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Properties

Label 380.2.i.b
Level $380$
Weight $2$
Character orbit 380.i
Analytic conductor $3.034$
Analytic rank $0$
Dimension $6$
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1783323.2
Defining polynomial: \( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{3}) q^{3} + (\beta_{4} + 1) q^{5} + ( - \beta_{3} + 1) q^{7} + ( - 2 \beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{3}) q^{3} + (\beta_{4} + 1) q^{5} + ( - \beta_{3} + 1) q^{7} + ( - 2 \beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{9} + ( - \beta_{3} + 2) q^{11} - \beta_{4} q^{13} - \beta_{5} q^{15} + (\beta_{3} - \beta_{2} - 2 \beta_1) q^{17} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{19} + (\beta_{5} - 6 \beta_{4} - \beta_{2} - 2 \beta_1 - 6) q^{21} + (2 \beta_{5} - 5 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1) q^{23} + \beta_{4} q^{25} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 - 3) q^{27} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{29} + ( - \beta_{2} + \beta_1 + 4) q^{31} + ( - 6 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 6) q^{33} + (\beta_{5} + \beta_{4} - \beta_{3} + 1) q^{35} + ( - \beta_{3} + 1) q^{37} + \beta_{3} q^{39} + ( - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 3) q^{41} + (\beta_{5} + 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{43} + (\beta_{2} - \beta_1 - 3) q^{45} + (3 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{47} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{49} + (9 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{51} + ( - 2 \beta_{5} - 4 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{53} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 2) q^{55} + ( - 2 \beta_{5} + 9 \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 + 9) q^{57} + ( - 3 \beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 3) q^{59} + (2 \beta_{5} + 3 \beta_{4}) q^{61} + 5 \beta_{5} q^{63} + q^{65} + (2 \beta_{5} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{67} + (7 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{69} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{71} + (7 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 7) q^{73} - \beta_{3} q^{75} + ( - 3 \beta_{3} - \beta_{2} + \beta_1 + 8) q^{77} + (2 \beta_{5} - 10 \beta_{4} - 2 \beta_{3} - 10) q^{79} + (5 \beta_{5} - 6 \beta_{4} - 5 \beta_{3} - 6) q^{81} + ( - \beta_{2} + \beta_1 - 10) q^{83} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{85} + ( - 7 \beta_{3} + \beta_{2} - \beta_1) q^{87} + (4 \beta_{5} + 2 \beta_{4}) q^{89} + ( - \beta_{5} - \beta_{4}) q^{91} + ( - 8 \beta_{5} + 3 \beta_{4} + 7 \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{93} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{95} + (5 \beta_{5} - 4 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 4) q^{97} + (3 \beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + 3 q^{5} + 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{3} + 3 q^{5} + 4 q^{7} - 8 q^{9} + 10 q^{11} + 3 q^{13} - q^{15} + 3 q^{17} - 16 q^{21} + 14 q^{23} - 3 q^{25} - 20 q^{27} - 6 q^{29} + 22 q^{31} - 15 q^{33} + 2 q^{35} + 4 q^{37} + 2 q^{39} - 6 q^{41} + 5 q^{43} - 16 q^{45} + 6 q^{47} - 6 q^{49} - 24 q^{51} + 13 q^{53} + 5 q^{55} + 25 q^{57} + 6 q^{59} - 7 q^{61} + 5 q^{63} + 6 q^{65} - 4 q^{67} + 30 q^{69} + 9 q^{71} + 18 q^{73} - 2 q^{75} + 40 q^{77} - 32 q^{79} - 23 q^{81} - 62 q^{83} - 3 q^{85} - 12 q^{87} - 2 q^{89} + 2 q^{91} + 14 q^{93} - 6 q^{95} - 11 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{5} + 15\nu^{4} + 8\nu^{3} + 57\nu^{2} + 47\nu + 180 ) / 83 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{5} - 20\nu^{4} + 17\nu^{3} - 76\nu^{2} + 131\nu - 240 ) / 83 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} + 25\nu^{4} - 42\nu^{3} + 95\nu^{2} - 60\nu + 300 ) / 83 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -20\nu^{5} + 17\nu^{4} - 85\nu^{3} - 35\nu^{2} - 323\nu - 45 ) / 249 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{5} + 3\nu^{4} + 68\nu^{3} + 28\nu^{2} + 358\nu + 36 ) / 83 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{5} - 9\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 - 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{3} - 5\beta_{2} + 5\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{5} + 12\beta_{4} - 2\beta_{3} - 4\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 18\beta_{5} + 9\beta_{4} + 5\beta_{3} - 23\beta_{2} - 46\beta _1 + 9 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−0.956115 + 1.65604i
1.09935 − 1.90412i
0.356769 − 0.617942i
−0.956115 − 1.65604i
1.09935 + 1.90412i
0.356769 + 0.617942i
0 −1.28442 − 2.22469i 0 0.500000 + 0.866025i 0 3.56885 0 −1.79949 + 3.11682i 0
121.2 0 0.182224 + 0.315621i 0 0.500000 + 0.866025i 0 0.635552 0 1.43359 − 2.48305i 0
121.3 0 1.60220 + 2.77509i 0 0.500000 + 0.866025i 0 −2.20440 0 −3.63409 + 6.29444i 0
201.1 0 −1.28442 + 2.22469i 0 0.500000 − 0.866025i 0 3.56885 0 −1.79949 − 3.11682i 0
201.2 0 0.182224 − 0.315621i 0 0.500000 − 0.866025i 0 0.635552 0 1.43359 + 2.48305i 0
201.3 0 1.60220 − 2.77509i 0 0.500000 − 0.866025i 0 −2.20440 0 −3.63409 − 6.29444i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.3
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 380.2.i.b ✓ 6
3.b odd 2 1 3420.2.t.v 6
4.b odd 2 1 1520.2.q.i 6
5.b even 2 1 1900.2.i.c 6
5.c odd 4 2 1900.2.s.c 12
19.c even 3 1 inner 380.2.i.b ✓ 6
19.c even 3 1 7220.2.a.n 3
19.d odd 6 1 7220.2.a.o 3
57.h odd 6 1 3420.2.t.v 6
76.g odd 6 1 1520.2.q.i 6
95.i even 6 1 1900.2.i.c 6
95.m odd 12 2 1900.2.s.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.b ✓ 6 1.a even 1 1 trivial
380.2.i.b ✓ 6 19.c even 3 1 inner
1520.2.q.i 6 4.b odd 2 1
1520.2.q.i 6 76.g odd 6 1
1900.2.i.c 6 5.b even 2 1
1900.2.i.c 6 95.i even 6 1
1900.2.s.c 12 5.c odd 4 2
1900.2.s.c 12 95.m odd 12 2
3420.2.t.v 6 3.b odd 2 1
3420.2.t.v 6 57.h odd 6 1
7220.2.a.n 3 19.c even 3 1
7220.2.a.o 3 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - T_{3}^{5} + 9T_{3}^{4} + 2T_{3}^{3} + 67T_{3}^{2} - 24T_{3} + 9 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} + 9 T^{4} + 2 T^{3} + 67 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T^{3} - 2 T^{2} - 7 T + 5)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 5 T^{2} + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$19$ \( T^{6} - 18 T^{4} + 7 T^{3} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} - 14 T^{5} + 157 T^{4} + \cdots + 2025 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} + 117 T^{4} + \cdots + 263169 \) Copy content Toggle raw display
$31$ \( (T^{3} - 11 T^{2} + 14 T + 71)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 2 T^{2} - 7 T + 5)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 6 T^{5} + 99 T^{4} + \cdots + 18225 \) Copy content Toggle raw display
$43$ \( T^{6} - 5 T^{5} + 87 T^{4} + \cdots + 11025 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + 117 T^{4} + \cdots + 263169 \) Copy content Toggle raw display
$53$ \( T^{6} - 13 T^{5} + 139 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{6} - 6 T^{5} + 117 T^{4} + \cdots + 263169 \) Copy content Toggle raw display
$61$ \( T^{6} + 7 T^{5} + 66 T^{4} + \cdots + 3969 \) Copy content Toggle raw display
$67$ \( T^{6} + 4 T^{5} + 108 T^{4} + \cdots + 28224 \) Copy content Toggle raw display
$71$ \( T^{6} - 9 T^{5} + 99 T^{4} + 108 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$73$ \( T^{6} - 18 T^{5} + 255 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$79$ \( T^{6} + 32 T^{5} + 716 T^{4} + \cdots + 732736 \) Copy content Toggle raw display
$83$ \( (T^{3} + 31 T^{2} + 294 T + 855)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 2 T^{5} + 136 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$97$ \( T^{6} + 11 T^{5} + 215 T^{4} + \cdots + 93025 \) Copy content Toggle raw display
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