Properties

Label 37.4.a.b
Level $37$
Weight $4$
Character orbit 37.a
Self dual yes
Analytic conductor $2.183$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 37.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.18307067021\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - x^{4} - 27 x^{3} + 3 x^{2} + 176 x + 144\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 3 - \beta_{2} ) q^{3} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{5} + ( 3 - \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{6} + ( 4 + 3 \beta_{1} + \beta_{2} - \beta_{4} ) q^{7} + ( 3 + \beta_{1} + 3 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} ) q^{8} + ( 9 + 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 3 - \beta_{2} ) q^{3} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{5} + ( 3 - \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{6} + ( 4 + 3 \beta_{1} + \beta_{2} - \beta_{4} ) q^{7} + ( 3 + \beta_{1} + 3 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} ) q^{8} + ( 9 + 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{9} + ( -14 + \beta_{2} + \beta_{3} - 5 \beta_{4} ) q^{10} + ( 12 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 6 \beta_{4} ) q^{11} + ( -11 - \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{12} + ( -3 - 5 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} + 7 \beta_{4} ) q^{13} + ( -24 - 4 \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} ) q^{14} + ( -23 + 5 \beta_{1} + 3 \beta_{2} + \beta_{3} - 10 \beta_{4} ) q^{15} + ( -37 - 7 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} ) q^{16} + ( 18 - 6 \beta_{1} + 6 \beta_{2} - 16 \beta_{3} + 2 \beta_{4} ) q^{17} + ( -27 - 7 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} + 15 \beta_{4} ) q^{18} + ( -10 - 10 \beta_{1} + 14 \beta_{2} - 8 \beta_{3} - 6 \beta_{4} ) q^{19} + ( -18 + 8 \beta_{1} + 7 \beta_{2} - 5 \beta_{3} - \beta_{4} ) q^{20} + ( -10 + 9 \beta_{1} - \beta_{2} - 6 \beta_{3} - 13 \beta_{4} ) q^{21} + ( -8 - 18 \beta_{1} - 22 \beta_{2} + 2 \beta_{3} + 11 \beta_{4} ) q^{22} + ( 19 + 9 \beta_{1} + 8 \beta_{2} + 28 \beta_{3} + 9 \beta_{4} ) q^{23} + ( -27 + 25 \beta_{1} - 6 \beta_{2} - 14 \beta_{3} - 11 \beta_{4} ) q^{24} + ( 18 - 9 \beta_{1} - 14 \beta_{2} + 22 \beta_{3} + 17 \beta_{4} ) q^{25} + ( 49 + 13 \beta_{1} - 28 \beta_{2} + 8 \beta_{3} + 11 \beta_{4} ) q^{26} + ( 75 + 24 \beta_{1} + 8 \beta_{2} - 33 \beta_{3} - 9 \beta_{4} ) q^{27} + ( -6 + 8 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{28} + ( 57 + 11 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 25 \beta_{4} ) q^{29} + ( -51 + 39 \beta_{1} + 27 \beta_{2} + 7 \beta_{3} - 20 \beta_{4} ) q^{30} + ( 72 - 18 \beta_{1} + 13 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{31} + ( 15 + 25 \beta_{1} + 9 \beta_{2} + 21 \beta_{3} - 2 \beta_{4} ) q^{32} + ( 53 - 50 \beta_{1} - 24 \beta_{2} - 5 \beta_{3} + 21 \beta_{4} ) q^{33} + ( 42 - 66 \beta_{1} + 22 \beta_{2} - 42 \beta_{3} ) q^{34} + ( 126 - 8 \beta_{2} + 14 \beta_{3} + 18 \beta_{4} ) q^{35} + ( -35 + 5 \beta_{1} - 22 \beta_{2} + 26 \beta_{3} + 15 \beta_{4} ) q^{36} -37 q^{37} + ( 86 - 22 \beta_{1} + 50 \beta_{2} - 54 \beta_{3} - 40 \beta_{4} ) q^{38} + ( 8 - 80 \beta_{1} + \beta_{2} + 35 \beta_{3} + 43 \beta_{4} ) q^{39} + ( 6 - 4 \beta_{1} - \beta_{2} - 25 \beta_{3} + 23 \beta_{4} ) q^{40} + ( 101 + 8 \beta_{1} - 43 \beta_{2} + 18 \beta_{3} - 10 \beta_{4} ) q^{41} + ( -86 + 26 \beta_{1} + 35 \beta_{2} + 19 \beta_{3} - 4 \beta_{4} ) q^{42} + ( -72 - 22 \beta_{1} - 12 \beta_{3} - 50 \beta_{4} ) q^{43} + ( 58 + 26 \beta_{1} - 23 \beta_{2} + 31 \beta_{3} + 27 \beta_{4} ) q^{44} + ( -161 + 29 \beta_{1} + 21 \beta_{2} - 29 \beta_{3} - 28 \beta_{4} ) q^{45} + ( -33 + 3 \beta_{1} - 56 \beta_{2} + 4 \beta_{3} - 43 \beta_{4} ) q^{46} + ( 50 + 37 \beta_{1} - 5 \beta_{2} - 6 \beta_{3} - 13 \beta_{4} ) q^{47} + ( -195 + 41 \beta_{1} + 48 \beta_{2} + 24 \beta_{3} - 3 \beta_{4} ) q^{48} + ( -193 + 19 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} + 15 \beta_{4} ) q^{49} + ( 118 + 20 \beta_{1} - 78 \beta_{2} + 38 \beta_{3} + 37 \beta_{4} ) q^{50} + ( -6 + 6 \beta_{1} - 48 \beta_{2} - 28 \beta_{3} + 26 \beta_{4} ) q^{51} + ( -63 + 41 \beta_{1} - 50 \beta_{2} + 30 \beta_{3} + 31 \beta_{4} ) q^{52} + ( -16 - 39 \beta_{1} + 57 \beta_{2} - 64 \beta_{3} + 33 \beta_{4} ) q^{53} + ( -213 - 139 \beta_{1} + 44 \beta_{2} - 24 \beta_{3} ) q^{54} + ( -227 + 111 \beta_{1} + 80 \beta_{2} - 30 \beta_{3} - 39 \beta_{4} ) q^{55} + ( 102 + 18 \beta_{1} - 6 \beta_{2} - 26 \beta_{3} + 10 \beta_{4} ) q^{56} + ( -322 + 26 \beta_{1} + 20 \beta_{2} + 40 \beta_{3} + 6 \beta_{4} ) q^{57} + ( -11 - 7 \beta_{1} + 64 \beta_{2} + 44 \beta_{3} - 9 \beta_{4} ) q^{58} + ( 54 - 82 \beta_{1} - 52 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} ) q^{59} + ( -203 + 11 \beta_{1} + 17 \beta_{2} - 23 \beta_{3} - 28 \beta_{4} ) q^{60} + ( -338 + 88 \beta_{1} + 13 \beta_{2} + 75 \beta_{3} + 15 \beta_{4} ) q^{61} + ( 256 - 94 \beta_{1} + 25 \beta_{2} - 55 \beta_{3} - 41 \beta_{4} ) q^{62} + ( -4 + 66 \beta_{1} - 4 \beta_{2} - 54 \beta_{3} - 52 \beta_{4} ) q^{63} + ( 107 + 69 \beta_{1} - 71 \beta_{2} + 45 \beta_{3} - 2 \beta_{4} ) q^{64} + ( -193 + 61 \beta_{1} + 49 \beta_{2} + 13 \beta_{3} - 58 \beta_{4} ) q^{65} + ( 501 - 57 \beta_{1} - 32 \beta_{2} - 4 \beta_{3} + 98 \beta_{4} ) q^{66} + ( 114 + 156 \beta_{1} - 27 \beta_{2} - 39 \beta_{3} + 37 \beta_{4} ) q^{67} + ( 474 - 122 \beta_{1} + 82 \beta_{2} - 46 \beta_{3} - 40 \beta_{4} ) q^{68} + ( -400 - 180 \beta_{1} + 27 \beta_{2} + 109 \beta_{3} + 9 \beta_{4} ) q^{69} + ( 118 - 118 \beta_{1} - 76 \beta_{2} + 20 \beta_{3} + 28 \beta_{4} ) q^{70} + ( 384 - 171 \beta_{1} - 35 \beta_{2} + 62 \beta_{3} + 23 \beta_{4} ) q^{71} + ( 153 + 157 \beta_{1} - 66 \beta_{2} - 14 \beta_{3} - 65 \beta_{4} ) q^{72} + ( -34 + 23 \beta_{1} - 100 \beta_{2} + 5 \beta_{3} + 44 \beta_{4} ) q^{73} + ( -37 + 37 \beta_{1} ) q^{74} + ( 193 - 186 \beta_{1} - 22 \beta_{2} + 73 \beta_{3} + 95 \beta_{4} ) q^{75} + ( 358 - 134 \beta_{1} + 134 \beta_{2} - 122 \beta_{3} - 88 \beta_{4} ) q^{76} + ( -124 + 121 \beta_{1} + 81 \beta_{2} - 10 \beta_{3} - 25 \beta_{4} ) q^{77} + ( 792 - 26 \beta_{1} - 83 \beta_{2} - 91 \beta_{3} + 5 \beta_{4} ) q^{78} + ( 75 + 87 \beta_{1} + 26 \beta_{2} - 20 \beta_{3} + 15 \beta_{4} ) q^{79} + ( 94 - 164 \beta_{1} - 97 \beta_{2} - 9 \beta_{3} + 59 \beta_{4} ) q^{80} + ( 42 + 99 \beta_{1} - 7 \beta_{2} - 99 \beta_{3} - 108 \beta_{4} ) q^{81} + ( 77 + 41 \beta_{1} - 39 \beta_{2} + 165 \beta_{3} + 101 \beta_{4} ) q^{82} + ( 530 - 53 \beta_{1} - 41 \beta_{2} + 92 \beta_{3} + 91 \beta_{4} ) q^{83} + ( -220 - 10 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} - 24 \beta_{4} ) q^{84} + ( -560 + 108 \beta_{1} + 106 \beta_{2} - 146 \beta_{3} - 22 \beta_{4} ) q^{85} + ( 224 + 148 \beta_{1} + 184 \beta_{2} + 16 \beta_{3} - 38 \beta_{4} ) q^{86} + ( 432 + 260 \beta_{1} - 19 \beta_{2} - 61 \beta_{3} - 133 \beta_{4} ) q^{87} + ( -130 + 140 \beta_{1} + 15 \beta_{2} + 83 \beta_{3} - 23 \beta_{4} ) q^{88} + ( -452 - 188 \beta_{1} - 20 \beta_{2} - 22 \beta_{3} - 22 \beta_{4} ) q^{89} + ( -453 + 117 \beta_{1} + 105 \beta_{2} - 35 \beta_{3} - 62 \beta_{4} ) q^{90} + ( -340 + 6 \beta_{1} + 64 \beta_{2} + 46 \beta_{3} - 20 \beta_{4} ) q^{91} + ( -121 + 167 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + 49 \beta_{4} ) q^{92} + ( -161 - 75 \beta_{1} - 55 \beta_{2} + 105 \beta_{3} + 58 \beta_{4} ) q^{93} + ( -306 - 26 \beta_{1} + 3 \beta_{2} + 59 \beta_{3} + 8 \beta_{4} ) q^{94} + ( 176 - 92 \beta_{1} - 50 \beta_{2} - 14 \beta_{3} + 78 \beta_{4} ) q^{95} + ( -335 - 47 \beta_{1} + 40 \beta_{2} + 36 \beta_{3} - 83 \beta_{4} ) q^{96} + ( -570 - 42 \beta_{1} + 70 \beta_{2} - 200 \beta_{3} - 54 \beta_{4} ) q^{97} + ( -405 + 181 \beta_{1} - 73 \beta_{2} + 23 \beta_{3} + 26 \beta_{4} ) q^{98} + ( 413 - 179 \beta_{1} - 75 \beta_{2} + 31 \beta_{3} + 72 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 4q^{2} + 13q^{3} + 18q^{4} + 11q^{5} + 9q^{6} + 24q^{7} + 30q^{8} + 46q^{9} + O(q^{10}) \) \( 5q + 4q^{2} + 13q^{3} + 18q^{4} + 11q^{5} + 9q^{6} + 24q^{7} + 30q^{8} + 46q^{9} - 75q^{10} + 61q^{11} - 65q^{12} - 37q^{13} - 128q^{14} - 116q^{15} - 182q^{16} + 130q^{17} - 159q^{18} - 22q^{19} - 59q^{20} - 44q^{21} - 95q^{22} + 73q^{23} - 105q^{24} + 26q^{25} + 197q^{26} + 472q^{27} - 2q^{28} + 271q^{29} - 196q^{30} + 363q^{31} + 74q^{32} + 198q^{33} + 272q^{34} + 604q^{35} - 251q^{36} - 185q^{37} + 576q^{38} - 65q^{39} + 97q^{40} + 381q^{41} - 376q^{42} - 408q^{43} + 235q^{44} - 704q^{45} - 325q^{46} + 276q^{47} - 889q^{48} - 949q^{49} + 415q^{50} - 38q^{51} - 403q^{52} + 156q^{53} - 1068q^{54} - 843q^{55} + 578q^{56} - 1618q^{57} - 31q^{58} + 100q^{59} - 952q^{60} - 1711q^{61} + 1305q^{62} + 94q^{63} + 370q^{64} - 890q^{65} + 2490q^{66} + 787q^{67} + 2464q^{68} - 2335q^{69} + 308q^{70} + 1578q^{71} + 753q^{72} - 313q^{73} - 148q^{74} + 684q^{75} + 2080q^{76} - 342q^{77} + 3955q^{78} + 569q^{79} + 189q^{80} + 385q^{81} + 119q^{82} + 2422q^{83} - 1098q^{84} - 2210q^{85} + 1566q^{86} + 2371q^{87} - 669q^{88} - 2466q^{89} - 1930q^{90} - 1678q^{91} - 373q^{92} - 1142q^{93} - 1660q^{94} + 794q^{95} - 1797q^{96} - 2406q^{97} - 2010q^{98} + 1746q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 27 x^{3} + 3 x^{2} + 176 x + 144\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 15 \nu^{2} + 3 \nu + 12 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 23 \nu^{2} + 11 \nu + 92 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{4} + 5 \nu^{3} + 23 \nu^{2} - 71 \nu - 140 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 10\)
\(\nu^{3}\)\(=\)\(2 \beta_{4} + 2 \beta_{3} + 15 \beta_{1} + 12\)
\(\nu^{4}\)\(=\)\(2 \beta_{4} - 13 \beta_{3} + 23 \beta_{2} + 27 \beta_{1} + 150\)

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
4.59736
3.56768
−0.960270
−2.40805
−3.79672
−3.59736 −4.28827 4.94099 11.8788 15.4264 17.7262 11.0044 −8.61076 −42.7323
1.2 −2.56768 9.45274 −1.40700 1.50802 −24.2716 12.6891 24.1542 62.3543 −3.87211
1.3 1.96027 3.37210 −4.15734 14.4391 6.61023 7.73331 −23.8317 −15.6289 28.3045
1.4 3.40805 7.32702 3.61479 −17.2892 24.9708 −15.1635 −14.9450 26.6852 −58.9226
1.5 4.79672 −2.86359 15.0086 0.463335 −13.7359 1.01485 33.6181 −18.7998 2.22249
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(37\) \(1\)

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 37.4.a.b 5
3.b odd 2 1 333.4.a.f 5
4.b odd 2 1 592.4.a.g 5
5.b even 2 1 925.4.a.b 5
7.b odd 2 1 1813.4.a.c 5
8.b even 2 1 2368.4.a.m 5
8.d odd 2 1 2368.4.a.r 5
37.b even 2 1 1369.4.a.d 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.4.a.b 5 1.a even 1 1 trivial
333.4.a.f 5 3.b odd 2 1
592.4.a.g 5 4.b odd 2 1
925.4.a.b 5 5.b even 2 1
1369.4.a.d 5 37.b even 2 1
1813.4.a.c 5 7.b odd 2 1
2368.4.a.m 5 8.b even 2 1
2368.4.a.r 5 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 4 T_{2}^{4} - 21 T_{2}^{3} + 74 T_{2}^{2} + 102 T_{2} - 296 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(37))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -296 + 102 T + 74 T^{2} - 21 T^{3} - 4 T^{4} + T^{5} \)
$3$ \( -2868 - 125 T + 419 T^{2} - 6 T^{3} - 13 T^{4} + T^{5} \)
$5$ \( 2072 - 6044 T + 3518 T^{2} - 265 T^{3} - 11 T^{4} + T^{5} \)
$7$ \( 26768 - 31692 T + 5358 T^{2} - 95 T^{3} - 24 T^{4} + T^{5} \)
$11$ \( 5135996 + 1640601 T + 68919 T^{2} - 1604 T^{3} - 61 T^{4} + T^{5} \)
$13$ \( -7098184 + 3328324 T - 114346 T^{2} - 4705 T^{3} + 37 T^{4} + T^{5} \)
$17$ \( 1123875456 - 78208768 T + 1700232 T^{2} - 7572 T^{3} - 130 T^{4} + T^{5} \)
$19$ \( -69604224 - 21414080 T - 1114968 T^{2} - 15140 T^{3} + 22 T^{4} + T^{5} \)
$23$ \( 14836947832 + 545396084 T + 2380382 T^{2} - 53485 T^{3} - 73 T^{4} + T^{5} \)
$29$ \( -45459799656 + 355008724 T + 8240054 T^{2} - 31001 T^{3} - 271 T^{4} + T^{5} \)
$31$ \( 11228647136 - 360398496 T + 2259006 T^{2} + 28525 T^{3} - 363 T^{4} + T^{5} \)
$37$ \( ( 37 + T )^{5} \)
$41$ \( -762162331986 + 2709102209 T + 36452569 T^{2} - 97656 T^{3} - 381 T^{4} + T^{5} \)
$43$ \( -965840235296 + 13076853312 T - 29526168 T^{2} - 180756 T^{3} + 408 T^{4} + T^{5} \)
$47$ \( -11399037456 + 92096020 T + 3688882 T^{2} - 21719 T^{3} - 276 T^{4} + T^{5} \)
$53$ \( -12174094047032 + 45944762364 T + 146651062 T^{2} - 557343 T^{3} - 156 T^{4} + T^{5} \)
$59$ \( 199509626624 - 7523341760 T + 92816520 T^{2} - 371968 T^{3} - 100 T^{4} + T^{5} \)
$61$ \( 5262207537472 - 100881205264 T - 180371922 T^{2} + 670783 T^{3} + 1711 T^{4} + T^{5} \)
$67$ \( -69671300293888 + 224685531712 T + 539987068 T^{2} - 883479 T^{3} - 787 T^{4} + T^{5} \)
$71$ \( -105494295472656 - 74998874428 T + 763989184 T^{2} - 2235 T^{3} - 1578 T^{4} + T^{5} \)
$73$ \( 24654270137062 + 100663626345 T - 158432169 T^{2} - 724148 T^{3} + 313 T^{4} + T^{5} \)
$79$ \( 429381365248 + 9143394496 T + 34996590 T^{2} - 207829 T^{3} - 569 T^{4} + T^{5} \)
$83$ \( 8663698210944 - 203104302320 T + 28555740 T^{2} + 1554649 T^{3} - 2422 T^{4} + T^{5} \)
$89$ \( -2194912910336 - 92968874496 T - 103478304 T^{2} + 1325320 T^{3} + 2466 T^{4} + T^{5} \)
$97$ \( 108205552171136 - 387360172992 T - 1160056152 T^{2} + 651324 T^{3} + 2406 T^{4} + T^{5} \)
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