# Properties

 Label 225.4.b.e 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 7 q^{4} + 24 i q^{7} + 15 i q^{8}+O(q^{10})$$ q + i * q^2 + 7 * q^4 + 24*i * q^7 + 15*i * q^8 $$q + i q^{2} + 7 q^{4} + 24 i q^{7} + 15 i q^{8} - 52 q^{11} + 22 i q^{13} - 24 q^{14} + 41 q^{16} - 14 i q^{17} + 20 q^{19} - 52 i q^{22} + 168 i q^{23} - 22 q^{26} + 168 i q^{28} + 230 q^{29} - 288 q^{31} + 161 i q^{32} + 14 q^{34} + 34 i q^{37} + 20 i q^{38} - 122 q^{41} - 188 i q^{43} - 364 q^{44} - 168 q^{46} + 256 i q^{47} - 233 q^{49} + 154 i q^{52} + 338 i q^{53} - 360 q^{56} + 230 i q^{58} + 100 q^{59} + 742 q^{61} - 288 i q^{62} + 167 q^{64} + 84 i q^{67} - 98 i q^{68} + 328 q^{71} - 38 i q^{73} - 34 q^{74} + 140 q^{76} - 1248 i q^{77} + 240 q^{79} - 122 i q^{82} - 1212 i q^{83} + 188 q^{86} - 780 i q^{88} + 330 q^{89} - 528 q^{91} + 1176 i q^{92} - 256 q^{94} - 866 i q^{97} - 233 i q^{98} +O(q^{100})$$ q + i * q^2 + 7 * q^4 + 24*i * q^7 + 15*i * q^8 - 52 * q^11 + 22*i * q^13 - 24 * q^14 + 41 * q^16 - 14*i * q^17 + 20 * q^19 - 52*i * q^22 + 168*i * q^23 - 22 * q^26 + 168*i * q^28 + 230 * q^29 - 288 * q^31 + 161*i * q^32 + 14 * q^34 + 34*i * q^37 + 20*i * q^38 - 122 * q^41 - 188*i * q^43 - 364 * q^44 - 168 * q^46 + 256*i * q^47 - 233 * q^49 + 154*i * q^52 + 338*i * q^53 - 360 * q^56 + 230*i * q^58 + 100 * q^59 + 742 * q^61 - 288*i * q^62 + 167 * q^64 + 84*i * q^67 - 98*i * q^68 + 328 * q^71 - 38*i * q^73 - 34 * q^74 + 140 * q^76 - 1248*i * q^77 + 240 * q^79 - 122*i * q^82 - 1212*i * q^83 + 188 * q^86 - 780*i * q^88 + 330 * q^89 - 528 * q^91 + 1176*i * q^92 - 256 * q^94 - 866*i * q^97 - 233*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{4}+O(q^{10})$$ 2 * q + 14 * q^4 $$2 q + 14 q^{4} - 104 q^{11} - 48 q^{14} + 82 q^{16} + 40 q^{19} - 44 q^{26} + 460 q^{29} - 576 q^{31} + 28 q^{34} - 244 q^{41} - 728 q^{44} - 336 q^{46} - 466 q^{49} - 720 q^{56} + 200 q^{59} + 1484 q^{61} + 334 q^{64} + 656 q^{71} - 68 q^{74} + 280 q^{76} + 480 q^{79} + 376 q^{86} + 660 q^{89} - 1056 q^{91} - 512 q^{94}+O(q^{100})$$ 2 * q + 14 * q^4 - 104 * q^11 - 48 * q^14 + 82 * q^16 + 40 * q^19 - 44 * q^26 + 460 * q^29 - 576 * q^31 + 28 * q^34 - 244 * q^41 - 728 * q^44 - 336 * q^46 - 466 * q^49 - 720 * q^56 + 200 * q^59 + 1484 * q^61 + 334 * q^64 + 656 * q^71 - 68 * q^74 + 280 * q^76 + 480 * q^79 + 376 * q^86 + 660 * q^89 - 1056 * q^91 - 512 * 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 24.0000i 15.0000i 0 0
199.2 1.00000i 0 7.00000 0 0 24.0000i 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.e 2
3.b odd 2 1 75.4.b.b 2
5.b even 2 1 inner 225.4.b.e 2
5.c odd 4 1 45.4.a.c 1
5.c odd 4 1 225.4.a.f 1
12.b even 2 1 1200.4.f.b 2
15.d odd 2 1 75.4.b.b 2
15.e even 4 1 15.4.a.a 1
15.e even 4 1 75.4.a.b 1
20.e even 4 1 720.4.a.n 1
35.f even 4 1 2205.4.a.l 1
45.k odd 12 2 405.4.e.i 2
45.l even 12 2 405.4.e.g 2
60.h even 2 1 1200.4.f.b 2
60.l odd 4 1 240.4.a.e 1
60.l odd 4 1 1200.4.a.t 1
105.k odd 4 1 735.4.a.e 1
120.q odd 4 1 960.4.a.ba 1
120.w even 4 1 960.4.a.b 1
165.l odd 4 1 1815.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 15.e even 4 1
45.4.a.c 1 5.c odd 4 1
75.4.a.b 1 15.e even 4 1
75.4.b.b 2 3.b odd 2 1
75.4.b.b 2 15.d odd 2 1
225.4.a.f 1 5.c odd 4 1
225.4.b.e 2 1.a even 1 1 trivial
225.4.b.e 2 5.b even 2 1 inner
240.4.a.e 1 60.l odd 4 1
405.4.e.g 2 45.l even 12 2
405.4.e.i 2 45.k odd 12 2
720.4.a.n 1 20.e even 4 1
735.4.a.e 1 105.k odd 4 1
960.4.a.b 1 120.w even 4 1
960.4.a.ba 1 120.q odd 4 1
1200.4.a.t 1 60.l odd 4 1
1200.4.f.b 2 12.b even 2 1
1200.4.f.b 2 60.h even 2 1
1815.4.a.e 1 165.l odd 4 1
2205.4.a.l 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 $$S_{4}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{7}^{2} + 576$$ T7^2 + 576 $$T_{11} + 52$$ T11 + 52

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 576$$
$11$ $$(T + 52)^{2}$$
$13$ $$T^{2} + 484$$
$17$ $$T^{2} + 196$$
$19$ $$(T - 20)^{2}$$
$23$ $$T^{2} + 28224$$
$29$ $$(T - 230)^{2}$$
$31$ $$(T + 288)^{2}$$
$37$ $$T^{2} + 1156$$
$41$ $$(T + 122)^{2}$$
$43$ $$T^{2} + 35344$$
$47$ $$T^{2} + 65536$$
$53$ $$T^{2} + 114244$$
$59$ $$(T - 100)^{2}$$
$61$ $$(T - 742)^{2}$$
$67$ $$T^{2} + 7056$$
$71$ $$(T - 328)^{2}$$
$73$ $$T^{2} + 1444$$
$79$ $$(T - 240)^{2}$$
$83$ $$T^{2} + 1468944$$
$89$ $$(T - 330)^{2}$$
$97$ $$T^{2} + 749956$$