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Properties

Label 420.2.q.c
Level $420$
Weight $2$
Character orbit 420.q
Analytic conductor $3.354$
Analytic rank $0$
Dimension $4$
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] = [420,2,Mod(121,420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(420, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("420.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.q (of order \(3\), 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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{2} - 1) q^{5} + \beta_1 q^{7} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{2} - 1) q^{5} + \beta_1 q^{7} + ( - \beta_{2} - 1) q^{9} + (\beta_{3} + \beta_{2} + \beta_1) q^{11} - \beta_{3} q^{13} + q^{15} + (\beta_{3} + \beta_{2} + \beta_1) q^{17} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{19} + \beta_{3} q^{21} + (\beta_{2} + \beta_1 + 1) q^{23} + \beta_{2} q^{25} + q^{27} + (\beta_{3} + 5) q^{29} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{31} + ( - \beta_{2} - \beta_1 - 1) q^{33} + ( - \beta_{3} - \beta_1) q^{35} + (2 \beta_{2} + \beta_1 + 2) q^{37} + (\beta_{3} + \beta_1) q^{39} + (3 \beta_{3} - 3) q^{41} + ( - 3 \beta_{3} + 2) q^{43} + \beta_{2} q^{45} + ( - 6 \beta_{2} - 6) q^{47} + 7 \beta_{2} q^{49} + ( - \beta_{2} - \beta_1 - 1) q^{51} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{53} + ( - \beta_{3} + 1) q^{55} + (2 \beta_{3} + 3) q^{57} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{59} + ( - 8 \beta_{2} - 8) q^{61} + ( - \beta_{3} - \beta_1) q^{63} - \beta_1 q^{65} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{67} + (\beta_{3} - 1) q^{69} + (\beta_{3} + 11) q^{71} + ( - 5 \beta_{3} - 5 \beta_1) q^{73} + ( - \beta_{2} - 1) q^{75} + (\beta_{3} - 7) q^{77} + (3 \beta_{2} + 2 \beta_1 + 3) q^{79} + \beta_{2} q^{81} + ( - 3 \beta_{3} - 3) q^{83} + ( - \beta_{3} + 1) q^{85} + ( - \beta_{3} + 5 \beta_{2} - \beta_1) q^{87} + (\beta_{2} - 5 \beta_1 + 1) q^{89} + (7 \beta_{2} + 7) q^{91} + ( - \beta_{2} - 2 \beta_1 - 1) q^{93} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{95} + 8 q^{97} + ( - \beta_{3} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 2 q^{5} - 2 q^{9} - 2 q^{11} + 4 q^{15} - 2 q^{17} - 6 q^{19} + 2 q^{23} - 2 q^{25} + 4 q^{27} + 20 q^{29} - 2 q^{31} - 2 q^{33} + 4 q^{37} - 12 q^{41} + 8 q^{43} - 2 q^{45} - 12 q^{47} - 14 q^{49} - 2 q^{51} + 4 q^{53} + 4 q^{55} + 12 q^{57} - 6 q^{59} - 16 q^{61} + 4 q^{67} - 4 q^{69} + 44 q^{71} - 2 q^{75} - 28 q^{77} + 6 q^{79} - 2 q^{81} - 12 q^{83} + 4 q^{85} - 10 q^{87} + 2 q^{89} + 14 q^{91} - 2 q^{93} - 6 q^{95} + 32 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\) \(1\) \(1\)

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
−1.32288 + 2.29129i
1.32288 − 2.29129i
−1.32288 − 2.29129i
1.32288 + 2.29129i
0 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0 −1.32288 + 2.29129i 0 −0.500000 + 0.866025i 0
121.2 0 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0 1.32288 − 2.29129i 0 −0.500000 + 0.866025i 0
361.1 0 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0 −1.32288 − 2.29129i 0 −0.500000 − 0.866025i 0
361.2 0 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0 1.32288 + 2.29129i 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.q.c ✓ 4
3.b odd 2 1 1260.2.s.f 4
4.b odd 2 1 1680.2.bg.q 4
5.b even 2 1 2100.2.q.h 4
5.c odd 4 2 2100.2.bc.e 8
7.b odd 2 1 2940.2.q.t 4
7.c even 3 1 inner 420.2.q.c ✓ 4
7.c even 3 1 2940.2.a.s 2
7.d odd 6 1 2940.2.a.m 2
7.d odd 6 1 2940.2.q.t 4
21.g even 6 1 8820.2.a.bj 2
21.h odd 6 1 1260.2.s.f 4
21.h odd 6 1 8820.2.a.be 2
28.g odd 6 1 1680.2.bg.q 4
35.j even 6 1 2100.2.q.h 4
35.l odd 12 2 2100.2.bc.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.c ✓ 4 1.a even 1 1 trivial
420.2.q.c ✓ 4 7.c even 3 1 inner
1260.2.s.f 4 3.b odd 2 1
1260.2.s.f 4 21.h odd 6 1
1680.2.bg.q 4 4.b odd 2 1
1680.2.bg.q 4 28.g odd 6 1
2100.2.q.h 4 5.b even 2 1
2100.2.q.h 4 35.j even 6 1
2100.2.bc.e 8 5.c odd 4 2
2100.2.bc.e 8 35.l odd 12 2
2940.2.a.m 2 7.d odd 6 1
2940.2.a.s 2 7.c even 3 1
2940.2.q.t 4 7.b odd 2 1
2940.2.q.t 4 7.d odd 6 1
8820.2.a.be 2 21.h odd 6 1
8820.2.a.bj 2 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 2T_{11}^{3} + 10T_{11}^{2} - 12T_{11} + 36 \) acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} - 10 T + 18)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 54)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T - 59)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + \cdots + 2916 \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$71$ \( (T^{2} - 22 T + 114)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 175 T^{2} + 30625 \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 54)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 2 T^{3} + \cdots + 30276 \) Copy content Toggle raw display
$97$ \( (T - 8)^{4} \) Copy content Toggle raw display
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