# Properties

 Label 1617.4.a.ba Level $1617$ Weight $4$ Character orbit 1617.a Self dual yes Analytic conductor $95.406$ Analytic rank $0$ Dimension $12$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,4,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1617.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1617 = 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1617.a (trivial)

## 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: yes Analytic conductor: $$95.4060884793$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 4 x^{11} - 79 x^{10} + 310 x^{9} + 2225 x^{8} - 8576 x^{7} - 26761 x^{6} + 101926 x^{5} + \cdots + 275328$$ x^12 - 4*x^11 - 79*x^10 + 310*x^9 + 2225*x^8 - 8576*x^7 - 26761*x^6 + 101926*x^5 + 129494*x^4 - 494728*x^3 - 195584*x^2 + 656256*x + 275328 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}\cdot 3\cdot 7$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

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

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 7) q^{4} + ( - \beta_{4} + \beta_1 - 2) q^{5} - 3 \beta_1 q^{6} + (\beta_{3} + 7 \beta_1 - 1) q^{8} + 9 q^{9}+O(q^{10})$$ q + b1 * q^2 - 3 * q^3 + (b2 + 7) * q^4 + (-b4 + b1 - 2) * q^5 - 3*b1 * q^6 + (b3 + 7*b1 - 1) * q^8 + 9 * q^9 $$q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 7) q^{4} + ( - \beta_{4} + \beta_1 - 2) q^{5} - 3 \beta_1 q^{6} + (\beta_{3} + 7 \beta_1 - 1) q^{8} + 9 q^{9} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \cdots + 9) q^{10}+ \cdots + 99 q^{99}+O(q^{100})$$ q + b1 * q^2 - 3 * q^3 + (b2 + 7) * q^4 + (-b4 + b1 - 2) * q^5 - 3*b1 * q^6 + (b3 + 7*b1 - 1) * q^8 + 9 * q^9 + (-b7 + b5 - b4 + b2 - b1 + 9) * q^10 + 11 * q^11 + (-3*b2 - 21) * q^12 + (b8 - 3) * q^13 + (3*b4 - 3*b1 + 6) * q^15 + (-b7 + b6 + 2*b5 + b4 + b3 + 8*b2 + b1 + 47) * q^16 + (-b11 + b7 - 2*b4 - b2 - 9) * q^17 + 9*b1 * q^18 + (b7 - b6 + b5 + b3 + b2 - b1) * q^19 + (-b10 - b8 + b7 + b6 + 2*b5 - 8*b4 + 4*b3 - 4*b2 + 16*b1 - 10) * q^20 + 11*b1 * q^22 + (-b11 - b10 + b9 + b4 + b3 + 2*b2 - 6*b1 + 45) * q^23 + (-3*b3 - 21*b1 + 3) * q^24 + (b11 + b8 - 2*b7 - b6 + b5 + 2*b4 - b3 + 4*b2 + 3*b1 + 43) * q^25 + (b10 - 2*b9 + 3*b8 - 4*b7 - b6 - 3*b5 + b4 - b3 - b2 - 3*b1 - 4) * q^26 - 27 * q^27 + (2*b10 - b9 + b7 - 2*b5 - 2*b2 + 7*b1 + 4) * q^29 + (3*b7 - 3*b5 + 3*b4 - 3*b2 + 3*b1 - 27) * q^30 + (b10 + b9 - b8 + 2*b7 + b6 + b5 - 4*b4 - 2*b3 + 6*b2 - 4*b1 - 14) * q^31 + (-b10 - 2*b9 - b8 - b7 + 2*b6 + b5 - 16*b4 + 6*b3 + 2*b2 + 63*b1 + 12) * q^32 - 33 * q^33 + (2*b11 + 2*b9 + 2*b8 - 4*b7 - 2*b6 - 2*b5 + 2*b4 - 4*b3 - 5*b2 - 18*b1 - 17) * q^34 + (9*b2 + 63) * q^36 + (b11 + b10 + 2*b9 + 2*b7 - 3*b6 + b5 + 4*b4 + 4*b3 + 7*b2 - 9*b1 + 99) * q^37 + (3*b11 + b10 + 2*b9 - 2*b8 - 6*b7 + 2*b5 - 5*b4 + 4*b2 + 6*b1 - 17) * q^38 + (-3*b8 + 9) * q^39 + (-2*b10 - 6*b8 - 9*b7 + 6*b6 + 11*b5 - 3*b4 + 4*b3 + 31*b2 - 19*b1 + 111) * q^40 + (-3*b11 - 2*b10 + b9 - 2*b8 + 4*b7 + 2*b6 + 2*b4 + 2*b3 - 7*b2 - b1 - 69) * q^41 + (2*b10 - b9 + b8 - b7 - 2*b6 - 2*b5 - 4*b4 - 2*b3 - 6*b2 + 11*b1 + 43) * q^43 + (11*b2 + 77) * q^44 + (-9*b4 + 9*b1 - 18) * q^45 + (-3*b11 - 4*b10 + 4*b9 - b8 + 7*b7 + b6 - 3*b4 + 6*b3 - 7*b2 + 69*b1 - 104) * q^46 + (b11 + 2*b10 + b9 - b8 + 4*b7 - 2*b5 - b4 - 2*b3 + 9*b2 - 28*b1 - 21) * q^47 + (3*b7 - 3*b6 - 6*b5 - 3*b4 - 3*b3 - 24*b2 - 3*b1 - 141) * q^48 + (-b11 - 2*b9 - b8 + 4*b7 + b6 - b5 - 22*b4 + 7*b3 - 4*b2 + 80*b1 + 54) * q^50 + (3*b11 - 3*b7 + 6*b4 + 3*b2 + 27) * q^51 + (3*b10 - 8*b9 + 7*b8 + 3*b7 - 9*b5 - 4*b4 - 7*b3 + 10*b2 - 6*b1 - 15) * q^52 + (-b10 - 2*b9 + 3*b8 - 2*b6 - 4*b5 - 8*b4 - 5*b3 + 11*b2 + 34*b1 + 80) * q^53 - 27*b1 * q^54 + (-11*b4 + 11*b1 - 22) * q^55 + (-3*b7 + 3*b6 - 3*b5 - 3*b3 - 3*b2 + 3*b1) * q^57 + (5*b11 + 6*b10 - 2*b9 + 7*b8 - 5*b7 - 4*b6 - 5*b5 + 27*b4 - 12*b3 + 24*b2 - 26*b1 + 127) * q^58 + (b11 + b9 + 7*b8 - 8*b6 - 10*b5 + 4*b4 - 2*b3 - 3*b2 + 49*b1 - 117) * q^59 + (3*b10 + 3*b8 - 3*b7 - 3*b6 - 6*b5 + 24*b4 - 12*b3 + 12*b2 - 48*b1 + 30) * q^60 + (-b10 - 5*b9 + 4*b8 - 3*b6 - 5*b5 - 11*b4 + 2*b3 + 12*b2 - 19*b1 + 87) * q^61 + (2*b11 + 2*b10 + 2*b9 - 2*b8 - 4*b7 - 6*b6 - 2*b5 + 6*b4 + 3*b3 - 26*b2 + 26*b1 - 77) * q^62 + (-2*b10 - 6*b9 - 2*b8 - 20*b7 + 4*b6 + 16*b5 - 26*b4 + 4*b3 + 49*b2 + 50*b1 + 455) * q^64 + (-8*b11 - 6*b10 + 2*b9 - 10*b8 + 8*b7 + 6*b6 + 2*b5 - 4*b4 - 4*b3 + 2*b2 - 32*b1 - 4) * q^65 - 33*b1 * q^66 + (-b11 - b10 - b9 + 8*b8 - 2*b7 + 2*b5 + 4*b4 - 5*b3 + 16*b2 + 25*b1 + 116) * q^67 + (-2*b11 - 2*b10 + 18*b7 - 2*b6 - 4*b5 - b3 - 30*b2 - 47*b1 - 171) * q^68 + (3*b11 + 3*b10 - 3*b9 - 3*b4 - 3*b3 - 6*b2 + 18*b1 - 135) * q^69 + (3*b11 - b10 + 7*b9 + 6*b8 + 8*b7 - 8*b6 - 6*b5 + 8*b4 - 7*b3 - 18*b2 - 7*b1 + 191) * q^71 + (9*b3 + 63*b1 - 9) * q^72 + (2*b10 + 2*b9 - 5*b8 + 2*b7 + 2*b6 + 2*b5 + 8*b4 - 4*b3 + 52*b2 - 14*b1 + 9) * q^73 + (3*b11 + 3*b10 + 8*b9 - 6*b8 + b7 - b6 + 6*b5 + 4*b4 + 8*b3 + 28*b2 + 152*b1 - 111) * q^74 + (-3*b11 - 3*b8 + 6*b7 + 3*b6 - 3*b5 - 6*b4 + 3*b3 - 12*b2 - 9*b1 - 129) * q^75 + (-9*b11 - 5*b10 + 2*b9 - 14*b8 + 19*b7 + 15*b6 + 16*b5 - 54*b4 + 17*b3 + 10*b2 + 58*b1 + 74) * q^76 + (-3*b10 + 6*b9 - 9*b8 + 12*b7 + 3*b6 + 9*b5 - 3*b4 + 3*b3 + 3*b2 + 9*b1 + 12) * q^78 + (-8*b11 - 5*b10 - 2*b9 - 6*b8 + 12*b7 + 6*b6 - 17*b4 + 11*b3 - 5*b2 + 43*b1 + 17) * q^79 + (-8*b11 - 11*b10 - 19*b8 + 17*b7 + 15*b6 + 20*b5 - 66*b4 + 48*b3 + 6*b2 + 302*b1 - 296) * q^80 + 81 * q^81 + (-b11 - 6*b10 + 8*b9 - 3*b8 + b7 - 3*b5 + 27*b4 - 6*b3 - 9*b2 - 130*b1 - 24) * q^82 + (-5*b11 - 4*b10 - 8*b9 + 5*b8 - 8*b7 + 7*b6 + 7*b5 + 12*b4 + 5*b3 + 2*b2 + 97*b1 + 66) * q^83 + (-3*b10 - b9 + 8*b8 - 2*b7 - 3*b6 + 3*b5 + 27*b4 - 16*b3 - 6*b2 - 51*b1 + 319) * q^85 + (5*b11 + 5*b10 + 8*b8 - b7 - 5*b6 - 4*b5 - 2*b4 - 17*b3 + 13*b2 - 9*b1 + 165) * q^86 + (-6*b10 + 3*b9 - 3*b7 + 6*b5 + 6*b2 - 21*b1 - 12) * q^87 + (11*b3 + 77*b1 - 11) * q^88 + (8*b11 + 6*b10 + 4*b9 + 2*b8 - 2*b6 - 6*b5 - 20*b4 - 2*b3 + 2*b2 - 6*b1 - 54) * q^89 + (-9*b7 + 9*b5 - 9*b4 + 9*b2 - 9*b1 + 81) * q^90 + (b11 - b10 + 6*b9 - 8*b8 - 8*b7 + b6 + 9*b5 + 31*b4 - 8*b3 + 64*b2 - 106*b1 + 613) * q^92 + (-3*b10 - 3*b9 + 3*b8 - 6*b7 - 3*b6 - 3*b5 + 12*b4 + 6*b3 - 18*b2 + 12*b1 + 42) * q^93 + (3*b11 + 7*b10 + 2*b9 - 4*b7 - 9*b6 - 5*b5 + 47*b4 - 3*b3 - 31*b2 + 23*b1 - 397) * q^94 + (-2*b11 + 10*b10 - 9*b9 - 9*b8 - 2*b7 + 13*b6 + 9*b5 - 18*b4 - 15*b3 + 75*b2 + 56*b1 + 163) * q^95 + (3*b10 + 6*b9 + 3*b8 + 3*b7 - 6*b6 - 3*b5 + 48*b4 - 18*b3 - 6*b2 - 189*b1 - 36) * q^96 + (12*b11 + 7*b10 + 9*b9 + 7*b8 - 15*b6 - 7*b5 + 14*b4 - 16*b3 + 20*b1 + 39) * q^97 + 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 4 q^{2} - 36 q^{3} + 78 q^{4} - 20 q^{5} - 12 q^{6} + 18 q^{8} + 108 q^{9}+O(q^{10})$$ 12 * q + 4 * q^2 - 36 * q^3 + 78 * q^4 - 20 * q^5 - 12 * q^6 + 18 * q^8 + 108 * q^9 $$12 q + 4 q^{2} - 36 q^{3} + 78 q^{4} - 20 q^{5} - 12 q^{6} + 18 q^{8} + 108 q^{9} + 100 q^{10} + 132 q^{11} - 234 q^{12} - 32 q^{13} + 60 q^{15} + 526 q^{16} - 100 q^{17} + 36 q^{18} - 6 q^{19} - 28 q^{20} + 44 q^{22} + 500 q^{23} - 54 q^{24} + 506 q^{25} - 38 q^{26} - 324 q^{27} + 96 q^{29} - 300 q^{30} - 226 q^{31} + 398 q^{32} - 396 q^{33} - 262 q^{34} + 702 q^{36} + 1114 q^{37} - 218 q^{38} + 96 q^{39} + 1068 q^{40} - 800 q^{41} + 604 q^{43} + 858 q^{44} - 180 q^{45} - 948 q^{46} - 428 q^{47} - 1578 q^{48} + 1010 q^{50} + 300 q^{51} - 224 q^{52} + 1028 q^{53} - 108 q^{54} - 220 q^{55} + 18 q^{57} + 1292 q^{58} - 1192 q^{59} + 84 q^{60} + 922 q^{61} - 674 q^{62} + 5414 q^{64} - 248 q^{65} - 132 q^{66} + 1424 q^{67} - 2074 q^{68} - 1500 q^{69} + 2332 q^{71} + 162 q^{72} - 284 q^{73} - 914 q^{74} - 1518 q^{75} + 1060 q^{76} + 114 q^{78} + 408 q^{79} - 2348 q^{80} + 972 q^{81} - 838 q^{82} + 1238 q^{83} + 3658 q^{85} + 1866 q^{86} - 288 q^{87} + 198 q^{88} - 700 q^{89} + 900 q^{90} + 6488 q^{92} + 678 q^{93} - 4488 q^{94} + 1762 q^{95} - 1194 q^{96} + 498 q^{97} + 1188 q^{99}+O(q^{100})$$ 12 * q + 4 * q^2 - 36 * q^3 + 78 * q^4 - 20 * q^5 - 12 * q^6 + 18 * q^8 + 108 * q^9 + 100 * q^10 + 132 * q^11 - 234 * q^12 - 32 * q^13 + 60 * q^15 + 526 * q^16 - 100 * q^17 + 36 * q^18 - 6 * q^19 - 28 * q^20 + 44 * q^22 + 500 * q^23 - 54 * q^24 + 506 * q^25 - 38 * q^26 - 324 * q^27 + 96 * q^29 - 300 * q^30 - 226 * q^31 + 398 * q^32 - 396 * q^33 - 262 * q^34 + 702 * q^36 + 1114 * q^37 - 218 * q^38 + 96 * q^39 + 1068 * q^40 - 800 * q^41 + 604 * q^43 + 858 * q^44 - 180 * q^45 - 948 * q^46 - 428 * q^47 - 1578 * q^48 + 1010 * q^50 + 300 * q^51 - 224 * q^52 + 1028 * q^53 - 108 * q^54 - 220 * q^55 + 18 * q^57 + 1292 * q^58 - 1192 * q^59 + 84 * q^60 + 922 * q^61 - 674 * q^62 + 5414 * q^64 - 248 * q^65 - 132 * q^66 + 1424 * q^67 - 2074 * q^68 - 1500 * q^69 + 2332 * q^71 + 162 * q^72 - 284 * q^73 - 914 * q^74 - 1518 * q^75 + 1060 * q^76 + 114 * q^78 + 408 * q^79 - 2348 * q^80 + 972 * q^81 - 838 * q^82 + 1238 * q^83 + 3658 * q^85 + 1866 * q^86 - 288 * q^87 + 198 * q^88 - 700 * q^89 + 900 * q^90 + 6488 * q^92 + 678 * q^93 - 4488 * q^94 + 1762 * q^95 - 1194 * q^96 + 498 * q^97 + 1188 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} - 79 x^{10} + 310 x^{9} + 2225 x^{8} - 8576 x^{7} - 26761 x^{6} + 101926 x^{5} + \cdots + 275328$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 15$$ v^2 - 15 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 23\nu + 1$$ v^3 - 23*v + 1 $$\beta_{4}$$ $$=$$ $$( - 1573 \nu^{11} + 3847 \nu^{10} + 130234 \nu^{9} - 285364 \nu^{8} - 3944969 \nu^{7} + \cdots + 169295232 ) / 9515520$$ (-1573*v^11 + 3847*v^10 + 130234*v^9 - 285364*v^8 - 3944969*v^7 + 7332911*v^6 + 53071300*v^5 - 74777098*v^4 - 300768392*v^3 + 234875776*v^2 + 507716864*v + 169295232) / 9515520 $$\beta_{5}$$ $$=$$ $$( - 13999 \nu^{11} + 19189 \nu^{10} + 1161646 \nu^{9} - 1341628 \nu^{8} - 34734875 \nu^{7} + \cdots + 1189817472 ) / 9515520$$ (-13999*v^11 + 19189*v^10 + 1161646*v^9 - 1341628*v^8 - 34734875*v^7 + 31977581*v^6 + 448650700*v^5 - 297853054*v^4 - 2332530008*v^3 + 861985216*v^2 + 3454572032*v + 1189817472) / 9515520 $$\beta_{6}$$ $$=$$ $$( 175 \nu^{11} - 249 \nu^{10} - 14522 \nu^{9} + 17468 \nu^{8} + 434139 \nu^{7} - 424609 \nu^{6} + \cdots + 1686912 ) / 113280$$ (175*v^11 - 249*v^10 - 14522*v^9 + 17468*v^8 + 434139*v^7 - 424609*v^6 - 5586200*v^5 + 4283910*v^4 + 28347856*v^3 - 17097264*v^2 - 36304320*v + 1686912) / 113280 $$\beta_{7}$$ $$=$$ $$( - 4957 \nu^{11} + 7103 \nu^{10} + 411226 \nu^{9} - 500436 \nu^{8} - 12315681 \nu^{7} + \cdots + 465506688 ) / 3171840$$ (-4957*v^11 + 7103*v^10 + 411226*v^9 - 500436*v^8 - 12315681*v^7 + 11873639*v^6 + 160377300*v^5 - 106716762*v^4 - 858364328*v^3 + 275724224*v^2 + 1385985536*v + 465506688) / 3171840 $$\beta_{8}$$ $$=$$ $$( 2487 \nu^{11} - 3629 \nu^{10} - 207054 \nu^{9} + 245068 \nu^{8} + 6263747 \nu^{7} + \cdots - 267421440 ) / 792960$$ (2487*v^11 - 3629*v^10 - 207054*v^9 + 245068*v^8 + 6263747*v^7 - 5495205*v^6 - 83353660*v^5 + 45601342*v^4 + 465631032*v^3 - 101435360*v^2 - 817119936*v - 267421440) / 792960 $$\beta_{9}$$ $$=$$ $$( 31127 \nu^{11} - 45917 \nu^{10} - 2580878 \nu^{9} + 3264284 \nu^{8} + 77544355 \nu^{7} + \cdots - 2495054976 ) / 9515520$$ (31127*v^11 - 45917*v^10 - 2580878*v^9 + 3264284*v^8 + 77544355*v^7 - 79504213*v^6 - 1020902780*v^5 + 762360302*v^4 + 5603262904*v^3 - 2318910848*v^2 - 9375636736*v - 2495054976) / 9515520 $$\beta_{10}$$ $$=$$ $$( - 18329 \nu^{11} + 14939 \nu^{10} + 1525466 \nu^{9} - 904628 \nu^{8} - 45993325 \nu^{7} + \cdots + 2726477952 ) / 4757760$$ (-18329*v^11 + 14939*v^10 + 1525466*v^9 - 904628*v^8 - 45993325*v^7 + 16323331*v^6 + 606215180*v^5 - 66754514*v^4 - 3307605688*v^3 - 360731584*v^2 + 5545044352*v + 2726477952) / 4757760 $$\beta_{11}$$ $$=$$ $$( - 49289 \nu^{11} + 85571 \nu^{10} + 4022642 \nu^{9} - 5931812 \nu^{8} - 118366717 \nu^{7} + \cdots + 4231245696 ) / 9515520$$ (-49289*v^11 + 85571*v^10 + 4022642*v^9 - 5931812*v^8 - 118366717*v^7 + 137986123*v^6 + 1515599300*v^5 - 1211044754*v^4 - 8041988296*v^3 + 3070780928*v^2 + 13158295552*v + 4231245696) / 9515520
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 15$$ b2 + 15 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 23\beta _1 - 1$$ b3 + 23*b1 - 1 $$\nu^{4}$$ $$=$$ $$-\beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} + 32\beta_{2} + \beta _1 + 343$$ -b7 + b6 + 2*b5 + b4 + b3 + 32*b2 + b1 + 343 $$\nu^{5}$$ $$=$$ $$- \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} - 16 \beta_{4} + 38 \beta_{3} + \cdots - 20$$ -b10 - 2*b9 - b8 - b7 + 2*b6 + b5 - 16*b4 + 38*b3 + 2*b2 + 607*b1 - 20 $$\nu^{6}$$ $$=$$ $$- 2 \beta_{10} - 6 \beta_{9} - 2 \beta_{8} - 60 \beta_{7} + 44 \beta_{6} + 96 \beta_{5} + 14 \beta_{4} + \cdots + 8927$$ -2*b10 - 6*b9 - 2*b8 - 60*b7 + 44*b6 + 96*b5 + 14*b4 + 44*b3 + 945*b2 + 90*b1 + 8927 $$\nu^{7}$$ $$=$$ $$- 60 \beta_{10} - 96 \beta_{9} - 56 \beta_{8} - 48 \beta_{7} + 116 \beta_{6} + 100 \beta_{5} + \cdots + 53$$ -60*b10 - 96*b9 - 56*b8 - 48*b7 + 116*b6 + 100*b5 - 916*b4 + 1243*b3 + 124*b2 + 16943*b1 + 53 $$\nu^{8}$$ $$=$$ $$8 \beta_{11} - 128 \beta_{10} - 312 \beta_{9} - 176 \beta_{8} - 2539 \beta_{7} + 1563 \beta_{6} + \cdots + 245683$$ 8*b11 - 128*b10 - 312*b9 - 176*b8 - 2539*b7 + 1563*b6 + 3622*b5 - 405*b4 + 1663*b3 + 27670*b2 + 4619*b1 + 245683 $$\nu^{9}$$ $$=$$ $$- 120 \beta_{11} - 2651 \beta_{10} - 3414 \beta_{9} - 2547 \beta_{8} - 1591 \beta_{7} + 4954 \beta_{6} + \cdots + 22598$$ -120*b11 - 2651*b10 - 3414*b9 - 2547*b8 - 1591*b7 + 4954*b6 + 5515*b5 - 38084*b4 + 39412*b3 + 6394*b2 + 486395*b1 + 22598 $$\nu^{10}$$ $$=$$ $$616 \beta_{11} - 6146 \beta_{10} - 11402 \beta_{9} - 9850 \beta_{8} - 93758 \beta_{7} + 52146 \beta_{6} + \cdots + 6963187$$ 616*b11 - 6146*b10 - 11402*b9 - 9850*b8 - 93758*b7 + 52146*b6 + 126128*b5 - 37368*b4 + 60358*b3 + 813971*b2 + 199428*b1 + 6963187 $$\nu^{11}$$ $$=$$ $$- 9880 \beta_{11} - 103882 \beta_{10} - 108628 \beta_{9} - 105654 \beta_{8} - 45938 \beta_{7} + \cdots + 1370863$$ -9880*b11 - 103882*b10 - 108628*b9 - 105654*b8 - 45938*b7 + 188276*b6 + 243386*b5 - 1401904*b4 + 1240073*b3 + 290284*b2 + 14204811*b1 + 1370863

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −5.45929 −4.98543 −3.63350 −2.95517 −1.14635 −0.428288 2.09996 2.60515 2.77685 4.42348 5.06724 5.63534
−5.45929 −3.00000 21.8038 −18.9501 16.3779 0 −75.3590 9.00000 103.454
1.2 −4.98543 −3.00000 16.8545 7.45523 14.9563 0 −44.1437 9.00000 −37.1676
1.3 −3.63350 −3.00000 5.20232 9.34572 10.9005 0 10.1654 9.00000 −33.9577
1.4 −2.95517 −3.00000 0.733054 −4.94230 8.86552 0 21.4751 9.00000 14.6054
1.5 −1.14635 −3.00000 −6.68589 −17.9065 3.43904 0 16.8351 9.00000 20.5270
1.6 −0.428288 −3.00000 −7.81657 −4.03964 1.28486 0 6.77404 9.00000 1.73013
1.7 2.09996 −3.00000 −3.59017 −11.8735 −6.29988 0 −24.3389 9.00000 −24.9339
1.8 2.60515 −3.00000 −1.21321 12.9406 −7.81544 0 −24.0018 9.00000 33.7122
1.9 2.77685 −3.00000 −0.289077 15.3328 −8.33056 0 −23.0176 9.00000 42.5771
1.10 4.42348 −3.00000 11.5672 −10.8900 −13.2705 0 15.7795 9.00000 −48.1715
1.11 5.06724 −3.00000 17.6769 −13.6349 −15.2017 0 49.0353 9.00000 −69.0911
1.12 5.63534 −3.00000 23.7571 17.1624 −16.9060 0 88.7965 9.00000 96.7160
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1617.4.a.ba 12
7.b odd 2 1 1617.4.a.bb 12
7.d odd 6 2 231.4.i.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.i.d 24 7.d odd 6 2
1617.4.a.ba 12 1.a even 1 1 trivial
1617.4.a.bb 12 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1617))$$:

 $$T_{2}^{12} - 4 T_{2}^{11} - 79 T_{2}^{10} + 310 T_{2}^{9} + 2225 T_{2}^{8} - 8576 T_{2}^{7} + \cdots + 275328$$ T2^12 - 4*T2^11 - 79*T2^10 + 310*T2^9 + 2225*T2^8 - 8576*T2^7 - 26761*T2^6 + 101926*T2^5 + 129494*T2^4 - 494728*T2^3 - 195584*T2^2 + 656256*T2 + 275328 $$T_{5}^{12} + 20 T_{5}^{11} - 803 T_{5}^{10} - 16668 T_{5}^{9} + 229764 T_{5}^{8} + 5157804 T_{5}^{7} + \cdots - 2833859960832$$ T5^12 + 20*T5^11 - 803*T5^10 - 16668*T5^9 + 229764*T5^8 + 5157804*T5^7 - 26811684*T5^6 - 729582816*T5^5 + 808951872*T5^4 + 46283211776*T5^3 + 54040171264*T5^2 - 1037166352384*T5 - 2833859960832

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 4 T^{11} + \cdots + 275328$$
$3$ $$(T + 3)^{12}$$
$5$ $$T^{12} + \cdots - 2833859960832$$
$7$ $$T^{12}$$
$11$ $$(T - 11)^{12}$$
$13$ $$T^{12} + \cdots - 20\!\cdots\!88$$
$17$ $$T^{12} + \cdots + 19\!\cdots\!24$$
$19$ $$T^{12} + \cdots + 12\!\cdots\!68$$
$23$ $$T^{12} + \cdots - 83\!\cdots\!28$$
$29$ $$T^{12} + \cdots - 11\!\cdots\!44$$
$31$ $$T^{12} + \cdots - 81\!\cdots\!08$$
$37$ $$T^{12} + \cdots - 20\!\cdots\!28$$
$41$ $$T^{12} + \cdots + 72\!\cdots\!60$$
$43$ $$T^{12} + \cdots + 76\!\cdots\!24$$
$47$ $$T^{12} + \cdots - 83\!\cdots\!16$$
$53$ $$T^{12} + \cdots + 22\!\cdots\!28$$
$59$ $$T^{12} + \cdots + 92\!\cdots\!48$$
$61$ $$T^{12} + \cdots - 15\!\cdots\!20$$
$67$ $$T^{12} + \cdots + 99\!\cdots\!08$$
$71$ $$T^{12} + \cdots + 16\!\cdots\!92$$
$73$ $$T^{12} + \cdots - 47\!\cdots\!72$$
$79$ $$T^{12} + \cdots - 13\!\cdots\!80$$
$83$ $$T^{12} + \cdots - 88\!\cdots\!68$$
$89$ $$T^{12} + \cdots + 15\!\cdots\!36$$
$97$ $$T^{12} + \cdots + 54\!\cdots\!04$$