# Properties

 Label 325.2.d.d Level $325$ Weight $2$ Character orbit 325.d Analytic conductor $2.595$ 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] = [325,2,Mod(324,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("325.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.d (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: $$2.59513806569$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + i q^{3} + 2 q^{4} + 2 i q^{6} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ q + 2 * q^2 + i * q^3 + 2 * q^4 + 2*i * q^6 + 2 * q^7 + 2 * q^9 $$q + 2 q^{2} + i q^{3} + 2 q^{4} + 2 i q^{6} + 2 q^{7} + 2 q^{9} + 2 i q^{12} + (3 i - 2) q^{13} + 4 q^{14} - 4 q^{16} - 2 i q^{17} + 4 q^{18} - 4 i q^{19} + 2 i q^{21} + i q^{23} + (6 i - 4) q^{26} + 5 i q^{27} + 4 q^{28} - 5 q^{29} - 10 i q^{31} - 8 q^{32} - 4 i q^{34} + 4 q^{36} + 2 q^{37} - 8 i q^{38} + ( - 2 i - 3) q^{39} - 10 i q^{41} + 4 i q^{42} + 11 i q^{43} + 2 i q^{46} - 8 q^{47} - 4 i q^{48} - 3 q^{49} + 2 q^{51} + (6 i - 4) q^{52} - 9 i q^{53} + 10 i q^{54} + 4 q^{57} - 10 q^{58} + 6 i q^{59} + 7 q^{61} - 20 i q^{62} + 4 q^{63} - 8 q^{64} + 12 q^{67} - 4 i q^{68} - q^{69} + 10 i q^{71} - 14 q^{73} + 4 q^{74} - 8 i q^{76} + ( - 4 i - 6) q^{78} + 5 q^{79} + q^{81} - 20 i q^{82} + 6 q^{83} + 4 i q^{84} + 22 i q^{86} - 5 i q^{87} + 6 i q^{89} + (6 i - 4) q^{91} + 2 i q^{92} + 10 q^{93} - 16 q^{94} - 8 i q^{96} + 2 q^{97} - 6 q^{98} +O(q^{100})$$ q + 2 * q^2 + i * q^3 + 2 * q^4 + 2*i * q^6 + 2 * q^7 + 2 * q^9 + 2*i * q^12 + (3*i - 2) * q^13 + 4 * q^14 - 4 * q^16 - 2*i * q^17 + 4 * q^18 - 4*i * q^19 + 2*i * q^21 + i * q^23 + (6*i - 4) * q^26 + 5*i * q^27 + 4 * q^28 - 5 * q^29 - 10*i * q^31 - 8 * q^32 - 4*i * q^34 + 4 * q^36 + 2 * q^37 - 8*i * q^38 + (-2*i - 3) * q^39 - 10*i * q^41 + 4*i * q^42 + 11*i * q^43 + 2*i * q^46 - 8 * q^47 - 4*i * q^48 - 3 * q^49 + 2 * q^51 + (6*i - 4) * q^52 - 9*i * q^53 + 10*i * q^54 + 4 * q^57 - 10 * q^58 + 6*i * q^59 + 7 * q^61 - 20*i * q^62 + 4 * q^63 - 8 * q^64 + 12 * q^67 - 4*i * q^68 - q^69 + 10*i * q^71 - 14 * q^73 + 4 * q^74 - 8*i * q^76 + (-4*i - 6) * q^78 + 5 * q^79 + q^81 - 20*i * q^82 + 6 * q^83 + 4*i * q^84 + 22*i * q^86 - 5*i * q^87 + 6*i * q^89 + (6*i - 4) * q^91 + 2*i * q^92 + 10 * q^93 - 16 * q^94 - 8*i * q^96 + 2 * q^97 - 6 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^9 $$2 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{9} - 4 q^{13} + 8 q^{14} - 8 q^{16} + 8 q^{18} - 8 q^{26} + 8 q^{28} - 10 q^{29} - 16 q^{32} + 8 q^{36} + 4 q^{37} - 6 q^{39} - 16 q^{47} - 6 q^{49} + 4 q^{51} - 8 q^{52} + 8 q^{57} - 20 q^{58} + 14 q^{61} + 8 q^{63} - 16 q^{64} + 24 q^{67} - 2 q^{69} - 28 q^{73} + 8 q^{74} - 12 q^{78} + 10 q^{79} + 2 q^{81} + 12 q^{83} - 8 q^{91} + 20 q^{93} - 32 q^{94} + 4 q^{97} - 12 q^{98}+O(q^{100})$$ 2 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^9 - 4 * q^13 + 8 * q^14 - 8 * q^16 + 8 * q^18 - 8 * q^26 + 8 * q^28 - 10 * q^29 - 16 * q^32 + 8 * q^36 + 4 * q^37 - 6 * q^39 - 16 * q^47 - 6 * q^49 + 4 * q^51 - 8 * q^52 + 8 * q^57 - 20 * q^58 + 14 * q^61 + 8 * q^63 - 16 * q^64 + 24 * q^67 - 2 * q^69 - 28 * q^73 + 8 * q^74 - 12 * q^78 + 10 * q^79 + 2 * q^81 + 12 * q^83 - 8 * q^91 + 20 * q^93 - 32 * q^94 + 4 * q^97 - 12 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
324.1
 − 1.00000i 1.00000i
2.00000 1.00000i 2.00000 0 2.00000i 2.00000 0 2.00000 0
324.2 2.00000 1.00000i 2.00000 0 2.00000i 2.00000 0 2.00000 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
65.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.d.d 2
5.b even 2 1 325.2.d.a 2
5.c odd 4 1 325.2.c.c 2
5.c odd 4 1 325.2.c.d yes 2
13.b even 2 1 325.2.d.a 2
65.d even 2 1 inner 325.2.d.d 2
65.f even 4 1 4225.2.a.a 1
65.f even 4 1 4225.2.a.c 1
65.h odd 4 1 325.2.c.c 2
65.h odd 4 1 325.2.c.d yes 2
65.k even 4 1 4225.2.a.o 1
65.k even 4 1 4225.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.c.c 2 5.c odd 4 1
325.2.c.c 2 65.h odd 4 1
325.2.c.d yes 2 5.c odd 4 1
325.2.c.d yes 2 65.h odd 4 1
325.2.d.a 2 5.b even 2 1
325.2.d.a 2 13.b even 2 1
325.2.d.d 2 1.a even 1 1 trivial
325.2.d.d 2 65.d even 2 1 inner
4225.2.a.a 1 65.f even 4 1
4225.2.a.c 1 65.f even 4 1
4225.2.a.o 1 65.k even 4 1
4225.2.a.q 1 65.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 2$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4T + 13$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2} + 16$$
$23$ $$T^{2} + 1$$
$29$ $$(T + 5)^{2}$$
$31$ $$T^{2} + 100$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} + 100$$
$43$ $$T^{2} + 121$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 81$$
$59$ $$T^{2} + 36$$
$61$ $$(T - 7)^{2}$$
$67$ $$(T - 12)^{2}$$
$71$ $$T^{2} + 100$$
$73$ $$(T + 14)^{2}$$
$79$ $$(T - 5)^{2}$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} + 36$$
$97$ $$(T - 2)^{2}$$