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Properties

Label 3700.1.dy.a
Level $3700$
Weight $1$
Character orbit 3700.dy
Analytic conductor $1.847$
Analytic rank $0$
Dimension $24$
Projective image $D_{90}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3700,1,Mod(139,3700)] 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("3700.139"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3700, base_ring=CyclotomicField(90)) chi = DirichletCharacter(H, H._module([45, 27, 85])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3700.dy (of order \(90\), degree \(24\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.84654054674\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{45})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{90}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{90} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{90}^{2} q^{2} + \zeta_{90}^{4} q^{4} + \zeta_{90}^{21} q^{5} - \zeta_{90}^{6} q^{8} + \zeta_{90}^{23} q^{9} - \zeta_{90}^{23} q^{10} + ( - \zeta_{90}^{19} - \zeta_{90}^{7}) q^{13} + \zeta_{90}^{8} q^{16} + \cdots + \zeta_{90}^{22} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 3 q^{5} - 3 q^{8} + 3 q^{17} + 3 q^{25} + 3 q^{26} + 12 q^{34} + 6 q^{36} - 6 q^{40} + 3 q^{41} + 3 q^{58} - 6 q^{61} + 3 q^{64} + 3 q^{74} - 24 q^{85} - 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1001\) \(1777\) \(1851\)
\(\chi(n)\) \(-\zeta_{90}^{40}\) \(\zeta_{90}^{9}\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
−0.374607 + 0.927184i
0.990268 − 0.139173i
−0.374607 − 0.927184i
0.990268 + 0.139173i
−0.615661 − 0.788011i
−0.997564 − 0.0697565i
0.0348995 − 0.999391i
0.438371 + 0.898794i
0.559193 − 0.829038i
−0.241922 − 0.970296i
0.961262 − 0.275637i
−0.719340 + 0.694658i
0.848048 + 0.529919i
−0.882948 + 0.469472i
0.961262 + 0.275637i
0.848048 − 0.529919i
0.559193 + 0.829038i
−0.882948 − 0.469472i
−0.241922 + 0.970296i
−0.719340 − 0.694658i
0.719340 + 0.694658i 0 0.0348995 + 0.999391i 0.978148 + 0.207912i 0 0 −0.669131 + 0.743145i −0.559193 − 0.829038i 0.559193 + 0.829038i
539.1 −0.961262 + 0.275637i 0 0.848048 − 0.529919i 0.978148 + 0.207912i 0 0 −0.669131 + 0.743145i 0.997564 − 0.0697565i −0.997564 + 0.0697565i
559.1 0.719340 − 0.694658i 0 0.0348995 − 0.999391i 0.978148 − 0.207912i 0 0 −0.669131 − 0.743145i −0.559193 + 0.829038i 0.559193 − 0.829038i
659.1 −0.961262 − 0.275637i 0 0.848048 + 0.529919i 0.978148 − 0.207912i 0 0 −0.669131 − 0.743145i 0.997564 + 0.0697565i −0.997564 − 0.0697565i
839.1 0.241922 − 0.970296i 0 −0.882948 − 0.469472i 0.978148 + 0.207912i 0 0 −0.669131 + 0.743145i −0.438371 + 0.898794i 0.438371 − 0.898794i
879.1 −0.990268 − 0.139173i 0 0.961262 + 0.275637i 0.104528 + 0.994522i 0 0 −0.913545 − 0.406737i −0.0348995 + 0.999391i 0.0348995 − 0.999391i
1039.1 0.997564 + 0.0697565i 0 0.990268 + 0.139173i −0.669131 + 0.743145i 0 0 0.978148 + 0.207912i 0.719340 − 0.694658i −0.719340 + 0.694658i
1279.1 0.615661 − 0.788011i 0 −0.241922 − 0.970296i 0.104528 + 0.994522i 0 0 −0.913545 − 0.406737i −0.848048 − 0.529919i 0.848048 + 0.529919i
1579.1 0.374607 + 0.927184i 0 −0.719340 + 0.694658i 0.104528 + 0.994522i 0 0 −0.913545 − 0.406737i 0.882948 − 0.469472i −0.882948 + 0.469472i
1619.1 0.882948 − 0.469472i 0 0.559193 − 0.829038i −0.913545 + 0.406737i 0 0 0.104528 − 0.994522i 0.615661 − 0.788011i −0.615661 + 0.788011i
1779.1 −0.848048 + 0.529919i 0 0.438371 − 0.898794i −0.913545 − 0.406737i 0 0 0.104528 + 0.994522i −0.990268 + 0.139173i 0.990268 − 0.139173i
2019.1 −0.0348995 + 0.999391i 0 −0.997564 − 0.0697565i −0.913545 + 0.406737i 0 0 0.104528 − 0.994522i 0.374607 + 0.927184i −0.374607 − 0.927184i
2039.1 −0.438371 − 0.898794i 0 −0.615661 + 0.788011i −0.669131 + 0.743145i 0 0 0.978148 + 0.207912i −0.961262 − 0.275637i 0.961262 + 0.275637i
2139.1 −0.559193 + 0.829038i 0 −0.374607 − 0.927184i −0.669131 + 0.743145i 0 0 0.978148 + 0.207912i 0.241922 + 0.970296i −0.241922 − 0.970296i
2319.1 −0.848048 − 0.529919i 0 0.438371 + 0.898794i −0.913545 + 0.406737i 0 0 0.104528 − 0.994522i −0.990268 − 0.139173i 0.990268 + 0.139173i
2359.1 −0.438371 + 0.898794i 0 −0.615661 − 0.788011i −0.669131 − 0.743145i 0 0 0.978148 − 0.207912i −0.961262 + 0.275637i 0.961262 − 0.275637i
2519.1 0.374607 − 0.927184i 0 −0.719340 − 0.694658i 0.104528 − 0.994522i 0 0 −0.913545 + 0.406737i 0.882948 + 0.469472i −0.882948 − 0.469472i
2759.1 −0.559193 − 0.829038i 0 −0.374607 + 0.927184i −0.669131 − 0.743145i 0 0 0.978148 − 0.207912i 0.241922 − 0.970296i −0.241922 + 0.970296i
2779.1 0.882948 + 0.469472i 0 0.559193 + 0.829038i −0.913545 − 0.406737i 0 0 0.104528 + 0.994522i 0.615661 + 0.788011i −0.615661 − 0.788011i
2879.1 −0.0348995 − 0.999391i 0 −0.997564 + 0.0697565i −0.913545 − 0.406737i 0 0 0.104528 + 0.994522i 0.374607 − 0.927184i −0.374607 + 0.927184i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.1
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.cb even 90 1 inner
3700.dy odd 90 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3700.1.dy.a ✓ 24
4.b odd 2 1 CM 3700.1.dy.a ✓ 24
25.e even 10 1 3700.1.dy.b yes 24
37.h even 18 1 3700.1.dy.b yes 24
100.h odd 10 1 3700.1.dy.b yes 24
148.o odd 18 1 3700.1.dy.b yes 24
925.cb even 90 1 inner 3700.1.dy.a ✓ 24
3700.dy odd 90 1 inner 3700.1.dy.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3700.1.dy.a ✓ 24 1.a even 1 1 trivial
3700.1.dy.a ✓ 24 4.b odd 2 1 CM
3700.1.dy.a ✓ 24 925.cb even 90 1 inner
3700.1.dy.a ✓ 24 3700.dy odd 90 1 inner
3700.1.dy.b yes 24 25.e even 10 1
3700.1.dy.b yes 24 37.h even 18 1
3700.1.dy.b yes 24 100.h odd 10 1
3700.1.dy.b yes 24 148.o odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{24} - 29 T_{13}^{21} + 345 T_{13}^{18} - 2141 T_{13}^{15} + 7314 T_{13}^{12} - 13421 T_{13}^{9} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(3700, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} + T^{21} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{24} \) Copy content Toggle raw display
$5$ \( (T^{8} + T^{7} - T^{5} + \cdots + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{24} \) Copy content Toggle raw display
$11$ \( T^{24} \) Copy content Toggle raw display
$13$ \( T^{24} - 29 T^{21} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{24} - 3 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{24} \) Copy content Toggle raw display
$23$ \( T^{24} \) Copy content Toggle raw display
$29$ \( T^{24} + 3 T^{22} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{24} \) Copy content Toggle raw display
$37$ \( T^{24} + T^{21} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{24} - 3 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{24} \) Copy content Toggle raw display
$47$ \( T^{24} \) Copy content Toggle raw display
$53$ \( T^{24} - 4 T^{21} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{24} \) Copy content Toggle raw display
$61$ \( T^{24} + 6 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{24} \) Copy content Toggle raw display
$71$ \( T^{24} \) Copy content Toggle raw display
$73$ \( (T^{8} - 3 T^{6} + 5 T^{5} + \cdots + 1)^{3} \) Copy content Toggle raw display
$79$ \( T^{24} \) Copy content Toggle raw display
$83$ \( T^{24} \) Copy content Toggle raw display
$89$ \( T^{24} + 24 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{24} - 3 T^{22} + \cdots + 1 \) Copy content Toggle raw display
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