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Properties

Label 252.8.k.c
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: \( x^{10} - x^{9} + 342 x^{8} + 2165 x^{7} + 113605 x^{6} + 319380 x^{5} + 1438128 x^{4} + 1705752 x^{3} + \cdots + 23619600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{8}\cdot 7^{5} \)
Twist minimal: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + 50 \beta_1 - 50) q^{5} + (\beta_{5} - \beta_{4} - \beta_{2} + \cdots + 129) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + 50 \beta_1 - 50) q^{5} + (\beta_{5} - \beta_{4} - \beta_{2} + \cdots + 129) q^{7}+ \cdots + ( - 2240 \beta_{9} - 2618 \beta_{8} + \cdots + 31440) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 249 q^{5} + 332 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 249 q^{5} + 332 q^{7} - 6399 q^{11} - 26988 q^{13} - 3609 q^{17} - 12403 q^{19} + 13959 q^{23} - 162364 q^{25} - 26148 q^{29} - 20181 q^{31} - 791715 q^{35} - 54763 q^{37} + 1824468 q^{41} - 1938424 q^{43} - 1217253 q^{47} + 1722058 q^{49} + 2349159 q^{53} + 3123494 q^{55} - 2188611 q^{59} + 6107445 q^{61} + 3022014 q^{65} - 3171581 q^{67} + 18938976 q^{71} + 8034853 q^{73} - 24100191 q^{77} - 7589559 q^{79} + 13452408 q^{83} - 48717670 q^{85} - 29864757 q^{89} + 34267512 q^{91} + 25812009 q^{95} + 285580 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 342 x^{8} + 2165 x^{7} + 113605 x^{6} + 319380 x^{5} + 1438128 x^{4} + 1705752 x^{3} + \cdots + 23619600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 41255617984 \nu^{9} - 107667922597 \nu^{8} + 14045699887767 \nu^{7} + 66486384588320 \nu^{6} + \cdots + 13\!\cdots\!40 ) / 63\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 38380564536913 \nu^{9} + 33729276619325 \nu^{8} + \cdots + 90\!\cdots\!00 ) / 20\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 89084817540212 \nu^{9} + \cdots + 35\!\cdots\!00 ) / 41\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 275746765304656 \nu^{9} + \cdots - 33\!\cdots\!00 ) / 83\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 53102187703213 \nu^{9} - 884170321273686 \nu^{8} + \cdots - 42\!\cdots\!80 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 170976780525575 \nu^{9} + \cdots + 21\!\cdots\!00 ) / 41\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8370298711069 \nu^{9} - 7698009493391 \nu^{8} + \cdots - 10\!\cdots\!28 ) / 92\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 410410936325427 \nu^{9} + \cdots - 68\!\cdots\!60 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 58\!\cdots\!26 \nu^{9} + \cdots - 51\!\cdots\!20 ) / 83\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{9} - 5 \beta_{8} - 2 \beta_{7} + 3 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} + \beta_{3} + \cdots + 101 ) / 504 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 16 \beta_{9} + 38 \beta_{8} + 136 \beta_{7} + 152 \beta_{6} - 28 \beta_{5} - 4 \beta_{4} + 5 \beta_{3} + \cdots - 11 ) / 504 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 285 \beta_{9} + 1248 \beta_{8} + 1862 \beta_{7} + 231 \beta_{5} - 570 \beta_{4} + 339 \beta_{3} + \cdots - 378700 ) / 504 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3991 \beta_{9} - 3671 \beta_{8} + 3991 \beta_{7} - 56307 \beta_{6} - 7982 \beta_{5} + \cdots - 23217469 ) / 504 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 134972 \beta_{9} + 30698 \beta_{8} - 592424 \beta_{7} - 727396 \beta_{6} - 466312 \beta_{5} + \cdots - 82835 ) / 504 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 513615 \beta_{9} - 4258572 \beta_{8} - 20569984 \beta_{7} + 3670131 \beta_{5} - 1027230 \beta_{4} + \cdots + 8354028674 ) / 504 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 78916229 \beta_{9} - 153780005 \beta_{8} - 78916229 \beta_{7} + 316957071 \beta_{6} + \cdots + 101008036277 ) / 504 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1065092422 \beta_{9} + 1208137316 \beta_{8} + 6997767292 \beta_{7} + 8062859714 \beta_{6} + \cdots - 71522447 ) / 504 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 7835868045 \beta_{9} + 58884610176 \beta_{8} + 141652552694 \beta_{7} - 5934700953 \beta_{5} + \cdots - 44667131209900 ) / 504 \) Copy content Toggle raw display

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

Character values

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

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

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
−1.38371 + 2.39666i
−1.17613 + 2.03713i
10.0194 − 17.3541i
−8.10827 + 14.0439i
1.14872 − 1.98964i
−1.38371 − 2.39666i
−1.17613 − 2.03713i
10.0194 + 17.3541i
−8.10827 − 14.0439i
1.14872 + 1.98964i
0 0 0 −274.614 + 475.646i 0 907.484 + 4.07000i 0 0 0
37.2 0 0 0 −89.0278 + 154.201i 0 −795.793 + 436.183i 0 0 0
37.3 0 0 0 −58.1649 + 100.745i 0 −276.518 − 864.338i 0 0 0
37.4 0 0 0 88.1161 − 152.621i 0 836.746 + 351.283i 0 0 0
37.5 0 0 0 209.191 − 362.329i 0 −505.918 − 753.386i 0 0 0
109.1 0 0 0 −274.614 − 475.646i 0 907.484 − 4.07000i 0 0 0
109.2 0 0 0 −89.0278 − 154.201i 0 −795.793 − 436.183i 0 0 0
109.3 0 0 0 −58.1649 − 100.745i 0 −276.518 + 864.338i 0 0 0
109.4 0 0 0 88.1161 + 152.621i 0 836.746 − 351.283i 0 0 0
109.5 0 0 0 209.191 + 362.329i 0 −505.918 + 753.386i 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.c 10
3.b odd 2 1 28.8.e.a ✓ 10
7.c even 3 1 inner 252.8.k.c 10
12.b even 2 1 112.8.i.d 10
21.c even 2 1 196.8.e.f 10
21.g even 6 1 196.8.a.e 5
21.g even 6 1 196.8.e.f 10
21.h odd 6 1 28.8.e.a ✓ 10
21.h odd 6 1 196.8.a.d 5
84.n even 6 1 112.8.i.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.8.e.a ✓ 10 3.b odd 2 1
28.8.e.a ✓ 10 21.h odd 6 1
112.8.i.d 10 12.b even 2 1
112.8.i.d 10 84.n even 6 1
196.8.a.d 5 21.h odd 6 1
196.8.a.e 5 21.g even 6 1
196.8.e.f 10 21.c even 2 1
196.8.e.f 10 21.g even 6 1
252.8.k.c 10 1.a even 1 1 trivial
252.8.k.c 10 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 249 T_{5}^{9} + 307495 T_{5}^{8} + 8629014 T_{5}^{7} + 62267917501 T_{5}^{6} + \cdots + 70\!\cdots\!25 \) 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
$5$ \( T^{10} + \cdots + 70\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 37\!\cdots\!43 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 23\!\cdots\!09 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots - 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 51\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 63\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 28\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 63\!\cdots\!29 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 14\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 90\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 47\!\cdots\!12)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 93\!\cdots\!89 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 76\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 27\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 31\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 21\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 78\!\cdots\!69 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 34\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 47\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 58\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 95\!\cdots\!40)^{2} \) Copy content Toggle raw display
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