Properties

Label 57.2.d.a
Level $57$
Weight $2$
Character orbit 57.d
Analytic conductor $0.455$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 57.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
Defining polynomial: \( x^{4} + 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 9 \) : Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1
−0.707107 + 1.58114i
−0.707107 1.58114i
0.707107 + 1.58114i
0.707107 1.58114i
−1.41421 0.707107 1.58114i 0 2.23607i −1.00000 + 2.23607i 1.00000 2.82843 −2.00000 2.23607i 3.16228i
56.2 −1.41421 0.707107 + 1.58114i 0 2.23607i −1.00000 2.23607i 1.00000 2.82843 −2.00000 + 2.23607i 3.16228i
56.3 1.41421 −0.707107 1.58114i 0 2.23607i −1.00000 2.23607i 1.00000 −2.82843 −2.00000 + 2.23607i 3.16228i
56.4 1.41421 −0.707107 + 1.58114i 0 2.23607i −1.00000 + 2.23607i 1.00000 −2.82843 −2.00000 2.23607i 3.16228i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.b odd 2 1 inner
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.d.a 4
3.b odd 2 1 inner 57.2.d.a 4
4.b odd 2 1 912.2.f.f 4
12.b even 2 1 912.2.f.f 4
19.b odd 2 1 inner 57.2.d.a 4
57.d even 2 1 inner 57.2.d.a 4
76.d even 2 1 912.2.f.f 4
228.b odd 2 1 912.2.f.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.d.a 4 1.a even 1 1 trivial
57.2.d.a 4 3.b odd 2 1 inner
57.2.d.a 4 19.b odd 2 1 inner
57.2.d.a 4 57.d even 2 1 inner
912.2.f.f 4 4.b odd 2 1
912.2.f.f 4 12.b even 2 1
912.2.f.f 4 76.d even 2 1
912.2.f.f 4 228.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(57, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 4T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 90)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$43$ \( (T + 5)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$73$ \( (T + 3)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 160)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
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