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Properties

Label 175.6.b.f
Level $175$
Weight $6$
Character orbit 175.b
Analytic conductor $28.067$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

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 = [8,0,0,-98] 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 76x^{5} + 2849x^{4} - 2030x^{3} + 1250x^{2} + 28000x + 313600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + 2 \beta_1) q^{2} + (\beta_{7} - \beta_{4} - 4 \beta_1) q^{3} + (5 \beta_{5} - 3 \beta_{3} - 15) q^{4} + ( - 2 \beta_{5} - 11 \beta_{3} + 29) q^{6} - 49 \beta_1 q^{7} + ( - 6 \beta_{7} - 7 \beta_{4} + \cdots - 112 \beta_1) q^{8}+ \cdots + (3650 \beta_{6} - 15730 \beta_{5} + \cdots + 106900) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 98 q^{4} + 272 q^{6} - 1548 q^{9} + 1540 q^{11} + 686 q^{14} - 1230 q^{16} - 1128 q^{19} - 1372 q^{21} + 12488 q^{24} - 17460 q^{26} - 16132 q^{29} - 11712 q^{31} - 6804 q^{34} - 57518 q^{36} - 114932 q^{39}+ \cdots + 841336 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 76x^{5} + 2849x^{4} - 2030x^{3} + 1250x^{2} + 28000x + 313600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 250477 \nu^{7} + 4616410 \nu^{6} - 7118026 \nu^{5} - 12818540 \nu^{4} - 436201837 \nu^{3} + \cdots - 3283036400 ) / 149603716080 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 257216 \nu^{7} - 413567 \nu^{6} + 388607 \nu^{5} + 17337946 \nu^{4} + 889448909 \nu^{3} + \cdots + 4909349200 ) / 9350232255 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 257216 \nu^{7} + 413567 \nu^{6} - 388607 \nu^{5} - 17337946 \nu^{4} - 889448909 \nu^{3} + \cdots - 14259581455 ) / 9350232255 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10120807 \nu^{7} - 10135627 \nu^{6} + 150671620 \nu^{5} + 600546536 \nu^{4} + \cdots + 163085980400 ) / 205705109610 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10469951 \nu^{7} - 14544047 \nu^{6} - 133041898 \nu^{5} + 795527956 \nu^{4} + \cdots - 66292850960 ) / 205705109610 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12688981 \nu^{7} + 17312197 \nu^{6} + 181672238 \nu^{5} - 4241619506 \nu^{4} + \cdots - 4741686029360 ) / 205705109610 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7047265 \nu^{7} + 71085889 \nu^{6} - 154621588 \nu^{5} - 388187432 \nu^{4} + \cdots - 102513194000 ) / 68568369870 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{4} + 32\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -22\beta_{5} - 22\beta_{4} - 41\beta_{3} + 41\beta_{2} + 32\beta _1 - 73 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -63\beta_{6} - 85\beta_{5} - 16\beta_{3} - 1504 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 76\beta_{7} - 76\beta_{6} - 1506\beta_{5} + 1506\beta_{4} - 2055\beta_{3} - 2055\beta_{2} - 3392\beta _1 - 5447 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3485\beta_{7} + 5751\beta_{4} - 2038\beta_{2} - 78416\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8608 \beta_{7} + 8608 \beta_{6} + 89810 \beta_{5} + 89810 \beta_{4} + 109781 \beta_{3} - 109781 \beta_{2} + \cdots + 387989 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.47839 − 2.47839i
5.48085 + 5.48085i
−4.82172 + 4.82172i
−2.13752 − 2.13752i
−2.13752 + 2.13752i
−4.82172 − 4.82172i
5.48085 − 5.48085i
2.47839 + 2.47839i
− 9.92431i 0.133419i −66.4919 0 1.32409 49.0000i 342.308i 242.982 0
99.2 − 6.05190i − 15.9755i −4.62555 0 −96.6824 − 49.0000i − 165.668i −12.2177 0
99.3 − 5.86448i 26.2268i −2.39209 0 153.807 49.0000i − 173.635i −444.847 0
99.4 − 2.73688i 28.3358i 24.5095 0 77.5516 − 49.0000i − 154.660i −559.917 0
99.5 2.73688i − 28.3358i 24.5095 0 77.5516 49.0000i 154.660i −559.917 0
99.6 5.86448i − 26.2268i −2.39209 0 153.807 − 49.0000i 173.635i −444.847 0
99.7 6.05190i 15.9755i −4.62555 0 −96.6824 49.0000i 165.668i −12.2177 0
99.8 9.92431i − 0.133419i −66.4919 0 1.32409 − 49.0000i − 342.308i 242.982 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.8
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.f 8
5.b even 2 1 inner 175.6.b.f 8
5.c odd 4 1 35.6.a.d ✓ 4
5.c odd 4 1 175.6.a.f 4
15.e even 4 1 315.6.a.l 4
20.e even 4 1 560.6.a.v 4
35.f even 4 1 245.6.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.d ✓ 4 5.c odd 4 1
175.6.a.f 4 5.c odd 4 1
175.6.b.f 8 1.a even 1 1 trivial
175.6.b.f 8 5.b even 2 1 inner
245.6.a.e 4 35.f even 4 1
315.6.a.l 4 15.e even 4 1
560.6.a.v 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 177T_{2}^{6} + 9524T_{2}^{4} + 185892T_{2}^{2} + 929296 \) acting on \(S_{6}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 177 T^{6} + \cdots + 929296 \) Copy content Toggle raw display
$3$ \( T^{8} + 1746 T^{6} + \cdots + 2509056 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2401)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 770 T^{3} + \cdots - 20510223536)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 5221683964480)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{4} + 8066 T^{3} + \cdots + 831691300)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 738407035699200)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 225655412958976)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 13\!\cdots\!40)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 25\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 17\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 89\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 21\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 57\!\cdots\!76 \) Copy content Toggle raw display
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