# Properties

 Label 525.2.bb.a Level 525 Weight 2 Character orbit 525.bb Analytic conductor 4.192 Analytic rank 0 Dimension 304 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.bb (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$304$$ Relative dimension: $$76$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$304q + 60q^{4} - 12q^{7} - 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$304q + 60q^{4} - 12q^{7} - 6q^{9} + 8q^{15} - 92q^{16} - 16q^{18} + 12q^{21} - 48q^{22} - 32q^{25} - 36q^{28} - 14q^{30} - 8q^{36} - 4q^{37} - 26q^{39} - 11q^{42} - 20q^{49} + 20q^{51} - 168q^{57} + 32q^{58} + 76q^{60} + 29q^{63} + 68q^{64} - 44q^{67} + 30q^{70} - 114q^{72} - 124q^{78} - 68q^{79} - 2q^{81} + 102q^{84} + 72q^{85} - 156q^{88} + 70q^{91} - 32q^{93} - 36q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1 −1.64760 + 2.26772i −1.16943 + 1.27766i −1.80996 5.57048i 0.899153 + 2.04732i −0.970632 4.75703i 2.30027 1.30720i 10.2826 + 3.34103i −0.264850 2.98829i −6.12420 1.33413i
41.2 −1.64760 + 2.26772i 1.16943 1.27766i −1.80996 5.57048i −0.899153 2.04732i 0.970632 + 4.75703i 2.30027 + 1.30720i 10.2826 + 3.34103i −0.264850 2.98829i 6.12420 + 1.33413i
41.3 −1.51682 + 2.08772i −0.356101 1.69505i −1.43980 4.43126i −1.66426 + 1.49340i 4.07893 + 1.82764i −2.38379 1.14785i 6.52662 + 2.12063i −2.74638 + 1.20722i −0.593408 5.73972i
41.4 −1.51682 + 2.08772i 0.356101 + 1.69505i −1.43980 4.43126i 1.66426 1.49340i −4.07893 1.82764i −2.38379 + 1.14785i 6.52662 + 2.12063i −2.74638 + 1.20722i 0.593408 + 5.73972i
41.5 −1.47080 + 2.02438i −1.52339 0.824183i −1.31684 4.05281i 1.94672 + 1.10013i 3.90907 1.87172i −1.44927 + 2.21351i 5.38165 + 1.74860i 1.64144 + 2.51111i −5.09032 + 2.32283i
41.6 −1.47080 + 2.02438i 1.52339 + 0.824183i −1.31684 4.05281i −1.94672 1.10013i −3.90907 + 1.87172i −1.44927 2.21351i 5.38165 + 1.74860i 1.64144 + 2.51111i 5.09032 2.32283i
41.7 −1.41289 + 1.94468i −1.73181 + 0.0287273i −1.16748 3.59315i −0.433145 2.19371i 2.39100 3.40841i −0.449509 2.60729i 4.06484 + 1.32074i 2.99835 0.0995005i 4.87807 + 2.25716i
41.8 −1.41289 + 1.94468i 1.73181 0.0287273i −1.16748 3.59315i 0.433145 + 2.19371i −2.39100 + 3.40841i −0.449509 + 2.60729i 4.06484 + 1.32074i 2.99835 0.0995005i −4.87807 2.25716i
41.9 −1.35055 + 1.85887i −1.51070 0.847225i −1.01339 3.11890i −2.19863 + 0.407466i 3.61516 1.66398i 2.35939 + 1.19720i 2.79580 + 0.908411i 1.56442 + 2.55980i 2.21193 4.63728i
41.10 −1.35055 + 1.85887i 1.51070 + 0.847225i −1.01339 3.11890i 2.19863 0.407466i −3.61516 + 1.66398i 2.35939 1.19720i 2.79580 + 0.908411i 1.56442 + 2.55980i −2.21193 + 4.63728i
41.11 −1.19114 + 1.63947i −0.207251 + 1.71961i −0.650999 2.00357i −2.21550 0.302612i −2.57237 2.38808i 1.91855 1.82186i 0.205599 + 0.0668032i −2.91409 0.712779i 3.13510 3.27178i
41.12 −1.19114 + 1.63947i 0.207251 1.71961i −0.650999 2.00357i 2.21550 + 0.302612i 2.57237 + 2.38808i 1.91855 + 1.82186i 0.205599 + 0.0668032i −2.91409 0.712779i −3.13510 + 3.27178i
41.13 −1.17599 + 1.61862i −0.838989 + 1.51529i −0.618923 1.90485i 0.0163558 2.23601i −1.46602 3.13997i 1.60206 + 2.10557i 0.00547699 + 0.00177958i −1.59219 2.54262i 3.60000 + 2.65600i
41.14 −1.17599 + 1.61862i 0.838989 1.51529i −0.618923 1.90485i −0.0163558 + 2.23601i 1.46602 + 3.13997i 1.60206 2.10557i 0.00547699 + 0.00177958i −1.59219 2.54262i −3.60000 2.65600i
41.15 −1.16868 + 1.60855i −0.154997 1.72510i −0.603587 1.85765i 0.146015 2.23130i 2.95606 + 1.76677i −2.32118 + 1.26970i −0.0884038 0.0287241i −2.95195 + 0.534770i 3.41851 + 2.84254i
41.16 −1.16868 + 1.60855i 0.154997 + 1.72510i −0.603587 1.85765i −0.146015 + 2.23130i −2.95606 1.76677i −2.32118 1.26970i −0.0884038 0.0287241i −2.95195 + 0.534770i −3.41851 2.84254i
41.17 −1.05761 + 1.45568i −1.54874 + 0.775505i −0.382420 1.17697i 2.23358 0.105508i 0.509079 3.07465i −2.28537 1.33307i −1.30476 0.423944i 1.79718 2.40211i −2.20867 + 3.36295i
41.18 −1.05761 + 1.45568i 1.54874 0.775505i −0.382420 1.17697i −2.23358 + 0.105508i −0.509079 + 3.07465i −2.28537 + 1.33307i −1.30476 0.423944i 1.79718 2.40211i 2.20867 3.36295i
41.19 −0.990929 + 1.36390i −1.48109 + 0.897991i −0.260241 0.800939i −1.04257 + 1.97814i 0.242884 2.90989i −0.769360 + 2.53142i −1.85644 0.603193i 1.38723 2.66000i −1.66486 3.38216i
41.20 −0.990929 + 1.36390i 1.48109 0.897991i −0.260241 0.800939i 1.04257 1.97814i −0.242884 + 2.90989i −0.769360 2.53142i −1.85644 0.603193i 1.38723 2.66000i 1.66486 + 3.38216i
See next 80 embeddings (of 304 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 461.76 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner
175.l odd 10 1 inner
525.bb 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.bb.a 304
3.b odd 2 1 inner 525.2.bb.a 304
7.b odd 2 1 inner 525.2.bb.a 304
21.c even 2 1 inner 525.2.bb.a 304
25.d even 5 1 inner 525.2.bb.a 304
75.j odd 10 1 inner 525.2.bb.a 304
175.l odd 10 1 inner 525.2.bb.a 304
525.bb even 10 1 inner 525.2.bb.a 304

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bb.a 304 1.a even 1 1 trivial
525.2.bb.a 304 3.b odd 2 1 inner
525.2.bb.a 304 7.b odd 2 1 inner
525.2.bb.a 304 21.c even 2 1 inner
525.2.bb.a 304 25.d even 5 1 inner
525.2.bb.a 304 75.j odd 10 1 inner
525.2.bb.a 304 175.l odd 10 1 inner
525.2.bb.a 304 525.bb even 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database