Properties

Label 231.2.a.b
Level 231
Weight 2
Character orbit 231.a
Self dual yes
Analytic conductor 1.845
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 231.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

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

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
2.79129
−1.79129
−2.79129 −1.00000 5.79129 3.00000 2.79129 1.00000 −10.5826 1.00000 −8.37386
1.2 1.79129 −1.00000 1.20871 3.00000 −1.79129 1.00000 −1.41742 1.00000 5.37386
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 231.2.a.b 2
3.b odd 2 1 693.2.a.j 2
4.b odd 2 1 3696.2.a.bl 2
5.b even 2 1 5775.2.a.bn 2
7.b odd 2 1 1617.2.a.o 2
11.b odd 2 1 2541.2.a.z 2
21.c even 2 1 4851.2.a.ba 2
33.d even 2 1 7623.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.b 2 1.a even 1 1 trivial
693.2.a.j 2 3.b odd 2 1
1617.2.a.o 2 7.b odd 2 1
2541.2.a.z 2 11.b odd 2 1
3696.2.a.bl 2 4.b odd 2 1
4851.2.a.ba 2 21.c even 2 1
5775.2.a.bn 2 5.b even 2 1
7623.2.a.bf 2 33.d even 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(231))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 - 3 T + 5 T^{2} )^{2} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( 1 - T + 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 22 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 + 4 T + 21 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 + 2 T + 26 T^{2} + 46 T^{3} + 529 T^{4} \)
$29$ \( 1 - 2 T - 25 T^{2} - 58 T^{3} + 841 T^{4} \)
$31$ \( 1 + 2 T + 42 T^{2} + 62 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - T + 37 T^{2} )^{2} \)
$41$ \( 1 + 4 T + 2 T^{2} + 164 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 6 T + 74 T^{2} + 258 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 12 T + 109 T^{2} - 564 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 10 T + 110 T^{2} + 530 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 97 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 10 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 8 T + 129 T^{2} - 536 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 4 T + 62 T^{2} - 284 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 - 7 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 4 T + 78 T^{2} + 316 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 14 T + 194 T^{2} + 1162 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 94 T^{2} + 7921 T^{4} \)
$97$ \( 1 + 14 T + 222 T^{2} + 1358 T^{3} + 9409 T^{4} \)
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