LMFDB - Newform orbit 3025.2.a.bl

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3025,2,Mod(1,3025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3025.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$3025 = 5^{2} \cdot 11^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3025.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$24.1547466114$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.1480160000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 9x^{6} + 27x^{4} - 31x^{2} + 11$ x^8 - 9x^6 + 27x^4 - 31*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{2} + (\beta_{7} + \beta_{6} + \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{5} + 2 \beta_{3} + \beta_{2} + 2) q^{6} - \beta_{4} q^{7} + (\beta_{6} + \beta_{4} - \beta_1) q^{8} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2}) q^{9}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{6} + \cdots - 5 \beta_1) q^{98}+O(q^{100})$ q + b1 * q^2 + (b7 + b6 + b1) * q^3 + b2 * q^4 + (b5 + 2b3 + b2 + 2) q^6 - b4 * q^7 + (b6 + b4 - b1) * q^8 + (-b5 + b3 + 2b2) q^9 + (2b6 + b4 + b1) q^12 + (2b6 + b1) q^13 + (2b5 - 2b2 + 1) * q^14 + (b3 - b2 - 2) * q^16 + (3b7 + b6 + 2b1) * q^17 + (b7 + 2b6 + 2b4 + 2b1) q^18 + (-b5 - b3 + 2b2) q^19 + (-2b5 - b3 - 1) q^21 + (b6 - 2b4 + 2b1) * q^23 + (-2b3 + b2 - 1) q^24 + (4b5 + 2b3 + b2 + 4) * q^26 + (-2b7 + b4) q^27 - b1 * q^28 + (b5 + b3 + 4) * q^29 + (-b5 + b3 - b2 + 2) * q^31 + (b7 - 2b6 - 3b4 - b1) * q^32 + (-b5 + 4b3 + 2b2 + 2) * q^34 + (b5 + b3 + 2b2 + 3) q^36 + (b7 - 3b6 - 2b4) * q^37 + (-b7 + 2b4 + 2b1) * q^38 + (-b5 + 5b2 + 2) q^39 + (-5b5 + 2b3 + 3b2 + 2) q^41 + (-b7 - 3b6 - b1) q^42 + (2b7 - 2b6 + 3b1) q^43 + (6b5 + b3 - 2b2 + 7) * q^46 + (b7 + b6 - 3b4 - b1) q^47 + (-2b7 - 5b6 - b4 - 2b1) q^48 + (b5 - 2b3 - b2 - 4) q^49 + (-2b5 + 3b3 + b2 + 7) * q^51 + (2b7 + 3b6 + b4 + 3b1) q^52 + (-2b7 - 3b6 - b4 + 3b1) q^53 + (-2b3 + 2b2 + 1) * q^54 + (-4b5 + 3b2 - 4) * q^56 + (b7 + 5b6 + b4 + b1) q^57 + (b7 + 2b6 + 4b1) * q^58 + (2b5 - 3b3 - 5b2) q^59 + (5b5 - 4b3 - 4b2 + 4) q^61 + (b7 - b6 - b4 + b1) * q^62 + (b7 + b4 - 2b1) q^63 + (b5 - 3b3 - 5b2 + 2) * q^64 + (3b7 + 2b6 + b4 + 4b1) q^67 + (-2b7 + 3b6 + 2b4) q^68 + (-3b5 + b3 + 4b2 + 2) * q^69 + (-3b5 - 3b3 + 3b2 - 9) q^71 + (-b7 - 2b4 + b1) q^72 + (b7 - 5b4 + 2b1) * q^73 + (-3b5 - 2b3 - 4b2 - 2) q^74 + (-b5 + b3 + 2b2 + 3) q^76 + (4b6 + 5b4 + 7b1) q^78 + (5b5 + b3 - 4b2 + 6) * q^79 + (7b5 - 2b3 - 4b2 - 1) q^81 + (2b7 + 3b4 + 5b1) q^82 + (b7 - 3b6 - b4 - b1) q^83 + (-b5 - 2b3 - b2 - 2) q^84 + (-6b5 + 3b2 + 2) * q^86 + (3b7 + 3b6 + b4 + 5b1) q^87 + (-2b5 - 6b3 - 2b2 - 5) q^89 + (-4b3 - 2b2 + 1) * q^91 + (b7 + 3b6 + 2b4 + b1) * q^92 + (3b7 - b6 - 2b4 + 2b1) q^93 + (7b5 + 2b3 - 7b2 + 1) q^94 + (-6b5 - 3b3 - 6b2 - 4) q^96 + (-3b7 - b4 + b1) q^97 + (-2b7 - 2b6 - b4 - 5b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 2 q^{4} + 6 q^{6} + 4 q^{9} - 4 q^{14} - 22 q^{16} + 12 q^{19} + 4 q^{21} + 2 q^{24} + 10 q^{26} + 24 q^{29} + 14 q^{31} + 8 q^{34} + 20 q^{36} + 30 q^{39} + 34 q^{41} + 24 q^{46} - 30 q^{49} + 54 q^{51}+ \cdots - 8 q^{96}+O(q^{100})$ 8 * q + 2 * q^4 + 6 * q^6 + 4 * q^9 - 4 * q^14 - 22 * q^16 + 12 * q^19 + 4 * q^21 + 2 * q^24 + 10 * q^26 + 24 * q^29 + 14 * q^31 + 8 * q^34 + 20 * q^36 + 30 * q^39 + 34 * q^41 + 24 * q^46 - 30 * q^49 + 54 * q^51 + 20 * q^54 - 10 * q^56 - 6 * q^59 + 20 * q^61 + 14 * q^64 + 32 * q^69 - 42 * q^71 - 4 * q^74 + 28 * q^76 + 16 * q^79 - 36 * q^81 - 6 * q^84 + 46 * q^86 - 12 * q^89 + 20 * q^91 - 42 * q^94 - 8 * q^96

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - 9x^{6} + 27x^{4} - 31x^{2} + 11$ x^8 - 9*x^6 + 27*x^4 - 31*x^2 + 11 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 2$ v^2 - 2
$\beta_{3}$	$=$	$\nu^{4} - 5\nu^{2} + 4$ v^4 - 5*v^2 + 4
$\beta_{4}$	$=$	$-\nu^{7} + 7\nu^{5} - 13\nu^{3} + 5\nu$ -v^7 + 7v^5 - 13v^3 + 5*v
$\beta_{5}$	$=$	$\nu^{6} - 7\nu^{4} + 14\nu^{2} - 8$ v^6 - 7v^4 + 14v^2 - 8
$\beta_{6}$	$=$	$\nu^{7} - 7\nu^{5} + 14\nu^{3} - 8\nu$ v^7 - 7v^5 + 14v^3 - 8*v
$\beta_{7}$	$=$	$-\nu^{7} + 8\nu^{5} - 19\nu^{3} + 12\nu$ -v^7 + 8v^5 - 19v^3 + 12*v

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 2$ b2 + 2
$\nu^{3}$	$=$	$\beta_{6} + \beta_{4} + 3\beta_1$ b6 + b4 + 3*b1
$\nu^{4}$	$=$	$\beta_{3} + 5\beta_{2} + 6$ b3 + 5*b2 + 6
$\nu^{5}$	$=$	$\beta_{7} + 6\beta_{6} + 5\beta_{4} + 11\beta_1$ b7 + 6b6 + 5b4 + 11*b1
$\nu^{6}$	$=$	$\beta_{5} + 7\beta_{3} + 21\beta_{2} + 22$ b5 + 7b3 + 21b2 + 22
$\nu^{7}$	$=$	$7\beta_{7} + 29\beta_{6} + 21\beta_{4} + 43\beta_1$ 7b7 + 29b6 + 21b4 + 43b1

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

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}

1.1

−2.02368
−1.65458
−1.23399
−0.802699
0.802699
1.23399
1.65458
2.02368

−2.02368

−2.62059

2.09529

5.30325

0.965823

−0.192845

3.86752

1.2

−1.65458

1.97479

0.737640

−3.26745

2.24307

2.08868

0.899788

1.3

−1.23399

0.363982

−0.477260

−0.449152

−2.58558

3.05692

−2.86752

1.4

−0.802699

−1.76074

−1.35567

1.41335

−0.592103

2.69360

0.100212

1.5

0.802699

1.76074

−1.35567

1.41335

0.592103

−2.69360

0.100212

1.6

1.23399

−0.363982

−0.477260

−0.449152

2.58558

−3.05692

−2.86752

1.7

1.65458

−1.97479

0.737640

−3.26745

−2.24307

−2.08868

0.899788

1.8

2.02368

2.62059

2.09529

5.30325

−0.965823

0.192845

3.86752

Atkin-Lehner signs

$p$	Sign
$5$	$-1$
$11$	$+1$

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	3025.2.a.bl		8
5.b	even	2	1	inner	3025.2.a.bl		8
5.c	odd	4	2		605.2.b.g		8
11.b	odd	2	1		3025.2.a.bk		8
11.c	even	5	2		275.2.h.d		16
55.d	odd	2	1		3025.2.a.bk		8
55.e	even	4	2		605.2.b.f		8
55.j	even	10	2		275.2.h.d		16
55.k	odd	20	4		55.2.j.a	✓	16
55.k	odd	20	4		605.2.j.h		16
55.l	even	20	4		605.2.j.d		16
55.l	even	20	4		605.2.j.g		16
165.v	even	20	4		495.2.ba.a		16
220.v	even	20	4		880.2.cd.c		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
55.2.j.a	✓	16	55.k	odd	20	4
275.2.h.d		16	11.c	even	5	2
275.2.h.d		16	55.j	even	10	2
495.2.ba.a		16	165.v	even	20	4
605.2.b.f		8	55.e	even	4	2
605.2.b.g		8	5.c	odd	4	2
605.2.j.d		16	55.l	even	20	4
605.2.j.g		16	55.l	even	20	4
605.2.j.h		16	55.k	odd	20	4
880.2.cd.c		16	220.v	even	20	4
3025.2.a.bk		8	11.b	odd	2	1
3025.2.a.bk		8	55.d	odd	2	1
3025.2.a.bl		8	1.a	even	1	1	trivial
3025.2.a.bl		8	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 $S_{2}^{\mathrm{new}}(\Gamma_0(3025))$ :

T_{2}^{8} - 9T_{2}^{6} + 27T_{2}^{4} - 31T_{2}^{2} + 11

T2^8 - 9*T2^6 + 27*T2^4 - 31*T2^2 + 11

T_{3}^{8} - 14T_{3}^{6} + 62T_{3}^{4} - 91T_{3}^{2} + 11

T3^8 - 14*T3^6 + 62*T3^4 - 91*T3^2 + 11

T_{19}^{4} - 6T_{19}^{3} - 4T_{19}^{2} + 39T_{19} + 11

T19^4 - 6*T19^3 - 4*T19^2 + 39*T19 + 11

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8} - 9 T^{6} + \cdots + 11$ T^8 - 9T^6 + 27T^4 - 31*T^2 + 11
$3$	$T^{8} - 14 T^{6} + \cdots + 11$ T^8 - 14T^6 + 62T^4 - 91*T^2 + 11
$5$	$T^{8}$ T^8
$7$	$T^{8} - 13 T^{6} + \cdots + 11$ T^8 - 13T^6 + 49T^4 - 47*T^2 + 11
$11$	$T^{8}$ T^8
$13$	$T^{8} - 45 T^{6} + \cdots + 6875$ T^8 - 45T^6 + 675T^4 - 3875*T^2 + 6875
$17$	$T^{8} - 81 T^{6} + \cdots + 40931$ T^8 - 81T^6 + 1842T^4 - 15524*T^2 + 40931
$19$	$(T^{4} - 6 T^{3} - 4 T^{2} + \cdots + 11)^{2}$ (T^4 - 6T^3 - 4T^2 + 39*T + 11)^2
$23$	$T^{8} - 99 T^{6} + \cdots + 26411$ T^8 - 99T^6 + 2232T^4 - 16366*T^2 + 26411
$29$	$(T^{4} - 12 T^{3} + \cdots + 11)^{2}$ (T^4 - 12T^3 + 48T^2 - 67*T + 11)^2
$31$	$(T^{4} - 7 T^{3} + 5 T^{2} + \cdots - 1)^{2}$ (T^4 - 7T^3 + 5T^2 + T - 1)^2
$37$	$T^{8} - 101 T^{6} + \cdots + 3971$ T^8 - 101T^6 + 2822T^4 - 15304*T^2 + 3971
$41$	$(T^{4} - 17 T^{3} + \cdots - 1969)^{2}$ (T^4 - 17T^3 + 48T^2 + 438*T - 1969)^2
$43$	$T^{8} - 173 T^{6} + \cdots + 212531$ T^8 - 173T^6 + 8319T^4 - 79707*T^2 + 212531
$47$	$T^{8} - 191 T^{6} + \cdots + 244211$ T^8 - 191T^6 + 7351T^4 - 90271*T^2 + 244211
$53$	$T^{8} - 249 T^{6} + \cdots + 489731$ T^8 - 249T^6 + 15237T^4 - 164921*T^2 + 489731
$59$	$(T^{4} + 3 T^{3} - 75 T^{2} + \cdots - 1)^{2}$ (T^4 + 3T^3 - 75T^2 - 29*T - 1)^2
$61$	$(T^{4} - 10 T^{3} + \cdots + 209)^{2}$ (T^4 - 10T^3 - 61T^2 + 10*T + 209)^2
$67$	$T^{8} - 149 T^{6} + \cdots + 18491$ T^8 - 149T^6 + 5736T^4 - 46104*T^2 + 18491
$71$	$(T^{4} + 21 T^{3} + \cdots - 2511)^{2}$ (T^4 + 21T^3 + 90T^2 - 432*T - 2511)^2
$73$	$T^{8} - 297 T^{6} + \cdots + 1279091$ T^8 - 297T^6 + 25504T^4 - 671988*T^2 + 1279091
$79$	$(T^{4} - 8 T^{3} + \cdots - 319)^{2}$ (T^4 - 8T^3 - 52T^2 + 377*T - 319)^2
$83$	$T^{8} - 111 T^{6} + \cdots + 212531$ T^8 - 111T^6 + 3931T^4 - 50451*T^2 + 212531
$89$	$(T^{4} + 6 T^{3} + \cdots + 1871)^{2}$ (T^4 + 6T^3 - 128T^2 - 486*T + 1871)^2
$97$	$T^{8} - 162 T^{6} + \cdots + 161051$ T^8 - 162T^6 + 5588T^4 - 55902*T^2 + 161051
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