Newspace parameters
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(18.5856016518\) |
Analytic rank: | \(0\) |
Dimension: | \(20\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{20} + 144 x^{18} + 8518 x^{16} + 269932 x^{14} + 5002289 x^{12} + 55478700 x^{10} + 361614704 x^{8} + 1303665216 x^{6} + 2335301632 x^{4} + \cdots + 603979776 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{26}\cdot 5^{2}\cdot 7^{8} \) |
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 \(1,\beta_1,\ldots,\beta_{19}\) 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 \( x^{20} + 144 x^{18} + 8518 x^{16} + 269932 x^{14} + 5002289 x^{12} + 55478700 x^{10} + 361614704 x^{8} + 1303665216 x^{6} + 2335301632 x^{4} + \cdots + 603979776 \) :
\(\beta_{1}\) | \(=\) | \( ( - 465021503 \nu^{18} - 77684214024 \nu^{16} - 5383318221626 \nu^{14} - 200420748745508 \nu^{12} + \cdots - 56\!\cdots\!96 ) / 11\!\cdots\!28 \) |
\(\beta_{2}\) | \(=\) | \( ( 3255150521 \nu^{18} + 543789498168 \nu^{16} + 37683227551382 \nu^{14} + \cdots + 50\!\cdots\!16 ) / 22\!\cdots\!56 \) |
\(\beta_{3}\) | \(=\) | \( ( - 717392106679 \nu^{18} - 103623757921560 \nu^{16} + \cdots - 42\!\cdots\!24 ) / 15\!\cdots\!92 \) |
\(\beta_{4}\) | \(=\) | \( ( 221279727269 \nu^{19} + 31774996598160 \nu^{17} + \cdots + 10\!\cdots\!96 \nu ) / 43\!\cdots\!52 \) |
\(\beta_{5}\) | \(=\) | \( ( 4351528823527 \nu^{19} + 637080822795696 \nu^{17} + \cdots + 44\!\cdots\!72 \nu ) / 30\!\cdots\!64 \) |
\(\beta_{6}\) | \(=\) | \( ( 4199576014411 \nu^{19} - 5465745867120 \nu^{18} + 590865472700928 \nu^{17} - 750679005920256 \nu^{16} + \cdots - 12\!\cdots\!16 ) / 12\!\cdots\!60 \) |
\(\beta_{7}\) | \(=\) | \( ( 4199576014411 \nu^{19} + 74855304483120 \nu^{18} + 590865472700928 \nu^{17} + \cdots + 23\!\cdots\!44 ) / 12\!\cdots\!60 \) |
\(\beta_{8}\) | \(=\) | \( ( - 4199576014411 \nu^{19} - 112932858139920 \nu^{18} - 590865472700928 \nu^{17} + \cdots - 43\!\cdots\!84 ) / 12\!\cdots\!60 \) |
\(\beta_{9}\) | \(=\) | \( ( - 1548958090883 \nu^{19} - 222424976187120 \nu^{17} + \cdots - 68\!\cdots\!08 \nu ) / 43\!\cdots\!52 \) |
\(\beta_{10}\) | \(=\) | \( ( - 35670797971781 \nu^{19} - 12416725219872 \nu^{18} + \cdots - 99\!\cdots\!96 ) / 75\!\cdots\!60 \) |
\(\beta_{11}\) | \(=\) | \( ( - 1459673962307 \nu^{19} - 207509607094512 \nu^{17} + \cdots - 57\!\cdots\!60 \nu ) / 14\!\cdots\!84 \) |
\(\beta_{12}\) | \(=\) | \( ( - 204494539235861 \nu^{19} + 690720097783488 \nu^{18} + \cdots + 34\!\cdots\!64 ) / 15\!\cdots\!20 \) |
\(\beta_{13}\) | \(=\) | \( ( 37912440093937 \nu^{19} + 201405300010944 \nu^{18} + \cdots + 11\!\cdots\!12 ) / 21\!\cdots\!60 \) |
\(\beta_{14}\) | \(=\) | \( ( - 24714338304557 \nu^{19} - 4684925028960 \nu^{18} + \cdots - 10\!\cdots\!28 ) / 10\!\cdots\!80 \) |
\(\beta_{15}\) | \(=\) | \( ( 114858141342889 \nu^{19} + 18625087829808 \nu^{18} + \cdots + 14\!\cdots\!44 ) / 37\!\cdots\!80 \) |
\(\beta_{16}\) | \(=\) | \( ( 162695106701881 \nu^{19} - 192189596923520 \nu^{18} + \cdots - 71\!\cdots\!16 ) / 50\!\cdots\!40 \) |
\(\beta_{17}\) | \(=\) | \( ( - 505350503918939 \nu^{19} - 690720097783488 \nu^{18} + \cdots - 34\!\cdots\!64 ) / 15\!\cdots\!20 \) |
\(\beta_{18}\) | \(=\) | \( ( 498958560849 \nu^{19} + 71241448072144 \nu^{17} + \cdots + 19\!\cdots\!60 \nu ) / 12\!\cdots\!32 \) |
\(\beta_{19}\) | \(=\) | \( ( - 150495286990293 \nu^{19} + 4138908406624 \nu^{18} + \cdots + 33\!\cdots\!32 ) / 25\!\cdots\!20 \) |
\(\nu\) | \(=\) | \( ( \beta_{9} + 7\beta_{4} ) / 7 \) |
\(\nu^{2}\) | \(=\) | \( ( 2\beta_{2} + 7\beta _1 - 98 ) / 7 \) |
\(\nu^{3}\) | \(=\) | \( ( -3\beta_{18} - 7\beta_{11} - 40\beta_{9} - 175\beta_{4} ) / 7 \) |
\(\nu^{4}\) | \(=\) | \( ( 4 \beta_{19} - 4 \beta_{17} + 8 \beta_{15} - 8 \beta_{13} + 4 \beta_{12} - 12 \beta_{10} + 7 \beta_{8} + 7 \beta_{7} - 12 \beta_{3} - 84 \beta_{2} - 245 \beta _1 + 2492 ) / 7 \) |
\(\nu^{5}\) | \(=\) | \( ( 150 \beta_{18} - 14 \beta_{17} - 5 \beta_{16} + 14 \beta_{15} + 20 \beta_{14} - 14 \beta_{12} + 364 \beta_{11} + 42 \beta_{10} + 1516 \beta_{9} - 5 \beta_{7} - 45 \beta_{6} + 25 \beta_{5} + 5376 \beta_{4} + 5 \beta _1 + 15 ) / 7 \) |
\(\nu^{6}\) | \(=\) | \( ( - 332 \beta_{19} + 224 \beta_{17} - 592 \beta_{15} + 520 \beta_{13} - 224 \beta_{12} + 1356 \beta_{10} - 399 \beta_{8} - 455 \beta_{7} + 56 \beta_{6} - 216 \beta_{4} + 696 \beta_{3} + 3160 \beta_{2} + 8925 \beta _1 - 77644 ) / 7 \) |
\(\nu^{7}\) | \(=\) | \( ( - 6482 \beta_{18} + 994 \beta_{17} + 441 \beta_{16} - 1106 \beta_{15} - 1372 \beta_{14} + 994 \beta_{12} - 16408 \beta_{11} - 3318 \beta_{10} - 57436 \beta_{9} + 441 \beta_{7} + 3577 \beta_{6} - 1421 \beta_{5} + \cdots - 1323 ) / 7 \) |
\(\nu^{8}\) | \(=\) | \( ( 18128 \beta_{19} - 112 \beta_{18} - 9664 \beta_{17} + 112 \beta_{16} + 30912 \beta_{15} - 112 \beta_{14} - 25568 \beta_{13} + 9664 \beta_{12} - 82896 \beta_{10} + 112 \beta_{9} + 18711 \beta_{8} + 23191 \beta_{7} + \cdots + 2698500 ) / 7 \) |
\(\nu^{9}\) | \(=\) | \( ( 224 \beta_{19} + 271386 \beta_{18} - 52542 \beta_{17} - 25185 \beta_{16} + 64078 \beta_{15} + 71844 \beta_{14} - 52542 \beta_{12} + 713720 \beta_{11} + 192458 \beta_{10} + 2217676 \beta_{9} + \cdots + 75555 ) / 7 \) |
\(\nu^{10}\) | \(=\) | \( ( - 853364 \beta_{19} + 12544 \beta_{18} + 387008 \beta_{17} - 12544 \beta_{16} - 1426064 \beta_{15} + 12544 \beta_{14} + 1145400 \beta_{13} - 387008 \beta_{12} + 4144852 \beta_{10} + \cdots - 100170588 ) / 7 \) |
\(\nu^{11}\) | \(=\) | \( ( - 27328 \beta_{19} - 11276210 \beta_{18} + 2481850 \beta_{17} + 1227897 \beta_{16} - 3263722 \beta_{15} - 3399308 \beta_{14} + 2481850 \beta_{12} - 30561832 \beta_{11} + \cdots - 3683691 ) / 7 \) |
\(\nu^{12}\) | \(=\) | \( ( 37677528 \beta_{19} - 882896 \beta_{18} - 15189504 \beta_{17} + 882896 \beta_{16} + 62348128 \beta_{15} - 882896 \beta_{14} - 49341200 \beta_{13} + 15189504 \beta_{12} + \cdots + 3874329620 ) / 7 \) |
\(\nu^{13}\) | \(=\) | \( ( 1995168 \beta_{19} + 467849642 \beta_{18} - 110984118 \beta_{17} - 55700593 \beta_{16} + 154800646 \beta_{15} + 152895444 \beta_{14} - 110984118 \beta_{12} + 1296152984 \beta_{11} + \cdots + 167101779 ) / 7 \) |
\(\nu^{14}\) | \(=\) | \( ( - 1613956284 \beta_{19} + 50425312 \beta_{18} + 596231616 \beta_{17} - 50425312 \beta_{16} - 2657566128 \beta_{15} + 50425312 \beta_{14} + 2087219688 \beta_{13} + \cdots - 153784357804 ) / 7 \) |
\(\nu^{15}\) | \(=\) | \( ( - 116063360 \beta_{19} - 19413185602 \beta_{18} + 4818281650 \beta_{17} + 2435298313 \beta_{16} - 7025613602 \beta_{15} - 6682189948 \beta_{14} + 4818281650 \beta_{12} + \cdots - 7305894939 ) / 7 \) |
\(\nu^{16}\) | \(=\) | \( ( 68110340384 \beta_{19} - 2566954544 \beta_{18} - 23583958016 \beta_{17} + 2566954544 \beta_{16} + 111867584640 \beta_{15} - 2566954544 \beta_{14} + \cdots + 6205216413732 ) / 7 \) |
\(\nu^{17}\) | \(=\) | \( ( 5969217632 \beta_{19} + 805865205370 \beta_{18} - 205666851502 \beta_{17} - 104342791233 \beta_{16} + 309846751550 \beta_{15} + 286980690564 \beta_{14} + \cdots + 313028373699 ) / 7 \) |
\(\nu^{18}\) | \(=\) | \( ( - 2852683485700 \beta_{19} + 121905022400 \beta_{18} + 941952442816 \beta_{17} - 121905022400 \beta_{16} - 4679265488080 \beta_{15} + \cdots - 253003481109372 ) / 7 \) |
\(\nu^{19}\) | \(=\) | \( ( - 285229507136 \beta_{19} - 33464516148818 \beta_{18} + 8690900154538 \beta_{17} + 4418848167705 \beta_{16} - 13406589714266 \beta_{15} + \cdots - 13256544503115 ) / 7 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(136\) | \(281\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
64.1 |
|
− | 5.44617i | 0 | −21.6607 | −10.6013 | + | 3.55149i | 0 | 7.00000i | 74.3987i | 0 | 19.3420 | + | 57.7363i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.2 | − | 5.44617i | 0 | −21.6607 | 10.6013 | + | 3.55149i | 0 | − | 7.00000i | 74.3987i | 0 | 19.3420 | − | 57.7363i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.3 | − | 4.31226i | 0 | −10.5956 | −7.55443 | − | 8.24200i | 0 | − | 7.00000i | 11.1930i | 0 | −35.5417 | + | 32.5767i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.4 | − | 4.31226i | 0 | −10.5956 | 7.55443 | − | 8.24200i | 0 | 7.00000i | 11.1930i | 0 | −35.5417 | − | 32.5767i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.5 | − | 3.73445i | 0 | −5.94609 | −5.59787 | + | 9.67801i | 0 | − | 7.00000i | − | 7.67022i | 0 | 36.1420 | + | 20.9049i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.6 | − | 3.73445i | 0 | −5.94609 | 5.59787 | + | 9.67801i | 0 | 7.00000i | − | 7.67022i | 0 | 36.1420 | − | 20.9049i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.7 | − | 2.18874i | 0 | 3.20940 | −10.6651 | + | 3.35486i | 0 | 7.00000i | − | 24.5345i | 0 | 7.34293 | + | 23.3432i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.8 | − | 2.18874i | 0 | 3.20940 | 10.6651 | + | 3.35486i | 0 | − | 7.00000i | − | 24.5345i | 0 | 7.34293 | − | 23.3432i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.9 | − | 0.0833494i | 0 | 7.99305 | −7.17375 | + | 8.57539i | 0 | − | 7.00000i | − | 1.33301i | 0 | 0.714753 | + | 0.597928i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.10 | − | 0.0833494i | 0 | 7.99305 | 7.17375 | + | 8.57539i | 0 | 7.00000i | − | 1.33301i | 0 | 0.714753 | − | 0.597928i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.11 | 0.0833494i | 0 | 7.99305 | −7.17375 | − | 8.57539i | 0 | 7.00000i | 1.33301i | 0 | 0.714753 | − | 0.597928i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.12 | 0.0833494i | 0 | 7.99305 | 7.17375 | − | 8.57539i | 0 | − | 7.00000i | 1.33301i | 0 | 0.714753 | + | 0.597928i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.13 | 2.18874i | 0 | 3.20940 | −10.6651 | − | 3.35486i | 0 | − | 7.00000i | 24.5345i | 0 | 7.34293 | − | 23.3432i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.14 | 2.18874i | 0 | 3.20940 | 10.6651 | − | 3.35486i | 0 | 7.00000i | 24.5345i | 0 | 7.34293 | + | 23.3432i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.15 | 3.73445i | 0 | −5.94609 | −5.59787 | − | 9.67801i | 0 | 7.00000i | 7.67022i | 0 | 36.1420 | − | 20.9049i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.16 | 3.73445i | 0 | −5.94609 | 5.59787 | − | 9.67801i | 0 | − | 7.00000i | 7.67022i | 0 | 36.1420 | + | 20.9049i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.17 | 4.31226i | 0 | −10.5956 | −7.55443 | + | 8.24200i | 0 | 7.00000i | − | 11.1930i | 0 | −35.5417 | − | 32.5767i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.18 | 4.31226i | 0 | −10.5956 | 7.55443 | + | 8.24200i | 0 | − | 7.00000i | − | 11.1930i | 0 | −35.5417 | + | 32.5767i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.19 | 5.44617i | 0 | −21.6607 | −10.6013 | − | 3.55149i | 0 | − | 7.00000i | − | 74.3987i | 0 | 19.3420 | − | 57.7363i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
64.20 | 5.44617i | 0 | −21.6607 | 10.6013 | − | 3.55149i | 0 | 7.00000i | − | 74.3987i | 0 | 19.3420 | + | 57.7363i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.4.d.d | ✓ | 20 |
3.b | odd | 2 | 1 | inner | 315.4.d.d | ✓ | 20 |
5.b | even | 2 | 1 | inner | 315.4.d.d | ✓ | 20 |
5.c | odd | 4 | 1 | 1575.4.a.bt | 10 | ||
5.c | odd | 4 | 1 | 1575.4.a.bu | 10 | ||
15.d | odd | 2 | 1 | inner | 315.4.d.d | ✓ | 20 |
15.e | even | 4 | 1 | 1575.4.a.bt | 10 | ||
15.e | even | 4 | 1 | 1575.4.a.bu | 10 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.4.d.d | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
315.4.d.d | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
315.4.d.d | ✓ | 20 | 5.b | even | 2 | 1 | inner |
315.4.d.d | ✓ | 20 | 15.d | odd | 2 | 1 | inner |
1575.4.a.bt | 10 | 5.c | odd | 4 | 1 | ||
1575.4.a.bt | 10 | 15.e | even | 4 | 1 | ||
1575.4.a.bu | 10 | 5.c | odd | 4 | 1 | ||
1575.4.a.bu | 10 | 15.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 67T_{2}^{8} + 1523T_{2}^{6} + 13569T_{2}^{4} + 36944T_{2}^{2} + 256 \)
acting on \(S_{4}^{\mathrm{new}}(315, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{10} + 67 T^{8} + 1523 T^{6} + 13569 T^{4} + \cdots + 256)^{2} \)
$3$
\( T^{20} \)
$5$
\( T^{20} - 214 T^{18} + \cdots + 93\!\cdots\!25 \)
$7$
\( (T^{2} + 49)^{10} \)
$11$
\( (T^{10} - 8988 T^{8} + \cdots - 27\!\cdots\!00)^{2} \)
$13$
\( (T^{10} + 15472 T^{8} + \cdots + 10\!\cdots\!24)^{2} \)
$17$
\( (T^{10} + 12880 T^{8} + \cdots + 48\!\cdots\!16)^{2} \)
$19$
\( (T^{5} + 18 T^{4} - 20572 T^{3} + \cdots + 1947671040)^{4} \)
$23$
\( (T^{10} + 73124 T^{8} + \cdots + 13\!\cdots\!00)^{2} \)
$29$
\( (T^{10} - 93332 T^{8} + \cdots - 10\!\cdots\!00)^{2} \)
$31$
\( (T^{5} + 12 T^{4} - 102304 T^{3} + \cdots + 76972365952)^{4} \)
$37$
\( (T^{10} + 292996 T^{8} + \cdots + 17\!\cdots\!56)^{2} \)
$41$
\( (T^{10} - 222504 T^{8} + \cdots - 10\!\cdots\!16)^{2} \)
$43$
\( (T^{10} + 503028 T^{8} + \cdots + 34\!\cdots\!56)^{2} \)
$47$
\( (T^{10} + 320468 T^{8} + \cdots + 24\!\cdots\!24)^{2} \)
$53$
\( (T^{10} + 891664 T^{8} + \cdots + 21\!\cdots\!76)^{2} \)
$59$
\( (T^{10} - 1281632 T^{8} + \cdots - 21\!\cdots\!16)^{2} \)
$61$
\( (T^{5} - 512 T^{4} + \cdots - 1370493512000)^{4} \)
$67$
\( (T^{10} + 1725796 T^{8} + \cdots + 40\!\cdots\!64)^{2} \)
$71$
\( (T^{10} - 2602184 T^{8} + \cdots - 90\!\cdots\!16)^{2} \)
$73$
\( (T^{10} + 2167376 T^{8} + \cdots + 24\!\cdots\!96)^{2} \)
$79$
\( (T^{5} - 388 T^{4} + \cdots - 8769624080384)^{4} \)
$83$
\( (T^{10} + 774896 T^{8} + \cdots + 39\!\cdots\!00)^{2} \)
$89$
\( (T^{10} - 4067576 T^{8} + \cdots - 11\!\cdots\!84)^{2} \)
$97$
\( (T^{10} + 4967664 T^{8} + \cdots + 41\!\cdots\!16)^{2} \)
show more
show less