Properties

Label 138.2.e.a
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.

\(f(q)\) \(=\) \( q - \zeta_{22} q^{2} + \zeta_{22}^{8} q^{3} + \zeta_{22}^{2} q^{4} + ( - \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{5} - \zeta_{22} + 1) q^{5} - \zeta_{22}^{9} q^{6} + (\zeta_{22}^{8} + \zeta_{22}^{6} - \zeta_{22}^{4} + \zeta_{22}^{2} + 1) q^{7} - \zeta_{22}^{3} q^{8} - \zeta_{22}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{22} q^{2} + \zeta_{22}^{8} q^{3} + \zeta_{22}^{2} q^{4} + ( - \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{5} - \zeta_{22} + 1) q^{5} - \zeta_{22}^{9} q^{6} + (\zeta_{22}^{8} + \zeta_{22}^{6} - \zeta_{22}^{4} + \zeta_{22}^{2} + 1) q^{7} - \zeta_{22}^{3} q^{8} - \zeta_{22}^{5} q^{9} + (\zeta_{22}^{8} + \zeta_{22}^{7} + \zeta_{22}^{6} + \zeta_{22}^{2} - \zeta_{22}) q^{10} + ( - 3 \zeta_{22}^{9} + 2 \zeta_{22}^{8} - 4 \zeta_{22}^{7} + 2 \zeta_{22}^{6} - 2 \zeta_{22}^{5} + 2 \zeta_{22}^{4} + \cdots + 3) q^{11} + \cdots + ( - 2 \zeta_{22}^{7} - \zeta_{22}^{5} - 2 \zeta_{22}^{4} - \zeta_{22}^{3} - 2 \zeta_{22}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} - q^{3} - q^{4} + 8 q^{5} - q^{6} + 8 q^{7} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} - q^{3} - q^{4} + 8 q^{5} - q^{6} + 8 q^{7} - q^{8} - q^{9} - 3 q^{10} + 7 q^{11} - q^{12} + 3 q^{13} - 3 q^{14} - 3 q^{15} - q^{16} + 4 q^{17} - q^{18} - 3 q^{20} - 3 q^{21} - 26 q^{22} - 12 q^{23} + 10 q^{24} - 15 q^{25} + 3 q^{26} - q^{27} - 3 q^{28} - 25 q^{29} - 3 q^{30} + 6 q^{31} - q^{32} - 4 q^{33} - 7 q^{34} + 2 q^{35} - q^{36} + 9 q^{37} + 11 q^{38} + 3 q^{39} - 3 q^{40} + 24 q^{41} + 8 q^{42} - 30 q^{43} + 7 q^{44} - 14 q^{45} + 21 q^{46} - 48 q^{47} - q^{48} + 9 q^{49} + 7 q^{50} + 15 q^{51} + 14 q^{52} + 15 q^{53} - q^{54} - 23 q^{55} + 8 q^{56} - 11 q^{57} - 3 q^{58} + 5 q^{59} + 8 q^{60} + 12 q^{61} + 28 q^{62} + 8 q^{63} - q^{64} - 13 q^{65} + 18 q^{66} + 18 q^{67} - 18 q^{68} - q^{69} + 2 q^{70} + 28 q^{71} - q^{72} + 19 q^{73} + 9 q^{74} + 7 q^{75} + 22 q^{76} - 12 q^{77} - 8 q^{78} - 52 q^{79} + 8 q^{80} - q^{81} - 20 q^{82} + 7 q^{83} - 3 q^{84} + 23 q^{85} + 14 q^{86} + 30 q^{87} - 4 q^{88} + 3 q^{89} + 8 q^{90} + 42 q^{91} - 23 q^{92} - 16 q^{93} + 29 q^{94} + 22 q^{95} - q^{96} + 51 q^{97} - 2 q^{98} - 4 q^{99}+O(q^{100}) \) 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 −0.614354 + 1.34525i −0.142315 0.989821i 3.07385 + 0.902563i 0.841254 0.540641i 0.415415 + 0.909632i 1.41899 0.416652i
25.1 0.841254 + 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i 1.18639 + 0.348356i 0.415415 0.909632i 0.968468 1.11767i −0.142315 + 0.989821i −0.959493 + 0.281733i 0.809721 + 0.934468i
31.1 −0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i 1.69894 + 1.96068i −0.959493 + 0.281733i −1.04019 0.668491i 0.415415 0.909632i −0.654861 + 0.755750i −2.18251 + 1.40261i
49.1 −0.142315 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i 1.69894 1.96068i −0.959493 0.281733i −1.04019 + 0.668491i 0.415415 + 0.909632i −0.654861 0.755750i −2.18251 1.40261i
55.1 −0.959493 + 0.281733i −0.654861 0.755750i 0.841254 0.540641i 0.455922 + 3.17101i 0.841254 + 0.540641i 0.628663 1.37658i −0.654861 + 0.755750i −0.142315 + 0.989821i −1.33083 2.91411i
73.1 0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 1.27310 + 0.818172i −0.654861 0.755750i 0.369215 + 2.56794i −0.959493 0.281733i 0.841254 0.540641i −0.215370 + 1.49793i
85.1 −0.654861 + 0.755750i 0.841254 0.540641i −0.142315 0.989821i −0.614354 1.34525i −0.142315 + 0.989821i 3.07385 0.902563i 0.841254 + 0.540641i 0.415415 0.909632i 1.41899 + 0.416652i
121.1 0.415415 0.909632i −0.959493 0.281733i −0.654861 0.755750i 1.27310 0.818172i −0.654861 + 0.755750i 0.369215 2.56794i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.215370 1.49793i
127.1 0.841254 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i 1.18639 0.348356i 0.415415 + 0.909632i 0.968468 + 1.11767i −0.142315 0.989821i −0.959493 0.281733i 0.809721 0.934468i
133.1 −0.959493 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.455922 3.17101i 0.841254 0.540641i 0.628663 + 1.37658i −0.654861 0.755750i −0.142315 0.989821i −1.33083 + 2.91411i
\(n\): 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.a 10
3.b odd 2 1 414.2.i.d 10
23.c even 11 1 inner 138.2.e.a 10
23.c even 11 1 3174.2.a.bc 5
23.d odd 22 1 3174.2.a.bd 5
69.g even 22 1 9522.2.a.bq 5
69.h odd 22 1 414.2.i.d 10
69.h odd 22 1 9522.2.a.bt 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.e.a 10 1.a even 1 1 trivial
138.2.e.a 10 23.c even 11 1 inner
414.2.i.d 10 3.b odd 2 1
414.2.i.d 10 69.h odd 22 1
3174.2.a.bc 5 23.c even 11 1
3174.2.a.bd 5 23.d odd 22 1
9522.2.a.bq 5 69.g even 22 1
9522.2.a.bt 5 69.h odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 8 T_{5}^{9} + 42 T_{5}^{8} - 149 T_{5}^{7} + 389 T_{5}^{6} - 736 T_{5}^{5} + 1092 T_{5}^{4} - 1465 T_{5}^{3} + 1754 T_{5}^{2} - 1426 T_{5} + 529 \) 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
$5$ \( T^{10} - 8 T^{9} + 42 T^{8} - 149 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$7$ \( T^{10} - 8 T^{9} + 31 T^{8} - 83 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$11$ \( T^{10} - 7 T^{9} + 5 T^{8} - 123 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{10} - 3 T^{9} + 31 T^{8} - 38 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{10} - 4 T^{9} + 5 T^{8} - 97 T^{7} + \cdots + 1849 \) Copy content Toggle raw display
$19$ \( T^{10} + 22 T^{8} + 165 T^{7} + \cdots + 64009 \) Copy content Toggle raw display
$23$ \( T^{10} + 12 T^{9} - 10 T^{8} + \cdots + 6436343 \) Copy content Toggle raw display
$29$ \( T^{10} + 25 T^{9} + 372 T^{8} + \cdots + 2866249 \) Copy content Toggle raw display
$31$ \( T^{10} - 6 T^{9} + 58 T^{8} + \cdots + 896809 \) Copy content Toggle raw display
$37$ \( T^{10} - 9 T^{9} + 26 T^{8} - 69 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$41$ \( T^{10} - 24 T^{9} + 301 T^{8} + \cdots + 529 \) Copy content Toggle raw display
$43$ \( T^{10} + 30 T^{9} + 427 T^{8} + \cdots + 192721 \) Copy content Toggle raw display
$47$ \( (T^{5} + 24 T^{4} + 83 T^{3} - 1520 T^{2} + \cdots - 10649)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 15 T^{9} + \cdots + 361342081 \) Copy content Toggle raw display
$59$ \( T^{10} - 5 T^{9} - 19 T^{8} + \cdots + 31730689 \) Copy content Toggle raw display
$61$ \( T^{10} - 12 T^{9} + \cdots + 2801902489 \) Copy content Toggle raw display
$67$ \( T^{10} - 18 T^{9} + \cdots + 1804635361 \) Copy content Toggle raw display
$71$ \( T^{10} - 28 T^{9} + \cdots + 1490654881 \) Copy content Toggle raw display
$73$ \( T^{10} - 19 T^{9} + \cdots + 236452129 \) Copy content Toggle raw display
$79$ \( T^{10} + 52 T^{9} + \cdots + 2417590561 \) Copy content Toggle raw display
$83$ \( T^{10} - 7 T^{9} + 269 T^{8} + \cdots + 659102929 \) Copy content Toggle raw display
$89$ \( T^{10} - 3 T^{9} + 97 T^{8} + \cdots + 92871769 \) Copy content Toggle raw display
$97$ \( T^{10} - 51 T^{9} + \cdots + 22314683161 \) Copy content Toggle raw display
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