# Properties

 Label 315.2.j.f Level 315 Weight 2 Character orbit 315.j Analytic conductor 2.515 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.j (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.4406832.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{2} + ( -2 + \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{4} -\beta_{4} q^{5} + ( \beta_{2} + \beta_{3} - \beta_{5} ) q^{7} + ( 2 - 2 \beta_{2} - 2 \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{2} + ( -2 + \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{4} -\beta_{4} q^{5} + ( \beta_{2} + \beta_{3} - \beta_{5} ) q^{7} + ( 2 - 2 \beta_{2} - 2 \beta_{3} ) q^{8} + ( -1 + \beta_{1} + \beta_{4} ) q^{10} + ( -3 + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{11} + ( 2 - \beta_{3} ) q^{13} + ( -1 - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{14} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{16} + ( 1 - \beta_{1} - \beta_{4} ) q^{17} + ( 3 \beta_{1} - 3 \beta_{2} + \beta_{4} - \beta_{5} ) q^{19} + ( 2 - \beta_{2} - \beta_{3} ) q^{20} + ( 2 - 5 \beta_{2} - \beta_{3} ) q^{22} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} ) q^{23} + ( -1 + \beta_{4} ) q^{25} + ( -4 \beta_{1} + 4 \beta_{2} - \beta_{4} + \beta_{5} ) q^{26} + ( 2 - 4 \beta_{1} + \beta_{2} - \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{28} + ( 1 + 2 \beta_{2} + \beta_{3} ) q^{29} + ( -1 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{31} + ( -8 + 2 \beta_{1} + 2 \beta_{3} + 8 \beta_{4} - 2 \beta_{5} ) q^{32} + ( -4 + \beta_{2} + \beta_{3} ) q^{34} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{35} + ( \beta_{1} - \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{37} + ( 9 - 3 \beta_{1} - 4 \beta_{3} - 9 \beta_{4} + 4 \beta_{5} ) q^{38} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{40} + ( 1 + 2 \beta_{2} + 3 \beta_{3} ) q^{41} + ( -\beta_{2} + 2 \beta_{3} ) q^{43} + ( -4 \beta_{1} + 4 \beta_{2} - 10 \beta_{4} + 4 \beta_{5} ) q^{44} + ( 4 - \beta_{1} - 5 \beta_{3} - 4 \beta_{4} + 5 \beta_{5} ) q^{46} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{47} + ( 1 - 3 \beta_{1} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{49} + ( 1 - \beta_{2} ) q^{50} + ( -8 + 3 \beta_{1} + 3 \beta_{3} + 8 \beta_{4} - 3 \beta_{5} ) q^{52} + ( -2 + 2 \beta_{1} + 2 \beta_{4} ) q^{53} + ( 3 - \beta_{3} ) q^{55} + ( -12 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 8 \beta_{4} ) q^{56} + ( \beta_{1} - \beta_{2} + 4 \beta_{4} - 3 \beta_{5} ) q^{58} + ( -1 + 4 \beta_{1} + 5 \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{59} + ( \beta_{1} - \beta_{2} + 8 \beta_{4} + \beta_{5} ) q^{61} + ( 9 + 3 \beta_{2} ) q^{62} + ( 8 - 4 \beta_{2} ) q^{64} + ( -2 \beta_{4} + \beta_{5} ) q^{65} + ( 6 - 2 \beta_{1} + \beta_{3} - 6 \beta_{4} - \beta_{5} ) q^{67} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} ) q^{68} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{70} + ( 7 - 4 \beta_{2} + \beta_{3} ) q^{71} + ( -4 + 5 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{73} + ( -3 - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{74} + ( -12 + 11 \beta_{2} + 5 \beta_{3} ) q^{76} + ( -1 - 4 \beta_{1} - \beta_{2} - 4 \beta_{4} + 2 \beta_{5} ) q^{77} + ( -3 \beta_{1} + 3 \beta_{2} + \beta_{4} - \beta_{5} ) q^{79} + ( -2 + 4 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{80} + ( 5 \beta_{1} - 5 \beta_{2} + 2 \beta_{4} - 5 \beta_{5} ) q^{82} + ( 5 + \beta_{2} + 2 \beta_{3} ) q^{83} + ( -1 + \beta_{2} ) q^{85} + ( 4 \beta_{1} - 4 \beta_{2} - 5 \beta_{4} - \beta_{5} ) q^{86} + ( -14 + 8 \beta_{1} + 6 \beta_{3} + 14 \beta_{4} - 6 \beta_{5} ) q^{88} + ( 2 \beta_{1} - 2 \beta_{2} + 11 \beta_{4} + \beta_{5} ) q^{89} + ( -4 - \beta_{1} + 4 \beta_{2} + \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{91} + ( -8 + 8 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 10 + 2 \beta_{1} - 10 \beta_{4} ) q^{94} + ( 1 - 3 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{95} + ( -\beta_{2} + 3 \beta_{3} ) q^{97} + ( -13 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 2q^{2} - 6q^{4} - 3q^{5} + q^{7} + 12q^{8} + O(q^{10})$$ $$6q - 2q^{2} - 6q^{4} - 3q^{5} + q^{7} + 12q^{8} - 2q^{10} - 10q^{11} + 14q^{13} + 2q^{14} - 4q^{16} + 2q^{17} + q^{19} + 12q^{20} + 4q^{22} - 10q^{23} - 3q^{25} - 8q^{28} + 8q^{29} + q^{31} - 24q^{32} - 24q^{34} + q^{35} - 11q^{37} + 28q^{38} - 6q^{40} + 4q^{41} - 6q^{43} - 30q^{44} + 16q^{46} + 2q^{47} - 3q^{49} + 4q^{50} - 24q^{52} - 4q^{53} + 20q^{55} - 42q^{56} + 14q^{58} - 4q^{59} + 22q^{61} + 60q^{62} + 40q^{64} - 7q^{65} + 15q^{67} + 10q^{68} + 14q^{70} + 32q^{71} - 9q^{73} - 6q^{74} - 60q^{76} - 26q^{77} + 7q^{79} - 4q^{80} + 6q^{82} + 28q^{83} - 4q^{85} - 18q^{86} - 40q^{88} + 30q^{89} - 3q^{91} - 36q^{92} + 32q^{94} + q^{95} - 8q^{97} - 68q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 6 x^{4} + 7 x^{3} + 24 x^{2} + 5 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 6 \nu^{4} - 36 \nu^{3} + 24 \nu^{2} + 5 \nu - 30$$$$)/149$$ $$\beta_{3}$$ $$=$$ $$($$$$-6 \nu^{5} + 36 \nu^{4} - 67 \nu^{3} + 144 \nu^{2} + 30 \nu + 416$$$$)/149$$ $$\beta_{4}$$ $$=$$ $$($$$$30 \nu^{5} - 31 \nu^{4} + 186 \nu^{3} + 174 \nu^{2} + 744 \nu + 155$$$$)/149$$ $$\beta_{5}$$ $$=$$ $$($$$$89 \nu^{5} - 87 \nu^{4} + 522 \nu^{3} + 695 \nu^{2} + 2088 \nu + 435$$$$)/149$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 3 \beta_{4} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 6 \beta_{2} - 4$$ $$\nu^{4}$$ $$=$$ $$-6 \beta_{5} + 19 \beta_{4} + 6 \beta_{3} - 12 \beta_{1} - 19$$ $$\nu^{5}$$ $$=$$ $$-12 \beta_{5} + 42 \beta_{4} + 43 \beta_{2} - 43 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1
 −0.827721 + 1.43366i −0.105378 + 0.182520i 1.43310 − 2.48220i −0.827721 − 1.43366i −0.105378 − 0.182520i 1.43310 + 2.48220i
−1.32772 2.29968i 0 −2.52569 + 4.37462i −0.500000 0.866025i 0 −2.35341 + 1.20891i 8.10275 0 −1.32772 + 2.29968i
46.2 −0.605378 1.04855i 0 0.267035 0.462518i −0.500000 0.866025i 0 1.16166 2.37709i −3.06814 0 −0.605378 + 1.04855i
46.3 0.933099 + 1.61618i 0 −0.741348 + 1.28405i −0.500000 0.866025i 0 1.69175 + 2.03420i 0.965392 0 0.933099 1.61618i
226.1 −1.32772 + 2.29968i 0 −2.52569 4.37462i −0.500000 + 0.866025i 0 −2.35341 1.20891i 8.10275 0 −1.32772 2.29968i
226.2 −0.605378 + 1.04855i 0 0.267035 + 0.462518i −0.500000 + 0.866025i 0 1.16166 + 2.37709i −3.06814 0 −0.605378 1.04855i
226.3 0.933099 1.61618i 0 −0.741348 1.28405i −0.500000 + 0.866025i 0 1.69175 2.03420i 0.965392 0 0.933099 + 1.61618i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 226.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.j.f 6
3.b odd 2 1 315.2.j.g yes 6
7.c even 3 1 inner 315.2.j.f 6
7.c even 3 1 2205.2.a.be 3
7.d odd 6 1 2205.2.a.bd 3
21.g even 6 1 2205.2.a.bc 3
21.h odd 6 1 315.2.j.g yes 6
21.h odd 6 1 2205.2.a.bb 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.j.f 6 1.a even 1 1 trivial
315.2.j.f 6 7.c even 3 1 inner
315.2.j.g yes 6 3.b odd 2 1
315.2.j.g yes 6 21.h odd 6 1
2205.2.a.bb 3 21.h odd 6 1
2205.2.a.bc 3 21.g even 6 1
2205.2.a.bd 3 7.d odd 6 1
2205.2.a.be 3 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 2 T_{2}^{5} + 8 T_{2}^{4} + 4 T_{2}^{3} + 28 T_{2}^{2} + 24 T_{2} + 36$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} - 4 T^{4} - 4 T^{5} - 4 T^{6} - 8 T^{7} - 16 T^{8} + 32 T^{10} + 64 T^{11} + 64 T^{12}$$
$3$ 
$5$ $$( 1 + T + T^{2} )^{3}$$
$7$ $$1 - T + 2 T^{2} + 23 T^{3} + 14 T^{4} - 49 T^{5} + 343 T^{6}$$
$11$ $$1 + 10 T + 41 T^{2} + 138 T^{3} + 572 T^{4} + 1564 T^{5} + 3347 T^{6} + 17204 T^{7} + 69212 T^{8} + 183678 T^{9} + 600281 T^{10} + 1610510 T^{11} + 1771561 T^{12}$$
$13$ $$( 1 - 7 T + 48 T^{2} - 171 T^{3} + 624 T^{4} - 1183 T^{5} + 2197 T^{6} )^{2}$$
$17$ $$1 - 2 T - 43 T^{2} + 30 T^{3} + 1286 T^{4} - 296 T^{5} - 25039 T^{6} - 5032 T^{7} + 371654 T^{8} + 147390 T^{9} - 3591403 T^{10} - 2839714 T^{11} + 24137569 T^{12}$$
$19$ $$1 - T - 15 T^{2} + 146 T^{3} - 129 T^{4} - 1049 T^{5} + 15054 T^{6} - 19931 T^{7} - 46569 T^{8} + 1001414 T^{9} - 1954815 T^{10} - 2476099 T^{11} + 47045881 T^{12}$$
$23$ $$1 + 10 T + 47 T^{2} + 126 T^{3} - 430 T^{4} - 8384 T^{5} - 52369 T^{6} - 192832 T^{7} - 227470 T^{8} + 1533042 T^{9} + 13152527 T^{10} + 64363430 T^{11} + 148035889 T^{12}$$
$29$ $$( 1 - 4 T + 73 T^{2} - 178 T^{3} + 2117 T^{4} - 3364 T^{5} + 24389 T^{6} )^{2}$$
$31$ $$1 - T - 23 T^{2} + 298 T^{3} - 329 T^{4} - 3337 T^{5} + 66686 T^{6} - 103447 T^{7} - 316169 T^{8} + 8877718 T^{9} - 21240983 T^{10} - 28629151 T^{11} + 887503681 T^{12}$$
$37$ $$1 + 11 T - 5 T^{2} - 224 T^{3} + 2359 T^{4} + 13925 T^{5} - 3730 T^{6} + 515225 T^{7} + 3229471 T^{8} - 11346272 T^{9} - 9370805 T^{10} + 762783527 T^{11} + 2565726409 T^{12}$$
$41$ $$( 1 - 2 T + 65 T^{2} - 182 T^{3} + 2665 T^{4} - 3362 T^{5} + 68921 T^{6} )^{2}$$
$43$ $$( 1 + 3 T + 88 T^{2} + 209 T^{3} + 3784 T^{4} + 5547 T^{5} + 79507 T^{6} )^{2}$$
$47$ $$1 - 2 T - 69 T^{2} - 106 T^{3} + 1858 T^{4} + 7846 T^{5} - 51253 T^{6} + 368762 T^{7} + 4104322 T^{8} - 11005238 T^{9} - 336697989 T^{10} - 458690014 T^{11} + 10779215329 T^{12}$$
$53$ $$1 + 4 T - 127 T^{2} - 180 T^{3} + 11366 T^{4} + 4372 T^{5} - 689611 T^{6} + 231716 T^{7} + 31927094 T^{8} - 26797860 T^{9} - 1002091087 T^{10} + 1672781972 T^{11} + 22164361129 T^{12}$$
$59$ $$1 + 4 T + 9 T^{2} - 472 T^{3} - 1364 T^{4} + 12370 T^{5} + 414671 T^{6} + 729830 T^{7} - 4748084 T^{8} - 96938888 T^{9} + 109056249 T^{10} + 2859697196 T^{11} + 42180533641 T^{12}$$
$61$ $$1 - 22 T + 157 T^{2} - 1322 T^{3} + 22390 T^{4} - 164266 T^{5} + 750425 T^{6} - 10020226 T^{7} + 83313190 T^{8} - 300068882 T^{9} + 2173797037 T^{10} - 18581118622 T^{11} + 51520374361 T^{12}$$
$67$ $$1 - 15 T - 13 T^{2} + 148 T^{3} + 12813 T^{4} - 31397 T^{5} - 747766 T^{6} - 2103599 T^{7} + 57517557 T^{8} + 44512924 T^{9} - 261964573 T^{10} - 20251876605 T^{11} + 90458382169 T^{12}$$
$71$ $$( 1 - 16 T + 187 T^{2} - 1390 T^{3} + 13277 T^{4} - 80656 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$1 + 9 T - 49 T^{2} - 1532 T^{3} - 4749 T^{4} + 52771 T^{5} + 937886 T^{6} + 3852283 T^{7} - 25307421 T^{8} - 595974044 T^{9} - 1391513809 T^{10} + 18657644337 T^{11} + 151334226289 T^{12}$$
$79$ $$1 - 7 T - 135 T^{2} + 1070 T^{3} + 11067 T^{4} - 57119 T^{5} - 604314 T^{6} - 4512401 T^{7} + 69069147 T^{8} + 527551730 T^{9} - 5258260935 T^{10} - 21539394793 T^{11} + 243087455521 T^{12}$$
$83$ $$( 1 - 14 T + 289 T^{2} - 2342 T^{3} + 23987 T^{4} - 96446 T^{5} + 571787 T^{6} )^{2}$$
$89$ $$1 - 30 T + 371 T^{2} - 4122 T^{3} + 56540 T^{4} - 520116 T^{5} + 3879065 T^{6} - 46290324 T^{7} + 447853340 T^{8} - 2905882218 T^{9} + 23277371411 T^{10} - 167521783470 T^{11} + 496981290961 T^{12}$$
$97$ $$( 1 + 4 T + 211 T^{2} + 564 T^{3} + 20467 T^{4} + 37636 T^{5} + 912673 T^{6} )^{2}$$