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: \( x^{2} - x - 5 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - q^{2} - \beta q^{3} + q^{4} - q^{5} + \beta q^{6} + ( - \beta + 1) q^{7} - q^{8} + (\beta + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta q^{3} + q^{4} - q^{5} + \beta q^{6} + ( - \beta + 1) q^{7} - q^{8} + (\beta + 2) q^{9} + q^{10} + ( - \beta + 2) q^{11} - \beta q^{12} + (\beta + 3) q^{13} + (\beta - 1) q^{14} + \beta q^{15} + q^{16} + (\beta - 2) q^{17} + ( - \beta - 2) q^{18} + (\beta + 3) q^{19} - q^{20} + 5 q^{21} + (\beta - 2) q^{22} + q^{23} + \beta q^{24} + q^{25} + ( - \beta - 3) q^{26} - 5 q^{27} + ( - \beta + 1) q^{28} + (2 \beta + 2) q^{29} - \beta q^{30} + ( - 3 \beta + 5) q^{31} - q^{32} + ( - \beta + 5) q^{33} + ( - \beta + 2) q^{34} + (\beta - 1) q^{35} + (\beta + 2) q^{36} - 4 q^{37} + ( - \beta - 3) q^{38} + ( - 4 \beta - 5) q^{39} + q^{40} + ( - \beta - 4) q^{41} - 5 q^{42} + 4 \beta q^{43} + ( - \beta + 2) q^{44} + ( - \beta - 2) q^{45} - q^{46} + (2 \beta - 10) q^{47} - \beta q^{48} + ( - \beta - 1) q^{49} - q^{50} + (\beta - 5) q^{51} + (\beta + 3) q^{52} + 6 q^{53} + 5 q^{54} + (\beta - 2) q^{55} + (\beta - 1) q^{56} + ( - 4 \beta - 5) q^{57} + ( - 2 \beta - 2) q^{58} + ( - 2 \beta - 8) q^{59} + \beta q^{60} + (3 \beta + 2) q^{61} + (3 \beta - 5) q^{62} + ( - 2 \beta - 3) q^{63} + q^{64} + ( - \beta - 3) q^{65} + (\beta - 5) q^{66} + 4 \beta q^{67} + (\beta - 2) q^{68} - \beta q^{69} + ( - \beta + 1) q^{70} + 3 \beta q^{71} + ( - \beta - 2) q^{72} + (6 \beta - 4) q^{73} + 4 q^{74} - \beta q^{75} + (\beta + 3) q^{76} + ( - 2 \beta + 7) q^{77} + (4 \beta + 5) q^{78} + 8 q^{79} - q^{80} + (2 \beta - 6) q^{81} + (\beta + 4) q^{82} - 6 q^{83} + 5 q^{84} + ( - \beta + 2) q^{85} - 4 \beta q^{86} + ( - 4 \beta - 10) q^{87} + (\beta - 2) q^{88} + (4 \beta + 4) q^{89} + (\beta + 2) q^{90} + ( - 3 \beta - 2) q^{91} + q^{92} + ( - 2 \beta + 15) q^{93} + ( - 2 \beta + 10) q^{94} + ( - \beta - 3) q^{95} + \beta q^{96} + ( - 5 \beta + 6) q^{97} + (\beta + 1) q^{98} + ( - \beta - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} + q^{6} + q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} + q^{6} + q^{7} - 2 q^{8} + 5 q^{9} + 2 q^{10} + 3 q^{11} - q^{12} + 7 q^{13} - q^{14} + q^{15} + 2 q^{16} - 3 q^{17} - 5 q^{18} + 7 q^{19} - 2 q^{20} + 10 q^{21} - 3 q^{22} + 2 q^{23} + q^{24} + 2 q^{25} - 7 q^{26} - 10 q^{27} + q^{28} + 6 q^{29} - q^{30} + 7 q^{31} - 2 q^{32} + 9 q^{33} + 3 q^{34} - q^{35} + 5 q^{36} - 8 q^{37} - 7 q^{38} - 14 q^{39} + 2 q^{40} - 9 q^{41} - 10 q^{42} + 4 q^{43} + 3 q^{44} - 5 q^{45} - 2 q^{46} - 18 q^{47} - q^{48} - 3 q^{49} - 2 q^{50} - 9 q^{51} + 7 q^{52} + 12 q^{53} + 10 q^{54} - 3 q^{55} - q^{56} - 14 q^{57} - 6 q^{58} - 18 q^{59} + q^{60} + 7 q^{61} - 7 q^{62} - 8 q^{63} + 2 q^{64} - 7 q^{65} - 9 q^{66} + 4 q^{67} - 3 q^{68} - q^{69} + q^{70} + 3 q^{71} - 5 q^{72} - 2 q^{73} + 8 q^{74} - q^{75} + 7 q^{76} + 12 q^{77} + 14 q^{78} + 16 q^{79} - 2 q^{80} - 10 q^{81} + 9 q^{82} - 12 q^{83} + 10 q^{84} + 3 q^{85} - 4 q^{86} - 24 q^{87} - 3 q^{88} + 12 q^{89} + 5 q^{90} - 7 q^{91} + 2 q^{92} + 28 q^{93} + 18 q^{94} - 7 q^{95} + q^{96} + 7 q^{97} + 3 q^{98} - 3 q^{99}+O(q^{100}) \) 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
2.79129
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(230))\). Copy content Toggle raw display

Hecke characteristic polynomials

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