Properties

Label 153.4.a.g
Level $153$
Weight $4$
Character orbit 153.a
Self dual yes
Analytic conductor $9.027$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.02729223088\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2636.1
Defining polynomial: \( x^{3} - 14x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{2} + ( - \beta_{2} - 3 \beta_1 + 8) q^{4} + ( - 2 \beta_{2} + 2) q^{5} + ( - 4 \beta_{2} - \beta_1 + 6) q^{7} + (9 \beta_{2} - 5 \beta_1 + 16) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{2} + ( - \beta_{2} - 3 \beta_1 + 8) q^{4} + ( - 2 \beta_{2} + 2) q^{5} + ( - 4 \beta_{2} - \beta_1 + 6) q^{7} + (9 \beta_{2} - 5 \beta_1 + 16) q^{8} + (8 \beta_{2} - 16) q^{10} + (2 \beta_{2} - 11 \beta_1 + 10) q^{11} + (6 \beta_{2} + 8 \beta_1 + 12) q^{13} + (20 \beta_{2} - 4 \beta_1 - 24) q^{14} + (7 \beta_{2} - 11 \beta_1 + 48) q^{16} + 17 q^{17} + ( - 8 \beta_{2} + 22 \beta_1 + 24) q^{19} + ( - 24 \beta_{2} + 8 \beta_1 + 48) q^{20} + (26 \beta_{2} - 34 \beta_1 + 104) q^{22} + (4 \beta_{2} + 39 \beta_1 - 46) q^{23} + ( - 20 \beta_{2} + 4 \beta_1 - 81) q^{25} + ( - 22 \beta_{2} - 2 \beta_1 - 16) q^{26} + ( - 44 \beta_{2} + 4 \beta_1 + 144) q^{28} + ( - 30 \beta_{2} + 16 \beta_1 + 142) q^{29} + (16 \beta_{2} + 39 \beta_1 + 82) q^{31} + ( - 23 \beta_{2} - 37 \beta_1 + 16) q^{32} + (17 \beta_{2} - 17 \beta_1) q^{34} + ( - 44 \beta_{2} + 10 \beta_1 + 96) q^{35} + ( - 50 \beta_{2} - 28 \beta_1 + 102) q^{37} + (4 \beta_{2} + 28 \beta_1 - 240) q^{38} + (40 \beta_{2} - 8 \beta_1 - 128) q^{40} + (60 \beta_{2} + 52 \beta_1 + 118) q^{41} + (56 \beta_{2} + 2 \beta_1 + 204) q^{43} + (78 \beta_{2} - 110 \beta_1 + 400) q^{44} + ( - 136 \beta_{2} + 120 \beta_1 - 280) q^{46} + ( - 44 \beta_{2} + 48 \beta_1 - 228) q^{47} + ( - 94 \beta_{2} + 20 \beta_1 - 121) q^{49} + ( - 29 \beta_{2} + 109 \beta_1 - 192) q^{50} + (6 \beta_{2} - 30 \beta_1 - 256) q^{52} + (8 \beta_{2} - 116 \beta_1 - 98) q^{53} + ( - 4 \beta_{2} + 18 \beta_1 + 24) q^{55} + (108 \beta_{2} - 60 \beta_1 - 192) q^{56} + (200 \beta_{2} - 80 \beta_1 - 368) q^{58} + (130 \beta_1 - 212) q^{59} + (78 \beta_{2} - 64 \beta_1 - 2) q^{61} + ( - 44 \beta_{2} - 20 \beta_1 - 184) q^{62} + (103 \beta_{2} + 21 \beta_1 - 272) q^{64} + (24 \beta_{2} - 28 \beta_1 - 128) q^{65} + ( - 132 \beta_{2} + 24 \beta_1 + 292) q^{67} + ( - 17 \beta_{2} - 51 \beta_1 + 136) q^{68} + (208 \beta_{2} - 32 \beta_1 - 432) q^{70} + ( - 72 \beta_{2} - 185 \beta_1 + 110) q^{71} + ( - 16 \beta_{2} + 16 \beta_1 + 274) q^{73} + (308 \beta_{2} - 108 \beta_1 - 176) q^{74} + ( - 244 \beta_{2} + 116 \beta_1 - 384) q^{76} + (18 \beta_{2} + 174) q^{77} + (180 \beta_{2} - 267 \beta_1 - 138) q^{79} + ( - 40 \beta_{2} + 8 \beta_1) q^{80} + ( - 166 \beta_{2} - 74 \beta_1 + 64) q^{82} + ( - 128 \beta_{2} - 82 \beta_1 + 756) q^{83} + ( - 34 \beta_{2} + 34) q^{85} + (32 \beta_{2} - 256 \beta_1 + 432) q^{86} + (178 \beta_{2} - 426 \beta_1 + 672) q^{88} + ( - 110 \beta_{2} + 276 \beta_1 + 20) q^{89} + (44 \beta_{2} - 64 \beta_1 - 324) q^{91} + ( - 144 \beta_{2} + 344 \beta_1 - 1680) q^{92} + ( - 192 \beta_{2} + 368 \beta_1 - 736) q^{94} + ( - 112 \beta_{2} - 28 \beta_1 + 120) q^{95} + (120 \beta_{2} + 140 \beta_1 - 50) q^{97} + (121 \beta_{2} + 255 \beta_1 - 912) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8} - 56 q^{10} + 28 q^{11} + 30 q^{13} - 92 q^{14} + 137 q^{16} + 51 q^{17} + 80 q^{19} + 168 q^{20} + 286 q^{22} - 142 q^{23} - 223 q^{25} - 26 q^{26} + 476 q^{28} + 456 q^{29} + 230 q^{31} + 71 q^{32} - 17 q^{34} + 332 q^{35} + 356 q^{37} - 724 q^{38} - 424 q^{40} + 294 q^{41} + 556 q^{43} + 1122 q^{44} - 704 q^{46} - 640 q^{47} - 269 q^{49} - 547 q^{50} - 774 q^{52} - 302 q^{53} + 76 q^{55} - 684 q^{56} - 1304 q^{58} - 636 q^{59} - 84 q^{61} - 508 q^{62} - 919 q^{64} - 408 q^{65} + 1008 q^{67} + 425 q^{68} - 1504 q^{70} + 402 q^{71} + 838 q^{73} - 836 q^{74} - 908 q^{76} + 504 q^{77} - 594 q^{79} + 40 q^{80} + 358 q^{82} + 2396 q^{83} + 136 q^{85} + 1264 q^{86} + 1838 q^{88} + 170 q^{89} - 1016 q^{91} - 4896 q^{92} - 2016 q^{94} + 472 q^{95} - 270 q^{97} - 2857 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 14x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 10 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + 10 \) 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
−0.287410
3.87707
−3.58966
−4.67129 0 13.8209 11.9174 0 26.1222 −27.1912 0 −55.6696
1.2 −1.36122 0 −6.14708 −3.03171 0 −7.94049 19.2573 0 4.12682
1.3 5.03251 0 17.3261 −0.885690 0 3.81828 46.9339 0 −4.45724
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(-1\)

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 153.4.a.g 3
3.b odd 2 1 17.4.a.b 3
4.b odd 2 1 2448.4.a.bi 3
12.b even 2 1 272.4.a.h 3
15.d odd 2 1 425.4.a.g 3
15.e even 4 2 425.4.b.f 6
21.c even 2 1 833.4.a.d 3
24.f even 2 1 1088.4.a.x 3
24.h odd 2 1 1088.4.a.v 3
33.d even 2 1 2057.4.a.e 3
51.c odd 2 1 289.4.a.b 3
51.f odd 4 2 289.4.b.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.b 3 3.b odd 2 1
153.4.a.g 3 1.a even 1 1 trivial
272.4.a.h 3 12.b even 2 1
289.4.a.b 3 51.c odd 2 1
289.4.b.b 6 51.f odd 4 2
425.4.a.g 3 15.d odd 2 1
425.4.b.f 6 15.e even 4 2
833.4.a.d 3 21.c even 2 1
1088.4.a.v 3 24.h odd 2 1
1088.4.a.x 3 24.f even 2 1
2057.4.a.e 3 33.d even 2 1
2448.4.a.bi 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(153))\):

\( T_{2}^{3} + T_{2}^{2} - 24T_{2} - 32 \) Copy content Toggle raw display
\( T_{5}^{3} - 8T_{5}^{2} - 44T_{5} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 24 T - 32 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 8 T^{2} - 44 T - 32 \) Copy content Toggle raw display
$7$ \( T^{3} - 22 T^{2} - 138 T + 792 \) Copy content Toggle raw display
$11$ \( T^{3} - 28 T^{2} - 1366 T + 4692 \) Copy content Toggle raw display
$13$ \( T^{3} - 30 T^{2} - 1472 T - 9392 \) Copy content Toggle raw display
$17$ \( (T - 17)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 80 T^{2} - 4632 T + 340128 \) Copy content Toggle raw display
$23$ \( T^{3} + 142 T^{2} - 15770 T - 1600544 \) Copy content Toggle raw display
$29$ \( T^{3} - 456 T^{2} + 53908 T - 1518624 \) Copy content Toggle raw display
$31$ \( T^{3} - 230 T^{2} - 11586 T - 81608 \) Copy content Toggle raw display
$37$ \( T^{3} - 356 T^{2} - 17964 T + 6176752 \) Copy content Toggle raw display
$41$ \( T^{3} - 294 T^{2} - 86564 T + 1638744 \) Copy content Toggle raw display
$43$ \( T^{3} - 556 T^{2} + 51096 T + 7270272 \) Copy content Toggle raw display
$47$ \( T^{3} + 640 T^{2} + 85328 T + 1671168 \) Copy content Toggle raw display
$53$ \( T^{3} + 302 T^{2} + \cdots - 18162072 \) Copy content Toggle raw display
$59$ \( T^{3} + 636 T^{2} + \cdots - 49419072 \) Copy content Toggle raw display
$61$ \( T^{3} + 84 T^{2} - 124412 T - 6792784 \) Copy content Toggle raw display
$67$ \( T^{3} - 1008 T^{2} + 65040 T - 765952 \) Copy content Toggle raw display
$71$ \( T^{3} - 402 T^{2} + \cdots + 274866016 \) Copy content Toggle raw display
$73$ \( T^{3} - 838 T^{2} + \cdots - 19957512 \) Copy content Toggle raw display
$79$ \( T^{3} + 594 T^{2} + \cdots - 742135824 \) Copy content Toggle raw display
$83$ \( T^{3} - 2396 T^{2} + \cdots - 142080704 \) Copy content Toggle raw display
$89$ \( T^{3} - 170 T^{2} + \cdots + 446571376 \) Copy content Toggle raw display
$97$ \( T^{3} + 270 T^{2} + \cdots - 206623000 \) Copy content Toggle raw display
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