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Properties

Label 675.2.h.d
Level $675$
Weight $2$
Character orbit 675.h
Analytic conductor $5.390$
Analytic rank $0$
Dimension $40$
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] = [675,2,Mod(136,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.h (of order \(5\), 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{2} - 10 q^{4} + 3 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{2} - 10 q^{4} + 3 q^{5} - 2 q^{7} + 6 q^{8} + q^{10} + 13 q^{11} - 6 q^{13} - 12 q^{14} - 22 q^{16} + 8 q^{17} + 3 q^{19} - 3 q^{20} - 6 q^{22} + 11 q^{23} + 3 q^{25} - 104 q^{26} + 25 q^{28} + 5 q^{29} + 3 q^{31} - 46 q^{32} + 22 q^{34} - 32 q^{35} + 26 q^{37} + 36 q^{38} - 34 q^{40} + 7 q^{41} + 6 q^{43} - 56 q^{44} - 22 q^{46} + 17 q^{47} + 26 q^{49} + q^{50} + 3 q^{52} + 4 q^{53} - 2 q^{55} + 21 q^{56} + 22 q^{58} - 35 q^{59} - 24 q^{61} - 16 q^{62} - 46 q^{64} + 74 q^{65} - 4 q^{67} - 82 q^{68} - 32 q^{70} + 43 q^{71} - 18 q^{73} + 62 q^{74} - 32 q^{76} + 37 q^{77} - 20 q^{79} + 103 q^{80} - 134 q^{82} + 15 q^{83} + 34 q^{85} + 40 q^{86} + 11 q^{88} - 24 q^{89} + 25 q^{91} + 59 q^{92} + 11 q^{94} - 75 q^{95} + 19 q^{97} + 60 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1 −2.08937 + 1.51802i 0 1.44307 − 4.44131i 0.241928 + 2.22294i 0 3.31002 2.13074 + 6.55776i 0 −3.87995 − 4.27731i
136.2 −1.56242 + 1.13517i 0 0.534526 − 1.64510i −1.77613 − 1.35844i 0 0.821650 −0.161276 − 0.496358i 0 4.31712 + 0.106253i
136.3 −0.931644 + 0.676879i 0 −0.208239 + 0.640893i 1.91059 + 1.16174i 0 0.106923 −0.951515 − 2.92846i 0 −2.56635 + 0.210916i
136.4 −0.479048 + 0.348048i 0 −0.509685 + 1.56865i 1.20059 − 1.88642i 0 0.0706522 −0.667762 − 2.05516i 0 0.0814283 + 1.32155i
136.5 −0.312726 + 0.227209i 0 −0.571860 + 1.76000i −1.93460 + 1.12130i 0 −4.46445 −0.459955 − 1.41560i 0 0.350230 − 0.790220i
136.6 0.477736 − 0.347096i 0 −0.510277 + 1.57047i 0.668239 + 2.13388i 0 1.91884 0.666284 + 2.05061i 0 1.05990 + 0.787490i
136.7 1.15596 − 0.839852i 0 0.0128520 − 0.0395544i 1.43396 − 1.71574i 0 2.45442 0.864710 + 2.66130i 0 0.216637 − 3.18764i
136.8 1.22778 − 0.892033i 0 0.0936819 − 0.288323i −1.63829 − 1.52184i 0 −3.37628 0.795766 + 2.44911i 0 −3.36899 − 0.407073i
136.9 1.95260 − 1.41865i 0 1.18206 − 3.63800i −2.14106 + 0.644872i 0 3.41720 −1.36130 − 4.18965i 0 −3.26579 + 4.29659i
136.10 2.17917 − 1.58326i 0 1.62404 − 4.99830i 2.22575 − 0.214514i 0 −3.64093 −2.70980 − 8.33990i 0 4.51067 − 3.99142i
271.1 −0.836162 + 2.57344i 0 −4.30541 − 3.12806i 1.50386 − 1.65481i 0 −2.29932 7.27171 − 5.28321i 0 3.00109 + 5.25379i
271.2 −0.660540 + 2.03293i 0 −2.07847 − 1.51010i −2.15422 − 0.599434i 0 3.24595 0.984216 − 0.715075i 0 2.64156 − 3.98344i
271.3 −0.626640 + 1.92860i 0 −1.70879 − 1.24151i 0.185663 + 2.22835i 0 −3.40931 0.184033 − 0.133708i 0 −4.41393 − 1.03830i
271.4 −0.343320 + 1.05663i 0 0.619432 + 0.450044i 2.21922 + 0.273959i 0 0.957964 −2.48584 + 1.80607i 0 −1.05138 + 2.25084i
271.5 −0.154608 + 0.475834i 0 1.41552 + 1.02843i −2.17376 + 0.524192i 0 0.782520 −1.51775 + 1.10271i 0 0.0866517 − 1.11539i
271.6 −0.0557583 + 0.171606i 0 1.59169 + 1.15643i −0.0196531 − 2.23598i 0 −4.11905 −0.579156 + 0.420781i 0 0.384805 + 0.121302i
271.7 0.307793 − 0.947288i 0 0.815415 + 0.592434i 0.0944270 − 2.23407i 0 4.21601 2.42381 − 1.76100i 0 −2.08725 − 0.777081i
271.8 0.365753 − 1.12567i 0 0.484669 + 0.352133i 1.06520 + 1.96605i 0 −1.42835 2.48877 − 1.80819i 0 2.60273 − 0.479973i
271.9 0.651149 − 2.00403i 0 −1.97411 − 1.43427i −1.44849 + 1.70349i 0 2.40057 −0.750308 + 0.545130i 0 2.47065 + 4.01205i
271.10 0.734301 − 2.25995i 0 −2.95012 − 2.14339i 2.03678 − 0.922790i 0 −1.96502 −3.16537 + 2.29978i 0 −0.589848 − 5.28061i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.h.d yes 40
3.b odd 2 1 675.2.h.a ✓ 40
25.d even 5 1 inner 675.2.h.d yes 40
75.j odd 10 1 675.2.h.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
675.2.h.a ✓ 40 3.b odd 2 1
675.2.h.a ✓ 40 75.j odd 10 1
675.2.h.d yes 40 1.a even 1 1 trivial
675.2.h.d yes 40 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 2 T_{2}^{39} + 17 T_{2}^{38} - 36 T_{2}^{37} + 193 T_{2}^{36} - 326 T_{2}^{35} + \cdots + 5776 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display