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 $2$

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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: \(x^{8} - 12 x^{6} + 95 x^{4} - 588 x^{2} + 2401\)
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.

\(f(q)\) \(=\) \( q + ( \beta_{2} - \beta_{6} ) q^{2} + ( -1 - \beta_{3} ) q^{3} + ( -2 + 2 \beta_{3} ) q^{4} + ( -2 \beta_{2} + \beta_{6} ) q^{6} + ( 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{7} ) q^{7} + 2 \beta_{6} q^{8} + 3 \beta_{3} q^{9} +O(q^{10})\) \( q + ( \beta_{2} - \beta_{6} ) q^{2} + ( -1 - \beta_{3} ) q^{3} + ( -2 + 2 \beta_{3} ) q^{4} + ( -2 \beta_{2} + \beta_{6} ) q^{6} + ( 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{7} ) q^{7} + 2 \beta_{6} q^{8} + 3 \beta_{3} q^{9} + ( -1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{11} + ( 4 - 2 \beta_{3} ) q^{12} + ( 1 + 2 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{13} + ( -1 - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{14} -4 \beta_{3} q^{16} + ( -2 - \beta_{2} - 2 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{17} + 3 \beta_{2} q^{18} + ( 12 + \beta_{1} + 8 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + 7 \beta_{6} ) q^{19} + ( -5 \beta_{1} - 4 \beta_{2} - 4 \beta_{7} ) q^{21} + ( 3 + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{22} + ( 2 \beta_{1} - \beta_{2} + 15 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} ) q^{23} + ( 2 \beta_{2} - 4 \beta_{6} ) q^{24} + ( -12 - 4 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{26} + ( 3 - 6 \beta_{3} ) q^{27} + ( 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{7} ) q^{28} + ( -3 + 4 \beta_{1} - 2 \beta_{4} - \beta_{5} - 15 \beta_{6} - 4 \beta_{7} ) q^{29} + ( 8 - 4 \beta_{2} + 8 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} + 10 \beta_{6} + 2 \beta_{7} ) q^{31} -4 \beta_{2} q^{32} + ( 2 + 3 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{33} + ( -1 + 10 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - 3 \beta_{6} + 10 \beta_{7} ) q^{34} -6 q^{36} + ( 2 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} + 10 \beta_{5} - 2 \beta_{6} ) q^{37} + ( -15 + 4 \beta_{2} - 15 \beta_{3} - \beta_{4} - \beta_{5} - 12 \beta_{6} - 4 \beta_{7} ) q^{38} + ( -3 - 4 \beta_{1} - 12 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{39} + ( -1 + 7 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} - 8 \beta_{5} - 6 \beta_{6} + 7 \beta_{7} ) q^{41} + ( -1 + 5 \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{42} + ( -14 + 8 \beta_{1} - 8 \beta_{4} - 4 \beta_{5} - 26 \beta_{6} - 8 \beta_{7} ) q^{43} + ( 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 4 \beta_{6} + 4 \beta_{7} ) q^{44} + ( 8 - 4 \beta_{1} + 12 \beta_{2} - 8 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{46} + ( 14 + 13 \beta_{1} + 5 \beta_{2} - 7 \beta_{3} + 10 \beta_{4} - 8 \beta_{6} ) q^{47} + ( -4 + 8 \beta_{3} ) q^{48} + ( -48 + 30 \beta_{3} - 5 \beta_{4} - 8 \beta_{5} ) q^{49} + ( 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} - 5 \beta_{4} - 10 \beta_{5} + 6 \beta_{7} ) q^{51} + ( 2 - 8 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 12 \beta_{6} - 4 \beta_{7} ) q^{52} + ( 11 + 18 \beta_{1} + 4 \beta_{2} - 11 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} ) q^{53} + ( -3 \beta_{2} - 3 \beta_{6} ) q^{54} + ( 6 - 2 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} ) q^{56} + ( -18 - \beta_{1} + 4 \beta_{4} + 2 \beta_{5} - 22 \beta_{6} + \beta_{7} ) q^{57} + ( -2 \beta_{1} - 6 \beta_{2} + 26 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{58} + ( 26 - 12 \beta_{2} + 26 \beta_{3} - 5 \beta_{4} - 5 \beta_{5} + 3 \beta_{6} - 21 \beta_{7} ) q^{59} + ( -34 - 10 \beta_{1} - 2 \beta_{2} + 17 \beta_{3} - 10 \beta_{4} + 8 \beta_{6} ) q^{61} + ( 10 - 8 \beta_{1} + 8 \beta_{2} - 20 \beta_{3} + 2 \beta_{5} - 4 \beta_{6} - 8 \beta_{7} ) q^{62} + ( 9 \beta_{1} + 3 \beta_{2} + 3 \beta_{7} ) q^{63} + 8 q^{64} + ( -3 + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{66} + ( 30 + 6 \beta_{1} + 21 \beta_{2} - 30 \beta_{3} - 3 \beta_{6} + 3 \beta_{7} ) q^{67} + ( 8 - 6 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 10 \beta_{4} + 4 \beta_{6} ) q^{68} + ( 15 - 6 \beta_{1} + 2 \beta_{2} - 30 \beta_{3} + 3 \beta_{5} - \beta_{6} - 6 \beta_{7} ) q^{69} + ( -8 + 4 \beta_{1} - 24 \beta_{4} - 12 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{71} + ( -6 \beta_{2} + 6 \beta_{6} ) q^{72} + ( 7 + 7 \beta_{3} - 24 \beta_{4} - 24 \beta_{5} + 10 \beta_{6} + 10 \beta_{7} ) q^{73} + ( -4 + 20 \beta_{1} + 9 \beta_{2} + 4 \beta_{3} - 10 \beta_{6} + 10 \beta_{7} ) q^{74} + ( -12 - 2 \beta_{1} - 32 \beta_{2} + 24 \beta_{3} - 4 \beta_{5} + 16 \beta_{6} - 2 \beta_{7} ) q^{76} + ( 23 + \beta_{1} - 2 \beta_{2} + 11 \beta_{3} + 4 \beta_{4} + 5 \beta_{5} - 2 \beta_{7} ) q^{77} + ( 18 + 4 \beta_{1} + 4 \beta_{4} + 2 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{78} + ( -\beta_{1} - 7 \beta_{2} - 36 \beta_{3} - 12 \beta_{4} - 24 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} ) q^{79} + ( -9 + 9 \beta_{3} ) q^{81} + ( -10 - 16 \beta_{1} - 7 \beta_{2} + 5 \beta_{3} - 7 \beta_{4} + 9 \beta_{6} ) q^{82} + ( -16 + 7 \beta_{1} + 40 \beta_{2} + 32 \beta_{3} + \beta_{5} - 20 \beta_{6} + 7 \beta_{7} ) q^{83} + ( 2 \beta_{1} + 10 \beta_{2} + 10 \beta_{7} ) q^{84} + ( -8 \beta_{1} - 26 \beta_{2} + 44 \beta_{3} - 8 \beta_{4} - 16 \beta_{5} + 18 \beta_{6} - 16 \beta_{7} ) q^{86} + ( 3 - 7 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 26 \beta_{6} + 12 \beta_{7} ) q^{87} + ( -6 - 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{88} + ( -56 + 11 \beta_{1} + 21 \beta_{2} + 28 \beta_{3} - \beta_{4} + 10 \beta_{6} ) q^{89} + ( -36 - 7 \beta_{2} - 2 \beta_{3} - 16 \beta_{4} - 6 \beta_{5} - 28 \beta_{6} + 7 \beta_{7} ) q^{91} + ( -30 + 4 \beta_{1} + 4 \beta_{4} + 2 \beta_{5} - 10 \beta_{6} - 4 \beta_{7} ) q^{92} + ( -2 \beta_{1} + 12 \beta_{2} - 24 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} - 14 \beta_{6} - 4 \beta_{7} ) q^{93} + ( 3 + 17 \beta_{2} + 3 \beta_{3} - 13 \beta_{4} - 13 \beta_{5} - 14 \beta_{6} + 20 \beta_{7} ) q^{94} + ( 4 \beta_{2} + 4 \beta_{6} ) q^{96} + ( -24 + 26 \beta_{1} + 48 \beta_{3} - 3 \beta_{5} + 26 \beta_{7} ) q^{97} + ( -16 \beta_{1} - 31 \beta_{2} + 56 \beta_{6} - 10 \beta_{7} ) q^{98} + ( -3 - 3 \beta_{1} - 3 \beta_{6} + 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} - 8 q^{4} + 12 q^{9} + O(q^{10}) \) \( 8 q - 12 q^{3} - 8 q^{4} + 12 q^{9} - 4 q^{11} + 24 q^{12} - 16 q^{14} - 16 q^{16} - 24 q^{17} + 72 q^{19} + 24 q^{22} + 60 q^{23} - 72 q^{26} - 24 q^{29} + 96 q^{31} + 12 q^{33} - 48 q^{36} - 24 q^{37} - 180 q^{38} - 12 q^{39} + 12 q^{42} - 112 q^{43} - 8 q^{44} + 32 q^{46} + 84 q^{47} - 264 q^{49} + 24 q^{51} + 24 q^{52} + 44 q^{53} + 40 q^{56} - 144 q^{57} + 104 q^{58} + 312 q^{59} - 204 q^{61} + 64 q^{64} - 36 q^{66} + 120 q^{67} + 48 q^{68} - 64 q^{71} + 84 q^{73} - 16 q^{74} + 228 q^{77} + 144 q^{78} - 144 q^{79} - 36 q^{81} - 60 q^{82} + 176 q^{86} + 36 q^{87} - 24 q^{88} - 336 q^{89} - 296 q^{91} - 240 q^{92} - 96 q^{93} + 36 q^{94} - 24 q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -11 \nu^{7} + 475 \nu^{5} - 1045 \nu^{3} + 6468 \nu \)\()/32585\)
\(\beta_{3}\)\(=\)\((\)\( -12 \nu^{6} + 95 \nu^{4} - 1140 \nu^{2} + 7056 \)\()/4655\)
\(\beta_{4}\)\(=\)\((\)\( -23 \nu^{6} + 570 \nu^{4} - 2185 \nu^{2} + 13524 \)\()/4655\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 18 \)\()/95\)
\(\beta_{6}\)\(=\)\((\)\( -12 \nu^{7} + 95 \nu^{5} - 209 \nu^{3} + 2401 \nu \)\()/6517\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} - 12 \nu^{5} + 95 \nu^{3} - 588 \nu \)\()/343\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} - 6 \beta_{3} + 6\)
\(\nu^{3}\)\(=\)\(5 \beta_{7} + 7 \beta_{6} + 5 \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(12 \beta_{4} - 23 \beta_{3}\)
\(\nu^{5}\)\(=\)\(11 \beta_{7} + 95 \beta_{2}\)
\(\nu^{6}\)\(=\)\(-95 \beta_{5} + 18\)
\(\nu^{7}\)\(=\)\(-665 \beta_{6} + 665 \beta_{2} + 113 \beta_{1}\)

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 \(S_{3}^{\mathrm{new}}(1050, [\chi])\):

\(T_{11}^{8} + \cdots\)
\(T_{17}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$3$ \( ( 3 + 3 T + T^{2} )^{4} \)
$5$ \( T^{8} \)
$7$ \( 5764801 + 316932 T^{2} + 8183 T^{4} + 132 T^{6} + T^{8} \)
$11$ \( 31684 - 16376 T + 15940 T^{2} + 2440 T^{3} + 1954 T^{4} + 16 T^{5} + 58 T^{6} + 4 T^{7} + T^{8} \)
$13$ \( 3500641 + 686940 T^{2} + 39206 T^{4} + 540 T^{6} + T^{8} \)
$17$ \( 4284749764 - 432808296 T - 31116836 T^{2} + 4615176 T^{3} + 368850 T^{4} - 16752 T^{5} - 506 T^{6} + 24 T^{7} + T^{8} \)
$19$ \( 648364369 - 647167608 T + 218991024 T^{2} - 3659904 T^{3} - 563785 T^{4} + 10368 T^{5} + 1584 T^{6} - 72 T^{7} + T^{8} \)
$23$ \( 258309184 - 113404032 T + 32879392 T^{2} - 5494272 T^{3} + 667272 T^{4} - 49008 T^{5} + 2548 T^{6} - 60 T^{7} + T^{8} \)
$29$ \( ( -109496 - 24384 T - 1324 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$31$ \( 641507584 - 150752256 T + 2082816 T^{2} + 2285568 T^{3} - 17680 T^{4} - 36864 T^{5} + 3456 T^{6} - 96 T^{7} + T^{8} \)
$37$ \( 749278940881 - 19673561352 T + 2031377734 T^{2} - 1775232 T^{3} + 2742363 T^{4} + 3456 T^{5} + 2326 T^{6} + 24 T^{7} + T^{8} \)
$41$ \( 16384512004 + 1721127432 T^{2} + 6279368 T^{4} + 4764 T^{6} + T^{8} \)
$43$ \( ( -6447728 - 367712 T - 4504 T^{2} + 56 T^{3} + T^{4} )^{2} \)
$47$ \( 1293619615876 - 285594611400 T + 25500598308 T^{2} - 989836200 T^{3} + 9645938 T^{4} + 331128 T^{5} - 1590 T^{6} - 84 T^{7} + T^{8} \)
$53$ \( 157223352196 + 52471490648 T + 18743330708 T^{2} - 376129960 T^{3} + 15073330 T^{4} - 128000 T^{5} + 5042 T^{6} - 44 T^{7} + T^{8} \)
$59$ \( 14372454374404 - 1578478393128 T + 35957161516 T^{2} + 2397423912 T^{3} - 6356190 T^{4} - 1796496 T^{5} + 38206 T^{6} - 312 T^{7} + T^{8} \)
$61$ \( 16806974336161 + 1334626672788 T + 30432207354 T^{2} - 388704312 T^{3} - 16611997 T^{4} + 243576 T^{5} + 15066 T^{6} + 204 T^{7} + T^{8} \)
$67$ \( 122387325921 + 7556522400 T + 1851922440 T^{2} - 169497360 T^{3} + 13439439 T^{4} - 432000 T^{5} + 10440 T^{6} - 120 T^{7} + T^{8} \)
$71$ \( ( 27042304 - 217856 T - 11488 T^{2} + 32 T^{3} + T^{4} )^{2} \)
$73$ \( 1984264241296561 + 13784208044964 T - 667171971502 T^{2} - 4856414136 T^{3} + 193092123 T^{4} + 1318296 T^{5} - 13342 T^{6} - 84 T^{7} + T^{8} \)
$79$ \( 334870712077729 - 11217723344784 T + 441949702432 T^{2} - 3053611296 T^{3} + 83049135 T^{4} + 705312 T^{5} + 24352 T^{6} + 144 T^{7} + T^{8} \)
$83$ \( 29997547204 + 12752786136 T^{2} + 24918152 T^{4} + 10932 T^{6} + T^{8} \)
$89$ \( 9916238788036 - 284959850952 T - 27595327092 T^{2} + 871437960 T^{3} + 106021010 T^{4} + 3235680 T^{5} + 47262 T^{6} + 336 T^{7} + T^{8} \)
$97$ \( 2745758363089 + 203590226420 T^{2} + 152995446 T^{4} + 29012 T^{6} + T^{8} \)
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