Properties

Label 525.2.bg.a
Level 525
Weight 2
Character orbit 525.bg
Analytic conductor 4.192
Analytic rank 0
Dimension 160
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.bg (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(20\) over \(\Q(\zeta_{15})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160q - 2q^{2} - 20q^{3} + 20q^{4} - 4q^{6} + 4q^{7} - 6q^{8} + 20q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 160q - 2q^{2} - 20q^{3} + 20q^{4} - 4q^{6} + 4q^{7} - 6q^{8} + 20q^{9} - 2q^{10} - 6q^{11} - 20q^{12} + 28q^{16} - 6q^{17} + 8q^{18} + 4q^{19} - 8q^{20} + 7q^{21} + 32q^{22} - 2q^{23} - 18q^{24} + 4q^{25} + 56q^{26} + 40q^{27} - 43q^{28} + 26q^{29} + 12q^{30} + 3q^{31} + 54q^{32} - 4q^{33} - 76q^{34} - 37q^{35} - 40q^{36} - 18q^{37} + 19q^{38} - 22q^{40} - 30q^{41} + 16q^{42} - 164q^{43} + 6q^{44} - 5q^{45} - 10q^{46} - 11q^{47} + 56q^{48} - 136q^{49} - 86q^{50} - 4q^{51} - 19q^{52} - 34q^{53} + 2q^{54} - 48q^{55} + 33q^{56} - 72q^{57} - 24q^{58} - 24q^{59} + 11q^{60} - 16q^{61} - 128q^{62} - 7q^{63} - 162q^{64} - 36q^{65} + q^{66} - 6q^{67} + 16q^{68} - 24q^{69} + 32q^{70} - 68q^{71} + 3q^{72} + 34q^{73} + 4q^{74} - 9q^{75} + 420q^{76} - 40q^{77} - 8q^{78} + 14q^{79} - 76q^{80} + 20q^{81} + 54q^{82} - 38q^{83} + 54q^{84} + 42q^{85} - 30q^{86} - 12q^{87} - 30q^{88} - 33q^{89} - 26q^{90} + 5q^{91} + 2q^{92} + 2q^{93} + 46q^{94} + 4q^{95} + 11q^{96} - 58q^{97} + 166q^{98} - 8q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.54817 + 1.13452i −0.669131 + 0.743145i 3.86776 4.29558i 2.21934 + 0.272971i 0.861946 2.65280i −1.05082 2.42812i −3.25839 + 10.0283i −0.104528 0.994522i −5.96495 + 1.82231i
16.2 −2.43994 + 1.08633i −0.669131 + 0.743145i 3.43493 3.81488i −1.94782 1.09817i 0.825337 2.54013i 0.181785 + 2.63950i −2.58614 + 7.95931i −0.104528 0.994522i 5.94555 + 0.563491i
16.3 −1.90213 + 0.846881i −0.669131 + 0.743145i 1.56262 1.73546i −1.16209 + 1.91038i 0.643416 1.98023i 1.98170 1.75295i −0.215734 + 0.663961i −0.104528 0.994522i 0.592568 4.61794i
16.4 −1.86259 + 0.829278i −0.669131 + 0.743145i 1.44327 1.60292i −0.348898 2.20868i 0.630042 1.93907i 1.58384 2.11930i −0.0988780 + 0.304315i −0.104528 0.994522i 2.48146 + 3.82453i
16.5 −1.73660 + 0.773186i −0.669131 + 0.743145i 1.07971 1.19914i 0.215068 + 2.22570i 0.587426 1.80791i −2.62167 + 0.356150i 0.226976 0.698561i −0.104528 0.994522i −2.09437 3.69887i
16.6 −1.50205 + 0.668755i −0.669131 + 0.743145i 0.470657 0.522717i 2.21252 + 0.323643i 0.508085 1.56372i −1.94373 + 1.79497i 0.658790 2.02755i −0.104528 0.994522i −3.53976 + 0.993509i
16.7 −1.30094 + 0.579215i −0.669131 + 0.743145i 0.0186901 0.0207574i −2.15784 + 0.586291i 0.440057 1.35436i 0.667942 + 2.56005i 0.867823 2.67088i −0.104528 0.994522i 2.46763 2.01258i
16.8 −0.766251 + 0.341157i −0.669131 + 0.743145i −0.867508 + 0.963466i 0.938720 2.02948i 0.259193 0.797714i 1.21717 + 2.34915i 0.854422 2.62964i −0.104528 0.994522i −0.0269227 + 1.87535i
16.9 −0.650196 + 0.289486i −0.669131 + 0.743145i −0.999309 + 1.10984i 2.08872 + 0.798271i 0.219936 0.676893i 0.659935 2.56213i 0.768334 2.36469i −0.104528 0.994522i −1.58917 + 0.0856230i
16.10 0.0671031 0.0298762i −0.669131 + 0.743145i −1.33465 + 1.48228i 1.65633 1.50219i −0.0226984 + 0.0698585i −2.32293 1.26649i −0.0906711 + 0.279057i −0.104528 0.994522i 0.0662647 0.150287i
16.11 0.104767 0.0466452i −0.669131 + 0.743145i −1.32946 + 1.47652i 0.449213 + 2.19048i −0.0354385 + 0.109069i 2.46049 + 0.972613i −0.141288 + 0.434840i −0.104528 0.994522i 0.149238 + 0.208536i
16.12 0.246594 0.109791i −0.669131 + 0.743145i −1.28951 + 1.43214i −1.94989 + 1.09450i −0.0834131 + 0.256719i −2.11046 + 1.59561i −0.327575 + 1.00817i −0.104528 0.994522i −0.360664 + 0.483977i
16.13 0.337888 0.150438i −0.669131 + 0.743145i −1.24672 + 1.38463i −1.99129 1.01724i −0.114295 + 0.351762i 1.72521 2.00591i −0.441543 + 1.35893i −0.104528 0.994522i −0.825864 0.0441485i
16.14 1.26263 0.562160i −0.669131 + 0.743145i −0.0600440 + 0.0666857i 1.57128 + 1.59093i −0.427100 + 1.31448i −2.04775 + 1.67532i −0.892525 + 2.74691i −0.104528 0.994522i 2.87831 + 1.12545i
16.15 1.27220 0.566420i −0.669131 + 0.743145i −0.0405990 + 0.0450897i −0.413100 2.19758i −0.430336 + 1.32444i −0.306993 + 2.62788i −0.886782 + 2.72924i −0.104528 0.994522i −1.77030 2.56177i
16.16 1.43006 0.636705i −0.669131 + 0.743145i 0.301423 0.334764i −1.95826 + 1.07945i −0.483734 + 1.48878i −0.280963 2.63079i −0.749561 + 2.30691i −0.104528 0.994522i −2.11314 + 2.79051i
16.17 1.63302 0.727066i −0.669131 + 0.743145i 0.799857 0.888331i 0.337843 2.21040i −0.552386 + 1.70007i −2.03088 1.69574i −0.444468 + 1.36793i −0.104528 0.994522i −1.05540 3.85525i
16.18 1.77124 0.788606i −0.669131 + 0.743145i 1.17712 1.30733i 2.19192 0.442138i −0.599141 + 1.84397i 2.64021 + 0.171129i −0.144284 + 0.444060i −0.104528 0.994522i 3.53374 2.51169i
16.19 2.34270 1.04304i −0.669131 + 0.743145i 3.06206 3.40076i 1.20119 + 1.88604i −0.792445 + 2.43890i −0.520919 2.59396i 2.04148 6.28303i −0.104528 0.994522i 4.78124 + 3.16554i
16.20 2.41357 1.07459i −0.669131 + 0.743145i 3.33230 3.70090i −2.13490 0.664998i −0.816416 + 2.51267i 2.61882 + 0.376541i 2.43296 7.48789i −0.104528 0.994522i −5.86731 + 0.689119i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 466.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
25.d even 5 1 inner
175.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bg.a 160
7.c even 3 1 inner 525.2.bg.a 160
25.d even 5 1 inner 525.2.bg.a 160
175.q even 15 1 inner 525.2.bg.a 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bg.a 160 1.a even 1 1 trivial
525.2.bg.a 160 7.c even 3 1 inner
525.2.bg.a 160 25.d even 5 1 inner
525.2.bg.a 160 175.q even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{160} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database