LMFDB - Newform orbit 1400.2.q.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,2,Mod(401,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1400.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1400.q (of order $3$, degree $2$, 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:	$11.1790562830$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 $\zeta_{18}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{18}^{4} + \cdots - \zeta_{18}^{2}) q^{3}+ \cdots + ( - \zeta_{18}^{5} + \cdots + 2 \zeta_{18}) q^{9}+O(q^{10}) $ q + (-z^4 + z^3 - z^2) * q^3 + (-3z^4 + 2z) * q^7 + (-z^5 - z^4 - z^2 + 2z) q^9	$ q + ( - \zeta_{18}^{4} + \cdots - \zeta_{18}^{2}) q^{3}+ \cdots + ( - 7 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + \cdots - 3) q^{99}+O(q^{100}) $ q + (-z^4 + z^3 - z^2) * q^3 + (-3z^4 + 2z) * q^7 + (-z^5 - z^4 - z^2 + 2z) q^9 + (2z^5 - z^4 + 2z^3 - z^2 + 2z) q^11 + (-3z^5 + 4z^4 - z^2 - z) * q^13 + (-3z^5 + 3z^4 - 2z^3 + 3z^2 - 3z) q^17 + (-2z^5 - 2z^4 + 2z^3 - z^2 + 3z - 2) * q^19 + (z^5 - z^4 + z^3 - 3z^2 + 3z - 3) * q^21 + (-3z^5 - 3z^4 - z^3 + z^2 + 2z + 1) q^23 + (3z^4 - 3z^2 - 3z) q^27 + (z^5 + z^4 - 2z^2 - 2z + 4) * q^29 + (3z^5 + z^4 + z^3 + z^2 + 3z) * q^31 + (-2z^5 - 2z^4 + 2z^3 - 3z^2 + 5z - 2) q^33 + (-z^5 - z^4 + 2z^3 + 4z^2 - 3z - 2) q^37 + (-7z^5 + 7z^4 - 2z^3 + 7z^2 - 7z) q^39 + (2z^5 + z^4 - 3z^2 - 3z + 1) q^41 + (-4z^5 + 3z^4 + z^2 + z + 3) * q^43 + (z^5 + z^4 - 6z^3 - 3z^2 + 2z + 6) q^47 + (-3z^5 - 5z^2) * q^49 + (2z^5 + 2z^4 - 5z^3 + 6z^2 - 8z + 5) q^51 + (-5z^5 - 4z^4 + 2z^3 - 4z^2 - 5z) q^53 + (-6z^5 + 4z^4 + 2z^2 + 2z - 7) * q^57 + (-3z^4 + z^3 - 3z^2) * q^59 + (6z^5 + 6z^4 + 3z^3 - z^2 - 5z - 3) * q^61 + (-5z^5 - z^3 + z^2 - 4) q^63 + (-6z^4 + 4z^3 - 6z^2) q^67 + (z^4 - z^2 - z - 4) * q^69 + (-6z^4 + 6z^2 + 6z + 3) q^71 + (-4z^5 - 4z^3 - 4z) q^73 + (-5z^5 - 2z^4 + 5z^3 + z^2 + 6z - 1) * q^77 + (-z^5 - z^4 - z^3 + z^2 + 1) * q^79 + (-3z^4 + 3z^3 - 3z^2) q^81 + (-5z^5 + 3z^4 + 2z^2 + 2z) * q^83 + (-4z^4 + 7z^3 - 4z^2) q^87 + (-7z^5 - 7z^4 + 2z^3 + z^2 + 6z - 2) * q^89 + (-z^5 - 5z^3 + 10z^2 - 6) * q^91 + (-z^5 - z^4 - 4z^3 - 2z^2 + 3z + 4) q^93 + (3z^5 - 6z^4 + 3z^2 + 3z + 7) * q^97 + (-7z^5 + 2z^4 + 5z^2 + 5z - 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3}+O(q^{10}) $ 6 * q + 3 * q^3	$ 6 q + 3 q^{3} + 6 q^{11} - 6 q^{17} - 6 q^{19} - 15 q^{21} + 3 q^{23} + 24 q^{29} + 3 q^{31} - 6 q^{33} - 6 q^{37} - 6 q^{39} + 6 q^{41} + 18 q^{43} + 18 q^{47} + 15 q^{51} + 6 q^{53} - 42 q^{57} + 3 q^{59} - 9 q^{61} - 27 q^{63} + 12 q^{67} - 24 q^{69} + 18 q^{71} - 12 q^{73} + 9 q^{77} + 3 q^{79} + 9 q^{81} + 21 q^{87} - 6 q^{89} - 51 q^{91} + 12 q^{93} + 42 q^{97} - 18 q^{99}+O(q^{100}) $ 6 * q + 3 * q^3 + 6 * q^11 - 6 * q^17 - 6 * q^19 - 15 * q^21 + 3 * q^23 + 24 * q^29 + 3 * q^31 - 6 * q^33 - 6 * q^37 - 6 * q^39 + 6 * q^41 + 18 * q^43 + 18 * q^47 + 15 * q^51 + 6 * q^53 - 42 * q^57 + 3 * q^59 - 9 * q^61 - 27 * q^63 + 12 * q^67 - 24 * q^69 + 18 * q^71 - 12 * q^73 + 9 * q^77 + 3 * q^79 + 9 * q^81 + 21 * q^87 - 6 * q^89 - 51 * q^91 + 12 * q^93 + 42 * q^97 - 18 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$-1 + \zeta_{18}^{3}$	$1$

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.

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} $

401.1

0.939693	+	0.342020i
−0.173648	−	0.984808i
−0.766044	+	0.642788i
0.939693	−	0.342020i
−0.173648	+	0.984808i
−0.766044	−	0.642788i

−0.439693

−

0.761570i

1.35844

−

2.27038i

1.11334

−

1.92836i

401.2

0.673648

1.16679i

−2.64543

−

0.0412527i

0.592396

−

1.02606i

401.3

1.26604

2.19285i

1.28699

2.31164i

−1.70574

2.95442i

1201.1

−0.439693

0.761570i

1.35844

2.27038i

1.11334

1.92836i

1201.2

0.673648

−

1.16679i

−2.64543

0.0412527i

0.592396

1.02606i

1201.3

1.26604

−

2.19285i

1.28699

−

2.31164i

−1.70574

−

2.95442i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1400.2.q.k	yes	6
5.b	even	2	1		1400.2.q.i	✓	6
5.c	odd	4	2		1400.2.bh.h		12
7.c	even	3	1	inner	1400.2.q.k	yes	6
7.c	even	3	1		9800.2.a.cc		3
7.d	odd	6	1		9800.2.a.ch		3
35.i	odd	6	1		9800.2.a.cb		3
35.j	even	6	1		1400.2.q.i	✓	6
35.j	even	6	1		9800.2.a.ci		3
35.l	odd	12	2		1400.2.bh.h		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1400.2.q.i	✓	6	5.b	even	2	1
1400.2.q.i	✓	6	35.j	even	6	1
1400.2.q.k	yes	6	1.a	even	1	1	trivial
1400.2.q.k	yes	6	7.c	even	3	1	inner
1400.2.bh.h		12	5.c	odd	4	2
1400.2.bh.h		12	35.l	odd	12	2
9800.2.a.cb		3	35.i	odd	6	1
9800.2.a.cc		3	7.c	even	3	1
9800.2.a.ch		3	7.d	odd	6	1
9800.2.a.ci		3	35.j	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1400, [\chi])$:

$ T_{3}^{6} - 3T_{3}^{5} + 9T_{3}^{4} - 6T_{3}^{3} + 9T_{3}^{2} + 9 $

$ T_{11}^{6} - 6T_{11}^{5} + 33T_{11}^{4} - 56T_{11}^{3} + 123T_{11}^{2} + 57T_{11} + 361 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - 3 T^{5} + \cdots + 9 $ T^6 - 3T^5 + 9T^4 - 6T^3 + 9T^2 + 9
$5$	$ T^{6} $ T^6
$7$	$ T^{6} + 37T^{3} + 343 $ T^6 + 37*T^3 + 343
$11$	$ T^{6} - 6 T^{5} + \cdots + 361 $ T^6 - 6T^5 + 33T^4 - 56T^3 + 123T^2 + 57*T + 361
$13$	$ (T^{3} - 39 T - 19)^{2} $ (T^3 - 39*T - 19)^2
$17$	$ T^{6} + 6 T^{5} + \cdots + 361 $ T^6 + 6T^5 + 51T^4 - 52T^3 + 339T^2 + 285*T + 361
$19$	$ T^{6} + 6 T^{5} + \cdots + 289 $ T^6 + 6T^5 + 45T^4 - 20T^3 + 183T^2 + 153*T + 289
$23$	$ T^{6} - 3 T^{5} + \cdots + 289 $ T^6 - 3T^5 + 27T^4 + 88T^3 + 273T^2 + 306*T + 289
$29$	$ (T^{3} - 12 T^{2} + \cdots - 19)^{2} $ (T^3 - 12T^2 + 39T - 19)^2
$31$	$ T^{6} - 3 T^{5} + \cdots + 361 $ T^6 - 3T^5 + 45T^4 + 70T^3 + 1353T^2 - 684*T + 361
$37$	$ T^{6} + 6 T^{5} + \cdots + 2601 $ T^6 + 6T^5 + 63T^4 - 60T^3 + 1035T^2 + 1377*T + 2601
$41$	$ (T^{3} - 3 T^{2} - 18 T + 57)^{2} $ (T^3 - 3T^2 - 18T + 57)^2
$43$	$ (T^{3} - 9 T^{2} + \cdots + 179)^{2} $ (T^3 - 9T^2 - 12T + 179)^2
$47$	$ T^{6} - 18 T^{5} + \cdots + 11449 $ T^6 - 18T^5 + 237T^4 - 1352T^3 + 5643T^2 - 9309*T + 11449
$53$	$ T^{6} - 6 T^{5} + \cdots + 1159929 $ T^6 - 6T^5 + 207T^4 - 1128T^3 + 35703T^2 - 184167*T + 1159929
$59$	$ T^{6} - 3 T^{5} + \cdots + 2809 $ T^6 - 3T^5 + 33T^4 - 34T^3 + 735T^2 - 1272*T + 2809
$61$	$ T^{6} + 9 T^{5} + \cdots + 1369 $ T^6 + 9T^5 + 147T^4 - 668T^3 + 4023T^2 - 2442*T + 1369
$67$	$ T^{6} - 12 T^{5} + \cdots + 341056 $ T^6 - 12T^5 + 204T^4 - 448T^3 + 10608T^2 - 35040*T + 341056
$71$	$ (T^{3} - 9 T^{2} + \cdots + 513)^{2} $ (T^3 - 9T^2 - 81T + 513)^2
$73$	$ T^{6} + 12 T^{5} + \cdots + 36864 $ T^6 + 12T^5 + 144T^4 + 384T^3 + 2304T^2 + 36864
$79$	$ T^{6} - 3 T^{5} + \cdots + 9 $ T^6 - 3T^5 + 9T^4 - 6T^3 + 9T^2 + 9
$83$	$ (T^{3} - 57 T + 163)^{2} $ (T^3 - 57*T + 163)^2
$89$	$ T^{6} + 6 T^{5} + \cdots + 488601 $ T^6 + 6T^5 + 153T^4 + 696T^3 + 17883T^2 + 81783*T + 488601
$97$	$ (T^{3} - 21 T^{2} + \cdots + 467)^{2} $ (T^3 - 21T^2 + 66T + 467)^2
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{18}^{4} + \cdots - \zeta_{18}^{2}) q^{3}+ \cdots + ( - \zeta_{18}^{5} + \cdots + 2 \zeta_{18}) q^{9}+O(q^{10}) \) q + (-z^4 + z^3 - z^2) * q^3 + (-3z^4 + 2z) * q^7 + (-z^5 - z^4 - z^2 + 2z) q^9	\( q + ( - \zeta_{18}^{4} + \cdots - \zeta_{18}^{2}) q^{3}+ \cdots + ( - 7 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + \cdots - 3) q^{99}+O(q^{100}) \) q + (-z^4 + z^3 - z^2) * q^3 + (-3z^4 + 2z) * q^7 + (-z^5 - z^4 - z^2 + 2z) q^9 + (2z^5 - z^4 + 2z^3 - z^2 + 2z) q^11 + (-3z^5 + 4z^4 - z^2 - z) * q^13 + (-3z^5 + 3z^4 - 2z^3 + 3z^2 - 3z) q^17 + (-2z^5 - 2z^4 + 2z^3 - z^2 + 3z - 2) * q^19 + (z^5 - z^4 + z^3 - 3z^2 + 3z - 3) * q^21 + (-3z^5 - 3z^4 - z^3 + z^2 + 2z + 1) q^23 + (3z^4 - 3z^2 - 3z) q^27 + (z^5 + z^4 - 2z^2 - 2z + 4) * q^29 + (3z^5 + z^4 + z^3 + z^2 + 3z) * q^31 + (-2z^5 - 2z^4 + 2z^3 - 3z^2 + 5z - 2) q^33 + (-z^5 - z^4 + 2z^3 + 4z^2 - 3z - 2) q^37 + (-7z^5 + 7z^4 - 2z^3 + 7z^2 - 7z) q^39 + (2z^5 + z^4 - 3z^2 - 3z + 1) q^41 + (-4z^5 + 3z^4 + z^2 + z + 3) * q^43 + (z^5 + z^4 - 6z^3 - 3z^2 + 2z + 6) q^47 + (-3z^5 - 5z^2) * q^49 + (2z^5 + 2z^4 - 5z^3 + 6z^2 - 8z + 5) q^51 + (-5z^5 - 4z^4 + 2z^3 - 4z^2 - 5z) q^53 + (-6z^5 + 4z^4 + 2z^2 + 2z - 7) * q^57 + (-3z^4 + z^3 - 3z^2) * q^59 + (6z^5 + 6z^4 + 3z^3 - z^2 - 5z - 3) * q^61 + (-5z^5 - z^3 + z^2 - 4) q^63 + (-6z^4 + 4z^3 - 6z^2) q^67 + (z^4 - z^2 - z - 4) * q^69 + (-6z^4 + 6z^2 + 6z + 3) q^71 + (-4z^5 - 4z^3 - 4z) q^73 + (-5z^5 - 2z^4 + 5z^3 + z^2 + 6z - 1) * q^77 + (-z^5 - z^4 - z^3 + z^2 + 1) * q^79 + (-3z^4 + 3z^3 - 3z^2) q^81 + (-5z^5 + 3z^4 + 2z^2 + 2z) * q^83 + (-4z^4 + 7z^3 - 4z^2) q^87 + (-7z^5 - 7z^4 + 2z^3 + z^2 + 6z - 2) * q^89 + (-z^5 - 5z^3 + 10z^2 - 6) * q^91 + (-z^5 - z^4 - 4z^3 - 2z^2 + 3z + 4) q^93 + (3z^5 - 6z^4 + 3z^2 + 3z + 7) * q^97 + (-7z^5 + 2z^4 + 5z^2 + 5z - 3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3}+O(q^{10}) \) 6 * q + 3 * q^3	\( 6 q + 3 q^{3} + 6 q^{11} - 6 q^{17} - 6 q^{19} - 15 q^{21} + 3 q^{23} + 24 q^{29} + 3 q^{31} - 6 q^{33} - 6 q^{37} - 6 q^{39} + 6 q^{41} + 18 q^{43} + 18 q^{47} + 15 q^{51} + 6 q^{53} - 42 q^{57} + 3 q^{59} - 9 q^{61} - 27 q^{63} + 12 q^{67} - 24 q^{69} + 18 q^{71} - 12 q^{73} + 9 q^{77} + 3 q^{79} + 9 q^{81} + 21 q^{87} - 6 q^{89} - 51 q^{91} + 12 q^{93} + 42 q^{97} - 18 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 + 6 * q^11 - 6 * q^17 - 6 * q^19 - 15 * q^21 + 3 * q^23 + 24 * q^29 + 3 * q^31 - 6 * q^33 - 6 * q^37 - 6 * q^39 + 6 * q^41 + 18 * q^43 + 18 * q^47 + 15 * q^51 + 6 * q^53 - 42 * q^57 + 3 * q^59 - 9 * q^61 - 27 * q^63 + 12 * q^67 - 24 * q^69 + 18 * q^71 - 12 * q^73 + 9 * q^77 + 3 * q^79 + 9 * q^81 + 21 * q^87 - 6 * q^89 - 51 * q^91 + 12 * q^93 + 42 * q^97 - 18 * q^99

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