LMFDB - Newform orbit 425.3.t.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,3,Mod(24,425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 11])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("425.24"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	425.t (of order $16$, degree $8$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,8,8,0,0,-8,-8,40,16,0,-8,-40,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.5804112353$
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}^{6} + \zeta_{16}^{5} + \cdots + 1) q^{2} + (\zeta_{16}^{7} + \zeta_{16}^{5} + \cdots + 1) q^{3} + ( - 2 \zeta_{16}^{7} + \cdots + 2 \zeta_{16}) q^{4} + (\zeta_{16}^{7} - \zeta_{16}^{6} + \cdots - 1) q^{6}+ \cdots + (16 \zeta_{16}^{7} + 27 \zeta_{16}^{6} + \cdots + 28) q^{99}+O(q^{100}) $ q + (z^6 + z^5 - z^4 + z^3 - z^2 + z + 1) * q^2 + (z^7 + z^5 + z^4 + z^2 - z + 1) * q^3 + (-2z^7 + 3z^6 - 2z^5 - 2z^3 + 2z) q^4 + (z^7 - z^6 + 4z^5 - 3z^4 + 3z^3 - 4z^2 + z - 1) * q^6 + (-z^7 + z^6 - z^5 + z^4 - z^3 + 5z^2 + 5z - 1) * q^7 + (3z^7 - 3z^6 + 3z^4 - 3z^3 + 5z^2 + z + 5) q^8 + (4z^7 - z^6 + z^4 - 4z^3 + 2z^2 + 3z + 2) * q^9 + (-z^7 - z^6 + 3z^5 - 6z^4 - 6z^3 + 3z^2 - z - 1) * q^11 + (-9z^7 + 9z^6 - 5z^5 + 6z^4 - z^3 - z^2 + 6z - 5) q^12 + (-10z^7 - 10z^5 - 2z^4 - 2z^3 + 2z + 2) q^13 + (13z^7 - 3z^6 + z^5 + z^4 - 3z^3 + 13z^2 + z - 1) * q^14 + (4z^7 - 8z^6 + 4z^5 + 3z^4 + 4z^3 - 8z^2 + 4z) q^16 - 17z^3 q^17 + (13z^7 - 3z^6 + 2z^5 - 7z^4 + 2z^3 - 3z^2 + 13z) q^18 + (4z^7 - 7z^5 + 6z^4 - 6z^2 + 7z) q^19 + (14z^6 + 8z^4 - 2z^3 + 2z - 8) * q^21 + (-3z^7 - 3z^6 + 7z^5 - 13z^4 - 6z^3 + 6z^2 + 13z - 7) q^22 + (-5z^7 + 3z^6 + 7z^5 + 7z^4 + 7z^3 + 7z^2 + 3z - 5) q^23 + (13z^7 + 3z^6 + 3z^5 + 13z^4 - 10z^3 + 15z^2 - 15z + 10) q^24 + (2z^7 - 6z^6 + 6z^4 - 2z^3 + 22z^2 - 12z + 22) * q^26 + (5z^7 + 10z^6 + 10z^5 + 5z^4 + 2z^3 - 3z^2 + 3z - 2) q^27 + (7z^7 - 15z^6 - z^5 - 19z^4 + 19z^3 + z^2 + 15z - 7) q^28 + (2z^7 + 2z^6 + 6z^5 + 8z^4 - 16z^3 + 16z^2 - 8z - 6) q^29 + (5z^7 - 5z^6 - 5z^5 - 5z^4 - 5z^3 + 5z^2 - 3z + 3) q^31 + (-17z^7 - 11z^6 - z^5 + 11z^4 - 11z^2 + z + 11) * q^32 + (-z^7 - 7z^6 + 2z^5 - 2z^3 + 7z^2 + z + 12) * q^33 + (17z^7 - 17z^6 + 17z^5 - 17z^4 - 17z^3 + 17z + 17) * q^34 + (-3z^7 + 16z^6 - 20z^5 - 3z^4 + 3z^2 + 20z - 16) * q^36 + (-2z^7 + 12z^6 + 4z^5 + 4z^4 + 12z^3 - 2z^2 - 16z + 16) q^37 + (18z^7 - 18z^5 + 15z^4 - 2z^3 - 2z^2 - 2z + 15) * q^38 + (-20z^7 + 20z^6 - 6z^5 + 22z^4 + 12z^3 + 12z^2 + 22z - 6) q^39 + (14z^7 + 6z^6 + 6z^5 + 14z^4 - 6z^3 - 5z^2 + 5z + 6) q^41 + (26z^7 - 2z^6 + 2z^4 - 26z^3 + 16z^2 - 26z + 16) * q^42 + (2z^7 + 29z^6 - 22z^5 + 29z^4 + 2z^3 - 14z^2 + 14) * q^43 + (3z^7 + 15z^6 - 15z^5 - 3z^4 - 15z^3 + 8z^2 + 8z - 15) q^44 + (z^7 + z^6 + 11z^5 + 17z^4 - 9z^3 + 9z^2 - 17z - 11) q^46 + (-8z^7 + 2z^6 - 8z^5 + 24z^4 + 22z^3 - 22z - 24) * q^47 + (12z^7 - 17z^6 + 17z^5 - 17z^4 + 17z^3 - 12z^2 + z + 1) * q^48 + (-2z^6 + 4z^5 + 10z^4 + 7z^3 + 10z^2 + 4z - 2) * q^49 + (-17z^7 - 17z^5 + 17z^4 - 17z^3 + 17z^2 + 17) q^51 + (6z^7 - 34z^6 + 22z^5 - 48z^4 + 22z^3 - 34z^2 + 6z) q^52 + (-16z^6 - 18z^5 + 4z^4 + 2z^3 + 4z^2 - 18z - 16) * q^53 + (8z^7 + 21z^6 - 21z^3 - 8z^2 - z - 1) * q^54 + (-33z^7 + 33z^6 - 15z^5 + 17z^4 + 25z^3 + 25z^2 + 17z - 15) q^56 + (-9z^7 + 10z^6 - 6z^5 + 3z^4 + 3z^3 - 6z^2 + 10z - 9) q^57 + (30z^7 - 46z^6 + 46z^5 - 30z^4 - 4z^3 - 4) q^58 + (10z^7 - 15z^6 + 42z^5 - 15z^4 + 10z^3 - 6z^2 + 6) * q^59 + (8z^7 - 36z^6 + 36z^5 - 8z^4 - 20z^3 - 12z^2 - 12z - 20) q^61 + (7z^7 - 15z^6 + z^5 - 21z^4 + 21z^3 - z^2 + 15z - 7) q^62 + (-15z^7 + 15z^6 - 21z^5 - 7z^4 + 23z^3 + 23z^2 - 7z - 21) q^63 + (-42z^7 + 42z^5 + 8z^4 + 10z^3 - 13z^2 + 10z + 8) * q^64 + (-z^6 + 23z^5 - 12z^4 + 20z^3 - 12z^2 + 23z - 1) q^66 + (26z^7 + 4z^6 - 33z^5 + 33z^3 - 4z^2 - 26z - 42) * q^67 + (34z^6 - 34z^4 - 34z^2 + 51z - 34) * q^68 + (8z^7 + 18z^6 + 2z^5 - 2z^3 - 18z^2 - 8z - 30) * q^69 + (43z^7 + 35z^6 + 11z^5 - 11z^4 - 35z^3 - 43z^2 + 5z + 5) q^71 + (z^7 - z^5 + 5z^4 - 24z^3 + 53z^2 - 24z + 5) * q^72 + (-25z^7 - 25z^6 + 6z^5 - 6z^4 + 26z^3 - 26z^2 + 6z - 6) q^73 + (-20z^7 + 26z^6 + 28z^5 - 24z^4 + 24z^3 - 28z^2 - 26z + 20) q^74 + (21z^7 - 20z^6 + 12z^5 - 20z^4 + 21z^3 - 10z^2 + 10) * q^76 + (8z^7 - 6z^6 - 66z^5 - 6z^4 + 8z^3 - 10z^2 + 10) * q^77 + (50z^7 + 8z^6 + 8z^5 + 50z^4 - 38z^3 + 64z^2 - 64z + 38) q^78 + (17z^7 - 41z^6 + 3z^5 + 21z^4 + 21z^3 + 3z^2 - 41z + 17) q^79 + (-7z^7 + 7z^5 - 53z^4 + 6z^3 - 3z^2 + 6z - 53) * q^81 + (34z^7 + 8z^6 + 8z^5 - 8z^4 - 8z^3 - 34z^2 + 17z + 17) q^82 + (-54z^7 - 42z^6 - 20z^5 + 33z^4 - 33z^2 + 20z + 42) * q^83 + (34z^7 - 24z^6 + 60z^5 - 50z^4 + 60z^3 - 24z^2 + 34z) q^84 + (66z^7 + 8z^6 + 3z^5 - 3z^3 - 8z^2 - 66z + 104) * q^86 + (10z^6 - 34z^5 + 26z^4 - 52z^3 + 26z^2 - 34z + 10) * q^87 + (9z^7 + z^6 - 33z^5 - 33z^4 + z^3 + 9z^2 - 40z + 40) q^88 + (29z^7 + 8z^6 + 29z^5 + 20z^4 - 46z^3 + 46z - 20) * q^89 + (-40z^7 - 68z^6 - 28z^4 + 28z^3 + 68z + 40) q^91 + (-3z^7 - 31z^6 + 31z^5 + 3z^4 + 23z^3 + 3z^2 + 3z + 23) q^92 + (8z^7 - 8z^6 + 8z^4 - 8z^3 + 26z^2 - 6z + 26) * q^93 + (-18z^7 - 54z^6 + 54z^4 + 18z^3 - 4z^2 - 110z - 4) * q^94 + (-7z^7 + 8z^6 + 11z^5 + 29z^4 + 29z^3 + 11z^2 + 8z - 7) q^96 + (-30z^7 + 30z^6 + 13z^5 + 6z^4 + 22z^3 + 22z^2 + 6z + 13) q^97 + (11z^7 - 9z^6 + 11z^5 + 15z^4 + 9z^3 - 9z - 15) * q^98 + (16z^7 + 27z^6 - 34z^5 - 34z^4 + 27z^3 + 16z^2 - 28z + 28) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{3} - 8 q^{6} - 8 q^{7} + 40 q^{8} + 16 q^{9} - 8 q^{11} - 40 q^{12} + 16 q^{13} - 8 q^{14} - 64 q^{21} - 56 q^{22} - 40 q^{23} + 80 q^{24} + 176 q^{26} - 16 q^{27} - 56 q^{28} - 48 q^{29}+ \cdots + 224 q^{99}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^3 - 8 * q^6 - 8 * q^7 + 40 * q^8 + 16 * q^9 - 8 * q^11 - 40 * q^12 + 16 * q^13 - 8 * q^14 - 64 * q^21 - 56 * q^22 - 40 * q^23 + 80 * q^24 + 176 * q^26 - 16 * q^27 - 56 * q^28 - 48 * q^29 + 24 * q^31 + 88 * q^32 + 96 * q^33 + 136 * q^34 - 128 * q^36 + 128 * q^37 + 120 * q^38 - 48 * q^39 + 48 * q^41 + 128 * q^42 + 112 * q^43 - 120 * q^44 - 88 * q^46 - 192 * q^47 + 8 * q^48 - 16 * q^49 + 136 * q^51 - 128 * q^53 - 8 * q^54 - 120 * q^56 - 72 * q^57 - 32 * q^58 + 48 * q^59 - 160 * q^61 - 56 * q^62 - 168 * q^63 + 64 * q^64 - 8 * q^66 - 336 * q^67 - 272 * q^68 - 240 * q^69 + 40 * q^71 + 40 * q^72 - 48 * q^73 + 160 * q^74 + 80 * q^76 + 80 * q^77 + 304 * q^78 + 136 * q^79 - 424 * q^81 + 136 * q^82 + 336 * q^83 + 832 * q^86 + 80 * q^87 + 320 * q^88 - 160 * q^89 + 320 * q^91 + 184 * q^92 + 208 * q^93 - 32 * q^94 - 56 * q^96 + 104 * q^97 - 120 * q^98 + 224 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$-1$	$\zeta_{16}$

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} $

24.1

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

2.03153

0.841487i

0.134381

0.0897902i

0.590587

0.590587i

0.197441

0.295491i

−6.12453

−

1.21824i

−2.66313

−

6.42935i

−3.43416

−

8.29078i

74.1

0.509666

−

1.23044i

−0.523336

−

2.63099i

1.57420

1.57420i

−3.50400

−

0.696990i

7.37170

−

4.92562i

7.66104

−

3.17331i

1.66671

−

0.690373i

99.1

−1.33809

−

3.23044i

3.93755

0.783227i

−5.81684

5.81684i

−2.73864

−

13.7681i

−3.71485

5.55967i

13.6527

5.65512i

6.57593

2.72384i

124.1

2.03153

−

0.841487i

0.134381

−

0.0897902i

0.590587

−

0.590587i

0.197441

−

0.295491i

−6.12453

1.21824i

−2.66313

6.42935i

−3.43416

8.29078i

199.1

2.79690

−

1.15851i

0.451406

0.675577i

3.65205

−

3.65205i

2.04520

1.36656i

−1.53233

−

7.70353i

1.34942

−

3.25778i

3.19151

−

7.70500i

224.1