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Properties

Label 520.2.bu.a
Level $520$
Weight $2$
Character orbit 520.bu
Analytic conductor $4.152$
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] = [520,2,Mod(121,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.bu (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: \(4.15222090511\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.58891012706304.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{11} q^{3} - \beta_{5} q^{5} + (\beta_{10} + \beta_{7} + \cdots - \beta_{3}) q^{7}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{3} - \beta_{5} q^{5} + (\beta_{10} + \beta_{7} + \cdots - \beta_{3}) q^{7}+ \cdots + (2 \beta_{11} + 6 \beta_{9} + 2 \beta_{6} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{7} + 2 q^{9} - 2 q^{13} - 8 q^{17} - 24 q^{19} - 2 q^{23} - 12 q^{25} - 12 q^{27} + 12 q^{29} - 4 q^{35} + 24 q^{37} + 28 q^{39} + 24 q^{41} - 18 q^{43} - 12 q^{45} + 24 q^{49} - 68 q^{53} + 2 q^{55} - 48 q^{59} + 18 q^{61} - 36 q^{63} + 16 q^{65} + 18 q^{67} - 8 q^{69} + 64 q^{77} - 12 q^{79} + 14 q^{81} + 18 q^{87} + 30 q^{89} + 76 q^{91} - 12 q^{93} - 10 q^{95} - 84 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 6 \nu^{8} - 13 \nu^{7} + 30 \nu^{6} - 6 \nu^{5} - 28 \nu^{4} + \cdots + 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 7 \nu^{9} - 2 \nu^{8} - 29 \nu^{7} + 10 \nu^{6} + 70 \nu^{5} - 68 \nu^{4} - 92 \nu^{3} + \cdots - 192 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3 \nu^{11} - 2 \nu^{10} + 11 \nu^{9} + 8 \nu^{8} - 21 \nu^{7} - 4 \nu^{6} + 42 \nu^{5} - 84 \nu^{3} + \cdots + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3 \nu^{11} + 4 \nu^{10} - 7 \nu^{9} - 26 \nu^{8} + 13 \nu^{7} + 50 \nu^{6} - 42 \nu^{5} - 92 \nu^{4} + \cdots - 192 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} + \cdots - 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{11} - 4 \nu^{10} - 15 \nu^{9} + 14 \nu^{8} + 37 \nu^{7} - 54 \nu^{6} - 50 \nu^{5} + 132 \nu^{4} + \cdots + 128 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2 \nu^{11} - \nu^{10} - 10 \nu^{9} + \nu^{8} + 24 \nu^{7} - 11 \nu^{6} - 38 \nu^{5} + 38 \nu^{4} + \cdots + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} - 4 \nu^{10} + 17 \nu^{9} + 6 \nu^{8} - 51 \nu^{7} - 6 \nu^{6} + 122 \nu^{5} - 68 \nu^{4} + \cdots - 192 ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} - 5 \nu^{9} + 13 \nu^{8} + 13 \nu^{7} - 35 \nu^{6} - 12 \nu^{5} + 70 \nu^{4} + \cdots + 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{11} - 4 \nu^{10} - 11 \nu^{9} + 30 \nu^{8} + 41 \nu^{7} - 70 \nu^{6} - 74 \nu^{5} + 204 \nu^{4} + \cdots + 416 ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7 \nu^{11} - 35 \nu^{9} + 2 \nu^{8} + 89 \nu^{7} - 34 \nu^{6} - 162 \nu^{5} + 140 \nu^{4} + 244 \nu^{3} + \cdots + 128 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{8} - \beta_{4} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{11} - \beta_{10} - 2\beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} - \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + 2\beta_{10} - 2\beta_{9} + 3\beta_{8} - \beta_{6} - 2\beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{8} - 2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - \beta_{4} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{10} - 8\beta_{9} + 2\beta_{8} + 2\beta_{7} - \beta_{6} - 2\beta_{5} - \beta_{4} + 2\beta_{2} - 3\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{11} - 2\beta_{9} + 3\beta_{8} + 4\beta_{7} + \beta_{6} + 6\beta_{3} + 3\beta_{2} + 6\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 9 \beta_{11} - 3 \beta_{10} - 10 \beta_{9} - 3 \beta_{8} - 6 \beta_{7} - 2 \beta_{5} - 7 \beta_{4} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4 \beta_{11} + 11 \beta_{10} + 6 \beta_{8} + 2 \beta_{7} - 13 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 15 \beta_{11} - 4 \beta_{10} - 6 \beta_{9} - 3 \beta_{8} - 24 \beta_{7} + 7 \beta_{6} - 8 \beta_{5} + \cdots - 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 9 \beta_{11} + 29 \beta_{10} - 26 \beta_{9} - 7 \beta_{8} - 6 \beta_{7} - 14 \beta_{6} - 18 \beta_{5} + \cdots - 22 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 18 \beta_{11} + 17 \beta_{10} + 12 \beta_{9} - 8 \beta_{8} + 2 \beta_{7} + 5 \beta_{6} - 42 \beta_{5} + \cdots - 14 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/520\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(261\) \(391\) \(417\)
\(\chi(n)\) \(-\beta_{9}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.34408 + 0.439820i
1.40744 + 0.138282i
−1.08105 + 0.911778i
−1.30089 + 0.554694i
0.759479 − 1.19298i
−1.12906 − 0.851598i
1.34408 − 0.439820i
1.40744 − 0.138282i
−1.08105 − 0.911778i
−1.30089 − 0.554694i
0.759479 + 1.19298i
−1.12906 + 0.851598i
0 −1.05294 + 1.82374i 0 − 1.00000i 0 −2.22378 + 1.28390i 0 −0.717351 − 1.24249i 0
121.2 0 −0.823474 + 1.42630i 0 1.00000i 0 1.33221 − 0.769155i 0 0.143781 + 0.249036i 0
121.3 0 −0.249100 + 0.431454i 0 − 1.00000i 0 1.15921 − 0.669268i 0 1.37590 + 2.38313i 0
121.4 0 0.170066 − 0.294562i 0 1.00000i 0 −4.07999 + 2.35558i 0 1.44216 + 2.49789i 0
121.5 0 0.653409 − 1.13174i 0 1.00000i 0 2.51573 − 1.45245i 0 0.646115 + 1.11910i 0
121.6 0 1.30204 − 2.25519i 0 − 1.00000i 0 4.29663 − 2.48066i 0 −1.89060 − 3.27461i 0
361.1 0 −1.05294 − 1.82374i 0 1.00000i 0 −2.22378 − 1.28390i 0 −0.717351 + 1.24249i 0
361.2 0 −0.823474 − 1.42630i 0 − 1.00000i 0 1.33221 + 0.769155i 0 0.143781 − 0.249036i 0
361.3 0 −0.249100 − 0.431454i 0 1.00000i 0 1.15921 + 0.669268i 0 1.37590 − 2.38313i 0
361.4 0 0.170066 + 0.294562i 0 − 1.00000i 0 −4.07999 − 2.35558i 0 1.44216 − 2.49789i 0
361.5 0 0.653409 + 1.13174i 0 − 1.00000i 0 2.51573 + 1.45245i 0 0.646115 − 1.11910i 0
361.6 0 1.30204 + 2.25519i 0 1.00000i 0 4.29663 + 2.48066i 0 −1.89060 + 3.27461i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.bu.a ✓ 12
4.b odd 2 1 1040.2.da.e 12
13.e even 6 1 inner 520.2.bu.a ✓ 12
13.f odd 12 1 6760.2.a.bg 6
13.f odd 12 1 6760.2.a.bj 6
52.i odd 6 1 1040.2.da.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.bu.a ✓ 12 1.a even 1 1 trivial
520.2.bu.a ✓ 12 13.e even 6 1 inner
1040.2.da.e 12 4.b odd 2 1
1040.2.da.e 12 52.i odd 6 1
6760.2.a.bg 6 13.f odd 12 1
6760.2.a.bj 6 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 8 T_{3}^{10} + 4 T_{3}^{9} + 51 T_{3}^{8} + 18 T_{3}^{7} + 104 T_{3}^{6} + 6 T_{3}^{5} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 8 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} + \cdots + 128881 \) Copy content Toggle raw display
$11$ \( T^{12} - 19 T^{10} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{12} + 2 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 8 T^{11} + \cdots + 37636 \) Copy content Toggle raw display
$19$ \( T^{12} + 24 T^{11} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 292273216 \) Copy content Toggle raw display
$29$ \( T^{12} - 12 T^{11} + \cdots + 13366336 \) Copy content Toggle raw display
$31$ \( T^{12} + 200 T^{10} + \cdots + 22429696 \) Copy content Toggle raw display
$37$ \( T^{12} - 24 T^{11} + \cdots + 40462321 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 126061922704 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 894967056 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 255488256 \) Copy content Toggle raw display
$53$ \( (T^{6} + 34 T^{5} + \cdots + 19792)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 237283216 \) Copy content Toggle raw display
$61$ \( T^{12} - 18 T^{11} + \cdots + 12096484 \) Copy content Toggle raw display
$67$ \( T^{12} - 18 T^{11} + \cdots + 15241216 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 418284304 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 53809153024 \) Copy content Toggle raw display
$79$ \( (T^{6} + 6 T^{5} + \cdots + 50272)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 220794732544 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 2387592769 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 8409256804 \) Copy content Toggle raw display
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