Properties

Label 390.2.bd.a
Level $390$
Weight $2$
Character orbit 390.bd
Analytic conductor $3.114$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.bd (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(\zeta_{24}^{6}\) \(\zeta_{24}^{2} - \zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
0.866025 + 0.500000i −0.258819 0.965926i 0.500000 + 0.866025i −2.12132 0.707107i 0.258819 0.965926i −1.88366 3.26260i 1.00000i −0.866025 + 0.500000i −1.48356 1.67303i
7.2 0.866025 + 0.500000i 0.258819 + 0.965926i 0.500000 + 0.866025i 2.12132 + 0.707107i −0.258819 + 0.965926i −0.848387 1.46945i 1.00000i −0.866025 + 0.500000i 1.48356 + 1.67303i
37.1 −0.866025 + 0.500000i −0.965926 0.258819i 0.500000 0.866025i 2.12132 + 0.707107i 0.965926 0.258819i −1.56583 + 2.71209i 1.00000i 0.866025 + 0.500000i −2.19067 + 0.448288i
37.2 −0.866025 + 0.500000i 0.965926 + 0.258819i 0.500000 0.866025i −2.12132 0.707107i −0.965926 + 0.258819i 2.29788 3.98004i 1.00000i 0.866025 + 0.500000i 2.19067 0.448288i
223.1 0.866025 0.500000i −0.258819 + 0.965926i 0.500000 0.866025i −2.12132 + 0.707107i 0.258819 + 0.965926i −1.88366 + 3.26260i 1.00000i −0.866025 0.500000i −1.48356 + 1.67303i
223.2 0.866025 0.500000i 0.258819 0.965926i 0.500000 0.866025i 2.12132 0.707107i −0.258819 0.965926i −0.848387 + 1.46945i 1.00000i −0.866025 0.500000i 1.48356 1.67303i
253.1 −0.866025 0.500000i −0.965926 + 0.258819i 0.500000 + 0.866025i 2.12132 0.707107i 0.965926 + 0.258819i −1.56583 2.71209i 1.00000i 0.866025 0.500000i −2.19067 0.448288i
253.2 −0.866025 0.500000i 0.965926 0.258819i 0.500000 + 0.866025i −2.12132 + 0.707107i −0.965926 0.258819i 2.29788 + 3.98004i 1.00000i 0.866025 0.500000i 2.19067 + 0.448288i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 253.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.bd.a 8
5.c odd 4 1 390.2.bn.a yes 8
13.f odd 12 1 390.2.bn.a yes 8
65.t even 12 1 inner 390.2.bd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bd.a 8 1.a even 1 1 trivial
390.2.bd.a 8 65.t even 12 1 inner
390.2.bn.a yes 8 5.c odd 4 1
390.2.bn.a yes 8 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 4T_{7}^{7} + 32T_{7}^{6} + 112T_{7}^{5} + 700T_{7}^{4} + 2144T_{7}^{3} + 6272T_{7}^{2} + 8096T_{7} + 8464 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} - 8 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + 32 T^{6} + \cdots + 8464 \) Copy content Toggle raw display
$11$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 36481 \) Copy content Toggle raw display
$13$ \( (T^{4} - 24 T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$19$ \( T^{8} + 24 T^{7} + 300 T^{6} + \cdots + 150544 \) Copy content Toggle raw display
$23$ \( T^{8} - 16 T^{7} + 98 T^{6} + \cdots + 5041 \) Copy content Toggle raw display
$29$ \( T^{8} + 24 T^{7} + 204 T^{6} + \cdots + 2211169 \) Copy content Toggle raw display
$31$ \( T^{8} + 4 T^{7} + 8 T^{6} - 92 T^{5} + \cdots + 529 \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} + 92 T^{6} + 112 T^{5} + \cdots + 529 \) Copy content Toggle raw display
$41$ \( T^{8} - 28 T^{7} + 344 T^{6} + \cdots + 8464 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + 80 T^{6} + 216 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$47$ \( (T^{4} - 16 T^{3} + 74 T^{2} - 32 T - 311)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 8 T^{3} + 32 T^{2} - 704 T + 7744)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 32 T^{7} + 464 T^{6} + \cdots + 3411409 \) Copy content Toggle raw display
$61$ \( T^{8} + 8 T^{7} + 188 T^{6} + \cdots + 234256 \) Copy content Toggle raw display
$67$ \( T^{8} - 68 T^{6} + 4620 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( T^{8} - 8 T^{7} + 128 T^{6} + \cdots + 37552384 \) Copy content Toggle raw display
$73$ \( T^{8} + 336 T^{6} + \cdots + 16289296 \) Copy content Toggle raw display
$79$ \( T^{8} + 324 T^{6} + 34470 T^{4} + \cdots + 7834401 \) Copy content Toggle raw display
$83$ \( (T^{4} + 16 T^{3} - 148 T^{2} + \cdots - 15452)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 4 T^{7} + 200 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$97$ \( T^{8} - 12 T^{7} + 16 T^{6} + \cdots + 150544 \) Copy content Toggle raw display
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