Properties

Label 165.4.a.e
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: 3.3.47528.1
Defining polynomial: \( x^{3} - x^{2} - 26x - 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{2} + 10) q^{4} - 5 q^{5} + (3 \beta_1 - 3) q^{6} + (2 \beta_{2} - 2 \beta_1 + 4) q^{7} + ( - 2 \beta_{2} + 9 \beta_1 + 3) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{2} + 10) q^{4} - 5 q^{5} + (3 \beta_1 - 3) q^{6} + (2 \beta_{2} - 2 \beta_1 + 4) q^{7} + ( - 2 \beta_{2} + 9 \beta_1 + 3) q^{8} + 9 q^{9} + ( - 5 \beta_1 + 5) q^{10} + 11 q^{11} + (3 \beta_{2} + 30) q^{12} + ( - 2 \beta_{2} + 38) q^{13} + ( - 6 \beta_{2} + 16 \beta_1 - 28) q^{14} - 15 q^{15} + (5 \beta_{2} - 2 \beta_1 + 60) q^{16} + ( - 2 \beta_{2} - 2 \beta_1 - 34) q^{17} + (9 \beta_1 - 9) q^{18} + ( - 8 \beta_{2} + 14 \beta_1 - 24) q^{19} + ( - 5 \beta_{2} - 50) q^{20} + (6 \beta_{2} - 6 \beta_1 + 12) q^{21} + (11 \beta_1 - 11) q^{22} + (4 \beta_{2} - 24 \beta_1 + 48) q^{23} + ( - 6 \beta_{2} + 27 \beta_1 + 9) q^{24} + 25 q^{25} + (4 \beta_{2} + 24 \beta_1 - 48) q^{26} + 27 q^{27} + (12 \beta_{2} - 38 \beta_1 + 238) q^{28} + ( - 34 \beta_1 - 62) q^{29} + ( - 15 \beta_1 + 15) q^{30} + ( - 8 \beta_{2} - 40 \beta_1 + 96) q^{31} + (4 \beta_{2} + 21 \beta_1 - 93) q^{32} + 33 q^{33} + (2 \beta_{2} - 50 \beta_1 - 10) q^{34} + ( - 10 \beta_{2} + 10 \beta_1 - 20) q^{35} + (9 \beta_{2} + 90) q^{36} + ( - 20 \beta_1 + 286) q^{37} + (30 \beta_{2} - 66 \beta_1 + 222) q^{38} + ( - 6 \beta_{2} + 114) q^{39} + (10 \beta_{2} - 45 \beta_1 - 15) q^{40} + (66 \beta_1 + 30) q^{41} + ( - 18 \beta_{2} + 48 \beta_1 - 84) q^{42} + ( - 14 \beta_{2} - 22 \beta_1 + 48) q^{43} + (11 \beta_{2} + 110) q^{44} - 45 q^{45} + ( - 32 \beta_{2} + 52 \beta_1 - 436) q^{46} + ( - 4 \beta_{2} + 168) q^{47} + (15 \beta_{2} - 6 \beta_1 + 180) q^{48} + ( - 72 \beta_1 + 117) q^{49} + (25 \beta_1 - 25) q^{50} + ( - 6 \beta_{2} - 6 \beta_1 - 102) q^{51} + (32 \beta_{2} + 4 \beta_1 + 172) q^{52} + (16 \beta_{2} - 96 \beta_1 + 126) q^{53} + (27 \beta_1 - 27) q^{54} - 55 q^{55} + ( - 14 \beta_{2} + 156 \beta_1 - 600) q^{56} + ( - 24 \beta_{2} + 42 \beta_1 - 72) q^{57} + ( - 34 \beta_{2} - 96 \beta_1 - 516) q^{58} + (8 \beta_{2} + 32 \beta_1 + 172) q^{59} + ( - 15 \beta_{2} - 150) q^{60} + (68 \beta_{2} + 12 \beta_1 + 134) q^{61} + ( - 24 \beta_{2} - 816) q^{62} + (18 \beta_{2} - 18 \beta_1 + 36) q^{63} + ( - 27 \beta_{2} - 28 \beta_1 - 10) q^{64} + (10 \beta_{2} - 190) q^{65} + (33 \beta_1 - 33) q^{66} + ( - 16 \beta_{2} + 100 \beta_1 - 176) q^{67} + ( - 38 \beta_{2} - 30 \beta_1 - 558) q^{68} + (12 \beta_{2} - 72 \beta_1 + 144) q^{69} + (30 \beta_{2} - 80 \beta_1 + 140) q^{70} + ( - 28 \beta_{2} + 60 \beta_1 - 324) q^{71} + ( - 18 \beta_{2} + 81 \beta_1 + 27) q^{72} + ( - 38 \beta_{2} + 36 \beta_1 + 194) q^{73} + ( - 20 \beta_{2} + 266 \beta_1 - 626) q^{74} + 75 q^{75} + ( - 62 \beta_{2} + 254 \beta_1 - 1002) q^{76} + (22 \beta_{2} - 22 \beta_1 + 44) q^{77} + (12 \beta_{2} + 72 \beta_1 - 144) q^{78} + (8 \beta_{2} + 94 \beta_1 - 212) q^{79} + ( - 25 \beta_{2} + 10 \beta_1 - 300) q^{80} + 81 q^{81} + (66 \beta_{2} + 96 \beta_1 + 1092) q^{82} + (54 \beta_{2} - 180 \beta_1 + 60) q^{83} + (36 \beta_{2} - 114 \beta_1 + 714) q^{84} + (10 \beta_{2} + 10 \beta_1 + 170) q^{85} + (6 \beta_{2} - 72 \beta_1 - 492) q^{86} + ( - 102 \beta_1 - 186) q^{87} + ( - 22 \beta_{2} + 99 \beta_1 + 33) q^{88} + (120 \beta_{2} + 28 \beta_1 + 254) q^{89} + ( - 45 \beta_1 + 45) q^{90} + (92 \beta_{2} - 40 \beta_1 - 244) q^{91} + (84 \beta_{2} - 416 \beta_1 + 776) q^{92} + ( - 24 \beta_{2} - 120 \beta_1 + 288) q^{93} + (8 \beta_{2} + 140 \beta_1 - 188) q^{94} + (40 \beta_{2} - 70 \beta_1 + 120) q^{95} + (12 \beta_{2} + 63 \beta_1 - 279) q^{96} + ( - 44 \beta_{2} + 100 \beta_1 + 658) q^{97} + ( - 72 \beta_{2} + 45 \beta_1 - 1341) q^{98} + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 9 q^{3} + 30 q^{4} - 15 q^{5} - 6 q^{6} + 10 q^{7} + 18 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 9 q^{3} + 30 q^{4} - 15 q^{5} - 6 q^{6} + 10 q^{7} + 18 q^{8} + 27 q^{9} + 10 q^{10} + 33 q^{11} + 90 q^{12} + 114 q^{13} - 68 q^{14} - 45 q^{15} + 178 q^{16} - 104 q^{17} - 18 q^{18} - 58 q^{19} - 150 q^{20} + 30 q^{21} - 22 q^{22} + 120 q^{23} + 54 q^{24} + 75 q^{25} - 120 q^{26} + 81 q^{27} + 676 q^{28} - 220 q^{29} + 30 q^{30} + 248 q^{31} - 258 q^{32} + 99 q^{33} - 80 q^{34} - 50 q^{35} + 270 q^{36} + 838 q^{37} + 600 q^{38} + 342 q^{39} - 90 q^{40} + 156 q^{41} - 204 q^{42} + 122 q^{43} + 330 q^{44} - 135 q^{45} - 1256 q^{46} + 504 q^{47} + 534 q^{48} + 279 q^{49} - 50 q^{50} - 312 q^{51} + 520 q^{52} + 282 q^{53} - 54 q^{54} - 165 q^{55} - 1644 q^{56} - 174 q^{57} - 1644 q^{58} + 548 q^{59} - 450 q^{60} + 414 q^{61} - 2448 q^{62} + 90 q^{63} - 58 q^{64} - 570 q^{65} - 66 q^{66} - 428 q^{67} - 1704 q^{68} + 360 q^{69} + 340 q^{70} - 912 q^{71} + 162 q^{72} + 618 q^{73} - 1612 q^{74} + 225 q^{75} - 2752 q^{76} + 110 q^{77} - 360 q^{78} - 542 q^{79} - 890 q^{80} + 243 q^{81} + 3372 q^{82} + 2028 q^{84} + 520 q^{85} - 1548 q^{86} - 660 q^{87} + 198 q^{88} + 790 q^{89} + 90 q^{90} - 772 q^{91} + 1912 q^{92} + 744 q^{93} - 424 q^{94} + 290 q^{95} - 774 q^{96} + 2074 q^{97} - 3978 q^{98} + 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 26x - 22 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 17 \) 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
−4.06484
−0.906392
5.97123
−5.06484 3.00000 17.6526 −5.00000 −15.1945 27.4348 −48.8887 9.00000 25.3242
1.2 −1.90639 3.00000 −4.36567 −5.00000 −5.71918 −22.9186 23.5738 9.00000 9.53196
1.3 4.97123 3.00000 16.7131 −5.00000 14.9137 5.48376 43.3148 9.00000 −24.8561
\(n\): 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.e 3
3.b odd 2 1 495.4.a.k 3
5.b even 2 1 825.4.a.r 3
5.c odd 4 2 825.4.c.k 6
11.b odd 2 1 1815.4.a.r 3
15.d odd 2 1 2475.4.a.t 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.e 3 1.a even 1 1 trivial
495.4.a.k 3 3.b odd 2 1
825.4.a.r 3 5.b even 2 1
825.4.c.k 6 5.c odd 4 2
1815.4.a.r 3 11.b odd 2 1
2475.4.a.t 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 2T_{2}^{2} - 25T_{2} - 48 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(165))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2 T^{2} - 25 T - 48 \) Copy content Toggle raw display
$3$ \( (T - 3)^{3} \) Copy content Toggle raw display
$5$ \( (T + 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 10 T^{2} - 604 T + 3448 \) Copy content Toggle raw display
$11$ \( (T - 11)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 114 T^{2} + 3712 T - 37216 \) Copy content Toggle raw display
$17$ \( T^{3} + 104 T^{2} + 2792 T + 8448 \) Copy content Toggle raw display
$19$ \( T^{3} + 58 T^{2} - 11496 T + 65520 \) Copy content Toggle raw display
$23$ \( T^{3} - 120 T^{2} - 10736 T + 148224 \) Copy content Toggle raw display
$29$ \( T^{3} + 220 T^{2} - 14308 T - 629760 \) Copy content Toggle raw display
$31$ \( T^{3} - 248 T^{2} - 38592 T + 9589248 \) Copy content Toggle raw display
$37$ \( T^{3} - 838 T^{2} + \cdots - 18607336 \) Copy content Toggle raw display
$41$ \( T^{3} - 156 T^{2} - 106596 T - 3013632 \) Copy content Toggle raw display
$43$ \( T^{3} - 122 T^{2} - 44940 T + 1445400 \) Copy content Toggle raw display
$47$ \( T^{3} - 504 T^{2} + 82192 T - 4372224 \) Copy content Toggle raw display
$53$ \( T^{3} - 282 T^{2} - 222068 T - 3654264 \) Copy content Toggle raw display
$59$ \( T^{3} - 548 T^{2} + 57584 T - 1206720 \) Copy content Toggle raw display
$61$ \( T^{3} - 414 T^{2} + \cdots + 342344792 \) Copy content Toggle raw display
$67$ \( T^{3} + 428 T^{2} - 206752 T - 8135552 \) Copy content Toggle raw display
$71$ \( T^{3} + 912 T^{2} + 97888 T - 2867712 \) Copy content Toggle raw display
$73$ \( T^{3} - 618 T^{2} + \cdots + 26458592 \) Copy content Toggle raw display
$79$ \( T^{3} + 542 T^{2} + \cdots - 88503440 \) Copy content Toggle raw display
$83$ \( T^{3} - 1091340 T - 434328048 \) Copy content Toggle raw display
$89$ \( T^{3} - 790 T^{2} + \cdots + 1941629400 \) Copy content Toggle raw display
$97$ \( T^{3} - 2074 T^{2} + \cdots + 98075336 \) Copy content Toggle raw display
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