Properties

Label 420.2.f.b
Level $420$
Weight $2$
Character orbit 420.f
Analytic conductor $3.354$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} -\beta_{2} q^{5} + ( \beta_{5} + \beta_{6} ) q^{7} + ( 1 - \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{6} q^{3} -\beta_{2} q^{5} + ( \beta_{5} + \beta_{6} ) q^{7} + ( 1 - \beta_{4} ) q^{9} -\beta_{4} q^{11} + ( \beta_{3} - \beta_{6} ) q^{13} + ( -\beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{15} + ( -2 \beta_{3} - 2 \beta_{6} ) q^{17} + ( -2 \beta_{1} - \beta_{4} ) q^{19} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{21} + ( \beta_{3} + \beta_{6} + 2 \beta_{7} ) q^{23} + 5 q^{25} + ( 3 \beta_{3} + 2 \beta_{6} ) q^{27} -2 \beta_{4} q^{29} + ( 2 \beta_{1} + \beta_{4} ) q^{31} + ( 3 \beta_{3} + \beta_{6} ) q^{33} + ( -3 \beta_{3} - 3 \beta_{6} - \beta_{7} ) q^{35} + ( -2 \beta_{3} - 4 \beta_{5} - 2 \beta_{6} ) q^{37} + ( -4 + \beta_{4} ) q^{39} -2 \beta_{2} q^{41} + ( \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{43} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{45} + ( -3 \beta_{3} - 3 \beta_{6} ) q^{47} + ( -3 - 2 \beta_{1} - \beta_{4} ) q^{49} + ( 4 + 2 \beta_{4} ) q^{51} + ( \beta_{3} + \beta_{6} + 2 \beta_{7} ) q^{53} + ( 2 \beta_{1} + \beta_{4} ) q^{55} + ( \beta_{3} + 4 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{57} + 4 \beta_{2} q^{59} + ( \beta_{3} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{63} + ( \beta_{3} + \beta_{6} + 2 \beta_{7} ) q^{65} + ( -\beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{67} + ( -2 \beta_{1} + 4 \beta_{2} - \beta_{4} ) q^{69} + 5 \beta_{4} q^{71} + ( -3 \beta_{3} + 3 \beta_{6} ) q^{73} + 5 \beta_{6} q^{75} + ( \beta_{3} + \beta_{6} - 2 \beta_{7} ) q^{77} -12 q^{79} + ( -7 - 2 \beta_{4} ) q^{81} + ( -5 \beta_{3} - 5 \beta_{6} ) q^{83} + ( 2 \beta_{3} + 4 \beta_{5} + 2 \beta_{6} ) q^{85} + ( 6 \beta_{3} + 2 \beta_{6} ) q^{87} + 2 \beta_{2} q^{89} + ( -4 + 2 \beta_{1} + \beta_{4} ) q^{91} + ( -\beta_{3} - 4 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{93} + 5 \beta_{4} q^{95} + ( -3 \beta_{3} + 3 \beta_{6} ) q^{97} + ( -8 - \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9} + O(q^{10}) \) \( 8 q + 8 q^{9} + 16 q^{21} + 40 q^{25} - 32 q^{39} - 24 q^{49} + 32 q^{51} - 96 q^{79} - 56 q^{81} - 32 q^{91} - 64 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{7} + 16 \nu^{3} + 6 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{4} + 7 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{7} - \nu^{6} - \nu^{5} + 13 \nu^{3} - 8 \nu^{2} - 5 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( 4 \nu^{7} + 2 \nu^{5} + 26 \nu^{3} + 10 \nu \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{6} - 10 \nu^{2} \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{7} - \nu^{6} + \nu^{5} - 13 \nu^{3} - 8 \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( -4 \nu^{7} + \nu^{6} + \nu^{5} - 29 \nu^{3} + 8 \nu^{2} + 11 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{3} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{6} + \beta_{5} - \beta_{3}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{7} + \beta_{6} - \beta_{4} - 3 \beta_{3} + 2 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{2} - 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{7} + 3 \beta_{6} + 3 \beta_{4} - 8 \beta_{3} - 5 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(5 \beta_{6} - 8 \beta_{5} + 5 \beta_{3}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(13 \beta_{7} - 8 \beta_{6} + 8 \beta_{4} + 21 \beta_{3} - 13 \beta_{1}\)\()/4\)

Character values

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

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−0.437016 0.437016i
1.14412 + 1.14412i
−0.437016 + 0.437016i
1.14412 1.14412i
0.437016 + 0.437016i
−1.14412 1.14412i
0.437016 0.437016i
−1.14412 + 1.14412i
0 −1.41421 1.00000i 0 −2.23607 0 −1.41421 2.23607i 0 1.00000 + 2.82843i 0
209.2 0 −1.41421 1.00000i 0 2.23607 0 −1.41421 + 2.23607i 0 1.00000 + 2.82843i 0
209.3 0 −1.41421 + 1.00000i 0 −2.23607 0 −1.41421 + 2.23607i 0 1.00000 2.82843i 0
209.4 0 −1.41421 + 1.00000i 0 2.23607 0 −1.41421 2.23607i 0 1.00000 2.82843i 0
209.5 0 1.41421 1.00000i 0 −2.23607 0 1.41421 2.23607i 0 1.00000 2.82843i 0
209.6 0 1.41421 1.00000i 0 2.23607 0 1.41421 + 2.23607i 0 1.00000 2.82843i 0
209.7 0 1.41421 + 1.00000i 0 −2.23607 0 1.41421 + 2.23607i 0 1.00000 + 2.82843i 0
209.8 0 1.41421 + 1.00000i 0 2.23607 0 1.41421 2.23607i 0 1.00000 + 2.82843i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.f.b 8
3.b odd 2 1 inner 420.2.f.b 8
4.b odd 2 1 1680.2.k.g 8
5.b even 2 1 inner 420.2.f.b 8
5.c odd 4 1 2100.2.d.f 4
5.c odd 4 1 2100.2.d.i 4
7.b odd 2 1 inner 420.2.f.b 8
12.b even 2 1 1680.2.k.g 8
15.d odd 2 1 inner 420.2.f.b 8
15.e even 4 1 2100.2.d.f 4
15.e even 4 1 2100.2.d.i 4
20.d odd 2 1 1680.2.k.g 8
21.c even 2 1 inner 420.2.f.b 8
28.d even 2 1 1680.2.k.g 8
35.c odd 2 1 inner 420.2.f.b 8
35.f even 4 1 2100.2.d.f 4
35.f even 4 1 2100.2.d.i 4
60.h even 2 1 1680.2.k.g 8
84.h odd 2 1 1680.2.k.g 8
105.g even 2 1 inner 420.2.f.b 8
105.k odd 4 1 2100.2.d.f 4
105.k odd 4 1 2100.2.d.i 4
140.c even 2 1 1680.2.k.g 8
420.o odd 2 1 1680.2.k.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.f.b 8 1.a even 1 1 trivial
420.2.f.b 8 3.b odd 2 1 inner
420.2.f.b 8 5.b even 2 1 inner
420.2.f.b 8 7.b odd 2 1 inner
420.2.f.b 8 15.d odd 2 1 inner
420.2.f.b 8 21.c even 2 1 inner
420.2.f.b 8 35.c odd 2 1 inner
420.2.f.b 8 105.g even 2 1 inner
1680.2.k.g 8 4.b odd 2 1
1680.2.k.g 8 12.b even 2 1
1680.2.k.g 8 20.d odd 2 1
1680.2.k.g 8 28.d even 2 1
1680.2.k.g 8 60.h even 2 1
1680.2.k.g 8 84.h odd 2 1
1680.2.k.g 8 140.c even 2 1
1680.2.k.g 8 420.o odd 2 1
2100.2.d.f 4 5.c odd 4 1
2100.2.d.f 4 15.e even 4 1
2100.2.d.f 4 35.f even 4 1
2100.2.d.f 4 105.k odd 4 1
2100.2.d.i 4 5.c odd 4 1
2100.2.d.i 4 15.e even 4 1
2100.2.d.i 4 35.f even 4 1
2100.2.d.i 4 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 9 - 2 T^{2} + T^{4} )^{2} \)
$5$ \( ( -5 + T^{2} )^{4} \)
$7$ \( ( 49 + 6 T^{2} + T^{4} )^{2} \)
$11$ \( ( 8 + T^{2} )^{4} \)
$13$ \( ( -8 + T^{2} )^{4} \)
$17$ \( ( 16 + T^{2} )^{4} \)
$19$ \( ( 40 + T^{2} )^{4} \)
$23$ \( ( -40 + T^{2} )^{4} \)
$29$ \( ( 32 + T^{2} )^{4} \)
$31$ \( ( 40 + T^{2} )^{4} \)
$37$ \( ( 80 + T^{2} )^{4} \)
$41$ \( ( -20 + T^{2} )^{4} \)
$43$ \( ( 20 + T^{2} )^{4} \)
$47$ \( ( 36 + T^{2} )^{4} \)
$53$ \( ( -40 + T^{2} )^{4} \)
$59$ \( ( -80 + T^{2} )^{4} \)
$61$ \( T^{8} \)
$67$ \( ( 20 + T^{2} )^{4} \)
$71$ \( ( 200 + T^{2} )^{4} \)
$73$ \( ( -72 + T^{2} )^{4} \)
$79$ \( ( 12 + T )^{8} \)
$83$ \( ( 100 + T^{2} )^{4} \)
$89$ \( ( -20 + T^{2} )^{4} \)
$97$ \( ( -72 + T^{2} )^{4} \)
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