Properties

Label 74.4.a.d
Level $74$
Weight $4$
Character orbit 74.a
Self dual yes
Analytic conductor $4.366$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 74.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.36614134042\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 96 x^{2} - 287 x + 330\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( 1 - \beta_{1} ) q^{3} + 4 q^{4} + ( 5 + \beta_{2} ) q^{5} + ( 2 - 2 \beta_{1} ) q^{6} + ( 6 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7} + 8 q^{8} + ( 24 + 5 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{9} +O(q^{10})\) \( q + 2 q^{2} + ( 1 - \beta_{1} ) q^{3} + 4 q^{4} + ( 5 + \beta_{2} ) q^{5} + ( 2 - 2 \beta_{1} ) q^{6} + ( 6 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7} + 8 q^{8} + ( 24 + 5 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{9} + ( 10 + 2 \beta_{2} ) q^{10} + ( -19 - 3 \beta_{1} + 3 \beta_{2} + 7 \beta_{3} ) q^{11} + ( 4 - 4 \beta_{1} ) q^{12} + ( 13 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{13} + ( 12 + 6 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{14} + ( -16 - 8 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{15} + 16 q^{16} + ( 2 - 4 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{17} + ( 48 + 10 \beta_{1} - 6 \beta_{2} - 10 \beta_{3} ) q^{18} + ( -4 + 6 \beta_{1} - 14 \beta_{2} - 4 \beta_{3} ) q^{19} + ( 20 + 4 \beta_{2} ) q^{20} + ( -114 - 15 \beta_{1} - 4 \beta_{2} + 21 \beta_{3} ) q^{21} + ( -38 - 6 \beta_{1} + 6 \beta_{2} + 14 \beta_{3} ) q^{22} + ( 19 + 4 \beta_{1} - 12 \beta_{2} - 19 \beta_{3} ) q^{23} + ( 8 - 8 \beta_{1} ) q^{24} + ( -42 - 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{25} + ( 26 + 8 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{26} + ( -130 - 33 \beta_{1} + 15 \beta_{2} + 8 \beta_{3} ) q^{27} + ( 24 + 12 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{28} + ( -7 + 2 \beta_{1} + 14 \beta_{2} - 7 \beta_{3} ) q^{29} + ( -32 - 16 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} ) q^{30} + ( 13 + 18 \beta_{1} - 17 \beta_{2} - 18 \beta_{3} ) q^{31} + 32 q^{32} + ( -16 + 49 \beta_{1} - 15 \beta_{2} + 10 \beta_{3} ) q^{33} + ( 4 - 8 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{34} + ( -44 + 28 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{35} + ( 96 + 20 \beta_{1} - 12 \beta_{2} - 20 \beta_{3} ) q^{36} + 37 q^{37} + ( -8 + 12 \beta_{1} - 28 \beta_{2} - 8 \beta_{3} ) q^{38} + ( -217 - 46 \beta_{1} + 25 \beta_{2} + 14 \beta_{3} ) q^{39} + ( 40 + 8 \beta_{2} ) q^{40} + ( -49 + 17 \beta_{1} - 2 \beta_{2} + 24 \beta_{3} ) q^{41} + ( -228 - 30 \beta_{1} - 8 \beta_{2} + 42 \beta_{3} ) q^{42} + ( -146 - 36 \beta_{1} + 34 \beta_{2} + 36 \beta_{3} ) q^{43} + ( -76 - 12 \beta_{1} + 12 \beta_{2} + 28 \beta_{3} ) q^{44} + ( 156 + 46 \beta_{1} - 23 \beta_{2} - 49 \beta_{3} ) q^{45} + ( 38 + 8 \beta_{1} - 24 \beta_{2} - 38 \beta_{3} ) q^{46} + ( -10 - 9 \beta_{1} - 36 \beta_{2} - 17 \beta_{3} ) q^{47} + ( 16 - 16 \beta_{1} ) q^{48} + ( 333 + 31 \beta_{1} - 30 \beta_{2} - 69 \beta_{3} ) q^{49} + ( -84 - 4 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} ) q^{50} + ( 82 - 14 \beta_{1} + 40 \beta_{2} - 44 \beta_{3} ) q^{51} + ( 52 + 16 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{52} + ( -10 - 67 \beta_{1} + 14 \beta_{2} + 29 \beta_{3} ) q^{53} + ( -260 - 66 \beta_{1} + 30 \beta_{2} + 16 \beta_{3} ) q^{54} + ( -131 - 30 \beta_{1} + 32 \beta_{2} + 69 \beta_{3} ) q^{55} + ( 48 + 24 \beta_{1} - 16 \beta_{2} + 8 \beta_{3} ) q^{56} + ( 38 - 2 \beta_{1} - 40 \beta_{2} + 28 \beta_{3} ) q^{57} + ( -14 + 4 \beta_{1} + 28 \beta_{2} - 14 \beta_{3} ) q^{58} + ( 90 + 4 \beta_{1} - 6 \beta_{3} ) q^{59} + ( -64 - 32 \beta_{1} + 20 \beta_{2} - 4 \beta_{3} ) q^{60} + ( 259 - 60 \beta_{1} + 51 \beta_{2} + 52 \beta_{3} ) q^{61} + ( 26 + 36 \beta_{1} - 34 \beta_{2} - 36 \beta_{3} ) q^{62} + ( 306 + 198 \beta_{1} - 74 \beta_{2} - 14 \beta_{3} ) q^{63} + 64 q^{64} + ( 286 + 28 \beta_{1} - 11 \beta_{2} + \beta_{3} ) q^{65} + ( -32 + 98 \beta_{1} - 30 \beta_{2} + 20 \beta_{3} ) q^{66} + ( 241 - 38 \beta_{1} + 9 \beta_{2} - 8 \beta_{3} ) q^{67} + ( 8 - 16 \beta_{1} + 32 \beta_{2} - 16 \beta_{3} ) q^{68} + ( 299 - 64 \beta_{1} + 9 \beta_{2} - 44 \beta_{3} ) q^{69} + ( -88 + 56 \beta_{1} + 4 \beta_{2} + 12 \beta_{3} ) q^{70} + ( 4 - 23 \beta_{1} - 4 \beta_{2} + 15 \beta_{3} ) q^{71} + ( 192 + 40 \beta_{1} - 24 \beta_{2} - 40 \beta_{3} ) q^{72} + ( 545 - 11 \beta_{1} - 23 \beta_{2} + 11 \beta_{3} ) q^{73} + 74 q^{74} + ( -62 + 51 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{75} + ( -16 + 24 \beta_{1} - 56 \beta_{2} - 16 \beta_{3} ) q^{76} + ( 228 - 177 \beta_{1} + 68 \beta_{2} - 47 \beta_{3} ) q^{77} + ( -434 - 92 \beta_{1} + 50 \beta_{2} + 28 \beta_{3} ) q^{78} + ( 143 - 4 \beta_{1} - 100 \beta_{2} - 23 \beta_{3} ) q^{79} + ( 80 + 16 \beta_{2} ) q^{80} + ( 461 + 172 \beta_{1} + 33 \beta_{2} - 13 \beta_{3} ) q^{81} + ( -98 + 34 \beta_{1} - 4 \beta_{2} + 48 \beta_{3} ) q^{82} + ( -336 + \beta_{1} + 6 \beta_{2} + 45 \beta_{3} ) q^{83} + ( -456 - 60 \beta_{1} - 16 \beta_{2} + 84 \beta_{3} ) q^{84} + ( 474 - 48 \beta_{1} - 14 \beta_{2} - 16 \beta_{3} ) q^{85} + ( -292 - 72 \beta_{1} + 68 \beta_{2} + 72 \beta_{3} ) q^{86} + ( -317 - 68 \beta_{1} + 97 \beta_{2} - 32 \beta_{3} ) q^{87} + ( -152 - 24 \beta_{1} + 24 \beta_{2} + 56 \beta_{3} ) q^{88} + ( -94 + 48 \beta_{1} + 58 \beta_{2} + 54 \beta_{3} ) q^{89} + ( 312 + 92 \beta_{1} - 46 \beta_{2} - 98 \beta_{3} ) q^{90} + ( 278 + 152 \beta_{1} + 6 \beta_{2} - 52 \beta_{3} ) q^{91} + ( 76 + 16 \beta_{1} - 48 \beta_{2} - 76 \beta_{3} ) q^{92} + ( -314 - 124 \beta_{1} + 23 \beta_{2} + 35 \beta_{3} ) q^{93} + ( -20 - 18 \beta_{1} - 72 \beta_{2} - 34 \beta_{3} ) q^{94} + ( -622 + 76 \beta_{1} - 38 \beta_{2} - 72 \beta_{3} ) q^{95} + ( 32 - 32 \beta_{1} ) q^{96} + ( 210 - 66 \beta_{1} - 44 \beta_{2} + 46 \beta_{3} ) q^{97} + ( 666 + 62 \beta_{1} - 60 \beta_{2} - 138 \beta_{3} ) q^{98} + ( -1758 - 122 \beta_{1} - 39 \beta_{2} + 111 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{2} + 4q^{3} + 16q^{4} + 21q^{5} + 8q^{6} + 23q^{7} + 32q^{8} + 88q^{9} + O(q^{10}) \) \( 4q + 8q^{2} + 4q^{3} + 16q^{4} + 21q^{5} + 8q^{6} + 23q^{7} + 32q^{8} + 88q^{9} + 42q^{10} - 66q^{11} + 16q^{12} + 53q^{13} + 46q^{14} - 60q^{15} + 64q^{16} + 12q^{17} + 176q^{18} - 34q^{19} + 84q^{20} - 439q^{21} - 132q^{22} + 45q^{23} + 32q^{24} - 161q^{25} + 106q^{26} - 497q^{27} + 92q^{28} - 21q^{29} - 120q^{30} + 17q^{31} + 128q^{32} - 69q^{33} + 24q^{34} - 168q^{35} + 352q^{36} + 148q^{37} - 68q^{38} - 829q^{39} + 168q^{40} - 174q^{41} - 878q^{42} - 514q^{43} - 264q^{44} + 552q^{45} + 90q^{46} - 93q^{47} + 64q^{48} + 1233q^{49} - 322q^{50} + 324q^{51} + 212q^{52} + 3q^{53} - 994q^{54} - 423q^{55} + 184q^{56} + 140q^{57} - 42q^{58} + 354q^{59} - 240q^{60} + 1139q^{61} + 34q^{62} + 1136q^{63} + 256q^{64} + 1134q^{65} - 138q^{66} + 965q^{67} + 48q^{68} + 1161q^{69} - 336q^{70} + 27q^{71} + 704q^{72} + 2168q^{73} + 296q^{74} - 245q^{75} - 136q^{76} + 933q^{77} - 1658q^{78} + 449q^{79} + 336q^{80} + 1864q^{81} - 348q^{82} - 1293q^{83} - 1756q^{84} + 1866q^{85} - 1028q^{86} - 1203q^{87} - 528q^{88} - 264q^{89} + 1104q^{90} + 1066q^{91} + 180q^{92} - 1198q^{93} - 186q^{94} - 2598q^{95} + 128q^{96} + 842q^{97} + 2466q^{98} - 6960q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 96 x^{2} - 287 x + 330\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{3} + 23 \nu^{2} + 364 \nu - 15 \)\()/51\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 8 \nu^{2} - 49 \nu + 173 \)\()/17\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-5 \beta_{3} - 3 \beta_{2} + 7 \beta_{1} + 50\)
\(\nu^{3}\)\(=\)\(-23 \beta_{3} - 24 \beta_{2} + 105 \beta_{1} + 227\)

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
10.9313
0.888144
−4.94318
−6.87626
2.00000 −9.93130 4.00000 8.55405 −19.8626 30.9585 8.00000 71.6306 17.1081
1.2 2.00000 0.111856 4.00000 11.3318 0.223712 3.28728 8.00000 −26.9875 22.6637
1.3 2.00000 5.94318 4.00000 −7.71328 11.8864 22.4175 8.00000 8.32142 −15.4266
1.4 2.00000 7.87626 4.00000 8.82739 15.7525 −33.6633 8.00000 35.0354 17.6548
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(37\) \(-1\)

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 74.4.a.d 4
3.b odd 2 1 666.4.a.q 4
4.b odd 2 1 592.4.a.d 4
5.b even 2 1 1850.4.a.j 4
8.b even 2 1 2368.4.a.h 4
8.d odd 2 1 2368.4.a.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.a.d 4 1.a even 1 1 trivial
592.4.a.d 4 4.b odd 2 1
666.4.a.q 4 3.b odd 2 1
1850.4.a.j 4 5.b even 2 1
2368.4.a.h 4 8.b even 2 1
2368.4.a.k 4 8.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(74))\):

\( T_{3}^{4} - 4 T_{3}^{3} - 90 T_{3}^{2} + 475 T_{3} - 52 \)
\( T_{5}^{4} - 21 T_{5}^{3} + 51 T_{5}^{2} + 1246 T_{5} - 6600 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{4} \)
$3$ \( -52 + 475 T - 90 T^{2} - 4 T^{3} + T^{4} \)
$5$ \( -6600 + 1246 T + 51 T^{2} - 21 T^{3} + T^{4} \)
$7$ \( -76800 + 26988 T - 1038 T^{2} - 23 T^{3} + T^{4} \)
$11$ \( -232956 - 152405 T - 2274 T^{2} + 66 T^{3} + T^{4} \)
$13$ \( -243816 + 62310 T - 1653 T^{2} - 53 T^{3} + T^{4} \)
$17$ \( 2259504 + 104880 T - 11088 T^{2} - 12 T^{3} + T^{4} \)
$19$ \( 11080000 - 540296 T - 15060 T^{2} + 34 T^{3} + T^{4} \)
$23$ \( -43345368 + 2372994 T - 31905 T^{2} - 45 T^{3} + T^{4} \)
$29$ \( -59893020 + 2985024 T - 36819 T^{2} + 21 T^{3} + T^{4} \)
$31$ \( -34964464 + 2242640 T - 36105 T^{2} - 17 T^{3} + T^{4} \)
$37$ \( ( -37 + T )^{4} \)
$41$ \( -206445798 + 9420027 T - 99666 T^{2} + 174 T^{3} + T^{4} \)
$43$ \( -4525901824 - 44221120 T - 45780 T^{2} + 514 T^{3} + T^{4} \)
$47$ \( 3502908000 - 16363700 T - 172230 T^{2} + 93 T^{3} + T^{4} \)
$53$ \( -8943617448 + 106261644 T - 345114 T^{2} - 3 T^{3} + T^{4} \)
$59$ \( 31770624 - 2010464 T + 42708 T^{2} - 354 T^{3} + T^{4} \)
$61$ \( -13276304296 + 73774334 T + 144891 T^{2} - 1139 T^{3} + T^{4} \)
$67$ \( -4730048496 + 13530024 T + 202149 T^{2} - 965 T^{3} + T^{4} \)
$71$ \( 112341888 + 3019764 T - 72072 T^{2} - 27 T^{3} + T^{4} \)
$73$ \( 61284714458 - 527930593 T + 1643880 T^{2} - 2168 T^{3} + T^{4} \)
$79$ \( 10190211672 - 155721102 T - 993057 T^{2} - 449 T^{3} + T^{4} \)
$83$ \( -1337884224 + 29276416 T + 420672 T^{2} + 1293 T^{3} + T^{4} \)
$89$ \( 261101734464 - 139143344 T - 1013196 T^{2} + 264 T^{3} + T^{4} \)
$97$ \( 5575773984 + 175888968 T - 842844 T^{2} - 842 T^{3} + T^{4} \)
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