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Properties

Label 240.4.h.a
Level $240$
Weight $4$
Character orbit 240.h
Analytic conductor $14.160$
Analytic rank $0$
Dimension $8$
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] = [240,4,Mod(191,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, 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("240.191");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.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: \(14.1604584014\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.222919710806016.95
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 15x^{6} + 641x^{4} + 7344x^{2} + 53824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{4} q^{5} + (2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{7} + (\beta_{5} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{4} q^{5} + (2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{7} + (\beta_{5} + 2) q^{9} + ( - 2 \beta_{7} - \beta_{3} + \cdots + \beta_1) q^{11}+ \cdots + (30 \beta_{7} + 135 \beta_{3} + \cdots + 33 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{9} - 184 q^{13} + 420 q^{21} - 200 q^{25} - 288 q^{33} + 376 q^{37} + 60 q^{45} + 560 q^{49} - 888 q^{57} - 1640 q^{61} - 468 q^{69} + 3392 q^{73} + 1440 q^{85} - 816 q^{93} - 3760 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 15x^{6} + 641x^{4} + 7344x^{2} + 53824 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 57\nu^{6} + 1087\nu^{4} + 87249\nu^{2} + 565432 ) / 131920 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2525 \nu^{7} - 20474 \nu^{6} - 18965 \nu^{5} - 121974 \nu^{4} + 803045 \nu^{3} - 9593258 \nu^{2} + \cdots - 50608944 ) / 15302720 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2525 \nu^{7} + 20474 \nu^{6} - 18965 \nu^{5} + 121974 \nu^{4} + 803045 \nu^{3} + 9593258 \nu^{2} + \cdots + 50608944 ) / 15302720 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -57\nu^{7} - 1087\nu^{5} - 25169\nu^{3} - 394712\nu ) / 180032 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9747 \nu^{7} - 27840 \nu^{6} + 185877 \nu^{5} - 1336320 \nu^{4} + 7004379 \nu^{3} + \cdots - 396979840 ) / 15302720 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9747 \nu^{7} + 27840 \nu^{6} + 185877 \nu^{5} + 1336320 \nu^{4} + 7004379 \nu^{3} + \cdots + 381677120 ) / 15302720 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24025 \nu^{7} + 3306 \nu^{6} + 122575 \nu^{5} + 63046 \nu^{4} + 13549825 \nu^{3} + \cdots + 32795056 ) / 15302720 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -6\beta_{7} + 5\beta_{6} + 5\beta_{5} + 6\beta_{4} + 15\beta_{3} + 15\beta_{2} + 3\beta _1 + 5 ) / 60 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} - 3\beta_{3} + 3\beta_{2} + 27\beta _1 - 45 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 72\beta_{7} + 25\beta_{6} + 25\beta_{5} + 270\beta_{4} - 180\beta_{3} - 180\beta_{2} - 36\beta _1 + 25 ) / 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 27\beta_{6} - 27\beta_{5} - 33\beta_{3} + 33\beta_{2} - 23\beta _1 - 1057 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 966 \beta_{7} - 1595 \beta_{6} - 1595 \beta_{5} - 11274 \beta_{4} - 9255 \beta_{3} - 9255 \beta_{2} + \cdots - 1595 ) / 60 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -7\beta_{6} + 7\beta_{5} + 3240\beta_{3} - 3240\beta_{2} - 6120\beta _1 + 5157 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 40458 \beta_{7} - 26285 \beta_{6} - 26285 \beta_{5} - 254502 \beta_{4} + 231585 \beta_{3} + \cdots - 26285 ) / 60 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−3.33665 + 3.42214i
−3.33665 − 3.42214i
1.27153 + 2.92214i
1.27153 − 2.92214i
−1.27153 − 2.92214i
−1.27153 + 2.92214i
3.33665 − 3.42214i
3.33665 + 3.42214i
0 −4.87508 − 1.79821i 0 5.00000i 0 − 2.55732i 0 20.5329 + 17.5329i 0
191.2 0 −4.87508 + 1.79821i 0 − 5.00000i 0 2.55732i 0 20.5329 − 17.5329i 0
191.3 0 −2.17568 − 4.71873i 0 − 5.00000i 0 23.2263i 0 −17.5329 + 20.5329i 0
191.4 0 −2.17568 + 4.71873i 0 5.00000i 0 − 23.2263i 0 −17.5329 − 20.5329i 0
191.5 0 2.17568 − 4.71873i 0 5.00000i 0 23.2263i 0 −17.5329 − 20.5329i 0
191.6 0 2.17568 + 4.71873i 0 − 5.00000i 0 − 23.2263i 0 −17.5329 + 20.5329i 0
191.7 0 4.87508 − 1.79821i 0 − 5.00000i 0 − 2.55732i 0 20.5329 − 17.5329i 0
191.8 0 4.87508 + 1.79821i 0 5.00000i 0 2.55732i 0 20.5329 + 17.5329i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.8
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 240.4.h.a ✓ 8
3.b odd 2 1 inner 240.4.h.a ✓ 8
4.b odd 2 1 inner 240.4.h.a ✓ 8
8.b even 2 1 960.4.h.a 8
8.d odd 2 1 960.4.h.a 8
12.b even 2 1 inner 240.4.h.a ✓ 8
24.f even 2 1 960.4.h.a 8
24.h odd 2 1 960.4.h.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.4.h.a ✓ 8 1.a even 1 1 trivial
240.4.h.a ✓ 8 3.b odd 2 1 inner
240.4.h.a ✓ 8 4.b odd 2 1 inner
240.4.h.a ✓ 8 12.b even 2 1 inner
960.4.h.a 8 8.b even 2 1
960.4.h.a 8 8.d odd 2 1
960.4.h.a 8 24.f even 2 1
960.4.h.a 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 546T_{7}^{2} + 3528 \) acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{6} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 546 T^{2} + 3528)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2724 T^{2} + 180000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 46 T - 920)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 14184 T^{2} + 20250000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 15684 T^{2} + 7200)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 41154 T^{2} + 144160200)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 72900 T^{2} + 1296000000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 34476 T^{2} + 24220800)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 94 T - 115160)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 69876 T^{2} + 1026561600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 210102 T^{2} + 10846877472)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 78450 T^{2} + 16245000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 150984 T^{2} + 4083210000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 175596 T^{2} + 6498000000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 410 T - 75344)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 530118 T^{2} + 2285420832)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 1094424 T^{2} + 107277120000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 848 T + 173980)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1256964 T^{2} + 365307328800)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1026 T^{2} + 145800)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 1831824 T^{2} + 817794662400)^{2} \) Copy content Toggle raw display
$97$ \( (T + 470)^{8} \) Copy content Toggle raw display
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