# Properties

 Label 3234.2.e.b Level $3234$ Weight $2$ Character orbit 3234.e Analytic conductor $25.824$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.8236200137$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8 x^{15} + 74 x^{14} - 378 x^{13} + 1878 x^{12} - 6718 x^{11} + 22086 x^{10} - 56904 x^{9} + 130215 x^{8} - 239606 x^{7} + 378750 x^{6} - 477124 x^{5} + \cdots + 13417$$ x^16 - 8*x^15 + 74*x^14 - 378*x^13 + 1878*x^12 - 6718*x^11 + 22086*x^10 - 56904*x^9 + 130215*x^8 - 239606*x^7 + 378750*x^6 - 477124*x^5 + 493030*x^4 - 386266*x^3 + 223844*x^2 - 82874*x + 13417 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 462) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{12} q^{2} - \beta_{12} q^{3} - q^{4} + (\beta_{13} - \beta_{11}) q^{5} + q^{6} - \beta_{12} q^{8} - q^{9}+O(q^{10})$$ q + b12 * q^2 - b12 * q^3 - q^4 + (b13 - b11) * q^5 + q^6 - b12 * q^8 - q^9 $$q + \beta_{12} q^{2} - \beta_{12} q^{3} - q^{4} + (\beta_{13} - \beta_{11}) q^{5} + q^{6} - \beta_{12} q^{8} - q^{9} + \beta_{3} q^{10} + (\beta_{14} - \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + 2 \beta_{4} + \beta_{2}) q^{11} + \beta_{12} q^{12} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{13} - \beta_{3} q^{15} + q^{16} + (\beta_{5} - \beta_{3} - \beta_{2}) q^{17} - \beta_{12} q^{18} + ( - \beta_{10} + \beta_{9} + \beta_{4} - 1) q^{19} + ( - \beta_{13} + \beta_{11}) q^{20} + ( - \beta_{8} + \beta_{7}) q^{22} + (\beta_{10} - \beta_{9} + \beta_{5} - \beta_{3} - 2 \beta_{2} + \beta_1) q^{23} - q^{24} + (2 \beta_{10} - 2 \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} - 3 \beta_{4} - \beta_{3} - \beta_{2} + \cdots - 1) q^{25}+ \cdots + ( - \beta_{14} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} - 2 \beta_{4} - \beta_{2}) q^{99}+O(q^{100})$$ q + b12 * q^2 - b12 * q^3 - q^4 + (b13 - b11) * q^5 + q^6 - b12 * q^8 - q^9 + b3 * q^10 + (b14 - b10 + b9 + b7 + b6 + 2*b4 + b2) * q^11 + b12 * q^12 + (-b4 + b3 + b2 + b1) * q^13 - b3 * q^15 + q^16 + (b5 - b3 - b2) * q^17 - b12 * q^18 + (-b10 + b9 + b4 - 1) * q^19 + (-b13 + b11) * q^20 + (-b8 + b7) * q^22 + (b10 - b9 + b5 - b3 - 2*b2 + b1) * q^23 - q^24 + (2*b10 - 2*b9 - b7 - b6 + b5 - 3*b4 - b3 - b2 + b1 - 1) * q^25 + (-b14 + b11 - b9 - b8 - b7 - b4) * q^26 + b12 * q^27 + (b14 + 2*b13 + b12 - 2*b11 + b9 + 2*b8 + b6 + b4) * q^29 + (b13 - b11) * q^30 + (-b7 + b6) * q^31 + b12 * q^32 + (b8 - b7) * q^33 + (-b15 - b10 + b7 + b4) * q^34 + q^36 + (b10 - b9 + b4 + 2*b1 - 3) * q^37 + (b14 - b12 - b10 + b7 + b4) * q^38 + (b14 - b11 + b9 + b8 + b7 + b4) * q^39 - b3 * q^40 + (-b5 + b4 - b3 + b2 - b1 - 2) * q^41 + (-b15 + b14 - 2*b13 - 2*b12 + 2*b11 - b10 + b9 + b7 + b6 + 2*b4) * q^43 + (-b14 + b10 - b9 - b7 - b6 - 2*b4 - b2) * q^44 + (-b13 + b11) * q^45 + (-b15 + b13 + b9 + b8 + b7 + b4) * q^46 + (-2*b15 - b13 - 3*b12 - 2*b10 - 3*b8 + 3*b7 - b6 + 2*b4) * q^47 - b12 * q^48 + (-b15 - b14 + b13 - b12 - b7 + b6) * q^50 + (b15 + b10 - b7 - b4) * q^51 + (b4 - b3 - b2 - b1) * q^52 + (b10 - b9 - b7 - b6 + 2*b5 - b4 - 3*b3 - b2 + b1) * q^53 - q^54 + (-b15 + b14 + 2*b13 - 2*b11 - b10 + 3*b9 - b8 + b7 - b5 + 2*b4 + b3 - 2*b1 + 2) * q^55 + (-b14 + b12 + b10 - b7 - b4) * q^57 + (-b10 + b9 + b7 + b6 + b4 + 2*b3 + 2*b2 - 1) * q^58 + (3*b14 + 2*b13 + 3*b12 - 2*b11 + 3*b9 + 2*b7 + b6 + 3*b4) * q^59 + b3 * q^60 + (-b7 - b6 - b4 + 2*b3 - b2 - b1 + 4) * q^61 + (-b10 + b9 + b7 + b6 + 2*b4) * q^62 - q^64 + (b15 - 3*b13 + b10 + 2*b8 - 2*b7 + b6 - b4) * q^65 + (b14 - b10 + b9 + b7 + b6 + 2*b4 + b2) * q^66 + (b10 - b9 - b7 - b6 + 2*b5 - b4 - 2*b3 - 2*b2 - 2*b1 + 2) * q^67 + (-b5 + b3 + b2) * q^68 + (b15 - b13 - b9 - b8 - b7 - b4) * q^69 + (-b10 + b9 + b7 + b6 + b5 + 2*b3 - b1 - 4) * q^71 + b12 * q^72 + (2*b10 - 2*b9 - 2*b7 - 2*b6 - 4*b4 - 2*b2 + 2*b1 - 4) * q^73 + (b14 + 2*b13 - 3*b12 + b10 + 2*b9 + b7 + b4) * q^74 + (b15 + b14 - b13 + b12 + b7 - b6) * q^75 + (b10 - b9 - b4 + 1) * q^76 + (-b4 + b3 + b2 + b1) * q^78 + (2*b15 + b14 + b13 + b10 - b8 + b7 - 2*b6 - b4) * q^79 + (b13 - b11) * q^80 + q^81 + (b15 + b14 + b13 - 2*b12 - 2*b11 + b10 + b9) * q^82 + (b10 - b9 - b7 - b6 + b5 - b4 - b3 - 3*b2 - b1) * q^83 + (2*b14 + b13 + 2*b12 + b11 - b10 + b9 + 3*b7 - b6 + 2*b4) * q^85 + (-b10 + b9 + b7 + b6 - b5 + b4 - b3 + b2 + 2) * q^86 + (b10 - b9 - b7 - b6 - b4 - 2*b3 - 2*b2 + 1) * q^87 + (b8 - b7) * q^88 + (2*b14 + 2*b9 - 2*b8 + 2*b6 + 2*b4) * q^89 - b3 * q^90 + (-b10 + b9 - b5 + b3 + 2*b2 - b1) * q^92 + (b10 - b9 - b7 - b6 - 2*b4) * q^93 + (b10 - b9 - b7 - b6 - 2*b5 - 2*b4 + 2*b3 - b2 + b1 + 3) * q^94 + (-2*b15 + 4*b13 + 2*b9 + 2*b7 + 2*b4) * q^95 + q^96 + (-b15 - 2*b14 + 4*b13 - 4*b12 + 2*b11 + b10 - 4*b8 - b6 - b4) * q^97 + (-b14 + b10 - b9 - b7 - b6 - 2*b4 - b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 16 q^{4} + 16 q^{6} - 16 q^{9}+O(q^{10})$$ 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9 $$16 q - 16 q^{4} + 16 q^{6} - 16 q^{9} + 4 q^{10} + 8 q^{11} - 4 q^{15} + 16 q^{16} - 20 q^{19} + 2 q^{22} + 8 q^{23} - 16 q^{24} - 20 q^{25} - 2 q^{33} + 16 q^{36} - 28 q^{37} - 4 q^{40} - 32 q^{41} - 8 q^{44} - 16 q^{54} + 14 q^{55} + 4 q^{60} + 56 q^{61} + 8 q^{62} - 16 q^{64} + 8 q^{66} + 32 q^{67} - 56 q^{71} - 88 q^{73} + 20 q^{76} + 16 q^{81} - 8 q^{83} + 24 q^{86} - 2 q^{88} - 4 q^{90} - 8 q^{92} - 8 q^{93} + 28 q^{94} + 16 q^{96} - 8 q^{99}+O(q^{100})$$ 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9 + 4 * q^10 + 8 * q^11 - 4 * q^15 + 16 * q^16 - 20 * q^19 + 2 * q^22 + 8 * q^23 - 16 * q^24 - 20 * q^25 - 2 * q^33 + 16 * q^36 - 28 * q^37 - 4 * q^40 - 32 * q^41 - 8 * q^44 - 16 * q^54 + 14 * q^55 + 4 * q^60 + 56 * q^61 + 8 * q^62 - 16 * q^64 + 8 * q^66 + 32 * q^67 - 56 * q^71 - 88 * q^73 + 20 * q^76 + 16 * q^81 - 8 * q^83 + 24 * q^86 - 2 * q^88 - 4 * q^90 - 8 * q^92 - 8 * q^93 + 28 * q^94 + 16 * q^96 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 74 x^{14} - 378 x^{13} + 1878 x^{12} - 6718 x^{11} + 22086 x^{10} - 56904 x^{9} + 130215 x^{8} - 239606 x^{7} + 378750 x^{6} - 477124 x^{5} + \cdots + 13417$$ :

 $$\beta_{1}$$ $$=$$ $$( 6983 \nu^{14} - 48881 \nu^{13} + 474100 \nu^{12} - 2209147 \nu^{11} + 11140707 \nu^{10} - 36618018 \nu^{9} + 118372755 \nu^{8} - 276086391 \nu^{7} + \cdots + 32362633 ) / 22807968$$ (6983*v^14 - 48881*v^13 + 474100*v^12 - 2209147*v^11 + 11140707*v^10 - 36618018*v^9 + 118372755*v^8 - 276086391*v^7 + 593880549*v^6 - 946931332*v^5 + 1294712939*v^4 - 1271162965*v^3 + 915620088*v^2 - 401151387*v + 32362633) / 22807968 $$\beta_{2}$$ $$=$$ $$( - 3325577 \nu^{14} + 23279039 \nu^{13} - 218220364 \nu^{12} + 1006694677 \nu^{11} - 4890679917 \nu^{10} + 15780182142 \nu^{9} + \cdots - 13304183911 ) / 3261539424$$ (-3325577*v^14 + 23279039*v^13 - 218220364*v^12 + 1006694677*v^11 - 4890679917*v^10 + 15780182142*v^9 - 48699356733*v^8 + 110238860505*v^7 - 222125567883*v^6 + 336742486204*v^5 - 415275809861*v^4 + 371398223899*v^3 - 230411582664*v^2 + 86434816533*v - 13304183911) / 3261539424 $$\beta_{3}$$ $$=$$ $$( - 189803 \nu^{14} + 1328621 \nu^{13} - 12040024 \nu^{12} + 54968071 \nu^{11} - 260705691 \nu^{10} + 831319938 \nu^{9} - 2534621871 \nu^{8} + \cdots - 1021023169 ) / 148251792$$ (-189803*v^14 + 1328621*v^13 - 12040024*v^12 + 54968071*v^11 - 260705691*v^10 + 831319938*v^9 - 2534621871*v^8 + 5700932979*v^7 - 11573124741*v^6 + 17710001596*v^5 - 22848161615*v^4 + 21441640105*v^3 - 14950384260*v^2 + 6439036695*v - 1021023169) / 148251792 $$\beta_{4}$$ $$=$$ $$( - 1252325 \nu^{14} + 8766275 \nu^{13} - 81315274 \nu^{12} + 373930069 \nu^{11} - 1818246999 \nu^{10} + 5872472250 \nu^{9} - 18432565677 \nu^{8} + \cdots - 14998863325 ) / 815384856$$ (-1252325*v^14 + 8766275*v^13 - 81315274*v^12 + 373930069*v^11 - 1818246999*v^10 + 5872472250*v^9 - 18432565677*v^8 + 42250495017*v^7 - 88964564157*v^6 + 139944185800*v^5 - 189878074001*v^4 + 185929817443*v^3 - 140966904954*v^2 + 65763256533*v - 14998863325) / 815384856 $$\beta_{5}$$ $$=$$ $$( - 250035 \nu^{14} + 1750245 \nu^{13} - 16051742 \nu^{12} + 73557267 \nu^{11} - 351447497 \nu^{10} + 1124676710 \nu^{9} - 3429869109 \nu^{8} + \cdots + 128107011 ) / 135897476$$ (-250035*v^14 + 1750245*v^13 - 16051742*v^12 + 73557267*v^11 - 351447497*v^10 + 1124676710*v^9 - 3429869109*v^8 + 7709036103*v^7 - 15429510803*v^6 + 23296503804*v^5 - 28630334953*v^4 + 25544328593*v^3 - 15305197790*v^2 + 5412809207*v + 128107011) / 135897476 $$\beta_{6}$$ $$=$$ $$( 31288770 \nu^{15} - 16748485 \nu^{14} + 797619205 \nu^{13} + 2631204646 \nu^{12} - 4029956479 \nu^{11} + 98833917597 \nu^{10} + \cdots - 181576943117 ) / 158184662064$$ (31288770*v^15 - 16748485*v^14 + 797619205*v^13 + 2631204646*v^12 - 4029956479*v^11 + 98833917597*v^10 - 253528285620*v^9 + 1180057442883*v^8 - 2297035581639*v^7 + 5676299627439*v^6 - 7256018389690*v^5 + 10030408696559*v^4 - 6580080409159*v^3 + 4269219214770*v^2 - 43198382379*v - 181576943117) / 158184662064 $$\beta_{7}$$ $$=$$ $$( - 31288770 \nu^{15} + 452583065 \nu^{14} - 3848461265 \nu^{13} + 25712532526 \nu^{12} - 126371519773 \nu^{11} + 533477938419 \nu^{10} + \cdots + 4642794315301 ) / 158184662064$$ (-31288770*v^15 + 452583065*v^14 - 3848461265*v^13 + 25712532526*v^12 - 126371519773*v^11 + 533477938419*v^10 - 1785395864580*v^9 + 5160916096233*v^8 - 12143081222517*v^7 + 24141818008761*v^6 - 38917872020806*v^5 + 50481947842229*v^4 - 50908209830125*v^3 + 37436097284178*v^2 - 18719983335993*v + 4642794315301) / 158184662064 $$\beta_{8}$$ $$=$$ $$( - 61841878 \nu^{15} + 463814085 \nu^{14} - 4332228469 \nu^{13} + 21124971426 \nu^{12} - 103973734141 \nu^{11} + 354956782059 \nu^{10} + \cdots - 306749906363 ) / 158184662064$$ (-61841878*v^15 + 463814085*v^14 - 4332228469*v^13 + 21124971426*v^12 - 103973734141*v^11 + 354956782059*v^10 - 1130840178408*v^9 + 2742004258845*v^8 - 5867164985181*v^7 + 9805433466473*v^6 - 13472523722594*v^5 + 14104341127729*v^4 - 10389357329569*v^3 + 4991902261014*v^2 - 438472848665*v - 306749906363) / 158184662064 $$\beta_{9}$$ $$=$$ $$( 124647224 \nu^{15} - 792790405 \nu^{14} + 7684579947 \nu^{13} - 32966828504 \nu^{12} + 164837589985 \nu^{11} - 499636537173 \nu^{10} + \cdots - 1247802075679 ) / 158184662064$$ (124647224*v^15 - 792790405*v^14 + 7684579947*v^13 - 32966828504*v^12 + 164837589985*v^11 - 499636537173*v^10 + 1593858572646*v^9 - 3402670735473*v^8 + 7208963912421*v^7 - 10318496609323*v^6 + 14096862838860*v^5 - 12094824539073*v^4 + 10307867859415*v^3 - 4833384190332*v^2 + 3394648790233*v - 1247802075679) / 158184662064 $$\beta_{10}$$ $$=$$ $$( 124647224 \nu^{15} - 1076917955 \nu^{14} + 9673472797 \nu^{13} - 51503270872 \nu^{12} + 250200637143 \nu^{11} - 916288554507 \nu^{10} + \cdots - 4344274484769 ) / 158184662064$$ (124647224*v^15 - 1076917955*v^14 + 9673472797*v^13 - 51503270872*v^12 + 250200637143*v^11 - 916288554507*v^10 + 2942026006626*v^9 - 7649444175615*v^8 + 16964786108547*v^7 - 30953383474645*v^6 + 46664778955628*v^5 - 56590192827103*v^4 + 54130606016001*v^3 - 38151775447740*v^2 + 18943545384919*v - 4344274484769) / 158184662064 $$\beta_{11}$$ $$=$$ $$( 15958 \nu^{15} - 119685 \nu^{14} + 1130119 \nu^{13} - 5530551 \nu^{12} + 27795985 \nu^{11} - 96035346 \nu^{10} + 317610396 \nu^{9} - 791678301 \nu^{8} + \cdots - 518381770 ) / 18627492$$ (15958*v^15 - 119685*v^14 + 1130119*v^13 - 5530551*v^12 + 27795985*v^11 - 96035346*v^10 + 317610396*v^9 - 791678301*v^8 + 1812360285*v^7 - 3211318028*v^6 + 5028988886*v^5 - 5993427763*v^4 + 5979185161*v^3 - 4172380467*v^2 + 2140166891*v - 518381770) / 18627492 $$\beta_{12}$$ $$=$$ $$( - 13342 \nu^{15} + 100065 \nu^{14} - 928497 \nu^{13} + 4517578 \nu^{12} - 22245945 \nu^{11} + 75998175 \nu^{10} - 244879016 \nu^{9} + 599354649 \nu^{8} + \cdots + 261797265 ) / 7787936$$ (-13342*v^15 + 100065*v^14 - 928497*v^13 + 4517578*v^12 - 22245945*v^11 + 75998175*v^10 - 244879016*v^9 + 599354649*v^8 - 1324712505*v^7 + 2282122421*v^6 - 3405656298*v^5 + 3894289773*v^4 - 3651650205*v^3 + 2418391182*v^2 - 1148282565*v + 261797265) / 7787936 $$\beta_{13}$$ $$=$$ $$( 15958 \nu^{15} - 119685 \nu^{14} + 1130119 \nu^{13} - 5530551 \nu^{12} + 27795985 \nu^{11} - 96035346 \nu^{10} + 317610396 \nu^{9} - 791678301 \nu^{8} + \cdots - 509068024 ) / 9313746$$ (15958*v^15 - 119685*v^14 + 1130119*v^13 - 5530551*v^12 + 27795985*v^11 - 96035346*v^10 + 317610396*v^9 - 791678301*v^8 + 1812360285*v^7 - 3211318028*v^6 + 5028988886*v^5 - 5993427763*v^4 + 5979185161*v^3 - 4172380467*v^2 + 2121539399*v - 509068024) / 9313746 $$\beta_{14}$$ $$=$$ $$( 575875964 \nu^{15} - 4436099745 \nu^{14} + 41319402047 \nu^{13} - 205410283332 \nu^{12} + 1021087579937 \nu^{11} + \cdots - 16963072781711 ) / 158184662064$$ (575875964*v^15 - 4436099745*v^14 + 41319402047*v^13 - 205410283332*v^12 + 1021087579937*v^11 - 3562268688861*v^10 + 11652271091298*v^9 - 29183922744189*v^8 + 65936755256061*v^7 - 116732328141163*v^6 + 179642141127088*v^5 - 212376054391829*v^4 + 207235157443403*v^3 - 142699154525280*v^2 + 71422281850849*v - 16963072781711) / 158184662064 $$\beta_{15}$$ $$=$$ $$( 158954387 \nu^{15} - 1172652900 \nu^{14} + 11041151095 \nu^{13} - 53296410441 \nu^{12} + 265930611712 \nu^{11} - 905854992267 \nu^{10} + \cdots - 3970555117103 ) / 26364110344$$ (158954387*v^15 - 1172652900*v^14 + 11041151095*v^13 - 53296410441*v^12 + 265930611712*v^11 - 905854992267*v^10 + 2960788040621*v^9 - 7259574921286*v^8 + 16337255299198*v^7 - 28374154378543*v^6 + 43439436758339*v^5 - 50569373700088*v^4 + 49211364580575*v^3 - 33565389721049*v^2 + 16733640083382*v - 3970555117103) / 26364110344
 $$\nu$$ $$=$$ $$( -\beta_{13} + 2\beta_{11} + 1 ) / 2$$ (-b13 + 2*b11 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( - \beta_{13} + 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 9 ) / 2$$ (-b13 + 2*b11 + 2*b10 - 2*b9 - 2*b7 - 2*b6 + 2*b5 - 4*b4 - 2*b3 - 4*b2 - 9) / 2 $$\nu^{3}$$ $$=$$ $$( - 2 \beta_{15} + \beta_{14} + 9 \beta_{13} - 2 \beta_{12} - 17 \beta_{11} + 3 \beta_{10} - 2 \beta_{8} - 4 \beta_{7} + \beta_{6} + 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} - 14 ) / 2$$ (-2*b15 + b14 + 9*b13 - 2*b12 - 17*b11 + 3*b10 - 2*b8 - 4*b7 + b6 + 3*b5 - 3*b4 - 3*b3 - 6*b2 - 14) / 2 $$\nu^{4}$$ $$=$$ $$( - 4 \beta_{15} + 2 \beta_{14} + 19 \beta_{13} - 4 \beta_{12} - 36 \beta_{11} - 22 \beta_{10} + 28 \beta_{9} - 4 \beta_{8} + 20 \beta_{7} + 30 \beta_{6} - 22 \beta_{5} + 46 \beta_{4} + 26 \beta_{3} + 42 \beta_{2} - 10 \beta _1 + 71 ) / 2$$ (-4*b15 + 2*b14 + 19*b13 - 4*b12 - 36*b11 - 22*b10 + 28*b9 - 4*b8 + 20*b7 + 30*b6 - 22*b5 + 46*b4 + 26*b3 + 42*b2 - 10*b1 + 71) / 2 $$\nu^{5}$$ $$=$$ $$( 19 \beta_{15} + 3 \beta_{14} - 83 \beta_{13} + 34 \beta_{12} + 145 \beta_{11} - 69 \beta_{10} + 35 \beta_{9} + 22 \beta_{8} + 87 \beta_{7} + 17 \beta_{6} - 60 \beta_{5} + 94 \beta_{4} + 70 \beta_{3} + 115 \beta_{2} - 25 \beta _1 + 201 ) / 2$$ (19*b15 + 3*b14 - 83*b13 + 34*b12 + 145*b11 - 69*b10 + 35*b9 + 22*b8 + 87*b7 + 17*b6 - 60*b5 + 94*b4 + 70*b3 + 115*b2 - 25*b1 + 201) / 2 $$\nu^{6}$$ $$=$$ $$( 67 \beta_{15} + 4 \beta_{14} - 297 \beta_{13} + 112 \beta_{12} + 526 \beta_{11} + 176 \beta_{10} - 293 \beta_{9} + 76 \beta_{8} - 97 \beta_{7} - 332 \beta_{6} + 207 \beta_{5} - 393 \beta_{4} - 259 \beta_{3} - 361 \beta_{2} + 141 \beta _1 - 586 ) / 2$$ (67*b15 + 4*b14 - 297*b13 + 112*b12 + 526*b11 + 176*b10 - 293*b9 + 76*b8 - 97*b7 - 332*b6 + 207*b5 - 393*b4 - 259*b3 - 361*b2 + 141*b1 - 586) / 2 $$\nu^{7}$$ $$=$$ $$( - 150 \beta_{15} - 121 \beta_{14} + 707 \beta_{13} - 360 \beta_{12} - 1095 \beta_{11} + 1072 \beta_{10} - 747 \beta_{9} - 175 \beta_{8} - 1169 \beta_{7} - 510 \beta_{6} + 938 \beta_{5} - 1518 \beta_{4} - 1155 \beta_{3} + \cdots - 2771 ) / 2$$ (-150*b15 - 121*b14 + 707*b13 - 360*b12 - 1095*b11 + 1072*b10 - 747*b9 - 175*b8 - 1169*b7 - 510*b6 + 938*b5 - 1518*b4 - 1155*b3 - 1673*b2 + 581*b1 - 2771) / 2 $$\nu^{8}$$ $$=$$ $$( - 922 \beta_{15} - 498 \beta_{14} + 4259 \beta_{13} - 1972 \beta_{12} - 6920 \beta_{11} - 826 \beta_{10} + 2686 \beta_{9} - 1064 \beta_{8} - 482 \beta_{7} + 3274 \beta_{6} - 1508 \beta_{5} + 2644 \beta_{4} + \cdots + 3975 ) / 2$$ (-922*b15 - 498*b14 + 4259*b13 - 1972*b12 - 6920*b11 - 826*b10 + 2686*b9 - 1064*b8 - 482*b7 + 3274*b6 - 1508*b5 + 2644*b4 + 1964*b3 + 2428*b2 - 1340*b1 + 3975) / 2 $$\nu^{9}$$ $$=$$ $$( 927 \beta_{15} + 1353 \beta_{14} - 4574 \beta_{13} + 2602 \beta_{12} + 5587 \beta_{11} - 14119 \beta_{10} + 11807 \beta_{9} + 992 \beta_{8} + 12937 \beta_{7} + 8909 \beta_{6} - 12672 \beta_{5} + 20172 \beta_{4} + \cdots + 35386 ) / 2$$ (927*b15 + 1353*b14 - 4574*b13 + 2602*b12 + 5587*b11 - 14119*b10 + 11807*b9 + 992*b8 + 12937*b7 + 8909*b6 - 12672*b5 + 20172*b4 + 16068*b3 + 21459*b2 - 9621*b1 + 35386) / 2 $$\nu^{10}$$ $$=$$ $$( 12039 \beta_{15} + 10518 \beta_{14} - 56988 \beta_{13} + 28604 \beta_{12} + 83700 \beta_{11} - 6230 \beta_{10} - 20129 \beta_{9} + 13492 \beta_{8} + 21505 \beta_{7} - 28500 \beta_{6} + 4915 \beta_{5} + \cdots - 11089 ) / 2$$ (12039*b15 + 10518*b14 - 56988*b13 + 28604*b12 + 83700*b11 - 6230*b10 - 20129*b9 + 13492*b8 + 21505*b7 - 28500*b6 + 4915*b5 - 8061*b4 - 7043*b3 - 6263*b2 + 7359*b1 - 11089) / 2 $$\nu^{11}$$ $$=$$ $$( - 50 \beta_{15} - 6198 \beta_{14} + 796 \beta_{13} - 3010 \beta_{12} + 18558 \beta_{11} + 166047 \beta_{10} - 163611 \beta_{9} + 1489 \beta_{8} - 124121 \beta_{7} - 128625 \beta_{6} + 154187 \beta_{5} + \cdots - 418131 ) / 2$$ (-50*b15 - 6198*b14 + 796*b13 - 3010*b12 + 18558*b11 + 166047*b10 - 163611*b9 + 1489*b8 - 124121*b7 - 128625*b6 + 154187*b5 - 241383*b4 - 199518*b3 - 250855*b2 + 135333*b1 - 418131) / 2 $$\nu^{12}$$ $$=$$ $$- 74741 \beta_{15} - 80579 \beta_{14} + 354385 \beta_{13} - 183886 \beta_{12} - 467854 \beta_{11} + 130150 \beta_{10} + 38814 \beta_{9} - 79386 \beta_{8} - 195571 \beta_{7} + 99599 \beta_{6} + \cdots - 141469$$ -74741*b15 - 80579*b14 + 354385*b13 - 183886*b12 - 467854*b11 + 130150*b10 + 38814*b9 - 79386*b8 - 195571*b7 + 99599*b6 + 48888*b5 - 81749*b4 - 59284*b3 - 89044*b2 + 24892*b1 - 141469 $$\nu^{13}$$ $$=$$ $$( - 144088 \beta_{15} - 108010 \beta_{14} + 697161 \beta_{13} - 343180 \beta_{12} - 1133136 \beta_{11} - 1750154 \beta_{10} + 2068612 \beta_{9} - 175276 \beta_{8} + 997188 \beta_{7} + \cdots + 4562117 ) / 2$$ (-144088*b15 - 108010*b14 + 697161*b13 - 343180*b12 - 1133136*b11 - 1750154*b10 + 2068612*b9 - 175276*b8 + 997188*b7 + 1658494*b6 - 1711866*b5 + 2645958*b4 + 2256254*b3 + 2688400*b2 - 1689844*b1 + 4562117) / 2 $$\nu^{14}$$ $$=$$ $$( 1733162 \beta_{15} + 2074966 \beta_{14} - 8106763 \beta_{13} + 4282548 \beta_{12} + 9602618 \beta_{11} - 5068490 \beta_{10} + 1147786 \beta_{9} + 1715280 \beta_{8} + 5625272 \beta_{7} + \cdots + 7785995 ) / 2$$ (1733162*b15 + 2074966*b14 - 8106763*b13 + 4282548*b12 + 9602618*b11 - 5068490*b10 + 1147786*b9 + 1715280*b8 + 5625272*b7 - 660412*b6 - 2864062*b5 + 4680930*b4 + 3676122*b3 + 4707416*b2 - 2421852*b1 + 7785995) / 2 $$\nu^{15}$$ $$=$$ $$( 3522094 \beta_{15} + 3795721 \beta_{14} - 16818405 \beta_{13} + 8683710 \beta_{12} + 22482349 \beta_{11} + 16006457 \beta_{10} - 24041258 \beta_{9} + 3807178 \beta_{8} + \cdots - 45325614 ) / 2$$ (3522094*b15 + 3795721*b14 - 16818405*b13 + 8683710*b12 + 22482349*b11 + 16006457*b10 - 24041258*b9 + 3807178*b8 - 5454972*b7 - 19608667*b6 + 17217975*b5 - 26269633*b4 - 23170387*b3 - 26098192*b2 + 18842846*b1 - 45325614) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3234\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$1079$$ $$2059$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2155.1
 0.5 + 2.40229i 0.5 + 3.32851i 0.5 + 1.35798i 0.5 − 1.56688i 0.5 + 0.0286340i 0.5 − 0.921602i 0.5 − 3.19339i 0.5 − 3.43554i 0.5 + 3.43554i 0.5 + 3.19339i 0.5 + 0.921602i 0.5 − 0.0286340i 0.5 + 1.56688i 0.5 − 1.35798i 0.5 − 3.32851i 0.5 − 2.40229i
1.00000i 1.00000i −1.00000 3.26832i 1.00000 0 1.00000i −1.00000 −3.26832
2155.2 1.00000i 1.00000i −1.00000 2.46248i 1.00000 0 1.00000i −1.00000 −2.46248
2155.3 1.00000i 1.00000i −1.00000 2.22400i 1.00000 0 1.00000i −1.00000 −2.22400
2155.4 1.00000i 1.00000i −1.00000 0.700858i 1.00000 0 1.00000i −1.00000 0.700858
2155.5 1.00000i 1.00000i −1.00000 0.837391i 1.00000 0 1.00000i −1.00000 0.837391
2155.6 1.00000i 1.00000i −1.00000 1.78763i 1.00000 0 1.00000i −1.00000 1.78763
2155.7 1.00000i 1.00000i −1.00000 2.32736i 1.00000 0 1.00000i −1.00000 2.32736
2155.8 1.00000i 1.00000i −1.00000 4.30156i 1.00000 0 1.00000i −1.00000 4.30156
2155.9 1.00000i 1.00000i −1.00000 4.30156i 1.00000 0 1.00000i −1.00000 4.30156
2155.10 1.00000i 1.00000i −1.00000 2.32736i 1.00000 0 1.00000i −1.00000 2.32736
2155.11 1.00000i 1.00000i −1.00000 1.78763i 1.00000 0 1.00000i −1.00000 1.78763
2155.12 1.00000i 1.00000i −1.00000 0.837391i 1.00000 0 1.00000i −1.00000 0.837391
2155.13 1.00000i 1.00000i −1.00000 0.700858i 1.00000 0 1.00000i −1.00000 0.700858
2155.14 1.00000i 1.00000i −1.00000 2.22400i 1.00000 0 1.00000i −1.00000 −2.22400
2155.15 1.00000i 1.00000i −1.00000 2.46248i 1.00000 0 1.00000i −1.00000 −2.46248
2155.16 1.00000i 1.00000i −1.00000 3.26832i 1.00000 0 1.00000i −1.00000 −3.26832
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2155.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3234.2.e.b 16
7.b odd 2 1 3234.2.e.a 16
7.c even 3 1 462.2.p.b yes 16
7.d odd 6 1 462.2.p.a 16
11.b odd 2 1 3234.2.e.a 16
21.g even 6 1 1386.2.bk.a 16
21.h odd 6 1 1386.2.bk.b 16
77.b even 2 1 inner 3234.2.e.b 16
77.h odd 6 1 462.2.p.a 16
77.i even 6 1 462.2.p.b yes 16
231.k odd 6 1 1386.2.bk.b 16
231.l even 6 1 1386.2.bk.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.p.a 16 7.d odd 6 1
462.2.p.a 16 77.h odd 6 1
462.2.p.b yes 16 7.c even 3 1
462.2.p.b yes 16 77.i even 6 1
1386.2.bk.a 16 21.g even 6 1
1386.2.bk.a 16 231.l even 6 1
1386.2.bk.b 16 21.h odd 6 1
1386.2.bk.b 16 231.k odd 6 1
3234.2.e.a 16 7.b odd 2 1
3234.2.e.a 16 11.b odd 2 1
3234.2.e.b 16 1.a even 1 1 trivial
3234.2.e.b 16 77.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3234, [\chi])$$:

 $$T_{5}^{16} + 50 T_{5}^{14} + 971 T_{5}^{12} + 9580 T_{5}^{10} + 52131 T_{5}^{8} + 156530 T_{5}^{6} + 240841 T_{5}^{4} + 158136 T_{5}^{2} + 35344$$ T5^16 + 50*T5^14 + 971*T5^12 + 9580*T5^10 + 52131*T5^8 + 156530*T5^6 + 240841*T5^4 + 158136*T5^2 + 35344 $$T_{13}^{8} - 64T_{13}^{6} + 8T_{13}^{5} + 836T_{13}^{4} - 1168T_{13}^{3} - 592T_{13}^{2} + 1216T_{13} - 128$$ T13^8 - 64*T13^6 + 8*T13^5 + 836*T13^4 - 1168*T13^3 - 592*T13^2 + 1216*T13 - 128

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{8}$$
$3$ $$(T^{2} + 1)^{8}$$
$5$ $$T^{16} + 50 T^{14} + 971 T^{12} + \cdots + 35344$$
$7$ $$T^{16}$$
$11$ $$T^{16} - 8 T^{15} + 33 T^{14} + \cdots + 214358881$$
$13$ $$(T^{8} - 64 T^{6} + 8 T^{5} + 836 T^{4} + \cdots - 128)^{2}$$
$17$ $$(T^{8} - 53 T^{6} - 68 T^{5} + 727 T^{4} + \cdots + 24)^{2}$$
$19$ $$(T^{8} + 10 T^{7} - 7 T^{6} - 300 T^{5} + \cdots - 1664)^{2}$$
$23$ $$(T^{8} - 4 T^{7} - 127 T^{6} + \cdots - 155048)^{2}$$
$29$ $$T^{16} + 244 T^{14} + \cdots + 12810617856$$
$31$ $$T^{16} + 110 T^{14} + 4321 T^{12} + \cdots + 262144$$
$37$ $$(T^{8} + 14 T^{7} - 67 T^{6} + \cdots - 354432)^{2}$$
$41$ $$(T^{8} + 16 T^{7} + 2 T^{6} - 952 T^{5} + \cdots + 256)^{2}$$
$43$ $$T^{16} + 318 T^{14} + \cdots + 23084548096$$
$47$ $$T^{16} + 690 T^{14} + \cdots + 1344737098384$$
$53$ $$(T^{8} - 235 T^{6} - 116 T^{5} + \cdots - 162752)^{2}$$
$59$ $$T^{16} + 532 T^{14} + \cdots + 109280491776$$
$61$ $$(T^{8} - 28 T^{7} + 164 T^{6} + \cdots - 59072)^{2}$$
$67$ $$(T^{8} - 16 T^{7} - 94 T^{6} + 1924 T^{5} + \cdots + 49024)^{2}$$
$71$ $$(T^{8} + 28 T^{7} + 97 T^{6} + \cdots + 4193232)^{2}$$
$73$ $$(T^{8} + 44 T^{7} + 620 T^{6} + \cdots - 346112)^{2}$$
$79$ $$T^{16} + 726 T^{14} + \cdots + 1130708969104$$
$83$ $$(T^{8} + 4 T^{7} - 141 T^{6} - 780 T^{5} + \cdots + 5604)^{2}$$
$89$ $$T^{16} + 784 T^{14} + \cdots + 96546188427264$$
$97$ $$T^{16} + 944 T^{14} + \cdots + 68597371921$$