# Properties

 Label 912.3.o.c Level $912$ Weight $3$ Character orbit 912.o Analytic conductor $24.850$ Analytic rank $0$ Dimension $6$ 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.o (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.8502001097$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.219615408.1 Defining polynomial: $$x^{6} - 3 x^{5} + 22 x^{4} - 39 x^{3} + 112 x^{2} - 93 x + 39$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}\cdot 3$$ Twist minimal: no (minimal twist has level 228) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} -\beta_{3} q^{5} -\beta_{1} q^{7} -3 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} -\beta_{3} q^{5} -\beta_{1} q^{7} -3 q^{9} + ( 4 - \beta_{1} ) q^{11} + ( 4 \beta_{2} - \beta_{4} - \beta_{5} ) q^{13} + \beta_{4} q^{15} + ( -8 - \beta_{1} - 2 \beta_{3} ) q^{17} + ( 1 - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{19} + \beta_{5} q^{21} + ( -6 - 3 \beta_{1} + \beta_{3} ) q^{23} + ( 5 + 3 \beta_{1} + 2 \beta_{3} ) q^{25} -3 \beta_{2} q^{27} + ( -8 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{29} + ( 8 \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{31} + ( 4 \beta_{2} + \beta_{5} ) q^{33} + ( -2 - \beta_{1} + 4 \beta_{3} ) q^{35} + ( 4 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} ) q^{37} + ( -12 - 3 \beta_{1} - 3 \beta_{3} ) q^{39} + ( -4 \beta_{2} + 4 \beta_{4} - 4 \beta_{5} ) q^{41} + ( 36 + \beta_{1} - 2 \beta_{3} ) q^{43} + 3 \beta_{3} q^{45} + ( -4 + 4 \beta_{1} + 5 \beta_{3} ) q^{47} + ( -7 - 5 \beta_{1} - 2 \beta_{3} ) q^{49} + ( -8 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{51} + 4 \beta_{4} q^{53} + ( -2 - \beta_{1} ) q^{55} + ( 6 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{57} + ( -24 \beta_{2} + 2 \beta_{4} + 6 \beta_{5} ) q^{59} + ( 32 - 7 \beta_{1} + 4 \beta_{3} ) q^{61} + 3 \beta_{1} q^{63} + ( -28 \beta_{2} + 10 \beta_{4} + 2 \beta_{5} ) q^{65} + ( 4 \beta_{2} + 4 \beta_{4} ) q^{67} + ( -6 \beta_{2} - \beta_{4} + 3 \beta_{5} ) q^{69} + ( -8 \beta_{2} - 6 \beta_{4} - 2 \beta_{5} ) q^{71} + ( 12 - \beta_{1} + 14 \beta_{3} ) q^{73} + ( 5 \beta_{2} - 2 \beta_{4} - 3 \beta_{5} ) q^{75} + ( 42 - 9 \beta_{1} - 2 \beta_{3} ) q^{77} + ( 24 \beta_{2} - 7 \beta_{4} + \beta_{5} ) q^{79} + 9 q^{81} + ( 66 + 4 \beta_{3} ) q^{83} + ( 58 + 5 \beta_{1} + 16 \beta_{3} ) q^{85} + ( 24 - 6 \beta_{1} - 6 \beta_{3} ) q^{87} + ( -8 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{89} + ( -40 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{91} + ( -24 - 3 \beta_{1} + 9 \beta_{3} ) q^{93} + ( -60 - 6 \beta_{1} + 32 \beta_{2} - 5 \beta_{3} - 4 \beta_{5} ) q^{95} + ( 24 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{97} + ( -12 + 3 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{5} + 2 q^{7} - 18 q^{9} + O(q^{10})$$ $$6 q - 2 q^{5} + 2 q^{7} - 18 q^{9} + 26 q^{11} - 50 q^{17} + 10 q^{19} - 28 q^{23} + 28 q^{25} - 2 q^{35} - 72 q^{39} + 210 q^{43} + 6 q^{45} - 22 q^{47} - 36 q^{49} - 10 q^{55} + 48 q^{57} + 214 q^{61} - 6 q^{63} + 102 q^{73} + 266 q^{77} + 54 q^{81} + 404 q^{83} + 370 q^{85} + 144 q^{87} - 120 q^{93} - 358 q^{95} - 78 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 22 x^{4} - 39 x^{3} + 112 x^{2} - 93 x + 39$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{4} + 2 \nu^{3} - 9 \nu^{2} + 8 \nu + 7$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 32 \nu^{3} - 43 \nu^{2} + 98 \nu - 42$$$$)/19$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{4} + 2 \nu^{3} - 13 \nu^{2} + 12 \nu - 17$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 108 \nu^{3} - 157 \nu^{2} + 934 \nu - 441$$$$)/38$$ $$\beta_{5}$$ $$=$$ $$($$$$-6 \nu^{5} + 15 \nu^{4} - 172 \nu^{3} + 243 \nu^{2} - 902 \nu + 411$$$$)/38$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{2} + 3$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - 3 \beta_{3} + \beta_{2} + 3 \beta_{1} - 33$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-19 \beta_{5} - 13 \beta_{4} - 9 \beta_{3} - 22 \beta_{2} + 9 \beta_{1} - 102$$$$)/12$$ $$\nu^{4}$$ $$=$$ $$($$$$-20 \beta_{5} - 14 \beta_{4} + 18 \beta_{3} - 23 \beta_{2} - 30 \beta_{1} + 261$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$149 \beta_{5} + 83 \beta_{4} + 105 \beta_{3} + 296 \beta_{2} - 165 \beta_{1} + 1476$$$$)/12$$

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
721.1
 0.5 + 2.79345i 0.5 − 3.19918i 0.5 − 0.460304i 0.5 − 2.79345i 0.5 + 3.19918i 0.5 + 0.460304i
0 1.73205i 0 −7.39180 0 −3.28502 0 −3.00000 0
721.2 0 1.73205i 0 0.556406 0 9.52587 0 −3.00000 0
721.3 0 1.73205i 0 5.83539 0 −5.24085 0 −3.00000 0
721.4 0 1.73205i 0 −7.39180 0 −3.28502 0 −3.00000 0
721.5 0 1.73205i 0 0.556406 0 9.52587 0 −3.00000 0
721.6 0 1.73205i 0 5.83539 0 −5.24085 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 721.6 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.o.c 6
3.b odd 2 1 2736.3.o.m 6
4.b odd 2 1 228.3.h.a 6
12.b even 2 1 684.3.h.e 6
19.b odd 2 1 inner 912.3.o.c 6
57.d even 2 1 2736.3.o.m 6
76.d even 2 1 228.3.h.a 6
228.b odd 2 1 684.3.h.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.h.a 6 4.b odd 2 1
228.3.h.a 6 76.d even 2 1
684.3.h.e 6 12.b even 2 1
684.3.h.e 6 228.b odd 2 1
912.3.o.c 6 1.a even 1 1 trivial
912.3.o.c 6 19.b odd 2 1 inner
2736.3.o.m 6 3.b odd 2 1
2736.3.o.m 6 57.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} + T_{5}^{2} - 44 T_{5} + 24$$ acting on $$S_{3}^{\mathrm{new}}(912, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 3 + T^{2} )^{3}$$
$5$ $$( 24 - 44 T + T^{2} + T^{3} )^{2}$$
$7$ $$( -164 - 64 T - T^{2} + T^{3} )^{2}$$
$11$ $$( 12 - 8 T - 13 T^{2} + T^{3} )^{2}$$
$13$ $$2495232 + 98064 T^{2} + 792 T^{4} + T^{6}$$
$17$ $$( -108 - 32 T + 25 T^{2} + T^{3} )^{2}$$
$19$ $$47045881 - 1303210 T + 146927 T^{2} - 7372 T^{3} + 407 T^{4} - 10 T^{5} + T^{6}$$
$23$ $$( -5136 - 560 T + 14 T^{2} + T^{3} )^{2}$$
$29$ $$322486272 + 2011392 T^{2} + 3168 T^{4} + T^{6}$$
$31$ $$62208 + 1009296 T^{2} + 3192 T^{4} + T^{6}$$
$37$ $$832267008 + 8883216 T^{2} + 5976 T^{4} + T^{6}$$
$41$ $$29104386048 + 35322624 T^{2} + 10896 T^{4} + T^{6}$$
$43$ $$( -35764 + 3432 T - 105 T^{2} + T^{3} )^{2}$$
$47$ $$( -25272 - 2084 T + 11 T^{2} + T^{3} )^{2}$$
$53$ $$63700992 + 4349952 T^{2} + 4272 T^{4} + T^{6}$$
$59$ $$207873257472 + 115418880 T^{2} + 19536 T^{4} + T^{6}$$
$61$ $$( 104276 - 64 T - 107 T^{2} + T^{3} )^{2}$$
$67$ $$511377408 + 4886784 T^{2} + 4320 T^{4} + T^{6}$$
$71$ $$48612814848 + 41654016 T^{2} + 11472 T^{4} + T^{6}$$
$73$ $$( 115492 - 7896 T - 51 T^{2} + T^{3} )^{2}$$
$79$ $$60723965952 + 68395536 T^{2} + 19848 T^{4} + T^{6}$$
$83$ $$( -259992 + 12892 T - 202 T^{2} + T^{3} )^{2}$$
$89$ $$956510208 + 3278592 T^{2} + 3408 T^{4} + T^{6}$$
$97$ $$168210432 + 16512768 T^{2} + 8400 T^{4} + T^{6}$$