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Properties

Label 378.4.g.e
Level $378$
Weight $4$
Character orbit 378.g
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.g (of order \(3\), degree \(2\), 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: \( x^{8} - x^{7} + 66x^{6} + 59x^{5} + 3770x^{4} + 721x^{3} + 29779x^{2} + 1374x + 209764 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
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.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} + ( - 4 \beta_1 - 4) q^{4} + (\beta_{6} + \beta_1) q^{5} + ( - \beta_{7} - \beta_{4} - \beta_{3} + 3) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} + ( - 4 \beta_1 - 4) q^{4} + (\beta_{6} + \beta_1) q^{5} + ( - \beta_{7} - \beta_{4} - \beta_{3} + 3) q^{7} - 8 q^{8} + (2 \beta_{2} + 2 \beta_1 + 2) q^{10} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} - 14 \beta_1 - 14) q^{11} + (\beta_{7} - 3 \beta_{6} + \beta_{5} + 3 \beta_{2} - 2) q^{13} + ( - 2 \beta_{4} - 6 \beta_1) q^{14} + 16 \beta_1 q^{16} + (3 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 30 \beta_1 + 30) q^{17} + (3 \beta_{7} + 2 \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 10 \beta_1) q^{19} + ( - 4 \beta_{6} + 4 \beta_{2} + 4) q^{20} + (4 \beta_{6} - 4 \beta_{5} + 2 \beta_{3} - 4 \beta_{2} - 28) q^{22} + (8 \beta_{7} - 2 \beta_{6} + 8 \beta_{4} + 48 \beta_1) q^{23} + (3 \beta_{5} - 8 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 24 \beta_1 - 24) q^{25} + (2 \beta_{7} - 6 \beta_{6} + 2 \beta_{4} + 4 \beta_1) q^{26} + (4 \beta_{7} + 4 \beta_{3} - 12 \beta_1 - 12) q^{28} + (6 \beta_{7} + 11 \beta_{6} - 2 \beta_{5} + 4 \beta_{3} - 11 \beta_{2} + 67) q^{29} + (7 \beta_{5} - 8 \beta_{4} - \beta_{3} - 12 \beta_{2} + 71 \beta_1 + 71) q^{31} + (32 \beta_1 + 32) q^{32} + (4 \beta_{7} - 4 \beta_{6} + 8 \beta_{5} - 2 \beta_{3} + 4 \beta_{2} + 60) q^{34} + (4 \beta_{7} - 7 \beta_{5} + 6 \beta_{4} + 4 \beta_{3} + 14 \beta_{2} + 20 \beta_1 - 26) q^{35} + ( - 3 \beta_{7} + 24 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 6 \beta_{3} + \cdots + 47 \beta_1) q^{37}+ \cdots + (2 \beta_{7} - 28 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - 42 \beta_{2} + 160 \beta_1 - 258) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8} + 8 q^{10} - 56 q^{11} - 18 q^{13} + 22 q^{14} - 64 q^{16} + 118 q^{17} + 37 q^{19} + 32 q^{20} - 224 q^{22} - 200 q^{23} - 104 q^{25} - 18 q^{26} - 56 q^{28} + 524 q^{29} + 276 q^{31} + 128 q^{32} + 472 q^{34} - 290 q^{35} - 185 q^{37} - 74 q^{38} + 32 q^{40} - 60 q^{41} - 1556 q^{43} - 224 q^{44} + 400 q^{46} - 30 q^{47} - 1159 q^{49} - 416 q^{50} + 36 q^{52} - 480 q^{53} + 1456 q^{55} - 200 q^{56} + 524 q^{58} + 296 q^{59} + 474 q^{61} + 1104 q^{62} + 512 q^{64} - 1542 q^{65} + 1319 q^{67} + 472 q^{68} - 32 q^{70} + 1852 q^{71} - 1423 q^{73} + 370 q^{74} - 296 q^{76} - 1228 q^{77} + 765 q^{79} - 64 q^{80} - 60 q^{82} + 1660 q^{83} - 584 q^{85} - 1556 q^{86} + 448 q^{88} + 864 q^{89} - 738 q^{91} + 1600 q^{92} + 60 q^{94} - 1766 q^{95} + 1088 q^{97} - 2704 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 66x^{6} + 59x^{5} + 3770x^{4} + 721x^{3} + 29779x^{2} + 1374x + 209764 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1049071 \nu^{7} + 115038401 \nu^{6} - 163698896 \nu^{5} + 6652953189 \nu^{4} + 3920788192 \nu^{3} + 430332705877 \nu^{2} + \cdots + 368827486562 ) / 3033579204498 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1696542203 \nu^{7} + 16739134507 \nu^{6} - 23819679472 \nu^{5} + 926039756877 \nu^{4} + 570510371744 \nu^{3} + \cdots + 495081144679420 ) / 63705163294458 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 863252840 \nu^{7} - 9761813387 \nu^{6} + 75308363320 \nu^{5} - 543267423074 \nu^{4} + 3215243331501 \nu^{3} + \cdots - 128122572731930 ) / 10617527215743 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7051508536 \nu^{7} + 31228732907 \nu^{6} - 412942482080 \nu^{5} + 868174142577 \nu^{4} - 20223346800686 \nu^{3} + \cdots - 39943270058320 ) / 63705163294458 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1208164720 \nu^{7} - 1739457099 \nu^{6} - 58942130264 \nu^{5} - 384930665533 \nu^{4} - 3607234572013 \nu^{3} + \cdots - 212042096779230 ) / 10617527215743 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8617899467 \nu^{7} + 10568597603 \nu^{6} + 583780276750 \nu^{5} + 1638688334415 \nu^{4} + 33815120349688 \nu^{3} + \cdots + 74164049534822 ) / 63705163294458 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9748790105 \nu^{7} + 20249178479 \nu^{6} + 500910335965 \nu^{5} + 2482301067297 \nu^{4} + 34135187416753 \nu^{3} + \cdots + 219625364013092 ) / 63705163294458 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} - 5\beta_{6} - 3\beta_{5} - 2\beta_{4} + 6\beta_{3} + 197\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 94\beta_{7} - 101\beta_{6} + 12\beta_{5} + 41\beta_{3} + 101\beta_{2} - 257 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -275\beta_{5} + 42\beta_{4} - 233\beta_{3} + 423\beta_{2} - 10311\beta _1 - 10311 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5158\beta_{7} + 6515\beta_{6} + 2431\beta_{5} - 5158\beta_{4} - 4862\beta_{3} - 25967\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3062\beta_{7} + 31417\beta_{6} + 29096\beta_{5} - 16079\beta_{3} - 31417\beta_{2} + 605185 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -140789\beta_{5} + 295406\beta_{4} + 154617\beta_{3} - 407365\beta_{2} + 2144401\beta _1 + 2144401 ) / 6 \) Copy content Toggle raw display

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
1.39378 − 2.41410i
−3.50329 + 6.06788i
−1.44566 + 2.50395i
4.05517 − 7.02376i
1.39378 + 2.41410i
−3.50329 − 6.06788i
−1.44566 − 2.50395i
4.05517 + 7.02376i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i −8.97703 − 15.5487i 0 13.2278 + 12.9624i −8.00000 0 17.9541 − 31.0973i
109.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −2.21575 − 3.83779i 0 −11.5959 − 14.4407i −8.00000 0 4.43150 − 7.67558i
109.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 1.18685 + 2.05569i 0 9.15909 − 16.0969i −8.00000 0 −2.37371 + 4.11138i
109.4 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 8.00592 + 13.8667i 0 1.70902 + 18.4412i −8.00000 0 −16.0118 + 27.7333i
163.1 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −8.97703 + 15.5487i 0 13.2278 − 12.9624i −8.00000 0 17.9541 + 31.0973i
163.2 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −2.21575 + 3.83779i 0 −11.5959 + 14.4407i −8.00000 0 4.43150 + 7.67558i
163.3 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 1.18685 − 2.05569i 0 9.15909 + 16.0969i −8.00000 0 −2.37371 − 4.11138i
163.4 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 8.00592 − 13.8667i 0 1.70902 − 18.4412i −8.00000 0 −16.0118 − 27.7333i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.g.e yes 8
3.b odd 2 1 378.4.g.d ✓ 8
7.c even 3 1 inner 378.4.g.e yes 8
21.h odd 6 1 378.4.g.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.4.g.d ✓ 8 3.b odd 2 1
378.4.g.d ✓ 8 21.h odd 6 1
378.4.g.e yes 8 1.a even 1 1 trivial
378.4.g.e yes 8 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 4T_{5}^{7} + 310T_{5}^{6} + 48T_{5}^{5} + 85860T_{5}^{4} + 155736T_{5}^{3} + 1263600T_{5}^{2} - 1850688T_{5} + 9144576 \) 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
$5$ \( T^{8} + 4 T^{7} + 310 T^{6} + \cdots + 9144576 \) Copy content Toggle raw display
$7$ \( T^{8} - 25 T^{7} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{8} + 56 T^{7} + \cdots + 943105130496 \) Copy content Toggle raw display
$13$ \( (T^{4} + 9 T^{3} - 2637 T^{2} + \cdots + 1313022)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 118 T^{7} + \cdots + 8822183086656 \) Copy content Toggle raw display
$19$ \( T^{8} - 37 T^{7} + \cdots + 96\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{8} + 200 T^{7} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{4} - 262 T^{3} - 27174 T^{2} + \cdots - 737431128)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 276 T^{7} + \cdots + 11\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( T^{8} + 185 T^{7} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} + 30 T^{3} - 329454 T^{2} + \cdots + 23111436456)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 778 T^{3} + 137328 T^{2} + \cdots - 309283361)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 30 T^{7} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + 480 T^{7} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{8} - 296 T^{7} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{8} - 474 T^{7} + \cdots + 37\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{8} - 1319 T^{7} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{4} - 926 T^{3} + \cdots - 55652552064)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 1423 T^{7} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{8} - 765 T^{7} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( (T^{4} - 830 T^{3} - 10644 T^{2} + \cdots + 331327584)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 864 T^{7} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{4} - 544 T^{3} + \cdots - 91533471583)^{2} \) Copy content Toggle raw display
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