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Properties

Label 975.2.bo.f
Level $975$
Weight $2$
Character orbit 975.bo
Analytic conductor $7.785$
Analytic rank $0$
Dimension $72$
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] = [975,2,Mod(176,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.176");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bo (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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{6} + 6 q^{9} + 36 q^{16} - 20 q^{18} + 24 q^{19} + 24 q^{21} - 20 q^{24} + 12 q^{27} - 80 q^{28} + 44 q^{31} - 40 q^{33} + 4 q^{34} + 66 q^{36} + 40 q^{37} + 26 q^{39} - 28 q^{42} - 48 q^{43} - 8 q^{46} - 4 q^{48} - 52 q^{52} - 40 q^{54} + 36 q^{57} + 48 q^{58} - 8 q^{61} - 34 q^{63} - 108 q^{66} - 88 q^{67} - 36 q^{69} - 144 q^{72} - 20 q^{76} - 30 q^{78} - 64 q^{79} - 18 q^{81} - 120 q^{82} - 40 q^{84} - 44 q^{87} - 72 q^{88} - 24 q^{91} + 120 q^{93} - 52 q^{94} - 182 q^{96} + 56 q^{97} - 46 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
176.1 −0.686196 − 2.56092i 0.584339 + 1.63051i −4.35539 + 2.51459i 0 3.77462 − 2.61529i 0.243015 + 0.0651156i 5.67887 + 5.67887i −2.31710 + 1.90554i 0
176.2 −0.662470 − 2.47237i 0.343204 − 1.69771i −3.94171 + 2.27575i 0 −4.42473 + 0.276153i 4.04534 + 1.08395i 4.61796 + 4.61796i −2.76442 − 1.16532i 0
176.3 −0.564413 − 2.10642i 1.61534 − 0.625045i −2.38638 + 1.37778i 0 −2.22832 − 3.04979i −4.09108 − 1.09620i 1.16507 + 1.16507i 2.21864 − 2.01932i 0
176.4 −0.503070 − 1.87748i −1.26951 + 1.17828i −1.53982 + 0.889013i 0 2.85085 + 1.79073i −0.796539 − 0.213432i −0.305084 − 0.305084i 0.223313 − 2.99168i 0
176.5 −0.498694 − 1.86115i −1.65193 − 0.520712i −1.48314 + 0.856293i 0 −0.145318 + 3.33416i 2.03910 + 0.546374i −0.391591 − 0.391591i 2.45772 + 1.72035i 0
176.6 −0.346888 − 1.29460i 1.71111 + 0.268541i 0.176384 − 0.101835i 0 −0.245908 − 2.30836i 3.87371 + 1.03796i −2.08845 − 2.08845i 2.85577 + 0.919005i 0
176.7 −0.202452 − 0.755562i 1.21060 − 1.23873i 1.20216 − 0.694070i 0 −1.18103 − 0.663897i −1.09187 − 0.292567i −1.87401 − 1.87401i −0.0689070 − 2.99921i 0
176.8 −0.202003 − 0.753885i 0.301278 + 1.70565i 1.20451 − 0.695427i 0 1.22500 − 0.571675i −2.54822 − 0.682794i −1.87135 − 1.87135i −2.81846 + 1.02775i 0
176.9 −0.102168 − 0.381297i −1.47401 − 0.909565i 1.59710 − 0.922087i 0 −0.196218 + 0.654962i −1.67345 − 0.448399i −1.07302 − 1.07302i 1.34538 + 2.68141i 0
176.10 0.102168 + 0.381297i 1.52471 + 0.821744i 1.59710 − 0.922087i 0 −0.157552 + 0.665323i −1.67345 − 0.448399i 1.07302 + 1.07302i 1.64947 + 2.50584i 0
176.11 0.202003 + 0.753885i −1.62777 + 0.591909i 1.20451 − 0.695427i 0 −0.775045 − 1.10759i −2.54822 − 0.682794i 1.87135 + 1.87135i 2.29929 − 1.92699i 0
176.12 0.202452 + 0.755562i 0.467473 − 1.66777i 1.20216 − 0.694070i 0 1.35475 + 0.0155606i −1.09187 − 0.292567i 1.87401 + 1.87401i −2.56294 − 1.55928i 0
176.13 0.346888 + 1.29460i −1.08812 − 1.34759i 0.176384 − 0.101835i 0 1.36714 − 1.87614i 3.87371 + 1.03796i 2.08845 + 2.08845i −0.632004 + 2.93267i 0
176.14 0.498694 + 1.86115i 1.27691 + 1.17025i −1.48314 + 0.856293i 0 −1.54123 + 2.96013i 2.03910 + 0.546374i 0.391591 + 0.391591i 0.261011 + 2.98862i 0
176.15 0.503070 + 1.87748i −0.385665 + 1.68857i −1.53982 + 0.889013i 0 −3.36428 + 0.125388i −0.796539 − 0.213432i 0.305084 + 0.305084i −2.70252 − 1.30244i 0
176.16 0.564413 + 2.10642i −0.266364 − 1.71145i −2.38638 + 1.37778i 0 3.45468 − 1.52704i −4.09108 − 1.09620i −1.16507 − 1.16507i −2.85810 + 0.911737i 0
176.17 0.662470 + 2.47237i 1.29866 − 1.14608i −3.94171 + 2.27575i 0 3.69385 + 2.45152i 4.04534 + 1.08395i −4.61796 − 4.61796i 0.373015 − 2.97672i 0
176.18 0.686196 + 2.56092i −1.70423 + 0.309200i −4.35539 + 2.51459i 0 −1.96127 − 4.15222i 0.243015 + 0.0651156i −5.67887 − 5.67887i 2.80879 − 1.05390i 0
401.1 −2.65007 − 0.710085i 1.72667 − 0.136461i 4.78662 + 2.76356i 0 −4.67270 − 0.864448i −0.00723386 − 0.0269971i −6.84257 − 6.84257i 2.96276 − 0.471247i 0
401.2 −2.42068 − 0.648620i −1.55204 + 0.768879i 3.70694 + 2.14020i 0 4.25570 − 0.854527i 0.525704 + 1.96195i −4.04102 − 4.04102i 1.81765 − 2.38666i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 176.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bo.f ✓ 72
3.b odd 2 1 inner 975.2.bo.f ✓ 72
5.b even 2 1 975.2.bo.g yes 72
5.c odd 4 1 975.2.bp.g 72
5.c odd 4 1 975.2.bp.j 72
13.f odd 12 1 inner 975.2.bo.f ✓ 72
15.d odd 2 1 975.2.bo.g yes 72
15.e even 4 1 975.2.bp.g 72
15.e even 4 1 975.2.bp.j 72
39.k even 12 1 inner 975.2.bo.f ✓ 72
65.o even 12 1 975.2.bp.j 72
65.s odd 12 1 975.2.bo.g yes 72
65.t even 12 1 975.2.bp.g 72
195.bc odd 12 1 975.2.bp.g 72
195.bh even 12 1 975.2.bo.g yes 72
195.bn odd 12 1 975.2.bp.j 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bo.f ✓ 72 1.a even 1 1 trivial
975.2.bo.f ✓ 72 3.b odd 2 1 inner
975.2.bo.f ✓ 72 13.f odd 12 1 inner
975.2.bo.f ✓ 72 39.k even 12 1 inner
975.2.bo.g yes 72 5.b even 2 1
975.2.bo.g yes 72 15.d odd 2 1
975.2.bo.g yes 72 65.s odd 12 1
975.2.bo.g yes 72 195.bh even 12 1
975.2.bp.g 72 5.c odd 4 1
975.2.bp.g 72 15.e even 4 1
975.2.bp.g 72 65.t even 12 1
975.2.bp.g 72 195.bc odd 12 1
975.2.bp.j 72 5.c odd 4 1
975.2.bp.j 72 15.e even 4 1
975.2.bp.j 72 65.o even 12 1
975.2.bp.j 72 195.bn odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{72} - 147 T_{2}^{68} + 13522 T_{2}^{64} + 1734 T_{2}^{62} - 768741 T_{2}^{60} + \cdots + 26454371904 \) Copy content Toggle raw display
\( T_{7}^{36} - 64 T_{7}^{33} - 508 T_{7}^{32} + 1232 T_{7}^{31} + 2048 T_{7}^{30} + 5264 T_{7}^{29} + \cdots + 16451136 \) Copy content Toggle raw display