Properties

Label 378.2.u.a
Level $378$
Weight $2$
Character orbit 378.u
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.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{18} q^{2} + ( - 2 \zeta_{18}^{4} + \zeta_{18}) q^{3} + \zeta_{18}^{2} q^{4} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 1) q^{5} + (2 \zeta_{18}^{5} - \zeta_{18}^{2}) q^{6} + ( - \zeta_{18}^{4} + \zeta_{18}) q^{7} - \zeta_{18}^{3} q^{8} - 3 \zeta_{18}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{18} q^{2} + ( - 2 \zeta_{18}^{4} + \zeta_{18}) q^{3} + \zeta_{18}^{2} q^{4} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 1) q^{5} + (2 \zeta_{18}^{5} - \zeta_{18}^{2}) q^{6} + ( - \zeta_{18}^{4} + \zeta_{18}) q^{7} - \zeta_{18}^{3} q^{8} - 3 \zeta_{18}^{2} q^{9} + (\zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18} + 1) q^{10} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{11} + ( - \zeta_{18}^{3} + 2) q^{12} + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{13} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{14} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} - \zeta_{18} + 1) q^{15} + \zeta_{18}^{4} q^{16} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18}) q^{17} + 3 \zeta_{18}^{3} q^{18} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3 \zeta_{18}) q^{19} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 1) q^{20} + ( - \zeta_{18}^{5} - \zeta_{18}^{2}) q^{21} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}) q^{22} + (2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 5 \zeta_{18}^{2} - 2 \zeta_{18} + 2) q^{23} + (\zeta_{18}^{4} - 2 \zeta_{18}) q^{24} + (\zeta_{18}^{5} - \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{25} + ( - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{26} + (3 \zeta_{18}^{3} - 6) q^{27} + q^{28} + (5 \zeta_{18}^{5} - 5 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{29} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 1) q^{30} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18} - 1) q^{31} - \zeta_{18}^{5} q^{32} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2 \zeta_{18} - 4) q^{33} + (2 \zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}^{2} - 2) q^{34} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2}) q^{35} - 3 \zeta_{18}^{4} q^{36} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 4 \zeta_{18} + 4) q^{37} + ( - 3 \zeta_{18}^{5} - \zeta_{18}^{4} + 3 \zeta_{18}^{2} - 3) q^{38} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{39} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} - \zeta_{18} - 1) q^{40} + (4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{2}) q^{41} + (2 \zeta_{18}^{3} - 1) q^{42} + ( - 5 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2) q^{43} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 1) q^{44} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3 \zeta_{18} - 3) q^{45} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{46} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 4 \zeta_{18}^{2} + \zeta_{18} + 2) q^{47} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{2}) q^{48} - \zeta_{18}^{5} q^{49} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{50} + ( - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2}) q^{51} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} - 1) q^{52} + ( - 7 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 5 \zeta_{18}^{2} + 5 \zeta_{18}) q^{53} + ( - 3 \zeta_{18}^{4} + 6 \zeta_{18}) q^{54} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 3) q^{55} - \zeta_{18} q^{56} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} - 6 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 2 \zeta_{18} + 3) q^{57} + (5 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 5) q^{58} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 5 \zeta_{18} - 3) q^{59} + (\zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 1) q^{60} + ( - \zeta_{18}^{5} + 6 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 6 \zeta_{18} + 1) q^{61} + (\zeta_{18}^{5} - \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{62} - 3 q^{63} + (\zeta_{18}^{3} - 1) q^{64} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} - 2) q^{65} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 4 \zeta_{18} - 1) q^{66} + ( - 8 \zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 8 \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{67} + ( - \zeta_{18}^{4} - \zeta_{18}^{3} + 2 \zeta_{18} + 2) q^{68} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2 \zeta_{18} - 10) q^{69} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3}) q^{70} + ( - 5 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - \zeta_{18}^{2} + 6 \zeta_{18} - 4) q^{71} + 3 \zeta_{18}^{5} q^{72} + (5 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 5 \zeta_{18}) q^{73} + ( - 2 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 4 \zeta_{18} + 2) q^{74} + ( - 4 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 3) q^{75} + (\zeta_{18}^{5} + 3 \zeta_{18} - 3) q^{76} + ( - 2 \zeta_{18}^{4} + \zeta_{18}^{2} - 2) q^{77} + ( - \zeta_{18}^{5} - \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 1) q^{78} + (\zeta_{18}^{5} - \zeta_{18}^{3} + 5 \zeta_{18}^{2} - 3 \zeta_{18} + 6) q^{79} + ( - \zeta_{18}^{5} + \zeta_{18}^{2} + \zeta_{18} - 1) q^{80} + 9 \zeta_{18}^{4} q^{81} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 4) q^{82} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 6 \zeta_{18} + 2) q^{83} + ( - 2 \zeta_{18}^{4} + \zeta_{18}) q^{84} - 3 \zeta_{18}^{2} q^{85} + (5 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}) q^{86} + ( - 4 \zeta_{18}^{5} + \zeta_{18}^{4} + 8 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 8 \zeta_{18} - 1) q^{87} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} - 2) q^{88} + (4 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 7 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 4 \zeta_{18}) q^{89} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{90} + (\zeta_{18}^{2} - \zeta_{18}) q^{91} + ( - 2 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 2 \zeta_{18}^{2} - 2) q^{92} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} - 6) q^{93} + (\zeta_{18}^{5} - 4 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18} - 2) q^{94} + ( - 2 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 6 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + \zeta_{18} + 1) q^{95} + ( - \zeta_{18}^{3} - 1) q^{96} + ( - 2 \zeta_{18}^{5} - 8 \zeta_{18}^{4} + \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3) q^{97} + (\zeta_{18}^{3} - 1) q^{98} + (6 \zeta_{18}^{5} + 6 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 6 \zeta_{18}^{2} + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{5} - 3 q^{8} + 3 q^{10} + 6 q^{11} + 9 q^{12} - 3 q^{13} + 9 q^{15} + 9 q^{18} + 3 q^{19} + 3 q^{20} + 6 q^{22} + 6 q^{23} + 9 q^{25} - 27 q^{27} + 6 q^{28} - 3 q^{29} + 9 q^{30} - 3 q^{31} - 18 q^{33} - 9 q^{34} + 3 q^{35} + 12 q^{37} - 18 q^{38} - 6 q^{40} - 12 q^{41} - 6 q^{43} - 3 q^{44} - 9 q^{45} + 15 q^{46} + 24 q^{47} - 9 q^{51} - 3 q^{52} - 18 q^{55} + 24 q^{58} - 24 q^{59} - 3 q^{61} + 9 q^{62} - 18 q^{63} - 3 q^{64} - 12 q^{65} - 9 q^{66} + 3 q^{67} + 9 q^{68} - 45 q^{69} + 3 q^{70} - 12 q^{71} + 3 q^{73} + 18 q^{75} - 18 q^{76} - 12 q^{77} - 9 q^{78} + 33 q^{79} - 6 q^{80} + 24 q^{82} - 6 q^{86} + 18 q^{87} - 12 q^{88} + 21 q^{89} + 18 q^{90} - 12 q^{92} - 27 q^{93} + 6 q^{94} - 12 q^{95} - 9 q^{96} - 15 q^{97} - 3 q^{98} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(\zeta_{18}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−0.766044 0.642788i
0.939693 + 0.342020i
0.939693 0.342020i
−0.766044 + 0.642788i
−0.173648 + 0.984808i
−0.173648 0.984808i
0.766044 + 0.642788i 1.11334 1.32683i 0.173648 + 0.984808i 2.37939 + 0.866025i 1.70574 0.300767i 0.173648 0.984808i −0.500000 + 0.866025i −0.520945 2.95442i 1.26604 + 2.19285i
85.1 −0.939693 0.342020i 0.592396 1.62760i 0.766044 + 0.642788i 0.152704 0.866025i −1.11334 + 1.32683i 0.766044 0.642788i −0.500000 0.866025i −2.29813 1.92836i −0.439693 + 0.761570i
169.1 −0.939693 + 0.342020i 0.592396 + 1.62760i 0.766044 0.642788i 0.152704 + 0.866025i −1.11334 1.32683i 0.766044 + 0.642788i −0.500000 + 0.866025i −2.29813 + 1.92836i −0.439693 0.761570i
211.1 0.766044 0.642788i 1.11334 + 1.32683i 0.173648 0.984808i 2.37939 0.866025i 1.70574 + 0.300767i 0.173648 + 0.984808i −0.500000 0.866025i −0.520945 + 2.95442i 1.26604 2.19285i
295.1 0.173648 0.984808i −1.70574 0.300767i −0.939693 0.342020i −1.03209 + 0.866025i −0.592396 + 1.62760i −0.939693 + 0.342020i −0.500000 + 0.866025i 2.81908 + 1.02606i 0.673648 + 1.16679i
337.1 0.173648 + 0.984808i −1.70574 + 0.300767i −0.939693 + 0.342020i −1.03209 0.866025i −0.592396 1.62760i −0.939693 0.342020i −0.500000 0.866025i 2.81908 1.02606i 0.673648 1.16679i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.u.a 6
27.e even 9 1 inner 378.2.u.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.u.a 6 1.a even 1 1 trivial
378.2.u.a 6 27.e even 9 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + 9T^{3} + 27 \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + 3 T^{3} + 9 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$11$ \( T^{6} - 6 T^{5} + 18 T^{4} - 3 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} + 33 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} - 36 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + 108 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$31$ \( T^{6} + 3 T^{5} + 24 T^{4} + 17 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( T^{6} - 12 T^{5} + 132 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( T^{6} + 12 T^{5} + 144 T^{4} + \cdots + 36864 \) Copy content Toggle raw display
$43$ \( T^{6} + 6 T^{5} - 6 T^{4} - 361 T^{3} + \cdots + 5329 \) Copy content Toggle raw display
$47$ \( T^{6} - 24 T^{5} + 306 T^{4} + \cdots + 47961 \) Copy content Toggle raw display
$53$ \( (T^{3} - 117 T + 153)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 24 T^{5} + 252 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$61$ \( T^{6} + 3 T^{5} + 114 T^{4} + \cdots + 16129 \) Copy content Toggle raw display
$67$ \( T^{6} - 3 T^{5} - 96 T^{4} + \cdots + 94249 \) Copy content Toggle raw display
$71$ \( T^{6} + 12 T^{5} + 189 T^{4} + \cdots + 106929 \) Copy content Toggle raw display
$73$ \( T^{6} - 3 T^{5} + 123 T^{4} + \cdots + 1369 \) Copy content Toggle raw display
$79$ \( T^{6} - 33 T^{5} + 492 T^{4} + \cdots + 73441 \) Copy content Toggle raw display
$83$ \( T^{6} + 144 T^{4} - 720 T^{3} + \cdots + 5184 \) Copy content Toggle raw display
$89$ \( T^{6} - 21 T^{5} + 405 T^{4} + \cdots + 12321 \) Copy content Toggle raw display
$97$ \( T^{6} + 15 T^{5} + 120 T^{4} + \cdots + 130321 \) Copy content Toggle raw display
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