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Properties

Label 252.3.y.c
Level $252$
Weight $3$
Character orbit 252.y
Analytic conductor $6.867$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,3,Mod(163,252)]
 
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([3, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.163");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 252.y (of order \(6\), 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: \(6.86650266188\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 4 x^{10} + 3 x^{9} + 86 x^{8} - 163 x^{7} + 155 x^{6} - 166 x^{5} + 164 x^{4} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{9} - \beta_{5} - 1) q^{4} + ( - \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{11} + 2 \beta_{10} - 2 \beta_{8} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{9} - \beta_{5} - 1) q^{4} + ( - \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{5}+ \cdots + (6 \beta_{11} - 12 \beta_{10} + \cdots + 46) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 4 q^{4} + 2 q^{5} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} - 4 q^{4} + 2 q^{5} + 8 q^{8} - 2 q^{10} - 24 q^{13} - 2 q^{14} + 16 q^{16} + 2 q^{17} - 152 q^{20} + 44 q^{22} - 56 q^{26} + 8 q^{28} - 72 q^{29} + 112 q^{32} - 316 q^{34} + 86 q^{37} + 2 q^{38} - 148 q^{40} - 8 q^{41} - 64 q^{44} + 162 q^{46} + 108 q^{49} - 208 q^{50} - 64 q^{52} + 74 q^{53} - 16 q^{56} - 176 q^{58} + 86 q^{61} + 532 q^{62} - 160 q^{64} + 140 q^{65} + 68 q^{68} + 90 q^{70} - 234 q^{73} - 290 q^{74} + 576 q^{76} + 262 q^{77} + 272 q^{82} + 268 q^{85} + 16 q^{86} - 188 q^{88} - 6 q^{89} + 448 q^{92} + 102 q^{94} + 744 q^{97} + 190 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 4 x^{10} + 3 x^{9} + 86 x^{8} - 163 x^{7} + 155 x^{6} - 166 x^{5} + 164 x^{4} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 548052 \nu^{11} + 1422116 \nu^{10} - 6191612 \nu^{9} - 21007396 \nu^{8} + 23867696 \nu^{7} + \cdots - 9243685 ) / 5128417 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2016555 \nu^{11} + 4608465 \nu^{10} + 11867940 \nu^{9} + 1459025 \nu^{8} - 175954938 \nu^{7} + \cdots + 13204764 ) / 10256834 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6080681 \nu^{11} + 174798 \nu^{10} - 64448111 \nu^{9} - 87898803 \nu^{8} + 490503169 \nu^{7} + \cdots + 199145960 ) / 20513668 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8092731 \nu^{11} + 13062376 \nu^{10} + 58450763 \nu^{9} + 38422753 \nu^{8} - 688812905 \nu^{7} + \cdots - 38518016 ) / 20513668 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8864783 \nu^{11} - 20455402 \nu^{10} - 48834261 \nu^{9} - 8785933 \nu^{8} + 751559539 \nu^{7} + \cdots - 70375800 ) / 20513668 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2781796 \nu^{11} + 6284192 \nu^{10} + 15510516 \nu^{9} + 3410783 \nu^{8} - 234558069 \nu^{7} + \cdots + 15160294 ) / 5128417 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11588911 \nu^{11} - 2778978 \nu^{10} - 108332469 \nu^{9} - 167711837 \nu^{8} + 897253275 \nu^{7} + \cdots + 51100076 ) / 20513668 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7577433 \nu^{11} - 23003531 \nu^{10} - 36322258 \nu^{9} + 36406939 \nu^{8} + 694698148 \nu^{7} + \cdots - 10748502 ) / 10256834 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18854063 \nu^{11} + 54731322 \nu^{10} + 82312181 \nu^{9} - 56986907 \nu^{8} - 1629791971 \nu^{7} + \cdots + 162375652 ) / 20513668 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13906733 \nu^{11} - 32333536 \nu^{10} - 77826224 \nu^{9} - 8906454 \nu^{8} + 1190477049 \nu^{7} + \cdots - 51147512 ) / 10256834 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6254493 \nu^{11} + 16328866 \nu^{10} + 30974575 \nu^{9} - 6021165 \nu^{8} - 537499493 \nu^{7} + \cdots + 44275732 ) / 2930524 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 2\beta_{10} + 2\beta_{6} + 3\beta_{5} + 2\beta_{4} + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{9} + 2\beta_{8} + 2\beta_{7} - 11\beta_{5} + 2\beta_{4} - \beta_{3} - 6\beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{11} + 18\beta_{10} + 2\beta_{8} + 6\beta_{6} - 9\beta_{3} + 6\beta_{2} + 4\beta _1 + 25 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{11} + 38\beta_{10} - 14\beta_{9} + 18\beta_{7} + 54\beta_{6} - 75\beta_{5} + 38\beta_{4} - 75 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -52\beta_{9} + 38\beta_{8} + 38\beta_{7} - 293\beta_{5} - 122\beta_{4} - 81\beta_{3} + 2\beta_{2} + 52\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 71\beta_{11} + 462\beta_{10} - 122\beta_{8} + 482\beta_{6} - 71\beta_{3} + 482\beta_{2} - 70\beta _1 - 379 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 651 \beta_{11} - 622 \beta_{10} - 532 \beta_{9} + 462 \beta_{7} + 462 \beta_{6} - 2987 \beta_{5} + \cdots - 2987 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 90 \beta_{9} - 622 \beta_{8} - 622 \beta_{7} + 391 \beta_{5} - 4734 \beta_{4} - 1283 \beta_{3} + \cdots - 90 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4513 \beta_{11} - 802 \beta_{10} - 4734 \beta_{8} + 7942 \beta_{6} + 4513 \beta_{3} + 7942 \beta_{2} + \cdots - 26985 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 15935 \beta_{11} - 42942 \beta_{10} - 4022 \beta_{9} - 802 \beta_{7} - 26154 \beta_{6} - 23505 \beta_{5} + \cdots - 23505 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 38920 \beta_{9} - 42942 \beta_{8} - 42942 \beta_{7} + 216715 \beta_{5} - 35802 \beta_{4} + \cdots - 38920 \beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{5}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.121721 + 0.507075i
−0.407369 + 0.812545i
0.378279 + 0.358951i
−2.29733 + 1.90372i
2.79733 − 1.03769i
0.907369 + 0.0534805i
0.121721 − 0.507075i
−0.407369 − 0.812545i
0.378279 − 0.358951i
−2.29733 − 1.90372i
2.79733 + 1.03769i
0.907369 − 0.0534805i
−1.78207 + 0.907869i 0 2.35155 − 3.23577i 1.12649 − 1.95113i 0 −6.84270 + 1.47562i −1.25297 + 7.90127i 0 −0.236107 + 4.49975i
163.2 −1.19654 − 1.60259i 0 −1.13656 + 3.83513i −3.25304 + 5.63443i 0 2.39669 − 6.57692i 7.50608 − 2.76746i 0 12.9221 − 1.52857i
163.3 0.104798 + 1.99725i 0 −3.97803 + 0.418616i 1.12649 − 1.95113i 0 6.84270 − 1.47562i −1.25297 − 7.90127i 0 4.01495 + 2.04540i
163.4 0.371518 − 1.96519i 0 −3.72395 − 1.46021i 2.62655 − 4.54932i 0 −5.86799 − 3.81663i −4.25310 + 6.77577i 0 −7.96447 − 6.85183i
163.5 1.51615 − 1.30434i 0 0.597396 − 3.95514i 2.62655 − 4.54932i 0 5.86799 + 3.81663i −4.25310 − 6.77577i 0 −1.95163 − 10.3234i
163.6 1.98615 + 0.234945i 0 3.88960 + 0.933271i −3.25304 + 5.63443i 0 −2.39669 + 6.57692i 7.50608 + 2.76746i 0 −7.78481 + 10.4265i
235.1 −1.78207 − 0.907869i 0 2.35155 + 3.23577i 1.12649 + 1.95113i 0 −6.84270 − 1.47562i −1.25297 − 7.90127i 0 −0.236107 − 4.49975i
235.2 −1.19654 + 1.60259i 0 −1.13656 − 3.83513i −3.25304 − 5.63443i 0 2.39669 + 6.57692i 7.50608 + 2.76746i 0 12.9221 + 1.52857i
235.3 0.104798 − 1.99725i 0 −3.97803 − 0.418616i 1.12649 + 1.95113i 0 6.84270 + 1.47562i −1.25297 + 7.90127i 0 4.01495 − 2.04540i
235.4 0.371518 + 1.96519i 0 −3.72395 + 1.46021i 2.62655 + 4.54932i 0 −5.86799 + 3.81663i −4.25310 − 6.77577i 0 −7.96447 + 6.85183i
235.5 1.51615 + 1.30434i 0 0.597396 + 3.95514i 2.62655 + 4.54932i 0 5.86799 − 3.81663i −4.25310 + 6.77577i 0 −1.95163 + 10.3234i
235.6 1.98615 − 0.234945i 0 3.88960 − 0.933271i −3.25304 − 5.63443i 0 −2.39669 − 6.57692i 7.50608 − 2.76746i 0 −7.78481 − 10.4265i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.3.y.c 12
3.b odd 2 1 28.3.g.a ✓ 12
4.b odd 2 1 inner 252.3.y.c 12
7.c even 3 1 inner 252.3.y.c 12
12.b even 2 1 28.3.g.a ✓ 12
21.c even 2 1 196.3.g.i 12
21.g even 6 1 196.3.c.h 6
21.g even 6 1 196.3.g.i 12
21.h odd 6 1 28.3.g.a ✓ 12
21.h odd 6 1 196.3.c.i 6
24.f even 2 1 448.3.r.h 12
24.h odd 2 1 448.3.r.h 12
28.g odd 6 1 inner 252.3.y.c 12
84.h odd 2 1 196.3.g.i 12
84.j odd 6 1 196.3.c.h 6
84.j odd 6 1 196.3.g.i 12
84.n even 6 1 28.3.g.a ✓ 12
84.n even 6 1 196.3.c.i 6
168.s odd 6 1 448.3.r.h 12
168.v even 6 1 448.3.r.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.3.g.a ✓ 12 3.b odd 2 1
28.3.g.a ✓ 12 12.b even 2 1
28.3.g.a ✓ 12 21.h odd 6 1
28.3.g.a ✓ 12 84.n even 6 1
196.3.c.h 6 21.g even 6 1
196.3.c.h 6 84.j odd 6 1
196.3.c.i 6 21.h odd 6 1
196.3.c.i 6 84.n even 6 1
196.3.g.i 12 21.c even 2 1
196.3.g.i 12 21.g even 6 1
196.3.g.i 12 84.h odd 2 1
196.3.g.i 12 84.j odd 6 1
252.3.y.c 12 1.a even 1 1 trivial
252.3.y.c 12 4.b odd 2 1 inner
252.3.y.c 12 7.c even 3 1 inner
252.3.y.c 12 28.g odd 6 1 inner
448.3.r.h 12 24.f even 2 1
448.3.r.h 12 24.h odd 2 1
448.3.r.h 12 168.s odd 6 1
448.3.r.h 12 168.v even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(252, [\chi])\):

\( T_{5}^{6} - T_{5}^{5} + 38T_{5}^{4} - 117T_{5}^{3} + 1446T_{5}^{2} - 2849T_{5} + 5929 \) Copy content Toggle raw display
\( T_{11}^{12} - 101T_{11}^{10} + 9694T_{11}^{8} - 50857T_{11}^{6} + 239374T_{11}^{4} - 88725T_{11}^{2} + 30625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - T^{5} + 38 T^{4} + \cdots + 5929)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{12} - 101 T^{10} + \cdots + 30625 \) Copy content Toggle raw display
$13$ \( (T^{3} + 6 T^{2} + \cdots - 280)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} - T^{5} + \cdots + 17935225)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 28722900390625 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 234433177489 \) Copy content Toggle raw display
$29$ \( (T^{3} + 18 T^{2} + \cdots + 4408)^{4} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( (T^{6} - 43 T^{5} + \cdots + 1926771025)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 2 T^{2} + \cdots - 44296)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 6976 T^{4} + \cdots + 1101463552)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 43\!\cdots\!89 \) Copy content Toggle raw display
$53$ \( (T^{6} - 37 T^{5} + \cdots + 417180625)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{6} - 43 T^{5} + \cdots + 5716418449)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 49\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( (T^{6} + 11648 T^{4} + \cdots + 6884147200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 117 T^{5} + \cdots + 1997643025)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{6} + 2944 T^{4} + \cdots + 275365888)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 3 T^{5} + \cdots + 52287361)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 186 T^{2} + \cdots + 994280)^{4} \) Copy content Toggle raw display
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