Properties

Label 380.2.c.a
Level $380$
Weight $2$
Character orbit 380.c
Analytic conductor $3.034$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{2} q^{5} + ( - \beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{2} q^{5} + ( - \beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9} + (\beta_{2} - 3) q^{11} + ( - \beta_{3} - 2 \beta_1) q^{13} + (\beta_{3} - 2 \beta_1) q^{15} + 2 \beta_{3} q^{17} + q^{19} + (2 \beta_{2} - 2) q^{21} + ( - \beta_{3} + 3 \beta_1) q^{23} + 5 q^{25} + (\beta_{3} + \beta_1) q^{27} + 2 \beta_{2} q^{29} - 4 q^{31} + (\beta_{3} - 5 \beta_1) q^{33} + ( - \beta_{3} - 3 \beta_1) q^{35} + 3 \beta_1 q^{37} + ( - \beta_{2} + 7) q^{39} - 6 q^{41} + (3 \beta_{3} - 3 \beta_1) q^{43} + 5 q^{45} + ( - 3 \beta_{3} + 3 \beta_1) q^{47} - q^{49} + ( - 2 \beta_{2} - 2) q^{51} + 3 \beta_1 q^{53} + ( - 3 \beta_{2} + 5) q^{55} + \beta_1 q^{57} + ( - 2 \beta_{2} + 6) q^{59} + ( - 3 \beta_{2} - 5) q^{61} + ( - \beta_{3} - 3 \beta_1) q^{63} + ( - 4 \beta_{3} + 3 \beta_1) q^{65} - 3 \beta_{3} q^{67} + (4 \beta_{2} - 8) q^{69} + ( - 2 \beta_{2} - 6) q^{71} + 6 \beta_1 q^{73} + 5 \beta_1 q^{75} + (2 \beta_{3} - 6 \beta_1) q^{77} + ( - 6 \beta_{2} - 2) q^{79} + (3 \beta_{2} - 4) q^{81} + (3 \beta_{3} + 3 \beta_1) q^{83} + (4 \beta_{3} + 2 \beta_1) q^{85} + (2 \beta_{3} - 4 \beta_1) q^{87} + ( - 4 \beta_{2} + 6) q^{89} + ( - 6 \beta_{2} - 2) q^{91} - 4 \beta_1 q^{93} + \beta_{2} q^{95} + ( - 4 \beta_{3} + \beta_1) q^{97} + ( - 3 \beta_{2} + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{11} + 4 q^{19} - 8 q^{21} + 20 q^{25} - 16 q^{31} + 28 q^{39} - 24 q^{41} + 20 q^{45} - 4 q^{49} - 8 q^{51} + 20 q^{55} + 24 q^{59} - 20 q^{61} - 32 q^{69} - 24 q^{71} - 8 q^{79} - 16 q^{81} + 24 q^{89} - 8 q^{91} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 5\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
2.28825i
0.874032i
0.874032i
2.28825i
0 2.28825i 0 −2.23607 0 2.82843i 0 −2.23607 0
229.2 0 0.874032i 0 2.23607 0 2.82843i 0 2.23607 0
229.3 0 0.874032i 0 2.23607 0 2.82843i 0 2.23607 0
229.4 0 2.28825i 0 −2.23607 0 2.82843i 0 −2.23607 0
\(n\): e.g. 2-40 or 990-1000
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 380.2.c.a 4
3.b odd 2 1 3420.2.f.a 4
4.b odd 2 1 1520.2.d.f 4
5.b even 2 1 inner 380.2.c.a 4
5.c odd 4 2 1900.2.a.j 4
15.d odd 2 1 3420.2.f.a 4
20.d odd 2 1 1520.2.d.f 4
20.e even 4 2 7600.2.a.ce 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.c.a 4 1.a even 1 1 trivial
380.2.c.a 4 5.b even 2 1 inner
1520.2.d.f 4 4.b odd 2 1
1520.2.d.f 4 20.d odd 2 1
1900.2.a.j 4 5.c odd 4 2
3420.2.f.a 4 3.b odd 2 1
3420.2.f.a 4 15.d odd 2 1
7600.2.a.ce 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 6T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 6T^{2} + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 46T^{2} + 484 \) Copy content Toggle raw display
$17$ \( T^{4} + 56T^{2} + 64 \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 56T^{2} + 64 \) Copy content Toggle raw display
$29$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 54T^{2} + 324 \) Copy content Toggle raw display
$41$ \( (T + 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 54T^{2} + 324 \) Copy content Toggle raw display
$59$ \( (T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T - 20)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 126T^{2} + 324 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 216T^{2} + 5184 \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 176)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 216T^{2} + 5184 \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T - 44)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 214T^{2} + 3844 \) Copy content Toggle raw display
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