LMFDB - Newform orbit 2900.2.c.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2900,2,Mod(349,2900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2900.349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2900.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,0,0,0,0,-14,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$23.1566165862$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.9689973693776896.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 22x^{8} + 179x^{6} + 639x^{4} + 847x^{2} + 9 $ x^10 + 22x^8 + 179x^6 + 639x^4 + 847x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 basis $1,\beta_1,\ldots,\beta_{9}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{7} + (\beta_{2} - 1) q^{9} + ( - \beta_{8} - \beta_{2}) q^{11} + (\beta_{6} + \beta_1) q^{13} + ( - \beta_{4} + \beta_1) q^{17} + (\beta_{9} + \beta_{8} - \beta_{2} - 1) q^{19}+ \cdots + ( - \beta_{9} + 3 \beta_{8} + \beta_{2} - 7) q^{99}+O(q^{100}) $ q + b1 * q^3 + (-b4 - b3 + b1) * q^7 + (b2 - 1) * q^9 + (-b8 - b2) * q^11 + (b6 + b1) * q^13 + (-b4 + b1) * q^17 + (b9 + b8 - b2 - 1) * q^19 + (-b5 + b2 - 2) * q^21 + (b6 - b3 + b1) * q^23 + b3 * q^27 - q^29 + (b8 - b5 - b2 - 2) * q^31 + (-b7 - b6 + b4 - 2b3 + b1) q^33 + (b7 - b6 + 2b1) q^37 + (-b9 - b8 + b5 + b2 - 2) * q^39 + (b9 - b8 - b2 - 2) * q^41 + (-b7 + b4 + 3b3 - 2b1) * q^43 + (-b7 - b6 + b4 + 2b3 - b1) q^47 + (b9 + b8 - b5 - b2 - 3) * q^49 + (b9 - b8 - b5 - 3) * q^51 + (-b7 - b6 + b4 - b3 + 2b1) q^53 + (2b7 + 2b6 - b4 - 3b3 + 3b1) * q^57 + (-b9 - b8 + b5 - b2 - 3) * q^59 + (-b8 + b2 - 1) * q^61 + (-b7 + 2b6 + b3 - 2b1) * q^63 + (3b4 - 2b1) * q^67 + (-2b9 + b5 + 2b2 - 1) * q^69 + (-b8 - 2b2 - 3) q^71 + (-2b7 + b6 + b1) q^73 + (-3b6 + b4 - b3 + b1) q^77 + (b5 - 1) * q^79 + (b9 - b8 + 2b2 - 4) q^81 + (2b6 - b4 + 2b3 + b1) * q^83 - b1 * q^87 + (-2b8 - b5 - 1) q^89 + (-2b9 - b8 + b5 - b2) q^91 + (3b6 + 2b4 + 3b3) q^93 + (b7 - b6 - b4 + 4b3 + b1) q^97 + (-b9 + 3b8 + b2 - 7) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 14 q^{9} - 4 q^{19} - 26 q^{21} - 10 q^{29} - 14 q^{31} - 24 q^{39} - 22 q^{41} - 26 q^{49} - 38 q^{51} - 26 q^{59} - 18 q^{61} - 12 q^{69} - 26 q^{71} - 8 q^{79} - 54 q^{81} - 20 q^{89} + 6 q^{91}+ \cdots - 60 q^{99}+O(q^{100}) $ 10 * q - 14 * q^9 - 4 * q^19 - 26 * q^21 - 10 * q^29 - 14 * q^31 - 24 * q^39 - 22 * q^41 - 26 * q^49 - 38 * q^51 - 26 * q^59 - 18 * q^61 - 12 * q^69 - 26 * q^71 - 8 * q^79 - 54 * q^81 - 20 * q^89 + 6 * q^91 - 60 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{10} + 22x^{8} + 179x^{6} + 639x^{4} + 847x^{2} + 9 $ x^10 + 22*x^8 + 179*x^6 + 639*x^4 + 847*x^2 + 9 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} + 4 $ v^2 + 4
$\beta_{3}$	$=$	$ \nu^{3} + 6\nu $ v^3 + 6*v
$\beta_{4}$	$=$	$ \nu^{5} + 11\nu^{3} + 29\nu $ v^5 + 11v^3 + 29v
$\beta_{5}$	$=$	$ \nu^{6} + 12\nu^{4} + 35\nu^{2} + 2 $ v^6 + 12v^4 + 35v^2 + 2
$\beta_{6}$	$=$	$ ( \nu^{9} + 16\nu^{7} + 80\nu^{5} + 108\nu^{3} - 83\nu ) / 3 $ (v^9 + 16v^7 + 80v^5 + 108v^3 - 83v) / 3
$\beta_{7}$	$=$	$ ( 2\nu^{9} + 35\nu^{7} + 205\nu^{5} + 429\nu^{3} + 152\nu ) / 3 $ (2v^9 + 35v^7 + 205v^5 + 429v^3 + 152*v) / 3
$\beta_{8}$	$=$	$ \nu^{8} + 17\nu^{6} + 94\nu^{4} + 169\nu^{2} + 1 $ v^8 + 17v^6 + 94v^4 + 169*v^2 + 1
$\beta_{9}$	$=$	$ \nu^{8} + 17\nu^{6} + 95\nu^{4} + 176\nu^{2} + 6 $ v^8 + 17v^6 + 95v^4 + 176*v^2 + 6

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} - 4 $ b2 - 4
$\nu^{3}$	$=$	$ \beta_{3} - 6\beta_1 $ b3 - 6*b1
$\nu^{4}$	$=$	$ \beta_{9} - \beta_{8} - 7\beta_{2} + 23 $ b9 - b8 - 7*b2 + 23
$\nu^{5}$	$=$	$ \beta_{4} - 11\beta_{3} + 37\beta_1 $ b4 - 11b3 + 37b1
$\nu^{6}$	$=$	$ -12\beta_{9} + 12\beta_{8} + \beta_{5} + 49\beta_{2} - 138 $ -12b9 + 12b8 + b5 + 49*b2 - 138
$\nu^{7}$	$=$	$ \beta_{7} - 2\beta_{6} - 15\beta_{4} + 94\beta_{3} - 235\beta_1 $ b7 - 2b6 - 15b4 + 94b3 - 235b1
$\nu^{8}$	$=$	$ 110\beta_{9} - 109\beta_{8} - 17\beta_{5} - 344\beta_{2} + 859 $ 110b9 - 109b8 - 17b5 - 344b2 + 859
$\nu^{9}$	$=$	$ -16\beta_{7} + 35\beta_{6} + 160\beta_{4} - 732\beta_{3} + 1531\beta_1 $ -16b7 + 35b6 + 160b4 - 732b3 + 1531*b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$1$	$-1$	$1$

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.

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} $

349.1

	−	2.67880i
	−	2.40772i
	−	2.20537i
	−	2.03778i
	−	0.103499i
		0.103499i
		2.03778i
		2.20537i
		2.40772i
		2.67880i

−

2.67880i

−

1.65303i

−4.17598

349.2

−

2.40772i

−

4.71632i

−2.79711

349.3

−

2.20537i

−

1.56292i

−1.86364

349.4

−

2.03778i

2.87946i

−1.15256

349.5

−

0.103499i

3.50567i

2.98929

349.6

0.103499i

−

3.50567i

2.98929

349.7

2.03778i

−

2.87946i

−1.15256

349.8

2.20537i

1.56292i

−1.86364

349.9

2.40772i

4.71632i

−2.79711

349.10

2.67880i

1.65303i

−4.17598

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2900.2.c.h		10
5.b	even	2	1	inner	2900.2.c.h		10
5.c	odd	4	1		2900.2.a.j	✓	5
5.c	odd	4	1		2900.2.a.l	yes	5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2900.2.a.j	✓	5	5.c	odd	4	1
2900.2.a.l	yes	5	5.c	odd	4	1
2900.2.c.h		10	1.a	even	1	1	trivial
2900.2.c.h		10	5.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2900, [\chi])$:

$ T_{3}^{10} + 22T_{3}^{8} + 179T_{3}^{6} + 639T_{3}^{4} + 847T_{3}^{2} + 9 $

T3^10 + 22*T3^8 + 179*T3^6 + 639*T3^4 + 847*T3^2 + 9

$ T_{7}^{10} + 48T_{7}^{8} + 788T_{7}^{6} + 5449T_{7}^{4} + 15466T_{7}^{2} + 15129 $

T7^10 + 48*T7^8 + 788*T7^6 + 5449*T7^4 + 15466*T7^2 + 15129

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} $ T^10
$3$	$ T^{10} + 22 T^{8} + \cdots + 9 $ T^10 + 22T^8 + 179T^6 + 639T^4 + 847T^2 + 9
$5$	$ T^{10} $ T^10
$7$	$ T^{10} + 48 T^{8} + \cdots + 15129 $ T^10 + 48T^8 + 788T^6 + 5449T^4 + 15466T^2 + 15129
$11$	$ (T^{5} - 39 T^{3} + 10 T^{2} + \cdots + 9)^{2} $ (T^5 - 39T^3 + 10T^2 + 328*T + 9)^2
$13$	$ T^{10} + 60 T^{8} + \cdots + 105625 $ T^10 + 60T^8 + 1352T^6 + 14049T^4 + 65650T^2 + 105625
$17$	$ T^{10} + 55 T^{8} + \cdots + 6561 $ T^10 + 55T^8 + 971T^6 + 6283T^4 + 12726T^2 + 6561
$19$	$ (T^{5} + 2 T^{4} + \cdots + 2725)^{2} $ (T^5 + 2T^4 - 83T^3 - 153T^2 + 1615T + 2725)^2
$23$	$ T^{10} + 85 T^{8} + \cdots + 145161 $ T^10 + 85T^8 + 2324T^6 + 24987T^4 + 104035T^2 + 145161
$29$	$ (T + 1)^{10} $ (T + 1)^10
$31$	$ (T^{5} + 7 T^{4} + \cdots - 1147)^{2} $ (T^5 + 7T^4 - 92T^3 - 343T^2 + 2649T - 1147)^2
$37$	$ T^{10} + 194 T^{8} + \cdots + 201601 $ T^10 + 194T^8 + 13515T^6 + 393503T^4 + 3894295T^2 + 201601
$41$	$ (T^{5} + 11 T^{4} + \cdots - 975)^{2} $ (T^5 + 11T^4 - 29T^3 - 586T^2 - 1495T - 975)^2
$43$	$ T^{10} + 286 T^{8} + \cdots + 157609 $ T^10 + 286T^8 + 22517T^6 + 346464T^4 + 866024T^2 + 157609
$47$	$ T^{10} + 243 T^{8} + \cdots + 27342441 $ T^10 + 243T^8 + 19923T^6 + 695212T^4 + 9599193T^2 + 27342441
$53$	$ T^{10} + 390 T^{8} + \cdots + 6305121 $ T^10 + 390T^8 + 51051T^6 + 2537263T^4 + 41162211T^2 + 6305121
$59$	$ (T^{5} + 13 T^{4} + \cdots - 16983)^{2} $ (T^5 + 13T^4 - 92T^3 - 2137T^2 - 10749T - 16983)^2
$61$	$ (T^{5} + 9 T^{4} - 5 T^{3} + \cdots - 61)^{2} $ (T^5 + 9T^4 - 5T^3 - 115T^2 - 172T - 61)^2
$67$	$ T^{10} + 403 T^{8} + \cdots + 26286129 $ T^10 + 403T^8 + 56723T^6 + 3340824T^4 + 70118041T^2 + 26286129
$71$	$ (T^{5} + 13 T^{4} + \cdots + 3051)^{2} $ (T^5 + 13T^4 - 16T^3 - 479T^2 + 57T + 3051)^2
$73$	$ T^{10} + \cdots + 174953529 $ T^10 + 340T^8 + 37688T^6 + 1647345T^4 + 29319694T^2 + 174953529
$79$	$ (T^{5} + 4 T^{4} - 56 T^{3} + \cdots + 27)^{2} $ (T^5 + 4T^4 - 56T^3 - 93T^2 + 36T + 27)^2
$83$	$ T^{10} + \cdots + 1247855625 $ T^10 + 387T^8 + 53435T^6 + 3442711T^4 + 105632350T^2 + 1247855625
$89$	$ (T^{5} + 10 T^{4} + \cdots + 1053)^{2} $ (T^5 + 10T^4 - 86T^3 - 311T^2 + 806T + 1053)^2
$97$	$ T^{10} + \cdots + 48745133089 $ T^10 + 811T^8 + 243807T^6 + 33656264T^4 + 2120132529T^2 + 48745133089
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Level:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{7} + (\beta_{2} - 1) q^{9} + ( - \beta_{8} - \beta_{2}) q^{11} + (\beta_{6} + \beta_1) q^{13} + ( - \beta_{4} + \beta_1) q^{17} + (\beta_{9} + \beta_{8} - \beta_{2} - 1) q^{19}+ \cdots + ( - \beta_{9} + 3 \beta_{8} + \beta_{2} - 7) q^{99}+O(q^{100}) \) q + b1 * q^3 + (-b4 - b3 + b1) * q^7 + (b2 - 1) * q^9 + (-b8 - b2) * q^11 + (b6 + b1) * q^13 + (-b4 + b1) * q^17 + (b9 + b8 - b2 - 1) * q^19 + (-b5 + b2 - 2) * q^21 + (b6 - b3 + b1) * q^23 + b3 * q^27 - q^29 + (b8 - b5 - b2 - 2) * q^31 + (-b7 - b6 + b4 - 2b3 + b1) q^33 + (b7 - b6 + 2b1) q^37 + (-b9 - b8 + b5 + b2 - 2) * q^39 + (b9 - b8 - b2 - 2) * q^41 + (-b7 + b4 + 3b3 - 2b1) * q^43 + (-b7 - b6 + b4 + 2b3 - b1) q^47 + (b9 + b8 - b5 - b2 - 3) * q^49 + (b9 - b8 - b5 - 3) * q^51 + (-b7 - b6 + b4 - b3 + 2b1) q^53 + (2b7 + 2b6 - b4 - 3b3 + 3b1) * q^57 + (-b9 - b8 + b5 - b2 - 3) * q^59 + (-b8 + b2 - 1) * q^61 + (-b7 + 2b6 + b3 - 2b1) * q^63 + (3b4 - 2b1) * q^67 + (-2b9 + b5 + 2b2 - 1) * q^69 + (-b8 - 2b2 - 3) q^71 + (-2b7 + b6 + b1) q^73 + (-3b6 + b4 - b3 + b1) q^77 + (b5 - 1) * q^79 + (b9 - b8 + 2b2 - 4) q^81 + (2b6 - b4 + 2b3 + b1) * q^83 - b1 * q^87 + (-2b8 - b5 - 1) q^89 + (-2b9 - b8 + b5 - b2) q^91 + (3b6 + 2b4 + 3b3) q^93 + (b7 - b6 - b4 + 4b3 + b1) q^97 + (-b9 + 3b8 + b2 - 7) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 14 q^{9} - 4 q^{19} - 26 q^{21} - 10 q^{29} - 14 q^{31} - 24 q^{39} - 22 q^{41} - 26 q^{49} - 38 q^{51} - 26 q^{59} - 18 q^{61} - 12 q^{69} - 26 q^{71} - 8 q^{79} - 54 q^{81} - 20 q^{89} + 6 q^{91}+ \cdots - 60 q^{99}+O(q^{100}) \) 10 * q - 14 * q^9 - 4 * q^19 - 26 * q^21 - 10 * q^29 - 14 * q^31 - 24 * q^39 - 22 * q^41 - 26 * q^49 - 38 * q^51 - 26 * q^59 - 18 * q^61 - 12 * q^69 - 26 * q^71 - 8 * q^79 - 54 * q^81 - 20 * q^89 + 6 * q^91 - 60 * q^99

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