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Properties

Label 400.3.h.a
Level $400$
Weight $3$
Character orbit 400.h
Analytic conductor $10.899$
Analytic rank $0$
Dimension $2$
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] = [400,3,Mod(399,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.399"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.h (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,-18] 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: \(10.8992105744\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 10i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 9 q^{9} + \beta q^{13} + 3 \beta q^{17} - 42 q^{29} + 7 \beta q^{37} + 18 q^{41} - 49 q^{49} + 9 \beta q^{53} - 22 q^{61} - 11 \beta q^{73} + 81 q^{81} + 78 q^{89} - 13 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} - 84 q^{29} + 36 q^{41} - 98 q^{49} - 44 q^{61} + 162 q^{81} + 156 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
399.1
− 1.00000i
1.00000i
0 0 0 0 0 0 0 −9.00000 0
399.2 0 0 0 0 0 0 0 −9.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.h.a 2
3.b odd 2 1 3600.3.j.a 2
4.b odd 2 1 CM 400.3.h.a 2
5.b even 2 1 inner 400.3.h.a 2
5.c odd 4 1 16.3.c.a ✓ 1
5.c odd 4 1 400.3.b.a 1
8.b even 2 1 1600.3.h.b 2
8.d odd 2 1 1600.3.h.b 2
12.b even 2 1 3600.3.j.a 2
15.d odd 2 1 3600.3.j.a 2
15.e even 4 1 144.3.g.a 1
15.e even 4 1 3600.3.e.c 1
20.d odd 2 1 inner 400.3.h.a 2
20.e even 4 1 16.3.c.a ✓ 1
20.e even 4 1 400.3.b.a 1
35.f even 4 1 784.3.d.b 1
35.k even 12 2 784.3.r.d 2
35.l odd 12 2 784.3.r.e 2
40.e odd 2 1 1600.3.h.b 2
40.f even 2 1 1600.3.h.b 2
40.i odd 4 1 64.3.c.a 1
40.i odd 4 1 1600.3.b.b 1
40.k even 4 1 64.3.c.a 1
40.k even 4 1 1600.3.b.b 1
45.k odd 12 2 1296.3.o.o 2
45.l even 12 2 1296.3.o.b 2
60.h even 2 1 3600.3.j.a 2
60.l odd 4 1 144.3.g.a 1
60.l odd 4 1 3600.3.e.c 1
80.i odd 4 1 256.3.d.b 2
80.j even 4 1 256.3.d.b 2
80.s even 4 1 256.3.d.b 2
80.t odd 4 1 256.3.d.b 2
120.q odd 4 1 576.3.g.b 1
120.w even 4 1 576.3.g.b 1
140.j odd 4 1 784.3.d.b 1
140.w even 12 2 784.3.r.e 2
140.x odd 12 2 784.3.r.d 2
180.v odd 12 2 1296.3.o.b 2
180.x even 12 2 1296.3.o.o 2
240.z odd 4 1 2304.3.b.f 2
240.bb even 4 1 2304.3.b.f 2
240.bd odd 4 1 2304.3.b.f 2
240.bf even 4 1 2304.3.b.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.3.c.a ✓ 1 5.c odd 4 1
16.3.c.a ✓ 1 20.e even 4 1
64.3.c.a 1 40.i odd 4 1
64.3.c.a 1 40.k even 4 1
144.3.g.a 1 15.e even 4 1
144.3.g.a 1 60.l odd 4 1
256.3.d.b 2 80.i odd 4 1
256.3.d.b 2 80.j even 4 1
256.3.d.b 2 80.s even 4 1
256.3.d.b 2 80.t odd 4 1
400.3.b.a 1 5.c odd 4 1
400.3.b.a 1 20.e even 4 1
400.3.h.a 2 1.a even 1 1 trivial
400.3.h.a 2 4.b odd 2 1 CM
400.3.h.a 2 5.b even 2 1 inner
400.3.h.a 2 20.d odd 2 1 inner
576.3.g.b 1 120.q odd 4 1
576.3.g.b 1 120.w even 4 1
784.3.d.b 1 35.f even 4 1
784.3.d.b 1 140.j odd 4 1
784.3.r.d 2 35.k even 12 2
784.3.r.d 2 140.x odd 12 2
784.3.r.e 2 35.l odd 12 2
784.3.r.e 2 140.w even 12 2
1296.3.o.b 2 45.l even 12 2
1296.3.o.b 2 180.v odd 12 2
1296.3.o.o 2 45.k odd 12 2
1296.3.o.o 2 180.x even 12 2
1600.3.b.b 1 40.i odd 4 1
1600.3.b.b 1 40.k even 4 1
1600.3.h.b 2 8.b even 2 1
1600.3.h.b 2 8.d odd 2 1
1600.3.h.b 2 40.e odd 2 1
1600.3.h.b 2 40.f even 2 1
2304.3.b.f 2 240.z odd 4 1
2304.3.b.f 2 240.bb even 4 1
2304.3.b.f 2 240.bd odd 4 1
2304.3.b.f 2 240.bf even 4 1
3600.3.e.c 1 15.e even 4 1
3600.3.e.c 1 60.l odd 4 1
3600.3.j.a 2 3.b odd 2 1
3600.3.j.a 2 12.b even 2 1
3600.3.j.a 2 15.d odd 2 1
3600.3.j.a 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 100 \) Copy content Toggle raw display
$17$ \( T^{2} + 900 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 42)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4900 \) Copy content Toggle raw display
$41$ \( (T - 18)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 8100 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 22)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 12100 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 78)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 16900 \) Copy content Toggle raw display
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