# Properties

 Label 600.3.l.g Level $600$ Weight $3$ Character orbit 600.l Analytic conductor $16.349$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# 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: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 4 x^{10} + 30 x^{8} - 216 x^{6} + 1080 x^{4} - 5184 x^{2} + 46656$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{18}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} -\beta_{5} q^{7} + ( -1 - \beta_{9} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} -\beta_{5} q^{7} + ( -1 - \beta_{9} ) q^{9} + ( -\beta_{7} + \beta_{9} ) q^{11} + ( \beta_{4} - \beta_{8} ) q^{13} + ( -\beta_{2} - \beta_{6} ) q^{17} + ( -\beta_{3} + \beta_{11} ) q^{19} + ( 1 - 2 \beta_{1} - \beta_{3} + 2 \beta_{7} - \beta_{9} ) q^{21} + ( -\beta_{2} + 3 \beta_{4} - 2 \beta_{6} - \beta_{10} ) q^{23} + ( 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} ) q^{27} + ( -3 \beta_{1} - \beta_{3} - 2 \beta_{7} - \beta_{11} ) q^{29} + ( -6 + \beta_{3} - 2 \beta_{7} - 2 \beta_{9} + \beta_{11} ) q^{31} + ( \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{8} + 2 \beta_{10} ) q^{33} + ( 5 \beta_{4} + 2 \beta_{5} + \beta_{8} + 2 \beta_{10} ) q^{37} + ( 9 - 5 \beta_{1} - 3 \beta_{7} - \beta_{9} + \beta_{11} ) q^{39} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{9} - \beta_{11} ) q^{41} + ( -9 \beta_{4} - 3 \beta_{10} ) q^{43} + ( -5 \beta_{2} + 3 \beta_{4} + 4 \beta_{6} - \beta_{10} ) q^{47} + ( 21 - 3 \beta_{3} + 3 \beta_{11} ) q^{49} + ( 4 + 2 \beta_{1} - \beta_{3} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{51} + ( -2 \beta_{2} - 12 \beta_{4} + 5 \beta_{6} + 4 \beta_{10} ) q^{53} + ( 8 \beta_{2} - 3 \beta_{4} - 4 \beta_{5} + 5 \beta_{6} - \beta_{8} ) q^{57} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{7} - 3 \beta_{9} - 2 \beta_{11} ) q^{59} + ( 12 - \beta_{3} + \beta_{11} ) q^{61} + ( 4 \beta_{2} - 3 \beta_{4} + \beta_{5} + 7 \beta_{6} + \beta_{8} - 4 \beta_{10} ) q^{63} + ( -5 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} - \beta_{10} ) q^{67} + ( -23 + 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{7} - \beta_{9} + \beta_{11} ) q^{69} + ( -12 \beta_{1} + 2 \beta_{7} - 2 \beta_{9} ) q^{71} + ( 14 \beta_{4} + 6 \beta_{5} + 4 \beta_{8} + 6 \beta_{10} ) q^{73} + ( -12 \beta_{2} - 12 \beta_{4} - 4 \beta_{6} + 4 \beta_{10} ) q^{77} + ( -40 + \beta_{3} - 4 \beta_{7} - 4 \beta_{9} + 3 \beta_{11} ) q^{79} + ( -17 - 2 \beta_{1} + \beta_{3} - 8 \beta_{7} + 4 \beta_{9} - \beta_{11} ) q^{81} + ( -2 \beta_{2} + 15 \beta_{4} + 2 \beta_{6} - 5 \beta_{10} ) q^{83} + ( 13 \beta_{2} + \beta_{4} + \beta_{5} - 8 \beta_{6} + 4 \beta_{8} - \beta_{10} ) q^{87} + ( -12 \beta_{1} + 6 \beta_{7} - 6 \beta_{9} ) q^{89} + ( 4 - 4 \beta_{3} + 4 \beta_{7} + 4 \beta_{9} ) q^{91} + ( 6 \beta_{2} - 3 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} + 6 \beta_{10} ) q^{93} + ( -8 \beta_{4} - 4 \beta_{8} - 4 \beta_{10} ) q^{97} + ( 50 + 2 \beta_{1} - 4 \beta_{3} + 5 \beta_{7} - \beta_{9} - 2 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 8 q^{9} + O(q^{10})$$ $$12 q - 8 q^{9} + 4 q^{21} - 48 q^{31} + 128 q^{39} + 252 q^{49} + 48 q^{51} + 144 q^{61} - 268 q^{69} - 432 q^{79} - 188 q^{81} + 560 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{10} + 30 x^{8} - 216 x^{6} + 1080 x^{4} - 5184 x^{2} + 46656$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{11} + 22 \nu^{9} + 150 \nu^{7} + 1260 \nu^{5} + 432 \nu^{3} - 7776 \nu$$$$)/23328$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{10} + 10 \nu^{8} + 126 \nu^{6} - 108 \nu^{4} + 864 \nu^{2} - 7776$$$$)/5832$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{11} - 22 \nu^{9} - 72 \nu^{8} + 138 \nu^{7} + 288 \nu^{6} - 684 \nu^{5} + 432 \nu^{4} + 1296 \nu^{3} + 5184 \nu^{2} - 2592 \nu - 23328$$$$)/7776$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{11} - 11 \nu^{10} - 10 \nu^{9} + 2 \nu^{8} + 90 \nu^{7} - 126 \nu^{6} - 756 \nu^{5} - 540 \nu^{4} + 5616 \nu^{3} - 9936 \nu^{2} - 38880 \nu + 7776$$$$)/46656$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{11} + 50 \nu^{9} - 18 \nu^{7} - 540 \nu^{5} - 4752 \nu^{3} + 23328 \nu$$$$)/23328$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{10} + 10 \nu^{8} - 36 \nu^{6} + 540 \nu^{4} - 3996 \nu^{2} + 9720$$$$)/2916$$ $$\beta_{7}$$ $$=$$ $$($$$$13 \nu^{11} - 24 \nu^{10} - 70 \nu^{9} + 168 \nu^{8} + 642 \nu^{7} - 1008 \nu^{6} - 2988 \nu^{5} + 4752 \nu^{4} + 15120 \nu^{3} + 23328 \nu + 93312$$$$)/46656$$ $$\beta_{8}$$ $$=$$ $$($$$$25 \nu^{11} - 11 \nu^{10} + 38 \nu^{9} + 2 \nu^{8} + 234 \nu^{7} - 126 \nu^{6} - 1620 \nu^{5} - 540 \nu^{4} - 30672 \nu^{3} - 9936 \nu^{2} - 69984 \nu + 7776$$$$)/46656$$ $$\beta_{9}$$ $$=$$ $$($$$$-7 \nu^{11} - 24 \nu^{10} - 62 \nu^{9} + 168 \nu^{8} + 186 \nu^{7} - 1008 \nu^{6} - 1116 \nu^{5} + 4752 \nu^{4} - 7344 \nu^{3} - 38880 \nu + 93312$$$$)/46656$$ $$\beta_{10}$$ $$=$$ $$($$$$\nu^{11} + 11 \nu^{10} - 10 \nu^{9} - 2 \nu^{8} + 90 \nu^{7} + 126 \nu^{6} - 756 \nu^{5} + 540 \nu^{4} + 5616 \nu^{3} + 9936 \nu^{2} - 38880 \nu - 7776$$$$)/15552$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{11} + 8 \nu^{10} - 22 \nu^{9} + 16 \nu^{8} + 138 \nu^{7} + 48 \nu^{6} - 684 \nu^{5} - 2016 \nu^{4} + 1296 \nu^{3} - 5184 \nu^{2} - 2592 \nu - 7776$$$$)/7776$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} - 7 \beta_{10} - 5 \beta_{9} - 3 \beta_{8} + 7 \beta_{7} + 3 \beta_{5} - 18 \beta_{4} + \beta_{3} - 3 \beta_{1}$$$$)/48$$ $$\nu^{2}$$ $$=$$ $$($$$$-4 \beta_{11} + 4 \beta_{10} + 5 \beta_{9} + 5 \beta_{7} - 12 \beta_{6} - 12 \beta_{4} - \beta_{3} + 3 \beta_{2} + 21$$$$)/24$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{11} + 7 \beta_{10} - 22 \beta_{9} - 15 \beta_{8} + 14 \beta_{7} - 3 \beta_{5} + 36 \beta_{4} - 4 \beta_{3}$$$$)/24$$ $$\nu^{4}$$ $$=$$ $$($$$$-14 \beta_{11} + 20 \beta_{10} + 4 \beta_{9} + 4 \beta_{7} + 30 \beta_{6} - 60 \beta_{4} + 10 \beta_{3} - 3 \beta_{2} - 84$$$$)/12$$ $$\nu^{5}$$ $$=$$ $$($$$$-13 \beta_{11} - 47 \beta_{10} + 5 \beta_{9} - 15 \beta_{8} - 31 \beta_{7} - 165 \beta_{5} - 126 \beta_{4} - 13 \beta_{3} + 207 \beta_{1}$$$$)/24$$ $$\nu^{6}$$ $$=$$ $$($$$$4 \beta_{11} + 20 \beta_{10} - 59 \beta_{9} - 59 \beta_{7} + 84 \beta_{6} - 60 \beta_{4} + 55 \beta_{3} + 375 \beta_{2} + 645$$$$)/12$$ $$\nu^{7}$$ $$=$$ $$($$$$142 \beta_{11} - 37 \beta_{10} + 124 \beta_{9} + 33 \beta_{8} + 160 \beta_{7} + 201 \beta_{5} - 144 \beta_{4} + 142 \beta_{3} + 738 \beta_{1}$$$$)/12$$ $$\nu^{8}$$ $$=$$ $$($$$$110 \beta_{11} + 172 \beta_{10} + 200 \beta_{9} + 200 \beta_{7} + 42 \beta_{6} - 516 \beta_{4} - 310 \beta_{3} + 795 \beta_{2} - 528$$$$)/6$$ $$\nu^{9}$$ $$=$$ $$($$$$-587 \beta_{11} + 935 \beta_{10} - 1901 \beta_{9} + 519 \beta_{8} + 727 \beta_{7} + 3117 \beta_{5} + 2286 \beta_{4} - 587 \beta_{3} + 1305 \beta_{1}$$$$)/12$$ $$\nu^{10}$$ $$=$$ $$($$$$1244 \beta_{11} + 2764 \beta_{10} - 853 \beta_{9} - 853 \beta_{7} + 1500 \beta_{6} - 8292 \beta_{4} - 391 \beta_{3} - 2607 \beta_{2} - 2229$$$$)/6$$ $$\nu^{11}$$ $$=$$ $$($$$$-1306 \beta_{11} + 265 \beta_{10} - 7552 \beta_{9} + 4755 \beta_{8} + 4940 \beta_{7} - 5853 \beta_{5} - 3960 \beta_{4} - 1306 \beta_{3} - 2142 \beta_{1}$$$$)/6$$

## 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
 1.05281 − 2.21170i 1.05281 + 2.21170i 1.79523 + 1.66648i 1.79523 − 1.66648i 2.38091 + 0.575548i 2.38091 − 0.575548i −2.38091 − 0.575548i −2.38091 + 0.575548i −1.79523 − 1.66648i −1.79523 + 1.66648i −1.05281 + 2.21170i −1.05281 − 2.21170i
0 −2.84952 0.938195i 0 0 0 6.81219 0 7.23958 + 5.34682i 0
401.2 0 −2.84952 + 0.938195i 0 0 0 6.81219 0 7.23958 5.34682i 0
401.3 0 −1.67109 2.49147i 0 0 0 −12.7692 0 −3.41489 + 8.32698i 0
401.4 0 −1.67109 + 2.49147i 0 0 0 −12.7692 0 −3.41489 8.32698i 0
401.5 0 −1.26002 2.72256i 0 0 0 0.735748 0 −5.82469 + 6.86097i 0
401.6 0 −1.26002 + 2.72256i 0 0 0 0.735748 0 −5.82469 6.86097i 0
401.7 0 1.26002 2.72256i 0 0 0 −0.735748 0 −5.82469 6.86097i 0
401.8 0 1.26002 + 2.72256i 0 0 0 −0.735748 0 −5.82469 + 6.86097i 0
401.9 0 1.67109 2.49147i 0 0 0 12.7692 0 −3.41489 8.32698i 0
401.10 0 1.67109 + 2.49147i 0 0 0 12.7692 0 −3.41489 + 8.32698i 0
401.11 0 2.84952 0.938195i 0 0 0 −6.81219 0 7.23958 5.34682i 0
401.12 0 2.84952 + 0.938195i 0 0 0 −6.81219 0 7.23958 + 5.34682i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 401.12 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
5.b even 2 1 inner
15.d 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.g 12
3.b odd 2 1 inner 600.3.l.g 12
4.b odd 2 1 1200.3.l.y 12
5.b even 2 1 inner 600.3.l.g 12
5.c odd 4 2 120.3.c.a 12
12.b even 2 1 1200.3.l.y 12
15.d odd 2 1 inner 600.3.l.g 12
15.e even 4 2 120.3.c.a 12
20.d odd 2 1 1200.3.l.y 12
20.e even 4 2 240.3.c.e 12
40.i odd 4 2 960.3.c.k 12
40.k even 4 2 960.3.c.j 12
60.h even 2 1 1200.3.l.y 12
60.l odd 4 2 240.3.c.e 12
120.q odd 4 2 960.3.c.j 12
120.w even 4 2 960.3.c.k 12

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{6} - 210 T_{7}^{4} + 7680 T_{7}^{2} - 4096$$ acting on $$S_{3}^{\mathrm{new}}(600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$531441 + 26244 T^{2} + 4455 T^{4} - 504 T^{6} + 55 T^{8} + 4 T^{10} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$( -4096 + 7680 T^{2} - 210 T^{4} + T^{6} )^{2}$$
$11$ $$( 1083392 + 34944 T^{2} + 336 T^{4} + T^{6} )^{2}$$
$13$ $$( -6553600 + 157824 T^{2} - 768 T^{4} + T^{6} )^{2}$$
$17$ $$( 165888 + 13968 T^{2} + 264 T^{4} + T^{6} )^{2}$$
$19$ $$( -5648 - 780 T + T^{3} )^{4}$$
$23$ $$( 13148192 + 289440 T^{2} + 1050 T^{4} + T^{6} )^{2}$$
$29$ $$( 807698432 + 4295808 T^{2} + 4416 T^{4} + T^{6} )^{2}$$
$31$ $$( -31104 - 1692 T + 12 T^{2} + T^{3} )^{4}$$
$37$ $$( -237899776 + 1828992 T^{2} - 2592 T^{4} + T^{6} )^{2}$$
$41$ $$( 772087808 + 4826112 T^{2} + 4356 T^{4} + T^{6} )^{2}$$
$43$ $$( -1224440064 + 4199040 T^{2} - 4050 T^{4} + T^{6} )^{2}$$
$47$ $$( 2393766432 + 8026272 T^{2} + 5754 T^{4} + T^{6} )^{2}$$
$53$ $$( 21525635072 + 34334352 T^{2} + 11400 T^{4} + T^{6} )^{2}$$
$59$ $$( 1313998848 + 22331520 T^{2} + 11856 T^{4} + T^{6} )^{2}$$
$61$ $$( 1984 - 348 T - 36 T^{2} + T^{3} )^{4}$$
$67$ $$( -1763584 + 4443264 T^{2} - 5106 T^{4} + T^{6} )^{2}$$
$71$ $$( 124856041472 + 105400320 T^{2} + 19200 T^{4} + T^{6} )^{2}$$
$73$ $$( -238331428864 + 163399680 T^{2} - 26400 T^{4} + T^{6} )^{2}$$
$79$ $$( -234400 - 3516 T + 108 T^{2} + T^{3} )^{4}$$
$83$ $$( 387755552 + 52514208 T^{2} + 14586 T^{4} + T^{6} )^{2}$$
$89$ $$( 278628139008 + 168708096 T^{2} + 27648 T^{4} + T^{6} )^{2}$$
$97$ $$( -16777216 + 1572864 T^{2} - 13344 T^{4} + T^{6} )^{2}$$