# Properties

 Label 378.2.l.a Level $378$ Weight $2$ Character orbit 378.l Analytic conductor $3.018$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.l (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + 2785 x^{8} - 2640 x^{7} - 2601 x^{6} + 10260 x^{5} - 10611 x^{4} - 1944 x^{3} + \cdots + 6561$$ x^16 - 8*x^15 + 23*x^14 - 8*x^13 - 131*x^12 + 380*x^11 - 289*x^10 - 880*x^9 + 2785*x^8 - 2640*x^7 - 2601*x^6 + 10260*x^5 - 10611*x^4 - 1944*x^3 + 16767*x^2 - 17496*x + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{4}$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{7} + \beta_1) q^{2} - q^{4} + ( - \beta_{12} - \beta_{10} - \beta_{5}) q^{5} + (\beta_{12} + \beta_{8} - \beta_{6} - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} - \beta_1) q^{8}+O(q^{10})$$ q + (b7 + b1) * q^2 - q^4 + (-b12 - b10 - b5) * q^5 + (b12 + b8 - b6 - b3 + b2) * q^7 + (-b7 - b1) * q^8 $$q + (\beta_{7} + \beta_1) q^{2} - q^{4} + ( - \beta_{12} - \beta_{10} - \beta_{5}) q^{5} + (\beta_{12} + \beta_{8} - \beta_{6} - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} - \beta_1) q^{8} + ( - \beta_{13} + \beta_{9} - \beta_{3} + \beta_{2}) q^{10} + (\beta_{15} + \beta_{11} + \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{11} + (\beta_{13} + \beta_{11} + \beta_{9} + \beta_{7} + \beta_{2} + 2 \beta_1 + 1) q^{13} + (\beta_{15} - \beta_{9}) q^{14} + q^{16} + (\beta_{15} + \beta_{13} + \beta_{11} - 2 \beta_{8} - \beta_{6} - \beta_{2} - \beta_1 + 2) q^{17} + (\beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{19} + (\beta_{12} + \beta_{10} + \beta_{5}) q^{20} + ( - \beta_{14} + \beta_{10} + \beta_{8} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{22} + ( - \beta_{13} + \beta_{9} + \beta_{8} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{23} + (\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - 2 \beta_{11} + \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + \cdots - 1) q^{25}+ \cdots + ( - 2 \beta_{14} - \beta_{12} - \beta_{10} - \beta_{9} + 6 \beta_{8} - \beta_{6} + \beta_{4} - 2 \beta_{3} + \cdots - 1) q^{98}+O(q^{100})$$ q + (b7 + b1) * q^2 - q^4 + (-b12 - b10 - b5) * q^5 + (b12 + b8 - b6 - b3 + b2) * q^7 + (-b7 - b1) * q^8 + (-b13 + b9 - b3 + b2) * q^10 + (b15 + b11 + b8 + b4 - b3 + b2 - 1) * q^11 + (b13 + b11 + b9 + b7 + b2 + 2*b1 + 1) * q^13 + (b15 - b9) * q^14 + q^16 + (b15 + b13 + b11 - 2*b8 - b6 - b2 - b1 + 2) * q^17 + (b15 - b14 - b13 - b12 - b9 - b7 + b6 - b5 + b4 + b2 + b1) * q^19 + (b12 + b10 + b5) * q^20 + (-b14 + b10 + b8 + b5 - b4 - b3 + b2 - 1) * q^22 + (-b13 + b9 + b8 - b4 - 2*b3 + b2 - 1) * q^23 + (b15 + b14 - b13 - b12 - 2*b11 + b9 - 2*b8 - 3*b7 - b6 - b5 - b3 - 3*b1 - 1) * q^25 + (-b14 + b12 + 2*b7 + b5 - 2*b3 + b2 + b1 - 1) * q^26 + (-b12 - b8 + b6 + b3 - b2) * q^28 + (b13 - b9 + b8 - b6 - b4 - 1) * q^29 + (b8 + 3*b7 - b6 - 2*b4 + b3 + b2 + 3*b1) * q^31 + (b7 + b1) * q^32 + (-b14 + b8 + 3*b7 - b6 + b5 - b3 + b2) * q^34 + (-2*b13 + b12 + b11 + 3*b8 + b7 + b6 - b5 + b4 + 2*b2 + 3*b1 - 2) * q^35 + (-2*b15 + b14 - b13 + b12 + b11 - 2*b10 + b9 + 3*b7 + 2*b6 - b5 + b4 - b3 + b2 + 3*b1 + 1) * q^37 + (-b13 - b12 - b11 + b10 + b9 - b8 - b7 + b6 - b3 - b1 - 1) * q^38 + (b13 - b9 + b3 - b2) * q^40 + (-b14 + b12 + b8 - b6 + b5 + b4 + 1) * q^41 + (-2*b15 + b14 - b13 - b12 - 2*b11 - b10 + b9 - b8 + b4 - b3 - b2) * q^43 + (-b15 - b11 - b8 - b4 + b3 - b2 + 1) * q^44 + (b12 + b10 - b8 - b6 + b5 - b4 + b3 - b2 + 1) * q^46 + (-b15 + b13 - b11 - b10 - b9 - 2*b8 + 2*b6 + b5 + b4 + 2*b3 - 3*b2 + 3) * q^47 + (b15 + 3*b13 + b12 + 2*b11 - b9 + b7 + b5 + b4 + 2*b3 - 3*b2 - 4*b1) * q^49 + (2*b14 + b13 + b12 + b11 - b10 + b9 - 2*b8 + b7 + b6 + 2*b5 + b4 - b2 - 2*b1 + 2) * q^50 + (-b13 - b11 - b9 - b7 - b2 - 2*b1 - 1) * q^52 + (b15 - b14 + b13 - 2*b12 - b11 + 2*b10 - 2*b9 + 2*b8 - 2*b7 - 2*b6 - b5 - b4 + 2*b3 - b1 + 1) * q^53 + (b15 + b13 + b11 + 2*b10 - b9 - 3*b8 - 6*b7 + b6 + 2*b5 + 2*b3 - 2*b2 - 6*b1 + 1) * q^55 + (-b15 + b9) * q^56 + (-b12 - b10 - b8 - b5 - b4 + b3 + 1) * q^58 + (-2*b15 + b14 + 2*b13 + b12 + 2*b7 - b4 + 2*b3 - b2 - 3) * q^59 + (-2*b14 + 2*b12 + b11 + b10 - b9 + 7*b8 + b7 + b6 + b5 + 2*b4 - 2*b3 + b2 + 2*b1 - 3) * q^61 + (-2*b8 - b7 + 2*b6 + b4 + 2*b3 - b2 + b1 - 1) * q^62 - q^64 + (-2*b15 + 3*b14 - 2*b13 - 3*b12 - 2*b11 - 3*b10 + 2*b9 - 6*b8 - 3*b7 - 3*b5 - b4 + b3 - 3*b1 + 3) * q^65 + (b10 + 2*b8 - 2*b6 - b5 - 3*b4 - b3 - 4) * q^67 + (-b15 - b13 - b11 + 2*b8 + b6 + b2 + b1 - 2) * q^68 + (2*b15 - b14 + 3*b10 - b9 + b8 - 5*b7 - b1 - 3) * q^70 + (-b15 + b14 - b13 - b12 - 3*b11 + b10 + 3*b9 - 2*b8 + b7 + b5 - b4 - 2*b3 + b2 - b1 - 1) * q^71 + (-b15 + b14 + b13 + b12 + b11 - b10 - b8 - 4*b7 + b6 + b4 + 2*b3 - b2 + b1 + 2) * q^73 + (2*b15 - b14 - b13 + b12 + b11 + 2*b10 - b9 + 3*b8 - b7 - 2*b6 - b5 - b4 - b3 + b2 - 3) * q^74 + (-b15 + b14 + b13 + b12 + b9 + b7 - b6 + b5 - b4 - b2 - b1) * q^76 + (b15 - 3*b14 - b12 + 2*b10 - 3*b9 - b8 - 4*b7 + b6 - b5 + 2*b4 + 4*b3 - 2*b2 + 2*b1 + 4) * q^77 + (-b15 - b14 + b13 - b12 - b11 - b10 - b9 - b8 + b6 + b5 - b4 + b3 - 4*b2 + 1) * q^79 + (-b12 - b10 - b5) * q^80 + (-b13 - b11 - b9 - b7 + b6 + 2*b4 - b3 + b2 + b1) * q^82 + (2*b15 - b14 + b13 + 2*b11 - b9 + b8 - b6 - b5 - 2*b4 + 2*b3) * q^83 + (4*b14 - b12 - 3*b10 - 2*b8 - 6*b7 - 4*b5 - b4 + b3 - 3*b1) * q^85 + (-b15 + 2*b14 + b12 + b11 - b10 + b9 + b7 - b6 - b5 - b4 + 2*b2 + b1) * q^86 + (b14 - b10 - b8 - b5 + b4 + b3 - b2 + 1) * q^88 + (-2*b15 + b14 + 2*b11 - b10 + 2*b8 - 2*b7 + 4*b6 - b5 + b4 + b3 - b2 + b1 + 1) * q^89 + (2*b15 - b14 + 2*b12 + 3*b10 - b9 - 3*b8 + 4*b7 - b6 + b5 + b4 + 3*b3 + 5*b1 + 2) * q^91 + (b13 - b9 - b8 + b4 + 2*b3 - b2 + 1) * q^92 + (-b15 + b14 - b13 - b12 - 2*b10 - b8 + b7 + b6 - 2*b5 + 2*b4 - b2 + 2*b1 + 1) * q^94 + (-2*b11 + 3*b10 + 2*b9 - 8*b8 - 5*b7 + 3*b5 + b4 - b3 - 2*b2 - 7*b1 + 2) * q^95 + (b15 + 2*b14 + b13 - b11 - 2*b9 + 4*b7 - b6 - 2*b5 + 2*b3 - b2 - b1) * q^97 + (-2*b14 - b12 - b10 - b9 + 6*b8 - b6 + b4 - 2*b3 + b2 - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 16 q^{4} + 2 q^{7}+O(q^{10})$$ 16 * q - 16 * q^4 + 2 * q^7 $$16 q - 16 q^{4} + 2 q^{7} - 12 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + 18 q^{17} + 6 q^{23} - 8 q^{25} - 12 q^{26} - 2 q^{28} - 6 q^{29} - 30 q^{35} - 2 q^{37} + 6 q^{41} - 2 q^{43} + 12 q^{44} + 6 q^{46} + 36 q^{47} - 8 q^{49} + 12 q^{50} - 6 q^{52} + 36 q^{53} - 6 q^{56} + 6 q^{58} - 60 q^{59} - 36 q^{62} - 16 q^{64} - 28 q^{67} - 18 q^{68} - 18 q^{70} - 18 q^{74} + 42 q^{77} + 32 q^{79} - 12 q^{85} - 24 q^{86} + 24 q^{89} - 12 q^{91} - 6 q^{92} + 6 q^{97} + 24 q^{98}+O(q^{100})$$ 16 * q - 16 * q^4 + 2 * q^7 - 12 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + 18 * q^17 + 6 * q^23 - 8 * q^25 - 12 * q^26 - 2 * q^28 - 6 * q^29 - 30 * q^35 - 2 * q^37 + 6 * q^41 - 2 * q^43 + 12 * q^44 + 6 * q^46 + 36 * q^47 - 8 * q^49 + 12 * q^50 - 6 * q^52 + 36 * q^53 - 6 * q^56 + 6 * q^58 - 60 * q^59 - 36 * q^62 - 16 * q^64 - 28 * q^67 - 18 * q^68 - 18 * q^70 - 18 * q^74 + 42 * q^77 + 32 * q^79 - 12 * q^85 - 24 * q^86 + 24 * q^89 - 12 * q^91 - 6 * q^92 + 6 * q^97 + 24 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + 2785 x^{8} - 2640 x^{7} - 2601 x^{6} + 10260 x^{5} - 10611 x^{4} - 1944 x^{3} + \cdots + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$( 50 \nu^{15} + 1352 \nu^{14} - 6827 \nu^{13} + 7676 \nu^{12} + 27422 \nu^{11} - 107246 \nu^{10} + 107467 \nu^{9} + 206194 \nu^{8} - 757363 \nu^{7} + 724572 \nu^{6} + \cdots + 2825604 ) / 142155$$ (50*v^15 + 1352*v^14 - 6827*v^13 + 7676*v^12 + 27422*v^11 - 107246*v^10 + 107467*v^9 + 206194*v^8 - 757363*v^7 + 724572*v^6 + 756198*v^5 - 2730942*v^4 + 2372247*v^3 + 1318518*v^2 - 4444713*v + 2825604) / 142155 $$\beta_{2}$$ $$=$$ $$( 169 \nu^{15} - 866 \nu^{14} + 1319 \nu^{13} + 2308 \nu^{12} - 13199 \nu^{11} + 19055 \nu^{10} + 12131 \nu^{9} - 90010 \nu^{8} + 128722 \nu^{7} + 13734 \nu^{6} - 313347 \nu^{5} + \cdots - 244944 ) / 47385$$ (169*v^15 - 866*v^14 + 1319*v^13 + 2308*v^12 - 13199*v^11 + 19055*v^10 + 12131*v^9 - 90010*v^8 + 128722*v^7 + 13734*v^6 - 313347*v^5 + 432351*v^4 - 55431*v^3 - 524799*v^2 + 629856*v - 244944) / 47385 $$\beta_{3}$$ $$=$$ $$( 154 \nu^{15} - 1325 \nu^{14} + 3608 \nu^{13} - 224 \nu^{12} - 22478 \nu^{11} + 55022 \nu^{10} - 23518 \nu^{9} - 159688 \nu^{8} + 382978 \nu^{7} - 226785 \nu^{6} + \cdots - 1285227 ) / 47385$$ (154*v^15 - 1325*v^14 + 3608*v^13 - 224*v^12 - 22478*v^11 + 55022*v^10 - 23518*v^9 - 159688*v^8 + 382978*v^7 - 226785*v^6 - 566733*v^5 + 1355805*v^4 - 871830*v^3 - 949725*v^2 + 2135727*v - 1285227) / 47385 $$\beta_{4}$$ $$=$$ $$( 1445 \nu^{15} - 9836 \nu^{14} + 21081 \nu^{13} + 15627 \nu^{12} - 172766 \nu^{11} + 334353 \nu^{10} + 13019 \nu^{9} - 1283242 \nu^{8} + 2419149 \nu^{7} - 693301 \nu^{6} + \cdots - 7117227 ) / 47385$$ (1445*v^15 - 9836*v^14 + 21081*v^13 + 15627*v^12 - 172766*v^11 + 334353*v^10 + 13019*v^9 - 1283242*v^8 + 2419149*v^7 - 693301*v^6 - 4779069*v^5 + 8827731*v^4 - 3909951*v^3 - 8165529*v^2 + 13935564*v - 7117227) / 47385 $$\beta_{5}$$ $$=$$ $$( 2858 \nu^{15} - 19265 \nu^{14} + 40866 \nu^{13} + 31392 \nu^{12} - 338066 \nu^{11} + 649239 \nu^{10} + 37829 \nu^{9} - 2521996 \nu^{8} + 4717386 \nu^{7} + \cdots - 14041269 ) / 47385$$ (2858*v^15 - 19265*v^14 + 40866*v^13 + 31392*v^12 - 338066*v^11 + 649239*v^10 + 37829*v^9 - 2521996*v^8 + 4717386*v^7 - 1293175*v^6 - 9430341*v^5 + 17293545*v^4 - 7519770*v^3 - 16197165*v^2 + 27464589*v - 14041269) / 47385 $$\beta_{6}$$ $$=$$ $$( - 11192 \nu^{15} + 70123 \nu^{14} - 136087 \nu^{13} - 145859 \nu^{12} + 1215277 \nu^{11} - 2154880 \nu^{10} - 494788 \nu^{9} + 9027170 \nu^{8} - 15633236 \nu^{7} + \cdots + 42998607 ) / 142155$$ (-11192*v^15 + 70123*v^14 - 136087*v^13 - 145859*v^12 + 1215277*v^11 - 2154880*v^10 - 494788*v^9 + 9027170*v^8 - 15633236*v^7 + 2544783*v^6 + 33710346*v^5 - 57047193*v^4 + 20576268*v^3 + 57644217*v^2 - 89475273*v + 42998607) / 142155 $$\beta_{7}$$ $$=$$ $$( 16898 \nu^{15} - 108472 \nu^{14} + 217033 \nu^{13} + 208526 \nu^{12} - 1883968 \nu^{11} + 3436840 \nu^{10} + 565132 \nu^{9} - 13978340 \nu^{8} + 24892859 \nu^{7} + \cdots - 69445998 ) / 142155$$ (16898*v^15 - 108472*v^14 + 217033*v^13 + 208526*v^12 - 1883968*v^11 + 3436840*v^10 + 565132*v^9 - 13978340*v^8 + 24892859*v^7 - 5100702*v^6 - 52089894*v^5 + 90692622*v^4 - 35165097*v^3 - 88915158*v^2 + 142069707*v - 69445998) / 142155 $$\beta_{8}$$ $$=$$ $$( - 4120 \nu^{15} + 25571 \nu^{14} - 48788 \nu^{13} - 55006 \nu^{12} + 441224 \nu^{11} - 771188 \nu^{10} - 200900 \nu^{9} + 3269857 \nu^{8} - 5585140 \nu^{7} + \cdots + 14963454 ) / 28431$$ (-4120*v^15 + 25571*v^14 - 48788*v^13 - 55006*v^12 + 441224*v^11 - 771188*v^10 - 200900*v^9 + 3269857*v^8 - 5585140*v^7 + 796053*v^6 + 12187116*v^5 - 20322468*v^4 + 7028856*v^3 + 20790351*v^2 - 31653180*v + 14963454) / 28431 $$\beta_{9}$$ $$=$$ $$( 8586 \nu^{15} - 53254 \nu^{14} + 101261 \nu^{13} + 115567 \nu^{12} - 919066 \nu^{11} + 1601225 \nu^{10} + 428149 \nu^{9} - 6806990 \nu^{8} + 11591968 \nu^{7} + \cdots - 30865131 ) / 47385$$ (8586*v^15 - 53254*v^14 + 101261*v^13 + 115567*v^12 - 919066*v^11 + 1601225*v^10 + 428149*v^9 - 6806990*v^8 + 11591968*v^7 - 1602394*v^6 - 25345758*v^5 + 42144849*v^4 - 14468679*v^3 - 43148781*v^2 + 65494089*v - 30865131) / 47385 $$\beta_{10}$$ $$=$$ $$( - 26068 \nu^{15} + 165707 \nu^{14} - 325403 \nu^{13} - 334861 \nu^{12} + 2873363 \nu^{11} - 5145350 \nu^{10} - 1063457 \nu^{9} + 21324130 \nu^{8} + \cdots + 102032298 ) / 142155$$ (-26068*v^15 + 165707*v^14 - 325403*v^13 - 334861*v^12 + 2873363*v^11 - 5145350*v^10 - 1063457*v^9 + 21324130*v^8 - 37270684*v^7 + 6581337*v^6 + 79507809*v^5 - 135722682*v^4 + 50026572*v^3 + 135776493*v^2 - 212064642*v + 102032298) / 142155 $$\beta_{11}$$ $$=$$ $$( - 31130 \nu^{15} + 194263 \nu^{14} - 374983 \nu^{13} - 405716 \nu^{12} + 3354958 \nu^{11} - 5936959 \nu^{10} - 1366837 \nu^{9} + 24841391 \nu^{8} + \cdots + 117273501 ) / 142155$$ (-31130*v^15 + 194263*v^14 - 374983*v^13 - 405716*v^12 + 3354958*v^11 - 5936959*v^10 - 1366837*v^9 + 24841391*v^8 - 43003982*v^7 + 7063563*v^6 + 92428227*v^5 - 156501153*v^4 + 56669868*v^3 + 157282722*v^2 - 244082322*v + 117273501) / 142155 $$\beta_{12}$$ $$=$$ $$( 33311 \nu^{15} - 207064 \nu^{14} + 396466 \nu^{13} + 441842 \nu^{12} - 3574291 \nu^{11} + 6268915 \nu^{10} + 1580329 \nu^{9} - 26474600 \nu^{8} + \cdots - 121811526 ) / 142155$$ (33311*v^15 - 207064*v^14 + 396466*v^13 + 441842*v^12 - 3574291*v^11 + 6268915*v^10 + 1580329*v^9 - 26474600*v^8 + 45384563*v^7 - 6735609*v^6 - 98586783*v^5 + 165029724*v^4 - 57791394*v^3 - 167924421*v^2 + 256798269*v - 121811526) / 142155 $$\beta_{13}$$ $$=$$ $$( 5368 \nu^{15} - 32959 \nu^{14} + 62025 \nu^{13} + 72930 \nu^{12} - 567820 \nu^{11} + 981351 \nu^{10} + 280045 \nu^{9} - 4204439 \nu^{8} + 7108452 \nu^{7} + \cdots - 18826182 ) / 15795$$ (5368*v^15 - 32959*v^14 + 62025*v^13 + 72930*v^12 - 567820*v^11 + 981351*v^10 + 280045*v^9 - 4204439*v^8 + 7108452*v^7 - 906329*v^6 - 15657912*v^5 + 25851399*v^4 - 8698959*v^3 - 26672436*v^2 + 40175190*v - 18826182) / 15795 $$\beta_{14}$$ $$=$$ $$( - 62668 \nu^{15} + 397070 \nu^{14} - 778856 \nu^{13} - 802747 \nu^{12} + 6878696 \nu^{11} - 12322064 \nu^{10} - 2527469 \nu^{9} + 51016561 \nu^{8} + \cdots + 244346949 ) / 142155$$ (-62668*v^15 + 397070*v^14 - 778856*v^13 - 802747*v^12 + 6878696*v^11 - 12322064*v^10 - 2527469*v^9 + 51016561*v^8 - 89230126*v^7 + 15905550*v^6 + 190088901*v^5 - 324842130*v^4 + 120144465*v^3 + 324340605*v^2 - 507326409*v + 244346949) / 142155 $$\beta_{15}$$ $$=$$ $$( 5105 \nu^{15} - 31804 \nu^{14} + 61039 \nu^{13} + 67583 \nu^{12} - 549514 \nu^{11} + 965497 \nu^{10} + 239686 \nu^{9} - 4072523 \nu^{8} + 6992981 \nu^{7} + \cdots - 18816948 ) / 10935$$ (5105*v^15 - 31804*v^14 + 61039*v^13 + 67583*v^12 - 549514*v^11 + 965497*v^10 + 239686*v^9 - 4072523*v^8 + 6992981*v^7 - 1054329*v^6 - 15172956*v^5 + 25447554*v^4 - 8949609*v^3 - 25866621*v^2 + 39638646*v - 18816948) / 10935
 $$\nu$$ $$=$$ $$( -\beta_{15} + 2\beta_{9} + \beta_{8} + \beta_{7} - 2\beta_{6} - \beta_{4} - \beta_{3} + 2\beta_{2} - \beta _1 + 2 ) / 3$$ (-b15 + 2*b9 + b8 + b7 - 2*b6 - b4 - b3 + 2*b2 - b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{14} - 2 \beta_{13} - \beta_{11} + \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - \beta_{6} - 2 \beta_{4} - \beta_{3} - 2 \beta _1 + 3 ) / 3$$ (b14 - 2*b13 - b11 + b10 + 3*b9 - 2*b8 + 2*b7 - b6 - 2*b4 - b3 - 2*b1 + 3) / 3 $$\nu^{3}$$ $$=$$ $$( - 2 \beta_{13} + \beta_{12} + 2 \beta_{11} + 3 \beta_{10} + 6 \beta_{9} + 2 \beta_{8} + 5 \beta_{7} + 2 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + 8 \beta_{2} + \beta _1 - 4 ) / 3$$ (-2*b13 + b12 + 2*b11 + 3*b10 + 6*b9 + 2*b8 + 5*b7 + 2*b5 - 3*b4 - 5*b3 + 8*b2 + b1 - 4) / 3 $$\nu^{4}$$ $$=$$ $$( 5 \beta_{15} - 3 \beta_{13} - \beta_{12} + 3 \beta_{11} + 6 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + \beta_{7} + 7 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 11 \beta _1 + 8 ) / 3$$ (5*b15 - 3*b13 - b12 + 3*b11 + 6*b10 + 2*b9 - 3*b8 + b7 + 7*b6 + 4*b5 + 2*b4 + 4*b3 - 2*b2 + 11*b1 + 8) / 3 $$\nu^{5}$$ $$=$$ $$( - 10 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + 9 \beta_{11} + 5 \beta_{10} + 6 \beta_{9} + 30 \beta_{8} - 3 \beta_{7} - 2 \beta_{6} + 6 \beta_{5} - 7 \beta_{4} + \beta_{3} + 16 \beta_{2} + 21 \beta _1 - 1 ) / 3$$ (-10*b14 + 3*b13 + 3*b12 + 9*b11 + 5*b10 + 6*b9 + 30*b8 - 3*b7 - 2*b6 + 6*b5 - 7*b4 + b3 + 16*b2 + 21*b1 - 1) / 3 $$\nu^{6}$$ $$=$$ $$( - 7 \beta_{15} - 2 \beta_{14} + 7 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} - 2 \beta_{10} + 8 \beta_{9} + 15 \beta_{8} + 30 \beta_{7} - 3 \beta_{6} - 4 \beta_{5} - 12 \beta_{4} + 32 \beta_{3} - 21 \beta_{2} + 51 \beta _1 + 20 ) / 3$$ (-7*b15 - 2*b14 + 7*b13 - 2*b12 + 11*b11 - 2*b10 + 8*b9 + 15*b8 + 30*b7 - 3*b6 - 4*b5 - 12*b4 + 32*b3 - 21*b2 + 51*b1 + 20) / 3 $$\nu^{7}$$ $$=$$ $$( - 36 \beta_{15} + 3 \beta_{14} + 6 \beta_{13} + 37 \beta_{12} + 24 \beta_{11} - 21 \beta_{10} + 33 \beta_{9} + 49 \beta_{8} + 93 \beta_{7} - \beta_{5} - 39 \beta_{4} - 17 \beta_{3} + 46 \beta_{2} + 90 \beta _1 - 3 ) / 3$$ (-36*b15 + 3*b14 + 6*b13 + 37*b12 + 24*b11 - 21*b10 + 33*b9 + 49*b8 + 93*b7 - b5 - 39*b4 - 17*b3 + 46*b2 + 90*b1 - 3) / 3 $$\nu^{8}$$ $$=$$ $$( - 42 \beta_{15} + 68 \beta_{14} + 10 \beta_{13} + 66 \beta_{12} + 50 \beta_{11} - 25 \beta_{10} + 42 \beta_{9} - 71 \beta_{8} + 179 \beta_{7} + 61 \beta_{6} + 18 \beta_{5} + 50 \beta_{4} + 31 \beta_{3} - 16 \beta_{2} + 163 \beta _1 + 94 ) / 3$$ (-42*b15 + 68*b14 + 10*b13 + 66*b12 + 50*b11 - 25*b10 + 42*b9 - 71*b8 + 179*b7 + 61*b6 + 18*b5 + 50*b4 + 31*b3 - 16*b2 + 163*b1 + 94) / 3 $$\nu^{9}$$ $$=$$ $$( - 44 \beta_{15} + 32 \beta_{14} + 216 \beta_{12} + 72 \beta_{11} + 14 \beta_{10} + 40 \beta_{9} + 50 \beta_{8} - 19 \beta_{7} + 66 \beta_{6} + 126 \beta_{5} + 66 \beta_{4} - 64 \beta_{3} + 110 \beta_{2} + 67 \beta _1 + 168 ) / 3$$ (-44*b15 + 32*b14 + 216*b12 + 72*b11 + 14*b10 + 40*b9 + 50*b8 - 19*b7 + 66*b6 + 126*b5 + 66*b4 - 64*b3 + 110*b2 + 67*b1 + 168) / 3 $$\nu^{10}$$ $$=$$ $$( - 58 \beta_{15} - 18 \beta_{14} + 90 \beta_{13} + 172 \beta_{12} + 126 \beta_{11} + 162 \beta_{10} + 62 \beta_{9} + 14 \beta_{8} - 161 \beta_{7} - 8 \beta_{6} + 167 \beta_{5} + 158 \beta_{4} + 153 \beta_{3} - 144 \beta_{2} - 61 \beta _1 + 188 ) / 3$$ (-58*b15 - 18*b14 + 90*b13 + 172*b12 + 126*b11 + 162*b10 + 62*b9 + 14*b8 - 161*b7 - 8*b6 + 167*b5 + 158*b4 + 153*b3 - 144*b2 - 61*b1 + 188) / 3 $$\nu^{11}$$ $$=$$ $$( - 126 \beta_{15} - 351 \beta_{14} - 100 \beta_{13} + 231 \beta_{12} + 13 \beta_{11} + 333 \beta_{10} + 48 \beta_{9} + 80 \beta_{8} - 482 \beta_{7} - 81 \beta_{6} + 141 \beta_{5} - 219 \beta_{4} - 138 \beta_{3} + 86 \beta_{2} + \cdots + 217 ) / 3$$ (-126*b15 - 351*b14 - 100*b13 + 231*b12 + 13*b11 + 333*b10 + 48*b9 + 80*b8 - 482*b7 - 81*b6 + 141*b5 - 219*b4 - 138*b3 + 86*b2 - 46*b1 + 217) / 3 $$\nu^{12}$$ $$=$$ $$( - 209 \beta_{15} - 418 \beta_{14} - 251 \beta_{13} - 170 \beta_{12} + 68 \beta_{11} + 416 \beta_{10} + 301 \beta_{9} - 320 \beta_{8} + 325 \beta_{7} - 36 \beta_{6} - 298 \beta_{5} + 153 \beta_{4} - 357 \beta_{3} + 376 \beta_{2} + \cdots - 97 ) / 3$$ (-209*b15 - 418*b14 - 251*b13 - 170*b12 + 68*b11 + 416*b10 + 301*b9 - 320*b8 + 325*b7 - 36*b6 - 298*b5 + 153*b4 - 357*b3 + 376*b2 + 1031*b1 - 97) / 3 $$\nu^{13}$$ $$=$$ $$( 385 \beta_{15} - 609 \beta_{14} - 1440 \beta_{13} + 402 \beta_{12} - 186 \beta_{11} + 414 \beta_{10} - 155 \beta_{9} - 478 \beta_{8} - 220 \beta_{7} + 98 \beta_{6} - 597 \beta_{5} + 277 \beta_{4} - 1199 \beta_{3} + 1540 \beta_{2} + \cdots + 1243 ) / 3$$ (385*b15 - 609*b14 - 1440*b13 + 402*b12 - 186*b11 + 414*b10 - 155*b9 - 478*b8 - 220*b7 + 98*b6 - 597*b5 + 277*b4 - 1199*b3 + 1540*b2 + 2320*b1 + 1243) / 3 $$\nu^{14}$$ $$=$$ $$( 1281 \beta_{15} - 511 \beta_{14} - 835 \beta_{13} + 960 \beta_{12} + 1042 \beta_{11} + 812 \beta_{10} - 27 \beta_{9} + 3809 \beta_{8} + 541 \beta_{7} - 1481 \beta_{6} - 882 \beta_{5} + 1178 \beta_{4} - 1454 \beta_{3} + \cdots - 1641 ) / 3$$ (1281*b15 - 511*b14 - 835*b13 + 960*b12 + 1042*b11 + 812*b10 - 27*b9 + 3809*b8 + 541*b7 - 1481*b6 - 882*b5 + 1178*b4 - 1454*b3 + 2715*b2 + 2486*b1 - 1641) / 3 $$\nu^{15}$$ $$=$$ $$( 2826 \beta_{15} - 1179 \beta_{14} - 1012 \beta_{13} + 2828 \beta_{12} + 1522 \beta_{11} + 2838 \beta_{10} - 2730 \beta_{9} + 6361 \beta_{8} - 11 \beta_{7} - 2559 \beta_{6} - 431 \beta_{5} - 2124 \beta_{4} + 809 \beta_{3} + \cdots + 208 ) / 3$$ (2826*b15 - 1179*b14 - 1012*b13 + 2828*b12 + 1522*b11 + 2838*b10 - 2730*b9 + 6361*b8 - 11*b7 - 2559*b6 - 431*b5 - 2124*b4 + 809*b3 + 757*b2 + 1787*b1 + 208) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$\beta_{8}$$ $$\beta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 1.73109 − 0.0577511i −1.68301 + 0.409224i 0.320287 + 1.70218i 0.765614 − 1.55365i 1.27866 − 1.16834i −1.70672 − 0.295146i 1.58110 + 0.707199i 1.71298 + 0.256290i 1.27866 + 1.16834i −1.70672 + 0.295146i 1.58110 − 0.707199i 1.71298 − 0.256290i 1.73109 + 0.0577511i −1.68301 − 0.409224i 0.320287 − 1.70218i 0.765614 + 1.55365i
1.00000i 0 −1.00000 −1.14095 + 1.97618i 0 1.42337 2.23025i 1.00000i 0 1.97618 + 1.14095i
143.2 1.00000i 0 −1.00000 −0.714925 + 1.23829i 0 0.327442 + 2.62541i 1.00000i 0 1.23829 + 0.714925i
143.3 1.00000i 0 −1.00000 0.0338034 0.0585493i 0 1.19767 + 2.35915i 1.00000i 0 −0.0585493 0.0338034i
143.4 1.00000i 0 −1.00000 1.82207 3.15592i 0 −1.58246 2.12034i 1.00000i 0 −3.15592 1.82207i
143.5 1.00000i 0 −1.00000 −1.77612 + 3.07634i 0 2.63804 + 0.201867i 1.00000i 0 −3.07634 1.77612i
143.6 1.00000i 0 −1.00000 −0.483662 + 0.837727i 0 −2.16249 1.52435i 1.00000i 0 −0.837727 0.483662i
143.7 1.00000i 0 −1.00000 0.450129 0.779646i 0 1.57151 2.12847i 1.00000i 0 0.779646 + 0.450129i
143.8 1.00000i 0 −1.00000 1.80966 3.13442i 0 −2.41308 + 1.08492i 1.00000i 0 3.13442 + 1.80966i
341.1 1.00000i 0 −1.00000 −1.77612 3.07634i 0 2.63804 0.201867i 1.00000i 0 −3.07634 + 1.77612i
341.2 1.00000i 0 −1.00000 −0.483662 0.837727i 0 −2.16249 + 1.52435i 1.00000i 0 −0.837727 + 0.483662i
341.3 1.00000i 0 −1.00000 0.450129 + 0.779646i 0 1.57151 + 2.12847i 1.00000i 0 0.779646 0.450129i
341.4 1.00000i 0 −1.00000 1.80966 + 3.13442i 0 −2.41308 1.08492i 1.00000i 0 3.13442 1.80966i
341.5 1.00000i 0 −1.00000 −1.14095 1.97618i 0 1.42337 + 2.23025i 1.00000i 0 1.97618 1.14095i
341.6 1.00000i 0 −1.00000 −0.714925 1.23829i 0 0.327442 2.62541i 1.00000i 0 1.23829 0.714925i
341.7 1.00000i 0 −1.00000 0.0338034 + 0.0585493i 0 1.19767 2.35915i 1.00000i 0 −0.0585493 + 0.0338034i
341.8 1.00000i 0 −1.00000 1.82207 + 3.15592i 0 −1.58246 + 2.12034i 1.00000i 0 −3.15592 + 1.82207i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 341.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.l.a 16
3.b odd 2 1 126.2.l.a 16
4.b odd 2 1 3024.2.ca.c 16
7.b odd 2 1 2646.2.l.a 16
7.c even 3 1 2646.2.m.b 16
7.c even 3 1 2646.2.t.b 16
7.d odd 6 1 378.2.t.a 16
7.d odd 6 1 2646.2.m.a 16
9.c even 3 1 126.2.t.a yes 16
9.c even 3 1 1134.2.k.b 16
9.d odd 6 1 378.2.t.a 16
9.d odd 6 1 1134.2.k.a 16
12.b even 2 1 1008.2.ca.c 16
21.c even 2 1 882.2.l.b 16
21.g even 6 1 126.2.t.a yes 16
21.g even 6 1 882.2.m.a 16
21.h odd 6 1 882.2.m.b 16
21.h odd 6 1 882.2.t.a 16
28.f even 6 1 3024.2.df.c 16
36.f odd 6 1 1008.2.df.c 16
36.h even 6 1 3024.2.df.c 16
63.g even 3 1 882.2.m.a 16
63.h even 3 1 882.2.l.b 16
63.i even 6 1 inner 378.2.l.a 16
63.j odd 6 1 2646.2.l.a 16
63.k odd 6 1 882.2.m.b 16
63.k odd 6 1 1134.2.k.a 16
63.l odd 6 1 882.2.t.a 16
63.n odd 6 1 2646.2.m.a 16
63.o even 6 1 2646.2.t.b 16
63.s even 6 1 1134.2.k.b 16
63.s even 6 1 2646.2.m.b 16
63.t odd 6 1 126.2.l.a 16
84.j odd 6 1 1008.2.df.c 16
252.r odd 6 1 3024.2.ca.c 16
252.bj even 6 1 1008.2.ca.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.l.a 16 3.b odd 2 1
126.2.l.a 16 63.t odd 6 1
126.2.t.a yes 16 9.c even 3 1
126.2.t.a yes 16 21.g even 6 1
378.2.l.a 16 1.a even 1 1 trivial
378.2.l.a 16 63.i even 6 1 inner
378.2.t.a 16 7.d odd 6 1
378.2.t.a 16 9.d odd 6 1
882.2.l.b 16 21.c even 2 1
882.2.l.b 16 63.h even 3 1
882.2.m.a 16 21.g even 6 1
882.2.m.a 16 63.g even 3 1
882.2.m.b 16 21.h odd 6 1
882.2.m.b 16 63.k odd 6 1
882.2.t.a 16 21.h odd 6 1
882.2.t.a 16 63.l odd 6 1
1008.2.ca.c 16 12.b even 2 1
1008.2.ca.c 16 252.bj even 6 1
1008.2.df.c 16 36.f odd 6 1
1008.2.df.c 16 84.j odd 6 1
1134.2.k.a 16 9.d odd 6 1
1134.2.k.a 16 63.k odd 6 1
1134.2.k.b 16 9.c even 3 1
1134.2.k.b 16 63.s even 6 1
2646.2.l.a 16 7.b odd 2 1
2646.2.l.a 16 63.j odd 6 1
2646.2.m.a 16 7.d odd 6 1
2646.2.m.a 16 63.n odd 6 1
2646.2.m.b 16 7.c even 3 1
2646.2.m.b 16 63.s even 6 1
2646.2.t.b 16 7.c even 3 1
2646.2.t.b 16 63.o even 6 1
3024.2.ca.c 16 4.b odd 2 1
3024.2.ca.c 16 252.r odd 6 1
3024.2.df.c 16 28.f even 6 1
3024.2.df.c 16 36.h even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(378, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{8}$$
$3$ $$T^{16}$$
$5$ $$T^{16} + 24 T^{14} + 24 T^{13} + 423 T^{12} + \cdots + 81$$
$7$ $$T^{16} - 2 T^{15} + 6 T^{14} + \cdots + 5764801$$
$11$ $$T^{16} + 12 T^{15} + 18 T^{14} + \cdots + 61732449$$
$13$ $$T^{16} - 6 T^{15} - 57 T^{14} + \cdots + 390971529$$
$17$ $$T^{16} - 18 T^{15} + 231 T^{14} + \cdots + 56070144$$
$19$ $$T^{16} - 72 T^{14} + 4167 T^{12} + \cdots + 9199089$$
$23$ $$T^{16} - 6 T^{15} - 54 T^{14} + \cdots + 187388721$$
$29$ $$T^{16} + 6 T^{15} - 36 T^{14} + \cdots + 1108809$$
$31$ $$T^{16} + 204 T^{14} + \cdots + 65610000$$
$37$ $$T^{16} + 2 T^{15} + \cdots + 32746159681$$
$41$ $$T^{16} - 6 T^{15} + 105 T^{14} + 210 T^{13} + \cdots + 81$$
$43$ $$T^{16} + 2 T^{15} + \cdots + 2999643361$$
$47$ $$(T^{8} - 18 T^{7} + 3 T^{6} + 1650 T^{5} + \cdots + 766944)^{2}$$
$53$ $$T^{16} - 36 T^{15} + \cdots + 36759242529$$
$59$ $$(T^{8} + 30 T^{7} + 228 T^{6} + \cdots + 465300)^{2}$$
$61$ $$T^{16} + 504 T^{14} + \cdots + 547560000$$
$67$ $$(T^{8} + 14 T^{7} - 101 T^{6} + \cdots + 51028)^{2}$$
$71$ $$T^{16} + 486 T^{14} + \cdots + 65610000$$
$73$ $$T^{16} - 150 T^{14} + \cdots + 71115489$$
$79$ $$(T^{8} - 16 T^{7} - 149 T^{6} + \cdots - 985100)^{2}$$
$83$ $$T^{16} + 177 T^{14} + \cdots + 953512641$$
$89$ $$T^{16} + \cdots + 131145120363321$$
$97$ $$T^{16} - 6 T^{15} + \cdots + 9120206721024$$