Properties

Label 930.2.s.a
Level $930$
Weight $2$
Character orbit 930.s
Analytic conductor $7.426$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
439.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
1.00000i −0.866025 0.500000i −1.00000 2.23205 0.133975i −0.500000 + 0.866025i −1.73205 1.00000i 1.00000i 0.500000 + 0.866025i −0.133975 2.23205i
439.2 1.00000i 0.866025 + 0.500000i −1.00000 −1.23205 + 1.86603i −0.500000 + 0.866025i 1.73205 + 1.00000i 1.00000i 0.500000 + 0.866025i −1.86603 1.23205i
769.1 1.00000i 0.866025 0.500000i −1.00000 −1.23205 1.86603i −0.500000 0.866025i 1.73205 1.00000i 1.00000i 0.500000 0.866025i −1.86603 + 1.23205i
769.2 1.00000i −0.866025 + 0.500000i −1.00000 2.23205 + 0.133975i −0.500000 0.866025i −1.73205 + 1.00000i 1.00000i 0.500000 0.866025i −0.133975 + 2.23205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.c even 3 1 inner
155.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.s.a 4
5.b even 2 1 inner 930.2.s.a 4
31.c even 3 1 inner 930.2.s.a 4
155.j even 6 1 inner 930.2.s.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.s.a 4 1.a even 1 1 trivial
930.2.s.a 4 5.b even 2 1 inner
930.2.s.a 4 31.c even 3 1 inner
930.2.s.a 4 155.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 4 T_{7}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( 25 - 10 T - T^{2} - 2 T^{3} + T^{4} \)
$7$ \( 16 - 4 T^{2} + T^{4} \)
$11$ \( ( 25 - 5 T + T^{2} )^{2} \)
$13$ \( 16 - 4 T^{2} + T^{4} \)
$17$ \( 81 - 9 T^{2} + T^{4} \)
$19$ \( ( 16 - 4 T + T^{2} )^{2} \)
$23$ \( ( 9 + T^{2} )^{2} \)
$29$ \( ( -2 + T )^{4} \)
$31$ \( ( 31 + 4 T + T^{2} )^{2} \)
$37$ \( 625 - 25 T^{2} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( 1 - T^{2} + T^{4} \)
$47$ \( ( 81 + T^{2} )^{2} \)
$53$ \( 4096 - 64 T^{2} + T^{4} \)
$59$ \( ( 16 - 4 T + T^{2} )^{2} \)
$61$ \( ( 2 + T )^{4} \)
$67$ \( 2401 - 49 T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 1296 - 36 T^{2} + T^{4} \)
$79$ \( ( 25 - 5 T + T^{2} )^{2} \)
$83$ \( 1296 - 36 T^{2} + T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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