Properties

Label 390.2.bn.a
Level $390$
Weight $2$
Character orbit 390.bn
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.bn (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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−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
0.500000 + 0.866025i −0.965926 0.258819i −0.500000 + 0.866025i 0.707107 2.12132i −0.258819 0.965926i −1.46945 0.848387i −1.00000 0.866025 + 0.500000i 2.19067 0.448288i
67.2 0.500000 + 0.866025i 0.965926 + 0.258819i −0.500000 + 0.866025i −0.707107 + 2.12132i 0.258819 + 0.965926i −3.26260 1.88366i −1.00000 0.866025 + 0.500000i −2.19067 + 0.448288i
97.1 0.500000 0.866025i −0.258819 0.965926i −0.500000 0.866025i −0.707107 + 2.12132i −0.965926 0.258819i −3.98004 + 2.29788i −1.00000 −0.866025 + 0.500000i 1.48356 + 1.67303i
97.2 0.500000 0.866025i 0.258819 + 0.965926i −0.500000 0.866025i 0.707107 2.12132i 0.965926 + 0.258819i 2.71209 1.56583i −1.00000 −0.866025 + 0.500000i −1.48356 1.67303i
163.1 0.500000 0.866025i −0.965926 + 0.258819i −0.500000 0.866025i 0.707107 + 2.12132i −0.258819 + 0.965926i −1.46945 + 0.848387i −1.00000 0.866025 0.500000i 2.19067 + 0.448288i
163.2 0.500000 0.866025i 0.965926 0.258819i −0.500000 0.866025i −0.707107 2.12132i 0.258819 0.965926i −3.26260 + 1.88366i −1.00000 0.866025 0.500000i −2.19067 0.448288i
193.1 0.500000 + 0.866025i −0.258819 + 0.965926i −0.500000 + 0.866025i −0.707107 2.12132i −0.965926 + 0.258819i −3.98004 2.29788i −1.00000 −0.866025 0.500000i 1.48356 1.67303i
193.2 0.500000 + 0.866025i 0.258819 0.965926i −0.500000 + 0.866025i 0.707107 + 2.12132i 0.965926 0.258819i 2.71209 + 1.56583i −1.00000 −0.866025 0.500000i −1.48356 + 1.67303i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.2
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 12T_{7}^{7} + 48T_{7}^{6} - 388T_{7}^{4} + 4800T_{7}^{2} + 11040T_{7} + 8464 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) 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
$7$ \( T^{8} + 12 T^{7} + 48 T^{6} + \cdots + 8464 \) 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
$23$ \( T^{8} + 4 T^{7} + 38 T^{6} + \cdots + 5041 \) 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
$37$ \( T^{8} - 24 T^{7} + 228 T^{6} + \cdots + 529 \) Copy content Toggle raw display
$41$ \( T^{8} - 28 T^{7} + 344 T^{6} + \cdots + 8464 \) Copy content Toggle raw display
$43$ \( T^{8} + 16 T^{7} + 80 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$47$ \( T^{8} + 108 T^{6} + 3830 T^{4} + \cdots + 96721 \) 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
$67$ \( (T^{4} + 8 T^{3} + 66 T^{2} - 16 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 8 T^{7} + 128 T^{6} + \cdots + 37552384 \) Copy content Toggle raw display
$73$ \( (T^{4} + 4 T^{3} - 160 T^{2} - 568 T + 4036)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 324 T^{6} + 34470 T^{4} + \cdots + 7834401 \) Copy content Toggle raw display
$83$ \( T^{8} + 552 T^{6} + \cdots + 238764304 \) Copy content Toggle raw display
$89$ \( T^{8} - 4 T^{7} + 200 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$97$ \( T^{8} + 4 T^{7} + 64 T^{6} + \cdots + 150544 \) Copy content Toggle raw display
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