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Properties

Label 209.4.e.b
Level $209$
Weight $4$
Character orbit 209.e
Analytic conductor $12.331$
Analytic rank $0$
Dimension $50$
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] = [209,4,Mod(45,209)] 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("209.45"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(209, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 209.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [50,2] 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: \(12.3313991912\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(25\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 50 q + 2 q^{2} + 6 q^{3} - 106 q^{4} + 31 q^{6} - 20 q^{7} - 42 q^{8} - 225 q^{9} + 77 q^{10} + 550 q^{11} - 80 q^{12} + 153 q^{13} + 205 q^{14} + 34 q^{15} - 402 q^{16} - 28 q^{17} + 26 q^{18} + 31 q^{19}+ \cdots - 2475 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −2.78389 − 4.82185i 2.67338 + 4.63043i −11.5001 + 19.9188i −4.39198 − 7.60712i 14.8848 − 25.7812i −33.7385 83.5182 −0.793921 + 1.37511i −24.4536 + 42.3548i
45.2 −2.49189 − 4.31607i −0.507200 − 0.878497i −8.41900 + 14.5821i 0.607904 + 1.05292i −2.52777 + 4.37823i 8.85382 44.0466 12.9855 − 22.4915i 3.02965 − 5.24751i
45.3 −2.27681 − 3.94355i 2.60654 + 4.51466i −6.36775 + 11.0293i 10.5237 + 18.2276i 11.8692 − 20.5580i −12.7767 21.5637 −0.0880757 + 0.152552i 47.9210 − 83.0016i
45.4 −2.09201 − 3.62347i −4.38265 − 7.59097i −4.75301 + 8.23245i 7.96036 + 13.7877i −18.3371 + 31.7608i 22.9244 6.30120 −24.9153 + 43.1545i 33.3063 − 57.6882i
45.5 −2.08212 − 3.60634i −3.28157 − 5.68385i −4.67045 + 8.08946i 1.03869 + 1.79906i −13.6653 + 23.6689i −17.8317 5.58385 −8.03745 + 13.9213i 4.32535 − 7.49172i
45.6 −1.70872 − 2.95958i 4.96092 + 8.59257i −1.83942 + 3.18597i −10.5624 − 18.2946i 16.9536 − 29.3645i 1.84752 −14.7673 −35.7215 + 61.8715i −36.0963 + 62.5206i
45.7 −1.58120 − 2.73871i 2.44619 + 4.23692i −1.00036 + 1.73267i −0.0588590 − 0.101947i 7.73580 − 13.3988i 14.7745 −18.9721 1.53235 − 2.65410i −0.186135 + 0.322396i
45.8 −1.34252 − 2.32530i −3.52750 − 6.10981i 0.395307 − 0.684691i −7.33775 − 12.7094i −9.47144 + 16.4050i −10.9438 −23.6031 −11.3865 + 19.7220i −19.7021 + 34.1250i
45.9 −1.10126 − 1.90744i 0.756330 + 1.31000i 1.57445 − 2.72703i −2.49070 − 4.31402i 1.66583 − 2.88530i −13.1385 −24.5557 12.3559 − 21.4011i −5.48582 + 9.50172i
45.10 −0.653734 − 1.13230i −0.508121 − 0.880091i 3.14526 − 5.44775i 9.45676 + 16.3796i −0.664352 + 1.15069i −29.5904 −18.6844 12.9836 − 22.4883i 12.3644 − 21.4158i
45.11 −0.443182 − 0.767613i −1.50351 − 2.60416i 3.60718 − 6.24782i −6.86123 − 11.8840i −1.33266 + 2.30823i 25.1095 −13.4855 8.97890 − 15.5519i −6.08155 + 10.5336i
45.12 −0.335466 − 0.581045i 4.79153 + 8.29917i 3.77492 − 6.53836i 8.18474 + 14.1764i 3.21479 − 5.56818i 30.6908 −10.4329 −32.4175 + 56.1487i 5.49141 − 9.51140i
45.13 0.207943 + 0.360167i 1.00861 + 1.74696i 3.91352 − 6.77842i 4.98855 + 8.64043i −0.419464 + 0.726533i 2.13461 6.58223 11.4654 − 19.8587i −2.07467 + 3.59343i
45.14 0.352709 + 0.610910i −1.86221 − 3.22544i 3.75119 − 6.49726i 0.361236 + 0.625679i 1.31364 − 2.27528i 24.7755 10.9357 6.56435 − 11.3698i −0.254822 + 0.441365i
45.15 0.528281 + 0.915009i 3.22295 + 5.58232i 3.44184 − 5.96144i −7.94641 − 13.7636i −3.40525 + 5.89806i −15.8425 15.7255 −7.27485 + 12.6004i 8.39587 − 14.5421i
45.16 0.754673 + 1.30713i −2.95121 − 5.11165i 2.86094 − 4.95529i −5.76887 − 9.99198i 4.45440 − 7.71524i −21.3209 20.7111 −3.91928 + 6.78840i 8.70722 − 15.0814i
45.17 1.23191 + 2.13372i 3.60775 + 6.24881i 0.964818 − 1.67111i 5.25500 + 9.10192i −8.88882 + 15.3959i −18.0778 24.4647 −12.5318 + 21.7057i −12.9473 + 22.4254i
45.18 1.47173 + 2.54911i −4.72271 − 8.17997i −0.331960 + 0.574972i 4.68001 + 8.10601i 13.9011 − 24.0774i 8.55796 21.5934 −31.1079 + 53.8805i −13.7754 + 23.8597i
45.19 1.58002 + 2.73668i 2.10215 + 3.64103i −0.992938 + 1.71982i −4.39103 − 7.60549i −6.64289 + 11.5058i 27.5987 19.0049 4.66192 − 8.07469i 13.8758 − 24.0337i
45.20 1.69842 + 2.94174i −1.24857 − 2.16258i −1.76923 + 3.06440i −0.245162 − 0.424633i 4.24117 − 7.34592i −16.8767 15.1551 10.3822 − 17.9824i 0.832774 − 1.44241i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.25
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.4.e.b ✓ 50
19.c even 3 1 inner 209.4.e.b ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.4.e.b ✓ 50 1.a even 1 1 trivial
209.4.e.b ✓ 50 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{50} - 2 T_{2}^{49} + 155 T_{2}^{48} - 272 T_{2}^{47} + 13477 T_{2}^{46} - 21509 T_{2}^{45} + \cdots + 10\!\cdots\!04 \) acting on \(S_{4}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display