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Properties

Label 1870.4.a.l
Level $1870$
Weight $4$
Character orbit 1870.a
Self dual yes
Analytic conductor $110.334$
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] = [1870,4,Mod(1,1870)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1870, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("1870.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1870 = 2 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1870.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,20,0] 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: \(110.333571711\)
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: \( x^{10} - 188 x^{8} - 48 x^{7} + 11155 x^{6} + 5616 x^{5} - 235675 x^{4} - 52830 x^{3} + 1553057 x^{2} + \cdots + 133376 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + 5 q^{5} + 2 \beta_1 q^{6} + (\beta_{4} + 2) q^{7} + 8 q^{8} + (\beta_{2} + 11) q^{9} + 10 q^{10} - 11 q^{11} + 4 \beta_1 q^{12} + (\beta_{9} + \beta_{7} + \beta_1 + 12) q^{13}+ \cdots + ( - 11 \beta_{2} - 121) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 20 q^{2} + 40 q^{4} + 50 q^{5} + 17 q^{7} + 80 q^{8} + 106 q^{9} + 100 q^{10} - 110 q^{11} + 118 q^{13} + 34 q^{14} + 160 q^{16} - 170 q^{17} + 212 q^{18} + 146 q^{19} + 200 q^{20} + 136 q^{21} - 220 q^{22}+ \cdots - 1166 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 188 x^{8} - 48 x^{7} + 11155 x^{6} + 5616 x^{5} - 235675 x^{4} - 52830 x^{3} + 1553057 x^{2} + \cdots + 133376 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 38 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1773930 \nu^{9} + 583064 \nu^{8} - 335363514 \nu^{7} - 187243343 \nu^{6} + 20039370762 \nu^{5} + \cdots - 1077615086506 ) / 3534148449 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3803673 \nu^{9} - 1359938 \nu^{8} + 716071656 \nu^{7} + 432004490 \nu^{6} + \cdots + 1683131049628 ) / 2718575730 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 76080479 \nu^{9} + 24849484 \nu^{8} - 14270825528 \nu^{7} - 8196264970 \nu^{6} + \cdots - 34775287143554 ) / 35341484490 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16304601 \nu^{9} + 6934298 \nu^{8} - 3066878538 \nu^{7} - 2064012074 \nu^{6} + \cdots - 6651207451114 ) / 7068296898 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 82457371 \nu^{9} + 26152696 \nu^{8} - 15477870112 \nu^{7} - 8890608400 \nu^{6} + \cdots - 36397724747156 ) / 35341484490 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 83931877 \nu^{9} + 19273912 \nu^{8} - 15761655664 \nu^{7} - 7777219510 \nu^{6} + \cdots - 38301482027642 ) / 35341484490 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 93365447 \nu^{9} - 22253572 \nu^{8} + 17510222654 \nu^{7} + 8801055760 \nu^{6} + \cdots + 40030180077632 ) / 35341484490 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 38 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 2\beta_{8} - 2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 69\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{9} - 9 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 15 \beta_{4} - 6 \beta_{3} + \cdots + 2643 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 173 \beta_{9} + 256 \beta_{8} - 169 \beta_{7} + 83 \beta_{6} + 116 \beta_{5} + 89 \beta_{4} + \cdots + 1621 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1221 \beta_{9} - 1353 \beta_{8} - 357 \beta_{7} - 573 \beta_{6} - 285 \beta_{5} - 2229 \beta_{4} + \cdots + 215785 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 20721 \beta_{9} + 26502 \beta_{8} - 12201 \beta_{7} + 5934 \beta_{6} + 11910 \beta_{5} + 7644 \beta_{4} + \cdots + 99918 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 128175 \beta_{9} - 156414 \beta_{8} - 29670 \beta_{7} - 73779 \beta_{6} - 22416 \beta_{5} + \cdots + 18796331 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2193790 \beta_{9} + 2584700 \beta_{8} - 880217 \beta_{7} + 414004 \beta_{6} + 1184503 \beta_{5} + \cdots + 2334191 \) 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
−9.79800
−6.56251
−5.20184
−4.33767
0.290515
0.315034
2.70982
4.61033
8.37973
9.59459
2.00000 −9.79800 4.00000 5.00000 −19.5960 −7.20737 8.00000 69.0008 10.0000
1.2 2.00000 −6.56251 4.00000 5.00000 −13.1250 24.4652 8.00000 16.0666 10.0000
1.3 2.00000 −5.20184 4.00000 5.00000 −10.4037 −20.4546 8.00000 0.0591775 10.0000
1.4 2.00000 −4.33767 4.00000 5.00000 −8.67534 17.6546 8.00000 −8.18460 10.0000
1.5 2.00000 0.290515 4.00000 5.00000 0.581029 17.1180 8.00000 −26.9156 10.0000
1.6 2.00000 0.315034 4.00000 5.00000 0.630069 −30.9763 8.00000 −26.9008 10.0000
1.7 2.00000 2.70982 4.00000 5.00000 5.41965 −19.6830 8.00000 −19.6569 10.0000
1.8 2.00000 4.61033 4.00000 5.00000 9.22067 11.8768 8.00000 −5.74483 10.0000
1.9 2.00000 8.37973 4.00000 5.00000 16.7595 26.9764 8.00000 43.2198 10.0000
1.10 2.00000 9.59459 4.00000 5.00000 19.1892 −2.76970 8.00000 65.0562 10.0000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(11\) \( +1 \)
\(17\) \( +1 \)

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 1870.4.a.l ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1870.4.a.l ✓ 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 188 T_{3}^{8} - 48 T_{3}^{7} + 11155 T_{3}^{6} + 5616 T_{3}^{5} - 235675 T_{3}^{4} + \cdots + 133376 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1870))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 188 T^{8} + \cdots + 133376 \) Copy content Toggle raw display
$5$ \( (T - 5)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 589744220160 \) Copy content Toggle raw display
$11$ \( (T + 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 19266617376 \) Copy content Toggle raw display
$17$ \( (T + 17)^{10} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 41\!\cdots\!50 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 53\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 31\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 56\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 23\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 39\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 19\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 42\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 31\!\cdots\!60 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 26\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 58\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 52\!\cdots\!50 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 49\!\cdots\!60 \) Copy content Toggle raw display
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