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: \( x^{2} - x - 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(74))\). Copy content Toggle raw display

Hecke characteristic polynomials

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