↑

Properties

Label 1120.2.x.f
Level $1120$
Weight $2$
Character orbit 1120.x
Analytic conductor $8.943$
Analytic rank $0$
Dimension $20$
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] = [1120,2,Mod(127,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 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("1120.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.x (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: \(8.94324502638\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 4 x^{18} + 16 x^{17} - 56 x^{16} - 24 x^{15} + 512 x^{14} + 856 x^{13} + 402 x^{12} + \cdots + 5000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{3} q^{5} - \beta_{2} q^{7} + (\beta_{15} + \beta_{14} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{3} q^{5} - \beta_{2} q^{7} + (\beta_{15} + \beta_{14} + \cdots - \beta_{2}) q^{9}+ \cdots + ( - \beta_{19} + \beta_{17} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{5} - 12 q^{13} + 8 q^{15} - 4 q^{17} - 24 q^{19} + 8 q^{23} + 4 q^{25} - 16 q^{33} - 4 q^{35} + 4 q^{37} + 8 q^{39} + 32 q^{41} - 8 q^{43} + 32 q^{45} + 24 q^{47} + 28 q^{53} + 24 q^{59} + 24 q^{61} - 8 q^{63} - 12 q^{65} - 8 q^{67} + 20 q^{73} + 72 q^{75} + 8 q^{77} - 24 q^{79} - 20 q^{81} - 48 q^{83} - 20 q^{85} + 56 q^{87} - 8 q^{93} - 8 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 4 x^{18} + 16 x^{17} - 56 x^{16} - 24 x^{15} + 512 x^{14} + 856 x^{13} + 402 x^{12} + \cdots + 5000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 44\!\cdots\!23 \nu^{19} + \cdots + 11\!\cdots\!00 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 53\!\cdots\!11 \nu^{19} + \cdots - 30\!\cdots\!40 ) / 98\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 33\!\cdots\!01 \nu^{19} + \cdots - 10\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 85\!\cdots\!87 \nu^{19} + \cdots + 59\!\cdots\!00 ) / 98\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!87 \nu^{19} + \cdots - 18\!\cdots\!00 ) / 98\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20\!\cdots\!39 \nu^{19} + \cdots - 16\!\cdots\!50 ) / 14\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 89\!\cdots\!13 \nu^{19} + \cdots - 72\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!93 \nu^{19} + \cdots - 89\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15\!\cdots\!41 \nu^{19} + \cdots - 86\!\cdots\!00 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!03 \nu^{19} + \cdots + 62\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 62\!\cdots\!91 \nu^{19} + \cdots - 45\!\cdots\!00 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!81 \nu^{19} + \cdots + 76\!\cdots\!00 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!81 \nu^{19} + \cdots + 12\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 22\!\cdots\!23 \nu^{19} + \cdots - 19\!\cdots\!00 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 16\!\cdots\!63 \nu^{19} + \cdots + 54\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 18\!\cdots\!51 \nu^{19} + \cdots - 17\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 18\!\cdots\!13 \nu^{19} + \cdots - 14\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 20\!\cdots\!87 \nu^{19} + \cdots - 18\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 24\!\cdots\!73 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{18} + 2 \beta_{17} + \beta_{15} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} + \cdots + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - 2\beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{6} - 8\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14 \beta_{19} + 9 \beta_{18} - 4 \beta_{17} + 4 \beta_{16} - \beta_{15} + 4 \beta_{14} + 4 \beta_{13} + \cdots - 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 23 \beta_{19} + 16 \beta_{18} - 3 \beta_{17} + 3 \beta_{16} - \beta_{14} + 23 \beta_{13} + \cdots + 14 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 60 \beta_{19} + 93 \beta_{18} + 44 \beta_{17} - 60 \beta_{16} + 27 \beta_{15} - 134 \beta_{14} + \cdots + 45 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 42 \beta_{19} + 100 \beta_{18} + 42 \beta_{17} - 256 \beta_{16} + 171 \beta_{15} - 256 \beta_{14} + \cdots - 139 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 444 \beta_{19} + 1065 \beta_{18} - 752 \beta_{17} - 1438 \beta_{16} + 1081 \beta_{15} - 752 \beta_{14} + \cdots - 3297 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 477 \beta_{19} + 2346 \beta_{18} - 2877 \beta_{17} - 339 \beta_{16} + 449 \beta_{14} - 477 \beta_{13} + \cdots - 7936 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6256 \beta_{19} + 12269 \beta_{18} - 16086 \beta_{17} + 6256 \beta_{16} - 12113 \beta_{15} + \cdots - 38901 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 4704 \beta_{19} - 4704 \beta_{17} + 5192 \beta_{16} - 22949 \beta_{15} - 5192 \beta_{14} + \cdots - 21529 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 182606 \beta_{19} - 139353 \beta_{18} + 65516 \beta_{17} - 45412 \beta_{16} - 77919 \beta_{15} + \cdots + 24829 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 369635 \beta_{19} - 316212 \beta_{18} + 56039 \beta_{17} - 38079 \beta_{16} - 56120 \beta_{15} + \cdots - 262762 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1261108 \beta_{19} - 1800405 \beta_{18} - 458204 \beta_{17} + 1261108 \beta_{16} - 958763 \beta_{15} + \cdots - 128229 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 302158 \beta_{19} - 2614460 \beta_{18} - 302158 \beta_{17} + 4207484 \beta_{16} - 3007463 \beta_{15} + \cdots + 3408367 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 4579116 \beta_{19} - 21530801 \beta_{18} + 14861432 \beta_{17} + 23869102 \beta_{16} - 17184289 \beta_{15} + \cdots + 62200657 ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 6504905 \beta_{19} - 42493934 \beta_{18} + 47981201 \beta_{17} + 10375679 \beta_{16} + \cdots + 137918744 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 69453880 \beta_{19} - 206838141 \beta_{18} + 273798982 \beta_{17} - 69453880 \beta_{16} + \cdots + 727024245 ) / 4 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 131128852 \beta_{19} + 131128852 \beta_{17} - 70004596 \beta_{16} + 395183809 \beta_{15} + \cdots + 526765533 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 3145665150 \beta_{19} + 2367133225 \beta_{18} - 688605348 \beta_{17} + 434621140 \beta_{16} + \cdots + 483619755 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(421\) \(801\) \(897\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-\beta_{6}\)

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
1.78526 − 0.739480i
0.318488 + 0.768898i
−0.345447 − 0.833983i
−0.532503 + 0.220570i
−0.696163 − 1.68069i
−1.87480 + 0.776568i
−1.55374 + 0.643579i
1.26451 + 3.05279i
3.17578 − 1.31545i
0.458613 + 1.10719i
1.78526 + 0.739480i
0.318488 − 0.768898i
−0.345447 + 0.833983i
−0.532503 − 0.220570i
−0.696163 + 1.68069i
−1.87480 − 0.776568i
−1.55374 − 0.643579i
1.26451 − 3.05279i
3.17578 + 1.31545i
0.458613 − 1.10719i
0 −2.25068 − 2.25068i 0 −1.19421 − 1.89047i 0 0.707107 − 0.707107i 0 7.13112i 0
127.2 0 −1.88757 − 1.88757i 0 1.23152 + 1.86638i 0 −0.707107 + 0.707107i 0 4.12587i 0
127.3 0 −1.03701 − 1.03701i 0 −2.08319 + 0.812608i 0 −0.707107 + 0.707107i 0 − 0.849212i 0
127.4 0 −0.767511 − 0.767511i 0 2.19409 − 0.431229i 0 0.707107 − 0.707107i 0 − 1.82185i 0
127.5 0 −0.766532 − 0.766532i 0 0.00511711 − 2.23606i 0 −0.707107 + 0.707107i 0 − 1.82486i 0
127.6 0 0.254210 + 0.254210i 0 −0.655120 + 2.13795i 0 0.707107 − 0.707107i 0 − 2.87075i 0
127.7 0 1.22913 + 1.22913i 0 0.613139 − 2.15036i 0 0.707107 − 0.707107i 0 0.0215020i 0
127.8 0 1.41189 + 1.41189i 0 −2.22768 − 0.193509i 0 −0.707107 + 0.707107i 0 0.986845i 0
127.9 0 1.53486 + 1.53486i 0 −1.95790 − 1.08010i 0 0.707107 − 0.707107i 0 1.71156i 0
127.10 0 2.27923 + 2.27923i 0 2.07423 − 0.835201i 0 −0.707107 + 0.707107i 0 7.38979i 0
1023.1 0 −2.25068 + 2.25068i 0 −1.19421 + 1.89047i 0 0.707107 + 0.707107i 0 − 7.13112i 0
1023.2 0 −1.88757 + 1.88757i 0 1.23152 − 1.86638i 0 −0.707107 − 0.707107i 0 − 4.12587i 0
1023.3 0 −1.03701 + 1.03701i 0 −2.08319 − 0.812608i 0 −0.707107 − 0.707107i 0 0.849212i 0
1023.4 0 −0.767511 + 0.767511i 0 2.19409 + 0.431229i 0 0.707107 + 0.707107i 0 1.82185i 0
1023.5 0 −0.766532 + 0.766532i 0 0.00511711 + 2.23606i 0 −0.707107 − 0.707107i 0 1.82486i 0
1023.6 0 0.254210 − 0.254210i 0 −0.655120 − 2.13795i 0 0.707107 + 0.707107i 0 2.87075i 0
1023.7 0 1.22913 − 1.22913i 0 0.613139 + 2.15036i 0 0.707107 + 0.707107i 0 − 0.0215020i 0
1023.8 0 1.41189 − 1.41189i 0 −2.22768 + 0.193509i 0 −0.707107 − 0.707107i 0 − 0.986845i 0
1023.9 0 1.53486 − 1.53486i 0 −1.95790 + 1.08010i 0 0.707107 + 0.707107i 0 − 1.71156i 0
1023.10 0 2.27923 − 2.27923i 0 2.07423 + 0.835201i 0 −0.707107 − 0.707107i 0 − 7.38979i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.10
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 1120.2.x.f yes 20
4.b odd 2 1 1120.2.x.e ✓ 20
5.c odd 4 1 1120.2.x.e ✓ 20
20.e even 4 1 inner 1120.2.x.f yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.x.e ✓ 20 4.b odd 2 1
1120.2.x.e ✓ 20 5.c odd 4 1
1120.2.x.f yes 20 1.a even 1 1 trivial
1120.2.x.f yes 20 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 158 T_{3}^{16} - 8 T_{3}^{15} + 56 T_{3}^{13} + 6077 T_{3}^{12} - 1080 T_{3}^{11} + \cdots + 16384 \) acting on \(S_{2}^{\mathrm{new}}(1120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 158 T^{16} + \cdots + 16384 \) Copy content Toggle raw display
$5$ \( T^{20} + 4 T^{19} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{20} + 140 T^{18} + \cdots + 1048576 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 1060935184 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 899040256 \) Copy content Toggle raw display
$19$ \( (T^{10} + 12 T^{9} + \cdots - 105344)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} - 8 T^{19} + \cdots + 67108864 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 10019209216 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 1073741824 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 845575880704 \) Copy content Toggle raw display
$41$ \( (T^{10} - 16 T^{9} + \cdots + 307460096)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 208578543616 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 1674974347264 \) Copy content Toggle raw display
$59$ \( (T^{10} - 12 T^{9} + \cdots - 569174912)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - 12 T^{9} + \cdots + 9526784)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 3030384640000 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 58856637988864 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{10} + 12 T^{9} + \cdots - 35249600)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 89\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 553925220499456 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
show more
show less