Properties

Label 165.6.a.b
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: 3.3.3368.1
Defining polynomial: \( x^{3} - x^{2} - 15x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + 9 q^{3} + (4 \beta_{2} + 8) q^{4} - 25 q^{5} + ( - 9 \beta_1 - 9) q^{6} + ( - 11 \beta_{2} + 14 \beta_1 - 69) q^{7} + ( - 8 \beta_{2} + 4 \beta_1 + 12) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + 9 q^{3} + (4 \beta_{2} + 8) q^{4} - 25 q^{5} + ( - 9 \beta_1 - 9) q^{6} + ( - 11 \beta_{2} + 14 \beta_1 - 69) q^{7} + ( - 8 \beta_{2} + 4 \beta_1 + 12) q^{8} + 81 q^{9} + (25 \beta_1 + 25) q^{10} + 121 q^{11} + (36 \beta_{2} + 72) q^{12} + ( - 31 \beta_{2} - 25 \beta_1 + 152) q^{13} + ( - 34 \beta_{2} + 138 \beta_1 - 444) q^{14} - 225 q^{15} + ( - 128 \beta_{2} + 32 \beta_1 - 400) q^{16} + ( - 57 \beta_{2} + 34 \beta_1 - 81) q^{17} + ( - 81 \beta_1 - 81) q^{18} + (136 \beta_{2} + 119 \beta_1 - 1351) q^{19} + ( - 100 \beta_{2} - 200) q^{20} + ( - 99 \beta_{2} + 126 \beta_1 - 621) q^{21} + ( - 121 \beta_1 - 121) q^{22} + (164 \beta_{2} + 372 \beta_1 - 2284) q^{23} + ( - 72 \beta_{2} + 36 \beta_1 + 108) q^{24} + 625 q^{25} + (162 \beta_{2} - 22 \beta_1 + 916) q^{26} + 729 q^{27} + ( - 132 \beta_{2} + 304 \beta_1 - 2628) q^{28} + (580 \beta_{2} + 93 \beta_1 + 1185) q^{29} + (225 \beta_1 + 225) q^{30} + (764 \beta_{2} - 920 \beta_1 - 764) q^{31} + (384 \beta_{2} + 944 \beta_1 - 848) q^{32} + 1089 q^{33} + ( - 22 \beta_{2} + 400 \beta_1 - 1074) q^{34} + (275 \beta_{2} - 350 \beta_1 + 1725) q^{35} + (324 \beta_{2} + 648) q^{36} + ( - 1312 \beta_{2} + 410 \beta_1 + 1324) q^{37} + ( - 748 \beta_{2} + 790 \beta_1 - 3698) q^{38} + ( - 279 \beta_{2} - 225 \beta_1 + 1368) q^{39} + (200 \beta_{2} - 100 \beta_1 - 300) q^{40} + (140 \beta_{2} - 521 \beta_1 + 3331) q^{41} + ( - 306 \beta_{2} + 1242 \beta_1 - 3996) q^{42} + ( - 37 \beta_{2} - 2 \beta_1 - 11831) q^{43} + (484 \beta_{2} + 968) q^{44} - 2025 q^{45} + ( - 1816 \beta_{2} + 1836 \beta_1 - 12716) q^{46} + (1004 \beta_{2} - 640 \beta_1 - 1248) q^{47} + ( - 1152 \beta_{2} + 288 \beta_1 - 3600) q^{48} + (1510 \beta_{2} - 3622 \beta_1 + 845) q^{49} + ( - 625 \beta_1 - 625) q^{50} + ( - 513 \beta_{2} + 306 \beta_1 - 729) q^{51} + (756 \beta_{2} - 948 \beta_1 - 5408) q^{52} + ( - 2746 \beta_{2} - 2226 \beta_1 - 4102) q^{53} + ( - 729 \beta_1 - 729) q^{54} - 3025 q^{55} + (136 \beta_{2} - 824 \beta_1 + 5376) q^{56} + (1224 \beta_{2} + 1071 \beta_1 - 12159) q^{57} + ( - 1532 \beta_{2} - 3992 \beta_1 - 6552) q^{58} + (844 \beta_{2} + 3292 \beta_1 - 26476) q^{59} + ( - 900 \beta_{2} - 1800) q^{60} + (916 \beta_{2} - 3326 \beta_1 - 22180) q^{61} + (2152 \beta_{2} - 3976 \beta_1 + 34352) q^{62} + ( - 891 \beta_{2} + 1134 \beta_1 - 5589) q^{63} + ( - 448 \beta_{2} - 1152 \beta_1 - 24320) q^{64} + (775 \beta_{2} + 625 \beta_1 - 3800) q^{65} + ( - 1089 \beta_1 - 1089) q^{66} + ( - 4474 \beta_{2} + 496 \beta_1 - 13726) q^{67} + (268 \beta_{2} + 496 \beta_1 - 11868) q^{68} + (1476 \beta_{2} + 3348 \beta_1 - 20556) q^{69} + (850 \beta_{2} - 3450 \beta_1 + 11100) q^{70} + ( - 26 \beta_{2} + 1080 \beta_1 - 21206) q^{71} + ( - 648 \beta_{2} + 324 \beta_1 + 972) q^{72} + (3773 \beta_{2} - 8443 \beta_1 - 3802) q^{73} + (984 \beta_{2} + 5646 \beta_1 - 13378) q^{74} + 5625 q^{75} + ( - 6016 \beta_{2} + 4420 \beta_1 + 18364) q^{76} + ( - 1331 \beta_{2} + 1694 \beta_1 - 8349) q^{77} + (1458 \beta_{2} - 198 \beta_1 + 8244) q^{78} + (10742 \beta_{2} - 5831 \beta_1 + 9101) q^{79} + (3200 \beta_{2} - 800 \beta_1 + 10000) q^{80} + 6561 q^{81} + (1804 \beta_{2} - 4552 \beta_1 + 16568) q^{82} + ( - 2095 \beta_{2} - 4559 \beta_1 - 34496) q^{83} + ( - 1188 \beta_{2} + 2736 \beta_1 - 23652) q^{84} + (1425 \beta_{2} - 850 \beta_1 + 2025) q^{85} + (82 \beta_{2} + 12014 \beta_1 + 12020) q^{86} + (5220 \beta_{2} + 837 \beta_1 + 10665) q^{87} + ( - 968 \beta_{2} + 484 \beta_1 + 1452) q^{88} + ( - 12390 \beta_{2} + 2204 \beta_1 + 24872) q^{89} + (2025 \beta_1 + 2025) q^{90} + ( - 2456 \beta_{2} + 4440 \beta_1 - 7224) q^{91} + ( - 8960 \beta_{2} + 11728 \beta_1 + 19648) q^{92} + (6876 \beta_{2} - 8280 \beta_1 - 6876) q^{93} + (552 \beta_{2} - 4412 \beta_1 + 23196) q^{94} + ( - 3400 \beta_{2} - 2975 \beta_1 + 33775) q^{95} + (3456 \beta_{2} + 8496 \beta_1 - 7632) q^{96} + (3142 \beta_{2} + 10816 \beta_1 - 104024) q^{97} + (11468 \beta_{2} - 12017 \beta_1 + 135883) q^{98} + 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 27 q^{3} + 28 q^{4} - 75 q^{5} - 18 q^{6} - 232 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 27 q^{3} + 28 q^{4} - 75 q^{5} - 18 q^{6} - 232 q^{7} + 24 q^{8} + 243 q^{9} + 50 q^{10} + 363 q^{11} + 252 q^{12} + 450 q^{13} - 1504 q^{14} - 675 q^{15} - 1360 q^{16} - 334 q^{17} - 162 q^{18} - 4036 q^{19} - 700 q^{20} - 2088 q^{21} - 242 q^{22} - 7060 q^{23} + 216 q^{24} + 1875 q^{25} + 2932 q^{26} + 2187 q^{27} - 8320 q^{28} + 4042 q^{29} + 450 q^{30} - 608 q^{31} - 3104 q^{32} + 3267 q^{33} - 3644 q^{34} + 5800 q^{35} + 2268 q^{36} + 2250 q^{37} - 12632 q^{38} + 4050 q^{39} - 600 q^{40} + 10654 q^{41} - 13536 q^{42} - 35528 q^{43} + 3388 q^{44} - 6075 q^{45} - 41800 q^{46} - 2100 q^{47} - 12240 q^{48} + 7667 q^{49} - 1250 q^{50} - 3006 q^{51} - 14520 q^{52} - 12826 q^{53} - 1458 q^{54} - 9075 q^{55} + 17088 q^{56} - 36324 q^{57} - 17196 q^{58} - 81876 q^{59} - 6300 q^{60} - 62298 q^{61} + 109184 q^{62} - 18792 q^{63} - 72256 q^{64} - 11250 q^{65} - 2178 q^{66} - 46148 q^{67} - 35832 q^{68} - 63540 q^{69} + 37600 q^{70} - 64724 q^{71} + 1944 q^{72} + 810 q^{73} - 44796 q^{74} + 16875 q^{75} + 44656 q^{76} - 28072 q^{77} + 26388 q^{78} + 43876 q^{79} + 34000 q^{80} + 19683 q^{81} + 56060 q^{82} - 101024 q^{83} - 74880 q^{84} + 8350 q^{85} + 24128 q^{86} + 36378 q^{87} + 2904 q^{88} + 60022 q^{89} + 4050 q^{90} - 28568 q^{91} + 38256 q^{92} - 5472 q^{93} + 74552 q^{94} + 100900 q^{95} - 27936 q^{96} - 319746 q^{97} + 431134 q^{98} + 29403 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 15x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 10 \) 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
4.03932
0.723686
−3.76300
−8.07863 9.00000 33.2643 −25.0000 −72.7077 −39.3760 −10.2141 81.0000 201.966
1.2 −1.44737 9.00000 −29.9051 −25.0000 −13.0263 41.5023 89.5997 81.0000 36.1843
1.3 7.52601 9.00000 24.6408 −25.0000 67.7341 −234.126 −55.3856 81.0000 −188.150
\(n\): 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.b 3
3.b odd 2 1 495.6.a.d 3
5.b even 2 1 825.6.a.i 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.b 3 1.a even 1 1 trivial
495.6.a.d 3 3.b odd 2 1
825.6.a.i 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 2T_{2}^{2} - 60T_{2} - 88 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(165))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2 T^{2} - 60 T - 88 \) Copy content Toggle raw display
$3$ \( (T - 9)^{3} \) Copy content Toggle raw display
$5$ \( (T + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 232 T^{2} - 2132 T - 382608 \) Copy content Toggle raw display
$11$ \( (T - 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 450 T^{2} + \cdots + 22659488 \) Copy content Toggle raw display
$17$ \( T^{3} + 334 T^{2} + \cdots - 57782448 \) Copy content Toggle raw display
$19$ \( T^{3} + 4036 T^{2} + \cdots - 1630951200 \) Copy content Toggle raw display
$23$ \( T^{3} + 7060 T^{2} + \cdots - 24275701568 \) Copy content Toggle raw display
$29$ \( T^{3} - 4042 T^{2} + \cdots + 65949214584 \) Copy content Toggle raw display
$31$ \( T^{3} + 608 T^{2} + \cdots - 211578448896 \) Copy content Toggle raw display
$37$ \( T^{3} - 2250 T^{2} + \cdots - 431879868536 \) Copy content Toggle raw display
$41$ \( T^{3} - 10654 T^{2} + \cdots - 7803557208 \) Copy content Toggle raw display
$43$ \( T^{3} + 35528 T^{2} + \cdots + 1659712050000 \) Copy content Toggle raw display
$47$ \( T^{3} + 2100 T^{2} + \cdots + 52162385088 \) Copy content Toggle raw display
$53$ \( T^{3} + 12826 T^{2} + \cdots + 2687939232856 \) Copy content Toggle raw display
$59$ \( T^{3} + 81876 T^{2} + \cdots - 3633753791296 \) Copy content Toggle raw display
$61$ \( T^{3} + 62298 T^{2} + \cdots - 12904038746056 \) Copy content Toggle raw display
$67$ \( T^{3} + 46148 T^{2} + \cdots - 40648408406912 \) Copy content Toggle raw display
$71$ \( T^{3} + 64724 T^{2} + \cdots + 8578136735360 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 144432126809632 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 351884592248992 \) Copy content Toggle raw display
$83$ \( T^{3} + 101024 T^{2} + \cdots + 5794291383408 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 246103360939432 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 179909862970168 \) Copy content Toggle raw display
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