Properties

Label 165.6.a.a
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: 3.3.34253.1
Defining polynomial: \( x^{3} - x^{2} - 52x + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{2} + 9 q^{3} + (\beta_{2} + 4 \beta_1 + 7) q^{4} + 25 q^{5} + ( - 9 \beta_1 - 18) q^{6} + (\beta_{2} + 11 \beta_1 - 61) q^{7} + ( - 7 \beta_{2} - 77) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 2) q^{2} + 9 q^{3} + (\beta_{2} + 4 \beta_1 + 7) q^{4} + 25 q^{5} + ( - 9 \beta_1 - 18) q^{6} + (\beta_{2} + 11 \beta_1 - 61) q^{7} + ( - 7 \beta_{2} - 77) q^{8} + 81 q^{9} + ( - 25 \beta_1 - 50) q^{10} - 121 q^{11} + (9 \beta_{2} + 36 \beta_1 + 63) q^{12} + ( - 28 \beta_{2} - 24 \beta_1 - 210) q^{13} + ( - 14 \beta_{2} + 22 \beta_1 - 250) q^{14} + 225 q^{15} + ( - 11 \beta_{2} + 68 \beta_1 - 161) q^{16} + (39 \beta_{2} + 37 \beta_1 - 801) q^{17} + ( - 81 \beta_1 - 162) q^{18} + ( - 29 \beta_{2} - 67 \beta_1 - 935) q^{19} + (25 \beta_{2} + 100 \beta_1 + 175) q^{20} + (9 \beta_{2} + 99 \beta_1 - 549) q^{21} + (121 \beta_1 + 242) q^{22} + ( - 20 \beta_{2} + 220 \beta_1 + 684) q^{23} + ( - 63 \beta_{2} - 693) q^{24} + 625 q^{25} + (108 \beta_{2} + 734 \beta_1 + 896) q^{26} + 729 q^{27} + ( - 12 \beta_{2} + 92 \beta_1 + 1500) q^{28} + (77 \beta_{2} + 107 \beta_1 - 2615) q^{29} + ( - 225 \beta_1 - 450) q^{30} + (38 \beta_{2} + 130 \beta_1 + 146) q^{31} + (189 \beta_{2} + 212 \beta_1 + 263) q^{32} - 1089 q^{33} + ( - 154 \beta_{2} + 64 \beta_1 + 814) q^{34} + (25 \beta_{2} + 275 \beta_1 - 1525) q^{35} + (81 \beta_{2} + 324 \beta_1 + 567) q^{36} + (272 \beta_{2} - 528 \beta_1 - 2866) q^{37} + (154 \beta_{2} + 1562 \beta_1 + 3838) q^{38} + ( - 252 \beta_{2} - 216 \beta_1 - 1890) q^{39} + ( - 175 \beta_{2} - 1925) q^{40} + ( - 473 \beta_{2} + 977 \beta_1 - 3245) q^{41} + ( - 126 \beta_{2} + 198 \beta_1 - 2250) q^{42} + (341 \beta_{2} - 809 \beta_1 - 4621) q^{43} + ( - 121 \beta_{2} - 484 \beta_1 - 847) q^{44} + 2025 q^{45} + ( - 160 \beta_{2} - 784 \beta_1 - 9328) q^{46} + ( - 422 \beta_{2} + 222 \beta_1 - 6538) q^{47} + ( - 99 \beta_{2} + 612 \beta_1 - 1449) q^{48} + (4 \beta_{2} - 964 \beta_1 - 8555) q^{49} + ( - 625 \beta_1 - 1250) q^{50} + (351 \beta_{2} + 333 \beta_1 - 7209) q^{51} + ( - 162 \beta_{2} - 3432 \beta_1 - 19358) q^{52} + (586 \beta_{2} - 962 \beta_1 - 1212) q^{53} + ( - 729 \beta_1 - 1458) q^{54} - 3025 q^{55} + (392 \beta_{2} - 2184 \beta_1 + 1624) q^{56} + ( - 261 \beta_{2} - 603 \beta_1 - 8415) q^{57} + ( - 338 \beta_{2} + 1092 \beta_1 + 2486) q^{58} + (356 \beta_{2} - 2748 \beta_1 - 2200) q^{59} + (225 \beta_{2} + 900 \beta_1 + 1575) q^{60} + (364 \beta_{2} - 3300 \beta_1 - 18926) q^{61} + ( - 244 \beta_{2} - 1052 \beta_1 - 4348) q^{62} + (81 \beta_{2} + 891 \beta_1 - 4941) q^{63} + ( - 427 \beta_{2} - 6076 \beta_1 - 337) q^{64} + ( - 700 \beta_{2} - 600 \beta_1 - 5250) q^{65} + (1089 \beta_1 + 2178) q^{66} + (680 \beta_{2} - 2144 \beta_1 - 12108) q^{67} + ( - 850 \beta_{2} + 492 \beta_1 + 19762) q^{68} + ( - 180 \beta_{2} + 1980 \beta_1 + 6156) q^{69} + ( - 350 \beta_{2} + 550 \beta_1 - 6250) q^{70} + (980 \beta_{2} + 2612 \beta_1 - 25548) q^{71} + ( - 567 \beta_{2} - 6237) q^{72} + (428 \beta_{2} + 2808 \beta_1 - 15750) q^{73} + ( - 288 \beta_{2} - 702 \beta_1 + 27748) q^{74} + 5625 q^{75} + ( - 1096 \beta_{2} - 7436 \beta_1 - 30424) q^{76} + ( - 121 \beta_{2} - 1331 \beta_1 + 7381) q^{77} + (972 \beta_{2} + 6606 \beta_1 + 8064) q^{78} + ( - 775 \beta_{2} - 5417 \beta_1 - 34233) q^{79} + ( - 275 \beta_{2} + 1700 \beta_1 - 4025) q^{80} + 6561 q^{81} + (442 \beta_{2} + 9332 \beta_1 - 33854) q^{82} + (1474 \beta_{2} + 14234 \beta_1 - 32042) q^{83} + ( - 108 \beta_{2} + 828 \beta_1 + 13500) q^{84} + (975 \beta_{2} + 925 \beta_1 - 20025) q^{85} + ( - 214 \beta_{2} + 442 \beta_1 + 41990) q^{86} + (693 \beta_{2} + 963 \beta_1 - 23535) q^{87} + (847 \beta_{2} + 9317) q^{88} + ( - 132 \beta_{2} - 2140 \beta_1 + 56494) q^{89} + ( - 2025 \beta_1 - 4050) q^{90} + (1378 \beta_{2} - 6602 \beta_1 - 8410) q^{91} + (1904 \beta_{2} + 6576 \beta_1 + 22128) q^{92} + (342 \beta_{2} + 1170 \beta_1 + 1314) q^{93} + (1044 \beta_{2} + 13268 \beta_1 - 180) q^{94} + ( - 725 \beta_{2} - 1675 \beta_1 - 23375) q^{95} + (1701 \beta_{2} + 1908 \beta_1 + 2367) q^{96} + ( - 1664 \beta_{2} - 8760 \beta_1 + 56154) q^{97} + (952 \beta_{2} + 10415 \beta_1 + 50902) q^{98} - 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 7 q^{2} + 27 q^{3} + 25 q^{4} + 75 q^{5} - 63 q^{6} - 172 q^{7} - 231 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 7 q^{2} + 27 q^{3} + 25 q^{4} + 75 q^{5} - 63 q^{6} - 172 q^{7} - 231 q^{8} + 243 q^{9} - 175 q^{10} - 363 q^{11} + 225 q^{12} - 654 q^{13} - 728 q^{14} + 675 q^{15} - 415 q^{16} - 2366 q^{17} - 567 q^{18} - 2872 q^{19} + 625 q^{20} - 1548 q^{21} + 847 q^{22} + 2272 q^{23} - 2079 q^{24} + 1875 q^{25} + 3422 q^{26} + 2187 q^{27} + 4592 q^{28} - 7738 q^{29} - 1575 q^{30} + 568 q^{31} + 1001 q^{32} - 3267 q^{33} + 2506 q^{34} - 4300 q^{35} + 2025 q^{36} - 9126 q^{37} + 13076 q^{38} - 5886 q^{39} - 5775 q^{40} - 8758 q^{41} - 6552 q^{42} - 14672 q^{43} - 3025 q^{44} + 6075 q^{45} - 28768 q^{46} - 19392 q^{47} - 3735 q^{48} - 26629 q^{49} - 4375 q^{50} - 21294 q^{51} - 61506 q^{52} - 4598 q^{53} - 5103 q^{54} - 9075 q^{55} + 2688 q^{56} - 25848 q^{57} + 8550 q^{58} - 9348 q^{59} + 5625 q^{60} - 60078 q^{61} - 14096 q^{62} - 13932 q^{63} - 7087 q^{64} - 16350 q^{65} + 7623 q^{66} - 38468 q^{67} + 59778 q^{68} + 20448 q^{69} - 18200 q^{70} - 74032 q^{71} - 18711 q^{72} - 44442 q^{73} + 82542 q^{74} + 16875 q^{75} - 98708 q^{76} + 20812 q^{77} + 30798 q^{78} - 108116 q^{79} - 10375 q^{80} + 19683 q^{81} - 92230 q^{82} - 81892 q^{83} + 41328 q^{84} - 59150 q^{85} + 126412 q^{86} - 69642 q^{87} + 27951 q^{88} + 167342 q^{89} - 14175 q^{90} - 31832 q^{91} + 72960 q^{92} + 5112 q^{93} + 12728 q^{94} - 71800 q^{95} + 9009 q^{96} + 159702 q^{97} + 163121 q^{98} - 29403 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 52x + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 35 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 35 \) 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
7.25531
0.921799
−7.17710
−9.25531 9.00000 53.6607 25.0000 −83.2977 36.4478 −200.476 81.0000 −231.383
1.2 −2.92180 9.00000 −23.4631 25.0000 −26.2962 −85.0105 162.052 81.0000 −73.0450
1.3 5.17710 9.00000 −5.19759 25.0000 46.5939 −123.437 −192.576 81.0000 129.428
\(n\): 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.a 3
3.b odd 2 1 495.6.a.e 3
5.b even 2 1 825.6.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.a 3 1.a even 1 1 trivial
495.6.a.e 3 3.b odd 2 1
825.6.a.j 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 7T_{2}^{2} - 36T_{2} - 140 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(165))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 7 T^{2} - 36 T - 140 \) Copy content Toggle raw display
$3$ \( (T - 9)^{3} \) Copy content Toggle raw display
$5$ \( (T - 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 172 T^{2} + 2896 T - 382464 \) Copy content Toggle raw display
$11$ \( (T + 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 654 T^{2} + \cdots - 317918392 \) Copy content Toggle raw display
$17$ \( T^{3} + 2366 T^{2} + \cdots - 137826264 \) Copy content Toggle raw display
$19$ \( T^{3} + 2872 T^{2} + \cdots + 11543616 \) Copy content Toggle raw display
$23$ \( T^{3} - 2272 T^{2} + \cdots + 3706904576 \) Copy content Toggle raw display
$29$ \( T^{3} + 7738 T^{2} + \cdots + 5220625848 \) Copy content Toggle raw display
$31$ \( T^{3} - 568 T^{2} + \cdots - 289787904 \) Copy content Toggle raw display
$37$ \( T^{3} + 9126 T^{2} + \cdots - 129972509048 \) Copy content Toggle raw display
$41$ \( T^{3} + 8758 T^{2} + \cdots - 1122652557432 \) Copy content Toggle raw display
$43$ \( T^{3} + 14672 T^{2} + \cdots - 518908872384 \) Copy content Toggle raw display
$47$ \( T^{3} + 19392 T^{2} + \cdots - 1508908531200 \) Copy content Toggle raw display
$53$ \( T^{3} + 4598 T^{2} + \cdots + 728896505288 \) Copy content Toggle raw display
$59$ \( T^{3} + 9348 T^{2} + \cdots - 6267836310080 \) Copy content Toggle raw display
$61$ \( T^{3} + 60078 T^{2} + \cdots - 13500896397400 \) Copy content Toggle raw display
$67$ \( T^{3} + 38468 T^{2} + \cdots - 8479952260160 \) Copy content Toggle raw display
$71$ \( T^{3} + 74032 T^{2} + \cdots - 17011990639616 \) Copy content Toggle raw display
$73$ \( T^{3} + 44442 T^{2} + \cdots - 9750515676328 \) Copy content Toggle raw display
$79$ \( T^{3} + 108116 T^{2} + \cdots + 9069370346752 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 739830059345664 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 158914472576552 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 352998320493112 \) Copy content Toggle raw display
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