Properties

Label 1476.1.o.b
Level $1476$
Weight $1$
Character orbit 1476.o
Analytic conductor $0.737$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -164
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1476.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.736619958646\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.13284.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.357286464.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} + q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{5} + \zeta_{6}^{2} q^{6} -\zeta_{6}^{2} q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{2} + q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{5} + \zeta_{6}^{2} q^{6} -\zeta_{6}^{2} q^{7} + q^{8} + q^{9} - q^{10} + 2 \zeta_{6}^{2} q^{11} -\zeta_{6} q^{12} + \zeta_{6} q^{14} + \zeta_{6} q^{15} + \zeta_{6}^{2} q^{16} + \zeta_{6}^{2} q^{18} - q^{19} -\zeta_{6}^{2} q^{20} -\zeta_{6}^{2} q^{21} -2 \zeta_{6} q^{22} + q^{24} + q^{27} - q^{28} - q^{30} -\zeta_{6} q^{32} + 2 \zeta_{6}^{2} q^{33} + q^{35} -\zeta_{6} q^{36} + 2 q^{37} -\zeta_{6}^{2} q^{38} + \zeta_{6} q^{40} -\zeta_{6} q^{41} + \zeta_{6} q^{42} + 2 q^{44} + \zeta_{6} q^{45} -\zeta_{6}^{2} q^{47} + \zeta_{6}^{2} q^{48} + \zeta_{6}^{2} q^{54} -2 q^{55} -\zeta_{6}^{2} q^{56} - q^{57} -\zeta_{6}^{2} q^{60} + 2 \zeta_{6}^{2} q^{61} -\zeta_{6}^{2} q^{63} + q^{64} -2 \zeta_{6} q^{66} -2 \zeta_{6} q^{67} + \zeta_{6}^{2} q^{70} - q^{71} + q^{72} - q^{73} + 2 \zeta_{6}^{2} q^{74} + \zeta_{6} q^{76} + 2 \zeta_{6} q^{77} -\zeta_{6}^{2} q^{79} - q^{80} + q^{81} + q^{82} - q^{84} + 2 \zeta_{6}^{2} q^{88} - q^{90} + \zeta_{6} q^{94} -\zeta_{6} q^{95} -\zeta_{6} q^{96} + 2 \zeta_{6}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} - q^{4} + q^{5} - q^{6} + q^{7} + 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q - q^{2} + 2 q^{3} - q^{4} + q^{5} - q^{6} + q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - q^{12} + q^{14} + q^{15} - q^{16} - q^{18} - 2 q^{19} + q^{20} + q^{21} - 2 q^{22} + 2 q^{24} + 2 q^{27} - 2 q^{28} - 2 q^{30} - q^{32} - 2 q^{33} + 2 q^{35} - q^{36} + 4 q^{37} + q^{38} + q^{40} - q^{41} + q^{42} + 4 q^{44} + q^{45} + q^{47} - q^{48} - q^{54} - 4 q^{55} + q^{56} - 2 q^{57} + q^{60} - 2 q^{61} + q^{63} + 2 q^{64} - 2 q^{66} - 2 q^{67} - q^{70} - 2 q^{71} + 2 q^{72} - 2 q^{73} - 2 q^{74} + q^{76} + 2 q^{77} + q^{79} - 2 q^{80} + 2 q^{81} + 2 q^{82} - 2 q^{84} - 2 q^{88} - 2 q^{90} + q^{94} - q^{95} - q^{96} - 2 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1476\mathbb{Z}\right)^\times\).

\(n\) \(739\) \(821\) \(1441\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
655.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i 1.00000 1.00000 −1.00000
1147.1 −0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i 1.00000 1.00000 −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
164.d odd 2 1 CM by \(\Q(\sqrt{-41}) \)
9.c even 3 1 inner
1476.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1476.1.o.b yes 2
4.b odd 2 1 1476.1.o.a 2
9.c even 3 1 inner 1476.1.o.b yes 2
36.f odd 6 1 1476.1.o.a 2
41.b even 2 1 1476.1.o.a 2
164.d odd 2 1 CM 1476.1.o.b yes 2
369.i even 6 1 1476.1.o.a 2
1476.o odd 6 1 inner 1476.1.o.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1476.1.o.a 2 4.b odd 2 1
1476.1.o.a 2 36.f odd 6 1
1476.1.o.a 2 41.b even 2 1
1476.1.o.a 2 369.i even 6 1
1476.1.o.b yes 2 1.a even 1 1 trivial
1476.1.o.b yes 2 9.c even 3 1 inner
1476.1.o.b yes 2 164.d odd 2 1 CM
1476.1.o.b yes 2 1476.o odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1476, [\chi])\):

\( T_{5}^{2} - T_{5} + 1 \)
\( T_{7}^{2} - T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( 4 + 2 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( 1 + T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( 1 - T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 4 + 2 T + T^{2} \)
$67$ \( 4 + 2 T + T^{2} \)
$71$ \( ( 1 + T )^{2} \)
$73$ \( ( 1 + T )^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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