Properties

Label 3332.1.o.g
Level $3332$
Weight $1$
Character orbit 3332.o
Analytic conductor $1.663$
Analytic rank $0$
Dimension $16$
Projective image $D_{10}$
CM discriminant -119
Inner twists $16$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,1,Mod(67,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.67");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.o (of order \(6\), degree \(2\), 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: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{60})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.205346735104.4

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{60}^{4} q^{2} + ( - \zeta_{60}^{29} - \zeta_{60}^{11}) q^{3} + \zeta_{60}^{8} q^{4} + ( - \zeta_{60}^{13} + \zeta_{60}^{7}) q^{5} + ( - \zeta_{60}^{15} + \zeta_{60}^{3}) q^{6} + \zeta_{60}^{12} q^{8} + ( - \zeta_{60}^{28} + \zeta_{60}^{22} - \zeta_{60}^{10}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{60}^{4} q^{2} + ( - \zeta_{60}^{29} - \zeta_{60}^{11}) q^{3} + \zeta_{60}^{8} q^{4} + ( - \zeta_{60}^{13} + \zeta_{60}^{7}) q^{5} + ( - \zeta_{60}^{15} + \zeta_{60}^{3}) q^{6} + \zeta_{60}^{12} q^{8} + ( - \zeta_{60}^{28} + \zeta_{60}^{22} - \zeta_{60}^{10}) q^{9} + ( - \zeta_{60}^{17} + \zeta_{60}^{11}) q^{10} + ( - \zeta_{60}^{19} + \zeta_{60}^{7}) q^{12} + (\zeta_{60}^{24} - \zeta_{60}^{18} - \zeta_{60}^{12} + \zeta_{60}^{6}) q^{15} + \zeta_{60}^{16} q^{16} - \zeta_{60}^{5} q^{17} + (\zeta_{60}^{26} - \zeta_{60}^{14} + \zeta_{60}^{2}) q^{18} + ( - \zeta_{60}^{21} + \zeta_{60}^{15}) q^{20} + ( - \zeta_{60}^{23} + \zeta_{60}^{11}) q^{24} + (\zeta_{60}^{26} - \zeta_{60}^{20} + \zeta_{60}^{14}) q^{25} + ( - \zeta_{60}^{27} + \zeta_{60}^{21} + \zeta_{60}^{9} + \zeta_{60}^{3}) q^{27} + (\zeta_{60}^{28} - \zeta_{60}^{22} - \zeta_{60}^{16} + \zeta_{60}^{10}) q^{30} + ( - \zeta_{60}^{23} - \zeta_{60}^{17}) q^{31} + \zeta_{60}^{20} q^{32} - \zeta_{60}^{9} q^{34} + ( - \zeta_{60}^{18} + \zeta_{60}^{6} - 1) q^{36} + ( - \zeta_{60}^{25} + \zeta_{60}^{19}) q^{40} + (\zeta_{60}^{21} + \zeta_{60}^{9}) q^{41} + ( - \zeta_{60}^{24} - \zeta_{60}^{6}) q^{43} + (\zeta_{60}^{29} + \zeta_{60}^{23} - \zeta_{60}^{17} - \zeta_{60}^{11} + 2 \zeta_{60}^{5}) q^{45} + ( - \zeta_{60}^{27} + \zeta_{60}^{15}) q^{48} + ( - \zeta_{60}^{24} + \zeta_{60}^{18} - 1) q^{50} + (\zeta_{60}^{16} - \zeta_{60}^{4}) q^{51} + (\zeta_{60}^{8} - \zeta_{60}^{2}) q^{53} + (\zeta_{60}^{25} - \zeta_{60}^{13} + \zeta_{60}^{7} + \zeta_{60}) q^{54} + ( - \zeta_{60}^{26} - \zeta_{60}^{20} + \zeta_{60}^{14} - \zeta_{60}^{2}) q^{60} + (\zeta_{60}^{19} - \zeta_{60}) q^{61} + ( - \zeta_{60}^{27} - \zeta_{60}^{21}) q^{62} + \zeta_{60}^{24} q^{64} + ( - \zeta_{60}^{8} - \zeta_{60}^{2}) q^{67} - \zeta_{60}^{13} q^{68} + ( - \zeta_{60}^{22} + \zeta_{60}^{10} - \zeta_{60}^{4}) q^{72} + (\zeta_{60}^{29} - \zeta_{60}^{11}) q^{73} + (\zeta_{60}^{25} - \zeta_{60}^{19} + \zeta_{60}^{13} + \zeta_{60}^{7} - \zeta_{60}) q^{75} + ( - \zeta_{60}^{29} + \zeta_{60}^{23}) q^{80} + ( - \zeta_{60}^{26} + \zeta_{60}^{20} - \zeta_{60}^{14} + \zeta_{60}^{8} - \zeta_{60}^{2}) q^{81} + (\zeta_{60}^{25} + \zeta_{60}^{13}) q^{82} + (\zeta_{60}^{18} - \zeta_{60}^{12}) q^{85} + ( - \zeta_{60}^{28} - \zeta_{60}^{10}) q^{86} + (\zeta_{60}^{27} - \zeta_{60}^{21} - \zeta_{60}^{15} + \zeta_{60}^{9} - \zeta_{60}^{3}) q^{90} + (\zeta_{60}^{28} - \zeta_{60}^{22} - \zeta_{60}^{16} - \zeta_{60}^{4}) q^{93} + (\zeta_{60}^{19} + \zeta_{60}) q^{96} + (\zeta_{60}^{27} + \zeta_{60}^{3}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 2 q^{4} - 4 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} + 2 q^{4} - 4 q^{8} - 12 q^{9} + 2 q^{16} - 2 q^{18} + 4 q^{25} + 10 q^{30} - 8 q^{32} - 16 q^{36} - 8 q^{50} + 4 q^{53} + 10 q^{60} - 4 q^{64} + 8 q^{72} - 8 q^{81} + 8 q^{85} - 10 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(-\zeta_{60}^{10}\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.743145 0.669131i
0.743145 + 0.669131i
0.406737 0.913545i
−0.406737 + 0.913545i
−0.207912 0.978148i
0.207912 + 0.978148i
−0.994522 0.104528i
0.994522 + 0.104528i
−0.743145 + 0.669131i
0.743145 0.669131i
0.406737 + 0.913545i
−0.406737 0.913545i
−0.207912 + 0.978148i
0.207912 0.978148i
−0.994522 + 0.104528i
0.994522 0.104528i
−0.978148 + 0.207912i −0.951057 + 1.64728i 0.913545 0.406737i −1.40126 + 0.809017i 0.587785 1.80902i 0 −0.809017 + 0.587785i −1.30902 2.26728i 1.20243 1.08268i
67.2 −0.978148 + 0.207912i 0.951057 1.64728i 0.913545 0.406737i 1.40126 0.809017i −0.587785 + 1.80902i 0 −0.809017 + 0.587785i −1.30902 2.26728i −1.20243 + 1.08268i
67.3 −0.104528 + 0.994522i −0.587785 + 1.01807i −0.978148 0.207912i 0.535233 0.309017i −0.951057 0.690983i 0 0.309017 0.951057i −0.190983 0.330792i 0.251377 + 0.564602i
67.4 −0.104528 + 0.994522i 0.587785 1.01807i −0.978148 0.207912i −0.535233 + 0.309017i 0.951057 + 0.690983i 0 0.309017 0.951057i −0.190983 0.330792i −0.251377 0.564602i
67.5 0.669131 0.743145i −0.951057 + 1.64728i −0.104528 0.994522i 1.40126 0.809017i 0.587785 + 1.80902i 0 −0.809017 0.587785i −1.30902 2.26728i 0.336408 1.58268i
67.6 0.669131 0.743145i 0.951057 1.64728i −0.104528 0.994522i −1.40126 + 0.809017i −0.587785 1.80902i 0 −0.809017 0.587785i −1.30902 2.26728i −0.336408 + 1.58268i
67.7 0.913545 + 0.406737i −0.587785 + 1.01807i 0.669131 + 0.743145i −0.535233 + 0.309017i −0.951057 + 0.690983i 0 0.309017 + 0.951057i −0.190983 0.330792i −0.614648 + 0.0646021i
67.8 0.913545 + 0.406737i 0.587785 1.01807i 0.669131 + 0.743145i 0.535233 0.309017i 0.951057 0.690983i 0 0.309017 + 0.951057i −0.190983 0.330792i 0.614648 0.0646021i
2039.1 −0.978148 0.207912i −0.951057 1.64728i 0.913545 + 0.406737i −1.40126 0.809017i 0.587785 + 1.80902i 0 −0.809017 0.587785i −1.30902 + 2.26728i 1.20243 + 1.08268i
2039.2 −0.978148 0.207912i 0.951057 + 1.64728i 0.913545 + 0.406737i 1.40126 + 0.809017i −0.587785 1.80902i 0 −0.809017 0.587785i −1.30902 + 2.26728i −1.20243 1.08268i
2039.3 −0.104528 0.994522i −0.587785 1.01807i −0.978148 + 0.207912i 0.535233 + 0.309017i −0.951057 + 0.690983i 0 0.309017 + 0.951057i −0.190983 + 0.330792i 0.251377 0.564602i
2039.4 −0.104528 0.994522i 0.587785 + 1.01807i −0.978148 + 0.207912i −0.535233 0.309017i 0.951057 0.690983i 0 0.309017 + 0.951057i −0.190983 + 0.330792i −0.251377 + 0.564602i
2039.5 0.669131 + 0.743145i −0.951057 1.64728i −0.104528 + 0.994522i 1.40126 + 0.809017i 0.587785 1.80902i 0 −0.809017 + 0.587785i −1.30902 + 2.26728i 0.336408 + 1.58268i
2039.6 0.669131 + 0.743145i 0.951057 + 1.64728i −0.104528 + 0.994522i −1.40126 0.809017i −0.587785 + 1.80902i 0 −0.809017 + 0.587785i −1.30902 + 2.26728i −0.336408 1.58268i
2039.7 0.913545 0.406737i −0.587785 1.01807i 0.669131 0.743145i −0.535233 0.309017i −0.951057 0.690983i 0 0.309017 0.951057i −0.190983 + 0.330792i −0.614648 0.0646021i
2039.8 0.913545 0.406737i 0.587785 + 1.01807i 0.669131 0.743145i 0.535233 + 0.309017i 0.951057 + 0.690983i 0 0.309017 0.951057i −0.190983 + 0.330792i 0.614648 + 0.0646021i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
119.d odd 2 1 CM by \(\Q(\sqrt{-119}) \)
4.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
17.b even 2 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
68.d odd 2 1 inner
119.h odd 6 1 inner
119.j even 6 1 inner
476.e even 2 1 inner
476.o odd 6 1 inner
476.q even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.o.g 16
4.b odd 2 1 inner 3332.1.o.g 16
7.b odd 2 1 inner 3332.1.o.g 16
7.c even 3 1 3332.1.g.i 8
7.c even 3 1 inner 3332.1.o.g 16
7.d odd 6 1 3332.1.g.i 8
7.d odd 6 1 inner 3332.1.o.g 16
17.b even 2 1 inner 3332.1.o.g 16
28.d even 2 1 inner 3332.1.o.g 16
28.f even 6 1 3332.1.g.i 8
28.f even 6 1 inner 3332.1.o.g 16
28.g odd 6 1 3332.1.g.i 8
28.g odd 6 1 inner 3332.1.o.g 16
68.d odd 2 1 inner 3332.1.o.g 16
119.d odd 2 1 CM 3332.1.o.g 16
119.h odd 6 1 3332.1.g.i 8
119.h odd 6 1 inner 3332.1.o.g 16
119.j even 6 1 3332.1.g.i 8
119.j even 6 1 inner 3332.1.o.g 16
476.e even 2 1 inner 3332.1.o.g 16
476.o odd 6 1 3332.1.g.i 8
476.o odd 6 1 inner 3332.1.o.g 16
476.q even 6 1 3332.1.g.i 8
476.q even 6 1 inner 3332.1.o.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.g.i 8 7.c even 3 1
3332.1.g.i 8 7.d odd 6 1
3332.1.g.i 8 28.f even 6 1
3332.1.g.i 8 28.g odd 6 1
3332.1.g.i 8 119.h odd 6 1
3332.1.g.i 8 119.j even 6 1
3332.1.g.i 8 476.o odd 6 1
3332.1.g.i 8 476.q even 6 1
3332.1.o.g 16 1.a even 1 1 trivial
3332.1.o.g 16 4.b odd 2 1 inner
3332.1.o.g 16 7.b odd 2 1 inner
3332.1.o.g 16 7.c even 3 1 inner
3332.1.o.g 16 7.d odd 6 1 inner
3332.1.o.g 16 17.b even 2 1 inner
3332.1.o.g 16 28.d even 2 1 inner
3332.1.o.g 16 28.f even 6 1 inner
3332.1.o.g 16 28.g odd 6 1 inner
3332.1.o.g 16 68.d odd 2 1 inner
3332.1.o.g 16 119.d odd 2 1 CM
3332.1.o.g 16 119.h odd 6 1 inner
3332.1.o.g 16 119.j even 6 1 inner
3332.1.o.g 16 476.e even 2 1 inner
3332.1.o.g 16 476.o odd 6 1 inner
3332.1.o.g 16 476.q even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\):

\( T_{3}^{8} + 5T_{3}^{6} + 20T_{3}^{4} + 25T_{3}^{2} + 25 \) Copy content Toggle raw display
\( T_{5}^{8} - 3T_{5}^{6} + 8T_{5}^{4} - 3T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 5 T^{6} + 20 T^{4} + 25 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 3 T^{6} + 8 T^{4} - 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( (T^{8} + 5 T^{6} + 20 T^{4} + 25 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( (T^{4} + 3 T^{2} + 1)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 5 T^{2} + 5)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{4} - T^{3} + 2 T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} - 3 T^{6} + 8 T^{4} - 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 5 T^{6} + 20 T^{4} - 25 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{8} - 3 T^{6} + 8 T^{4} - 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( (T^{4} + 3 T^{2} + 1)^{4} \) Copy content Toggle raw display
show more
show less