Properties

Label 420.2.w.a
Level $420$
Weight $2$
Character orbit 420.w
Analytic conductor $3.354$
Analytic rank $0$
Dimension $8$
CM discriminant -84
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.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.12745506816.1
Defining polynomial: \(x^{8} + 23 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} ) q^{2} + \beta_{1} q^{3} -2 \beta_{2} q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{7} ) q^{6} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{7} + ( 2 + 2 \beta_{2} ) q^{8} + 3 \beta_{2} q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} ) q^{2} + \beta_{1} q^{3} -2 \beta_{2} q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{7} ) q^{6} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{7} + ( 2 + 2 \beta_{2} ) q^{8} + 3 \beta_{2} q^{9} + ( \beta_{3} + \beta_{4} + \beta_{7} ) q^{10} + ( \beta_{5} - \beta_{6} ) q^{11} + 2 \beta_{7} q^{12} + ( -\beta_{1} - 2 \beta_{3} + \beta_{7} ) q^{14} + ( -1 - \beta_{2} + \beta_{6} ) q^{15} -4 q^{16} + ( \beta_{1} + \beta_{3} - \beta_{4} - 4 \beta_{7} ) q^{17} + ( -3 - 3 \beta_{2} ) q^{18} + ( \beta_{1} - 2 \beta_{3} - \beta_{7} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{7} ) q^{20} + ( -1 + \beta_{5} - \beta_{6} ) q^{21} + ( 2 \beta_{2} + 2 \beta_{6} ) q^{22} + ( 1 - \beta_{2} - 2 \beta_{5} ) q^{23} + ( 2 \beta_{1} - 2 \beta_{7} ) q^{24} + ( -2 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{25} -3 \beta_{7} q^{27} + ( 2 \beta_{3} + 2 \beta_{4} ) q^{28} + ( 3 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{30} + ( \beta_{1} - 4 \beta_{4} - 3 \beta_{7} ) q^{31} + ( 4 - 4 \beta_{2} ) q^{32} + ( \beta_{1} - 3 \beta_{3} - 3 \beta_{4} ) q^{33} + ( -4 \beta_{1} - 2 \beta_{3} + 4 \beta_{7} ) q^{34} + ( 4 + 4 \beta_{2} + \beta_{6} ) q^{35} + 6 q^{36} + ( 5 - 3 \beta_{2} + 2 \beta_{6} ) q^{37} + ( -4 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{38} + ( -4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{40} + ( -\beta_{1} + 2 \beta_{4} + \beta_{7} ) q^{41} + ( 1 + \beta_{2} + 2 \beta_{6} ) q^{42} + ( -4 \beta_{2} - 2 \beta_{5} - 2 \beta_{6} ) q^{44} + ( -3 \beta_{1} + 3 \beta_{4} + 3 \beta_{7} ) q^{45} + ( -2 + 2 \beta_{5} - 2 \beta_{6} ) q^{46} -4 \beta_{1} q^{48} -7 \beta_{2} q^{49} + ( 1 - \beta_{2} + 2 \beta_{5} ) q^{50} + ( -10 + \beta_{5} - \beta_{6} ) q^{51} + ( -3 \beta_{1} + 3 \beta_{7} ) q^{54} + ( \beta_{1} - \beta_{3} + 3 \beta_{4} - 2 \beta_{7} ) q^{55} + ( 2 \beta_{1} - 4 \beta_{4} - 2 \beta_{7} ) q^{56} + ( -5 + 7 \beta_{2} + 2 \beta_{6} ) q^{57} + ( -4 + 4 \beta_{2} + 2 \beta_{5} ) q^{60} + ( -4 \beta_{1} - 4 \beta_{3} + 4 \beta_{4} + 6 \beta_{7} ) q^{62} + ( -3 \beta_{3} - 3 \beta_{4} ) q^{63} + 8 \beta_{2} q^{64} + ( -4 \beta_{1} + 6 \beta_{4} + 2 \beta_{7} ) q^{66} + ( 6 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{68} + ( 3 \beta_{1} + 6 \beta_{3} - 3 \beta_{7} ) q^{69} + ( -7 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{70} + ( 10 + \beta_{5} - \beta_{6} ) q^{71} + ( -6 + 6 \beta_{2} ) q^{72} + ( 6 \beta_{2} - 2 \beta_{5} - 2 \beta_{6} ) q^{74} + ( \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{7} ) q^{75} + ( 6 \beta_{1} - 4 \beta_{4} + 2 \beta_{7} ) q^{76} + ( \beta_{1} + \beta_{3} - \beta_{4} + 6 \beta_{7} ) q^{77} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{80} -9 q^{81} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{82} + ( -2 \beta_{2} - 2 \beta_{5} - 2 \beta_{6} ) q^{84} + ( 10 - 2 \beta_{2} - 3 \beta_{5} + \beta_{6} ) q^{85} + ( 4 \beta_{2} + 4 \beta_{5} ) q^{88} + ( -\beta_{1} + \beta_{7} ) q^{89} + ( 6 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{7} ) q^{90} + ( 2 + 2 \beta_{2} + 4 \beta_{6} ) q^{92} + ( -5 - \beta_{2} + 4 \beta_{5} ) q^{93} + ( -5 - 7 \beta_{2} - 3 \beta_{5} + \beta_{6} ) q^{95} + ( 4 \beta_{1} + 4 \beta_{7} ) q^{96} + ( 7 + 7 \beta_{2} ) q^{98} + ( 6 \beta_{2} + 3 \beta_{5} + 3 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 16 q^{8} + O(q^{10}) \) \( 8 q - 8 q^{2} + 16 q^{8} + 8 q^{11} - 12 q^{15} - 32 q^{16} - 24 q^{18} - 8 q^{22} - 16 q^{25} + 24 q^{30} + 32 q^{32} + 28 q^{35} + 48 q^{36} + 32 q^{37} + 16 q^{50} - 72 q^{51} - 48 q^{57} - 24 q^{60} - 56 q^{70} + 88 q^{71} - 48 q^{72} - 72 q^{81} + 64 q^{85} + 16 q^{88} - 24 q^{93} - 56 q^{95} + 56 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 23 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 19 \nu \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 24 \nu^{2} \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 24 \nu^{3} + 5 \nu \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} - 24 \nu^{3} + 24 \nu \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{6} - \nu^{4} - 67 \nu^{2} - 9 \)\()/5\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{6} + \nu^{4} - 67 \nu^{2} + 9 \)\()/5\)
\(\beta_{7}\)\(=\)\((\)\( -4 \nu^{7} - 91 \nu^{3} \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + \beta_{5} + 6 \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{7} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{6} - 5 \beta_{5} - 18\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-19 \beta_{4} - 19 \beta_{3} + 29 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(-12 \beta_{6} - 12 \beta_{5} - 67 \beta_{2}\)
\(\nu^{7}\)\(=\)\((\)\(-48 \beta_{7} + 91 \beta_{4} - 91 \beta_{3} - 91 \beta_{1}\)\()/2\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
1.54779 + 1.54779i
−0.323042 0.323042i
0.323042 + 0.323042i
−1.54779 1.54779i
−0.323042 + 0.323042i
1.54779 1.54779i
−1.54779 + 1.54779i
0.323042 0.323042i
−1.00000 + 1.00000i −1.22474 1.22474i 2.00000i 1.22474 1.87083i 2.44949 −1.87083 + 1.87083i 2.00000 + 2.00000i 3.00000i 0.646084 + 3.09557i
83.2 −1.00000 + 1.00000i −1.22474 1.22474i 2.00000i 1.22474 + 1.87083i 2.44949 1.87083 1.87083i 2.00000 + 2.00000i 3.00000i −3.09557 0.646084i
83.3 −1.00000 + 1.00000i 1.22474 + 1.22474i 2.00000i −1.22474 1.87083i −2.44949 −1.87083 + 1.87083i 2.00000 + 2.00000i 3.00000i 3.09557 + 0.646084i
83.4 −1.00000 + 1.00000i 1.22474 + 1.22474i 2.00000i −1.22474 + 1.87083i −2.44949 1.87083 1.87083i 2.00000 + 2.00000i 3.00000i −0.646084 3.09557i
167.1 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 1.22474 1.87083i 2.44949 1.87083 + 1.87083i 2.00000 2.00000i 3.00000i −3.09557 + 0.646084i
167.2 −1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 1.22474 + 1.87083i 2.44949 −1.87083 1.87083i 2.00000 2.00000i 3.00000i 0.646084 3.09557i
167.3 −1.00000 1.00000i 1.22474 1.22474i 2.00000i −1.22474 1.87083i −2.44949 1.87083 + 1.87083i 2.00000 2.00000i 3.00000i −0.646084 + 3.09557i
167.4 −1.00000 1.00000i 1.22474 1.22474i 2.00000i −1.22474 + 1.87083i −2.44949 −1.87083 1.87083i 2.00000 2.00000i 3.00000i 3.09557 0.646084i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 167.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
5.c odd 4 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
35.f even 4 1 inner
60.l odd 4 1 inner
420.w even 4 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} - 2 T_{11} - 20 \) acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + 2 T + T^{2} )^{4} \)
$3$ \( ( 9 + T^{4} )^{2} \)
$5$ \( ( 25 + 4 T^{2} + T^{4} )^{2} \)
$7$ \( ( 49 + T^{4} )^{2} \)
$11$ \( ( -20 - 2 T + T^{2} )^{4} \)
$13$ \( T^{8} \)
$17$ \( 160000 + 3824 T^{4} + T^{8} \)
$19$ \( ( 100 + 76 T^{2} + T^{4} )^{2} \)
$23$ \( ( 1764 + T^{4} )^{2} \)
$29$ \( T^{8} \)
$31$ \( ( 2500 - 124 T^{2} + T^{4} )^{2} \)
$37$ \( ( 100 + 160 T + 128 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$41$ \( ( -14 + T^{2} )^{4} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( T^{8} \)
$71$ \( ( 100 - 22 T + T^{2} )^{4} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( 6 + T^{2} )^{4} \)
$97$ \( T^{8} \)
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