LMFDB - Newform orbit 289.3.e.k

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:	$\Z[a_1, \ldots, a_{4}]$
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^5 - z^4 - z^3 + z^2) * q^3 + (2z^7 - 3z^6 + 2z^5 - 2z^3 + 2z) q^4 + (-2z^7 + 2z^6 - 2z^5 - 2z^3 - 2z^2 - 2) q^5 + (z^7 - 3z^6 + z^5 + 4z^4 - 4z^3 - z^2 + 3z - 1) * q^6 + (z^7 - z^6 - z^5 + z^4 - 5z^3 - z^2 + z + 5) 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 + (34 \zeta_{16}^{7} + 28 \zeta_{16}^{6} + \cdots + 16) 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^5 - z^4 - z^3 + z^2) * q^3 + (2z^7 - 3z^6 + 2z^5 - 2z^3 + 2z) q^4 + (-2z^7 + 2z^6 - 2z^5 - 2z^3 - 2z^2 - 2) q^5 + (z^7 - 3z^6 + z^5 + 4z^4 - 4z^3 - z^2 + 3z - 1) * q^6 + (z^7 - z^6 - z^5 + z^4 - 5z^3 - z^2 + z + 5) 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 + (-4z^7 + 6z^6 - 6z^5 + 4z^4 - 6z^3 - 2z^2 - 2z - 6) q^10 + (-z^7 - 6z^6 - z^5 - 3z^4 - 3z^3 - z^2 - 6z - 1) * q^11 + (6z^7 + 6z^6 - 9z^5 + z^4 + 5z^3 - 5z^2 - z + 9) q^12 + (10z^7 - 10z^5 - 2z^4 + 2z^3 + 2z - 2) q^13 + (-z^7 - z^6 - 3z^5 - 3z^4 - z^3 - z^2 - 13z + 13) 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 + (-4z^7 + 10z^6 - 10z^5 + 10z^4 - 10z^3 + 4z^2 - 2z - 2) q^20 + (-2z^7 + 14z^6 - 2z^5 - 8z^4 + 8) * q^21 + (13z^7 - 13z^6 - 3z^5 - 6z^4 - 7z^3 - 7z^2 - 6z - 3) q^22 + (-7z^7 - 5z^6 + 7z^5 + 3z^4 - 3z^3 - 7z^2 + 5z + 7) q^23 + (-15z^7 + 13z^6 + 13z^5 - 15z^4 - 3z^3 + 10z^2 - 10z + 3) 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 + (-10z^7 - 2z^6 + 2z^5 + 10z^4 - 3z^3 - 5z^2 - 5z - 3) q^27 + (z^7 - 7z^6 + 19z^5 - 15z^4 - 15z^3 + 19z^2 - 7z + 1) * q^28 + (6z^7 + 6z^6 + 16z^5 - 2z^4 - 8z^3 + 8z^2 + 2z - 16) q^29 + (-2z^7 + 2z^5 - 2z^4 + 4z^3 - 8z^2 + 4z - 2) * q^30 + (-5z^7 - 3z^6 + 5z^5 + 5z^4 - 3z^3 - 5z^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 + (16z^7 - 4z^6 + 2z^5 - 2z^4 + 4z^3 - 16z^2 - 12z - 12) q^37 + (-18z^7 - 2z^6 - 18z^5 - 15z^4 + 2z^3 - 2z + 15) * q^38 + (-6z^7 + 6z^6 - 12z^5 - 20z^4 + 22z^3 + 22z^2 - 20z - 12) q^39 + (-14z^7 - 2z^6 - 8z^5 + 6z^4 - 6z^3 + 8z^2 + 2z + 14) q^40 + (-5z^7 - 14z^6 - 14z^5 - 5z^4 + 6z^3 - 6z^2 + 6z - 6) 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 + (8z^7 - 3z^6 + 3z^5 - 8z^4 + 15z^3 - 15z^2 - 15z + 15) q^44 + (-14z^7 + 20z^6 + 16z^4 + 16z^3 + 20z - 14) q^45 + (-11z^7 - 11z^6 - 9z^5 + z^4 + 17z^3 - 17z^2 - z + 9) q^46 + (-8z^7 + 8z^5 - 24z^4 + 22z^3 + 2z^2 + 22z - 24) * q^47 + (z^7 - 17z^6 + 12z^5 + 12z^4 - 17z^3 + z^2 + 17z - 17) 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^6 - 21z^5 + 21z^4 + z^3 - 8z - 8) q^54 + (2z^7 + 32z^6 + 2z^5 + 28z^4 + 4z^3 - 4z - 28) * q^55 + (15z^7 - 15z^6 + 25z^5 + 33z^4 - 17z^3 - 17z^2 + 33z + 25) q^56 + (-6z^7 + 9z^6 - 3z^5 - 10z^4 + 10z^3 + 3z^2 - 9z + 6) q^57 + (46z^7 + 4z^6 + 4z^5 + 46z^4 - 30z^2 + 30z) * 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 + (12z^7 + 8z^6 - 8z^5 - 12z^4 + 36z^3 + 20z^2 + 20z + 36) q^61 + (z^7 + 7z^6 - 21z^5 + 15z^4 + 15z^3 - 21z^2 + 7z + 1) * q^62 + (7z^7 + 7z^6 + 15z^5 + 23z^4 - 21z^3 + 21z^2 - 23z - 15) q^63 + (-10z^7 + 10z^5 + 8z^4 - 42z^3 + 13z^2 - 42z + 8) * q^64 + (12z^7 + 12z^6 + 12z^3 + 12z^2 + 8z - 8) 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 + (11z^7 - 5z^6 + 35z^5 - 35z^4 + 5z^3 - 11z^2 + 43z + 43) q^71 + (z^7 - 53z^6 + z^5 + 5z^4 - 24z^3 + 24z - 5) * q^72 + (-6z^7 + 6z^6 + 25z^5 - 26z^4 + 6z^3 + 6z^2 - 26z + 25) q^73 + (26z^7 + 24z^6 + 20z^5 - 28z^4 + 28z^3 - 20z^2 - 24z - 26) q^74 + (-17z^7 - 17z^4 + 13z^3 + 37z^2 - 37z - 13) 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 + (8z^7 - 38z^6 + 38z^5 - 8z^4 - 64z^3 + 50z^2 + 50z - 64) q^78 + (41z^7 - 21z^6 - 17z^5 + 3z^4 + 3z^3 - 17z^2 - 21z + 41) q^79 + (-30z^7 - 30z^6 - 8z^5 - 30z^4 + 14z^3 - 14z^2 + 30z + 8) q^80 + (6z^7 - 6z^5 + 53z^4 + 7z^3 - 3z^2 + 7z + 53) * q^81 + (-17z^7 + 8z^6 - 34z^5 - 34z^4 + 8z^3 - 17z^2 + 8z - 8) 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 + (-40z^7 - 33z^6 + 9z^5 - 9z^4 + 33z^3 + 40z^2 + z + 1) * q^88 + (46z^7 - 8z^6 + 46z^5 + 20z^4 + 29z^3 - 29z - 20) * q^89 + (-56z^7 + 56z^6 - 28z^5 + 48z^4 + 18z^3 + 18z^2 + 48z - 28) q^90 + (68z^7 - 28z^6 - 40z^5 + 40z^2 + 28z - 68) q^91 + (-31z^7 + 23z^6 + 23z^5 - 31z^4 - 3z^3 - 3z^2 + 3z + 3) 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 + 2z^6 - 2z^5 - 24z^4 - 24z^3 - 34z^2 - 34z - 24) q^95 + (8z^7 + 29z^6 - 7z^5 - 11z^4 - 11z^3 - 7z^2 + 29z + 8) q^96 + (6z^7 + 6z^6 - 30z^5 - 22z^4 - 13z^3 + 13z^2 + 22z + 30) q^97 + (-11z^7 + 11z^5 + 15z^4 - 9z^3 + 9z^2 - 9z + 15) * q^98 + (34z^7 + 28z^6 + 27z^5 + 27z^4 + 28z^3 + 34z^2 - 16z + 16) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 16 q^{5} - 8 q^{6} + 40 q^{7} + 40 q^{8} - 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 16 * q^5 - 8 * q^6 + 40 * q^7 + 40 * q^8 - 8 * q^9	\( 8 q + 8 q^{2} - 16 q^{5} - 8 q^{6} + 40 q^{7} + 40 q^{8} - 8 q^{9} - 48 q^{10} - 8 q^{11} + 72 q^{12} - 16 q^{13} + 104 q^{14} + 56 q^{18} + 48 q^{19} - 16 q^{20} + 64 q^{21} - 24 q^{22} + 56 q^{23} + 24 q^{24} + 32 q^{25} + 48 q^{26} - 24 q^{27} + 8 q^{28} - 128 q^{29} - 16 q^{30} - 40 q^{31} + 88 q^{32} - 160 q^{35} + 24 q^{36} - 96 q^{37} + 120 q^{38} - 96 q^{39} + 112 q^{40} - 48 q^{41} - 128 q^{42} + 112 q^{43} + 120 q^{44} - 112 q^{45} + 72 q^{46} - 192 q^{47} - 136 q^{48} + 80 q^{49} - 384 q^{52} + 128 q^{53} - 64 q^{54} - 224 q^{55} + 200 q^{56} + 48 q^{57} + 120 q^{59} - 48 q^{60} + 288 q^{61} + 8 q^{62} - 120 q^{63} + 64 q^{64} - 64 q^{65} + 96 q^{66} + 240 q^{69} - 480 q^{70} + 344 q^{71} - 40 q^{72} + 200 q^{73} - 208 q^{74} - 104 q^{75} - 160 q^{76} - 80 q^{77} - 512 q^{78} + 328 q^{79} + 64 q^{80} + 424 q^{81} - 64 q^{82} - 336 q^{83} + 832 q^{86} + 80 q^{87} + 8 q^{88} - 160 q^{89} - 224 q^{90} - 544 q^{91} + 24 q^{92} + 208 q^{93} - 432 q^{94} - 192 q^{95} + 64 q^{96} + 240 q^{97} + 120 q^{98} + 128 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 16 * q^5 - 8 * q^6 + 40 * q^7 + 40 * q^8 - 8 * q^9 - 48 * q^10 - 8 * q^11 + 72 * q^12 - 16 * q^13 + 104 * q^14 + 56 * q^18 + 48 * q^19 - 16 * q^20 + 64 * q^21 - 24 * q^22 + 56 * q^23 + 24 * q^24 + 32 * q^25 + 48 * q^26 - 24 * q^27 + 8 * q^28 - 128 * q^29 - 16 * q^30 - 40 * q^31 + 88 * q^32 - 160 * q^35 + 24 * q^36 - 96 * q^37 + 120 * q^38 - 96 * q^39 + 112 * q^40 - 48 * q^41 - 128 * q^42 + 112 * q^43 + 120 * q^44 - 112 * q^45 + 72 * q^46 - 192 * q^47 - 136 * q^48 + 80 * q^49 - 384 * q^52 + 128 * q^53 - 64 * q^54 - 224 * q^55 + 200 * q^56 + 48 * q^57 + 120 * q^59 - 48 * q^60 + 288 * q^61 + 8 * q^62 - 120 * q^63 + 64 * q^64 - 64 * q^65 + 96 * q^66 + 240 * q^69 - 480 * q^70 + 344 * q^71 - 40 * q^72 + 200 * q^73 - 208 * q^74 - 104 * q^75 - 160 * q^76 - 80 * q^77 - 512 * q^78 + 328 * q^79 + 64 * q^80 + 424 * q^81 - 64 * q^82 - 336 * q^83 + 832 * q^86 + 80 * q^87 + 8 * q^88 - 160 * q^89 - 224 * q^90 - 544 * q^91 + 24 * q^92 + 208 * q^93 - 432 * q^94 - 192 * q^95 + 64 * q^96 + 240 * q^97 + 120 * q^98 + 128 * 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

2.03153

0.841487i

0.158513

−

0.0315301i

0.590587

0.590587i

−2.98067

4.46088i

0.348555

0.0693320i

3.46927

5.19212i

−2.66313

−

6.42935i

−8.29078

3.43416i

−9.80910

6.55423i

65.1

2.79690

−

1.15851i

−0.158513

0.796897i

3.65205

−

3.65205i

−6.67619

4.46088i

0.479872

2.41248i

6.53073

4.36370i

1.34942

−

3.25778i

7.70500

3.19151i

−13.5046

20.2111i

75.1

−1.33809

3.23044i

−2.23044

3.33809i

−5.81684

−

5.81684i

0.0630603

0.317025i

−7.79898

−

11.6720i

1.30448

−

6.55807i

13.6527

−

5.65512i

−2.72384

−

6.57593i

−1.10851

−

0.220497i

131.1

0.509666

1.23044i

2.23044

−

1.49033i

1.57420

−

1.57420i

1.59379

0.317025i

2.97055

1.98486i

8.69552

−

1.72965i

7.66104

3.17331i

−0.690373

1.66671i

0.422221

2.12265i

158.1

−1.33809

−

3.23044i

−2.23044

−

3.33809i

−5.81684

5.81684i

0.0630603

−

0.317025i

−7.79898

11.6720i

1.30448

6.55807i

13.6527

5.65512i

−2.72384

6.57593i

−1.10851

0.220497i

214.1

0.509666

−

1.23044i

2.23044

1.49033i

1.57420

1.57420i

1.59379

−

0.317025i

2.97055

−

1.98486i

8.69552

1.72965i

7.66104

−

3.17331i

−0.690373

−

1.66671i

0.422221

−

2.12265i

224.1

2.03153

−

0.841487i

0.158513

0.0315301i

0.590587

−

0.590587i

−2.98067

−

4.46088i

0.348555

−

0.0693320i

3.46927

−

5.19212i

−2.66313

6.42935i

−8.29078

−

3.43416i

−9.80910

−

6.55423i

249.1

2.79690

1.15851i

−0.158513

−

0.796897i

3.65205

3.65205i

−6.67619

−

4.46088i

0.479872

−

2.41248i

6.53073

−

4.36370i

1.34942

3.25778i

7.70500

−

3.19151i

−13.5046

−

20.2111i

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.k		8
17.b	even	2	1		289.3.e.l		8
17.c	even	4	1		289.3.e.b		8
17.c	even	4	1		289.3.e.d		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		289.3.e.d		8
17.e	odd	16	1		289.3.e.i		8
17.e	odd	16	1	inner	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.a		8
85.m	even	8	1		425.3.u.b		8
85.n	odd	8	1		425.3.t.c		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.c	even	4	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	17.c	even	4	1
289.3.e.d		8	17.e	odd	16	1
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	1.a	even	1	1	trivial
289.3.e.k		8	17.e	odd	16	1	inner
289.3.e.l		8	17.b	even	2	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.k	odd	8	1
425.3.t.a		8	85.r	even	16	1
425.3.t.c		8	85.n	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} + 4T_{3}^{6} - 40T_{3}^{5} + 118T_{3}^{4} - 24T_{3}^{3} + 76T_{3}^{2} - 24T_{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} + 4 T^{6} + \cdots + 2 $ T^8 + 4T^6 - 40T^5 + 118T^4 - 24T^3 + 76T^2 - 24T + 2
$5$	$ T^{8} + 16 T^{7} + \cdots + 512 $ T^8 + 16T^7 + 112T^6 + 256T^5 - 160T^4 - 3840T^3 + 5376T^2 - 1024*T + 512
$7$	$ T^{8} - 40 T^{7} + \cdots + 8454272 $ T^8 - 40T^7 + 760T^6 - 9120T^5 + 76592T^4 - 464512T^3 + 1999232T^2 - 5658112*T + 8454272
$11$	$ T^{8} + 8 T^{7} + \cdots + 27572738 $ T^8 + 8T^7 + 184T^6 - 536T^5 - 13742T^4 - 142816T^3 + 229868T^2 + 6267544*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} - 56 T^{7} + \cdots + 859299968 $ T^8 - 56T^7 + 1240T^6 - 10496T^5 + 98576T^4 - 1817472T^3 + 10338816T^2 - 45104128*T + 859299968
$29$	$ T^{8} + \cdots + 4240836608 $ T^8 + 128T^7 + 6640T^6 + 180224T^5 + 3025600T^4 + 44551680T^3 + 591369216T^2 + 1862549504*T + 4240836608
$31$	$ T^{8} + 40 T^{7} + \cdots + 14536832 $ T^8 + 40T^7 + 600T^6 + 4800T^5 + 90512T^4 + 1677952T^3 + 10760960T^2 + 7074304*T + 14536832
$37$	$ T^{8} + \cdots + 120057840128 $ T^8 + 96T^7 + 3088T^6 + 64000T^5 + 1999456T^4 + 30240000T^3 + 310505728T^2 + 5221610496*T + 120057840128
$41$	$ T^{8} + \cdots + 59058658562 $ T^8 + 48T^7 + 868T^6 + 34912T^5 + 1622150T^4 + 34972120T^3 + 419681284T^2 + 5071371592*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 - 288T^7 + 37760T^6 - 3020288T^5 + 178450944T^4 - 8681603072T^3 + 320410402816T^2 - 7744200245248*T + 106680004247552
$67$	$ T^{8} + \cdots + 83643718741636 $ T^8 + 21304T^6 + 118064724T^4 + 200330319344*T^2 + 83643718741636
$71$	$ T^{8} + \cdots + 71246079996032 $ T^8 - 344T^7 + 62296T^6 - 7376256T^5 + 601037072T^4 - 30495378560T^3 + 822692609024T^2 - 9844498245632*T + 71246079996032
$73$	$ T^{8} + \cdots + 51301810562 $ T^8 - 200T^7 + 22940T^6 - 1745000T^5 + 86172742T^4 - 2340373992T^3 + 24720959132T^2 + 37821868168*T + 51301810562
$79$	$ T^{8} + \cdots + 3948319764608 $ T^8 - 328T^7 + 42664T^6 - 2603328T^5 + 69066896T^4 - 1274091520T^3 + 64809024256T^2 + 507076203008*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 - 240T^7 + 31936T^6 - 2846504T^5 + 176051042T^4 - 7151829088T^3 + 182929540332T^2 - 2823963108152*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^5 - z^4 - z^3 + z^2) * q^3 + (2z^7 - 3z^6 + 2z^5 - 2z^3 + 2z) q^4 + (-2z^7 + 2z^6 - 2z^5 - 2z^3 - 2z^2 - 2) q^5 + (z^7 - 3z^6 + z^5 + 4z^4 - 4z^3 - z^2 + 3z - 1) * q^6 + (z^7 - z^6 - z^5 + z^4 - 5z^3 - z^2 + z + 5) 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	\( q + ( - \zeta_{16}^{7} - \zeta_{16}^{6} + \cdots + 1) q^{2}+ \cdots + (34 \zeta_{16}^{7} + 28 \zeta_{16}^{6} + \cdots + 16) q^{99}+O(q^{100}) \) q + (-z^7 - z^6 + z^5 - z^4 + z^2 - z + 1) * q^2 + (-z^7 + z^6 + z^5 - z^4 - z^3 + z^2) * q^3 + (2z^7 - 3z^6 + 2z^5 - 2z^3 + 2z) q^4 + (-2z^7 + 2z^6 - 2z^5 - 2z^3 - 2z^2 - 2) q^5 + (z^7 - 3z^6 + z^5 + 4z^4 - 4z^3 - z^2 + 3z - 1) * q^6 + (z^7 - z^6 - z^5 + z^4 - 5z^3 - z^2 + z + 5) 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 + (-4z^7 + 6z^6 - 6z^5 + 4z^4 - 6z^3 - 2z^2 - 2z - 6) q^10 + (-z^7 - 6z^6 - z^5 - 3z^4 - 3z^3 - z^2 - 6z - 1) * q^11 + (6z^7 + 6z^6 - 9z^5 + z^4 + 5z^3 - 5z^2 - z + 9) q^12 + (10z^7 - 10z^5 - 2z^4 + 2z^3 + 2z - 2) q^13 + (-z^7 - z^6 - 3z^5 - 3z^4 - z^3 - z^2 - 13z + 13) 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 + (-4z^7 + 10z^6 - 10z^5 + 10z^4 - 10z^3 + 4z^2 - 2z - 2) q^20 + (-2z^7 + 14z^6 - 2z^5 - 8z^4 + 8) * q^21 + (13z^7 - 13z^6 - 3z^5 - 6z^4 - 7z^3 - 7z^2 - 6z - 3) q^22 + (-7z^7 - 5z^6 + 7z^5 + 3z^4 - 3z^3 - 7z^2 + 5z + 7) q^23 + (-15z^7 + 13z^6 + 13z^5 - 15z^4 - 3z^3 + 10z^2 - 10z + 3) 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 + (-10z^7 - 2z^6 + 2z^5 + 10z^4 - 3z^3 - 5z^2 - 5z - 3) q^27 + (z^7 - 7z^6 + 19z^5 - 15z^4 - 15z^3 + 19z^2 - 7z + 1) * q^28 + (6z^7 + 6z^6 + 16z^5 - 2z^4 - 8z^3 + 8z^2 + 2z - 16) q^29 + (-2z^7 + 2z^5 - 2z^4 + 4z^3 - 8z^2 + 4z - 2) * q^30 + (-5z^7 - 3z^6 + 5z^5 + 5z^4 - 3z^3 - 5z^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 + (16z^7 - 4z^6 + 2z^5 - 2z^4 + 4z^3 - 16z^2 - 12z - 12) q^37 + (-18z^7 - 2z^6 - 18z^5 - 15z^4 + 2z^3 - 2z + 15) * q^38 + (-6z^7 + 6z^6 - 12z^5 - 20z^4 + 22z^3 + 22z^2 - 20z - 12) q^39 + (-14z^7 - 2z^6 - 8z^5 + 6z^4 - 6z^3 + 8z^2 + 2z + 14) q^40 + (-5z^7 - 14z^6 - 14z^5 - 5z^4 + 6z^3 - 6z^2 + 6z - 6) 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 + (8z^7 - 3z^6 + 3z^5 - 8z^4 + 15z^3 - 15z^2 - 15z + 15) q^44 + (-14z^7 + 20z^6 + 16z^4 + 16z^3 + 20z - 14) q^45 + (-11z^7 - 11z^6 - 9z^5 + z^4 + 17z^3 - 17z^2 - z + 9) q^46 + (-8z^7 + 8z^5 - 24z^4 + 22z^3 + 2z^2 + 22z - 24) * q^47 + (z^7 - 17z^6 + 12z^5 + 12z^4 - 17z^3 + z^2 + 17z - 17) 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^6 - 21z^5 + 21z^4 + z^3 - 8z - 8) q^54 + (2z^7 + 32z^6 + 2z^5 + 28z^4 + 4z^3 - 4z - 28) * q^55 + (15z^7 - 15z^6 + 25z^5 + 33z^4 - 17z^3 - 17z^2 + 33z + 25) q^56 + (-6z^7 + 9z^6 - 3z^5 - 10z^4 + 10z^3 + 3z^2 - 9z + 6) q^57 + (46z^7 + 4z^6 + 4z^5 + 46z^4 - 30z^2 + 30z) * 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 + (12z^7 + 8z^6 - 8z^5 - 12z^4 + 36z^3 + 20z^2 + 20z + 36) q^61 + (z^7 + 7z^6 - 21z^5 + 15z^4 + 15z^3 - 21z^2 + 7z + 1) * q^62 + (7z^7 + 7z^6 + 15z^5 + 23z^4 - 21z^3 + 21z^2 - 23z - 15) q^63 + (-10z^7 + 10z^5 + 8z^4 - 42z^3 + 13z^2 - 42z + 8) * q^64 + (12z^7 + 12z^6 + 12z^3 + 12z^2 + 8z - 8) 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 + (11z^7 - 5z^6 + 35z^5 - 35z^4 + 5z^3 - 11z^2 + 43z + 43) q^71 + (z^7 - 53z^6 + z^5 + 5z^4 - 24z^3 + 24z - 5) * q^72 + (-6z^7 + 6z^6 + 25z^5 - 26z^4 + 6z^3 + 6z^2 - 26z + 25) q^73 + (26z^7 + 24z^6 + 20z^5 - 28z^4 + 28z^3 - 20z^2 - 24z - 26) q^74 + (-17z^7 - 17z^4 + 13z^3 + 37z^2 - 37z - 13) 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 + (8z^7 - 38z^6 + 38z^5 - 8z^4 - 64z^3 + 50z^2 + 50z - 64) q^78 + (41z^7 - 21z^6 - 17z^5 + 3z^4 + 3z^3 - 17z^2 - 21z + 41) q^79 + (-30z^7 - 30z^6 - 8z^5 - 30z^4 + 14z^3 - 14z^2 + 30z + 8) q^80 + (6z^7 - 6z^5 + 53z^4 + 7z^3 - 3z^2 + 7z + 53) * q^81 + (-17z^7 + 8z^6 - 34z^5 - 34z^4 + 8z^3 - 17z^2 + 8z - 8) 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 + (-40z^7 - 33z^6 + 9z^5 - 9z^4 + 33z^3 + 40z^2 + z + 1) * q^88 + (46z^7 - 8z^6 + 46z^5 + 20z^4 + 29z^3 - 29z - 20) * q^89 + (-56z^7 + 56z^6 - 28z^5 + 48z^4 + 18z^3 + 18z^2 + 48z - 28) q^90 + (68z^7 - 28z^6 - 40z^5 + 40z^2 + 28z - 68) q^91 + (-31z^7 + 23z^6 + 23z^5 - 31z^4 - 3z^3 - 3z^2 + 3z + 3) 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 + 2z^6 - 2z^5 - 24z^4 - 24z^3 - 34z^2 - 34z - 24) q^95 + (8z^7 + 29z^6 - 7z^5 - 11z^4 - 11z^3 - 7z^2 + 29z + 8) q^96 + (6z^7 + 6z^6 - 30z^5 - 22z^4 - 13z^3 + 13z^2 + 22z + 30) q^97 + (-11z^7 + 11z^5 + 15z^4 - 9z^3 + 9z^2 - 9z + 15) * q^98 + (34z^7 + 28z^6 + 27z^5 + 27z^4 + 28z^3 + 34z^2 - 16z + 16) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 16 q^{5} - 8 q^{6} + 40 q^{7} + 40 q^{8} - 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 16 * q^5 - 8 * q^6 + 40 * q^7 + 40 * q^8 - 8 * q^9	\( 8 q + 8 q^{2} - 16 q^{5} - 8 q^{6} + 40 q^{7} + 40 q^{8} - 8 q^{9} - 48 q^{10} - 8 q^{11} + 72 q^{12} - 16 q^{13} + 104 q^{14} + 56 q^{18} + 48 q^{19} - 16 q^{20} + 64 q^{21} - 24 q^{22} + 56 q^{23} + 24 q^{24} + 32 q^{25} + 48 q^{26} - 24 q^{27} + 8 q^{28} - 128 q^{29} - 16 q^{30} - 40 q^{31} + 88 q^{32} - 160 q^{35} + 24 q^{36} - 96 q^{37} + 120 q^{38} - 96 q^{39} + 112 q^{40} - 48 q^{41} - 128 q^{42} + 112 q^{43} + 120 q^{44} - 112 q^{45} + 72 q^{46} - 192 q^{47} - 136 q^{48} + 80 q^{49} - 384 q^{52} + 128 q^{53} - 64 q^{54} - 224 q^{55} + 200 q^{56} + 48 q^{57} + 120 q^{59} - 48 q^{60} + 288 q^{61} + 8 q^{62} - 120 q^{63} + 64 q^{64} - 64 q^{65} + 96 q^{66} + 240 q^{69} - 480 q^{70} + 344 q^{71} - 40 q^{72} + 200 q^{73} - 208 q^{74} - 104 q^{75} - 160 q^{76} - 80 q^{77} - 512 q^{78} + 328 q^{79} + 64 q^{80} + 424 q^{81} - 64 q^{82} - 336 q^{83} + 832 q^{86} + 80 q^{87} + 8 q^{88} - 160 q^{89} - 224 q^{90} - 544 q^{91} + 24 q^{92} + 208 q^{93} - 432 q^{94} - 192 q^{95} + 64 q^{96} + 240 q^{97} + 120 q^{98} + 128 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 16 * q^5 - 8 * q^6 + 40 * q^7 + 40 * q^8 - 8 * q^9 - 48 * q^10 - 8 * q^11 + 72 * q^12 - 16 * q^13 + 104 * q^14 + 56 * q^18 + 48 * q^19 - 16 * q^20 + 64 * q^21 - 24 * q^22 + 56 * q^23 + 24 * q^24 + 32 * q^25 + 48 * q^26 - 24 * q^27 + 8 * q^28 - 128 * q^29 - 16 * q^30 - 40 * q^31 + 88 * q^32 - 160 * q^35 + 24 * q^36 - 96 * q^37 + 120 * q^38 - 96 * q^39 + 112 * q^40 - 48 * q^41 - 128 * q^42 + 112 * q^43 + 120 * q^44 - 112 * q^45 + 72 * q^46 - 192 * q^47 - 136 * q^48 + 80 * q^49 - 384 * q^52 + 128 * q^53 - 64 * q^54 - 224 * q^55 + 200 * q^56 + 48 * q^57 + 120 * q^59 - 48 * q^60 + 288 * q^61 + 8 * q^62 - 120 * q^63 + 64 * q^64 - 64 * q^65 + 96 * q^66 + 240 * q^69 - 480 * q^70 + 344 * q^71 - 40 * q^72 + 200 * q^73 - 208 * q^74 - 104 * q^75 - 160 * q^76 - 80 * q^77 - 512 * q^78 + 328 * q^79 + 64 * q^80 + 424 * q^81 - 64 * q^82 - 336 * q^83 + 832 * q^86 + 80 * q^87 + 8 * q^88 - 160 * q^89 - 224 * q^90 - 544 * q^91 + 24 * q^92 + 208 * q^93 - 432 * q^94 - 192 * q^95 + 64 * q^96 + 240 * q^97 + 120 * q^98 + 128 * 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.k

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:	\(\Z[a_1, \ldots, a_{4}]\)
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
$3$	\( T^{8} + 4 T^{6} + \cdots + 2 \) T^8 + 4T^6 - 40T^5 + 118T^4 - 24T^3 + 76T^2 - 24T + 2
$5$	\( T^{8} + 16 T^{7} + \cdots + 512 \) T^8 + 16T^7 + 112T^6 + 256T^5 - 160T^4 - 3840T^3 + 5376T^2 - 1024*T + 512
$7$	\( T^{8} - 40 T^{7} + \cdots + 8454272 \) T^8 - 40T^7 + 760T^6 - 9120T^5 + 76592T^4 - 464512T^3 + 1999232T^2 - 5658112*T + 8454272
$11$	\( T^{8} + 8 T^{7} + \cdots + 27572738 \) T^8 + 8T^7 + 184T^6 - 536T^5 - 13742T^4 - 142816T^3 + 229868T^2 + 6267544*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
$23$	\( T^{8} - 56 T^{7} + \cdots + 859299968 \) T^8 - 56T^7 + 1240T^6 - 10496T^5 + 98576T^4 - 1817472T^3 + 10338816T^2 - 45104128*T + 859299968
$29$	\( T^{8} + \cdots + 4240836608 \) T^8 + 128T^7 + 6640T^6 + 180224T^5 + 3025600T^4 + 44551680T^3 + 591369216T^2 + 1862549504*T + 4240836608
$31$	\( T^{8} + 40 T^{7} + \cdots + 14536832 \) T^8 + 40T^7 + 600T^6 + 4800T^5 + 90512T^4 + 1677952T^3 + 10760960T^2 + 7074304*T + 14536832
$37$	\( T^{8} + \cdots + 120057840128 \) T^8 + 96T^7 + 3088T^6 + 64000T^5 + 1999456T^4 + 30240000T^3 + 310505728T^2 + 5221610496*T + 120057840128
$41$	\( T^{8} + \cdots + 59058658562 \) T^8 + 48T^7 + 868T^6 + 34912T^5 + 1622150T^4 + 34972120T^3 + 419681284T^2 + 5071371592*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 - 288T^7 + 37760T^6 - 3020288T^5 + 178450944T^4 - 8681603072T^3 + 320410402816T^2 - 7744200245248*T + 106680004247552
$67$	\( T^{8} + \cdots + 83643718741636 \) T^8 + 21304T^6 + 118064724T^4 + 200330319344*T^2 + 83643718741636
$71$	\( T^{8} + \cdots + 71246079996032 \) T^8 - 344T^7 + 62296T^6 - 7376256T^5 + 601037072T^4 - 30495378560T^3 + 822692609024T^2 - 9844498245632*T + 71246079996032
$73$	\( T^{8} + \cdots + 51301810562 \) T^8 - 200T^7 + 22940T^6 - 1745000T^5 + 86172742T^4 - 2340373992T^3 + 24720959132T^2 + 37821868168*T + 51301810562
$79$	\( T^{8} + \cdots + 3948319764608 \) T^8 - 328T^7 + 42664T^6 - 2603328T^5 + 69066896T^4 - 1274091520T^3 + 64809024256T^2 + 507076203008*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 - 240T^7 + 31936T^6 - 2846504T^5 + 176051042T^4 - 7151829088T^3 + 182929540332T^2 - 2823963108152*T + 21682310986562
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\(n\)	\(3\)
\(\chi(n)\)	\(\zeta_{16}\)