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Properties

Label 229.3.f.a
Level $229$
Weight $3$
Character orbit 229.f
Analytic conductor $6.240$
Analytic rank $0$
Dimension $148$
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] = [229,3,Mod(18,229)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(229, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("229.18");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 229 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 229.f (of order \(12\), degree \(4\), 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: \(6.23979805385\)
Analytic rank: \(0\)
Dimension: \(148\)
Relative dimension: \(37\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 148 q - 2 q^{3} - 24 q^{5} - 20 q^{7} - 12 q^{8} - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 148 q - 2 q^{3} - 24 q^{5} - 20 q^{7} - 12 q^{8} - 216 q^{9} + 10 q^{10} - 48 q^{12} + 26 q^{13} - 10 q^{14} - 440 q^{16} + 28 q^{17} - 42 q^{18} - 76 q^{19} + 38 q^{20} + 50 q^{21} + 12 q^{22} + 2 q^{23} + 68 q^{24} + 318 q^{25} + 88 q^{27} - 120 q^{28} + 58 q^{29} - 390 q^{30} + 28 q^{31} - 118 q^{32} + 240 q^{33} - 160 q^{34} + 160 q^{35} - 246 q^{36} - 162 q^{37} - 198 q^{38} - 84 q^{39} + 352 q^{40} - 130 q^{41} + 464 q^{42} + 168 q^{43} + 72 q^{44} + 276 q^{45} + 330 q^{46} - 170 q^{47} + 320 q^{48} - 312 q^{49} + 90 q^{50} + 68 q^{51} + 4 q^{52} - 88 q^{53} + 484 q^{54} + 206 q^{55} - 342 q^{56} - 68 q^{57} + 564 q^{58} - 98 q^{59} - 608 q^{60} + 116 q^{61} + 66 q^{62} - 102 q^{63} - 16 q^{65} + 258 q^{66} - 202 q^{67} - 620 q^{69} + 138 q^{70} - 78 q^{71} - 10 q^{72} - 218 q^{73} + 108 q^{74} + 116 q^{75} - 294 q^{76} + 134 q^{77} + 48 q^{78} + 444 q^{79} - 510 q^{80} - 510 q^{81} + 66 q^{82} + 16 q^{83} + 1064 q^{84} - 244 q^{86} - 440 q^{87} + 290 q^{88} - 26 q^{89} + 1654 q^{90} - 490 q^{91} + 1264 q^{92} + 20 q^{93} + 218 q^{94} - 450 q^{95} - 96 q^{96} + 222 q^{97} - 1666 q^{98} + 234 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1 −2.69728 − 2.69728i −1.18816 + 2.05795i 10.5506i −1.56247 + 0.902095i 8.75565 − 2.34607i −0.177432 − 0.0475429i 17.6688 − 17.6688i 1.67656 + 2.90389i 6.64762 + 1.78122i
18.2 −2.56329 − 2.56329i 0.218861 − 0.379078i 9.14096i 5.63027 − 3.25064i −1.53269 + 0.410684i −4.97739 − 1.33369i 13.1778 − 13.1778i 4.40420 + 7.62830i −22.7644 − 6.09970i
18.3 −2.47918 − 2.47918i 2.45087 − 4.24502i 8.29267i −3.66088 + 2.11361i −16.6003 + 4.44804i −5.72950 − 1.53521i 10.6423 − 10.6423i −7.51348 − 13.0137i 14.3160 + 3.83596i
18.4 −2.40108 − 2.40108i 1.79109 − 3.10226i 7.53035i 3.54131 − 2.04457i −11.7493 + 3.14822i 6.32771 + 1.69550i 8.47666 − 8.47666i −1.91601 − 3.31863i −13.4121 − 3.59377i
18.5 −2.26175 − 2.26175i −2.48286 + 4.30044i 6.23106i 0.966574 − 0.558052i 15.3421 − 4.11092i 10.0275 + 2.68686i 5.04610 − 5.04610i −7.82918 − 13.5605i −3.44833 − 0.923976i
18.6 −2.24489 − 2.24489i −0.964314 + 1.67024i 6.07903i −5.97058 + 3.44711i 5.91427 − 1.58473i −3.95883 − 1.06077i 4.66719 − 4.66719i 2.64020 + 4.57296i 21.1416 + 5.66489i
18.7 −1.81755 − 1.81755i 1.09896 − 1.90345i 2.60694i −6.89884 + 3.98305i −5.45700 + 1.46220i 1.46117 + 0.391519i −2.53195 + 2.53195i 2.08459 + 3.61062i 19.7783 + 5.29959i
18.8 −1.72805 − 1.72805i −0.443547 + 0.768246i 1.97230i 2.74640 − 1.58564i 2.09404 − 0.561095i −5.46179 − 1.46348i −3.50396 + 3.50396i 4.10653 + 7.11272i −7.48598 − 2.00586i
18.9 −1.69229 − 1.69229i −1.86367 + 3.22798i 1.72770i 7.10361 − 4.10127i 8.61656 − 2.30880i −3.31065 − 0.887087i −3.84539 + 3.84539i −2.44656 − 4.23756i −18.9619 − 5.08083i
18.10 −1.62377 − 1.62377i 0.463186 − 0.802262i 1.27325i 0.0177847 − 0.0102680i −2.05479 + 0.550581i 10.5164 + 2.81787i −4.42762 + 4.42762i 4.07092 + 7.05104i −0.0455511 − 0.0122054i
18.11 −1.48063 − 1.48063i −2.76236 + 4.78455i 0.384535i −5.44170 + 3.14176i 11.1742 − 2.99411i −10.1531 − 2.72050i −5.35317 + 5.35317i −10.7613 − 18.6391i 12.7089 + 3.40535i
18.12 −1.23827 − 1.23827i 2.39582 − 4.14968i − 0.933371i 6.15853 − 3.55563i −8.10511 + 2.17176i −5.36526 − 1.43762i −6.10885 + 6.10885i −6.97992 − 12.0896i −12.0288 − 3.22310i
18.13 −1.04290 − 1.04290i 0.692017 − 1.19861i − 1.82471i −0.930178 + 0.537039i −1.97173 + 0.528325i −12.8766 − 3.45026i −6.07460 + 6.07460i 3.54223 + 6.13532i 1.53016 + 0.410006i
18.14 −0.986215 − 0.986215i 2.43954 − 4.22540i − 2.05476i −2.88300 + 1.66450i −6.57306 + 1.76125i 1.62097 + 0.434336i −5.97129 + 5.97129i −7.40268 − 12.8218i 4.48481 + 1.20170i
18.15 −0.647232 − 0.647232i −1.66525 + 2.88429i − 3.16218i −7.40440 + 4.27493i 2.94461 − 0.789006i 11.0076 + 2.94949i −4.63559 + 4.63559i −1.04610 − 1.81191i 7.55924 + 2.02549i
18.16 −0.505155 − 0.505155i −2.14014 + 3.70683i − 3.48964i 1.39560 − 0.805751i 2.95363 − 0.791422i 0.209785 + 0.0562117i −3.78343 + 3.78343i −4.66040 − 8.07204i −1.11202 − 0.297966i
18.17 −0.458722 − 0.458722i −0.972096 + 1.68372i − 3.57915i −2.34442 + 1.35355i 1.21828 − 0.326438i 0.452280 + 0.121188i −3.47672 + 3.47672i 2.61006 + 4.52075i 1.69634 + 0.454534i
18.18 −0.272964 − 0.272964i 0.265661 − 0.460139i − 3.85098i 7.01098 − 4.04779i −0.198118 + 0.0530855i 6.17467 + 1.65450i −2.14304 + 2.14304i 4.35885 + 7.54975i −3.01865 − 0.808845i
18.19 0.0420104 + 0.0420104i 0.836061 − 1.44810i − 3.99647i −1.50542 + 0.869156i 0.0959586 − 0.0257120i −5.35127 − 1.43387i 0.335935 − 0.335935i 3.10200 + 5.37283i −0.0997571 − 0.0267298i
18.20 0.314300 + 0.314300i 2.41109 − 4.17613i − 3.80243i 1.04723 − 0.604621i 2.07036 − 0.554752i 9.60901 + 2.57473i 2.45230 − 2.45230i −7.12671 − 12.3438i 0.519178 + 0.139113i
See next 80 embeddings (of 148 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.37
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
229.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 229.3.f.a ✓ 148
229.f odd 12 1 inner 229.3.f.a ✓ 148
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.3.f.a ✓ 148 1.a even 1 1 trivial
229.3.f.a ✓ 148 229.f odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(229, [\chi])\).