# Properties

 Label 725.2.b.c 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 145) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1) q^{3} + ( - \beta_{3} - 1) q^{4} + (\beta_{3} + 2) q^{6} + 2 \beta_{2} q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} - q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b2 - b1) * q^3 + (-b3 - 1) * q^4 + (b3 + 2) * q^6 + 2*b2 * q^7 + (-b2 - 2*b1) * q^8 - q^9 $$q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1) q^{3} + ( - \beta_{3} - 1) q^{4} + (\beta_{3} + 2) q^{6} + 2 \beta_{2} q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} - q^{9} + ( - \beta_{3} - 2) q^{11} + ( - \beta_{2} + 3 \beta_1) q^{12} + ( - \beta_{2} - \beta_1) q^{13} + 2 q^{14} + 3 q^{16} + ( - \beta_{2} + \beta_1) q^{17} - \beta_1 q^{18} + ( - \beta_{3} + 2) q^{19} + ( - 2 \beta_{3} + 4) q^{21} + ( - \beta_{2} - 5 \beta_1) q^{22} + ( - 2 \beta_{2} - 4 \beta_1) q^{23} + ( - \beta_{3} - 6) q^{24} + (\beta_{3} + 2) q^{26} + ( - 2 \beta_{2} - 2 \beta_1) q^{27} + (4 \beta_{2} + 2 \beta_1) q^{28} - q^{29} + ( - 3 \beta_{3} - 2) q^{31} + ( - 2 \beta_{2} - \beta_1) q^{32} + 4 \beta_1 q^{33} + ( - \beta_{3} - 4) q^{34} + (\beta_{3} + 1) q^{36} + (3 \beta_{2} - 3 \beta_1) q^{37} + ( - \beta_{2} - \beta_1) q^{38} - 4 q^{39} - 6 q^{41} + ( - 2 \beta_{2} - 2 \beta_1) q^{42} + ( - 3 \beta_{2} - 3 \beta_1) q^{43} + (3 \beta_{3} + 10) q^{44} + (4 \beta_{3} + 10) q^{46} + (5 \beta_{2} + \beta_1) q^{47} + ( - 3 \beta_{2} - 3 \beta_1) q^{48} + (4 \beta_{3} - 5) q^{49} + 2 \beta_{3} q^{51} + ( - \beta_{2} + 3 \beta_1) q^{52} + ( - \beta_{2} + 3 \beta_1) q^{53} + (2 \beta_{3} + 4) q^{54} + ( - 2 \beta_{3} + 2) q^{56} - 4 \beta_{2} q^{57} - \beta_1 q^{58} + (2 \beta_{3} + 2) q^{61} + ( - 3 \beta_{2} - 11 \beta_1) q^{62} - 2 \beta_{2} q^{63} + (\beta_{3} + 7) q^{64} + ( - 4 \beta_{3} - 12) q^{66} + ( - 2 \beta_{2} + 4 \beta_1) q^{67} + ( - 3 \beta_{2} - 5 \beta_1) q^{68} + ( - 2 \beta_{3} - 12) q^{69} + (4 \beta_{3} - 4) q^{71} + (\beta_{2} + 2 \beta_1) q^{72} + (3 \beta_{2} - 3 \beta_1) q^{73} + (3 \beta_{3} + 12) q^{74} + ( - \beta_{3} + 6) q^{76} + (2 \beta_{2} + 2 \beta_1) q^{77} - 4 \beta_1 q^{78} + ( - 3 \beta_{3} - 6) q^{79} - 11 q^{81} - 6 \beta_1 q^{82} + (4 \beta_{2} + 6 \beta_1) q^{83} + ( - 2 \beta_{3} + 12) q^{84} + (3 \beta_{3} + 6) q^{86} + (\beta_{2} + \beta_1) q^{87} + (\beta_{2} + 9 \beta_1) q^{88} + ( - 2 \beta_{3} + 2) q^{89} + ( - 2 \beta_{3} + 4) q^{91} + 14 \beta_1 q^{92} + ( - 4 \beta_{2} + 8 \beta_1) q^{93} + ( - \beta_{3} + 2) q^{94} + (\beta_{3} - 6) q^{96} + (5 \beta_{2} - \beta_1) q^{97} + (4 \beta_{2} + 7 \beta_1) q^{98} + (\beta_{3} + 2) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b2 - b1) * q^3 + (-b3 - 1) * q^4 + (b3 + 2) * q^6 + 2*b2 * q^7 + (-b2 - 2*b1) * q^8 - q^9 + (-b3 - 2) * q^11 + (-b2 + 3*b1) * q^12 + (-b2 - b1) * q^13 + 2 * q^14 + 3 * q^16 + (-b2 + b1) * q^17 - b1 * q^18 + (-b3 + 2) * q^19 + (-2*b3 + 4) * q^21 + (-b2 - 5*b1) * q^22 + (-2*b2 - 4*b1) * q^23 + (-b3 - 6) * q^24 + (b3 + 2) * q^26 + (-2*b2 - 2*b1) * q^27 + (4*b2 + 2*b1) * q^28 - q^29 + (-3*b3 - 2) * q^31 + (-2*b2 - b1) * q^32 + 4*b1 * q^33 + (-b3 - 4) * q^34 + (b3 + 1) * q^36 + (3*b2 - 3*b1) * q^37 + (-b2 - b1) * q^38 - 4 * q^39 - 6 * q^41 + (-2*b2 - 2*b1) * q^42 + (-3*b2 - 3*b1) * q^43 + (3*b3 + 10) * q^44 + (4*b3 + 10) * q^46 + (5*b2 + b1) * q^47 + (-3*b2 - 3*b1) * q^48 + (4*b3 - 5) * q^49 + 2*b3 * q^51 + (-b2 + 3*b1) * q^52 + (-b2 + 3*b1) * q^53 + (2*b3 + 4) * q^54 + (-2*b3 + 2) * q^56 - 4*b2 * q^57 - b1 * q^58 + (2*b3 + 2) * q^61 + (-3*b2 - 11*b1) * q^62 - 2*b2 * q^63 + (b3 + 7) * q^64 + (-4*b3 - 12) * q^66 + (-2*b2 + 4*b1) * q^67 + (-3*b2 - 5*b1) * q^68 + (-2*b3 - 12) * q^69 + (4*b3 - 4) * q^71 + (b2 + 2*b1) * q^72 + (3*b2 - 3*b1) * q^73 + (3*b3 + 12) * q^74 + (-b3 + 6) * q^76 + (2*b2 + 2*b1) * q^77 - 4*b1 * q^78 + (-3*b3 - 6) * q^79 - 11 * q^81 - 6*b1 * q^82 + (4*b2 + 6*b1) * q^83 + (-2*b3 + 12) * q^84 + (3*b3 + 6) * q^86 + (b2 + b1) * q^87 + (b2 + 9*b1) * q^88 + (-2*b3 + 2) * q^89 + (-2*b3 + 4) * q^91 + 14*b1 * q^92 + (-4*b2 + 8*b1) * q^93 + (-b3 + 2) * q^94 + (b3 - 6) * q^96 + (5*b2 - b1) * q^97 + (4*b2 + 7*b1) * q^98 + (b3 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 + 8 * q^6 - 4 * q^9 $$4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} - 8 q^{11} + 8 q^{14} + 12 q^{16} + 8 q^{19} + 16 q^{21} - 24 q^{24} + 8 q^{26} - 4 q^{29} - 8 q^{31} - 16 q^{34} + 4 q^{36} - 16 q^{39} - 24 q^{41} + 40 q^{44} + 40 q^{46} - 20 q^{49} + 16 q^{54} + 8 q^{56} + 8 q^{61} + 28 q^{64} - 48 q^{66} - 48 q^{69} - 16 q^{71} + 48 q^{74} + 24 q^{76} - 24 q^{79} - 44 q^{81} + 48 q^{84} + 24 q^{86} + 8 q^{89} + 16 q^{91} + 8 q^{94} - 24 q^{96} + 8 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 + 8 * q^6 - 4 * q^9 - 8 * q^11 + 8 * q^14 + 12 * q^16 + 8 * q^19 + 16 * q^21 - 24 * q^24 + 8 * q^26 - 4 * q^29 - 8 * q^31 - 16 * q^34 + 4 * q^36 - 16 * q^39 - 24 * q^41 + 40 * q^44 + 40 * q^46 - 20 * q^49 + 16 * q^54 + 8 * q^56 + 8 * q^61 + 28 * q^64 - 48 * q^66 - 48 * q^69 - 16 * q^71 + 48 * q^74 + 24 * q^76 - 24 * q^79 - 44 * q^81 + 48 * q^84 + 24 * q^86 + 8 * q^89 + 16 * q^91 + 8 * q^94 - 24 * q^96 + 8 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}$$ v^3 + v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}$$ -v^3 + v^2 - v $$\beta_{3}$$ $$=$$ $$-2\zeta_{8}^{3} + 2\zeta_{8}$$ -2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (b3 - b2 + b1) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (-b3 - b2 + b1) / 4

## 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 2.00000i −3.82843 0 4.82843 0.828427i 4.41421i −1.00000 0
349.2 0.414214i 2.00000i 1.82843 0 −0.828427 4.82843i 1.58579i −1.00000 0
349.3 0.414214i 2.00000i 1.82843 0 −0.828427 4.82843i 1.58579i −1.00000 0
349.4 2.41421i 2.00000i −3.82843 0 4.82843 0.828427i 4.41421i −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
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.c 4
5.b even 2 1 inner 725.2.b.c 4
5.c odd 4 1 145.2.a.b 2
5.c odd 4 1 725.2.a.c 2
15.e even 4 1 1305.2.a.n 2
15.e even 4 1 6525.2.a.p 2
20.e even 4 1 2320.2.a.k 2
35.f even 4 1 7105.2.a.e 2
40.i odd 4 1 9280.2.a.be 2
40.k even 4 1 9280.2.a.w 2
145.h odd 4 1 4205.2.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.b 2 5.c odd 4 1
725.2.a.c 2 5.c odd 4 1
725.2.b.c 4 1.a even 1 1 trivial
725.2.b.c 4 5.b even 2 1 inner
1305.2.a.n 2 15.e even 4 1
2320.2.a.k 2 20.e even 4 1
4205.2.a.d 2 145.h odd 4 1
6525.2.a.p 2 15.e even 4 1
7105.2.a.e 2 35.f even 4 1
9280.2.a.w 2 40.k even 4 1
9280.2.a.be 2 40.i 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 $$T_{3}^{2} + 4$$ T3^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$(T^{2} + 4)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 24T^{2} + 16$$
$11$ $$(T^{2} + 4 T - 4)^{2}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$(T^{2} + 8)^{2}$$
$19$ $$(T^{2} - 4 T - 4)^{2}$$
$23$ $$T^{4} + 88T^{2} + 784$$
$29$ $$(T + 1)^{4}$$
$31$ $$(T^{2} + 4 T - 68)^{2}$$
$37$ $$(T^{2} + 72)^{2}$$
$41$ $$(T + 6)^{4}$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$T^{4} + 136T^{2} + 16$$
$53$ $$T^{4} + 72T^{2} + 784$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 4 T - 28)^{2}$$
$67$ $$T^{4} + 152T^{2} + 4624$$
$71$ $$(T^{2} + 8 T - 112)^{2}$$
$73$ $$(T^{2} + 72)^{2}$$
$79$ $$(T^{2} + 12 T - 36)^{2}$$
$83$ $$T^{4} + 216T^{2} + 8464$$
$89$ $$(T^{2} - 4 T - 28)^{2}$$
$97$ $$T^{4} + 176T^{2} + 3136$$