Properties

Label 378.4.f.b
Level $378$
Weight $4$
Character orbit 378.f
Analytic conductor $22.303$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 378.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.3027219822\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2x^{7} - 2x^{6} + 53x^{5} + 38x^{4} - 166x^{3} + 7x^{2} + 1543x + 2707 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} + 2) q^{2} - 4 \beta_{2} q^{4} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{5} + ( - 7 \beta_{2} + 7) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} + 2) q^{2} - 4 \beta_{2} q^{4} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{5} + ( - 7 \beta_{2} + 7) q^{7} - 8 q^{8} + ( - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{10} + (5 \beta_{7} - 3 \beta_{6} - \beta_{5} + \beta_{2} - 1) q^{11} + ( - 2 \beta_{7} - 3 \beta_{6} - \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + \beta_1) q^{13} - 14 \beta_{2} q^{14} + (16 \beta_{2} - 16) q^{16} + (\beta_{4} + 3 \beta_{3} + 6 \beta_1 + 9) q^{17} + ( - 4 \beta_{4} + 11 \beta_{3} - 3 \beta_1 - 26) q^{19} + ( - 4 \beta_{7} + 4 \beta_{6} + 4 \beta_{5}) q^{20} + (10 \beta_{7} - 6 \beta_{6} - 2 \beta_{5} - 6 \beta_{4} - 10 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{22} + (6 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} - 94 \beta_{2} - 3 \beta_1) q^{23} + ( - 6 \beta_{7} + 7 \beta_{6} + 16 \beta_{5} + 4 \beta_{2} - 4) q^{25} + ( - 6 \beta_{4} + 4 \beta_{3} + 2 \beta_1 + 10) q^{26} - 28 q^{28} + (8 \beta_{7} - 7 \beta_{6} - 6 \beta_{5} + 69 \beta_{2} - 69) q^{29} + ( - 11 \beta_{7} - 9 \beta_{6} + 13 \beta_{5} - 9 \beta_{4} + 11 \beta_{3} + \cdots - 13 \beta_1) q^{31}+ \cdots - 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8} - 4 q^{10} - 5 q^{11} + 21 q^{13} - 56 q^{14} - 64 q^{16} + 46 q^{17} - 188 q^{19} - 4 q^{20} + 10 q^{22} - 374 q^{23} - 41 q^{25} + 84 q^{26} - 224 q^{28} - 271 q^{29} + 243 q^{31} + 128 q^{32} + 46 q^{34} - 14 q^{35} - 362 q^{37} - 188 q^{38} + 8 q^{40} + 213 q^{41} + 238 q^{43} + 40 q^{44} - 1496 q^{46} - 675 q^{47} - 196 q^{49} + 82 q^{50} + 84 q^{52} - 108 q^{53} - 2828 q^{55} - 224 q^{56} + 542 q^{58} - 202 q^{59} + 1212 q^{61} + 972 q^{62} + 512 q^{64} - 549 q^{65} - 139 q^{67} - 92 q^{68} - 14 q^{70} + 2590 q^{71} - 4000 q^{73} - 362 q^{74} + 376 q^{76} + 35 q^{77} + 1545 q^{79} + 32 q^{80} + 852 q^{82} + 142 q^{83} + 793 q^{85} - 476 q^{86} + 40 q^{88} + 264 q^{89} + 294 q^{91} - 1496 q^{92} + 1350 q^{94} - 1244 q^{95} + 638 q^{97} - 784 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 2x^{6} + 53x^{5} + 38x^{4} - 166x^{3} + 7x^{2} + 1543x + 2707 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -45\nu^{7} - 152\nu^{6} + 809\nu^{5} - 1570\nu^{4} - 15799\nu^{3} + 13322\nu^{2} + 104222\nu - 75818 ) / 22491 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 956 \nu^{7} - 4527 \nu^{6} + 9062 \nu^{5} + 27893 \nu^{4} - 46669 \nu^{3} - 58960 \nu^{2} + 109080 \nu + 1139585 ) / 247401 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -65\nu^{7} + 314\nu^{6} - 394\nu^{5} - 3814\nu^{4} + 7091\nu^{3} + 2915\nu^{2} - 19694\nu - 97401 ) / 14553 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2066 \nu^{7} + 4665 \nu^{6} - 491 \nu^{5} - 89666 \nu^{4} + 16261 \nu^{3} + 115456 \nu^{2} + 337857 \nu - 635990 ) / 247401 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 872\nu^{7} - 4184\nu^{6} + 10588\nu^{5} + 23098\nu^{4} - 47180\nu^{3} + 23353\nu^{2} + 234695\nu + 890847 ) / 35343 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8335 \nu^{7} + 36450 \nu^{6} - 74695 \nu^{5} - 269326 \nu^{4} + 276374 \nu^{3} + 446765 \nu^{2} - 2126268 \nu - 10262284 ) / 247401 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9432 \nu^{7} - 40945 \nu^{6} + 80167 \nu^{5} + 302416 \nu^{4} - 345233 \nu^{3} - 780263 \nu^{2} + 1668493 \nu + 10687202 ) / 247401 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - 2\beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} + 2\beta _1 + 3 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} - 2\beta_{6} - \beta_{4} + 10\beta_{2} - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{7} - 17\beta_{6} - 4\beta_{5} + 11\beta_{4} - 4\beta_{3} + 18\beta_{2} - 13\beta _1 - 168 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12\beta_{7} + 8\beta_{6} + 9\beta_{5} + 19\beta_{4} - 39\beta_{3} - 121\beta_{2} - 21\beta _1 - 139 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 401\beta_{7} + 334\beta_{6} + 230\beta_{5} + 233\beta_{4} - 241\beta_{3} - 2394\beta_{2} - 205\beta _1 + 57 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 157\beta_{7} + 131\beta_{6} + 73\beta_{5} - 20\beta_{4} + 7\beta_{3} - 906\beta_{2} + 4\beta _1 + 994 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 764\beta_{7} + 745\beta_{6} + 62\beta_{5} - 4258\beta_{4} + 5573\beta_{3} - 270\beta_{2} + 3089\beta _1 + 34347 ) / 9 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−1.30108 1.53317i
2.17384 1.62361i
2.52060 + 3.08349i
−2.39336 + 0.0732839i
−1.30108 + 1.53317i
2.17384 + 1.62361i
2.52060 3.08349i
−2.39336 0.0732839i
1.00000 1.73205i 0 −2.00000 3.46410i −9.75052 16.8884i 0 3.50000 6.06218i −8.00000 0 −39.0021
127.2 1.00000 1.73205i 0 −2.00000 3.46410i 0.721413 + 1.24952i 0 3.50000 6.06218i −8.00000 0 2.88565
127.3 1.00000 1.73205i 0 −2.00000 3.46410i 2.98310 + 5.16688i 0 3.50000 6.06218i −8.00000 0 11.9324
127.4 1.00000 1.73205i 0 −2.00000 3.46410i 5.54600 + 9.60595i 0 3.50000 6.06218i −8.00000 0 22.1840
253.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −9.75052 + 16.8884i 0 3.50000 + 6.06218i −8.00000 0 −39.0021
253.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 0.721413 1.24952i 0 3.50000 + 6.06218i −8.00000 0 2.88565
253.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.98310 5.16688i 0 3.50000 + 6.06218i −8.00000 0 11.9324
253.4 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 5.54600 9.60595i 0 3.50000 + 6.06218i −8.00000 0 22.1840
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 253.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.4.f.b 8
3.b odd 2 1 126.4.f.b 8
9.c even 3 1 inner 378.4.f.b 8
9.c even 3 1 1134.4.a.l 4
9.d odd 6 1 126.4.f.b 8
9.d odd 6 1 1134.4.a.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.f.b 8 3.b odd 2 1
126.4.f.b 8 9.d odd 6 1
378.4.f.b 8 1.a even 1 1 trivial
378.4.f.b 8 9.c even 3 1 inner
1134.4.a.l 4 9.c even 3 1
1134.4.a.o 4 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + T_{5}^{7} + 271T_{5}^{6} - 3620T_{5}^{5} + 73087T_{5}^{4} - 448526T_{5}^{3} + 2302885T_{5}^{2} - 3118850T_{5} + 3467044 \) acting on \(S_{4}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + T^{7} + 271 T^{6} + \cdots + 3467044 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7 T + 49)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 5 T^{7} + \cdots + 97721886025 \) Copy content Toggle raw display
$13$ \( T^{8} - 21 T^{7} + \cdots + 6395840676 \) Copy content Toggle raw display
$17$ \( (T^{4} - 23 T^{3} - 9759 T^{2} + \cdots + 20106721)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 94 T^{3} - 13617 T^{2} + \cdots + 36369301)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 374 T^{7} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + 271 T^{7} + \cdots + 39\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{8} - 243 T^{7} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + 181 T^{3} - 159768 T^{2} + \cdots - 2370848354)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 213 T^{7} + \cdots + 28\!\cdots\!09 \) Copy content Toggle raw display
$43$ \( T^{8} - 238 T^{7} + \cdots + 35\!\cdots\!49 \) Copy content Toggle raw display
$47$ \( T^{8} + 675 T^{7} + \cdots + 50\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{4} + 54 T^{3} - 358917 T^{2} + \cdots + 7638055812)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 202 T^{7} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{8} - 1212 T^{7} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{8} + 139 T^{7} + \cdots + 58\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{4} - 1295 T^{3} + \cdots - 17389723904)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 2000 T^{3} + \cdots - 41833011071)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 1545 T^{7} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{8} - 142 T^{7} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{4} - 132 T^{3} + \cdots - 127304987850)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 638 T^{7} + \cdots + 42\!\cdots\!69 \) Copy content Toggle raw display
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