Properties

Label 99.4.a.f
Level $99$
Weight $4$
Character orbit 99.a
Self dual yes
Analytic conductor $5.841$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 99.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.84118909057\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{97})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta + 16) q^{4} + (2 \beta + 6) q^{5} + ( - 4 \beta + 14) q^{7} + ( - 9 \beta - 24) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta + 16) q^{4} + (2 \beta + 6) q^{5} + ( - 4 \beta + 14) q^{7} + ( - 9 \beta - 24) q^{8} + ( - 8 \beta - 48) q^{10} + 11 q^{11} + (2 \beta + 14) q^{13} + ( - 10 \beta + 96) q^{14} + (25 \beta + 88) q^{16} + (14 \beta - 60) q^{17} + ( - 2 \beta + 26) q^{19} + (40 \beta + 144) q^{20} - 11 \beta q^{22} + (10 \beta - 72) q^{23} + (28 \beta + 7) q^{25} + ( - 16 \beta - 48) q^{26} + ( - 54 \beta + 128) q^{28} + (6 \beta + 96) q^{29} + (8 \beta + 176) q^{31} + ( - 41 \beta - 408) q^{32} + (46 \beta - 336) q^{34} + ( - 4 \beta - 108) q^{35} + (52 \beta - 190) q^{37} + ( - 24 \beta + 48) q^{38} + ( - 120 \beta - 576) q^{40} + (14 \beta + 384) q^{41} + ( - 26 \beta + 206) q^{43} + (11 \beta + 176) q^{44} + (62 \beta - 240) q^{46} + ( - 74 \beta - 96) q^{47} + ( - 96 \beta + 237) q^{49} + ( - 35 \beta - 672) q^{50} + (48 \beta + 272) q^{52} + (54 \beta + 234) q^{53} + (22 \beta + 66) q^{55} + (6 \beta + 528) q^{56} + ( - 102 \beta - 144) q^{58} + (100 \beta + 36) q^{59} + (34 \beta - 406) q^{61} + ( - 184 \beta - 192) q^{62} + (249 \beta + 280) q^{64} + (44 \beta + 180) q^{65} + ( - 96 \beta - 340) q^{67} + (178 \beta - 624) q^{68} + (112 \beta + 96) q^{70} + ( - 54 \beta - 288) q^{71} + ( - 28 \beta + 662) q^{73} + (138 \beta - 1248) q^{74} + ( - 8 \beta + 368) q^{76} + ( - 44 \beta + 154) q^{77} + (144 \beta + 254) q^{79} + (376 \beta + 1728) q^{80} + ( - 398 \beta - 336) q^{82} + ( - 156 \beta + 240) q^{83} + ( - 8 \beta + 312) q^{85} + ( - 180 \beta + 624) q^{86} + ( - 99 \beta - 264) q^{88} + ( - 72 \beta + 414) q^{89} + ( - 36 \beta + 4) q^{91} + (98 \beta - 912) q^{92} + (170 \beta + 1776) q^{94} + (36 \beta + 60) q^{95} + (192 \beta - 322) q^{97} + ( - 141 \beta + 2304) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 33 q^{4} + 14 q^{5} + 24 q^{7} - 57 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 33 q^{4} + 14 q^{5} + 24 q^{7} - 57 q^{8} - 104 q^{10} + 22 q^{11} + 30 q^{13} + 182 q^{14} + 201 q^{16} - 106 q^{17} + 50 q^{19} + 328 q^{20} - 11 q^{22} - 134 q^{23} + 42 q^{25} - 112 q^{26} + 202 q^{28} + 198 q^{29} + 360 q^{31} - 857 q^{32} - 626 q^{34} - 220 q^{35} - 328 q^{37} + 72 q^{38} - 1272 q^{40} + 782 q^{41} + 386 q^{43} + 363 q^{44} - 418 q^{46} - 266 q^{47} + 378 q^{49} - 1379 q^{50} + 592 q^{52} + 522 q^{53} + 154 q^{55} + 1062 q^{56} - 390 q^{58} + 172 q^{59} - 778 q^{61} - 568 q^{62} + 809 q^{64} + 404 q^{65} - 776 q^{67} - 1070 q^{68} + 304 q^{70} - 630 q^{71} + 1296 q^{73} - 2358 q^{74} + 728 q^{76} + 264 q^{77} + 652 q^{79} + 3832 q^{80} - 1070 q^{82} + 324 q^{83} + 616 q^{85} + 1068 q^{86} - 627 q^{88} + 756 q^{89} - 28 q^{91} - 1726 q^{92} + 3722 q^{94} + 156 q^{95} - 452 q^{97} + 4467 q^{98}+O(q^{100}) \) 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.

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
−5.42443 0 21.4244 16.8489 0 −7.69772 −72.8199 0 −91.3954
1.2 4.42443 0 11.5756 −2.84886 0 31.6977 15.8199 0 −12.6046
\(n\): 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 99.4.a.f 2
3.b odd 2 1 33.4.a.c 2
4.b odd 2 1 1584.4.a.bj 2
5.b even 2 1 2475.4.a.p 2
11.b odd 2 1 1089.4.a.u 2
12.b even 2 1 528.4.a.p 2
15.d odd 2 1 825.4.a.l 2
15.e even 4 2 825.4.c.h 4
21.c even 2 1 1617.4.a.k 2
24.f even 2 1 2112.4.a.bg 2
24.h odd 2 1 2112.4.a.bn 2
33.d even 2 1 363.4.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.c 2 3.b odd 2 1
99.4.a.f 2 1.a even 1 1 trivial
363.4.a.i 2 33.d even 2 1
528.4.a.p 2 12.b even 2 1
825.4.a.l 2 15.d odd 2 1
825.4.c.h 4 15.e even 4 2
1089.4.a.u 2 11.b odd 2 1
1584.4.a.bj 2 4.b odd 2 1
1617.4.a.k 2 21.c even 2 1
2112.4.a.bg 2 24.f even 2 1
2112.4.a.bn 2 24.h odd 2 1
2475.4.a.p 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 24 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 24 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 14T - 48 \) Copy content Toggle raw display
$7$ \( T^{2} - 24T - 244 \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 30T + 128 \) Copy content Toggle raw display
$17$ \( T^{2} + 106T - 1944 \) Copy content Toggle raw display
$19$ \( T^{2} - 50T + 528 \) Copy content Toggle raw display
$23$ \( T^{2} + 134T + 2064 \) Copy content Toggle raw display
$29$ \( T^{2} - 198T + 8928 \) Copy content Toggle raw display
$31$ \( T^{2} - 360T + 30848 \) Copy content Toggle raw display
$37$ \( T^{2} + 328T - 38676 \) Copy content Toggle raw display
$41$ \( T^{2} - 782T + 148128 \) Copy content Toggle raw display
$43$ \( T^{2} - 386T + 20856 \) Copy content Toggle raw display
$47$ \( T^{2} + 266T - 115104 \) Copy content Toggle raw display
$53$ \( T^{2} - 522T - 2592 \) Copy content Toggle raw display
$59$ \( T^{2} - 172T - 235104 \) Copy content Toggle raw display
$61$ \( T^{2} + 778T + 123288 \) Copy content Toggle raw display
$67$ \( T^{2} + 776T - 72944 \) Copy content Toggle raw display
$71$ \( T^{2} + 630T + 28512 \) Copy content Toggle raw display
$73$ \( T^{2} - 1296 T + 400892 \) Copy content Toggle raw display
$79$ \( T^{2} - 652T - 396572 \) Copy content Toggle raw display
$83$ \( T^{2} - 324T - 563904 \) Copy content Toggle raw display
$89$ \( T^{2} - 756T + 17172 \) Copy content Toggle raw display
$97$ \( T^{2} + 452T - 842876 \) Copy content Toggle raw display
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