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Properties

Label 4675.2.a.bt
Level 46754675
Weight 22
Character orbit 4675.a
Self dual yes
Analytic conductor 37.33037.330
Analytic rank 11
Dimension 2222
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4675,2,Mod(1,4675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4675, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4675.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 4675=52⋅11⋅17 4675 = 5^{2} \cdot 11 \cdot 17
Weight: k k == 2 2
Character orbit: [χ][\chi] == 4675.a (trivial)

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: yes
Analytic conductor: 37.330062944937.3300629449
Analytic rank: 11
Dimension: 2222
Twist minimal: no (minimal twist has level 935)
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

The algebraic qq-expansion of this newform has not been computed, but we have computed the trace expansion.

Tr⁡(f)(q)=\operatorname{Tr}(f)(q) = 22q−9q2−4q3+25q4−2q6−16q7−27q8+20q9+22q11−12q12−20q13+8q14+31q16−22q17−24q18+6q19−9q22−22q23+q24+⋯+20q99+O(q100) 22 q - 9 q^{2} - 4 q^{3} + 25 q^{4} - 2 q^{6} - 16 q^{7} - 27 q^{8} + 20 q^{9} + 22 q^{11} - 12 q^{12} - 20 q^{13} + 8 q^{14} + 31 q^{16} - 22 q^{17} - 24 q^{18} + 6 q^{19} - 9 q^{22} - 22 q^{23} + q^{24}+ \cdots + 20 q^{99}+O(q^{100}) Copy content Toggle raw display

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
1.1 −2.77068 0.753069 5.67669 0 −2.08652 2.41960 −10.1870 −2.43289 0
1.2 −2.76245 −1.22272 5.63113 0 3.37771 −5.08336 −10.0308 −1.50495 0
1.3 −2.59711 3.33544 4.74498 0 −8.66252 −2.91886 −7.12903 8.12518 0
1.4 −2.52105 −2.76985 4.35570 0 6.98294 2.60133 −5.93886 4.67207 0
1.5 −2.18930 −1.82824 2.79305 0 4.00257 −2.62629 −1.73622 0.342452 0
1.6 −1.97325 1.62502 1.89373 0 −3.20657 −2.37978 0.209701 −0.359323 0
1.7 −1.82925 −2.88143 1.34616 0 5.27086 −1.51037 1.19604 5.30266 0
1.8 −1.70614 1.08909 0.910926 0 −1.85815 2.74439 1.85812 −1.81388 0
1.9 −1.19964 −0.567918 −0.560863 0 0.681297 1.91514 3.07211 −2.67747 0
1.10 −0.785778 0.0417035 −1.38255 0 −0.0327697 −1.34686 2.65793 −2.99826 0
1.11 −0.594738 −2.84214 −1.64629 0 1.69033 −3.13953 2.16859 5.07777 0
1.12 −0.576411 2.17667 −1.66775 0 −1.25466 3.08854 2.11413 1.73789 0
1.13 −0.306294 2.62915 −1.90618 0 −0.805293 −2.16273 1.19644 3.91245 0
1.14 0.137127 −0.548266 −1.98120 0 −0.0751820 −4.85019 −0.545929 −2.69940 0
1.15 0.843676 0.498989 −1.28821 0 0.420986 1.65405 −2.77419 −2.75101 0
1.16 0.890316 2.31907 −1.20734 0 2.06470 −0.253528 −2.85554 2.37807 0
1.17 0.977188 −2.29468 −1.04510 0 −2.24233 1.08001 −2.97564 2.26555 0
1.18 1.66800 −3.11445 0.782241 0 −5.19492 −3.03771 −2.03123 6.69981 0
1.19 1.80850 −1.99036 1.27067 0 −3.59956 4.37746 −1.31899 0.961531 0
1.20 1.85769 2.05778 1.45101 0 3.82271 −4.43429 −1.01986 1.23446 0
See all 22 embeddings
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
55 −1 -1
1111 −1 -1
1717 +1 +1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4675.2.a.bt 22
5.b even 2 1 4675.2.a.bu 22
5.c odd 4 2 935.2.b.b ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
935.2.b.b ✓ 44 5.c odd 4 2
4675.2.a.bt 22 1.a even 1 1 trivial
4675.2.a.bu 22 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(4675))S_{2}^{\mathrm{new}}(\Gamma_0(4675)):

T222+9T221+6T220−168T219−429T218+1071T217+⋯−200 T_{2}^{22} + 9 T_{2}^{21} + 6 T_{2}^{20} - 168 T_{2}^{19} - 429 T_{2}^{18} + 1071 T_{2}^{17} + \cdots - 200 Copy content Toggle raw display
T322+4T321−35T320−150T319+488T318+2308T317+⋯−88 T_{3}^{22} + 4 T_{3}^{21} - 35 T_{3}^{20} - 150 T_{3}^{19} + 488 T_{3}^{18} + 2308 T_{3}^{17} + \cdots - 88 Copy content Toggle raw display
T722+16T721+39T720−644T719−3794T718+6050T717+⋯+8650752 T_{7}^{22} + 16 T_{7}^{21} + 39 T_{7}^{20} - 644 T_{7}^{19} - 3794 T_{7}^{18} + 6050 T_{7}^{17} + \cdots + 8650752 Copy content Toggle raw display