Properties

Label 99.8.a.f
Level $99$
Weight $8$
Character orbit 99.a
Self dual yes
Analytic conductor $30.926$
Analytic rank $1$
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: \(1\)
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: \( x^{4} - x^{3} - 510x^{2} - 1544x + 28880 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 4) q^{2} + (\beta_{2} - 2 \beta_1 + 142) q^{4} + ( - \beta_{3} - 10 \beta_1 - 74) q^{5} + ( - 2 \beta_{3} - 2 \beta_{2} - 32 \beta_1 + 230) q^{7} + (6 \beta_{3} - \beta_{2} + 126 \beta_1 - 646) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 4) q^{2} + (\beta_{2} - 2 \beta_1 + 142) q^{4} + ( - \beta_{3} - 10 \beta_1 - 74) q^{5} + ( - 2 \beta_{3} - 2 \beta_{2} - 32 \beta_1 + 230) q^{7} + (6 \beta_{3} - \beta_{2} + 126 \beta_1 - 646) q^{8} + (14 \beta_{3} - 20 \beta_{2} - 70 \beta_1 - 2224) q^{10} - 1331 q^{11} + (15 \beta_{3} - 28 \beta_{2} + 206 \beta_1 - 514) q^{13} + (16 \beta_{3} - 54 \beta_{2} - 18 \beta_1 - 8844) q^{14} + ( - 90 \beta_{3} + 57 \beta_{2} - 398 \beta_1 + 16374) q^{16} + (21 \beta_{3} + 14 \beta_{2} - 842 \beta_1 - 8000) q^{17} + (43 \beta_{3} - 126 \beta_{2} + 810 \beta_1 - 3430) q^{19} + ( - 188 \beta_{3} + 50 \beta_{2} - 3740 \beta_1 + 1948) q^{20} + ( - 1331 \beta_1 + 5324) q^{22} + (149 \beta_{3} - 12 \beta_{2} - 3414 \beta_1 - 27864) q^{23} + (106 \beta_{3} + 280 \beta_{2} - 1260 \beta_1 + 18599) q^{25} + ( - 378 \beta_{3} + 328 \beta_{2} - 3710 \beta_1 + 56376) q^{26} + ( - 292 \beta_{3} + 344 \beta_{2} - 11432 \beta_1 + 5472) q^{28} + (349 \beta_{3} + 110 \beta_{2} + 3254 \beta_1 + 25452) q^{29} + (532 \beta_{3} + 744 \beta_{2} + 3832 \beta_1 - 7016) q^{31} + (834 \beta_{3} - 1113 \beta_{2} + 8222 \beta_1 - 86774) q^{32} + ( - 210 \beta_{3} - 618 \beta_{2} - 8564 \beta_1 - 183436) q^{34} + ( - 234 \beta_{3} + 780 \beta_{2} + 100 \beta_1 + 180884) q^{35} + ( - 610 \beta_{3} - 364 \beta_{2} + 8708 \beta_1 - 126982) q^{37} + ( - 1358 \beta_{3} + 1114 \beta_{2} - 17458 \beta_1 + 228932) q^{38} + (1140 \beta_{3} - 3010 \beta_{2} + 13740 \beta_1 - 673420) q^{40} + (1477 \beta_{3} - 310 \beta_{2} - 13602 \beta_1 - 180316) q^{41} + (119 \beta_{3} + 2814 \beta_{2} + 25034 \beta_1 - 56034) q^{43} + ( - 1331 \beta_{2} + 2662 \beta_1 - 189002) q^{44} + ( - 2158 \beta_{3} - 1936 \beta_{2} - 39660 \beta_1 - 757696) q^{46} + ( - 1677 \beta_{3} - 3428 \beta_{2} - 11674 \beta_1 - 496912) q^{47} + ( - 1336 \beta_{3} + 540 \beta_{2} + 4216 \beta_1 - 193675) q^{49} + (196 \beta_{3} + 80 \beta_{2} + 46015 \beta_1 - 419516) q^{50} + (5340 \beta_{3} - 3578 \beta_{2} + 69708 \beta_1 - 1121388) q^{52} + ( - 1883 \beta_{3} + 3908 \beta_{2} + 74378 \beta_1 - 250110) q^{53} + (1331 \beta_{3} + 13310 \beta_1 + 98494) q^{55} + (4104 \beta_{3} - 7096 \beta_{2} + 31824 \beta_1 - 1815952) q^{56} + ( - 4226 \beta_{3} + 6854 \beta_{2} + 36344 \beta_1 + 708708) q^{58} + ( - 582 \beta_{3} + 11944 \beta_{2} + 54068 \beta_1 - 348820) q^{59} + ( - 1057 \beta_{3} + 56 \beta_{2} - 66010 \beta_1 + 1016210) q^{61} + ( - 2984 \beta_{3} + 9896 \beta_{2} + 74184 \beta_1 + 929744) q^{62} + ( - 6834 \beta_{3} + 8153 \beta_{2} - 168510 \beta_1 + 414198) q^{64} + (1322 \beta_{3} - 5600 \beta_{2} + 100100 \beta_1 - 1679452) q^{65} + (3760 \beta_{3} - 1048 \beta_{2} - 36688 \beta_1 + 439412) q^{67} + ( - 3456 \beta_{3} - 13074 \beta_{2} - 159436 \beta_1 - 362636) q^{68} + (7956 \beta_{3} - 1460 \beta_{2} + 277180 \beta_1 - 757416) q^{70} + (7581 \beta_{3} - 4004 \beta_{2} - 62262 \beta_1 - 1378400) q^{71} + ( - 6122 \beta_{3} + 8684 \beta_{2} - 94460 \beta_1 + 1425918) q^{73} + (6356 \beta_{3} + 2244 \beta_{2} - 137150 \beta_1 + 2761808) q^{74} + (20192 \beta_{3} - 13796 \beta_{2} + 252152 \beta_1 - 4975208) q^{76} + (2662 \beta_{3} + 2662 \beta_{2} + 42592 \beta_1 - 306130) q^{77} + (16 \beta_{3} - 10722 \beta_{2} - 18508 \beta_1 - 1925610) q^{79} + ( - 9956 \beta_{3} + 15730 \beta_{2} - 543740 \beta_1 + 6158316) q^{80} + ( - 22538 \beta_{3} + 858 \beta_{2} - 278928 \beta_1 - 2737764) q^{82} + ( - 16670 \beta_{3} - 19220 \beta_{2} + 186716 \beta_1 - 909448) q^{83} + ( - 3556 \beta_{3} + 15300 \beta_{2} + 202160 \beta_1 + 1430656) q^{85} + (15218 \beta_{3} + 29038 \beta_{2} + 317602 \beta_1 + 6349644) q^{86} + ( - 7986 \beta_{3} + 1331 \beta_{2} - 167706 \beta_1 + 859826) q^{88} + ( - 6580 \beta_{3} + 18280 \beta_{2} - 107288 \beta_1 - 1121490) q^{89} + (6982 \beta_{3} - 26036 \beta_{2} + 243332 \beta_1 - 2325516) q^{91} + ( - 476 \beta_{3} - 61640 \beta_{2} - 572808 \beta_1 - 3274352) q^{92} + (2910 \beta_{3} - 31872 \beta_{2} - 877660 \beta_1 - 662912) q^{94} + (9130 \beta_{3} - 20900 \beta_{2} + 398540 \beta_1 - 5561860) q^{95} + (27384 \beta_{3} - 13116 \beta_{2} - 269112 \beta_1 + 414422) q^{97} + (21944 \beta_{3} - 8604 \beta_{2} - 90539 \beta_1 + 1828004) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 15 q^{2} + 565 q^{4} - 306 q^{5} + 890 q^{7} - 2457 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 15 q^{2} + 565 q^{4} - 306 q^{5} + 890 q^{7} - 2457 q^{8} - 8946 q^{10} - 5324 q^{11} - 1822 q^{13} - 35340 q^{14} + 65041 q^{16} - 32856 q^{17} - 12784 q^{19} + 4002 q^{20} + 19965 q^{22} - 114858 q^{23} + 72856 q^{25} + 221466 q^{26} + 10112 q^{28} + 104952 q^{29} - 24976 q^{31} - 337761 q^{32} - 741690 q^{34} + 722856 q^{35} - 498856 q^{37} + 897156 q^{38} - 2676930 q^{40} - 734556 q^{41} - 201916 q^{43} - 752015 q^{44} - 3068508 q^{46} - 1995894 q^{47} - 771024 q^{49} - 1632129 q^{50} - 4412266 q^{52} - 929970 q^{53} + 407286 q^{55} - 7224888 q^{56} + 2864322 q^{58} - 1353156 q^{59} + 3998774 q^{61} + 3783264 q^{62} + 1480129 q^{64} - 6612108 q^{65} + 1722008 q^{67} - 1596906 q^{68} - 2751024 q^{70} - 5571858 q^{71} + 5600528 q^{73} + 10907838 q^{74} - 19634884 q^{76} - 1184590 q^{77} - 7710226 q^{79} + 24073794 q^{80} - 11230842 q^{82} - 3431856 q^{83} + 5909484 q^{85} + 25687140 q^{86} + 3270267 q^{88} - 4611528 q^{89} - 9032696 q^{91} - 13608576 q^{92} - 3497436 q^{94} - 21828000 q^{95} + 1401692 q^{97} + 7230081 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 510x^{2} - 1544x + 28880 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 6\nu - 254 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 11\nu^{2} - 340\nu + 1352 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 6\beta _1 + 254 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{3} + 11\beta_{2} + 406\beta _1 + 1442 \) 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
−17.6807
−11.0316
6.33327
23.3791
−21.6807 0 342.055 369.874 0 1000.53 −4640.87 0 −8019.14
1.2 −15.0316 0 97.9505 −367.278 0 −91.9512 451.694 0 5520.80
1.3 2.33327 0 −122.556 27.4165 0 860.612 −584.614 0 63.9699
1.4 19.3791 0 247.551 −336.012 0 −879.192 2316.79 0 −6511.62
\(n\): 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.f 4
3.b odd 2 1 33.8.a.e 4
12.b even 2 1 528.8.a.r 4
33.d even 2 1 363.8.a.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.a.e 4 3.b odd 2 1
99.8.a.f 4 1.a even 1 1 trivial
363.8.a.f 4 33.d even 2 1
528.8.a.r 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 15T_{2}^{3} - 426T_{2}^{2} - 5416T_{2} + 14736 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 15 T^{3} - 426 T^{2} + \cdots + 14736 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 306 T^{3} + \cdots + 1251456000 \) Copy content Toggle raw display
$7$ \( T^{4} - 890 T^{3} + \cdots + 69611139776 \) Copy content Toggle raw display
$11$ \( (T + 1331)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots - 161846909982208 \) Copy content Toggle raw display
$17$ \( T^{4} + 32856 T^{3} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{4} + 12784 T^{3} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + 114858 T^{3} + \cdots - 49\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{4} - 104952 T^{3} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + 24976 T^{3} + \cdots + 59\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{4} + 498856 T^{3} + \cdots + 37\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{4} + 734556 T^{3} + \cdots - 49\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{4} + 201916 T^{3} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{4} + 1995894 T^{3} + \cdots - 25\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{4} + 929970 T^{3} + \cdots - 80\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{4} + 1353156 T^{3} + \cdots + 12\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{4} - 3998774 T^{3} + \cdots - 13\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{4} - 1722008 T^{3} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{4} + 5571858 T^{3} + \cdots - 50\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{4} - 5600528 T^{3} + \cdots + 45\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{4} + 7710226 T^{3} + \cdots - 88\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{4} + 3431856 T^{3} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{4} + 4611528 T^{3} + \cdots - 11\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{4} - 1401692 T^{3} + \cdots - 11\!\cdots\!36 \) Copy content Toggle raw display
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