# Properties

 Label 33.4.e.c Level $33$ Weight $4$ Character orbit 33.e Analytic conductor $1.947$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,4,Mod(4,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("33.4");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 33.e (of order $$5$$, degree $$4$$, 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: $$1.94706303019$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 3 x^{11} + 21 x^{10} - 26 x^{9} + 281 x^{8} + 486 x^{7} + 3506 x^{6} + 15102 x^{5} + 46669 x^{4} + 41850 x^{3} + 16292 x^{2} + 616 x + 1936$$ x^12 - 3*x^11 + 21*x^10 - 26*x^9 + 281*x^8 + 486*x^7 + 3506*x^6 + 15102*x^5 + 46669*x^4 + 41850*x^3 + 16292*x^2 + 616*x + 1936 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 1) q^{2} - 3 \beta_{6} q^{3} + ( - \beta_{9} - \beta_{8} + 2 \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{4} + ( - \beta_{11} + 2 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + 4 \beta_{6} - \beta_{4} + 4 \beta_{3} + \cdots + 4) q^{5}+ \cdots + ( - 9 \beta_{6} - 9 \beta_{5} - 9 \beta_{3} - 9) q^{9}+O(q^{10})$$ q + (-b7 - b6 - b5 - b3 - 1) * q^2 - 3*b6 * q^3 + (-b9 - b8 + 2*b5 - b3 + b2 - b1 - 1) * q^4 + (-b11 + 2*b10 - b9 + b8 - b7 + 4*b6 - b4 + 4*b3 + b1 + 4) * q^5 + (-3*b3 - 3*b1) * q^6 + (2*b10 - b9 - b8 + 2*b5 - b4 + 2*b3 - 2*b2 + 2*b1 + 2) * q^7 + (3*b11 + b10 - b9 + 3*b8 + 10*b6 + 14*b5 - 2*b4 + 14*b3 + 2*b1) * q^8 + (-9*b6 - 9*b5 - 9*b3 - 9) * q^9 $$q + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 1) q^{2} - 3 \beta_{6} q^{3} + ( - \beta_{9} - \beta_{8} + 2 \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{4} + ( - \beta_{11} + 2 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + 4 \beta_{6} - \beta_{4} + 4 \beta_{3} + \cdots + 4) q^{5}+ \cdots + ( - 9 \beta_{11} + 18 \beta_{10} - 36 \beta_{9} - 45 \beta_{8} + 81 \beta_{7} + \cdots + 27) q^{99}+O(q^{100})$$ q + (-b7 - b6 - b5 - b3 - 1) * q^2 - 3*b6 * q^3 + (-b9 - b8 + 2*b5 - b3 + b2 - b1 - 1) * q^4 + (-b11 + 2*b10 - b9 + b8 - b7 + 4*b6 - b4 + 4*b3 + b1 + 4) * q^5 + (-3*b3 - 3*b1) * q^6 + (2*b10 - b9 - b8 + 2*b5 - b4 + 2*b3 - 2*b2 + 2*b1 + 2) * q^7 + (3*b11 + b10 - b9 + 3*b8 + 10*b6 + 14*b5 - 2*b4 + 14*b3 + 2*b1) * q^8 + (-9*b6 - 9*b5 - 9*b3 - 9) * q^9 + (3*b11 + 4*b9 - 4*b7 - 10*b2 + 4*b1 + 8) * q^10 + (-4*b11 - 4*b10 + b9 - b8 + 2*b7 + 6*b6 - 5*b5 - 3*b4 - b3 + 6*b2 - 2*b1 - 4) * q^11 + (-3*b9 + 3*b7 + 3*b6 + 3*b5 + 3*b2 - 3*b1 - 6) * q^12 + (-6*b11 - 6*b10 - 9*b8 + 6*b7 - 15*b6 - 14*b5 + 9*b4 - 15*b3 + 9*b2 - 14) * q^13 + (b11 - 6*b10 + 6*b9 + b8 - 14*b6 - 20*b5 + 6*b4 - 20*b3 - 6*b1) * q^14 + (-3*b10 + 6*b9 + 6*b8 + 3*b4 + 12*b3 + 3*b2 - 3*b1 + 12) * q^15 + (4*b11 - 2*b10 + 4*b9 + 2*b8 + 10*b7 + 34*b6 + 10*b4 + 3*b3 + 7*b1 + 34) * q^16 + (5*b10 + 5*b8 + 14*b7 - 5*b6 + 14*b4 + 6*b3 - b1 - 5) * q^17 + (9*b5 - 9*b4) * q^18 + (-b11 - 5*b10 + 5*b9 - b8 - 9*b7 + 5*b6 + 48*b5 - 14*b4 + 48*b3 - 9*b2 + 14*b1) * q^19 + (-2*b11 - 2*b10 - 8*b8 - 25*b7 - 87*b6 - 64*b5 - 9*b4 - 87*b3 - 9*b2 - 64) * q^20 + (-6*b11 - 3*b9 - 6*b7 - 6*b6 - 6*b5 - 3*b2 + 6*b1 - 6) * q^21 + (7*b11 + 10*b10 - 8*b9 + b8 - 3*b7 + 87*b6 + 42*b5 - 23*b4 + 62*b3 - 6*b2 + 19*b1 + 31) * q^22 + (7*b11 - 4*b9 + 25*b7 + 23*b6 + 23*b5 - 4*b2 - 25*b1 - 8) * q^23 + (9*b11 + 9*b10 + 12*b8 + 30*b6 - 12*b5 + 6*b4 + 30*b3 + 6*b2 - 12) * q^24 + (-4*b11 + 19*b10 - 19*b9 - 4*b8 + 12*b7 + 46*b6 - 42*b5 + 25*b4 - 42*b3 + 12*b2 - 25*b1) * q^25 + (3*b10 - 6*b9 - 6*b8 + 44*b5 - 14*b4 + 65*b3 + 19*b2 - 19*b1 + 65) * q^26 - 27*b3 * q^27 + (-2*b11 - 6*b10 - 2*b9 - 8*b8 - 15*b7 - 61*b6 - 15*b4 - 34*b3 + b1 - 61) * q^28 + (6*b10 + 3*b9 + 3*b8 + 125*b5 - 11*b4 + 44*b3 - 14*b2 + 14*b1 + 44) * q^29 + (-12*b11 + 9*b10 - 9*b9 - 12*b8 - 30*b7 - 24*b6 - 18*b4 - 30*b2 + 18*b1) * q^30 + (2*b11 + 2*b10 + 27*b8 - 36*b7 - 57*b6 - 9*b5 + 11*b4 - 57*b3 + 11*b2 - 9) * q^31 + (-10*b11 - 11*b9 - 35*b7 - 27*b6 - 27*b5 - 9*b2 + 35*b1 + 80) * q^32 + (6*b11 - 12*b10 + 9*b9 - 6*b8 + 18*b7 + 30*b6 + 21*b5 + 12*b4 + 18*b3 + 27*b2 - 12*b1 + 33) * q^33 + (-14*b11 + 23*b9 + b7 - 149*b6 - 149*b5 - 40*b2 - b1 + 21) * q^34 + (7*b11 + 7*b10 + b8 + 9*b7 - 53*b6 - 186*b5 + 36*b4 - 53*b3 + 36*b2 - 186) * q^35 + (9*b11 + 9*b8 + 9*b7 + 27*b6 + 9*b5 + 9*b3 + 9*b2) * q^36 + (9*b10 + 72*b5 - 9*b4 + 103*b3 + 39*b2 - 39*b1 + 103) * q^37 + (-5*b11 - 11*b10 - 5*b9 - 16*b8 + 18*b7 + 76*b6 + 18*b4 - 99*b3 - 17*b1 + 76) * q^38 + (-9*b11 - 18*b10 - 9*b9 - 27*b8 + 27*b7 - 3*b6 + 27*b4 - 45*b3 - 9*b1 - 3) * q^39 + (-45*b10 + 3*b9 + 3*b8 + 167*b5 - 3*b4 - 141*b3 + 27*b2 - 27*b1 - 141) * q^40 + (2*b11 + b10 - b9 + 2*b8 - 24*b7 + 225*b6 + 147*b5 - 43*b4 + 147*b3 - 24*b2 + 43*b1) * q^41 + (3*b11 + 3*b10 - 15*b8 - 42*b6 + 18*b5 - 18*b4 - 42*b3 - 18*b2 + 18) * q^42 + (19*b11 + 39*b7 - 117*b6 - 117*b5 - 26*b2 - 39*b1) * q^43 + (b11 + 16*b10 + 20*b9 + 20*b8 + 21*b7 + 177*b6 - 20*b5 + 31*b4 + 92*b3 + 9*b2 + 27*b1 + 112) * q^44 + (9*b11 + 18*b9 + 9*b7 - 36*b6 - 36*b5 - 9*b1) * q^45 + (4*b11 + 4*b10 + 43*b8 + 79*b7 + 53*b6 - 210*b5 + 19*b4 + 53*b3 + 19*b2 - 210) * q^46 + (-9*b11 - 26*b10 + 26*b9 - 9*b8 - 83*b7 + 104*b6 - b5 - 73*b4 - b3 - 83*b2 + 73*b1) * q^47 + (12*b10 - 6*b9 - 6*b8 + 93*b5 + 21*b4 + 102*b3 - 30*b2 + 30*b1 + 102) * q^48 + (-b11 + 7*b10 - b9 + 6*b8 - 29*b7 - 123*b6 - 29*b4 - 203*b3 + 56*b1 - 123) * q^49 + (25*b11 + 4*b10 + 25*b9 + 29*b8 + 36*b7 - 140*b6 + 36*b4 + 194*b3 - 140) * q^50 + (15*b9 + 15*b8 - 33*b5 - 3*b4 - 15*b3 - 42*b2 + 42*b1 - 15) * q^51 + (-3*b11 - 11*b10 + 11*b9 - 3*b8 + 67*b7 + 223*b6 + 146*b5 + 49*b4 + 146*b3 + 67*b2 - 49*b1) * q^52 + (6*b11 + 6*b10 - 15*b8 + 74*b7 - 133*b6 + 83*b5 + 9*b4 - 133*b3 + 9*b2 + 83) * q^53 + (27*b2 - 27) * q^54 + (-23*b11 - 8*b10 - 25*b9 - 16*b8 - 39*b7 + 167*b6 + 136*b5 - 62*b4 + 19*b3 - 67*b2 - 43*b1 + 378) * q^55 + (-29*b11 - 34*b9 + 24*b7 + 70*b6 + 70*b5 + 75*b2 - 24*b1 - 37) * q^56 + (-3*b11 - 3*b10 - 18*b8 - 27*b7 + 15*b6 - 129*b5 + 15*b4 + 15*b3 + 15*b2 - 129) * q^57 + (-9*b11 - 26*b10 + 26*b9 - 9*b8 + 49*b7 + 7*b6 - 188*b5 + 103*b4 - 188*b3 + 49*b2 - 103*b1) * q^58 + (-54*b10 - 14*b9 - 14*b8 - 187*b5 + 76*b4 + 61*b3 + 61) * q^59 + (-18*b11 - 6*b10 - 18*b9 - 24*b8 - 27*b7 - 69*b6 - 27*b4 - 261*b3 - 48*b1 - 69) * q^60 + (5*b11 + 39*b10 + 5*b9 + 44*b8 - 123*b7 - 86*b6 - 123*b4 + 157*b3 + 26*b1 - 86) * q^61 + (61*b10 - 32*b9 - 32*b8 + 369*b5 + 5*b4 + 46*b3 + 18*b2 - 18*b1 + 46) * q^62 + (9*b11 - 18*b10 + 18*b9 + 9*b8 - 9*b7 - 18*b5 + 9*b4 - 18*b3 - 9*b2 - 9*b1) * q^63 + (-3*b11 - 3*b10 - 26*b8 + 28*b7 - 194*b6 - 138*b5 + 60*b4 - 194*b3 + 60*b2 - 138) * q^64 + (33*b11 + 29*b9 + 53*b7 - 246*b6 - 246*b5 + 37*b2 - 53*b1 + 277) * q^65 + (-3*b11 + 21*b10 - 18*b9 + 27*b8 - 18*b7 + 168*b6 + 75*b5 + 39*b4 + 261*b3 + 51*b2 + 9*b1 + 135) * q^66 + (33*b11 - 63*b9 - 201*b7 - 168*b6 - 168*b5 - 69*b2 + 201*b1 - 217) * q^67 + (-46*b11 - 46*b10 - 73*b8 - 171*b7 - 655*b6 - 352*b5 - 143*b4 - 655*b3 - 143*b2 - 352) * q^68 + (12*b11 + 21*b10 - 21*b9 + 12*b8 - 12*b7 + 93*b6 + 69*b5 - 87*b4 + 69*b3 - 12*b2 + 87*b1) * q^69 + (24*b10 + 23*b9 + 23*b8 + 398*b5 - 90*b4 + 481*b3 - 95*b2 + 95*b1 + 481) * q^70 + (b11 + 39*b10 + b9 + 40*b8 - 57*b7 + 48*b6 - 57*b4 - 101*b3 - 78*b1 + 48) * q^71 + (9*b11 + 27*b10 + 9*b9 + 36*b8 + 18*b7 + 126*b6 + 18*b4 + 90*b3 - 18*b1 + 126) * q^72 + (94*b10 - 29*b9 - 29*b8 + 43*b5 + 73*b4 - 78*b3 + 14*b2 - 14*b1 - 78) * q^73 + (48*b11 + 21*b10 - 21*b9 + 48*b8 - b7 + 461*b6 + 287*b5 + 120*b4 + 287*b3 - b2 - 120*b1) * q^74 + (-12*b11 - 12*b10 + 45*b8 + 36*b7 + 138*b6 + 264*b5 - 39*b4 + 138*b3 - 39*b2 + 264) * q^75 + (-48*b11 - 13*b9 + 7*b7 + 183*b6 + 183*b5 + 58*b2 - 7*b1 + 13) * q^76 + (-4*b11 - 13*b10 - 13*b9 - 24*b8 + 18*b7 + 97*b6 - 232*b5 + 75*b4 - 408*b3 + 24*b2 - 81*b1 - 95) * q^77 + (-9*b11 - 18*b9 + 57*b7 - 195*b6 - 195*b5 + 99*b2 - 57*b1 - 132) * q^78 + (17*b11 + 17*b10 - 64*b8 - 13*b7 + 232*b6 + 97*b5 - 193*b4 + 232*b3 - 193*b2 + 97) * q^79 + (8*b11 + 69*b10 - 69*b9 + 8*b8 + 168*b7 + 62*b6 + 266*b5 - 36*b4 + 266*b3 + 168*b2 + 36*b1) * q^80 + 81*b5 * q^81 + (21*b11 - 20*b10 + 21*b9 + b8 + 109*b7 + 301*b6 + 109*b4 + 15*b3 + 144*b1 + 301) * q^82 + (-23*b11 - 21*b10 - 23*b9 - 44*b8 + 51*b7 - 447*b6 + 51*b4 + 94*b3 - 36*b1 - 447) * q^83 + (-6*b10 - 18*b9 - 18*b8 - 81*b5 + 3*b4 - 183*b3 + 45*b2 - 45*b1 - 183) * q^84 + (29*b10 - 29*b9 + 260*b7 + 355*b6 + 119*b5 + 131*b4 + 119*b3 + 260*b2 - 131*b1) * q^85 + (-26*b11 - 26*b10 + 51*b8 - 59*b7 - 281*b6 - 508*b5 - 125*b4 - 281*b3 - 125*b2 - 508) * q^86 + (-18*b11 + 9*b9 - 42*b7 - 132*b6 - 132*b5 - 9*b2 + 42*b1 - 375) * q^87 + (18*b11 - 71*b10 + 88*b9 - 23*b8 - 126*b7 - 8*b6 - 104*b5 - 2*b4 + 232*b3 - 241*b2 + 82*b1 + 127) * q^88 + (-b11 + 90*b9 - 27*b7 + 297*b6 + 297*b5 + 128*b2 + 27*b1 + 140) * q^89 + (-36*b11 - 36*b10 - 9*b8 - 90*b7 - 72*b6 - 72*b5 - 36*b4 - 72*b3 - 36*b2 - 72) * q^90 + (34*b11 - 45*b10 + 45*b9 + 34*b8 - 56*b7 + 115*b6 - 85*b5 + 45*b4 - 85*b3 - 56*b2 - 45*b1) * q^91 + (41*b10 + 55*b9 + 55*b8 - 367*b5 - 85*b4 + 380*b3 - 244*b2 + 244*b1 + 380) * q^92 + (75*b11 + 6*b10 + 75*b9 + 81*b8 + 33*b7 - 144*b6 + 33*b4 - 171*b3 - 141*b1 - 144) * q^93 + (-62*b11 - 101*b10 - 62*b9 - 163*b8 - 161*b7 - 227*b6 - 161*b4 - 850*b3 + 193*b1 - 227) * q^94 + (-208*b10 + 83*b9 + 83*b8 + 281*b5 + 101*b4 + 231*b3 - 28*b2 + 28*b1 + 231) * q^95 + (33*b11 - 30*b10 + 30*b9 + 33*b8 - 27*b7 - 321*b6 - 81*b5 + 78*b4 - 81*b3 - 27*b2 - 78*b1) * q^96 + (6*b11 + 6*b10 - 115*b8 + 314*b7 + 178*b6 + 326*b5 + 171*b4 + 178*b3 + 171*b2 + 326) * q^97 + (31*b11 + 41*b9 + 63*b7 + 337*b6 + 337*b5 + 205*b2 - 63*b1 + 400) * q^98 + (-9*b11 + 18*b10 - 36*b9 - 45*b8 + 81*b7 - 9*b6 + 36*b5 + 45*b4 + 90*b3 + 45*b2 - 27*b1 + 27) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 9 q^{3} - 16 q^{4} + 28 q^{5} + 12 q^{7} - 112 q^{8} - 27 q^{9}+O(q^{10})$$ 12 * q + 9 * q^3 - 16 * q^4 + 28 * q^5 + 12 * q^7 - 112 * q^8 - 27 * q^9 $$12 q + 9 q^{3} - 16 q^{4} + 28 q^{5} + 12 q^{7} - 112 q^{8} - 27 q^{9} + 100 q^{10} - 54 q^{11} - 102 q^{12} - 18 q^{13} + 156 q^{14} + 111 q^{15} + 308 q^{16} - 80 q^{17} - 45 q^{18} - 280 q^{19} - 15 q^{20} - 6 q^{21} - 193 q^{22} - 392 q^{23} - 264 q^{24} + 77 q^{25} + 406 q^{26} + 81 q^{27} - 429 q^{28} + 13 q^{29} + 120 q^{30} + 413 q^{31} + 1314 q^{32} + 177 q^{33} + 1060 q^{34} - 1239 q^{35} - 144 q^{36} + 654 q^{37} + 912 q^{38} + 54 q^{39} - 1803 q^{40} - 1490 q^{41} + 342 q^{42} + 416 q^{43} + 695 q^{44} + 162 q^{45} - 2369 q^{46} - 150 q^{47} + 711 q^{48} - 301 q^{49} - 1878 q^{50} - 1661 q^{52} + 1359 q^{53} - 270 q^{54} + 3300 q^{55} - 858 q^{56} - 1110 q^{57} + 955 q^{58} + 1262 q^{59} + 45 q^{60} - 1044 q^{61} - 701 q^{62} + 108 q^{63} + 78 q^{64} + 4556 q^{65} + 369 q^{66} - 528 q^{67} + 703 q^{68} - 594 q^{69} + 3050 q^{70} + 558 q^{71} + 792 q^{72} - 699 q^{73} - 3224 q^{74} + 1284 q^{75} - 868 q^{76} + 390 q^{77} - 558 q^{78} - 1252 q^{79} - 1914 q^{80} - 243 q^{81} + 2987 q^{82} - 4464 q^{83} - 1443 q^{84} - 2170 q^{85} - 3209 q^{86} - 3474 q^{87} + 1302 q^{88} + 316 q^{89} - 90 q^{90} + 176 q^{91} + 4595 q^{92} - 1239 q^{93} + 1247 q^{94} + 1466 q^{95} + 1398 q^{96} + 1608 q^{97} + 2810 q^{98} - 171 q^{99}+O(q^{100})$$ 12 * q + 9 * q^3 - 16 * q^4 + 28 * q^5 + 12 * q^7 - 112 * q^8 - 27 * q^9 + 100 * q^10 - 54 * q^11 - 102 * q^12 - 18 * q^13 + 156 * q^14 + 111 * q^15 + 308 * q^16 - 80 * q^17 - 45 * q^18 - 280 * q^19 - 15 * q^20 - 6 * q^21 - 193 * q^22 - 392 * q^23 - 264 * q^24 + 77 * q^25 + 406 * q^26 + 81 * q^27 - 429 * q^28 + 13 * q^29 + 120 * q^30 + 413 * q^31 + 1314 * q^32 + 177 * q^33 + 1060 * q^34 - 1239 * q^35 - 144 * q^36 + 654 * q^37 + 912 * q^38 + 54 * q^39 - 1803 * q^40 - 1490 * q^41 + 342 * q^42 + 416 * q^43 + 695 * q^44 + 162 * q^45 - 2369 * q^46 - 150 * q^47 + 711 * q^48 - 301 * q^49 - 1878 * q^50 - 1661 * q^52 + 1359 * q^53 - 270 * q^54 + 3300 * q^55 - 858 * q^56 - 1110 * q^57 + 955 * q^58 + 1262 * q^59 + 45 * q^60 - 1044 * q^61 - 701 * q^62 + 108 * q^63 + 78 * q^64 + 4556 * q^65 + 369 * q^66 - 528 * q^67 + 703 * q^68 - 594 * q^69 + 3050 * q^70 + 558 * q^71 + 792 * q^72 - 699 * q^73 - 3224 * q^74 + 1284 * q^75 - 868 * q^76 + 390 * q^77 - 558 * q^78 - 1252 * q^79 - 1914 * q^80 - 243 * q^81 + 2987 * q^82 - 4464 * q^83 - 1443 * q^84 - 2170 * q^85 - 3209 * q^86 - 3474 * q^87 + 1302 * q^88 + 316 * q^89 - 90 * q^90 + 176 * q^91 + 4595 * q^92 - 1239 * q^93 + 1247 * q^94 + 1466 * q^95 + 1398 * q^96 + 1608 * q^97 + 2810 * q^98 - 171 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 21 x^{10} - 26 x^{9} + 281 x^{8} + 486 x^{7} + 3506 x^{6} + 15102 x^{5} + 46669 x^{4} + 41850 x^{3} + 16292 x^{2} + 616 x + 1936$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 69\!\cdots\!13 \nu^{11} + \cdots - 78\!\cdots\!98 ) / 18\!\cdots\!89$$ (-6986944582847976513*v^11 + 23163569380150890824*v^10 - 153931767460290190961*v^9 + 235140221660543649730*v^8 - 2069964535083095660825*v^7 - 2495959479343529156658*v^6 - 24989208660214669470346*v^5 - 94664257433687411316282*v^4 - 296324885413782065156597*v^3 - 189062407549799154437530*v^2 + 15961334394692549325760*v - 78558043089841259336898) / 186805902146170459744889 $$\beta_{3}$$ $$=$$ $$( 16\!\cdots\!34 \nu^{11} + \cdots + 11\!\cdots\!56 ) / 37\!\cdots\!78$$ (16531513000088361634*v^11 - 44963669399566691738*v^10 + 348915737436588292851*v^9 - 389366357308863855609*v^8 + 4870370919031939910121*v^7 + 8818141577887473112246*v^6 + 64754961937344144101221*v^5 + 269893071475894974094208*v^4 + 894540942164531756734248*v^3 + 1091817446741743200538922*v^2 + 1086067427049355865922117*v + 114186350384130631498056) / 373611804292340919489778 $$\beta_{4}$$ $$=$$ $$( - 46\!\cdots\!57 \nu^{11} + \cdots + 12\!\cdots\!44 ) / 74\!\cdots\!56$$ (-46187384624125069357*v^11 + 186960657265634921981*v^10 - 1135246953413097226587*v^9 + 2286911209821250340128*v^8 - 14734418497538625039305*v^7 - 7812521628419409462428*v^6 - 146176049454948748163474*v^5 - 526488943354278938492894*v^4 - 1563782666952777918802773*v^3 + 130778948334237759252744*v^2 + 333828271431495750175728*v + 128217599677938392211744) / 747223608584681838979556 $$\beta_{5}$$ $$=$$ $$( 12\!\cdots\!25 \nu^{11} + \cdots - 10\!\cdots\!24 ) / 14\!\cdots\!12$$ (126037762753186476125*v^11 - 544907092573386951849*v^10 + 3199007029650462175031*v^9 - 6940741800865098547612*v^8 + 40971492462543808505353*v^7 + 12090494382928494599052*v^6 + 379113767684947340266462*v^5 + 1330219876312765163559874*v^4 + 3579637355896770456213173*v^3 - 1823378556747139397651160*v^2 - 2464478201962668760690208*v - 1037472490797173716634624) / 1494447217169363677959112 $$\beta_{6}$$ $$=$$ $$( 66\!\cdots\!54 \nu^{11} + \cdots - 29\!\cdots\!64 ) / 74\!\cdots\!56$$ (66228099007199582754*v^11 - 152496912397473678905*v^10 + 1203829421885556315853*v^9 - 586683620774091925017*v^8 + 16323184611201832413746*v^7 + 46921274615037622257749*v^6 + 240008236747661146597952*v^5 + 1146352800661676846914382*v^4 + 3617288095921276266039320*v^3 + 4335428610404080457057673*v^2 + 948209240691057842975424*v - 293031762443060807199264) / 747223608584681838979556 $$\beta_{7}$$ $$=$$ $$( 83\!\cdots\!37 \nu^{11} + \cdots + 12\!\cdots\!00 ) / 74\!\cdots\!56$$ (83396902156913761737*v^11 - 276107005916773088203*v^10 + 1831879984641125084181*v^9 - 2777440564449204357114*v^8 + 24581929157560066398849*v^7 + 31387314263862222513894*v^6 + 286601208392928499439938*v^5 + 1151209497015844599032226*v^4 + 3549029463983996711741205*v^3 + 2258942716368791414859354*v^2 + 557555876326568292963812*v + 122004554345084508889000) / 747223608584681838979556 $$\beta_{8}$$ $$=$$ $$( 11\!\cdots\!73 \nu^{11} + \cdots + 12\!\cdots\!84 ) / 74\!\cdots\!56$$ (115983252017734326873*v^11 - 907731642419289179416*v^10 + 4260581331073491155112*v^9 - 14982870263321782726327*v^8 + 48978231317019716063651*v^7 - 96960610820782954517143*v^6 + 153626675667707795867104*v^5 - 37492592919423453345032*v^4 - 2612100325686493587623925*v^3 - 18180721886271700252456903*v^2 - 12600118248645652827302862*v + 1254998591973740725948584) / 747223608584681838979556 $$\beta_{9}$$ $$=$$ $$( 53\!\cdots\!95 \nu^{11} + \cdots - 13\!\cdots\!12 ) / 14\!\cdots\!12$$ (533015920636260583195*v^11 - 1748513562934959788065*v^10 + 11720122199125039606691*v^9 - 17844911247530639011218*v^8 + 157628596649528102797719*v^7 + 200252009113898715815570*v^6 + 1848747365963578049544906*v^5 + 7368699656691330615509042*v^4 + 22745306836813671960695191*v^3 + 14480156264424739008193918*v^2 - 1216349985487171824694944*v - 13878174711647099254746312) / 1494447217169363677959112 $$\beta_{10}$$ $$=$$ $$( - 55\!\cdots\!27 \nu^{11} + \cdots - 42\!\cdots\!12 ) / 74\!\cdots\!56$$ (-558862071085917679827*v^11 + 2077472016341165596055*v^10 - 13353277182944826227147*v^9 + 24273942814112362823198*v^8 - 176155127320564137754001*v^7 - 147926364176448287773920*v^6 - 1872569628726162495698116*v^5 - 7250437516214319138907242*v^4 - 21294618303526191256564867*v^3 - 10899947865824080520193532*v^2 - 9169360122713995076075770*v - 4297246498670572018907412) / 747223608584681838979556 $$\beta_{11}$$ $$=$$ $$( - 18\!\cdots\!45 \nu^{11} + \cdots + 40\!\cdots\!04 ) / 14\!\cdots\!12$$ (-1822543589778456797545*v^11 + 5971100475229906293843*v^10 - 39301103063307384239393*v^9 + 55671505655896899767166*v^8 - 511206662952446781194181*v^7 - 780310465174366349243918*v^6 - 5956967152038617789901382*v^5 - 25822713828465960937971662*v^4 - 75807033214760810401670173*v^3 - 47868070482912537731555626*v^2 + 3945481256433326717417376*v + 4059780316857578394660304) / 1494447217169363677959112
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{11} + \beta_{8} - \beta_{7} + 10\beta_{6} + \beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} + 2\beta_1$$ b11 + b8 - b7 + 10*b6 + b5 - 2*b4 + b3 - b2 + 2*b1 $$\nu^{3}$$ $$=$$ $$-\beta_{10} - 16\beta_{5} - 17\beta_{4} - 11\beta_{3} - \beta_{2} + \beta _1 - 11$$ -b10 - 16*b5 - 17*b4 - 11*b3 - b2 + b1 - 11 $$\nu^{4}$$ $$=$$ $$- 16 \beta_{11} - 16 \beta_{10} - 16 \beta_{8} + 21 \beta_{7} - 162 \beta_{6} - 154 \beta_{5} - 32 \beta_{4} - 162 \beta_{3} - 32 \beta_{2} - 154$$ -16*b11 - 16*b10 - 16*b8 + 21*b7 - 162*b6 - 154*b5 - 32*b4 - 162*b3 - 32*b2 - 154 $$\nu^{5}$$ $$=$$ $$-32\beta_{11} - 5\beta_{9} + 61\beta_{7} - 315\beta_{6} - 315\beta_{5} - 279\beta_{2} - 61\beta _1 - 429$$ -32*b11 - 5*b9 + 61*b7 - 315*b6 - 315*b5 - 279*b2 - 61*b1 - 429 $$\nu^{6}$$ $$=$$ $$- 29 \beta_{11} + 274 \beta_{10} - 29 \beta_{9} + 245 \beta_{8} + 807 \beta_{7} - 253 \beta_{6} + 807 \beta_{4} + 2805 \beta_{3} - 1249 \beta _1 - 253$$ -29*b11 + 274*b10 - 29*b9 + 245*b8 + 807*b7 - 253*b6 + 807*b4 + 2805*b3 - 1249*b1 - 253 $$\nu^{7}$$ $$=$$ $$- 168 \beta_{11} + 778 \beta_{10} - 778 \beta_{9} - 168 \beta_{8} + 5121 \beta_{7} - 2604 \beta_{6} + 7614 \beta_{5} + 7243 \beta_{4} + 7614 \beta_{3} + 5121 \beta_{2} - 7243 \beta_1$$ -168*b11 + 778*b10 - 778*b9 - 168*b8 + 5121*b7 - 2604*b6 + 7614*b5 + 7243*b4 + 7614*b3 + 5121*b2 - 7243*b1 $$\nu^{8}$$ $$=$$ $$1344 \beta_{10} - 4953 \beta_{9} - 4953 \beta_{8} + 64453 \beta_{5} + 28886 \beta_{4} + 13703 \beta_{3} + 19145 \beta_{2} - 19145 \beta _1 + 13703$$ 1344*b10 - 4953*b9 - 4953*b8 + 64453*b5 + 28886*b4 + 13703*b3 + 19145*b2 - 19145*b1 + 13703 $$\nu^{9}$$ $$=$$ $$4788 \beta_{11} + 4788 \beta_{10} - 13013 \beta_{8} - 98104 \beta_{7} + 62485 \beta_{6} + 238564 \beta_{5} + 60813 \beta_{4} + 62485 \beta_{3} + 60813 \beta_{2} + 238564$$ 4788*b11 + 4788*b10 - 13013*b8 - 98104*b7 + 62485*b6 + 238564*b5 + 60813*b4 + 62485*b3 + 60813*b2 + 238564 $$\nu^{10}$$ $$=$$ $$43012 \beta_{11} + 93316 \beta_{9} - 444424 \beta_{7} + 445776 \beta_{6} + 445776 \beta_{5} + 222753 \beta_{2} + 444424 \beta _1 + 1400254$$ 43012*b11 + 93316*b9 - 444424*b7 + 445776*b6 + 445776*b5 + 222753*b2 + 444424*b1 + 1400254 $$\nu^{11}$$ $$=$$ $$401412 \beta_{11} - 129437 \beta_{10} + 401412 \beta_{9} + 271975 \beta_{8} - 1593097 \beta_{7} + 4013999 \beta_{6} - 1593097 \beta_{4} - 1538878 \beta_{3} + 3545004 \beta _1 + 4013999$$ 401412*b11 - 129437*b10 + 401412*b9 + 271975*b8 - 1593097*b7 + 4013999*b6 - 1593097*b4 - 1538878*b3 + 3545004*b1 + 4013999

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$\beta_{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 −1.92306 + 1.39719i −0.654357 + 0.475418i 3.88644 − 2.82366i −1.17727 + 3.62326i 0.0936861 − 0.288336i 1.27457 − 3.92271i −1.92306 − 1.39719i −0.654357 − 0.475418i 3.88644 + 2.82366i −1.17727 − 3.62326i 0.0936861 + 0.288336i 1.27457 + 3.92271i
−2.73208 1.98497i −0.927051 + 2.85317i 1.05201 + 3.23775i −11.3314 + 8.23272i 8.19624 5.95492i 7.21292 + 22.1991i −4.79582 + 14.7600i −7.28115 5.29007i 47.2999
4.2 −1.46337 1.06320i −0.927051 + 2.85317i −1.46107 4.49672i 17.3626 12.6147i 4.39012 3.18961i −5.78498 17.8043i −7.11451 + 21.8962i −7.28115 5.29007i −38.8200
4.3 3.07742 + 2.23588i −0.927051 + 2.85317i 1.99923 + 6.15301i 2.08679 1.51614i −9.23226 + 6.70763i 0.454025 + 1.39734i 1.79887 5.53636i −7.28115 5.29007i 9.81184
16.1 −0.868251 2.67220i 2.42705 + 1.76336i 0.0853305 0.0619962i 3.76992 11.6026i 2.60475 8.01661i −5.42571 + 3.94201i −18.4246 13.3863i 2.78115 + 8.55951i −34.2777
16.2 0.402703 + 1.23939i 2.42705 + 1.76336i 5.09821 3.70407i −2.50364 + 7.70540i −1.20811 + 3.71818i −0.439746 + 0.319494i 15.0782 + 10.9549i 2.78115 + 8.55951i −10.5582
16.3 1.58358 + 4.87376i 2.42705 + 1.76336i −14.7737 + 10.7337i 4.61569 14.2056i −4.75075 + 14.6213i 9.98349 7.25343i −42.5421 30.9086i 2.78115 + 8.55951i 76.5442
25.1 −2.73208 + 1.98497i −0.927051 2.85317i 1.05201 3.23775i −11.3314 8.23272i 8.19624 + 5.95492i 7.21292 22.1991i −4.79582 14.7600i −7.28115 + 5.29007i 47.2999
25.2 −1.46337 + 1.06320i −0.927051 2.85317i −1.46107 + 4.49672i 17.3626 + 12.6147i 4.39012 + 3.18961i −5.78498 + 17.8043i −7.11451 21.8962i −7.28115 + 5.29007i −38.8200
25.3 3.07742 2.23588i −0.927051 2.85317i 1.99923 6.15301i 2.08679 + 1.51614i −9.23226 6.70763i 0.454025 1.39734i 1.79887 + 5.53636i −7.28115 + 5.29007i 9.81184
31.1 −0.868251 + 2.67220i 2.42705 1.76336i 0.0853305 + 0.0619962i 3.76992 + 11.6026i 2.60475 + 8.01661i −5.42571 3.94201i −18.4246 + 13.3863i 2.78115 8.55951i −34.2777
31.2 0.402703 1.23939i 2.42705 1.76336i 5.09821 + 3.70407i −2.50364 7.70540i −1.20811 3.71818i −0.439746 0.319494i 15.0782 10.9549i 2.78115 8.55951i −10.5582
31.3 1.58358 4.87376i 2.42705 1.76336i −14.7737 10.7337i 4.61569 + 14.2056i −4.75075 14.6213i 9.98349 + 7.25343i −42.5421 + 30.9086i 2.78115 8.55951i 76.5442
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.4.e.c 12
3.b odd 2 1 99.4.f.d 12
11.c even 5 1 inner 33.4.e.c 12
11.c even 5 1 363.4.a.v 6
11.d odd 10 1 363.4.a.u 6
33.f even 10 1 1089.4.a.bk 6
33.h odd 10 1 99.4.f.d 12
33.h odd 10 1 1089.4.a.bi 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.e.c 12 1.a even 1 1 trivial
33.4.e.c 12 11.c even 5 1 inner
99.4.f.d 12 3.b odd 2 1
99.4.f.d 12 33.h odd 10 1
363.4.a.u 6 11.d odd 10 1
363.4.a.v 6 11.c even 5 1
1089.4.a.bi 6 33.h odd 10 1
1089.4.a.bk 6 33.f even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 20 T_{2}^{10} + 64 T_{2}^{9} + 225 T_{2}^{8} + 70 T_{2}^{7} + 3191 T_{2}^{6} + 11905 T_{2}^{5} + 53975 T_{2}^{4} + 92394 T_{2}^{3} + 119980 T_{2}^{2} + 109000 T_{2} + 190096$$ acting on $$S_{4}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 20 T^{10} + 64 T^{9} + \cdots + 190096$$
$3$ $$(T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81)^{3}$$
$5$ $$T^{12} - 28 T^{11} + \cdots + 1310363273521$$
$7$ $$T^{12} - 12 T^{11} + \cdots + 834112161$$
$11$ $$T^{12} + 54 T^{11} + \cdots + 55\!\cdots\!81$$
$13$ $$T^{12} + 18 T^{11} + \cdots + 49\!\cdots\!36$$
$17$ $$T^{12} + 80 T^{11} + \cdots + 25\!\cdots\!76$$
$19$ $$T^{12} + 280 T^{11} + \cdots + 47\!\cdots\!00$$
$23$ $$(T^{6} + 196 T^{5} - 22204 T^{4} + \cdots - 3716146836)^{2}$$
$29$ $$T^{12} - 13 T^{11} + \cdots + 95\!\cdots\!00$$
$31$ $$T^{12} - 413 T^{11} + \cdots + 13\!\cdots\!61$$
$37$ $$T^{12} - 654 T^{11} + \cdots + 18\!\cdots\!76$$
$41$ $$T^{12} + 1490 T^{11} + \cdots + 40\!\cdots\!96$$
$43$ $$(T^{6} - 208 T^{5} + \cdots + 92003355120644)^{2}$$
$47$ $$T^{12} + 150 T^{11} + \cdots + 61\!\cdots\!56$$
$53$ $$T^{12} - 1359 T^{11} + \cdots + 35\!\cdots\!41$$
$59$ $$T^{12} - 1262 T^{11} + \cdots + 22\!\cdots\!25$$
$61$ $$T^{12} + 1044 T^{11} + \cdots + 24\!\cdots\!36$$
$67$ $$(T^{6} + 264 T^{5} + \cdots - 66\!\cdots\!84)^{2}$$
$71$ $$T^{12} - 558 T^{11} + \cdots + 12\!\cdots\!96$$
$73$ $$T^{12} + 699 T^{11} + \cdots + 16\!\cdots\!56$$
$79$ $$T^{12} + 1252 T^{11} + \cdots + 32\!\cdots\!25$$
$83$ $$T^{12} + 4464 T^{11} + \cdots + 18\!\cdots\!81$$
$89$ $$(T^{6} - 158 T^{5} + \cdots - 28\!\cdots\!20)^{2}$$
$97$ $$T^{12} - 1608 T^{11} + \cdots + 11\!\cdots\!21$$