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Properties

Label 600.2.k.d
Level $600$
Weight $2$
Character orbit 600.k
Analytic conductor $4.791$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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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: \(8\)
Coefficient field: 8.0.214798336.3
Defining polynomial: \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{2} q^{3} + (\beta_{6} - \beta_{5} - \beta_{4}) q^{4} + \beta_{4} q^{6} + ( - \beta_{6} - \beta_{3} + \beta_1 + 1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{2} + 1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{2} q^{3} + (\beta_{6} - \beta_{5} - \beta_{4}) q^{4} + \beta_{4} q^{6} + ( - \beta_{6} - \beta_{3} + \beta_1 + 1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{2} + 1) q^{8} - q^{9} + ( - \beta_{5} + \beta_{4} - \beta_{3} - 1) q^{11} + (\beta_{2} + \beta_1) q^{12} + (\beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 - 1) q^{13} + (\beta_{6} + \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{14} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{16} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} + \beta_1) q^{17} - \beta_{3} q^{18} + ( - 2 \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 1) q^{19} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} + 1) q^{21} + ( - \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{2} + \beta_1 + 2) q^{22} + ( - 2 \beta_{7} + \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 2 \beta_1 + 1) q^{23} + (\beta_{7} + \beta_{6} + \beta_{2} - 1) q^{24} + (2 \beta_{7} - \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{26} - \beta_{2} q^{27} + ( - 2 \beta_{5} - 2 \beta_{3} - 3 \beta_{2} + \beta_1 - 2) q^{28} + (\beta_{7} - 2 \beta_{6} + \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - \beta_1) q^{29} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_1 + 1) q^{31} + (2 \beta_{7} + 2 \beta_1 + 4) q^{32} + (\beta_{7} - \beta_{4} - 2 \beta_{3} + \beta_1) q^{33} + (3 \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{34} + ( - \beta_{6} + \beta_{5} + \beta_{4}) q^{36} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{3}) q^{37} + ( - 3 \beta_{6} + 3 \beta_{5} - 3 \beta_{2} - \beta_1 + 6) q^{38} + ( - \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_1) q^{39} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + \beta_1) q^{41} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 2) q^{42} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} + 1) q^{43} + (2 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} - 2) q^{44} + (\beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_{2} + 3 \beta_1 + 2) q^{46} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4}) q^{47} + ( - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{48} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} - 6 \beta_{3} + 4 \beta_1) q^{49} + (2 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 1) q^{51} + (\beta_{6} - \beta_{5} + 3 \beta_{4} + 4 \beta_{2} - 4) q^{52} + (3 \beta_{7} + 3 \beta_{4} + 2 \beta_{2} + 3 \beta_1) q^{53} - \beta_{4} q^{54} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 5 \beta_{2} + 3) q^{56} + (\beta_{7} - 2 \beta_{5} - 3 \beta_{4} + \beta_1 + 1) q^{57} + ( - 4 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{58} + (2 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 4) q^{59} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} - \beta_1 + 1) q^{61} + (3 \beta_{6} - \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 4) q^{62} + (\beta_{6} + \beta_{3} - \beta_1 - 1) q^{63} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 4) q^{64} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 2) q^{66} + (2 \beta_{7} - 4 \beta_{6} + 3 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{67} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 4) q^{68} + ( - \beta_{7} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1 - 2) q^{69} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{3} + 2 \beta_1 - 5) q^{71} + ( - \beta_{7} + \beta_{6} - \beta_{2} - 1) q^{72} + (2 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} - 6 \beta_{4} + 2 \beta_{3} - 2) q^{73} + ( - 4 \beta_{7} + 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2}) q^{74} + ( - 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + 6 \beta_{3} - 7 \beta_{2} - 3 \beta_1 - 2) q^{76} + ( - 3 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} + \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + \cdots - 4) q^{77}+ \cdots + (\beta_{5} - \beta_{4} + \beta_{3} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 4 q^{4} + 2 q^{6} + 8 q^{7} + 4 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 4 q^{4} + 2 q^{6} + 8 q^{7} + 4 q^{8} - 8 q^{9} - 6 q^{14} + 8 q^{16} + 2 q^{18} + 12 q^{22} + 8 q^{23} - 8 q^{24} - 2 q^{26} - 4 q^{28} + 8 q^{31} + 28 q^{32} + 12 q^{34} - 4 q^{36} + 30 q^{38} - 6 q^{42} - 12 q^{44} + 20 q^{46} - 8 q^{48} - 20 q^{52} - 2 q^{54} + 8 q^{56} + 8 q^{57} + 12 q^{58} + 30 q^{62} - 8 q^{63} - 32 q^{64} - 20 q^{66} - 28 q^{68} - 40 q^{71} - 4 q^{72} - 16 q^{73} + 8 q^{74} - 20 q^{76} - 22 q^{78} - 16 q^{79} + 8 q^{81} - 24 q^{82} + 24 q^{84} - 18 q^{86} + 24 q^{87} - 8 q^{88} - 36 q^{92} - 4 q^{94} + 12 q^{96} - 8 q^{97} - 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 2\nu^{4} + 3\nu^{3} - 6\nu^{2} - 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} + 2\nu^{6} + 4\nu^{5} + 18\nu^{4} - 21\nu^{3} - 12\nu^{2} - 20\nu + 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{7} - 4\nu^{6} - 4\nu^{5} - 18\nu^{4} + 25\nu^{3} + 10\nu^{2} + 24\nu - 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} - 2\nu^{6} - 2\nu^{5} - 10\nu^{4} + 15\nu^{3} + 8\nu^{2} + 10\nu - 36 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 2\nu^{6} + 4\nu^{5} + 10\nu^{4} - 15\nu^{3} - 12\nu^{2} - 8\nu + 36 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{7} + 3\nu^{6} + 4\nu^{5} + 12\nu^{4} - 18\nu^{3} - 9\nu^{2} - 10\nu + 44 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} - 4\nu^{6} - 6\nu^{5} - 22\nu^{4} + 31\nu^{3} + 18\nu^{2} + 26\nu - 80 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 2\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 2\beta_{3} + 5\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + 2\beta_{5} + 7\beta_{4} - \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4\beta_{7} + 3\beta_{6} + \beta_{5} - \beta_{4} - 8\beta_{3} - 5\beta_{2} + 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{7} - 5\beta_{6} + 8\beta_{5} - \beta_{4} - 11\beta_{3} + 3\beta_{2} - 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
1.41216 − 0.0762223i
1.41216 + 0.0762223i
−0.565036 − 1.29643i
−0.565036 + 1.29643i
−1.08003 − 0.912978i
−1.08003 + 0.912978i
1.23291 − 0.692769i
1.23291 + 0.692769i
−1.29150 − 0.576222i 1.00000i 1.33594 + 1.48838i 0 0.576222 − 1.29150i −1.97676 −0.867721 − 2.69204i −1.00000 0
301.2 −1.29150 + 0.576222i − 1.00000i 1.33594 − 1.48838i 0 0.576222 + 1.29150i −1.97676 −0.867721 + 2.69204i −1.00000 0
301.3 −1.16863 − 0.796431i − 1.00000i 0.731395 + 1.86147i 0 −0.796431 + 1.16863i 4.72294 0.627801 − 2.75787i −1.00000 0
301.4 −1.16863 + 0.796431i 1.00000i 0.731395 − 1.86147i 0 −0.796431 − 1.16863i 4.72294 0.627801 + 2.75787i −1.00000 0
301.5 0.0591148 − 1.41298i 1.00000i −1.99301 − 0.167056i 0 1.41298 + 0.0591148i 1.33411 −0.353863 + 2.80620i −1.00000 0
301.6 0.0591148 + 1.41298i − 1.00000i −1.99301 + 0.167056i 0 1.41298 − 0.0591148i 1.33411 −0.353863 − 2.80620i −1.00000 0
301.7 1.40101 − 0.192769i − 1.00000i 1.92568 − 0.540143i 0 −0.192769 − 1.40101i −0.0802864 2.59378 − 1.12796i −1.00000 0
301.8 1.40101 + 0.192769i 1.00000i 1.92568 + 0.540143i 0 −0.192769 + 1.40101i −0.0802864 2.59378 + 1.12796i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.8
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.d ✓ 8
3.b odd 2 1 1800.2.k.t 8
4.b odd 2 1 2400.2.k.d 8
5.b even 2 1 600.2.k.e yes 8
5.c odd 4 1 600.2.d.g 8
5.c odd 4 1 600.2.d.h 8
8.b even 2 1 inner 600.2.k.d ✓ 8
8.d odd 2 1 2400.2.k.d 8
12.b even 2 1 7200.2.k.r 8
15.d odd 2 1 1800.2.k.q 8
15.e even 4 1 1800.2.d.s 8
15.e even 4 1 1800.2.d.t 8
20.d odd 2 1 2400.2.k.e 8
20.e even 4 1 2400.2.d.g 8
20.e even 4 1 2400.2.d.h 8
24.f even 2 1 7200.2.k.r 8
24.h odd 2 1 1800.2.k.t 8
40.e odd 2 1 2400.2.k.e 8
40.f even 2 1 600.2.k.e yes 8
40.i odd 4 1 600.2.d.g 8
40.i odd 4 1 600.2.d.h 8
40.k even 4 1 2400.2.d.g 8
40.k even 4 1 2400.2.d.h 8
60.h even 2 1 7200.2.k.s 8
60.l odd 4 1 7200.2.d.s 8
60.l odd 4 1 7200.2.d.t 8
120.i odd 2 1 1800.2.k.q 8
120.m even 2 1 7200.2.k.s 8
120.q odd 4 1 7200.2.d.s 8
120.q odd 4 1 7200.2.d.t 8
120.w even 4 1 1800.2.d.s 8
120.w even 4 1 1800.2.d.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.d.g 8 5.c odd 4 1
600.2.d.g 8 40.i odd 4 1
600.2.d.h 8 5.c odd 4 1
600.2.d.h 8 40.i odd 4 1
600.2.k.d ✓ 8 1.a even 1 1 trivial
600.2.k.d ✓ 8 8.b even 2 1 inner
600.2.k.e yes 8 5.b even 2 1
600.2.k.e yes 8 40.f even 2 1
1800.2.d.s 8 15.e even 4 1
1800.2.d.s 8 120.w even 4 1
1800.2.d.t 8 15.e even 4 1
1800.2.d.t 8 120.w even 4 1
1800.2.k.q 8 15.d odd 2 1
1800.2.k.q 8 120.i odd 2 1
1800.2.k.t 8 3.b odd 2 1
1800.2.k.t 8 24.h odd 2 1
2400.2.d.g 8 20.e even 4 1
2400.2.d.g 8 40.k even 4 1
2400.2.d.h 8 20.e even 4 1
2400.2.d.h 8 40.k even 4 1
2400.2.k.d 8 4.b odd 2 1
2400.2.k.d 8 8.d odd 2 1
2400.2.k.e 8 20.d odd 2 1
2400.2.k.e 8 40.e odd 2 1
7200.2.d.s 8 60.l odd 4 1
7200.2.d.s 8 120.q odd 4 1
7200.2.d.t 8 60.l odd 4 1
7200.2.d.t 8 120.q odd 4 1
7200.2.k.r 8 12.b even 2 1
7200.2.k.r 8 24.f even 2 1
7200.2.k.s 8 60.h even 2 1
7200.2.k.s 8 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 4T_{7}^{3} - 6T_{7}^{2} + 12T_{7} + 1 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} - 4 T^{5} - 6 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} - 6 T^{2} + 12 T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 32 T^{6} + 336 T^{4} + \cdots + 1600 \) Copy content Toggle raw display
$13$ \( T^{8} + 44 T^{6} + 502 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( (T^{4} - 40 T^{2} + 104 T - 24)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 116 T^{6} + 4662 T^{4} + \cdots + 380689 \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} - 56 T^{2} + 152 T - 88)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 144 T^{6} + 6288 T^{4} + \cdots + 627264 \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} - 70 T^{2} + 204 T + 673)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 128 T^{6} + 4224 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( (T^{4} - 64 T^{2} + 56 T + 328)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 244 T^{6} + 18614 T^{4} + \cdots + 4363921 \) Copy content Toggle raw display
$47$ \( (T^{4} - 72 T^{2} - 256 T - 176)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 256 T^{6} + 19536 T^{4} + \cdots + 23104 \) Copy content Toggle raw display
$59$ \( T^{8} + 432 T^{6} + \cdots + 31181056 \) Copy content Toggle raw display
$61$ \( T^{8} + 236 T^{6} + 14838 T^{4} + \cdots + 3025 \) Copy content Toggle raw display
$67$ \( T^{8} + 372 T^{6} + \cdots + 25979409 \) Copy content Toggle raw display
$71$ \( (T^{4} + 20 T^{3} + 96 T^{2} - 72 T - 536)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} - 168 T^{2} - 864 T - 432)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} - 184 T^{2} - 864 T + 8080)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 368 T^{6} + 44256 T^{4} + \cdots + 3873024 \) Copy content Toggle raw display
$89$ \( (T^{4} - 224 T^{2} + 64 T + 10880)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4 T^{3} - 202 T^{2} - 348 T + 8881)^{2} \) Copy content Toggle raw display
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