Properties

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

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.o (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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
1.00000i −1.50000 + 0.866025i −1.00000 0.866025 + 0.500000i 0.866025 + 1.50000i 2.00000 + 3.46410i 1.00000i 1.50000 2.59808i 0.500000 0.866025i
161.2 1.00000i −1.50000 + 0.866025i −1.00000 −0.866025 0.500000i −0.866025 1.50000i 2.00000 + 3.46410i 1.00000i 1.50000 2.59808i 0.500000 0.866025i
491.1 1.00000i −1.50000 0.866025i −1.00000 −0.866025 + 0.500000i −0.866025 + 1.50000i 2.00000 3.46410i 1.00000i 1.50000 + 2.59808i 0.500000 + 0.866025i
491.2 1.00000i −1.50000 0.866025i −1.00000 0.866025 0.500000i 0.866025 1.50000i 2.00000 3.46410i 1.00000i 1.50000 + 2.59808i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.e odd 6 1 inner
93.g 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.o.a 4
3.b odd 2 1 inner 930.2.o.a 4
31.e odd 6 1 inner 930.2.o.a 4
93.g even 6 1 inner 930.2.o.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.o.a 4 1.a even 1 1 trivial
930.2.o.a 4 3.b odd 2 1 inner
930.2.o.a 4 31.e odd 6 1 inner
930.2.o.a 4 93.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{2} - 4 T_{7} + 16 \)
\( T_{11}^{4} + 27 T_{11}^{2} + 729 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 3 + 3 T + T^{2} )^{2} \)
$5$ \( 1 - T^{2} + T^{4} \)
$7$ \( ( 16 - 4 T + T^{2} )^{2} \)
$11$ \( 729 + 27 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 729 + 27 T^{2} + T^{4} \)
$19$ \( ( 4 - 2 T + T^{2} )^{2} \)
$23$ \( ( -27 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( 31 - 4 T + T^{2} )^{2} \)
$37$ \( ( 27 + 9 T + T^{2} )^{2} \)
$41$ \( 1296 - 36 T^{2} + T^{4} \)
$43$ \( ( 75 + 15 T + T^{2} )^{2} \)
$47$ \( ( 81 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( 20736 - 144 T^{2} + T^{4} \)
$61$ \( ( 192 + T^{2} )^{2} \)
$67$ \( ( 49 + 7 T + T^{2} )^{2} \)
$71$ \( 1296 - 36 T^{2} + T^{4} \)
$73$ \( ( 192 - 24 T + T^{2} )^{2} \)
$79$ \( ( 3 + 3 T + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( ( -108 + T^{2} )^{2} \)
$97$ \( ( 4 + T )^{4} \)
show more
show less