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: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
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.

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

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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

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

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

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