# Properties

 Label 912.3.be.g Level $912$ Weight $3$ Character orbit 912.be Analytic conductor $24.850$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 912.be (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.8502001097$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.520060207104.10 Defining polynomial: $$x^{8} - 44x^{6} + 664x^{4} - 3528x^{2} + 8100$$ x^8 - 44*x^6 + 664*x^4 - 3528*x^2 + 8100 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 114) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{3} + ( - \beta_{4} - \beta_1 + 1) q^{5} + ( - \beta_{6} + \beta_{2} + 3) q^{7} + 3 \beta_1 q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^3 + (-b4 - b1 + 1) * q^5 + (-b6 + b2 + 3) * q^7 + 3*b1 * q^9 $$q + ( - \beta_1 - 1) q^{3} + ( - \beta_{4} - \beta_1 + 1) q^{5} + ( - \beta_{6} + \beta_{2} + 3) q^{7} + 3 \beta_1 q^{9} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{11} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{13} + (\beta_{7} + 2 \beta_{4} + \beta_1 - 2) q^{15} + (4 \beta_{6} - \beta_{5} + \beta_{4} - 3 \beta_{3} + \beta_{2} + 5 \beta_1 - 5) q^{17} + (3 \beta_{6} - 5 \beta_{5} + 8 \beta_1 - 7) q^{19} + (2 \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - 3 \beta_1 - 3) q^{21} + (3 \beta_{6} - 2 \beta_{5} + 3 \beta_{3} + \beta_{2} - 10 \beta_1) q^{23} + (3 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{3} - 3 \beta_{2} - 22 \beta_1) q^{25} + ( - 6 \beta_1 + 3) q^{27} + ( - 2 \beta_{6} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 8) q^{29} + (6 \beta_{6} - 3 \beta_{5} + 3 \beta_{3} + 6 \beta_{2} + 14 \beta_1 - 7) q^{31} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{33} + (\beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} - 5 \beta_{2} + 8 \beta_1 - 8) q^{35} + ( - 2 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + \cdots + 5) q^{37}+ \cdots + (3 \beta_{7} + 3 \beta_{5} + 3 \beta_{2} + 3 \beta_1) q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^3 + (-b4 - b1 + 1) * q^5 + (-b6 + b2 + 3) * q^7 + 3*b1 * q^9 + (b7 - b6 + b5 + b4 + b3 + b2 + 1) * q^11 + (b7 + b6 - 2*b5 + 2*b4 - b3 + b2 - 2*b1 + 4) * q^13 + (b7 + 2*b4 + b1 - 2) * q^15 + (4*b6 - b5 + b4 - 3*b3 + b2 + 5*b1 - 5) * q^17 + (3*b6 - 5*b5 + 8*b1 - 7) * q^19 + (2*b6 - b5 + b3 - b2 - 3*b1 - 3) * q^21 + (3*b6 - 2*b5 + 3*b3 + b2 - 10*b1) * q^23 + (3*b7 - b6 - 2*b5 - b3 - 3*b2 - 22*b1) * q^25 + (-6*b1 + 3) * q^27 + (-2*b6 + 2*b3 + 2*b2 + 4*b1 - 8) * q^29 + (6*b6 - 3*b5 + 3*b3 + 6*b2 + 14*b1 - 7) * q^31 + (-2*b7 + b6 - 2*b5 - b4 - b3 - 2*b2 - b1 - 1) * q^33 + (b6 + 5*b5 - 2*b4 - 6*b3 - 5*b2 + 8*b1 - 8) * q^35 + (-2*b7 - 4*b6 + 4*b5 + 2*b4 - 4*b3 - 4*b2 - 10*b1 + 5) * q^37 + (-3*b7 + 3*b5 - 3*b4 + 3*b3 - 6) * q^39 + (-2*b7 - 4*b6 + 7*b5 - b4 + 3*b3 + 7*b2 + 5*b1 + 5) * q^41 + (b6 - 11*b5 - 6*b4 + 10*b3 + 11*b2 - 29*b1 + 29) * q^43 + (-3*b7 - 3*b4 + 3) * q^45 + (-5*b7 - 7*b6 + 11*b5 - 7*b3 + 4*b2 + 17*b1) * q^47 + (3*b7 - 8*b6 + b5 + 3*b4 + b3 + 8*b2 - 15) * q^49 + (-b7 - 7*b6 + 5*b5 - 2*b4 + 7*b3 + 2*b2 - 5*b1 + 10) * q^51 + (-2*b7 - 5*b6 + 3*b5 - 4*b4 + 5*b3 + 2*b2 + 14*b1 - 28) * q^53 + (-3*b6 + 10*b5 - 7*b3 - 10*b2 + 58*b1 - 58) * q^55 + (-b6 + 8*b5 - 9*b1 + 15) * q^57 + (2*b7 - 13*b6 - 4*b5 + b4 - 17*b3 - 4*b2 + 13*b1 + 13) * q^59 + (3*b7 - 5*b6 - 5*b3 - 5*b2 + 18*b1) * q^61 + (-3*b6 + 3*b5 - 3*b3 + 9*b1) * q^63 + (-2*b7 + 17*b6 - 4*b5 + 2*b4 + 4*b3 + 17*b2 + 84*b1 - 42) * q^65 + (4*b7 - b6 - 5*b5 + 8*b4 + b3 + 6*b2 - 9*b1 + 18) * q^67 + (-4*b6 + 5*b5 - 5*b3 - 4*b2 + 20*b1 - 10) * q^69 + (2*b7 + 2*b6 + 3*b5 + b4 + 5*b3 + 3*b2 - 37*b1 - 37) * q^71 + (8*b6 - 10*b5 - 12*b4 + 2*b3 + 10*b2 + 17*b1 - 17) * q^73 + (-3*b7 + 4*b6 + b5 + 3*b4 - b3 + 4*b2 + 44*b1 - 22) * q^75 + (5*b7 - 14*b6 + 4*b5 + 5*b4 + 4*b3 + 14*b2 + 37) * q^77 + (3*b6 + 5*b5 + 8*b3 + 5*b2 - 35*b1 - 35) * q^79 + (9*b1 - 9) * q^81 + (3*b7 + 5*b6 + 6*b5 + 3*b4 + 6*b3 - 5*b2 - 53) * q^83 + (-12*b7 + 40*b6 - 12*b5 + 40*b3 + 28*b2 + 10*b1) * q^85 + (4*b6 - 2*b5 - 2*b3 - 4*b2 + 12) * q^87 + (-3*b7 - 16*b6 + 4*b5 - 6*b4 + 16*b3 + 12*b2 + 35*b1 - 70) * q^89 + (2*b7 - 7*b6 - 10*b5 + 4*b4 + 7*b3 + 17*b2 - 19*b1 + 38) * q^91 + (-9*b6 + 9*b5 - 9*b2 - 21*b1 + 21) * q^93 + (-11*b7 + 4*b6 - 13*b5 + b4 + 4*b3 + 32*b2 - 2*b1 + 35) * q^95 + (6*b7 + 7*b6 - 7*b5 + 3*b4 - 7*b2 + 13*b1 + 13) * q^97 + (3*b7 + 3*b5 + 3*b2 + 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 12 q^{3} + 4 q^{5} + 24 q^{7} + 12 q^{9}+O(q^{10})$$ 8 * q - 12 * q^3 + 4 * q^5 + 24 * q^7 + 12 * q^9 $$8 q - 12 q^{3} + 4 q^{5} + 24 q^{7} + 12 q^{9} + 8 q^{11} + 24 q^{13} - 12 q^{15} - 20 q^{17} - 24 q^{19} - 36 q^{21} - 40 q^{23} - 88 q^{25} - 48 q^{29} - 12 q^{33} - 32 q^{35} - 48 q^{39} + 60 q^{41} + 116 q^{43} + 24 q^{45} + 68 q^{47} - 120 q^{49} + 60 q^{51} - 168 q^{53} - 232 q^{55} + 84 q^{57} + 156 q^{59} + 72 q^{61} + 36 q^{63} + 108 q^{67} - 444 q^{71} - 68 q^{73} + 296 q^{77} - 420 q^{79} - 36 q^{81} - 424 q^{83} + 40 q^{85} + 96 q^{87} - 420 q^{89} + 228 q^{91} + 84 q^{93} + 272 q^{95} + 156 q^{97} + 12 q^{99}+O(q^{100})$$ 8 * q - 12 * q^3 + 4 * q^5 + 24 * q^7 + 12 * q^9 + 8 * q^11 + 24 * q^13 - 12 * q^15 - 20 * q^17 - 24 * q^19 - 36 * q^21 - 40 * q^23 - 88 * q^25 - 48 * q^29 - 12 * q^33 - 32 * q^35 - 48 * q^39 + 60 * q^41 + 116 * q^43 + 24 * q^45 + 68 * q^47 - 120 * q^49 + 60 * q^51 - 168 * q^53 - 232 * q^55 + 84 * q^57 + 156 * q^59 + 72 * q^61 + 36 * q^63 + 108 * q^67 - 444 * q^71 - 68 * q^73 + 296 * q^77 - 420 * q^79 - 36 * q^81 - 424 * q^83 + 40 * q^85 + 96 * q^87 - 420 * q^89 + 228 * q^91 + 84 * q^93 + 272 * q^95 + 156 * q^97 + 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 44x^{6} + 664x^{4} - 3528x^{2} + 8100$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} + 44\nu^{5} - 574\nu^{3} + 1548\nu + 1080 ) / 2160$$ (-v^7 + 44*v^5 - 574*v^3 + 1548*v + 1080) / 2160 $$\beta_{2}$$ $$=$$ $$( -7\nu^{7} - 15\nu^{6} + 263\nu^{5} - 105\nu^{4} - 3208\nu^{3} + 9840\nu^{2} + 1566\nu - 61830 ) / 14040$$ (-7*v^7 - 15*v^6 + 263*v^5 - 105*v^4 - 3208*v^3 + 9840*v^2 + 1566*v - 61830) / 14040 $$\beta_{3}$$ $$=$$ $$( -7\nu^{7} + 15\nu^{6} + 263\nu^{5} + 105\nu^{4} - 3208\nu^{3} - 9840\nu^{2} + 1566\nu + 61830 ) / 14040$$ (-7*v^7 + 15*v^6 + 263*v^5 + 105*v^4 - 3208*v^3 - 9840*v^2 + 1566*v + 61830) / 14040 $$\beta_{4}$$ $$=$$ $$( -11\nu^{7} + 270\nu^{6} + 664\nu^{5} - 9810\nu^{4} - 14234\nu^{3} + 122400\nu^{2} + 86868\nu - 403380 ) / 28080$$ (-11*v^7 + 270*v^6 + 664*v^5 - 9810*v^4 - 14234*v^3 + 122400*v^2 + 86868*v - 403380) / 28080 $$\beta_{5}$$ $$=$$ $$( 14\nu^{7} + 30\nu^{6} - 526\nu^{5} - 1545\nu^{4} + 6416\nu^{3} + 18930\nu^{2} - 24192\nu - 34290 ) / 14040$$ (14*v^7 + 30*v^6 - 526*v^5 - 1545*v^4 + 6416*v^3 + 18930*v^2 - 24192*v - 34290) / 14040 $$\beta_{6}$$ $$=$$ $$( -14\nu^{7} + 30\nu^{6} + 526\nu^{5} - 1545\nu^{4} - 6416\nu^{3} + 18930\nu^{2} + 24192\nu - 34290 ) / 14040$$ (-14*v^7 + 30*v^6 + 526*v^5 - 1545*v^4 - 6416*v^3 + 18930*v^2 + 24192*v - 34290) / 14040 $$\beta_{7}$$ $$=$$ $$( 31\nu^{7} - 300\nu^{6} - 914\nu^{5} + 10770\nu^{4} + 5014\nu^{3} - 128460\nu^{2} + 35352\nu + 385020 ) / 28080$$ (31*v^7 - 300*v^6 - 914*v^5 + 10770*v^4 + 5014*v^3 - 128460*v^2 + 35352*v + 385020) / 28080
 $$\nu$$ $$=$$ $$( \beta_{6} - \beta_{5} - 2\beta_{3} - 2\beta_{2} ) / 3$$ (b6 - b5 - 2*b3 - 2*b2) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{7} - 4\beta_{6} - 3\beta_{5} + \beta_{4} - 3\beta_{3} + 2\beta_{2} + 33 ) / 3$$ (-b7 - 4*b6 - 3*b5 + b4 - 3*b3 + 2*b2 + 33) / 3 $$\nu^{3}$$ $$=$$ $$( -9\beta_{7} + \beta_{6} - 4\beta_{5} - 9\beta_{4} - 38\beta_{3} - 35\beta_{2} + 54\beta _1 - 27 ) / 3$$ (-9*b7 + b6 - 4*b5 - 9*b4 - 38*b3 - 35*b2 + 54*b1 - 27) / 3 $$\nu^{4}$$ $$=$$ $$( -22\beta_{7} - 100\beta_{6} - 78\beta_{5} + 22\beta_{4} - 42\beta_{3} + 20\beta_{2} + 456 ) / 3$$ (-22*b7 - 100*b6 - 78*b5 + 22*b4 - 42*b3 + 20*b2 + 456) / 3 $$\nu^{5}$$ $$=$$ $$( -162\beta_{7} - 188\beta_{6} + 134\beta_{5} - 162\beta_{4} - 740\beta_{3} - 686\beta_{2} + 1980\beta _1 - 990 ) / 3$$ (-162*b7 - 188*b6 + 134*b5 - 162*b4 - 740*b3 - 686*b2 + 1980*b1 - 990) / 3 $$\nu^{6}$$ $$=$$ $$( -502\beta_{7} - 1924\beta_{6} - 1422\beta_{5} + 502\beta_{4} - 270\beta_{3} - 232\beta_{2} + 6090 ) / 3$$ (-502*b7 - 1924*b6 - 1422*b5 + 502*b4 - 270*b3 - 232*b2 + 6090) / 3 $$\nu^{7}$$ $$=$$ $$( - 1962 \beta_{7} - 7298 \beta_{6} + 6644 \beta_{5} - 1962 \beta_{4} - 13844 \beta_{3} - 13190 \beta_{2} + 49644 \beta _1 - 24822 ) / 3$$ (-1962*b7 - 7298*b6 + 6644*b5 - 1962*b4 - 13844*b3 - 13190*b2 + 49644*b1 - 24822) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$\beta_{1}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 −4.34148 − 0.707107i 2.03753 + 0.707107i 4.34148 − 0.707107i −2.03753 + 0.707107i −4.34148 + 0.707107i 2.03753 − 0.707107i 4.34148 + 0.707107i −2.03753 − 0.707107i
0 −1.50000 + 0.866025i 0 −3.25884 5.64448i 0 11.0157 0 1.50000 2.59808i 0
145.2 0 −1.50000 + 0.866025i 0 −2.40185 4.16012i 0 −2.71177 0 1.50000 2.59808i 0
145.3 0 −1.50000 + 0.866025i 0 3.03409 + 5.25521i 0 2.33275 0 1.50000 2.59808i 0
145.4 0 −1.50000 + 0.866025i 0 4.62659 + 8.01349i 0 1.36330 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 −3.25884 + 5.64448i 0 11.0157 0 1.50000 + 2.59808i 0
673.2 0 −1.50000 0.866025i 0 −2.40185 + 4.16012i 0 −2.71177 0 1.50000 + 2.59808i 0
673.3 0 −1.50000 0.866025i 0 3.03409 5.25521i 0 2.33275 0 1.50000 + 2.59808i 0
673.4 0 −1.50000 0.866025i 0 4.62659 8.01349i 0 1.36330 0 1.50000 + 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 673.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.be.g 8
4.b odd 2 1 114.3.f.b 8
12.b even 2 1 342.3.m.b 8
19.d odd 6 1 inner 912.3.be.g 8
76.f even 6 1 114.3.f.b 8
228.n odd 6 1 342.3.m.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.3.f.b 8 4.b odd 2 1
114.3.f.b 8 76.f even 6 1
342.3.m.b 8 12.b even 2 1
342.3.m.b 8 228.n odd 6 1
912.3.be.g 8 1.a even 1 1 trivial
912.3.be.g 8 19.d odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{8} - 4T_{5}^{7} + 102T_{5}^{6} + 32T_{5}^{5} + 6262T_{5}^{4} + 648T_{5}^{3} + 175524T_{5}^{2} + 274248T_{5} + 3090564$$ T5^8 - 4*T5^7 + 102*T5^6 + 32*T5^5 + 6262*T5^4 + 648*T5^3 + 175524*T5^2 + 274248*T5 + 3090564 $$T_{7}^{4} - 12T_{7}^{3} + 4T_{7}^{2} + 84T_{7} - 95$$ T7^4 - 12*T7^3 + 4*T7^2 + 84*T7 - 95

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} + 3 T + 3)^{4}$$
$5$ $$T^{8} - 4 T^{7} + 102 T^{6} + \cdots + 3090564$$
$7$ $$(T^{4} - 12 T^{3} + 4 T^{2} + 84 T - 95)^{2}$$
$11$ $$(T^{4} - 4 T^{3} - 230 T^{2} - 60 T + 1902)^{2}$$
$13$ $$T^{8} - 24 T^{7} + \cdots + 845588241$$
$17$ $$T^{8} + 20 T^{7} + \cdots + 42571243584$$
$19$ $$T^{8} + 24 T^{7} + \cdots + 16983563041$$
$23$ $$T^{8} + 40 T^{7} + 1266 T^{6} + \cdots + 1726596$$
$29$ $$T^{8} + 48 T^{7} + \cdots + 269485056$$
$31$ $$T^{8} + 3288 T^{6} + \cdots + 157608206001$$
$37$ $$T^{8} + 3084 T^{6} + \cdots + 3853057329$$
$41$ $$T^{8} - 60 T^{7} + \cdots + 4588472453184$$
$43$ $$T^{8} + \cdots + 196975358413681$$
$47$ $$T^{8} - 68 T^{7} + \cdots + 791744040000$$
$53$ $$T^{8} + 168 T^{7} + \cdots + 984988761156$$
$59$ $$T^{8} - 156 T^{7} + \cdots + 76622956930116$$
$61$ $$T^{8} - 72 T^{7} + \cdots + 2354429392225$$
$67$ $$T^{8} - 108 T^{7} + \cdots + 421690488129$$
$71$ $$T^{8} + \cdots + 158714643240000$$
$73$ $$T^{8} + 68 T^{7} + \cdots + 14\!\cdots\!21$$
$79$ $$T^{8} + 420 T^{7} + \cdots + 36966995842401$$
$83$ $$(T^{4} + 212 T^{3} + 13252 T^{2} + \cdots - 740232)^{2}$$
$89$ $$T^{8} + \cdots + 341891268051204$$
$97$ $$T^{8} - 156 T^{7} + \cdots + 312852286224$$