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Properties

Label 175.10.b.a
Level $175$
Weight $10$
Character orbit 175.b
Analytic conductor $90.131$
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] = [175,10,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, 10); 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, 10, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
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,-544] 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: \(90.1312713287\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 28 i q^{2} + 116 i q^{3} - 272 q^{4} - 3248 q^{6} + 2401 i q^{7} + 6720 i q^{8} + 6227 q^{9} - 25548 q^{11} - 31552 i q^{12} + 42306 i q^{13} - 67228 q^{14} - 327424 q^{16} - 526342 i q^{17} + 174356 i q^{18} + \cdots - 159087396 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 544 q^{4} - 6496 q^{6} + 12454 q^{9} - 51096 q^{11} - 134456 q^{14} - 654848 q^{16} + 700120 q^{19} - 557032 q^{21} - 1559040 q^{24} - 2369136 q^{26} - 13440860 q^{29} - 12824416 q^{31} + 29475152 q^{34}+ \cdots - 318174792 q^{99}+O(q^{100}) \) 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
− 28.0000i − 116.000i −272.000 0 −3248.00 − 2401.00i − 6720.00i 6227.00 0
99.2 28.0000i 116.000i −272.000 0 −3248.00 2401.00i 6720.00i 6227.00 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 175.10.b.a 2
5.b even 2 1 inner 175.10.b.a 2
5.c odd 4 1 35.10.a.a ✓ 1
5.c odd 4 1 175.10.a.a 1
15.e even 4 1 315.10.a.a 1
35.f even 4 1 245.10.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.a.a ✓ 1 5.c odd 4 1
175.10.a.a 1 5.c odd 4 1
175.10.b.a 2 1.a even 1 1 trivial
175.10.b.a 2 5.b even 2 1 inner
245.10.a.b 1 35.f even 4 1
315.10.a.a 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 784 \) acting on \(S_{10}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 784 \) Copy content Toggle raw display
$3$ \( T^{2} + 13456 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 5764801 \) Copy content Toggle raw display
$11$ \( (T + 25548)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1789797636 \) Copy content Toggle raw display
$17$ \( T^{2} + 277035900964 \) Copy content Toggle raw display
$19$ \( (T - 350060)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 386854144576 \) Copy content Toggle raw display
$29$ \( (T + 6720430)^{2} \) Copy content Toggle raw display
$31$ \( (T + 6412208)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 5371649853124 \) Copy content Toggle raw display
$41$ \( (T + 10224678)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 906853236912016 \) Copy content Toggle raw display
$47$ \( T^{2} + 559081863487744 \) Copy content Toggle raw display
$53$ \( T^{2} + 32\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T + 84934780)^{2} \) Copy content Toggle raw display
$61$ \( (T - 14677822)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 59\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T - 61901952)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 80\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T + 276107480)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 53\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T - 896368470)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 14\!\cdots\!84 \) Copy content Toggle raw display
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