Properties

Label 165.4.a.g
Level $165$
Weight $4$
Character orbit 165.a
Self dual yes
Analytic conductor $9.735$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.73531515095\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1957.1
Defining polynomial: \( x^{3} - x^{2} - 9x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{3} + (2 \beta_{2} + \beta_1 + 6) q^{4} + 5 q^{5} + 3 \beta_1 q^{6} + (\beta_{2} - 3 \beta_1 + 1) q^{7} + (2 \beta_{2} - 3 \beta_1 - 2) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 3 q^{3} + (2 \beta_{2} + \beta_1 + 6) q^{4} + 5 q^{5} + 3 \beta_1 q^{6} + (\beta_{2} - 3 \beta_1 + 1) q^{7} + (2 \beta_{2} - 3 \beta_1 - 2) q^{8} + 9 q^{9} - 5 \beta_1 q^{10} + 11 q^{11} + ( - 6 \beta_{2} - 3 \beta_1 - 18) q^{12} + ( - 11 \beta_{2} - 13 \beta_1 - 11) q^{13} + (8 \beta_{2} + 48) q^{14} - 15 q^{15} + ( - 6 \beta_{2} - 7 \beta_1 + 6) q^{16} + (7 \beta_{2} - 5 \beta_1 + 9) q^{17} - 9 \beta_1 q^{18} + ( - 2 \beta_{2} - 8 \beta_1 + 36) q^{19} + (10 \beta_{2} + 5 \beta_1 + 30) q^{20} + ( - 3 \beta_{2} + 9 \beta_1 - 3) q^{21} - 11 \beta_1 q^{22} + ( - 16 \beta_{2} + 80) q^{23} + ( - 6 \beta_{2} + 9 \beta_1 + 6) q^{24} + 25 q^{25} + (4 \beta_{2} + 46 \beta_1 + 116) q^{26} - 27 q^{27} + (8 \beta_{2} - 40 \beta_1 + 40) q^{28} + (26 \beta_1 + 88) q^{29} + 15 \beta_1 q^{30} + ( - 22 \beta_{2} + 34 \beta_1 + 42) q^{31} + ( - 14 \beta_{2} + 37 \beta_1 + 78) q^{32} - 33 q^{33} + (24 \beta_{2} - 18 \beta_1 + 112) q^{34} + (5 \beta_{2} - 15 \beta_1 + 5) q^{35} + (18 \beta_{2} + 9 \beta_1 + 54) q^{36} + (42 \beta_{2} + 66 \beta_1 - 8) q^{37} + (12 \beta_{2} - 24 \beta_1 + 100) q^{38} + (33 \beta_{2} + 39 \beta_1 + 33) q^{39} + (10 \beta_{2} - 15 \beta_1 - 10) q^{40} + (22 \beta_1 - 8) q^{41} + ( - 24 \beta_{2} - 144) q^{42} + ( - 27 \beta_{2} - 19 \beta_1 - 51) q^{43} + (22 \beta_{2} + 11 \beta_1 + 66) q^{44} + 45 q^{45} + ( - 32 \beta_{2} - 48 \beta_1 - 96) q^{46} + ( - 34 \beta_{2} + 58 \beta_1 - 54) q^{47} + (18 \beta_{2} + 21 \beta_1 - 18) q^{48} + (30 \beta_{2} - 14 \beta_1 - 157) q^{49} - 25 \beta_1 q^{50} + ( - 21 \beta_{2} + 15 \beta_1 - 27) q^{51} + (4 \beta_{2} - 66 \beta_1 - 532) q^{52} + ( - 56 \beta_{2} - 12 \beta_1 - 270) q^{53} + 27 \beta_1 q^{54} + 55 q^{55} + (32 \beta_{2} - 16 \beta_1 + 224) q^{56} + (6 \beta_{2} + 24 \beta_1 - 108) q^{57} + ( - 52 \beta_{2} - 114 \beta_1 - 364) q^{58} + ( - 8 \beta_{2} + 60 \beta_1 + 432) q^{59} + ( - 30 \beta_{2} - 15 \beta_1 - 90) q^{60} + (22 \beta_{2} + 78 \beta_1 + 140) q^{61} + ( - 112 \beta_{2} - 32 \beta_1 - 608) q^{62} + (9 \beta_{2} - 27 \beta_1 + 9) q^{63} + ( - 54 \beta_{2} - 31 \beta_1 - 650) q^{64} + ( - 55 \beta_{2} - 65 \beta_1 - 55) q^{65} + 33 \beta_1 q^{66} + (104 \beta_{2} + 88 \beta_1 + 284) q^{67} + (28 \beta_{2} - 102 \beta_1 + 324) q^{68} + (48 \beta_{2} - 240) q^{69} + (40 \beta_{2} + 240) q^{70} + (28 \beta_{2} - 16 \beta_1 + 600) q^{71} + (18 \beta_{2} - 27 \beta_1 - 18) q^{72} + (23 \beta_{2} - 43 \beta_1 + 19) q^{73} + ( - 48 \beta_{2} - 142 \beta_1 - 672) q^{74} - 75 q^{75} + (88 \beta_{2} - 36 \beta_1 + 120) q^{76} + (11 \beta_{2} - 33 \beta_1 + 11) q^{77} + ( - 12 \beta_{2} - 138 \beta_1 - 348) q^{78} + ( - 154 \beta_{2} - 24 \beta_1 - 40) q^{79} + ( - 30 \beta_{2} - 35 \beta_1 + 30) q^{80} + 81 q^{81} + ( - 44 \beta_{2} - 14 \beta_1 - 308) q^{82} + (195 \beta_{2} + 9 \beta_1 + 289) q^{83} + ( - 24 \beta_{2} + 120 \beta_1 - 120) q^{84} + (35 \beta_{2} - 25 \beta_1 + 45) q^{85} + ( - 16 \beta_{2} + 124 \beta_1 + 104) q^{86} + ( - 78 \beta_1 - 264) q^{87} + (22 \beta_{2} - 33 \beta_1 - 22) q^{88} + ( - 292 \beta_1 + 182) q^{89} - 45 \beta_1 q^{90} + (38 \beta_{2} + 154 \beta_1 + 162) q^{91} + (160 \beta_{2} + 208 \beta_1 - 160) q^{92} + (66 \beta_{2} - 102 \beta_1 - 126) q^{93} + ( - 184 \beta_{2} + 64 \beta_1 - 1016) q^{94} + ( - 10 \beta_{2} - 40 \beta_1 + 180) q^{95} + (42 \beta_{2} - 111 \beta_1 - 234) q^{96} + (148 \beta_{2} + 200 \beta_1 - 374) q^{97} + (88 \beta_{2} + 111 \beta_1 + 376) q^{98} + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 27 q^{9} + 5 q^{10} + 33 q^{11} - 51 q^{12} - 20 q^{13} + 144 q^{14} - 45 q^{15} + 25 q^{16} + 32 q^{17} + 9 q^{18} + 116 q^{19} + 85 q^{20} - 18 q^{21} + 11 q^{22} + 240 q^{23} + 9 q^{24} + 75 q^{25} + 302 q^{26} - 81 q^{27} + 160 q^{28} + 238 q^{29} - 15 q^{30} + 92 q^{31} + 197 q^{32} - 99 q^{33} + 354 q^{34} + 30 q^{35} + 153 q^{36} - 90 q^{37} + 324 q^{38} + 60 q^{39} - 15 q^{40} - 46 q^{41} - 432 q^{42} - 134 q^{43} + 187 q^{44} + 135 q^{45} - 240 q^{46} - 220 q^{47} - 75 q^{48} - 457 q^{49} + 25 q^{50} - 96 q^{51} - 1530 q^{52} - 798 q^{53} - 27 q^{54} + 165 q^{55} + 688 q^{56} - 348 q^{57} - 978 q^{58} + 1236 q^{59} - 255 q^{60} + 342 q^{61} - 1792 q^{62} + 54 q^{63} - 1919 q^{64} - 100 q^{65} - 33 q^{66} + 764 q^{67} + 1074 q^{68} - 720 q^{69} + 720 q^{70} + 1816 q^{71} - 27 q^{72} + 100 q^{73} - 1874 q^{74} - 225 q^{75} + 396 q^{76} + 66 q^{77} - 906 q^{78} - 96 q^{79} + 125 q^{80} + 243 q^{81} - 910 q^{82} + 858 q^{83} - 480 q^{84} + 160 q^{85} + 188 q^{86} - 714 q^{87} - 33 q^{88} + 838 q^{89} + 45 q^{90} + 332 q^{91} - 688 q^{92} - 276 q^{93} - 3112 q^{94} + 580 q^{95} - 591 q^{96} - 1322 q^{97} + 1017 q^{98} + 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 9x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + \nu + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{2} + \beta _1 + 13 ) / 2 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.91150
−3.04096
1.12946
−4.38835 −3.00000 11.2577 5.00000 13.1651 −11.7304 −14.2958 9.00000 −21.9418
1.2 0.793499 −3.00000 −7.37036 5.00000 −2.38050 −2.90793 −12.1964 9.00000 3.96749
1.3 4.59486 −3.00000 13.1127 5.00000 −13.7846 20.6383 23.4921 9.00000 22.9743
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.4.a.g 3
3.b odd 2 1 495.4.a.i 3
5.b even 2 1 825.4.a.p 3
5.c odd 4 2 825.4.c.m 6
11.b odd 2 1 1815.4.a.q 3
15.d odd 2 1 2475.4.a.z 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.g 3 1.a even 1 1 trivial
495.4.a.i 3 3.b odd 2 1
825.4.a.p 3 5.b even 2 1
825.4.c.m 6 5.c odd 4 2
1815.4.a.q 3 11.b odd 2 1
2475.4.a.z 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - T_{2}^{2} - 20T_{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(165))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 20 T + 16 \) Copy content Toggle raw display
$3$ \( (T + 3)^{3} \) Copy content Toggle raw display
$5$ \( (T - 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 6 T^{2} - 268 T - 704 \) Copy content Toggle raw display
$11$ \( (T - 11)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 20 T^{2} - 4920 T - 78104 \) Copy content Toggle raw display
$17$ \( T^{3} - 32 T^{2} - 2680 T - 22424 \) Copy content Toggle raw display
$19$ \( T^{3} - 116 T^{2} + 3356 T - 80 \) Copy content Toggle raw display
$23$ \( T^{3} - 240 T^{2} + 9728 T + 180224 \) Copy content Toggle raw display
$29$ \( T^{3} - 238 T^{2} + 5136 T + 428416 \) Copy content Toggle raw display
$31$ \( T^{3} - 92 T^{2} - 53552 T + 6769664 \) Copy content Toggle raw display
$37$ \( T^{3} + 90 T^{2} - 95700 T - 6364168 \) Copy content Toggle raw display
$41$ \( T^{3} + 46 T^{2} - 9136 T - 245888 \) Copy content Toggle raw display
$43$ \( T^{3} + 134 T^{2} - 18068 T - 2381360 \) Copy content Toggle raw display
$47$ \( T^{3} + 220 T^{2} + \cdots + 10980224 \) Copy content Toggle raw display
$53$ \( T^{3} + 798 T^{2} + \cdots - 17262968 \) Copy content Toggle raw display
$59$ \( T^{3} - 1236 T^{2} + \cdots - 32923904 \) Copy content Toggle raw display
$61$ \( T^{3} - 342 T^{2} - 68308 T - 2655176 \) Copy content Toggle raw display
$67$ \( T^{3} - 764 T^{2} + \cdots + 153685184 \) Copy content Toggle raw display
$71$ \( T^{3} - 1816 T^{2} + \cdots - 198158720 \) Copy content Toggle raw display
$73$ \( T^{3} - 100 T^{2} - 73616 T - 5132984 \) Copy content Toggle raw display
$79$ \( T^{3} + 96 T^{2} + \cdots - 167159872 \) Copy content Toggle raw display
$83$ \( T^{3} - 858 T^{2} + \cdots + 542136176 \) Copy content Toggle raw display
$89$ \( T^{3} - 838 T^{2} + \cdots + 693013592 \) Copy content Toggle raw display
$97$ \( T^{3} + 1322 T^{2} + \cdots - 354601256 \) Copy content Toggle raw display
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