Properties

Label 99.4.e.a
Level $99$
Weight $4$
Character orbit 99.e
Analytic conductor $5.841$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 99.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.84118909057\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 4 \zeta_{6} + 4) q^{2} + (3 \zeta_{6} - 6) q^{3} - 8 \zeta_{6} q^{4} + 19 \zeta_{6} q^{5} + (24 \zeta_{6} - 12) q^{6} + ( - 26 \zeta_{6} + 26) q^{7} + ( - 27 \zeta_{6} + 27) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 \zeta_{6} + 4) q^{2} + (3 \zeta_{6} - 6) q^{3} - 8 \zeta_{6} q^{4} + 19 \zeta_{6} q^{5} + (24 \zeta_{6} - 12) q^{6} + ( - 26 \zeta_{6} + 26) q^{7} + ( - 27 \zeta_{6} + 27) q^{9} + 76 q^{10} + (11 \zeta_{6} - 11) q^{11} + (24 \zeta_{6} + 24) q^{12} + 56 \zeta_{6} q^{13} - 104 \zeta_{6} q^{14} + ( - 57 \zeta_{6} - 57) q^{15} + ( - 64 \zeta_{6} + 64) q^{16} + 104 q^{17} - 108 \zeta_{6} q^{18} - 96 q^{19} + ( - 152 \zeta_{6} + 152) q^{20} + (156 \zeta_{6} - 78) q^{21} + 44 \zeta_{6} q^{22} - 40 \zeta_{6} q^{23} + (236 \zeta_{6} - 236) q^{25} + 224 q^{26} + (162 \zeta_{6} - 81) q^{27} - 208 q^{28} + (18 \zeta_{6} - 18) q^{29} + (228 \zeta_{6} - 456) q^{30} - 49 \zeta_{6} q^{31} - 256 \zeta_{6} q^{32} + ( - 66 \zeta_{6} + 33) q^{33} + ( - 416 \zeta_{6} + 416) q^{34} + 494 q^{35} - 216 q^{36} + 75 q^{37} + (384 \zeta_{6} - 384) q^{38} + ( - 168 \zeta_{6} - 168) q^{39} - 296 \zeta_{6} q^{41} + (312 \zeta_{6} + 312) q^{42} + (372 \zeta_{6} - 372) q^{43} + 88 q^{44} + 513 q^{45} - 160 q^{46} + ( - 149 \zeta_{6} + 149) q^{47} + (384 \zeta_{6} - 192) q^{48} - 333 \zeta_{6} q^{49} + 944 \zeta_{6} q^{50} + (312 \zeta_{6} - 624) q^{51} + ( - 448 \zeta_{6} + 448) q^{52} - 417 q^{53} + (324 \zeta_{6} + 324) q^{54} - 209 q^{55} + ( - 288 \zeta_{6} + 576) q^{57} + 72 \zeta_{6} q^{58} + 17 \zeta_{6} q^{59} + (912 \zeta_{6} - 456) q^{60} + (90 \zeta_{6} - 90) q^{61} - 196 q^{62} - 702 \zeta_{6} q^{63} - 512 q^{64} + (1064 \zeta_{6} - 1064) q^{65} + ( - 132 \zeta_{6} - 132) q^{66} - 1073 \zeta_{6} q^{67} - 832 \zeta_{6} q^{68} + (120 \zeta_{6} + 120) q^{69} + ( - 1976 \zeta_{6} + 1976) q^{70} - 285 q^{71} - 962 q^{73} + ( - 300 \zeta_{6} + 300) q^{74} + ( - 1416 \zeta_{6} + 708) q^{75} + 768 \zeta_{6} q^{76} + 286 \zeta_{6} q^{77} + (672 \zeta_{6} - 1344) q^{78} + (596 \zeta_{6} - 596) q^{79} + 1216 q^{80} - 729 \zeta_{6} q^{81} - 1184 q^{82} + ( - 498 \zeta_{6} + 498) q^{83} + ( - 624 \zeta_{6} + 1248) q^{84} + 1976 \zeta_{6} q^{85} + 1488 \zeta_{6} q^{86} + ( - 108 \zeta_{6} + 54) q^{87} + 1230 q^{89} + ( - 2052 \zeta_{6} + 2052) q^{90} + 1456 q^{91} + (320 \zeta_{6} - 320) q^{92} + (147 \zeta_{6} + 147) q^{93} - 596 \zeta_{6} q^{94} - 1824 \zeta_{6} q^{95} + (768 \zeta_{6} + 768) q^{96} + ( - 331 \zeta_{6} + 331) q^{97} - 1332 q^{98} + 297 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 9 q^{3} - 8 q^{4} + 19 q^{5} + 26 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 9 q^{3} - 8 q^{4} + 19 q^{5} + 26 q^{7} + 27 q^{9} + 152 q^{10} - 11 q^{11} + 72 q^{12} + 56 q^{13} - 104 q^{14} - 171 q^{15} + 64 q^{16} + 208 q^{17} - 108 q^{18} - 192 q^{19} + 152 q^{20} + 44 q^{22} - 40 q^{23} - 236 q^{25} + 448 q^{26} - 416 q^{28} - 18 q^{29} - 684 q^{30} - 49 q^{31} - 256 q^{32} + 416 q^{34} + 988 q^{35} - 432 q^{36} + 150 q^{37} - 384 q^{38} - 504 q^{39} - 296 q^{41} + 936 q^{42} - 372 q^{43} + 176 q^{44} + 1026 q^{45} - 320 q^{46} + 149 q^{47} - 333 q^{49} + 944 q^{50} - 936 q^{51} + 448 q^{52} - 834 q^{53} + 972 q^{54} - 418 q^{55} + 864 q^{57} + 72 q^{58} + 17 q^{59} - 90 q^{61} - 392 q^{62} - 702 q^{63} - 1024 q^{64} - 1064 q^{65} - 396 q^{66} - 1073 q^{67} - 832 q^{68} + 360 q^{69} + 1976 q^{70} - 570 q^{71} - 1924 q^{73} + 300 q^{74} + 768 q^{76} + 286 q^{77} - 2016 q^{78} - 596 q^{79} + 2432 q^{80} - 729 q^{81} - 2368 q^{82} + 498 q^{83} + 1872 q^{84} + 1976 q^{85} + 1488 q^{86} + 2460 q^{89} + 2052 q^{90} + 2912 q^{91} - 320 q^{92} + 441 q^{93} - 596 q^{94} - 1824 q^{95} + 2304 q^{96} + 331 q^{97} - 2664 q^{98} + 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
0.500000 + 0.866025i
0.500000 0.866025i
2.00000 3.46410i −4.50000 + 2.59808i −4.00000 6.92820i 9.50000 + 16.4545i 20.7846i 13.0000 22.5167i 0 13.5000 23.3827i 76.0000
67.1 2.00000 + 3.46410i −4.50000 2.59808i −4.00000 + 6.92820i 9.50000 16.4545i 20.7846i 13.0000 + 22.5167i 0 13.5000 + 23.3827i 76.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.4.e.a 2
3.b odd 2 1 297.4.e.a 2
9.c even 3 1 inner 99.4.e.a 2
9.c even 3 1 891.4.a.a 1
9.d odd 6 1 297.4.e.a 2
9.d odd 6 1 891.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.4.e.a 2 1.a even 1 1 trivial
99.4.e.a 2 9.c even 3 1 inner
297.4.e.a 2 3.b odd 2 1
297.4.e.a 2 9.d odd 6 1
891.4.a.a 1 9.c even 3 1
891.4.a.d 1 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 4T_{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$5$ \( T^{2} - 19T + 361 \) Copy content Toggle raw display
$7$ \( T^{2} - 26T + 676 \) Copy content Toggle raw display
$11$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$13$ \( T^{2} - 56T + 3136 \) Copy content Toggle raw display
$17$ \( (T - 104)^{2} \) Copy content Toggle raw display
$19$ \( (T + 96)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 40T + 1600 \) Copy content Toggle raw display
$29$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
$31$ \( T^{2} + 49T + 2401 \) Copy content Toggle raw display
$37$ \( (T - 75)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 296T + 87616 \) Copy content Toggle raw display
$43$ \( T^{2} + 372T + 138384 \) Copy content Toggle raw display
$47$ \( T^{2} - 149T + 22201 \) Copy content Toggle raw display
$53$ \( (T + 417)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$61$ \( T^{2} + 90T + 8100 \) Copy content Toggle raw display
$67$ \( T^{2} + 1073 T + 1151329 \) Copy content Toggle raw display
$71$ \( (T + 285)^{2} \) Copy content Toggle raw display
$73$ \( (T + 962)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 596T + 355216 \) Copy content Toggle raw display
$83$ \( T^{2} - 498T + 248004 \) Copy content Toggle raw display
$89$ \( (T - 1230)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 331T + 109561 \) Copy content Toggle raw display
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