LMFDB - Newform orbit 289.3.e.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,3,Mod(40,289)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([15]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("289.40");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 289 = 17^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	289.e (of order $16$, degree $8$, 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:	$7.87467964001$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-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 primitive root of unity $\zeta_{16}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{16}^{7} + \zeta_{16}^{6} + \cdots - 1) q^{2}+ \cdots + (4 \zeta_{16}^{7} + 2 \zeta_{16}^{6} + \cdots + 1) q^{9}+O(q^{10}) $ q + (-z^7 + z^6 + z^5 + z^4 - z^2 - z - 1) * q^2 + (z^7 + z^6 - z^3 - z^2 + z + 1) * q^3 + (-2z^7 - 3z^6 - 2z^5 + 2z^3 - 2z) q^4 + (2z^7 - 2z^6 - 2z^4 - 2z^3 - 2z^2 - 2z) * q^5 + (-4z^7 + z^6 + 3z^5 + z^4 - z^3 - 3z^2 - z + 4) q^6 + (-5z^7 + z^6 + z^5 - 5z^4 - z^3 - z^2 + z + 1) * q^7 + (3z^7 - 3z^6 + z^5 - 3z^4 + 3z^3 + 5z^2 - 5) q^8 + (4z^7 + 2z^6 - 2z^4 - 4z^3 + z^2 - 3z + 1) q^9	$ q + ( - \zeta_{16}^{7} + \zeta_{16}^{6} + \cdots - 1) q^{2}+ \cdots + ( - 28 \zeta_{16}^{7} + 34 \zeta_{16}^{6} + \cdots - 27) q^{99}+O(q^{100}) $ q + (-z^7 + z^6 + z^5 + z^4 - z^2 - z - 1) * q^2 + (z^7 + z^6 - z^3 - z^2 + z + 1) * q^3 + (-2z^7 - 3z^6 - 2z^5 + 2z^3 - 2z) q^4 + (2z^7 - 2z^6 - 2z^4 - 2z^3 - 2z^2 - 2z) * q^5 + (-4z^7 + z^6 + 3z^5 + z^4 - z^3 - 3z^2 - z + 4) q^6 + (-5z^7 + z^6 + z^5 - 5z^4 - z^3 - z^2 + z + 1) * q^7 + (3z^7 - 3z^6 + z^5 - 3z^4 + 3z^3 + 5z^2 - 5) q^8 + (4z^7 + 2z^6 - 2z^4 - 4z^3 + z^2 - 3z + 1) q^9 + (-6z^7 + 2z^6 - 2z^5 + 6z^4 + 4z^3 + 6z^2 + 6z + 4) q^10 + (-3z^7 + z^6 - 6z^5 + z^4 + z^3 - 6z^2 + z - 3) q^11 + (-5z^7 - 5z^6 + z^5 + 9z^4 + 6z^3 - 6z^2 - 9z - 1) * q^12 + (-10z^7 + 10z^5 - 2z^4 - 2z^3 - 2z - 2) q^13 + (z^7 - z^6 + 13z^5 + 13z^4 - z^3 + z^2 - 3z + 3) q^14 + (2z^5 + 2z^4 + 2z^2 + 2z) * q^15 + (4z^7 + 8z^6 - 4z^5 + 3z^4 - 4z^3 + 8z^2 + 4z) q^16 + (-13z^7 + 3z^6 + 2z^5 - 2z^3 - 3z^2 + 13z + 7) * q^18 + (-4z^7 + 6z^6 - 7z^5 + 7z - 6) * q^19 + (10z^7 + 4z^6 + 2z^5 - 2z^4 - 4z^3 - 10z^2 - 10z - 10) q^20 + (2z^7 + 14z^6 + 2z^5 - 8z^4 + 8) * q^21 + (7z^7 - 7z^6 + 6z^5 - 3z^4 + 13z^3 + 13z^2 - 3z + 6) q^22 + (-3z^7 + 7z^6 + 5z^5 - 7z^4 + 7z^3 - 5z^2 - 7z + 3) q^23 + (-3z^7 - 10z^6 - 10z^5 - 3z^4 + 15z^3 + 13z^2 - 13z - 15) q^24 + (16z^7 + 8z^6 - z^5 + 8z^4 + 16z^3 + 4z^2 - 4) q^25 + (-2z^7 - 22z^6 + 22z^4 + 2z^3 - 6z^2 - 12z - 6) * q^26 + (-3z^7 + 5z^6 - 5z^5 + 3z^4 + 10z^3 - 2z^2 - 2z + 10) q^27 + (-15z^7 - 19z^6 - 7z^5 - z^4 - z^3 - 7z^2 - 19z - 15) q^28 + (8z^7 + 8z^6 - 2z^5 - 16z^4 + 6z^3 - 6z^2 + 16z + 2) q^29 + (2z^7 - 2z^5 - 2z^4 - 4z^3 - 8z^2 - 4z - 2) * q^30 + (3z^7 - 5z^6 - 5z^5 - 5z^4 - 5z^3 + 3z^2 + 5z - 5) q^31 + (-11z^6 + z^5 - 11z^4 - 17z^3 - 11z^2 + z - 11) * q^32 + (z^7 - 7z^6 + 2z^5 + 12z^4 + 2z^3 - 7z^2 + z) q^33 + (8z^7 + 20z^6 - 20z^2 - 8z - 20) * q^35 + (-3z^7 + 3z^6 + 20z^5 + 16z^4 - 16z^2 - 20z - 3) * q^36 + (-4z^7 - 16z^6 + 12z^5 - 12z^4 + 16z^3 + 4z^2 + 2z + 2) q^37 + (18z^7 - 2z^6 + 18z^5 - 15z^4 - 2z^3 + 2z + 15) * q^38 + (-22z^7 + 22z^6 + 20z^5 - 12z^4 - 6z^3 - 6z^2 - 12z + 20) q^39 + (-6z^7 - 8z^6 + 2z^5 - 14z^4 + 14z^3 - 2z^2 + 8z + 6) q^40 + (6z^7 + 6z^6 + 6z^5 + 6z^4 + 5z^3 - 14z^2 + 14z - 5) q^41 + (-26z^7 + 2z^6 + 26z^5 + 2z^4 - 26z^3 - 16z^2 + 16) * q^42 + (-2z^7 + 29z^6 - 29z^4 + 2z^3 - 14z^2 + 22z - 14) * q^43 + (15z^7 + 15z^6 - 15z^5 - 15z^4 - 8z^3 - 3z^2 - 3z - 8) q^44 + (16z^7 + 20z^5 + 14z^4 + 14z^3 + 20z^2 + 16) q^45 + (-17z^7 - 17z^6 + z^5 + 9z^4 - 11z^3 + 11z^2 - 9z - 1) * q^46 + (8z^7 - 8z^5 - 24z^4 - 22z^3 + 2z^2 - 22z - 24) * q^47 + (17z^7 + z^6 - 17z^5 - 17z^4 + z^3 + 17z^2 + 12z - 12) q^48 + (-10z^6 + 4z^5 - 2z^4 - 7z^3 - 2z^2 + 4z - 10) * q^49 + (z^7 + z^6 - 35z^5 - 57z^4 - 35z^3 + z^2 + z) q^50 + (6z^7 - 34z^6 - 22z^5 + 22z^3 + 34z^2 - 6z - 48) * q^52 + (2z^7 + 16z^6 + 18z^5 + 4z^4 - 4z^2 - 18z - 16) * q^53 + (-z^7 + 8z^5 - 8z^4 + z^2 - 21z - 21) q^54 + (-2z^7 + 32z^6 - 2z^5 + 28z^4 - 4z^3 + 4z - 28) * q^55 + (17z^7 - 17z^6 - 33z^5 + 25z^4 + 15z^3 + 15z^2 + 25z - 33) q^56 + (10z^7 - 3z^6 - 9z^5 - 6z^4 + 6z^3 + 9z^2 + 3z - 10) q^57 + (30z^6 + 30z^5 - 46z^3 + 4z^2 - 4z + 46) q^58 + (10z^7 - 6z^6 - 42z^5 - 6z^4 + 10z^3 + 15z^2 - 15) * q^59 + (-6z^7 - 8z^6 + 8z^4 + 6z^3 + 6z^2 + 8z + 6) * q^60 + (36z^7 - 20z^6 + 20z^5 - 36z^4 - 12z^3 + 8z^2 + 8z - 12) q^61 + (15z^7 + 21z^6 + 7z^5 - z^4 - z^3 + 7z^2 + 21z + 15) q^62 + (21z^7 + 21z^6 + 23z^5 - 15z^4 + 7z^3 - 7z^2 + 15z - 23) q^63 + (10z^7 - 10z^5 + 8z^4 + 42z^3 + 13z^2 + 42z + 8) * q^64 + (-12z^7 + 12z^6 - 8z^5 - 8z^4 + 12z^3 - 12z^2) * q^65 + (-12z^6 - 23z^5 + z^4 + 20z^3 + z^2 - 23z - 12) * q^66 + (26z^7 - 4z^6 + 33z^5 + 42z^4 + 33z^3 - 4z^2 + 26z) q^67 + (2z^7 + 18z^6 - 8z^5 + 8z^3 - 18z^2 - 2z + 30) * q^69 + (-36z^7 - 60z^6 - 16z^5 - 28z^4 + 28z^2 + 16z + 60) * q^70 + (-5z^7 - 11z^6 - 43z^5 + 43z^4 + 11z^3 + 5z^2 + 35z + 35) q^71 + (-z^7 - 53z^6 - z^5 + 5z^4 + 24z^3 - 24z - 5) * q^72 + (-6z^7 + 6z^6 + 26z^5 + 25z^4 - 6z^3 - 6z^2 + 25z + 26) q^73 + (28z^7 + 20z^6 - 24z^5 + 26z^4 - 26z^3 + 24z^2 - 20z - 28) q^74 + (13z^7 - 37z^6 - 37z^5 + 13z^4 + 17z^3 - 17) q^75 + (-21z^7 + 10z^6 + 12z^5 + 10z^4 - 21z^3 - 20z^2 + 20) * q^76 + (8z^7 + 6z^6 - 6z^4 - 8z^3 + 10z^2 - 66z + 10) * q^77 + (-64z^7 - 50z^6 + 50z^5 + 64z^4 - 8z^3 - 38z^2 - 38z - 8) q^78 + (3z^7 + 17z^6 - 21z^5 - 41z^4 - 41z^3 - 21z^2 + 17z + 3) q^79 + (-14z^7 - 14z^6 - 30z^5 + 8z^4 - 30z^3 + 30z^2 - 8z + 30) q^80 + (-6z^7 + 6z^5 + 53z^4 - 7z^3 - 3z^2 - 7z + 53) * q^81 + (-8z^7 - 17z^6 - 8z^5 - 8z^4 - 17z^3 - 8z^2 - 34z + 34) q^82 + (42z^6 - 20z^5 + 33z^4 + 54z^3 + 33z^2 - 20z + 42) * q^83 + (-60z^7 - 24z^6 + 34z^5 + 50z^4 + 34z^3 - 24z^2 - 60z) q^84 + (-3z^7 - 8z^6 + 66z^5 - 66z^3 + 8z^2 + 3z + 104) * q^86 + (52z^7 + 10z^6 - 34z^5 - 26z^4 + 26z^2 + 34z - 10) * q^87 + (-33z^7 + 40z^6 - z^5 + z^4 - 40z^3 + 33z^2 + 9z + 9) q^88 + (-46z^7 - 8z^6 - 46z^5 + 20z^4 - 29z^3 + 29z - 20) * q^89 + (-18z^7 + 18z^6 - 48z^5 - 28z^4 - 56z^3 - 56z^2 - 28z - 48) q^90 + (-40z^6 + 28z^5 + 68z^4 - 68z^3 - 28z^2 + 40z) * q^91 + (-3z^7 + 3z^6 + 3z^5 - 3z^4 + 31z^3 + 23z^2 - 23z - 31) q^92 + (8z^7 - 8z^6 - 6z^5 - 8z^4 + 8z^3 + 26z^2 - 26) * q^93 + (-18z^7 - 4z^6 + 4z^4 + 18z^3 + 54z^2 + 110z + 54) * q^94 + (-24z^7 + 34z^6 - 34z^5 + 24z^4 - 24z^3 + 2z^2 + 2z - 24) q^95 + (-11z^7 + 7z^6 + 29z^5 - 8z^4 - 8z^3 + 29z^2 + 7z - 11) q^96 + (13z^7 + 13z^6 - 22z^5 + 30z^4 + 6z^3 - 6z^2 - 30z + 22) q^97 + (11z^7 - 11z^5 + 15z^4 + 9z^3 + 9z^2 + 9z + 15) * q^98 + (-28z^7 + 34z^6 + 16z^5 + 16z^4 + 34z^3 - 28z^2 + 27z - 27) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{3} + 32 q^{6} + 8 q^{7} - 40 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^3 + 32 * q^6 + 8 * q^7 - 40 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 8 q^{3} + 32 q^{6} + 8 q^{7} - 40 q^{8} + 8 q^{9} + 32 q^{10} - 24 q^{11} - 8 q^{12} - 16 q^{13} + 24 q^{14} + 56 q^{18} - 48 q^{19} - 80 q^{20} + 64 q^{21} + 48 q^{22} + 24 q^{23} - 120 q^{24} - 32 q^{25} - 48 q^{26} + 80 q^{27} - 120 q^{28} + 16 q^{29} - 16 q^{30} - 40 q^{31} - 88 q^{32} - 160 q^{35} - 24 q^{36} + 16 q^{37} + 120 q^{38} + 160 q^{39} + 48 q^{40} - 40 q^{41} + 128 q^{42} - 112 q^{43} - 64 q^{44} + 128 q^{45} - 8 q^{46} - 192 q^{47} - 96 q^{48} - 80 q^{49} - 384 q^{52} - 128 q^{53} - 168 q^{54} - 224 q^{55} - 264 q^{56} - 80 q^{57} + 368 q^{58} - 120 q^{59} + 48 q^{60} - 96 q^{61} + 120 q^{62} - 184 q^{63} + 64 q^{64} - 96 q^{66} + 240 q^{69} + 480 q^{70} + 280 q^{71} - 40 q^{72} + 208 q^{73} - 224 q^{74} - 136 q^{75} + 160 q^{76} + 80 q^{77} - 64 q^{78} + 24 q^{79} + 240 q^{80} + 424 q^{81} + 272 q^{82} + 336 q^{83} + 832 q^{86} - 80 q^{87} + 72 q^{88} - 160 q^{89} - 384 q^{90} - 248 q^{92} - 208 q^{93} + 432 q^{94} - 192 q^{95} - 88 q^{96} + 176 q^{97} + 120 q^{98} - 216 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 32 * q^6 + 8 * q^7 - 40 * q^8 + 8 * q^9 + 32 * q^10 - 24 * q^11 - 8 * q^12 - 16 * q^13 + 24 * q^14 + 56 * q^18 - 48 * q^19 - 80 * q^20 + 64 * q^21 + 48 * q^22 + 24 * q^23 - 120 * q^24 - 32 * q^25 - 48 * q^26 + 80 * q^27 - 120 * q^28 + 16 * q^29 - 16 * q^30 - 40 * q^31 - 88 * q^32 - 160 * q^35 - 24 * q^36 + 16 * q^37 + 120 * q^38 + 160 * q^39 + 48 * q^40 - 40 * q^41 + 128 * q^42 - 112 * q^43 - 64 * q^44 + 128 * q^45 - 8 * q^46 - 192 * q^47 - 96 * q^48 - 80 * q^49 - 384 * q^52 - 128 * q^53 - 168 * q^54 - 224 * q^55 - 264 * q^56 - 80 * q^57 + 368 * q^58 - 120 * q^59 + 48 * q^60 - 96 * q^61 + 120 * q^62 - 184 * q^63 + 64 * q^64 - 96 * q^66 + 240 * q^69 + 480 * q^70 + 280 * q^71 - 40 * q^72 + 208 * q^73 - 224 * q^74 - 136 * q^75 + 160 * q^76 + 80 * q^77 - 64 * q^78 + 24 * q^79 + 240 * q^80 + 424 * q^81 + 272 * q^82 + 336 * q^83 + 832 * q^86 - 80 * q^87 + 72 * q^88 - 160 * q^89 - 384 * q^90 - 248 * q^92 - 208 * q^93 + 432 * q^94 - 192 * q^95 - 88 * q^96 + 176 * q^97 + 120 * q^98 - 216 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/289\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$\zeta_{16}$

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} $

40.1

0.923880	−	0.382683i
−0.923880	−	0.382683i
−0.382683	−	0.923880i
0.382683	−	0.923880i
−0.382683	+	0.923880i
0.382683	+	0.923880i
0.923880	+	0.382683i
−0.923880	+	0.382683i

−2.79690

−

1.15851i

−0.796897

0.158513i

3.65205

3.65205i

−4.46088

6.67619i

2.41248

0.479872i

4.36370

6.53073i

−1.34942

−

3.25778i

−7.70500

3.19151i

20.2111

−

13.5046i

65.1

−2.03153

0.841487i

−0.0315301

0.158513i

0.590587

−

0.590587i

4.46088

−

2.98067i

−0.0693320

−

0.348555i

−5.19212

−

3.46927i

2.66313

−

6.42935i

8.29078

3.43416i

−6.55423

9.80910i

75.1

−0.509666

1.23044i

1.49033

−

2.23044i

1.57420

1.57420i

−0.317025

−

1.59379i

1.98486

2.97055i

−1.72965

8.69552i

−7.66104

3.17331i

0.690373

1.66671i

2.12265

0.422221i

131.1

1.33809

3.23044i

3.33809

−

2.23044i

−5.81684

5.81684i

0.317025

0.0630603i

11.6720

7.79898i

6.55807

−

1.30448i

−13.6527

−

5.65512i

2.72384

−

6.57593i

0.220497

1.10851i

158.1

−0.509666

−

1.23044i

1.49033

2.23044i

1.57420

−

1.57420i

−0.317025

1.59379i

1.98486

−

2.97055i

−1.72965

−

8.69552i

−7.66104

−

3.17331i

0.690373

−

1.66671i

2.12265

−

0.422221i

214.1

1.33809

−

3.23044i

3.33809

2.23044i

−5.81684

−

5.81684i

0.317025

−

0.0630603i

11.6720

−

7.79898i

6.55807

1.30448i

−13.6527

5.65512i

2.72384

6.57593i

0.220497

−

1.10851i

224.1

−2.79690

1.15851i

−0.796897

−

0.158513i

3.65205

−

3.65205i

−4.46088

−

6.67619i

2.41248

−

0.479872i

4.36370

−

6.53073i

−1.34942

3.25778i

−7.70500

−

3.19151i

20.2111

13.5046i

249.1

−2.03153

−

0.841487i

−0.0315301

−

0.158513i

0.590587

0.590587i

4.46088

2.98067i

−0.0693320

0.348555i

−5.19212

3.46927i

2.66313

6.42935i

8.29078

−

3.43416i

−6.55423

−

9.80910i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.e	odd	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	289.3.e.d		8
17.b	even	2	1		289.3.e.b		8
17.c	even	4	1		289.3.e.k		8
17.c	even	4	1		289.3.e.l		8
17.d	even	8	1		17.3.e.a	✓	8
17.d	even	8	1		289.3.e.c		8
17.d	even	8	1		289.3.e.i		8
17.d	even	8	1		289.3.e.m		8
17.e	odd	16	1		17.3.e.a	✓	8
17.e	odd	16	1		289.3.e.b		8
17.e	odd	16	1		289.3.e.c		8
17.e	odd	16	1	inner	289.3.e.d		8
17.e	odd	16	1		289.3.e.i		8
17.e	odd	16	1		289.3.e.k		8
17.e	odd	16	1		289.3.e.l		8
17.e	odd	16	1		289.3.e.m		8
51.g	odd	8	1		153.3.p.b		8
51.i	even	16	1		153.3.p.b		8
68.g	odd	8	1		272.3.bh.c		8
68.i	even	16	1		272.3.bh.c		8
85.k	odd	8	1		425.3.t.c		8
85.m	even	8	1		425.3.u.b		8
85.n	odd	8	1		425.3.t.a		8
85.o	even	16	1		425.3.t.c		8
85.p	odd	16	1		425.3.u.b		8
85.r	even	16	1		425.3.t.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
17.3.e.a	✓	8	17.d	even	8	1
17.3.e.a	✓	8	17.e	odd	16	1
153.3.p.b		8	51.g	odd	8	1
153.3.p.b		8	51.i	even	16	1
272.3.bh.c		8	68.g	odd	8	1
272.3.bh.c		8	68.i	even	16	1
289.3.e.b		8	17.b	even	2	1
289.3.e.b		8	17.e	odd	16	1
289.3.e.c		8	17.d	even	8	1
289.3.e.c		8	17.e	odd	16	1
289.3.e.d		8	1.a	even	1	1	trivial
289.3.e.d		8	17.e	odd	16	1	inner
289.3.e.i		8	17.d	even	8	1
289.3.e.i		8	17.e	odd	16	1
289.3.e.k		8	17.c	even	4	1
289.3.e.k		8	17.e	odd	16	1
289.3.e.l		8	17.c	even	4	1
289.3.e.l		8	17.e	odd	16	1
289.3.e.m		8	17.d	even	8	1
289.3.e.m		8	17.e	odd	16	1
425.3.t.a		8	85.n	odd	8	1
425.3.t.a		8	85.r	even	16	1
425.3.t.c		8	85.k	odd	8	1
425.3.t.c		8	85.o	even	16	1
425.3.u.b		8	85.m	even	8	1
425.3.u.b		8	85.p	odd	16	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(289, [\chi])$:

$ T_{2}^{8} + 8T_{2}^{7} + 32T_{2}^{6} + 120T_{2}^{5} + 448T_{2}^{4} + 1144T_{2}^{3} + 1792T_{2}^{2} + 1736T_{2} + 961 $

$ T_{3}^{8} - 8T_{3}^{7} + 28T_{3}^{6} - 32T_{3}^{5} - 10T_{3}^{4} + 120T_{3}^{3} + 84T_{3}^{2} + 8T_{3} + 2 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 8 T^{7} + \cdots + 961 $ T^8 + 8T^7 + 32T^6 + 120T^5 + 448T^4 + 1144T^3 + 1792T^2 + 1736*T + 961
$3$	$ T^{8} - 8 T^{7} + \cdots + 2 $ T^8 - 8T^7 + 28T^6 - 32T^5 - 10T^4 + 120T^3 + 84T^2 + 8*T + 2
$5$	$ T^{8} + 16 T^{6} + \cdots + 512 $ T^8 + 16T^6 - 320T^5 + 1888T^4 - 768T^3 + 4864T^2 - 3072T + 512
$7$	$ T^{8} - 8 T^{7} + \cdots + 8454272 $ T^8 - 8T^7 + 72T^6 - 544T^5 + 2352T^4 - 2816T^3 - 40320T^2 - 1052672*T + 8454272
$11$	$ T^{8} + 24 T^{7} + \cdots + 27572738 $ T^8 + 24T^7 + 136T^6 + 160T^5 + 2066T^4 - 2464T^3 + 972468T^2 + 5198200*T + 27572738
$13$	$ T^{8} + 16 T^{7} + \cdots + 9048064 $ T^8 + 16T^7 + 128T^6 - 2688T^5 + 129408T^4 + 1129472T^3 + 5120000T^2 - 9625600*T + 9048064
$17$	$ T^{8} $ T^8
$19$	$ T^{8} + 48 T^{7} + \cdots + 929296 $ T^8 + 48T^7 + 784T^6 + 4032T^5 + 21632T^4 - 294144T^3 + 863936T^2 - 1017984*T + 929296
$23$	$ T^{8} - 24 T^{7} + \cdots + 859299968 $ T^8 - 24T^7 + 616T^6 + 1984T^5 + 42256T^4 + 352512T^3 - 8233472T^2 + 39797760*T + 859299968
$29$	$ T^{8} + \cdots + 4240836608 $ T^8 - 16T^7 + 1680T^6 - 29952T^5 + 1069760T^4 - 47741440T^3 + 676834304T^2 - 2699517952*T + 4240836608
$31$	$ T^{8} + 40 T^{7} + \cdots + 14536832 $ T^8 + 40T^7 + 1000T^6 + 15424T^5 + 195472T^4 + 1916160T^3 + 13694208T^2 + 25709056*T + 14536832
$37$	$ T^{8} + \cdots + 120057840128 $ T^8 - 16T^7 + 1648T^6 - 448T^5 - 727456T^4 - 26123520T^3 + 83138304T^2 + 8200907776*T + 120057840128
$41$	$ T^{8} + \cdots + 59058658562 $ T^8 + 40T^7 + 1084T^6 + 16680T^5 + 402758T^4 - 6674744T^3 - 104090084T^2 - 3399702344*T + 59058658562
$43$	$ T^{8} + \cdots + 6201305218564 $ T^8 + 112T^7 + 5252T^6 + 377208T^5 + 29495944T^4 + 659093648T^3 + 11251238296T^2 + 373556221936*T + 6201305218564
$47$	$ T^{8} + \cdots + 2259754549504 $ T^8 + 192T^7 + 18432T^6 + 907072T^5 + 25209440T^4 + 277465600T^3 + 2803712T^2 + 3559691264*T + 2259754549504
$53$	$ T^{8} + \cdots + 26754490624 $ T^8 + 128T^7 + 6432T^6 + 169600T^5 + 4121088T^4 + 47191552T^3 + 1236934144T^2 + 10007744512*T + 26754490624
$59$	$ T^{8} + \cdots + 2675455605124 $ T^8 + 120T^7 + 2364T^6 - 120360T^5 + 6149448T^4 - 202344096T^3 + 5715853032T^2 - 153610167984*T + 2675455605124
$61$	$ T^{8} + \cdots + 106680004247552 $ T^8 + 96T^7 + 8320T^6 + 834560T^5 + 67023360T^4 + 4955521024T^3 + 268994297856T^2 + 8110656847872*T + 106680004247552
$67$	$ T^{8} + \cdots + 83643718741636 $ T^8 + 21304T^6 + 118064724T^4 + 200330319344*T^2 + 83643718741636
$71$	$ T^{8} + \cdots + 71246079996032 $ T^8 - 280T^7 + 36072T^6 - 2760512T^5 + 126349584T^4 - 5585860096T^3 + 246660350976T^2 + 7447165054976*T + 71246079996032
$73$	$ T^{8} + \cdots + 51301810562 $ T^8 - 208T^7 + 18692T^6 - 1013344T^5 + 36374662T^4 - 999045256T^3 + 30382900100T^2 - 64987397112*T + 51301810562
$79$	$ T^{8} + \cdots + 3948319764608 $ T^8 - 24T^7 + 11416T^6 + 15424T^5 + 59751056T^4 + 3663717760T^3 + 25152949504T^2 - 673254040064*T + 3948319764608
$83$	$ T^{8} + \cdots + 10\!\cdots\!04 $ T^8 - 336T^7 + 56196T^6 - 5630952T^5 + 313068168T^4 - 2524583856T^3 - 173221055208T^2 + 31481546945712*T + 1042256526864004
$89$	$ T^{8} + \cdots + 117588822570244 $ T^8 + 160T^7 + 12800T^6 - 64128T^5 + 72896012T^4 + 9469447872T^3 + 584098906112T^2 + 11720367113216*T + 117588822570244
$97$	$ T^{8} + \cdots + 21682310986562 $ T^8 - 176T^7 + 12352T^6 - 448864T^5 + 5558242T^4 - 123290672T^3 + 16219072820T^2 + 813164614088*T + 21682310986562
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	289.e (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{16}^{7} + \zeta_{16}^{6} + \cdots - 1) q^{2}+ \cdots + (4 \zeta_{16}^{7} + 2 \zeta_{16}^{6} + \cdots + 1) q^{9}+O(q^{10}) \) q + (-z^7 + z^6 + z^5 + z^4 - z^2 - z - 1) * q^2 + (z^7 + z^6 - z^3 - z^2 + z + 1) * q^3 + (-2z^7 - 3z^6 - 2z^5 + 2z^3 - 2z) q^4 + (2z^7 - 2z^6 - 2z^4 - 2z^3 - 2z^2 - 2z) * q^5 + (-4z^7 + z^6 + 3z^5 + z^4 - z^3 - 3z^2 - z + 4) q^6 + (-5z^7 + z^6 + z^5 - 5z^4 - z^3 - z^2 + z + 1) * q^7 + (3z^7 - 3z^6 + z^5 - 3z^4 + 3z^3 + 5z^2 - 5) q^8 + (4z^7 + 2z^6 - 2z^4 - 4z^3 + z^2 - 3z + 1) q^9	\( q + ( - \zeta_{16}^{7} + \zeta_{16}^{6} + \cdots - 1) q^{2}+ \cdots + ( - 28 \zeta_{16}^{7} + 34 \zeta_{16}^{6} + \cdots - 27) q^{99}+O(q^{100}) \) q + (-z^7 + z^6 + z^5 + z^4 - z^2 - z - 1) * q^2 + (z^7 + z^6 - z^3 - z^2 + z + 1) * q^3 + (-2z^7 - 3z^6 - 2z^5 + 2z^3 - 2z) q^4 + (2z^7 - 2z^6 - 2z^4 - 2z^3 - 2z^2 - 2z) * q^5 + (-4z^7 + z^6 + 3z^5 + z^4 - z^3 - 3z^2 - z + 4) q^6 + (-5z^7 + z^6 + z^5 - 5z^4 - z^3 - z^2 + z + 1) * q^7 + (3z^7 - 3z^6 + z^5 - 3z^4 + 3z^3 + 5z^2 - 5) q^8 + (4z^7 + 2z^6 - 2z^4 - 4z^3 + z^2 - 3z + 1) q^9 + (-6z^7 + 2z^6 - 2z^5 + 6z^4 + 4z^3 + 6z^2 + 6z + 4) q^10 + (-3z^7 + z^6 - 6z^5 + z^4 + z^3 - 6z^2 + z - 3) q^11 + (-5z^7 - 5z^6 + z^5 + 9z^4 + 6z^3 - 6z^2 - 9z - 1) * q^12 + (-10z^7 + 10z^5 - 2z^4 - 2z^3 - 2z - 2) q^13 + (z^7 - z^6 + 13z^5 + 13z^4 - z^3 + z^2 - 3z + 3) q^14 + (2z^5 + 2z^4 + 2z^2 + 2z) * q^15 + (4z^7 + 8z^6 - 4z^5 + 3z^4 - 4z^3 + 8z^2 + 4z) q^16 + (-13z^7 + 3z^6 + 2z^5 - 2z^3 - 3z^2 + 13z + 7) * q^18 + (-4z^7 + 6z^6 - 7z^5 + 7z - 6) * q^19 + (10z^7 + 4z^6 + 2z^5 - 2z^4 - 4z^3 - 10z^2 - 10z - 10) q^20 + (2z^7 + 14z^6 + 2z^5 - 8z^4 + 8) * q^21 + (7z^7 - 7z^6 + 6z^5 - 3z^4 + 13z^3 + 13z^2 - 3z + 6) q^22 + (-3z^7 + 7z^6 + 5z^5 - 7z^4 + 7z^3 - 5z^2 - 7z + 3) q^23 + (-3z^7 - 10z^6 - 10z^5 - 3z^4 + 15z^3 + 13z^2 - 13z - 15) q^24 + (16z^7 + 8z^6 - z^5 + 8z^4 + 16z^3 + 4z^2 - 4) q^25 + (-2z^7 - 22z^6 + 22z^4 + 2z^3 - 6z^2 - 12z - 6) * q^26 + (-3z^7 + 5z^6 - 5z^5 + 3z^4 + 10z^3 - 2z^2 - 2z + 10) q^27 + (-15z^7 - 19z^6 - 7z^5 - z^4 - z^3 - 7z^2 - 19z - 15) q^28 + (8z^7 + 8z^6 - 2z^5 - 16z^4 + 6z^3 - 6z^2 + 16z + 2) q^29 + (2z^7 - 2z^5 - 2z^4 - 4z^3 - 8z^2 - 4z - 2) * q^30 + (3z^7 - 5z^6 - 5z^5 - 5z^4 - 5z^3 + 3z^2 + 5z - 5) q^31 + (-11z^6 + z^5 - 11z^4 - 17z^3 - 11z^2 + z - 11) * q^32 + (z^7 - 7z^6 + 2z^5 + 12z^4 + 2z^3 - 7z^2 + z) q^33 + (8z^7 + 20z^6 - 20z^2 - 8z - 20) * q^35 + (-3z^7 + 3z^6 + 20z^5 + 16z^4 - 16z^2 - 20z - 3) * q^36 + (-4z^7 - 16z^6 + 12z^5 - 12z^4 + 16z^3 + 4z^2 + 2z + 2) q^37 + (18z^7 - 2z^6 + 18z^5 - 15z^4 - 2z^3 + 2z + 15) * q^38 + (-22z^7 + 22z^6 + 20z^5 - 12z^4 - 6z^3 - 6z^2 - 12z + 20) q^39 + (-6z^7 - 8z^6 + 2z^5 - 14z^4 + 14z^3 - 2z^2 + 8z + 6) q^40 + (6z^7 + 6z^6 + 6z^5 + 6z^4 + 5z^3 - 14z^2 + 14z - 5) q^41 + (-26z^7 + 2z^6 + 26z^5 + 2z^4 - 26z^3 - 16z^2 + 16) * q^42 + (-2z^7 + 29z^6 - 29z^4 + 2z^3 - 14z^2 + 22z - 14) * q^43 + (15z^7 + 15z^6 - 15z^5 - 15z^4 - 8z^3 - 3z^2 - 3z - 8) q^44 + (16z^7 + 20z^5 + 14z^4 + 14z^3 + 20z^2 + 16) q^45 + (-17z^7 - 17z^6 + z^5 + 9z^4 - 11z^3 + 11z^2 - 9z - 1) * q^46 + (8z^7 - 8z^5 - 24z^4 - 22z^3 + 2z^2 - 22z - 24) * q^47 + (17z^7 + z^6 - 17z^5 - 17z^4 + z^3 + 17z^2 + 12z - 12) q^48 + (-10z^6 + 4z^5 - 2z^4 - 7z^3 - 2z^2 + 4z - 10) * q^49 + (z^7 + z^6 - 35z^5 - 57z^4 - 35z^3 + z^2 + z) q^50 + (6z^7 - 34z^6 - 22z^5 + 22z^3 + 34z^2 - 6z - 48) * q^52 + (2z^7 + 16z^6 + 18z^5 + 4z^4 - 4z^2 - 18z - 16) * q^53 + (-z^7 + 8z^5 - 8z^4 + z^2 - 21z - 21) q^54 + (-2z^7 + 32z^6 - 2z^5 + 28z^4 - 4z^3 + 4z - 28) * q^55 + (17z^7 - 17z^6 - 33z^5 + 25z^4 + 15z^3 + 15z^2 + 25z - 33) q^56 + (10z^7 - 3z^6 - 9z^5 - 6z^4 + 6z^3 + 9z^2 + 3z - 10) q^57 + (30z^6 + 30z^5 - 46z^3 + 4z^2 - 4z + 46) q^58 + (10z^7 - 6z^6 - 42z^5 - 6z^4 + 10z^3 + 15z^2 - 15) * q^59 + (-6z^7 - 8z^6 + 8z^4 + 6z^3 + 6z^2 + 8z + 6) * q^60 + (36z^7 - 20z^6 + 20z^5 - 36z^4 - 12z^3 + 8z^2 + 8z - 12) q^61 + (15z^7 + 21z^6 + 7z^5 - z^4 - z^3 + 7z^2 + 21z + 15) q^62 + (21z^7 + 21z^6 + 23z^5 - 15z^4 + 7z^3 - 7z^2 + 15z - 23) q^63 + (10z^7 - 10z^5 + 8z^4 + 42z^3 + 13z^2 + 42z + 8) * q^64 + (-12z^7 + 12z^6 - 8z^5 - 8z^4 + 12z^3 - 12z^2) * q^65 + (-12z^6 - 23z^5 + z^4 + 20z^3 + z^2 - 23z - 12) * q^66 + (26z^7 - 4z^6 + 33z^5 + 42z^4 + 33z^3 - 4z^2 + 26z) q^67 + (2z^7 + 18z^6 - 8z^5 + 8z^3 - 18z^2 - 2z + 30) * q^69 + (-36z^7 - 60z^6 - 16z^5 - 28z^4 + 28z^2 + 16z + 60) * q^70 + (-5z^7 - 11z^6 - 43z^5 + 43z^4 + 11z^3 + 5z^2 + 35z + 35) q^71 + (-z^7 - 53z^6 - z^5 + 5z^4 + 24z^3 - 24z - 5) * q^72 + (-6z^7 + 6z^6 + 26z^5 + 25z^4 - 6z^3 - 6z^2 + 25z + 26) q^73 + (28z^7 + 20z^6 - 24z^5 + 26z^4 - 26z^3 + 24z^2 - 20z - 28) q^74 + (13z^7 - 37z^6 - 37z^5 + 13z^4 + 17z^3 - 17) q^75 + (-21z^7 + 10z^6 + 12z^5 + 10z^4 - 21z^3 - 20z^2 + 20) * q^76 + (8z^7 + 6z^6 - 6z^4 - 8z^3 + 10z^2 - 66z + 10) * q^77 + (-64z^7 - 50z^6 + 50z^5 + 64z^4 - 8z^3 - 38z^2 - 38z - 8) q^78 + (3z^7 + 17z^6 - 21z^5 - 41z^4 - 41z^3 - 21z^2 + 17z + 3) q^79 + (-14z^7 - 14z^6 - 30z^5 + 8z^4 - 30z^3 + 30z^2 - 8z + 30) q^80 + (-6z^7 + 6z^5 + 53z^4 - 7z^3 - 3z^2 - 7z + 53) * q^81 + (-8z^7 - 17z^6 - 8z^5 - 8z^4 - 17z^3 - 8z^2 - 34z + 34) q^82 + (42z^6 - 20z^5 + 33z^4 + 54z^3 + 33z^2 - 20z + 42) * q^83 + (-60z^7 - 24z^6 + 34z^5 + 50z^4 + 34z^3 - 24z^2 - 60z) q^84 + (-3z^7 - 8z^6 + 66z^5 - 66z^3 + 8z^2 + 3z + 104) * q^86 + (52z^7 + 10z^6 - 34z^5 - 26z^4 + 26z^2 + 34z - 10) * q^87 + (-33z^7 + 40z^6 - z^5 + z^4 - 40z^3 + 33z^2 + 9z + 9) q^88 + (-46z^7 - 8z^6 - 46z^5 + 20z^4 - 29z^3 + 29z - 20) * q^89 + (-18z^7 + 18z^6 - 48z^5 - 28z^4 - 56z^3 - 56z^2 - 28z - 48) q^90 + (-40z^6 + 28z^5 + 68z^4 - 68z^3 - 28z^2 + 40z) * q^91 + (-3z^7 + 3z^6 + 3z^5 - 3z^4 + 31z^3 + 23z^2 - 23z - 31) q^92 + (8z^7 - 8z^6 - 6z^5 - 8z^4 + 8z^3 + 26z^2 - 26) * q^93 + (-18z^7 - 4z^6 + 4z^4 + 18z^3 + 54z^2 + 110z + 54) * q^94 + (-24z^7 + 34z^6 - 34z^5 + 24z^4 - 24z^3 + 2z^2 + 2z - 24) q^95 + (-11z^7 + 7z^6 + 29z^5 - 8z^4 - 8z^3 + 29z^2 + 7z - 11) q^96 + (13z^7 + 13z^6 - 22z^5 + 30z^4 + 6z^3 - 6z^2 - 30z + 22) q^97 + (11z^7 - 11z^5 + 15z^4 + 9z^3 + 9z^2 + 9z + 15) * q^98 + (-28z^7 + 34z^6 + 16z^5 + 16z^4 + 34z^3 - 28z^2 + 27z - 27) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{3} + 32 q^{6} + 8 q^{7} - 40 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^3 + 32 * q^6 + 8 * q^7 - 40 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 8 q^{3} + 32 q^{6} + 8 q^{7} - 40 q^{8} + 8 q^{9} + 32 q^{10} - 24 q^{11} - 8 q^{12} - 16 q^{13} + 24 q^{14} + 56 q^{18} - 48 q^{19} - 80 q^{20} + 64 q^{21} + 48 q^{22} + 24 q^{23} - 120 q^{24} - 32 q^{25} - 48 q^{26} + 80 q^{27} - 120 q^{28} + 16 q^{29} - 16 q^{30} - 40 q^{31} - 88 q^{32} - 160 q^{35} - 24 q^{36} + 16 q^{37} + 120 q^{38} + 160 q^{39} + 48 q^{40} - 40 q^{41} + 128 q^{42} - 112 q^{43} - 64 q^{44} + 128 q^{45} - 8 q^{46} - 192 q^{47} - 96 q^{48} - 80 q^{49} - 384 q^{52} - 128 q^{53} - 168 q^{54} - 224 q^{55} - 264 q^{56} - 80 q^{57} + 368 q^{58} - 120 q^{59} + 48 q^{60} - 96 q^{61} + 120 q^{62} - 184 q^{63} + 64 q^{64} - 96 q^{66} + 240 q^{69} + 480 q^{70} + 280 q^{71} - 40 q^{72} + 208 q^{73} - 224 q^{74} - 136 q^{75} + 160 q^{76} + 80 q^{77} - 64 q^{78} + 24 q^{79} + 240 q^{80} + 424 q^{81} + 272 q^{82} + 336 q^{83} + 832 q^{86} - 80 q^{87} + 72 q^{88} - 160 q^{89} - 384 q^{90} - 248 q^{92} - 208 q^{93} + 432 q^{94} - 192 q^{95} - 88 q^{96} + 176 q^{97} + 120 q^{98} - 216 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 32 * q^6 + 8 * q^7 - 40 * q^8 + 8 * q^9 + 32 * q^10 - 24 * q^11 - 8 * q^12 - 16 * q^13 + 24 * q^14 + 56 * q^18 - 48 * q^19 - 80 * q^20 + 64 * q^21 + 48 * q^22 + 24 * q^23 - 120 * q^24 - 32 * q^25 - 48 * q^26 + 80 * q^27 - 120 * q^28 + 16 * q^29 - 16 * q^30 - 40 * q^31 - 88 * q^32 - 160 * q^35 - 24 * q^36 + 16 * q^37 + 120 * q^38 + 160 * q^39 + 48 * q^40 - 40 * q^41 + 128 * q^42 - 112 * q^43 - 64 * q^44 + 128 * q^45 - 8 * q^46 - 192 * q^47 - 96 * q^48 - 80 * q^49 - 384 * q^52 - 128 * q^53 - 168 * q^54 - 224 * q^55 - 264 * q^56 - 80 * q^57 + 368 * q^58 - 120 * q^59 + 48 * q^60 - 96 * q^61 + 120 * q^62 - 184 * q^63 + 64 * q^64 - 96 * q^66 + 240 * q^69 + 480 * q^70 + 280 * q^71 - 40 * q^72 + 208 * q^73 - 224 * q^74 - 136 * q^75 + 160 * q^76 + 80 * q^77 - 64 * q^78 + 24 * q^79 + 240 * q^80 + 424 * q^81 + 272 * q^82 + 336 * q^83 + 832 * q^86 - 80 * q^87 + 72 * q^88 - 160 * q^89 - 384 * q^90 - 248 * q^92 - 208 * q^93 + 432 * q^94 - 192 * q^95 - 88 * q^96 + 176 * q^97 + 120 * q^98 - 216 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	289.3.e.d

Level	$289$
Weight	$3$
Character orbit	289.e
Analytic conductor	$7.875$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(7.87467964001\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{16})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 1 \) x^8 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 40.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} + 8 T^{7} + \cdots + 961 \) T^8 + 8T^7 + 32T^6 + 120T^5 + 448T^4 + 1144T^3 + 1792T^2 + 1736*T + 961
$3$	\( T^{8} - 8 T^{7} + \cdots + 2 \) T^8 - 8T^7 + 28T^6 - 32T^5 - 10T^4 + 120T^3 + 84T^2 + 8*T + 2
$5$	\( T^{8} + 16 T^{6} + \cdots + 512 \) T^8 + 16T^6 - 320T^5 + 1888T^4 - 768T^3 + 4864T^2 - 3072T + 512
$7$	\( T^{8} - 8 T^{7} + \cdots + 8454272 \) T^8 - 8T^7 + 72T^6 - 544T^5 + 2352T^4 - 2816T^3 - 40320T^2 - 1052672*T + 8454272
$11$	\( T^{8} + 24 T^{7} + \cdots + 27572738 \) T^8 + 24T^7 + 136T^6 + 160T^5 + 2066T^4 - 2464T^3 + 972468T^2 + 5198200*T + 27572738
$13$	\( T^{8} + 16 T^{7} + \cdots + 9048064 \) T^8 + 16T^7 + 128T^6 - 2688T^5 + 129408T^4 + 1129472T^3 + 5120000T^2 - 9625600*T + 9048064
$17$	\( T^{8} \) T^8
$19$	\( T^{8} + 48 T^{7} + \cdots + 929296 \) T^8 + 48T^7 + 784T^6 + 4032T^5 + 21632T^4 - 294144T^3 + 863936T^2 - 1017984*T + 929296
$23$	\( T^{8} - 24 T^{7} + \cdots + 859299968 \) T^8 - 24T^7 + 616T^6 + 1984T^5 + 42256T^4 + 352512T^3 - 8233472T^2 + 39797760*T + 859299968
$29$	\( T^{8} + \cdots + 4240836608 \) T^8 - 16T^7 + 1680T^6 - 29952T^5 + 1069760T^4 - 47741440T^3 + 676834304T^2 - 2699517952*T + 4240836608
$31$	\( T^{8} + 40 T^{7} + \cdots + 14536832 \) T^8 + 40T^7 + 1000T^6 + 15424T^5 + 195472T^4 + 1916160T^3 + 13694208T^2 + 25709056*T + 14536832
$37$	\( T^{8} + \cdots + 120057840128 \) T^8 - 16T^7 + 1648T^6 - 448T^5 - 727456T^4 - 26123520T^3 + 83138304T^2 + 8200907776*T + 120057840128
$41$	\( T^{8} + \cdots + 59058658562 \) T^8 + 40T^7 + 1084T^6 + 16680T^5 + 402758T^4 - 6674744T^3 - 104090084T^2 - 3399702344*T + 59058658562
$43$	\( T^{8} + \cdots + 6201305218564 \) T^8 + 112T^7 + 5252T^6 + 377208T^5 + 29495944T^4 + 659093648T^3 + 11251238296T^2 + 373556221936*T + 6201305218564
$47$	\( T^{8} + \cdots + 2259754549504 \) T^8 + 192T^7 + 18432T^6 + 907072T^5 + 25209440T^4 + 277465600T^3 + 2803712T^2 + 3559691264*T + 2259754549504
$53$	\( T^{8} + \cdots + 26754490624 \) T^8 + 128T^7 + 6432T^6 + 169600T^5 + 4121088T^4 + 47191552T^3 + 1236934144T^2 + 10007744512*T + 26754490624
$59$	\( T^{8} + \cdots + 2675455605124 \) T^8 + 120T^7 + 2364T^6 - 120360T^5 + 6149448T^4 - 202344096T^3 + 5715853032T^2 - 153610167984*T + 2675455605124
$61$	\( T^{8} + \cdots + 106680004247552 \) T^8 + 96T^7 + 8320T^6 + 834560T^5 + 67023360T^4 + 4955521024T^3 + 268994297856T^2 + 8110656847872*T + 106680004247552
$67$	\( T^{8} + \cdots + 83643718741636 \) T^8 + 21304T^6 + 118064724T^4 + 200330319344*T^2 + 83643718741636
$71$	\( T^{8} + \cdots + 71246079996032 \) T^8 - 280T^7 + 36072T^6 - 2760512T^5 + 126349584T^4 - 5585860096T^3 + 246660350976T^2 + 7447165054976*T + 71246079996032
$73$	\( T^{8} + \cdots + 51301810562 \) T^8 - 208T^7 + 18692T^6 - 1013344T^5 + 36374662T^4 - 999045256T^3 + 30382900100T^2 - 64987397112*T + 51301810562
$79$	\( T^{8} + \cdots + 3948319764608 \) T^8 - 24T^7 + 11416T^6 + 15424T^5 + 59751056T^4 + 3663717760T^3 + 25152949504T^2 - 673254040064*T + 3948319764608
$83$	\( T^{8} + \cdots + 10\!\cdots\!04 \) T^8 - 336T^7 + 56196T^6 - 5630952T^5 + 313068168T^4 - 2524583856T^3 - 173221055208T^2 + 31481546945712*T + 1042256526864004
$89$	\( T^{8} + \cdots + 117588822570244 \) T^8 + 160T^7 + 12800T^6 - 64128T^5 + 72896012T^4 + 9469447872T^3 + 584098906112T^2 + 11720367113216*T + 117588822570244
$97$	\( T^{8} + \cdots + 21682310986562 \) T^8 - 176T^7 + 12352T^6 - 448864T^5 + 5558242T^4 - 123290672T^3 + 16219072820T^2 + 813164614088*T + 21682310986562
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\(n\)	\(3\)
\(\chi(n)\)	\(\zeta_{16}\)