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Properties

Label 1120.2.x.c
Level $1120$
Weight $2$
Character orbit 1120.x
Analytic conductor $8.943$
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] = [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: \(12\)
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: \( x^{12} - 4 x^{11} + 2 x^{10} + 8 x^{9} + 2 x^{8} + 12 x^{7} + 40 x^{6} + 4 x^{5} + 49 x^{4} - 56 x^{3} + \cdots + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + (\beta_{10} + \beta_{8} + \beta_{6} + \cdots - 1) q^{3}+ \cdots + ( - 2 \beta_{9} - \beta_{8} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} + \beta_{8} + \beta_{6} + \cdots - 1) q^{3}+ \cdots + ( - 2 \beta_{11} - 6 \beta_{9} + \cdots - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} + 4 q^{5} + 16 q^{13} - 28 q^{15} + 16 q^{17} + 8 q^{19} - 4 q^{21} - 16 q^{23} - 8 q^{25} - 4 q^{27} + 8 q^{33} - 4 q^{35} + 24 q^{37} - 28 q^{39} - 16 q^{41} + 4 q^{43} - 4 q^{45} - 4 q^{47} + 44 q^{53} - 4 q^{55} - 36 q^{57} + 72 q^{59} - 16 q^{63} - 24 q^{65} - 12 q^{67} - 24 q^{73} + 4 q^{75} - 4 q^{77} + 28 q^{79} - 20 q^{81} - 24 q^{83} + 12 q^{85} + 44 q^{87} - 36 q^{93} + 12 q^{95} - 16 q^{97} - 112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 2 x^{10} + 8 x^{9} + 2 x^{8} + 12 x^{7} + 40 x^{6} + 4 x^{5} + 49 x^{4} - 56 x^{3} + \cdots + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4686360 \nu^{11} - 64933539 \nu^{10} + 134652862 \nu^{9} + 144360164 \nu^{8} - 336315907 \nu^{7} + \cdots + 2211301 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 46188099 \nu^{11} + 125280142 \nu^{10} + 106869284 \nu^{9} - 345688627 \nu^{8} + \cdots - 9372720 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 57365257 \nu^{11} + 234079984 \nu^{10} - 135078180 \nu^{9} - 426165507 \nu^{8} + \cdots + 956458910 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 73568234 \nu^{11} - 402512652 \nu^{10} + 571430133 \nu^{9} + 425684114 \nu^{8} + \cdots - 2737878295 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 203253 \nu^{11} - 752988 \nu^{10} + 169207 \nu^{9} + 1739574 \nu^{8} + 876160 \nu^{7} + \cdots - 746360 ) / 915349 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 428933359 \nu^{11} - 1325315590 \nu^{10} - 546155470 \nu^{9} + 3582028695 \nu^{8} + \cdots - 1242538333 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 373180 \nu^{11} + 1289467 \nu^{10} + 6628 \nu^{9} - 3154647 \nu^{8} - 2485934 \nu^{7} + \cdots + 798369 ) / 915349 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 478229455 \nu^{11} - 1855552563 \nu^{10} + 722378926 \nu^{9} + 3960913820 \nu^{8} + \cdots - 3124247761 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 831002630 \nu^{11} - 2922008653 \nu^{10} + 109898070 \nu^{9} + 7202178267 \nu^{8} + \cdots + 258971265 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 889604042 \nu^{11} + 3537526831 \nu^{10} - 1545841877 \nu^{9} - 7687837151 \nu^{8} + \cdots + 4781830419 ) / 1092011357 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{5} - \beta_{4} - 2\beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{11} - 2 \beta_{10} - \beta_{9} - 3 \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{10} + \beta_{9} - 17 \beta_{8} - 7 \beta_{7} + 3 \beta_{6} - 9 \beta_{5} - 10 \beta_{4} + \cdots + 10 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{11} - 10 \beta_{10} + 13 \beta_{9} - 36 \beta_{8} - 23 \beta_{7} + 17 \beta_{6} - 23 \beta_{5} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 60 \beta_{11} + 77 \beta_{9} - 93 \beta_{8} - 83 \beta_{7} + 36 \beta_{6} - 60 \beta_{5} - 96 \beta_{4} + \cdots - 93 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 229 \beta_{11} + 96 \beta_{10} + 265 \beta_{9} - 147 \beta_{8} - 229 \beta_{7} + 93 \beta_{6} + \cdots - 361 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 781 \beta_{11} + 555 \beta_{10} + 874 \beta_{9} - 555 \beta_{7} + 147 \beta_{6} - 361 \beta_{4} + \cdots - 1247 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2205 \beta_{11} + 2205 \beta_{10} + 2352 \beta_{9} + 1436 \beta_{8} - 916 \beta_{7} + 916 \beta_{5} + \cdots - 3482 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5242 \beta_{11} + 7405 \beta_{10} + 5242 \beta_{9} + 8279 \beta_{8} - 1436 \beta_{6} + 5242 \beta_{5} + \cdots - 8279 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8724 \beta_{11} + 21047 \beta_{10} + 7288 \beta_{9} + 33253 \beta_{8} + 8724 \beta_{7} - 8279 \beta_{6} + \cdots - 13759 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(351\) \(421\) \(801\) \(897\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-\beta_{8}\)

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
−0.445186 + 1.07477i
−1.48663 − 0.615783i
2.84980 + 1.18043i
0.568642 − 1.37282i
0.343932 + 0.142462i
0.169437 − 0.409058i
−0.445186 − 1.07477i
−1.48663 + 0.615783i
2.84980 − 1.18043i
0.568642 + 1.37282i
0.343932 − 0.142462i
0.169437 + 0.409058i
0 −2.12313 − 2.12313i 0 1.75546 − 1.38505i 0 −0.707107 + 0.707107i 0 6.01534i 0
127.2 0 −1.33633 − 1.33633i 0 1.24501 + 1.85741i 0 0.707107 − 0.707107i 0 0.571561i 0
127.3 0 −0.824549 − 0.824549i 0 −1.06299 − 1.96725i 0 0.707107 − 0.707107i 0 − 1.64024i 0
127.4 0 −0.120712 − 0.120712i 0 2.20064 − 0.396460i 0 −0.707107 + 0.707107i 0 − 2.97086i 0
127.5 0 0.453774 + 0.453774i 0 −1.30334 + 1.81695i 0 0.707107 − 0.707107i 0 − 2.58818i 0
127.6 0 1.95095 + 1.95095i 0 −0.834781 + 2.07440i 0 −0.707107 + 0.707107i 0 4.61238i 0
1023.1 0 −2.12313 + 2.12313i 0 1.75546 + 1.38505i 0 −0.707107 − 0.707107i 0 − 6.01534i 0
1023.2 0 −1.33633 + 1.33633i 0 1.24501 − 1.85741i 0 0.707107 + 0.707107i 0 − 0.571561i 0
1023.3 0 −0.824549 + 0.824549i 0 −1.06299 + 1.96725i 0 0.707107 + 0.707107i 0 1.64024i 0
1023.4 0 −0.120712 + 0.120712i 0 2.20064 + 0.396460i 0 −0.707107 − 0.707107i 0 2.97086i 0
1023.5 0 0.453774 − 0.453774i 0 −1.30334 − 1.81695i 0 0.707107 + 0.707107i 0 2.58818i 0
1023.6 0 1.95095 − 1.95095i 0 −0.834781 − 2.07440i 0 −0.707107 − 0.707107i 0 − 4.61238i 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 1120.2.x.c ✓ 12
4.b odd 2 1 1120.2.x.d yes 12
5.c odd 4 1 1120.2.x.d yes 12
20.e even 4 1 inner 1120.2.x.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.x.c ✓ 12 1.a even 1 1 trivial
1120.2.x.c ✓ 12 20.e even 4 1 inner
1120.2.x.d yes 12 4.b odd 2 1
1120.2.x.d yes 12 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 4 T_{3}^{11} + 8 T_{3}^{10} + 4 T_{3}^{9} + 61 T_{3}^{8} + 232 T_{3}^{7} + 448 T_{3}^{6} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(1120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} + 54 T^{10} + \cdots + 12544 \) Copy content Toggle raw display
$13$ \( T^{12} - 16 T^{11} + \cdots + 42436 \) Copy content Toggle raw display
$17$ \( T^{12} - 16 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{6} - 4 T^{5} + \cdots + 892)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 16 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 3683033344 \) Copy content Toggle raw display
$31$ \( T^{12} + 268 T^{10} + \cdots + 27541504 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 456933376 \) Copy content Toggle raw display
$41$ \( (T^{6} + 8 T^{5} + \cdots - 3808)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 4 T^{11} + \cdots + 8111104 \) Copy content Toggle raw display
$47$ \( T^{12} + 4 T^{11} + \cdots + 295936 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 3961947136 \) Copy content Toggle raw display
$59$ \( (T^{6} - 36 T^{5} + \cdots - 74372)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 300 T^{4} + \cdots - 112712)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 12 T^{11} + \cdots + 1721344 \) Copy content Toggle raw display
$71$ \( T^{12} + 136 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 129777664 \) Copy content Toggle raw display
$79$ \( (T^{6} - 14 T^{5} + \cdots - 103300)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 5778432256 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 192876544 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 42845688064 \) Copy content Toggle raw display
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