Properties

Label 378.5.b.b
Level $378$
Weight $5$
Character orbit 378.b
Analytic conductor $39.074$
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: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 378.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.0738460457\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 92x^{6} + 2949x^{4} + 37548x^{2} + 142884 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 8 q^{4} + ( - \beta_{2} - \beta_1) q^{5} + \beta_{3} q^{7} + 8 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 8 q^{4} + ( - \beta_{2} - \beta_1) q^{5} + \beta_{3} q^{7} + 8 \beta_1 q^{8} + ( - \beta_{4} + \beta_{3} - 4) q^{10} + (\beta_{7} + \beta_{2} + 13 \beta_1) q^{11} + ( - \beta_{5} - 2 \beta_{3} + 37) q^{13} + \beta_{6} q^{14} + 64 q^{16} + ( - \beta_{7} + \beta_{6} + 2 \beta_{2} + 15 \beta_1) q^{17} + ( - \beta_{5} - 12 \beta_{3} - 10) q^{19} + (8 \beta_{2} + 8 \beta_1) q^{20} + ( - \beta_{5} + \beta_{4} - 4 \beta_{3} + 100) q^{22} + ( - \beta_{7} + \beta_{6} - 7 \beta_{2} - 72 \beta_1) q^{23} + (\beta_{5} - 14 \beta_{3} - 183) q^{25} + ( - 8 \beta_{7} + \beta_{6} - 37 \beta_1) q^{26} - 8 \beta_{3} q^{28} + (7 \beta_{7} - 9 \beta_{6} - 10 \beta_{2} + 41 \beta_1) q^{29} + ( - 6 \beta_{4} - 11 \beta_{3} - 128) q^{31} - 64 \beta_1 q^{32} + (\beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 112) q^{34} + (7 \beta_{7} - 2 \beta_{6} - 7 \beta_{2} + 21 \beta_1) q^{35} + (\beta_{5} - 51 \beta_{3} + 113) q^{37} + ( - 8 \beta_{7} - 9 \beta_{6} + 10 \beta_1) q^{38} + (8 \beta_{4} - 8 \beta_{3} + 32) q^{40} + (12 \beta_{7} - 17 \beta_{6} + 19 \beta_{2} - 46 \beta_1) q^{41} + (11 \beta_{5} + 6 \beta_{4} - 36 \beta_{3} - 403) q^{43} + ( - 8 \beta_{7} - 8 \beta_{2} - 104 \beta_1) q^{44} + (\beta_{5} - 7 \beta_{4} + 2 \beta_{3} - 548) q^{46} + ( - \beta_{7} - 45 \beta_{6} - 18 \beta_{2} - 15 \beta_1) q^{47} + 343 q^{49} + (8 \beta_{7} - 17 \beta_{6} + 183 \beta_1) q^{50} + (8 \beta_{5} + 16 \beta_{3} - 296) q^{52} + (28 \beta_{7} - 34 \beta_{6} + 38 \beta_{2} + 364 \beta_1) q^{53} + ( - 5 \beta_{5} + 12 \beta_{4} - 97 \beta_{3} + 1039) q^{55} - 8 \beta_{6} q^{56} + ( - 7 \beta_{5} - 10 \beta_{4} + 61 \beta_{3} + 368) q^{58} + ( - 45 \beta_{7} - 35 \beta_{6} + 36 \beta_{2} + 31 \beta_1) q^{59} + (2 \beta_{5} + 48 \beta_{4} - 140 \beta_{3} + 16) q^{61} + ( - 17 \beta_{6} + 48 \beta_{2} + 152 \beta_1) q^{62} - 512 q^{64} + ( - 25 \beta_{7} - 89 \beta_{6} - 44 \beta_{2} - 199 \beta_1) q^{65} + ( - 11 \beta_{5} - 42 \beta_{4} + 56 \beta_{3} - 1269) q^{67} + (8 \beta_{7} - 8 \beta_{6} - 16 \beta_{2} - 120 \beta_1) q^{68} + ( - 7 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} + 196) q^{70} + ( - 10 \beta_{7} - 34 \beta_{6} - 87 \beta_{2} + 263 \beta_1) q^{71} + ( - 11 \beta_{5} - 6 \beta_{4} - 142 \beta_{3} + 1329) q^{73} + (8 \beta_{7} - 54 \beta_{6} - 113 \beta_1) q^{74} + (8 \beta_{5} + 96 \beta_{3} + 80) q^{76} + ( - 14 \beta_{7} - 4 \beta_{6} - 35 \beta_{2} - 189 \beta_1) q^{77} + (\beta_{5} - 60 \beta_{4} - 26 \beta_{3} - 509) q^{79} + ( - 64 \beta_{2} - 64 \beta_1) q^{80} + ( - 12 \beta_{5} + 19 \beta_{4} + 81 \beta_{3} - 444) q^{82} + ( - 37 \beta_{7} - 107 \beta_{6} + 92 \beta_{2} - 645 \beta_1) q^{83} + ( - 5 \beta_{5} + 6 \beta_{4} + 116 \beta_{3} + 1681) q^{85} + (88 \beta_{7} - 63 \beta_{6} - 48 \beta_{2} + 379 \beta_1) q^{86} + (8 \beta_{5} - 8 \beta_{4} + 32 \beta_{3} - 800) q^{88} + (66 \beta_{7} + \beta_{6} - 5 \beta_{2} - 1144 \beta_1) q^{89} + (7 \beta_{5} - 42 \beta_{4} + 52 \beta_{3} - 833) q^{91} + (8 \beta_{7} - 8 \beta_{6} + 56 \beta_{2} + 576 \beta_1) q^{92} + (\beta_{5} - 18 \beta_{4} + 381 \beta_{3} - 48) q^{94} + ( - 95 \beta_{7} - 69 \beta_{6} + 73 \beta_{2} - 362 \beta_1) q^{95} + ( - 28 \beta_{5} + 12 \beta_{4} - 94 \beta_{3} + 4452) q^{97} - 343 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 64 q^{4} - 32 q^{10} + 296 q^{13} + 512 q^{16} - 80 q^{19} + 800 q^{22} - 1464 q^{25} - 1024 q^{31} + 896 q^{34} + 904 q^{37} + 256 q^{40} - 3224 q^{43} - 4384 q^{46} + 2744 q^{49} - 2368 q^{52} + 8312 q^{55} + 2944 q^{58} + 128 q^{61} - 4096 q^{64} - 10152 q^{67} + 1568 q^{70} + 10632 q^{73} + 640 q^{76} - 4072 q^{79} - 3552 q^{82} + 13448 q^{85} - 6400 q^{88} - 6664 q^{91} - 384 q^{94} + 35616 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 92x^{6} + 2949x^{4} + 37548x^{2} + 142884 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 34\nu^{5} - 3351\nu^{3} - 37674\nu ) / 15876 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -25\nu^{7} - 1502\nu^{5} - 23241\nu^{3} + 77490\nu ) / 21168 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{6} - 334\nu^{4} - 6429\nu^{2} - 30366 ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 86\nu^{4} - 2073\nu^{2} - 11862 ) / 24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{6} - 334\nu^{4} - 6237\nu^{2} - 25950 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} + 334\nu^{5} + 6177\nu^{3} + 24066\nu ) / 162 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 29\nu^{7} + 1954\nu^{5} + 37914\nu^{3} + 182385\nu ) / 441 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + 28\beta_{2} + 35\beta_1 ) / 168 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 9\beta_{3} - 552 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{7} - 21\beta_{6} - 108\beta_{2} - 87\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -41\beta_{5} - 30\beta_{4} + 387\beta_{3} + 15396 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -364\beta_{7} + 903\beta_{6} + 3228\beta_{2} - 1929\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1453\beta_{5} + 2004\beta_{4} - 14625\beta_{3} - 464448 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 14432\beta_{7} - 34287\beta_{6} - 101460\beta_{2} + 212271\beta_1 ) / 24 \) 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
4.82885i
5.12618i
2.54824i
5.99257i
5.99257i
2.54824i
5.12618i
4.82885i
2.82843i 0 −8.00000 30.1368i 0 18.5203 22.6274i 0 −85.2398
323.2 2.82843i 0 −8.00000 28.1791i 0 −18.5203 22.6274i 0 −79.7026
323.3 2.82843i 0 −8.00000 17.8674i 0 −18.5203 22.6274i 0 50.5366
323.4 2.82843i 0 −8.00000 34.7917i 0 18.5203 22.6274i 0 98.4059
323.5 2.82843i 0 −8.00000 34.7917i 0 18.5203 22.6274i 0 98.4059
323.6 2.82843i 0 −8.00000 17.8674i 0 −18.5203 22.6274i 0 50.5366
323.7 2.82843i 0 −8.00000 28.1791i 0 −18.5203 22.6274i 0 −79.7026
323.8 2.82843i 0 −8.00000 30.1368i 0 18.5203 22.6274i 0 −85.2398
\(n\): 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 378.5.b.b 8
3.b odd 2 1 inner 378.5.b.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.5.b.b 8 1.a even 1 1 trivial
378.5.b.b 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 3232T_{5}^{6} + 3711634T_{5}^{4} + 1761033888T_{5}^{2} + 278692135569 \) acting on \(S_{5}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 3232 T^{6} + \cdots + 278692135569 \) Copy content Toggle raw display
$7$ \( (T^{2} - 343)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 31504 T^{6} + \cdots + 7401278039841 \) Copy content Toggle raw display
$13$ \( (T^{4} - 148 T^{3} - 75032 T^{2} + \cdots + 1130516676)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 42760 T^{6} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{4} + 40 T^{3} - 184566 T^{2} + \cdots - 274761143)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 330784 T^{6} + \cdots + 44\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( T^{8} + 1794088 T^{6} + \cdots + 40\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( (T^{4} + 512 T^{3} + \cdots + 21940230321)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 452 T^{3} + \cdots + 685159025593)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 5484096 T^{6} + \cdots + 59\!\cdots\!69 \) Copy content Toggle raw display
$43$ \( (T^{4} + 1612 T^{3} + \cdots + 19781954641348)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 25013736 T^{6} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{8} + 28831680 T^{6} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{8} + 72276952 T^{6} + \cdots + 65\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{4} - 64 T^{3} + \cdots + 37203857115408)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 5076 T^{3} + \cdots - 106304126923004)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 51389008 T^{6} + \cdots + 87\!\cdots\!41 \) Copy content Toggle raw display
$73$ \( (T^{4} - 5316 T^{3} + \cdots - 33552318651836)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2036 T^{3} + \cdots + 451461470055876)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 214925080 T^{6} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{8} + 136948752 T^{6} + \cdots + 55\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( (T^{4} - 17808 T^{3} + \cdots + 47506778460304)^{2} \) Copy content Toggle raw display
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