Properties

Label 3528.1.fb.a
Level $3528$
Weight $1$
Character orbit 3528.fb
Analytic conductor $1.761$
Analytic rank $0$
Dimension $24$
Projective image $D_{42}$
CM discriminant -24
Inner twists $8$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.fb (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.76070136457\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{42})\)
Coefficient field: \(\Q(\zeta_{84})\)
Defining polynomial: \(x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{42}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{84}^{31} q^{2} -\zeta_{84}^{20} q^{4} + ( -\zeta_{84}^{33} + \zeta_{84}^{41} ) q^{5} -\zeta_{84}^{26} q^{7} -\zeta_{84}^{9} q^{8} +O(q^{10})\) \( q -\zeta_{84}^{31} q^{2} -\zeta_{84}^{20} q^{4} + ( -\zeta_{84}^{33} + \zeta_{84}^{41} ) q^{5} -\zeta_{84}^{26} q^{7} -\zeta_{84}^{9} q^{8} + ( -\zeta_{84}^{22} + \zeta_{84}^{30} ) q^{10} + ( \zeta_{84}^{37} - \zeta_{84}^{39} ) q^{11} -\zeta_{84}^{15} q^{14} + \zeta_{84}^{40} q^{16} + ( -\zeta_{84}^{11} + \zeta_{84}^{19} ) q^{20} + ( \zeta_{84}^{26} - \zeta_{84}^{28} ) q^{22} + ( -\zeta_{84}^{24} + \zeta_{84}^{32} - \zeta_{84}^{40} ) q^{25} -\zeta_{84}^{4} q^{28} + ( -\zeta_{84}^{5} - \zeta_{84}^{25} ) q^{29} + ( \zeta_{84}^{12} - \zeta_{84}^{16} ) q^{31} + \zeta_{84}^{29} q^{32} + ( -\zeta_{84}^{17} + \zeta_{84}^{25} ) q^{35} + ( -1 + \zeta_{84}^{8} ) q^{40} + ( \zeta_{84}^{15} - \zeta_{84}^{17} ) q^{44} -\zeta_{84}^{10} q^{49} + ( -\zeta_{84}^{13} + \zeta_{84}^{21} - \zeta_{84}^{29} ) q^{50} + \zeta_{84}^{41} q^{53} + ( \zeta_{84}^{28} - \zeta_{84}^{30} - \zeta_{84}^{36} + \zeta_{84}^{38} ) q^{55} + \zeta_{84}^{35} q^{56} + ( -\zeta_{84}^{14} + \zeta_{84}^{36} ) q^{58} + ( \zeta_{84}^{19} + \zeta_{84}^{33} ) q^{59} + ( \zeta_{84} - \zeta_{84}^{5} ) q^{62} + \zeta_{84}^{18} q^{64} + ( -\zeta_{84}^{6} + \zeta_{84}^{14} ) q^{70} + ( \zeta_{84}^{2} + \zeta_{84}^{20} ) q^{73} + ( \zeta_{84}^{21} - \zeta_{84}^{23} ) q^{77} + ( \zeta_{84}^{2} - \zeta_{84}^{12} ) q^{79} + ( \zeta_{84}^{31} - \zeta_{84}^{39} ) q^{80} + ( \zeta_{84}^{7} + \zeta_{84}^{29} ) q^{83} + ( \zeta_{84}^{4} - \zeta_{84}^{6} ) q^{88} + ( -\zeta_{84}^{8} - \zeta_{84}^{34} ) q^{97} + \zeta_{84}^{41} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 2q^{4} + 2q^{7} + O(q^{10}) \) \( 24q - 2q^{4} + 2q^{7} + 6q^{10} + 2q^{16} + 10q^{22} + 4q^{25} - 2q^{28} - 6q^{31} - 22q^{40} + 2q^{49} - 14q^{55} - 16q^{58} + 4q^{64} + 8q^{70} + 2q^{79} - 2q^{88} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(-\zeta_{84}^{4}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
397.1
−0.294755 + 0.955573i
0.294755 0.955573i
0.930874 + 0.365341i
−0.930874 0.365341i
−0.294755 0.955573i
0.294755 + 0.955573i
0.680173 0.733052i
−0.680173 + 0.733052i
0.563320 0.826239i
−0.563320 + 0.826239i
0.563320 + 0.826239i
−0.563320 0.826239i
−0.997204 + 0.0747301i
0.997204 0.0747301i
0.930874 0.365341i
−0.930874 + 0.365341i
0.680173 + 0.733052i
−0.680173 0.733052i
0.149042 + 0.988831i
−0.149042 0.988831i
−0.149042 0.988831i 0 −0.955573 + 0.294755i −0.139129 + 1.85654i 0 0.0747301 + 0.997204i 0.433884 + 0.900969i 0 1.85654 0.139129i
397.2 0.149042 + 0.988831i 0 −0.955573 + 0.294755i 0.139129 1.85654i 0 0.0747301 + 0.997204i −0.433884 0.900969i 0 1.85654 0.139129i
829.1 −0.563320 + 0.826239i 0 −0.365341 0.930874i −1.90580 + 0.587862i 0 0.955573 + 0.294755i 0.974928 + 0.222521i 0 0.587862 1.90580i
829.2 0.563320 0.826239i 0 −0.365341 0.930874i 1.90580 0.587862i 0 0.955573 + 0.294755i −0.974928 0.222521i 0 0.587862 1.90580i
1333.1 −0.149042 + 0.988831i 0 −0.955573 0.294755i −0.139129 1.85654i 0 0.0747301 0.997204i 0.433884 0.900969i 0 1.85654 + 0.139129i
1333.2 0.149042 0.988831i 0 −0.955573 0.294755i 0.139129 + 1.85654i 0 0.0747301 0.997204i −0.433884 + 0.900969i 0 1.85654 + 0.139129i
1405.1 −0.930874 + 0.365341i 0 0.733052 0.680173i −0.246289 + 0.167917i 0 0.826239 + 0.563320i −0.433884 + 0.900969i 0 0.167917 0.246289i
1405.2 0.930874 0.365341i 0 0.733052 0.680173i 0.246289 0.167917i 0 0.826239 + 0.563320i 0.433884 0.900969i 0 0.167917 0.246289i
1837.1 −0.294755 0.955573i 0 −0.826239 + 0.563320i −1.34515 0.202749i 0 −0.988831 + 0.149042i 0.781831 + 0.623490i 0 0.202749 + 1.34515i
1837.2 0.294755 + 0.955573i 0 −0.826239 + 0.563320i 1.34515 + 0.202749i 0 −0.988831 + 0.149042i −0.781831 0.623490i 0 0.202749 + 1.34515i
1909.1 −0.294755 + 0.955573i 0 −0.826239 0.563320i −1.34515 + 0.202749i 0 −0.988831 0.149042i 0.781831 0.623490i 0 0.202749 1.34515i
1909.2 0.294755 0.955573i 0 −0.826239 0.563320i 1.34515 0.202749i 0 −0.988831 0.149042i −0.781831 + 0.623490i 0 0.202749 1.34515i
2341.1 −0.680173 0.733052i 0 −0.0747301 + 0.997204i 0.215372 0.548760i 0 0.365341 + 0.930874i 0.781831 0.623490i 0 −0.548760 + 0.215372i
2341.2 0.680173 + 0.733052i 0 −0.0747301 + 0.997204i −0.215372 + 0.548760i 0 0.365341 + 0.930874i −0.781831 + 0.623490i 0 −0.548760 + 0.215372i
2413.1 −0.563320 0.826239i 0 −0.365341 + 0.930874i −1.90580 0.587862i 0 0.955573 0.294755i 0.974928 0.222521i 0 0.587862 + 1.90580i
2413.2 0.563320 + 0.826239i 0 −0.365341 + 0.930874i 1.90580 + 0.587862i 0 0.955573 0.294755i −0.974928 + 0.222521i 0 0.587862 + 1.90580i
2845.1 −0.930874 0.365341i 0 0.733052 + 0.680173i −0.246289 0.167917i 0 0.826239 0.563320i −0.433884 0.900969i 0 0.167917 + 0.246289i
2845.2 0.930874 + 0.365341i 0 0.733052 + 0.680173i 0.246289 + 0.167917i 0 0.826239 0.563320i 0.433884 + 0.900969i 0 0.167917 + 0.246289i
2917.1 −0.997204 0.0747301i 0 0.988831 + 0.149042i 0.825886 + 0.766310i 0 −0.733052 + 0.680173i −0.974928 0.222521i 0 −0.766310 0.825886i
2917.2 0.997204 + 0.0747301i 0 0.988831 + 0.149042i −0.825886 0.766310i 0 −0.733052 + 0.680173i 0.974928 + 0.222521i 0 −0.766310 0.825886i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3421.2
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.h odd 42 1 inner
147.o even 42 1 inner
392.bf odd 42 1 inner
1176.cc even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.fb.a 24
3.b odd 2 1 inner 3528.1.fb.a 24
8.b even 2 1 inner 3528.1.fb.a 24
24.h odd 2 1 CM 3528.1.fb.a 24
49.h odd 42 1 inner 3528.1.fb.a 24
147.o even 42 1 inner 3528.1.fb.a 24
392.bf odd 42 1 inner 3528.1.fb.a 24
1176.cc even 42 1 inner 3528.1.fb.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.fb.a 24 1.a even 1 1 trivial
3528.1.fb.a 24 3.b odd 2 1 inner
3528.1.fb.a 24 8.b even 2 1 inner
3528.1.fb.a 24 24.h odd 2 1 CM
3528.1.fb.a 24 49.h odd 42 1 inner
3528.1.fb.a 24 147.o even 42 1 inner
3528.1.fb.a 24 392.bf odd 42 1 inner
3528.1.fb.a 24 1176.cc even 42 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
$3$ \( T^{24} \)
$5$ \( 1 - 5 T^{2} + 105 T^{4} + 363 T^{6} + 663 T^{8} - 728 T^{10} + 610 T^{12} - 427 T^{14} + 24 T^{16} + 83 T^{18} - 14 T^{20} - 3 T^{22} + T^{24} \)
$7$ \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
$11$ \( 1 + 71 T^{2} + 1785 T^{4} - 5881 T^{6} + 2267 T^{8} + 3416 T^{10} + 1198 T^{12} + 525 T^{14} + 132 T^{16} - T^{18} + 14 T^{20} + T^{22} + T^{24} \)
$13$ \( T^{24} \)
$17$ \( T^{24} \)
$19$ \( T^{24} \)
$23$ \( T^{24} \)
$29$ \( 1 - 23 T^{2} + 2068 T^{4} - 6220 T^{6} + 5388 T^{8} + 785 T^{10} + 1421 T^{12} - 195 T^{14} + 173 T^{16} - 39 T^{18} + 3 T^{20} - 2 T^{22} + T^{24} \)
$31$ \( ( 1 - 6 T + 10 T^{2} + 12 T^{3} - 24 T^{4} - 21 T^{5} + 36 T^{6} + 42 T^{7} + 2 T^{8} - 12 T^{9} - T^{10} + 3 T^{11} + T^{12} )^{2} \)
$37$ \( T^{24} \)
$41$ \( T^{24} \)
$43$ \( T^{24} \)
$47$ \( T^{24} \)
$53$ \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
$59$ \( 531441 - 177147 T^{2} + 19683 T^{6} - 6561 T^{8} + 729 T^{12} - 81 T^{16} + 27 T^{18} - 3 T^{22} + T^{24} \)
$61$ \( T^{24} \)
$67$ \( T^{24} \)
$71$ \( T^{24} \)
$73$ \( ( 49 + 49 T + 49 T^{2} + 98 T^{3} + 49 T^{4} + 63 T^{6} + 7 T^{7} + 14 T^{9} + T^{12} )^{2} \)
$79$ \( ( 1 - 8 T + 56 T^{2} - 76 T^{3} + 118 T^{4} - 49 T^{5} + 78 T^{6} - 28 T^{7} + 34 T^{8} - 6 T^{9} + 7 T^{10} - T^{11} + T^{12} )^{2} \)
$83$ \( 1 + 10 T^{2} + 159 T^{4} + 668 T^{6} + 927 T^{8} - 108 T^{10} + 301 T^{12} + 408 T^{14} + 132 T^{16} + 10 T^{18} + 13 T^{20} - T^{22} + T^{24} \)
$89$ \( T^{24} \)
$97$ \( ( 1 + 16 T^{2} + 60 T^{4} + 78 T^{6} + 44 T^{8} + 11 T^{10} + T^{12} )^{2} \)
show more
show less