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: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_1 + 1) q^{5} + \beta_1 q^{6} + \beta_{3} q^{7} - q^{8} + (\beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_1 + 1) q^{5} + \beta_1 q^{6} + \beta_{3} q^{7} - q^{8} + (\beta_1 - 1) q^{9} + (\beta_1 - 1) q^{10} + (5 \beta_1 - 5) q^{11} - \beta_1 q^{12} + (2 \beta_1 - 2) q^{13} - \beta_{3} q^{14} - q^{15} + q^{16} + ( - \beta_{3} + 3 \beta_1) q^{17} + ( - \beta_1 + 1) q^{18} - 2 \beta_{3} q^{19} + ( - \beta_1 + 1) q^{20} + ( - \beta_{3} + \beta_{2} + 1) q^{21} + ( - 5 \beta_1 + 5) q^{22} + ( - 3 \beta_{2} - 2) q^{23} + \beta_1 q^{24} - \beta_1 q^{25} + ( - 2 \beta_1 + 2) q^{26} + q^{27} + \beta_{3} q^{28} + ( - \beta_{2} - 7) q^{29} + q^{30} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{31} - q^{32} + 5 q^{33} + (\beta_{3} - 3 \beta_1) q^{34} + (\beta_{2} + 1) q^{35} + (\beta_1 - 1) q^{36} + (3 \beta_{3} - 3 \beta_1) q^{37} + 2 \beta_{3} q^{38} + 2 q^{39} + (\beta_1 - 1) q^{40} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{41} + (\beta_{3} - \beta_{2} - 1) q^{42} + (\beta_{3} - \beta_1) q^{43} + (5 \beta_1 - 5) q^{44} + \beta_1 q^{45} + (3 \beta_{2} + 2) q^{46} + ( - 3 \beta_{2} + 2) q^{47} - \beta_1 q^{48} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{49} + \beta_1 q^{50} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{51} + (2 \beta_1 - 2) q^{52} + ( - \beta_{3} + \beta_{2} - 4 \beta_1 + 5) q^{53} - q^{54} + 5 \beta_1 q^{55} - \beta_{3} q^{56} + (2 \beta_{3} - 2 \beta_{2} - 2) q^{57} + (\beta_{2} + 7) q^{58} + ( - 3 \beta_{3} + 4 \beta_1) q^{59} - q^{60} - 2 q^{61} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{62} + ( - \beta_{2} - 1) q^{63} + q^{64} + 2 \beta_1 q^{65} - 5 q^{66} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{67} + ( - \beta_{3} + 3 \beta_1) q^{68} + (3 \beta_{3} - \beta_1) q^{69} + ( - \beta_{2} - 1) q^{70} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{71} + ( - \beta_1 + 1) q^{72} + (14 \beta_1 - 14) q^{73} + ( - 3 \beta_{3} + 3 \beta_1) q^{74} + (\beta_1 - 1) q^{75} - 2 \beta_{3} q^{76} + ( - 5 \beta_{2} - 5) q^{77} - 2 q^{78} + ( - \beta_{3} + \beta_1) q^{79} + ( - \beta_1 + 1) q^{80} - \beta_1 q^{81} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{82} + ( - \beta_{3} + \beta_{2} + 1) q^{83} + ( - \beta_{3} + \beta_{2} + 1) q^{84} + ( - \beta_{2} + 2) q^{85} + ( - \beta_{3} + \beta_1) q^{86} + (\beta_{3} + 6 \beta_1) q^{87} + ( - 5 \beta_1 + 5) q^{88} - \beta_1 q^{90} + ( - 2 \beta_{2} - 2) q^{91} + ( - 3 \beta_{2} - 2) q^{92} + (2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{93} + (3 \beta_{2} - 2) q^{94} + ( - 2 \beta_{2} - 2) q^{95} + \beta_1 q^{96} + ( - \beta_{2} + 13) q^{97} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{98} - 5 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} - 4 q^{8} - 2 q^{9} - 2 q^{10} - 10 q^{11} - 2 q^{12} - 4 q^{13} - q^{14} - 4 q^{15} + 4 q^{16} + 5 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{20} + q^{21} + 10 q^{22} - 2 q^{23} + 2 q^{24} - 2 q^{25} + 4 q^{26} + 4 q^{27} + q^{28} - 26 q^{29} + 4 q^{30} + 5 q^{31} - 4 q^{32} + 20 q^{33} - 5 q^{34} + 2 q^{35} - 2 q^{36} - 3 q^{37} + 2 q^{38} + 8 q^{39} - 2 q^{40} - 2 q^{41} - q^{42} - q^{43} - 10 q^{44} + 2 q^{45} + 2 q^{46} + 14 q^{47} - 2 q^{48} - 3 q^{49} + 2 q^{50} + 5 q^{51} - 4 q^{52} + 9 q^{53} - 4 q^{54} + 10 q^{55} - q^{56} - 2 q^{57} + 26 q^{58} + 5 q^{59} - 4 q^{60} - 8 q^{61} - 5 q^{62} - 2 q^{63} + 4 q^{64} + 4 q^{65} - 20 q^{66} + 3 q^{67} + 5 q^{68} + q^{69} - 2 q^{70} - 6 q^{71} + 2 q^{72} - 28 q^{73} + 3 q^{74} - 2 q^{75} - 2 q^{76} - 10 q^{77} - 8 q^{78} + q^{79} + 2 q^{80} - 2 q^{81} + 2 q^{82} + q^{83} + q^{84} + 10 q^{85} + q^{86} + 13 q^{87} + 10 q^{88} - 2 q^{90} - 4 q^{91} - 2 q^{92} - 10 q^{93} - 14 q^{94} - 4 q^{95} + 2 q^{96} + 54 q^{97} + 3 q^{98} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

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

Hecke characteristic polynomials

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