# Properties

 Label 725.2.b.e Level $725$ Weight $2$ Character orbit 725.b Analytic conductor $5.789$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.350464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + ( - \beta_{4} + \beta_{2}) q^{3} + \beta_1 q^{4} + (\beta_1 - 1) q^{6} + ( - \beta_{5} - \beta_{4}) q^{7} + \beta_{5} q^{8} + (\beta_{3} - \beta_1 - 1) q^{9}+O(q^{10})$$ q - b4 * q^2 + (-b4 + b2) * q^3 + b1 * q^4 + (b1 - 1) * q^6 + (-b5 - b4) * q^7 + b5 * q^8 + (b3 - b1 - 1) * q^9 $$q - \beta_{4} q^{2} + ( - \beta_{4} + \beta_{2}) q^{3} + \beta_1 q^{4} + (\beta_1 - 1) q^{6} + ( - \beta_{5} - \beta_{4}) q^{7} + \beta_{5} q^{8} + (\beta_{3} - \beta_1 - 1) q^{9} + (\beta_{3} + 1) q^{11} + (\beta_{5} + \beta_{4} + 2 \beta_{2}) q^{12} - 2 \beta_{4} q^{13} + (\beta_{3} + 2 \beta_1 - 1) q^{14} + ( - \beta_{3} + \beta_1 - 1) q^{16} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{17} + (\beta_{5} - \beta_{4} + \beta_{2}) q^{18} + ( - \beta_1 + 3) q^{19} + (\beta_{3} + \beta_1 - 2) q^{21} + (2 \beta_{5} - \beta_{4} + \beta_{2}) q^{22} + (2 \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{23} + ( - \beta_{3} + 1) q^{24} + (2 \beta_1 - 4) q^{26} + ( - 2 \beta_{4} - 2 \beta_{2}) q^{27} + (2 \beta_{5} + 3 \beta_{4} + \beta_{2}) q^{28} + q^{29} + ( - \beta_1 - 5) q^{31} + (\beta_{5} + 3 \beta_{4} - \beta_{2}) q^{32} + (\beta_{5} - \beta_{2}) q^{33} + (\beta_{3} + 5) q^{34} + (\beta_{3} - 2 \beta_1 - 4) q^{36} + (3 \beta_{5} - \beta_{4}) q^{37} + ( - \beta_{5} - 5 \beta_{4}) q^{38} + (2 \beta_1 - 2) q^{39} + ( - 3 \beta_{3} - \beta_1 - 2) q^{41} + (3 \beta_{5} + 4 \beta_{4} + \beta_{2}) q^{42} + (2 \beta_{5} + 5 \beta_{4} - 3 \beta_{2}) q^{43} + ( - \beta_1 - 1) q^{44} + ( - 2 \beta_{3} - 3 \beta_1 + 3) q^{46} + ( - 2 \beta_{5} + \beta_{4} - 3 \beta_{2}) q^{47} + (\beta_{4} + 3 \beta_{2}) q^{48} + (\beta_{3} + 3 \beta_1 + 3) q^{49} + (3 \beta_{3} - 5 \beta_1 - 6) q^{51} + (2 \beta_{5} + 4 \beta_{4}) q^{52} + 2 \beta_{2} q^{53} + (2 \beta_1 - 6) q^{54} + ( - \beta_1 + 3) q^{56} + ( - \beta_{5} - 4 \beta_{4} + \beta_{2}) q^{57} - \beta_{4} q^{58} + ( - 3 \beta_{3} - \beta_1 - 4) q^{59} + (3 \beta_{3} + 3 \beta_1) q^{61} + ( - \beta_{5} + 3 \beta_{4}) q^{62} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{63} + ( - 3 \beta_{3} - 2 \beta_1 + 2) q^{64} + ( - \beta_{3} - \beta_1 - 2) q^{66} + ( - 4 \beta_{5} - \beta_{4} - 5 \beta_{2}) q^{67} + ( - 3 \beta_{4} + 5 \beta_{2}) q^{68} + (\beta_{3} - 7 \beta_1 - 6) q^{69} + ( - 3 \beta_{3} - \beta_1 + 8) q^{71} + (2 \beta_{5} - 2 \beta_{4} + 3 \beta_{2}) q^{72} + ( - 6 \beta_{5} - \beta_{4} + \beta_{2}) q^{73} + ( - 3 \beta_{3} - 2 \beta_1 - 5) q^{74} + (\beta_{3} + 4 \beta_1 - 3) q^{76} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{77} + (2 \beta_{5} + 6 \beta_{4}) q^{78} + (\beta_{3} - 4 \beta_1 + 1) q^{79} + (\beta_{3} + 3 \beta_1 + 1) q^{81} + ( - 7 \beta_{5} - 3 \beta_{2}) q^{82} + (\beta_{5} - 3 \beta_{4} + 4 \beta_{2}) q^{83} + ( - \beta_{3} - 5 \beta_1 + 2) q^{84} + ( - 2 \beta_{3} - 7 \beta_1 + 5) q^{86} + ( - \beta_{4} + \beta_{2}) q^{87} + (3 \beta_{5} - 3 \beta_{4} + 2 \beta_{2}) q^{88} + (2 \beta_1 + 4) q^{89} + (2 \beta_{3} + 4 \beta_1 - 2) q^{91} + ( - 3 \beta_{5} - 7 \beta_{4} + 4 \beta_{2}) q^{92} + ( - \beta_{5} + 4 \beta_{4} - 7 \beta_{2}) q^{93} + (2 \beta_{3} + \beta_1 + 1) q^{94} + ( - 2 \beta_{3} - \beta_1 + 7) q^{96} + ( - 2 \beta_{5} - 5 \beta_{4} + \beta_{2}) q^{97} + (5 \beta_{5} + 3 \beta_{4} + \beta_{2}) q^{98} + (\beta_{3} + 2 \beta_1 + 7) q^{99}+O(q^{100})$$ q - b4 * q^2 + (-b4 + b2) * q^3 + b1 * q^4 + (b1 - 1) * q^6 + (-b5 - b4) * q^7 + b5 * q^8 + (b3 - b1 - 1) * q^9 + (b3 + 1) * q^11 + (b5 + b4 + 2*b2) * q^12 - 2*b4 * q^13 + (b3 + 2*b1 - 1) * q^14 + (-b3 + b1 - 1) * q^16 + (-b5 + b4 + 2*b2) * q^17 + (b5 - b4 + b2) * q^18 + (-b1 + 3) * q^19 + (b3 + b1 - 2) * q^21 + (2*b5 - b4 + b2) * q^22 + (2*b5 + b4 + 3*b2) * q^23 + (-b3 + 1) * q^24 + (2*b1 - 4) * q^26 + (-2*b4 - 2*b2) * q^27 + (2*b5 + 3*b4 + b2) * q^28 + q^29 + (-b1 - 5) * q^31 + (b5 + 3*b4 - b2) * q^32 + (b5 - b2) * q^33 + (b3 + 5) * q^34 + (b3 - 2*b1 - 4) * q^36 + (3*b5 - b4) * q^37 + (-b5 - 5*b4) * q^38 + (2*b1 - 2) * q^39 + (-3*b3 - b1 - 2) * q^41 + (3*b5 + 4*b4 + b2) * q^42 + (2*b5 + 5*b4 - 3*b2) * q^43 + (-b1 - 1) * q^44 + (-2*b3 - 3*b1 + 3) * q^46 + (-2*b5 + b4 - 3*b2) * q^47 + (b4 + 3*b2) * q^48 + (b3 + 3*b1 + 3) * q^49 + (3*b3 - 5*b1 - 6) * q^51 + (2*b5 + 4*b4) * q^52 + 2*b2 * q^53 + (2*b1 - 6) * q^54 + (-b1 + 3) * q^56 + (-b5 - 4*b4 + b2) * q^57 - b4 * q^58 + (-3*b3 - b1 - 4) * q^59 + (3*b3 + 3*b1) * q^61 + (-b5 + 3*b4) * q^62 + (-b5 + b4 - 2*b2) * q^63 + (-3*b3 - 2*b1 + 2) * q^64 + (-b3 - b1 - 2) * q^66 + (-4*b5 - b4 - 5*b2) * q^67 + (-3*b4 + 5*b2) * q^68 + (b3 - 7*b1 - 6) * q^69 + (-3*b3 - b1 + 8) * q^71 + (2*b5 - 2*b4 + 3*b2) * q^72 + (-6*b5 - b4 + b2) * q^73 + (-3*b3 - 2*b1 - 5) * q^74 + (b3 + 4*b1 - 3) * q^76 + (-b5 + 2*b4 - b2) * q^77 + (2*b5 + 6*b4) * q^78 + (b3 - 4*b1 + 1) * q^79 + (b3 + 3*b1 + 1) * q^81 + (-7*b5 - 3*b2) * q^82 + (b5 - 3*b4 + 4*b2) * q^83 + (-b3 - 5*b1 + 2) * q^84 + (-2*b3 - 7*b1 + 5) * q^86 + (-b4 + b2) * q^87 + (3*b5 - 3*b4 + 2*b2) * q^88 + (2*b1 + 4) * q^89 + (2*b3 + 4*b1 - 2) * q^91 + (-3*b5 - 7*b4 + 4*b2) * q^92 + (-b5 + 4*b4 - 7*b2) * q^93 + (2*b3 + b1 + 1) * q^94 + (-2*b3 - b1 + 7) * q^96 + (-2*b5 - 5*b4 + b2) * q^97 + (5*b5 + 3*b4 + b2) * q^98 + (b3 + 2*b1 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{4} - 8 q^{6} - 6 q^{9}+O(q^{10})$$ 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9 $$6 q - 2 q^{4} - 8 q^{6} - 6 q^{9} + 4 q^{11} - 12 q^{14} - 6 q^{16} + 20 q^{19} - 16 q^{21} + 8 q^{24} - 28 q^{26} + 6 q^{29} - 28 q^{31} + 28 q^{34} - 22 q^{36} - 16 q^{39} - 4 q^{41} - 4 q^{44} + 28 q^{46} + 10 q^{49} - 32 q^{51} - 40 q^{54} + 20 q^{56} - 16 q^{59} - 12 q^{61} + 22 q^{64} - 8 q^{66} - 24 q^{69} + 56 q^{71} - 20 q^{74} - 28 q^{76} + 12 q^{79} - 2 q^{81} + 24 q^{84} + 48 q^{86} + 20 q^{89} - 24 q^{91} + 48 q^{96} + 36 q^{99}+O(q^{100})$$ 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9 + 4 * q^11 - 12 * q^14 - 6 * q^16 + 20 * q^19 - 16 * q^21 + 8 * q^24 - 28 * q^26 + 6 * q^29 - 28 * q^31 + 28 * q^34 - 22 * q^36 - 16 * q^39 - 4 * q^41 - 4 * q^44 + 28 * q^46 + 10 * q^49 - 32 * q^51 - 40 * q^54 + 20 * q^56 - 16 * q^59 - 12 * q^61 + 22 * q^64 - 8 * q^66 - 24 * q^69 + 56 * q^71 - 20 * q^74 - 28 * q^76 + 12 * q^79 - 2 * q^81 + 24 * q^84 + 48 * q^86 + 20 * q^89 - 24 * q^91 + 48 * q^96 + 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23$$ (-4*v^5 + 9*v^4 - 16*v^3 - 4*v^2 + 8*v - 9) / 23 $$\beta_{2}$$ $$=$$ $$( -7\nu^{5} + 10\nu^{4} - 5\nu^{3} - 7\nu^{2} - 32\nu + 13 ) / 23$$ (-7*v^5 + 10*v^4 - 5*v^3 - 7*v^2 - 32*v + 13) / 23 $$\beta_{3}$$ $$=$$ $$( -6\nu^{5} + 25\nu^{4} - 24\nu^{3} - 6\nu^{2} + 12\nu + 67 ) / 23$$ (-6*v^5 + 25*v^4 - 24*v^3 - 6*v^2 + 12*v + 67) / 23 $$\beta_{4}$$ $$=$$ $$( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23$$ (-11*v^5 + 19*v^4 - 21*v^3 - 11*v^2 - 70*v + 27) / 23 $$\beta_{5}$$ $$=$$ $$( 21\nu^{5} - 30\nu^{4} + 15\nu^{3} + 67\nu^{2} + 96\nu - 39 ) / 23$$ (21*v^5 - 30*v^4 + 15*v^3 + 67*v^2 + 96*v - 39) / 23
 $$\nu$$ $$=$$ $$( -\beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2$$ (-b4 + b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 3\beta_{2} ) / 2$$ (b5 + 3*b2) / 2 $$\nu^{3}$$ $$=$$ $$( -2\beta_{4} + \beta_{3} + 4\beta_{2} - 3\beta _1 - 4 ) / 2$$ (-2*b4 + b3 + 4*b2 - 3*b1 - 4) / 2 $$\nu^{4}$$ $$=$$ $$2\beta_{3} - 3\beta _1 - 7$$ 2*b3 - 3*b1 - 7 $$\nu^{5}$$ $$=$$ $$( -\beta_{5} + 6\beta_{4} + 5\beta_{3} - 17\beta_{2} - 11\beta _1 - 18 ) / 2$$ (-b5 + 6*b4 + 5*b3 - 17*b2 - 11*b1 - 18) / 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
 −0.854638 − 0.854638i 0.403032 − 0.403032i 1.45161 + 1.45161i 1.45161 − 1.45161i 0.403032 + 0.403032i −0.854638 + 0.854638i
2.17009i 1.70928i −2.70928 0 −3.70928 3.70928i 1.53919i 0.0783777 0
349.2 1.48119i 0.806063i −0.193937 0 −1.19394 1.19394i 2.67513i 2.35026 0
349.3 0.311108i 2.90321i 1.90321 0 0.903212 0.903212i 1.21432i −5.42864 0
349.4 0.311108i 2.90321i 1.90321 0 0.903212 0.903212i 1.21432i −5.42864 0
349.5 1.48119i 0.806063i −0.193937 0 −1.19394 1.19394i 2.67513i 2.35026 0
349.6 2.17009i 1.70928i −2.70928 0 −3.70928 3.70928i 1.53919i 0.0783777 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.6 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.e 6
5.b even 2 1 inner 725.2.b.e 6
5.c odd 4 1 145.2.a.c 3
5.c odd 4 1 725.2.a.e 3
15.e even 4 1 1305.2.a.p 3
15.e even 4 1 6525.2.a.be 3
20.e even 4 1 2320.2.a.n 3
35.f even 4 1 7105.2.a.o 3
40.i odd 4 1 9280.2.a.bj 3
40.k even 4 1 9280.2.a.br 3
145.h odd 4 1 4205.2.a.f 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.c 3 5.c odd 4 1
725.2.a.e 3 5.c odd 4 1
725.2.b.e 6 1.a even 1 1 trivial
725.2.b.e 6 5.b even 2 1 inner
1305.2.a.p 3 15.e even 4 1
2320.2.a.n 3 20.e even 4 1
4205.2.a.f 3 145.h odd 4 1
6525.2.a.be 3 15.e even 4 1
7105.2.a.o 3 35.f even 4 1
9280.2.a.bj 3 40.i odd 4 1
9280.2.a.br 3 40.k even 4 1

## Hecke kernels

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

 $$T_{2}^{6} + 7T_{2}^{4} + 11T_{2}^{2} + 1$$ T2^6 + 7*T2^4 + 11*T2^2 + 1 $$T_{3}^{6} + 12T_{3}^{4} + 32T_{3}^{2} + 16$$ T3^6 + 12*T3^4 + 32*T3^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 7 T^{4} + \cdots + 1$$
$3$ $$T^{6} + 12 T^{4} + \cdots + 16$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 16 T^{4} + \cdots + 16$$
$11$ $$(T^{3} - 2 T^{2} - 8 T - 4)^{2}$$
$13$ $$T^{6} + 28 T^{4} + \cdots + 64$$
$17$ $$T^{6} + 96 T^{4} + \cdots + 4624$$
$19$ $$(T^{3} - 10 T^{2} + \cdots - 20)^{2}$$
$23$ $$T^{6} + 104 T^{4} + \cdots + 8464$$
$29$ $$(T - 1)^{6}$$
$31$ $$(T^{3} + 14 T^{2} + \cdots + 76)^{2}$$
$37$ $$T^{6} + 112 T^{4} + \cdots + 8464$$
$41$ $$(T^{3} + 2 T^{2} + \cdots + 232)^{2}$$
$43$ $$T^{6} + 268 T^{4} + \cdots + 16$$
$47$ $$T^{6} + 76 T^{4} + \cdots + 5776$$
$53$ $$T^{6} + 44 T^{4} + \cdots + 64$$
$59$ $$(T^{3} + 8 T^{2} - 64 T + 80)^{2}$$
$61$ $$(T^{3} + 6 T^{2} + \cdots - 216)^{2}$$
$67$ $$T^{6} + 280 T^{4} + \cdots + 512656$$
$71$ $$(T^{3} - 28 T^{2} + \cdots + 272)^{2}$$
$73$ $$T^{6} + 456 T^{4} + \cdots + 2890000$$
$79$ $$(T^{3} - 6 T^{2} + \cdots + 460)^{2}$$
$83$ $$T^{6} + 144 T^{4} + \cdots + 21904$$
$89$ $$(T^{3} - 10 T^{2} + \cdots + 40)^{2}$$
$97$ $$T^{6} + 200 T^{4} + \cdots + 5776$$