Properties

Label 378.5.b.a
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: 8.0.5443747577856.29
Defining polynomial: \( x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 8 q^{4} + (\beta_{7} + \beta_{3} - 4 \beta_1) q^{5} + \beta_{4} q^{7} + 8 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 8 q^{4} + (\beta_{7} + \beta_{3} - 4 \beta_1) q^{5} + \beta_{4} q^{7} + 8 \beta_1 q^{8} + (\beta_{5} + \beta_{4} - \beta_{2} - 28) q^{10} + ( - 2 \beta_{7} + 3 \beta_{6} - \beta_{3} - 15 \beta_1) q^{11} + ( - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 47) q^{13} + ( - \beta_{7} - \beta_{6} + \beta_{3}) q^{14} + 64 q^{16} + (16 \beta_{7} - 4 \beta_{6} - \beta_{3} - 38 \beta_1) q^{17} + (9 \beta_{5} + 4 \beta_{4} - 2 \beta_{2} + 140) q^{19} + ( - 8 \beta_{7} - 8 \beta_{3} + 32 \beta_1) q^{20} + (2 \beta_{5} + 12 \beta_{4} + 5 \beta_{2} - 140) q^{22} + ( - 17 \beta_{7} - 4 \beta_{6} - 28 \beta_{3} + 4 \beta_1) q^{23} + (9 \beta_{5} + 30 \beta_{4} - 10 \beta_{2} + 99) q^{25} + ( - 11 \beta_{7} + 5 \beta_{6} + 19 \beta_{3} + 55 \beta_1) q^{26} - 8 \beta_{4} q^{28} + (18 \beta_{7} + 6 \beta_{6} + 55 \beta_{3} + 112 \beta_1) q^{29} + ( - 12 \beta_{5} + 37 \beta_{4} + 6 \beta_{2} - 110) q^{31} - 64 \beta_1 q^{32} + ( - 5 \beta_{5} + 13 \beta_{4} - 20 \beta_{2} - 224) q^{34} + ( - 22 \beta_{7} - \beta_{6} - 13 \beta_{3} + 35 \beta_1) q^{35} + ( - 33 \beta_{5} - 13 \beta_{4} + 26 \beta_{2} + 197) q^{37} + ( - 25 \beta_{7} - 9 \beta_{6} - 63 \beta_{3} - 132 \beta_1) q^{38} + ( - 8 \beta_{5} - 8 \beta_{4} + 8 \beta_{2} + 224) q^{40} + (52 \beta_{7} - 13 \beta_{6} + 126 \beta_{3} - 85 \beta_1) q^{41} + (21 \beta_{5} + 76 \beta_{4} + 28 \beta_{2} - 721) q^{43} + (16 \beta_{7} - 24 \beta_{6} + 8 \beta_{3} + 120 \beta_1) q^{44} + ( - 32 \beta_{5} - 26 \beta_{4} + 13 \beta_{2} - 20) q^{46} + (22 \beta_{7} - 42 \beta_{6} - 27 \beta_{3} + 40 \beta_1) q^{47} + 343 q^{49} + ( - 99 \beta_{7} - 19 \beta_{6} - 53 \beta_{3} - 59 \beta_1) q^{50} + (24 \beta_{5} - 16 \beta_{4} + 16 \beta_{2} + 376) q^{52} + ( - 92 \beta_{7} - 50 \beta_{6} + 72 \beta_{3} + 78 \beta_1) q^{53} + (39 \beta_{5} + 7 \beta_{4} - 34 \beta_{2} + 61) q^{55} + (8 \beta_{7} + 8 \beta_{6} - 8 \beta_{3}) q^{56} + (61 \beta_{5} + 11 \beta_{4} - 12 \beta_{2} + 944) q^{58} + ( - 10 \beta_{7} - 76 \beta_{6} + 73 \beta_{3} - 488 \beta_1) q^{59} + ( - 30 \beta_{5} + 44 \beta_{4} - 92 \beta_{2} - 320) q^{61} + (11 \beta_{7} - 37 \beta_{6} + 133 \beta_{3} + 86 \beta_1) q^{62} - 512 q^{64} + ( - 124 \beta_{7} + 2 \beta_{6} - 27 \beta_{3} + 246 \beta_1) q^{65} + ( - 33 \beta_{5} + 64 \beta_{4} + 104 \beta_{2} + 2973) q^{67} + ( - 128 \beta_{7} + 32 \beta_{6} + 8 \beta_{3} + 304 \beta_1) q^{68} + ( - 14 \beta_{5} - 36 \beta_{4} + 21 \beta_{2} + 196) q^{70} + ( - 295 \beta_{7} - 92 \beta_{6} - 109 \beta_{3} - 756 \beta_1) q^{71} + ( - 3 \beta_{5} + 106 \beta_{4} + 44 \beta_{2} - 1647) q^{73} + (202 \beta_{7} - 6 \beta_{6} + 270 \beta_{3} - 301 \beta_1) q^{74} + ( - 72 \beta_{5} - 32 \beta_{4} + 16 \beta_{2} - 1120) q^{76} + (5 \beta_{7} - 2 \beta_{6} + 121 \beta_{3} + 560 \beta_1) q^{77} + ( - 15 \beta_{5} + 18 \beta_{4} - 94 \beta_{2} - 1199) q^{79} + (64 \beta_{7} + 64 \beta_{3} - 256 \beta_1) q^{80} + (113 \beta_{5} - 87 \beta_{4} - 65 \beta_{2} - 420) q^{82} + ( - 122 \beta_{7} + 8 \beta_{6} - 199 \beta_{3} - 416 \beta_1) q^{83} + (57 \beta_{5} + 364 \beta_{4} - 124 \beta_{2} - 4985) q^{85} + (71 \beta_{7} - 153 \beta_{6} - 15 \beta_{3} + 609 \beta_1) q^{86} + ( - 16 \beta_{5} - 96 \beta_{4} - 40 \beta_{2} + 1120) q^{88} + (16 \beta_{7} - 253 \beta_{6} - 12 \beta_{3} + 851 \beta_1) q^{89} + ( - 63 \beta_{5} - 48 \beta_{4} - 28 \beta_{2} + 637) q^{91} + (136 \beta_{7} + 32 \beta_{6} + 224 \beta_{3} - 32 \beta_1) q^{92} + ( - 69 \beta_{5} - 139 \beta_{4} - 64 \beta_{2} + 576) q^{94} + ( - 137 \beta_{7} - 96 \beta_{6} - 34 \beta_{3} + 1608 \beta_1) q^{95} + (132 \beta_{5} - 78 \beta_{4} + 172 \beta_{2} - 1752) 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 \( 8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 18\nu^{5} - 66\nu^{3} + 4\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 10\nu^{4} + 38\nu^{2} + 186 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{7} + 102\nu^{5} + 522\nu^{3} + 316\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{6} - 133\nu^{4} - 616\nu^{2} - 588 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{6} + 247\nu^{4} + 1072\nu^{2} + 660 ) / 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\nu^{7} + 207\nu^{5} + 534\nu^{3} - 646\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -22\nu^{7} - 441\nu^{5} - 2280\nu^{3} - 2846\nu ) / 36 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{7} - 2\beta_{6} - 12\beta_{3} - 21\beta_1 ) / 84 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{5} - 13\beta_{4} - 504 ) / 84 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 31\beta_{6} + 81\beta_{3} + 371\beta_1 ) / 84 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 49\beta_{5} + 87\beta_{4} + 14\beta_{2} + 2268 ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{7} - 220\beta_{6} - 354\beta_{3} - 2359\beta_1 ) / 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -623\beta_{5} - 1117\beta_{4} - 266\beta_{2} - 24444 ) / 42 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -25\beta_{7} + 419\beta_{6} + 525\beta_{3} + 4203\beta_1 ) / 6 \) 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
0.982345i
2.14697i
3.56118i
2.39656i
2.39656i
3.56118i
2.14697i
0.982345i
2.82843i 0 −8.00000 42.2558i 0 −18.5203 22.6274i 0 −119.518
323.2 2.82843i 0 −8.00000 9.19100i 0 18.5203 22.6274i 0 −25.9961
323.3 2.82843i 0 −8.00000 3.12468i 0 18.5203 22.6274i 0 −8.83793
323.4 2.82843i 0 −8.00000 14.9735i 0 −18.5203 22.6274i 0 42.3515
323.5 2.82843i 0 −8.00000 14.9735i 0 −18.5203 22.6274i 0 42.3515
323.6 2.82843i 0 −8.00000 3.12468i 0 18.5203 22.6274i 0 −8.83793
323.7 2.82843i 0 −8.00000 9.19100i 0 18.5203 22.6274i 0 −25.9961
323.8 2.82843i 0 −8.00000 42.2558i 0 −18.5203 22.6274i 0 −119.518
\(n\): 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.a 8
3.b odd 2 1 inner 378.5.b.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.5.b.a 8 1.a even 1 1 trivial
378.5.b.a 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2104T_{5}^{6} + 590554T_{5}^{4} + 39384216T_{5}^{2} + 330185241 \) 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
$5$ \( T^{8} + 2104 T^{6} + \cdots + 330185241 \) Copy content Toggle raw display
$7$ \( (T^{2} - 343)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 71752 T^{6} + \cdots + 16\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( (T^{4} + 188 T^{3} - 40088 T^{2} + \cdots - 317096316)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 509464 T^{6} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{4} - 560 T^{3} + \cdots + 11482979377)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 957688 T^{6} + \cdots + 13\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( T^{8} + 3390232 T^{6} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{4} + 440 T^{3} + \cdots + 88737781689)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 788 T^{3} + \cdots + 1491560880553)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 15718392 T^{6} + \cdots + 54\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( (T^{4} + 2884 T^{3} + \cdots - 16528157070236)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 13041720 T^{6} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + 44978400 T^{6} + \cdots + 96\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{8} + 52796776 T^{6} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{4} + 1280 T^{3} + \cdots + 394607747749392)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 11892 T^{3} + \cdots - 650727964094204)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 216381064 T^{6} + \cdots + 22\!\cdots\!69 \) Copy content Toggle raw display
$73$ \( (T^{4} + 6588 T^{3} + \cdots - 20805309026012)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4796 T^{3} + \cdots + 345776930946468)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 49790920 T^{6} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{8} + 452517768 T^{6} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( (T^{4} + 7008 T^{3} + \cdots + 44\!\cdots\!52)^{2} \) Copy content Toggle raw display
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