# Properties

 Label 1638.2.bj.g Level $1638$ Weight $2$ Character orbit 1638.bj Analytic conductor $13.079$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.bj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + 3842 x^{4} - 3394 x^{3} + 2141 x^{2} - 832 x + 169$$ x^12 - 6*x^11 + 39*x^10 - 140*x^9 + 460*x^8 - 1066*x^7 + 2127*x^6 - 3172*x^5 + 3842*x^4 - 3394*x^3 + 2141*x^2 - 832*x + 169 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 182) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + \beta_{7} q^{4} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_1) q^{5} - \beta_{5} q^{7} + (\beta_{6} - \beta_{5}) q^{8}+O(q^{10})$$ q + b6 * q^2 + b7 * q^4 + (b9 - b8 - b7 + b6 - b5 + b1) * q^5 - b5 * q^7 + (b6 - b5) * q^8 $$q + \beta_{6} q^{2} + \beta_{7} q^{4} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_1) q^{5} - \beta_{5} q^{7} + (\beta_{6} - \beta_{5}) q^{8} + (\beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} - \beta_{3} - 1) q^{10} + (\beta_{11} - \beta_{8} + \beta_{7} + \beta_{3} + 1) q^{11} + ( - \beta_{10} - \beta_{9} + \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{13} - q^{14} + (\beta_{7} - 1) q^{16} + (\beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{10} + 2 \beta_{9} - 2 \beta_{7} - \beta_{4} + \beta_{2} + 4) q^{19} + (\beta_{9} - \beta_{5} + \beta_{4}) q^{20} + (\beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{7} + 3 \beta_{6} - \beta_{5}) q^{22} + (\beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_1) q^{23} + (\beta_{9} + \beta_{8} - \beta_{3} - 2 \beta_{2} - 2) q^{25} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{3} - \beta_1 + 1) q^{26} - \beta_{6} q^{28} + ( - \beta_{10} + 2 \beta_{9} - \beta_{8} - 3 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} + \cdots + 2) q^{29}+ \cdots + \beta_{5} q^{98}+O(q^{100})$$ q + b6 * q^2 + b7 * q^4 + (b9 - b8 - b7 + b6 - b5 + b1) * q^5 - b5 * q^7 + (b6 - b5) * q^8 + (b11 + b10 + b7 + b6 - b3 - 1) * q^10 + (b11 - b8 + b7 + b3 + 1) * q^11 + (-b10 - b9 + b5 + b3 - b2 - 1) * q^13 - q^14 + (b7 - 1) * q^16 + (b11 + b10 + b7 + b6 + b4 + b2 - 2*b1) * q^17 + (-b10 + 2*b9 - 2*b7 - b4 + b2 + 4) * q^19 + (b9 - b5 + b4) * q^20 + (b11 + b9 - 2*b8 - b7 + 3*b6 - b5) * q^22 + (b11 - b10 - b7 + b6 + b4 - b3 + b1) * q^23 + (b9 + b8 - b3 - 2*b2 - 2) * q^25 + (-b8 + b7 - b6 + b3 - b1 + 1) * q^26 - b6 * q^28 + (-b10 + 2*b9 - b8 - 3*b7 - b6 + 2*b5 + b4 + b1 + 2) * q^29 + (2*b11 + b9 - b8 - 2*b7 + b6 + b5 - b3 + 1) * q^31 - b5 * q^32 + (-2*b10 + b9 - b8 - b7 - b6 + b5 - b2 + b1) * q^34 + (-b11 - b10 - b7 - b6 - b2) * q^35 + (2*b11 + b10 - b8 + b7 + b6 + b4 + 2*b3 + 2*b2 - b1) * q^37 + (-b7 + 2*b6 + 2*b5 - 2*b4 - 2*b3 + b2 + b1) * q^38 + (-b3 - b2 - 1) * q^40 + (-b11 - b10 + 2*b8 + 2*b7 - b6 + b4 - b3 - 2*b2 - b1 + 1) * q^41 + (b11 - 2*b10 + b9 - 2*b8 + 3*b7 - b6 + b5 - 2*b2) * q^43 + (2*b11 + b9 - b8 + 2*b7 + b6 + b5 - b3 - 1) * q^44 + (b10 + b9 - b4 - b2) * q^46 + (-2*b9 + 2*b8 - 4*b7 - 2*b6 + 2*b5 + 2) * q^47 + (-b7 + 1) * q^49 + (-b11 + b8 + 2*b7 - 3*b6 + 2*b4 - b3 - 2*b1) * q^50 + (b11 - b10 - b8 - b7 + 2*b6) * q^52 + (-b9 - b8 + b7 - 3*b6 - 3*b5 + 2*b4 - 3*b2 - b1 - 6) * q^53 + (b11 + 2*b10 + 2*b9 - b8 + b7 + 6*b6 - 10*b5 + 2*b4 - b3 + 2*b1 - 3) * q^55 - b7 * q^56 + (b11 + b10 - b7 + b6 + 2*b5 - b4 - 2*b3 - b2 + 2) * q^58 + (b10 - 2*b4 - b2) * q^59 + (-4*b10 + b9 - 2*b8 - 6*b7 + 2*b6 - b5 + b4 - 4*b2 - 2*b1) * q^61 + (b11 + 2*b9 - b8 - b7 + 2*b5 - b3 + 1) * q^62 - q^64 + (2*b11 + 4*b10 - 2*b9 + 4*b7 + 5*b6 - b5 - 2*b4 - b3 + b2 + b1) * q^65 + (-b11 + b8 - b7 - 2*b6 + 2*b4 - b3 - 2*b1 - 3) * q^67 + (b11 + b10 + b6 - b4 - b3 - b1 + 1) * q^68 + (-b9 + b8 + b7 - b6 + b5 - b1) * q^70 + (-b11 + b10 + 3*b7 - b6 - 3*b5 + b4 + 2*b3 - b2 - 6) * q^71 + (2*b11 + 2*b10 - 2*b9 + 2*b8 + 9*b7 + 4*b6 - 2*b5 - b3 + b2 - 3*b1 - 3) * q^73 + (b11 - b10 + 2*b9 - 4*b8 - 3*b7 + b6 - b4 - b2 + 2*b1) * q^74 + (b10 + 2*b8 + b7 - b4 + 2*b2 + b1 + 2) * q^76 + (b9 + b8 - b6 - b5 - b3 + 1) * q^77 + (b9 + b8 - b6 - b5 - 3*b3 - 2*b2 + 3) * q^79 + (b8 + b7 - b6 + b4 - b1) * q^80 + (-2*b11 - b10 - b9 + 2*b8 + 2*b7 - b5 + b4 - b2 - 2*b1) * q^82 + (-2*b11 - 2*b10 - 3*b9 + 3*b8 + 5*b7 - b6 - b5 + b3 - b2 + b1 - 3) * q^83 + (-b10 - b9 - 2*b7 - 5*b5 + 3*b4 + b2 + 4) * q^85 + (2*b11 + b9 - b8 + 5*b6 - 3*b5 - b3 - 2*b1 + 1) * q^86 + (b11 + 2*b9 - b8 - b7 + 2*b6 - 2*b5 - b3 + 1) * q^88 + (2*b11 - 2*b8 - 2*b7 + 2*b6 - 2*b4 + 2*b3 + 2*b1) * q^89 + (b11 + b9 + b6 + b5 - b4 - b3 + b1 - 1) * q^91 + (b7 + 2*b4 - b3 + b2 - b1) * q^92 + (-2*b11 - 2*b7 - 4*b6 + 4*b5 + 2*b3 + 2) * q^94 + (3*b11 - 2*b10 + b9 - 2*b8 - 6*b7 + 11*b6 - 4*b5 - 2*b2) * q^95 + (-2*b9 - 4*b7 - 4*b4 + 8) * q^97 + b5 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{4}+O(q^{10})$$ 12 * q + 6 * q^4 $$12 q + 6 q^{4} - 2 q^{10} + 18 q^{11} - 8 q^{13} - 12 q^{14} - 6 q^{16} - 4 q^{17} + 12 q^{19} - 2 q^{22} + 6 q^{23} - 24 q^{25} + 14 q^{26} + 10 q^{29} - 2 q^{35} - 6 q^{37} - 8 q^{38} - 4 q^{40} + 24 q^{41} + 26 q^{43} - 6 q^{46} + 6 q^{49} + 12 q^{50} - 4 q^{52} - 36 q^{53} - 6 q^{55} - 6 q^{56} + 24 q^{58} - 6 q^{59} - 28 q^{61} + 2 q^{62} - 12 q^{64} + 34 q^{65} - 42 q^{67} + 4 q^{68} - 48 q^{71} + 12 q^{76} + 4 q^{77} + 44 q^{79} + 6 q^{82} + 54 q^{85} + 2 q^{88} - 12 q^{89} - 16 q^{91} + 12 q^{92} + 8 q^{94} - 32 q^{95} + 60 q^{97}+O(q^{100})$$ 12 * q + 6 * q^4 - 2 * q^10 + 18 * q^11 - 8 * q^13 - 12 * q^14 - 6 * q^16 - 4 * q^17 + 12 * q^19 - 2 * q^22 + 6 * q^23 - 24 * q^25 + 14 * q^26 + 10 * q^29 - 2 * q^35 - 6 * q^37 - 8 * q^38 - 4 * q^40 + 24 * q^41 + 26 * q^43 - 6 * q^46 + 6 * q^49 + 12 * q^50 - 4 * q^52 - 36 * q^53 - 6 * q^55 - 6 * q^56 + 24 * q^58 - 6 * q^59 - 28 * q^61 + 2 * q^62 - 12 * q^64 + 34 * q^65 - 42 * q^67 + 4 * q^68 - 48 * q^71 + 12 * q^76 + 4 * q^77 + 44 * q^79 + 6 * q^82 + 54 * q^85 + 2 * q^88 - 12 * q^89 - 16 * q^91 + 12 * q^92 + 8 * q^94 - 32 * q^95 + 60 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + 3842 x^{4} - 3394 x^{3} + 2141 x^{2} - 832 x + 169$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 24 \nu^{10} - 120 \nu^{9} + 751 \nu^{8} - 2284 \nu^{7} + 6728 \nu^{6} - 12694 \nu^{5} + 20323 \nu^{4} - 21914 \nu^{3} + 17046 \nu^{2} - 7860 \nu + 2041 ) / 286$$ (24*v^10 - 120*v^9 + 751*v^8 - 2284*v^7 + 6728*v^6 - 12694*v^5 + 20323*v^4 - 21914*v^3 + 17046*v^2 - 7860*v + 2041) / 286 $$\beta_{3}$$ $$=$$ $$( - 31 \nu^{10} + 155 \nu^{9} - 976 \nu^{8} + 2974 \nu^{7} - 8881 \nu^{6} + 16885 \nu^{5} - 28044 \nu^{4} + 31106 \nu^{3} - 27273 \nu^{2} + 14085 \nu - 5252 ) / 286$$ (-31*v^10 + 155*v^9 - 976*v^8 + 2974*v^7 - 8881*v^6 + 16885*v^5 - 28044*v^4 + 31106*v^3 - 27273*v^2 + 14085*v - 5252) / 286 $$\beta_{4}$$ $$=$$ $$( 162 \nu^{11} + 186 \nu^{10} - 51 \nu^{9} + 15887 \nu^{8} - 51264 \nu^{7} + 192268 \nu^{6} - 390377 \nu^{5} + 670405 \nu^{4} - 762542 \nu^{3} + 592138 \nu^{2} - 270385 \nu + 66227 ) / 7898$$ (162*v^11 + 186*v^10 - 51*v^9 + 15887*v^8 - 51264*v^7 + 192268*v^6 - 390377*v^5 + 670405*v^4 - 762542*v^3 + 592138*v^2 - 270385*v + 66227) / 7898 $$\beta_{5}$$ $$=$$ $$( 2323 \nu^{11} - 22649 \nu^{10} + 132943 \nu^{9} - 590285 \nu^{8} + 1852053 \nu^{7} - 4762085 \nu^{6} + 9077085 \nu^{5} - 13790463 \nu^{4} + 15194803 \nu^{3} + \cdots - 1517893 ) / 102674$$ (2323*v^11 - 22649*v^10 + 132943*v^9 - 590285*v^8 + 1852053*v^7 - 4762085*v^6 + 9077085*v^5 - 13790463*v^4 + 15194803*v^3 - 12124347*v^2 + 5858917*v - 1517893) / 102674 $$\beta_{6}$$ $$=$$ $$( - 2323 \nu^{11} + 2904 \nu^{10} - 34218 \nu^{9} - 29708 \nu^{8} + 35569 \nu^{7} - 841546 \nu^{6} + 1541776 \nu^{5} - 3573290 \nu^{4} + 3839377 \nu^{3} + \cdots - 689598 ) / 102674$$ (-2323*v^11 + 2904*v^10 - 34218*v^9 - 29708*v^8 + 35569*v^7 - 841546*v^6 + 1541776*v^5 - 3573290*v^4 + 3839377*v^3 - 3683500*v^2 + 1916664*v - 689598) / 102674 $$\beta_{7}$$ $$=$$ $$( - 324 \nu^{11} + 1782 \nu^{10} - 10668 \nu^{9} + 34641 \nu^{8} - 98512 \nu^{7} + 195608 \nu^{6} - 301272 \nu^{5} + 336079 \nu^{4} - 238324 \nu^{3} + 98790 \nu^{2} + \cdots - 3573 ) / 7898$$ (-324*v^11 + 1782*v^10 - 10668*v^9 + 34641*v^8 - 98512*v^7 + 195608*v^6 - 301272*v^5 + 336079*v^4 - 238324*v^3 + 98790*v^2 - 2756*v - 3573) / 7898 $$\beta_{8}$$ $$=$$ $$( - 5977 \nu^{11} + 33053 \nu^{10} - 223218 \nu^{9} + 741326 \nu^{8} - 2485739 \nu^{7} + 5177975 \nu^{6} - 10201456 \nu^{5} + 12482628 \nu^{4} - 13546965 \nu^{3} + \cdots - 527956 ) / 102674$$ (-5977*v^11 + 33053*v^10 - 223218*v^9 + 741326*v^8 - 2485739*v^7 + 5177975*v^6 - 10201456*v^5 + 12482628*v^4 - 13546965*v^3 + 7075047*v^2 - 2116390*v - 527956) / 102674 $$\beta_{9}$$ $$=$$ $$( 5977 \nu^{11} - 32694 \nu^{10} + 221423 \nu^{9} - 766456 \nu^{8} + 2597029 \nu^{7} - 6035626 \nu^{6} + 12377355 \nu^{5} - 19119820 \nu^{4} + 23328279 \nu^{3} + \cdots - 3597672 ) / 102674$$ (5977*v^11 - 32694*v^10 + 221423*v^9 - 766456*v^8 + 2597029*v^7 - 6035626*v^6 + 12377355*v^5 - 19119820*v^4 + 23328279*v^3 - 21335604*v^2 + 11829853*v - 3597672) / 102674 $$\beta_{10}$$ $$=$$ $$( - 6388 \nu^{11} + 30826 \nu^{10} - 187767 \nu^{9} + 543572 \nu^{8} - 1501380 \nu^{7} + 2562258 \nu^{6} - 3406537 \nu^{5} + 2547170 \nu^{4} - 212336 \nu^{3} + \cdots - 435708 ) / 102674$$ (-6388*v^11 + 30826*v^10 - 187767*v^9 + 543572*v^8 - 1501380*v^7 + 2562258*v^6 - 3406537*v^5 + 2547170*v^4 - 212336*v^3 - 1550640*v^2 + 1319919*v - 435708) / 102674 $$\beta_{11}$$ $$=$$ $$( - 9240 \nu^{11} + 55128 \nu^{10} - 344643 \nu^{9} + 1207618 \nu^{8} - 3759676 \nu^{7} + 8280896 \nu^{6} - 15228345 \nu^{5} + 20588490 \nu^{4} + \cdots + 1667016 ) / 102674$$ (-9240*v^11 + 55128*v^10 - 344643*v^9 + 1207618*v^8 - 3759676*v^7 + 8280896*v^6 - 15228345*v^5 + 20588490*v^4 - 21887598*v^3 + 15896592*v^2 - 7811231*v + 1667016) / 102674
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + \beta _1 - 4$$ b6 + b5 - b3 + b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$- \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} + 6 \beta_{6} - 4 \beta_{5} - \beta_{3} + 2 \beta_{2} - 5 \beta _1 - 2$$ -b11 + b10 + b9 - b8 - 2*b7 + 6*b6 - 4*b5 - b3 + 2*b2 - 5*b1 - 2 $$\nu^{4}$$ $$=$$ $$- 2 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - \beta_{8} - 6 \beta_{7} - \beta_{6} - 21 \beta_{5} - 4 \beta_{4} + 6 \beta_{3} - 9 \beta_{2} - 9 \beta _1 + 29$$ -2*b11 + 2*b10 + 3*b9 - b8 - 6*b7 - b6 - 21*b5 - 4*b4 + 6*b3 - 9*b2 - 9*b1 + 29 $$\nu^{5}$$ $$=$$ $$8 \beta_{11} - 15 \beta_{10} - 8 \beta_{9} + 13 \beta_{8} + 25 \beta_{7} - 73 \beta_{6} + 26 \beta_{5} - 10 \beta_{4} + 11 \beta_{3} - 35 \beta_{2} + 33 \beta _1 + 34$$ 8*b11 - 15*b10 - 8*b9 + 13*b8 + 25*b7 - 73*b6 + 26*b5 - 10*b4 + 11*b3 - 35*b2 + 33*b1 + 34 $$\nu^{6}$$ $$=$$ $$29 \beta_{11} - 50 \beta_{10} - 48 \beta_{9} + 25 \beta_{8} + 124 \beta_{7} - 81 \beta_{6} + 266 \beta_{5} + 48 \beta_{4} - 41 \beta_{3} + 42 \beta_{2} + 88 \beta _1 - 247$$ 29*b11 - 50*b10 - 48*b9 + 25*b8 + 124*b7 - 81*b6 + 266*b5 + 48*b4 - 41*b3 + 42*b2 + 88*b1 - 247 $$\nu^{7}$$ $$=$$ $$- 60 \beta_{11} + 135 \beta_{10} + 17 \beta_{9} - 115 \beta_{8} - 145 \beta_{7} + 668 \beta_{6} + 14 \beta_{5} + 203 \beta_{4} - 117 \beta_{3} + 400 \beta_{2} - 244 \beta _1 - 526$$ -60*b11 + 135*b10 + 17*b9 - 115*b8 - 145*b7 + 668*b6 + 14*b5 + 203*b4 - 117*b3 + 400*b2 - 244*b1 - 526 $$\nu^{8}$$ $$=$$ $$- 380 \beta_{11} + 778 \beta_{10} + 515 \beta_{9} - 363 \beta_{8} - 1632 \beta_{7} + 1517 \beta_{6} - 2765 \beta_{5} - 340 \beta_{4} + 304 \beta_{3} + 91 \beta_{2} - 949 \beta _1 + 2008$$ -380*b11 + 778*b10 + 515*b9 - 363*b8 - 1632*b7 + 1517*b6 - 2765*b5 - 340*b4 + 304*b3 + 91*b2 - 949*b1 + 2008 $$\nu^{9}$$ $$=$$ $$264 \beta_{11} - 673 \beta_{10} + 454 \beta_{9} + 839 \beta_{8} - 287 \beta_{7} - 5265 \beta_{6} - 3066 \beta_{5} - 2790 \beta_{4} + 1329 \beta_{3} - 3791 \beta_{2} + 1710 \beta _1 + 7116$$ 264*b11 - 673*b10 + 454*b9 + 839*b8 - 287*b7 - 5265*b6 - 3066*b5 - 2790*b4 + 1329*b3 - 3791*b2 + 1710*b1 + 7116 $$\nu^{10}$$ $$=$$ $$4383 \beta_{11} - 9560 \beta_{10} - 4630 \beta_{9} + 4411 \beta_{8} + 17550 \beta_{7} - 20512 \beta_{6} + 25127 \beta_{5} + 650 \beta_{4} - 1974 \beta_{3} - 5273 \beta_{2} + 10483 \beta _1 - 13713$$ 4383*b11 - 9560*b10 - 4630*b9 + 4411*b8 + 17550*b7 - 20512*b6 + 25127*b5 + 650*b4 - 1974*b3 - 5273*b2 + 10483*b1 - 13713 $$\nu^{11}$$ $$=$$ $$1759 \beta_{11} - 3186 \beta_{10} - 9684 \beta_{9} - 4506 \beta_{8} + 22413 \beta_{7} + 33458 \beta_{6} + 57148 \beta_{5} + 31493 \beta_{4} - 14866 \beta_{3} + 31330 \beta_{2} - 8841 \beta _1 - 85518$$ 1759*b11 - 3186*b10 - 9684*b9 - 4506*b8 + 22413*b7 + 33458*b6 + 57148*b5 + 31493*b4 - 14866*b3 + 31330*b2 - 8841*b1 - 85518

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$$.

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$\beta_{7}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.5 − 0.399480i 0.5 − 3.15681i 0.5 + 1.69027i 0.5 − 1.73154i 0.5 − 0.613147i 0.5 + 2.47866i 0.5 − 1.69027i 0.5 + 3.15681i 0.5 + 0.399480i 0.5 − 2.47866i 0.5 + 0.613147i 0.5 + 1.73154i
−0.866025 + 0.500000i 0 0.500000 0.866025i 3.38938i 0 0.866025 + 0.500000i 1.00000i 0 1.69469 + 2.93529i
127.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.901839i 0 0.866025 + 0.500000i 1.00000i 0 −0.450919 0.781015i
127.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 3.48754i 0 0.866025 + 0.500000i 1.00000i 0 −1.74377 3.02030i
127.4 0.866025 0.500000i 0 0.500000 0.866025i 3.71131i 0 −0.866025 0.500000i 1.00000i 0 −1.85566 3.21409i
127.5 0.866025 0.500000i 0 0.500000 0.866025i 1.14776i 0 −0.866025 0.500000i 1.00000i 0 0.573878 + 0.993985i
127.6 0.866025 0.500000i 0 0.500000 0.866025i 1.56356i 0 −0.866025 0.500000i 1.00000i 0 0.781779 + 1.35408i
1135.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 3.48754i 0 0.866025 0.500000i 1.00000i 0 −1.74377 + 3.02030i
1135.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.901839i 0 0.866025 0.500000i 1.00000i 0 −0.450919 + 0.781015i
1135.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 3.38938i 0 0.866025 0.500000i 1.00000i 0 1.69469 2.93529i
1135.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.56356i 0 −0.866025 + 0.500000i 1.00000i 0 0.781779 1.35408i
1135.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.14776i 0 −0.866025 + 0.500000i 1.00000i 0 0.573878 0.993985i
1135.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i 3.71131i 0 −0.866025 + 0.500000i 1.00000i 0 −1.85566 + 3.21409i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1135.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.bj.g 12
3.b odd 2 1 182.2.m.b 12
12.b even 2 1 1456.2.cc.d 12
13.e even 6 1 inner 1638.2.bj.g 12
21.c even 2 1 1274.2.m.c 12
21.g even 6 1 1274.2.o.e 12
21.g even 6 1 1274.2.v.d 12
21.h odd 6 1 1274.2.o.d 12
21.h odd 6 1 1274.2.v.e 12
39.h odd 6 1 182.2.m.b 12
39.h odd 6 1 2366.2.d.r 12
39.i odd 6 1 2366.2.d.r 12
39.k even 12 1 2366.2.a.bf 6
39.k even 12 1 2366.2.a.bh 6
156.r even 6 1 1456.2.cc.d 12
273.u even 6 1 1274.2.m.c 12
273.x odd 6 1 1274.2.o.d 12
273.y even 6 1 1274.2.o.e 12
273.bp odd 6 1 1274.2.v.e 12
273.br even 6 1 1274.2.v.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.m.b 12 3.b odd 2 1
182.2.m.b 12 39.h odd 6 1
1274.2.m.c 12 21.c even 2 1
1274.2.m.c 12 273.u even 6 1
1274.2.o.d 12 21.h odd 6 1
1274.2.o.d 12 273.x odd 6 1
1274.2.o.e 12 21.g even 6 1
1274.2.o.e 12 273.y even 6 1
1274.2.v.d 12 21.g even 6 1
1274.2.v.d 12 273.br even 6 1
1274.2.v.e 12 21.h odd 6 1
1274.2.v.e 12 273.bp odd 6 1
1456.2.cc.d 12 12.b even 2 1
1456.2.cc.d 12 156.r even 6 1
1638.2.bj.g 12 1.a even 1 1 trivial
1638.2.bj.g 12 13.e even 6 1 inner
2366.2.a.bf 6 39.k even 12 1
2366.2.a.bh 6 39.k even 12 1
2366.2.d.r 12 39.h odd 6 1
2366.2.d.r 12 39.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1638, [\chi])$$:

 $$T_{5}^{12} + 42T_{5}^{10} + 643T_{5}^{8} + 4292T_{5}^{6} + 11827T_{5}^{4} + 13306T_{5}^{2} + 5041$$ T5^12 + 42*T5^10 + 643*T5^8 + 4292*T5^6 + 11827*T5^4 + 13306*T5^2 + 5041 $$T_{11}^{12} - 18 T_{11}^{11} + 118 T_{11}^{10} - 180 T_{11}^{9} - 1292 T_{11}^{8} + 2976 T_{11}^{7} + 32288 T_{11}^{6} - 196992 T_{11}^{5} + 425344 T_{11}^{4} - 239616 T_{11}^{3} - 390144 T_{11}^{2} + 270336 T_{11} + 495616$$ T11^12 - 18*T11^11 + 118*T11^10 - 180*T11^9 - 1292*T11^8 + 2976*T11^7 + 32288*T11^6 - 196992*T11^5 + 425344*T11^4 - 239616*T11^3 - 390144*T11^2 + 270336*T11 + 495616

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{3}$$
$3$ $$T^{12}$$
$5$ $$T^{12} + 42 T^{10} + 643 T^{8} + \cdots + 5041$$
$7$ $$(T^{4} - T^{2} + 1)^{3}$$
$11$ $$T^{12} - 18 T^{11} + 118 T^{10} + \cdots + 495616$$
$13$ $$T^{12} + 8 T^{11} + 15 T^{10} + \cdots + 4826809$$
$17$ $$T^{12} + 4 T^{11} + 55 T^{10} + \cdots + 30976$$
$19$ $$T^{12} - 12 T^{11} + \cdots + 369869824$$
$23$ $$T^{12} - 6 T^{11} + 104 T^{10} + \cdots + 9872164$$
$29$ $$T^{12} - 10 T^{11} + 139 T^{10} + \cdots + 135424$$
$31$ $$T^{12} + 88 T^{10} + 1732 T^{8} + \cdots + 1024$$
$37$ $$T^{12} + 6 T^{11} - 137 T^{10} + \cdots + 1024$$
$41$ $$T^{12} - 24 T^{11} + 175 T^{10} + \cdots + 1024$$
$43$ $$T^{12} - 26 T^{11} + 462 T^{10} + \cdots + 8667136$$
$47$ $$T^{12} + 272 T^{10} + \cdots + 31719424$$
$53$ $$(T^{6} + 18 T^{5} - 51 T^{4} - 1656 T^{3} + \cdots + 44928)^{2}$$
$59$ $$T^{12} + 6 T^{11} - 66 T^{10} - 468 T^{9} + \cdots + 4$$
$61$ $$T^{12} + 28 T^{11} + \cdots + 80364879169$$
$67$ $$T^{12} + 42 T^{11} + 742 T^{10} + \cdots + 1024$$
$71$ $$T^{12} + 48 T^{11} + \cdots + 750321664$$
$73$ $$T^{12} + 610 T^{10} + \cdots + 2708994304$$
$79$ $$(T^{6} - 22 T^{5} + 62 T^{4} + 1584 T^{3} + \cdots + 8032)^{2}$$
$83$ $$T^{12} + 464 T^{10} + \cdots + 879478336$$
$89$ $$T^{12} + 12 T^{11} + \cdots + 10303062016$$
$97$ $$T^{12} - 60 T^{11} + \cdots + 6400000000$$