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Properties

Label 2960.2.p.j
Level $2960$
Weight $2$
Character orbit 2960.p
Analytic conductor $23.636$
Analytic rank $0$
Dimension $12$
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] = [2960,2,Mod(961,2960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2960.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.p (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: \(23.6357189983\)
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: \( x^{12} + 17x^{10} + 88x^{8} + 176x^{6} + 132x^{4} + 40x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1480)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{3} + \beta_{7} q^{5} + (\beta_{8} - \beta_{5} + \beta_{3} + 2) q^{7} + ( - \beta_{5} + 2 \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{3} + \beta_{7} q^{5} + (\beta_{8} - \beta_{5} + \beta_{3} + 2) q^{7} + ( - \beta_{5} + 2 \beta_{3} + 1) q^{9} + (\beta_{11} + \beta_{5} - \beta_{4} + \cdots - 1) q^{11}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{4} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} + 10 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} + 10 q^{7} - 2 q^{9} - 6 q^{11} + 10 q^{21} - 12 q^{25} + 30 q^{27} - 6 q^{33} - 2 q^{37} + 6 q^{41} - 2 q^{47} + 6 q^{49} - 10 q^{53} + 56 q^{63} - 56 q^{67} - 14 q^{71} + 54 q^{73} - 6 q^{75} - 54 q^{77} + 28 q^{81} - 2 q^{83} + 12 q^{85} - 4 q^{95} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 17x^{10} + 88x^{8} + 176x^{6} + 132x^{4} + 40x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{11} - 49\nu^{9} - 231\nu^{7} - 369\nu^{5} - 130\nu^{3} - 6\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{10} - 182\nu^{8} - 885\nu^{6} - 1530\nu^{4} - 746\nu^{2} - 98 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -8\nu^{10} - 132\nu^{8} - 638\nu^{6} - 1089\nu^{4} - 511\nu^{2} - 61 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -11\nu^{10} - 182\nu^{8} - 885\nu^{6} - 1530\nu^{4} - 745\nu^{2} - 94 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{11} - 380\nu^{9} - 1842\nu^{7} - 3163\nu^{5} - 1506\nu^{3} - 176\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -24\nu^{11} - 397\nu^{9} - 1930\nu^{7} - 3339\nu^{5} - 1638\nu^{3} - 214\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -25\nu^{10} - 413\nu^{8} - 2002\nu^{6} - 3442\nu^{4} - 1654\nu^{2} - 206 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -28\nu^{11} - 462\nu^{9} - 2233\nu^{7} - 3811\nu^{5} - 1784\nu^{3} - 210\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -37\nu^{11} - 612\nu^{9} - 2975\nu^{7} - 5148\nu^{5} - 2530\nu^{3} - 328\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 37\nu^{10} + 612\nu^{8} + 2975\nu^{6} + 5148\nu^{4} + 2530\nu^{2} + 330 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 2\beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{10} - \beta_{9} - 3\beta_{7} + \beta_{6} + \beta_{2} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + 4\beta_{8} - 9\beta_{5} - 3\beta_{4} + 20\beta_{3} + 33 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -25\beta_{10} + 13\beta_{9} + 34\beta_{7} - 9\beta_{6} - 16\beta_{2} + 63\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -25\beta_{11} - 54\beta_{8} + 85\beta_{5} + 41\beta_{4} - 191\beta_{3} - 305 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 257\beta_{10} - 139\beta_{9} - 338\beta_{7} + 85\beta_{6} + 180\beta_{2} - 593\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 257\beta_{11} + 576\beta_{8} - 817\beta_{5} - 437\beta_{4} + 1825\beta_{3} + 2893 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -2519\beta_{10} + 1393\beta_{9} + 3270\beta_{7} - 817\beta_{6} - 1830\beta_{2} + 5653\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2519\beta_{11} - 5742\beta_{8} + 7863\beta_{5} + 4349\beta_{4} - 17475\beta_{3} - 27655 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 24343\beta_{10} - 13605\beta_{9} - 31436\beta_{7} + 7863\beta_{6} + 17954\beta_{2} - 54119\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2960\mathbb{Z}\right)^\times\).

\(n\) \(741\) \(1777\) \(2481\) \(2591\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
1.57934i
− 1.57934i
0.690667i
− 0.690667i
0.448699i
− 0.448699i
− 0.679423i
0.679423i
− 1.94150i
1.94150i
− 3.09779i
3.09779i
0 −1.57934 0 − 1.00000i 0 3.26216 0 −0.505673 0
961.2 0 −1.57934 0 1.00000i 0 3.26216 0 −0.505673 0
961.3 0 −0.690667 0 − 1.00000i 0 −1.67702 0 −2.52298 0
961.4 0 −0.690667 0 1.00000i 0 −1.67702 0 −2.52298 0
961.5 0 −0.448699 0 − 1.00000i 0 1.22531 0 −2.79867 0
961.6 0 −0.448699 0 1.00000i 0 1.22531 0 −2.79867 0
961.7 0 0.679423 0 − 1.00000i 0 0.161247 0 −2.53838 0
961.8 0 0.679423 0 1.00000i 0 0.161247 0 −2.53838 0
961.9 0 1.94150 0 − 1.00000i 0 −2.72494 0 0.769426 0
961.10 0 1.94150 0 1.00000i 0 −2.72494 0 0.769426 0
961.11 0 3.09779 0 − 1.00000i 0 4.75324 0 6.59628 0
961.12 0 3.09779 0 1.00000i 0 4.75324 0 6.59628 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.2.p.j 12
4.b odd 2 1 1480.2.p.c ✓ 12
37.b even 2 1 inner 2960.2.p.j 12
148.b odd 2 1 1480.2.p.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.p.c ✓ 12 4.b odd 2 1
1480.2.p.c ✓ 12 148.b odd 2 1
2960.2.p.j 12 1.a even 1 1 trivial
2960.2.p.j 12 37.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2960, [\chi])\):

\( T_{3}^{6} - 3T_{3}^{5} - 4T_{3}^{4} + 10T_{3}^{3} + 6T_{3}^{2} - 4T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{6} - 5T_{7}^{5} - 10T_{7}^{4} + 52T_{7}^{3} + 24T_{7}^{2} - 92T_{7} + 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 3 T^{5} - 4 T^{4} + \cdots - 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} - 5 T^{5} - 10 T^{4} + \cdots + 14)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 3 T^{5} - 24 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 72 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{12} + 60 T^{10} + \cdots + 53824 \) Copy content Toggle raw display
$19$ \( T^{12} + 156 T^{10} + \cdots + 10471696 \) Copy content Toggle raw display
$23$ \( T^{12} + 116 T^{10} + \cdots + 50176 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 112869376 \) Copy content Toggle raw display
$31$ \( T^{12} + 152 T^{10} + \cdots + 2999824 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( (T^{6} - 3 T^{5} + \cdots + 9008)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 244 T^{10} + \cdots + 78570496 \) Copy content Toggle raw display
$47$ \( (T^{6} + T^{5} - 92 T^{4} + \cdots - 2594)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 5 T^{5} + \cdots + 35968)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 2590402816 \) Copy content Toggle raw display
$61$ \( T^{12} + 248 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$67$ \( (T^{6} + 28 T^{5} + \cdots + 22328)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 7 T^{5} + \cdots - 16768)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 27 T^{5} + \cdots - 48704)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 67487726656 \) Copy content Toggle raw display
$83$ \( (T^{6} + T^{5} + \cdots - 39022)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 433532698624 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 2400216064 \) Copy content Toggle raw display
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