Properties

Label 99.5.c.c
Level $99$
Weight $5$
Character orbit 99.c
Analytic conductor $10.234$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 99.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.2336263453\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 102 x^{6} + 2913 x^{4} + 23292 x^{2} + 41364\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -10 - \beta_{2} - \beta_{4} + \beta_{5} ) q^{4} + ( 5 - \beta_{2} - \beta_{5} ) q^{5} + ( -6 \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{7} + ( -14 \beta_{1} - 2 \beta_{3} + \beta_{6} + \beta_{7} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -10 - \beta_{2} - \beta_{4} + \beta_{5} ) q^{4} + ( 5 - \beta_{2} - \beta_{5} ) q^{5} + ( -6 \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{7} + ( -14 \beta_{1} - 2 \beta_{3} + \beta_{6} + \beta_{7} ) q^{8} + ( 15 \beta_{1} - 2 \beta_{3} - \beta_{7} ) q^{10} + ( -4 + 3 \beta_{1} - 4 \beta_{2} - \beta_{3} - 7 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{11} + ( -9 \beta_{1} - 2 \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{13} + ( 143 + 17 \beta_{2} + 16 \beta_{4} - \beta_{5} ) q^{14} + ( 178 + 39 \beta_{2} + 33 \beta_{4} - 3 \beta_{5} ) q^{16} + ( 23 \beta_{1} - 8 \beta_{3} + \beta_{6} - 5 \beta_{7} ) q^{17} + ( 3 \beta_{1} + 9 \beta_{3} + 5 \beta_{6} + 2 \beta_{7} ) q^{19} + ( -321 + 9 \beta_{2} - 10 \beta_{4} + 9 \beta_{5} ) q^{20} + ( -100 - 42 \beta_{1} + \beta_{2} - 19 \beta_{3} + 19 \beta_{4} + 5 \beta_{5} + 7 \beta_{6} - \beta_{7} ) q^{22} + ( -77 - 29 \beta_{2} - 2 \beta_{4} + 25 \beta_{5} ) q^{23} + ( -279 - 48 \beta_{2} - 6 \beta_{4} - 12 \beta_{5} ) q^{25} + ( 199 + 43 \beta_{2} + 56 \beta_{4} - 17 \beta_{5} ) q^{26} + ( 189 \beta_{1} + 32 \beta_{3} - 17 \beta_{7} ) q^{28} + ( -105 \beta_{1} + 16 \beta_{3} + 13 \beta_{6} - 11 \beta_{7} ) q^{29} + ( 338 - 96 \beta_{2} + 18 \beta_{4} + 12 \beta_{5} ) q^{31} + ( 266 \beta_{1} + 70 \beta_{3} - 17 \beta_{6} + 13 \beta_{7} ) q^{32} + ( -648 + 126 \beta_{2} + 66 \beta_{5} ) q^{34} + ( -20 \beta_{1} - 14 \beta_{3} - 8 \beta_{6} + 16 \beta_{7} ) q^{35} + ( 670 + 76 \beta_{2} + 34 \beta_{4} - 16 \beta_{5} ) q^{37} + ( -66 - 228 \beta_{2} + 12 \beta_{4} - 42 \beta_{5} ) q^{38} + ( -231 \beta_{1} - 34 \beta_{3} + 10 \beta_{6} - 7 \beta_{7} ) q^{40} + ( -103 \beta_{1} - 84 \beta_{3} - 3 \beta_{6} + 3 \beta_{7} ) q^{41} + ( 99 \beta_{1} - 31 \beta_{3} - 27 \beta_{6} + 10 \beta_{7} ) q^{43} + ( 835 + 4 \beta_{1} + 343 \beta_{2} + 28 \beta_{3} + 138 \beta_{4} - 35 \beta_{5} + 13 \beta_{6} - 11 \beta_{7} ) q^{44} + ( -447 \beta_{1} - 8 \beta_{3} + 2 \beta_{6} + 25 \beta_{7} ) q^{46} + ( -43 - 19 \beta_{2} - 156 \beta_{4} - 19 \beta_{5} ) q^{47} + ( -799 + 218 \beta_{2} - 220 \beta_{4} - 110 \beta_{5} ) q^{49} + ( -267 \beta_{1} - 72 \beta_{3} + 6 \beta_{6} - 12 \beta_{7} ) q^{50} + ( 681 \beta_{1} + 106 \beta_{3} - 24 \beta_{6} - \beta_{7} ) q^{52} + ( -489 + 453 \beta_{2} - 100 \beta_{4} + 93 \beta_{5} ) q^{53} + ( 486 + 87 \beta_{1} - 292 \beta_{2} - 40 \beta_{3} - 52 \beta_{4} - 26 \beta_{5} - 16 \beta_{6} + 9 \beta_{7} ) q^{55} + ( -2351 - 689 \beta_{2} - 310 \beta_{4} + 211 \beta_{5} ) q^{56} + ( 2758 - 382 \beta_{2} + 50 \beta_{4} - 110 \beta_{5} ) q^{58} + ( 2156 - 112 \beta_{2} - 90 \beta_{4} - 154 \beta_{5} ) q^{59} + ( -603 \beta_{1} + 38 \beta_{3} - 28 \beta_{6} + 63 \beta_{7} ) q^{61} + ( 110 \beta_{1} - 48 \beta_{3} - 18 \beta_{6} + 12 \beta_{7} ) q^{62} + ( -3464 - 1011 \beta_{2} - 315 \beta_{4} + 51 \beta_{5} ) q^{64} + ( -184 \beta_{1} - 52 \beta_{3} + 2 \beta_{6} - 10 \beta_{7} ) q^{65} + ( -544 + 510 \beta_{2} - 342 \beta_{4} + 174 \beta_{5} ) q^{67} + ( -820 \beta_{1} + 64 \beta_{3} + 16 \beta_{6} - 14 \beta_{7} ) q^{68} + ( 446 + 448 \beta_{2} + 166 \beta_{4} - 64 \beta_{5} ) q^{70} + ( 1737 - 747 \beta_{2} + 320 \beta_{4} - 171 \beta_{5} ) q^{71} + ( -588 \beta_{1} - 62 \beta_{6} + 16 \beta_{7} ) q^{73} + ( 1218 \beta_{1} + 128 \beta_{3} - 34 \beta_{6} - 16 \beta_{7} ) q^{74} + ( 102 \beta_{1} - 102 \beta_{3} + 68 \beta_{6} - 10 \beta_{7} ) q^{76} + ( -2837 + 765 \beta_{1} + 421 \beta_{2} + 64 \beta_{3} - 148 \beta_{4} - 143 \beta_{5} - 71 \beta_{6} + 49 \beta_{7} ) q^{77} + ( -324 \beta_{1} + 191 \beta_{3} - 95 \beta_{6} - 91 \beta_{7} ) q^{79} + ( 563 + 1025 \beta_{2} + 366 \beta_{4} - 7 \beta_{5} ) q^{80} + ( 2108 + 1984 \beta_{2} + 682 \beta_{4} + 56 \beta_{5} ) q^{82} + ( -302 \beta_{1} + 240 \beta_{3} - 48 \beta_{6} + 150 \beta_{7} ) q^{83} + ( 912 \beta_{1} - 216 \beta_{3} + 10 \beta_{6} + 4 \beta_{7} ) q^{85} + ( -2578 + 812 \beta_{2} - 116 \beta_{4} + 182 \beta_{5} ) q^{86} + ( -1592 + 2097 \beta_{1} - 739 \beta_{2} + 280 \beta_{3} + 161 \beta_{4} + 55 \beta_{5} - 26 \beta_{6} - 51 \beta_{7} ) q^{88} + ( -2044 - 946 \beta_{2} + 156 \beta_{4} + 206 \beta_{5} ) q^{89} + ( -2560 - 554 \beta_{2} - 302 \beta_{4} + 182 \beta_{5} ) q^{91} + ( 10241 + 245 \beta_{2} + 720 \beta_{4} - 187 \beta_{5} ) q^{92} + ( -789 \beta_{1} - 350 \beta_{3} + 156 \beta_{6} - 19 \beta_{7} ) q^{94} + ( -536 \beta_{1} + 158 \beta_{3} - 82 \beta_{6} + 26 \beta_{7} ) q^{95} + ( 938 - 1098 \beta_{2} + 492 \beta_{4} + 30 \beta_{5} ) q^{97} + ( -363 \beta_{1} - 332 \beta_{3} + 220 \beta_{6} - 110 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 76q^{4} + 36q^{5} + O(q^{10}) \) \( 8q - 76q^{4} + 36q^{5} - 36q^{11} + 1140q^{14} + 1412q^{16} - 2532q^{20} - 780q^{22} - 516q^{23} - 2280q^{25} + 1524q^{26} + 2752q^{31} - 4920q^{34} + 5296q^{37} - 696q^{38} + 6540q^{44} - 420q^{47} - 6832q^{49} - 3540q^{53} + 3784q^{55} - 17964q^{56} + 21624q^{58} + 16632q^{59} - 27508q^{64} - 3656q^{67} + 3312q^{70} + 13212q^{71} - 23268q^{77} + 4476q^{80} + 17088q^{82} - 19896q^{86} - 12516q^{88} - 15528q^{89} - 19752q^{91} + 81180q^{92} + 7624q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 102 x^{6} + 2913 x^{4} + 23292 x^{2} + 41364\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{6} - 444 \nu^{4} - 9669 \nu^{2} - 38034 \)\()/3216\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{7} - 444 \nu^{5} - 9669 \nu^{3} - 44466 \nu \)\()/3216\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{6} + 320 \nu^{4} + 8535 \nu^{2} + 31182 \)\()/1608\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} + 196 \nu^{4} + 10617 \nu^{2} + 107946 \)\()/3216\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{7} - 191 \nu^{5} - 4551 \nu^{3} - 16500 \nu \)\()/402\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{7} + 320 \nu^{5} + 10143 \nu^{3} + 95502 \nu \)\()/1608\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} - \beta_{2} - 26\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} - 2 \beta_{3} - 46 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-51 \beta_{5} + 81 \beta_{4} + 87 \beta_{2} + 1170\)
\(\nu^{5}\)\(=\)\(-51 \beta_{7} - 81 \beta_{6} + 198 \beta_{3} + 2442 \beta_{1}\)
\(\nu^{6}\)\(=\)\(2595 \beta_{5} - 5259 \beta_{4} - 6435 \beta_{2} - 61224\)
\(\nu^{7}\)\(=\)\(2595 \beta_{7} + 5259 \beta_{6} - 14358 \beta_{3} - 136788 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
7.70102i
5.58567i
3.00247i
1.57474i
1.57474i
3.00247i
5.58567i
7.70102i
7.70102i 0 −43.3057 12.5296 0 63.6540i 210.281i 0 96.4909i
10.2 5.58567i 0 −15.1997 29.7487 0 12.9420i 4.47031i 0 166.166i
10.3 3.00247i 0 6.98517 −8.72578 0 1.45810i 69.0123i 0 26.1989i
10.4 1.57474i 0 13.5202 −15.5526 0 93.8006i 46.4867i 0 24.4913i
10.5 1.57474i 0 13.5202 −15.5526 0 93.8006i 46.4867i 0 24.4913i
10.6 3.00247i 0 6.98517 −8.72578 0 1.45810i 69.0123i 0 26.1989i
10.7 5.58567i 0 −15.1997 29.7487 0 12.9420i 4.47031i 0 166.166i
10.8 7.70102i 0 −43.3057 12.5296 0 63.6540i 210.281i 0 96.4909i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.5.c.c 8
3.b odd 2 1 33.5.c.a 8
11.b odd 2 1 inner 99.5.c.c 8
12.b even 2 1 528.5.j.a 8
33.d even 2 1 33.5.c.a 8
132.d odd 2 1 528.5.j.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.5.c.a 8 3.b odd 2 1
33.5.c.a 8 33.d even 2 1
99.5.c.c 8 1.a even 1 1 trivial
99.5.c.c 8 11.b odd 2 1 inner
528.5.j.a 8 12.b even 2 1
528.5.j.a 8 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 102 T_{2}^{6} + 2913 T_{2}^{4} + 23292 T_{2}^{2} + 41364 \) acting on \(S_{5}^{\mathrm{new}}(99, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 41364 + 23292 T^{2} + 2913 T^{4} + 102 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 50584 + 3312 T - 518 T^{2} - 18 T^{3} + T^{4} )^{2} \)
$7$ \( 12695273424 + 6051633840 T^{2} + 37830264 T^{4} + 13020 T^{6} + T^{8} \)
$11$ \( 45949729863572161 + 112983421561956 T - 1032352370896 T^{2} + 16417538940 T^{3} + 160159230 T^{4} + 1121340 T^{5} - 4816 T^{6} + 36 T^{7} + T^{8} \)
$13$ \( 1164901648483584 + 2284214269248 T^{2} + 1142069460 T^{4} + 66480 T^{6} + T^{8} \)
$17$ \( 5763713931173964096 + 770067814668192 T^{2} + 27408756564 T^{4} + 326700 T^{6} + T^{8} \)
$19$ \( 1148548870541318976 + 433340712892512 T^{2} + 42562524276 T^{4} + 467172 T^{6} + T^{8} \)
$23$ \( ( 2066510872 - 154896312 T - 458558 T^{2} + 258 T^{3} + T^{4} )^{2} \)
$29$ \( \)\(32\!\cdots\!84\)\( + 668949031877815296 T^{2} + 3572518482852 T^{4} + 3652884 T^{6} + T^{8} \)
$31$ \( ( 6236194624 + 171603616 T - 14412 T^{2} - 1376 T^{3} + T^{4} )^{2} \)
$37$ \( ( 26585618752 - 398290592 T + 1748388 T^{2} - 2648 T^{3} + T^{4} )^{2} \)
$41$ \( \)\(10\!\cdots\!76\)\( + \)\(19\!\cdots\!24\)\( T^{2} + 107439635858532 T^{4} + 18633108 T^{6} + T^{8} \)
$43$ \( \)\(25\!\cdots\!84\)\( + 27608350007521242720 T^{2} + 26793750661620 T^{4} + 8964516 T^{6} + T^{8} \)
$47$ \( ( 12403328125408 - 291605376 T - 7278002 T^{2} + 210 T^{3} + T^{4} )^{2} \)
$53$ \( ( -15377555198312 - 31141326432 T - 14617718 T^{2} + 1770 T^{3} + T^{4} )^{2} \)
$59$ \( ( -39877809797408 + 30362839632 T + 7759660 T^{2} - 8316 T^{3} + T^{4} )^{2} \)
$61$ \( \)\(11\!\cdots\!36\)\( + \)\(94\!\cdots\!12\)\( T^{2} + 1388503615694388 T^{4} + 65474352 T^{6} + T^{8} \)
$67$ \( ( -192515994273728 - 221754664832 T - 53637708 T^{2} + 1828 T^{3} + T^{4} )^{2} \)
$71$ \( ( -855560872809632 + 445427720832 T - 46867514 T^{2} - 6606 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(16\!\cdots\!84\)\( + \)\(86\!\cdots\!80\)\( T^{2} + 1203338659893120 T^{4} + 62146512 T^{6} + T^{8} \)
$79$ \( \)\(28\!\cdots\!84\)\( + \)\(16\!\cdots\!20\)\( T^{2} + 35826082579997880 T^{4} + 318678684 T^{6} + T^{8} \)
$83$ \( \)\(76\!\cdots\!44\)\( + \)\(30\!\cdots\!48\)\( T^{2} + 16776000917143872 T^{4} + 244092288 T^{6} + T^{8} \)
$89$ \( ( -1270313049471344 - 705522821424 T - 69427640 T^{2} + 7764 T^{3} + T^{4} )^{2} \)
$97$ \( ( -466619407059968 + 494162140288 T - 121159848 T^{2} - 3812 T^{3} + T^{4} )^{2} \)
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