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Properties

Label 276.6.a.d
Level $276$
Weight $6$
Character orbit 276.a
Self dual yes
Analytic conductor $44.266$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [276,6,Mod(1,276)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(276, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("276.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 276.a (trivial)

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: yes
Analytic conductor: \(44.2659342684\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 3740x^{4} - 50049x^{3} + 3200252x^{2} + 86063268x + 576646848 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + ( - \beta_{2} + 8) q^{5} + ( - \beta_{4} - 26) q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + ( - \beta_{2} + 8) q^{5} + ( - \beta_{4} - 26) q^{7} + 81 q^{9} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots + 11) q^{11}+ \cdots + (81 \beta_{5} - 81 \beta_{4} + \cdots + 891) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 54 q^{3} + 50 q^{5} - 154 q^{7} + 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 54 q^{3} + 50 q^{5} - 154 q^{7} + 486 q^{9} + 68 q^{11} + 1076 q^{13} + 450 q^{15} + 1650 q^{17} - 226 q^{19} - 1386 q^{21} + 3174 q^{23} + 14994 q^{25} + 4374 q^{27} + 12760 q^{29} + 10980 q^{31} + 612 q^{33} + 8492 q^{35} + 18012 q^{37} + 9684 q^{39} + 15396 q^{41} - 7146 q^{43} + 4050 q^{45} + 23588 q^{47} + 56182 q^{49} + 14850 q^{51} + 9310 q^{53} + 52048 q^{55} - 2034 q^{57} + 5260 q^{59} + 16932 q^{61} - 12474 q^{63} + 30852 q^{65} + 24858 q^{67} + 28566 q^{69} + 101832 q^{71} + 33100 q^{73} + 134946 q^{75} + 182232 q^{77} + 100570 q^{79} + 39366 q^{81} + 222524 q^{83} + 23824 q^{85} + 114840 q^{87} + 28374 q^{89} + 18204 q^{91} + 98820 q^{93} + 481996 q^{95} + 47632 q^{97} + 5508 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 3740x^{4} - 50049x^{3} + 3200252x^{2} + 86063268x + 576646848 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 150779 \nu^{5} + 3549096 \nu^{4} + 482651188 \nu^{3} - 2678832805 \nu^{2} + \cdots - 4210410850416 ) / 2814707340 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 150779 \nu^{5} - 3549096 \nu^{4} - 482651188 \nu^{3} + 2678832805 \nu^{2} + \cdots + 4204781435736 ) / 2814707340 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 206191 \nu^{5} - 2237844 \nu^{4} - 750043772 \nu^{3} - 2710622575 \nu^{2} + \cdots + 9436247653284 ) / 2814707340 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 233369 \nu^{5} + 3547536 \nu^{4} + 849698668 \nu^{3} - 141497695 \nu^{2} + \cdots - 9341088280056 ) / 2814707340 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 137261 \nu^{5} + 2333814 \nu^{4} + 474717472 \nu^{3} + 359082965 \nu^{2} + \cdots - 5976753938964 ) / 1407353670 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{5} - \beta_{4} + 2\beta_{3} + 13\beta_{2} + 10\beta _1 + 2502 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 165\beta_{5} + 274\beta_{4} + 317\beta_{3} + 2288\beta_{2} + 1997\beta _1 + 116583 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 25174\beta_{5} + 4556\beta_{4} + 24962\beta_{3} + 90989\beta_{2} + 72239\beta _1 + 10478888 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 489298\beta_{5} + 509931\beta_{4} + 765616\beta_{3} + 3152760\beta_{2} + 2482299\beta _1 + 206923014 ) / 2 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−26.9361
56.6655
40.9459
−14.3197
−36.8883
−17.4672
0 9.00000 0 −103.091 0 −186.665 0 81.0000 0
1.2 0 9.00000 0 −55.3592 0 −89.4396 0 81.0000 0
1.3 0 9.00000 0 −13.3377 0 170.308 0 81.0000 0
1.4 0 9.00000 0 27.8362 0 58.7071 0 81.0000 0
1.5 0 9.00000 0 84.9274 0 141.499 0 81.0000 0
1.6 0 9.00000 0 109.024 0 −248.409 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 276.6.a.d ✓ 6
3.b odd 2 1 828.6.a.e 6
4.b odd 2 1 1104.6.a.s 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.6.a.d ✓ 6 1.a even 1 1 trivial
828.6.a.e 6 3.b odd 2 1
1104.6.a.s 6 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 50T_{5}^{5} - 15622T_{5}^{4} + 601980T_{5}^{3} + 53472576T_{5}^{2} - 899873680T_{5} - 19618704480 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(276))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T - 9)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots - 19618704480 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots - 5867339963520 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 570133261440000 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 290392537175552 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 66\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 24\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( (T - 529)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 43\!\cdots\!92 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 23\!\cdots\!80 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 14\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 37\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 35\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 54\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 20\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 78\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 14\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 21\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 91\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 50\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 23\!\cdots\!80 \) Copy content Toggle raw display
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