↑

Properties

Label 1920.2.w.i
Level $1920$
Weight $2$
Character orbit 1920.w
Analytic conductor $15.331$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1920,2,Mod(127,1920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1920, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("1920.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1920.w (of order \(4\), degree \(2\), 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: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 14x^{10} + 71x^{8} + 158x^{6} + 149x^{4} + 52x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} - \beta_{3} q^{5} + ( - \beta_{11} + \beta_{10} - \beta_{8} + \cdots - 1) q^{7}+ \cdots + \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} - \beta_{3} q^{5} + ( - \beta_{11} + \beta_{10} - \beta_{8} + \cdots - 1) q^{7}+ \cdots + (\beta_{10} + \beta_{9} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{13} - 4 q^{15} - 20 q^{17} + 8 q^{19} + 8 q^{21} - 4 q^{25} + 8 q^{35} - 20 q^{37} - 8 q^{39} + 16 q^{41} + 16 q^{43} + 4 q^{45} + 40 q^{47} + 4 q^{53} - 24 q^{55} - 16 q^{57} + 16 q^{61} - 12 q^{65} - 8 q^{67} + 4 q^{73} + 16 q^{75} - 48 q^{77} - 16 q^{79} - 12 q^{81} - 40 q^{83} - 28 q^{85} + 8 q^{87} + 16 q^{93} - 72 q^{95} - 52 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 14x^{10} + 71x^{8} + 158x^{6} + 149x^{4} + 52x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} + 12\nu^{9} + 47\nu^{7} + 60\nu^{5} - 7\nu^{3} - 26\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} - \nu^{10} - 14 \nu^{9} - 14 \nu^{8} - 71 \nu^{7} - 69 \nu^{6} - 156 \nu^{5} - 138 \nu^{4} + \cdots - 14 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} - 14 \nu^{9} - 2 \nu^{8} - 71 \nu^{7} - 22 \nu^{6} - 160 \nu^{5} - 78 \nu^{4} - 163 \nu^{3} + \cdots - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} - 14 \nu^{9} + 2 \nu^{8} - 71 \nu^{7} + 22 \nu^{6} - 160 \nu^{5} + 78 \nu^{4} - 163 \nu^{3} + \cdots + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - \nu^{10} + 14 \nu^{9} - 14 \nu^{8} + 71 \nu^{7} - 69 \nu^{6} + 156 \nu^{5} - 138 \nu^{4} + \cdots - 14 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3 \nu^{11} - 3 \nu^{10} - 40 \nu^{9} - 40 \nu^{8} - 189 \nu^{7} - 185 \nu^{6} - 376 \nu^{5} + \cdots - 26 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{11} + \nu^{10} + 40 \nu^{9} + 12 \nu^{8} + 189 \nu^{7} + 47 \nu^{6} + 376 \nu^{5} + 60 \nu^{4} + \cdots - 18 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{11} - \nu^{10} + 40 \nu^{9} - 12 \nu^{8} + 189 \nu^{7} - 47 \nu^{6} + 376 \nu^{5} - 60 \nu^{4} + \cdots + 18 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2 \nu^{11} - 27 \nu^{9} - \nu^{8} - 128 \nu^{7} - 9 \nu^{6} - 248 \nu^{5} - 21 \nu^{4} - 167 \nu^{3} + \cdots + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2\nu^{11} + 27\nu^{9} - \nu^{8} + 128\nu^{7} - 9\nu^{6} + 248\nu^{5} - 21\nu^{4} + 167\nu^{3} - 4\nu^{2} + 26\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3 \nu^{11} - 3 \nu^{10} + 40 \nu^{9} - 40 \nu^{8} + 189 \nu^{7} - 185 \nu^{6} + 376 \nu^{5} - 344 \nu^{4} + \cdots - 26 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{11} - \beta_{8} - \beta_{7} + 3\beta_{6} + 4\beta_{5} - 4\beta_{4} - 4\beta_{3} - 4\beta_{2} + 8\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{11} - 4\beta_{8} + 4\beta_{7} - \beta_{6} + 7\beta_{5} + 5\beta_{4} - 5\beta_{3} + 7\beta_{2} + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 12 \beta_{11} + 7 \beta_{8} + 7 \beta_{7} - 12 \beta_{6} - 21 \beta_{5} + 17 \beta_{4} + 17 \beta_{3} + \cdots - 38 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 15 \beta_{8} - 15 \beta_{7} + 9 \beta_{6} - 42 \beta_{5} + \cdots - 76 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 51 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 40 \beta_{8} - 40 \beta_{7} + 51 \beta_{6} + \cdots + 186 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 60 \beta_{11} - 22 \beta_{10} - 22 \beta_{9} - 55 \beta_{8} + 55 \beta_{7} - 60 \beta_{6} + \cdots + 380 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 221 \beta_{11} + 24 \beta_{10} - 24 \beta_{9} + 215 \beta_{8} + 215 \beta_{7} - 221 \beta_{6} + \cdots - 928 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 357 \beta_{11} + 170 \beta_{10} + 170 \beta_{9} + 194 \beta_{8} - 194 \beta_{7} + 357 \beta_{6} + \cdots - 1940 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 970 \beta_{11} - 194 \beta_{10} + 194 \beta_{9} - 1127 \beta_{8} - 1127 \beta_{7} + 970 \beta_{6} + \cdots + 4694 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(511\) \(641\) \(901\) \(1537\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
− 0.741252i
2.02068i
− 2.27943i
1.04757i
− 0.324536i
− 1.72303i
0.741252i
− 2.02068i
2.27943i
− 1.04757i
0.324536i
1.72303i
0 −0.707107 − 0.707107i 0 −1.80858 − 1.31492i 0 −3.60601 + 3.60601i 0 1.00000i 0
127.2 0 −0.707107 − 0.707107i 0 −0.118826 − 2.23291i 0 2.68963 − 2.68963i 0 1.00000i 0
127.3 0 −0.707107 − 0.707107i 0 1.92741 + 1.13362i 0 −0.497835 + 0.497835i 0 1.00000i 0
127.4 0 0.707107 + 0.707107i 0 −2.15008 + 0.614127i 0 1.55919 − 1.55919i 0 1.00000i 0
127.5 0 0.707107 + 0.707107i 0 0.893756 − 2.04968i 0 −0.804999 + 0.804999i 0 1.00000i 0
127.6 0 0.707107 + 0.707107i 0 1.25633 + 1.84977i 0 0.660026 − 0.660026i 0 1.00000i 0
1663.1 0 −0.707107 + 0.707107i 0 −1.80858 + 1.31492i 0 −3.60601 − 3.60601i 0 − 1.00000i 0
1663.2 0 −0.707107 + 0.707107i 0 −0.118826 + 2.23291i 0 2.68963 + 2.68963i 0 − 1.00000i 0
1663.3 0 −0.707107 + 0.707107i 0 1.92741 − 1.13362i 0 −0.497835 − 0.497835i 0 − 1.00000i 0
1663.4 0 0.707107 − 0.707107i 0 −2.15008 − 0.614127i 0 1.55919 + 1.55919i 0 − 1.00000i 0
1663.5 0 0.707107 − 0.707107i 0 0.893756 + 2.04968i 0 −0.804999 − 0.804999i 0 − 1.00000i 0
1663.6 0 0.707107 − 0.707107i 0 1.25633 − 1.84977i 0 0.660026 + 0.660026i 0 − 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1920.2.w.i ✓ 12
4.b odd 2 1 1920.2.w.j yes 12
5.c odd 4 1 1920.2.w.j yes 12
8.b even 2 1 1920.2.w.k yes 12
8.d odd 2 1 1920.2.w.l yes 12
20.e even 4 1 inner 1920.2.w.i ✓ 12
40.i odd 4 1 1920.2.w.l yes 12
40.k even 4 1 1920.2.w.k yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1920.2.w.i ✓ 12 1.a even 1 1 trivial
1920.2.w.i ✓ 12 20.e even 4 1 inner
1920.2.w.j yes 12 4.b odd 2 1
1920.2.w.j yes 12 5.c odd 4 1
1920.2.w.k yes 12 8.b even 2 1
1920.2.w.k yes 12 40.k even 4 1
1920.2.w.l yes 12 8.d odd 2 1
1920.2.w.l yes 12 40.i odd 4 1

Hecke kernels

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

\( T_{7}^{12} - 32 T_{7}^{9} + 456 T_{7}^{8} - 736 T_{7}^{7} + 512 T_{7}^{6} + 1152 T_{7}^{5} + 2064 T_{7}^{4} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{13}^{12} + 4 T_{13}^{11} + 8 T_{13}^{10} - 72 T_{13}^{9} + 292 T_{13}^{8} + 256 T_{13}^{7} + \cdots + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} + 2 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} - 32 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( T^{12} + 56 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{12} + 4 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( T^{12} + 20 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( (T^{6} - 4 T^{5} + \cdots - 4096)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 192 T^{9} + \cdots + 15745024 \) Copy content Toggle raw display
$29$ \( T^{12} + 152 T^{10} + \cdots + 65536 \) Copy content Toggle raw display
$31$ \( T^{12} + 256 T^{10} + \cdots + 43454464 \) Copy content Toggle raw display
$37$ \( T^{12} + 20 T^{11} + \cdots + 8111104 \) Copy content Toggle raw display
$41$ \( (T^{6} - 8 T^{5} + \cdots + 39808)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 16 T^{11} + \cdots + 262144 \) Copy content Toggle raw display
$47$ \( T^{12} - 40 T^{11} + \cdots + 62980096 \) Copy content Toggle raw display
$53$ \( T^{12} - 4 T^{11} + \cdots + 58491904 \) Copy content Toggle raw display
$59$ \( (T^{6} - 172 T^{4} + \cdots - 76144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 8 T^{5} + \cdots + 25472)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 163840000 \) Copy content Toggle raw display
$71$ \( T^{12} + 336 T^{10} + \cdots + 4194304 \) Copy content Toggle raw display
$73$ \( T^{12} - 4 T^{11} + \cdots + 399424 \) Copy content Toggle raw display
$79$ \( (T^{6} + 8 T^{5} + \cdots - 634432)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 40 T^{11} + \cdots + 62980096 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 597704704 \) Copy content Toggle raw display
$97$ \( T^{12} + 52 T^{11} + \cdots + 2534464 \) Copy content Toggle raw display
show more
show less