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Properties

Label 252.5.g.c
Level $252$
Weight $5$
Character orbit 252.g
Analytic conductor $26.049$
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] = [252,5,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: \( x^{12} + 16x^{10} + 195x^{8} + 1900x^{6} + 12480x^{4} + 65536x^{2} + 262144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{21}\cdot 7^{6} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + 5) q^{4} + (\beta_{8} - 4 \beta_1) q^{5} - \beta_{2} q^{7} + (2 \beta_{8} + \beta_{6} + 6 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + 5) q^{4} + (\beta_{8} - 4 \beta_1) q^{5} - \beta_{2} q^{7} + (2 \beta_{8} + \beta_{6} + 6 \beta_1) q^{8} + (\beta_{9} + \beta_{5} + 2 \beta_{3} + \cdots - 70) q^{10}+ \cdots - 343 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 62 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 62 q^{4} - 836 q^{10} + 264 q^{13} - 302 q^{16} - 628 q^{22} + 3156 q^{25} - 490 q^{28} - 1612 q^{34} + 6392 q^{37} + 8996 q^{40} + 7452 q^{46} - 4116 q^{49} + 3020 q^{52} - 4568 q^{58} - 9736 q^{61} + 6086 q^{64} + 9996 q^{70} - 30936 q^{73} - 11592 q^{76} + 13964 q^{82} + 33760 q^{85} + 4756 q^{88} + 32088 q^{94} + 85416 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 16x^{10} + 195x^{8} + 1900x^{6} + 12480x^{4} + 65536x^{2} + 262144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 23\nu^{11} + 688\nu^{9} + 3461\nu^{7} + 28276\nu^{5} + 118848\nu^{3} + 151552\nu ) / 974848 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\nu^{10} + 688\nu^{8} + 3461\nu^{6} + 28276\nu^{4} + 118848\nu^{2} - 335872 ) / 69632 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{10} + 16\nu^{8} + 241\nu^{6} + 2628\nu^{4} + 51648\nu^{2} + 170368 ) / 15232 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 261\nu^{10} - 10736\nu^{8} - 193841\nu^{6} - 2002340\nu^{4} - 10741568\nu^{2} - 62726144 ) / 487424 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -361\nu^{10} - 4176\nu^{8} - 75515\nu^{6} - 275596\nu^{4} - 1934272\nu^{2} - 7528448 ) / 487424 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 229\nu^{11} + 1552\nu^{9} + 2671\nu^{7} + 244444\nu^{5} - 2514752\nu^{3} - 13283328\nu ) / 1949696 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 111\nu^{11} - 464\nu^{9} - 6003\nu^{7} - 29292\nu^{5} - 257472\nu^{3} + 4091904\nu ) / 557056 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -1145\nu^{11} - 7760\nu^{9} - 13355\nu^{7} - 247372\nu^{5} - 99264\nu^{3} + 23523328\nu ) / 3899392 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2183\nu^{10} + 24240\nu^{8} + 264917\nu^{6} + 2030772\nu^{4} + 12292160\nu^{2} + 44752896 ) / 487424 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 139\nu^{11} + 1840\nu^{9} + 22241\nu^{7} + 185124\nu^{5} + 943424\nu^{3} + 4619264\nu ) / 121856 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 597\nu^{11} + 4944\nu^{9} + 58047\nu^{7} + 186588\nu^{5} - 95296\nu^{3} - 3059712\nu ) / 487424 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} + 4\beta_1 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + \beta_{5} + \beta_{4} + 10\beta_{3} - 3\beta_{2} - 74 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{11} + 3\beta_{10} - 33\beta_{8} - 15\beta_{7} - 27\beta_{6} - 87\beta_1 ) / 56 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{9} + 31\beta_{5} - 11\beta_{4} - 54\beta_{3} - 7\beta_{2} - 642 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -52\beta_{11} + 39\beta_{10} - 29\beta_{8} - 19\beta_{7} + 209\beta_{6} - 427\beta_1 ) / 56 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -61\beta_{9} - 369\beta_{5} - 19\beta_{4} - 78\beta_{3} - 47\beta_{2} - 1898 ) / 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 84\beta_{11} + \beta_{10} + 197\beta_{8} - 117\beta_{7} - 41\beta_{6} - 765\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -261\beta_{9} + 103\beta_{5} + 229\beta_{4} + 1226\beta_{3} + 4617\beta_{2} + 63582 ) / 28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -372\beta_{11} - 2409\beta_{10} + 3795\beta_{8} + 4349\beta_{7} - 1503\beta_{6} + 153765\beta_1 ) / 56 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 8131\beta_{9} + 9167\beta_{5} + 4365\beta_{4} - 10222\beta_{3} - 22159\beta_{2} - 35786 ) / 28 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7244\beta_{11} + 7559\beta_{10} - 141213\beta_{8} + 67661\beta_{7} - 29279\beta_{6} - 551147\beta_1 ) / 56 \) 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
−0.458818 − 2.79097i
−0.458818 + 2.79097i
−2.16402 + 1.82127i
−2.16402 − 1.82127i
1.76253 − 2.21212i
1.76253 + 2.21212i
−1.76253 − 2.21212i
−1.76253 + 2.21212i
2.16402 + 1.82127i
2.16402 − 1.82127i
0.458818 − 2.79097i
0.458818 + 2.79097i
−3.92151 − 0.788523i 0 14.7565 + 6.18440i 12.8744 0 18.5203i −52.9911 − 35.8880i 0 −50.4871 − 10.1518i
127.2 −3.92151 + 0.788523i 0 14.7565 − 6.18440i 12.8744 0 − 18.5203i −52.9911 + 35.8880i 0 −50.4871 + 10.1518i
127.3 −3.49132 − 1.95209i 0 8.37865 + 13.6308i 18.1143 0 − 18.5203i −2.64399 − 63.9454i 0 −63.2427 − 35.3607i
127.4 −3.49132 + 1.95209i 0 8.37865 − 13.6308i 18.1143 0 18.5203i −2.64399 + 63.9454i 0 −63.2427 + 35.3607i
127.5 −2.04510 − 3.43767i 0 −7.63512 + 14.0608i 46.5846 0 18.5203i 63.9508 − 2.50871i 0 −95.2702 − 160.142i
127.6 −2.04510 + 3.43767i 0 −7.63512 − 14.0608i 46.5846 0 − 18.5203i 63.9508 + 2.50871i 0 −95.2702 + 160.142i
127.7 2.04510 − 3.43767i 0 −7.63512 − 14.0608i −46.5846 0 − 18.5203i −63.9508 − 2.50871i 0 −95.2702 + 160.142i
127.8 2.04510 + 3.43767i 0 −7.63512 + 14.0608i −46.5846 0 18.5203i −63.9508 + 2.50871i 0 −95.2702 − 160.142i
127.9 3.49132 − 1.95209i 0 8.37865 − 13.6308i −18.1143 0 18.5203i 2.64399 − 63.9454i 0 −63.2427 + 35.3607i
127.10 3.49132 + 1.95209i 0 8.37865 + 13.6308i −18.1143 0 − 18.5203i 2.64399 + 63.9454i 0 −63.2427 − 35.3607i
127.11 3.92151 − 0.788523i 0 14.7565 − 6.18440i −12.8744 0 − 18.5203i 52.9911 − 35.8880i 0 −50.4871 + 10.1518i
127.12 3.92151 + 0.788523i 0 14.7565 + 6.18440i −12.8744 0 18.5203i 52.9911 + 35.8880i 0 −50.4871 − 10.1518i
\(n\): 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 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.c ✓ 12
3.b odd 2 1 inner 252.5.g.c ✓ 12
4.b odd 2 1 inner 252.5.g.c ✓ 12
12.b even 2 1 inner 252.5.g.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.5.g.c ✓ 12 1.a even 1 1 trivial
252.5.g.c ✓ 12 3.b odd 2 1 inner
252.5.g.c ✓ 12 4.b odd 2 1 inner
252.5.g.c ✓ 12 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 2664T_{5}^{4} + 1126160T_{5}^{2} - 118026496 \) acting on \(S_{5}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 31 T^{10} + \cdots + 16777216 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - 2664 T^{4} + \cdots - 118026496)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 343)^{6} \) Copy content Toggle raw display
$11$ \( (T^{6} + 5912 T^{4} + \cdots + 900804352)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 66 T^{2} + \cdots + 3801224)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 8803041528064)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 423136 T^{4} + \cdots + 215886856192)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 25\!\cdots\!52)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 1598 T^{2} + \cdots + 1217612536)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 62\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 37\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 67\!\cdots\!72)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 24\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 79\!\cdots\!48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 2434 T^{2} + \cdots - 7000405384)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 76\!\cdots\!48)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 57\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 7734 T^{2} + \cdots - 105119525560)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 18\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 84\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 70\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 21354 T^{2} + \cdots + 147242265800)^{4} \) Copy content Toggle raw display
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