Properties

Label 3703.1.l.f
Level $3703$
Weight $1$
Character orbit 3703.l
Analytic conductor $1.848$
Analytic rank $0$
Dimension $10$
Projective image $D_{11}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3703,1,Mod(118,3703)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3703, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 16]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3703.118");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3703 = 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3703.l (of order \(22\), degree \(10\), 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: \(1.84803774178\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 161)
Projective image: \(D_{11}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{11} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{22}^{7} - \zeta_{22}) q^{2} + (\zeta_{22}^{8} - \zeta_{22}^{3} + \zeta_{22}^{2}) q^{4} + \zeta_{22}^{5} q^{7} + (\zeta_{22}^{10} - \zeta_{22}^{9} - \zeta_{22}^{4} - \zeta_{22}^{3}) q^{8} - \zeta_{22}^{9} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{22}^{7} - \zeta_{22}) q^{2} + (\zeta_{22}^{8} - \zeta_{22}^{3} + \zeta_{22}^{2}) q^{4} + \zeta_{22}^{5} q^{7} + (\zeta_{22}^{10} - \zeta_{22}^{9} - \zeta_{22}^{4} - \zeta_{22}^{3}) q^{8} - \zeta_{22}^{9} q^{9} + ( - \zeta_{22}^{8} - \zeta_{22}^{6}) q^{11} + ( - \zeta_{22}^{6} + \zeta_{22}) q^{14} + (\zeta_{22}^{10} + \zeta_{22}^{6} + \zeta_{22}^{5} + \zeta_{22}^{4} - 1) q^{16} + (\zeta_{22}^{10} - \zeta_{22}^{5}) q^{18} + (\zeta_{22}^{9} + \zeta_{22}^{7} - \zeta_{22}^{4} - \zeta_{22}^{2}) q^{22} + \zeta_{22}^{4} q^{25} + ( - \zeta_{22}^{8} + \zeta_{22}^{7} - \zeta_{22}^{2}) q^{28} + (\zeta_{22}^{10} - \zeta_{22}^{7}) q^{29} + (\zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{5} + \zeta_{22}^{2} + \zeta_{22} - 1) q^{32} + (\zeta_{22}^{6} - \zeta_{22} + 1) q^{36} + ( - \zeta_{22}^{4} + \zeta_{22}^{3}) q^{37} + ( - \zeta_{22}^{8} + \zeta_{22}) q^{43} + ( - \zeta_{22}^{10} + \zeta_{22}^{9} - \zeta_{22}^{8} + \zeta_{22}^{5} + \zeta_{22}^{3} - 1) q^{44} + \zeta_{22}^{10} q^{49} + ( - \zeta_{22}^{5} + 1) q^{50} + (\zeta_{22}^{9} + \zeta_{22}) q^{53} + (\zeta_{22}^{9} - \zeta_{22}^{8} - \zeta_{22}^{4} + \zeta_{22}^{3}) q^{56} + (\zeta_{22}^{8} + \zeta_{22}^{6} - \zeta_{22}^{3} + 1) q^{58} + \zeta_{22}^{3} q^{63} + ( - \zeta_{22}^{9} + \zeta_{22}^{8} + \zeta_{22}^{7} + \zeta_{22}^{6} + \zeta_{22}^{3} - \zeta_{22}^{2} + \zeta_{22}) q^{64} + (\zeta_{22}^{7} + \zeta_{22}) q^{67} + ( - \zeta_{22}^{5} - \zeta_{22}^{3}) q^{71} + (\zeta_{22}^{8} - \zeta_{22}^{7} + \zeta_{22}^{2} - \zeta_{22}) q^{72} + ( - \zeta_{22}^{10} + \zeta_{22}^{5} - \zeta_{22}^{4} - 1) q^{74} + (\zeta_{22}^{2} + 1) q^{77} + ( - \zeta_{22}^{10} - \zeta_{22}^{2}) q^{79} - \zeta_{22}^{7} q^{81} + (\zeta_{22}^{9} - \zeta_{22}^{8} - \zeta_{22}^{4} - \zeta_{22}^{2}) q^{86} + ( - \zeta_{22}^{10} + \zeta_{22}^{9} + \zeta_{22}^{7} - \zeta_{22}^{6} + \zeta_{22}^{5} - \zeta_{22}^{4} + \zeta_{22} - 1) q^{88} + (\zeta_{22}^{6} + 1) q^{98} + ( - \zeta_{22}^{6} - \zeta_{22}^{4}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} - 3 q^{4} + q^{7} - 4 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{2} - 3 q^{4} + q^{7} - 4 q^{8} - q^{9} + 2 q^{11} + 2 q^{14} + 6 q^{16} - 2 q^{18} + 4 q^{22} - q^{25} + 3 q^{28} - 2 q^{29} + 5 q^{32} + 8 q^{36} + 2 q^{37} + 2 q^{43} - 5 q^{44} - q^{49} + 9 q^{50} + 2 q^{53} + 4 q^{56} + 7 q^{58} + q^{63} - 7 q^{64} + 2 q^{67} - 2 q^{71} - 4 q^{72} - 7 q^{74} + 9 q^{77} + 2 q^{79} - q^{81} + 4 q^{86} - 3 q^{88} + 9 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3703\mathbb{Z}\right)^\times\).

\(n\) \(2117\) \(3179\)
\(\chi(n)\) \(-1\) \(\zeta_{22}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
−0.415415 0.909632i
0.654861 0.755750i
−0.841254 0.540641i
0.142315 + 0.989821i
0.959493 0.281733i
0.959493 + 0.281733i
0.654861 + 0.755750i
−0.841254 + 0.540641i
−0.415415 + 0.909632i
0.142315 0.989821i
0.273100 + 1.89945i 0 −2.57385 + 0.755750i 0 0 −0.841254 + 0.540641i −1.34125 2.93694i −0.654861 0.755750i 0
699.1 −1.61435 + 0.474017i 0 1.54019 0.989821i 0 0 −0.415415 + 0.909632i −0.915415 + 1.05645i −0.142315 + 0.989821i 0
706.1 0.186393 0.215109i 0 0.130785 + 0.909632i 0 0 0.959493 0.281733i 0.459493 + 0.295298i 0.415415 0.909632i 0
1392.1 0.698939 0.449181i 0 −0.128663 + 0.281733i 0 0 0.654861 + 0.755750i 0.154861 + 1.07708i −0.959493 0.281733i 0
2582.1 −0.544078 + 1.19136i 0 −0.468468 0.540641i 0 0 0.142315 0.989821i −0.357685 + 0.105026i 0.841254 + 0.540641i 0
2603.1 −0.544078 1.19136i 0 −0.468468 + 0.540641i 0 0 0.142315 + 0.989821i −0.357685 0.105026i 0.841254 0.540641i 0
2617.1 −1.61435 0.474017i 0 1.54019 + 0.989821i 0 0 −0.415415 0.909632i −0.915415 1.05645i −0.142315 0.989821i 0
2911.1 0.186393 + 0.215109i 0 0.130785 0.909632i 0 0 0.959493 + 0.281733i 0.459493 0.295298i 0.415415 + 0.909632i 0
3044.1 0.273100 1.89945i 0 −2.57385 0.755750i 0 0 −0.841254 0.540641i −1.34125 + 2.93694i −0.654861 + 0.755750i 0
3429.1 0.698939 + 0.449181i 0 −0.128663 0.281733i 0 0 0.654861 0.755750i 0.154861 1.07708i −0.959493 + 0.281733i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
23.c even 11 1 inner
161.l odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3703.1.l.f 10
7.b odd 2 1 CM 3703.1.l.f 10
23.b odd 2 1 161.1.l.a 10
23.c even 11 1 3703.1.b.b 5
23.c even 11 2 3703.1.l.d 10
23.c even 11 2 3703.1.l.e 10
23.c even 11 1 inner 3703.1.l.f 10
23.c even 11 2 3703.1.l.g 10
23.c even 11 2 3703.1.l.i 10
23.d odd 22 1 161.1.l.a 10
23.d odd 22 1 3703.1.b.c 5
23.d odd 22 2 3703.1.l.a 10
23.d odd 22 2 3703.1.l.b 10
23.d odd 22 2 3703.1.l.c 10
23.d odd 22 2 3703.1.l.h 10
69.c even 2 1 1449.1.bq.a 10
69.g even 22 1 1449.1.bq.a 10
92.b even 2 1 2576.1.cj.a 10
92.h even 22 1 2576.1.cj.a 10
161.c even 2 1 161.1.l.a 10
161.f odd 6 2 1127.1.v.a 20
161.g even 6 2 1127.1.v.a 20
161.k even 22 1 161.1.l.a 10
161.k even 22 1 3703.1.b.c 5
161.k even 22 2 3703.1.l.a 10
161.k even 22 2 3703.1.l.b 10
161.k even 22 2 3703.1.l.c 10
161.k even 22 2 3703.1.l.h 10
161.l odd 22 1 3703.1.b.b 5
161.l odd 22 2 3703.1.l.d 10
161.l odd 22 2 3703.1.l.e 10
161.l odd 22 1 inner 3703.1.l.f 10
161.l odd 22 2 3703.1.l.g 10
161.l odd 22 2 3703.1.l.i 10
161.o even 66 2 1127.1.v.a 20
161.p odd 66 2 1127.1.v.a 20
483.c odd 2 1 1449.1.bq.a 10
483.w odd 22 1 1449.1.bq.a 10
644.h odd 2 1 2576.1.cj.a 10
644.r odd 22 1 2576.1.cj.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.1.l.a 10 23.b odd 2 1
161.1.l.a 10 23.d odd 22 1
161.1.l.a 10 161.c even 2 1
161.1.l.a 10 161.k even 22 1
1127.1.v.a 20 161.f odd 6 2
1127.1.v.a 20 161.g even 6 2
1127.1.v.a 20 161.o even 66 2
1127.1.v.a 20 161.p odd 66 2
1449.1.bq.a 10 69.c even 2 1
1449.1.bq.a 10 69.g even 22 1
1449.1.bq.a 10 483.c odd 2 1
1449.1.bq.a 10 483.w odd 22 1
2576.1.cj.a 10 92.b even 2 1
2576.1.cj.a 10 92.h even 22 1
2576.1.cj.a 10 644.h odd 2 1
2576.1.cj.a 10 644.r odd 22 1
3703.1.b.b 5 23.c even 11 1
3703.1.b.b 5 161.l odd 22 1
3703.1.b.c 5 23.d odd 22 1
3703.1.b.c 5 161.k even 22 1
3703.1.l.a 10 23.d odd 22 2
3703.1.l.a 10 161.k even 22 2
3703.1.l.b 10 23.d odd 22 2
3703.1.l.b 10 161.k even 22 2
3703.1.l.c 10 23.d odd 22 2
3703.1.l.c 10 161.k even 22 2
3703.1.l.d 10 23.c even 11 2
3703.1.l.d 10 161.l odd 22 2
3703.1.l.e 10 23.c even 11 2
3703.1.l.e 10 161.l odd 22 2
3703.1.l.f 10 1.a even 1 1 trivial
3703.1.l.f 10 7.b odd 2 1 CM
3703.1.l.f 10 23.c even 11 1 inner
3703.1.l.f 10 161.l odd 22 1 inner
3703.1.l.g 10 23.c even 11 2
3703.1.l.g 10 161.l odd 22 2
3703.1.l.h 10 23.d odd 22 2
3703.1.l.h 10 161.k even 22 2
3703.1.l.i 10 23.c even 11 2
3703.1.l.i 10 161.l odd 22 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3703, [\chi])\):

\( T_{2}^{10} + 2T_{2}^{9} + 4T_{2}^{8} + 8T_{2}^{7} + 5T_{2}^{6} - T_{2}^{5} - 2T_{2}^{4} - 4T_{2}^{3} + 14T_{2}^{2} - 5T_{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{10} - 2T_{11}^{9} + 4T_{11}^{8} + 3T_{11}^{7} - 6T_{11}^{6} + 12T_{11}^{5} + 9T_{11}^{4} - 7T_{11}^{3} + 14T_{11}^{2} - 6T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{10} \) Copy content Toggle raw display
$17$ \( T^{10} \) Copy content Toggle raw display
$19$ \( T^{10} \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{10} \) Copy content Toggle raw display
$37$ \( T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{10} \) Copy content Toggle raw display
$43$ \( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{10} \) Copy content Toggle raw display
$53$ \( T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{10} \) Copy content Toggle raw display
$61$ \( T^{10} \) Copy content Toggle raw display
$67$ \( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{10} + 2 T^{9} + 4 T^{8} - 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{10} \) Copy content Toggle raw display
$79$ \( T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{10} \) Copy content Toggle raw display
$89$ \( T^{10} \) Copy content Toggle raw display
$97$ \( T^{10} \) Copy content Toggle raw display
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