# Properties

 Label 1200.3.l.x Level $1200$ Weight $3$ Character orbit 1200.l Analytic conductor $32.698$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1200.l (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.6976317232$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.681615360000.5 Defining polynomial: $$x^{8} - 4 x^{7} - 2 x^{6} + 20 x^{5} + 49 x^{4} - 136 x^{3} + 168 x^{2} - 96 x + 864$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{7}\cdot 3$$ Twist minimal: no (minimal twist has level 120) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( 2 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{7} + ( 2 + \beta_{5} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( 2 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{7} + ( 2 + \beta_{5} - \beta_{6} ) q^{9} + ( -1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{11} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{13} + ( 2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{17} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{19} + ( 5 - \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{21} + ( 1 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} ) q^{23} + ( 5 + 3 \beta_{2} + 3 \beta_{4} + \beta_{5} + 5 \beta_{6} ) q^{27} + ( 1 + \beta_{1} + 3 \beta_{2} + \beta_{4} + 5 \beta_{6} + 2 \beta_{7} ) q^{29} + ( -13 + \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + \beta_{4} + 4 \beta_{5} - 2 \beta_{7} ) q^{31} + ( 14 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 7 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} ) q^{33} + ( 2 + \beta_{1} + 7 \beta_{2} + \beta_{3} + 3 \beta_{4} + 5 \beta_{5} - \beta_{7} ) q^{37} + ( 8 + 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{39} + ( -5 - 5 \beta_{1} - 9 \beta_{2} - 5 \beta_{4} + 4 \beta_{6} - 4 \beta_{7} ) q^{41} + ( -43 + \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{43} + ( -1 - \beta_{1} - 4 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{47} + ( 4 + 3 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 4 \beta_{7} ) q^{49} + ( -6 - 3 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 7 \beta_{6} - 5 \beta_{7} ) q^{51} + ( -6 - 6 \beta_{1} - 4 \beta_{3} - 10 \beta_{4} - 7 \beta_{6} + 2 \beta_{7} ) q^{53} + ( -8 - \beta_{1} - 3 \beta_{2} + 9 \beta_{3} + \beta_{4} + \beta_{5} + 9 \beta_{6} - 5 \beta_{7} ) q^{57} + ( 3 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} + 11 \beta_{6} - 5 \beta_{7} ) q^{59} + ( 9 - 7 \beta_{1} + 9 \beta_{2} + 8 \beta_{3} - 7 \beta_{4} - 6 \beta_{5} - 8 \beta_{7} ) q^{61} + ( 12 + 5 \beta_{1} + 7 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} + 4 \beta_{7} ) q^{63} + ( 27 - 3 \beta_{1} + 13 \beta_{2} - \beta_{3} + 6 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{67} + ( 18 + 4 \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 15 \beta_{7} ) q^{69} + ( 4 + 4 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 10 \beta_{4} + 6 \beta_{6} - 2 \beta_{7} ) q^{71} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{7} ) q^{73} + ( -12 - 12 \beta_{1} - 20 \beta_{2} + 12 \beta_{3} - 10 \beta_{6} + 4 \beta_{7} ) q^{77} + ( -15 + \beta_{1} - 7 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{79} + ( 26 - 9 \beta_{1} - 3 \beta_{2} + 12 \beta_{3} - 9 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} ) q^{81} + ( -3 - 3 \beta_{1} - 13 \beta_{2} + 3 \beta_{3} - 20 \beta_{6} - 7 \beta_{7} ) q^{83} + ( -25 - 9 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} - 13 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} - 5 \beta_{7} ) q^{87} + ( 2 + 2 \beta_{1} + 26 \beta_{2} - 8 \beta_{3} - 6 \beta_{4} - 4 \beta_{6} + 16 \beta_{7} ) q^{89} + ( -58 - 10 \beta_{1} + 14 \beta_{2} + 6 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} - 6 \beta_{7} ) q^{91} + ( 42 - 7 \beta_{1} - 15 \beta_{2} + 3 \beta_{3} + 7 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} ) q^{93} + ( -10 - 8 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} - 4 \beta_{7} ) q^{97} + ( -1 - 5 \beta_{1} + 20 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} - 15 \beta_{6} + 23 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{3} + 16 q^{7} + 20 q^{9} + O(q^{10})$$ $$8 q - 4 q^{3} + 16 q^{7} + 20 q^{9} + 8 q^{13} + 8 q^{19} + 28 q^{21} + 20 q^{27} - 120 q^{31} + 112 q^{33} - 8 q^{37} + 72 q^{39} - 328 q^{43} + 64 q^{49} - 64 q^{51} - 72 q^{57} + 8 q^{61} + 88 q^{63} + 152 q^{67} + 100 q^{69} - 32 q^{73} - 88 q^{79} + 224 q^{81} - 152 q^{87} - 560 q^{91} + 368 q^{93} - 144 q^{97} - 32 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 2 x^{6} + 20 x^{5} + 49 x^{4} - 136 x^{3} + 168 x^{2} - 96 x + 864$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-9 \nu^{7} - 511 \nu^{6} + 1235 \nu^{5} + 2685 \nu^{4} - 3206 \nu^{3} - 34634 \nu^{2} + 25656 \nu - 66288$$$$)/18600$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{7} + 7 \nu^{6} + 85 \nu^{5} - 230 \nu^{4} - 523 \nu^{3} + 243 \nu^{2} + 4668 \nu - 2124$$$$)/3100$$ $$\beta_{3}$$ $$=$$ $$($$$$-26 \nu^{7} - 64 \nu^{6} + 795 \nu^{5} - 45 \nu^{4} - 4939 \nu^{3} - 1181 \nu^{2} + 23484 \nu - 23892$$$$)/9300$$ $$\beta_{4}$$ $$=$$ $$($$$$-59 \nu^{7} + 129 \nu^{6} + 415 \nu^{5} + 1275 \nu^{4} - 7136 \nu^{3} - 11664 \nu^{2} - 15744 \nu + 64752$$$$)/18600$$ $$\beta_{5}$$ $$=$$ $$($$$$-37 \nu^{7} + 362 \nu^{6} - 675 \nu^{5} - 1310 \nu^{4} + 322 \nu^{3} + 16818 \nu^{2} - 1992 \nu + 24876$$$$)/9300$$ $$\beta_{6}$$ $$=$$ $$($$$$8 \nu^{7} - 28 \nu^{6} - 30 \nu^{5} + 145 \nu^{4} + 232 \nu^{3} - 507 \nu^{2} + 1788 \nu - 804$$$$)/930$$ $$\beta_{7}$$ $$=$$ $$($$$$82 \nu^{7} - 287 \nu^{6} - 385 \nu^{5} + 1680 \nu^{4} + 5943 \nu^{3} - 8413 \nu^{2} + 13212 \nu - 5916$$$$)/9300$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{7} + 3 \beta_{5} + \beta_{3} + 3 \beta_{2} + 2 \beta_{1} + 5$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{4} + 2 \beta_{3} - 3 \beta_{2} - \beta_{1} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$6 \beta_{7} - 6 \beta_{6} - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_{1} + 3$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$16 \beta_{7} - 8 \beta_{6} + 2 \beta_{5} + 11 \beta_{4} + 16 \beta_{3} - 17 \beta_{2} - 9 \beta_{1} - 43$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$54 \beta_{7} - 50 \beta_{6} - 28 \beta_{5} + 23 \beta_{4} + 38 \beta_{3} + 3 \beta_{2} - 49 \beta_{1} - 89$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$188 \beta_{7} - 124 \beta_{6} - 26 \beta_{5} + 179 \beta_{4} + 36 \beta_{3} + 143 \beta_{2} - 137 \beta_{1} - 839$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$322 \beta_{7} - 70 \beta_{6} - 344 \beta_{5} + 479 \beta_{4} + 10 \beta_{3} + 547 \beta_{2} - 609 \beta_{1} - 2373$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 3.22255 + 1.41421i 3.22255 − 1.41421i −0.542939 − 1.41421i −0.542939 + 1.41421i −2.22255 − 1.41421i −2.22255 + 1.41421i 1.54294 − 1.41421i 1.54294 + 1.41421i
0 −2.87275 0.864473i 0 0 0 9.02416 0 7.50537 + 4.96683i 0
401.2 0 −2.87275 + 0.864473i 0 0 0 9.02416 0 7.50537 4.96683i 0
401.3 0 −2.40140 1.79813i 0 0 0 −10.2132 0 2.53346 + 8.63606i 0
401.4 0 −2.40140 + 1.79813i 0 0 0 −10.2132 0 2.53346 8.63606i 0
401.5 0 0.291610 2.98579i 0 0 0 4.46268 0 −8.82993 1.74137i 0
401.6 0 0.291610 + 2.98579i 0 0 0 4.46268 0 −8.82993 + 1.74137i 0
401.7 0 2.98254 0.323191i 0 0 0 4.72640 0 8.79110 1.92786i 0
401.8 0 2.98254 + 0.323191i 0 0 0 4.72640 0 8.79110 + 1.92786i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 401.8 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 1200.3.l.x 8
3.b odd 2 1 inner 1200.3.l.x 8
4.b odd 2 1 600.3.l.f 8
5.b even 2 1 240.3.l.d 8
5.c odd 4 2 1200.3.c.m 16
12.b even 2 1 600.3.l.f 8
15.d odd 2 1 240.3.l.d 8
15.e even 4 2 1200.3.c.m 16
20.d odd 2 1 120.3.l.a 8
20.e even 4 2 600.3.c.d 16
40.e odd 2 1 960.3.l.h 8
40.f even 2 1 960.3.l.g 8
60.h even 2 1 120.3.l.a 8
60.l odd 4 2 600.3.c.d 16
120.i odd 2 1 960.3.l.g 8
120.m even 2 1 960.3.l.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.l.a 8 20.d odd 2 1
120.3.l.a 8 60.h even 2 1
240.3.l.d 8 5.b even 2 1
240.3.l.d 8 15.d odd 2 1
600.3.c.d 16 20.e even 4 2
600.3.c.d 16 60.l odd 4 2
600.3.l.f 8 4.b odd 2 1
600.3.l.f 8 12.b even 2 1
960.3.l.g 8 40.f even 2 1
960.3.l.g 8 120.i odd 2 1
960.3.l.h 8 40.e odd 2 1
960.3.l.h 8 120.m even 2 1
1200.3.c.m 16 5.c odd 4 2
1200.3.c.m 16 15.e even 4 2
1200.3.l.x 8 1.a even 1 1 trivial
1200.3.l.x 8 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{4} - 8 T_{7}^{3} - 82 T_{7}^{2} + 872 T_{7} - 1944$$ $$T_{11}^{8} + 888 T_{11}^{6} + 226128 T_{11}^{4} + 14951168 T_{11}^{2} + 232989696$$ $$T_{13}^{4} - 4 T_{13}^{3} - 380 T_{13}^{2} + 3648 T_{13} - 3456$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$6561 + 2916 T - 162 T^{2} - 324 T^{3} - 102 T^{4} - 36 T^{5} - 2 T^{6} + 4 T^{7} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( -1944 + 872 T - 82 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$11$ $$232989696 + 14951168 T^{2} + 226128 T^{4} + 888 T^{6} + T^{8}$$
$13$ $$( -3456 + 3648 T - 380 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$17$ $$15872256 + 25369856 T^{2} + 312672 T^{4} + 1104 T^{6} + T^{8}$$
$19$ $$( 16736 + 3424 T - 708 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$23$ $$93650688576 + 885144416 T^{2} + 2697012 T^{4} + 2964 T^{6} + T^{8}$$
$29$ $$3474395136 + 145293824 T^{2} + 1160592 T^{4} + 2376 T^{6} + T^{8}$$
$31$ $$( -151296 - 71360 T - 924 T^{2} + 60 T^{3} + T^{4} )^{2}$$
$37$ $$( -31104 - 39744 T - 3228 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$41$ $$43961355472896 + 77260718592 T^{2} + 46587024 T^{4} + 11528 T^{6} + T^{8}$$
$43$ $$( 1582656 + 220576 T + 9482 T^{2} + 164 T^{3} + T^{4} )^{2}$$
$47$ $$13517317696 + 266492768 T^{2} + 1541364 T^{4} + 2612 T^{6} + T^{8}$$
$53$ $$6801580544256 + 88193961216 T^{2} + 82087776 T^{4} + 16976 T^{6} + T^{8}$$
$59$ $$15563214360576 + 64836527616 T^{2} + 49411216 T^{4} + 12616 T^{6} + T^{8}$$
$61$ $$( 30631296 + 12544 T - 12508 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$67$ $$( -668224 + 236224 T - 4854 T^{2} - 76 T^{3} + T^{4} )^{2}$$
$71$ $$35499479924736 + 145459224576 T^{2} + 83806272 T^{4} + 16304 T^{6} + T^{8}$$
$73$ $$( 2938896 - 29376 T - 3528 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$79$ $$( -12384 + 8256 T - 1268 T^{2} + 44 T^{3} + T^{4} )^{2}$$
$83$ $$336130569170496 + 2057217800544 T^{2} + 532895092 T^{4} + 41716 T^{6} + T^{8}$$
$89$ $$1334603390386176 + 2220134105088 T^{2} + 649648384 T^{4} + 49312 T^{6} + T^{8}$$
$97$ $$( 10270096 - 535392 T - 10376 T^{2} + 72 T^{3} + T^{4} )^{2}$$