Properties

Label 165.4.a.c
Level $165$
Weight $4$
Character orbit 165.a
Self dual yes
Analytic conductor $9.735$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + 3 q^{3} + (\beta - 4) q^{4} - 5 q^{5} - 3 \beta q^{6} + (4 \beta - 4) q^{7} + (11 \beta - 4) q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + 3 q^{3} + (\beta - 4) q^{4} - 5 q^{5} - 3 \beta q^{6} + (4 \beta - 4) q^{7} + (11 \beta - 4) q^{8} + 9 q^{9} + 5 \beta q^{10} - 11 q^{11} + (3 \beta - 12) q^{12} + ( - 2 \beta - 44) q^{13} - 16 q^{14} - 15 q^{15} + ( - 15 \beta - 12) q^{16} + (44 \beta - 30) q^{17} - 9 \beta q^{18} + ( - 22 \beta - 74) q^{19} + ( - 5 \beta + 20) q^{20} + (12 \beta - 12) q^{21} + 11 \beta q^{22} + ( - 60 \beta - 32) q^{23} + (33 \beta - 12) q^{24} + 25 q^{25} + (46 \beta + 8) q^{26} + 27 q^{27} + ( - 16 \beta + 32) q^{28} + (34 \beta - 96) q^{29} + 15 \beta q^{30} + ( - 12 \beta + 36) q^{31} + ( - 61 \beta + 92) q^{32} - 33 q^{33} + ( - 14 \beta - 176) q^{34} + ( - 20 \beta + 20) q^{35} + (9 \beta - 36) q^{36} + ( - 112 \beta - 130) q^{37} + (96 \beta + 88) q^{38} + ( - 6 \beta - 132) q^{39} + ( - 55 \beta + 20) q^{40} + ( - 154 \beta + 96) q^{41} - 48 q^{42} + ( - 124 \beta - 196) q^{43} + ( - 11 \beta + 44) q^{44} - 45 q^{45} + (92 \beta + 240) q^{46} + (216 \beta + 4) q^{47} + ( - 45 \beta - 36) q^{48} + ( - 16 \beta - 263) q^{49} - 25 \beta q^{50} + (132 \beta - 90) q^{51} + ( - 38 \beta + 168) q^{52} + ( - 196 \beta + 334) q^{53} - 27 \beta q^{54} + 55 q^{55} + ( - 16 \beta + 192) q^{56} + ( - 66 \beta - 222) q^{57} + (62 \beta - 136) q^{58} + (240 \beta + 4) q^{59} + ( - 15 \beta + 60) q^{60} + (364 \beta - 146) q^{61} + ( - 24 \beta + 48) q^{62} + (36 \beta - 36) q^{63} + (89 \beta + 340) q^{64} + (10 \beta + 220) q^{65} + 33 \beta q^{66} + (16 \beta - 380) q^{67} + ( - 162 \beta + 296) q^{68} + ( - 180 \beta - 96) q^{69} + 80 q^{70} + (44 \beta + 1008) q^{71} + (99 \beta - 36) q^{72} + (58 \beta - 272) q^{73} + (242 \beta + 448) q^{74} + 75 q^{75} + ( - 8 \beta + 208) q^{76} + ( - 44 \beta + 44) q^{77} + (138 \beta + 24) q^{78} + ( - 306 \beta + 474) q^{79} + (75 \beta + 60) q^{80} + 81 q^{81} + (58 \beta + 616) q^{82} + ( - 426 \beta + 70) q^{83} + ( - 48 \beta + 96) q^{84} + ( - 220 \beta + 150) q^{85} + (320 \beta + 496) q^{86} + (102 \beta - 288) q^{87} + ( - 121 \beta + 44) q^{88} + ( - 128 \beta + 186) q^{89} + 45 \beta q^{90} + ( - 176 \beta + 144) q^{91} + (148 \beta - 112) q^{92} + ( - 36 \beta + 108) q^{93} + ( - 220 \beta - 864) q^{94} + (110 \beta + 370) q^{95} + ( - 183 \beta + 276) q^{96} + (428 \beta - 298) q^{97} + (279 \beta + 64) q^{98} - 99 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 6 q^{3} - 7 q^{4} - 10 q^{5} - 3 q^{6} - 4 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 6 q^{3} - 7 q^{4} - 10 q^{5} - 3 q^{6} - 4 q^{7} + 3 q^{8} + 18 q^{9} + 5 q^{10} - 22 q^{11} - 21 q^{12} - 90 q^{13} - 32 q^{14} - 30 q^{15} - 39 q^{16} - 16 q^{17} - 9 q^{18} - 170 q^{19} + 35 q^{20} - 12 q^{21} + 11 q^{22} - 124 q^{23} + 9 q^{24} + 50 q^{25} + 62 q^{26} + 54 q^{27} + 48 q^{28} - 158 q^{29} + 15 q^{30} + 60 q^{31} + 123 q^{32} - 66 q^{33} - 366 q^{34} + 20 q^{35} - 63 q^{36} - 372 q^{37} + 272 q^{38} - 270 q^{39} - 15 q^{40} + 38 q^{41} - 96 q^{42} - 516 q^{43} + 77 q^{44} - 90 q^{45} + 572 q^{46} + 224 q^{47} - 117 q^{48} - 542 q^{49} - 25 q^{50} - 48 q^{51} + 298 q^{52} + 472 q^{53} - 27 q^{54} + 110 q^{55} + 368 q^{56} - 510 q^{57} - 210 q^{58} + 248 q^{59} + 105 q^{60} + 72 q^{61} + 72 q^{62} - 36 q^{63} + 769 q^{64} + 450 q^{65} + 33 q^{66} - 744 q^{67} + 430 q^{68} - 372 q^{69} + 160 q^{70} + 2060 q^{71} + 27 q^{72} - 486 q^{73} + 1138 q^{74} + 150 q^{75} + 408 q^{76} + 44 q^{77} + 186 q^{78} + 642 q^{79} + 195 q^{80} + 162 q^{81} + 1290 q^{82} - 286 q^{83} + 144 q^{84} + 80 q^{85} + 1312 q^{86} - 474 q^{87} - 33 q^{88} + 244 q^{89} + 45 q^{90} + 112 q^{91} - 76 q^{92} + 180 q^{93} - 1948 q^{94} + 850 q^{95} + 369 q^{96} - 168 q^{97} + 407 q^{98} - 198 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
−2.56155 3.00000 −1.43845 −5.00000 −7.68466 6.24621 24.1771 9.00000 12.8078
1.2 1.56155 3.00000 −5.56155 −5.00000 4.68466 −10.2462 −21.1771 9.00000 −7.80776
\(n\): 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.c 2
3.b odd 2 1 495.4.a.d 2
5.b even 2 1 825.4.a.m 2
5.c odd 4 2 825.4.c.j 4
11.b odd 2 1 1815.4.a.n 2
15.d odd 2 1 2475.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.c 2 1.a even 1 1 trivial
495.4.a.d 2 3.b odd 2 1
825.4.a.m 2 5.b even 2 1
825.4.c.j 4 5.c odd 4 2
1815.4.a.n 2 11.b odd 2 1
2475.4.a.n 2 15.d odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 90T + 2008 \) Copy content Toggle raw display
$17$ \( T^{2} + 16T - 8164 \) Copy content Toggle raw display
$19$ \( T^{2} + 170T + 5168 \) Copy content Toggle raw display
$23$ \( T^{2} + 124T - 11456 \) Copy content Toggle raw display
$29$ \( T^{2} + 158T + 1328 \) Copy content Toggle raw display
$31$ \( T^{2} - 60T + 288 \) Copy content Toggle raw display
$37$ \( T^{2} + 372T - 18716 \) Copy content Toggle raw display
$41$ \( T^{2} - 38T - 100432 \) Copy content Toggle raw display
$43$ \( T^{2} + 516T + 1216 \) Copy content Toggle raw display
$47$ \( T^{2} - 224T - 185744 \) Copy content Toggle raw display
$53$ \( T^{2} - 472T - 107572 \) Copy content Toggle raw display
$59$ \( T^{2} - 248T - 229424 \) Copy content Toggle raw display
$61$ \( T^{2} - 72T - 561812 \) Copy content Toggle raw display
$67$ \( T^{2} + 744T + 137296 \) Copy content Toggle raw display
$71$ \( T^{2} - 2060 T + 1052672 \) Copy content Toggle raw display
$73$ \( T^{2} + 486T + 44752 \) Copy content Toggle raw display
$79$ \( T^{2} - 642T - 294912 \) Copy content Toggle raw display
$83$ \( T^{2} + 286T - 750824 \) Copy content Toggle raw display
$89$ \( T^{2} - 244T - 54748 \) Copy content Toggle raw display
$97$ \( T^{2} + 168T - 771476 \) Copy content Toggle raw display
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