Properties

Label 99.8.a.b
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: \(\Q(\sqrt{97}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
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 \(\beta = \frac{1}{2}(1 + 3\sqrt{97})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta + 90) q^{4} + (14 \beta + 90) q^{5} + ( - 42 \beta - 188) q^{7} + (37 \beta - 218) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta + 90) q^{4} + (14 \beta + 90) q^{5} + ( - 42 \beta - 188) q^{7} + (37 \beta - 218) q^{8} + ( - 104 \beta - 3052) q^{10} - 1331 q^{11} + (38 \beta - 6642) q^{13} + (230 \beta + 9156) q^{14} + (53 \beta - 19586) q^{16} + ( - 180 \beta + 5218) q^{17} + (2372 \beta + 5912) q^{19} + (1364 \beta + 11152) q^{20} + 1331 \beta q^{22} + (2050 \beta + 5808) q^{23} + (2716 \beta - 27297) q^{25} + (6604 \beta - 8284) q^{26} + ( - 4010 \beta - 26076) q^{28} + (4564 \beta - 9938) q^{29} + (184 \beta - 24112) q^{31} + (14797 \beta + 16350) q^{32} + ( - 5038 \beta + 39240) q^{34} + ( - 7000 \beta - 145104) q^{35} + (13104 \beta + 130494) q^{37} + ( - 8284 \beta - 517096) q^{38} + (796 \beta + 93304) q^{40} + ( - 29712 \beta - 363062) q^{41} + ( - 7216 \beta - 848440) q^{43} + ( - 1331 \beta - 119790) q^{44} + ( - 7858 \beta - 446900) q^{46} + ( - 42642 \beta + 612368) q^{47} + (17556 \beta - 403647) q^{49} + (24581 \beta - 592088) q^{50} + ( - 3184 \beta - 589496) q^{52} + (26758 \beta + 1064818) q^{53} + ( - 18634 \beta - 119790) q^{55} + (646 \beta - 297788) q^{56} + (5374 \beta - 994952) q^{58} + ( - 76380 \beta - 425476) q^{59} + ( - 172662 \beta - 444666) q^{61} + (23928 \beta - 40112) q^{62} + ( - 37931 \beta - 718738) q^{64} + ( - 89036 \beta - 481804) q^{65} + (2392 \beta - 1631172) q^{67} + ( - 11162 \beta + 430380) q^{68} + (152104 \beta + 1526000) q^{70} + ( - 3574 \beta + 2749544) q^{71} + ( - 118912 \beta - 2665950) q^{73} + ( - 143598 \beta - 2856672) q^{74} + (221764 \beta + 1049176) q^{76} + (55902 \beta + 250228) q^{77} + ( - 43494 \beta - 746548) q^{79} + ( - 268692 \beta - 1600984) q^{80} + (392774 \beta + 6477216) q^{82} + (255344 \beta - 4553116) q^{83} + (54332 \beta - 79740) q^{85} + (855656 \beta + 1573088) q^{86} + ( - 49247 \beta + 290158) q^{88} + (72992 \beta - 3441562) q^{89} + (270224 \beta + 900768) q^{91} + (192358 \beta + 969620) q^{92} + ( - 569726 \beta + 9295956) q^{94} + (329456 \beta + 7771424) q^{95} + ( - 481620 \beta + 5189498) q^{97} + (386091 \beta - 3827208) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 181 q^{4} + 194 q^{5} - 418 q^{7} - 399 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 181 q^{4} + 194 q^{5} - 418 q^{7} - 399 q^{8} - 6208 q^{10} - 2662 q^{11} - 13246 q^{13} + 18542 q^{14} - 39119 q^{16} + 10256 q^{17} + 14196 q^{19} + 23668 q^{20} + 1331 q^{22} + 13666 q^{23} - 51878 q^{25} - 9964 q^{26} - 56162 q^{28} - 15312 q^{29} - 48040 q^{31} + 47497 q^{32} + 73442 q^{34} - 297208 q^{35} + 274092 q^{37} - 1042476 q^{38} + 187404 q^{40} - 755836 q^{41} - 1704096 q^{43} - 240911 q^{44} - 901658 q^{46} + 1182094 q^{47} - 789738 q^{49} - 1159595 q^{50} - 1182176 q^{52} + 2156394 q^{53} - 258214 q^{55} - 594930 q^{56} - 1984530 q^{58} - 927332 q^{59} - 1061994 q^{61} - 56296 q^{62} - 1475407 q^{64} - 1052644 q^{65} - 3259952 q^{67} + 849598 q^{68} + 3204104 q^{70} + 5495514 q^{71} - 5450812 q^{73} - 5856942 q^{74} + 2320116 q^{76} + 556358 q^{77} - 1536590 q^{79} - 3470660 q^{80} + 13347206 q^{82} - 8850888 q^{83} - 105148 q^{85} + 4001832 q^{86} + 531069 q^{88} - 6810132 q^{89} + 2071760 q^{91} + 2131598 q^{92} + 18022186 q^{94} + 15872304 q^{95} + 9897376 q^{97} - 7268325 q^{98}+O(q^{100}) \) 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
5.42443
−4.42443
−15.2733 0 105.273 303.826 0 −829.478 347.112 0 −4640.42
1.2 14.2733 0 75.7267 −109.826 0 411.478 −746.112 0 −1567.58
\(n\): 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.b 2
3.b odd 2 1 33.8.a.c 2
12.b even 2 1 528.8.a.h 2
33.d even 2 1 363.8.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.a.c 2 3.b odd 2 1
99.8.a.b 2 1.a even 1 1 trivial
363.8.a.c 2 33.d even 2 1
528.8.a.h 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 218 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 218 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 194T - 33368 \) Copy content Toggle raw display
$7$ \( T^{2} + 418T - 341312 \) Copy content Toggle raw display
$11$ \( (T + 1331)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 13246 T + 43548976 \) Copy content Toggle raw display
$17$ \( T^{2} - 10256 T + 19225084 \) Copy content Toggle raw display
$19$ \( T^{2} - 14196 T - 1177576704 \) Copy content Toggle raw display
$23$ \( T^{2} - 13666 T - 870505736 \) Copy content Toggle raw display
$29$ \( T^{2} + 15312 T - 4487554116 \) Copy content Toggle raw display
$31$ \( T^{2} + 48040 T + 569571328 \) Copy content Toggle raw display
$37$ \( T^{2} - 274092 T - 18695152476 \) Copy content Toggle raw display
$41$ \( T^{2} + 755836 T - 49849727804 \) Copy content Toggle raw display
$43$ \( T^{2} + 1704096 T + 714621373632 \) Copy content Toggle raw display
$47$ \( T^{2} - 1182094 T - 47516184584 \) Copy content Toggle raw display
$53$ \( T^{2} - 2156394 T + 1006243830216 \) Copy content Toggle raw display
$59$ \( T^{2} + 927332 T - 1058263475744 \) Copy content Toggle raw display
$61$ \( T^{2} + 1061994 T - 6224547468744 \) Copy content Toggle raw display
$67$ \( T^{2} + 3259952 T + 2655573007408 \) Copy content Toggle raw display
$71$ \( T^{2} - 5495514 T + 7547380719912 \) Copy content Toggle raw display
$73$ \( T^{2} + 5450812 T + 4341768952708 \) Copy content Toggle raw display
$79$ \( T^{2} + 1536590 T + 177407563168 \) Copy content Toggle raw display
$83$ \( T^{2} + 8850888 T + 5354532740304 \) Copy content Toggle raw display
$89$ \( T^{2} + 6810132 T + 10431675116388 \) Copy content Toggle raw display
$97$ \( T^{2} - 9897376 T - 26135282253956 \) Copy content Toggle raw display
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