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Properties

Label 975.2.bu.i
Level $975$
Weight $2$
Character orbit 975.bu
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(7,975)]
 
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, 3, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bu (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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 28 q^{4} + 8 q^{11} - 16 q^{12} - 28 q^{16} + 20 q^{17} + 8 q^{18} + 24 q^{19} - 8 q^{21} + 28 q^{22} + 8 q^{23} - 8 q^{31} + 60 q^{32} - 12 q^{33} - 4 q^{34} + 20 q^{37} - 8 q^{39} - 4 q^{41} + 60 q^{42} + 12 q^{43} - 40 q^{44} - 8 q^{46} - 24 q^{47} - 16 q^{48} - 44 q^{49} - 8 q^{52} - 4 q^{53} + 72 q^{56} + 4 q^{58} + 64 q^{59} + 16 q^{61} - 108 q^{62} - 56 q^{64} - 16 q^{66} - 12 q^{68} - 8 q^{69} - 16 q^{71} + 12 q^{72} - 36 q^{74} + 112 q^{76} - 48 q^{77} + 8 q^{78} + 28 q^{81} + 52 q^{82} - 104 q^{83} + 32 q^{84} - 64 q^{86} - 36 q^{87} + 24 q^{88} + 12 q^{89} - 40 q^{91} + 64 q^{92} + 24 q^{94} + 108 q^{97} + 96 q^{98} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.21322 − 1.27780i 0.258819 + 0.965926i 2.26557 + 3.92408i 0 0.661440 − 2.46853i 2.30261 + 3.98824i − 6.46858i −0.866025 + 0.500000i 0
7.2 −2.13750 − 1.23409i −0.258819 − 0.965926i 2.04594 + 3.54367i 0 −0.638810 + 2.38407i 0.204703 + 0.354557i − 5.16311i −0.866025 + 0.500000i 0
7.3 −1.96880 − 1.13669i 0.258819 + 0.965926i 1.58411 + 2.74376i 0 0.588392 − 2.19591i −1.06779 − 1.84947i − 2.65581i −0.866025 + 0.500000i 0
7.4 −1.49213 − 0.861480i −0.258819 − 0.965926i 0.484294 + 0.838822i 0 −0.445935 + 1.66425i −0.243108 − 0.421075i 1.77708i −0.866025 + 0.500000i 0
7.5 −0.975669 − 0.563303i −0.258819 − 0.965926i −0.365380 − 0.632856i 0 −0.291587 + 1.08822i −1.23258 − 2.13490i 3.07649i −0.866025 + 0.500000i 0
7.6 −0.835096 − 0.482143i 0.258819 + 0.965926i −0.535077 − 0.926780i 0 0.249575 − 0.931428i 0.676930 + 1.17248i 2.96050i −0.866025 + 0.500000i 0
7.7 −0.579821 − 0.334760i 0.258819 + 0.965926i −0.775871 − 1.34385i 0 0.173285 − 0.646707i −0.0297909 − 0.0515993i 2.37796i −0.866025 + 0.500000i 0
7.8 −0.138788 − 0.0801292i −0.258819 − 0.965926i −0.987159 − 1.70981i 0 −0.0414779 + 0.154798i 2.22106 + 3.84699i 0.636918i −0.866025 + 0.500000i 0
7.9 0.545471 + 0.314928i −0.258819 − 0.965926i −0.801641 − 1.38848i 0 0.163019 − 0.608393i −1.84227 − 3.19090i − 2.26955i −0.866025 + 0.500000i 0
7.10 1.08442 + 0.626089i 0.258819 + 0.965926i −0.216026 − 0.374168i 0 −0.324087 + 1.20951i 1.37566 + 2.38272i − 3.04536i −0.866025 + 0.500000i 0
7.11 1.47770 + 0.853152i 0.258819 + 0.965926i 0.455736 + 0.789358i 0 −0.441624 + 1.64816i −2.46357 − 4.26702i − 1.85736i −0.866025 + 0.500000i 0
7.12 1.59037 + 0.918199i −0.258819 − 0.965926i 0.686179 + 1.18850i 0 0.475295 − 1.77382i 1.31456 + 2.27688i − 1.15260i −0.866025 + 0.500000i 0
7.13 1.74222 + 1.00587i −0.258819 − 0.965926i 1.02355 + 1.77285i 0 0.520677 − 1.94319i 0.0952740 + 0.165019i 0.0947706i −0.866025 + 0.500000i 0
7.14 2.16879 + 1.25215i 0.258819 + 0.965926i 2.13577 + 3.69927i 0 −0.648162 + 2.41897i −1.31169 − 2.27192i 5.68865i −0.866025 + 0.500000i 0
232.1 −2.30953 + 1.33341i −0.965926 − 0.258819i 2.55595 − 4.42704i 0 2.57595 − 0.690223i 2.11365 − 3.66094i 8.29887i 0.866025 + 0.500000i 0
232.2 −2.11806 + 1.22286i 0.965926 + 0.258819i 1.99078 − 3.44813i 0 −2.36239 + 0.633000i −2.22495 + 3.85373i 4.84635i 0.866025 + 0.500000i 0
232.3 −1.48323 + 0.856342i −0.965926 − 0.258819i 0.466643 − 0.808249i 0 1.65433 − 0.443275i −0.341866 + 0.592129i − 1.82694i 0.866025 + 0.500000i 0
232.4 −0.757319 + 0.437238i 0.965926 + 0.258819i −0.617645 + 1.06979i 0 −0.844680 + 0.226331i 1.09235 − 1.89201i − 2.82919i 0.866025 + 0.500000i 0
232.5 −0.722106 + 0.416908i 0.965926 + 0.258819i −0.652375 + 1.12995i 0 −0.805405 + 0.215808i −2.36958 + 4.10424i − 2.75555i 0.866025 + 0.500000i 0
232.6 −0.483002 + 0.278861i −0.965926 − 0.258819i −0.844473 + 1.46267i 0 0.538719 − 0.144349i −1.04015 + 1.80160i − 2.05741i 0.866025 + 0.500000i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.14
Significant digits:
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Inner twists

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

Hecke kernels

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

\( T_{2}^{56} - 42 T_{2}^{54} + 987 T_{2}^{52} - 12 T_{2}^{51} - 15946 T_{2}^{50} + 432 T_{2}^{49} + \cdots + 85264 \) Copy content Toggle raw display
\( T_{7}^{56} + 120 T_{7}^{54} - 64 T_{7}^{53} + 8134 T_{7}^{52} - 7148 T_{7}^{51} + 379120 T_{7}^{50} + \cdots + 239172411040000 \) Copy content Toggle raw display