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Properties

Label 1520.2.d.i
Level $1520$
Weight $2$
Character orbit 1520.d
Analytic conductor $12.137$
Analytic rank $0$
Dimension $6$
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] = [1520,2,Mod(609,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1520.609");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.d (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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.14077504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 14x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{5} q^{5} + ( - \beta_{4} - \beta_{3}) q^{7} + ( - \beta_{5} + \beta_{3} - \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{5} q^{5} + ( - \beta_{4} - \beta_{3}) q^{7} + ( - \beta_{5} + \beta_{3} - \beta_1 - 2) q^{9} + ( - \beta_1 - 3) q^{11} + (\beta_{5} - \beta_{4} + \beta_{2}) q^{13} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + \cdots + 2) q^{15}+ \cdots + (6 \beta_{5} - 6 \beta_{3} + 3 \beta_1 + 17) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{5} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{5} - 10 q^{9} - 18 q^{11} + 10 q^{15} + 6 q^{19} + 4 q^{21} - 5 q^{25} + 4 q^{29} - 8 q^{31} + 13 q^{35} - 4 q^{39} + 4 q^{41} - 27 q^{45} - 12 q^{49} - 36 q^{51} - q^{55} + 28 q^{59} + 30 q^{61} - 12 q^{65} + 32 q^{69} - 44 q^{71} + 46 q^{75} + 24 q^{79} + 50 q^{81} - 15 q^{85} - 8 q^{89} - 24 q^{91} - q^{95} + 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 9x^{4} + 14x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 9\nu^{3} + 14\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} - \nu^{4} - 8\nu^{3} - 8\nu^{2} - 8\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 8\nu^{2} - 4\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 8\nu^{2} - 8\nu + 6 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{5} - 3\beta_{4} + \beta_{3} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - \beta_{3} - 8\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -29\beta_{5} + 20\beta_{4} - 9\beta_{3} - 16\beta_{2} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
609.1
− 0.608430i
− 1.23277i
− 2.66648i
2.66648i
1.23277i
0.608430i
0 − 3.28715i 0 1.58777 + 1.57448i 0 − 1.93210i 0 −7.80536 0
609.2 0 − 1.62236i 0 −1.92411 − 1.13921i 0 4.74397i 0 0.367938 0
609.3 0 − 0.750054i 0 −0.163664 + 2.23007i 0 0.872810i 0 2.43742 0
609.4 0 0.750054i 0 −0.163664 − 2.23007i 0 − 0.872810i 0 2.43742 0
609.5 0 1.62236i 0 −1.92411 + 1.13921i 0 − 4.74397i 0 0.367938 0
609.6 0 3.28715i 0 1.58777 − 1.57448i 0 1.93210i 0 −7.80536 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 609.6
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 1520.2.d.i 6
4.b odd 2 1 380.2.c.b ✓ 6
5.b even 2 1 inner 1520.2.d.i 6
5.c odd 4 2 7600.2.a.cj 6
12.b even 2 1 3420.2.f.c 6
20.d odd 2 1 380.2.c.b ✓ 6
20.e even 4 2 1900.2.a.k 6
60.h even 2 1 3420.2.f.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.c.b ✓ 6 4.b odd 2 1
380.2.c.b ✓ 6 20.d odd 2 1
1520.2.d.i 6 1.a even 1 1 trivial
1520.2.d.i 6 5.b even 2 1 inner
1900.2.a.k 6 20.e even 4 2
3420.2.f.c 6 12.b even 2 1
3420.2.f.c 6 60.h even 2 1
7600.2.a.cj 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1520, [\chi])\):

\( T_{3}^{6} + 14T_{3}^{4} + 36T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{6} + 27T_{7}^{4} + 104T_{7}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 14 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 27 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T^{3} + 9 T^{2} + 14 T - 28)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 26 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{6} + 51 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 44 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} - 84 T - 88)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 4 T^{2} + \cdots - 256)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 14 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$41$ \( (T^{3} - 2 T^{2} + \cdots + 488)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 227 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$47$ \( T^{6} + 243 T^{4} + \cdots + 118336 \) Copy content Toggle raw display
$53$ \( T^{6} + 114 T^{4} + \cdots + 23104 \) Copy content Toggle raw display
$59$ \( (T^{3} - 14 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 15 T^{2} + \cdots - 44)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 378 T^{4} + \cdots + 222784 \) Copy content Toggle raw display
$71$ \( (T^{3} + 22 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 251 T^{4} + \cdots + 118336 \) Copy content Toggle raw display
$79$ \( (T - 4)^{6} \) Copy content Toggle raw display
$83$ \( T^{6} + 44 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( (T^{3} + 4 T^{2} - 44 T - 64)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 114 T^{4} + \cdots + 23104 \) Copy content Toggle raw display
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