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Properties

Label 1400.2.q.e
Level $1400$
Weight $2$
Character orbit 1400.q
Analytic conductor $11.179$
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] = [1400,2,Mod(401,1400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1400, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) 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("1400.401"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,0,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{3} + ( - \zeta_{6} + 3) q^{7} + 2 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{11} + ( - 4 \zeta_{6} + 4) q^{17} + 2 \zeta_{6} q^{19} + ( - 3 \zeta_{6} + 2) q^{21} + \zeta_{6} q^{23}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 5 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{17} + 2 q^{19} + q^{21} + q^{23} + 10 q^{27} + 18 q^{29} - 4 q^{31} + 2 q^{33} + 4 q^{37} + 2 q^{41} - 18 q^{43} + 11 q^{49} - 4 q^{51} - 10 q^{53} + 4 q^{57}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 0 0 2.50000 + 0.866025i 0 1.00000 − 1.73205i 0
1201.1 0 0.500000 − 0.866025i 0 0 0 2.50000 − 0.866025i 0 1.00000 + 1.73205i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.q.e 2
5.b even 2 1 280.2.q.a ✓ 2
5.c odd 4 2 1400.2.bh.b 4
7.c even 3 1 inner 1400.2.q.e 2
7.c even 3 1 9800.2.a.r 1
7.d odd 6 1 9800.2.a.bc 1
15.d odd 2 1 2520.2.bi.a 2
20.d odd 2 1 560.2.q.h 2
35.c odd 2 1 1960.2.q.k 2
35.i odd 6 1 1960.2.a.e 1
35.i odd 6 1 1960.2.q.k 2
35.j even 6 1 280.2.q.a ✓ 2
35.j even 6 1 1960.2.a.i 1
35.l odd 12 2 1400.2.bh.b 4
105.o odd 6 1 2520.2.bi.a 2
140.p odd 6 1 560.2.q.h 2
140.p odd 6 1 3920.2.a.m 1
140.s even 6 1 3920.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.a ✓ 2 5.b even 2 1
280.2.q.a ✓ 2 35.j even 6 1
560.2.q.h 2 20.d odd 2 1
560.2.q.h 2 140.p odd 6 1
1400.2.q.e 2 1.a even 1 1 trivial
1400.2.q.e 2 7.c even 3 1 inner
1400.2.bh.b 4 5.c odd 4 2
1400.2.bh.b 4 35.l odd 12 2
1960.2.a.e 1 35.i odd 6 1
1960.2.a.i 1 35.j even 6 1
1960.2.q.k 2 35.c odd 2 1
1960.2.q.k 2 35.i odd 6 1
2520.2.bi.a 2 15.d odd 2 1
2520.2.bi.a 2 105.o odd 6 1
3920.2.a.m 1 140.p odd 6 1
3920.2.a.y 1 140.s even 6 1
9800.2.a.r 1 7.c even 3 1
9800.2.a.bc 1 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$29$ \( (T - 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$41$ \( (T - 1)^{2} \) Copy content Toggle raw display
$43$ \( (T + 9)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$59$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$61$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$71$ \( (T - 14)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$83$ \( (T + 11)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$97$ \( (T - 18)^{2} \) Copy content Toggle raw display
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