# Properties

 Label 225.4.b.f Level $225$ Weight $4$ Character orbit 225.b Analytic conductor $13.275$ Analytic rank $0$ Dimension $2$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 7 q^{4} - 6 i q^{7} + 15 i q^{8} +O(q^{10})$$ q + i * q^2 + 7 * q^4 - 6*i * q^7 + 15*i * q^8 $$q + i q^{2} + 7 q^{4} - 6 i q^{7} + 15 i q^{8} + 43 q^{11} - 28 i q^{13} + 6 q^{14} + 41 q^{16} + 91 i q^{17} + 35 q^{19} + 43 i q^{22} - 162 i q^{23} + 28 q^{26} - 42 i q^{28} + 160 q^{29} + 42 q^{31} + 161 i q^{32} - 91 q^{34} + 314 i q^{37} + 35 i q^{38} + 203 q^{41} + 92 i q^{43} + 301 q^{44} + 162 q^{46} + 196 i q^{47} + 307 q^{49} - 196 i q^{52} - 82 i q^{53} + 90 q^{56} + 160 i q^{58} - 280 q^{59} - 518 q^{61} + 42 i q^{62} + 167 q^{64} - 141 i q^{67} + 637 i q^{68} - 412 q^{71} - 763 i q^{73} - 314 q^{74} + 245 q^{76} - 258 i q^{77} - 510 q^{79} + 203 i q^{82} - 777 i q^{83} - 92 q^{86} + 645 i q^{88} - 945 q^{89} - 168 q^{91} - 1134 i q^{92} - 196 q^{94} - 1246 i q^{97} + 307 i q^{98} +O(q^{100})$$ q + i * q^2 + 7 * q^4 - 6*i * q^7 + 15*i * q^8 + 43 * q^11 - 28*i * q^13 + 6 * q^14 + 41 * q^16 + 91*i * q^17 + 35 * q^19 + 43*i * q^22 - 162*i * q^23 + 28 * q^26 - 42*i * q^28 + 160 * q^29 + 42 * q^31 + 161*i * q^32 - 91 * q^34 + 314*i * q^37 + 35*i * q^38 + 203 * q^41 + 92*i * q^43 + 301 * q^44 + 162 * q^46 + 196*i * q^47 + 307 * q^49 - 196*i * q^52 - 82*i * q^53 + 90 * q^56 + 160*i * q^58 - 280 * q^59 - 518 * q^61 + 42*i * q^62 + 167 * q^64 - 141*i * q^67 + 637*i * q^68 - 412 * q^71 - 763*i * q^73 - 314 * q^74 + 245 * q^76 - 258*i * q^77 - 510 * q^79 + 203*i * q^82 - 777*i * q^83 - 92 * q^86 + 645*i * q^88 - 945 * q^89 - 168 * q^91 - 1134*i * q^92 - 196 * q^94 - 1246*i * q^97 + 307*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{4}+O(q^{10})$$ 2 * q + 14 * q^4 $$2 q + 14 q^{4} + 86 q^{11} + 12 q^{14} + 82 q^{16} + 70 q^{19} + 56 q^{26} + 320 q^{29} + 84 q^{31} - 182 q^{34} + 406 q^{41} + 602 q^{44} + 324 q^{46} + 614 q^{49} + 180 q^{56} - 560 q^{59} - 1036 q^{61} + 334 q^{64} - 824 q^{71} - 628 q^{74} + 490 q^{76} - 1020 q^{79} - 184 q^{86} - 1890 q^{89} - 336 q^{91} - 392 q^{94}+O(q^{100})$$ 2 * q + 14 * q^4 + 86 * q^11 + 12 * q^14 + 82 * q^16 + 70 * q^19 + 56 * q^26 + 320 * q^29 + 84 * q^31 - 182 * q^34 + 406 * q^41 + 602 * q^44 + 324 * q^46 + 614 * q^49 + 180 * q^56 - 560 * q^59 - 1036 * q^61 + 334 * q^64 - 824 * q^71 - 628 * q^74 + 490 * q^76 - 1020 * q^79 - 184 * q^86 - 1890 * q^89 - 336 * q^91 - 392 * 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
1.00000i 0 7.00000 0 0 6.00000i 15.0000i 0 0
199.2 1.00000i 0 7.00000 0 0 6.00000i 15.0000i 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.f 2
3.b odd 2 1 25.4.b.b 2
5.b even 2 1 inner 225.4.b.f 2
5.c odd 4 1 225.4.a.c 1
5.c odd 4 1 225.4.a.e 1
12.b even 2 1 400.4.c.e 2
15.d odd 2 1 25.4.b.b 2
15.e even 4 1 25.4.a.a 1
15.e even 4 1 25.4.a.b yes 1
60.h even 2 1 400.4.c.e 2
60.l odd 4 1 400.4.a.c 1
60.l odd 4 1 400.4.a.s 1
105.k odd 4 1 1225.4.a.h 1
105.k odd 4 1 1225.4.a.i 1
120.q odd 4 1 1600.4.a.h 1
120.q odd 4 1 1600.4.a.bs 1
120.w even 4 1 1600.4.a.i 1
120.w even 4 1 1600.4.a.bt 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 15.e even 4 1
25.4.a.b yes 1 15.e even 4 1
25.4.b.b 2 3.b odd 2 1
25.4.b.b 2 15.d odd 2 1
225.4.a.c 1 5.c odd 4 1
225.4.a.e 1 5.c odd 4 1
225.4.b.f 2 1.a even 1 1 trivial
225.4.b.f 2 5.b even 2 1 inner
400.4.a.c 1 60.l odd 4 1
400.4.a.s 1 60.l odd 4 1
400.4.c.e 2 12.b even 2 1
400.4.c.e 2 60.h even 2 1
1225.4.a.h 1 105.k odd 4 1
1225.4.a.i 1 105.k odd 4 1
1600.4.a.h 1 120.q odd 4 1
1600.4.a.i 1 120.w even 4 1
1600.4.a.bs 1 120.q odd 4 1
1600.4.a.bt 1 120.w even 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{7}^{2} + 36$$ T7^2 + 36 $$T_{11} - 43$$ T11 - 43

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 36$$
$11$ $$(T - 43)^{2}$$
$13$ $$T^{2} + 784$$
$17$ $$T^{2} + 8281$$
$19$ $$(T - 35)^{2}$$
$23$ $$T^{2} + 26244$$
$29$ $$(T - 160)^{2}$$
$31$ $$(T - 42)^{2}$$
$37$ $$T^{2} + 98596$$
$41$ $$(T - 203)^{2}$$
$43$ $$T^{2} + 8464$$
$47$ $$T^{2} + 38416$$
$53$ $$T^{2} + 6724$$
$59$ $$(T + 280)^{2}$$
$61$ $$(T + 518)^{2}$$
$67$ $$T^{2} + 19881$$
$71$ $$(T + 412)^{2}$$
$73$ $$T^{2} + 582169$$
$79$ $$(T + 510)^{2}$$
$83$ $$T^{2} + 603729$$
$89$ $$(T + 945)^{2}$$
$97$ $$T^{2} + 1552516$$