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Properties

Label 228.2.w.b
Level $228$
Weight $2$
Character orbit 228.w
Analytic conductor $1.821$
Analytic rank $0$
Dimension $60$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [228,2,Mod(67,228)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(228, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 0, 17])) 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("228.67"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 228.w (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60,3] 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: \(1.82058916609\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 3 q^{2} - 3 q^{4} + 3 q^{6} - 3 q^{8} - 6 q^{10} - 6 q^{13} - 9 q^{14} + 21 q^{16} + 18 q^{19} + 30 q^{20} - 6 q^{21} - 12 q^{22} - 18 q^{24} - 30 q^{27} - 18 q^{28} - 12 q^{31} + 33 q^{32} - 15 q^{34}+ \cdots - 105 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −1.41419 + 0.00878762i 0.766044 − 0.642788i 1.99985 − 0.0248547i 3.39140 + 1.23437i −1.07768 + 0.915753i 1.70277 + 0.983096i −2.82794 + 0.0527230i 0.173648 − 0.984808i −4.80692 − 1.71582i
67.2 −1.18161 + 0.777041i 0.766044 − 0.642788i 0.792416 − 1.83632i −1.25586 − 0.457097i −0.405696 + 1.35477i 1.18216 + 0.682521i 0.490568 + 2.78556i 0.173648 − 0.984808i 1.83913 − 0.435746i
67.3 −1.05000 − 0.947369i 0.766044 − 0.642788i 0.204983 + 1.98947i 1.27204 + 0.462984i −1.41330 − 0.0508027i −3.79268 − 2.18970i 1.66953 − 2.28313i 0.173648 − 0.984808i −0.897018 − 1.69122i
67.4 −0.203595 + 1.39948i 0.766044 − 0.642788i −1.91710 − 0.569856i −2.74412 − 0.998777i 0.743606 + 1.20293i −3.79172 − 2.18915i 1.18781 − 2.56692i 0.173648 − 0.984808i 1.95646 − 3.63700i
67.5 −0.109693 + 1.40995i 0.766044 − 0.642788i −1.97593 − 0.309324i 0.378363 + 0.137713i 0.822271 + 1.15060i 3.35222 + 1.93540i 0.652879 − 2.75204i 0.173648 − 0.984808i −0.235673 + 0.518369i
67.6 0.0625414 − 1.41283i 0.766044 − 0.642788i −1.99218 − 0.176721i −3.72560 − 1.35601i −0.860240 − 1.12249i −0.752955 − 0.434719i −0.374270 + 2.80356i 0.173648 − 0.984808i −2.14881 + 5.17883i
67.7 0.803261 − 1.16395i 0.766044 − 0.642788i −0.709544 − 1.86991i 1.35681 + 0.493837i −0.132837 − 1.40796i 0.113130 + 0.0653155i −2.74642 − 0.676151i 0.173648 − 0.984808i 1.66467 − 1.18257i
67.8 0.814750 + 1.15593i 0.766044 − 0.642788i −0.672365 + 1.88359i 3.29537 + 1.19942i 1.36715 + 0.361785i −2.21173 − 1.27694i −2.72512 + 0.757450i 0.173648 − 0.984808i 1.29846 + 4.78645i
67.9 1.19665 + 0.753681i 0.766044 − 0.642788i 0.863929 + 1.80378i −2.00728 − 0.730589i 1.40114 − 0.191837i 3.36529 + 1.94295i −0.325658 + 2.80962i 0.173648 − 0.984808i −1.85137 − 2.38710i
67.10 1.40824 − 0.129898i 0.766044 − 0.642788i 1.96625 − 0.365855i 0.0388803 + 0.0141513i 0.995274 − 1.00470i −1.39317 − 0.804345i 2.72142 − 0.770623i 0.173648 − 0.984808i 0.0565908 + 0.0148778i
79.1 −1.37514 + 0.330144i −0.939693 + 0.342020i 1.78201 − 0.907986i 0.432248 − 2.45140i 1.17929 − 0.780559i −1.95860 + 1.13080i −2.15074 + 1.83693i 0.766044 − 0.642788i 0.214913 + 3.51371i
79.2 −1.11501 + 0.869917i −0.939693 + 0.342020i 0.486490 − 1.93993i −0.379809 + 2.15400i 0.750237 − 1.19881i 3.95398 − 2.28283i 1.14514 + 2.58625i 0.766044 − 0.642788i −1.45031 − 2.73213i
79.3 −1.02960 − 0.969499i −0.939693 + 0.342020i 0.120142 + 1.99639i 0.162908 − 0.923898i 1.29909 + 0.558888i −2.27351 + 1.31261i 1.81180 − 2.17195i 0.766044 − 0.642788i −1.06345 + 0.793304i
79.4 −0.626457 + 1.26789i −0.939693 + 0.342020i −1.21510 − 1.58856i 0.316000 − 1.79212i 0.155032 − 1.40569i −0.258690 + 0.149354i 2.77533 − 0.545458i 0.766044 − 0.642788i 2.07426 + 1.52334i
79.5 −0.526616 − 1.31251i −0.939693 + 0.342020i −1.44535 + 1.38238i −0.0217506 + 0.123354i 0.943762 + 1.05324i 3.23383 − 1.86705i 2.57552 + 1.16905i 0.766044 − 0.642788i 0.173357 − 0.0364123i
79.6 0.260406 + 1.39003i −0.939693 + 0.342020i −1.86438 + 0.723946i −0.175493 + 0.995269i −0.720121 − 1.21714i −2.28954 + 1.32186i −1.49180 − 2.40302i 0.766044 − 0.642788i −1.42916 + 0.0152335i
79.7 0.436096 − 1.34530i −0.939693 + 0.342020i −1.61964 − 1.17336i −0.711633 + 4.03587i 0.0503222 + 1.41332i −1.83842 + 1.06142i −2.28483 + 1.66720i 0.766044 − 0.642788i 5.11910 + 2.71738i
79.8 0.947134 − 1.05021i −0.939693 + 0.342020i −0.205875 − 1.98938i 0.234529 − 1.33008i −0.530822 + 1.31081i 1.38264 − 0.798268i −2.28425 − 1.66799i 0.766044 − 0.642788i −1.17473 − 1.50607i
79.9 1.37128 + 0.345809i −0.939693 + 0.342020i 1.76083 + 0.948403i −0.540387 + 3.06469i −1.40686 + 0.144052i −1.31815 + 0.761035i 2.08663 + 1.90944i 0.766044 − 0.642788i −1.80082 + 4.01568i
79.10 1.39185 + 0.250479i −0.939693 + 0.342020i 1.87452 + 0.697261i 0.683388 − 3.87569i −1.39358 + 0.240669i 0.181667 − 0.104885i 2.43441 + 1.44001i 0.766044 − 0.642788i 1.92196 − 5.22322i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.k even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 228.2.w.b yes 60
3.b odd 2 1 684.2.cf.b 60
4.b odd 2 1 228.2.w.a ✓ 60
12.b even 2 1 684.2.cf.c 60
19.f odd 18 1 228.2.w.a ✓ 60
57.j even 18 1 684.2.cf.c 60
76.k even 18 1 inner 228.2.w.b yes 60
228.u odd 18 1 684.2.cf.b 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.w.a ✓ 60 4.b odd 2 1
228.2.w.a ✓ 60 19.f odd 18 1
228.2.w.b yes 60 1.a even 1 1 trivial
228.2.w.b yes 60 76.k even 18 1 inner
684.2.cf.b 60 3.b odd 2 1
684.2.cf.b 60 228.u odd 18 1
684.2.cf.c 60 12.b even 2 1
684.2.cf.c 60 57.j even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{60} - 120 T_{7}^{58} + 8037 T_{7}^{56} + 234 T_{7}^{55} - 370070 T_{7}^{54} + \cdots + 54\!\cdots\!89 \) acting on \(S_{2}^{\mathrm{new}}(228, [\chi])\). Copy content Toggle raw display