Properties

Label 930.2.h.a
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: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} -\beta_{3} q^{3} - q^{4} -\beta_{2} q^{5} + \beta_{1} q^{6} -4 q^{7} -\beta_{2} q^{8} -3 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} -\beta_{3} q^{3} - q^{4} -\beta_{2} q^{5} + \beta_{1} q^{6} -4 q^{7} -\beta_{2} q^{8} -3 \beta_{2} q^{9} + q^{10} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{11} + \beta_{3} q^{12} + ( -\beta_{1} - \beta_{3} ) q^{13} -4 \beta_{2} q^{14} -\beta_{1} q^{15} + q^{16} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{17} + 3 q^{18} -2 q^{19} + \beta_{2} q^{20} + 4 \beta_{3} q^{21} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{22} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{23} -\beta_{1} q^{24} - q^{25} + ( \beta_{1} - \beta_{3} ) q^{26} -3 \beta_{1} q^{27} + 4 q^{28} -\beta_{3} q^{30} + ( 5 - \beta_{1} - \beta_{3} ) q^{31} + \beta_{2} q^{32} + ( 6 - 6 \beta_{2} ) q^{33} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{34} + 4 \beta_{2} q^{35} + 3 \beta_{2} q^{36} + ( \beta_{1} + \beta_{3} ) q^{37} -2 \beta_{2} q^{38} + ( -3 - 3 \beta_{2} ) q^{39} - q^{40} -12 \beta_{2} q^{41} -4 \beta_{1} q^{42} + ( \beta_{1} + \beta_{3} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{44} -3 q^{45} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{46} -12 \beta_{2} q^{47} -\beta_{3} q^{48} + 9 q^{49} -\beta_{2} q^{50} + ( -9 + 9 \beta_{2} ) q^{51} + ( \beta_{1} + \beta_{3} ) q^{52} + ( -\beta_{1} + \beta_{3} ) q^{53} -3 \beta_{3} q^{54} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{55} + 4 \beta_{2} q^{56} + 2 \beta_{3} q^{57} + 6 \beta_{2} q^{59} + \beta_{1} q^{60} + ( \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{62} + 12 \beta_{2} q^{63} - q^{64} + ( -\beta_{1} + \beta_{3} ) q^{65} + ( 6 + 6 \beta_{2} ) q^{66} + 4 q^{67} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{68} + ( -9 + 9 \beta_{2} ) q^{69} -4 q^{70} -3 q^{72} + ( \beta_{1} + \beta_{3} ) q^{73} + ( -\beta_{1} + \beta_{3} ) q^{74} + \beta_{3} q^{75} + 2 q^{76} + ( -8 \beta_{1} + 8 \beta_{3} ) q^{77} + ( 3 - 3 \beta_{2} ) q^{78} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{79} -\beta_{2} q^{80} -9 q^{81} + 12 q^{82} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{83} -4 \beta_{3} q^{84} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{85} + ( -\beta_{1} + \beta_{3} ) q^{86} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{88} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{89} -3 \beta_{2} q^{90} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{91} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{92} + ( -3 - 3 \beta_{2} - 5 \beta_{3} ) q^{93} + 12 q^{94} + 2 \beta_{2} q^{95} + \beta_{1} q^{96} + 14 q^{97} + 9 \beta_{2} q^{98} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 16q^{7} + O(q^{10}) \) \( 4q - 4q^{4} - 16q^{7} + 4q^{10} + 4q^{16} + 12q^{18} - 8q^{19} - 4q^{25} + 16q^{28} + 20q^{31} + 24q^{33} - 12q^{39} - 4q^{40} - 12q^{45} + 36q^{49} - 36q^{51} - 4q^{64} + 24q^{66} + 16q^{67} - 36q^{69} - 16q^{70} - 12q^{72} + 8q^{76} + 12q^{78} - 36q^{81} + 48q^{82} - 12q^{93} + 48q^{94} + 56q^{97} + O(q^{100}) \)

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

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

Character values

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

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

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
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
1.00000i −1.22474 1.22474i −1.00000 1.00000i −1.22474 + 1.22474i −4.00000 1.00000i 3.00000i 1.00000
371.2 1.00000i 1.22474 + 1.22474i −1.00000 1.00000i 1.22474 1.22474i −4.00000 1.00000i 3.00000i 1.00000
371.3 1.00000i −1.22474 + 1.22474i −1.00000 1.00000i −1.22474 1.22474i −4.00000 1.00000i 3.00000i 1.00000
371.4 1.00000i 1.22474 1.22474i −1.00000 1.00000i 1.22474 + 1.22474i −4.00000 1.00000i 3.00000i 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.a 4
3.b odd 2 1 inner 930.2.h.a 4
31.b odd 2 1 inner 930.2.h.a 4
93.c even 2 1 inner 930.2.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.h.a 4 1.a even 1 1 trivial
930.2.h.a 4 3.b odd 2 1 inner
930.2.h.a 4 31.b odd 2 1 inner
930.2.h.a 4 93.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 4 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 9 + T^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( 4 + T )^{4} \)
$11$ \( ( -24 + T^{2} )^{2} \)
$13$ \( ( 6 + T^{2} )^{2} \)
$17$ \( ( -54 + T^{2} )^{2} \)
$19$ \( ( 2 + T )^{4} \)
$23$ \( ( -54 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( 31 - 10 T + T^{2} )^{2} \)
$37$ \( ( 6 + T^{2} )^{2} \)
$41$ \( ( 144 + T^{2} )^{2} \)
$43$ \( ( 6 + T^{2} )^{2} \)
$47$ \( ( 144 + T^{2} )^{2} \)
$53$ \( ( -6 + T^{2} )^{2} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( -4 + T )^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 6 + T^{2} )^{2} \)
$79$ \( ( 24 + T^{2} )^{2} \)
$83$ \( ( -150 + T^{2} )^{2} \)
$89$ \( ( -24 + T^{2} )^{2} \)
$97$ \( ( -14 + T )^{4} \)
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