Properties

Label 637.4.a.d
Level $637$
Weight $4$
Character orbit 637.a
Self dual yes
Analytic conductor $37.584$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.5842166737\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5364412.1
Defining polynomial: \( x^{4} - 27x^{2} - 24x + 76 \) 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 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + (\beta_{3} - \beta_1 + 7) q^{4} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 10) q^{5} + (2 \beta_{3} - 3 \beta_{2} + \beta_1 + 13) q^{6} + ( - \beta_{3} + 4 \beta_{2} + 5 \beta_1 - 9) q^{8} + (2 \beta_{3} + \beta_{2} + \beta_1 + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + (\beta_{3} - \beta_1 + 7) q^{4} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 10) q^{5} + (2 \beta_{3} - 3 \beta_{2} + \beta_1 + 13) q^{6} + ( - \beta_{3} + 4 \beta_{2} + 5 \beta_1 - 9) q^{8} + (2 \beta_{3} + \beta_{2} + \beta_1 + 6) q^{9} + ( - \beta_{3} + 10 \beta_{2} + 16 \beta_1 + 8) q^{10} + (2 \beta_{3} + \beta_{2} - 5 \beta_1 - 23) q^{11} + ( - 2 \beta_{3} + 9 \beta_{2} + 17 \beta_1 + 1) q^{12} + 13 q^{13} + (2 \beta_{3} + 4 \beta_{2} + 28 \beta_1 - 4) q^{15} + (\beta_{3} - 16 \beta_{2} - 7 \beta_1 + 19) q^{16} + (2 \beta_{2} - 20 \beta_1 + 36) q^{17} + (2 \beta_{3} + 5 \beta_{2} + 18 \beta_1 + 16) q^{18} + (3 \beta_{3} - 10 \beta_{2} - 9 \beta_1 + 16) q^{19} + (18 \beta_{3} - 18 \beta_{2} - 6 \beta_1 + 132) q^{20} + ( - 4 \beta_{3} + 5 \beta_{2} - 11 \beta_1 - 39) q^{22} + ( - \beta_{3} - 7 \beta_{2} - 18 \beta_1 - 29) q^{23} + (10 \beta_{3} - 11 \beta_{2} - 19 \beta_1 + 125) q^{24} + (19 \beta_{3} - 4 \beta_{2} + 11 \beta_1 + 137) q^{25} + (13 \beta_1 - 13) q^{26} + (2 \beta_{3} - 21 \beta_{2} + 3 \beta_1 + 27) q^{27} + ( - 9 \beta_{3} - 18 \beta_{2} + 29 \beta_1 - 110) q^{29} + (32 \beta_{3} - 4 \beta_{2} + 8 \beta_1 + 404) q^{30} + (9 \beta_{3} + \beta_{2} - 30 \beta_1 + 75) q^{31} + ( - 15 \beta_{3} + 20 \beta_{2} - 15 \beta_1 - 41) q^{32} + ( - 10 \beta_{3} - 11 \beta_{2} - 11 \beta_1 - 59) q^{33} + ( - 18 \beta_{3} - 6 \beta_{2} + 36 \beta_1 - 316) q^{34} + (7 \beta_{3} - 15 \beta_{2} + 20 \beta_1 + 196) q^{36} + ( - 18 \beta_{3} + 7 \beta_{2} - 39 \beta_1 - 63) q^{37} + ( - 19 \beta_{3} + 42 \beta_{2} + 34 \beta_1 - 130) q^{38} + (13 \beta_{2} + 13 \beta_1 + 13) q^{39} + ( - 16 \beta_{3} + 46 \beta_{2} + 112 \beta_1 - 208) q^{40} + ( - 2 \beta_{3} + 77 \beta_{2} + 37 \beta_1 + 3) q^{41} + (9 \beta_{3} + 8 \beta_{2} + 21 \beta_1 + 134) q^{43} + ( - 22 \beta_{3} - 39 \beta_{2} - 23 \beta_1 + 53) q^{44} + (29 \beta_{3} + 2 \beta_{2} + 41 \beta_1 + 206) q^{45} + ( - 25 \beta_{3} + 17 \beta_{2} - 35 \beta_1 - 227) q^{46} + (7 \beta_{3} - 31 \beta_{2} + 4 \beta_1 + 207) q^{47} + ( - 14 \beta_{3} + \beta_{2} + 49 \beta_1 - 359) q^{48} + (7 \beta_{3} + 88 \beta_{2} + 251 \beta_1 + 93) q^{50} + ( - 40 \beta_{3} + 80 \beta_{2} - 8 \beta_1 - 208) q^{51} + (13 \beta_{3} - 13 \beta_1 + 91) q^{52} + (27 \beta_{3} - 54 \beta_{2} - 59 \beta_1 - 58) q^{53} + ( - 18 \beta_{3} + 71 \beta_{2} + 39 \beta_1 + 23) q^{54} + (12 \beta_{2} - 90 \beta_1 - 192) q^{55} + ( - 18 \beta_{3} + 14 \beta_{2} + 54 \beta_1 - 266) q^{57} + (11 \beta_{3} + 18 \beta_{2} - 164 \beta_1 + 480) q^{58} + ( - 48 \beta_{3} + 50 \beta_{2} + 26 \beta_1 + 194) q^{59} + ( - 12 \beta_{3} + 108 \beta_{2} + 372 \beta_1 - 132) q^{60} + (6 \beta_{3} + 13 \beta_{2} + 9 \beta_1 + 35) q^{61} + ( - 29 \beta_{3} + 33 \beta_{2} + 129 \beta_1 - 459) q^{62} + ( - 3 \beta_{3} + 8 \beta_{2} - 75 \beta_1 - 381) q^{64} + (13 \beta_{3} - 26 \beta_{2} + 13 \beta_1 + 130) q^{65} + ( - 22 \beta_{3} - 7 \beta_{2} - 119 \beta_1 - 135) q^{66} + ( - 8 \beta_{3} - 31 \beta_{2} + 11 \beta_1 - 127) q^{67} + (30 \beta_{3} - 70 \beta_{2} - 264 \beta_1 + 460) q^{68} + ( - 36 \beta_{3} - 7 \beta_{2} - 63 \beta_1 - 415) q^{69} + (18 \beta_{3} + 32 \beta_{2} + 32 \beta_1 + 368) q^{71} + ( - 11 \beta_{3} + 33 \beta_{2} + 94 \beta_1 - 16) q^{72} + ( - 75 \beta_{3} + 5 \beta_{2} + 78 \beta_1 - 27) q^{73} + ( - 32 \beta_{3} - 93 \beta_{2} - 171 \beta_1 - 555) q^{74} + (22 \beta_{3} + 107 \beta_{2} + 395 \beta_1 + 371) q^{75} + (52 \beta_{3} - 122 \beta_{2} - 172 \beta_1 + 402) q^{76} + (26 \beta_{3} - 39 \beta_{2} + 13 \beta_1 + 169) q^{78} + ( - 37 \beta_{3} + 3 \beta_{2} + 64 \beta_1 + 237) q^{79} + (14 \beta_{3} - 58 \beta_{2} - 256 \beta_1 + 656) q^{80} + ( - 48 \beta_{3} - 48 \beta_{2} + 72 \beta_1 - 455) q^{81} + (114 \beta_{3} - 239 \beta_{2} - 9 \beta_1 + 507) q^{82} + ( - 47 \beta_{3} - 48 \beta_{2} - 13 \beta_1 + 286) q^{83} + (38 \beta_{3} - 236 \beta_{2} - 284 \beta_1 - 64) q^{85} + (29 \beta_{3} + 12 \beta_{2} + 188 \beta_1 + 196) q^{86} + (58 \beta_{3} - 204 \beta_{2} - 124 \beta_1 - 100) q^{87} + ( - 30 \beta_{3} - 11 \beta_{2} + 9 \beta_1 - 151) q^{88} + ( - 33 \beta_{3} + 16 \beta_{2} - 7 \beta_1 + 720) q^{89} + (43 \beta_{3} + 110 \beta_{2} + 380 \beta_1 + 484) q^{90} + ( - 10 \beta_{3} - 95 \beta_{2} - 233 \beta_1 - 131) q^{92} + ( - 60 \beta_{3} + 137 \beta_{2} + 121 \beta_1 - 255) q^{93} + ( - 27 \beta_{3} + 121 \beta_{2} + 249 \beta_1 - 123) q^{94} + (39 \beta_{3} - 72 \beta_{2} - 225 \beta_1 + 558) q^{95} + ( - 30 \beta_{3} + 29 \beta_{2} - 291 \beta_1 - 11) q^{96} + (13 \beta_{3} + 83 \beta_{2} + 224 \beta_1 - 693) q^{97} + ( - 76 \beta_{3} - 86 \beta_{2} - 44 \beta_1 + 130) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 5 q^{3} + 26 q^{4} + 36 q^{5} + 45 q^{6} - 30 q^{8} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 5 q^{3} + 26 q^{4} + 36 q^{5} + 45 q^{6} - 30 q^{8} + 21 q^{9} + 44 q^{10} - 95 q^{11} + 17 q^{12} + 52 q^{13} - 16 q^{15} + 58 q^{16} + 146 q^{17} + 65 q^{18} + 48 q^{19} + 474 q^{20} - 143 q^{22} - 121 q^{23} + 469 q^{24} + 506 q^{25} - 52 q^{26} + 83 q^{27} - 440 q^{29} + 1548 q^{30} + 283 q^{31} - 114 q^{32} - 227 q^{33} - 1234 q^{34} + 755 q^{36} - 209 q^{37} - 440 q^{38} + 65 q^{39} - 754 q^{40} + 93 q^{41} + 526 q^{43} + 217 q^{44} + 768 q^{45} - 841 q^{46} + 783 q^{47} - 1407 q^{48} + 446 q^{50} - 672 q^{51} + 338 q^{52} - 340 q^{53} + 199 q^{54} - 756 q^{55} - 1014 q^{57} + 1916 q^{58} + 922 q^{59} - 396 q^{60} + 141 q^{61} - 1745 q^{62} - 1510 q^{64} + 468 q^{65} - 503 q^{66} - 523 q^{67} + 1710 q^{68} - 1595 q^{69} + 1468 q^{71} - 9 q^{72} + 47 q^{73} - 2249 q^{74} + 1547 q^{75} + 1382 q^{76} + 585 q^{78} + 1025 q^{79} + 2538 q^{80} - 1772 q^{81} + 1561 q^{82} + 1190 q^{83} - 568 q^{85} + 738 q^{86} - 720 q^{87} - 555 q^{88} + 2962 q^{89} + 1960 q^{90} - 599 q^{92} - 763 q^{93} - 317 q^{94} + 2082 q^{95} + 45 q^{96} - 2715 q^{97} + 586 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 27x^{2} - 24x + 76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 19\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 14 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 4\beta_{2} + 21\beta _1 + 18 \) 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
−4.05873
−2.63459
1.32361
5.36970
−5.05873 −6.23157 17.5908 18.8190 31.5238 0 −48.5170 11.8325 −95.2001
1.2 −3.63459 5.33748 5.21021 −11.0031 −19.3995 0 10.1397 1.48874 39.9917
1.3 0.323612 −1.75980 −7.89528 5.91876 −0.569491 0 −5.14391 −23.9031 1.91538
1.4 4.36970 7.65388 11.0943 22.2654 33.4452 0 13.5212 31.5819 97.2930
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(-1\)

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 637.4.a.d 4
7.b odd 2 1 91.4.a.b 4
21.c even 2 1 819.4.a.h 4
28.d even 2 1 1456.4.a.s 4
35.c odd 2 1 2275.4.a.h 4
91.b odd 2 1 1183.4.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.4.a.b 4 7.b odd 2 1
637.4.a.d 4 1.a even 1 1 trivial
819.4.a.h 4 21.c even 2 1
1183.4.a.e 4 91.b odd 2 1
1456.4.a.s 4 28.d even 2 1
2275.4.a.h 4 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(637))\):

\( T_{2}^{4} + 4T_{2}^{3} - 21T_{2}^{2} - 74T_{2} + 26 \) Copy content Toggle raw display
\( T_{3}^{4} - 5T_{3}^{3} - 52T_{3}^{2} + 184T_{3} + 448 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4 T^{3} - 21 T^{2} - 74 T + 26 \) Copy content Toggle raw display
$3$ \( T^{4} - 5 T^{3} - 52 T^{2} + 184 T + 448 \) Copy content Toggle raw display
$5$ \( T^{4} - 36 T^{3} + 145 T^{2} + \cdots - 27288 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 95 T^{3} + 2128 T^{2} + \cdots - 151632 \) Copy content Toggle raw display
$13$ \( (T - 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 146 T^{3} - 3120 T^{2} + \cdots - 1065472 \) Copy content Toggle raw display
$19$ \( T^{4} - 48 T^{3} - 5327 T^{2} + \cdots + 317232 \) Copy content Toggle raw display
$23$ \( T^{4} + 121 T^{3} - 5241 T^{2} + \cdots - 2384104 \) Copy content Toggle raw display
$29$ \( T^{4} + 440 T^{3} + \cdots - 484339768 \) Copy content Toggle raw display
$31$ \( T^{4} - 283 T^{3} - 3281 T^{2} + \cdots - 1026856 \) Copy content Toggle raw display
$37$ \( T^{4} + 209 T^{3} + \cdots + 328158128 \) Copy content Toggle raw display
$41$ \( T^{4} - 93 T^{3} + \cdots + 12096773224 \) Copy content Toggle raw display
$43$ \( T^{4} - 526 T^{3} + \cdots + 18583856 \) Copy content Toggle raw display
$47$ \( T^{4} - 783 T^{3} + \cdots - 1054241384 \) Copy content Toggle raw display
$53$ \( T^{4} + 340 T^{3} + \cdots - 11218230832 \) Copy content Toggle raw display
$59$ \( T^{4} - 922 T^{3} + \cdots + 10047112192 \) Copy content Toggle raw display
$61$ \( T^{4} - 141 T^{3} - 9038 T^{2} + \cdots + 3710376 \) Copy content Toggle raw display
$67$ \( T^{4} + 523 T^{3} + \cdots - 951710544 \) Copy content Toggle raw display
$71$ \( T^{4} - 1468 T^{3} + \cdots + 2887158784 \) Copy content Toggle raw display
$73$ \( T^{4} - 47 T^{3} + \cdots + 38124898514 \) Copy content Toggle raw display
$79$ \( T^{4} - 1025 T^{3} + \cdots - 13183278632 \) Copy content Toggle raw display
$83$ \( T^{4} - 1190 T^{3} + \cdots - 11400717312 \) Copy content Toggle raw display
$89$ \( T^{4} - 2962 T^{3} + \cdots + 205066944356 \) Copy content Toggle raw display
$97$ \( T^{4} + 2715 T^{3} + \cdots - 914822530202 \) Copy content Toggle raw display
show more
show less