Properties

Label 570.2.d.d
Level $570$
Weight $2$
Character orbit 570.d
Analytic conductor $4.551$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{1}\)\(=\) \( ( -7\nu^{5} + 10\nu^{4} - 5\nu^{3} - 30\nu^{2} - 32\nu + 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{5} + 3\nu^{4} + 10\nu^{3} - 32\nu^{2} - 74\nu - 3 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -10\nu^{5} + 11\nu^{4} - 17\nu^{3} - 10\nu^{2} - 72\nu - 11 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{5} - 27\nu^{4} + 25\nu^{3} + 12\nu^{2} + 68\nu - 65 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -19\nu^{5} + 37\nu^{4} - 30\nu^{3} - 42\nu^{2} - 54\nu + 55 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{2} - 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} - 3\beta_{3} + 3\beta_{2} - 4\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} - 5\beta_{4} - 5\beta_{3} + \beta_{2} - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{5} - 11\beta_{4} - 5\beta_{3} - 5\beta_{2} + 18\beta _1 - 18 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(211\) \(457\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1
0.403032 + 0.403032i
1.45161 + 1.45161i
−0.854638 0.854638i
0.403032 0.403032i
1.45161 1.45161i
−0.854638 + 0.854638i
1.00000i 1.00000i −1.00000 −1.48119 + 1.67513i 1.00000 3.35026i 1.00000i −1.00000 1.67513 + 1.48119i
229.2 1.00000i 1.00000i −1.00000 0.311108 2.21432i 1.00000 4.42864i 1.00000i −1.00000 −2.21432 0.311108i
229.3 1.00000i 1.00000i −1.00000 2.17009 + 0.539189i 1.00000 1.07838i 1.00000i −1.00000 0.539189 2.17009i
229.4 1.00000i 1.00000i −1.00000 −1.48119 1.67513i 1.00000 3.35026i 1.00000i −1.00000 1.67513 1.48119i
229.5 1.00000i 1.00000i −1.00000 0.311108 + 2.21432i 1.00000 4.42864i 1.00000i −1.00000 −2.21432 + 0.311108i
229.6 1.00000i 1.00000i −1.00000 2.17009 0.539189i 1.00000 1.07838i 1.00000i −1.00000 0.539189 + 2.17009i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.d.d 6
3.b odd 2 1 1710.2.d.e 6
5.b even 2 1 inner 570.2.d.d 6
5.c odd 4 1 2850.2.a.bk 3
5.c odd 4 1 2850.2.a.bn 3
15.d odd 2 1 1710.2.d.e 6
15.e even 4 1 8550.2.a.cf 3
15.e even 4 1 8550.2.a.cr 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.d.d 6 1.a even 1 1 trivial
570.2.d.d 6 5.b even 2 1 inner
1710.2.d.e 6 3.b odd 2 1
1710.2.d.e 6 15.d odd 2 1
2850.2.a.bk 3 5.c odd 4 1
2850.2.a.bn 3 5.c odd 4 1
8550.2.a.cf 3 15.e even 4 1
8550.2.a.cr 3 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7}^{6} + 32T_{7}^{4} + 256T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{3} + 4T_{11}^{2} - 16T_{11} - 32 \) Copy content Toggle raw display
\( T_{13}^{6} + 44T_{13}^{4} + 112T_{13}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + 3 T^{4} - 12 T^{3} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{3} + 4 T^{2} - 16 T - 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 44 T^{4} + 112 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{6} + 60 T^{4} + 560 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 44 T^{4} + 112 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( (T^{3} + 10 T^{2} + 12 T - 40)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 14 T^{2} + 28 T + 152)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 172 T^{4} + 6128 T^{2} + \cdots + 53824 \) Copy content Toggle raw display
$41$ \( (T^{3} - 16 T - 16)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 108 T^{4} + 176 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( T^{6} + 108 T^{4} + 3440 T^{2} + \cdots + 33856 \) Copy content Toggle raw display
$53$ \( T^{6} + 172 T^{4} + 4400 T^{2} + \cdots + 23104 \) Copy content Toggle raw display
$59$ \( (T^{3} + 14 T^{2} + 52 T + 40)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{6} \) Copy content Toggle raw display
$67$ \( T^{6} + 128 T^{4} + 3072 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$71$ \( (T^{3} + 4 T^{2} - 80 T - 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 192 T^{4} + 8192 T^{2} + \cdots + 65536 \) Copy content Toggle raw display
$79$ \( (T^{3} + 22 T^{2} + 124 T + 200)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 240 T^{4} + 14400 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( (T^{3} + 4 T^{2} - 48 T + 80)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 396 T^{4} + 34992 T^{2} + \cdots + 46656 \) Copy content Toggle raw display
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