# Properties

 Label 304.5.e.e Level $304$ Weight $5$ Character orbit 304.e Analytic conductor $31.424$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 304.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.4244687775$$ 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{7}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( 2 - \beta_{5} ) q^{5} + ( 20 - \beta_{4} ) q^{7} + ( -33 - \beta_{3} + \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( 2 - \beta_{5} ) q^{5} + ( 20 - \beta_{4} ) q^{7} + ( -33 - \beta_{3} + \beta_{4} + \beta_{5} ) q^{9} + ( 2 + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{11} + ( 7 \beta_{1} + \beta_{6} ) q^{13} + ( -12 \beta_{1} - 3 \beta_{2} + 2 \beta_{6} ) q^{15} + ( 64 + 7 \beta_{3} + \beta_{4} ) q^{17} + ( 1 - 14 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{19} + ( -47 \beta_{1} + \beta_{2} + \beta_{6} - 2 \beta_{7} ) q^{21} + ( 48 - 14 \beta_{3} + \beta_{4} - 7 \beta_{5} ) q^{23} + ( 431 + 12 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{25} + ( -3 \beta_{1} + 11 \beta_{2} - 4 \beta_{6} + 2 \beta_{7} ) q^{27} + ( 15 \beta_{1} - \beta_{2} - \beta_{6} + 4 \beta_{7} ) q^{29} + ( 30 \beta_{1} - 21 \beta_{2} ) q^{31} + ( 66 \beta_{1} - 11 \beta_{2} - 6 \beta_{6} + 4 \beta_{7} ) q^{33} + ( -128 + 58 \beta_{3} - 2 \beta_{4} - 9 \beta_{5} ) q^{35} + ( -76 \beta_{1} - 13 \beta_{2} + 4 \beta_{6} - 2 \beta_{7} ) q^{37} + ( 834 - 6 \beta_{3} - 3 \beta_{4} + 45 \beta_{5} ) q^{39} + ( 42 \beta_{1} - 25 \beta_{2} - 4 \beta_{6} + 4 \beta_{7} ) q^{41} + ( 1078 - 48 \beta_{3} - 15 \beta_{5} ) q^{43} + ( -1278 - 74 \beta_{3} + 20 \beta_{4} + 35 \beta_{5} ) q^{45} + ( -398 + 24 \beta_{3} - 18 \beta_{4} + 31 \beta_{5} ) q^{47} + ( 1137 - 57 \beta_{3} - 39 \beta_{4} - 24 \beta_{5} ) q^{49} + ( -93 \beta_{1} - 64 \beta_{2} + 6 \beta_{6} + 2 \beta_{7} ) q^{51} + ( -183 \beta_{1} + 5 \beta_{2} + 11 \beta_{6} + 4 \beta_{7} ) q^{53} + ( -2160 - 144 \beta_{3} - 22 \beta_{4} - 45 \beta_{5} ) q^{55} + ( -1758 - 28 \beta_{1} - 37 \beta_{2} - 69 \beta_{3} - 21 \beta_{4} + 27 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} ) q^{57} + ( -159 \beta_{1} + 53 \beta_{2} - 2 \beta_{6} - 4 \beta_{7} ) q^{59} + ( 178 + 150 \beta_{3} - 20 \beta_{4} + 75 \beta_{5} ) q^{61} + ( -3714 - 34 \beta_{3} + 40 \beta_{4} + 73 \beta_{5} ) q^{63} + ( -408 \beta_{1} - 20 \beta_{2} - 22 \beta_{6} + 4 \beta_{7} ) q^{65} + ( 275 \beta_{1} + 56 \beta_{2} + 10 \beta_{6} + 16 \beta_{7} ) q^{67} + ( 21 \beta_{1} + 104 \beta_{2} - \beta_{6} + 2 \beta_{7} ) q^{69} + ( -66 \beta_{1} + 72 \beta_{2} - 34 \beta_{6} + 16 \beta_{7} ) q^{71} + ( 2972 - 69 \beta_{3} + 39 \beta_{4} + 150 \beta_{5} ) q^{73} + ( -611 \beta_{1} - 115 \beta_{2} + 20 \beta_{6} - 4 \beta_{7} ) q^{75} + ( -5564 + 6 \beta_{3} + 60 \beta_{4} - 5 \beta_{5} ) q^{77} + ( -218 \beta_{1} - 61 \beta_{2} - 18 \beta_{6} + 16 \beta_{7} ) q^{79} + ( -2571 + 86 \beta_{3} - 2 \beta_{4} - 98 \beta_{5} ) q^{81} + ( 1362 + 64 \beta_{3} + 70 \beta_{4} + 158 \beta_{5} ) q^{83} + ( 2592 - 30 \beta_{3} - 110 \beta_{4} - 159 \beta_{5} ) q^{85} + ( 1746 - 12 \beta_{3} - 159 \beta_{4} - 15 \beta_{5} ) q^{87} + ( 480 \beta_{1} + 153 \beta_{2} + 10 \beta_{6} + 8 \beta_{7} ) q^{89} + ( 389 \beta_{1} - 255 \beta_{2} + 14 \beta_{6} + 18 \beta_{7} ) q^{91} + ( 2412 - 222 \beta_{3} - 30 \beta_{4} - 30 \beta_{5} ) q^{93} + ( 4310 - 582 \beta_{1} + 144 \beta_{2} - 10 \beta_{3} - 52 \beta_{4} - 51 \beta_{5} + 46 \beta_{6} - 10 \beta_{7} ) q^{95} + ( 194 \beta_{1} - 272 \beta_{2} - 24 \beta_{6} + 20 \beta_{7} ) q^{97} + ( 7062 + 146 \beta_{3} - 68 \beta_{4} - 83 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 18q^{5} + 162q^{7} - 268q^{9} + O(q^{10})$$ $$8q + 18q^{5} + 162q^{7} - 268q^{9} + 6q^{11} + 510q^{17} + 12q^{19} + 396q^{23} + 3458q^{25} - 1002q^{35} + 6588q^{39} + 8654q^{43} - 10334q^{45} - 3210q^{47} + 9222q^{49} - 17146q^{55} - 14076q^{57} + 1314q^{61} - 29938q^{63} + 23398q^{73} - 44622q^{77} - 20368q^{81} + 10440q^{83} + 21274q^{85} + 14316q^{87} + 19416q^{93} + 34686q^{95} + 56798q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} - 450 \nu^{5} - 59427 \nu^{3} - 3270662 \nu$$$$)/1249884$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{7} - 900 \nu^{5} - 118854 \nu^{3} - 4041556 \nu$$$$)/104157$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{4} + 225 \nu^{2} + 8802$$$$)/71$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} - 352 \nu^{4} - 33685 \nu^{2} - 733602$$$$)/3408$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} + 352 \nu^{4} + 37093 \nu^{2} + 1115298$$$$)/3408$$ $$\beta_{6}$$ $$=$$ $$($$$$-56 \nu^{7} - 20799 \nu^{5} - 2337687 \nu^{3} - 75675850 \nu$$$$)/312471$$ $$\beta_{7}$$ $$=$$ $$($$$$-475 \nu^{7} - 169740 \nu^{5} - 17075691 \nu^{3} - 401259422 \nu$$$$)/2499768$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 24 \beta_{1}$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} - 112$$ $$\nu^{3}$$ $$=$$ $$($$$$24 \beta_{7} - 30 \beta_{6} - 31 \beta_{2} + 1764 \beta_{1}$$$$)/12$$ $$\nu^{4}$$ $$=$$ $$-225 \beta_{5} - 225 \beta_{4} + 71 \beta_{3} + 16398$$ $$\nu^{5}$$ $$=$$ $$($$$$-2700 \beta_{7} + 3801 \beta_{6} - 2618 \beta_{2} - 147342 \beta_{1}$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$45515 \beta_{5} + 42107 \beta_{4} - 24992 \beta_{3} - 2732978$$ $$\nu^{7}$$ $$=$$ $$($$$$501876 \beta_{7} - 819045 \beta_{6} + 1281553 \beta_{2} + 26013954 \beta_{1}$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 − 13.9305i − 12.2418i − 6.38941i − 8.07810i 8.07810i 6.38941i 12.2418i 13.9305i
0 15.3447i 0 41.6240 0 62.4342 0 −154.460 0
113.2 0 10.8276i 0 −26.2296 0 86.0910 0 −36.2364 0
113.3 0 7.80363i 0 −33.0971 0 −16.0783 0 20.1034 0
113.4 0 6.66389i 0 26.7027 0 −51.4469 0 36.5926 0
113.5 0 6.66389i 0 26.7027 0 −51.4469 0 36.5926 0
113.6 0 7.80363i 0 −33.0971 0 −16.0783 0 20.1034 0
113.7 0 10.8276i 0 −26.2296 0 86.0910 0 −36.2364 0
113.8 0 15.3447i 0 41.6240 0 62.4342 0 −154.460 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 113.8 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 304.5.e.e 8
4.b odd 2 1 38.5.b.a 8
12.b even 2 1 342.5.d.a 8
19.b odd 2 1 inner 304.5.e.e 8
76.d even 2 1 38.5.b.a 8
228.b odd 2 1 342.5.d.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.5.b.a 8 4.b odd 2 1
38.5.b.a 8 76.d even 2 1
304.5.e.e 8 1.a even 1 1 trivial
304.5.e.e 8 19.b odd 2 1 inner
342.5.d.a 8 12.b even 2 1
342.5.d.a 8 228.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(304, [\chi])$$:

 $$T_{3}^{8} + 458 T_{3}^{6} + 67449 T_{3}^{4} + 3860640 T_{3}^{2} + 74649600$$ $$T_{5}^{4} - 9 T_{5}^{3} - 2074 T_{5}^{2} + 6624 T_{5} + 964896$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$74649600 + 3860640 T^{2} + 67449 T^{4} + 458 T^{6} + T^{8}$$
$5$ $$( 964896 + 6624 T - 2074 T^{2} - 9 T^{3} + T^{4} )^{2}$$
$7$ $$( 4446118 + 240093 T - 3827 T^{2} - 81 T^{3} + T^{4} )^{2}$$
$11$ $$( 76301016 - 2109492 T - 37690 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$13$ $$57445091172827136 + 34129514947104 T^{2} + 5282600121 T^{4} + 140586 T^{6} + T^{8}$$
$17$ $$( 403423998 + 14985243 T - 67555 T^{2} - 255 T^{3} + T^{4} )^{2}$$
$19$ $$28\!\cdots\!81$$$$- 26559779028793932 T + 5155530396725960 T^{2} + 2634460387644 T^{3} + 46020337038 T^{4} + 20215164 T^{5} + 303560 T^{6} - 12 T^{7} + T^{8}$$
$23$ $$( 18350234964 + 14380020 T - 326059 T^{2} - 198 T^{3} + T^{4} )^{2}$$
$29$ $$26\!\cdots\!00$$$$+ 1941717474110086560 T^{2} + 4545080369001 T^{4} + 3774042 T^{6} + T^{8}$$
$31$ $$13\!\cdots\!16$$$$+ 290480962963064832 T^{2} + 1470564161424 T^{4} + 2202408 T^{6} + T^{8}$$
$37$ $$61\!\cdots\!36$$$$+ 68770887827876352 T^{2} + 1219007010960 T^{4} + 6003528 T^{6} + T^{8}$$
$41$ $$14\!\cdots\!04$$$$+ 5375602779897139200 T^{2} + 14417398812816 T^{4} + 7187688 T^{6} + T^{8}$$
$43$ $$( 407751532960 + 2817187520 T + 3365214 T^{2} - 4327 T^{3} + T^{4} )^{2}$$
$47$ $$( -98774187816 + 1482029532 T - 3671746 T^{2} + 1605 T^{3} + T^{4} )^{2}$$
$53$ $$82\!\cdots\!84$$$$+$$$$33\!\cdots\!60$$$$T^{2} + 356018538506121 T^{4} + 36847098 T^{6} + T^{8}$$
$59$ $$20\!\cdots\!00$$$$+ 17355507167396547360 T^{2} + 42239279002761 T^{4} + 25952346 T^{6} + T^{8}$$
$61$ $$( 195962902247296 + 64039717152 T - 41732066 T^{2} - 657 T^{3} + T^{4} )^{2}$$
$67$ $$13\!\cdots\!96$$$$+$$$$17\!\cdots\!60$$$$T^{2} + 7964319437470377 T^{4} + 149621370 T^{6} + T^{8}$$
$71$ $$22\!\cdots\!56$$$$+$$$$26\!\cdots\!40$$$$T^{2} + 10801833589824912 T^{4} + 175804200 T^{6} + T^{8}$$
$73$ $$( -1571012619241250 + 629910296275 T - 25007871 T^{2} - 11699 T^{3} + T^{4} )^{2}$$
$79$ $$50\!\cdots\!76$$$$+$$$$58\!\cdots\!16$$$$T^{2} + 1855389240525456 T^{4} + 113259624 T^{6} + T^{8}$$
$83$ $$( 57645106800768 + 4083728832 T - 57758860 T^{2} - 5220 T^{3} + T^{4} )^{2}$$
$89$ $$70\!\cdots\!96$$$$+$$$$80\!\cdots\!60$$$$T^{2} + 25689634455831552 T^{4} + 282427200 T^{6} + T^{8}$$
$97$ $$11\!\cdots\!56$$$$+$$$$10\!\cdots\!88$$$$T^{2} + 51581376150900624 T^{4} + 436011432 T^{6} + T^{8}$$