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Properties

Label 3700.1.dq.a
Level $3700$
Weight $1$
Character orbit 3700.dq
Analytic conductor $1.847$
Analytic rank $0$
Dimension $16$
Projective image $D_{60}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3700,1,Mod(103,3700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3700, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([30, 21, 35]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3700.103");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3700.dq (of order \(60\), degree \(16\), 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: \(1.84654054674\)
Analytic rank: \(0\)
Dimension: \(16\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{60}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{60} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{60}^{26} q^{2} - \zeta_{60}^{22} q^{4} - \zeta_{60}^{8} q^{5} + \zeta_{60}^{18} q^{8} + \zeta_{60}^{29} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{60}^{26} q^{2} - \zeta_{60}^{22} q^{4} - \zeta_{60}^{8} q^{5} + \zeta_{60}^{18} q^{8} + \zeta_{60}^{29} q^{9} + \zeta_{60}^{4} q^{10} + ( - \zeta_{60}^{7} - \zeta_{60}) q^{13} - \zeta_{60}^{14} q^{16} + (\zeta_{60}^{9} - \zeta_{60}^{7}) q^{17} - \zeta_{60}^{25} q^{18} - q^{20} + \zeta_{60}^{16} q^{25} + ( - \zeta_{60}^{27} + \zeta_{60}^{3}) q^{26} + ( - \zeta_{60}^{26} + \zeta_{60}^{13}) q^{29} + \zeta_{60}^{10} q^{32} + ( - \zeta_{60}^{5} + \zeta_{60}^{3}) q^{34} + \zeta_{60}^{21} q^{36} + \zeta_{60}^{10} q^{37} - \zeta_{60}^{26} q^{40} + (\zeta_{60}^{26} + \zeta_{60}^{12}) q^{41} + \zeta_{60}^{7} q^{45} + \zeta_{60}^{5} q^{49} - \zeta_{60}^{12} q^{50} + (\zeta_{60}^{29} + \zeta_{60}^{23}) q^{52} + ( - \zeta_{60}^{29} + \zeta_{60}^{20}) q^{53} + (\zeta_{60}^{22} - \zeta_{60}^{9}) q^{58} + ( - \zeta_{60}^{24} - \zeta_{60}^{23}) q^{61} - \zeta_{60}^{6} q^{64} + (\zeta_{60}^{15} + \zeta_{60}^{9}) q^{65} + (\zeta_{60}^{29} + \zeta_{60}) q^{68} - \zeta_{60}^{17} q^{72} + (\zeta_{60}^{24} - \zeta_{60}^{3}) q^{73} - \zeta_{60}^{6} q^{74} + \zeta_{60}^{22} q^{80} - \zeta_{60}^{28} q^{81} + ( - \zeta_{60}^{22} - \zeta_{60}^{8}) q^{82} + ( - \zeta_{60}^{17} + \zeta_{60}^{15}) q^{85} + (\zeta_{60}^{27} + \zeta_{60}^{10}) q^{89} - \zeta_{60}^{3} q^{90} + ( - \zeta_{60}^{28} - \zeta_{60}^{26}) q^{97} - \zeta_{60} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{8} + 2 q^{10} + 2 q^{16} - 16 q^{20} + 2 q^{25} + 2 q^{29} + 8 q^{32} + 8 q^{37} + 2 q^{40} - 6 q^{41} + 4 q^{50} - 8 q^{53} - 2 q^{58} + 4 q^{61} - 4 q^{64} - 4 q^{73} - 4 q^{74} - 2 q^{80} - 2 q^{81} + 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\) \(1851\)
\(\chi(n)\) \(-\zeta_{60}^{25}\) \(-\zeta_{60}^{27}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1
0.743145 − 0.669131i
0.207912 − 0.978148i
0.743145 + 0.669131i
−0.743145 − 0.669131i
−0.207912 + 0.978148i
−0.743145 + 0.669131i
−0.207912 − 0.978148i
0.406737 − 0.913545i
−0.994522 + 0.104528i
0.406737 + 0.913545i
−0.994522 − 0.104528i
0.994522 + 0.104528i
−0.406737 − 0.913545i
0.994522 − 0.104528i
−0.406737 + 0.913545i
0.207912 + 0.978148i
0.978148 − 0.207912i 0 0.913545 − 0.406737i −0.913545 − 0.406737i 0 0 0.809017 − 0.587785i −0.743145 − 0.669131i −0.978148 − 0.207912i
267.1 −0.669131 + 0.743145i 0 −0.104528 − 0.994522i 0.104528 − 0.994522i 0 0 0.809017 + 0.587785i −0.207912 − 0.978148i 0.669131 + 0.743145i
467.1 0.978148 + 0.207912i 0 0.913545 + 0.406737i −0.913545 + 0.406737i 0 0 0.809017 + 0.587785i −0.743145 + 0.669131i −0.978148 + 0.207912i
1383.1 0.978148 + 0.207912i 0 0.913545 + 0.406737i −0.913545 + 0.406737i 0 0 0.809017 + 0.587785i 0.743145 − 0.669131i −0.978148 + 0.207912i
1583.1 −0.669131 + 0.743145i 0 −0.104528 − 0.994522i 0.104528 − 0.994522i 0 0 0.809017 + 0.587785i 0.207912 + 0.978148i 0.669131 + 0.743145i
1747.1 0.978148 − 0.207912i 0 0.913545 − 0.406737i −0.913545 − 0.406737i 0 0 0.809017 − 0.587785i 0.743145 + 0.669131i −0.978148 − 0.207912i
1947.1 −0.669131 − 0.743145i 0 −0.104528 + 0.994522i 0.104528 + 0.994522i 0 0 0.809017 − 0.587785i 0.207912 − 0.978148i 0.669131 − 0.743145i
2123.1 0.104528 + 0.994522i 0 −0.978148 + 0.207912i 0.978148 + 0.207912i 0 0 −0.309017 − 0.951057i −0.406737 − 0.913545i −0.104528 + 0.994522i
2323.1 −0.913545 − 0.406737i 0 0.669131 + 0.743145i −0.669131 + 0.743145i 0 0 −0.309017 − 0.951057i 0.994522 + 0.104528i 0.913545 − 0.406737i
2487.1 0.104528 − 0.994522i 0 −0.978148 − 0.207912i 0.978148 − 0.207912i 0 0 −0.309017 + 0.951057i −0.406737 + 0.913545i −0.104528 − 0.994522i
2687.1 −0.913545 + 0.406737i 0 0.669131 − 0.743145i −0.669131 − 0.743145i 0 0 −0.309017 + 0.951057i 0.994522 − 0.104528i 0.913545 + 0.406737i
2863.1 −0.913545 + 0.406737i 0 0.669131 − 0.743145i −0.669131 − 0.743145i 0 0 −0.309017 + 0.951057i −0.994522 + 0.104528i 0.913545 + 0.406737i
3063.1 0.104528 − 0.994522i 0 −0.978148 − 0.207912i 0.978148 − 0.207912i 0 0 −0.309017 + 0.951057i 0.406737 − 0.913545i −0.104528 − 0.994522i
3227.1 −0.913545 − 0.406737i 0 0.669131 + 0.743145i −0.669131 + 0.743145i 0 0 −0.309017 − 0.951057i −0.994522 − 0.104528i 0.913545 − 0.406737i
3427.1 0.104528 + 0.994522i 0 −0.978148 + 0.207912i 0.978148 + 0.207912i 0 0 −0.309017 − 0.951057i 0.406737 + 0.913545i −0.104528 + 0.994522i
3603.1 −0.669131 − 0.743145i 0 −0.104528 + 0.994522i 0.104528 + 0.994522i 0 0 0.809017 − 0.587785i −0.207912 + 0.978148i 0.669131 − 0.743145i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
925.by even 60 1 inner
3700.dq odd 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3700.1.dq.a yes 16
4.b odd 2 1 CM 3700.1.dq.a yes 16
25.f odd 20 1 3700.1.dn.a ✓ 16
37.g odd 12 1 3700.1.dn.a ✓ 16
100.l even 20 1 3700.1.dn.a ✓ 16
148.l even 12 1 3700.1.dn.a ✓ 16
925.by even 60 1 inner 3700.1.dq.a yes 16
3700.dq odd 60 1 inner 3700.1.dq.a yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3700.1.dn.a ✓ 16 25.f odd 20 1
3700.1.dn.a ✓ 16 37.g odd 12 1
3700.1.dn.a ✓ 16 100.l even 20 1
3700.1.dn.a ✓ 16 148.l even 12 1
3700.1.dq.a yes 16 1.a even 1 1 trivial
3700.1.dq.a yes 16 4.b odd 2 1 CM
3700.1.dq.a yes 16 925.by even 60 1 inner
3700.1.dq.a yes 16 3700.dq odd 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3700, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + T^{7} - T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + T^{7} - T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 10 T^{12} + \cdots + 625 \) Copy content Toggle raw display
$17$ \( T^{16} + T^{14} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$41$ \( (T^{8} + 3 T^{7} + 6 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} + 8 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} - 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} - 8 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( (T^{8} - 3 T^{6} - 5 T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
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