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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,4,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [27,6,-18,132,-135] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(85.2577599583\)
Analytic rank: \(0\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 27 q + 6 q^{2} - 18 q^{3} + 132 q^{4} - 135 q^{5} + 45 q^{6} - 42 q^{7} + 123 q^{8} + 297 q^{9} - 30 q^{10} - 30 q^{11} - 216 q^{12} + 132 q^{13} - 162 q^{14} + 90 q^{15} + 720 q^{16} + 462 q^{18} + 507 q^{19}+ \cdots - 4926 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.63567 −6.80303 23.7607 −5.00000 38.3396 −31.0113 −88.8222 19.2812 28.1783
1.2 −4.68652 −8.76453 13.9635 −5.00000 41.0751 7.59444 −27.9478 49.8171 23.4326
1.3 −4.53513 −4.01236 12.5674 −5.00000 18.1966 35.0730 −20.7136 −10.9010 22.6756
1.4 −4.30394 3.53588 10.5239 −5.00000 −15.2182 14.9453 −10.8628 −14.4976 21.5197
1.5 −4.09563 6.97271 8.77422 −5.00000 −28.5577 18.0715 −3.17092 21.6187 20.4782
1.6 −3.96275 5.08638 7.70340 −5.00000 −20.1561 −13.0902 1.17537 −1.12873 19.8138
1.7 −3.77110 −6.02141 6.22122 −5.00000 22.7074 −34.9027 6.70798 9.25737 18.8555
1.8 −2.26853 0.787610 −2.85377 −5.00000 −1.78672 −18.9115 24.6221 −26.3797 11.3426
1.9 −1.87578 −3.59652 −4.48145 −5.00000 6.74629 17.1177 23.4125 −14.0650 9.37890
1.10 −1.79246 5.09341 −4.78709 −5.00000 −9.12973 −28.8838 22.9203 −1.05721 8.96230
1.11 −1.46968 7.72790 −5.84003 −5.00000 −11.3576 26.6494 20.3405 32.7204 7.34842
1.12 −1.12124 −9.72959 −6.74281 −5.00000 10.9093 0.398766 16.5303 67.6650 5.60622
1.13 −0.425515 −2.40580 −7.81894 −5.00000 1.02370 8.34930 6.73119 −21.2121 2.12757
1.14 −0.254096 −6.08990 −7.93544 −5.00000 1.54742 11.8189 4.04913 10.0869 1.27048
1.15 1.51308 2.94106 −5.71059 −5.00000 4.45007 −22.2692 −20.7452 −18.3501 −7.56541
1.16 1.58849 −9.06360 −5.47671 −5.00000 −14.3974 2.45708 −21.4076 55.1488 −7.94244
1.17 1.63564 6.58071 −5.32468 −5.00000 10.7637 14.1747 −21.7944 16.3057 −8.17820
1.18 2.34429 1.03758 −2.50431 −5.00000 2.43240 1.70165 −24.6251 −25.9234 −11.7214
1.19 2.41055 5.82977 −2.18926 −5.00000 14.0529 −27.4139 −24.5617 6.98625 −12.0527
1.20 3.16884 1.41392 2.04153 −5.00000 4.48047 34.0951 −18.8814 −25.0008 −15.8442
See all 27 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.27
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(17\) \( +1 \)

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 1445.4.a.w ✓ 27
17.b even 2 1 1445.4.a.x yes 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1445.4.a.w ✓ 27 1.a even 1 1 trivial
1445.4.a.x yes 27 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1445))\):

\( T_{2}^{27} - 6 T_{2}^{26} - 156 T_{2}^{25} + 935 T_{2}^{24} + 10662 T_{2}^{23} - 63651 T_{2}^{22} + \cdots - 160646823936 \) Copy content Toggle raw display
\( T_{3}^{27} + 18 T_{3}^{26} - 351 T_{3}^{25} - 7722 T_{3}^{24} + 46476 T_{3}^{23} + 1435500 T_{3}^{22} + \cdots - 46\!\cdots\!16 \) Copy content Toggle raw display