Properties

Label 380.2.k.a
Level $380$
Weight $2$
Character orbit 380.k
Analytic conductor $3.034$
Analytic rank $0$
Dimension $2$
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.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + i ) q^{2} + ( -1 - i ) q^{3} -2 i q^{4} + ( -2 - i ) q^{5} + 2 q^{6} + ( -2 + 2 i ) q^{7} + ( 2 + 2 i ) q^{8} -i q^{9} +O(q^{10})\) \( q + ( -1 + i ) q^{2} + ( -1 - i ) q^{3} -2 i q^{4} + ( -2 - i ) q^{5} + 2 q^{6} + ( -2 + 2 i ) q^{7} + ( 2 + 2 i ) q^{8} -i q^{9} + ( 3 - i ) q^{10} + ( -2 + 2 i ) q^{12} -4 i q^{14} + ( 1 + 3 i ) q^{15} -4 q^{16} + ( 5 + 5 i ) q^{17} + ( 1 + i ) q^{18} - q^{19} + ( -2 + 4 i ) q^{20} + 4 q^{21} + ( 4 + 4 i ) q^{23} -4 i q^{24} + ( 3 + 4 i ) q^{25} + ( -4 + 4 i ) q^{27} + ( 4 + 4 i ) q^{28} + 6 i q^{29} + ( -4 - 2 i ) q^{30} + ( 4 - 4 i ) q^{32} -10 q^{34} + ( 6 - 2 i ) q^{35} -2 q^{36} + ( 1 - i ) q^{38} + ( -2 - 6 i ) q^{40} + 2 q^{41} + ( -4 + 4 i ) q^{42} + ( -6 - 6 i ) q^{43} + ( -1 + 2 i ) q^{45} -8 q^{46} + ( -2 + 2 i ) q^{47} + ( 4 + 4 i ) q^{48} -i q^{49} + ( -7 - i ) q^{50} -10 i q^{51} + ( -10 + 10 i ) q^{53} -8 i q^{54} -8 q^{56} + ( 1 + i ) q^{57} + ( -6 - 6 i ) q^{58} + 10 q^{59} + ( 6 - 2 i ) q^{60} + 2 q^{61} + ( 2 + 2 i ) q^{63} + 8 i q^{64} + ( 3 - 3 i ) q^{67} + ( 10 - 10 i ) q^{68} -8 i q^{69} + ( -4 + 8 i ) q^{70} + ( 2 - 2 i ) q^{72} + ( 5 - 5 i ) q^{73} + ( 1 - 7 i ) q^{75} + 2 i q^{76} -10 q^{79} + ( 8 + 4 i ) q^{80} + 5 q^{81} + ( -2 + 2 i ) q^{82} + ( 4 + 4 i ) q^{83} -8 i q^{84} + ( -5 - 15 i ) q^{85} + 12 q^{86} + ( 6 - 6 i ) q^{87} + 6 i q^{89} + ( -1 - 3 i ) q^{90} + ( 8 - 8 i ) q^{92} -4 i q^{94} + ( 2 + i ) q^{95} -8 q^{96} + ( -10 - 10 i ) q^{97} + ( 1 + i ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} - 4q^{5} + 4q^{6} - 4q^{7} + 4q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} - 4q^{5} + 4q^{6} - 4q^{7} + 4q^{8} + 6q^{10} - 4q^{12} + 2q^{15} - 8q^{16} + 10q^{17} + 2q^{18} - 2q^{19} - 4q^{20} + 8q^{21} + 8q^{23} + 6q^{25} - 8q^{27} + 8q^{28} - 8q^{30} + 8q^{32} - 20q^{34} + 12q^{35} - 4q^{36} + 2q^{38} - 4q^{40} + 4q^{41} - 8q^{42} - 12q^{43} - 2q^{45} - 16q^{46} - 4q^{47} + 8q^{48} - 14q^{50} - 20q^{53} - 16q^{56} + 2q^{57} - 12q^{58} + 20q^{59} + 12q^{60} + 4q^{61} + 4q^{63} + 6q^{67} + 20q^{68} - 8q^{70} + 4q^{72} + 10q^{73} + 2q^{75} - 20q^{79} + 16q^{80} + 10q^{81} - 4q^{82} + 8q^{83} - 10q^{85} + 24q^{86} + 12q^{87} - 2q^{90} + 16q^{92} + 4q^{95} - 16q^{96} - 20q^{97} + 2q^{98} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
267.1
1.00000i
1.00000i
−1.00000 + 1.00000i −1.00000 1.00000i 2.00000i −2.00000 1.00000i 2.00000 −2.00000 + 2.00000i 2.00000 + 2.00000i 1.00000i 3.00000 1.00000i
343.1 −1.00000 1.00000i −1.00000 + 1.00000i 2.00000i −2.00000 + 1.00000i 2.00000 −2.00000 2.00000i 2.00000 2.00000i 1.00000i 3.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.k.a 2
4.b odd 2 1 380.2.k.b yes 2
5.c odd 4 1 380.2.k.b yes 2
20.e even 4 1 inner 380.2.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.k.a 2 1.a even 1 1 trivial
380.2.k.a 2 20.e even 4 1 inner
380.2.k.b yes 2 4.b odd 2 1
380.2.k.b yes 2 5.c odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + 2 T + T^{2} \)
$3$ \( 2 + 2 T + T^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( 8 + 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( 50 - 10 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 32 - 8 T + T^{2} \)
$29$ \( 36 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( 72 + 12 T + T^{2} \)
$47$ \( 8 + 4 T + T^{2} \)
$53$ \( 200 + 20 T + T^{2} \)
$59$ \( ( -10 + T )^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 18 - 6 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 50 - 10 T + T^{2} \)
$79$ \( ( 10 + T )^{2} \)
$83$ \( 32 - 8 T + T^{2} \)
$89$ \( 36 + T^{2} \)
$97$ \( 200 + 20 T + T^{2} \)
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