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Properties

Label 961.2.d.s
Level $961$
Weight $2$
Character orbit 961.d
Analytic conductor $7.674$
Analytic rank $0$
Dimension $64$
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] = [961,2,Mod(374,961)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) 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("961.374"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.d (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64,8,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.67362363425\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 8 q^{2} - 8 q^{4} - 64 q^{5} + 16 q^{7} - 8 q^{10} + 8 q^{14} + 8 q^{16} + 24 q^{18} + 32 q^{19} + 24 q^{20} + 8 q^{28} - 32 q^{32} + 32 q^{33} + 16 q^{35} - 160 q^{36} + 24 q^{38} + 32 q^{39} + 32 q^{41}+ \cdots + 64 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
374.1 −1.75058 − 1.27187i −0.0997723 + 0.0724888i 0.828835 + 2.55089i −2.65612 0.266855 −0.943380 − 2.90342i 0.456138 − 1.40385i −0.922351 + 2.83870i 4.64975 + 3.37824i
374.2 −1.75058 − 1.27187i 0.0997723 − 0.0724888i 0.828835 + 2.55089i −2.65612 −0.266855 −0.943380 − 2.90342i 0.456138 − 1.40385i −0.922351 + 2.83870i 4.64975 + 3.37824i
374.3 −0.984188 − 0.715055i −0.405559 + 0.294656i −0.160710 − 0.494616i 1.24784 0.609842 −0.605971 − 1.86499i −0.947361 + 2.91568i −0.849395 + 2.61417i −1.22811 − 0.892276i
374.4 −0.984188 − 0.715055i 0.405559 − 0.294656i −0.160710 − 0.494616i 1.24784 −0.609842 −0.605971 − 1.86499i −0.947361 + 2.91568i −0.849395 + 2.61417i −1.22811 − 0.892276i
374.5 −0.100459 − 0.0729880i −1.19674 + 0.869483i −0.613269 − 1.88745i −1.22944 0.183686 0.824157 + 2.53649i −0.152897 + 0.470567i −0.250863 + 0.772076i 0.123509 + 0.0897347i
374.6 −0.100459 − 0.0729880i 1.19674 − 0.869483i −0.613269 − 1.88745i −1.22944 −0.183686 0.824157 + 2.53649i −0.152897 + 0.470567i −0.250863 + 0.772076i 0.123509 + 0.0897347i
374.7 0.273247 + 0.198525i −2.39630 + 1.74101i −0.582783 − 1.79362i −1.25850 −1.00042 −0.755059 − 2.32383i 0.405578 − 1.24824i 1.78407 − 5.49079i −0.343882 − 0.249845i
374.8 0.273247 + 0.198525i 2.39630 − 1.74101i −0.582783 − 1.79362i −1.25850 1.00042 −0.755059 − 2.32383i 0.405578 − 1.24824i 1.78407 − 5.49079i −0.343882 − 0.249845i
374.9 1.06472 + 0.773568i −0.998704 + 0.725601i −0.0828021 − 0.254839i 0.00727825 −1.62465 −0.245643 − 0.756011i 0.922351 − 2.83870i −0.456138 + 1.40385i 0.00774933 + 0.00563022i
374.10 1.06472 + 0.773568i 0.998704 − 0.725601i −0.0828021 − 0.254839i 0.00727825 1.62465 −0.245643 − 0.756011i 0.922351 − 2.83870i −0.456138 + 1.40385i 0.00774933 + 0.00563022i
374.11 1.17401 + 0.852969i −1.99250 + 1.44764i 0.0327114 + 0.100675i −3.68139 −3.57401 −0.248574 − 0.765033i 0.849395 − 2.61417i 0.947361 − 2.91568i −4.32199 − 3.14011i
374.12 1.17401 + 0.852969i 1.99250 − 1.44764i 0.0327114 + 0.100675i −3.68139 3.57401 −0.248574 − 0.765033i 0.849395 − 2.61417i 0.947361 − 2.91568i −4.32199 − 3.14011i
374.13 1.52867 + 1.11064i −1.51240 + 1.09883i 0.485270 + 1.49351i 2.49142 −3.53237 −1.20568 − 3.71070i 0.250863 − 0.772076i 0.152897 − 0.470567i 3.80856 + 2.76708i
374.14 1.52867 + 1.11064i 1.51240 − 1.09883i 0.485270 + 1.49351i 2.49142 3.53237 −1.20568 − 3.71070i 0.250863 − 0.772076i 0.152897 − 0.470567i 3.80856 + 2.76708i
374.15 2.03064 + 1.47535i −1.05095 + 0.763561i 1.32882 + 4.08967i −2.92108 −3.26062 0.708014 + 2.17904i −1.78407 + 5.49079i −0.405578 + 1.24824i −5.93165 − 4.30960i
374.16 2.03064 + 1.47535i 1.05095 − 0.763561i 1.32882 + 4.08967i −2.92108 3.26062 0.708014 + 2.17904i −1.78407 + 5.49079i −0.405578 + 1.24824i −5.93165 − 4.30960i
388.1 −1.75058 + 1.27187i −0.0997723 − 0.0724888i 0.828835 − 2.55089i −2.65612 0.266855 −0.943380 + 2.90342i 0.456138 + 1.40385i −0.922351 − 2.83870i 4.64975 − 3.37824i
388.2 −1.75058 + 1.27187i 0.0997723 + 0.0724888i 0.828835 − 2.55089i −2.65612 −0.266855 −0.943380 + 2.90342i 0.456138 + 1.40385i −0.922351 − 2.83870i 4.64975 − 3.37824i
388.3 −0.984188 + 0.715055i −0.405559 − 0.294656i −0.160710 + 0.494616i 1.24784 0.609842 −0.605971 + 1.86499i −0.947361 − 2.91568i −0.849395 − 2.61417i −1.22811 + 0.892276i
388.4 −0.984188 + 0.715055i 0.405559 + 0.294656i −0.160710 + 0.494616i 1.24784 −0.609842 −0.605971 + 1.86499i −0.947361 − 2.91568i −0.849395 − 2.61417i −1.22811 + 0.892276i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 374.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner
31.d even 5 3 inner
31.f odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.d.s 64
31.b odd 2 1 inner 961.2.d.s 64
31.c even 3 2 961.2.g.w 128
31.d even 5 1 961.2.a.l ✓ 16
31.d even 5 3 inner 961.2.d.s 64
31.e odd 6 2 961.2.g.w 128
31.f odd 10 1 961.2.a.l ✓ 16
31.f odd 10 3 inner 961.2.d.s 64
31.g even 15 2 961.2.c.l 32
31.g even 15 6 961.2.g.w 128
31.h odd 30 2 961.2.c.l 32
31.h odd 30 6 961.2.g.w 128
93.k even 10 1 8649.2.a.bs 16
93.l odd 10 1 8649.2.a.bs 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
961.2.a.l ✓ 16 31.d even 5 1
961.2.a.l ✓ 16 31.f odd 10 1
961.2.c.l 32 31.g even 15 2
961.2.c.l 32 31.h odd 30 2
961.2.d.s 64 1.a even 1 1 trivial
961.2.d.s 64 31.b odd 2 1 inner
961.2.d.s 64 31.d even 5 3 inner
961.2.d.s 64 31.f odd 10 3 inner
961.2.g.w 128 31.c even 3 2
961.2.g.w 128 31.e odd 6 2
961.2.g.w 128 31.g even 15 6
961.2.g.w 128 31.h odd 30 6
8649.2.a.bs 16 93.k even 10 1
8649.2.a.bs 16 93.l odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(961, [\chi])\):

\( T_{2}^{32} - 4 T_{2}^{31} + 18 T_{2}^{30} - 56 T_{2}^{29} + 183 T_{2}^{28} - 428 T_{2}^{27} + 1142 T_{2}^{26} + \cdots + 1 \) Copy content Toggle raw display
\( T_{3}^{64} + 24 T_{3}^{62} + 356 T_{3}^{60} + 4256 T_{3}^{58} + 45266 T_{3}^{56} + 388592 T_{3}^{54} + \cdots + 256 \) Copy content Toggle raw display