Properties

Label 138.2.e.d
Level $138$
Weight $2$
Character orbit 138.e
Analytic conductor $1.102$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 138.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.10193554789\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{22}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(47\) \(97\)
\(\chi(n)\) \(1\) \(-\zeta_{22}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.654861 + 0.755750i
−0.841254 0.540641i
0.142315 0.989821i
0.142315 + 0.989821i
0.959493 0.281733i
−0.415415 0.909632i
0.654861 0.755750i
−0.415415 + 0.909632i
−0.841254 + 0.540641i
0.959493 + 0.281733i
0.654861 + 0.755750i −0.841254 0.540641i −0.142315 + 0.989821i −1.37787 + 3.01713i −0.142315 0.989821i 3.66820 + 1.07708i −0.841254 + 0.540641i 0.415415 + 0.909632i −3.18251 + 0.934468i
25.1 −0.841254 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i 1.78074 + 0.522874i 0.415415 0.909632i −2.54487 + 2.93694i 0.142315 0.989821i −0.959493 + 0.281733i −1.21537 1.40261i
31.1 0.142315 0.989821i −0.415415 0.909632i −0.959493 0.281733i −1.81440 2.09393i −0.959493 + 0.281733i 0.163423 + 0.105026i −0.415415 + 0.909632i −0.654861 + 0.755750i −2.33083 + 1.49793i
49.1 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.81440 + 2.09393i −0.959493 0.281733i 0.163423 0.105026i −0.415415 0.909632i −0.654861 0.755750i −2.33083 1.49793i
55.1 0.959493 0.281733i 0.654861 + 0.755750i 0.841254 0.540641i −0.0651865 0.453382i 0.841254 + 0.540641i −0.134858 + 0.295298i 0.654861 0.755750i −0.142315 + 0.989821i −0.190279 0.416652i
73.1 −0.415415 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i 2.47672 + 1.59169i −0.654861 0.755750i −0.151894 1.05645i 0.959493 + 0.281733i 0.841254 0.540641i 0.418986 2.91411i
85.1 0.654861 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i −1.37787 3.01713i −0.142315 + 0.989821i 3.66820 1.07708i −0.841254 0.540641i 0.415415 0.909632i −3.18251 0.934468i
121.1 −0.415415 + 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i 2.47672 1.59169i −0.654861 + 0.755750i −0.151894 + 1.05645i 0.959493 0.281733i 0.841254 + 0.540641i 0.418986 + 2.91411i
127.1 −0.841254 + 0.540641i 0.142315 0.989821i 0.415415 0.909632i 1.78074 0.522874i 0.415415 + 0.909632i −2.54487 2.93694i 0.142315 + 0.989821i −0.959493 0.281733i −1.21537 + 1.40261i
133.1 0.959493 + 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i −0.0651865 + 0.453382i 0.841254 0.540641i −0.134858 0.295298i 0.654861 + 0.755750i −0.142315 0.989821i −0.190279 + 0.416652i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.2.e.d 10
3.b odd 2 1 414.2.i.a 10
23.c even 11 1 inner 138.2.e.d 10
23.c even 11 1 3174.2.a.x 5
23.d odd 22 1 3174.2.a.w 5
69.g even 22 1 9522.2.a.by 5
69.h odd 22 1 414.2.i.a 10
69.h odd 22 1 9522.2.a.bx 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.e.d 10 1.a even 1 1 trivial
138.2.e.d 10 23.c even 11 1 inner
414.2.i.a 10 3.b odd 2 1
414.2.i.a 10 69.h odd 22 1
3174.2.a.w 5 23.d odd 22 1
3174.2.a.x 5 23.c even 11 1
9522.2.a.bx 5 69.h odd 22 1
9522.2.a.by 5 69.g even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 2 T_{5}^{9} + 4 T_{5}^{8} - 41 T_{5}^{7} + 137 T_{5}^{6} - 76 T_{5}^{5} + 460 T_{5}^{4} - 2185 T_{5}^{3} + 2324 T_{5}^{2} - 138 T_{5} + 529 \) acting on \(S_{2}^{\mathrm{new}}(138, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + 4 T^{8} - 41 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$7$ \( T^{10} - 2 T^{9} - 7 T^{8} - 41 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} + 5 T^{9} + 25 T^{8} + 37 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$13$ \( T^{10} - 13 T^{9} + 125 T^{8} + \cdots + 529 \) Copy content Toggle raw display
$17$ \( T^{10} + 55 T^{8} - 121 T^{7} + \cdots + 7745089 \) Copy content Toggle raw display
$19$ \( T^{10} - 33 T^{7} - 165 T^{6} + \cdots + 64009 \) Copy content Toggle raw display
$23$ \( T^{10} + 32 T^{9} + 496 T^{8} + \cdots + 6436343 \) Copy content Toggle raw display
$29$ \( T^{10} - 27 T^{9} + 322 T^{8} + \cdots + 7921 \) Copy content Toggle raw display
$31$ \( T^{10} + 8 T^{9} + 108 T^{8} + \cdots + 8300161 \) Copy content Toggle raw display
$37$ \( T^{10} + 11 T^{9} + 66 T^{8} + \cdots + 64009 \) Copy content Toggle raw display
$41$ \( T^{10} + 10 T^{9} + 23 T^{8} + \cdots + 1515361 \) Copy content Toggle raw display
$43$ \( T^{10} - 34 T^{9} + 595 T^{8} + \cdots + 109561 \) Copy content Toggle raw display
$47$ \( (T^{5} - 4 T^{4} - 75 T^{3} - 28 T^{2} + \cdots + 857)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 9 T^{9} + 103 T^{8} + \cdots + 8427409 \) Copy content Toggle raw display
$59$ \( T^{10} + 21 T^{9} + 199 T^{8} + \cdots + 978121 \) Copy content Toggle raw display
$61$ \( T^{10} + 4 T^{9} + 5 T^{8} + \cdots + 9078169 \) Copy content Toggle raw display
$67$ \( T^{10} + 32 T^{9} + 628 T^{8} + \cdots + 83375161 \) Copy content Toggle raw display
$71$ \( T^{10} - 22 T^{9} + 198 T^{8} + \cdots + 64009 \) Copy content Toggle raw display
$73$ \( T^{10} - 43 T^{9} + 1013 T^{8} + \cdots + 21132409 \) Copy content Toggle raw display
$79$ \( T^{10} + 16 T^{9} + 14 T^{8} + \cdots + 72471169 \) Copy content Toggle raw display
$83$ \( T^{10} + 3 T^{9} - 79 T^{8} + \cdots + 16752649 \) Copy content Toggle raw display
$89$ \( T^{10} + 11 T^{9} + \cdots + 517972081 \) Copy content Toggle raw display
$97$ \( T^{10} - 39 T^{9} + 773 T^{8} + \cdots + 5650129 \) Copy content Toggle raw display
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