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

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

 $$f(q)$$ $$=$$ $$q + ( - \beta_{6} + \beta_{2}) q^{2} + ( - \beta_{3} - 1) q^{3} + (2 \beta_{3} - 2) q^{4} + (\beta_{6} - 2 \beta_{2}) q^{6} + (3 \beta_{7} + 3 \beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{6} q^{8} + 3 \beta_{3} q^{9}+O(q^{10})$$ q + (-b6 + b2) * q^2 + (-b3 - 1) * q^3 + (2*b3 - 2) * q^4 + (b6 - 2*b2) * q^6 + (3*b7 + 3*b2 + 2*b1) * q^7 + 2*b6 * q^8 + 3*b3 * q^9 $$q + ( - \beta_{6} + \beta_{2}) q^{2} + ( - \beta_{3} - 1) q^{3} + (2 \beta_{3} - 2) q^{4} + (\beta_{6} - 2 \beta_{2}) q^{6} + (3 \beta_{7} + 3 \beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{6} q^{8} + 3 \beta_{3} q^{9} + ( - \beta_{7} + \beta_{6} + \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 1) q^{11} + ( - 2 \beta_{3} + 4) q^{12} + (2 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + 8 \beta_{2} + 2 \beta_1 + 1) q^{13} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 1) q^{14} - 4 \beta_{3} q^{16} + ( - 3 \beta_{7} - \beta_{6} + 5 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{17} + 3 \beta_{2} q^{18} + (7 \beta_{6} - 2 \beta_{4} - 6 \beta_{3} + 8 \beta_{2} + \beta_1 + 12) q^{19} + ( - 4 \beta_{7} - 4 \beta_{2} - 5 \beta_1) q^{21} + (\beta_{6} + \beta_{5} + 2 \beta_{4} + 3) q^{22} + (4 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - \beta_{4} + 15 \beta_{3} - \beta_{2} + 2 \beta_1) q^{23} + ( - 4 \beta_{6} + 2 \beta_{2}) q^{24} + (\beta_{6} - 2 \beta_{4} + 6 \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 12) q^{26} + ( - 6 \beta_{3} + 3) q^{27} + ( - 4 \beta_{7} - 4 \beta_{2} + 2 \beta_1) q^{28} + ( - 4 \beta_{7} - 15 \beta_{6} - \beta_{5} - 2 \beta_{4} + 4 \beta_1 - 3) q^{29} + (2 \beta_{7} + 10 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} + 8 \beta_{3} - 4 \beta_{2} + \cdots + 8) q^{31}+ \cdots + (3 \beta_{7} - 3 \beta_{6} - 3 \beta_1 - 3) q^{99}+O(q^{100})$$ q + (-b6 + b2) * q^2 + (-b3 - 1) * q^3 + (2*b3 - 2) * q^4 + (b6 - 2*b2) * q^6 + (3*b7 + 3*b2 + 2*b1) * q^7 + 2*b6 * q^8 + 3*b3 * q^9 + (-b7 + b6 + b3 - 3*b2 - 2*b1 - 1) * q^11 + (-2*b3 + 4) * q^12 + (2*b7 - 4*b6 - 2*b5 - 2*b3 + 8*b2 + 2*b1 + 1) * q^13 + (b5 - 2*b4 - 2*b3 - 1) * q^14 - 4*b3 * q^16 + (-3*b7 - b6 + 5*b5 + 5*b4 - 2*b3 - b2 - 2) * q^17 + 3*b2 * q^18 + (7*b6 - 2*b4 - 6*b3 + 8*b2 + b1 + 12) * q^19 + (-4*b7 - 4*b2 - 5*b1) * q^21 + (b6 + b5 + 2*b4 + 3) * q^22 + (4*b7 + 3*b6 - 2*b5 - b4 + 15*b3 - b2 + 2*b1) * q^23 + (-4*b6 + 2*b2) * q^24 + (b6 - 2*b4 + 6*b3 - 3*b2 - 4*b1 - 12) * q^26 + (-6*b3 + 3) * q^27 + (-4*b7 - 4*b2 + 2*b1) * q^28 + (-4*b7 - 15*b6 - b5 - 2*b4 + 4*b1 - 3) * q^29 + (2*b7 + 10*b6 - 4*b5 - 4*b4 + 8*b3 - 4*b2 + 8) * q^31 - 4*b2 * q^32 + (-b3 + 3*b2 + 3*b1 + 2) * q^33 + (10*b7 - 3*b6 - 3*b5 + 2*b3 + 6*b2 + 10*b1 - 1) * q^34 - 6 * q^36 + (-2*b6 + 10*b5 + 5*b4 - 6*b3 + 2*b2) * q^37 + (-4*b7 - 12*b6 - b5 - b4 - 15*b3 + 4*b2 - 15) * q^38 + (-2*b7 + 2*b6 + 2*b5 - 2*b4 + 3*b3 - 12*b2 - 4*b1 - 3) * q^39 + (7*b7 - 6*b6 - 8*b5 + 2*b3 + 12*b2 + 7*b1 - 1) * q^41 + (b5 + 5*b4 + 5*b3 - 1) * q^42 + (-8*b7 - 26*b6 - 4*b5 - 8*b4 + 8*b1 - 14) * q^43 + (4*b7 - 4*b6 - 2*b3 + 6*b2 + 2*b1) * q^44 + (-2*b7 + 2*b6 + 2*b5 - 2*b4 - 8*b3 + 12*b2 - 4*b1 + 8) * q^46 + (-8*b6 + 10*b4 - 7*b3 + 5*b2 + 13*b1 + 14) * q^47 + (8*b3 - 4) * q^48 + (-8*b5 - 5*b4 + 30*b3 - 48) * q^49 + (6*b7 - 10*b5 - 5*b4 + 6*b3 + 3*b2 + 3*b1) * q^51 + (-4*b7 + 12*b6 + 4*b5 + 4*b4 + 2*b3 - 8*b2 + 2) * q^52 + (9*b7 - 9*b6 - 2*b5 + 2*b4 - 11*b3 + 4*b2 + 18*b1 + 11) * q^53 + (-3*b6 - 3*b2) * q^54 + (-6*b5 - 2*b4 - 2*b3 + 6) * q^56 + (b7 - 22*b6 + 2*b5 + 4*b4 - b1 - 18) * q^57 + (-4*b7 + 4*b6 - 8*b5 - 4*b4 + 26*b3 - 6*b2 - 2*b1) * q^58 + (-21*b7 + 3*b6 - 5*b5 - 5*b4 + 26*b3 - 12*b2 + 26) * q^59 + (8*b6 - 10*b4 + 17*b3 - 2*b2 - 10*b1 - 34) * q^61 + (-8*b7 - 4*b6 + 2*b5 - 20*b3 + 8*b2 - 8*b1 + 10) * q^62 + (3*b7 + 3*b2 + 9*b1) * q^63 + 8 * q^64 + (-2*b6 - 3*b5 - 3*b4 - 3*b3 + b2 - 3) * q^66 + (3*b7 - 3*b6 - 30*b3 + 21*b2 + 6*b1 + 30) * q^67 + (4*b6 - 10*b4 - 4*b3 - 2*b2 - 6*b1 + 8) * q^68 + (-6*b7 - b6 + 3*b5 - 30*b3 + 2*b2 - 6*b1 + 15) * q^69 + (-4*b7 - 4*b6 - 12*b5 - 24*b4 + 4*b1 - 8) * q^71 + (6*b6 - 6*b2) * q^72 + (10*b7 + 10*b6 - 24*b5 - 24*b4 + 7*b3 + 7) * q^73 + (10*b7 - 10*b6 + 4*b3 + 9*b2 + 20*b1 - 4) * q^74 + (-2*b7 + 16*b6 - 4*b5 + 24*b3 - 32*b2 - 2*b1 - 12) * q^76 + (-2*b7 + 5*b5 + 4*b4 + 11*b3 - 2*b2 + b1 + 23) * q^77 + (-4*b7 + b6 + 2*b5 + 4*b4 + 4*b1 + 18) * q^78 + (-2*b7 + 6*b6 - 24*b5 - 12*b4 - 36*b3 - 7*b2 - b1) * q^79 + (9*b3 - 9) * q^81 + (9*b6 - 7*b4 + 5*b3 - 7*b2 - 16*b1 - 10) * q^82 + (7*b7 - 20*b6 + b5 + 32*b3 + 40*b2 + 7*b1 - 16) * q^83 + (10*b7 + 10*b2 + 2*b1) * q^84 + (-16*b7 + 18*b6 - 16*b5 - 8*b4 + 44*b3 - 26*b2 - 8*b1) * q^86 + (12*b7 + 26*b6 + 3*b5 + 3*b4 + 3*b3 - 7*b2 + 3) * q^87 + (2*b5 - 2*b4 + 6*b3 - 2*b2 - 6) * q^88 + (10*b6 - b4 + 28*b3 + 21*b2 + 11*b1 - 56) * q^89 + (7*b7 - 28*b6 - 6*b5 - 16*b4 - 2*b3 - 7*b2 - 36) * q^91 + (-4*b7 - 10*b6 + 2*b5 + 4*b4 + 4*b1 - 30) * q^92 + (-4*b7 - 14*b6 + 8*b5 + 4*b4 - 24*b3 + 12*b2 - 2*b1) * q^93 + (20*b7 - 14*b6 - 13*b5 - 13*b4 + 3*b3 + 17*b2 + 3) * q^94 + (4*b6 + 4*b2) * q^96 + (26*b7 - 3*b5 + 48*b3 + 26*b1 - 24) * q^97 + (-10*b7 + 56*b6 - 31*b2 - 16*b1) * q^98 + (3*b7 - 3*b6 - 3*b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 12 q^{3} - 8 q^{4} + 12 q^{9}+O(q^{10})$$ 8 * q - 12 * q^3 - 8 * q^4 + 12 * q^9 $$8 q - 12 q^{3} - 8 q^{4} + 12 q^{9} - 4 q^{11} + 24 q^{12} - 16 q^{14} - 16 q^{16} - 24 q^{17} + 72 q^{19} + 24 q^{22} + 60 q^{23} - 72 q^{26} - 24 q^{29} + 96 q^{31} + 12 q^{33} - 48 q^{36} - 24 q^{37} - 180 q^{38} - 12 q^{39} + 12 q^{42} - 112 q^{43} - 8 q^{44} + 32 q^{46} + 84 q^{47} - 264 q^{49} + 24 q^{51} + 24 q^{52} + 44 q^{53} + 40 q^{56} - 144 q^{57} + 104 q^{58} + 312 q^{59} - 204 q^{61} + 64 q^{64} - 36 q^{66} + 120 q^{67} + 48 q^{68} - 64 q^{71} + 84 q^{73} - 16 q^{74} + 228 q^{77} + 144 q^{78} - 144 q^{79} - 36 q^{81} - 60 q^{82} + 176 q^{86} + 36 q^{87} - 24 q^{88} - 336 q^{89} - 296 q^{91} - 240 q^{92} - 96 q^{93} + 36 q^{94} - 24 q^{99}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -11\nu^{7} + 475\nu^{5} - 1045\nu^{3} + 6468\nu ) / 32585$$ (-11*v^7 + 475*v^5 - 1045*v^3 + 6468*v) / 32585 $$\beta_{3}$$ $$=$$ $$( -12\nu^{6} + 95\nu^{4} - 1140\nu^{2} + 7056 ) / 4655$$ (-12*v^6 + 95*v^4 - 1140*v^2 + 7056) / 4655 $$\beta_{4}$$ $$=$$ $$( -23\nu^{6} + 570\nu^{4} - 2185\nu^{2} + 13524 ) / 4655$$ (-23*v^6 + 570*v^4 - 2185*v^2 + 13524) / 4655 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + 18 ) / 95$$ (-v^6 + 18) / 95 $$\beta_{6}$$ $$=$$ $$( -12\nu^{7} + 95\nu^{5} - 209\nu^{3} + 2401\nu ) / 6517$$ (-12*v^7 + 95*v^5 - 209*v^3 + 2401*v) / 6517 $$\beta_{7}$$ $$=$$ $$( \nu^{7} - 12\nu^{5} + 95\nu^{3} - 588\nu ) / 343$$ (v^7 - 12*v^5 + 95*v^3 - 588*v) / 343
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} - 6\beta_{3} + 6$$ b5 + b4 - 6*b3 + 6 $$\nu^{3}$$ $$=$$ $$5\beta_{7} + 7\beta_{6} + 5\beta_{2} + 5\beta_1$$ 5*b7 + 7*b6 + 5*b2 + 5*b1 $$\nu^{4}$$ $$=$$ $$12\beta_{4} - 23\beta_{3}$$ 12*b4 - 23*b3 $$\nu^{5}$$ $$=$$ $$11\beta_{7} + 95\beta_{2}$$ 11*b7 + 95*b2 $$\nu^{6}$$ $$=$$ $$-95\beta_{5} + 18$$ -95*b5 + 18 $$\nu^{7}$$ $$=$$ $$-665\beta_{6} + 665\beta_{2} + 113\beta_1$$ -665*b6 + 665*b2 + 113*b1

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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

 $$T_{11}^{8} + 4T_{11}^{7} + 58T_{11}^{6} + 16T_{11}^{5} + 1954T_{11}^{4} + 2440T_{11}^{3} + 15940T_{11}^{2} - 16376T_{11} + 31684$$ T11^8 + 4*T11^7 + 58*T11^6 + 16*T11^5 + 1954*T11^4 + 2440*T11^3 + 15940*T11^2 - 16376*T11 + 31684 $$T_{17}^{8} + 24 T_{17}^{7} - 506 T_{17}^{6} - 16752 T_{17}^{5} + 368850 T_{17}^{4} + 4615176 T_{17}^{3} - 31116836 T_{17}^{2} - 432808296 T_{17} + 4284749764$$ T17^8 + 24*T17^7 - 506*T17^6 - 16752*T17^5 + 368850*T17^4 + 4615176*T17^3 - 31116836*T17^2 - 432808296*T17 + 4284749764

Hecke characteristic polynomials

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