Properties

Label 99.8.a.c
Level $99$
Weight $8$
Character orbit 99.a
Self dual yes
Analytic conductor $30.926$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{15}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 11)
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 \(\beta = 2\sqrt{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 4) q^{2} + (8 \beta - 52) q^{4} + (20 \beta + 235) q^{5} + ( - 82 \beta - 614) q^{7} + ( - 148 \beta - 240) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 4) q^{2} + (8 \beta - 52) q^{4} + (20 \beta + 235) q^{5} + ( - 82 \beta - 614) q^{7} + ( - 148 \beta - 240) q^{8} + (315 \beta + 2140) q^{10} - 1331 q^{11} + ( - 518 \beta + 172) q^{13} + ( - 942 \beta - 7376) q^{14} + ( - 1856 \beta - 3184) q^{16} + ( - 3666 \beta + 4234) q^{17} + (2982 \beta - 17640) q^{19} + (840 \beta - 2620) q^{20} + ( - 1331 \beta - 5324) q^{22} + (4290 \beta + 30743) q^{23} + (9400 \beta + 1100) q^{25} + ( - 1900 \beta - 30392) q^{26} + ( - 648 \beta - 7432) q^{28} + ( - 11468 \beta - 89520) q^{29} + ( - 20210 \beta - 28583) q^{31} + (8336 \beta - 93376) q^{32} + ( - 10430 \beta - 203024) q^{34} + ( - 31550 \beta - 242690) q^{35} + (3748 \beta - 438849) q^{37} + ( - 5712 \beta + 108360) q^{38} + ( - 39580 \beta - 234000) q^{40} + (68870 \beta + 141808) q^{41} + ( - 12760 \beta + 137742) q^{43} + ( - 10648 \beta + 69212) q^{44} + (47903 \beta + 380372) q^{46} + ( - 15252 \beta - 831256) q^{47} + (100696 \beta - 43107) q^{49} + (38700 \beta + 568400) q^{50} + (28312 \beta - 257584) q^{52} + (66388 \beta - 808242) q^{53} + ( - 26620 \beta - 312785) q^{55} + (110552 \beta + 875520) q^{56} + ( - 135392 \beta - 1046160) q^{58} + ( - 147078 \beta + 1227065) q^{59} + (28900 \beta - 3009588) q^{61} + ( - 109423 \beta - 1326932) q^{62} + (177536 \beta + 534208) q^{64} + ( - 118290 \beta - 581180) q^{65} + (392590 \beta - 87349) q^{67} + (224504 \beta - 1979848) q^{68} + ( - 368890 \beta - 2863760) q^{70} + (452890 \beta + 575733) q^{71} + (195234 \beta + 442972) q^{73} + ( - 423857 \beta - 1530516) q^{74} + ( - 296184 \beta + 2348640) q^{76} + (109142 \beta + 817234) q^{77} + ( - 323896 \beta + 1900730) q^{79} + ( - 499840 \beta - 2975440) q^{80} + (417288 \beta + 4699432) q^{82} + ( - 175068 \beta + 1141458) q^{83} + ( - 776830 \beta - 3404210) q^{85} + (86702 \beta - 214632) q^{86} + (196988 \beta + 319440) q^{88} + (201740 \beta + 6740985) q^{89} + (303948 \beta + 2442952) q^{91} + (22864 \beta + 460564) q^{92} + ( - 892264 \beta - 4240144) q^{94} + (347970 \beta - 567000) q^{95} + ( - 174936 \beta - 34039) q^{97} + (359677 \beta + 5869332) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} - 104 q^{4} + 470 q^{5} - 1228 q^{7} - 480 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} - 104 q^{4} + 470 q^{5} - 1228 q^{7} - 480 q^{8} + 4280 q^{10} - 2662 q^{11} + 344 q^{13} - 14752 q^{14} - 6368 q^{16} + 8468 q^{17} - 35280 q^{19} - 5240 q^{20} - 10648 q^{22} + 61486 q^{23} + 2200 q^{25} - 60784 q^{26} - 14864 q^{28} - 179040 q^{29} - 57166 q^{31} - 186752 q^{32} - 406048 q^{34} - 485380 q^{35} - 877698 q^{37} + 216720 q^{38} - 468000 q^{40} + 283616 q^{41} + 275484 q^{43} + 138424 q^{44} + 760744 q^{46} - 1662512 q^{47} - 86214 q^{49} + 1136800 q^{50} - 515168 q^{52} - 1616484 q^{53} - 625570 q^{55} + 1751040 q^{56} - 2092320 q^{58} + 2454130 q^{59} - 6019176 q^{61} - 2653864 q^{62} + 1068416 q^{64} - 1162360 q^{65} - 174698 q^{67} - 3959696 q^{68} - 5727520 q^{70} + 1151466 q^{71} + 885944 q^{73} - 3061032 q^{74} + 4697280 q^{76} + 1634468 q^{77} + 3801460 q^{79} - 5950880 q^{80} + 9398864 q^{82} + 2282916 q^{83} - 6808420 q^{85} - 429264 q^{86} + 638880 q^{88} + 13481970 q^{89} + 4885904 q^{91} + 921128 q^{92} - 8480288 q^{94} - 1134000 q^{95} - 68078 q^{97} + 11738664 q^{98}+O(q^{100}) \) 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
−3.87298
3.87298
−3.74597 0 −113.968 80.0807 0 21.1693 906.403 0 −299.980
1.2 11.7460 0 9.96773 389.919 0 −1249.17 −1386.40 0 4579.98
\(n\): 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.c 2
3.b odd 2 1 11.8.a.a 2
12.b even 2 1 176.8.a.d 2
15.d odd 2 1 275.8.a.a 2
21.c even 2 1 539.8.a.a 2
33.d even 2 1 121.8.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.8.a.a 2 3.b odd 2 1
99.8.a.c 2 1.a even 1 1 trivial
121.8.a.b 2 33.d even 2 1
176.8.a.d 2 12.b even 2 1
275.8.a.a 2 15.d odd 2 1
539.8.a.a 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 8T_{2} - 44 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 8T - 44 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 470T + 31225 \) Copy content Toggle raw display
$7$ \( T^{2} + 1228T - 26444 \) Copy content Toggle raw display
$11$ \( (T + 1331)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 344 T - 16069856 \) Copy content Toggle raw display
$17$ \( T^{2} - 8468 T - 788446604 \) Copy content Toggle raw display
$19$ \( T^{2} + 35280 T - 222369840 \) Copy content Toggle raw display
$23$ \( T^{2} - 61486 T - 159113951 \) Copy content Toggle raw display
$29$ \( T^{2} + 179040 T + 122928960 \) Copy content Toggle raw display
$31$ \( T^{2} + 57166 T - 23689658111 \) Copy content Toggle raw display
$37$ \( T^{2} + 877698 T + 191745594561 \) Copy content Toggle raw display
$41$ \( T^{2} - 283616 T - 264475105136 \) Copy content Toggle raw display
$43$ \( T^{2} - 275484 T + 9203802564 \) Copy content Toggle raw display
$47$ \( T^{2} + 1662512 T + 677029127296 \) Copy content Toggle raw display
$53$ \( T^{2} + 1616484 T + 388813137924 \) Copy content Toggle raw display
$59$ \( T^{2} - 2454130 T + 207772229185 \) Copy content Toggle raw display
$61$ \( T^{2} + 6019176 T + 9007507329744 \) Copy content Toggle raw display
$67$ \( T^{2} + 174698 T - 9239984638199 \) Copy content Toggle raw display
$71$ \( T^{2} - 1151466 T - 11975092638711 \) Copy content Toggle raw display
$73$ \( T^{2} - 885944 T - 2090754692576 \) Copy content Toggle raw display
$79$ \( T^{2} - 3801460 T - 2681742596060 \) Copy content Toggle raw display
$83$ \( T^{2} - 2282916 T - 536001911676 \) Copy content Toggle raw display
$89$ \( T^{2} - 13481970 T + 42998937114225 \) Copy content Toggle raw display
$97$ \( T^{2} + 68078 T - 1834997592239 \) Copy content Toggle raw display
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