Properties

Label 3528.1.dh.a
Level $3528$
Weight $1$
Character orbit 3528.dh
Analytic conductor $1.761$
Analytic rank $0$
Dimension $12$
Projective image $D_{14}$
CM discriminant -24
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.dh (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.76070136457\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
Defining polynomial: \(x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{14}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{14} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{28} q^{2} + \zeta_{28}^{2} q^{4} + ( \zeta_{28}^{9} + \zeta_{28}^{11} ) q^{5} + \zeta_{28}^{4} q^{7} + \zeta_{28}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{28} q^{2} + \zeta_{28}^{2} q^{4} + ( \zeta_{28}^{9} + \zeta_{28}^{11} ) q^{5} + \zeta_{28}^{4} q^{7} + \zeta_{28}^{3} q^{8} + ( \zeta_{28}^{10} + \zeta_{28}^{12} ) q^{10} + ( -\zeta_{28}^{3} + \zeta_{28}^{13} ) q^{11} + \zeta_{28}^{5} q^{14} + \zeta_{28}^{4} q^{16} + ( \zeta_{28}^{11} + \zeta_{28}^{13} ) q^{20} + ( -1 - \zeta_{28}^{4} ) q^{22} + ( -\zeta_{28}^{4} - \zeta_{28}^{6} - \zeta_{28}^{8} ) q^{25} + \zeta_{28}^{6} q^{28} + ( \zeta_{28}^{11} - \zeta_{28}^{13} ) q^{29} + ( \zeta_{28}^{4} + \zeta_{28}^{10} ) q^{31} + \zeta_{28}^{5} q^{32} + ( -\zeta_{28} + \zeta_{28}^{13} ) q^{35} + ( -1 + \zeta_{28}^{12} ) q^{40} + ( -\zeta_{28} - \zeta_{28}^{5} ) q^{44} + \zeta_{28}^{8} q^{49} + ( -\zeta_{28}^{5} - \zeta_{28}^{7} - \zeta_{28}^{9} ) q^{50} -2 \zeta_{28}^{9} q^{53} + ( 1 - \zeta_{28}^{8} - \zeta_{28}^{10} - \zeta_{28}^{12} ) q^{55} + \zeta_{28}^{7} q^{56} + ( 1 + \zeta_{28}^{12} ) q^{58} + ( \zeta_{28}^{5} + \zeta_{28}^{11} ) q^{62} + \zeta_{28}^{6} q^{64} + ( -1 - \zeta_{28}^{2} ) q^{70} + ( -\zeta_{28}^{2} + \zeta_{28}^{10} ) q^{73} + ( -\zeta_{28}^{3} - \zeta_{28}^{7} ) q^{77} + ( -\zeta_{28}^{4} + \zeta_{28}^{10} ) q^{79} + ( -\zeta_{28} + \zeta_{28}^{13} ) q^{80} + ( \zeta_{28}^{5} + \zeta_{28}^{7} ) q^{83} + ( -\zeta_{28}^{2} - \zeta_{28}^{6} ) q^{88} + ( -\zeta_{28}^{2} - \zeta_{28}^{12} ) q^{97} + \zeta_{28}^{9} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 2q^{4} - 2q^{7} + O(q^{10}) \) \( 12q + 2q^{4} - 2q^{7} - 2q^{16} - 10q^{22} + 2q^{25} + 2q^{28} - 14q^{40} - 2q^{49} + 14q^{55} + 10q^{58} + 2q^{64} - 14q^{70} + 4q^{79} - 4q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(\zeta_{28}^{6}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
−0.974928 0.222521i
0.974928 + 0.222521i
−0.974928 + 0.222521i
0.974928 0.222521i
−0.433884 0.900969i
0.433884 + 0.900969i
−0.781831 + 0.623490i
0.781831 0.623490i
−0.781831 0.623490i
0.781831 + 0.623490i
−0.433884 + 0.900969i
0.433884 0.900969i
−0.974928 0.222521i 0 0.900969 + 0.433884i 1.21572 1.52446i 0 0.623490 + 0.781831i −0.781831 0.623490i 0 −1.52446 + 1.21572i
181.2 0.974928 + 0.222521i 0 0.900969 + 0.433884i −1.21572 + 1.52446i 0 0.623490 + 0.781831i 0.781831 + 0.623490i 0 −1.52446 + 1.21572i
1189.1 −0.974928 + 0.222521i 0 0.900969 0.433884i 1.21572 + 1.52446i 0 0.623490 0.781831i −0.781831 + 0.623490i 0 −1.52446 1.21572i
1189.2 0.974928 0.222521i 0 0.900969 0.433884i −1.21572 1.52446i 0 0.623490 0.781831i 0.781831 0.623490i 0 −1.52446 1.21572i
1693.1 −0.433884 0.900969i 0 −0.623490 + 0.781831i −0.193096 + 0.846011i 0 −0.222521 0.974928i 0.974928 + 0.222521i 0 0.846011 0.193096i
1693.2 0.433884 + 0.900969i 0 −0.623490 + 0.781831i 0.193096 0.846011i 0 −0.222521 0.974928i −0.974928 0.222521i 0 0.846011 0.193096i
2197.1 −0.781831 + 0.623490i 0 0.222521 0.974928i −1.40881 + 0.678448i 0 −0.900969 0.433884i 0.433884 + 0.900969i 0 0.678448 1.40881i
2197.2 0.781831 0.623490i 0 0.222521 0.974928i 1.40881 0.678448i 0 −0.900969 0.433884i −0.433884 0.900969i 0 0.678448 1.40881i
2701.1 −0.781831 0.623490i 0 0.222521 + 0.974928i −1.40881 0.678448i 0 −0.900969 + 0.433884i 0.433884 0.900969i 0 0.678448 + 1.40881i
2701.2 0.781831 + 0.623490i 0 0.222521 + 0.974928i 1.40881 + 0.678448i 0 −0.900969 + 0.433884i −0.433884 + 0.900969i 0 0.678448 + 1.40881i
3205.1 −0.433884 + 0.900969i 0 −0.623490 0.781831i −0.193096 0.846011i 0 −0.222521 + 0.974928i 0.974928 0.222521i 0 0.846011 + 0.193096i
3205.2 0.433884 0.900969i 0 −0.623490 0.781831i 0.193096 + 0.846011i 0 −0.222521 + 0.974928i −0.974928 + 0.222521i 0 0.846011 + 0.193096i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3205.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
49.f odd 14 1 inner
147.k even 14 1 inner
392.r odd 14 1 inner
1176.bo even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.dh.a 12
3.b odd 2 1 inner 3528.1.dh.a 12
8.b even 2 1 inner 3528.1.dh.a 12
24.h odd 2 1 CM 3528.1.dh.a 12
49.f odd 14 1 inner 3528.1.dh.a 12
147.k even 14 1 inner 3528.1.dh.a 12
392.r odd 14 1 inner 3528.1.dh.a 12
1176.bo even 14 1 inner 3528.1.dh.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.dh.a 12 1.a even 1 1 trivial
3528.1.dh.a 12 3.b odd 2 1 inner
3528.1.dh.a 12 8.b even 2 1 inner
3528.1.dh.a 12 24.h odd 2 1 CM
3528.1.dh.a 12 49.f odd 14 1 inner
3528.1.dh.a 12 147.k even 14 1 inner
3528.1.dh.a 12 392.r odd 14 1 inner
3528.1.dh.a 12 1176.bo even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3528, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
$3$ \( T^{12} \)
$5$ \( 49 + 98 T^{2} + 49 T^{4} - 14 T^{6} + 14 T^{8} + T^{12} \)
$7$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$11$ \( 1 - 2 T^{2} + 25 T^{4} + 6 T^{6} + 2 T^{8} - 4 T^{10} + T^{12} \)
$13$ \( T^{12} \)
$17$ \( T^{12} \)
$19$ \( T^{12} \)
$23$ \( T^{12} \)
$29$ \( 1 + 5 T^{2} + 18 T^{4} - 29 T^{6} + 16 T^{8} - 4 T^{10} + T^{12} \)
$31$ \( ( 7 + 14 T^{2} + 7 T^{4} + T^{6} )^{2} \)
$37$ \( T^{12} \)
$41$ \( T^{12} \)
$43$ \( T^{12} \)
$47$ \( T^{12} \)
$53$ \( 4096 - 1024 T^{2} + 256 T^{4} - 64 T^{6} + 16 T^{8} - 4 T^{10} + T^{12} \)
$59$ \( T^{12} \)
$61$ \( T^{12} \)
$67$ \( T^{12} \)
$71$ \( T^{12} \)
$73$ \( ( 7 - 7 T + 7 T^{3} + T^{6} )^{2} \)
$79$ \( ( 1 - 2 T - T^{2} + T^{3} )^{4} \)
$83$ \( 49 - 49 T^{2} + 49 T^{4} + 35 T^{6} + 21 T^{8} + 7 T^{10} + T^{12} \)
$89$ \( T^{12} \)
$97$ \( ( 7 + 14 T^{2} + 7 T^{4} + T^{6} )^{2} \)
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