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Properties

Label 975.2.bl.i
Level $975$
Weight $2$
Character orbit 975.bl
Analytic conductor $7.785$
Analytic rank $0$
Dimension $56$
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] = [975,2,Mod(193,975)]
 
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([0, 9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bl (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: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 195)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{2} - 28 q^{4} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{2} - 28 q^{4} + 24 q^{8} + 8 q^{11} + 16 q^{12} + 12 q^{13} - 28 q^{16} - 8 q^{17} - 24 q^{19} - 8 q^{21} + 28 q^{22} - 8 q^{23} - 8 q^{31} - 8 q^{32} + 4 q^{33} + 4 q^{34} + 24 q^{37} + 8 q^{39} - 4 q^{41} - 60 q^{42} - 12 q^{43} + 40 q^{44} - 8 q^{46} - 16 q^{48} + 44 q^{49} + 24 q^{52} - 4 q^{53} + 72 q^{56} - 120 q^{58} - 64 q^{59} + 16 q^{61} - 36 q^{62} + 56 q^{64} - 16 q^{66} - 64 q^{67} - 48 q^{68} + 8 q^{69} - 16 q^{71} + 80 q^{73} + 36 q^{74} + 112 q^{76} + 48 q^{77} + 40 q^{78} + 28 q^{81} + 32 q^{82} - 32 q^{84} - 64 q^{86} + 60 q^{87} + 120 q^{88} - 12 q^{89} - 40 q^{91} + 64 q^{92} - 24 q^{94} + 12 q^{97} - 108 q^{98} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1 −1.34748 − 2.33390i 0.258819 − 0.965926i −2.63140 + 4.55772i 0 −2.60313 + 0.697507i −1.16997 − 0.675482i 8.79312 −0.866025 − 0.500000i 0
193.2 −1.28821 − 2.23125i −0.258819 + 0.965926i −2.31898 + 4.01660i 0 2.48864 − 0.666828i −0.841701 − 0.485956i 6.79653 −0.866025 − 0.500000i 0
193.3 −1.04920 − 1.81727i 0.258819 − 0.965926i −1.20166 + 2.08133i 0 −2.02691 + 0.543108i 3.44089 + 1.98660i 0.846322 −0.866025 − 0.500000i 0
193.4 −0.771447 − 1.33619i −0.258819 + 0.965926i −0.190261 + 0.329542i 0 1.49032 − 0.399330i −1.21357 − 0.700657i −2.49868 −0.866025 − 0.500000i 0
193.5 −0.665414 − 1.15253i 0.258819 − 0.965926i 0.114447 − 0.198228i 0 −1.28548 + 0.344444i −1.63545 − 0.944230i −2.96628 −0.866025 − 0.500000i 0
193.6 −0.642122 − 1.11219i −0.258819 + 0.965926i 0.175358 − 0.303729i 0 1.24049 − 0.332387i 2.60612 + 1.50465i −3.01889 −0.866025 − 0.500000i 0
193.7 0.0934870 + 0.161924i 0.258819 − 0.965926i 0.982520 − 1.70178i 0 0.180603 − 0.0483924i −2.71431 − 1.56711i 0.741359 −0.866025 − 0.500000i 0
193.8 0.124774 + 0.216115i −0.258819 + 0.965926i 0.968863 − 1.67812i 0 −0.241045 + 0.0645879i −3.27074 − 1.88836i 0.982653 −0.866025 − 0.500000i 0
193.9 0.278861 + 0.483002i 0.258819 − 0.965926i 0.844473 − 1.46267i 0 0.538719 − 0.144349i 1.80160 + 1.04015i 2.05741 −0.866025 − 0.500000i 0
193.10 0.416908 + 0.722106i −0.258819 + 0.965926i 0.652375 − 1.12995i 0 −0.805405 + 0.215808i 4.10424 + 2.36958i 2.75555 −0.866025 − 0.500000i 0
193.11 0.437238 + 0.757319i −0.258819 + 0.965926i 0.617645 − 1.06979i 0 −0.844680 + 0.226331i −1.89201 − 1.09235i 2.82919 −0.866025 − 0.500000i 0
193.12 0.856342 + 1.48323i 0.258819 − 0.965926i −0.466643 + 0.808249i 0 1.65433 − 0.443275i 0.592129 + 0.341866i 1.82694 −0.866025 − 0.500000i 0
193.13 1.22286 + 2.11806i −0.258819 + 0.965926i −1.99078 + 3.44813i 0 −2.36239 + 0.633000i 3.85373 + 2.22495i −4.84635 −0.866025 − 0.500000i 0
193.14 1.33341 + 2.30953i 0.258819 − 0.965926i −2.55595 + 4.42704i 0 2.57595 − 0.690223i −3.66094 − 2.11365i −8.29887 −0.866025 − 0.500000i 0
457.1 −1.27780 − 2.21322i −0.965926 − 0.258819i −2.26557 + 3.92408i 0 0.661440 + 2.46853i 3.98824 + 2.30261i 6.46858 0.866025 + 0.500000i 0
457.2 −1.23409 − 2.13750i 0.965926 + 0.258819i −2.04594 + 3.54367i 0 −0.638810 − 2.38407i 0.354557 + 0.204703i 5.16311 0.866025 + 0.500000i 0
457.3 −1.13669 − 1.96880i −0.965926 − 0.258819i −1.58411 + 2.74376i 0 0.588392 + 2.19591i −1.84947 − 1.06779i 2.65581 0.866025 + 0.500000i 0
457.4 −0.861480 − 1.49213i 0.965926 + 0.258819i −0.484294 + 0.838822i 0 −0.445935 − 1.66425i −0.421075 − 0.243108i −1.77708 0.866025 + 0.500000i 0
457.5 −0.563303 − 0.975669i 0.965926 + 0.258819i 0.365380 − 0.632856i 0 −0.291587 − 1.08822i −2.13490 − 1.23258i −3.07649 0.866025 + 0.500000i 0
457.6 −0.482143 − 0.835096i −0.965926 − 0.258819i 0.535077 − 0.926780i 0 0.249575 + 0.931428i 1.17248 + 0.676930i −2.96050 0.866025 + 0.500000i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.o 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.bl.i 56
5.b even 2 1 195.2.bd.a ✓ 56
5.c odd 4 1 195.2.bm.a yes 56
5.c odd 4 1 975.2.bu.i 56
13.f odd 12 1 975.2.bu.i 56
15.d odd 2 1 585.2.cf.b 56
15.e even 4 1 585.2.dp.c 56
65.o even 12 1 inner 975.2.bl.i 56
65.s odd 12 1 195.2.bm.a yes 56
65.t even 12 1 195.2.bd.a ✓ 56
195.bc odd 12 1 585.2.cf.b 56
195.bh even 12 1 585.2.dp.c 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bd.a ✓ 56 5.b even 2 1
195.2.bd.a ✓ 56 65.t even 12 1
195.2.bm.a yes 56 5.c odd 4 1
195.2.bm.a yes 56 65.s odd 12 1
585.2.cf.b 56 15.d odd 2 1
585.2.cf.b 56 195.bc odd 12 1
585.2.dp.c 56 15.e even 4 1
585.2.dp.c 56 195.bh even 12 1
975.2.bl.i 56 1.a even 1 1 trivial
975.2.bl.i 56 65.o even 12 1 inner
975.2.bu.i 56 5.c odd 4 1
975.2.bu.i 56 13.f odd 12 1

Hecke kernels

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

\( T_{2}^{56} + 4 T_{2}^{55} + 50 T_{2}^{54} + 160 T_{2}^{53} + 1259 T_{2}^{52} + 3508 T_{2}^{51} + \cdots + 85264 \) Copy content Toggle raw display
\( T_{7}^{56} - 120 T_{7}^{54} + 8134 T_{7}^{52} + 1092 T_{7}^{51} - 377072 T_{7}^{50} + \cdots + 239172411040000 \) Copy content Toggle raw display