Properties

Label 420.2.q.a
Level $420$
Weight $2$
Character orbit 420.q
Analytic conductor $3.354$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{11} + q^{13} - q^{15} + ( 4 - 4 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + ( 3 - 2 \zeta_{6} ) q^{21} -4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} - q^{27} + ( 5 - 5 \zeta_{6} ) q^{31} -2 \zeta_{6} q^{33} + ( 1 - 3 \zeta_{6} ) q^{35} + 5 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{39} + 2 q^{41} -9 q^{43} + ( -1 + \zeta_{6} ) q^{45} + 2 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} -4 \zeta_{6} q^{51} + ( -12 + 12 \zeta_{6} ) q^{53} -2 q^{55} + q^{57} + ( 8 - 8 \zeta_{6} ) q^{59} + 14 \zeta_{6} q^{61} + ( 1 - 3 \zeta_{6} ) q^{63} -\zeta_{6} q^{65} + ( -9 + 9 \zeta_{6} ) q^{67} -4 q^{69} + 2 q^{71} + ( -1 + \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + ( 6 - 4 \zeta_{6} ) q^{77} + 3 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -18 q^{83} -4 q^{85} + 4 \zeta_{6} q^{89} + ( 2 + \zeta_{6} ) q^{91} -5 \zeta_{6} q^{93} + ( 1 - \zeta_{6} ) q^{95} + 10 q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - q^{5} + 5 q^{7} - q^{9} + O(q^{10}) \) \( 2 q + q^{3} - q^{5} + 5 q^{7} - q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} + 4 q^{17} + q^{19} + 4 q^{21} - 4 q^{23} - q^{25} - 2 q^{27} + 5 q^{31} - 2 q^{33} - q^{35} + 5 q^{37} + q^{39} + 4 q^{41} - 18 q^{43} - q^{45} + 2 q^{47} + 11 q^{49} - 4 q^{51} - 12 q^{53} - 4 q^{55} + 2 q^{57} + 8 q^{59} + 14 q^{61} - q^{63} - q^{65} - 9 q^{67} - 8 q^{69} + 4 q^{71} - q^{73} + q^{75} + 8 q^{77} + 3 q^{79} - q^{81} - 36 q^{83} - 8 q^{85} + 4 q^{89} + 5 q^{91} - 5 q^{93} + q^{95} + 20 q^{97} - 4 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 2.50000 0.866025i 0 −0.500000 + 0.866025i 0
361.1 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 2.50000 + 0.866025i 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.a 2
3.b odd 2 1 1260.2.s.d 2
4.b odd 2 1 1680.2.bg.a 2
5.b even 2 1 2100.2.q.a 2
5.c odd 4 2 2100.2.bc.c 4
7.b odd 2 1 2940.2.q.h 2
7.c even 3 1 inner 420.2.q.a 2
7.c even 3 1 2940.2.a.d 1
7.d odd 6 1 2940.2.a.h 1
7.d odd 6 1 2940.2.q.h 2
21.g even 6 1 8820.2.a.y 1
21.h odd 6 1 1260.2.s.d 2
21.h odd 6 1 8820.2.a.j 1
28.g odd 6 1 1680.2.bg.a 2
35.j even 6 1 2100.2.q.a 2
35.l odd 12 2 2100.2.bc.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.a 2 1.a even 1 1 trivial
420.2.q.a 2 7.c even 3 1 inner
1260.2.s.d 2 3.b odd 2 1
1260.2.s.d 2 21.h odd 6 1
1680.2.bg.a 2 4.b odd 2 1
1680.2.bg.a 2 28.g odd 6 1
2100.2.q.a 2 5.b even 2 1
2100.2.q.a 2 35.j even 6 1
2100.2.bc.c 4 5.c odd 4 2
2100.2.bc.c 4 35.l odd 12 2
2940.2.a.d 1 7.c even 3 1
2940.2.a.h 1 7.d odd 6 1
2940.2.q.h 2 7.b odd 2 1
2940.2.q.h 2 7.d odd 6 1
8820.2.a.j 1 21.h odd 6 1
8820.2.a.y 1 21.g even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 7 - 5 T + T^{2} \)
$11$ \( 4 - 2 T + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( 16 - 4 T + T^{2} \)
$19$ \( 1 - T + T^{2} \)
$23$ \( 16 + 4 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( 25 - 5 T + T^{2} \)
$37$ \( 25 - 5 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( ( 9 + T )^{2} \)
$47$ \( 4 - 2 T + T^{2} \)
$53$ \( 144 + 12 T + T^{2} \)
$59$ \( 64 - 8 T + T^{2} \)
$61$ \( 196 - 14 T + T^{2} \)
$67$ \( 81 + 9 T + T^{2} \)
$71$ \( ( -2 + T )^{2} \)
$73$ \( 1 + T + T^{2} \)
$79$ \( 9 - 3 T + T^{2} \)
$83$ \( ( 18 + T )^{2} \)
$89$ \( 16 - 4 T + T^{2} \)
$97$ \( ( -10 + T )^{2} \)
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