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Properties

Label 1870.4.a.m
Level $1870$
Weight $4$
Character orbit 1870.a
Self dual yes
Analytic conductor $110.334$
Analytic rank $1$
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 = [10,20,1] 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: \(1\)
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: \( x^{10} - x^{9} - 184 x^{8} + 198 x^{7} + 11125 x^{6} - 13769 x^{5} - 258561 x^{4} + 275849 x^{3} + \cdots - 2234586 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + \beta_1 q^{3} + 4 q^{4} - 5 q^{5} + 2 \beta_1 q^{6} + ( - \beta_{2} - \beta_1 + 2) q^{7} + 8 q^{8} + (\beta_{6} + \beta_{5} + \beta_{2} + 10) q^{9} - 10 q^{10} - 11 q^{11} + 4 \beta_1 q^{12}+ \cdots + ( - 11 \beta_{6} - 11 \beta_{5} + \cdots - 110) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 20 q^{2} + q^{3} + 40 q^{4} - 50 q^{5} + 2 q^{6} + 19 q^{7} + 80 q^{8} + 99 q^{9} - 100 q^{10} - 110 q^{11} + 4 q^{12} - 41 q^{13} + 38 q^{14} - 5 q^{15} + 160 q^{16} - 170 q^{17} + 198 q^{18} - 157 q^{19}+ \cdots - 1089 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 184 x^{8} + 198 x^{7} + 11125 x^{6} - 13769 x^{5} - 258561 x^{4} + 275849 x^{3} + \cdots - 2234586 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 237874538 \nu^{9} - 6414013997 \nu^{8} + 37211212156 \nu^{7} + 845225995205 \nu^{6} + \cdots - 12\!\cdots\!37 ) / 58288323092805 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 303308364 \nu^{9} - 3600705601 \nu^{8} + 47202995748 \nu^{7} + 560489083550 \nu^{6} + \cdots - 12\!\cdots\!61 ) / 19429441030935 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1011956425 \nu^{9} - 23793683347 \nu^{8} + 218731339598 \nu^{7} + 3427878017365 \nu^{6} + \cdots - 16\!\cdots\!81 ) / 58288323092805 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 758794499 \nu^{9} - 435305038 \nu^{8} + 117128388766 \nu^{7} + 132038642430 \nu^{6} + \cdots - 445863377693832 ) / 19429441030935 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2514258035 \nu^{9} + 7719929111 \nu^{8} - 388596378454 \nu^{7} - 1241341922495 \nu^{6} + \cdots + 444566934892848 ) / 58288323092805 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4227526523 \nu^{9} + 15224439326 \nu^{8} - 626207424367 \nu^{7} - 2339946903875 \nu^{6} + \cdots + 779899530896139 ) / 58288323092805 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7484515372 \nu^{9} + 10592790358 \nu^{8} - 1235424994499 \nu^{7} - 1324307630305 \nu^{6} + \cdots + 728811432900528 ) / 58288323092805 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13832130209 \nu^{9} + 71896028654 \nu^{8} - 2182028556430 \nu^{7} - 10695029486990 \nu^{6} + \cdots + 72\!\cdots\!40 ) / 58288323092805 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{2} + 37 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{9} - \beta_{8} - 5\beta_{7} - 3\beta_{6} + 3\beta_{4} + 2\beta_{3} + 2\beta_{2} + 66\beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{9} + 3 \beta_{8} - 13 \beta_{7} + 94 \beta_{6} + 102 \beta_{5} + 12 \beta_{4} - 24 \beta_{3} + \cdots + 2323 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 310 \beta_{9} - 87 \beta_{8} - 519 \beta_{7} - 238 \beta_{6} + 140 \beta_{5} + 333 \beta_{4} + \cdots - 803 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 561 \beta_{9} + 343 \beta_{8} - 1757 \beta_{7} + 7843 \beta_{6} + 9375 \beta_{5} + 1644 \beta_{4} + \cdots + 172908 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27655 \beta_{9} - 6996 \beta_{8} - 46409 \beta_{7} - 13600 \beta_{6} + 23550 \beta_{5} + 31752 \beta_{4} + \cdots + 37135 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 74582 \beta_{9} + 28482 \beta_{8} - 194926 \beta_{7} + 644612 \beta_{6} + 835129 \beta_{5} + \cdots + 13851549 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2409472 \beta_{9} - 557704 \beta_{8} - 4067986 \beta_{7} - 516241 \beta_{6} + 2892996 \beta_{5} + \cdots + 14650811 \) 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
−8.67405
−8.58060
−4.75094
−3.52052
−0.834326
1.41104
4.67926
5.70109
5.92899
9.64005
2.00000 −8.67405 4.00000 −5.00000 −17.3481 20.5364 8.00000 48.2392 −10.0000
1.2 2.00000 −8.58060 4.00000 −5.00000 −17.1612 −16.9869 8.00000 46.6267 −10.0000
1.3 2.00000 −4.75094 4.00000 −5.00000 −9.50188 −10.3584 8.00000 −4.42858 −10.0000
1.4 2.00000 −3.52052 4.00000 −5.00000 −7.04104 19.1629 8.00000 −14.6059 −10.0000
1.5 2.00000 −0.834326 4.00000 −5.00000 −1.66865 34.7753 8.00000 −26.3039 −10.0000
1.6 2.00000 1.41104 4.00000 −5.00000 2.82208 −12.1008 8.00000 −25.0090 −10.0000
1.7 2.00000 4.67926 4.00000 −5.00000 9.35853 2.04701 8.00000 −5.10449 −10.0000
1.8 2.00000 5.70109 4.00000 −5.00000 11.4022 7.84778 8.00000 5.50244 −10.0000
1.9 2.00000 5.92899 4.00000 −5.00000 11.8580 −1.15801 8.00000 8.15298 −10.0000
1.10 2.00000 9.64005 4.00000 −5.00000 19.2801 −24.7653 8.00000 65.9305 −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.m ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1870.4.a.m ✓ 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - T_{3}^{9} - 184 T_{3}^{8} + 198 T_{3}^{7} + 11125 T_{3}^{6} - 13769 T_{3}^{5} + \cdots - 2234586 \) 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
$3$ \( T^{10} - T^{9} + \cdots - 2234586 \) Copy content Toggle raw display
$5$ \( (T + 5)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 13424546960 \) Copy content Toggle raw display
$11$ \( (T + 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 558882264805232 \) Copy content Toggle raw display
$17$ \( (T + 17)^{10} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 12\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 88\!\cdots\!55 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 53\!\cdots\!40 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 26\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 11\!\cdots\!90 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 11\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 93\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 19\!\cdots\!47 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 11\!\cdots\!34 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 41\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 19\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 77\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 11\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 87\!\cdots\!65 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 46\!\cdots\!40 \) Copy content Toggle raw display
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