Properties

Label 99.2.e.d
Level $99$
Weight $2$
Character orbit 99.e
Analytic conductor $0.791$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{2} + (\beta_{4} + \beta_{2}) q^{3} + \beta_{5} q^{4} + (\beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{6} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{4} - 1) q^{8} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{2} + (\beta_{4} + \beta_{2}) q^{3} + \beta_{5} q^{4} + (\beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{6} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{4} - 1) q^{8} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{9} + (2 \beta_{4} + \beta_{3} - 2) q^{10} + \beta_1 q^{11} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{12} + ( - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{13} + (\beta_{5} - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{14} + ( - 2 \beta_{5} + \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{15} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{16} + ( - \beta_{4} - 2 \beta_{3} - 1) q^{17} + (\beta_{5} - 2 \beta_{4} - \beta_{2} + 3 \beta_1) q^{18} + (\beta_{4} - \beta_{3} - 3) q^{19} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{20} + (\beta_{4} - \beta_{3} - \beta_1 + 2) q^{21} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{22} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{23} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{24} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{25} + ( - 5 \beta_{4} - 2 \beta_{3} + 4) q^{26} + (3 \beta_1 - 6) q^{27} - q^{28} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{29} + ( - \beta_{5} - 3 \beta_{4} - \beta_{2} + 3) q^{30} + (2 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{31} + (2 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 3) q^{32} + ( - \beta_{5} + \beta_{4}) q^{33} + \beta_{2} q^{34} + \beta_{4} q^{35} + (\beta_{5} + \beta_{4} + 2 \beta_{2} + 3 \beta_1) q^{36} + ( - 2 \beta_{4} + \beta_{3} - 4) q^{37} + ( - 4 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{38} + (5 \beta_{5} - 2 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 2) q^{39} + (\beta_{5} + \beta_1 - 1) q^{40} + ( - \beta_{5} + 6 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 2) q^{41} + ( - \beta_{4} + 2 \beta_{2} - 3 \beta_1) q^{42} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} - \beta_1) q^{43} + (\beta_{4} + \beta_{3}) q^{44} + (4 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{45} + ( - 3 \beta_{4} + \beta_{3}) q^{46} + ( - 5 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - \beta_{2} - 4 \beta_1) q^{47} + ( - \beta_{5} + 4 \beta_{4} + \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 4) q^{48} + ( - \beta_{5} + 3 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 4) q^{49} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 5 \beta_1 + 5) q^{50} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 3 \beta_1) q^{51} + (5 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{52} + (7 \beta_{4} + 6 \beta_{3} + 1) q^{53} + ( - 3 \beta_{5} + 6 \beta_{4} + 3 \beta_{3} - 3 \beta_{2}) q^{54} + (\beta_{4} - 1) q^{55} + (\beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{56} + (\beta_{5} - 5 \beta_{4} - 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{57} + (4 \beta_{5} - 7 \beta_{3} + 7 \beta_{2} - \beta_1 + 1) q^{58} + (3 \beta_{5} + 3 \beta_{3} - 3 \beta_{2} + 7 \beta_1 - 7) q^{59} + (2 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{60} + (\beta_{2} + 7 \beta_1) q^{61} + ( - \beta_{4} + 3 \beta_{3} - 5) q^{62} + (2 \beta_{5} - \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 3) q^{63} + (\beta_{4} - 3 \beta_{3} - 1) q^{64} + (7 \beta_{5} - 7 \beta_{4} - 7 \beta_{3} + 5 \beta_{2} + 7 \beta_1) q^{65} + ( - \beta_{5} + \beta_{3} - 2 \beta_1 + 1) q^{66} + ( - 7 \beta_{5} + 6 \beta_{3} - 6 \beta_{2} + \beta_1 - 1) q^{67} + ( - 2 \beta_{5} - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{68} + (3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - 3) q^{69} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_1) q^{70} + (4 \beta_{4} + 3 \beta_{3} + 4) q^{71} + ( - 3 \beta_{2} + 3) q^{72} + ( - 7 \beta_{4} - 2 \beta_{3} - 2) q^{73} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{74} + ( - \beta_{4} - 5 \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 1) q^{75} + ( - 5 \beta_{5} + \beta_{3} - \beta_{2}) q^{76} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{77} + (2 \beta_{5} + 7 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 8) q^{78} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_1) q^{79} + (6 \beta_{4} + 2 \beta_{3} - 3) q^{80} + ( - 3 \beta_{5} - 3 \beta_{4} - 6 \beta_{2}) q^{81} + ( - 3 \beta_{4} - 6 \beta_{3} + 7) q^{82} + 3 \beta_{2} q^{83} + ( - \beta_{4} - \beta_{2}) q^{84} + (\beta_{5} - \beta_1 + 1) q^{85} + ( - 4 \beta_{5} + 9 \beta_{3} - 9 \beta_{2} + 2 \beta_1 - 2) q^{86} + (\beta_{5} + 4 \beta_{4} - \beta_{3} - 2 \beta_{2} - 7 \beta_1 + 8) q^{87} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{88} + ( - 2 \beta_{4} - 8 \beta_{3} - 4) q^{89} + (2 \beta_{5} + 5 \beta_{4} + \beta_{2} + 3 \beta_1 - 6) q^{90} + ( - 2 \beta_{4} - \beta_{3} + 1) q^{91} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1) q^{92} + (\beta_{5} - \beta_{4} + \beta_{3} + 4 \beta_{2} - 5 \beta_1 + 1) q^{93} + ( - 9 \beta_{5} + 8 \beta_{3} - 8 \beta_{2} - 6 \beta_1 + 6) q^{94} + ( - 5 \beta_{5} + 4 \beta_{3} - 4 \beta_{2} - 6 \beta_1 + 6) q^{95} + ( - 5 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 1) q^{96} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + \beta_1) q^{97} + ( - 5 \beta_{4} - 3 \beta_{3} + 4) q^{98} + (\beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 9 q^{6} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{5} - 9 q^{6} + 3 q^{7} - 6 q^{8} - 12 q^{10} + 3 q^{11} + 9 q^{12} + 9 q^{13} + 3 q^{14} + 6 q^{16} - 6 q^{17} + 9 q^{18} - 18 q^{19} - 3 q^{20} + 9 q^{21} + 6 q^{23} - 9 q^{24} + 6 q^{25} + 24 q^{26} - 27 q^{27} - 6 q^{28} + 6 q^{29} + 18 q^{30} + 9 q^{31} - 9 q^{32} + 9 q^{36} - 24 q^{37} - 9 q^{38} - 3 q^{40} + 6 q^{41} - 9 q^{42} - 3 q^{43} - 9 q^{45} - 12 q^{47} + 9 q^{48} + 12 q^{49} + 15 q^{50} - 9 q^{51} + 6 q^{52} + 6 q^{53} - 6 q^{55} - 6 q^{56} + 9 q^{57} + 3 q^{58} - 21 q^{59} + 9 q^{60} + 21 q^{61} - 30 q^{62} + 9 q^{63} - 6 q^{64} + 21 q^{65} - 3 q^{67} - 9 q^{68} - 18 q^{69} - 6 q^{70} + 24 q^{71} + 18 q^{72} - 12 q^{73} + 15 q^{74} + 18 q^{75} - 3 q^{77} - 45 q^{78} + 9 q^{79} - 18 q^{80} + 42 q^{82} + 3 q^{85} - 6 q^{86} + 27 q^{87} - 3 q^{88} - 24 q^{89} - 27 q^{90} + 6 q^{91} + 9 q^{92} - 9 q^{93} + 18 q^{94} + 18 q^{95} - 9 q^{96} + 3 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{18}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{18}^{5} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{18}^{5} + \zeta_{18}^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \) Copy content Toggle raw display
\(\zeta_{18}\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{2}\)\(=\) \( ( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{18}^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{5}\)\(=\) \( ( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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
−0.173648 + 0.984808i
0.939693 0.342020i
−0.766044 0.642788i
−0.173648 0.984808i
0.939693 + 0.342020i
−0.766044 + 0.642788i
−0.766044 + 1.32683i 0.592396 + 1.62760i −0.173648 0.300767i 0.266044 + 0.460802i −2.61334 0.460802i 1.43969 2.49362i −2.53209 −2.29813 + 1.92836i −0.815207
34.2 −0.173648 + 0.300767i 1.11334 1.32683i 0.939693 + 1.62760i −0.326352 0.565258i 0.205737 + 0.565258i −0.266044 + 0.460802i −1.34730 −0.520945 2.95442i 0.226682
34.3 0.939693 1.62760i −1.70574 0.300767i −0.766044 1.32683i −1.43969 2.49362i −2.09240 + 2.49362i 0.326352 0.565258i 0.879385 2.81908 + 1.02606i −5.41147
67.1 −0.766044 1.32683i 0.592396 1.62760i −0.173648 + 0.300767i 0.266044 0.460802i −2.61334 + 0.460802i 1.43969 + 2.49362i −2.53209 −2.29813 1.92836i −0.815207
67.2 −0.173648 0.300767i 1.11334 + 1.32683i 0.939693 1.62760i −0.326352 + 0.565258i 0.205737 0.565258i −0.266044 0.460802i −1.34730 −0.520945 + 2.95442i 0.226682
67.3 0.939693 + 1.62760i −1.70574 + 0.300767i −0.766044 + 1.32683i −1.43969 + 2.49362i −2.09240 2.49362i 0.326352 + 0.565258i 0.879385 2.81908 1.02606i −5.41147
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.3
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 99.2.e.d 6
3.b odd 2 1 297.2.e.d 6
9.c even 3 1 inner 99.2.e.d 6
9.c even 3 1 891.2.a.l 3
9.d odd 6 1 297.2.e.d 6
9.d odd 6 1 891.2.a.k 3
11.b odd 2 1 1089.2.e.h 6
99.g even 6 1 9801.2.a.bd 3
99.h odd 6 1 1089.2.e.h 6
99.h odd 6 1 9801.2.a.be 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.e.d 6 1.a even 1 1 trivial
99.2.e.d 6 9.c even 3 1 inner
297.2.e.d 6 3.b odd 2 1
297.2.e.d 6 9.d odd 6 1
891.2.a.k 3 9.d odd 6 1
891.2.a.l 3 9.c even 3 1
1089.2.e.h 6 11.b odd 2 1
1089.2.e.h 6 99.h odd 6 1
9801.2.a.bd 3 99.g even 6 1
9801.2.a.be 3 99.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 3T_{2}^{4} + 2T_{2}^{3} + 9T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + 9T^{3} + 27 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + 9 T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + 9 T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{6} - 9 T^{5} + 66 T^{4} - 137 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( (T^{3} + 3 T^{2} - 6 T + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 9 T^{2} + 18 T - 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} + 33 T^{4} - 56 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$29$ \( T^{6} - 6 T^{5} + 81 T^{4} + \cdots + 45369 \) Copy content Toggle raw display
$31$ \( T^{6} - 9 T^{5} + 75 T^{4} - 92 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( (T^{3} + 12 T^{2} + 27 T - 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 6 T^{5} + 117 T^{4} + \cdots + 218089 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} + 153 T^{4} + \cdots + 294849 \) Copy content Toggle raw display
$47$ \( T^{6} + 12 T^{5} + 159 T^{4} + \cdots + 32041 \) Copy content Toggle raw display
$53$ \( (T^{3} - 3 T^{2} - 126 T + 57)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 21 T^{5} + 375 T^{4} + \cdots + 218089 \) Copy content Toggle raw display
$61$ \( T^{6} - 21 T^{5} + 297 T^{4} + \cdots + 103041 \) Copy content Toggle raw display
$67$ \( T^{6} + 3 T^{5} + 135 T^{4} + \cdots + 332929 \) Copy content Toggle raw display
$71$ \( (T^{3} - 12 T^{2} + 9 T + 111)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 6 T^{2} - 105 T - 703)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 9 T^{5} + 57 T^{4} - 178 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$83$ \( T^{6} + 27 T^{4} - 54 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$89$ \( (T^{3} + 12 T^{2} - 108 T - 408)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 3 T^{5} + 150 T^{4} + \cdots + 516961 \) Copy content Toggle raw display
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