Properties

Label 165.4.a.d
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: 3.3.23612.1
Defining polynomial: \( x^{3} - x^{2} - 20x + 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} - 3 q^{3} + (\beta_{2} + \beta_1 + 7) q^{4} - 5 q^{5} + (3 \beta_1 + 3) q^{6} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{7} + ( - 4 \beta_{2} - 3 \beta_1 - 15) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} - 3 q^{3} + (\beta_{2} + \beta_1 + 7) q^{4} - 5 q^{5} + (3 \beta_1 + 3) q^{6} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{7} + ( - 4 \beta_{2} - 3 \beta_1 - 15) q^{8} + 9 q^{9} + (5 \beta_1 + 5) q^{10} + 11 q^{11} + ( - 3 \beta_{2} - 3 \beta_1 - 21) q^{12} + (2 \beta_{2} + 12 \beta_1 - 4) q^{13} + (4 \beta_{2} + 10 \beta_1 - 22) q^{14} + 15 q^{15} + (7 \beta_{2} + 23 \beta_1 + 9) q^{16} + (6 \beta_{2} + 10 \beta_1 - 76) q^{17} + ( - 9 \beta_1 - 9) q^{18} + (4 \beta_{2} + 2 \beta_1 + 48) q^{19} + ( - 5 \beta_{2} - 5 \beta_1 - 35) q^{20} + (6 \beta_{2} - 6 \beta_1 + 6) q^{21} + ( - 11 \beta_1 - 11) q^{22} + ( - 8 \beta_{2} - 20 \beta_1 - 60) q^{23} + (12 \beta_{2} + 9 \beta_1 + 45) q^{24} + 25 q^{25} + ( - 18 \beta_{2} - 4 \beta_1 - 168) q^{26} - 27 q^{27} + ( - 6 \beta_{2} - 10 \beta_1 - 110) q^{28} + (20 \beta_{2} - 34 \beta_1 + 34) q^{29} + ( - 15 \beta_1 - 15) q^{30} + ( - 16 \beta_{2} + 4 \beta_1 - 24) q^{31} + ( - 12 \beta_{2} - 13 \beta_1 - 225) q^{32} - 33 q^{33} + ( - 28 \beta_{2} + 52 \beta_1 - 76) q^{34} + (10 \beta_{2} - 10 \beta_1 + 10) q^{35} + (9 \beta_{2} + 9 \beta_1 + 63) q^{36} + ( - 4 \beta_{2} - 24 \beta_1 - 122) q^{37} + ( - 14 \beta_{2} - 64 \beta_1 - 84) q^{38} + ( - 6 \beta_{2} - 36 \beta_1 + 12) q^{39} + (20 \beta_{2} + 15 \beta_1 + 75) q^{40} + ( - 20 \beta_{2} + 2 \beta_1 - 66) q^{41} + ( - 12 \beta_{2} - 30 \beta_1 + 66) q^{42} + (14 \beta_{2} - 74 \beta_1 - 150) q^{43} + (11 \beta_{2} + 11 \beta_1 + 77) q^{44} - 45 q^{45} + (44 \beta_{2} + 92 \beta_1 + 356) q^{46} + (28 \beta_{2} + 48 \beta_1 - 36) q^{47} + ( - 21 \beta_{2} - 69 \beta_1 - 27) q^{48} + ( - 20 \beta_{2} - 4 \beta_1 - 51) q^{49} + ( - 25 \beta_1 - 25) q^{50} + ( - 18 \beta_{2} - 30 \beta_1 + 228) q^{51} + (42 \beta_{2} + 144 \beta_1 + 292) q^{52} + (52 \beta_{2} - 32 \beta_1 - 42) q^{53} + (27 \beta_1 + 27) q^{54} - 55 q^{55} + ( - 4 \beta_{2} + 54 \beta_1 + 438) q^{56} + ( - 12 \beta_{2} - 6 \beta_1 - 144) q^{57} + ( - 26 \beta_{2} - 114 \beta_1 + 402) q^{58} + ( - 4 \beta_{2} - 120 \beta_1 - 308) q^{59} + (15 \beta_{2} + 15 \beta_1 + 105) q^{60} + ( - 44 \beta_{2} - 12 \beta_1 + 218) q^{61} + (44 \beta_{2} + 88 \beta_1) q^{62} + ( - 18 \beta_{2} + 18 \beta_1 - 18) q^{63} + ( - 7 \beta_{2} + 89 \beta_1 + 359) q^{64} + ( - 10 \beta_{2} - 60 \beta_1 + 20) q^{65} + (33 \beta_1 + 33) q^{66} + ( - 28 \beta_{2} + 148 \beta_1 - 128) q^{67} + ( - 16 \beta_{2} + 108 \beta_1 + 12) q^{68} + (24 \beta_{2} + 60 \beta_1 + 180) q^{69} + ( - 20 \beta_{2} - 50 \beta_1 + 110) q^{70} + ( - 16 \beta_{2} + 104 \beta_1 - 216) q^{71} + ( - 36 \beta_{2} - 27 \beta_1 - 135) q^{72} + (14 \beta_{2} - 168 \beta_1 + 356) q^{73} + (36 \beta_{2} + 138 \beta_1 + 466) q^{74} - 75 q^{75} + (74 \beta_{2} + 124 \beta_1 + 624) q^{76} + ( - 22 \beta_{2} + 22 \beta_1 - 22) q^{77} + (54 \beta_{2} + 12 \beta_1 + 504) q^{78} + ( - 52 \beta_{2} - 14 \beta_1 - 524) q^{79} + ( - 35 \beta_{2} - 115 \beta_1 - 45) q^{80} + 81 q^{81} + (58 \beta_{2} + 146 \beta_1 + 78) q^{82} + (70 \beta_{2} + 164 \beta_1 - 582) q^{83} + (18 \beta_{2} + 30 \beta_1 + 330) q^{84} + ( - 30 \beta_{2} - 50 \beta_1 + 380) q^{85} + (32 \beta_{2} + 94 \beta_1 + 1158) q^{86} + ( - 60 \beta_{2} + 102 \beta_1 - 102) q^{87} + ( - 44 \beta_{2} - 33 \beta_1 - 165) q^{88} + (68 \beta_1 - 730) q^{89} + (45 \beta_1 + 45) q^{90} + (4 \beta_{2} - 176 \beta_1 + 56) q^{91} + ( - 160 \beta_{2} - 372 \beta_1 - 1252) q^{92} + (48 \beta_{2} - 12 \beta_1 + 72) q^{93} + ( - 132 \beta_{2} - 76 \beta_1 - 692) q^{94} + ( - 20 \beta_{2} - 10 \beta_1 - 240) q^{95} + (36 \beta_{2} + 39 \beta_1 + 675) q^{96} + (64 \beta_{2} - 240 \beta_1 + 286) q^{97} + (64 \beta_{2} + 131 \beta_1 + 147) q^{98} + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} + 12 q^{6} - 4 q^{7} - 48 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} + 12 q^{6} - 4 q^{7} - 48 q^{8} + 27 q^{9} + 20 q^{10} + 33 q^{11} - 66 q^{12} - 56 q^{14} + 45 q^{15} + 50 q^{16} - 218 q^{17} - 36 q^{18} + 146 q^{19} - 110 q^{20} + 12 q^{21} - 44 q^{22} - 200 q^{23} + 144 q^{24} + 75 q^{25} - 508 q^{26} - 81 q^{27} - 340 q^{28} + 68 q^{29} - 60 q^{30} - 68 q^{31} - 688 q^{32} - 99 q^{33} - 176 q^{34} + 20 q^{35} + 198 q^{36} - 390 q^{37} - 316 q^{38} + 240 q^{40} - 196 q^{41} + 168 q^{42} - 524 q^{43} + 242 q^{44} - 135 q^{45} + 1160 q^{46} - 60 q^{47} - 150 q^{48} - 157 q^{49} - 100 q^{50} + 654 q^{51} + 1020 q^{52} - 158 q^{53} + 108 q^{54} - 165 q^{55} + 1368 q^{56} - 438 q^{57} + 1092 q^{58} - 1044 q^{59} + 330 q^{60} + 642 q^{61} + 88 q^{62} - 36 q^{63} + 1166 q^{64} + 132 q^{66} - 236 q^{67} + 144 q^{68} + 600 q^{69} + 280 q^{70} - 544 q^{71} - 432 q^{72} + 900 q^{73} + 1536 q^{74} - 225 q^{75} + 1996 q^{76} - 44 q^{77} + 1524 q^{78} - 1586 q^{79} - 250 q^{80} + 243 q^{81} + 380 q^{82} - 1582 q^{83} + 1020 q^{84} + 1090 q^{85} + 3568 q^{86} - 204 q^{87} - 528 q^{88} - 2122 q^{89} + 180 q^{90} - 8 q^{91} - 4128 q^{92} + 204 q^{93} - 2152 q^{94} - 730 q^{95} + 2064 q^{96} + 618 q^{97} + 572 q^{98} + 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 20x + 26 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 14 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 14 \) 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
4.26150
1.32906
−4.59056
−5.26150 −3.00000 19.6833 −5.00000 15.7845 −10.3207 −61.4719 9.00000 26.3075
1.2 −2.32906 −3.00000 −2.57547 −5.00000 6.98719 22.4672 24.6309 9.00000 11.6453
1.3 3.59056 −3.00000 4.89212 −5.00000 −10.7717 −16.1465 −11.1590 9.00000 −17.9528
\(n\): 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.d 3
3.b odd 2 1 495.4.a.l 3
5.b even 2 1 825.4.a.s 3
5.c odd 4 2 825.4.c.l 6
11.b odd 2 1 1815.4.a.s 3
15.d odd 2 1 2475.4.a.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.d 3 1.a even 1 1 trivial
495.4.a.l 3 3.b odd 2 1
825.4.a.s 3 5.b even 2 1
825.4.c.l 6 5.c odd 4 2
1815.4.a.s 3 11.b odd 2 1
2475.4.a.s 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 4T_{2}^{2} - 15T_{2} - 44 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(165))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 4 T^{2} - 15 T - 44 \) Copy content Toggle raw display
$3$ \( (T + 3)^{3} \) Copy content Toggle raw display
$5$ \( (T + 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 4 T^{2} - 428 T - 3744 \) Copy content Toggle raw display
$11$ \( (T - 11)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 3560T - 34144 \) Copy content Toggle raw display
$17$ \( T^{3} + 218 T^{2} + 9680 T - 235104 \) Copy content Toggle raw display
$19$ \( T^{3} - 146 T^{2} + 5376 T - 30960 \) Copy content Toggle raw display
$23$ \( T^{3} + 200 T^{2} - 2672 T + 1664 \) Copy content Toggle raw display
$29$ \( T^{3} - 68 T^{2} - 54364 T + 3163056 \) Copy content Toggle raw display
$31$ \( T^{3} + 68 T^{2} - 23232 T - 1812096 \) Copy content Toggle raw display
$37$ \( T^{3} + 390 T^{2} + 36460 T + 618952 \) Copy content Toggle raw display
$41$ \( T^{3} + 196 T^{2} - 26076 T - 4364208 \) Copy content Toggle raw display
$43$ \( T^{3} + 524 T^{2} + \cdots - 31273920 \) Copy content Toggle raw display
$47$ \( T^{3} + 60 T^{2} - 135920 T - 20966976 \) Copy content Toggle raw display
$53$ \( T^{3} + 158 T^{2} + \cdots + 39574952 \) Copy content Toggle raw display
$59$ \( T^{3} + 1044 T^{2} + \cdots - 84227264 \) Copy content Toggle raw display
$61$ \( T^{3} - 642 T^{2} + \cdots + 22757384 \) Copy content Toggle raw display
$67$ \( T^{3} + 236 T^{2} + \cdots + 87537664 \) Copy content Toggle raw display
$71$ \( T^{3} + 544 T^{2} - 129728 T + 6553600 \) Copy content Toggle raw display
$73$ \( T^{3} - 900 T^{2} - 299576 T - 5609344 \) Copy content Toggle raw display
$79$ \( T^{3} + 1586 T^{2} + \cdots - 14694992 \) Copy content Toggle raw display
$83$ \( T^{3} + 1582 T^{2} + \cdots - 924645384 \) Copy content Toggle raw display
$89$ \( T^{3} + 2122 T^{2} + \cdots + 293444632 \) Copy content Toggle raw display
$97$ \( T^{3} - 618 T^{2} + \cdots - 223543736 \) Copy content Toggle raw display
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