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Properties

Label 4080.2.m.r
Level $4080$
Weight $2$
Character orbit 4080.m
Analytic conductor $32.579$
Analytic rank $0$
Dimension $10$
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] = [4080,2,Mod(2449,4080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("4080.2449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4080.m (of order \(2\), degree \(1\), not 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: \(32.5789640247\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 2x^{8} + 14x^{7} + 42x^{6} + 2x^{5} + 10x^{4} + 54x^{3} + 121x^{2} + 44x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1020)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} - \beta_{7} q^{5} + \beta_{9} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} - \beta_{7} q^{5} + \beta_{9} q^{7} - q^{9} + (\beta_{3} + \beta_{2} - \beta_1) q^{11} + (\beta_{9} + \beta_{8} + \cdots - \beta_{3}) q^{13}+ \cdots + ( - \beta_{3} - \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{9} - 12 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 12 q^{25} - 12 q^{29} - 12 q^{31} - 4 q^{35} + 28 q^{41} - 30 q^{49} + 10 q^{51} - 26 q^{55} + 48 q^{59} - 28 q^{61} + 48 q^{65} - 12 q^{69} - 8 q^{75} + 44 q^{79} + 10 q^{81} + 4 q^{85} - 36 q^{89} - 40 q^{91} - 44 q^{95} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 2x^{8} + 14x^{7} + 42x^{6} + 2x^{5} + 10x^{4} + 54x^{3} + 121x^{2} + 44x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 69233 \nu^{9} + 148323 \nu^{8} - 272095 \nu^{7} - 739833 \nu^{6} - 2791917 \nu^{5} + \cdots - 3757168 ) / 2838228 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 79825 \nu^{9} + 138516 \nu^{8} - 77795 \nu^{7} - 1414236 \nu^{6} - 3036789 \nu^{5} + \cdots - 7864340 ) / 2838228 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 81071 \nu^{9} - 192696 \nu^{8} + 143365 \nu^{7} + 1374462 \nu^{6} + 2586483 \nu^{5} + \cdots - 4709108 ) / 2838228 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 49556 \nu^{9} + 217521 \nu^{8} - 384652 \nu^{7} - 363408 \nu^{6} - 431820 \nu^{5} + \cdots + 6466400 ) / 1419114 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 144253 \nu^{9} - 317079 \nu^{8} + 358985 \nu^{7} + 1900851 \nu^{6} + 5758887 \nu^{5} + \cdots + 3239660 ) / 2838228 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 91922 \nu^{9} + 172891 \nu^{8} - 164666 \nu^{7} - 1272575 \nu^{6} - 4146236 \nu^{5} + \cdots - 2758580 ) / 946076 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 120515 \nu^{9} + 297390 \nu^{8} - 387673 \nu^{7} - 1481320 \nu^{6} - 4386569 \nu^{5} + \cdots - 2155756 ) / 946076 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 152228 \nu^{9} - 334325 \nu^{8} + 327646 \nu^{7} + 2159179 \nu^{6} + 5888058 \nu^{5} + \cdots + 900608 ) / 946076 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 368954 \nu^{9} - 799764 \nu^{8} + 828550 \nu^{7} + 5196051 \nu^{6} + 14329410 \nu^{5} + \cdots + 8056936 ) / 1419114 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 4\beta_{5} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11 \beta_{9} - 3 \beta_{8} - 7 \beta_{7} + 15 \beta_{6} - 19 \beta_{5} + 11 \beta_{4} - \beta_{3} + \cdots - 16 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -13\beta_{7} + 13\beta_{6} + 12\beta_{4} + 4\beta_{3} + 4\beta_{2} - \beta _1 - 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 121 \beta_{9} + 17 \beta_{8} - 169 \beta_{7} + 81 \beta_{6} + 241 \beta_{5} + 121 \beta_{4} + \cdots - 224 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -139\beta_{9} + 14\beta_{8} - 44\beta_{7} - 44\beta_{6} + 314\beta_{5} + 135\beta_{3} - 149\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1347 \beta_{9} + 155 \beta_{8} + 943 \beta_{7} - 1887 \beta_{6} + 2819 \beta_{5} - 1347 \beta_{4} + \cdots + 2664 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1683\beta_{7} - 1683\beta_{6} - 1582\beta_{4} - 696\beta_{3} - 696\beta_{2} + 167\beta _1 + 3274 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 15121 \beta_{9} - 1657 \beta_{8} + 21185 \beta_{7} - 10769 \beta_{6} - 32161 \beta_{5} - 15121 \beta_{4} + \cdots + 30504 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4080\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(511\) \(817\) \(1361\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
2.37468 + 2.37468i
−1.29737 − 1.29737i
−0.200038 − 0.200038i
−0.841453 − 0.841453i
0.964178 + 0.964178i
2.37468 − 2.37468i
−1.29737 + 1.29737i
−0.200038 + 0.200038i
−0.841453 + 0.841453i
0.964178 − 0.964178i
0 − 1.00000i 0 −2.16715 − 0.550869i 0 2.94396i 0 −1.00000 0
2449.2 0 − 1.00000i 0 −1.55795 + 1.60399i 0 − 3.74956i 0 −1.00000 0
2449.3 0 − 1.00000i 0 −0.327045 − 2.21202i 0 − 4.57406i 0 −1.00000 0
2449.4 0 − 1.00000i 0 1.85384 − 1.25031i 0 1.12377i 0 −1.00000 0
2449.5 0 − 1.00000i 0 2.19831 + 0.409208i 0 2.25589i 0 −1.00000 0
2449.6 0 1.00000i 0 −2.16715 + 0.550869i 0 − 2.94396i 0 −1.00000 0
2449.7 0 1.00000i 0 −1.55795 − 1.60399i 0 3.74956i 0 −1.00000 0
2449.8 0 1.00000i 0 −0.327045 + 2.21202i 0 4.57406i 0 −1.00000 0
2449.9 0 1.00000i 0 1.85384 + 1.25031i 0 − 1.12377i 0 −1.00000 0
2449.10 0 1.00000i 0 2.19831 − 0.409208i 0 − 2.25589i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2449.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4080.2.m.r 10
4.b odd 2 1 1020.2.g.d ✓ 10
5.b even 2 1 inner 4080.2.m.r 10
12.b even 2 1 3060.2.g.g 10
20.d odd 2 1 1020.2.g.d ✓ 10
20.e even 4 1 5100.2.a.bc 5
20.e even 4 1 5100.2.a.bd 5
60.h even 2 1 3060.2.g.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1020.2.g.d ✓ 10 4.b odd 2 1
1020.2.g.d ✓ 10 20.d odd 2 1
3060.2.g.g 10 12.b even 2 1
3060.2.g.g 10 60.h even 2 1
4080.2.m.r 10 1.a even 1 1 trivial
4080.2.m.r 10 5.b even 2 1 inner
5100.2.a.bc 5 20.e even 4 1
5100.2.a.bd 5 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4080, [\chi])\):

\( T_{7}^{10} + 50T_{7}^{8} + 881T_{7}^{6} + 6624T_{7}^{4} + 20032T_{7}^{2} + 16384 \) Copy content Toggle raw display
\( T_{11}^{5} + 6T_{11}^{4} - 18T_{11}^{3} - 126T_{11}^{2} - 83T_{11} + 148 \) Copy content Toggle raw display
\( T_{23}^{10} + 130T_{23}^{8} + 5009T_{23}^{6} + 55248T_{23}^{4} + 193088T_{23}^{2} + 65536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} - 6 T^{8} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 50 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$11$ \( (T^{5} + 6 T^{4} + \cdots + 148)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 118 T^{8} + \cdots + 495616 \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$19$ \( (T^{5} - 12 T^{4} + \cdots + 836)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 130 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$29$ \( (T^{5} + 6 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 6 T^{4} + \cdots + 1504)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 198 T^{8} + \cdots + 2611456 \) Copy content Toggle raw display
$41$ \( (T^{5} - 14 T^{4} + \cdots - 568)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 878055424 \) Copy content Toggle raw display
$47$ \( T^{10} + 318 T^{8} + \cdots + 1420864 \) Copy content Toggle raw display
$53$ \( T^{10} + 206 T^{8} + \cdots + 2849344 \) Copy content Toggle raw display
$59$ \( (T^{5} - 24 T^{4} + \cdots + 21568)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 14 T^{4} + \cdots + 83584)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 465178624 \) Copy content Toggle raw display
$71$ \( (T^{5} - 236 T^{3} + \cdots + 22528)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 414 T^{8} + \cdots + 11235904 \) Copy content Toggle raw display
$79$ \( (T^{5} - 22 T^{4} + \cdots - 35296)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 3071819776 \) Copy content Toggle raw display
$89$ \( (T^{5} + 18 T^{4} + \cdots - 29632)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 460 T^{8} + \cdots + 23658496 \) Copy content Toggle raw display
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