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Properties

Label 228.3.j.d
Level $228$
Weight $3$
Character orbit 228.j
Analytic conductor $6.213$
Analytic rank $0$
Dimension $38$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [228,3,Mod(7,228)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(228, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("228.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 228.j (of order \(6\), degree \(2\), 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: \(6.21255002741\)
Analytic rank: \(0\)
Dimension: \(38\)
Relative dimension: \(19\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 38 q - q^{2} + 57 q^{3} - 7 q^{4} + 4 q^{5} - 3 q^{6} - 22 q^{8} + 57 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - q^{2} + 57 q^{3} - 7 q^{4} + 4 q^{5} - 3 q^{6} - 22 q^{8} + 57 q^{9} + 12 q^{10} - 12 q^{12} - 11 q^{13} - 15 q^{14} + 12 q^{15} - 51 q^{16} - 32 q^{17} - 6 q^{18} + 38 q^{19} + 78 q^{20} + 15 q^{21} - 102 q^{22} + 108 q^{23} - 51 q^{24} - 109 q^{25} + 46 q^{26} + 165 q^{28} + 52 q^{29} + 24 q^{30} - 111 q^{32} + 6 q^{33} - 108 q^{34} + 12 q^{35} - 15 q^{36} + 42 q^{37} + 223 q^{38} - 140 q^{40} - 36 q^{41} - 39 q^{42} + 177 q^{43} - 318 q^{44} + 24 q^{45} - 136 q^{46} + 210 q^{47} - 81 q^{48} - 348 q^{49} - 382 q^{50} - 96 q^{51} - 11 q^{52} - 52 q^{53} - 9 q^{54} + 168 q^{55} + 96 q^{56} + 93 q^{57} + 108 q^{58} + 54 q^{59} + 162 q^{60} + 95 q^{61} - 171 q^{62} + 45 q^{63} - 190 q^{64} + 216 q^{65} - 150 q^{66} + 171 q^{67} - 240 q^{68} + 216 q^{69} - 60 q^{70} + 78 q^{71} - 87 q^{72} + 65 q^{73} - 63 q^{74} - 271 q^{76} + 44 q^{77} + 51 q^{78} - 93 q^{79} + 400 q^{80} - 171 q^{81} + 412 q^{82} + 228 q^{84} - 288 q^{85} + 149 q^{86} + 560 q^{88} + 82 q^{89} + 36 q^{90} - 591 q^{91} - 42 q^{92} - 105 q^{93} + 700 q^{94} - 52 q^{95} - 150 q^{96} + 218 q^{97} - 666 q^{98} + 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.99810 + 0.0871454i 1.50000 − 0.866025i 3.98481 − 0.348251i 0.525524 + 0.910235i −2.92168 + 1.86112i 8.41699i −7.93171 + 1.04310i 1.50000 − 2.59808i −1.12937 − 1.77294i
7.2 −1.99728 + 0.104319i 1.50000 − 0.866025i 3.97824 − 0.416707i 1.73763 + 3.00967i −2.90557 + 1.88617i − 11.4833i −7.90217 + 1.24728i 1.50000 − 2.59808i −3.78450 − 5.82988i
7.3 −1.66348 − 1.11034i 1.50000 − 0.866025i 1.53431 + 3.69403i −3.96819 − 6.87311i −3.45679 − 0.224890i − 2.27366i 1.54933 − 7.84854i 1.50000 − 2.59808i −1.03046 + 15.8393i
7.4 −1.65044 − 1.12962i 1.50000 − 0.866025i 1.44793 + 3.72874i 1.45787 + 2.52510i −3.45394 − 0.265098i − 1.23360i 1.82231 − 7.78968i 1.50000 − 2.59808i 0.446267 − 5.81438i
7.5 −1.58814 + 1.21565i 1.50000 − 0.866025i 1.04437 − 3.86125i −1.53218 − 2.65381i −1.32942 + 3.19885i 1.58139i 3.03534 + 7.40180i 1.50000 − 2.59808i 5.65943 + 2.35202i
7.6 −1.28466 + 1.53286i 1.50000 − 0.866025i −0.699290 − 3.93840i 3.79250 + 6.56881i −0.599501 + 3.41183i 7.95866i 6.93535 + 3.98760i 1.50000 − 2.59808i −14.9411 − 2.62534i
7.7 −0.958739 − 1.75523i 1.50000 − 0.866025i −2.16164 + 3.36561i −0.220249 − 0.381482i −2.95818 − 1.80255i 3.14365i 7.97985 + 0.567422i 1.50000 − 2.59808i −0.458426 + 0.752328i
7.8 −0.433118 − 1.95254i 1.50000 − 0.866025i −3.62482 + 1.69136i 2.14913 + 3.72240i −2.34063 − 2.55372i 6.16880i 4.87242 + 6.34504i 1.50000 − 2.59808i 6.33730 − 5.80849i
7.9 −0.363851 + 1.96662i 1.50000 − 0.866025i −3.73522 − 1.43112i −3.76352 − 6.51861i 1.15737 + 3.26504i 11.3890i 4.17354 − 6.82507i 1.50000 − 2.59808i 14.1890 − 5.02963i
7.10 −0.154040 + 1.99406i 1.50000 − 0.866025i −3.95254 − 0.614331i −0.639810 − 1.10818i 1.49585 + 3.12449i − 4.57176i 1.83386 − 7.78697i 1.50000 − 2.59808i 2.30834 − 1.10511i
7.11 0.321570 − 1.97398i 1.50000 − 0.866025i −3.79319 − 1.26955i −2.28347 − 3.95508i −1.22716 − 3.23946i − 6.17600i −3.72583 + 7.07942i 1.50000 − 2.59808i −8.54155 + 3.23568i
7.12 0.709174 + 1.87005i 1.50000 − 0.866025i −2.99414 + 2.65238i 4.05729 + 7.02743i 2.68327 + 2.19091i − 2.19484i −7.08343 − 3.71819i 1.50000 − 2.59808i −10.2643 + 12.5710i
7.13 0.928138 − 1.77160i 1.50000 − 0.866025i −2.27712 − 3.28857i 3.75967 + 6.51193i −0.142042 − 3.46119i − 7.36641i −7.93951 + 0.981891i 1.50000 − 2.59808i 15.0260 − 0.616646i
7.14 1.15859 + 1.63024i 1.50000 − 0.866025i −1.31534 + 3.77755i −4.34173 − 7.52009i 3.14971 + 1.44199i − 11.0258i −7.68224 + 2.23231i 1.50000 − 2.59808i 7.22925 − 15.7907i
7.15 1.39290 − 1.43521i 1.50000 − 0.866025i −0.119678 − 3.99821i −3.80581 − 6.59185i 0.846413 − 3.35910i 11.2361i −5.90498 − 5.39733i 1.50000 − 2.59808i −14.7618 − 3.71962i
7.16 1.62474 + 1.16629i 1.50000 − 0.866025i 1.27956 + 3.78982i −0.0175343 − 0.0303703i 3.44714 + 0.342362i 9.17511i −2.34106 + 7.64980i 1.50000 − 2.59808i 0.00693177 − 0.0697939i
7.17 1.69571 − 1.06046i 1.50000 − 0.866025i 1.75086 − 3.59646i −1.00634 − 1.74304i 1.62518 − 3.05921i − 7.56784i −0.844949 − 7.95525i 1.50000 − 2.59808i −3.55488 − 1.88850i
7.18 1.83710 − 0.790613i 1.50000 − 0.866025i 2.74986 − 2.90487i 3.27567 + 5.67363i 2.07096 − 2.77689i 10.0678i 2.75514 − 7.51061i 1.50000 − 2.59808i 10.5034 + 7.83322i
7.19 1.92393 + 0.546330i 1.50000 − 0.866025i 3.40305 + 2.10220i 2.82355 + 4.89053i 3.35904 − 0.846682i − 6.58406i 5.39874 + 5.90369i 1.50000 − 2.59808i 2.76048 + 10.9517i
163.1 −1.99810 − 0.0871454i 1.50000 + 0.866025i 3.98481 + 0.348251i 0.525524 − 0.910235i −2.92168 − 1.86112i − 8.41699i −7.93171 − 1.04310i 1.50000 + 2.59808i −1.12937 + 1.77294i
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.19
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 228.3.j.d yes 38
4.b odd 2 1 228.3.j.c ✓ 38
19.c even 3 1 228.3.j.c ✓ 38
76.g odd 6 1 inner 228.3.j.d yes 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.j.c ✓ 38 4.b odd 2 1
228.3.j.c ✓ 38 19.c even 3 1
228.3.j.d yes 38 1.a even 1 1 trivial
228.3.j.d yes 38 76.g odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(228, [\chi])\):

\( T_{5}^{38} - 4 T_{5}^{37} + 300 T_{5}^{36} - 1280 T_{5}^{35} + 53140 T_{5}^{34} - 229368 T_{5}^{33} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
\( T_{23}^{38} - 108 T_{23}^{37} + 436 T_{23}^{36} + 372816 T_{23}^{35} - 7792236 T_{23}^{34} + \cdots + 23\!\cdots\!48 \) Copy content Toggle raw display