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Properties

Label 315.3.ca.b
Level $315$
Weight $3$
Character orbit 315.ca
Analytic conductor $8.583$
Analytic rank $0$
Dimension $64$
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] = [315,3,Mod(37,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.ca (of order \(12\), degree \(4\), 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: \(8.58312832735\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 4 q^{5} - 4 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 4 q^{5} - 4 q^{7} - 24 q^{8} - 16 q^{10} - 16 q^{11} + 80 q^{16} - 56 q^{17} - 96 q^{22} - 72 q^{23} - 4 q^{25} + 288 q^{26} - 380 q^{28} - 136 q^{31} + 48 q^{32} - 76 q^{35} - 28 q^{37} + 68 q^{38} + 164 q^{40} - 128 q^{41} + 344 q^{43} + 240 q^{46} - 412 q^{47} + 72 q^{50} + 388 q^{52} + 40 q^{53} - 8 q^{55} + 864 q^{56} + 56 q^{58} - 216 q^{61} + 912 q^{62} - 20 q^{65} - 368 q^{67} + 492 q^{68} + 416 q^{70} - 784 q^{71} - 316 q^{73} - 32 q^{76} - 844 q^{77} - 908 q^{80} + 556 q^{82} - 1408 q^{83} - 536 q^{85} - 1024 q^{86} + 372 q^{88} - 1064 q^{91} + 1704 q^{92} - 260 q^{95} + 352 q^{97} - 272 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −3.52852 + 0.945463i 0 8.09242 − 4.67216i −3.83301 − 3.21062i 0 −6.80203 + 1.65300i −13.8047 + 13.8047i 0 16.5603 + 7.70474i
37.2 −3.41166 + 0.914152i 0 7.33966 − 4.23756i 4.98672 − 0.364163i 0 −6.35750 − 2.92953i −11.1766 + 11.1766i 0 −16.6801 + 5.80102i
37.3 −2.88660 + 0.773463i 0 4.27013 − 2.46536i −0.160937 + 4.99741i 0 5.19830 − 4.68804i −1.96674 + 1.96674i 0 −3.40075 − 14.5500i
37.4 −2.56059 + 0.686107i 0 2.62176 − 1.51367i −1.32982 + 4.81992i 0 0.158915 + 6.99820i 1.82323 − 1.82323i 0 0.0981374 − 13.2542i
37.5 −2.13226 + 0.571337i 0 0.756006 − 0.436480i −2.78563 − 4.15214i 0 2.73668 − 6.44287i 4.88107 − 4.88107i 0 8.31195 + 7.26192i
37.6 −0.984292 + 0.263740i 0 −2.56483 + 1.48081i 4.04958 + 2.93273i 0 −5.81621 + 3.89508i 5.01620 − 5.01620i 0 −4.75945 − 1.81862i
37.7 −0.901161 + 0.241465i 0 −2.71032 + 1.56480i −2.44472 − 4.36158i 0 5.41162 + 4.44009i 4.70337 − 4.70337i 0 3.25625 + 3.34017i
37.8 −0.232416 + 0.0622758i 0 −3.41396 + 1.97105i 2.65242 − 4.23847i 0 −4.06422 + 5.69931i 1.35127 − 1.35127i 0 −0.352512 + 1.15027i
37.9 0.435396 − 0.116664i 0 −3.28814 + 1.89841i −3.62365 + 3.44517i 0 6.87993 + 1.29096i −2.48509 + 2.48509i 0 −1.17579 + 1.92276i
37.10 0.445275 − 0.119311i 0 −3.28007 + 1.89375i −4.59550 + 1.97013i 0 −0.171680 − 6.99789i −2.53844 + 2.53844i 0 −1.81120 + 1.42554i
37.11 1.84280 − 0.493776i 0 −0.312015 + 0.180142i 1.82066 + 4.65674i 0 −6.63712 − 2.22456i −5.88211 + 5.88211i 0 5.65448 + 7.68243i
37.12 1.91023 − 0.511845i 0 −0.0771041 + 0.0445161i 4.99333 + 0.258093i 0 6.99587 + 0.240410i −5.71805 + 5.71805i 0 9.67053 − 2.06280i
37.13 2.38023 − 0.637781i 0 1.79464 − 1.03614i −2.91424 + 4.06291i 0 −4.25379 + 5.55925i −3.35897 + 3.35897i 0 −4.34532 + 11.5293i
37.14 2.87375 − 0.770020i 0 4.20143 − 2.42570i −3.78688 − 3.26489i 0 −1.65502 − 6.80154i 1.79111 − 1.79111i 0 −13.3966 − 6.46653i
37.15 3.18498 − 0.853412i 0 5.95167 − 3.43620i 4.91224 − 0.932701i 0 5.00478 + 4.89410i 6.69718 − 6.69718i 0 14.8494 − 7.16279i
37.16 3.56483 − 0.955194i 0 8.33154 − 4.81021i 1.05941 − 4.88648i 0 −6.28878 + 3.07429i 14.6673 − 14.6673i 0 −0.890898 − 18.4314i
163.1 −0.955194 − 3.56483i 0 −8.33154 + 4.81021i 3.70210 − 3.36072i 0 3.07429 + 6.28878i 14.6673 + 14.6673i 0 −15.5166 − 9.98725i
163.2 −0.853412 − 3.18498i 0 −5.95167 + 3.43620i −1.64838 − 4.72047i 0 4.89410 − 5.00478i 6.69718 + 6.69718i 0 −13.6279 + 9.27855i
163.3 −0.770020 − 2.87375i 0 −4.20143 + 2.42570i 4.72092 + 1.64709i 0 −6.80154 + 1.65502i 1.79111 + 1.79111i 0 1.09812 − 14.8351i
163.4 −0.637781 − 2.38023i 0 −1.79464 + 1.03614i −2.06146 + 4.55526i 0 5.55925 + 4.25379i −3.35897 − 3.35897i 0 12.1573 + 2.00150i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.c even 3 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.3.ca.b 64
3.b odd 2 1 105.3.v.a ✓ 64
5.c odd 4 1 inner 315.3.ca.b 64
7.c even 3 1 inner 315.3.ca.b 64
15.e even 4 1 105.3.v.a ✓ 64
21.h odd 6 1 105.3.v.a ✓ 64
35.l odd 12 1 inner 315.3.ca.b 64
105.x even 12 1 105.3.v.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.v.a ✓ 64 3.b odd 2 1
105.3.v.a ✓ 64 15.e even 4 1
105.3.v.a ✓ 64 21.h odd 6 1
105.3.v.a ✓ 64 105.x even 12 1
315.3.ca.b 64 1.a even 1 1 trivial
315.3.ca.b 64 5.c odd 4 1 inner
315.3.ca.b 64 7.c even 3 1 inner
315.3.ca.b 64 35.l odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} + 8 T_{2}^{61} - 468 T_{2}^{60} - 48 T_{2}^{59} + 32 T_{2}^{58} - 3248 T_{2}^{57} + \cdots + 169835630410000 \) acting on \(S_{3}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display