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Properties

Label 3520.2.f.h
Level $3520$
Weight $2$
Character orbit 3520.f
Analytic conductor $28.107$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3520,2,Mod(2111,3520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3520.2111");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3520.f (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: \(28.1073415115\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.170772624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 880)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - q^{5} + \beta_{2} q^{7} + ( - \beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - q^{5} + \beta_{2} q^{7} + ( - \beta_{4} - 1) q^{9} + ( - \beta_{5} - \beta_{3} - \beta_{2}) q^{11} + \beta_{7} q^{13} - \beta_{3} q^{15} + \beta_{6} q^{17} + ( - \beta_{2} - \beta_1) q^{19} + ( - \beta_{7} - \beta_{6}) q^{21} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{23} + q^{25} + (2 \beta_{5} + \beta_{3}) q^{27} + (\beta_{7} + \beta_{6}) q^{29} - 3 \beta_{3} q^{31} + (\beta_{7} + \beta_{6} + 2) q^{33} - \beta_{2} q^{35} + (3 \beta_{4} + 2) q^{37} + 2 \beta_{2} q^{39} + (2 \beta_{2} + \beta_1) q^{43} + (\beta_{4} + 1) q^{45} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{47} + (\beta_{4} + 1) q^{49} + (3 \beta_{2} + \beta_1) q^{51} + (\beta_{4} + 2) q^{53} + (\beta_{5} + \beta_{3} + \beta_{2}) q^{55} + (\beta_{7} + 3 \beta_{6}) q^{57} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{59} + (\beta_{7} - \beta_{6}) q^{61} + ( - 2 \beta_{2} - \beta_1) q^{63} - \beta_{7} q^{65} + 2 \beta_{5} q^{67} + ( - 4 \beta_{4} - 12) q^{69} + ( - 4 \beta_{5} - 5 \beta_{3}) q^{71} + 3 \beta_{7} q^{73} + \beta_{3} q^{75} + (\beta_{7} - \beta_{4} - 8) q^{77} + ( - 2 \beta_{4} - 3) q^{81} - \beta_1 q^{83} - \beta_{6} q^{85} + (5 \beta_{2} + \beta_1) q^{87} + ( - \beta_{4} - 14) q^{89} + ( - 4 \beta_{5} - 6 \beta_{3}) q^{91} + (3 \beta_{4} + 12) q^{93} + (\beta_{2} + \beta_1) q^{95} + ( - 4 \beta_{4} + 2) q^{97} + ( - 3 \beta_{5} - \beta_{3} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} - 4 q^{9} + 8 q^{25} + 16 q^{33} + 4 q^{37} + 4 q^{45} + 4 q^{49} + 12 q^{53} - 80 q^{69} - 60 q^{77} - 16 q^{81} - 108 q^{89} + 84 q^{93} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 3\nu^{4} + 6\nu^{2} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 2\nu^{5} - \nu^{4} - 10\nu^{2} + 12\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 4\nu^{5} + 3\nu^{4} - 2\nu^{3} + 6\nu^{2} - 12\nu + 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 4\nu^{6} + 6\nu^{5} - 5\nu^{4} + 6\nu^{3} - 14\nu^{2} + 20\nu - 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 5\nu^{6} - 7\nu^{5} + 6\nu^{4} - 10\nu^{3} + 24\nu^{2} - 28\nu + 28 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 4\nu^{6} + 12\nu^{5} - 9\nu^{4} + 12\nu^{3} - 30\nu^{2} + 36\nu - 56 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 4\nu^{6} - 8\nu^{5} + 5\nu^{4} - 8\nu^{3} + 22\nu^{2} - 20\nu + 32 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} + 2\beta_{6} - 4\beta_{5} - 4\beta_{4} - 4\beta_{3} + 2\beta_{2} + \beta _1 + 4 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{6} - 4\beta_{5} - 4\beta_{4} - 12\beta_{3} - 6\beta_{2} + \beta _1 - 4 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{5} + 4\beta_{3} - 6\beta_{2} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{7} - 2\beta_{6} + 4\beta_{5} - 4\beta_{4} + 12\beta_{3} + 6\beta_{2} + 7\beta _1 - 4 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9\beta_{7} + 2\beta_{6} + 20\beta_{5} + 4\beta_{4} - 4\beta_{3} - 2\beta_{2} + 3\beta _1 + 68 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{7} + 10\beta_{6} - 12\beta_{4} + 20 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{7} - 10\beta_{6} - 28\beta_{5} - 52\beta_{4} - 52\beta_{3} - 10\beta_{2} - \beta _1 - 20 ) / 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(1541\) \(2751\) \(2817\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2111.1
−1.02187 − 0.977642i
0.335728 + 1.37379i
1.41203 + 0.0786378i
0.774115 + 1.18353i
1.41203 − 0.0786378i
0.774115 − 1.18353i
−1.02187 + 0.977642i
0.335728 − 1.37379i
0 − 2.52434i 0 −1.00000 0 −3.22060 0 −3.37228 0
2111.2 0 − 2.52434i 0 −1.00000 0 3.22060 0 −3.37228 0
2111.3 0 − 0.792287i 0 −1.00000 0 −2.15121 0 2.37228 0
2111.4 0 − 0.792287i 0 −1.00000 0 2.15121 0 2.37228 0
2111.5 0 0.792287i 0 −1.00000 0 −2.15121 0 2.37228 0
2111.6 0 0.792287i 0 −1.00000 0 2.15121 0 2.37228 0
2111.7 0 2.52434i 0 −1.00000 0 −3.22060 0 −3.37228 0
2111.8 0 2.52434i 0 −1.00000 0 3.22060 0 −3.37228 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2111.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.b odd 2 1 inner
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3520.2.f.h 8
4.b odd 2 1 inner 3520.2.f.h 8
8.b even 2 1 880.2.f.e ✓ 8
8.d odd 2 1 880.2.f.e ✓ 8
11.b odd 2 1 inner 3520.2.f.h 8
44.c even 2 1 inner 3520.2.f.h 8
88.b odd 2 1 880.2.f.e ✓ 8
88.g even 2 1 880.2.f.e ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.f.e ✓ 8 8.b even 2 1
880.2.f.e ✓ 8 8.d odd 2 1
880.2.f.e ✓ 8 88.b odd 2 1
880.2.f.e ✓ 8 88.g even 2 1
3520.2.f.h 8 1.a even 1 1 trivial
3520.2.f.h 8 4.b odd 2 1 inner
3520.2.f.h 8 11.b odd 2 1 inner
3520.2.f.h 8 44.c 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 \(S_{2}^{\mathrm{new}}(3520, [\chi])\):

\( T_{3}^{4} + 7T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 15T_{7}^{2} + 48 \) Copy content Toggle raw display
\( T_{19}^{4} - 111T_{19}^{2} + 3072 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 15 T^{2} + 48)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 16 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} + 36 T^{2} + 192)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 45 T^{2} + 432)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 111 T^{2} + 3072)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 76 T^{2} + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 69 T^{2} + 192)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 63 T^{2} + 324)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - T - 74)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} - 144 T^{2} + 3072)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 28 T^{2} + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 3 T - 6)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 28 T^{2} + 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 93 T^{2} + 768)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 151 T^{2} + 3844)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 324 T^{2} + 15552)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 108 T^{2} + 1728)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 27 T + 174)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T - 116)^{4} \) Copy content Toggle raw display
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