Properties

Label 285.2.v.a
Level $285$
Weight $2$
Character orbit 285.v
Analytic conductor $2.276$
Analytic rank $0$
Dimension $144$
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [285,2,Mod(68,285)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("285.68"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(285, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.27573645761\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(36\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 2 q^{3} - 4 q^{6} - 16 q^{7} - 4 q^{10} - 28 q^{12} + 4 q^{13} - 8 q^{15} + 36 q^{16} - 60 q^{18} - 4 q^{21} + 12 q^{22} - 12 q^{25} - 20 q^{27} + 28 q^{28} - 28 q^{30} - 64 q^{31} - 8 q^{33} - 20 q^{36}+ \cdots + 4 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 −2.62964 + 0.704610i −1.18652 − 1.26182i 4.68649 − 2.70575i 1.19478 + 1.89011i 4.00920 + 2.48209i 0.272212 − 0.272212i −6.56723 + 6.56723i −0.184358 + 2.99433i −4.47363 − 4.12845i
68.2 −2.48752 + 0.666529i 1.68237 + 0.411869i 4.01145 − 2.31601i 1.61992 − 1.54138i −4.45945 + 0.0968144i −3.42757 + 3.42757i −4.79290 + 4.79290i 2.66073 + 1.38583i −3.00222 + 4.91394i
68.3 −2.43603 + 0.652733i −1.59584 + 0.673263i 3.77615 − 2.18016i −1.16898 − 1.90617i 3.44807 − 2.68175i 0.385325 − 0.385325i −4.20915 + 4.20915i 2.09343 − 2.14884i 4.09189 + 3.88047i
68.4 −2.25121 + 0.603210i −0.104855 + 1.72887i 2.97204 − 1.71591i −1.10578 + 1.94352i −0.806824 − 3.95531i −1.20844 + 1.20844i −2.35963 + 2.35963i −2.97801 − 0.362561i 1.31699 − 5.04228i
68.5 −1.98480 + 0.531825i 0.630634 − 1.61316i 1.92453 − 1.11113i 2.00780 − 0.984246i −0.393760 + 3.53719i 2.07924 − 2.07924i −0.322933 + 0.322933i −2.20460 − 2.03463i −3.46163 + 3.02133i
68.6 −1.90409 + 0.510198i 1.01087 + 1.40646i 1.63319 − 0.942922i −1.48283 − 1.67368i −2.64236 − 2.16228i 1.11398 − 1.11398i 0.159120 − 0.159120i −0.956271 + 2.84351i 3.67734 + 2.43030i
68.7 −1.85455 + 0.496924i −0.549511 − 1.64257i 1.46036 − 0.843139i −2.20866 + 0.349002i 1.83533 + 2.77316i −0.952461 + 0.952461i 0.425916 − 0.425916i −2.39608 + 1.80522i 3.92264 − 1.74478i
68.8 −1.65531 + 0.443540i 1.65064 + 0.524783i 0.811283 − 0.468394i 1.58928 + 1.57295i −2.96508 − 0.136557i 2.04416 − 2.04416i 1.28837 − 1.28837i 2.44921 + 1.73245i −3.32843 − 1.89882i
68.9 −1.61023 + 0.431460i −1.51727 − 0.835403i 0.674637 − 0.389502i 1.22777 − 1.86884i 2.80360 + 0.690552i −1.45719 + 1.45719i 1.43928 − 1.43928i 1.60420 + 2.53506i −1.17066 + 3.53901i
68.10 −1.51172 + 0.405064i −1.64008 + 0.556903i 0.389164 − 0.224684i 0.718854 + 2.11737i 2.25376 − 1.50622i 2.61601 − 2.61601i 1.71601 − 1.71601i 2.37972 − 1.82673i −1.94437 − 2.90968i
68.11 −1.17498 + 0.314835i 1.73083 + 0.0649881i −0.450597 + 0.260152i −2.20070 + 0.396155i −2.05415 + 0.468566i −0.245890 + 0.245890i 2.16782 − 2.16782i 2.99155 + 0.224967i 2.46105 − 1.15833i
68.12 −1.01730 + 0.272584i −0.859232 + 1.50390i −0.771457 + 0.445401i 1.26772 − 1.84198i 0.464156 − 1.76413i 1.17470 − 1.17470i 2.15282 − 2.15282i −1.52344 − 2.58440i −0.787559 + 2.21940i
68.13 −0.943124 + 0.252709i 0.327283 + 1.70085i −0.906430 + 0.523327i 2.07916 + 0.822857i −0.738489 − 1.52140i −2.17155 + 2.17155i 2.10346 − 2.10346i −2.78577 + 1.11332i −2.16885 − 0.250633i
68.14 −0.863769 + 0.231446i −1.72219 + 0.184575i −1.03952 + 0.600168i −1.45426 + 1.69857i 1.44485 − 0.558024i −2.61961 + 2.61961i 2.02365 − 2.02365i 2.93186 − 0.635745i 0.863014 − 1.80375i
68.15 −0.649148 + 0.173939i 1.42932 − 0.978290i −1.34091 + 0.774176i −0.275535 − 2.21903i −0.757676 + 0.883669i −1.17228 + 1.17228i 1.68621 − 1.68621i 1.08590 − 2.79657i 0.564838 + 1.39255i
68.16 −0.510023 + 0.136660i 0.0963563 − 1.72937i −1.49060 + 0.860600i 1.62543 + 1.53557i 0.187192 + 0.895186i −2.68311 + 2.68311i 1.38936 − 1.38936i −2.98143 − 0.333271i −1.03886 − 0.561043i
68.17 −0.337876 + 0.0905336i 0.786398 − 1.54324i −1.62609 + 0.938822i −1.34607 + 1.78552i −0.125990 + 0.592618i 2.93818 − 2.93818i 0.959106 − 0.959106i −1.76316 − 2.42720i 0.293156 − 0.725150i
68.18 −0.0971046 + 0.0260191i −0.704314 + 1.58239i −1.72330 + 0.994947i −2.22847 − 0.184215i 0.0272198 − 0.171982i 1.31431 − 1.31431i 0.283623 − 0.283623i −2.00788 − 2.22899i 0.221187 − 0.0400945i
68.19 0.0971046 − 0.0260191i −1.40115 − 1.01823i −1.72330 + 0.994947i 2.22847 + 0.184215i −0.162551 − 0.0624181i 1.31431 − 1.31431i −0.283623 + 0.283623i 0.926420 + 2.85337i 0.221187 − 0.0400945i
68.20 0.337876 − 0.0905336i 1.45266 + 0.943283i −1.62609 + 0.938822i 1.34607 − 1.78552i 0.576217 + 0.187198i 2.93818 − 2.93818i −0.959106 + 0.959106i 1.22044 + 2.74054i 0.293156 − 0.725150i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 68.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner
95.m odd 12 1 inner
285.v even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.2.v.a ✓ 144
3.b odd 2 1 inner 285.2.v.a ✓ 144
5.c odd 4 1 inner 285.2.v.a ✓ 144
15.e even 4 1 inner 285.2.v.a ✓ 144
19.c even 3 1 inner 285.2.v.a ✓ 144
57.h odd 6 1 inner 285.2.v.a ✓ 144
95.m odd 12 1 inner 285.2.v.a ✓ 144
285.v even 12 1 inner 285.2.v.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.v.a ✓ 144 1.a even 1 1 trivial
285.2.v.a ✓ 144 3.b odd 2 1 inner
285.2.v.a ✓ 144 5.c odd 4 1 inner
285.2.v.a ✓ 144 15.e even 4 1 inner
285.2.v.a ✓ 144 19.c even 3 1 inner
285.2.v.a ✓ 144 57.h odd 6 1 inner
285.2.v.a ✓ 144 95.m odd 12 1 inner
285.2.v.a ✓ 144 285.v even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(285, [\chi])\).