Properties

Label 99.8.a.h
Level $99$
Weight $8$
Character orbit 99.a
Self dual yes
Analytic conductor $30.926$
Analytic rank $1$
Dimension $5$
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: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 382x^{3} + 558x^{2} + 23640x + 53488 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{2} + ( - \beta_{3} + 4 \beta_1 + 35) q^{4} + ( - \beta_{4} + \beta_{3} - 13 \beta_1 - 105) q^{5} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 25 \beta_1 + 98) q^{7} + (\beta_{4} + 10 \beta_{3} + 5 \beta_{2} - 37 \beta_1 - 567) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 2) q^{2} + ( - \beta_{3} + 4 \beta_1 + 35) q^{4} + ( - \beta_{4} + \beta_{3} - 13 \beta_1 - 105) q^{5} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 25 \beta_1 + 98) q^{7} + (\beta_{4} + 10 \beta_{3} + 5 \beta_{2} - 37 \beta_1 - 567) q^{8} + (3 \beta_{4} - 19 \beta_{3} + \beta_{2} + 245 \beta_1 + 2404) q^{10} + 1331 q^{11} + (29 \beta_{4} + 41 \beta_{3} + 28 \beta_{2} - 129 \beta_1 - 3399) q^{13} + (5 \beta_{4} + 61 \beta_{3} - 25 \beta_{2} - 149 \beta_1 - 4182) q^{14} + ( - 34 \beta_{4} - 19 \beta_{3} - 26 \beta_{2} + 1182 \beta_1 + 3487) q^{16} + ( - 29 \beta_{4} - 4 \beta_{3} - 37 \beta_{2} + 752 \beta_1 + 687) q^{17} + ( - 47 \beta_{4} + 44 \beta_{3} - 23 \beta_{2} + 1712 \beta_1 - 1399) q^{19} + (131 \beta_{4} + 221 \beta_{3} + 83 \beta_{2} - 3559 \beta_1 - 32618) q^{20} + ( - 1331 \beta_1 - 2662) q^{22} + ( - 85 \beta_{4} - 333 \beta_{3} + 96 \beta_{2} + 2229 \beta_1 + 8105) q^{23} + ( - 122 \beta_{4} - 112 \beta_{3} - 26 \beta_{2} + 6768 \beta_1 + 23621) q^{25} + ( - 269 \beta_{4} - 655 \beta_{3} - 211 \beta_{2} + 7911 \beta_1 + 30640) q^{26} + ( - 237 \beta_{4} - 137 \beta_{3} - 101 \beta_{2} + 9637 \beta_1 + 27648) q^{28} + (75 \beta_{4} - 438 \beta_{3} + 385 \beta_{2} - 2878 \beta_1 - 52829) q^{29} + (552 \beta_{4} + 354 \beta_{3} + 338 \beta_{2} + 4006 \beta_1 - 7608) q^{31} + (131 \beta_{4} + 276 \beta_{3} - 497 \beta_{2} - 2795 \beta_1 - 123127) q^{32} + (268 \beta_{4} + 1146 \beta_{3} - 28 \beta_{2} - 1616 \beta_1 - 119566) q^{34} + (448 \beta_{4} + 322 \beta_{3} + 350 \beta_{2} - 7266 \beta_1 - 195840) q^{35} + (1126 \beta_{4} - 486 \beta_{3} + 852 \beta_{2} + 2998 \beta_1 + 79812) q^{37} + (236 \beta_{4} + 1678 \beta_{3} - 76 \beta_{2} + 3772 \beta_1 - 262826) q^{38} + ( - 1461 \beta_{4} - 3283 \beta_{3} - 1521 \beta_{2} + \cdots + 344466) q^{40}+ \cdots + (12108 \beta_{4} - 2296 \beta_{3} - 2668 \beta_{2} + \cdots + 5272762) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{2} + 166 q^{4} - 500 q^{5} + 446 q^{7} - 2754 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 8 q^{2} + 166 q^{4} - 500 q^{5} + 446 q^{7} - 2754 q^{8} + 11516 q^{10} + 6655 q^{11} - 16666 q^{13} - 20516 q^{14} + 15010 q^{16} + 1906 q^{17} - 10446 q^{19} - 155572 q^{20} - 10648 q^{22} + 35468 q^{23} + 104239 q^{25} + 136396 q^{26} + 118456 q^{28} - 259062 q^{29} - 44932 q^{31} - 609010 q^{32} - 592888 q^{34} - 963800 q^{35} + 393978 q^{37} - 1319448 q^{38} + 1650684 q^{40} - 1554074 q^{41} + 263118 q^{43} + 220946 q^{44} - 1948268 q^{46} - 2481904 q^{47} + 498177 q^{49} - 5560984 q^{50} - 4530056 q^{52} - 2325840 q^{53} - 665500 q^{55} - 5221992 q^{56} + 2479968 q^{58} - 4613452 q^{59} + 529362 q^{61} - 3077192 q^{62} + 1437674 q^{64} + 2187496 q^{65} - 4612664 q^{67} + 2362916 q^{68} + 7327832 q^{70} - 5583864 q^{71} + 980054 q^{73} - 3387480 q^{74} + 1023396 q^{76} + 593626 q^{77} + 2804758 q^{79} - 10053364 q^{80} - 110080 q^{82} + 2451324 q^{83} - 4240456 q^{85} + 5937408 q^{86} - 3665574 q^{88} - 4025472 q^{89} - 12525436 q^{91} + 44606548 q^{92} + 1592764 q^{94} - 4273320 q^{95} - 234806 q^{97} + 26686536 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 382x^{3} + 558x^{2} + 23640x + 53488 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 46\nu^{4} - 59\nu^{3} - 14435\nu^{2} + 54086\nu + 375282 ) / 19138 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 75\nu^{4} + 1776\nu^{3} - 20415\nu^{2} - 502182\nu + 460433 ) / 9569 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 335\nu^{4} - 3550\nu^{3} - 129463\nu^{2} + 1014208\nu + 6063460 ) / 19138 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 180\nu^{4} - 1479\nu^{3} - 77703\nu^{2} + 586080\nu + 4542224 ) / 9569 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_{3} - \beta_{2} + 10\beta _1 + 11 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -19\beta_{4} + 2\beta_{3} - 11\beta_{2} + 170\beta _1 + 5581 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 347\beta_{4} - 634\beta_{3} - 113\beta_{2} + 2270\beta _1 - 2921 ) / 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2231\beta_{4} + 722\beta_{3} - 807\beta_{2} + 19826\beta _1 + 480321 ) / 12 \) 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
16.1140
−2.78521
−18.2221
11.5540
−5.66065
−20.4574 0 290.506 −501.309 0 829.954 −3324.45 0 10255.5
1.2 −8.09815 0 −62.4199 −308.036 0 97.1968 1542.05 0 2494.52
1.3 −3.32273 0 −116.959 363.595 0 −794.809 813.934 0 −1208.13
1.4 8.34840 0 −58.3042 −113.711 0 1428.97 −1555.34 0 −949.305
1.5 15.5299 0 113.178 59.4610 0 −1115.32 −230.190 0 923.423
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.h 5
3.b odd 2 1 99.8.a.i yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.8.a.h 5 1.a even 1 1 trivial
99.8.a.i yes 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 8T_{2}^{4} - 371T_{2}^{3} - 1538T_{2}^{2} + 20636T_{2} + 71368 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 8 T^{4} - 371 T^{3} + \cdots + 71368 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + 500 T^{4} + \cdots + 379629184000 \) Copy content Toggle raw display
$7$ \( T^{5} + \cdots - 102186002826976 \) Copy content Toggle raw display
$11$ \( (T - 1331)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + 16666 T^{4} + \cdots - 17\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{5} - 1906 T^{4} + \cdots - 66\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{5} + 10446 T^{4} + \cdots + 40\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{5} - 35468 T^{4} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{5} + 259062 T^{4} + \cdots - 70\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{5} + 44932 T^{4} + \cdots - 47\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{5} - 393978 T^{4} + \cdots - 58\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{5} + 1554074 T^{4} + \cdots - 62\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{5} - 263118 T^{4} + \cdots - 99\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{5} + 2481904 T^{4} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{5} + 2325840 T^{4} + \cdots - 30\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{5} + 4613452 T^{4} + \cdots - 22\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{5} - 529362 T^{4} + \cdots - 30\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{5} + 4612664 T^{4} + \cdots - 68\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{5} + 5583864 T^{4} + \cdots + 13\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{5} - 980054 T^{4} + \cdots - 59\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{5} - 2804758 T^{4} + \cdots - 47\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{5} - 2451324 T^{4} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{5} + 4025472 T^{4} + \cdots - 44\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{5} + 234806 T^{4} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
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