Properties

Label 570.2.i.h
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: \(\Q(\sqrt{-3}, \sqrt{19})\)
Defining polynomial: \( x^{4} + 19x^{2} + 361 \) 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} - 3 q^{11} + q^{12} - 4 \beta_{2} q^{13} - \beta_1 q^{14} - \beta_{2} q^{15} + ( - \beta_{2} - 1) q^{16} + (\beta_{2} - \beta_1 + 1) q^{17} - q^{18} - \beta_1 q^{19} - q^{20} + \beta_1 q^{21} + ( - 3 \beta_{2} - 3) q^{22} - 3 \beta_{2} q^{23} + (\beta_{2} + 1) q^{24} + \beta_{2} q^{25} + 4 q^{26} + q^{27} + ( - \beta_{3} - \beta_1) q^{28} + (\beta_{3} + 5 \beta_{2} + \beta_1) q^{29} + q^{30} + (2 \beta_{3} - 2) q^{31} - \beta_{2} q^{32} + (3 \beta_{2} + 3) q^{33} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{34} - \beta_1 q^{35} + ( - \beta_{2} - 1) q^{36} - 7 q^{37} + ( - \beta_{3} - \beta_1) q^{38} - 4 q^{39} + ( - \beta_{2} - 1) q^{40} + (2 \beta_{2} + \beta_1 + 2) q^{41} + (\beta_{3} + \beta_1) q^{42} + (10 \beta_{2} + 10) q^{43} - 3 \beta_{2} q^{44} - q^{45} + 3 q^{46} - 6 \beta_{2} q^{47} + \beta_{2} q^{48} + 12 q^{49} - q^{50} + (\beta_{3} - \beta_{2} + \beta_1) q^{51} + (4 \beta_{2} + 4) q^{52} + ( - 3 \beta_{3} - 3 \beta_1) q^{53} + (\beta_{2} + 1) q^{54} + ( - 3 \beta_{2} - 3) q^{55} - \beta_{3} q^{56} + (\beta_{3} + \beta_1) q^{57} + (\beta_{3} - 5) q^{58} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{59} + (\beta_{2} + 1) q^{60} + (\beta_{3} + \beta_{2} + \beta_1) q^{61} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{62} + ( - \beta_{3} - \beta_1) q^{63} + q^{64} + 4 q^{65} + 3 \beta_{2} q^{66} + (\beta_{3} + 7 \beta_{2} + \beta_1) q^{67} + ( - \beta_{3} - 1) q^{68} - 3 q^{69} + ( - \beta_{3} - \beta_1) q^{70} + ( - 3 \beta_{2} + 3 \beta_1 - 3) q^{71} - \beta_{2} q^{72} + ( - 9 \beta_{2} + \beta_1 - 9) q^{73} + ( - 7 \beta_{2} - 7) q^{74} + q^{75} - \beta_{3} q^{76} - 3 \beta_{3} q^{77} + ( - 4 \beta_{2} - 4) q^{78} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{79} - \beta_{2} q^{80} + ( - \beta_{2} - 1) q^{81} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{82} - 6 q^{83} + \beta_{3} q^{84} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{85} + 10 \beta_{2} q^{86} + ( - \beta_{3} + 5) q^{87} + 3 q^{88} + (\beta_{3} - 4 \beta_{2} + \beta_1) q^{89} + ( - \beta_{2} - 1) q^{90} + (4 \beta_{3} + 4 \beta_1) q^{91} + (3 \beta_{2} + 3) q^{92} + (2 \beta_{2} + 2 \beta_1 + 2) q^{93} + 6 q^{94} + ( - \beta_{3} - \beta_1) q^{95} - q^{96} + (\beta_{2} + 3 \beta_1 + 1) q^{97} + (12 \beta_{2} + 12) q^{98} - 3 \beta_{2} 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 \( 4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} + 2 q^{6} - 4 q^{8} - 2 q^{9} - 2 q^{10} - 12 q^{11} + 4 q^{12} + 8 q^{13} + 2 q^{15} - 2 q^{16} + 2 q^{17} - 4 q^{18} - 4 q^{20} - 6 q^{22} + 6 q^{23} + 2 q^{24} - 2 q^{25} + 16 q^{26} + 4 q^{27} - 10 q^{29} + 4 q^{30} - 8 q^{31} + 2 q^{32} + 6 q^{33} - 2 q^{34} - 2 q^{36} - 28 q^{37} - 16 q^{39} - 2 q^{40} + 4 q^{41} + 20 q^{43} + 6 q^{44} - 4 q^{45} + 12 q^{46} + 12 q^{47} - 2 q^{48} + 48 q^{49} - 4 q^{50} + 2 q^{51} + 8 q^{52} + 2 q^{54} - 6 q^{55} - 20 q^{58} - 8 q^{59} + 2 q^{60} - 2 q^{61} - 4 q^{62} + 4 q^{64} + 16 q^{65} - 6 q^{66} - 14 q^{67} - 4 q^{68} - 12 q^{69} - 6 q^{71} + 2 q^{72} - 18 q^{73} - 14 q^{74} + 4 q^{75} - 8 q^{78} - 8 q^{79} + 2 q^{80} - 2 q^{81} - 4 q^{82} - 24 q^{83} - 2 q^{85} - 20 q^{86} + 20 q^{87} + 12 q^{88} + 8 q^{89} - 2 q^{90} + 6 q^{92} + 4 q^{93} + 24 q^{94} - 4 q^{96} + 2 q^{97} + 24 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 19x^{2} + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 19 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 19\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 19\beta_{3} \) 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
2.17945 + 3.77492i
−2.17945 3.77492i
2.17945 3.77492i
−2.17945 + 3.77492i
0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i −4.35890 −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 4.35890 −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 −4.35890 −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 4.35890 −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.h 4
3.b odd 2 1 1710.2.l.k 4
19.c even 3 1 inner 570.2.i.h 4
57.h odd 6 1 1710.2.l.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.h 4 1.a even 1 1 trivial
570.2.i.h 4 19.c even 3 1 inner
1710.2.l.k 4 3.b odd 2 1
1710.2.l.k 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 \(S_{2}^{\mathrm{new}}(570, [\chi])\):

\( T_{7}^{2} - 19 \) 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
$7$ \( (T^{2} - 19)^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + 22 T^{2} + 36 T + 324 \) Copy content Toggle raw display
$19$ \( T^{4} + 19T^{2} + 361 \) Copy content Toggle raw display
$23$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{3} + 94 T^{2} + 60 T + 36 \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 72)^{2} \) Copy content Toggle raw display
$37$ \( (T + 7)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 4 T^{3} + 31 T^{2} + 60 T + 225 \) Copy content Toggle raw display
$43$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 171 T^{2} + 29241 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} + 124 T^{2} + \cdots + 3600 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + 22 T^{2} - 36 T + 324 \) Copy content Toggle raw display
$67$ \( T^{4} + 14 T^{3} + 166 T^{2} + \cdots + 900 \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} + 198 T^{2} + \cdots + 26244 \) Copy content Toggle raw display
$73$ \( T^{4} + 18 T^{3} + 262 T^{2} + \cdots + 3844 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} + 124 T^{2} + \cdots + 3600 \) Copy content Toggle raw display
$83$ \( (T + 6)^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 8 T^{3} + 67 T^{2} + 24 T + 9 \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} + 174 T^{2} + \cdots + 28900 \) Copy content Toggle raw display
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