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Properties

Label 344.1.be.a
Level $344$
Weight $1$
Character orbit 344.be
Analytic conductor $0.172$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [344,1,Mod(67,344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(344, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 21, 40]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("344.67");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 344 = 2^{3} \cdot 43 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 344.be (of order \(42\), degree \(12\), 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: \(0.171678364346\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{21}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{42}^{15} q^{2} + ( - \zeta_{42}^{11} + \zeta_{42}^{8}) q^{3} - \zeta_{42}^{9} q^{4} + ( - \zeta_{42}^{5} + \zeta_{42}^{2}) q^{6} - \zeta_{42}^{3} q^{8} + ( - \zeta_{42}^{19} + \cdots - \zeta_{42}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{42}^{15} q^{2} + ( - \zeta_{42}^{11} + \zeta_{42}^{8}) q^{3} - \zeta_{42}^{9} q^{4} + ( - \zeta_{42}^{5} + \zeta_{42}^{2}) q^{6} - \zeta_{42}^{3} q^{8} + ( - \zeta_{42}^{19} + \cdots - \zeta_{42}) q^{9} + \cdots + (\zeta_{42}^{20} + \zeta_{42}^{18} + \cdots + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} - 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} - 2 q^{8} + 3 q^{9} + 2 q^{11} + 2 q^{12} - 2 q^{16} - q^{17} + 3 q^{18} - 8 q^{19} - 5 q^{22} - 5 q^{24} + q^{25} - 8 q^{27} - 2 q^{32} - 9 q^{33} + 13 q^{34} - 4 q^{36} - q^{38} + 2 q^{41} + q^{43} + 2 q^{44} + 2 q^{48} - 6 q^{49} - 6 q^{50} - 3 q^{51} + 6 q^{54} - 2 q^{57} + 2 q^{59} - 2 q^{64} - 2 q^{66} - q^{67} - q^{68} + 3 q^{72} - q^{73} + 10 q^{75} - q^{76} - 2 q^{81} + 2 q^{82} - 5 q^{83} + q^{86} - 5 q^{88} - q^{89} + 2 q^{96} + 2 q^{97} + q^{98} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/344\mathbb{Z}\right)^\times\).

\(n\) \(87\) \(89\) \(173\)
\(\chi(n)\) \(-1\) \(-\zeta_{42}^{19}\) \(-1\)

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
−0.988831 − 0.149042i
0.0747301 + 0.997204i
−0.733052 − 0.680173i
−0.733052 + 0.680173i
0.365341 + 0.930874i
0.365341 − 0.930874i
0.826239 + 0.563320i
−0.988831 + 0.149042i
0.955573 + 0.294755i
0.826239 − 0.563320i
0.0747301 − 0.997204i
0.955573 − 0.294755i
0.623490 − 0.781831i 0.440071 − 0.0663300i −0.222521 − 0.974928i 0 0.222521 − 0.385418i 0 −0.900969 − 0.433884i −0.766310 + 0.236375i 0
83.1 −0.900969 − 0.433884i 0.0931869 − 1.24349i 0.623490 + 0.781831i 0 −0.623490 + 1.07992i 0 −0.222521 − 0.974928i −0.548760 − 0.0827122i 0
99.1 −0.222521 + 0.974928i 1.32091 − 1.22563i −0.900969 − 0.433884i 0 0.900969 + 1.56052i 0 0.623490 − 0.781831i 0.167917 − 2.24070i 0
139.1 −0.222521 − 0.974928i 1.32091 + 1.22563i −0.900969 + 0.433884i 0 0.900969 − 1.56052i 0 0.623490 + 0.781831i 0.167917 + 2.24070i 0
187.1 0.623490 − 0.781831i −0.162592 + 0.414278i −0.222521 − 0.974928i 0 0.222521 + 0.385418i 0 −0.900969 − 0.433884i 0.587862 + 0.545456i 0
195.1 0.623490 + 0.781831i −0.162592 − 0.414278i −0.222521 + 0.974928i 0 0.222521 − 0.385418i 0 −0.900969 + 0.433884i 0.587862 − 0.545456i 0
203.1 −0.900969 + 0.433884i 1.03030 − 0.702449i 0.623490 − 0.781831i 0 −0.623490 + 1.07992i 0 −0.222521 + 0.974928i 0.202749 − 0.516596i 0
267.1 0.623490 + 0.781831i 0.440071 + 0.0663300i −0.222521 + 0.974928i 0 0.222521 + 0.385418i 0 −0.900969 + 0.433884i −0.766310 − 0.236375i 0
275.1 −0.222521 − 0.974928i −1.72188 + 0.531130i −0.900969 + 0.433884i 0 0.900969 + 1.56052i 0 0.623490 + 0.781831i 1.85654 − 1.26577i 0
283.1 −0.900969 − 0.433884i 1.03030 + 0.702449i 0.623490 + 0.781831i 0 −0.623490 − 1.07992i 0 −0.222521 − 0.974928i 0.202749 + 0.516596i 0
315.1 −0.900969 + 0.433884i 0.0931869 + 1.24349i 0.623490 − 0.781831i 0 −0.623490 − 1.07992i 0 −0.222521 + 0.974928i −0.548760 + 0.0827122i 0
339.1 −0.222521 + 0.974928i −1.72188 − 0.531130i −0.900969 − 0.433884i 0 0.900969 − 1.56052i 0 0.623490 − 0.781831i 1.85654 + 1.26577i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
43.g even 21 1 inner
344.be odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 344.1.be.a ✓ 12
3.b odd 2 1 3096.1.eq.a 12
4.b odd 2 1 1376.1.by.a 12
8.b even 2 1 1376.1.by.a 12
8.d odd 2 1 CM 344.1.be.a ✓ 12
24.f even 2 1 3096.1.eq.a 12
43.g even 21 1 inner 344.1.be.a ✓ 12
129.o odd 42 1 3096.1.eq.a 12
172.o odd 42 1 1376.1.by.a 12
344.z even 42 1 1376.1.by.a 12
344.be odd 42 1 inner 344.1.be.a ✓ 12
1032.ci even 42 1 3096.1.eq.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
344.1.be.a ✓ 12 1.a even 1 1 trivial
344.1.be.a ✓ 12 8.d odd 2 1 CM
344.1.be.a ✓ 12 43.g even 21 1 inner
344.1.be.a ✓ 12 344.be odd 42 1 inner
1376.1.by.a 12 4.b odd 2 1
1376.1.by.a 12 8.b even 2 1
1376.1.by.a 12 172.o odd 42 1
1376.1.by.a 12 344.z even 42 1
3096.1.eq.a 12 3.b odd 2 1
3096.1.eq.a 12 24.f even 2 1
3096.1.eq.a 12 129.o odd 42 1
3096.1.eq.a 12 1032.ci even 42 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(344, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{12} + 8 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} + 5 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
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