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Properties

Label 315.2.bo.b
Level $315$
Weight $2$
Character orbit 315.bo
Analytic conductor $2.515$
Analytic rank $0$
Dimension $84$
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] = [315,2,Mod(4,315)]
 
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([2, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.bo (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: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(42\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 44 q^{4} - 6 q^{5} + 6 q^{6} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 44 q^{4} - 6 q^{5} + 6 q^{6} - 14 q^{9} + 6 q^{10} - 24 q^{11} - 10 q^{14} + 4 q^{15} - 36 q^{16} + 8 q^{19} - 10 q^{20} - 14 q^{21} + 18 q^{24} + 10 q^{25} - 40 q^{26} - 10 q^{29} - 28 q^{30} - 6 q^{31} - 12 q^{34} + 4 q^{35} - 6 q^{36} + 4 q^{39} - 8 q^{40} - 30 q^{41} - 4 q^{44} - 30 q^{45} + 4 q^{46} + 8 q^{49} + 42 q^{50} + 14 q^{51} + 18 q^{54} - 54 q^{55} + 48 q^{56} + 42 q^{59} + 66 q^{60} + 22 q^{61} - 28 q^{64} + 8 q^{65} - 38 q^{66} - 32 q^{69} - 26 q^{70} - 4 q^{71} - 108 q^{74} + 6 q^{75} + 24 q^{76} + 24 q^{79} - 9 q^{80} - 106 q^{81} - 64 q^{84} + q^{85} - 92 q^{86} + 46 q^{89} + 17 q^{90} - 44 q^{91} - 8 q^{94} - 25 q^{95} + 54 q^{96} - 102 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.35408 − 1.35913i −0.595272 + 1.62655i 2.69445 + 4.66693i 0.317086 + 2.21347i 3.61200 − 3.01996i 1.81310 − 1.92683i − 9.21190i −2.29130 − 1.93648i 2.26194 − 5.64164i
4.2 −2.26777 − 1.30930i 1.21970 − 1.22977i 2.42851 + 4.20630i 2.23375 − 0.101685i −4.37613 + 1.19187i −0.957411 + 2.46645i − 7.48135i −0.0246457 − 2.99990i −5.19877 − 2.69405i
4.3 −2.26367 − 1.30693i 1.21834 + 1.23112i 2.41613 + 4.18486i −1.53331 − 1.62756i −1.14893 − 4.37913i −2.64512 + 0.0579738i − 7.40313i −0.0313079 + 2.99984i 1.34379 + 5.68819i
4.4 −2.10397 − 1.21472i −0.601157 − 1.62438i 1.95111 + 3.37943i −2.23603 − 0.0133001i −0.708362 + 4.14788i 1.65650 + 2.06301i − 4.62136i −2.27722 + 1.95301i 4.68837 + 2.74414i
4.5 −2.05276 − 1.18516i −1.39727 + 1.02354i 1.80922 + 3.13366i −0.218973 − 2.22532i 4.08132 − 0.445097i 1.63234 + 2.08218i − 3.83621i 0.904722 − 2.86033i −2.18786 + 4.82757i
4.6 −1.90125 − 1.09768i −1.70434 − 0.308593i 1.40982 + 2.44189i −1.91379 + 1.15647i 2.90163 + 2.45754i −2.51786 − 0.812625i − 1.79943i 2.80954 + 1.05190i 4.90802 − 0.0979947i
4.7 −1.84157 − 1.06323i −0.0962482 − 1.72937i 1.26093 + 2.18399i 1.02419 + 1.98772i −1.66148 + 3.28710i 0.441480 − 2.60866i − 1.10970i −2.98147 + 0.332898i 0.227292 − 4.74948i
4.8 −1.75373 − 1.01252i 1.36024 − 1.07227i 1.05039 + 1.81932i −0.846932 − 2.06947i −3.47118 + 0.503210i 1.36749 − 2.26495i − 0.204070i 0.700480 − 2.91708i −0.610083 + 4.48683i
4.9 −1.56714 − 0.904786i 1.72657 + 0.137634i 0.637277 + 1.10380i −0.0315838 + 2.23584i −2.58125 − 1.77787i −2.37354 − 1.16889i 1.31275i 2.96211 + 0.475270i 2.07246 − 3.47530i
4.10 −1.56370 − 0.902800i 0.880417 + 1.49160i 0.630096 + 1.09136i 1.94086 + 1.11043i −0.0300880 − 3.12725i 0.480299 + 2.60179i 1.33580i −1.44973 + 2.62646i −2.03243 − 3.48858i
4.11 −1.42338 − 0.821790i −1.53987 − 0.792964i 0.350678 + 0.607392i 2.06509 − 0.857547i 1.54018 + 2.39414i 2.61685 − 0.390007i 2.13443i 1.74242 + 2.44213i −3.64414 − 0.476457i
4.12 −1.40440 − 0.810833i −0.306778 + 1.70467i 0.314901 + 0.545425i 1.75547 − 1.38504i 1.81304 − 2.14529i −1.09563 − 2.40824i 2.22200i −2.81177 − 1.04591i −3.58842 + 0.521762i
4.13 −1.17913 − 0.680771i 0.973105 + 1.43285i −0.0731024 − 0.126617i −2.23088 − 0.152304i −0.171973 − 2.35198i 2.52908 − 0.777026i 2.92215i −1.10613 + 2.78863i 2.52681 + 1.69830i
4.14 −0.941848 − 0.543776i −1.56380 + 0.744658i −0.408615 − 0.707741i −1.28643 + 1.82896i 1.87779 + 0.149005i 2.64561 − 0.0270009i 3.06389i 1.89097 − 2.32900i 2.20617 − 1.02307i
4.15 −0.931541 − 0.537825i 1.13575 − 1.30770i −0.421488 − 0.730038i −2.11877 + 0.714718i −1.76131 + 0.607338i −1.08220 + 2.41430i 3.05805i −0.420144 − 2.97043i 2.35811 + 0.473738i
4.16 −0.755931 − 0.436437i −1.14385 + 1.30062i −0.619046 − 1.07222i −1.55427 − 1.60756i 1.43231 − 0.483960i −2.14141 + 1.55382i 2.82644i −0.383218 − 2.97542i 0.473324 + 1.89354i
4.17 −0.627169 − 0.362096i −1.09783 − 1.33969i −0.737773 − 1.27786i 0.851637 + 2.06754i 0.203428 + 1.23773i −0.722021 + 2.54533i 2.51696i −0.589540 + 2.94150i 0.214528 − 1.60507i
4.18 −0.582215 − 0.336142i −1.56590 − 0.740246i −0.774017 − 1.34064i −1.25057 − 1.85366i 0.662861 + 0.957346i −0.680768 − 2.55667i 2.38529i 1.90407 + 2.31830i 0.105006 + 1.49960i
4.19 −0.248464 − 0.143451i 0.701026 − 1.58384i −0.958844 − 1.66077i 2.21083 − 0.335020i −0.401384 + 0.292966i −2.25543 − 1.38312i 1.12399i −2.01713 − 2.22063i −0.597370 − 0.233905i
4.20 −0.223223 − 0.128878i 1.72666 − 0.136523i −0.966781 − 1.67451i 1.91223 + 1.15903i −0.403026 − 0.192053i 2.07085 − 1.64669i 1.01390i 2.96272 − 0.471459i −0.277481 − 0.505168i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
63.g even 3 1 inner
315.bo even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bo.b yes 84
3.b odd 2 1 945.2.bo.b 84
5.b even 2 1 inner 315.2.bo.b yes 84
7.c even 3 1 315.2.r.b ✓ 84
9.c even 3 1 315.2.r.b ✓ 84
9.d odd 6 1 945.2.r.b 84
15.d odd 2 1 945.2.bo.b 84
21.h odd 6 1 945.2.r.b 84
35.j even 6 1 315.2.r.b ✓ 84
45.h odd 6 1 945.2.r.b 84
45.j even 6 1 315.2.r.b ✓ 84
63.g even 3 1 inner 315.2.bo.b yes 84
63.n odd 6 1 945.2.bo.b 84
105.o odd 6 1 945.2.r.b 84
315.v odd 6 1 945.2.bo.b 84
315.bo even 6 1 inner 315.2.bo.b yes 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.r.b ✓ 84 7.c even 3 1
315.2.r.b ✓ 84 9.c even 3 1
315.2.r.b ✓ 84 35.j even 6 1
315.2.r.b ✓ 84 45.j even 6 1
315.2.bo.b yes 84 1.a even 1 1 trivial
315.2.bo.b yes 84 5.b even 2 1 inner
315.2.bo.b yes 84 63.g even 3 1 inner
315.2.bo.b yes 84 315.bo even 6 1 inner
945.2.r.b 84 9.d odd 6 1
945.2.r.b 84 21.h odd 6 1
945.2.r.b 84 45.h odd 6 1
945.2.r.b 84 105.o odd 6 1
945.2.bo.b 84 3.b odd 2 1
945.2.bo.b 84 15.d odd 2 1
945.2.bo.b 84 63.n odd 6 1
945.2.bo.b 84 315.v odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{84} - 64 T_{2}^{82} + 2207 T_{2}^{80} - 52654 T_{2}^{78} + 962504 T_{2}^{76} - 14220262 T_{2}^{74} + \cdots + 5764801 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display