Properties

Label 930.2.a.r
Level $930$
Weight $2$
Character orbit 930.a
Self dual yes
Analytic conductor $7.426$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + \beta q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + \beta q^{7} + q^{8} + q^{9} + q^{10} + ( - \beta + 4) q^{11} + q^{12} - 2 q^{13} + \beta q^{14} + q^{15} + q^{16} + ( - 2 \beta + 2) q^{17} + q^{18} + (\beta - 4) q^{19} + q^{20} + \beta q^{21} + ( - \beta + 4) q^{22} + \beta q^{23} + q^{24} + q^{25} - 2 q^{26} + q^{27} + \beta q^{28} + ( - 2 \beta - 2) q^{29} + q^{30} - q^{31} + q^{32} + ( - \beta + 4) q^{33} + ( - 2 \beta + 2) q^{34} + \beta q^{35} + q^{36} + ( - 2 \beta + 6) q^{37} + (\beta - 4) q^{38} - 2 q^{39} + q^{40} + (2 \beta - 6) q^{41} + \beta q^{42} + ( - \beta + 4) q^{43} + ( - \beta + 4) q^{44} + q^{45} + \beta q^{46} - 2 \beta q^{47} + q^{48} + (\beta + 1) q^{49} + q^{50} + ( - 2 \beta + 2) q^{51} - 2 q^{52} + (\beta - 2) q^{53} + q^{54} + ( - \beta + 4) q^{55} + \beta q^{56} + (\beta - 4) q^{57} + ( - 2 \beta - 2) q^{58} + (2 \beta - 4) q^{59} + q^{60} + (4 \beta - 2) q^{61} - q^{62} + \beta q^{63} + q^{64} - 2 q^{65} + ( - \beta + 4) q^{66} + (2 \beta + 4) q^{67} + ( - 2 \beta + 2) q^{68} + \beta q^{69} + \beta q^{70} + \beta q^{71} + q^{72} + ( - 3 \beta + 2) q^{73} + ( - 2 \beta + 6) q^{74} + q^{75} + (\beta - 4) q^{76} + (3 \beta - 8) q^{77} - 2 q^{78} + (\beta - 8) q^{79} + q^{80} + q^{81} + (2 \beta - 6) q^{82} - 12 q^{83} + \beta q^{84} + ( - 2 \beta + 2) q^{85} + ( - \beta + 4) q^{86} + ( - 2 \beta - 2) q^{87} + ( - \beta + 4) q^{88} + ( - \beta + 2) q^{89} + q^{90} - 2 \beta q^{91} + \beta q^{92} - q^{93} - 2 \beta q^{94} + (\beta - 4) q^{95} + q^{96} + 2 q^{97} + (\beta + 1) q^{98} + ( - \beta + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 7 q^{11} + 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 7 q^{19} + 2 q^{20} + q^{21} + 7 q^{22} + q^{23} + 2 q^{24} + 2 q^{25} - 4 q^{26} + 2 q^{27} + q^{28} - 6 q^{29} + 2 q^{30} - 2 q^{31} + 2 q^{32} + 7 q^{33} + 2 q^{34} + q^{35} + 2 q^{36} + 10 q^{37} - 7 q^{38} - 4 q^{39} + 2 q^{40} - 10 q^{41} + q^{42} + 7 q^{43} + 7 q^{44} + 2 q^{45} + q^{46} - 2 q^{47} + 2 q^{48} + 3 q^{49} + 2 q^{50} + 2 q^{51} - 4 q^{52} - 3 q^{53} + 2 q^{54} + 7 q^{55} + q^{56} - 7 q^{57} - 6 q^{58} - 6 q^{59} + 2 q^{60} - 2 q^{62} + q^{63} + 2 q^{64} - 4 q^{65} + 7 q^{66} + 10 q^{67} + 2 q^{68} + q^{69} + q^{70} + q^{71} + 2 q^{72} + q^{73} + 10 q^{74} + 2 q^{75} - 7 q^{76} - 13 q^{77} - 4 q^{78} - 15 q^{79} + 2 q^{80} + 2 q^{81} - 10 q^{82} - 24 q^{83} + q^{84} + 2 q^{85} + 7 q^{86} - 6 q^{87} + 7 q^{88} + 3 q^{89} + 2 q^{90} - 2 q^{91} + q^{92} - 2 q^{93} - 2 q^{94} - 7 q^{95} + 2 q^{96} + 4 q^{97} + 3 q^{98} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.37228
3.37228
1.00000 1.00000 1.00000 1.00000 1.00000 −2.37228 1.00000 1.00000 1.00000
1.2 1.00000 1.00000 1.00000 1.00000 1.00000 3.37228 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(31\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.a.r 2
3.b odd 2 1 2790.2.a.bd 2
4.b odd 2 1 7440.2.a.bg 2
5.b even 2 1 4650.2.a.by 2
5.c odd 4 2 4650.2.d.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.r 2 1.a even 1 1 trivial
2790.2.a.bd 2 3.b odd 2 1
4650.2.a.by 2 5.b even 2 1
4650.2.d.bh 4 5.c odd 4 2
7440.2.a.bg 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(930))\):

\( T_{7}^{2} - T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 7T_{11} + 4 \) Copy content Toggle raw display
\( T_{19}^{2} + 7T_{19} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$11$ \( T^{2} - 7T + 4 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$31$ \( (T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 10T - 8 \) Copy content Toggle raw display
$41$ \( T^{2} + 10T - 8 \) Copy content Toggle raw display
$43$ \( T^{2} - 7T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$61$ \( T^{2} - 132 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T - 8 \) Copy content Toggle raw display
$71$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$73$ \( T^{2} - T - 74 \) Copy content Toggle raw display
$79$ \( T^{2} + 15T + 48 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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