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Properties

Label 196.3.o.a
Level $196$
Weight $3$
Character orbit 196.o
Analytic conductor $5.341$
Analytic rank $0$
Dimension $648$
Inner twists $4$

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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] = [196,3,Mod(11,196)] 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("196.11"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(196, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([21, 40])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 196.o (of order \(42\), degree \(12\), 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: \(5.34061318146\)
Analytic rank: \(0\)
Dimension: \(648\)
Relative dimension: \(54\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 648 q - 13 q^{2} - 13 q^{4} - 26 q^{5} - 2 q^{6} - 16 q^{8} - 176 q^{9} - 12 q^{10} + 45 q^{12} - 4 q^{13} + 89 q^{14} - 25 q^{16} - 26 q^{17} - 48 q^{18} - 166 q^{20} + 50 q^{21} + 10 q^{22} - 96 q^{24}+ \cdots + 771 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.99834 + 0.0813931i −0.890429 − 5.90761i 3.98675 − 0.325303i −0.693008 + 1.76576i 2.26022 + 11.7330i −1.01134 + 6.92656i −7.94042 + 0.974561i −25.5068 + 7.86781i 1.24115 − 3.58499i
11.2 −1.99274 − 0.170222i −0.201068 − 1.33400i 3.94205 + 0.678419i 3.01252 − 7.67577i 0.173600 + 2.69254i −6.53373 − 2.51206i −7.74001 − 2.02294i 6.86104 − 2.11635i −7.30976 + 14.7830i
11.3 −1.98074 + 0.276918i 0.395705 + 2.62533i 3.84663 − 1.09700i 0.136996 − 0.349061i −1.51079 − 5.09051i 5.15618 − 4.73433i −7.31539 + 3.23807i 1.86438 − 0.575086i −0.174692 + 0.729334i
11.4 −1.96966 + 0.347058i 0.640356 + 4.24848i 3.75910 − 1.36717i −2.20043 + 5.60659i −2.73575 − 8.14581i −5.66464 − 4.11241i −6.92965 + 3.99749i −9.03938 + 2.78828i 2.38827 − 11.8067i
11.5 −1.92990 − 0.524869i −0.378415 − 2.51062i 3.44902 + 2.02589i −1.86555 + 4.75335i −0.587445 + 5.04386i 4.39216 − 5.45059i −5.59294 − 5.72005i 2.44014 − 0.752684i 6.09521 − 8.19431i
11.6 −1.92562 − 0.540347i 0.290399 + 1.92667i 3.41605 + 2.08101i 0.0455030 − 0.115940i 0.481873 − 3.86696i 1.95514 + 6.72141i −5.45356 − 5.85309i 4.97241 − 1.53379i −0.150269 + 0.198669i
11.7 −1.78844 + 0.895263i −0.378329 − 2.51005i 2.39701 − 3.20224i 1.68959 − 4.30500i 2.92377 + 4.15036i 6.58919 + 2.36275i −1.42005 + 7.87296i 2.44295 − 0.753551i 0.832384 + 9.21184i
11.8 −1.76409 + 0.942337i −0.177932 − 1.18050i 2.22400 − 3.32473i −0.392454 + 0.999956i 1.42632 + 1.91484i −5.97561 + 3.64583i −0.790316 + 7.96087i 7.23823 − 2.23270i −0.249973 − 2.13383i
11.9 −1.76080 + 0.948456i 0.805611 + 5.34488i 2.20086 − 3.34009i 2.78087 − 7.08553i −6.48790 − 8.64719i 0.694514 + 6.96546i −0.707355 + 7.96867i −19.3185 + 5.95898i 1.82376 + 15.1138i
11.10 −1.73035 − 1.00295i −0.252232 − 1.67345i 1.98819 + 3.47089i −2.78151 + 7.08717i −1.24194 + 3.14862i −6.65862 + 2.15935i 0.0408774 − 7.99990i 5.86334 − 1.80860i 11.9210 − 9.47354i
11.11 −1.61823 + 1.17531i −0.467323 − 3.10048i 1.23732 − 3.80382i −2.64152 + 6.73049i 4.40025 + 4.46804i −1.27884 − 6.88219i 2.46839 + 7.60967i −0.794458 + 0.245058i −3.63580 − 13.9961i
11.12 −1.59593 − 1.20540i 0.684930 + 4.54422i 1.09400 + 3.84749i 2.99145 − 7.62209i 4.38452 − 8.07788i 3.24915 − 6.20025i 2.89184 − 7.45904i −11.5806 + 3.57215i −13.9619 + 8.55843i
11.13 −1.57366 − 1.23434i −0.519055 − 3.44371i 0.952796 + 3.88487i 1.36400 − 3.47542i −3.43390 + 6.05991i 6.93624 − 0.942679i 3.29588 − 7.28952i −2.98954 + 0.922152i −6.43634 + 3.78548i
11.14 −1.36423 − 1.46249i 0.747940 + 4.96226i −0.277728 + 3.99035i −3.24635 + 8.27157i 6.23686 − 7.86353i 6.73049 + 1.92368i 6.21471 − 5.03759i −15.4644 + 4.77014i 16.5258 − 6.53662i
11.15 −1.35234 − 1.47349i 0.262954 + 1.74459i −0.342373 + 3.98532i −0.459540 + 1.17089i 2.21504 − 2.74673i −4.77835 − 5.11541i 6.33535 − 4.88501i 5.62572 − 1.73530i 2.34675 − 0.906306i
11.16 −1.29295 + 1.52587i 0.467323 + 3.10048i −0.656566 − 3.94575i −2.64152 + 6.73049i −5.33516 − 3.29569i 1.27884 + 6.88219i 6.86961 + 4.09981i −0.794458 + 0.245058i −6.85451 − 12.7328i
11.17 −1.10062 − 1.66992i −0.743880 − 4.93532i −1.57726 + 3.67590i 0.861874 − 2.19602i −7.42286 + 6.67414i −6.10896 − 3.41770i 7.87443 − 1.41188i −15.2039 + 4.68977i −4.61577 + 0.977727i
11.18 −1.07739 + 1.68500i −0.805611 − 5.34488i −1.67846 − 3.63081i 2.78087 − 7.08553i 9.87409 + 4.40106i −0.694514 − 6.96546i 7.92628 + 1.08358i −19.3185 + 5.95898i 8.94306 + 12.3196i
11.19 −1.07153 + 1.68873i 0.177932 + 1.18050i −1.70364 − 3.61906i −0.392454 + 0.999956i −2.18421 − 0.964466i 5.97561 − 3.64583i 7.93713 + 1.00096i 7.23823 − 2.23270i −1.26813 − 1.73423i
11.20 −1.02641 + 1.71653i 0.378329 + 2.51005i −1.89297 − 3.52373i 1.68959 − 4.30500i −4.69690 − 1.92692i −6.58919 − 2.36275i 7.99156 + 0.367447i 2.44295 − 0.753551i 5.65546 + 7.31892i
See next 80 embeddings (of 648 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.54
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
49.g even 21 1 inner
196.o odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.3.o.a ✓ 648
4.b odd 2 1 inner 196.3.o.a ✓ 648
49.g even 21 1 inner 196.3.o.a ✓ 648
196.o odd 42 1 inner 196.3.o.a ✓ 648
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.3.o.a ✓ 648 1.a even 1 1 trivial
196.3.o.a ✓ 648 4.b odd 2 1 inner
196.3.o.a ✓ 648 49.g even 21 1 inner
196.3.o.a ✓ 648 196.o odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(196, [\chi])\).