Properties

Label 230.2.a.c
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
1.00000 −0.618034 1.00000 1.00000 −0.618034 1.61803 1.00000 −2.61803 1.00000
1.2 1.00000 1.61803 1.00000 1.00000 1.61803 −0.618034 1.00000 −0.381966 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.c 2
3.b odd 2 1 2070.2.a.u 2
4.b odd 2 1 1840.2.a.l 2
5.b even 2 1 1150.2.a.j 2
5.c odd 4 2 1150.2.b.i 4
8.b even 2 1 7360.2.a.bh 2
8.d odd 2 1 7360.2.a.bn 2
20.d odd 2 1 9200.2.a.bu 2
23.b odd 2 1 5290.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.c 2 1.a even 1 1 trivial
1150.2.a.j 2 5.b even 2 1
1150.2.b.i 4 5.c odd 4 2
1840.2.a.l 2 4.b odd 2 1
2070.2.a.u 2 3.b odd 2 1
5290.2.a.o 2 23.b odd 2 1
7360.2.a.bh 2 8.b even 2 1
7360.2.a.bn 2 8.d odd 2 1
9200.2.a.bu 2 20.d odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -1 - T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -1 - T + T^{2} \)
$11$ \( -11 - T + T^{2} \)
$13$ \( -29 + 3 T + T^{2} \)
$17$ \( -31 - T + T^{2} \)
$19$ \( -9 + 3 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 44 + 14 T + T^{2} \)
$31$ \( -19 - 7 T + T^{2} \)
$37$ \( -16 - 4 T + T^{2} \)
$41$ \( -41 + 9 T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( -36 - 6 T + T^{2} \)
$53$ \( -4 + 8 T + T^{2} \)
$59$ \( -20 + 10 T + T^{2} \)
$61$ \( -59 + 3 T + T^{2} \)
$67$ \( 80 - 20 T + T^{2} \)
$71$ \( -29 - 3 T + T^{2} \)
$73$ \( -4 - 2 T + T^{2} \)
$79$ \( 16 - 12 T + T^{2} \)
$83$ \( -76 - 4 T + T^{2} \)
$89$ \( 16 + 12 T + T^{2} \)
$97$ \( 181 - 27 T + T^{2} \)
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