Newspace parameters
Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 99.g (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.790518980011\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
Defining polynomial: |
\( x^{16} + 15x^{14} + 150x^{12} + 837x^{10} + 3372x^{8} + 8010x^{6} + 13761x^{4} + 13392x^{2} + 8649 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{16} + 15x^{14} + 150x^{12} + 837x^{10} + 3372x^{8} + 8010x^{6} + 13761x^{4} + 13392x^{2} + 8649 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 2126395 \nu^{14} - 29523650 \nu^{12} - 288594750 \nu^{10} - 1489331264 \nu^{8} - 5823700650 \nu^{6} - 11995771800 \nu^{4} + \cdots - 22396351233 ) / 13131109983 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 2126395 \nu^{15} + 29523650 \nu^{13} + 288594750 \nu^{11} + 1489331264 \nu^{9} + 5823700650 \nu^{7} + 11995771800 \nu^{5} + \cdots + 9265241250 \nu ) / 13131109983 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 4873266 \nu^{15} - 160857665 \nu^{13} - 1890026425 \nu^{11} - 14824095809 \nu^{9} - 66244476882 \nu^{7} - 206255555451 \nu^{5} + \cdots - 290195282991 \nu ) / 13131109983 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 47500907 \nu^{15} + 812018893 \nu^{13} + 8261503794 \nu^{11} + 49665627558 \nu^{9} + 195642483900 \nu^{7} + 477061297716 \nu^{5} + \cdots + 551339033751 \nu ) / 39393329949 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 49032332 \nu^{14} - 864786042 \nu^{12} - 8872595664 \nu^{10} - 54770706318 \nu^{8} - 220344752319 \nu^{6} - 565719472953 \nu^{4} + \cdots - 671327583345 ) / 39393329949 \)
|
\(\beta_{7}\) | \(=\) |
\( ( - 54617732 \nu^{14} - 903112393 \nu^{12} - 9132887847 \nu^{10} - 53705137428 \nu^{8} - 210752440350 \nu^{6} - 495377443209 \nu^{4} + \cdots - 448939285020 ) / 39393329949 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 54617732 \nu^{15} + 903112393 \nu^{13} + 9132887847 \nu^{11} + 53705137428 \nu^{9} + 210752440350 \nu^{7} + 495377443209 \nu^{5} + \cdots + 409545955071 \nu ) / 39393329949 \)
|
\(\beta_{9}\) | \(=\) |
\( ( - 58397849 \nu^{14} - 683729698 \nu^{12} - 6054561519 \nu^{10} - 23065315299 \nu^{8} - 66793461054 \nu^{6} - 2243892963 \nu^{4} + \cdots + 208063920222 ) / 39393329949 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 64777034 \nu^{15} + 772300648 \nu^{13} + 6920345769 \nu^{11} + 27533309091 \nu^{9} + 84264563004 \nu^{7} + 38231208363 \nu^{5} + \cdots - 180268196472 \nu ) / 39393329949 \)
|
\(\beta_{11}\) | \(=\) |
\( ( 70362434 \nu^{15} + 810626999 \nu^{13} + 7180637952 \nu^{11} + 26467740201 \nu^{9} + 74672251035 \nu^{7} - 32110821381 \nu^{5} + \cdots - 442049824746 \nu ) / 39393329949 \)
|
\(\beta_{12}\) | \(=\) |
\( ( - 93878026 \nu^{14} - 1321129241 \nu^{12} - 12590096616 \nu^{10} - 63366471351 \nu^{8} - 225132595554 \nu^{6} - 389659389972 \nu^{4} + \cdots - 157488193548 ) / 39393329949 \)
|
\(\beta_{13}\) | \(=\) |
\( ( 105898756 \nu^{14} + 1495748591 \nu^{12} + 14316065313 \nu^{10} + 72730942857 \nu^{8} + 262435944954 \nu^{6} + 479305190679 \nu^{4} + \cdots + 303881783580 ) / 39393329949 \)
|
\(\beta_{14}\) | \(=\) |
\( ( - 131099776 \nu^{14} - 1868178342 \nu^{12} - 17994696072 \nu^{10} - 92965220511 \nu^{8} - 341617204785 \nu^{6} + \cdots - 446373561123 ) / 39393329949 \)
|
\(\beta_{15}\) | \(=\) |
\( ( 145719574 \nu^{15} + 2350751337 \nu^{13} + 23664775347 \nu^{11} + 137437507938 \nu^{9} + 540350635431 \nu^{7} + \cdots + 1277566080147 \nu ) / 39393329949 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( -\beta_{12} + \beta_{9} + \beta_{7} - 3\beta_{2} - 3 \)
|
\(\nu^{3}\) | \(=\) |
\( -\beta_{10} + \beta_{5} + \beta_{4} + 5\beta_{3} \)
|
\(\nu^{4}\) | \(=\) |
\( \beta_{14} + 9\beta_{13} + 8\beta_{12} - \beta_{7} + \beta_{6} + 12\beta_{2} \)
|
\(\nu^{5}\) | \(=\) |
\( -\beta_{15} + \beta_{11} + 8\beta_{10} - 8\beta_{8} + \beta_{5} - 9\beta_{4} - 29\beta_{3} - 29\beta_1 \)
|
\(\nu^{6}\) | \(=\) |
\( 13\beta_{14} - 67\beta_{13} - 14\beta_{12} - 68\beta_{9} - 41\beta_{7} - 26\beta_{6} + 57 \)
|
\(\nu^{7}\) | \(=\) |
\( -13\beta_{15} - 26\beta_{11} + 27\beta_{10} + 81\beta_{8} - 53\beta_{5} + \beta_{4} + 13\beta_{3} + 166\beta_1 \)
|
\(\nu^{8}\) | \(=\) |
\( -240\beta_{14} - 15\beta_{13} - 234\beta_{12} + 474\beta_{9} + 369\beta_{7} + 120\beta_{6} - 294\beta_{2} - 294 \)
|
\(\nu^{9}\) | \(=\) |
\( 240 \beta_{15} + 120 \beta_{11} - 609 \beta_{10} - 255 \beta_{8} + 219 \beta_{5} + 474 \beta_{4} + 1017 \beta_{3} + 120 \beta_1 \)
|
\(\nu^{10}\) | \(=\) |
\( 969\beta_{14} + 3438\beta_{13} + 2469\beta_{12} + 150\beta_{9} - 1119\beta_{7} + 969\beta_{6} + 1584\beta_{2} \)
|
\(\nu^{11}\) | \(=\) |
\( - 969 \beta_{15} + 969 \beta_{11} + 2319 \beta_{10} - 2319 \beta_{8} + 969 \beta_{5} - 3438 \beta_{4} - 7341 \beta_{3} - 7341 \beta_1 \)
|
\(\nu^{12}\) | \(=\) |
\( 7314\beta_{14} - 22581\beta_{13} - 8583\beta_{12} - 23850\beta_{9} - 7953\beta_{7} - 14628\beta_{6} + 8802 \)
|
\(\nu^{13}\) | \(=\) |
\( - 7314 \beta_{15} - 14628 \beta_{11} + 15897 \beta_{10} + 31164 \beta_{8} - 13998 \beta_{5} + 1269 \beta_{4} + 7314 \beta_{3} + 40605 \beta_1 \)
|
\(\nu^{14}\) | \(=\) |
\( - 106212 \beta_{14} - 9852 \beta_{13} - 47928 \beta_{12} + 154140 \beta_{9} + 110886 \beta_{7} + 53106 \beta_{6} - 50265 \beta_{2} - 50265 \)
|
\(\nu^{15}\) | \(=\) |
\( 106212 \beta_{15} + 53106 \beta_{11} - 217098 \beta_{10} - 116064 \beta_{8} + 38076 \beta_{5} + 154140 \beta_{4} + 262185 \beta_{3} + 53106 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).
\(n\) | \(46\) | \(56\) |
\(\chi(n)\) | \(-1\) | \(1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
32.1 |
|
−1.29716 | − | 2.24675i | −1.72785 | + | 0.120512i | −2.36526 | + | 4.09675i | −0.137404 | − | 0.0793301i | 2.51207 | + | 3.72573i | −2.91814 | + | 1.68479i | 7.08384 | 2.97095 | − | 0.416454i | 0.411616i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
32.2 | −1.10617 | − | 1.91594i | 1.59999 | − | 0.663339i | −1.44722 | + | 2.50665i | −2.54721 | − | 1.47063i | −3.04078 | − | 2.33173i | 1.72189 | − | 0.994132i | 1.97879 | 2.11996 | − | 2.12268i | 6.50707i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
32.3 | −0.679041 | − | 1.17613i | −0.323333 | − | 1.70160i | 0.0778064 | − | 0.134765i | 0.901139 | + | 0.520273i | −1.78176 | + | 1.53574i | 0.600962 | − | 0.346965i | −2.92750 | −2.79091 | + | 1.10037i | − | 1.41315i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
32.4 | −0.618600 | − | 1.07145i | −1.04881 | + | 1.37841i | 0.234668 | − | 0.406456i | 1.78348 | + | 1.02969i | 2.12568 | + | 0.271060i | 3.59283 | − | 2.07432i | −3.05506 | −0.800005 | − | 2.89137i | − | 2.54787i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
32.5 | 0.618600 | + | 1.07145i | −1.04881 | + | 1.37841i | 0.234668 | − | 0.406456i | 1.78348 | + | 1.02969i | −2.12568 | − | 0.271060i | −3.59283 | + | 2.07432i | 3.05506 | −0.800005 | − | 2.89137i | 2.54787i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
32.6 | 0.679041 | + | 1.17613i | −0.323333 | − | 1.70160i | 0.0778064 | − | 0.134765i | 0.901139 | + | 0.520273i | 1.78176 | − | 1.53574i | −0.600962 | + | 0.346965i | 2.92750 | −2.79091 | + | 1.10037i | 1.41315i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
32.7 | 1.10617 | + | 1.91594i | 1.59999 | − | 0.663339i | −1.44722 | + | 2.50665i | −2.54721 | − | 1.47063i | 3.04078 | + | 2.33173i | −1.72189 | + | 0.994132i | −1.97879 | 2.11996 | − | 2.12268i | − | 6.50707i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
32.8 | 1.29716 | + | 2.24675i | −1.72785 | + | 0.120512i | −2.36526 | + | 4.09675i | −0.137404 | − | 0.0793301i | −2.51207 | − | 3.72573i | 2.91814 | − | 1.68479i | −7.08384 | 2.97095 | − | 0.416454i | − | 0.411616i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
65.1 | −1.29716 | + | 2.24675i | −1.72785 | − | 0.120512i | −2.36526 | − | 4.09675i | −0.137404 | + | 0.0793301i | 2.51207 | − | 3.72573i | −2.91814 | − | 1.68479i | 7.08384 | 2.97095 | + | 0.416454i | − | 0.411616i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
65.2 | −1.10617 | + | 1.91594i | 1.59999 | + | 0.663339i | −1.44722 | − | 2.50665i | −2.54721 | + | 1.47063i | −3.04078 | + | 2.33173i | 1.72189 | + | 0.994132i | 1.97879 | 2.11996 | + | 2.12268i | − | 6.50707i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
65.3 | −0.679041 | + | 1.17613i | −0.323333 | + | 1.70160i | 0.0778064 | + | 0.134765i | 0.901139 | − | 0.520273i | −1.78176 | − | 1.53574i | 0.600962 | + | 0.346965i | −2.92750 | −2.79091 | − | 1.10037i | 1.41315i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
65.4 | −0.618600 | + | 1.07145i | −1.04881 | − | 1.37841i | 0.234668 | + | 0.406456i | 1.78348 | − | 1.02969i | 2.12568 | − | 0.271060i | 3.59283 | + | 2.07432i | −3.05506 | −0.800005 | + | 2.89137i | 2.54787i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
65.5 | 0.618600 | − | 1.07145i | −1.04881 | − | 1.37841i | 0.234668 | + | 0.406456i | 1.78348 | − | 1.02969i | −2.12568 | + | 0.271060i | −3.59283 | − | 2.07432i | 3.05506 | −0.800005 | + | 2.89137i | − | 2.54787i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
65.6 | 0.679041 | − | 1.17613i | −0.323333 | + | 1.70160i | 0.0778064 | + | 0.134765i | 0.901139 | − | 0.520273i | 1.78176 | + | 1.53574i | −0.600962 | − | 0.346965i | 2.92750 | −2.79091 | − | 1.10037i | − | 1.41315i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
65.7 | 1.10617 | − | 1.91594i | 1.59999 | + | 0.663339i | −1.44722 | − | 2.50665i | −2.54721 | + | 1.47063i | 3.04078 | − | 2.33173i | −1.72189 | − | 0.994132i | −1.97879 | 2.11996 | + | 2.12268i | 6.50707i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
65.8 | 1.29716 | − | 2.24675i | −1.72785 | − | 0.120512i | −2.36526 | − | 4.09675i | −0.137404 | + | 0.0793301i | −2.51207 | + | 3.72573i | 2.91814 | + | 1.68479i | −7.08384 | 2.97095 | + | 0.416454i | 0.411616i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
11.b | odd | 2 | 1 | inner |
99.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 99.2.g.b | ✓ | 16 |
3.b | odd | 2 | 1 | 297.2.g.b | 16 | ||
9.c | even | 3 | 1 | 297.2.g.b | 16 | ||
9.c | even | 3 | 1 | 891.2.d.b | 16 | ||
9.d | odd | 6 | 1 | inner | 99.2.g.b | ✓ | 16 |
9.d | odd | 6 | 1 | 891.2.d.b | 16 | ||
11.b | odd | 2 | 1 | inner | 99.2.g.b | ✓ | 16 |
33.d | even | 2 | 1 | 297.2.g.b | 16 | ||
99.g | even | 6 | 1 | inner | 99.2.g.b | ✓ | 16 |
99.g | even | 6 | 1 | 891.2.d.b | 16 | ||
99.h | odd | 6 | 1 | 297.2.g.b | 16 | ||
99.h | odd | 6 | 1 | 891.2.d.b | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
99.2.g.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
99.2.g.b | ✓ | 16 | 9.d | odd | 6 | 1 | inner |
99.2.g.b | ✓ | 16 | 11.b | odd | 2 | 1 | inner |
99.2.g.b | ✓ | 16 | 99.g | even | 6 | 1 | inner |
297.2.g.b | 16 | 3.b | odd | 2 | 1 | ||
297.2.g.b | 16 | 9.c | even | 3 | 1 | ||
297.2.g.b | 16 | 33.d | even | 2 | 1 | ||
297.2.g.b | 16 | 99.h | odd | 6 | 1 | ||
891.2.d.b | 16 | 9.c | even | 3 | 1 | ||
891.2.d.b | 16 | 9.d | odd | 6 | 1 | ||
891.2.d.b | 16 | 99.g | even | 6 | 1 | ||
891.2.d.b | 16 | 99.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 15T_{2}^{14} + 150T_{2}^{12} + 837T_{2}^{10} + 3372T_{2}^{8} + 8010T_{2}^{6} + 13761T_{2}^{4} + 13392T_{2}^{2} + 8649 \)
acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} + 15 T^{14} + 150 T^{12} + \cdots + 8649 \)
$3$
\( (T^{8} + 3 T^{7} + 3 T^{6} - 3 T^{5} - 15 T^{4} + \cdots + 81)^{2} \)
$5$
\( (T^{8} - 7 T^{6} + 48 T^{4} - 63 T^{3} + \cdots + 1)^{2} \)
$7$
\( T^{16} - 33 T^{14} + 765 T^{12} + \cdots + 138384 \)
$11$
\( T^{16} + 12 T^{15} + \cdots + 214358881 \)
$13$
\( T^{16} - 51 T^{14} + 2046 T^{12} + \cdots + 138384 \)
$17$
\( (T^{8} - 93 T^{6} + 3078 T^{4} + \cdots + 214272)^{2} \)
$19$
\( (T^{8} + 63 T^{6} + 1152 T^{4} + \cdots + 3348)^{2} \)
$23$
\( (T^{8} - 6 T^{7} - 25 T^{6} + 222 T^{5} + \cdots + 119716)^{2} \)
$29$
\( T^{16} + 30 T^{14} + 585 T^{12} + \cdots + 2214144 \)
$31$
\( (T^{8} + 2 T^{7} + 19 T^{6} - 16 T^{5} + \cdots + 1)^{2} \)
$37$
\( (T^{4} + 7 T^{3} - 51 T^{2} + 52 T + 31)^{4} \)
$41$
\( T^{16} + 132 T^{14} + \cdots + 18034341264 \)
$43$
\( T^{16} - 183 T^{14} + \cdots + 54056250000 \)
$47$
\( (T^{8} + 15 T^{7} + 50 T^{6} - 375 T^{5} + \cdots + 6889)^{2} \)
$53$
\( (T^{8} + 146 T^{6} + 5619 T^{4} + \cdots + 105625)^{2} \)
$59$
\( (T^{8} + 18 T^{7} + 71 T^{6} + \cdots + 265225)^{2} \)
$61$
\( T^{16} - 477 T^{14} + \cdots + 16\!\cdots\!44 \)
$67$
\( (T^{8} - 4 T^{7} + 103 T^{6} + 794 T^{5} + \cdots + 18769)^{2} \)
$71$
\( (T^{8} + 53 T^{6} + 735 T^{4} + 1382 T^{2} + \cdots + 361)^{2} \)
$73$
\( (T^{8} + 408 T^{6} + 58167 T^{4} + \cdots + 64686708)^{2} \)
$79$
\( T^{16} - 279 T^{14} + \cdots + 9069133824 \)
$83$
\( T^{16} + \cdots + 162096184890000 \)
$89$
\( (T^{8} + 308 T^{6} + 23664 T^{4} + \cdots + 1893376)^{2} \)
$97$
\( (T^{8} + 2 T^{7} + 124 T^{6} + \cdots + 564001)^{2} \)
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