Properties

Label 304.5.r.d
Level $304$
Weight $5$
Character orbit 304.r
Analytic conductor $31.424$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,5,Mod(65,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 304.r (of order \(6\), degree \(2\), not 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: \(31.4244687775\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 12 q^{3} + 32 q^{7} + 624 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 12 q^{3} + 32 q^{7} + 624 q^{9} - 24 q^{11} + 264 q^{13} - 624 q^{15} + 216 q^{17} + 652 q^{19} - 216 q^{21} + 1296 q^{23} - 3044 q^{25} + 288 q^{29} - 6660 q^{33} - 360 q^{35} - 3184 q^{39} + 1260 q^{41} - 632 q^{43} + 256 q^{45} - 1248 q^{47} + 16696 q^{49} + 8064 q^{51} - 3672 q^{53} + 3408 q^{55} - 4552 q^{57} - 12492 q^{59} + 2720 q^{61} - 12472 q^{63} - 16260 q^{67} + 504 q^{71} + 9220 q^{73} - 14688 q^{77} + 28944 q^{79} - 1660 q^{81} + 39192 q^{83} - 18632 q^{85} + 34400 q^{87} + 3456 q^{89} - 54432 q^{91} - 17208 q^{93} - 44520 q^{95} - 30540 q^{97} + 10096 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0 −14.4186 8.32460i 0 10.1950 17.6583i 0 −80.1603 0 98.0979 + 169.911i 0
65.2 0 −13.5253 7.80881i 0 14.4872 25.0925i 0 26.1842 0 81.4552 + 141.085i 0
65.3 0 −12.4606 7.19410i 0 −20.6420 + 35.7530i 0 −1.47481 0 63.0102 + 109.137i 0
65.4 0 −10.6119 6.12677i 0 3.27153 5.66646i 0 87.4966 0 34.5747 + 59.8851i 0
65.5 0 −9.48048 5.47356i 0 −15.3284 + 26.5496i 0 −33.8970 0 19.4197 + 33.6359i 0
65.6 0 −7.37289 4.25674i 0 7.14367 12.3732i 0 35.7108 0 −4.26032 7.37909i 0
65.7 0 −5.77348 3.33332i 0 −1.87999 + 3.25623i 0 40.7434 0 −18.2780 31.6584i 0
65.8 0 −5.09915 2.94399i 0 0.570982 0.988970i 0 −80.7369 0 −23.1658 40.1243i 0
65.9 0 −2.52583 1.45829i 0 22.1213 38.3151i 0 11.1904 0 −36.2468 62.7813i 0
65.10 0 −1.93870 1.11931i 0 −15.6195 + 27.0538i 0 6.74912 0 −37.9943 65.8080i 0
65.11 0 −0.620551 0.358275i 0 16.9339 29.3305i 0 −38.8265 0 −40.2433 69.7034i 0
65.12 0 0.962404 + 0.555644i 0 −13.4131 + 23.2322i 0 62.8585 0 −39.8825 69.0785i 0
65.13 0 5.28375 + 3.05057i 0 1.28449 2.22480i 0 6.16432 0 −21.8880 37.9111i 0
65.14 0 5.59346 + 3.22938i 0 −4.68830 + 8.12037i 0 −33.5286 0 −19.6422 34.0212i 0
65.15 0 7.81020 + 4.50922i 0 15.3964 26.6673i 0 70.8202 0 0.166103 + 0.287698i 0
65.16 0 9.62676 + 5.55801i 0 −23.6855 + 41.0244i 0 −82.2157 0 21.2830 + 36.8632i 0
65.17 0 10.7629 + 6.21398i 0 20.5828 35.6505i 0 −44.5677 0 36.7272 + 63.6134i 0
65.18 0 11.3222 + 6.53690i 0 −19.8361 + 34.3572i 0 76.8642 0 44.9621 + 77.8766i 0
65.19 0 12.1979 + 7.04247i 0 0.573898 0.994020i 0 40.4604 0 58.6927 + 101.659i 0
65.20 0 14.2678 + 8.23749i 0 2.53183 4.38526i 0 −53.8347 0 95.2125 + 164.913i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.5.r.d 40
4.b odd 2 1 152.5.n.a 40
19.d odd 6 1 inner 304.5.r.d 40
76.f even 6 1 152.5.n.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.5.n.a 40 4.b odd 2 1
152.5.n.a 40 76.f even 6 1
304.5.r.d 40 1.a even 1 1 trivial
304.5.r.d 40 19.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 12 T_{3}^{39} - 1050 T_{3}^{38} - 13176 T_{3}^{37} + 653455 T_{3}^{36} + 8261832 T_{3}^{35} - 273020834 T_{3}^{34} - 3478133916 T_{3}^{33} + 85744483199 T_{3}^{32} + \cdots + 16\!\cdots\!21 \) acting on \(S_{5}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display