# Properties

 Label 156.3.j.a Level $156$ Weight $3$ Character orbit 156.j Analytic conductor $4.251$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 156.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.25069212402$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.1579585536.10 Defining polynomial: $$x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484$$ x^8 - 2*x^7 - 4*x^6 + 28*x^5 - 38*x^4 + 8*x^3 + 200*x^2 - 352*x + 484 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + (\beta_{5} + \beta_{4} - \beta_1 + 1) q^{5} + (2 \beta_{6} + 2 \beta_{5} + \beta_{2} - \beta_1 - 2) q^{7} + 3 q^{9}+O(q^{10})$$ q + b2 * q^3 + (b5 + b4 - b1 + 1) * q^5 + (2*b6 + 2*b5 + b2 - b1 - 2) * q^7 + 3 * q^9 $$q + \beta_{2} q^{3} + (\beta_{5} + \beta_{4} - \beta_1 + 1) q^{5} + (2 \beta_{6} + 2 \beta_{5} + \beta_{2} - \beta_1 - 2) q^{7} + 3 q^{9} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{3} - 3 \beta_1) q^{11} + (3 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{13} + ( - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{15} + (6 \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} - 1) q^{17} + ( - 3 \beta_{7} + 10 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 10) q^{19} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{3} - 2 \beta_1 - 1) q^{21} + (6 \beta_{7} - 2 \beta_{6} + 6 \beta_{5} - 2 \beta_{4} + 10 \beta_1 + 2) q^{23} + ( - 2 \beta_{7} + 4 \beta_{6} - 7 \beta_{5} + 4 \beta_{4} - 12 \beta_1 - 4) q^{25} + 3 \beta_{2} q^{27} + ( - 3 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 8 \beta_{2} + 3) q^{29} + ( - 2 \beta_{7} - 4 \beta_{4} - 2 \beta_{3} + 7 \beta_{2} - 3 \beta_1) q^{31} + ( - 5 \beta_{7} - \beta_{6} + 4 \beta_{5} + 5 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{33} + (5 \beta_{6} + 5 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} - 27) q^{35} + ( - 7 \beta_{7} - 10 \beta_{6} - 9 \beta_{5} + 7 \beta_{3} + 3 \beta_{2} + 13 \beta_1 + 9) q^{37} + ( - 3 \beta_{7} - \beta_{6} - 4 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} + 5 \beta_{2} + \cdots + 12) q^{39}+ \cdots + (6 \beta_{7} + 9 \beta_{6} - 6 \beta_{3} - 9 \beta_1) q^{99}+O(q^{100})$$ q + b2 * q^3 + (b5 + b4 - b1 + 1) * q^5 + (2*b6 + 2*b5 + b2 - b1 - 2) * q^7 + 3 * q^9 + (2*b7 + 3*b6 - 2*b3 - 3*b1) * q^11 + (3*b7 - 2*b5 + 2*b3 + 3*b2 + 2*b1 + 3) * q^13 + (-b7 + b5 + b4 - b3 + b2 - 2*b1 + 1) * q^15 + (6*b7 + b6 - 3*b5 + b4 - 1) * q^17 + (-3*b7 + 10*b5 + 2*b4 - 3*b3 - 2*b1 + 10) * q^19 + (-2*b7 + 2*b6 + b5 + 2*b3 - 2*b1 - 1) * q^21 + (6*b7 - 2*b6 + 6*b5 - 2*b4 + 10*b1 + 2) * q^23 + (-2*b7 + 4*b6 - 7*b5 + 4*b4 - 12*b1 - 4) * q^25 + 3*b2 * q^27 + (-3*b6 - 3*b5 + 3*b4 - 4*b3 - 8*b2 + 3) * q^29 + (-2*b7 - 4*b4 - 2*b3 + 7*b2 - 3*b1) * q^31 + (-5*b7 - b6 + 4*b5 + 5*b3 + b2 + 2*b1 - 4) * q^33 + (5*b6 + 5*b5 - 5*b4 + 2*b3 - 6*b2 - 27) * q^35 + (-7*b7 - 10*b6 - 9*b5 + 7*b3 + 3*b2 + 13*b1 + 9) * q^37 + (-3*b7 - b6 - 4*b5 - 5*b4 - 2*b3 + 5*b2 + 5*b1 + 12) * q^39 + (4*b7 - 21*b5 - 3*b4 + 4*b3 - 4*b2 + 7*b1 - 21) * q^41 + (-10*b7 - 4*b6 + 6*b5 - 4*b4 - 4*b1 + 4) * q^43 + (3*b5 + 3*b4 - 3*b1 + 3) * q^45 + (4*b7 - 7*b6 - 8*b5 - 4*b3 - 18*b2 - 11*b1 + 8) * q^47 + (-4*b6 - b5 - 4*b4 + 24*b1 + 4) * q^49 + (-8*b7 - 5*b6 - 3*b5 - 5*b4 + 8*b1 + 5) * q^51 + (-7*b6 - 7*b5 + 7*b4 - 6*b3 - 26*b2 - 9) * q^53 + (4*b6 + 4*b5 - 4*b4 + 2*b3 - 30) * q^55 + (b7 - b5 + 8*b4 + b3 + 7*b2 - 15*b1 - 1) * q^57 + (2*b7 - b6 + 28*b5 - 2*b3 + 2*b2 + 3*b1 - 28) * q^59 + (-6*b6 - 6*b5 + 6*b4 + 8*b3 - 6*b2 - 24) * q^61 + (6*b6 + 6*b5 + 3*b2 - 3*b1 - 6) * q^63 + (4*b7 - 7*b6 - 10*b5 + 4*b4 - 6*b3 + 8*b2 + 17*b1 + 18) * q^65 + (-b7 + 4*b5 - 4*b4 - b3 + 2*b2 + 2*b1 + 4) * q^67 + (-2*b7 - 8*b6 - 24*b5 - 8*b4 + 2*b1 + 8) * q^69 + (6*b7 + 13*b5 - 17*b4 + 6*b3 - 10*b2 + 27*b1 + 13) * q^71 + (-6*b7 + 10*b6 - 15*b5 + 6*b3 - 18*b2 - 28*b1 + 15) * q^73 + (-6*b7 + 6*b6 + 24*b5 + 6*b4 + b1 - 6) * q^75 + (2*b7 - 3*b6 - 57*b5 - 3*b4 + 8*b1 + 3) * q^77 + (6*b6 + 6*b5 - 6*b4 - 10*b3 - 22*b2 + 6) * q^79 + 9 * q^81 + (8*b7 + 3*b5 - 3*b4 + 8*b3 + 40*b2 - 37*b1 + 3) * q^83 + (6*b7 - 6*b6 - 6*b5 - 6*b3 + 16*b2 + 22*b1 + 6) * q^85 + (-7*b6 - 7*b5 + 7*b4 - 2*b3 - 4*b2 - 15) * q^87 + (4*b7 + 19*b6 + 2*b5 - 4*b3 + 32*b2 + 13*b1 - 2) * q^89 + (-11*b7 + 12*b6 + 10*b5 + 8*b4 - 3*b3 + 36*b2 - 22*b1 + 4) * q^91 + (6*b7 + 15*b5 + 6*b3 - 2*b2 + 2*b1 + 15) * q^93 + (-10*b7 + 19*b6 + 69*b5 + 19*b4 - 64*b1 - 19) * q^95 + (6*b7 + 39*b5 + 26*b4 + 6*b3 - 18*b2 - 8*b1 + 39) * q^97 + (6*b7 + 9*b6 - 6*b3 - 9*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 12 q^{5} - 8 q^{7} + 24 q^{9}+O(q^{10})$$ 8 * q + 12 * q^5 - 8 * q^7 + 24 * q^9 $$8 q + 12 q^{5} - 8 q^{7} + 24 q^{9} + 12 q^{11} + 24 q^{13} + 12 q^{15} + 88 q^{19} + 24 q^{29} - 16 q^{31} - 36 q^{33} - 216 q^{35} + 32 q^{37} + 72 q^{39} - 180 q^{41} + 36 q^{45} + 36 q^{47} - 72 q^{53} - 240 q^{55} + 24 q^{57} - 228 q^{59} - 192 q^{61} - 24 q^{63} + 132 q^{65} + 16 q^{67} + 36 q^{71} + 160 q^{73} + 48 q^{79} + 72 q^{81} + 12 q^{83} + 24 q^{85} - 120 q^{87} + 60 q^{89} + 112 q^{91} + 120 q^{93} + 416 q^{97} + 36 q^{99}+O(q^{100})$$ 8 * q + 12 * q^5 - 8 * q^7 + 24 * q^9 + 12 * q^11 + 24 * q^13 + 12 * q^15 + 88 * q^19 + 24 * q^29 - 16 * q^31 - 36 * q^33 - 216 * q^35 + 32 * q^37 + 72 * q^39 - 180 * q^41 + 36 * q^45 + 36 * q^47 - 72 * q^53 - 240 * q^55 + 24 * q^57 - 228 * q^59 - 192 * q^61 - 24 * q^63 + 132 * q^65 + 16 * q^67 + 36 * q^71 + 160 * q^73 + 48 * q^79 + 72 * q^81 + 12 * q^83 + 24 * q^85 - 120 * q^87 + 60 * q^89 + 112 * q^91 + 120 * q^93 + 416 * q^97 + 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} - 46\nu^{6} + 150\nu^{5} - 27\nu^{4} - 720\nu^{3} + 1768\nu^{2} - 2792\nu + 1881 ) / 935$$ (v^7 - 46*v^6 + 150*v^5 - 27*v^4 - 720*v^3 + 1768*v^2 - 2792*v + 1881) / 935 $$\beta_{2}$$ $$=$$ $$( 118\nu^{7} + 607\nu^{6} - 490\nu^{5} + 384\nu^{4} + 8710\nu^{3} - 19856\nu^{2} + 35024\nu + 91398 ) / 68510$$ (118*v^7 + 607*v^6 - 490*v^5 + 384*v^4 + 8710*v^3 - 19856*v^2 + 35024*v + 91398) / 68510 $$\beta_{3}$$ $$=$$ $$( 3448 \nu^{7} - 21163 \nu^{6} - 45670 \nu^{5} + 200494 \nu^{4} - 126360 \nu^{3} - 635936 \nu^{2} + 1327644 \nu - 2421122 ) / 753610$$ (3448*v^7 - 21163*v^6 - 45670*v^5 + 200494*v^4 - 126360*v^3 - 635936*v^2 + 1327644*v - 2421122) / 753610 $$\beta_{4}$$ $$=$$ $$( - 3448 \nu^{7} + 21163 \nu^{6} + 45670 \nu^{5} - 200494 \nu^{4} + 126360 \nu^{3} + 635936 \nu^{2} + 179576 \nu + 2421122 ) / 753610$$ (-3448*v^7 + 21163*v^6 + 45670*v^5 - 200494*v^4 + 126360*v^3 + 635936*v^2 + 179576*v + 2421122) / 753610 $$\beta_{5}$$ $$=$$ $$( 86\nu^{7} - 227\nu^{6} + 162\nu^{5} + 758\nu^{4} - 1794\nu^{3} + 8124\nu^{2} - 8408\nu + 13662 ) / 8866$$ (86*v^7 - 227*v^6 + 162*v^5 + 758*v^4 - 1794*v^3 + 8124*v^2 - 8408*v + 13662) / 8866 $$\beta_{6}$$ $$=$$ $$( 3777 \nu^{7} - 11052 \nu^{6} - 45875 \nu^{5} + 222466 \nu^{4} - 199615 \nu^{3} - 521764 \nu^{2} + 1944346 \nu - 3038343 ) / 376805$$ (3777*v^7 - 11052*v^6 - 45875*v^5 + 222466*v^4 - 199615*v^3 - 521764*v^2 + 1944346*v - 3038343) / 376805 $$\beta_{7}$$ $$=$$ $$( -1123\nu^{7} + 3803\nu^{6} - 3465\nu^{5} - 12944\nu^{4} + 51805\nu^{3} - 107134\nu^{2} + 75416\nu - 40743 ) / 34255$$ (-1123*v^7 + 3803*v^6 - 3465*v^5 - 12944*v^4 + 51805*v^3 - 107134*v^2 + 75416*v - 40743) / 34255
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} ) / 2$$ (b4 + b3) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{6} + 4\beta_{5} + \beta_{4} - \beta_{3} - 4\beta_{2} + 2 ) / 2$$ (b7 + b6 + 4*b5 + b4 - b3 - 4*b2 + 2) / 2 $$\nu^{3}$$ $$=$$ $$2\beta_{7} + 6\beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} + \beta _1 - 7$$ 2*b7 + 6*b5 + b4 + 2*b3 + b2 + b1 - 7 $$\nu^{4}$$ $$=$$ $$5\beta_{7} + 5\beta_{6} + 13\beta_{5} - 3\beta_{4} - 7\beta_{3} + \beta_{2} + 4\beta _1 + 4$$ 5*b7 + 5*b6 + 13*b5 - 3*b4 - 7*b3 + b2 + 4*b1 + 4 $$\nu^{5}$$ $$=$$ $$5\beta_{7} + 2\beta_{6} + 7\beta_{5} + \beta_{4} + 2\beta_{3} + 34\beta_{2} + 12\beta _1 - 55$$ 5*b7 + 2*b6 + 7*b5 + b4 + 2*b3 + 34*b2 + 12*b1 - 55 $$\nu^{6}$$ $$=$$ $$22\beta_{6} - 4\beta_{5} - 22\beta_{4} - 70\beta_{3} + 22\beta_{2}$$ 22*b6 - 4*b5 - 22*b4 - 70*b3 + 22*b2 $$\nu^{7}$$ $$=$$ $$-59\beta_{7} - 37\beta_{6} - 99\beta_{5} - 11\beta_{4} + 11\beta_{3} + 195\beta_{2} - 37\beta _1 - 331$$ -59*b7 - 37*b6 - 99*b5 - 11*b4 + 11*b3 + 195*b2 - 37*b1 - 331

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/156\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{5}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1
 −2.59436 + 0.0368949i 2.22833 + 1.32913i 0.252411 − 1.79004i 1.11361 + 1.42401i −2.59436 − 0.0368949i 2.22833 − 1.32913i 0.252411 + 1.79004i 1.11361 − 1.42401i
0 −1.73205 0 −0.658261 0.658261i 0 1.58447 1.58447i 0 3.00000 0
73.2 0 −1.73205 0 1.92621 + 1.92621i 0 −3.58447 + 3.58447i 0 3.00000 0
73.3 0 1.73205 0 −0.848026 0.848026i 0 5.42810 5.42810i 0 3.00000 0
73.4 0 1.73205 0 5.58008 + 5.58008i 0 −7.42810 + 7.42810i 0 3.00000 0
109.1 0 −1.73205 0 −0.658261 + 0.658261i 0 1.58447 + 1.58447i 0 3.00000 0
109.2 0 −1.73205 0 1.92621 1.92621i 0 −3.58447 3.58447i 0 3.00000 0
109.3 0 1.73205 0 −0.848026 + 0.848026i 0 5.42810 + 5.42810i 0 3.00000 0
109.4 0 1.73205 0 5.58008 5.58008i 0 −7.42810 7.42810i 0 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.3.j.a 8
3.b odd 2 1 468.3.m.d 8
4.b odd 2 1 624.3.ba.d 8
13.d odd 4 1 inner 156.3.j.a 8
39.f even 4 1 468.3.m.d 8
52.f even 4 1 624.3.ba.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.j.a 8 1.a even 1 1 trivial
156.3.j.a 8 13.d odd 4 1 inner
468.3.m.d 8 3.b odd 2 1
468.3.m.d 8 39.f even 4 1
624.3.ba.d 8 4.b odd 2 1
624.3.ba.d 8 52.f even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(156, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} - 3)^{4}$$
$5$ $$T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 576$$
$7$ $$T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 839056$$
$11$ $$T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 90782784$$
$13$ $$T^{8} - 24 T^{7} + \cdots + 815730721$$
$17$ $$T^{8} + 1296 T^{6} + \cdots + 427993344$$
$19$ $$T^{8} - 88 T^{7} + 3872 T^{6} + \cdots + 2972176$$
$23$ $$T^{8} + 2880 T^{6} + \cdots + 8422834176$$
$29$ $$(T^{4} - 12 T^{3} - 960 T^{2} + \cdots - 75504)^{2}$$
$31$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 10061584$$
$37$ $$T^{8} - 32 T^{7} + \cdots + 149561639824$$
$41$ $$T^{8} + 180 T^{7} + \cdots + 28199813184$$
$43$ $$T^{8} + 5088 T^{6} + \cdots + 619986161664$$
$47$ $$T^{8} - 36 T^{7} + \cdots + 4080820170816$$
$53$ $$(T^{4} + 36 T^{3} - 5184 T^{2} + \cdots - 976944)^{2}$$
$59$ $$T^{8} + 228 T^{7} + \cdots + 4746925417536$$
$61$ $$(T^{4} + 96 T^{3} + 1152 T^{2} + \cdots - 249024)^{2}$$
$67$ $$T^{8} - 16 T^{7} + \cdots + 929274256$$
$71$ $$T^{8} + \cdots + 448998131059776$$
$73$ $$T^{8} + \cdots + 601874565729424$$
$79$ $$(T^{4} - 24 T^{3} - 7296 T^{2} + \cdots + 613248)^{2}$$
$83$ $$T^{8} - 12 T^{7} + \cdots + 31\!\cdots\!84$$
$89$ $$T^{8} - 60 T^{7} + \cdots + 1057162337856$$
$97$ $$T^{8} - 416 T^{7} + \cdots + 63914204772496$$