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Properties

Label 48.2.k.a
Level $48$
Weight $2$
Character orbit 48.k
Analytic conductor $0.383$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,2,Mod(11,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 48.k (of order \(4\), degree \(2\), 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: \(0.383281929702\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.163368480538624.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{10} - 2x^{8} + 16x^{6} - 8x^{4} - 32x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} - \beta_{10} q^{3} + (\beta_{11} + \beta_{10} - \beta_1) q^{4} + (\beta_{10} - \beta_{8} + \cdots + \beta_{3}) q^{5}+ \cdots + ( - \beta_{11} - \beta_{10} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} - \beta_{10} q^{3} + (\beta_{11} + \beta_{10} - \beta_1) q^{4} + (\beta_{10} - \beta_{8} + \cdots + \beta_{3}) q^{5}+ \cdots + (\beta_{9} + 2 \beta_{8} + \beta_{7} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} - 4 q^{4} - 8 q^{6} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} - 4 q^{4} - 8 q^{6} - 8 q^{7} - 8 q^{12} - 4 q^{13} + 16 q^{16} + 4 q^{18} - 12 q^{19} - 8 q^{21} + 16 q^{22} + 24 q^{24} + 10 q^{27} - 8 q^{28} + 28 q^{30} - 4 q^{33} - 8 q^{34} + 20 q^{36} - 4 q^{37} + 20 q^{39} - 40 q^{40} - 24 q^{42} + 12 q^{43} - 12 q^{45} - 40 q^{46} - 48 q^{48} - 20 q^{49} + 24 q^{51} - 16 q^{52} - 52 q^{54} + 24 q^{55} + 32 q^{58} - 16 q^{60} + 12 q^{61} + 56 q^{64} + 28 q^{66} + 28 q^{67} + 4 q^{69} + 40 q^{70} + 40 q^{72} - 34 q^{75} + 56 q^{76} + 60 q^{78} - 4 q^{81} - 16 q^{82} + 16 q^{84} + 32 q^{85} - 60 q^{87} - 64 q^{88} - 16 q^{90} - 56 q^{91} + 28 q^{93} - 48 q^{94} - 56 q^{96} - 8 q^{97} - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{10} - 2x^{8} + 16x^{6} - 8x^{4} - 32x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} - 2\nu^{4} + 4\nu^{2} + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{11} + 2\nu^{9} + 2\nu^{7} + 8\nu^{3} ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 2\nu^{9} - 2\nu^{7} + 16\nu^{5} - 8\nu^{3} - 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - \nu^{10} - 6\nu^{7} + 10\nu^{6} + 4\nu^{5} - 12\nu^{4} + 8\nu^{3} - 32\nu + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{11} + \nu^{10} + 8\nu^{7} - 2\nu^{6} - 16\nu^{5} - 4\nu^{4} - 24\nu^{3} + 16\nu^{2} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} + 2\nu^{9} + 10\nu^{7} - 16\nu^{5} - 8\nu^{3} + 64\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{11} + \nu^{10} + 6\nu^{7} - 10\nu^{6} - 4\nu^{5} + 12\nu^{4} - 8\nu^{3} + 32\nu^{2} + 32\nu - 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{9} + 2\nu^{5} - 4\nu^{3} - 16\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{11} - \nu^{10} + 6\nu^{7} + 10\nu^{6} - 4\nu^{5} - 12\nu^{4} - 8\nu^{3} - 32\nu^{2} + 32\nu + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -2\nu^{11} - \nu^{10} + 8\nu^{7} + 2\nu^{6} - 16\nu^{5} + 4\nu^{4} - 24\nu^{3} - 16\nu^{2} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2\nu^{11} + 3\nu^{10} - 8\nu^{7} - 6\nu^{6} + 16\nu^{5} + 20\nu^{4} + 24\nu^{3} + 16\nu^{2} - 32\nu - 32 ) / 32 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 2\beta_{10} + \beta_{7} - \beta_{5} + \beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{11} + 2\beta_{10} + 2\beta_{9} + 2\beta_{7} + 4\beta_{4} - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{10} + 4\beta_{8} + 6\beta_{6} - 2\beta_{5} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{11} + 4\beta_{10} - 2\beta_{7} - 2\beta_{5} - 2\beta_{4} + 8\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4\beta_{10} - 10\beta_{9} - 6\beta_{8} - 10\beta_{7} + 6\beta_{6} + 4\beta_{5} + 12\beta_{3} + 10\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 8\beta_{11} - 4\beta_{10} + 4\beta_{9} - 8\beta_{7} + 12\beta_{5} - 4\beta_{4} - 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -4\beta_{10} - 12\beta_{9} + 4\beta_{8} - 12\beta_{7} + 24\beta_{6} - 4\beta_{5} + 8\beta_{3} - 8\beta_{2} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.204810 − 1.39930i
1.27715 − 0.607364i
−1.35164 − 0.416001i
1.35164 + 0.416001i
−1.27715 + 0.607364i
−0.204810 + 1.39930i
0.204810 + 1.39930i
1.27715 + 0.607364i
−1.35164 + 0.416001i
1.35164 − 0.416001i
−1.27715 − 0.607364i
−0.204810 − 1.39930i
−1.39930 + 0.204810i −0.814141 − 1.52878i 1.91611 − 0.573183i 2.08397 − 2.08397i 1.45234 + 1.97249i −1.14637 −2.56382 + 1.19449i −1.67435 + 2.48929i −2.48929 + 3.34292i
11.2 −0.607364 + 1.27715i 1.73003 + 0.0835731i −1.26222 − 1.55139i −0.431733 + 0.431733i −1.15749 + 2.15875i −3.10278 2.74798 − 0.669785i 2.98603 + 0.289169i −0.289169 − 0.813607i
11.3 −0.416001 − 1.35164i 0.966579 − 1.43726i −1.65389 + 1.12457i −1.57184 + 1.57184i −2.34477 − 0.708570i 2.24914 2.20804 + 1.76765i −1.13145 − 2.77846i 2.77846 + 1.47068i
11.4 0.416001 + 1.35164i −1.43726 + 0.966579i −1.65389 + 1.12457i 1.57184 − 1.57184i −1.90437 − 1.54057i 2.24914 −2.20804 − 1.76765i 1.13145 − 2.77846i 2.77846 + 1.47068i
11.5 0.607364 − 1.27715i 0.0835731 + 1.73003i −1.26222 − 1.55139i 0.431733 − 0.431733i 2.26027 + 0.944024i −3.10278 −2.74798 + 0.669785i −2.98603 + 0.289169i −0.289169 − 0.813607i
11.6 1.39930 − 0.204810i −1.52878 − 0.814141i 1.91611 − 0.573183i −2.08397 + 2.08397i −2.30598 − 0.826122i −1.14637 2.56382 − 1.19449i 1.67435 + 2.48929i −2.48929 + 3.34292i
35.1 −1.39930 − 0.204810i −0.814141 + 1.52878i 1.91611 + 0.573183i 2.08397 + 2.08397i 1.45234 − 1.97249i −1.14637 −2.56382 − 1.19449i −1.67435 − 2.48929i −2.48929 − 3.34292i
35.2 −0.607364 − 1.27715i 1.73003 − 0.0835731i −1.26222 + 1.55139i −0.431733 − 0.431733i −1.15749 − 2.15875i −3.10278 2.74798 + 0.669785i 2.98603 − 0.289169i −0.289169 + 0.813607i
35.3 −0.416001 + 1.35164i 0.966579 + 1.43726i −1.65389 − 1.12457i −1.57184 − 1.57184i −2.34477 + 0.708570i 2.24914 2.20804 − 1.76765i −1.13145 + 2.77846i 2.77846 − 1.47068i
35.4 0.416001 − 1.35164i −1.43726 − 0.966579i −1.65389 − 1.12457i 1.57184 + 1.57184i −1.90437 + 1.54057i 2.24914 −2.20804 + 1.76765i 1.13145 + 2.77846i 2.77846 − 1.47068i
35.5 0.607364 + 1.27715i 0.0835731 − 1.73003i −1.26222 + 1.55139i 0.431733 + 0.431733i 2.26027 − 0.944024i −3.10278 −2.74798 − 0.669785i −2.98603 − 0.289169i −0.289169 + 0.813607i
35.6 1.39930 + 0.204810i −1.52878 + 0.814141i 1.91611 + 0.573183i −2.08397 − 2.08397i −2.30598 + 0.826122i −1.14637 2.56382 + 1.19449i 1.67435 − 2.48929i −2.48929 − 3.34292i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.2.k.a ✓ 12
3.b odd 2 1 inner 48.2.k.a ✓ 12
4.b odd 2 1 192.2.k.a 12
8.b even 2 1 384.2.k.b 12
8.d odd 2 1 384.2.k.a 12
12.b even 2 1 192.2.k.a 12
16.e even 4 1 192.2.k.a 12
16.e even 4 1 384.2.k.a 12
16.f odd 4 1 inner 48.2.k.a ✓ 12
16.f odd 4 1 384.2.k.b 12
24.f even 2 1 384.2.k.a 12
24.h odd 2 1 384.2.k.b 12
48.i odd 4 1 192.2.k.a 12
48.i odd 4 1 384.2.k.a 12
48.k even 4 1 inner 48.2.k.a ✓ 12
48.k even 4 1 384.2.k.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.k.a ✓ 12 1.a even 1 1 trivial
48.2.k.a ✓ 12 3.b odd 2 1 inner
48.2.k.a ✓ 12 16.f odd 4 1 inner
48.2.k.a ✓ 12 48.k even 4 1 inner
192.2.k.a 12 4.b odd 2 1
192.2.k.a 12 12.b even 2 1
192.2.k.a 12 16.e even 4 1
192.2.k.a 12 48.i odd 4 1
384.2.k.a 12 8.d odd 2 1
384.2.k.a 12 16.e even 4 1
384.2.k.a 12 24.f even 2 1
384.2.k.a 12 48.i odd 4 1
384.2.k.b 12 8.b even 2 1
384.2.k.b 12 16.f odd 4 1
384.2.k.b 12 24.h odd 2 1
384.2.k.b 12 48.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 100 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{3} + 2 T^{2} - 6 T - 8)^{4} \) Copy content Toggle raw display
$11$ \( T^{12} + 356 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + 2 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 40 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 6 T^{5} + \cdots + 128)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 52 T^{4} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 1316 T^{8} + \cdots + 71639296 \) Copy content Toggle raw display
$31$ \( (T^{6} + 36 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 2 T^{5} + \cdots + 2312)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 108 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 6 T^{5} + \cdots + 128)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 112 T^{4} + \cdots - 32768)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 24356 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( T^{12} + 27748 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$61$ \( (T^{6} - 6 T^{5} + \cdots + 95048)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 14 T^{5} + \cdots + 32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 196 T^{4} + \cdots + 147968)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 272 T^{4} + \cdots + 369664)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 116 T^{4} + \cdots + 8464)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 5473632256 \) Copy content Toggle raw display
$89$ \( (T^{6} - 212 T^{4} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 2 T^{2} + \cdots - 608)^{4} \) Copy content Toggle raw display
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