Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} -\beta_{1} q^{3} + q^{4} + ( 1 - \beta_{1} ) q^{5} + \beta_{1} q^{6} + \beta_{3} q^{7} - q^{8} + ( -1 + \beta_{1} ) q^{9} +O(q^{10})\) \( q - q^{2} -\beta_{1} q^{3} + q^{4} + ( 1 - \beta_{1} ) q^{5} + \beta_{1} q^{6} + \beta_{3} q^{7} - q^{8} + ( -1 + \beta_{1} ) q^{9} + ( -1 + \beta_{1} ) q^{10} + ( -5 + 5 \beta_{1} ) q^{11} -\beta_{1} q^{12} + ( -2 + 2 \beta_{1} ) q^{13} -\beta_{3} q^{14} - q^{15} + q^{16} + ( 3 \beta_{1} - \beta_{3} ) q^{17} + ( 1 - \beta_{1} ) q^{18} -2 \beta_{3} q^{19} + ( 1 - \beta_{1} ) q^{20} + ( 1 + \beta_{2} - \beta_{3} ) q^{21} + ( 5 - 5 \beta_{1} ) q^{22} + ( -2 - 3 \beta_{2} ) q^{23} + \beta_{1} q^{24} -\beta_{1} q^{25} + ( 2 - 2 \beta_{1} ) q^{26} + q^{27} + \beta_{3} q^{28} + ( -7 - \beta_{2} ) q^{29} + q^{30} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{31} - q^{32} + 5 q^{33} + ( -3 \beta_{1} + \beta_{3} ) q^{34} + ( 1 + \beta_{2} ) q^{35} + ( -1 + \beta_{1} ) q^{36} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{37} + 2 \beta_{3} q^{38} + 2 q^{39} + ( -1 + \beta_{1} ) q^{40} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -1 - \beta_{2} + \beta_{3} ) q^{42} + ( -\beta_{1} + \beta_{3} ) q^{43} + ( -5 + 5 \beta_{1} ) q^{44} + \beta_{1} q^{45} + ( 2 + 3 \beta_{2} ) q^{46} + ( 2 - 3 \beta_{2} ) q^{47} -\beta_{1} q^{48} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{49} + \beta_{1} q^{50} + ( 2 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{51} + ( -2 + 2 \beta_{1} ) q^{52} + ( 5 - 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{53} - q^{54} + 5 \beta_{1} q^{55} -\beta_{3} q^{56} + ( -2 - 2 \beta_{2} + 2 \beta_{3} ) q^{57} + ( 7 + \beta_{2} ) q^{58} + ( 4 \beta_{1} - 3 \beta_{3} ) q^{59} - q^{60} -2 q^{61} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{62} + ( -1 - \beta_{2} ) q^{63} + q^{64} + 2 \beta_{1} q^{65} -5 q^{66} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{67} + ( 3 \beta_{1} - \beta_{3} ) q^{68} + ( -\beta_{1} + 3 \beta_{3} ) q^{69} + ( -1 - \beta_{2} ) q^{70} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 1 - \beta_{1} ) q^{72} + ( -14 + 14 \beta_{1} ) q^{73} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{74} + ( -1 + \beta_{1} ) q^{75} -2 \beta_{3} q^{76} + ( -5 - 5 \beta_{2} ) q^{77} -2 q^{78} + ( \beta_{1} - \beta_{3} ) q^{79} + ( 1 - \beta_{1} ) q^{80} -\beta_{1} q^{81} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{82} + ( 1 + \beta_{2} - \beta_{3} ) q^{83} + ( 1 + \beta_{2} - \beta_{3} ) q^{84} + ( 2 - \beta_{2} ) q^{85} + ( \beta_{1} - \beta_{3} ) q^{86} + ( 6 \beta_{1} + \beta_{3} ) q^{87} + ( 5 - 5 \beta_{1} ) q^{88} -\beta_{1} q^{90} + ( -2 - 2 \beta_{2} ) q^{91} + ( -2 - 3 \beta_{2} ) q^{92} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{93} + ( -2 + 3 \beta_{2} ) q^{94} + ( -2 - 2 \beta_{2} ) q^{95} + \beta_{1} q^{96} + ( 13 - \beta_{2} ) q^{97} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{98} -5 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 2q^{3} + 4q^{4} + 2q^{5} + 2q^{6} + q^{7} - 4q^{8} - 2q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 2q^{3} + 4q^{4} + 2q^{5} + 2q^{6} + q^{7} - 4q^{8} - 2q^{9} - 2q^{10} - 10q^{11} - 2q^{12} - 4q^{13} - q^{14} - 4q^{15} + 4q^{16} + 5q^{17} + 2q^{18} - 2q^{19} + 2q^{20} + q^{21} + 10q^{22} - 2q^{23} + 2q^{24} - 2q^{25} + 4q^{26} + 4q^{27} + q^{28} - 26q^{29} + 4q^{30} + 5q^{31} - 4q^{32} + 20q^{33} - 5q^{34} + 2q^{35} - 2q^{36} - 3q^{37} + 2q^{38} + 8q^{39} - 2q^{40} - 2q^{41} - q^{42} - q^{43} - 10q^{44} + 2q^{45} + 2q^{46} + 14q^{47} - 2q^{48} - 3q^{49} + 2q^{50} + 5q^{51} - 4q^{52} + 9q^{53} - 4q^{54} + 10q^{55} - q^{56} - 2q^{57} + 26q^{58} + 5q^{59} - 4q^{60} - 8q^{61} - 5q^{62} - 2q^{63} + 4q^{64} + 4q^{65} - 20q^{66} + 3q^{67} + 5q^{68} + q^{69} - 2q^{70} - 6q^{71} + 2q^{72} - 28q^{73} + 3q^{74} - 2q^{75} - 2q^{76} - 10q^{77} - 8q^{78} + q^{79} + 2q^{80} - 2q^{81} + 2q^{82} + q^{83} + q^{84} + 10q^{85} + q^{86} + 13q^{87} + 10q^{88} - 2q^{90} - 4q^{91} - 2q^{92} - 10q^{93} - 14q^{94} - 4q^{95} + 2q^{96} + 54q^{97} + 3q^{98} - 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + \nu^{2} + 2 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11\)\()/3\)

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
−1.18614 1.26217i
1.68614 + 0.396143i
−1.18614 + 1.26217i
1.68614 0.396143i
−1.00000 −0.500000 0.866025i 1.00000 0.500000 0.866025i 0.500000 + 0.866025i −1.18614 2.05446i −1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
211.2 −1.00000 −0.500000 0.866025i 1.00000 0.500000 0.866025i 0.500000 + 0.866025i 1.68614 + 2.92048i −1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
811.1 −1.00000 −0.500000 + 0.866025i 1.00000 0.500000 + 0.866025i 0.500000 0.866025i −1.18614 + 2.05446i −1.00000 −0.500000 0.866025i −0.500000 0.866025i
811.2 −1.00000 −0.500000 + 0.866025i 1.00000 0.500000 + 0.866025i 0.500000 0.866025i 1.68614 2.92048i −1.00000 −0.500000 0.866025i −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
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.i.k 4
31.c even 3 1 inner 930.2.i.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.i.k 4 1.a even 1 1 trivial
930.2.i.k 4 31.c even 3 1 inner

Hecke kernels

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

\( T_{7}^{4} - T_{7}^{3} + 9 T_{7}^{2} + 8 T_{7} + 64 \)
\( T_{11}^{2} + 5 T_{11} + 25 \)
\( T_{13}^{2} + 2 T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 64 + 8 T + 9 T^{2} - T^{3} + T^{4} \)
$11$ \( ( 25 + 5 T + T^{2} )^{2} \)
$13$ \( ( 4 + 2 T + T^{2} )^{2} \)
$17$ \( 4 + 10 T + 27 T^{2} - 5 T^{3} + T^{4} \)
$19$ \( 1024 - 64 T + 36 T^{2} + 2 T^{3} + T^{4} \)
$23$ \( ( -74 + T + T^{2} )^{2} \)
$29$ \( ( 34 + 13 T + T^{2} )^{2} \)
$31$ \( 961 - 155 T - 6 T^{2} - 5 T^{3} + T^{4} \)
$37$ \( 5184 - 216 T + 81 T^{2} + 3 T^{3} + T^{4} \)
$41$ \( 1024 - 64 T + 36 T^{2} + 2 T^{3} + T^{4} \)
$43$ \( 64 - 8 T + 9 T^{2} + T^{3} + T^{4} \)
$47$ \( ( -62 - 7 T + T^{2} )^{2} \)
$53$ \( 144 - 108 T + 69 T^{2} - 9 T^{3} + T^{4} \)
$59$ \( 4624 + 340 T + 93 T^{2} - 5 T^{3} + T^{4} \)
$61$ \( ( 2 + T )^{4} \)
$67$ \( 5184 + 216 T + 81 T^{2} - 3 T^{3} + T^{4} \)
$71$ \( 576 - 144 T + 60 T^{2} + 6 T^{3} + T^{4} \)
$73$ \( ( 196 + 14 T + T^{2} )^{2} \)
$79$ \( 64 + 8 T + 9 T^{2} - T^{3} + T^{4} \)
$83$ \( 64 + 8 T + 9 T^{2} - T^{3} + T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( 174 - 27 T + T^{2} )^{2} \)
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