Properties

Label 1104.6.a.y
Level $1104$
Weight $6$
Character orbit 1104.a
Self dual yes
Analytic conductor $177.064$
Analytic rank $0$
Dimension $7$
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] = [1104,6,Mod(1,1104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1104, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1104.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1104.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: \(177.063737074\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 2429x^{5} - 54929x^{4} - 436970x^{3} - 1590048x^{2} - 2711880x - 1760400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3 \)
Twist minimal: no (minimal twist has level 552)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + (\beta_{2} + 11) q^{5} + (\beta_{2} - \beta_1 - 7) q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + (\beta_{2} + 11) q^{5} + (\beta_{2} - \beta_1 - 7) q^{7} + 81 q^{9} + ( - \beta_{6} + \beta_{5} - \beta_{3} + 3 \beta_{2} - 19) q^{11} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 + 138) q^{13} + (9 \beta_{2} + 99) q^{15} + (\beta_{3} + 5 \beta_{2} + 9 \beta_1 + 91) q^{17} + ( - 2 \beta_{6} - 2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 6 \beta_1 + 77) q^{19} + (9 \beta_{2} - 9 \beta_1 - 63) q^{21} - 529 q^{23} + ( - \beta_{6} + 11 \beta_{3} + 24 \beta_{2} + 5 \beta_1 + 728) q^{25} + 729 q^{27} + ( - 4 \beta_{6} + \beta_{5} - 9 \beta_{4} - 6 \beta_{3} + 38 \beta_{2} - 8 \beta_1 + 731) q^{29} + ( - 9 \beta_{6} + 3 \beta_{5} + 6 \beta_{4} - 16 \beta_{3} - 15 \beta_{2} + \cdots + 389) q^{31}+ \cdots + ( - 81 \beta_{6} + 81 \beta_{5} - 81 \beta_{3} + 243 \beta_{2} + \cdots - 1539) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 63 q^{3} + 80 q^{5} - 46 q^{7} + 567 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 63 q^{3} + 80 q^{5} - 46 q^{7} + 567 q^{9} - 124 q^{11} + 966 q^{13} + 720 q^{15} + 652 q^{17} + 542 q^{19} - 414 q^{21} - 3703 q^{23} + 5165 q^{25} + 5103 q^{27} + 5222 q^{29} + 2660 q^{31} - 1116 q^{33} + 17088 q^{35} - 202 q^{37} + 8694 q^{39} - 706 q^{41} + 7086 q^{43} + 6480 q^{45} + 37696 q^{47} - 34469 q^{49} + 5868 q^{51} + 30180 q^{53} + 51104 q^{55} + 4878 q^{57} + 12284 q^{59} + 53534 q^{61} - 3726 q^{63} + 46920 q^{65} - 47750 q^{67} - 33327 q^{69} + 2552 q^{71} - 19474 q^{73} + 46485 q^{75} + 36264 q^{77} + 4510 q^{79} + 45927 q^{81} + 47580 q^{83} + 224944 q^{85} + 46998 q^{87} + 150664 q^{89} - 57460 q^{91} + 23940 q^{93} + 21648 q^{95} + 392426 q^{97} - 10044 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 2429x^{5} - 54929x^{4} - 436970x^{3} - 1590048x^{2} - 2711880x - 1760400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2261 \nu^{6} + 7367 \nu^{5} + 5478391 \nu^{4} + 111793711 \nu^{3} + 728356012 \nu^{2} + 1841655564 \nu + 1558654440 ) / 421710 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 44786 \nu^{6} + 166685 \nu^{5} + 108339331 \nu^{4} + 2165125561 \nu^{3} + 13658024797 \nu^{2} + 33701592756 \nu + 28337175090 ) / 3795390 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 44786 \nu^{6} + 166685 \nu^{5} + 108339331 \nu^{4} + 2165125561 \nu^{3} + 13658024797 \nu^{2} + 33716774316 \nu + 28333379700 ) / 1897695 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 71245 \nu^{6} + 245734 \nu^{5} + 172447184 \nu^{4} + 3491080334 \nu^{3} + 22594092569 \nu^{2} + 58218737280 \nu + 52166110140 ) / 1265130 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 233713 \nu^{6} - 1050622 \nu^{5} - 564128408 \nu^{4} - 10865044718 \nu^{3} - 63881346221 \nu^{2} - 143695550976 \nu - 108775910790 ) / 3795390 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 228833 \nu^{6} + 773750 \nu^{5} + 553985998 \nu^{4} + 11250448048 \nu^{3} + 73218728701 \nu^{2} + 189692394708 \nu + 169447175595 ) / 1897695 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 2\beta_{2} + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 8\beta_{6} - 3\beta_{5} - 15\beta_{4} + 39\beta_{3} - 112\beta_{2} + 18\beta _1 + 5599 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 202\beta_{6} - 41\beta_{5} - 435\beta_{4} + 3005\beta_{3} - 6274\beta_{2} + 136\beta _1 + 199277 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 20870\beta_{6} - 7441\beta_{5} - 39915\beta_{4} + 155895\beta_{3} - 393582\beta_{2} + 44416\beta _1 + 15908517 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 953940 \beta_{6} - 271935 \beta_{5} - 1928875 \beta_{4} + 10042601 \beta_{3} - 22667572 \beta_{2} + 1365250 \beta _1 + 817462147 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 66240988 \beta_{6} - 21909183 \beta_{5} - 129339055 \beta_{4} + 572412759 \beta_{3} - 1375427252 \beta_{2} + 124589558 \beta _1 + 52873707739 ) / 8 \) Copy content Toggle raw display

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
59.5190
−2.47495
−2.10756
−33.0467
−13.6663
−2.91227
−4.31122
0 9.00000 0 −71.9101 0 −83.2222 0 81.0000 0
1.2 0 9.00000 0 −40.6905 0 8.51571 0 81.0000 0
1.3 0 9.00000 0 −32.8770 0 11.9330 0 81.0000 0
1.4 0 9.00000 0 −10.8093 0 −159.351 0 81.0000 0
1.5 0 9.00000 0 62.5293 0 199.683 0 81.0000 0
1.6 0 9.00000 0 84.6740 0 60.6620 0 81.0000 0
1.7 0 9.00000 0 89.0835 0 −84.2212 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 1104.6.a.y 7
4.b odd 2 1 552.6.a.f 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.6.a.f 7 4.b odd 2 1
1104.6.a.y 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{7} - 80 T_{5}^{6} - 10320 T_{5}^{5} + 635432 T_{5}^{4} + 38621312 T_{5}^{3} - 1023911424 T_{5}^{2} - 60024250368 T_{5} - 490461308928 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1104))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( (T - 9)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 80 T^{6} + \cdots - 490461308928 \) Copy content Toggle raw display
$7$ \( T^{7} + 46 T^{6} + \cdots + 1374810163200 \) Copy content Toggle raw display
$11$ \( T^{7} + 124 T^{6} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{7} - 966 T^{6} + \cdots - 14\!\cdots\!20 \) Copy content Toggle raw display
$17$ \( T^{7} - 652 T^{6} + \cdots - 61\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{7} - 542 T^{6} + \cdots + 29\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( (T + 529)^{7} \) Copy content Toggle raw display
$29$ \( T^{7} - 5222 T^{6} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{7} - 2660 T^{6} + \cdots + 23\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{7} + 202 T^{6} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{7} + 706 T^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{7} - 7086 T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{7} - 37696 T^{6} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{7} - 30180 T^{6} + \cdots - 27\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{7} - 12284 T^{6} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{7} - 53534 T^{6} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{7} + 47750 T^{6} + \cdots - 12\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{7} - 2552 T^{6} + \cdots - 89\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{7} + 19474 T^{6} + \cdots + 23\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{7} - 4510 T^{6} + \cdots - 14\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{7} - 47580 T^{6} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{7} - 150664 T^{6} + \cdots - 24\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{7} - 392426 T^{6} + \cdots + 61\!\cdots\!88 \) Copy content Toggle raw display
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