Properties

Label 115.2.b.a
Level $115$
Weight $2$
Character orbit 115.b
Analytic conductor $0.918$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
1.00000i
1.00000i
2.00000i 2.00000i −2.00000 2.00000 1.00000i 4.00000 1.00000i 0 −1.00000 −2.00000 4.00000i
24.2 2.00000i 2.00000i −2.00000 2.00000 + 1.00000i 4.00000 1.00000i 0 −1.00000 −2.00000 + 4.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.2.b.a 2
3.b odd 2 1 1035.2.b.a 2
4.b odd 2 1 1840.2.e.b 2
5.b even 2 1 inner 115.2.b.a 2
5.c odd 4 1 575.2.a.a 1
5.c odd 4 1 575.2.a.e 1
15.d odd 2 1 1035.2.b.a 2
15.e even 4 1 5175.2.a.a 1
15.e even 4 1 5175.2.a.z 1
20.d odd 2 1 1840.2.e.b 2
20.e even 4 1 9200.2.a.g 1
20.e even 4 1 9200.2.a.bg 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.a 2 1.a even 1 1 trivial
115.2.b.a 2 5.b even 2 1 inner
575.2.a.a 1 5.c odd 4 1
575.2.a.e 1 5.c odd 4 1
1035.2.b.a 2 3.b odd 2 1
1035.2.b.a 2 15.d odd 2 1
1840.2.e.b 2 4.b odd 2 1
1840.2.e.b 2 20.d odd 2 1
5175.2.a.a 1 15.e even 4 1
5175.2.a.z 1 15.e even 4 1
9200.2.a.g 1 20.e even 4 1
9200.2.a.bg 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(115, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( 5 - 4 T + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 25 + T^{2} \)
$19$ \( ( 8 + T )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( ( -5 + T )^{2} \)
$31$ \( ( 5 + T )^{2} \)
$37$ \( 49 + T^{2} \)
$41$ \( ( 7 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 4 + T^{2} \)
$53$ \( 1 + T^{2} \)
$59$ \( ( 3 + T )^{2} \)
$61$ \( ( 6 + T )^{2} \)
$67$ \( 169 + T^{2} \)
$71$ \( ( -13 + T )^{2} \)
$73$ \( 64 + T^{2} \)
$79$ \( ( -14 + T )^{2} \)
$83$ \( 9 + T^{2} \)
$89$ \( ( -14 + T )^{2} \)
$97$ \( 196 + T^{2} \)
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