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Properties

Label 252.12.k.c
Level $252$
Weight $12$
Character orbit 252.k
Analytic conductor $193.622$
Analytic rank $0$
Dimension $14$
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] = [252,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.37");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(193.622481501\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 21425626 x^{12} + 1160400425 x^{11} + 153589314802829 x^{10} + \cdots + 71\!\cdots\!52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{42}\cdot 3^{9}\cdot 7^{11} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{2} + 531 \beta_1 - 531) q^{5} + ( - \beta_{9} - 2 \beta_{4} + \cdots - 6806) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{2} + 531 \beta_1 - 531) q^{5} + ( - \beta_{9} - 2 \beta_{4} + \cdots - 6806) q^{7}+ \cdots + (199760 \beta_{12} + 70560 \beta_{11} + \cdots - 6871930380) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3719 q^{5} - 83356 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 3719 q^{5} - 83356 q^{7} + 72905 q^{11} - 2193572 q^{13} + 261533 q^{17} + 13374085 q^{19} + 11167331 q^{23} - 2989084 q^{25} + 30906484 q^{29} - 256257785 q^{31} + 64940645 q^{35} + 83028715 q^{37} - 979709316 q^{41} + 172751992 q^{43} + 1276652103 q^{47} + 766357550 q^{49} - 521040303 q^{53} + 511903374 q^{55} - 5829711443 q^{59} + 1751145379 q^{61} + 412905234 q^{65} + 6447767179 q^{67} + 24715621280 q^{71} + 10539220351 q^{73} - 188362769 q^{77} - 16799842619 q^{79} - 37081512184 q^{83} - 64191070650 q^{85} - 36562951407 q^{89} + 104005007176 q^{91} + 30995705909 q^{95} - 96236737820 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 21425626 x^{12} + 1160400425 x^{11} + 153589314802829 x^{10} + \cdots + 71\!\cdots\!52 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 39\!\cdots\!77 \nu^{13} + \cdots + 42\!\cdots\!04 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 39\!\cdots\!77 \nu^{13} + \cdots - 42\!\cdots\!04 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 38\!\cdots\!71 \nu^{13} + \cdots + 39\!\cdots\!72 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12\!\cdots\!27 \nu^{13} + \cdots - 18\!\cdots\!80 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 52\!\cdots\!31 \nu^{13} + \cdots + 56\!\cdots\!12 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19\!\cdots\!41 \nu^{13} + \cdots - 23\!\cdots\!84 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 41\!\cdots\!99 \nu^{13} + \cdots - 93\!\cdots\!40 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 50\!\cdots\!29 \nu^{13} + \cdots - 54\!\cdots\!64 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!61 \nu^{13} + \cdots + 12\!\cdots\!68 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 38\!\cdots\!31 \nu^{13} + \cdots + 36\!\cdots\!04 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 46\!\cdots\!93 \nu^{13} + \cdots - 47\!\cdots\!08 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 52\!\cdots\!99 \nu^{13} + \cdots + 59\!\cdots\!12 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!01 \nu^{13} + \cdots - 15\!\cdots\!20 ) / 15\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 4\beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 16 \beta_{12} + 2 \beta_{11} - 120 \beta_{10} + 193 \beta_{9} - 33 \beta_{8} + 201 \beta_{6} + \cdots + 24486422 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 396 \beta_{13} - 468265 \beta_{12} + 74395 \beta_{11} + 345433 \beta_{10} - 87482 \beta_{9} + \cdots - 7967630220 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 270568 \beta_{13} + 310913455 \beta_{12} + 190692435 \beta_{11} - 4530618159 \beta_{10} + \cdots + 694315656624444 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5319192460 \beta_{13} - 5148134718014 \beta_{12} + 819391355267 \beta_{11} + 3927273532835 \beta_{10} + \cdots - 42\!\cdots\!11 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 6034388239272 \beta_{13} + \cdots + 55\!\cdots\!91 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 30\!\cdots\!10 \beta_{13} + \cdots - 30\!\cdots\!39 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 46\!\cdots\!68 \beta_{13} + \cdots + 22\!\cdots\!43 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 67\!\cdots\!04 \beta_{13} + \cdots - 67\!\cdots\!86 ) / 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12\!\cdots\!36 \beta_{13} + \cdots + 39\!\cdots\!68 ) / 32 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 71\!\cdots\!64 \beta_{13} + \cdots - 70\!\cdots\!53 ) / 32 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 15\!\cdots\!00 \beta_{13} + \cdots + 33\!\cdots\!01 ) / 32 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 37\!\cdots\!98 \beta_{13} + \cdots - 35\!\cdots\!71 ) / 16 \) 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
2656.57 + 0.866025i
2018.45 + 0.866025i
95.3211 + 0.866025i
−374.521 + 0.866025i
−658.395 + 0.866025i
−700.345 + 0.866025i
−3036.58 + 0.866025i
2656.57 − 0.866025i
2018.45 − 0.866025i
95.3211 − 0.866025i
−374.521 − 0.866025i
−658.395 − 0.866025i
−700.345 − 0.866025i
−3036.58 − 0.866025i
0 0 0 −5578.64 + 9662.49i 0 −41531.9 − 15888.1i 0 0 0
37.2 0 0 0 −4302.39 + 7451.96i 0 17422.8 + 40911.8i 0 0 0
37.3 0 0 0 −456.142 + 790.062i 0 −41854.1 − 15018.7i 0 0 0
37.4 0 0 0 483.543 − 837.520i 0 19347.3 + 40037.6i 0 0 0
37.5 0 0 0 1051.29 − 1820.89i 0 39765.4 − 19900.8i 0 0 0
37.6 0 0 0 1135.19 − 1966.21i 0 2206.92 − 44412.3i 0 0 0
37.7 0 0 0 5807.66 − 10059.2i 0 −37034.4 + 24612.6i 0 0 0
109.1 0 0 0 −5578.64 − 9662.49i 0 −41531.9 + 15888.1i 0 0 0
109.2 0 0 0 −4302.39 − 7451.96i 0 17422.8 − 40911.8i 0 0 0
109.3 0 0 0 −456.142 − 790.062i 0 −41854.1 + 15018.7i 0 0 0
109.4 0 0 0 483.543 + 837.520i 0 19347.3 − 40037.6i 0 0 0
109.5 0 0 0 1051.29 + 1820.89i 0 39765.4 + 19900.8i 0 0 0
109.6 0 0 0 1135.19 + 1966.21i 0 2206.92 + 44412.3i 0 0 0
109.7 0 0 0 5807.66 + 10059.2i 0 −37034.4 − 24612.6i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.7
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 252.12.k.c 14
3.b odd 2 1 28.12.e.a ✓ 14
7.c even 3 1 inner 252.12.k.c 14
12.b even 2 1 112.12.i.d 14
21.c even 2 1 196.12.e.h 14
21.g even 6 1 196.12.a.f 7
21.g even 6 1 196.12.e.h 14
21.h odd 6 1 28.12.e.a ✓ 14
21.h odd 6 1 196.12.a.e 7
84.n even 6 1 112.12.i.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.12.e.a ✓ 14 3.b odd 2 1
28.12.e.a ✓ 14 21.h odd 6 1
112.12.i.d 14 12.b even 2 1
112.12.i.d 14 84.n even 6 1
196.12.a.e 7 21.h odd 6 1
196.12.a.f 7 21.g even 6 1
196.12.e.h 14 21.c even 2 1
196.12.e.h 14 21.g even 6 1
252.12.k.c 14 1.a even 1 1 trivial
252.12.k.c 14 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 3719 T_{5}^{13} + 179308460 T_{5}^{12} + 357832606589 T_{5}^{11} + \cdots + 22\!\cdots\!25 \) acting on \(S_{12}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 11\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 26\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( (T^{7} + \cdots + 11\!\cdots\!28)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 74\!\cdots\!29 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 54\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 87\!\cdots\!49 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots + 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 91\!\cdots\!29 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 13\!\cdots\!09 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots + 84\!\cdots\!20)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 66\!\cdots\!60)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 16\!\cdots\!69 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 15\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 52\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 25\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 43\!\cdots\!49 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots - 40\!\cdots\!28)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
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