Properties

Label 74.2.e.b
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: \(x^{4} - x^{2} + 1\)
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.

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

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 0.366025 0.633975i 0.500000 0.866025i 1.50000 + 0.866025i 0.732051i 1.00000 1.73205i 1.00000i 1.23205 + 2.13397i −1.73205
11.2 0.866025 0.500000i −1.36603 + 2.36603i 0.500000 0.866025i 1.50000 + 0.866025i 2.73205i 1.00000 1.73205i 1.00000i −2.23205 3.86603i 1.73205
27.1 −0.866025 0.500000i 0.366025 + 0.633975i 0.500000 + 0.866025i 1.50000 0.866025i 0.732051i 1.00000 + 1.73205i 1.00000i 1.23205 2.13397i −1.73205
27.2 0.866025 + 0.500000i −1.36603 2.36603i 0.500000 + 0.866025i 1.50000 0.866025i 2.73205i 1.00000 + 1.73205i 1.00000i −2.23205 + 3.86603i 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.b 4
3.b odd 2 1 666.2.s.a 4
4.b odd 2 1 592.2.w.e 4
37.e even 6 1 inner 74.2.e.b 4
37.g odd 12 1 2738.2.a.e 2
37.g odd 12 1 2738.2.a.i 2
111.h odd 6 1 666.2.s.a 4
148.j odd 6 1 592.2.w.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.e.b 4 1.a even 1 1 trivial
74.2.e.b 4 37.e even 6 1 inner
592.2.w.e 4 4.b odd 2 1
592.2.w.e 4 148.j odd 6 1
666.2.s.a 4 3.b odd 2 1
666.2.s.a 4 111.h odd 6 1
2738.2.a.e 2 37.g odd 12 1
2738.2.a.i 2 37.g odd 12 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( ( 3 - 3 T + T^{2} )^{2} \)
$7$ \( ( 4 - 2 T + T^{2} )^{2} \)
$11$ \( ( 6 + 6 T + T^{2} )^{2} \)
$13$ \( ( 12 + 6 T + T^{2} )^{2} \)
$17$ \( 1089 + 198 T - 21 T^{2} - 6 T^{3} + T^{4} \)
$19$ \( 36 - 36 T + 6 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( 36 + 24 T^{2} + T^{4} \)
$29$ \( ( 75 + T^{2} )^{2} \)
$31$ \( 36 + 24 T^{2} + T^{4} \)
$37$ \( 1369 + 74 T - 33 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( 1521 - 234 T + 75 T^{2} + 6 T^{3} + T^{4} \)
$43$ \( 144 + 168 T^{2} + T^{4} \)
$47$ \( ( 6 - 6 T + T^{2} )^{2} \)
$53$ \( 576 + 288 T + 120 T^{2} + 12 T^{3} + T^{4} \)
$59$ \( 576 - 288 T + 24 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( ( 3 - 3 T + T^{2} )^{2} \)
$67$ \( 4 + 20 T + 102 T^{2} - 10 T^{3} + T^{4} \)
$71$ \( 144 + 12 T^{2} + T^{4} \)
$73$ \( ( -4 + T )^{4} \)
$79$ \( 49284 + 1332 T - 210 T^{2} - 6 T^{3} + T^{4} \)
$83$ \( 4356 - 396 T + 102 T^{2} + 6 T^{3} + T^{4} \)
$89$ \( 13689 + 2106 T - 9 T^{2} - 18 T^{3} + T^{4} \)
$97$ \( 1089 + 78 T^{2} + T^{4} \)
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