Properties

Label 315.4.bj.b
Level $315$
Weight $4$
Character orbit 315.bj
Analytic conductor $18.586$
Analytic rank $0$
Dimension $32$
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] = [315,4,Mod(26,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.26");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.bj (of order \(6\), 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: \(18.5856016518\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} + 80 q^{5} - 44 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} + 80 q^{5} - 44 q^{7} + 72 q^{11} - 108 q^{14} - 196 q^{16} + 396 q^{19} + 640 q^{20} - 480 q^{22} + 36 q^{23} - 400 q^{25} - 120 q^{26} - 316 q^{28} - 300 q^{31} + 1620 q^{32} - 20 q^{35} + 980 q^{37} - 888 q^{38} - 1296 q^{41} + 8 q^{43} + 4068 q^{44} + 540 q^{46} - 120 q^{47} - 1396 q^{49} - 4452 q^{52} + 576 q^{53} - 4260 q^{56} + 828 q^{58} - 96 q^{59} + 336 q^{61} - 4152 q^{62} - 5120 q^{64} + 1020 q^{65} + 700 q^{67} + 60 q^{68} + 540 q^{70} - 756 q^{73} + 5796 q^{74} - 2952 q^{77} + 916 q^{79} + 980 q^{80} + 5832 q^{82} - 2352 q^{83} + 3276 q^{86} - 1212 q^{88} + 1608 q^{89} - 960 q^{91} + 5112 q^{94} + 1980 q^{95} - 5184 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −4.79668 2.76937i 0 11.3388 + 19.6393i 2.50000 4.33013i 0 −3.51757 18.1831i 81.2950i 0 −23.9834 + 13.8468i
26.2 −4.28337 2.47301i 0 8.23152 + 14.2574i 2.50000 4.33013i 0 −12.4677 + 13.6952i 41.8583i 0 −21.4169 + 12.3650i
26.3 −3.10158 1.79070i 0 2.41320 + 4.17978i 2.50000 4.33013i 0 18.5132 0.511080i 11.3659i 0 −15.5079 + 8.95349i
26.4 −2.82648 1.63187i 0 1.32598 + 2.29666i 2.50000 4.33013i 0 −18.2872 + 2.92918i 17.4546i 0 −14.1324 + 8.15933i
26.5 −2.71573 1.56793i 0 0.916790 + 1.58793i 2.50000 4.33013i 0 15.0868 10.7418i 19.3370i 0 −13.5786 + 7.83964i
26.6 −1.57393 0.908710i 0 −2.34849 4.06771i 2.50000 4.33013i 0 −16.6185 8.17469i 23.0758i 0 −7.86966 + 4.54355i
26.7 −1.45507 0.840086i 0 −2.58851 4.48343i 2.50000 4.33013i 0 2.62579 + 18.3332i 22.1397i 0 −7.27536 + 4.20043i
26.8 −0.452924 0.261496i 0 −3.86324 6.69133i 2.50000 4.33013i 0 1.23679 + 18.4789i 8.22481i 0 −2.26462 + 1.30748i
26.9 −0.0946322 0.0546359i 0 −3.99403 6.91786i 2.50000 4.33013i 0 2.69161 18.3236i 1.74705i 0 −0.473161 + 0.273180i
26.10 1.23944 + 0.715594i 0 −2.97585 5.15433i 2.50000 4.33013i 0 16.7856 7.82585i 19.9675i 0 6.19722 3.57797i
26.11 2.02851 + 1.17116i 0 −1.25677 2.17679i 2.50000 4.33013i 0 −7.09670 + 17.1066i 24.6261i 0 10.1425 5.85580i
26.12 2.89621 + 1.67213i 0 1.59202 + 2.75747i 2.50000 4.33013i 0 −11.0741 14.8447i 16.1058i 0 14.4811 8.36064i
26.13 2.91070 + 1.68049i 0 1.64813 + 2.85464i 2.50000 4.33013i 0 −14.9406 + 10.9443i 15.8092i 0 14.5535 8.40247i
26.14 3.50335 + 2.02266i 0 4.18231 + 7.24398i 2.50000 4.33013i 0 −5.54534 17.6706i 1.47503i 0 17.5168 10.1133i
26.15 4.24064 + 2.44833i 0 7.98866 + 13.8368i 2.50000 4.33013i 0 16.8273 + 7.73571i 39.0623i 0 21.2032 12.2417i
26.16 4.48155 + 2.58742i 0 9.38951 + 16.2631i 2.50000 4.33013i 0 −6.21952 + 17.4447i 55.7798i 0 22.4077 12.9371i
206.1 −4.79668 + 2.76937i 0 11.3388 19.6393i 2.50000 + 4.33013i 0 −3.51757 + 18.1831i 81.2950i 0 −23.9834 13.8468i
206.2 −4.28337 + 2.47301i 0 8.23152 14.2574i 2.50000 + 4.33013i 0 −12.4677 13.6952i 41.8583i 0 −21.4169 12.3650i
206.3 −3.10158 + 1.79070i 0 2.41320 4.17978i 2.50000 + 4.33013i 0 18.5132 + 0.511080i 11.3659i 0 −15.5079 8.95349i
206.4 −2.82648 + 1.63187i 0 1.32598 2.29666i 2.50000 + 4.33013i 0 −18.2872 2.92918i 17.4546i 0 −14.1324 8.15933i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.4.bj.b yes 32
3.b odd 2 1 315.4.bj.a 32
7.d odd 6 1 315.4.bj.a 32
21.g even 6 1 inner 315.4.bj.b yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.bj.a 32 3.b odd 2 1
315.4.bj.a 32 7.d odd 6 1
315.4.bj.b yes 32 1.a even 1 1 trivial
315.4.bj.b yes 32 21.g even 6 1 inner