Properties

Label 165.4.a.f
Level $165$
Weight $4$
Character orbit 165.a
Self dual yes
Analytic conductor $9.735$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.788.1
Defining polynomial: \( x^{3} - x^{2} - 7x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - 3 q^{3} + ( - \beta_1 - 2) q^{4} + 5 q^{5} - 3 \beta_{2} q^{6} + ( - 2 \beta_{2} + 5 \beta_1 - 3) q^{7} + ( - 7 \beta_{2} - \beta_1 + 1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - 3 q^{3} + ( - \beta_1 - 2) q^{4} + 5 q^{5} - 3 \beta_{2} q^{6} + ( - 2 \beta_{2} + 5 \beta_1 - 3) q^{7} + ( - 7 \beta_{2} - \beta_1 + 1) q^{8} + 9 q^{9} + 5 \beta_{2} q^{10} - 11 q^{11} + (3 \beta_1 + 6) q^{12} + ( - 20 \beta_{2} - 11 \beta_1 - 11) q^{13} + ( - 18 \beta_{2} + 7 \beta_1 - 17) q^{14} - 15 q^{15} + (4 \beta_{2} + 14 \beta_1 - 25) q^{16} + (14 \beta_{2} - 9 \beta_1 - 19) q^{17} + 9 \beta_{2} q^{18} + ( - 26 \beta_{2} - 10 \beta_1 - 88) q^{19} + ( - 5 \beta_1 - 10) q^{20} + (6 \beta_{2} - 15 \beta_1 + 9) q^{21} - 11 \beta_{2} q^{22} + (36 \beta_{2} - 14 \beta_1 - 54) q^{23} + (21 \beta_{2} + 3 \beta_1 - 3) q^{24} + 25 q^{25} + (22 \beta_{2} + 9 \beta_1 - 109) q^{26} - 27 q^{27} + ( - 22 \beta_{2} - 15 \beta_1 - 91) q^{28} + (18 \beta_{2} + 44 \beta_1 - 88) q^{29} - 15 \beta_{2} q^{30} + (32 \beta_{2} + 22 \beta_1 - 134) q^{31} + ( - 11 \beta_{2} + 18 \beta_1 + 2) q^{32} + 33 q^{33} + (8 \beta_{2} - 23 \beta_1 + 93) q^{34} + ( - 10 \beta_{2} + 25 \beta_1 - 15) q^{35} + ( - 9 \beta_1 - 18) q^{36} + ( - 56 \beta_{2} - 32 \beta_1 + 198) q^{37} + ( - 58 \beta_{2} + 16 \beta_1 - 146) q^{38} + (60 \beta_{2} + 33 \beta_1 + 33) q^{39} + ( - 35 \beta_{2} - 5 \beta_1 + 5) q^{40} + (110 \beta_{2} - 44 \beta_1 - 272) q^{41} + (54 \beta_{2} - 21 \beta_1 + 51) q^{42} + ( - 22 \beta_{2} + 85 \beta_1 + 13) q^{43} + (11 \beta_1 + 22) q^{44} + 45 q^{45} + ( - 12 \beta_{2} - 50 \beta_1 + 230) q^{46} + (112 \beta_{2} - 38 \beta_1 + 38) q^{47} + ( - 12 \beta_{2} - 42 \beta_1 + 75) q^{48} + (172 \beta_{2} - 4 \beta_1 + 185) q^{49} + 25 \beta_{2} q^{50} + ( - 42 \beta_{2} + 27 \beta_1 + 57) q^{51} + (24 \beta_{2} + 75 \beta_1 + 211) q^{52} + ( - 60 \beta_{2} - 38 \beta_1 - 24) q^{53} - 27 \beta_{2} q^{54} - 55 q^{55} + (98 \beta_{2} - 49 \beta_1 + 19) q^{56} + (78 \beta_{2} + 30 \beta_1 + 264) q^{57} + ( - 220 \beta_{2} + 26 \beta_1 + 64) q^{58} + ( - 196 \beta_{2} - 106 \beta_1 - 174) q^{59} + (15 \beta_1 + 30) q^{60} + ( - 100 \beta_{2} - 22 \beta_1 - 168) q^{61} + ( - 200 \beta_{2} - 10 \beta_1 + 170) q^{62} + ( - 18 \beta_{2} + 45 \beta_1 - 27) q^{63} + ( - 84 \beta_{2} - 83 \beta_1 + 116) q^{64} + ( - 100 \beta_{2} - 55 \beta_1 - 55) q^{65} + 33 \beta_{2} q^{66} + (272 \beta_{2} + 58 \beta_1 + 242) q^{67} + (50 \beta_{2} + 41 \beta_1 + 223) q^{68} + ( - 108 \beta_{2} + 42 \beta_1 + 162) q^{69} + ( - 90 \beta_{2} + 35 \beta_1 - 85) q^{70} + (96 \beta_{2} + 74 \beta_1 - 546) q^{71} + ( - 63 \beta_{2} - 9 \beta_1 + 9) q^{72} + (16 \beta_{2} - 17 \beta_1 + 235) q^{73} + (294 \beta_{2} + 24 \beta_1 - 304) q^{74} - 75 q^{75} + (14 \beta_{2} + 154 \beta_1 + 340) q^{76} + (22 \beta_{2} - 55 \beta_1 + 33) q^{77} + ( - 66 \beta_{2} - 27 \beta_1 + 327) q^{78} + ( - 130 \beta_{2} + 22 \beta_1 + 92) q^{79} + (20 \beta_{2} + 70 \beta_1 - 125) q^{80} + 81 q^{81} + ( - 140 \beta_{2} - 154 \beta_1 + 704) q^{82} + ( - 148 \beta_{2} + 7 \beta_1 - 359) q^{83} + (66 \beta_{2} + 45 \beta_1 + 273) q^{84} + (70 \beta_{2} - 45 \beta_1 - 95) q^{85} + ( - 242 \beta_{2} + 107 \beta_1 - 217) q^{86} + ( - 54 \beta_{2} - 132 \beta_1 + 264) q^{87} + (77 \beta_{2} + 11 \beta_1 - 11) q^{88} + ( - 236 \beta_{2} + 132 \beta_1 + 402) q^{89} + 45 \beta_{2} q^{90} + (96 \beta_{2} - 250 \beta_1 - 694) q^{91} + (92 \beta_{2} + 74 \beta_1 + 410) q^{92} + ( - 96 \beta_{2} - 66 \beta_1 + 402) q^{93} + (152 \beta_{2} - 150 \beta_1 + 710) q^{94} + ( - 130 \beta_{2} - 50 \beta_1 - 440) q^{95} + (33 \beta_{2} - 54 \beta_1 - 6) q^{96} + ( - 16 \beta_{2} + 278 \beta_1 + 68) q^{97} + (197 \beta_{2} - 176 \beta_1 + 1036) q^{98} - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 9 q^{3} - 5 q^{4} + 15 q^{5} - 3 q^{6} - 16 q^{7} - 3 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - 9 q^{3} - 5 q^{4} + 15 q^{5} - 3 q^{6} - 16 q^{7} - 3 q^{8} + 27 q^{9} + 5 q^{10} - 33 q^{11} + 15 q^{12} - 42 q^{13} - 76 q^{14} - 45 q^{15} - 85 q^{16} - 34 q^{17} + 9 q^{18} - 280 q^{19} - 25 q^{20} + 48 q^{21} - 11 q^{22} - 112 q^{23} + 9 q^{24} + 75 q^{25} - 314 q^{26} - 81 q^{27} - 280 q^{28} - 290 q^{29} - 15 q^{30} - 392 q^{31} - 23 q^{32} + 99 q^{33} + 310 q^{34} - 80 q^{35} - 45 q^{36} + 570 q^{37} - 512 q^{38} + 126 q^{39} - 15 q^{40} - 662 q^{41} + 228 q^{42} - 68 q^{43} + 55 q^{44} + 135 q^{45} + 728 q^{46} + 264 q^{47} + 255 q^{48} + 731 q^{49} + 25 q^{50} + 102 q^{51} + 582 q^{52} - 94 q^{53} - 27 q^{54} - 165 q^{55} + 204 q^{56} + 840 q^{57} - 54 q^{58} - 612 q^{59} + 75 q^{60} - 582 q^{61} + 320 q^{62} - 144 q^{63} + 347 q^{64} - 210 q^{65} + 33 q^{66} + 940 q^{67} + 678 q^{68} + 336 q^{69} - 380 q^{70} - 1616 q^{71} - 27 q^{72} + 738 q^{73} - 642 q^{74} - 225 q^{75} + 880 q^{76} + 176 q^{77} + 942 q^{78} + 124 q^{79} - 425 q^{80} + 243 q^{81} + 2126 q^{82} - 1232 q^{83} + 840 q^{84} - 170 q^{85} - 1000 q^{86} + 870 q^{87} + 33 q^{88} + 838 q^{89} + 45 q^{90} - 1736 q^{91} + 1248 q^{92} + 1176 q^{93} + 2432 q^{94} - 1400 q^{95} + 69 q^{96} - 90 q^{97} + 3481 q^{98} - 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 7x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 5 \) 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
−0.476452
3.35386
−1.87740
−2.82009 −3.00000 −0.0470959 5.00000 8.46027 −7.12434 22.6935 9.00000 −14.1004
1.2 0.540637 −3.00000 −7.70771 5.00000 −1.62191 24.4573 −8.49217 9.00000 2.70319
1.3 3.27945 −3.00000 2.75481 5.00000 −9.83836 −33.3329 −17.2014 9.00000 16.3973
\(n\): 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.f 3
3.b odd 2 1 495.4.a.g 3
5.b even 2 1 825.4.a.n 3
5.c odd 4 2 825.4.c.o 6
11.b odd 2 1 1815.4.a.p 3
15.d odd 2 1 2475.4.a.w 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.f 3 1.a even 1 1 trivial
495.4.a.g 3 3.b odd 2 1
825.4.a.n 3 5.b even 2 1
825.4.c.o 6 5.c odd 4 2
1815.4.a.p 3 11.b odd 2 1
2475.4.a.w 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - T_{2}^{2} - 9T_{2} + 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(165))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 9T + 5 \) Copy content Toggle raw display
$3$ \( (T + 3)^{3} \) Copy content Toggle raw display
$5$ \( (T - 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 16 T^{2} - 752 T - 5808 \) Copy content Toggle raw display
$11$ \( (T + 11)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 42 T^{2} - 5228 T - 137416 \) Copy content Toggle raw display
$17$ \( T^{3} + 34 T^{2} - 4660 T - 179064 \) Copy content Toggle raw display
$19$ \( T^{3} + 280 T^{2} + 18624 T - 97056 \) Copy content Toggle raw display
$23$ \( T^{3} + 112 T^{2} - 17024 T - 1916288 \) Copy content Toggle raw display
$29$ \( T^{3} + 290 T^{2} - 26500 T - 9251496 \) Copy content Toggle raw display
$31$ \( T^{3} + 392 T^{2} + 32160 T - 316800 \) Copy content Toggle raw display
$37$ \( T^{3} - 570 T^{2} + 60940 T + 1039624 \) Copy content Toggle raw display
$41$ \( T^{3} + 662 T^{2} + \cdots - 68561784 \) Copy content Toggle raw display
$43$ \( T^{3} + 68 T^{2} - 227376 T - 20491056 \) Copy content Toggle raw display
$47$ \( T^{3} - 264 T^{2} + \cdots - 14121600 \) Copy content Toggle raw display
$53$ \( T^{3} + 94 T^{2} - 57812 T - 2403992 \) Copy content Toggle raw display
$59$ \( T^{3} + 612 T^{2} + \cdots - 162128320 \) Copy content Toggle raw display
$61$ \( T^{3} + 582 T^{2} + \cdots - 21355000 \) Copy content Toggle raw display
$67$ \( T^{3} - 940 T^{2} + \cdots + 394498240 \) Copy content Toggle raw display
$71$ \( T^{3} + 1616 T^{2} + \cdots + 40198784 \) Copy content Toggle raw display
$73$ \( T^{3} - 738 T^{2} + \cdots - 12046264 \) Copy content Toggle raw display
$79$ \( T^{3} - 124 T^{2} + \cdots + 26871328 \) Copy content Toggle raw display
$83$ \( T^{3} + 1232 T^{2} + \cdots - 15656400 \) Copy content Toggle raw display
$89$ \( T^{3} - 838 T^{2} + \cdots + 831946232 \) Copy content Toggle raw display
$97$ \( T^{3} + 90 T^{2} + \cdots - 924170696 \) Copy content Toggle raw display
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