Properties

Label 165.2.c.a
Level $165$
Weight $2$
Character orbit 165.c
Analytic conductor $1.318$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 165.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.31753163335\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(67\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
0.403032 0.403032i
−0.854638 + 0.854638i
1.45161 + 1.45161i
1.45161 1.45161i
−0.854638 0.854638i
0.403032 + 0.403032i
2.67513i 1.00000i −5.15633 −1.48119 + 1.67513i −2.67513 2.80606i 8.44358i −1.00000 4.48119 + 3.96239i
34.2 1.53919i 1.00000i −0.369102 2.17009 + 0.539189i −1.53919 0.290725i 2.51026i −1.00000 0.829914 3.34017i
34.3 1.21432i 1.00000i 0.525428 0.311108 + 2.21432i 1.21432 4.90321i 3.06668i −1.00000 2.68889 0.377784i
34.4 1.21432i 1.00000i 0.525428 0.311108 2.21432i 1.21432 4.90321i 3.06668i −1.00000 2.68889 + 0.377784i
34.5 1.53919i 1.00000i −0.369102 2.17009 0.539189i −1.53919 0.290725i 2.51026i −1.00000 0.829914 + 3.34017i
34.6 2.67513i 1.00000i −5.15633 −1.48119 1.67513i −2.67513 2.80606i 8.44358i −1.00000 4.48119 3.96239i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.c.a 6
3.b odd 2 1 495.2.c.d 6
4.b odd 2 1 2640.2.d.i 6
5.b even 2 1 inner 165.2.c.a 6
5.c odd 4 1 825.2.a.h 3
5.c odd 4 1 825.2.a.n 3
11.b odd 2 1 1815.2.c.d 6
15.d odd 2 1 495.2.c.d 6
15.e even 4 1 2475.2.a.y 3
15.e even 4 1 2475.2.a.be 3
20.d odd 2 1 2640.2.d.i 6
55.d odd 2 1 1815.2.c.d 6
55.e even 4 1 9075.2.a.cc 3
55.e even 4 1 9075.2.a.ck 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.c.a 6 1.a even 1 1 trivial
165.2.c.a 6 5.b even 2 1 inner
495.2.c.d 6 3.b odd 2 1
495.2.c.d 6 15.d odd 2 1
825.2.a.h 3 5.c odd 4 1
825.2.a.n 3 5.c odd 4 1
1815.2.c.d 6 11.b odd 2 1
1815.2.c.d 6 55.d odd 2 1
2475.2.a.y 3 15.e even 4 1
2475.2.a.be 3 15.e even 4 1
2640.2.d.i 6 4.b odd 2 1
2640.2.d.i 6 20.d odd 2 1
9075.2.a.cc 3 55.e even 4 1
9075.2.a.ck 3 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 11T_{2}^{4} + 31T_{2}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(165, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 11 T^{4} + 31 T^{2} + 25 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + 3 T^{4} - 12 T^{3} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 32 T^{4} + 192 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 92 T^{4} + 2560 T^{2} + \cdots + 21904 \) Copy content Toggle raw display
$17$ \( T^{6} + 72 T^{4} + 1712 T^{2} + \cdots + 13456 \) Copy content Toggle raw display
$19$ \( (T^{3} - 2 T^{2} - 52 T + 184)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{3} \) Copy content Toggle raw display
$29$ \( (T^{3} - 8 T^{2} + 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 8 T^{2} + 8 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( (T^{3} + 8 T^{2} - 32)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 32 T^{4} + 192 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( T^{6} + 96 T^{4} + 2816 T^{2} + \cdots + 25600 \) Copy content Toggle raw display
$53$ \( T^{6} + 128 T^{4} + 5376 T^{2} + \cdots + 73984 \) Copy content Toggle raw display
$59$ \( (T^{3} - 8 T^{2} - 64 T - 80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 2 T^{2} - 52 T + 40)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 176 T^{4} + 7936 T^{2} + \cdots + 102400 \) Copy content Toggle raw display
$71$ \( (T^{3} + 12 T^{2} + 32 T - 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 204 T^{4} + 6912 T^{2} + \cdots + 8464 \) Copy content Toggle raw display
$79$ \( (T^{3} - 6 T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{4} + 32 T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T^{3} + 2 T^{2} - 12 T - 8)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 128 T^{4} + 3072 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
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