Properties

Label 99.8.a.d
Level $99$
Weight $8$
Character orbit 99.a
Self dual yes
Analytic conductor $30.926$
Analytic rank $0$
Dimension $2$
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] = [99,8,Mod(1,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, 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("99.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 99.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: \(30.9261175229\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 10) q^{2} + ( - 19 \beta + 16) q^{4} + ( - 38 \beta + 36) q^{5} + ( - 154 \beta - 6) q^{7} + ( - 59 \beta - 284) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 10) q^{2} + ( - 19 \beta + 16) q^{4} + ( - 38 \beta + 36) q^{5} + ( - 154 \beta - 6) q^{7} + ( - 59 \beta - 284) q^{8} + ( - 378 \beta + 2032) q^{10} + 1331 q^{11} + ( - 878 \beta - 5896) q^{13} + ( - 1380 \beta + 6716) q^{14} + (2185 \beta - 2292) q^{16} + ( - 996 \beta + 31670) q^{17} + ( - 852 \beta - 18564) q^{19} + ( - 570 \beta + 32344) q^{20} + ( - 1331 \beta + 13310) q^{22} + (6330 \beta + 42178) q^{23} + ( - 1292 \beta - 13293) q^{25} + ( - 2006 \beta - 20328) q^{26} + (576 \beta + 128648) q^{28} + ( - 29644 \beta + 7826) q^{29} + (17608 \beta + 113696) q^{31} + (29509 \beta - 82708) q^{32} + ( - 40634 \beta + 360524) q^{34} + (536 \beta + 257272) q^{35} + (27472 \beta + 150190) q^{37} + (10896 \beta - 148152) q^{38} + (10910 \beta + 88424) q^{40} + ( - 87896 \beta + 181914) q^{41} + ( - 70936 \beta - 86584) q^{43} + ( - 25289 \beta + 21296) q^{44} + (14792 \beta + 143260) q^{46} + (74886 \beta - 305906) q^{47} + (25564 \beta + 219997) q^{49} + (1665 \beta - 76082) q^{50} + (114658 \beta + 639672) q^{52} + (98834 \beta + 861524) q^{53} + ( - 50578 \beta + 47916) q^{55} + (53176 \beta + 401488) q^{56} + ( - 274622 \beta + 1382596) q^{58} + (189540 \beta - 1345784) q^{59} + (201934 \beta - 1196516) q^{61} + (44776 \beta + 362208) q^{62} + (68609 \beta - 1832100) q^{64} + (225804 \beta + 1255760) q^{65} + (464104 \beta + 605340) q^{67} + ( - 598742 \beta + 1339376) q^{68} + ( - 252448 \beta + 2549136) q^{70} + ( - 36334 \beta + 202322) q^{71} + (436992 \beta - 1886798) q^{73} + (97058 \beta + 293132) q^{74} + (355272 \beta + 415248) q^{76} + ( - 204974 \beta - 7986) q^{77} + ( - 708646 \beta - 486986) q^{79} + (82726 \beta - 3735832) q^{80} + ( - 972978 \beta + 5686564) q^{82} + ( - 461376 \beta + 4418940) q^{83} + ( - 1201468 \beta + 2805432) q^{85} + ( - 551840 \beta + 2255344) q^{86} + ( - 78529 \beta - 378004) q^{88} + (903568 \beta - 4965386) q^{89} + (1048464 \beta + 5984704) q^{91} + ( - 820372 \beta - 4617032) q^{92} + (979880 \beta - 6354044) q^{94} + (707136 \beta + 756240) q^{95} + ( - 2748 \beta - 8350402) q^{97} + (10079 \beta + 1075154) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 19 q^{2} + 13 q^{4} + 34 q^{5} - 166 q^{7} - 627 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 19 q^{2} + 13 q^{4} + 34 q^{5} - 166 q^{7} - 627 q^{8} + 3686 q^{10} + 2662 q^{11} - 12670 q^{13} + 12052 q^{14} - 2399 q^{16} + 62344 q^{17} - 37980 q^{19} + 64118 q^{20} + 25289 q^{22} + 90686 q^{23} - 27878 q^{25} - 42662 q^{26} + 257872 q^{28} - 13992 q^{29} + 245000 q^{31} - 135907 q^{32} + 680414 q^{34} + 515080 q^{35} + 327852 q^{37} - 285408 q^{38} + 187758 q^{40} + 275932 q^{41} - 244104 q^{43} + 17303 q^{44} + 301312 q^{46} - 536926 q^{47} + 465558 q^{49} - 150499 q^{50} + 1394002 q^{52} + 1821882 q^{53} + 45254 q^{55} + 856152 q^{56} + 2490570 q^{58} - 2502028 q^{59} - 2191098 q^{61} + 769192 q^{62} - 3595591 q^{64} + 2737324 q^{65} + 1674784 q^{67} + 2080010 q^{68} + 4845824 q^{70} + 368310 q^{71} - 3336604 q^{73} + 683322 q^{74} + 1185768 q^{76} - 220946 q^{77} - 1682618 q^{79} - 7388938 q^{80} + 10400150 q^{82} + 8376504 q^{83} + 4409396 q^{85} + 3958848 q^{86} - 834537 q^{88} - 9027204 q^{89} + 13017872 q^{91} - 10054436 q^{92} - 11728208 q^{94} + 2219616 q^{95} - 16703552 q^{97} + 2160387 q^{98}+O(q^{100}) \) 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
7.15207
−6.15207
2.84793 0 −119.889 −235.779 0 −1107.42 −705.972 0 −671.481
1.2 16.1521 0 132.889 269.779 0 941.418 78.9720 0 4357.48
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 99.8.a.d 2
3.b odd 2 1 33.8.a.b 2
12.b even 2 1 528.8.a.f 2
33.d even 2 1 363.8.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.a.b 2 3.b odd 2 1
99.8.a.d 2 1.a even 1 1 trivial
363.8.a.d 2 33.d even 2 1
528.8.a.f 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 19T_{2} + 46 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 19T + 46 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 34T - 63608 \) Copy content Toggle raw display
$7$ \( T^{2} + 166 T - 1042544 \) Copy content Toggle raw display
$11$ \( (T - 1331)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 12670 T + 6020608 \) Copy content Toggle raw display
$17$ \( T^{2} - 62344 T + 927796876 \) Copy content Toggle raw display
$19$ \( T^{2} + 37980 T + 328498848 \) Copy content Toggle raw display
$23$ \( T^{2} - 90686 T + 282938824 \) Copy content Toggle raw display
$29$ \( T^{2} + 13992 T - 38836484052 \) Copy content Toggle raw display
$31$ \( T^{2} - 245000 T + 1286906368 \) Copy content Toggle raw display
$37$ \( T^{2} - 327852 T - 6524218716 \) Copy content Toggle raw display
$41$ \( T^{2} - 275932 T - 322827909452 \) Copy content Toggle raw display
$43$ \( T^{2} + 244104 T - 207765596544 \) Copy content Toggle raw display
$47$ \( T^{2} + 536926 T - 176077767704 \) Copy content Toggle raw display
$53$ \( T^{2} - 1821882 T + 397572445128 \) Copy content Toggle raw display
$59$ \( T^{2} + 2502028 T - 24663435104 \) Copy content Toggle raw display
$61$ \( T^{2} + 2191098 T - 604169699352 \) Copy content Toggle raw display
$67$ \( T^{2} - 1674784 T - 8829893772944 \) Copy content Toggle raw display
$71$ \( T^{2} - 368310 T - 24503996328 \) Copy content Toggle raw display
$73$ \( T^{2} + 3336604 T - 5666837293628 \) Copy content Toggle raw display
$79$ \( T^{2} + 1682618 T - 21513626700752 \) Copy content Toggle raw display
$83$ \( T^{2} - 8376504 T + 8122054073616 \) Copy content Toggle raw display
$89$ \( T^{2} + 9027204 T - 15754651515708 \) Copy content Toggle raw display
$97$ \( T^{2} + 16703552 T + 69751828200124 \) Copy content Toggle raw display
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