Properties

Label 74.2.e.a
Level $74$
Weight $2$
Character orbit 74.e
Analytic conductor $0.591$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(39\)
\(\chi(n)\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i −1.36603 + 2.36603i 0.500000 0.866025i −1.50000 0.866025i 2.73205i −2.00000 + 3.46410i 1.00000i −2.23205 3.86603i 1.73205
11.2 0.866025 0.500000i 0.366025 0.633975i 0.500000 0.866025i −1.50000 0.866025i 0.732051i −2.00000 + 3.46410i 1.00000i 1.23205 + 2.13397i −1.73205
27.1 −0.866025 0.500000i −1.36603 2.36603i 0.500000 + 0.866025i −1.50000 + 0.866025i 2.73205i −2.00000 3.46410i 1.00000i −2.23205 + 3.86603i 1.73205
27.2 0.866025 + 0.500000i 0.366025 + 0.633975i 0.500000 + 0.866025i −1.50000 + 0.866025i 0.732051i −2.00000 3.46410i 1.00000i 1.23205 2.13397i −1.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.e.a 4
3.b odd 2 1 666.2.s.e 4
4.b odd 2 1 592.2.w.d 4
37.e even 6 1 inner 74.2.e.a 4
37.g odd 12 1 2738.2.a.f 2
37.g odd 12 1 2738.2.a.j 2
111.h odd 6 1 666.2.s.e 4
148.j odd 6 1 592.2.w.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.e.a 4 1.a even 1 1 trivial
74.2.e.a 4 37.e even 6 1 inner
592.2.w.d 4 4.b odd 2 1
592.2.w.d 4 148.j odd 6 1
666.2.s.e 4 3.b odd 2 1
666.2.s.e 4 111.h odd 6 1
2738.2.a.f 2 37.g odd 12 1
2738.2.a.j 2 37.g odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$17$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$23$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$29$ \( T^{4} + 78T^{2} + 1089 \) Copy content Toggle raw display
$31$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$37$ \( (T^{2} + T + 37)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} + 39 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$43$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T - 66)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + 24 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$61$ \( T^{4} - 30 T^{3} + 339 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676 \) Copy content Toggle raw display
$71$ \( T^{4} + 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$73$ \( (T^{2} + 4 T - 104)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 18 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} + 102 T^{2} + \cdots + 4356 \) Copy content Toggle raw display
$89$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
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