Properties

Label 74.2.a.b
Level $74$
Weight $2$
Character orbit 74.a
Self dual yes
Analytic conductor $0.591$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} -\beta q^{3} + q^{4} + ( -1 + 3 \beta ) q^{5} -\beta q^{6} -2 \beta q^{7} + q^{8} + ( -2 + \beta ) q^{9} +O(q^{10})\) \( q + q^{2} -\beta q^{3} + q^{4} + ( -1 + 3 \beta ) q^{5} -\beta q^{6} -2 \beta q^{7} + q^{8} + ( -2 + \beta ) q^{9} + ( -1 + 3 \beta ) q^{10} + ( -3 + \beta ) q^{11} -\beta q^{12} + ( 2 - 3 \beta ) q^{13} -2 \beta q^{14} + ( -3 - 2 \beta ) q^{15} + q^{16} + ( 2 - 4 \beta ) q^{17} + ( -2 + \beta ) q^{18} + ( -2 + 4 \beta ) q^{19} + ( -1 + 3 \beta ) q^{20} + ( 2 + 2 \beta ) q^{21} + ( -3 + \beta ) q^{22} + ( -2 + 3 \beta ) q^{23} -\beta q^{24} + ( 5 + 3 \beta ) q^{25} + ( 2 - 3 \beta ) q^{26} + ( -1 + 4 \beta ) q^{27} -2 \beta q^{28} + ( 2 - 7 \beta ) q^{29} + ( -3 - 2 \beta ) q^{30} + ( 9 - \beta ) q^{31} + q^{32} + ( -1 + 2 \beta ) q^{33} + ( 2 - 4 \beta ) q^{34} + ( -6 - 4 \beta ) q^{35} + ( -2 + \beta ) q^{36} - q^{37} + ( -2 + 4 \beta ) q^{38} + ( 3 + \beta ) q^{39} + ( -1 + 3 \beta ) q^{40} + ( 8 + \beta ) q^{41} + ( 2 + 2 \beta ) q^{42} + ( -2 - 2 \beta ) q^{43} + ( -3 + \beta ) q^{44} + ( 5 - 4 \beta ) q^{45} + ( -2 + 3 \beta ) q^{46} + ( 2 - 2 \beta ) q^{47} -\beta q^{48} + ( -3 + 4 \beta ) q^{49} + ( 5 + 3 \beta ) q^{50} + ( 4 + 2 \beta ) q^{51} + ( 2 - 3 \beta ) q^{52} + ( -6 + 4 \beta ) q^{53} + ( -1 + 4 \beta ) q^{54} + ( 6 - 7 \beta ) q^{55} -2 \beta q^{56} + ( -4 - 2 \beta ) q^{57} + ( 2 - 7 \beta ) q^{58} + ( -8 + 2 \beta ) q^{59} + ( -3 - 2 \beta ) q^{60} + ( 9 + \beta ) q^{61} + ( 9 - \beta ) q^{62} + ( -2 + 2 \beta ) q^{63} + q^{64} -11 q^{65} + ( -1 + 2 \beta ) q^{66} + ( -7 + 5 \beta ) q^{67} + ( 2 - 4 \beta ) q^{68} + ( -3 - \beta ) q^{69} + ( -6 - 4 \beta ) q^{70} + ( -10 + 8 \beta ) q^{71} + ( -2 + \beta ) q^{72} + ( -1 + 5 \beta ) q^{73} - q^{74} + ( -3 - 8 \beta ) q^{75} + ( -2 + 4 \beta ) q^{76} + ( -2 + 4 \beta ) q^{77} + ( 3 + \beta ) q^{78} + ( 6 - 9 \beta ) q^{79} + ( -1 + 3 \beta ) q^{80} + ( 2 - 6 \beta ) q^{81} + ( 8 + \beta ) q^{82} + ( -8 - 4 \beta ) q^{83} + ( 2 + 2 \beta ) q^{84} + ( -14 - 2 \beta ) q^{85} + ( -2 - 2 \beta ) q^{86} + ( 7 + 5 \beta ) q^{87} + ( -3 + \beta ) q^{88} + ( -8 + 4 \beta ) q^{89} + ( 5 - 4 \beta ) q^{90} + ( 6 + 2 \beta ) q^{91} + ( -2 + 3 \beta ) q^{92} + ( 1 - 8 \beta ) q^{93} + ( 2 - 2 \beta ) q^{94} + ( 14 + 2 \beta ) q^{95} -\beta q^{96} + ( 6 - 4 \beta ) q^{97} + ( -3 + 4 \beta ) q^{98} + ( 7 - 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - q^{3} + 2q^{4} + q^{5} - q^{6} - 2q^{7} + 2q^{8} - 3q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - q^{3} + 2q^{4} + q^{5} - q^{6} - 2q^{7} + 2q^{8} - 3q^{9} + q^{10} - 5q^{11} - q^{12} + q^{13} - 2q^{14} - 8q^{15} + 2q^{16} - 3q^{18} + q^{20} + 6q^{21} - 5q^{22} - q^{23} - q^{24} + 13q^{25} + q^{26} + 2q^{27} - 2q^{28} - 3q^{29} - 8q^{30} + 17q^{31} + 2q^{32} - 16q^{35} - 3q^{36} - 2q^{37} + 7q^{39} + q^{40} + 17q^{41} + 6q^{42} - 6q^{43} - 5q^{44} + 6q^{45} - q^{46} + 2q^{47} - q^{48} - 2q^{49} + 13q^{50} + 10q^{51} + q^{52} - 8q^{53} + 2q^{54} + 5q^{55} - 2q^{56} - 10q^{57} - 3q^{58} - 14q^{59} - 8q^{60} + 19q^{61} + 17q^{62} - 2q^{63} + 2q^{64} - 22q^{65} - 9q^{67} - 7q^{69} - 16q^{70} - 12q^{71} - 3q^{72} + 3q^{73} - 2q^{74} - 14q^{75} + 7q^{78} + 3q^{79} + q^{80} - 2q^{81} + 17q^{82} - 20q^{83} + 6q^{84} - 30q^{85} - 6q^{86} + 19q^{87} - 5q^{88} - 12q^{89} + 6q^{90} + 14q^{91} - q^{92} - 6q^{93} + 2q^{94} + 30q^{95} - q^{96} + 8q^{97} - 2q^{98} + 10q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
1.00000 −1.61803 1.00000 3.85410 −1.61803 −3.23607 1.00000 −0.381966 3.85410
1.2 1.00000 0.618034 1.00000 −2.85410 0.618034 1.23607 1.00000 −2.61803 −2.85410
\(n\): 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 74.2.a.b 2
3.b odd 2 1 666.2.a.i 2
4.b odd 2 1 592.2.a.g 2
5.b even 2 1 1850.2.a.t 2
5.c odd 4 2 1850.2.b.j 4
7.b odd 2 1 3626.2.a.s 2
8.b even 2 1 2368.2.a.y 2
8.d odd 2 1 2368.2.a.u 2
11.b odd 2 1 8954.2.a.j 2
12.b even 2 1 5328.2.a.bc 2
37.b even 2 1 2738.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.b 2 1.a even 1 1 trivial
592.2.a.g 2 4.b odd 2 1
666.2.a.i 2 3.b odd 2 1
1850.2.a.t 2 5.b even 2 1
1850.2.b.j 4 5.c odd 4 2
2368.2.a.u 2 8.d odd 2 1
2368.2.a.y 2 8.b even 2 1
2738.2.a.g 2 37.b even 2 1
3626.2.a.s 2 7.b odd 2 1
5328.2.a.bc 2 12.b even 2 1
8954.2.a.j 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(74))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -1 + T + T^{2} \)
$5$ \( -11 - T + T^{2} \)
$7$ \( -4 + 2 T + T^{2} \)
$11$ \( 5 + 5 T + T^{2} \)
$13$ \( -11 - T + T^{2} \)
$17$ \( -20 + T^{2} \)
$19$ \( -20 + T^{2} \)
$23$ \( -11 + T + T^{2} \)
$29$ \( -59 + 3 T + T^{2} \)
$31$ \( 71 - 17 T + T^{2} \)
$37$ \( ( 1 + T )^{2} \)
$41$ \( 71 - 17 T + T^{2} \)
$43$ \( 4 + 6 T + T^{2} \)
$47$ \( -4 - 2 T + T^{2} \)
$53$ \( -4 + 8 T + T^{2} \)
$59$ \( 44 + 14 T + T^{2} \)
$61$ \( 89 - 19 T + T^{2} \)
$67$ \( -11 + 9 T + T^{2} \)
$71$ \( -44 + 12 T + T^{2} \)
$73$ \( -29 - 3 T + T^{2} \)
$79$ \( -99 - 3 T + T^{2} \)
$83$ \( 80 + 20 T + T^{2} \)
$89$ \( 16 + 12 T + T^{2} \)
$97$ \( -4 - 8 T + T^{2} \)
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