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Properties

Label 700.4.e.c
Level $700$
Weight $4$
Character orbit 700.e
Analytic conductor $41.301$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [700,4,Mod(449,700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("700.449"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 700.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-74,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(41.3013370040\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 i q^{3} - 7 i q^{7} - 37 q^{9} + 28 q^{11} + 82 i q^{13} + 46 i q^{17} - 8 q^{19} + 56 q^{21} - 128 i q^{23} - 80 i q^{27} - 174 q^{29} - 152 q^{31} + 224 i q^{33} + 290 i q^{37} - 656 q^{39} + 50 q^{41} + \cdots - 1036 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 74 q^{9} + 56 q^{11} - 16 q^{19} + 112 q^{21} - 348 q^{29} - 304 q^{31} - 1312 q^{39} + 100 q^{41} - 98 q^{49} - 736 q^{51} + 544 q^{59} - 1324 q^{61} + 2048 q^{69} - 1760 q^{71} + 1200 q^{79} - 718 q^{81}+ \cdots - 2072 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
− 1.00000i
1.00000i
0 − 8.00000i 0 0 0 7.00000i 0 −37.0000 0
449.2 0 8.00000i 0 0 0 − 7.00000i 0 −37.0000 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 700.4.e.c 2
5.b even 2 1 inner 700.4.e.c 2
5.c odd 4 1 140.4.a.e ✓ 1
5.c odd 4 1 700.4.a.b 1
15.e even 4 1 1260.4.a.j 1
20.e even 4 1 560.4.a.b 1
35.f even 4 1 980.4.a.b 1
35.k even 12 2 980.4.i.q 2
35.l odd 12 2 980.4.i.b 2
40.i odd 4 1 2240.4.a.c 1
40.k even 4 1 2240.4.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.4.a.e ✓ 1 5.c odd 4 1
560.4.a.b 1 20.e even 4 1
700.4.a.b 1 5.c odd 4 1
700.4.e.c 2 1.a even 1 1 trivial
700.4.e.c 2 5.b even 2 1 inner
980.4.a.b 1 35.f even 4 1
980.4.i.b 2 35.l odd 12 2
980.4.i.q 2 35.k even 12 2
1260.4.a.j 1 15.e even 4 1
2240.4.a.c 1 40.i odd 4 1
2240.4.a.bj 1 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(700, [\chi])\):

\( T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{11} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T - 28)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 6724 \) Copy content Toggle raw display
$17$ \( T^{2} + 2116 \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16384 \) Copy content Toggle raw display
$29$ \( (T + 174)^{2} \) Copy content Toggle raw display
$31$ \( (T + 152)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 84100 \) Copy content Toggle raw display
$41$ \( (T - 50)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 156816 \) Copy content Toggle raw display
$47$ \( T^{2} + 87616 \) Copy content Toggle raw display
$53$ \( T^{2} + 324900 \) Copy content Toggle raw display
$59$ \( (T - 272)^{2} \) Copy content Toggle raw display
$61$ \( (T + 662)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 767376 \) Copy content Toggle raw display
$71$ \( (T + 880)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 407044 \) Copy content Toggle raw display
$79$ \( (T - 600)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 389376 \) Copy content Toggle raw display
$89$ \( (T + 698)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 568516 \) Copy content Toggle raw display
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