Properties

Label 74.2.c.b
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + 2 \zeta_{6} q^{3} - \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + 2 q^{6} + 4 \zeta_{6} q^{7} - q^{8} + (\zeta_{6} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + 2 \zeta_{6} q^{3} - \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + 2 q^{6} + 4 \zeta_{6} q^{7} - q^{8} + (\zeta_{6} - 1) q^{9} - 3 q^{10} - 6 q^{11} + ( - 2 \zeta_{6} + 2) q^{12} - 2 \zeta_{6} q^{13} + 4 q^{14} + ( - 6 \zeta_{6} + 6) q^{15} + (\zeta_{6} - 1) q^{16} + (3 \zeta_{6} - 3) q^{17} + \zeta_{6} q^{18} - 2 \zeta_{6} q^{19} + (3 \zeta_{6} - 3) q^{20} + (8 \zeta_{6} - 8) q^{21} + (6 \zeta_{6} - 6) q^{22} + 6 q^{23} - 2 \zeta_{6} q^{24} + (4 \zeta_{6} - 4) q^{25} - 2 q^{26} + 4 q^{27} + ( - 4 \zeta_{6} + 4) q^{28} + 3 q^{29} - 6 \zeta_{6} q^{30} + 2 q^{31} + \zeta_{6} q^{32} - 12 \zeta_{6} q^{33} + 3 \zeta_{6} q^{34} + ( - 12 \zeta_{6} + 12) q^{35} + q^{36} + (3 \zeta_{6} + 4) q^{37} - 2 q^{38} + ( - 4 \zeta_{6} + 4) q^{39} + 3 \zeta_{6} q^{40} - 3 \zeta_{6} q^{41} + 8 \zeta_{6} q^{42} - 4 q^{43} + 6 \zeta_{6} q^{44} + 3 q^{45} + ( - 6 \zeta_{6} + 6) q^{46} - 6 q^{47} - 2 q^{48} + (9 \zeta_{6} - 9) q^{49} + 4 \zeta_{6} q^{50} - 6 q^{51} + (2 \zeta_{6} - 2) q^{52} + ( - 6 \zeta_{6} + 6) q^{53} + ( - 4 \zeta_{6} + 4) q^{54} + 18 \zeta_{6} q^{55} - 4 \zeta_{6} q^{56} + ( - 4 \zeta_{6} + 4) q^{57} + ( - 3 \zeta_{6} + 3) q^{58} - 6 q^{60} + \zeta_{6} q^{61} + ( - 2 \zeta_{6} + 2) q^{62} - 4 q^{63} + q^{64} + (6 \zeta_{6} - 6) q^{65} - 12 q^{66} - 2 \zeta_{6} q^{67} + 3 q^{68} + 12 \zeta_{6} q^{69} - 12 \zeta_{6} q^{70} + 12 \zeta_{6} q^{71} + ( - \zeta_{6} + 1) q^{72} - 10 q^{73} + ( - 4 \zeta_{6} + 7) q^{74} - 8 q^{75} + (2 \zeta_{6} - 2) q^{76} - 24 \zeta_{6} q^{77} - 4 \zeta_{6} q^{78} - 14 \zeta_{6} q^{79} + 3 q^{80} + 11 \zeta_{6} q^{81} - 3 q^{82} + (6 \zeta_{6} - 6) q^{83} + 8 q^{84} + 9 q^{85} + (4 \zeta_{6} - 4) q^{86} + 6 \zeta_{6} q^{87} + 6 q^{88} + (3 \zeta_{6} - 3) q^{89} + ( - 3 \zeta_{6} + 3) q^{90} + ( - 8 \zeta_{6} + 8) q^{91} - 6 \zeta_{6} q^{92} + 4 \zeta_{6} q^{93} + (6 \zeta_{6} - 6) q^{94} + (6 \zeta_{6} - 6) q^{95} + (2 \zeta_{6} - 2) q^{96} - 13 q^{97} + 9 \zeta_{6} q^{98} + ( - 6 \zeta_{6} + 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} - q^{4} - 3 q^{5} + 4 q^{6} + 4 q^{7} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 2 q^{3} - q^{4} - 3 q^{5} + 4 q^{6} + 4 q^{7} - 2 q^{8} - q^{9} - 6 q^{10} - 12 q^{11} + 2 q^{12} - 2 q^{13} + 8 q^{14} + 6 q^{15} - q^{16} - 3 q^{17} + q^{18} - 2 q^{19} - 3 q^{20} - 8 q^{21} - 6 q^{22} + 12 q^{23} - 2 q^{24} - 4 q^{25} - 4 q^{26} + 8 q^{27} + 4 q^{28} + 6 q^{29} - 6 q^{30} + 4 q^{31} + q^{32} - 12 q^{33} + 3 q^{34} + 12 q^{35} + 2 q^{36} + 11 q^{37} - 4 q^{38} + 4 q^{39} + 3 q^{40} - 3 q^{41} + 8 q^{42} - 8 q^{43} + 6 q^{44} + 6 q^{45} + 6 q^{46} - 12 q^{47} - 4 q^{48} - 9 q^{49} + 4 q^{50} - 12 q^{51} - 2 q^{52} + 6 q^{53} + 4 q^{54} + 18 q^{55} - 4 q^{56} + 4 q^{57} + 3 q^{58} - 12 q^{60} + q^{61} + 2 q^{62} - 8 q^{63} + 2 q^{64} - 6 q^{65} - 24 q^{66} - 2 q^{67} + 6 q^{68} + 12 q^{69} - 12 q^{70} + 12 q^{71} + q^{72} - 20 q^{73} + 10 q^{74} - 16 q^{75} - 2 q^{76} - 24 q^{77} - 4 q^{78} - 14 q^{79} + 6 q^{80} + 11 q^{81} - 6 q^{82} - 6 q^{83} + 16 q^{84} + 18 q^{85} - 4 q^{86} + 6 q^{87} + 12 q^{88} - 3 q^{89} + 3 q^{90} + 8 q^{91} - 6 q^{92} + 4 q^{93} - 6 q^{94} - 6 q^{95} - 2 q^{96} - 26 q^{97} + 9 q^{98} + 6 q^{99}+O(q^{100}) \) 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 −1.50000 2.59808i 2.00000 2.00000 + 3.46410i −1.00000 −0.500000 + 0.866025i −3.00000
63.1 0.500000 + 0.866025i 1.00000 1.73205i −0.500000 + 0.866025i −1.50000 + 2.59808i 2.00000 2.00000 3.46410i −1.00000 −0.500000 0.866025i −3.00000
\(n\): 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.b 2
3.b odd 2 1 666.2.f.d 2
4.b odd 2 1 592.2.i.a 2
37.c even 3 1 inner 74.2.c.b 2
37.c even 3 1 2738.2.a.a 1
37.e even 6 1 2738.2.a.c 1
111.i odd 6 1 666.2.f.d 2
148.i odd 6 1 592.2.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.c.b 2 1.a even 1 1 trivial
74.2.c.b 2 37.c even 3 1 inner
592.2.i.a 2 4.b odd 2 1
592.2.i.a 2 148.i odd 6 1
666.2.f.d 2 3.b odd 2 1
666.2.f.d 2 111.i odd 6 1
2738.2.a.a 1 37.c even 3 1
2738.2.a.c 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
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$11$ \( (T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) 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
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 37 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$97$ \( (T + 13)^{2} \) Copy content Toggle raw display
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