# Properties

 Label 2268.2.i.n Level $2268$ Weight $2$ Character orbit 2268.i Analytic conductor $18.110$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2268 = 2^{2} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2268.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.1100711784$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 9x^{14} + 31x^{12} - 282x^{10} + 1695x^{8} - 3318x^{6} + 4606x^{4} - 4116x^{2} + 2401$$ x^16 - 9*x^14 + 31*x^12 - 282*x^10 + 1695*x^8 - 3318*x^6 + 4606*x^4 - 4116*x^2 + 2401 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3^{7}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{13} q^{5} + \beta_{3} q^{7}+O(q^{10})$$ q - b13 * q^5 + b3 * q^7 $$q - \beta_{13} q^{5} + \beta_{3} q^{7} - \beta_{12} q^{11} + (\beta_{9} - \beta_{8} - 2 \beta_{4} + 2) q^{13} + ( - \beta_{13} - \beta_{11} - \beta_{6}) q^{17} + (\beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{19} + \beta_{11} q^{23} + ( - \beta_{9} + \beta_{8} + \beta_{4} + \beta_{2} + \beta_1 - 1) q^{25} + (2 \beta_{13} + 2 \beta_{11} + \beta_{6} + 2 \beta_{5}) q^{29} + ( - \beta_{7} - \beta_{4} + \beta_{2} - \beta_1 + 2) q^{31} + ( - \beta_{14} + 2 \beta_{13} + \beta_{11} + 2 \beta_{10} + \beta_{6} + \beta_{5}) q^{35} + ( - \beta_{9} + \beta_{8} + \beta_{4} - \beta_{2} - \beta_1 - 1) q^{37} + ( - \beta_{15} + 3 \beta_{14} - \beta_{12} + \beta_{11} + \beta_{6}) q^{41} + ( - 2 \beta_{8} + \beta_{7} - 3 \beta_{4} - 2 \beta_{3} - \beta_1 + 1) q^{43} + (2 \beta_{15} - \beta_{14} - \beta_{12} + 3 \beta_{10} - \beta_{6} - \beta_{5}) q^{47} + ( - 2 \beta_{9} + \beta_{8} - 3 \beta_{7} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 2) q^{49} + (3 \beta_{13} - \beta_{6}) q^{53} + (\beta_{9} + \beta_{7} - \beta_{3} - \beta_{2} - 2) q^{55} + ( - \beta_{14} - 2 \beta_{12} + \beta_{10} - \beta_{6} - 2 \beta_{5}) q^{59} + (2 \beta_{9} + 3 \beta_{7} + \beta_{4} - \beta_{2} + \beta_1 - 3) q^{61} + (\beta_{15} - 3 \beta_{14} + 4 \beta_{10} - 3 \beta_{6}) q^{65} + (\beta_{9} - \beta_{7} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{67} + (3 \beta_{15} - 2 \beta_{14} + \beta_{12} - 2 \beta_{6} + \beta_{5}) q^{71} + ( - \beta_{8} + \beta_{7} + 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 2) q^{73} + ( - \beta_{15} + 2 \beta_{14} - \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + 5 \beta_{6} - \beta_{5}) q^{77} + (\beta_{9} - \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 5) q^{79} + ( - 2 \beta_{13} + 3 \beta_{11} - 2 \beta_{6} + 3 \beta_{5}) q^{83} + ( - \beta_{9} + \beta_{8} - \beta_{7} + 6 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 6) q^{85} + (2 \beta_{15} - 3 \beta_{14} - 3 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} - 3 \beta_{10} + \cdots - 2 \beta_{6}) q^{89}+ \cdots + (\beta_{8} + 3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{97}+O(q^{100})$$ q - b13 * q^5 + b3 * q^7 - b12 * q^11 + (b9 - b8 - 2*b4 + 2) * q^13 + (-b13 - b11 - b6) * q^17 + (b9 - b8 + b7 - 2*b4 - b3 - b2 - b1 + 2) * q^19 + b11 * q^23 + (-b9 + b8 + b4 + b2 + b1 - 1) * q^25 + (2*b13 + 2*b11 + b6 + 2*b5) * q^29 + (-b7 - b4 + b2 - b1 + 2) * q^31 + (-b14 + 2*b13 + b11 + 2*b10 + b6 + b5) * q^35 + (-b9 + b8 + b4 - b2 - b1 - 1) * q^37 + (-b15 + 3*b14 - b12 + b11 + b6) * q^41 + (-2*b8 + b7 - 3*b4 - 2*b3 - b1 + 1) * q^43 + (2*b15 - b14 - b12 + 3*b10 - b6 - b5) * q^47 + (-2*b9 + b8 - 3*b7 + 2*b4 + b3 + b2 + b1 - 2) * q^49 + (3*b13 - b6) * q^53 + (b9 + b7 - b3 - b2 - 2) * q^55 + (-b14 - 2*b12 + b10 - b6 - 2*b5) * q^59 + (2*b9 + 3*b7 + b4 - b2 + b1 - 3) * q^61 + (b15 - 3*b14 + 4*b10 - 3*b6) * q^65 + (b9 - b7 - 2*b4 - 2*b3 - 2*b1 - 1) * q^67 + (3*b15 - 2*b14 + b12 - 2*b6 + b5) * q^71 + (-b8 + b7 + 2*b4 + 2*b3 - 3*b2 + 2*b1 - 2) * q^73 + (-b15 + 2*b14 - b13 - 2*b12 + 2*b11 + 5*b6 - b5) * q^77 + (b9 - b4 + b3 + 2*b2 - b1 + 5) * q^79 + (-2*b13 + 3*b11 - 2*b6 + 3*b5) * q^83 + (-b9 + b8 - b7 + 6*b4 + b3 + 2*b2 + 2*b1 - 6) * q^85 + (2*b15 - 3*b14 - 3*b13 + 2*b12 - 2*b11 - 3*b10 - 2*b6) * q^89 + (-3*b9 + 2*b8 + 3*b3 - b2 + 3*b1 - 2) * q^91 + (-b15 - 4*b14 - 2*b12 + 5*b10 - 4*b6 - 2*b5) * q^95 + (b8 + 3*b4 - 2*b3 + 3*b2 - 3*b1 + 3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 6 q^{7}+O(q^{10})$$ 16 * q - 6 * q^7 $$16 q - 6 q^{7} + 10 q^{13} + 8 q^{19} + 16 q^{31} - 4 q^{37} - 10 q^{43} + 10 q^{49} - 32 q^{55} - 56 q^{61} - 36 q^{67} + 40 q^{79} - 38 q^{85} - 2 q^{91} + 42 q^{97}+O(q^{100})$$ 16 * q - 6 * q^7 + 10 * q^13 + 8 * q^19 + 16 * q^31 - 4 * q^37 - 10 * q^43 + 10 * q^49 - 32 * q^55 - 56 * q^61 - 36 * q^67 + 40 * q^79 - 38 * q^85 - 2 * q^91 + 42 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 9x^{14} + 31x^{12} - 282x^{10} + 1695x^{8} - 3318x^{6} + 4606x^{4} - 4116x^{2} + 2401$$ :

 $$\beta_{1}$$ $$=$$ $$( 92249 \nu^{14} - 22952 \nu^{12} - 2800495 \nu^{10} - 21569239 \nu^{8} - 19347710 \nu^{6} + 586034029 \nu^{4} + 1081706164 \nu^{2} + 170475459 ) / 982062165$$ (92249*v^14 - 22952*v^12 - 2800495*v^10 - 21569239*v^8 - 19347710*v^6 + 586034029*v^4 + 1081706164*v^2 + 170475459) / 982062165 $$\beta_{2}$$ $$=$$ $$( 244351 \nu^{14} - 3103286 \nu^{12} + 15156266 \nu^{10} - 92194120 \nu^{8} + 665006344 \nu^{6} - 2126767580 \nu^{4} + 3126524037 \nu^{2} + \cdots - 4446117606 ) / 982062165$$ (244351*v^14 - 3103286*v^12 + 15156266*v^10 - 92194120*v^8 + 665006344*v^6 - 2126767580*v^4 + 3126524037*v^2 - 4446117606) / 982062165 $$\beta_{3}$$ $$=$$ $$( 50804 \nu^{14} - 431490 \nu^{12} + 910001 \nu^{10} - 11273818 \nu^{8} + 75102634 \nu^{6} - 34785107 \nu^{4} - 145125260 \nu^{2} - 188652058 ) / 140294595$$ (50804*v^14 - 431490*v^12 + 910001*v^10 - 11273818*v^8 + 75102634*v^6 - 34785107*v^4 - 145125260*v^2 - 188652058) / 140294595 $$\beta_{4}$$ $$=$$ $$( 95779 \nu^{14} - 763661 \nu^{12} + 2336349 \nu^{10} - 25351497 \nu^{8} + 138865326 \nu^{6} - 211337511 \nu^{4} + 367714228 \nu^{2} + \cdots - 129671150 ) / 196412433$$ (95779*v^14 - 763661*v^12 + 2336349*v^10 - 25351497*v^8 + 138865326*v^6 - 211337511*v^4 + 367714228*v^2 - 129671150) / 196412433 $$\beta_{5}$$ $$=$$ $$( - 784170 \nu^{15} + 214456 \nu^{13} + 30604253 \nu^{11} + 93415788 \nu^{9} + 434366602 \nu^{7} - 7116987318 \nu^{5} + 3388669823 \nu^{3} + \cdots - 6997483661 \nu ) / 6874435155$$ (-784170*v^15 + 214456*v^13 + 30604253*v^11 + 93415788*v^9 + 434366602*v^7 - 7116987318*v^5 + 3388669823*v^3 - 6997483661*v) / 6874435155 $$\beta_{6}$$ $$=$$ $$( 202439 \nu^{15} - 3779851 \nu^{13} + 20969988 \nu^{11} - 90097992 \nu^{9} + 810563151 \nu^{7} - 3169832022 \nu^{5} + 2394515438 \nu^{3} + \cdots - 2189789518 \nu ) / 1374887031$$ (202439*v^15 - 3779851*v^13 + 20969988*v^11 - 90097992*v^9 + 810563151*v^7 - 3169832022*v^5 + 2394515438*v^3 - 2189789518*v) / 1374887031 $$\beta_{7}$$ $$=$$ $$( - 1700901 \nu^{14} + 13906205 \nu^{12} - 39713699 \nu^{10} + 439930867 \nu^{8} - 2478376396 \nu^{6} + 3214460858 \nu^{4} - 4135709270 \nu^{2} + \cdots + 401714397 ) / 982062165$$ (-1700901*v^14 + 13906205*v^12 - 39713699*v^10 + 439930867*v^8 - 2478376396*v^6 + 3214460858*v^4 - 4135709270*v^2 + 401714397) / 982062165 $$\beta_{8}$$ $$=$$ $$( - 53324 \nu^{14} + 366096 \nu^{12} - 722548 \nu^{10} + 12291821 \nu^{8} - 61209692 \nu^{6} + 7818769 \nu^{4} - 19057962 \nu^{2} + 6024452 ) / 22838655$$ (-53324*v^14 + 366096*v^12 - 722548*v^10 + 12291821*v^8 - 61209692*v^6 + 7818769*v^4 - 19057962*v^2 + 6024452) / 22838655 $$\beta_{9}$$ $$=$$ $$( 160457 \nu^{14} - 1409701 \nu^{12} + 4498055 \nu^{10} - 43172212 \nu^{8} + 259237135 \nu^{6} - 437984288 \nu^{4} + 445839877 \nu^{2} + \cdots - 305714528 ) / 42698355$$ (160457*v^14 - 1409701*v^12 + 4498055*v^10 - 43172212*v^8 + 259237135*v^6 - 437984288*v^4 + 445839877*v^2 - 305714528) / 42698355 $$\beta_{10}$$ $$=$$ $$( - 635345 \nu^{15} + 5091886 \nu^{13} - 13365107 \nu^{11} + 152655738 \nu^{9} - 878690623 \nu^{7} + 875750037 \nu^{5} + 504845978 \nu^{3} + \cdots - 2003509256 \nu ) / 982062165$$ (-635345*v^15 + 5091886*v^13 - 13365107*v^11 + 152655738*v^9 - 878690623*v^7 + 875750037*v^5 + 504845978*v^3 - 2003509256*v) / 982062165 $$\beta_{11}$$ $$=$$ $$( 751964 \nu^{15} - 2837787 \nu^{13} - 5745335 \nu^{11} - 140264304 \nu^{9} + 307950965 \nu^{7} + 2566998474 \nu^{5} - 571100796 \nu^{3} + \cdots + 3200801814 \nu ) / 982062165$$ (751964*v^15 - 2837787*v^13 - 5745335*v^11 - 140264304*v^9 + 307950965*v^7 + 2566998474*v^5 - 571100796*v^3 + 3200801814*v) / 982062165 $$\beta_{12}$$ $$=$$ $$( 7006527 \nu^{15} - 54909609 \nu^{13} + 140871131 \nu^{11} - 1753147101 \nu^{9} + 9658554559 \nu^{7} - 8917693029 \nu^{5} + \cdots - 6552594825 \nu ) / 6874435155$$ (7006527*v^15 - 54909609*v^13 + 140871131*v^11 - 1753147101*v^9 + 9658554559*v^7 - 8917693029*v^5 + 11250666413*v^3 - 6552594825*v) / 6874435155 $$\beta_{13}$$ $$=$$ $$( 7841464 \nu^{15} - 71750653 \nu^{13} + 246634027 \nu^{11} - 2192423802 \nu^{9} + 13373548133 \nu^{7} - 26134593933 \nu^{5} + \cdots - 8080133320 \nu ) / 6874435155$$ (7841464*v^15 - 71750653*v^13 + 246634027*v^11 - 2192423802*v^9 + 13373548133*v^7 - 26134593933*v^5 + 29297334416*v^3 - 8080133320*v) / 6874435155 $$\beta_{14}$$ $$=$$ $$( 1635500 \nu^{15} - 12087451 \nu^{13} + 27574530 \nu^{11} - 390459270 \nu^{9} + 2074772181 \nu^{7} - 1221325980 \nu^{5} + 1133172530 \nu^{3} + \cdots - 504509782 \nu ) / 1374887031$$ (1635500*v^15 - 12087451*v^13 + 27574530*v^11 - 390459270*v^9 + 2074772181*v^7 - 1221325980*v^5 + 1133172530*v^3 - 504509782*v) / 1374887031 $$\beta_{15}$$ $$=$$ $$( - 12267456 \nu^{15} + 107332095 \nu^{13} - 343863724 \nu^{11} + 3308249367 \nu^{9} - 19848020141 \nu^{7} + 33437101488 \nu^{5} + \cdots + 38302053342 \nu ) / 6874435155$$ (-12267456*v^15 + 107332095*v^13 - 343863724*v^11 + 3308249367*v^9 - 19848020141*v^7 + 33437101488*v^5 - 35352763285*v^3 + 38302053342*v) / 6874435155
 $$\nu$$ $$=$$ $$( \beta_{15} + \beta_{13} + \beta_{12} - \beta_{6} + 2\beta_{5} ) / 3$$ (b15 + b13 + b12 - b6 + 2*b5) / 3 $$\nu^{2}$$ $$=$$ $$( -2\beta_{9} + \beta_{8} - 3\beta_{7} + 7\beta_{4} + 3\beta_{3} + \beta_{2} - \beta_1 ) / 3$$ (-2*b9 + b8 - 3*b7 + 7*b4 + 3*b3 + b2 - b1) / 3 $$\nu^{3}$$ $$=$$ $$( 5\beta_{15} + 9\beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + 6\beta_{10} + 9\beta_{6} + \beta_{5} ) / 3$$ (5*b15 + 9*b14 + b13 - b12 + b11 + 6*b10 + 9*b6 + b5) / 3 $$\nu^{4}$$ $$=$$ $$( -6\beta_{9} - 6\beta_{8} - 13\beta_{7} - 31\beta_{4} - 8\beta_{3} + 20\beta_{2} - 10\beta _1 + 25 ) / 3$$ (-6*b9 - 6*b8 - 13*b7 - 31*b4 - 8*b3 + 20*b2 - 10*b1 + 25) / 3 $$\nu^{5}$$ $$=$$ $$( 10 \beta_{15} - 16 \beta_{14} - 21 \beta_{13} + 44 \beta_{12} + 17 \beta_{11} + 2 \beta_{10} + 46 \beta_{6} + 22 \beta_{5} ) / 3$$ (10*b15 - 16*b14 - 21*b13 + 44*b12 + 17*b11 + 2*b10 + 46*b6 + 22*b5) / 3 $$\nu^{6}$$ $$=$$ $$( 11\beta_{9} - 22\beta_{8} + 71\beta_{7} + 10\beta_{4} + 31\beta_{3} + 49\beta_{2} + 21\beta _1 + 322 ) / 3$$ (11*b9 - 22*b8 + 71*b7 + 10*b4 + 31*b3 + 49*b2 + 21*b1 + 322) / 3 $$\nu^{7}$$ $$=$$ $$( 70 \beta_{15} - 12 \beta_{14} + 23 \beta_{13} + 178 \beta_{12} + 14 \beta_{11} + 33 \beta_{10} - 117 \beta_{6} + 356 \beta_{5} ) / 3$$ (70*b15 - 12*b14 + 23*b13 + 178*b12 + 14*b11 + 33*b10 - 117*b6 + 356*b5) / 3 $$\nu^{8}$$ $$=$$ $$( -234\beta_{9} + 117\beta_{8} - 169\beta_{7} + 1409\beta_{4} + 538\beta_{3} + 26\beta_{2} - 304\beta _1 + 187 ) / 3$$ (-234*b9 + 117*b8 - 169*b7 + 1409*b4 + 538*b3 + 26*b2 - 304*b1 + 187) / 3 $$\nu^{9}$$ $$=$$ $$( 804 \beta_{15} + 1814 \beta_{14} - 25 \beta_{13} - 467 \beta_{12} + 221 \beta_{11} + 902 \beta_{10} + 1691 \beta_{6} + 467 \beta_{5} ) / 3$$ (804*b15 + 1814*b14 - 25*b13 - 467*b12 + 221*b11 + 902*b10 + 1691*b6 + 467*b5) / 3 $$\nu^{10}$$ $$=$$ $$( - 1148 \beta_{9} - 1148 \beta_{8} - 2191 \beta_{7} - 3861 \beta_{4} - 2165 \beta_{3} + 3234 \beta_{2} - 3182 \beta _1 + 2713 ) / 3$$ (-1148*b9 - 1148*b8 - 2191*b7 - 3861*b4 - 2165*b3 + 3234*b2 - 3182*b1 + 2713) / 3 $$\nu^{11}$$ $$=$$ $$( 1322 \beta_{15} - 2061 \beta_{14} - 5174 \beta_{13} + 5600 \beta_{12} + 4591 \beta_{11} + 156 \beta_{10} + 12972 \beta_{6} + 2800 \beta_{5} ) / 3$$ (1322*b15 - 2061*b14 - 5174*b13 + 5600*b12 + 4591*b11 + 156*b10 + 12972*b6 + 2800*b5) / 3 $$\nu^{12}$$ $$=$$ $$1284\beta_{9} - 2568\beta_{8} + 5956\beta_{7} - 1481\beta_{4} - 1678\beta_{3} + 3388\beta_{2} - 197\beta _1 + 19573$$ 1284*b9 - 2568*b8 + 5956*b7 - 1481*b4 - 1678*b3 + 3388*b2 - 197*b1 + 19573 $$\nu^{13}$$ $$=$$ $$( 3218 \beta_{15} - 21675 \beta_{14} - 5929 \beta_{13} + 34676 \beta_{12} + 13164 \beta_{11} - 4017 \beta_{10} - 12365 \beta_{6} + 69352 \beta_{5} ) / 3$$ (3218*b15 - 21675*b14 - 5929*b13 + 34676*b12 + 13164*b11 - 4017*b10 - 12365*b6 + 69352*b5) / 3 $$\nu^{14}$$ $$=$$ $$( - 24730 \beta_{9} + 12365 \beta_{8} + 31071 \beta_{7} + 328079 \beta_{4} + 80961 \beta_{3} - 21718 \beta_{2} - 56231 \beta _1 + 43866 ) / 3$$ (-24730*b9 + 12365*b8 + 31071*b7 + 328079*b4 + 80961*b3 - 21718*b2 - 56231*b1 + 43866) / 3 $$\nu^{15}$$ $$=$$ $$( 108952 \beta_{15} + 307191 \beta_{14} - 7798 \beta_{13} - 141578 \beta_{12} + 66890 \beta_{11} + 138498 \beta_{10} + 269847 \beta_{6} + 141578 \beta_{5} ) / 3$$ (108952*b15 + 307191*b14 - 7798*b13 - 141578*b12 + 66890*b11 + 138498*b10 + 269847*b6 + 141578*b5) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1135$$ $$1541$$ $$\chi(n)$$ $$-1 + \beta_{4}$$ $$1$$ $$-1 + \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 0.817131 − 0.735533i −2.40332 + 0.123797i 1.04556 − 0.339889i 1.30887 − 2.01944i −1.30887 + 2.01944i −1.04556 + 0.339889i 2.40332 − 0.123797i −0.817131 + 0.735533i 0.817131 + 0.735533i −2.40332 − 0.123797i 1.04556 + 0.339889i 1.30887 + 2.01944i −1.30887 − 2.01944i −1.04556 − 0.339889i 2.40332 + 0.123797i −0.817131 − 0.735533i
0 0 0 −1.83843 + 3.18426i 0 −1.55575 + 2.14001i 0 0 0
865.2 0 0 0 −1.15101 + 1.99360i 0 2.41508 + 1.08045i 0 0 0
865.3 0 0 0 −0.515559 + 0.892975i 0 −2.63118 + 0.277320i 0 0 0
865.4 0 0 0 −0.171869 + 0.297685i 0 0.271847 2.63175i 0 0 0
865.5 0 0 0 0.171869 0.297685i 0 0.271847 2.63175i 0 0 0
865.6 0 0 0 0.515559 0.892975i 0 −2.63118 + 0.277320i 0 0 0
865.7 0 0 0 1.15101 1.99360i 0 2.41508 + 1.08045i 0 0 0
865.8 0 0 0 1.83843 3.18426i 0 −1.55575 + 2.14001i 0 0 0
2053.1 0 0 0 −1.83843 3.18426i 0 −1.55575 2.14001i 0 0 0
2053.2 0 0 0 −1.15101 1.99360i 0 2.41508 1.08045i 0 0 0
2053.3 0 0 0 −0.515559 0.892975i 0 −2.63118 0.277320i 0 0 0
2053.4 0 0 0 −0.171869 0.297685i 0 0.271847 + 2.63175i 0 0 0
2053.5 0 0 0 0.171869 + 0.297685i 0 0.271847 + 2.63175i 0 0 0
2053.6 0 0 0 0.515559 + 0.892975i 0 −2.63118 0.277320i 0 0 0
2053.7 0 0 0 1.15101 + 1.99360i 0 2.41508 1.08045i 0 0 0
2053.8 0 0 0 1.83843 + 3.18426i 0 −1.55575 2.14001i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2053.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.h even 3 1 inner
63.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.i.n 16
3.b odd 2 1 inner 2268.2.i.n 16
7.c even 3 1 2268.2.l.n 16
9.c even 3 1 2268.2.k.g 16
9.c even 3 1 2268.2.l.n 16
9.d odd 6 1 2268.2.k.g 16
9.d odd 6 1 2268.2.l.n 16
21.h odd 6 1 2268.2.l.n 16
63.g even 3 1 2268.2.k.g 16
63.h even 3 1 inner 2268.2.i.n 16
63.j odd 6 1 inner 2268.2.i.n 16
63.n odd 6 1 2268.2.k.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.2.i.n 16 1.a even 1 1 trivial
2268.2.i.n 16 3.b odd 2 1 inner
2268.2.i.n 16 63.h even 3 1 inner
2268.2.i.n 16 63.j odd 6 1 inner
2268.2.k.g 16 9.c even 3 1
2268.2.k.g 16 9.d odd 6 1
2268.2.k.g 16 63.g even 3 1
2268.2.k.g 16 63.n odd 6 1
2268.2.l.n 16 7.c even 3 1
2268.2.l.n 16 9.c even 3 1
2268.2.l.n 16 9.d odd 6 1
2268.2.l.n 16 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2268, [\chi])$$:

 $$T_{5}^{16} + 20T_{5}^{14} + 306T_{5}^{12} + 1706T_{5}^{10} + 7087T_{5}^{8} + 7818T_{5}^{6} + 6723T_{5}^{4} + 783T_{5}^{2} + 81$$ T5^16 + 20*T5^14 + 306*T5^12 + 1706*T5^10 + 7087*T5^8 + 7818*T5^6 + 6723*T5^4 + 783*T5^2 + 81 $$T_{13}^{8} - 5T_{13}^{7} + 45T_{13}^{6} - 152T_{13}^{5} + 1177T_{13}^{4} - 3990T_{13}^{3} + 12936T_{13}^{2} - 18522T_{13} + 21609$$ T13^8 - 5*T13^7 + 45*T13^6 - 152*T13^5 + 1177*T13^4 - 3990*T13^3 + 12936*T13^2 - 18522*T13 + 21609 $$T_{19}^{8} - 4T_{19}^{7} + 40T_{19}^{6} + 110T_{19}^{5} + 541T_{19}^{4} + 224T_{19}^{3} + 217T_{19}^{2} - 49T_{19} + 49$$ T19^8 - 4*T19^7 + 40*T19^6 + 110*T19^5 + 541*T19^4 + 224*T19^3 + 217*T19^2 - 49*T19 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$T^{16} + 20 T^{14} + 306 T^{12} + \cdots + 81$$
$7$ $$(T^{8} + 3 T^{7} + 2 T^{6} - 3 T^{5} + \cdots + 2401)^{2}$$
$11$ $$T^{16} + 47 T^{14} + 1599 T^{12} + \cdots + 194481$$
$13$ $$(T^{8} - 5 T^{7} + 45 T^{6} - 152 T^{5} + \cdots + 21609)^{2}$$
$17$ $$T^{16} + 62 T^{14} + 2589 T^{12} + \cdots + 194481$$
$19$ $$(T^{8} - 4 T^{7} + 40 T^{6} + 110 T^{5} + \cdots + 49)^{2}$$
$23$ $$T^{16} + 44 T^{14} + 1590 T^{12} + \cdots + 50625$$
$29$ $$T^{16} + 212 T^{14} + \cdots + 10016218555281$$
$31$ $$(T^{4} - 4 T^{3} - 17 T^{2} + 42 T + 63)^{4}$$
$37$ $$(T^{8} + 2 T^{7} + 64 T^{6} + 254 T^{5} + \cdots + 1)^{2}$$
$41$ $$T^{16} + 111 T^{14} + \cdots + 9845600625$$
$43$ $$(T^{8} + 5 T^{7} + 139 T^{6} + \cdots + 3200521)^{2}$$
$47$ $$(T^{8} - 429 T^{6} + 62541 T^{4} + \cdots + 62583921)^{2}$$
$53$ $$T^{16} + 192 T^{14} + \cdots + 1315703055681$$
$59$ $$(T^{8} - 200 T^{6} + 13942 T^{4} + \cdots + 3272481)^{2}$$
$61$ $$(T^{4} + 14 T^{3} - 95 T^{2} - 1596 T - 2649)^{4}$$
$67$ $$(T^{4} + 9 T^{3} - 62 T^{2} - 672 T - 917)^{4}$$
$71$ $$(T^{8} - 305 T^{6} + 22657 T^{4} + \cdots + 505521)^{2}$$
$73$ $$(T^{8} + 128 T^{6} - 210 T^{5} + \cdots + 7458361)^{2}$$
$79$ $$(T^{4} - 10 T^{3} - 50 T^{2} + 669 T - 1497)^{4}$$
$83$ $$T^{16} + 731 T^{14} + \cdots + 60\!\cdots\!21$$
$89$ $$T^{16} + 300 T^{14} + \cdots + 4631487063921$$
$97$ $$(T^{8} - 21 T^{7} + 449 T^{6} + \cdots + 8614225)^{2}$$