Properties

Label 138.2.d.a
Level $138$
Weight $2$
Character orbit 138.d
Analytic conductor $1.102$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 138.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.10193554789\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.34447360000.3
Defining polynomial: \( x^{8} + 63x^{4} + 841 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{7} + \beta_1) q^{3} - q^{4} + \beta_{3} q^{5} + (\beta_{4} - 1) q^{6} - \beta_{5} q^{7} - \beta_1 q^{8} + ( - \beta_{7} + \beta_{4} + 2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{7} + \beta_1) q^{3} - q^{4} + \beta_{3} q^{5} + (\beta_{4} - 1) q^{6} - \beta_{5} q^{7} - \beta_1 q^{8} + ( - \beta_{7} + \beta_{4} + 2 \beta_1 - 1) q^{9} + \beta_{2} q^{10} + (\beta_{6} - \beta_{3}) q^{11} + ( - \beta_{7} - \beta_1) q^{12} + (\beta_{7} - \beta_{4}) q^{13} - \beta_{6} q^{14} + ( - \beta_{6} + \beta_{5} + \beta_{2}) q^{15} + q^{16} + \beta_{6} q^{17} + ( - \beta_{7} - \beta_{4} - \beta_1 - 2) q^{18} - \beta_{2} q^{19} - \beta_{3} q^{20} + (\beta_{5} + \beta_{3} - \beta_{2}) q^{21} + ( - \beta_{5} - \beta_{2}) q^{22} + ( - \beta_{7} - \beta_{6} - \beta_{4} - 3 \beta_1) q^{23} + ( - \beta_{4} + 1) q^{24} + (\beta_{7} - \beta_{4} + 7) q^{25} + (\beta_{7} + \beta_{4}) q^{26} + (\beta_{4} - 3 \beta_1 - 4) q^{27} + \beta_{5} q^{28} + (3 \beta_{7} + 3 \beta_{4} + 2 \beta_1) q^{29} + (\beta_{6} + \beta_{5} - \beta_{3}) q^{30} + 2 q^{31} + \beta_1 q^{32} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2}) q^{33} - \beta_{5} q^{34} + ( - 5 \beta_{7} - 5 \beta_{4} - 2 \beta_1) q^{35} + (\beta_{7} - \beta_{4} - 2 \beta_1 + 1) q^{36} + (\beta_{5} + \beta_{2}) q^{37} + \beta_{3} q^{38} + ( - \beta_{7} + \beta_{4} + 2 \beta_1 + 2) q^{39} - \beta_{2} q^{40} + ( - 2 \beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{41} + (\beta_{6} + \beta_{3} + \beta_{2}) q^{42} + \beta_{2} q^{43} + ( - \beta_{6} + \beta_{3}) q^{44} + (2 \beta_{6} - \beta_{3} + 2 \beta_{2}) q^{45} + (\beta_{7} + \beta_{5} - \beta_{4} + 3) q^{46} + (\beta_{7} + \beta_{4} - 6 \beta_1) q^{47} + (\beta_{7} + \beta_1) q^{48} + (4 \beta_{7} - 4 \beta_{4} - 3) q^{49} + (\beta_{7} + \beta_{4} + 7 \beta_1) q^{50} + ( - \beta_{6} - \beta_{3} - \beta_{2}) q^{51} + ( - \beta_{7} + \beta_{4}) q^{52} - \beta_{3} q^{53} + ( - \beta_{7} - 4 \beta_1 + 3) q^{54} + ( - 6 \beta_{7} + 6 \beta_{4} - 14) q^{55} + \beta_{6} q^{56} + ( - \beta_{6} - \beta_{5} + \beta_{3}) q^{57} + ( - 3 \beta_{7} + 3 \beta_{4} - 2) q^{58} + (4 \beta_{7} + 4 \beta_{4} + 8 \beta_1) q^{59} + (\beta_{6} - \beta_{5} - \beta_{2}) q^{60} + ( - \beta_{5} - 3 \beta_{2}) q^{61} + 2 \beta_1 q^{62} + ( - 2 \beta_{6} - \beta_{5} + 2 \beta_{2}) q^{63} - q^{64} - 2 \beta_{6} q^{65} + ( - \beta_{6} + 2 \beta_{3} - \beta_{2}) q^{66} + 3 \beta_{2} q^{67} - \beta_{6} q^{68} + (\beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 5) q^{69} + (5 \beta_{7} - 5 \beta_{4} + 2) q^{70} + (\beta_{7} + \beta_{4} - 4 \beta_1) q^{71} + (\beta_{7} + \beta_{4} + \beta_1 + 2) q^{72} + ( - 3 \beta_{7} + 3 \beta_{4} - 4) q^{73} + (\beta_{6} - \beta_{3}) q^{74} + (6 \beta_{7} + \beta_{4} + 9 \beta_1 + 2) q^{75} + \beta_{2} q^{76} + (\beta_{7} + \beta_{4} + 12 \beta_1) q^{77} + ( - \beta_{7} - \beta_{4} + 2 \beta_1 - 2) q^{78} + (\beta_{5} + 2 \beta_{2}) q^{79} + \beta_{3} q^{80} + ( - 4 \beta_{7} - 4 \beta_{4} - 4 \beta_1 + 1) q^{81} + (2 \beta_{7} - 2 \beta_{4} + 2) q^{82} + (\beta_{6} + \beta_{3}) q^{83} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{84} + ( - 5 \beta_{7} + 5 \beta_{4} - 2) q^{85} - \beta_{3} q^{86} + ( - 3 \beta_{7} - \beta_{4} + 6 \beta_1 - 8) q^{87} + (\beta_{5} + \beta_{2}) q^{88} + (\beta_{6} + 2 \beta_{3}) q^{89} + ( - 2 \beta_{5} - 2 \beta_{3} - \beta_{2}) q^{90} + (2 \beta_{5} - 2 \beta_{2}) q^{91} + (\beta_{7} + \beta_{6} + \beta_{4} + 3 \beta_1) q^{92} + (2 \beta_{7} + 2 \beta_1) q^{93} + ( - \beta_{7} + \beta_{4} + 6) q^{94} + ( - \beta_{7} - \beta_{4} - 12 \beta_1) q^{95} + (\beta_{4} - 1) q^{96} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{97} + (4 \beta_{7} + 4 \beta_{4} - 3 \beta_1) q^{98} + ( - \beta_{6} - 2 \beta_{5} + 3 \beta_{3} - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 8 q^{4} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} - 8 q^{4} - 4 q^{6} + 4 q^{12} - 8 q^{13} + 8 q^{16} - 16 q^{18} + 4 q^{24} + 48 q^{25} - 28 q^{27} + 16 q^{31} + 24 q^{39} + 16 q^{46} - 4 q^{48} - 56 q^{49} + 8 q^{52} + 28 q^{54} - 64 q^{55} + 8 q^{58} - 8 q^{64} + 28 q^{69} - 24 q^{70} + 16 q^{72} - 8 q^{73} - 4 q^{75} - 16 q^{78} + 8 q^{81} + 24 q^{85} - 56 q^{87} - 8 q^{93} + 56 q^{94} - 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 63x^{4} + 841 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 92\nu^{2} ) / 319 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 92\nu^{3} + 319\nu ) / 319 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 92\nu^{3} + 319\nu ) / 319 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{6} + 29\nu^{4} - 233\nu^{2} + 1073 ) / 319 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{7} - 29\nu^{5} - 233\nu^{3} - 1073\nu ) / 319 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -6\nu^{7} + 29\nu^{5} - 233\nu^{3} + 1073\nu ) / 319 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{6} - 29\nu^{4} - 233\nu^{2} - 1073 ) / 319 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{4} + 12\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} + \beta_{5} - 6\beta_{3} + 6\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11\beta_{7} + 11\beta_{4} - 74 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{6} - 11\beta_{5} - 37\beta_{3} - 37\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -46\beta_{7} - 46\beta_{4} - 233\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -92\beta_{6} - 92\beta_{5} + 233\beta_{3} - 233\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1
−1.48020 + 1.48020i
1.48020 1.48020i
−1.81907 + 1.81907i
1.81907 1.81907i
−1.48020 1.48020i
1.48020 + 1.48020i
−1.81907 1.81907i
1.81907 + 1.81907i
1.00000i −1.61803 + 0.618034i −1.00000 −2.96039 0.618034 + 1.61803i 4.79002i 1.00000i 2.23607 2.00000i 2.96039i
137.2 1.00000i −1.61803 + 0.618034i −1.00000 2.96039 0.618034 + 1.61803i 4.79002i 1.00000i 2.23607 2.00000i 2.96039i
137.3 1.00000i 0.618034 1.61803i −1.00000 −3.63814 −1.61803 0.618034i 2.24849i 1.00000i −2.23607 2.00000i 3.63814i
137.4 1.00000i 0.618034 1.61803i −1.00000 3.63814 −1.61803 0.618034i 2.24849i 1.00000i −2.23607 2.00000i 3.63814i
137.5 1.00000i −1.61803 0.618034i −1.00000 −2.96039 0.618034 1.61803i 4.79002i 1.00000i 2.23607 + 2.00000i 2.96039i
137.6 1.00000i −1.61803 0.618034i −1.00000 2.96039 0.618034 1.61803i 4.79002i 1.00000i 2.23607 + 2.00000i 2.96039i
137.7 1.00000i 0.618034 + 1.61803i −1.00000 −3.63814 −1.61803 + 0.618034i 2.24849i 1.00000i −2.23607 + 2.00000i 3.63814i
137.8 1.00000i 0.618034 + 1.61803i −1.00000 3.63814 −1.61803 + 0.618034i 2.24849i 1.00000i −2.23607 + 2.00000i 3.63814i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.b odd 2 1 inner
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.2.d.a 8
3.b odd 2 1 inner 138.2.d.a 8
4.b odd 2 1 1104.2.m.d 8
12.b even 2 1 1104.2.m.d 8
23.b odd 2 1 inner 138.2.d.a 8
69.c even 2 1 inner 138.2.d.a 8
92.b even 2 1 1104.2.m.d 8
276.h odd 2 1 1104.2.m.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.d.a 8 1.a even 1 1 trivial
138.2.d.a 8 3.b odd 2 1 inner
138.2.d.a 8 23.b odd 2 1 inner
138.2.d.a 8 69.c even 2 1 inner
1104.2.m.d 8 4.b odd 2 1
1104.2.m.d 8 12.b even 2 1
1104.2.m.d 8 92.b even 2 1
1104.2.m.d 8 276.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(138, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{3} + 2 T^{2} + 6 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 22 T^{2} + 116)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 28 T^{2} + 116)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 38 T^{2} + 116)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T - 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 28 T^{2} + 116)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 22 T^{2} + 116)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 20 T^{6} - 122 T^{4} + \cdots + 279841 \) Copy content Toggle raw display
$29$ \( (T^{4} + 92 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 38 T^{2} + 116)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 22 T^{2} + 116)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 108 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 22 T^{2} + 116)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 192 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 190 T^{2} + 2900)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 198 T^{2} + 9396)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 60 T^{2} + 400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 44)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 92 T^{2} + 116)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 62 T^{2} + 116)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 140 T^{2} + 2900)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 248 T^{2} + 1856)^{2} \) Copy content Toggle raw display
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