Properties

Label 55.8.a.a
Level $55$
Weight $8$
Character orbit 55.a
Self dual yes
Analytic conductor $17.181$
Analytic rank $1$
Dimension $4$
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] = [55,8,Mod(1,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 55.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: \(17.1811764016\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 200x^{2} + 471x + 2765 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 5) q^{2} + (\beta_{2} - \beta_1) q^{3} + (7 \beta_{3} + \beta_{2} + 5 \beta_1 + 44) q^{4} - 125 q^{5} + ( - 3 \beta_{3} - 11 \beta_{2} + \cdots + 108) q^{6}+ \cdots + ( - 15 \beta_{3} - 42 \beta_{2} + \cdots - 678) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 5) q^{2} + (\beta_{2} - \beta_1) q^{3} + (7 \beta_{3} + \beta_{2} + 5 \beta_1 + 44) q^{4} - 125 q^{5} + ( - 3 \beta_{3} - 11 \beta_{2} + \cdots + 108) q^{6}+ \cdots + ( - 19965 \beta_{3} - 55902 \beta_{2} + \cdots - 902418) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 21 q^{2} + 189 q^{4} - 500 q^{5} + 411 q^{6} + 1262 q^{7} - 2367 q^{8} - 2706 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 21 q^{2} + 189 q^{4} - 500 q^{5} + 411 q^{6} + 1262 q^{7} - 2367 q^{8} - 2706 q^{9} + 2625 q^{10} + 5324 q^{11} + 4353 q^{12} + 2696 q^{13} - 16259 q^{14} + 7633 q^{16} - 24824 q^{17} - 13470 q^{18} - 22730 q^{19} - 23625 q^{20} + 5544 q^{21} - 27951 q^{22} - 160 q^{23} - 179787 q^{24} + 62500 q^{25} - 343346 q^{26} - 264060 q^{27} - 249037 q^{28} - 433924 q^{29} - 51375 q^{30} - 517576 q^{31} - 376727 q^{32} + 371131 q^{34} - 157750 q^{35} - 353298 q^{36} + 523762 q^{37} - 126695 q^{38} - 144696 q^{39} + 295875 q^{40} - 277004 q^{41} + 54717 q^{42} - 251596 q^{43} + 251559 q^{44} + 338250 q^{45} - 399654 q^{46} - 711180 q^{47} + 1790325 q^{48} + 241462 q^{49} - 328125 q^{50} - 2200542 q^{51} + 5757946 q^{52} - 940654 q^{53} + 1284201 q^{54} - 665500 q^{55} + 2594823 q^{56} + 551700 q^{57} + 5209629 q^{58} + 1014028 q^{59} - 544125 q^{60} + 3634268 q^{61} + 5520995 q^{62} - 87120 q^{63} + 1670385 q^{64} - 337000 q^{65} + 547041 q^{66} - 163644 q^{67} - 2116383 q^{68} + 602232 q^{69} + 2032375 q^{70} - 5547896 q^{71} + 10064430 q^{72} - 3614404 q^{73} - 3922253 q^{74} + 599375 q^{76} + 1679722 q^{77} - 5685402 q^{78} - 8995368 q^{79} - 954125 q^{80} + 561564 q^{81} + 3995374 q^{82} - 5010356 q^{83} - 6175065 q^{84} + 3103000 q^{85} - 9767128 q^{86} + 5173554 q^{87} - 3150477 q^{88} - 8759946 q^{89} + 1683750 q^{90} - 18295460 q^{91} + 23023838 q^{92} - 17520930 q^{93} + 4317674 q^{94} + 2841250 q^{95} - 7304979 q^{96} + 11731552 q^{97} + 16303462 q^{98} - 3601686 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 200x^{2} + 471x + 2765 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{3} - 2\nu^{2} + 850\nu - 981 ) / 184 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{3} - 38\nu^{2} + 1062\nu + 2981 ) / 184 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{3} - 38\nu^{2} + 326\nu + 3165 ) / 184 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -15\beta_{3} - 5\beta_{2} + 12\beta _1 + 403 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -41\beta_{3} + 43\beta_{2} - 38\beta _1 - 194 \) 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
5.54688
−14.3208
12.5479
−2.77399
−20.3206 23.7588 284.925 −125.000 −482.793 −57.2967 −3188.81 −1622.52 2540.07
1.2 −11.0932 −67.0170 −4.94146 −125.000 743.431 381.767 1474.74 2304.28 1386.65
1.3 −2.23601 26.6592 −123.000 −125.000 −59.6105 1682.67 561.240 −1476.28 279.502
1.4 12.6498 16.5989 32.0163 −125.000 209.972 −745.139 −1214.17 −1911.48 −1581.22
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(11\) \( -1 \)

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 55.8.a.a 4
3.b odd 2 1 495.8.a.c 4
5.b even 2 1 275.8.a.c 4
11.b odd 2 1 605.8.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.8.a.a 4 1.a even 1 1 trivial
275.8.a.c 4 5.b even 2 1
495.8.a.c 4 3.b odd 2 1
605.8.a.e 4 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 21T_{2}^{3} - 130T_{2}^{2} - 3236T_{2} - 6376 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(55))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 21 T^{3} + \cdots - 6376 \) Copy content Toggle raw display
$3$ \( T^{4} - 3021 T^{2} + \cdots - 704592 \) Copy content Toggle raw display
$5$ \( (T + 125)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 27426094884 \) Copy content Toggle raw display
$11$ \( (T - 1331)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 60\!\cdots\!08 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots - 327142625760000 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 93\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 81\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 27\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 88\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 89\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 13\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 76\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 54\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 34\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 14\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 13\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 12\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 89\!\cdots\!88 \) Copy content Toggle raw display
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