Properties

Label 230.2.a.a
Level $230$
Weight $2$
Character orbit 230.a
Self dual yes
Analytic conductor $1.837$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \(x^{2} - x - 5\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.79129
−1.79129
−1.00000 −2.79129 1.00000 −1.00000 2.79129 −1.79129 −1.00000 4.79129 1.00000
1.2 −1.00000 1.79129 1.00000 −1.00000 −1.79129 2.79129 −1.00000 0.208712 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(23\) \(-1\)

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 230.2.a.a 2
3.b odd 2 1 2070.2.a.x 2
4.b odd 2 1 1840.2.a.n 2
5.b even 2 1 1150.2.a.o 2
5.c odd 4 2 1150.2.b.g 4
8.b even 2 1 7360.2.a.bq 2
8.d odd 2 1 7360.2.a.bk 2
20.d odd 2 1 9200.2.a.bs 2
23.b odd 2 1 5290.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.a 2 1.a even 1 1 trivial
1150.2.a.o 2 5.b even 2 1
1150.2.b.g 4 5.c odd 4 2
1840.2.a.n 2 4.b odd 2 1
2070.2.a.x 2 3.b odd 2 1
5290.2.a.e 2 23.b odd 2 1
7360.2.a.bk 2 8.d odd 2 1
7360.2.a.bq 2 8.b even 2 1
9200.2.a.bs 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -5 + T + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -5 - T + T^{2} \)
$11$ \( -3 - 3 T + T^{2} \)
$13$ \( 7 - 7 T + T^{2} \)
$17$ \( -3 + 3 T + T^{2} \)
$19$ \( 7 - 7 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( -12 - 6 T + T^{2} \)
$31$ \( -35 - 7 T + T^{2} \)
$37$ \( ( 4 + T )^{2} \)
$41$ \( 15 + 9 T + T^{2} \)
$43$ \( -80 - 4 T + T^{2} \)
$47$ \( 60 + 18 T + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 60 + 18 T + T^{2} \)
$61$ \( -35 - 7 T + T^{2} \)
$67$ \( -80 - 4 T + T^{2} \)
$71$ \( -45 - 3 T + T^{2} \)
$73$ \( -188 + 2 T + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( -48 - 12 T + T^{2} \)
$97$ \( -119 - 7 T + T^{2} \)
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