# Properties

 Label 405.4.e.q Level $405$ Weight $4$ Character orbit 405.e Analytic conductor $23.896$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 405.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.8957735523$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.84779568.3 Defining polynomial: $$x^{6} - x^{5} + 13 x^{4} - 4 x^{3} + 152 x^{2} - 96 x + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 135) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{2} + ( -7 + 7 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{4} + ( -5 + 5 \beta_{3} ) q^{5} + ( -\beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{7} + ( 29 + 7 \beta_{1} + 5 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{2} + ( -7 + 7 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{4} + ( -5 + 5 \beta_{3} ) q^{5} + ( -\beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{7} + ( 29 + 7 \beta_{1} + 5 \beta_{2} ) q^{8} + ( 10 + 5 \beta_{1} ) q^{10} + ( 9 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} - 9 \beta_{5} ) q^{11} + ( -5 + 5 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} ) q^{13} + ( -13 + 13 \beta_{3} + 5 \beta_{4} + 16 \beta_{5} ) q^{14} + ( -37 \beta_{1} - 9 \beta_{2} - 69 \beta_{3} + 9 \beta_{4} + 37 \beta_{5} ) q^{16} + ( 51 + 3 \beta_{1} - 5 \beta_{2} ) q^{17} + ( -28 - 33 \beta_{1} - \beta_{2} ) q^{19} + ( -15 \beta_{1} - 5 \beta_{2} - 35 \beta_{3} + 5 \beta_{4} + 15 \beta_{5} ) q^{20} + ( 95 - 95 \beta_{3} + 19 \beta_{4} + 37 \beta_{5} ) q^{22} + ( -85 + 85 \beta_{3} + 5 \beta_{4} + 25 \beta_{5} ) q^{23} -25 \beta_{3} q^{25} + ( 72 + 21 \beta_{1} + 10 \beta_{2} ) q^{26} + ( -124 - 36 \beta_{1} - 2 \beta_{2} ) q^{28} + ( -\beta_{1} - 25 \beta_{2} - 47 \beta_{3} + 25 \beta_{4} + \beta_{5} ) q^{29} + ( 39 - 39 \beta_{3} - 17 \beta_{4} + 19 \beta_{5} ) q^{31} + ( -295 + 295 \beta_{3} - 15 \beta_{4} - 95 \beta_{5} ) q^{32} + ( -29 \beta_{1} + 7 \beta_{2} - 145 \beta_{3} - 7 \beta_{4} + 29 \beta_{5} ) q^{34} + ( -10 + 5 \beta_{1} - 15 \beta_{2} ) q^{35} + ( -138 - 55 \beta_{1} + 25 \beta_{2} ) q^{37} + ( 66 \beta_{1} + 35 \beta_{2} + 417 \beta_{3} - 35 \beta_{4} - 66 \beta_{5} ) q^{38} + ( -145 + 145 \beta_{3} - 25 \beta_{4} - 35 \beta_{5} ) q^{40} + ( -188 + 188 \beta_{3} - 10 \beta_{4} + 26 \beta_{5} ) q^{41} + ( 17 \beta_{1} + 29 \beta_{2} + 281 \beta_{3} - 29 \beta_{4} - 17 \beta_{5} ) q^{43} + ( -535 - 155 \beta_{1} - 35 \beta_{2} ) q^{44} + ( -95 + 35 \beta_{1} - 35 \beta_{2} ) q^{46} + ( -123 \beta_{1} + 5 \beta_{2} + 9 \beta_{3} - 5 \beta_{4} + 123 \beta_{5} ) q^{47} + ( -161 + 161 \beta_{3} + 41 \beta_{4} + 53 \beta_{5} ) q^{49} + ( -50 + 50 \beta_{3} - 25 \beta_{5} ) q^{50} + ( -95 \beta_{1} - 25 \beta_{2} - 315 \beta_{3} + 25 \beta_{4} + 95 \beta_{5} ) q^{52} + ( -136 - 38 \beta_{1} + 30 \beta_{2} ) q^{53} + ( -15 - 45 \beta_{1} - 25 \beta_{2} ) q^{55} + ( 42 \beta_{1} + 744 \beta_{3} - 42 \beta_{5} ) q^{56} + ( -55 + 55 \beta_{3} - 51 \beta_{4} - 173 \beta_{5} ) q^{58} + ( -104 + 104 \beta_{3} + 68 \beta_{5} ) q^{59} + ( -43 \beta_{1} - 31 \beta_{2} + 26 \beta_{3} + 31 \beta_{4} + 43 \beta_{5} ) q^{61} + ( -321 + 27 \beta_{1} + 15 \beta_{2} ) q^{62} + ( 1053 + 169 \beta_{1} + 53 \beta_{2} ) q^{64} + ( -30 \beta_{1} - 10 \beta_{2} - 25 \beta_{3} + 10 \beta_{4} + 30 \beta_{5} ) q^{65} + ( -40 + 40 \beta_{3} + 3 \beta_{4} - 121 \beta_{5} ) q^{67} + ( -215 + 215 \beta_{3} - 55 \beta_{4} - 115 \beta_{5} ) q^{68} + ( 80 \beta_{1} + 25 \beta_{2} - 65 \beta_{3} - 25 \beta_{4} - 80 \beta_{5} ) q^{70} + ( 64 + 182 \beta_{1} - 30 \beta_{2} ) q^{71} + ( -306 + 59 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 68 \beta_{1} + 5 \beta_{2} + 931 \beta_{3} - 5 \beta_{4} - 68 \beta_{5} ) q^{74} + ( 1266 - 1266 \beta_{3} + 128 \beta_{4} + 394 \beta_{5} ) q^{76} + ( -658 + 658 \beta_{3} + 80 \beta_{4} - 104 \beta_{5} ) q^{77} + ( 38 \beta_{1} - 54 \beta_{2} - 343 \beta_{3} + 54 \beta_{4} - 38 \beta_{5} ) q^{79} + ( 345 + 185 \beta_{1} + 45 \beta_{2} ) q^{80} + ( 70 + 212 \beta_{1} - 6 \beta_{2} ) q^{82} + ( 24 \beta_{1} - 60 \beta_{2} - 102 \beta_{3} + 60 \beta_{4} - 24 \beta_{5} ) q^{83} + ( -255 + 255 \beta_{3} + 25 \beta_{4} - 15 \beta_{5} ) q^{85} + ( 691 - 691 \beta_{3} + 75 \beta_{4} + 443 \beta_{5} ) q^{86} + ( 569 \beta_{1} + 73 \beta_{2} + 1945 \beta_{3} - 73 \beta_{4} - 569 \beta_{5} ) q^{88} + ( 360 + 60 \beta_{2} ) q^{89} + ( -230 - 81 \beta_{1} + 23 \beta_{2} ) q^{91} + ( 35 \beta_{1} - 5 \beta_{2} + 415 \beta_{3} + 5 \beta_{4} - 35 \beta_{5} ) q^{92} + ( -1345 + 1345 \beta_{3} - 113 \beta_{4} - 89 \beta_{5} ) q^{94} + ( 140 - 140 \beta_{3} + 5 \beta_{4} + 165 \beta_{5} ) q^{95} + ( 113 \beta_{1} + \beta_{2} - 202 \beta_{3} - \beta_{4} - 113 \beta_{5} ) q^{97} + ( -179 - 97 \beta_{1} - 135 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 5 q^{2} - 17 q^{4} - 15 q^{5} + 4 q^{7} + 150 q^{8} + O(q^{10})$$ $$6 q - 5 q^{2} - 17 q^{4} - 15 q^{5} + 4 q^{7} + 150 q^{8} + 50 q^{10} - 5 q^{11} - 7 q^{13} - 60 q^{14} - 161 q^{16} + 310 q^{17} - 100 q^{19} - 85 q^{20} + 229 q^{22} - 285 q^{23} - 75 q^{25} + 370 q^{26} - 668 q^{28} - 115 q^{29} + 115 q^{31} - 775 q^{32} - 413 q^{34} - 40 q^{35} - 768 q^{37} + 1150 q^{38} - 375 q^{40} - 580 q^{41} + 797 q^{43} - 2830 q^{44} - 570 q^{46} + 145 q^{47} - 577 q^{49} - 125 q^{50} - 825 q^{52} - 800 q^{53} + 50 q^{55} + 2190 q^{56} + 59 q^{58} - 380 q^{59} + 152 q^{61} - 2010 q^{62} + 5874 q^{64} - 35 q^{65} - 2 q^{67} - 475 q^{68} - 300 q^{70} + 80 q^{71} - 1960 q^{73} + 2720 q^{74} + 3276 q^{76} - 1950 q^{77} - 1013 q^{79} + 1610 q^{80} + 8 q^{82} - 270 q^{83} - 775 q^{85} + 1555 q^{86} + 5193 q^{88} + 2040 q^{89} - 1264 q^{91} + 1215 q^{92} - 3833 q^{94} + 250 q^{95} - 720 q^{97} - 610 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 13 x^{4} - 4 x^{3} + 152 x^{2} - 96 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-11 \nu^{5} + 143 \nu^{4} + 21 \nu^{3} + 1672 \nu^{2} - 1056 \nu + 13728$$$$)/3760$$ $$\beta_{2}$$ $$=$$ $$($$$$-17 \nu^{5} + 221 \nu^{4} - 993 \nu^{3} + 2584 \nu^{2} - 1632 \nu + 17456$$$$)/3760$$ $$\beta_{3}$$ $$=$$ $$($$$$39 \nu^{5} - 37 \nu^{4} + 481 \nu^{3} + 182 \nu^{2} + 5624 \nu + 208$$$$)/3760$$ $$\beta_{4}$$ $$=$$ $$($$$$-61 \nu^{5} + 88 \nu^{4} - 1144 \nu^{3} + 1047 \nu^{2} - 13376 \nu + 8448$$$$)/1880$$ $$\beta_{5}$$ $$=$$ $$($$$$-31 \nu^{5} + 27 \nu^{4} - 351 \nu^{3} + 200 \nu^{2} - 4104 \nu + 2592$$$$)/752$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$5 \beta_{5} + \beta_{4} + 23 \beta_{3} - 23$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-11 \beta_{2} + 17 \beta_{1} - 11$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$-23 \beta_{5} - 3 \beta_{4} - 93 \beta_{3} + 3 \beta_{2} + 23 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$-233 \beta_{5} + 131 \beta_{4} - 227 \beta_{3} + 227$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$$.

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
136.1
 0.327167 − 0.566669i −1.66402 + 2.88216i 1.83685 − 3.18152i 0.327167 + 0.566669i −1.66402 − 2.88216i 1.83685 + 3.18152i
−2.72938 + 4.72742i 0 −10.8990 18.8776i −2.50000 4.33013i 0 5.90326 10.2247i 75.3201 0 27.2938
136.2 −1.06306 + 1.84127i 0 1.73981 + 3.01344i −2.50000 4.33013i 0 −15.3500 + 26.5870i −24.4070 0 10.6306
136.3 1.29244 2.23857i 0 0.659207 + 1.14178i −2.50000 4.33013i 0 11.4468 19.8264i 24.0869 0 −12.9244
271.1 −2.72938 4.72742i 0 −10.8990 + 18.8776i −2.50000 + 4.33013i 0 5.90326 + 10.2247i 75.3201 0 27.2938
271.2 −1.06306 1.84127i 0 1.73981 3.01344i −2.50000 + 4.33013i 0 −15.3500 26.5870i −24.4070 0 10.6306
271.3 1.29244 + 2.23857i 0 0.659207 1.14178i −2.50000 + 4.33013i 0 11.4468 + 19.8264i 24.0869 0 −12.9244
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 271.3 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 405.4.e.q 6
3.b odd 2 1 405.4.e.v 6
9.c even 3 1 135.4.a.h yes 3
9.c even 3 1 inner 405.4.e.q 6
9.d odd 6 1 135.4.a.e 3
9.d odd 6 1 405.4.e.v 6
36.f odd 6 1 2160.4.a.bq 3
36.h even 6 1 2160.4.a.bi 3
45.h odd 6 1 675.4.a.s 3
45.j even 6 1 675.4.a.p 3
45.k odd 12 2 675.4.b.n 6
45.l even 12 2 675.4.b.m 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.e 3 9.d odd 6 1
135.4.a.h yes 3 9.c even 3 1
405.4.e.q 6 1.a even 1 1 trivial
405.4.e.q 6 9.c even 3 1 inner
405.4.e.v 6 3.b odd 2 1
405.4.e.v 6 9.d odd 6 1
675.4.a.p 3 45.j even 6 1
675.4.a.s 3 45.h odd 6 1
675.4.b.m 6 45.l even 12 2
675.4.b.n 6 45.k odd 12 2
2160.4.a.bi 3 36.h even 6 1
2160.4.a.bq 3 36.f odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{6} + 5 T_{2}^{5} + 33 T_{2}^{4} + 20 T_{2}^{3} + 214 T_{2}^{2} + 240 T_{2} + 900$$ $$T_{7}^{6} - 4 T_{7}^{5} + 811 T_{7}^{4} - 13416 T_{7}^{3} + 665217 T_{7}^{2} - 6596910 T_{7} + 68856804$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$900 + 240 T + 214 T^{2} + 20 T^{3} + 33 T^{4} + 5 T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$( 25 + 5 T + T^{2} )^{3}$$
$7$ $$68856804 - 6596910 T + 665217 T^{2} - 13416 T^{3} + 811 T^{4} - 4 T^{5} + T^{6}$$
$11$ $$977187600 + 90278880 T + 8496844 T^{2} + 48080 T^{3} + 2913 T^{4} + 5 T^{5} + T^{6}$$
$13$ $$41280625 - 4940825 T + 546386 T^{2} - 18233 T^{3} + 818 T^{4} + 7 T^{5} + T^{6}$$
$17$ $$( -41760 + 5608 T - 155 T^{2} + T^{3} )^{2}$$
$19$ $$( -368012 - 16663 T + 50 T^{2} + T^{3} )^{2}$$
$23$ $$306362250000 - 8966700000 T + 420187500 T^{2} + 5724000 T^{3} + 65025 T^{4} + 285 T^{5} + T^{6}$$
$29$ $$41477979315600 + 305323638720 T + 2988157564 T^{2} + 7428760 T^{3} + 60633 T^{4} + 115 T^{5} + T^{6}$$
$31$ $$880414396416 - 27428502528 T + 962414784 T^{2} + 1485072 T^{3} + 42457 T^{4} - 115 T^{5} + T^{6}$$
$37$ $$( -22667198 - 67923 T + 384 T^{2} + T^{3} )^{2}$$
$41$ $$15345082598400 + 351818751360 T + 5794172944 T^{2} + 44256400 T^{3} + 246588 T^{4} + 580 T^{5} + T^{6}$$
$43$ $$28707478180096 + 763613038720 T + 24582225392 T^{2} - 124304312 T^{3} + 492689 T^{4} - 797 T^{5} + T^{6}$$
$47$ $$207021450297600 + 3584858772480 T + 59990424304 T^{2} + 64903520 T^{3} + 270177 T^{4} - 145 T^{5} + T^{6}$$
$53$ $$( -12658320 - 58172 T + 400 T^{2} + T^{3} )^{2}$$
$59$ $$27093274214400 + 142578647040 T + 2728267264 T^{2} + 1280 T^{3} + 171792 T^{4} + 380 T^{5} + T^{6}$$
$61$ $$25695430112356 - 443416548350 T + 8422373657 T^{2} + 3158068 T^{3} + 110579 T^{4} - 152 T^{5} + T^{6}$$
$67$ $$431363822490000 + 5067688430700 T + 59577050601 T^{2} + 41050602 T^{3} + 244003 T^{4} + 2 T^{5} + T^{6}$$
$71$ $$( 216071280 - 677372 T - 40 T^{2} + T^{3} )^{2}$$
$73$ $$( 16447954 + 264533 T + 980 T^{2} + T^{3} )^{2}$$
$79$ $$8207802819455625 - 4730518438875 T + 94501091250 T^{2} + 234087645 T^{3} + 973954 T^{4} + 1013 T^{5} + T^{6}$$
$83$ $$7146869573505600 + 25482469920480 T + 113684412384 T^{2} + 87692760 T^{3} + 374328 T^{4} + 270 T^{5} + T^{6}$$
$89$ $$( 125064000 + 46800 T - 1020 T^{2} + T^{3} )^{2}$$
$97$ $$752435512191364 + 948082376154 T + 20944602729 T^{2} + 29975756 T^{3} + 552963 T^{4} + 720 T^{5} + T^{6}$$