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Properties

Label 99.3.k.b
Level $99$
Weight $3$
Character orbit 99.k
Analytic conductor $2.698$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 99.k (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.69755461717\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 21x^{14} + 227x^{12} - 1488x^{10} + 24225x^{8} - 62832x^{6} + 64372x^{4} + 7986x^{2} + 14641 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( - \beta_{12} + \beta_{11} - \beta_{6} - \beta_{4} - 3 \beta_{2} - 1) q^{4} + (\beta_{15} - \beta_{14} + \beta_{9} + \beta_{8} + \beta_{3} - \beta_1) q^{5} + (\beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} - 2 \beta_{6} + 3) q^{7} + ( - 2 \beta_{14} + 3 \beta_{9} + 5 \beta_{8} + \beta_{3} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + ( - \beta_{12} + \beta_{11} - \beta_{6} - \beta_{4} - 3 \beta_{2} - 1) q^{4} + (\beta_{15} - \beta_{14} + \beta_{9} + \beta_{8} + \beta_{3} - \beta_1) q^{5} + (\beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} - 2 \beta_{6} + 3) q^{7} + ( - 2 \beta_{14} + 3 \beta_{9} + 5 \beta_{8} + \beta_{3} - 2 \beta_1) q^{8} + ( - 2 \beta_{11} - 2 \beta_{7} + 3 \beta_{6} + 3 \beta_{2}) q^{10} + (2 \beta_{15} - \beta_{14} + \beta_{8} + 2 \beta_{3} - 3 \beta_1) q^{11} + ( - \beta_{11} + \beta_{10} + 3 \beta_{7} - 2 \beta_{6} + \beta_{4} + 2 \beta_{2} - 2) q^{13} + ( - \beta_{15} + \beta_{13} + 3 \beta_{9} + 6 \beta_{5} - \beta_{3}) q^{14} + ( - 3 \beta_{11} + 6 \beta_{10} - 6 \beta_{7} + 13 \beta_{6} + 8 \beta_{4} + \cdots - 13) q^{16}+ \cdots + ( - 6 \beta_{15} - 2 \beta_{14} + 5 \beta_{13} - 13 \beta_{9} + 14 \beta_{8} + \cdots - 14 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} + 30 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{4} + 30 q^{7} - 30 q^{13} - 176 q^{16} + 90 q^{22} - 74 q^{25} - 50 q^{28} + 130 q^{31} + 328 q^{34} + 90 q^{37} + 450 q^{40} - 370 q^{46} - 54 q^{49} - 790 q^{52} - 476 q^{55} - 630 q^{58} + 210 q^{61} + 1104 q^{64} + 300 q^{67} + 268 q^{70} - 170 q^{73} + 30 q^{79} + 90 q^{82} - 610 q^{85} - 600 q^{88} - 402 q^{91} + 1030 q^{94} + 870 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 21x^{14} + 227x^{12} - 1488x^{10} + 24225x^{8} - 62832x^{6} + 64372x^{4} + 7986x^{2} + 14641 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 95551278 \nu^{14} - 126822633412 \nu^{12} + 2507387635134 \nu^{10} - 25685652894720 \nu^{8} + 157339424200095 \nu^{6} + \cdots - 29\!\cdots\!58 ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 32130978280 \nu^{15} + 31948404670 \nu^{13} + 5924614934588 \nu^{11} - 93193803417051 \nu^{9} + 133796302845113 \nu^{7} + \cdots - 43\!\cdots\!77 \nu ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5854282560 \nu^{14} + 125428263227 \nu^{12} - 1388833610400 \nu^{10} + 9410848624160 \nu^{8} - 146776049383200 \nu^{6} + \cdots + 88541185404320 ) / 355575559329595 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5854282560 \nu^{15} + 125428263227 \nu^{13} - 1388833610400 \nu^{11} + 9410848624160 \nu^{9} - 146776049383200 \nu^{7} + \cdots + 88541185404320 \nu ) / 355575559329595 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7530891822 \nu^{14} - 161922628410 \nu^{12} + 1773071437239 \nu^{10} - 11760003820535 \nu^{8} + 185084618505135 \nu^{6} + \cdots + 243765923221832 ) / 355575559329595 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 115291499597 \nu^{14} - 2410283199611 \nu^{12} + 25350978529100 \nu^{10} - 157438624000303 \nu^{8} + \cdots + 13\!\cdots\!07 ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 147141366924 \nu^{15} + 3034037174595 \nu^{13} - 32273567888895 \nu^{11} + 207193723996925 \nu^{9} + \cdots - 753747908220945 \nu ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13385174382 \nu^{15} - 287350891637 \nu^{13} + 3161905047639 \nu^{11} - 21170852444695 \nu^{9} + 331860667888335 \nu^{7} + \cdots + 155224737817512 \nu ) / 355575559329595 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 298410148268 \nu^{14} + 6273343728321 \nu^{12} - 67973023670350 \nu^{10} + 448617912246188 \nu^{8} + \cdots + 12\!\cdots\!73 ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 312289527222 \nu^{14} + 6157499382802 \nu^{12} - 62836567910801 \nu^{10} + 380954707458452 \nu^{8} + \cdots - 13\!\cdots\!26 ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 88180182 \nu^{14} + 1926186986 \nu^{12} - 21449350233 \nu^{10} + 145597396588 \nu^{8} - 2222070588469 \nu^{6} + \cdots - 484613427771 ) / 924883223605 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 598241835319 \nu^{15} - 13289199325505 \nu^{13} + 150985563543754 \nu^{11} + \cdots - 26\!\cdots\!82 \nu ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 689220968556 \nu^{15} + 13753077377678 \nu^{13} - 141872714227952 \nu^{11} + 873330891688747 \nu^{9} + \cdots - 27\!\cdots\!29 \nu ) / 39\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 699614355640 \nu^{15} - 14626335417890 \nu^{13} + 156853985883961 \nu^{11} + \cdots + 42\!\cdots\!97 \nu ) / 39\!\cdots\!45 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + \beta_{11} - \beta_{10} + 2\beta_{7} - 2\beta_{6} - 5\beta_{4} - \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - 2\beta_{13} - \beta_{9} - 11\beta_{5} + 2\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{12} + 6\beta_{11} - 3\beta_{10} + 18\beta_{7} - 82\beta_{6} - 19\beta_{4} - 25\beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30\beta_{15} + 3\beta_{14} - 18\beta_{13} - 149\beta_{9} - 37\beta_{8} - 146\beta_{5} + 33\beta_{3} + 37\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -212\beta_{12} + 212\beta_{11} + 70\beta_{10} + 70\beta_{7} - 282\beta_{6} - 364\beta_{4} - 1191\beta_{2} - 434 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 282 \beta_{15} + 282 \beta_{14} - 70 \beta_{13} - 2321 \beta_{9} - 2321 \beta_{8} - 714 \beta_{5} + 212 \beta_{3} + 1607 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2556 \beta_{12} + 1749 \beta_{11} + 4305 \beta_{10} - 1278 \beta_{7} + 6413 \beta_{6} - 7691 \beta_{4} - 15755 \beta_{2} - 18311 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1278\beta_{15} + 6054\beta_{14} + 1278\beta_{13} - 18759\beta_{9} - 35195\beta_{8} - 3027\beta_{3} + 6054\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 43805 \beta_{12} - 22490 \beta_{11} + 87610 \beta_{10} - 65120 \beta_{7} + 124340 \beta_{6} - 80535 \beta_{2} - 291103 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 21315 \beta_{15} + 65120 \beta_{14} + 65120 \beta_{13} - 65120 \beta_{9} - 274720 \beta_{8} + 209600 \beta_{5} - 108925 \beta_{3} - 147798 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 300528 \beta_{12} - 640368 \beta_{11} + 980208 \beta_{10} - 1280736 \beta_{7} + 2603456 \beta_{6} + 1360625 \beta_{4} + 640368 \beta_{2} - 2603456 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 980208 \beta_{15} + 339840 \beta_{14} + 1280736 \beta_{13} + 2982608 \beta_{9} - 339840 \beta_{8} + 6185713 \beta_{5} - 1960416 \beta_{3} - 2982608 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 4144001 \beta_{12} - 10565728 \beta_{11} + 5282864 \beta_{10} - 14709729 \beta_{7} + 37812551 \beta_{6} + 25058897 \beta_{4} + 35624625 \beta_{2} - 5282864 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 18853730 \beta_{15} - 5282864 \beta_{14} + 14709729 \beta_{13} + 96434907 \beta_{9} + 56756081 \beta_{8} + 91152043 \beta_{5} - 24136594 \beta_{3} - 56756081 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
3.67414 + 1.19380i
1.32111 + 0.429256i
−1.32111 − 0.429256i
−3.67414 − 1.19380i
−1.83190 − 2.52140i
−0.386583 − 0.532086i
0.386583 + 0.532086i
1.83190 + 2.52140i
−1.83190 + 2.52140i
−0.386583 + 0.532086i
0.386583 − 0.532086i
1.83190 − 2.52140i
3.67414 − 1.19380i
1.32111 − 0.429256i
−1.32111 + 0.429256i
−3.67414 + 1.19380i
−2.27075 − 3.12541i 0 −3.37586 + 10.3898i 3.35774 + 2.43954i 0 7.08028 + 2.30052i 25.4416 − 8.26648i 0 − 16.0339i
19.2 −0.816494 − 1.12381i 0 0.639787 − 1.96906i −3.53770 − 2.57029i 0 0.582836 + 0.189375i −8.01969 + 2.60575i 0 6.07432i
19.3 0.816494 + 1.12381i 0 0.639787 − 1.96906i 3.53770 + 2.57029i 0 0.582836 + 0.189375i 8.01969 − 2.60575i 0 6.07432i
19.4 2.27075 + 3.12541i 0 −3.37586 + 10.3898i −3.35774 − 2.43954i 0 7.08028 + 2.30052i −25.4416 + 8.26648i 0 − 16.0339i
28.1 −2.96408 + 0.963089i 0 4.62219 − 3.35821i −0.439256 + 1.35189i 0 2.23863 + 3.08121i −3.13867 + 4.32000i 0 − 4.43016i
28.2 −0.625505 + 0.203239i 0 −2.88612 + 2.09689i 2.50346 − 7.70484i 0 −2.40175 − 3.30573i 2.92545 − 4.02653i 0 5.32822i
28.3 0.625505 − 0.203239i 0 −2.88612 + 2.09689i −2.50346 + 7.70484i 0 −2.40175 − 3.30573i −2.92545 + 4.02653i 0 5.32822i
28.4 2.96408 − 0.963089i 0 4.62219 − 3.35821i 0.439256 − 1.35189i 0 2.23863 + 3.08121i 3.13867 − 4.32000i 0 − 4.43016i
46.1 −2.96408 − 0.963089i 0 4.62219 + 3.35821i −0.439256 − 1.35189i 0 2.23863 − 3.08121i −3.13867 − 4.32000i 0 4.43016i
46.2 −0.625505 − 0.203239i 0 −2.88612 − 2.09689i 2.50346 + 7.70484i 0 −2.40175 + 3.30573i 2.92545 + 4.02653i 0 − 5.32822i
46.3 0.625505 + 0.203239i 0 −2.88612 − 2.09689i −2.50346 − 7.70484i 0 −2.40175 + 3.30573i −2.92545 − 4.02653i 0 − 5.32822i
46.4 2.96408 + 0.963089i 0 4.62219 + 3.35821i 0.439256 + 1.35189i 0 2.23863 − 3.08121i 3.13867 + 4.32000i 0 4.43016i
73.1 −2.27075 + 3.12541i 0 −3.37586 − 10.3898i 3.35774 − 2.43954i 0 7.08028 − 2.30052i 25.4416 + 8.26648i 0 16.0339i
73.2 −0.816494 + 1.12381i 0 0.639787 + 1.96906i −3.53770 + 2.57029i 0 0.582836 − 0.189375i −8.01969 − 2.60575i 0 − 6.07432i
73.3 0.816494 − 1.12381i 0 0.639787 + 1.96906i 3.53770 − 2.57029i 0 0.582836 − 0.189375i 8.01969 + 2.60575i 0 − 6.07432i
73.4 2.27075 − 3.12541i 0 −3.37586 − 10.3898i −3.35774 + 2.43954i 0 7.08028 − 2.30052i −25.4416 − 8.26648i 0 16.0339i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.3.k.b ✓ 16
3.b odd 2 1 inner 99.3.k.b ✓ 16
11.c even 5 1 1089.3.c.l 16
11.d odd 10 1 inner 99.3.k.b ✓ 16
11.d odd 10 1 1089.3.c.l 16
33.f even 10 1 inner 99.3.k.b ✓ 16
33.f even 10 1 1089.3.c.l 16
33.h odd 10 1 1089.3.c.l 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.k.b ✓ 16 1.a even 1 1 trivial
99.3.k.b ✓ 16 3.b odd 2 1 inner
99.3.k.b ✓ 16 11.d odd 10 1 inner
99.3.k.b ✓ 16 33.f even 10 1 inner
1089.3.c.l 16 11.c even 5 1
1089.3.c.l 16 11.d odd 10 1
1089.3.c.l 16 33.f even 10 1
1089.3.c.l 16 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 6T_{2}^{14} + 172T_{2}^{12} - 2568T_{2}^{10} + 20265T_{2}^{8} + 1848T_{2}^{6} + 71027T_{2}^{4} - 51909T_{2}^{2} + 14641 \) acting on \(S_{3}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 6 T^{14} + 172 T^{12} + \cdots + 14641 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 87 T^{14} + \cdots + 1908029761 \) Copy content Toggle raw display
$7$ \( (T^{8} - 15 T^{7} + 77 T^{6} - 200 T^{5} + \cdots + 5041)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 417 T^{14} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( (T^{8} + 15 T^{7} - 403 T^{6} + \cdots + 441168016)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 803437664440576 \) Copy content Toggle raw display
$19$ \( (T^{8} - 640 T^{6} - 16235 T^{5} + \cdots + 5628000400)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 1430 T^{6} + \cdots + 1301766400)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} - 1115 T^{14} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} - 65 T^{7} + \cdots + 350813367025)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 45 T^{7} + 1615 T^{6} + \cdots + 474368400)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 5155 T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{8} + 5102 T^{6} + \cdots + 431696305296)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 2455 T^{14} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + 12905 T^{14} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{16} + 4320 T^{14} + \cdots + 41\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{8} - 105 T^{7} + \cdots + 3114589632400)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 75 T^{3} - 3655 T^{2} + \cdots - 3482380)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} + 1575 T^{14} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + 85 T^{7} + \cdots + 494466201667216)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 15 T^{7} + \cdots + 436852010010025)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 23291 T^{14} + \cdots + 21\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( (T^{8} - 29246 T^{6} + \cdots + 33083847444736)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 435 T^{7} + \cdots + 20\!\cdots\!25)^{2} \) Copy content Toggle raw display
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