# Properties

 Label 600.2.d.e Level 600 Weight 2 Character orbit 600.d Analytic conductor 4.791 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 600.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.79102412128$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.399424.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} - q^{3} -\beta_{2} q^{4} -\beta_{3} q^{6} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} - q^{3} -\beta_{2} q^{4} -\beta_{3} q^{6} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{8} + q^{9} + ( \beta_{4} - 2 \beta_{5} ) q^{11} + \beta_{2} q^{12} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{13} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{5} ) q^{14} + ( 2 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{16} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{17} + \beta_{3} q^{18} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{19} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{21} + ( -3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{22} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{23} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{24} + ( 4 - 2 \beta_{2} ) q^{26} - q^{27} + ( -3 + \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{28} -\beta_{4} q^{29} + ( -2 + \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{31} + ( 1 - 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{32} + ( -\beta_{4} + 2 \beta_{5} ) q^{33} + ( -2 + 2 \beta_{4} + 2 \beta_{5} ) q^{34} -\beta_{2} q^{36} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{37} + ( -4 - 2 \beta_{2} ) q^{38} + ( -\beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{39} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{42} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{43} + ( 1 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{4} - \beta_{5} ) q^{44} + ( -4 - 2 \beta_{2} ) q^{46} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{47} + ( -2 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{48} + ( -5 + 2 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} ) q^{49} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{51} + ( 2 + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{52} -2 q^{53} -\beta_{3} q^{54} + ( -1 - 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{5} ) q^{56} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{57} + ( 1 + \beta_{1} - \beta_{5} ) q^{58} + ( -\beta_{4} - 2 \beta_{5} ) q^{59} + ( 2 \beta_{4} - 4 \beta_{5} ) q^{61} + ( 4 - 2 \beta_{2} - 2 \beta_{3} ) q^{62} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{63} + ( -4 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{64} + ( 3 + \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{66} -4 q^{67} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{68} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{69} + ( 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{71} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{72} + 3 \beta_{4} q^{73} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{74} + ( 2 - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{76} + ( 8 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{77} + ( -4 + 2 \beta_{2} ) q^{78} + ( -6 - \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{79} + q^{81} + ( 2 - 2 \beta_{1} - 6 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{82} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( 3 - \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{84} + ( -2 + 2 \beta_{1} + 4 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{86} + \beta_{4} q^{87} + ( 11 + \beta_{1} + 2 \beta_{2} - 3 \beta_{5} ) q^{88} + ( 6 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( -4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{91} + ( 2 - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{92} + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{93} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{94} + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{96} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} ) q^{97} + ( 8 - 4 \beta_{2} - 5 \beta_{3} ) q^{98} + ( \beta_{4} - 2 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{3} + 2q^{4} + 6q^{8} + 6q^{9} + O(q^{10})$$ $$6q - 6q^{3} + 2q^{4} + 6q^{8} + 6q^{9} - 2q^{12} + 16q^{14} + 10q^{16} - 12q^{22} - 6q^{24} + 28q^{26} - 6q^{27} - 20q^{28} - 12q^{31} + 10q^{32} - 12q^{34} + 2q^{36} - 8q^{37} - 20q^{38} - 20q^{41} - 16q^{42} + 8q^{43} + 4q^{44} - 20q^{46} - 10q^{48} - 30q^{49} + 12q^{52} - 12q^{53} + 4q^{56} + 4q^{58} + 28q^{62} - 22q^{64} + 12q^{66} - 24q^{67} - 8q^{71} + 6q^{72} - 12q^{74} + 12q^{76} + 32q^{77} - 28q^{78} - 36q^{79} + 6q^{81} + 16q^{82} - 32q^{83} + 20q^{84} - 16q^{86} + 60q^{88} + 28q^{89} + 12q^{92} + 12q^{93} - 4q^{94} - 10q^{96} + 56q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + \nu^{3} - 4$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - \nu^{3} + 3 \nu^{2} + 2$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 3 \nu^{3} + 3 \nu^{2} - 2 \nu + 6$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} - 3 \nu^{3} + 4 \nu^{2} - 2 \nu + 8$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - \nu^{4} + 2 \nu^{3} - 3 \nu^{2} + 3 \nu - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{3} + \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} - \beta_{3} + \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{5} - 3 \beta_{3} + \beta_{2} + 3$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_{1} + 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{5} + 3 \beta_{3} - \beta_{2} + 4 \beta_{1} + 5$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1
 −0.671462 − 1.24464i −0.671462 + 1.24464i 1.40680 + 0.144584i 1.40680 − 0.144584i 0.264658 − 1.38923i 0.264658 + 1.38923i
−1.24464 0.671462i −1.00000 1.09828 + 1.67146i 0 1.24464 + 0.671462i 4.68585i −0.244644 2.81783i 1.00000 0
349.2 −1.24464 + 0.671462i −1.00000 1.09828 1.67146i 0 1.24464 0.671462i 4.68585i −0.244644 + 2.81783i 1.00000 0
349.3 −0.144584 1.40680i −1.00000 −1.95819 + 0.406803i 0 0.144584 + 1.40680i 3.62721i 0.855416 + 2.69597i 1.00000 0
349.4 −0.144584 + 1.40680i −1.00000 −1.95819 0.406803i 0 0.144584 1.40680i 3.62721i 0.855416 2.69597i 1.00000 0
349.5 1.38923 0.264658i −1.00000 1.85991 0.735342i 0 −1.38923 + 0.264658i 0.941367i 2.38923 1.51380i 1.00000 0
349.6 1.38923 + 0.264658i −1.00000 1.85991 + 0.735342i 0 −1.38923 0.264658i 0.941367i 2.38923 + 1.51380i 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.d.e 6
3.b odd 2 1 1800.2.d.q 6
4.b odd 2 1 2400.2.d.f 6
5.b even 2 1 600.2.d.f 6
5.c odd 4 1 120.2.k.b 6
5.c odd 4 1 600.2.k.c 6
8.b even 2 1 600.2.d.f 6
8.d odd 2 1 2400.2.d.e 6
12.b even 2 1 7200.2.d.q 6
15.d odd 2 1 1800.2.d.r 6
15.e even 4 1 360.2.k.f 6
15.e even 4 1 1800.2.k.p 6
20.d odd 2 1 2400.2.d.e 6
20.e even 4 1 480.2.k.b 6
20.e even 4 1 2400.2.k.c 6
24.f even 2 1 7200.2.d.r 6
24.h odd 2 1 1800.2.d.r 6
40.e odd 2 1 2400.2.d.f 6
40.f even 2 1 inner 600.2.d.e 6
40.i odd 4 1 120.2.k.b 6
40.i odd 4 1 600.2.k.c 6
40.k even 4 1 480.2.k.b 6
40.k even 4 1 2400.2.k.c 6
60.h even 2 1 7200.2.d.r 6
60.l odd 4 1 1440.2.k.f 6
60.l odd 4 1 7200.2.k.p 6
80.i odd 4 1 3840.2.a.bq 3
80.j even 4 1 3840.2.a.br 3
80.s even 4 1 3840.2.a.bo 3
80.t odd 4 1 3840.2.a.bp 3
120.i odd 2 1 1800.2.d.q 6
120.m even 2 1 7200.2.d.q 6
120.q odd 4 1 1440.2.k.f 6
120.q odd 4 1 7200.2.k.p 6
120.w even 4 1 360.2.k.f 6
120.w even 4 1 1800.2.k.p 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.b 6 5.c odd 4 1
120.2.k.b 6 40.i odd 4 1
360.2.k.f 6 15.e even 4 1
360.2.k.f 6 120.w even 4 1
480.2.k.b 6 20.e even 4 1
480.2.k.b 6 40.k even 4 1
600.2.d.e 6 1.a even 1 1 trivial
600.2.d.e 6 40.f even 2 1 inner
600.2.d.f 6 5.b even 2 1
600.2.d.f 6 8.b even 2 1
600.2.k.c 6 5.c odd 4 1
600.2.k.c 6 40.i odd 4 1
1440.2.k.f 6 60.l odd 4 1
1440.2.k.f 6 120.q odd 4 1
1800.2.d.q 6 3.b odd 2 1
1800.2.d.q 6 120.i odd 2 1
1800.2.d.r 6 15.d odd 2 1
1800.2.d.r 6 24.h odd 2 1
1800.2.k.p 6 15.e even 4 1
1800.2.k.p 6 120.w even 4 1
2400.2.d.e 6 8.d odd 2 1
2400.2.d.e 6 20.d odd 2 1
2400.2.d.f 6 4.b odd 2 1
2400.2.d.f 6 40.e odd 2 1
2400.2.k.c 6 20.e even 4 1
2400.2.k.c 6 40.k even 4 1
3840.2.a.bo 3 80.s even 4 1
3840.2.a.bp 3 80.t odd 4 1
3840.2.a.bq 3 80.i odd 4 1
3840.2.a.br 3 80.j even 4 1
7200.2.d.q 6 12.b even 2 1
7200.2.d.q 6 120.m even 2 1
7200.2.d.r 6 24.f even 2 1
7200.2.d.r 6 60.h even 2 1
7200.2.k.p 6 60.l odd 4 1
7200.2.k.p 6 120.q odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(600, [\chi])$$:

 $$T_{7}^{6} + 36 T_{7}^{4} + 320 T_{7}^{2} + 256$$ $$T_{11}^{6} + 64 T_{11}^{4} + 1088 T_{11}^{2} + 4096$$ $$T_{13}^{3} - 28 T_{13} - 16$$ $$T_{37}^{3} + 4 T_{37}^{2} - 60 T_{37} - 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} - 2 T^{3} - 2 T^{4} + 8 T^{6}$$
$3$ $$( 1 + T )^{6}$$
$5$ 1
$7$ $$1 - 6 T^{2} + 47 T^{4} - 500 T^{6} + 2303 T^{8} - 14406 T^{10} + 117649 T^{12}$$
$11$ $$1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 10527 T^{8} - 29282 T^{10} + 1771561 T^{12}$$
$13$ $$( 1 + 11 T^{2} - 16 T^{3} + 143 T^{4} + 2197 T^{6} )^{2}$$
$17$ $$1 - 34 T^{2} + 351 T^{4} - 1084 T^{6} + 101439 T^{8} - 2839714 T^{10} + 24137569 T^{12}$$
$19$ $$1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 955567 T^{8} - 9643754 T^{10} + 47045881 T^{12}$$
$23$ $$1 - 98 T^{2} + 4527 T^{4} - 128636 T^{6} + 2394783 T^{8} - 27424418 T^{10} + 148035889 T^{12}$$
$29$ $$( 1 - 54 T^{2} + 841 T^{4} )^{3}$$
$31$ $$( 1 + 6 T + 77 T^{2} + 308 T^{3} + 2387 T^{4} + 5766 T^{5} + 29791 T^{6} )^{2}$$
$37$ $$( 1 + 4 T + 51 T^{2} + 40 T^{3} + 1887 T^{4} + 5476 T^{5} + 50653 T^{6} )^{2}$$
$41$ $$( 1 + 10 T + 87 T^{2} + 588 T^{3} + 3567 T^{4} + 16810 T^{5} + 68921 T^{6} )^{2}$$
$43$ $$( 1 - 4 T + 65 T^{2} - 216 T^{3} + 2795 T^{4} - 7396 T^{5} + 79507 T^{6} )^{2}$$
$47$ $$1 - 82 T^{2} + 7967 T^{4} - 348252 T^{6} + 17599103 T^{8} - 400133842 T^{10} + 10779215329 T^{12}$$
$53$ $$( 1 + 2 T + 53 T^{2} )^{6}$$
$59$ $$1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 118044191 T^{8} - 3320156914 T^{10} + 42180533641 T^{12}$$
$61$ $$1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 40034239 T^{8} - 1523042510 T^{10} + 51520374361 T^{12}$$
$67$ $$( 1 + 4 T + 67 T^{2} )^{6}$$
$71$ $$( 1 + 4 T + 101 T^{2} + 632 T^{3} + 7171 T^{4} + 20164 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$( 1 - 16 T + 73 T^{2} )^{3}( 1 + 16 T + 73 T^{2} )^{3}$$
$79$ $$( 1 + 18 T + 317 T^{2} + 2908 T^{3} + 25043 T^{4} + 112338 T^{5} + 493039 T^{6} )^{2}$$
$83$ $$( 1 + 16 T + 265 T^{2} + 2400 T^{3} + 21995 T^{4} + 110224 T^{5} + 571787 T^{6} )^{2}$$
$89$ $$( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 23407 T^{4} - 110894 T^{5} + 704969 T^{6} )^{2}$$
$97$ $$1 - 250 T^{2} + 24143 T^{4} - 1697004 T^{6} + 227161487 T^{8} - 22132320250 T^{10} + 832972004929 T^{12}$$