Properties

 Label 495.2.n.d Level $495$ Weight $2$ Character orbit 495.n Analytic conductor $3.953$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 495.n (of order $$5$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.95259490005$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.13140625.1 Defining polynomial: $$x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1$$ x^8 - 3*x^7 + 5*x^6 - 3*x^5 + 4*x^4 + 3*x^3 + 5*x^2 + 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{2} + (\beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{4} + \beta_{7} q^{5} + (\beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{2} + 1) q^{7} + (3 \beta_{7} + \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{8}+O(q^{10})$$ q + (b6 + b5 - b4 + b3 - 1) * q^2 + (b6 + 2*b5 + b4 - 2*b2) * q^4 + b7 * q^5 + (b7 - 2*b6 + b5 - b2 + 1) * q^7 + (3*b7 + b6 - 2*b5 + 3*b4 - b3 - b2 + 2*b1 + 1) * q^8 $$q + (\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{2} + (\beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{4} + \beta_{7} q^{5} + (\beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{2} + 1) q^{7} + (3 \beta_{7} + \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{8} + ( - \beta_{7} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{10} + ( - \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{11} + (\beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 2) q^{13} + ( - 2 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + \cdots - 1) q^{14}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 10) q^{98}+O(q^{100})$$ q + (b6 + b5 - b4 + b3 - 1) * q^2 + (b6 + 2*b5 + b4 - 2*b2) * q^4 + b7 * q^5 + (b7 - 2*b6 + b5 - b2 + 1) * q^7 + (3*b7 + b6 - 2*b5 + 3*b4 - b3 - b2 + 2*b1 + 1) * q^8 + (-b7 + b3 - b2 - b1 - 1) * q^10 + (-b7 + 2*b6 + 3*b5 - 2*b4 + b3 - 3*b2 - 2*b1) * q^11 + (b6 + b5 + 2*b4 - 2*b3 - 2*b1 + 2) * q^13 + (-2*b7 - 3*b6 - 5*b5 - 2*b4 + b3 + 3*b2 + 5*b1 - 1) * q^14 + (5*b6 - 5*b2 - 5*b1) * q^16 - 5*b7 * q^17 + (-b7 - 3*b6 - 5*b5 - b4 + b3 + 3*b2 + 5*b1 - 1) * q^19 + (-2*b6 - 2*b5 - b3 + b1) * q^20 + (3*b7 - b6 - b5 + 3*b4 - 2*b3 - 2*b2 + 3*b1 - 1) * q^22 + (-2*b7 + 2*b3 - b2 - b1 + 1) * q^23 + (-b7 - b4 + b3 - 1) * q^25 + (b7 + 2*b5 - 2*b4 - 2*b2 + 1) * q^26 + (-4*b4 + 4*b3 + 5*b2) * q^28 + (-b7 + b6 + 4*b5 - 4*b4 - 4*b2 - 1) * q^29 + (-b4 + 3*b1 - 1) * q^31 + (b7 - 4*b5 - b3 - 4*b2 - 4*b1 - 3) * q^32 + (5*b7 - 5*b3 + 5*b2 + 5*b1 + 5) * q^34 + (-b6 - b5 - b4 + b3 + 3*b1 - 1) * q^35 + (-4*b7 - b6 - b5 - 3*b4 + b2 - 4) * q^37 + (-b7 + 3*b6 - 2*b4 + 2*b3 + 7*b2 - 3*b1) * q^38 + (-2*b7 - b6 - 3*b5 - 3*b4 + 3*b2 - 2) * q^40 + (6*b7 - 4*b6 - 3*b5 + 6*b4 - 7*b3 + 4*b2 + 3*b1 + 7) * q^41 + (-6*b7 - b5 + 6*b3 + b2 + b1 - 4) * q^43 + (5*b7 + 2*b6 - 5*b5 - 3*b3 - 2*b2 - b1 + 3) * q^44 + (b6 + b5 - 2*b4 - 2*b3 + 2*b1 - 2) * q^46 + (-6*b7 + 4*b6 + 5*b5 - 6*b4 + 5*b3 - 4*b2 - 5*b1 - 5) * q^47 + (-b7 - b6 - 6*b4 + 6*b3 - 2*b2 + b1) * q^49 + (-b6 + b4 - b3 + b2 + b1) * q^50 + (b7 + b6 + b5 + b4 - 2*b3 - b2 - b1 + 2) * q^52 + (-3*b6 - 3*b5 + b4 + 4*b3 - 3*b1 + 1) * q^53 + (b7 - 3*b6 - b5 + b4 + b3 + b1) * q^55 + (-7*b7 + 4*b5 + 7*b3 + 4*b2 + 4*b1 - 10) * q^56 + (6*b7 - 3*b6 - 7*b5 + 6*b4 - 2*b3 + 3*b2 + 7*b1 + 2) * q^58 + (-7*b7 + b6 + b5 - 3*b4 - b2 - 7) * q^59 + (-8*b7 + 6*b6 - 5*b4 + 5*b3 - 3*b2 - 6*b1) * q^61 + (3*b7 + 4*b6 - b5 + 4*b4 + b2 + 3) * q^62 + (-2*b6 - 2*b5 - 5*b4 - 6*b3 - 9*b1 - 5) * q^64 + (2*b7 + 2*b5 - 2*b3 - b2 - b1 + 2) * q^65 + (4*b7 + 10*b5 - 4*b3 - 5*b2 - 5*b1) * q^67 + (10*b6 + 10*b5 + 5*b3 - 5*b1) * q^68 + (b7 + 3*b6 - 2*b5 + 2*b4 + 2*b2 + 1) * q^70 + (7*b7 - 2*b6 + b4 - b3 - 3*b2 + 2*b1) * q^71 + (b7 + 3*b6 - 5*b5 + 8*b4 + 5*b2 + 1) * q^73 + (2*b7 - 9*b6 - 4*b5 + 2*b4 - 4*b3 + 9*b2 + 4*b1 + 4) * q^74 + (-8*b7 + 12*b5 + 8*b3 - b2 - b1 - 8) * q^76 + (3*b7 - 3*b6 - 2*b5 - 4*b4 + 3*b3 + b2 + 4*b1 + 6) * q^77 + (2*b6 + 2*b5 + 5*b4 - 4*b3 + 5) * q^79 + (-5*b6 + 5*b2) * q^80 + (-4*b7 + 4*b6 - 11*b4 + 11*b3 + 3*b2 - 4*b1) * q^82 + (-9*b7 + 2*b6 + 3*b4 - 3*b3 - 5*b2 - 2*b1) * q^83 + (5*b7 + 5*b4 - 5*b3 + 5) * q^85 + (4*b6 + 4*b5 + 4*b4 - 9*b3 + 4*b1 + 4) * q^86 + (-b7 - 2*b6 - 6*b5 - b4 + 8*b3 - 2*b2 - 9*b1 - 7) * q^88 + (5*b7 + 2*b5 - 5*b3 - 2) * q^89 + (5*b7 + 2*b6 + 7*b5 + 5*b4 - 11*b3 - 2*b2 - 7*b1 + 11) * q^91 + (b7 + 2*b6 + 1) * q^92 + (4*b7 - 8*b6 + 9*b4 - 9*b3 - b2 + 8*b1) * q^94 + (3*b6 - 2*b5 + b4 + 2*b2) * q^95 + (3*b6 + 3*b5 - b4 - 2*b3 + b1 - 1) * q^97 + (-3*b7 + 3*b3 + 3*b2 + 3*b1 - 10) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{4} - 2 q^{5} + q^{7} - 5 q^{8}+O(q^{10})$$ 8 * q - 2 * q^4 - 2 * q^5 + q^7 - 5 * q^8 $$8 q - 2 q^{4} - 2 q^{5} + q^{7} - 5 q^{8} - 10 q^{10} + 3 q^{11} + 6 q^{13} + 10 q^{14} - 20 q^{16} + 10 q^{17} + 6 q^{19} - 7 q^{20} - 25 q^{22} + 10 q^{23} - 2 q^{25} + 8 q^{26} + 31 q^{28} + 3 q^{31} - 60 q^{32} + 50 q^{34} + q^{35} - 19 q^{37} + 28 q^{38} - 5 q^{40} + 25 q^{41} - 4 q^{43} - 7 q^{44} - 6 q^{46} - 15 q^{47} + 21 q^{49} + 6 q^{52} - 7 q^{53} - 7 q^{55} - 20 q^{56} - 2 q^{58} - 35 q^{59} + 21 q^{61} + 19 q^{62} - 77 q^{64} + 6 q^{65} - 26 q^{67} + 35 q^{68} + 10 q^{70} - 25 q^{71} + q^{73} + 29 q^{74} - 14 q^{76} + 61 q^{77} + 30 q^{79} + 5 q^{80} + 57 q^{82} - 11 q^{83} + 10 q^{85} + 34 q^{86} - 85 q^{88} - 32 q^{89} + 37 q^{91} + 10 q^{92} - 39 q^{94} + 6 q^{95} + 5 q^{97} - 50 q^{98}+O(q^{100})$$ 8 * q - 2 * q^4 - 2 * q^5 + q^7 - 5 * q^8 - 10 * q^10 + 3 * q^11 + 6 * q^13 + 10 * q^14 - 20 * q^16 + 10 * q^17 + 6 * q^19 - 7 * q^20 - 25 * q^22 + 10 * q^23 - 2 * q^25 + 8 * q^26 + 31 * q^28 + 3 * q^31 - 60 * q^32 + 50 * q^34 + q^35 - 19 * q^37 + 28 * q^38 - 5 * q^40 + 25 * q^41 - 4 * q^43 - 7 * q^44 - 6 * q^46 - 15 * q^47 + 21 * q^49 + 6 * q^52 - 7 * q^53 - 7 * q^55 - 20 * q^56 - 2 * q^58 - 35 * q^59 + 21 * q^61 + 19 * q^62 - 77 * q^64 + 6 * q^65 - 26 * q^67 + 35 * q^68 + 10 * q^70 - 25 * q^71 + q^73 + 29 * q^74 - 14 * q^76 + 61 * q^77 + 30 * q^79 + 5 * q^80 + 57 * q^82 - 11 * q^83 + 10 * q^85 + 34 * q^86 - 85 * q^88 - 32 * q^89 + 37 * q^91 + 10 * q^92 - 39 * q^94 + 6 * q^95 + 5 * q^97 - 50 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8$$ (-v^7 + 2*v^6 - 3*v^5 - 4*v^3 - 7*v^2 - 12*v - 7) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8$$ (v^7 - 7*v^5 + 20*v^4 - 16*v^3 + 19*v^2 + 6*v + 9) / 8 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8$$ (-v^7 + 4*v^6 - 9*v^5 + 12*v^4 - 16*v^3 + 13*v^2 - 10*v - 1) / 8 $$\beta_{5}$$ $$=$$ $$( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8$$ (-3*v^7 + 10*v^6 - 17*v^5 + 8*v^4 - 4*v^3 - 13*v^2 - 8*v - 5) / 8 $$\beta_{6}$$ $$=$$ $$( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8$$ (3*v^7 - 12*v^6 + 23*v^5 - 20*v^4 + 16*v^3 + v^2 + 6*v - 1) / 8 $$\beta_{7}$$ $$=$$ $$( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8$$ (-5*v^7 + 18*v^6 - 35*v^5 + 32*v^4 - 28*v^3 - 11*v^2 - 12*v - 7) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}$$ b6 + b5 + b4 - b2 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1$$ b7 + 3*b6 + 2*b5 + b4 - 3*b2 - 2*b1 $$\nu^{4}$$ $$=$$ $$3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1$$ 3*b7 + 4*b6 + b4 - b3 - 5*b2 - 4*b1 $$\nu^{5}$$ $$=$$ $$4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1$$ 4*b7 - 6*b5 - 4*b3 - 6*b2 - 6*b1 - 1 $$\nu^{6}$$ $$=$$ $$-16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6$$ -16*b6 - 16*b5 - 6*b4 - 6*b3 - 7*b1 - 6 $$\nu^{7}$$ $$=$$ $$-16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16$$ -16*b7 - 51*b6 - 29*b5 - 23*b4 + 29*b2 - 16

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$397$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 −0.386111 + 0.280526i 1.69513 − 1.23158i −0.386111 − 0.280526i 1.69513 + 1.23158i 0.418926 + 1.28932i −0.227943 − 0.701538i 0.418926 − 1.28932i −0.227943 + 0.701538i
−1.12474 + 0.817172i 0 −0.0207616 + 0.0638975i −0.809017 0.587785i 0 0.394797 1.21506i −0.888090 2.73326i 0 1.39026
91.2 2.24278 1.62947i 0 1.75683 5.40697i −0.809017 0.587785i 0 −0.703814 + 2.16612i −3.15700 9.71623i 0 −2.77222
136.1 −1.12474 0.817172i 0 −0.0207616 0.0638975i −0.809017 + 0.587785i 0 0.394797 + 1.21506i −0.888090 + 2.73326i 0 1.39026
136.2 2.24278 + 1.62947i 0 1.75683 + 5.40697i −0.809017 + 0.587785i 0 −0.703814 2.16612i −3.15700 + 9.71623i 0 −2.77222
181.1 −0.758911 2.33569i 0 −3.26145 + 2.36959i 0.309017 0.951057i 0 −2.65911 + 1.93196i 4.03606 + 2.93237i 0 −2.45589
181.2 −0.359123 1.10527i 0 0.525387 0.381716i 0.309017 0.951057i 0 3.46813 2.51974i −2.49097 1.80980i 0 −1.16215
361.1 −0.758911 + 2.33569i 0 −3.26145 2.36959i 0.309017 + 0.951057i 0 −2.65911 1.93196i 4.03606 2.93237i 0 −2.45589
361.2 −0.359123 + 1.10527i 0 0.525387 + 0.381716i 0.309017 + 0.951057i 0 3.46813 + 2.51974i −2.49097 + 1.80980i 0 −1.16215
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.2 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 495.2.n.d 8
3.b odd 2 1 165.2.m.a 8
11.c even 5 1 inner 495.2.n.d 8
11.c even 5 1 5445.2.a.be 4
11.d odd 10 1 5445.2.a.bv 4
15.d odd 2 1 825.2.n.k 8
15.e even 4 2 825.2.bx.h 16
33.f even 10 1 1815.2.a.o 4
33.h odd 10 1 165.2.m.a 8
33.h odd 10 1 1815.2.a.x 4
165.o odd 10 1 825.2.n.k 8
165.o odd 10 1 9075.2.a.cl 4
165.r even 10 1 9075.2.a.dj 4
165.v even 20 2 825.2.bx.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.a 8 3.b odd 2 1
165.2.m.a 8 33.h odd 10 1
495.2.n.d 8 1.a even 1 1 trivial
495.2.n.d 8 11.c even 5 1 inner
825.2.n.k 8 15.d odd 2 1
825.2.n.k 8 165.o odd 10 1
825.2.bx.h 16 15.e even 4 2
825.2.bx.h 16 165.v even 20 2
1815.2.a.o 4 33.f even 10 1
1815.2.a.x 4 33.h odd 10 1
5445.2.a.be 4 11.c even 5 1
5445.2.a.bv 4 11.d odd 10 1
9075.2.a.cl 4 165.o odd 10 1
9075.2.a.dj 4 165.r even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 3T_{2}^{6} - 5T_{2}^{5} + 24T_{2}^{4} + 85T_{2}^{3} + 177T_{2}^{2} + 165T_{2} + 121$$ acting on $$S_{2}^{\mathrm{new}}(495, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 3 T^{6} - 5 T^{5} + 24 T^{4} + \cdots + 121$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$7$ $$T^{8} - T^{7} - 3 T^{6} + 7 T^{5} + \cdots + 1681$$
$11$ $$T^{8} - 3 T^{7} + 8 T^{6} + \cdots + 14641$$
$13$ $$T^{8} - 6 T^{7} + 39 T^{6} - 87 T^{5} + \cdots + 1$$
$17$ $$(T^{4} - 5 T^{3} + 25 T^{2} - 125 T + 625)^{2}$$
$19$ $$T^{8} - 6 T^{7} + 9 T^{6} + 123 T^{5} + \cdots + 961$$
$23$ $$(T^{4} - 5 T^{3} - T^{2} + 5 T - 1)^{2}$$
$29$ $$T^{8} + 17 T^{6} - 95 T^{5} + \cdots + 290521$$
$31$ $$T^{8} - 3 T^{7} + 11 T^{6} + \cdots + 19321$$
$37$ $$T^{8} + 19 T^{7} + 255 T^{6} + \cdots + 185761$$
$41$ $$T^{8} - 25 T^{7} + 327 T^{6} + \cdots + 4289041$$
$43$ $$(T^{4} + 2 T^{3} - 92 T^{2} - 63 T + 1861)^{2}$$
$47$ $$T^{8} + 15 T^{7} + 123 T^{6} + \cdots + 121$$
$53$ $$T^{8} + 7 T^{7} - 14 T^{6} + \cdots + 1615441$$
$59$ $$T^{8} + 35 T^{7} + 633 T^{6} + \cdots + 5285401$$
$61$ $$T^{8} - 21 T^{7} + 242 T^{6} + \cdots + 3575881$$
$67$ $$(T^{4} + 13 T^{3} - 136 T^{2} - 1768 T - 3379)^{2}$$
$71$ $$T^{8} + 25 T^{7} + 347 T^{6} + \cdots + 5527201$$
$73$ $$T^{8} - T^{7} + 30 T^{6} + \cdots + 84621601$$
$79$ $$T^{8} - 30 T^{7} + 487 T^{6} + \cdots + 1437601$$
$83$ $$T^{8} + 11 T^{7} + \cdots + 149352841$$
$89$ $$(T^{4} + 16 T^{3} + 45 T^{2} - 132 T - 271)^{2}$$
$97$ $$T^{8} - 5 T^{7} - 20 T^{6} + 225 T^{5} + \cdots + 625$$