Properties

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

Newform invariants

Self dual: yes
Analytic conductor: \(26.4633302691\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.788.1
Defining polynomial: \( x^{3} - x^{2} - 7x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} - 9 q^{3} + ( - 4 \beta_1 - 8) q^{4} + 25 q^{5} + ( - 9 \beta_{2} - 9) q^{6} + ( - 7 \beta_{2} - 31 \beta_1 + 38) q^{7} + ( - 28 \beta_{2} - 8 \beta_1 - 20) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} - 9 q^{3} + ( - 4 \beta_1 - 8) q^{4} + 25 q^{5} + ( - 9 \beta_{2} - 9) q^{6} + ( - 7 \beta_{2} - 31 \beta_1 + 38) q^{7} + ( - 28 \beta_{2} - 8 \beta_1 - 20) q^{8} + 81 q^{9} + (25 \beta_{2} + 25) q^{10} - 121 q^{11} + (36 \beta_1 + 72) q^{12} + (2 \beta_{2} - 77 \beta_1 - 207) q^{13} + (138 \beta_{2} - 34 \beta_1 + 32) q^{14} - 225 q^{15} + (32 \beta_{2} + 224 \beta_1 - 368) q^{16} + (119 \beta_{2} - 219 \beta_1 - 138) q^{17} + (81 \beta_{2} + 81) q^{18} + (149 \beta_{2} + 206 \beta_1 + 721) q^{19} + ( - 100 \beta_1 - 200) q^{20} + (63 \beta_{2} + 279 \beta_1 - 342) q^{21} + ( - 121 \beta_{2} - 121) q^{22} + (150 \beta_{2} - 190 \beta_1 + 1416) q^{23} + (252 \beta_{2} + 72 \beta_1 + 180) q^{24} + 625 q^{25} + (22 \beta_{2} - 162 \beta_1 + 224) q^{26} - 729 q^{27} + (220 \beta_{2} + 372 \beta_1 + 2160) q^{28} + (129 \beta_{2} - 308 \beta_1 + 1801) q^{29} + ( - 225 \beta_{2} - 225) q^{30} + ( - 32 \beta_{2} - 242 \beta_1 + 2018) q^{31} + ( - 176 \beta_{2} + 576 \beta_1 - 112) q^{32} + 1089 q^{33} + (400 \beta_{2} - 914 \beta_1 + 3694) q^{34} + ( - 175 \beta_{2} - 775 \beta_1 + 950) q^{35} + ( - 324 \beta_1 - 648) q^{36} + (1076 \beta_{2} + 928 \beta_1 + 6290) q^{37} + ( - 46 \beta_{2} - 184 \beta_1 + 3118) q^{38} + ( - 18 \beta_{2} + 693 \beta_1 + 1863) q^{39} + ( - 700 \beta_{2} - 200 \beta_1 - 500) q^{40} + ( - 781 \beta_{2} + 1892 \beta_1 + 8131) q^{41} + ( - 1242 \beta_{2} + 306 \beta_1 - 288) q^{42} + (1903 \beta_{2} + 949 \beta_1 + 7808) q^{43} + (484 \beta_1 + 968) q^{44} + 2025 q^{45} + (1836 \beta_{2} - 980 \beta_1 + 5816) q^{46} + (520 \beta_{2} - 598 \beta_1 + 1118) q^{47} + ( - 288 \beta_{2} - 2016 \beta_1 + 3312) q^{48} + ( - 10 \beta_{2} - 196 \beta_1 + 3775) q^{49} + (625 \beta_{2} + 625) q^{50} + ( - 1071 \beta_{2} + 1971 \beta_1 + 1242) q^{51} + (624 \beta_{2} + 2052 \beta_1 + 8164) q^{52} + ( - 3174 \beta_{2} - 2146 \beta_1 + 3606) q^{53} + ( - 729 \beta_{2} - 729) q^{54} - 3025 q^{55} + ( - 3592 \beta_{2} + 952 \beta_1 + 4336) q^{56} + ( - 1341 \beta_{2} - 1854 \beta_1 - 6489) q^{57} + (2596 \beta_{2} - 1132 \beta_1 + 6308) q^{58} + ( - 1822 \beta_{2} - 2954 \beta_1 + 5732) q^{59} + (900 \beta_1 + 1800) q^{60} + ( - 2786 \beta_{2} + 1694 \beta_1 + 2410) q^{61} + (2776 \beta_{2} - 356 \beta_1 + 2492) q^{62} + ( - 567 \beta_{2} - 2511 \beta_1 + 3078) q^{63} + ( - 2688 \beta_{2} - 5312 \beta_1 + 4736) q^{64} + (50 \beta_{2} - 1925 \beta_1 - 5175) q^{65} + (1089 \beta_{2} + 1089) q^{66} + ( - 5936 \beta_{2} - 4430 \beta_1 - 31566) q^{67} + (2228 \beta_{2} + 3580 \beta_1 + 21880) q^{68} + ( - 1350 \beta_{2} + 1710 \beta_1 - 12744) q^{69} + (3450 \beta_{2} - 850 \beta_1 + 800) q^{70} + ( - 9528 \beta_{2} + 3010 \beta_1 + 14670) q^{71} + ( - 2268 \beta_{2} - 648 \beta_1 - 1620) q^{72} + (1244 \beta_{2} + 12157 \beta_1 - 13479) q^{73} + (2430 \beta_{2} - 2448 \beta_1 + 26398) q^{74} - 5625 q^{75} + ( - 1052 \beta_{2} - 6776 \beta_1 - 20092) q^{76} + (847 \beta_{2} + 3751 \beta_1 - 4598) q^{77} + ( - 198 \beta_{2} + 1458 \beta_1 - 2016) q^{78} + (3973 \beta_{2} - 770 \beta_1 + 3189) q^{79} + (800 \beta_{2} + 5600 \beta_1 - 9200) q^{80} + 6561 q^{81} + (3236 \beta_{2} + 6908 \beta_1 - 19292) q^{82} + (1382 \beta_{2} + 10151 \beta_1 + 35935) q^{83} + ( - 1980 \beta_{2} - 3348 \beta_1 - 19440) q^{84} + (2975 \beta_{2} - 5475 \beta_1 - 3450) q^{85} + (3058 \beta_{2} - 5714 \beta_1 + 46832) q^{86} + ( - 1161 \beta_{2} + 2772 \beta_1 - 16209) q^{87} + (3388 \beta_{2} + 968 \beta_1 + 2420) q^{88} + ( - 10502 \beta_{2} + 6468 \beta_1 - 14324) q^{89} + (2025 \beta_{2} + 2025) q^{90} + (4896 \beta_{2} + 8798 \beta_1 + 39554) q^{91} + (2120 \beta_{2} - 3224 \beta_1 + 7632) q^{92} + (288 \beta_{2} + 2178 \beta_1 - 18162) q^{93} + (2392 \beta_{2} - 3276 \beta_1 + 16068) q^{94} + (3725 \beta_{2} + 5150 \beta_1 + 18025) q^{95} + (1584 \beta_{2} - 5184 \beta_1 + 1008) q^{96} + (3184 \beta_{2} - 20686 \beta_1 - 40248) q^{97} + (4373 \beta_{2} - 352 \beta_1 + 4525) q^{98} - 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 27 q^{3} - 20 q^{4} + 75 q^{5} - 18 q^{6} + 152 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} - 27 q^{3} - 20 q^{4} + 75 q^{5} - 18 q^{6} + 152 q^{7} - 24 q^{8} + 243 q^{9} + 50 q^{10} - 363 q^{11} + 180 q^{12} - 546 q^{13} - 8 q^{14} - 675 q^{15} - 1360 q^{16} - 314 q^{17} + 162 q^{18} + 1808 q^{19} - 500 q^{20} - 1368 q^{21} - 242 q^{22} + 4288 q^{23} + 216 q^{24} + 1875 q^{25} + 812 q^{26} - 2187 q^{27} + 5888 q^{28} + 5582 q^{29} - 450 q^{30} + 6328 q^{31} - 736 q^{32} + 3267 q^{33} + 11596 q^{34} + 3800 q^{35} - 1620 q^{36} + 16866 q^{37} + 9584 q^{38} + 4914 q^{39} - 600 q^{40} + 23282 q^{41} + 72 q^{42} + 20572 q^{43} + 2420 q^{44} + 6075 q^{45} + 16592 q^{46} + 3432 q^{47} + 12240 q^{48} + 11531 q^{49} + 1250 q^{50} + 2826 q^{51} + 21816 q^{52} + 16138 q^{53} - 1458 q^{54} - 9075 q^{55} + 15648 q^{56} - 16272 q^{57} + 17460 q^{58} + 21972 q^{59} + 4500 q^{60} + 8322 q^{61} + 5056 q^{62} + 12312 q^{63} + 22208 q^{64} - 13650 q^{65} + 2178 q^{66} - 84332 q^{67} + 59832 q^{68} - 38592 q^{69} - 200 q^{70} + 50528 q^{71} - 1944 q^{72} - 53838 q^{73} + 79212 q^{74} - 16875 q^{75} - 52448 q^{76} - 18392 q^{77} - 7308 q^{78} + 6364 q^{79} - 34000 q^{80} + 19683 q^{81} - 68020 q^{82} + 96272 q^{83} - 52992 q^{84} - 7850 q^{85} + 143152 q^{86} - 50238 q^{87} + 2904 q^{88} - 38938 q^{89} + 4050 q^{90} + 104968 q^{91} + 24000 q^{92} - 56952 q^{93} + 49088 q^{94} + 45200 q^{95} + 6624 q^{96} - 103242 q^{97} + 9554 q^{98} - 29403 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 4\nu - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 2\beta _1 + 11 ) / 2 \) 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
−5.64018 −9.00000 −0.188384 25.0000 50.7616 145.021 181.548 81.0000 −141.004
1.2 1.08127 −9.00000 −30.8308 25.0000 −9.73147 −139.508 −67.9374 81.0000 27.0319
1.3 6.55890 −9.00000 11.0192 25.0000 −59.0301 146.487 −137.611 81.0000 163.973
\(n\): 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.d 3
3.b odd 2 1 495.6.a.c 3
5.b even 2 1 825.6.a.h 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.d 3 1.a even 1 1 trivial
495.6.a.c 3 3.b odd 2 1
825.6.a.h 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 2T_{2}^{2} - 36T_{2} + 40 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(165))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 2 T^{2} - 36 T + 40 \) Copy content Toggle raw display
$3$ \( (T + 9)^{3} \) Copy content Toggle raw display
$5$ \( (T - 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 152 T^{2} - 19424 T + 2963664 \) Copy content Toggle raw display
$11$ \( (T + 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 546 T^{2} - 76748 T - 7691848 \) Copy content Toggle raw display
$17$ \( T^{3} + 314 T^{2} + \cdots - 1079472216 \) Copy content Toggle raw display
$19$ \( T^{3} - 1808 T^{2} + \cdots + 729480096 \) Copy content Toggle raw display
$23$ \( T^{3} - 4288 T^{2} + \cdots - 857355136 \) Copy content Toggle raw display
$29$ \( T^{3} - 5582 T^{2} + \cdots - 329440872 \) Copy content Toggle raw display
$31$ \( T^{3} - 6328 T^{2} + \cdots - 5126546304 \) Copy content Toggle raw display
$37$ \( T^{3} - 16866 T^{2} + \cdots + 244700027368 \) Copy content Toggle raw display
$41$ \( T^{3} - 23282 T^{2} + \cdots + 945181300968 \) Copy content Toggle raw display
$43$ \( T^{3} - 20572 T^{2} + \cdots + 1240285492944 \) Copy content Toggle raw display
$47$ \( T^{3} - 3432 T^{2} + \cdots - 18016103040 \) Copy content Toggle raw display
$53$ \( T^{3} - 16138 T^{2} + \cdots - 985333601848 \) Copy content Toggle raw display
$59$ \( T^{3} - 21972 T^{2} + \cdots + 2567903224000 \) Copy content Toggle raw display
$61$ \( T^{3} - 8322 T^{2} + \cdots + 4408473611240 \) Copy content Toggle raw display
$67$ \( T^{3} + 84332 T^{2} + \cdots - 41154720036800 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 117803610062464 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 163971863295832 \) Copy content Toggle raw display
$79$ \( T^{3} - 6364 T^{2} + \cdots - 554174036768 \) Copy content Toggle raw display
$83$ \( T^{3} - 96272 T^{2} + \cdots + 3029676562224 \) Copy content Toggle raw display
$89$ \( T^{3} + 38938 T^{2} + \cdots + 96218735089528 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 251540556847112 \) Copy content Toggle raw display
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