Properties

Label 165.6.a.c
Level $165$
Weight $6$
Character orbit 165.a
Self dual yes
Analytic conductor $26.463$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 165.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.4633302691\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.18257.1
Defining polynomial: \( x^{3} - x^{2} - 26x + 8 \) 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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - 9 q^{3} + (\beta_{2} - \beta_1 - 14) q^{4} + 25 q^{5} + (9 \beta_1 - 9) q^{6} + ( - 6 \beta_{2} - 20 \beta_1 - 14) q^{7} + (2 \beta_{2} + 37 \beta_1 - 21) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} - 9 q^{3} + (\beta_{2} - \beta_1 - 14) q^{4} + 25 q^{5} + (9 \beta_1 - 9) q^{6} + ( - 6 \beta_{2} - 20 \beta_1 - 14) q^{7} + (2 \beta_{2} + 37 \beta_1 - 21) q^{8} + 81 q^{9} + ( - 25 \beta_1 + 25) q^{10} + 121 q^{11} + ( - 9 \beta_{2} + 9 \beta_1 + 126) q^{12} + (44 \beta_{2} - 24 \beta_1 + 90) q^{13} + (14 \beta_{2} + 68 \beta_1 + 278) q^{14} - 225 q^{15} + ( - 67 \beta_{2} + 35 \beta_1 - 186) q^{16} + ( - 102 \beta_{2} + 236 \beta_1 + 100) q^{17} + ( - 81 \beta_1 + 81) q^{18} + ( - 46 \beta_{2} + 172 \beta_1 - 994) q^{19} + (25 \beta_{2} - 25 \beta_1 - 350) q^{20} + (54 \beta_{2} + 180 \beta_1 + 126) q^{21} + ( - 121 \beta_1 + 121) q^{22} + (64 \beta_{2} + 688 \beta_1 - 464) q^{23} + ( - 18 \beta_{2} - 333 \beta_1 + 189) q^{24} + 625 q^{25} + (68 \beta_{2} - 486 \beta_1 + 850) q^{26} - 729 q^{27} + (138 \beta_{2} + 236 \beta_1 - 318) q^{28} + (330 \beta_{2} + 652 \beta_1 - 1840) q^{29} + (225 \beta_1 - 225) q^{30} + ( - 44 \beta_{2} - 56 \beta_1 - 4956) q^{31} + ( - 166 \beta_{2} - 395 \beta_1 - 645) q^{32} - 1089 q^{33} + ( - 338 \beta_{2} + 818 \beta_1 - 4728) q^{34} + ( - 150 \beta_{2} - 500 \beta_1 - 350) q^{35} + (81 \beta_{2} - 81 \beta_1 - 1134) q^{36} + (648 \beta_{2} - 448 \beta_1 - 2130) q^{37} + ( - 218 \beta_{2} + 1408 \beta_1 - 4286) q^{38} + ( - 396 \beta_{2} + 216 \beta_1 - 810) q^{39} + (50 \beta_{2} + 925 \beta_1 - 525) q^{40} + ( - 330 \beta_{2} - 1804 \beta_1 - 2264) q^{41} + ( - 126 \beta_{2} - 612 \beta_1 - 2502) q^{42} + (618 \beta_{2} + 1340 \beta_1 - 11850) q^{43} + (121 \beta_{2} - 121 \beta_1 - 1694) q^{44} + 2025 q^{45} + ( - 624 \beta_{2} - 112 \beta_1 - 11648) q^{46} + (364 \beta_{2} - 1544 \beta_1 - 7820) q^{47} + (603 \beta_{2} - 315 \beta_1 + 1674) q^{48} + (280 \beta_{2} + 2832 \beta_1 - 5935) q^{49} + ( - 625 \beta_1 + 625) q^{50} + (918 \beta_{2} - 2124 \beta_1 - 900) q^{51} + ( - 854 \beta_{2} - 694 \beta_1 + 6776) q^{52} + ( - 1196 \beta_{2} - 360 \beta_1 - 7126) q^{53} + (729 \beta_1 - 729) q^{54} + 3025 q^{55} + ( - 546 \beta_{2} - 3100 \beta_1 - 12122) q^{56} + (414 \beta_{2} - 1548 \beta_1 + 8946) q^{57} + ( - 322 \beta_{2} - 1130 \beta_1 - 10284) q^{58} + ( - 2864 \beta_{2} - 4928 \beta_1 - 1988) q^{59} + ( - 225 \beta_{2} + 225 \beta_1 + 3150) q^{60} + ( - 616 \beta_{2} + 432 \beta_1 + 8262) q^{61} + (12 \beta_{2} + 5352 \beta_1 - 4356) q^{62} + ( - 486 \beta_{2} - 1620 \beta_1 - 1134) q^{63} + (2373 \beta_{2} + 1019 \beta_1 + 10694) q^{64} + (1100 \beta_{2} - 600 \beta_1 + 2250) q^{65} + (1089 \beta_1 - 1089) q^{66} + ( - 64 \beta_{2} + 3360 \beta_1 + 4524) q^{67} + (2108 \beta_{2} + 218 \beta_1 - 24538) q^{68} + ( - 576 \beta_{2} - 6192 \beta_1 + 4176) q^{69} + (350 \beta_{2} + 1700 \beta_1 + 6950) q^{70} + (3576 \beta_{2} - 3376 \beta_1 + 1552) q^{71} + (162 \beta_{2} + 2997 \beta_1 - 1701) q^{72} + (1028 \beta_{2} - 1656 \beta_1 + 846) q^{73} + (1096 \beta_{2} - 3702 \beta_1 + 10670) q^{74} - 5625 q^{75} + ( - 154 \beta_{2} + 744 \beta_1 + 1842) q^{76} + ( - 726 \beta_{2} - 2420 \beta_1 - 1694) q^{77} + ( - 612 \beta_{2} + 4374 \beta_1 - 7650) q^{78} + ( - 4202 \beta_{2} - 8252 \beta_1 - 8130) q^{79} + ( - 1675 \beta_{2} + 875 \beta_1 - 4650) q^{80} + 6561 q^{81} + (1474 \beta_{2} + 5234 \beta_1 + 25764) q^{82} + (2800 \beta_{2} - 5744 \beta_1 - 15284) q^{83} + ( - 1242 \beta_{2} - 2124 \beta_1 + 2862) q^{84} + ( - 2550 \beta_{2} + 5900 \beta_1 + 2500) q^{85} + ( - 722 \beta_{2} + 6288 \beta_1 - 29686) q^{86} + ( - 2970 \beta_{2} - 5868 \beta_1 + 16560) q^{87} + (242 \beta_{2} + 4477 \beta_1 - 2541) q^{88} + (2640 \beta_{2} + 8896 \beta_1 - 66838) q^{89} + ( - 2025 \beta_1 + 2025) q^{90} + (1436 \beta_{2} - 5496 \beta_1 - 29716) q^{91} + ( - 2560 \beta_{2} - 4752 \beta_1 + 112) q^{92} + (396 \beta_{2} + 504 \beta_1 + 44604) q^{93} + (1908 \beta_{2} + 4544 \beta_1 + 21340) q^{94} + ( - 1150 \beta_{2} + 4300 \beta_1 - 24850) q^{95} + (1494 \beta_{2} + 3555 \beta_1 + 5805) q^{96} + ( - 48 \beta_{2} - 13696 \beta_1 + 87090) q^{97} + ( - 2552 \beta_{2} + 3415 \beta_1 - 51839) q^{98} + 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 27 q^{3} - 42 q^{4} + 75 q^{5} - 18 q^{6} - 68 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} - 27 q^{3} - 42 q^{4} + 75 q^{5} - 18 q^{6} - 68 q^{7} - 24 q^{8} + 243 q^{9} + 50 q^{10} + 363 q^{11} + 378 q^{12} + 290 q^{13} + 916 q^{14} - 675 q^{15} - 590 q^{16} + 434 q^{17} + 162 q^{18} - 2856 q^{19} - 1050 q^{20} + 612 q^{21} + 242 q^{22} - 640 q^{23} + 216 q^{24} + 1875 q^{25} + 2132 q^{26} - 2187 q^{27} - 580 q^{28} - 4538 q^{29} - 450 q^{30} - 14968 q^{31} - 2496 q^{32} - 3267 q^{33} - 13704 q^{34} - 1700 q^{35} - 3402 q^{36} - 6190 q^{37} - 11668 q^{38} - 2610 q^{39} - 600 q^{40} - 8926 q^{41} - 8244 q^{42} - 33592 q^{43} - 5082 q^{44} + 6075 q^{45} - 35680 q^{46} - 24640 q^{47} + 5310 q^{48} - 14693 q^{49} + 1250 q^{50} - 3906 q^{51} + 18780 q^{52} - 22934 q^{53} - 1458 q^{54} + 9075 q^{55} - 40012 q^{56} + 25704 q^{57} - 32304 q^{58} - 13756 q^{59} + 9450 q^{60} + 24602 q^{61} - 7704 q^{62} - 5508 q^{63} + 35474 q^{64} + 7250 q^{65} - 2178 q^{66} + 16868 q^{67} - 71288 q^{68} + 5760 q^{69} + 22900 q^{70} + 4856 q^{71} - 1944 q^{72} + 1910 q^{73} + 29404 q^{74} - 16875 q^{75} + 6116 q^{76} - 8228 q^{77} - 19188 q^{78} - 36844 q^{79} - 14750 q^{80} + 19683 q^{81} + 84000 q^{82} - 48796 q^{83} + 5220 q^{84} + 10850 q^{85} - 83492 q^{86} + 40842 q^{87} - 2904 q^{88} - 188978 q^{89} + 4050 q^{90} - 93208 q^{91} - 6976 q^{92} + 134712 q^{93} + 70472 q^{94} - 71400 q^{95} + 22464 q^{96} + 247526 q^{97} - 154654 q^{98} + 29403 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 26x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 17 \) 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
5.47894
0.305203
−4.78415
−4.47894 −9.00000 −11.9391 25.0000 40.3105 −168.818 196.801 81.0000 −111.974
1.2 0.694797 −9.00000 −31.5173 25.0000 −6.25317 83.1683 −44.1316 81.0000 17.3699
1.3 5.78415 −9.00000 1.45634 25.0000 −52.0573 17.6498 −176.669 81.0000 144.604
\(n\): 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 165.6.a.c 3
3.b odd 2 1 495.6.a.b 3
5.b even 2 1 825.6.a.g 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.c 3 1.a even 1 1 trivial
495.6.a.b 3 3.b odd 2 1
825.6.a.g 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 2T_{2}^{2} - 25T_{2} + 18 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(165))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 2 T^{2} - 25 T + 18 \) Copy content Toggle raw display
$3$ \( (T + 9)^{3} \) Copy content Toggle raw display
$5$ \( (T - 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 68 T^{2} - 15552 T + 247808 \) Copy content Toggle raw display
$11$ \( (T - 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 290 T^{2} + \cdots + 132063592 \) Copy content Toggle raw display
$17$ \( T^{3} - 434 T^{2} + \cdots + 2547052488 \) Copy content Toggle raw display
$19$ \( T^{3} + 2856 T^{2} + \cdots + 137703680 \) Copy content Toggle raw display
$23$ \( T^{3} + 640 T^{2} + \cdots - 15777349632 \) Copy content Toggle raw display
$29$ \( T^{3} + 4538 T^{2} + \cdots - 44413548456 \) Copy content Toggle raw display
$31$ \( T^{3} + 14968 T^{2} + \cdots + 121645522944 \) Copy content Toggle raw display
$37$ \( T^{3} + 6190 T^{2} + \cdots + 28013661736 \) Copy content Toggle raw display
$41$ \( T^{3} + 8926 T^{2} + \cdots + 119305168392 \) Copy content Toggle raw display
$43$ \( T^{3} + 33592 T^{2} + \cdots - 38997547520 \) Copy content Toggle raw display
$47$ \( T^{3} + 24640 T^{2} + \cdots - 679997104128 \) Copy content Toggle raw display
$53$ \( T^{3} + 22934 T^{2} + \cdots - 4393759072056 \) Copy content Toggle raw display
$59$ \( T^{3} + 13756 T^{2} + \cdots - 20798004639936 \) Copy content Toggle raw display
$61$ \( T^{3} - 24602 T^{2} + \cdots + 43064794504 \) Copy content Toggle raw display
$67$ \( T^{3} - 16868 T^{2} + \cdots + 1826752720192 \) Copy content Toggle raw display
$71$ \( T^{3} - 4856 T^{2} + \cdots + 34155066048000 \) Copy content Toggle raw display
$73$ \( T^{3} - 1910 T^{2} + \cdots - 163103734088 \) Copy content Toggle raw display
$79$ \( T^{3} + 36844 T^{2} + \cdots - 70772253539328 \) Copy content Toggle raw display
$83$ \( T^{3} + 48796 T^{2} + \cdots - 70386077185728 \) Copy content Toggle raw display
$89$ \( T^{3} + 188978 T^{2} + \cdots - 16088649675432 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 148869121092488 \) Copy content Toggle raw display
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