# Properties

 Label 405.4.b.c Level $405$ Weight $4$ Character orbit 405.b Analytic conductor $23.896$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 405.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.8957735523$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 44 x^{6} + 567 x^{4} + 2024 x^{2} + 1900$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -3 + \beta_{2} ) q^{4} + ( 2 - \beta_{3} ) q^{5} -\beta_{6} q^{7} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -3 + \beta_{2} ) q^{4} + ( 2 - \beta_{3} ) q^{5} -\beta_{6} q^{7} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} ) q^{8} + ( 1 + 3 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{10} + ( \beta_{2} + \beta_{7} ) q^{11} + ( -2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{13} + ( -5 + 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{14} + ( 1 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{16} + ( -8 \beta_{1} + 3 \beta_{5} ) q^{17} + ( 15 - 4 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{19} + ( -16 + 4 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} ) q^{20} + ( -9 \beta_{1} - 6 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{22} + ( 4 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - 5 \beta_{6} ) q^{23} + ( 3 - 12 \beta_{1} + 5 \beta_{2} - \beta_{3} - 6 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{25} + ( -3 + \beta_{2} - 6 \beta_{3} - 6 \beta_{4} - 5 \beta_{7} ) q^{26} + ( -24 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - 5 \beta_{6} ) q^{28} + ( 40 - 13 \beta_{2} - \beta_{3} - \beta_{4} + 5 \beta_{7} ) q^{29} + ( -54 - \beta_{2} + 8 \beta_{3} + 8 \beta_{4} - 5 \beta_{7} ) q^{31} + ( -16 \beta_{1} + 8 \beta_{3} - 8 \beta_{4} + 3 \beta_{5} + 9 \beta_{6} ) q^{32} + ( 82 - 14 \beta_{2} - 9 \beta_{3} - 9 \beta_{4} - 3 \beta_{7} ) q^{34} + ( 25 - 12 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - \beta_{5} - 10 \beta_{6} - \beta_{7} ) q^{35} + ( -12 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 13 \beta_{5} + 10 \beta_{6} ) q^{37} + ( 49 \beta_{1} + 13 \beta_{3} - 13 \beta_{4} - 10 \beta_{5} - 4 \beta_{6} ) q^{38} + ( -35 - 30 \beta_{1} + 10 \beta_{2} + 15 \beta_{4} + 5 \beta_{6} ) q^{40} + ( 63 - 16 \beta_{2} - 9 \beta_{3} - 9 \beta_{4} - 4 \beta_{7} ) q^{41} + ( -12 \beta_{1} + 15 \beta_{3} - 15 \beta_{4} + 9 \beta_{5} - 4 \beta_{6} ) q^{43} + ( 110 - 10 \beta_{2} - 14 \beta_{3} - 14 \beta_{4} - 7 \beta_{7} ) q^{44} + ( -73 + 17 \beta_{2} - 7 \beta_{3} - 7 \beta_{4} + \beta_{7} ) q^{46} + ( -12 \beta_{1} + 14 \beta_{3} - 14 \beta_{4} + 9 \beta_{6} ) q^{47} + ( 38 - 35 \beta_{2} + 23 \beta_{3} + 23 \beta_{4} - 5 \beta_{7} ) q^{49} + ( 144 - 23 \beta_{1} - 35 \beta_{2} + 5 \beta_{3} - 14 \beta_{4} - 4 \beta_{5} + 5 \beta_{6} + \beta_{7} ) q^{50} + ( 12 \beta_{1} + 20 \beta_{3} - 20 \beta_{4} - 13 \beta_{5} - 7 \beta_{6} ) q^{52} + ( 64 \beta_{1} + 14 \beta_{3} - 14 \beta_{4} - 4 \beta_{5} + 17 \beta_{6} ) q^{53} + ( -20 + 72 \beta_{1} + 25 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{55} + ( 195 + 13 \beta_{2} - 15 \beta_{3} - 15 \beta_{4} + \beta_{7} ) q^{56} + ( 123 \beta_{1} - 29 \beta_{3} + 29 \beta_{4} - 5 \beta_{5} - 13 \beta_{6} ) q^{58} + ( -139 + 76 \beta_{2} - 17 \beta_{3} - 17 \beta_{4} - 2 \beta_{7} ) q^{59} + ( -42 + 14 \beta_{2} - 31 \beta_{3} - 31 \beta_{4} - 2 \beta_{7} ) q^{61} + ( -53 \beta_{1} + 22 \beta_{3} - 22 \beta_{4} + 5 \beta_{5} - \beta_{6} ) q^{62} + ( 207 - 41 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} + 5 \beta_{7} ) q^{64} + ( -227 - 72 \beta_{1} + 30 \beta_{2} - 6 \beta_{3} - 11 \beta_{4} + 4 \beta_{5} - 5 \beta_{6} - 6 \beta_{7} ) q^{65} + ( -60 \beta_{1} - 26 \beta_{3} + 26 \beta_{4} - 8 \beta_{5} + 8 \beta_{6} ) q^{67} + ( 140 \beta_{1} + 27 \beta_{3} - 27 \beta_{4} - 14 \beta_{5} - 14 \beta_{6} ) q^{68} + ( 80 + 20 \beta_{2} + 5 \beta_{3} - 15 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} ) q^{70} + ( -216 - 3 \beta_{2} + 21 \beta_{7} ) q^{71} + ( -204 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 13 \beta_{5} - 10 \beta_{6} ) q^{73} + ( 160 - 68 \beta_{2} - 31 \beta_{3} - 31 \beta_{4} - 17 \beta_{7} ) q^{74} + ( -445 + 49 \beta_{2} + 31 \beta_{3} + 31 \beta_{4} + 20 \beta_{7} ) q^{76} + ( -36 \beta_{1} - 7 \beta_{3} + 7 \beta_{4} + 5 \beta_{5} - 22 \beta_{6} ) q^{77} + ( 332 - 32 \beta_{2} - 30 \beta_{3} - 30 \beta_{4} + 12 \beta_{7} ) q^{79} + ( 242 - 88 \beta_{1} - 5 \beta_{2} + 6 \beta_{3} - 19 \beta_{4} + \beta_{5} + 10 \beta_{6} + \beta_{7} ) q^{80} + ( 201 \beta_{1} + 33 \beta_{3} - 33 \beta_{4} - 42 \beta_{5} - 16 \beta_{6} ) q^{82} + ( 64 \beta_{1} - 19 \beta_{3} + 19 \beta_{4} + 41 \beta_{5} + 2 \beta_{6} ) q^{83} + ( -32 - 132 \beta_{1} + 45 \beta_{2} - 6 \beta_{3} + 29 \beta_{4} - \beta_{5} - 15 \beta_{6} - \beta_{7} ) q^{85} + ( 64 - 18 \beta_{2} - 16 \beta_{3} - 16 \beta_{4} + 21 \beta_{7} ) q^{86} + ( 150 \beta_{1} + 8 \beta_{3} - 8 \beta_{4} - 28 \beta_{5} - 2 \beta_{6} ) q^{88} + ( 134 + 5 \beta_{2} + 7 \beta_{3} + 7 \beta_{4} + 35 \beta_{7} ) q^{89} + ( -325 - 31 \beta_{2} + 13 \beta_{3} + 13 \beta_{4} - 13 \beta_{7} ) q^{91} + ( -148 \beta_{1} + 9 \beta_{3} - 9 \beta_{4} + 13 \beta_{5} - 23 \beta_{6} ) q^{92} + ( 149 - 39 \beta_{2} + 23 \beta_{3} + 23 \beta_{4} + 28 \beta_{7} ) q^{94} + ( 210 - 164 \beta_{1} - 55 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 8 \beta_{5} + 5 \beta_{6} - 7 \beta_{7} ) q^{95} + ( -48 \beta_{1} + 29 \beta_{3} - 29 \beta_{4} + 29 \beta_{5} + 34 \beta_{6} ) q^{97} + ( 247 \beta_{1} + 7 \beta_{3} - 7 \beta_{4} + \beta_{5} - 35 \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 24 q^{4} + 15 q^{5} + O(q^{10})$$ $$8 q - 24 q^{4} + 15 q^{5} + 7 q^{10} - 42 q^{14} + 4 q^{16} + 118 q^{19} - 129 q^{20} + 17 q^{25} - 36 q^{26} + 318 q^{29} - 416 q^{31} + 638 q^{34} + 192 q^{35} - 265 q^{40} + 486 q^{41} + 852 q^{44} - 598 q^{46} + 350 q^{49} + 1143 q^{50} - 162 q^{55} + 1530 q^{56} - 1146 q^{59} - 398 q^{61} + 1640 q^{64} - 1833 q^{65} + 630 q^{70} - 1728 q^{71} + 1218 q^{74} - 3498 q^{76} + 2596 q^{79} + 1923 q^{80} - 233 q^{85} + 480 q^{86} + 1086 q^{89} - 2574 q^{91} + 1238 q^{94} + 1674 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 44 x^{6} + 567 x^{4} + 2024 x^{2} + 1900$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 11$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} - 5 \nu^{6} - 117 \nu^{5} - 225 \nu^{4} - 1206 \nu^{3} - 2700 \nu^{2} - 2112 \nu - 5080$$$$)/180$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{7} - 5 \nu^{6} + 117 \nu^{5} - 225 \nu^{4} + 1206 \nu^{3} - 2700 \nu^{2} + 2112 \nu - 5080$$$$)/180$$ $$\beta_{5}$$ $$=$$ $$($$$$-4 \nu^{7} - 171 \nu^{5} - 1953 \nu^{3} - 3506 \nu$$$$)/90$$ $$\beta_{6}$$ $$=$$ $$($$$$4 \nu^{7} + 171 \nu^{5} + 2043 \nu^{3} + 5126 \nu$$$$)/90$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + 36 \nu^{4} + 333 \nu^{2} + 548$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 11$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} - 18 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{7} - 2 \beta_{4} - 2 \beta_{3} - 23 \beta_{2} + 201$$ $$\nu^{5}$$ $$=$$ $$-23 \beta_{6} - 29 \beta_{5} - 8 \beta_{4} + 8 \beta_{3} + 368 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$45 \beta_{7} + 72 \beta_{4} + 72 \beta_{3} + 495 \beta_{2} - 4121$$ $$\nu^{7}$$ $$=$$ $$495 \beta_{6} + 729 \beta_{5} + 342 \beta_{4} - 342 \beta_{3} - 7820 \beta_{1}$$

## Character values

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

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
244.1
 − 4.80346i − 3.99806i − 1.85698i − 1.22227i 1.22227i 1.85698i 3.99806i 4.80346i
4.80346i 0 −15.0732 8.38048 + 7.40051i 0 5.92463i 33.9759i 0 35.5481 40.2553i
244.2 3.99806i 0 −7.98447 −3.61252 10.5806i 0 7.79840i 0.0620886i 0 −42.3020 + 14.4431i
244.3 1.85698i 0 4.55164 −7.77825 + 8.03112i 0 1.02402i 23.3081i 0 14.9136 + 14.4440i
244.4 1.22227i 0 6.50606 10.5103 3.81233i 0 33.1668i 17.7303i 0 −4.65969 12.8464i
244.5 1.22227i 0 6.50606 10.5103 + 3.81233i 0 33.1668i 17.7303i 0 −4.65969 + 12.8464i
244.6 1.85698i 0 4.55164 −7.77825 8.03112i 0 1.02402i 23.3081i 0 14.9136 14.4440i
244.7 3.99806i 0 −7.98447 −3.61252 + 10.5806i 0 7.79840i 0.0620886i 0 −42.3020 14.4431i
244.8 4.80346i 0 −15.0732 8.38048 7.40051i 0 5.92463i 33.9759i 0 35.5481 + 40.2553i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 244.8 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 405.4.b.c yes 8
3.b odd 2 1 405.4.b.b 8
5.b even 2 1 inner 405.4.b.c yes 8
5.c odd 4 2 2025.4.a.bd 8
15.d odd 2 1 405.4.b.b 8
15.e even 4 2 2025.4.a.bc 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.b.b 8 3.b odd 2 1
405.4.b.b 8 15.d odd 2 1
405.4.b.c yes 8 1.a even 1 1 trivial
405.4.b.c yes 8 5.b even 2 1 inner
2025.4.a.bc 8 15.e even 4 2
2025.4.a.bd 8 5.c odd 4 2

## Hecke kernels

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

 $$T_{2}^{8} + 44 T_{2}^{6} + 567 T_{2}^{4} + 2024 T_{2}^{2} + 1900$$ $$T_{11}^{4} - 1755 T_{11}^{2} + 7038 T_{11} + 502920$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1900 + 2024 T^{2} + 567 T^{4} + 44 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$244140625 - 29296875 T + 1625000 T^{2} - 230625 T^{3} + 34350 T^{4} - 1845 T^{5} + 104 T^{6} - 15 T^{7} + T^{8}$$
$7$ $$2462400 + 2461104 T^{2} + 108900 T^{4} + 1197 T^{6} + T^{8}$$
$11$ $$( 502920 + 7038 T - 1755 T^{2} + T^{4} )^{2}$$
$13$ $$68838460416 + 7505488512 T^{2} + 13643379 T^{4} + 7092 T^{6} + T^{8}$$
$17$ $$24363576945664 + 118398768560 T^{2} + 78381864 T^{4} + 16211 T^{6} + T^{8}$$
$19$ $$( 2964730 - 113597 T - 9027 T^{2} - 59 T^{3} + T^{4} )^{2}$$
$23$ $$533400694426624 + 659629787264 T^{2} + 274020876 T^{4} + 40685 T^{6} + T^{8}$$
$29$ $$( -16354800 + 4022847 T - 48753 T^{2} - 159 T^{3} + T^{4} )^{2}$$
$31$ $$( -27072584 - 8679338 T - 48627 T^{2} + 208 T^{3} + T^{4} )^{2}$$
$37$ $$5375194471745553600 + 655413776331984 T^{2} + 23795411796 T^{4} + 280863 T^{6} + T^{8}$$
$41$ $$( 54333414 - 6557373 T - 103095 T^{2} - 243 T^{3} + T^{4} )^{2}$$
$43$ $$578275432952599296 + 1209275819455104 T^{2} + 46101716928 T^{4} + 406296 T^{6} + T^{8}$$
$47$ $$1865773798652194816 + 321649464215696 T^{2} + 15208554840 T^{4} + 249761 T^{6} + T^{8}$$
$53$ $$21\!\cdots\!24$$$$+ 9837833153758352 T^{2} + 143875639800 T^{4} + 684473 T^{6} + T^{8}$$
$59$ $$( 77712038244 - 412300269 T - 817749 T^{2} + 573 T^{3} + T^{4} )^{2}$$
$61$ $$( 37684911880 - 73735532 T - 449262 T^{2} + 199 T^{3} + T^{4} )^{2}$$
$67$ $$68\!\cdots\!56$$$$+ 52528406620483584 T^{2} + 429045004800 T^{4} + 1167084 T^{6} + T^{8}$$
$71$ $$( 17785914552 - 318162330 T - 370899 T^{2} + 864 T^{3} + T^{4} )^{2}$$
$73$ $$37\!\cdots\!04$$$$+ 403391598989898960 T^{2} + 1443625928244 T^{4} + 2070639 T^{6} + T^{8}$$
$79$ $$( -256210144256 + 791238016 T - 188712 T^{2} - 1298 T^{3} + T^{4} )^{2}$$
$83$ $$50\!\cdots\!44$$$$+ 1025821137189220352 T^{2} + 3158604717840 T^{4} + 3204488 T^{6} + T^{8}$$
$89$ $$( 438625777050 + 203849379 T - 1780695 T^{2} - 543 T^{3} + T^{4} )^{2}$$
$97$ $$31\!\cdots\!36$$$$+ 364061499235557120 T^{2} + 1434827084688 T^{4} + 2218680 T^{6} + T^{8}$$