Properties

 Label 1680.3.bd.a Level $1680$ Weight $3$ Character orbit 1680.bd Analytic conductor $45.777$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1680.bd (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$45.7766844125$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + 362298 x^{8} - 722060 x^{7} + 1208164 x^{6} - 1570812 x^{5} + \cdots + 33750$$ x^16 - 8*x^15 + 96*x^14 - 532*x^13 + 3236*x^12 - 12864*x^11 + 49526*x^10 - 141436*x^9 + 362298*x^8 - 722060*x^7 + 1208164*x^6 - 1570812*x^5 + 1591101*x^4 - 1183860*x^3 + 619650*x^2 - 202500*x + 33750 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{12}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 210) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - \beta_{6} q^{5} + (\beta_{9} + \beta_{2} - \beta_1) q^{7} + 3 q^{9}+O(q^{10})$$ q - b1 * q^3 - b6 * q^5 + (b9 + b2 - b1) * q^7 + 3 * q^9 $$q - \beta_1 q^{3} - \beta_{6} q^{5} + (\beta_{9} + \beta_{2} - \beta_1) q^{7} + 3 q^{9} + ( - \beta_{12} + \beta_{11} + \beta_{3} - 6) q^{11} + (\beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{2} - 3 \beta_1) q^{13} + (\beta_{11} + \beta_{3} + 2) q^{15} + (\beta_{6} + \beta_{5} - 2 \beta_{2} - \beta_1) q^{17} + (\beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_1) q^{19} + ( - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{3} + 2) q^{21} + ( - 2 \beta_{15} + 2 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} - 4 \beta_{10} + 3) q^{23} + (\beta_{15} + \beta_{14} - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - 3 \beta_{3} + \beta_1 + 2) q^{25} - 3 \beta_1 q^{27} + (2 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} - 4 \beta_{11} + 2) q^{29} + (2 \beta_{15} + 2 \beta_{13} + \beta_{6} - \beta_{5} + 5 \beta_{4} - \beta_1) q^{31} + ( - \beta_{9} - \beta_{8} - 2 \beta_{6} - 2 \beta_{5} - \beta_{2} + 6 \beta_1) q^{33} + ( - 3 \beta_{15} - 2 \beta_{14} + \beta_{13} + 5 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + \cdots + 4) q^{35}+ \cdots + ( - 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{3} - 18) q^{99}+O(q^{100})$$ q - b1 * q^3 - b6 * q^5 + (b9 + b2 - b1) * q^7 + 3 * q^9 + (-b12 + b11 + b3 - 6) * q^11 + (b9 + b8 - b6 - b5 - b2 - 3*b1) * q^13 + (b11 + b3 + 2) * q^15 + (b6 + b5 - 2*b2 - b1) * q^17 + (b9 - b8 - 2*b7 + b6 - b5 + b4 - 2*b1) * q^19 + (-b14 + b13 + b12 + b3 + 2) * q^21 + (-2*b15 + 2*b13 + 3*b12 + 3*b11 - 4*b10 + 3) * q^23 + (b15 + b14 - b13 + b11 + b10 - b9 + b8 - 3*b3 + b1 + 2) * q^25 - 3*b1 * q^27 + (2*b15 + 4*b14 - 2*b13 - 4*b11 + 2) * q^29 + (2*b15 + 2*b13 + b6 - b5 + 5*b4 - b1) * q^31 + (-b9 - b8 - 2*b6 - 2*b5 - b2 + 6*b1) * q^33 + (-3*b15 - 2*b14 + b13 + 5*b12 + 2*b11 - 2*b10 - 2*b9 + b7 - b6 + b5 - 3*b4 + 6*b1 + 4) * q^35 + (-3*b15 + 3*b13 + 4*b12 + 4*b11 - 4*b10 - 4*b9 + 4*b8 + 4*b1 + 4) * q^37 + (2*b15 + 4*b14 - 2*b13 - 2*b12 - 2*b11 + b3 + 7) * q^39 + (-b15 - b13 - b6 + b5 + 4*b4 + b1) * q^41 + (2*b15 - 2*b13 - 2*b12 - 2*b11 + 8*b10 - 6*b9 + 6*b8 + 6*b1 - 2) * q^43 - 3*b6 * q^45 + (5*b9 + 5*b8 - 4*b6 - 4*b5 + 2*b2 + 13*b1) * q^47 + (-b15 - 2*b14 + b13 - 2*b12 + 4*b11 - 2*b6 + 2*b5 - 12*b4 + 6*b3 + 2*b1 + 15) * q^49 + (2*b15 + 4*b14 - 2*b13 - b12 - 3*b11 - 4*b3 + 1) * q^51 + (4*b15 - 4*b13 + b12 + b11 - b10 - 4*b9 + 4*b8 + 4*b1 + 1) * q^53 + (-b15 - b13 - 4*b9 - 3*b8 + b7 + 7*b6 + b5 + 7*b4 - 5*b2 - 17*b1) * q^55 + (-3*b15 + 3*b13 + b12 + b11 - b10 + 1) * q^57 + (-2*b15 - 2*b13 - 2*b9 + 2*b8 + 4*b7 - 6*b6 + 6*b5 - 14*b4 + 8*b1) * q^59 + (-2*b15 - 2*b13 + 2*b9 - 2*b8 - 4*b7 - 8*b6 + 8*b5 + 4*b4 + 6*b1) * q^61 + (3*b9 + 3*b2 - 3*b1) * q^63 + (6*b15 + 6*b14 - 6*b13 - 5*b12 - 3*b11 - 9*b10 + 4*b9 - 4*b8 - 2*b3 - 4*b1 + 19) * q^65 + (2*b15 - 2*b13 - 6*b12 - 6*b11 - 4*b10 + 7*b9 - 7*b8 - 7*b1 - 6) * q^67 + (2*b9 - 2*b8 - 4*b7 - 5*b6 + 5*b5 - 8*b4 + 3*b1) * q^69 + (-4*b15 - 8*b14 + 4*b13 + b12 + 7*b11 - 3*b3 + 28) * q^71 + (-7*b9 - 7*b8 - 7*b6 - 7*b5 - 7*b2 - 3*b1) * q^73 + (-b15 - b13 - 4*b9 - 3*b8 + b7 - 2*b6 + 6*b5 + 2*b4 - 5*b2 + 3*b1) * q^75 + (2*b15 - 2*b13 - b12 - b11 + b10 - 10*b9 + 6*b8 + 5*b6 + 5*b5 + 23*b1 - 1) * q^77 + (-2*b15 - 4*b14 + 2*b13 + 4*b11 - 4*b3 + 40) * q^79 + 9 * q^81 + (-3*b9 - 3*b8 - 4*b6 - 4*b5 - 10*b2 + 5*b1) * q^83 + (5*b15 + 5*b14 - 5*b13 - 10*b12 - 5*b11 - 5*b10 + 5*b9 - 5*b8 + 5*b3 - 5*b1 - 35) * q^85 + (6*b9 + 6*b8 - 4*b1) * q^87 + (-6*b15 - 6*b13 - 5*b6 + 5*b5 + 12*b4 + 5*b1) * q^89 + (4*b15 + 8*b14 - 4*b13 + 4*b12 - 12*b11 - b9 + b8 + 2*b7 - 7*b6 + 7*b5 - 15*b4 - 2*b3 + 8*b1 - 18) * q^91 + (-b12 - b11 + 5*b10 + 6*b9 - 6*b8 - 6*b1 - 1) * q^93 + (-2*b15 + 8*b14 + 2*b13 - 12*b10 - 3*b9 + 3*b8 + 3*b3 + 3*b1 + 35) * q^95 + (-5*b9 - 5*b8 + 3*b6 + 3*b5 - 5*b2 - 33*b1) * q^97 + (-3*b12 + 3*b11 + 3*b3 - 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 48 q^{9}+O(q^{10})$$ 16 * q + 48 * q^9 $$16 q + 48 q^{9} - 96 q^{11} + 24 q^{15} + 24 q^{21} + 24 q^{25} + 64 q^{29} + 8 q^{35} + 144 q^{39} + 224 q^{49} + 48 q^{51} + 368 q^{65} + 384 q^{71} + 608 q^{79} + 144 q^{81} - 440 q^{85} - 224 q^{91} + 560 q^{95} - 288 q^{99}+O(q^{100})$$ 16 * q + 48 * q^9 - 96 * q^11 + 24 * q^15 + 24 * q^21 + 24 * q^25 + 64 * q^29 + 8 * q^35 + 144 * q^39 + 224 * q^49 + 48 * q^51 + 368 * q^65 + 384 * q^71 + 608 * q^79 + 144 * q^81 - 440 * q^85 - 224 * q^91 + 560 * q^95 - 288 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + 362298 x^{8} - 722060 x^{7} + 1208164 x^{6} - 1570812 x^{5} + \cdots + 33750$$ :

 $$\beta_{1}$$ $$=$$ $$( 1304 \nu^{14} - 9128 \nu^{13} + 115446 \nu^{12} - 574012 \nu^{11} + 3591842 \nu^{10} - 12914984 \nu^{9} + 49995515 \nu^{8} - 128433344 \nu^{7} + 320844898 \nu^{6} + \cdots + 99070875 ) / 560625$$ (1304*v^14 - 9128*v^13 + 115446*v^12 - 574012*v^11 + 3591842*v^10 - 12914984*v^9 + 49995515*v^8 - 128433344*v^7 + 320844898*v^6 - 561338382*v^5 + 866468654*v^4 - 924341564*v^3 + 755470905*v^2 - 368877150*v + 99070875) / 560625 $$\beta_{2}$$ $$=$$ $$( - 19306 \nu^{14} + 135142 \nu^{13} - 1709079 \nu^{12} + 8497628 \nu^{11} - 53165468 \nu^{10} + 191153301 \nu^{9} - 739735340 \nu^{8} + \cdots - 1406219625 ) / 3027375$$ (-19306*v^14 + 135142*v^13 - 1709079*v^12 + 8497628*v^11 - 53165468*v^10 + 191153301*v^9 - 739735340*v^8 + 1899973946*v^7 - 4742613417*v^6 + 8293098988*v^5 - 12773014036*v^4 + 13602334971*v^3 - 11039578230*v^2 + 5354640900*v - 1406219625) / 3027375 $$\beta_{3}$$ $$=$$ $$( - 181 \nu^{14} + 1267 \nu^{13} - 16034 \nu^{12} + 79733 \nu^{11} - 499338 \nu^{10} + 1796001 \nu^{9} - 6961130 \nu^{8} + 17893811 \nu^{7} - 44790412 \nu^{6} + \cdots - 14271525 ) / 26325$$ (-181*v^14 + 1267*v^13 - 16034*v^12 + 79733*v^11 - 499338*v^10 + 1796001*v^9 - 6961130*v^8 + 17893811*v^7 - 44790412*v^6 + 78463393*v^5 - 121536126*v^4 + 129995121*v^3 - 106932015*v^2 + 52505910*v - 14271525) / 26325 $$\beta_{4}$$ $$=$$ $$( 1984 \nu^{15} - 14880 \nu^{14} + 171958 \nu^{13} - 892047 \nu^{12} + 4997626 \nu^{11} - 18170922 \nu^{10} + 58997141 \nu^{9} - 142860075 \nu^{8} + \cdots + 575687250 ) / 85899825$$ (1984*v^15 - 14880*v^14 + 171958*v^13 - 892047*v^12 + 4997626*v^11 - 18170922*v^10 + 58997141*v^9 - 142860075*v^8 + 233442698*v^7 - 259430001*v^6 - 304752040*v^5 + 1154379162*v^4 - 2711291745*v^3 + 2854944621*v^2 - 2020897980*v + 575687250) / 85899825 $$\beta_{5}$$ $$=$$ $$( 447494447 \nu^{15} - 3678104460 \nu^{14} + 43437004320 \nu^{13} - 245298593960 \nu^{12} + 1472808594649 \nu^{11} + \cdots - 41918527948500 ) / 83752329375$$ (447494447*v^15 - 3678104460*v^14 + 43437004320*v^13 - 245298593960*v^12 + 1472808594649*v^11 - 5935461073300*v^10 + 22561339527721*v^9 - 65003385595077*v^8 + 163843095257880*v^7 - 326952778883473*v^6 + 532146803968220*v^5 - 679957700632208*v^4 + 645721048309461*v^3 - 443220376967070*v^2 + 188538895443600*v - 41918527948500) / 83752329375 $$\beta_{6}$$ $$=$$ $$( - 447494447 \nu^{15} + 3229118109 \nu^{14} - 40294099863 \nu^{13} + 205531523111 \nu^{12} - 1275063927496 \nu^{11} + \cdots + 5890964889375 ) / 83752329375$$ (-447494447*v^15 + 3229118109*v^14 - 40294099863*v^13 + 205531523111*v^12 - 1275063927496*v^11 + 4697341630102*v^10 - 18108495871075*v^9 + 47750119577067*v^8 - 119500311883419*v^7 + 216005057687986*v^6 - 337841799571937*v^5 + 379071089785232*v^4 - 323942441246295*v^3 + 177871284272175*v^2 - 57903995250000*v + 5890964889375) / 83752329375 $$\beta_{7}$$ $$=$$ $$( 472182246 \nu^{15} - 3541366845 \nu^{14} + 43501742065 \nu^{13} - 229050592940 \nu^{12} + 1408375647077 \nu^{11} + \cdots - 20360648194875 ) / 83752329375$$ (472182246*v^15 - 3541366845*v^14 + 43501742065*v^13 - 229050592940*v^12 + 1408375647077*v^11 - 5344673143645*v^10 + 20553576453618*v^9 - 55932172518036*v^8 + 140858730851255*v^7 - 264716842677244*v^6 + 423427752337405*v^5 - 504073092105359*v^4 + 458352419961288*v^3 - 285479170396935*v^2 + 111855010015800*v - 20360648194875) / 83752329375 $$\beta_{8}$$ $$=$$ $$( - 755997166 \nu^{15} + 5904909925 \nu^{14} - 71246353625 \nu^{13} + 387215467260 \nu^{12} - 2354325589857 \nu^{11} + \cdots + 47103825674250 ) / 83752329375$$ (-755997166*v^15 + 5904909925*v^14 - 71246353625*v^13 + 387215467260*v^12 - 2354325589857*v^11 + 9185644498910*v^10 - 35117010838133*v^9 + 98161804769156*v^8 - 247143493283185*v^7 + 477997246153434*v^6 - 770301863592495*v^5 + 949203739326694*v^4 - 880602978178053*v^3 + 575157357209685*v^2 - 234220819182300*v + 47103825674250) / 83752329375 $$\beta_{9}$$ $$=$$ $$( 755997166 \nu^{15} - 5240241701 \nu^{14} + 66593676057 \nu^{13} - 328372443834 \nu^{12} + 2061752257685 \nu^{11} + \cdots + 2190542081625 ) / 83752329375$$ (755997166*v^15 - 5240241701*v^14 + 66593676057*v^13 - 328372443834*v^12 + 2061752257685*v^11 - 7355029073458*v^10 + 28534967107079*v^9 - 72686930748241*v^8 + 181708048044021*v^7 - 314610293664696*v^6 + 484539187579153*v^5 - 508676272046420*v^4 + 411133559676519*v^3 - 193000679236230*v^2 + 48331533796650*v + 2190542081625) / 83752329375 $$\beta_{10}$$ $$=$$ $$( 5163308 \nu^{15} - 38724810 \nu^{14} + 475369020 \nu^{13} - 2502572345 \nu^{12} + 15373400596 \nu^{11} - 58317525310 \nu^{10} + \cdots - 200352316500 ) / 404600625$$ (5163308*v^15 - 38724810*v^14 + 475369020*v^13 - 2502572345*v^12 + 15373400596*v^11 - 58317525310*v^10 + 223952035489*v^9 - 608926339653*v^8 + 1529887949640*v^7 - 2869994011087*v^6 + 4569918940190*v^5 - 5417408634332*v^4 + 4875031263099*v^3 - 3000039760455*v^2 + 1143288079650*v - 200352316500) / 404600625 $$\beta_{11}$$ $$=$$ $$( 49652491 \nu^{15} - 376911377 \nu^{14} + 4602946389 \nu^{13} - 24466123483 \nu^{12} + 149826974288 \nu^{11} - 573281443581 \nu^{10} + \cdots - 2269018608750 ) / 3641405625$$ (49652491*v^15 - 376911377*v^14 + 4602946389*v^13 - 24466123483*v^12 + 149826974288*v^11 - 573281443581*v^10 + 2198472882950*v^9 - 6029737893451*v^8 + 15159159986457*v^7 - 28719249343958*v^6 + 45904925982811*v^5 - 55129564525971*v^4 + 50105046928710*v^3 - 31492498380525*v^2 + 12264277563000*v - 2269018608750) / 3641405625 $$\beta_{12}$$ $$=$$ $$( 49652491 \nu^{15} - 367875988 \nu^{14} + 4539698666 \nu^{13} - 23665127672 \nu^{12} + 145843219821 \nu^{11} - 548311933059 \nu^{10} + \cdots - 1551811091625 ) / 3641405625$$ (49652491*v^15 - 367875988*v^14 + 4539698666*v^13 - 23665127672*v^12 + 145843219821*v^11 - 548311933059*v^10 + 2108635675556*v^9 - 5681213642486*v^8 + 14262849049078*v^7 - 26473494439015*v^6 + 41968541249574*v^5 - 49027516733757*v^4 + 43575030605286*v^3 - 26118496273095*v^2 + 9624765169350*v - 1551811091625) / 3641405625 $$\beta_{13}$$ $$=$$ $$( - 1309724492 \nu^{15} + 9822933690 \nu^{14} - 120588431330 \nu^{13} + 634843642680 \nu^{12} - 3900144384104 \nu^{11} + \cdots + 51303336994125 ) / 83752329375$$ (-1309724492*v^15 + 9822933690*v^14 - 120588431330*v^13 + 634843642680*v^12 - 3900144384104*v^11 + 14795272597215*v^10 - 56822838416461*v^9 + 154510808740347*v^8 - 388258083600460*v^7 + 728436475517238*v^6 - 1160169126957310*v^5 + 1375610500307643*v^4 - 1238330659376901*v^3 + 762341466277845*v^2 - 291343113113850*v + 51303336994125) / 83752329375 $$\beta_{14}$$ $$=$$ $$( 65666397 \nu^{15} - 477141136 \nu^{14} + 5938288402 \nu^{13} - 30467772959 \nu^{12} + 188760325727 \nu^{11} - 699352435848 \nu^{10} + \cdots - 1356819889500 ) / 3641405625$$ (65666397*v^15 - 477141136*v^14 + 5938288402*v^13 - 30467772959*v^12 + 188760325727*v^11 - 699352435848*v^10 + 2696041238592*v^9 - 7154487354962*v^8 + 17941292124116*v^7 - 32706466477060*v^6 + 51472689963028*v^5 - 58600430423934*v^4 + 50974428840402*v^3 - 29077711700715*v^2 + 10067817170700*v - 1356819889500) / 3641405625 $$\beta_{15}$$ $$=$$ $$( - 2046364168 \nu^{15} + 15347731260 \nu^{14} - 188413764520 \nu^{13} + 991915545270 \nu^{12} - 6093859902766 \nu^{11} + \cdots + 80293733362125 ) / 83752329375$$ (-2046364168*v^15 + 15347731260*v^14 - 188413764520*v^13 + 991915545270*v^12 - 6093859902766*v^11 + 23117261100285*v^10 - 88784958486419*v^9 + 241422098648613*v^8 - 606652691866790*v^7 + 1138182954778752*v^6 - 1812786534666740*v^5 + 2149443611275047*v^4 - 1935249651865779*v^3 + 1191644101481355*v^2 - 455646600367650*v + 80293733362125) / 83752329375
 $$\nu$$ $$=$$ $$( - \beta_{15} - \beta_{13} - 2 \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 3 \beta _1 + 6 ) / 12$$ (-b15 - b13 - 2*b10 + b9 - b8 - 2*b7 + 2*b6 - 2*b5 + 2*b4 - 3*b1 + 6) / 12 $$\nu^{2}$$ $$=$$ $$( - 7 \beta_{15} - 12 \beta_{14} + 5 \beta_{13} + 6 \beta_{12} + 6 \beta_{11} - 2 \beta_{10} + 7 \beta_{9} + 5 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 6 \beta_{2} - 3 \beta _1 - 90 ) / 12$$ (-7*b15 - 12*b14 + 5*b13 + 6*b12 + 6*b11 - 2*b10 + 7*b9 + 5*b8 - 2*b7 + 2*b6 - 2*b5 + 2*b4 - 6*b3 + 6*b2 - 3*b1 - 90) / 12 $$\nu^{3}$$ $$=$$ $$( 11 \beta_{15} - 18 \beta_{14} + 11 \beta_{13} - 9 \beta_{12} - 9 \beta_{11} + 61 \beta_{10} - 14 \beta_{9} + 32 \beta_{8} + 28 \beta_{7} - 46 \beta_{6} + 46 \beta_{5} - 79 \beta_{4} - 9 \beta_{3} + 9 \beta_{2} + 69 \beta _1 - 156 ) / 12$$ (11*b15 - 18*b14 + 11*b13 - 9*b12 - 9*b11 + 61*b10 - 14*b9 + 32*b8 + 28*b7 - 46*b6 + 46*b5 - 79*b4 - 9*b3 + 9*b2 + 69*b1 - 156) / 12 $$\nu^{4}$$ $$=$$ $$( 143 \beta_{15} + 204 \beta_{14} - 97 \beta_{13} - 108 \beta_{12} - 168 \beta_{11} + 124 \beta_{10} - 227 \beta_{9} - 133 \beta_{8} + 58 \beta_{7} - 106 \beta_{6} + 82 \beta_{5} - 160 \beta_{4} + 132 \beta_{3} - 144 \beta_{2} + \cdots + 1386 ) / 12$$ (143*b15 + 204*b14 - 97*b13 - 108*b12 - 168*b11 + 124*b10 - 227*b9 - 133*b8 + 58*b7 - 106*b6 + 82*b5 - 160*b4 + 132*b3 - 144*b2 + 321*b1 + 1386) / 12 $$\nu^{5}$$ $$=$$ $$( - 88 \beta_{15} + 540 \beta_{14} - 328 \beta_{13} + 255 \beta_{12} + 105 \beta_{11} - 1493 \beta_{10} + 37 \beta_{9} - 967 \beta_{8} - 404 \beta_{7} + 878 \beta_{6} - 938 \beta_{5} + 1937 \beta_{4} + 345 \beta_{3} + \cdots + 4236 ) / 12$$ (-88*b15 + 540*b14 - 328*b13 + 255*b12 + 105*b11 - 1493*b10 + 37*b9 - 967*b8 - 404*b7 + 878*b6 - 938*b5 + 1937*b4 + 345*b3 - 375*b2 - 960*b1 + 4236) / 12 $$\nu^{6}$$ $$=$$ $$( - 1496 \beta_{15} - 1815 \beta_{14} + 814 \beta_{13} + 1095 \beta_{12} + 2160 \beta_{11} - 2395 \beta_{10} + 2960 \beta_{9} + 1336 \beta_{8} - 679 \beta_{7} + 1432 \beta_{6} - 1528 \beta_{5} + 3106 \beta_{4} + \cdots - 11916 ) / 6$$ (-1496*b15 - 1815*b14 + 814*b13 + 1095*b12 + 2160*b11 - 2395*b10 + 2960*b9 + 1336*b8 - 679*b7 + 1432*b6 - 1528*b5 + 3106*b4 - 1305*b3 + 1482*b2 - 5667*b1 - 11916) / 6 $$\nu^{7}$$ $$=$$ $$( - 1450 \beta_{15} - 14616 \beta_{14} + 9932 \beta_{13} - 6321 \beta_{12} + 1659 \beta_{11} + 31705 \beta_{10} + 6625 \beta_{9} + 26723 \beta_{8} + 5944 \beta_{7} - 18136 \beta_{6} + 17674 \beta_{5} + \cdots - 111504 ) / 12$$ (-1450*b15 - 14616*b14 + 9932*b13 - 6321*b12 + 1659*b11 + 31705*b10 + 6625*b9 + 26723*b8 + 5944*b7 - 18136*b6 + 17674*b5 - 43381*b4 - 10353*b3 + 11697*b2 + 2754*b1 - 111504) / 12 $$\nu^{8}$$ $$=$$ $$( 61883 \beta_{15} + 65724 \beta_{14} - 21481 \beta_{13} - 50412 \beta_{12} - 106032 \beta_{11} + 149464 \beta_{10} - 138761 \beta_{9} - 42991 \beta_{8} + 30250 \beta_{7} - 76510 \beta_{6} + \cdots + 413190 ) / 12$$ (61883*b15 + 65724*b14 - 21481*b13 - 50412*b12 - 106032*b11 + 149464*b10 - 138761*b9 - 42991*b8 + 30250*b7 - 76510*b6 + 94798*b5 - 202888*b4 + 50412*b3 - 55128*b2 + 313719*b1 + 413190) / 12 $$\nu^{9}$$ $$=$$ $$( 103439 \beta_{15} + 385758 \beta_{14} - 270907 \beta_{13} + 143883 \beta_{12} - 154917 \beta_{11} - 611849 \beta_{10} - 321620 \beta_{9} - 700294 \beta_{8} - 85022 \beta_{7} + 385532 \beta_{6} + \cdots + 2878206 ) / 12$$ (103439*b15 + 385758*b14 - 270907*b13 + 143883*b12 - 154917*b11 - 611849*b10 - 321620*b9 - 700294*b8 - 85022*b7 + 385532*b6 - 300716*b5 + 904241*b4 + 290439*b3 - 319851*b2 + 431439*b1 + 2878206) / 12 $$\nu^{10}$$ $$=$$ $$( - 1245547 \beta_{15} - 1144488 \beta_{14} + 95411 \beta_{13} + 1245744 \beta_{12} + 2473284 \beta_{11} - 4214378 \beta_{10} + 3045589 \beta_{9} + 410735 \beta_{8} - 661784 \beta_{7} + \cdots - 6713352 ) / 12$$ (-1245547*b15 - 1144488*b14 + 95411*b13 + 1245744*b12 + 2473284*b11 - 4214378*b10 + 3045589*b9 + 410735*b8 - 661784*b7 + 2100980*b6 - 2657456*b5 + 6087152*b4 - 924384*b3 + 938016*b2 - 7665195*b1 - 6713352) / 12 $$\nu^{11}$$ $$=$$ $$( - 3766714 \beta_{15} - 9997614 \beta_{14} + 6779426 \beta_{13} - 3036957 \beta_{12} + 6542943 \beta_{11} + 10534555 \beta_{10} + 11151253 \beta_{9} + 17603231 \beta_{8} + \cdots - 73078230 ) / 12$$ (-3766714*b15 - 9997614*b14 + 6779426*b13 - 3036957*b12 + 6542943*b11 + 10534555*b10 + 11151253*b9 + 17603231*b8 + 1095814*b7 - 7955410*b6 + 4112890*b5 - 16908991*b4 - 7864197*b3 + 8223897*b2 - 21684786*b1 - 73078230) / 12 $$\nu^{12}$$ $$=$$ $$( 12045634 \beta_{15} + 8786955 \beta_{14} + 3666955 \beta_{13} - 15943449 \beta_{12} - 27259074 \beta_{11} + 56069540 \beta_{10} - 31399507 \beta_{9} + 3274735 \beta_{8} + \cdots + 44325837 ) / 6$$ (12045634*b15 + 8786955*b14 + 3666955*b13 - 15943449*b12 - 27259074*b11 + 56069540*b10 - 31399507*b9 + 3274735*b8 + 7192238*b7 - 28946339*b6 + 34909931*b5 - 85949792*b4 + 7439175*b3 - 6871764*b2 + 85300266*b1 + 44325837) / 6 $$\nu^{13}$$ $$=$$ $$( 114248852 \beta_{15} + 254014020 \beta_{14} - 158151874 \beta_{13} + 57387759 \beta_{12} - 221707941 \beta_{11} - 144608033 \beta_{10} - 338639183 \beta_{9} + \cdots + 1821064524 ) / 12$$ (114248852*b15 + 254014020*b14 - 158151874*b13 + 57387759*b12 - 221707941*b11 - 144608033*b10 - 338639183*b9 - 426549553*b8 - 9984236*b7 + 150806456*b6 - 21289250*b5 + 255360275*b4 + 206303955*b3 - 204360897*b2 + 764757972*b1 + 1821064524) / 12 $$\nu^{14}$$ $$=$$ $$( - 436459717 \beta_{15} - 180750738 \beta_{14} - 388881655 \beta_{13} + 824613468 \beta_{12} + 1120617078 \beta_{11} - 2864827628 \beta_{10} + 1192678201 \beta_{9} + \cdots - 387024930 ) / 12$$ (-436459717*b15 - 180750738*b14 - 388881655*b13 + 824613468*b12 + 1120617078*b11 - 2864827628*b10 + 1192678201*b9 - 569575879*b8 - 312438986*b7 + 1568012156*b6 - 1745259920*b5 + 4612914080*b4 - 164515434*b3 + 128280642*b2 - 3454007043*b1 - 387024930) / 12 $$\nu^{15}$$ $$=$$ $$( - 3176368585 \beta_{15} - 6304996872 \beta_{14} + 3451153529 \beta_{13} - 864267417 \beta_{12} + 6713172693 \beta_{11} + 715237729 \beta_{10} + 9540516082 \beta_{9} + \cdots - 44352101334 ) / 12$$ (-3176368585*b15 - 6304996872*b14 + 3451153529*b13 - 864267417*b12 + 6713172693*b11 + 715237729*b10 + 9540516082*b9 + 9961028114*b8 - 59019596*b7 - 2390965066*b6 - 1389971618*b5 - 1780800109*b4 - 5238916821*b3 + 4953318759*b2 - 23347761285*b1 - 44352101334) / 12

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
769.1
 0.5 + 2.68650i 0.5 − 2.68650i 0.5 − 0.971291i 0.5 + 0.971291i 0.5 + 4.96598i 0.5 − 4.96598i 0.5 + 0.422343i 0.5 − 0.422343i 0.5 + 1.83656i 0.5 − 1.83656i 0.5 + 3.55177i 0.5 − 3.55177i 0.5 + 0.442923i 0.5 − 0.442923i 0.5 + 4.10071i 0.5 − 4.10071i
0 −1.73205 0 −4.59769 1.96501i 0 6.81963 + 1.57881i 0 3.00000 0
769.2 0 −1.73205 0 −4.59769 + 1.96501i 0 6.81963 1.57881i 0 3.00000 0
769.3 0 −1.73205 0 −2.40341 4.38447i 0 −6.94781 0.853218i 0 3.00000 0
769.4 0 −1.73205 0 −2.40341 + 4.38447i 0 −6.94781 + 0.853218i 0 3.00000 0
769.5 0 −1.73205 0 −1.38028 4.80571i 0 −5.24961 + 4.63050i 0 3.00000 0
769.6 0 −1.73205 0 −1.38028 + 4.80571i 0 −5.24961 4.63050i 0 3.00000 0
769.7 0 −1.73205 0 4.91728 0.905717i 0 1.91369 + 6.73333i 0 3.00000 0
769.8 0 −1.73205 0 4.91728 + 0.905717i 0 1.91369 6.73333i 0 3.00000 0
769.9 0 1.73205 0 −4.91728 0.905717i 0 −1.91369 6.73333i 0 3.00000 0
769.10 0 1.73205 0 −4.91728 + 0.905717i 0 −1.91369 + 6.73333i 0 3.00000 0
769.11 0 1.73205 0 1.38028 4.80571i 0 5.24961 4.63050i 0 3.00000 0
769.12 0 1.73205 0 1.38028 + 4.80571i 0 5.24961 + 4.63050i 0 3.00000 0
769.13 0 1.73205 0 2.40341 4.38447i 0 6.94781 + 0.853218i 0 3.00000 0
769.14 0 1.73205 0 2.40341 + 4.38447i 0 6.94781 0.853218i 0 3.00000 0
769.15 0 1.73205 0 4.59769 1.96501i 0 −6.81963 1.57881i 0 3.00000 0
769.16 0 1.73205 0 4.59769 + 1.96501i 0 −6.81963 + 1.57881i 0 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 769.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.3.bd.a 16
4.b odd 2 1 210.3.h.a 16
5.b even 2 1 inner 1680.3.bd.a 16
7.b odd 2 1 inner 1680.3.bd.a 16
12.b even 2 1 630.3.h.e 16
20.d odd 2 1 210.3.h.a 16
20.e even 4 2 1050.3.f.e 16
28.d even 2 1 210.3.h.a 16
35.c odd 2 1 inner 1680.3.bd.a 16
60.h even 2 1 630.3.h.e 16
84.h odd 2 1 630.3.h.e 16
140.c even 2 1 210.3.h.a 16
140.j odd 4 2 1050.3.f.e 16
420.o odd 2 1 630.3.h.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.h.a 16 4.b odd 2 1
210.3.h.a 16 20.d odd 2 1
210.3.h.a 16 28.d even 2 1
210.3.h.a 16 140.c even 2 1
630.3.h.e 16 12.b even 2 1
630.3.h.e 16 60.h even 2 1
630.3.h.e 16 84.h odd 2 1
630.3.h.e 16 420.o odd 2 1
1050.3.f.e 16 20.e even 4 2
1050.3.f.e 16 140.j odd 4 2
1680.3.bd.a 16 1.a even 1 1 trivial
1680.3.bd.a 16 5.b even 2 1 inner
1680.3.bd.a 16 7.b odd 2 1 inner
1680.3.bd.a 16 35.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} + 24T_{11}^{3} + 28T_{11}^{2} - 1776T_{11} - 4844$$ acting on $$S_{3}^{\mathrm{new}}(1680, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$(T^{2} - 3)^{8}$$
$5$ $$T^{16} - 12 T^{14} + \cdots + 152587890625$$
$7$ $$T^{16} - 112 T^{14} + \cdots + 33232930569601$$
$11$ $$(T^{4} + 24 T^{3} + 28 T^{2} - 1776 T - 4844)^{4}$$
$13$ $$(T^{8} - 1044 T^{6} + 313732 T^{4} + \cdots + 586802176)^{2}$$
$17$ $$(T^{8} - 996 T^{6} + 270436 T^{4} + \cdots + 338560000)^{2}$$
$19$ $$(T^{8} + 1796 T^{6} + \cdots + 2149991424)^{2}$$
$23$ $$(T^{8} + 2932 T^{6} + \cdots + 11186869824)^{2}$$
$29$ $$(T^{4} - 16 T^{3} - 2088 T^{2} + \cdots + 19600)^{4}$$
$31$ $$(T^{8} + 3740 T^{6} + 1609684 T^{4} + \cdots + 747256896)^{2}$$
$37$ $$(T^{8} + 8536 T^{6} + \cdots + 3617786594304)^{2}$$
$41$ $$(T^{8} + 932 T^{6} + 280612 T^{4} + \cdots + 1141899264)^{2}$$
$43$ $$(T^{8} + 12880 T^{6} + \cdots + 9151544623104)^{2}$$
$47$ $$(T^{8} - 10904 T^{6} + \cdots + 14545741676544)^{2}$$
$53$ $$(T^{8} + 15604 T^{6} + \cdots + 38389325511744)^{2}$$
$59$ $$(T^{8} + 16112 T^{6} + \cdots + 203235065856)^{2}$$
$61$ $$(T^{8} + 13824 T^{6} + \cdots + 72666906624)^{2}$$
$67$ $$(T^{8} + 17224 T^{6} + \cdots + 404129746944)^{2}$$
$71$ $$(T^{4} - 96 T^{3} - 6116 T^{2} + \cdots - 18715436)^{4}$$
$73$ $$(T^{8} - 24020 T^{6} + \cdots + 105841627140096)^{2}$$
$79$ $$(T^{4} - 152 T^{3} + 3952 T^{2} + \cdots - 12612096)^{4}$$
$83$ $$(T^{8} - 15480 T^{6} + \cdots + 13129665286144)^{2}$$
$89$ $$(T^{8} + 24860 T^{6} + \cdots + 135150669186624)^{2}$$
$97$ $$(T^{8} - 20468 T^{6} + \cdots + 5898136817664)^{2}$$