Properties

Label 2057.4.a.e
Level $2057$
Weight $4$
Character orbit 2057.a
Self dual yes
Analytic conductor $121.367$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2057 = 11^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2057.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(121.366928882\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2636.1
Defining polynomial: \( x^{3} - 14x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{2} + (2 \beta_{2} - \beta_1 + 2) q^{3} + ( - \beta_{2} - 3 \beta_1 + 8) q^{4} + (2 \beta_{2} - 2) q^{5} + ( - 2 \beta_{2} - 6 \beta_1 + 24) q^{6} + (4 \beta_{2} + \beta_1 - 6) q^{7} + (9 \beta_{2} - 5 \beta_1 + 16) q^{8} + ( - 2 \beta_{2} - 8 \beta_1 + 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{2} + (2 \beta_{2} - \beta_1 + 2) q^{3} + ( - \beta_{2} - 3 \beta_1 + 8) q^{4} + (2 \beta_{2} - 2) q^{5} + ( - 2 \beta_{2} - 6 \beta_1 + 24) q^{6} + (4 \beta_{2} + \beta_1 - 6) q^{7} + (9 \beta_{2} - 5 \beta_1 + 16) q^{8} + ( - 2 \beta_{2} - 8 \beta_1 + 19) q^{9} + ( - 8 \beta_{2} + 16) q^{10} + (26 \beta_{2} - 26 \beta_1 + 16) q^{12} + ( - 6 \beta_{2} - 8 \beta_1 - 12) q^{13} + ( - 20 \beta_{2} + 4 \beta_1 + 24) q^{14} + ( - 12 \beta_{2} + 2 \beta_1 + 32) q^{15} + (7 \beta_{2} - 11 \beta_1 + 48) q^{16} + 17 q^{17} + (41 \beta_{2} - 33 \beta_1 + 48) q^{18} + (8 \beta_{2} - 22 \beta_1 - 24) q^{19} + (24 \beta_{2} - 8 \beta_1 - 48) q^{20} + ( - 30 \beta_{2} + 12 \beta_1 + 54) q^{21} + ( - 4 \beta_{2} - 39 \beta_1 + 46) q^{23} + (6 \beta_{2} - 46 \beta_1 + 224) q^{24} + ( - 20 \beta_{2} + 4 \beta_1 - 81) q^{25} + (22 \beta_{2} + 2 \beta_1 + 16) q^{26} + (8 \beta_{2} - 40 \beta_1 - 4) q^{27} + (44 \beta_{2} - 4 \beta_1 - 144) q^{28} + ( - 30 \beta_{2} + 16 \beta_1 + 142) q^{29} + (64 \beta_{2} - 16 \beta_1 - 112) q^{30} + (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} + (7 \beta_{2} - 91 \beta_1 + 440) q^{36} + ( - 50 \beta_{2} - 28 \beta_1 + 102) q^{37} + ( - 4 \beta_{2} - 28 \beta_1 + 240) q^{38} + (16 \beta_{2} - 36 \beta_1 - 84) q^{39} + ( - 40 \beta_{2} + 8 \beta_1 + 128) q^{40} + (60 \beta_{2} + 52 \beta_1 + 118) q^{41} + (120 \beta_{2} - 336) q^{42} + ( - 56 \beta_{2} - 2 \beta_1 - 204) q^{43} + (54 \beta_{2} - 20 \beta_1 - 110) q^{45} + (136 \beta_{2} - 120 \beta_1 + 280) q^{46} + (44 \beta_{2} - 48 \beta_1 + 228) q^{47} + (90 \beta_{2} - 114 \beta_1 + 288) q^{48} + ( - 94 \beta_{2} + 20 \beta_1 - 121) q^{49} + ( - 29 \beta_{2} + 109 \beta_1 - 192) q^{50} + (34 \beta_{2} - 17 \beta_1 + 34) q^{51} + ( - 6 \beta_{2} + 30 \beta_1 + 256) q^{52} + ( - 8 \beta_{2} + 116 \beta_1 + 98) q^{53} + (52 \beta_{2} - 84 \beta_1 + 384) q^{54} + ( - 108 \beta_{2} + 60 \beta_1 + 192) q^{56} + ( - 36 \beta_{2} - 108 \beta_1 + 228) q^{57} + (200 \beta_{2} - 80 \beta_1 - 368) q^{58} + ( - 130 \beta_1 + 212) q^{59} + ( - 176 \beta_{2} + 384) q^{60} + ( - 78 \beta_{2} + 64 \beta_1 + 2) q^{61} + ( - 44 \beta_{2} - 20 \beta_1 - 184) q^{62} + (96 \beta_{2} - 9 \beta_1 - 342) q^{63} + (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} + (186 \beta_{2} - 280 \beta_1 + 254) q^{69} + (208 \beta_{2} - 32 \beta_1 - 432) q^{70} + (72 \beta_{2} + 185 \beta_1 - 110) q^{71} + (273 \beta_{2} - 365 \beta_1 + 400) q^{72} + (16 \beta_{2} - 16 \beta_1 - 274) q^{73} + (308 \beta_{2} - 108 \beta_1 - 176) q^{74} + ( - 90 \beta_{2} + 105 \beta_1 - 546) q^{75} + (244 \beta_{2} - 116 \beta_1 + 384) q^{76} + ( - 60 \beta_{2} - 4 \beta_1 + 416) q^{78} + ( - 180 \beta_{2} + 267 \beta_1 + 138) q^{79} + (40 \beta_{2} - 8 \beta_1) q^{80} + (94 \beta_{2} - 20 \beta_1 - 137) q^{81} + ( - 166 \beta_{2} - 74 \beta_1 + 64) q^{82} + ( - 128 \beta_{2} - 82 \beta_1 + 756) q^{83} + ( - 456 \beta_{2} + 120 \beta_1 + 528) q^{84} + (34 \beta_{2} - 34) q^{85} + ( - 32 \beta_{2} + 256 \beta_1 - 432) q^{86} + (372 \beta_{2} - 46 \beta_1 - 352) q^{87} + (110 \beta_{2} - 276 \beta_1 - 20) q^{89} + ( - 232 \beta_{2} + 16 \beta_1 + 592) q^{90} + (44 \beta_{2} - 64 \beta_1 - 324) q^{91} + (144 \beta_{2} - 344 \beta_1 + 1680) q^{92} + (22 \beta_{2} + 152 \beta_1 + 218) q^{93} + (192 \beta_{2} - 368 \beta_1 + 736) q^{94} + ( - 112 \beta_{2} - 28 \beta_1 + 120) q^{95} + (198 \beta_{2} - 238 \beta_1 - 160) q^{96} + (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
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 4 q^{3} + 25 q^{4} - 8 q^{5} + 74 q^{6} - 22 q^{7} + 39 q^{8} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 4 q^{3} + 25 q^{4} - 8 q^{5} + 74 q^{6} - 22 q^{7} + 39 q^{8} + 59 q^{9} + 56 q^{10} + 22 q^{12} - 30 q^{13} + 92 q^{14} + 108 q^{15} + 137 q^{16} + 51 q^{17} + 103 q^{18} - 80 q^{19} - 168 q^{20} + 192 q^{21} + 142 q^{23} + 666 q^{24} - 223 q^{25} + 26 q^{26} - 20 q^{27} - 476 q^{28} + 456 q^{29} - 400 q^{30} + 230 q^{31} + 71 q^{32} - 17 q^{34} + 332 q^{35} + 1313 q^{36} + 356 q^{37} + 724 q^{38} - 268 q^{39} + 424 q^{40} + 294 q^{41} - 1128 q^{42} - 556 q^{43} - 384 q^{45} + 704 q^{46} + 640 q^{47} + 774 q^{48} - 269 q^{49} - 547 q^{50} + 68 q^{51} + 774 q^{52} + 302 q^{53} + 1100 q^{54} + 684 q^{56} + 720 q^{57} - 1304 q^{58} + 636 q^{59} + 1328 q^{60} + 84 q^{61} - 508 q^{62} - 1122 q^{63} - 919 q^{64} - 408 q^{65} + 1008 q^{67} + 425 q^{68} + 576 q^{69} - 1504 q^{70} - 402 q^{71} + 927 q^{72} - 838 q^{73} - 836 q^{74} - 1548 q^{75} + 908 q^{76} + 1308 q^{78} + 594 q^{79} - 40 q^{80} - 505 q^{81} + 358 q^{82} + 2396 q^{83} + 2040 q^{84} - 136 q^{85} - 1264 q^{86} - 1428 q^{87} - 170 q^{89} + 2008 q^{90} - 1016 q^{91} + 4896 q^{92} + 632 q^{93} + 2016 q^{94} + 472 q^{95} - 678 q^{96} - 270 q^{97} - 2857 q^{98}+O(q^{100}) \) 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 −7.62999 13.8209 −11.9174 35.6419 −26.1222 −27.1912 31.2167 55.6696
1.2 −1.36122 3.15463 −6.14708 3.03171 −4.29415 7.94049 19.2573 −17.0483 −4.12682
1.3 5.03251 8.47535 17.3261 0.885690 42.6523 −3.81828 46.9339 44.8316 4.45724
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(17\) \(-1\)

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 2057.4.a.e 3
11.b odd 2 1 17.4.a.b 3
33.d even 2 1 153.4.a.g 3
44.c even 2 1 272.4.a.h 3
55.d odd 2 1 425.4.a.g 3
55.e even 4 2 425.4.b.f 6
77.b even 2 1 833.4.a.d 3
88.b odd 2 1 1088.4.a.v 3
88.g even 2 1 1088.4.a.x 3
132.d odd 2 1 2448.4.a.bi 3
187.b odd 2 1 289.4.a.b 3
187.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 11.b odd 2 1
153.4.a.g 3 33.d even 2 1
272.4.a.h 3 44.c even 2 1
289.4.a.b 3 187.b odd 2 1
289.4.b.b 6 187.f odd 4 2
425.4.a.g 3 55.d odd 2 1
425.4.b.f 6 55.e even 4 2
833.4.a.d 3 77.b even 2 1
1088.4.a.v 3 88.b odd 2 1
1088.4.a.x 3 88.g even 2 1
2057.4.a.e 3 1.a even 1 1 trivial
2448.4.a.bi 3 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):

\( T_{2}^{3} + T_{2}^{2} - 24T_{2} - 32 \) Copy content Toggle raw display
\( T_{3}^{3} - 4T_{3}^{2} - 62T_{3} + 204 \) 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
$3$ \( T^{3} - 4 T^{2} - 62 T + 204 \) 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
$11$ \( T^{3} \) 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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