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Properties

Label 315.2.bl.j
Level $315$
Weight $2$
Character orbit 315.bl
Analytic conductor $2.515$
Analytic rank $0$
Dimension $24$
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] = [315,2,Mod(41,315)]
 
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([5, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.bl (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: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{2} + 5 q^{3} + 18 q^{4} - 12 q^{5} + q^{6} + 9 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{2} + 5 q^{3} + 18 q^{4} - 12 q^{5} + q^{6} + 9 q^{7} + q^{9} + 9 q^{11} - 18 q^{12} + 3 q^{13} + 18 q^{14} + 2 q^{15} - 18 q^{16} + 18 q^{17} + 2 q^{18} + 18 q^{20} + 8 q^{21} - 9 q^{22} + 9 q^{23} - 7 q^{24} - 12 q^{25} - 18 q^{26} - 4 q^{27} - 9 q^{28} + 9 q^{29} - 5 q^{30} - 42 q^{31} + 18 q^{32} + 13 q^{33} - 39 q^{34} - 9 q^{35} - 21 q^{36} - 12 q^{38} - 21 q^{39} - 6 q^{40} - 33 q^{41} - 65 q^{42} + 18 q^{43} + q^{45} - 30 q^{46} - 17 q^{48} + 9 q^{49} - 6 q^{50} - 12 q^{51} + 129 q^{52} + 52 q^{54} - 9 q^{56} + 6 q^{57} - 15 q^{58} + 12 q^{59} + 15 q^{60} - 15 q^{61} + 12 q^{62} + 46 q^{63} - 60 q^{64} - 3 q^{65} + 29 q^{66} - 15 q^{67} + 9 q^{68} + 61 q^{69} - 9 q^{70} + 61 q^{72} - 18 q^{74} - 7 q^{75} + 54 q^{76} - 45 q^{77} - 66 q^{78} + 21 q^{79} + 36 q^{80} + q^{81} - 30 q^{83} + 15 q^{84} - 9 q^{85} - 102 q^{86} + 10 q^{87} - 9 q^{88} + 102 q^{89} - 37 q^{90} + 42 q^{91} - 3 q^{92} - 6 q^{93} - 156 q^{94} - 18 q^{95} - 42 q^{96} - 45 q^{97} + 3 q^{98} + 23 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −2.29184 + 1.32320i −1.70895 + 0.281919i 2.50170 − 4.33307i −0.500000 + 0.866025i 3.54362 − 2.90740i −1.33893 − 2.28195i 7.94816i 2.84104 − 0.963574i − 2.64639i
41.2 −2.21303 + 1.27769i 0.920618 − 1.46713i 2.26501 − 3.92311i −0.500000 + 0.866025i −0.162815 + 4.42307i 1.04917 + 2.42884i 6.46517i −1.30493 − 2.70133i − 2.55539i
41.3 −1.41561 + 0.817305i 1.73161 − 0.0392468i 0.335974 − 0.581925i −0.500000 + 0.866025i −2.41921 + 1.47081i 2.11432 − 1.59049i − 2.17085i 2.99692 − 0.135920i − 1.63461i
41.4 −1.13192 + 0.653515i 0.957531 − 1.44331i −0.145837 + 0.252598i −0.500000 + 0.866025i −0.140627 + 2.25947i −2.56224 − 0.659483i − 2.99529i −1.16627 − 2.76402i − 1.30703i
41.5 −0.334847 + 0.193324i 0.454500 + 1.67136i −0.925251 + 1.60258i −0.500000 + 0.866025i −0.475302 − 0.471783i 1.06999 + 2.41974i − 1.48879i −2.58686 + 1.51926i − 0.386648i
41.6 0.552767 − 0.319140i −0.662870 − 1.60019i −0.796299 + 1.37923i −0.500000 + 0.866025i −0.877097 − 0.672983i −1.77287 + 1.96391i 2.29308i −2.12121 + 2.12143i 0.638280i
41.7 0.963349 − 0.556190i 1.73166 + 0.0368078i −0.381305 + 0.660440i −0.500000 + 0.866025i 1.68867 − 0.927673i 2.64533 − 0.0472671i 3.07307i 2.99729 + 0.127477i 1.11238i
41.8 1.02342 − 0.590871i −1.63294 − 0.577510i −0.301744 + 0.522636i −0.500000 + 0.866025i −2.01241 + 0.373820i 2.45705 − 0.981280i 3.07665i 2.33296 + 1.88607i 1.18174i
41.9 1.58089 − 0.912729i −0.791176 + 1.54079i 0.666147 − 1.15380i −0.500000 + 0.866025i 0.155560 + 3.15796i 0.317742 + 2.62660i 1.21887i −1.74808 − 2.43808i 1.82546i
41.10 1.94805 − 1.12471i 0.913713 + 1.47144i 1.52994 − 2.64993i −0.500000 + 0.866025i 3.43490 + 1.83878i 0.315547 − 2.62687i − 2.38412i −1.33026 + 2.68894i 2.24942i
41.11 1.99878 − 1.15400i 1.43832 − 0.965005i 1.66342 − 2.88113i −0.500000 + 0.866025i 1.76127 − 3.58865i −2.42508 + 1.05782i − 3.06234i 1.13753 − 2.77597i 2.30799i
41.12 2.32000 − 1.33945i −0.852012 − 1.50800i 2.58825 − 4.48298i −0.500000 + 0.866025i −3.99656 − 2.35733i 2.62997 + 0.288507i − 8.50953i −1.54815 + 2.56968i 2.67890i
146.1 −2.29184 − 1.32320i −1.70895 − 0.281919i 2.50170 + 4.33307i −0.500000 − 0.866025i 3.54362 + 2.90740i −1.33893 + 2.28195i − 7.94816i 2.84104 + 0.963574i 2.64639i
146.2 −2.21303 − 1.27769i 0.920618 + 1.46713i 2.26501 + 3.92311i −0.500000 − 0.866025i −0.162815 − 4.42307i 1.04917 − 2.42884i − 6.46517i −1.30493 + 2.70133i 2.55539i
146.3 −1.41561 − 0.817305i 1.73161 + 0.0392468i 0.335974 + 0.581925i −0.500000 − 0.866025i −2.41921 − 1.47081i 2.11432 + 1.59049i 2.17085i 2.99692 + 0.135920i 1.63461i
146.4 −1.13192 − 0.653515i 0.957531 + 1.44331i −0.145837 − 0.252598i −0.500000 − 0.866025i −0.140627 − 2.25947i −2.56224 + 0.659483i 2.99529i −1.16627 + 2.76402i 1.30703i
146.5 −0.334847 − 0.193324i 0.454500 − 1.67136i −0.925251 − 1.60258i −0.500000 − 0.866025i −0.475302 + 0.471783i 1.06999 − 2.41974i 1.48879i −2.58686 − 1.51926i 0.386648i
146.6 0.552767 + 0.319140i −0.662870 + 1.60019i −0.796299 − 1.37923i −0.500000 − 0.866025i −0.877097 + 0.672983i −1.77287 − 1.96391i − 2.29308i −2.12121 − 2.12143i − 0.638280i
146.7 0.963349 + 0.556190i 1.73166 − 0.0368078i −0.381305 − 0.660440i −0.500000 − 0.866025i 1.68867 + 0.927673i 2.64533 + 0.0472671i − 3.07307i 2.99729 − 0.127477i − 1.11238i
146.8 1.02342 + 0.590871i −1.63294 + 0.577510i −0.301744 − 0.522636i −0.500000 − 0.866025i −2.01241 − 0.373820i 2.45705 + 0.981280i − 3.07665i 2.33296 − 1.88607i − 1.18174i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.o 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.bl.j yes 24
3.b odd 2 1 945.2.bl.j 24
7.b odd 2 1 315.2.bl.i ✓ 24
9.c even 3 1 945.2.bl.i 24
9.d odd 6 1 315.2.bl.i ✓ 24
21.c even 2 1 945.2.bl.i 24
63.l odd 6 1 945.2.bl.j 24
63.o even 6 1 inner 315.2.bl.j yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bl.i ✓ 24 7.b odd 2 1
315.2.bl.i ✓ 24 9.d odd 6 1
315.2.bl.j yes 24 1.a even 1 1 trivial
315.2.bl.j yes 24 63.o even 6 1 inner
945.2.bl.i 24 9.c even 3 1
945.2.bl.i 24 21.c even 2 1
945.2.bl.j 24 3.b odd 2 1
945.2.bl.j 24 63.l odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{24} - 6 T_{2}^{23} - 3 T_{2}^{22} + 90 T_{2}^{21} - 48 T_{2}^{20} - 954 T_{2}^{19} + 1571 T_{2}^{18} + \cdots + 14161 \) Copy content Toggle raw display
\( T_{11}^{24} - 9 T_{11}^{23} - 36 T_{11}^{22} + 567 T_{11}^{21} + 1074 T_{11}^{20} - 21777 T_{11}^{19} + \cdots + 207475216 \) Copy content Toggle raw display
\( T_{13}^{24} - 3 T_{13}^{23} - 69 T_{13}^{22} + 216 T_{13}^{21} + 3150 T_{13}^{20} - 11916 T_{13}^{19} + \cdots + 10732176 \) Copy content Toggle raw display
\( T_{17}^{12} - 9 T_{17}^{11} - 57 T_{17}^{10} + 681 T_{17}^{9} + 174 T_{17}^{8} - 15678 T_{17}^{7} + \cdots + 410004 \) Copy content Toggle raw display