Properties

Label 74.2.c.a
Level $74$
Weight $2$
Character orbit 74.c
Analytic conductor $0.591$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i −1.00000 1.73205i −0.500000 0.866025i 0.500000 + 0.866025i −2.00000 0 −1.00000 −0.500000 + 0.866025i 1.00000
63.1 0.500000 + 0.866025i −1.00000 + 1.73205i −0.500000 + 0.866025i 0.500000 0.866025i −2.00000 0 −1.00000 −0.500000 0.866025i 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.c.a 2
3.b odd 2 1 666.2.f.b 2
4.b odd 2 1 592.2.i.d 2
37.c even 3 1 inner 74.2.c.a 2
37.c even 3 1 2738.2.a.b 1
37.e even 6 1 2738.2.a.d 1
111.i odd 6 1 666.2.f.b 2
148.i odd 6 1 592.2.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.c.a 2 1.a even 1 1 trivial
74.2.c.a 2 37.c even 3 1 inner
592.2.i.d 2 4.b odd 2 1
592.2.i.d 2 148.i odd 6 1
666.2.f.b 2 3.b odd 2 1
666.2.f.b 2 111.i odd 6 1
2738.2.a.b 1 37.c even 3 1
2738.2.a.d 1 37.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + T + 37 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( (T - 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$61$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$83$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$89$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$97$ \( (T - 3)^{2} \) Copy content Toggle raw display
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