Newspace parameters
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.bf (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
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: | \( x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} - 14 x^{12} + 38 x^{11} - 40 x^{10} + 64 x^{9} + 291 x^{8} - 630 x^{7} + 584 x^{6} - 800 x^{5} + 734 x^{4} - 188 x^{3} + 32 x^{2} - 8 x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 105) |
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 \( x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} - 14 x^{12} + 38 x^{11} - 40 x^{10} + 64 x^{9} + 291 x^{8} - 630 x^{7} + 584 x^{6} - 800 x^{5} + 734 x^{4} - 188 x^{3} + 32 x^{2} - 8 x + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( - 38414168495 \nu^{15} + 46999484762 \nu^{14} - 56685616158 \nu^{13} + 131873366880 \nu^{12} + 642956778028 \nu^{11} + \cdots - 33746305280705 ) / 11431248166457 \) |
\(\beta_{2}\) | \(=\) | \( ( 145306983361 \nu^{15} - 177522986678 \nu^{14} + 213949914242 \nu^{13} - 520543533819 \nu^{12} - 2426061321940 \nu^{11} + \cdots + 31632456480199 ) / 11431248166457 \) |
\(\beta_{3}\) | \(=\) | \( ( 760172030748 \nu^{15} - 826140524550 \nu^{14} + 311805690170 \nu^{13} - 1979665829518 \nu^{12} - 13114010371696 \nu^{11} + \cdots - 575089767434 ) / 11431248166457 \) |
\(\beta_{4}\) | \(=\) | \( ( - 1159675947687 \nu^{15} + 2123382923210 \nu^{14} - 1790860341485 \nu^{13} + 4132499467647 \nu^{12} + \cdots + 29672440598565 ) / 11431248166457 \) |
\(\beta_{5}\) | \(=\) | \( ( - 2241001108620 \nu^{15} + 2440547176294 \nu^{14} - 921260080696 \nu^{13} + 5836557871079 \nu^{12} + 38651783264660 \nu^{11} + \cdots + 1695440450074 ) / 11431248166457 \) |
\(\beta_{6}\) | \(=\) | \( ( 4058097157982 \nu^{15} - 7568755097298 \nu^{14} + 7059730047127 \nu^{13} - 15184509679358 \nu^{12} - 58946431470254 \nu^{11} + \cdots - 16137847630635 ) / 11431248166457 \) |
\(\beta_{7}\) | \(=\) | \( ( - 2575730834 \nu^{15} + 4699915310 \nu^{14} - 4379029892 \nu^{13} + 9614611682 \nu^{12} + 37669504970 \nu^{11} - 91091321784 \nu^{10} + \cdots + 17015828908 ) / 7008735847 \) |
\(\beta_{8}\) | \(=\) | \( ( 1063280569755 \nu^{15} - 2138087245431 \nu^{14} + 2140603616666 \nu^{13} - 4270022226214 \nu^{12} - 14845214238260 \nu^{11} + \cdots - 8602022998880 ) / 1633035452351 \) |
\(\beta_{9}\) | \(=\) | \( ( 7502690467678 \nu^{15} - 13694199839130 \nu^{14} + 12758334474904 \nu^{13} - 28102531513444 \nu^{12} + \cdots - 29172119880252 ) / 11431248166457 \) |
\(\beta_{10}\) | \(=\) | \( ( - 8183751109447 \nu^{15} + 14956739476040 \nu^{14} - 13922497684986 \nu^{13} + 30578923581156 \nu^{12} + \cdots + 54263991142284 ) / 11431248166457 \) |
\(\beta_{11}\) | \(=\) | \( ( - 9494932205673 \nu^{15} + 17203785936492 \nu^{14} - 15830728042254 \nu^{13} + 35235059970569 \nu^{12} + \cdots + 61766681609962 ) / 11431248166457 \) |
\(\beta_{12}\) | \(=\) | \( ( 10583106332007 \nu^{15} - 18562866254784 \nu^{14} + 16212768119001 \nu^{13} - 37702760224023 \nu^{12} + \cdots - 62361843284805 ) / 11431248166457 \) |
\(\beta_{13}\) | \(=\) | \( ( 10663789073454 \nu^{15} - 18661163594876 \nu^{14} + 16331067749359 \nu^{13} - 37987756392193 \nu^{12} + \cdots - 32056382963606 ) / 11431248166457 \) |
\(\beta_{14}\) | \(=\) | \( ( 11015276267169 \nu^{15} - 18856066985592 \nu^{14} + 16454339422594 \nu^{13} - 39194391629605 \nu^{12} + \cdots - 62916861144830 ) / 11431248166457 \) |
\(\beta_{15}\) | \(=\) | \( ( 13399447651346 \nu^{15} - 23430444102746 \nu^{14} + 20519536606234 \nu^{13} - 47737126077397 \nu^{12} + \cdots - 46488804269431 ) / 11431248166457 \) |
\(\nu\) | \(=\) | \( ( -\beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( -\beta_{14} - \beta_{11} + 2\beta_{3} \) |
\(\nu^{3}\) | \(=\) | \( ( 5\beta_{15} - 8\beta_{13} + 3\beta_{12} - 2\beta_{6} + 3\beta_{4} - 2\beta_{3} - 3\beta_{2} - 3\beta _1 + 3 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( 5\beta_{10} + \beta_{9} - 8\beta_{7} + 5\beta _1 + 13 \) |
\(\nu^{5}\) | \(=\) | \( ( - 23 \beta_{14} + 11 \beta_{11} - 11 \beta_{10} + 23 \beta_{9} - 13 \beta_{7} - 23 \beta_{5} - 10 \beta_{3} - 23 \beta_{2} + 23 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( 7\beta_{15} - \beta_{13} - 23\beta_{12} + 28\beta_{6} - 22\beta_{4} + 28\beta_{3} - \beta_{2} + 1 \) |
\(\nu^{7}\) | \(=\) | \( ( 103 \beta_{15} - 43 \beta_{14} - 148 \beta_{13} + 43 \beta_{12} - 43 \beta_{11} - 43 \beta_{10} + 103 \beta_{9} - 148 \beta_{8} - 61 \beta_{7} - 42 \beta_{6} + 103 \beta_{5} + 43 \beta_{4} - 148 \beta_{2} - 43 \beta _1 + 166 ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( -10\beta_{14} + 94\beta_{10} + 40\beta_{9} - 10\beta_{8} - 157\beta_{7} - 40\beta_{2} + 40 \) |
\(\nu^{9}\) | \(=\) | \( ( 459\beta_{15} - 479\beta_{12} - 168\beta_{6} + 171\beta_{4} - 168\beta_{3} - 459\beta_{2} + 171\beta _1 + 339 ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( 214 \beta_{15} + 469 \beta_{14} - 70 \beta_{13} - 469 \beta_{12} + 399 \beta_{11} - 70 \beta_{8} + 496 \beta_{6} + 214 \beta_{5} - 399 \beta_{4} - 70 \beta_{2} + 70 \) |
\(\nu^{11}\) | \(=\) | \( ( - 681 \beta_{14} - 681 \beta_{11} - 681 \beta_{10} + 2047 \beta_{9} - 2868 \beta_{8} - 1389 \beta_{7} + 2047 \beta_{5} + 658 \beta_{3} - 2047 \beta_{2} + 2047 ) / 2 \) |
\(\nu^{12}\) | \(=\) | \( 424\beta_{13} - 424\beta_{12} - 683\beta_{2} - 1693\beta _1 - 4232 \) |
\(\nu^{13}\) | \(=\) | \( ( 9149 \beta_{15} + 9997 \beta_{14} - 9997 \beta_{12} - 2701 \beta_{11} + 2701 \beta_{10} - 9149 \beta_{9} + 6611 \beta_{7} - 2538 \beta_{6} + 9149 \beta_{5} + 2701 \beta_{4} + 2701 \beta _1 - 3910 ) / 2 \) |
\(\nu^{14}\) | \(=\) | \( 9573\beta_{14} + 7194\beta_{11} - 2379\beta_{8} + 5603\beta_{5} - 9050\beta_{3} \) |
\(\nu^{15}\) | \(=\) | \( ( - 40993 \beta_{15} + 56392 \beta_{13} - 10641 \beta_{12} + 9630 \beta_{6} - 10641 \beta_{4} + 9630 \beta_{3} + 15399 \beta_{2} + 10641 \beta _1 - 36121 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(136\) | \(281\) |
\(\chi(n)\) | \(-1\) | \(-1 + \beta_{7}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
109.1 |
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−2.17942 | + | 1.25829i | 0 | 2.16659 | − | 3.75264i | 1.11685 | − | 1.93717i | 0 | −1.31340 | + | 2.29673i | 5.87162i | 0 | 0.00343282 | + | 5.62724i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
109.2 | −1.54296 | + | 0.890827i | 0 | 0.587145 | − | 1.01696i | 1.33920 | + | 1.79069i | 0 | −2.40898 | + | 1.09398i | − | 1.47113i | 0 | −3.66151 | − | 1.56996i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
109.3 | −1.34443 | + | 0.776205i | 0 | 0.204988 | − | 0.355049i | −1.88899 | + | 1.19655i | 0 | −0.478401 | − | 2.60214i | − | 2.46837i | 0 | 1.61083 | − | 3.07491i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
109.4 | −0.248840 | + | 0.143668i | 0 | −0.958719 | + | 1.66055i | −0.717839 | − | 2.11771i | 0 | −1.11487 | − | 2.39939i | − | 1.12562i | 0 | 0.482874 | + | 0.423842i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
109.5 | 0.248840 | − | 0.143668i | 0 | −0.958719 | + | 1.66055i | −1.47507 | − | 1.68052i | 0 | 1.11487 | + | 2.39939i | 1.12562i | 0 | −0.608495 | − | 0.206261i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
109.6 | 1.34443 | − | 0.776205i | 0 | 0.204988 | − | 0.355049i | 1.98074 | − | 1.03763i | 0 | 0.478401 | + | 2.60214i | 2.46837i | 0 | 1.85754 | − | 2.93248i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
109.7 | 1.54296 | − | 0.890827i | 0 | 0.587145 | − | 1.01696i | 0.881181 | + | 2.05512i | 0 | 2.40898 | − | 1.09398i | 1.47113i | 0 | 3.19038 | + | 2.38598i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
109.8 | 2.17942 | − | 1.25829i | 0 | 2.16659 | − | 3.75264i | −2.23607 | − | 0.00136408i | 0 | 1.31340 | − | 2.29673i | − | 5.87162i | 0 | −4.87505 | + | 2.81065i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
289.1 | −2.17942 | − | 1.25829i | 0 | 2.16659 | + | 3.75264i | 1.11685 | + | 1.93717i | 0 | −1.31340 | − | 2.29673i | − | 5.87162i | 0 | 0.00343282 | − | 5.62724i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
289.2 | −1.54296 | − | 0.890827i | 0 | 0.587145 | + | 1.01696i | 1.33920 | − | 1.79069i | 0 | −2.40898 | − | 1.09398i | 1.47113i | 0 | −3.66151 | + | 1.56996i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
289.3 | −1.34443 | − | 0.776205i | 0 | 0.204988 | + | 0.355049i | −1.88899 | − | 1.19655i | 0 | −0.478401 | + | 2.60214i | 2.46837i | 0 | 1.61083 | + | 3.07491i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
289.4 | −0.248840 | − | 0.143668i | 0 | −0.958719 | − | 1.66055i | −0.717839 | + | 2.11771i | 0 | −1.11487 | + | 2.39939i | 1.12562i | 0 | 0.482874 | − | 0.423842i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
289.5 | 0.248840 | + | 0.143668i | 0 | −0.958719 | − | 1.66055i | −1.47507 | + | 1.68052i | 0 | 1.11487 | − | 2.39939i | − | 1.12562i | 0 | −0.608495 | + | 0.206261i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
289.6 | 1.34443 | + | 0.776205i | 0 | 0.204988 | + | 0.355049i | 1.98074 | + | 1.03763i | 0 | 0.478401 | − | 2.60214i | − | 2.46837i | 0 | 1.85754 | + | 2.93248i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
289.7 | 1.54296 | + | 0.890827i | 0 | 0.587145 | + | 1.01696i | 0.881181 | − | 2.05512i | 0 | 2.40898 | + | 1.09398i | − | 1.47113i | 0 | 3.19038 | − | 2.38598i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
289.8 | 2.17942 | + | 1.25829i | 0 | 2.16659 | + | 3.75264i | −2.23607 | + | 0.00136408i | 0 | 1.31340 | + | 2.29673i | 5.87162i | 0 | −4.87505 | − | 2.81065i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
35.j | even | 6 | 1 | inner |
Twists
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.2.q.a | ✓ | 16 | 3.b | odd | 2 | 1 | |
105.2.q.a | ✓ | 16 | 15.d | odd | 2 | 1 | |
105.2.q.a | ✓ | 16 | 21.h | odd | 6 | 1 | |
105.2.q.a | ✓ | 16 | 105.o | odd | 6 | 1 | |
315.2.bf.b | 16 | 1.a | even | 1 | 1 | trivial | |
315.2.bf.b | 16 | 5.b | even | 2 | 1 | inner | |
315.2.bf.b | 16 | 7.c | even | 3 | 1 | inner | |
315.2.bf.b | 16 | 35.j | even | 6 | 1 | inner | |
525.2.i.h | 8 | 15.e | even | 4 | 1 | ||
525.2.i.h | 8 | 105.x | even | 12 | 1 | ||
525.2.i.k | 8 | 15.e | even | 4 | 1 | ||
525.2.i.k | 8 | 105.x | even | 12 | 1 | ||
735.2.d.d | 8 | 21.h | odd | 6 | 1 | ||
735.2.d.d | 8 | 105.o | odd | 6 | 1 | ||
735.2.d.e | 8 | 21.g | even | 6 | 1 | ||
735.2.d.e | 8 | 105.p | even | 6 | 1 | ||
735.2.q.g | 16 | 21.c | even | 2 | 1 | ||
735.2.q.g | 16 | 21.g | even | 6 | 1 | ||
735.2.q.g | 16 | 105.g | even | 2 | 1 | ||
735.2.q.g | 16 | 105.p | even | 6 | 1 | ||
1680.2.di.d | 16 | 12.b | even | 2 | 1 | ||
1680.2.di.d | 16 | 60.h | even | 2 | 1 | ||
1680.2.di.d | 16 | 84.n | even | 6 | 1 | ||
1680.2.di.d | 16 | 420.ba | even | 6 | 1 | ||
2205.2.d.o | 8 | 7.d | odd | 6 | 1 | ||
2205.2.d.o | 8 | 35.i | odd | 6 | 1 | ||
2205.2.d.s | 8 | 7.c | even | 3 | 1 | ||
2205.2.d.s | 8 | 35.j | even | 6 | 1 | ||
3675.2.a.bn | 4 | 105.w | odd | 12 | 1 | ||
3675.2.a.bp | 4 | 105.x | even | 12 | 1 | ||
3675.2.a.bz | 4 | 105.x | even | 12 | 1 | ||
3675.2.a.cb | 4 | 105.w | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 12T_{2}^{14} + 100T_{2}^{12} - 424T_{2}^{10} + 1308T_{2}^{8} - 2192T_{2}^{6} + 2528T_{2}^{4} - 208T_{2}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} - 12 T^{14} + 100 T^{12} + \cdots + 16 \)
$3$
\( T^{16} \)
$5$
\( T^{16} + 2 T^{15} + 4 T^{14} + \cdots + 390625 \)
$7$
\( T^{16} + 20 T^{14} + 202 T^{12} + \cdots + 5764801 \)
$11$
\( (T^{8} + 18 T^{6} + 28 T^{5} + 294 T^{4} + \cdots + 900)^{2} \)
$13$
\( (T^{8} + 60 T^{6} + 1182 T^{4} + \cdots + 16129)^{2} \)
$17$
\( T^{16} - 60 T^{14} + \cdots + 322417936 \)
$19$
\( (T^{8} + 12 T^{7} + 106 T^{6} + 376 T^{5} + \cdots + 81)^{2} \)
$23$
\( T^{16} - 164 T^{14} + \cdots + 5143987297296 \)
$29$
\( (T^{4} + 6 T^{3} - 38 T^{2} - 190 T - 22)^{4} \)
$31$
\( (T^{8} - 8 T^{7} + 70 T^{6} - 144 T^{5} + \cdots + 3721)^{2} \)
$37$
\( T^{16} - 168 T^{14} + \cdots + 676751377201 \)
$41$
\( (T^{4} + 4 T^{3} - 50 T^{2} - 146 T - 10)^{4} \)
$43$
\( (T^{8} + 128 T^{6} + 3390 T^{4} + \cdots + 2401)^{2} \)
$47$
\( T^{16} - 68 T^{14} + \cdots + 207360000 \)
$53$
\( T^{16} - 160 T^{14} + \cdots + 84934656 \)
$59$
\( (T^{8} + 2 T^{7} + 10 T^{6} + 38 T^{4} + \cdots + 100)^{2} \)
$61$
\( (T^{8} - 8 T^{7} + 164 T^{6} + \cdots + 250000)^{2} \)
$67$
\( T^{16} - 180 T^{14} + \cdots + 670801950625 \)
$71$
\( (T^{4} - 14 T^{3} - 90 T^{2} + 1334 T - 3202)^{4} \)
$73$
\( T^{16} - 272 T^{14} + \cdots + 3722279521041 \)
$79$
\( (T^{8} + 8 T^{7} + 218 T^{6} + \cdots + 50140561)^{2} \)
$83$
\( (T^{8} + 148 T^{6} + 6716 T^{4} + \cdots + 131044)^{2} \)
$89$
\( (T^{8} + 8 T^{7} + 258 T^{6} + \cdots + 285156)^{2} \)
$97$
\( (T^{8} + 20 T^{6} + 120 T^{4} + 192 T^{2} + \cdots + 16)^{2} \)
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