# Properties

 Label 342.5.d.a Level $342$ Weight $5$ Character orbit 342.d Analytic conductor $35.353$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 342.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.3525273747$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 450 x^{6} + 68229 x^{4} + 4001228 x^{2} + 77475204$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 38) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} -8 q^{4} + ( -2 + \beta_{6} ) q^{5} + ( -20 + \beta_{5} ) q^{7} -8 \beta_{2} q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} -8 q^{4} + ( -2 + \beta_{6} ) q^{5} + ( -20 + \beta_{5} ) q^{7} -8 \beta_{2} q^{8} + ( -3 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{10} + ( 2 + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{11} + ( -7 \beta_{1} - 6 \beta_{2} - \beta_{3} - \beta_{7} ) q^{13} + ( -\beta_{1} - 21 \beta_{2} + \beta_{7} ) q^{14} + 64 q^{16} + ( -64 - 7 \beta_{4} - \beta_{5} ) q^{17} + ( -1 - 14 \beta_{1} - 57 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{19} + ( 16 - 8 \beta_{6} ) q^{20} + ( 5 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{7} ) q^{22} + ( 48 - 14 \beta_{4} + \beta_{5} - 7 \beta_{6} ) q^{23} + ( 431 + 12 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{25} + ( 28 + 11 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} ) q^{26} + ( 160 - 8 \beta_{5} ) q^{28} + ( 15 \beta_{1} + 24 \beta_{2} - 9 \beta_{3} + 3 \beta_{7} ) q^{29} + ( 30 \beta_{1} - 222 \beta_{2} ) q^{31} + 64 \beta_{2} q^{32} + ( -55 \beta_{1} - 91 \beta_{2} - \beta_{7} ) q^{34} + ( -128 + 58 \beta_{4} - 2 \beta_{5} - 9 \beta_{6} ) q^{35} + ( 76 \beta_{1} + 246 \beta_{2} - 8 \beta_{3} - 2 \beta_{7} ) q^{37} + ( 404 - 32 \beta_{1} - 18 \beta_{2} + 3 \beta_{3} + 19 \beta_{4} - 8 \beta_{5} + 16 \beta_{6} - \beta_{7} ) q^{38} + ( 24 \beta_{1} + 32 \beta_{2} - 8 \beta_{3} ) q^{40} + ( 42 \beta_{1} - 234 \beta_{2} - 12 \beta_{3} ) q^{41} + ( -1078 + 48 \beta_{4} + 15 \beta_{6} ) q^{43} + ( -16 - 16 \beta_{4} - 16 \beta_{5} - 24 \beta_{6} ) q^{44} + ( -92 \beta_{1} + 5 \beta_{2} - 7 \beta_{3} + \beta_{7} ) q^{46} + ( -398 + 24 \beta_{4} - 18 \beta_{5} + 31 \beta_{6} ) q^{47} + ( 1137 - 57 \beta_{4} - 39 \beta_{5} - 24 \beta_{6} ) q^{49} + ( 107 \beta_{1} + 487 \beta_{2} - 3 \beta_{3} - 2 \beta_{7} ) q^{50} + ( 56 \beta_{1} + 48 \beta_{2} + 8 \beta_{3} + 8 \beta_{7} ) q^{52} + ( -183 \beta_{1} - 114 \beta_{2} + 3 \beta_{3} + 15 \beta_{7} ) q^{53} + ( 2160 + 144 \beta_{4} + 22 \beta_{5} + 45 \beta_{6} ) q^{55} + ( 8 \beta_{1} + 168 \beta_{2} - 8 \beta_{7} ) q^{56} + ( -108 + 9 \beta_{4} - 24 \beta_{5} + 72 \beta_{6} ) q^{58} + ( 159 \beta_{1} - 459 \beta_{2} - 6 \beta_{3} + 6 \beta_{7} ) q^{59} + ( 178 + 150 \beta_{4} - 20 \beta_{5} + 75 \beta_{6} ) q^{61} + ( 1896 - 30 \beta_{4} ) q^{62} -512 q^{64} + ( -408 \beta_{1} - 606 \beta_{2} - 30 \beta_{3} - 18 \beta_{7} ) q^{65} + ( 275 \beta_{1} + 1017 \beta_{2} - 22 \beta_{3} + 26 \beta_{7} ) q^{67} + ( 512 + 56 \beta_{4} + 8 \beta_{5} ) q^{68} + ( 493 \beta_{1} + 124 \beta_{2} - 9 \beta_{3} - 2 \beta_{7} ) q^{70} + ( 66 \beta_{1} - 912 \beta_{2} + 66 \beta_{3} + 18 \beta_{7} ) q^{71} + ( 2972 - 69 \beta_{4} + 39 \beta_{5} + 150 \beta_{6} ) q^{73} + ( -1624 - 50 \beta_{4} + 16 \beta_{5} + 64 \beta_{6} ) q^{74} + ( 8 + 112 \beta_{1} + 456 \beta_{2} + 16 \beta_{3} + 24 \beta_{4} + 8 \beta_{5} - 24 \beta_{6} - 8 \beta_{7} ) q^{76} + ( 5564 - 6 \beta_{4} - 60 \beta_{5} + 5 \beta_{6} ) q^{77} + ( -218 \beta_{1} - 852 \beta_{2} - 50 \beta_{3} - 2 \beta_{7} ) q^{79} + ( -128 + 64 \beta_{6} ) q^{80} + ( 2088 - 6 \beta_{4} + 96 \beta_{6} ) q^{82} + ( 1362 + 64 \beta_{4} + 70 \beta_{5} + 158 \beta_{6} ) q^{83} + ( 2592 - 30 \beta_{4} - 110 \beta_{5} - 159 \beta_{6} ) q^{85} + ( 339 \beta_{1} - 916 \beta_{2} + 15 \beta_{3} ) q^{86} + ( -40 \beta_{1} - 16 \beta_{2} - 24 \beta_{3} - 16 \beta_{7} ) q^{88} + ( 480 \beta_{1} + 2346 \beta_{2} - 6 \beta_{3} + 18 \beta_{7} ) q^{89} + ( 389 \beta_{1} - 2595 \beta_{2} - 22 \beta_{3} + 32 \beta_{7} ) q^{91} + ( -384 + 112 \beta_{4} - 8 \beta_{5} + 56 \beta_{6} ) q^{92} + ( 117 \beta_{1} - 346 \beta_{2} + 31 \beta_{3} - 18 \beta_{7} ) q^{94} + ( 4310 + 582 \beta_{1} - 1050 \beta_{2} - 66 \beta_{3} - 10 \beta_{4} - 52 \beta_{5} - 51 \beta_{6} - 36 \beta_{7} ) q^{95} + ( -194 \beta_{1} + 2946 \beta_{2} + 64 \beta_{3} + 4 \beta_{7} ) q^{97} + ( -345 \beta_{1} + 996 \beta_{2} - 24 \beta_{3} - 39 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 64q^{4} - 18q^{5} - 162q^{7} + O(q^{10})$$ $$8q - 64q^{4} - 18q^{5} - 162q^{7} + 6q^{11} + 512q^{16} - 510q^{17} - 12q^{19} + 144q^{20} + 396q^{23} + 3458q^{25} + 192q^{26} + 1296q^{28} - 1002q^{35} + 3216q^{38} - 8654q^{43} - 48q^{44} - 3210q^{47} + 9222q^{49} + 17146q^{55} - 960q^{58} + 1314q^{61} + 15168q^{62} - 4096q^{64} + 4080q^{68} + 23398q^{73} - 13152q^{74} + 96q^{76} + 44622q^{77} - 1152q^{80} + 16512q^{82} + 10440q^{83} + 21274q^{85} - 3168q^{92} + 34686q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 450 x^{6} + 68229 x^{4} + 4001228 x^{2} + 77475204$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} - 450 \nu^{5} - 59427 \nu^{3} - 770894 \nu$$$$)/1249884$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 450 \nu^{5} + 59427 \nu^{3} + 2020778 \nu$$$$)/624942$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 2484 \nu^{5} + 1017351 \nu^{3} + 84215350 \nu$$$$)/2499768$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + 225 \nu^{2} + 8802$$$$)/71$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} - 352 \nu^{4} - 33685 \nu^{2} - 733602$$$$)/3408$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + 352 \nu^{4} + 37093 \nu^{2} + 1115298$$$$)/3408$$ $$\beta_{7}$$ $$=$$ $$($$$$151 \nu^{7} + 55236 \nu^{5} + 5973951 \nu^{3} + 176424854 \nu$$$$)/833256$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} - 112$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + 13 \beta_{3} - 257 \beta_{2} - 294 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-225 \beta_{6} - 225 \beta_{5} + 71 \beta_{4} + 16398$$ $$\nu^{5}$$ $$=$$ $$($$$$-367 \beta_{7} - 3067 \beta_{3} + 65353 \beta_{2} + 49114 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$45515 \beta_{6} + 42107 \beta_{5} - 24992 \beta_{4} - 2732978$$ $$\nu^{7}$$ $$=$$ $$($$$$105723 \beta_{7} + 607599 \beta_{3} - 14907005 \beta_{2} - 8671318 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$325$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 − 13.9305i 8.07810i 12.2418i − 6.38941i 13.9305i − 8.07810i − 12.2418i 6.38941i
2.82843i 0 −8.00000 −41.6240 0 −62.4342 22.6274i 0 117.730i
37.2 2.82843i 0 −8.00000 −26.7027 0 51.4469 22.6274i 0 75.5266i
37.3 2.82843i 0 −8.00000 26.2296 0 −86.0910 22.6274i 0 74.1886i
37.4 2.82843i 0 −8.00000 33.0971 0 16.0783 22.6274i 0 93.6127i
37.5 2.82843i 0 −8.00000 −41.6240 0 −62.4342 22.6274i 0 117.730i
37.6 2.82843i 0 −8.00000 −26.7027 0 51.4469 22.6274i 0 75.5266i
37.7 2.82843i 0 −8.00000 26.2296 0 −86.0910 22.6274i 0 74.1886i
37.8 2.82843i 0 −8.00000 33.0971 0 16.0783 22.6274i 0 93.6127i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.5.d.a 8
3.b odd 2 1 38.5.b.a 8
12.b even 2 1 304.5.e.e 8
19.b odd 2 1 inner 342.5.d.a 8
57.d even 2 1 38.5.b.a 8
228.b odd 2 1 304.5.e.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.5.b.a 8 3.b odd 2 1
38.5.b.a 8 57.d even 2 1
304.5.e.e 8 12.b even 2 1
304.5.e.e 8 228.b odd 2 1
342.5.d.a 8 1.a even 1 1 trivial
342.5.d.a 8 19.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 9 T_{5}^{3} - 2074 T_{5}^{2} - 6624 T_{5} + 964896$$ acting on $$S_{5}^{\mathrm{new}}(342, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 8 + T^{2} )^{4}$$
$3$ $$T^{8}$$
$5$ $$( 964896 - 6624 T - 2074 T^{2} + 9 T^{3} + T^{4} )^{2}$$
$7$ $$( 4446118 - 240093 T - 3827 T^{2} + 81 T^{3} + T^{4} )^{2}$$
$11$ $$( 76301016 - 2109492 T - 37690 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$13$ $$57445091172827136 + 34129514947104 T^{2} + 5282600121 T^{4} + 140586 T^{6} + T^{8}$$
$17$ $$( 403423998 - 14985243 T - 67555 T^{2} + 255 T^{3} + T^{4} )^{2}$$
$19$ $$28\!\cdots\!81$$$$+ 26559779028793932 T + 5155530396725960 T^{2} - 2634460387644 T^{3} + 46020337038 T^{4} - 20215164 T^{5} + 303560 T^{6} + 12 T^{7} + T^{8}$$
$23$ $$( 18350234964 + 14380020 T - 326059 T^{2} - 198 T^{3} + T^{4} )^{2}$$
$29$ $$26\!\cdots\!00$$$$+ 1941717474110086560 T^{2} + 4545080369001 T^{4} + 3774042 T^{6} + T^{8}$$
$31$ $$13\!\cdots\!16$$$$+ 290480962963064832 T^{2} + 1470564161424 T^{4} + 2202408 T^{6} + T^{8}$$
$37$ $$61\!\cdots\!36$$$$+ 68770887827876352 T^{2} + 1219007010960 T^{4} + 6003528 T^{6} + T^{8}$$
$41$ $$14\!\cdots\!04$$$$+ 5375602779897139200 T^{2} + 14417398812816 T^{4} + 7187688 T^{6} + T^{8}$$
$43$ $$( 407751532960 - 2817187520 T + 3365214 T^{2} + 4327 T^{3} + T^{4} )^{2}$$
$47$ $$( -98774187816 + 1482029532 T - 3671746 T^{2} + 1605 T^{3} + T^{4} )^{2}$$
$53$ $$82\!\cdots\!84$$$$+$$$$33\!\cdots\!60$$$$T^{2} + 356018538506121 T^{4} + 36847098 T^{6} + T^{8}$$
$59$ $$20\!\cdots\!00$$$$+ 17355507167396547360 T^{2} + 42239279002761 T^{4} + 25952346 T^{6} + T^{8}$$
$61$ $$( 195962902247296 + 64039717152 T - 41732066 T^{2} - 657 T^{3} + T^{4} )^{2}$$
$67$ $$13\!\cdots\!96$$$$+$$$$17\!\cdots\!60$$$$T^{2} + 7964319437470377 T^{4} + 149621370 T^{6} + T^{8}$$
$71$ $$22\!\cdots\!56$$$$+$$$$26\!\cdots\!40$$$$T^{2} + 10801833589824912 T^{4} + 175804200 T^{6} + T^{8}$$
$73$ $$( -1571012619241250 + 629910296275 T - 25007871 T^{2} - 11699 T^{3} + T^{4} )^{2}$$
$79$ $$50\!\cdots\!76$$$$+$$$$58\!\cdots\!16$$$$T^{2} + 1855389240525456 T^{4} + 113259624 T^{6} + T^{8}$$
$83$ $$( 57645106800768 + 4083728832 T - 57758860 T^{2} - 5220 T^{3} + T^{4} )^{2}$$
$89$ $$70\!\cdots\!96$$$$+$$$$80\!\cdots\!60$$$$T^{2} + 25689634455831552 T^{4} + 282427200 T^{6} + T^{8}$$
$97$ $$11\!\cdots\!56$$$$+$$$$10\!\cdots\!88$$$$T^{2} + 51581376150900624 T^{4} + 436011432 T^{6} + T^{8}$$