Properties

 Label 832.2.w.j Level $832$ Weight $2$ Character orbit 832.w Analytic conductor $6.644$ Analytic rank $0$ Dimension $12$ Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [832,2,Mod(257,832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(832, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("832.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$832 = 2^{6} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 832.w (of order $$6$$, degree $$2$$, 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: $$6.64355344817$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.18092737797525504.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 12x^{10} + 108x^{8} - 430x^{6} + 1284x^{4} - 36x^{2} + 1$$ x^12 - 12*x^10 + 108*x^8 - 430*x^6 + 1284*x^4 - 36*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 416) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{9} q^{3} + ( - \beta_{6} + \beta_{5} + 2 \beta_{4}) q^{5} + \beta_{3} q^{7} + ( - \beta_{7} + 2 \beta_{4}) q^{9}+O(q^{10})$$ q - b9 * q^3 + (-b6 + b5 + 2*b4) * q^5 + b3 * q^7 + (-b7 + 2*b4) * q^9 $$q - \beta_{9} q^{3} + ( - \beta_{6} + \beta_{5} + 2 \beta_{4}) q^{5} + \beta_{3} q^{7} + ( - \beta_{7} + 2 \beta_{4}) q^{9} + ( - \beta_{10} + \beta_{8}) q^{11} + ( - \beta_{6} - \beta_{4} - \beta_{2} - 2) q^{13} + (\beta_{11} + \beta_{10} + \cdots + \beta_{8}) q^{15}+ \cdots + (2 \beta_{11} + 2 \beta_{10} + \cdots - 2 \beta_1) q^{99}+O(q^{100})$$ q - b9 * q^3 + (-b6 + b5 + 2*b4) * q^5 + b3 * q^7 + (-b7 + 2*b4) * q^9 + (-b10 + b8) * q^11 + (-b6 - b4 - b2 - 2) * q^13 + (b11 + b10 - b9 + b8) * q^15 + b4 * q^17 + (-b9 - b8) * q^19 + (-2*b7 + 2*b6 - 2*b5 + 2*b4 + b2 + 3) * q^21 + (b11 - b9 - 2*b1) * q^23 + (-2*b6 - 2*b5 - b2 - 1) * q^25 + (-b10 + b8 - b3) * q^27 + (b7 + b6 - 2*b5 - b2 + 2) * q^29 + (-b11 - b10 + 2*b9 + b3 + b1) * q^31 + (b7 + 2*b5 - b4 - 2*b2 - 4) * q^33 + (2*b9 - 2*b8) * q^35 + (-b7 - b6 - b2) * q^37 + (b10 + 3*b9 - 2*b8 - b3 + b1) * q^39 + (-2*b6 - b4 + 1) * q^41 + (2*b10 - 2*b9 - b3) * q^43 + (-3*b5 - b4 + 1) * q^45 + (-b11 + b10 + 2*b9 - 2*b8 - b3 + b1) * q^47 + (-b7 + 2*b6 - 4*b5 + 2*b4 + b2 + 6) * q^49 + b8 * q^51 + (b6 + b5 + 4) * q^53 - 2*b9 * q^55 + (-2*b7 + 10*b4 + b2 + 5) * q^57 + (2*b9 + 2*b8 + b3) * q^59 + (b7 - 2*b6 + b5 + 4*b4 - 1) * q^61 + (-2*b11 - 2*b10 - 2*b9 + 6*b8) * q^63 + (b7 + 4*b6 - 4*b5 - 6*b4 - b2) * q^65 + (-2*b11 + 2*b10 + b9 - 4*b8) * q^67 + (-b7 - 8*b6 + 4*b5 + 5*b4 - 4) * q^69 + (-b9 - b8 - b1) * q^71 + (-2*b7 - 4*b6 + 4*b5 + b2 - 4) * q^73 + (-2*b11 - b10 + 4*b9 + b8 + 2*b3 + 4*b1) * q^75 + (2*b6 + 2*b5 - b2 - 11) * q^77 + (-2*b10 - 2*b8 - 2*b3) * q^79 + (-2*b6 + 4*b5 + 3*b4 - 1) * q^81 + (2*b11 - 2*b10 + 2*b8 + 2*b3 - 2*b1) * q^83 + (-b5 - b4 - 1) * q^85 + (-2*b11 + b9 - b8 + b1) * q^87 + (-b7 + 4*b6 - 3*b4 - b2 + 3) * q^89 + (2*b10 + 2*b9 - 2*b8 - b3 - 2*b1) * q^91 + (6*b6 - 4*b4 + 4) * q^93 + (2*b11 + 2*b10 - 3*b9 + b8 - b3 - b1) * q^95 + (b7 - 5*b4 - 2*b2 - 10) * q^97 + (2*b11 + 2*b10 + 4*b9 - 4*b8 - 2*b3 - 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{9}+O(q^{10})$$ 12 * q - 12 * q^9 $$12 q - 12 q^{9} - 12 q^{13} - 6 q^{17} - 12 q^{25} + 6 q^{29} - 30 q^{33} + 6 q^{37} + 30 q^{41} + 24 q^{49} + 48 q^{53} - 18 q^{61} - 12 q^{65} - 6 q^{69} - 132 q^{77} + 6 q^{81} - 12 q^{85} + 30 q^{89} + 36 q^{93} - 90 q^{97}+O(q^{100})$$ 12 * q - 12 * q^9 - 12 * q^13 - 6 * q^17 - 12 * q^25 + 6 * q^29 - 30 * q^33 + 6 * q^37 + 30 * q^41 + 24 * q^49 + 48 * q^53 - 18 * q^61 - 12 * q^65 - 6 * q^69 - 132 * q^77 + 6 * q^81 - 12 * q^85 + 30 * q^89 + 36 * q^93 - 90 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 12x^{10} + 108x^{8} - 430x^{6} + 1284x^{4} - 36x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{11} + 9\nu^{9} - 81\nu^{7} + 107\nu^{5} - 3\nu^{3} - 5733\nu ) / 240$$ (-v^11 + 9*v^9 - 81*v^7 + 107*v^5 - 3*v^3 - 5733*v) / 240 $$\beta_{2}$$ $$=$$ $$( \nu^{10} - 9\nu^{8} + 61\nu^{6} - 107\nu^{4} + 3\nu^{2} + 913 ) / 240$$ (v^10 - 9*v^8 + 61*v^6 - 107*v^4 + 3*v^2 + 913) / 240 $$\beta_{3}$$ $$=$$ $$( -\nu^{11} + 9\nu^{9} - 71\nu^{7} + 107\nu^{5} - 3\nu^{3} - 3443\nu ) / 120$$ (-v^11 + 9*v^9 - 71*v^7 + 107*v^5 - 3*v^3 - 3443*v) / 120 $$\beta_{4}$$ $$=$$ $$( 27\nu^{10} - 323\nu^{8} + 2907\nu^{6} - 11529\nu^{4} + 34561\nu^{2} - 969 ) / 960$$ (27*v^10 - 323*v^8 + 2907*v^6 - 11529*v^4 + 34561*v^2 - 969) / 960 $$\beta_{5}$$ $$=$$ $$( -17\nu^{10} + 203\nu^{8} - 1847\nu^{6} + 7339\nu^{4} - 22201\nu^{2} - 251 ) / 360$$ (-17*v^10 + 203*v^8 - 1847*v^6 + 7339*v^4 - 22201*v^2 - 251) / 360 $$\beta_{6}$$ $$=$$ $$( 71\nu^{10} - 839\nu^{8} + 7511\nu^{6} - 29677\nu^{4} + 88813\nu^{2} - 4957 ) / 1440$$ (71*v^10 - 839*v^8 + 7511*v^6 - 29677*v^4 + 88813*v^2 - 4957) / 1440 $$\beta_{7}$$ $$=$$ $$( -101\nu^{10} + 1229\nu^{8} - 11061\nu^{6} + 44407\nu^{4} - 131503\nu^{2} + 3687 ) / 960$$ (-101*v^10 + 1229*v^8 - 11061*v^6 + 44407*v^4 - 131503*v^2 + 3687) / 960 $$\beta_{8}$$ $$=$$ $$( -409\nu^{11} + 4921\nu^{9} - 44329\nu^{7} + 177203\nu^{5} - 530387\nu^{3} + 29603\nu ) / 1440$$ (-409*v^11 + 4921*v^9 - 44329*v^7 + 177203*v^5 - 530387*v^3 + 29603*v) / 1440 $$\beta_{9}$$ $$=$$ $$( 421\nu^{11} - 5029\nu^{9} + 45181\nu^{7} - 178487\nu^{5} + 530423\nu^{3} + 14593\nu ) / 1440$$ (421*v^11 - 5029*v^9 + 45181*v^7 - 178487*v^5 + 530423*v^3 + 14593*v) / 1440 $$\beta_{10}$$ $$=$$ $$( 749\nu^{11} - 8981\nu^{9} + 80789\nu^{7} - 321103\nu^{5} + 957127\nu^{3} - 12103\nu ) / 1440$$ (749*v^11 - 8981*v^9 + 80789*v^7 - 321103*v^5 + 957127*v^3 - 12103*v) / 1440 $$\beta_{11}$$ $$=$$ $$( 161\nu^{11} - 1929\nu^{9} + 17361\nu^{7} - 69067\nu^{5} + 206403\nu^{3} - 5787\nu ) / 240$$ (161*v^11 - 1929*v^9 + 17361*v^7 - 69067*v^5 + 206403*v^3 - 5787*v) / 240
 $$\nu$$ $$=$$ $$( \beta_{9} + \beta_{8} + \beta_{3} ) / 2$$ (b9 + b8 + b3) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} - \beta_{6} + 2\beta_{5} + 9\beta_{4} - \beta_{2} + 7 ) / 2$$ (b7 - b6 + 2*b5 + 9*b4 - b2 + 7) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{11} - 12\beta_{10} + 24\beta_{9} + 12\beta_{3} + \beta_1 ) / 4$$ (-b11 - 12*b10 + 24*b9 + 12*b3 + b1) / 4 $$\nu^{4}$$ $$=$$ $$( 5\beta_{7} - 14\beta_{6} + 7\beta_{5} + 55\beta_{4} - 7 ) / 2$$ (5*b7 - 14*b6 + 7*b5 + 55*b4 - 7) / 2 $$\nu^{5}$$ $$=$$ $$( -11\beta_{11} - 70\beta_{10} + 74\beta_{9} - 78\beta_{8} ) / 4$$ (-11*b11 - 70*b10 + 74*b9 - 78*b8) / 4 $$\nu^{6}$$ $$=$$ $$-24\beta_{6} - 24\beta_{5} + 12\beta_{2} - 145$$ -24*b6 - 24*b5 + 12*b2 - 145 $$\nu^{7}$$ $$=$$ $$( -229\beta_{9} - 229\beta_{8} - 205\beta_{3} - 48\beta_1 ) / 2$$ (-229*b9 - 229*b8 - 205*b3 - 48*b1) / 2 $$\nu^{8}$$ $$=$$ $$( -109\beta_{7} + 325\beta_{6} - 650\beta_{5} - 2085\beta_{4} + 109\beta_{2} - 1435 ) / 2$$ (-109*b7 + 325*b6 - 650*b5 - 2085*b4 + 109*b2 - 1435) / 2 $$\nu^{9}$$ $$=$$ $$( 757\beta_{11} + 2412\beta_{10} - 5688\beta_{9} + 432\beta_{8} - 2412\beta_{3} - 757\beta_1 ) / 4$$ (757*b11 + 2412*b10 - 5688*b9 + 432*b8 - 2412*b3 - 757*b1) / 4 $$\nu^{10}$$ $$=$$ $$( -449\beta_{7} + 4358\beta_{6} - 2179\beta_{5} - 12907\beta_{4} + 2179 ) / 2$$ (-449*b7 + 4358*b6 - 2179*b5 - 12907*b4 + 2179) / 2 $$\nu^{11}$$ $$=$$ $$( 5639\beta_{11} + 14254\beta_{10} - 17714\beta_{9} + 21174\beta_{8} ) / 4$$ (5639*b11 + 14254*b10 - 17714*b9 + 21174*b8) / 4

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$703$$ $$769$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{4}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 0.145015 + 0.0837246i 2.19011 + 1.26446i 2.04509 + 1.18073i −2.04509 − 1.18073i −2.19011 − 1.26446i −0.145015 − 0.0837246i 0.145015 − 0.0837246i 2.19011 − 1.26446i 2.04509 − 1.18073i −2.04509 + 1.18073i −2.19011 + 1.26446i −0.145015 + 0.0837246i
0 −1.48362 2.56970i 0 1.23519i 0 −4.16083 2.40226i 0 −2.90226 + 5.02685i 0
257.2 0 −1.13286 1.96217i 0 3.99777i 0 0.981633 + 0.566746i 0 −1.06675 + 1.84766i 0
257.3 0 −0.515266 0.892467i 0 0.701519i 0 2.54439 + 1.46900i 0 0.969002 1.67836i 0
257.4 0 0.515266 + 0.892467i 0 0.701519i 0 −2.54439 1.46900i 0 0.969002 1.67836i 0
257.5 0 1.13286 + 1.96217i 0 3.99777i 0 −0.981633 0.566746i 0 −1.06675 + 1.84766i 0
257.6 0 1.48362 + 2.56970i 0 1.23519i 0 4.16083 + 2.40226i 0 −2.90226 + 5.02685i 0
641.1 0 −1.48362 + 2.56970i 0 1.23519i 0 −4.16083 + 2.40226i 0 −2.90226 5.02685i 0
641.2 0 −1.13286 + 1.96217i 0 3.99777i 0 0.981633 0.566746i 0 −1.06675 1.84766i 0
641.3 0 −0.515266 + 0.892467i 0 0.701519i 0 2.54439 1.46900i 0 0.969002 + 1.67836i 0
641.4 0 0.515266 0.892467i 0 0.701519i 0 −2.54439 + 1.46900i 0 0.969002 + 1.67836i 0
641.5 0 1.13286 1.96217i 0 3.99777i 0 −0.981633 + 0.566746i 0 −1.06675 1.84766i 0
641.6 0 1.48362 2.56970i 0 1.23519i 0 4.16083 2.40226i 0 −2.90226 5.02685i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.e even 6 1 inner
52.i odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 832.2.w.j 12
4.b odd 2 1 inner 832.2.w.j 12
8.b even 2 1 416.2.w.d 12
8.d odd 2 1 416.2.w.d 12
13.e even 6 1 inner 832.2.w.j 12
52.i odd 6 1 inner 832.2.w.j 12
104.p odd 6 1 416.2.w.d 12
104.s even 6 1 416.2.w.d 12
104.u even 12 1 5408.2.a.bm 6
104.u even 12 1 5408.2.a.bn 6
104.x odd 12 1 5408.2.a.bm 6
104.x odd 12 1 5408.2.a.bn 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.w.d 12 8.b even 2 1
416.2.w.d 12 8.d odd 2 1
416.2.w.d 12 104.p odd 6 1
416.2.w.d 12 104.s even 6 1
832.2.w.j 12 1.a even 1 1 trivial
832.2.w.j 12 4.b odd 2 1 inner
832.2.w.j 12 13.e even 6 1 inner
832.2.w.j 12 52.i odd 6 1 inner
5408.2.a.bm 6 104.u even 12 1
5408.2.a.bm 6 104.x odd 12 1
5408.2.a.bn 6 104.u even 12 1
5408.2.a.bn 6 104.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 15T_{3}^{10} + 165T_{3}^{8} + 804T_{3}^{6} + 2880T_{3}^{4} + 2880T_{3}^{2} + 2304$$ acting on $$S_{2}^{\mathrm{new}}(832, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 15 T^{10} + \cdots + 2304$$
$5$ $$(T^{6} + 18 T^{4} + \cdots + 12)^{2}$$
$7$ $$T^{12} - 33 T^{10} + \cdots + 65536$$
$11$ $$T^{12} - 33 T^{10} + \cdots + 65536$$
$13$ $$(T^{6} + 6 T^{5} + \cdots + 2197)^{2}$$
$17$ $$(T^{2} + T + 1)^{6}$$
$19$ $$T^{12} - 45 T^{10} + \cdots + 1679616$$
$23$ $$T^{12} + \cdots + 2127792384$$
$29$ $$(T^{6} - 3 T^{5} + \cdots + 961)^{2}$$
$31$ $$(T^{6} + 96 T^{4} + \cdots + 576)^{2}$$
$37$ $$(T^{6} - 3 T^{5} + \cdots + 5043)^{2}$$
$41$ $$(T^{6} - 15 T^{5} + \cdots + 6627)^{2}$$
$43$ $$T^{12} + \cdots + 150994944$$
$47$ $$(T^{6} + 108 T^{4} + \cdots + 43264)^{2}$$
$53$ $$(T^{3} - 12 T^{2} + 27 T + 4)^{4}$$
$59$ $$T^{12} + \cdots + 10485760000$$
$61$ $$(T^{6} + 9 T^{5} + \cdots + 56169)^{2}$$
$67$ $$T^{12} + \cdots + 385571451136$$
$71$ $$T^{12} - 69 T^{10} + \cdots + 160000$$
$73$ $$(T^{6} + 234 T^{4} + \cdots + 288300)^{2}$$
$79$ $$(T^{6} - 396 T^{4} + \cdots - 786432)^{2}$$
$83$ $$(T^{6} + 228 T^{4} + \cdots + 16384)^{2}$$
$89$ $$(T^{6} - 15 T^{5} + \cdots + 2157312)^{2}$$
$97$ $$(T^{6} + 45 T^{5} + \cdots + 62208)^{2}$$