Properties

Label 175.10.a.i
Level $175$
Weight $10$
Character orbit 175.a
Self dual yes
Analytic conductor $90.131$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [175,10,Mod(1,175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) 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("175.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,27] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(90.1312713287\)
Analytic rank: \(1\)
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: \( x^{8} - 3 x^{7} - 3503 x^{6} + 21249 x^{5} + 3516263 x^{4} - 28277577 x^{3} - 915460405 x^{2} + \cdots + 31328928252 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 5^{3} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 + 3) q^{2} + (\beta_{2} - \beta_1 - 8) q^{3} + (\beta_{4} + 2 \beta_{2} + \beta_1 + 376) q^{4} + ( - \beta_{6} - 3 \beta_{4} + \cdots - 1045) q^{6} - 2401 q^{7} + (\beta_{7} - 2 \beta_{6} + \beta_{5} + \cdots + 279) q^{8}+ \cdots + (33613 \beta_{7} - 90328 \beta_{6} + \cdots - 377883492) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 27 q^{2} - 69 q^{3} + 3009 q^{4} - 8285 q^{6} - 19208 q^{7} + 3591 q^{8} - 3269 q^{9} - 77952 q^{11} + 194763 q^{12} - 101596 q^{13} - 64827 q^{14} + 1525505 q^{16} + 467007 q^{17} - 505144 q^{18}+ \cdots - 2991181349 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3 x^{7} - 3503 x^{6} + 21249 x^{5} + 3516263 x^{4} - 28277577 x^{3} - 915460405 x^{2} + \cdots + 31328928252 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11703 \nu^{7} + 169138 \nu^{6} - 36445607 \nu^{5} - 339509136 \nu^{4} + 30022928145 \nu^{3} + \cdots - 6079442587044 ) / 120848501760 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23011 \nu^{7} - 5491334 \nu^{6} - 100903539 \nu^{5} + 16258418928 \nu^{4} + \cdots + 10\!\cdots\!92 ) / 120848501760 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11703 \nu^{7} - 169138 \nu^{6} + 36445607 \nu^{5} + 339509136 \nu^{4} + \cdots - 47033473936476 ) / 60424250880 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 138249 \nu^{7} - 15735694 \nu^{6} + 115769241 \nu^{5} + 42342014448 \nu^{4} + \cdots + 17\!\cdots\!52 ) / 120848501760 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 84569 \nu^{7} - 2021294 \nu^{6} + 239424681 \nu^{5} + 5054685168 \nu^{4} + \cdots + 271847649208092 ) / 60424250880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 74605 \nu^{7} + 2021734 \nu^{6} - 194383517 \nu^{5} - 4364356848 \nu^{4} + \cdots + 61558726929588 ) / 24169700352 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 2\beta_{2} - 5\beta _1 + 879 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{6} + \beta_{5} - 3\beta_{4} - \beta_{3} - 53\beta_{2} + 1471\beta _1 - 4587 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 29\beta_{7} + 90\beta_{6} + 5\beta_{5} + 1954\beta_{4} - 13\beta_{3} + 4369\beta_{2} - 14823\beta _1 + 1299656 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2610 \beta_{7} - 6180 \beta_{6} + 2274 \beta_{5} - 8658 \beta_{4} - 1554 \beta_{3} - 159906 \beta_{2} + \cdots - 13461294 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 74622 \beta_{7} + 285708 \beta_{6} + 6558 \beta_{5} + 3664663 \beta_{4} - 54174 \beta_{3} + \cdots + 2207376309 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5325511 \beta_{7} - 15633278 \beta_{6} + 4566583 \beta_{5} - 24062445 \beta_{4} - 1868263 \beta_{3} + \cdots - 31320715677 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−45.4708
−36.5367
−16.4566
−4.02881
9.22118
25.6654
28.6308
41.9756
−42.4708 194.204 1291.77 0 −8248.01 −2401.00 −33117.6 18032.2 0
1.2 −33.5367 −53.1722 612.712 0 1783.22 −2401.00 −3377.55 −16855.7 0
1.3 −13.4566 −121.784 −330.919 0 1638.80 −2401.00 11342.8 −4851.66 0
1.4 −1.02881 101.525 −510.942 0 −104.450 −2401.00 1052.41 −9375.68 0
1.5 12.2212 −205.690 −362.643 0 −2513.78 −2401.00 −10689.2 22625.6 0
1.6 28.6654 179.033 309.707 0 5132.05 −2401.00 −5798.81 12369.7 0
1.7 31.6308 −102.165 488.504 0 −3231.54 −2401.00 −743.183 −9245.41 0
1.8 44.9756 −60.9506 1510.81 0 −2741.29 −2401.00 44922.0 −15968.0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.10.a.i yes 8
5.b even 2 1 175.10.a.h 8
5.c odd 4 2 175.10.b.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.10.a.h 8 5.b even 2 1
175.10.a.i yes 8 1.a even 1 1 trivial
175.10.b.h 16 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 27 T_{2}^{7} - 3188 T_{2}^{6} + 82224 T_{2}^{5} + 2733128 T_{2}^{4} - 66690816 T_{2}^{3} + \cdots + 9827481600 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(175))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 9827481600 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots - 29\!\cdots\!64 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T + 2401)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 72\!\cdots\!11 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 82\!\cdots\!48 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots - 93\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 61\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 60\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 78\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 75\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 16\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 13\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 50\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
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