Properties

Label 930.2.a.p
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.56155
2.56155
1.00000 −1.00000 1.00000 1.00000 −1.00000 −1.56155 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 2.56155 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.p 2
3.b odd 2 1 2790.2.a.be 2
4.b odd 2 1 7440.2.a.bl 2
5.b even 2 1 4650.2.a.ce 2
5.c odd 4 2 4650.2.d.bd 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.p 2 1.a even 1 1 trivial
2790.2.a.be 2 3.b odd 2 1
4650.2.a.ce 2 5.b even 2 1
4650.2.d.bd 4 5.c odd 4 2
7440.2.a.bl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(930))\):

\( T_{7}^{2} - T_{7} - 4 \)
\( T_{11}^{2} - T_{11} - 4 \)
\( T_{19}^{2} + 3 T_{19} - 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -4 - T + T^{2} \)
$11$ \( -4 - T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -16 - 2 T + T^{2} \)
$19$ \( -36 + 3 T + T^{2} \)
$23$ \( 8 - 7 T + T^{2} \)
$29$ \( -8 - 6 T + T^{2} \)
$31$ \( ( -1 + T )^{2} \)
$37$ \( -16 - 2 T + T^{2} \)
$41$ \( -8 - 6 T + T^{2} \)
$43$ \( -100 - 5 T + T^{2} \)
$47$ \( -16 - 2 T + T^{2} \)
$53$ \( 86 - 19 T + T^{2} \)
$59$ \( 64 + 18 T + T^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( -144 + 6 T + T^{2} \)
$71$ \( -36 + 3 T + T^{2} \)
$73$ \( -106 + T + T^{2} \)
$79$ \( 72 + 21 T + T^{2} \)
$83$ \( -32 - 12 T + T^{2} \)
$89$ \( 18 + 15 T + T^{2} \)
$97$ \( ( 6 + T )^{2} \)
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