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Properties

Label 99.4.f.b
Level $99$
Weight $4$
Character orbit 99.f
Analytic conductor $5.841$
Analytic rank $0$
Dimension $8$
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.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.84118909057\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.682515625.5
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{5} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{2} + ( - 4 \beta_{7} - 3 \beta_{6} + 3 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 2) q^{4} + (2 \beta_{6} - 3 \beta_{5} - 4 \beta_{3} + 7 \beta_{2} - 4) q^{5} + (6 \beta_{6} - 6 \beta_{4} - 4 \beta_{2} - 15 \beta_1 + 4) q^{7} + (2 \beta_{7} + 8 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{5} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{2} + ( - 4 \beta_{7} - 3 \beta_{6} + 3 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 2) q^{4} + (2 \beta_{6} - 3 \beta_{5} - 4 \beta_{3} + 7 \beta_{2} - 4) q^{5} + (6 \beta_{6} - 6 \beta_{4} - 4 \beta_{2} - 15 \beta_1 + 4) q^{7} + (2 \beta_{7} + 8 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{8}+ \cdots + ( - 97 \beta_{7} + 401 \beta_{5} - \beta_{4} - 498 \beta_{2} - 401 \beta_1 - 445) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{2} - 16 q^{4} - 9 q^{5} + 3 q^{7} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{2} - 16 q^{4} - 9 q^{5} + 3 q^{7} - 36 q^{8} + 8 q^{10} + 87 q^{11} + 171 q^{13} - 12 q^{14} + 44 q^{16} - 36 q^{17} + 324 q^{19} + 87 q^{20} - 521 q^{22} + 84 q^{23} + 263 q^{25} + 774 q^{26} + 387 q^{28} - 393 q^{29} + 15 q^{31} - 102 q^{32} - 712 q^{34} - 1002 q^{35} - 747 q^{37} + 36 q^{38} + 41 q^{40} - 159 q^{41} - 644 q^{43} - 219 q^{44} + 753 q^{46} + 351 q^{47} - 1967 q^{49} - 330 q^{50} + 2871 q^{52} + 531 q^{53} - 716 q^{55} - 1470 q^{56} - 1205 q^{58} + 1002 q^{59} + 1449 q^{61} - 99 q^{62} - 1118 q^{64} + 954 q^{65} - 518 q^{67} - 873 q^{68} + 26 q^{70} - 429 q^{71} + 2547 q^{73} - 468 q^{74} - 2276 q^{76} + 2697 q^{77} + 2805 q^{79} + 1620 q^{80} - 1631 q^{82} + 2553 q^{83} - 197 q^{85} + 1713 q^{86} + 2866 q^{88} - 1788 q^{89} + 2885 q^{91} - 423 q^{92} + 1159 q^{94} - 3009 q^{95} + 9 q^{97} - 5550 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 528 \nu^{7} + 2098 \nu^{6} - 15725 \nu^{5} + 33439 \nu^{4} + 71401 \nu^{3} - 332708 \nu^{2} + 319181 \nu + 440220 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5794 \nu^{7} - 9973 \nu^{6} - 30517 \nu^{5} + 195125 \nu^{4} - 61888 \nu^{3} + 104068 \nu^{2} + 501961 \nu + 528473 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7409 \nu^{7} - 59487 \nu^{6} + 183537 \nu^{5} - 171974 \nu^{4} - 58164 \nu^{3} - 77439 \nu^{2} + 18601 \nu - 701074 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8817 \nu^{7} + 16927 \nu^{6} - 106264 \nu^{5} + 200474 \nu^{4} + 521745 \nu^{3} + 380907 \nu^{2} + 2179908 \nu + 2809884 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11971 \nu^{7} + 3536 \nu^{6} + 58156 \nu^{5} - 228404 \nu^{4} - 102852 \nu^{3} - 979996 \nu^{2} - 1085964 \nu - 2305776 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13790 \nu^{7} + 57068 \nu^{6} - 113608 \nu^{5} + 65418 \nu^{4} - 266949 \nu^{3} + 6060 \nu^{2} - 742824 \nu + 665808 ) / 1168519 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{3} - 5\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 6\beta_{6} + 6\beta_{5} + 2\beta_{4} + 4\beta_{3} - 10\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{7} + 10\beta_{6} + 13\beta_{5} + 13\beta_{4} + 14\beta_{3} - 13\beta_{2} - 10\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 43\beta_{7} + 25\beta_{5} + 49\beta_{4} + 18\beta_{2} - 25\beta _1 - 62 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 97\beta_{7} - 92\beta_{6} + 92\beta_{4} - 97\beta_{3} + 221\beta_{2} - 44\beta _1 - 221 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -449\beta_{6} - 260\beta_{5} - 412\beta_{3} + 896\beta_{2} - 412 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1 + \beta_{2} - \beta_{3} + \beta_{7}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−0.390899 − 1.20306i
0.581882 + 1.79085i
−1.20316 − 0.874145i
2.51217 + 1.82520i
−1.20316 + 0.874145i
2.51217 − 1.82520i
−0.390899 + 1.20306i
0.581882 − 1.79085i
−0.523388 − 0.380264i 0 −2.34280 − 7.21040i −9.01441 + 6.54935i 0 8.07696 + 24.8583i −3.11499 + 9.58696i 0 7.20851
37.2 2.02339 + 1.47008i 0 −0.539165 − 1.65938i 8.44146 − 6.13308i 0 −10.1220 − 31.1524i 7.53140 − 23.1793i 0 26.0964
64.1 0.0404346 − 0.124445i 0 6.45828 + 4.69222i −2.06705 − 6.36172i 0 11.6029 + 8.43002i 1.69194 − 1.22926i 0 −0.875265
64.2 1.45957 − 4.49208i 0 −11.5763 − 8.41069i −1.86000 − 5.72450i 0 −8.05785 − 5.85437i −24.1083 + 17.5157i 0 −28.4297
82.1 0.0404346 + 0.124445i 0 6.45828 − 4.69222i −2.06705 + 6.36172i 0 11.6029 − 8.43002i 1.69194 + 1.22926i 0 −0.875265
82.2 1.45957 + 4.49208i 0 −11.5763 + 8.41069i −1.86000 + 5.72450i 0 −8.05785 + 5.85437i −24.1083 − 17.5157i 0 −28.4297
91.1 −0.523388 + 0.380264i 0 −2.34280 + 7.21040i −9.01441 − 6.54935i 0 8.07696 − 24.8583i −3.11499 − 9.58696i 0 7.20851
91.2 2.02339 − 1.47008i 0 −0.539165 + 1.65938i 8.44146 + 6.13308i 0 −10.1220 + 31.1524i 7.53140 + 23.1793i 0 26.0964
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.4.f.b 8
3.b odd 2 1 33.4.e.b ✓ 8
11.c even 5 1 inner 99.4.f.b 8
11.c even 5 1 1089.4.a.bg 4
11.d odd 10 1 1089.4.a.z 4
33.f even 10 1 363.4.a.t 4
33.h odd 10 1 33.4.e.b ✓ 8
33.h odd 10 1 363.4.a.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.e.b ✓ 8 3.b odd 2 1
33.4.e.b ✓ 8 33.h odd 10 1
99.4.f.b 8 1.a even 1 1 trivial
99.4.f.b 8 11.c even 5 1 inner
363.4.a.p 4 33.h odd 10 1
363.4.a.t 4 33.f even 10 1
1089.4.a.z 4 11.d odd 10 1
1089.4.a.bg 4 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 6T_{2}^{7} + 34T_{2}^{6} - 72T_{2}^{5} + 49T_{2}^{4} + 96T_{2}^{3} + 51T_{2}^{2} - 3T_{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{7} + 34 T^{6} - 72 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 9 T^{7} + 34 T^{6} + \cdots + 21911761 \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + \cdots + 14957045401 \) Copy content Toggle raw display
$11$ \( T^{8} - 87 T^{7} + \cdots + 3138428376721 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 153370490192656 \) Copy content Toggle raw display
$17$ \( T^{8} + 36 T^{7} + \cdots + 16499324068096 \) Copy content Toggle raw display
$19$ \( T^{8} - 324 T^{7} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{4} - 42 T^{3} - 20241 T^{2} + \cdots - 46471644)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 393 T^{7} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{8} - 15 T^{7} + \cdots + 2860289355121 \) Copy content Toggle raw display
$37$ \( T^{8} + 747 T^{7} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{8} + 159 T^{7} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{4} + 322 T^{3} - 136785 T^{2} + \cdots + 5520039844)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 351 T^{7} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{8} - 531 T^{7} + \cdots + 27\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{8} - 1002 T^{7} + \cdots + 36\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{8} - 1449 T^{7} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{4} + 259 T^{3} - 86025 T^{2} + \cdots + 1798706704)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 429 T^{7} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{8} - 2547 T^{7} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{8} - 2805 T^{7} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{8} - 2553 T^{7} + \cdots + 82\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( (T^{4} + 894 T^{3} + \cdots - 245710544796)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 9 T^{7} + \cdots + 98\!\cdots\!81 \) Copy content Toggle raw display
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