Properties

Label 74.2.a.a
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: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.30278
2.30278
−1.00000 −0.302776 1.00000 1.30278 0.302776 4.60555 −1.00000 −2.90833 −1.30278
1.2 −1.00000 3.30278 1.00000 −2.30278 −3.30278 −2.60555 −1.00000 7.90833 2.30278
\(n\): 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.a 2
3.b odd 2 1 666.2.a.j 2
4.b odd 2 1 592.2.a.f 2
5.b even 2 1 1850.2.a.u 2
5.c odd 4 2 1850.2.b.i 4
7.b odd 2 1 3626.2.a.a 2
8.b even 2 1 2368.2.a.s 2
8.d odd 2 1 2368.2.a.ba 2
11.b odd 2 1 8954.2.a.p 2
12.b even 2 1 5328.2.a.bf 2
37.b even 2 1 2738.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 1.a even 1 1 trivial
592.2.a.f 2 4.b odd 2 1
666.2.a.j 2 3.b odd 2 1
1850.2.a.u 2 5.b even 2 1
1850.2.b.i 4 5.c odd 4 2
2368.2.a.s 2 8.b even 2 1
2368.2.a.ba 2 8.d odd 2 1
2738.2.a.l 2 37.b even 2 1
3626.2.a.a 2 7.b odd 2 1
5328.2.a.bf 2 12.b even 2 1
8954.2.a.p 2 11.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -1 - 3 T + T^{2} \)
$5$ \( -3 + T + T^{2} \)
$7$ \( -12 - 2 T + T^{2} \)
$11$ \( -3 + T + T^{2} \)
$13$ \( -3 + T + T^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( -27 + 3 T + T^{2} \)
$29$ \( -27 - 3 T + T^{2} \)
$31$ \( -1 - 3 T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( -9 - 9 T + T^{2} \)
$43$ \( -4 + 6 T + T^{2} \)
$47$ \( -12 - 2 T + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 36 - 14 T + T^{2} \)
$61$ \( -79 + 3 T + T^{2} \)
$67$ \( -51 - 11 T + T^{2} \)
$71$ \( ( -6 + T )^{2} \)
$73$ \( 107 + 21 T + T^{2} \)
$79$ \( -147 + 7 T + T^{2} \)
$83$ \( 48 - 20 T + T^{2} \)
$89$ \( -48 + 4 T + T^{2} \)
$97$ \( -204 + 4 T + T^{2} \)
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