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Properties

Label 304.2.m.a
Level $304$
Weight $2$
Character orbit 304.m
Analytic conductor $2.427$
Analytic rank $0$
Dimension $76$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,2,Mod(75,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.75");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 304.m (of order \(4\), degree \(2\), 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: \(2.42745222145\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 4 q^{4} - 4 q^{5} - 4 q^{6} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q - 4 q^{4} - 4 q^{5} - 4 q^{6} - 8 q^{7} - 4 q^{11} + 4 q^{16} - 8 q^{17} + 6 q^{19} - 32 q^{20} - 8 q^{23} - 12 q^{24} - 8 q^{26} - 40 q^{30} + 44 q^{36} - 12 q^{38} - 8 q^{39} + 12 q^{42} - 4 q^{43} - 76 q^{44} + 4 q^{45} + 44 q^{49} + 96 q^{54} - 8 q^{55} + 24 q^{58} + 28 q^{61} + 20 q^{62} - 64 q^{64} + 12 q^{66} - 80 q^{74} + 12 q^{76} - 32 q^{77} - 16 q^{80} - 52 q^{81} + 68 q^{82} + 36 q^{83} - 56 q^{85} - 120 q^{87} - 8 q^{92} - 16 q^{93} + 48 q^{96} + 76 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
75.1 −1.41289 − 0.0612206i −1.42780 + 1.42780i 1.99250 + 0.172996i 0.391081 + 0.391081i 2.10473 − 1.92991i −3.36082 −2.80459 − 0.366406i − 1.07723i −0.528611 − 0.576495i
75.2 −1.40550 + 0.156739i 1.02920 − 1.02920i 1.95087 − 0.440592i −2.13462 − 2.13462i −1.28522 + 1.60785i 4.26829 −2.67289 + 0.925029i 0.881504i 3.33478 + 2.66563i
75.3 −1.32973 − 0.481474i 2.33030 − 2.33030i 1.53637 + 1.28046i 1.33252 + 1.33252i −4.22065 + 1.97669i 2.39899 −1.42644 − 2.44239i − 7.86060i −1.13032 − 2.41347i
75.4 −1.32380 − 0.497543i 0.105438 − 0.105438i 1.50490 + 1.31730i 0.199404 + 0.199404i −0.192039 + 0.0871190i 0.290770 −1.33678 − 2.49259i 2.97777i −0.164759 − 0.363183i
75.5 −1.28847 + 0.582956i 0.623611 − 0.623611i 1.32033 − 1.50225i 2.23509 + 2.23509i −0.439968 + 1.16704i −0.609161 −0.825461 + 2.70529i 2.22222i −4.18282 − 1.57690i
75.6 −1.24208 + 0.676199i 2.19409 − 2.19409i 1.08551 − 1.67978i −1.29836 − 1.29836i −1.24159 + 4.20887i −4.71096 −0.212419 + 2.82044i − 6.62806i 2.49061 + 0.734713i
75.7 −1.22234 − 0.711259i −1.96187 + 1.96187i 0.988221 + 1.73880i −3.06749 − 3.06749i 3.79347 − 1.00267i 1.97168 0.0287966 − 2.82828i − 4.69788i 1.56773 + 5.93129i
75.8 −1.21730 + 0.719852i −1.08919 + 1.08919i 0.963625 − 1.75255i −0.770023 − 0.770023i 0.541810 − 2.10992i 1.47744 0.0885582 + 2.82704i 0.627346i 1.49165 + 0.383044i
75.9 −1.18010 − 0.779343i 1.07779 − 1.07779i 0.785249 + 1.83940i −1.37400 − 1.37400i −2.11187 + 0.431928i −4.00321 0.506854 − 2.78264i 0.676721i 0.550634 + 2.69227i
75.10 −0.927697 + 1.06742i −0.599104 + 0.599104i −0.278756 − 1.98048i −1.69205 − 1.69205i −0.0837063 − 1.19528i −1.26000 2.37260 + 1.53974i 2.28215i 3.37583 − 0.236412i
75.11 −0.869188 − 1.11558i −2.03444 + 2.03444i −0.489024 + 1.93929i 2.23466 + 2.23466i 4.03789 + 0.501263i −0.725567 2.58848 − 1.14007i − 5.27790i 0.550596 − 4.43528i
75.12 −0.726793 − 1.21317i 1.34292 − 1.34292i −0.943543 + 1.76344i 2.53450 + 2.53450i −2.60521 − 0.653159i −2.70706 2.82511 − 0.136983i − 0.606865i 1.23271 − 4.91682i
75.13 −0.657664 − 1.25199i 0.308215 − 0.308215i −1.13496 + 1.64678i 0.114298 + 0.114298i −0.588585 − 0.183180i 4.05189 2.80817 + 0.337930i 2.81001i 0.0679305 − 0.218270i
75.14 −0.643641 + 1.25926i 1.63122 − 1.63122i −1.17145 − 1.62102i 0.837183 + 0.837183i 1.00421 + 3.10405i 1.95985 2.79527 − 0.431807i − 2.32176i −1.59307 + 0.515383i
75.15 −0.514042 + 1.31748i −1.03801 + 1.03801i −1.47152 − 1.35448i 2.13321 + 2.13321i −0.833978 − 1.90114i 3.87204 2.54093 − 1.24244i 0.845075i −3.90703 + 1.71391i
75.16 −0.352269 − 1.36964i −0.101486 + 0.101486i −1.75181 + 0.964961i −2.62302 − 2.62302i 0.174749 + 0.103248i −2.14594 1.93876 + 2.05942i 2.97940i −2.66857 + 4.51659i
75.17 −0.204512 + 1.39935i −1.96469 + 1.96469i −1.91635 − 0.572367i −0.884107 − 0.884107i −2.34749 − 3.15109i −0.721104 1.19286 − 2.56458i − 4.72004i 1.41798 − 1.05636i
75.18 −0.117772 − 1.40930i −1.65943 + 1.65943i −1.97226 + 0.331952i −0.668279 − 0.668279i 2.53408 + 2.14321i 1.43332 0.700097 + 2.74041i − 2.50744i −0.863102 + 1.02051i
75.19 −0.0324855 + 1.41384i 0.549728 − 0.549728i −1.99789 − 0.0918587i 1.49999 + 1.49999i 0.759369 + 0.795085i −3.48044 0.194776 − 2.82171i 2.39560i −2.16948 + 2.07202i
75.20 0.0324855 − 1.41384i −0.549728 + 0.549728i −1.99789 − 0.0918587i 1.49999 + 1.49999i 0.759369 + 0.795085i −3.48044 −0.194776 + 2.82171i 2.39560i 2.16948 − 2.07202i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner
19.b odd 2 1 inner
304.m even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.2.m.a ✓ 76
4.b odd 2 1 1216.2.m.a 76
16.e even 4 1 1216.2.m.a 76
16.f odd 4 1 inner 304.2.m.a ✓ 76
19.b odd 2 1 inner 304.2.m.a ✓ 76
76.d even 2 1 1216.2.m.a 76
304.j odd 4 1 1216.2.m.a 76
304.m even 4 1 inner 304.2.m.a ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.m.a ✓ 76 1.a even 1 1 trivial
304.2.m.a ✓ 76 16.f odd 4 1 inner
304.2.m.a ✓ 76 19.b odd 2 1 inner
304.2.m.a ✓ 76 304.m even 4 1 inner
1216.2.m.a 76 4.b odd 2 1
1216.2.m.a 76 16.e even 4 1
1216.2.m.a 76 76.d even 2 1
1216.2.m.a 76 304.j odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(304, [\chi])\).