↑

Properties

Label 296.2.q.a
Level $296$
Weight $2$
Character orbit 296.q
Analytic conductor $2.364$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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] = [296,2,Mod(85,296)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(296, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 5])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("296.85"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 296 = 2^{3} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 296.q (of order \(6\), degree \(2\), 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: \(2.36357189983\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 6 q^{2} - 2 q^{4} + 2 q^{7} + 30 q^{9} - 12 q^{12} - 6 q^{15} - 6 q^{16} - 12 q^{17} - 48 q^{18} + 18 q^{20} - 12 q^{22} - 6 q^{24} - 32 q^{25} - 16 q^{26} + 10 q^{28} - 14 q^{30} - 6 q^{32} + 4 q^{33}+ \cdots + 48 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1 −1.39280 − 0.245152i −0.0574215 − 0.0331523i 1.87980 + 0.682898i −0.0588225 + 0.101884i 0.0718494 + 0.0602516i 1.40284 − 2.42979i −2.45078 − 1.41198i −1.49780 − 2.59427i 0.106905 − 0.127483i
85.2 −1.38029 − 0.307877i −1.31634 − 0.759988i 1.81042 + 0.849922i 0.762054 − 1.31992i 1.58295 + 1.45428i −2.35964 + 4.08701i −2.23725 − 1.73053i −0.344836 − 0.597274i −1.45823 + 1.58725i
85.3 −1.36412 + 0.373059i 2.16163 + 1.24802i 1.72165 − 1.01780i 0.890777 − 1.54287i −3.41431 − 0.896032i −1.76150 + 3.05101i −1.96885 + 2.03068i 1.61510 + 2.79744i −0.639545 + 2.43698i
85.4 −1.34606 + 0.433734i −2.63548 − 1.52159i 1.62375 − 1.16766i 1.82516 − 3.16127i 4.20748 + 0.905059i 1.55620 − 2.69542i −1.67921 + 2.27602i 3.13050 + 5.42218i −1.08562 + 5.04689i
85.5 −1.33470 − 0.467534i 2.44339 + 1.41069i 1.56282 + 1.24803i 0.380208 − 0.658539i −2.60164 − 3.02522i 1.51623 − 2.62619i −1.50239 − 2.39642i 2.48012 + 4.29569i −0.815351 + 0.701189i
85.6 −1.26611 + 0.630052i 0.593819 + 0.342841i 1.20607 − 1.59543i 1.02589 − 1.77690i −0.967848 − 0.0599385i 0.355330 − 0.615450i −0.521814 + 2.77988i −1.26492 − 2.19090i −0.179356 + 2.89612i
85.7 −1.26355 − 0.635167i −2.54245 − 1.46788i 1.19312 + 1.60513i −2.01329 + 3.48712i 2.28016 + 3.46962i 0.477860 − 0.827678i −0.488046 − 2.78600i 2.80936 + 4.86595i 4.75880 − 3.12738i
85.8 −1.17870 + 0.781458i −0.593819 − 0.342841i 0.778648 − 1.84220i −1.02589 + 1.77690i 0.967848 − 0.0599385i 0.355330 − 0.615450i 0.521814 + 2.77988i −1.26492 − 2.19090i −0.179356 − 2.89612i
85.9 −1.07776 − 0.915657i 1.20571 + 0.696117i 0.323144 + 1.97372i −1.27097 + 2.20138i −0.662065 − 1.85427i −0.577919 + 1.00099i 1.45898 − 2.42309i −0.530841 − 0.919444i 3.38552 − 1.20880i
85.10 −1.04865 + 0.948854i 2.63548 + 1.52159i 0.199351 − 1.99004i −1.82516 + 3.16127i −4.20748 + 0.905059i 1.55620 − 2.69542i 1.67921 + 2.27602i 3.13050 + 5.42218i −1.08562 − 5.04689i
85.11 −1.00514 + 0.994834i −2.16163 − 1.24802i 0.0206105 − 1.99989i −0.890777 + 1.54287i 3.41431 − 0.896032i −1.76150 + 3.05101i 1.96885 + 2.03068i 1.61510 + 2.79744i −0.639545 − 2.43698i
85.12 −0.913372 − 1.07970i 1.03067 + 0.595057i −0.331504 + 1.97233i 1.89805 − 3.28751i −0.298901 − 1.65632i 0.213384 − 0.369592i 2.43232 − 1.44355i −0.791813 − 1.37146i −5.28315 + 0.953401i
85.13 −0.833130 − 1.14276i −1.81291 − 1.04668i −0.611788 + 1.90413i 1.03003 − 1.78406i 0.314285 + 2.94374i 1.21757 − 2.10889i 2.68566 − 0.887265i 0.691097 + 1.19702i −2.89690 + 0.309284i
85.14 −0.484093 + 1.32878i 0.0574215 + 0.0331523i −1.53131 − 1.28651i 0.0588225 − 0.101884i −0.0718494 + 0.0602516i 1.40284 − 2.42979i 2.45078 − 1.41198i −1.49780 − 2.59427i 0.106905 + 0.127483i
85.15 −0.423518 + 1.34931i 1.31634 + 0.759988i −1.64127 − 1.14291i −0.762054 + 1.31992i −1.58295 + 1.45428i −2.35964 + 4.08701i 2.23725 − 1.73053i −0.344836 − 0.597274i −1.45823 − 1.58725i
85.16 −0.410393 − 1.35336i −1.78951 − 1.03318i −1.66315 + 1.11082i −0.216072 + 0.374248i −0.663852 + 2.84586i −0.770459 + 1.33447i 2.18588 + 1.79497i 0.634902 + 1.09968i 0.595166 + 0.138834i
85.17 −0.288005 − 1.38458i 2.75173 + 1.58871i −1.83411 + 0.797529i −0.352717 + 0.610924i 1.40718 − 4.26753i −1.43310 + 2.48220i 1.63247 + 2.30977i 3.54800 + 6.14532i 0.947455 + 0.312415i
85.18 −0.269353 − 1.38833i 0.280448 + 0.161917i −1.85490 + 0.747900i −1.57735 + 2.73205i 0.149254 − 0.432966i 2.58572 − 4.47860i 1.53795 + 2.37375i −1.44757 − 2.50726i 4.21784 + 1.45399i
85.19 −0.262451 + 1.38965i −2.44339 − 1.41069i −1.86224 − 0.729429i −0.380208 + 0.658539i 2.60164 − 3.02522i 1.51623 − 2.62619i 1.50239 − 2.39642i 2.48012 + 4.29569i −0.815351 − 0.701189i
85.20 −0.0817045 + 1.41185i 2.54245 + 1.46788i −1.98665 − 0.230709i 2.01329 − 3.48712i −2.28016 + 3.46962i 0.477860 − 0.827678i 0.488046 − 2.78600i 2.80936 + 4.86595i 4.75880 + 3.12738i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
37.e even 6 1 inner
296.q even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 296.2.q.a ✓ 72
4.b odd 2 1 1184.2.y.a 72
8.b even 2 1 inner 296.2.q.a ✓ 72
8.d odd 2 1 1184.2.y.a 72
37.e even 6 1 inner 296.2.q.a ✓ 72
148.j odd 6 1 1184.2.y.a 72
296.n odd 6 1 1184.2.y.a 72
296.q even 6 1 inner 296.2.q.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
296.2.q.a ✓ 72 1.a even 1 1 trivial
296.2.q.a ✓ 72 8.b even 2 1 inner
296.2.q.a ✓ 72 37.e even 6 1 inner
296.2.q.a ✓ 72 296.q even 6 1 inner
1184.2.y.a 72 4.b odd 2 1
1184.2.y.a 72 8.d odd 2 1
1184.2.y.a 72 148.j odd 6 1
1184.2.y.a 72 296.n odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(296, [\chi])\).