# Properties

 Label 925.2.d.a Level $925$ Weight $2$ Character orbit 925.d Analytic conductor $7.386$ 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] = [925,2,Mod(924,925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(925, 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("925.924");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$925 = 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 925.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: $$7.38616218697$$ 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 37) 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/925\mathbb{Z}\right)^\times$$.

 $$n$$ $$76$$ $$852$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
924.1
 1.00000i − 1.00000i
−2.00000 1.00000i 2.00000 0 2.00000i 3.00000i 0 2.00000 0
924.2 −2.00000 1.00000i 2.00000 0 2.00000i 3.00000i 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
185.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.d.a 2
5.b even 2 1 925.2.d.d 2
5.c odd 4 1 37.2.b.a 2
5.c odd 4 1 925.2.c.b 2
15.e even 4 1 333.2.c.a 2
20.e even 4 1 592.2.g.b 2
37.b even 2 1 925.2.d.d 2
40.i odd 4 1 2368.2.g.f 2
40.k even 4 1 2368.2.g.b 2
60.l odd 4 1 5328.2.h.c 2
185.d even 2 1 inner 925.2.d.a 2
185.f even 4 1 1369.2.a.f 1
185.h odd 4 1 37.2.b.a 2
185.h odd 4 1 925.2.c.b 2
185.k even 4 1 1369.2.a.a 1
555.n even 4 1 333.2.c.a 2
740.m even 4 1 592.2.g.b 2
1480.x odd 4 1 2368.2.g.f 2
1480.bh even 4 1 2368.2.g.b 2
2220.bf odd 4 1 5328.2.h.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.b.a 2 5.c odd 4 1
37.2.b.a 2 185.h odd 4 1
333.2.c.a 2 15.e even 4 1
333.2.c.a 2 555.n even 4 1
592.2.g.b 2 20.e even 4 1
592.2.g.b 2 740.m even 4 1
925.2.c.b 2 5.c odd 4 1
925.2.c.b 2 185.h odd 4 1
925.2.d.a 2 1.a even 1 1 trivial
925.2.d.a 2 185.d even 2 1 inner
925.2.d.d 2 5.b even 2 1
925.2.d.d 2 37.b even 2 1
1369.2.a.a 1 185.k even 4 1
1369.2.a.f 1 185.f even 4 1
2368.2.g.b 2 40.k even 4 1
2368.2.g.b 2 1480.bh even 4 1
2368.2.g.f 2 40.i odd 4 1
2368.2.g.f 2 1480.x odd 4 1
5328.2.h.c 2 60.l odd 4 1
5328.2.h.c 2 2220.bf odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(925, [\chi])$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T + 3)^{2}$$
$13$ $$(T + 6)^{2}$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2} + 16$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 12T + 37$$
$41$ $$(T + 3)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} + 9$$
$53$ $$T^{2} + 81$$
$59$ $$T^{2} + 16$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T + 3)^{2}$$
$73$ $$T^{2} + 81$$
$79$ $$T^{2} + 36$$
$83$ $$T^{2} + 81$$
$89$ $$T^{2} + 196$$
$97$ $$(T + 12)^{2}$$