# Properties

 Label 912.2.ci.c Level $912$ Weight $2$ Character orbit 912.ci Analytic conductor $7.282$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.ci (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{18})$$ Coefficient field: 12.0.101559956668416.1 Defining polynomial: $$x^{12} - 8x^{6} + 64$$ x^12 - 8*x^6 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q - \beta_{8} q^{3} + ( - \beta_{10} - \beta_{9} - \beta_{6} + \beta_{4} + \beta_1 + 1) q^{5} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 1) q^{7} + (\beta_{10} - \beta_{4}) q^{9}+O(q^{10})$$ q - b8 * q^3 + (-b10 - b9 - b6 + b4 + b1 + 1) * q^5 + (b11 - b10 + b8 - b7 - b6 - b5 + b4 - 1) * q^7 + (b10 - b4) * q^9 $$q - \beta_{8} q^{3} + ( - \beta_{10} - \beta_{9} - \beta_{6} + \beta_{4} + \beta_1 + 1) q^{5} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 1) q^{7} + (\beta_{10} - \beta_{4}) q^{9} + ( - \beta_{11} + 2 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_1) q^{11} + ( - \beta_{11} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2} + 1) q^{13} + (\beta_{11} - \beta_{9} - \beta_{6} - \beta_{5} - \beta_{2}) q^{15} + (\beta_{11} - \beta_{10} + \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{17} + ( - \beta_{9} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{2} + \beta_1) q^{19} + ( - \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2}) q^{21} + (2 \beta_{11} + 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{2} + \cdots + 2) q^{23}+ \cdots + (\beta_{11} + \beta_{9} - 2 \beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{99}+O(q^{100})$$ q - b8 * q^3 + (-b10 - b9 - b6 + b4 + b1 + 1) * q^5 + (b11 - b10 + b8 - b7 - b6 - b5 + b4 - 1) * q^7 + (b10 - b4) * q^9 + (-b11 + 2*b9 - b8 - b7 + b5 + b4 - 2*b3 + b1) * q^11 + (-b11 - b8 + b6 + b5 + b3 + 2*b2 + 1) * q^13 + (b11 - b9 - b6 - b5 - b2) * q^15 + (b11 - b10 + b6 - 2*b5 + b4 - b3 - b1) * q^17 + (-b9 - 2*b8 - b7 - 2*b6 + 2*b5 + b2 + b1) * q^19 + (-b10 + b9 + 2*b8 + b7 - b6 + b4 - b3 - b2) * q^21 + (2*b11 + 2*b9 - 2*b8 - 2*b7 - b6 - 2*b5 + b2 + 2*b1 + 2) * q^23 + (-2*b11 - b10 - b8 - 4*b7 - b6 + 2*b5 - 2*b3 + 3*b2 + 2*b1 - 2) * q^25 + b6 * q^27 + (-b11 + 2*b10 - b9 - b8 - b7 - b6 - b5 + 2*b4 - b3) * q^29 + (2*b11 - 2*b10 + 4*b9 - 2*b8 + 2*b7 + 3*b6 - b5 + b4 - 2*b3 + b2 - b1 - 3) * q^31 + (b10 - b7 - b6 + 2*b5 - b4 - b3 + 1) * q^33 + (-b11 + 2*b10 + 4*b9 - b8 - 2*b7 + 2*b6 + b5 - 4*b4 - 2*b3 + 2*b2 - 2*b1 - 4) * q^35 + (2*b11 + 2*b10 - b9 + 2*b7 - 2*b6 - 2*b4 + 2*b2 + 1) * q^37 + (-b11 - b10 - 2*b8 - b7 - b4 + b2) * q^39 + (2*b11 - 2*b10 - b8 - b7 + 3*b6 - 2*b5 - b4 + b3 + 2*b2 + b1 - 1) * q^41 + (b10 + 3*b9 - b8 - 3*b7 - b6 - b4 - 3*b3 - b2 + 3*b1 + 1) * q^43 + (b11 + b10 + b8 + b7 - b5 - b2) * q^45 + (-2*b11 + b10 - b8 - b6 + b4 - 2*b3) * q^47 + (-2*b11 + b10 - 3*b8 + 2*b7 + 2*b6 + 2*b5 - 3*b4 + b2 - 2*b1) * q^49 + (b11 + b9 - b8 + 2*b7 - b6 + b2 - b1) * q^51 + (3*b11 - 2*b10 + 3*b9 + 2*b8 - 2*b6 - 2*b5 + b4 - b3 + 2*b2 - 2) * q^53 + (-b11 + 3*b10 - 4*b8 + 3*b6 - 5*b4 + b3 + 2*b2 - b1 + 2) * q^55 + (b11 + b10 + 2*b8 - 2*b7 - b5 - 2*b4 - b3 - 2*b2 + 2*b1) * q^57 + (-b10 - b9 + b7 + b6 + b4 + b3) * q^59 + (b8 + 3*b7 - 3*b6 - 3*b5 - b4 - 3*b3 - 3*b2 + 3*b1 + 1) * q^61 + (-b10 - b9 + b8 - b6 + b5 + 2*b4 + b3 - b2) * q^63 + (-2*b11 - b10 - 5*b9 - 4*b8 + b7 - 4*b6 - b5 + 4*b4 + 5*b3 + b2 + 2*b1 + 8) * q^65 + (b11 - 4*b10 + 3*b9 + b8 + b7 - b6 - 3*b5 + 4*b4 - b3 - 2*b1) * q^67 + (-2*b11 + b10 - b8 + 2*b7 + 2*b5 - 2*b4 - 2*b3 - b2) * q^69 + (b10 - b8 - 7*b6 + 7*b4 + b2 - 1) * q^71 + (b10 + 2*b9 + 5*b8 + 4*b6 + 3*b4 - 2*b3 + 2*b1 - 3) * q^73 + (2*b11 - 2*b10 + 2*b9 + 3*b8 - 2*b7 - b4 - 4*b3 - b2 - 1) * q^75 + (-b11 - 2*b9 - 2*b8 + 5*b7 - 4*b5 + 2*b4 + 4*b3 + 2*b2 - 4*b1 - 3) * q^77 + (2*b11 - 2*b9 - 3*b8 - 3*b7 - b6 + 2*b5 - b4 - b3 + b1 + 1) * q^79 + (-b8 + b2) * q^81 + (-4*b11 - b10 + b8 + 4*b7 - b6 + 3*b5 + 2*b4 + 5*b3 + b2 - b1 - 1) * q^83 + (2*b11 + b10 + b9 + 3*b8 + b7 - b5 - b4 - 2*b3 - 3*b2 - 2*b1 - 3) * q^85 + (2*b11 + b10 + b9 + b8 + b7 - 2*b6 - b5 - b4 - b3 - b2 - 2*b1 + 4) * q^87 + (-b11 - 5*b10 - 2*b8 + 2*b6 + b5 + 10*b4 + b3 + 3*b2 + 1) * q^89 + (3*b11 - 3*b9 + 2*b8 - 6*b6 - 4*b5 + b4 - b3 - 6*b2 + 2) * q^91 + (-2*b11 + b10 - b9 + b7 - b6 + 4*b5 - 2*b4 + 2*b3 + 3*b2 + b1 - 1) * q^93 + (-2*b11 - 2*b10 - 7*b8 - 4*b7 + 2*b6 - 4*b4 + 5*b2 - 3) * q^95 + (6*b11 - 2*b10 - 2*b9 + 4*b8 - 2*b7 - 2*b6 - b4 - 2*b2 + 2*b1 + 3) * q^97 + (b11 + b9 - 2*b7 + b6 - b3 - b2 + 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{5} - 18 q^{7}+O(q^{10})$$ 12 * q + 6 * q^5 - 18 * q^7 $$12 q + 6 q^{5} - 18 q^{7} + 18 q^{13} - 6 q^{15} + 6 q^{17} - 12 q^{19} - 6 q^{21} + 18 q^{23} - 30 q^{25} + 6 q^{27} - 6 q^{29} - 18 q^{31} + 6 q^{33} - 36 q^{35} + 6 q^{41} + 6 q^{43} - 6 q^{47} + 12 q^{49} - 6 q^{51} - 36 q^{53} + 42 q^{55} + 6 q^{59} - 6 q^{61} - 6 q^{63} + 72 q^{65} - 6 q^{67} - 54 q^{71} - 12 q^{73} - 12 q^{75} - 36 q^{77} + 6 q^{79} - 18 q^{83} - 36 q^{85} + 36 q^{87} + 24 q^{89} - 12 q^{91} - 18 q^{93} - 24 q^{95} + 24 q^{97} + 6 q^{99}+O(q^{100})$$ 12 * q + 6 * q^5 - 18 * q^7 + 18 * q^13 - 6 * q^15 + 6 * q^17 - 12 * q^19 - 6 * q^21 + 18 * q^23 - 30 * q^25 + 6 * q^27 - 6 * q^29 - 18 * q^31 + 6 * q^33 - 36 * q^35 + 6 * q^41 + 6 * q^43 - 6 * q^47 + 12 * q^49 - 6 * q^51 - 36 * q^53 + 42 * q^55 + 6 * q^59 - 6 * q^61 - 6 * q^63 + 72 * q^65 - 6 * q^67 - 54 * q^71 - 12 * q^73 - 12 * q^75 - 36 * q^77 + 6 * q^79 - 18 * q^83 - 36 * q^85 + 36 * q^87 + 24 * q^89 - 12 * q^91 - 18 * q^93 - 24 * q^95 + 24 * q^97 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 8x^{6} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{4} ) / 4$$ (v^4) / 4 $$\beta_{5}$$ $$=$$ $$( \nu^{5} ) / 4$$ (v^5) / 4 $$\beta_{6}$$ $$=$$ $$( \nu^{6} ) / 8$$ (v^6) / 8 $$\beta_{7}$$ $$=$$ $$( \nu^{7} ) / 8$$ (v^7) / 8 $$\beta_{8}$$ $$=$$ $$( \nu^{8} ) / 16$$ (v^8) / 16 $$\beta_{9}$$ $$=$$ $$( \nu^{9} ) / 16$$ (v^9) / 16 $$\beta_{10}$$ $$=$$ $$( \nu^{10} ) / 32$$ (v^10) / 32 $$\beta_{11}$$ $$=$$ $$( \nu^{11} ) / 32$$ (v^11) / 32
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3 $$\nu^{4}$$ $$=$$ $$4\beta_{4}$$ 4*b4 $$\nu^{5}$$ $$=$$ $$4\beta_{5}$$ 4*b5 $$\nu^{6}$$ $$=$$ $$8\beta_{6}$$ 8*b6 $$\nu^{7}$$ $$=$$ $$8\beta_{7}$$ 8*b7 $$\nu^{8}$$ $$=$$ $$16\beta_{8}$$ 16*b8 $$\nu^{9}$$ $$=$$ $$16\beta_{9}$$ 16*b9 $$\nu^{10}$$ $$=$$ $$32\beta_{10}$$ 32*b10 $$\nu^{11}$$ $$=$$ $$32\beta_{11}$$ 32*b11

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$-\beta_{4}$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.483690 + 1.32893i 0.483690 − 1.32893i −0.483690 − 1.32893i 0.483690 + 1.32893i −1.39273 + 0.245576i 1.39273 − 0.245576i −1.39273 − 0.245576i 1.39273 + 0.245576i −0.909039 + 1.08335i 0.909039 − 1.08335i −0.909039 − 1.08335i 0.909039 + 1.08335i
0 0.939693 0.342020i 0 −0.749734 + 4.25195i 0 −3.63109 + 2.09641i 0 0.766044 0.642788i 0
79.2 0 0.939693 0.342020i 0 0.217645 1.23433i 0 −2.78039 + 1.60526i 0 0.766044 0.642788i 0
127.1 0 0.939693 + 0.342020i 0 −0.749734 4.25195i 0 −3.63109 2.09641i 0 0.766044 + 0.642788i 0
127.2 0 0.939693 + 0.342020i 0 0.217645 + 1.23433i 0 −2.78039 1.60526i 0 0.766044 + 0.642788i 0
223.1 0 −0.173648 + 0.984808i 0 0.0469641 + 0.0394076i 0 1.48976 0.860113i 0 −0.939693 0.342020i 0
223.2 0 −0.173648 + 0.984808i 0 2.83242 + 2.37668i 0 −2.26308 + 1.30659i 0 −0.939693 0.342020i 0
319.1 0 −0.173648 0.984808i 0 0.0469641 0.0394076i 0 1.48976 + 0.860113i 0 −0.939693 + 0.342020i 0
319.2 0 −0.173648 0.984808i 0 2.83242 2.37668i 0 −2.26308 1.30659i 0 −0.939693 + 0.342020i 0
751.1 0 −0.766044 + 0.642788i 0 −0.582687 0.212081i 0 1.39416 + 0.804921i 0 0.173648 0.984808i 0
751.2 0 −0.766044 + 0.642788i 0 1.23539 + 0.449645i 0 −3.20937 1.85293i 0 0.173648 0.984808i 0
895.1 0 −0.766044 0.642788i 0 −0.582687 + 0.212081i 0 1.39416 0.804921i 0 0.173648 + 0.984808i 0
895.2 0 −0.766044 0.642788i 0 1.23539 0.449645i 0 −3.20937 + 1.85293i 0 0.173648 + 0.984808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 895.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.k even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.ci.c 12
4.b odd 2 1 912.2.ci.d yes 12
19.f odd 18 1 912.2.ci.d yes 12
76.k even 18 1 inner 912.2.ci.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.ci.c 12 1.a even 1 1 trivial
912.2.ci.c 12 76.k even 18 1 inner
912.2.ci.d yes 12 4.b odd 2 1
912.2.ci.d yes 12 19.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{12} - 6 T_{5}^{11} + 33 T_{5}^{10} - 136 T_{5}^{9} + 450 T_{5}^{8} - 612 T_{5}^{7} + 432 T_{5}^{6} - 90 T_{5}^{5} - 342 T_{5}^{4} + 296 T_{5}^{3} + 240 T_{5}^{2} - 24 T_{5} + 1$$ T5^12 - 6*T5^11 + 33*T5^10 - 136*T5^9 + 450*T5^8 - 612*T5^7 + 432*T5^6 - 90*T5^5 - 342*T5^4 + 296*T5^3 + 240*T5^2 - 24*T5 + 1 $$T_{7}^{12} + 18 T_{7}^{11} + 135 T_{7}^{10} + 486 T_{7}^{9} + 531 T_{7}^{8} - 1782 T_{7}^{7} - 4616 T_{7}^{6} + 6570 T_{7}^{5} + 29457 T_{7}^{4} - 71478 T_{7}^{2} + 130321$$ T7^12 + 18*T7^11 + 135*T7^10 + 486*T7^9 + 531*T7^8 - 1782*T7^7 - 4616*T7^6 + 6570*T7^5 + 29457*T7^4 - 71478*T7^2 + 130321

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$(T^{6} - T^{3} + 1)^{2}$$
$5$ $$T^{12} - 6 T^{11} + 33 T^{10} - 136 T^{9} + \cdots + 1$$
$7$ $$T^{12} + 18 T^{11} + 135 T^{10} + \cdots + 130321$$
$11$ $$T^{12} - 42 T^{10} + 1515 T^{8} + \cdots + 5329$$
$13$ $$T^{12} - 18 T^{11} + 147 T^{10} - 756 T^{9} + \cdots + 1$$
$17$ $$T^{12} - 6 T^{11} - 21 T^{10} + \cdots + 26569$$
$19$ $$T^{12} + 12 T^{11} + 48 T^{10} + \cdots + 47045881$$
$23$ $$T^{12} - 18 T^{11} + 249 T^{10} + \cdots + 3988009$$
$29$ $$T^{12} + 6 T^{11} + 39 T^{10} + \cdots + 151117849$$
$31$ $$T^{12} + 18 T^{11} + \cdots + 254625849$$
$37$ $$T^{12} + 174 T^{10} + \cdots + 166126321$$
$41$ $$T^{12} - 6 T^{11} + 93 T^{10} + \cdots + 4422609$$
$43$ $$T^{12} - 6 T^{11} - 51 T^{10} + \cdots + 71014329$$
$47$ $$T^{12} + 6 T^{11} - 21 T^{10} + \cdots + 3052009$$
$53$ $$T^{12} + 36 T^{11} + \cdots + 4583154601$$
$59$ $$T^{12} - 6 T^{11} + 15 T^{10} - 28 T^{9} + \cdots + 361$$
$61$ $$T^{12} + 6 T^{11} + \cdots + 129665528281$$
$67$ $$T^{12} + 6 T^{11} - 45 T^{10} + \cdots + 34023889$$
$71$ $$(T^{6} + 27 T^{5} + 369 T^{4} + \cdots + 104329)^{2}$$
$73$ $$T^{12} + 12 T^{11} + \cdots + 749390625$$
$79$ $$T^{12} - 6 T^{11} + \cdots + 4267094329$$
$83$ $$T^{12} + 18 T^{11} + \cdots + 5079555441$$
$89$ $$T^{12} - 24 T^{11} + \cdots + 21849865489$$
$97$ $$T^{12} - 24 T^{11} + \cdots + 1138995001$$