Newspace parameters
Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 912.m (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(24.8502001097\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 2 x^{11} - 2 x^{10} - 28 x^{9} - 400 x^{8} - 520 x^{7} + 17067 x^{6} - 3250 x^{5} - 195494 x^{4} + 302996 x^{3} + 602332 x^{2} - 2263536 x + 2052928 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{14} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} - 2 x^{11} - 2 x^{10} - 28 x^{9} - 400 x^{8} - 520 x^{7} + 17067 x^{6} - 3250 x^{5} - 195494 x^{4} + 302996 x^{3} + 602332 x^{2} - 2263536 x + 2052928 \) :
\(\beta_{1}\) | \(=\) | \( ( 651569787185 \nu^{11} + 1209190969407 \nu^{10} + 4989114003428 \nu^{9} - 6889549986684 \nu^{8} - 277551169868310 \nu^{7} + \cdots + 13\!\cdots\!02 ) / 21\!\cdots\!98 \) |
\(\beta_{2}\) | \(=\) | \( ( 51\!\cdots\!21 \nu^{11} + \cdots - 73\!\cdots\!44 ) / 64\!\cdots\!08 \) |
\(\beta_{3}\) | \(=\) | \( ( 64\!\cdots\!87 \nu^{11} + \cdots - 12\!\cdots\!48 ) / 64\!\cdots\!08 \) |
\(\beta_{4}\) | \(=\) | \( ( - 99\!\cdots\!53 \nu^{11} + \cdots + 13\!\cdots\!44 ) / 64\!\cdots\!08 \) |
\(\beta_{5}\) | \(=\) | \( ( 157524610518805 \nu^{11} + 6067399667026 \nu^{10} - 201453765859666 \nu^{9} + \cdots - 15\!\cdots\!52 ) / 80\!\cdots\!12 \) |
\(\beta_{6}\) | \(=\) | \( ( - 78\!\cdots\!35 \nu^{11} + \cdots + 98\!\cdots\!14 ) / 32\!\cdots\!54 \) |
\(\beta_{7}\) | \(=\) | \( ( - 16\!\cdots\!15 \nu^{11} + \cdots + 69\!\cdots\!96 ) / 64\!\cdots\!08 \) |
\(\beta_{8}\) | \(=\) | \( ( 237830692375 \nu^{11} + 6844355458 \nu^{10} - 326852665758 \nu^{9} - 7584570552796 \nu^{8} - 109922313318736 \nu^{7} + \cdots - 23\!\cdots\!12 ) / 48\!\cdots\!12 \) |
\(\beta_{9}\) | \(=\) | \( ( 88\!\cdots\!41 \nu^{11} + \cdots - 61\!\cdots\!92 ) / 12\!\cdots\!16 \) |
\(\beta_{10}\) | \(=\) | \( ( - 88\!\cdots\!89 \nu^{11} + \cdots + 76\!\cdots\!96 ) / 64\!\cdots\!08 \) |
\(\beta_{11}\) | \(=\) | \( ( 74\!\cdots\!88 \nu^{11} + \cdots - 68\!\cdots\!28 ) / 32\!\cdots\!54 \) |
\(\nu\) | \(=\) | \( ( -\beta_{9} + \beta_{8} - \beta_{7} + \beta_{4} + \beta_{2} + 1 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( - \beta_{11} - 3 \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{7} - 5 \beta_{6} - 4 \beta_{5} + 5 \beta_{4} + \beta_{3} - 10 \beta_{2} - \beta_1 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( 2 \beta_{11} + 7 \beta_{10} + 10 \beta_{9} - 23 \beta_{8} + 7 \beta_{7} - 4 \beta_{6} + 61 \beta_{5} + 6 \beta_{4} - 6 \beta_{2} - 6 \beta _1 + 16 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( - 19 \beta_{11} - 55 \beta_{10} - \beta_{9} - 49 \beta_{8} + 42 \beta_{7} + 165 \beta_{6} + 38 \beta_{5} - 75 \beta_{4} + 13 \beta_{3} + 306 \beta_{2} - 13 \beta _1 + 718 ) / 4 \) |
\(\nu^{5}\) | \(=\) | \( ( - 173 \beta_{11} - 461 \beta_{10} - 943 \beta_{9} + 979 \beta_{8} - 898 \beta_{7} - 83 \beta_{6} - 1512 \beta_{5} - 43 \beta_{4} - 55 \beta_{3} - 246 \beta_{2} - 363 \beta _1 + 2464 ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( ( - 20 \beta_{11} - 244 \beta_{10} - 852 \beta_{9} - 80 \beta_{8} - 840 \beta_{7} - 1907 \beta_{6} + 628 \beta_{5} + 1917 \beta_{4} + 316 \beta_{3} - 2841 \beta_{2} + 774 \beta _1 - 13560 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( - 989 \beta_{11} + 1759 \beta_{10} + 24805 \beta_{9} - 33945 \beta_{8} + 28894 \beta_{7} - 7801 \beta_{6} + 62032 \beta_{5} + 5227 \beta_{4} + 127 \beta_{3} - 13858 \beta_{2} - 1065 \beta _1 - 20864 ) / 4 \) |
\(\nu^{8}\) | \(=\) | \( ( 10239 \beta_{11} + 23999 \beta_{10} + 27437 \beta_{9} - 92599 \beta_{8} + 19282 \beta_{7} + 133925 \beta_{6} + 211174 \beta_{5} - 108035 \beta_{4} - 10767 \beta_{3} + 221390 \beta_{2} + \cdots + 798374 ) / 4 \) |
\(\nu^{9}\) | \(=\) | \( ( - 73222 \beta_{11} - 213437 \beta_{10} - 430818 \beta_{9} + 552189 \beta_{8} - 392265 \beta_{7} + 162346 \beta_{6} - 1043371 \beta_{5} - 92634 \beta_{4} + 2770 \beta_{3} + 303036 \beta_{2} + \cdots + 437110 ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( - 263331 \beta_{11} - 835819 \beta_{10} - 1880729 \beta_{9} + 1984355 \beta_{8} - 1756674 \beta_{7} - 4759831 \beta_{6} - 3536502 \beta_{5} + 3235969 \beta_{4} + 132573 \beta_{3} + \cdots - 21212802 ) / 4 \) |
\(\nu^{11}\) | \(=\) | \( ( 3045003 \beta_{11} + 10000603 \beta_{10} + 25128717 \beta_{9} - 36690833 \beta_{8} + 24330242 \beta_{7} - 8278379 \beta_{6} + 71479104 \beta_{5} + 7683217 \beta_{4} + \cdots - 66587876 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).
\(n\) | \(97\) | \(229\) | \(305\) | \(799\) |
\(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
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799.1 |
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0 | − | 1.73205i | 0 | −6.54540 | 0 | − | 0.687253i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
799.2 | 0 | − | 1.73205i | 0 | −6.08630 | 0 | 8.85163i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
799.3 | 0 | − | 1.73205i | 0 | −4.65777 | 0 | − | 4.49506i | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
799.4 | 0 | − | 1.73205i | 0 | 2.67107 | 0 | − | 11.9371i | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
799.5 | 0 | − | 1.73205i | 0 | 4.69015 | 0 | 3.50408i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
799.6 | 0 | − | 1.73205i | 0 | 4.92825 | 0 | 9.95984i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
799.7 | 0 | 1.73205i | 0 | −6.54540 | 0 | 0.687253i | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
799.8 | 0 | 1.73205i | 0 | −6.08630 | 0 | − | 8.85163i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
799.9 | 0 | 1.73205i | 0 | −4.65777 | 0 | 4.49506i | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
799.10 | 0 | 1.73205i | 0 | 2.67107 | 0 | 11.9371i | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
799.11 | 0 | 1.73205i | 0 | 4.69015 | 0 | − | 3.50408i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
799.12 | 0 | 1.73205i | 0 | 4.92825 | 0 | − | 9.95984i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 912.3.m.b | ✓ | 12 |
3.b | odd | 2 | 1 | 2736.3.m.e | 12 | ||
4.b | odd | 2 | 1 | inner | 912.3.m.b | ✓ | 12 |
12.b | even | 2 | 1 | 2736.3.m.e | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
912.3.m.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
912.3.m.b | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
2736.3.m.e | 12 | 3.b | odd | 2 | 1 | ||
2736.3.m.e | 12 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 5T_{5}^{5} - 65T_{5}^{4} - 245T_{5}^{3} + 1468T_{5}^{2} + 2964T_{5} - 11456 \)
acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( (T^{2} + 3)^{6} \)
$5$
\( (T^{6} + 5 T^{5} - 65 T^{4} - 245 T^{3} + \cdots - 11456)^{2} \)
$7$
\( T^{12} + 353 T^{10} + \cdots + 129777664 \)
$11$
\( T^{12} + 717 T^{10} + \cdots + 67566724096 \)
$13$
\( (T^{6} - 18 T^{5} - 104 T^{4} + 1568 T^{3} + \cdots - 7808)^{2} \)
$17$
\( (T^{6} - 7 T^{5} - 1443 T^{4} + \cdots - 65017064)^{2} \)
$19$
\( (T^{2} + 19)^{6} \)
$23$
\( T^{12} + 6520 T^{10} + \cdots + 75\!\cdots\!76 \)
$29$
\( (T^{6} + 54 T^{5} - 2384 T^{4} + \cdots + 351804928)^{2} \)
$31$
\( T^{12} + 10576 T^{10} + \cdots + 29\!\cdots\!16 \)
$37$
\( (T^{6} + 8 T^{5} - 3532 T^{4} + \cdots + 343318528)^{2} \)
$41$
\( (T^{6} - 8 T^{5} - 5820 T^{4} + \cdots - 622678016)^{2} \)
$43$
\( T^{12} + 7105 T^{10} + \cdots + 14\!\cdots\!56 \)
$47$
\( T^{12} + 17145 T^{10} + \cdots + 15\!\cdots\!24 \)
$53$
\( (T^{6} - 110 T^{5} - 4184 T^{4} + \cdots - 5005174784)^{2} \)
$59$
\( T^{12} + 16688 T^{10} + \cdots + 34\!\cdots\!04 \)
$61$
\( (T^{6} - 183 T^{5} + \cdots + 21645030952)^{2} \)
$67$
\( T^{12} + 33328 T^{10} + \cdots + 28\!\cdots\!16 \)
$71$
\( T^{12} + 30160 T^{10} + \cdots + 53\!\cdots\!44 \)
$73$
\( (T^{6} + 79 T^{5} - 10287 T^{4} + \cdots + 30337444648)^{2} \)
$79$
\( T^{12} + 35772 T^{10} + \cdots + 31\!\cdots\!96 \)
$83$
\( T^{12} + 21676 T^{10} + \cdots + 14\!\cdots\!24 \)
$89$
\( (T^{6} + 198 T^{5} + \cdots - 12103330176)^{2} \)
$97$
\( (T^{6} - 112 T^{5} + \cdots - 12366236864)^{2} \)
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