# Properties

 Label 338.10.a.a Level $338$ Weight $10$ Character orbit 338.a Self dual yes Analytic conductor $174.082$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,10,Mod(1,338)]

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([0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 338.a (trivial)

## 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: yes Analytic conductor: $$174.082112623$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 16 q^{2} - 156 q^{3} + 256 q^{4} - 870 q^{5} + 2496 q^{6} + 952 q^{7} - 4096 q^{8} + 4653 q^{9}+O(q^{10})$$ q - 16 * q^2 - 156 * q^3 + 256 * q^4 - 870 * q^5 + 2496 * q^6 + 952 * q^7 - 4096 * q^8 + 4653 * q^9 $$q - 16 q^{2} - 156 q^{3} + 256 q^{4} - 870 q^{5} + 2496 q^{6} + 952 q^{7} - 4096 q^{8} + 4653 q^{9} + 13920 q^{10} + 56148 q^{11} - 39936 q^{12} - 15232 q^{14} + 135720 q^{15} + 65536 q^{16} - 247662 q^{17} - 74448 q^{18} - 315380 q^{19} - 222720 q^{20} - 148512 q^{21} - 898368 q^{22} + 204504 q^{23} + 638976 q^{24} - 1196225 q^{25} + 2344680 q^{27} + 243712 q^{28} - 3840450 q^{29} - 2171520 q^{30} + 1309408 q^{31} - 1048576 q^{32} - 8759088 q^{33} + 3962592 q^{34} - 828240 q^{35} + 1191168 q^{36} - 4307078 q^{37} + 5046080 q^{38} + 3563520 q^{40} - 1512042 q^{41} + 2376192 q^{42} + 33670604 q^{43} + 14373888 q^{44} - 4048110 q^{45} - 3272064 q^{46} + 10581072 q^{47} - 10223616 q^{48} - 39447303 q^{49} + 19139600 q^{50} + 38635272 q^{51} + 16616214 q^{53} - 37514880 q^{54} - 48848760 q^{55} - 3899392 q^{56} + 49199280 q^{57} + 61447200 q^{58} - 112235100 q^{59} + 34744320 q^{60} - 33197218 q^{61} - 20950528 q^{62} + 4429656 q^{63} + 16777216 q^{64} + 140145408 q^{66} + 121372252 q^{67} - 63401472 q^{68} - 31902624 q^{69} + 13251840 q^{70} + 387172728 q^{71} - 19058688 q^{72} - 255240074 q^{73} + 68913248 q^{74} + 186611100 q^{75} - 80737280 q^{76} + 53452896 q^{77} + 492101840 q^{79} - 57016320 q^{80} - 457355079 q^{81} + 24192672 q^{82} + 457420236 q^{83} - 38019072 q^{84} + 215465940 q^{85} - 538729664 q^{86} + 599110200 q^{87} - 229982208 q^{88} + 31809510 q^{89} + 64769760 q^{90} + 52353024 q^{92} - 204267648 q^{93} - 169297152 q^{94} + 274380600 q^{95} + 163577856 q^{96} + 673532062 q^{97} + 631156848 q^{98} + 261256644 q^{99}+O(q^{100})$$ q - 16 * q^2 - 156 * q^3 + 256 * q^4 - 870 * q^5 + 2496 * q^6 + 952 * q^7 - 4096 * q^8 + 4653 * q^9 + 13920 * q^10 + 56148 * q^11 - 39936 * q^12 - 15232 * q^14 + 135720 * q^15 + 65536 * q^16 - 247662 * q^17 - 74448 * q^18 - 315380 * q^19 - 222720 * q^20 - 148512 * q^21 - 898368 * q^22 + 204504 * q^23 + 638976 * q^24 - 1196225 * q^25 + 2344680 * q^27 + 243712 * q^28 - 3840450 * q^29 - 2171520 * q^30 + 1309408 * q^31 - 1048576 * q^32 - 8759088 * q^33 + 3962592 * q^34 - 828240 * q^35 + 1191168 * q^36 - 4307078 * q^37 + 5046080 * q^38 + 3563520 * q^40 - 1512042 * q^41 + 2376192 * q^42 + 33670604 * q^43 + 14373888 * q^44 - 4048110 * q^45 - 3272064 * q^46 + 10581072 * q^47 - 10223616 * q^48 - 39447303 * q^49 + 19139600 * q^50 + 38635272 * q^51 + 16616214 * q^53 - 37514880 * q^54 - 48848760 * q^55 - 3899392 * q^56 + 49199280 * q^57 + 61447200 * q^58 - 112235100 * q^59 + 34744320 * q^60 - 33197218 * q^61 - 20950528 * q^62 + 4429656 * q^63 + 16777216 * q^64 + 140145408 * q^66 + 121372252 * q^67 - 63401472 * q^68 - 31902624 * q^69 + 13251840 * q^70 + 387172728 * q^71 - 19058688 * q^72 - 255240074 * q^73 + 68913248 * q^74 + 186611100 * q^75 - 80737280 * q^76 + 53452896 * q^77 + 492101840 * q^79 - 57016320 * q^80 - 457355079 * q^81 + 24192672 * q^82 + 457420236 * q^83 - 38019072 * q^84 + 215465940 * q^85 - 538729664 * q^86 + 599110200 * q^87 - 229982208 * q^88 + 31809510 * q^89 + 64769760 * q^90 + 52353024 * q^92 - 204267648 * q^93 - 169297152 * q^94 + 274380600 * q^95 + 163577856 * q^96 + 673532062 * q^97 + 631156848 * q^98 + 261256644 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−16.0000 −156.000 256.000 −870.000 2496.00 952.000 −4096.00 4653.00 13920.0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$13$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.10.a.a 1
13.b even 2 1 2.10.a.a 1
39.d odd 2 1 18.10.a.a 1
52.b odd 2 1 16.10.a.d 1
65.d even 2 1 50.10.a.c 1
65.h odd 4 2 50.10.b.a 2
91.b odd 2 1 98.10.a.c 1
91.r even 6 2 98.10.c.c 2
91.s odd 6 2 98.10.c.b 2
104.e even 2 1 64.10.a.h 1
104.h odd 2 1 64.10.a.b 1
117.n odd 6 2 162.10.c.i 2
117.t even 6 2 162.10.c.b 2
143.d odd 2 1 242.10.a.a 1
156.h even 2 1 144.10.a.d 1
208.o odd 4 2 256.10.b.e 2
208.p even 4 2 256.10.b.g 2
260.g odd 2 1 400.10.a.b 1
260.p even 4 2 400.10.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.10.a.a 1 13.b even 2 1
16.10.a.d 1 52.b odd 2 1
18.10.a.a 1 39.d odd 2 1
50.10.a.c 1 65.d even 2 1
50.10.b.a 2 65.h odd 4 2
64.10.a.b 1 104.h odd 2 1
64.10.a.h 1 104.e even 2 1
98.10.a.c 1 91.b odd 2 1
98.10.c.b 2 91.s odd 6 2
98.10.c.c 2 91.r even 6 2
144.10.a.d 1 156.h even 2 1
162.10.c.b 2 117.t even 6 2
162.10.c.i 2 117.n odd 6 2
242.10.a.a 1 143.d odd 2 1
256.10.b.e 2 208.o odd 4 2
256.10.b.g 2 208.p even 4 2
338.10.a.a 1 1.a even 1 1 trivial
400.10.a.b 1 260.g odd 2 1
400.10.c.d 2 260.p even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(338))$$:

 $$T_{3} + 156$$ T3 + 156 $$T_{5} + 870$$ T5 + 870

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 16$$
$3$ $$T + 156$$
$5$ $$T + 870$$
$7$ $$T - 952$$
$11$ $$T - 56148$$
$13$ $$T$$
$17$ $$T + 247662$$
$19$ $$T + 315380$$
$23$ $$T - 204504$$
$29$ $$T + 3840450$$
$31$ $$T - 1309408$$
$37$ $$T + 4307078$$
$41$ $$T + 1512042$$
$43$ $$T - 33670604$$
$47$ $$T - 10581072$$
$53$ $$T - 16616214$$
$59$ $$T + 112235100$$
$61$ $$T + 33197218$$
$67$ $$T - 121372252$$
$71$ $$T - 387172728$$
$73$ $$T + 255240074$$
$79$ $$T - 492101840$$
$83$ $$T - 457420236$$
$89$ $$T - 31809510$$
$97$ $$T - 673532062$$