Properties

Label 96.4.f.b
Level $96$
Weight $4$
Character orbit 96.f
Analytic conductor $5.664$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [96,4,Mod(47,96)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(96, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.47");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 96.f (of order \(2\), degree \(1\), not 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: \(5.66418336055\)
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: \( x^{8} - 10x^{6} + 120x^{4} - 640x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} + \beta_{4} q^{5} - \beta_{6} q^{7} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} + \beta_{4} q^{5} - \beta_{6} q^{7} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 9) q^{9} + (\beta_{3} + \beta_{2} + 2 \beta_1) q^{11} + (\beta_{7} - \beta_{6}) q^{13} + (\beta_{7} - \beta_{5} + 2 \beta_{4}) q^{15} + (4 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{17} + ( - 11 \beta_{3} + 11 \beta_{2} - 34) q^{19} + ( - \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \beta_{4}) q^{21} + (2 \beta_{5} + 4 \beta_{4}) q^{23} + ( - 2 \beta_{3} + 2 \beta_{2} + 35) q^{25} + (9 \beta_{3} - 12 \beta_{2} + 18 \beta_1 + 51) q^{27} + (4 \beta_{5} - 3 \beta_{4}) q^{29} + ( - 2 \beta_{7} + \beta_{6}) q^{31} + (9 \beta_{3} - \beta_{2} + 9 \beta_1 - 28) q^{33} + ( - 2 \beta_{3} - 2 \beta_{2} - 42 \beta_1) q^{35} + ( - \beta_{7} + 9 \beta_{6}) q^{37} + ( - \beta_{7} + 3 \beta_{6} - 5 \beta_{5} - 14 \beta_{4}) q^{39} + ( - 28 \beta_{3} - 28 \beta_{2} - 32 \beta_1) q^{41} + (39 \beta_{3} - 39 \beta_{2} + 58) q^{43} + (12 \beta_{6} - 6 \beta_{5} - 3 \beta_{4}) q^{45} + (8 \beta_{5} - 24 \beta_{4}) q^{47} + (46 \beta_{3} - 46 \beta_{2} - 73) q^{49} + ( - 24 \beta_{3} + 16 \beta_{2} + 6 \beta_1 - 152) q^{51} + (8 \beta_{5} - 7 \beta_{4}) q^{53} + (2 \beta_{7} + 8 \beta_{6}) q^{55} + ( - 33 \beta_{3} - 23 \beta_{2} + 33 \beta_1 + 142) q^{57} + ( - 11 \beta_{3} - 11 \beta_{2} + 8 \beta_1) q^{59} + ( - 7 \beta_{7} - 17 \beta_{6}) q^{61} + ( - 6 \beta_{7} - 3 \beta_{6} - 6 \beta_{5} + 36 \beta_{4}) q^{63} + (76 \beta_{3} + 76 \beta_{2} - 36 \beta_1) q^{65} + ( - 9 \beta_{3} + 9 \beta_{2} + 178) q^{67} + (8 \beta_{7} - 12 \beta_{6} - 2 \beta_{5} - 8 \beta_{4}) q^{69} + (2 \beta_{5} + 44 \beta_{4}) q^{71} + ( - 148 \beta_{3} + 148 \beta_{2} - 14) q^{73} + ( - 6 \beta_{3} + 37 \beta_{2} + 6 \beta_1 + 67) q^{75} + ( - 8 \beta_{5} + 26 \beta_{4}) q^{77} + (14 \beta_{7} - 7 \beta_{6}) q^{79} + (144 \beta_{3} + 18 \beta_1 + 9) q^{81} + (21 \beta_{3} + 21 \beta_{2} + 62 \beta_1) q^{83} + (8 \beta_{7} - 8 \beta_{6}) q^{85} + (5 \beta_{7} - 24 \beta_{6} + 7 \beta_{5} - 38 \beta_{4}) q^{87} + ( - 112 \beta_{3} - 112 \beta_{2} + 106 \beta_1) q^{89} + ( - 116 \beta_{3} + 116 \beta_{2} - 512) q^{91} + (\beta_{7} - 9 \beta_{6} + 8 \beta_{5} + 29 \beta_{4}) q^{93} + ( - 22 \beta_{5} - 12 \beta_{4}) q^{95} + (90 \beta_{3} - 90 \beta_{2} - 394) q^{97} + (57 \beta_{3} - 39 \beta_{2} + 24 \beta_1 - 216) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{3} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{3} - 48 q^{9} - 184 q^{19} + 296 q^{25} + 324 q^{27} - 264 q^{33} + 152 q^{43} - 952 q^{49} - 1056 q^{51} + 1176 q^{57} + 1496 q^{67} + 1072 q^{73} + 708 q^{75} - 504 q^{81} - 3168 q^{91} - 3872 q^{97} - 2112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 10x^{6} + 120x^{4} - 640x^{2} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 10\nu^{5} - 120\nu^{3} + 128\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 4\nu^{5} - 10\nu^{4} - 8\nu^{3} + 56\nu^{2} + 160\nu - 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 4\nu^{5} + 10\nu^{4} - 8\nu^{3} - 56\nu^{2} + 160\nu + 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 14\nu^{5} - 72\nu^{3} + 256\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 10\nu^{5} - 120\nu^{3} + 1152\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 22\nu^{4} + 136\nu^{2} - 1280 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{6} - 18\nu^{4} + 600\nu^{2} - 1280 ) / 64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - 4\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 4\beta_{3} - 4\beta_{2} + 24 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{5} + 8\beta_{4} + 8\beta_{3} + 8\beta_{2} - 20\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} - 5\beta_{6} + 20\beta_{3} - 20\beta_{2} - 120 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{5} + 8\beta_{4} + 72\beta_{3} + 72\beta_{2} + 20\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{7} - 39\beta_{6} - 84\beta_{3} + 84\beta_{2} - 424 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -9\beta_{5} - 200\beta_{4} + 120\beta_{3} + 120\beta_{2} + 124\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−2.58576 1.14624i
2.58576 1.14624i
−2.58576 + 1.14624i
2.58576 + 1.14624i
1.95291 + 2.04601i
−1.95291 + 2.04601i
1.95291 2.04601i
−1.95291 2.04601i
0 −1.37228 5.01167i 0 −12.2683 0 14.0624i 0 −23.2337 + 13.7548i 0
47.2 0 −1.37228 5.01167i 0 12.2683 0 14.0624i 0 −23.2337 + 13.7548i 0
47.3 0 −1.37228 + 5.01167i 0 −12.2683 0 14.0624i 0 −23.2337 13.7548i 0
47.4 0 −1.37228 + 5.01167i 0 12.2683 0 14.0624i 0 −23.2337 13.7548i 0
47.5 0 4.37228 2.80770i 0 −13.1715 0 26.9490i 0 11.2337 24.5521i 0
47.6 0 4.37228 2.80770i 0 13.1715 0 26.9490i 0 11.2337 24.5521i 0
47.7 0 4.37228 + 2.80770i 0 −13.1715 0 26.9490i 0 11.2337 + 24.5521i 0
47.8 0 4.37228 + 2.80770i 0 13.1715 0 26.9490i 0 11.2337 + 24.5521i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.4.f.b 8
3.b odd 2 1 inner 96.4.f.b 8
4.b odd 2 1 24.4.f.b 8
8.b even 2 1 24.4.f.b 8
8.d odd 2 1 inner 96.4.f.b 8
12.b even 2 1 24.4.f.b 8
16.e even 4 2 768.4.c.v 16
16.f odd 4 2 768.4.c.v 16
24.f even 2 1 inner 96.4.f.b 8
24.h odd 2 1 24.4.f.b 8
48.i odd 4 2 768.4.c.v 16
48.k even 4 2 768.4.c.v 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.f.b 8 4.b odd 2 1
24.4.f.b 8 8.b even 2 1
24.4.f.b 8 12.b even 2 1
24.4.f.b 8 24.h odd 2 1
96.4.f.b 8 1.a even 1 1 trivial
96.4.f.b 8 3.b odd 2 1 inner
96.4.f.b 8 8.d odd 2 1 inner
96.4.f.b 8 24.f even 2 1 inner
768.4.c.v 16 16.e even 4 2
768.4.c.v 16 16.f odd 4 2
768.4.c.v 16 48.i odd 4 2
768.4.c.v 16 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 324T_{5}^{2} + 26112 \) acting on \(S_{4}^{\mathrm{new}}(96, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 6 T^{3} + 30 T^{2} - 162 T + 729)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 324 T^{2} + 26112)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 924 T^{2} + 143616)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 484 T^{2} + 352)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 5808 T^{2} + 574464)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 2464 T^{2} + 90112)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 46 T - 3464)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 17472 T^{2} + 1671168)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 43620 T^{2} + \cdots + 448108032)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 20988 T^{2} + 41505024)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 77616 T^{2} + \cdots + 1494180864)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 193600 T^{2} + \cdots + 3151126528)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 38 T - 49832)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 321792 T^{2} + \cdots + 427819008)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 177156 T^{2} + \cdots + 7033554432)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 21604 T^{2} + 296032)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 550704 T^{2} + \cdots + 18820015104)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 374 T + 32296)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 654912 T^{2} + \cdots + 103614087168)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 268 T - 704876)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1028412 T^{2} + \cdots + 99653562624)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 396484 T^{2} + \cdots + 2128431712)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 2644576 T^{2} + \cdots + 147293673472)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 968 T - 33044)^{4} \) Copy content Toggle raw display
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