Properties

Label 58.4.a.d
Level $58$
Weight $4$
Character orbit 58.a
Self dual yes
Analytic conductor $3.422$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 58.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.42211078033\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.19816.1
Defining polynomial: \( x^{3} - x^{2} - 42x - 54 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta_1 + 1) q^{3} + 4 q^{4} + (\beta_{2} + 2 \beta_1 + 6) q^{5} + ( - 2 \beta_1 + 2) q^{6} + ( - 4 \beta_{2} + 8) q^{7} + 8 q^{8} + (3 \beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + ( - \beta_1 + 1) q^{3} + 4 q^{4} + (\beta_{2} + 2 \beta_1 + 6) q^{5} + ( - 2 \beta_1 + 2) q^{6} + ( - 4 \beta_{2} + 8) q^{7} + 8 q^{8} + (3 \beta_{2} + 2 \beta_1 + 1) q^{9} + (2 \beta_{2} + 4 \beta_1 + 12) q^{10} + ( - 2 \beta_{2} + \beta_1 + 3) q^{11} + ( - 4 \beta_1 + 4) q^{12} + (11 \beta_{2} + 8 \beta_1 - 4) q^{13} + ( - 8 \beta_{2} + 16) q^{14} + ( - 2 \beta_{2} - 13 \beta_1 - 39) q^{15} + 16 q^{16} + ( - 16 \beta_{2} - 12 \beta_1 - 18) q^{17} + (6 \beta_{2} + 4 \beta_1 + 2) q^{18} + (10 \beta_{2} - 8 \beta_1 - 52) q^{19} + (4 \beta_{2} + 8 \beta_1 + 24) q^{20} + ( - 16 \beta_{2} - 4 \beta_1 - 28) q^{21} + ( - 4 \beta_{2} + 2 \beta_1 + 6) q^{22} + (12 \beta_{2} - 6 \beta_1 - 66) q^{23} + ( - 8 \beta_1 + 8) q^{24} + (11 \beta_{2} + 40 \beta_1 + 13) q^{25} + (22 \beta_{2} + 16 \beta_1 - 8) q^{26} + (6 \beta_{2} + 17 \beta_1 - 53) q^{27} + ( - 16 \beta_{2} + 32) q^{28} + 29 q^{29} + ( - 4 \beta_{2} - 26 \beta_1 - 78) q^{30} + ( - 36 \beta_{2} - 11 \beta_1 - 25) q^{31} + 32 q^{32} + ( - 11 \beta_{2} - 4 \beta_1 - 42) q^{33} + ( - 32 \beta_{2} - 24 \beta_1 - 36) q^{34} + (12 \beta_{2} + 24 \beta_1) q^{35} + (12 \beta_{2} + 8 \beta_1 + 4) q^{36} + (38 \beta_{2} - 12 \beta_1 - 10) q^{37} + (20 \beta_{2} - 16 \beta_1 - 104) q^{38} + (20 \beta_{2} - 31 \beta_1 - 121) q^{39} + (8 \beta_{2} + 16 \beta_1 + 48) q^{40} + ( - 6 \beta_{2} + 4 \beta_1 + 186) q^{41} + ( - 32 \beta_{2} - 8 \beta_1 - 56) q^{42} + (2 \beta_{2} - 15 \beta_1 + 11) q^{43} + ( - 8 \beta_{2} + 4 \beta_1 + 12) q^{44} + (4 \beta_{2} + 26 \beta_1 + 132) q^{45} + (24 \beta_{2} - 12 \beta_1 - 132) q^{46} + (2 \beta_{2} + 33 \beta_1 + 207) q^{47} + ( - 16 \beta_1 + 16) q^{48} + ( - 80 \beta_{2} - 64 \beta_1 + 201) q^{49} + (22 \beta_{2} + 80 \beta_1 + 26) q^{50} + ( - 28 \beta_{2} + 70 \beta_1 + 162) q^{51} + (44 \beta_{2} + 32 \beta_1 - 16) q^{52} + ( - 27 \beta_{2} - 116 \beta_1 + 276) q^{53} + (12 \beta_{2} + 34 \beta_1 - 106) q^{54} + (8 \beta_{2} + 25 \beta_1 + 39) q^{55} + ( - 32 \beta_{2} + 64) q^{56} + (64 \beta_{2} + 66 \beta_1 + 254) q^{57} + 58 q^{58} + (4 \beta_{2} - 86 \beta_1 + 90) q^{59} + ( - 8 \beta_{2} - 52 \beta_1 - 156) q^{60} + (40 \beta_{2} - 80 \beta_1 + 134) q^{61} + ( - 72 \beta_{2} - 22 \beta_1 - 50) q^{62} + (56 \beta_{2} + 56 \beta_1 - 280) q^{63} + 64 q^{64} + (9 \beta_{2} + 90 \beta_1 + 468) q^{65} + ( - 22 \beta_{2} - 8 \beta_1 - 84) q^{66} + ( - 72 \beta_{2} + 36 \beta_1 - 88) q^{67} + ( - 64 \beta_{2} - 48 \beta_1 - 72) q^{68} + (66 \beta_{2} + 72 \beta_1 + 204) q^{69} + (24 \beta_{2} + 48 \beta_1) q^{70} + ( - 4 \beta_{2} + 2 \beta_1 - 18) q^{71} + (24 \beta_{2} + 16 \beta_1 + 8) q^{72} + (30 \beta_{2} + 40 \beta_1 - 178) q^{73} + (76 \beta_{2} - 24 \beta_1 - 20) q^{74} + ( - 76 \beta_{2} - 144 \beta_1 - 968) q^{75} + (40 \beta_{2} - 32 \beta_1 - 208) q^{76} + ( - 24 \beta_{2} - 28 \beta_1 + 300) q^{77} + (40 \beta_{2} - 62 \beta_1 - 242) q^{78} + ( - 38 \beta_{2} - 37 \beta_1 - 691) q^{79} + (16 \beta_{2} + 32 \beta_1 + 96) q^{80} + ( - 108 \beta_{2} - 58 \beta_1 - 485) q^{81} + ( - 12 \beta_{2} + 8 \beta_1 + 372) q^{82} + (12 \beta_{2} + 162 \beta_1 - 150) q^{83} + ( - 64 \beta_{2} - 16 \beta_1 - 112) q^{84} + ( - 38 \beta_{2} - 184 \beta_1 - 840) q^{85} + (4 \beta_{2} - 30 \beta_1 + 22) q^{86} + ( - 29 \beta_1 + 29) q^{87} + ( - 16 \beta_{2} + 8 \beta_1 + 24) q^{88} + (154 \beta_{2} + 68 \beta_1 + 282) q^{89} + (8 \beta_{2} + 52 \beta_1 + 264) q^{90} + (244 \beta_{2} + 208 \beta_1 - 1064) q^{91} + (48 \beta_{2} - 24 \beta_1 - 264) q^{92} + ( - 111 \beta_{2} + 94 \beta_1 - 52) q^{93} + (4 \beta_{2} + 66 \beta_1 + 414) q^{94} + ( - 86 \beta_{2} - 244 \beta_1 - 552) q^{95} + ( - 32 \beta_1 + 32) q^{96} + ( - 34 \beta_{2} + 176 \beta_1 + 14) q^{97} + ( - 160 \beta_{2} - 128 \beta_1 + 402) q^{98} + (22 \beta_{2} + 38 \beta_1 - 114) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{2} + 2 q^{3} + 12 q^{4} + 20 q^{5} + 4 q^{6} + 24 q^{7} + 24 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{2} + 2 q^{3} + 12 q^{4} + 20 q^{5} + 4 q^{6} + 24 q^{7} + 24 q^{8} + 5 q^{9} + 40 q^{10} + 10 q^{11} + 8 q^{12} - 4 q^{13} + 48 q^{14} - 130 q^{15} + 48 q^{16} - 66 q^{17} + 10 q^{18} - 164 q^{19} + 80 q^{20} - 88 q^{21} + 20 q^{22} - 204 q^{23} + 16 q^{24} + 79 q^{25} - 8 q^{26} - 142 q^{27} + 96 q^{28} + 87 q^{29} - 260 q^{30} - 86 q^{31} + 96 q^{32} - 130 q^{33} - 132 q^{34} + 24 q^{35} + 20 q^{36} - 42 q^{37} - 328 q^{38} - 394 q^{39} + 160 q^{40} + 562 q^{41} - 176 q^{42} + 18 q^{43} + 40 q^{44} + 422 q^{45} - 408 q^{46} + 654 q^{47} + 32 q^{48} + 539 q^{49} + 158 q^{50} + 556 q^{51} - 16 q^{52} + 712 q^{53} - 284 q^{54} + 142 q^{55} + 192 q^{56} + 828 q^{57} + 174 q^{58} + 184 q^{59} - 520 q^{60} + 322 q^{61} - 172 q^{62} - 784 q^{63} + 192 q^{64} + 1494 q^{65} - 260 q^{66} - 228 q^{67} - 264 q^{68} + 684 q^{69} + 48 q^{70} - 52 q^{71} + 40 q^{72} - 494 q^{73} - 84 q^{74} - 3048 q^{75} - 656 q^{76} + 872 q^{77} - 788 q^{78} - 2110 q^{79} + 320 q^{80} - 1513 q^{81} + 1124 q^{82} - 288 q^{83} - 352 q^{84} - 2704 q^{85} + 36 q^{86} + 58 q^{87} + 80 q^{88} + 914 q^{89} + 844 q^{90} - 2984 q^{91} - 816 q^{92} - 62 q^{93} + 1308 q^{94} - 1900 q^{95} + 64 q^{96} + 218 q^{97} + 1078 q^{98} - 304 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 42x - 54 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 4\nu - 27 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} + 4\beta _1 + 27 \) 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
7.53003
−1.39712
−5.13291
2.00000 −6.53003 4.00000 20.9205 −13.0601 8.55839 8.00000 15.6413 41.8409
1.2 2.00000 2.39712 4.00000 −3.28077 4.79424 33.9461 8.00000 −21.2538 −6.56153
1.3 2.00000 6.13291 4.00000 2.36031 12.2658 −18.5045 8.00000 10.6126 4.72062
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(29\) \(-1\)

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 58.4.a.d 3
3.b odd 2 1 522.4.a.k 3
4.b odd 2 1 464.4.a.i 3
5.b even 2 1 1450.4.a.h 3
8.b even 2 1 1856.4.a.r 3
8.d odd 2 1 1856.4.a.s 3
29.b even 2 1 1682.4.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.d 3 1.a even 1 1 trivial
464.4.a.i 3 4.b odd 2 1
522.4.a.k 3 3.b odd 2 1
1450.4.a.h 3 5.b even 2 1
1682.4.a.d 3 29.b even 2 1
1856.4.a.r 3 8.b even 2 1
1856.4.a.s 3 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 2T_{3}^{2} - 41T_{3} + 96 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(58))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 2 T^{2} - 41 T + 96 \) Copy content Toggle raw display
$5$ \( T^{3} - 20 T^{2} - 27 T + 162 \) Copy content Toggle raw display
$7$ \( T^{3} - 24 T^{2} - 496 T + 5376 \) Copy content Toggle raw display
$11$ \( T^{3} - 10 T^{2} - 233 T + 2424 \) Copy content Toggle raw display
$13$ \( T^{3} + 4 T^{2} - 5619 T + 131706 \) Copy content Toggle raw display
$17$ \( T^{3} + 66 T^{2} - 10660 T - 679368 \) Copy content Toggle raw display
$19$ \( T^{3} + 164 T^{2} - 124 T - 664448 \) Copy content Toggle raw display
$23$ \( T^{3} + 204 T^{2} + 4284 T - 677376 \) Copy content Toggle raw display
$29$ \( (T - 29)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} + 86 T^{2} - 48089 T - 4766172 \) Copy content Toggle raw display
$37$ \( T^{3} + 42 T^{2} - 79456 T - 7684896 \) Copy content Toggle raw display
$41$ \( T^{3} - 562 T^{2} + 102432 T - 5982048 \) Copy content Toggle raw display
$43$ \( T^{3} - 18 T^{2} - 10369 T + 196488 \) Copy content Toggle raw display
$47$ \( T^{3} - 654 T^{2} + 98015 T - 3425124 \) Copy content Toggle raw display
$53$ \( T^{3} - 712 T^{2} + \cdots + 252120546 \) Copy content Toggle raw display
$59$ \( T^{3} - 184 T^{2} + \cdots + 57362928 \) Copy content Toggle raw display
$61$ \( T^{3} - 322 T^{2} - 388372 T - 5254424 \) Copy content Toggle raw display
$67$ \( T^{3} + 228 T^{2} + \cdots + 47608192 \) Copy content Toggle raw display
$71$ \( T^{3} + 52 T^{2} - 164 T - 672 \) Copy content Toggle raw display
$73$ \( T^{3} + 494 T^{2} + 6112 T - 9410208 \) Copy content Toggle raw display
$79$ \( T^{3} + 2110 T^{2} + \cdots + 285187172 \) Copy content Toggle raw display
$83$ \( T^{3} + 288 T^{2} + \cdots - 437606064 \) Copy content Toggle raw display
$89$ \( T^{3} - 914 T^{2} + \cdots + 598011552 \) Copy content Toggle raw display
$97$ \( T^{3} - 218 T^{2} + \cdots - 17006112 \) Copy content Toggle raw display
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