Properties

Label 570.2.i.i
Level $570$
Weight $2$
Character orbit 570.i
Analytic conductor $4.551$
Analytic rank $0$
Dimension $4$
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.32288 + 2.29129i
−1.32288 2.29129i
1.32288 2.29129i
−1.32288 + 2.29129i
0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −2.64575 −1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
121.2 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 2.64575 −1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
391.1 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i −2.64575 −1.00000 −0.500000 0.866025i −0.500000 0.866025i
391.2 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 2.64575 −1.00000 −0.500000 0.866025i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.i.i 4
3.b odd 2 1 1710.2.l.j 4
19.c even 3 1 inner 570.2.i.i 4
57.h odd 6 1 1710.2.l.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.i 4 1.a even 1 1 trivial
570.2.i.i 4 19.c even 3 1 inner
1710.2.l.j 4 3.b odd 2 1
1710.2.l.j 4 57.h odd 6 1

Hecke kernels

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

\( T_{7}^{2} - 7 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 27)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 34 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + 67 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36 \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T - 19)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + 115 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4 T^{3} + 124 T^{2} + \cdots + 11664 \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} + 19 T^{2} + 12 T + 9 \) Copy content Toggle raw display
$59$ \( T^{4} + 28T^{2} + 784 \) Copy content Toggle raw display
$61$ \( T^{4} - 2 T^{3} + 66 T^{2} + \cdots + 3844 \) Copy content Toggle raw display
$67$ \( T^{4} - 14 T^{3} + 154 T^{2} + \cdots + 1764 \) Copy content Toggle raw display
$71$ \( T^{4} - 22 T^{3} + 370 T^{2} + \cdots + 12996 \) Copy content Toggle raw display
$73$ \( T^{4} + 22 T^{3} + 370 T^{2} + \cdots + 12996 \) Copy content Toggle raw display
$79$ \( T^{4} + 28T^{2} + 784 \) Copy content Toggle raw display
$83$ \( (T^{2} - 12 T - 76)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 175 T^{2} + 30625 \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36 \) Copy content Toggle raw display
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