# Properties

 Label 4650.2.d.bh Level $4650$ Weight $2$ Character orbit 4650.d Analytic conductor $37.130$ 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] = [4650,2,Mod(3349,4650)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4650.3349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.d (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: $$37.1304369399$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 17x^{2} + 64$$ x^4 + 17*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + \beta_{2} q^{3} - q^{4} + q^{6} + \beta_1 q^{7} + \beta_{2} q^{8} - q^{9}+O(q^{10})$$ q - b2 * q^2 + b2 * q^3 - q^4 + q^6 + b1 * q^7 + b2 * q^8 - q^9 $$q - \beta_{2} q^{2} + \beta_{2} q^{3} - q^{4} + q^{6} + \beta_1 q^{7} + \beta_{2} q^{8} - q^{9} + (\beta_{3} + 3) q^{11} - \beta_{2} q^{12} - 2 \beta_{2} q^{13} + (\beta_{3} - 1) q^{14} + q^{16} + ( - 2 \beta_{2} - 2 \beta_1) q^{17} + \beta_{2} q^{18} + (\beta_{3} + 3) q^{19} + ( - \beta_{3} + 1) q^{21} + ( - 4 \beta_{2} - \beta_1) q^{22} - \beta_1 q^{23} - q^{24} - 2 q^{26} - \beta_{2} q^{27} - \beta_1 q^{28} + ( - 2 \beta_{3} + 4) q^{29} - q^{31} - \beta_{2} q^{32} + (4 \beta_{2} + \beta_1) q^{33} - 2 \beta_{3} q^{34} + q^{36} + ( - 6 \beta_{2} - 2 \beta_1) q^{37} + ( - 4 \beta_{2} - \beta_1) q^{38} + 2 q^{39} + ( - 2 \beta_{3} - 4) q^{41} + \beta_1 q^{42} + (4 \beta_{2} + \beta_1) q^{43} + ( - \beta_{3} - 3) q^{44} + ( - \beta_{3} + 1) q^{46} - 2 \beta_1 q^{47} + \beta_{2} q^{48} + (\beta_{3} - 2) q^{49} + 2 \beta_{3} q^{51} + 2 \beta_{2} q^{52} + ( - 2 \beta_{2} - \beta_1) q^{53} - q^{54} + ( - \beta_{3} + 1) q^{56} + (4 \beta_{2} + \beta_1) q^{57} + ( - 2 \beta_{2} + 2 \beta_1) q^{58} + (2 \beta_{3} + 2) q^{59} + ( - 4 \beta_{3} + 2) q^{61} + \beta_{2} q^{62} - \beta_1 q^{63} - q^{64} + (\beta_{3} + 3) q^{66} + ( - 4 \beta_{2} + 2 \beta_1) q^{67} + (2 \beta_{2} + 2 \beta_1) q^{68} + (\beta_{3} - 1) q^{69} + ( - \beta_{3} + 1) q^{71} - \beta_{2} q^{72} + (2 \beta_{2} + 3 \beta_1) q^{73} + ( - 2 \beta_{3} - 4) q^{74} + ( - \beta_{3} - 3) q^{76} + (8 \beta_{2} + 3 \beta_1) q^{77} - 2 \beta_{2} q^{78} + (\beta_{3} + 7) q^{79} + q^{81} + (6 \beta_{2} + 2 \beta_1) q^{82} - 12 \beta_{2} q^{83} + (\beta_{3} - 1) q^{84} + (\beta_{3} + 3) q^{86} + (2 \beta_{2} - 2 \beta_1) q^{87} + (4 \beta_{2} + \beta_1) q^{88} + ( - \beta_{3} - 1) q^{89} + (2 \beta_{3} - 2) q^{91} + \beta_1 q^{92} - \beta_{2} q^{93} + ( - 2 \beta_{3} + 2) q^{94} + q^{96} - 2 \beta_{2} q^{97} + (\beta_{2} - \beta_1) q^{98} + ( - \beta_{3} - 3) q^{99}+O(q^{100})$$ q - b2 * q^2 + b2 * q^3 - q^4 + q^6 + b1 * q^7 + b2 * q^8 - q^9 + (b3 + 3) * q^11 - b2 * q^12 - 2*b2 * q^13 + (b3 - 1) * q^14 + q^16 + (-2*b2 - 2*b1) * q^17 + b2 * q^18 + (b3 + 3) * q^19 + (-b3 + 1) * q^21 + (-4*b2 - b1) * q^22 - b1 * q^23 - q^24 - 2 * q^26 - b2 * q^27 - b1 * q^28 + (-2*b3 + 4) * q^29 - q^31 - b2 * q^32 + (4*b2 + b1) * q^33 - 2*b3 * q^34 + q^36 + (-6*b2 - 2*b1) * q^37 + (-4*b2 - b1) * q^38 + 2 * q^39 + (-2*b3 - 4) * q^41 + b1 * q^42 + (4*b2 + b1) * q^43 + (-b3 - 3) * q^44 + (-b3 + 1) * q^46 - 2*b1 * q^47 + b2 * q^48 + (b3 - 2) * q^49 + 2*b3 * q^51 + 2*b2 * q^52 + (-2*b2 - b1) * q^53 - q^54 + (-b3 + 1) * q^56 + (4*b2 + b1) * q^57 + (-2*b2 + 2*b1) * q^58 + (2*b3 + 2) * q^59 + (-4*b3 + 2) * q^61 + b2 * q^62 - b1 * q^63 - q^64 + (b3 + 3) * q^66 + (-4*b2 + 2*b1) * q^67 + (2*b2 + 2*b1) * q^68 + (b3 - 1) * q^69 + (-b3 + 1) * q^71 - b2 * q^72 + (2*b2 + 3*b1) * q^73 + (-2*b3 - 4) * q^74 + (-b3 - 3) * q^76 + (8*b2 + 3*b1) * q^77 - 2*b2 * q^78 + (b3 + 7) * q^79 + q^81 + (6*b2 + 2*b1) * q^82 - 12*b2 * q^83 + (b3 - 1) * q^84 + (b3 + 3) * q^86 + (2*b2 - 2*b1) * q^87 + (4*b2 + b1) * q^88 + (-b3 - 1) * q^89 + (2*b3 - 2) * q^91 + b1 * q^92 - b2 * q^93 + (-2*b3 + 2) * q^94 + q^96 - 2*b2 * q^97 + (b2 - b1) * q^98 + (-b3 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 14 q^{11} - 2 q^{14} + 4 q^{16} + 14 q^{19} + 2 q^{21} - 4 q^{24} - 8 q^{26} + 12 q^{29} - 4 q^{31} - 4 q^{34} + 4 q^{36} + 8 q^{39} - 20 q^{41} - 14 q^{44} + 2 q^{46} - 6 q^{49} + 4 q^{51} - 4 q^{54} + 2 q^{56} + 12 q^{59} - 4 q^{64} + 14 q^{66} - 2 q^{69} + 2 q^{71} - 20 q^{74} - 14 q^{76} + 30 q^{79} + 4 q^{81} - 2 q^{84} + 14 q^{86} - 6 q^{89} - 4 q^{91} + 4 q^{94} + 4 q^{96} - 14 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 14 * q^11 - 2 * q^14 + 4 * q^16 + 14 * q^19 + 2 * q^21 - 4 * q^24 - 8 * q^26 + 12 * q^29 - 4 * q^31 - 4 * q^34 + 4 * q^36 + 8 * q^39 - 20 * q^41 - 14 * q^44 + 2 * q^46 - 6 * q^49 + 4 * q^51 - 4 * q^54 + 2 * q^56 + 12 * q^59 - 4 * q^64 + 14 * q^66 - 2 * q^69 + 2 * q^71 - 20 * q^74 - 14 * q^76 + 30 * q^79 + 4 * q^81 - 2 * q^84 + 14 * q^86 - 6 * q^89 - 4 * q^91 + 4 * q^94 + 4 * q^96 - 14 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 17x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 9\nu ) / 8$$ (v^3 + 9*v) / 8 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 9$$ v^2 + 9
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 9$$ b3 - 9 $$\nu^{3}$$ $$=$$ $$8\beta_{2} - 9\beta_1$$ 8*b2 - 9*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$$.

 $$n$$ $$1801$$ $$2977$$ $$3101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3349.1
 − 3.37228i 2.37228i − 2.37228i 3.37228i
1.00000i 1.00000i −1.00000 0 1.00000 3.37228i 1.00000i −1.00000 0
3349.2 1.00000i 1.00000i −1.00000 0 1.00000 2.37228i 1.00000i −1.00000 0
3349.3 1.00000i 1.00000i −1.00000 0 1.00000 2.37228i 1.00000i −1.00000 0
3349.4 1.00000i 1.00000i −1.00000 0 1.00000 3.37228i 1.00000i −1.00000 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 4650.2.d.bh 4
5.b even 2 1 inner 4650.2.d.bh 4
5.c odd 4 1 930.2.a.r 2
5.c odd 4 1 4650.2.a.by 2
15.e even 4 1 2790.2.a.bd 2
20.e even 4 1 7440.2.a.bg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.r 2 5.c odd 4 1
2790.2.a.bd 2 15.e even 4 1
4650.2.a.by 2 5.c odd 4 1
4650.2.d.bh 4 1.a even 1 1 trivial
4650.2.d.bh 4 5.b even 2 1 inner
7440.2.a.bg 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4650, [\chi])$$:

 $$T_{7}^{4} + 17T_{7}^{2} + 64$$ T7^4 + 17*T7^2 + 64 $$T_{11}^{2} - 7T_{11} + 4$$ T11^2 - 7*T11 + 4 $$T_{13}^{2} + 4$$ T13^2 + 4 $$T_{17}^{4} + 68T_{17}^{2} + 1024$$ T17^4 + 68*T17^2 + 1024 $$T_{19}^{2} - 7T_{19} + 4$$ T19^2 - 7*T19 + 4 $$T_{29}^{2} - 6T_{29} - 24$$ T29^2 - 6*T29 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 17T^{2} + 64$$
$11$ $$(T^{2} - 7 T + 4)^{2}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$T^{4} + 68T^{2} + 1024$$
$19$ $$(T^{2} - 7 T + 4)^{2}$$
$23$ $$T^{4} + 17T^{2} + 64$$
$29$ $$(T^{2} - 6 T - 24)^{2}$$
$31$ $$(T + 1)^{4}$$
$37$ $$T^{4} + 116T^{2} + 64$$
$41$ $$(T^{2} + 10 T - 8)^{2}$$
$43$ $$T^{4} + 41T^{2} + 16$$
$47$ $$T^{4} + 68T^{2} + 1024$$
$53$ $$T^{4} + 21T^{2} + 36$$
$59$ $$(T^{2} - 6 T - 24)^{2}$$
$61$ $$(T^{2} - 132)^{2}$$
$67$ $$T^{4} + 116T^{2} + 64$$
$71$ $$(T^{2} - T - 8)^{2}$$
$73$ $$T^{4} + 149T^{2} + 5476$$
$79$ $$(T^{2} - 15 T + 48)^{2}$$
$83$ $$(T^{2} + 144)^{2}$$
$89$ $$(T^{2} + 3 T - 6)^{2}$$
$97$ $$(T^{2} + 4)^{2}$$