Properties

Label 420.2.l.f
Level $420$
Weight $2$
Character orbit 420.l
Analytic conductor $3.354$
Analytic rank $0$
Dimension $8$
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.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.386672896.3
Defining polynomial: \( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{2} q^{3} + ( - \beta_{6} - \beta_{4} + 1) q^{4} + (\beta_{7} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{5} + (\beta_{5} + \beta_1 + 1) q^{6} + q^{7} + (\beta_{7} - \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{8} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{2} q^{3} + ( - \beta_{6} - \beta_{4} + 1) q^{4} + (\beta_{7} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{5} + (\beta_{5} + \beta_1 + 1) q^{6} + q^{7} + (\beta_{7} - \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{8} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_1) q^{9} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - \beta_1) q^{10} + (\beta_{6} - \beta_{2}) q^{11} + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{12} + (\beta_{6} - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{13} + \beta_{3} q^{14} + ( - \beta_{7} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{15} + ( - \beta_{6} + \beta_{4} + 2 \beta_1 - 1) q^{16} + ( - \beta_{6} + 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{17} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 2) q^{18} + ( - 2 \beta_{7} - 2 \beta_{4}) q^{19} + (\beta_{7} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 3) q^{20} - \beta_{2} q^{21} + ( - \beta_{7} + \beta_{5} + \beta_{2} + \beta_1 + 1) q^{22} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{2}) q^{23} + (\beta_{6} - \beta_{4} - 2 \beta_{3} + 2 \beta_1 + 1) q^{24} + (2 \beta_{7} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 3) q^{25} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1 + 1) q^{26} + ( - \beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{27} + ( - \beta_{6} - \beta_{4} + 1) q^{28} + ( - \beta_{6} + 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{29} + (\beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{30} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{31} + (\beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 - 3) q^{32} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_1 + 3) q^{33} + (\beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_1 + 5) q^{34} + (\beta_{7} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{35} + (\beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 3) q^{36} + (4 \beta_{7} + 2 \beta_{3} + 2 \beta_1 + 2) q^{37} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{38} + (\beta_{7} - 3 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_1 + 3) q^{39} + ( - 2 \beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 3) q^{40} + (2 \beta_{7} - 2 \beta_{4} + 2) q^{41} + (\beta_{5} + \beta_1 + 1) q^{42} + (2 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{43} + ( - 2 \beta_{4} - 2 \beta_1 - 2) q^{44} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{45}+ \cdots + ( - \beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 2 q^{4} + 8 q^{5} + 4 q^{6} + 8 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} + 2 q^{4} + 8 q^{5} + 4 q^{6} + 8 q^{7} - 6 q^{8} + 2 q^{9} - 4 q^{10} + 4 q^{11} - 14 q^{12} - 6 q^{15} - 6 q^{16} - 4 q^{17} - 6 q^{18} + 10 q^{20} + 2 q^{21} + 6 q^{22} + 6 q^{24} - 24 q^{25} + 26 q^{26} + 8 q^{27} + 2 q^{28} - 16 q^{30} - 30 q^{32} + 26 q^{33} + 30 q^{34} + 8 q^{35} + 10 q^{36} - 20 q^{38} + 18 q^{39} - 14 q^{40} + 4 q^{42} - 8 q^{43} - 24 q^{44} - 14 q^{45} + 16 q^{46} - 38 q^{48} + 8 q^{49} - 8 q^{50} - 14 q^{51} + 16 q^{52} + 8 q^{54} + 4 q^{55} - 6 q^{56} + 20 q^{57} - 26 q^{58} + 8 q^{59} + 10 q^{60} - 16 q^{61} - 40 q^{62} + 2 q^{63} + 26 q^{64} + 32 q^{65} - 6 q^{66} - 24 q^{67} + 12 q^{68} + 24 q^{69} - 4 q^{70} - 64 q^{71} + 22 q^{72} - 4 q^{74} - 22 q^{75} - 28 q^{76} + 4 q^{77} + 42 q^{78} - 38 q^{80} + 2 q^{81} + 4 q^{82} - 14 q^{84} - 4 q^{85} - 24 q^{86} - 18 q^{87} + 24 q^{88} - 6 q^{90} + 36 q^{92} + 32 q^{93} - 2 q^{94} + 48 q^{95} - 14 q^{96} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{5} + 2\nu^{4} + 2\nu^{3} - 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - \nu^{5} - 2\nu^{4} + 2\nu^{3} - 4\nu^{2} - 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + \nu^{5} - 2\nu^{4} - 2\nu^{3} + 8\nu^{2} + 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - \nu^{4} + 2\nu^{3} + 2\nu^{2} - 4\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 2\nu^{6} - \nu^{5} - 2\nu^{3} - 4\nu^{2} - 12\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 3\nu^{5} + 2\nu^{3} - 4\nu^{2} - 12\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{5} - \beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{4} - 2\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{6} + 2\beta_{5} - \beta_{4} - 2\beta_{3} - \beta_{2} + 4\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} + \beta_{6} - 3\beta_{5} + 5\beta_{4} + 4\beta_{3} + 6\beta_{2} + 4\beta_1 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
1.40961 0.114062i
1.40961 + 0.114062i
0.621372 1.27039i
0.621372 + 1.27039i
−0.835949 1.14070i
−0.835949 + 1.14070i
−1.19503 0.756243i
−1.19503 + 0.756243i
−1.40961 0.114062i −1.47398 + 0.909606i 1.97398 + 0.321565i 1.00000 2.00000i 2.18148 1.11406i 1.00000 −2.74586 0.678435i 1.34523 2.68148i −1.63773 + 2.70515i
239.2 −1.40961 + 0.114062i −1.47398 0.909606i 1.97398 0.321565i 1.00000 + 2.00000i 2.18148 + 1.11406i 1.00000 −2.74586 + 0.678435i 1.34523 + 2.68148i −1.63773 2.70515i
239.3 −0.621372 1.27039i 1.72779 + 0.121372i −1.22779 + 1.57877i 1.00000 2.00000i −0.919412 2.27039i 1.00000 2.76858 + 0.578773i 2.97054 + 0.419412i −3.16216 0.0276478i
239.4 −0.621372 + 1.27039i 1.72779 0.121372i −1.22779 1.57877i 1.00000 + 2.00000i −0.919412 + 2.27039i 1.00000 2.76858 0.578773i 2.97054 0.419412i −3.16216 + 0.0276478i
239.5 0.835949 1.14070i 1.10238 + 1.33595i −0.602380 1.90713i 1.00000 + 2.00000i 2.44545 0.140697i 1.00000 −2.67901 0.907128i −0.569517 + 2.94545i 3.11734 + 0.531200i
239.6 0.835949 + 1.14070i 1.10238 1.33595i −0.602380 + 1.90713i 1.00000 2.00000i 2.44545 + 0.140697i 1.00000 −2.67901 + 0.907128i −0.569517 2.94545i 3.11734 0.531200i
239.7 1.19503 0.756243i −0.356193 1.69503i 0.856193 1.80747i 1.00000 2.00000i −1.70752 1.75624i 1.00000 −0.343707 2.80747i −2.74625 + 1.20752i −0.317456 3.14630i
239.8 1.19503 + 0.756243i −0.356193 + 1.69503i 0.856193 + 1.80747i 1.00000 + 2.00000i −1.70752 + 1.75624i 1.00000 −0.343707 + 2.80747i −2.74625 1.20752i −0.317456 + 3.14630i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
60.h 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.l.f yes 8
3.b odd 2 1 420.2.l.e yes 8
4.b odd 2 1 420.2.l.d yes 8
5.b even 2 1 420.2.l.c 8
12.b even 2 1 420.2.l.c 8
15.d odd 2 1 420.2.l.d yes 8
20.d odd 2 1 420.2.l.e yes 8
60.h even 2 1 inner 420.2.l.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.l.c 8 5.b even 2 1
420.2.l.c 8 12.b even 2 1
420.2.l.d yes 8 4.b odd 2 1
420.2.l.d yes 8 15.d odd 2 1
420.2.l.e yes 8 3.b odd 2 1
420.2.l.e yes 8 20.d odd 2 1
420.2.l.f yes 8 1.a even 1 1 trivial
420.2.l.f yes 8 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\):

\( T_{11}^{4} - 2T_{11}^{3} - 11T_{11}^{2} + 16T_{11} + 16 \) Copy content Toggle raw display
\( T_{17}^{4} + 2T_{17}^{3} - 35T_{17}^{2} - 52T_{17} + 100 \) Copy content Toggle raw display
\( T_{43}^{4} + 4T_{43}^{3} - 56T_{43}^{2} - 192T_{43} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{6} + 2 T^{5} + 2 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + T^{6} - 2 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T + 5)^{4} \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 11 T^{2} + 16 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 70 T^{6} + 1321 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$17$ \( (T^{4} + 2 T^{3} - 35 T^{2} - 52 T + 100)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 88 T^{6} + 1104 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{8} + 96 T^{6} + 2656 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$29$ \( T^{8} + 70 T^{6} + 1321 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$31$ \( T^{8} + 192 T^{6} + 11488 T^{4} + \cdots + 12544 \) Copy content Toggle raw display
$37$ \( T^{8} + 136 T^{6} + 3984 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( T^{8} + 104 T^{6} + 3472 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} - 56 T^{2} - 192 T - 128)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 22 T^{6} + 145 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( (T^{4} - 200 T^{2} - 256 T + 5392)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4 T^{3} - 140 T^{2} + 1152 T - 2240)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8 T^{3} - 148 T^{2} - 1232 T - 1312)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 12 T^{3} - 124 T^{2} - 2112 T - 6848)^{2} \) Copy content Toggle raw display
$71$ \( (T + 8)^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} + 502 T^{6} + 66289 T^{4} + \cdots + 753424 \) Copy content Toggle raw display
$83$ \( T^{8} + 432 T^{6} + \cdots + 29073664 \) Copy content Toggle raw display
$89$ \( T^{8} + 352 T^{6} + 26176 T^{4} + \cdots + 6553600 \) Copy content Toggle raw display
$97$ \( T^{8} + 694 T^{6} + \cdots + 16777216 \) Copy content Toggle raw display
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