# Properties

 Label 325.2.k.a Level $325$ Weight $2$ Character orbit 325.k Analytic conductor $2.595$ Analytic rank $1$ 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(57,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1, 3]))

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

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

chi := DirichletCharacter("325.57");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.k (of order $$4$$, degree $$2$$, 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: $$1$$ 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: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( - i - 1) q^{3} - q^{4} + (i + 1) q^{6} - 2 i q^{7} + 3 q^{8} - i q^{9} +O(q^{10})$$ q - q^2 + (-i - 1) * q^3 - q^4 + (i + 1) * q^6 - 2*i * q^7 + 3 * q^8 - i * q^9 $$q - q^{2} + ( - i - 1) q^{3} - q^{4} + (i + 1) q^{6} - 2 i q^{7} + 3 q^{8} - i q^{9} + (i - 1) q^{11} + (i + 1) q^{12} + (2 i - 3) q^{13} + 2 i q^{14} - q^{16} + ( - i - 1) q^{17} + i q^{18} + (5 i - 5) q^{19} + (2 i - 2) q^{21} + ( - i + 1) q^{22} + (3 i - 3) q^{23} + ( - 3 i - 3) q^{24} + ( - 2 i + 3) q^{26} + (4 i - 4) q^{27} + 2 i q^{28} + (5 i + 5) q^{31} - 5 q^{32} + 2 q^{33} + (i + 1) q^{34} + i q^{36} + ( - 5 i + 5) q^{38} + (i + 5) q^{39} + ( - 7 i - 7) q^{41} + ( - 2 i + 2) q^{42} + ( - i + 1) q^{43} + ( - i + 1) q^{44} + ( - 3 i + 3) q^{46} + 6 i q^{47} + (i + 1) q^{48} + 3 q^{49} + 2 i q^{51} + ( - 2 i + 3) q^{52} + ( - 5 i - 5) q^{53} + ( - 4 i + 4) q^{54} - 6 i q^{56} + 10 q^{57} + ( - 7 i - 7) q^{59} - 14 q^{61} + ( - 5 i - 5) q^{62} - 2 q^{63} + 7 q^{64} - 2 q^{66} + 4 q^{67} + (i + 1) q^{68} + 6 q^{69} + (i + 1) q^{71} - 3 i q^{72} + 10 q^{73} + ( - 5 i + 5) q^{76} + (2 i + 2) q^{77} + ( - i - 5) q^{78} + 2 i q^{79} + 5 q^{81} + (7 i + 7) q^{82} - 6 i q^{83} + ( - 2 i + 2) q^{84} + (i - 1) q^{86} + (3 i - 3) q^{88} + (5 i + 5) q^{89} + (6 i + 4) q^{91} + ( - 3 i + 3) q^{92} - 10 i q^{93} - 6 i q^{94} + (5 i + 5) q^{96} - 2 q^{97} - 3 q^{98} + (i + 1) q^{99} +O(q^{100})$$ q - q^2 + (-i - 1) * q^3 - q^4 + (i + 1) * q^6 - 2*i * q^7 + 3 * q^8 - i * q^9 + (i - 1) * q^11 + (i + 1) * q^12 + (2*i - 3) * q^13 + 2*i * q^14 - q^16 + (-i - 1) * q^17 + i * q^18 + (5*i - 5) * q^19 + (2*i - 2) * q^21 + (-i + 1) * q^22 + (3*i - 3) * q^23 + (-3*i - 3) * q^24 + (-2*i + 3) * q^26 + (4*i - 4) * q^27 + 2*i * q^28 + (5*i + 5) * q^31 - 5 * q^32 + 2 * q^33 + (i + 1) * q^34 + i * q^36 + (-5*i + 5) * q^38 + (i + 5) * q^39 + (-7*i - 7) * q^41 + (-2*i + 2) * q^42 + (-i + 1) * q^43 + (-i + 1) * q^44 + (-3*i + 3) * q^46 + 6*i * q^47 + (i + 1) * q^48 + 3 * q^49 + 2*i * q^51 + (-2*i + 3) * q^52 + (-5*i - 5) * q^53 + (-4*i + 4) * q^54 - 6*i * q^56 + 10 * q^57 + (-7*i - 7) * q^59 - 14 * q^61 + (-5*i - 5) * q^62 - 2 * q^63 + 7 * q^64 - 2 * q^66 + 4 * q^67 + (i + 1) * q^68 + 6 * q^69 + (i + 1) * q^71 - 3*i * q^72 + 10 * q^73 + (-5*i + 5) * q^76 + (2*i + 2) * q^77 + (-i - 5) * q^78 + 2*i * q^79 + 5 * q^81 + (7*i + 7) * q^82 - 6*i * q^83 + (-2*i + 2) * q^84 + (i - 1) * q^86 + (3*i - 3) * q^88 + (5*i + 5) * q^89 + (6*i + 4) * q^91 + (-3*i + 3) * q^92 - 10*i * q^93 - 6*i * q^94 + (5*i + 5) * q^96 - 2 * q^97 - 3 * q^98 + (i + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 6 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 6 * q^8 $$2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 6 q^{8} - 2 q^{11} + 2 q^{12} - 6 q^{13} - 2 q^{16} - 2 q^{17} - 10 q^{19} - 4 q^{21} + 2 q^{22} - 6 q^{23} - 6 q^{24} + 6 q^{26} - 8 q^{27} + 10 q^{31} - 10 q^{32} + 4 q^{33} + 2 q^{34} + 10 q^{38} + 10 q^{39} - 14 q^{41} + 4 q^{42} + 2 q^{43} + 2 q^{44} + 6 q^{46} + 2 q^{48} + 6 q^{49} + 6 q^{52} - 10 q^{53} + 8 q^{54} + 20 q^{57} - 14 q^{59} - 28 q^{61} - 10 q^{62} - 4 q^{63} + 14 q^{64} - 4 q^{66} + 8 q^{67} + 2 q^{68} + 12 q^{69} + 2 q^{71} + 20 q^{73} + 10 q^{76} + 4 q^{77} - 10 q^{78} + 10 q^{81} + 14 q^{82} + 4 q^{84} - 2 q^{86} - 6 q^{88} + 10 q^{89} + 8 q^{91} + 6 q^{92} + 10 q^{96} - 4 q^{97} - 6 q^{98} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 6 * q^8 - 2 * q^11 + 2 * q^12 - 6 * q^13 - 2 * q^16 - 2 * q^17 - 10 * q^19 - 4 * q^21 + 2 * q^22 - 6 * q^23 - 6 * q^24 + 6 * q^26 - 8 * q^27 + 10 * q^31 - 10 * q^32 + 4 * q^33 + 2 * q^34 + 10 * q^38 + 10 * q^39 - 14 * q^41 + 4 * q^42 + 2 * q^43 + 2 * q^44 + 6 * q^46 + 2 * q^48 + 6 * q^49 + 6 * q^52 - 10 * q^53 + 8 * q^54 + 20 * q^57 - 14 * q^59 - 28 * q^61 - 10 * q^62 - 4 * q^63 + 14 * q^64 - 4 * q^66 + 8 * q^67 + 2 * q^68 + 12 * q^69 + 2 * q^71 + 20 * q^73 + 10 * q^76 + 4 * q^77 - 10 * q^78 + 10 * q^81 + 14 * q^82 + 4 * q^84 - 2 * q^86 - 6 * q^88 + 10 * q^89 + 8 * q^91 + 6 * q^92 + 10 * q^96 - 4 * q^97 - 6 * q^98 + 2 * q^99

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1
 − 1.00000i 1.00000i
−1.00000 −1.00000 + 1.00000i −1.00000 0 1.00000 1.00000i 2.00000i 3.00000 1.00000i 0
268.1 −1.00000 −1.00000 1.00000i −1.00000 0 1.00000 + 1.00000i 2.00000i 3.00000 1.00000i 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.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.k.a 2
5.b even 2 1 65.2.k.a yes 2
5.c odd 4 1 65.2.f.a 2
5.c odd 4 1 325.2.f.a 2
13.d odd 4 1 325.2.f.a 2
15.d odd 2 1 585.2.w.b 2
15.e even 4 1 585.2.n.c 2
20.d odd 2 1 1040.2.bg.a 2
20.e even 4 1 1040.2.cd.b 2
65.d even 2 1 845.2.k.a 2
65.f even 4 1 65.2.k.a yes 2
65.g odd 4 1 65.2.f.a 2
65.g odd 4 1 845.2.f.a 2
65.h odd 4 1 845.2.f.a 2
65.k even 4 1 inner 325.2.k.a 2
65.k even 4 1 845.2.k.a 2
65.l even 6 2 845.2.o.b 4
65.n even 6 2 845.2.o.a 4
65.o even 12 2 845.2.o.b 4
65.q odd 12 2 845.2.t.a 4
65.r odd 12 2 845.2.t.b 4
65.s odd 12 2 845.2.t.a 4
65.s odd 12 2 845.2.t.b 4
65.t even 12 2 845.2.o.a 4
195.n even 4 1 585.2.n.c 2
195.u odd 4 1 585.2.w.b 2
260.l odd 4 1 1040.2.bg.a 2
260.u even 4 1 1040.2.cd.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.a 2 5.c odd 4 1
65.2.f.a 2 65.g odd 4 1
65.2.k.a yes 2 5.b even 2 1
65.2.k.a yes 2 65.f even 4 1
325.2.f.a 2 5.c odd 4 1
325.2.f.a 2 13.d odd 4 1
325.2.k.a 2 1.a even 1 1 trivial
325.2.k.a 2 65.k even 4 1 inner
585.2.n.c 2 15.e even 4 1
585.2.n.c 2 195.n even 4 1
585.2.w.b 2 15.d odd 2 1
585.2.w.b 2 195.u odd 4 1
845.2.f.a 2 65.g odd 4 1
845.2.f.a 2 65.h odd 4 1
845.2.k.a 2 65.d even 2 1
845.2.k.a 2 65.k even 4 1
845.2.o.a 4 65.n even 6 2
845.2.o.a 4 65.t even 12 2
845.2.o.b 4 65.l even 6 2
845.2.o.b 4 65.o even 12 2
845.2.t.a 4 65.q odd 12 2
845.2.t.a 4 65.s odd 12 2
845.2.t.b 4 65.r odd 12 2
845.2.t.b 4 65.s odd 12 2
1040.2.bg.a 2 20.d odd 2 1
1040.2.bg.a 2 260.l odd 4 1
1040.2.cd.b 2 20.e even 4 1
1040.2.cd.b 2 260.u 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

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + 2T + 2$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 2T + 2$$
$13$ $$T^{2} + 6T + 13$$
$17$ $$T^{2} + 2T + 2$$
$19$ $$T^{2} + 10T + 50$$
$23$ $$T^{2} + 6T + 18$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 10T + 50$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 14T + 98$$
$43$ $$T^{2} - 2T + 2$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 10T + 50$$
$59$ $$T^{2} + 14T + 98$$
$61$ $$(T + 14)^{2}$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} - 2T + 2$$
$73$ $$(T - 10)^{2}$$
$79$ $$T^{2} + 4$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2} - 10T + 50$$
$97$ $$(T + 2)^{2}$$