# Properties

 Label 725.2.b.b Level $725$ Weight $2$ Character orbit 725.b Analytic conductor $5.789$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(349,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, 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("725.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.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: $$5.78915414654$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 29) 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.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$$.

 $$n$$ $$176$$ $$552$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 2.41421i −3.82843 0 −5.82843 2.82843i 4.41421i −2.82843 0
349.2 0.414214i 0.414214i 1.82843 0 −0.171573 2.82843i 1.58579i 2.82843 0
349.3 0.414214i 0.414214i 1.82843 0 −0.171573 2.82843i 1.58579i 2.82843 0
349.4 2.41421i 2.41421i −3.82843 0 −5.82843 2.82843i 4.41421i −2.82843 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 725.2.b.b 4
5.b even 2 1 inner 725.2.b.b 4
5.c odd 4 1 29.2.a.a 2
5.c odd 4 1 725.2.a.b 2
15.e even 4 1 261.2.a.d 2
15.e even 4 1 6525.2.a.o 2
20.e even 4 1 464.2.a.h 2
35.f even 4 1 1421.2.a.j 2
40.i odd 4 1 1856.2.a.r 2
40.k even 4 1 1856.2.a.w 2
55.e even 4 1 3509.2.a.j 2
60.l odd 4 1 4176.2.a.bq 2
65.h odd 4 1 4901.2.a.g 2
85.g odd 4 1 8381.2.a.e 2
145.e even 4 1 841.2.b.a 4
145.h odd 4 1 841.2.a.d 2
145.j even 4 1 841.2.b.a 4
145.o even 28 6 841.2.e.k 24
145.p odd 28 6 841.2.d.j 12
145.q odd 28 6 841.2.d.f 12
145.t even 28 6 841.2.e.k 24
435.p even 4 1 7569.2.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 5.c odd 4 1
261.2.a.d 2 15.e even 4 1
464.2.a.h 2 20.e even 4 1
725.2.a.b 2 5.c odd 4 1
725.2.b.b 4 1.a even 1 1 trivial
725.2.b.b 4 5.b even 2 1 inner
841.2.a.d 2 145.h odd 4 1
841.2.b.a 4 145.e even 4 1
841.2.b.a 4 145.j even 4 1
841.2.d.f 12 145.q odd 28 6
841.2.d.j 12 145.p odd 28 6
841.2.e.k 24 145.o even 28 6
841.2.e.k 24 145.t even 28 6
1421.2.a.j 2 35.f even 4 1
1856.2.a.r 2 40.i odd 4 1
1856.2.a.w 2 40.k even 4 1
3509.2.a.j 2 55.e even 4 1
4176.2.a.bq 2 60.l odd 4 1
4901.2.a.g 2 65.h odd 4 1
6525.2.a.o 2 15.e even 4 1
7569.2.a.c 2 435.p even 4 1
8381.2.a.e 2 85.g odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(725, [\chi])$$:

 $$T_{2}^{4} + 6T_{2}^{2} + 1$$ T2^4 + 6*T2^2 + 1 $$T_{3}^{4} + 6T_{3}^{2} + 1$$ T3^4 + 6*T3^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$T^{4} + 6T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 8)^{2}$$
$11$ $$(T^{2} - 2 T - 1)^{2}$$
$13$ $$T^{4} + 18T^{2} + 49$$
$17$ $$T^{4} + 24T^{2} + 16$$
$19$ $$(T + 6)^{4}$$
$23$ $$T^{4} + 72T^{2} + 784$$
$29$ $$(T + 1)^{4}$$
$31$ $$(T^{2} - 6 T - 41)^{2}$$
$37$ $$(T^{2} + 16)^{2}$$
$41$ $$(T^{2} - 8 T - 56)^{2}$$
$43$ $$T^{4} + 54T^{2} + 529$$
$47$ $$T^{4} + 38T^{2} + 289$$
$53$ $$T^{4} + 146T^{2} + 5041$$
$59$ $$(T^{2} + 4 T - 28)^{2}$$
$61$ $$(T^{2} + 4 T - 4)^{2}$$
$67$ $$(T^{2} + 32)^{2}$$
$71$ $$(T^{2} + 12 T + 28)^{2}$$
$73$ $$(T^{2} + 16)^{2}$$
$79$ $$(T^{2} - 2 T - 1)^{2}$$
$83$ $$T^{4} + 72T^{2} + 784$$
$89$ $$(T^{2} - 8 T - 56)^{2}$$
$97$ $$T^{4} + 176T^{2} + 3136$$