Properties

Label 99.4.f.a
Level $99$
Weight $4$
Character orbit 99.f
Analytic conductor $5.841$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{2} - 12 \zeta_{10}^{3} q^{4} + (11 \zeta_{10}^{2} - 12 \zeta_{10} + 11) q^{5} + (19 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{7} + (16 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 16 \zeta_{10}) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{2} - 12 \zeta_{10}^{3} q^{4} + (11 \zeta_{10}^{2} - 12 \zeta_{10} + 11) q^{5} + (19 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{7} + (16 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 16 \zeta_{10}) q^{8} + ( - 26 \zeta_{10}^{3} + 26 \zeta_{10}^{2} - 42) q^{10} + (10 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 35 \zeta_{10} - 20) q^{11} + (33 \zeta_{10}^{3} - 11 \zeta_{10}^{2} + 11 \zeta_{10} - 33) q^{13} + ( - 88 \zeta_{10}^{3} + 74 \zeta_{10}^{2} - 88 \zeta_{10}) q^{14} + 16 \zeta_{10} q^{16} + ( - 57 \zeta_{10}^{2} - 21 \zeta_{10} - 57) q^{17} + ( - 45 \zeta_{10}^{3} - 38 \zeta_{10}^{2} - 45 \zeta_{10}) q^{19} + (12 \zeta_{10}^{3} - 144 \zeta_{10}^{2} + 144 \zeta_{10} - 12) q^{20} + (60 \zeta_{10}^{3} - 184 \zeta_{10}^{2} + 12 \zeta_{10} - 54) q^{22} + (44 \zeta_{10}^{3} - 44 \zeta_{10}^{2} + 15) q^{23} + ( - 143 \zeta_{10}^{3} + 140 \zeta_{10}^{2} - 143 \zeta_{10}) q^{25} + ( - 154 \zeta_{10}^{3} - 88 \zeta_{10} + 88) q^{26} + ( - 72 \zeta_{10}^{2} + 300 \zeta_{10} - 72) q^{28} + (162 \zeta_{10}^{3} + 192 \zeta_{10} - 192) q^{29} + (44 \zeta_{10}^{3} + 33 \zeta_{10}^{2} - 33 \zeta_{10} - 44) q^{31} + (192 \zeta_{10}^{3} - 192 \zeta_{10}^{2} - 96) q^{32} + ( - 198 \zeta_{10}^{3} + 198 \zeta_{10}^{2} + 384) q^{34} + ( - 85 \zeta_{10}^{3} + 366 \zeta_{10}^{2} - 366 \zeta_{10} + 85) q^{35} + (267 \zeta_{10}^{3} + 22 \zeta_{10} - 22) q^{37} + (242 \zeta_{10}^{2} + 104 \zeta_{10} + 242) q^{38} + (168 \zeta_{10}^{3} - 104 \zeta_{10} + 104) q^{40} + (34 \zeta_{10}^{3} - 61 \zeta_{10}^{2} + 34 \zeta_{10}) q^{41} + (84 \zeta_{10}^{3} - 84 \zeta_{10}^{2} - 75) q^{43} + ( - 180 \zeta_{10}^{3} + 420 \zeta_{10}^{2} - 300 \zeta_{10} + 492) q^{44} + ( - 146 \zeta_{10}^{3} + 294 \zeta_{10}^{2} - 294 \zeta_{10} + 146) q^{46} + ( - 55 \zeta_{10}^{3} - 145 \zeta_{10}^{2} - 55 \zeta_{10}) q^{47} + (264 \zeta_{10}^{2} - 318 \zeta_{10} + 264) q^{49} + ( - 274 \zeta_{10}^{2} + 852 \zeta_{10} - 274) q^{50} + (264 \zeta_{10}^{3} + 132 \zeta_{10}^{2} + 264 \zeta_{10}) q^{52} + ( - 98 \zeta_{10}^{3} + 241 \zeta_{10}^{2} - 241 \zeta_{10} + 98) q^{53} + (369 \zeta_{10}^{3} - 520 \zeta_{10}^{2} + 571 \zeta_{10} - 276) q^{55} + (352 \zeta_{10}^{3} - 352 \zeta_{10}^{2} - 56) q^{56} + ( - 264 \zeta_{10}^{3} - 828 \zeta_{10}^{2} - 264 \zeta_{10}) q^{58} + ( - 33 \zeta_{10}^{3} - 451 \zeta_{10} + 451) q^{59} + (279 \zeta_{10}^{2} + 438 \zeta_{10} + 279) q^{61} + ( - 396 \zeta_{10}^{3} - 22 \zeta_{10} + 22) q^{62} + ( - 832 \zeta_{10}^{3} + 832 \zeta_{10}^{2} - 832 \zeta_{10} + 832) q^{64} + (99 \zeta_{10}^{3} - 99 \zeta_{10}^{2} - 209) q^{65} + (561 \zeta_{10}^{3} - 561 \zeta_{10}^{2} - 243) q^{67} + (936 \zeta_{10}^{3} - 252 \zeta_{10}^{2} + 252 \zeta_{10} - 936) q^{68} + ( - 954 \zeta_{10}^{3} + 902 \zeta_{10} - 902) q^{70} + ( - 275 \zeta_{10}^{2} + 708 \zeta_{10} - 275) q^{71} + ( - 28 \zeta_{10}^{3} + 318 \zeta_{10} - 318) q^{73} + ( - 1024 \zeta_{10}^{3} + 402 \zeta_{10}^{2} - 1024 \zeta_{10}) q^{74} + (540 \zeta_{10}^{3} - 540 \zeta_{10}^{2} - 996) q^{76} + (249 \zeta_{10}^{3} - 779 \zeta_{10}^{2} + 745 \zeta_{10} - 839) q^{77} + (65 \zeta_{10}^{3} + 637 \zeta_{10}^{2} - 637 \zeta_{10} - 65) q^{79} + (176 \zeta_{10}^{3} - 192 \zeta_{10}^{2} + 176 \zeta_{10}) q^{80} + (176 \zeta_{10}^{2} - 258 \zeta_{10} + 176) q^{82} + (466 \zeta_{10}^{2} - 431 \zeta_{10} + 466) q^{83} + ( - 174 \zeta_{10}^{3} - 375 \zeta_{10}^{2} - 174 \zeta_{10}) q^{85} + ( - 486 \zeta_{10}^{3} + 354 \zeta_{10}^{2} - 354 \zeta_{10} + 486) q^{86} + (168 \zeta_{10}^{3} + 48 \zeta_{10}^{2} + 192 \zeta_{10} - 688) q^{88} + (154 \zeta_{10}^{3} - 154 \zeta_{10}^{2} + 1279) q^{89} + ( - 352 \zeta_{10}^{3} - 143 \zeta_{10}^{2} - 352 \zeta_{10}) q^{91} + ( - 180 \zeta_{10}^{3} + 528 \zeta_{10} - 528) q^{92} + (690 \zeta_{10}^{2} - 70 \zeta_{10} + 690) q^{94} + ( - 412 \zeta_{10}^{3} - 373 \zeta_{10} + 373) q^{95} + ( - 609 \zeta_{10}^{3} - 282 \zeta_{10}^{2} + 282 \zeta_{10} + 609) q^{97} + ( - 744 \zeta_{10}^{3} + 744 \zeta_{10}^{2} - 948) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{2} - 12 q^{4} + 21 q^{5} + 37 q^{7} + 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{2} - 12 q^{4} + 21 q^{5} + 37 q^{7} + 40 q^{8} - 220 q^{10} - 41 q^{11} - 77 q^{13} - 250 q^{14} + 16 q^{16} - 192 q^{17} - 52 q^{19} + 252 q^{20} + 40 q^{22} + 148 q^{23} - 426 q^{25} + 110 q^{26} + 84 q^{28} - 414 q^{29} - 198 q^{31} + 1140 q^{34} - 477 q^{35} + 201 q^{37} + 830 q^{38} + 480 q^{40} + 129 q^{41} - 132 q^{43} + 1068 q^{44} - 150 q^{46} + 35 q^{47} + 474 q^{49} + 30 q^{50} + 396 q^{52} - 188 q^{53} + 356 q^{55} + 480 q^{56} + 300 q^{58} + 1320 q^{59} + 1275 q^{61} - 330 q^{62} + 832 q^{64} - 638 q^{65} + 150 q^{67} - 2304 q^{68} - 3660 q^{70} - 117 q^{71} - 982 q^{73} - 2450 q^{74} - 2904 q^{76} - 1583 q^{77} - 1469 q^{79} + 544 q^{80} + 270 q^{82} + 967 q^{83} + 27 q^{85} + 750 q^{86} - 2440 q^{88} + 5424 q^{89} - 561 q^{91} - 1764 q^{92} + 2000 q^{94} + 707 q^{95} + 2391 q^{97} - 5280 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(1\)

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
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−3.61803 2.62866i 0 3.70820 + 11.4127i 4.69098 3.40820i 0 −4.72542 14.5434i 5.52786 17.0130i 0 −25.9311
64.1 −1.38197 + 4.25325i 0 −9.70820 7.05342i 5.80902 + 17.8783i 0 23.2254 + 16.8743i 14.4721 10.5146i 0 −84.0689
82.1 −1.38197 4.25325i 0 −9.70820 + 7.05342i 5.80902 17.8783i 0 23.2254 16.8743i 14.4721 + 10.5146i 0 −84.0689
91.1 −3.61803 + 2.62866i 0 3.70820 11.4127i 4.69098 + 3.40820i 0 −4.72542 + 14.5434i 5.52786 + 17.0130i 0 −25.9311
\(n\): e.g. 2-40 or 990-1000
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.a 4
3.b odd 2 1 33.4.e.a 4
11.c even 5 1 inner 99.4.f.a 4
11.c even 5 1 1089.4.a.p 2
11.d odd 10 1 1089.4.a.q 2
33.f even 10 1 363.4.a.o 2
33.h odd 10 1 33.4.e.a 4
33.h odd 10 1 363.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.e.a 4 3.b odd 2 1
33.4.e.a 4 33.h odd 10 1
99.4.f.a 4 1.a even 1 1 trivial
99.4.f.a 4 11.c even 5 1 inner
363.4.a.n 2 33.h odd 10 1
363.4.a.o 2 33.f even 10 1
1089.4.a.p 2 11.c even 5 1
1089.4.a.q 2 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 10T_{2}^{3} + 60T_{2}^{2} + 200T_{2} + 400 \) acting on \(S_{4}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 21 T^{3} + 496 T^{2} + \cdots + 11881 \) Copy content Toggle raw display
$7$ \( T^{4} - 37 T^{3} + 619 T^{2} + \cdots + 192721 \) Copy content Toggle raw display
$11$ \( T^{4} + 41 T^{3} + 1881 T^{2} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{4} + 77 T^{3} + 2299 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$17$ \( T^{4} + 192 T^{3} + 14634 T^{2} + \cdots + 2595321 \) Copy content Toggle raw display
$19$ \( T^{4} + 52 T^{3} + 11254 T^{2} + \cdots + 1274641 \) Copy content Toggle raw display
$23$ \( (T^{2} - 74 T - 1051)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 414 T^{3} + \cdots + 1740892176 \) Copy content Toggle raw display
$31$ \( T^{4} + 198 T^{3} + \cdots + 54479161 \) Copy content Toggle raw display
$37$ \( T^{4} - 201 T^{3} + \cdots + 4216034761 \) Copy content Toggle raw display
$41$ \( T^{4} - 129 T^{3} + 6271 T^{2} + \cdots + 241081 \) Copy content Toggle raw display
$43$ \( (T^{2} + 66 T - 7731)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 35 T^{3} + \cdots + 674700625 \) Copy content Toggle raw display
$53$ \( T^{4} + 188 T^{3} + \cdots + 10042561 \) Copy content Toggle raw display
$59$ \( T^{4} - 1320 T^{3} + \cdots + 47173668025 \) Copy content Toggle raw display
$61$ \( T^{4} - 1275 T^{3} + \cdots + 55792802025 \) Copy content Toggle raw display
$67$ \( (T^{2} - 75 T - 391995)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 117 T^{3} + \cdots + 53332821721 \) Copy content Toggle raw display
$73$ \( T^{4} + 982 T^{3} + \cdots + 8360542096 \) Copy content Toggle raw display
$79$ \( T^{4} + 1469 T^{3} + \cdots + 285379255681 \) Copy content Toggle raw display
$83$ \( T^{4} - 967 T^{3} + \cdots + 53935882081 \) Copy content Toggle raw display
$89$ \( (T^{2} - 2712 T + 1809091)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 2391 T^{3} + \cdots + 932420053161 \) Copy content Toggle raw display
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