LMFDB - Embedded newform 384.9.g.a.127.32

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			127.32
Character	$\chi$	$=$	384.127
Dual form			384.9.g.a.127.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} -779.371 q^{5} +4180.82i q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} -779.371 q^{5} +4180.82i q^{7} -2187.00 q^{9} -21024.0i q^{11} -19757.9 q^{13} -36447.6i q^{15} +1153.64 q^{17} +119133. i q^{19} -195518. q^{21} +539371. i q^{23} +216794. q^{25} -102276. i q^{27} -1.12540e6 q^{29} -1.73617e6i q^{31} +983197. q^{33} -3.25841e6i q^{35} -2.56469e6 q^{37} -923984. i q^{39} -104638. q^{41} -4.46811e6i q^{43} +1.70448e6 q^{45} -2.02273e6i q^{47} -1.17145e7 q^{49} +53950.2i q^{51} +5.27300e6 q^{53} +1.63855e7i q^{55} -5.57132e6 q^{57} -1.25906e7i q^{59} -6.80863e6 q^{61} -9.14346e6i q^{63} +1.53987e7 q^{65} +2.86278e6i q^{67} -2.52239e7 q^{69} +3.64993e7i q^{71} -4.82305e7 q^{73} +1.01385e7i q^{75} +8.78978e7 q^{77} +2.45732e6i q^{79} +4.78297e6 q^{81} +6.55150e7i q^{83} -899111. q^{85} -5.26298e7i q^{87} -3.96123e7 q^{89} -8.26041e7i q^{91} +8.11926e7 q^{93} -9.28491e7i q^{95} +9.06436e7 q^{97} +4.59796e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−779.371	−1.24699	−0.623497	−	0.781826i	$-0.714289\pi$
$5$	−779.371	−1.24699	−0.623497	+	0.781826i	$0.714289\pi$
$6$	0	0
$7$	4180.82i	1.74128i	0.491917	+	0.870642i	$0.336296\pi$
$7$	4180.82i	1.74128i	−0.491917	+	0.870642i	$0.663704\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	− 21024.0i	− 1.43597i	−0.696058	−	0.717985i	$-0.745064\pi$
$11$	− 21024.0i	− 1.43597i	0.696058	−	0.717985i	$-0.254936\pi$
$12$	0	0
$13$	−19757.9	−0.691778	−0.345889	−	0.938276i	$-0.612423\pi$
$13$	−19757.9	−0.691778	−0.345889	+	0.938276i	$0.612423\pi$
$14$	0	0
$15$	− 36447.6i	− 0.719952i
$16$	0	0
$17$	1153.64	0.0138125	0.00690627	−	0.999976i	$-0.497802\pi$
$17$	1153.64	0.0138125	0.00690627	+	0.999976i	$0.497802\pi$
$18$	0	0
$19$	119133.i	0.914153i	0.889427	+	0.457077i	$0.151104\pi$
$19$	119133.i	0.914153i	−0.889427	+	0.457077i	$0.848896\pi$
$20$	0	0
$21$	−195518.	−1.00533
$22$	0	0
$23$	539371.i	1.92742i	0.266950	+	0.963710i	$0.413984\pi$
$23$	539371.i	1.92742i	−0.266950	+	0.963710i	$0.586016\pi$
$24$	0	0
$25$	216794.	0.554993
$26$	0	0
$27$	− 102276.i	− 0.192450i
$28$	0	0
$29$	−1.12540e6	−1.59116	−0.795582	−	0.605846i	$-0.792835\pi$
$29$	−1.12540e6	−1.59116	−0.795582	+	0.605846i	$0.792835\pi$
$30$	0	0
$31$	− 1.73617e6i	− 1.87995i	−0.341248	−	0.939973i	$-0.610850\pi$
$31$	− 1.73617e6i	− 1.87995i	0.341248	−	0.939973i	$-0.389150\pi$
$32$	0	0
$33$	983197.	0.829058
$34$	0	0
$35$	− 3.25841e6i	− 2.17137i
$36$	0	0
$37$	−2.56469e6	−1.36845	−0.684224	−	0.729272i	$-0.739859\pi$
$37$	−2.56469e6	−1.36845	−0.684224	+	0.729272i	$0.739859\pi$
$38$	0	0
$39$	− 923984.i	− 0.399398i
$40$	0	0
$41$	−104638.	−0.0370299	−0.0185149	−	0.999829i	$-0.505894\pi$
$41$	−104638.	−0.0370299	−0.0185149	+	0.999829i	$0.505894\pi$
$42$	0	0
$43$	− 4.46811e6i	− 1.30692i	−0.756960	−	0.653462i	$-0.773316\pi$
$43$	− 4.46811e6i	− 1.30692i	0.756960	−	0.653462i	$-0.226684\pi$
$44$	0	0
$45$	1.70448e6	0.415665
$46$	0	0
$47$	− 2.02273e6i	− 0.414521i	−0.978286	−	0.207260i	$-0.933545\pi$
$47$	− 2.02273e6i	− 0.414521i	0.978286	−	0.207260i	$-0.0664547\pi$
$48$	0	0
$49$	−1.17145e7	−2.03207
$50$	0	0
$51$	53950.2i	0.00797467i
$52$	0	0
$53$	5.27300e6	0.668274	0.334137	−	0.942525i	$-0.391555\pi$
$53$	5.27300e6	0.668274	0.334137	+	0.942525i	$0.391555\pi$
$54$	0	0
$55$	1.63855e7i	1.79065i
$56$	0	0
$57$	−5.57132e6	−0.527787
$58$	0	0
$59$	− 1.25906e7i	− 1.03906i	−0.854453	−	0.519529i	$-0.826108\pi$
$59$	− 1.25906e7i	− 1.03906i	0.854453	−	0.519529i	$-0.173892\pi$
$60$	0	0
$61$	−6.80863e6	−0.491746	−0.245873	−	0.969302i	$-0.579075\pi$
$61$	−6.80863e6	−0.491746	−0.245873	+	0.969302i	$0.579075\pi$
$62$	0	0
$63$	− 9.14346e6i	− 0.580428i
$64$	0	0
$65$	1.53987e7	0.862642
$66$	0	0
$67$	2.86278e6i	0.142065i	0.997474	+	0.0710327i	$0.0226295\pi$
$67$	2.86278e6i	0.142065i	−0.997474	+	0.0710327i	$0.977371\pi$
$68$	0	0
$69$	−2.52239e7	−1.11280
$70$	0	0
$71$	3.64993e7i	1.43632i	0.695878	+	0.718160i	$0.255016\pi$
$71$	3.64993e7i	1.43632i	−0.695878	+	0.718160i	$0.744984\pi$
$72$	0	0
$73$	−4.82305e7	−1.69836	−0.849181	−	0.528101i	$-0.822904\pi$
$73$	−4.82305e7	−1.69836	−0.849181	+	0.528101i	$0.822904\pi$
$74$	0	0
$75$	1.01385e7i	0.320425i
$76$	0	0
$77$	8.78978e7	2.50043
$78$	0	0
$79$	2.45732e6i	0.0630890i	0.999502	+	0.0315445i	$0.0100426\pi$
$79$	2.45732e6i	0.0630890i	−0.999502	+	0.0315445i	$0.989957\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	6.55150e7i	1.38047i	0.723584	+	0.690237i	$0.242494\pi$
$83$	6.55150e7i	1.38047i	−0.723584	+	0.690237i	$0.757506\pi$
$84$	0	0
$85$	−899111.	−0.0172241
$86$	0	0
$87$	− 5.26298e7i	− 0.918659i
$88$	0	0
$89$	−3.96123e7	−0.631349	−0.315675	−	0.948868i	$-0.602231\pi$
$89$	−3.96123e7	−0.631349	−0.315675	+	0.948868i	$0.602231\pi$
$90$	0	0
$91$	− 8.26041e7i	− 1.20458i
$92$	0	0
$93$	8.11926e7	1.08539
$94$	0	0
$95$	− 9.28491e7i	− 1.13994i
$96$	0	0
$97$	9.06436e7	1.02388	0.511941	−	0.859020i	$-0.328926\pi$
$97$	9.06436e7	1.02388	0.511941	+	0.859020i	$0.328926\pi$
$98$	0	0
$99$	4.59796e7i	0.478657i
$100$	0	0
$101$	6.71660e7	0.645452	0.322726	−	0.946492i	$-0.395401\pi$
$101$	6.71660e7	0.645452	0.322726	+	0.946492i	$0.395401\pi$
$102$	0	0
$103$	− 6.99806e7i	− 0.621768i	−0.950448	−	0.310884i	$-0.899375\pi$
$103$	− 6.99806e7i	− 0.621768i	0.950448	−	0.310884i	$-0.100625\pi$
$104$	0	0
$105$	1.52381e8	1.25364
$106$	0	0
$107$	3.25033e6i	0.0247966i	0.999923	+	0.0123983i	$0.00394660\pi$
$107$	3.25033e6i	0.0247966i	−0.999923	+	0.0123983i	$0.996053\pi$
$108$	0	0
$109$	−7.76372e7	−0.550002	−0.275001	−	0.961444i	$-0.588678\pi$
$109$	−7.76372e7	−0.550002	−0.275001	+	0.961444i	$0.588678\pi$
$110$	0	0
$111$	− 1.19939e8i	− 0.790074i
$112$	0	0
$113$	4.79258e7	0.293938	0.146969	−	0.989141i	$-0.453048\pi$
$113$	4.79258e7	0.293938	0.146969	+	0.989141i	$0.453048\pi$
$114$	0	0
$115$	− 4.20370e8i	− 2.40348i
$116$	0	0
$117$	4.32104e7	0.230593
$118$	0	0
$119$	4.82315e6i	0.0240515i
$120$	0	0
$121$	−2.27652e8	−1.06201
$122$	0	0
$123$	− 4.89342e6i	− 0.0213792i
$124$	0	0
$125$	1.35479e8	0.554921
$126$	0	0
$127$	1.74138e8i	0.669390i	0.942327	+	0.334695i	$0.108633\pi$
$127$	1.74138e8i	0.669390i	−0.942327	+	0.334695i	$0.891367\pi$
$128$	0	0
$129$	2.08953e8	0.754552
$130$	0	0
$131$	1.20083e8i	0.407751i	0.978997	+	0.203876i	$0.0653538\pi$
$131$	1.20083e8i	0.407751i	−0.978997	+	0.203876i	$0.934646\pi$
$132$	0	0
$133$	−4.98076e8	−1.59180
$134$	0	0
$135$	7.97108e7i	0.239984i
$136$	0	0
$137$	3.44017e8	0.976558	0.488279	−	0.872688i	$-0.337625\pi$
$137$	3.44017e8	0.976558	0.488279	+	0.872688i	$0.337625\pi$
$138$	0	0
$139$	1.60623e8i	0.430278i	0.976583	+	0.215139i	$0.0690204\pi$
$139$	1.60623e8i	0.430278i	−0.976583	+	0.215139i	$0.930980\pi$
$140$	0	0
$141$	9.45937e7	0.239324
$142$	0	0
$143$	4.15390e8i	0.993372i
$144$	0	0
$145$	8.77104e8	1.98417
$146$	0	0
$147$	− 5.47832e8i	− 1.17322i
$148$	0	0
$149$	6.31905e8	1.28206	0.641028	−	0.767518i	$-0.278508\pi$
$149$	6.31905e8	1.28206	0.641028	+	0.767518i	$0.278508\pi$
$150$	0	0
$151$	6.25604e8i	1.20335i	0.798741	+	0.601675i	$0.205500\pi$
$151$	6.25604e8i	1.20335i	−0.798741	+	0.601675i	$0.794500\pi$
$152$	0	0
$153$	−2.52300e6	−0.00460418
$154$	0	0
$155$	1.35312e9i	2.34428i
$156$	0	0
$157$	2.22210e8	0.365734	0.182867	−	0.983138i	$-0.441462\pi$
$157$	2.22210e8	0.365734	0.182867	+	0.983138i	$0.441462\pi$
$158$	0	0
$159$	2.46594e8i	0.385828i
$160$	0	0
$161$	−2.25502e9	−3.35619
$162$	0	0
$163$	− 1.23489e9i	− 1.74935i	−0.484706	−	0.874677i	$-0.661073\pi$
$163$	− 1.23489e9i	− 1.74935i	0.484706	−	0.874677i	$-0.338927\pi$
$164$	0	0
$165$	−7.66275e8	−1.03383
$166$	0	0
$167$	1.39931e9i	1.79907i	0.436849	+	0.899535i	$0.356094\pi$
$167$	1.39931e9i	1.79907i	−0.436849	+	0.899535i	$0.643906\pi$
$168$	0	0
$169$	−4.25358e8	−0.521444
$170$	0	0
$171$	− 2.60545e8i	− 0.304718i
$172$	0	0
$173$	−2.21565e8	−0.247352	−0.123676	−	0.992323i	$-0.539468\pi$
$173$	−2.21565e8	−0.247352	−0.123676	+	0.992323i	$0.539468\pi$
$174$	0	0
$175$	9.06378e8i	0.966401i
$176$	0	0
$177$	5.88806e8	0.599900
$178$	0	0
$179$	− 7.31420e8i	− 0.712450i	−0.934400	−	0.356225i	$-0.884064\pi$
$179$	− 7.31420e8i	− 0.712450i	0.934400	−	0.356225i	$-0.115936\pi$
$180$	0	0
$181$	1.86140e9	1.73431	0.867153	−	0.498042i	$-0.165947\pi$
$181$	1.86140e9	1.73431	0.867153	+	0.498042i	$0.165947\pi$
$182$	0	0
$183$	− 3.18408e8i	− 0.283909i
$184$	0	0
$185$	1.99885e9	1.70645
$186$	0	0
$187$	− 2.42541e7i	− 0.0198344i
$188$	0	0
$189$	4.27597e8	0.335110
$190$	0	0
$191$	7.79607e8i	0.585790i	0.956145	+	0.292895i	$0.0946187\pi$
$191$	7.79607e8i	0.585790i	−0.956145	+	0.292895i	$0.905381\pi$
$192$	0	0
$193$	6.29345e8	0.453586	0.226793	−	0.973943i	$-0.427176\pi$
$193$	6.29345e8	0.453586	0.226793	+	0.973943i	$0.427176\pi$
$194$	0	0
$195$	7.20126e8i	0.498047i
$196$	0	0
$197$	7.95205e8	0.527976	0.263988	−	0.964526i	$-0.414962\pi$
$197$	7.95205e8	0.527976	0.263988	+	0.964526i	$0.414962\pi$
$198$	0	0
$199$	− 1.19534e9i	− 0.762218i	−0.924530	−	0.381109i	$-0.875542\pi$
$199$	− 1.19534e9i	− 0.762218i	0.924530	−	0.381109i	$-0.124458\pi$
$200$	0	0
$201$	−1.33879e8	−0.0820215
$202$	0	0
$203$	− 4.70510e9i	− 2.77067i
$204$	0	0
$205$	8.15515e7	0.0461760
$206$	0	0
$207$	− 1.17961e9i	− 0.642474i
$208$	0	0
$209$	2.50467e9	1.31270
$210$	0	0
$211$	2.39181e9i	1.20669i	0.797479	+	0.603347i	$0.206166\pi$
$211$	2.39181e9i	1.20669i	−0.797479	+	0.603347i	$0.793834\pi$
$212$	0	0
$213$	−1.70690e9	−0.829260
$214$	0	0
$215$	3.48232e9i	1.62972i
$216$	0	0
$217$	7.25862e9	3.27352
$218$	0	0
$219$	− 2.25552e9i	− 0.980550i
$220$	0	0
$221$	−2.27934e7	−0.00955520
$222$	0	0
$223$	− 2.75771e9i	− 1.11514i	−0.830130	−	0.557570i	$-0.811734\pi$
$223$	− 2.75771e9i	− 1.11514i	0.830130	−	0.557570i	$-0.188266\pi$
$224$	0	0
$225$	−4.74129e8	−0.184998
$226$	0	0
$227$	− 5.46107e8i	− 0.205671i	−0.994698	−	0.102836i	$-0.967208\pi$
$227$	− 5.46107e8i	− 0.205671i	0.994698	−	0.102836i	$-0.0327916\pi$
$228$	0	0
$229$	1.72082e9	0.625739	0.312869	−	0.949796i	$-0.398710\pi$
$229$	1.72082e9	0.625739	0.312869	+	0.949796i	$0.398710\pi$
$230$	0	0
$231$	4.11057e9i	1.44363i
$232$	0	0
$233$	3.90278e9	1.32419	0.662095	−	0.749420i	$-0.269668\pi$
$233$	3.90278e9	1.32419	0.662095	+	0.749420i	$0.269668\pi$
$234$	0	0
$235$	1.57646e9i	0.516905i
$236$	0	0
$237$	−1.14917e8	−0.0364244
$238$	0	0
$239$	3.48186e8i	0.106714i	0.998576	+	0.0533568i	$0.0169921\pi$
$239$	3.48186e8i	0.106714i	−0.998576	+	0.0533568i	$0.983008\pi$
$240$	0	0
$241$	1.22288e9	0.362508	0.181254	−	0.983436i	$-0.441984\pi$
$241$	1.22288e9	0.362508	0.181254	+	0.983436i	$0.441984\pi$
$242$	0	0
$243$	2.23677e8i	0.0641500i
$244$	0	0
$245$	9.12993e9	2.53398
$246$	0	0
$247$	− 2.35382e9i	− 0.632391i
$248$	0	0
$249$	−3.06383e9	−0.797017
$250$	0	0
$251$	− 3.48885e7i	− 0.00878996i	−0.999990	−	0.00439498i	$-0.998601\pi$
$251$	− 3.48885e7i	− 0.00878996i	0.999990	−	0.00439498i	$-0.00139897\pi$
$252$	0	0
$253$	1.13398e10	2.76772
$254$	0	0
$255$	− 4.20472e7i	− 0.00994436i
$256$	0	0
$257$	−3.69389e9	−0.846742	−0.423371	−	0.905956i	$-0.639153\pi$
$257$	−3.69389e9	−0.846742	−0.423371	+	0.905956i	$0.639153\pi$
$258$	0	0
$259$	− 1.07225e10i	− 2.38286i
$260$	0	0
$261$	2.46125e9	0.530388
$262$	0	0
$263$	− 1.15684e9i	− 0.241797i	−0.992665	−	0.120898i	$-0.961423\pi$
$263$	− 1.15684e9i	− 0.241797i	0.992665	−	0.120898i	$-0.0385775\pi$
$264$	0	0
$265$	−4.10962e9	−0.833333
$266$	0	0
$267$	− 1.85248e9i	− 0.364510i
$268$	0	0
$269$	−2.75413e9	−0.525987	−0.262993	−	0.964798i	$-0.584710\pi$
$269$	−2.75413e9	−0.525987	−0.262993	+	0.964798i	$0.584710\pi$
$270$	0	0
$271$	8.09075e9i	1.50007i	0.661398	+	0.750035i	$0.269963\pi$
$271$	8.09075e9i	1.50007i	−0.661398	+	0.750035i	$0.730037\pi$
$272$	0	0
$273$	3.86301e9	0.695465
$274$	0	0
$275$	− 4.55789e9i	− 0.796954i
$276$	0	0
$277$	−8.17325e9	−1.38827	−0.694137	−	0.719842i	$-0.744214\pi$
$277$	−8.17325e9	−1.38827	−0.694137	+	0.719842i	$0.744214\pi$
$278$	0	0
$279$	3.79700e9i	0.626649i
$280$	0	0
$281$	3.90452e9	0.626242	0.313121	−	0.949713i	$-0.398625\pi$
$281$	3.90452e9	0.626242	0.313121	+	0.949713i	$0.398625\pi$
$282$	0	0
$283$	− 8.28627e8i	− 0.129185i	−0.997912	−	0.0645926i	$-0.979425\pi$
$283$	− 8.28627e8i	− 0.129185i	0.997912	−	0.0645926i	$-0.0205748\pi$
$284$	0	0
$285$	4.34212e9	0.658147
$286$	0	0
$287$	− 4.37471e8i	− 0.0644796i
$288$	0	0
$289$	−6.97443e9	−0.999809
$290$	0	0
$291$	4.23898e9i	0.591139i
$292$	0	0
$293$	−3.83542e9	−0.520406	−0.260203	−	0.965554i	$-0.583790\pi$
$293$	−3.83542e9	−0.520406	−0.260203	+	0.965554i	$0.583790\pi$
$294$	0	0
$295$	9.81277e9i	1.29570i
$296$	0	0
$297$	−2.15025e9	−0.276353
$298$	0	0
$299$	− 1.06568e10i	− 1.33335i
$300$	0	0
$301$	1.86804e10	2.27572
$302$	0	0
$303$	3.14104e9i	0.372652i
$304$	0	0
$305$	5.30645e9	0.613204
$306$	0	0
$307$	− 3.79157e9i	− 0.426841i	−0.976960	−	0.213420i	$-0.931540\pi$
$307$	− 3.79157e9i	− 0.426841i	0.976960	−	0.213420i	$-0.0684604\pi$
$308$	0	0
$309$	3.27267e9	0.358978
$310$	0	0
$311$	3.37947e9i	0.361249i	0.983552	+	0.180625i	$0.0578119\pi$
$311$	3.37947e9i	0.361249i	−0.983552	+	0.180625i	$0.942188\pi$
$312$	0	0
$313$	−1.35946e9	−0.141641	−0.0708207	−	0.997489i	$-0.522562\pi$
$313$	−1.35946e9	−0.141641	−0.0708207	+	0.997489i	$0.522562\pi$
$314$	0	0
$315$	7.12615e9i	0.723790i
$316$	0	0
$317$	−2.10991e9	−0.208942	−0.104471	−	0.994528i	$-0.533315\pi$
$317$	−2.10991e9	−0.208942	−0.104471	+	0.994528i	$0.533315\pi$
$318$	0	0
$319$	2.36605e10i	2.28486i
$320$	0	0
$321$	−1.52003e8	−0.0143163
$322$	0	0
$323$	1.37437e8i	0.0126268i
$324$	0	0
$325$	−4.28339e9	−0.383932
$326$	0	0
$327$	− 3.63073e9i	− 0.317544i
$328$	0	0
$329$	8.45667e9	0.721798
$330$	0	0
$331$	− 1.21939e10i	− 1.01586i	−0.861399	−	0.507928i	$-0.830412\pi$
$331$	− 1.21939e10i	− 1.01586i	0.861399	−	0.507928i	$-0.169588\pi$
$332$	0	0
$333$	5.60898e9	0.456150
$334$	0	0
$335$	− 2.23116e9i	− 0.177155i
$336$	0	0
$337$	1.31318e10	1.01813	0.509065	−	0.860728i	$-0.329991\pi$
$337$	1.31318e10	1.01813	0.509065	+	0.860728i	$0.329991\pi$
$338$	0	0
$339$	2.24127e9i	0.169705i
$340$	0	0
$341$	−3.65013e10	−2.69955
$342$	0	0
$343$	− 2.48746e10i	− 1.79713i
$344$	0	0
$345$	1.96588e10	1.38765
$346$	0	0
$347$	− 2.97499e9i	− 0.205195i	−0.994723	−	0.102598i	$-0.967285\pi$
$347$	− 2.97499e9i	− 0.205195i	0.994723	−	0.102598i	$-0.0327154\pi$
$348$	0	0
$349$	−8.51646e9	−0.574060	−0.287030	−	0.957922i	$-0.592668\pi$
$349$	−8.51646e9	−0.574060	−0.287030	+	0.957922i	$0.592668\pi$
$350$	0	0
$351$	2.02075e9i	0.133133i
$352$	0	0
$353$	8.89883e9	0.573105	0.286553	−	0.958065i	$-0.407491\pi$
$353$	8.89883e9	0.573105	0.286553	+	0.958065i	$0.407491\pi$
$354$	0	0
$355$	− 2.84465e10i	− 1.79108i
$356$	0	0
$357$	−2.25556e8	−0.0138862
$358$	0	0
$359$	− 2.88120e10i	− 1.73459i	−0.497797	−	0.867293i	$-0.665858\pi$
$359$	− 2.88120e10i	− 1.73459i	0.497797	−	0.867293i	$-0.334142\pi$
$360$	0	0
$361$	2.79080e9	0.164323
$362$	0	0
$363$	− 1.06462e10i	− 0.613153i
$364$	0	0
$365$	3.75895e10	2.11785
$366$	0	0
$367$	1.94007e10i	1.06943i	0.845032	+	0.534715i	$0.179581\pi$
$367$	1.94007e10i	1.06943i	−0.845032	+	0.534715i	$0.820419\pi$
$368$	0	0
$369$	2.28842e8	0.0123433
$370$	0	0
$371$	2.20455e10i	1.16365i
$372$	0	0
$373$	8.75251e8	0.0452165	0.0226083	−	0.999744i	$-0.492803\pi$
$373$	8.75251e8	0.0452165	0.0226083	+	0.999744i	$0.492803\pi$
$374$	0	0
$375$	6.33571e9i	0.320384i
$376$	0	0
$377$	2.22355e10	1.10073
$378$	0	0
$379$	3.19857e10i	1.55024i	0.631815	+	0.775119i	$0.282310\pi$
$379$	3.19857e10i	1.55024i	−0.631815	+	0.775119i	$0.717690\pi$
$380$	0	0
$381$	−8.14363e9	−0.386472
$382$	0	0
$383$	− 1.01822e9i	− 0.0473202i	−0.999720	−	0.0236601i	$-0.992468\pi$
$383$	− 1.01822e9i	− 0.0473202i	0.999720	−	0.0236601i	$-0.00753195\pi$
$384$	0	0
$385$	−6.85050e10	−3.11802
$386$	0	0
$387$	9.77176e9i	0.435641i
$388$	0	0
$389$	9.77379e9	0.426840	0.213420	−	0.976961i	$-0.431540\pi$
$389$	9.77379e9	0.426840	0.213420	+	0.976961i	$0.431540\pi$
$390$	0	0
$391$	6.22238e8i	0.0266226i
$392$	0	0
$393$	−5.61571e9	−0.235415
$394$	0	0
$395$	− 1.91516e9i	− 0.0786715i
$396$	0	0
$397$	−2.65817e10	−1.07009	−0.535046	−	0.844823i	$-0.679706\pi$
$397$	−2.65817e10	−1.07009	−0.535046	+	0.844823i	$0.679706\pi$
$398$	0	0
$399$	− 2.32927e10i	− 0.919027i
$400$	0	0
$401$	2.28108e10	0.882192	0.441096	−	0.897460i	$-0.354590\pi$
$401$	2.28108e10	0.882192	0.441096	+	0.897460i	$0.354590\pi$
$402$	0	0
$403$	3.43030e10i	1.30050i
$404$	0	0
$405$	−3.72771e9	−0.138555
$406$	0	0
$407$	5.39202e10i	1.96505i
$408$	0	0
$409$	1.95240e10	0.697711	0.348855	−	0.937177i	$-0.386570\pi$
$409$	1.95240e10	0.697711	0.348855	+	0.937177i	$0.386570\pi$
$410$	0	0
$411$	1.60881e10i	0.563816i
$412$	0	0
$413$	5.26392e10	1.80929
$414$	0	0
$415$	− 5.10605e10i	− 1.72144i
$416$	0	0
$417$	−7.51160e9	−0.248421
$418$	0	0
$419$	3.55916e10i	1.15476i	0.816476	+	0.577379i	$0.195925\pi$
$419$	3.55916e10i	1.15476i	−0.816476	+	0.577379i	$0.804075\pi$
$420$	0	0
$421$	2.65923e10	0.846500	0.423250	−	0.906013i	$-0.360889\pi$
$421$	2.65923e10	0.846500	0.423250	+	0.906013i	$0.360889\pi$
$422$	0	0
$423$	4.42371e9i	0.138174i
$424$	0	0
$425$	2.50102e8	0.00766586
$426$	0	0
$427$	− 2.84657e10i	− 0.856269i
$428$	0	0
$429$	−1.94259e10	−0.573524
$430$	0	0
$431$	− 8.12717e9i	− 0.235521i	−0.993042	−	0.117761i	$-0.962428\pi$
$431$	− 8.12717e9i	− 0.235521i	0.993042	−	0.117761i	$-0.0375716\pi$
$432$	0	0
$433$	−2.99625e10	−0.852367	−0.426184	−	0.904637i	$-0.640142\pi$
$433$	−2.99625e10	−0.852367	−0.426184	+	0.904637i	$0.640142\pi$
$434$	0	0
$435$	4.10181e10i	1.14556i
$436$	0	0
$437$	−6.42571e10	−1.76196
$438$	0	0
$439$	− 2.93355e10i	− 0.789834i	−0.918717	−	0.394917i	$-0.870773\pi$
$439$	− 2.93355e10i	− 0.789834i	0.918717	−	0.394917i	$-0.129227\pi$
$440$	0	0
$441$	2.56196e10	0.677357
$442$	0	0
$443$	3.52014e10i	0.913997i	0.889467	+	0.456999i	$0.151076\pi$
$443$	3.52014e10i	0.913997i	−0.889467	+	0.456999i	$0.848924\pi$
$444$	0	0
$445$	3.08727e10	0.787288
$446$	0	0
$447$	2.95513e10i	0.740195i
$448$	0	0
$449$	−7.80346e10	−1.92000	−0.960001	−	0.279997i	$-0.909666\pi$
$449$	−7.80346e10	−1.92000	−0.960001	+	0.279997i	$0.909666\pi$
$450$	0	0
$451$	2.19991e9i	0.0531738i
$452$	0	0
$453$	−2.92566e10	−0.694754
$454$	0	0
$455$	6.43793e10i	1.50211i
$456$	0	0
$457$	4.08090e10	0.935601	0.467801	−	0.883834i	$-0.345047\pi$
$457$	4.08090e10	0.935601	0.467801	+	0.883834i	$0.345047\pi$
$458$	0	0
$459$	− 1.17989e8i	− 0.00265822i
$460$	0	0
$461$	−5.52346e10	−1.22295	−0.611473	−	0.791265i	$-0.709423\pi$
$461$	−5.52346e10	−1.22295	−0.611473	+	0.791265i	$0.709423\pi$
$462$	0	0
$463$	− 5.92351e10i	− 1.28901i	−0.764601	−	0.644504i	$-0.777064\pi$
$463$	− 5.92351e10i	− 1.28901i	0.764601	−	0.644504i	$-0.222936\pi$
$464$	0	0
$465$	−6.32792e10	−1.35347
$466$	0	0
$467$	− 1.38274e10i	− 0.290719i	−0.989379	−	0.145359i	$-0.953566\pi$
$467$	− 1.38274e10i	− 0.290719i	0.989379	−	0.145359i	$-0.0464338\pi$
$468$	0	0
$469$	−1.19688e10	−0.247376
$470$	0	0
$471$	1.03918e10i	0.211157i
$472$	0	0
$473$	−9.39377e10	−1.87670
$474$	0	0
$475$	2.58274e10i	0.507349i
$476$	0	0
$477$	−1.15321e10	−0.222758
$478$	0	0
$479$	4.05068e10i	0.769461i	0.923029	+	0.384730i	$0.125706\pi$
$479$	4.05068e10i	0.769461i	−0.923029	+	0.384730i	$0.874294\pi$
$480$	0	0
$481$	5.06729e10	0.946662
$482$	0	0
$483$	− 1.05457e11i	− 1.93770i
$484$	0	0
$485$	−7.06450e10	−1.27678
$486$	0	0
$487$	− 4.94953e10i	− 0.879931i	−0.898015	−	0.439965i	$-0.854991\pi$
$487$	− 4.94953e10i	− 0.879931i	0.898015	−	0.439965i	$-0.145009\pi$
$488$	0	0
$489$	5.77501e10	1.00999
$490$	0	0
$491$	− 3.97280e10i	− 0.683550i	−0.939782	−	0.341775i	$-0.888972\pi$
$491$	− 3.97280e10i	− 0.683550i	0.939782	−	0.341775i	$-0.111028\pi$
$492$	0	0
$493$	−1.29830e9	−0.0219780
$494$	0	0
$495$	− 3.58352e10i	− 0.596882i
$496$	0	0
$497$	−1.52597e11	−2.50104
$498$	0	0
$499$	− 7.81917e10i	− 1.26113i	−0.776138	−	0.630563i	$-0.782824\pi$
$499$	− 7.81917e10i	− 1.26113i	0.776138	−	0.630563i	$-0.217176\pi$
$500$	0	0
$501$	−6.54392e10	−1.03869
$502$	0	0
$503$	− 1.18256e11i	− 1.84736i	−0.383166	−	0.923680i	$-0.625166\pi$
$503$	− 1.18256e11i	− 1.84736i	0.383166	−	0.923680i	$-0.374834\pi$
$504$	0	0
$505$	−5.23472e10	−0.804874
$506$	0	0
$507$	− 1.98920e10i	− 0.301056i
$508$	0	0
$509$	1.84193e10	0.274411	0.137206	−	0.990543i	$-0.456188\pi$
$509$	1.84193e10	0.274411	0.137206	+	0.990543i	$0.456188\pi$
$510$	0	0
$511$	− 2.01643e11i	− 2.95733i
$512$	0	0
$513$	1.21845e10	0.175929
$514$	0	0
$515$	5.45408e10i	0.775341i
$516$	0	0
$517$	−4.25259e10	−0.595240
$518$	0	0
$519$	− 1.03616e10i	− 0.142809i
$520$	0	0
$521$	8.82546e10	1.19781	0.598903	−	0.800822i	$-0.295604\pi$
$521$	8.82546e10	1.19781	0.598903	+	0.800822i	$0.295604\pi$
$522$	0	0
$523$	− 9.36319e10i	− 1.25146i	−0.780040	−	0.625730i	$-0.784801\pi$
$523$	− 9.36319e10i	− 1.25146i	0.780040	−	0.625730i	$-0.215199\pi$
$524$	0	0
$525$	−4.23871e10	−0.557952
$526$	0	0
$527$	− 2.00291e9i	− 0.0259668i
$528$	0	0
$529$	−2.12610e11	−2.71495
$530$	0	0
$531$	2.75357e10i	0.346352i
$532$	0	0
$533$	2.06742e9	0.0256164
$534$	0	0
$535$	− 2.53321e9i	− 0.0309212i
$536$	0	0
$537$	3.42051e10	0.411333
$538$	0	0
$539$	2.46286e11i	2.91799i
$540$	0	0
$541$	1.05140e11	1.22738	0.613690	−	0.789547i	$-0.289684\pi$
$541$	1.05140e11	1.22738	0.613690	+	0.789547i	$0.289684\pi$
$542$	0	0
$543$	8.70491e10i	1.00130i
$544$	0	0
$545$	6.05082e10	0.685848
$546$	0	0
$547$	1.24531e11i	1.39101i	0.718523	+	0.695503i	$0.244819\pi$
$547$	1.24531e11i	1.39101i	−0.718523	+	0.695503i	$0.755181\pi$
$548$	0	0
$549$	1.48905e10	0.163915
$550$	0	0
$551$	− 1.34073e11i	− 1.45457i
$552$	0	0
$553$	−1.02736e10	−0.109856
$554$	0	0
$555$	9.34769e10i	0.985218i
$556$	0	0
$557$	2.50738e9	0.0260495	0.0130247	−	0.999915i	$-0.495854\pi$
$557$	2.50738e9	0.0260495	0.0130247	+	0.999915i	$0.495854\pi$
$558$	0	0
$559$	8.82803e10i	0.904100i
$560$	0	0
$561$	1.13425e9	0.0114514
$562$	0	0
$563$	− 5.23350e10i	− 0.520905i	−0.965487	−	0.260453i	$-0.916128\pi$
$563$	− 5.23350e10i	− 0.520905i	0.965487	−	0.260453i	$-0.0838718\pi$
$564$	0	0
$565$	−3.73520e10	−0.366539
$566$	0	0
$567$	1.99967e10i	0.193476i
$568$	0	0
$569$	1.77568e10	0.169400	0.0847002	−	0.996406i	$-0.473007\pi$
$569$	1.77568e10	0.169400	0.0847002	+	0.996406i	$0.473007\pi$
$570$	0	0
$571$	2.12437e10i	0.199842i	0.994995	+	0.0999209i	$0.0318590\pi$
$571$	2.12437e10i	0.199842i	−0.994995	+	0.0999209i	$0.968141\pi$
$572$	0	0
$573$	−3.64586e10	−0.338206
$574$	0	0
$575$	1.16933e11i	1.06971i
$576$	0	0
$577$	−8.18997e10	−0.738889	−0.369444	−	0.929253i	$-0.620452\pi$
$577$	−8.18997e10	−0.738889	−0.369444	+	0.929253i	$0.620452\pi$
$578$	0	0
$579$	2.94316e10i	0.261878i
$580$	0	0
$581$	−2.73906e11	−2.40380
$582$	0	0
$583$	− 1.10860e11i	− 0.959621i
$584$	0	0
$585$	−3.36770e10	−0.287547
$586$	0	0
$587$	1.58677e11i	1.33648i	0.743947	+	0.668238i	$0.232951\pi$
$587$	1.58677e11i	1.33648i	−0.743947	+	0.668238i	$0.767049\pi$
$588$	0	0
$589$	2.06836e11	1.71856
$590$	0	0
$591$	3.71880e10i	0.304827i
$592$	0	0
$593$	−6.36216e9	−0.0514500	−0.0257250	−	0.999669i	$-0.508189\pi$
$593$	−6.36216e9	−0.0514500	−0.0257250	+	0.999669i	$0.508189\pi$
$594$	0	0
$595$	− 3.75902e9i	− 0.0299921i
$596$	0	0
$597$	5.59005e10	0.440067
$598$	0	0
$599$	7.58957e10i	0.589536i	0.955569	+	0.294768i	$0.0952423\pi$
$599$	7.58957e10i	0.589536i	−0.955569	+	0.294768i	$0.904758\pi$
$600$	0	0
$601$	5.32835e10	0.408409	0.204204	−	0.978928i	$-0.434539\pi$
$601$	5.32835e10	0.408409	0.204204	+	0.978928i	$0.434539\pi$
$602$	0	0
$603$	− 6.26089e9i	− 0.0473551i
$604$	0	0
$605$	1.77425e11	1.32432
$606$	0	0
$607$	− 1.50399e11i	− 1.10787i	−0.832560	−	0.553935i	$-0.813126\pi$
$607$	− 1.50399e11i	− 1.10787i	0.832560	−	0.553935i	$-0.186874\pi$
$608$	0	0
$609$	2.20036e11	1.59965
$610$	0	0
$611$	3.99648e10i	0.286756i
$612$	0	0
$613$	−8.69995e10	−0.616134	−0.308067	−	0.951365i	$-0.599682\pi$
$613$	−8.69995e10	−0.616134	−0.308067	+	0.951365i	$0.599682\pi$
$614$	0	0
$615$	3.81379e9i	0.0266597i
$616$	0	0
$617$	−5.61787e9	−0.0387642	−0.0193821	−	0.999812i	$-0.506170\pi$
$617$	−5.61787e9	−0.0387642	−0.0193821	+	0.999812i	$0.506170\pi$
$618$	0	0
$619$	1.43212e11i	0.975475i	0.872990	+	0.487738i	$0.162178\pi$
$619$	1.43212e11i	0.975475i	−0.872990	+	0.487738i	$0.837822\pi$
$620$	0	0
$621$	5.51647e10	0.370932
$622$	0	0
$623$	− 1.65612e11i	− 1.09936i
$624$	0	0
$625$	−1.90273e11	−1.24698
$626$	0	0
$627$	1.17132e11i	0.757886i
$628$	0	0
$629$	−2.95872e9	−0.0189017
$630$	0	0
$631$	1.18575e11i	0.747957i	0.927437	+	0.373979i	$0.122007\pi$
$631$	1.18575e11i	0.747957i	−0.927437	+	0.373979i	$0.877993\pi$
$632$	0	0
$633$	−1.11854e11	−0.696685
$634$	0	0
$635$	− 1.35718e11i	− 0.834725i
$636$	0	0
$637$	2.31453e11	1.40574
$638$	0	0
$639$	− 7.98240e10i	− 0.478774i
$640$	0	0
$641$	1.99839e11	1.18372	0.591859	−	0.806041i	$-0.298394\pi$
$641$	1.99839e11	1.18372	0.591859	+	0.806041i	$0.298394\pi$
$642$	0	0
$643$	− 3.11107e11i	− 1.81998i	−0.414635	−	0.909988i	$-0.636091\pi$
$643$	− 3.11107e11i	− 1.81998i	0.414635	−	0.909988i	$-0.363909\pi$
$644$	0	0
$645$	−1.62852e11	−0.940922
$646$	0	0
$647$	1.59820e11i	0.912038i	0.889970	+	0.456019i	$0.150725\pi$
$647$	1.59820e11i	0.912038i	−0.889970	+	0.456019i	$0.849275\pi$
$648$	0	0
$649$	−2.64706e11	−1.49206
$650$	0	0
$651$	3.39452e11i	1.88997i
$652$	0	0
$653$	−2.87808e11	−1.58289	−0.791445	−	0.611240i	$-0.790671\pi$
$653$	−2.87808e11	−1.58289	−0.791445	+	0.611240i	$0.790671\pi$
$654$	0	0
$655$	− 9.35890e10i	− 0.508463i
$656$	0	0
$657$	1.05480e11	0.566121
$658$	0	0
$659$	6.95274e10i	0.368650i	0.982865	+	0.184325i	$0.0590099\pi$
$659$	6.95274e10i	0.368650i	−0.982865	+	0.184325i	$0.940990\pi$
$660$	0	0
$661$	−1.04720e11	−0.548558	−0.274279	−	0.961650i	$-0.588439\pi$
$661$	−1.04720e11	−0.548558	−0.274279	+	0.961650i	$0.588439\pi$
$662$	0	0
$663$	− 1.06594e9i	− 0.00551670i
$664$	0	0
$665$	3.88186e11	1.98497
$666$	0	0
$667$	− 6.07009e11i	− 3.06684i
$668$	0	0
$669$	1.28965e11	0.643827
$670$	0	0
$671$	1.43145e11i	0.706132i
$672$	0	0
$673$	−2.14643e11	−1.04630	−0.523150	−	0.852241i	$-0.675243\pi$
$673$	−2.14643e11	−1.04630	−0.523150	+	0.852241i	$0.675243\pi$
$674$	0	0
$675$	− 2.21728e10i	− 0.106808i
$676$	0	0
$677$	3.51692e11	1.67420	0.837101	−	0.547048i	$-0.184248\pi$
$677$	3.51692e11	1.67420	0.837101	+	0.547048i	$0.184248\pi$
$678$	0	0
$679$	3.78965e11i	1.78287i
$680$	0	0
$681$	2.55389e10	0.118744
$682$	0	0
$683$	− 4.32985e11i	− 1.98971i	−0.101294	−	0.994857i	$-0.532298\pi$
$683$	− 4.32985e11i	− 1.98971i	0.101294	−	0.994857i	$-0.467702\pi$
$684$	0	0
$685$	−2.68117e11	−1.21776
$686$	0	0
$687$	8.04747e10i	0.361270i
$688$	0	0
$689$	−1.04183e11	−0.462297
$690$	0	0
$691$	1.38328e11i	0.606735i	0.952874	+	0.303368i	$0.0981110\pi$
$691$	1.38328e11i	0.606735i	−0.952874	+	0.303368i	$0.901889\pi$
$692$	0	0
$693$	−1.92233e11	−0.833478
$694$	0	0
$695$	− 1.25185e11i	− 0.536554i
$696$	0	0
$697$	−1.20714e8	−0.000511476 0
$698$	0	0
$699$	1.82515e11i	0.764521i
$700$	0	0
$701$	−3.38464e11	−1.40165	−0.700827	−	0.713331i	$-0.747186\pi$
$701$	−3.38464e11	−1.40165	−0.700827	+	0.713331i	$0.747186\pi$
$702$	0	0
$703$	− 3.05541e11i	− 1.25097i
$704$	0	0
$705$	−7.37236e10	−0.298435
$706$	0	0
$707$	2.80809e11i	1.12391i
$708$	0	0
$709$	−1.26673e11	−0.501302	−0.250651	−	0.968077i	$-0.580645\pi$
$709$	−1.26673e11	−0.501302	−0.250651	+	0.968077i	$0.580645\pi$
$710$	0	0
$711$	− 5.37416e9i	− 0.0210297i
$712$	0	0
$713$	9.36440e11	3.62345
$714$	0	0
$715$	− 3.23743e11i	− 1.23873i
$716$	0	0
$717$	−1.62830e10	−0.0616111
$718$	0	0
$719$	1.89365e11i	0.708574i	0.935137	+	0.354287i	$0.115276\pi$
$719$	1.89365e11i	0.708574i	−0.935137	+	0.354287i	$0.884724\pi$
$720$	0	0
$721$	2.92576e11	1.08268
$722$	0	0
$723$	5.71886e10i	0.209294i
$724$	0	0
$725$	−2.43980e11	−0.883085
$726$	0	0
$727$	− 1.52166e11i	− 0.544729i	−0.962194	−	0.272365i	$-0.912194\pi$
$727$	− 1.52166e11i	− 0.544729i	0.962194	−	0.272365i	$-0.0878057\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	− 5.15457e9i	− 0.0180519i
$732$	0	0
$733$	3.89373e11	1.34881	0.674404	−	0.738363i	$-0.264401\pi$
$733$	3.89373e11	1.34881	0.674404	+	0.738363i	$0.264401\pi$
$734$	0	0
$735$	4.26965e11i	1.46299i
$736$	0	0
$737$	6.01871e10	0.204002
$738$	0	0
$739$	− 1.91793e11i	− 0.643067i	−0.946898	−	0.321533i	$-0.895802\pi$
$739$	− 1.91793e11i	− 0.643067i	0.946898	−	0.321533i	$-0.104198\pi$
$740$	0	0
$741$	1.10077e11	0.365111
$742$	0	0
$743$	− 1.23830e11i	− 0.406321i	−0.979145	−	0.203161i	$-0.934879\pi$
$743$	− 1.23830e11i	− 0.406321i	0.979145	−	0.203161i	$-0.0651214\pi$
$744$	0	0
$745$	−4.92489e11	−1.59872
$746$	0	0
$747$	− 1.43281e11i	− 0.460158i
$748$	0	0
$749$	−1.35891e10	−0.0431779
$750$	0	0
$751$	3.91995e11i	1.23231i	0.787624	+	0.616157i	$0.211311\pi$
$751$	3.91995e11i	1.23231i	−0.787624	+	0.616157i	$0.788689\pi$
$752$	0	0
$753$	1.63157e9	0.00507489
$754$	0	0
$755$	− 4.87578e11i	− 1.50057i
$756$	0	0
$757$	5.29589e11	1.61271	0.806353	−	0.591435i	$-0.201438\pi$
$757$	5.29589e11	1.61271	0.806353	+	0.591435i	$0.201438\pi$
$758$	0	0
$759$	5.30308e11i	1.59794i
$760$	0	0
$761$	−4.17795e11	−1.24573	−0.622865	−	0.782329i	$-0.714032\pi$
$761$	−4.17795e11	−1.24573	−0.622865	+	0.782329i	$0.714032\pi$
$762$	0	0
$763$	− 3.24587e11i	− 0.957709i
$764$	0	0
$765$	1.96636e9	0.00574138
$766$	0	0
$767$	2.48764e11i	0.718797i
$768$	0	0
$769$	−4.01895e11	−1.14923	−0.574615	−	0.818424i	$-0.694848\pi$
$769$	−4.01895e11	−1.14923	−0.574615	+	0.818424i	$0.694848\pi$
$770$	0	0
$771$	− 1.72746e11i	− 0.488867i
$772$	0	0
$773$	3.26865e11	0.915484	0.457742	−	0.889085i	$-0.348658\pi$
$773$	3.26865e11	0.915484	0.457742	+	0.889085i	$0.348658\pi$
$774$	0	0
$775$	− 3.76392e11i	− 1.04336i
$776$	0	0
$777$	5.01443e11	1.37574
$778$	0	0
$779$	− 1.24658e10i	− 0.0338510i
$780$	0	0
$781$	7.67364e11	2.06251
$782$	0	0
$783$	1.15101e11i	0.306220i
$784$	0	0
$785$	−1.73184e11	−0.456069
$786$	0	0
$787$	1.18744e11i	0.309538i	0.987951	+	0.154769i	$0.0494633\pi$
$787$	1.18744e11i	0.309538i	−0.987951	+	0.154769i	$0.950537\pi$
$788$	0	0
$789$	5.41000e10	0.139601
$790$	0	0
$791$	2.00369e11i	0.511830i
$792$	0	0
$793$	1.34524e11	0.340179
$794$	0	0
$795$	− 1.92188e11i	− 0.481125i
$796$	0	0
$797$	−1.54317e11	−0.382455	−0.191228	−	0.981546i	$-0.561247\pi$
$797$	−1.54317e11	−0.382455	−0.191228	+	0.981546i	$0.561247\pi$
$798$	0	0
$799$	− 2.33349e9i	− 0.00572558i
$800$	0	0
$801$	8.66320e10	0.210450
$802$	0	0
$803$	1.01400e12i	2.43880i
$804$	0	0
$805$	1.75749e12	4.18514
$806$	0	0
$807$	− 1.28798e11i	− 0.303678i
$808$	0	0
$809$	−3.13015e11	−0.730753	−0.365376	−	0.930860i	$-0.619060\pi$
$809$	−3.13015e11	−0.730753	−0.365376	+	0.930860i	$0.619060\pi$
$810$	0	0
$811$	6.76735e11i	1.56436i	0.623056	+	0.782178i	$0.285891\pi$
$811$	6.76735e11i	1.56436i	−0.623056	+	0.782178i	$0.714109\pi$
$812$	0	0
$813$	−3.78367e11	−0.866065
$814$	0	0
$815$	9.62438e11i	2.18143i
$816$	0	0
$817$	5.32301e11	1.19473
$818$	0	0
$819$	1.80655e11i	0.401527i
$820$	0	0
$821$	1.54523e11	0.340111	0.170056	−	0.985434i	$-0.445605\pi$
$821$	1.54523e11	0.340111	0.170056	+	0.985434i	$0.445605\pi$
$822$	0	0
$823$	− 6.00452e11i	− 1.30882i	−0.756141	−	0.654409i	$-0.772918\pi$
$823$	− 6.00452e11i	− 1.30882i	0.756141	−	0.654409i	$-0.227082\pi$
$824$	0	0
$825$	2.13151e11	0.460122
$826$	0	0
$827$	− 2.50374e11i	− 0.535264i	−0.963521	−	0.267632i	$-0.913759\pi$
$827$	− 2.50374e11i	− 0.535264i	0.963521	−	0.267632i	$-0.0862411\pi$
$828$	0	0
$829$	3.87352e11	0.820139	0.410069	−	0.912054i	$-0.365504\pi$
$829$	3.87352e11	0.820139	0.410069	+	0.912054i	$0.365504\pi$
$830$	0	0
$831$	− 3.82225e11i	− 0.801521i
$832$	0	0
$833$	−1.35143e10	−0.0280680
$834$	0	0
$835$	− 1.09058e12i	− 2.24343i
$836$	0	0
$837$	−1.77568e11	−0.361796
$838$	0	0
$839$	− 2.06378e11i	− 0.416500i	−0.978076	−	0.208250i	$-0.933223\pi$
$839$	− 2.06378e11i	− 0.416500i	0.978076	−	0.208250i	$-0.0667768\pi$
$840$	0	0
$841$	7.66279e11	1.53180
$842$	0	0
$843$	1.82596e11i	0.361561i
$844$	0	0
$845$	3.31511e11	0.650237
$846$	0	0
$847$	− 9.51771e11i	− 1.84926i
$848$	0	0
$849$	3.87510e10	0.0745852
$850$	0	0
$851$	− 1.38332e12i	− 2.63758i
$852$	0	0
$853$	1.09299e10	0.0206452	0.0103226	−	0.999947i	$-0.496714\pi$
$853$	1.09299e10	0.0206452	0.0103226	+	0.999947i	$0.496714\pi$
$854$	0	0
$855$	2.03061e11i	0.379981i
$856$	0	0
$857$	6.02367e11	1.11670	0.558352	−	0.829604i	$-0.311434\pi$
$857$	6.02367e11	1.11670	0.558352	+	0.829604i	$0.311434\pi$
$858$	0	0
$859$	− 2.58905e11i	− 0.475520i	−0.971324	−	0.237760i	$-0.923587\pi$
$859$	− 2.58905e11i	− 0.475520i	0.971324	−	0.237760i	$-0.0764131\pi$
$860$	0	0
$861$	2.04585e10	0.0372273
$862$	0	0
$863$	− 3.39353e11i	− 0.611799i	−0.952064	−	0.305899i	$-0.901043\pi$
$863$	− 3.39353e11i	− 0.611799i	0.952064	−	0.305899i	$-0.0989571\pi$
$864$	0	0
$865$	1.72681e11	0.308447
$866$	0	0
$867$	− 3.26162e11i	− 0.577240i
$868$	0	0
$869$	5.16628e10	0.0905939
$870$	0	0
$871$	− 5.65623e10i	− 0.0982776i
$872$	0	0
$873$	−1.98238e11	−0.341294
$874$	0	0
$875$	5.66412e11i	0.966275i
$876$	0	0
$877$	−7.77818e11	−1.31486	−0.657430	−	0.753516i	$-0.728356\pi$
$877$	−7.77818e11	−1.31486	−0.657430	+	0.753516i	$0.728356\pi$
$878$	0	0
$879$	− 1.79365e11i	− 0.300457i
$880$	0	0
$881$	8.88969e11	1.47565	0.737825	−	0.674993i	$-0.235853\pi$
$881$	8.88969e11	1.47565	0.737825	+	0.674993i	$0.235853\pi$
$882$	0	0
$883$	− 7.07920e11i	− 1.16450i	−0.813008	−	0.582252i	$-0.802172\pi$
$883$	− 7.07920e11i	− 1.16450i	0.813008	−	0.582252i	$-0.197828\pi$
$884$	0	0
$885$	−4.58898e11	−0.748071
$886$	0	0
$887$	8.17754e11i	1.32108i	0.750792	+	0.660538i	$0.229672\pi$
$887$	8.17754e11i	1.32108i	−0.750792	+	0.660538i	$0.770328\pi$
$888$	0	0
$889$	−7.28041e11	−1.16560
$890$	0	0
$891$	− 1.00557e11i	− 0.159552i
$892$	0	0
$893$	2.40975e11	0.378936
$894$	0	0
$895$	5.70047e11i	0.888421i
$896$	0	0
$897$	4.98370e11	0.769808
$898$	0	0
$899$	1.95389e12i	2.99130i
$900$	0	0
$901$	6.08313e9	0.00923055
$902$	0	0
$903$	8.73595e11i	1.31389i
$904$	0	0
$905$	−1.45072e12	−2.16267
$906$	0	0
$907$	3.56754e11i	0.527157i	0.964638	+	0.263579i	$0.0849029\pi$
$907$	3.56754e11i	0.527157i	−0.964638	+	0.263579i	$0.915097\pi$
$908$	0	0
$909$	−1.46892e11	−0.215151
$910$	0	0
$911$	3.09811e11i	0.449803i	0.974381	+	0.224902i	$0.0722061\pi$
$911$	3.09811e11i	0.449803i	−0.974381	+	0.224902i	$0.927794\pi$
$912$	0	0
$913$	1.37739e12	1.98232
$914$	0	0
$915$	2.48158e11i	0.354033i
$916$	0	0
$917$	−5.02045e11	−0.710011
$918$	0	0
$919$	− 3.12251e11i	− 0.437766i	−0.975751	−	0.218883i	$-0.929759\pi$
$919$	− 3.12251e11i	− 0.437766i	0.975751	−	0.218883i	$-0.0702412\pi$
$920$	0	0
$921$	1.77314e11	0.246437
$922$	0	0
$923$	− 7.21149e11i	− 0.993615i
$924$	0	0
$925$	−5.56011e11	−0.759480
$926$	0	0
$927$	1.53048e11i	0.207256i
$928$	0	0
$929$	1.06463e12	1.42934	0.714669	−	0.699463i	$-0.246577\pi$
$929$	1.06463e12	1.42934	0.714669	+	0.699463i	$0.246577\pi$
$930$	0	0
$931$	− 1.39559e12i	− 1.85762i
$932$	0	0
$933$	−1.58042e11	−0.208567
$934$	0	0
$935$	1.89029e10i	0.0247334i
$936$	0	0
$937$	−8.70853e11	−1.12976	−0.564880	−	0.825173i	$-0.691078\pi$
$937$	−8.70853e11	−1.12976	−0.564880	+	0.825173i	$0.691078\pi$
$938$	0	0
$939$	− 6.35758e10i	− 0.0817767i
$940$	0	0
$941$	−7.16472e11	−0.913778	−0.456889	−	0.889524i	$-0.651036\pi$
$941$	−7.16472e11	−0.913778	−0.456889	+	0.889524i	$0.651036\pi$
$942$	0	0
$943$	− 5.64385e10i	− 0.0713722i
$944$	0	0
$945$	−3.33257e11	−0.417880
$946$	0	0
$947$	− 5.75789e11i	− 0.715918i	−0.933737	−	0.357959i	$-0.883473\pi$
$947$	− 5.75789e11i	− 0.715918i	0.933737	−	0.357959i	$-0.116527\pi$
$948$	0	0
$949$	9.52932e11	1.17489
$950$	0	0
$951$	− 9.86706e10i	− 0.120633i
$952$	0	0
$953$	5.37994e11	0.652237	0.326119	−	0.945329i	$-0.394259\pi$
$953$	5.37994e11	0.652237	0.326119	+	0.945329i	$0.394259\pi$
$954$	0	0
$955$	− 6.07603e11i	− 0.730477i
$956$	0	0
$957$	−1.10649e12	−1.31917
$958$	0	0
$959$	1.43828e12i	1.70046i
$960$	0	0
$961$	−2.16140e12	−2.53420
$962$	0	0
$963$	− 7.10847e9i	− 0.00826554i
$964$	0	0
$965$	−4.90493e11	−0.565619
$966$	0	0
$967$	− 1.40296e12i	− 1.60449i	−0.596992	−	0.802247i	$-0.703638\pi$
$967$	− 1.40296e12i	− 1.60449i	0.596992	−	0.802247i	$-0.296362\pi$
$968$	0	0
$969$	−6.42727e9	−0.00729007
$970$	0	0
$971$	− 1.92974e11i	− 0.217082i	−0.994092	−	0.108541i	$-0.965382\pi$
$971$	− 1.92974e11i	− 0.217082i	0.994092	−	0.108541i	$-0.0346178\pi$
$972$	0	0
$973$	−6.71537e11	−0.749236
$974$	0	0
$975$	− 2.00314e11i	− 0.221663i
$976$	0	0
$977$	4.51891e11	0.495970	0.247985	−	0.968764i	$-0.420232\pi$
$977$	4.51891e11	0.495970	0.247985	+	0.968764i	$0.420232\pi$
$978$	0	0
$979$	8.32810e11i	0.906599i
$980$	0	0
$981$	1.69793e11	0.183334
$982$	0	0
$983$	2.10421e11i	0.225359i	0.993631	+	0.112679i	$0.0359433\pi$
$983$	2.10421e11i	0.225359i	−0.993631	+	0.112679i	$0.964057\pi$
$984$	0	0
$985$	−6.19760e11	−0.658383
$986$	0	0
$987$	3.95479e11i	0.416731i
$988$	0	0
$989$	2.40997e12	2.51899
$990$	0	0
$991$	− 8.99896e11i	− 0.933034i	−0.884512	−	0.466517i	$-0.845509\pi$
$991$	− 8.99896e11i	− 0.933034i	0.884512	−	0.466517i	$-0.154491\pi$
$992$	0	0
$993$	5.70254e11	0.586505
$994$	0	0
$995$	9.31613e11i	0.950481i
$996$	0	0
$997$	2.79453e11	0.282832	0.141416	−	0.989950i	$-0.454834\pi$
$997$	2.79453e11	0.282832	0.141416	+	0.989950i	$0.454834\pi$
$998$	0	0
$999$	2.62306e11i	0.263358i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.g.a.127.32	yes	32
4.3	odd	2	inner	384.9.g.a.127.31	✓	32
8.3	odd	2		384.9.g.b.127.2	yes	32
8.5	even	2		384.9.g.b.127.1	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.g.a.127.31	✓	32	4.3	odd	2	inner
384.9.g.a.127.32	yes	32	1.1	even	1	trivial
384.9.g.b.127.1	yes	32	8.5	even	2
384.9.g.b.127.2	yes	32	8.3	odd	2

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)

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} -779.371 q^{5} +4180.82i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} -779.371 q^{5} +4180.82i q^{7} -2187.00 q^{9} -21024.0i q^{11} -19757.9 q^{13} -36447.6i q^{15} +1153.64 q^{17} +119133. i q^{19} -195518. q^{21} +539371. i q^{23} +216794. q^{25} -102276. i q^{27} -1.12540e6 q^{29} -1.73617e6i q^{31} +983197. q^{33} -3.25841e6i q^{35} -2.56469e6 q^{37} -923984. i q^{39} -104638. q^{41} -4.46811e6i q^{43} +1.70448e6 q^{45} -2.02273e6i q^{47} -1.17145e7 q^{49} +53950.2i q^{51} +5.27300e6 q^{53} +1.63855e7i q^{55} -5.57132e6 q^{57} -1.25906e7i q^{59} -6.80863e6 q^{61} -9.14346e6i q^{63} +1.53987e7 q^{65} +2.86278e6i q^{67} -2.52239e7 q^{69} +3.64993e7i q^{71} -4.82305e7 q^{73} +1.01385e7i q^{75} +8.78978e7 q^{77} +2.45732e6i q^{79} +4.78297e6 q^{81} +6.55150e7i q^{83} -899111. q^{85} -5.26298e7i q^{87} -3.96123e7 q^{89} -8.26041e7i q^{91} +8.11926e7 q^{93} -9.28491e7i q^{95} +9.06436e7 q^{97} +4.59796e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 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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