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Properties

Label 252.2.i.b
Level $252$
Weight $2$
Character orbit 252.i
Analytic conductor $2.012$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,2,Mod(25,252)]
 
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, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.i (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: \(2.01223013094\)
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: \( x^{14} - 5x^{12} - 3x^{11} + 7x^{10} + 30x^{9} - 117x^{7} + 270x^{5} + 189x^{4} - 243x^{3} - 1215x^{2} + 2187 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{3} - \beta_{9} q^{5} + \beta_{10} q^{7} + (\beta_{12} - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1) q^{3} - \beta_{9} q^{5} + \beta_{10} q^{7} + (\beta_{12} - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{9}+ \cdots + (2 \beta_{13} + \beta_{12} - \beta_{11} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 3 q^{3} - 2 q^{5} + 6 q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 3 q^{3} - 2 q^{5} + 6 q^{7} - 5 q^{9} + 2 q^{11} + 2 q^{13} + 7 q^{15} + 2 q^{17} + 7 q^{19} - 11 q^{21} + 11 q^{23} - 9 q^{25} + 9 q^{27} + q^{29} + 2 q^{31} - 4 q^{33} - 19 q^{35} + 10 q^{37} - 2 q^{39} - 33 q^{41} + 7 q^{43} - 10 q^{45} + 6 q^{47} - 4 q^{49} - 13 q^{51} - 15 q^{53} - 28 q^{55} - 18 q^{57} + 28 q^{59} + 20 q^{61} + 33 q^{63} - 30 q^{65} - 12 q^{67} - 43 q^{69} + 2 q^{71} + 21 q^{73} - 44 q^{75} - 47 q^{77} + 20 q^{79} - 29 q^{81} - 25 q^{83} + 8 q^{85} + 28 q^{87} - 6 q^{89} + 2 q^{91} + 22 q^{93} + 56 q^{95} - 18 q^{97} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 5x^{12} - 3x^{11} + 7x^{10} + 30x^{9} - 117x^{7} + 270x^{5} + 189x^{4} - 243x^{3} - 1215x^{2} + 2187 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35 \nu^{13} - 72 \nu^{12} - 157 \nu^{11} - 312 \nu^{10} + 290 \nu^{9} + 1383 \nu^{8} - 1143 \nu^{7} + \cdots + 46656 ) / 43011 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8 \nu^{13} + 2 \nu^{12} - 23 \nu^{11} + 5 \nu^{10} + 37 \nu^{9} - 127 \nu^{8} + 78 \nu^{7} + \cdots - 8505 ) / 4779 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 28 \nu^{13} - 15 \nu^{12} - 191 \nu^{11} - 135 \nu^{10} + 289 \nu^{9} + 546 \nu^{8} + 498 \nu^{7} + \cdots - 40581 ) / 14337 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26 \nu^{13} + 7 \nu^{12} - 70 \nu^{11} - 266 \nu^{10} - 301 \nu^{9} + 955 \nu^{8} + 846 \nu^{7} + \cdots - 30618 ) / 14337 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 56 \nu^{13} + 441 \nu^{12} - 62 \nu^{11} - 1227 \nu^{10} - 761 \nu^{9} + 165 \nu^{8} + \cdots - 293058 ) / 43011 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 83 \nu^{13} + 423 \nu^{12} + 406 \nu^{11} - 354 \nu^{10} - 2237 \nu^{9} - 2364 \nu^{8} + \cdots - 237654 ) / 43011 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 97 \nu^{13} - 675 \nu^{12} + \nu^{11} + 1491 \nu^{10} + 2002 \nu^{9} + 1209 \nu^{8} - 10584 \nu^{7} + \cdots + 454896 ) / 43011 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 73 \nu^{13} - 360 \nu^{12} - 40 \nu^{11} + 804 \nu^{10} + 1703 \nu^{9} - 417 \nu^{8} + \cdots + 239841 ) / 43011 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5 \nu^{13} + 203 \nu^{12} + 68 \nu^{11} - 463 \nu^{10} - 760 \nu^{9} - 247 \nu^{8} + 3780 \nu^{7} + \cdots - 116397 ) / 14337 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 239 \nu^{13} - 324 \nu^{12} - 385 \nu^{11} + 93 \nu^{10} + 1133 \nu^{9} + 3687 \nu^{8} + \cdots + 267543 ) / 43011 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 179 \nu^{13} + 471 \nu^{12} - 769 \nu^{11} - 2352 \nu^{10} - 763 \nu^{9} + 5184 \nu^{8} + \cdots - 324405 ) / 43011 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 500 \nu^{13} - 81 \nu^{12} - 1096 \nu^{11} - 1392 \nu^{10} - 145 \nu^{9} + 10167 \nu^{8} + \cdots + 97686 ) / 43011 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{12} - \beta_{11} + \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5 \beta_{13} - 9 \beta_{12} - 2 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} - \beta_{8} + 6 \beta_{6} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3 \beta_{13} + 5 \beta_{12} - 4 \beta_{11} - 2 \beta_{10} - 18 \beta_{9} + 10 \beta_{8} - 10 \beta_{7} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13 \beta_{13} - 13 \beta_{12} - 14 \beta_{11} - 22 \beta_{10} - \beta_{9} - 8 \beta_{8} + 2 \beta_{7} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 9 \beta_{13} + 10 \beta_{12} + 14 \beta_{11} + 10 \beta_{10} + 22 \beta_{9} + 19 \beta_{8} - 9 \beta_{7} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 21 \beta_{13} + 22 \beta_{12} - 65 \beta_{11} - 46 \beta_{10} - 28 \beta_{9} - 6 \beta_{8} - 32 \beta_{7} + \cdots + 57 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 54 \beta_{13} + 66 \beta_{12} - 10 \beta_{11} - 53 \beta_{10} - 115 \beta_{9} + 31 \beta_{8} + \cdots - 242 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 77 \beta_{13} + 81 \beta_{12} + 7 \beta_{11} - 58 \beta_{10} + 167 \beta_{9} - 19 \beta_{8} + 135 \beta_{7} + \cdots - 109 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 51 \beta_{13} - 310 \beta_{12} + 329 \beta_{11} + 367 \beta_{10} - 117 \beta_{9} - 62 \beta_{8} + \cdots - 457 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(-1 + \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
1.64515 − 0.541745i
−0.473632 − 1.66604i
1.68442 + 0.403398i
1.13119 + 1.31165i
−1.73040 − 0.0755709i
−1.58203 + 0.705117i
−0.674693 + 1.59524i
1.64515 + 0.541745i
−0.473632 + 1.66604i
1.68442 − 0.403398i
1.13119 − 1.31165i
−1.73040 + 0.0755709i
−1.58203 − 0.705117i
−0.674693 − 1.59524i
0 −1.29174 − 1.15387i 0 0.381918 − 0.661502i 0 2.62892 − 0.297968i 0 0.337180 + 2.98099i 0
25.2 0 −1.20601 + 1.24319i 0 0.951504 − 1.64805i 0 2.11495 + 1.58965i 0 −0.0910656 − 2.99862i 0
25.3 0 −0.492857 − 1.66045i 0 −1.80173 + 3.12069i 0 −1.02133 + 2.44067i 0 −2.51418 + 1.63673i 0
25.4 0 0.570327 − 1.63546i 0 0.764702 − 1.32450i 0 −1.91978 − 1.82056i 0 −2.34945 − 1.86549i 0
25.5 0 0.799754 + 1.53636i 0 −0.483929 + 0.838189i 0 −1.52054 + 2.16517i 0 −1.72079 + 2.45742i 0
25.6 0 1.40166 + 1.01752i 0 1.26013 − 2.18261i 0 0.527655 − 2.59260i 0 0.929318 + 2.85243i 0
25.7 0 1.71886 − 0.213318i 0 −2.07260 + 3.58985i 0 2.19013 − 1.48437i 0 2.90899 − 0.733330i 0
121.1 0 −1.29174 + 1.15387i 0 0.381918 + 0.661502i 0 2.62892 + 0.297968i 0 0.337180 − 2.98099i 0
121.2 0 −1.20601 − 1.24319i 0 0.951504 + 1.64805i 0 2.11495 − 1.58965i 0 −0.0910656 + 2.99862i 0
121.3 0 −0.492857 + 1.66045i 0 −1.80173 − 3.12069i 0 −1.02133 − 2.44067i 0 −2.51418 − 1.63673i 0
121.4 0 0.570327 + 1.63546i 0 0.764702 + 1.32450i 0 −1.91978 + 1.82056i 0 −2.34945 + 1.86549i 0
121.5 0 0.799754 − 1.53636i 0 −0.483929 − 0.838189i 0 −1.52054 − 2.16517i 0 −1.72079 − 2.45742i 0
121.6 0 1.40166 − 1.01752i 0 1.26013 + 2.18261i 0 0.527655 + 2.59260i 0 0.929318 − 2.85243i 0
121.7 0 1.71886 + 0.213318i 0 −2.07260 − 3.58985i 0 2.19013 + 1.48437i 0 2.90899 + 0.733330i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.i.b ✓ 14
3.b odd 2 1 756.2.i.b 14
4.b odd 2 1 1008.2.q.j 14
7.b odd 2 1 1764.2.i.i 14
7.c even 3 1 252.2.l.b yes 14
7.c even 3 1 1764.2.j.g 14
7.d odd 6 1 1764.2.j.h 14
7.d odd 6 1 1764.2.l.i 14
9.c even 3 1 252.2.l.b yes 14
9.c even 3 1 2268.2.k.e 14
9.d odd 6 1 756.2.l.b 14
9.d odd 6 1 2268.2.k.f 14
12.b even 2 1 3024.2.q.j 14
21.c even 2 1 5292.2.i.i 14
21.g even 6 1 5292.2.j.g 14
21.g even 6 1 5292.2.l.i 14
21.h odd 6 1 756.2.l.b 14
21.h odd 6 1 5292.2.j.h 14
28.g odd 6 1 1008.2.t.j 14
36.f odd 6 1 1008.2.t.j 14
36.h even 6 1 3024.2.t.j 14
63.g even 3 1 1764.2.j.g 14
63.g even 3 1 2268.2.k.e 14
63.h even 3 1 inner 252.2.i.b ✓ 14
63.i even 6 1 5292.2.i.i 14
63.j odd 6 1 756.2.i.b 14
63.k odd 6 1 1764.2.j.h 14
63.l odd 6 1 1764.2.l.i 14
63.n odd 6 1 2268.2.k.f 14
63.n odd 6 1 5292.2.j.h 14
63.o even 6 1 5292.2.l.i 14
63.s even 6 1 5292.2.j.g 14
63.t odd 6 1 1764.2.i.i 14
84.n even 6 1 3024.2.t.j 14
252.u odd 6 1 1008.2.q.j 14
252.bb even 6 1 3024.2.q.j 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.b ✓ 14 1.a even 1 1 trivial
252.2.i.b ✓ 14 63.h even 3 1 inner
252.2.l.b yes 14 7.c even 3 1
252.2.l.b yes 14 9.c even 3 1
756.2.i.b 14 3.b odd 2 1
756.2.i.b 14 63.j odd 6 1
756.2.l.b 14 9.d odd 6 1
756.2.l.b 14 21.h odd 6 1
1008.2.q.j 14 4.b odd 2 1
1008.2.q.j 14 252.u odd 6 1
1008.2.t.j 14 28.g odd 6 1
1008.2.t.j 14 36.f odd 6 1
1764.2.i.i 14 7.b odd 2 1
1764.2.i.i 14 63.t odd 6 1
1764.2.j.g 14 7.c even 3 1
1764.2.j.g 14 63.g even 3 1
1764.2.j.h 14 7.d odd 6 1
1764.2.j.h 14 63.k odd 6 1
1764.2.l.i 14 7.d odd 6 1
1764.2.l.i 14 63.l odd 6 1
2268.2.k.e 14 9.c even 3 1
2268.2.k.e 14 63.g even 3 1
2268.2.k.f 14 9.d odd 6 1
2268.2.k.f 14 63.n odd 6 1
3024.2.q.j 14 12.b even 2 1
3024.2.q.j 14 252.bb even 6 1
3024.2.t.j 14 36.h even 6 1
3024.2.t.j 14 84.n even 6 1
5292.2.i.i 14 21.c even 2 1
5292.2.i.i 14 63.i even 6 1
5292.2.j.g 14 21.g even 6 1
5292.2.j.g 14 63.s even 6 1
5292.2.j.h 14 21.h odd 6 1
5292.2.j.h 14 63.n odd 6 1
5292.2.l.i 14 21.g even 6 1
5292.2.l.i 14 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 2 T_{5}^{13} + 24 T_{5}^{12} - 16 T_{5}^{11} + 295 T_{5}^{10} - 357 T_{5}^{9} + 2670 T_{5}^{8} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 3 T^{13} + \cdots + 2187 \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( T^{14} - 6 T^{13} + \cdots + 823543 \) Copy content Toggle raw display
$11$ \( T^{14} - 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 150626529 \) Copy content Toggle raw display
$17$ \( T^{14} - 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$19$ \( T^{14} - 7 T^{13} + \cdots + 4084441 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 105822369 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 145660761 \) Copy content Toggle raw display
$31$ \( (T^{7} - T^{6} + \cdots - 117504)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 1566893056 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 1108290681 \) Copy content Toggle raw display
$43$ \( T^{14} - 7 T^{13} + \cdots + 4084441 \) Copy content Toggle raw display
$47$ \( (T^{7} - 3 T^{6} + \cdots - 11664)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 952401321 \) Copy content Toggle raw display
$59$ \( (T^{7} - 14 T^{6} + \cdots + 26244)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 10 T^{6} + \cdots - 12192)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} + 6 T^{6} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$71$ \( (T^{7} - T^{6} - 116 T^{5} + \cdots - 972)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 2748590329 \) Copy content Toggle raw display
$79$ \( (T^{7} - 10 T^{6} + \cdots - 233232)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 901054679121 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 16524331209 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 767677849 \) Copy content Toggle raw display
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