Properties

Label 384.2.f.a.191.1
Level $384$
Weight $2$
Character 384.191
Analytic conductor $3.066$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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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] = [384,2,Mod(191,384)] 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("384.191"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 384.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-12,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 191.1
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 384.191
Dual form 384.2.f.a.191.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.73205i q^{3} -2.82843 q^{5} +4.89898i q^{7} -3.00000 q^{9} +3.46410i q^{11} +4.89898i q^{15} +8.48528 q^{21} +3.00000 q^{25} +5.19615i q^{27} -2.82843 q^{29} +4.89898i q^{31} +6.00000 q^{33} -13.8564i q^{35} +8.48528 q^{45} -17.0000 q^{49} -14.1421 q^{53} -9.79796i q^{55} +10.3923i q^{59} -14.6969i q^{63} +14.0000 q^{73} -5.19615i q^{75} -16.9706 q^{77} -14.6969i q^{79} +9.00000 q^{81} +17.3205i q^{83} +4.89898i q^{87} +8.48528 q^{93} +2.00000 q^{97} -10.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 12 q^{25} + 24 q^{33} - 68 q^{49} + 56 q^{73} + 36 q^{81} + 8 q^{97}+O(q^{100}) \) 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
\(3\) − 1.73205i − 1.00000i
\(4\) 0 0
\(5\) −2.82843 −1.26491 −0.632456 − 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) 4.89898i 1.85164i 0.377964 + 0.925820i \(0.376624\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 4.89898i 1.26491i
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 8.48528 1.85164
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) −2.82843 −0.525226 −0.262613 − 0.964901i \(-0.584584\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 0 0
\(31\) 4.89898i 0.879883i 0.898027 + 0.439941i \(0.145001\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) − 13.8564i − 2.34216i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 8.48528 1.26491
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −17.0000 −2.42857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −14.1421 −1.94257 −0.971286 − 0.237915i \(-0.923536\pi\)
−0.971286 + 0.237915i \(0.923536\pi\)
\(54\) 0 0
\(55\) − 9.79796i − 1.32116i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.3923i 1.35296i 0.736460 + 0.676481i \(0.236496\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) − 14.6969i − 1.85164i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) − 5.19615i − 0.600000i
\(76\) 0 0
\(77\) −16.9706 −1.93398
\(78\) 0 0
\(79\) − 14.6969i − 1.65353i −0.562544 − 0.826767i \(-0.690177\pi\)
0.562544 − 0.826767i \(-0.309823\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 17.3205i 1.90117i 0.310460 + 0.950586i \(0.399517\pi\)
−0.310460 + 0.950586i \(0.600483\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.89898i 0.525226i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.48528 0.879883
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) − 10.3923i − 1.04447i
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 384.2.f.a.191.1 ✓ 4
3.2 odd 2 inner 384.2.f.a.191.4 yes 4
4.3 odd 2 inner 384.2.f.a.191.3 yes 4
8.3 odd 2 inner 384.2.f.a.191.2 yes 4
8.5 even 2 inner 384.2.f.a.191.4 yes 4
12.11 even 2 inner 384.2.f.a.191.2 yes 4
16.3 odd 4 768.2.c.j.767.2 4
16.5 even 4 768.2.c.j.767.1 4
16.11 odd 4 768.2.c.j.767.3 4
16.13 even 4 768.2.c.j.767.4 4
24.5 odd 2 CM 384.2.f.a.191.1 ✓ 4
24.11 even 2 inner 384.2.f.a.191.3 yes 4
48.5 odd 4 768.2.c.j.767.4 4
48.11 even 4 768.2.c.j.767.2 4
48.29 odd 4 768.2.c.j.767.1 4
48.35 even 4 768.2.c.j.767.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.f.a.191.1 ✓ 4 1.1 even 1 trivial
384.2.f.a.191.1 ✓ 4 24.5 odd 2 CM
384.2.f.a.191.2 yes 4 8.3 odd 2 inner
384.2.f.a.191.2 yes 4 12.11 even 2 inner
384.2.f.a.191.3 yes 4 4.3 odd 2 inner
384.2.f.a.191.3 yes 4 24.11 even 2 inner
384.2.f.a.191.4 yes 4 3.2 odd 2 inner
384.2.f.a.191.4 yes 4 8.5 even 2 inner
768.2.c.j.767.1 4 16.5 even 4
768.2.c.j.767.1 4 48.29 odd 4
768.2.c.j.767.2 4 16.3 odd 4
768.2.c.j.767.2 4 48.11 even 4
768.2.c.j.767.3 4 16.11 odd 4
768.2.c.j.767.3 4 48.35 even 4
768.2.c.j.767.4 4 16.13 even 4
768.2.c.j.767.4 4 48.5 odd 4