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: \(x^{3} - x^{2} - 9 x + 12\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 7\)

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 \( T_{3}^{3} - T_{3}^{2} - 9 T_{3} + 12 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

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