# Properties

 Label 325.2.b.e Level $325$ Weight $2$ Character orbit 325.b Analytic conductor $2.595$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(274,325)]

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, 0]))

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

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.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: $$2.59513806569$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 65) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} - q^{4} + ( - \beta_{3} - 3) q^{6} + 2 \beta_1 q^{7} - \beta_{2} q^{8} + ( - 2 \beta_{3} - 1) q^{9}+O(q^{10})$$ q - b2 * q^2 + (-b2 - b1) * q^3 - q^4 + (-b3 - 3) * q^6 + 2*b1 * q^7 - b2 * q^8 + (-2*b3 - 1) * q^9 $$q - \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} - q^{4} + ( - \beta_{3} - 3) q^{6} + 2 \beta_1 q^{7} - \beta_{2} q^{8} + ( - 2 \beta_{3} - 1) q^{9} + ( - \beta_{3} - 3) q^{11} + (\beta_{2} + \beta_1) q^{12} - \beta_1 q^{13} + 2 \beta_{3} q^{14} - 5 q^{16} - 2 \beta_{2} q^{17} + (\beta_{2} + 6 \beta_1) q^{18} + (3 \beta_{3} + 1) q^{19} + (2 \beta_{3} + 2) q^{21} + (3 \beta_{2} + 3 \beta_1) q^{22} + (\beta_{2} - 3 \beta_1) q^{23} + ( - \beta_{3} - 3) q^{24} - \beta_{3} q^{26} + 4 \beta_1 q^{27} - 2 \beta_1 q^{28} + ( - 2 \beta_{3} + 6) q^{29} + (3 \beta_{3} + 5) q^{31} + 3 \beta_{2} q^{32} + (4 \beta_{2} + 6 \beta_1) q^{33} - 6 q^{34} + (2 \beta_{3} + 1) q^{36} - 4 \beta_1 q^{37} + ( - \beta_{2} - 9 \beta_1) q^{38} + ( - \beta_{3} - 1) q^{39} + 2 \beta_{3} q^{41} + ( - 2 \beta_{2} - 6 \beta_1) q^{42} + (3 \beta_{2} - 5 \beta_1) q^{43} + (\beta_{3} + 3) q^{44} + ( - 3 \beta_{3} + 3) q^{46} + 6 \beta_1 q^{47} + (5 \beta_{2} + 5 \beta_1) q^{48} + 3 q^{49} + ( - 2 \beta_{3} - 6) q^{51} + \beta_1 q^{52} - 6 \beta_{2} q^{53} + 4 \beta_{3} q^{54} + 2 \beta_{3} q^{56} + ( - 4 \beta_{2} - 10 \beta_1) q^{57} + ( - 6 \beta_{2} + 6 \beta_1) q^{58} + ( - 7 \beta_{3} + 3) q^{59} + ( - 6 \beta_{3} + 2) q^{61} + ( - 5 \beta_{2} - 9 \beta_1) q^{62} + ( - 4 \beta_{2} - 2 \beta_1) q^{63} - q^{64} + (6 \beta_{3} + 12) q^{66} + (6 \beta_{2} - 4 \beta_1) q^{67} + 2 \beta_{2} q^{68} - 2 \beta_{3} q^{69} + (\beta_{3} + 3) q^{71} + (\beta_{2} + 6 \beta_1) q^{72} + 4 \beta_1 q^{73} - 4 \beta_{3} q^{74} + ( - 3 \beta_{3} - 1) q^{76} + ( - 2 \beta_{2} - 6 \beta_1) q^{77} + (\beta_{2} + 3 \beta_1) q^{78} + (6 \beta_{3} - 2) q^{79} + ( - 2 \beta_{3} + 1) q^{81} - 6 \beta_1 q^{82} + 6 \beta_1 q^{83} + ( - 2 \beta_{3} - 2) q^{84} + ( - 5 \beta_{3} + 9) q^{86} - 4 \beta_{2} q^{87} + (3 \beta_{2} + 3 \beta_1) q^{88} + (4 \beta_{3} + 6) q^{89} + 2 q^{91} + ( - \beta_{2} + 3 \beta_1) q^{92} + ( - 8 \beta_{2} - 14 \beta_1) q^{93} + 6 \beta_{3} q^{94} + (3 \beta_{3} + 9) q^{96} + 2 \beta_1 q^{97} - 3 \beta_{2} q^{98} + (7 \beta_{3} + 9) q^{99}+O(q^{100})$$ q - b2 * q^2 + (-b2 - b1) * q^3 - q^4 + (-b3 - 3) * q^6 + 2*b1 * q^7 - b2 * q^8 + (-2*b3 - 1) * q^9 + (-b3 - 3) * q^11 + (b2 + b1) * q^12 - b1 * q^13 + 2*b3 * q^14 - 5 * q^16 - 2*b2 * q^17 + (b2 + 6*b1) * q^18 + (3*b3 + 1) * q^19 + (2*b3 + 2) * q^21 + (3*b2 + 3*b1) * q^22 + (b2 - 3*b1) * q^23 + (-b3 - 3) * q^24 - b3 * q^26 + 4*b1 * q^27 - 2*b1 * q^28 + (-2*b3 + 6) * q^29 + (3*b3 + 5) * q^31 + 3*b2 * q^32 + (4*b2 + 6*b1) * q^33 - 6 * q^34 + (2*b3 + 1) * q^36 - 4*b1 * q^37 + (-b2 - 9*b1) * q^38 + (-b3 - 1) * q^39 + 2*b3 * q^41 + (-2*b2 - 6*b1) * q^42 + (3*b2 - 5*b1) * q^43 + (b3 + 3) * q^44 + (-3*b3 + 3) * q^46 + 6*b1 * q^47 + (5*b2 + 5*b1) * q^48 + 3 * q^49 + (-2*b3 - 6) * q^51 + b1 * q^52 - 6*b2 * q^53 + 4*b3 * q^54 + 2*b3 * q^56 + (-4*b2 - 10*b1) * q^57 + (-6*b2 + 6*b1) * q^58 + (-7*b3 + 3) * q^59 + (-6*b3 + 2) * q^61 + (-5*b2 - 9*b1) * q^62 + (-4*b2 - 2*b1) * q^63 - q^64 + (6*b3 + 12) * q^66 + (6*b2 - 4*b1) * q^67 + 2*b2 * q^68 - 2*b3 * q^69 + (b3 + 3) * q^71 + (b2 + 6*b1) * q^72 + 4*b1 * q^73 - 4*b3 * q^74 + (-3*b3 - 1) * q^76 + (-2*b2 - 6*b1) * q^77 + (b2 + 3*b1) * q^78 + (6*b3 - 2) * q^79 + (-2*b3 + 1) * q^81 - 6*b1 * q^82 + 6*b1 * q^83 + (-2*b3 - 2) * q^84 + (-5*b3 + 9) * q^86 - 4*b2 * q^87 + (3*b2 + 3*b1) * q^88 + (4*b3 + 6) * q^89 + 2 * q^91 + (-b2 + 3*b1) * q^92 + (-8*b2 - 14*b1) * q^93 + 6*b3 * q^94 + (3*b3 + 9) * q^96 + 2*b1 * q^97 - 3*b2 * q^98 + (7*b3 + 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 12 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 12 * q^6 - 4 * q^9 $$4 q - 4 q^{4} - 12 q^{6} - 4 q^{9} - 12 q^{11} - 20 q^{16} + 4 q^{19} + 8 q^{21} - 12 q^{24} + 24 q^{29} + 20 q^{31} - 24 q^{34} + 4 q^{36} - 4 q^{39} + 12 q^{44} + 12 q^{46} + 12 q^{49} - 24 q^{51} + 12 q^{59} + 8 q^{61} - 4 q^{64} + 48 q^{66} + 12 q^{71} - 4 q^{76} - 8 q^{79} + 4 q^{81} - 8 q^{84} + 36 q^{86} + 24 q^{89} + 8 q^{91} + 36 q^{96} + 36 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 12 * q^6 - 4 * q^9 - 12 * q^11 - 20 * q^16 + 4 * q^19 + 8 * q^21 - 12 * q^24 + 24 * q^29 + 20 * q^31 - 24 * q^34 + 4 * q^36 - 4 * q^39 + 12 * q^44 + 12 * q^46 + 12 * q^49 - 24 * q^51 + 12 * q^59 + 8 * q^61 - 4 * q^64 + 48 * q^66 + 12 * q^71 - 4 * q^76 - 8 * q^79 + 4 * q^81 - 8 * q^84 + 36 * q^86 + 24 * q^89 + 8 * q^91 + 36 * q^96 + 36 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
274.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
1.73205i 2.73205i −1.00000 0 −4.73205 2.00000i 1.73205i −4.46410 0
274.2 1.73205i 0.732051i −1.00000 0 −1.26795 2.00000i 1.73205i 2.46410 0
274.3 1.73205i 0.732051i −1.00000 0 −1.26795 2.00000i 1.73205i 2.46410 0
274.4 1.73205i 2.73205i −1.00000 0 −4.73205 2.00000i 1.73205i −4.46410 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 325.2.b.e 4
3.b odd 2 1 2925.2.c.v 4
5.b even 2 1 inner 325.2.b.e 4
5.c odd 4 1 65.2.a.c 2
5.c odd 4 1 325.2.a.g 2
15.d odd 2 1 2925.2.c.v 4
15.e even 4 1 585.2.a.k 2
15.e even 4 1 2925.2.a.z 2
20.e even 4 1 1040.2.a.h 2
20.e even 4 1 5200.2.a.ca 2
35.f even 4 1 3185.2.a.k 2
40.i odd 4 1 4160.2.a.y 2
40.k even 4 1 4160.2.a.bj 2
55.e even 4 1 7865.2.a.h 2
60.l odd 4 1 9360.2.a.cm 2
65.f even 4 1 845.2.c.e 4
65.h odd 4 1 845.2.a.d 2
65.h odd 4 1 4225.2.a.w 2
65.k even 4 1 845.2.c.e 4
65.o even 12 1 845.2.m.a 4
65.o even 12 1 845.2.m.c 4
65.q odd 12 2 845.2.e.e 4
65.r odd 12 2 845.2.e.f 4
65.t even 12 1 845.2.m.a 4
65.t even 12 1 845.2.m.c 4
195.s even 4 1 7605.2.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.c 2 5.c odd 4 1
325.2.a.g 2 5.c odd 4 1
325.2.b.e 4 1.a even 1 1 trivial
325.2.b.e 4 5.b even 2 1 inner
585.2.a.k 2 15.e even 4 1
845.2.a.d 2 65.h odd 4 1
845.2.c.e 4 65.f even 4 1
845.2.c.e 4 65.k even 4 1
845.2.e.e 4 65.q odd 12 2
845.2.e.f 4 65.r odd 12 2
845.2.m.a 4 65.o even 12 1
845.2.m.a 4 65.t even 12 1
845.2.m.c 4 65.o even 12 1
845.2.m.c 4 65.t even 12 1
1040.2.a.h 2 20.e even 4 1
2925.2.a.z 2 15.e even 4 1
2925.2.c.v 4 3.b odd 2 1
2925.2.c.v 4 15.d odd 2 1
3185.2.a.k 2 35.f even 4 1
4160.2.a.y 2 40.i odd 4 1
4160.2.a.bj 2 40.k even 4 1
4225.2.a.w 2 65.h odd 4 1
5200.2.a.ca 2 20.e even 4 1
7605.2.a.be 2 195.s even 4 1
7865.2.a.h 2 55.e even 4 1
9360.2.a.cm 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$:

 $$T_{2}^{2} + 3$$ T2^2 + 3 $$T_{3}^{4} + 8T_{3}^{2} + 4$$ T3^4 + 8*T3^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 3)^{2}$$
$3$ $$T^{4} + 8T^{2} + 4$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} + 6 T + 6)^{2}$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$(T^{2} + 12)^{2}$$
$19$ $$(T^{2} - 2 T - 26)^{2}$$
$23$ $$T^{4} + 24T^{2} + 36$$
$29$ $$(T^{2} - 12 T + 24)^{2}$$
$31$ $$(T^{2} - 10 T - 2)^{2}$$
$37$ $$(T^{2} + 16)^{2}$$
$41$ $$(T^{2} - 12)^{2}$$
$43$ $$T^{4} + 104T^{2} + 4$$
$47$ $$(T^{2} + 36)^{2}$$
$53$ $$(T^{2} + 108)^{2}$$
$59$ $$(T^{2} - 6 T - 138)^{2}$$
$61$ $$(T^{2} - 4 T - 104)^{2}$$
$67$ $$T^{4} + 248T^{2} + 8464$$
$71$ $$(T^{2} - 6 T + 6)^{2}$$
$73$ $$(T^{2} + 16)^{2}$$
$79$ $$(T^{2} + 4 T - 104)^{2}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$(T^{2} - 12 T - 12)^{2}$$
$97$ $$(T^{2} + 4)^{2}$$