Properties

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

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
301.1
1.40680 + 0.144584i
1.40680 0.144584i
0.264658 + 1.38923i
0.264658 1.38923i
−0.671462 + 1.24464i
−0.671462 1.24464i
−1.40680 0.144584i 1.00000i 1.95819 + 0.406803i 0 0.144584 1.40680i 3.62721 −2.69597 0.855416i −1.00000 0
301.2 −1.40680 + 0.144584i 1.00000i 1.95819 0.406803i 0 0.144584 + 1.40680i 3.62721 −2.69597 + 0.855416i −1.00000 0
301.3 −0.264658 1.38923i 1.00000i −1.85991 + 0.735342i 0 −1.38923 + 0.264658i −0.941367 1.51380 + 2.38923i −1.00000 0
301.4 −0.264658 + 1.38923i 1.00000i −1.85991 0.735342i 0 −1.38923 0.264658i −0.941367 1.51380 2.38923i −1.00000 0
301.5 0.671462 1.24464i 1.00000i −1.09828 1.67146i 0 1.24464 + 0.671462i −4.68585 −2.81783 + 0.244644i −1.00000 0
301.6 0.671462 + 1.24464i 1.00000i −1.09828 + 1.67146i 0 1.24464 0.671462i −4.68585 −2.81783 0.244644i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.6
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{3} + 2 T_{7}^{2} - 16 T_{7} - 16 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 3 T^{2} + 6 T^{3} + 6 T^{4} + 8 T^{5} + 8 T^{6} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ 1
$7$ \( ( 1 + 2 T + 5 T^{2} + 12 T^{3} + 35 T^{4} + 98 T^{5} + 343 T^{6} )^{2} \)
$11$ \( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 10527 T^{8} - 29282 T^{10} + 1771561 T^{12} \)
$13$ \( 1 - 22 T^{2} + 407 T^{4} - 7284 T^{6} + 68783 T^{8} - 628342 T^{10} + 4826809 T^{12} \)
$17$ \( ( 1 + 6 T + 35 T^{2} + 172 T^{3} + 595 T^{4} + 1734 T^{5} + 4913 T^{6} )^{2} \)
$19$ \( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 955567 T^{8} - 9643754 T^{10} + 47045881 T^{12} \)
$23$ \( ( 1 - 4 T + 57 T^{2} - 168 T^{3} + 1311 T^{4} - 2116 T^{5} + 12167 T^{6} )^{2} \)
$29$ \( ( 1 - 54 T^{2} + 841 T^{4} )^{3} \)
$31$ \( ( 1 + 6 T + 77 T^{2} + 308 T^{3} + 2387 T^{4} + 5766 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( 1 - 86 T^{2} + 6055 T^{4} - 248372 T^{6} + 8289295 T^{8} - 161177846 T^{10} + 2565726409 T^{12} \)
$41$ \( ( 1 + 10 T + 87 T^{2} + 588 T^{3} + 3567 T^{4} + 16810 T^{5} + 68921 T^{6} )^{2} \)
$43$ \( 1 - 114 T^{2} + 8087 T^{4} - 416540 T^{6} + 14952863 T^{8} - 389743314 T^{10} + 6321363049 T^{12} \)
$47$ \( ( 1 + 4 T + 49 T^{2} - 120 T^{3} + 2303 T^{4} + 8836 T^{5} + 103823 T^{6} )^{2} \)
$53$ \( ( 1 - 102 T^{2} + 2809 T^{4} )^{3} \)
$59$ \( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 118044191 T^{8} - 3320156914 T^{10} + 42180533641 T^{12} \)
$61$ \( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 40034239 T^{8} - 1523042510 T^{10} + 51520374361 T^{12} \)
$67$ \( ( 1 - 118 T^{2} + 4489 T^{4} )^{3} \)
$71$ \( ( 1 + 4 T + 101 T^{2} + 632 T^{3} + 7171 T^{4} + 20164 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( ( 1 - 6 T + 73 T^{2} )^{6} \)
$79$ \( ( 1 - 18 T + 317 T^{2} - 2908 T^{3} + 25043 T^{4} - 112338 T^{5} + 493039 T^{6} )^{2} \)
$83$ \( 1 - 274 T^{2} + 37415 T^{4} - 3513756 T^{6} + 257751935 T^{8} - 13003579954 T^{10} + 326940373369 T^{12} \)
$89$ \( ( 1 + 14 T + 263 T^{2} + 2308 T^{3} + 23407 T^{4} + 110894 T^{5} + 704969 T^{6} )^{2} \)
$97$ \( ( 1 + 18 T + 287 T^{2} + 3164 T^{3} + 27839 T^{4} + 169362 T^{5} + 912673 T^{6} )^{2} \)
show more
show less