Properties

Label 930.2.j.a
Level $930$
Weight $2$
Character orbit 930.j
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.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{2} + ( - \zeta_{8}^{2} + \zeta_{8} + 1) q^{3} - \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} + 2 \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8} - 1) q^{6} + \zeta_{8} q^{8} + ( - 2 \zeta_{8}^{3} - \zeta_{8}^{2} + 2 \zeta_{8}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8}^{3} q^{2} + ( - \zeta_{8}^{2} + \zeta_{8} + 1) q^{3} - \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} + 2 \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8} - 1) q^{6} + \zeta_{8} q^{8} + ( - 2 \zeta_{8}^{3} - \zeta_{8}^{2} + 2 \zeta_{8}) q^{9} + (\zeta_{8}^{2} - 2) q^{10} + (3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{11} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} - 1) q^{12} + (3 \zeta_{8}^{2} - 3) q^{13} + ( - 3 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + \zeta_{8} + 1) q^{15} - q^{16} + 2 \zeta_{8}^{3} q^{17} + (2 \zeta_{8}^{2} + \zeta_{8} - 2) q^{18} + 4 \zeta_{8}^{2} q^{19} + ( - 2 \zeta_{8}^{3} - \zeta_{8}) q^{20} + ( - 3 \zeta_{8}^{2} - 3) q^{22} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{24} + (3 \zeta_{8}^{2} + 4) q^{25} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{26} + ( - 5 \zeta_{8}^{3} + \zeta_{8}^{2} + 1) q^{27} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{29} + (\zeta_{8}^{3} + 3 \zeta_{8}^{2} - 2 \zeta_{8} - 1) q^{30} - q^{31} - \zeta_{8}^{3} q^{32} + (3 \zeta_{8}^{2} + 6 \zeta_{8} - 3) q^{33} - 2 \zeta_{8}^{2} q^{34} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} - 1) q^{36} + ( - 5 \zeta_{8}^{2} - 5) q^{37} - 4 \zeta_{8} q^{38} + (3 \zeta_{8}^{3} + 6 \zeta_{8}^{2} - 3 \zeta_{8}) q^{39} + (2 \zeta_{8}^{2} + 1) q^{40} + ( - 8 \zeta_{8}^{2} + 8) q^{43} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{44} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - \zeta_{8} + 6) q^{45} - 6 \zeta_{8}^{3} q^{47} + (\zeta_{8}^{2} - \zeta_{8} - 1) q^{48} - 7 \zeta_{8}^{2} q^{49} + (4 \zeta_{8}^{3} - 3 \zeta_{8}) q^{50} + (2 \zeta_{8}^{3} + 2 \zeta_{8} - 2) q^{51} + (3 \zeta_{8}^{2} + 3) q^{52} + 6 \zeta_{8} q^{53} + (\zeta_{8}^{3} + 5 \zeta_{8}^{2} - \zeta_{8}) q^{54} + (9 \zeta_{8}^{2} - 3) q^{55} + (4 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 4) q^{57} + (2 \zeta_{8}^{2} - 2) q^{58} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{59} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} - 3 \zeta_{8} + 2) q^{60} - 8 q^{61} - \zeta_{8}^{3} q^{62} + \zeta_{8}^{2} q^{64} + (9 \zeta_{8}^{3} - 3 \zeta_{8}) q^{65} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8} - 6) q^{66} + ( - \zeta_{8}^{2} - 1) q^{67} + 2 \zeta_{8} q^{68} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{71} + ( - \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2) q^{72} + ( - 2 \zeta_{8}^{2} + 2) q^{73} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{74} + (3 \zeta_{8}^{3} - \zeta_{8}^{2} + 4 \zeta_{8} + 7) q^{75} + 4 q^{76} + ( - 3 \zeta_{8}^{2} - 6 \zeta_{8} + 3) q^{78} - 10 \zeta_{8}^{2} q^{79} + (\zeta_{8}^{3} - 2 \zeta_{8}) q^{80} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8} + 7) q^{81} - 18 \zeta_{8} q^{83} + (2 \zeta_{8}^{2} - 4) q^{85} + (8 \zeta_{8}^{3} + 8 \zeta_{8}) q^{86} + ( - 4 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2) q^{87} + (3 \zeta_{8}^{2} - 3) q^{88} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{89} + (6 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2 \zeta_{8} + 1) q^{90} + (\zeta_{8}^{2} - \zeta_{8} - 1) q^{93} + 6 \zeta_{8}^{2} q^{94} + (8 \zeta_{8}^{3} + 4 \zeta_{8}) q^{95} + ( - \zeta_{8}^{3} - \zeta_{8} + 1) q^{96} + (3 \zeta_{8}^{2} + 3) q^{97} + 7 \zeta_{8} q^{98} + ( - 3 \zeta_{8}^{3} + 12 \zeta_{8}^{2} + 3 \zeta_{8}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{6} - 8 q^{10} - 4 q^{12} - 12 q^{13} + 4 q^{15} - 4 q^{16} - 8 q^{18} - 12 q^{22} + 16 q^{25} + 4 q^{27} - 4 q^{30} - 4 q^{31} - 12 q^{33} - 4 q^{36} - 20 q^{37} + 4 q^{40} + 32 q^{43} + 24 q^{45} - 4 q^{48} - 8 q^{51} + 12 q^{52} - 12 q^{55} + 16 q^{57} - 8 q^{58} + 8 q^{60} - 32 q^{61} - 24 q^{66} - 4 q^{67} + 8 q^{72} + 8 q^{73} + 28 q^{75} + 16 q^{76} + 12 q^{78} + 28 q^{81} - 16 q^{85} + 8 q^{87} - 12 q^{88} + 4 q^{90} - 4 q^{93} + 4 q^{96} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(\zeta_{8}^{2}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
497.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 + 0.707107i 1.70711 0.292893i 1.00000i 2.12132 + 0.707107i −1.00000 + 1.41421i 0 0.707107 + 0.707107i 2.82843 1.00000i −2.00000 + 1.00000i
497.2 0.707107 0.707107i 0.292893 1.70711i 1.00000i −2.12132 0.707107i −1.00000 1.41421i 0 −0.707107 0.707107i −2.82843 1.00000i −2.00000 + 1.00000i
683.1 −0.707107 0.707107i 1.70711 + 0.292893i 1.00000i 2.12132 0.707107i −1.00000 1.41421i 0 0.707107 0.707107i 2.82843 + 1.00000i −2.00000 1.00000i
683.2 0.707107 + 0.707107i 0.292893 + 1.70711i 1.00000i −2.12132 + 0.707107i −1.00000 + 1.41421i 0 −0.707107 + 0.707107i −2.82843 + 1.00000i −2.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.j.a 4
3.b odd 2 1 inner 930.2.j.a 4
5.c odd 4 1 inner 930.2.j.a 4
15.e even 4 1 inner 930.2.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.j.a 4 1.a even 1 1 trivial
930.2.j.a 4 3.b odd 2 1 inner
930.2.j.a 4 5.c odd 4 1 inner
930.2.j.a 4 15.e even 4 1 inner

Hecke kernels

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

\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} + 18 \) Copy content Toggle raw display
\( T_{17}^{4} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + 8 T^{2} - 12 T + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - 8T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$31$ \( (T + 1)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 16 T + 128)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 1296 \) Copy content Toggle raw display
$53$ \( T^{4} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$61$ \( (T + 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 104976 \) Copy content Toggle raw display
$89$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
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