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Properties

Label 315.2.k.c
Level $315$
Weight $2$
Character orbit 315.k
Analytic conductor $2.515$
Analytic rank $0$
Dimension $36$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [315,2,Mod(16,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - q^{3} - 22 q^{4} + 36 q^{5} - 4 q^{6} - q^{7} + 3 q^{9} - 2 q^{11} + 5 q^{12} + 2 q^{13} - 6 q^{14} - q^{15} - 30 q^{16} - 5 q^{17} + 3 q^{18} - 2 q^{19} - 22 q^{20} - 11 q^{21} - 19 q^{22} + 6 q^{23}+ \cdots - 89 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.37850 − 2.38764i −1.12222 − 1.31932i −2.80055 + 4.85069i 1.00000 −1.60307 + 4.49816i 2.15266 + 1.53820i 9.92824 −0.481224 + 2.96115i −1.37850 − 2.38764i
16.2 −1.29282 − 2.23923i 1.72292 − 0.177629i −2.34278 + 4.05782i 1.00000 −2.62518 − 3.62838i −2.13970 − 1.55618i 6.94392 2.93690 − 0.612079i −1.29282 − 2.23923i
16.3 −1.04330 − 1.80705i 0.853418 + 1.50721i −1.17696 + 2.03855i 1.00000 1.83323 − 3.11464i 2.35024 − 1.21507i 0.738471 −1.54336 + 2.57256i −1.04330 − 1.80705i
16.4 −0.949927 − 1.64532i −1.72850 + 0.110854i −0.804724 + 1.39382i 1.00000 1.82434 + 2.73864i 0.880383 − 2.49498i −0.741992 2.97542 − 0.383223i −0.949927 − 1.64532i
16.5 −0.926279 − 1.60436i −0.957532 + 1.44331i −0.715985 + 1.24012i 1.00000 3.20253 + 0.199324i −2.45383 + 0.989310i −1.05231 −1.16627 − 2.76402i −0.926279 − 1.60436i
16.6 −0.750300 − 1.29956i −0.590164 − 1.62841i −0.125900 + 0.218065i 1.00000 −1.67341 + 1.98874i −2.32993 + 1.25356i −2.62335 −2.30341 + 1.92205i −0.750300 − 1.29956i
16.7 −0.415616 − 0.719867i 1.70907 + 0.281183i 0.654527 − 1.13367i 1.00000 −0.507903 − 1.34717i 0.762251 + 2.53357i −2.75059 2.84187 + 0.961126i −0.415616 − 0.719867i
16.8 −0.349525 − 0.605394i 1.42094 − 0.990413i 0.755665 − 1.30885i 1.00000 −1.09625 − 0.514057i −1.62523 − 2.08773i −2.45459 1.03816 − 2.81464i −0.349525 − 0.605394i
16.9 −0.0127903 − 0.0221535i −1.71905 − 0.211800i 0.999673 − 1.73148i 1.00000 0.0172951 + 0.0407920i 0.170066 + 2.64028i −0.102306 2.91028 + 0.728190i −0.0127903 − 0.0221535i
16.10 0.129832 + 0.224875i −0.458568 − 1.67024i 0.966288 − 1.67366i 1.00000 0.316059 − 0.319971i 2.52961 − 0.775284i 1.02114 −2.57943 + 1.53184i 0.129832 + 0.224875i
16.11 0.195497 + 0.338610i 0.811590 + 1.53014i 0.923562 − 1.59966i 1.00000 −0.359457 + 0.573950i 0.0590728 − 2.64509i 1.50420 −1.68264 + 2.48369i 0.195497 + 0.338610i
16.12 0.588830 + 1.01988i −0.114571 + 1.72826i 0.306558 − 0.530973i 1.00000 −1.83009 + 0.900802i −1.15510 + 2.38028i 3.07736 −2.97375 − 0.396015i 0.588830 + 1.01988i
16.13 0.712894 + 1.23477i 1.38205 − 1.04400i −0.0164351 + 0.0284664i 1.00000 2.27435 + 0.962251i −0.526228 + 2.59289i 2.80471 0.820129 − 2.88572i 0.712894 + 1.23477i
16.14 0.792206 + 1.37214i −1.71008 + 0.274985i −0.255182 + 0.441987i 1.00000 −1.73206 − 2.12863i 2.64012 − 0.172564i 2.36020 2.84877 − 0.940495i 0.792206 + 1.37214i
16.15 0.845194 + 1.46392i 0.216636 − 1.71845i −0.428706 + 0.742540i 1.00000 2.69877 − 1.13529i −2.15586 − 1.53371i 1.93142 −2.90614 − 0.744556i 0.845194 + 1.46392i
16.16 1.17969 + 2.04328i 1.66846 + 0.465033i −1.78334 + 3.08884i 1.00000 1.01807 + 3.95772i −0.0799028 − 2.64454i −3.69640 2.56749 + 1.55177i 1.17969 + 2.04328i
16.17 1.32610 + 2.29687i −0.540247 + 1.64564i −2.51707 + 4.35969i 1.00000 −4.49624 + 0.941405i 2.58884 + 0.545800i −8.04712 −2.41627 − 1.77810i 1.32610 + 2.29687i
16.18 1.34882 + 2.33623i −1.34415 − 1.09237i −2.63865 + 4.57028i 1.00000 0.739006 − 4.61365i −2.16746 + 1.51728i −8.84100 0.613465 + 2.93661i 1.34882 + 2.33623i
256.1 −1.37850 + 2.38764i −1.12222 + 1.31932i −2.80055 − 4.85069i 1.00000 −1.60307 − 4.49816i 2.15266 − 1.53820i 9.92824 −0.481224 − 2.96115i −1.37850 + 2.38764i
256.2 −1.29282 + 2.23923i 1.72292 + 0.177629i −2.34278 − 4.05782i 1.00000 −2.62518 + 3.62838i −2.13970 + 1.55618i 6.94392 2.93690 + 0.612079i −1.29282 + 2.23923i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.k.c ✓ 36
3.b odd 2 1 945.2.k.c 36
7.c even 3 1 315.2.l.c yes 36
9.c even 3 1 315.2.l.c yes 36
9.d odd 6 1 945.2.l.c 36
21.h odd 6 1 945.2.l.c 36
63.g even 3 1 inner 315.2.k.c ✓ 36
63.n odd 6 1 945.2.k.c 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.k.c ✓ 36 1.a even 1 1 trivial
315.2.k.c ✓ 36 63.g even 3 1 inner
315.2.l.c yes 36 7.c even 3 1
315.2.l.c yes 36 9.c even 3 1
945.2.k.c 36 3.b odd 2 1
945.2.k.c 36 63.n odd 6 1
945.2.l.c 36 9.d odd 6 1
945.2.l.c 36 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 29 T_{2}^{34} + 497 T_{2}^{32} - 2 T_{2}^{31} + 5658 T_{2}^{30} - 74 T_{2}^{29} + 47976 T_{2}^{28} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display