# Properties

 Label 225.4.b.c 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: $$2$$ Twist minimal: no (minimal twist has level 5) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} - 8 q^{4} + 3 i q^{7}+O(q^{10})$$ q + 2*i * q^2 - 8 * q^4 + 3*i * q^7 $$q + 2 i q^{2} - 8 q^{4} + 3 i q^{7} - 32 q^{11} + 19 i q^{13} - 24 q^{14} - 64 q^{16} - 13 i q^{17} - 100 q^{19} - 64 i q^{22} - 39 i q^{23} - 152 q^{26} - 24 i q^{28} - 50 q^{29} - 108 q^{31} - 128 i q^{32} + 104 q^{34} + 133 i q^{37} - 200 i q^{38} - 22 q^{41} - 221 i q^{43} + 256 q^{44} + 312 q^{46} + 257 i q^{47} + 307 q^{49} - 152 i q^{52} + i q^{53} - 100 i q^{58} + 500 q^{59} - 518 q^{61} - 216 i q^{62} + 512 q^{64} + 63 i q^{67} + 104 i q^{68} - 412 q^{71} + 439 i q^{73} - 1064 q^{74} + 800 q^{76} - 96 i q^{77} - 600 q^{79} - 44 i q^{82} + 141 i q^{83} + 1768 q^{86} - 150 q^{89} - 228 q^{91} + 312 i q^{92} - 2056 q^{94} + 193 i q^{97} + 614 i q^{98} +O(q^{100})$$ q + 2*i * q^2 - 8 * q^4 + 3*i * q^7 - 32 * q^11 + 19*i * q^13 - 24 * q^14 - 64 * q^16 - 13*i * q^17 - 100 * q^19 - 64*i * q^22 - 39*i * q^23 - 152 * q^26 - 24*i * q^28 - 50 * q^29 - 108 * q^31 - 128*i * q^32 + 104 * q^34 + 133*i * q^37 - 200*i * q^38 - 22 * q^41 - 221*i * q^43 + 256 * q^44 + 312 * q^46 + 257*i * q^47 + 307 * q^49 - 152*i * q^52 + i * q^53 - 100*i * q^58 + 500 * q^59 - 518 * q^61 - 216*i * q^62 + 512 * q^64 + 63*i * q^67 + 104*i * q^68 - 412 * q^71 + 439*i * q^73 - 1064 * q^74 + 800 * q^76 - 96*i * q^77 - 600 * q^79 - 44*i * q^82 + 141*i * q^83 + 1768 * q^86 - 150 * q^89 - 228 * q^91 + 312*i * q^92 - 2056 * q^94 + 193*i * q^97 + 614*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{4}+O(q^{10})$$ 2 * q - 16 * q^4 $$2 q - 16 q^{4} - 64 q^{11} - 48 q^{14} - 128 q^{16} - 200 q^{19} - 304 q^{26} - 100 q^{29} - 216 q^{31} + 208 q^{34} - 44 q^{41} + 512 q^{44} + 624 q^{46} + 614 q^{49} + 1000 q^{59} - 1036 q^{61} + 1024 q^{64} - 824 q^{71} - 2128 q^{74} + 1600 q^{76} - 1200 q^{79} + 3536 q^{86} - 300 q^{89} - 456 q^{91} - 4112 q^{94}+O(q^{100})$$ 2 * q - 16 * q^4 - 64 * q^11 - 48 * q^14 - 128 * q^16 - 200 * q^19 - 304 * q^26 - 100 * q^29 - 216 * q^31 + 208 * q^34 - 44 * q^41 + 512 * q^44 + 624 * q^46 + 614 * q^49 + 1000 * q^59 - 1036 * q^61 + 1024 * q^64 - 824 * q^71 - 2128 * q^74 + 1600 * q^76 - 1200 * q^79 + 3536 * q^86 - 300 * q^89 - 456 * q^91 - 4112 * 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
4.00000i 0 −8.00000 0 0 6.00000i 0 0 0
199.2 4.00000i 0 −8.00000 0 0 6.00000i 0 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.c 2
3.b odd 2 1 25.4.b.a 2
5.b even 2 1 inner 225.4.b.c 2
5.c odd 4 1 45.4.a.d 1
5.c odd 4 1 225.4.a.b 1
12.b even 2 1 400.4.c.k 2
15.d odd 2 1 25.4.b.a 2
15.e even 4 1 5.4.a.a 1
15.e even 4 1 25.4.a.c 1
20.e even 4 1 720.4.a.u 1
35.f even 4 1 2205.4.a.q 1
45.k odd 12 2 405.4.e.c 2
45.l even 12 2 405.4.e.l 2
60.h even 2 1 400.4.c.k 2
60.l odd 4 1 80.4.a.d 1
60.l odd 4 1 400.4.a.m 1
105.k odd 4 1 245.4.a.a 1
105.k odd 4 1 1225.4.a.k 1
105.w odd 12 2 245.4.e.g 2
105.x even 12 2 245.4.e.f 2
120.q odd 4 1 320.4.a.h 1
120.q odd 4 1 1600.4.a.s 1
120.w even 4 1 320.4.a.g 1
120.w even 4 1 1600.4.a.bi 1
165.l odd 4 1 605.4.a.d 1
195.s even 4 1 845.4.a.b 1
240.z odd 4 1 1280.4.d.l 2
240.bb even 4 1 1280.4.d.e 2
240.bd odd 4 1 1280.4.d.l 2
240.bf even 4 1 1280.4.d.e 2
255.o even 4 1 1445.4.a.a 1
285.j odd 4 1 1805.4.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 15.e even 4 1
25.4.a.c 1 15.e even 4 1
25.4.b.a 2 3.b odd 2 1
25.4.b.a 2 15.d odd 2 1
45.4.a.d 1 5.c odd 4 1
80.4.a.d 1 60.l odd 4 1
225.4.a.b 1 5.c odd 4 1
225.4.b.c 2 1.a even 1 1 trivial
225.4.b.c 2 5.b even 2 1 inner
245.4.a.a 1 105.k odd 4 1
245.4.e.f 2 105.x even 12 2
245.4.e.g 2 105.w odd 12 2
320.4.a.g 1 120.w even 4 1
320.4.a.h 1 120.q odd 4 1
400.4.a.m 1 60.l odd 4 1
400.4.c.k 2 12.b even 2 1
400.4.c.k 2 60.h even 2 1
405.4.e.c 2 45.k odd 12 2
405.4.e.l 2 45.l even 12 2
605.4.a.d 1 165.l odd 4 1
720.4.a.u 1 20.e even 4 1
845.4.a.b 1 195.s even 4 1
1225.4.a.k 1 105.k odd 4 1
1280.4.d.e 2 240.bb even 4 1
1280.4.d.e 2 240.bf even 4 1
1280.4.d.l 2 240.z odd 4 1
1280.4.d.l 2 240.bd odd 4 1
1445.4.a.a 1 255.o even 4 1
1600.4.a.s 1 120.q odd 4 1
1600.4.a.bi 1 120.w even 4 1
1805.4.a.h 1 285.j odd 4 1
2205.4.a.q 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 $$S_{4}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{2} + 16$$ T2^2 + 16 $$T_{7}^{2} + 36$$ T7^2 + 36 $$T_{11} + 32$$ T11 + 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 36$$
$11$ $$(T + 32)^{2}$$
$13$ $$T^{2} + 1444$$
$17$ $$T^{2} + 676$$
$19$ $$(T + 100)^{2}$$
$23$ $$T^{2} + 6084$$
$29$ $$(T + 50)^{2}$$
$31$ $$(T + 108)^{2}$$
$37$ $$T^{2} + 70756$$
$41$ $$(T + 22)^{2}$$
$43$ $$T^{2} + 195364$$
$47$ $$T^{2} + 264196$$
$53$ $$T^{2} + 4$$
$59$ $$(T - 500)^{2}$$
$61$ $$(T + 518)^{2}$$
$67$ $$T^{2} + 15876$$
$71$ $$(T + 412)^{2}$$
$73$ $$T^{2} + 770884$$
$79$ $$(T + 600)^{2}$$
$83$ $$T^{2} + 79524$$
$89$ $$(T + 150)^{2}$$
$97$ $$T^{2} + 148996$$