Properties

Label 252.4.b.d
Level $252$
Weight $4$
Character orbit 252.b
Analytic conductor $14.868$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,4,Mod(55,252)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.55");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.b (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: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 30x^{6} + 84x^{5} + 493x^{4} - 464x^{3} - 3172x^{2} + 1072x + 8978 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} + (\beta_{5} + \beta_{4} - \beta_{2} - 2) q^{4} + \beta_{7} q^{5} + (\beta_{6} + \beta_{4} + 3 \beta_{2}) q^{7} + (4 \beta_{5} + 4 \beta_{2} + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} + (\beta_{5} + \beta_{4} - \beta_{2} - 2) q^{4} + \beta_{7} q^{5} + (\beta_{6} + \beta_{4} + 3 \beta_{2}) q^{7} + (4 \beta_{5} + 4 \beta_{2} + 4) q^{8} + (2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_1) q^{10} + ( - 5 \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{11} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + 4 \beta_{3} - 2 \beta_1) q^{13} + (2 \beta_{7} + \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 19) q^{14} + (4 \beta_{5} - 12 \beta_{4} - 4 \beta_{2} + 8) q^{16} + (2 \beta_{6} + \beta_{5} + 4 \beta_{3} - 2 \beta_1) q^{17} + ( - 2 \beta_{6} + \beta_{5} + \beta_1) q^{19} + (2 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + 4 \beta_{3} + 8 \beta_1) q^{20} + ( - 12 \beta_{5} + 8 \beta_{4} + 2 \beta_{2} + 30) q^{22} + ( - 19 \beta_{5} - 7 \beta_{4} - 21 \beta_{2}) q^{23} + ( - 19 \beta_{4} + 19 \beta_{2} - 59) q^{25} + (6 \beta_{7} + 3 \beta_{6} - \beta_{5} + \beta_{3} + 23 \beta_1) q^{26} + (6 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} - 25 \beta_{2} + \cdots - 6) q^{28}+ \cdots + ( - 8 \beta_{7} + 22 \beta_{6} + 36 \beta_{5} + 52 \beta_{4} - 10 \beta_{3} + \cdots + 393) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 16 q^{4} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 16 q^{4} + 32 q^{8} + 152 q^{14} + 64 q^{16} + 240 q^{22} - 472 q^{25} - 48 q^{28} + 592 q^{29} - 1152 q^{32} + 1392 q^{37} + 1184 q^{44} - 816 q^{46} + 1480 q^{49} - 1688 q^{50} + 1168 q^{53} - 800 q^{56} - 560 q^{58} - 3328 q^{64} - 448 q^{65} - 3200 q^{70} + 496 q^{74} - 368 q^{77} + 1024 q^{85} - 240 q^{86} + 3776 q^{88} + 3808 q^{92} + 3144 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 30x^{6} + 84x^{5} + 493x^{4} - 464x^{3} - 3172x^{2} + 1072x + 8978 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2107780 \nu^{7} - 28020178 \nu^{6} + 12013496 \nu^{5} + 856630098 \nu^{4} - 1820971716 \nu^{3} - 7812762598 \nu^{2} + 31061346042 \nu + 26653239984 ) / 11759822513 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2107780 \nu^{7} + 28020178 \nu^{6} - 12013496 \nu^{5} - 856630098 \nu^{4} + 1820971716 \nu^{3} + 7812762598 \nu^{2} + \cdots - 38413062497 ) / 11759822513 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5280 \nu^{7} + 27552 \nu^{6} + 648572 \nu^{5} - 3328540 \nu^{4} - 9022980 \nu^{3} + 48793686 \nu^{2} + 78029576 \nu - 204544903 ) / 7201361 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 37410820 \nu^{7} - 178195062 \nu^{6} - 946363840 \nu^{5} + 4456400722 \nu^{4} + 10555104876 \nu^{3} - 33992527302 \nu^{2} + \cdots + 158760439399 ) / 11759822513 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 71287048 \nu^{7} + 411792664 \nu^{6} + 1067149048 \nu^{5} - 6385173864 \nu^{4} - 14881408792 \nu^{3} + 32020502452 \nu^{2} + \cdots - 65434616690 ) / 11759822513 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 85892144 \nu^{7} - 485124166 \nu^{6} - 2515029448 \nu^{5} + 14797753670 \nu^{4} + 29088392180 \nu^{3} - 121788747936 \nu^{2} + \cdots + 390648894103 ) / 11759822513 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2876 \nu^{7} + 28790 \nu^{6} - 17704 \nu^{5} - 509450 \nu^{4} + 374248 \nu^{3} + 4834142 \nu^{2} - 1479810 \nu - 15131816 ) / 379019 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} + \beta_{5} - 2\beta_{4} + 2\beta_{3} + 6\beta_{2} + 2\beta _1 + 38 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + 6\beta_{6} + 10\beta_{5} + 3\beta_{4} + 6\beta_{3} + 79\beta_{2} + 17\beta _1 + 86 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -16\beta_{7} + 48\beta_{6} + 83\beta_{5} + 39\beta_{4} + 60\beta_{3} + 305\beta_{2} + 32\beta _1 + 330 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -250\beta_{7} + 330\beta_{6} + 965\beta_{5} + 834\beta_{4} + 560\beta_{3} + 3838\beta_{2} - 68\beta _1 + 1764 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 590 \beta_{7} + 754 \beta_{6} + 2877 \beta_{5} + 3229 \beta_{4} + 1576 \beta_{3} + 8337 \beta_{2} - 928 \beta _1 - 2086 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6440 \beta_{7} + 4344 \beta_{6} + 28219 \beta_{5} + 39754 \beta_{4} + 15134 \beta_{3} + 73906 \beta_{2} - 20408 \beta _1 - 63556 ) / 8 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
4.98105 + 1.39897i
−2.56684 + 1.39897i
−2.56684 1.39897i
4.98105 1.39897i
−2.60656 + 0.736813i
2.19234 + 0.736813i
2.19234 0.736813i
−2.60656 0.736813i
−0.414214 2.79793i 0 −7.65685 + 2.31788i 8.74756i 0 13.8008 + 12.3507i 9.65685 + 20.4633i 0 −24.4751 + 3.62336i
55.2 −0.414214 2.79793i 0 −7.65685 + 2.31788i 8.74756i 0 −13.8008 + 12.3507i 9.65685 + 20.4633i 0 24.4751 3.62336i
55.3 −0.414214 + 2.79793i 0 −7.65685 2.31788i 8.74756i 0 −13.8008 12.3507i 9.65685 20.4633i 0 24.4751 + 3.62336i
55.4 −0.414214 + 2.79793i 0 −7.65685 2.31788i 8.74756i 0 13.8008 12.3507i 9.65685 20.4633i 0 −24.4751 3.62336i
55.5 2.41421 1.47363i 0 3.65685 7.11529i 17.0728i 0 18.3722 + 2.33686i −1.65685 22.5667i 0 −25.1589 41.2174i
55.6 2.41421 1.47363i 0 3.65685 7.11529i 17.0728i 0 −18.3722 + 2.33686i −1.65685 22.5667i 0 25.1589 + 41.2174i
55.7 2.41421 + 1.47363i 0 3.65685 + 7.11529i 17.0728i 0 −18.3722 2.33686i −1.65685 + 22.5667i 0 25.1589 41.2174i
55.8 2.41421 + 1.47363i 0 3.65685 + 7.11529i 17.0728i 0 18.3722 2.33686i −1.65685 + 22.5667i 0 −25.1589 + 41.2174i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.b.d 8
3.b odd 2 1 28.4.d.b 8
4.b odd 2 1 inner 252.4.b.d 8
7.b odd 2 1 inner 252.4.b.d 8
12.b even 2 1 28.4.d.b 8
21.c even 2 1 28.4.d.b 8
21.g even 6 2 196.4.f.c 16
21.h odd 6 2 196.4.f.c 16
24.f even 2 1 448.4.f.d 8
24.h odd 2 1 448.4.f.d 8
28.d even 2 1 inner 252.4.b.d 8
84.h odd 2 1 28.4.d.b 8
84.j odd 6 2 196.4.f.c 16
84.n even 6 2 196.4.f.c 16
168.e odd 2 1 448.4.f.d 8
168.i even 2 1 448.4.f.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.d.b 8 3.b odd 2 1
28.4.d.b 8 12.b even 2 1
28.4.d.b 8 21.c even 2 1
28.4.d.b 8 84.h odd 2 1
196.4.f.c 16 21.g even 6 2
196.4.f.c 16 21.h odd 6 2
196.4.f.c 16 84.j odd 6 2
196.4.f.c 16 84.n even 6 2
252.4.b.d 8 1.a even 1 1 trivial
252.4.b.d 8 4.b odd 2 1 inner
252.4.b.d 8 7.b odd 2 1 inner
252.4.b.d 8 28.d even 2 1 inner
448.4.f.d 8 24.f even 2 1
448.4.f.d 8 24.h odd 2 1
448.4.f.d 8 168.e odd 2 1
448.4.f.d 8 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(252, [\chi])\):

\( T_{5}^{4} + 368T_{5}^{2} + 22304 \) Copy content Toggle raw display
\( T_{11}^{4} + 1720T_{11}^{2} + 272 \) Copy content Toggle raw display
\( T_{19}^{4} - 2128T_{19}^{2} + 694048 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{3} + 12 T^{2} - 32 T + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 368 T^{2} + 22304)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 740 T^{6} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{4} + 1720 T^{2} + 272)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 7792 T^{2} + 11798816)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 7936 T^{2} + 5709824)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2128 T^{2} + 694048)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 23800 T^{2} + \cdots + 132140048)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 148 T - 4892)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 23040 T^{2} + \cdots + 108822528)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 348 T + 24004)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 58304 T^{2} + \cdots + 342946304)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 34360 T^{2} + \cdots + 141396752)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 103680 T^{2} + \cdots + 1789861888)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 292 T - 163516)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 570320 T^{2} + \cdots + 67995188512)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 606832 T^{2} + 6445856)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 343672 T^{2} + \cdots + 12017669648)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 275576 T^{2} + \cdots + 9248152592)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 92864 T^{2} + \cdots + 1798951424)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1466680 T^{2} + \cdots + 108115639568)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1267920 T^{2} + \cdots + 340971272992)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 1846976 T^{2} + \cdots + 661026681344)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 3065344 T^{2} + \cdots + 1135775350784)^{2} \) Copy content Toggle raw display
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