# Properties

 Label 225.4.p.b Level $225$ Weight $4$ Character orbit 225.p Analytic conductor $13.275$ Analytic rank $0$ Dimension $64$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(32,225)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([10, 3]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("225.32");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.p (of order $$12$$, 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: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$16$$ over $$\Q(\zeta_{12})$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$64 q + 6 q^{2} - 24 q^{6} + 2 q^{7}+O(q^{10})$$ 64 * q + 6 * q^2 - 24 * q^6 + 2 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$64 q + 6 q^{2} - 24 q^{6} + 2 q^{7} - 36 q^{11} + 138 q^{12} + 2 q^{13} + 316 q^{16} + 480 q^{18} + 480 q^{21} + 34 q^{22} - 306 q^{23} - 180 q^{27} + 232 q^{28} - 4 q^{31} + 1770 q^{32} + 294 q^{33} - 216 q^{36} - 136 q^{37} - 114 q^{38} + 1992 q^{41} - 1698 q^{42} + 2 q^{43} - 952 q^{46} - 3462 q^{47} - 4326 q^{48} - 2496 q^{51} + 242 q^{52} - 7128 q^{56} + 2544 q^{57} - 534 q^{58} + 32 q^{61} + 4038 q^{63} + 2892 q^{66} - 610 q^{67} + 2694 q^{68} + 1854 q^{72} + 8 q^{73} + 1368 q^{76} + 6486 q^{77} - 1434 q^{78} + 3012 q^{81} + 3784 q^{82} - 2814 q^{83} + 12480 q^{86} - 4830 q^{87} + 1338 q^{88} + 992 q^{91} - 13152 q^{92} - 8310 q^{93} - 7932 q^{96} - 358 q^{97}+O(q^{100})$$ 64 * q + 6 * q^2 - 24 * q^6 + 2 * q^7 - 36 * q^11 + 138 * q^12 + 2 * q^13 + 316 * q^16 + 480 * q^18 + 480 * q^21 + 34 * q^22 - 306 * q^23 - 180 * q^27 + 232 * q^28 - 4 * q^31 + 1770 * q^32 + 294 * q^33 - 216 * q^36 - 136 * q^37 - 114 * q^38 + 1992 * q^41 - 1698 * q^42 + 2 * q^43 - 952 * q^46 - 3462 * q^47 - 4326 * q^48 - 2496 * q^51 + 242 * q^52 - 7128 * q^56 + 2544 * q^57 - 534 * q^58 + 32 * q^61 + 4038 * q^63 + 2892 * q^66 - 610 * q^67 + 2694 * q^68 + 1854 * q^72 + 8 * q^73 + 1368 * q^76 + 6486 * q^77 - 1434 * q^78 + 3012 * q^81 + 3784 * q^82 - 2814 * q^83 + 12480 * q^86 - 4830 * q^87 + 1338 * q^88 + 992 * q^91 - 13152 * q^92 - 8310 * q^93 - 7932 * q^96 - 358 * q^97

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1 −4.88245 1.30825i 1.83715 + 4.86054i 15.1986 + 8.77490i 0 −2.61101 26.1348i 7.62587 28.4601i −34.1329 34.1329i −20.2497 + 17.8591i 0
32.2 −4.29910 1.15194i 4.54171 2.52446i 10.2271 + 5.90461i 0 −22.4333 + 5.62113i −1.91190 + 7.13532i −11.9883 11.9883i 14.2542 22.9307i 0
32.3 −4.14355 1.11026i −1.80338 4.87317i 9.00816 + 5.20086i 0 2.06189 + 22.1945i −3.90791 + 14.5845i −7.28515 7.28515i −20.4957 + 17.5763i 0
32.4 −3.56967 0.956489i −4.00002 + 3.31660i 4.89944 + 2.82870i 0 17.4510 8.01320i −1.11867 + 4.17492i 6.12165 + 6.12165i 5.00028 26.5329i 0
32.5 −2.15214 0.576664i −4.75638 2.09209i −2.62904 1.51788i 0 9.02996 + 7.24530i 3.70051 13.8105i 17.3866 + 17.3866i 18.2463 + 19.9016i 0
32.6 −1.43923 0.385641i 2.38802 + 4.61491i −5.00553 2.88994i 0 −1.65722 7.56285i −6.62758 + 24.7344i 14.5184 + 14.5184i −15.5947 + 22.0410i 0
32.7 −1.25351 0.335877i 5.15697 + 0.636882i −5.46973 3.15795i 0 −6.25041 2.53045i 0.803840 2.99997i 13.1367 + 13.1367i 26.1888 + 6.56877i 0
32.8 −0.145161 0.0388957i 0.490448 5.17295i −6.90864 3.98871i 0 −0.272400 + 0.731834i 4.62436 17.2584i 1.69784 + 1.69784i −26.5189 5.07413i 0
32.9 0.471331 + 0.126293i −3.69630 + 3.65204i −6.72200 3.88095i 0 −2.20341 + 1.25450i 1.53228 5.71856i −5.43846 5.43846i 0.325220 26.9980i 0
32.10 1.60134 + 0.429078i 1.67949 4.91725i −4.54802 2.62580i 0 4.79931 7.15356i −8.15082 + 30.4193i −15.5344 15.5344i −21.3587 16.5169i 0
32.11 2.57169 + 0.689083i 0.974542 + 5.10395i −0.789441 0.455784i 0 −1.01082 + 13.7973i 0.400135 1.49332i −16.7770 16.7770i −25.1005 + 9.94802i 0
32.12 2.79258 + 0.748271i −5.18419 0.352314i 0.310415 + 0.179218i 0 −14.2137 4.86305i −5.03374 + 18.7862i −15.6218 15.6218i 26.7517 + 3.65293i 0
32.13 2.87860 + 0.771317i 5.17956 + 0.414910i 0.763180 + 0.440622i 0 14.5898 + 5.18944i 8.39055 31.3140i −15.0012 15.0012i 26.6557 + 4.29810i 0
32.14 3.92041 + 1.05047i −3.82410 3.51799i 7.33795 + 4.23656i 0 −11.2965 17.8091i 6.13411 22.8928i 1.35786 + 1.35786i 2.24743 + 26.9063i 0
32.15 4.73973 + 1.27001i 4.68975 2.23746i 13.9239 + 8.03899i 0 25.0698 4.64896i −5.69629 + 21.2589i 28.0284 + 28.0284i 16.9875 20.9863i 0
32.16 5.27515 + 1.41347i −1.94123 + 4.81992i 18.9011 + 10.9125i 0 −17.0531 + 22.6819i −1.13077 + 4.22008i 53.3880 + 53.3880i −19.4633 18.7131i 0
68.1 −1.30825 + 4.88245i −4.86054 + 1.83715i −15.1986 8.77490i 0 −2.61101 26.1348i −28.4601 7.62587i 34.1329 34.1329i 20.2497 17.8591i 0
68.2 −1.15194 + 4.29910i 2.52446 + 4.54171i −10.2271 5.90461i 0 −22.4333 + 5.62113i 7.13532 + 1.91190i 11.9883 11.9883i −14.2542 + 22.9307i 0
68.3 −1.11026 + 4.14355i 4.87317 1.80338i −9.00816 5.20086i 0 2.06189 + 22.1945i 14.5845 + 3.90791i 7.28515 7.28515i 20.4957 17.5763i 0
68.4 −0.956489 + 3.56967i −3.31660 4.00002i −4.89944 2.82870i 0 17.4510 8.01320i 4.17492 + 1.11867i −6.12165 + 6.12165i −5.00028 + 26.5329i 0
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 32.16 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.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.p.b 64
5.b even 2 1 45.4.l.a 64
5.c odd 4 1 45.4.l.a 64
5.c odd 4 1 inner 225.4.p.b 64
9.d odd 6 1 inner 225.4.p.b 64
15.d odd 2 1 135.4.m.a 64
15.e even 4 1 135.4.m.a 64
45.h odd 6 1 45.4.l.a 64
45.j even 6 1 135.4.m.a 64
45.k odd 12 1 135.4.m.a 64
45.l even 12 1 45.4.l.a 64
45.l even 12 1 inner 225.4.p.b 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.l.a 64 5.b even 2 1
45.4.l.a 64 5.c odd 4 1
45.4.l.a 64 45.h odd 6 1
45.4.l.a 64 45.l even 12 1
135.4.m.a 64 15.d odd 2 1
135.4.m.a 64 15.e even 4 1
135.4.m.a 64 45.j even 6 1
135.4.m.a 64 45.k odd 12 1
225.4.p.b 64 1.a even 1 1 trivial
225.4.p.b 64 5.c odd 4 1 inner
225.4.p.b 64 9.d odd 6 1 inner
225.4.p.b 64 45.l even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{64} - 6 T_{2}^{63} + 18 T_{2}^{62} - 36 T_{2}^{61} - 1721 T_{2}^{60} + 9540 T_{2}^{59} - 25614 T_{2}^{58} + 73356 T_{2}^{57} + 1833411 T_{2}^{56} - 9945624 T_{2}^{55} + 25455636 T_{2}^{54} + \cdots + 47\!\cdots\!16$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.