3.20503e10	0.0490674
$900$	0	0
$901$	− 4.00061e11i	− 0.607054i
$902$	0	0
$903$	6.64050e10i	0.0998735i
$904$	0	0
$905$	1.62919e11	0.242872
$906$	0	0
$907$	6.10957e11	0.902779	0.451390	−	0.892327i	$-0.350928\pi$
$907$	6.10957e11	0.902779	0.451390	+	0.892327i	$0.350928\pi$
$908$	0	0
$909$	− 1.67182e11i	− 0.244869i
$910$	0	0
$911$	− 9.42463e11i	− 1.36833i	−0.729328	−	0.684165i	$-0.760167\pi$
$911$	− 9.42463e11i	− 1.36833i	0.729328	−	0.684165i	$-0.239833\pi$
$912$	0	0
$913$	9.61994e11	1.38449
$914$	0	0
$915$	−2.12487e11	−0.303143
$916$	0	0
$917$	1.21810e11i	0.172269i
$918$	0	0
$919$	8.39974e11i	1.17762i	0.808273	+	0.588808i	$0.200403\pi$
$919$	8.39974e11i	1.17762i	−0.808273	+	0.588808i	$0.799597\pi$
$920$	0	0
$921$	−6.54270e11	−0.909323
$922$	0	0
$923$	2.71521e11	0.374107
$924$	0	0
$925$	− 1.00527e12i	− 1.37315i
$926$	0	0
$927$	− 4.18933e11i	− 0.567316i
$928$	0	0
$929$	−1.01174e12	−1.35834	−0.679169	−	0.733982i	$-0.737660\pi$
$929$	−1.01174e12	−1.35834	−0.679169	+	0.733982i	$0.737660\pi$
$930$	0	0
$931$	−1.76760e11	−0.235280
$932$	0	0
$933$	− 2.73428e11i	− 0.360841i
$934$	0	0
$935$	− 2.17065e11i	− 0.284016i
$936$	0	0
$937$	1.06278e12	1.37875	0.689374	−	0.724406i	$-0.257886\pi$
$937$	1.06278e12	1.37875	0.689374	+	0.724406i	$0.257886\pi$
$938$	0	0
$939$	−3.86989e11	−0.497778
$940$	0	0
$941$	2.62998e11i	0.335424i	0.985836	+	0.167712i	$0.0536379\pi$
$941$	2.62998e11i	0.335424i	−0.985836	+	0.167712i	$0.946362\pi$
$942$	0	0
$943$	− 4.76334e8i	0 0.000602372i
$944$	0	0
$945$	−7.65809e10	−0.0960269
$946$	0	0
$947$	−4.60464e11	−0.572526	−0.286263	−	0.958151i	$-0.592413\pi$
$947$	−4.60464e11	−0.572526	−0.286263	+	0.958151i	$0.592413\pi$
$948$	0	0
$949$	− 4.11673e10i	− 0.0507560i
$950$	0	0
$951$	6.44066e11i	0.787423i
$952$	0	0
$953$	−1.73103e11	−0.209862	−0.104931	−	0.994480i	$-0.533462\pi$
$953$	−1.73103e11	−0.209862	−0.104931	+	0.994480i	$0.533462\pi$
$954$	0	0
$955$	−3.83694e11	−0.461288
$956$	0	0
$957$	5.58795e11i	0.666200i
$958$	0	0
$959$	2.60070e12i	3.07480i
$960$	0	0
$961$	8.50937e11	0.997709
$962$	0	0
$963$	−1.19150e11	−0.138545
$964$	0	0
$965$	− 1.78747e11i	− 0.206125i
$966$	0	0
$967$	− 6.52116e11i	− 0.745794i	−0.927873	−	0.372897i	$-0.878365\pi$
$967$	− 6.52116e11i	− 0.745794i	0.927873	−	0.372897i	$-0.121635\pi$
$968$	0	0
$969$	−6.39250e10	−0.0725063
$970$	0	0
$971$	7.61429e11	0.856549	0.428275	−	0.903649i	$-0.359122\pi$
$971$	7.61429e11	0.856549	0.428275	+	0.903649i	$0.359122\pi$
$972$	0	0
$973$	2.20481e12i	2.45991i
$974$	0	0
$975$	1.82729e11i	0.202203i
$976$	0	0
$977$	−1.01203e12	−1.11075	−0.555374	−	0.831600i	$-0.687425\pi$
$977$	−1.01203e12	−1.11075	−0.555374	+	0.831600i	$0.687425\pi$
$978$	0	0
$979$	−1.35180e12	−1.47158
$980$	0	0
$981$	− 3.78219e11i	− 0.408383i
$982$	0	0
$983$	− 1.43816e12i	− 1.54026i	−0.637886	−	0.770131i	$-0.720191\pi$
$983$	− 1.43816e12i	− 1.54026i	0.637886	−	0.770131i	$-0.279809\pi$
$984$	0	0
$985$	−4.28509e11	−0.455214
$986$	0	0
$987$	3.75210e11	0.395372
$988$	0	0
$989$	− 1.66565e10i	− 0.0174100i
$990$	0	0
$991$	− 1.76628e12i	− 1.83132i	−0.401953	−	0.915660i	$-0.631669\pi$
$991$	− 1.76628e12i	− 1.83132i	0.401953	−	0.915660i	$-0.368331\pi$
$992$	0	0
$993$	1.27770e11	0.131411
$994$	0	0
$995$	1.05229e11	0.107360
$996$	0	0
$997$	− 9.33822e11i	− 0.945112i	−0.881301	−	0.472556i	$-0.843331\pi$
$997$	− 9.33822e11i	− 0.945112i	0.881301	−	0.472556i	$-0.156669\pi$
$998$	0	0
$999$	2.92357e11i	0.293529i

Twists

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

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

\(f(q)\)	\(=\)	\(q-46.7654 q^{3} -197.353i q^{5} +3794.06i q^{7} +2187.00 q^{9} +O(q^{10})\)	\(q-46.7654 q^{3} -197.353i q^{5} +3794.06i q^{7} +2187.00 q^{9} +16480.4 q^{11} -11110.6i q^{13} +9229.26i q^{15} +66738.9 q^{17} +20481.8 q^{19} -177431. i q^{21} +44505.2i q^{23} +351677. q^{25} -102276. q^{27} -725037. i q^{29} +44205.0i q^{31} -770713. q^{33} +748768. q^{35} -2.85851e6i q^{37} +519592. i q^{39} -10702.9 q^{41} -374259. q^{43} -431610. i q^{45} +2.11469e6i q^{47} -8.63010e6 q^{49} -3.12107e6 q^{51} -5.99442e6i q^{53} -3.25245e6i q^{55} -957837. q^{57} +1.11289e7 q^{59} +2.30232e7i q^{61} +8.29761e6i q^{63} -2.19271e6 q^{65} -1.77776e7 q^{67} -2.08130e6i q^{69} +2.44380e7i q^{71} +3.70522e6 q^{73} -1.64463e7 q^{75} +6.25277e7i q^{77} -5.98403e7i q^{79} +4.78297e6 q^{81} +5.83719e7 q^{83} -1.31711e7i q^{85} +3.39066e7i q^{87} -8.20248e7 q^{89} +4.21544e7 q^{91} -2.06726e6i q^{93} -4.04213e6i q^{95} -5.81980e7 q^{97} +3.60427e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 17496 q^{9}+O(q^{10}) \) 8 * q + 17496 * q^9	\( 8 q + 17496 q^{9} + 276848 q^{17} - 2841976 q^{25} + 775008 q^{33} + 5826832 q^{41} - 16599928 q^{49} - 42366240 q^{57} - 132963072 q^{65} - 57759760 q^{73} + 38263752 q^{81} - 258778768 q^{89} - 771489424 q^{97}+O(q^{100}) \) 8 * q + 17496 * q^9 + 276848 * q^17 - 2841976 * q^25 + 775008 * q^33 + 5826832 * q^41 - 16599928 * q^49 - 42366240 * q^57 - 132963072 * q^65 - 57759760 * q^73 + 38263752 * q^81 - 258778768 * q^89 - 771489424 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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