# Properties

 Label 135.4.e.b Level $135$ Weight $4$ Character orbit 135.e Analytic conductor $7.965$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 135.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.96525785077$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.15759792.1 Defining polynomial: $$x^{6} - 3 x^{5} + 16 x^{4} - 27 x^{3} + 52 x^{2} - 39 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 45) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{4} ) q^{2} + ( -\beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{4} -5 \beta_{3} q^{5} + ( 14 - \beta_{1} + 14 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{7} + ( 8 + 2 \beta_{1} + \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{4} ) q^{2} + ( -\beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{4} -5 \beta_{3} q^{5} + ( 14 - \beta_{1} + 14 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{7} + ( 8 + 2 \beta_{1} + \beta_{2} ) q^{8} -5 \beta_{1} q^{10} + ( 11 \beta_{1} + 11 \beta_{4} + 3 \beta_{5} ) q^{11} + ( \beta_{2} + 18 \beta_{3} - 13 \beta_{4} - \beta_{5} ) q^{13} + ( -\beta_{2} + 17 \beta_{3} - 25 \beta_{4} + \beta_{5} ) q^{14} + ( 5 - 8 \beta_{1} + 5 \beta_{3} - 8 \beta_{4} + 6 \beta_{5} ) q^{16} + ( 56 - 3 \beta_{1} + \beta_{2} ) q^{17} + ( -58 + 7 \beta_{1} + 3 \beta_{2} ) q^{19} + ( 15 + 5 \beta_{1} + 15 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} ) q^{20} + ( 11 \beta_{2} - 112 \beta_{3} - 29 \beta_{4} - 11 \beta_{5} ) q^{22} + ( -12 \beta_{2} - 60 \beta_{3} - 3 \beta_{4} + 12 \beta_{5} ) q^{23} + ( -25 - 25 \beta_{3} ) q^{25} + ( -140 - \beta_{1} - 13 \beta_{2} ) q^{26} + ( -166 - 10 \beta_{1} - 9 \beta_{2} ) q^{28} + ( -107 - 4 \beta_{1} - 107 \beta_{3} - 4 \beta_{4} - 10 \beta_{5} ) q^{29} + ( -11 \beta_{2} - 138 \beta_{3} + 51 \beta_{4} + 11 \beta_{5} ) q^{31} + ( 42 \beta_{3} - 49 \beta_{4} ) q^{32} + ( 36 - 59 \beta_{1} + 36 \beta_{3} - 59 \beta_{4} + 3 \beta_{5} ) q^{34} + ( 70 - 5 \beta_{1} + 10 \beta_{2} ) q^{35} + ( 126 + 7 \beta_{1} + 17 \beta_{2} ) q^{37} + ( -68 + 33 \beta_{1} - 68 \beta_{3} + 33 \beta_{4} - 7 \beta_{5} ) q^{38} + ( 5 \beta_{2} - 40 \beta_{3} - 10 \beta_{4} - 5 \beta_{5} ) q^{40} + ( 23 \beta_{2} - 49 \beta_{3} - 17 \beta_{4} - 23 \beta_{5} ) q^{41} + ( 198 + 37 \beta_{1} + 198 \beta_{3} + 37 \beta_{4} - 29 \beta_{5} ) q^{43} + ( -286 - 119 \beta_{1} - 5 \beta_{2} ) q^{44} + ( -69 + 9 \beta_{1} - 3 \beta_{2} ) q^{46} + ( 202 + 54 \beta_{1} + 202 \beta_{3} + 54 \beta_{4} + 5 \beta_{5} ) q^{47} + ( -37 \beta_{2} + 152 \beta_{3} - 63 \beta_{4} + 37 \beta_{5} ) q^{49} + 25 \beta_{4} q^{50} + ( 116 + 115 \beta_{1} + 116 \beta_{3} + 115 \beta_{4} - 7 \beta_{5} ) q^{52} + ( 220 + 62 \beta_{1} + 8 \beta_{2} ) q^{53} + ( 55 \beta_{1} + 15 \beta_{2} ) q^{55} + ( 219 + 30 \beta_{1} + 219 \beta_{3} + 30 \beta_{4} + 18 \beta_{5} ) q^{56} + ( -4 \beta_{2} + 14 \beta_{3} + 171 \beta_{4} + 4 \beta_{5} ) q^{58} + ( 36 \beta_{2} + 72 \beta_{3} + 118 \beta_{4} - 36 \beta_{5} ) q^{59} + ( 473 + 45 \beta_{1} + 473 \beta_{3} + 45 \beta_{4} - 25 \beta_{5} ) q^{61} + ( 528 - 21 \beta_{1} + 51 \beta_{2} ) q^{62} + ( -499 - 71 \beta_{1} - \beta_{2} ) q^{64} + ( 90 - 65 \beta_{1} + 90 \beta_{3} - 65 \beta_{4} - 5 \beta_{5} ) q^{65} + ( 7 \beta_{2} - 630 \beta_{3} + 48 \beta_{4} - 7 \beta_{5} ) q^{67} + ( -51 \beta_{2} + 210 \beta_{3} + 29 \beta_{4} + 51 \beta_{5} ) q^{68} + ( 85 - 125 \beta_{1} + 85 \beta_{3} - 125 \beta_{4} + 5 \beta_{5} ) q^{70} + ( 2 + 7 \beta_{1} - 83 \beta_{2} ) q^{71} + ( -108 + 20 \beta_{1} - 64 \beta_{2} ) q^{73} + ( -26 - 235 \beta_{1} - 26 \beta_{3} - 235 \beta_{4} - 7 \beta_{5} ) q^{74} + ( 57 \beta_{2} + 80 \beta_{3} + 21 \beta_{4} - 57 \beta_{5} ) q^{76} + ( -\beta_{2} + 236 \beta_{3} + 239 \beta_{4} + \beta_{5} ) q^{77} + ( 90 + 48 \beta_{1} + 90 \beta_{3} + 48 \beta_{4} + 64 \beta_{5} ) q^{79} + ( 25 - 40 \beta_{1} + 30 \beta_{2} ) q^{80} + ( -118 - 204 \beta_{1} - 17 \beta_{2} ) q^{82} + ( -242 + 13 \beta_{1} - 242 \beta_{3} + 13 \beta_{4} - 118 \beta_{5} ) q^{83} + ( 5 \beta_{2} - 280 \beta_{3} + 15 \beta_{4} - 5 \beta_{5} ) q^{85} + ( 37 \beta_{2} - 494 \beta_{3} - 61 \beta_{4} - 37 \beta_{5} ) q^{86} + ( 398 + 203 \beta_{1} + 398 \beta_{3} + 203 \beta_{4} + 31 \beta_{5} ) q^{88} + ( -629 + 88 \beta_{1} + 80 \beta_{2} ) q^{89} + ( -332 + 331 \beta_{1} - 45 \beta_{2} ) q^{91} + ( -588 + 54 \beta_{1} - 588 \beta_{3} + 54 \beta_{4} + 87 \beta_{5} ) q^{92} + ( 54 \beta_{2} - 579 \beta_{3} - 286 \beta_{4} - 54 \beta_{5} ) q^{94} + ( 15 \beta_{2} + 290 \beta_{3} - 35 \beta_{4} - 15 \beta_{5} ) q^{95} + ( 120 - 58 \beta_{1} + 120 \beta_{3} - 58 \beta_{4} - 20 \beta_{5} ) q^{97} + ( -804 + 311 \beta_{1} - 63 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{2} - 11 q^{4} + 15 q^{5} + 43 q^{7} + 54 q^{8} + O(q^{10})$$ $$6 q - q^{2} - 11 q^{4} + 15 q^{5} + 43 q^{7} + 54 q^{8} - 10 q^{10} + 14 q^{11} - 40 q^{13} - 27 q^{14} + 13 q^{16} + 332 q^{17} - 328 q^{19} + 55 q^{20} + 376 q^{22} + 171 q^{23} - 75 q^{25} - 868 q^{26} - 1034 q^{28} - 335 q^{29} + 352 q^{31} - 77 q^{32} + 52 q^{34} + 430 q^{35} + 804 q^{37} - 178 q^{38} + 135 q^{40} + 187 q^{41} + 602 q^{43} - 1964 q^{44} - 402 q^{46} + 665 q^{47} - 430 q^{49} - 25 q^{50} + 456 q^{52} + 1460 q^{53} + 140 q^{55} + 705 q^{56} - 217 q^{58} - 298 q^{59} + 1439 q^{61} + 3228 q^{62} - 3138 q^{64} + 200 q^{65} + 1849 q^{67} - 710 q^{68} + 135 q^{70} - 140 q^{71} - 736 q^{73} - 320 q^{74} - 204 q^{76} - 948 q^{77} + 382 q^{79} + 130 q^{80} - 1150 q^{82} - 831 q^{83} + 830 q^{85} + 1580 q^{86} + 1428 q^{88} - 3438 q^{89} - 1420 q^{91} - 1623 q^{92} + 2077 q^{94} - 820 q^{95} + 282 q^{97} - 4328 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 16 x^{4} - 27 x^{3} + 52 x^{2} - 39 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{2} + \nu - 4$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} + 10 \nu^{2} - 9 \nu + 9$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 30 \nu^{3} - 40 \nu^{2} + 88 \nu - 39$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-8 \nu^{5} + 20 \nu^{4} - 117 \nu^{3} + 157 \nu^{2} - 334 \nu + 147$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$5 \nu^{5} - 12 \nu^{4} + 72 \nu^{3} - 92 \nu^{2} + 198 \nu - 81$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{5} - 4 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_{1} + 4$$$$)/9$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{5} - 4 \beta_{4} - \beta_{3} + \beta_{2} - 11 \beta_{1} - 32$$$$)/9$$ $$\nu^{3}$$ $$=$$ $$($$$$10 \beta_{5} + 29 \beta_{4} + 41 \beta_{3} - 5 \beta_{2} + \beta_{1} - 29$$$$)/9$$ $$\nu^{4}$$ $$=$$ $$($$$$22 \beta_{5} + 62 \beta_{4} + 83 \beta_{3} - 2 \beta_{2} + 94 \beta_{1} + 217$$$$)/9$$ $$\nu^{5}$$ $$=$$ $$($$$$-47 \beta_{5} - 184 \beta_{4} - 370 \beta_{3} + 46 \beta_{2} + 88 \beta_{1} + 337$$$$)/9$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$$.

 $$n$$ $$56$$ $$82$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$

## 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
 0.5 + 2.88506i 0.5 − 1.98116i 0.5 − 0.0378788i 0.5 − 2.88506i 0.5 + 1.98116i 0.5 + 0.0378788i
−2.28679 + 3.96084i 0 −6.45882 11.1870i 2.50000 + 4.33013i 0 10.0573 17.4197i 22.4912 0 −22.8679
46.2 −0.0874923 + 0.151541i 0 3.98469 + 6.90169i 2.50000 + 4.33013i 0 −4.23186 + 7.32979i −2.79440 0 −0.874923
46.3 1.87428 3.24635i 0 −3.02587 5.24096i 2.50000 + 4.33013i 0 15.6746 27.1492i 7.30318 0 18.7428
91.1 −2.28679 3.96084i 0 −6.45882 + 11.1870i 2.50000 4.33013i 0 10.0573 + 17.4197i 22.4912 0 −22.8679
91.2 −0.0874923 0.151541i 0 3.98469 6.90169i 2.50000 4.33013i 0 −4.23186 7.32979i −2.79440 0 −0.874923
91.3 1.87428 + 3.24635i 0 −3.02587 + 5.24096i 2.50000 4.33013i 0 15.6746 + 27.1492i 7.30318 0 18.7428
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.4.e.b 6
3.b odd 2 1 45.4.e.b 6
9.c even 3 1 inner 135.4.e.b 6
9.c even 3 1 405.4.a.j 3
9.d odd 6 1 45.4.e.b 6
9.d odd 6 1 405.4.a.h 3
15.d odd 2 1 225.4.e.c 6
15.e even 4 2 225.4.k.c 12
45.h odd 6 1 225.4.e.c 6
45.h odd 6 1 2025.4.a.s 3
45.j even 6 1 2025.4.a.q 3
45.l even 12 2 225.4.k.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.b 6 3.b odd 2 1
45.4.e.b 6 9.d odd 6 1
135.4.e.b 6 1.a even 1 1 trivial
135.4.e.b 6 9.c even 3 1 inner
225.4.e.c 6 15.d odd 2 1
225.4.e.c 6 45.h odd 6 1
225.4.k.c 12 15.e even 4 2
225.4.k.c 12 45.l even 12 2
405.4.a.h 3 9.d odd 6 1
405.4.a.j 3 9.c even 3 1
2025.4.a.q 3 45.j even 6 1
2025.4.a.s 3 45.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + T_{2}^{5} + 18 T_{2}^{4} - 11 T_{2}^{3} + 292 T_{2}^{2} + 51 T_{2} + 9$$ acting on $$S_{4}^{\mathrm{new}}(135, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 51 T + 292 T^{2} - 11 T^{3} + 18 T^{4} + T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$( 25 - 5 T + T^{2} )^{3}$$
$7$ $$28483569 + 1040715 T + 267516 T^{2} - 19059 T^{3} + 1654 T^{4} - 43 T^{5} + T^{6}$$
$11$ $$1896428304 + 122631168 T + 7320184 T^{2} + 126520 T^{3} + 3012 T^{4} - 14 T^{5} + T^{6}$$
$13$ $$5679732496 + 184792528 T + 9026864 T^{2} + 52648 T^{3} + 4052 T^{4} + 40 T^{5} + T^{6}$$
$17$ $$( -156324 + 8920 T - 166 T^{2} + T^{3} )^{2}$$
$19$ $$( 57316 + 7292 T + 164 T^{2} + T^{3} )^{2}$$
$23$ $$3746541681 - 295823097 T + 33824628 T^{2} + 704025 T^{3} + 34074 T^{4} - 171 T^{5} + T^{6}$$
$29$ $$11463342489 - 2926248177 T + 782851006 T^{2} + 9370019 T^{3} + 84894 T^{4} + 335 T^{5} + T^{6}$$
$31$ $$97238137683600 - 137382616080 T + 3665151504 T^{2} - 14817816 T^{3} + 137836 T^{4} - 352 T^{5} + T^{6}$$
$37$ $$( 3335284 + 24708 T - 402 T^{2} + T^{3} )^{2}$$
$41$ $$52247959475625 - 322359380175 T + 3340579834 T^{2} - 6116911 T^{3} + 79566 T^{4} - 187 T^{5} + T^{6}$$
$43$ $$238533321586576 + 153765680944 T + 9396725384 T^{2} - 36882560 T^{3} + 352448 T^{4} - 602 T^{5} + T^{6}$$
$47$ $$6187571675289 - 237009867723 T + 7424292766 T^{2} - 58386899 T^{3} + 346944 T^{4} - 665 T^{5} + T^{6}$$
$53$ $$( -3250536 + 106300 T - 730 T^{2} + T^{3} )^{2}$$
$59$ $$16224618881802816 + 45173097261024 T + 163730383744 T^{2} + 149067880 T^{3} + 443448 T^{4} + 298 T^{5} + T^{6}$$
$61$ $$3092861048569009 - 32773423076579 T + 267254918066 T^{2} - 736785779 T^{3} + 1481414 T^{4} - 1439 T^{5} + T^{6}$$
$67$ $$43587484566793281 - 228333683246643 T + 810102262338 T^{2} - 1604656455 T^{3} + 2325124 T^{4} - 1849 T^{5} + T^{6}$$
$71$ $$( -223775052 - 685460 T + 70 T^{2} + T^{3} )^{2}$$
$73$ $$( -134927744 - 372928 T + 368 T^{2} + T^{3} )^{2}$$
$79$ $$19133137244437056 - 53658650075616 T + 203324256864 T^{2} - 128458200 T^{3} + 533848 T^{4} - 382 T^{5} + T^{6}$$
$83$ $$913637410143880041 + 1109708868397833 T + 2142164521980 T^{2} + 946919079 T^{3} + 1851534 T^{4} + 831 T^{5} + T^{6}$$
$89$ $$( -125506395 + 238491 T + 1719 T^{2} + T^{3} )^{2}$$
$97$ $$285551326213696 - 1143471728352 T + 9344268672 T^{2} - 14714152 T^{3} + 147192 T^{4} - 282 T^{5} + T^{6}$$