Properties

Label 228.2.v.a
Level $228$
Weight $2$
Character orbit 228.v
Analytic conductor $1.821$
Analytic rank $0$
Dimension $216$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 228.v (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.82058916609\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(36\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216q - 18q^{4} - 6q^{6} - 18q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 216q - 18q^{4} - 6q^{6} - 18q^{9} - 18q^{10} - 3q^{12} - 36q^{13} - 6q^{16} - 12q^{18} - 30q^{21} - 18q^{24} - 24q^{25} - 36q^{28} + 12q^{33} + 30q^{34} - 66q^{36} - 48q^{37} - 42q^{40} - 117q^{42} - 6q^{45} - 6q^{46} - 81q^{48} + 24q^{49} + 12q^{52} + 33q^{54} - 12q^{57} + 48q^{58} - 18q^{60} - 48q^{61} + 36q^{64} - 18q^{66} - 6q^{69} - 18q^{70} + 12q^{72} - 132q^{73} + 60q^{76} - 39q^{78} + 18q^{81} + 132q^{82} - 45q^{84} - 144q^{85} + 114q^{88} + 72q^{90} - 30q^{93} + 138q^{96} - 60q^{97} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.41413 0.0157672i 0.174671 1.72322i 1.99950 + 0.0445937i −0.894860 1.06645i −0.274177 + 2.43410i −3.30990 1.91097i −2.82685 0.0945877i −2.93898 0.601994i 1.24863 + 1.52221i
23.2 −1.40660 0.146591i 0.0483859 + 1.73137i 1.95702 + 0.412388i 2.52711 + 3.01169i 0.185744 2.44244i 1.60040 + 0.923992i −2.69229 0.866945i −2.99532 + 0.167548i −3.11313 4.60668i
23.3 −1.39258 0.246404i −1.60968 + 0.639475i 1.87857 + 0.686276i −0.942247 1.12293i 2.39918 0.493889i 0.536440 + 0.309714i −2.44696 1.41858i 2.18214 2.05870i 1.03546 + 1.79594i
23.4 −1.32091 + 0.505162i −1.37188 1.05734i 1.48962 1.33455i −0.118249 0.140923i 2.34625 + 0.703629i 1.88835 + 1.09024i −1.29350 + 2.51532i 0.764084 + 2.90106i 0.227385 + 0.126413i
23.5 −1.22517 0.706378i 1.73132 + 0.0503667i 1.00206 + 1.73086i 0.942247 + 1.12293i −2.08557 1.28467i −0.536440 0.309714i −0.00504802 2.82842i 2.99493 + 0.174402i −0.361198 2.04135i
23.6 −1.18680 + 0.769092i 1.18231 1.26576i 0.816994 1.82552i 2.15444 + 2.56756i −0.429685 + 2.41151i 0.213070 + 0.123016i 0.434383 + 2.79487i −0.204279 2.99304i −4.53159 1.39022i
23.7 −1.17174 0.791847i 0.546697 + 1.64351i 0.745957 + 1.85568i −2.52711 3.01169i 0.660820 2.35867i −1.60040 0.923992i 0.595347 2.76506i −2.40224 + 1.79700i 0.576319 + 5.53000i
23.8 −1.16179 + 0.806381i 0.524795 + 1.65063i 0.699499 1.87369i −1.25702 1.49806i −1.94074 1.49450i 2.57863 + 1.48877i 0.698237 + 2.74089i −2.44918 + 1.73249i 2.66839 + 0.726786i
23.9 −1.09342 0.896904i −0.753513 1.55956i 0.391126 + 1.96138i 0.894860 + 1.06645i −0.574868 + 2.38108i 3.30990 + 1.91097i 1.33151 2.49541i −1.86444 + 2.35029i −0.0219504 1.96868i
23.10 −1.08345 + 0.908925i −1.12016 + 1.32107i 0.347710 1.96954i 0.218260 + 0.260112i 0.0128810 2.44946i −3.52551 2.03545i 1.41344 + 2.44993i −0.490465 2.95964i −0.472896 0.0834354i
23.11 −0.709055 + 1.22362i 1.68017 0.420760i −0.994483 1.73522i −2.09742 2.49961i −0.676481 + 2.35422i −2.63219 1.51969i 2.82839 + 0.0135021i 2.64592 1.41389i 4.54576 0.794084i
23.12 −0.687167 1.23604i 0.927511 1.46278i −1.05560 + 1.69873i 0.118249 + 0.140923i −2.44541 0.141271i −1.88835 1.09024i 2.82509 + 0.137458i −1.27945 2.71349i 0.0929307 0.242998i
23.13 −0.557050 + 1.29988i −1.38187 1.04424i −1.37939 1.44820i 1.43000 + 1.70421i 2.12716 1.21457i −1.04816 0.605157i 2.65088 0.986326i 0.819119 + 2.88601i −3.01185 + 0.909504i
23.14 −0.414780 1.35202i −1.54392 0.785048i −1.65592 + 1.12158i −2.15444 2.56756i −0.421012 + 2.41304i −0.213070 0.123016i 2.20324 + 1.77362i 1.76740 + 2.42411i −2.57778 + 3.97782i
23.15 −0.408823 + 1.35383i 1.38187 + 1.04424i −1.66573 1.10696i 1.43000 + 1.70421i −1.97867 + 1.44391i 1.04816 + 0.605157i 2.17962 1.80257i 0.819119 + 2.88601i −2.89183 + 1.23926i
23.16 −0.371649 1.36451i 0.0714037 + 1.73058i −1.72375 + 1.01423i 1.25702 + 1.49806i 2.33485 0.740598i −2.57863 1.48877i 2.02456 + 1.97513i −2.98980 + 0.247139i 1.57694 2.27196i
23.17 −0.245722 1.39270i 1.50444 + 0.858283i −1.87924 + 0.684435i −0.218260 0.260112i 0.825659 2.30614i 3.52551 + 2.03545i 1.41498 + 2.44904i 1.52670 + 2.58248i −0.308628 + 0.367887i
23.18 −0.243359 + 1.39312i −1.68017 + 0.420760i −1.88155 0.678056i −2.09742 2.49961i −0.177284 2.44307i 2.63219 + 1.51969i 1.40250 2.45621i 2.64592 1.41389i 3.99268 2.31366i
23.19 0.243359 1.39312i −1.72275 + 0.179266i −1.88155 0.678056i 2.09742 + 2.49961i −0.169508 + 2.44362i 2.63219 + 1.51969i −1.40250 + 2.45621i 2.93573 0.617661i 3.99268 2.31366i
23.20 0.245722 + 1.39270i 1.12016 1.32107i −1.87924 + 0.684435i 0.218260 + 0.260112i 2.11511 + 1.23544i 3.52551 + 2.03545i −1.41498 2.44904i −0.490465 2.95964i −0.308628 + 0.367887i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 215.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
19.e even 9 1 inner
57.l odd 18 1 inner
76.l odd 18 1 inner
228.v 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.v.a 216
3.b odd 2 1 inner 228.2.v.a 216
4.b odd 2 1 inner 228.2.v.a 216
12.b even 2 1 inner 228.2.v.a 216
19.e even 9 1 inner 228.2.v.a 216
57.l odd 18 1 inner 228.2.v.a 216
76.l odd 18 1 inner 228.2.v.a 216
228.v even 18 1 inner 228.2.v.a 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.v.a 216 1.a even 1 1 trivial
228.2.v.a 216 3.b odd 2 1 inner
228.2.v.a 216 4.b odd 2 1 inner
228.2.v.a 216 12.b even 2 1 inner
228.2.v.a 216 19.e even 9 1 inner
228.2.v.a 216 57.l odd 18 1 inner
228.2.v.a 216 76.l odd 18 1 inner
228.2.v.a 216 228.v even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(228, [\chi])\).