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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [192,2,Mod(95,192)] 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("192.95"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 192.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] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.53312771881\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 95.4
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 192.95
Dual form 192.2.f.a.95.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} +3.46410 q^{5} -2.00000i q^{7} -3.00000 q^{9} +3.46410i q^{11} +6.00000i q^{15} +3.46410 q^{21} +7.00000 q^{25} -5.19615i q^{27} -10.3923 q^{29} -10.0000i q^{31} -6.00000 q^{33} -6.92820i q^{35} -10.3923 q^{45} +3.00000 q^{49} +3.46410 q^{53} +12.0000i q^{55} -10.3923i q^{59} +6.00000i q^{63} -14.0000 q^{73} +12.1244i q^{75} +6.92820 q^{77} -10.0000i q^{79} +9.00000 q^{81} +17.3205i q^{83} -18.0000i q^{87} +17.3205 q^{93} +2.00000 q^{97} -10.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 28 q^{25} - 24 q^{33} + 12 q^{49} - 56 q^{73} + 36 q^{81} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(5\) 3.46410 1.54919 0.774597 − 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) 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
\(15\) 6.00000i 1.54919i
\(16\) 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
\(21\) 3.46410 0.755929
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) − 5.19615i − 1.00000i
\(28\) 0 0
\(29\) −10.3923 −1.92980 −0.964901 − 0.262613i \(-0.915416\pi\)
−0.964901 + 0.262613i \(0.915416\pi\)
\(30\) 0 0
\(31\) − 10.0000i − 1.79605i −0.439941 − 0.898027i \(-0.645001\pi\)
0.439941 − 0.898027i \(-0.354999\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) − 6.92820i − 1.17108i
\(36\) 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
\(45\) −10.3923 −1.54919
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.46410 0.475831 0.237915 − 0.971286i \(-0.423536\pi\)
0.237915 + 0.971286i \(0.423536\pi\)
\(54\) 0 0
\(55\) 12.0000i 1.61808i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 10.3923i − 1.35296i −0.736460 − 0.676481i \(-0.763504\pi\)
0.736460 − 0.676481i \(-0.236496\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 6.00000i 0.755929i
\(64\) 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 \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 12.1244i 1.40000i
\(76\) 0 0
\(77\) 6.92820 0.789542
\(78\) 0 0
\(79\) − 10.0000i − 1.12509i −0.826767 − 0.562544i \(-0.809823\pi\)
0.826767 − 0.562544i \(-0.190177\pi\)
\(80\) 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
\(87\) − 18.0000i − 1.92980i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 17.3205 1.79605
\(94\) 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 192.2.f.a.95.4 yes 4
3.2 odd 2 inner 192.2.f.a.95.1 ✓ 4
4.3 odd 2 inner 192.2.f.a.95.2 yes 4
8.3 odd 2 inner 192.2.f.a.95.3 yes 4
8.5 even 2 inner 192.2.f.a.95.1 ✓ 4
12.11 even 2 inner 192.2.f.a.95.3 yes 4
16.3 odd 4 768.2.c.i.767.3 4
16.5 even 4 768.2.c.i.767.4 4
16.11 odd 4 768.2.c.i.767.2 4
16.13 even 4 768.2.c.i.767.1 4
24.5 odd 2 CM 192.2.f.a.95.4 yes 4
24.11 even 2 inner 192.2.f.a.95.2 yes 4
48.5 odd 4 768.2.c.i.767.1 4
48.11 even 4 768.2.c.i.767.3 4
48.29 odd 4 768.2.c.i.767.4 4
48.35 even 4 768.2.c.i.767.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
192.2.f.a.95.1 ✓ 4 3.2 odd 2 inner
192.2.f.a.95.1 ✓ 4 8.5 even 2 inner
192.2.f.a.95.2 yes 4 4.3 odd 2 inner
192.2.f.a.95.2 yes 4 24.11 even 2 inner
192.2.f.a.95.3 yes 4 8.3 odd 2 inner
192.2.f.a.95.3 yes 4 12.11 even 2 inner
192.2.f.a.95.4 yes 4 1.1 even 1 trivial
192.2.f.a.95.4 yes 4 24.5 odd 2 CM
768.2.c.i.767.1 4 16.13 even 4
768.2.c.i.767.1 4 48.5 odd 4
768.2.c.i.767.2 4 16.11 odd 4
768.2.c.i.767.2 4 48.35 even 4
768.2.c.i.767.3 4 16.3 odd 4
768.2.c.i.767.3 4 48.11 even 4
768.2.c.i.767.4 4 16.5 even 4
768.2.c.i.767.4 4 48.29 odd 4