Properties

Label 600.2.r.f
Level $600$
Weight $2$
Character orbit 600.r
Analytic conductor $4.791$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.6040479020157644046336.1
Defining polynomial: \( x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{3} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{7} + (\beta_{11} + \beta_{7}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{3} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{7} + (\beta_{11} + \beta_{7}) q^{9} + (\beta_{8} + \beta_1) q^{11} + (\beta_{14} - 3 \beta_{10} + 3 \beta_{6}) q^{13} + ( - \beta_{12} + \beta_{5} + \beta_{3}) q^{17} + (2 \beta_{11} + 2 \beta_{9} + \beta_{7}) q^{19} + ( - \beta_{8} - \beta_1 + 1) q^{21} + ( - \beta_{15} - 2 \beta_{10} - 2 \beta_{6}) q^{23} + (\beta_{12} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3}) q^{27} + ( - 3 \beta_{13} + \beta_{11} + 2 \beta_{9}) q^{29} + (\beta_{2} + \beta_1 - 3) q^{31} + (3 \beta_{14} - 2 \beta_{10} + 3 \beta_{6}) q^{33} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{37} + (\beta_{13} + 2 \beta_{11} - \beta_{9} - 5 \beta_{7}) q^{39} + ( - 3 \beta_{8} - 2 \beta_{2} - \beta_1) q^{41} + (5 \beta_{14} - \beta_{10} + \beta_{6}) q^{43} - 2 \beta_{12} q^{47} + (\beta_{11} + \beta_{9} - 4 \beta_{7}) q^{49} + (\beta_{8} + 3 \beta_{2} + \beta_1 - 4) q^{51} + ( - \beta_{15} + 2 \beta_{10} + 2 \beta_{6}) q^{53} + (2 \beta_{12} + 2 \beta_{5} - 2 \beta_{4} + \beta_{3}) q^{57} + ( - 2 \beta_{11} + 2 \beta_{9}) q^{59} + ( - \beta_{2} - \beta_1 - 1) q^{61} + ( - 3 \beta_{14} + \beta_{10} - 3 \beta_{6}) q^{63} - \beta_{4} q^{67} + ( - \beta_{13} + \beta_{11} - 2 \beta_{9} + 8 \beta_{7}) q^{69} + (3 \beta_{8} + \beta_{2} + 2 \beta_1) q^{71} + ( - 6 \beta_{14} + \beta_{10} - \beta_{6}) q^{73} + ( - \beta_{12} - 2 \beta_{5} - 2 \beta_{3}) q^{77} + ( - 2 \beta_{11} - 2 \beta_{9} + 4 \beta_{7}) q^{79} + ( - 2 \beta_{8} - 3 \beta_{2} - 4 \beta_1 + 3) q^{81} + (2 \beta_{15} - \beta_{10} - \beta_{6}) q^{83} + (\beta_{12} + 4 \beta_{5} + 8 \beta_{4} - 4 \beta_{3}) q^{87} + (2 \beta_{13} - 3 \beta_{11} + \beta_{9}) q^{89} + (\beta_{2} + \beta_1 + 7) q^{91} + (\beta_{15} + \beta_{14} + 3 \beta_{10} - \beta_{6}) q^{93} + (5 \beta_{5} + 3 \beta_{4} - 5 \beta_{3}) q^{97} + (3 \beta_{13} - \beta_{11} - 3 \beta_{9} - 4 \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{21} - 40 q^{31} - 52 q^{51} - 24 q^{61} + 28 q^{81} + 120 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 31\nu^{12} + 512\nu^{8} - 992\nu^{4} - 30537 ) / 12960 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{12} - 32\nu^{8} + 512\nu^{4} + 5121 ) / 1440 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{13} - 1079\nu ) / 480 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{13} + 4\nu^{9} + 296\nu^{5} - 6885\nu ) / 4860 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{13} + 32\nu^{9} - 224\nu^{5} - 6561\nu ) / 2592 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -31\nu^{15} - 512\nu^{11} + 992\nu^{7} + 17577\nu^{3} ) / 116640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{14} + 32\nu^{10} - 62\nu^{6} - 6561\nu^{2} ) / 7290 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 155\nu^{12} - 32\nu^{8} - 2368\nu^{4} - 103437 ) / 12960 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 203\nu^{14} + 928\nu^{10} - 9088\nu^{6} - 73629\nu^{2} ) / 116640 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -49\nu^{15} - 224\nu^{11} - 1024\nu^{7} + 45927\nu^{3} ) / 69984 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 293\nu^{14} - 512\nu^{10} + 992\nu^{6} - 215379\nu^{2} ) / 116640 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 37\nu^{13} - 16\nu^{9} - 1184\nu^{5} - 20979\nu ) / 4860 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 313\nu^{14} + 320\nu^{10} - 15200\nu^{6} - 211167\nu^{2} ) / 116640 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 43\nu^{15} + 104\nu^{11} - 2024\nu^{7} - 28593\nu^{3} ) / 43740 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{15} - 359\nu^{3} ) / 540 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{12} + 4\beta_{4} - 8\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{13} + 3\beta_{11} + 10\beta_{9} - 12\beta_{7} ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} + 2\beta_{10} - 2\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{8} + 31\beta_{2} + 18\beta _1 + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -15\beta_{12} - 8\beta_{5} + 60\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{13} + 6\beta_{11} + 4\beta_{9} + 5\beta_{7} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 35\beta_{15} - 140\beta_{14} - 104\beta_{10} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -31\beta_{8} + 31\beta_{2} + 218\beta _1 + 156 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -17\beta_{12} + 74\beta_{5} + 37\beta_{4} - 74\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -315\beta_{13} - 315\beta_{11} + 382\beta_{9} + 1012\beta_{7} ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -93\beta_{15} - 372\beta_{14} - 2024\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 100\beta_{8} + 60\beta_{2} + 40\beta _1 + 679 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1079\beta_{12} + 4316\beta_{4} - 4792\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -3955\beta_{13} + 4677\beta_{11} + 7910\beta_{9} - 7188\beta_{7} ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 899\beta_{15} - 359\beta_{14} + 718\beta_{10} - 718\beta_{6} \) Copy content Toggle raw display

Character values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−0.912166 + 1.47240i
0.0537601 + 1.73122i
−1.73122 0.0537601i
1.47240 0.912166i
−1.47240 + 0.912166i
1.73122 + 0.0537601i
−0.0537601 1.73122i
0.912166 1.47240i
−0.912166 1.47240i
0.0537601 1.73122i
−1.73122 + 0.0537601i
1.47240 + 0.912166i
−1.47240 0.912166i
1.73122 0.0537601i
−0.0537601 + 1.73122i
0.912166 + 1.47240i
0 −1.73122 + 0.0537601i 0 0 0 −0.560232 0.560232i 0 2.99422 0.186141i 0
257.2 0 −1.47240 0.912166i 0 0 0 −1.78498 1.78498i 0 1.33591 + 2.68614i 0
257.3 0 −0.912166 1.47240i 0 0 0 1.78498 + 1.78498i 0 −1.33591 + 2.68614i 0
257.4 0 −0.0537601 + 1.73122i 0 0 0 −0.560232 0.560232i 0 −2.99422 0.186141i 0
257.5 0 0.0537601 1.73122i 0 0 0 0.560232 + 0.560232i 0 −2.99422 0.186141i 0
257.6 0 0.912166 + 1.47240i 0 0 0 −1.78498 1.78498i 0 −1.33591 + 2.68614i 0
257.7 0 1.47240 + 0.912166i 0 0 0 1.78498 + 1.78498i 0 1.33591 + 2.68614i 0
257.8 0 1.73122 0.0537601i 0 0 0 0.560232 + 0.560232i 0 2.99422 0.186141i 0
593.1 0 −1.73122 0.0537601i 0 0 0 −0.560232 + 0.560232i 0 2.99422 + 0.186141i 0
593.2 0 −1.47240 + 0.912166i 0 0 0 −1.78498 + 1.78498i 0 1.33591 2.68614i 0
593.3 0 −0.912166 + 1.47240i 0 0 0 1.78498 1.78498i 0 −1.33591 2.68614i 0
593.4 0 −0.0537601 1.73122i 0 0 0 −0.560232 + 0.560232i 0 −2.99422 + 0.186141i 0
593.5 0 0.0537601 + 1.73122i 0 0 0 0.560232 0.560232i 0 −2.99422 + 0.186141i 0
593.6 0 0.912166 1.47240i 0 0 0 −1.78498 + 1.78498i 0 −1.33591 2.68614i 0
593.7 0 1.47240 0.912166i 0 0 0 1.78498 1.78498i 0 1.33591 2.68614i 0
593.8 0 1.73122 + 0.0537601i 0 0 0 0.560232 0.560232i 0 2.99422 + 0.186141i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.8
Significant digits:
Format:

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
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.r.f 16
3.b odd 2 1 inner 600.2.r.f 16
4.b odd 2 1 1200.2.v.m 16
5.b even 2 1 inner 600.2.r.f 16
5.c odd 4 2 inner 600.2.r.f 16
12.b even 2 1 1200.2.v.m 16
15.d odd 2 1 inner 600.2.r.f 16
15.e even 4 2 inner 600.2.r.f 16
20.d odd 2 1 1200.2.v.m 16
20.e even 4 2 1200.2.v.m 16
60.h even 2 1 1200.2.v.m 16
60.l odd 4 2 1200.2.v.m 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.r.f 16 1.a even 1 1 trivial
600.2.r.f 16 3.b odd 2 1 inner
600.2.r.f 16 5.b even 2 1 inner
600.2.r.f 16 5.c odd 4 2 inner
600.2.r.f 16 15.d odd 2 1 inner
600.2.r.f 16 15.e even 4 2 inner
1200.2.v.m 16 4.b odd 2 1
1200.2.v.m 16 12.b even 2 1
1200.2.v.m 16 20.d odd 2 1
1200.2.v.m 16 20.e even 4 2
1200.2.v.m 16 60.h even 2 1
1200.2.v.m 16 60.l odd 4 2

Hecke kernels

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

\( T_{7}^{8} + 41T_{7}^{4} + 16 \) Copy content Toggle raw display
\( T_{17}^{8} + 1649T_{17}^{4} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 7 T^{12} - 32 T^{8} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 41 T^{4} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 19 T^{2} + 16)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 1449 T^{4} + 331776)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1649 T^{4} + 256)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 74 T^{2} + 841)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 5904 T^{4} + 331776)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 112 T^{2} + 1024)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 5 T - 2)^{8} \) Copy content Toggle raw display
$37$ \( (T^{8} + 656 T^{4} + 4096)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 139 T^{2} + 4624)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 9401 T^{4} + 11316496)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 4096)^{4} \) Copy content Toggle raw display
$53$ \( (T^{8} + 13328 T^{4} + 4096)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 172 T^{2} + 4096)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 3 T - 6)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 9)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 112 T^{2} + 1024)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 19481 T^{4} + 59969536)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 84 T^{2} + 576)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 17729 T^{4} + 16777216)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 259 T^{2} + 64)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 10073 T^{4} + 21381376)^{2} \) Copy content Toggle raw display
show more
show less