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Properties

Label 232.3.u.a
Level $232$
Weight $3$
Character orbit 232.u
Analytic conductor $6.322$
Analytic rank $0$
Dimension $696$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [232,3,Mod(21,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([0, 14, 17]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.21");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 232.u (of order \(28\), degree \(12\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.32154213316\)
Analytic rank: \(0\)
Dimension: \(696\)
Relative dimension: \(58\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 696 q - 10 q^{2} - 14 q^{4} - 14 q^{6} - 20 q^{7} - 40 q^{8} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 696 q - 10 q^{2} - 14 q^{4} - 14 q^{6} - 20 q^{7} - 40 q^{8} - 28 q^{9} - 4 q^{10} - 24 q^{12} + 10 q^{14} + 12 q^{15} + 10 q^{16} - 32 q^{17} + 84 q^{18} + 54 q^{20} - 14 q^{22} - 20 q^{23} - 46 q^{24} - 568 q^{25} - 68 q^{26} + 176 q^{30} - 24 q^{31} + 60 q^{32} - 28 q^{33} - 14 q^{34} + 58 q^{36} - 14 q^{38} + 204 q^{39} - 32 q^{40} + 16 q^{41} - 14 q^{42} + 194 q^{44} - 548 q^{46} - 24 q^{47} + 848 q^{48} - 664 q^{49} - 490 q^{50} + 586 q^{52} - 424 q^{54} + 76 q^{55} + 68 q^{56} - 74 q^{58} - 232 q^{60} + 476 q^{62} - 28 q^{63} - 896 q^{64} + 180 q^{65} + 1156 q^{66} - 740 q^{68} + 1506 q^{70} - 28 q^{71} - 928 q^{72} - 144 q^{73} - 292 q^{74} + 206 q^{76} - 478 q^{78} - 344 q^{79} - 14 q^{80} + 672 q^{81} + 50 q^{82} + 284 q^{84} - 408 q^{87} + 688 q^{88} - 320 q^{89} - 844 q^{90} - 14 q^{92} + 134 q^{94} + 76 q^{95} - 812 q^{96} - 424 q^{97} - 1428 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −1.99230 − 0.175308i 0.187853 − 1.66724i 3.93853 + 0.698533i 6.07243 − 2.92433i −0.666539 + 3.28871i −5.31671 − 6.66694i −7.72429 − 2.08214i 6.02996 + 1.37630i −12.6108 + 4.76160i
21.2 −1.97814 − 0.294898i −0.222527 + 1.97498i 3.82607 + 1.16670i −0.990384 + 0.476944i 1.02261 − 3.84117i 2.92628 + 3.66944i −7.22444 − 3.43620i 4.92330 + 1.12371i 2.09977 − 0.651399i
21.3 −1.96273 + 0.384327i 0.0795738 − 0.706237i 3.70459 − 1.50866i −5.87881 + 2.83108i 0.115245 + 1.41673i −0.260311 − 0.326420i −6.69127 + 4.38485i 8.28191 + 1.89029i 10.4504 − 7.81603i
21.4 −1.93873 + 0.491237i 0.372501 − 3.30604i 3.51737 − 1.90475i 1.76986 − 0.852320i 0.901867 + 6.59251i 8.03538 + 10.0761i −5.88356 + 5.42067i −2.01678 − 0.460318i −3.01260 + 2.52184i
21.5 −1.88877 − 0.657695i 0.618515 − 5.48947i 3.13487 + 2.48447i 2.50759 − 1.20759i −4.77863 + 9.96154i −0.956997 − 1.20004i −4.28702 − 6.75437i −20.9774 − 4.78796i −5.53048 + 0.631627i
21.6 −1.87486 − 0.696353i −0.514755 + 4.56857i 3.03019 + 2.61112i −3.18903 + 1.53575i 4.14643 − 8.20698i −5.74986 − 7.21010i −3.86290 − 7.00557i −11.8325 − 2.70070i 7.04840 − 0.658634i
21.7 −1.83396 + 0.797857i −0.382248 + 3.39255i 2.72685 − 2.92648i 3.35698 − 1.61664i −2.00574 − 6.52678i −5.14960 − 6.45740i −2.66602 + 7.54270i −2.58890 − 0.590899i −4.86673 + 5.64324i
21.8 −1.81384 + 0.842614i −0.327034 + 2.90250i 2.58000 − 3.05673i 3.82578 − 1.84240i −1.85251 − 5.54023i 1.92478 + 2.41359i −2.10406 + 7.71835i 0.456779 + 0.104257i −5.38691 + 6.56546i
21.9 −1.81283 − 0.844767i 0.335694 − 2.97937i 2.57274 + 3.06285i −6.67610 + 3.21504i −3.12543 + 5.11751i −1.52899 − 1.91730i −2.07655 − 7.72580i 0.0104181 + 0.00237786i 14.8186 − 0.188584i
21.10 −1.77173 − 0.927895i −0.135126 + 1.19928i 2.27802 + 3.28795i 7.93179 − 3.81975i 1.35221 − 1.99941i 4.50824 + 5.65315i −0.985153 − 7.93911i 7.35434 + 1.67858i −17.5973 − 0.592325i
21.11 −1.73824 + 0.989207i 0.587956 − 5.21826i 2.04294 − 3.43895i −3.88695 + 1.87185i 4.13993 + 9.65218i −2.62188 − 3.28773i −0.149270 + 7.99861i −18.1102 − 4.13353i 4.90478 − 7.09872i
21.12 −1.73323 + 0.997958i −0.605138 + 5.37075i 2.00816 − 3.45938i −8.21504 + 3.95615i −4.31094 − 9.91263i 5.69854 + 7.14574i −0.0282810 + 7.99995i −19.7044 − 4.49740i 10.2905 − 15.0552i
21.13 −1.39065 − 1.43739i −0.426691 + 3.78699i −0.132183 + 3.99782i −2.13810 + 1.02966i 6.03676 − 4.65305i 4.99964 + 6.26935i 5.93024 − 5.36957i −5.38485 − 1.22906i 4.45337 + 1.64140i
21.14 −1.37743 + 1.45006i 0.330463 − 2.93294i −0.205363 − 3.99472i 6.44856 − 3.10546i 3.79776 + 4.51912i −0.208934 − 0.261995i 6.07548 + 5.20467i 0.281399 + 0.0642276i −4.37933 + 13.6284i
21.15 −1.34418 − 1.48094i 0.305806 − 2.71411i −0.386370 + 3.98130i 1.50225 − 0.723444i −4.43049 + 3.19536i 2.73605 + 3.43089i 6.41541 − 4.77938i 1.50150 + 0.342708i −3.09066 − 1.25230i
21.16 −1.30262 + 1.51762i −0.0183559 + 0.162914i −0.606351 − 3.95378i −4.52760 + 2.18038i −0.223330 − 0.240072i −6.47097 − 8.11434i 6.79018 + 4.23006i 8.74815 + 1.99671i 2.58876 − 9.71139i
21.17 −1.29550 − 1.52370i 0.0300519 − 0.266718i −0.643338 + 3.94793i −0.733086 + 0.353036i −0.445331 + 0.299744i −7.42818 − 9.31465i 6.84891 − 4.13430i 8.70412 + 1.98666i 1.48764 + 0.659646i
21.18 −1.16786 + 1.62361i 0.000169458 − 0.00150398i −1.27220 − 3.79229i −2.88784 + 1.39071i 0.00224397 + 0.00203157i 2.33654 + 2.92992i 7.64296 + 2.36331i 8.77435 + 2.00269i 1.11463 − 6.31288i
21.19 −1.12935 − 1.65063i −0.483691 + 4.29288i −1.44913 + 3.72827i 8.08680 − 3.89440i 7.63219 − 4.04977i −4.37110 − 5.48119i 7.79056 − 1.81856i −9.42048 − 2.15016i −15.5610 − 8.95014i
21.20 −0.882048 + 1.79499i −0.563099 + 4.99765i −2.44398 − 3.16654i 0.385869 − 0.185825i −8.47404 − 5.41892i −5.38108 − 6.74766i 7.83961 − 1.59388i −15.8850 − 3.62565i −0.00680155 + 0.856537i
See next 80 embeddings (of 696 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
29.f odd 28 1 inner
232.u odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.3.u.a ✓ 696
8.b even 2 1 inner 232.3.u.a ✓ 696
29.f odd 28 1 inner 232.3.u.a ✓ 696
232.u odd 28 1 inner 232.3.u.a ✓ 696
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.3.u.a ✓ 696 1.a even 1 1 trivial
232.3.u.a ✓ 696 8.b even 2 1 inner
232.3.u.a ✓ 696 29.f odd 28 1 inner
232.3.u.a ✓ 696 232.u odd 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(232, [\chi])\).