Properties

Label 637.4.a.b
Level $637$
Weight $4$
Character orbit 637.a
Self dual yes
Analytic conductor $37.584$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (3 \beta - 4) q^{3} + (\beta - 4) q^{4} + ( - \beta + 2) q^{5} + ( - \beta + 12) q^{6} + ( - 11 \beta + 4) q^{8} + ( - 15 \beta + 25) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (3 \beta - 4) q^{3} + (\beta - 4) q^{4} + ( - \beta + 2) q^{5} + ( - \beta + 12) q^{6} + ( - 11 \beta + 4) q^{8} + ( - 15 \beta + 25) q^{9} + (\beta - 4) q^{10} + (12 \beta + 34) q^{11} + ( - 13 \beta + 28) q^{12} + 13 q^{13} + (7 \beta - 20) q^{15} + ( - 15 \beta - 12) q^{16} + (17 \beta - 18) q^{17} + (10 \beta - 60) q^{18} + (32 \beta + 26) q^{19} + (5 \beta - 12) q^{20} + (46 \beta + 48) q^{22} + ( - 12 \beta + 104) q^{23} + (23 \beta - 148) q^{24} + ( - 3 \beta - 117) q^{25} + 13 \beta q^{26} + (9 \beta - 172) q^{27} + (96 \beta - 70) q^{29} + ( - 13 \beta + 28) q^{30} + (34 \beta + 26) q^{31} + (61 \beta - 92) q^{32} + (90 \beta + 8) q^{33} + ( - \beta + 68) q^{34} + (70 \beta - 160) q^{36} + (5 \beta + 102) q^{37} + (58 \beta + 128) q^{38} + (39 \beta - 52) q^{39} + ( - 15 \beta + 52) q^{40} + ( - 22 \beta + 126) q^{41} + (143 \beta + 72) q^{43} + ( - 2 \beta - 88) q^{44} + ( - 40 \beta + 110) q^{45} + (92 \beta - 48) q^{46} + (121 \beta - 278) q^{47} + ( - 21 \beta - 132) q^{48} + ( - 120 \beta - 12) q^{50} + ( - 71 \beta + 276) q^{51} + (13 \beta - 52) q^{52} + (30 \beta - 74) q^{53} + ( - 163 \beta + 36) q^{54} + ( - 22 \beta + 20) q^{55} + (46 \beta + 280) q^{57} + (26 \beta + 384) q^{58} + ( - 124 \beta + 246) q^{59} + ( - 41 \beta + 108) q^{60} + (190 \beta + 434) q^{61} + (60 \beta + 136) q^{62} + (89 \beta + 340) q^{64} + ( - 13 \beta + 26) q^{65} + (98 \beta + 360) q^{66} + ( - 232 \beta + 150) q^{67} + ( - 69 \beta + 140) q^{68} + (324 \beta - 560) q^{69} + ( - 231 \beta + 50) q^{71} + ( - 170 \beta + 760) q^{72} + ( - 260 \beta - 98) q^{73} + (107 \beta + 20) q^{74} + ( - 348 \beta + 432) q^{75} + ( - 70 \beta + 24) q^{76} + ( - 13 \beta + 156) q^{78} + (40 \beta - 524) q^{79} + ( - 3 \beta + 36) q^{80} + ( - 120 \beta + 121) q^{81} + (104 \beta - 88) q^{82} + (182 \beta - 1070) q^{83} + (35 \beta - 104) q^{85} + (215 \beta + 572) q^{86} + ( - 306 \beta + 1432) q^{87} + ( - 458 \beta - 392) q^{88} + (388 \beta + 166) q^{89} + (70 \beta - 160) q^{90} + (140 \beta - 464) q^{92} + (44 \beta + 304) q^{93} + ( - 157 \beta + 484) q^{94} + (6 \beta - 76) q^{95} + ( - 337 \beta + 1100) q^{96} + ( - 508 \beta + 718) q^{97} + ( - 390 \beta + 130) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} - 3 q^{8} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} - 3 q^{8} + 35 q^{9} - 7 q^{10} + 80 q^{11} + 43 q^{12} + 26 q^{13} - 33 q^{15} - 39 q^{16} - 19 q^{17} - 110 q^{18} + 84 q^{19} - 19 q^{20} + 142 q^{22} + 196 q^{23} - 273 q^{24} - 237 q^{25} + 13 q^{26} - 335 q^{27} - 44 q^{29} + 43 q^{30} + 86 q^{31} - 123 q^{32} + 106 q^{33} + 135 q^{34} - 250 q^{36} + 209 q^{37} + 314 q^{38} - 65 q^{39} + 89 q^{40} + 230 q^{41} + 287 q^{43} - 178 q^{44} + 180 q^{45} - 4 q^{46} - 435 q^{47} - 285 q^{48} - 144 q^{50} + 481 q^{51} - 91 q^{52} - 118 q^{53} - 91 q^{54} + 18 q^{55} + 606 q^{57} + 794 q^{58} + 368 q^{59} + 175 q^{60} + 1058 q^{61} + 332 q^{62} + 769 q^{64} + 39 q^{65} + 818 q^{66} + 68 q^{67} + 211 q^{68} - 796 q^{69} - 131 q^{71} + 1350 q^{72} - 456 q^{73} + 147 q^{74} + 516 q^{75} - 22 q^{76} + 299 q^{78} - 1008 q^{79} + 69 q^{80} + 122 q^{81} - 72 q^{82} - 1958 q^{83} - 173 q^{85} + 1359 q^{86} + 2558 q^{87} - 1242 q^{88} + 720 q^{89} - 250 q^{90} - 788 q^{92} + 652 q^{93} + 811 q^{94} - 146 q^{95} + 1863 q^{96} + 928 q^{97} - 130 q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
−1.56155 −8.68466 −5.56155 3.56155 13.5616 0 21.1771 48.4233 −5.56155
1.2 2.56155 3.68466 −1.43845 −0.561553 9.43845 0 −24.1771 −13.4233 −1.43845
\(n\): 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.b 2
7.b odd 2 1 13.4.a.b 2
21.c even 2 1 117.4.a.d 2
28.d even 2 1 208.4.a.h 2
35.c odd 2 1 325.4.a.f 2
35.f even 4 2 325.4.b.e 4
56.e even 2 1 832.4.a.z 2
56.h odd 2 1 832.4.a.s 2
77.b even 2 1 1573.4.a.b 2
84.h odd 2 1 1872.4.a.bb 2
91.b odd 2 1 169.4.a.g 2
91.i even 4 2 169.4.b.f 4
91.n odd 6 2 169.4.c.g 4
91.t odd 6 2 169.4.c.j 4
91.bc even 12 4 169.4.e.f 8
273.g even 2 1 1521.4.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 7.b odd 2 1
117.4.a.d 2 21.c even 2 1
169.4.a.g 2 91.b odd 2 1
169.4.b.f 4 91.i even 4 2
169.4.c.g 4 91.n odd 6 2
169.4.c.j 4 91.t odd 6 2
169.4.e.f 8 91.bc even 12 4
208.4.a.h 2 28.d even 2 1
325.4.a.f 2 35.c odd 2 1
325.4.b.e 4 35.f even 4 2
637.4.a.b 2 1.a even 1 1 trivial
832.4.a.s 2 56.h odd 2 1
832.4.a.z 2 56.e even 2 1
1521.4.a.r 2 273.g even 2 1
1573.4.a.b 2 77.b even 2 1
1872.4.a.bb 2 84.h odd 2 1

Hecke kernels

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

\( T_{2}^{2} - T_{2} - 4 \) Copy content Toggle raw display
\( T_{3}^{2} + 5T_{3} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 5T - 32 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 80T + 988 \) Copy content Toggle raw display
$13$ \( (T - 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 19T - 1138 \) Copy content Toggle raw display
$19$ \( T^{2} - 84T - 2588 \) Copy content Toggle raw display
$23$ \( T^{2} - 196T + 8992 \) Copy content Toggle raw display
$29$ \( T^{2} + 44T - 38684 \) Copy content Toggle raw display
$31$ \( T^{2} - 86T - 3064 \) Copy content Toggle raw display
$37$ \( T^{2} - 209T + 10814 \) Copy content Toggle raw display
$41$ \( T^{2} - 230T + 11168 \) Copy content Toggle raw display
$43$ \( T^{2} - 287T - 66316 \) Copy content Toggle raw display
$47$ \( T^{2} + 435T - 14918 \) Copy content Toggle raw display
$53$ \( T^{2} + 118T - 344 \) Copy content Toggle raw display
$59$ \( T^{2} - 368T - 31492 \) Copy content Toggle raw display
$61$ \( T^{2} - 1058 T + 126416 \) Copy content Toggle raw display
$67$ \( T^{2} - 68T - 227596 \) Copy content Toggle raw display
$71$ \( T^{2} + 131T - 222494 \) Copy content Toggle raw display
$73$ \( T^{2} + 456T - 235316 \) Copy content Toggle raw display
$79$ \( T^{2} + 1008 T + 247216 \) Copy content Toggle raw display
$83$ \( T^{2} + 1958 T + 817664 \) Copy content Toggle raw display
$89$ \( T^{2} - 720T - 510212 \) Copy content Toggle raw display
$97$ \( T^{2} - 928T - 881476 \) Copy content Toggle raw display
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