# Properties

 Label 225.4.b.d Level $225$ Weight $4$ Character orbit 225.b Analytic conductor $13.275$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{2} - q^{4} - 20 i q^{7} + 21 i q^{8} +O(q^{10})$$ q + 3*i * q^2 - q^4 - 20*i * q^7 + 21*i * q^8 $$q + 3 i q^{2} - q^{4} - 20 i q^{7} + 21 i q^{8} + 24 q^{11} + 74 i q^{13} + 60 q^{14} - 71 q^{16} + 54 i q^{17} + 124 q^{19} + 72 i q^{22} + 120 i q^{23} - 222 q^{26} + 20 i q^{28} - 78 q^{29} + 200 q^{31} - 45 i q^{32} - 162 q^{34} + 70 i q^{37} + 372 i q^{38} - 330 q^{41} + 92 i q^{43} - 24 q^{44} - 360 q^{46} - 24 i q^{47} - 57 q^{49} - 74 i q^{52} - 450 i q^{53} + 420 q^{56} - 234 i q^{58} + 24 q^{59} - 322 q^{61} + 600 i q^{62} - 433 q^{64} + 196 i q^{67} - 54 i q^{68} + 288 q^{71} - 430 i q^{73} - 210 q^{74} - 124 q^{76} - 480 i q^{77} + 520 q^{79} - 990 i q^{82} - 156 i q^{83} - 276 q^{86} + 504 i q^{88} + 1026 q^{89} + 1480 q^{91} - 120 i q^{92} + 72 q^{94} + 286 i q^{97} - 171 i q^{98} +O(q^{100})$$ q + 3*i * q^2 - q^4 - 20*i * q^7 + 21*i * q^8 + 24 * q^11 + 74*i * q^13 + 60 * q^14 - 71 * q^16 + 54*i * q^17 + 124 * q^19 + 72*i * q^22 + 120*i * q^23 - 222 * q^26 + 20*i * q^28 - 78 * q^29 + 200 * q^31 - 45*i * q^32 - 162 * q^34 + 70*i * q^37 + 372*i * q^38 - 330 * q^41 + 92*i * q^43 - 24 * q^44 - 360 * q^46 - 24*i * q^47 - 57 * q^49 - 74*i * q^52 - 450*i * q^53 + 420 * q^56 - 234*i * q^58 + 24 * q^59 - 322 * q^61 + 600*i * q^62 - 433 * q^64 + 196*i * q^67 - 54*i * q^68 + 288 * q^71 - 430*i * q^73 - 210 * q^74 - 124 * q^76 - 480*i * q^77 + 520 * q^79 - 990*i * q^82 - 156*i * q^83 - 276 * q^86 + 504*i * q^88 + 1026 * q^89 + 1480 * q^91 - 120*i * q^92 + 72 * q^94 + 286*i * q^97 - 171*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} + 48 q^{11} + 120 q^{14} - 142 q^{16} + 248 q^{19} - 444 q^{26} - 156 q^{29} + 400 q^{31} - 324 q^{34} - 660 q^{41} - 48 q^{44} - 720 q^{46} - 114 q^{49} + 840 q^{56} + 48 q^{59} - 644 q^{61} - 866 q^{64} + 576 q^{71} - 420 q^{74} - 248 q^{76} + 1040 q^{79} - 552 q^{86} + 2052 q^{89} + 2960 q^{91} + 144 q^{94}+O(q^{100})$$ 2 * q - 2 * q^4 + 48 * q^11 + 120 * q^14 - 142 * q^16 + 248 * q^19 - 444 * q^26 - 156 * q^29 + 400 * q^31 - 324 * q^34 - 660 * q^41 - 48 * q^44 - 720 * q^46 - 114 * q^49 + 840 * q^56 + 48 * q^59 - 644 * q^61 - 866 * q^64 + 576 * q^71 - 420 * q^74 - 248 * q^76 + 1040 * q^79 - 552 * q^86 + 2052 * q^89 + 2960 * q^91 + 144 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 1.00000i 1.00000i
3.00000i 0 −1.00000 0 0 20.0000i 21.0000i 0 0
199.2 3.00000i 0 −1.00000 0 0 20.0000i 21.0000i 0 0
 $$n$$: e.g. 2-40 or 990-1000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.b.d 2
3.b odd 2 1 75.4.b.a 2
5.b even 2 1 inner 225.4.b.d 2
5.c odd 4 1 45.4.a.b 1
5.c odd 4 1 225.4.a.g 1
12.b even 2 1 1200.4.f.m 2
15.d odd 2 1 75.4.b.a 2
15.e even 4 1 15.4.a.b 1
15.e even 4 1 75.4.a.a 1
20.e even 4 1 720.4.a.r 1
35.f even 4 1 2205.4.a.c 1
45.k odd 12 2 405.4.e.k 2
45.l even 12 2 405.4.e.d 2
60.h even 2 1 1200.4.f.m 2
60.l odd 4 1 240.4.a.f 1
60.l odd 4 1 1200.4.a.o 1
105.k odd 4 1 735.4.a.i 1
120.q odd 4 1 960.4.a.l 1
120.w even 4 1 960.4.a.bi 1
165.l odd 4 1 1815.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 15.e even 4 1
45.4.a.b 1 5.c odd 4 1
75.4.a.a 1 15.e even 4 1
75.4.b.a 2 3.b odd 2 1
75.4.b.a 2 15.d odd 2 1
225.4.a.g 1 5.c odd 4 1
225.4.b.d 2 1.a even 1 1 trivial
225.4.b.d 2 5.b even 2 1 inner
240.4.a.f 1 60.l odd 4 1
405.4.e.d 2 45.l even 12 2
405.4.e.k 2 45.k odd 12 2
720.4.a.r 1 20.e even 4 1
735.4.a.i 1 105.k odd 4 1
960.4.a.l 1 120.q odd 4 1
960.4.a.bi 1 120.w even 4 1
1200.4.a.o 1 60.l odd 4 1
1200.4.f.m 2 12.b even 2 1
1200.4.f.m 2 60.h even 2 1
1815.4.a.a 1 165.l odd 4 1
2205.4.a.c 1 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{2} + 9$$ T2^2 + 9 $$T_{7}^{2} + 400$$ T7^2 + 400 $$T_{11} - 24$$ T11 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 9$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 400$$
$11$ $$(T - 24)^{2}$$
$13$ $$T^{2} + 5476$$
$17$ $$T^{2} + 2916$$
$19$ $$(T - 124)^{2}$$
$23$ $$T^{2} + 14400$$
$29$ $$(T + 78)^{2}$$
$31$ $$(T - 200)^{2}$$
$37$ $$T^{2} + 4900$$
$41$ $$(T + 330)^{2}$$
$43$ $$T^{2} + 8464$$
$47$ $$T^{2} + 576$$
$53$ $$T^{2} + 202500$$
$59$ $$(T - 24)^{2}$$
$61$ $$(T + 322)^{2}$$
$67$ $$T^{2} + 38416$$
$71$ $$(T - 288)^{2}$$
$73$ $$T^{2} + 184900$$
$79$ $$(T - 520)^{2}$$
$83$ $$T^{2} + 24336$$
$89$ $$(T - 1026)^{2}$$
$97$ $$T^{2} + 81796$$