# Properties

 Label 15.5.d.c Level $15$ Weight $5$ Character orbit 15.d Analytic conductor $1.551$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.55054944626$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{10}, \sqrt{-26})$$ Defining polynomial: $$x^{4} + 8x^{2} + 81$$ x^4 + 8*x^2 + 81 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} - 6 q^{4} + ( - \beta_{3} + 2 \beta_{2}) q^{5} + (\beta_{3} + 15) q^{6} + (\beta_{2} - 2 \beta_1) q^{7} - 22 \beta_{2} q^{8} + (3 \beta_{3} - 36) q^{9}+O(q^{10})$$ q + b2 * q^2 + (b2 + b1) * q^3 - 6 * q^4 + (-b3 + 2*b2) * q^5 + (b3 + 15) * q^6 + (b2 - 2*b1) * q^7 - 22*b2 * q^8 + (3*b3 - 36) * q^9 $$q + \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} - 6 q^{4} + ( - \beta_{3} + 2 \beta_{2}) q^{5} + (\beta_{3} + 15) q^{6} + (\beta_{2} - 2 \beta_1) q^{7} - 22 \beta_{2} q^{8} + (3 \beta_{3} - 36) q^{9} + (5 \beta_{2} - 10 \beta_1 + 20) q^{10} - 4 \beta_{3} q^{11} + ( - 6 \beta_{2} - 6 \beta_1) q^{12} + ( - 16 \beta_{2} + 32 \beta_1) q^{13} - 2 \beta_{3} q^{14} + (2 \beta_{3} + 66 \beta_{2} - 15 \beta_1 + 30) q^{15} - 124 q^{16} + 88 \beta_{2} q^{17} + ( - 51 \beta_{2} + 30 \beta_1) q^{18} + 308 q^{19} + (6 \beta_{3} - 12 \beta_{2}) q^{20} + ( - 3 \beta_{3} + 117) q^{21} + (20 \beta_{2} - 40 \beta_1) q^{22} - 131 \beta_{2} q^{23} + ( - 22 \beta_{3} - 330) q^{24} + (20 \beta_{2} - 40 \beta_1 - 545) q^{25} + 32 \beta_{3} q^{26} + ( - 234 \beta_{2} + 9 \beta_1) q^{27} + ( - 6 \beta_{2} + 12 \beta_1) q^{28} + 8 \beta_{3} q^{29} + ( - 15 \beta_{3} + 20 \beta_{2} + 20 \beta_1 + 585) q^{30} + 32 q^{31} + 228 \beta_{2} q^{32} + (264 \beta_{2} - 60 \beta_1) q^{33} + 880 q^{34} + ( - 4 \beta_{3} - 117 \beta_{2}) q^{35} + ( - 18 \beta_{3} + 216) q^{36} + ( - 84 \beta_{2} + 168 \beta_1) q^{37} + 308 \beta_{2} q^{38} + (48 \beta_{3} - 1872) q^{39} + ( - 110 \beta_{2} + 220 \beta_1 - 440) q^{40} - 86 \beta_{3} q^{41} + (132 \beta_{2} - 30 \beta_1) q^{42} + (169 \beta_{2} - 338 \beta_1) q^{43} + 24 \beta_{3} q^{44} + (36 \beta_{3} - 102 \beta_{2} + 60 \beta_1 + 1755) q^{45} - 1310 q^{46} - 773 \beta_{2} q^{47} + ( - 124 \beta_{2} - 124 \beta_1) q^{48} + 2167 q^{49} + ( - 40 \beta_{3} - 545 \beta_{2}) q^{50} + (88 \beta_{3} + 1320) q^{51} + (96 \beta_{2} - 192 \beta_1) q^{52} + 448 \beta_{2} q^{53} + (9 \beta_{3} - 2295) q^{54} + (40 \beta_{2} - 80 \beta_1 - 2340) q^{55} + 44 \beta_{3} q^{56} + (308 \beta_{2} + 308 \beta_1) q^{57} + ( - 40 \beta_{2} + 80 \beta_1) q^{58} - 164 \beta_{3} q^{59} + ( - 12 \beta_{3} - 396 \beta_{2} + 90 \beta_1 - 180) q^{60} - 928 q^{61} + 32 \beta_{2} q^{62} + (315 \beta_{2} + 72 \beta_1) q^{63} + 4264 q^{64} + (64 \beta_{3} + 1872 \beta_{2}) q^{65} + ( - 60 \beta_{3} + 2340) q^{66} + ( - 169 \beta_{2} + 338 \beta_1) q^{67} - 528 \beta_{2} q^{68} + ( - 131 \beta_{3} - 1965) q^{69} + (20 \beta_{2} - 40 \beta_1 - 1170) q^{70} + 192 \beta_{3} q^{71} + (1122 \beta_{2} - 660 \beta_1) q^{72} + ( - 276 \beta_{2} + 552 \beta_1) q^{73} + 168 \beta_{3} q^{74} + ( - 60 \beta_{3} - 545 \beta_{2} - 545 \beta_1 + 2340) q^{75} - 1848 q^{76} - 468 \beta_{2} q^{77} + ( - 2112 \beta_{2} + 480 \beta_1) q^{78} + 8 q^{79} + (124 \beta_{3} - 248 \beta_{2}) q^{80} + ( - 216 \beta_{3} - 3969) q^{81} + (430 \beta_{2} - 860 \beta_1) q^{82} - 1403 \beta_{2} q^{83} + (18 \beta_{3} - 702) q^{84} + (440 \beta_{2} - 880 \beta_1 + 1760) q^{85} - 338 \beta_{3} q^{86} + ( - 528 \beta_{2} + 120 \beta_1) q^{87} + ( - 440 \beta_{2} + 880 \beta_1) q^{88} + 384 \beta_{3} q^{89} + (60 \beta_{3} + 1575 \beta_{2} + 360 \beta_1 - 720) q^{90} + 3744 q^{91} + 786 \beta_{2} q^{92} + (32 \beta_{2} + 32 \beta_1) q^{93} - 7730 q^{94} + ( - 308 \beta_{3} + 616 \beta_{2}) q^{95} + (228 \beta_{3} + 3420) q^{96} + ( - 164 \beta_{2} + 328 \beta_1) q^{97} + 2167 \beta_{2} q^{98} + (144 \beta_{3} + 7020) q^{99}+O(q^{100})$$ q + b2 * q^2 + (b2 + b1) * q^3 - 6 * q^4 + (-b3 + 2*b2) * q^5 + (b3 + 15) * q^6 + (b2 - 2*b1) * q^7 - 22*b2 * q^8 + (3*b3 - 36) * q^9 + (5*b2 - 10*b1 + 20) * q^10 - 4*b3 * q^11 + (-6*b2 - 6*b1) * q^12 + (-16*b2 + 32*b1) * q^13 - 2*b3 * q^14 + (2*b3 + 66*b2 - 15*b1 + 30) * q^15 - 124 * q^16 + 88*b2 * q^17 + (-51*b2 + 30*b1) * q^18 + 308 * q^19 + (6*b3 - 12*b2) * q^20 + (-3*b3 + 117) * q^21 + (20*b2 - 40*b1) * q^22 - 131*b2 * q^23 + (-22*b3 - 330) * q^24 + (20*b2 - 40*b1 - 545) * q^25 + 32*b3 * q^26 + (-234*b2 + 9*b1) * q^27 + (-6*b2 + 12*b1) * q^28 + 8*b3 * q^29 + (-15*b3 + 20*b2 + 20*b1 + 585) * q^30 + 32 * q^31 + 228*b2 * q^32 + (264*b2 - 60*b1) * q^33 + 880 * q^34 + (-4*b3 - 117*b2) * q^35 + (-18*b3 + 216) * q^36 + (-84*b2 + 168*b1) * q^37 + 308*b2 * q^38 + (48*b3 - 1872) * q^39 + (-110*b2 + 220*b1 - 440) * q^40 - 86*b3 * q^41 + (132*b2 - 30*b1) * q^42 + (169*b2 - 338*b1) * q^43 + 24*b3 * q^44 + (36*b3 - 102*b2 + 60*b1 + 1755) * q^45 - 1310 * q^46 - 773*b2 * q^47 + (-124*b2 - 124*b1) * q^48 + 2167 * q^49 + (-40*b3 - 545*b2) * q^50 + (88*b3 + 1320) * q^51 + (96*b2 - 192*b1) * q^52 + 448*b2 * q^53 + (9*b3 - 2295) * q^54 + (40*b2 - 80*b1 - 2340) * q^55 + 44*b3 * q^56 + (308*b2 + 308*b1) * q^57 + (-40*b2 + 80*b1) * q^58 - 164*b3 * q^59 + (-12*b3 - 396*b2 + 90*b1 - 180) * q^60 - 928 * q^61 + 32*b2 * q^62 + (315*b2 + 72*b1) * q^63 + 4264 * q^64 + (64*b3 + 1872*b2) * q^65 + (-60*b3 + 2340) * q^66 + (-169*b2 + 338*b1) * q^67 - 528*b2 * q^68 + (-131*b3 - 1965) * q^69 + (20*b2 - 40*b1 - 1170) * q^70 + 192*b3 * q^71 + (1122*b2 - 660*b1) * q^72 + (-276*b2 + 552*b1) * q^73 + 168*b3 * q^74 + (-60*b3 - 545*b2 - 545*b1 + 2340) * q^75 - 1848 * q^76 - 468*b2 * q^77 + (-2112*b2 + 480*b1) * q^78 + 8 * q^79 + (124*b3 - 248*b2) * q^80 + (-216*b3 - 3969) * q^81 + (430*b2 - 860*b1) * q^82 - 1403*b2 * q^83 + (18*b3 - 702) * q^84 + (440*b2 - 880*b1 + 1760) * q^85 - 338*b3 * q^86 + (-528*b2 + 120*b1) * q^87 + (-440*b2 + 880*b1) * q^88 + 384*b3 * q^89 + (60*b3 + 1575*b2 + 360*b1 - 720) * q^90 + 3744 * q^91 + 786*b2 * q^92 + (32*b2 + 32*b1) * q^93 - 7730 * q^94 + (-308*b3 + 616*b2) * q^95 + (228*b3 + 3420) * q^96 + (-164*b2 + 328*b1) * q^97 + 2167*b2 * q^98 + (144*b3 + 7020) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 24 q^{4} + 60 q^{6} - 144 q^{9}+O(q^{10})$$ 4 * q - 24 * q^4 + 60 * q^6 - 144 * q^9 $$4 q - 24 q^{4} + 60 q^{6} - 144 q^{9} + 80 q^{10} + 120 q^{15} - 496 q^{16} + 1232 q^{19} + 468 q^{21} - 1320 q^{24} - 2180 q^{25} + 2340 q^{30} + 128 q^{31} + 3520 q^{34} + 864 q^{36} - 7488 q^{39} - 1760 q^{40} + 7020 q^{45} - 5240 q^{46} + 8668 q^{49} + 5280 q^{51} - 9180 q^{54} - 9360 q^{55} - 720 q^{60} - 3712 q^{61} + 17056 q^{64} + 9360 q^{66} - 7860 q^{69} - 4680 q^{70} + 9360 q^{75} - 7392 q^{76} + 32 q^{79} - 15876 q^{81} - 2808 q^{84} + 7040 q^{85} - 2880 q^{90} + 14976 q^{91} - 30920 q^{94} + 13680 q^{96} + 28080 q^{99}+O(q^{100})$$ 4 * q - 24 * q^4 + 60 * q^6 - 144 * q^9 + 80 * q^10 + 120 * q^15 - 496 * q^16 + 1232 * q^19 + 468 * q^21 - 1320 * q^24 - 2180 * q^25 + 2340 * q^30 + 128 * q^31 + 3520 * q^34 + 864 * q^36 - 7488 * q^39 - 1760 * q^40 + 7020 * q^45 - 5240 * q^46 + 8668 * q^49 + 5280 * q^51 - 9180 * q^54 - 9360 * q^55 - 720 * q^60 - 3712 * q^61 + 17056 * q^64 + 9360 * q^66 - 7860 * q^69 - 4680 * q^70 + 9360 * q^75 - 7392 * q^76 + 32 * q^79 - 15876 * q^81 - 2808 * q^84 + 7040 * q^85 - 2880 * q^90 + 14976 * q^91 - 30920 * q^94 + 13680 * q^96 + 28080 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 26\nu ) / 9$$ (v^3 + 26*v) / 9 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu ) / 9$$ (-v^3 + v) / 9 $$\beta_{3}$$ $$=$$ $$3\nu^{2} + 12$$ 3*v^2 + 12
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 3$$ (b2 + b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 12 ) / 3$$ (b3 - 12) / 3 $$\nu^{3}$$ $$=$$ $$( -26\beta_{2} + \beta_1 ) / 3$$ (-26*b2 + b1) / 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1
 −1.58114 − 2.54951i −1.58114 + 2.54951i 1.58114 − 2.54951i 1.58114 + 2.54951i
−3.16228 −4.74342 7.64853i −6.00000 −6.32456 24.1868i 15.0000 + 24.1868i 15.2971i 69.5701 −36.0000 + 72.5603i 20.0000 + 76.4853i
14.2 −3.16228 −4.74342 + 7.64853i −6.00000 −6.32456 + 24.1868i 15.0000 24.1868i 15.2971i 69.5701 −36.0000 72.5603i 20.0000 76.4853i
14.3 3.16228 4.74342 7.64853i −6.00000 6.32456 + 24.1868i 15.0000 24.1868i 15.2971i −69.5701 −36.0000 72.5603i 20.0000 + 76.4853i
14.4 3.16228 4.74342 + 7.64853i −6.00000 6.32456 24.1868i 15.0000 + 24.1868i 15.2971i −69.5701 −36.0000 + 72.5603i 20.0000 76.4853i
 $$n$$: 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
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 15.5.d.c 4
3.b odd 2 1 inner 15.5.d.c 4
4.b odd 2 1 240.5.c.c 4
5.b even 2 1 inner 15.5.d.c 4
5.c odd 4 2 75.5.c.h 4
12.b even 2 1 240.5.c.c 4
15.d odd 2 1 inner 15.5.d.c 4
15.e even 4 2 75.5.c.h 4
20.d odd 2 1 240.5.c.c 4
60.h even 2 1 240.5.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.5.d.c 4 1.a even 1 1 trivial
15.5.d.c 4 3.b odd 2 1 inner
15.5.d.c 4 5.b even 2 1 inner
15.5.d.c 4 15.d odd 2 1 inner
75.5.c.h 4 5.c odd 4 2
75.5.c.h 4 15.e even 4 2
240.5.c.c 4 4.b odd 2 1
240.5.c.c 4 12.b even 2 1
240.5.c.c 4 20.d odd 2 1
240.5.c.c 4 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 10$$ acting on $$S_{5}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 10)^{2}$$
$3$ $$T^{4} + 72T^{2} + 6561$$
$5$ $$T^{4} + 1090 T^{2} + 390625$$
$7$ $$(T^{2} + 234)^{2}$$
$11$ $$(T^{2} + 9360)^{2}$$
$13$ $$(T^{2} + 59904)^{2}$$
$17$ $$(T^{2} - 77440)^{2}$$
$19$ $$(T - 308)^{4}$$
$23$ $$(T^{2} - 171610)^{2}$$
$29$ $$(T^{2} + 37440)^{2}$$
$31$ $$(T - 32)^{4}$$
$37$ $$(T^{2} + 1651104)^{2}$$
$41$ $$(T^{2} + 4326660)^{2}$$
$43$ $$(T^{2} + 6683274)^{2}$$
$47$ $$(T^{2} - 5975290)^{2}$$
$53$ $$(T^{2} - 2007040)^{2}$$
$59$ $$(T^{2} + 15734160)^{2}$$
$61$ $$(T + 928)^{4}$$
$67$ $$(T^{2} + 6683274)^{2}$$
$71$ $$(T^{2} + 21565440)^{2}$$
$73$ $$(T^{2} + 17825184)^{2}$$
$79$ $$(T - 8)^{4}$$
$83$ $$(T^{2} - 19684090)^{2}$$
$89$ $$(T^{2} + 86261760)^{2}$$
$97$ $$(T^{2} + 6293664)^{2}$$