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

\(f(q)\) \(=\) \( q - q^{2} + (\beta + 1) q^{3} + q^{4} + q^{5} + ( - \beta - 1) q^{6} + ( - \beta + 2) q^{7} - q^{8} + (3 \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta + 1) q^{3} + q^{4} + q^{5} + ( - \beta - 1) q^{6} + ( - \beta + 2) q^{7} - q^{8} + (3 \beta + 1) q^{9} - q^{10} + ( - \beta - 3) q^{11} + (\beta + 1) q^{12} + ( - \beta + 2) q^{13} + (\beta - 2) q^{14} + (\beta + 1) q^{15} + q^{16} + ( - 3 \beta + 3) q^{17} + ( - 3 \beta - 1) q^{18} + ( - 3 \beta + 2) q^{19} + q^{20} - q^{21} + (\beta + 3) q^{22} - q^{23} + ( - \beta - 1) q^{24} + q^{25} + (\beta - 2) q^{26} + (4 \beta + 7) q^{27} + ( - \beta + 2) q^{28} + 2 \beta q^{29} + ( - \beta - 1) q^{30} + (3 \beta - 4) q^{31} - q^{32} + ( - 5 \beta - 6) q^{33} + (3 \beta - 3) q^{34} + ( - \beta + 2) q^{35} + (3 \beta + 1) q^{36} + 8 q^{37} + (3 \beta - 2) q^{38} - q^{39} - q^{40} + ( - 3 \beta - 3) q^{41} + q^{42} + (4 \beta - 4) q^{43} + ( - \beta - 3) q^{44} + (3 \beta + 1) q^{45} + q^{46} + 2 \beta q^{47} + (\beta + 1) q^{48} - 3 \beta q^{49} - q^{50} + ( - 3 \beta - 6) q^{51} + ( - \beta + 2) q^{52} + (4 \beta - 6) q^{53} + ( - 4 \beta - 7) q^{54} + ( - \beta - 3) q^{55} + (\beta - 2) q^{56} + ( - 4 \beta - 7) q^{57} - 2 \beta q^{58} + ( - 2 \beta - 6) q^{59} + (\beta + 1) q^{60} + ( - 5 \beta + 5) q^{61} + ( - 3 \beta + 4) q^{62} + (2 \beta - 7) q^{63} + q^{64} + ( - \beta + 2) q^{65} + (5 \beta + 6) q^{66} - 4 q^{67} + ( - 3 \beta + 3) q^{68} + ( - \beta - 1) q^{69} + (\beta - 2) q^{70} + (\beta - 15) q^{71} + ( - 3 \beta - 1) q^{72} + (6 \beta + 2) q^{73} - 8 q^{74} + (\beta + 1) q^{75} + ( - 3 \beta + 2) q^{76} + (2 \beta - 3) q^{77} + q^{78} + (8 \beta - 4) q^{79} + q^{80} + (6 \beta + 16) q^{81} + (3 \beta + 3) q^{82} + ( - 4 \beta + 6) q^{83} - q^{84} + ( - 3 \beta + 3) q^{85} + ( - 4 \beta + 4) q^{86} + (4 \beta + 6) q^{87} + (\beta + 3) q^{88} + ( - 3 \beta - 1) q^{90} + ( - 3 \beta + 7) q^{91} - q^{92} + (2 \beta + 5) q^{93} - 2 \beta q^{94} + ( - 3 \beta + 2) q^{95} + ( - \beta - 1) q^{96} + ( - \beta + 5) q^{97} + 3 \beta q^{98} + ( - 13 \beta - 12) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} + 2 q^{5} - 3 q^{6} + 3 q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} + 2 q^{5} - 3 q^{6} + 3 q^{7} - 2 q^{8} + 5 q^{9} - 2 q^{10} - 7 q^{11} + 3 q^{12} + 3 q^{13} - 3 q^{14} + 3 q^{15} + 2 q^{16} + 3 q^{17} - 5 q^{18} + q^{19} + 2 q^{20} - 2 q^{21} + 7 q^{22} - 2 q^{23} - 3 q^{24} + 2 q^{25} - 3 q^{26} + 18 q^{27} + 3 q^{28} + 2 q^{29} - 3 q^{30} - 5 q^{31} - 2 q^{32} - 17 q^{33} - 3 q^{34} + 3 q^{35} + 5 q^{36} + 16 q^{37} - q^{38} - 2 q^{39} - 2 q^{40} - 9 q^{41} + 2 q^{42} - 4 q^{43} - 7 q^{44} + 5 q^{45} + 2 q^{46} + 2 q^{47} + 3 q^{48} - 3 q^{49} - 2 q^{50} - 15 q^{51} + 3 q^{52} - 8 q^{53} - 18 q^{54} - 7 q^{55} - 3 q^{56} - 18 q^{57} - 2 q^{58} - 14 q^{59} + 3 q^{60} + 5 q^{61} + 5 q^{62} - 12 q^{63} + 2 q^{64} + 3 q^{65} + 17 q^{66} - 8 q^{67} + 3 q^{68} - 3 q^{69} - 3 q^{70} - 29 q^{71} - 5 q^{72} + 10 q^{73} - 16 q^{74} + 3 q^{75} + q^{76} - 4 q^{77} + 2 q^{78} + 2 q^{80} + 38 q^{81} + 9 q^{82} + 8 q^{83} - 2 q^{84} + 3 q^{85} + 4 q^{86} + 16 q^{87} + 7 q^{88} - 5 q^{90} + 11 q^{91} - 2 q^{92} + 12 q^{93} - 2 q^{94} + q^{95} - 3 q^{96} + 9 q^{97} + 3 q^{98} - 37 q^{99}+O(q^{100}) \) 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
−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 \( T_{3}^{2} - 3T_{3} - 1 \) 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
$3$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 7T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T - 27 \) Copy content Toggle raw display
$19$ \( T^{2} - T - 29 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$31$ \( T^{2} + 5T - 23 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 9T - 9 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 5T - 75 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 29T + 207 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T - 92 \) Copy content Toggle raw display
$79$ \( T^{2} - 208 \) Copy content Toggle raw display
$83$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 9T + 17 \) Copy content Toggle raw display
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