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Properties

Label 725.2.y.a
Level $725$
Weight $2$
Character orbit 725.y
Analytic conductor $5.789$
Analytic rank $0$
Dimension $120$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(18,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([21, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.18");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.y (of order \(28\), degree \(12\), 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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 20 q^{4} + 28 q^{9} - 12 q^{11} + 20 q^{16} + 4 q^{19} + 4 q^{21} + 12 q^{29} - 32 q^{31} - 40 q^{34} - 16 q^{36} - 184 q^{39} - 4 q^{41} + 36 q^{44} + 76 q^{46} - 84 q^{49} + 112 q^{51} + 168 q^{54}+ \cdots - 208 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1 −2.43176 − 1.17107i −1.54099 − 1.22890i 3.29505 + 4.13187i 0 2.30818 + 4.79299i 0.367535 + 3.26196i −1.97287 − 8.64369i 0.196894 + 0.862650i 0
18.2 −1.61621 − 0.778326i 0.677001 + 0.539890i 0.759368 + 0.952217i 0 −0.673965 − 1.39950i −0.00559911 − 0.0496935i 0.312179 + 1.36775i −0.500714 − 2.19377i 0
18.3 −1.34275 − 0.646635i −1.78889 − 1.42659i 0.137866 + 0.172879i 0 1.47955 + 3.07231i −0.298557 − 2.64976i 0.589934 + 2.58467i 0.497396 + 2.17923i 0
18.4 −0.368695 − 0.177554i 2.43448 + 1.94143i −1.14257 − 1.43274i 0 −0.552871 − 1.14805i −0.376408 − 3.34071i 0.348992 + 1.52903i 1.48996 + 6.52795i 0
18.5 −0.359005 − 0.172888i 0.705151 + 0.562339i −1.14799 − 1.43953i 0 −0.155931 − 0.323795i 0.486105 + 4.31430i 0.340590 + 1.49222i −0.486550 − 2.13172i 0
18.6 0.359005 + 0.172888i −0.705151 − 0.562339i −1.14799 − 1.43953i 0 −0.155931 − 0.323795i −0.486105 − 4.31430i −0.340590 − 1.49222i −0.486550 − 2.13172i 0
18.7 0.368695 + 0.177554i −2.43448 − 1.94143i −1.14257 − 1.43274i 0 −0.552871 − 1.14805i 0.376408 + 3.34071i −0.348992 − 1.52903i 1.48996 + 6.52795i 0
18.8 1.34275 + 0.646635i 1.78889 + 1.42659i 0.137866 + 0.172879i 0 1.47955 + 3.07231i 0.298557 + 2.64976i −0.589934 − 2.58467i 0.497396 + 2.17923i 0
18.9 1.61621 + 0.778326i −0.677001 − 0.539890i 0.759368 + 0.952217i 0 −0.673965 − 1.39950i 0.00559911 + 0.0496935i −0.312179 − 1.36775i −0.500714 − 2.19377i 0
18.10 2.43176 + 1.17107i 1.54099 + 1.22890i 3.29505 + 4.13187i 0 2.30818 + 4.79299i −0.367535 − 3.26196i 1.97287 + 8.64369i 0.196894 + 0.862650i 0
32.1 −0.574590 − 2.51744i 0.00163787 − 0.00340106i −4.20544 + 2.02523i 0 −0.00950310 − 0.00216902i −0.286827 − 0.819705i 4.29488 + 5.38561i 1.87046 + 2.34548i 0
32.2 −0.445524 − 1.95197i −1.46808 + 3.04850i −1.80976 + 0.871532i 0 6.60465 + 1.50747i 0.560599 + 1.60210i 0.0108322 + 0.0135831i −5.26764 − 6.60541i 0
32.3 −0.305062 − 1.33657i 0.793823 − 1.64839i 0.108593 − 0.0522958i 0 −2.44535 − 0.558135i −1.11129 − 3.17588i −1.81256 − 2.27287i −0.216568 − 0.271568i 0
32.4 −0.199843 − 0.875569i −0.586819 + 1.21854i 1.07525 − 0.517815i 0 1.18419 + 0.270283i 0.0189641 + 0.0541962i −1.78816 − 2.24228i 0.729981 + 0.915367i 0
32.5 −0.0605112 − 0.265117i 0.814352 − 1.69102i 1.73531 − 0.835682i 0 −0.497595 − 0.113573i 1.58480 + 4.52910i −0.665656 − 0.834707i −0.325909 − 0.408678i 0
32.6 0.0605112 + 0.265117i −0.814352 + 1.69102i 1.73531 − 0.835682i 0 −0.497595 − 0.113573i −1.58480 − 4.52910i 0.665656 + 0.834707i −0.325909 − 0.408678i 0
32.7 0.199843 + 0.875569i 0.586819 − 1.21854i 1.07525 − 0.517815i 0 1.18419 + 0.270283i −0.0189641 − 0.0541962i 1.78816 + 2.24228i 0.729981 + 0.915367i 0
32.8 0.305062 + 1.33657i −0.793823 + 1.64839i 0.108593 − 0.0522958i 0 −2.44535 − 0.558135i 1.11129 + 3.17588i 1.81256 + 2.27287i −0.216568 − 0.271568i 0
32.9 0.445524 + 1.95197i 1.46808 − 3.04850i −1.80976 + 0.871532i 0 6.60465 + 1.50747i −0.560599 − 1.60210i −0.0108322 − 0.0135831i −5.26764 − 6.60541i 0
32.10 0.574590 + 2.51744i −0.00163787 + 0.00340106i −4.20544 + 2.02523i 0 −0.00950310 − 0.00216902i 0.286827 + 0.819705i −4.29488 − 5.38561i 1.87046 + 2.34548i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
145.o even 28 1 inner
145.t even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.y.a ✓ 120
5.b even 2 1 inner 725.2.y.a ✓ 120
5.c odd 4 2 725.2.bd.a yes 120
29.f odd 28 1 725.2.bd.a yes 120
145.o even 28 1 inner 725.2.y.a ✓ 120
145.s odd 28 1 725.2.bd.a yes 120
145.t even 28 1 inner 725.2.y.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.y.a ✓ 120 1.a even 1 1 trivial
725.2.y.a ✓ 120 5.b even 2 1 inner
725.2.y.a ✓ 120 145.o even 28 1 inner
725.2.y.a ✓ 120 145.t even 28 1 inner
725.2.bd.a yes 120 5.c odd 4 2
725.2.bd.a yes 120 29.f odd 28 1
725.2.bd.a yes 120 145.s odd 28 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} + 30 T_{2}^{118} + 515 T_{2}^{116} + 6936 T_{2}^{114} + 82423 T_{2}^{112} + \cdots + 1355457106081 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display