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Properties

Label 975.2.bn.d
Level $975$
Weight $2$
Character orbit 975.bn
Analytic conductor $7.785$
Analytic rank $0$
Dimension $96$
Inner twists $8$

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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] = [975,2,Mod(218,975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 10])) 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("975.218"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bn (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0,2,0,0,-12,12,0,0,0,0,12,24,0,0,16,0,0,0,0,0,-20,0,0,0,0, 32,36,0,0,0,0,-30,0,0,-4,84,0,0,0,0,48,-8,0,0,0,0,28,0,0,-16,-28,0,0,0, 0,0,-84,0,0,-32,0,-90,0,0,0,-36,0,0,0,0,-90,0,0,0,72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(76)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 2 q^{3} - 12 q^{6} + 12 q^{7} + 12 q^{12} + 24 q^{13} + 16 q^{16} - 20 q^{22} + 32 q^{27} + 36 q^{28} - 30 q^{33} - 4 q^{36} + 84 q^{37} + 48 q^{42} - 8 q^{43} + 28 q^{48} - 16 q^{51} - 28 q^{52}+ \cdots + 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
218.1 −2.61871 − 0.701681i 1.68401 − 0.405108i 4.63323 + 2.67499i 0 −4.69419 − 0.120777i −1.53607 + 0.411589i −6.42202 − 6.42202i 2.67177 − 1.36441i 0
218.2 −2.27913 − 0.610690i 0.0442801 + 1.73148i 3.08942 + 1.78368i 0 0.956480 − 3.97331i 3.71897 − 0.996495i −2.61503 − 2.61503i −2.99608 + 0.153341i 0
218.3 −2.12039 − 0.568157i −1.72421 + 0.164580i 2.44121 + 1.40943i 0 3.74952 + 0.630650i 3.37093 − 0.903238i −1.27107 − 1.27107i 2.94583 − 0.567543i 0
218.4 −2.09662 − 0.561788i −0.298187 − 1.70619i 2.34816 + 1.35571i 0 −0.333332 + 3.74475i 0.403690 − 0.108168i −1.09192 − 1.09192i −2.82217 + 1.01753i 0
218.5 −1.85180 − 0.496187i 1.59450 + 0.676428i 1.45090 + 0.837676i 0 −2.61706 − 2.04378i −1.97020 + 0.527913i 0.440097 + 0.440097i 2.08489 + 2.15713i 0
218.6 −1.65038 − 0.442218i 0.744530 − 1.56387i 0.796141 + 0.459652i 0 −1.92033 + 2.25172i −0.109428 + 0.0293212i 1.30565 + 1.30565i −1.89135 − 2.32869i 0
218.7 −1.42342 − 0.381405i −1.13110 + 1.31172i 0.148609 + 0.0857994i 0 2.11032 − 1.43573i −3.97287 + 1.06453i 1.90523 + 1.90523i −0.441245 − 2.96737i 0
218.8 −1.05989 − 0.283997i −1.49614 − 0.872682i −0.689337 − 0.397989i 0 1.33790 + 1.34985i 0.160495 − 0.0430045i 2.16938 + 2.16938i 1.47685 + 2.61130i 0
218.9 −1.01047 − 0.270756i 0.981641 + 1.42702i −0.784302 − 0.452817i 0 −0.605549 − 1.70775i −0.396013 + 0.106111i 2.14935 + 2.14935i −1.07276 + 2.80164i 0
218.10 −0.615188 − 0.164839i −0.956716 + 1.44385i −1.38077 − 0.797186i 0 0.826563 − 0.730534i 2.17570 − 0.582977i 1.61872 + 1.61872i −1.16939 − 2.76270i 0
218.11 −0.254998 − 0.0683265i −1.56121 − 0.750075i −1.67170 − 0.965154i 0 0.346856 + 0.297940i −3.34011 + 0.894980i 0.733676 + 0.733676i 1.87477 + 2.34206i 0
218.12 −0.107570 − 0.0288234i 1.48257 − 0.895534i −1.72131 − 0.993799i 0 −0.185293 + 0.0536002i 3.86093 − 1.03453i 0.314011 + 0.314011i 1.39604 − 2.65539i 0
218.13 0.107570 + 0.0288234i −0.836177 − 1.51684i −1.72131 − 0.993799i 0 −0.0462274 − 0.187269i 3.86093 − 1.03453i −0.314011 − 0.314011i −1.60161 + 2.53670i 0
218.14 0.254998 + 0.0683265i 1.72709 + 0.131022i −1.67170 − 0.965154i 0 0.431452 + 0.151416i −3.34011 + 0.894980i −0.733676 − 0.733676i 2.96567 + 0.452574i 0
218.15 0.615188 + 0.164839i 0.106617 + 1.72877i −1.38077 − 0.797186i 0 −0.219379 + 1.08109i 2.17570 − 0.582977i −1.61872 − 1.61872i −2.97727 + 0.368632i 0
218.16 1.01047 + 0.270756i −1.56363 + 0.745014i −0.784302 − 0.452817i 0 −1.78173 + 0.329454i −0.396013 + 0.106111i −2.14935 − 2.14935i 1.88991 − 2.32986i 0
218.17 1.05989 + 0.283997i 1.73203 − 0.00769600i −0.689337 − 0.397989i 0 1.83795 + 0.483735i 0.160495 − 0.0430045i −2.16938 − 2.16938i 2.99988 − 0.0266595i 0
218.18 1.42342 + 0.381405i 0.323695 + 1.70154i 0.148609 + 0.0857994i 0 −0.188219 + 2.54546i −3.97287 + 1.06453i −1.90523 − 1.90523i −2.79044 + 1.10156i 0
218.19 1.65038 + 0.442218i 0.137150 − 1.72661i 0.796141 + 0.459652i 0 0.989888 − 2.78891i −0.109428 + 0.0293212i −1.30565 − 1.30565i −2.96238 − 0.473611i 0
218.20 1.85180 + 0.496187i −1.71910 − 0.211449i 1.45090 + 0.837676i 0 −3.07850 − 1.24455i −1.97020 + 0.527913i −0.440097 − 0.440097i 2.91058 + 0.727001i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 218.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
13.e even 6 1 inner
15.e even 4 1 inner
39.h odd 6 1 inner
65.r odd 12 1 inner
195.bf even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bn.d 96
3.b odd 2 1 inner 975.2.bn.d 96
5.b even 2 1 195.2.bf.a ✓ 96
5.c odd 4 1 195.2.bf.a ✓ 96
5.c odd 4 1 inner 975.2.bn.d 96
13.e even 6 1 inner 975.2.bn.d 96
15.d odd 2 1 195.2.bf.a ✓ 96
15.e even 4 1 195.2.bf.a ✓ 96
15.e even 4 1 inner 975.2.bn.d 96
39.h odd 6 1 inner 975.2.bn.d 96
65.l even 6 1 195.2.bf.a ✓ 96
65.r odd 12 1 195.2.bf.a ✓ 96
65.r odd 12 1 inner 975.2.bn.d 96
195.y odd 6 1 195.2.bf.a ✓ 96
195.bf even 12 1 195.2.bf.a ✓ 96
195.bf even 12 1 inner 975.2.bn.d 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bf.a ✓ 96 5.b even 2 1
195.2.bf.a ✓ 96 5.c odd 4 1
195.2.bf.a ✓ 96 15.d odd 2 1
195.2.bf.a ✓ 96 15.e even 4 1
195.2.bf.a ✓ 96 65.l even 6 1
195.2.bf.a ✓ 96 65.r odd 12 1
195.2.bf.a ✓ 96 195.y odd 6 1
195.2.bf.a ✓ 96 195.bf even 12 1
975.2.bn.d 96 1.a even 1 1 trivial
975.2.bn.d 96 3.b odd 2 1 inner
975.2.bn.d 96 5.c odd 4 1 inner
975.2.bn.d 96 13.e even 6 1 inner
975.2.bn.d 96 15.e even 4 1 inner
975.2.bn.d 96 39.h odd 6 1 inner
975.2.bn.d 96 65.r odd 12 1 inner
975.2.bn.d 96 195.bf even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{96} - 160 T_{2}^{92} + 15396 T_{2}^{88} - 955300 T_{2}^{84} + 43571992 T_{2}^{80} - 1488515276 T_{2}^{76} + \cdots + 10000 \) Copy content Toggle raw display
\( T_{7}^{48} - 6 T_{7}^{47} + 18 T_{7}^{46} - 36 T_{7}^{45} - 497 T_{7}^{44} + 2886 T_{7}^{43} + \cdots + 104976 \) Copy content Toggle raw display