# Properties

 Label 1450.2.b.f 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.

 $$f(q)$$ $$=$$ $$q - i q^{2} + 3 i q^{3} - q^{4} + 3 q^{6} - 2 i q^{7} + i q^{8} - 6 q^{9} +O(q^{10})$$ q - i * q^2 + 3*i * q^3 - q^4 + 3 * q^6 - 2*i * q^7 + i * q^8 - 6 * q^9 $$q - i q^{2} + 3 i q^{3} - q^{4} + 3 q^{6} - 2 i q^{7} + i q^{8} - 6 q^{9} - q^{11} - 3 i q^{12} - 3 i q^{13} - 2 q^{14} + q^{16} - 4 i q^{17} + 6 i q^{18} + 8 q^{19} + 6 q^{21} + i q^{22} - 3 q^{24} - 3 q^{26} - 9 i q^{27} + 2 i q^{28} + q^{29} + 3 q^{31} - i q^{32} - 3 i q^{33} - 4 q^{34} + 6 q^{36} - 8 i q^{37} - 8 i q^{38} + 9 q^{39} - 2 q^{41} - 6 i q^{42} - 7 i q^{43} + q^{44} + 11 i q^{47} + 3 i q^{48} + 3 q^{49} + 12 q^{51} + 3 i q^{52} - i q^{53} - 9 q^{54} + 2 q^{56} + 24 i q^{57} - i q^{58} + 4 q^{59} + 4 q^{61} - 3 i q^{62} + 12 i q^{63} - q^{64} - 3 q^{66} - 4 i q^{67} + 4 i q^{68} - 2 q^{71} - 6 i q^{72} + 12 i q^{73} - 8 q^{74} - 8 q^{76} + 2 i q^{77} - 9 i q^{78} + 7 q^{79} + 9 q^{81} + 2 i q^{82} - 6 q^{84} - 7 q^{86} + 3 i q^{87} - i q^{88} + 6 q^{89} - 6 q^{91} + 9 i q^{93} + 11 q^{94} + 3 q^{96} - 6 i q^{97} - 3 i q^{98} + 6 q^{99} +O(q^{100})$$ q - i * q^2 + 3*i * q^3 - q^4 + 3 * q^6 - 2*i * q^7 + i * q^8 - 6 * q^9 - q^11 - 3*i * q^12 - 3*i * q^13 - 2 * q^14 + q^16 - 4*i * q^17 + 6*i * q^18 + 8 * q^19 + 6 * q^21 + i * q^22 - 3 * q^24 - 3 * q^26 - 9*i * q^27 + 2*i * q^28 + q^29 + 3 * q^31 - i * q^32 - 3*i * q^33 - 4 * q^34 + 6 * q^36 - 8*i * q^37 - 8*i * q^38 + 9 * q^39 - 2 * q^41 - 6*i * q^42 - 7*i * q^43 + q^44 + 11*i * q^47 + 3*i * q^48 + 3 * q^49 + 12 * q^51 + 3*i * q^52 - i * q^53 - 9 * q^54 + 2 * q^56 + 24*i * q^57 - i * q^58 + 4 * q^59 + 4 * q^61 - 3*i * q^62 + 12*i * q^63 - q^64 - 3 * q^66 - 4*i * q^67 + 4*i * q^68 - 2 * q^71 - 6*i * q^72 + 12*i * q^73 - 8 * q^74 - 8 * q^76 + 2*i * q^77 - 9*i * q^78 + 7 * q^79 + 9 * q^81 + 2*i * q^82 - 6 * q^84 - 7 * q^86 + 3*i * q^87 - i * q^88 + 6 * q^89 - 6 * q^91 + 9*i * q^93 + 11 * q^94 + 3 * q^96 - 6*i * q^97 - 3*i * q^98 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 6 q^{6} - 12 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 6 * q^6 - 12 * q^9 $$2 q - 2 q^{4} + 6 q^{6} - 12 q^{9} - 2 q^{11} - 4 q^{14} + 2 q^{16} + 16 q^{19} + 12 q^{21} - 6 q^{24} - 6 q^{26} + 2 q^{29} + 6 q^{31} - 8 q^{34} + 12 q^{36} + 18 q^{39} - 4 q^{41} + 2 q^{44} + 6 q^{49} + 24 q^{51} - 18 q^{54} + 4 q^{56} + 8 q^{59} + 8 q^{61} - 2 q^{64} - 6 q^{66} - 4 q^{71} - 16 q^{74} - 16 q^{76} + 14 q^{79} + 18 q^{81} - 12 q^{84} - 14 q^{86} + 12 q^{89} - 12 q^{91} + 22 q^{94} + 6 q^{96} + 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 6 * q^6 - 12 * q^9 - 2 * q^11 - 4 * q^14 + 2 * q^16 + 16 * q^19 + 12 * q^21 - 6 * q^24 - 6 * q^26 + 2 * q^29 + 6 * q^31 - 8 * q^34 + 12 * q^36 + 18 * q^39 - 4 * q^41 + 2 * q^44 + 6 * q^49 + 24 * q^51 - 18 * q^54 + 4 * q^56 + 8 * q^59 + 8 * q^61 - 2 * q^64 - 6 * q^66 - 4 * q^71 - 16 * q^74 - 16 * q^76 + 14 * q^79 + 18 * q^81 - 12 * q^84 - 14 * q^86 + 12 * q^89 - 12 * q^91 + 22 * q^94 + 6 * 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 3.00000i −1.00000 0 3.00000 2.00000i 1.00000i −6.00000 0
349.2 1.00000i 3.00000i −1.00000 0 3.00000 2.00000i 1.00000i −6.00000 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.f 2
5.b even 2 1 inner 1450.2.b.f 2
5.c odd 4 1 58.2.a.a 1
5.c odd 4 1 1450.2.a.i 1
15.e even 4 1 522.2.a.k 1
20.e even 4 1 464.2.a.f 1
35.f even 4 1 2842.2.a.d 1
40.i odd 4 1 1856.2.a.p 1
40.k even 4 1 1856.2.a.b 1
55.e even 4 1 7018.2.a.c 1
60.l odd 4 1 4176.2.a.bh 1
65.h odd 4 1 9802.2.a.d 1
145.e even 4 1 1682.2.b.e 2
145.h odd 4 1 1682.2.a.j 1
145.j even 4 1 1682.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.a.a 1 5.c odd 4 1
464.2.a.f 1 20.e even 4 1
522.2.a.k 1 15.e even 4 1
1450.2.a.i 1 5.c odd 4 1
1450.2.b.f 2 1.a even 1 1 trivial
1450.2.b.f 2 5.b even 2 1 inner
1682.2.a.j 1 145.h odd 4 1
1682.2.b.e 2 145.e even 4 1
1682.2.b.e 2 145.j even 4 1
1856.2.a.b 1 40.k even 4 1
1856.2.a.p 1 40.i odd 4 1
2842.2.a.d 1 35.f even 4 1
4176.2.a.bh 1 60.l odd 4 1
7018.2.a.c 1 55.e even 4 1
9802.2.a.d 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 $$S_{2}^{\mathrm{new}}(1450, [\chi])$$:

 $$T_{3}^{2} + 9$$ T3^2 + 9 $$T_{7}^{2} + 4$$ T7^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 9$$
$17$ $$T^{2} + 16$$
$19$ $$(T - 8)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T - 3)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 49$$
$47$ $$T^{2} + 121$$
$53$ $$T^{2} + 1$$
$59$ $$(T - 4)^{2}$$
$61$ $$(T - 4)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T + 2)^{2}$$
$73$ $$T^{2} + 144$$
$79$ $$(T - 7)^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 36$$