# Properties

 Label 1450.2.b.d Level $1450$ Weight $2$ Character orbit 1450.b Analytic conductor $11.578$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

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: $$2$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 290) 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.

 $$f(q)$$ $$=$$ $$q - i q^{2} - q^{4} - 2 i q^{7} + i q^{8} + 3 q^{9} +O(q^{10})$$ q - i * q^2 - q^4 - 2*i * q^7 + i * q^8 + 3 * q^9 $$q - i q^{2} - q^{4} - 2 i q^{7} + i q^{8} + 3 q^{9} + 2 q^{11} + 6 i q^{13} - 2 q^{14} + q^{16} + 2 i q^{17} - 3 i q^{18} + 2 q^{19} - 2 i q^{22} + 6 i q^{23} + 6 q^{26} + 2 i q^{28} + q^{29} - 6 q^{31} - i q^{32} + 2 q^{34} - 3 q^{36} - 2 i q^{37} - 2 i q^{38} + 10 q^{41} + 8 i q^{43} - 2 q^{44} + 6 q^{46} - 4 i q^{47} + 3 q^{49} - 6 i q^{52} - 10 i q^{53} + 2 q^{56} - i q^{58} - 8 q^{59} + 10 q^{61} + 6 i q^{62} - 6 i q^{63} - q^{64} + 2 i q^{67} - 2 i q^{68} + 4 q^{71} + 3 i q^{72} - 6 i q^{73} - 2 q^{74} - 2 q^{76} - 4 i q^{77} + 10 q^{79} + 9 q^{81} - 10 i q^{82} + 6 i q^{83} + 8 q^{86} + 2 i q^{88} + 6 q^{89} + 12 q^{91} - 6 i q^{92} - 4 q^{94} + 6 i q^{97} - 3 i q^{98} + 6 q^{99} +O(q^{100})$$ q - i * q^2 - q^4 - 2*i * q^7 + i * q^8 + 3 * q^9 + 2 * q^11 + 6*i * q^13 - 2 * q^14 + q^16 + 2*i * q^17 - 3*i * q^18 + 2 * q^19 - 2*i * q^22 + 6*i * q^23 + 6 * q^26 + 2*i * q^28 + q^29 - 6 * q^31 - i * q^32 + 2 * q^34 - 3 * q^36 - 2*i * q^37 - 2*i * q^38 + 10 * q^41 + 8*i * q^43 - 2 * q^44 + 6 * q^46 - 4*i * q^47 + 3 * q^49 - 6*i * q^52 - 10*i * q^53 + 2 * q^56 - i * q^58 - 8 * q^59 + 10 * q^61 + 6*i * q^62 - 6*i * q^63 - q^64 + 2*i * q^67 - 2*i * q^68 + 4 * q^71 + 3*i * q^72 - 6*i * q^73 - 2 * q^74 - 2 * q^76 - 4*i * q^77 + 10 * q^79 + 9 * q^81 - 10*i * q^82 + 6*i * q^83 + 8 * q^86 + 2*i * q^88 + 6 * q^89 + 12 * q^91 - 6*i * q^92 - 4 * q^94 + 6*i * q^97 - 3*i * q^98 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 6 * q^9 $$2 q - 2 q^{4} + 6 q^{9} + 4 q^{11} - 4 q^{14} + 2 q^{16} + 4 q^{19} + 12 q^{26} + 2 q^{29} - 12 q^{31} + 4 q^{34} - 6 q^{36} + 20 q^{41} - 4 q^{44} + 12 q^{46} + 6 q^{49} + 4 q^{56} - 16 q^{59} + 20 q^{61} - 2 q^{64} + 8 q^{71} - 4 q^{74} - 4 q^{76} + 20 q^{79} + 18 q^{81} + 16 q^{86} + 12 q^{89} + 24 q^{91} - 8 q^{94} + 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 6 * q^9 + 4 * q^11 - 4 * q^14 + 2 * q^16 + 4 * q^19 + 12 * q^26 + 2 * q^29 - 12 * q^31 + 4 * q^34 - 6 * q^36 + 20 * q^41 - 4 * q^44 + 12 * q^46 + 6 * q^49 + 4 * q^56 - 16 * q^59 + 20 * q^61 - 2 * q^64 + 8 * q^71 - 4 * q^74 - 4 * q^76 + 20 * q^79 + 18 * q^81 + 16 * q^86 + 12 * q^89 + 24 * q^91 - 8 * q^94 + 12 * q^99

## 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
 1.00000i − 1.00000i
1.00000i 0 −1.00000 0 0 2.00000i 1.00000i 3.00000 0
349.2 1.00000i 0 −1.00000 0 0 2.00000i 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.d 2
5.b even 2 1 inner 1450.2.b.d 2
5.c odd 4 1 290.2.a.a 1
5.c odd 4 1 1450.2.a.g 1
15.e even 4 1 2610.2.a.l 1
20.e even 4 1 2320.2.a.d 1
40.i odd 4 1 9280.2.a.k 1
40.k even 4 1 9280.2.a.q 1
145.h odd 4 1 8410.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.a.a 1 5.c odd 4 1
1450.2.a.g 1 5.c odd 4 1
1450.2.b.d 2 1.a even 1 1 trivial
1450.2.b.d 2 5.b even 2 1 inner
2320.2.a.d 1 20.e even 4 1
2610.2.a.l 1 15.e even 4 1
8410.2.a.i 1 145.h odd 4 1
9280.2.a.k 1 40.i odd 4 1
9280.2.a.q 1 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 $$S_{2}^{\mathrm{new}}(1450, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T + 6)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 10)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 16$$
$53$ $$T^{2} + 100$$
$59$ $$(T + 8)^{2}$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$(T - 4)^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 36$$
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