Properties

 Label 400.2.u.f Level $400$ Weight $2$ Character orbit 400.u Analytic conductor $3.194$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 400.u (of order $$5$$, degree $$4$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.19401608085$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 3 x^{11} + 13 x^{10} - 24 x^{9} + 93 x^{8} - 6 x^{7} + 342 x^{6} + 786 x^{5} + 1473 x^{4} + 84 x^{3} + 688 x^{2} + 168 x + 16$$ x^12 - 3*x^11 + 13*x^10 - 24*x^9 + 93*x^8 - 6*x^7 + 342*x^6 + 786*x^5 + 1473*x^4 + 84*x^3 + 688*x^2 + 168*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 100) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} + ( - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{5} + (\beta_{11} - \beta_{10} + \beta_{6} + \beta_{5} - \beta_1 + 1) q^{7} + (\beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{9}+O(q^{10})$$ q - b3 * q^3 + (-b11 - b10 - b9 + b8 + b7 - 2*b5 + b3 + b2 - b1) * q^5 + (b11 - b10 + b6 + b5 - b1 + 1) * q^7 + (b11 + b10 + b9 - 2*b7 + b5 - b4 + b2 - 1) * q^9 $$q - \beta_{3} q^{3} + ( - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{5} + (\beta_{11} - \beta_{10} + \beta_{6} + \beta_{5} - \beta_1 + 1) q^{7} + (\beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{9} + ( - \beta_{11} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{11} + ( - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} - \beta_{3} + 1) q^{13} + (\beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{6} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{15} + ( - 3 \beta_{8} + 3 \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} + 2) q^{17} + ( - \beta_{11} + \beta_{9} + \beta_{5} + \beta_{4} + \beta_{2} + 2) q^{19} + (\beta_{10} - 2 \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1 + 1) q^{21} + ( - \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 1) q^{23} + ( - \beta_{10} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_1 - 1) q^{25} + (\beta_{11} - 2 \beta_{8} + \beta_{7} - 3 \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1 + 2) q^{27} + (2 \beta_{11} + \beta_{10} + \beta_{9} - 3 \beta_{8} - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \cdots - 2) q^{29}+ \cdots + ( - \beta_{11} + \beta_{10} - 2 \beta_{9} - 2 \beta_{7} - 7 \beta_{6} - 7 \beta_{5} - 6 \beta_{4} + \cdots - 9) q^{99}+O(q^{100})$$ q - b3 * q^3 + (-b11 - b10 - b9 + b8 + b7 - 2*b5 + b3 + b2 - b1) * q^5 + (b11 - b10 + b6 + b5 - b1 + 1) * q^7 + (b11 + b10 + b9 - 2*b7 + b5 - b4 + b2 - 1) * q^9 + (-b11 - b7 - b5 - b4 + b2 + b1) * q^11 + (-b10 + b9 - b8 - b6 + b5 - b3 + 1) * q^13 + (b10 + b9 - 2*b8 - b6 + b3 + b2 - b1 - 1) * q^15 + (-3*b8 + 3*b6 + 2*b5 - b3 - b2 + 2) * q^17 + (-b11 + b9 + b5 + b4 + b2 + 2) * q^19 + (b10 - 2*b8 + b7 + b6 + 2*b4 - 2*b3 + 2*b1 + 1) * q^21 + (-b11 - b10 - 2*b9 + b7 + b6 - b5 + b3 + b2 + 1) * q^23 + (-b10 + b6 + b5 + 2*b4 - b3 + b1 - 1) * q^25 + (b11 - 2*b8 + b7 - 3*b5 + b4 - b2 + 2*b1 + 2) * q^27 + (2*b11 + b10 + b9 - 3*b8 - b7 - 2*b6 + 2*b5 - 2*b4 - b3 - b2 - b1 - 2) * q^29 + (-2*b8 + 2*b6 + b4 - 2*b3 - 2*b2) * q^31 + (-b11 + b9 + 4*b8 - 4*b6 - 7*b5 - 2*b4 + b3 + 2*b2 - 6) * q^33 + (b11 + b10 + b9 - 5*b6 + 3*b5 - b4 + b3 + b2 + b1 - 1) * q^35 + (b10 - b9 + b8 - 4*b6 - b5 + 2*b3 + b2 - b1 - 1) * q^37 + (2*b8 - 2*b5 - b4 + b2 - 3*b1 - 2) * q^39 + (-b11 - 2*b9 - b8 + 2*b7 + 4*b6 + b4 + 2*b3 + b2 - b1) * q^41 + (-b9 - b7 - 2*b6 - 2*b5 - 2*b4 + 2*b3 + b2 - b1 - 6) * q^43 + (-b11 - b10 - b9 - 3*b8 + b7 + b6 + b5 + b4 + 2*b3 - 2*b2 + b1 + 7) * q^45 + (2*b11 + b10 + b9 - b7 - b6 + 2*b5 - b4 - 2*b3 - b2 - 1) * q^47 + (-b9 - b7 + 2*b6 + 2*b5 - 2*b4 + 2*b3 + b2 - b1 + 3) * q^49 + (-b9 - b7 + 4*b6 + 4*b5 + b4 - b3 - b2 + b1) * q^51 + (-b10 + 2*b8 - b7 + 2*b6 - 2*b4 - 2*b1 + 2) * q^53 + (-3*b11 - 2*b10 - 2*b9 - 4*b8 + 3*b7 + 8*b6 + 3*b5 + 3*b4 - b2 - b1 + 6) * q^55 + (2*b11 - 2*b10 + b4 - b3 - 2*b2 + 6) * q^57 + (2*b11 + 2*b10 + 2*b9 - 4*b8 - 4*b7 - 2*b6 + 6*b5 - 2*b4 - 3*b3 - 2*b2 + 3*b1 - 2) * q^59 + (-2*b11 + 3*b8 - 2*b7 + b5 + b4 - b2 + b1 - 3) * q^61 + (-2*b10 + 2*b9 + 6*b8 + 4*b6 - 6*b5 - 2*b3 - 2*b2 + 2) * q^63 + (-b11 + 2*b10 + 2*b9 + 4*b8 + b7 - b6 + 3*b5 + 2*b2 - 1) * q^65 + (-2*b10 - 2*b9 + 8*b8 + b7 - 6*b6 - 2*b5 + 2*b4 + b3 - b1) * q^67 + (-b11 + 2*b10 + 3*b9 + 2*b8 - b7 - 4*b6 - 5*b5 + b4 - 2*b3 + b1 - 6) * q^69 + (-2*b11 + b10 - b9 + 4*b8 + 3*b7 - 3*b6 - 2*b5 + b4 + 2*b3 + b2 - 3) * q^71 + (-2*b11 - 2*b10 - 4*b9 + b8 + 2*b7 + 2*b6 - 5*b5 + 3*b4 + 2*b3 - b2 + b1 + 1) * q^73 + (b11 - b10 + 2*b9 + 6*b8 - b7 - b6 - 5*b5 + b4 - 2*b3 - 2*b2 + 2*b1 + 3) * q^75 + (2*b11 + 8*b8 + 2*b7 - 2*b5 + 2*b1 - 8) * q^77 + (4*b11 + 3*b10 + 2*b9 - 4*b8 - b7 + b6 + 4*b5 - 2*b4 + b3 - 2*b2 + 1) * q^79 + (b11 - b9 + 5*b8 - 5*b6 - 2*b5 - 2*b4 - 3*b3 - 4*b2 - 3) * q^81 + (-2*b11 + 2*b9 + 4*b8 - 4*b6 + 2*b5 - b4 + 2*b2 + 4) * q^83 + (2*b11 + b10 + 2*b9 - 3*b8 + b7 + b6 - 2*b4 + b2 - 2*b1) * q^85 + (2*b10 - 2*b9 - 6*b8 + 6*b6 + 6*b5 + 3*b3 - b1 - 2) * q^87 + (b11 + b10 + 2*b9 - 3*b8 - b7 - b6 + 9*b5 + b4 - b3 - 2*b2 + 2) * q^89 + (-2*b11 + b10 - 5*b9 + 6*b8 + 4*b7 - 3*b6 - 8*b5 + 2*b4 + 4*b3 + 3*b2 - b1 - 1) * q^91 + (b11 - b10 - 2*b9 - 2*b7 + 9*b6 + 9*b5 - b4 + b3 - b1 + 5) * q^93 + (-2*b11 - b10 - b9 + 8*b8 + 2*b7 + b6 - 6*b5 + b4 + 2*b3 - b2 + b1 - 3) * q^95 + (4*b11 + 3*b10 + 2*b9 - 3*b8 - b7 - 2*b6 + 4*b5 + b4 - 4*b3 - 2*b2 + 3*b1 - 2) * q^97 + (-b11 + b10 - 2*b9 - 2*b7 - 7*b6 - 7*b5 - 6*b4 + 6*b3 + 3*b2 - 2*b1 - 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 2 q^{3} - 4 q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10})$$ 12 * q - 2 * q^3 - 4 * q^5 + 2 * q^7 - 3 * q^9 $$12 q - 2 q^{3} - 4 q^{5} + 2 q^{7} - 3 q^{9} + 5 q^{11} - 2 q^{13} - 18 q^{15} + q^{17} + 8 q^{19} + 2 q^{21} + 6 q^{23} - 26 q^{25} + 34 q^{27} - 18 q^{29} - 12 q^{31} - 35 q^{33} + 3 q^{35} + 13 q^{37} - 22 q^{39} - 23 q^{41} - 50 q^{43} + 71 q^{45} - q^{47} + 34 q^{49} - 14 q^{51} + 21 q^{53} - 5 q^{55} + 72 q^{57} - 9 q^{59} - 26 q^{61} + 32 q^{63} - 18 q^{65} + 37 q^{67} - 44 q^{69} - 21 q^{71} + 18 q^{73} + 73 q^{75} - 60 q^{77} + 24 q^{79} + 18 q^{81} + 46 q^{83} - 16 q^{85} - 57 q^{87} - 2 q^{89} + 32 q^{91} + 22 q^{93} - 6 q^{95} - 7 q^{97} - 40 q^{99}+O(q^{100})$$ 12 * q - 2 * q^3 - 4 * q^5 + 2 * q^7 - 3 * q^9 + 5 * q^11 - 2 * q^13 - 18 * q^15 + q^17 + 8 * q^19 + 2 * q^21 + 6 * q^23 - 26 * q^25 + 34 * q^27 - 18 * q^29 - 12 * q^31 - 35 * q^33 + 3 * q^35 + 13 * q^37 - 22 * q^39 - 23 * q^41 - 50 * q^43 + 71 * q^45 - q^47 + 34 * q^49 - 14 * q^51 + 21 * q^53 - 5 * q^55 + 72 * q^57 - 9 * q^59 - 26 * q^61 + 32 * q^63 - 18 * q^65 + 37 * q^67 - 44 * q^69 - 21 * q^71 + 18 * q^73 + 73 * q^75 - 60 * q^77 + 24 * q^79 + 18 * q^81 + 46 * q^83 - 16 * q^85 - 57 * q^87 - 2 * q^89 + 32 * q^91 + 22 * q^93 - 6 * q^95 - 7 * q^97 - 40 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 13 x^{10} - 24 x^{9} + 93 x^{8} - 6 x^{7} + 342 x^{6} + 786 x^{5} + 1473 x^{4} + 84 x^{3} + 688 x^{2} + 168 x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 260325772 \nu^{11} - 576792708 \nu^{10} + 3020952123 \nu^{9} - 4900209927 \nu^{8} + 25045472937 \nu^{7} + 3519097812 \nu^{6} + \cdots + 112273499910 ) / 438461548785$$ (260325772*v^11 - 576792708*v^10 + 3020952123*v^9 - 4900209927*v^8 + 25045472937*v^7 + 3519097812*v^6 + 129671476953*v^5 + 224325622533*v^4 + 677052260712*v^3 + 442357548822*v^2 + 521633846125*v + 112273499910) / 438461548785 $$\beta_{3}$$ $$=$$ $$( 3457672033 \nu^{11} - 11783661403 \nu^{10} + 48283397793 \nu^{9} - 97783045044 \nu^{8} + 339987222837 \nu^{7} + \cdots + 178985069048 ) / 1753846195140$$ (3457672033*v^11 - 11783661403*v^10 + 48283397793*v^9 - 97783045044*v^8 + 339987222837*v^7 - 127593738606*v^6 + 1125752172738*v^5 + 2207600053386*v^4 + 3903365483097*v^3 - 2849443497756*v^2 + 2016280256236*v + 178985069048) / 1753846195140 $$\beta_{4}$$ $$=$$ $$( 5369632561 \nu^{11} - 15433734165 \nu^{10} + 67292419131 \nu^{9} - 119576878026 \nu^{8} + 481102476429 \nu^{7} + \cdots + 391154012184 ) / 1753846195140$$ (5369632561*v^11 - 15433734165*v^10 + 67292419131*v^9 - 119576878026*v^8 + 481102476429*v^7 + 23885891676*v^6 + 1832724711006*v^5 + 4299712819554*v^4 + 8668716529509*v^3 + 489856121406*v^2 + 2657911506232*v + 391154012184) / 1753846195140 $$\beta_{5}$$ $$=$$ $$( 22373133631 \nu^{11} - 74034744959 \nu^{10} + 314418060009 \nu^{9} - 633522002730 \nu^{8} + 2276267517771 \nu^{7} + \cdots - 273874062464 ) / 3507692390280$$ (22373133631*v^11 - 74034744959*v^10 + 314418060009*v^9 - 633522002730*v^8 + 2276267517771*v^7 - 814213247460*v^6 + 7906799179014*v^5 + 15333778688490*v^4 + 28540425731691*v^3 - 5927387741190*v^2 + 21091602933640*v - 273874062464) / 3507692390280 $$\beta_{6}$$ $$=$$ $$( - 29615632063 \nu^{11} + 87105581309 \nu^{10} - 382317412959 \nu^{9} + 696924743820 \nu^{8} - 2749867795311 \nu^{7} + \cdots - 5593310307856 ) / 3507692390280$$ (-29615632063*v^11 + 87105581309*v^10 - 382317412959*v^9 + 696924743820*v^8 - 2749867795311*v^7 + 95827068690*v^6 - 10403352643614*v^5 - 23654460062430*v^4 - 46013446580871*v^3 - 6601997100420*v^2 - 23515293707092*v - 5593310307856) / 3507692390280 $$\beta_{7}$$ $$=$$ $$( 99321102659 \nu^{11} - 361617020679 \nu^{10} + 1484218111509 \nu^{9} - 3218484118998 \nu^{8} + 10803833121471 \nu^{7} + \cdots - 17878470653712 ) / 8769230975700$$ (99321102659*v^11 - 361617020679*v^10 + 1484218111509*v^9 - 3218484118998*v^8 + 10803833121471*v^7 - 6581106786312*v^6 + 34599514729674*v^5 + 56181193046322*v^4 + 96648914614131*v^3 - 80865862575282*v^2 + 67498588603448*v - 17878470653712) / 8769230975700 $$\beta_{8}$$ $$=$$ $$( 48894251523 \nu^{11} - 157422019691 \nu^{10} + 666492738129 \nu^{9} - 1308046874814 \nu^{8} + 4786319147691 \nu^{7} + \cdots + 2898411243400 ) / 3507692390280$$ (48894251523*v^11 - 157422019691*v^10 + 666492738129*v^9 - 1308046874814*v^8 + 4786319147691*v^7 - 1255570461996*v^6 + 16674062237514*v^5 + 34765432275066*v^4 + 63421806854271*v^3 - 13230315931086*v^2 + 32659532805012*v + 2898411243400) / 3507692390280 $$\beta_{9}$$ $$=$$ $$( - 167923324379 \nu^{11} + 478101391109 \nu^{10} - 2090376253539 \nu^{9} + 3656650156068 \nu^{8} - 14810975764191 \nu^{7} + \cdots - 37003359759088 ) / 8769230975700$$ (-167923324379*v^11 + 478101391109*v^10 - 2090376253539*v^9 + 3656650156068*v^8 - 14810975764191*v^7 - 1748332186158*v^6 - 55756593480954*v^5 - 140798466578802*v^4 - 260834151846351*v^3 - 39698554795488*v^2 - 90978016936148*v - 37003359759088) / 8769230975700 $$\beta_{10}$$ $$=$$ $$( 542891618887 \nu^{11} - 1717504490237 \nu^{10} + 7294073305527 \nu^{9} - 14065746308484 \nu^{8} + 52158480741663 \nu^{7} + \cdots + 77595685775224 ) / 17538461951400$$ (542891618887*v^11 - 1717504490237*v^10 + 7294073305527*v^9 - 14065746308484*v^8 + 52158480741663*v^7 - 10432981854546*v^6 + 182680683235422*v^5 + 399600830367126*v^4 + 719350254826743*v^3 - 94750243258356*v^2 + 336336630781804*v + 77595685775224) / 17538461951400 $$\beta_{11}$$ $$=$$ $$( 572643183207 \nu^{11} - 1777970044487 \nu^{10} + 7631251682697 \nu^{9} - 14590245523914 \nu^{8} + 54915725330643 \nu^{7} + \cdots + 26266096925344 ) / 17538461951400$$ (572643183207*v^11 - 1777970044487*v^10 + 7631251682697*v^9 - 14590245523914*v^8 + 54915725330643*v^7 - 9699921955716*v^6 + 197679060702342*v^5 + 426485463435546*v^4 + 796406422106523*v^3 - 43808899935726*v^2 + 363489467079144*v + 26266096925344) / 17538461951400
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{10} - 4\beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta _1 - 1$$ b10 - 4*b8 + b7 - b6 + b4 + b3 + b1 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{11} + 2\beta_{10} + \beta_{9} - 6\beta_{8} - \beta_{7} + 4\beta_{6} + 5\beta_{5} + 7\beta_{4} + \beta _1 + 2$$ b11 + 2*b10 + b9 - 6*b8 - b7 + 4*b6 + 5*b5 + 7*b4 + b1 + 2 $$\nu^{4}$$ $$=$$ $$8 \beta_{11} + 8 \beta_{10} + 8 \beta_{9} - 2 \beta_{8} - 16 \beta_{7} + 24 \beta_{6} + 10 \beta_{5} - 8 \beta_{4} + 4 \beta_{3} + 15 \beta_{2} - 4 \beta _1 - 8$$ 8*b11 + 8*b10 + 8*b9 - 2*b8 - 16*b7 + 24*b6 + 10*b5 - 8*b4 + 4*b3 + 15*b2 - 4*b1 - 8 $$\nu^{5}$$ $$=$$ $$- 3 \beta_{11} + 3 \beta_{10} - 12 \beta_{9} - 12 \beta_{7} + 29 \beta_{6} + 29 \beta_{5} - 69 \beta_{4} + 69 \beta_{3} + 78 \beta_{2} - 75 \beta _1 - 47$$ -3*b11 + 3*b10 - 12*b9 - 12*b7 + 29*b6 + 29*b5 - 69*b4 + 69*b3 + 78*b2 - 75*b1 - 47 $$\nu^{6}$$ $$=$$ $$- 75 \beta_{11} - 66 \beta_{10} - 132 \beta_{9} + 54 \beta_{8} + 57 \beta_{7} + 66 \beta_{6} + 177 \beta_{5} - 71 \beta_{4} + 66 \beta_{3} + 137 \beta_{2} - 244 \beta _1 + 12$$ -75*b11 - 66*b10 - 132*b9 + 54*b8 + 57*b7 + 66*b6 + 177*b5 - 71*b4 + 66*b3 + 137*b2 - 244*b1 + 12 $$\nu^{7}$$ $$=$$ $$- 256 \beta_{11} - 178 \beta_{10} - 128 \beta_{9} + 844 \beta_{8} + 78 \beta_{7} - 220 \beta_{6} - 256 \beta_{5} - 184 \beta_{4} - 491 \beta_{3} + 128 \beta_{2} - 312 \beta _1 - 220$$ -256*b11 - 178*b10 - 128*b9 + 844*b8 + 78*b7 - 220*b6 - 256*b5 - 184*b4 - 491*b3 + 128*b2 - 312*b1 - 220 $$\nu^{8}$$ $$=$$ $$- 753 \beta_{11} - 368 \beta_{10} + 385 \beta_{9} + 3112 \beta_{8} + 184 \beta_{7} - 2744 \beta_{6} - 2973 \beta_{5} - 1037 \beta_{4} - 1357 \beta_{3} - 788 \beta_{2} - 184 \beta _1 - 1852$$ -753*b11 - 368*b10 + 385*b9 + 3112*b8 + 184*b7 - 2744*b6 - 2973*b5 - 1037*b4 - 1357*b3 - 788*b2 - 184*b1 - 1852 $$\nu^{9}$$ $$=$$ $$- 652 \beta_{11} - 2009 \beta_{10} + 705 \beta_{9} + 4658 \beta_{8} + 1304 \beta_{7} - 7165 \beta_{6} - 5310 \beta_{5} + 652 \beta_{4} - 3405 \beta_{3} - 7800 \beta_{2} + 2048 \beta _1 + 2009$$ -652*b11 - 2009*b10 + 705*b9 + 4658*b8 + 1304*b7 - 7165*b6 - 5310*b5 + 652*b4 - 3405*b3 - 7800*b2 + 2048*b1 + 2009 $$\nu^{10}$$ $$=$$ $$5100 \beta_{11} - 5100 \beta_{10} + 2700 \beta_{9} + 2700 \beta_{7} - 7110 \beta_{6} - 7110 \beta_{5} + 12915 \beta_{4} - 12915 \beta_{3} - 27435 \beta_{2} + 22335 \beta _1 + 25394$$ 5100*b11 - 5100*b10 + 2700*b9 + 2700*b7 - 7110*b6 - 7110*b5 + 12915*b4 - 12915*b3 - 27435*b2 + 22335*b1 + 25394 $$\nu^{11}$$ $$=$$ $$22335 \beta_{11} + 7815 \beta_{10} + 15630 \beta_{9} - 53280 \beta_{8} + 6705 \beta_{7} - 7815 \beta_{6} - 23925 \beta_{5} + 41550 \beta_{4} - 7815 \beta_{3} - 49365 \beta_{2} + 90074 \beta _1 + 45465$$ 22335*b11 + 7815*b10 + 15630*b9 - 53280*b8 + 6705*b7 - 7815*b6 - 23925*b5 + 41550*b4 - 7815*b3 - 49365*b2 + 90074*b1 + 45465

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-\beta_{8}$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.838695 + 2.58124i 0.213831 + 0.658105i −0.861543 − 2.65156i 2.67982 − 1.94700i −0.120753 + 0.0877319i −1.25005 + 0.908212i 2.67982 + 1.94700i −0.120753 − 0.0877319i −1.25005 − 0.908212i 0.838695 − 2.58124i 0.213831 − 0.658105i −0.861543 + 2.65156i
0 −2.19573 + 1.59529i 0 0.548020 + 2.16787i 0 4.40288 0 1.34923 4.15250i 0
81.2 0 −0.559818 + 0.406731i 0 −0.463031 2.18760i 0 −3.25686 0 −0.779086 + 2.39778i 0
81.3 0 2.25555 1.63875i 0 −2.20302 + 0.383000i 0 2.70809 0 1.47494 4.53940i 0
161.1 0 −1.02360 + 3.15031i 0 −1.34720 + 1.78467i 0 −1.69984 0 −6.44966 4.68596i 0
161.2 0 0.0461234 0.141953i 0 0.383646 2.20291i 0 −3.58696 0 2.40903 + 1.75026i 0
161.3 0 0.477475 1.46952i 0 1.08159 + 1.95708i 0 2.43270 0 0.495552 + 0.360039i 0
241.1 0 −1.02360 3.15031i 0 −1.34720 1.78467i 0 −1.69984 0 −6.44966 + 4.68596i 0
241.2 0 0.0461234 + 0.141953i 0 0.383646 + 2.20291i 0 −3.58696 0 2.40903 1.75026i 0
241.3 0 0.477475 + 1.46952i 0 1.08159 1.95708i 0 2.43270 0 0.495552 0.360039i 0
321.1 0 −2.19573 1.59529i 0 0.548020 2.16787i 0 4.40288 0 1.34923 + 4.15250i 0
321.2 0 −0.559818 0.406731i 0 −0.463031 + 2.18760i 0 −3.25686 0 −0.779086 2.39778i 0
321.3 0 2.25555 + 1.63875i 0 −2.20302 0.383000i 0 2.70809 0 1.47494 + 4.53940i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 321.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.u.f 12
4.b odd 2 1 100.2.g.a 12
12.b even 2 1 900.2.n.c 12
20.d odd 2 1 500.2.g.a 12
20.e even 4 2 500.2.i.b 24
25.d even 5 1 inner 400.2.u.f 12
25.d even 5 1 10000.2.a.bc 6
25.e even 10 1 10000.2.a.bd 6
100.h odd 10 1 500.2.g.a 12
100.h odd 10 1 2500.2.a.c 6
100.j odd 10 1 100.2.g.a 12
100.j odd 10 1 2500.2.a.d 6
100.l even 20 2 500.2.i.b 24
100.l even 20 2 2500.2.c.c 12
300.n even 10 1 900.2.n.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.g.a 12 4.b odd 2 1
100.2.g.a 12 100.j odd 10 1
400.2.u.f 12 1.a even 1 1 trivial
400.2.u.f 12 25.d even 5 1 inner
500.2.g.a 12 20.d odd 2 1
500.2.g.a 12 100.h odd 10 1
500.2.i.b 24 20.e even 4 2
500.2.i.b 24 100.l even 20 2
900.2.n.c 12 12.b even 2 1
900.2.n.c 12 300.n even 10 1
2500.2.a.c 6 100.h odd 10 1
2500.2.a.d 6 100.j odd 10 1
2500.2.c.c 12 100.l even 20 2
10000.2.a.bc 6 25.d even 5 1
10000.2.a.bd 6 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 2 T_{3}^{11} + 8 T_{3}^{10} - 4 T_{3}^{9} + 23 T_{3}^{8} + 124 T_{3}^{7} + 637 T_{3}^{6} + 281 T_{3}^{5} + 1403 T_{3}^{4} + 1414 T_{3}^{3} + 608 T_{3}^{2} - 32 T_{3} + 16$$ acting on $$S_{2}^{\mathrm{new}}(400, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 2 T^{11} + 8 T^{10} - 4 T^{9} + \cdots + 16$$
$5$ $$T^{12} + 4 T^{11} + 21 T^{10} + \cdots + 15625$$
$7$ $$(T^{6} - T^{5} - 29 T^{4} + 18 T^{3} + \cdots - 576)^{2}$$
$11$ $$T^{12} - 5 T^{11} + 40 T^{10} + \cdots + 160000$$
$13$ $$T^{12} + 2 T^{11} + 23 T^{10} + \cdots + 32761$$
$17$ $$T^{12} - T^{11} + 42 T^{10} - 187 T^{9} + \cdots + 1296$$
$19$ $$T^{12} - 8 T^{11} + 51 T^{10} + \cdots + 10471696$$
$23$ $$T^{12} - 6 T^{11} + 82 T^{10} + \cdots + 4096$$
$29$ $$T^{12} + 18 T^{11} + 241 T^{10} + \cdots + 1042441$$
$31$ $$T^{12} + 12 T^{11} + \cdots + 102495376$$
$37$ $$T^{12} - 13 T^{11} + 78 T^{10} + \cdots + 2128681$$
$41$ $$T^{12} + 23 T^{11} + 306 T^{10} + \cdots + 41422096$$
$43$ $$(T^{6} + 25 T^{5} + 185 T^{4} + 140 T^{3} + \cdots + 6400)^{2}$$
$47$ $$T^{12} + T^{11} + 72 T^{10} - 43 T^{9} + \cdots + 256$$
$53$ $$T^{12} - 21 T^{11} + 272 T^{10} + \cdots + 1745041$$
$59$ $$T^{12} + 9 T^{11} + 184 T^{10} + \cdots + 130051216$$
$61$ $$T^{12} + 26 T^{11} + \cdots + 1661296081$$
$67$ $$T^{12} - 37 T^{11} + \cdots + 58285547776$$
$71$ $$T^{12} + 21 T^{11} + \cdots + 5547866256$$
$73$ $$T^{12} - 18 T^{11} + \cdots + 4336354201$$
$79$ $$T^{12} - 24 T^{11} + \cdots + 56261942416$$
$83$ $$T^{12} - 46 T^{11} + 1162 T^{10} + \cdots + 36048016$$
$89$ $$T^{12} + 2 T^{11} + 301 T^{10} + \cdots + 63744256$$
$97$ $$T^{12} + 7 T^{11} + \cdots + 121807282081$$