Properties

Label 74.4.a.a
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

\(f(q)\) \(=\) \( q - 2 q^{2} - 5 q^{3} + 4 q^{4} + 12 q^{5} + 10 q^{6} - 7 q^{7} - 8 q^{8} - 2 q^{9} + O(q^{10}) \) \( q - 2 q^{2} - 5 q^{3} + 4 q^{4} + 12 q^{5} + 10 q^{6} - 7 q^{7} - 8 q^{8} - 2 q^{9} - 24 q^{10} - 63 q^{11} - 20 q^{12} - 28 q^{13} + 14 q^{14} - 60 q^{15} + 16 q^{16} + 6 q^{17} + 4 q^{18} - 70 q^{19} + 48 q^{20} + 35 q^{21} + 126 q^{22} - 6 q^{23} + 40 q^{24} + 19 q^{25} + 56 q^{26} + 145 q^{27} - 28 q^{28} - 42 q^{29} + 120 q^{30} - 292 q^{31} - 32 q^{32} + 315 q^{33} - 12 q^{34} - 84 q^{35} - 8 q^{36} + 37 q^{37} + 140 q^{38} + 140 q^{39} - 96 q^{40} + 351 q^{41} - 70 q^{42} + 32 q^{43} - 252 q^{44} - 24 q^{45} + 12 q^{46} + 357 q^{47} - 80 q^{48} - 294 q^{49} - 38 q^{50} - 30 q^{51} - 112 q^{52} + 57 q^{53} - 290 q^{54} - 756 q^{55} + 56 q^{56} + 350 q^{57} + 84 q^{58} + 432 q^{59} - 240 q^{60} - 340 q^{61} + 584 q^{62} + 14 q^{63} + 64 q^{64} - 336 q^{65} - 630 q^{66} - 1012 q^{67} + 24 q^{68} + 30 q^{69} + 168 q^{70} - 609 q^{71} + 16 q^{72} + 539 q^{73} - 74 q^{74} - 95 q^{75} - 280 q^{76} + 441 q^{77} - 280 q^{78} + 818 q^{79} + 192 q^{80} - 671 q^{81} - 702 q^{82} + 1299 q^{83} + 140 q^{84} + 72 q^{85} - 64 q^{86} + 210 q^{87} + 504 q^{88} - 390 q^{89} + 48 q^{90} + 196 q^{91} - 24 q^{92} + 1460 q^{93} - 714 q^{94} - 840 q^{95} + 160 q^{96} + 1772 q^{97} + 588 q^{98} + 126 q^{99} + O(q^{100}) \)

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 12.0000 10.0000 −7.00000 −8.00000 −2.00000 −24.0000
\(n\): 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.a 1
3.b odd 2 1 666.4.a.e 1
4.b odd 2 1 592.4.a.b 1
5.b even 2 1 1850.4.a.g 1
8.b even 2 1 2368.4.a.c 1
8.d odd 2 1 2368.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.a.a 1 1.a even 1 1 trivial
592.4.a.b 1 4.b odd 2 1
666.4.a.e 1 3.b odd 2 1
1850.4.a.g 1 5.b even 2 1
2368.4.a.a 1 8.d odd 2 1
2368.4.a.c 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(74))\):

\( T_{3} + 5 \)
\( T_{5} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( 5 + T \)
$5$ \( -12 + T \)
$7$ \( 7 + T \)
$11$ \( 63 + T \)
$13$ \( 28 + T \)
$17$ \( -6 + T \)
$19$ \( 70 + T \)
$23$ \( 6 + T \)
$29$ \( 42 + T \)
$31$ \( 292 + T \)
$37$ \( -37 + T \)
$41$ \( -351 + T \)
$43$ \( -32 + T \)
$47$ \( -357 + T \)
$53$ \( -57 + T \)
$59$ \( -432 + T \)
$61$ \( 340 + T \)
$67$ \( 1012 + T \)
$71$ \( 609 + T \)
$73$ \( -539 + T \)
$79$ \( -818 + T \)
$83$ \( -1299 + T \)
$89$ \( 390 + T \)
$97$ \( -1772 + T \)
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