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Properties

Label 2175.4.a.o
Level $2175$
Weight $4$
Character orbit 2175.a
Self dual yes
Analytic conductor $128.329$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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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] = [2175,4,Mod(1,2175)] 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("2175.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,4,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(128.329154262\)
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: \( x^{8} - 4x^{7} - 43x^{6} + 117x^{5} + 586x^{4} - 701x^{3} - 1792x^{2} + 924x + 1424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 435)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} + 5) q^{4} + ( - 3 \beta_1 + 3) q^{6} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots - 1) q^{7} + (\beta_{6} - \beta_{5} + \beta_{2} + \cdots + 5) q^{8}+ \cdots + (9 \beta_{7} + 9 \beta_{6} + 9 \beta_{4} + \cdots + 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 24 q^{3} + 38 q^{4} + 12 q^{6} + q^{7} + 9 q^{8} + 72 q^{9} + 11 q^{11} + 114 q^{12} + 97 q^{13} - 69 q^{14} + 242 q^{16} + 119 q^{17} + 36 q^{18} + 220 q^{19} + 3 q^{21} + 129 q^{22} + 262 q^{23}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 43x^{6} + 117x^{5} + 586x^{4} - 701x^{3} - 1792x^{2} + 924x + 1424 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} + 5\nu^{6} + 85\nu^{5} - 98\nu^{4} - 1081\nu^{3} - 143\nu^{2} + 2426\nu + 1128 ) / 136 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{7} + 43\nu^{6} + 204\nu^{5} - 1193\nu^{4} - 1837\nu^{3} + 7838\nu^{2} + 3544\nu - 9040 ) / 272 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 21\nu^{6} + 204\nu^{5} - 619\nu^{4} - 2371\nu^{3} + 3578\nu^{2} + 1968\nu - 3232 ) / 136 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 21\nu^{6} + 204\nu^{5} - 619\nu^{4} - 2507\nu^{3} + 3850\nu^{2} + 4824\nu - 4320 ) / 136 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} - 9\nu^{6} - 374\nu^{5} - 31\nu^{4} + 5883\nu^{3} + 6952\nu^{2} - 14220\nu - 15440 ) / 272 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{5} + 2\beta_{2} + 25\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} - 3\beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{3} + 28\beta_{2} + 86\beta _1 + 273 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} - 26\beta_{6} + 28\beta_{5} - 12\beta_{3} + 89\beta_{2} + 700\beta _1 + 780 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -48\beta_{7} - 119\beta_{6} + 93\beta_{5} + 44\beta_{4} - 96\beta_{3} + 780\beta_{2} + 3065\beta _1 + 7324 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -241\beta_{7} - 715\beta_{6} + 784\beta_{5} + 61\beta_{4} - 769\beta_{3} + 3208\beta_{2} + 20756\beta _1 + 29141 \) Copy content Toggle raw display

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

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
5.75837
5.42731
1.81879
1.26223
−0.859765
−1.46965
−3.76263
−4.17466
−4.75837 3.00000 14.6421 0 −14.2751 11.0144 −31.6057 9.00000 0
1.2 −4.42731 3.00000 11.6011 0 −13.2819 2.85439 −15.9430 9.00000 0
1.3 −0.818792 3.00000 −7.32958 0 −2.45638 −13.8632 12.5517 9.00000 0
1.4 −0.262226 3.00000 −7.93124 0 −0.786679 −12.1027 4.17759 9.00000 0
1.5 1.85976 3.00000 −4.54128 0 5.57929 0.560700 −23.3238 9.00000 0
1.6 2.46965 3.00000 −1.90084 0 7.40894 36.5630 −24.4516 9.00000 0
1.7 4.76263 3.00000 14.6826 0 14.2879 −35.2196 31.8267 9.00000 0
1.8 5.17466 3.00000 18.7771 0 15.5240 11.1931 55.7681 9.00000 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
\(3\) \( -1 \)
\(5\) \( +1 \)
\(29\) \( -1 \)

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 2175.4.a.o 8
5.b even 2 1 435.4.a.k ✓ 8
15.d odd 2 1 1305.4.a.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.4.a.k ✓ 8 5.b even 2 1
1305.4.a.p 8 15.d odd 2 1
2175.4.a.o 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2175))\):

\( T_{2}^{8} - 4T_{2}^{7} - 43T_{2}^{6} + 169T_{2}^{5} + 456T_{2}^{4} - 1869T_{2}^{3} + 90T_{2}^{2} + 2112T_{2} + 512 \) Copy content Toggle raw display
\( T_{7}^{8} - T_{7}^{7} - 1585 T_{7}^{6} + 413 T_{7}^{5} + 403612 T_{7}^{4} - 688840 T_{7}^{3} + \cdots - 42630912 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{7} + \cdots + 512 \) Copy content Toggle raw display
$3$ \( (T - 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + \cdots - 42630912 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 109041848480 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 791998533632 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 13781129317504 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 66835834880 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots - 22\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T - 29)^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 33\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 22\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 10\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 11\!\cdots\!40 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 48\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 79\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 17\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 19\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 21\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 43\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots - 15\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 89\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 73\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 70\!\cdots\!80 \) Copy content Toggle raw display
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