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Properties

Label 252.8.k.d
Level $252$
Weight $8$
Character orbit 252.k
Analytic conductor $78.721$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,8,Mod(37,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.37");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), degree \(2\), 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: \(78.7210264220\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 8083 x^{8} - 434726 x^{7} + 60654392 x^{6} - 1915889576 x^{5} + 88369975648 x^{4} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 40 \beta_1) q^{5} + ( - \beta_{8} - \beta_{4} + 65 \beta_1 + 40) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 40 \beta_1) q^{5} + ( - \beta_{8} - \beta_{4} + 65 \beta_1 + 40) q^{7} + (\beta_{9} + 4 \beta_{8} - \beta_{7} + \cdots + 1460) q^{11}+ \cdots + (1719 \beta_{9} + 298 \beta_{8} + \cdots - 1432384) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 198 q^{5} + 71 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 198 q^{5} + 71 q^{7} + 7296 q^{11} + 28134 q^{13} - 12300 q^{17} - 28837 q^{19} + 9876 q^{23} - 144301 q^{25} - 293304 q^{29} + 150387 q^{31} + 122154 q^{35} - 18145 q^{37} + 305244 q^{41} + 2429330 q^{43} + 399018 q^{47} - 917993 q^{49} - 2247636 q^{53} - 2981224 q^{55} + 1602990 q^{59} - 1452750 q^{61} - 5437914 q^{65} + 3540085 q^{67} - 2921892 q^{71} - 5064251 q^{73} + 22073064 q^{77} - 654891 q^{79} - 22171632 q^{83} + 34560104 q^{85} + 14925528 q^{89} - 25905393 q^{91} - 5825550 q^{95} - 14399960 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 8083 x^{8} - 434726 x^{7} + 60654392 x^{6} - 1915889576 x^{5} + 88369975648 x^{4} + \cdots + 18\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13\!\cdots\!43 \nu^{9} + \cdots - 10\!\cdots\!00 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 72\!\cdots\!19 \nu^{9} + \cdots + 71\!\cdots\!00 ) / 46\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23\!\cdots\!21 \nu^{9} + \cdots + 61\!\cdots\!40 ) / 62\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!87 \nu^{9} + \cdots - 64\!\cdots\!20 ) / 50\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 90\!\cdots\!53 \nu^{9} + \cdots - 96\!\cdots\!40 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15\!\cdots\!07 \nu^{9} + \cdots - 37\!\cdots\!56 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!99 \nu^{9} + \cdots + 46\!\cdots\!80 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 55\!\cdots\!31 \nu^{9} + \cdots + 85\!\cdots\!60 ) / 50\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!43 \nu^{9} + \cdots + 90\!\cdots\!00 ) / 31\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 9\beta_{9} - 29\beta_{8} + \beta_{7} - 7\beta_{6} - 14\beta_{4} + 7\beta_{3} - 42\beta_{2} - 145\beta _1 + 3 ) / 756 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 268 \beta_{9} + 748 \beta_{8} - 160 \beta_{7} + 1239 \beta_{5} + 217 \beta_{4} - 427 \beta_{3} + \cdots - 1222400 ) / 378 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 532 \beta_{9} - 552 \beta_{8} + 256 \beta_{7} + 509 \beta_{6} - 4041 \beta_{5} + 1044 \beta_{4} + \cdots + 769101 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 205290 \beta_{9} - 3980854 \beta_{8} + 2298362 \beta_{7} - 4053581 \beta_{6} - 2430925 \beta_{4} + \cdots - 1960509 ) / 378 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 195124858 \beta_{9} + 1034462458 \beta_{8} - 279779200 \beta_{7} + 2212231119 \beta_{5} + \cdots - 602759867510 ) / 378 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 358633616 \beta_{9} - 661983980 \beta_{8} + 27641626 \beta_{7} + 586747371 \beta_{6} + \cdots + 1136060576117 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 572903375598 \beta_{9} - 5605787245006 \beta_{8} + 1943538559106 \beta_{7} - 3333334073489 \beta_{6} + \cdots - 816892138785 ) / 378 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 196189032196474 \beta_{9} + 877903877784394 \beta_{8} - 227238281862640 \beta_{7} + \cdots - 65\!\cdots\!50 ) / 378 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 372394774238160 \beta_{9} - 598786671265180 \beta_{8} + 73001438605570 \beta_{7} + \cdots + 97\!\cdots\!33 ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
11.3573 − 19.6714i
−48.9652 + 84.8102i
19.4288 − 33.6517i
31.2797 − 54.1780i
−12.6006 + 21.8249i
11.3573 + 19.6714i
−48.9652 − 84.8102i
19.4288 + 33.6517i
31.2797 + 54.1780i
−12.6006 − 21.8249i
0 0 0 −240.336 + 416.273i 0 −183.244 − 888.800i 0 0 0
37.2 0 0 0 −147.307 + 255.143i 0 −280.992 + 862.894i 0 0 0
37.3 0 0 0 −30.5185 + 52.8597i 0 657.281 + 625.719i 0 0 0
37.4 0 0 0 124.884 − 216.306i 0 718.848 − 553.896i 0 0 0
37.5 0 0 0 194.277 − 336.497i 0 −876.393 + 235.540i 0 0 0
109.1 0 0 0 −240.336 − 416.273i 0 −183.244 + 888.800i 0 0 0
109.2 0 0 0 −147.307 − 255.143i 0 −280.992 − 862.894i 0 0 0
109.3 0 0 0 −30.5185 − 52.8597i 0 657.281 − 625.719i 0 0 0
109.4 0 0 0 124.884 + 216.306i 0 718.848 + 553.896i 0 0 0
109.5 0 0 0 194.277 + 336.497i 0 −876.393 − 235.540i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.8.k.d 10
3.b odd 2 1 84.8.i.b ✓ 10
7.c even 3 1 inner 252.8.k.d 10
21.c even 2 1 588.8.i.p 10
21.g even 6 1 588.8.a.k 5
21.g even 6 1 588.8.i.p 10
21.h odd 6 1 84.8.i.b ✓ 10
21.h odd 6 1 588.8.a.l 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.8.i.b ✓ 10 3.b odd 2 1
84.8.i.b ✓ 10 21.h odd 6 1
252.8.k.d 10 1.a even 1 1 trivial
252.8.k.d 10 7.c even 3 1 inner
588.8.a.k 5 21.g even 6 1
588.8.a.l 5 21.h odd 6 1
588.8.i.p 10 21.c even 2 1
588.8.i.p 10 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 198 T_{5}^{9} + 287065 T_{5}^{8} + 12509742 T_{5}^{7} + 54713791801 T_{5}^{6} + \cdots + 70\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 37\!\cdots\!43 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 31\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 89\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 14\!\cdots\!84)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 33\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 81\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 62\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 87\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 34\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 54\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 68\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 21\!\cdots\!40)^{2} \) Copy content Toggle raw display
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