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Properties

Label 252.7.d.a
Level $252$
Weight $7$
Character orbit 252.d
Analytic conductor $57.974$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,7,Mod(181,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.181");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 252.d (of order \(2\), degree \(1\), 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: \(57.9736290722\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.903168.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 24x^{2} + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 28)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 7) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 7) q^{7} + (4 \beta_{3} + 54) q^{11} + ( - 26 \beta_{2} + 25 \beta_1) q^{13} + (210 \beta_{2} + 26 \beta_1) q^{17} + (221 \beta_{2} + 14 \beta_1) q^{19} + ( - 16 \beta_{3} + 3438) q^{23} + (46 \beta_{3} - 887) q^{25} + ( - 68 \beta_{3} - 8730) q^{29} + ( - 1266 \beta_{2} + 66 \beta_1) q^{31} + (34 \beta_{3} - 705 \beta_{2} + \cdots - 16128) q^{35}+ \cdots + ( - 34886 \beta_{2} + 1990 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 28 q^{7} + 216 q^{11} + 13752 q^{23} - 3548 q^{25} - 34920 q^{29} - 64512 q^{35} - 158936 q^{37} + 215080 q^{43} + 320068 q^{49} + 158040 q^{53} + 1631232 q^{65} - 511000 q^{67} - 1648584 q^{71} - 1582056 q^{77} - 2827768 q^{79} + 1878528 q^{85} + 1752576 q^{91} + 1094400 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 24x^{2} + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 156\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{3} - 44\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 28\nu^{2} + 336 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} + 3\beta_1 ) / 112 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 336 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -39\beta_{2} - 33\beta_1 ) / 56 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
4.81923i
0.880357i
− 0.880357i
− 4.81923i
0 0 0 − 175.982i 0 307.299 − 152.369i 0 0 0
181.2 0 0 0 − 45.3237i 0 −321.299 − 120.066i 0 0 0
181.3 0 0 0 45.3237i 0 −321.299 + 120.066i 0 0 0
181.4 0 0 0 175.982i 0 307.299 + 152.369i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.7.d.a 4
3.b odd 2 1 28.7.b.a ✓ 4
7.b odd 2 1 inner 252.7.d.a 4
12.b even 2 1 112.7.c.d 4
15.d odd 2 1 700.7.d.b 4
15.e even 4 2 700.7.h.a 8
21.c even 2 1 28.7.b.a ✓ 4
21.g even 6 2 196.7.h.b 8
21.h odd 6 2 196.7.h.b 8
24.f even 2 1 448.7.c.g 4
24.h odd 2 1 448.7.c.f 4
84.h odd 2 1 112.7.c.d 4
105.g even 2 1 700.7.d.b 4
105.k odd 4 2 700.7.h.a 8
168.e odd 2 1 448.7.c.g 4
168.i even 2 1 448.7.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.7.b.a ✓ 4 3.b odd 2 1
28.7.b.a ✓ 4 21.c even 2 1
112.7.c.d 4 12.b even 2 1
112.7.c.d 4 84.h odd 2 1
196.7.h.b 8 21.g even 6 2
196.7.h.b 8 21.h odd 6 2
252.7.d.a 4 1.a even 1 1 trivial
252.7.d.a 4 7.b odd 2 1 inner
448.7.c.f 4 24.h odd 2 1
448.7.c.f 4 168.i even 2 1
448.7.c.g 4 24.f even 2 1
448.7.c.g 4 168.e odd 2 1
700.7.d.b 4 15.d odd 2 1
700.7.d.b 4 105.g even 2 1
700.7.h.a 8 15.e even 4 2
700.7.h.a 8 105.k odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 33024T_{5}^{2} + 63619200 \) acting on \(S_{7}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 33024 T^{2} + 63619200 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{2} - 108 T - 1577628)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 74197833257088 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{2} - 6876 T - 13468860)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 17460 T - 380564316)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{2} + 79468 T + 1564565860)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} - 107540 T - 4853452700)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 91\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( (T^{2} - 79020 T - 11241366300)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{2} + 255500 T + 15687844900)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 824292 T + 115477411140)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 48\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( (T^{2} + 1413884 T + 454357567108)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 83\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 73\!\cdots\!68 \) Copy content Toggle raw display
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