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Properties

Label 2178.4.a.cf
Level $2178$
Weight $4$
Character orbit 2178.a
Self dual yes
Analytic conductor $128.506$
Analytic rank $1$
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] = [2178,4,Mod(1,2178)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2178, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2178.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2178.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: \(128.506159993\)
Analytic rank: \(1\)
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: \( x^{6} - x^{5} - 331x^{4} + 48x^{3} + 23386x^{2} - 36820x - 100804 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 11 \)
Twist minimal: no (minimal twist has level 198)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + (\beta_{3} - 3) q^{5} + (\beta_{4} + \beta_{3} - 1) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + (\beta_{3} - 3) q^{5} + (\beta_{4} + \beta_{3} - 1) q^{7} + 8 q^{8} + (2 \beta_{3} - 6) q^{10} + ( - \beta_{5} + \beta_{4} - 4 \beta_{3} + \cdots - 10) q^{13}+ \cdots + ( - 22 \beta_{5} + 20 \beta_{4} + \cdots - 142) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{2} + 24 q^{4} - 17 q^{5} - 7 q^{7} + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 12 q^{2} + 24 q^{4} - 17 q^{5} - 7 q^{7} + 48 q^{8} - 34 q^{10} - 68 q^{13} - 14 q^{14} + 96 q^{16} + 42 q^{17} - 98 q^{19} - 68 q^{20} - 210 q^{23} + 47 q^{25} - 136 q^{26} - 28 q^{28} + 13 q^{29} - 125 q^{31} + 192 q^{32} + 84 q^{34} + 534 q^{35} + 282 q^{37} - 196 q^{38} - 136 q^{40} - 170 q^{41} - 868 q^{43} - 420 q^{46} - 782 q^{47} - 439 q^{49} + 94 q^{50} - 272 q^{52} - 645 q^{53} - 56 q^{56} + 26 q^{58} - 507 q^{59} - 1772 q^{61} - 250 q^{62} + 384 q^{64} - 1856 q^{65} + 686 q^{67} + 168 q^{68} + 1068 q^{70} - 2782 q^{71} - 335 q^{73} + 564 q^{74} - 392 q^{76} - 127 q^{79} - 272 q^{80} - 340 q^{82} - 9 q^{83} - 370 q^{85} - 1736 q^{86} - 2526 q^{89} + 296 q^{91} - 840 q^{92} - 1564 q^{94} - 1194 q^{95} + 89 q^{97} - 878 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 331x^{4} + 48x^{3} + 23386x^{2} - 36820x - 100804 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 346\nu^{5} - 2737\nu^{4} - 90849\nu^{3} + 401487\nu^{2} + 7941654\nu - 7079204 ) / 3296150 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9699\nu^{5} + 137622\nu^{4} - 3589806\nu^{3} - 42041297\nu^{2} + 244100926\nu + 1328646374 ) / 62626850 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -409\nu^{5} + 4188\nu^{4} + 58806\nu^{3} - 608913\nu^{2} + 3379664\nu + 6878256 ) / 1977690 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5971\nu^{5} - 38938\nu^{4} + 2366724\nu^{3} + 7918413\nu^{2} - 186805804\nu + 171838304 ) / 17080050 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 346\nu^{5} - 2737\nu^{4} - 90849\nu^{3} + 401487\nu^{2} + 4645504\nu - 5880604 ) / 898950 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{5} + 11\beta _1 + 4 ) / 11 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -36\beta_{5} - 33\beta_{4} - 33\beta_{3} - 44\beta_{2} + 22\beta _1 + 1192 ) / 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -949\beta_{5} - 77\beta_{4} - 572\beta_{3} - 33\beta_{2} + 2145\beta _1 + 1863 ) / 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -13269\beta_{5} - 8580\beta_{4} - 10989\beta_{3} - 7887\beta_{2} + 10065\beta _1 + 226681 ) / 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -243510\beta_{5} - 49797\beta_{4} - 198825\beta_{3} - 19998\beta_{2} + 469612\beta _1 + 1032398 ) / 11 \) 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
−11.9730
16.3255
−1.44840
3.35806
−13.2590
7.99688
2.00000 0 4.00000 −20.7546 0 −9.94881 8.00000 0 −41.5093
1.2 2.00000 0 4.00000 −13.7077 0 −21.0653 8.00000 0 −27.4155
1.3 2.00000 0 4.00000 −2.72287 0 25.7229 8.00000 0 −5.44574
1.4 2.00000 0 4.00000 4.05148 0 −10.5792 8.00000 0 8.10296
1.5 2.00000 0 4.00000 4.57651 0 −7.03954 8.00000 0 9.15301
1.6 2.00000 0 4.00000 11.5573 0 15.9100 8.00000 0 23.1145
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(11\) \( +1 \)

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 2178.4.a.cf 6
3.b odd 2 1 2178.4.a.ce 6
11.b odd 2 1 2178.4.a.cd 6
11.c even 5 2 198.4.f.g ✓ 12
33.d even 2 1 2178.4.a.cg 6
33.h odd 10 2 198.4.f.h yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
198.4.f.g ✓ 12 11.c even 5 2
198.4.f.h yes 12 33.h odd 10 2
2178.4.a.cd 6 11.b odd 2 1
2178.4.a.ce 6 3.b odd 2 1
2178.4.a.cf 6 1.a even 1 1 trivial
2178.4.a.cg 6 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2178))\):

\( T_{5}^{6} + 17T_{5}^{5} - 254T_{5}^{4} - 2679T_{5}^{3} + 21136T_{5}^{2} + 10535T_{5} - 166001 \) Copy content Toggle raw display
\( T_{7}^{6} + 7T_{7}^{5} - 785T_{7}^{4} - 8670T_{7}^{3} + 105595T_{7}^{2} + 1806607T_{7} + 6387469 \) Copy content Toggle raw display
\( T_{17}^{6} - 42T_{17}^{5} - 16741T_{17}^{4} + 660134T_{17}^{3} + 62978959T_{17}^{2} - 2121675142T_{17} + 5321458276 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 17 T^{5} + \cdots - 166001 \) Copy content Toggle raw display
$7$ \( T^{6} + 7 T^{5} + \cdots + 6387469 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 68 T^{5} + \cdots + 29354480 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 5321458276 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 132637201516 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 383709146084 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 580276239280 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 559103921759 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 35490237317820 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 67838856345196 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 1672466963904 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 833264191349524 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 403443973196039 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 191319629200771 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 183894027103620 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 19\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 21\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 19\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 54\!\cdots\!09 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 83\!\cdots\!45 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 68\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 12\!\cdots\!19 \) Copy content Toggle raw display
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