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Properties

Label 1700.2.e.c
Level $1700$
Weight $2$
Character orbit 1700.e
Analytic conductor $13.575$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1700,2,Mod(749,1700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1700.749"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1700.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.5745683436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 68)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_{2} + \beta_1) q^{7} + (2 \beta_{3} - 1) q^{9} + ( - \beta_{3} - 3) q^{11} + ( - 2 \beta_{2} + 2 \beta_1) q^{13} + \beta_1 q^{17} + ( - 2 \beta_{3} - 2) q^{19}+ \cdots + ( - 5 \beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} - 12 q^{11} - 8 q^{19} - 16 q^{21} - 4 q^{31} - 32 q^{39} - 24 q^{41} + 12 q^{49} - 4 q^{51} - 24 q^{59} - 16 q^{61} - 12 q^{71} + 28 q^{79} + 4 q^{81} - 24 q^{89} - 32 q^{91} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1700\mathbb{Z}\right)^\times\).

\(n\) \(477\) \(851\) \(1601\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 − 2.73205i 0 0 0 − 2.73205i 0 −4.46410 0
749.2 0 − 0.732051i 0 0 0 − 0.732051i 0 2.46410 0
749.3 0 0.732051i 0 0 0 0.732051i 0 2.46410 0
749.4 0 2.73205i 0 0 0 2.73205i 0 −4.46410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1700.2.e.c 4
5.b even 2 1 inner 1700.2.e.c 4
5.c odd 4 1 68.2.a.a ✓ 2
5.c odd 4 1 1700.2.a.d 2
15.e even 4 1 612.2.a.e 2
20.e even 4 1 272.2.a.e 2
20.e even 4 1 6800.2.a.bh 2
35.f even 4 1 3332.2.a.h 2
40.i odd 4 1 1088.2.a.p 2
40.k even 4 1 1088.2.a.t 2
55.e even 4 1 8228.2.a.k 2
60.l odd 4 1 2448.2.a.y 2
85.f odd 4 1 1156.2.b.c 4
85.g odd 4 1 1156.2.a.a 2
85.i odd 4 1 1156.2.b.c 4
85.k odd 8 2 1156.2.e.d 8
85.n odd 8 2 1156.2.e.d 8
85.o even 16 4 1156.2.h.f 16
85.r even 16 4 1156.2.h.f 16
120.q odd 4 1 9792.2.a.cs 2
120.w even 4 1 9792.2.a.cr 2
340.r even 4 1 4624.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.a.a ✓ 2 5.c odd 4 1
272.2.a.e 2 20.e even 4 1
612.2.a.e 2 15.e even 4 1
1088.2.a.p 2 40.i odd 4 1
1088.2.a.t 2 40.k even 4 1
1156.2.a.a 2 85.g odd 4 1
1156.2.b.c 4 85.f odd 4 1
1156.2.b.c 4 85.i odd 4 1
1156.2.e.d 8 85.k odd 8 2
1156.2.e.d 8 85.n odd 8 2
1156.2.h.f 16 85.o even 16 4
1156.2.h.f 16 85.r even 16 4
1700.2.a.d 2 5.c odd 4 1
1700.2.e.c 4 1.a even 1 1 trivial
1700.2.e.c 4 5.b even 2 1 inner
2448.2.a.y 2 60.l odd 4 1
3332.2.a.h 2 35.f even 4 1
4624.2.a.x 2 340.r even 4 1
6800.2.a.bh 2 20.e even 4 1
8228.2.a.k 2 55.e even 4 1
9792.2.a.cr 2 120.w even 4 1
9792.2.a.cs 2 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 8T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1700, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 152T^{2} + 2704 \) Copy content Toggle raw display
$41$ \( (T + 6)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 224 T^{2} + 10816 \) Copy content Toggle raw display
$47$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 224T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 22)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
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