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Properties

Label 2160.4.h.e
Level $2160$
Weight $4$
Character orbit 2160.h
Analytic conductor $127.444$
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] = [2160,4,Mod(431,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.431");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.h (of order \(2\), degree \(1\), 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: \(127.444125612\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 68x^{10} + 3468x^{8} - 70158x^{6} + 1049036x^{4} - 4884100x^{2} + 17850625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{12}\cdot 5^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + (\beta_{4} + \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} + (\beta_{4} + \beta_{2}) q^{7} + \beta_{7} q^{11} + ( - \beta_{5} + 27) q^{13} + (\beta_{10} - \beta_{3}) q^{17} + ( - \beta_{6} + 2 \beta_{4} - \beta_{2}) q^{19} + ( - 2 \beta_{8} + \beta_{7}) q^{23} - 25 q^{25} + (2 \beta_{11} + \beta_{10} + 17 \beta_{3}) q^{29} + (2 \beta_{6} - \beta_{4} - 8 \beta_{2}) q^{31} + (\beta_{9} + 4 \beta_{7}) q^{35} + (2 \beta_{5} + \beta_1 + 80) q^{37} + ( - 3 \beta_{11} + 2 \beta_{10} + 40 \beta_{3}) q^{41} + ( - 2 \beta_{6} - 7 \beta_{4} - 4 \beta_{2}) q^{43} + ( - 4 \beta_{9} + 3 \beta_{8} + 6 \beta_{7}) q^{47} + (3 \beta_{5} + 5 \beta_1 - 171) q^{49} + (11 \beta_{11} - 2 \beta_{10} - 4 \beta_{3}) q^{53} - 5 \beta_{4} q^{55} + ( - 2 \beta_{9} + \beta_{8} - 21 \beta_{7}) q^{59} + ( - 5 \beta_{5} - 7 \beta_1 - 40) q^{61} + (5 \beta_{11} - 25 \beta_{3}) q^{65} + (4 \beta_{6} + 3 \beta_{4} - 11 \beta_{2}) q^{67} + (4 \beta_{9} - 3 \beta_{8} + 9 \beta_{7}) q^{71} + ( - 4 \beta_{5} - \beta_1 + 122) q^{73} + ( - 6 \beta_{11} + 2 \beta_{10} + 76 \beta_{3}) q^{77} + (5 \beta_{6} - 11 \beta_{4} - 23 \beta_{2}) q^{79} + ( - 8 \beta_{9} + 7 \beta_{8} - 31 \beta_{7}) q^{83} + ( - 5 \beta_1 - 35) q^{85} + ( - 5 \beta_{11} + 6 \beta_{10} - 66 \beta_{3}) q^{89} + (\beta_{6} + 46 \beta_{4} - 7 \beta_{2}) q^{91} + ( - 3 \beta_{9} + 5 \beta_{8} + 13 \beta_{7}) q^{95} + (12 \beta_{5} - 3 \beta_1 + 628) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 324 q^{13} - 300 q^{25} + 960 q^{37} - 2052 q^{49} - 480 q^{61} + 1464 q^{73} - 420 q^{85} + 7536 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 68x^{10} + 3468x^{8} - 70158x^{6} + 1049036x^{4} - 4884100x^{2} + 17850625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{10} - 153\nu^{8} + 14562\nu^{6} - 46281\nu^{4} + 215475\nu^{2} + 243288110 ) / 3319420 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7803 \nu^{10} - 492579 \nu^{8} + 25121529 \nu^{6} - 448539849 \nu^{4} + 7599016233 \nu^{2} - 21786461775 ) / 1510336100 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -867\nu^{11} + 54731\nu^{9} - 2791281\nu^{7} + 49837761\nu^{5} - 755491837\nu^{3} + 910381875\nu ) / 1154962900 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19639 \nu^{10} + 1285467 \nu^{8} - 65217112 \nu^{6} + 1240987877 \nu^{4} + \cdots + 52993296200 ) / 1510336100 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -174\nu^{10} + 8874\nu^{8} - 405261\nu^{6} + 2684298\nu^{4} - 12497550\nu^{2} - 742771445 ) / 11617970 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4089 \nu^{10} + 312177 \nu^{8} - 15188802 \nu^{6} + 322914987 \nu^{4} - 3322215579 \nu^{2} + 10628164950 ) / 215762300 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 147963 \nu^{11} + 12030334 \nu^{9} - 635757859 \nu^{7} + 16323567529 \nu^{5} + \cdots + 2058554697225 \nu ) / 98171846500 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 168843 \nu^{11} - 19035624 \nu^{9} + 1037449299 \nu^{7} - 34226352969 \nu^{5} + \cdots - 4080707973225 \nu ) / 98171846500 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 325167 \nu^{11} - 20142506 \nu^{9} + 1005056981 \nu^{7} - 18869261261 \nu^{5} + \cdots - 2563073768775 \nu ) / 98171846500 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 235518 \nu^{11} - 15128819 \nu^{9} + 758243274 \nu^{7} - 13538281194 \nu^{5} + \cdots - 247302558750 \nu ) / 19634369300 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 655656 \nu^{11} - 42696033 \nu^{9} + 2110865208 \nu^{7} - 37689073848 \nu^{5} + \cdots - 688462905000 \nu ) / 49085923250 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -5\beta_{11} + 10\beta_{10} - 2\beta_{9} + 15\beta_{8} + 47\beta_{7} + 2\beta_{3} ) / 360 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{6} + 3\beta_{5} + 45\beta_{4} + 112\beta_{2} - 6\beta _1 + 816 ) / 72 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -85\beta_{11} + 170\beta_{10} + 1204\beta_{3} ) / 90 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -297\beta_{6} - 37\beta_{5} + 2115\beta_{4} + 4068\beta_{2} + 334\beta _1 - 27744 ) / 72 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4805\beta_{11} + 13510\beta_{10} - 13418\beta_{9} - 18315\beta_{8} - 53227\beta_{7} + 136082\beta_{3} ) / 360 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 238\beta_{5} + 3944\beta _1 - 273849 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 151345 \beta_{11} - 567890 \beta_{10} - 682022 \beta_{9} - 719235 \beta_{8} - 2271283 \beta_{7} - 6478378 \beta_{3} ) / 360 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 806847\beta_{6} + 94833\beta_{5} - 3987585\beta_{4} - 6926128\beta_{2} + 712014\beta _1 - 45862464 ) / 72 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2727565\beta_{11} - 12217730\beta_{10} - 146696356\beta_{3} ) / 90 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 36782853 \beta_{6} - 5191807 \beta_{5} - 173970855 \beta_{4} - 298519892 \beta_{2} - 31591046 \beta _1 + 1969588176 ) / 72 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 216295145 \beta_{11} - 1062210190 \beta_{10} + 1422342802 \beta_{9} + 1278505335 \beta_{8} + \cdots - 13036645898 \beta_{3} ) / 360 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
−5.73275 + 3.30981i
−1.94353 − 1.12210i
3.78922 − 2.18771i
−3.78922 − 2.18771i
1.94353 − 1.12210i
5.73275 + 3.30981i
5.73275 − 3.30981i
1.94353 + 1.12210i
−3.78922 + 2.18771i
3.78922 + 2.18771i
−1.94353 + 1.12210i
−5.73275 − 3.30981i
0 0 0 − 5.00000i 0 − 33.2548i 0 0 0
431.2 0 0 0 − 5.00000i 0 − 18.5665i 0 0 0
431.3 0 0 0 − 5.00000i 0 − 9.56038i 0 0 0
431.4 0 0 0 − 5.00000i 0 9.56038i 0 0 0
431.5 0 0 0 − 5.00000i 0 18.5665i 0 0 0
431.6 0 0 0 − 5.00000i 0 33.2548i 0 0 0
431.7 0 0 0 5.00000i 0 − 33.2548i 0 0 0
431.8 0 0 0 5.00000i 0 − 18.5665i 0 0 0
431.9 0 0 0 5.00000i 0 − 9.56038i 0 0 0
431.10 0 0 0 5.00000i 0 9.56038i 0 0 0
431.11 0 0 0 5.00000i 0 18.5665i 0 0 0
431.12 0 0 0 5.00000i 0 33.2548i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.4.h.e ✓ 12
3.b odd 2 1 inner 2160.4.h.e ✓ 12
4.b odd 2 1 inner 2160.4.h.e ✓ 12
12.b even 2 1 inner 2160.4.h.e ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2160.4.h.e ✓ 12 1.a even 1 1 trivial
2160.4.h.e ✓ 12 3.b odd 2 1 inner
2160.4.h.e ✓ 12 4.b odd 2 1 inner
2160.4.h.e ✓ 12 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 1542T_{7}^{4} + 513801T_{7}^{2} + 34843392 \) acting on \(S_{4}^{\mathrm{new}}(2160, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} + 1542 T^{4} + \cdots + 34843392)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 1515 T^{4} + \cdots - 13230000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 81 T^{2} + \cdots + 54425)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 23121 T^{4} + \cdots + 280052640000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 24078 T^{4} + \cdots + 245154679488)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 11645300545200)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 15365302419600)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 8850655393200)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 240 T^{2} + \cdots + 150526)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 57366081921600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 8943649873200)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 294339 T^{4} + \cdots - 621839635200)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 671591513779200)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 120 T^{2} + \cdots - 119148722)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 819529310407500)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 366 T^{2} + \cdots + 10525424)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 587767762935675)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 1884 T^{2} + \cdots + 479345426)^{4} \) Copy content Toggle raw display
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