Properties

Label 1445.4.a.z
Level $1445$
Weight $4$
Character orbit 1445.a
Self dual yes
Analytic conductor $85.258$
Analytic rank $0$
Dimension $36$
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] = [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 = [36,8,24,144,180] 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: \(36\)
Twist minimal: no (minimal twist has level 85)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 8 q^{2} + 24 q^{3} + 144 q^{4} + 180 q^{5} + 48 q^{6} + 112 q^{7} + 96 q^{8} + 324 q^{9} + 40 q^{10} + 176 q^{11} + 192 q^{12} + 104 q^{13} - 104 q^{14} + 120 q^{15} + 576 q^{16} - 8 q^{18} + 720 q^{20}+ \cdots + 5664 q^{99}+O(q^{100}) \) 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 −5.36920 0.213922 20.8283 5.00000 −1.14859 −3.77662 −68.8778 −26.9542 −26.8460
1.2 −5.26104 −7.83182 19.6785 5.00000 41.2035 21.2775 −61.4411 34.3373 −26.3052
1.3 −4.64294 3.18357 13.5569 5.00000 −14.7811 23.6690 −25.8002 −16.8649 −23.2147
1.4 −4.57503 5.28260 12.9309 5.00000 −24.1681 −10.7957 −22.5588 0.905908 −22.8751
1.5 −4.41809 −6.69255 11.5195 5.00000 29.5683 33.0069 −15.5496 17.7902 −22.0905
1.6 −3.83981 −0.841682 6.74413 5.00000 3.23190 −10.2656 4.82230 −26.2916 −19.1990
1.7 −3.70979 9.38478 5.76251 5.00000 −34.8155 22.5293 8.30059 61.0741 −18.5489
1.8 −3.61978 9.56829 5.10278 5.00000 −34.6351 −19.6676 10.4873 64.5522 −18.0989
1.9 −3.37646 −8.03242 3.40048 5.00000 27.1211 −5.12157 15.5301 37.5197 −16.8823
1.10 −2.74575 8.37951 −0.460830 5.00000 −23.0081 −14.5191 23.2314 43.2161 −13.7288
1.11 −2.39244 2.19287 −2.27622 5.00000 −5.24631 33.6600 24.5853 −22.1913 −11.9622
1.12 −1.99727 −1.92099 −4.01093 5.00000 3.83674 −16.8524 23.9890 −23.3098 −9.98633
1.13 −1.85859 −6.71273 −4.54565 5.00000 12.4762 0.771178 23.3172 18.0608 −9.29293
1.14 −1.39768 0.421250 −6.04648 5.00000 −0.588774 −23.0410 19.6325 −26.8225 −6.98841
1.15 −1.04430 5.98947 −6.90943 5.00000 −6.25482 23.8041 15.5700 8.87379 −5.22151
1.16 −0.603721 −0.773577 −7.63552 5.00000 0.467025 3.72731 9.43950 −26.4016 −3.01861
1.17 0.171201 −8.57038 −7.97069 5.00000 −1.46726 2.01211 −2.73420 46.4514 0.856004
1.18 0.440524 −7.06319 −7.80594 5.00000 −3.11151 8.04204 −6.96290 22.8887 2.20262
1.19 0.555316 −2.26972 −7.69162 5.00000 −1.26041 30.5357 −8.71382 −21.8484 2.77658
1.20 0.778487 6.27665 −7.39396 5.00000 4.88629 3.80638 −11.9840 12.3963 3.89244
See all 36 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.36
Significant digits:
Format:

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.z 36
17.b even 2 1 1445.4.a.y 36
17.e odd 16 2 85.4.l.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.4.l.a ✓ 72 17.e odd 16 2
1445.4.a.y 36 17.b even 2 1
1445.4.a.z 36 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1445))\):

\( T_{2}^{36} - 8 T_{2}^{35} - 184 T_{2}^{34} + 1568 T_{2}^{33} + 15024 T_{2}^{32} - 138752 T_{2}^{31} + \cdots + 33651927040000 \) Copy content Toggle raw display
\( T_{3}^{36} - 24 T_{3}^{35} - 360 T_{3}^{34} + 12528 T_{3}^{33} + 34056 T_{3}^{32} + \cdots - 41\!\cdots\!92 \) Copy content Toggle raw display