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Properties

Label 1520.2.bq.o
Level $1520$
Weight $2$
Character orbit 1520.bq
Analytic conductor $12.137$
Analytic rank $0$
Dimension $8$
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(31,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.bq (of order \(6\), 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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.1121513121.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 4x^{6} - 4x^{5} + 8x^{4} + 16x^{3} + 18x^{2} + 28x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{2}) q^{3} + (\beta_{6} - 1) q^{5} + ( - 2 \beta_{7} - \beta_{5} + \beta_{4} + \cdots + 1) q^{7}+ \cdots + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{2}) q^{3} + (\beta_{6} - 1) q^{5} + ( - 2 \beta_{7} - \beta_{5} + \beta_{4} + \cdots + 1) q^{7}+ \cdots + ( - 7 \beta_{6} - \beta_{5} + \cdots + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{5} - 2 q^{9} - 15 q^{13} - 3 q^{17} - 3 q^{19} - 12 q^{21} - 12 q^{23} - 4 q^{25} + 18 q^{27} - 12 q^{29} - 42 q^{31} - 12 q^{33} - 6 q^{35} - 9 q^{41} + 3 q^{43} + 4 q^{45} - 18 q^{47} - 44 q^{49} - 18 q^{51} + 15 q^{53} + 6 q^{55} - 25 q^{57} - 15 q^{59} + 25 q^{61} + 63 q^{63} - 27 q^{67} + 21 q^{71} + 21 q^{73} - 88 q^{77} + 6 q^{79} - 4 q^{81} - 3 q^{85} + 30 q^{89} - 18 q^{91} - 18 q^{93} - 12 q^{95} + 18 q^{97} + 57 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 4x^{6} - 4x^{5} + 8x^{4} + 16x^{3} + 18x^{2} + 28x + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 2\nu^{5} - 2\nu^{4} + 6\nu^{3} + 22\nu^{2} + 40\nu + 41 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{7} + 26\nu^{6} - 44\nu^{5} + 53\nu^{4} - 99\nu^{3} - 13\nu^{2} - 100\nu - 86 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 11\nu^{6} - 20\nu^{5} + 29\nu^{4} - 50\nu^{3} - 14\nu^{2} - 48\nu - 58 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11\nu^{7} + 43\nu^{6} - 79\nu^{5} + 97\nu^{4} - 150\nu^{3} - 56\nu^{2} - 107\nu - 169 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\nu^{7} - 50\nu^{6} + 95\nu^{5} - 131\nu^{4} + 201\nu^{3} + 58\nu^{2} + 178\nu + 257 ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35\nu^{7} - 130\nu^{6} + 229\nu^{5} - 292\nu^{4} + 477\nu^{3} + 227\nu^{2} + 437\nu + 610 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - 3\beta_{6} - 4\beta_{5} - 4\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{7} - 6\beta_{6} - 11\beta_{5} - 7\beta_{4} + \beta_{3} + 6\beta_{2} + 2\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 6\beta_{6} - 23\beta_{5} - 11\beta_{4} + 2\beta_{3} + 20\beta_{2} - 3\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{7} + 2\beta_{6} - 29\beta_{5} - 12\beta_{4} - 3\beta_{3} + 51\beta_{2} - 36\beta _1 - 26 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -14\beta_{7} + 44\beta_{6} + 12\beta_{5} + 30\beta_{4} - 36\beta_{3} + 95\beta_{2} - 130\beta _1 - 93 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(-1\) \(\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.177457 − 1.52428i
2.26669 + 1.18132i
−0.731824 − 0.0280585i
−0.212319 + 1.23705i
0.177457 + 1.52428i
2.26669 − 1.18132i
−0.731824 + 0.0280585i
−0.212319 − 1.23705i
0 −1.40879 − 2.44010i 0 −0.500000 − 0.866025i 0 − 3.90855i 0 −2.46939 + 4.27711i 0
31.2 0 −0.110293 − 0.191033i 0 −0.500000 − 0.866025i 0 − 1.12206i 0 1.47567 − 2.55594i 0
31.3 0 0.341612 + 0.591690i 0 −0.500000 − 0.866025i 0 4.79806i 0 1.26660 − 2.19382i 0
31.4 0 1.17747 + 2.03944i 0 −0.500000 − 0.866025i 0 − 3.23155i 0 −1.27288 + 2.20470i 0
1471.1 0 −1.40879 + 2.44010i 0 −0.500000 + 0.866025i 0 3.90855i 0 −2.46939 − 4.27711i 0
1471.2 0 −0.110293 + 0.191033i 0 −0.500000 + 0.866025i 0 1.12206i 0 1.47567 + 2.55594i 0
1471.3 0 0.341612 − 0.591690i 0 −0.500000 + 0.866025i 0 − 4.79806i 0 1.26660 + 2.19382i 0
1471.4 0 1.17747 − 2.03944i 0 −0.500000 + 0.866025i 0 3.23155i 0 −1.27288 − 2.20470i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.bq.o ✓ 8
4.b odd 2 1 1520.2.bq.p yes 8
19.d odd 6 1 1520.2.bq.p yes 8
76.f even 6 1 inner 1520.2.bq.o ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1520.2.bq.o ✓ 8 1.a even 1 1 trivial
1520.2.bq.o ✓ 8 76.f even 6 1 inner
1520.2.bq.p yes 8 4.b odd 2 1
1520.2.bq.p yes 8 19.d odd 6 1

Hecke kernels

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

\( T_{3}^{8} + 7T_{3}^{6} - 6T_{3}^{5} + 48T_{3}^{4} - 21T_{3}^{3} + 16T_{3}^{2} + 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} + 50T_{7}^{6} + 813T_{7}^{4} + 4619T_{7}^{2} + 4624 \) Copy content Toggle raw display
\( T_{11}^{8} + 50T_{11}^{6} + 327T_{11}^{4} + 569T_{11}^{2} + 169 \) Copy content Toggle raw display
\( T_{13}^{8} + 15T_{13}^{7} + 88T_{13}^{6} + 195T_{13}^{5} - 96T_{13}^{4} - 897T_{13}^{3} + 547T_{13}^{2} + 5520T_{13} + 6400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 7 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 50 T^{6} + \cdots + 4624 \) Copy content Toggle raw display
$11$ \( T^{8} + 50 T^{6} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{8} + 15 T^{7} + \cdots + 6400 \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{7} + \cdots + 8100 \) Copy content Toggle raw display
$19$ \( T^{8} + 3 T^{7} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + \cdots + 28900 \) Copy content Toggle raw display
$29$ \( T^{8} + 12 T^{7} + \cdots + 5184 \) Copy content Toggle raw display
$31$ \( (T^{4} + 21 T^{3} + \cdots + 252)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 154 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( T^{8} + 9 T^{7} + \cdots + 164025 \) Copy content Toggle raw display
$43$ \( T^{8} - 3 T^{7} + \cdots + 396900 \) Copy content Toggle raw display
$47$ \( T^{8} + 18 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$53$ \( T^{8} - 15 T^{7} + \cdots + 2676496 \) Copy content Toggle raw display
$59$ \( T^{8} + 15 T^{7} + \cdots + 337561 \) Copy content Toggle raw display
$61$ \( T^{8} - 25 T^{7} + \cdots + 62500 \) Copy content Toggle raw display
$67$ \( T^{8} + 27 T^{7} + \cdots + 279023616 \) Copy content Toggle raw display
$71$ \( T^{8} - 21 T^{7} + \cdots + 321269776 \) Copy content Toggle raw display
$73$ \( T^{8} - 21 T^{7} + \cdots + 3143529 \) Copy content Toggle raw display
$79$ \( T^{8} - 6 T^{7} + \cdots + 1327104 \) Copy content Toggle raw display
$83$ \( T^{8} + 414 T^{6} + \cdots + 21986721 \) Copy content Toggle raw display
$89$ \( T^{8} - 30 T^{7} + \cdots + 82944 \) Copy content Toggle raw display
$97$ \( T^{8} - 18 T^{7} + \cdots + 52780225 \) Copy content Toggle raw display
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