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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 = [21,-6,18,60,-105] 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: \(1\)
Dimension: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q - 6 q^{2} + 18 q^{3} + 60 q^{4} - 105 q^{5} - 45 q^{6} + 42 q^{7} - 21 q^{8} + 135 q^{9} + 30 q^{10} + 30 q^{11} + 216 q^{12} - 132 q^{13} - 162 q^{14} - 90 q^{15} + 48 q^{16} - 126 q^{18} - 405 q^{19}+ \cdots - 9066 q^{99}+O(q^{100}) \) 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 −5.42745 5.99066 21.4572 −5.00000 −32.5140 9.50765 −73.0382 8.88806 27.1372
1.2 −4.97020 −4.72455 16.7029 −5.00000 23.4820 3.35700 −43.2553 −4.67865 24.8510
1.3 −4.50797 1.62074 12.3218 −5.00000 −7.30623 −23.4604 −19.4823 −24.3732 22.5398
1.4 −3.68222 7.90959 5.55874 −5.00000 −29.1248 −8.59035 8.98927 35.5616 18.4111
1.5 −3.10779 7.50370 1.65834 −5.00000 −23.3199 26.4890 19.7085 29.3056 15.5389
1.6 −2.86454 1.34318 0.205604 −5.00000 −3.84761 32.5260 22.3274 −25.1959 14.3227
1.7 −2.85654 0.322579 0.159821 −5.00000 −0.921460 −6.32418 22.3958 −26.8959 14.2827
1.8 −2.75215 −6.49307 −0.425663 −5.00000 17.8699 13.1194 23.1887 15.1600 13.7608
1.9 −1.98792 10.2526 −4.04818 −5.00000 −20.3813 −3.29700 23.9508 78.1153 9.93959
1.10 −1.05636 −9.10271 −6.88410 −5.00000 9.61576 33.1316 15.7230 55.8593 5.28181
1.11 −0.360698 0.346877 −7.86990 −5.00000 −0.125118 −19.6621 5.72423 −26.8797 1.80349
1.12 −0.0188046 −2.57336 −7.99965 −5.00000 0.0483910 −22.8583 0.300867 −20.3778 0.0940230
1.13 1.38171 −4.83398 −6.09088 −5.00000 −6.67916 −19.2556 −19.4695 −3.63259 −6.90854
1.14 1.39951 4.91180 −6.04137 −5.00000 6.87411 22.4872 −19.6511 −2.87427 −6.99756
1.15 1.80233 4.83362 −4.75160 −5.00000 8.71179 11.7626 −22.9826 −3.63611 −9.01166
1.16 2.19524 −6.07103 −3.18093 −5.00000 −13.3274 15.4107 −24.5448 9.85746 −10.9762
1.17 3.08151 7.81581 1.49572 −5.00000 24.0845 −20.6713 −20.0430 34.0869 −15.4076
1.18 3.54744 −6.90695 4.58430 −5.00000 −24.5020 8.86098 −12.1170 20.7059 −17.7372
1.19 4.56903 8.01532 12.8760 −5.00000 36.6222 −30.9023 22.2787 37.2454 −22.8451
1.20 4.66766 −1.54092 13.7871 −5.00000 −7.19249 13.8059 27.0121 −24.6256 −23.3383
See all 21 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.21
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.v yes 21
17.b even 2 1 1445.4.a.u ✓ 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1445.4.a.u ✓ 21 17.b even 2 1
1445.4.a.v yes 21 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1445))\):

\( T_{2}^{21} + 6 T_{2}^{20} - 96 T_{2}^{19} - 609 T_{2}^{18} + 3690 T_{2}^{17} + 25551 T_{2}^{16} + \cdots - 3939648 \) Copy content Toggle raw display
\( T_{3}^{21} - 18 T_{3}^{20} - 189 T_{3}^{19} + 4806 T_{3}^{18} + 8385 T_{3}^{17} - 521346 T_{3}^{16} + \cdots + 183749193728 \) Copy content Toggle raw display