Properties

Label 570.2.i.f
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: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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.

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

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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 4\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
−0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0.500000 0.866025i −1.44949 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 3.44949 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 −1.44949 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 3.44949 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.f 4
3.b odd 2 1 1710.2.l.n 4
19.c even 3 1 inner 570.2.i.f 4
57.h odd 6 1 1710.2.l.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.f 4 1.a even 1 1 trivial
570.2.i.f 4 19.c even 3 1 inner
1710.2.l.n 4 3.b odd 2 1
1710.2.l.n 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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7}^{2} - 2T_{7} - 5 \) Copy content Toggle raw display
\( T_{11} + 3 \) 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
$7$ \( (T^{2} - 2 T - 5)^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + 54 T^{2} - 80 T + 100 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} - 15 T^{2} + 38 T + 361 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + 27 T^{2} + 46 T + 529 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} + 66 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 10 T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} + 81 T^{2} + \cdots + 2025 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} + 108 T^{2} + \cdots + 8464 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + 132 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$53$ \( T^{4} - 10 T^{3} + 81 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$59$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 20 T^{3} + 306 T^{2} + \cdots + 8836 \) Copy content Toggle raw display
$67$ \( T^{4} + 54T^{2} + 2916 \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} + 102 T^{2} + \cdots + 1444 \) Copy content Toggle raw display
$73$ \( T^{4} + 16 T^{3} + 198 T^{2} + \cdots + 3364 \) Copy content Toggle raw display
$79$ \( T^{4} + 4 T^{3} + 36 T^{2} - 80 T + 400 \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T - 92)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 6 T^{3} + 33 T^{2} - 18 T + 9 \) Copy content Toggle raw display
$97$ \( T^{4} - 32 T^{3} + 774 T^{2} + \cdots + 62500 \) Copy content Toggle raw display
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