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Properties

Label 352.3.bd.a
Level $352$
Weight $3$
Character orbit 352.bd
Analytic conductor $9.591$
Analytic rank $0$
Dimension $1504$
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] = [352,3,Mod(3,352)] 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("352.3"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(352, base_ring=CyclotomicField(40)) chi = DirichletCharacter(H, H._module([20, 15, 32])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 352.bd (of order \(40\), degree \(16\), 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: \(9.59130530548\)
Analytic rank: \(0\)
Dimension: \(1504\)
Relative dimension: \(94\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1504 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 12 q^{8} - 12 q^{9} - 32 q^{10} - 16 q^{11} + 64 q^{12} - 12 q^{13} + 4 q^{14} - 24 q^{15} - 52 q^{16} + 168 q^{18} - 12 q^{19}+ \cdots - 236 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.00000 + 0.00170451i −2.16824 − 0.170644i 3.99999 − 0.00681805i 0.825892 − 3.44009i 4.33676 + 0.337592i 0.628023 − 0.319994i −7.99997 + 0.0204541i −4.21707 − 0.667918i −1.64592 + 6.88158i
3.2 −1.99884 + 0.0682260i −4.03092 − 0.317241i 3.99069 − 0.272745i 0.846392 − 3.52548i 8.07880 + 0.359098i 2.60804 − 1.32886i −7.95813 + 0.817442i 7.25850 + 1.14963i −1.45127 + 7.10459i
3.3 −1.99825 − 0.0836136i 4.04375 + 0.318250i 3.98602 + 0.334162i −0.892478 + 3.71744i −8.05383 − 0.974057i −1.84319 + 0.939151i −7.93712 − 1.00103i 7.36147 + 1.16594i 2.09422 − 7.35375i
3.4 −1.99672 + 0.114517i 2.10420 + 0.165605i 3.97377 − 0.457317i −0.241222 + 1.00476i −4.22047 − 0.0896982i −6.26879 + 3.19411i −7.88213 + 1.36820i −4.48894 − 0.710979i 0.366590 − 2.03385i
3.5 −1.97607 − 0.308457i 5.88724 + 0.463336i 3.80971 + 1.21907i 1.44244 − 6.00821i −11.4907 − 2.73155i −1.48409 + 0.756180i −7.15222 − 3.58410i 25.5557 + 4.04762i −4.70365 + 11.4277i
3.6 −1.96393 + 0.378107i −4.93597 − 0.388470i 3.71407 − 1.48515i −1.74330 + 7.26138i 9.84081 − 1.10340i 7.47227 − 3.80731i −6.73264 + 4.32106i 15.3237 + 2.42704i 0.678151 − 14.9200i
3.7 −1.93619 − 0.501168i 1.57113 + 0.123650i 3.49766 + 1.94071i 1.65598 − 6.89765i −2.98003 − 1.02681i 10.7453 − 5.47502i −5.79951 − 5.51051i −6.43605 − 1.01937i −6.66317 + 12.5252i
3.8 −1.92233 − 0.551956i −3.60514 − 0.283731i 3.39069 + 2.12208i −0.669464 + 2.78852i 6.77366 + 2.53530i −11.2633 + 5.73891i −5.34672 − 5.95084i 4.02734 + 0.637868i 2.82607 − 4.99093i
3.9 −1.88541 + 0.667251i 1.07842 + 0.0848735i 3.10955 − 2.51609i −0.265119 + 1.10430i −2.08990 + 0.559555i 3.73689 − 1.90404i −4.18392 + 6.81871i −7.73341 − 1.22485i −0.236987 − 2.25896i
3.10 −1.87841 − 0.686706i 0.940388 + 0.0740101i 3.05687 + 2.57984i −2.00429 + 8.34845i −1.71561 − 0.784791i 3.78972 − 1.93096i −3.97047 − 6.94517i −8.01034 − 1.26871i 9.49781 − 14.3055i
3.11 −1.86737 + 0.716197i 4.81165 + 0.378685i 2.97412 − 2.67481i −1.39598 + 5.81467i −9.25633 + 2.73894i 11.2722 − 5.74348i −3.63810 + 7.12490i 14.1193 + 2.23629i −1.55764 − 11.8579i
3.12 −1.86049 + 0.733889i −1.77292 − 0.139532i 2.92281 − 2.73078i −2.20693 + 9.19251i 3.40089 − 1.04153i −9.78317 + 4.98478i −3.43376 + 7.22560i −5.76543 − 0.913154i −2.64033 − 18.7222i
3.13 −1.84943 − 0.761328i 0.815381 + 0.0641719i 2.84076 + 2.81604i 1.73462 − 7.22523i −1.45913 − 0.739454i −10.4618 + 5.33054i −3.10984 − 7.37081i −8.22847 − 1.30326i −8.70883 + 12.0419i
3.14 −1.83243 − 0.801377i −0.132848 − 0.0104554i 2.71559 + 2.93693i −0.888171 + 3.69950i 0.235056 + 0.125620i 4.12005 − 2.09927i −2.62253 − 7.55793i −8.87166 − 1.40513i 4.59221 − 6.06731i
3.15 −1.78054 + 0.910871i 2.77027 + 0.218025i 2.34063 − 3.24368i 1.73303 − 7.21860i −5.13116 + 2.13515i 4.72492 − 2.40747i −1.21300 + 7.90751i −1.26235 − 0.199937i 3.48949 + 14.4316i
3.16 −1.75070 − 0.966985i −3.94361 − 0.310369i 2.12988 + 3.38580i −1.15080 + 4.79342i 6.60395 + 4.35678i 3.66205 − 1.86591i −0.454763 − 7.98706i 6.56656 + 1.04004i 6.64986 − 7.27902i
3.17 −1.72404 + 1.01375i −5.86315 − 0.461440i 1.94463 − 3.49549i 1.02945 − 4.28796i 10.5761 − 5.14822i −6.03672 + 3.07586i 0.190932 + 7.99772i 25.2744 + 4.00307i 2.57211 + 8.43622i
3.18 −1.69421 + 1.06285i −2.07316 − 0.163161i 1.74071 − 3.60138i 0.131783 − 0.548916i 3.68579 − 1.92703i −3.13697 + 1.59837i 0.878588 + 7.95161i −4.61782 − 0.731390i 0.360146 + 1.07005i
3.19 −1.65926 + 1.11663i 2.79419 + 0.219908i 1.50629 − 3.70555i 1.34188 − 5.58935i −4.88184 + 2.75519i −8.03334 + 4.09319i 1.63840 + 7.83043i −1.13006 − 0.178983i 4.01469 + 10.7726i
3.20 −1.62585 − 1.16474i −3.74557 − 0.294783i 1.28676 + 3.78738i 1.19855 − 4.99231i 5.74638 + 4.84189i 8.72930 − 4.44780i 2.31923 − 7.65645i 5.05320 + 0.800349i −7.76340 + 6.72074i
See next 80 embeddings (of 1504 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.94
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
32.h odd 8 1 inner
352.bd odd 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 352.3.bd.a ✓ 1504
11.c even 5 1 inner 352.3.bd.a ✓ 1504
32.h odd 8 1 inner 352.3.bd.a ✓ 1504
352.bd odd 40 1 inner 352.3.bd.a ✓ 1504
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.3.bd.a ✓ 1504 1.a even 1 1 trivial
352.3.bd.a ✓ 1504 11.c even 5 1 inner
352.3.bd.a ✓ 1504 32.h odd 8 1 inner
352.3.bd.a ✓ 1504 352.bd odd 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(352, [\chi])\).