# Properties

 Label 117.4.q.f Level $117$ Weight $4$ Character orbit 117.q Analytic conductor $6.903$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 117.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.90322347067$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 32x^{10} + 823x^{8} - 5964x^{6} + 32913x^{4} - 47034x^{2} + 54756$$ x^12 - 32*x^10 + 823*x^8 - 5964*x^6 + 32913*x^4 - 47034*x^2 + 54756 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3^{5}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{8} - 3 \beta_{3}) q^{4} + (\beta_{9} - \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{11} - \beta_{5} - \beta_{3} - 2) q^{7} + ( - \beta_{10} + \beta_{4} + 3 \beta_{2}) q^{8}+O(q^{10})$$ q + (b2 + b1) * q^2 + (-b8 - 3*b3) * q^4 + (b9 - b6 + b2) * q^5 + (-b11 - b5 - b3 - 2) * q^7 + (-b10 + b4 + 3*b2) * q^8 $$q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{8} - 3 \beta_{3}) q^{4} + (\beta_{9} - \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{11} - \beta_{5} - \beta_{3} - 2) q^{7} + ( - \beta_{10} + \beta_{4} + 3 \beta_{2}) q^{8} + ( - 2 \beta_{11} - 2 \beta_{8} - \beta_{7} + \beta_{5} - 10 \beta_{3} - 10) q^{10} + ( - \beta_{10} - \beta_{9} + 4 \beta_{2} + 4 \beta_1) q^{11} + ( - 4 \beta_{11} - \beta_{8} - \beta_{7} + \beta_{5} - 4 \beta_{3} - 16) q^{13} + ( - \beta_{9} - \beta_{6} + 2 \beta_{2} + 4 \beta_1) q^{14} + ( - 4 \beta_{11} - 5 \beta_{8} - 2 \beta_{7} + 3 \beta_{5} - 17 \beta_{3} + \cdots - 17) q^{16}+ \cdots + (12 \beta_{6} + 9 \beta_{4} - 289 \beta_1) q^{98}+O(q^{100})$$ q + (b2 + b1) * q^2 + (-b8 - 3*b3) * q^4 + (b9 - b6 + b2) * q^5 + (-b11 - b5 - b3 - 2) * q^7 + (-b10 + b4 + 3*b2) * q^8 + (-2*b11 - 2*b8 - b7 + b5 - 10*b3 - 10) * q^10 + (-b10 - b9 + 4*b2 + 4*b1) * q^11 + (-4*b11 - b8 - b7 + b5 - 4*b3 - 16) * q^13 + (-b9 - b6 + 2*b2 + 4*b1) * q^14 + (-4*b11 - 5*b8 - 2*b7 + 3*b5 - 17*b3 - 17) * q^16 + (2*b10 + 4*b9 - 2*b6 - b4 - 6*b2 - 3*b1) * q^17 + (-5*b11 - b8 - 3*b5 - 3*b3 - 6) * q^19 + (5*b6 + 2*b4 - 9*b1) * q^20 + (-b11 - 13*b8 - 2*b7 + b5 - 53*b3) * q^22 + (3*b10 - b9 + 2*b6 - 6*b4 - 4*b2 + 4*b1) * q^23 + (-4*b11 + 4*b7 - 13*b5 - 6) * q^25 + (3*b10 - 2*b9 - 5*b6 + b4 - 30*b2 - 27*b1) * q^26 + (-b8 - 5*b7 + 4*b5 - 31*b3 + 31) * q^28 + (-9*b9 + 18*b6 - 17*b2 + 17*b1) * q^29 + (4*b11 - 18*b8 + 4*b7 + 9*b5 - 20*b3 - 10) * q^31 + (-6*b6 - 3*b4 - 15*b1) * q^32 + (22*b8 - 11*b5 + 86*b3 + 43) * q^34 + (-6*b10 - 14*b9 + 7*b6 + 3*b4 - 24*b2 - 12*b1) * q^35 + (12*b8 - b7 + 13*b5 + 24*b3 - 24) * q^37 + (b10 - 5*b9 - 5*b6 + b4 + 4*b2 + 8*b1) * q^38 + (9*b11 - 9*b7 + 26*b5 - 40) * q^40 + (3*b10 + 28*b9 - 15*b2 - 15*b1) * q^41 + (9*b11 + 24*b8 + 18*b7 - 9*b5 + 143*b3) * q^43 + (-5*b10 + 11*b9 - 11*b6 + 5*b4 + 116*b2) * q^44 + (21*b11 + 35*b8 - 49*b5 + 67*b3 + 134) * q^46 + (-b10 + 19*b9 - 19*b6 + b4 - 24*b2) * q^47 + (8*b11 - 9*b8 + 4*b7 + 13*b5 - 181*b3 - 181) * q^49 + (-5*b10 - 12*b9 + 70*b2 + 70*b1) * q^50 + (9*b11 + 38*b8 - b7 + 14*b5 + 256*b3 + 62) * q^52 + (2*b10 - 11*b9 - 11*b6 + 2*b4 + 79*b2 + 158*b1) * q^53 + (22*b11 - 17*b8 + 11*b7 + 28*b5 + 71*b3 + 71) * q^55 + (-2*b10 + 26*b9 - 13*b6 + b4 + 16*b2 + 8*b1) * q^56 + (27*b11 + 26*b8 - 25*b5 + 178*b3 + 356) * q^58 + (8*b6 - 8*b4 - 20*b1) * q^59 + (11*b11 - 31*b8 + 22*b7 - 11*b5 - 226*b3) * q^61 + (-9*b10 - 4*b9 + 8*b6 + 18*b4 + 94*b2 - 94*b1) * q^62 + (16*b11 - 16*b7 - 17*b5 + 1) * q^64 + (-13*b10 - 26*b9 - 117*b2 - 65*b1) * q^65 + (26*b8 + 9*b7 + 17*b5 - 139*b3 + 139) * q^67 + (3*b10 - 16*b9 + 32*b6 - 6*b4 - 107*b2 + 107*b1) * q^68 + (3*b11 - 22*b8 + 3*b7 + 11*b5 + 202*b3 + 101) * q^70 + (-9*b6 + 7*b4 - 168*b1) * q^71 + (-6*b11 + 16*b8 - 6*b7 - 8*b5 + 78*b3 + 39) * q^73 + (24*b10 + 2*b9 - b6 - 12*b4 - 246*b2 - 123*b1) * q^74 + (3*b8 - 19*b7 + 22*b5 - 65*b3 + 65) * q^76 + (2*b10 + 18*b9 + 18*b6 + 2*b4 + 24*b2 + 48*b1) * q^77 + (-36*b11 + 36*b7 - 17*b5 - 550) * q^79 + (-8*b10 - 13*b9 - 113*b2 - 113*b1) * q^80 + (-22*b11 + 17*b8 - 44*b7 + 22*b5 + 217*b3) * q^82 + (23*b10 - b9 + b6 - 23*b4 + 200*b2) * q^83 + (-45*b11 + 22*b8 - 89*b5 - 202*b3 - 404) * q^85 + (24*b10 - 27*b9 + 27*b6 - 24*b4 - 254*b2) * q^86 + (-26*b11 - 73*b8 - 13*b7 + 60*b5 - 881*b3 - 881) * q^88 + (34*b10 - 4*b9 + 126*b2 + 126*b1) * q^89 + (5*b11 - 41*b8 + 11*b7 + 28*b5 + 486*b3 + 267) * q^91 + (-11*b10 + 13*b9 + 13*b6 - 11*b4 + 252*b2 + 504*b1) * q^92 + (-42*b11 - 5*b8 - 21*b7 - 16*b5 + 275*b3 + 275) * q^94 + (-26*b10 - 50*b9 + 25*b6 + 13*b4 - 128*b2 - 64*b1) * q^95 + (-64*b11 - 59*b8 + 54*b5 + 74*b3 + 148) * q^97 + (12*b6 + 9*b4 - 289*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 16 q^{4} - 18 q^{7}+O(q^{10})$$ 12 * q + 16 * q^4 - 18 * q^7 $$12 q + 16 q^{4} - 18 q^{7} - 56 q^{10} - 154 q^{13} - 92 q^{16} - 48 q^{19} + 292 q^{22} - 92 q^{25} + 552 q^{28} - 360 q^{37} - 448 q^{40} - 810 q^{43} + 996 q^{46} - 1068 q^{49} - 700 q^{52} + 460 q^{55} + 3048 q^{58} + 1294 q^{61} - 184 q^{64} + 2658 q^{67} + 1188 q^{76} - 6380 q^{79} - 1268 q^{82} - 3768 q^{85} - 5140 q^{88} + 342 q^{91} + 1660 q^{94} + 1686 q^{97}+O(q^{100})$$ 12 * q + 16 * q^4 - 18 * q^7 - 56 * q^10 - 154 * q^13 - 92 * q^16 - 48 * q^19 + 292 * q^22 - 92 * q^25 + 552 * q^28 - 360 * q^37 - 448 * q^40 - 810 * q^43 + 996 * q^46 - 1068 * q^49 - 700 * q^52 + 460 * q^55 + 3048 * q^58 + 1294 * q^61 - 184 * q^64 + 2658 * q^67 + 1188 * q^76 - 6380 * q^79 - 1268 * q^82 - 3768 * q^85 - 5140 * q^88 + 342 * q^91 + 1660 * q^94 + 1686 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 32x^{10} + 823x^{8} - 5964x^{6} + 32913x^{4} - 47034x^{2} + 54756$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 55141 \nu^{11} + 1684640 \nu^{9} - 43326835 \nu^{7} + 276029262 \nu^{5} - 1732704885 \nu^{3} + 480684672 \nu ) / 1995420258$$ (-55141*v^11 + 1684640*v^9 - 43326835*v^7 + 276029262*v^5 - 1732704885*v^3 + 480684672*v) / 1995420258 $$\beta_{3}$$ $$=$$ $$( 55141 \nu^{10} - 1684640 \nu^{8} + 43326835 \nu^{6} - 276029262 \nu^{4} + 1732704885 \nu^{2} - 2476104930 ) / 1995420258$$ (55141*v^10 - 1684640*v^8 + 43326835*v^6 - 276029262*v^4 + 1732704885*v^2 - 2476104930) / 1995420258 $$\beta_{4}$$ $$=$$ $$( 1024\nu^{11} - 26336\nu^{9} + 677329\nu^{7} - 1053216\nu^{5} + 1505088\nu^{3} + 447354927\nu ) / 25582311$$ (1024*v^11 - 26336*v^9 + 677329*v^7 - 1053216*v^5 + 1505088*v^3 + 447354927*v) / 25582311 $$\beta_{5}$$ $$=$$ $$( 1024\nu^{10} - 26336\nu^{8} + 677329\nu^{6} - 1053216\nu^{4} + 1505088\nu^{2} + 242696439 ) / 25582311$$ (1024*v^10 - 26336*v^8 + 677329*v^6 - 1053216*v^4 + 1505088*v^2 + 242696439) / 25582311 $$\beta_{6}$$ $$=$$ $$( 23264\nu^{11} - 598321\nu^{9} + 14588621\nu^{7} - 23927751\nu^{5} + 34193718\nu^{3} + 2003386986\nu ) / 153493866$$ (23264*v^11 - 598321*v^9 + 14588621*v^7 - 23927751*v^5 + 34193718*v^3 + 2003386986*v) / 153493866 $$\beta_{7}$$ $$=$$ $$( 328789 \nu^{10} - 12986243 \nu^{8} + 344382751 \nu^{6} - 3733955064 \nu^{4} + 18916912152 \nu^{2} - 28627945398 ) / 1995420258$$ (328789*v^10 - 12986243*v^8 + 344382751*v^6 - 3733955064*v^4 + 18916912152*v^2 - 28627945398) / 1995420258 $$\beta_{8}$$ $$=$$ $$( - 526679 \nu^{10} + 16476832 \nu^{8} - 423763523 \nu^{6} + 2954171034 \nu^{4} - 16946936613 \nu^{2} + 24217853634 ) / 1995420258$$ (-526679*v^10 + 16476832*v^8 - 423763523*v^6 + 2954171034*v^4 - 16946936613*v^2 + 24217853634) / 1995420258 $$\beta_{9}$$ $$=$$ $$( - 238924 \nu^{11} + 7921376 \nu^{9} - 203727889 \nu^{7} + 1596984255 \nu^{5} - 8147382759 \nu^{3} + 11642937462 \nu ) / 665140086$$ (-238924*v^11 + 7921376*v^9 - 203727889*v^7 + 1596984255*v^5 - 8147382759*v^3 + 11642937462*v) / 665140086 $$\beta_{10}$$ $$=$$ $$( - 967807 \nu^{11} + 29953952 \nu^{9} - 770378203 \nu^{7} + 5162405130 \nu^{5} - 30808575693 \nu^{3} + 44026693074 \nu ) / 1995420258$$ (-967807*v^11 + 29953952*v^9 - 770378203*v^7 + 5162405130*v^5 - 30808575693*v^3 + 44026693074*v) / 1995420258 $$\beta_{11}$$ $$=$$ $$( 1076341 \nu^{10} - 32212346 \nu^{8} + 807675646 \nu^{6} - 4502835657 \nu^{4} + 20015673426 \nu^{2} - 6315279750 ) / 1995420258$$ (1076341*v^10 - 32212346*v^8 + 807675646*v^6 - 4502835657*v^4 + 20015673426*v^2 - 6315279750) / 1995420258
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{5} + 11\beta_{3} + 11$$ b8 - b5 + 11*b3 + 11 $$\nu^{3}$$ $$=$$ $$\beta_{10} - \beta_{4} - 19\beta_{2}$$ b10 - b4 - 19*b2 $$\nu^{4}$$ $$=$$ $$2\beta_{11} + 29\beta_{8} + 4\beta_{7} - 2\beta_{5} + 217\beta_{3}$$ 2*b11 + 29*b8 + 4*b7 - 2*b5 + 217*b3 $$\nu^{5}$$ $$=$$ $$29\beta_{10} - 6\beta_{9} - 431\beta_{2} - 431\beta_1$$ 29*b10 - 6*b9 - 431*b2 - 431*b1 $$\nu^{6}$$ $$=$$ $$-64\beta_{11} + 64\beta_{7} + 599\beta_{5} - 4967$$ -64*b11 + 64*b7 + 599*b5 - 4967 $$\nu^{7}$$ $$=$$ $$-192\beta_{6} + 727\beta_{4} - 10207\beta_1$$ -192*b6 + 727*b4 - 10207*b1 $$\nu^{8}$$ $$=$$ $$-3292\beta_{11} - 17669\beta_{8} - 1646\beta_{7} + 16023\beta_{5} - 117901\beta_{3} - 117901$$ -3292*b11 - 17669*b8 - 1646*b7 + 16023*b5 - 117901*b3 - 117901 $$\nu^{9}$$ $$=$$ $$-17669\beta_{10} + 4938\beta_{9} - 4938\beta_{6} + 17669\beta_{4} + 244439\beta_{2}$$ -17669*b10 + 4938*b9 - 4938*b6 + 17669*b4 + 244439*b2 $$\nu^{10}$$ $$=$$ $$-40276\beta_{11} - 426067\beta_{8} - 80552\beta_{7} + 40276\beta_{5} - 2825243\beta_{3}$$ -40276*b11 - 426067*b8 - 80552*b7 + 40276*b5 - 2825243*b3 $$\nu^{11}$$ $$=$$ $$-426067\beta_{10} + 120828\beta_{9} + 5871295\beta_{2} + 5871295\beta_1$$ -426067*b10 + 120828*b9 + 5871295*b2 + 5871295*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 −4.24667 − 2.45182i −2.19846 − 1.26928i −1.06422 − 0.614429i 1.06422 + 0.614429i 2.19846 + 1.26928i 4.24667 + 2.45182i −4.24667 + 2.45182i −2.19846 + 1.26928i −1.06422 + 0.614429i 1.06422 − 0.614429i 2.19846 − 1.26928i 4.24667 − 2.45182i
−4.24667 + 2.45182i 0 8.02279 13.8959i 5.61325i 0 2.78293 + 1.60672i 39.4526i 0 −13.7627 23.8376i
10.2 −2.19846 + 1.26928i 0 −0.777841 + 1.34726i 8.18941i 0 9.33900 + 5.39187i 24.2577i 0 10.3947 + 18.0041i
10.3 −1.06422 + 0.614429i 0 −3.24495 + 5.62043i 17.3039i 0 −16.6219 9.59667i 17.8060i 0 −10.6320 18.4152i
10.4 1.06422 0.614429i 0 −3.24495 + 5.62043i 17.3039i 0 −16.6219 9.59667i 17.8060i 0 −10.6320 18.4152i
10.5 2.19846 1.26928i 0 −0.777841 + 1.34726i 8.18941i 0 9.33900 + 5.39187i 24.2577i 0 10.3947 + 18.0041i
10.6 4.24667 2.45182i 0 8.02279 13.8959i 5.61325i 0 2.78293 + 1.60672i 39.4526i 0 −13.7627 23.8376i
82.1 −4.24667 2.45182i 0 8.02279 + 13.8959i 5.61325i 0 2.78293 1.60672i 39.4526i 0 −13.7627 + 23.8376i
82.2 −2.19846 1.26928i 0 −0.777841 1.34726i 8.18941i 0 9.33900 5.39187i 24.2577i 0 10.3947 18.0041i
82.3 −1.06422 0.614429i 0 −3.24495 5.62043i 17.3039i 0 −16.6219 + 9.59667i 17.8060i 0 −10.6320 + 18.4152i
82.4 1.06422 + 0.614429i 0 −3.24495 5.62043i 17.3039i 0 −16.6219 + 9.59667i 17.8060i 0 −10.6320 + 18.4152i
82.5 2.19846 + 1.26928i 0 −0.777841 1.34726i 8.18941i 0 9.33900 5.39187i 24.2577i 0 10.3947 18.0041i
82.6 4.24667 + 2.45182i 0 8.02279 + 13.8959i 5.61325i 0 2.78293 1.60672i 39.4526i 0 −13.7627 + 23.8376i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 82.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.q.f 12
3.b odd 2 1 inner 117.4.q.f 12
13.e even 6 1 inner 117.4.q.f 12
13.f odd 12 2 1521.4.a.bl 12
39.h odd 6 1 inner 117.4.q.f 12
39.k even 12 2 1521.4.a.bl 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.q.f 12 1.a even 1 1 trivial
117.4.q.f 12 3.b odd 2 1 inner
117.4.q.f 12 13.e even 6 1 inner
117.4.q.f 12 39.h odd 6 1 inner
1521.4.a.bl 12 13.f odd 12 2
1521.4.a.bl 12 39.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 32T_{2}^{10} + 823T_{2}^{8} - 5964T_{2}^{6} + 32913T_{2}^{4} - 47034T_{2}^{2} + 54756$$ acting on $$S_{4}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 32 T^{10} + 823 T^{8} + \cdots + 54756$$
$3$ $$T^{12}$$
$5$ $$(T^{6} + 398 T^{4} + 31629 T^{2} + \cdots + 632736)^{2}$$
$7$ $$(T^{6} + 9 T^{5} - 207 T^{4} + \cdots + 442368)^{2}$$
$11$ $$T^{12} - 3092 T^{10} + \cdots + 60\!\cdots\!56$$
$13$ $$(T^{6} + 77 T^{5} + 1430 T^{4} + \cdots + 10604499373)^{2}$$
$17$ $$T^{12} + 11850 T^{10} + \cdots + 58\!\cdots\!84$$
$19$ $$(T^{6} + 24 T^{5} - 4740 T^{4} + \cdots + 231475968)^{2}$$
$23$ $$T^{12} + 60252 T^{10} + \cdots + 17\!\cdots\!04$$
$29$ $$T^{12} + 114954 T^{10} + \cdots + 28\!\cdots\!24$$
$31$ $$(T^{6} + 98019 T^{4} + \cdots + 30656351232)^{2}$$
$37$ $$(T^{6} + 180 T^{5} + \cdots + 4986821583792)^{2}$$
$41$ $$T^{12} - 321494 T^{10} + \cdots + 56\!\cdots\!96$$
$43$ $$(T^{6} + 405 T^{5} + \cdots + 131524015065664)^{2}$$
$47$ $$(T^{6} + 159044 T^{4} + \cdots + 35969118142464)^{2}$$
$53$ $$(T^{6} - 769410 T^{4} + \cdots - 581060481251328)^{2}$$
$59$ $$T^{12} - 169088 T^{10} + \cdots + 62\!\cdots\!16$$
$61$ $$(T^{6} - 647 T^{5} + \cdots + 88\!\cdots\!09)^{2}$$
$67$ $$(T^{6} - 1329 T^{5} + \cdots + 74\!\cdots\!68)^{2}$$
$71$ $$T^{12} - 1058468 T^{10} + \cdots + 21\!\cdots\!16$$
$73$ $$(T^{6} + 112065 T^{4} + \cdots + 13909107863187)^{2}$$
$79$ $$(T^{3} + 1595 T^{2} + 72716 T - 515213504)^{4}$$
$83$ $$(T^{6} + 2259428 T^{4} + \cdots + 13\!\cdots\!04)^{2}$$
$89$ $$T^{12} - 2817944 T^{10} + \cdots + 52\!\cdots\!16$$
$97$ $$(T^{6} - 843 T^{5} + \cdots + 587965056519168)^{2}$$