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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [145,2,Mod(12,145)] 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("145.12"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(145, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.15783082931\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 4 q^{3} - 22 q^{4} - 4 q^{7} + 10 q^{9} - 8 q^{10} - 8 q^{11} - 8 q^{12} - 14 q^{13} + 4 q^{14} - 10 q^{15} + 6 q^{16} - 20 q^{20} + 16 q^{21} + 8 q^{22} - 4 q^{23} + 10 q^{25} + 6 q^{26} - 4 q^{27}+ \cdots - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 − 2.73482i 1.63186 −5.47925 0.903087 − 2.04559i − 4.46285i 1.05887 + 1.05887i 9.51511i −0.337028 −5.59432 − 2.46978i
12.2 − 2.41122i −1.85973 −3.81399 −1.59267 + 1.56952i 4.48422i −0.291676 − 0.291676i 4.37394i 0.458586 3.78447 + 3.84028i
12.3 − 1.82099i 2.59340 −1.31599 −1.75450 + 1.38626i − 4.72256i −0.820621 − 0.820621i − 1.24557i 3.72575 2.52437 + 3.19492i
12.4 − 1.77873i −1.38965 −1.16389 2.14366 − 0.636182i 2.47181i −3.41296 − 3.41296i − 1.48721i −1.06888 −1.13160 − 3.81300i
12.5 − 1.36192i −0.228160 0.145179 1.87852 + 1.21291i 0.310736i 3.45046 + 3.45046i − 2.92156i −2.94794 1.65188 − 2.55840i
12.6 − 0.839004i 0.711801 1.29607 −1.24503 − 1.85739i − 0.597203i 1.13987 + 1.13987i − 2.76542i −2.49334 −1.55836 + 1.04459i
12.7 − 0.342532i −2.64611 1.88267 −1.82632 − 1.29018i 0.906377i −1.55474 − 1.55474i − 1.32994i 4.00189 −0.441927 + 0.625573i
12.8 0.222351i 1.02589 1.95056 0.621451 + 2.14798i 0.228107i −2.35964 − 2.35964i 0.878412i −1.94756 −0.477605 + 0.138180i
12.9 0.895351i −2.11245 1.19835 1.71183 − 1.43862i − 1.89138i 1.38465 + 1.38465i 2.86364i 1.46244 1.28807 + 1.53269i
12.10 1.26373i −0.913274 0.402981 −1.78577 + 1.34575i − 1.15413i 2.03055 + 2.03055i 3.03672i −2.16593 −1.70067 − 2.25673i
12.11 1.41066i 2.58872 0.0100302 −1.93694 − 1.11726i 3.65181i −0.510628 − 0.510628i 2.83547i 3.70148 1.57607 − 2.73237i
12.12 2.23019i 1.25170 −2.97373 2.19887 − 0.406147i 2.79153i 0.483409 + 0.483409i − 2.17160i −1.43324 0.905783 + 4.90390i
12.13 2.26693i −2.65401 −3.13899 0.683805 + 2.12895i − 6.01647i −2.59753 − 2.59753i − 2.58201i 4.04378 −4.82618 + 1.55014i
133.1 − 2.26693i −2.65401 −3.13899 0.683805 − 2.12895i 6.01647i −2.59753 + 2.59753i 2.58201i 4.04378 −4.82618 − 1.55014i
133.2 − 2.23019i 1.25170 −2.97373 2.19887 + 0.406147i − 2.79153i 0.483409 − 0.483409i 2.17160i −1.43324 0.905783 − 4.90390i
133.3 − 1.41066i 2.58872 0.0100302 −1.93694 + 1.11726i − 3.65181i −0.510628 + 0.510628i − 2.83547i 3.70148 1.57607 + 2.73237i
133.4 − 1.26373i −0.913274 0.402981 −1.78577 − 1.34575i 1.15413i 2.03055 − 2.03055i − 3.03672i −2.16593 −1.70067 + 2.25673i
133.5 − 0.895351i −2.11245 1.19835 1.71183 + 1.43862i 1.89138i 1.38465 − 1.38465i − 2.86364i 1.46244 1.28807 − 1.53269i
133.6 − 0.222351i 1.02589 1.95056 0.621451 − 2.14798i − 0.228107i −2.35964 + 2.35964i − 0.878412i −1.94756 −0.477605 − 0.138180i
133.7 0.342532i −2.64611 1.88267 −1.82632 + 1.29018i − 0.906377i −1.55474 + 1.55474i 1.32994i 4.00189 −0.441927 − 0.625573i
See all 26 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 12.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.2.e.a ✓ 26
5.b even 2 1 725.2.e.c 26
5.c odd 4 1 145.2.j.a yes 26
5.c odd 4 1 725.2.j.c 26
29.c odd 4 1 145.2.j.a yes 26
145.e even 4 1 inner 145.2.e.a ✓ 26
145.f odd 4 1 725.2.j.c 26
145.j even 4 1 725.2.e.c 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.e.a ✓ 26 1.a even 1 1 trivial
145.2.e.a ✓ 26 145.e even 4 1 inner
145.2.j.a yes 26 5.c odd 4 1
145.2.j.a yes 26 29.c odd 4 1
725.2.e.c 26 5.b even 2 1
725.2.e.c 26 145.j even 4 1
725.2.j.c 26 5.c odd 4 1
725.2.j.c 26 145.f odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(145, [\chi])\).