# Properties

 Label 336.3.bh.g Level $336$ Weight $3$ Character orbit 336.bh Analytic conductor $9.155$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 336.bh (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.15533688251$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.35911766016.9 Defining polynomial: $$x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529$$ x^8 - 2*x^7 - 7*x^6 - 2*x^5 + 78*x^4 - 18*x^3 - 153*x^2 - 230*x + 529 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 7$$ Twist minimal: no (minimal twist has level 168) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{3} + ( - \beta_{6} + \beta_1 - 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{7} + 3 \beta_1 q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^3 + (-b6 + b1 - 1) * q^5 + (b4 - b3 - b2 - b1 - 1) * q^7 + 3*b1 * q^9 $$q + ( - \beta_1 - 1) q^{3} + ( - \beta_{6} + \beta_1 - 1) q^{5} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{7} + 3 \beta_1 q^{9} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - 6 \beta_1 + 5) q^{11} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} + 3 \beta_1 - 1) q^{13} + ( - \beta_{7} + \beta_{6} - \beta_1 + 2) q^{15} + (\beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 2) q^{17} + (5 \beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 6) q^{19} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{21} + ( - 2 \beta_{7} - \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + \beta_{2} - 10 \beta_1 - 1) q^{23} + (\beta_{7} + 2 \beta_{6} + 5 \beta_{5} - 11 \beta_1 + 10) q^{25} + ( - 6 \beta_1 + 3) q^{27} + (3 \beta_{5} - \beta_{4} + 5 \beta_{2} + 2 \beta_1 + 9) q^{29} + (4 \beta_{7} + 3 \beta_{5} - 6 \beta_{3} + 7 \beta_1 + 3) q^{31} + ( - 3 \beta_{6} - \beta_{5} - \beta_{3} + 7 \beta_1 - 11) q^{33} + ( - 2 \beta_{7} + \beta_{6} + 15 \beta_{5} - \beta_{4} - 5 \beta_{3} + 4 \beta_{2} + \cdots + 4) q^{35}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{6} + 3 \beta_{3} - 3 \beta_1 + 18) q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^3 + (-b6 + b1 - 1) * q^5 + (b4 - b3 - b2 - b1 - 1) * q^7 + 3*b1 * q^9 + (b7 + 2*b6 + b5 - 6*b1 + 5) * q^11 + (-b7 - b6 + 2*b5 - b3 + 3*b1 - 1) * q^13 + (-b7 + b6 - b1 + 2) * q^15 + (b7 - b5 + b4 + 2*b3 + 2*b2 + 5*b1 + 2) * q^17 + (5*b5 - 2*b4 + b3 + 3*b2 + 4*b1 - 6) * q^19 + (-b5 - b4 + b3 + 2*b2 + 3*b1) * q^21 + (-2*b7 - b6 - 4*b5 + 4*b4 + 2*b3 + b2 - 10*b1 - 1) * q^23 + (b7 + 2*b6 + 5*b5 - 11*b1 + 10) * q^25 + (-6*b1 + 3) * q^27 + (3*b5 - b4 + 5*b2 + 2*b1 + 9) * q^29 + (4*b7 + 3*b5 - 6*b3 + 7*b1 + 3) * q^31 + (-3*b6 - b5 - b3 + 7*b1 - 11) * q^33 + (-2*b7 + b6 + 15*b5 - b4 - 5*b3 + 4*b2 - 2*b1 + 4) * q^35 + (-2*b7 - b6 + 9*b5 - 9*b3 - 30*b1 + 1) * q^37 + (b7 + 2*b6 - 3*b5 - 5*b1 + 4) * q^39 + (-2*b7 - 2*b6 + 6*b5 + 6*b4 - 4*b3 - 2*b2 + 14*b1 - 8) * q^41 + (4*b7 - 4*b6 + 3*b5 - b4 - b3 + 5*b2 + 6*b1 + 10) * q^43 + (3*b7 - 3) * q^45 + (-3*b6 + 2*b5 + 2*b4 + 6*b3 - 3*b2 + 1) * q^47 + (-3*b7 + 5*b6 - 8*b5 + 5*b3 - 24*b1 + 9) * q^49 + (-2*b7 - b6 + 6*b5 - 4*b4 - 4*b3 - b2 - 12*b1 + 3) * q^51 + (3*b7 + 6*b6 + 16*b5 - 4*b1 + 1) * q^53 + (-8*b7 - 8*b6 - 13*b5 + 3*b4 + 6*b3 - b2 + 76*b1 - 35) * q^55 + (-3*b5 + b4 - 5*b3 - 5*b2 - 2*b1 + 10) * q^57 + (8*b5 + 3*b4 - 16*b3 + 6*b2 + 9*b1 + 3) * q^59 + (12*b6 - 8*b5 - 8*b3 - 8*b1 + 4) * q^61 + (3*b5 - 3*b2 - 6*b1 + 3) * q^63 + (2*b7 + b6 + 2*b5 + 4*b4 - 4*b3 + b2 - 34*b1 - 3) * q^65 + (-2*b7 - 4*b6 - 14*b5 - 5*b4 - 2*b3 + 4*b2 + 6*b1 - 2) * q^67 + (3*b7 + 3*b6 + 11*b5 - 9*b4 - 4*b3 + 3*b2 + 21*b1 - 9) * q^69 + (-b7 + b6 - 3*b5 + b4 + 28*b3 - 5*b2 - 3*b1 + 49) * q^71 + (-b7 - 17*b5 - 2*b4 + 34*b3 - 4*b2 - 15*b1 - 10) * q^73 + (-3*b6 - 5*b5 - 5*b3 + 12*b1 - 21) * q^75 + (-2*b7 - 6*b6 - 32*b5 + 8*b4 + 14*b3 - 7*b2 + 13*b1 - 24) * q^77 + (12*b7 + 6*b6 - 23*b5 - 8*b4 + 27*b3 - 2*b2 - 43*b1 - 2) * q^79 + (9*b1 - 9) * q^81 + (4*b7 + 4*b6 + 7*b5 - 3*b4 - 3*b3 + b2 + 12*b1 - 7) * q^83 + (4*b7 - 4*b6 - 6*b5 + 2*b4 - 22*b3 - 10*b2 + 22) * q^85 + (b5 - 3*b4 - 2*b3 - 6*b2 - 13*b1 - 7) * q^87 + (-8*b6 + 12*b5 - 4*b4 + 4*b3 + 6*b2 + 10*b1 - 8) * q^89 + (-4*b7 + 2*b6 + 13*b5 - 6*b4 - 13*b3 + b2 + 6*b1 - 12) * q^91 + (-8*b7 - 4*b6 - 9*b5 + 9*b3 - 17*b1 + 4) * q^93 + (b7 + 2*b6 - 47*b5 + 15*b4 + 6*b3 - 12*b2 + b1 - 8) * q^95 + (-7*b7 - 7*b6 - 10*b5 - 12*b4 + 7*b3 + 4*b2 + 107*b1 - 46) * q^97 + (-3*b7 + 3*b6 + 3*b3 - 3*b1 + 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 12 q^{3} - 6 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10})$$ 8 * q - 12 * q^3 - 6 * q^5 - 8 * q^7 + 12 * q^9 $$8 q - 12 q^{3} - 6 q^{5} - 8 q^{7} + 12 q^{9} + 22 q^{11} + 12 q^{15} + 36 q^{17} - 42 q^{19} + 6 q^{21} - 48 q^{23} + 42 q^{25} + 68 q^{29} + 60 q^{31} - 66 q^{33} + 12 q^{35} - 118 q^{37} + 18 q^{39} + 92 q^{43} - 18 q^{45} + 12 q^{47} - 20 q^{49} - 36 q^{51} + 10 q^{53} + 84 q^{57} + 54 q^{59} + 24 q^{61} + 6 q^{63} - 148 q^{65} - 22 q^{67} + 392 q^{71} - 138 q^{73} - 126 q^{75} - 126 q^{77} - 164 q^{79} - 36 q^{81} + 200 q^{85} - 102 q^{87} - 60 q^{89} - 90 q^{91} - 60 q^{93} + 132 q^{99}+O(q^{100})$$ 8 * q - 12 * q^3 - 6 * q^5 - 8 * q^7 + 12 * q^9 + 22 * q^11 + 12 * q^15 + 36 * q^17 - 42 * q^19 + 6 * q^21 - 48 * q^23 + 42 * q^25 + 68 * q^29 + 60 * q^31 - 66 * q^33 + 12 * q^35 - 118 * q^37 + 18 * q^39 + 92 * q^43 - 18 * q^45 + 12 * q^47 - 20 * q^49 - 36 * q^51 + 10 * q^53 + 84 * q^57 + 54 * q^59 + 24 * q^61 + 6 * q^63 - 148 * q^65 - 22 * q^67 + 392 * q^71 - 138 * q^73 - 126 * q^75 - 126 * q^77 - 164 * q^79 - 36 * q^81 + 200 * q^85 - 102 * q^87 - 60 * q^89 - 90 * q^91 - 60 * q^93 + 132 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529$$ :

 $$\beta_{1}$$ $$=$$ $$( 4244 \nu^{7} + 873 \nu^{6} - 33756 \nu^{5} - 71462 \nu^{4} + 213594 \nu^{3} + 469674 \nu^{2} - 193196 \nu - 1034839 ) / 658490$$ (4244*v^7 + 873*v^6 - 33756*v^5 - 71462*v^4 + 213594*v^3 + 469674*v^2 - 193196*v - 1034839) / 658490 $$\beta_{2}$$ $$=$$ $$( 3371 \nu^{7} + 6667 \nu^{6} - 38294 \nu^{5} - 96948 \nu^{4} + 94716 \nu^{3} + 623641 \nu^{2} + 796111 \nu - 1188686 ) / 329245$$ (3371*v^7 + 6667*v^6 - 38294*v^5 - 96948*v^4 + 94716*v^3 + 623641*v^2 + 796111*v - 1188686) / 329245 $$\beta_{3}$$ $$=$$ $$( 33\nu^{7} + 41\nu^{6} - 222\nu^{5} - 414\nu^{4} + 488\nu^{3} + 1608\nu^{2} - 207\nu + 3637 ) / 2045$$ (33*v^7 + 41*v^6 - 222*v^5 - 414*v^4 + 488*v^3 + 1608*v^2 - 207*v + 3637) / 2045 $$\beta_{4}$$ $$=$$ $$( 6088 \nu^{7} - 4954 \nu^{6} - 27942 \nu^{5} - 30829 \nu^{4} + 324398 \nu^{3} - 31292 \nu^{2} + 378248 \nu - 816983 ) / 329245$$ (6088*v^7 - 4954*v^6 - 27942*v^5 - 30829*v^4 + 324398*v^3 - 31292*v^2 + 378248*v - 816983) / 329245 $$\beta_{5}$$ $$=$$ $$( - 6546 \nu^{7} - 1812 \nu^{6} + 70064 \nu^{5} + 173218 \nu^{4} - 443336 \nu^{3} - 974856 \nu^{2} + 215444 \nu + 3514676 ) / 329245$$ (-6546*v^7 - 1812*v^6 + 70064*v^5 + 173218*v^4 - 443336*v^3 - 974856*v^2 + 215444*v + 3514676) / 329245 $$\beta_{6}$$ $$=$$ $$( - 11533 \nu^{7} + 30909 \nu^{6} + 121832 \nu^{5} - 29696 \nu^{4} - 997968 \nu^{3} - 162453 \nu^{2} + 2146717 \nu + 2603393 ) / 329245$$ (-11533*v^7 + 30909*v^6 + 121832*v^5 - 29696*v^4 - 997968*v^3 - 162453*v^2 + 2146717*v + 2603393) / 329245 $$\beta_{7}$$ $$=$$ $$( - 45244 \nu^{7} - 5583 \nu^{6} + 215876 \nu^{5} + 916372 \nu^{4} - 1365974 \nu^{3} - 3003654 \nu^{2} - 3864944 \nu + 10829159 ) / 658490$$ (-45244*v^7 - 5583*v^6 + 215876*v^5 + 916372*v^4 - 1365974*v^3 - 3003654*v^2 - 3864944*v + 10829159) / 658490
 $$\nu$$ $$=$$ $$( \beta_{5} + 2\beta_{4} - 2\beta_{3} + 4\beta_{2} - 4\beta _1 + 6 ) / 14$$ (b5 + 2*b4 - 2*b3 + 4*b2 - 4*b1 + 6) / 14 $$\nu^{2}$$ $$=$$ $$( 6\beta_{5} - 2\beta_{4} - 5\beta_{3} + 3\beta_{2} + 32\beta _1 + 1 ) / 7$$ (6*b5 - 2*b4 - 5*b3 + 3*b2 + 32*b1 + 1) / 7 $$\nu^{3}$$ $$=$$ $$( 7\beta_{7} + 7\beta_{6} - 18\beta_{5} + 27\beta_{4} - 6\beta_{3} - 9\beta_{2} + 9\beta _1 + 81 ) / 14$$ (7*b7 + 7*b6 - 18*b5 + 27*b4 - 6*b3 - 9*b2 + 9*b1 + 81) / 14 $$\nu^{4}$$ $$=$$ $$( 7\beta_{7} + 33\beta_{5} + 10\beta_{4} - 10\beta_{3} + 20\beta_{2} + 141\beta _1 - 131 ) / 7$$ (7*b7 + 33*b5 + 10*b4 - 10*b3 + 20*b2 + 141*b1 - 131) / 7 $$\nu^{5}$$ $$=$$ $$( 49\beta_{6} + 75\beta_{5} + 66\beta_{4} - 108\beta_{3} - 99\beta_{2} + 736\beta _1 - 33 ) / 14$$ (49*b6 + 75*b5 + 66*b4 - 108*b3 - 99*b2 + 736*b1 - 33) / 14 $$\nu^{6}$$ $$=$$ $$( 91\beta_{7} + 91\beta_{6} - 150\beta_{5} + 225\beta_{4} + 160\beta_{3} - 75\beta_{2} + 75\beta _1 - 494 ) / 7$$ (91*b7 + 91*b6 - 150*b5 + 225*b4 + 160*b3 - 75*b2 + 75*b1 - 494) / 7 $$\nu^{7}$$ $$=$$ $$( -22\beta_{7} + 199\beta_{5} - 8\beta_{4} + 8\beta_{3} - 16\beta_{2} + 718\beta _1 - 726 ) / 2$$ (-22*b7 + 199*b5 - 8*b4 + 8*b3 - 16*b2 + 718*b1 - 726) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 1.83172 − 0.480194i −1.90015 + 1.67440i 2.40015 − 0.808379i −1.33172 + 1.34622i 1.83172 + 0.480194i −1.90015 − 1.67440i 2.40015 + 0.808379i −1.33172 − 1.34622i
0 −1.50000 + 0.866025i 0 −6.80550 3.92916i 0 −6.99187 0.337312i 0 1.50000 2.59808i 0
145.2 0 −1.50000 + 0.866025i 0 −4.68140 2.70281i 0 6.12873 + 3.38210i 0 1.50000 2.59808i 0
145.3 0 −1.50000 + 0.866025i 0 3.18140 + 1.83678i 0 −2.47188 6.54903i 0 1.50000 2.59808i 0
145.4 0 −1.50000 + 0.866025i 0 5.30550 + 3.06313i 0 −0.664986 + 6.96834i 0 1.50000 2.59808i 0
241.1 0 −1.50000 0.866025i 0 −6.80550 + 3.92916i 0 −6.99187 + 0.337312i 0 1.50000 + 2.59808i 0
241.2 0 −1.50000 0.866025i 0 −4.68140 + 2.70281i 0 6.12873 3.38210i 0 1.50000 + 2.59808i 0
241.3 0 −1.50000 0.866025i 0 3.18140 1.83678i 0 −2.47188 + 6.54903i 0 1.50000 + 2.59808i 0
241.4 0 −1.50000 0.866025i 0 5.30550 3.06313i 0 −0.664986 6.96834i 0 1.50000 + 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 241.4 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 336.3.bh.g 8
3.b odd 2 1 1008.3.cg.p 8
4.b odd 2 1 168.3.z.b 8
7.c even 3 1 2352.3.f.g 8
7.d odd 6 1 inner 336.3.bh.g 8
7.d odd 6 1 2352.3.f.g 8
12.b even 2 1 504.3.by.c 8
21.g even 6 1 1008.3.cg.p 8
28.d even 2 1 1176.3.z.c 8
28.f even 6 1 168.3.z.b 8
28.f even 6 1 1176.3.f.c 8
28.g odd 6 1 1176.3.f.c 8
28.g odd 6 1 1176.3.z.c 8
84.j odd 6 1 504.3.by.c 8
84.j odd 6 1 3528.3.f.b 8
84.n even 6 1 3528.3.f.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.z.b 8 4.b odd 2 1
168.3.z.b 8 28.f even 6 1
336.3.bh.g 8 1.a even 1 1 trivial
336.3.bh.g 8 7.d odd 6 1 inner
504.3.by.c 8 12.b even 2 1
504.3.by.c 8 84.j odd 6 1
1008.3.cg.p 8 3.b odd 2 1
1008.3.cg.p 8 21.g even 6 1
1176.3.f.c 8 28.f even 6 1
1176.3.f.c 8 28.g odd 6 1
1176.3.z.c 8 28.d even 2 1
1176.3.z.c 8 28.g odd 6 1
2352.3.f.g 8 7.c even 3 1
2352.3.f.g 8 7.d odd 6 1
3528.3.f.b 8 84.j odd 6 1
3528.3.f.b 8 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 6T_{5}^{7} - 53T_{5}^{6} - 390T_{5}^{5} + 2861T_{5}^{4} + 13260T_{5}^{3} - 48268T_{5}^{2} - 195024T_{5} + 913936$$ acting on $$S_{3}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} + 3 T + 3)^{4}$$
$5$ $$T^{8} + 6 T^{7} - 53 T^{6} + \cdots + 913936$$
$7$ $$T^{8} + 8 T^{7} + 42 T^{6} + \cdots + 5764801$$
$11$ $$T^{8} - 22 T^{7} + 527 T^{6} + \cdots + 17875984$$
$13$ $$T^{8} + 262 T^{6} + 19817 T^{4} + \cdots + 2408704$$
$17$ $$T^{8} - 36 T^{7} + \cdots + 3470623744$$
$19$ $$T^{8} + 42 T^{7} + 43 T^{6} + \cdots + 17272336$$
$23$ $$T^{8} + 48 T^{7} + \cdots + 22620160000$$
$29$ $$(T^{4} - 34 T^{3} - 1063 T^{2} + \cdots + 224128)^{2}$$
$31$ $$T^{8} - 60 T^{7} + \cdots + 25912950625$$
$37$ $$T^{8} + 118 T^{7} + \cdots + 17069945104$$
$41$ $$T^{8} + 7280 T^{6} + \cdots + 580010189056$$
$43$ $$(T^{4} - 46 T^{3} - 3723 T^{2} + \cdots + 1658308)^{2}$$
$47$ $$T^{8} - 12 T^{7} + \cdots + 52408029184$$
$53$ $$T^{8} - 10 T^{7} + \cdots + 1491466217536$$
$59$ $$T^{8} - 54 T^{7} + \cdots + 1048985640000$$
$61$ $$T^{8} - 24 T^{7} + \cdots + 43785853599744$$
$67$ $$T^{8} + 22 T^{7} + \cdots + 95387087104$$
$71$ $$(T^{4} - 196 T^{3} + 2460 T^{2} + \cdots - 13209344)^{2}$$
$73$ $$T^{8} + \cdots + 622497709609216$$
$79$ $$T^{8} + 164 T^{7} + \cdots + 26\!\cdots\!25$$
$83$ $$T^{8} + 4430 T^{6} + \cdots + 839297841424$$
$89$ $$T^{8} + 60 T^{7} + \cdots + 723343446016$$
$97$ $$T^{8} + 65270 T^{6} + \cdots + 22\!\cdots\!16$$