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Properties

Label 175.6.b.b
Level $175$
Weight $6$
Character orbit 175.b
Analytic conductor $28.067$
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] = [175,6,Mod(99,175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("175.99"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.0671684673\)
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 35)
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.

\(f(q)\) \(=\) \( q + 8 i q^{2} + i q^{3} - 32 q^{4} - 8 q^{6} - 49 i q^{7} + 242 q^{9} - 453 q^{11} - 32 i q^{12} - 969 i q^{13} + 392 q^{14} - 1024 q^{16} - 1637 i q^{17} + 1936 i q^{18} + 1550 q^{19} + 49 q^{21} - 3624 i q^{22} + \cdots - 109626 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 64 q^{4} - 16 q^{6} + 484 q^{9} - 906 q^{11} + 784 q^{14} - 2048 q^{16} + 3100 q^{19} + 98 q^{21} + 15504 q^{26} + 9970 q^{29} + 2384 q^{31} + 26192 q^{34} - 15488 q^{36} + 1938 q^{39} - 3456 q^{41}+ \cdots - 219252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
− 1.00000i
1.00000i
− 8.00000i − 1.00000i −32.0000 0 −8.00000 49.0000i 0 242.000 0
99.2 8.00000i 1.00000i −32.0000 0 −8.00000 − 49.0000i 0 242.000 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 175.6.b.b 2
5.b even 2 1 inner 175.6.b.b 2
5.c odd 4 1 35.6.a.a ✓ 1
5.c odd 4 1 175.6.a.a 1
15.e even 4 1 315.6.a.a 1
20.e even 4 1 560.6.a.c 1
35.f even 4 1 245.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.a ✓ 1 5.c odd 4 1
175.6.a.a 1 5.c odd 4 1
175.6.b.b 2 1.a even 1 1 trivial
175.6.b.b 2 5.b even 2 1 inner
245.6.a.a 1 35.f even 4 1
315.6.a.a 1 15.e even 4 1
560.6.a.c 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 64 \) acting on \(S_{6}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 64 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T + 453)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 938961 \) Copy content Toggle raw display
$17$ \( T^{2} + 2679769 \) Copy content Toggle raw display
$19$ \( (T - 1550)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2735716 \) Copy content Toggle raw display
$29$ \( (T - 4985)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1192)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 121396324 \) Copy content Toggle raw display
$41$ \( (T + 1728)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 116942596 \) Copy content Toggle raw display
$47$ \( T^{2} + 688380169 \) Copy content Toggle raw display
$53$ \( T^{2} + 672676096 \) Copy content Toggle raw display
$59$ \( (T - 4580)^{2} \) Copy content Toggle raw display
$61$ \( (T + 12488)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 251159104 \) Copy content Toggle raw display
$71$ \( (T - 51792)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 23483716 \) Copy content Toggle raw display
$79$ \( (T + 62765)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 559038736 \) Copy content Toggle raw display
$89$ \( (T - 147300)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 69605649 \) Copy content Toggle raw display
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