# Properties

 Label 325.2.d.b 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: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + 2 i q^{3} - q^{4} - 2 i q^{6} + 5 q^{7} + 3 q^{8} - q^{9} +O(q^{10})$$ q - q^2 + 2*i * q^3 - q^4 - 2*i * q^6 + 5 * q^7 + 3 * q^8 - q^9 $$q - q^{2} + 2 i q^{3} - q^{4} - 2 i q^{6} + 5 q^{7} + 3 q^{8} - q^{9} - 3 i q^{11} - 2 i q^{12} + (3 i - 2) q^{13} - 5 q^{14} - q^{16} + 5 i q^{17} + q^{18} + 4 i q^{19} + 10 i q^{21} + 3 i q^{22} - 4 i q^{23} + 6 i q^{24} + ( - 3 i + 2) q^{26} + 4 i q^{27} - 5 q^{28} + q^{29} + i q^{31} - 5 q^{32} + 6 q^{33} - 5 i q^{34} + q^{36} - 4 q^{37} - 4 i q^{38} + ( - 4 i - 6) q^{39} - 8 i q^{41} - 10 i q^{42} + 4 i q^{43} + 3 i q^{44} + 4 i q^{46} + 7 q^{47} - 2 i q^{48} + 18 q^{49} - 10 q^{51} + ( - 3 i + 2) q^{52} + 3 i q^{53} - 4 i q^{54} + 15 q^{56} - 8 q^{57} - q^{58} - 3 i q^{59} + q^{61} - i q^{62} - 5 q^{63} + 7 q^{64} - 6 q^{66} - 3 q^{67} - 5 i q^{68} + 8 q^{69} + 8 i q^{71} - 3 q^{72} + 4 q^{73} + 4 q^{74} - 4 i q^{76} - 15 i q^{77} + (4 i + 6) q^{78} - 10 q^{79} - 11 q^{81} + 8 i q^{82} - 9 q^{83} - 10 i q^{84} - 4 i q^{86} + 2 i q^{87} - 9 i q^{88} - 18 i q^{89} + (15 i - 10) q^{91} + 4 i q^{92} - 2 q^{93} - 7 q^{94} - 10 i q^{96} + 14 q^{97} - 18 q^{98} + 3 i q^{99} +O(q^{100})$$ q - q^2 + 2*i * q^3 - q^4 - 2*i * q^6 + 5 * q^7 + 3 * q^8 - q^9 - 3*i * q^11 - 2*i * q^12 + (3*i - 2) * q^13 - 5 * q^14 - q^16 + 5*i * q^17 + q^18 + 4*i * q^19 + 10*i * q^21 + 3*i * q^22 - 4*i * q^23 + 6*i * q^24 + (-3*i + 2) * q^26 + 4*i * q^27 - 5 * q^28 + q^29 + i * q^31 - 5 * q^32 + 6 * q^33 - 5*i * q^34 + q^36 - 4 * q^37 - 4*i * q^38 + (-4*i - 6) * q^39 - 8*i * q^41 - 10*i * q^42 + 4*i * q^43 + 3*i * q^44 + 4*i * q^46 + 7 * q^47 - 2*i * q^48 + 18 * q^49 - 10 * q^51 + (-3*i + 2) * q^52 + 3*i * q^53 - 4*i * q^54 + 15 * q^56 - 8 * q^57 - q^58 - 3*i * q^59 + q^61 - i * q^62 - 5 * q^63 + 7 * q^64 - 6 * q^66 - 3 * q^67 - 5*i * q^68 + 8 * q^69 + 8*i * q^71 - 3 * q^72 + 4 * q^73 + 4 * q^74 - 4*i * q^76 - 15*i * q^77 + (4*i + 6) * q^78 - 10 * q^79 - 11 * q^81 + 8*i * q^82 - 9 * q^83 - 10*i * q^84 - 4*i * q^86 + 2*i * q^87 - 9*i * q^88 - 18*i * q^89 + (15*i - 10) * q^91 + 4*i * q^92 - 2 * q^93 - 7 * q^94 - 10*i * q^96 + 14 * q^97 - 18 * q^98 + 3*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{4} + 10 q^{7} + 6 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^4 + 10 * q^7 + 6 * q^8 - 2 * q^9 $$2 q - 2 q^{2} - 2 q^{4} + 10 q^{7} + 6 q^{8} - 2 q^{9} - 4 q^{13} - 10 q^{14} - 2 q^{16} + 2 q^{18} + 4 q^{26} - 10 q^{28} + 2 q^{29} - 10 q^{32} + 12 q^{33} + 2 q^{36} - 8 q^{37} - 12 q^{39} + 14 q^{47} + 36 q^{49} - 20 q^{51} + 4 q^{52} + 30 q^{56} - 16 q^{57} - 2 q^{58} + 2 q^{61} - 10 q^{63} + 14 q^{64} - 12 q^{66} - 6 q^{67} + 16 q^{69} - 6 q^{72} + 8 q^{73} + 8 q^{74} + 12 q^{78} - 20 q^{79} - 22 q^{81} - 18 q^{83} - 20 q^{91} - 4 q^{93} - 14 q^{94} + 28 q^{97} - 36 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^4 + 10 * q^7 + 6 * q^8 - 2 * q^9 - 4 * q^13 - 10 * q^14 - 2 * q^16 + 2 * q^18 + 4 * q^26 - 10 * q^28 + 2 * q^29 - 10 * q^32 + 12 * q^33 + 2 * q^36 - 8 * q^37 - 12 * q^39 + 14 * q^47 + 36 * q^49 - 20 * q^51 + 4 * q^52 + 30 * q^56 - 16 * q^57 - 2 * q^58 + 2 * q^61 - 10 * q^63 + 14 * q^64 - 12 * q^66 - 6 * q^67 + 16 * q^69 - 6 * q^72 + 8 * q^73 + 8 * q^74 + 12 * q^78 - 20 * q^79 - 22 * q^81 - 18 * q^83 - 20 * q^91 - 4 * q^93 - 14 * q^94 + 28 * q^97 - 36 * 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
−1.00000 2.00000i −1.00000 0 2.00000i 5.00000 3.00000 −1.00000 0
324.2 −1.00000 2.00000i −1.00000 0 2.00000i 5.00000 3.00000 −1.00000 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.b 2
5.b even 2 1 325.2.d.c 2
5.c odd 4 1 325.2.c.a 2
5.c odd 4 1 325.2.c.f yes 2
13.b even 2 1 325.2.d.c 2
65.d even 2 1 inner 325.2.d.b 2
65.f even 4 1 4225.2.a.l 1
65.f even 4 1 4225.2.a.n 1
65.h odd 4 1 325.2.c.a 2
65.h odd 4 1 325.2.c.f yes 2
65.k even 4 1 4225.2.a.d 1
65.k even 4 1 4225.2.a.f 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$(T - 5)^{2}$$
$11$ $$T^{2} + 9$$
$13$ $$T^{2} + 4T + 13$$
$17$ $$T^{2} + 25$$
$19$ $$T^{2} + 16$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 1)^{2}$$
$31$ $$T^{2} + 1$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} + 64$$
$43$ $$T^{2} + 16$$
$47$ $$(T - 7)^{2}$$
$53$ $$T^{2} + 9$$
$59$ $$T^{2} + 9$$
$61$ $$(T - 1)^{2}$$
$67$ $$(T + 3)^{2}$$
$71$ $$T^{2} + 64$$
$73$ $$(T - 4)^{2}$$
$79$ $$(T + 10)^{2}$$
$83$ $$(T + 9)^{2}$$
$89$ $$T^{2} + 324$$
$97$ $$(T - 14)^{2}$$