Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.y (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{5} + ( -2 - 2 \zeta_{12}^{3} ) q^{7} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{5} + ( -2 - 2 \zeta_{12}^{3} ) q^{7} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{9} + q^{11} + ( 2 - \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{13} + ( -2 - 3 \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{15} + ( 1 + \zeta_{12} - \zeta_{12}^{2} ) q^{17} + ( 5 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{19} + ( -4 - 4 \zeta_{12}^{2} ) q^{21} + ( -4 \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{25} + ( 5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{29} + ( 1 - 2 \zeta_{12}^{2} ) q^{31} + ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{33} + ( -6 + 2 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{35} + ( -2 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{37} -6 \zeta_{12}^{3} q^{39} + ( 2 + 2 \zeta_{12}^{2} ) q^{41} + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{43} + ( -6 - 3 \zeta_{12}^{3} ) q^{45} + ( -3 \zeta_{12} + 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{47} + \zeta_{12}^{3} q^{49} + ( 4 - 2 \zeta_{12}^{2} ) q^{51} + ( 6 - 3 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{53} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{55} + ( 7 + 8 \zeta_{12} - 8 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{57} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{59} -13 \zeta_{12}^{2} q^{61} + ( -6 \zeta_{12} - 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{63} + ( -1 - 6 \zeta_{12} + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{65} + ( 10 + 5 \zeta_{12} - 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{67} + ( 7 + 7 \zeta_{12}^{2} ) q^{71} + ( 10 - 10 \zeta_{12} - 10 \zeta_{12}^{2} ) q^{73} + ( -7 - 2 \zeta_{12} + 14 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{75} + ( -2 - 2 \zeta_{12}^{3} ) q^{77} + ( -7 \zeta_{12} + 14 \zeta_{12}^{3} ) q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + ( -4 + 4 \zeta_{12}^{3} ) q^{83} + ( -\zeta_{12} - 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{85} + ( 15 + 15 \zeta_{12}^{3} ) q^{87} + ( 9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{89} + ( -8 + 4 \zeta_{12}^{2} ) q^{91} + ( 3 - 3 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{93} + ( -6 + 2 \zeta_{12} - 4 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{95} + ( -8 + 8 \zeta_{12} - 8 \zeta_{12}^{2} - 16 \zeta_{12}^{3} ) q^{97} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{3} + 2q^{5} - 8q^{7} + O(q^{10}) \) \( 4q + 6q^{3} + 2q^{5} - 8q^{7} + 4q^{11} + 6q^{13} - 6q^{15} + 2q^{17} - 24q^{21} + 6q^{25} + 6q^{33} - 12q^{35} + 12q^{41} - 8q^{43} - 24q^{45} + 6q^{47} + 12q^{51} + 18q^{53} + 2q^{55} + 12q^{57} - 26q^{61} - 12q^{63} + 30q^{67} + 42q^{71} + 20q^{73} - 8q^{77} - 18q^{81} - 16q^{83} - 6q^{85} + 60q^{87} - 24q^{91} + 6q^{93} - 32q^{95} - 48q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(\zeta_{12}^{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 2.36603 0.633975i 0 −1.23205 1.86603i 0 −2.00000 2.00000i 0 2.59808 1.50000i 0
293.1 0 0.633975 + 2.36603i 0 2.23205 + 0.133975i 0 −2.00000 + 2.00000i 0 −2.59808 + 1.50000i 0
297.1 0 0.633975 2.36603i 0 2.23205 0.133975i 0 −2.00000 2.00000i 0 −2.59808 1.50000i 0
373.1 0 2.36603 + 0.633975i 0 −1.23205 + 1.86603i 0 −2.00000 + 2.00000i 0 2.59808 + 1.50000i 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.c odd 4 1 inner
19.d odd 6 1 inner
95.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.y.a 4
5.c odd 4 1 inner 380.2.y.a 4
19.d odd 6 1 inner 380.2.y.a 4
95.l even 12 1 inner 380.2.y.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.y.a 4 1.a even 1 1 trivial
380.2.y.a 4 5.c odd 4 1 inner
380.2.y.a 4 19.d odd 6 1 inner
380.2.y.a 4 95.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 6 T_{3}^{3} + 18 T_{3}^{2} - 36 T_{3} + 36 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 36 - 36 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$5$ \( 25 - 10 T - T^{2} - 2 T^{3} + T^{4} \)
$7$ \( ( 8 + 4 T + T^{2} )^{2} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( 36 - 36 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$17$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( 361 - 37 T^{2} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( 5625 + 75 T^{2} + T^{4} \)
$31$ \( ( 3 + T^{2} )^{2} \)
$37$ \( 576 + T^{4} \)
$41$ \( ( 12 - 6 T + T^{2} )^{2} \)
$43$ \( 1024 + 256 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$47$ \( 324 - 108 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( 2916 - 972 T + 162 T^{2} - 18 T^{3} + T^{4} \)
$59$ \( 9 + 3 T^{2} + T^{4} \)
$61$ \( ( 169 + 13 T + T^{2} )^{2} \)
$67$ \( 22500 - 4500 T + 450 T^{2} - 30 T^{3} + T^{4} \)
$71$ \( ( 147 - 21 T + T^{2} )^{2} \)
$73$ \( 40000 - 4000 T + 200 T^{2} - 20 T^{3} + T^{4} \)
$79$ \( 21609 + 147 T^{2} + T^{4} \)
$83$ \( ( 32 + 8 T + T^{2} )^{2} \)
$89$ \( 59049 + 243 T^{2} + T^{4} \)
$97$ \( 147456 + 18432 T + 1152 T^{2} + 48 T^{3} + T^{4} \)
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