↑

Properties

Label 252.5.g.a
Level $252$
Weight $5$
Character orbit 252.g
Analytic conductor $26.049$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,5,Mod(127,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.127");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 252.g (of order \(2\), degree \(1\), 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: \(26.0492306971\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 10 x^{10} - 29 x^{9} + 174 x^{8} - 96 x^{7} + 88 x^{6} - 3030 x^{5} - 399 x^{4} + \cdots + 117656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{3} - 3) q^{4} + ( - \beta_{9} - 2) q^{5} + (\beta_{5} - \beta_{2}) q^{7} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots - 15) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{3} - 3) q^{4} + ( - \beta_{9} - 2) q^{5} + (\beta_{5} - \beta_{2}) q^{7} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots - 15) q^{8}+ \cdots + 343 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 31 q^{4} - 24 q^{5} - 171 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} - 31 q^{4} - 24 q^{5} - 171 q^{8} + 52 q^{10} + 120 q^{13} - 147 q^{14} - 911 q^{16} + 648 q^{17} - 912 q^{20} - 628 q^{22} + 2340 q^{25} + 2280 q^{26} + 245 q^{28} - 24 q^{29} - 1443 q^{32} + 1634 q^{34} - 424 q^{37} - 2970 q^{38} - 4936 q^{40} - 7320 q^{41} + 2244 q^{44} - 7512 q^{46} - 4116 q^{49} + 5451 q^{50} - 3484 q^{52} + 7080 q^{53} - 3087 q^{56} + 4666 q^{58} + 9752 q^{61} + 5964 q^{62} + 6329 q^{64} + 11280 q^{65} + 7314 q^{68} - 7644 q^{70} + 13752 q^{73} + 2838 q^{74} + 11202 q^{76} - 4704 q^{77} + 49272 q^{80} - 11950 q^{82} + 23152 q^{85} - 30612 q^{86} + 20740 q^{88} + 12552 q^{89} - 47880 q^{92} + 71340 q^{94} + 19704 q^{97} + 1029 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 10 x^{10} - 29 x^{9} + 174 x^{8} - 96 x^{7} + 88 x^{6} - 3030 x^{5} - 399 x^{4} + \cdots + 117656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 19 \nu^{11} - 265 \nu^{10} + 1793 \nu^{9} + 544 \nu^{8} - 10314 \nu^{7} - 21082 \nu^{6} + \cdots - 18232184 ) / 376832 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13 \nu^{11} + 89 \nu^{10} - 129 \nu^{9} - 144 \nu^{8} - 262 \nu^{7} - 1526 \nu^{6} + \cdots - 461320 ) / 94208 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{11} - 11 \nu^{10} - 49 \nu^{9} + 268 \nu^{8} - 314 \nu^{7} + 1742 \nu^{6} - 5858 \nu^{5} + \cdots + 237080 ) / 11776 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 173 \nu^{11} + 961 \nu^{10} - 481 \nu^{9} - 1128 \nu^{8} - 16382 \nu^{7} - 27614 \nu^{6} + \cdots - 12261704 ) / 376832 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 137 \nu^{11} - 1317 \nu^{10} + 4685 \nu^{9} - 8080 \nu^{8} + 16270 \nu^{7} - 21602 \nu^{6} + \cdots + 4389480 ) / 376832 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8 \nu^{11} + 91 \nu^{10} - 396 \nu^{9} + 827 \nu^{8} - 2197 \nu^{7} + 7925 \nu^{6} + \cdots + 856648 ) / 23552 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 55 \nu^{11} + 411 \nu^{10} - 1331 \nu^{9} + 3760 \nu^{8} - 9842 \nu^{7} + 1502 \nu^{6} + \cdots - 4219288 ) / 94208 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 233 \nu^{11} - 2053 \nu^{10} + 7981 \nu^{9} - 25232 \nu^{8} + 73934 \nu^{7} - 87586 \nu^{6} + \cdots + 7651688 ) / 376832 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 281 \nu^{11} - 2941 \nu^{10} + 12413 \nu^{9} - 35160 \nu^{8} + 111782 \nu^{7} - 271802 \nu^{6} + \cdots - 10514904 ) / 376832 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 309 \nu^{11} - 1933 \nu^{10} + 2425 \nu^{9} - 796 \nu^{8} + 38442 \nu^{7} - 31662 \nu^{6} + \cdots + 9335336 ) / 188416 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 109 \nu^{11} + 781 \nu^{10} - 1777 \nu^{9} + 3588 \nu^{8} - 18898 \nu^{7} + 25654 \nu^{6} + \cdots - 3596584 ) / 47104 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} - 7\beta_{2} + 16 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{11} + 8 \beta_{9} + 2 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} - 7 \beta_{5} + 16 \beta_{4} + \cdots + 90 ) / 56 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - \beta_{10} + 2 \beta_{9} + 13 \beta_{8} - 15 \beta_{5} + 25 \beta_{4} + 30 \beta_{3} + \cdots + 310 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6 \beta_{11} - 6 \beta_{10} + 5 \beta_{9} + 57 \beta_{8} + 35 \beta_{7} - 7 \beta_{6} - 12 \beta_{5} + \cdots + 194 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 57 \beta_{11} - 66 \beta_{10} - 28 \beta_{9} + 384 \beta_{8} + 373 \beta_{7} + 19 \beta_{6} + \cdots - 4510 ) / 56 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 99 \beta_{11} - 82 \beta_{10} - 146 \beta_{9} + 334 \beta_{8} + 423 \beta_{7} - 45 \beta_{6} + \cdots - 6918 ) / 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 107 \beta_{11} - 45 \beta_{10} - 193 \beta_{9} + 248 \beta_{8} + 503 \beta_{7} - 119 \beta_{6} + \cdots - 7924 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4489 \beta_{11} + 2296 \beta_{10} - 10776 \beta_{9} + 4306 \beta_{8} + 20287 \beta_{7} + \cdots - 421278 ) / 56 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4645 \beta_{11} + 11939 \beta_{10} - 19398 \beta_{9} - 6555 \beta_{8} + 14938 \beta_{7} + \cdots - 710550 ) / 28 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2792 \beta_{11} + 68886 \beta_{10} - 66155 \beta_{9} - 66767 \beta_{8} - 33747 \beta_{7} + \cdots - 1975818 ) / 28 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 54987 \beta_{11} + 626758 \beta_{10} - 439852 \beta_{9} - 703792 \beta_{8} - 874983 \beta_{7} + \cdots - 10069398 ) / 56 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−0.686439 + 2.27067i
−0.686439 − 2.27067i
−1.59179 + 1.13253i
−1.59179 − 1.13253i
3.16763 − 1.81697i
3.16763 + 1.81697i
−1.31008 − 1.63203i
−1.31008 + 1.63203i
3.63313 + 1.03277i
3.63313 − 1.03277i
−0.212454 − 2.55538i
−0.212454 + 2.55538i
−3.84703 − 1.09562i 0 13.5992 + 8.42973i 9.41925 0 − 18.5203i −43.0809 − 47.3290i 0 −36.2361 − 10.3199i
127.2 −3.84703 + 1.09562i 0 13.5992 − 8.42973i 9.41925 0 18.5203i −43.0809 + 47.3290i 0 −36.2361 + 10.3199i
127.3 −2.79410 − 2.86235i 0 −0.386047 + 15.9953i −36.3145 0 − 18.5203i 46.8629 − 43.5875i 0 101.466 + 103.945i
127.4 −2.79410 + 2.86235i 0 −0.386047 − 15.9953i −36.3145 0 18.5203i 46.8629 + 43.5875i 0 101.466 − 103.945i
127.5 −1.31981 − 3.77599i 0 −12.5162 + 9.96716i 0.440227 0 18.5203i 54.1549 + 34.1064i 0 −0.581015 − 1.66229i
127.6 −1.31981 + 3.77599i 0 −12.5162 − 9.96716i 0.440227 0 − 18.5203i 54.1549 − 34.1064i 0 −0.581015 + 1.66229i
127.7 1.00393 − 3.87197i 0 −13.9843 − 7.77433i 46.3749 0 − 18.5203i −44.1411 + 46.3418i 0 46.5569 − 179.562i
127.8 1.00393 + 3.87197i 0 −13.9843 + 7.77433i 46.3749 0 18.5203i −44.1411 − 46.3418i 0 46.5569 + 179.562i
127.9 2.68279 − 2.96693i 0 −1.60531 − 15.9193i −36.6030 0 18.5203i −51.5380 − 37.9452i 0 −98.1982 + 108.599i
127.10 2.68279 + 2.96693i 0 −1.60531 + 15.9193i −36.6030 0 − 18.5203i −51.5380 + 37.9452i 0 −98.1982 − 108.599i
127.11 2.77422 − 2.88161i 0 −0.607410 − 15.9885i 4.68319 0 − 18.5203i −47.7577 − 42.6052i 0 12.9922 − 13.4951i
127.12 2.77422 + 2.88161i 0 −0.607410 + 15.9885i 4.68319 0 18.5203i −47.7577 + 42.6052i 0 12.9922 + 13.4951i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.5.g.a 12
3.b odd 2 1 28.5.c.a ✓ 12
4.b odd 2 1 inner 252.5.g.a 12
12.b even 2 1 28.5.c.a ✓ 12
21.c even 2 1 196.5.c.f 12
24.f even 2 1 448.5.d.e 12
24.h odd 2 1 448.5.d.e 12
84.h odd 2 1 196.5.c.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.5.c.a ✓ 12 3.b odd 2 1
28.5.c.a ✓ 12 12.b even 2 1
196.5.c.f 12 21.c even 2 1
196.5.c.f 12 84.h odd 2 1
252.5.g.a 12 1.a even 1 1 trivial
252.5.g.a 12 4.b odd 2 1 inner
448.5.d.e 12 24.f even 2 1
448.5.d.e 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 12T_{5}^{5} - 2388T_{5}^{4} - 30480T_{5}^{3} + 792656T_{5}^{2} - 3062016T_{5} + 1197056 \) acting on \(S_{5}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{11} + \cdots + 16777216 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 12 T^{5} + \cdots + 1197056)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 343)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{6} - 60 T^{5} + \cdots + 73212423424)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 497696355050944)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 43\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 67\!\cdots\!92)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 23\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 58\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 29\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 25\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 88\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 57\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 75\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 23\!\cdots\!12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 98\!\cdots\!36)^{2} \) Copy content Toggle raw display
show more
show less