# Properties

 Label 15.4.b.a Level 15 Weight 4 Character orbit 15.b Analytic conductor 0.885 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 15.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.885028650086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ Defining polynomial: $$x^{4} + 21 x^{2} + 100$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{2} q^{3} + ( -5 + \beta_{3} ) q^{4} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + ( 5 - \beta_{3} ) q^{6} + ( 6 \beta_{1} - 4 \beta_{2} ) q^{7} + ( \beta_{1} + 8 \beta_{2} ) q^{8} -9 q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{2} q^{3} + ( -5 + \beta_{3} ) q^{4} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + ( 5 - \beta_{3} ) q^{6} + ( 6 \beta_{1} - 4 \beta_{2} ) q^{7} + ( \beta_{1} + 8 \beta_{2} ) q^{8} -9 q^{9} + ( 3 - 6 \beta_{1} - 8 \beta_{2} + \beta_{3} ) q^{10} + ( -22 + 2 \beta_{3} ) q^{11} -9 \beta_{1} q^{12} + ( -6 \beta_{1} + 16 \beta_{2} ) q^{13} + ( 58 - 2 \beta_{3} ) q^{14} + ( 13 + 9 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{15} + ( 13 - \beta_{3} ) q^{16} + ( -10 \beta_{1} - 12 \beta_{2} ) q^{17} + 9 \beta_{1} q^{18} + ( -24 - 8 \beta_{3} ) q^{19} + ( -102 + 9 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} ) q^{20} + ( 6 + 6 \beta_{3} ) q^{21} + ( 30 \beta_{1} + 16 \beta_{2} ) q^{22} + ( 8 \beta_{1} - 12 \beta_{2} ) q^{23} + ( -77 + \beta_{3} ) q^{24} + ( 67 - 24 \beta_{1} + 28 \beta_{2} - 6 \beta_{3} ) q^{25} + ( 2 - 10 \beta_{3} ) q^{26} -9 \beta_{2} q^{27} + ( -18 \beta_{1} - 48 \beta_{2} ) q^{28} + ( 152 + 14 \beta_{3} ) q^{29} + ( 102 - 9 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} ) q^{30} + ( 24 + 4 \beta_{3} ) q^{31} + ( -9 \beta_{1} + 56 \beta_{2} ) q^{32} + ( -18 \beta_{1} - 12 \beta_{2} ) q^{33} + ( -190 + 22 \beta_{3} ) q^{34} + ( -70 + 60 \beta_{2} - 10 \beta_{3} ) q^{35} + ( 45 - 9 \beta_{3} ) q^{36} + ( -54 \beta_{1} - 24 \beta_{2} ) q^{37} + ( -8 \beta_{1} - 64 \beta_{2} ) q^{38} + ( -114 - 6 \beta_{3} ) q^{39} + ( 101 + 78 \beta_{1} - 16 \beta_{2} + 7 \beta_{3} ) q^{40} + ( -206 + 4 \beta_{3} ) q^{41} + ( 18 \beta_{1} + 48 \beta_{2} ) q^{42} + ( 96 \beta_{1} - 28 \beta_{2} ) q^{43} + ( 294 - 30 \beta_{3} ) q^{44} + ( -18 - 9 \beta_{1} + 18 \beta_{2} + 9 \beta_{3} ) q^{45} + ( 44 + 4 \beta_{3} ) q^{46} + ( 92 \beta_{1} - 76 \beta_{2} ) q^{47} + ( 9 \beta_{1} + 8 \beta_{2} ) q^{48} + ( -29 - 12 \beta_{3} ) q^{49} + ( -172 - 91 \beta_{1} - 48 \beta_{2} - 4 \beta_{3} ) q^{50} + ( 158 - 10 \beta_{3} ) q^{51} + ( -90 \beta_{1} + 48 \beta_{2} ) q^{52} + ( -82 \beta_{1} + 108 \beta_{2} ) q^{53} + ( -45 + 9 \beta_{3} ) q^{54} + ( -228 + 6 \beta_{1} + 8 \beta_{2} + 24 \beta_{3} ) q^{55} + ( -10 + 50 \beta_{3} ) q^{56} + ( 72 \beta_{1} - 64 \beta_{2} ) q^{57} + ( -96 \beta_{1} + 112 \beta_{2} ) q^{58} + ( 94 - 2 \beta_{3} ) q^{59} + ( 27 - 54 \beta_{1} - 72 \beta_{2} + 9 \beta_{3} ) q^{60} + ( 186 - 32 \beta_{3} ) q^{61} + ( -8 \beta_{1} + 32 \beta_{2} ) q^{62} + ( -54 \beta_{1} + 36 \beta_{2} ) q^{63} + ( 267 - 55 \beta_{3} ) q^{64} + ( 226 + 108 \beta_{1} - 96 \beta_{2} + 22 \beta_{3} ) q^{65} + ( -294 + 30 \beta_{3} ) q^{66} + ( -60 \beta_{1} - 92 \beta_{2} ) q^{67} + ( 198 \beta_{1} + 80 \beta_{2} ) q^{68} + ( 68 + 8 \beta_{3} ) q^{69} + ( 300 + 30 \beta_{1} - 80 \beta_{2} - 60 \beta_{3} ) q^{70} + ( 12 - 60 \beta_{3} ) q^{71} + ( -9 \beta_{1} - 72 \beta_{2} ) q^{72} + ( 108 \beta_{1} + 168 \beta_{2} ) q^{73} + ( -822 + 78 \beta_{3} ) q^{74} + ( -132 + 54 \beta_{1} + 37 \beta_{2} - 24 \beta_{3} ) q^{75} + ( -616 + 8 \beta_{3} ) q^{76} + ( -108 \beta_{1} - 48 \beta_{2} ) q^{77} + ( 90 \beta_{1} - 48 \beta_{2} ) q^{78} + ( -240 + 100 \beta_{3} ) q^{79} + ( 118 - \beta_{1} - 8 \beta_{2} - 14 \beta_{3} ) q^{80} + 81 q^{81} + ( 222 \beta_{1} + 32 \beta_{2} ) q^{82} + ( -208 \beta_{1} - 60 \beta_{2} ) q^{83} + ( 522 - 18 \beta_{3} ) q^{84} + ( -126 - 168 \beta_{1} - 44 \beta_{2} - 2 \beta_{3} ) q^{85} + ( 1108 - 68 \beta_{3} ) q^{86} + ( -126 \beta_{1} + 222 \beta_{2} ) q^{87} + ( -174 \beta_{1} - 112 \beta_{2} ) q^{88} + ( -534 - 48 \beta_{3} ) q^{89} + ( -27 + 54 \beta_{1} + 72 \beta_{2} - 9 \beta_{3} ) q^{90} + ( 444 + 84 \beta_{3} ) q^{91} + ( 36 \beta_{1} - 64 \beta_{2} ) q^{92} + ( -36 \beta_{1} + 44 \beta_{2} ) q^{93} + ( 816 - 16 \beta_{3} ) q^{94} + ( 688 - 136 \beta_{1} + 192 \beta_{2} + 16 \beta_{3} ) q^{95} + ( -459 - 9 \beta_{3} ) q^{96} + ( 240 \beta_{1} + 8 \beta_{2} ) q^{97} + ( -19 \beta_{1} - 96 \beta_{2} ) q^{98} + ( 198 - 18 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 18q^{4} + 6q^{5} + 18q^{6} - 36q^{9} + O(q^{10})$$ $$4q - 18q^{4} + 6q^{5} + 18q^{6} - 36q^{9} + 14q^{10} - 84q^{11} + 228q^{14} + 54q^{15} + 50q^{16} - 112q^{19} - 396q^{20} + 36q^{21} - 306q^{24} + 256q^{25} - 12q^{26} + 636q^{29} + 396q^{30} + 104q^{31} - 716q^{34} - 300q^{35} + 162q^{36} - 468q^{39} + 418q^{40} - 816q^{41} + 1116q^{44} - 54q^{45} + 184q^{46} - 140q^{49} - 696q^{50} + 612q^{51} - 162q^{54} - 864q^{55} + 60q^{56} + 372q^{59} + 126q^{60} + 680q^{61} + 958q^{64} + 948q^{65} - 1116q^{66} + 288q^{69} + 1080q^{70} - 72q^{71} - 3132q^{74} - 576q^{75} - 2448q^{76} - 760q^{79} + 444q^{80} + 324q^{81} + 2052q^{84} - 508q^{85} + 4296q^{86} - 2232q^{89} - 126q^{90} + 1944q^{91} + 3232q^{94} + 2784q^{95} - 1854q^{96} + 756q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 21 x^{2} + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + \nu$$$$)/10$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{3} + 33 \nu$$$$)/10$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{2} + 32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 3 \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 32$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 33 \beta_{1}$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 − 3.70156i − 2.70156i 2.70156i 3.70156i
4.70156i 3.00000i −14.1047 11.1047 1.29844i 14.1047 16.2094i 28.7016i −9.00000 −6.10469 52.2094i
4.2 1.70156i 3.00000i 5.10469 −8.10469 + 7.70156i −5.10469 22.2094i 22.2984i −9.00000 13.1047 + 13.7906i
4.3 1.70156i 3.00000i 5.10469 −8.10469 7.70156i −5.10469 22.2094i 22.2984i −9.00000 13.1047 13.7906i
4.4 4.70156i 3.00000i −14.1047 11.1047 + 1.29844i 14.1047 16.2094i 28.7016i −9.00000 −6.10469 + 52.2094i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.4.b.a 4
3.b odd 2 1 45.4.b.b 4
4.b odd 2 1 240.4.f.f 4
5.b even 2 1 inner 15.4.b.a 4
5.c odd 4 1 75.4.a.c 2
5.c odd 4 1 75.4.a.f 2
8.b even 2 1 960.4.f.q 4
8.d odd 2 1 960.4.f.p 4
12.b even 2 1 720.4.f.j 4
15.d odd 2 1 45.4.b.b 4
15.e even 4 1 225.4.a.i 2
15.e even 4 1 225.4.a.o 2
20.d odd 2 1 240.4.f.f 4
20.e even 4 1 1200.4.a.bn 2
20.e even 4 1 1200.4.a.bt 2
40.e odd 2 1 960.4.f.p 4
40.f even 2 1 960.4.f.q 4
60.h even 2 1 720.4.f.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 1.a even 1 1 trivial
15.4.b.a 4 5.b even 2 1 inner
45.4.b.b 4 3.b odd 2 1
45.4.b.b 4 15.d odd 2 1
75.4.a.c 2 5.c odd 4 1
75.4.a.f 2 5.c odd 4 1
225.4.a.i 2 15.e even 4 1
225.4.a.o 2 15.e even 4 1
240.4.f.f 4 4.b odd 2 1
240.4.f.f 4 20.d odd 2 1
720.4.f.j 4 12.b even 2 1
720.4.f.j 4 60.h even 2 1
960.4.f.p 4 8.d odd 2 1
960.4.f.p 4 40.e odd 2 1
960.4.f.q 4 8.b even 2 1
960.4.f.q 4 40.f even 2 1
1200.4.a.bn 2 20.e even 4 1
1200.4.a.bt 2 20.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 7 T^{2} + 48 T^{4} - 448 T^{6} + 4096 T^{8}$$
$3$ $$( 1 + 9 T^{2} )^{2}$$
$5$ $$1 - 6 T - 110 T^{2} - 750 T^{3} + 15625 T^{4}$$
$7$ $$1 - 616 T^{2} + 316878 T^{4} - 72471784 T^{6} + 13841287201 T^{8}$$
$11$ $$( 1 + 42 T + 2734 T^{2} + 55902 T^{3} + 1771561 T^{4} )^{2}$$
$13$ $$1 - 5008 T^{2} + 13678638 T^{4} - 24172659472 T^{6} + 23298085122481 T^{8}$$
$17$ $$1 - 12400 T^{2} + 76051038 T^{4} - 299305855600 T^{6} + 582622237229761 T^{8}$$
$19$ $$( 1 + 56 T + 8598 T^{2} + 384104 T^{3} + 47045881 T^{4} )^{2}$$
$23$ $$1 - 46204 T^{2} + 828262758 T^{4} - 6839850215356 T^{6} + 21914624432020321 T^{8}$$
$29$ $$( 1 - 318 T + 55978 T^{2} - 7755702 T^{3} + 594823321 T^{4} )^{2}$$
$31$ $$( 1 - 52 T + 58782 T^{2} - 1549132 T^{3} + 887503681 T^{4} )^{2}$$
$37$ $$1 - 96016 T^{2} + 4637534478 T^{4} - 246350786886544 T^{6} + 6582952005840035281 T^{8}$$
$41$ $$( 1 + 408 T + 177982 T^{2} + 28119768 T^{3} + 4750104241 T^{4} )^{2}$$
$43$ $$1 - 121900 T^{2} + 14997346998 T^{4} - 770574155673100 T^{6} + 39959630797262576401 T^{8}$$
$47$ $$1 - 225580 T^{2} + 31469120358 T^{4} - 2431575393915820 T^{6} +$$$$11\!\cdots\!41$$$$T^{8}$$
$53$ $$1 - 376864 T^{2} + 68697431598 T^{4} - 8352949792519456 T^{6} +$$$$49\!\cdots\!41$$$$T^{8}$$
$59$ $$( 1 - 186 T + 419038 T^{2} - 38200494 T^{3} + 42180533641 T^{4} )^{2}$$
$61$ $$( 1 - 340 T + 388398 T^{2} - 77173540 T^{3} + 51520374361 T^{4} )^{2}$$
$67$ $$1 - 861340 T^{2} + 346621507638 T^{4} - 77915422897446460 T^{6} +$$$$81\!\cdots\!61$$$$T^{8}$$
$71$ $$( 1 + 36 T + 384046 T^{2} + 12884796 T^{3} + 128100283921 T^{4} )^{2}$$
$73$ $$1 - 429844 T^{2} + 136740794118 T^{4} - 65050109164968916 T^{6} +$$$$22\!\cdots\!21$$$$T^{8}$$
$79$ $$( 1 + 380 T + 99678 T^{2} + 187354820 T^{3} + 243087455521 T^{4} )^{2}$$
$83$ $$1 - 916108 T^{2} + 434315280918 T^{4} - 299512691566327852 T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$89$ $$( 1 + 1116 T + 1508758 T^{2} + 786745404 T^{3} + 496981290961 T^{4} )^{2}$$
$97$ $$1 - 2174980 T^{2} + 2500346420358 T^{4} - 1811697451280476420 T^{6} +$$$$69\!\cdots\!41$$$$T^{8}$$