Properties

Label 230.2.a.d
Level $230$
Weight $2$
Character orbit 230.a
Self dual yes
Analytic conductor $1.837$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.1101.1
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + ( - \beta_{2} - \beta_1 + 1) q^{7} + q^{8} + (\beta_{2} - \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + ( - \beta_{2} - \beta_1 + 1) q^{7} + q^{8} + (\beta_{2} - \beta_1 + 4) q^{9} - q^{10} + (2 \beta_{2} - \beta_1 + 2) q^{11} + \beta_1 q^{12} + ( - \beta_{2} + \beta_1 - 1) q^{13} + ( - \beta_{2} - \beta_1 + 1) q^{14} - \beta_1 q^{15} + q^{16} + ( - \beta_1 - 2) q^{17} + (\beta_{2} - \beta_1 + 4) q^{18} + ( - \beta_{2} - \beta_1 + 1) q^{19} - q^{20} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{21} + (2 \beta_{2} - \beta_1 + 2) q^{22} - q^{23} + \beta_1 q^{24} + q^{25} + ( - \beta_{2} + \beta_1 - 1) q^{26} + (\beta_{2} + 2 \beta_1 - 5) q^{27} + ( - \beta_{2} - \beta_1 + 1) q^{28} + (2 \beta_1 - 2) q^{29} - \beta_1 q^{30} + (\beta_{2} - \beta_1 - 1) q^{31} + q^{32} + (3 \beta_{2} + 3 \beta_1 - 3) q^{33} + ( - \beta_1 - 2) q^{34} + (\beta_{2} + \beta_1 - 1) q^{35} + (\beta_{2} - \beta_1 + 4) q^{36} + 2 \beta_{2} q^{37} + ( - \beta_{2} - \beta_1 + 1) q^{38} + ( - \beta_{2} - 2 \beta_1 + 5) q^{39} - q^{40} + ( - 2 \beta_{2} - \beta_1) q^{41} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{42} + 8 q^{43} + (2 \beta_{2} - \beta_1 + 2) q^{44} + ( - \beta_{2} + \beta_1 - 4) q^{45} - q^{46} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{47} + \beta_1 q^{48} + (2 \beta_{2} - \beta_1 + 11) q^{49} + q^{50} + ( - \beta_{2} - \beta_1 - 7) q^{51} + ( - \beta_{2} + \beta_1 - 1) q^{52} - 6 q^{53} + (\beta_{2} + 2 \beta_1 - 5) q^{54} + ( - 2 \beta_{2} + \beta_1 - 2) q^{55} + ( - \beta_{2} - \beta_1 + 1) q^{56} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{57} + (2 \beta_1 - 2) q^{58} + (2 \beta_1 + 4) q^{59} - \beta_1 q^{60} + (4 \beta_{2} - \beta_1 + 2) q^{61} + (\beta_{2} - \beta_1 - 1) q^{62} + ( - \beta_{2} - 8 \beta_1 + 5) q^{63} + q^{64} + (\beta_{2} - \beta_1 + 1) q^{65} + (3 \beta_{2} + 3 \beta_1 - 3) q^{66} + (4 \beta_{2} + 4) q^{67} + ( - \beta_1 - 2) q^{68} - \beta_1 q^{69} + (\beta_{2} + \beta_1 - 1) q^{70} + ( - \beta_1 + 4) q^{71} + (\beta_{2} - \beta_1 + 4) q^{72} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{73} + 2 \beta_{2} q^{74} + \beta_1 q^{75} + ( - \beta_{2} - \beta_1 + 1) q^{76} + (\beta_{2} - 8 \beta_1 - 5) q^{77} + ( - \beta_{2} - 2 \beta_1 + 5) q^{78} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{79} - q^{80} + (\beta_{2} - 4 \beta_1 + 4) q^{81} + ( - 2 \beta_{2} - \beta_1) q^{82} + ( - 2 \beta_{2} + 2) q^{83} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{84} + (\beta_1 + 2) q^{85} + 8 q^{86} + (2 \beta_{2} - 4 \beta_1 + 14) q^{87} + (2 \beta_{2} - \beta_1 + 2) q^{88} + ( - 2 \beta_{2} + 4 \beta_1 + 4) q^{89} + ( - \beta_{2} + \beta_1 - 4) q^{90} + ( - 2 \beta_{2} + 5 \beta_1 - 2) q^{91} - q^{92} + (\beta_{2} - 5) q^{93} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{94} + (\beta_{2} + \beta_1 - 1) q^{95} + \beta_1 q^{96} + ( - 3 \beta_1 - 10) q^{97} + (2 \beta_{2} - \beta_1 + 11) q^{98} + (3 \beta_{2} - 3 \beta_1 + 21) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + q^{3} + 3 q^{4} - 3 q^{5} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + q^{3} + 3 q^{4} - 3 q^{5} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9} - 3 q^{10} + 3 q^{11} + q^{12} - q^{13} + 3 q^{14} - q^{15} + 3 q^{16} - 7 q^{17} + 10 q^{18} + 3 q^{19} - 3 q^{20} - 22 q^{21} + 3 q^{22} - 3 q^{23} + q^{24} + 3 q^{25} - q^{26} - 14 q^{27} + 3 q^{28} - 4 q^{29} - q^{30} - 5 q^{31} + 3 q^{32} - 9 q^{33} - 7 q^{34} - 3 q^{35} + 10 q^{36} - 2 q^{37} + 3 q^{38} + 14 q^{39} - 3 q^{40} + q^{41} - 22 q^{42} + 24 q^{43} + 3 q^{44} - 10 q^{45} - 3 q^{46} - 14 q^{47} + q^{48} + 30 q^{49} + 3 q^{50} - 21 q^{51} - q^{52} - 18 q^{53} - 14 q^{54} - 3 q^{55} + 3 q^{56} - 22 q^{57} - 4 q^{58} + 14 q^{59} - q^{60} + q^{61} - 5 q^{62} + 8 q^{63} + 3 q^{64} + q^{65} - 9 q^{66} + 8 q^{67} - 7 q^{68} - q^{69} - 3 q^{70} + 11 q^{71} + 10 q^{72} - 8 q^{73} - 2 q^{74} + q^{75} + 3 q^{76} - 24 q^{77} + 14 q^{78} - 4 q^{79} - 3 q^{80} + 7 q^{81} + q^{82} + 8 q^{83} - 22 q^{84} + 7 q^{85} + 24 q^{86} + 36 q^{87} + 3 q^{88} + 18 q^{89} - 10 q^{90} + q^{91} - 3 q^{92} - 16 q^{93} - 14 q^{94} - 3 q^{95} + q^{96} - 33 q^{97} + 30 q^{98} + 57 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 9x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 7 \) 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
−3.11903
1.43163
2.68740
1.00000 −3.11903 1.00000 −1.00000 −3.11903 4.50973 1.00000 6.72833 −1.00000
1.2 1.00000 1.43163 1.00000 −1.00000 1.43163 3.08719 1.00000 −0.950444 −1.00000
1.3 1.00000 2.68740 1.00000 −1.00000 2.68740 −4.59692 1.00000 4.22212 −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.d 3
3.b odd 2 1 2070.2.a.z 3
4.b odd 2 1 1840.2.a.r 3
5.b even 2 1 1150.2.a.q 3
5.c odd 4 2 1150.2.b.j 6
8.b even 2 1 7360.2.a.bz 3
8.d odd 2 1 7360.2.a.ce 3
20.d odd 2 1 9200.2.a.cf 3
23.b odd 2 1 5290.2.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.d 3 1.a even 1 1 trivial
1150.2.a.q 3 5.b even 2 1
1150.2.b.j 6 5.c odd 4 2
1840.2.a.r 3 4.b odd 2 1
2070.2.a.z 3 3.b odd 2 1
5290.2.a.r 3 23.b odd 2 1
7360.2.a.bz 3 8.b even 2 1
7360.2.a.ce 3 8.d odd 2 1
9200.2.a.cf 3 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - T_{3}^{2} - 9T_{3} + 12 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(230))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 9T + 12 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 3 T^{2} - 21 T + 64 \) Copy content Toggle raw display
$11$ \( T^{3} - 3 T^{2} - 39 T + 144 \) Copy content Toggle raw display
$13$ \( T^{3} + T^{2} - 15 T - 18 \) Copy content Toggle raw display
$17$ \( T^{3} + 7 T^{2} + 7 T - 18 \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} - 21 T + 64 \) Copy content Toggle raw display
$23$ \( (T + 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} - 32 T + 24 \) Copy content Toggle raw display
$31$ \( T^{3} + 5 T^{2} - 7 T - 8 \) Copy content Toggle raw display
$37$ \( T^{3} + 2 T^{2} - 40 T - 32 \) Copy content Toggle raw display
$41$ \( T^{3} - T^{2} - 59 T + 186 \) Copy content Toggle raw display
$43$ \( (T - 8)^{3} \) Copy content Toggle raw display
$47$ \( T^{3} + 14 T^{2} + 4 T - 288 \) Copy content Toggle raw display
$53$ \( (T + 6)^{3} \) Copy content Toggle raw display
$59$ \( T^{3} - 14 T^{2} + 28 T + 144 \) Copy content Toggle raw display
$61$ \( T^{3} - T^{2} - 157 T + 526 \) Copy content Toggle raw display
$67$ \( T^{3} - 8 T^{2} - 144 T + 384 \) Copy content Toggle raw display
$71$ \( T^{3} - 11 T^{2} + 31 T - 24 \) Copy content Toggle raw display
$73$ \( T^{3} + 8 T^{2} - 40 T - 248 \) Copy content Toggle raw display
$79$ \( T^{3} + 4 T^{2} - 240 T - 1152 \) Copy content Toggle raw display
$83$ \( T^{3} - 8 T^{2} - 20 T + 96 \) Copy content Toggle raw display
$89$ \( T^{3} - 18 T^{2} - 48 T + 1152 \) Copy content Toggle raw display
$97$ \( T^{3} + 33 T^{2} + 279 T + 166 \) Copy content Toggle raw display
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