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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 $56$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3481,1,Mod(506,3481)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3481, base_ring=CyclotomicField(58))
 
chi = DirichletCharacter(H, H._module([41]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3481.506");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3481 = 59^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3481.d (of order \(58\), degree \(28\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.73724530898\)
Analytic rank: \(0\)
Dimension: \(28\)
Coefficient field: \(\Q(\zeta_{58})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} + \cdots + 1 \) Copy content Toggle raw display
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: $S_3\times C_{29}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{87} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 506.1
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 \(S_{1}^{\mathrm{new}}(3481, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{28} \) Copy content Toggle raw display
$3$ \( T^{28} - T^{27} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{28} - T^{27} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{28} - T^{27} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{28} \) Copy content Toggle raw display
$13$ \( T^{28} \) Copy content Toggle raw display
$17$ \( T^{28} + \cdots + 268435456 \) Copy content Toggle raw display
$19$ \( T^{28} - T^{27} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{28} \) Copy content Toggle raw display
$29$ \( T^{28} - T^{27} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{28} \) Copy content Toggle raw display
$37$ \( T^{28} \) Copy content Toggle raw display
$41$ \( T^{28} - T^{27} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{28} \) Copy content Toggle raw display
$47$ \( T^{28} \) Copy content Toggle raw display
$53$ \( T^{28} - T^{27} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{28} \) Copy content Toggle raw display
$61$ \( T^{28} \) Copy content Toggle raw display
$67$ \( T^{28} \) Copy content Toggle raw display
$71$ \( T^{28} + \cdots + 268435456 \) Copy content Toggle raw display
$73$ \( T^{28} \) Copy content Toggle raw display
$79$ \( T^{28} - T^{27} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{28} \) Copy content Toggle raw display
$89$ \( T^{28} \) Copy content Toggle raw display
$97$ \( T^{28} \) Copy content Toggle raw display
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