# Properties

 Label 3332.2.b.b Level $3332$ Weight $2$ Character orbit 3332.b Analytic conductor $26.606$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(2549,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("3332.2549");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3332.b (of order $$2$$, degree $$1$$, 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: $$26.6061539535$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.980441344.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 8x^{6} + 18x^{4} + 9x^{2} + 1$$ x^8 + 8*x^6 + 18*x^4 + 9*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 476) 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{7} + \beta_{4}) q^{3} + \beta_{3} q^{5} + ( - \beta_{5} - \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10})$$ q + (b7 + b4) * q^3 + b3 * q^5 + (-b5 - b2 - b1 - 1) * q^9 $$q + (\beta_{7} + \beta_{4}) q^{3} + \beta_{3} q^{5} + ( - \beta_{5} - \beta_{2} - \beta_1 - 1) q^{9} + ( - \beta_{7} - \beta_{6} + 2 \beta_{4}) q^{11} + ( - \beta_{5} - \beta_1 - 2) q^{13} + ( - 2 \beta_{5} - 2 \beta_{2} + 1) q^{15} + (\beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{17} + (\beta_{5} - \beta_{2} - 2) q^{19} + (\beta_{7} - 2 \beta_{6} + \beta_{3}) q^{23} + (\beta_{5} - \beta_{2} - \beta_1 - 2) q^{25} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4} - 3 \beta_{3}) q^{27} + (\beta_{7} - \beta_{6} + 2 \beta_{4}) q^{29} + (2 \beta_{4} - \beta_{3}) q^{31} + (\beta_{5} - 3 \beta_{2}) q^{33} + ( - \beta_{7} - 2 \beta_{4} - \beta_{3}) q^{37} + ( - 5 \beta_{7} + \beta_{6} - 2 \beta_{4} - 2 \beta_{3}) q^{39} + (\beta_{7} - 2 \beta_{6} - \beta_{4}) q^{41} + ( - \beta_{5} + 3 \beta_{2} + \beta_1 - 1) q^{43} + ( - 3 \beta_{7} + 2 \beta_{6} - 3 \beta_{4} - 3 \beta_{3}) q^{45} + (2 \beta_{5} + 3 \beta_{2} + \beta_1 - 4) q^{47} + ( - \beta_{7} - 3 \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{51} + (\beta_{5} + 3 \beta_{2} - \beta_1 + 2) q^{53} + ( - 3 \beta_{5} + 2 \beta_{2} - \beta_1 + 2) q^{55} + (\beta_{6} - 6 \beta_{4} + \beta_{3}) q^{57} + ( - 2 \beta_{5} + 4 \beta_{2} + 4) q^{59} + (\beta_{7} - 2 \beta_{6} + \beta_{4} + 2 \beta_{3}) q^{61} + ( - \beta_{7} + 2 \beta_{6} - 4 \beta_{4} - 3 \beta_{3}) q^{65} + ( - \beta_{5} + \beta_{2} - 3 \beta_1) q^{67} + ( - 3 \beta_{5} - 4 \beta_{2} - 3 \beta_1 - 4) q^{69} + (3 \beta_{7} - 4 \beta_{4} - \beta_{3}) q^{71} + ( - 5 \beta_{7} - 2 \beta_{6} + 3 \beta_{4} - 2 \beta_{3}) q^{73} + ( - \beta_{7} + 2 \beta_{6} - 7 \beta_{4} + \beta_{3}) q^{75} + ( - \beta_{7} + 3 \beta_{6} + 4 \beta_{4} - 4 \beta_{3}) q^{79} + (4 \beta_{5} + 6 \beta_{2}) q^{81} + ( - \beta_{5} + 2 \beta_{2} - 3 \beta_1) q^{83} + (2 \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 - 6) q^{85} + ( - \beta_{5} - 3 \beta_{2} - 2 \beta_1 - 6) q^{87} + ( - 4 \beta_{5} - \beta_{2} + \beta_1 + 2) q^{89} + (2 \beta_{5} - 3) q^{93} + ( - \beta_{7} - \beta_{6} + 8 \beta_{4} - 4 \beta_{3}) q^{95} + ( - 2 \beta_{6} - 4 \beta_{4} - \beta_{3}) q^{97} + ( - \beta_{7} - 4 \beta_{4} - \beta_{3}) q^{99}+O(q^{100})$$ q + (b7 + b4) * q^3 + b3 * q^5 + (-b5 - b2 - b1 - 1) * q^9 + (-b7 - b6 + 2*b4) * q^11 + (-b5 - b1 - 2) * q^13 + (-2*b5 - 2*b2 + 1) * q^15 + (b7 - b4 + b3 + b2 - b1) * q^17 + (b5 - b2 - 2) * q^19 + (b7 - 2*b6 + b3) * q^23 + (b5 - b2 - b1 - 2) * q^25 + (-b7 + 2*b6 - b4 - 3*b3) * q^27 + (b7 - b6 + 2*b4) * q^29 + (2*b4 - b3) * q^31 + (b5 - 3*b2) * q^33 + (-b7 - 2*b4 - b3) * q^37 + (-5*b7 + b6 - 2*b4 - 2*b3) * q^39 + (b7 - 2*b6 - b4) * q^41 + (-b5 + 3*b2 + b1 - 1) * q^43 + (-3*b7 + 2*b6 - 3*b4 - 3*b3) * q^45 + (2*b5 + 3*b2 + b1 - 4) * q^47 + (-b7 - 3*b5 + 2*b4 + b3 - b2 - b1 - 1) * q^51 + (b5 + 3*b2 - b1 + 2) * q^53 + (-3*b5 + 2*b2 - b1 + 2) * q^55 + (b6 - 6*b4 + b3) * q^57 + (-2*b5 + 4*b2 + 4) * q^59 + (b7 - 2*b6 + b4 + 2*b3) * q^61 + (-b7 + 2*b6 - 4*b4 - 3*b3) * q^65 + (-b5 + b2 - 3*b1) * q^67 + (-3*b5 - 4*b2 - 3*b1 - 4) * q^69 + (3*b7 - 4*b4 - b3) * q^71 + (-5*b7 - 2*b6 + 3*b4 - 2*b3) * q^73 + (-b7 + 2*b6 - 7*b4 + b3) * q^75 + (-b7 + 3*b6 + 4*b4 - 4*b3) * q^79 + (4*b5 + 6*b2) * q^81 + (-b5 + 2*b2 - 3*b1) * q^83 + (2*b7 + b6 + b5 + 2*b4 - b3 - 3*b2 - b1 - 6) * q^85 + (-b5 - 3*b2 - 2*b1 - 6) * q^87 + (-4*b5 - b2 + b1 + 2) * q^89 + (2*b5 - 3) * q^93 + (-b7 - b6 + 8*b4 - 4*b3) * q^95 + (-2*b6 - 4*b4 - b3) * q^97 + (-b7 - 4*b4 - b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 12 q^{9}+O(q^{10})$$ 8 * q - 12 * q^9 $$8 q - 12 q^{9} - 20 q^{13} - 12 q^{19} - 12 q^{25} + 4 q^{33} - 12 q^{43} - 24 q^{47} - 20 q^{51} + 20 q^{53} + 4 q^{55} + 24 q^{59} - 4 q^{67} - 44 q^{69} + 16 q^{81} - 4 q^{83} - 44 q^{85} - 52 q^{87} - 16 q^{93}+O(q^{100})$$ 8 * q - 12 * q^9 - 20 * q^13 - 12 * q^19 - 12 * q^25 + 4 * q^33 - 12 * q^43 - 24 * q^47 - 20 * q^51 + 20 * q^53 + 4 * q^55 + 24 * q^59 - 4 * q^67 - 44 * q^69 + 16 * q^81 - 4 * q^83 - 44 * q^85 - 52 * q^87 - 16 * q^93

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 8x^{6} + 18x^{4} + 9x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$-\nu^{4} - 3\nu^{2} + 1$$ -v^4 - 3*v^2 + 1 $$\beta_{2}$$ $$=$$ $$\nu^{4} + 5\nu^{2} + 3$$ v^4 + 5*v^2 + 3 $$\beta_{3}$$ $$=$$ $$\nu^{5} + 6\nu^{3} + 8\nu$$ v^5 + 6*v^3 + 8*v $$\beta_{4}$$ $$=$$ $$\nu^{7} + 8\nu^{5} + 17\nu^{3} + 5\nu$$ v^7 + 8*v^5 + 17*v^3 + 5*v $$\beta_{5}$$ $$=$$ $$2\nu^{6} + 15\nu^{4} + 29\nu^{2} + 7$$ 2*v^6 + 15*v^4 + 29*v^2 + 7 $$\beta_{6}$$ $$=$$ $$-2\nu^{7} - 15\nu^{5} - 30\nu^{3} - 8\nu$$ -2*v^7 - 15*v^5 - 30*v^3 - 8*v $$\beta_{7}$$ $$=$$ $$2\nu^{7} + 15\nu^{5} + 30\nu^{3} + 10\nu$$ 2*v^7 + 15*v^5 + 30*v^3 + 10*v
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{6} ) / 2$$ (b7 + b6) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 - 4 ) / 2$$ (b2 + b1 - 4) / 2 $$\nu^{3}$$ $$=$$ $$( -3\beta_{7} - 4\beta_{6} - 2\beta_{4} + \beta_{3} ) / 2$$ (-3*b7 - 4*b6 - 2*b4 + b3) / 2 $$\nu^{4}$$ $$=$$ $$( -3\beta_{2} - 5\beta _1 + 14 ) / 2$$ (-3*b2 - 5*b1 + 14) / 2 $$\nu^{5}$$ $$=$$ $$5\beta_{7} + 8\beta_{6} + 6\beta_{4} - 2\beta_{3}$$ 5*b7 + 8*b6 + 6*b4 - 2*b3 $$\nu^{6}$$ $$=$$ $$( \beta_{5} + 8\beta_{2} + 23\beta _1 - 54 ) / 2$$ (b5 + 8*b2 + 23*b1 - 54) / 2 $$\nu^{7}$$ $$=$$ $$( -34\beta_{7} - 65\beta_{6} - 60\beta_{4} + 15\beta_{3} ) / 2$$ (-34*b7 - 65*b6 - 60*b4 + 15*b3) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2549.1
 − 0.396339i − 1.76401i 0.693822i − 2.06150i 2.06150i − 0.693822i 1.76401i 0.396339i
0 3.23925i 0 2.80694i 0 0 0 −7.49277 0
2549.2 0 1.87576i 0 1.74199i 0 0 0 −0.518489 0
2549.3 0 1.82479i 0 3.70737i 0 0 0 −0.329851 0
2549.4 0 0.811721i 0 1.15845i 0 0 0 2.34111 0
2549.5 0 0.811721i 0 1.15845i 0 0 0 2.34111 0
2549.6 0 1.82479i 0 3.70737i 0 0 0 −0.329851 0
2549.7 0 1.87576i 0 1.74199i 0 0 0 −0.518489 0
2549.8 0 3.23925i 0 2.80694i 0 0 0 −7.49277 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2549.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.2.b.b 8
7.b odd 2 1 476.2.b.a 8
17.b even 2 1 inner 3332.2.b.b 8
21.c even 2 1 4284.2.d.e 8
28.d even 2 1 1904.2.c.f 8
119.d odd 2 1 476.2.b.a 8
119.f odd 4 1 8092.2.a.q 4
119.f odd 4 1 8092.2.a.r 4
357.c even 2 1 4284.2.d.e 8
476.e even 2 1 1904.2.c.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.2.b.a 8 7.b odd 2 1
476.2.b.a 8 119.d odd 2 1
1904.2.c.f 8 28.d even 2 1
1904.2.c.f 8 476.e even 2 1
3332.2.b.b 8 1.a even 1 1 trivial
3332.2.b.b 8 17.b even 2 1 inner
4284.2.d.e 8 21.c even 2 1
4284.2.d.e 8 357.c even 2 1
8092.2.a.q 4 119.f odd 4 1
8092.2.a.r 4 119.f odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3332, [\chi])$$:

 $$T_{3}^{8} + 18T_{3}^{6} + 95T_{3}^{4} + 178T_{3}^{2} + 81$$ T3^8 + 18*T3^6 + 95*T3^4 + 178*T3^2 + 81 $$T_{13}^{4} + 10T_{13}^{3} + 20T_{13}^{2} - 24T_{13} - 16$$ T13^4 + 10*T13^3 + 20*T13^2 - 24*T13 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + 18 T^{6} + 95 T^{4} + 178 T^{2} + \cdots + 81$$
$5$ $$T^{8} + 26 T^{6} + 207 T^{4} + \cdots + 441$$
$7$ $$T^{8}$$
$11$ $$T^{8} + 48 T^{6} + 608 T^{4} + \cdots + 2304$$
$13$ $$(T^{4} + 10 T^{3} + 20 T^{2} - 24 T - 16)^{2}$$
$17$ $$T^{8} + 4 T^{6} - 64 T^{5} + \cdots + 83521$$
$19$ $$(T^{4} + 6 T^{3} - 8 T^{2} - 32 T - 16)^{2}$$
$23$ $$T^{8} + 124 T^{6} + 3984 T^{4} + \cdots + 256$$
$29$ $$T^{8} + 64 T^{6} + 1184 T^{4} + \cdots + 256$$
$31$ $$T^{8} + 34 T^{6} + 47 T^{4} + 18 T^{2} + \cdots + 1$$
$37$ $$T^{8} + 60 T^{6} + 528 T^{4} + \cdots + 256$$
$41$ $$T^{8} + 138 T^{6} + 4559 T^{4} + \cdots + 196249$$
$43$ $$(T^{4} + 6 T^{3} - 101 T^{2} - 314 T + 2429)^{2}$$
$47$ $$(T^{4} + 12 T^{3} - 56 T^{2} - 1160 T - 3664)^{2}$$
$53$ $$(T^{4} - 10 T^{3} - 41 T^{2} + 350 T + 557)^{2}$$
$59$ $$(T^{4} - 12 T^{3} - 124 T^{2} + 1312 T - 1104)^{2}$$
$61$ $$T^{8} + 178 T^{6} + 5831 T^{4} + \cdots + 47961$$
$67$ $$(T^{4} + 2 T^{3} - 101 T^{2} - 586 T - 787)^{2}$$
$71$ $$T^{8} + 244 T^{6} + 7904 T^{4} + \cdots + 256$$
$73$ $$T^{8} + 514 T^{6} + \cdots + 154828249$$
$79$ $$T^{8} + 480 T^{6} + \cdots + 16128256$$
$83$ $$(T^{4} + 2 T^{3} - 112 T^{2} - 432 T - 368)^{2}$$
$89$ $$(T^{4} - 240 T^{2} - 1576 T - 1776)^{2}$$
$97$ $$T^{8} + 242 T^{6} + 10791 T^{4} + \cdots + 978121$$