# Properties

 Label 1450.2.b.b Level $1450$ Weight $2$ Character orbit 1450.b Analytic conductor $11.578$ Analytic rank $0$ Dimension $2$ 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: $$2$$ 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 58) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} - q^{4} - q^{6} - 2 i q^{7} - i q^{8} + 2 q^{9} +O(q^{10})$$ q + i * q^2 + i * q^3 - q^4 - q^6 - 2*i * q^7 - i * q^8 + 2 * q^9 $$q + i q^{2} + i q^{3} - q^{4} - q^{6} - 2 i q^{7} - i q^{8} + 2 q^{9} - 3 q^{11} - i q^{12} + i q^{13} + 2 q^{14} + q^{16} + 8 i q^{17} + 2 i q^{18} + 2 q^{21} - 3 i q^{22} - 4 i q^{23} + q^{24} - q^{26} + 5 i q^{27} + 2 i q^{28} + q^{29} - 3 q^{31} + i q^{32} - 3 i q^{33} - 8 q^{34} - 2 q^{36} + 8 i q^{37} - q^{39} + 2 q^{41} + 2 i q^{42} + 11 i q^{43} + 3 q^{44} + 4 q^{46} + 13 i q^{47} + i q^{48} + 3 q^{49} - 8 q^{51} - i q^{52} + 11 i q^{53} - 5 q^{54} - 2 q^{56} + i q^{58} - 8 q^{61} - 3 i q^{62} - 4 i q^{63} - q^{64} + 3 q^{66} - 12 i q^{67} - 8 i q^{68} + 4 q^{69} + 2 q^{71} - 2 i q^{72} - 4 i q^{73} - 8 q^{74} + 6 i q^{77} - i q^{78} - 15 q^{79} + q^{81} + 2 i q^{82} - 4 i q^{83} - 2 q^{84} - 11 q^{86} + i q^{87} + 3 i q^{88} + 10 q^{89} + 2 q^{91} + 4 i q^{92} - 3 i q^{93} - 13 q^{94} - q^{96} - 2 i q^{97} + 3 i q^{98} - 6 q^{99} +O(q^{100})$$ q + i * q^2 + i * q^3 - q^4 - q^6 - 2*i * q^7 - i * q^8 + 2 * q^9 - 3 * q^11 - i * q^12 + i * q^13 + 2 * q^14 + q^16 + 8*i * q^17 + 2*i * q^18 + 2 * q^21 - 3*i * q^22 - 4*i * q^23 + q^24 - q^26 + 5*i * q^27 + 2*i * q^28 + q^29 - 3 * q^31 + i * q^32 - 3*i * q^33 - 8 * q^34 - 2 * q^36 + 8*i * q^37 - q^39 + 2 * q^41 + 2*i * q^42 + 11*i * q^43 + 3 * q^44 + 4 * q^46 + 13*i * q^47 + i * q^48 + 3 * q^49 - 8 * q^51 - i * q^52 + 11*i * q^53 - 5 * q^54 - 2 * q^56 + i * q^58 - 8 * q^61 - 3*i * q^62 - 4*i * q^63 - q^64 + 3 * q^66 - 12*i * q^67 - 8*i * q^68 + 4 * q^69 + 2 * q^71 - 2*i * q^72 - 4*i * q^73 - 8 * q^74 + 6*i * q^77 - i * q^78 - 15 * q^79 + q^81 + 2*i * q^82 - 4*i * q^83 - 2 * q^84 - 11 * q^86 + i * q^87 + 3*i * q^88 + 10 * q^89 + 2 * q^91 + 4*i * q^92 - 3*i * q^93 - 13 * q^94 - q^96 - 2*i * q^97 + 3*i * q^98 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 $$2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} - 6 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{21} + 2 q^{24} - 2 q^{26} + 2 q^{29} - 6 q^{31} - 16 q^{34} - 4 q^{36} - 2 q^{39} + 4 q^{41} + 6 q^{44} + 8 q^{46} + 6 q^{49} - 16 q^{51} - 10 q^{54} - 4 q^{56} - 16 q^{61} - 2 q^{64} + 6 q^{66} + 8 q^{69} + 4 q^{71} - 16 q^{74} - 30 q^{79} + 2 q^{81} - 4 q^{84} - 22 q^{86} + 20 q^{89} + 4 q^{91} - 26 q^{94} - 2 q^{96} - 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 - 6 * q^11 + 4 * q^14 + 2 * q^16 + 4 * q^21 + 2 * q^24 - 2 * q^26 + 2 * q^29 - 6 * q^31 - 16 * q^34 - 4 * q^36 - 2 * q^39 + 4 * q^41 + 6 * q^44 + 8 * q^46 + 6 * q^49 - 16 * q^51 - 10 * q^54 - 4 * q^56 - 16 * q^61 - 2 * q^64 + 6 * q^66 + 8 * q^69 + 4 * q^71 - 16 * q^74 - 30 * q^79 + 2 * q^81 - 4 * q^84 - 22 * q^86 + 20 * q^89 + 4 * q^91 - 26 * q^94 - 2 * q^96 - 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 1.00000i −1.00000 0 −1.00000 2.00000i 1.00000i 2.00000 0
349.2 1.00000i 1.00000i −1.00000 0 −1.00000 2.00000i 1.00000i 2.00000 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.b 2
5.b even 2 1 inner 1450.2.b.b 2
5.c odd 4 1 58.2.a.b 1
5.c odd 4 1 1450.2.a.c 1
15.e even 4 1 522.2.a.b 1
20.e even 4 1 464.2.a.e 1
35.f even 4 1 2842.2.a.e 1
40.i odd 4 1 1856.2.a.k 1
40.k even 4 1 1856.2.a.f 1
55.e even 4 1 7018.2.a.a 1
60.l odd 4 1 4176.2.a.n 1
65.h odd 4 1 9802.2.a.a 1
145.e even 4 1 1682.2.b.a 2
145.h odd 4 1 1682.2.a.d 1
145.j even 4 1 1682.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.a.b 1 5.c odd 4 1
464.2.a.e 1 20.e even 4 1
522.2.a.b 1 15.e even 4 1
1450.2.a.c 1 5.c odd 4 1
1450.2.b.b 2 1.a even 1 1 trivial
1450.2.b.b 2 5.b even 2 1 inner
1682.2.a.d 1 145.h odd 4 1
1682.2.b.a 2 145.e even 4 1
1682.2.b.a 2 145.j even 4 1
1856.2.a.f 1 40.k even 4 1
1856.2.a.k 1 40.i odd 4 1
2842.2.a.e 1 35.f even 4 1
4176.2.a.n 1 60.l odd 4 1
7018.2.a.a 1 55.e even 4 1
9802.2.a.a 1 65.h odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 4$$ T7^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 64$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 121$$
$47$ $$T^{2} + 169$$
$53$ $$T^{2} + 121$$
$59$ $$T^{2}$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T - 2)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T + 15)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 4$$