Properties

Label 740.1.bw.a
Level $740$
Weight $1$
Character orbit 740.bw
Analytic conductor $0.369$
Analytic rank $0$
Dimension $12$
Projective image $D_{36}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 740.bw (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.369308109348\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{36})\)
Defining polynomial: \(x^{12} - x^{6} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{36}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{36} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{36}^{7} q^{2} + \zeta_{36}^{14} q^{4} + \zeta_{36}^{15} q^{5} + \zeta_{36}^{3} q^{8} -\zeta_{36}^{13} q^{9} +O(q^{10})\) \( q -\zeta_{36}^{7} q^{2} + \zeta_{36}^{14} q^{4} + \zeta_{36}^{15} q^{5} + \zeta_{36}^{3} q^{8} -\zeta_{36}^{13} q^{9} + \zeta_{36}^{4} q^{10} + ( \zeta_{36}^{2} + \zeta_{36}^{8} ) q^{13} -\zeta_{36}^{10} q^{16} + ( \zeta_{36}^{12} - \zeta_{36}^{14} ) q^{17} -\zeta_{36}^{2} q^{18} -\zeta_{36}^{11} q^{20} -\zeta_{36}^{12} q^{25} + ( -\zeta_{36}^{9} - \zeta_{36}^{15} ) q^{26} + ( -\zeta_{36}^{16} - \zeta_{36}^{17} ) q^{29} + \zeta_{36}^{17} q^{32} + ( \zeta_{36} - \zeta_{36}^{3} ) q^{34} + \zeta_{36}^{9} q^{36} + \zeta_{36}^{5} q^{37} - q^{40} + ( -\zeta_{36}^{4} - \zeta_{36}^{6} ) q^{41} + \zeta_{36}^{10} q^{45} + \zeta_{36}^{7} q^{49} -\zeta_{36} q^{50} + ( -\zeta_{36}^{4} + \zeta_{36}^{16} ) q^{52} + ( \zeta_{36}^{13} + \zeta_{36}^{16} ) q^{53} + ( -\zeta_{36}^{5} - \zeta_{36}^{6} ) q^{58} + ( -1 + \zeta_{36} ) q^{61} + \zeta_{36}^{6} q^{64} + ( -\zeta_{36}^{5} + \zeta_{36}^{17} ) q^{65} + ( -\zeta_{36}^{8} + \zeta_{36}^{10} ) q^{68} -\zeta_{36}^{16} q^{72} + ( -\zeta_{36}^{3} + \zeta_{36}^{6} ) q^{73} -\zeta_{36}^{12} q^{74} + \zeta_{36}^{7} q^{80} -\zeta_{36}^{8} q^{81} + ( \zeta_{36}^{11} + \zeta_{36}^{13} ) q^{82} + ( -\zeta_{36}^{9} + \zeta_{36}^{11} ) q^{85} + ( -\zeta_{36}^{2} + \zeta_{36}^{9} ) q^{89} -\zeta_{36}^{17} q^{90} + ( -\zeta_{36} - \zeta_{36}^{11} ) q^{97} -\zeta_{36}^{14} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q - 6q^{17} + 6q^{25} - 12q^{40} - 6q^{41} - 6q^{58} - 12q^{61} + 6q^{64} + 6q^{73} + 6q^{74} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/740\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(-\zeta_{36}^{5}\) \(-\zeta_{36}^{9}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
183.1
0.342020 0.939693i
−0.342020 + 0.939693i
−0.642788 + 0.766044i
−0.642788 0.766044i
−0.342020 0.939693i
−0.984808 + 0.173648i
0.984808 + 0.173648i
−0.984808 0.173648i
0.984808 0.173648i
0.342020 + 0.939693i
0.642788 + 0.766044i
0.642788 0.766044i
0.642788 + 0.766044i 0 −0.173648 + 0.984808i 0.866025 + 0.500000i 0 0 −0.866025 + 0.500000i 0.984808 0.173648i 0.173648 + 0.984808i
187.1 −0.642788 0.766044i 0 −0.173648 + 0.984808i −0.866025 0.500000i 0 0 0.866025 0.500000i −0.984808 + 0.173648i 0.173648 + 0.984808i
383.1 0.984808 + 0.173648i 0 0.939693 + 0.342020i −0.866025 + 0.500000i 0 0 0.866025 + 0.500000i 0.342020 + 0.939693i −0.939693 + 0.342020i
427.1 0.984808 0.173648i 0 0.939693 0.342020i −0.866025 0.500000i 0 0 0.866025 0.500000i 0.342020 0.939693i −0.939693 0.342020i
463.1 −0.642788 + 0.766044i 0 −0.173648 0.984808i −0.866025 + 0.500000i 0 0 0.866025 + 0.500000i −0.984808 0.173648i 0.173648 0.984808i
503.1 0.342020 0.939693i 0 −0.766044 0.642788i 0.866025 + 0.500000i 0 0 −0.866025 + 0.500000i −0.642788 0.766044i 0.766044 0.642788i
523.1 −0.342020 0.939693i 0 −0.766044 + 0.642788i −0.866025 + 0.500000i 0 0 0.866025 + 0.500000i 0.642788 0.766044i 0.766044 + 0.642788i
587.1 0.342020 + 0.939693i 0 −0.766044 + 0.642788i 0.866025 0.500000i 0 0 −0.866025 0.500000i −0.642788 + 0.766044i 0.766044 + 0.642788i
607.1 −0.342020 + 0.939693i 0 −0.766044 0.642788i −0.866025 0.500000i 0 0 0.866025 0.500000i 0.642788 + 0.766044i 0.766044 0.642788i
647.1 0.642788 0.766044i 0 −0.173648 0.984808i 0.866025 0.500000i 0 0 −0.866025 0.500000i 0.984808 + 0.173648i 0.173648 0.984808i
683.1 −0.984808 + 0.173648i 0 0.939693 0.342020i 0.866025 + 0.500000i 0 0 −0.866025 + 0.500000i −0.342020 + 0.939693i −0.939693 0.342020i
727.1 −0.984808 0.173648i 0 0.939693 + 0.342020i 0.866025 0.500000i 0 0 −0.866025 0.500000i −0.342020 0.939693i −0.939693 + 0.342020i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 727.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
185.bc even 36 1 inner
740.bw odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.1.bw.a 12
4.b odd 2 1 CM 740.1.bw.a 12
5.b even 2 1 3700.1.cy.a 12
5.c odd 4 1 740.1.cb.a yes 12
5.c odd 4 1 3700.1.dd.a 12
20.d odd 2 1 3700.1.cy.a 12
20.e even 4 1 740.1.cb.a yes 12
20.e even 4 1 3700.1.dd.a 12
37.i odd 36 1 740.1.cb.a yes 12
148.q even 36 1 740.1.cb.a yes 12
185.z even 36 1 3700.1.cy.a 12
185.ba odd 36 1 3700.1.dd.a 12
185.bc even 36 1 inner 740.1.bw.a 12
740.bw odd 36 1 inner 740.1.bw.a 12
740.ca even 36 1 3700.1.dd.a 12
740.cb odd 36 1 3700.1.cy.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.bw.a 12 1.a even 1 1 trivial
740.1.bw.a 12 4.b odd 2 1 CM
740.1.bw.a 12 185.bc even 36 1 inner
740.1.bw.a 12 740.bw odd 36 1 inner
740.1.cb.a yes 12 5.c odd 4 1
740.1.cb.a yes 12 20.e even 4 1
740.1.cb.a yes 12 37.i odd 36 1
740.1.cb.a yes 12 148.q even 36 1
3700.1.cy.a 12 5.b even 2 1
3700.1.cy.a 12 20.d odd 2 1
3700.1.cy.a 12 185.z even 36 1
3700.1.cy.a 12 740.cb odd 36 1
3700.1.dd.a 12 5.c odd 4 1
3700.1.dd.a 12 20.e even 4 1
3700.1.dd.a 12 185.ba odd 36 1
3700.1.dd.a 12 740.ca even 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(740, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{6} + T^{12} \)
$3$ \( T^{12} \)
$5$ \( ( 1 - T^{2} + T^{4} )^{3} \)
$7$ \( T^{12} \)
$11$ \( T^{12} \)
$13$ \( ( 27 + 9 T^{3} + T^{6} )^{2} \)
$17$ \( ( 1 - 3 T + 3 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$19$ \( T^{12} \)
$23$ \( T^{12} \)
$29$ \( 1 + 6 T + 45 T^{2} + 110 T^{3} + 105 T^{4} + 36 T^{5} + 2 T^{6} - 6 T^{7} - 9 T^{8} - 2 T^{9} + T^{12} \)
$31$ \( T^{12} \)
$37$ \( 1 - T^{6} + T^{12} \)
$41$ \( ( 3 + 9 T + 9 T^{2} + 6 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$43$ \( T^{12} \)
$47$ \( T^{12} \)
$53$ \( 1 - 14 T^{3} + 53 T^{6} - 4 T^{9} + T^{12} \)
$59$ \( T^{12} \)
$61$ \( 1 + 6 T + 51 T^{2} + 200 T^{3} + 480 T^{4} + 786 T^{5} + 923 T^{6} + 792 T^{7} + 495 T^{8} + 220 T^{9} + 66 T^{10} + 12 T^{11} + T^{12} \)
$67$ \( T^{12} \)
$71$ \( T^{12} \)
$73$ \( ( 1 + 2 T + 2 T^{2} - 2 T^{3} + T^{4} )^{3} \)
$79$ \( T^{12} \)
$83$ \( T^{12} \)
$89$ \( 1 - 12 T + 39 T^{2} - 14 T^{3} - 12 T^{4} - 12 T^{5} + 23 T^{6} + 15 T^{8} + 2 T^{9} + 6 T^{10} + T^{12} \)
$97$ \( 9 + 27 T^{2} + 63 T^{4} + 48 T^{6} + 27 T^{8} + 6 T^{10} + T^{12} \)
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