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Properties

Label 384.3.h.f.65.2
Level $384$
Weight $3$
Character 384.65
Analytic conductor $10.463$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [384,3,Mod(65,384)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("384.65"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 65.2
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 384.65
Dual form 384.3.h.f.65.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.23607 + 2.00000i) q^{3} +4.00000 q^{5} -8.94427 q^{7} +(1.00000 - 8.94427i) q^{9} +4.47214 q^{11} -17.8885i q^{13} +(-8.94427 + 8.00000i) q^{15} +17.8885i q^{17} -20.0000i q^{19} +(20.0000 - 17.8885i) q^{21} -16.0000i q^{23} -9.00000 q^{25} +(15.6525 + 22.0000i) q^{27} +52.0000 q^{29} +26.8328 q^{31} +(-10.0000 + 8.94427i) q^{33} -35.7771 q^{35} -53.6656i q^{37} +(35.7771 + 40.0000i) q^{39} -35.7771i q^{41} +36.0000i q^{43} +(4.00000 - 35.7771i) q^{45} -64.0000i q^{47} +31.0000 q^{49} +(-35.7771 - 40.0000i) q^{51} +20.0000 q^{53} +17.8885 q^{55} +(40.0000 + 44.7214i) q^{57} +102.859 q^{59} -17.8885i q^{61} +(-8.94427 + 80.0000i) q^{63} -71.5542i q^{65} -44.0000i q^{67} +(32.0000 + 35.7771i) q^{69} -80.0000i q^{71} -50.0000 q^{73} +(20.1246 - 18.0000i) q^{75} -40.0000 q^{77} -80.4984 q^{79} +(-79.0000 - 17.8885i) q^{81} -102.859 q^{83} +71.5542i q^{85} +(-116.276 + 104.000i) q^{87} +160.997i q^{89} +160.000i q^{91} +(-60.0000 + 53.6656i) q^{93} -80.0000i q^{95} +50.0000 q^{97} +(4.47214 - 40.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{5} + 4 q^{9} + 80 q^{21} - 36 q^{25} + 208 q^{29} - 40 q^{33} + 16 q^{45} + 124 q^{49} + 80 q^{53} + 160 q^{57} + 128 q^{69} - 200 q^{73} - 160 q^{77} - 316 q^{81} - 240 q^{93} + 200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.23607 + 2.00000i −0.745356 + 0.666667i
\(4\) 0 0
\(5\) 4.00000 0.800000 0.400000 − 0.916515i \(-0.369010\pi\)
0.400000 + 0.916515i \(0.369010\pi\)
\(6\) 0 0
\(7\) −8.94427 −1.27775 −0.638877 − 0.769309i \(-0.720601\pi\)
−0.638877 + 0.769309i \(0.720601\pi\)
\(8\) 0 0
\(9\) 1.00000 − 8.94427i 0.111111 − 0.993808i
\(10\) 0 0
\(11\) 4.47214 0.406558 0.203279 − 0.979121i \(-0.434840\pi\)
0.203279 + 0.979121i \(0.434840\pi\)
\(12\) 0 0
\(13\) − 17.8885i − 1.37604i −0.725691 − 0.688021i \(-0.758480\pi\)
0.725691 − 0.688021i \(-0.241520\pi\)
\(14\) 0 0
\(15\) −8.94427 + 8.00000i −0.596285 + 0.533333i
\(16\) 0 0
\(17\) 17.8885i 1.05227i 0.850402 + 0.526134i \(0.176359\pi\)
−0.850402 + 0.526134i \(0.823641\pi\)
\(18\) 0 0
\(19\) − 20.0000i − 1.05263i −0.850289 − 0.526316i \(-0.823573\pi\)
0.850289 − 0.526316i \(-0.176427\pi\)
\(20\) 0 0
\(21\) 20.0000 − 17.8885i 0.952381 − 0.851835i
\(22\) 0 0
\(23\) − 16.0000i − 0.695652i −0.937559 − 0.347826i \(-0.886920\pi\)
0.937559 − 0.347826i \(-0.113080\pi\)
\(24\) 0 0
\(25\) −9.00000 −0.360000
\(26\) 0 0
\(27\) 15.6525 + 22.0000i 0.579721 + 0.814815i
\(28\) 0 0
\(29\) 52.0000 1.79310 0.896552 − 0.442939i \(-0.146064\pi\)
0.896552 + 0.442939i \(0.146064\pi\)
\(30\) 0 0
\(31\) 26.8328 0.865575 0.432787 − 0.901496i \(-0.357530\pi\)
0.432787 + 0.901496i \(0.357530\pi\)
\(32\) 0 0
\(33\) −10.0000 + 8.94427i −0.303030 + 0.271039i
\(34\) 0 0
\(35\) −35.7771 −1.02220
\(36\) 0 0
\(37\) − 53.6656i − 1.45042i −0.688526 − 0.725211i \(-0.741742\pi\)
0.688526 − 0.725211i \(-0.258258\pi\)
\(38\) 0 0
\(39\) 35.7771 + 40.0000i 0.917361 + 1.02564i
\(40\) 0 0
\(41\) − 35.7771i − 0.872612i −0.899798 − 0.436306i \(-0.856287\pi\)
0.899798 − 0.436306i \(-0.143713\pi\)
\(42\) 0 0
\(43\) 36.0000i 0.837209i 0.908169 + 0.418605i \(0.137481\pi\)
−0.908169 + 0.418605i \(0.862519\pi\)
\(44\) 0 0
\(45\) 4.00000 − 35.7771i 0.0888889 − 0.795046i
\(46\) 0 0
\(47\) − 64.0000i − 1.36170i −0.732422 − 0.680851i \(-0.761610\pi\)
0.732422 − 0.680851i \(-0.238390\pi\)
\(48\) 0 0
\(49\) 31.0000 0.632653
\(50\) 0 0
\(51\) −35.7771 − 40.0000i −0.701512 − 0.784314i
\(52\) 0 0
\(53\) 20.0000 0.377358 0.188679 − 0.982039i \(-0.439579\pi\)
0.188679 + 0.982039i \(0.439579\pi\)
\(54\) 0 0
\(55\) 17.8885 0.325246
\(56\) 0 0
\(57\) 40.0000 + 44.7214i 0.701754 + 0.784585i
\(58\) 0 0
\(59\) 102.859 1.74338 0.871688 − 0.490062i \(-0.163026\pi\)
0.871688 + 0.490062i \(0.163026\pi\)
\(60\) 0 0
\(61\) − 17.8885i − 0.293255i −0.989192 − 0.146627i \(-0.953158\pi\)
0.989192 − 0.146627i \(-0.0468418\pi\)
\(62\) 0 0
\(63\) −8.94427 + 80.0000i −0.141973 + 1.26984i
\(64\) 0 0
\(65\) − 71.5542i − 1.10083i
\(66\) 0 0
\(67\) − 44.0000i − 0.656716i −0.944553 − 0.328358i \(-0.893505\pi\)
0.944553 − 0.328358i \(-0.106495\pi\)
\(68\) 0 0
\(69\) 32.0000 + 35.7771i 0.463768 + 0.518509i
\(70\) 0 0
\(71\) − 80.0000i − 1.12676i −0.826198 − 0.563380i \(-0.809501\pi\)
0.826198 − 0.563380i \(-0.190499\pi\)
\(72\) 0 0
\(73\) −50.0000 −0.684932 −0.342466 − 0.939530i \(-0.611262\pi\)
−0.342466 + 0.939530i \(0.611262\pi\)
\(74\) 0 0
\(75\) 20.1246 − 18.0000i 0.268328 − 0.240000i
\(76\) 0 0
\(77\) −40.0000 −0.519481
\(78\) 0 0
\(79\) −80.4984 −1.01897 −0.509484 − 0.860480i \(-0.670164\pi\)
−0.509484 + 0.860480i \(0.670164\pi\)
\(80\) 0 0
\(81\) −79.0000 − 17.8885i −0.975309 − 0.220846i
\(82\) 0 0
\(83\) −102.859 −1.23927 −0.619633 − 0.784891i \(-0.712719\pi\)
−0.619633 + 0.784891i \(0.712719\pi\)
\(84\) 0 0
\(85\) 71.5542i 0.841814i
\(86\) 0 0
\(87\) −116.276 + 104.000i −1.33650 + 1.19540i
\(88\) 0 0
\(89\) 160.997i 1.80895i 0.426523 + 0.904477i \(0.359738\pi\)
−0.426523 + 0.904477i \(0.640262\pi\)
\(90\) 0 0
\(91\) 160.000i 1.75824i
\(92\) 0 0
\(93\) −60.0000 + 53.6656i −0.645161 + 0.577050i
\(94\) 0 0
\(95\) − 80.0000i − 0.842105i
\(96\) 0 0
\(97\) 50.0000 0.515464 0.257732 − 0.966216i \(-0.417025\pi\)
0.257732 + 0.966216i \(0.417025\pi\)
\(98\) 0 0
\(99\) 4.47214 − 40.0000i 0.0451731 − 0.404040i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 384.3.h.f.65.2 yes 4
3.2 odd 2 384.3.h.e.65.4 yes 4
4.3 odd 2 inner 384.3.h.f.65.3 yes 4
8.3 odd 2 384.3.h.e.65.2 yes 4
8.5 even 2 384.3.h.e.65.3 yes 4
12.11 even 2 384.3.h.e.65.1 ✓ 4
16.3 odd 4 768.3.e.m.257.3 4
16.5 even 4 768.3.e.m.257.4 4
16.11 odd 4 768.3.e.h.257.2 4
16.13 even 4 768.3.e.h.257.1 4
24.5 odd 2 inner 384.3.h.f.65.1 yes 4
24.11 even 2 inner 384.3.h.f.65.4 yes 4
48.5 odd 4 768.3.e.m.257.1 4
48.11 even 4 768.3.e.h.257.3 4
48.29 odd 4 768.3.e.h.257.4 4
48.35 even 4 768.3.e.m.257.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.h.e.65.1 ✓ 4 12.11 even 2
384.3.h.e.65.2 yes 4 8.3 odd 2
384.3.h.e.65.3 yes 4 8.5 even 2
384.3.h.e.65.4 yes 4 3.2 odd 2
384.3.h.f.65.1 yes 4 24.5 odd 2 inner
384.3.h.f.65.2 yes 4 1.1 even 1 trivial
384.3.h.f.65.3 yes 4 4.3 odd 2 inner
384.3.h.f.65.4 yes 4 24.11 even 2 inner
768.3.e.h.257.1 4 16.13 even 4
768.3.e.h.257.2 4 16.11 odd 4
768.3.e.h.257.3 4 48.11 even 4
768.3.e.h.257.4 4 48.29 odd 4
768.3.e.m.257.1 4 48.5 odd 4
768.3.e.m.257.2 4 48.35 even 4
768.3.e.m.257.3 4 16.3 odd 4
768.3.e.m.257.4 4 16.5 even 4