Properties

Label 230.2.a.b
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: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.30278
2.30278
−1.00000 −0.302776 1.00000 1.00000 0.302776 3.30278 −1.00000 −2.90833 −1.00000
1.2 −1.00000 3.30278 1.00000 1.00000 −3.30278 −0.302776 −1.00000 7.90833 −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.b 2
3.b odd 2 1 2070.2.a.w 2
4.b odd 2 1 1840.2.a.j 2
5.b even 2 1 1150.2.a.m 2
5.c odd 4 2 1150.2.b.f 4
8.b even 2 1 7360.2.a.bc 2
8.d odd 2 1 7360.2.a.bu 2
20.d odd 2 1 9200.2.a.ca 2
23.b odd 2 1 5290.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.b 2 1.a even 1 1 trivial
1150.2.a.m 2 5.b even 2 1
1150.2.b.f 4 5.c odd 4 2
1840.2.a.j 2 4.b odd 2 1
2070.2.a.w 2 3.b odd 2 1
5290.2.a.j 2 23.b odd 2 1
7360.2.a.bc 2 8.b even 2 1
7360.2.a.bu 2 8.d odd 2 1
9200.2.a.ca 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3 T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -1 - 3 T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -1 - 3 T + T^{2} \)
$11$ \( 9 + 7 T + T^{2} \)
$13$ \( -1 - 3 T + T^{2} \)
$17$ \( -27 - 3 T + T^{2} \)
$19$ \( -29 - T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -12 - 2 T + T^{2} \)
$31$ \( -23 + 5 T + T^{2} \)
$37$ \( ( -8 + T )^{2} \)
$41$ \( -9 + 9 T + T^{2} \)
$43$ \( -48 + 4 T + T^{2} \)
$47$ \( -12 - 2 T + T^{2} \)
$53$ \( -36 + 8 T + T^{2} \)
$59$ \( 36 + 14 T + T^{2} \)
$61$ \( -75 - 5 T + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( 207 + 29 T + T^{2} \)
$73$ \( -92 - 10 T + T^{2} \)
$79$ \( -208 + T^{2} \)
$83$ \( -36 - 8 T + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 17 - 9 T + T^{2} \)
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