# Properties

 Label 1450.2.b.l Level $1450$ Weight $2$ Character orbit 1450.b Analytic conductor $11.578$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.24681024.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 12x^{4} + 36x^{2} + 9$$ x^6 + 12*x^4 + 36*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} - q^{4} + ( - \beta_{3} + 1) q^{6} + ( - \beta_{5} - \beta_{2} + \beta_1) q^{7} + \beta_{2} q^{8} + (\beta_{4} + \beta_{3} - 2) q^{9}+O(q^{10})$$ q - b2 * q^2 + (b2 + b1) * q^3 - q^4 + (-b3 + 1) * q^6 + (-b5 - b2 + b1) * q^7 + b2 * q^8 + (b4 + b3 - 2) * q^9 $$q - \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} - q^{4} + ( - \beta_{3} + 1) q^{6} + ( - \beta_{5} - \beta_{2} + \beta_1) q^{7} + \beta_{2} q^{8} + (\beta_{4} + \beta_{3} - 2) q^{9} + 2 \beta_{3} q^{11} + ( - \beta_{2} - \beta_1) q^{12} + ( - 2 \beta_{5} + \beta_{2} - \beta_1) q^{13} + ( - \beta_{4} - \beta_{3} - 1) q^{14} + q^{16} + (\beta_{5} - \beta_{2} + \beta_1) q^{17} + ( - \beta_{5} + 2 \beta_{2} + \beta_1) q^{18} + ( - 2 \beta_{4} + 2 \beta_{3}) q^{19} + (3 \beta_{4} - 4) q^{21} + 2 \beta_1 q^{22} + (2 \beta_{5} - 5 \beta_{2} + \beta_1) q^{23} + (\beta_{3} - 1) q^{24} + ( - 2 \beta_{4} + \beta_{3} + 1) q^{26} + (3 \beta_{5} - 4 \beta_{2}) q^{27} + (\beta_{5} + \beta_{2} - \beta_1) q^{28} + q^{29} + ( - \beta_{4} + \beta_{3} - 3) q^{31} - \beta_{2} q^{32} + (2 \beta_{5} - 8 \beta_{2}) q^{33} + (\beta_{4} - \beta_{3} - 1) q^{34} + ( - \beta_{4} - \beta_{3} + 2) q^{36} + ( - 2 \beta_{5} + 2 \beta_1) q^{37} + (2 \beta_{5} + 2 \beta_1) q^{38} + (3 \beta_{4} + 3 \beta_{3} + 1) q^{39} + ( - 2 \beta_{4} - 2 \beta_{3} - 4) q^{41} + ( - 3 \beta_{5} + 4 \beta_{2}) q^{42} + ( - \beta_{5} + 3 \beta_{2} - \beta_1) q^{43} - 2 \beta_{3} q^{44} + (2 \beta_{4} - \beta_{3} - 5) q^{46} - 4 \beta_1 q^{47} + (\beta_{2} + \beta_1) q^{48} + (\beta_{3} - 6) q^{49} + ( - \beta_{4} - 2 \beta_{3} - 2) q^{51} + (2 \beta_{5} - \beta_{2} + \beta_1) q^{52} + ( - 3 \beta_{5} + 3 \beta_{2} - 3 \beta_1) q^{53} + (3 \beta_{4} - 4) q^{54} + (\beta_{4} + \beta_{3} + 1) q^{56} + ( - 2 \beta_{5} - 6 \beta_{2} + 2 \beta_1) q^{57} - \beta_{2} q^{58} + ( - 2 \beta_{4} + 3 \beta_{3} + 5) q^{59} + ( - 3 \beta_{3} + 5) q^{61} + (\beta_{5} + 3 \beta_{2} + \beta_1) q^{62} + (3 \beta_{5} - 10 \beta_{2} - 4 \beta_1) q^{63} - q^{64} + (2 \beta_{4} - 8) q^{66} + ( - 2 \beta_{5} - 6 \beta_{2} - 4 \beta_1) q^{67} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{68} + ( - 3 \beta_{4} - 7 \beta_{3} + 3) q^{69} - 12 q^{71} + (\beta_{5} - 2 \beta_{2} - \beta_1) q^{72} + ( - \beta_{5} - 3 \beta_{2} - \beta_1) q^{73} + ( - 2 \beta_{4} - 2 \beta_{3}) q^{74} + (2 \beta_{4} - 2 \beta_{3}) q^{76} + (4 \beta_{5} - 10 \beta_{2} + 2 \beta_1) q^{77} + ( - 3 \beta_{5} - \beta_{2} + 3 \beta_1) q^{78} + (4 \beta_{4} - 3 \beta_{3} - 3) q^{79} + ( - 3 \beta_{4} - 4 \beta_{3} + 1) q^{81} + (2 \beta_{5} + 4 \beta_{2} - 2 \beta_1) q^{82} + (2 \beta_{5} + 4 \beta_{2} + 2 \beta_1) q^{83} + ( - 3 \beta_{4} + 4) q^{84} + ( - \beta_{4} + \beta_{3} + 3) q^{86} + (\beta_{2} + \beta_1) q^{87} - 2 \beta_1 q^{88} - 2 \beta_{3} q^{89} + ( - 3 \beta_{4} + 8 \beta_{3} - 8) q^{91} + ( - 2 \beta_{5} + 5 \beta_{2} - \beta_1) q^{92} + ( - \beta_{5} - 6 \beta_{2} - 2 \beta_1) q^{93} + 4 \beta_{3} q^{94} + ( - \beta_{3} + 1) q^{96} + ( - 2 \beta_{5} - 3 \beta_{2} - \beta_1) q^{97} + (6 \beta_{2} + \beta_1) q^{98} + ( - 4 \beta_{4} - 4 \beta_{3} + 10) q^{99}+O(q^{100})$$ q - b2 * q^2 + (b2 + b1) * q^3 - q^4 + (-b3 + 1) * q^6 + (-b5 - b2 + b1) * q^7 + b2 * q^8 + (b4 + b3 - 2) * q^9 + 2*b3 * q^11 + (-b2 - b1) * q^12 + (-2*b5 + b2 - b1) * q^13 + (-b4 - b3 - 1) * q^14 + q^16 + (b5 - b2 + b1) * q^17 + (-b5 + 2*b2 + b1) * q^18 + (-2*b4 + 2*b3) * q^19 + (3*b4 - 4) * q^21 + 2*b1 * q^22 + (2*b5 - 5*b2 + b1) * q^23 + (b3 - 1) * q^24 + (-2*b4 + b3 + 1) * q^26 + (3*b5 - 4*b2) * q^27 + (b5 + b2 - b1) * q^28 + q^29 + (-b4 + b3 - 3) * q^31 - b2 * q^32 + (2*b5 - 8*b2) * q^33 + (b4 - b3 - 1) * q^34 + (-b4 - b3 + 2) * q^36 + (-2*b5 + 2*b1) * q^37 + (2*b5 + 2*b1) * q^38 + (3*b4 + 3*b3 + 1) * q^39 + (-2*b4 - 2*b3 - 4) * q^41 + (-3*b5 + 4*b2) * q^42 + (-b5 + 3*b2 - b1) * q^43 - 2*b3 * q^44 + (2*b4 - b3 - 5) * q^46 - 4*b1 * q^47 + (b2 + b1) * q^48 + (b3 - 6) * q^49 + (-b4 - 2*b3 - 2) * q^51 + (2*b5 - b2 + b1) * q^52 + (-3*b5 + 3*b2 - 3*b1) * q^53 + (3*b4 - 4) * q^54 + (b4 + b3 + 1) * q^56 + (-2*b5 - 6*b2 + 2*b1) * q^57 - b2 * q^58 + (-2*b4 + 3*b3 + 5) * q^59 + (-3*b3 + 5) * q^61 + (b5 + 3*b2 + b1) * q^62 + (3*b5 - 10*b2 - 4*b1) * q^63 - q^64 + (2*b4 - 8) * q^66 + (-2*b5 - 6*b2 - 4*b1) * q^67 + (-b5 + b2 - b1) * q^68 + (-3*b4 - 7*b3 + 3) * q^69 - 12 * q^71 + (b5 - 2*b2 - b1) * q^72 + (-b5 - 3*b2 - b1) * q^73 + (-2*b4 - 2*b3) * q^74 + (2*b4 - 2*b3) * q^76 + (4*b5 - 10*b2 + 2*b1) * q^77 + (-3*b5 - b2 + 3*b1) * q^78 + (4*b4 - 3*b3 - 3) * q^79 + (-3*b4 - 4*b3 + 1) * q^81 + (2*b5 + 4*b2 - 2*b1) * q^82 + (2*b5 + 4*b2 + 2*b1) * q^83 + (-3*b4 + 4) * q^84 + (-b4 + b3 + 3) * q^86 + (b2 + b1) * q^87 - 2*b1 * q^88 - 2*b3 * q^89 + (-3*b4 + 8*b3 - 8) * q^91 + (-2*b5 + 5*b2 - b1) * q^92 + (-b5 - 6*b2 - 2*b1) * q^93 + 4*b3 * q^94 + (-b3 + 1) * q^96 + (-2*b5 - 3*b2 - b1) * q^97 + (6*b2 + b1) * q^98 + (-4*b4 - 4*b3 + 10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} + 6 q^{6} - 12 q^{9}+O(q^{10})$$ 6 * q - 6 * q^4 + 6 * q^6 - 12 * q^9 $$6 q - 6 q^{4} + 6 q^{6} - 12 q^{9} - 6 q^{14} + 6 q^{16} - 24 q^{21} - 6 q^{24} + 6 q^{26} + 6 q^{29} - 18 q^{31} - 6 q^{34} + 12 q^{36} + 6 q^{39} - 24 q^{41} - 30 q^{46} - 36 q^{49} - 12 q^{51} - 24 q^{54} + 6 q^{56} + 30 q^{59} + 30 q^{61} - 6 q^{64} - 48 q^{66} + 18 q^{69} - 72 q^{71} - 18 q^{79} + 6 q^{81} + 24 q^{84} + 18 q^{86} - 48 q^{91} + 6 q^{96} + 60 q^{99}+O(q^{100})$$ 6 * q - 6 * q^4 + 6 * q^6 - 12 * q^9 - 6 * q^14 + 6 * q^16 - 24 * q^21 - 6 * q^24 + 6 * q^26 + 6 * q^29 - 18 * q^31 - 6 * q^34 + 12 * q^36 + 6 * q^39 - 24 * q^41 - 30 * q^46 - 36 * q^49 - 12 * q^51 - 24 * q^54 + 6 * q^56 + 30 * q^59 + 30 * q^61 - 6 * q^64 - 48 * q^66 + 18 * q^69 - 72 * q^71 - 18 * q^79 + 6 * q^81 + 24 * q^84 + 18 * q^86 - 48 * q^91 + 6 * q^96 + 60 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 12x^{4} + 36x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 6\nu ) / 3$$ (v^3 + 6*v) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{4} + 6\nu^{2} ) / 3$$ (v^4 + 6*v^2) / 3 $$\beta_{4}$$ $$=$$ $$( \nu^{4} + 9\nu^{2} + 12 ) / 3$$ (v^4 + 9*v^2 + 12) / 3 $$\beta_{5}$$ $$=$$ $$( \nu^{5} + 10\nu^{3} + 21\nu ) / 3$$ (v^5 + 10*v^3 + 21*v) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} - 4$$ b4 - b3 - 4 $$\nu^{3}$$ $$=$$ $$3\beta_{2} - 6\beta_1$$ 3*b2 - 6*b1 $$\nu^{4}$$ $$=$$ $$-6\beta_{4} + 9\beta_{3} + 24$$ -6*b4 + 9*b3 + 24 $$\nu^{5}$$ $$=$$ $$3\beta_{5} - 30\beta_{2} + 39\beta_1$$ 3*b5 - 30*b2 + 39*b1

## 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
 − 2.66908i 0.523976i 2.14510i − 2.14510i − 0.523976i 2.66908i
1.00000i 1.66908i −1.00000 0 −1.66908 3.21417i 1.00000i 0.214175 0
349.2 1.00000i 1.52398i −1.00000 0 1.52398 3.67750i 1.00000i 0.677496 0
349.3 1.00000i 3.14510i −1.00000 0 3.14510 3.89167i 1.00000i −6.89167 0
349.4 1.00000i 3.14510i −1.00000 0 3.14510 3.89167i 1.00000i −6.89167 0
349.5 1.00000i 1.52398i −1.00000 0 1.52398 3.67750i 1.00000i 0.677496 0
349.6 1.00000i 1.66908i −1.00000 0 −1.66908 3.21417i 1.00000i 0.214175 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 1450.2.b.l 6
5.b even 2 1 inner 1450.2.b.l 6
5.c odd 4 1 290.2.a.e 3
5.c odd 4 1 1450.2.a.p 3
15.e even 4 1 2610.2.a.x 3
20.e even 4 1 2320.2.a.l 3
40.i odd 4 1 9280.2.a.bf 3
40.k even 4 1 9280.2.a.by 3
145.h odd 4 1 8410.2.a.v 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.a.e 3 5.c odd 4 1
1450.2.a.p 3 5.c odd 4 1
1450.2.b.l 6 1.a even 1 1 trivial
1450.2.b.l 6 5.b even 2 1 inner
2320.2.a.l 3 20.e even 4 1
2610.2.a.x 3 15.e even 4 1
8410.2.a.v 3 145.h odd 4 1
9280.2.a.bf 3 40.i odd 4 1
9280.2.a.by 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 $$S_{2}^{\mathrm{new}}(1450, [\chi])$$:

 $$T_{3}^{6} + 15T_{3}^{4} + 57T_{3}^{2} + 64$$ T3^6 + 15*T3^4 + 57*T3^2 + 64 $$T_{7}^{6} + 39T_{7}^{4} + 501T_{7}^{2} + 2116$$ T7^6 + 39*T7^4 + 501*T7^2 + 2116

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{3}$$
$3$ $$T^{6} + 15 T^{4} + \cdots + 64$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 39 T^{4} + \cdots + 2116$$
$11$ $$(T^{3} - 24 T - 24)^{2}$$
$13$ $$T^{6} + 75 T^{4} + \cdots + 13924$$
$17$ $$T^{6} + 27 T^{4} + \cdots + 324$$
$19$ $$(T^{3} - 48 T + 56)^{2}$$
$23$ $$T^{6} + 147 T^{4} + \cdots + 19044$$
$29$ $$(T - 1)^{6}$$
$31$ $$(T^{3} + 9 T^{2} + 15 T - 2)^{2}$$
$37$ $$T^{6} + 144 T^{4} + \cdots + 53824$$
$41$ $$(T^{3} + 12 T^{2} + \cdots - 456)^{2}$$
$43$ $$T^{6} + 51 T^{4} + \cdots + 256$$
$47$ $$T^{6} + 192 T^{4} + \cdots + 36864$$
$53$ $$T^{6} + 243 T^{4} + \cdots + 236196$$
$59$ $$(T^{3} - 15 T^{2} + \cdots + 168)^{2}$$
$61$ $$(T^{3} - 15 T^{2} + \cdots + 226)^{2}$$
$67$ $$T^{6} + 324 T^{4} + \cdots + 541696$$
$71$ $$(T + 12)^{6}$$
$73$ $$T^{6} + 51 T^{4} + \cdots + 4$$
$79$ $$(T^{3} + 9 T^{2} + \cdots - 1102)^{2}$$
$83$ $$T^{6} + 144 T^{4} + \cdots + 5184$$
$89$ $$(T^{3} - 24 T + 24)^{2}$$
$97$ $$T^{6} + 99 T^{4} + \cdots + 4$$