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Properties

Label 356.3.n.c
Level $356$
Weight $3$
Character orbit 356.n
Analytic conductor $9.700$
Analytic rank $0$
Dimension $1720$
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] = [356,3,Mod(47,356)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(356, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([22, 27]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("356.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 356 = 2^{2} \cdot 89 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 356.n (of order \(44\), degree \(20\), 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: \(9.70029741123\)
Analytic rank: \(0\)
Dimension: \(1720\)
Relative dimension: \(86\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1720 q - 18 q^{2} - 2 q^{4} - 44 q^{5} - 28 q^{6} - 18 q^{8} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1720 q - 18 q^{2} - 2 q^{4} - 44 q^{5} - 28 q^{6} - 18 q^{8} - 44 q^{9} - 22 q^{10} - 52 q^{12} - 72 q^{13} + 280 q^{14} + 94 q^{16} - 44 q^{17} - 22 q^{18} - 22 q^{20} - 44 q^{21} - 34 q^{22} - 74 q^{24} + 488 q^{25} + 90 q^{26} - 96 q^{28} - 24 q^{29} + 800 q^{30} - 18 q^{32} - 76 q^{33} - 22 q^{36} - 200 q^{37} + 98 q^{38} + 968 q^{40} - 40 q^{41} - 22 q^{42} - 82 q^{44} - 236 q^{45} - 44 q^{46} - 398 q^{48} - 44 q^{49} - 66 q^{50} - 174 q^{52} - 44 q^{53} - 366 q^{54} + 36 q^{56} - 148 q^{57} - 254 q^{58} + 170 q^{60} + 104 q^{61} - 304 q^{62} + 478 q^{64} - 252 q^{65} + 928 q^{66} - 22 q^{68} - 396 q^{69} - 128 q^{70} - 2684 q^{72} - 788 q^{73} - 872 q^{74} + 120 q^{76} - 332 q^{77} - 758 q^{78} + 2090 q^{80} + 536 q^{81} - 448 q^{82} - 22 q^{84} - 172 q^{85} - 388 q^{86} - 248 q^{88} - 200 q^{89} + 852 q^{90} - 126 q^{92} + 828 q^{93} - 22 q^{94} + 320 q^{96} + 108 q^{97} - 1012 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −1.99469 + 0.145643i 2.49190 + 0.929429i 3.95758 − 0.581025i 5.99867 − 0.862479i −5.10593 − 1.49100i 10.9403 + 8.18979i −7.80952 + 1.73536i −1.45604 − 1.26166i −11.8399 + 2.59404i
47.2 −1.99139 − 0.185342i −3.87755 − 1.44625i 3.93130 + 0.738176i −0.726361 + 0.104435i 7.45367 + 3.59872i −0.386287 − 0.289171i −7.69195 − 2.19863i 6.14198 + 5.32206i 1.46583 − 0.0733462i
47.3 −1.99033 + 0.196401i 1.10287 + 0.411349i 3.92285 − 0.781805i −8.16351 + 1.17374i −2.27587 − 0.602118i 4.26176 + 3.19031i −7.65424 + 2.32650i −5.75463 − 4.98642i 16.0176 − 3.93944i
47.4 −1.97150 − 0.336441i −0.965266 − 0.360026i 3.77361 + 1.32659i −2.90581 + 0.417793i 1.78189 + 1.03454i −9.42613 − 7.05631i −6.99336 − 3.88496i −5.99963 − 5.19871i 5.86937 + 0.153957i
47.5 −1.96873 + 0.352268i 1.09427 + 0.408141i 3.75182 − 1.38704i 3.94353 − 0.566994i −2.29810 − 0.418045i 0.0857543 + 0.0641949i −6.89771 + 4.05236i −5.77090 − 5.00051i −7.56402 + 2.50544i
47.6 −1.95843 − 0.405674i 4.64039 + 1.73078i 3.67086 + 1.58896i 2.07314 − 0.298073i −8.38573 − 5.27208i 0.226474 + 0.169536i −6.54450 − 4.60104i 11.7359 + 10.1692i −4.18102 − 0.257266i
47.7 −1.93745 − 0.496266i −3.70116 − 1.38046i 3.50744 + 1.92298i −0.924696 + 0.132951i 6.48574 + 4.51134i 10.7122 + 8.01908i −5.84118 − 5.46631i 4.99117 + 4.32487i 1.85753 + 0.201309i
47.8 −1.91409 + 0.579889i 5.03681 + 1.87863i 3.32746 − 2.21992i −7.15436 + 1.02864i −10.7303 − 0.675073i −6.73981 − 5.04536i −5.08173 + 6.17867i 15.0384 + 13.0309i 13.0976 − 6.11765i
47.9 −1.91082 − 0.590568i −1.60566 − 0.598879i 3.30246 + 2.25694i 7.97305 − 1.14635i 2.71444 + 2.09260i −3.63603 − 2.72190i −4.97753 − 6.26293i −4.58227 − 3.97056i −15.9121 − 2.51816i
47.10 −1.89801 + 0.630529i −2.88152 − 1.07475i 3.20487 − 2.39350i −3.65920 + 0.526114i 6.14680 + 0.223004i 1.81087 + 1.35560i −4.57369 + 6.56364i 0.346303 + 0.300073i 6.61347 − 3.30580i
47.11 −1.89736 + 0.632476i −5.25481 − 1.95994i 3.19995 − 2.40007i 9.16137 − 1.31721i 11.2099 + 0.395177i −1.12706 − 0.843709i −4.55347 + 6.57768i 16.9699 + 14.7045i −16.5493 + 8.29356i
47.12 −1.87092 − 0.706854i 2.90602 + 1.08389i 3.00071 + 2.64494i −3.10895 + 0.447000i −4.67079 − 4.08200i −1.76080 − 1.31812i −3.74452 − 7.06955i 0.468372 + 0.405847i 6.13258 + 1.36127i
47.13 −1.83073 + 0.805244i 2.43036 + 0.906479i 2.70316 − 2.94837i 5.50395 − 0.791349i −5.17928 + 0.297516i −7.98904 − 5.98052i −2.57461 + 7.57439i −1.71679 − 1.48760i −9.43903 + 5.88077i
47.14 −1.79628 + 0.879425i −3.96774 − 1.47989i 2.45322 − 3.15938i −2.91686 + 0.419381i 8.42860 − 0.831034i −8.67849 − 6.49664i −1.62823 + 7.83255i 6.75111 + 5.84987i 4.87067 − 3.31848i
47.15 −1.66134 − 1.11353i 0.00373440 + 0.00139286i 1.52009 + 3.69991i −2.84743 + 0.409399i −0.00465311 − 0.00647239i 4.46409 + 3.34177i 1.59459 − 7.83947i −6.80173 − 5.89374i 5.18643 + 2.49056i
47.16 −1.59971 − 1.20038i −2.20220 − 0.821378i 1.11817 + 3.84053i −9.56902 + 1.37582i 2.53692 + 3.95745i −3.59327 − 2.68989i 2.82135 − 7.48598i −2.62672 − 2.27607i 16.9592 + 9.28555i
47.17 −1.54713 − 1.26743i −5.25182 − 1.95883i 0.787233 + 3.92177i 0.156744 − 0.0225364i 5.64258 + 9.68689i −4.44153 − 3.32489i 3.75262 − 7.06526i 16.9429 + 14.6811i −0.271067 − 0.163796i
47.18 −1.54615 + 1.26863i 3.96774 + 1.47989i 0.781181 − 3.92298i −2.91686 + 0.419381i −8.01215 + 2.74543i 8.67849 + 6.49664i 3.76896 + 7.05655i 6.75111 + 5.84987i 3.97788 − 4.34883i
47.19 −1.53656 − 1.28023i 0.0681756 + 0.0254282i 0.722029 + 3.93429i 4.96505 − 0.713867i −0.0722019 − 0.126352i 4.36063 + 3.26433i 3.92736 − 6.96964i −6.79774 − 5.89028i −8.54301 − 5.25951i
47.20 −1.49299 + 1.33078i −2.43036 − 0.906479i 0.458036 − 3.97369i 5.50395 − 0.791349i 4.83483 − 1.88092i 7.98904 + 5.98052i 4.60427 + 6.54222i −1.71679 − 1.48760i −7.16423 + 8.50603i
See next 80 embeddings (of 1720 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.86
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
89.g even 44 1 inner
356.n odd 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 356.3.n.c ✓ 1720
4.b odd 2 1 inner 356.3.n.c ✓ 1720
89.g even 44 1 inner 356.3.n.c ✓ 1720
356.n odd 44 1 inner 356.3.n.c ✓ 1720
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
356.3.n.c ✓ 1720 1.a even 1 1 trivial
356.3.n.c ✓ 1720 4.b odd 2 1 inner
356.3.n.c ✓ 1720 89.g even 44 1 inner
356.3.n.c ✓ 1720 356.n odd 44 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(356, [\chi])\):

\( T_{3}^{1720} + 22 T_{3}^{1718} - 10503 T_{3}^{1716} - 219868 T_{3}^{1714} + 58965014 T_{3}^{1712} + \cdots + 69\!\cdots\!04 \) Copy content Toggle raw display
\( T_{13}^{860} + 36 T_{13}^{859} + 659 T_{13}^{858} + 16146 T_{13}^{857} - 635608 T_{13}^{856} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display