Properties

Label 930.2.h.b
Level $930$
Weight $2$
Character orbit 930.h
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

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

$q$-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.

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

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
371.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
1.00000i −1.73205 −1.00000 1.00000i 1.73205i 2.00000 1.00000i 3.00000 −1.00000
371.2 1.00000i 1.73205 −1.00000 1.00000i 1.73205i 2.00000 1.00000i 3.00000 −1.00000
371.3 1.00000i −1.73205 −1.00000 1.00000i 1.73205i 2.00000 1.00000i 3.00000 −1.00000
371.4 1.00000i 1.73205 −1.00000 1.00000i 1.73205i 2.00000 1.00000i 3.00000 −1.00000
\(n\): 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.b odd 2 1 inner
93.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.h.b 4
3.b odd 2 1 inner 930.2.h.b 4
31.b odd 2 1 inner 930.2.h.b 4
93.c even 2 1 inner 930.2.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.h.b 4 1.a even 1 1 trivial
930.2.h.b 4 3.b odd 2 1 inner
930.2.h.b 4 31.b odd 2 1 inner
930.2.h.b 4 93.c even 2 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( -2 + T )^{4} \)
$11$ \( ( -12 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( -48 + T^{2} )^{2} \)
$19$ \( ( -4 + T )^{4} \)
$23$ \( ( -12 + T^{2} )^{2} \)
$29$ \( ( -12 + T^{2} )^{2} \)
$31$ \( ( 31 - 4 T + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( ( 144 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( 8 + T )^{4} \)
$71$ \( ( 144 + T^{2} )^{2} \)
$73$ \( ( 108 + T^{2} )^{2} \)
$79$ \( ( 108 + T^{2} )^{2} \)
$83$ \( ( -12 + T^{2} )^{2} \)
$89$ \( ( -192 + T^{2} )^{2} \)
$97$ \( ( 10 + T )^{4} \)
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