# Properties

 Label 9196.2.a.n Level $9196$ Weight $2$ Character orbit 9196.a Self dual yes Analytic conductor $73.430$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9196,2,Mod(1,9196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9196, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9196.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9196 = 2^{2} \cdot 11^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9196.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: $$73.4304296988$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.744786576.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 17x^{4} + 13x^{3} + 69x^{2} - 21x - 30$$ x^6 - x^5 - 17*x^4 + 13*x^3 + 69*x^2 - 21*x - 30 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 836) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{5} + 1) q^{5} + (\beta_{2} - \beta_1 + 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 3) q^{9}+O(q^{10})$$ q + b1 * q^3 + (b5 + 1) * q^5 + (b2 - b1 + 1) * q^7 + (b5 - b4 + b3 + 3) * q^9 $$q + \beta_1 q^{3} + (\beta_{5} + 1) q^{5} + (\beta_{2} - \beta_1 + 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 3) q^{9} + (\beta_{5} + \beta_{3} - 1) q^{13} + (\beta_{4} + \beta_{2} + \beta_1 + 2) q^{15} - 2 \beta_{4} q^{17} + q^{19} + (3 \beta_{4} + \beta_{2} - 2) q^{21} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{23} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 3) q^{25} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + 3 \beta_1 + 1) q^{27} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{29} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{31} + (2 \beta_{5} - \beta_{4} + \beta_{3} + 1) q^{35} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{37} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + 3) q^{39} + (\beta_{5} - \beta_{3} - 1) q^{41} + (2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{43} + (2 \beta_{3} + 3 \beta_{2} - \beta_1 + 9) q^{45} + (2 \beta_{5} - 2 \beta_{4}) q^{47} + ( - 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{49} + ( - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{2} + 4 \beta_1 - 4) q^{51} + (2 \beta_{5} - 2 \beta_1 + 4) q^{53} + \beta_1 q^{57} + (\beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 + 1) q^{59} + (2 \beta_{5} + 2 \beta_1 - 4) q^{61} + (4 \beta_{5} - \beta_{4} + \beta_{3} + 4 \beta_{2} - 6 \beta_1 + 7) q^{63} + ( - 4 \beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 6) q^{65} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 + 4) q^{67} + ( - \beta_{5} + 3 \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_1 - 7) q^{69} + (2 \beta_{5} + 2 \beta_{2} - \beta_1) q^{71} + ( - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 4) q^{73} + (3 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 7) q^{75} + ( - 2 \beta_{4} + 2 \beta_1 + 2) q^{79} + (2 \beta_{5} - 3 \beta_{4} + \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 12) q^{81} + (\beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{83} + ( - 2 \beta_{4} - 2 \beta_1 - 2) q^{85} + ( - 4 \beta_{4} - 5 \beta_{2} - \beta_1 + 3) q^{87} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{89} + (3 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{91} + ( - 3 \beta_{5} - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 12) q^{93} + (\beta_{5} + 1) q^{95} + ( - 3 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 5) q^{97}+O(q^{100})$$ q + b1 * q^3 + (b5 + 1) * q^5 + (b2 - b1 + 1) * q^7 + (b5 - b4 + b3 + 3) * q^9 + (b5 + b3 - 1) * q^13 + (b4 + b2 + b1 + 2) * q^15 - 2*b4 * q^17 + q^19 + (3*b4 + b2 - 2) * q^21 + (-b5 - b4 - b3 + b2 - b1 + 1) * q^23 + (-b5 + b4 - b3 + 2*b2 + 3) * q^25 + (-b5 + 2*b4 + b3 + 3*b1 + 1) * q^27 + (-b5 - b4 - b3 - b2 + 2*b1 - 3) * q^29 + (-b4 - b2 - b1) * q^31 + (2*b5 - b4 + b3 + 1) * q^35 + (b5 + b4 - b3 - b2 - b1 + 1) * q^37 + (b4 + b3 + 2*b2 + 3) * q^39 + (b5 - b3 - 1) * q^41 + (2*b5 - b4 - b3 + b2 - b1 - 2) * q^43 + (2*b3 + 3*b2 - b1 + 9) * q^45 + (2*b5 - 2*b4) * q^47 + (-3*b4 + b3 - b2 + b1 + 1) * q^49 + (-2*b5 + 2*b4 - 4*b2 + 4*b1 - 4) * q^51 + (2*b5 - 2*b1 + 4) * q^53 + b1 * q^57 + (b5 - b4 + b3 - 3*b2 + b1 + 1) * q^59 + (2*b5 + 2*b1 - 4) * q^61 + (4*b5 - b4 + b3 + 4*b2 - 6*b1 + 7) * q^63 + (-4*b5 + b4 + 2*b3 + 3*b2 + 6) * q^65 + (-b4 + 2*b3 - b2 - b1 + 4) * q^67 + (-b5 + 3*b4 - b3 - 3*b2 + b1 - 7) * q^69 + (2*b5 + 2*b2 - b1) * q^71 + (-2*b4 + 2*b3 + 2*b1 + 4) * q^73 + (3*b5 + 2*b4 + b3 + 2*b2 - 2*b1 + 7) * q^75 + (-2*b4 + 2*b1 + 2) * q^79 + (2*b5 - 3*b4 + b3 + 4*b2 - 2*b1 + 12) * q^81 + (b4 + b3 - b2 + b1) * q^83 + (-2*b4 - 2*b1 - 2) * q^85 + (-4*b4 - 5*b2 - b1 + 3) * q^87 + (-b5 + 3*b4 + b3 - b2 + b1 + 1) * q^89 + (3*b5 - b4 + b3 - b2 + 2*b1 + 1) * q^91 + (-3*b5 - 2*b3 - 3*b2 + 3*b1 - 12) * q^93 + (b5 + 1) * q^95 + (-3*b5 + b4 - b3 - b2 + b1 + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10})$$ 6 * q + q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9 $$6 q + q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} - 8 q^{13} + 9 q^{15} + 2 q^{17} + 6 q^{19} - 18 q^{21} + 5 q^{23} + 13 q^{25} + 7 q^{27} - 10 q^{29} + 3 q^{31} + 4 q^{35} + 7 q^{37} + 10 q^{39} - 6 q^{41} - 16 q^{43} + 42 q^{45} + 12 q^{49} - 8 q^{51} + 20 q^{53} + q^{57} + 15 q^{59} - 24 q^{61} + 20 q^{63} + 28 q^{65} + 25 q^{67} - 33 q^{69} - 9 q^{71} + 26 q^{73} + 28 q^{75} + 16 q^{79} + 58 q^{81} + 2 q^{83} - 12 q^{85} + 36 q^{87} + 7 q^{89} + 8 q^{91} - 55 q^{93} + 5 q^{95} + 37 q^{97}+O(q^{100})$$ 6 * q + q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9 - 8 * q^13 + 9 * q^15 + 2 * q^17 + 6 * q^19 - 18 * q^21 + 5 * q^23 + 13 * q^25 + 7 * q^27 - 10 * q^29 + 3 * q^31 + 4 * q^35 + 7 * q^37 + 10 * q^39 - 6 * q^41 - 16 * q^43 + 42 * q^45 + 12 * q^49 - 8 * q^51 + 20 * q^53 + q^57 + 15 * q^59 - 24 * q^61 + 20 * q^63 + 28 * q^65 + 25 * q^67 - 33 * q^69 - 9 * q^71 + 26 * q^73 + 28 * q^75 + 16 * q^79 + 58 * q^81 + 2 * q^83 - 12 * q^85 + 36 * q^87 + 7 * q^89 + 8 * q^91 - 55 * q^93 + 5 * q^95 + 37 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 17x^{4} + 13x^{3} + 69x^{2} - 21x - 30$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 24\nu^{4} + \nu^{3} - 253\nu^{2} - 48\nu + 137 ) / 97$$ (v^5 + 24*v^4 + v^3 - 253*v^2 - 48*v + 137) / 97 $$\beta_{3}$$ $$=$$ $$( -4\nu^{5} + \nu^{4} + 93\nu^{3} + 42\nu^{2} - 487\nu - 354 ) / 97$$ (-4*v^5 + v^4 + 93*v^3 + 42*v^2 - 487*v - 354) / 97 $$\beta_{4}$$ $$=$$ $$( 8\nu^{5} - 2\nu^{4} - 89\nu^{3} + 13\nu^{2} + 101\nu + 29 ) / 97$$ (8*v^5 - 2*v^4 - 89*v^3 + 13*v^2 + 101*v + 29) / 97 $$\beta_{5}$$ $$=$$ $$( 12\nu^{5} - 3\nu^{4} - 182\nu^{3} + 68\nu^{2} + 588\nu - 199 ) / 97$$ (12*v^5 - 3*v^4 - 182*v^3 + 68*v^2 + 588*v - 199) / 97
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + \beta_{3} + 6$$ b5 - b4 + b3 + 6 $$\nu^{3}$$ $$=$$ $$-\beta_{5} + 2\beta_{4} + \beta_{3} + 9\beta _1 + 1$$ -b5 + 2*b4 + b3 + 9*b1 + 1 $$\nu^{4}$$ $$=$$ $$11\beta_{5} - 12\beta_{4} + 10\beta_{3} + 4\beta_{2} - 2\beta _1 + 57$$ 11*b5 - 12*b4 + 10*b3 + 4*b2 - 2*b1 + 57 $$\nu^{5}$$ $$=$$ $$-10\beta_{5} + 33\beta_{4} + 12\beta_{3} + \beta_{2} + 87\beta _1 + 12$$ -10*b5 + 33*b4 + 12*b3 + b2 + 87*b1 + 12

## 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
 −3.31717 −2.05861 −0.566270 0.837309 2.82559 3.27915
0 −3.31717 0 1.60797 0 4.11176 0 8.00361 0
1.2 0 −2.05861 0 0.680036 0 −1.59123 0 1.23787 0
1.3 0 −0.566270 0 −3.92909 0 2.44546 0 −2.67934 0
1.4 0 0.837309 0 3.44987 0 −0.535976 0 −2.29891 0
1.5 0 2.82559 0 −0.343563 0 −4.77460 0 4.98394 0
1.6 0 3.27915 0 3.53478 0 2.34459 0 7.75283 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$-1$$
$$19$$ $$-1$$

## 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 9196.2.a.n 6
11.b odd 2 1 836.2.a.e 6
33.d even 2 1 7524.2.a.q 6
44.c even 2 1 3344.2.a.w 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
836.2.a.e 6 11.b odd 2 1
3344.2.a.w 6 44.c even 2 1
7524.2.a.q 6 33.d even 2 1
9196.2.a.n 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9196))$$:

 $$T_{3}^{6} - T_{3}^{5} - 17T_{3}^{4} + 13T_{3}^{3} + 69T_{3}^{2} - 21T_{3} - 30$$ T3^6 - T3^5 - 17*T3^4 + 13*T3^3 + 69*T3^2 - 21*T3 - 30 $$T_{5}^{6} - 5T_{5}^{5} - 9T_{5}^{4} + 77T_{5}^{3} - 99T_{5}^{2} + 9T_{5} + 18$$ T5^6 - 5*T5^5 - 9*T5^4 + 77*T5^3 - 99*T5^2 + 9*T5 + 18 $$T_{7}^{6} - 2T_{7}^{5} - 25T_{7}^{4} + 58T_{7}^{3} + 81T_{7}^{2} - 156T_{7} - 96$$ T7^6 - 2*T7^5 - 25*T7^4 + 58*T7^3 + 81*T7^2 - 156*T7 - 96 $$T_{13}^{6} + 8T_{13}^{5} - 23T_{13}^{4} - 246T_{13}^{3} - 191T_{13}^{2} + 254T_{13} - 40$$ T13^6 + 8*T13^5 - 23*T13^4 - 246*T13^3 - 191*T13^2 + 254*T13 - 40

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - T^{5} - 17 T^{4} + 13 T^{3} + \cdots - 30$$
$5$ $$T^{6} - 5 T^{5} - 9 T^{4} + 77 T^{3} + \cdots + 18$$
$7$ $$T^{6} - 2 T^{5} - 25 T^{4} + 58 T^{3} + \cdots - 96$$
$11$ $$T^{6}$$
$13$ $$T^{6} + 8 T^{5} - 23 T^{4} - 246 T^{3} + \cdots - 40$$
$17$ $$T^{6} - 2 T^{5} - 72 T^{4} + 48 T^{3} + \cdots - 256$$
$19$ $$(T - 1)^{6}$$
$23$ $$T^{6} - 5 T^{5} - 88 T^{4} + \cdots - 7232$$
$29$ $$T^{6} + 10 T^{5} - 91 T^{4} + \cdots - 11424$$
$31$ $$T^{6} - 3 T^{5} - 79 T^{4} + 387 T^{3} + \cdots - 142$$
$37$ $$T^{6} - 7 T^{5} - 110 T^{4} + \cdots - 5856$$
$41$ $$T^{6} + 6 T^{5} - 43 T^{4} - 464 T^{3} + \cdots - 896$$
$43$ $$T^{6} + 16 T^{5} - 49 T^{4} + \cdots + 39880$$
$47$ $$T^{6} - 124 T^{4} + 176 T^{3} + \cdots + 10752$$
$53$ $$T^{6} - 20 T^{5} + 52 T^{4} + \cdots + 32960$$
$59$ $$T^{6} - 15 T^{5} - 106 T^{4} + \cdots - 28064$$
$61$ $$T^{6} + 24 T^{5} + 60 T^{4} + \cdots + 25152$$
$67$ $$T^{6} - 25 T^{5} + 77 T^{4} + \cdots - 56022$$
$71$ $$T^{6} + 9 T^{5} - 157 T^{4} + \cdots - 82154$$
$73$ $$T^{6} - 26 T^{5} + 20 T^{4} + \cdots + 60672$$
$79$ $$T^{6} - 16 T^{5} - 20 T^{4} + \cdots - 4096$$
$83$ $$T^{6} - 2 T^{5} - 77 T^{4} + \cdots - 7968$$
$89$ $$T^{6} - 7 T^{5} - 226 T^{4} + \cdots + 1152$$
$97$ $$T^{6} - 37 T^{5} + 334 T^{4} + \cdots + 405664$$