Properties

Label 1104.6.a.w
Level $1104$
Weight $6$
Character orbit 1104.a
Self dual yes
Analytic conductor $177.064$
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] = [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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 327x^{4} - 1112x^{3} + 5863x^{2} + 9144x + 2799 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + (\beta_{2} + 7) q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{2} - \beta_1 + 14) q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + (\beta_{2} + 7) q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{2} - \beta_1 + 14) q^{7} + 81 q^{9} + (2 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 96) q^{11} + (2 \beta_{5} - 4 \beta_{4} - \beta_{3} + 7 \beta_{2} - 302) q^{13} + (9 \beta_{2} + 63) q^{15} + ( - 4 \beta_{5} - 11 \beta_{3} - 4 \beta_{2} - \beta_1 - 18) q^{17} + (3 \beta_{5} + 23 \beta_{4} - 7 \beta_{3} + 22 \beta_{2} - 10 \beta_1 - 211) q^{19} + ( - 9 \beta_{5} - 9 \beta_{4} - 9 \beta_{2} - 9 \beta_1 + 126) q^{21} - 529 q^{23} + ( - 15 \beta_{5} - 18 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 752) q^{25} + 729 q^{27} + ( - \beta_{5} + 4 \beta_{4} - 16 \beta_{3} - 48 \beta_{2} - 24 \beta_1 + 1583) q^{29} + ( - 8 \beta_{5} - 23 \beta_{4} - 13 \beta_{3} + 69 \beta_{2} + 38 \beta_1 + 1171) q^{31} + (18 \beta_{5} - 27 \beta_{4} - 18 \beta_{3} + 36 \beta_{2} + 27 \beta_1 - 864) q^{33} + (40 \beta_{5} - 64 \beta_{4} + 11 \beta_{3} + 137 \beta_{2} + 30 \beta_1 - 306) q^{35} + (56 \beta_{5} - 15 \beta_{4} + 2 \beta_{3} + 14 \beta_{2} - 7 \beta_1 + 1340) q^{37} + (18 \beta_{5} - 36 \beta_{4} - 9 \beta_{3} + 63 \beta_{2} - 2718) q^{39} + ( - 47 \beta_{5} + 2 \beta_{4} + 110 \beta_{3} + 84 \beta_{2} - 76 \beta_1 + 877) q^{41} + ( - 12 \beta_{5} - 53 \beta_{4} + 27 \beta_{3} - 134 \beta_{2} + \cdots + 2470) q^{43}+ \cdots + (162 \beta_{5} - 243 \beta_{4} - 162 \beta_{3} + 324 \beta_{2} + \cdots - 7776) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 54 q^{3} + 42 q^{5} + 82 q^{7} + 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 54 q^{3} + 42 q^{5} + 82 q^{7} + 486 q^{9} - 560 q^{11} - 1800 q^{13} + 378 q^{15} - 118 q^{17} - 1326 q^{19} + 738 q^{21} - 3174 q^{23} - 4498 q^{25} + 4374 q^{27} + 9440 q^{29} + 7132 q^{31} - 5040 q^{33} - 1568 q^{35} + 8168 q^{37} - 16200 q^{39} + 5012 q^{41} + 14762 q^{43} + 3402 q^{45} - 15904 q^{47} + 75346 q^{49} - 1062 q^{51} + 58414 q^{53} + 43320 q^{55} - 11934 q^{57} + 38640 q^{59} + 25192 q^{61} + 6642 q^{63} + 93732 q^{65} + 50446 q^{67} - 28566 q^{69} + 8464 q^{71} - 63132 q^{73} - 40482 q^{75} - 140072 q^{77} + 239558 q^{79} + 39366 q^{81} + 120464 q^{83} - 5116 q^{85} + 84960 q^{87} - 33298 q^{89} + 572 q^{91} + 64188 q^{93} + 299608 q^{95} + 92376 q^{97} - 45360 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 327x^{4} - 1112x^{3} + 5863x^{2} + 9144x + 2799 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 297\nu^{5} - 86\nu^{4} - 96206\nu^{3} - 294426\nu^{2} + 1541749\nu + 583276 ) / 14860 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 57\nu^{5} - 9\nu^{4} - 18794\nu^{3} - 59478\nu^{2} + 381749\nu + 380375 ) / 2972 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -501\nu^{5} + 783\nu^{4} + 163938\nu^{3} + 303478\nu^{2} - 3773017\nu - 1081213 ) / 14860 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -69\nu^{5} + 50\nu^{4} + 22516\nu^{3} + 60972\nu^{2} - 453123\nu - 369456 ) / 1486 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -464\nu^{5} + 347\nu^{4} + 150852\nu^{3} + 405492\nu^{2} - 2779608\nu - 2016077 ) / 7430 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 3\beta_{5} - 4\beta_{4} - 2\beta_{2} + 2\beta _1 - 3 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 15\beta_{5} + 4\beta_{4} - 12\beta_{3} + 2\beta_{2} + 34\beta _1 + 2601 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 795\beta_{5} - 1156\beta_{4} + 72\beta_{3} - 986\beta_{2} + 866\beta _1 + 25749 ) / 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3135\beta_{5} - 484\beta_{4} - 1584\beta_{3} + 22\beta_{2} + 5978\beta _1 + 377613 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 275319\beta_{5} - 346324\beta_{4} - 2304\beta_{3} - 305018\beta_{2} + 346874\beta _1 + 13856361 ) / 48 \) 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
−6.34672
19.1762
−0.925381
3.73291
−15.1990
−0.438035
0 9.00000 0 −72.1293 0 242.314 0 81.0000 0
1.2 0 9.00000 0 −30.2941 0 −238.307 0 81.0000 0
1.3 0 9.00000 0 3.98074 0 −76.6731 0 81.0000 0
1.4 0 9.00000 0 19.9796 0 115.537 0 81.0000 0
1.5 0 9.00000 0 45.0506 0 −123.069 0 81.0000 0
1.6 0 9.00000 0 75.4124 0 162.199 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 1104.6.a.w 6
4.b odd 2 1 552.6.a.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.6.a.e 6 4.b odd 2 1
1104.6.a.w 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 42T_{5}^{5} - 6244T_{5}^{4} + 245184T_{5}^{3} + 4855680T_{5}^{2} - 171131392T_{5} + 590425088 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1104))\). 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
$5$ \( T^{6} - 42 T^{5} + \cdots + 590425088 \) Copy content Toggle raw display
$7$ \( T^{6} - 82 T^{5} + \cdots - 10211152303168 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 617873698758656 \) Copy content Toggle raw display
$13$ \( T^{6} + 1800 T^{5} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{6} + 118 T^{5} + \cdots - 21\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{6} + 1326 T^{5} + \cdots - 53\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T + 529)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} - 9440 T^{5} + \cdots - 66\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{6} - 7132 T^{5} + \cdots - 19\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{6} - 8168 T^{5} + \cdots - 69\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{6} - 5012 T^{5} + \cdots + 55\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( T^{6} - 14762 T^{5} + \cdots + 45\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{6} + 15904 T^{5} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{6} - 58414 T^{5} + \cdots - 43\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{6} - 38640 T^{5} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} - 25192 T^{5} + \cdots + 13\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{6} - 50446 T^{5} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{6} - 8464 T^{5} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{6} + 63132 T^{5} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{6} - 239558 T^{5} + \cdots - 12\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{6} - 120464 T^{5} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{6} + 33298 T^{5} + \cdots - 15\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{6} - 92376 T^{5} + \cdots + 48\!\cdots\!76 \) Copy content Toggle raw display
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