# Properties

 Label 338.8.b.e Level $338$ Weight $8$ Character orbit 338.b Analytic conductor $105.586$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,8,Mod(337,338)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 338.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$105.586138614$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{105})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 53x^{2} + 676$$ x^4 + 53*x^2 + 676 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 8 \beta_1 q^{2} + (7 \beta_{3} - 6) q^{3} - 64 q^{4} + (36 \beta_{2} - 73 \beta_1) q^{5} + ( - 56 \beta_{2} + 48 \beta_1) q^{6} + ( - 27 \beta_{2} + 890 \beta_1) q^{7} + 512 \beta_1 q^{8} + ( - 84 \beta_{3} + 2994) q^{9}+O(q^{10})$$ q - 8*b1 * q^2 + (7*b3 - 6) * q^3 - 64 * q^4 + (36*b2 - 73*b1) * q^5 + (-56*b2 + 48*b1) * q^6 + (-27*b2 + 890*b1) * q^7 + 512*b1 * q^8 + (-84*b3 + 2994) * q^9 $$q - 8 \beta_1 q^{2} + (7 \beta_{3} - 6) q^{3} - 64 q^{4} + (36 \beta_{2} - 73 \beta_1) q^{5} + ( - 56 \beta_{2} + 48 \beta_1) q^{6} + ( - 27 \beta_{2} + 890 \beta_1) q^{7} + 512 \beta_1 q^{8} + ( - 84 \beta_{3} + 2994) q^{9} + (288 \beta_{3} - 584) q^{10} + ( - 90 \beta_{2} - 5452 \beta_1) q^{11} + ( - 448 \beta_{3} + 384) q^{12} + ( - 216 \beta_{3} + 7120) q^{14} + ( - 727 \beta_{2} + 26898 \beta_1) q^{15} + 4096 q^{16} + ( - 2304 \beta_{3} + 7059) q^{17} + (672 \beta_{2} - 23952 \beta_1) q^{18} + (1386 \beta_{2} + 27204 \beta_1) q^{19} + ( - 2304 \beta_{2} + 4672 \beta_1) q^{20} + (6392 \beta_{2} - 25185 \beta_1) q^{21} + ( - 720 \beta_{3} - 43616) q^{22} + (3564 \beta_{3} + 78232) q^{23} + (3584 \beta_{2} - 3072 \beta_1) q^{24} + (5256 \beta_{3} - 63284) q^{25} + (6153 \beta_{3} - 66582) q^{27} + (1728 \beta_{2} - 56960 \beta_1) q^{28} + (14544 \beta_{3} - 72934) q^{29} + ( - 5816 \beta_{3} + 215184) q^{30} + (2808 \beta_{2} + 22900 \beta_1) q^{31} - 32768 \beta_1 q^{32} + ( - 37624 \beta_{2} - 33438 \beta_1) q^{33} + (18432 \beta_{2} - 56472 \beta_1) q^{34} + ( - 34011 \beta_{3} + 167030) q^{35} + (5376 \beta_{3} - 191616) q^{36} + (14544 \beta_{2} - 174279 \beta_1) q^{37} + (11088 \beta_{3} + 217632) q^{38} + ( - 18432 \beta_{3} + 37376) q^{40} + ( - 9864 \beta_{2} - 78120 \beta_1) q^{41} + (51136 \beta_{3} - 201480) q^{42} + (36891 \beta_{3} - 368626) q^{43} + (5760 \beta_{2} + 348928 \beta_1) q^{44} + (113916 \beta_{2} - 536082 \beta_1) q^{45} + ( - 28512 \beta_{2} - 625856 \beta_1) q^{46} + (44037 \beta_{2} + 307598 \beta_1) q^{47} + (28672 \beta_{3} - 24576) q^{48} + (48060 \beta_{3} - 45102) q^{49} + ( - 42048 \beta_{2} + 506272 \beta_1) q^{50} + (63237 \beta_{3} - 1735794) q^{51} + ( - 76104 \beta_{3} - 768012) q^{53} + ( - 49224 \beta_{2} + 532656 \beta_1) q^{54} + (189702 \beta_{3} - 57796) q^{55} + (13824 \beta_{3} - 455680) q^{56} + (182112 \beta_{2} + 855486 \beta_1) q^{57} + ( - 116352 \beta_{2} + 583472 \beta_1) q^{58} + (202734 \beta_{2} + 881236 \beta_1) q^{59} + (46528 \beta_{2} - 1721472 \beta_1) q^{60} + (90864 \beta_{3} - 2230136) q^{61} + (22464 \beta_{3} + 183200) q^{62} + ( - 155598 \beta_{2} + 2902800 \beta_1) q^{63} - 262144 q^{64} + ( - 300992 \beta_{3} - 267504) q^{66} + (3114 \beta_{2} + 963340 \beta_1) q^{67} + (147456 \beta_{3} - 451776) q^{68} + (526240 \beta_{3} + 2150148) q^{69} + (272088 \beta_{2} - 1336240 \beta_1) q^{70} + (388305 \beta_{2} + 1252570 \beta_1) q^{71} + ( - 43008 \beta_{2} + 1532928 \beta_1) q^{72} + (219960 \beta_{2} - 1862702 \beta_1) q^{73} + (116352 \beta_{3} - 1394232) q^{74} + ( - 474524 \beta_{3} + 4242864) q^{75} + ( - 88704 \beta_{2} - 1741056 \beta_1) q^{76} + ( - 67104 \beta_{3} + 4597130) q^{77} + ( - 169956 \beta_{3} + 1955696) q^{79} + (147456 \beta_{2} - 299008 \beta_1) q^{80} + ( - 319284 \beta_{3} - 1625931) q^{81} + ( - 78912 \beta_{3} - 624960) q^{82} + (203472 \beta_{2} + 4441680 \beta_1) q^{83} + ( - 409088 \beta_{2} + 1611840 \beta_1) q^{84} + (422316 \beta_{2} - 9224427 \beta_1) q^{85} + ( - 295128 \beta_{2} + 2949008 \beta_1) q^{86} + ( - 597802 \beta_{3} + 11127444) q^{87} + (46080 \beta_{3} + 2791424) q^{88} + ( - 50904 \beta_{2} + 8360274 \beta_1) q^{89} + (911328 \beta_{3} - 4288656) q^{90} + ( - 228096 \beta_{3} - 5006848) q^{92} + (143452 \beta_{2} + 1926480 \beta_1) q^{93} + (352296 \beta_{3} + 2460784) q^{94} + ( - 878166 \beta_{3} - 3253188) q^{95} + ( - 229376 \beta_{2} + 196608 \beta_1) q^{96} + (900504 \beta_{2} + 5658566 \beta_1) q^{97} + ( - 384480 \beta_{2} + 360816 \beta_1) q^{98} + (188508 \beta_{2} - 15529488 \beta_1) q^{99}+O(q^{100})$$ q - 8*b1 * q^2 + (7*b3 - 6) * q^3 - 64 * q^4 + (36*b2 - 73*b1) * q^5 + (-56*b2 + 48*b1) * q^6 + (-27*b2 + 890*b1) * q^7 + 512*b1 * q^8 + (-84*b3 + 2994) * q^9 + (288*b3 - 584) * q^10 + (-90*b2 - 5452*b1) * q^11 + (-448*b3 + 384) * q^12 + (-216*b3 + 7120) * q^14 + (-727*b2 + 26898*b1) * q^15 + 4096 * q^16 + (-2304*b3 + 7059) * q^17 + (672*b2 - 23952*b1) * q^18 + (1386*b2 + 27204*b1) * q^19 + (-2304*b2 + 4672*b1) * q^20 + (6392*b2 - 25185*b1) * q^21 + (-720*b3 - 43616) * q^22 + (3564*b3 + 78232) * q^23 + (3584*b2 - 3072*b1) * q^24 + (5256*b3 - 63284) * q^25 + (6153*b3 - 66582) * q^27 + (1728*b2 - 56960*b1) * q^28 + (14544*b3 - 72934) * q^29 + (-5816*b3 + 215184) * q^30 + (2808*b2 + 22900*b1) * q^31 - 32768*b1 * q^32 + (-37624*b2 - 33438*b1) * q^33 + (18432*b2 - 56472*b1) * q^34 + (-34011*b3 + 167030) * q^35 + (5376*b3 - 191616) * q^36 + (14544*b2 - 174279*b1) * q^37 + (11088*b3 + 217632) * q^38 + (-18432*b3 + 37376) * q^40 + (-9864*b2 - 78120*b1) * q^41 + (51136*b3 - 201480) * q^42 + (36891*b3 - 368626) * q^43 + (5760*b2 + 348928*b1) * q^44 + (113916*b2 - 536082*b1) * q^45 + (-28512*b2 - 625856*b1) * q^46 + (44037*b2 + 307598*b1) * q^47 + (28672*b3 - 24576) * q^48 + (48060*b3 - 45102) * q^49 + (-42048*b2 + 506272*b1) * q^50 + (63237*b3 - 1735794) * q^51 + (-76104*b3 - 768012) * q^53 + (-49224*b2 + 532656*b1) * q^54 + (189702*b3 - 57796) * q^55 + (13824*b3 - 455680) * q^56 + (182112*b2 + 855486*b1) * q^57 + (-116352*b2 + 583472*b1) * q^58 + (202734*b2 + 881236*b1) * q^59 + (46528*b2 - 1721472*b1) * q^60 + (90864*b3 - 2230136) * q^61 + (22464*b3 + 183200) * q^62 + (-155598*b2 + 2902800*b1) * q^63 - 262144 * q^64 + (-300992*b3 - 267504) * q^66 + (3114*b2 + 963340*b1) * q^67 + (147456*b3 - 451776) * q^68 + (526240*b3 + 2150148) * q^69 + (272088*b2 - 1336240*b1) * q^70 + (388305*b2 + 1252570*b1) * q^71 + (-43008*b2 + 1532928*b1) * q^72 + (219960*b2 - 1862702*b1) * q^73 + (116352*b3 - 1394232) * q^74 + (-474524*b3 + 4242864) * q^75 + (-88704*b2 - 1741056*b1) * q^76 + (-67104*b3 + 4597130) * q^77 + (-169956*b3 + 1955696) * q^79 + (147456*b2 - 299008*b1) * q^80 + (-319284*b3 - 1625931) * q^81 + (-78912*b3 - 624960) * q^82 + (203472*b2 + 4441680*b1) * q^83 + (-409088*b2 + 1611840*b1) * q^84 + (422316*b2 - 9224427*b1) * q^85 + (-295128*b2 + 2949008*b1) * q^86 + (-597802*b3 + 11127444) * q^87 + (46080*b3 + 2791424) * q^88 + (-50904*b2 + 8360274*b1) * q^89 + (911328*b3 - 4288656) * q^90 + (-228096*b3 - 5006848) * q^92 + (143452*b2 + 1926480*b1) * q^93 + (352296*b3 + 2460784) * q^94 + (-878166*b3 - 3253188) * q^95 + (-229376*b2 + 196608*b1) * q^96 + (900504*b2 + 5658566*b1) * q^97 + (-384480*b2 + 360816*b1) * q^98 + (188508*b2 - 15529488*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 24 q^{3} - 256 q^{4} + 11976 q^{9}+O(q^{10})$$ 4 * q - 24 * q^3 - 256 * q^4 + 11976 * q^9 $$4 q - 24 q^{3} - 256 q^{4} + 11976 q^{9} - 2336 q^{10} + 1536 q^{12} + 28480 q^{14} + 16384 q^{16} + 28236 q^{17} - 174464 q^{22} + 312928 q^{23} - 253136 q^{25} - 266328 q^{27} - 291736 q^{29} + 860736 q^{30} + 668120 q^{35} - 766464 q^{36} + 870528 q^{38} + 149504 q^{40} - 805920 q^{42} - 1474504 q^{43} - 98304 q^{48} - 180408 q^{49} - 6943176 q^{51} - 3072048 q^{53} - 231184 q^{55} - 1822720 q^{56} - 8920544 q^{61} + 732800 q^{62} - 1048576 q^{64} - 1070016 q^{66} - 1807104 q^{68} + 8600592 q^{69} - 5576928 q^{74} + 16971456 q^{75} + 18388520 q^{77} + 7822784 q^{79} - 6503724 q^{81} - 2499840 q^{82} + 44509776 q^{87} + 11165696 q^{88} - 17154624 q^{90} - 20027392 q^{92} + 9843136 q^{94} - 13012752 q^{95}+O(q^{100})$$ 4 * q - 24 * q^3 - 256 * q^4 + 11976 * q^9 - 2336 * q^10 + 1536 * q^12 + 28480 * q^14 + 16384 * q^16 + 28236 * q^17 - 174464 * q^22 + 312928 * q^23 - 253136 * q^25 - 266328 * q^27 - 291736 * q^29 + 860736 * q^30 + 668120 * q^35 - 766464 * q^36 + 870528 * q^38 + 149504 * q^40 - 805920 * q^42 - 1474504 * q^43 - 98304 * q^48 - 180408 * q^49 - 6943176 * q^51 - 3072048 * q^53 - 231184 * q^55 - 1822720 * q^56 - 8920544 * q^61 + 732800 * q^62 - 1048576 * q^64 - 1070016 * q^66 - 1807104 * q^68 + 8600592 * q^69 - 5576928 * q^74 + 16971456 * q^75 + 18388520 * q^77 + 7822784 * q^79 - 6503724 * q^81 - 2499840 * q^82 + 44509776 * q^87 + 11165696 * q^88 - 17154624 * q^90 - 20027392 * q^92 + 9843136 * q^94 - 13012752 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 53x^{2} + 676$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 27\nu ) / 26$$ (v^3 + 27*v) / 26 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 79\nu ) / 26$$ (v^3 + 79*v) / 26 $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 53$$ 2*v^2 + 53
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 53 ) / 2$$ (b3 - 53) / 2 $$\nu^{3}$$ $$=$$ $$( -27\beta_{2} + 79\beta_1 ) / 2$$ (-27*b2 + 79*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$\chi(n)$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 5.62348i 4.62348i 5.62348i − 4.62348i
8.00000i −77.7287 −64.0000 441.890i 621.829i 1166.67i 512.000i 3854.74 −3535.12
337.2 8.00000i 65.7287 −64.0000 295.890i 525.829i 613.332i 512.000i 2133.26 2367.12
337.3 8.00000i −77.7287 −64.0000 441.890i 621.829i 1166.67i 512.000i 3854.74 −3535.12
337.4 8.00000i 65.7287 −64.0000 295.890i 525.829i 613.332i 512.000i 2133.26 2367.12
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.8.b.e 4
13.b even 2 1 inner 338.8.b.e 4
13.d odd 4 1 26.8.a.d 2
13.d odd 4 1 338.8.a.f 2
39.f even 4 1 234.8.a.l 2
52.f even 4 1 208.8.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.a.d 2 13.d odd 4 1
208.8.a.h 2 52.f even 4 1
234.8.a.l 2 39.f even 4 1
338.8.a.f 2 13.d odd 4 1
338.8.b.e 4 1.a even 1 1 trivial
338.8.b.e 4 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 12T_{3} - 5109$$ acting on $$S_{8}^{\mathrm{new}}(338, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 64)^{2}$$
$3$ $$(T^{2} + 12 T - 5109)^{2}$$
$5$ $$T^{4} + \cdots + 17095824001$$
$7$ $$T^{4} + \cdots + 512018958025$$
$11$ $$T^{4} + \cdots + 833696557430416$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 14118 T - 507554199)^{2}$$
$19$ $$T^{4} + \cdots + 28\!\cdots\!96$$
$23$ $$(T^{2} - 156464 T + 4786525744)^{2}$$
$29$ $$(T^{2} + 145868 T - 16891064924)^{2}$$
$31$ $$T^{4} + \cdots + 92\!\cdots\!00$$
$37$ $$T^{4} + \cdots + 66\!\cdots\!21$$
$41$ $$T^{4} + \cdots + 16\!\cdots\!00$$
$43$ $$(T^{2} + 737252 T - 7014189629)^{2}$$
$47$ $$T^{4} + \cdots + 11\!\cdots\!81$$
$53$ $$(T^{2} + 1536024 T - 18298543536)^{2}$$
$59$ $$T^{4} + \cdots + 12\!\cdots\!56$$
$61$ $$(T^{2} + \cdots + 4106598596416)^{2}$$
$67$ $$T^{4} + \cdots + 85\!\cdots\!00$$
$71$ $$T^{4} + \cdots + 20\!\cdots\!25$$
$73$ $$T^{4} + \cdots + 25\!\cdots\!16$$
$79$ $$(T^{2} - 3911392 T + 791817441136)^{2}$$
$83$ $$T^{4} + \cdots + 23\!\cdots\!00$$
$89$ $$T^{4} + \cdots + 48\!\cdots\!16$$
$97$ $$T^{4} + \cdots + 28\!\cdots\!76$$