Properties

Label 99.8.a.e
Level $99$
Weight $8$
Character orbit 99.a
Self dual yes
Analytic conductor $30.926$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(3\)
Coefficient field: 3.3.115512.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 70x - 194 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 3) q^{2} + ( - \beta_{2} + 13 \beta_1 - 5) q^{4} + (6 \beta_{2} - 14 \beta_1 + 148) q^{5} + (28 \beta_{2} + 84 \beta_1 + 538) q^{7} + (9 \beta_{2} - 5 \beta_1 - 1051) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 3) q^{2} + ( - \beta_{2} + 13 \beta_1 - 5) q^{4} + (6 \beta_{2} - 14 \beta_1 + 148) q^{5} + (28 \beta_{2} + 84 \beta_1 + 538) q^{7} + (9 \beta_{2} - 5 \beta_1 - 1051) q^{8} + (10 \beta_{2} + 40 \beta_1 + 960) q^{10} + 1331 q^{11} + ( - 186 \beta_{2} + 466 \beta_1 + 6924) q^{13} + (196 \beta_{2} - 1154 \beta_1 - 12086) q^{14} + (159 \beta_{2} - 491 \beta_1 + 4075) q^{16} + ( - 542 \beta_{2} + 1462 \beta_1 + 4846) q^{17} + ( - 526 \beta_{2} - 2074 \beta_1 + 8164) q^{19} + ( - 688 \beta_{2} + 512 \beta_1 - 26704) q^{20} + ( - 1331 \beta_1 - 3993) q^{22} + (2030 \beta_{2} - 310 \beta_1 - 11698) q^{23} + (3044 \beta_{2} - 4596 \beta_1 + 9707) q^{25} + ( - 278 \beta_{2} - 13072 \beta_1 - 67944) q^{26} + ( - 3954 \beta_{2} + 14442 \beta_1 + 92678) q^{28} + ( - 2278 \beta_{2} - 7634 \beta_1 + 59954) q^{29} + (2728 \beta_{2} - 11080 \beta_1 + 96296) q^{31} + ( - 1007 \beta_{2} + 2747 \beta_1 + 173189) q^{32} + ( - 706 \beta_{2} - 23802 \beta_1 - 163862) q^{34} + (9108 \beta_{2} - 19012 \beta_1 + 177624) q^{35} + (1996 \beta_{2} + 5828 \beta_1 + 35854) q^{37} + ( - 4178 \beta_{2} + 8368 \beta_1 + 228776) q^{38} + ( - 3520 \beta_{2} + 10960 \beta_1 - 79120) q^{40} + ( - 2462 \beta_{2} + 4694 \beta_1 + 45066) q^{41} + ( - 7718 \beta_{2} - 18594 \beta_1 + 64512) q^{43} + ( - 1331 \beta_{2} + 17303 \beta_1 - 6655) q^{44} + (7810 \beta_{2} + 31038 \beta_1 + 5474) q^{46} + ( - 1294 \beta_{2} + 65462 \beta_1 + 197162) q^{47} + ( - 3584 \beta_{2} + 33152 \beta_1 + 1487053) q^{49} + (7580 \beta_{2} + 60605 \beta_1 + 397415) q^{50} + (9624 \beta_{2} + 136792 \beta_1 + 816664) q^{52} + ( - 9102 \beta_{2} + 72182 \beta_1 - 26348) q^{53} + (7986 \beta_{2} - 18634 \beta_1 + 196988) q^{55} + ( - 26462 \beta_{2} - 121018 \beta_1 - 250886) q^{56} + ( - 16746 \beta_{2} - 1838 \beta_1 + 763310) q^{58} + (37356 \beta_{2} - 43644 \beta_1 - 844256) q^{59} + (7574 \beta_{2} - 115710 \beta_1 + 2226264) q^{61} + ( - 168 \beta_{2} + 36328 \beta_1 + 886936) q^{62} + ( - 21633 \beta_{2} - 145867 \beta_1 - 1322101) q^{64} + ( - 26188 \beta_{2} - 18628 \beta_1 - 1063944) q^{65} + ( - 2480 \beta_{2} + 31184 \beta_1 + 2383452) q^{67} + (42750 \beta_{2} + 209098 \beta_1 + 2607318) q^{68} + (17420 \beta_{2} + 85360 \beta_1 + 1343040) q^{70} + ( - 40050 \beta_{2} - 315766 \beta_1 - 463466) q^{71} + ( - 30404 \beta_{2} - 58412 \beta_1 - 2143038) q^{73} + (13812 \beta_{2} - 78166 \beta_1 - 835826) q^{74} + (58984 \beta_{2} - 80408 \beta_1 - 2551576) q^{76} + (37268 \beta_{2} + 111804 \beta_1 + 716078) q^{77} + (95960 \beta_{2} - 163544 \beta_1 + 2291062) q^{79} + (84944 \beta_{2} - 124176 \beta_1 + 2518672) q^{80} + ( - 5154 \beta_{2} - 111702 \beta_1 - 591530) q^{82} + (2892 \beta_{2} + 376356 \beta_1 - 2168532) q^{83} + ( - 171208 \beta_{2} + 160552 \beta_1 - 5515344) q^{85} + ( - 49466 \beta_{2} + 59684 \beta_1 + 2173156) q^{86} + (11979 \beta_{2} - 6655 \beta_1 - 1398881) q^{88} + ( - 109592 \beta_{2} - 103240 \beta_1 + 2947654) q^{89} + (31028 \beta_{2} + 1585244 \beta_1 + 1022216) q^{91} + ( - 197562 \beta_{2} - 213694 \beta_1 - 2307330) q^{92} + (60286 \beta_{2} - 862134 \beta_1 - 8012746) q^{94} + ( - 47588 \beta_{2} + 100372 \beta_1 + 63656) q^{95} + ( - 47392 \beta_{2} + 148640 \beta_1 - 588258) q^{97} + (18816 \beta_{2} - 1847245 \beta_1 - 8125799) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 9 q^{2} - 15 q^{4} + 444 q^{5} + 1614 q^{7} - 3153 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 9 q^{2} - 15 q^{4} + 444 q^{5} + 1614 q^{7} - 3153 q^{8} + 2880 q^{10} + 3993 q^{11} + 20772 q^{13} - 36258 q^{14} + 12225 q^{16} + 14538 q^{17} + 24492 q^{19} - 80112 q^{20} - 11979 q^{22} - 35094 q^{23} + 29121 q^{25} - 203832 q^{26} + 278034 q^{28} + 179862 q^{29} + 288888 q^{31} + 519567 q^{32} - 491586 q^{34} + 532872 q^{35} + 107562 q^{37} + 686328 q^{38} - 237360 q^{40} + 135198 q^{41} + 193536 q^{43} - 19965 q^{44} + 16422 q^{46} + 591486 q^{47} + 4461159 q^{49} + 1192245 q^{50} + 2449992 q^{52} - 79044 q^{53} + 590964 q^{55} - 752658 q^{56} + 2289930 q^{58} - 2532768 q^{59} + 6678792 q^{61} + 2660808 q^{62} - 3966303 q^{64} - 3191832 q^{65} + 7150356 q^{67} + 7821954 q^{68} + 4029120 q^{70} - 1390398 q^{71} - 6429114 q^{73} - 2507478 q^{74} - 7654728 q^{76} + 2148234 q^{77} + 6873186 q^{79} + 7556016 q^{80} - 1774590 q^{82} - 6505596 q^{83} - 16546032 q^{85} + 6519468 q^{86} - 4196643 q^{88} + 8842962 q^{89} + 3066648 q^{91} - 6921990 q^{92} - 24038238 q^{94} + 190968 q^{95} - 1764774 q^{97} - 24377397 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 70x - 194 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 6\nu - 45 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 6\nu - 92 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 2\beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 47 \) 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
−5.30133
9.97132
−3.66999
−17.9120 0 192.840 −84.6717 0 1679.06 −1161.42 0 1516.64
1.2 2.40077 0 −122.236 505.769 0 1401.07 −600.759 0 1214.23
1.3 6.51124 0 −85.6037 22.9029 0 −1466.13 −1390.83 0 149.126
\(n\): 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.e 3
3.b odd 2 1 33.8.a.d 3
12.b even 2 1 528.8.a.o 3
33.d even 2 1 363.8.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.a.d 3 3.b odd 2 1
99.8.a.e 3 1.a even 1 1 trivial
363.8.a.e 3 33.d even 2 1
528.8.a.o 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 9T_{2}^{2} - 144T_{2} + 280 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 9 T^{2} - 144 T + 280 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 444 T^{2} - 33180 T + 980800 \) Copy content Toggle raw display
$7$ \( T^{3} - 1614 T^{2} + \cdots + 3449053112 \) Copy content Toggle raw display
$11$ \( (T - 1331)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 20772 T^{2} + \cdots + 665759180384 \) Copy content Toggle raw display
$17$ \( T^{3} - 14538 T^{2} + \cdots + 11730861043168 \) Copy content Toggle raw display
$19$ \( T^{3} - 24492 T^{2} + \cdots - 5608943166816 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 200462606267008 \) Copy content Toggle raw display
$29$ \( T^{3} - 179862 T^{2} + \cdots - 61407931779072 \) Copy content Toggle raw display
$31$ \( T^{3} - 288888 T^{2} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{3} - 107562 T^{2} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{3} - 135198 T^{2} + \cdots + 12\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{3} - 193536 T^{2} + \cdots - 20\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{3} - 591486 T^{2} + \cdots - 94\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{3} + 79044 T^{2} + \cdots - 29\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{3} + 2532768 T^{2} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} - 6678792 T^{2} + \cdots - 45\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{3} - 7150356 T^{2} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{3} + 1390398 T^{2} + \cdots - 13\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{3} + 6429114 T^{2} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{3} - 6873186 T^{2} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{3} + 6505596 T^{2} + \cdots - 81\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{3} - 8842962 T^{2} + \cdots + 26\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{3} + 1764774 T^{2} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
show more
show less