# Properties

 Label 1450.2.b.i Level $1450$ Weight $2$ Character orbit 1450.b Analytic conductor $11.578$ 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] = [1450,2,Mod(349,1450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1450, 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("1450.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1450 = 2 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1450.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: $$11.5783082931$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1277$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
 1.22474 − 1.22474i −1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
1.00000i 2.44949i −1.00000 0 −2.44949 4.44949i 1.00000i −3.00000 0
349.2 1.00000i 2.44949i −1.00000 0 2.44949 0.449490i 1.00000i −3.00000 0
349.3 1.00000i 2.44949i −1.00000 0 2.44949 0.449490i 1.00000i −3.00000 0
349.4 1.00000i 2.44949i −1.00000 0 −2.44949 4.44949i 1.00000i −3.00000 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 1450.2.b.i 4
5.b even 2 1 inner 1450.2.b.i 4
5.c odd 4 1 1450.2.a.j 2
5.c odd 4 1 1450.2.a.o yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1450.2.a.j 2 5.c odd 4 1
1450.2.a.o yes 2 5.c odd 4 1
1450.2.b.i 4 1.a even 1 1 trivial
1450.2.b.i 4 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1450, [\chi])$$:

 $$T_{3}^{2} + 6$$ T3^2 + 6 $$T_{7}^{4} + 20T_{7}^{2} + 4$$ T7^4 + 20*T7^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} + 6)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 20T^{2} + 4$$
$11$ $$(T - 2)^{4}$$
$13$ $$(T^{2} + 6)^{2}$$
$17$ $$(T^{2} + 4)^{2}$$
$19$ $$(T^{2} + 8 T + 10)^{2}$$
$23$ $$(T^{2} + 6)^{2}$$
$29$ $$(T - 1)^{4}$$
$31$ $$(T + 3)^{4}$$
$37$ $$T^{4} + 14T^{2} + 25$$
$41$ $$(T^{2} - 8 T - 38)^{2}$$
$43$ $$T^{4} + 80T^{2} + 64$$
$47$ $$T^{4} + 50T^{2} + 529$$
$53$ $$T^{4} + 140T^{2} + 3364$$
$59$ $$(T^{2} + 22 T + 115)^{2}$$
$61$ $$(T^{2} - 2 T - 5)^{2}$$
$67$ $$T^{4} + 254 T^{2} + 13225$$
$71$ $$(T^{2} - 8 T - 38)^{2}$$
$73$ $$T^{4} + 180T^{2} + 324$$
$79$ $$(T^{2} + 4 T - 20)^{2}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$(T^{2} + 12 T + 30)^{2}$$
$97$ $$(T^{2} + 54)^{2}$$