# Properties

 Label 9200.2.a.cu Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.a (trivial)

## 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: yes Analytic conductor: $$73.4623698596$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.13955077.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 14x^{3} - x^{2} + 32x + 16$$ x^5 - 14*x^3 - x^2 + 32*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{3}) q^{7} + (\beta_{4} + \beta_{2} + 2) q^{9}+O(q^{10})$$ q + b1 * q^3 + (-b4 - b3) * q^7 + (b4 + b2 + 2) * q^9 $$q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{3}) q^{7} + (\beta_{4} + \beta_{2} + 2) q^{9} + (\beta_{2} - \beta_1) q^{11} + (\beta_{4} - \beta_{2} - 1) q^{13} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{17} + ( - \beta_{4} - 2 \beta_1 - 1) q^{19} + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{21} - q^{23} + ( - \beta_{4} + 2 \beta_{3} + 3 \beta_1 + 1) q^{27} + (\beta_{3} - \beta_{2} + 1) q^{29} + (\beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{31} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 3) q^{33} + (\beta_{3} - 3) q^{37} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{39} + (\beta_{3} - \beta_1 + 5) q^{41} + ( - 2 \beta_{3} - \beta_{2} - 2) q^{47} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 6) q^{49} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{51} + (2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{53} + ( - \beta_{4} - \beta_{2} - 3 \beta_1 - 9) q^{57} + ( - 2 \beta_{4} + \beta_{3} + 1) q^{59} + ( - 2 \beta_{4} - \beta_{2} - \beta_1) q^{61} + (\beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 - 8) q^{63} + (2 \beta_{4} + \beta_{3} - 2 \beta_1 + 1) q^{67} - \beta_1 q^{69} + (2 \beta_{4} - \beta_{3} + 3 \beta_1 - 1) q^{71} + (2 \beta_{3} + \beta_{2}) q^{73} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{77} + 2 \beta_{2} q^{79} + (\beta_{4} + 5 \beta_{2} - 3 \beta_1 + 10) q^{81} + (\beta_{3} - 9) q^{83} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{87} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 2) q^{89} + (4 \beta_{4} + \beta_{2} + 5 \beta_1 - 4) q^{91} + (\beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 7) q^{93} + (2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 6) q^{97} + (3 \beta_{4} + 4 \beta_{2} - 4 \beta_1 + 11) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-b4 - b3) * q^7 + (b4 + b2 + 2) * q^9 + (b2 - b1) * q^11 + (b4 - b2 - 1) * q^13 + (-b3 + b2 - b1 - 1) * q^17 + (-b4 - 2*b1 - 1) * q^19 + (b4 - b2 - b1 + 1) * q^21 - q^23 + (-b4 + 2*b3 + 3*b1 + 1) * q^27 + (b3 - b2 + 1) * q^29 + (b4 + b3 - b2 + 2*b1 - 4) * q^31 + (-b4 + 2*b3 + 2*b1 - 3) * q^33 + (b3 - 3) * q^37 + (-b4 - 2*b3 - 2*b2 - b1 - 3) * q^39 + (b3 - b1 + 5) * q^41 + (-2*b3 - b2 - 2) * q^47 + (-2*b4 + b3 - b2 - b1 + 6) * q^49 + (-b4 + 2*b3 - 2*b2 + 2*b1 - 3) * q^51 + (2*b4 + b3 - 2*b2 + 2*b1 - 1) * q^53 + (-b4 - b2 - 3*b1 - 9) * q^57 + (-2*b4 + b3 + 1) * q^59 + (-2*b4 - b2 - b1) * q^61 + (b4 + b3 - 3*b2 + b1 - 8) * q^63 + (2*b4 + b3 - 2*b1 + 1) * q^67 - b1 * q^69 + (2*b4 - b3 + 3*b1 - 1) * q^71 + (2*b3 + b2) * q^73 + (-b4 + 2*b3 - b2 - b1 - 3) * q^77 + 2*b2 * q^79 + (b4 + 5*b2 - 3*b1 + 10) * q^81 + (b3 - 9) * q^83 + (-2*b3 + b2 - 2*b1 - 2) * q^87 + (-2*b4 - 2*b3 - 2*b1 + 2) * q^89 + (4*b4 + b2 + 5*b1 - 4) * q^91 + (b4 - 2*b3 + 2*b2 - 5*b1 + 7) * q^93 + (2*b4 - 2*b3 + b2 - b1 - 6) * q^97 + (3*b4 + 4*b2 - 4*b1 + 11) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 2 q^{7} + 13 q^{9}+O(q^{10})$$ 5 * q - 2 * q^7 + 13 * q^9 $$5 q - 2 q^{7} + 13 q^{9} + q^{11} - 4 q^{13} - 4 q^{17} - 7 q^{19} + 6 q^{21} - 5 q^{23} + 3 q^{27} + 4 q^{29} - 19 q^{31} - 17 q^{33} - 15 q^{37} - 19 q^{39} + 25 q^{41} - 11 q^{47} + 25 q^{49} - 19 q^{51} - 3 q^{53} - 48 q^{57} + q^{59} - 5 q^{61} - 41 q^{63} + 9 q^{67} - q^{71} + q^{73} - 18 q^{77} + 2 q^{79} + 57 q^{81} - 45 q^{83} - 9 q^{87} + 6 q^{89} - 11 q^{91} + 39 q^{93} - 25 q^{97} + 65 q^{99}+O(q^{100})$$ 5 * q - 2 * q^7 + 13 * q^9 + q^11 - 4 * q^13 - 4 * q^17 - 7 * q^19 + 6 * q^21 - 5 * q^23 + 3 * q^27 + 4 * q^29 - 19 * q^31 - 17 * q^33 - 15 * q^37 - 19 * q^39 + 25 * q^41 - 11 * q^47 + 25 * q^49 - 19 * q^51 - 3 * q^53 - 48 * q^57 + q^59 - 5 * q^61 - 41 * q^63 + 9 * q^67 - q^71 + q^73 - 18 * q^77 + 2 * q^79 + 57 * q^81 - 45 * q^83 - 9 * q^87 + 6 * q^89 - 11 * q^91 + 39 * q^93 - 25 * q^97 + 65 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 14x^{3} - x^{2} + 32x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} - 10\nu^{2} + 3\nu + 4 ) / 4$$ (v^4 - 10*v^2 + 3*v + 4) / 4 $$\beta_{3}$$ $$=$$ $$( -\nu^{4} + 4\nu^{3} + 14\nu^{2} - 39\nu - 28 ) / 8$$ (-v^4 + 4*v^3 + 14*v^2 - 39*v - 28) / 8 $$\beta_{4}$$ $$=$$ $$( -\nu^{4} + 14\nu^{2} - 3\nu - 24 ) / 4$$ (-v^4 + 14*v^2 - 3*v - 24) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{2} + 5$$ b4 + b2 + 5 $$\nu^{3}$$ $$=$$ $$-\beta_{4} + 2\beta_{3} + 9\beta _1 + 1$$ -b4 + 2*b3 + 9*b1 + 1 $$\nu^{4}$$ $$=$$ $$10\beta_{4} + 14\beta_{2} - 3\beta _1 + 46$$ 10*b4 + 14*b2 - 3*b1 + 46

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.36002 −1.31091 −0.568386 1.93283 3.30649
0 −3.36002 0 0 0 −1.90754 0 8.28974 0
1.2 0 −1.31091 0 0 0 −4.66212 0 −1.28151 0
1.3 0 −0.568386 0 0 0 4.73770 0 −2.67694 0
1.4 0 1.93283 0 0 0 2.38236 0 0.735829 0
1.5 0 3.30649 0 0 0 −2.55040 0 7.93288 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$23$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9200.2.a.cu 5
4.b odd 2 1 4600.2.a.be 5
5.b even 2 1 1840.2.a.v 5
20.d odd 2 1 920.2.a.j 5
20.e even 4 2 4600.2.e.u 10
40.e odd 2 1 7360.2.a.co 5
40.f even 2 1 7360.2.a.cp 5
60.h even 2 1 8280.2.a.bs 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.j 5 20.d odd 2 1
1840.2.a.v 5 5.b even 2 1
4600.2.a.be 5 4.b odd 2 1
4600.2.e.u 10 20.e even 4 2
7360.2.a.co 5 40.e odd 2 1
7360.2.a.cp 5 40.f even 2 1
8280.2.a.bs 5 60.h even 2 1
9200.2.a.cu 5 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3}^{5} - 14T_{3}^{3} - T_{3}^{2} + 32T_{3} + 16$$ T3^5 - 14*T3^3 - T3^2 + 32*T3 + 16 $$T_{7}^{5} + 2T_{7}^{4} - 28T_{7}^{3} - 57T_{7}^{2} + 128T_{7} + 256$$ T7^5 + 2*T7^4 - 28*T7^3 - 57*T7^2 + 128*T7 + 256 $$T_{11}^{5} - T_{11}^{4} - 35T_{11}^{3} + 28T_{11}^{2} + 172T_{11} - 64$$ T11^5 - T11^4 - 35*T11^3 + 28*T11^2 + 172*T11 - 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - 14 T^{3} - T^{2} + 32 T + 16$$
$5$ $$T^{5}$$
$7$ $$T^{5} + 2 T^{4} - 28 T^{3} - 57 T^{2} + \cdots + 256$$
$11$ $$T^{5} - T^{4} - 35 T^{3} + 28 T^{2} + \cdots - 64$$
$13$ $$T^{5} + 4 T^{4} - 46 T^{3} - 75 T^{2} + \cdots - 500$$
$17$ $$T^{5} + 4 T^{4} - 46 T^{3} - 157 T^{2} + \cdots - 32$$
$19$ $$T^{5} + 7 T^{4} - 41 T^{3} - 180 T^{2} + \cdots - 512$$
$23$ $$(T + 1)^{5}$$
$29$ $$T^{5} - 4 T^{4} - 41 T^{3} - 36 T^{2} + \cdots + 8$$
$31$ $$T^{5} + 19 T^{4} + 72 T^{3} + \cdots + 128$$
$37$ $$T^{5} + 15 T^{4} + 64 T^{3} + 36 T^{2} + \cdots - 64$$
$41$ $$T^{5} - 25 T^{4} + 212 T^{3} + \cdots + 2182$$
$43$ $$T^{5}$$
$47$ $$T^{5} + 11 T^{4} - 110 T^{3} + \cdots + 512$$
$53$ $$T^{5} + 3 T^{4} - 160 T^{3} + \cdots + 20272$$
$59$ $$T^{5} - T^{4} - 142 T^{3} + \cdots - 13568$$
$61$ $$T^{5} + 5 T^{4} - 115 T^{3} + \cdots + 7664$$
$67$ $$T^{5} - 9 T^{4} - 126 T^{3} + \cdots + 8192$$
$71$ $$T^{5} + T^{4} - 214 T^{3} - 407 T^{2} + \cdots + 3968$$
$73$ $$T^{5} - T^{4} - 158 T^{3} + 272 T^{2} + \cdots + 1328$$
$79$ $$T^{5} - 2 T^{4} - 128 T^{3} + \cdots + 1024$$
$83$ $$T^{5} + 45 T^{4} + 784 T^{3} + \cdots + 41216$$
$89$ $$T^{5} - 6 T^{4} - 136 T^{3} + \cdots - 8192$$
$97$ $$T^{5} + 25 T^{4} - 39 T^{3} + \cdots + 49616$$