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Properties

Label 3120.2.l.p
Level $3120$
Weight $2$
Character orbit 3120.l
Analytic conductor $24.913$
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] = [3120,2,Mod(1249,3120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3120, 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("3120.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.l (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: \(24.9133254306\)
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: \( x^{10} + 16x^{8} + 90x^{6} + 208x^{4} + 169x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 195)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{5} q^{5} + ( - \beta_{9} - \beta_{2}) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{5} q^{5} + ( - \beta_{9} - \beta_{2}) q^{7} - q^{9} + (\beta_{8} + \beta_{7} + \beta_{4} - 1) q^{11} - \beta_{2} q^{13} - \beta_{7} q^{15} + ( - \beta_{9} - \beta_{2} - \beta_1) q^{17} + ( - \beta_{8} - \beta_{7} - \beta_{6} + \cdots + 2) q^{19}+ \cdots + ( - \beta_{8} - \beta_{7} - \beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{5} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{5} - 10 q^{9} - 10 q^{11} + 16 q^{19} + 10 q^{21} + 10 q^{25} - 16 q^{29} - 24 q^{31} - 12 q^{35} + 10 q^{39} + 10 q^{41} + 2 q^{45} - 44 q^{49} + 10 q^{51} - 2 q^{55} + 16 q^{59} + 26 q^{61} + 14 q^{69} - 10 q^{71} + 2 q^{79} + 10 q^{81} - 4 q^{85} - 38 q^{89} - 10 q^{91} + 48 q^{95} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 16x^{8} + 90x^{6} + 208x^{4} + 169x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 8\nu^{3} + 13\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 8\nu^{4} - 13\nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 10\nu^{4} + 25\nu^{2} + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} + 13\nu^{6} + 53\nu^{4} + 4\nu^{3} + 73\nu^{2} + 20\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{8} - 13\nu^{6} - 53\nu^{4} + 4\nu^{3} - 73\nu^{2} + 20\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{8} - 2\nu^{7} - 13\nu^{6} - 22\nu^{5} - 53\nu^{4} - 70\nu^{3} - 65\nu^{2} - 58\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{8} + 2\nu^{7} - 13\nu^{6} + 22\nu^{5} - 53\nu^{4} + 70\nu^{3} - 65\nu^{2} + 58\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} + 14\nu^{7} + 65\nu^{5} + 110\nu^{3} + 44\nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{6} + 2\beta_{5} - 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{8} - 3\beta_{7} + 3\beta_{6} - 3\beta_{5} + \beta_{4} + \beta_{3} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -16\beta_{6} - 16\beta_{5} + 8\beta_{2} + 27\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 35\beta_{8} + 35\beta_{7} - 35\beta_{6} + 35\beta_{5} - 16\beta_{4} - 20\beta_{3} - 146 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4\beta_{8} - 4\beta_{7} + 106\beta_{6} + 106\beta_{5} - 88\beta_{2} - 151\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -105\beta_{8} - 105\beta_{7} + 101\beta_{6} - 101\beta_{5} + 51\beta_{4} + 77\beta_{3} + 402 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8\beta_{9} - 56\beta_{8} + 56\beta_{7} - 664\beta_{6} - 664\beta_{5} + 712\beta_{2} + 865\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times\).

\(n\) \(1951\) \(2081\) \(2341\) \(2497\) \(2641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
2.26036i
− 2.47948i
− 1.22308i
1.77159i
− 0.329386i
− 2.26036i
2.47948i
1.22308i
− 1.77159i
0.329386i
0 − 1.00000i 0 −2.23266 − 0.123438i 0 4.96953i 0 −1.00000 0
1249.2 0 − 1.00000i 0 −1.72481 + 1.42303i 0 0.949959i 0 −1.00000 0
1249.3 0 − 1.00000i 0 −0.638796 − 2.14288i 0 − 4.18388i 0 −1.00000 0
1249.4 0 − 1.00000i 0 1.51036 + 1.64888i 0 − 0.437634i 0 −1.00000 0
1249.5 0 − 1.00000i 0 2.08591 − 0.805596i 0 3.70203i 0 −1.00000 0
1249.6 0 1.00000i 0 −2.23266 + 0.123438i 0 − 4.96953i 0 −1.00000 0
1249.7 0 1.00000i 0 −1.72481 − 1.42303i 0 − 0.949959i 0 −1.00000 0
1249.8 0 1.00000i 0 −0.638796 + 2.14288i 0 4.18388i 0 −1.00000 0
1249.9 0 1.00000i 0 1.51036 − 1.64888i 0 0.437634i 0 −1.00000 0
1249.10 0 1.00000i 0 2.08591 + 0.805596i 0 − 3.70203i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.10
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 3120.2.l.p 10
4.b odd 2 1 195.2.c.b ✓ 10
5.b even 2 1 inner 3120.2.l.p 10
12.b even 2 1 585.2.c.c 10
20.d odd 2 1 195.2.c.b ✓ 10
20.e even 4 1 975.2.a.r 5
20.e even 4 1 975.2.a.s 5
60.h even 2 1 585.2.c.c 10
60.l odd 4 1 2925.2.a.bl 5
60.l odd 4 1 2925.2.a.bm 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.c.b ✓ 10 4.b odd 2 1
195.2.c.b ✓ 10 20.d odd 2 1
585.2.c.c 10 12.b even 2 1
585.2.c.c 10 60.h even 2 1
975.2.a.r 5 20.e even 4 1
975.2.a.s 5 20.e even 4 1
2925.2.a.bl 5 60.l odd 4 1
2925.2.a.bm 5 60.l odd 4 1
3120.2.l.p 10 1.a even 1 1 trivial
3120.2.l.p 10 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3120, [\chi])\):

\( T_{7}^{10} + 57T_{7}^{8} + 1072T_{7}^{6} + 7040T_{7}^{4} + 6656T_{7}^{2} + 1024 \) Copy content Toggle raw display
\( T_{11}^{5} + 5T_{11}^{4} - 24T_{11}^{3} - 100T_{11}^{2} + 156T_{11} + 452 \) 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
$5$ \( T^{10} + 2 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 57 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( (T^{5} + 5 T^{4} + \cdots + 452)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{10} + 73 T^{8} + \cdots + 6400 \) Copy content Toggle raw display
$19$ \( (T^{5} - 8 T^{4} + \cdots - 1280)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 161 T^{8} + \cdots + 102400 \) Copy content Toggle raw display
$29$ \( (T^{5} + 8 T^{4} + \cdots + 640)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 12 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 209 T^{8} + \cdots + 350464 \) Copy content Toggle raw display
$41$ \( (T^{5} - 5 T^{4} + \cdots - 196)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 128 T^{8} + \cdots + 200704 \) Copy content Toggle raw display
$47$ \( T^{10} + 204 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$53$ \( T^{10} + 217 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( (T^{5} - 8 T^{4} + \cdots - 400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 13 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 320 T^{8} + \cdots + 15745024 \) Copy content Toggle raw display
$71$ \( (T^{5} + 5 T^{4} + \cdots + 28868)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 223323136 \) Copy content Toggle raw display
$79$ \( (T^{5} - T^{4} - 248 T^{3} + \cdots - 400)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 3685946944 \) Copy content Toggle raw display
$89$ \( (T^{5} + 19 T^{4} + \cdots - 4900)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 1260534016 \) Copy content Toggle raw display
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