# Properties

 Label 1450.2.b.h 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: $$\Q(i, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + \beta_1 q^{3} - q^{4} + (\beta_{3} - 1) q^{6} + (\beta_{2} - \beta_1) q^{7} + \beta_{2} q^{8} + (\beta_{3} - 1) q^{9}+O(q^{10})$$ q - b2 * q^2 + b1 * q^3 - q^4 + (b3 - 1) * q^6 + (b2 - b1) * q^7 + b2 * q^8 + (b3 - 1) * q^9 $$q - \beta_{2} q^{2} + \beta_1 q^{3} - q^{4} + (\beta_{3} - 1) q^{6} + (\beta_{2} - \beta_1) q^{7} + \beta_{2} q^{8} + (\beta_{3} - 1) q^{9} + 2 \beta_{3} q^{11} - \beta_1 q^{12} - \beta_1 q^{13} + ( - \beta_{3} + 2) q^{14} + q^{16} + ( - \beta_{2} - \beta_1) q^{17} - \beta_1 q^{18} + ( - 2 \beta_{3} - 2) q^{19} + ( - 2 \beta_{3} + 5) q^{21} + ( - 2 \beta_{2} - 2 \beta_1) q^{22} - 3 \beta_1 q^{23} + ( - \beta_{3} + 1) q^{24} + ( - \beta_{3} + 1) q^{26} + (3 \beta_{2} + 2 \beta_1) q^{27} + ( - \beta_{2} + \beta_1) q^{28} + q^{29} + (\beta_{3} + 8) q^{31} - \beta_{2} q^{32} + 6 \beta_{2} q^{33} - \beta_{3} q^{34} + ( - \beta_{3} + 1) q^{36} + (4 \beta_{2} + 2 \beta_1) q^{37} + (4 \beta_{2} + 2 \beta_1) q^{38} + ( - \beta_{3} + 4) q^{39} + ( - 2 \beta_{3} + 6) q^{41} + ( - 3 \beta_{2} + 2 \beta_1) q^{42} + ( - \beta_{2} + \beta_1) q^{43} - 2 \beta_{3} q^{44} + ( - 3 \beta_{3} + 3) q^{46} + ( - 4 \beta_{2} - 4 \beta_1) q^{47} + \beta_1 q^{48} + 3 \beta_{3} q^{49} + 3 q^{51} + \beta_1 q^{52} + (11 \beta_{2} - \beta_1) q^{53} + (2 \beta_{3} + 1) q^{54} + (\beta_{3} - 2) q^{56} + ( - 6 \beta_{2} - 2 \beta_1) q^{57} - \beta_{2} q^{58} + (\beta_{3} + 9) q^{59} + ( - \beta_{3} - 1) q^{61} + ( - 9 \beta_{2} - \beta_1) q^{62} + ( - 3 \beta_{2} + 2 \beta_1) q^{63} - q^{64} + 6 q^{66} - 4 \beta_{2} q^{67} + (\beta_{2} + \beta_1) q^{68} + ( - 3 \beta_{3} + 12) q^{69} + 4 \beta_{3} q^{71} + \beta_1 q^{72} + ( - 3 \beta_{2} - 7 \beta_1) q^{73} + (2 \beta_{3} + 2) q^{74} + (2 \beta_{3} + 2) q^{76} + ( - 4 \beta_{2} + 2 \beta_1) q^{77} + ( - 3 \beta_{2} + \beta_1) q^{78} + (3 \beta_{3} - 5) q^{79} + (2 \beta_{3} - 8) q^{81} + ( - 4 \beta_{2} + 2 \beta_1) q^{82} - 6 \beta_1 q^{83} + (2 \beta_{3} - 5) q^{84} + (\beta_{3} - 2) q^{86} + \beta_1 q^{87} + (2 \beta_{2} + 2 \beta_1) q^{88} + 6 \beta_{3} q^{89} + (2 \beta_{3} - 5) q^{91} + 3 \beta_1 q^{92} + (3 \beta_{2} + 8 \beta_1) q^{93} - 4 \beta_{3} q^{94} + (\beta_{3} - 1) q^{96} + 7 \beta_1 q^{97} + ( - 3 \beta_{2} - 3 \beta_1) q^{98} + 6 q^{99}+O(q^{100})$$ q - b2 * q^2 + b1 * q^3 - q^4 + (b3 - 1) * q^6 + (b2 - b1) * q^7 + b2 * q^8 + (b3 - 1) * q^9 + 2*b3 * q^11 - b1 * q^12 - b1 * q^13 + (-b3 + 2) * q^14 + q^16 + (-b2 - b1) * q^17 - b1 * q^18 + (-2*b3 - 2) * q^19 + (-2*b3 + 5) * q^21 + (-2*b2 - 2*b1) * q^22 - 3*b1 * q^23 + (-b3 + 1) * q^24 + (-b3 + 1) * q^26 + (3*b2 + 2*b1) * q^27 + (-b2 + b1) * q^28 + q^29 + (b3 + 8) * q^31 - b2 * q^32 + 6*b2 * q^33 - b3 * q^34 + (-b3 + 1) * q^36 + (4*b2 + 2*b1) * q^37 + (4*b2 + 2*b1) * q^38 + (-b3 + 4) * q^39 + (-2*b3 + 6) * q^41 + (-3*b2 + 2*b1) * q^42 + (-b2 + b1) * q^43 - 2*b3 * q^44 + (-3*b3 + 3) * q^46 + (-4*b2 - 4*b1) * q^47 + b1 * q^48 + 3*b3 * q^49 + 3 * q^51 + b1 * q^52 + (11*b2 - b1) * q^53 + (2*b3 + 1) * q^54 + (b3 - 2) * q^56 + (-6*b2 - 2*b1) * q^57 - b2 * q^58 + (b3 + 9) * q^59 + (-b3 - 1) * q^61 + (-9*b2 - b1) * q^62 + (-3*b2 + 2*b1) * q^63 - q^64 + 6 * q^66 - 4*b2 * q^67 + (b2 + b1) * q^68 + (-3*b3 + 12) * q^69 + 4*b3 * q^71 + b1 * q^72 + (-3*b2 - 7*b1) * q^73 + (2*b3 + 2) * q^74 + (2*b3 + 2) * q^76 + (-4*b2 + 2*b1) * q^77 + (-3*b2 + b1) * q^78 + (3*b3 - 5) * q^79 + (2*b3 - 8) * q^81 + (-4*b2 + 2*b1) * q^82 - 6*b1 * q^83 + (2*b3 - 5) * q^84 + (b3 - 2) * q^86 + b1 * q^87 + (2*b2 + 2*b1) * q^88 + 6*b3 * q^89 + (2*b3 - 5) * q^91 + 3*b1 * q^92 + (3*b2 + 8*b1) * q^93 - 4*b3 * q^94 + (b3 - 1) * q^96 + 7*b1 * q^97 + (-3*b2 - 3*b1) * q^98 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 2 * q^6 - 2 * q^9 $$4 q - 4 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{11} + 6 q^{14} + 4 q^{16} - 12 q^{19} + 16 q^{21} + 2 q^{24} + 2 q^{26} + 4 q^{29} + 34 q^{31} - 2 q^{34} + 2 q^{36} + 14 q^{39} + 20 q^{41} - 4 q^{44} + 6 q^{46} + 6 q^{49} + 12 q^{51} + 8 q^{54} - 6 q^{56} + 38 q^{59} - 6 q^{61} - 4 q^{64} + 24 q^{66} + 42 q^{69} + 8 q^{71} + 12 q^{74} + 12 q^{76} - 14 q^{79} - 28 q^{81} - 16 q^{84} - 6 q^{86} + 12 q^{89} - 16 q^{91} - 8 q^{94} - 2 q^{96} + 24 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^11 + 6 * q^14 + 4 * q^16 - 12 * q^19 + 16 * q^21 + 2 * q^24 + 2 * q^26 + 4 * q^29 + 34 * q^31 - 2 * q^34 + 2 * q^36 + 14 * q^39 + 20 * q^41 - 4 * q^44 + 6 * q^46 + 6 * q^49 + 12 * q^51 + 8 * q^54 - 6 * q^56 + 38 * q^59 - 6 * q^61 - 4 * q^64 + 24 * q^66 + 42 * q^69 + 8 * q^71 + 12 * q^74 + 12 * q^76 - 14 * q^79 - 28 * q^81 - 16 * q^84 - 6 * q^86 + 12 * q^89 - 16 * q^91 - 8 * q^94 - 2 * q^96 + 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 3$$ (v^3 + 4*v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ b3 - 4 $$\nu^{3}$$ $$=$$ $$3\beta_{2} - 4\beta_1$$ 3*b2 - 4*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
 − 2.30278i 1.30278i − 1.30278i 2.30278i
1.00000i 2.30278i −1.00000 0 −2.30278 3.30278i 1.00000i −2.30278 0
349.2 1.00000i 1.30278i −1.00000 0 1.30278 0.302776i 1.00000i 1.30278 0
349.3 1.00000i 1.30278i −1.00000 0 1.30278 0.302776i 1.00000i 1.30278 0
349.4 1.00000i 2.30278i −1.00000 0 −2.30278 3.30278i 1.00000i −2.30278 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.h 4
5.b even 2 1 inner 1450.2.b.h 4
5.c odd 4 1 290.2.a.c 2
5.c odd 4 1 1450.2.a.n 2
15.e even 4 1 2610.2.a.s 2
20.e even 4 1 2320.2.a.j 2
40.i odd 4 1 9280.2.a.x 2
40.k even 4 1 9280.2.a.bb 2
145.h odd 4 1 8410.2.a.s 2

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

 $$T_{3}^{4} + 7T_{3}^{2} + 9$$ T3^4 + 7*T3^2 + 9 $$T_{7}^{4} + 11T_{7}^{2} + 1$$ T7^4 + 11*T7^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} + 7T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 11T^{2} + 1$$
$11$ $$(T^{2} - 2 T - 12)^{2}$$
$13$ $$T^{4} + 7T^{2} + 9$$
$17$ $$T^{4} + 7T^{2} + 9$$
$19$ $$(T^{2} + 6 T - 4)^{2}$$
$23$ $$T^{4} + 63T^{2} + 729$$
$29$ $$(T - 1)^{4}$$
$31$ $$(T^{2} - 17 T + 69)^{2}$$
$37$ $$T^{4} + 44T^{2} + 16$$
$41$ $$(T^{2} - 10 T + 12)^{2}$$
$43$ $$T^{4} + 11T^{2} + 1$$
$47$ $$T^{4} + 112T^{2} + 2304$$
$53$ $$T^{4} + 271 T^{2} + 16641$$
$59$ $$(T^{2} - 19 T + 87)^{2}$$
$61$ $$(T^{2} + 3 T - 1)^{2}$$
$67$ $$(T^{2} + 16)^{2}$$
$71$ $$(T^{2} - 4 T - 48)^{2}$$
$73$ $$T^{4} + 319 T^{2} + 25281$$
$79$ $$(T^{2} + 7 T - 17)^{2}$$
$83$ $$T^{4} + 252 T^{2} + 11664$$
$89$ $$(T^{2} - 6 T - 108)^{2}$$
$97$ $$T^{4} + 343 T^{2} + 21609$$