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Properties

Label 400.10.c.h
Level $400$
Weight $10$
Character orbit 400.c
Analytic conductor $206.014$
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] = [400,10,Mod(49,400)] 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([0, 0, 1])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.49"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,34758] 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: \(206.014334466\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 20)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 24 \beta q^{3} - 266 \beta q^{7} + 17379 q^{9} + 33180 q^{11} - 49841 \beta q^{13} + 221727 \beta q^{17} - 357244 q^{19} + 25536 q^{21} + 71478 \beta q^{23} + 889488 \beta q^{27} - 1527966 q^{29} - 7323416 q^{31} + \cdots + 576635220 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 34758 q^{9} + 66360 q^{11} - 714488 q^{19} + 51072 q^{21} - 3055932 q^{29} - 14646832 q^{31} + 9569472 q^{39} - 15878028 q^{41} + 80141166 q^{49} - 42571584 q^{51} + 241250424 q^{59} + 187153084 q^{61}+ \cdots + 1153270440 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
− 1.00000i
1.00000i
0 − 48.0000i 0 0 0 532.000i 0 17379.0 0
49.2 0 48.0000i 0 0 0 − 532.000i 0 17379.0 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 400.10.c.h 2
4.b odd 2 1 100.10.c.b 2
5.b even 2 1 inner 400.10.c.h 2
5.c odd 4 1 80.10.a.c 1
5.c odd 4 1 400.10.a.e 1
20.d odd 2 1 100.10.c.b 2
20.e even 4 1 20.10.a.a ✓ 1
20.e even 4 1 100.10.a.b 1
40.i odd 4 1 320.10.a.d 1
40.k even 4 1 320.10.a.g 1
60.l odd 4 1 180.10.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.10.a.a ✓ 1 20.e even 4 1
80.10.a.c 1 5.c odd 4 1
100.10.a.b 1 20.e even 4 1
100.10.c.b 2 4.b odd 2 1
100.10.c.b 2 20.d odd 2 1
180.10.a.b 1 60.l odd 4 1
320.10.a.d 1 40.i odd 4 1
320.10.a.g 1 40.k even 4 1
400.10.a.e 1 5.c odd 4 1
400.10.c.h 2 1.a even 1 1 trivial
400.10.c.h 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2304 \) acting on \(S_{10}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2304 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 283024 \) Copy content Toggle raw display
$11$ \( (T - 33180)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 9936501124 \) Copy content Toggle raw display
$17$ \( T^{2} + 196651450116 \) Copy content Toggle raw display
$19$ \( (T + 357244)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 20436417936 \) Copy content Toggle raw display
$29$ \( (T + 1527966)^{2} \) Copy content Toggle raw display
$31$ \( (T + 7323416)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 7112046252964 \) Copy content Toggle raw display
$41$ \( (T + 7939014)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 448360297230400 \) Copy content Toggle raw display
$47$ \( T^{2} + 257911908452496 \) Copy content Toggle raw display
$53$ \( T^{2} + 77\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T - 120625212)^{2} \) Copy content Toggle raw display
$61$ \( (T - 93576542)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 37\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T + 417763488)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 20\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T + 91425472)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 42\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T - 170059206)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 119837290668484 \) Copy content Toggle raw display
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