Properties

Label 1050.3.q.b
Level $1050$
Weight $3$
Character orbit 1050.q
Analytic conductor $28.610$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.22986704741655040229376.1
Defining polynomial: \( x^{16} - 31x^{12} + 880x^{8} - 2511x^{4} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} + ( - \beta_{9} - \beta_1) q^{3} + (2 \beta_{3} + 2) q^{4} + ( - 2 \beta_{12} - \beta_{6}) q^{6} + (\beta_{15} - 2 \beta_{8} - 2 \beta_{7}) q^{7} + 2 \beta_{7} q^{8} + 3 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{2} + ( - \beta_{9} - \beta_1) q^{3} + (2 \beta_{3} + 2) q^{4} + ( - 2 \beta_{12} - \beta_{6}) q^{6} + (\beta_{15} - 2 \beta_{8} - 2 \beta_{7}) q^{7} + 2 \beta_{7} q^{8} + 3 \beta_{3} q^{9} + ( - \beta_{14} - \beta_{12} + \beta_{10} - \beta_{3} - 1) q^{11} + ( - 4 \beta_{9} + 2 \beta_1) q^{12} + (6 \beta_{11} + \beta_{9} + 2 \beta_{8} + 4 \beta_{7} - 2 \beta_{2} - 2 \beta_1) q^{13} + (\beta_{13} + 3 \beta_{5} - 2 \beta_{3} + 1) q^{14} + 4 \beta_{3} q^{16} + ( - 3 \beta_{15} - 2 \beta_{11} + 2 \beta_{9} - \beta_{7} - 5 \beta_{4} - 5 \beta_{2} + 2 \beta_1) q^{17} + (3 \beta_{11} + 3 \beta_{7}) q^{18} + ( - \beta_{14} - 7 \beta_{12} + 8 \beta_{6} + 2 \beta_{5} - 6 \beta_{3} - 12) q^{19} + ( - \beta_{14} + \beta_{12} + 4 \beta_{10} + \beta_{6}) q^{21} + ( - 3 \beta_{9} - \beta_{7} - 2 \beta_{4} - \beta_{2}) q^{22} + ( - 2 \beta_{15} - 5 \beta_{11} + 2 \beta_{8} - \beta_{4} - 2 \beta_{2} + 15 \beta_1) q^{23} + ( - 2 \beta_{12} - 4 \beta_{6}) q^{24} + ( - 4 \beta_{14} + \beta_{12} + 3 \beta_{6} - 2 \beta_{5} - 6 \beta_{3} - 12) q^{26} + ( - 3 \beta_{9} + 6 \beta_1) q^{27} + (6 \beta_{15} + 2 \beta_{8} + 2 \beta_{7}) q^{28} + ( - 8 \beta_{14} - \beta_{13} + 4 \beta_{12} - 4 \beta_{10} - 7 \beta_{6} + \beta_{5} + 3) q^{29} + ( - 2 \beta_{14} + 4 \beta_{13} - 4 \beta_{12} - 2 \beta_{10} - 8 \beta_{6} - 8 \beta_{3} + \cdots + 8) q^{31}+ \cdots + (6 \beta_{14} - 3 \beta_{12} + 3 \beta_{10} - 9 \beta_{6} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} - 24 q^{9} - 8 q^{11} + 32 q^{14} - 32 q^{16} - 144 q^{19} - 144 q^{26} + 48 q^{29} + 192 q^{31} - 96 q^{36} + 24 q^{39} + 16 q^{44} + 64 q^{46} + 528 q^{49} + 48 q^{51} + 80 q^{56} - 624 q^{59} - 408 q^{61} - 128 q^{64} - 72 q^{66} - 128 q^{71} + 32 q^{74} + 288 q^{79} - 72 q^{81} + 352 q^{86} + 672 q^{89} - 592 q^{91} - 72 q^{94} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 31x^{12} + 880x^{8} - 2511x^{4} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} + 32689\nu^{2} ) / 55440 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{12} + 11129 ) / 3080 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31\nu^{12} - 880\nu^{8} + 27280\nu^{4} - 77841 ) / 71280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 799\nu^{12} - 27280\nu^{8} + 703120\nu^{4} - 2006289 ) / 498960 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{14} + 16849\nu^{2} ) / 7920 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{15} + 127\nu^{13} - 4180\nu^{9} + 111760\nu^{5} - 169461\nu^{3} - 318897\nu ) / 374220 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{15} + 127\nu^{13} - 4180\nu^{9} + 111760\nu^{5} + 169461\nu^{3} - 318897\nu ) / 374220 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{15} + 31\nu^{13} - 880\nu^{9} + 27280\nu^{5} - 74307\nu^{3} - 77841\nu ) / 71280 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 31\nu^{14} - 880\nu^{10} + 25498\nu^{6} - 6561\nu^{2} ) / 112266 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -3\nu^{15} - 31\nu^{13} + 880\nu^{9} - 27280\nu^{5} - 74307\nu^{3} + 77841\nu ) / 71280 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2683 \nu^{15} - 5301 \nu^{13} - 85360 \nu^{11} + 150480 \nu^{9} + 2361040 \nu^{7} - 4023360 \nu^{5} - 6737013 \nu^{3} + 1121931 \nu ) / 13471920 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3007\nu^{15} + 729\nu^{13} - 85360\nu^{11} + 2361040\nu^{7} - 636417\nu^{3} + 10358361\nu ) / 13471920 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 601\nu^{14} - 19360\nu^{10} + 528880\nu^{6} - 1509111\nu^{2} ) / 641520 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1159 \nu^{15} + 3720 \nu^{13} + 35200 \nu^{11} - 105600 \nu^{9} - 1019920 \nu^{7} + 3059760 \nu^{5} + 2910249 \nu^{3} - 787320 \nu ) / 4490640 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1240\nu^{15} + 243\nu^{13} - 35200\nu^{11} + 1019920\nu^{7} - 262440\nu^{3} + 7943427\nu ) / 4490640 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{10} - 4\beta_{8} - 7\beta_{7} + 7\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{4} + 31\beta_{3} - 7\beta_{2} + 31 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19\beta_{15} + 19\beta_{14} - 40\beta_{12} + 40\beta_{11} + 19\beta_{8} + 19\beta_{7} - 40\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{13} + 77\beta_{9} - 20\beta_{5} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 97\beta_{15} - 97\beta_{14} - 120\beta_{12} - 120\beta_{11} - 97\beta_{10} - 120\beta_{7} + 97\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -217\beta_{4} + 799\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -508\beta_{10} + 508\beta_{8} - 651\beta_{7} - 651\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -1159\beta_{13} + 4207\beta_{9} - 4207\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2683 \beta_{15} - 2683 \beta_{14} - 3477 \beta_{12} - 3477 \beta_{11} + 2683 \beta_{8} + 2683 \beta_{7} - 3477 \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3080\beta_{2} - 11129 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 14209 \beta_{15} - 14209 \beta_{14} + 32689 \beta_{12} - 32689 \beta_{11} - 14209 \beta_{10} - 32689 \beta_{7} + 14209 \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 32689\beta_{5} - 117943\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 75316\beta_{10} + 75316\beta_{8} + 173383\beta_{7} - 173383\beta_{6} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(-\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−1.25838 + 0.337183i
2.22431 0.596002i
0.596002 + 2.22431i
−0.337183 1.25838i
−2.22431 + 0.596002i
1.25838 0.337183i
0.337183 + 1.25838i
−0.596002 2.22431i
−1.25838 0.337183i
2.22431 + 0.596002i
0.596002 2.22431i
−0.337183 + 1.25838i
−2.22431 0.596002i
1.25838 + 0.337183i
0.337183 1.25838i
−0.596002 + 2.22431i
−1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −6.98615 + 0.440173i 2.82843i −1.50000 2.59808i 0
199.2 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 5.76140 3.97571i 2.82843i −1.50000 2.59808i 0
199.3 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −6.98615 + 0.440173i 2.82843i −1.50000 2.59808i 0
199.4 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 5.76140 3.97571i 2.82843i −1.50000 2.59808i 0
199.5 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −5.76140 + 3.97571i 2.82843i −1.50000 2.59808i 0
199.6 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 6.98615 0.440173i 2.82843i −1.50000 2.59808i 0
199.7 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −5.76140 + 3.97571i 2.82843i −1.50000 2.59808i 0
199.8 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 6.98615 0.440173i 2.82843i −1.50000 2.59808i 0
649.1 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −6.98615 0.440173i 2.82843i −1.50000 + 2.59808i 0
649.2 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i 5.76140 + 3.97571i 2.82843i −1.50000 + 2.59808i 0
649.3 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −6.98615 0.440173i 2.82843i −1.50000 + 2.59808i 0
649.4 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 5.76140 + 3.97571i 2.82843i −1.50000 + 2.59808i 0
649.5 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −5.76140 3.97571i 2.82843i −1.50000 + 2.59808i 0
649.6 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i 6.98615 + 0.440173i 2.82843i −1.50000 + 2.59808i 0
649.7 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −5.76140 3.97571i 2.82843i −1.50000 + 2.59808i 0
649.8 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 6.98615 + 0.440173i 2.82843i −1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.q.b 16
5.b even 2 1 inner 1050.3.q.b 16
5.c odd 4 1 1050.3.p.c 8
5.c odd 4 1 1050.3.p.d yes 8
7.d odd 6 1 inner 1050.3.q.b 16
35.i odd 6 1 inner 1050.3.q.b 16
35.k even 12 1 1050.3.p.c 8
35.k even 12 1 1050.3.p.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.3.p.c 8 5.c odd 4 1
1050.3.p.c 8 35.k even 12 1
1050.3.p.d yes 8 5.c odd 4 1
1050.3.p.d yes 8 35.k even 12 1
1050.3.q.b 16 1.a even 1 1 trivial
1050.3.q.b 16 5.b even 2 1 inner
1050.3.q.b 16 7.d odd 6 1 inner
1050.3.q.b 16 35.i odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} + 4T_{11}^{7} + 58T_{11}^{6} + 16T_{11}^{5} + 1954T_{11}^{4} + 2440T_{11}^{3} + 15940T_{11}^{2} - 16376T_{11} + 31684 \) acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 132 T^{6} + 8183 T^{4} + \cdots + 5764801)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 4 T^{7} + 58 T^{6} + 16 T^{5} + \cdots + 31684)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 540 T^{6} + 39206 T^{4} + \cdots + 3500641)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 1588 T^{14} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{8} + 72 T^{7} + 1584 T^{6} + \cdots + 648364369)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 1496 T^{14} + \cdots + 66\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} - 12 T^{3} - 1324 T^{2} + \cdots - 109496)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 96 T^{7} + 3456 T^{6} + \cdots + 641507584)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 4076 T^{14} + \cdots + 56\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( (T^{8} + 4764 T^{6} + \cdots + 16384512004)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 12144 T^{6} + \cdots + 41573196361984)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 10236 T^{14} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} - 8148 T^{14} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{8} + 312 T^{7} + \cdots + 14372454374404)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 204 T^{7} + \cdots + 16806974336161)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} - 6480 T^{14} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{4} + 32 T^{3} - 11488 T^{2} + \cdots + 27042304)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + 33740 T^{14} + \cdots + 39\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( (T^{8} - 144 T^{7} + \cdots + 334870712077729)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 10932 T^{6} + \cdots + 29997547204)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 336 T^{7} + \cdots + 9916238788036)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 29012 T^{6} + \cdots + 2745758363089)^{2} \) Copy content Toggle raw display
show more
show less