Properties

Label 114.2.b.a
Level $114$
Weight $2$
Character orbit 114.b
Analytic conductor $0.910$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 114.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.910294583043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
1.41421i
1.41421i
−1.00000 −1.00000 1.41421i 1.00000 1.41421i 1.00000 + 1.41421i −4.00000 −1.00000 −1.00000 + 2.82843i 1.41421i
113.2 −1.00000 −1.00000 + 1.41421i 1.00000 1.41421i 1.00000 1.41421i −4.00000 −1.00000 −1.00000 2.82843i 1.41421i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.b.a 2
3.b odd 2 1 114.2.b.d yes 2
4.b odd 2 1 912.2.f.d 2
12.b even 2 1 912.2.f.b 2
19.b odd 2 1 114.2.b.d yes 2
57.d even 2 1 inner 114.2.b.a 2
76.d even 2 1 912.2.f.b 2
228.b odd 2 1 912.2.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.b.a 2 1.a even 1 1 trivial
114.2.b.a 2 57.d even 2 1 inner
114.2.b.d yes 2 3.b odd 2 1
114.2.b.d yes 2 19.b odd 2 1
912.2.f.b 2 12.b even 2 1
912.2.f.b 2 76.d even 2 1
912.2.f.d 2 4.b odd 2 1
912.2.f.d 2 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(114, [\chi])\):

\( T_{5}^{2} + 2 \)
\( T_{29} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( 3 + 2 T + T^{2} \)
$5$ \( 2 + T^{2} \)
$7$ \( ( 4 + T )^{2} \)
$11$ \( 32 + T^{2} \)
$13$ \( 18 + T^{2} \)
$17$ \( 8 + T^{2} \)
$19$ \( 19 - 2 T + T^{2} \)
$23$ \( 2 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( 18 + T^{2} \)
$37$ \( 18 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( 2 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( ( 4 + T )^{2} \)
$79$ \( 18 + T^{2} \)
$83$ \( 8 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 72 + T^{2} \)
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