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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{3} q^{3} - q^{4} - \beta_1 q^{5} + \beta_{2} q^{6} + 2 q^{7} + \beta_1 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{3} q^{3} - q^{4} - \beta_1 q^{5} + \beta_{2} q^{6} + 2 q^{7} + \beta_1 q^{8} + 3 q^{9} - q^{10} - 2 \beta_{3} q^{11} + \beta_{3} q^{12} - 2 \beta_1 q^{14} + \beta_{2} q^{15} + q^{16} + 4 \beta_{3} q^{17} - 3 \beta_1 q^{18} + 4 q^{19} + \beta_1 q^{20} - 2 \beta_{3} q^{21} + 2 \beta_{2} q^{22} - 2 \beta_{3} q^{23} - \beta_{2} q^{24} - q^{25} - 3 \beta_{3} q^{27} - 2 q^{28} + 2 \beta_{3} q^{29} + \beta_{3} q^{30} + ( - 3 \beta_{2} + 2) q^{31} - \beta_1 q^{32} + 6 q^{33} - 4 \beta_{2} q^{34} - 2 \beta_1 q^{35} - 3 q^{36} - 4 \beta_1 q^{38} + q^{40} - 12 \beta_1 q^{41} + 2 \beta_{2} q^{42} + 2 \beta_{3} q^{44} - 3 \beta_1 q^{45} + 2 \beta_{2} q^{46} - \beta_{3} q^{48} - 3 q^{49} + \beta_1 q^{50} - 12 q^{51} + 3 \beta_{2} q^{54} + 2 \beta_{2} q^{55} + 2 \beta_1 q^{56} - 4 \beta_{3} q^{57} - 2 \beta_{2} q^{58} - 6 \beta_1 q^{59} - \beta_{2} q^{60} + ( - 3 \beta_{3} - 2 \beta_1) q^{62} + 6 q^{63} - q^{64} - 6 \beta_1 q^{66} - 8 q^{67} - 4 \beta_{3} q^{68} + 6 q^{69} - 2 q^{70} - 12 \beta_1 q^{71} + 3 \beta_1 q^{72} - 6 \beta_{2} q^{73} + \beta_{3} q^{75} - 4 q^{76} - 4 \beta_{3} q^{77} - 6 \beta_{2} q^{79} - \beta_1 q^{80} + 9 q^{81} - 12 q^{82} - 2 \beta_{3} q^{83} + 2 \beta_{3} q^{84} - 4 \beta_{2} q^{85} - 6 q^{87} - 2 \beta_{2} q^{88} + 8 \beta_{3} q^{89} - 3 q^{90} + 2 \beta_{3} q^{92} + ( - 2 \beta_{3} + 9 \beta_1) q^{93} - 4 \beta_1 q^{95} + \beta_{2} q^{96} - 10 q^{97} + 3 \beta_1 q^{98} - 6 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{7} + 12 q^{9} - 4 q^{10} + 4 q^{16} + 16 q^{19} - 4 q^{25} - 8 q^{28} + 8 q^{31} + 24 q^{33} - 12 q^{36} + 4 q^{40} - 12 q^{49} - 48 q^{51} + 24 q^{63} - 4 q^{64} - 32 q^{67} + 24 q^{69} - 8 q^{70} - 16 q^{76} + 36 q^{81} - 48 q^{82} - 24 q^{87} - 12 q^{90} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

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

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

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 \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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