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Properties

Label 5472.2.g.c
Level $5472$
Weight $2$
Character orbit 5472.g
Analytic conductor $43.694$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5472,2,Mod(2737,5472)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5472, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5472.2737");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5472.g (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: \(43.6941399860\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{16} - 4 x^{15} - 4 x^{14} + 8 x^{13} + 18 x^{12} + 12 x^{11} - 20 x^{10} - 64 x^{9} + \cdots + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: no (minimal twist has level 456)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{5} + ( - \beta_{12} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{5} + ( - \beta_{12} - 1) q^{7} - \beta_{6} q^{11} + ( - \beta_{9} + \beta_{7} + \cdots - \beta_{3}) q^{13}+ \cdots + ( - \beta_{16} + \beta_{14} + \beta_{13} + \cdots - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 20 q^{7} - 4 q^{17} + 20 q^{23} - 22 q^{25} + 32 q^{31} + 4 q^{41} - 52 q^{47} + 18 q^{49} - 40 q^{55} + 16 q^{65} + 16 q^{71} + 12 q^{73} + 16 q^{79} - 36 q^{89} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{16} - 4 x^{15} - 4 x^{14} + 8 x^{13} + 18 x^{12} + 12 x^{11} - 20 x^{10} - 64 x^{9} + \cdots + 512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2 \nu^{17} + \nu^{16} - 2 \nu^{15} - 5 \nu^{14} - 4 \nu^{13} + 8 \nu^{12} + 20 \nu^{11} + \cdots + 256 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{17} + 10 \nu^{16} - 3 \nu^{15} - 22 \nu^{14} - 16 \nu^{13} + 62 \nu^{11} + 88 \nu^{10} + \cdots + 2304 ) / 256 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{17} - 10 \nu^{16} + 5 \nu^{15} + 30 \nu^{14} + 24 \nu^{13} - 16 \nu^{12} - 98 \nu^{11} + \cdots - 2304 ) / 256 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{17} + 16 \nu^{16} - 5 \nu^{15} - 36 \nu^{14} - 32 \nu^{13} + 24 \nu^{12} + 130 \nu^{11} + \cdots + 2816 ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{17} - 16 \nu^{16} + 7 \nu^{15} + 40 \nu^{14} + 20 \nu^{13} - 28 \nu^{12} - 142 \nu^{11} + \cdots - 4096 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{17} + 9 \nu^{16} - \nu^{15} - 21 \nu^{14} - 24 \nu^{13} - 4 \nu^{12} + 74 \nu^{11} + 110 \nu^{10} + \cdots + 1280 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{17} - 16 \nu^{16} + 11 \nu^{15} + 44 \nu^{14} + 16 \nu^{13} - 40 \nu^{12} - 158 \nu^{11} + \cdots - 4352 ) / 256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9 \nu^{17} + 4 \nu^{16} + 29 \nu^{15} + 16 \nu^{14} - 32 \nu^{13} - 104 \nu^{12} - 82 \nu^{11} + \cdots - 1280 ) / 256 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{17} + 18 \nu^{16} - 7 \nu^{15} - 50 \nu^{14} - 28 \nu^{13} + 28 \nu^{12} + 158 \nu^{11} + \cdots + 4608 ) / 256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13 \nu^{17} + 33 \nu^{15} + 28 \nu^{14} - 112 \nu^{12} - 154 \nu^{11} + 52 \nu^{10} + 332 \nu^{9} + \cdots - 256 ) / 256 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - \nu^{17} + 7 \nu^{16} - 3 \nu^{15} - 17 \nu^{14} - 8 \nu^{13} + 10 \nu^{12} + 58 \nu^{11} + \cdots + 1920 ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 9 \nu^{17} + \nu^{16} + 25 \nu^{15} + 23 \nu^{14} - 8 \nu^{13} - 88 \nu^{12} - 114 \nu^{11} + \cdots - 640 ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19 \nu^{17} + 6 \nu^{16} + 55 \nu^{15} + 46 \nu^{14} - 32 \nu^{13} - 200 \nu^{12} - 230 \nu^{11} + \cdots - 1536 ) / 256 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 8 \nu^{17} - \nu^{16} - 22 \nu^{15} - 19 \nu^{14} + 10 \nu^{13} + 76 \nu^{12} + 96 \nu^{11} + \cdots + 640 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 31 \nu^{17} - 6 \nu^{16} - 91 \nu^{15} - 70 \nu^{14} + 56 \nu^{13} + 320 \nu^{12} + 366 \nu^{11} + \cdots + 3328 ) / 256 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 33 \nu^{17} + 6 \nu^{16} + 93 \nu^{15} + 78 \nu^{14} - 48 \nu^{13} - 336 \nu^{12} - 402 \nu^{11} + \cdots - 3328 ) / 256 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 2 \nu^{17} - 37 \nu^{16} + 16 \nu^{15} + 95 \nu^{14} + 50 \nu^{13} - 58 \nu^{12} - 324 \nu^{11} + \cdots - 9600 ) / 128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{16} + \beta_{15} - \beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{17} - \beta_{14} - \beta_{13} + \beta_{11} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{17} + 2 \beta_{15} - \beta_{14} + \beta_{13} - \beta_{11} + 4 \beta_{9} + \beta_{8} - \beta_{7} + \cdots + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{17} + \beta_{14} + \beta_{13} + \beta_{11} + 2 \beta_{10} - \beta_{8} - 3 \beta_{7} + \beta_{6} + \cdots + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{17} - 3 \beta_{14} - \beta_{13} - \beta_{11} - 2 \beta_{10} - 5 \beta_{8} - \beta_{7} + 3 \beta_{6} + \cdots - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3 \beta_{17} - 4 \beta_{16} - \beta_{14} - \beta_{13} + 4 \beta_{12} + 3 \beta_{11} + 2 \beta_{10} + \cdots - 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3 \beta_{17} - 2 \beta_{16} + 6 \beta_{15} - 7 \beta_{14} + 7 \beta_{13} - 12 \beta_{12} - 5 \beta_{11} + \cdots + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3 \beta_{17} - 4 \beta_{15} + 5 \beta_{14} - 3 \beta_{13} + 4 \beta_{12} - 11 \beta_{11} + 2 \beta_{10} + \cdots - 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9 \beta_{17} + 2 \beta_{16} - 10 \beta_{15} - \beta_{14} - 15 \beta_{13} + 4 \beta_{12} + \beta_{11} + \cdots - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3 \beta_{17} + 8 \beta_{16} + 12 \beta_{15} - 5 \beta_{14} + 3 \beta_{13} - 28 \beta_{12} - 5 \beta_{11} + \cdots - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 19 \beta_{17} + 2 \beta_{16} + 14 \beta_{15} + 5 \beta_{14} + 3 \beta_{13} - 12 \beta_{12} + \cdots + 37 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 11 \beta_{17} + 24 \beta_{16} - 12 \beta_{15} - 11 \beta_{14} - 35 \beta_{13} - 4 \beta_{12} + \cdots + 51 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11 \beta_{17} + 46 \beta_{16} - 22 \beta_{15} + 59 \beta_{14} - 35 \beta_{13} + 28 \beta_{12} + \cdots - 21 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 59 \beta_{17} - 8 \beta_{16} + 76 \beta_{15} - 53 \beta_{14} + 67 \beta_{13} - 44 \beta_{12} + \cdots + 221 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 19 \beta_{17} - 54 \beta_{16} + 14 \beta_{15} - 3 \beta_{14} + 27 \beta_{13} + 84 \beta_{12} + \cdots + 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 69 \beta_{17} + 56 \beta_{16} + 36 \beta_{15} - 59 \beta_{14} + 13 \beta_{13} + 76 \beta_{12} + \cdots - 365 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 131 \beta_{17} - 202 \beta_{16} - 158 \beta_{15} + 67 \beta_{14} + 69 \beta_{13} + 268 \beta_{12} + \cdots + 499 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times\).

\(n\) \(1217\) \(2053\) \(3745\) \(4447\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2737.1
0.119195 + 1.40918i
−0.818660 + 1.15317i
1.21683 − 0.720636i
1.40520 + 0.159415i
−0.283402 − 1.38553i
−1.25114 + 0.659284i
−1.39438 − 0.236040i
1.29816 − 0.561056i
−0.291812 + 1.38378i
−0.291812 − 1.38378i
1.29816 + 0.561056i
−1.39438 + 0.236040i
−1.25114 − 0.659284i
−0.283402 + 1.38553i
1.40520 − 0.159415i
1.21683 + 0.720636i
−0.818660 − 1.15317i
0.119195 − 1.40918i
0 0 0 − 4.28166i 0 −0.394232 0 0 0
2737.2 0 0 0 − 3.71695i 0 −4.73089 0 0 0
2737.3 0 0 0 − 3.36719i 0 0.205725 0 0 0
2737.4 0 0 0 − 2.56804i 0 −4.12447 0 0 0
2737.5 0 0 0 − 1.97608i 0 −1.31060 0 0 0
2737.6 0 0 0 − 1.08708i 0 3.65550 0 0 0
2737.7 0 0 0 − 0.721820i 0 −1.43246 0 0 0
2737.8 0 0 0 − 0.540373i 0 −3.53540 0 0 0
2737.9 0 0 0 − 0.138759i 0 1.66683 0 0 0
2737.10 0 0 0 0.138759i 0 1.66683 0 0 0
2737.11 0 0 0 0.540373i 0 −3.53540 0 0 0
2737.12 0 0 0 0.721820i 0 −1.43246 0 0 0
2737.13 0 0 0 1.08708i 0 3.65550 0 0 0
2737.14 0 0 0 1.97608i 0 −1.31060 0 0 0
2737.15 0 0 0 2.56804i 0 −4.12447 0 0 0
2737.16 0 0 0 3.36719i 0 0.205725 0 0 0
2737.17 0 0 0 3.71695i 0 −4.73089 0 0 0
2737.18 0 0 0 4.28166i 0 −0.394232 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2737.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5472.2.g.c 18
3.b odd 2 1 1824.2.g.a 18
4.b odd 2 1 1368.2.g.d 18
8.b even 2 1 inner 5472.2.g.c 18
8.d odd 2 1 1368.2.g.d 18
12.b even 2 1 456.2.g.b ✓ 18
24.f even 2 1 456.2.g.b ✓ 18
24.h odd 2 1 1824.2.g.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.g.b ✓ 18 12.b even 2 1
456.2.g.b ✓ 18 24.f even 2 1
1368.2.g.d 18 4.b odd 2 1
1368.2.g.d 18 8.d odd 2 1
1824.2.g.a 18 3.b odd 2 1
1824.2.g.a 18 24.h odd 2 1
5472.2.g.c 18 1.a even 1 1 trivial
5472.2.g.c 18 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 56 T_{5}^{16} + 1210 T_{5}^{14} + 12756 T_{5}^{12} + 68441 T_{5}^{10} + 179004 T_{5}^{8} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(5472, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + 56 T^{16} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{9} + 10 T^{8} + \cdots - 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} + 100 T^{16} + \cdots + 2359296 \) Copy content Toggle raw display
$13$ \( T^{18} + 192 T^{16} + \cdots + 262144 \) Copy content Toggle raw display
$17$ \( (T^{9} + 2 T^{8} + \cdots - 6176)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$23$ \( (T^{9} - 10 T^{8} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 309171585024 \) Copy content Toggle raw display
$31$ \( (T^{9} - 16 T^{8} + \cdots - 12032)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 17179869184 \) Copy content Toggle raw display
$41$ \( (T^{9} - 2 T^{8} + \cdots - 16384)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 447122243584 \) Copy content Toggle raw display
$47$ \( (T^{9} + 26 T^{8} + \cdots - 342528)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 608849362944 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 1670008274944 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 3131031158784 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 7044820107264 \) Copy content Toggle raw display
$71$ \( (T^{9} - 8 T^{8} + \cdots - 30572544)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} - 6 T^{8} + \cdots - 3713088)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} - 8 T^{8} + \cdots + 3352192)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 1073741824 \) Copy content Toggle raw display
$89$ \( (T^{9} + 18 T^{8} + \cdots + 503650816)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} + 2 T^{8} + \cdots + 5659904)^{2} \) Copy content Toggle raw display
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