Properties

Label 3481.1.d.a
Level $3481$
Weight $1$
Character orbit 3481.d
Analytic conductor $1.737$
Analytic rank $0$
Dimension $28$
Projective image $D_{3}$
CM discriminant -59
Inner twists $56$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3481 = 59^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3481.d (of order \(58\), degree \(28\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.73724530898\)
Analytic rank: \(0\)
Dimension: \(28\)
Coefficient field: \(\Q(\zeta_{58})\)
Defining polynomial: \(x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 59)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.59.1
Artin image $C_{29}\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{87} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{58}^{17} q^{3} -\zeta_{58}^{3} q^{4} + \zeta_{58}^{9} q^{5} + \zeta_{58}^{27} q^{7} +O(q^{10})\) \( q + \zeta_{58}^{17} q^{3} -\zeta_{58}^{3} q^{4} + \zeta_{58}^{9} q^{5} + \zeta_{58}^{27} q^{7} -\zeta_{58}^{20} q^{12} + \zeta_{58}^{26} q^{15} + \zeta_{58}^{6} q^{16} + 2 \zeta_{58}^{2} q^{17} -\zeta_{58}^{28} q^{19} -\zeta_{58}^{12} q^{20} -\zeta_{58}^{15} q^{21} + \zeta_{58}^{22} q^{27} + \zeta_{58} q^{28} + \zeta_{58}^{13} q^{29} -\zeta_{58}^{7} q^{35} + \zeta_{58}^{21} q^{41} + \zeta_{58}^{23} q^{48} + 2 \zeta_{58}^{19} q^{51} -\zeta_{58}^{4} q^{53} + \zeta_{58}^{16} q^{57} + q^{60} -\zeta_{58}^{9} q^{64} -2 \zeta_{58}^{5} q^{68} + 2 \zeta_{58}^{20} q^{71} -\zeta_{58}^{2} q^{76} -\zeta_{58}^{12} q^{79} + \zeta_{58}^{15} q^{80} -\zeta_{58}^{10} q^{81} + \zeta_{58}^{18} q^{84} + 2 \zeta_{58}^{11} q^{85} -\zeta_{58} q^{87} + \zeta_{58}^{8} q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + q^{3} - q^{4} + q^{5} + q^{7} + O(q^{10}) \) \( 28q + q^{3} - q^{4} + q^{5} + q^{7} + q^{12} - q^{15} - q^{16} - 2q^{17} + q^{19} + q^{20} - q^{21} - q^{27} + q^{28} + q^{29} - q^{35} + q^{41} + q^{48} + 2q^{51} + q^{53} - q^{57} + 28q^{60} - q^{64} - 2q^{68} - 2q^{71} + q^{76} + q^{79} + q^{80} + q^{81} - q^{84} + 2q^{85} - q^{87} - q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3481\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{58}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
506.1
−0.907575 0.419889i
0.856857 + 0.515554i
0.161782 + 0.986827i
0.725995 0.687699i
−0.796093 + 0.605174i
0.994138 + 0.108119i
−0.468408 0.883512i
0.370138 + 0.928977i
0.561187 + 0.827689i
−0.0541389 + 0.998533i
−0.976621 + 0.214970i
−0.468408 + 0.883512i
−0.647386 0.762162i
0.856857 0.515554i
−0.0541389 0.998533i
−0.267528 0.963550i
0.370138 0.928977i
−0.796093 0.605174i
0.161782 0.986827i
−0.647386 + 0.762162i
0 −0.468408 0.883512i 0.267528 + 0.963550i 0.725995 + 0.687699i 0 −0.647386 + 0.762162i 0 0 0
672.1 0 −0.976621 + 0.214970i 0.0541389 0.998533i 0.161782 0.986827i 0 −0.468408 + 0.883512i 0 0 0
805.1 0 0.370138 0.928977i 0.468408 + 0.883512i 0.994138 + 0.108119i 0 0.947653 + 0.319302i 0 0 0
806.1 0 0.947653 0.319302i 0.647386 + 0.762162i 0.856857 0.515554i 0 −0.0541389 0.998533i 0 0 0
809.1 0 −0.0541389 0.998533i −0.370138 0.928977i −0.907575 0.419889i 0 −0.267528 0.963550i 0 0 0
893.1 0 −0.267528 + 0.963550i −0.947653 0.319302i 0.561187 + 0.827689i 0 −0.976621 + 0.214970i 0 0 0
946.1 0 −0.907575 + 0.419889i −0.994138 0.108119i 0.947653 + 0.319302i 0 0.561187 + 0.827689i 0 0 0
1105.1 0 0.161782 + 0.986827i 0.907575 + 0.419889i −0.267528 0.963550i 0 0.725995 + 0.687699i 0 0 0
1106.1 0 −0.647386 0.762162i 0.976621 0.214970i −0.796093 + 0.605174i 0 0.370138 + 0.928977i 0 0 0
1311.1 0 −0.796093 + 0.605174i −0.161782 + 0.986827i −0.468408 + 0.883512i 0 0.994138 0.108119i 0 0 0
1404.1 0 0.856857 0.515554i 0.796093 0.605174i 0.370138 + 0.928977i 0 −0.907575 0.419889i 0 0 0
1505.1 0 −0.907575 0.419889i −0.994138 + 0.108119i 0.947653 0.319302i 0 0.561187 0.827689i 0 0 0
1558.1 0 0.561187 0.827689i −0.856857 + 0.515554i −0.0541389 0.998533i 0 0.161782 + 0.986827i 0 0 0
1611.1 0 −0.976621 0.214970i 0.0541389 + 0.998533i 0.161782 + 0.986827i 0 −0.468408 0.883512i 0 0 0
1702.1 0 −0.796093 0.605174i −0.161782 0.986827i −0.468408 0.883512i 0 0.994138 + 0.108119i 0 0 0
1839.1 0 0.994138 + 0.108119i −0.725995 0.687699i −0.647386 + 0.762162i 0 0.856857 + 0.515554i 0 0 0
2076.1 0 0.161782 0.986827i 0.907575 0.419889i −0.267528 + 0.963550i 0 0.725995 0.687699i 0 0 0
2117.1 0 −0.0541389 + 0.998533i −0.370138 + 0.928977i −0.907575 + 0.419889i 0 −0.267528 + 0.963550i 0 0 0
2374.1 0 0.370138 + 0.928977i 0.468408 0.883512i 0.994138 0.108119i 0 0.947653 0.319302i 0 0 0
2451.1 0 0.561187 + 0.827689i −0.856857 0.515554i −0.0541389 + 0.998533i 0 0.161782 0.986827i 0 0 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3428.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.b odd 2 1 CM by \(\Q(\sqrt{-59}) \)
59.c even 29 27 inner
59.d odd 58 27 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3481.1.d.a 28
59.b odd 2 1 CM 3481.1.d.a 28
59.c even 29 1 59.1.b.a 1
59.c even 29 27 inner 3481.1.d.a 28
59.d odd 58 1 59.1.b.a 1
59.d odd 58 27 inner 3481.1.d.a 28
177.f even 58 1 531.1.c.a 1
177.h odd 58 1 531.1.c.a 1
236.g even 58 1 944.1.h.a 1
236.h odd 58 1 944.1.h.a 1
295.h odd 58 1 1475.1.c.b 1
295.j even 58 1 1475.1.c.b 1
295.k odd 116 2 1475.1.d.a 2
295.l even 116 2 1475.1.d.a 2
413.j odd 58 1 2891.1.c.e 1
413.l even 58 1 2891.1.c.e 1
413.m even 87 2 2891.1.g.d 2
413.n even 174 2 2891.1.g.b 2
413.o odd 174 2 2891.1.g.d 2
413.p odd 174 2 2891.1.g.b 2
472.k odd 58 1 3776.1.h.a 1
472.l even 58 1 3776.1.h.a 1
472.o odd 58 1 3776.1.h.b 1
472.p even 58 1 3776.1.h.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.1.b.a 1 59.c even 29 1
59.1.b.a 1 59.d odd 58 1
531.1.c.a 1 177.f even 58 1
531.1.c.a 1 177.h odd 58 1
944.1.h.a 1 236.g even 58 1
944.1.h.a 1 236.h odd 58 1
1475.1.c.b 1 295.h odd 58 1
1475.1.c.b 1 295.j even 58 1
1475.1.d.a 2 295.k odd 116 2
1475.1.d.a 2 295.l even 116 2
2891.1.c.e 1 413.j odd 58 1
2891.1.c.e 1 413.l even 58 1
2891.1.g.b 2 413.n even 174 2
2891.1.g.b 2 413.p odd 174 2
2891.1.g.d 2 413.m even 87 2
2891.1.g.d 2 413.o odd 174 2
3481.1.d.a 28 1.a even 1 1 trivial
3481.1.d.a 28 59.b odd 2 1 CM
3481.1.d.a 28 59.c even 29 27 inner
3481.1.d.a 28 59.d odd 58 27 inner
3776.1.h.a 1 472.k odd 58 1
3776.1.h.a 1 472.l even 58 1
3776.1.h.b 1 472.o odd 58 1
3776.1.h.b 1 472.p even 58 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(3481, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{28} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} \)
$5$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} \)
$7$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} \)
$11$ \( T^{28} \)
$13$ \( T^{28} \)
$17$ \( 268435456 + 134217728 T + 67108864 T^{2} + 33554432 T^{3} + 16777216 T^{4} + 8388608 T^{5} + 4194304 T^{6} + 2097152 T^{7} + 1048576 T^{8} + 524288 T^{9} + 262144 T^{10} + 131072 T^{11} + 65536 T^{12} + 32768 T^{13} + 16384 T^{14} + 8192 T^{15} + 4096 T^{16} + 2048 T^{17} + 1024 T^{18} + 512 T^{19} + 256 T^{20} + 128 T^{21} + 64 T^{22} + 32 T^{23} + 16 T^{24} + 8 T^{25} + 4 T^{26} + 2 T^{27} + T^{28} \)
$19$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} \)
$23$ \( T^{28} \)
$29$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} \)
$31$ \( T^{28} \)
$37$ \( T^{28} \)
$41$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} \)
$43$ \( T^{28} \)
$47$ \( T^{28} \)
$53$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} \)
$59$ \( T^{28} \)
$61$ \( T^{28} \)
$67$ \( T^{28} \)
$71$ \( 268435456 + 134217728 T + 67108864 T^{2} + 33554432 T^{3} + 16777216 T^{4} + 8388608 T^{5} + 4194304 T^{6} + 2097152 T^{7} + 1048576 T^{8} + 524288 T^{9} + 262144 T^{10} + 131072 T^{11} + 65536 T^{12} + 32768 T^{13} + 16384 T^{14} + 8192 T^{15} + 4096 T^{16} + 2048 T^{17} + 1024 T^{18} + 512 T^{19} + 256 T^{20} + 128 T^{21} + 64 T^{22} + 32 T^{23} + 16 T^{24} + 8 T^{25} + 4 T^{26} + 2 T^{27} + T^{28} \)
$73$ \( T^{28} \)
$79$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} \)
$83$ \( T^{28} \)
$89$ \( T^{28} \)
$97$ \( T^{28} \)
show more
show less