# Properties

 Label 154.2.i.a Level $154$ Weight $2$ Character orbit 154.i Analytic conductor $1.230$ Analytic rank $0$ Dimension $16$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [154,2,Mod(87,154)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(154, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1, 3]))

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

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

chi := DirichletCharacter("154.87");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$154 = 2 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 154.i (of order $$6$$, degree $$2$$, 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: $$1.22969619113$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1$$ x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 34*x^12 + 18*x^11 - 72*x^10 + 132*x^9 - 93*x^8 - 102*x^7 + 144*x^6 - 432*x^5 + 502*x^4 + 288*x^3 + 72*x^2 + 12*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{13} + \beta_{4} - \beta_{3}) q^{3} + \beta_{10} q^{4} + ( - \beta_{10} - \beta_{8}) q^{5} + (\beta_{14} - \beta_{6}) q^{6} + (\beta_{15} + \beta_{14} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{13} - 4 \beta_{10} + \cdots + 3) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b13 + b4 - b3) * q^3 + b10 * q^4 + (-b10 - b8) * q^5 + (b14 - b6) * q^6 + (b15 + b14 - 2*b11 + b9 - b6 - b5 - b2 - b1) * q^7 - b11 * q^8 + (-b13 - 4*b10 - b8 + 2*b7 + b3 + 3) * q^9 $$q + \beta_1 q^{2} + (\beta_{13} + \beta_{4} - \beta_{3}) q^{3} + \beta_{10} q^{4} + ( - \beta_{10} - \beta_{8}) q^{5} + (\beta_{14} - \beta_{6}) q^{6} + (\beta_{15} + \beta_{14} + \cdots - \beta_1) q^{7}+ \cdots + ( - 6 \beta_{15} + \beta_{14} + \cdots + 4) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b13 + b4 - b3) * q^3 + b10 * q^4 + (-b10 - b8) * q^5 + (b14 - b6) * q^6 + (b15 + b14 - 2*b11 + b9 - b6 - b5 - b2 - b1) * q^7 - b11 * q^8 + (-b13 - 4*b10 - b8 + 2*b7 + b3 + 3) * q^9 + (b11 + b5) * q^10 + (b14 + b13 - b11 - b6 + b4 + b3 - b2 - b1) * q^11 + (b12 + b4) * q^12 + (-b15 - b14 - b11 + b6 + b2 - 3*b1) * q^13 + (b13 + b12 - b7 + b4 + b3 - 1) * q^14 + (-3*b13 - 3*b12 - 3*b4 + 1) * q^15 + (b10 - 1) * q^16 + (-3*b15 - 2*b14 - 2*b11 + b6 - b5 - b2 - 2*b1) * q^17 + (-b14 + 4*b11 - 2*b9 + b6 + b5 + b2 + 3*b1) * q^18 + (b15 - b14 - 2*b11 + 2*b9 - b6 - 2*b5) * q^19 + (-2*b10 - b8 + b7 + 1) * q^20 + (3*b15 + 3*b11 - 2*b9 + b5 + 2*b2 + 4*b1) * q^21 + (b14 + b13 + b12 + b6 + b4 - 1) * q^22 + (b13 + b12 + b10 + b8 - 2*b7 - b4 - b3) * q^23 + (b15 - b6 + b2 + b1) * q^24 + (b12 + b10 + 2*b8 - b7 + b4 + b3 - 1) * q^25 + (-b13 - b12 - b10 - b4 - b3 - 1) * q^26 + (b13 - b12 + 8*b10 + 2*b8 - 2*b7 - 3*b4 - 4*b3 - 4) * q^27 + (b15 + b14 + b9) * q^28 + (-b15 + b14 - b11 + b6 - b2 - b1) * q^29 + (-3*b15 - 3*b14 + 3*b6 - 2*b1) * q^30 + (-3*b13 - 2*b12 - 2*b10 + 3*b7 - b4 + b3 + 1) * q^31 + (-b11 - b1) * q^32 + (b15 - b14 + b13 + 2*b12 + 6*b11 - b10 - 2*b9 + b8 + 2*b5 + 2*b4 + b3 + b2 + 4*b1 - 2) * q^33 + (b13 - b12 + 4*b10 + b8 - b7 - 2*b4 - 3*b3 - 2) * q^34 + (-b15 + 5*b11 - b9 + b6 + b5 - 2*b2 + 2*b1) * q^35 + (-b13 - b12 + b8 + b7 - b4 + 2) * q^36 + (b12 + 3*b10 - b4 - 3) * q^37 + (b12 + b10 - 2*b7 - b4 + b3) * q^38 + (b15 - 3*b14 - 2*b11 + 2*b9 + 2*b6 - b5 + 2*b2) * q^39 + (2*b11 - b9 + b5 + b1) * q^40 + (-2*b15 + b14 + b11 - b6 + 2*b2) * q^41 + (-2*b13 - b10 + b8 + b7 + 3*b3 + 1) * q^42 + (-2*b15 + 3*b14 + 6*b11 - b9 + 3*b6 + 2*b5 - 2*b2 - b1) * q^43 + (b15 + b14 - b12 - b6 + b4) * q^44 + (b13 + 6*b10 - 3*b7 + b4 - b3 - 9) * q^45 + (b15 + b14 - b11 + 2*b9 - 2*b6 - b5 - 2*b2 + b1) * q^46 + (2*b13 + 2*b12 + 2*b4 + 2*b3) * q^47 + (-b13 + b12 + b3) * q^48 + (-4*b13 + b12 + b10 - b7 - 3*b4 + b3 + 1) * q^49 + (b15 - b11 + b9 - 2*b5 + b2) * q^50 + (3*b15 + 3*b14 + 3*b9 - 4*b6 + 3*b5 + b2 - 3*b1) * q^51 + (-b15 - b14 + b11 - 2*b1) * q^52 + (2*b13 - 7*b10 - 2*b8 + b7 + 2*b4 + 2*b3 + 1) * q^53 + (-b15 + b14 - 8*b11 + 2*b9 - 3*b6 - 2*b5 - 4*b2 - 5*b1) * q^54 + (-b15 - 3*b14 + b13 - b12 - 6*b10 - 2*b8 + 2*b7 + 3*b6 + b4 + b2 - b1 + 3) * q^55 + (-2*b10 - b8 + b4 + b3 + 1) * q^56 + (2*b15 + b14 - 3*b11 + b6 + 2*b2 + 2*b1) * q^57 + (b13 - b12 + b10 + b4 - b3 - 1) * q^58 + (4*b13 + b12 - 2*b10 + 3*b4 - 3*b3 + 4) * q^59 + (-3*b12 + b10 - 3*b4 - 3*b3) * q^60 + (b15 - b14 + 2*b11 - 3*b9 - 2*b6 + 3*b5 - b2 + 2*b1) * q^61 + (-2*b15 - 3*b14 + 2*b11 - 3*b9 + 3*b6 + 2*b2 - b1) * q^62 + (-3*b15 + 3*b14 - 6*b11 - 2*b5 - 4*b2 - b1) * q^63 - q^64 + (b15 + b14 - 2*b9 - 2*b6 - 2*b5 + b2 + 4*b1) * q^65 + (2*b15 + b14 - b13 + b11 - 3*b10 + 2*b7 - b6 - b5 - b4 + b3 + b2 + 4) * q^66 + (-2*b13 - b12 + 4*b10 - 3*b4 - 3*b3) * q^67 + (-b15 + b14 - 4*b11 + b9 - 2*b6 - b5 - 3*b2 - 3*b1) * q^68 + (-6*b10 - 3*b8 + 3*b7 + 2*b4 + 2*b3 + 3) * q^69 + (2*b13 - b12 - 2*b10 + b7 - b3 + 4) * q^70 + (-b13 - b12 - 2*b8 - 2*b7 + b4 - 2*b3 + 3) * q^71 + (-b15 - b14 - b9 + b6 - b5 + b1) * q^72 + (-2*b15 - 2*b14 - 5*b11 - b5 + 2*b1) * q^73 + (b15 - 3*b11 - b6 - b2 - 2*b1) * q^74 + (3*b13 + 4*b12 + 2*b10 + 2*b8 + 4*b4 + 3*b3) * q^75 + (b15 - b11 + 2*b9 - b2 + b1) * q^76 + (3*b15 - 2*b13 + b12 - 5*b11 - 2*b10 + 2*b9 + b7 - 2*b6 + b5 + b4 + 2*b3 + 2*b1 + 4) * q^77 + (-2*b13 - 2*b12 - b8 - b7 - 3*b4 + b3) * q^78 + (-2*b15 - 2*b14 - 3*b9 + b6 - 3*b5 + b2 - 4*b1) * q^79 + (-b10 + b7 + 1) * q^80 + (-2*b13 + 5*b12 - 3*b10 - 2*b8 + b7 + 3*b4 + 3*b3 + 1) * q^81 + (-2*b13 + b12 + b10 + b4 - 2*b3 + 1) * q^82 + (-2*b15 + b14 + 2*b11 + b9 - b6 + 2*b2 + 3*b1) * q^83 + (-2*b14 + b11 - b9 + 3*b6 - b5 + 2*b2 + b1) * q^84 + (6*b15 - 2*b14 + 2*b11 - 2*b6 + 6*b2 + 6*b1) * q^85 + (2*b13 - 3*b12 - 6*b10 - b8 + 2*b7 + 3*b4 - 2*b3 + 5) * q^86 + (3*b15 + b14 + 4*b11 - 2*b6 + b5 + 2*b2) * q^87 + (-b15 + b12 - b10 + b6 + b4 + b3 + b2 - b1) * q^88 + (-b13 - 4*b12 + 4*b10 - b8 - 4*b4 - b3 + 5) * q^89 + (b14 - 6*b11 + 3*b9 - b6 - 9*b1) * q^90 + (-b13 - 4*b12 - 6*b10 - 2*b8 + 4*b7 - b4 - 2*b3 + 2) * q^91 + (2*b13 + 2*b12 - b8 - b7 + b4 + b3 + 1) * q^92 + (6*b13 + b12 + 14*b10 + 3*b8 - 6*b7 - b4 - 6*b3 - 11) * q^93 + (2*b15 + 2*b14 + 2*b1) * q^94 + (b15 + 3*b14 + 4*b11 - 4*b6 - 4*b2 + 5*b1) * q^95 + (b15 - b14 + b2 + b1) * q^96 + (2*b13 - 2*b12 + 6*b10 + 2*b8 - 2*b7 + 4*b4 + 2*b3 - 3) * q^97 + (b15 - 4*b14 - b11 + b9 + b2 + 2*b1) * q^98 + (-6*b15 + b14 - 2*b13 - 2*b12 - 6*b11 + b9 + 2*b8 + 2*b7 + b6 - 2*b5 - b4 - b3 - 6*b2 - 7*b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4} - 12 q^{5} + 16 q^{9}+O(q^{10})$$ 16 * q + 8 * q^4 - 12 * q^5 + 16 * q^9 $$16 q + 8 q^{4} - 12 q^{5} + 16 q^{9} + 8 q^{11} - 8 q^{14} - 8 q^{15} - 8 q^{16} - 8 q^{22} + 16 q^{23} - 36 q^{26} - 12 q^{31} - 24 q^{33} + 32 q^{36} - 16 q^{37} + 12 q^{38} + 12 q^{42} - 8 q^{44} - 108 q^{45} + 24 q^{47} + 8 q^{49} - 28 q^{53} - 4 q^{56} - 12 q^{58} + 60 q^{59} - 4 q^{60} - 16 q^{64} + 48 q^{66} + 12 q^{67} + 60 q^{70} + 8 q^{71} + 60 q^{75} + 44 q^{77} - 16 q^{78} + 12 q^{80} - 8 q^{81} + 20 q^{86} - 4 q^{88} + 96 q^{89} - 36 q^{91} + 32 q^{92} - 44 q^{93} + 56 q^{99}+O(q^{100})$$ 16 * q + 8 * q^4 - 12 * q^5 + 16 * q^9 + 8 * q^11 - 8 * q^14 - 8 * q^15 - 8 * q^16 - 8 * q^22 + 16 * q^23 - 36 * q^26 - 12 * q^31 - 24 * q^33 + 32 * q^36 - 16 * q^37 + 12 * q^38 + 12 * q^42 - 8 * q^44 - 108 * q^45 + 24 * q^47 + 8 * q^49 - 28 * q^53 - 4 * q^56 - 12 * q^58 + 60 * q^59 - 4 * q^60 - 16 * q^64 + 48 * q^66 + 12 * q^67 + 60 * q^70 + 8 * q^71 + 60 * q^75 + 44 * q^77 - 16 * q^78 + 12 * q^80 - 8 * q^81 + 20 * q^86 - 4 * q^88 + 96 * q^89 - 36 * q^91 + 32 * q^92 - 44 * q^93 + 56 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( - 80029143512 \nu^{15} + 385788744870 \nu^{14} - 820783926284 \nu^{13} + 848040618120 \nu^{12} + \cdots + 33432180594 ) / 3707507912227$$ (-80029143512*v^15 + 385788744870*v^14 - 820783926284*v^13 + 848040618120*v^12 + 1720995499366*v^11 - 6827412737484*v^10 + 6402779061545*v^9 - 3349022853273*v^8 - 9000312527194*v^7 + 24987667997140*v^6 - 8831452106135*v^5 + 17087672692002*v^4 + 9515651467064*v^3 - 97367215891956*v^2 + 394928996361*v + 33432180594) / 3707507912227 $$\beta_{2}$$ $$=$$ $$( - 115452774644 \nu^{15} + 886173093256 \nu^{14} - 3342656190846 \nu^{13} + \cdots - 1622048373702 ) / 3707507912227$$ (-115452774644*v^15 + 886173093256*v^14 - 3342656190846*v^13 + 8285369121684*v^12 - 12911491993004*v^11 + 8714136666371*v^10 + 7289551050986*v^9 - 30002334611460*v^8 + 43949161545424*v^7 - 21576071160793*v^6 - 23813610618566*v^5 + 85339924434804*v^4 - 157011910468036*v^3 + 111746321740448*v^2 - 13202743167632*v - 1622048373702) / 3707507912227 $$\beta_{3}$$ $$=$$ $$( 229876192869 \nu^{15} - 1434169055319 \nu^{14} + 4481995762636 \nu^{13} + \cdots + 7097366139528 ) / 3707507912227$$ (229876192869*v^15 - 1434169055319*v^14 + 4481995762636*v^13 - 9349199101054*v^12 + 10037306027966*v^11 + 1831974981075*v^10 - 17299825871464*v^9 + 35017964444230*v^8 - 30103090682590*v^7 - 16716169888734*v^6 + 38997521702488*v^5 - 110456015651282*v^4 + 142750731668897*v^3 + 33344856580986*v^2 + 5189891590886*v + 7097366139528) / 3707507912227 $$\beta_{4}$$ $$=$$ $$( 234407101203 \nu^{15} - 1463062841541 \nu^{14} + 4568245377402 \nu^{13} + \cdots - 5702240414038 ) / 3707507912227$$ (234407101203*v^15 - 1463062841541*v^14 + 4568245377402*v^13 - 9516224014418*v^12 + 10205971274842*v^11 + 1827352370160*v^10 - 17224931483234*v^9 + 34568461621396*v^8 - 29494836526278*v^7 - 17068584063939*v^6 + 37448827738712*v^5 - 108439817187702*v^4 + 143867520986011*v^3 + 33621690891801*v^2 + 5235429778548*v - 5702240414038) / 3707507912227 $$\beta_{5}$$ $$=$$ $$( - 323659513794 \nu^{15} + 2126436132890 \nu^{14} - 7021940502570 \nu^{13} + \cdots - 2750714974023 ) / 3707507912227$$ (-323659513794*v^15 + 2126436132890*v^14 - 7021940502570*v^13 + 15520046894136*v^12 - 19351307844336*v^11 + 4000851706865*v^10 + 22846579328323*v^9 - 56754662154696*v^8 + 61179136551334*v^7 + 2104319424462*v^6 - 54144940005777*v^5 + 172779950825427*v^4 - 256920959600440*v^3 + 39533236474234*v^2 - 22206981366431*v - 2750714974023) / 3707507912227 $$\beta_{6}$$ $$=$$ $$( - 394538957168 \nu^{15} + 2231811901383 \nu^{14} - 6199106801734 \nu^{13} + \cdots - 1505816495286 ) / 3707507912227$$ (-394538957168*v^15 + 2231811901383*v^14 - 6199106801734*v^13 + 11192177985162*v^12 - 6728239255060*v^11 - 15500105696193*v^10 + 30089262246160*v^9 - 41574777060864*v^8 + 11544594088212*v^7 + 67008107672279*v^6 - 55357249671040*v^5 + 144140956476978*v^4 - 124089191799984*v^3 - 224470101825857*v^2 - 11832162962962*v - 1505816495286) / 3707507912227 $$\beta_{7}$$ $$=$$ $$( 401980321088 \nu^{15} - 2507531613006 \nu^{14} + 7838812215514 \nu^{13} + \cdots + 6420259797322 ) / 3707507912227$$ (401980321088*v^15 - 2507531613006*v^14 + 7838812215514*v^13 - 16353473148736*v^12 + 17544133384043*v^11 + 3227680818897*v^10 - 30296941112790*v^9 + 61133313639343*v^8 - 52069085549277*v^7 - 29293132185474*v^6 + 66949942975780*v^5 - 191025350183983*v^4 + 246377240843999*v^3 + 57553292205336*v^2 + 8958074916496*v + 6420259797322) / 3707507912227 $$\beta_{8}$$ $$=$$ $$( - 403027374657 \nu^{15} + 2521501247748 \nu^{14} - 7885517844470 \nu^{13} + \cdots + 4012225523467 ) / 3707507912227$$ (-403027374657*v^15 + 2521501247748*v^14 - 7885517844470*v^13 + 16440266855031*v^12 - 17668207398569*v^11 - 3186799527882*v^10 + 30146969929770*v^9 - 60331564786329*v^8 + 51795261997335*v^7 + 29466424871154*v^6 - 65993485499734*v^5 + 187112766068823*v^4 - 248327455373448*v^3 - 58021984313211*v^2 - 9033094079978*v + 4012225523467) / 3707507912227 $$\beta_{9}$$ $$=$$ $$( 556158552240 \nu^{15} - 3247720931080 \nu^{14} + 9404820278882 \nu^{13} + \cdots + 2653688275785 ) / 3707507912227$$ (556158552240*v^15 - 3247720931080*v^14 + 9404820278882*v^13 - 17969597678127*v^12 + 14300654206114*v^11 + 15947399137709*v^10 - 41579756460341*v^9 + 66454318847001*v^8 - 34746646803794*v^7 - 75102453822786*v^6 + 80010681773455*v^5 - 222201806649708*v^4 + 229349441750990*v^3 + 234407277756901*v^2 + 21061506068817*v + 2653688275785) / 3707507912227 $$\beta_{10}$$ $$=$$ $$( - 785074438 \nu^{15} + 4907305158 \nu^{14} - 15343416116 \nu^{13} + 31997236762 \nu^{12} + \cdots - 1370541375 ) / 1973128213$$ (-785074438*v^15 + 4907305158*v^14 - 15343416116*v^13 + 31997236762*v^12 - 34368654754*v^11 - 6224799624*v^10 + 58813815140*v^9 - 118164117069*v^8 + 101294734438*v^7 + 57313765188*v^6 - 129597823592*v^5 + 370531312158*v^4 - 484158220100*v^3 - 113116110792*v^2 - 17609140908*v - 1370541375) / 1973128213 $$\beta_{11}$$ $$=$$ $$( - 1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} + \cdots - 10233974547 ) / 3810388399$$ (-1616321538*v^15 + 9938916556*v^14 - 30594542475*v^13 + 62871582510*v^12 - 64714538406*v^11 - 18641341380*v^10 + 118271636061*v^9 - 231117756606*v^8 + 186162436800*v^7 + 134499720676*v^6 - 250255719435*v^5 + 736991238906*v^4 - 924410484114*v^3 - 318408624800*v^2 - 82003950372*v - 10233974547) / 3810388399 $$\beta_{12}$$ $$=$$ $$( 2432602484212 \nu^{15} - 15210362430228 \nu^{14} + 47566230428388 \nu^{13} + \cdots + 3660982645558 ) / 3707507912227$$ (2432602484212*v^15 - 15210362430228*v^14 + 47566230428388*v^13 - 99202432432724*v^12 + 106574496442407*v^11 + 19314445911990*v^10 - 182513504406276*v^9 + 366427539327546*v^8 - 314133045516125*v^7 - 177707059051737*v^6 + 401666792306712*v^5 - 1146623846023684*v^4 + 1499508109685491*v^3 + 350334214614267*v^2 + 54537167624894*v + 3660982645558) / 3707507912227 $$\beta_{13}$$ $$=$$ $$( - 2667083154356 \nu^{15} + 16670596841871 \nu^{14} - 52123769377554 \nu^{13} + \cdots - 5233663502032 ) / 3707507912227$$ (-2667083154356*v^15 + 16670596841871*v^14 - 52123769377554*v^13 + 108698529181646*v^12 - 116744726696199*v^11 - 21159726133215*v^10 + 199797208529346*v^9 - 401304270681864*v^8 + 343678264658725*v^7 + 194739830729736*v^6 - 439361137765482*v^5 + 1256589457116658*v^4 - 1642628604419603*v^3 - 383777392568439*v^2 - 59744172756124*v - 5233663502032) / 3707507912227 $$\beta_{14}$$ $$=$$ $$( 2737320095452 \nu^{15} - 16635640723011 \nu^{14} + 50529646182960 \nu^{13} + \cdots + 16350688060416 ) / 3707507912227$$ (2737320095452*v^15 - 16635640723011*v^14 + 50529646182960*v^13 - 102286748592894*v^12 + 100494628369536*v^11 + 42467570312879*v^10 - 201266372887674*v^9 + 376503294081456*v^8 - 282140218575116*v^7 - 260583517395552*v^6 + 416079703454118*v^5 - 1212268368579630*v^4 + 1465363584459148*v^3 + 686420342024835*v^2 + 130791312035158*v + 16350688060416) / 3707507912227 $$\beta_{15}$$ $$=$$ $$( - 2891771622084 \nu^{15} + 17710086389784 \nu^{14} - 54266401205842 \nu^{13} + \cdots - 17956801097484 ) / 3707507912227$$ (-2891771622084*v^15 + 17710086389784*v^14 - 54266401205842*v^13 + 110934804723696*v^12 - 112385854122694*v^11 - 37485437796620*v^10 + 212147297949199*v^9 - 408031002582501*v^8 + 320685750244910*v^7 + 252628621076411*v^6 - 444942596806753*v^5 + 1305146306980602*v^4 - 1617999730660324*v^3 - 622496304087051*v^2 - 143808261987203*v - 17956801097484) / 3707507912227
 $$\nu$$ $$=$$ $$( -\beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_1 ) / 2$$ (-b15 - b14 + b13 + b12 - b1) / 2 $$\nu^{2}$$ $$=$$ $$-\beta_{15} - \beta_{14} + \beta_{6} - 3\beta_1$$ -b15 - b14 + b6 - 3*b1 $$\nu^{3}$$ $$=$$ $$( - 3 \beta_{15} - 4 \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{11} + 4 \beta_{10} + \beta_{9} + \cdots - 2 ) / 2$$ (-3*b15 - 4*b14 - b13 + b12 - 2*b11 + 4*b10 + b9 + b8 - b7 + 4*b6 + 3*b4 + 4*b3 + 3*b2 - 6*b1 - 2) / 2 $$\nu^{4}$$ $$=$$ $$5\beta_{12} + 9\beta_{10} + 2\beta_{8} - \beta_{7} + 5\beta_{4} + 5\beta_{3} - 1$$ 5*b12 + 9*b10 + 2*b8 - b7 + 5*b4 + 5*b3 - 1 $$\nu^{5}$$ $$=$$ $$( - 8 \beta_{15} + 8 \beta_{14} + 11 \beta_{13} + 19 \beta_{12} + 28 \beta_{11} + 14 \beta_{10} + \cdots + 6 ) / 2$$ (-8*b15 + 8*b14 + 11*b13 + 19*b12 + 28*b11 + 14*b10 - 8*b9 + 8*b8 - 11*b6 + 8*b5 + 19*b4 + 11*b3 - 19*b2 + 6*b1 + 6) / 2 $$\nu^{6}$$ $$=$$ $$-26\beta_{15} + \beta_{14} + 46\beta_{11} - 8\beta_{9} + \beta_{6} + 16\beta_{5} - 26\beta_{2} - 18\beta_1$$ -26*b15 + b14 + 46*b11 - 8*b9 + b6 + 16*b5 - 26*b2 - 18*b1 $$\nu^{7}$$ $$=$$ $$( - 96 \beta_{15} - 47 \beta_{14} - 96 \beta_{13} - 47 \beta_{12} + 82 \beta_{11} + 82 \beta_{10} + \cdots - 113 ) / 2$$ (-96*b15 - 47*b14 - 96*b13 - 47*b12 + 82*b11 + 82*b10 - 51*b7 + 49*b6 + 51*b5 - 49*b4 + 49*b3 - 49*b2 - 127*b1 - 113) / 2 $$\nu^{8}$$ $$=$$ $$-138\beta_{13} + 10\beta_{12} + 245\beta_{10} + 50\beta_{8} - 100\beta_{7} - 10\beta_{4} + 138\beta_{3} - 195$$ -138*b13 + 10*b12 + 245*b10 + 50*b8 - 100*b7 - 10*b4 + 138*b3 - 195 $$\nu^{9}$$ $$=$$ $$( 223 \beta_{15} + 501 \beta_{14} - 278 \beta_{13} + 278 \beta_{12} + 456 \beta_{11} + 912 \beta_{10} + \cdots - 456 ) / 2$$ (223*b15 + 501*b14 - 278*b13 + 278*b12 + 456*b11 + 912*b10 - 298*b9 + 298*b8 - 298*b7 - 501*b6 + 223*b4 + 501*b3 - 223*b2 + 837*b1 - 456) / 2 $$\nu^{10}$$ $$=$$ $$- 70 \beta_{15} + 739 \beta_{14} + 1320 \beta_{11} - 576 \beta_{9} - 669 \beta_{6} + 288 \beta_{5} + \cdots + 962 \beta_1$$ -70*b15 + 739*b14 + 1320*b11 - 576*b9 - 669*b6 + 288*b5 - 669*b2 + 962*b1 $$\nu^{11}$$ $$=$$ $$( - 1533 \beta_{15} + 1533 \beta_{14} - 1125 \beta_{13} - 2658 \beta_{12} + 4980 \beta_{11} - 2490 \beta_{10} + \cdots - 817 ) / 2$$ (-1533*b15 + 1533*b14 - 1125*b13 - 2658*b12 + 4980*b11 - 2490*b10 - 1673*b9 - 1673*b8 - 1125*b6 + 1673*b5 - 2658*b4 - 1125*b3 - 2658*b2 + 957*b1 - 817) / 2 $$\nu^{12}$$ $$=$$ $$-3545\beta_{13} - 3545\beta_{12} - 1603\beta_{8} - 1603\beta_{7} - 3973\beta_{4} + 428\beta_{3} - 3923$$ -3545*b13 - 3545*b12 - 1603*b8 - 1603*b7 - 3973*b4 + 428*b3 - 3923 $$\nu^{13}$$ $$=$$ $$( 14219 \beta_{15} + 5865 \beta_{14} - 14219 \beta_{13} - 5865 \beta_{12} - 13502 \beta_{11} + \cdots - 17794 ) / 2$$ (14219*b15 + 5865*b14 - 14219*b13 - 5865*b12 - 13502*b11 + 13502*b10 - 9210*b7 - 8354*b6 - 9210*b5 - 8354*b4 + 8354*b3 + 8354*b2 + 18511*b1 - 17794) / 2 $$\nu^{14}$$ $$=$$ $$18939\beta_{15} + 18939\beta_{14} - 8782\beta_{9} - 21398\beta_{6} - 8782\beta_{5} + 2459\beta_{2} + 39879\beta_1$$ 18939*b15 + 18939*b14 - 8782*b9 - 21398*b6 - 8782*b5 + 2459*b2 + 39879*b1 $$\nu^{15}$$ $$=$$ $$( 31097 \beta_{15} + 76382 \beta_{14} + 45285 \beta_{13} - 45285 \beta_{12} + 73006 \beta_{11} + \cdots + 73006 ) / 2$$ (31097*b15 + 76382*b14 + 45285*b13 - 45285*b12 + 73006*b11 - 146012*b10 - 50203*b9 - 50203*b8 + 50203*b7 - 76382*b6 - 31097*b4 - 76382*b3 - 31097*b2 + 126906*b1 + 73006) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/154\mathbb{Z}\right)^\times$$.

 $$n$$ $$45$$ $$57$$ $$\chi(n)$$ $$1 - \beta_{10}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
87.1
 −0.186243 + 0.0499037i 1.60599 − 0.430324i −1.29724 + 0.347596i 2.24352 − 0.601150i −0.0499037 − 0.186243i 0.430324 + 1.60599i −0.347596 − 1.29724i 0.601150 + 2.24352i −0.186243 − 0.0499037i 1.60599 + 0.430324i −1.29724 − 0.347596i 2.24352 + 0.601150i −0.0499037 + 0.186243i 0.430324 − 1.60599i −0.347596 + 1.29724i 0.601150 − 2.24352i
−0.866025 + 0.500000i −2.70809 1.56352i 0.500000 0.866025i −1.90775 + 1.10144i 3.12703 2.24014 + 1.40775i 1.00000i 3.38916 + 5.87020i 1.10144 1.90775i
87.2 −0.866025 + 0.500000i −0.889740 0.513691i 0.500000 0.866025i 1.09005 0.629341i 1.02738 −2.11465 1.59005i 1.00000i −0.972242 1.68397i −0.629341 + 1.09005i
87.3 −0.866025 + 0.500000i 1.35034 + 0.779618i 0.500000 0.866025i 0.882559 0.509546i −1.55924 2.25578 1.38256i 1.00000i −0.284392 0.492581i −0.509546 + 0.882559i
87.4 −0.866025 + 0.500000i 2.24749 + 1.29759i 0.500000 0.866025i −3.06486 + 1.76950i −2.59518 −0.649221 + 2.56486i 1.00000i 1.86747 + 3.23456i 1.76950 3.06486i
87.5 0.866025 0.500000i −2.70809 1.56352i 0.500000 0.866025i −1.90775 + 1.10144i −3.12703 −2.24014 1.40775i 1.00000i 3.38916 + 5.87020i −1.10144 + 1.90775i
87.6 0.866025 0.500000i −0.889740 0.513691i 0.500000 0.866025i 1.09005 0.629341i −1.02738 2.11465 + 1.59005i 1.00000i −0.972242 1.68397i 0.629341 1.09005i
87.7 0.866025 0.500000i 1.35034 + 0.779618i 0.500000 0.866025i 0.882559 0.509546i 1.55924 −2.25578 + 1.38256i 1.00000i −0.284392 0.492581i 0.509546 0.882559i
87.8 0.866025 0.500000i 2.24749 + 1.29759i 0.500000 0.866025i −3.06486 + 1.76950i 2.59518 0.649221 2.56486i 1.00000i 1.86747 + 3.23456i −1.76950 + 3.06486i
131.1 −0.866025 0.500000i −2.70809 + 1.56352i 0.500000 + 0.866025i −1.90775 1.10144i 3.12703 2.24014 1.40775i 1.00000i 3.38916 5.87020i 1.10144 + 1.90775i
131.2 −0.866025 0.500000i −0.889740 + 0.513691i 0.500000 + 0.866025i 1.09005 + 0.629341i 1.02738 −2.11465 + 1.59005i 1.00000i −0.972242 + 1.68397i −0.629341 1.09005i
131.3 −0.866025 0.500000i 1.35034 0.779618i 0.500000 + 0.866025i 0.882559 + 0.509546i −1.55924 2.25578 + 1.38256i 1.00000i −0.284392 + 0.492581i −0.509546 0.882559i
131.4 −0.866025 0.500000i 2.24749 1.29759i 0.500000 + 0.866025i −3.06486 1.76950i −2.59518 −0.649221 2.56486i 1.00000i 1.86747 3.23456i 1.76950 + 3.06486i
131.5 0.866025 + 0.500000i −2.70809 + 1.56352i 0.500000 + 0.866025i −1.90775 1.10144i −3.12703 −2.24014 + 1.40775i 1.00000i 3.38916 5.87020i −1.10144 1.90775i
131.6 0.866025 + 0.500000i −0.889740 + 0.513691i 0.500000 + 0.866025i 1.09005 + 0.629341i −1.02738 2.11465 1.59005i 1.00000i −0.972242 + 1.68397i 0.629341 + 1.09005i
131.7 0.866025 + 0.500000i 1.35034 0.779618i 0.500000 + 0.866025i 0.882559 + 0.509546i 1.55924 −2.25578 1.38256i 1.00000i −0.284392 + 0.492581i 0.509546 + 0.882559i
131.8 0.866025 + 0.500000i 2.24749 1.29759i 0.500000 + 0.866025i −3.06486 1.76950i 2.59518 0.649221 + 2.56486i 1.00000i 1.86747 3.23456i −1.76950 3.06486i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 87.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.b odd 2 1 inner
77.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.2.i.a 16
3.b odd 2 1 1386.2.bk.c 16
4.b odd 2 1 1232.2.bn.b 16
7.b odd 2 1 1078.2.i.c 16
7.c even 3 1 1078.2.c.b 16
7.c even 3 1 1078.2.i.c 16
7.d odd 6 1 inner 154.2.i.a 16
7.d odd 6 1 1078.2.c.b 16
11.b odd 2 1 inner 154.2.i.a 16
21.g even 6 1 1386.2.bk.c 16
28.f even 6 1 1232.2.bn.b 16
33.d even 2 1 1386.2.bk.c 16
44.c even 2 1 1232.2.bn.b 16
77.b even 2 1 1078.2.i.c 16
77.h odd 6 1 1078.2.c.b 16
77.h odd 6 1 1078.2.i.c 16
77.i even 6 1 inner 154.2.i.a 16
77.i even 6 1 1078.2.c.b 16
231.k odd 6 1 1386.2.bk.c 16
308.m odd 6 1 1232.2.bn.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.i.a 16 1.a even 1 1 trivial
154.2.i.a 16 7.d odd 6 1 inner
154.2.i.a 16 11.b odd 2 1 inner
154.2.i.a 16 77.i even 6 1 inner
1078.2.c.b 16 7.c even 3 1
1078.2.c.b 16 7.d odd 6 1
1078.2.c.b 16 77.h odd 6 1
1078.2.c.b 16 77.i even 6 1
1078.2.i.c 16 7.b odd 2 1
1078.2.i.c 16 7.c even 3 1
1078.2.i.c 16 77.b even 2 1
1078.2.i.c 16 77.h odd 6 1
1232.2.bn.b 16 4.b odd 2 1
1232.2.bn.b 16 28.f even 6 1
1232.2.bn.b 16 44.c even 2 1
1232.2.bn.b 16 308.m odd 6 1
1386.2.bk.c 16 3.b odd 2 1
1386.2.bk.c 16 21.g even 6 1
1386.2.bk.c 16 33.d even 2 1
1386.2.bk.c 16 231.k odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(154, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{4}$$
$3$ $$(T^{8} - 10 T^{6} + \cdots + 169)^{2}$$
$5$ $$(T^{8} + 6 T^{7} + \cdots + 100)^{2}$$
$7$ $$T^{16} - 4 T^{14} + \cdots + 5764801$$
$11$ $$T^{16} + \cdots + 214358881$$
$13$ $$(T^{8} - 44 T^{6} + \cdots + 25)^{2}$$
$17$ $$T^{16} + \cdots + 6146560000$$
$19$ $$T^{16} + 80 T^{14} + \cdots + 16$$
$23$ $$(T^{8} - 8 T^{7} + \cdots + 196)^{2}$$
$29$ $$(T^{8} + 36 T^{6} + 366 T^{4} + \cdots + 9)^{2}$$
$31$ $$(T^{8} + 6 T^{7} + \cdots + 3136)^{2}$$
$37$ $$(T^{8} + 8 T^{7} + 52 T^{6} + \cdots + 4)^{2}$$
$41$ $$(T^{8} - 200 T^{6} + \cdots + 1674436)^{2}$$
$43$ $$(T^{8} + 244 T^{6} + \cdots + 1201216)^{2}$$
$47$ $$(T^{8} - 12 T^{7} + \cdots + 4096)^{2}$$
$53$ $$(T^{8} + 14 T^{7} + \cdots + 1170724)^{2}$$
$59$ $$(T^{8} - 30 T^{7} + \cdots + 2356225)^{2}$$
$61$ $$T^{16} + \cdots + 1706808989601$$
$67$ $$(T^{8} - 6 T^{7} + \cdots + 173889)^{2}$$
$71$ $$(T^{4} - 2 T^{3} + \cdots + 3262)^{4}$$
$73$ $$T^{16} + \cdots + 15704099856$$
$79$ $$T^{16} + \cdots + 11\!\cdots\!41$$
$83$ $$(T^{8} - 260 T^{6} + \cdots + 107584)^{2}$$
$89$ $$(T^{8} - 48 T^{7} + \cdots + 49617936)^{2}$$
$97$ $$(T^{8} + 644 T^{6} + \cdots + 496532089)^{2}$$