# Properties

 Label 600.3.l.f Level $600$ Weight $3$ Character orbit 600.l Analytic conductor $16.349$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 600.l (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.3488158616$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{7} + ( 2 + \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{7} + ( 2 + \beta_{6} + \beta_{7} ) q^{9} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{11} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{13} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{17} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{19} + ( 6 - \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{21} + ( -1 - \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - 4 \beta_{5} ) q^{23} + ( -5 + 3 \beta_{2} - 3 \beta_{5} - \beta_{6} + 5 \beta_{7} ) q^{27} + ( 3 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{5} - 5 \beta_{7} ) q^{29} + ( 15 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 4 \beta_{6} ) q^{31} + ( 8 - \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + \beta_{4} - 7 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{33} + ( 1 + \beta_{1} - 7 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} + 5 \beta_{6} ) q^{37} + ( -8 - 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{39} + ( -9 - 5 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} - 5 \beta_{5} - 4 \beta_{7} ) q^{41} + ( 44 - \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{43} + ( 5 + \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{47} + ( 8 + 3 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{49} + ( 11 - 3 \beta_{2} + 5 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 7 \beta_{7} ) q^{51} + ( -4 - 6 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - 10 \beta_{5} + 7 \beta_{7} ) q^{53} + ( -13 - \beta_{1} + 3 \beta_{2} - 5 \beta_{3} - 9 \beta_{4} + \beta_{5} + \beta_{6} - 9 \beta_{7} ) q^{57} + ( 2 - 3 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 3 \beta_{5} + 11 \beta_{7} ) q^{59} + ( 1 - 7 \beta_{1} - 9 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} - 7 \beta_{5} - 6 \beta_{6} ) q^{61} + ( -16 - 5 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{63} + ( -28 + 3 \beta_{1} + 13 \beta_{2} - \beta_{3} - \beta_{4} - 6 \beta_{5} - 2 \beta_{6} ) q^{67} + ( 3 + 4 \beta_{1} + 4 \beta_{2} - 15 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{69} + ( -2 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 10 \beta_{5} + 6 \beta_{7} ) q^{71} + ( -2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} ) q^{73} + ( -8 - 12 \beta_{1} + 20 \beta_{2} + 4 \beta_{3} - 12 \beta_{4} + 10 \beta_{7} ) q^{77} + ( 13 - \beta_{1} - 7 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{79} + ( 20 - 9 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} - 12 \beta_{4} - 9 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{81} + ( 10 + 3 \beta_{1} - 13 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} - 20 \beta_{7} ) q^{83} + ( 30 + 9 \beta_{1} - 6 \beta_{2} + 5 \beta_{3} + 4 \beta_{4} + 13 \beta_{5} + 2 \beta_{6} - 8 \beta_{7} ) q^{87} + ( 18 + 2 \beta_{1} - 26 \beta_{2} + 16 \beta_{3} + 8 \beta_{4} - 6 \beta_{5} + 4 \beta_{7} ) q^{89} + ( 64 + 10 \beta_{1} + 14 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} ) q^{91} + ( 37 - 7 \beta_{1} + 15 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} + 7 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} ) q^{93} + ( -14 - 8 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 8 \beta_{6} ) q^{97} + ( -22 + 5 \beta_{1} + 20 \beta_{2} - 23 \beta_{3} - 6 \beta_{4} - 5 \beta_{5} + 4 \beta_{6} - 15 \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 $$x^{8} - 4 x^{7} - 2 x^{6} + 20 x^{5} + 49 x^{4} - 136 x^{3} + 168 x^{2} - 96 x + 864$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$7 \nu^{7} + 53 \nu^{6} - 530 \nu^{5} - 280 \nu^{4} + 3613 \nu^{3} + 12557 \nu^{2} - 20988 \nu - 26976$$$$)/9300$$ $$\beta_{2}$$ $$=$$ $$($$$$69 \nu^{7} - 319 \nu^{6} - 375 \nu^{5} + 2045 \nu^{4} + 6186 \nu^{3} - 16366 \nu^{2} + 1704 \nu - 3912$$$$)/18600$$ $$\beta_{3}$$ $$=$$ $$($$$$-83 \nu^{7} + 213 \nu^{6} - 115 \nu^{5} + 65 \nu^{4} - 2562 \nu^{3} - 998 \nu^{2} - 52728 \nu + 2064$$$$)/18600$$ $$\beta_{4}$$ $$=$$ $$($$$$21 \nu^{7} - 89 \nu^{6} - 71 \nu^{5} + 927 \nu^{4} + 950 \nu^{3} - 5698 \nu^{2} + 2136 \nu + 10584$$$$)/3720$$ $$\beta_{5}$$ $$=$$ $$($$$$27 \nu^{7} - 17 \nu^{6} - 295 \nu^{5} + 5 \nu^{4} + 2488 \nu^{3} + 672 \nu^{2} + 1152 \nu + 5424$$$$)/3720$$ $$\beta_{6}$$ $$=$$ $$($$$$-77 \nu^{7} + 192 \nu^{6} + 405 \nu^{5} - 20 \nu^{4} - 6418 \nu^{3} - 5602 \nu^{2} + 16968 \nu + 27036$$$$)/9300$$ $$\beta_{7}$$ $$=$$ $$($$$$8 \nu^{7} - 28 \nu^{6} - 30 \nu^{5} + 145 \nu^{4} + 232 \nu^{3} - 507 \nu^{2} + 1788 \nu - 804$$$$)/930$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{6} - \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 2 \beta_{1} + 1$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + 5 \beta_{1} + 13$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-6 \beta_{7} - \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 5 \beta_{2} + 3 \beta_{1} + 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-24 \beta_{7} + 6 \beta_{6} - 15 \beta_{5} + 64 \beta_{4} - 32 \beta_{3} - 45 \beta_{2} + 37 \beta_{1} - 97$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$-150 \beta_{7} - 84 \beta_{6} - 51 \beta_{5} + 146 \beta_{4} - 130 \beta_{3} - 117 \beta_{2} - 7 \beta_{1} - 257$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$-124 \beta_{7} - 26 \beta_{6} + 75 \beta_{5} + 180 \beta_{4} - 44 \beta_{3} - 239 \beta_{2} - 25 \beta_{1} - 771$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-210 \beta_{7} - 1032 \beta_{6} + 765 \beta_{5} + 1342 \beta_{4} + 346 \beta_{3} - 3093 \beta_{2} - 1127 \beta_{1} - 6073$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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
 1.54294 − 1.41421i 1.54294 + 1.41421i −2.22255 − 1.41421i −2.22255 + 1.41421i −0.542939 − 1.41421i −0.542939 + 1.41421i 3.22255 + 1.41421i 3.22255 − 1.41421i
0 −2.98254 0.323191i 0 0 0 −4.72640 0 8.79110 + 1.92786i 0
401.2 0 −2.98254 + 0.323191i 0 0 0 −4.72640 0 8.79110 1.92786i 0
401.3 0 −0.291610 2.98579i 0 0 0 −4.46268 0 −8.82993 + 1.74137i 0
401.4 0 −0.291610 + 2.98579i 0 0 0 −4.46268 0 −8.82993 1.74137i 0
401.5 0 2.40140 1.79813i 0 0 0 10.2132 0 2.53346 8.63606i 0
401.6 0 2.40140 + 1.79813i 0 0 0 10.2132 0 2.53346 + 8.63606i 0
401.7 0 2.87275 0.864473i 0 0 0 −9.02416 0 7.50537 4.96683i 0
401.8 0 2.87275 + 0.864473i 0 0 0 −9.02416 0 7.50537 + 4.96683i 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 600.3.l.f 8
3.b odd 2 1 inner 600.3.l.f 8
4.b odd 2 1 1200.3.l.x 8
5.b even 2 1 120.3.l.a 8
5.c odd 4 2 600.3.c.d 16
12.b even 2 1 1200.3.l.x 8
15.d odd 2 1 120.3.l.a 8
15.e even 4 2 600.3.c.d 16
20.d odd 2 1 240.3.l.d 8
20.e even 4 2 1200.3.c.m 16
40.e odd 2 1 960.3.l.g 8
40.f even 2 1 960.3.l.h 8
60.h even 2 1 240.3.l.d 8
60.l odd 4 2 1200.3.c.m 16
120.i odd 2 1 960.3.l.h 8
120.m even 2 1 960.3.l.g 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 8 T_{7}^{3} - 82 T_{7}^{2} - 872 T_{7} - 1944$$ acting on $$S_{3}^{\mathrm{new}}(600, [\chi])$$.

## Hecke characteristic polynomials

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