Properties

Label 930.2.d.g
Level $930$
Weight $2$
Character orbit 930.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_1 q^{3} - q^{4} + (\beta_1 + 2) q^{5} - q^{6} + (\beta_{2} + 2 \beta_1) q^{7} - \beta_1 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_1 q^{3} - q^{4} + (\beta_1 + 2) q^{5} - q^{6} + (\beta_{2} + 2 \beta_1) q^{7} - \beta_1 q^{8} - q^{9} + (2 \beta_1 - 1) q^{10} + (\beta_{3} + 2) q^{11} - \beta_1 q^{12} + ( - \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{3} - 2) q^{14} + (2 \beta_1 - 1) q^{15} + q^{16} + (\beta_{2} + 2 \beta_1) q^{17} - \beta_1 q^{18} + ( - \beta_1 - 2) q^{20} + ( - \beta_{3} - 2) q^{21} + (\beta_{2} + 2 \beta_1) q^{22} - 3 \beta_{2} q^{23} + q^{24} + (4 \beta_1 + 3) q^{25} + (\beta_{3} - 2) q^{26} - \beta_1 q^{27} + ( - \beta_{2} - 2 \beta_1) q^{28} + (2 \beta_{3} - 4) q^{29} + ( - \beta_1 - 2) q^{30} + q^{31} + \beta_1 q^{32} + (\beta_{2} + 2 \beta_1) q^{33} + ( - \beta_{3} - 2) q^{34} + ( - \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{35} + q^{36} + ( - 3 \beta_{2} + 2 \beta_1) q^{37} + (\beta_{3} - 2) q^{39} + ( - 2 \beta_1 + 1) q^{40} + ( - 2 \beta_{3} + 2) q^{41} + ( - \beta_{2} - 2 \beta_1) q^{42} + (2 \beta_{2} - 4 \beta_1) q^{43} + ( - \beta_{3} - 2) q^{44} + ( - \beta_1 - 2) q^{45} + 3 \beta_{3} q^{46} - 2 \beta_{2} q^{47} + \beta_1 q^{48} + ( - 4 \beta_{3} - 5) q^{49} + (3 \beta_1 - 4) q^{50} + ( - \beta_{3} - 2) q^{51} + (\beta_{2} - 2 \beta_1) q^{52} + (2 \beta_{2} + 6 \beta_1) q^{53} + q^{54} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{55} + (\beta_{3} + 2) q^{56} + (2 \beta_{2} - 4 \beta_1) q^{58} + ( - \beta_{3} - 6) q^{59} + ( - 2 \beta_1 + 1) q^{60} + ( - \beta_{3} - 2) q^{61} + \beta_1 q^{62} + ( - \beta_{2} - 2 \beta_1) q^{63} - q^{64} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 - 2) q^{65} + ( - \beta_{3} - 2) q^{66} + ( - \beta_{2} - 12 \beta_1) q^{67} + ( - \beta_{2} - 2 \beta_1) q^{68} + 3 \beta_{3} q^{69} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{70} + \beta_{3} q^{71} + \beta_1 q^{72} + (2 \beta_{2} - 8 \beta_1) q^{73} + (3 \beta_{3} - 2) q^{74} + (3 \beta_1 - 4) q^{75} + (4 \beta_{2} + 12 \beta_1) q^{77} + (\beta_{2} - 2 \beta_1) q^{78} + 4 \beta_{3} q^{79} + (\beta_1 + 2) q^{80} + q^{81} + ( - 2 \beta_{2} + 2 \beta_1) q^{82} + (2 \beta_{2} + 4 \beta_1) q^{83} + (\beta_{3} + 2) q^{84} + ( - \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{85} + ( - 2 \beta_{3} + 4) q^{86} + (2 \beta_{2} - 4 \beta_1) q^{87} + ( - \beta_{2} - 2 \beta_1) q^{88} + (\beta_{3} - 2) q^{89} + ( - 2 \beta_1 + 1) q^{90} + 4 q^{91} + 3 \beta_{2} q^{92} + \beta_1 q^{93} + 2 \beta_{3} q^{94} - q^{96} - 4 \beta_{2} q^{97} + ( - 4 \beta_{2} - 5 \beta_1) q^{98} + ( - \beta_{3} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{5} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{5} - 4 q^{6} - 4 q^{9} - 4 q^{10} + 8 q^{11} - 8 q^{14} - 4 q^{15} + 4 q^{16} - 8 q^{20} - 8 q^{21} + 4 q^{24} + 12 q^{25} - 8 q^{26} - 16 q^{29} - 8 q^{30} + 4 q^{31} - 8 q^{34} - 8 q^{35} + 4 q^{36} - 8 q^{39} + 4 q^{40} + 8 q^{41} - 8 q^{44} - 8 q^{45} - 20 q^{49} - 16 q^{50} - 8 q^{51} + 4 q^{54} + 16 q^{55} + 8 q^{56} - 24 q^{59} + 4 q^{60} - 8 q^{61} - 4 q^{64} - 8 q^{65} - 8 q^{66} - 16 q^{70} - 8 q^{74} - 16 q^{75} + 8 q^{80} + 4 q^{81} + 8 q^{84} - 8 q^{85} + 16 q^{86} - 8 q^{89} + 4 q^{90} + 16 q^{91} - 4 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 4 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
1.00000i 1.00000i −1.00000 2.00000 1.00000i −1.00000 4.82843i 1.00000i −1.00000 −1.00000 2.00000i
559.2 1.00000i 1.00000i −1.00000 2.00000 1.00000i −1.00000 0.828427i 1.00000i −1.00000 −1.00000 2.00000i
559.3 1.00000i 1.00000i −1.00000 2.00000 + 1.00000i −1.00000 0.828427i 1.00000i −1.00000 −1.00000 + 2.00000i
559.4 1.00000i 1.00000i −1.00000 2.00000 + 1.00000i −1.00000 4.82843i 1.00000i −1.00000 −1.00000 + 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.d.g 4
3.b odd 2 1 2790.2.d.i 4
5.b even 2 1 inner 930.2.d.g 4
5.c odd 4 1 4650.2.a.cc 2
5.c odd 4 1 4650.2.a.cf 2
15.d odd 2 1 2790.2.d.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.g 4 1.a even 1 1 trivial
930.2.d.g 4 5.b even 2 1 inner
2790.2.d.i 4 3.b odd 2 1
2790.2.d.i 4 15.d odd 2 1
4650.2.a.cc 2 5.c odd 4 1
4650.2.a.cf 2 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\( T_{7}^{4} + 24T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 4 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 152T^{2} + 4624 \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 96T^{2} + 256 \) Copy content Toggle raw display
$47$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 28)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 304 T^{2} + 18496 \) Copy content Toggle raw display
$71$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 192T^{2} + 1024 \) Copy content Toggle raw display
$79$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 96T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 128)^{2} \) Copy content Toggle raw display
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