Properties

Label 99.4.d.b
Level $99$
Weight $4$
Character orbit 99.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.84118909057\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 3 q^{2} + q^{4} + 14 \beta q^{5} + 12 \beta q^{7} - 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} + q^{4} + 14 \beta q^{5} + 12 \beta q^{7} - 21 q^{8} + 42 \beta q^{10} + ( - 11 \beta + 33) q^{11} - 21 \beta q^{13} + 36 \beta q^{14} - 71 q^{16} + 126 q^{17} + 63 \beta q^{19} + 14 \beta q^{20} + ( - 33 \beta + 99) q^{22} - 85 \beta q^{23} - 267 q^{25} - 63 \beta q^{26} + 12 \beta q^{28} - 24 q^{29} - 70 q^{31} - 45 q^{32} + 378 q^{34} - 336 q^{35} + 182 q^{37} + 189 \beta q^{38} - 294 \beta q^{40} + 294 q^{41} + 3 \beta q^{43} + ( - 11 \beta + 33) q^{44} - 255 \beta q^{46} + 77 \beta q^{47} + 55 q^{49} - 801 q^{50} - 21 \beta q^{52} + 104 \beta q^{53} + (462 \beta + 308) q^{55} - 252 \beta q^{56} - 72 q^{58} - 364 \beta q^{59} - 231 \beta q^{61} - 210 q^{62} + 433 q^{64} + 588 q^{65} - 880 q^{67} + 126 q^{68} - 1008 q^{70} + 239 \beta q^{71} - 126 \beta q^{73} + 546 q^{74} + 63 \beta q^{76} + (396 \beta + 264) q^{77} - 546 \beta q^{79} - 994 \beta q^{80} + 882 q^{82} + 1218 q^{83} + 1764 \beta q^{85} + 9 \beta q^{86} + (231 \beta - 693) q^{88} + 1085 \beta q^{89} + 504 q^{91} - 85 \beta q^{92} + 231 \beta q^{94} - 1764 q^{95} - 196 q^{97} + 165 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 2 q^{4} - 42 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{2} + 2 q^{4} - 42 q^{8} + 66 q^{11} - 142 q^{16} + 252 q^{17} + 198 q^{22} - 534 q^{25} - 48 q^{29} - 140 q^{31} - 90 q^{32} + 756 q^{34} - 672 q^{35} + 364 q^{37} + 588 q^{41} + 66 q^{44} + 110 q^{49} - 1602 q^{50} + 616 q^{55} - 144 q^{58} - 420 q^{62} + 866 q^{64} + 1176 q^{65} - 1760 q^{67} + 252 q^{68} - 2016 q^{70} + 1092 q^{74} + 528 q^{77} + 1764 q^{82} + 2436 q^{83} - 1386 q^{88} + 1008 q^{91} - 3528 q^{95} - 392 q^{97} + 330 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
98.1
1.41421i
1.41421i
3.00000 0 1.00000 19.7990i 0 16.9706i −21.0000 0 59.3970i
98.2 3.00000 0 1.00000 19.7990i 0 16.9706i −21.0000 0 59.3970i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.4.d.b yes 2
3.b odd 2 1 99.4.d.a 2
4.b odd 2 1 1584.4.b.a 2
11.b odd 2 1 99.4.d.a 2
12.b even 2 1 1584.4.b.b 2
33.d even 2 1 inner 99.4.d.b yes 2
44.c even 2 1 1584.4.b.b 2
132.d odd 2 1 1584.4.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.4.d.a 2 3.b odd 2 1
99.4.d.a 2 11.b odd 2 1
99.4.d.b yes 2 1.a even 1 1 trivial
99.4.d.b yes 2 33.d even 2 1 inner
1584.4.b.a 2 4.b odd 2 1
1584.4.b.a 2 132.d odd 2 1
1584.4.b.b 2 12.b even 2 1
1584.4.b.b 2 44.c even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 392 \) Copy content Toggle raw display
$7$ \( T^{2} + 288 \) Copy content Toggle raw display
$11$ \( T^{2} - 66T + 1331 \) Copy content Toggle raw display
$13$ \( T^{2} + 882 \) Copy content Toggle raw display
$17$ \( (T - 126)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 7938 \) Copy content Toggle raw display
$23$ \( T^{2} + 14450 \) Copy content Toggle raw display
$29$ \( (T + 24)^{2} \) Copy content Toggle raw display
$31$ \( (T + 70)^{2} \) Copy content Toggle raw display
$37$ \( (T - 182)^{2} \) Copy content Toggle raw display
$41$ \( (T - 294)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 18 \) Copy content Toggle raw display
$47$ \( T^{2} + 11858 \) Copy content Toggle raw display
$53$ \( T^{2} + 21632 \) Copy content Toggle raw display
$59$ \( T^{2} + 264992 \) Copy content Toggle raw display
$61$ \( T^{2} + 106722 \) Copy content Toggle raw display
$67$ \( (T + 880)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 114242 \) Copy content Toggle raw display
$73$ \( T^{2} + 31752 \) Copy content Toggle raw display
$79$ \( T^{2} + 596232 \) Copy content Toggle raw display
$83$ \( (T - 1218)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 2354450 \) Copy content Toggle raw display
$97$ \( (T + 196)^{2} \) Copy content Toggle raw display
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