Properties

Label 114.2.h.f
Level $114$
Weight $2$
Character orbit 114.h
Analytic conductor $0.910$
Analytic rank $0$
Dimension $4$
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.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.910294583043\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(77\) \(97\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
0.500000 0.866025i −0.724745 + 1.57313i −0.500000 0.866025i 1.22474 + 0.707107i 1.00000 + 1.41421i 4.44949 −1.00000 −1.94949 2.28024i 1.22474 0.707107i
65.2 0.500000 0.866025i 1.72474 + 0.158919i −0.500000 0.866025i −1.22474 0.707107i 1.00000 1.41421i −0.449490 −1.00000 2.94949 + 0.548188i −1.22474 + 0.707107i
107.1 0.500000 + 0.866025i −0.724745 1.57313i −0.500000 + 0.866025i 1.22474 0.707107i 1.00000 1.41421i 4.44949 −1.00000 −1.94949 + 2.28024i 1.22474 + 0.707107i
107.2 0.500000 + 0.866025i 1.72474 0.158919i −0.500000 + 0.866025i −1.22474 + 0.707107i 1.00000 + 1.41421i −0.449490 −1.00000 2.94949 0.548188i −1.22474 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.h.f yes 4
3.b odd 2 1 114.2.h.e 4
4.b odd 2 1 912.2.bn.h 4
12.b even 2 1 912.2.bn.g 4
19.d odd 6 1 114.2.h.e 4
57.f even 6 1 inner 114.2.h.f yes 4
76.f even 6 1 912.2.bn.g 4
228.n odd 6 1 912.2.bn.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.h.e 4 3.b odd 2 1
114.2.h.e 4 19.d odd 6 1
114.2.h.f yes 4 1.a even 1 1 trivial
114.2.h.f yes 4 57.f even 6 1 inner
912.2.bn.g 4 12.b even 2 1
912.2.bn.g 4 76.f even 6 1
912.2.bn.h 4 4.b odd 2 1
912.2.bn.h 4 228.n odd 6 1

Hecke kernels

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

\( T_{5}^{4} - 2 T_{5}^{2} + 4 \)
\( T_{7}^{2} - 4 T_{7} - 2 \)
\( T_{17}^{4} + 12 T_{17}^{3} + 52 T_{17}^{2} + 48 T_{17} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( 9 - 6 T + T^{2} - 2 T^{3} + T^{4} \)
$5$ \( 4 - 2 T^{2} + T^{4} \)
$7$ \( ( -2 - 4 T + T^{2} )^{2} \)
$11$ \( 1 + 10 T^{2} + T^{4} \)
$13$ \( ( 12 + 6 T + T^{2} )^{2} \)
$17$ \( 16 + 48 T + 52 T^{2} + 12 T^{3} + T^{4} \)
$19$ \( 361 + 38 T - 15 T^{2} + 2 T^{3} + T^{4} \)
$23$ \( 2500 - 50 T^{2} + T^{4} \)
$29$ \( 36 + 6 T^{2} + T^{4} \)
$31$ \( ( 18 + T^{2} )^{2} \)
$37$ \( 36 + 60 T^{2} + T^{4} \)
$41$ \( ( 9 + 3 T + T^{2} )^{2} \)
$43$ \( 64 + 64 T + 72 T^{2} - 8 T^{3} + T^{4} \)
$47$ \( 7396 + 1032 T - 38 T^{2} - 12 T^{3} + T^{4} \)
$53$ \( 144 + 144 T + 132 T^{2} + 12 T^{3} + T^{4} \)
$59$ \( 5625 - 1350 T + 249 T^{2} - 18 T^{3} + T^{4} \)
$61$ \( 100 - 80 T + 54 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( 225 - 90 T - 3 T^{2} + 6 T^{3} + T^{4} \)
$71$ \( ( 36 - 6 T + T^{2} )^{2} \)
$73$ \( 9025 + 190 T + 99 T^{2} - 2 T^{3} + T^{4} \)
$79$ \( 5184 - 72 T^{2} + T^{4} \)
$83$ \( 58081 + 490 T^{2} + T^{4} \)
$89$ \( 14400 - 2880 T + 456 T^{2} - 24 T^{3} + T^{4} \)
$97$ \( 2025 - 810 T + 63 T^{2} + 18 T^{3} + T^{4} \)
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