Properties

Label 378.2.e.d
Level $378$
Weight $2$
Character orbit 378.e
Analytic conductor $3.018$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - \beta_{2} q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - \beta_{2} q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{7} + q^{8} - \beta_{2} q^{10} + ( - \beta_{3} - \beta_{2}) q^{11} + ( - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{14} + q^{16} + (2 \beta_{5} - 2 \beta_{4} + 2) q^{17} + (3 \beta_{5} - 2 \beta_{4} - 3 \beta_1) q^{19} - \beta_{2} q^{20} + ( - \beta_{3} - \beta_{2}) q^{22} + ( - \beta_{5} - 2 \beta_{4} + 2) q^{23} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{25} + ( - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{26} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{28} + ( - 3 \beta_{5} + 2 \beta_{2}) q^{29} + (\beta_{3} + \beta_1 - 6) q^{31} + q^{32} + (2 \beta_{5} - 2 \beta_{4} + 2) q^{34} + ( - 3 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{35} + \beta_{4} q^{37} + (3 \beta_{5} - 2 \beta_{4} - 3 \beta_1) q^{38} - \beta_{2} q^{40} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{41} + (3 \beta_{5} + 2 \beta_{4} + 3 \beta_{2} - 2) q^{43} + ( - \beta_{3} - \beta_{2}) q^{44} + ( - \beta_{5} - 2 \beta_{4} + 2) q^{46} + ( - 3 \beta_{3} + 3 \beta_1 - 3) q^{47} + ( - 3 \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{49} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{50} + ( - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{52} + (\beta_{5} + 5 \beta_{4} + \beta_{2} - 5) q^{53} + (2 \beta_{3} - \beta_1 - 4) q^{55} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{56} + ( - 3 \beta_{5} + 2 \beta_{2}) q^{58} + ( - 2 \beta_{3} + \beta_1 - 5) q^{59} + ( - 2 \beta_{3} + \beta_1 - 3) q^{61} + (\beta_{3} + \beta_1 - 6) q^{62} + q^{64} + (5 \beta_{3} + \beta_1 - 2) q^{65} + ( - 4 \beta_{3} - \beta_1 - 2) q^{67} + (2 \beta_{5} - 2 \beta_{4} + 2) q^{68} + ( - 3 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{70} + (2 \beta_{3} - 7 \beta_1 - 4) q^{71} + (\beta_{5} - 8 \beta_{4} - 4 \beta_{2} + 8) q^{73} + \beta_{4} q^{74} + (3 \beta_{5} - 2 \beta_{4} - 3 \beta_1) q^{76} + (\beta_{5} - 4 \beta_{4} + 2 \beta_{3} + \beta_{2} + 1) q^{77} + (\beta_{3} + 4 \beta_1) q^{79} - \beta_{2} q^{80} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{82} + (2 \beta_{5} + \beta_{4} + 3 \beta_{2} - 1) q^{83} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{85} + (3 \beta_{5} + 2 \beta_{4} + 3 \beta_{2} - 2) q^{86} + ( - \beta_{3} - \beta_{2}) q^{88} + (4 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 4 \beta_1) q^{89} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + \beta_1 - 8) q^{91} + ( - \beta_{5} - 2 \beta_{4} + 2) q^{92} + ( - 3 \beta_{3} + 3 \beta_1 - 3) q^{94} + ( - 2 \beta_{3} - 3 \beta_1 - 3) q^{95} + ( - 2 \beta_{5} - 8 \beta_{4} + 2 \beta_{2} + 8) q^{97} + ( - 3 \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} - q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 6 q^{4} - q^{5} + 2 q^{7} + 6 q^{8} - q^{10} + q^{11} + 8 q^{13} + 2 q^{14} + 6 q^{16} + 4 q^{17} - 3 q^{19} - q^{20} + q^{22} + 7 q^{23} + 2 q^{25} + 8 q^{26} + 2 q^{28} + 5 q^{29} - 40 q^{31} + 6 q^{32} + 4 q^{34} + 13 q^{35} + 3 q^{37} - 3 q^{38} - q^{40} - 6 q^{43} + q^{44} + 7 q^{46} - 18 q^{47} + 12 q^{49} + 2 q^{50} + 8 q^{52} - 15 q^{53} - 26 q^{55} + 2 q^{56} + 5 q^{58} - 28 q^{59} - 16 q^{61} - 40 q^{62} + 6 q^{64} - 24 q^{65} - 2 q^{67} + 4 q^{68} + 13 q^{70} - 14 q^{71} + 19 q^{73} + 3 q^{74} - 3 q^{76} - 10 q^{77} - 10 q^{79} - q^{80} - 2 q^{83} - 2 q^{85} - 6 q^{86} + q^{88} + 9 q^{89} - 46 q^{91} + 7 q^{92} - 18 q^{94} - 8 q^{95} + 28 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-\beta_{4}\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 1.41036i
0.500000 + 2.05195i
0.500000 + 0.224437i
0.500000 + 1.41036i
0.500000 2.05195i
0.500000 0.224437i
1.00000 0 1.00000 −1.59097 2.75564i 0 −2.56238 0.658939i 1.00000 0 −1.59097 2.75564i
37.2 1.00000 0 1.00000 0.296790 + 0.514055i 0 2.32383 + 1.26483i 1.00000 0 0.296790 + 0.514055i
37.3 1.00000 0 1.00000 0.794182 + 1.37556i 0 1.23855 2.33795i 1.00000 0 0.794182 + 1.37556i
235.1 1.00000 0 1.00000 −1.59097 + 2.75564i 0 −2.56238 + 0.658939i 1.00000 0 −1.59097 + 2.75564i
235.2 1.00000 0 1.00000 0.296790 0.514055i 0 2.32383 1.26483i 1.00000 0 0.296790 0.514055i
235.3 1.00000 0 1.00000 0.794182 1.37556i 0 1.23855 + 2.33795i 1.00000 0 0.794182 1.37556i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.e.d 6
3.b odd 2 1 126.2.e.c 6
4.b odd 2 1 3024.2.q.g 6
7.b odd 2 1 2646.2.e.p 6
7.c even 3 1 378.2.h.c 6
7.c even 3 1 2646.2.f.l 6
7.d odd 6 1 2646.2.f.m 6
7.d odd 6 1 2646.2.h.o 6
9.c even 3 1 378.2.h.c 6
9.c even 3 1 1134.2.g.l 6
9.d odd 6 1 126.2.h.d yes 6
9.d odd 6 1 1134.2.g.m 6
12.b even 2 1 1008.2.q.g 6
21.c even 2 1 882.2.e.o 6
21.g even 6 1 882.2.f.o 6
21.g even 6 1 882.2.h.p 6
21.h odd 6 1 126.2.h.d yes 6
21.h odd 6 1 882.2.f.n 6
28.g odd 6 1 3024.2.t.h 6
36.f odd 6 1 3024.2.t.h 6
36.h even 6 1 1008.2.t.h 6
63.g even 3 1 1134.2.g.l 6
63.g even 3 1 2646.2.f.l 6
63.h even 3 1 inner 378.2.e.d 6
63.h even 3 1 7938.2.a.ca 3
63.i even 6 1 882.2.e.o 6
63.i even 6 1 7938.2.a.bw 3
63.j odd 6 1 126.2.e.c 6
63.j odd 6 1 7938.2.a.bv 3
63.k odd 6 1 2646.2.f.m 6
63.l odd 6 1 2646.2.h.o 6
63.n odd 6 1 882.2.f.n 6
63.n odd 6 1 1134.2.g.m 6
63.o even 6 1 882.2.h.p 6
63.s even 6 1 882.2.f.o 6
63.t odd 6 1 2646.2.e.p 6
63.t odd 6 1 7938.2.a.bz 3
84.n even 6 1 1008.2.t.h 6
252.u odd 6 1 3024.2.q.g 6
252.bb even 6 1 1008.2.q.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 3.b odd 2 1
126.2.e.c 6 63.j odd 6 1
126.2.h.d yes 6 9.d odd 6 1
126.2.h.d yes 6 21.h odd 6 1
378.2.e.d 6 1.a even 1 1 trivial
378.2.e.d 6 63.h even 3 1 inner
378.2.h.c 6 7.c even 3 1
378.2.h.c 6 9.c even 3 1
882.2.e.o 6 21.c even 2 1
882.2.e.o 6 63.i even 6 1
882.2.f.n 6 21.h odd 6 1
882.2.f.n 6 63.n odd 6 1
882.2.f.o 6 21.g even 6 1
882.2.f.o 6 63.s even 6 1
882.2.h.p 6 21.g even 6 1
882.2.h.p 6 63.o even 6 1
1008.2.q.g 6 12.b even 2 1
1008.2.q.g 6 252.bb even 6 1
1008.2.t.h 6 36.h even 6 1
1008.2.t.h 6 84.n even 6 1
1134.2.g.l 6 9.c even 3 1
1134.2.g.l 6 63.g even 3 1
1134.2.g.m 6 9.d odd 6 1
1134.2.g.m 6 63.n odd 6 1
2646.2.e.p 6 7.b odd 2 1
2646.2.e.p 6 63.t odd 6 1
2646.2.f.l 6 7.c even 3 1
2646.2.f.l 6 63.g even 3 1
2646.2.f.m 6 7.d odd 6 1
2646.2.f.m 6 63.k odd 6 1
2646.2.h.o 6 7.d odd 6 1
2646.2.h.o 6 63.l odd 6 1
3024.2.q.g 6 4.b odd 2 1
3024.2.q.g 6 252.u odd 6 1
3024.2.t.h 6 28.g odd 6 1
3024.2.t.h 6 36.f odd 6 1
7938.2.a.bv 3 63.j odd 6 1
7938.2.a.bw 3 63.i even 6 1
7938.2.a.bz 3 63.t odd 6 1
7938.2.a.ca 3 63.h even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + T_{5}^{5} + 7T_{5}^{4} - 12T_{5}^{3} + 33T_{5}^{2} - 18T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} + 7 T^{4} - 12 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} - 4 T^{4} + 31 T^{3} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} - T^{5} + 7 T^{4} + 12 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} - 8 T^{5} + 63 T^{4} + \cdots + 4761 \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} + 28 T^{4} + 240 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} + 45 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( T^{6} - 7 T^{5} + 37 T^{4} - 78 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{6} - 5 T^{5} + 91 T^{4} + \cdots + 131769 \) Copy content Toggle raw display
$31$ \( (T^{3} + 20 T^{2} + 121 T + 201)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{6} + 33 T^{4} - 18 T^{3} + 1089 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} + 6 T^{5} + 105 T^{4} + \cdots + 16129 \) Copy content Toggle raw display
$47$ \( (T^{3} + 9 T^{2} - 54 T - 189)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 15 T^{5} + 159 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( (T^{3} + 14 T^{2} + 39 T - 63)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 8 T^{2} - 5 T - 93)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + T^{2} - 112 T - 211)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 7 T^{2} - 198 T - 1593)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 19 T^{5} + 353 T^{4} + \cdots + 398161 \) Copy content Toggle raw display
$79$ \( (T^{3} + 5 T^{2} - 74 T - 321)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 2 T^{5} + 67 T^{4} + \cdots + 21609 \) Copy content Toggle raw display
$89$ \( T^{6} - 9 T^{5} + 123 T^{4} + 396 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$97$ \( T^{6} - 28 T^{5} + 572 T^{4} + \cdots + 61504 \) Copy content Toggle raw display
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