Properties

Label 378.2.e.c
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
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 2 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 4\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} - 10\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(16 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} - 17 \beta_{1} + 5\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-14 \beta_{5} - 16 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 23 \beta_{1} + 47\)\()/3\)

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 + 2.05195i
0.500000 1.41036i
0.500000 + 0.224437i
0.500000 2.05195i
0.500000 + 1.41036i
0.500000 0.224437i
−1.00000 0 1.00000 −0.230252 0.398809i 0 0.0665372 + 2.64491i −1.00000 0 0.230252 + 0.398809i
37.2 −1.00000 0 1.00000 0.880438 + 1.52496i 0 −0.710533 2.54856i −1.00000 0 −0.880438 1.52496i
37.3 −1.00000 0 1.00000 1.84981 + 3.20397i 0 2.64400 0.0963576i −1.00000 0 −1.84981 3.20397i
235.1 −1.00000 0 1.00000 −0.230252 + 0.398809i 0 0.0665372 2.64491i −1.00000 0 0.230252 0.398809i
235.2 −1.00000 0 1.00000 0.880438 1.52496i 0 −0.710533 + 2.54856i −1.00000 0 −0.880438 + 1.52496i
235.3 −1.00000 0 1.00000 1.84981 3.20397i 0 2.64400 + 0.0963576i −1.00000 0 −1.84981 + 3.20397i
\(n\): 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.c 6
3.b odd 2 1 126.2.e.d 6
4.b odd 2 1 3024.2.q.h 6
7.b odd 2 1 2646.2.e.o 6
7.c even 3 1 378.2.h.d 6
7.c even 3 1 2646.2.f.o 6
7.d odd 6 1 2646.2.f.n 6
7.d odd 6 1 2646.2.h.p 6
9.c even 3 1 378.2.h.d 6
9.c even 3 1 1134.2.g.n 6
9.d odd 6 1 126.2.h.c yes 6
9.d odd 6 1 1134.2.g.k 6
12.b even 2 1 1008.2.q.h 6
21.c even 2 1 882.2.e.p 6
21.g even 6 1 882.2.f.m 6
21.g even 6 1 882.2.h.o 6
21.h odd 6 1 126.2.h.c yes 6
21.h odd 6 1 882.2.f.l 6
28.g odd 6 1 3024.2.t.g 6
36.f odd 6 1 3024.2.t.g 6
36.h even 6 1 1008.2.t.g 6
63.g even 3 1 1134.2.g.n 6
63.g even 3 1 2646.2.f.o 6
63.h even 3 1 inner 378.2.e.c 6
63.h even 3 1 7938.2.a.bu 3
63.i even 6 1 882.2.e.p 6
63.i even 6 1 7938.2.a.by 3
63.j odd 6 1 126.2.e.d 6
63.j odd 6 1 7938.2.a.cb 3
63.k odd 6 1 2646.2.f.n 6
63.l odd 6 1 2646.2.h.p 6
63.n odd 6 1 882.2.f.l 6
63.n odd 6 1 1134.2.g.k 6
63.o even 6 1 882.2.h.o 6
63.s even 6 1 882.2.f.m 6
63.t odd 6 1 2646.2.e.o 6
63.t odd 6 1 7938.2.a.bx 3
84.n even 6 1 1008.2.t.g 6
252.u odd 6 1 3024.2.q.h 6
252.bb even 6 1 1008.2.q.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.d 6 3.b odd 2 1
126.2.e.d 6 63.j odd 6 1
126.2.h.c yes 6 9.d odd 6 1
126.2.h.c yes 6 21.h odd 6 1
378.2.e.c 6 1.a even 1 1 trivial
378.2.e.c 6 63.h even 3 1 inner
378.2.h.d 6 7.c even 3 1
378.2.h.d 6 9.c even 3 1
882.2.e.p 6 21.c even 2 1
882.2.e.p 6 63.i even 6 1
882.2.f.l 6 21.h odd 6 1
882.2.f.l 6 63.n odd 6 1
882.2.f.m 6 21.g even 6 1
882.2.f.m 6 63.s even 6 1
882.2.h.o 6 21.g even 6 1
882.2.h.o 6 63.o even 6 1
1008.2.q.h 6 12.b even 2 1
1008.2.q.h 6 252.bb even 6 1
1008.2.t.g 6 36.h even 6 1
1008.2.t.g 6 84.n even 6 1
1134.2.g.k 6 9.d odd 6 1
1134.2.g.k 6 63.n odd 6 1
1134.2.g.n 6 9.c even 3 1
1134.2.g.n 6 63.g even 3 1
2646.2.e.o 6 7.b odd 2 1
2646.2.e.o 6 63.t odd 6 1
2646.2.f.n 6 7.d odd 6 1
2646.2.f.n 6 63.k odd 6 1
2646.2.f.o 6 7.c even 3 1
2646.2.f.o 6 63.g even 3 1
2646.2.h.p 6 7.d odd 6 1
2646.2.h.p 6 63.l odd 6 1
3024.2.q.h 6 4.b odd 2 1
3024.2.q.h 6 252.u odd 6 1
3024.2.t.g 6 28.g odd 6 1
3024.2.t.g 6 36.f odd 6 1
7938.2.a.bu 3 63.h even 3 1
7938.2.a.bx 3 63.t odd 6 1
7938.2.a.by 3 63.i even 6 1
7938.2.a.cb 3 63.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 5 T_{5}^{5} + 21 T_{5}^{4} - 26 T_{5}^{3} + 31 T_{5}^{2} + 12 T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{6} \)
$3$ \( \)
$5$ \( 1 - 5 T + 6 T^{2} - T^{3} + 31 T^{4} - 68 T^{5} + 29 T^{6} - 340 T^{7} + 775 T^{8} - 125 T^{9} + 3750 T^{10} - 15625 T^{11} + 15625 T^{12} \)
$7$ \( 1 - 4 T + 14 T^{2} - 55 T^{3} + 98 T^{4} - 196 T^{5} + 343 T^{6} \)
$11$ \( 1 - T - 6 T^{2} + 103 T^{3} - 83 T^{4} - 352 T^{5} + 5027 T^{6} - 3872 T^{7} - 10043 T^{8} + 137093 T^{9} - 87846 T^{10} - 161051 T^{11} + 1771561 T^{12} \)
$13$ \( 1 + 2 T - 32 T^{2} - 26 T^{3} + 730 T^{4} + 230 T^{5} - 10729 T^{6} + 2990 T^{7} + 123370 T^{8} - 57122 T^{9} - 913952 T^{10} + 742586 T^{11} + 4826809 T^{12} \)
$17$ \( 1 - 4 T + 9 T^{2} - 92 T^{3} + 58 T^{4} + 20 T^{5} + 5393 T^{6} + 340 T^{7} + 16762 T^{8} - 451996 T^{9} + 751689 T^{10} - 5679428 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 3 T - 42 T^{2} - 61 T^{3} + 1311 T^{4} + 726 T^{5} - 27501 T^{6} + 13794 T^{7} + 473271 T^{8} - 418399 T^{9} - 5473482 T^{10} + 7428297 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - 7 T - 24 T^{2} + 127 T^{3} + 1417 T^{4} - 3484 T^{5} - 22393 T^{6} - 80132 T^{7} + 749593 T^{8} + 1545209 T^{9} - 6716184 T^{10} - 45054401 T^{11} + 148035889 T^{12} \)
$29$ \( 1 - 5 T - 30 T^{2} + 371 T^{3} - 185 T^{4} - 6020 T^{5} + 44357 T^{6} - 174580 T^{7} - 155585 T^{8} + 9048319 T^{9} - 21218430 T^{10} - 102555745 T^{11} + 594823321 T^{12} \)
$31$ \( ( 1 - 14 T + 138 T^{2} - 841 T^{3} + 4278 T^{4} - 13454 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( 1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 180375 T^{7} + 1770117 T^{8} - 13575004 T^{9} - 39357381 T^{10} + 624095613 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 - 12 T - 18 T^{2} + 78 T^{3} + 7470 T^{4} - 24546 T^{5} - 158105 T^{6} - 1006386 T^{7} + 12557070 T^{8} + 5375838 T^{9} - 50863698 T^{10} - 1390274412 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 - 18 T + 114 T^{2} - 682 T^{3} + 7188 T^{4} - 33492 T^{5} + 63039 T^{6} - 1440156 T^{7} + 13290612 T^{8} - 54223774 T^{9} + 389743314 T^{10} - 2646151974 T^{11} + 6321363049 T^{12} \)
$47$ \( ( 1 - 3 T + 117 T^{2} - 309 T^{3} + 5499 T^{4} - 6627 T^{5} + 103823 T^{6} )^{2} \)
$53$ \( 1 + 9 T - 36 T^{2} - 873 T^{3} - 1179 T^{4} + 26334 T^{5} + 272077 T^{6} + 1395702 T^{7} - 3311811 T^{8} - 129969621 T^{9} - 284057316 T^{10} + 3763759437 T^{11} + 22164361129 T^{12} \)
$59$ \( ( 1 - 4 T + 76 T^{2} - 649 T^{3} + 4484 T^{4} - 13924 T^{5} + 205379 T^{6} )^{2} \)
$61$ \( ( 1 + 4 T + 48 T^{2} - 229 T^{3} + 2928 T^{4} + 14884 T^{5} + 226981 T^{6} )^{2} \)
$67$ \( ( 1 + 5 T + 143 T^{2} + 521 T^{3} + 9581 T^{4} + 22445 T^{5} + 300763 T^{6} )^{2} \)
$71$ \( ( 1 + 7 T + 163 T^{2} + 895 T^{3} + 11573 T^{4} + 35287 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( 1 + 25 T + 254 T^{2} + 2073 T^{3} + 20533 T^{4} + 115046 T^{5} + 366817 T^{6} + 8398358 T^{7} + 109420357 T^{8} + 806432241 T^{9} + 7213153214 T^{10} + 51826789825 T^{11} + 151334226289 T^{12} \)
$79$ \( ( 1 + 7 T + 93 T^{2} + 335 T^{3} + 7347 T^{4} + 43687 T^{5} + 493039 T^{6} )^{2} \)
$83$ \( 1 + 8 T - 180 T^{2} - 518 T^{3} + 29404 T^{4} + 32420 T^{5} - 2713585 T^{6} + 2690860 T^{7} + 202564156 T^{8} - 296185666 T^{9} - 8542497780 T^{10} + 31512325144 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 - 9 T - 180 T^{2} + 729 T^{3} + 31041 T^{4} - 54846 T^{5} - 2925911 T^{6} - 4881294 T^{7} + 245875761 T^{8} + 513922401 T^{9} - 11293603380 T^{10} - 50256535041 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 + 28 T + 257 T^{2} + 2820 T^{3} + 59506 T^{4} + 545924 T^{5} + 3126001 T^{6} + 52954628 T^{7} + 559891954 T^{8} + 2573737860 T^{9} + 22752025217 T^{10} + 240445527196 T^{11} + 832972004929 T^{12} \)
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