Newspace parameters
Level: | \( N \) | \(=\) | \( 480 = 2^{5} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 480.l (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(13.0790526893\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2434 x^{12} - 8780 x^{11} + 25532 x^{10} - 57568 x^{9} + 104312 x^{8} - 150688 x^{7} + 173858 x^{6} - 157972 x^{5} + \cdots + 486 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
Coefficient ring index: | \( 2^{26}\cdot 3^{2} \) |
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 \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2434 x^{12} - 8780 x^{11} + 25532 x^{10} - 57568 x^{9} + 104312 x^{8} - 150688 x^{7} + 173858 x^{6} - 157972 x^{5} + \cdots + 486 \) :
\(\beta_{1}\) | \(=\) | \( ( - 231098 \nu^{14} + 1617686 \nu^{13} - 18599526 \nu^{12} + 90567238 \nu^{11} - 462307312 \nu^{10} + 1519891728 \nu^{9} - 4141077862 \nu^{8} + \cdots - 175521384 ) / 2058507 \) |
\(\beta_{2}\) | \(=\) | \( ( - 997042 \nu^{15} - 1058097 \nu^{14} - 24294277 \nu^{13} - 253966185 \nu^{12} + 1130769125 \nu^{11} - 9404278026 \nu^{10} + 34415887678 \nu^{9} + \cdots - 9776463642 ) / 34807482 \) |
\(\beta_{3}\) | \(=\) | \( ( 997042 \nu^{15} - 7971087 \nu^{14} + 87498565 \nu^{13} - 472682805 \nu^{12} + 2407469071 \nu^{11} - 8656182288 \nu^{10} + 24958932626 \nu^{9} + \cdots - 1183644576 ) / 34807482 \) |
\(\beta_{4}\) | \(=\) | \( ( 36066962 \nu^{15} - 252771207 \nu^{14} + 2913494285 \nu^{13} - 14209516638 \nu^{12} + 72902822759 \nu^{11} - 240460854195 \nu^{10} + \cdots + 1698468912 ) / 382882302 \) |
\(\beta_{5}\) | \(=\) | \( ( 116950 \nu^{15} - 877125 \nu^{14} + 9857083 \nu^{13} - 50767977 \nu^{12} + 259594045 \nu^{11} - 898586238 \nu^{10} + 2544488606 \nu^{9} + \cdots - 121233510 ) / 1200258 \) |
\(\beta_{6}\) | \(=\) | \( ( 14487080 \nu^{15} - 102235232 \nu^{14} + 1175477618 \nu^{13} - 5765621501 \nu^{12} + 29578100336 \nu^{11} - 98063343419 \nu^{10} + \cdots - 1819222686 ) / 127627434 \) |
\(\beta_{7}\) | \(=\) | \( ( - 129904 \nu^{15} + 974280 \nu^{14} - 10963756 \nu^{13} + 56487834 \nu^{12} - 289508632 \nu^{11} + 1003439778 \nu^{10} - 2854366820 \nu^{9} + \cdots + 174608460 ) / 801009 \) |
\(\beta_{8}\) | \(=\) | \( ( - 2122398 \nu^{15} + 15288809 \nu^{14} - 174045611 \nu^{13} + 867405085 \nu^{12} - 4427792221 \nu^{11} + 14859790638 \nu^{10} - 41080092030 \nu^{9} + \cdots - 431238924 ) / 11602494 \) |
\(\beta_{9}\) | \(=\) | \( ( 79528202 \nu^{15} - 581078013 \nu^{14} + 6591134909 \nu^{13} - 33262336089 \nu^{12} + 170207152445 \nu^{11} - 579154679256 \nu^{10} + \cdots - 74336559138 ) / 382882302 \) |
\(\beta_{10}\) | \(=\) | \( ( - 79528202 \nu^{15} + 597984111 \nu^{14} - 6709477595 \nu^{13} + 34620925101 \nu^{12} - 176820231599 \nu^{11} + 612816576096 \nu^{10} + \cdots + 76625566650 ) / 382882302 \) |
\(\beta_{11}\) | \(=\) | \( ( - 8761702 \nu^{15} + 75149661 \nu^{14} - 803217547 \nu^{13} + 4555430433 \nu^{12} - 23050484461 \nu^{11} + 85806746574 \nu^{10} + \cdots + 17030226186 ) / 34807482 \) |
\(\beta_{12}\) | \(=\) | \( ( - 79528202 \nu^{15} + 593416323 \nu^{14} - 6677503079 \nu^{13} + 34246129614 \nu^{12} - 174987127385 \nu^{11} + 603156615690 \nu^{10} + \cdots + 64965778326 ) / 191441151 \) |
\(\beta_{13}\) | \(=\) | \( ( 374895190 \nu^{15} - 2813236521 \nu^{14} + 31639994869 \nu^{13} - 163071292497 \nu^{12} + 835190424757 \nu^{11} - 2894814185856 \nu^{10} + \cdots - 430899272910 ) / 382882302 \) |
\(\beta_{14}\) | \(=\) | \( ( 11787550 \nu^{15} - 88406625 \nu^{14} + 992387359 \nu^{13} - 5109684021 \nu^{12} + 26078689825 \nu^{11} - 90176104194 \nu^{10} + \cdots - 10066168566 ) / 11602494 \) |
\(\beta_{15}\) | \(=\) | \( ( 45521818 \nu^{15} - 331976739 \nu^{14} + 3766027165 \nu^{13} - 18969909795 \nu^{12} + 96978712123 \nu^{11} - 329159543010 \nu^{10} + \cdots - 26056975698 ) / 34807482 \) |
\(\nu\) | \(=\) | \( ( 4 \beta_{15} - 4 \beta_{14} - 3 \beta_{11} + 12 \beta_{10} + \beta_{8} - 6 \beta_{7} - 12 \beta_{6} + \beta_{5} + 12 \beta_{4} - 2 \beta_{3} + \beta_{2} + 25 ) / 48 \) |
\(\nu^{2}\) | \(=\) | \( ( - 4 \beta_{15} + 4 \beta_{14} - 4 \beta_{12} - 3 \beta_{11} - 28 \beta_{10} - 36 \beta_{9} - 7 \beta_{8} - 6 \beta_{7} - 24 \beta_{6} + 17 \beta_{5} - 34 \beta_{3} - 7 \beta_{2} - 18 \beta _1 - 319 ) / 48 \) |
\(\nu^{3}\) | \(=\) | \( ( - 116 \beta_{15} + 176 \beta_{14} + 18 \beta_{13} + 36 \beta_{12} + 78 \beta_{11} - 132 \beta_{10} - 54 \beta_{9} - 38 \beta_{8} + 111 \beta_{7} + 240 \beta_{6} - 182 \beta_{5} - 276 \beta_{4} + 4 \beta_{3} - 38 \beta_{2} - 27 \beta _1 - 518 ) / 48 \) |
\(\nu^{4}\) | \(=\) | \( ( 8 \beta_{15} + 112 \beta_{14} + 36 \beta_{13} - 28 \beta_{12} + 117 \beta_{11} + 1172 \beta_{10} + 912 \beta_{9} + 209 \beta_{8} + 228 \beta_{7} + 720 \beta_{6} - 895 \beta_{5} - 336 \beta_{4} + 902 \beta_{3} + 41 \beta_{2} + \cdots + 7121 ) / 48 \) |
\(\nu^{5}\) | \(=\) | \( ( 3178 \beta_{15} - 5134 \beta_{14} - 834 \beta_{13} - 1512 \beta_{12} - 2232 \beta_{11} + 4200 \beta_{10} + 2370 \beta_{9} + 1156 \beta_{8} - 3645 \beta_{7} - 6072 \beta_{6} + 7156 \beta_{5} + 7092 \beta_{4} + \cdots + 19240 ) / 48 \) |
\(\nu^{6}\) | \(=\) | \( ( 2606 \beta_{15} - 8774 \beta_{14} - 2592 \beta_{13} + 3296 \beta_{12} - 4335 \beta_{11} - 37552 \beta_{10} - 22512 \beta_{9} - 6619 \beta_{8} - 11508 \beta_{7} - 24384 \beta_{6} + 40181 \beta_{5} + \cdots - 174823 ) / 48 \) |
\(\nu^{7}\) | \(=\) | \( ( - 87110 \beta_{15} + 144158 \beta_{14} + 28734 \beta_{13} + 60024 \beta_{12} + 66108 \beta_{11} - 155904 \beta_{10} - 87150 \beta_{9} - 38852 \beta_{8} + 112785 \beta_{7} + 153288 \beta_{6} + \cdots - 691424 ) / 48 \) |
\(\nu^{8}\) | \(=\) | \( ( - 153724 \beta_{15} + 410980 \beta_{14} + 127116 \beta_{13} - 114628 \beta_{12} + 170745 \beta_{11} + 1111724 \beta_{10} + 546624 \beta_{9} + 197021 \beta_{8} + 505380 \beta_{7} + \cdots + 4249301 ) / 48 \) |
\(\nu^{9}\) | \(=\) | \( ( 2366866 \beta_{15} - 3913510 \beta_{14} - 854118 \beta_{13} - 2238876 \beta_{12} - 1929630 \beta_{11} + 5792844 \beta_{10} + 2992770 \beta_{9} + 1355218 \beta_{8} - 3174585 \beta_{7} + \cdots + 23582830 ) / 48 \) |
\(\nu^{10}\) | \(=\) | \( ( 6656930 \beta_{15} - 16363346 \beta_{14} - 5242284 \beta_{13} + 2533868 \beta_{12} - 6655041 \beta_{11} - 30741676 \beta_{10} - 12806088 \beta_{9} - 5402113 \beta_{8} + \cdots - 99691693 ) / 48 \) |
\(\nu^{11}\) | \(=\) | \( ( - 63144098 \beta_{15} + 101700302 \beta_{14} + 22525446 \beta_{13} + 78307596 \beta_{12} + 54514944 \beta_{11} - 208618320 \beta_{10} - 98852226 \beta_{9} + \cdots - 776690336 ) / 48 \) |
\(\nu^{12}\) | \(=\) | \( ( - 255584704 \beta_{15} + 598074904 \beta_{14} + 194972832 \beta_{13} - 18430144 \beta_{12} + 249887163 \beta_{11} + 783849344 \beta_{10} + 279158208 \beta_{9} + \cdots + 2177062355 ) / 48 \) |
\(\nu^{13}\) | \(=\) | \( ( 1632969982 \beta_{15} - 2482933414 \beta_{14} - 516343602 \beta_{13} - 2585548212 \beta_{12} - 1470833718 \beta_{11} + 7231885752 \beta_{10} + 3174598518 \beta_{9} + \cdots + 24914694430 ) / 48 \) |
\(\nu^{14}\) | \(=\) | \( ( 9174874982 \beta_{15} - 20661944774 \beta_{14} - 6755761692 \beta_{13} - 1791403060 \beta_{12} - 8997167163 \beta_{11} - 17770036420 \beta_{10} + \cdots - 40707438727 ) / 48 \) |
\(\nu^{15}\) | \(=\) | \( ( - 40159943810 \beta_{15} + 54788157086 \beta_{14} + 9230988438 \beta_{13} + 81124707744 \beta_{12} + 37227162516 \beta_{11} - 241420340724 \beta_{10} + \cdots - 780594138464 ) / 48 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(97\) | \(161\) | \(421\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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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161.1 |
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0 | −2.87012 | − | 0.873151i | 0 | − | 2.23607i | 0 | −11.2043 | 0 | 7.47522 | + | 5.01210i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.2 | 0 | −2.87012 | + | 0.873151i | 0 | 2.23607i | 0 | −11.2043 | 0 | 7.47522 | − | 5.01210i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.3 | 0 | −2.37394 | − | 1.83423i | 0 | 2.23607i | 0 | 3.76207 | 0 | 2.27118 | + | 8.70872i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.4 | 0 | −2.37394 | + | 1.83423i | 0 | − | 2.23607i | 0 | 3.76207 | 0 | 2.27118 | − | 8.70872i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.5 | 0 | −0.956507 | − | 2.84343i | 0 | − | 2.23607i | 0 | −3.99659 | 0 | −7.17019 | + | 5.43952i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.6 | 0 | −0.956507 | + | 2.84343i | 0 | 2.23607i | 0 | −3.99659 | 0 | −7.17019 | − | 5.43952i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.7 | 0 | −0.460323 | − | 2.96447i | 0 | − | 2.23607i | 0 | 10.9698 | 0 | −8.57621 | + | 2.72923i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.8 | 0 | −0.460323 | + | 2.96447i | 0 | 2.23607i | 0 | 10.9698 | 0 | −8.57621 | − | 2.72923i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.9 | 0 | 0.460323 | − | 2.96447i | 0 | 2.23607i | 0 | −10.9698 | 0 | −8.57621 | − | 2.72923i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.10 | 0 | 0.460323 | + | 2.96447i | 0 | − | 2.23607i | 0 | −10.9698 | 0 | −8.57621 | + | 2.72923i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.11 | 0 | 0.956507 | − | 2.84343i | 0 | 2.23607i | 0 | 3.99659 | 0 | −7.17019 | − | 5.43952i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.12 | 0 | 0.956507 | + | 2.84343i | 0 | − | 2.23607i | 0 | 3.99659 | 0 | −7.17019 | + | 5.43952i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.13 | 0 | 2.37394 | − | 1.83423i | 0 | − | 2.23607i | 0 | −3.76207 | 0 | 2.27118 | − | 8.70872i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.14 | 0 | 2.37394 | + | 1.83423i | 0 | 2.23607i | 0 | −3.76207 | 0 | 2.27118 | + | 8.70872i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.15 | 0 | 2.87012 | − | 0.873151i | 0 | 2.23607i | 0 | 11.2043 | 0 | 7.47522 | − | 5.01210i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
161.16 | 0 | 2.87012 | + | 0.873151i | 0 | − | 2.23607i | 0 | 11.2043 | 0 | 7.47522 | + | 5.01210i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 480.3.l.a | ✓ | 16 |
3.b | odd | 2 | 1 | inner | 480.3.l.a | ✓ | 16 |
4.b | odd | 2 | 1 | inner | 480.3.l.a | ✓ | 16 |
8.b | even | 2 | 1 | 960.3.l.i | 16 | ||
8.d | odd | 2 | 1 | 960.3.l.i | 16 | ||
12.b | even | 2 | 1 | inner | 480.3.l.a | ✓ | 16 |
24.f | even | 2 | 1 | 960.3.l.i | 16 | ||
24.h | odd | 2 | 1 | 960.3.l.i | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
480.3.l.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
480.3.l.a | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
480.3.l.a | ✓ | 16 | 4.b | odd | 2 | 1 | inner |
480.3.l.a | ✓ | 16 | 12.b | even | 2 | 1 | inner |
960.3.l.i | 16 | 8.b | even | 2 | 1 | ||
960.3.l.i | 16 | 8.d | odd | 2 | 1 | ||
960.3.l.i | 16 | 24.f | even | 2 | 1 | ||
960.3.l.i | 16 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - 276T_{7}^{6} + 22740T_{7}^{4} - 510688T_{7}^{2} + 3415104 \)
acting on \(S_{3}^{\mathrm{new}}(480, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} + 12 T^{14} + 24 T^{12} + \cdots + 43046721 \)
$5$
\( (T^{2} + 5)^{8} \)
$7$
\( (T^{8} - 276 T^{6} + 22740 T^{4} + \cdots + 3415104)^{2} \)
$11$
\( (T^{8} + 408 T^{6} + 39888 T^{4} + \cdots + 3326976)^{2} \)
$13$
\( (T^{4} + 4 T^{3} - 468 T^{2} - 2784 T - 864)^{4} \)
$17$
\( (T^{8} + 960 T^{6} + 227424 T^{4} + \cdots + 2214144)^{2} \)
$19$
\( (T^{8} - 1192 T^{6} + 400080 T^{4} + \cdots + 827597824)^{2} \)
$23$
\( (T^{8} + 1508 T^{6} + 628404 T^{4} + \cdots + 554319936)^{2} \)
$29$
\( (T^{8} + 2888 T^{6} + \cdots + 22562443264)^{2} \)
$31$
\( (T^{8} - 4984 T^{6} + \cdots + 312025022464)^{2} \)
$37$
\( (T^{4} - 4 T^{3} - 4788 T^{2} + \cdots + 2941856)^{4} \)
$41$
\( (T^{8} + 6920 T^{6} + \cdots + 286473293824)^{2} \)
$43$
\( (T^{8} - 7324 T^{6} + \cdots + 608824393984)^{2} \)
$47$
\( (T^{8} + 9668 T^{6} + \cdots + 6335208456256)^{2} \)
$53$
\( (T^{8} + 18896 T^{6} + \cdots + 110324940730624)^{2} \)
$59$
\( (T^{8} + 18344 T^{6} + \cdots + 2358116499456)^{2} \)
$61$
\( (T^{4} + 92 T^{3} + 1380 T^{2} + \cdots + 7424)^{4} \)
$67$
\( (T^{8} - 18204 T^{6} + \cdots + 9194285355264)^{2} \)
$71$
\( (T^{8} + 10352 T^{6} + \cdots + 71430045696)^{2} \)
$73$
\( (T^{4} - 64 T^{3} - 11208 T^{2} + \cdots + 27175696)^{4} \)
$79$
\( (T^{8} - 42600 T^{6} + \cdots + 17\!\cdots\!84)^{2} \)
$83$
\( (T^{8} + 21332 T^{6} + \cdots + 572716500798016)^{2} \)
$89$
\( (T^{8} + 33440 T^{6} + \cdots + 370293356888064)^{2} \)
$97$
\( (T^{4} + 136 T^{3} - 21768 T^{2} + \cdots - 178613104)^{4} \)
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