Properties

 Label 3332.2.b.d Level $3332$ Weight $2$ Character orbit 3332.b Analytic conductor $26.606$ Analytic rank $0$ Dimension $12$ 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: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 22x^{10} + 174x^{8} + 585x^{6} + 756x^{4} + 325x^{2} + 9$$ x^12 + 22*x^10 + 174*x^8 + 585*x^6 + 756*x^4 + 325*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + \beta_{8} q^{5} + (\beta_{2} - 1) q^{9}+O(q^{10})$$ q + b1 * q^3 + b8 * q^5 + (b2 - 1) * q^9 $$q + \beta_1 q^{3} + \beta_{8} q^{5} + (\beta_{2} - 1) q^{9} + \beta_{3} q^{11} + \beta_{9} q^{13} + ( - \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2}) q^{15} + \beta_{4} q^{17} - \beta_{7} q^{19} + (\beta_{8} - \beta_{6} - \beta_{3}) q^{23} + (\beta_{11} + \beta_{9} - \beta_{7} - 2) q^{25} + \beta_{3} q^{27} + (\beta_{10} - \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{29} + ( - \beta_{10} - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_1) q^{31} + (\beta_{9} - \beta_{7} - \beta_{2} + 1) q^{33} + ( - \beta_{8} - \beta_{6} - \beta_1) q^{37} + (\beta_{8} - \beta_{3}) q^{39} + ( - \beta_{10} - \beta_{3}) q^{41} + (\beta_{7} + \beta_{5} + \beta_{4}) q^{43} + (\beta_{10} + \beta_{6} + \beta_{3} + \beta_1) q^{45} + ( - \beta_{7} + 2 \beta_{2} - 2) q^{47} + (\beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} - 1) q^{51} + ( - \beta_{11} + \beta_{9} + 2) q^{53} + (\beta_{11} - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} - 3) q^{55} + ( - \beta_{10} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{57} + ( - \beta_{11} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{59} + ( - \beta_{8} + 2 \beta_{3}) q^{61} + (\beta_{10} - \beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1) q^{65} + ( - \beta_{11} - 2 \beta_{9} - \beta_{2} + 3) q^{67} + ( - \beta_{11} - 3 \beta_{9} + 3 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{69} + ( - 2 \beta_{10} - \beta_{8} - 3 \beta_1) q^{71} + ( - \beta_{10} + 2 \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{73} + (\beta_{8} - 3 \beta_{6} - 2 \beta_{3}) q^{75} + ( - \beta_{10} + 2 \beta_{8} + \beta_{6} - \beta_1) q^{79} + (\beta_{9} - \beta_{7} + 2 \beta_{2} - 2) q^{81} + (\beta_{9} - 3 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{2} + 2) q^{83} + (\beta_{10} + \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{85} + (\beta_{11} + 2 \beta_{9} - \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + \beta_{2} - 1) q^{87} + (\beta_{11} + \beta_{9} + \beta_{2} - 3) q^{89} + ( - \beta_{11} - \beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{2} + 6) q^{93} + ( - \beta_{8} - 2 \beta_{3}) q^{95} + (\beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1) q^{97} + ( - \beta_{10} + \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} + 4 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^3 + b8 * q^5 + (b2 - 1) * q^9 + b3 * q^11 + b9 * q^13 + (-b9 + b7 + b5 + b4 - b2) * q^15 + b4 * q^17 - b7 * q^19 + (b8 - b6 - b3) * q^23 + (b11 + b9 - b7 - 2) * q^25 + b3 * q^27 + (b10 - b8 + b6 + b5 - b4 + b3 + b1) * q^29 + (-b10 - b6 - b5 + b4 - 2*b1) * q^31 + (b9 - b7 - b2 + 1) * q^33 + (-b8 - b6 - b1) * q^37 + (b8 - b3) * q^39 + (-b10 - b3) * q^41 + (b7 + b5 + b4) * q^43 + (b10 + b6 + b3 + b1) * q^45 + (-b7 + 2*b2 - 2) * q^47 + (b9 - b8 - 2*b7 + b6 - b4 + b2 - 1) * q^51 + (-b11 + b9 + 2) * q^53 + (b11 - b7 + b5 + b4 - b2 - 3) * q^55 + (-b10 + b6 + b5 - b4 - b3 + b1) * q^57 + (-b11 - b7 - b5 - b4 - b2 + 1) * q^59 + (-b8 + 2*b3) * q^61 + (b10 - b8 + b5 - b4 + b3 + 2*b1) * q^65 + (-b11 - 2*b9 - b2 + 3) * q^67 + (-b11 - 3*b9 + 3*b7 + 2*b5 + 2*b4 - b2) * q^69 + (-2*b10 - b8 - 3*b1) * q^71 + (-b10 + 2*b8 - b6 - b5 + b4 - b3 + b1) * q^73 + (b8 - 3*b6 - 2*b3) * q^75 + (-b10 + 2*b8 + b6 - b1) * q^79 + (b9 - b7 + 2*b2 - 2) * q^81 + (b9 - 3*b7 - 2*b5 - 2*b4 + 3*b2 + 2) * q^83 + (b10 + b7 - b6 + 2*b5 + b4 - b3 - 2*b2 - b1 + 1) * q^85 + (b11 + 2*b9 - b7 - 2*b5 - 2*b4 + b2 - 1) * q^87 + (b11 + b9 + b2 - 3) * q^89 + (-b11 - b7 + b5 + b4 - 2*b2 + 6) * q^93 + (-b8 - 2*b3) * q^95 + (b8 + b6 + b5 - b4 + 2*b3 + b1) * q^97 + (-b10 + b8 + b6 + b5 - b4 + 4*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 8 q^{9}+O(q^{10})$$ 12 * q - 8 * q^9 $$12 q - 8 q^{9} - 4 q^{13} + 2 q^{15} + 3 q^{17} + 4 q^{19} - 18 q^{25} + 8 q^{33} + 2 q^{43} - 12 q^{47} - 7 q^{51} + 14 q^{53} - 24 q^{55} + 34 q^{67} + 2 q^{69} - 16 q^{81} + 32 q^{83} + 9 q^{85} - 18 q^{87} - 30 q^{89} + 68 q^{93}+O(q^{100})$$ 12 * q - 8 * q^9 - 4 * q^13 + 2 * q^15 + 3 * q^17 + 4 * q^19 - 18 * q^25 + 8 * q^33 + 2 * q^43 - 12 * q^47 - 7 * q^51 + 14 * q^53 - 24 * q^55 + 34 * q^67 + 2 * q^69 - 16 * q^81 + 32 * q^83 + 9 * q^85 - 18 * q^87 - 30 * q^89 + 68 * q^93

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 22x^{10} + 174x^{8} + 585x^{6} + 756x^{4} + 325x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 6\nu$$ v^3 + 6*v $$\beta_{4}$$ $$=$$ $$( \nu^{11} + \nu^{10} + 19 \nu^{9} + 11 \nu^{8} + 117 \nu^{7} + 5 \nu^{6} + 210 \nu^{5} - 166 \nu^{4} - 210 \nu^{3} - 82 \nu^{2} - 365 \nu + 75 ) / 24$$ (v^11 + v^10 + 19*v^9 + 11*v^8 + 117*v^7 + 5*v^6 + 210*v^5 - 166*v^4 - 210*v^3 - 82*v^2 - 365*v + 75) / 24 $$\beta_{5}$$ $$=$$ $$( - \nu^{11} + \nu^{10} - 19 \nu^{9} + 11 \nu^{8} - 117 \nu^{7} + 5 \nu^{6} - 210 \nu^{5} - 166 \nu^{4} + 210 \nu^{3} - 82 \nu^{2} + 365 \nu + 75 ) / 24$$ (-v^11 + v^10 - 19*v^9 + 11*v^8 - 117*v^7 + 5*v^6 - 210*v^5 - 166*v^4 + 210*v^3 - 82*v^2 + 365*v + 75) / 24 $$\beta_{6}$$ $$=$$ $$( \nu^{11} + 19\nu^{9} + 123\nu^{7} + 306\nu^{5} + 216\nu^{3} + \nu ) / 6$$ (v^11 + 19*v^9 + 123*v^7 + 306*v^5 + 216*v^3 + v) / 6 $$\beta_{7}$$ $$=$$ $$( \nu^{10} + 23\nu^{8} + 185\nu^{6} + 578\nu^{4} + 482\nu^{2} + 39 ) / 12$$ (v^10 + 23*v^8 + 185*v^6 + 578*v^4 + 482*v^2 + 39) / 12 $$\beta_{8}$$ $$=$$ $$( \nu^{11} + 23\nu^{9} + 185\nu^{7} + 590\nu^{5} + 578\nu^{3} + 147\nu ) / 12$$ (v^11 + 23*v^9 + 185*v^7 + 590*v^5 + 578*v^3 + 147*v) / 12 $$\beta_{9}$$ $$=$$ $$( \nu^{10} + 23\nu^{8} + 185\nu^{6} + 590\nu^{4} + 566\nu^{2} + 75 ) / 12$$ (v^10 + 23*v^8 + 185*v^6 + 590*v^4 + 566*v^2 + 75) / 12 $$\beta_{10}$$ $$=$$ $$( \nu^{11} + 21\nu^{9} + 157\nu^{7} + 490\nu^{5} + 556\nu^{3} + 173\nu ) / 6$$ (v^11 + 21*v^9 + 157*v^7 + 490*v^5 + 556*v^3 + 173*v) / 6 $$\beta_{11}$$ $$=$$ $$( -5\nu^{10} - 91\nu^{8} - 553\nu^{6} - 1258\nu^{4} - 826\nu^{2} - 15 ) / 12$$ (-5*v^10 - 91*v^8 - 553*v^6 - 1258*v^4 - 826*v^2 - 15) / 12
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ b2 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 6\beta_1$$ b3 - 6*b1 $$\nu^{4}$$ $$=$$ $$\beta_{9} - \beta_{7} - 7\beta_{2} + 25$$ b9 - b7 - 7*b2 + 25 $$\nu^{5}$$ $$=$$ $$-\beta_{10} + \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} - 9\beta_{3} + 40\beta_1$$ -b10 + b8 + b6 + b5 - b4 - 9*b3 + 40*b1 $$\nu^{6}$$ $$=$$ $$\beta_{11} - 12\beta_{9} + 15\beta_{7} + 2\beta_{5} + 2\beta_{4} + 46\beta_{2} - 169$$ b11 - 12*b9 + 15*b7 + 2*b5 + 2*b4 + 46*b2 - 169 $$\nu^{7}$$ $$=$$ $$16\beta_{10} - 16\beta_{8} - 15\beta_{6} - 14\beta_{5} + 14\beta_{4} + 73\beta_{3} - 275\beta_1$$ 16*b10 - 16*b8 - 15*b6 - 14*b5 + 14*b4 + 73*b3 - 275*b1 $$\nu^{8}$$ $$=$$ $$-15\beta_{11} + 118\beta_{9} - 162\beta_{7} - 31\beta_{5} - 31\beta_{4} - 303\beta_{2} + 1176$$ -15*b11 + 118*b9 - 162*b7 - 31*b5 - 31*b4 - 303*b2 + 1176 $$\nu^{9}$$ $$=$$ $$-177\beta_{10} + 180\beta_{8} + 160\beta_{6} + 146\beta_{5} - 146\beta_{4} - 583\beta_{3} + 1929\beta_1$$ -177*b10 + 180*b8 + 160*b6 + 146*b5 - 146*b4 - 583*b3 + 1929*b1 $$\nu^{10}$$ $$=$$ $$160\beta_{11} - 1072\beta_{9} + 1541\beta_{7} + 343\beta_{5} + 343\beta_{4} + 2023\beta_{2} - 8344$$ 160*b11 - 1072*b9 + 1541*b7 + 343*b5 + 343*b4 + 2023*b2 - 8344 $$\nu^{11}$$ $$=$$ $$1701\beta_{10} - 1758\beta_{8} - 1495\beta_{6} - 1358\beta_{5} + 1358\beta_{4} + 4636\beta_{3} - 13771\beta_1$$ 1701*b10 - 1758*b8 - 1495*b6 - 1358*b5 + 1358*b4 + 4636*b3 - 13771*b1

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
 − 2.80235i − 2.47053i − 2.42743i − 1.13505i − 0.912626i − 0.172328i 0.172328i 0.912626i 1.13505i 2.42743i 2.47053i 2.80235i
0 2.80235i 0 1.23560i 0 0 0 −4.85318 0
2549.2 0 2.47053i 0 3.53416i 0 0 0 −3.10351 0
2549.3 0 2.42743i 0 0.208554i 0 0 0 −2.89240 0
2549.4 0 1.13505i 0 4.33588i 0 0 0 1.71167 0
2549.5 0 0.912626i 0 1.62343i 0 0 0 2.16711 0
2549.6 0 0.172328i 0 1.87193i 0 0 0 2.97030 0
2549.7 0 0.172328i 0 1.87193i 0 0 0 2.97030 0
2549.8 0 0.912626i 0 1.62343i 0 0 0 2.16711 0
2549.9 0 1.13505i 0 4.33588i 0 0 0 1.71167 0
2549.10 0 2.42743i 0 0.208554i 0 0 0 −2.89240 0
2549.11 0 2.47053i 0 3.53416i 0 0 0 −3.10351 0
2549.12 0 2.80235i 0 1.23560i 0 0 0 −4.85318 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2549.12 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.d 12
7.b odd 2 1 3332.2.b.e 12
7.d odd 6 2 476.2.t.a 24
17.b even 2 1 inner 3332.2.b.d 12
119.d odd 2 1 3332.2.b.e 12
119.h odd 6 2 476.2.t.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.2.t.a 24 7.d odd 6 2
476.2.t.a 24 119.h odd 6 2
3332.2.b.d 12 1.a even 1 1 trivial
3332.2.b.d 12 17.b even 2 1 inner
3332.2.b.e 12 7.b odd 2 1
3332.2.b.e 12 119.d odd 2 1

Hecke kernels

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

 $$T_{3}^{12} + 22T_{3}^{10} + 174T_{3}^{8} + 585T_{3}^{6} + 756T_{3}^{4} + 325T_{3}^{2} + 9$$ T3^12 + 22*T3^10 + 174*T3^8 + 585*T3^6 + 756*T3^4 + 325*T3^2 + 9 $$T_{13}^{6} + 2T_{13}^{5} - 38T_{13}^{4} - 9T_{13}^{3} + 127T_{13}^{2} + 95T_{13} + 2$$ T13^6 + 2*T13^5 - 38*T13^4 - 9*T13^3 + 127*T13^2 + 95*T13 + 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 22 T^{10} + 174 T^{8} + 585 T^{6} + \cdots + 9$$
$5$ $$T^{12} + 39 T^{10} + 495 T^{8} + \cdots + 144$$
$7$ $$T^{12}$$
$11$ $$T^{12} + 79 T^{10} + 2100 T^{8} + \cdots + 81$$
$13$ $$(T^{6} + 2 T^{5} - 38 T^{4} - 9 T^{3} + 127 T^{2} + \cdots + 2)^{2}$$
$17$ $$T^{12} - 3 T^{11} + 32 T^{10} + \cdots + 24137569$$
$19$ $$(T^{6} - 2 T^{5} - 52 T^{4} - 34 T^{3} + \cdots + 1040)^{2}$$
$23$ $$T^{12} + 143 T^{10} + 6836 T^{8} + \cdots + 2458624$$
$29$ $$T^{12} + 215 T^{10} + \cdots + 10240000$$
$31$ $$T^{12} + 259 T^{10} + 22949 T^{8} + \cdots + 6411024$$
$37$ $$T^{12} + 144 T^{10} + 6939 T^{8} + \cdots + 5531904$$
$41$ $$T^{12} + 142 T^{10} + 5547 T^{8} + \cdots + 256$$
$43$ $$(T^{6} - T^{5} - 73 T^{4} - 123 T^{3} + \cdots + 3920)^{2}$$
$47$ $$(T^{6} + 6 T^{5} - 104 T^{4} - 566 T^{3} + \cdots + 240)^{2}$$
$53$ $$(T^{6} - 7 T^{5} - 176 T^{4} + \cdots + 162045)^{2}$$
$59$ $$(T^{6} - 247 T^{4} - 425 T^{3} + \cdots - 41148)^{2}$$
$61$ $$T^{12} + 307 T^{10} + 29675 T^{8} + \cdots + 20736$$
$67$ $$(T^{6} - 17 T^{5} - 87 T^{4} + \cdots + 305552)^{2}$$
$71$ $$T^{12} + 419 T^{10} + \cdots + 1492276900$$
$73$ $$T^{12} + 358 T^{10} + \cdots + 19306546704$$
$79$ $$T^{12} + 312 T^{10} + \cdots + 177023025$$
$83$ $$(T^{6} - 16 T^{5} - 247 T^{4} + \cdots - 26304)^{2}$$
$89$ $$(T^{6} + 15 T^{5} - 45 T^{4} - 1231 T^{3} + \cdots + 7272)^{2}$$
$97$ $$T^{12} + 497 T^{10} + \cdots + 6635079936$$