# Properties

 Label 1840.2.e.g Level $1840$ Weight $2$ Character orbit 1840.e Analytic conductor $14.692$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{14} - 2 x^{11} + 39 x^{10} - 10 x^{9} + 2 x^{8} - 26 x^{7} + 297 x^{6} - 116 x^{5} + 24 x^{4} + \cdots + 8$$ x^14 - 2*x^11 + 39*x^10 - 10*x^9 + 2*x^8 - 26*x^7 + 297*x^6 - 116*x^5 + 24*x^4 - 20*x^3 + 64*x^2 - 32*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 920) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{8} q^{3} - \beta_{5} q^{5} + ( - \beta_{13} + \beta_{7}) q^{7} + ( - \beta_{12} - \beta_{10} + \beta_{5} + \cdots + 1) q^{9}+O(q^{10})$$ q - b8 * q^3 - b5 * q^5 + (-b13 + b7) * q^7 + (-b12 - b10 + b5 - b3 - b2 + 1) * q^9 $$q - \beta_{8} q^{3} - \beta_{5} q^{5} + ( - \beta_{13} + \beta_{7}) q^{7} + ( - \beta_{12} - \beta_{10} + \beta_{5} + \cdots + 1) q^{9}+ \cdots + ( - 4 \beta_{12} - 4 \beta_{10} + \cdots - 2) q^{99}+O(q^{100})$$ q - b8 * q^3 - b5 * q^5 + (-b13 + b7) * q^7 + (-b12 - b10 + b5 - b3 - b2 + 1) * q^9 + (b12 + b10 + b6 - b5 + b3 + b2) * q^11 + (b13 - b8 - b5) * q^13 + (-b13 - b12 - b10 - b8 + b5 + b4 - b3 - b2 + b1 + 2) * q^15 + (-b13 + b8 + 2*b7 - b5 - b4) * q^17 + (2*b12 + b10 - b9 + b6 - b5 + b3 - b2 - 2*b1 - 2) * q^19 + (-b12 - b10 + b5 - b3 + b2 + b1) * q^21 + b4 * q^23 + (b13 + b12 + b11 + b10 - b9 + b6 + b4 - b2 + b1) * q^25 + (b13 - b11 - b10 + b9 + b8 - b7 + b5 - b4 + b3) * q^27 + (3*b12 + b10 - 2*b9 - 2*b2 - 2*b1 + 1) * q^29 + (b12 - b9 - 3*b6 - b5 + b3 + 2*b2 - b1 - 1) * q^31 + (-2*b13 + b11 + b10 - b9 - 5*b8 - 2*b3) * q^33 + (b11 - b10 - b8 - b6 + b4 + b3 - b2 - b1) * q^35 + (-b11 - 2*b10 + 2*b9 + 2*b8 + b4) * q^37 + (-2*b12 - 2*b10 + b6 + b5 - b3 - 3*b2 + b1) * q^39 + (b6 + 2*b5 - 2*b3 + b1 - 1) * q^41 + (b13 + b10 - b9 + b8 + b7 - b5 - b4 + b3) * q^43 + (-b12 - b11 - 3*b10 + 2*b9 + 2*b8 - 2*b7 - b6 + b5 + b4 - b3 + b2 - 2) * q^45 + (3*b11 - b8 + b7 + b5 + b4 + 2*b3) * q^47 + (-b12 + b9 + b2 + 2) * q^49 + (-2*b12 - 2*b10 + b6 + b5 - b3 + 3*b1 + 4) * q^51 + (3*b13 - 3*b11 - b10 + b9 + b8 - b7 - 2*b5 - 2*b4) * q^53 + (b13 + b12 + 2*b10 - 2*b9 - b8 + 3*b7 + b6 - 3*b5 + 2*b3 + b1 + 3) * q^55 + (-4*b13 + 2*b11 - b8 + b7 + b5 + b4 - 2*b3) * q^57 + (-2*b12 - b10 + b9 - 3*b6 - 2*b5 + 2*b3 + 5*b2 - b1 - 5) * q^59 + (5*b12 + b10 - 4*b9 - b6 - 2*b5 + 2*b3 + b2 - 2*b1 - 2) * q^61 + (-b11 + 3*b8 + b7 - 4*b4 + b3) * q^63 + (-b13 - b12 - b9 + b7 + b5 + 2*b4 - 2*b3 - 2*b2 + 2*b1) * q^65 + (2*b13 - 3*b11 - 2*b10 + 2*b9 - 2*b8 - 4*b7 + 3*b4 - 2*b3) * q^67 + b2 * q^69 + (-b10 - b9 + 3*b6 + b5 - b3 - 2*b2 + b1 + 5) * q^71 + (-b11 - b10 + b9 + b8 + b7 - 3*b4 + b3) * q^73 + (-b13 - b12 + b11 - 3*b10 + 3*b9 + b8 - 2*b7 + b6 + 2*b5 + b4 - b2 + b1 + 3) * q^75 + (-b13 + 2*b11 - b10 + b9 - 2*b8 + 2*b4 - b3) * q^77 + (4*b12 + b10 - 3*b9 - 2*b6 - b5 + b3 - b2 - b1 - 6) * q^79 + (b12 + b10 - 4*b1 - 3) * q^81 + (-b13 - b11 - b10 + b9 + 5*b8 + b7 + b5 - 4*b4 + b3) * q^83 + (b13 + b12 + 2*b11 - 2*b9 + b7 - 2*b6 - b5 + 2*b4 + 2*b3 - 2*b2 - 2*b1 - 4) * q^85 + (-2*b13 + 3*b11 - 2*b10 + 2*b9 - b8 + b7 + b5 + b4) * q^87 + (-2*b12 - b10 + b9 - b5 + b3 + 3*b2 - b1 + 2) * q^89 + (b12 - b10 - 2*b9 - b6 + b5 - b3 - b2 + 2) * q^91 + (3*b13 + b11 + 2*b10 - 2*b9 - 3*b8 + 3*b7 - 4*b5 - b4 + 2*b3) * q^93 + (4*b13 + 3*b12 - 3*b11 + b10 - 2*b9 - b8 - b7 - 2*b5 + 3*b4 + b3 - b2 + 2) * q^95 + (5*b13 - 3*b11 + b8 - 4*b7 + b3) * q^97 + (-4*b12 - 4*b10 + b6 + 4*b5 - 4*b3 - 4*b2 + 3*b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q + 2 q^{5} - 4 q^{9}+O(q^{10})$$ 14 * q + 2 * q^5 - 4 * q^9 $$14 q + 2 q^{5} - 4 q^{9} + 14 q^{11} + 6 q^{15} - 14 q^{19} - 12 q^{21} - 14 q^{25} + 22 q^{29} + 20 q^{31} + 2 q^{35} - 48 q^{39} - 32 q^{41} - 26 q^{45} + 34 q^{49} + 14 q^{51} + 38 q^{55} - 22 q^{59} + 10 q^{61} - 38 q^{65} + 6 q^{69} + 28 q^{71} + 24 q^{75} - 64 q^{79} - 10 q^{81} - 50 q^{85} + 48 q^{89} + 14 q^{91} + 30 q^{95} - 122 q^{99}+O(q^{100})$$ 14 * q + 2 * q^5 - 4 * q^9 + 14 * q^11 + 6 * q^15 - 14 * q^19 - 12 * q^21 - 14 * q^25 + 22 * q^29 + 20 * q^31 + 2 * q^35 - 48 * q^39 - 32 * q^41 - 26 * q^45 + 34 * q^49 + 14 * q^51 + 38 * q^55 - 22 * q^59 + 10 * q^61 - 38 * q^65 + 6 * q^69 + 28 * q^71 + 24 * q^75 - 64 * q^79 - 10 * q^81 - 50 * q^85 + 48 * q^89 + 14 * q^91 + 30 * q^95 - 122 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 2 x^{11} + 39 x^{10} - 10 x^{9} + 2 x^{8} - 26 x^{7} + 297 x^{6} - 116 x^{5} + 24 x^{4} + \cdots + 8$$ :

 $$\beta_{1}$$ $$=$$ $$( 224809861 \nu^{13} + 40698207136 \nu^{12} + 13894530424 \nu^{11} + 2084956408 \nu^{10} + \cdots + 1689305356572 ) / 732246302876$$ (224809861*v^13 + 40698207136*v^12 + 13894530424*v^11 + 2084956408*v^10 - 68154663521*v^9 + 1559311271490*v^8 + 134953581114*v^7 + 28178713856*v^6 - 814144658387*v^5 + 11737106209660*v^4 - 402160406420*v^3 + 22675714854*v^2 + 32069661208*v + 1689305356572) / 732246302876 $$\beta_{2}$$ $$=$$ $$( 9599816455 \nu^{13} - 39917569898 \nu^{12} - 9824626316 \nu^{11} - 19980270148 \nu^{10} + \cdots - 2008645915756 ) / 732246302876$$ (9599816455*v^13 - 39917569898*v^12 - 9824626316*v^11 - 19980270148*v^10 + 450158077433*v^9 - 1634888076832*v^8 + 36371917978*v^7 - 253853562284*v^6 + 3717676805391*v^5 - 12743422120058*v^4 + 1957301556084*v^3 - 143702096254*v^2 - 142401498584*v - 2008645915756) / 732246302876 $$\beta_{3}$$ $$=$$ $$( - 14185734445 \nu^{13} + 33377065138 \nu^{12} + 22691432254 \nu^{11} + 49350875674 \nu^{10} + \cdots + 2294924313380 ) / 732246302876$$ (-14185734445*v^13 + 33377065138*v^12 + 22691432254*v^11 + 49350875674*v^10 - 612714646635*v^9 + 1401214875276*v^8 + 503279347384*v^7 + 1031632771786*v^6 - 4962097019689*v^5 + 11022845347922*v^4 + 2505098929650*v^3 + 4594222559616*v^2 - 1305274242300*v + 2294924313380) / 732246302876 $$\beta_{4}$$ $$=$$ $$( 19958784949 \nu^{13} + 4912313158 \nu^{12} + 390318619 \nu^{11} - 37882617844 \nu^{10} + \cdots - 327723885618 ) / 366123151438$$ (19958784949*v^13 + 4912313158*v^12 + 390318619*v^11 - 37882617844*v^10 + 769444956141*v^9 - 8586142534*v^8 + 2129167227*v^7 - 433265659128*v^6 + 5814921705639*v^5 - 863452980582*v^4 - 24147116423*v^3 + 378394875852*v^2 + 1216849046036*v - 327723885618) / 366123151438 $$\beta_{5}$$ $$=$$ $$( - 40039329007 \nu^{13} - 48607321956 \nu^{12} - 16903078854 \nu^{11} + 91091509614 \nu^{10} + \cdots - 1596402425868 ) / 732246302876$$ (-40039329007*v^13 - 48607321956*v^12 - 16903078854*v^11 + 91091509614*v^10 - 1462932942489*v^9 - 1462473090366*v^8 - 274689612780*v^7 + 1553791371930*v^6 - 10634500106239*v^5 - 9375688488996*v^4 - 615006277590*v^3 + 4857036973804*v^2 - 1092034671500*v - 1596402425868) / 732246302876 $$\beta_{6}$$ $$=$$ $$( 51158604007 \nu^{13} + 95455044155 \nu^{12} + 57732941236 \nu^{11} - 86278427068 \nu^{10} + \cdots + 982823726512 ) / 732246302876$$ (51158604007*v^13 + 95455044155*v^12 + 57732941236*v^11 - 86278427068*v^10 + 1799215942067*v^9 + 3072231013943*v^8 + 1361220760372*v^7 - 1082405618534*v^6 + 12498302486205*v^5 + 20232610940731*v^4 + 6809297955588*v^3 - 557261024458*v^2 - 472073315000*v + 982823726512) / 732246302876 $$\beta_{7}$$ $$=$$ $$( - 61621117035 \nu^{13} - 58787127771 \nu^{12} - 866985826 \nu^{11} + 134629395734 \nu^{10} + \cdots - 1737381486820 ) / 732246302876$$ (-61621117035*v^13 - 58787127771*v^12 - 866985826*v^11 + 134629395734*v^10 - 2271165378531*v^9 - 1671410135475*v^8 + 385087447514*v^7 + 1922577441112*v^6 - 16317016456265*v^5 - 10243935308455*v^4 + 4086990094642*v^3 + 3333018633676*v^2 - 65148969984*v - 1737381486820) / 732246302876 $$\beta_{8}$$ $$=$$ $$( - 66772094683 \nu^{13} - 15046471791 \nu^{12} - 4521994539 \nu^{11} + 135188822801 \nu^{10} + \cdots + 1230815216570 ) / 366123151438$$ (-66772094683*v^13 - 15046471791*v^12 - 4521994539*v^11 + 135188822801*v^10 - 2575176731663*v^9 + 89277697645*v^8 - 162746449503*v^7 + 1819608044077*v^6 - 19510883885937*v^5 + 3382407351091*v^4 - 1242235247009*v^3 + 2137159584915*v^2 - 4712322126264*v + 1230815216570) / 366123151438 $$\beta_{9}$$ $$=$$ $$( 134719081377 \nu^{13} + 166717918 \nu^{12} - 18595581648 \nu^{11} - 277803595150 \nu^{10} + \cdots - 2143125728988 ) / 732246302876$$ (134719081377*v^13 + 166717918*v^12 - 18595581648*v^11 - 277803595150*v^10 + 5252053841299*v^9 - 1301055484020*v^8 - 460738908234*v^7 - 3645608595402*v^6 + 39984967995781*v^5 - 14884213113334*v^4 - 2574515483268*v^3 - 3062961226104*v^2 + 8601416001444*v - 2143125728988) / 732246302876 $$\beta_{10}$$ $$=$$ $$( - 159011401463 \nu^{13} - 74011296796 \nu^{12} - 17491051676 \nu^{11} + 317230204266 \nu^{10} + \cdots + 2488082785940 ) / 732246302876$$ (-159011401463*v^13 - 74011296796*v^12 - 17491051676*v^11 + 317230204266*v^10 - 6056933221517*v^9 - 1259279988702*v^8 - 257699180354*v^7 + 4139003457850*v^6 - 45390689597815*v^5 - 2875620487424*v^4 - 394657434132*v^3 + 3312052402844*v^2 - 8398081464788*v + 2488082785940) / 732246302876 $$\beta_{11}$$ $$=$$ $$( 161362783524 \nu^{13} + 45302004725 \nu^{12} - 17266584944 \nu^{11} - 359941008954 \nu^{10} + \cdots - 2076944162144 ) / 732246302876$$ (161362783524*v^13 + 45302004725*v^12 - 17266584944*v^11 - 359941008954*v^10 + 6182243364690*v^9 + 182163846443*v^8 - 731148740058*v^7 - 5300077616756*v^6 + 46296345417282*v^5 - 4899429442643*v^4 - 5587980665236*v^3 - 9490182700870*v^2 + 7131200437916*v - 2076944162144) / 732246302876 $$\beta_{12}$$ $$=$$ $$( 189780871574 \nu^{13} + 115971282613 \nu^{12} + 47907054048 \nu^{11} - 369867084278 \nu^{10} + \cdots - 88794449364 ) / 732246302876$$ (189780871574*v^13 + 115971282613*v^12 + 47907054048*v^11 - 369867084278*v^10 + 7164617072456*v^9 + 2529556762223*v^8 + 1049309031078*v^7 - 4799923558176*v^6 + 53149974193276*v^5 + 11212902114921*v^4 + 4612113196920*v^3 - 3654530344182*v^2 + 8104580101084*v - 88794449364) / 732246302876 $$\beta_{13}$$ $$=$$ $$( - 225258249983 \nu^{13} - 97924060431 \nu^{12} - 6621708034 \nu^{11} + 462876963284 \nu^{10} + \cdots + 1060356158372 ) / 732246302876$$ (-225258249983*v^13 - 97924060431*v^12 - 6621708034*v^11 + 462876963284*v^10 - 8578405340227*v^9 - 1551259260459*v^8 + 226661713146*v^7 + 6202649999094*v^6 - 64065095492413*v^5 - 2758530555507*v^4 + 3205954983134*v^3 + 6729521670402*v^2 - 10569330546208*v + 1060356158372) / 732246302876
 $$\nu$$ $$=$$ $$( \beta_{13} - \beta_{8} - \beta_{7} + \beta_{4} - \beta_{2} - \beta_1 ) / 2$$ (b13 - b8 - b7 + b4 - b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{10} - \beta_{9} - \beta_{5} + 6\beta_{4} - \beta_{3} ) / 2$$ (b10 - b9 - b5 + 6*b4 - b3) / 2 $$\nu^{3}$$ $$=$$ $$( 5 \beta_{13} - \beta_{10} + \beta_{9} - 3 \beta_{8} - 5 \beta_{7} + \beta_{5} + 3 \beta_{4} - \beta_{3} + \cdots + 2 ) / 2$$ (5*b13 - b10 + b9 - 3*b8 - 5*b7 + b5 + 3*b4 - b3 + 3*b2 + 5*b1 + 2) / 2 $$\nu^{4}$$ $$=$$ $$( -2\beta_{12} + 5\beta_{10} + 7\beta_{9} + 2\beta_{6} - 5\beta_{5} + 5\beta_{3} - 26 ) / 2$$ (-2*b12 + 5*b10 + 7*b9 + 2*b6 - 5*b5 + 5*b3 - 26) / 2 $$\nu^{5}$$ $$=$$ $$( - 26 \beta_{13} + 2 \beta_{12} + 9 \beta_{10} - 9 \beta_{9} + 12 \beta_{8} + 24 \beta_{7} + 9 \beta_{5} + \cdots + 14 ) / 2$$ (-26*b13 + 2*b12 + 9*b10 - 9*b9 + 12*b8 + 24*b7 + 9*b5 - 10*b4 - 9*b3 + 12*b2 + 24*b1 + 14) / 2 $$\nu^{6}$$ $$=$$ $$10 \beta_{13} + 9 \beta_{11} - 13 \beta_{10} + 13 \beta_{9} - \beta_{8} - \beta_{7} + \cdots + 21 \beta_{3}$$ 10*b13 + 9*b11 - 13*b10 + 13*b9 - b8 - b7 + 12*b5 - 61*b4 + 21*b3 $$\nu^{7}$$ $$=$$ $$( - 139 \beta_{13} - 20 \beta_{12} - 2 \beta_{11} + 44 \beta_{10} - 40 \beta_{9} + 53 \beta_{8} + \cdots - 90 ) / 2$$ (-139*b13 - 20*b12 - 2*b11 + 44*b10 - 40*b9 + 53*b8 + 119*b7 + 2*b6 - 44*b5 - 29*b4 + 40*b3 - 53*b2 - 119*b1 - 90) / 2 $$\nu^{8}$$ $$=$$ $$( 124 \beta_{12} - 117 \beta_{10} - 241 \beta_{9} - 120 \beta_{6} + 145 \beta_{5} - 145 \beta_{3} + \cdots + 630 ) / 2$$ (124*b12 - 117*b10 - 241*b9 - 120*b6 + 145*b5 - 145*b3 + 22*b2 + 30*b1 + 630) / 2 $$\nu^{9}$$ $$=$$ $$( 757 \beta_{13} - 148 \beta_{12} + 28 \beta_{11} - 361 \beta_{10} + 419 \beta_{9} - 251 \beta_{8} + \cdots - 564 ) / 2$$ (757*b13 - 148*b12 + 28*b11 - 361*b10 + 419*b9 - 251*b8 - 609*b7 + 28*b6 - 361*b5 + 45*b4 + 419*b3 - 251*b2 - 609*b1 - 564) / 2 $$\nu^{10}$$ $$=$$ $$( - 1072 \beta_{13} - 722 \beta_{11} + 843 \beta_{10} - 843 \beta_{9} + 178 \beta_{8} + \cdots - 1359 \beta_{3} ) / 2$$ (-1072*b13 - 722*b11 + 843*b10 - 843*b9 + 178*b8 + 294*b7 - 581*b5 + 3056*b4 - 1359*b3) / 2 $$\nu^{11}$$ $$=$$ $$( 4180 \beta_{13} + 984 \beta_{12} + 262 \beta_{11} - 1653 \beta_{10} + 1097 \beta_{9} - 1256 \beta_{8} + \cdots + 3484 ) / 2$$ (4180*b13 + 984*b12 + 262*b11 - 1653*b10 + 1097*b9 - 1256*b8 - 3196*b7 - 262*b6 + 1653*b5 - 288*b4 - 1097*b3 + 1256*b2 + 3196*b1 + 3484) / 2 $$\nu^{12}$$ $$=$$ $$- 2343 \beta_{12} + 1467 \beta_{10} + 3810 \beta_{9} + 2081 \beta_{6} - 2499 \beta_{5} + 2499 \beta_{3} + \cdots - 9188$$ -2343*b12 + 1467*b10 + 3810*b9 + 2081*b6 - 2499*b5 + 2499*b3 - 641*b2 - 1197*b1 - 9188 $$\nu^{13}$$ $$=$$ $$( - 23331 \beta_{13} + 6226 \beta_{12} - 2064 \beta_{11} + 11782 \beta_{10} - 16240 \beta_{9} + \cdots + 21310 ) / 2$$ (-23331*b13 + 6226*b12 - 2064*b11 + 11782*b10 - 16240*b9 + 6551*b8 + 17105*b7 - 2064*b6 + 11782*b5 + 4205*b4 - 16240*b3 + 6551*b2 + 17105*b1 + 21310) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$$.

 $$n$$ $$737$$ $$1151$$ $$1201$$ $$1381$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
369.1
 −0.503254 − 0.503254i 0.285770 − 0.285770i −1.27121 − 1.27121i −1.71470 − 1.71470i 0.416087 − 0.416087i 1.55369 + 1.55369i 1.23362 + 1.23362i 1.23362 − 1.23362i 1.55369 − 1.55369i 0.416087 + 0.416087i −1.71470 + 1.71470i −1.27121 + 1.27121i 0.285770 + 0.285770i −0.503254 + 0.503254i
0 2.98707i 0 0.274289 2.21918i 0 0.980560i 0 −5.92257 0
369.2 0 2.49931i 0 2.11714 + 0.719533i 0 2.92777i 0 −3.24657 0
369.3 0 1.78665i 0 1.46089 + 1.69287i 0 1.75578i 0 −0.192116 0
369.4 0 1.58319i 0 −2.09277 + 0.787606i 0 2.84620i 0 0.493499 0
369.5 0 1.40334i 0 0.466981 + 2.18676i 0 1.57117i 0 1.03063 0
369.6 0 0.356372i 0 0.376266 + 2.20418i 0 2.46376i 0 2.87300 0
369.7 0 0.189375i 0 −1.60280 1.55918i 0 1.65661i 0 2.96414 0
369.8 0 0.189375i 0 −1.60280 + 1.55918i 0 1.65661i 0 2.96414 0
369.9 0 0.356372i 0 0.376266 2.20418i 0 2.46376i 0 2.87300 0
369.10 0 1.40334i 0 0.466981 2.18676i 0 1.57117i 0 1.03063 0
369.11 0 1.58319i 0 −2.09277 0.787606i 0 2.84620i 0 0.493499 0
369.12 0 1.78665i 0 1.46089 1.69287i 0 1.75578i 0 −0.192116 0
369.13 0 2.49931i 0 2.11714 0.719533i 0 2.92777i 0 −3.24657 0
369.14 0 2.98707i 0 0.274289 + 2.21918i 0 0.980560i 0 −5.92257 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 369.14 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.e.g 14
4.b odd 2 1 920.2.e.b 14
5.b even 2 1 inner 1840.2.e.g 14
5.c odd 4 1 9200.2.a.cz 7
5.c odd 4 1 9200.2.a.dc 7
20.d odd 2 1 920.2.e.b 14
20.e even 4 1 4600.2.a.bh 7
20.e even 4 1 4600.2.a.bi 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.e.b 14 4.b odd 2 1
920.2.e.b 14 20.d odd 2 1
1840.2.e.g 14 1.a even 1 1 trivial
1840.2.e.g 14 5.b even 2 1 inner
4600.2.a.bh 7 20.e even 4 1
4600.2.a.bi 7 20.e even 4 1
9200.2.a.cz 7 5.c odd 4 1
9200.2.a.dc 7 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1840, [\chi])$$:

 $$T_{3}^{14} + 23T_{3}^{12} + 195T_{3}^{10} + 766T_{3}^{8} + 1431T_{3}^{6} + 1095T_{3}^{4} + 149T_{3}^{2} + 4$$ T3^14 + 23*T3^12 + 195*T3^10 + 766*T3^8 + 1431*T3^6 + 1095*T3^4 + 149*T3^2 + 4 $$T_{7}^{14} + 32T_{7}^{12} + 412T_{7}^{10} + 2745T_{7}^{8} + 10156T_{7}^{6} + 20760T_{7}^{4} + 21488T_{7}^{2} + 8464$$ T7^14 + 32*T7^12 + 412*T7^10 + 2745*T7^8 + 10156*T7^6 + 20760*T7^4 + 21488*T7^2 + 8464

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14}$$
$3$ $$T^{14} + 23 T^{12} + \cdots + 4$$
$5$ $$T^{14} - 2 T^{13} + \cdots + 78125$$
$7$ $$T^{14} + 32 T^{12} + \cdots + 8464$$
$11$ $$(T^{7} - 7 T^{6} + \cdots - 128)^{2}$$
$13$ $$T^{14} + 75 T^{12} + \cdots + 4$$
$17$ $$T^{14} + 116 T^{12} + \cdots + 2704$$
$19$ $$(T^{7} + 7 T^{6} + \cdots + 1936)^{2}$$
$23$ $$(T^{2} + 1)^{7}$$
$29$ $$(T^{7} - 11 T^{6} + \cdots - 9244)^{2}$$
$31$ $$(T^{7} - 10 T^{6} + \cdots + 225251)^{2}$$
$37$ $$T^{14} + \cdots + 1834580224$$
$41$ $$(T^{7} + 16 T^{6} + \cdots + 110153)^{2}$$
$43$ $$T^{14} + 96 T^{12} + \cdots + 4096$$
$47$ $$T^{14} + 392 T^{12} + \cdots + 33085504$$
$53$ $$T^{14} + \cdots + 24445947904$$
$59$ $$(T^{7} + 11 T^{6} + \cdots + 486592)^{2}$$
$61$ $$(T^{7} - 5 T^{6} + \cdots + 2669336)^{2}$$
$67$ $$T^{14} + \cdots + 214213460224$$
$71$ $$(T^{7} - 14 T^{6} + \cdots - 632317)^{2}$$
$73$ $$T^{14} + 160 T^{12} + \cdots + 256$$
$79$ $$(T^{7} + 32 T^{6} + \cdots + 473312)^{2}$$
$83$ $$T^{14} + \cdots + 143616256$$
$89$ $$(T^{7} - 24 T^{6} + \cdots - 311456)^{2}$$
$97$ $$T^{14} + \cdots + 16312357788736$$