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Properties

Label 525.2.z.b
Level $525$
Weight $2$
Character orbit 525.z
Analytic conductor $4.192$
Analytic rank $0$
Dimension $72$
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] = [525,2,Mod(64,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.z (of order \(10\), 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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 24 q^{4} - 2 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 24 q^{4} - 2 q^{5} + 18 q^{9} - 28 q^{10} - 12 q^{11} - 20 q^{13} - 24 q^{16} + 10 q^{19} + 10 q^{20} - 18 q^{21} + 50 q^{22} - 10 q^{23} + 12 q^{25} + 36 q^{26} + 20 q^{28} - 2 q^{29} + 10 q^{30} - 16 q^{31} - 10 q^{33} + 24 q^{34} - 10 q^{35} - 24 q^{36} + 10 q^{37} - 100 q^{38} + 16 q^{39} - 14 q^{40} - 16 q^{41} - 18 q^{44} + 2 q^{45} - 44 q^{46} + 20 q^{47} - 72 q^{49} + 86 q^{50} + 32 q^{51} - 80 q^{52} + 70 q^{53} + 46 q^{55} - 40 q^{58} + 44 q^{59} - 62 q^{60} + 4 q^{61} - 50 q^{62} + 48 q^{64} + 38 q^{65} - 16 q^{66} - 20 q^{67} + 4 q^{69} + 10 q^{70} - 8 q^{71} - 20 q^{73} - 116 q^{74} - 8 q^{75} + 92 q^{76} + 20 q^{77} + 90 q^{78} + 28 q^{79} + 114 q^{80} - 18 q^{81} + 30 q^{83} + 24 q^{84} - 122 q^{85} + 40 q^{86} - 40 q^{87} - 270 q^{88} + 2 q^{89} - 12 q^{90} - 16 q^{91} - 100 q^{92} + 22 q^{94} + 116 q^{95} + 10 q^{96} + 190 q^{97} - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −1.56611 − 2.15556i 0.951057 + 0.309017i −1.57573 + 4.84959i 0.673446 + 2.13225i −0.823352 − 2.53402i − 1.00000i 7.85332 − 2.55170i 0.809017 + 0.587785i 3.54150 − 4.79098i
64.2 −1.36398 − 1.87736i −0.951057 − 0.309017i −1.04600 + 3.21926i −1.94902 + 1.09605i 0.717088 + 2.20697i 1.00000i 3.05649 − 0.993115i 0.809017 + 0.587785i 4.71611 + 2.16402i
64.3 −1.32302 − 1.82098i −0.951057 − 0.309017i −0.947556 + 2.91628i 0.0287119 − 2.23588i 0.695553 + 2.14069i 1.00000i 2.28274 − 0.741706i 0.809017 + 0.587785i −4.10949 + 2.90584i
64.4 −1.30460 − 1.79562i 0.951057 + 0.309017i −0.904259 + 2.78302i 1.97142 − 1.05523i −0.685868 − 2.11088i − 1.00000i 1.95519 − 0.635280i 0.809017 + 0.587785i −4.46670 − 2.16328i
64.5 −1.12883 − 1.55370i 0.951057 + 0.309017i −0.521699 + 1.60562i −1.49468 − 1.66311i −0.593462 − 1.82649i − 1.00000i −0.569402 + 0.185010i 0.809017 + 0.587785i −0.896746 + 4.19966i
64.6 −0.680773 − 0.937004i −0.951057 − 0.309017i 0.203510 − 0.626339i 2.06906 − 0.847924i 0.357904 + 1.10151i 1.00000i −2.92845 + 0.951512i 0.809017 + 0.587785i −2.20307 − 1.36148i
64.7 −0.558675 − 0.768950i −0.951057 − 0.309017i 0.338868 − 1.04293i 1.56006 + 1.60194i 0.293713 + 0.903955i 1.00000i −2.79918 + 0.909510i 0.809017 + 0.587785i 0.360249 − 2.09457i
64.8 −0.433614 − 0.596818i 0.951057 + 0.309017i 0.449863 − 1.38454i −1.72003 + 1.42881i −0.227964 − 0.701602i − 1.00000i −2.42459 + 0.787796i 0.809017 + 0.587785i 1.59857 + 0.406995i
64.9 −0.227844 − 0.313601i 0.951057 + 0.309017i 0.571602 − 1.75921i 1.55920 − 1.60277i −0.119785 − 0.368660i − 1.00000i −1.41925 + 0.461141i 0.809017 + 0.587785i −0.857887 − 0.123785i
64.10 −0.0671497 − 0.0924237i −0.951057 − 0.309017i 0.614001 − 1.88970i −0.767321 + 2.10029i 0.0353027 + 0.108651i 1.00000i −0.433184 + 0.140750i 0.809017 + 0.587785i 0.245642 − 0.0701153i
64.11 0.455284 + 0.626644i −0.951057 − 0.309017i 0.432634 − 1.33151i 1.16342 − 1.90957i −0.239357 − 0.736665i 1.00000i 2.50468 − 0.813821i 0.809017 + 0.587785i 1.72631 − 0.140349i
64.12 0.643331 + 0.885469i 0.951057 + 0.309017i 0.247853 − 0.762814i 0.921770 + 2.03724i 0.338219 + 1.04093i − 1.00000i 2.91676 − 0.947713i 0.809017 + 0.587785i −1.21091 + 2.12682i
64.13 0.919804 + 1.26600i −0.951057 − 0.309017i −0.138687 + 0.426835i −2.14595 − 0.628400i −0.483570 − 1.48827i 1.00000i 2.30861 − 0.750113i 0.809017 + 0.587785i −1.17830 − 3.29479i
64.14 0.972332 + 1.33830i 0.951057 + 0.309017i −0.227584 + 0.700432i −0.366564 − 2.20582i 0.511185 + 1.57327i − 1.00000i 1.98786 − 0.645894i 0.809017 + 0.587785i 2.59562 − 2.63536i
64.15 1.01102 + 1.39156i −0.951057 − 0.309017i −0.296223 + 0.911681i 1.48832 + 1.66881i −0.531527 − 1.63587i 1.00000i 1.70360 − 0.553533i 0.809017 + 0.587785i −0.817513 + 3.75828i
64.16 1.43938 + 1.98113i 0.951057 + 0.309017i −1.23505 + 3.80109i 1.94961 + 1.09500i 0.756726 + 2.32896i − 1.00000i −4.65025 + 1.51096i 0.809017 + 0.587785i 0.636883 + 5.43856i
64.17 1.60595 + 2.21041i 0.951057 + 0.309017i −1.68877 + 5.19751i −2.23410 + 0.0937129i 0.844300 + 2.59849i − 1.00000i −9.00374 + 2.92549i 0.809017 + 0.587785i −3.79501 − 4.78778i
64.18 1.60749 + 2.21252i −0.951057 − 0.309017i −1.69318 + 5.21107i −2.08931 + 0.796731i −0.845106 − 2.60097i 1.00000i −9.04941 + 2.94033i 0.809017 + 0.587785i −5.12132 − 3.34190i
169.1 −2.59414 + 0.842888i 0.587785 + 0.809017i 4.40108 − 3.19758i 1.06976 + 1.96357i −2.20671 − 1.60327i 1.00000i −5.51531 + 7.59117i −0.309017 + 0.951057i −4.43019 − 4.19209i
169.2 −2.47820 + 0.805216i −0.587785 − 0.809017i 3.87506 − 2.81540i 1.64699 − 1.51242i 2.10808 + 1.53161i − 1.00000i −4.27295 + 5.88121i −0.309017 + 0.951057i −2.86374 + 5.07427i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.z.b ✓ 72
25.e even 10 1 inner 525.2.z.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.z.b ✓ 72 1.a even 1 1 trivial
525.2.z.b ✓ 72 25.e even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 30 T_{2}^{70} + 528 T_{2}^{68} - 7152 T_{2}^{66} - 130 T_{2}^{65} + 82992 T_{2}^{64} + \cdots + 8994001 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display