Properties

Label 74.4.a.b
Level $74$
Weight $4$
Character orbit 74.a
Self dual yes
Analytic conductor $4.366$
Analytic rank $1$
Dimension $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{2} - 5 q^{3} + 4 q^{4} - 14 q^{5} - 10 q^{6} - 19 q^{7} + 8 q^{8} - 2 q^{9} + O(q^{10}) \) \( q + 2 q^{2} - 5 q^{3} + 4 q^{4} - 14 q^{5} - 10 q^{6} - 19 q^{7} + 8 q^{8} - 2 q^{9} - 28 q^{10} + 5 q^{11} - 20 q^{12} + 6 q^{13} - 38 q^{14} + 70 q^{15} + 16 q^{16} - 72 q^{17} - 4 q^{18} - 44 q^{19} - 56 q^{20} + 95 q^{21} + 10 q^{22} + 182 q^{23} - 40 q^{24} + 71 q^{25} + 12 q^{26} + 145 q^{27} - 76 q^{28} + 10 q^{29} + 140 q^{30} - 244 q^{31} + 32 q^{32} - 25 q^{33} - 144 q^{34} + 266 q^{35} - 8 q^{36} - 37 q^{37} - 88 q^{38} - 30 q^{39} - 112 q^{40} - 225 q^{41} + 190 q^{42} - 2 q^{43} + 20 q^{44} + 28 q^{45} + 364 q^{46} + 221 q^{47} - 80 q^{48} + 18 q^{49} + 142 q^{50} + 360 q^{51} + 24 q^{52} - 659 q^{53} + 290 q^{54} - 70 q^{55} - 152 q^{56} + 220 q^{57} + 20 q^{58} + 156 q^{59} + 280 q^{60} - 620 q^{61} - 488 q^{62} + 38 q^{63} + 64 q^{64} - 84 q^{65} - 50 q^{66} + 416 q^{67} - 288 q^{68} - 910 q^{69} + 532 q^{70} - 1125 q^{71} - 16 q^{72} - 641 q^{73} - 74 q^{74} - 355 q^{75} - 176 q^{76} - 95 q^{77} - 60 q^{78} - 484 q^{79} - 224 q^{80} - 671 q^{81} - 450 q^{82} + 1239 q^{83} + 380 q^{84} + 1008 q^{85} - 4 q^{86} - 50 q^{87} + 40 q^{88} + 1304 q^{89} + 56 q^{90} - 114 q^{91} + 728 q^{92} + 1220 q^{93} + 442 q^{94} + 616 q^{95} - 160 q^{96} - 560 q^{97} + 36 q^{98} - 10 q^{99} + O(q^{100}) \)

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
0
2.00000 −5.00000 4.00000 −14.0000 −10.0000 −19.0000 8.00000 −2.00000 −28.0000
\(n\): 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.b 1
3.b odd 2 1 666.4.a.c 1
4.b odd 2 1 592.4.a.a 1
5.b even 2 1 1850.4.a.d 1
8.b even 2 1 2368.4.a.d 1
8.d odd 2 1 2368.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.a.b 1 1.a even 1 1 trivial
592.4.a.a 1 4.b odd 2 1
666.4.a.c 1 3.b odd 2 1
1850.4.a.d 1 5.b even 2 1
2368.4.a.b 1 8.d odd 2 1
2368.4.a.d 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(74))\):

\( T_{3} + 5 \)
\( T_{5} + 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( 5 + T \)
$5$ \( 14 + T \)
$7$ \( 19 + T \)
$11$ \( -5 + T \)
$13$ \( -6 + T \)
$17$ \( 72 + T \)
$19$ \( 44 + T \)
$23$ \( -182 + T \)
$29$ \( -10 + T \)
$31$ \( 244 + T \)
$37$ \( 37 + T \)
$41$ \( 225 + T \)
$43$ \( 2 + T \)
$47$ \( -221 + T \)
$53$ \( 659 + T \)
$59$ \( -156 + T \)
$61$ \( 620 + T \)
$67$ \( -416 + T \)
$71$ \( 1125 + T \)
$73$ \( 641 + T \)
$79$ \( 484 + T \)
$83$ \( -1239 + T \)
$89$ \( -1304 + T \)
$97$ \( 560 + T \)
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