# Properties

 Label 1225.2.b.h Level $1225$ Weight $2$ Character orbit 1225.b Analytic conductor $9.782$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(99,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1225.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1225.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: $$9.78167424761$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 35) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{2} q^{3} + ( - \beta_{3} - 1) q^{4} + q^{6} + ( - \beta_{2} - 2 \beta_1) q^{8} + \beta_{3} q^{9}+O(q^{10})$$ q + b1 * q^2 + b2 * q^3 + (-b3 - 1) * q^4 + q^6 + (-b2 - 2*b1) * q^8 + b3 * q^9 $$q + \beta_1 q^{2} + \beta_{2} q^{3} + ( - \beta_{3} - 1) q^{4} + q^{6} + ( - \beta_{2} - 2 \beta_1) q^{8} + \beta_{3} q^{9} + ( - \beta_{3} + 2) q^{11} + (2 \beta_{2} + \beta_1) q^{12} - 2 \beta_1 q^{13} + 3 q^{16} - 2 \beta_1 q^{17} + (\beta_{2} + 3 \beta_1) q^{18} + \beta_{3} q^{19} + ( - \beta_{2} - \beta_1) q^{22} + \beta_{2} q^{23} + ( - \beta_{3} + 1) q^{24} + (2 \beta_{3} + 6) q^{26} - \beta_1 q^{27} + q^{29} + 6 q^{31} + ( - 2 \beta_{2} - \beta_1) q^{32} + (5 \beta_{2} + \beta_1) q^{33} + (2 \beta_{3} + 6) q^{34} + ( - \beta_{3} - 8) q^{36} + (\beta_{2} + 3 \beta_1) q^{38} - 2 q^{39} + (\beta_{3} + 5) q^{41} + ( - 3 \beta_{2} - 2 \beta_1) q^{43} + ( - \beta_{3} + 6) q^{44} + q^{46} + ( - \beta_{2} - \beta_1) q^{47} + 3 \beta_{2} q^{48} - 2 q^{51} + (2 \beta_{2} + 8 \beta_1) q^{52} + (3 \beta_{2} + \beta_1) q^{53} + (\beta_{3} + 3) q^{54} + ( - 3 \beta_{2} - \beta_1) q^{57} + \beta_1 q^{58} + (3 \beta_{3} - 4) q^{59} + ( - 3 \beta_{3} + 3) q^{61} + 6 \beta_1 q^{62} + (\beta_{3} + 7) q^{64} + ( - \beta_{3} + 2) q^{66} + (6 \beta_{2} + 5 \beta_1) q^{67} + (2 \beta_{2} + 8 \beta_1) q^{68} + (\beta_{3} - 3) q^{69} + (3 \beta_{3} - 4) q^{71} + (\beta_{2} - 5 \beta_1) q^{72} + 2 \beta_{2} q^{73} + ( - \beta_{3} - 8) q^{76} - 2 \beta_1 q^{78} + ( - \beta_{3} - 12) q^{79} + (3 \beta_{3} - 1) q^{81} + (\beta_{2} + 8 \beta_1) q^{82} + ( - 4 \beta_{2} + 5 \beta_1) q^{83} + (2 \beta_{3} + 3) q^{86} + \beta_{2} q^{87} + ( - 3 \beta_{2} + \beta_1) q^{88} + ( - 2 \beta_{3} - 3) q^{89} + (2 \beta_{2} + \beta_1) q^{92} + 6 \beta_{2} q^{93} + (\beta_{3} + 2) q^{94} + ( - 2 \beta_{3} + 5) q^{96} + ( - \beta_{2} - 5 \beta_1) q^{97} + (2 \beta_{3} - 8) q^{99}+O(q^{100})$$ q + b1 * q^2 + b2 * q^3 + (-b3 - 1) * q^4 + q^6 + (-b2 - 2*b1) * q^8 + b3 * q^9 + (-b3 + 2) * q^11 + (2*b2 + b1) * q^12 - 2*b1 * q^13 + 3 * q^16 - 2*b1 * q^17 + (b2 + 3*b1) * q^18 + b3 * q^19 + (-b2 - b1) * q^22 + b2 * q^23 + (-b3 + 1) * q^24 + (2*b3 + 6) * q^26 - b1 * q^27 + q^29 + 6 * q^31 + (-2*b2 - b1) * q^32 + (5*b2 + b1) * q^33 + (2*b3 + 6) * q^34 + (-b3 - 8) * q^36 + (b2 + 3*b1) * q^38 - 2 * q^39 + (b3 + 5) * q^41 + (-3*b2 - 2*b1) * q^43 + (-b3 + 6) * q^44 + q^46 + (-b2 - b1) * q^47 + 3*b2 * q^48 - 2 * q^51 + (2*b2 + 8*b1) * q^52 + (3*b2 + b1) * q^53 + (b3 + 3) * q^54 + (-3*b2 - b1) * q^57 + b1 * q^58 + (3*b3 - 4) * q^59 + (-3*b3 + 3) * q^61 + 6*b1 * q^62 + (b3 + 7) * q^64 + (-b3 + 2) * q^66 + (6*b2 + 5*b1) * q^67 + (2*b2 + 8*b1) * q^68 + (b3 - 3) * q^69 + (3*b3 - 4) * q^71 + (b2 - 5*b1) * q^72 + 2*b2 * q^73 + (-b3 - 8) * q^76 - 2*b1 * q^78 + (-b3 - 12) * q^79 + (3*b3 - 1) * q^81 + (b2 + 8*b1) * q^82 + (-4*b2 + 5*b1) * q^83 + (2*b3 + 3) * q^86 + b2 * q^87 + (-3*b2 + b1) * q^88 + (-2*b3 - 3) * q^89 + (2*b2 + b1) * q^92 + 6*b2 * q^93 + (b3 + 2) * q^94 + (-2*b3 + 5) * q^96 + (-b2 - 5*b1) * q^97 + (2*b3 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6}+O(q^{10})$$ 4 * q - 4 * q^4 + 4 * q^6 $$4 q - 4 q^{4} + 4 q^{6} + 8 q^{11} + 12 q^{16} + 4 q^{24} + 24 q^{26} + 4 q^{29} + 24 q^{31} + 24 q^{34} - 32 q^{36} - 8 q^{39} + 20 q^{41} + 24 q^{44} + 4 q^{46} - 8 q^{51} + 12 q^{54} - 16 q^{59} + 12 q^{61} + 28 q^{64} + 8 q^{66} - 12 q^{69} - 16 q^{71} - 32 q^{76} - 48 q^{79} - 4 q^{81} + 12 q^{86} - 12 q^{89} + 8 q^{94} + 20 q^{96} - 32 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^6 + 8 * q^11 + 12 * q^16 + 4 * q^24 + 24 * q^26 + 4 * q^29 + 24 * q^31 + 24 * q^34 - 32 * q^36 - 8 * q^39 + 20 * q^41 + 24 * q^44 + 4 * q^46 - 8 * q^51 + 12 * q^54 - 16 * q^59 + 12 * q^61 + 28 * q^64 + 8 * q^66 - 12 * q^69 - 16 * q^71 - 32 * q^76 - 48 * q^79 - 4 * q^81 + 12 * q^86 - 12 * q^89 + 8 * q^94 + 20 * q^96 - 32 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}$$ v^3 + v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}$$ -v^3 + v^2 - v $$\beta_{3}$$ $$=$$ $$-2\zeta_{8}^{3} + 2\zeta_{8}$$ -2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (b3 - b2 + b1) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (-b3 - b2 + b1) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 0.414214i −3.82843 0 1.00000 0 4.41421i 2.82843 0
99.2 0.414214i 2.41421i 1.82843 0 1.00000 0 1.58579i −2.82843 0
99.3 0.414214i 2.41421i 1.82843 0 1.00000 0 1.58579i −2.82843 0
99.4 2.41421i 0.414214i −3.82843 0 1.00000 0 4.41421i 2.82843 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 1225.2.b.h 4
5.b even 2 1 inner 1225.2.b.h 4
5.c odd 4 1 245.2.a.g 2
5.c odd 4 1 1225.2.a.m 2
7.b odd 2 1 1225.2.b.g 4
7.d odd 6 2 175.2.k.a 8
15.e even 4 1 2205.2.a.q 2
20.e even 4 1 3920.2.a.bv 2
35.c odd 2 1 1225.2.b.g 4
35.f even 4 1 245.2.a.h 2
35.f even 4 1 1225.2.a.k 2
35.i odd 6 2 175.2.k.a 8
35.k even 12 2 35.2.e.a 4
35.k even 12 2 175.2.e.c 4
35.l odd 12 2 245.2.e.e 4
105.k odd 4 1 2205.2.a.n 2
105.w odd 12 2 315.2.j.e 4
140.j odd 4 1 3920.2.a.bq 2
140.x odd 12 2 560.2.q.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 35.k even 12 2
175.2.e.c 4 35.k even 12 2
175.2.k.a 8 7.d odd 6 2
175.2.k.a 8 35.i odd 6 2
245.2.a.g 2 5.c odd 4 1
245.2.a.h 2 35.f even 4 1
245.2.e.e 4 35.l odd 12 2
315.2.j.e 4 105.w odd 12 2
560.2.q.k 4 140.x odd 12 2
1225.2.a.k 2 35.f even 4 1
1225.2.a.m 2 5.c odd 4 1
1225.2.b.g 4 7.b odd 2 1
1225.2.b.g 4 35.c odd 2 1
1225.2.b.h 4 1.a even 1 1 trivial
1225.2.b.h 4 5.b even 2 1 inner
2205.2.a.n 2 105.k odd 4 1
2205.2.a.q 2 15.e even 4 1
3920.2.a.bq 2 140.j odd 4 1
3920.2.a.bv 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1225, [\chi])$$:

 $$T_{2}^{4} + 6T_{2}^{2} + 1$$ T2^4 + 6*T2^2 + 1 $$T_{19}^{2} - 8$$ T19^2 - 8 $$T_{31} - 6$$ T31 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$T^{4} + 6T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 4 T - 4)^{2}$$
$13$ $$T^{4} + 24T^{2} + 16$$
$17$ $$T^{4} + 24T^{2} + 16$$
$19$ $$(T^{2} - 8)^{2}$$
$23$ $$T^{4} + 6T^{2} + 1$$
$29$ $$(T - 1)^{4}$$
$31$ $$(T - 6)^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} - 10 T + 17)^{2}$$
$43$ $$T^{4} + 54T^{2} + 529$$
$47$ $$(T^{2} + 4)^{2}$$
$53$ $$T^{4} + 48T^{2} + 64$$
$59$ $$(T^{2} + 8 T - 56)^{2}$$
$61$ $$(T^{2} - 6 T - 63)^{2}$$
$67$ $$T^{4} + 246 T^{2} + 14161$$
$71$ $$(T^{2} + 8 T - 56)^{2}$$
$73$ $$T^{4} + 24T^{2} + 16$$
$79$ $$(T^{2} + 24 T + 136)^{2}$$
$83$ $$T^{4} + 326 T^{2} + 25921$$
$89$ $$(T^{2} + 6 T - 23)^{2}$$
$97$ $$T^{4} + 136T^{2} + 16$$