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Properties

Label 525.6.d.a
Level $525$
Weight $6$
Character orbit 525.d
Analytic conductor $84.202$
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] = [525,6,Mod(274,525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("525.274"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-136,0,180] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(84.2015054018\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + 10 i q^{2} - 9 i q^{3} - 68 q^{4} + 90 q^{6} - 49 i q^{7} - 360 i q^{8} - 81 q^{9} + 92 q^{11} + 612 i q^{12} - 670 i q^{13} + 490 q^{14} + 1424 q^{16} - 222 i q^{17} - 810 i q^{18} + 908 q^{19} - 441 q^{21} + \cdots - 7452 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 136 q^{4} + 180 q^{6} - 162 q^{9} + 184 q^{11} + 980 q^{14} + 2848 q^{16} + 1816 q^{19} - 882 q^{21} - 6480 q^{24} + 13400 q^{26} - 2236 q^{29} + 7392 q^{31} + 4440 q^{34} + 11016 q^{36} - 12060 q^{39}+ \cdots - 14904 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
− 1.00000i
1.00000i
− 10.0000i 9.00000i −68.0000 0 90.0000 49.0000i 360.000i −81.0000 0
274.2 10.0000i − 9.00000i −68.0000 0 90.0000 − 49.0000i − 360.000i −81.0000 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 525.6.d.a 2
5.b even 2 1 inner 525.6.d.a 2
5.c odd 4 1 21.6.a.d ✓ 1
5.c odd 4 1 525.6.a.a 1
15.e even 4 1 63.6.a.a 1
20.e even 4 1 336.6.a.a 1
35.f even 4 1 147.6.a.g 1
35.k even 12 2 147.6.e.b 2
35.l odd 12 2 147.6.e.a 2
60.l odd 4 1 1008.6.a.bc 1
105.k odd 4 1 441.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.d ✓ 1 5.c odd 4 1
63.6.a.a 1 15.e even 4 1
147.6.a.g 1 35.f even 4 1
147.6.e.a 2 35.l odd 12 2
147.6.e.b 2 35.k even 12 2
336.6.a.a 1 20.e even 4 1
441.6.a.b 1 105.k odd 4 1
525.6.a.a 1 5.c odd 4 1
525.6.d.a 2 1.a even 1 1 trivial
525.6.d.a 2 5.b even 2 1 inner
1008.6.a.bc 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 100 \) acting on \(S_{6}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 100 \) Copy content Toggle raw display
$3$ \( T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T - 92)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 448900 \) Copy content Toggle raw display
$17$ \( T^{2} + 49284 \) Copy content Toggle raw display
$19$ \( (T - 908)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1382976 \) Copy content Toggle raw display
$29$ \( (T + 1118)^{2} \) Copy content Toggle raw display
$31$ \( (T - 3696)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 17489124 \) Copy content Toggle raw display
$41$ \( (T + 6662)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 13690000 \) Copy content Toggle raw display
$47$ \( T^{2} + 49787136 \) Copy content Toggle raw display
$53$ \( T^{2} + 1412106084 \) Copy content Toggle raw display
$59$ \( (T + 32700)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10802)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4224480016 \) Copy content Toggle raw display
$71$ \( (T + 61320)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1514922084 \) Copy content Toggle raw display
$79$ \( (T - 88096)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 5168459664 \) Copy content Toggle raw display
$89$ \( (T + 111818)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 22754515716 \) Copy content Toggle raw display
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