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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Newspace parameters

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

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{10})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 72q + 24q^{4} - 2q^{5} + 18q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 72q + 24q^{4} - 2q^{5} + 18q^{9} - 28q^{10} - 12q^{11} - 20q^{13} - 24q^{16} + 10q^{19} + 10q^{20} - 18q^{21} + 50q^{22} - 10q^{23} + 12q^{25} + 36q^{26} + 20q^{28} - 2q^{29} + 10q^{30} - 16q^{31} - 10q^{33} + 24q^{34} - 10q^{35} - 24q^{36} + 10q^{37} - 100q^{38} + 16q^{39} - 14q^{40} - 16q^{41} - 18q^{44} + 2q^{45} - 44q^{46} + 20q^{47} - 72q^{49} + 86q^{50} + 32q^{51} - 80q^{52} + 70q^{53} + 46q^{55} - 40q^{58} + 44q^{59} - 62q^{60} + 4q^{61} - 50q^{62} + 48q^{64} + 38q^{65} - 16q^{66} - 20q^{67} + 4q^{69} + 10q^{70} - 8q^{71} - 20q^{73} - 116q^{74} - 8q^{75} + 92q^{76} + 20q^{77} + 90q^{78} + 28q^{79} + 114q^{80} - 18q^{81} + 30q^{83} + 24q^{84} - 122q^{85} + 40q^{86} - 40q^{87} - 270q^{88} + 2q^{89} - 12q^{90} - 16q^{91} - 100q^{92} + 22q^{94} + 116q^{95} + 10q^{96} + 190q^{97} - 48q^{99} + O(q^{100}) \)

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.

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 484.18
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 \(T_{2}^{72} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database