Properties

Label 1421.4.a.e
Level $1421$
Weight $4$
Character orbit 1421.a
Self dual yes
Analytic conductor $83.842$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1421 = 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1421.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(83.8417141182\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.13458092.1
Defining polynomial: \( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{4} + \beta_{3} + \beta_1 - 1) q^{3} + (\beta_{4} + 2 \beta_{3} + 2 \beta_1 + 6) q^{4} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{5} + ( - \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 6) q^{6} + ( - 4 \beta_{4} - 4 \beta_{3} - 8 \beta_{2} - 9 \beta_1 - 16) q^{8} + ( - 6 \beta_{4} - \beta_{3} + 5 \beta_{2} + \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{4} + \beta_{3} + \beta_1 - 1) q^{3} + (\beta_{4} + 2 \beta_{3} + 2 \beta_1 + 6) q^{4} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{5} + ( - \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 6) q^{6} + ( - 4 \beta_{4} - 4 \beta_{3} - 8 \beta_{2} - 9 \beta_1 - 16) q^{8} + ( - 6 \beta_{4} - \beta_{3} + 5 \beta_{2} + \beta_1 + 2) q^{9} + ( - 6 \beta_{4} - 4 \beta_{2} - \beta_1 + 12) q^{10} + (9 \beta_{4} + 5 \beta_{3} + 10 \beta_{2} + 7 \beta_1 + 3) q^{11} + (2 \beta_{4} + 8 \beta_{3} + 8 \beta_{2} + 13 \beta_1 + 44) q^{12} + (6 \beta_{4} - 5 \beta_{3} + 9 \beta_{2} + 3 \beta_1 - 5) q^{13} + (11 \beta_{4} - \beta_{3} - 14 \beta_{2} + 3 \beta_1 - 5) q^{15} + (9 \beta_{4} + 18 \beta_{3} + 16 \beta_{2} + 30 \beta_1 + 30) q^{16} + ( - 4 \beta_{3} - 10 \beta_{2} - 10) q^{17} + ( - 16 \beta_{4} - 12 \beta_{3} + 4 \beta_{2} - 6 \beta_1 - 32) q^{18} + ( - 4 \beta_{4} + 2 \beta_{3} - 8 \beta_1 - 44) q^{19} + (9 \beta_{4} + 2 \beta_{3} - 8 \beta_{2} - 8 \beta_1 + 6) q^{20} + ( - 9 \beta_{4} - 34 \beta_{3} - 20 \beta_{2} - 28 \beta_1 - 22) q^{22} + (24 \beta_{4} + 14 \beta_{3} - 2 \beta_{2} + 46) q^{23} + ( - 25 \beta_{4} - 26 \beta_{3} - 68 \beta_1 - 78) q^{24} + ( - 2 \beta_{4} - 21 \beta_{3} + 11 \beta_{2} - 17 \beta_1 + 32) q^{25} + (10 \beta_{4} - 24 \beta_{3} + 20 \beta_{2} + 25 \beta_1 - 20) q^{26} + (27 \beta_{4} - 11 \beta_{3} - 32 \beta_{2} + 11 \beta_1 - 51) q^{27} - 29 q^{29} + (35 \beta_{4} + 22 \beta_{3} + 4 \beta_{2} + 14 \beta_1 - 30) q^{30} + ( - 5 \beta_{4} - 7 \beta_{3} + 14 \beta_{2} + 19 \beta_1 - 93) q^{31} + ( - 32 \beta_{4} - 60 \beta_{3} - 8 \beta_{2} - 81 \beta_1 - 152) q^{32} + ( - 14 \beta_{4} - \beta_{3} - 9 \beta_{2} + 39 \beta_1 + 113) q^{33} + (18 \beta_{4} + 20 \beta_{3} + 16 \beta_{2} + 26 \beta_1 - 36) q^{34} + (42 \beta_{4} + 12 \beta_{3} + 8 \beta_{2} + 68 \beta_1 - 36) q^{36} + ( - 28 \beta_{4} - 22 \beta_{3} + 30 \beta_{2} + 22 \beta_1 + 48) q^{37} + ( - 4 \beta_{4} + 16 \beta_{3} - 8 \beta_{2} + 48 \beta_1 + 104) q^{38} + ( - 11 \beta_{4} - 11 \beta_{3} - 16 \beta_{2} + 11 \beta_1 - 75) q^{39} + (78 \beta_{4} + 32 \beta_{3} + 24 \beta_{2} + 19 \beta_1 + 44) q^{40} + ( - 44 \beta_{4} - 22 \beta_{3} - 16 \beta_{2} - 18 \beta_1 + 216) q^{41} + ( - 11 \beta_{4} + 25 \beta_{3} + 22 \beta_{2} - 29 \beta_1 - 49) q^{43} + (26 \beta_{4} + 56 \beta_{3} + 56 \beta_{2} + 149 \beta_1 + 156) q^{44} + ( - 56 \beta_{4} - 48 \beta_{3} + 78 \beta_{2} - 42 \beta_1 + 238) q^{45} + (22 \beta_{4} + 4 \beta_{3} - 56 \beta_{2} - 78 \beta_1 + 148) q^{46} + (7 \beta_{4} + 41 \beta_{3} + 16 \beta_{2} - 75 \beta_1 - 45) q^{47} + (54 \beta_{4} + 72 \beta_{3} + 40 \beta_{2} + 189 \beta_1 + 396) q^{48} + (44 \beta_{4} + 12 \beta_{3} + 84 \beta_{2} + 84 \beta_1 + 168) q^{50} + ( - 40 \beta_{4} - 10 \beta_{3} + 54 \beta_{2} - 44 \beta_1 + 6) q^{51} + ( - 25 \beta_{4} - 50 \beta_{3} + 24 \beta_{2} + 52 \beta_1 - 326) q^{52} + (10 \beta_{4} + 23 \beta_{3} + 17 \beta_{2} + 75 \beta_1 - 109) q^{53} + (97 \beta_{4} + 42 \beta_{3} + 44 \beta_{2} + 100 \beta_1 - 154) q^{54} + ( - 101 \beta_{4} - 41 \beta_{3} - 112 \beta_{2} + 13 \beta_1 - 113) q^{55} + ( - 28 \beta_{4} - 52 \beta_{3} - 42 \beta_{2} - 62 \beta_1 + 4) q^{57} + 29 \beta_1 q^{58} + (16 \beta_{4} - 30 \beta_{3} + 6 \beta_{2} - 52 \beta_1 - 90) q^{59} + ( - 80 \beta_{4} - 28 \beta_{3} + 24 \beta_{2} - 75 \beta_1 + 80) q^{60} + ( - 12 \beta_{4} - 60 \beta_{3} - 66 \beta_{2} + 20 \beta_1 - 114) q^{61} + ( - 29 \beta_{4} - 66 \beta_{3} + 28 \beta_{2} + 78 \beta_1 - 286) q^{62} + (73 \beta_{4} + 34 \beta_{3} + 112 \beta_{2} + 282 \beta_1 + 510) q^{64} + ( - 146 \beta_{4} - 15 \beta_{3} - 49 \beta_{2} + 51 \beta_1 - 373) q^{65} + ( - 56 \beta_{4} - 60 \beta_{3} + 4 \beta_{2} - 201 \beta_1 - 624) q^{66} + ( - 44 \beta_{3} - 140 \beta_{2} - 88 \beta_1 + 280) q^{67} + ( - 46 \beta_{4} - 52 \beta_{3} - 78 \beta_1 - 100) q^{68} + ( - 104 \beta_{4} - 2 \beta_{3} + 44 \beta_{2} - 18 \beta_1 + 334) q^{69} + (6 \beta_{4} - 122 \beta_{3} - 96 \beta_{2} - 58 \beta_1 - 122) q^{71} + (112 \beta_{4} - 56 \beta_{3} - 80 \beta_{2} - 58 \beta_1 - 464) q^{72} + (32 \beta_{4} + 38 \beta_{3} + 66 \beta_{2} + 106 \beta_1 + 384) q^{73} + ( - 64 \beta_{4} - 104 \beta_{3} + 88 \beta_{2} - 32 \beta_1 - 448) q^{74} + (60 \beta_{4} + 2 \beta_{3} - 106 \beta_{2} - 18 \beta_1 - 548) q^{75} + ( - 48 \beta_{4} - 96 \beta_{3} - 64 \beta_{2} - 204 \beta_1 - 288) q^{76} + (5 \beta_{4} + 10 \beta_{3} + 44 \beta_{2} + 86 \beta_1 - 274) q^{78} + ( - 177 \beta_{4} - 63 \beta_{3} - 140 \beta_{2} - 139 \beta_1 + 27) q^{79} + ( - 23 \beta_{4} - 102 \beta_{3} - 64 \beta_{2} - 68 \beta_1 + 174) q^{80} + ( - 136 \beta_{4} - 56 \beta_{3} + 134 \beta_{2} - 222 \beta_1 + 241) q^{81} + ( - 10 \beta_{4} + 68 \beta_{3} + 88 \beta_{2} - 136 \beta_1 - 44) q^{82} + ( - 60 \beta_{4} - 14 \beta_{3} + 42 \beta_{2} + 96 \beta_1 - 146) q^{83} + (136 \beta_{4} + 66 \beta_{3} + 20 \beta_{2} + 50 \beta_1 - 266) q^{85} + ( - 65 \beta_{4} + 14 \beta_{3} - 100 \beta_{2} - 4 \beta_1 + 506) q^{86} + ( - 29 \beta_{4} - 29 \beta_{3} - 29 \beta_1 + 29) q^{87} + ( - 193 \beta_{4} - 138 \beta_{3} - 64 \beta_{2} - 428 \beta_1 - 1470) q^{88} + ( - 152 \beta_{4} - 10 \beta_{3} - 100 \beta_{2} - 66 \beta_1 - 196) q^{89} + ( - 52 \beta_{4} - 72 \beta_{3} + 192 \beta_{2} - 18 \beta_1 + 328) q^{90} + ( - 22 \beta_{4} + 156 \beta_{3} + 14 \beta_1 + 716) q^{92} + ( - 2 \beta_{4} - 45 \beta_{3} + 3 \beta_{2} + 33 \beta_1 - 19) q^{93} + ( - 9 \beta_{4} + 118 \beta_{3} - 164 \beta_{2} + 38 \beta_1 + 1274) q^{94} + (64 \beta_{4} - 68 \beta_{3} - 38 \beta_{2} + 2 \beta_1 + 476) q^{95} + ( - 65 \beta_{4} - 250 \beta_{3} - 288 \beta_{2} - 464 \beta_1 - 1438) q^{96} + (32 \beta_{4} - 78 \beta_{3} - 48 \beta_{2} - 34 \beta_1 - 296) q^{97} + ( - 34 \beta_{4} + 84 \beta_{3} - 96 \beta_{2} + 94 \beta_1 - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{3} + 26 q^{4} - 10 q^{5} - 34 q^{6} - 84 q^{8} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 8 q^{3} + 26 q^{4} - 10 q^{5} - 34 q^{6} - 84 q^{8} + 33 q^{9} + 64 q^{10} + 12 q^{11} + 224 q^{12} - 14 q^{13} - 74 q^{15} + 146 q^{16} - 66 q^{17} - 108 q^{18} - 214 q^{19} - 6 q^{20} - 98 q^{22} + 164 q^{23} - 314 q^{24} + 207 q^{25} - 56 q^{26} - 362 q^{27} - 145 q^{29} - 234 q^{30} - 420 q^{31} - 652 q^{32} + 576 q^{33} - 204 q^{34} - 260 q^{36} + 378 q^{37} + 496 q^{38} - 374 q^{39} + 80 q^{40} + 1158 q^{41} - 204 q^{43} + 784 q^{44} + 1506 q^{45} + 580 q^{46} - 248 q^{47} + 1880 q^{48} + 908 q^{50} + 228 q^{51} - 1482 q^{52} - 554 q^{53} - 918 q^{54} - 546 q^{55} + 44 q^{57} - 440 q^{59} + 636 q^{60} - 618 q^{61} - 1250 q^{62} + 2594 q^{64} - 1656 q^{65} - 2940 q^{66} + 1164 q^{67} - 356 q^{68} + 1968 q^{69} - 692 q^{71} - 2648 q^{72} + 1950 q^{73} - 1832 q^{74} - 3074 q^{75} - 1376 q^{76} - 1302 q^{78} + 272 q^{79} + 890 q^{80} + 1801 q^{81} - 92 q^{82} - 512 q^{83} - 1628 q^{85} + 2446 q^{86} + 232 q^{87} - 6954 q^{88} - 866 q^{89} + 2200 q^{90} + 3468 q^{92} - 40 q^{93} + 5942 q^{94} + 2244 q^{95} - 7386 q^{96} - 1562 q^{97} - 238 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu^{2} - 8\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} - \nu^{2} + 12\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 13\nu^{2} + 6\nu + 14 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + 4\beta_{2} + 6\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} + 13\beta_{3} - 3\beta_{2} - 3\beta _1 + 64 \) 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
3.03898
−0.957567
2.27399
0.328194
−3.68360
−5.49488 6.46343 22.1937 −2.14270 −35.5158 0 −77.9928 14.7760 11.7739
1.2 −2.84972 −4.64574 0.120922 −12.8729 13.2391 0 22.4532 −5.41713 36.6841
1.3 1.63099 −9.87991 −5.33986 16.8209 −16.1141 0 −21.7572 70.6126 27.4348
1.4 2.24125 −1.84328 −2.97681 −18.3339 −4.13124 0 −24.6017 −23.6023 −41.0908
1.5 4.47236 1.90549 12.0020 6.52855 8.52204 0 17.8986 −23.3691 29.1981
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(29\) \(1\)

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 1421.4.a.e 5
7.b odd 2 1 29.4.a.b 5
21.c even 2 1 261.4.a.f 5
28.d even 2 1 464.4.a.l 5
35.c odd 2 1 725.4.a.c 5
56.e even 2 1 1856.4.a.bb 5
56.h odd 2 1 1856.4.a.y 5
203.c odd 2 1 841.4.a.b 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.4.a.b 5 7.b odd 2 1
261.4.a.f 5 21.c even 2 1
464.4.a.l 5 28.d even 2 1
725.4.a.c 5 35.c odd 2 1
841.4.a.b 5 203.c odd 2 1
1421.4.a.e 5 1.a even 1 1 trivial
1856.4.a.y 5 56.h odd 2 1
1856.4.a.bb 5 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1421))\):

\( T_{2}^{5} - 33T_{2}^{3} + 28T_{2}^{2} + 192T_{2} - 256 \) Copy content Toggle raw display
\( T_{3}^{5} + 8T_{3}^{4} - 52T_{3}^{3} - 322T_{3}^{2} + 187T_{3} + 1042 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 33 T^{3} + 28 T^{2} + \cdots - 256 \) Copy content Toggle raw display
$3$ \( T^{5} + 8 T^{4} - 52 T^{3} + \cdots + 1042 \) Copy content Toggle raw display
$5$ \( T^{5} + 10 T^{4} - 366 T^{3} + \cdots + 55534 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 12 T^{4} - 4892 T^{3} + \cdots + 30997958 \) Copy content Toggle raw display
$13$ \( T^{5} + 14 T^{4} - 7558 T^{3} + \cdots + 13078418 \) Copy content Toggle raw display
$17$ \( T^{5} + 66 T^{4} - 2444 T^{3} + \cdots - 19935872 \) Copy content Toggle raw display
$19$ \( T^{5} + 214 T^{4} + \cdots - 19441152 \) Copy content Toggle raw display
$23$ \( T^{5} - 164 T^{4} + \cdots - 7938109184 \) Copy content Toggle raw display
$29$ \( (T + 29)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} + 420 T^{4} + 45552 T^{3} + \cdots + 2094346 \) Copy content Toggle raw display
$37$ \( T^{5} - 378 T^{4} + \cdots + 23564115968 \) Copy content Toggle raw display
$41$ \( T^{5} - 1158 T^{4} + \cdots - 59613728000 \) Copy content Toggle raw display
$43$ \( T^{5} + 204 T^{4} + \cdots + 198643410886 \) Copy content Toggle raw display
$47$ \( T^{5} + 248 T^{4} + \cdots + 203435244846 \) Copy content Toggle raw display
$53$ \( T^{5} + 554 T^{4} + \cdots - 786854101018 \) Copy content Toggle raw display
$59$ \( T^{5} + 440 T^{4} + \cdots - 109032704000 \) Copy content Toggle raw display
$61$ \( T^{5} + 618 T^{4} + \cdots - 2140697762176 \) Copy content Toggle raw display
$67$ \( T^{5} - 1164 T^{4} + \cdots - 39308070146048 \) Copy content Toggle raw display
$71$ \( T^{5} + 692 T^{4} + \cdots + 98341318953856 \) Copy content Toggle raw display
$73$ \( T^{5} - 1950 T^{4} + \cdots - 7201878016 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 240961986300538 \) Copy content Toggle raw display
$83$ \( T^{5} + 512 T^{4} + \cdots - 6057622580224 \) Copy content Toggle raw display
$89$ \( T^{5} + 866 T^{4} + \cdots - 21549994365568 \) Copy content Toggle raw display
$97$ \( T^{5} + 1562 T^{4} + \cdots + 20480102175488 \) Copy content Toggle raw display
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