Properties

Label 378.3.b.c
Level $378$
Weight $3$
Character orbit 378.b
Analytic conductor $10.300$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.2997539928\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - 2 q^{4} + ( - \beta_{6} + \beta_{4} - \beta_1) q^{5} - \beta_{3} q^{7} + 2 \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - 2 q^{4} + ( - \beta_{6} + \beta_{4} - \beta_1) q^{5} - \beta_{3} q^{7} + 2 \beta_{4} q^{8} + ( - \beta_{5} + 2 \beta_{3} + 2) q^{10} + (\beta_{6} - \beta_{4} + \beta_{2} - \beta_1) q^{11} + ( - \beta_{7} + \beta_{5} - \beta_{3} - 5) q^{13} - \beta_{6} q^{14} + 4 q^{16} + (\beta_{6} - \beta_{4} + 2 \beta_{2} + \beta_1) q^{17} + ( - \beta_{7} + \beta_{5} + 3 \beta_{3} + 5) q^{19} + (2 \beta_{6} - 2 \beta_{4} + 2 \beta_1) q^{20} + ( - \beta_{7} - \beta_{5} - 2 \beta_{3} - 2) q^{22} + ( - 2 \beta_{6} - 4 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{23} + ( - 2 \beta_{7} + 2 \beta_{5} - 4 \beta_{3}) q^{25} + ( - \beta_{6} + 5 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{26} + 2 \beta_{3} q^{28} + (5 \beta_{6} + 7 \beta_{4} + \beta_{2} + 3 \beta_1) q^{29} + ( - 3 \beta_{7} - 3 \beta_{5} - \beta_{3} - 5) q^{31} - 4 \beta_{4} q^{32} + ( - 2 \beta_{7} + \beta_{5} - 2 \beta_{3} - 2) q^{34} + (\beta_{6} - 7 \beta_{4} + \beta_{2}) q^{35} + ( - 2 \beta_{7} + 2 \beta_{5} + 6 \beta_{3} - 1) q^{37} + (3 \beta_{6} - 5 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{38} + (2 \beta_{5} - 4 \beta_{3} - 4) q^{40} + ( - 9 \beta_{6} + 9 \beta_{4} + \beta_1) q^{41} + ( - \beta_{7} - 11 \beta_{5} + 5) q^{43} + ( - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{44} + ( - 2 \beta_{7} + 4 \beta_{5} + 4 \beta_{3} - 8) q^{46} + (6 \beta_{6} + 18 \beta_{4} - 4 \beta_{2} - 13 \beta_1) q^{47} + 7 q^{49} + ( - 4 \beta_{6} - 4 \beta_{2} - 4 \beta_1) q^{50} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 10) q^{52} + ( - 6 \beta_{6} - 18 \beta_{4} + 4 \beta_{2} - 6 \beta_1) q^{53} + (\beta_{7} - 7 \beta_{5} + 13 \beta_{3} + 7) q^{55} + 2 \beta_{6} q^{56} + ( - \beta_{7} + 3 \beta_{5} - 10 \beta_{3} + 14) q^{58} + ( - 10 \beta_{6} - 2 \beta_{4} - 9 \beta_1) q^{59} + ( - \beta_{7} + 13 \beta_{5} - 13 \beta_{3} - 11) q^{61} + ( - \beta_{6} + 5 \beta_{4} - 6 \beta_{2} + 6 \beta_1) q^{62} - 8 q^{64} + (15 \beta_{6} - 21 \beta_{4} + 5 \beta_{2} + 21 \beta_1) q^{65} + (4 \beta_{7} + 8 \beta_{5} + 10 \beta_{3} + 30) q^{67} + ( - 2 \beta_{6} + 2 \beta_{4} - 4 \beta_{2} - 2 \beta_1) q^{68} + ( - \beta_{7} - 2 \beta_{3} - 14) q^{70} + ( - 8 \beta_{6} - 4 \beta_{4} - 4 \beta_{2} + 8 \beta_1) q^{71} + ( - 2 \beta_{7} - 4 \beta_{5} - 14 \beta_{3} - 36) q^{73} + (6 \beta_{6} + \beta_{4} - 4 \beta_{2} - 4 \beta_1) q^{74} + (2 \beta_{7} - 2 \beta_{5} - 6 \beta_{3} - 10) q^{76} + ( - \beta_{6} + 7 \beta_{4} + \beta_{2} - 7 \beta_1) q^{77} + (\beta_{7} + 11 \beta_{5} - 10 \beta_{3} - 53) q^{79} + ( - 4 \beta_{6} + 4 \beta_{4} - 4 \beta_1) q^{80} + (\beta_{5} + 18 \beta_{3} + 18) q^{82} + (22 \beta_{6} + 2 \beta_{4} + 2 \beta_{2} + 7 \beta_1) q^{83} + (4 \beta_{7} - 16 \beta_{5} + 22 \beta_{3} + 25) q^{85} + ( - 5 \beta_{4} - 2 \beta_{2} + 22 \beta_1) q^{86} + (2 \beta_{7} + 2 \beta_{5} + 4 \beta_{3} + 4) q^{88} + ( - 6 \beta_{6} + 30 \beta_{4} + 6 \beta_{2} - 14 \beta_1) q^{89} + (\beta_{7} - 7 \beta_{5} + 5 \beta_{3} + 7) q^{91} + (4 \beta_{6} + 8 \beta_{4} - 4 \beta_{2} - 8 \beta_1) q^{92} + (4 \beta_{7} - 13 \beta_{5} - 12 \beta_{3} + 36) q^{94} + (\beta_{6} + 17 \beta_{4} + \beta_{2} + 11 \beta_1) q^{95} + (5 \beta_{7} - 11 \beta_{5} - 23 \beta_{3} + 75) q^{97} - 7 \beta_{4} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 16 q^{10} - 40 q^{13} + 32 q^{16} + 40 q^{19} - 16 q^{22} - 40 q^{31} - 16 q^{34} - 8 q^{37} - 32 q^{40} + 40 q^{43} - 64 q^{46} + 56 q^{49} + 80 q^{52} + 56 q^{55} + 112 q^{58} - 88 q^{61} - 64 q^{64} + 240 q^{67} - 112 q^{70} - 288 q^{73} - 80 q^{76} - 424 q^{79} + 144 q^{82} + 200 q^{85} + 32 q^{88} + 56 q^{91} + 288 q^{94} + 600 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 5\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{4} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 3\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 4\nu^{5} - 7\nu^{3} + 4\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 4\nu^{5} - 7\nu^{3} - 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} + 4\nu^{5} + 13\nu^{3} + 44\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 4\nu^{5} - 13\nu^{3} + 44\nu ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 3\beta_{6} - \beta_{5} + 3\beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} - 5\beta_{5} - 15\beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{2} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} + 3\beta_{6} + 11\beta_{5} - 33\beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{3} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} + 21\beta_{6} + 13\beta_{5} + 39\beta_{4} ) / 12 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1
0.581861 + 1.28897i
−0.581861 + 1.28897i
−1.28897 0.581861i
1.28897 0.581861i
1.28897 + 0.581861i
−1.28897 + 0.581861i
−0.581861 1.28897i
0.581861 1.28897i
1.41421i 0 −2.00000 5.32744i 0 2.64575 2.82843i 0 −7.53414
323.2 1.41421i 0 −2.00000 0.672556i 0 2.64575 2.82843i 0 0.951138
323.3 1.41421i 0 −2.00000 2.15587i 0 −2.64575 2.82843i 0 3.04886
323.4 1.41421i 0 −2.00000 8.15587i 0 −2.64575 2.82843i 0 11.5341
323.5 1.41421i 0 −2.00000 8.15587i 0 −2.64575 2.82843i 0 11.5341
323.6 1.41421i 0 −2.00000 2.15587i 0 −2.64575 2.82843i 0 3.04886
323.7 1.41421i 0 −2.00000 0.672556i 0 2.64575 2.82843i 0 0.951138
323.8 1.41421i 0 −2.00000 5.32744i 0 2.64575 2.82843i 0 −7.53414
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.3.b.c 8
3.b odd 2 1 inner 378.3.b.c 8
4.b odd 2 1 3024.3.d.h 8
9.c even 3 2 1134.3.q.f 16
9.d odd 6 2 1134.3.q.f 16
12.b even 2 1 3024.3.d.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.3.b.c 8 1.a even 1 1 trivial
378.3.b.c 8 3.b odd 2 1 inner
1134.3.q.f 16 9.c even 3 2
1134.3.q.f 16 9.d odd 6 2
3024.3.d.h 8 4.b odd 2 1
3024.3.d.h 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 100T_{5}^{6} + 2374T_{5}^{4} + 9828T_{5}^{2} + 3969 \) acting on \(S_{3}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 100 T^{6} + 2374 T^{4} + \cdots + 3969 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 352 T^{6} + 38536 T^{4} + \cdots + 63504 \) Copy content Toggle raw display
$13$ \( (T^{4} + 20 T^{3} - 152 T^{2} - 1512 T + 7812)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 1108 T^{6} + \cdots + 2793862449 \) Copy content Toggle raw display
$19$ \( (T^{4} - 20 T^{3} - 264 T^{2} + 616 T + 2884)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 1936 T^{6} + \cdots + 14876193024 \) Copy content Toggle raw display
$29$ \( T^{8} + 2368 T^{6} + \cdots + 29253313296 \) Copy content Toggle raw display
$31$ \( (T^{4} + 20 T^{3} - 2456 T^{2} + \cdots + 816804)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} - 1650 T^{2} + \cdots - 66023)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 5220 T^{6} + \cdots + 849230442369 \) Copy content Toggle raw display
$43$ \( (T^{4} - 20 T^{3} - 4458 T^{2} + \cdots + 4096129)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 125526008337921 \) Copy content Toggle raw display
$53$ \( T^{8} + 9936 T^{6} + \cdots + 1049760000 \) Copy content Toggle raw display
$59$ \( T^{8} + 8548 T^{6} + \cdots + 173256570081 \) Copy content Toggle raw display
$61$ \( (T^{4} + 44 T^{3} - 7976 T^{2} + \cdots + 3242628)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 120 T^{3} - 2336 T^{2} + \cdots - 19027904)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 10048 T^{6} + \cdots + 949829566464 \) Copy content Toggle raw display
$73$ \( (T^{4} + 144 T^{3} + 3448 T^{2} + \cdots - 8237936)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 212 T^{3} + 10846 T^{2} + \cdots - 13387527)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 29908 T^{6} + \cdots + 12\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( T^{8} + 25344 T^{6} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{4} - 300 T^{3} + 15688 T^{2} + \cdots - 190394300)^{2} \) Copy content Toggle raw display
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