Properties

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

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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: 8.0.619810816.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - q^{5} + ( - \beta_{7} + \beta_{2}) q^{7} + ( - \beta_{6} - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - q^{5} + ( - \beta_{7} + \beta_{2}) q^{7} + ( - \beta_{6} - \beta_{3} - 1) q^{9} + (\beta_{6} - \beta_{4} + \beta_{3} - \beta_1) q^{11} + ( - \beta_{7} + \beta_{3} - \beta_1) q^{13} + \beta_1 q^{15} + (\beta_{7} + \beta_{6} - \beta_{4}) q^{17} + ( - \beta_{6} - \beta_{5} - \beta_{4} - 2) q^{19} + (\beta_{5} + \beta_{4} + \cdots - \beta_{2}) q^{21}+ \cdots + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} + \cdots + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 8 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 8 q^{5} - 4 q^{9} - 2 q^{15} - 8 q^{19} - 4 q^{21} + 12 q^{23} + 8 q^{25} + 14 q^{27} + 8 q^{29} - 8 q^{33} - 28 q^{39} - 36 q^{43} + 4 q^{45} - 4 q^{47} + 20 q^{51} + 16 q^{63} - 28 q^{67} - 12 q^{69} + 24 q^{71} + 32 q^{73} + 2 q^{75} - 32 q^{77} + 8 q^{81} - 12 q^{87} - 40 q^{91} - 24 q^{93} + 8 q^{95} + 16 q^{97} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -59\nu^{7} + 65\nu^{6} + 96\nu^{5} + 142\nu^{4} - 950\nu^{3} + 1032\nu^{2} + 459\nu - 402 ) / 319 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -126\nu^{7} + 128\nu^{6} + 32\nu^{5} + 260\nu^{4} - 2018\nu^{3} + 2896\nu^{2} - 804\nu - 134 ) / 319 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -129\nu^{7} - 112\nu^{6} - 28\nu^{5} + 251\nu^{4} - 1504\nu^{3} - 301\nu^{2} + 225\nu - 441 ) / 319 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -129\nu^{7} - 112\nu^{6} - 28\nu^{5} + 251\nu^{4} - 1504\nu^{3} - 301\nu^{2} + 863\nu - 441 ) / 319 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -156\nu^{7} - 39\nu^{6} + 70\nu^{5} + 489\nu^{4} - 1982\nu^{3} + 593\nu^{2} + 554\nu + 305 ) / 319 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -17\nu^{7} + 3\nu^{6} + 8\nu^{5} + 36\nu^{4} - 258\nu^{3} + 144\nu^{2} + 31\nu - 48 ) / 29 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -322\nu^{7} + 79\nu^{6} - 60\nu^{5} + 629\nu^{4} - 4590\nu^{3} + 3821\nu^{2} - 1842\nu - 307 ) / 319 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{3} + 2\beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 3\beta_{6} + \beta_{3} - \beta_{2} + 4\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{6} + 4\beta_{5} + \beta_{4} - 5\beta_{3} - 5\beta _1 - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} - \beta_{5} - 11\beta_{4} + 17\beta_{3} + 6\beta_{2} + 4\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 17\beta_{7} - 7\beta_{6} + 7\beta_{4} - 22\beta_{3} - 28\beta_{2} + 22\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -23\beta_{7} + 43\beta_{6} + 8\beta_{5} - 15\beta_{3} + 30\beta_{2} - 74\beta _1 - 60 ) / 2 \) Copy content Toggle raw display

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
−1.49094 + 1.49094i
−1.49094 1.49094i
0.561103 + 0.561103i
0.561103 0.561103i
1.18254 1.18254i
1.18254 + 1.18254i
−0.252709 0.252709i
−0.252709 + 0.252709i
0 −1.15558 1.29021i 0 −1.00000 0 4.31116i 0 −0.329281 + 2.98187i 0
191.2 0 −1.15558 + 1.29021i 0 −1.00000 0 4.31116i 0 −0.329281 2.98187i 0
191.3 0 −0.329998 1.70032i 0 −1.00000 0 2.66000i 0 −2.78220 + 1.12221i 0
191.4 0 −0.329998 + 1.70032i 0 −1.00000 0 2.66000i 0 −2.78220 1.12221i 0
191.5 0 0.759725 1.55654i 0 −1.00000 0 0.480550i 0 −1.84564 2.36509i 0
191.6 0 0.759725 + 1.55654i 0 −1.00000 0 0.480550i 0 −1.84564 + 2.36509i 0
191.7 0 1.72585 0.146426i 0 −1.00000 0 1.45170i 0 2.95712 0.505418i 0
191.8 0 1.72585 + 0.146426i 0 −1.00000 0 1.45170i 0 2.95712 + 0.505418i 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
24.f even 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.g yes 8
3.b odd 2 1 1920.2.b.h yes 8
4.b odd 2 1 1920.2.b.a 8
8.b even 2 1 1920.2.b.b yes 8
8.d odd 2 1 1920.2.b.h yes 8
12.b even 2 1 1920.2.b.b yes 8
24.f even 2 1 inner 1920.2.b.g yes 8
24.h odd 2 1 1920.2.b.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1920.2.b.a 8 4.b odd 2 1
1920.2.b.a 8 24.h odd 2 1
1920.2.b.b yes 8 8.b even 2 1
1920.2.b.b yes 8 12.b even 2 1
1920.2.b.g yes 8 1.a even 1 1 trivial
1920.2.b.g yes 8 24.f even 2 1 inner
1920.2.b.h yes 8 3.b odd 2 1
1920.2.b.h yes 8 8.d odd 2 1

Hecke kernels

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

\( T_{7}^{8} + 28T_{7}^{6} + 192T_{7}^{4} + 320T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{19}^{4} + 4T_{19}^{3} - 40T_{19}^{2} - 16T_{19} + 160 \) Copy content Toggle raw display
\( T_{23}^{4} - 6T_{23}^{3} - 24T_{23}^{2} + 208T_{23} - 328 \) Copy content Toggle raw display
\( T_{29}^{4} - 4T_{29}^{3} - 64T_{29}^{2} + 400T_{29} - 592 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 28 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{8} + 64 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{8} + 48 T^{6} + \cdots + 6400 \) Copy content Toggle raw display
$17$ \( T^{8} + 64 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} + \cdots + 160)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 6 T^{3} + \cdots - 328)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{3} + \cdots - 592)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 80 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$37$ \( T^{8} + 208 T^{6} + \cdots + 3041536 \) Copy content Toggle raw display
$41$ \( T^{8} + 224 T^{6} + \cdots + 5017600 \) Copy content Toggle raw display
$43$ \( (T^{4} + 18 T^{3} + 56 T^{2} + \cdots + 8)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 2 T^{3} - 56 T^{2} + \cdots + 56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 88 T^{2} + \cdots + 464)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 176 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$61$ \( T^{8} + 288 T^{6} + \cdots + 5607424 \) Copy content Toggle raw display
$67$ \( (T^{4} + 14 T^{3} + \cdots + 200)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 12 T^{3} + \cdots - 4736)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 16 T^{3} + \cdots - 2864)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 352 T^{6} + \cdots + 3564544 \) Copy content Toggle raw display
$83$ \( T^{8} + 188 T^{6} + \cdots + 732736 \) Copy content Toggle raw display
$89$ \( T^{8} + 192 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$97$ \( (T^{4} - 8 T^{3} + \cdots + 656)^{2} \) Copy content Toggle raw display
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