Properties

Label 1050.3.p.c
Level $1050$
Weight $3$
Character orbit 1050.p
Analytic conductor $28.610$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.151613669376.6
Defining polynomial: \( x^{8} - 12x^{6} + 95x^{4} - 588x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{2}) q^{2} + ( - \beta_{3} - 1) q^{3} + (2 \beta_{3} - 2) q^{4} + (\beta_{6} - 2 \beta_{2}) q^{6} + (3 \beta_{7} + 3 \beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{6} q^{8} + 3 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{2}) q^{2} + ( - \beta_{3} - 1) q^{3} + (2 \beta_{3} - 2) q^{4} + (\beta_{6} - 2 \beta_{2}) q^{6} + (3 \beta_{7} + 3 \beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{6} q^{8} + 3 \beta_{3} q^{9} + ( - \beta_{7} + \beta_{6} + \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 1) q^{11} + ( - 2 \beta_{3} + 4) q^{12} + (2 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + 8 \beta_{2} + 2 \beta_1 + 1) q^{13} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 1) q^{14} - 4 \beta_{3} q^{16} + ( - 3 \beta_{7} - \beta_{6} + 5 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{17} + 3 \beta_{2} q^{18} + (7 \beta_{6} - 2 \beta_{4} - 6 \beta_{3} + 8 \beta_{2} + \beta_1 + 12) q^{19} + ( - 4 \beta_{7} - 4 \beta_{2} - 5 \beta_1) q^{21} + (\beta_{6} + \beta_{5} + 2 \beta_{4} + 3) q^{22} + (4 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - \beta_{4} + 15 \beta_{3} - \beta_{2} + 2 \beta_1) q^{23} + ( - 4 \beta_{6} + 2 \beta_{2}) q^{24} + (\beta_{6} - 2 \beta_{4} + 6 \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 12) q^{26} + ( - 6 \beta_{3} + 3) q^{27} + ( - 4 \beta_{7} - 4 \beta_{2} + 2 \beta_1) q^{28} + ( - 4 \beta_{7} - 15 \beta_{6} - \beta_{5} - 2 \beta_{4} + 4 \beta_1 - 3) q^{29} + (2 \beta_{7} + 10 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} + 8 \beta_{3} - 4 \beta_{2} + \cdots + 8) q^{31}+ \cdots + (3 \beta_{7} - 3 \beta_{6} - 3 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} - 8 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{3} - 8 q^{4} + 12 q^{9} - 4 q^{11} + 24 q^{12} - 16 q^{14} - 16 q^{16} - 24 q^{17} + 72 q^{19} + 24 q^{22} + 60 q^{23} - 72 q^{26} - 24 q^{29} + 96 q^{31} + 12 q^{33} - 48 q^{36} - 24 q^{37} - 180 q^{38} - 12 q^{39} + 12 q^{42} - 112 q^{43} - 8 q^{44} + 32 q^{46} + 84 q^{47} - 264 q^{49} + 24 q^{51} + 24 q^{52} + 44 q^{53} + 40 q^{56} - 144 q^{57} + 104 q^{58} + 312 q^{59} - 204 q^{61} + 64 q^{64} - 36 q^{66} + 120 q^{67} + 48 q^{68} - 64 q^{71} + 84 q^{73} - 16 q^{74} + 228 q^{77} + 144 q^{78} - 144 q^{79} - 36 q^{81} - 60 q^{82} + 176 q^{86} + 36 q^{87} - 24 q^{88} - 336 q^{89} - 296 q^{91} - 240 q^{92} - 96 q^{93} + 36 q^{94} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 12x^{6} + 95x^{4} - 588x^{2} + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{7} + 475\nu^{5} - 1045\nu^{3} + 6468\nu ) / 32585 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -12\nu^{6} + 95\nu^{4} - 1140\nu^{2} + 7056 ) / 4655 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -23\nu^{6} + 570\nu^{4} - 2185\nu^{2} + 13524 ) / 4655 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 18 ) / 95 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -12\nu^{7} + 95\nu^{5} - 209\nu^{3} + 2401\nu ) / 6517 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 12\nu^{5} + 95\nu^{3} - 588\nu ) / 343 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - 6\beta_{3} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{7} + 7\beta_{6} + 5\beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{4} - 23\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{7} + 95\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -95\beta_{5} + 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -665\beta_{6} + 665\beta_{2} + 113\beta_1 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
2.56149 0.662382i
−1.85439 + 1.88713i
1.85439 1.88713i
−2.56149 + 0.662382i
2.56149 + 0.662382i
−1.85439 1.88713i
1.85439 + 1.88713i
−2.56149 0.662382i
−0.707107 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i −0.440173 6.98615i 2.82843 1.50000 + 2.59808i 0
451.2 −0.707107 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i 3.97571 + 5.76140i 2.82843 1.50000 + 2.59808i 0
451.3 0.707107 + 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i −3.97571 5.76140i −2.82843 1.50000 + 2.59808i 0
451.4 0.707107 + 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i 0.440173 + 6.98615i −2.82843 1.50000 + 2.59808i 0
901.1 −0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i −0.440173 + 6.98615i 2.82843 1.50000 2.59808i 0
901.2 −0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i 3.97571 5.76140i 2.82843 1.50000 2.59808i 0
901.3 0.707107 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i −3.97571 + 5.76140i −2.82843 1.50000 2.59808i 0
901.4 0.707107 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i 0.440173 6.98615i −2.82843 1.50000 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.4
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{8} + 4T_{11}^{7} + 58T_{11}^{6} + 16T_{11}^{5} + 1954T_{11}^{4} + 2440T_{11}^{3} + 15940T_{11}^{2} - 16376T_{11} + 31684 \) Copy content Toggle raw display
\( T_{17}^{8} + 24 T_{17}^{7} - 506 T_{17}^{6} - 16752 T_{17}^{5} + 368850 T_{17}^{4} + 4615176 T_{17}^{3} - 31116836 T_{17}^{2} - 432808296 T_{17} + 4284749764 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 132 T^{6} + 8183 T^{4} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{8} + 4 T^{7} + 58 T^{6} + \cdots + 31684 \) Copy content Toggle raw display
$13$ \( T^{8} + 540 T^{6} + 39206 T^{4} + \cdots + 3500641 \) Copy content Toggle raw display
$17$ \( T^{8} + 24 T^{7} + \cdots + 4284749764 \) Copy content Toggle raw display
$19$ \( T^{8} - 72 T^{7} + \cdots + 648364369 \) Copy content Toggle raw display
$23$ \( T^{8} - 60 T^{7} + \cdots + 258309184 \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{3} - 1324 T^{2} + \cdots - 109496)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 96 T^{7} + \cdots + 641507584 \) Copy content Toggle raw display
$37$ \( T^{8} + 24 T^{7} + \cdots + 749278940881 \) Copy content Toggle raw display
$41$ \( T^{8} + 4764 T^{6} + \cdots + 16384512004 \) Copy content Toggle raw display
$43$ \( (T^{4} + 56 T^{3} - 4504 T^{2} + \cdots - 6447728)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 84 T^{7} + \cdots + 1293619615876 \) Copy content Toggle raw display
$53$ \( T^{8} - 44 T^{7} + \cdots + 157223352196 \) Copy content Toggle raw display
$59$ \( T^{8} - 312 T^{7} + \cdots + 14372454374404 \) Copy content Toggle raw display
$61$ \( T^{8} + 204 T^{7} + \cdots + 16806974336161 \) Copy content Toggle raw display
$67$ \( T^{8} - 120 T^{7} + \cdots + 122387325921 \) Copy content Toggle raw display
$71$ \( (T^{4} + 32 T^{3} - 11488 T^{2} + \cdots + 27042304)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 84 T^{7} + \cdots + 19\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 334870712077729 \) Copy content Toggle raw display
$83$ \( T^{8} + 10932 T^{6} + \cdots + 29997547204 \) Copy content Toggle raw display
$89$ \( T^{8} + 336 T^{7} + \cdots + 9916238788036 \) Copy content Toggle raw display
$97$ \( T^{8} + 29012 T^{6} + \cdots + 2745758363089 \) Copy content Toggle raw display
show more
show less