Properties

Label 115.2.a.c
Level $115$
Weight $2$
Character orbit 115.a
Self dual yes
Analytic conductor $0.918$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 2\)

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
−1.69353
−0.329727
1.32973
2.69353
−1.69353 −2.56155 0.868028 1.00000 4.33805 −0.819031 1.91702 3.56155 −1.69353
1.2 −0.329727 1.56155 −1.89128 1.00000 −0.514886 4.06562 1.28306 −0.561553 −0.329727
1.3 1.32973 1.56155 −0.231826 1.00000 2.07644 −3.50407 −2.96772 −0.561553 1.32973
1.4 2.69353 −2.56155 5.25508 1.00000 −6.89961 −2.74252 8.76763 3.56155 2.69353
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(23\) \(1\)

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 115.2.a.c 4
3.b odd 2 1 1035.2.a.o 4
4.b odd 2 1 1840.2.a.u 4
5.b even 2 1 575.2.a.h 4
5.c odd 4 2 575.2.b.e 8
7.b odd 2 1 5635.2.a.v 4
8.b even 2 1 7360.2.a.cj 4
8.d odd 2 1 7360.2.a.cg 4
15.d odd 2 1 5175.2.a.bx 4
20.d odd 2 1 9200.2.a.cl 4
23.b odd 2 1 2645.2.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.a.c 4 1.a even 1 1 trivial
575.2.a.h 4 5.b even 2 1
575.2.b.e 8 5.c odd 4 2
1035.2.a.o 4 3.b odd 2 1
1840.2.a.u 4 4.b odd 2 1
2645.2.a.m 4 23.b odd 2 1
5175.2.a.bx 4 15.d odd 2 1
5635.2.a.v 4 7.b odd 2 1
7360.2.a.cg 4 8.d odd 2 1
7360.2.a.cj 4 8.b even 2 1
9200.2.a.cl 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2 T_{2}^{3} - 4 T_{2}^{2} + 5 T_{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(115))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + 5 T - 4 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( ( -4 + T + T^{2} )^{2} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( -32 - 52 T - 14 T^{2} + 3 T^{3} + T^{4} \)
$11$ \( 32 + 40 T - 16 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( 212 - 41 T^{2} + T^{4} \)
$17$ \( 32 - 24 T - 18 T^{2} + T^{3} + T^{4} \)
$19$ \( 32 - 40 T - 16 T^{2} + 4 T^{3} + T^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( 202 - 269 T + 117 T^{2} - 19 T^{3} + T^{4} \)
$31$ \( 2144 + 11 T - 101 T^{2} + T^{3} + T^{4} \)
$37$ \( 2008 + 16 T - 116 T^{2} + 3 T^{3} + T^{4} \)
$41$ \( -94 - 3 T + 45 T^{2} - 13 T^{3} + T^{4} \)
$43$ \( 128 - 16 T - 36 T^{2} + 6 T^{3} + T^{4} \)
$47$ \( -128 + 548 T - 83 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( -8776 + 2092 T - 34 T^{2} - 19 T^{3} + T^{4} \)
$59$ \( -3136 + 560 T + 100 T^{2} - 23 T^{3} + T^{4} \)
$61$ \( -32 + 136 T - 56 T^{2} + T^{4} \)
$67$ \( 2032 - 212 T - 98 T^{2} + 3 T^{3} + T^{4} \)
$71$ \( -8 - 535 T - 149 T^{2} + 3 T^{3} + T^{4} \)
$73$ \( 1684 + 1392 T + 343 T^{2} + 32 T^{3} + T^{4} \)
$79$ \( 512 - 352 T - 140 T^{2} - 2 T^{3} + T^{4} \)
$83$ \( -1216 - 224 T + 96 T^{2} + 21 T^{3} + T^{4} \)
$89$ \( -2752 - 1496 T - 216 T^{2} + T^{4} \)
$97$ \( -1072 - 200 T + 72 T^{2} + 18 T^{3} + T^{4} \)
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