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Properties

Label 380.2.i.c
Level $380$
Weight $2$
Character orbit 380.i
Analytic conductor $3.034$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} + 9x^{6} + 2x^{5} + 65x^{4} - 20x^{3} + 25x^{2} + 6x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{4} q^{5} + \beta_{7} q^{7} + (\beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{4} q^{5} + \beta_{7} q^{7} + (\beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{9} + ( - \beta_{7} + 2 \beta_{3} + 1) q^{11} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 + 2) q^{13} + (\beta_{3} + \beta_1) q^{15} + (\beta_{6} + \beta_1) q^{17} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{19} + (\beta_{6} - 2 \beta_{4}) q^{21} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{2}) q^{23} + ( - \beta_{4} - 1) q^{25} + ( - \beta_{7} - 2 \beta_{3} - \beta_{2} + 2) q^{27} + (2 \beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{29} + (\beta_{7} - 2 \beta_{3} + \beta_{2} - 3) q^{31} + (\beta_{6} - 6 \beta_{4} + \beta_1) q^{33} - \beta_{5} q^{35} + ( - \beta_{7} + 2 \beta_{3} - 6) q^{37} + (3 \beta_{3} + 2 \beta_{2} - 6) q^{39} + (\beta_{5} + 2 \beta_{4}) q^{41} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_1) q^{43} + ( - \beta_{3} - \beta_{2} + 1) q^{45} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + 4 \beta_{4} - 2 \beta_{2} + 4) q^{47} + ( - \beta_{7} + 2 \beta_{3} + \beta_{2} + 3) q^{49} + ( - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - \beta_{2} - 4 \beta_1 + 2) q^{51} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{53} + (\beta_{5} + \beta_{4} + 2 \beta_1) q^{55} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + 3 \beta_1 + 4) q^{57} + (2 \beta_{5} - 3 \beta_{4} - \beta_1) q^{59} + (2 \beta_{7} - 2 \beta_{5} + 3 \beta_{4} + 3) q^{61} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} - 5 \beta_{3} - 5 \beta_1 - 2) q^{63} + (\beta_{7} + \beta_{3} - 2) q^{65} + ( - 2 \beta_{6} - 2 \beta_{2}) q^{67} + ( - \beta_{7} - 3 \beta_{3} - \beta_{2}) q^{69} + ( - \beta_{6} - 3 \beta_{4} + 2 \beta_1) q^{71} + ( - \beta_{6} - \beta_{5} - 4 \beta_1) q^{73} - \beta_{3} q^{75} + (2 \beta_{7} - 2 \beta_{3} - 3 \beta_{2} - 6) q^{77} + ( - 2 \beta_{6} - \beta_{5} - 3 \beta_{4} + \beta_1) q^{79} + (\beta_{5} + 5 \beta_{4} - 4 \beta_1) q^{81} + (3 \beta_{3} + \beta_{2} + 2) q^{83} + ( - \beta_{6} - \beta_{3} - \beta_{2} - \beta_1) q^{85} + (2 \beta_{3} - \beta_{2}) q^{87} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{89} + (3 \beta_{7} - 3 \beta_{5} - 12 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 12) q^{91} + ( - \beta_{6} - \beta_{5} + 8 \beta_{4} + 4 \beta_1) q^{93} + (\beta_{6} - \beta_{3} - 1) q^{95} + ( - 2 \beta_{6} - \beta_1) q^{97} + ( - 4 \beta_{7} - \beta_{6} + 4 \beta_{5} + 5 \beta_{4} - 4 \beta_{3} - \beta_{2} - 4 \beta_1 + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} - 4 q^{5} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{3} - 4 q^{5} - 5 q^{9} + 4 q^{11} + 9 q^{13} - q^{15} + q^{17} + 3 q^{19} + 8 q^{21} - 4 q^{25} + 20 q^{27} + 5 q^{29} - 20 q^{31} + 25 q^{33} - 52 q^{37} - 54 q^{39} - 8 q^{41} + 7 q^{43} + 10 q^{45} + 16 q^{47} + 20 q^{49} + 12 q^{51} + 5 q^{53} - 2 q^{55} + 27 q^{57} + 11 q^{59} + 12 q^{61} - 3 q^{63} - 18 q^{65} + 6 q^{69} + 14 q^{71} - 4 q^{73} + 2 q^{75} - 44 q^{77} + 13 q^{79} - 24 q^{81} + 10 q^{83} + q^{85} - 4 q^{87} + 5 q^{89} - 46 q^{91} - 28 q^{93} - 6 q^{95} - q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 9x^{6} + 2x^{5} + 65x^{4} - 20x^{3} + 25x^{2} + 6x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -304\nu^{7} - 31\nu^{6} - 2546\nu^{5} - 1710\nu^{4} - 23348\nu^{3} - 1178\nu^{2} - 380\nu + 48866 ) / 13629 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 536\nu^{7} - 304\nu^{6} + 4489\nu^{5} + 3015\nu^{4} + 36145\nu^{3} + 2077\nu^{2} + 670\nu + 3506 ) / 13629 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1753\nu^{7} - 2825\nu^{6} + 16385\nu^{5} - 5472\nu^{4} + 107915\nu^{3} - 107350\nu^{2} + 39671\nu - 18080 ) / 27258 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1418 \nu^{7} - 2635 \nu^{6} + 15283 \nu^{5} - 9060 \nu^{4} + 100657 \nu^{3} - 100130 \nu^{2} + 131248 \nu - 16864 ) / 13629 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3274 \nu^{7} - 5315 \nu^{6} + 30827 \nu^{5} - 12249 \nu^{4} + 203033 \nu^{3} - 201970 \nu^{2} + 65423 \nu - 34016 ) / 13629 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -1328\nu^{7} + 821\nu^{6} - 11122\nu^{5} - 7470\nu^{4} - 84061\nu^{3} - 5146\nu^{2} - 1660\nu - 16552 ) / 4543 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 4\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 8\beta_{3} + \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{6} + \beta_{5} + 32\beta_{4} - 13\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{7} - 14\beta_{6} + 9\beta_{5} + 36\beta_{4} - 72\beta_{3} - 14\beta_{2} - 72\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -14\beta_{7} - 150\beta_{3} - 81\beta_{2} + 278 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 164\beta_{6} - 81\beta_{5} - 466\beta_{4} + 671\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
1.58253 + 2.74101i
0.354609 + 0.614201i
−0.176725 − 0.306096i
−1.26041 − 2.18309i
1.58253 − 2.74101i
0.354609 − 0.614201i
−0.176725 + 0.306096i
−1.26041 + 2.18309i
0 −1.58253 − 2.74101i 0 −0.500000 − 0.866025i 0 −1.53315 0 −3.50877 + 6.07738i 0
121.2 0 −0.354609 − 0.614201i 0 −0.500000 − 0.866025i 0 3.11079 0 1.24850 − 2.16247i 0
121.3 0 0.176725 + 0.306096i 0 −0.500000 − 0.866025i 0 −4.30507 0 1.43754 − 2.48989i 0
121.4 0 1.26041 + 2.18309i 0 −0.500000 − 0.866025i 0 2.72743 0 −1.67727 + 2.90511i 0
201.1 0 −1.58253 + 2.74101i 0 −0.500000 + 0.866025i 0 −1.53315 0 −3.50877 − 6.07738i 0
201.2 0 −0.354609 + 0.614201i 0 −0.500000 + 0.866025i 0 3.11079 0 1.24850 + 2.16247i 0
201.3 0 0.176725 − 0.306096i 0 −0.500000 + 0.866025i 0 −4.30507 0 1.43754 + 2.48989i 0
201.4 0 1.26041 − 2.18309i 0 −0.500000 + 0.866025i 0 2.72743 0 −1.67727 − 2.90511i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.4
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.c ✓ 8
3.b odd 2 1 3420.2.t.w 8
4.b odd 2 1 1520.2.q.m 8
5.b even 2 1 1900.2.i.d 8
5.c odd 4 2 1900.2.s.d 16
19.c even 3 1 inner 380.2.i.c ✓ 8
19.c even 3 1 7220.2.a.r 4
19.d odd 6 1 7220.2.a.p 4
57.h odd 6 1 3420.2.t.w 8
76.g odd 6 1 1520.2.q.m 8
95.i even 6 1 1900.2.i.d 8
95.m odd 12 2 1900.2.s.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.c ✓ 8 1.a even 1 1 trivial
380.2.i.c ✓ 8 19.c even 3 1 inner
1520.2.q.m 8 4.b odd 2 1
1520.2.q.m 8 76.g odd 6 1
1900.2.i.d 8 5.b even 2 1
1900.2.i.d 8 95.i even 6 1
1900.2.s.d 16 5.c odd 4 2
1900.2.s.d 16 95.m odd 12 2
3420.2.t.w 8 3.b odd 2 1
3420.2.t.w 8 57.h odd 6 1
7220.2.a.p 4 19.d odd 6 1
7220.2.a.r 4 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + T_{3}^{7} + 9T_{3}^{6} - 2T_{3}^{5} + 65T_{3}^{4} + 20T_{3}^{3} + 25T_{3}^{2} - 6T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + 9 T^{6} - 2 T^{5} + 65 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 19 T^{2} + 11 T + 56)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 35 T^{2} + 21 T + 267)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 9 T^{7} + 86 T^{6} + \cdots + 2704 \) Copy content Toggle raw display
$17$ \( T^{8} - T^{7} + 35 T^{6} + 52 T^{5} + \cdots + 36864 \) Copy content Toggle raw display
$19$ \( T^{8} - 3 T^{7} - 43 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} + 39 T^{6} + 198 T^{5} + \cdots + 2916 \) Copy content Toggle raw display
$29$ \( T^{8} - 5 T^{7} + 84 T^{6} + \cdots + 74529 \) Copy content Toggle raw display
$31$ \( (T^{4} + 10 T^{3} - 29 T^{2} - 303 T - 277)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 26 T^{3} + 217 T^{2} + 609 T + 414)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + 59 T^{6} + 106 T^{5} + \cdots + 324 \) Copy content Toggle raw display
$43$ \( T^{8} - 7 T^{7} + 81 T^{6} + \cdots + 31684 \) Copy content Toggle raw display
$47$ \( T^{8} - 16 T^{7} + 281 T^{6} + \cdots + 1382976 \) Copy content Toggle raw display
$53$ \( T^{8} - 5 T^{7} + 143 T^{6} + \cdots + 1483524 \) Copy content Toggle raw display
$59$ \( T^{8} - 11 T^{7} + 176 T^{6} + \cdots + 2518569 \) Copy content Toggle raw display
$61$ \( T^{8} - 12 T^{7} + 166 T^{6} + \cdots + 841 \) Copy content Toggle raw display
$67$ \( T^{8} + 124 T^{6} + \cdots + 10863616 \) Copy content Toggle raw display
$71$ \( T^{8} - 14 T^{7} + 197 T^{6} + \cdots + 5049009 \) Copy content Toggle raw display
$73$ \( T^{8} + 4 T^{7} + 153 T^{6} + \cdots + 311364 \) Copy content Toggle raw display
$79$ \( T^{8} - 13 T^{7} + 297 T^{6} + \cdots + 9096256 \) Copy content Toggle raw display
$83$ \( (T^{4} - 5 T^{3} - 82 T^{2} + 669 T - 1302)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 5 T^{7} + 147 T^{6} + \cdots + 1382976 \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} + 123 T^{6} + \cdots + 13351716 \) Copy content Toggle raw display
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