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

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{3} q^{3} - q^{4} - \beta_{2} q^{5} + \beta_1 q^{6} - 4 q^{7} - \beta_{2} q^{8} - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 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_{3} + 2 \beta_1) q^{11} + \beta_{3} q^{12} + ( - \beta_{3} - \beta_1) q^{13} - 4 \beta_{2} q^{14} - \beta_1 q^{15} + q^{16} + (3 \beta_{3} - 3 \beta_1) q^{17} + 3 q^{18} - 2 q^{19} + \beta_{2} q^{20} + 4 \beta_{3} q^{21} + (2 \beta_{3} + 2 \beta_1) q^{22} + (3 \beta_{3} - 3 \beta_1) q^{23} - \beta_1 q^{24} - q^{25} + ( - \beta_{3} + \beta_1) q^{26} - 3 \beta_1 q^{27} + 4 q^{28} - \beta_{3} q^{30} + ( - \beta_{3} - \beta_1 + 5) q^{31} + \beta_{2} q^{32} + ( - 6 \beta_{2} + 6) q^{33} + ( - 3 \beta_{3} - 3 \beta_1) q^{34} + 4 \beta_{2} q^{35} + 3 \beta_{2} q^{36} + (\beta_{3} + \beta_1) q^{37} - 2 \beta_{2} q^{38} + ( - 3 \beta_{2} - 3) q^{39} - q^{40} - 12 \beta_{2} q^{41} - 4 \beta_1 q^{42} + (\beta_{3} + \beta_1) q^{43} + (2 \beta_{3} - 2 \beta_1) q^{44} - 3 q^{45} + ( - 3 \beta_{3} - 3 \beta_1) q^{46} - 12 \beta_{2} q^{47} - \beta_{3} q^{48} + 9 q^{49} - \beta_{2} q^{50} + (9 \beta_{2} - 9) q^{51} + (\beta_{3} + \beta_1) q^{52} + (\beta_{3} - \beta_1) q^{53} - 3 \beta_{3} q^{54} + ( - 2 \beta_{3} - 2 \beta_1) q^{55} + 4 \beta_{2} q^{56} + 2 \beta_{3} q^{57} + 6 \beta_{2} q^{59} + \beta_1 q^{60} + ( - \beta_{3} + 5 \beta_{2} + \beta_1) q^{62} + 12 \beta_{2} q^{63} - q^{64} + (\beta_{3} - \beta_1) q^{65} + (6 \beta_{2} + 6) q^{66} + 4 q^{67} + ( - 3 \beta_{3} + 3 \beta_1) q^{68} + (9 \beta_{2} - 9) q^{69} - 4 q^{70} - 3 q^{72} + (\beta_{3} + \beta_1) q^{73} + (\beta_{3} - \beta_1) q^{74} + \beta_{3} q^{75} + 2 q^{76} + (8 \beta_{3} - 8 \beta_1) q^{77} + ( - 3 \beta_{2} + 3) q^{78} + ( - 2 \beta_{3} - 2 \beta_1) q^{79} - \beta_{2} q^{80} - 9 q^{81} + 12 q^{82} + ( - 5 \beta_{3} + 5 \beta_1) q^{83} - 4 \beta_{3} q^{84} + (3 \beta_{3} + 3 \beta_1) q^{85} + (\beta_{3} - \beta_1) q^{86} + ( - 2 \beta_{3} - 2 \beta_1) q^{88} + (2 \beta_{3} - 2 \beta_1) q^{89} - 3 \beta_{2} q^{90} + (4 \beta_{3} + 4 \beta_1) q^{91} + ( - 3 \beta_{3} + 3 \beta_1) q^{92} + ( - 5 \beta_{3} - 3 \beta_{2} - 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_{3} - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 16 q^{7} + 4 q^{10} + 4 q^{16} + 12 q^{18} - 8 q^{19} - 4 q^{25} + 16 q^{28} + 20 q^{31} + 24 q^{33} - 12 q^{39} - 4 q^{40} - 12 q^{45} + 36 q^{49} - 36 q^{51} - 4 q^{64} + 24 q^{66} + 16 q^{67} - 36 q^{69} - 16 q^{70} - 12 q^{72} + 8 q^{76} + 12 q^{78} - 36 q^{81} + 48 q^{82} - 12 q^{93} + 48 q^{94} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

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

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

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

Hecke characteristic polynomials

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