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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(115, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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