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Properties

Label 1875.4.a.b
Level $1875$
Weight $4$
Character orbit 1875.a
Self dual yes
Analytic conductor $110.629$
Analytic rank $0$
Dimension $10$
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] = [1875,4,Mod(1,1875)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1875, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1875.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-1,30] 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: \(110.628581261\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 53x^{8} + 56x^{7} + 923x^{6} - 1003x^{5} - 6071x^{4} + 5492x^{3} + 14476x^{2} - 8352x - 5184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 5 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 3 q^{3} + ( - \beta_{3} + \beta_{2} + 2) q^{4} - 3 \beta_1 q^{6} + (\beta_{9} - \beta_{6} + \beta_{3} + \cdots + 3) q^{7} + ( - \beta_{7} + \beta_{5} + 3 \beta_{4} + \cdots + 4) q^{8}+ \cdots + (9 \beta_{9} - 9 \beta_{8} - 9 \beta_{7} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 30 q^{3} + 27 q^{4} - 3 q^{6} + 21 q^{7} + 24 q^{8} + 90 q^{9} + 36 q^{11} + 81 q^{12} + 41 q^{13} + 76 q^{14} - 13 q^{16} + 70 q^{17} - 9 q^{18} + 29 q^{19} + 63 q^{21} + 115 q^{22} + 292 q^{23}+ \cdots + 324 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 53x^{8} + 56x^{7} + 923x^{6} - 1003x^{5} - 6071x^{4} + 5492x^{3} + 14476x^{2} - 8352x - 5184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 8261 \nu^{9} + 54017 \nu^{8} + 405181 \nu^{7} - 2698612 \nu^{6} - 5611351 \nu^{5} + \cdots - 340653456 ) / 38799360 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8261 \nu^{9} + 54017 \nu^{8} + 405181 \nu^{7} - 2698612 \nu^{6} - 5611351 \nu^{5} + \cdots + 47340144 ) / 38799360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1271 \nu^{9} - 907 \nu^{8} - 62111 \nu^{7} + 55932 \nu^{6} + 881341 \nu^{5} - 975769 \nu^{4} + \cdots - 1189584 ) / 1077760 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47857 \nu^{9} - 175549 \nu^{8} - 2300297 \nu^{7} + 8240804 \nu^{6} + 33439547 \nu^{5} + \cdots - 292698288 ) / 19399680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 166447 \nu^{9} - 196579 \nu^{8} - 8563607 \nu^{7} + 8664284 \nu^{6} + 138346277 \nu^{5} + \cdots + 474635952 ) / 38799360 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24969 \nu^{9} - 43893 \nu^{8} - 1211489 \nu^{7} + 2202628 \nu^{6} + 17383619 \nu^{5} + \cdots - 56814896 ) / 4311040 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 126467 \nu^{9} + 150361 \nu^{8} - 6172747 \nu^{7} - 6110036 \nu^{6} + 94746817 \nu^{5} + \cdots + 40876272 ) / 12933120 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 475511 \nu^{9} - 587147 \nu^{8} - 21962911 \nu^{7} + 30557692 \nu^{6} + 288875581 \nu^{5} + \cdots - 210438864 ) / 38799360 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{5} - 3\beta_{4} - \beta_{3} + 19\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{7} + 2\beta_{6} - 4\beta_{5} - 8\beta_{4} - 44\beta_{3} + 24\beta_{2} - \beta _1 + 163 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + 2 \beta_{8} + 27 \beta_{7} + 2 \beta_{6} - 32 \beta_{5} - 114 \beta_{4} - 81 \beta_{3} + \cdots - 153 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 31 \beta_{9} + 6 \beta_{8} - 24 \beta_{7} + 58 \beta_{6} - 151 \beta_{5} - 301 \beta_{4} - 1401 \beta_{3} + \cdots + 3139 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 68 \beta_{9} + 74 \beta_{8} + 636 \beta_{7} + 68 \beta_{6} - 862 \beta_{5} - 3360 \beta_{4} + \cdots - 3880 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 800 \beta_{9} + 284 \beta_{8} - 446 \beta_{7} + 1408 \beta_{6} - 4528 \beta_{5} - 8794 \beta_{4} + \cdots + 65676 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2604 \beta_{9} + 2168 \beta_{8} + 14753 \beta_{7} + 1924 \beta_{6} - 22017 \beta_{5} - 90745 \beta_{4} + \cdots - 83620 \) 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
5.02794
3.56920
3.02547
2.42773
0.978680
−0.401257
−2.04301
−2.25282
−4.55862
−4.77330
−5.02794 3.00000 17.2802 0 −15.0838 −8.42899 −46.6601 9.00000 0
1.2 −3.56920 3.00000 4.73917 0 −10.7076 −23.3745 11.6386 9.00000 0
1.3 −3.02547 3.00000 1.15344 0 −9.07640 35.2831 20.7140 9.00000 0
1.4 −2.42773 3.00000 −2.10613 0 −7.28319 14.7223 24.5349 9.00000 0
1.5 −0.978680 3.00000 −7.04219 0 −2.93604 −26.6128 14.7215 9.00000 0
1.6 0.401257 3.00000 −7.83899 0 1.20377 7.62396 −6.35550 9.00000 0
1.7 2.04301 3.00000 −3.82612 0 6.12903 26.7857 −24.1609 9.00000 0
1.8 2.25282 3.00000 −2.92481 0 6.75846 −14.0753 −24.6116 9.00000 0
1.9 4.55862 3.00000 12.7810 0 13.6759 12.8690 21.7949 9.00000 0
1.10 4.77330 3.00000 14.7844 0 14.3199 −3.79243 32.3842 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)

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 1875.4.a.b ✓ 10
5.b even 2 1 1875.4.a.c yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.4.a.b ✓ 10 1.a even 1 1 trivial
1875.4.a.c yes 10 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + T_{2}^{9} - 53 T_{2}^{8} - 56 T_{2}^{7} + 923 T_{2}^{6} + 1003 T_{2}^{5} - 6071 T_{2}^{4} + \cdots - 5184 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1875))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + \cdots - 5184 \) Copy content Toggle raw display
$3$ \( (T - 3)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 382078740880 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 685413333353664 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 25903784712439 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 92\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 77\!\cdots\!75 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 22\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 16\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 43\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 28\!\cdots\!89 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 82\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 37\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 91\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 25\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 20\!\cdots\!69 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 18\!\cdots\!75 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 52\!\cdots\!71 \) Copy content Toggle raw display
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