# Properties

 Label 912.3.be.d Level $912$ Weight $3$ Character orbit 912.be Analytic conductor $24.850$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

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## 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.6967728.1 Defining polynomial: $$x^{6} - x^{5} + 8 x^{4} + 5 x^{3} + 50 x^{2} - 7 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 57) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 - \beta_{3} ) q^{3} + ( -\beta_{2} + \beta_{3} ) q^{5} + ( -4 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{7} + ( 3 + 3 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -2 - \beta_{3} ) q^{3} + ( -\beta_{2} + \beta_{3} ) q^{5} + ( -4 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{7} + ( 3 + 3 \beta_{3} ) q^{9} + ( -1 + 3 \beta_{1} ) q^{11} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{13} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{15} + ( -2 + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{17} + ( -2 + 3 \beta_{1} + 2 \beta_{2} - 16 \beta_{3} - 2 \beta_{5} ) q^{19} + ( 7 + 2 \beta_{1} + \beta_{2} + 5 \beta_{3} - 3 \beta_{5} ) q^{21} + ( 11 - 3 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{23} + ( 7 + 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{25} + ( -3 - 6 \beta_{3} ) q^{27} + ( 8 + 2 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} - 4 \beta_{4} ) q^{29} + ( -2 - 3 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{31} + ( 2 - 6 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{33} + ( 9 \beta_{2} - 5 \beta_{3} ) q^{35} + ( -3 - 24 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} ) q^{37} + ( 7 - 6 \beta_{1} + \beta_{4} + \beta_{5} ) q^{39} + ( 12 - 12 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} + 8 \beta_{5} ) q^{41} + ( -\beta_{2} + 4 \beta_{3} ) q^{43} + ( -3 + 3 \beta_{1} ) q^{45} + ( 4 - 4 \beta_{1} - 4 \beta_{2} + 16 \beta_{3} + 6 \beta_{4} - 12 \beta_{5} ) q^{47} + ( 14 + 6 \beta_{1} - 4 \beta_{4} - 4 \beta_{5} ) q^{49} + ( 4 + 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 6 \beta_{4} ) q^{51} + ( 17 - 3 \beta_{1} + 3 \beta_{2} - 17 \beta_{3} + 12 \beta_{4} ) q^{53} + ( 6 - 2 \beta_{2} - 64 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} ) q^{55} + ( -12 - 4 \beta_{1} - 5 \beta_{2} + 16 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{57} + ( -28 - 10 \beta_{1} - 5 \beta_{2} - 21 \beta_{3} + 14 \beta_{5} ) q^{59} + ( -48 - 12 \beta_{1} - 12 \beta_{2} - 38 \beta_{3} + 5 \beta_{4} - 10 \beta_{5} ) q^{61} + ( -9 - 3 \beta_{1} - 3 \beta_{2} - 15 \beta_{3} - 3 \beta_{4} + 6 \beta_{5} ) q^{63} + ( -27 - \beta_{1} - 2 \beta_{2} - 74 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} ) q^{65} + ( 4 + \beta_{1} - \beta_{2} - 4 \beta_{3} + 8 \beta_{4} ) q^{67} + ( -13 + 3 \beta_{1} + 6 \beta_{2} - 14 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{69} + ( 86 - 4 \beta_{1} - 2 \beta_{2} + 40 \beta_{3} + 6 \beta_{5} ) q^{71} + ( -10 + 20 \beta_{2} - 31 \beta_{3} + 20 \beta_{4} - 10 \beta_{5} ) q^{73} + ( -9 - 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{75} + ( 7 - 29 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{77} + ( 29 + 2 \beta_{1} + \beta_{2} + 11 \beta_{3} + 7 \beta_{5} ) q^{79} + 9 \beta_{3} q^{81} + ( 42 - 2 \beta_{4} - 2 \beta_{5} ) q^{83} + ( 30 + 14 \beta_{1} + 14 \beta_{2} + 38 \beta_{3} + 4 \beta_{4} - 8 \beta_{5} ) q^{85} + ( -20 - 6 \beta_{1} + 4 \beta_{4} + 4 \beta_{5} ) q^{87} + ( 71 + \beta_{1} - \beta_{2} - 71 \beta_{3} ) q^{89} + ( -4 - 15 \beta_{1} + 15 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{91} + ( 1 + 9 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{93} + ( 42 - 12 \beta_{1} - 19 \beta_{2} + 9 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{95} + ( 14 - 16 \beta_{1} - 8 \beta_{2} - 10 \beta_{3} + 34 \beta_{5} ) q^{97} + ( -3 + 9 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 9 q^{3} - 2 q^{5} - 26 q^{7} + 9 q^{9} + O(q^{10})$$ $$6 q - 9 q^{3} - 2 q^{5} - 26 q^{7} + 9 q^{9} - 15 q^{13} + 6 q^{15} - 10 q^{17} + 46 q^{19} + 39 q^{21} + 24 q^{23} + 15 q^{25} + 66 q^{29} + 6 q^{35} + 30 q^{39} + 24 q^{41} - 11 q^{43} - 12 q^{45} + 26 q^{47} + 96 q^{49} + 30 q^{51} + 180 q^{53} + 176 q^{55} - 141 q^{57} - 162 q^{59} - 141 q^{61} - 39 q^{63} + 63 q^{67} + 372 q^{71} + 103 q^{73} - 16 q^{77} + 123 q^{79} - 27 q^{81} + 252 q^{83} + 116 q^{85} - 132 q^{87} + 642 q^{89} - 87 q^{91} - 21 q^{93} + 214 q^{95} - 12 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 8 x^{4} + 5 x^{3} + 50 x^{2} - 7 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{5} - 16 \nu^{4} + 128 \nu^{3} - 100 \nu^{2} + 14 \nu + 281$$$$)/393$$ $$\beta_{2}$$ $$=$$ $$($$$$-56 \nu^{5} + 55 \nu^{4} - 440 \nu^{3} - 344 \nu^{2} - 1964 \nu - 8$$$$)/393$$ $$\beta_{3}$$ $$=$$ $$($$$$-56 \nu^{5} + 55 \nu^{4} - 440 \nu^{3} - 344 \nu^{2} - 2750 \nu - 8$$$$)/393$$ $$\beta_{4}$$ $$=$$ $$($$$$196 \nu^{5} - 127 \nu^{4} + 1540 \nu^{3} + 1466 \nu^{2} + 10018 \nu + 1469$$$$)/393$$ $$\beta_{5}$$ $$=$$ $$($$$$-210 \nu^{5} + 239 \nu^{4} - 1650 \nu^{3} - 766 \nu^{2} - 10116 \nu + 2459$$$$)/393$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{5} - \beta_{4} - 11 \beta_{3} - 9$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + 7 \beta_{1} - 15$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-4 \beta_{5} + 8 \beta_{4} + 46 \beta_{3} - 3 \beta_{2} - 4$$ $$\nu^{5}$$ $$=$$ $$($$$$-28 \beta_{5} + 14 \beta_{4} + 193 \beta_{3} - 55 \beta_{2} - 55 \beta_{1} + 165$$$$)/2$$

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$1 + \beta_{3}$$ $$1$$ $$1$$ $$1$$

## 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
 1.56632 + 2.71294i 0.0702177 + 0.121621i −1.13654 − 1.96854i 1.56632 − 2.71294i 0.0702177 − 0.121621i −1.13654 + 1.96854i
0 −1.50000 + 0.866025i 0 −3.13264 5.42589i 0 −8.36156 0 1.50000 2.59808i 0
145.2 0 −1.50000 + 0.866025i 0 −0.140435 0.243241i 0 5.24143 0 1.50000 2.59808i 0
145.3 0 −1.50000 + 0.866025i 0 2.27307 + 3.93708i 0 −9.87987 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 −3.13264 + 5.42589i 0 −8.36156 0 1.50000 + 2.59808i 0
673.2 0 −1.50000 0.866025i 0 −0.140435 + 0.243241i 0 5.24143 0 1.50000 + 2.59808i 0
673.3 0 −1.50000 0.866025i 0 2.27307 3.93708i 0 −9.87987 0 1.50000 + 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 673.3 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.d 6
4.b odd 2 1 57.3.g.a 6
12.b even 2 1 171.3.p.e 6
19.d odd 6 1 inner 912.3.be.d 6
76.f even 6 1 57.3.g.a 6
228.n odd 6 1 171.3.p.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.g.a 6 4.b odd 2 1
57.3.g.a 6 76.f even 6 1
171.3.p.e 6 12.b even 2 1
171.3.p.e 6 228.n odd 6 1
912.3.be.d 6 1.a even 1 1 trivial
912.3.be.d 6 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 $$S_{3}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{6} + 2 T_{5}^{5} + 32 T_{5}^{4} - 40 T_{5}^{3} + 800 T_{5}^{2} + 224 T_{5} + 64$$ $$T_{7}^{3} + 13 T_{7}^{2} - 13 T_{7} - 433$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 3 + 3 T + T^{2} )^{3}$$
$5$ $$64 + 224 T + 800 T^{2} - 40 T^{3} + 32 T^{4} + 2 T^{5} + T^{6}$$
$7$ $$( -433 - 13 T + 13 T^{2} + T^{3} )^{2}$$
$11$ $$( 304 - 264 T + T^{3} )^{2}$$
$13$ $$3518667 + 802503 T + 44764 T^{2} - 3705 T^{3} - 172 T^{4} + 15 T^{5} + T^{6}$$
$17$ $$5184 + 26784 T + 139104 T^{2} - 3576 T^{3} + 472 T^{4} + 10 T^{5} + T^{6}$$
$19$ $$47045881 - 5994766 T + 499263 T^{2} - 31844 T^{3} + 1383 T^{4} - 46 T^{5} + T^{6}$$
$23$ $$1517824 - 147840 T + 43968 T^{2} + 416 T^{3} + 696 T^{4} - 24 T^{5} + T^{6}$$
$29$ $$19293888 + 517344 T - 162752 T^{2} - 4488 T^{3} + 1520 T^{4} - 66 T^{5} + T^{6}$$
$31$ $$124768803 + 1160107 T^{2} + 2033 T^{4} + T^{6}$$
$37$ $$1689765867 + 7315515 T^{2} + 5913 T^{4} + T^{6}$$
$41$ $$1907539968 - 186396672 T + 5466112 T^{2} + 59136 T^{3} - 2272 T^{4} - 24 T^{5} + T^{6}$$
$43$ $$2209 - 517 T + 638 T^{2} + 215 T^{3} + 110 T^{4} + 11 T^{5} + T^{6}$$
$47$ $$2266521664 + 240134752 T + 24204128 T^{2} + 226360 T^{3} + 5720 T^{4} - 26 T^{5} + T^{6}$$
$53$ $$2495232 + 590976 T - 117504 T^{2} - 38880 T^{3} + 11016 T^{4} - 180 T^{5} + T^{6}$$
$59$ $$48350430912 + 623080416 T - 17889728 T^{2} - 265032 T^{3} + 7112 T^{4} + 162 T^{5} + T^{6}$$
$61$ $$558907255201 + 2509696557 T + 116681190 T^{2} + 1021865 T^{3} + 23238 T^{4} + 141 T^{5} + T^{6}$$
$67$ $$2349816507 - 179928423 T + 2829268 T^{2} + 135009 T^{3} - 820 T^{4} - 63 T^{5} + T^{6}$$
$71$ $$100019515392 - 7966854144 T + 279452160 T^{2} - 5410368 T^{3} + 60672 T^{4} - 372 T^{5} + T^{6}$$
$73$ $$214090364601 - 3052425303 T + 91178406 T^{2} - 245907 T^{3} + 17206 T^{4} - 103 T^{5} + T^{6}$$
$79$ $$2549342403 - 32095251 T - 3450884 T^{2} + 45141 T^{3} + 4676 T^{4} - 123 T^{5} + T^{6}$$
$83$ $$( -57704 + 4908 T - 126 T^{2} + T^{3} )^{2}$$
$89$ $$3516172210368 - 148452636384 T + 2784260736 T^{2} - 29344536 T^{3} + 183096 T^{4} - 642 T^{5} + T^{6}$$
$97$ $$3000768049152 + 82138512384 T + 737443840 T^{2} - 328512 T^{3} - 27328 T^{4} + 12 T^{5} + T^{6}$$
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