Properties

Label 138.2.e.c
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.

\(f(q)\) \(=\) \( q + \zeta_{22} q^{2} + \zeta_{22}^{8} q^{3} + \zeta_{22}^{2} q^{4} + (2 \zeta_{22}^{8} - \zeta_{22}^{7} + 3 \zeta_{22}^{6} - \zeta_{22}^{5} + 2 \zeta_{22}^{4} - \zeta_{22} + 1) q^{5} + \zeta_{22}^{9} q^{6} + (\zeta_{22}^{8} - 2 \zeta_{22}^{7} + \zeta_{22}^{6} - 2 \zeta_{22}^{5} + \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \zeta_{22}^{2} - 2 \zeta_{22} + 1) q^{7} + \zeta_{22}^{3} q^{8} - \zeta_{22}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{22} q^{2} + \zeta_{22}^{8} q^{3} + \zeta_{22}^{2} q^{4} + (2 \zeta_{22}^{8} - \zeta_{22}^{7} + 3 \zeta_{22}^{6} - \zeta_{22}^{5} + 2 \zeta_{22}^{4} - \zeta_{22} + 1) q^{5} + \zeta_{22}^{9} q^{6} + (\zeta_{22}^{8} - 2 \zeta_{22}^{7} + \zeta_{22}^{6} - 2 \zeta_{22}^{5} + \zeta_{22}^{4} - 2 \zeta_{22}^{3} + \zeta_{22}^{2} - 2 \zeta_{22} + 1) q^{7} + \zeta_{22}^{3} q^{8} - \zeta_{22}^{5} q^{9} + (2 \zeta_{22}^{9} - \zeta_{22}^{8} + 3 \zeta_{22}^{7} - \zeta_{22}^{6} + 2 \zeta_{22}^{5} - \zeta_{22}^{2} + \zeta_{22}) q^{10} + (\zeta_{22}^{9} - 4 \zeta_{22}^{8} + 2 \zeta_{22}^{7} - 2 \zeta_{22}^{6} + 2 \zeta_{22}^{5} - 2 \zeta_{22}^{4} + 2 \zeta_{22}^{3} + \cdots - 1) q^{11} + \cdots + ( - 2 \zeta_{22}^{6} - \zeta_{22}^{5} + 2 \zeta_{22}^{4} - \zeta_{22}^{3} - 2 \zeta_{22}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{7} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{7} + q^{8} - q^{9} + 11 q^{10} + 11 q^{11} - q^{12} - 13 q^{13} + 13 q^{14} - 11 q^{15} - q^{16} + q^{18} - 2 q^{19} - 11 q^{20} + 9 q^{21} - 22 q^{22} - 10 q^{23} - 10 q^{24} + 5 q^{25} - 9 q^{26} - q^{27} - 13 q^{28} - 27 q^{29} + 11 q^{30} - 18 q^{31} + q^{32} + 33 q^{34} + 44 q^{35} - q^{36} - q^{37} + 13 q^{38} - 13 q^{39} - 11 q^{40} - 16 q^{41} + 2 q^{42} + 20 q^{43} + 11 q^{44} + 22 q^{45} - q^{46} - q^{48} - 19 q^{49} - 27 q^{50} - 11 q^{51} - 2 q^{52} - q^{53} + q^{54} + 33 q^{55} + 2 q^{56} - 13 q^{57} - 17 q^{58} - q^{59} - 34 q^{61} - 4 q^{62} - 2 q^{63} - q^{64} + 11 q^{65} + 8 q^{67} + 22 q^{68} + 23 q^{69} + 22 q^{70} - 22 q^{71} + q^{72} + 31 q^{73} + q^{74} + 27 q^{75} - 2 q^{76} + 22 q^{77} + 2 q^{78} + 32 q^{79} - q^{81} - 28 q^{82} + 33 q^{83} + 9 q^{84} - 11 q^{85} - 20 q^{86} + 6 q^{87} + 22 q^{88} - 23 q^{89} + 18 q^{91} + 23 q^{92} + 4 q^{93} + 11 q^{94} - 22 q^{95} + q^{96} - q^{97} - 14 q^{98}+O(q^{100}) \) 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 0.810827 1.77546i 0.142315 + 0.989821i 0.439490 + 0.129046i −0.841254 + 0.540641i 0.415415 + 0.909632i 1.87279 0.549899i
25.1 −0.841254 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i −4.24593 1.24672i −0.415415 + 0.909632i −2.17208 + 2.50672i 0.142315 0.989821i −0.959493 + 0.281733i 2.89788 + 3.34433i
31.1 0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i 1.59283 + 1.83823i 0.959493 0.281733i 1.56130 + 1.00339i −0.415415 + 0.909632i −0.654861 + 0.755750i 2.04620 1.31501i
49.1 0.142315 + 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i 1.59283 1.83823i 0.959493 + 0.281733i 1.56130 1.00339i −0.415415 0.909632i −0.654861 0.755750i 2.04620 + 1.31501i
55.1 0.959493 0.281733i −0.654861 0.755750i 0.841254 0.540641i −0.592229 4.11904i −0.841254 0.540641i −1.22301 + 2.67803i 0.654861 0.755750i −0.142315 + 0.989821i −1.72871 3.78534i
73.1 −0.415415 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 2.43450 + 1.56456i 0.654861 + 0.755750i 0.394306 + 2.74246i 0.959493 + 0.281733i 0.841254 0.540641i 0.411844 2.86444i
85.1 0.654861 0.755750i 0.841254 0.540641i −0.142315 0.989821i 0.810827 + 1.77546i 0.142315 0.989821i 0.439490 0.129046i −0.841254 0.540641i 0.415415 0.909632i 1.87279 + 0.549899i
121.1 −0.415415 + 0.909632i −0.959493 0.281733i −0.654861 0.755750i 2.43450 1.56456i 0.654861 0.755750i 0.394306 2.74246i 0.959493 0.281733i 0.841254 + 0.540641i 0.411844 + 2.86444i
127.1 −0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i −4.24593 + 1.24672i −0.415415 0.909632i −2.17208 2.50672i 0.142315 + 0.989821i −0.959493 0.281733i 2.89788 3.34433i
133.1 0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.592229 + 4.11904i −0.841254 + 0.540641i −1.22301 2.67803i 0.654861 + 0.755750i −0.142315 0.989821i −1.72871 + 3.78534i
\(n\): 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.c 10
3.b odd 2 1 414.2.i.b 10
23.c even 11 1 inner 138.2.e.c 10
23.c even 11 1 3174.2.a.z 5
23.d odd 22 1 3174.2.a.y 5
69.g even 22 1 9522.2.a.ca 5
69.h odd 22 1 414.2.i.b 10
69.h odd 22 1 9522.2.a.bv 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.e.c 10 1.a even 1 1 trivial
138.2.e.c 10 23.c even 11 1 inner
414.2.i.b 10 3.b odd 2 1
414.2.i.b 10 69.h odd 22 1
3174.2.a.y 5 23.d odd 22 1
3174.2.a.z 5 23.c even 11 1
9522.2.a.bv 5 69.h odd 22 1
9522.2.a.ca 5 69.g even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 33T_{5}^{7} - 165T_{5}^{6} + 506T_{5}^{5} + 2178T_{5}^{4} - 14399T_{5}^{3} + 45012T_{5}^{2} - 66792T_{5} + 64009 \) 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
$5$ \( T^{10} + 33 T^{7} - 165 T^{6} + \cdots + 64009 \) Copy content Toggle raw display
$7$ \( T^{10} + 2 T^{9} + 15 T^{8} - 3 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$11$ \( T^{10} - 11 T^{9} + 99 T^{8} + \cdots + 64009 \) Copy content Toggle raw display
$13$ \( T^{10} + 13 T^{9} + 59 T^{8} + 96 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{10} - 11 T^{8} + 253 T^{7} + \cdots + 33860761 \) Copy content Toggle raw display
$19$ \( T^{10} + 2 T^{9} + 26 T^{8} + \cdots + 978121 \) Copy content Toggle raw display
$23$ \( T^{10} + 10 T^{9} + 100 T^{8} + \cdots + 6436343 \) Copy content Toggle raw display
$29$ \( T^{10} + 27 T^{9} + 344 T^{8} + \cdots + 591361 \) Copy content Toggle raw display
$31$ \( T^{10} + 18 T^{9} + 148 T^{8} + \cdots + 529 \) Copy content Toggle raw display
$37$ \( T^{10} + T^{9} + 34 T^{8} + \cdots + 157609 \) Copy content Toggle raw display
$41$ \( T^{10} + 16 T^{9} + 223 T^{8} + \cdots + 38809 \) Copy content Toggle raw display
$43$ \( T^{10} - 20 T^{9} + 191 T^{8} + \cdots + 978121 \) Copy content Toggle raw display
$47$ \( (T^{5} - 55 T^{3} - 22 T^{2} + 352 T + 253)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + T^{9} + 67 T^{8} + \cdots + 31956409 \) Copy content Toggle raw display
$59$ \( T^{10} + T^{9} - 21 T^{8} - 65 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{10} + 34 T^{9} + 507 T^{8} + \cdots + 39601 \) Copy content Toggle raw display
$67$ \( T^{10} - 8 T^{9} + 64 T^{8} + \cdots + 10042561 \) Copy content Toggle raw display
$71$ \( T^{10} + 22 T^{9} + 374 T^{8} + \cdots + 64009 \) Copy content Toggle raw display
$73$ \( T^{10} - 31 T^{9} + 389 T^{8} + \cdots + 20151121 \) Copy content Toggle raw display
$79$ \( T^{10} - 32 T^{9} + \cdots + 650199001 \) Copy content Toggle raw display
$83$ \( T^{10} - 33 T^{9} + \cdots + 6146089609 \) Copy content Toggle raw display
$89$ \( T^{10} + 23 T^{9} + \cdots + 4197614521 \) Copy content Toggle raw display
$97$ \( T^{10} + T^{9} + 221 T^{8} + \cdots + 16124682289 \) Copy content Toggle raw display
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