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Properties

Label 315.2.r.b
Level $315$
Weight $2$
Character orbit 315.r
Analytic conductor $2.515$
Analytic rank $0$
Dimension $84$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [315,2,Mod(184,315)] 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([2, 3, 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("315.184"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [84] 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: \(84\)
Relative dimension: \(42\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 88 q^{4} + 3 q^{5} + 6 q^{6} - 2 q^{9} + 6 q^{10} + 12 q^{11} + 8 q^{14} + 4 q^{15} + 72 q^{16} + 8 q^{19} - 10 q^{20} - 26 q^{21} + 18 q^{24} - 5 q^{25} - 40 q^{26} - 10 q^{29} + 5 q^{30} + 12 q^{31}+ \cdots - 102 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
184.1 − 2.71825i 1.11099 + 1.32879i −5.38890 −2.07547 + 0.832131i 3.61200 − 3.01996i −2.57523 + 0.606776i 9.21190i −0.531386 + 2.95256i 2.26194 + 5.64164i
184.2 − 2.61859i −0.455156 − 1.67118i −4.85701 −1.02882 − 1.98533i −4.37613 + 1.19187i 2.61471 + 0.404081i 7.48135i −2.58567 + 1.52129i −5.19877 + 2.69405i
184.3 − 2.61386i 1.67535 − 0.439552i −4.83226 2.17617 + 0.514103i −1.14893 − 4.37913i 1.37276 − 2.26175i 7.40313i 2.61359 − 1.47280i 1.34379 − 5.68819i
184.4 − 2.42945i −1.70733 − 0.291573i −3.90223 1.12953 + 1.92981i −0.708362 + 4.14788i 0.958372 + 2.46607i 4.62136i 2.82997 + 0.995624i 4.68837 − 2.74414i
184.5 − 2.37032i 0.187779 + 1.72184i −3.61843 2.03667 − 0.923024i 4.08132 − 0.445097i 0.987053 + 2.45474i 3.83621i −2.92948 + 0.646652i −2.18786 − 4.82757i
184.6 − 2.19537i −1.11942 + 1.32170i −2.81965 −0.0446370 + 2.23562i 2.90163 + 2.45754i 0.555178 − 2.58685i 1.79943i −0.493802 − 2.95908i 4.90802 + 0.0979947i
184.7 − 2.12646i −1.54581 − 0.781334i −2.52185 −2.23351 + 0.106887i −1.66148 + 3.28710i −2.47990 − 0.921996i 1.10970i 1.77903 + 2.41558i 0.227292 + 4.74948i
184.8 − 2.02504i −0.248494 − 1.71413i −2.10077 2.21568 − 0.301270i −3.47118 + 0.503210i −2.64524 + 0.0518041i 0.204070i −2.87650 + 0.851904i −0.610083 − 4.48683i
184.9 − 1.80957i 0.982481 − 1.42644i −1.27455 −1.92051 + 1.14527i −2.58125 − 1.77787i 0.174487 − 2.63999i − 1.31275i −1.06946 − 2.80290i 2.07246 + 3.47530i
184.10 − 1.80560i 1.73197 − 0.0166637i −1.26019 −1.93209 − 1.12562i −0.0300880 − 3.12725i 2.01307 + 1.71685i − 1.33580i 2.99944 − 0.0577222i −2.03243 + 3.48858i
184.11 − 1.64358i −1.45666 + 0.937087i −0.701356 −0.289890 − 2.21720i 1.54018 + 2.39414i −1.64618 + 2.07125i − 2.13443i 1.24373 − 2.73004i −3.64414 + 0.476457i
184.12 − 1.62167i 1.32290 + 1.11801i −0.629802 0.321745 − 2.21280i 1.81304 − 2.14529i −1.53778 − 2.15296i − 2.22200i 0.500103 + 2.95802i −3.58842 − 0.521762i
184.13 − 1.36154i 1.72744 − 0.126307i 0.146205 1.24734 + 1.85584i −0.171973 − 2.35198i −1.93746 + 1.80173i − 2.92215i 2.96809 − 0.436376i 2.52681 − 1.69830i
184.14 − 1.08755i −0.137009 + 1.72662i 0.817229 −0.940709 + 2.02856i 1.87779 + 0.149005i −1.34619 + 2.27767i − 3.06389i −2.96246 − 0.473127i 2.20617 + 1.02307i
184.15 − 1.07565i −0.564624 − 1.63744i 0.842975 0.440420 + 2.19227i −1.76131 + 0.607338i 2.63194 + 0.269937i − 3.05805i −2.36240 + 1.84907i 2.35811 − 0.473738i
184.16 − 0.872874i 0.554444 + 1.64091i 1.23809 2.16932 + 0.542260i 1.43231 − 0.483960i 2.41635 − 1.07761i − 2.82644i −2.38518 + 1.81959i 0.473324 − 1.89354i
184.17 − 0.724192i −1.70912 + 0.280903i 1.47555 −2.21636 + 0.296230i 0.203428 + 1.23773i 2.56533 + 0.647375i − 2.51696i 2.84219 − 0.960196i 0.214528 + 1.60507i
184.18 − 0.672284i −1.42402 + 0.985984i 1.54803 2.23061 + 0.156193i 0.662861 + 0.957346i −1.87376 − 1.86790i − 2.38529i 1.05567 − 2.80812i 0.105006 − 1.49960i
184.19 − 0.286902i −1.02114 − 1.39903i 1.91769 −0.815278 − 2.08214i −0.401384 + 0.292966i −0.0701001 − 2.64482i − 1.12399i −0.914560 + 2.85720i −0.597370 + 0.233905i
184.20 − 0.257756i 0.745098 − 1.56359i 1.93356 −1.95987 − 1.07653i −0.403026 − 0.192053i −2.46150 + 0.970066i − 1.01390i −1.88966 − 2.33006i −0.277481 + 0.505168i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 184.42
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
63.h even 3 1 inner
315.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.r.b ✓ 84
3.b odd 2 1 945.2.r.b 84
5.b even 2 1 inner 315.2.r.b ✓ 84
7.c even 3 1 315.2.bo.b yes 84
9.c even 3 1 315.2.bo.b yes 84
9.d odd 6 1 945.2.bo.b 84
15.d odd 2 1 945.2.r.b 84
21.h odd 6 1 945.2.bo.b 84
35.j even 6 1 315.2.bo.b yes 84
45.h odd 6 1 945.2.bo.b 84
45.j even 6 1 315.2.bo.b yes 84
63.h even 3 1 inner 315.2.r.b ✓ 84
63.j odd 6 1 945.2.r.b 84
105.o odd 6 1 945.2.bo.b 84
315.r even 6 1 inner 315.2.r.b ✓ 84
315.br odd 6 1 945.2.r.b 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.r.b ✓ 84 1.a even 1 1 trivial
315.2.r.b ✓ 84 5.b even 2 1 inner
315.2.r.b ✓ 84 63.h even 3 1 inner
315.2.r.b ✓ 84 315.r even 6 1 inner
315.2.bo.b yes 84 7.c even 3 1
315.2.bo.b yes 84 9.c even 3 1
315.2.bo.b yes 84 35.j even 6 1
315.2.bo.b yes 84 45.j even 6 1
945.2.r.b 84 3.b odd 2 1
945.2.r.b 84 15.d odd 2 1
945.2.r.b 84 63.j odd 6 1
945.2.r.b 84 315.br odd 6 1
945.2.bo.b 84 9.d odd 6 1
945.2.bo.b 84 21.h odd 6 1
945.2.bo.b 84 45.h odd 6 1
945.2.bo.b 84 105.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} + 64 T_{2}^{40} + 1889 T_{2}^{38} + 34121 T_{2}^{36} + 422073 T_{2}^{34} + 3791037 T_{2}^{32} + \cdots + 2401 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display