Properties

Label 420.2.q.d
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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Newspace parameters

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

Self dual: no
Analytic conductor: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} ) q^{3} -\beta_{1} q^{5} + ( \beta_{1} + \beta_{3} ) q^{7} + \beta_{1} q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{1} ) q^{3} -\beta_{1} q^{5} + ( \beta_{1} + \beta_{3} ) q^{7} + \beta_{1} q^{9} + ( \beta_{2} + \beta_{3} ) q^{11} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{13} - q^{15} + ( -\beta_{2} - \beta_{3} ) q^{17} -7 \beta_{1} q^{19} + ( 1 - \beta_{2} ) q^{21} + ( -2 \beta_{2} + \beta_{3} ) q^{23} + ( -1 - \beta_{1} ) q^{25} + q^{27} + ( -6 - \beta_{2} + 2 \beta_{3} ) q^{29} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -2 \beta_{2} + \beta_{3} ) q^{33} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{35} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{37} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{39} + ( \beta_{2} - 2 \beta_{3} ) q^{41} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{43} + ( 1 + \beta_{1} ) q^{45} -6 \beta_{1} q^{47} + ( 5 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{49} + ( 2 \beta_{2} - \beta_{3} ) q^{51} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -\beta_{2} + 2 \beta_{3} ) q^{55} -7 q^{57} + ( 6 + 6 \beta_{1} + \beta_{2} + \beta_{3} ) q^{59} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{63} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{65} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{67} + ( \beta_{2} - 2 \beta_{3} ) q^{69} + ( 3 \beta_{2} - 6 \beta_{3} ) q^{71} + ( -5 - 5 \beta_{1} - \beta_{2} - \beta_{3} ) q^{73} + \beta_{1} q^{75} + ( 6 + 12 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{77} + 11 \beta_{1} q^{79} + ( -1 - \beta_{1} ) q^{81} + ( -6 + \beta_{2} - 2 \beta_{3} ) q^{83} + ( \beta_{2} - 2 \beta_{3} ) q^{85} + ( 6 + 6 \beta_{1} - \beta_{2} - \beta_{3} ) q^{87} + ( -6 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{89} + ( 12 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{91} + ( -\beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{93} + ( -7 - 7 \beta_{1} ) q^{95} + ( 8 + 2 \beta_{2} - 4 \beta_{3} ) q^{97} + ( \beta_{2} - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{5} - 2 q^{7} - 2 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{3} + 2 q^{5} - 2 q^{7} - 2 q^{9} - 4 q^{13} - 4 q^{15} + 14 q^{19} + 4 q^{21} - 2 q^{25} + 4 q^{27} - 24 q^{29} + 2 q^{31} + 2 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{43} + 2 q^{45} + 12 q^{47} + 10 q^{49} - 28 q^{57} + 12 q^{59} + 8 q^{61} - 2 q^{63} - 2 q^{65} + 2 q^{67} - 10 q^{73} - 2 q^{75} - 22 q^{79} - 2 q^{81} - 24 q^{83} + 12 q^{87} + 12 q^{89} + 38 q^{91} + 2 q^{93} - 14 q^{95} + 32 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{3} + 2 \beta_{2}\)\()/3\)

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 −0.500000 0.866025i 0 0.500000 0.866025i 0 −2.62132 0.358719i 0 −0.500000 + 0.866025i 0
121.2 0 −0.500000 0.866025i 0 0.500000 0.866025i 0 1.62132 + 2.09077i 0 −0.500000 + 0.866025i 0
361.1 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 −2.62132 + 0.358719i 0 −0.500000 0.866025i 0
361.2 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 1.62132 2.09077i 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.d 4
3.b odd 2 1 1260.2.s.e 4
4.b odd 2 1 1680.2.bg.t 4
5.b even 2 1 2100.2.q.k 4
5.c odd 4 2 2100.2.bc.f 8
7.b odd 2 1 2940.2.q.q 4
7.c even 3 1 inner 420.2.q.d 4
7.c even 3 1 2940.2.a.r 2
7.d odd 6 1 2940.2.a.p 2
7.d odd 6 1 2940.2.q.q 4
21.g even 6 1 8820.2.a.bf 2
21.h odd 6 1 1260.2.s.e 4
21.h odd 6 1 8820.2.a.bk 2
28.g odd 6 1 1680.2.bg.t 4
35.j even 6 1 2100.2.q.k 4
35.l odd 12 2 2100.2.bc.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.d 4 1.a even 1 1 trivial
420.2.q.d 4 7.c even 3 1 inner
1260.2.s.e 4 3.b odd 2 1
1260.2.s.e 4 21.h odd 6 1
1680.2.bg.t 4 4.b odd 2 1
1680.2.bg.t 4 28.g odd 6 1
2100.2.q.k 4 5.b even 2 1
2100.2.q.k 4 35.j even 6 1
2100.2.bc.f 8 5.c odd 4 2
2100.2.bc.f 8 35.l odd 12 2
2940.2.a.p 2 7.d odd 6 1
2940.2.a.r 2 7.c even 3 1
2940.2.q.q 4 7.b odd 2 1
2940.2.q.q 4 7.d odd 6 1
8820.2.a.bf 2 21.g even 6 1
8820.2.a.bk 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 18 T_{11}^{2} + 324 \) acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 49 + 14 T - 3 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( 324 + 18 T^{2} + T^{4} \)
$13$ \( ( -17 + 2 T + T^{2} )^{2} \)
$17$ \( 324 + 18 T^{2} + T^{4} \)
$19$ \( ( 49 - 7 T + T^{2} )^{2} \)
$23$ \( 324 + 18 T^{2} + T^{4} \)
$29$ \( ( 18 + 12 T + T^{2} )^{2} \)
$31$ \( 5041 + 142 T + 75 T^{2} - 2 T^{3} + T^{4} \)
$37$ \( 289 + 34 T + 21 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( ( -18 + T^{2} )^{2} \)
$43$ \( ( -17 + 2 T + T^{2} )^{2} \)
$47$ \( ( 36 - 6 T + T^{2} )^{2} \)
$53$ \( 5184 + 72 T^{2} + T^{4} \)
$59$ \( 324 - 216 T + 126 T^{2} - 12 T^{3} + T^{4} \)
$61$ \( 3136 + 448 T + 120 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( 289 + 34 T + 21 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( ( -162 + T^{2} )^{2} \)
$73$ \( 49 + 70 T + 93 T^{2} + 10 T^{3} + T^{4} \)
$79$ \( ( 121 + 11 T + T^{2} )^{2} \)
$83$ \( ( 18 + 12 T + T^{2} )^{2} \)
$89$ \( 324 - 216 T + 126 T^{2} - 12 T^{3} + T^{4} \)
$97$ \( ( -8 - 16 T + T^{2} )^{2} \)
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