Properties

Label 570.2.q.a
Level $570$
Weight $2$
Character orbit 570.q
Analytic conductor $4.551$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1) q^{2} + ( - \beta_{4} + \beta_1) q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{5} + ( - \beta_{2} + 1) q^{6} + 3 \beta_{4} q^{7} - \beta_{4} q^{8} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1) q^{2} + ( - \beta_{4} + \beta_1) q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{5} + ( - \beta_{2} + 1) q^{6} + 3 \beta_{4} q^{7} - \beta_{4} q^{8} + ( - \beta_{2} + 1) q^{9} + (\beta_{5} - \beta_{2} + \beta_1 + 1) q^{10} + 2 q^{11} - \beta_{4} q^{12} + (\beta_{6} + \beta_{5} - \beta_1) q^{13} + 3 \beta_{2} q^{14} + (\beta_{5} - \beta_{2} + \beta_1 + 1) q^{15} - \beta_{2} q^{16} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3}) q^{17} - \beta_{4} q^{18} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{3} + 1) q^{19} + ( - \beta_{4} + \beta_{3} + 1) q^{20} + 3 \beta_{2} q^{21} + ( - 2 \beta_{4} + 2 \beta_1) q^{22} + (\beta_{6} + \beta_{5}) q^{23} - \beta_{2} q^{24} + ( - 2 \beta_{6} + 2 \beta_{5} - \beta_1) q^{25} + (\beta_{7} - \beta_{5} + \beta_{3} - 1) q^{26} - \beta_{4} q^{27} + 3 \beta_1 q^{28} + (3 \beta_{6} - 3 \beta_{5} + 2 \beta_{2} - 2) q^{29} + ( - \beta_{4} + \beta_{3} + 1) q^{30} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 3) q^{31} - \beta_1 q^{32} + ( - 2 \beta_{4} + 2 \beta_1) q^{33} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{34} + ( - 3 \beta_{7} + 3 \beta_{4} + 3 \beta_{2} - 3 \beta_1) q^{35} - \beta_{2} q^{36} + (\beta_{7} - \beta_{5} - 7 \beta_{4} - \beta_{3}) q^{37} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{38} + (\beta_{7} - \beta_{5} + \beta_{3} - 1) q^{39} + (\beta_{7} - \beta_{4} - \beta_{2} + \beta_1) q^{40} + (\beta_{7} - \beta_{6} + \beta_{3}) q^{41} + 3 \beta_1 q^{42} + ( - \beta_{7} - \beta_{6} + 3 \beta_{4} + \beta_{3} - 3 \beta_1) q^{43} + ( - 2 \beta_{2} + 2) q^{44} + ( - \beta_{4} + \beta_{3} + 1) q^{45} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{46} + ( - \beta_{6} - \beta_{5} - 6 \beta_1) q^{47} - \beta_1 q^{48} - 2 q^{49} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} - 1) q^{50} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{51} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_1) q^{52} + ( - 2 \beta_{6} - 2 \beta_{5} - 8 \beta_1) q^{53} - \beta_{2} q^{54} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{55} + 3 q^{56} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{57} + (3 \beta_{7} - 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3}) q^{58} + ( - 4 \beta_{7} + 4 \beta_{6} - 4 \beta_{3} + 2 \beta_{2}) q^{59} + (\beta_{7} - \beta_{4} - \beta_{2} + \beta_1) q^{60} + (3 \beta_{6} - 3 \beta_{5} - 3 \beta_{2} + 3) q^{61} + (2 \beta_{7} + 2 \beta_{6} + 3 \beta_{4} - 2 \beta_{3} - 3 \beta_1) q^{62} + 3 \beta_1 q^{63} - q^{64} + (\beta_{7} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 4) q^{65} + ( - 2 \beta_{2} + 2) q^{66} + (\beta_{6} + \beta_{5} + \beta_1) q^{67} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3}) q^{68} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{69} + ( - 3 \beta_{6} + 3 \beta_{2} + 3 \beta_1 - 3) q^{70} + (\beta_{7} - \beta_{6} + \beta_{3} + 6 \beta_{2}) q^{71} - \beta_1 q^{72} + ( - 2 \beta_{7} - 2 \beta_{6} - 7 \beta_{4} + 2 \beta_{3} + 7 \beta_1) q^{73} + ( - \beta_{7} + \beta_{6} - \beta_{3} - 7 \beta_{2}) q^{74} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} - 1) q^{75} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_{2} + 1) q^{76} + 6 \beta_{4} q^{77} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_1) q^{78} + ( - 4 \beta_{7} + 4 \beta_{6} - 4 \beta_{3} + 7 \beta_{2}) q^{79} + (\beta_{6} - \beta_{2} - \beta_1 + 1) q^{80} - \beta_{2} q^{81} + (\beta_{6} + \beta_{5}) q^{82} + ( - 3 \beta_{7} + 3 \beta_{5} + 6 \beta_{4} + 3 \beta_{3}) q^{83} + 3 q^{84} + ( - 4 \beta_{6} - 6 \beta_{2} - 6 \beta_1 + 6) q^{85} + ( - \beta_{6} + \beta_{5} + 3 \beta_{2} - 3) q^{86} + (3 \beta_{7} - 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3}) q^{87} - 2 \beta_{4} q^{88} + ( - \beta_{6} + \beta_{5} + 14 \beta_{2} - 14) q^{89} + (\beta_{7} - \beta_{4} - \beta_{2} + \beta_1) q^{90} + ( - 3 \beta_{6} + 3 \beta_{5} - 3 \beta_{2} + 3) q^{91} + (\beta_{7} + \beta_{6} - \beta_{3}) q^{92} + (2 \beta_{7} + 2 \beta_{6} + 3 \beta_{4} - 2 \beta_{3} - 3 \beta_1) q^{93} + ( - \beta_{7} + \beta_{5} - \beta_{3} - 6) q^{94} + (2 \beta_{7} - \beta_{6} - 4 \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 6) q^{95} - q^{96} + (6 \beta_{7} + 6 \beta_{6} + 4 \beta_{4} - 6 \beta_{3} - 4 \beta_1) q^{97} + (2 \beta_{4} - 2 \beta_1) q^{98} + ( - 2 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{9} + 4 q^{10} + 16 q^{11} + 12 q^{14} + 4 q^{15} - 4 q^{16} + 8 q^{19} + 8 q^{20} + 12 q^{21} - 4 q^{24} - 8 q^{26} - 8 q^{29} + 8 q^{30} - 24 q^{31} + 12 q^{35} - 4 q^{36} - 8 q^{39} - 4 q^{40} + 8 q^{44} + 8 q^{45} - 16 q^{49} - 8 q^{50} - 4 q^{54} + 8 q^{55} + 24 q^{56} + 8 q^{59} - 4 q^{60} + 12 q^{61} - 8 q^{64} - 32 q^{65} + 8 q^{66} - 12 q^{70} + 24 q^{71} - 28 q^{74} - 8 q^{75} + 4 q^{76} + 28 q^{79} + 4 q^{80} - 4 q^{81} + 24 q^{84} + 24 q^{85} - 12 q^{86} - 56 q^{89} - 4 q^{90} + 12 q^{91} - 48 q^{94} + 40 q^{95} - 8 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −2.03906 0.917738i 0.500000 + 0.866025i 3.00000i 1.00000i 0.500000 + 0.866025i 1.30701 + 1.81431i
49.2 −0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 1.30701 1.81431i 0.500000 + 0.866025i 3.00000i 1.00000i 0.500000 + 0.866025i −2.03906 + 0.917738i
49.3 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0.917738 2.03906i 0.500000 + 0.866025i 3.00000i 1.00000i 0.500000 + 0.866025i 1.81431 1.30701i
49.4 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 1.81431 + 1.30701i 0.500000 + 0.866025i 3.00000i 1.00000i 0.500000 + 0.866025i 0.917738 + 2.03906i
349.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −2.03906 + 0.917738i 0.500000 0.866025i 3.00000i 1.00000i 0.500000 0.866025i 1.30701 1.81431i
349.2 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 1.30701 + 1.81431i 0.500000 0.866025i 3.00000i 1.00000i 0.500000 0.866025i −2.03906 0.917738i
349.3 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0.917738 + 2.03906i 0.500000 0.866025i 3.00000i 1.00000i 0.500000 0.866025i 1.81431 + 1.30701i
349.4 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 1.81431 1.30701i 0.500000 0.866025i 3.00000i 1.00000i 0.500000 0.866025i 0.917738 2.03906i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.q.a 8
3.b odd 2 1 1710.2.t.a 8
5.b even 2 1 inner 570.2.q.a 8
15.d odd 2 1 1710.2.t.a 8
19.c even 3 1 inner 570.2.q.a 8
57.h odd 6 1 1710.2.t.a 8
95.i even 6 1 inner 570.2.q.a 8
285.n odd 6 1 1710.2.t.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.q.a 8 1.a even 1 1 trivial
570.2.q.a 8 5.b even 2 1 inner
570.2.q.a 8 19.c even 3 1 inner
570.2.q.a 8 95.i even 6 1 inner
1710.2.t.a 8 3.b odd 2 1
1710.2.t.a 8 15.d odd 2 1
1710.2.t.a 8 57.h odd 6 1
1710.2.t.a 8 285.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + 8 T^{6} + 8 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$11$ \( (T - 2)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 14 T^{6} + 171 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$17$ \( (T^{4} - 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T + 19)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + 66 T^{2} - 200 T + 2500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T - 15)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 110 T^{2} + 1849)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 30 T^{6} + 891 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$47$ \( T^{8} - 84 T^{6} + 6156 T^{4} + \cdots + 810000 \) Copy content Toggle raw display
$53$ \( T^{8} - 176 T^{6} + 29376 T^{4} + \cdots + 2560000 \) Copy content Toggle raw display
$59$ \( (T^{4} - 4 T^{3} + 108 T^{2} + 368 T + 8464)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 6 T^{3} + 81 T^{2} + 270 T + 2025)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 14 T^{6} + 171 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$71$ \( (T^{4} - 12 T^{3} + 114 T^{2} - 360 T + 900)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 146 T^{6} + 20691 T^{4} + \cdots + 390625 \) Copy content Toggle raw display
$79$ \( (T^{4} - 14 T^{3} + 243 T^{2} + 658 T + 2209)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 180 T^{2} + 324)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 28 T^{3} + 594 T^{2} + \cdots + 36100)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 464 T^{6} + \cdots + 1600000000 \) Copy content Toggle raw display
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