Properties

Label 276.2.o.a
Level $276$
Weight $2$
Character orbit 276.o
Analytic conductor $2.204$
Analytic rank $0$
Dimension $440$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 276.o (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.20387109579\)
Analytic rank: \(0\)
Dimension: \(440\)
Relative dimension: \(44\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 440q - 22q^{4} - 8q^{6} - 18q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 440q - 22q^{4} - 8q^{6} - 18q^{9} - 30q^{10} - 18q^{12} - 36q^{13} - 30q^{16} - 10q^{18} - 22q^{21} - 48q^{22} - 14q^{24} - 8q^{25} - 34q^{28} - 21q^{30} - 6q^{33} - 32q^{34} + 22q^{36} - 36q^{37} - 168q^{40} + q^{42} - 44q^{45} - 110q^{46} - 6q^{48} - 16q^{49} - 94q^{52} + 21q^{54} - 46q^{57} - 116q^{58} - 47q^{60} - 68q^{61} - 28q^{64} - 20q^{66} - 22q^{69} - 44q^{70} - 23q^{72} - 36q^{73} - 74q^{76} + 134q^{78} - 130q^{81} + 36q^{82} + 105q^{84} - 60q^{85} + 38q^{88} + 162q^{90} - 332q^{93} - 40q^{94} + 225q^{96} - 36q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −1.41343 + 0.0471244i −1.43832 + 0.965005i 1.99556 0.133214i 1.15864 + 3.94597i 1.98749 1.43174i −1.89500 + 1.64203i −2.81430 + 0.282328i 1.13753 2.77597i −1.82361 5.52274i
35.2 −1.41336 0.0490491i 1.04993 1.37755i 1.99519 + 0.138648i 0.512900 + 1.74678i −1.55150 + 1.89548i 3.04781 2.64094i −2.81312 0.293822i −0.795302 2.89266i −0.639236 2.49399i
35.3 −1.39810 + 0.212891i 1.33549 + 1.10293i 1.90935 0.595285i −0.168792 0.574851i −2.10195 1.25769i −2.73269 + 2.36789i −2.54273 + 1.23875i 0.567081 + 2.94592i 0.358368 + 0.767764i
35.4 −1.38733 + 0.274446i −1.69492 0.356694i 1.84936 0.761494i −0.803645 2.73696i 2.44931 + 0.0296852i 0.489256 0.423943i −2.35668 + 1.56399i 2.74554 + 1.20914i 1.86607 + 3.57651i
35.5 −1.34998 0.421356i −0.574135 + 1.63413i 1.64492 + 1.13765i −0.494860 1.68534i 1.46362 1.96413i 1.60533 1.39102i −1.74126 2.22890i −2.34074 1.87642i −0.0420733 + 2.48370i
35.6 −1.31697 0.515366i −0.275571 1.70999i 1.46880 + 1.35744i 0.241048 + 0.820935i −0.518352 + 2.39402i −2.26976 + 1.96676i −1.23478 2.54467i −2.84812 + 0.942447i 0.105630 1.20537i
35.7 −1.20977 + 0.732426i −0.914165 1.47116i 0.927105 1.77214i 0.605787 + 2.06312i 2.18345 + 1.11021i 0.738126 0.639590i 0.176372 + 2.82292i −1.32861 + 2.68976i −2.24395 2.05222i
35.8 −1.16880 + 0.796188i 1.29161 1.15401i 0.732168 1.86116i −0.605787 2.06312i −0.590813 + 2.37717i −0.738126 + 0.639590i 0.626082 + 2.75826i 0.336500 2.98107i 2.35068 + 1.92905i
35.9 −1.12871 0.852070i 1.60391 0.653824i 0.547953 + 1.92347i 0.250263 + 0.852318i −2.36744 0.628665i −1.73353 + 1.50211i 1.02046 2.63793i 2.14503 2.09735i 0.443761 1.17526i
35.10 −1.09972 0.889160i 0.954066 + 1.44560i 0.418787 + 1.95566i 0.982607 + 3.34645i 0.236161 2.43808i 0.453842 0.393256i 1.27835 2.52306i −1.17952 + 2.75839i 1.89494 4.55387i
35.11 −1.00830 0.991633i −1.72667 0.136433i 0.0333293 + 1.99972i −0.252117 0.858632i 1.60571 + 1.84979i −1.92220 + 1.66560i 1.94938 2.04937i 2.96277 + 0.471148i −0.597239 + 1.11576i
35.12 −0.932383 1.06333i −0.322706 1.70172i −0.261325 + 1.98285i −1.00656 3.42802i −1.50860 + 1.92980i 3.30235 2.86150i 2.35207 1.57091i −2.79172 + 1.09831i −2.70660 + 4.26652i
35.13 −0.825962 + 1.14795i 1.72676 + 0.135270i −0.635573 1.89632i 0.803645 + 2.73696i −1.58152 + 1.87050i −0.489256 + 0.423943i 2.70184 + 0.836687i 2.96340 + 0.467159i −3.80567 1.33808i
35.14 −0.774443 + 1.18332i −1.59213 + 0.682004i −0.800475 1.83282i 0.168792 + 0.574851i 0.425987 2.41216i 2.73269 2.36789i 2.78873 + 0.472203i 2.06974 2.17167i −0.810951 0.245456i
35.15 −0.630025 + 1.26612i 1.10819 + 1.33114i −1.20614 1.59538i −1.15864 3.94597i −2.38357 + 0.564449i 1.89500 1.64203i 2.77984 0.521989i −0.543851 + 2.95029i 5.72606 + 1.01908i
35.16 −0.579910 1.28985i 0.322706 + 1.70172i −1.32741 + 1.49599i −1.00656 3.42802i 2.00782 1.40309i −3.30235 + 2.86150i 2.69937 + 0.844617i −2.79172 + 1.09831i −3.83790 + 3.28624i
35.17 −0.542515 + 1.30602i −0.619297 1.61755i −1.41135 1.41707i −0.512900 1.74678i 2.44853 + 0.0687346i −3.04781 + 2.64094i 2.61639 1.07447i −2.23294 + 2.00349i 2.55957 + 0.277798i
35.18 −0.483159 1.32912i 1.72667 + 0.136433i −1.53312 + 1.28435i −0.252117 0.858632i −0.652920 2.36087i 1.92220 1.66560i 2.44779 + 1.41715i 2.96277 + 0.471148i −1.01941 + 0.749950i
35.19 −0.351967 1.36972i −0.954066 1.44560i −1.75224 + 0.964188i 0.982607 + 3.34645i −1.64426 + 1.81560i −0.453842 + 0.393256i 1.93739 + 2.06071i −1.17952 + 2.75839i 4.23784 2.52373i
35.20 −0.306189 1.38067i −1.60391 + 0.653824i −1.81250 + 0.845492i 0.250263 + 0.852318i 1.39381 + 2.01427i 1.73353 1.50211i 1.72231 + 2.24358i 2.14503 2.09735i 1.10014 0.606501i
See next 80 embeddings (of 440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.44
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
23.c even 11 1 inner
69.h odd 22 1 inner
92.g odd 22 1 inner
276.o even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 276.2.o.a 440
3.b odd 2 1 inner 276.2.o.a 440
4.b odd 2 1 inner 276.2.o.a 440
12.b even 2 1 inner 276.2.o.a 440
23.c even 11 1 inner 276.2.o.a 440
69.h odd 22 1 inner 276.2.o.a 440
92.g odd 22 1 inner 276.2.o.a 440
276.o even 22 1 inner 276.2.o.a 440
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.o.a 440 1.a even 1 1 trivial
276.2.o.a 440 3.b odd 2 1 inner
276.2.o.a 440 4.b odd 2 1 inner
276.2.o.a 440 12.b even 2 1 inner
276.2.o.a 440 23.c even 11 1 inner
276.2.o.a 440 69.h odd 22 1 inner
276.2.o.a 440 92.g odd 22 1 inner
276.2.o.a 440 276.o even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(276, [\chi])\).