↑

Properties

Label 252.3.g.b
Level $252$
Weight $3$
Character orbit 252.g
Analytic conductor $6.867$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
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: \(6.86650266188\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.489494783471841.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 7 x^{10} - 11 x^{9} + 18 x^{8} - 22 x^{7} + 33 x^{6} - 44 x^{5} + 72 x^{4} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{19} \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{5} q^{4} + (\beta_{8} - \beta_{3}) q^{5} - \beta_1 q^{7} + ( - \beta_{7} + \beta_{2} - \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{5} q^{4} + (\beta_{8} - \beta_{3}) q^{5} - \beta_1 q^{7} + ( - \beta_{7} + \beta_{2} - \beta_1 + 1) q^{8} + ( - \beta_{11} - \beta_{10} + \beta_{8} + \cdots + 2) q^{10}+ \cdots + 7 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + 2 q^{4} - 8 q^{5} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} + 2 q^{4} - 8 q^{5} + 10 q^{8} + 28 q^{10} - 24 q^{13} + 14 q^{14} - 14 q^{16} + 40 q^{17} + 20 q^{20} - 88 q^{22} + 180 q^{25} - 100 q^{26} + 14 q^{28} - 72 q^{29} - 142 q^{32} - 100 q^{34} - 88 q^{37} - 128 q^{38} - 28 q^{40} + 200 q^{41} + 40 q^{44} - 24 q^{46} - 84 q^{49} + 346 q^{50} - 364 q^{52} - 104 q^{53} - 98 q^{56} + 148 q^{58} + 104 q^{61} - 64 q^{62} - 70 q^{64} - 176 q^{65} - 188 q^{68} - 84 q^{70} + 312 q^{73} - 4 q^{74} + 432 q^{76} + 224 q^{77} + 564 q^{80} + 332 q^{82} + 352 q^{85} - 160 q^{86} + 328 q^{88} + 552 q^{89} - 232 q^{92} - 144 q^{94} - 264 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 7 x^{10} - 11 x^{9} + 18 x^{8} - 22 x^{7} + 33 x^{6} - 44 x^{5} + 72 x^{4} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 31 \nu^{11} + 23 \nu^{10} + 89 \nu^{9} + 43 \nu^{8} - 66 \nu^{7} + 386 \nu^{6} + 551 \nu^{5} + \cdots + 2608 ) / 1264 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 33 \nu^{11} - 475 \nu^{10} + 635 \nu^{9} - 2207 \nu^{8} + 1734 \nu^{7} - 4482 \nu^{6} + 2401 \nu^{5} + \cdots - 20832 ) / 1264 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 265 \nu^{11} + 731 \nu^{10} - 1403 \nu^{9} + 2191 \nu^{8} - 2606 \nu^{7} + 4162 \nu^{6} + \cdots + 14688 ) / 2528 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 191 \nu^{11} - 157 \nu^{10} - 319 \nu^{9} - 637 \nu^{8} - 628 \nu^{7} - 1206 \nu^{6} - 2243 \nu^{5} + \cdots - 8240 ) / 1264 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 423 \nu^{11} + 1521 \nu^{10} - 2193 \nu^{9} + 4877 \nu^{8} - 5134 \nu^{7} + 9534 \nu^{6} + \cdots + 39968 ) / 2528 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 443 \nu^{11} + 135 \nu^{10} + 1017 \nu^{9} + 747 \nu^{8} + 1330 \nu^{7} + 2458 \nu^{6} + 3083 \nu^{5} + \cdots + 13824 ) / 2528 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 799 \nu^{11} - 1609 \nu^{10} + 2457 \nu^{9} - 4437 \nu^{8} + 6046 \nu^{7} - 8318 \nu^{6} + \cdots - 27744 ) / 2528 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 897 \nu^{11} + 2311 \nu^{10} - 4879 \nu^{9} + 6931 \nu^{8} - 10506 \nu^{7} + 12378 \nu^{6} + \cdots + 39968 ) / 2528 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 973 \nu^{11} - 1959 \nu^{10} + 3823 \nu^{9} - 6163 \nu^{8} + 9070 \nu^{7} - 10290 \nu^{6} + \cdots - 33248 ) / 2528 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32 \nu^{11} - 68 \nu^{10} + 125 \nu^{9} - 213 \nu^{8} + 281 \nu^{7} - 389 \nu^{6} + 528 \nu^{5} + \cdots - 1449 ) / 79 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 443 \nu^{11} + 734 \nu^{10} - 1728 \nu^{9} + 2176 \nu^{8} - 3937 \nu^{7} + 3704 \nu^{6} + \cdots + 12088 ) / 632 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - 2\beta_{10} + \beta_{8} - 2\beta_{5} - \beta_{3} + \beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{8} - 2\beta_{6} + 2\beta_{5} - 3\beta_{3} + \beta _1 - 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{11} + \beta_{10} - 3 \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{4} + \cdots - 1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{11} - 4 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + \cdots - 12 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4 \beta_{11} + 3 \beta_{10} + \beta_{9} - 4 \beta_{8} - 3 \beta_{6} - \beta_{5} - 5 \beta_{4} + \cdots - 15 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - \beta_{11} + 2 \beta_{10} + 3 \beta_{9} + \beta_{8} - 4 \beta_{7} + \beta_{6} - \beta_{5} + 5 \beta_{4} + \cdots - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2 \beta_{11} + \beta_{10} + 5 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} + \beta_{6} + 3 \beta_{5} + \cdots + 11 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3 \beta_{11} + 6 \beta_{10} - 22 \beta_{9} - 5 \beta_{8} - 4 \beta_{6} - 12 \beta_{5} - 10 \beta_{4} + \cdots + 16 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -\beta_{11} + \beta_{9} + 4\beta_{8} - 11\beta_{7} + 3\beta_{6} - 33\beta_{3} - 2\beta_{2} - 5\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 10 \beta_{11} - 5 \beta_{10} - 5 \beta_{9} + 16 \beta_{8} - 10 \beta_{7} + 21 \beta_{6} + 7 \beta_{5} + \cdots + 85 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7 \beta_{11} - 34 \beta_{10} + 10 \beta_{9} + 33 \beta_{8} + 52 \beta_{7} + 54 \beta_{6} - 24 \beta_{5} + \cdots + 2 ) / 8 \) 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
0.754714 − 1.19600i
0.754714 + 1.19600i
−1.15503 + 0.816025i
−1.15503 − 0.816025i
−0.434363 + 1.34586i
−0.434363 − 1.34586i
1.10978 − 0.876576i
1.10978 + 0.876576i
0.0311486 − 1.41387i
0.0311486 + 1.41387i
1.19375 + 0.758257i
1.19375 − 0.758257i
−1.98491 − 0.245189i 0 3.87976 + 0.973359i −0.424396 0 − 2.64575i −7.46234 − 2.88331i 0 0.842389 + 0.104057i
127.2 −1.98491 + 0.245189i 0 3.87976 − 0.973359i −0.424396 0 2.64575i −7.46234 + 2.88331i 0 0.842389 − 0.104057i
127.3 −1.42562 − 1.40272i 0 0.0647610 + 3.99948i −1.94731 0 − 2.64575i 5.51781 − 5.79256i 0 2.77611 + 2.73152i
127.4 −1.42562 + 1.40272i 0 0.0647610 − 3.99948i −1.94731 0 2.64575i 5.51781 + 5.79256i 0 2.77611 − 2.73152i
127.5 −0.913644 − 1.77912i 0 −2.33051 + 3.25096i −8.22808 0 2.64575i 7.91309 + 1.17604i 0 7.51753 + 14.6387i
127.6 −0.913644 + 1.77912i 0 −2.33051 − 3.25096i −8.22808 0 − 2.64575i 7.91309 − 1.17604i 0 7.51753 − 14.6387i
127.7 −0.0345996 − 1.99970i 0 −3.99761 + 0.138378i 6.29204 0 2.64575i 0.415030 + 7.98923i 0 −0.217702 − 12.5822i
127.8 −0.0345996 + 1.99970i 0 −3.99761 − 0.138378i 6.29204 0 − 2.64575i 0.415030 − 7.98923i 0 −0.217702 + 12.5822i
127.9 1.51951 − 1.30042i 0 0.617841 − 3.95200i −7.86764 0 2.64575i −4.20042 − 6.80856i 0 −11.9550 + 10.2312i
127.10 1.51951 + 1.30042i 0 0.617841 + 3.95200i −7.86764 0 − 2.64575i −4.20042 + 6.80856i 0 −11.9550 − 10.2312i
127.11 1.83926 − 0.785573i 0 2.76575 − 2.88975i 8.17539 0 − 2.64575i 2.81683 − 7.48769i 0 15.0367 − 6.42236i
127.12 1.83926 + 0.785573i 0 2.76575 + 2.88975i 8.17539 0 2.64575i 2.81683 + 7.48769i 0 15.0367 + 6.42236i
\(n\): 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 252.3.g.b 12
3.b odd 2 1 84.3.g.a ✓ 12
4.b odd 2 1 inner 252.3.g.b 12
12.b even 2 1 84.3.g.a ✓ 12
21.c even 2 1 588.3.g.d 12
24.f even 2 1 1344.3.m.e 12
24.h odd 2 1 1344.3.m.e 12
84.h odd 2 1 588.3.g.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.g.a ✓ 12 3.b odd 2 1
84.3.g.a ✓ 12 12.b even 2 1
252.3.g.b 12 1.a even 1 1 trivial
252.3.g.b 12 4.b odd 2 1 inner
588.3.g.d 12 21.c even 2 1
588.3.g.d 12 84.h odd 2 1
1344.3.m.e 12 24.f even 2 1
1344.3.m.e 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 4T_{5}^{5} - 112T_{5}^{4} - 384T_{5}^{3} + 2976T_{5}^{2} + 7808T_{5} + 2752 \) acting on \(S_{3}^{\mathrm{new}}(252, [\chi])\). 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
$5$ \( (T^{6} + 4 T^{5} + \cdots + 2752)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 8305770496 \) Copy content Toggle raw display
$13$ \( (T^{6} + 12 T^{5} + \cdots - 1102016)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 20 T^{5} + \cdots + 39448000)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 4294967296 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{6} + 36 T^{5} + \cdots - 12566720)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 128544076201984 \) Copy content Toggle raw display
$37$ \( (T^{6} + 44 T^{5} + \cdots - 155225024)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 100 T^{5} + \cdots + 8374720)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 49\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + 52 T^{5} + \cdots - 11239563200)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 90\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{6} - 52 T^{5} + \cdots + 44445760)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} - 156 T^{5} + \cdots + 11256899392)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 84\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} - 276 T^{5} + \cdots + 733352728000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 132 T^{5} + \cdots - 290193606848)^{2} \) Copy content Toggle raw display
show more
show less