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Properties

Label 1407.1.cz.a
Level $1407$
Weight $1$
Character orbit 1407.cz
Analytic conductor $0.702$
Analytic rank $0$
Dimension $20$
Projective image $D_{33}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1407,1,Mod(23,1407)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1407.23"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1407, base_ring=CyclotomicField(66)) chi = DirichletCharacter(H, H._module([33, 22, 28])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1407 = 3 \cdot 7 \cdot 67 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1407.cz (of order \(66\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.702184472775\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{33}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{33} - \cdots)\)

$q$-expansion

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

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

\(f(q)\) \(=\) \( q + \zeta_{66}^{4} q^{3} - \zeta_{66}^{27} q^{4} - \zeta_{66}^{13} q^{7} + \zeta_{66}^{8} q^{9} - \zeta_{66}^{31} q^{12} + (\zeta_{66}^{28} - \zeta_{66}) q^{13} - \zeta_{66}^{21} q^{16} + ( - \zeta_{66}^{29} + \zeta_{66}^{10}) q^{19} + \cdots + (\zeta_{66}^{26} + \zeta_{66}^{18}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{3} - 2 q^{4} + q^{7} + q^{9} + q^{12} + 2 q^{13} - 2 q^{16} + 2 q^{19} + q^{21} + q^{25} - 2 q^{27} + q^{28} - 4 q^{31} + q^{36} + 2 q^{37} + 2 q^{39} - 4 q^{43} + q^{48} + q^{49} - 9 q^{52}+ \cdots - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1407\mathbb{Z}\right)^\times\).

\(n\) \(337\) \(470\) \(1207\)
\(\chi(n)\) \(-\zeta_{66}^{5}\) \(-1\) \(-\zeta_{66}^{11}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
−0.995472 + 0.0950560i
−0.327068 + 0.945001i
0.981929 + 0.189251i
0.928368 − 0.371662i
−0.327068 − 0.945001i
−0.786053 − 0.618159i
0.235759 + 0.971812i
0.235759 − 0.971812i
0.0475819 − 0.998867i
−0.888835 + 0.458227i
−0.786053 + 0.618159i
0.928368 + 0.371662i
0.580057 + 0.814576i
0.981929 − 0.189251i
−0.995472 − 0.0950560i
0.723734 − 0.690079i
−0.888835 − 0.458227i
0.580057 − 0.814576i
0.723734 + 0.690079i
0.0475819 + 0.998867i
0 0.928368 − 0.371662i 0.841254 + 0.540641i 0 0 −0.327068 + 0.945001i 0 0.723734 − 0.690079i 0
65.1 0 0.235759 + 0.971812i 0.415415 + 0.909632i 0 0 0.928368 − 0.371662i 0 −0.888835 + 0.458227i 0
86.1 0 0.723734 + 0.690079i 0.415415 − 0.909632i 0 0 −0.786053 + 0.618159i 0 0.0475819 + 0.998867i 0
317.1 0 0.0475819 − 0.998867i −0.654861 + 0.755750i 0 0 0.235759 + 0.971812i 0 −0.995472 − 0.0950560i 0
368.1 0 0.235759 − 0.971812i 0.415415 − 0.909632i 0 0 0.928368 + 0.371662i 0 −0.888835 − 0.458227i 0
473.1 0 −0.888835 + 0.458227i −0.654861 + 0.755750i 0 0 0.723734 − 0.690079i 0 0.580057 − 0.814576i 0
485.1 0 0.580057 − 0.814576i −0.142315 − 0.989821i 0 0 0.0475819 − 0.998867i 0 −0.327068 − 0.945001i 0
557.1 0 0.580057 + 0.814576i −0.142315 + 0.989821i 0 0 0.0475819 + 0.998867i 0 −0.327068 + 0.945001i 0
590.1 0 0.981929 + 0.189251i −0.959493 + 0.281733i 0 0 0.580057 − 0.814576i 0 0.928368 + 0.371662i 0
725.1 0 −0.327068 − 0.945001i −0.959493 + 0.281733i 0 0 −0.995472 − 0.0950560i 0 −0.786053 + 0.618159i 0
821.1 0 −0.888835 − 0.458227i −0.654861 − 0.755750i 0 0 0.723734 + 0.690079i 0 0.580057 + 0.814576i 0
830.1 0 0.0475819 + 0.998867i −0.654861 − 0.755750i 0 0 0.235759 − 0.971812i 0 −0.995472 + 0.0950560i 0
851.1 0 −0.786053 − 0.618159i 0.841254 + 0.540641i 0 0 0.981929 − 0.189251i 0 0.235759 + 0.971812i 0
998.1 0 0.723734 − 0.690079i 0.415415 + 0.909632i 0 0 −0.786053 − 0.618159i 0 0.0475819 − 0.998867i 0
1040.1 0 0.928368 + 0.371662i 0.841254 − 0.540641i 0 0 −0.327068 − 0.945001i 0 0.723734 + 0.690079i 0
1061.1 0 −0.995472 − 0.0950560i −0.142315 − 0.989821i 0 0 −0.888835 + 0.458227i 0 0.981929 + 0.189251i 0
1178.1 0 −0.327068 + 0.945001i −0.959493 − 0.281733i 0 0 −0.995472 + 0.0950560i 0 −0.786053 − 0.618159i 0
1283.1 0 −0.786053 + 0.618159i 0.841254 − 0.540641i 0 0 0.981929 + 0.189251i 0 0.235759 − 0.971812i 0
1346.1 0 −0.995472 + 0.0950560i −0.142315 + 0.989821i 0 0 −0.888835 − 0.458227i 0 0.981929 − 0.189251i 0
1376.1 0 0.981929 − 0.189251i −0.959493 − 0.281733i 0 0 0.580057 + 0.814576i 0 0.928368 − 0.371662i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
469.y even 33 1 inner
1407.cz odd 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1407.1.cz.a yes 20
3.b odd 2 1 CM 1407.1.cz.a yes 20
7.c even 3 1 1407.1.cg.a ✓ 20
21.h odd 6 1 1407.1.cg.a ✓ 20
67.g even 33 1 1407.1.cg.a ✓ 20
201.o odd 66 1 1407.1.cg.a ✓ 20
469.y even 33 1 inner 1407.1.cz.a yes 20
1407.cz odd 66 1 inner 1407.1.cz.a yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1407.1.cg.a ✓ 20 7.c even 3 1
1407.1.cg.a ✓ 20 21.h odd 6 1
1407.1.cg.a ✓ 20 67.g even 33 1
1407.1.cg.a ✓ 20 201.o odd 66 1
1407.1.cz.a yes 20 1.a even 1 1 trivial
1407.1.cz.a yes 20 3.b odd 2 1 CM
1407.1.cz.a yes 20 469.y even 33 1 inner
1407.1.cz.a yes 20 1407.cz odd 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1407, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - T^{19} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - T^{19} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} - 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} - 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{20} \) Copy content Toggle raw display
$29$ \( T^{20} \) Copy content Toggle raw display
$31$ \( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} - T^{9} - 10 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} \) Copy content Toggle raw display
$43$ \( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} \) Copy content Toggle raw display
$53$ \( T^{20} \) Copy content Toggle raw display
$59$ \( T^{20} \) Copy content Toggle raw display
$61$ \( T^{20} - 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{20} - T^{19} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{20} \) Copy content Toggle raw display
$73$ \( T^{20} + 12 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{20} + 12 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{20} \) Copy content Toggle raw display
$89$ \( T^{20} \) Copy content Toggle raw display
$97$ \( T^{20} + T^{19} + \cdots + 1 \) Copy content Toggle raw display
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