Properties

Label 99.8.a.g
Level $99$
Weight $8$
Character orbit 99.a
Self dual yes
Analytic conductor $30.926$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 341x^{2} + 1417x - 1412 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{3} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 + 151) q^{4} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 135) q^{5} + ( - 6 \beta_{3} - 4 \beta_{2} + 24 \beta_1 + 42) q^{7} + ( - 14 \beta_{3} + 14 \beta_{2} + 120 \beta_1 + 98) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 + 151) q^{4} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 135) q^{5} + ( - 6 \beta_{3} - 4 \beta_{2} + 24 \beta_1 + 42) q^{7} + ( - 14 \beta_{3} + 14 \beta_{2} + 120 \beta_1 + 98) q^{8} + (14 \beta_{3} + 28 \beta_{2} - 61 \beta_1 + 364) q^{10} + 1331 q^{11} + (50 \beta_{3} - 20 \beta_{2} + 280 \beta_1 + 1080) q^{13} + (112 \beta_{3} - 28 \beta_{2} - 398 \beta_1 + 7532) q^{14} + (174 \beta_{3} + 174 \beta_{2} + 314 \beta_1 + 13186) q^{16} + ( - 116 \beta_{3} - 24 \beta_{2} + 332 \beta_1 - 13598) q^{17} + ( - 220 \beta_{3} + 40 \beta_{2} + 500 \beta_1 + 16896) q^{19} + ( - 157 \beta_{3} + 47 \beta_{2} + 2175 \beta_1 - 4023) q^{20} + 1331 \beta_1 q^{22} + (263 \beta_{3} - 158 \beta_{2} + 3134 \beta_1 + 2359) q^{23} + (255 \beta_{3} - 822 \beta_{2} + 1586 \beta_1 + 5902) q^{25} + ( - 400 \beta_{3} + 20 \beta_{2} + 2240 \beta_1 + 77820) q^{26} + ( - 1170 \beta_{3} - 250 \beta_{2} + 6974 \beta_1 - 119190) q^{28} + ( - 22 \beta_{3} - 1252 \beta_{2} + 388 \beta_1 - 58212) q^{29} + ( - 511 \beta_{3} - 66 \beta_{2} - 3966 \beta_1 + 47353) q^{31} + ( - 504 \beta_{3} + 784 \beta_{2} + 14844 \beta_1 + 43568) q^{32} + (1980 \beta_{3} + 20 \beta_{2} - 19250 \beta_1 + 102124) q^{34} + (462 \beta_{3} + 1860 \beta_{2} - 10376 \beta_1 - 38874) q^{35} + ( - 1185 \beta_{3} + 1890 \beta_{2} + 15458 \beta_1 + 31205) q^{37} + (3540 \beta_{3} + 1020 \beta_{2} + 10836 \beta_1 + 146820) q^{38} + (2534 \beta_{3} - 798 \beta_{2} + 2312 \beta_1 + 563150) q^{40} + ( - 2458 \beta_{3} + 292 \beta_{2} - 10328 \beta_1 - 73148) q^{41} + ( - 4094 \beta_{3} + 1004 \beta_{2} + 5036 \beta_1 + 174958) q^{43} + (1331 \beta_{3} + 1331 \beta_{2} + 1331 \beta_1 + 200981) q^{44} + ( - 390 \beta_{3} + 1080 \beta_{2} + 7165 \beta_1 + 879408) q^{46} + ( - 2104 \beta_{3} - 1536 \beta_{2} + 6520 \beta_1 + 432128) q^{47} + (4060 \beta_{3} - 1960 \beta_{2} - 32760 \beta_1 + 36485) q^{49} + ( - 1162 \beta_{3} - 9100 \beta_{2} - 28344 \beta_1 + 530964) q^{50} + (1420 \beta_{3} + 5060 \beta_{2} + 29340 \beta_1 + 506620) q^{52} + (740 \beta_{3} + 2280 \beta_{2} - 47296 \beta_1 - 275050) q^{53} + ( - 1331 \beta_{3} + 2662 \beta_{2} + 2662 \beta_1 - 179685) q^{55} + (9268 \beta_{3} + 7308 \beta_{2} - 122072 \beta_1 + 1078420) q^{56} + (1948 \beta_{3} - 15888 \beta_{2} - 128816 \beta_1 + 265984) q^{58} + ( - 8599 \beta_{3} - 9434 \beta_{2} - 13054 \beta_1 + 1166653) q^{59} + (3882 \beta_{3} - 2468 \beta_{2} - 55548 \beta_1 + 79240) q^{61} + (3254 \beta_{3} - 4824 \beta_{2} + 19251 \beta_1 - 1069648) q^{62} + ( - 1156 \beta_{3} + 2764 \beta_{2} + 41964 \beta_1 + 2383892) q^{64} + (160 \beta_{3} + 16640 \beta_{2} - 9500 \beta_1 - 1135660) q^{65} + (7747 \beta_{3} - 3662 \beta_{2} - 31930 \beta_1 - 839209) q^{67} + ( - 32142 \beta_{3} - 15918 \beta_{2} + 120698 \beta_1 - 3743586) q^{68} + ( - 18704 \beta_{3} + 13804 \beta_{2} + 73390 \beta_1 - 3153276) q^{70} + ( - 9283 \beta_{3} - 1178 \beta_{2} - 189438 \beta_1 + 852109) q^{71} + ( - 9602 \beta_{3} - 1868 \beta_{2} + 116928 \beta_1 + 2864624) q^{73} + (30158 \beta_{3} + 40028 \beta_{2} + 105103 \beta_1 + 4142892) q^{74} + ( - 11584 \beta_{3} + 18976 \beta_{2} + 292376 \beta_1 + 534816) q^{76} + ( - 7986 \beta_{3} - 5324 \beta_{2} + 31944 \beta_1 + 55902) q^{77} + ( - 14902 \beta_{3} - 5972 \beta_{2} - 174612 \beta_1 + 139338) q^{79} + ( - 12270 \beta_{3} - 14078 \beta_{2} + 343734 \beta_1 + 1117838) q^{80} + (23792 \beta_{3} - 6532 \beta_{2} - 165444 \beta_1 - 2780364) q^{82} + (28086 \beta_{3} - 34396 \beta_{2} - 159940 \beta_1 + 1070818) q^{83} + (22774 \beta_{3} - 12396 \beta_{2} - 151824 \beta_1 + 2587018) q^{85} + (61348 \beta_{3} + 18088 \beta_{2} + 72458 \beta_1 + 1508808) q^{86} + ( - 18634 \beta_{3} + 18634 \beta_{2} + 159720 \beta_1 + 130438) q^{88} + (26089 \beta_{3} - 20186 \beta_{2} + 147254 \beta_1 - 4554729) q^{89} + (28200 \beta_{3} - 34080 \beta_{2} + 162360 \beta_1 - 267800) q^{91} + ( - 22119 \beta_{3} + 41429 \beta_{2} + 530301 \beta_1 + 1583923) q^{92} + (37512 \beta_{3} - 13448 \beta_{2} + 268472 \beta_1 + 2128904) q^{94} + (744 \beta_{3} + 32832 \beta_{2} - 83148 \beta_1 + 1407720) q^{95} + (49773 \beta_{3} + 61502 \beta_{2} - 14314 \beta_1 + 2853399) q^{97} + ( - 87640 \beta_{3} - 58240 \beta_{2} + 56365 \beta_1 - 9122400) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 604 q^{4} - 537 q^{5} + 170 q^{7} + 420 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 604 q^{4} - 537 q^{5} + 170 q^{7} + 420 q^{8} + 1470 q^{10} + 5324 q^{11} + 4250 q^{13} + 29988 q^{14} + 52744 q^{16} - 54300 q^{17} + 67844 q^{19} - 15888 q^{20} + 9015 q^{23} + 22531 q^{25} + 311700 q^{26} - 475840 q^{28} - 234078 q^{29} + 189857 q^{31} + 175560 q^{32} + 406536 q^{34} - 154098 q^{35} + 127895 q^{37} + 584760 q^{38} + 2249268 q^{40} - 289842 q^{41} + 704930 q^{43} + 803924 q^{44} + 3519102 q^{46} + 1729080 q^{47} + 139920 q^{49} + 2115918 q^{50} + 2030120 q^{52} - 1098660 q^{53} - 714747 q^{55} + 4311720 q^{56} + 1046100 q^{58} + 4665777 q^{59} + 310610 q^{61} - 4286670 q^{62} + 9539488 q^{64} - 4526160 q^{65} - 3368245 q^{67} - 14958120 q^{68} - 12580596 q^{70} + 3416541 q^{71} + 11466230 q^{73} + 16581438 q^{74} + 2169824 q^{76} + 226270 q^{77} + 566282 q^{79} + 4469544 q^{80} - 11151780 q^{82} + 4220790 q^{83} + 10312902 q^{85} + 5991972 q^{86} + 559020 q^{88} - 18265191 q^{89} - 1133480 q^{91} + 6399240 q^{92} + 8464656 q^{94} + 5662968 q^{95} + 11425325 q^{97} - 36460200 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 341x^{2} + 1417x - 1412 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 321\nu + 204 ) / 28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} - 4\nu^{2} + 573\nu - 260 ) / 28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} - \nu^{2} + 339\nu - 721 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 2 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 2\beta_{2} + 26\beta _1 + 513 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{3} + 115\beta_{2} + 87\beta _1 - 2450 ) / 3 \) 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
2.58394
1.64802
−19.8969
16.6649
−20.7673 0 303.281 −223.515 0 −1410.66 −3640.12 0 4641.80
1.2 −11.0598 0 −5.68000 60.1766 0 698.069 1478.48 0 −665.544
1.3 10.6261 0 −15.0863 −512.130 0 973.904 −1520.45 0 −5441.94
1.4 21.2011 0 321.485 138.468 0 −91.3115 4102.09 0 2935.68
\(n\): 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.g 4
3.b odd 2 1 11.8.a.b 4
12.b even 2 1 176.8.a.j 4
15.d odd 2 1 275.8.a.b 4
21.c even 2 1 539.8.a.b 4
33.d even 2 1 121.8.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.8.a.b 4 3.b odd 2 1
99.8.a.g 4 1.a even 1 1 trivial
121.8.a.c 4 33.d even 2 1
176.8.a.j 4 12.b even 2 1
275.8.a.b 4 15.d odd 2 1
539.8.a.b 4 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 558T_{2}^{2} - 140T_{2} + 51744 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 558 T^{2} - 140 T + 51744 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 537 T^{3} + \cdots + 953818350 \) Copy content Toggle raw display
$7$ \( T^{4} - 170 T^{3} + \cdots + 87571440704 \) Copy content Toggle raw display
$11$ \( (T - 1331)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots - 518474088880000 \) Copy content Toggle raw display
$17$ \( T^{4} + 54300 T^{3} + \cdots - 58\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{4} - 67844 T^{3} + \cdots - 31\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{4} - 9015 T^{3} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{4} + 234078 T^{3} + \cdots - 41\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{4} - 189857 T^{3} + \cdots - 61\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{4} - 127895 T^{3} + \cdots + 41\!\cdots\!94 \) Copy content Toggle raw display
$41$ \( T^{4} + 289842 T^{3} + \cdots - 66\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{4} - 704930 T^{3} + \cdots + 28\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{4} - 1729080 T^{3} + \cdots - 74\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{4} + 1098660 T^{3} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{4} - 4665777 T^{3} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{4} - 310610 T^{3} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{4} + 3368245 T^{3} + \cdots - 84\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{4} - 3416541 T^{3} + \cdots + 59\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{4} - 11466230 T^{3} + \cdots - 60\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{4} - 566282 T^{3} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{4} - 4220790 T^{3} + \cdots + 70\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{4} + 18265191 T^{3} + \cdots - 24\!\cdots\!34 \) Copy content Toggle raw display
$97$ \( T^{4} - 11425325 T^{3} + \cdots + 46\!\cdots\!46 \) Copy content Toggle raw display
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