# Properties

 Label 156.3.d.a Level $156$ Weight $3$ Character orbit 156.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.25069212402$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2x^{7} + 15x^{6} + 10x^{5} + 97x^{4} + 252x^{3} + 700x^{2} + 1696x + 3792$$ x^8 - 2*x^7 + 15*x^6 + 10*x^5 + 97*x^4 + 252*x^3 + 700*x^2 + 1696*x + 3792 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{3} + \beta_{3} q^{5} + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{7} + ( - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^3 + b3 * q^5 + (b6 + b4 + b3 + b2 - 1) * q^7 + (-b5 + b3 + b2 - b1 - 3) * q^9 $$q + ( - \beta_1 + 1) q^{3} + \beta_{3} q^{5} + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{7} + ( - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{9} + (\beta_{6} - \beta_{4} - \beta_{2} + 1) q^{11} - \beta_{5} q^{13} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{15} + ( - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 1) q^{17} + (2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_1 - 2) q^{19} + ( - 2 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_1 + 2) q^{21} + ( - 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2) q^{23} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{25} + ( - \beta_{7} - 5 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{27} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 2) q^{29} + (10 \beta_{5} + 12) q^{31} + (4 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + \cdots + 3) q^{33}+ \cdots + (16 \beta_{7} - 9 \beta_{6} + 4 \beta_{5} + 7 \beta_{4} + 5 \beta_{2} - 2 \beta_1 + 37) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^3 + b3 * q^5 + (b6 + b4 + b3 + b2 - 1) * q^7 + (-b5 + b3 + b2 - b1 - 3) * q^9 + (b6 - b4 - b2 + 1) * q^11 - b5 * q^13 + (-3*b5 + b4 + b3 - b2 + 1) * q^15 + (-3*b7 + 2*b6 - 2*b4 + 2*b3 + b2 - 1) * q^17 + (2*b7 - 2*b6 + 4*b5 - 2*b4 - 2*b3 - 4*b1 - 2) * q^19 + (-2*b7 + 3*b6 - 3*b5 - 2*b4 + b3 + 2*b1 + 2) * q^21 + (-2*b6 + 2*b4 + 2*b3 + 2*b2 - 2) * q^23 + (-b7 + 2*b6 + 2*b4 + 2*b3 + b2 + 2*b1 - 4) * q^25 + (-b7 - 5*b5 + 2*b4 + b3 + 2*b2 + 4*b1 + 2) * q^27 + (2*b7 - 2*b6 + 2*b4 + 2*b3 - 2*b2 - 4*b1 + 2) * q^29 + (10*b5 + 12) * q^31 + (4*b7 - 3*b6 - 2*b5 - 3*b4 - 4*b3 - 3*b2 - 2*b1 + 3) * q^33 + (3*b7 - 8*b3 - 3*b2 + 3) * q^35 + (2*b7 + b6 + 4*b5 + b4 + b3 + 3*b2 - 4*b1 - 11) * q^37 + (b7 - b5 + b4 - b3 + b2) * q^39 + (-2*b7 - b6 + b4 - 4*b3 - b2 - 8*b1 + 1) * q^41 + (-5*b7 - 2*b6 + 6*b5 - 2*b4 - 2*b3 - 7*b2 + 10*b1 + 7) * q^43 + (4*b7 + 3*b6 - 5*b5 + 4*b4 + 2*b2 - 2*b1 - 14) * q^45 + (2*b7 - 2*b6 + 2*b4 - 3*b3 + 4*b2 + 8*b1 - 4) * q^47 + (-3*b7 + 6*b5 - 3*b2 + 6*b1 + 24) * q^49 + (-4*b7 - 6*b6 - 7*b5 - 4*b4 - 9*b3 - 5*b2 - b1 - 10) * q^51 + (2*b6 - 2*b4 + 2*b3 + 4*b2 + 12*b1 - 4) * q^53 + (-4*b6 + 14*b5 - 4*b4 - 4*b3 - 4*b2 - 10) * q^55 + (-6*b6 + 4*b5 + 8*b3 + 2*b2 + 4*b1 + 18) * q^57 + (2*b7 - b6 + b4 - 6*b3 + 3*b2 + 8*b1 - 3) * q^59 + (2*b7 - 2*b6 - 6*b5 - 2*b4 - 2*b3 - 4*b1 - 12) * q^61 + (4*b7 - 6*b6 - 7*b5 - 5*b4 - 16*b3 - 5*b2 - 4*b1 + 7) * q^63 + (b7 + b6 - b4 + 4*b1) * q^65 + (-4*b7 + 4*b5 - 4*b2 + 8*b1 - 50) * q^67 + (-8*b7 + 6*b6 - 2*b5 + 8*b4 + 10*b3 + 4*b2 + 4*b1 - 4) * q^69 + (6*b7 - 7*b3 - 6*b2 + 6) * q^71 + (2*b7 + 2*b2 - 4*b1) * q^73 + (-4*b7 + 6*b6 - 3*b5 - 4*b4 - b3 - 3*b2 + 4*b1 - 11) * q^75 + (-6*b7 + 4*b6 - 4*b4 + 6*b3 - 10*b2 - 24*b1 + 10) * q^77 + (2*b7 + 6*b5 + 2*b2 - 4*b1 + 42) * q^79 + (-3*b7 + 6*b6 - b5 + 6*b4 - 5*b3 + 4*b2 - b1 + 6) * q^81 + (3*b6 - 3*b4 + 9*b2 + 24*b1 - 9) * q^83 + (10*b7 + 5*b6 - 2*b5 + 5*b4 + 5*b3 + 15*b2 - 20*b1 - 3) * q^85 + (4*b7 + 6*b6 - 8*b5 + 8*b4 + 16*b3 + 4*b2 - 2*b1 - 22) * q^87 + (6*b7 - 7*b6 + 7*b4 + 8*b3 + b2 - 1) * q^89 + (3*b7 - 2*b6 + 2*b5 - 2*b4 - 2*b3 + b2 - 6*b1 - 5) * q^91 + (-10*b7 + 10*b5 - 10*b4 + 10*b3 - 10*b2 - 12*b1 + 12) * q^93 + (-2*b7 - 10*b6 + 10*b4 + 8*b3 - 4*b2 - 32*b1 + 4) * q^95 + (-2*b6 - 28*b5 - 2*b4 - 2*b3 - 2*b2 + 52) * q^97 + (16*b7 - 9*b6 + 4*b5 + 7*b4 + 5*b2 - 2*b1 + 37) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 6 q^{3} - 8 q^{7} - 22 q^{9}+O(q^{10})$$ 8 * q + 6 * q^3 - 8 * q^7 - 22 * q^9 $$8 q + 6 q^{3} - 8 q^{7} - 22 q^{9} + 4 q^{15} - 24 q^{19} + 16 q^{21} - 28 q^{25} + 36 q^{27} + 96 q^{31} + 4 q^{33} - 96 q^{37} + 76 q^{43} - 136 q^{45} + 204 q^{49} - 62 q^{51} - 80 q^{55} + 184 q^{57} - 104 q^{61} + 36 q^{63} - 384 q^{67} - 8 q^{73} - 100 q^{75} + 328 q^{79} + 50 q^{81} - 64 q^{85} - 204 q^{87} - 52 q^{91} + 72 q^{93} + 416 q^{97} + 284 q^{99}+O(q^{100})$$ 8 * q + 6 * q^3 - 8 * q^7 - 22 * q^9 + 4 * q^15 - 24 * q^19 + 16 * q^21 - 28 * q^25 + 36 * q^27 + 96 * q^31 + 4 * q^33 - 96 * q^37 + 76 * q^43 - 136 * q^45 + 204 * q^49 - 62 * q^51 - 80 * q^55 + 184 * q^57 - 104 * q^61 + 36 * q^63 - 384 * q^67 - 8 * q^73 - 100 * q^75 + 328 * q^79 + 50 * q^81 - 64 * q^85 - 204 * q^87 - 52 * q^91 + 72 * q^93 + 416 * q^97 + 284 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 15x^{6} + 10x^{5} + 97x^{4} + 252x^{3} + 700x^{2} + 1696x + 3792$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{7} + 28\nu^{6} - 95\nu^{5} + 516\nu^{4} - 553\nu^{3} + 3758\nu^{2} + 736\nu + 13836 ) / 2916$$ (-v^7 + 28*v^6 - 95*v^5 + 516*v^4 - 553*v^3 + 3758*v^2 + 736*v + 13836) / 2916 $$\beta_{3}$$ $$=$$ $$( -5\nu^{7} - 4\nu^{6} + 11\nu^{5} - 408\nu^{4} + 511\nu^{3} - 1676\nu^{2} - 1972\nu - 1560 ) / 2916$$ (-5*v^7 - 4*v^6 + 11*v^5 - 408*v^4 + 511*v^3 - 1676*v^2 - 1972*v - 1560) / 2916 $$\beta_{4}$$ $$=$$ $$( -5\nu^{7} + 14\nu^{6} + 11\nu^{5} - 156\nu^{4} - 263\nu^{3} - 272\nu^{2} - 1144\nu - 6204 ) / 2916$$ (-5*v^7 + 14*v^6 + 11*v^5 - 156*v^4 - 263*v^3 - 272*v^2 - 1144*v - 6204) / 2916 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + 4\nu^{6} - 14\nu^{5} + 18\nu^{4} - 7\nu^{3} - 139\nu^{2} + 280\nu + 102 ) / 486$$ (-v^7 + 4*v^6 - 14*v^5 + 18*v^4 - 7*v^3 - 139*v^2 + 280*v + 102) / 486 $$\beta_{6}$$ $$=$$ $$( 7\nu^{7} - 52\nu^{6} + 179\nu^{5} - 138\nu^{4} + 109\nu^{3} - 494\nu^{2} + 2444\nu + 132 ) / 2916$$ (7*v^7 - 52*v^6 + 179*v^5 - 138*v^4 + 109*v^3 - 494*v^2 + 2444*v + 132) / 2916 $$\beta_{7}$$ $$=$$ $$( 13\nu^{7} - 40\nu^{6} + 263\nu^{5} - 228\nu^{4} + 2005\nu^{3} + 718\nu^{2} + 9224\nu + 13560 ) / 2916$$ (13*v^7 - 40*v^6 + 263*v^5 - 228*v^4 + 2005*v^3 + 718*v^2 + 9224*v + 13560) / 2916
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{3} + \beta_{2} + \beta _1 - 4$$ -b5 + b3 + b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2\beta_{5} - 2\beta_{4} + 2\beta_{3} + \beta_{2} - 4\beta _1 - 13$$ b7 + 2*b5 - 2*b4 + 2*b3 + b2 - 4*b1 - 13 $$\nu^{4}$$ $$=$$ $$\beta_{7} + 6\beta_{6} + 13\beta_{5} - 2\beta_{4} - 3\beta_{3} + 2\beta_{2} - 19\beta _1 - 23$$ b7 + 6*b6 + 13*b5 - 2*b4 - 3*b3 + 2*b2 - 19*b1 - 23 $$\nu^{5}$$ $$=$$ $$8\beta_{7} + 12\beta_{6} + 19\beta_{5} + 32\beta_{4} - 17\beta_{3} - \beta_{2} - 45\beta _1 + 22$$ 8*b7 + 12*b6 + 19*b5 + 32*b4 - 17*b3 - b2 - 45*b1 + 22 $$\nu^{6}$$ $$=$$ $$29\beta_{7} - 84\beta_{6} - 18\beta_{5} + 104\beta_{4} - 112\beta_{3} - 63\beta_{2} - 30\beta _1 + 333$$ 29*b7 - 84*b6 - 18*b5 + 104*b4 - 112*b3 - 63*b2 - 30*b1 + 333 $$\nu^{7}$$ $$=$$ $$15\beta_{7} - 396\beta_{6} - 465\beta_{5} - 54\beta_{4} - 417\beta_{3} - 348\beta_{2} + 337\beta _1 + 1359$$ 15*b7 - 396*b6 - 465*b5 - 54*b4 - 417*b3 - 348*b2 + 337*b1 + 1359

## Character values

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

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 2.33932 + 2.68444i 2.33932 − 2.68444i 1.12162 + 2.99753i 1.12162 − 2.99753i −0.621622 + 2.52395i −0.621622 − 2.52395i −1.83932 + 0.968627i −1.83932 − 0.968627i
0 −1.33932 2.68444i 0 5.74349i 0 −9.51181 0 −5.41242 + 7.19067i 0
53.2 0 −1.33932 + 2.68444i 0 5.74349i 0 −9.51181 0 −5.41242 7.19067i 0
53.3 0 −0.121622 2.99753i 0 0.347906i 0 10.5746 0 −8.97042 + 0.729130i 0
53.4 0 −0.121622 + 2.99753i 0 0.347906i 0 10.5746 0 −8.97042 0.729130i 0
53.5 0 1.62162 2.52395i 0 7.04754i 0 −8.96906 0 −3.74069 8.18580i 0
53.6 0 1.62162 + 2.52395i 0 7.04754i 0 −8.96906 0 −3.74069 + 8.18580i 0
53.7 0 2.83932 0.968627i 0 5.58779i 0 3.90626 0 7.12352 5.50049i 0
53.8 0 2.83932 + 0.968627i 0 5.58779i 0 3.90626 0 7.12352 + 5.50049i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 53.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.3.d.a 8
3.b odd 2 1 inner 156.3.d.a 8
4.b odd 2 1 624.3.f.a 8
12.b even 2 1 624.3.f.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.d.a 8 1.a even 1 1 trivial
156.3.d.a 8 3.b odd 2 1 inner
624.3.f.a 8 4.b odd 2 1
624.3.f.a 8 12.b even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - 6 T^{7} + 29 T^{6} + \cdots + 6561$$
$5$ $$T^{8} + 114 T^{6} + 4233 T^{4} + \cdots + 6192$$
$7$ $$(T^{4} + 4 T^{3} - 141 T^{2} - 472 T + 3524)^{2}$$
$11$ $$T^{8} + 612 T^{6} + \cdots + 128397312$$
$13$ $$(T^{2} - 13)^{4}$$
$17$ $$T^{8} + 2342 T^{6} + \cdots + 72223488$$
$19$ $$(T^{4} + 12 T^{3} - 1216 T^{2} + \cdots - 46224)^{2}$$
$23$ $$T^{8} + 2584 T^{6} + \cdots + 2054356992$$
$29$ $$T^{8} + 4288 T^{6} + \cdots + 120750538752$$
$31$ $$(T^{2} - 24 T - 1156)^{4}$$
$37$ $$(T^{4} + 48 T^{3} - 673 T^{2} + 2784 T - 3708)^{2}$$
$41$ $$T^{8} + 6380 T^{6} + \cdots + 406257120000$$
$43$ $$(T^{4} - 38 T^{3} - 5943 T^{2} + \cdots - 4083628)^{2}$$
$47$ $$T^{8} + 7098 T^{6} + \cdots + 181027138608$$
$53$ $$T^{8} + 8840 T^{6} + \cdots + 722234880000$$
$59$ $$T^{8} + 8244 T^{6} + \cdots + 717961310208$$
$61$ $$(T^{4} + 52 T^{3} - 1296 T^{2} + \cdots + 7856)^{2}$$
$67$ $$(T^{4} + 192 T^{3} + 10456 T^{2} + \cdots + 953488)^{2}$$
$71$ $$T^{8} + 16218 T^{6} + \cdots + 12770012803248$$
$73$ $$(T^{4} + 4 T^{3} - 732 T^{2} - 1472 T + 68032)^{2}$$
$79$ $$(T^{4} - 164 T^{3} + 8412 T^{2} + \cdots - 966848)^{2}$$
$83$ $$T^{8} + 36036 T^{6} + \cdots + 28\!\cdots\!12$$
$89$ $$T^{8} + 29420 T^{6} + \cdots + 10668283884288$$
$97$ $$(T^{4} - 208 T^{3} - 6204 T^{2} + \cdots + 57925568)^{2}$$