Properties

Label 930.2.i.j
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: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( 2 \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)\(/2\)

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
−1.32288 2.29129i
1.32288 + 2.29129i
−1.32288 + 2.29129i
1.32288 2.29129i
−1.00000 −0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −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 −0.500000 0.866025i −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 −0.500000 + 0.866025i −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 −0.500000 + 0.866025i −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.j 4
31.c even 3 1 inner 930.2.i.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.i.j 4 1.a even 1 1 trivial
930.2.i.j 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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{2} + T_{7} + 1 \)
\( T_{11}^{2} - T_{11} + 1 \)
\( T_{13}^{2} + 2 T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( ( 1 + T + T^{2} )^{2} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( ( 4 + 2 T + T^{2} )^{2} \)
$17$ \( 576 + 96 T + 40 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( 576 - 96 T + 40 T^{2} + 4 T^{3} + T^{4} \)
$23$ \( ( -24 - 4 T + T^{2} )^{2} \)
$29$ \( ( -27 + 2 T + T^{2} )^{2} \)
$31$ \( 961 + 186 T + 43 T^{2} + 6 T^{3} + T^{4} \)
$37$ \( 144 + 96 T + 76 T^{2} - 8 T^{3} + T^{4} \)
$41$ \( 784 + 28 T^{2} + T^{4} \)
$43$ \( ( 16 - 4 T + T^{2} )^{2} \)
$47$ \( ( 8 + 12 T + T^{2} )^{2} \)
$53$ \( ( 9 + 3 T + T^{2} )^{2} \)
$59$ \( ( 49 - 7 T + T^{2} )^{2} \)
$61$ \( ( -24 - 4 T + T^{2} )^{2} \)
$67$ \( 144 + 96 T + 76 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( 9216 - 768 T + 160 T^{2} + 8 T^{3} + T^{4} \)
$73$ \( 5776 + 912 T + 220 T^{2} - 12 T^{3} + T^{4} \)
$79$ \( ( 64 - 8 T + T^{2} )^{2} \)
$83$ \( 361 - 114 T + 55 T^{2} + 6 T^{3} + T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( -3 + 10 T + T^{2} )^{2} \)
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