Newspace parameters
Level: | \( N \) | \(=\) | \( 125 = 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 125.d (of order \(5\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.998130025266\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{5})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
Defining polynomial: |
\( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 5^{2} \) |
Twist minimal: | no (minimal twist has level 25) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \)
:
\(\beta_{1}\) | \(=\) |
\( ( 22849 \nu^{15} - 2422021 \nu^{13} + 6573709 \nu^{11} - 1538146 \nu^{9} + 97097069 \nu^{7} - 420964990 \nu^{5} - 210825325 \nu^{3} - 35093875 \nu ) / 858900625 \)
|
\(\beta_{2}\) | \(=\) |
\( ( 948392 \nu^{14} - 2081693 \nu^{12} - 1547103 \nu^{10} - 35207443 \nu^{8} + 136185777 \nu^{6} + 77053830 \nu^{4} + 27131400 \nu^{2} + \cdots + 181387875 ) / 171780125 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 122987 \nu^{14} + 97172 \nu^{12} - 701513 \nu^{10} - 5799603 \nu^{8} + 3932417 \nu^{6} + 54634025 \nu^{4} + 77779875 \nu^{2} + 52412250 ) / 15616375 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 1379032 \nu^{14} - 312743 \nu^{12} - 4990978 \nu^{10} - 61900293 \nu^{8} + 94590252 \nu^{6} + 424939915 \nu^{4} + 771306350 \nu^{2} + \cdots + 563878625 ) / 171780125 \)
|
\(\beta_{5}\) | \(=\) |
\( ( - 281280 \nu^{14} + 69219 \nu^{12} + 939009 \nu^{10} + 12381889 \nu^{8} - 18275316 \nu^{6} - 83450271 \nu^{4} - 152170830 \nu^{2} - 167096475 ) / 34356025 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 1157122 \nu^{15} + 1428203 \nu^{13} + 8678488 \nu^{11} + 43919978 \nu^{9} - 137248467 \nu^{7} - 474206065 \nu^{5} + 222668250 \nu^{3} + \cdots + 846395125 \nu ) / 858900625 \)
|
\(\beta_{7}\) | \(=\) |
\( ( - 441862 \nu^{14} + 108648 \nu^{12} + 1544473 \nu^{10} + 20140738 \nu^{8} - 29602957 \nu^{6} - 135351515 \nu^{4} - 246968035 \nu^{2} + \cdots - 215410900 ) / 34356025 \)
|
\(\beta_{8}\) | \(=\) |
\( ( - 2817591 \nu^{14} + 2379174 \nu^{12} + 8884104 \nu^{10} + 118381374 \nu^{8} - 262226761 \nu^{6} - 716617430 \nu^{4} - 884417625 \nu^{2} + \cdots - 860602375 ) / 171780125 \)
|
\(\beta_{9}\) | \(=\) |
\( ( - 626638 \nu^{14} + 144621 \nu^{12} + 2783831 \nu^{10} + 27217946 \nu^{8} - 41997039 \nu^{6} - 210462216 \nu^{4} - 281904075 \nu^{2} + \cdots - 222107450 ) / 34356025 \)
|
\(\beta_{10}\) | \(=\) |
\( ( - 257689 \nu^{15} - 260409 \nu^{13} + 1184711 \nu^{11} + 13501191 \nu^{9} - 4440499 \nu^{7} - 113117275 \nu^{5} - 254856700 \nu^{3} + \cdots - 189940875 \nu ) / 78081875 \)
|
\(\beta_{11}\) | \(=\) |
\( ( 3711283 \nu^{15} + 2442853 \nu^{13} - 18411087 \nu^{11} - 175485672 \nu^{9} + 137417033 \nu^{7} + 1546922530 \nu^{5} + 2678832450 \nu^{3} + \cdots + 1573606875 \nu ) / 858900625 \)
|
\(\beta_{12}\) | \(=\) |
\( ( 11296137 \nu^{15} - 2500148 \nu^{13} - 48798533 \nu^{11} - 491452273 \nu^{9} + 752782897 \nu^{7} + 3788808280 \nu^{5} + 5075395150 \nu^{3} + \cdots + 4625990250 \nu ) / 858900625 \)
|
\(\beta_{13}\) | \(=\) |
\( ( 1033909 \nu^{15} - 651386 \nu^{13} - 3517956 \nu^{11} - 44316636 \nu^{9} + 84515054 \nu^{7} + 285294535 \nu^{5} + 432663800 \nu^{3} + \cdots + 416073000 \nu ) / 78081875 \)
|
\(\beta_{14}\) | \(=\) |
\( ( 1468861 \nu^{15} - 201274 \nu^{13} - 5426879 \nu^{11} - 66369199 \nu^{9} + 90901286 \nu^{7} + 466969935 \nu^{5} + 843562400 \nu^{3} + \cdots + 722601875 \nu ) / 78081875 \)
|
\(\beta_{15}\) | \(=\) |
\( ( 18299323 \nu^{15} - 8550017 \nu^{13} - 66698807 \nu^{11} - 789938892 \nu^{9} + 1385305413 \nu^{7} + 5454301045 \nu^{5} + 7862490225 \nu^{3} + \cdots + 7417196625 \nu ) / 858900625 \)
|
\(\nu\) | \(=\) |
\( ( -\beta_{15} - 5\beta_{14} + 4\beta_{13} + 4\beta_{12} - 5\beta_{10} + 5\beta_{6} + \beta_1 ) / 5 \)
|
\(\nu^{2}\) | \(=\) |
\( ( -2\beta_{9} + 2\beta_{8} + 3\beta_{7} + 2\beta_{5} + 11\beta_{4} - 4\beta_{3} - \beta_{2} + 4 ) / 5 \)
|
\(\nu^{3}\) | \(=\) |
\( ( -6\beta_{15} + 14\beta_{13} - 6\beta_{12} + 5\beta_{11} - 9\beta_1 ) / 5 \)
|
\(\nu^{4}\) | \(=\) |
\( ( -7\beta_{9} - 13\beta_{8} + 3\beta_{7} + 7\beta_{5} + 6\beta_{4} - 19\beta_{3} - 26\beta_{2} + 14 ) / 5 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 39\beta_{15} - 20\beta_{14} - 6\beta_{13} - 36\beta_{12} - 30\beta_{10} - 29\beta_1 ) / 5 \)
|
\(\nu^{6}\) | \(=\) |
\( ( 23\beta_{9} - 23\beta_{8} + 63\beta_{7} + 52\beta_{5} + 156\beta_{4} + 11\beta_{3} - 11\beta_{2} + 144 ) / 5 \)
|
\(\nu^{7}\) | \(=\) |
\( ( - 26 \beta_{15} - 190 \beta_{14} + 184 \beta_{13} + 49 \beta_{12} + 190 \beta_{11} - 115 \beta_{10} + 115 \beta_{6} + 66 \beta_1 ) / 5 \)
|
\(\nu^{8}\) | \(=\) |
\( ( -57\beta_{9} - 18\beta_{8} + 363\beta_{7} - 208\beta_{5} + 381\beta_{4} - 114\beta_{3} - 96\beta_{2} + 39 ) / 5 \)
|
\(\nu^{9}\) | \(=\) |
\( ( -\beta_{15} + 284\beta_{13} - 286\beta_{12} + 285\beta_{11} + 285\beta_{10} + 190\beta_{6} - 479\beta_1 ) / 5 \)
|
\(\nu^{10}\) | \(=\) |
\( ( 208\beta_{9} - 688\beta_{8} + 208\beta_{7} - 493\beta_{5} - 149\beta_{4} - 344\beta_{3} - 896\beta_{2} - 606 ) / 5 \)
|
\(\nu^{11}\) | \(=\) |
\( ( 2344 \beta_{15} - 290 \beta_{14} - 1651 \beta_{13} - 1846 \beta_{12} + 195 \beta_{11} - 290 \beta_{10} + 95 \beta_{6} - 624 \beta_1 ) / 5 \)
|
\(\nu^{12}\) | \(=\) |
\( ( 2208 \beta_{9} - 683 \beta_{8} + 2888 \beta_{7} - 683 \beta_{5} + 3956 \beta_{4} + 2891 \beta_{3} + 1104 \beta_{2} + 1784 ) / 5 \)
|
\(\nu^{13}\) | \(=\) |
\( ( - 906 \beta_{15} - 4675 \beta_{14} + 2364 \beta_{13} + 3769 \beta_{12} + 7530 \beta_{11} + 4675 \beta_{6} + 5166 \beta_1 ) / 5 \)
|
\(\nu^{14}\) | \(=\) |
\( ( 393 \beta_{9} + 1012 \beta_{8} + 10778 \beta_{7} - 18118 \beta_{5} - 619 \beta_{4} + 1631 \beta_{3} + 2024 \beta_{2} - 18511 ) / 5 \)
|
\(\nu^{15}\) | \(=\) |
\( ( 1639\beta_{15} + 19130\beta_{14} - 12431\beta_{13} - 9336\beta_{12} + 30920\beta_{10} - 13429\beta_1 ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/125\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
26.1 |
|
−0.644389 | − | 1.98322i | 1.77862 | − | 1.29224i | −1.89991 | + | 1.38036i | 0 | −3.70892 | − | 2.69469i | 0.992398 | 0.587785 | + | 0.427051i | 0.566541 | − | 1.74363i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
26.2 | −0.0566033 | − | 0.174207i | 1.19083 | − | 0.865190i | 1.59089 | − | 1.15585i | 0 | −0.218127 | − | 0.158479i | −3.26086 | −0.587785 | − | 0.427051i | −0.257524 | + | 0.792578i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
26.3 | 0.0566033 | + | 0.174207i | −1.19083 | + | 0.865190i | 1.59089 | − | 1.15585i | 0 | −0.218127 | − | 0.158479i | 3.26086 | 0.587785 | + | 0.427051i | −0.257524 | + | 0.792578i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
26.4 | 0.644389 | + | 1.98322i | −1.77862 | + | 1.29224i | −1.89991 | + | 1.38036i | 0 | −3.70892 | − | 2.69469i | −0.992398 | −0.587785 | − | 0.427051i | 0.566541 | − | 1.74363i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
51.1 | −1.86824 | + | 1.35736i | 0.146753 | − | 0.451659i | 1.02988 | − | 3.16963i | 0 | 0.338893 | + | 1.04301i | 3.03582 | 0.951057 | + | 2.92705i | 2.24459 | + | 1.63079i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
51.2 | −0.917186 | + | 0.666375i | −0.804303 | + | 2.47539i | −0.220859 | + | 0.679734i | 0 | −0.911842 | − | 2.80636i | 0.407162 | −0.951057 | − | 2.92705i | −3.05361 | − | 2.21858i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
51.3 | 0.917186 | − | 0.666375i | 0.804303 | − | 2.47539i | −0.220859 | + | 0.679734i | 0 | −0.911842 | − | 2.80636i | −0.407162 | 0.951057 | + | 2.92705i | −3.05361 | − | 2.21858i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
51.4 | 1.86824 | − | 1.35736i | −0.146753 | + | 0.451659i | 1.02988 | − | 3.16963i | 0 | 0.338893 | + | 1.04301i | −3.03582 | −0.951057 | − | 2.92705i | 2.24459 | + | 1.63079i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
76.1 | −1.86824 | − | 1.35736i | 0.146753 | + | 0.451659i | 1.02988 | + | 3.16963i | 0 | 0.338893 | − | 1.04301i | 3.03582 | 0.951057 | − | 2.92705i | 2.24459 | − | 1.63079i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
76.2 | −0.917186 | − | 0.666375i | −0.804303 | − | 2.47539i | −0.220859 | − | 0.679734i | 0 | −0.911842 | + | 2.80636i | 0.407162 | −0.951057 | + | 2.92705i | −3.05361 | + | 2.21858i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
76.3 | 0.917186 | + | 0.666375i | 0.804303 | + | 2.47539i | −0.220859 | − | 0.679734i | 0 | −0.911842 | + | 2.80636i | −0.407162 | 0.951057 | − | 2.92705i | −3.05361 | + | 2.21858i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
76.4 | 1.86824 | + | 1.35736i | −0.146753 | − | 0.451659i | 1.02988 | + | 3.16963i | 0 | 0.338893 | − | 1.04301i | −3.03582 | −0.951057 | + | 2.92705i | 2.24459 | − | 1.63079i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
101.1 | −0.644389 | + | 1.98322i | 1.77862 | + | 1.29224i | −1.89991 | − | 1.38036i | 0 | −3.70892 | + | 2.69469i | 0.992398 | 0.587785 | − | 0.427051i | 0.566541 | + | 1.74363i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
101.2 | −0.0566033 | + | 0.174207i | 1.19083 | + | 0.865190i | 1.59089 | + | 1.15585i | 0 | −0.218127 | + | 0.158479i | −3.26086 | −0.587785 | + | 0.427051i | −0.257524 | − | 0.792578i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
101.3 | 0.0566033 | − | 0.174207i | −1.19083 | − | 0.865190i | 1.59089 | + | 1.15585i | 0 | −0.218127 | + | 0.158479i | 3.26086 | 0.587785 | − | 0.427051i | −0.257524 | − | 0.792578i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
101.4 | 0.644389 | − | 1.98322i | −1.77862 | − | 1.29224i | −1.89991 | − | 1.38036i | 0 | −3.70892 | + | 2.69469i | −0.992398 | −0.587785 | + | 0.427051i | 0.566541 | + | 1.74363i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
25.e | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 125.2.d.b | 16 | |
5.b | even | 2 | 1 | inner | 125.2.d.b | 16 | |
5.c | odd | 4 | 1 | 25.2.e.a | ✓ | 8 | |
5.c | odd | 4 | 1 | 125.2.e.b | 8 | ||
15.e | even | 4 | 1 | 225.2.m.a | 8 | ||
20.e | even | 4 | 1 | 400.2.y.c | 8 | ||
25.d | even | 5 | 1 | inner | 125.2.d.b | 16 | |
25.d | even | 5 | 1 | 625.2.a.f | 8 | ||
25.d | even | 5 | 2 | 625.2.d.o | 16 | ||
25.e | even | 10 | 1 | inner | 125.2.d.b | 16 | |
25.e | even | 10 | 1 | 625.2.a.f | 8 | ||
25.e | even | 10 | 2 | 625.2.d.o | 16 | ||
25.f | odd | 20 | 1 | 25.2.e.a | ✓ | 8 | |
25.f | odd | 20 | 1 | 125.2.e.b | 8 | ||
25.f | odd | 20 | 2 | 625.2.b.c | 8 | ||
25.f | odd | 20 | 2 | 625.2.e.a | 8 | ||
25.f | odd | 20 | 2 | 625.2.e.i | 8 | ||
75.h | odd | 10 | 1 | 5625.2.a.x | 8 | ||
75.j | odd | 10 | 1 | 5625.2.a.x | 8 | ||
75.l | even | 20 | 1 | 225.2.m.a | 8 | ||
100.h | odd | 10 | 1 | 10000.2.a.bj | 8 | ||
100.j | odd | 10 | 1 | 10000.2.a.bj | 8 | ||
100.l | even | 20 | 1 | 400.2.y.c | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
25.2.e.a | ✓ | 8 | 5.c | odd | 4 | 1 | |
25.2.e.a | ✓ | 8 | 25.f | odd | 20 | 1 | |
125.2.d.b | 16 | 1.a | even | 1 | 1 | trivial | |
125.2.d.b | 16 | 5.b | even | 2 | 1 | inner | |
125.2.d.b | 16 | 25.d | even | 5 | 1 | inner | |
125.2.d.b | 16 | 25.e | even | 10 | 1 | inner | |
125.2.e.b | 8 | 5.c | odd | 4 | 1 | ||
125.2.e.b | 8 | 25.f | odd | 20 | 1 | ||
225.2.m.a | 8 | 15.e | even | 4 | 1 | ||
225.2.m.a | 8 | 75.l | even | 20 | 1 | ||
400.2.y.c | 8 | 20.e | even | 4 | 1 | ||
400.2.y.c | 8 | 100.l | even | 20 | 1 | ||
625.2.a.f | 8 | 25.d | even | 5 | 1 | ||
625.2.a.f | 8 | 25.e | even | 10 | 1 | ||
625.2.b.c | 8 | 25.f | odd | 20 | 2 | ||
625.2.d.o | 16 | 25.d | even | 5 | 2 | ||
625.2.d.o | 16 | 25.e | even | 10 | 2 | ||
625.2.e.a | 8 | 25.f | odd | 20 | 2 | ||
625.2.e.i | 8 | 25.f | odd | 20 | 2 | ||
5625.2.a.x | 8 | 75.h | odd | 10 | 1 | ||
5625.2.a.x | 8 | 75.j | odd | 10 | 1 | ||
10000.2.a.bj | 8 | 100.h | odd | 10 | 1 | ||
10000.2.a.bj | 8 | 100.j | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 3T_{2}^{14} + 23T_{2}^{12} + 126T_{2}^{10} + 475T_{2}^{8} - 174T_{2}^{6} + 878T_{2}^{4} + 48T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(125, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} + 3 T^{14} + 23 T^{12} + 126 T^{10} + \cdots + 1 \)
$3$
\( T^{16} + 7 T^{14} + 33 T^{12} + 119 T^{10} + \cdots + 256 \)
$5$
\( T^{16} \)
$7$
\( (T^{8} - 21 T^{6} + 121 T^{4} - 116 T^{2} + \cdots + 16)^{2} \)
$11$
\( (T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16)^{4} \)
$13$
\( T^{16} + 17 T^{14} + 108 T^{12} - 101 T^{10} + \cdots + 1 \)
$17$
\( T^{16} - 12 T^{14} + 698 T^{12} + \cdots + 3748096 \)
$19$
\( (T^{8} - 5 T^{7} + 30 T^{6} - 40 T^{5} + \cdots + 400)^{2} \)
$23$
\( T^{16} + 27 T^{14} + 333 T^{12} + \cdots + 65536 \)
$29$
\( (T^{8} - 5 T^{7} + 30 T^{6} + 5 T^{5} + \cdots + 483025)^{2} \)
$31$
\( (T^{8} + 9 T^{7} + 117 T^{6} + 917 T^{5} + \cdots + 1936)^{2} \)
$37$
\( T^{16} + 88 T^{14} + \cdots + 13521270961 \)
$41$
\( (T^{8} + 4 T^{7} + 52 T^{6} + 457 T^{5} + \cdots + 13456)^{2} \)
$43$
\( (T^{8} - 129 T^{6} + 4421 T^{4} + \cdots + 246016)^{2} \)
$47$
\( T^{16} - 32 T^{14} + \cdots + 4294967296 \)
$53$
\( T^{16} + 112 T^{14} + \cdots + 76661949773761 \)
$59$
\( (T^{8} - 15 T^{5} + 5635 T^{4} + \cdots + 4080400)^{2} \)
$61$
\( (T^{8} + 9 T^{7} - 43 T^{6} - 1068 T^{5} + \cdots + 116281)^{2} \)
$67$
\( T^{16} + 168 T^{14} + \cdots + 60523872256 \)
$71$
\( (T^{8} - 6 T^{7} + 142 T^{6} + \cdots + 24245776)^{2} \)
$73$
\( T^{16} + 127 T^{14} + 6183 T^{12} + \cdots + 1 \)
$79$
\( (T^{8} + 15 T^{7} + 100 T^{6} + \cdots + 33408400)^{2} \)
$83$
\( T^{16} + 127 T^{14} + \cdots + 9971220736 \)
$89$
\( (T^{8} - 25 T^{7} + 520 T^{6} + \cdots + 1392400)^{2} \)
$97$
\( T^{16} + 328 T^{14} + \cdots + 90\!\cdots\!61 \)
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