# Properties

 Label 475.2.l.f Level $475$ Weight $2$ Character orbit 475.l Analytic conductor $3.793$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.l (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$8$$ over $$\Q(\zeta_{9})$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48 q + 18 q^{4} - 6 q^{6} + 12 q^{9}+O(q^{10})$$ 48 * q + 18 * q^4 - 6 * q^6 + 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q + 18 q^{4} - 6 q^{6} + 12 q^{9} - 12 q^{11} - 6 q^{14} - 42 q^{16} - 12 q^{19} - 54 q^{21} - 24 q^{24} + 12 q^{26} - 42 q^{31} + 36 q^{34} + 18 q^{36} + 48 q^{39} + 6 q^{41} + 6 q^{44} - 6 q^{46} - 12 q^{49} + 108 q^{51} - 24 q^{54} + 36 q^{56} + 36 q^{59} + 48 q^{61} + 180 q^{66} - 66 q^{69} - 24 q^{71} - 84 q^{74} + 66 q^{76} - 48 q^{79} - 78 q^{81} + 54 q^{84} - 42 q^{86} + 12 q^{89} - 30 q^{91} + 72 q^{94} - 240 q^{96} + 36 q^{99}+O(q^{100})$$ 48 * q + 18 * q^4 - 6 * q^6 + 12 * q^9 - 12 * q^11 - 6 * q^14 - 42 * q^16 - 12 * q^19 - 54 * q^21 - 24 * q^24 + 12 * q^26 - 42 * q^31 + 36 * q^34 + 18 * q^36 + 48 * q^39 + 6 * q^41 + 6 * q^44 - 6 * q^46 - 12 * q^49 + 108 * q^51 - 24 * q^54 + 36 * q^56 + 36 * q^59 + 48 * q^61 + 180 * q^66 - 66 * q^69 - 24 * q^71 - 84 * q^74 + 66 * q^76 - 48 * q^79 - 78 * q^81 + 54 * q^84 - 42 * q^86 + 12 * q^89 - 30 * q^91 + 72 * q^94 - 240 * q^96 + 36 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1 −2.05812 + 1.72697i −2.01713 0.734175i 0.906145 5.13900i 0 5.41939 1.97250i 1.39152 2.41018i 4.32326 + 7.48810i 1.23166 + 1.03349i 0
101.2 −1.36714 + 1.14717i 2.24001 + 0.815296i 0.205786 1.16707i 0 −3.99769 + 1.45504i 2.11955 3.67118i −0.727188 1.25953i 2.05480 + 1.72418i 0
101.3 −1.24028 + 1.04072i 1.11689 + 0.406515i 0.107903 0.611947i 0 −1.80832 + 0.658176i −1.11595 + 1.93288i −1.11604 1.93303i −1.21594 1.02030i 0
101.4 −0.344240 + 0.288852i −1.83883 0.669279i −0.312230 + 1.77075i 0 0.826320 0.300756i 1.03040 1.78470i −0.853374 1.47809i 0.635222 + 0.533015i 0
101.5 0.344240 0.288852i 1.83883 + 0.669279i −0.312230 + 1.77075i 0 0.826320 0.300756i −1.03040 + 1.78470i 0.853374 + 1.47809i 0.635222 + 0.533015i 0
101.6 1.24028 1.04072i −1.11689 0.406515i 0.107903 0.611947i 0 −1.80832 + 0.658176i 1.11595 1.93288i 1.11604 + 1.93303i −1.21594 1.02030i 0
101.7 1.36714 1.14717i −2.24001 0.815296i 0.205786 1.16707i 0 −3.99769 + 1.45504i −2.11955 + 3.67118i 0.727188 + 1.25953i 2.05480 + 1.72418i 0
101.8 2.05812 1.72697i 2.01713 + 0.734175i 0.906145 5.13900i 0 5.41939 1.97250i −1.39152 + 2.41018i −4.32326 7.48810i 1.23166 + 1.03349i 0
176.1 −2.22798 + 0.810919i −0.396806 + 2.25040i 2.77422 2.32785i 0 −0.940815 5.33563i 0.818386 + 1.41749i −1.92225 + 3.32944i −2.08777 0.759885i 0
176.2 −2.18443 + 0.795068i 0.199449 1.13113i 2.60752 2.18797i 0 0.463643 + 2.62945i 0.0716510 + 0.124103i −1.63174 + 2.82626i 1.57940 + 0.574856i 0
176.3 −0.984236 + 0.358233i −0.0922859 + 0.523379i −0.691698 + 0.580404i 0 −0.0966605 0.548189i −1.37016 2.37320i 1.52028 2.63320i 2.55367 + 0.929459i 0
176.4 −0.234689 + 0.0854197i −0.399662 + 2.26659i −1.48431 + 1.24548i 0 −0.0998158 0.566083i 1.98021 + 3.42983i 0.491712 0.851670i −2.15864 0.785682i 0
176.5 0.234689 0.0854197i 0.399662 2.26659i −1.48431 + 1.24548i 0 −0.0998158 0.566083i −1.98021 3.42983i −0.491712 + 0.851670i −2.15864 0.785682i 0
176.6 0.984236 0.358233i 0.0922859 0.523379i −0.691698 + 0.580404i 0 −0.0966605 0.548189i 1.37016 + 2.37320i −1.52028 + 2.63320i 2.55367 + 0.929459i 0
176.7 2.18443 0.795068i −0.199449 + 1.13113i 2.60752 2.18797i 0 0.463643 + 2.62945i −0.0716510 0.124103i 1.63174 2.82626i 1.57940 + 0.574856i 0
176.8 2.22798 0.810919i 0.396806 2.25040i 2.77422 2.32785i 0 −0.940815 5.33563i −0.818386 1.41749i 1.92225 3.32944i −2.08777 0.759885i 0
226.1 −0.340658 + 1.93197i −0.143793 0.120656i −1.73706 0.632239i 0 0.282088 0.236700i 0.338534 + 0.586358i −0.148561 + 0.257316i −0.514826 2.91972i 0
226.2 −0.256855 + 1.45670i 1.90225 + 1.59617i −0.176607 0.0642796i 0 −2.81374 + 2.36101i 1.62494 + 2.81448i −1.34017 + 2.32124i 0.549824 + 3.11821i 0
226.3 −0.212126 + 1.20303i −0.616222 0.517072i 0.477108 + 0.173653i 0 0.752768 0.631647i −1.89590 3.28379i −1.53170 + 2.65299i −0.408578 2.31716i 0
226.4 −0.0424530 + 0.240763i −2.09771 1.76019i 1.82322 + 0.663598i 0 0.512843 0.430326i 0.970136 + 1.68032i −0.481648 + 0.834240i 0.781184 + 4.43031i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 351.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.e even 9 1 inner
95.p even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.l.f 48
5.b even 2 1 inner 475.2.l.f 48
5.c odd 4 2 95.2.p.a 48
15.e even 4 2 855.2.da.b 48
19.e even 9 1 inner 475.2.l.f 48
19.e even 9 1 9025.2.a.cu 24
19.f odd 18 1 9025.2.a.ct 24
95.o odd 18 1 9025.2.a.ct 24
95.p even 18 1 inner 475.2.l.f 48
95.p even 18 1 9025.2.a.cu 24
95.q odd 36 2 95.2.p.a 48
95.q odd 36 2 1805.2.b.k 24
95.r even 36 2 1805.2.b.l 24
285.bi even 36 2 855.2.da.b 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.p.a 48 5.c odd 4 2
95.2.p.a 48 95.q odd 36 2
475.2.l.f 48 1.a even 1 1 trivial
475.2.l.f 48 5.b even 2 1 inner
475.2.l.f 48 19.e even 9 1 inner
475.2.l.f 48 95.p even 18 1 inner
855.2.da.b 48 15.e even 4 2
855.2.da.b 48 285.bi even 36 2
1805.2.b.k 24 95.q odd 36 2
1805.2.b.l 24 95.r even 36 2
9025.2.a.ct 24 19.f odd 18 1
9025.2.a.ct 24 95.o odd 18 1
9025.2.a.cu 24 19.e even 9 1
9025.2.a.cu 24 95.p even 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{48} - 9 T_{2}^{46} + 78 T_{2}^{44} - 181 T_{2}^{42} - 255 T_{2}^{40} + 10179 T_{2}^{38} - 5028 T_{2}^{36} - 24393 T_{2}^{34} + 653217 T_{2}^{32} + 527040 T_{2}^{30} + 5285235 T_{2}^{28} + 31702815 T_{2}^{26} + \cdots + 361$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.