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Properties

Label 1920.2.b.f
Level $1920$
Weight $2$
Character orbit 1920.b
Analytic conductor $15.331$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1920,2,Mod(191,1920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1920, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1920.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1920.b (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: \(15.3312771881\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3288334336.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 8x^{6} - 8x^{5} + 14x^{4} + 8x^{3} - 16x^{2} + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + q^{5} - \beta_1 q^{7} + ( - \beta_{7} + \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + q^{5} - \beta_1 q^{7} + ( - \beta_{7} + \beta_{5} + 1) q^{9} - 2 \beta_1 q^{11} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{13} - \beta_{6} q^{15} + ( - \beta_{7} - \beta_{3}) q^{17} + ( - \beta_{6} + \beta_{4}) q^{19} - \beta_{3} q^{21} + (\beta_{4} + \beta_{2}) q^{23} + q^{25} + ( - 2 \beta_{6} + \beta_{4} + \cdots + \beta_1) q^{27}+ \cdots + (2 \beta_{4} - 2 \beta_{2} - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} + 8 q^{9} + 8 q^{25} - 16 q^{29} + 8 q^{45} + 24 q^{49} + 16 q^{53} + 16 q^{57} - 16 q^{69} - 16 q^{73} - 64 q^{77} - 24 q^{81} + 16 q^{93} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 8x^{6} - 8x^{5} + 14x^{4} + 8x^{3} - 16x^{2} + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 510\nu^{7} + 1999\nu^{6} + 5184\nu^{5} + 13117\nu^{4} - 1256\nu^{3} + 29943\nu^{2} + 5516\nu - 12187 ) / 8561 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 720\nu^{7} - 703\nu^{6} + 6815\nu^{5} - 10690\nu^{4} + 24917\nu^{3} - 6072\nu^{2} - 5306\nu + 18046 ) / 8561 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -118\nu^{7} - 357\nu^{6} - 998\nu^{5} - 1951\nu^{4} + 540\nu^{3} - 6669\nu^{2} - 3540\nu + 2743 ) / 1223 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1196\nu^{7} + 592\nu^{6} + 8229\nu^{5} - 5867\nu^{4} - 2509\nu^{3} + 23587\nu^{2} - 42392\nu - 3031 ) / 8561 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -1444\nu^{7} + 316\nu^{6} - 11052\nu^{5} + 13354\nu^{4} - 19676\nu^{3} - 14076\nu^{2} + 39844\nu + 2142 ) / 8561 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -1508\nu^{7} - 2\nu^{6} - 12609\nu^{5} + 13353\nu^{4} - 25125\nu^{3} + 3015\nu^{2} + 11018\nu + 6055 ) / 8561 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2228\nu^{7} - 701\nu^{6} + 19424\nu^{5} - 24043\nu^{4} + 50042\nu^{3} - 9087\nu^{2} + 798\nu + 11991 ) / 8561 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 3\beta_{2} - \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{7} - 6\beta_{6} - 2\beta_{5} - 3\beta_{4} - 2\beta_{3} + 3\beta_{2} - 2\beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{7} + 16\beta_{6} - 11\beta_{5} - 6\beta_{4} + 8\beta_{3} - 26\beta_{2} + 10\beta _1 + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 18\beta_{7} + 30\beta_{6} + 30\beta_{5} + 30\beta_{4} + 9\beta_{3} + 20\beta_{2} + 11\beta _1 - 90 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -163\beta_{7} - 186\beta_{6} + 58\beta_{5} + 9\beta_{4} - 69\beta_{3} + 193\beta_{2} - 79\beta _1 - 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 80\beta_{7} - 13\beta_{6} - 313\beta_{5} - 252\beta_{4} + 27\beta_{3} - 449\beta_{2} + 28\beta _1 + 812 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(641\) \(901\) \(1537\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
0.707107 − 0.349249i
0.707107 + 0.349249i
−0.707107 − 0.179070i
−0.707107 + 0.179070i
−0.707107 + 2.79220i
−0.707107 − 2.79220i
0.707107 + 1.43164i
0.707107 − 1.43164i
0 −1.64533 − 0.541196i 0 1.00000 0 2.61313i 0 2.41421 + 1.78089i 0
191.2 0 −1.64533 + 0.541196i 0 1.00000 0 − 2.61313i 0 2.41421 − 1.78089i 0
191.3 0 −1.13705 − 1.30656i 0 1.00000 0 − 1.08239i 0 −0.414214 + 2.97127i 0
191.4 0 −1.13705 + 1.30656i 0 1.00000 0 1.08239i 0 −0.414214 − 2.97127i 0
191.5 0 1.13705 − 1.30656i 0 1.00000 0 − 1.08239i 0 −0.414214 − 2.97127i 0
191.6 0 1.13705 + 1.30656i 0 1.00000 0 1.08239i 0 −0.414214 + 2.97127i 0
191.7 0 1.64533 − 0.541196i 0 1.00000 0 2.61313i 0 2.41421 − 1.78089i 0
191.8 0 1.64533 + 0.541196i 0 1.00000 0 − 2.61313i 0 2.41421 + 1.78089i 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
4.b odd 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1920.2.b.f yes 8
3.b odd 2 1 1920.2.b.d ✓ 8
4.b odd 2 1 inner 1920.2.b.f yes 8
8.b even 2 1 1920.2.b.d ✓ 8
8.d odd 2 1 1920.2.b.d ✓ 8
12.b even 2 1 1920.2.b.d ✓ 8
24.f even 2 1 inner 1920.2.b.f yes 8
24.h odd 2 1 inner 1920.2.b.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1920.2.b.d ✓ 8 3.b odd 2 1
1920.2.b.d ✓ 8 8.b even 2 1
1920.2.b.d ✓ 8 8.d odd 2 1
1920.2.b.d ✓ 8 12.b even 2 1
1920.2.b.f yes 8 1.a even 1 1 trivial
1920.2.b.f yes 8 4.b odd 2 1 inner
1920.2.b.f yes 8 24.f even 2 1 inner
1920.2.b.f yes 8 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1920, [\chi])\):

\( T_{7}^{4} + 8T_{7}^{2} + 8 \) Copy content Toggle raw display
\( T_{19}^{4} - 32T_{19}^{2} + 224 \) Copy content Toggle raw display
\( T_{23}^{4} - 32T_{23}^{2} + 56 \) Copy content Toggle raw display
\( T_{29}^{2} + 4T_{29} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 8 T^{2} + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 32 T^{2} + 128)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 40 T^{2} + 112)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 24 T^{2} + 112)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 32 T^{2} + 224)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 32 T^{2} + 56)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 4)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 16 T^{2} + 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 104 T^{2} + 112)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 80 T^{2} + 448)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 112 T^{2} + 2744)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 128 T^{2} + 2744)^{2} \) Copy content Toggle raw display
$53$ \( (T - 2)^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 128 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 304 T^{2} + 21952)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 176 T^{2} + 2744)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 256 T^{2} + 896)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4 T - 28)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 208 T^{2} + 9248)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 104 T^{2} + 392)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 160 T^{2} + 1792)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4 T - 124)^{4} \) Copy content Toggle raw display
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