LMFDB - Newform orbit 7225.2.a.ba

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7225 = 5^{2} \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7225.a (trivial)

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:	yes
Analytic conductor:	$57.6919154604$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.93924352.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 17x^{4} + 73x^{2} - 67$ x^6 - 17x^4 + 73x^2 - 67
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 425)
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_{2} q^{2} - \beta_1 q^{3} + ( - \beta_{3} - \beta_{2}) q^{4} + ( - \beta_{5} + \beta_{4}) q^{6} - \beta_{4} q^{7} + (\beta_{3} - 1) q^{8} + ( - 2 \beta_{3} + \beta_{2} + 3) q^{9} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{11}+ \cdots + (\beta_{5} + 6 \beta_1) q^{99}+O(q^{100})$ q + b2 * q^2 - b1 * q^3 + (-b3 - b2) * q^4 + (-b5 + b4) * q^6 - b4 * q^7 + (b3 - 1) * q^8 + (-2b3 + b2 + 3) q^9 + (-b5 - b4 + b1) * q^11 + (b5 - b1) * q^12 + (-b3 - 2b2 - 1) q^13 + b1 * q^14 + (2b3 - 1) q^16 + (-b3 + 4b2 + 4) q^18 + (-b3 - 3b2 - 3) q^19 + (2b3 - 2b2 - 1) * q^21 + (2b5 - b4 - b1) q^22 + (b5 - b4 - b1) * q^23 + (-b4 + 2b1) q^24 + (2b3 + 2b2 - 3) * q^26 + (-b5 + 3b4 - 2b1) * q^27 + (b5 + b4) * q^28 + (b5 + 2b1) q^29 + (2b4 + b1) q^31 + (-2b3 - 3b2) * q^32 + (5b3 + 2b2 - 4) * q^33 + (-b2 + 3) * q^36 + (-b5 + 2b4 + 2b1) * q^37 + (3b3 + b2 - 5) q^38 + (2b5 - b4) q^39 + (-b5 + 2b4) q^41 + (2b3 - b2 - 6) q^42 + (5b3 + b2 - 1) q^43 + (-b5 + 3b4 + 3b1) * q^44 + (-2b5 + b4 + 3b1) * q^46 + (2b2 - 6) q^47 + (-2b4 + 3b1) * q^48 + (3b3 + 2b2) * q^49 + (-3b2 + 4) q^52 + (b3 + 5b2) q^53 + (-b5 + 2b4 - 5b1) * q^54 + (-b5 - b1) * q^56 + (3b5 - 2b4 + 2b1) q^57 + (b5 - 2b4 + 2b1) * q^58 + (-3b3 + b2 + 1) q^59 + (2b5 + 2b1) * q^61 + (b5 - b4 - 2b1) q^62 + (2b5 - b4 + 3b1) * q^63 + (-b3 + 5b2 - 2) q^64 + (-2b3 - 11b2 - 1) * q^66 + (2b3 - 4) q^67 + (-b3 - 6b2 + 2) q^69 + (b4 - 2b1) q^71 + (3b3 - 4b2 - 10) * q^72 + b5 * q^73 + (3b5 - 2b4 - 4b1) q^74 + (b3 - 3b2 + 5) q^76 + (6b3 + 5b2 + 9) * q^77 + (-2b5 + 5b1) * q^78 + (b5 - b4 + 2b1) q^79 + (-3b3 + 10b2 + 9) * q^81 + (b5 - 4b1) q^82 + (-6b3 - 2b2 - 4) * q^83 + (-3b3 - 3b2 - 2) * q^84 + (-b3 - 7b2 - 3) q^86 + (3b3 - 7b2 - 15) * q^87 + (-b4 - 3b1) q^88 + (-b3 + b2 - 1) * q^89 + (b5 + 2b4 - b1) q^91 + (3b5 - b4 - 3b1) * q^92 + (-2b3 + 3b2 - 4) * q^93 + (-2b3 - 8b2 + 4) * q^94 + (3b5 - b4 - 2b1) * q^96 + (-2b4 - 4b1) * q^97 + (-2b3 - 5b2 + 1) * q^98 + (b5 + 6b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 2 q^{2} + 2 q^{4} - 6 q^{8} + 16 q^{9} - 2 q^{13} - 6 q^{16} + 16 q^{18} - 12 q^{19} - 2 q^{21} - 22 q^{26} + 6 q^{32} - 28 q^{33} + 20 q^{36} - 32 q^{38} - 34 q^{42} - 8 q^{43} - 40 q^{47} - 4 q^{49}+ \cdots + 16 q^{98}+O(q^{100})$ 6 * q - 2 * q^2 + 2 * q^4 - 6 * q^8 + 16 * q^9 - 2 * q^13 - 6 * q^16 + 16 * q^18 - 12 * q^19 - 2 * q^21 - 22 * q^26 + 6 * q^32 - 28 * q^33 + 20 * q^36 - 32 * q^38 - 34 * q^42 - 8 * q^43 - 40 * q^47 - 4 * q^49 + 30 * q^52 - 10 * q^53 + 4 * q^59 - 22 * q^64 + 16 * q^66 - 24 * q^67 + 24 * q^69 - 52 * q^72 + 36 * q^76 + 44 * q^77 + 34 * q^81 - 20 * q^83 - 6 * q^84 - 4 * q^86 - 76 * q^87 - 8 * q^89 - 30 * q^93 + 40 * q^94 + 16 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{6} - 17x^{4} + 73x^{2} - 67$ x^6 - 17*x^4 + 73*x^2 - 67 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 2\nu^{4} - 21\nu^{2} + 18 ) / 17$ (2v^4 - 21v^2 + 18) / 17
$\beta_{3}$	$=$	$( \nu^{4} - 19\nu^{2} + 60 ) / 17$ (v^4 - 19*v^2 + 60) / 17
$\beta_{4}$	$=$	$( \nu^{5} - 19\nu^{3} + 77\nu ) / 17$ (v^5 - 19v^3 + 77v) / 17
$\beta_{5}$	$=$	$( 3\nu^{5} - 40\nu^{3} + 95\nu ) / 17$ (3v^5 - 40v^3 + 95*v) / 17

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$-2\beta_{3} + \beta_{2} + 6$ -2*b3 + b2 + 6
$\nu^{3}$	$=$	$\beta_{5} - 3\beta_{4} + 8\beta_1$ b5 - 3b4 + 8b1
$\nu^{4}$	$=$	$-21\beta_{3} + 19\beta_{2} + 54$ -21b3 + 19b2 + 54
$\nu^{5}$	$=$	$19\beta_{5} - 40\beta_{4} + 75\beta_1$ 19b5 - 40b4 + 75*b1

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

2.21547
−2.21547
1.12261
−1.12261
3.29112
−3.29112

−2.17009

−2.21547

2.70928

4.80775

−1.02091

−1.53919

1.90829

1.2

−2.17009

2.21547

2.70928

−4.80775

1.02091

−1.53919

1.90829

1.3

−0.311108

−1.12261

−1.90321

0.349253

−3.60843

1.21432

−1.73975

1.4

−0.311108

1.12261

−1.90321

−0.349253

3.60843

1.21432

−1.73975

1.5

1.48119

−3.29112

0.193937

−4.87478

2.22194

−2.67513

7.83146

1.6

1.48119

3.29112

0.193937

4.87478

−2.22194

−2.67513

7.83146

Atkin-Lehner signs

$p$	Sign
$5$	$+1$
$17$	$+1$

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7225.2.a.ba		6
5.b	even	2	1		7225.2.a.bg		6
17.b	even	2	1	inner	7225.2.a.ba		6
17.c	even	4	2		425.2.d.b	yes	6
85.c	even	2	1		7225.2.a.bg		6
85.f	odd	4	2		425.2.c.c		12
85.i	odd	4	2		425.2.c.c		12
85.j	even	4	2		425.2.d.a	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
425.2.c.c		12	85.f	odd	4	2
425.2.c.c		12	85.i	odd	4	2
425.2.d.a	✓	6	85.j	even	4	2
425.2.d.b	yes	6	17.c	even	4	2
7225.2.a.ba		6	1.a	even	1	1	trivial
7225.2.a.ba		6	17.b	even	2	1	inner
7225.2.a.bg		6	5.b	even	2	1
7225.2.a.bg		6	85.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(7225))$ :

T_{2}^{3} + T_{2}^{2} - 3T_{2} - 1

T_{3}^{6} - 17T_{3}^{4} + 73T_{3}^{2} - 67

T_{7}^{6} - 19T_{7}^{4} + 83T_{7}^{2} - 67

T_{11}^{6} - 66T_{11}^{4} + 1292T_{11}^{2} - 6700

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{3} + T^{2} - 3 T - 1)^{2}$ (T^3 + T^2 - 3*T - 1)^2
$3$	$T^{6} - 17 T^{4} + \cdots - 67$ T^6 - 17T^4 + 73T^2 - 67
$5$	$T^{6}$ T^6
$7$	$T^{6} - 19 T^{4} + \cdots - 67$ T^6 - 19T^4 + 83T^2 - 67
$11$	$T^{6} - 66 T^{4} + \cdots - 6700$ T^6 - 66T^4 + 1292T^2 - 6700
$13$	$(T^{3} + T^{2} - 13 T - 23)^{2}$ (T^3 + T^2 - 13*T - 23)^2
$17$	$T^{6}$ T^6
$19$	$(T^{3} + 6 T^{2} + \cdots - 100)^{2}$ (T^3 + 6T^2 - 16T - 100)^2
$23$	$T^{6} - 54 T^{4} + \cdots - 268$ T^6 - 54T^4 + 236T^2 - 268
$29$	$T^{6} - 116 T^{4} + \cdots - 1072$ T^6 - 116T^4 + 2632T^2 - 1072
$31$	$T^{6} - 89 T^{4} + \cdots - 1675$ T^6 - 89T^4 + 1377T^2 - 1675
$37$	$T^{6} - 144 T^{4} + \cdots - 1072$ T^6 - 144T^4 + 4088T^2 - 1072
$41$	$T^{6} - 100 T^{4} + \cdots - 26800$ T^6 - 100T^4 + 3032T^2 - 26800
$43$	$(T^{3} + 4 T^{2} + \cdots - 452)^{2}$ (T^3 + 4T^2 - 88T - 452)^2
$47$	$(T^{3} + 20 T^{2} + \cdots + 208)^{2}$ (T^3 + 20T^2 + 120T + 208)^2
$53$	$(T^{3} + 5 T^{2} - 69 T + 43)^{2}$ (T^3 + 5T^2 - 69T + 43)^2
$59$	$(T^{3} - 2 T^{2} - 44 T + 20)^{2}$ (T^3 - 2T^2 - 44T + 20)^2
$61$	$T^{6} - 228 T^{4} + \cdots - 107200$ T^6 - 228T^4 + 13008T^2 - 107200
$67$	$(T^{3} + 12 T^{2} + \cdots - 16)^{2}$ (T^3 + 12T^2 + 32T - 16)^2
$71$	$T^{6} - 91 T^{4} + \cdots - 1675$ T^6 - 91T^4 + 1067T^2 - 1675
$73$	$T^{6} - 32 T^{4} + \cdots - 1072$ T^6 - 32T^4 + 328T^2 - 1072
$79$	$T^{6} - 135 T^{4} + \cdots - 67$ T^6 - 135T^4 + 491T^2 - 67
$83$	$(T^{3} + 10 T^{2} + \cdots + 184)^{2}$ (T^3 + 10T^2 - 100T + 184)^2
$89$	$(T^{3} + 4 T^{2} - 4 T - 20)^{2}$ (T^3 + 4T^2 - 4T - 20)^2
$97$	$T^{6} - 332 T^{4} + \cdots - 1239232$ T^6 - 332T^4 + 35760T^2 - 1239232
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