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Properties

Label 363.2.i.a
Level $363$
Weight $2$
Character orbit 363.i
Analytic conductor $2.899$
Analytic rank $0$
Dimension $110$
Inner twists $2$

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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] = [363,2,Mod(34,363)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 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("363.34"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [110,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(110\)
Relative dimension: \(11\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 110 q - 3 q^{2} + 110 q^{3} - 17 q^{4} - 6 q^{5} - 3 q^{6} - 8 q^{7} - 15 q^{8} + 110 q^{9} - 7 q^{10} - 17 q^{12} + 8 q^{13} + 4 q^{14} - 6 q^{15} - 37 q^{16} - 18 q^{17} - 3 q^{18} - 20 q^{19} - 32 q^{20}+ \cdots - 83 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −0.376853 − 2.62107i 1.00000 −4.80902 + 1.41206i −1.49341 + 3.27011i −0.376853 − 2.62107i 1.28940 + 0.828649i 3.31334 + 7.25519i 1.00000 9.13399 + 2.68198i
34.2 −0.330553 − 2.29905i 1.00000 −3.25737 + 0.956450i 1.06578 − 2.33374i −0.330553 − 2.29905i 3.57344 + 2.29651i 1.34590 + 2.94710i 1.00000 −5.71768 − 1.67886i
34.3 −0.281562 − 1.95831i 1.00000 −1.83670 + 0.539304i −0.612450 + 1.34108i −0.281562 − 1.95831i −3.83832 − 2.46674i −0.0704841 − 0.154339i 1.00000 2.79869 + 0.821768i
34.4 −0.215462 − 1.49857i 1.00000 −0.280297 + 0.0823026i 1.26386 − 2.76746i −0.215462 − 1.49857i −0.870717 − 0.559576i −1.07413 − 2.35202i 1.00000 −4.41954 − 1.29769i
34.5 −0.124489 − 0.865842i 1.00000 1.18480 − 0.347889i −1.60595 + 3.51653i −0.124489 − 0.865842i 1.93251 + 1.24195i −1.17548 − 2.57394i 1.00000 3.24469 + 0.952726i
34.6 −0.0568188 − 0.395183i 1.00000 1.76604 − 0.518557i −0.433191 + 0.948555i −0.0568188 − 0.395183i −0.717772 − 0.461284i −0.636976 − 1.39478i 1.00000 0.399466 + 0.117294i
34.7 0.0394089 + 0.274095i 1.00000 1.84541 − 0.541862i 0.254412 − 0.557084i 0.0394089 + 0.274095i 3.50795 + 2.25442i 0.451315 + 0.988243i 1.00000 0.162720 + 0.0477789i
34.8 0.0866573 + 0.602715i 1.00000 1.56323 − 0.459006i 1.19980 − 2.62720i 0.0866573 + 0.602715i −2.25620 − 1.44997i 0.918018 + 2.01018i 1.00000 1.68742 + 0.495471i
34.9 0.246669 + 1.71562i 1.00000 −0.963518 + 0.282914i 0.586690 − 1.28467i 0.246669 + 1.71562i 0.292439 + 0.187939i 0.717003 + 1.57002i 1.00000 2.34873 + 0.689648i
34.10 0.249246 + 1.73354i 1.00000 −1.02406 + 0.300690i −1.04073 + 2.27889i 0.249246 + 1.73354i −1.67784 − 1.07828i 0.678588 + 1.48590i 1.00000 −4.20994 − 1.23615i
34.11 0.381997 + 2.65685i 1.00000 −4.99393 + 1.46635i −1.32324 + 2.89750i 0.381997 + 2.65685i 3.57319 + 2.29635i −3.57344 − 7.82475i 1.00000 −8.20368 − 2.40882i
67.1 −2.62777 + 0.771584i 1.00000 4.62734 − 2.97381i −1.65159 − 1.90604i −2.62777 + 0.771584i 0.238411 + 0.522048i −6.27811 + 7.24533i 1.00000 5.81068 + 3.73430i
67.2 −1.91396 + 0.561990i 1.00000 1.66491 − 1.06997i −0.318763 − 0.367872i −1.91396 + 0.561990i −0.203664 − 0.445963i 0.0273229 − 0.0315323i 1.00000 0.816841 + 0.524952i
67.3 −1.66022 + 0.487484i 1.00000 0.836175 − 0.537377i 0.840806 + 0.970341i −1.66022 + 0.487484i 1.37086 + 3.00177i 1.13995 − 1.31557i 1.00000 −1.86895 − 1.20110i
67.4 −0.703082 + 0.206444i 1.00000 −1.23080 + 0.790988i −2.65913 − 3.06880i −0.703082 + 0.206444i 1.84052 + 4.03017i 1.66178 − 1.91779i 1.00000 2.50312 + 1.60866i
67.5 −0.488277 + 0.143371i 1.00000 −1.46465 + 0.941272i 2.84817 + 3.28696i −0.488277 + 0.143371i −0.361141 − 0.790788i 1.24671 − 1.43878i 1.00000 −1.86195 − 1.19660i
67.6 −0.462869 + 0.135910i 1.00000 −1.48673 + 0.955464i −0.614517 − 0.709190i −0.462869 + 0.135910i −1.47089 − 3.22079i 1.19013 − 1.37348i 1.00000 0.380827 + 0.244743i
67.7 −0.188159 + 0.0552484i 1.00000 −1.65016 + 1.06049i −0.807975 − 0.932452i −0.188159 + 0.0552484i −1.17818 − 2.57985i 0.508740 − 0.587118i 1.00000 0.203544 + 0.130810i
67.8 0.747690 − 0.219542i 1.00000 −1.17166 + 0.752983i 1.02919 + 1.18775i 0.747690 − 0.219542i 0.462266 + 1.01222i −1.73134 + 1.99807i 1.00000 1.03028 + 0.662118i
67.9 1.24406 − 0.365288i 1.00000 −0.268265 + 0.172403i 0.476429 + 0.549828i 1.24406 − 0.365288i 1.81181 + 3.96731i −1.96892 + 2.27225i 1.00000 0.793551 + 0.509984i
See next 80 embeddings (of 110 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.i.a ✓ 110
121.e even 11 1 inner 363.2.i.a ✓ 110
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.i.a ✓ 110 1.a even 1 1 trivial
363.2.i.a ✓ 110 121.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{110} + 3 T_{2}^{109} + 24 T_{2}^{108} + 72 T_{2}^{107} + 365 T_{2}^{106} + 1029 T_{2}^{105} + \cdots + 529 \) acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display