# Properties

 Label 338.8.b.h Level $338$ Weight $8$ Character orbit 338.b Analytic conductor $105.586$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 13962x^{6} + 63111321x^{4} + 101620417536x^{2} + 51674244710400$$ x^8 + 13962*x^6 + 63111321*x^4 + 101620417536*x^2 + 51674244710400 Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 8 \beta_{2} q^{2} + \beta_{3} q^{3} - 64 q^{4} + (\beta_{7} + 70 \beta_{2} - \beta_1) q^{5} - 8 \beta_1 q^{6} + ( - 2 \beta_{5} - 138 \beta_{2} - 7 \beta_1) q^{7} + 512 \beta_{2} q^{8} + ( - 3 \beta_{6} + 6 \beta_{4} + \cdots + 1302) q^{9}+O(q^{10})$$ q - 8*b2 * q^2 + b3 * q^3 - 64 * q^4 + (b7 + 70*b2 - b1) * q^5 - 8*b1 * q^6 + (-2*b5 - 138*b2 - 7*b1) * q^7 + 512*b2 * q^8 + (-3*b6 + 6*b4 + 9*b3 + 1302) * q^9 $$q - 8 \beta_{2} q^{2} + \beta_{3} q^{3} - 64 q^{4} + (\beta_{7} + 70 \beta_{2} - \beta_1) q^{5} - 8 \beta_1 q^{6} + ( - 2 \beta_{5} - 138 \beta_{2} - 7 \beta_1) q^{7} + 512 \beta_{2} q^{8} + ( - 3 \beta_{6} + 6 \beta_{4} + \cdots + 1302) q^{9}+ \cdots + ( - 7152 \beta_{7} + \cdots + 136674 \beta_1) q^{99}+O(q^{100})$$ q - 8*b2 * q^2 + b3 * q^3 - 64 * q^4 + (b7 + 70*b2 - b1) * q^5 - 8*b1 * q^6 + (-2*b5 - 138*b2 - 7*b1) * q^7 + 512*b2 * q^8 + (-3*b6 + 6*b4 + 9*b3 + 1302) * q^9 + (8*b4 - 8*b3 + 552) * q^10 + (8*b5 - 1844*b2 + 19*b1) * q^11 - 64*b3 * q^12 + (16*b6 - 56*b3 - 1104) * q^14 + (-48*b7 + 12*b5 - 3900*b2 + 206*b1) * q^15 + 4096 * q^16 + (14*b6 + 34*b4 + 196*b3 + 7055) * q^17 + (-48*b7 - 24*b5 - 10464*b2 - 72*b1) * q^18 + (56*b7 + 25000*b2 + 259*b1) * q^19 + (-64*b7 - 4480*b2 + 64*b1) * q^20 + (-126*b7 - 87*b5 - 22875*b2 - 547*b1) * q^21 + (-64*b6 + 152*b3 - 14752) * q^22 + (-74*b6 - 176*b4 + 399*b3 - 8222) * q^23 + 512*b1 * q^24 + (43*b6 - 168*b4 + 1561*b3 - 21657) * q^25 + (-216*b6 - 72*b4 + 591*b3 + 26712) * q^27 + (128*b5 + 8832*b2 + 448*b1) * q^28 + (-187*b6 + 365*b4 + 600*b3 - 23374) * q^29 + (-96*b6 - 384*b4 + 1648*b3 - 30816) * q^30 + (-100*b7 + 88*b5 + 77784*b2 - 204*b1) * q^31 - 32768*b2 * q^32 + (450*b7 + 321*b5 + 60045*b2 - 289*b1) * q^33 + (-272*b7 + 112*b5 - 56712*b2 - 1568*b1) * q^34 + (12*b6 + 64*b4 + 2918*b3 - 35424) * q^35 + (192*b6 - 384*b4 - 576*b3 - 83328) * q^36 + (-281*b7 - 3*b5 - 2551*b2 - 868*b1) * q^37 + (448*b4 + 2072*b3 + 199552) * q^38 + (-512*b4 + 512*b3 - 35328) * q^40 + (-862*b7 - 70*b5 - 21189*b2 + 3560*b1) * q^41 + (696*b6 - 1008*b4 - 4376*b3 - 181992) * q^42 + (560*b6 - 1080*b4 + 5285*b3 - 142056) * q^43 + (-512*b5 + 118016*b2 - 1216*b1) * q^44 + (1569*b7 + 294*b5 + 576780*b2 - 4743*b1) * q^45 + (1408*b7 - 592*b5 + 67184*b2 - 3192*b1) * q^46 + (-1080*b7 + 692*b5 + 143356*b2 - 8648*b1) * q^47 + 4096*b3 * q^48 + (231*b6 - 462*b4 - 14049*b3 - 179334) * q^49 + (1344*b7 + 344*b5 + 174600*b2 - 12488*b1) * q^50 + (-636*b6 - 840*b4 + 11361*b3 + 683220) * q^51 + (2368*b6 + 689*b4 - 4153*b3 + 307309) * q^53 + (576*b7 - 1728*b5 - 213120*b2 - 4728*b1) * q^54 + (-156*b6 + 1708*b4 - 12214*b3 + 272352) * q^55 + (-1024*b6 + 3584*b3 + 70656) * q^56 + (-798*b7 + 1617*b5 + 882525*b2 + 35451*b1) * q^57 + (-2920*b7 - 1496*b5 + 184072*b2 - 4800*b1) * q^58 + (-5712*b7 - 1568*b5 + 54236*b2 - 12799*b1) * q^59 + (3072*b7 - 768*b5 + 249600*b2 - 13184*b1) * q^60 + (2353*b6 + 729*b4 + 43060*b3 - 172962) * q^61 + (-704*b6 - 800*b4 - 1632*b3 + 623072) * q^62 + (-1644*b7 - 2028*b5 - 1489764*b2 - 45810*b1) * q^63 - 262144 * q^64 + (-2568*b6 + 3600*b4 - 2312*b3 + 476760) * q^66 + (1568*b7 + 2176*b5 - 815892*b2 - 8379*b1) * q^67 + (-896*b6 - 2176*b4 - 12544*b3 - 451520) * q^68 + (-999*b6 + 12894*b4 - 17525*b3 + 1394061) * q^69 + (-512*b7 + 96*b5 + 282880*b2 - 23344*b1) * q^70 + (-3940*b7 + 1038*b5 + 42302*b2 - 39029*b1) * q^71 + (3072*b7 + 1536*b5 + 669696*b2 + 4608*b1) * q^72 + (1266*b7 - 2457*b5 - 1785068*b2 - 40803*b1) * q^73 + (24*b6 - 2248*b4 - 6944*b3 - 18160) * q^74 + (-744*b6 + 14616*b4 - 39575*b3 + 5543304) * q^75 + (-3584*b7 - 1600000*b2 - 16576*b1) * q^76 + (4651*b6 + 714*b4 + 34501*b3 + 3476119) * q^77 + (-3516*b6 - 9732*b4 + 35196*b3 - 1756864) * q^79 + (4096*b7 + 286720*b2 - 4096*b1) * q^80 + (-1260*b6 + 2520*b4 + 39204*b3 - 940131) * q^81 + (560*b6 - 6896*b4 + 28480*b3 - 162616) * q^82 + (14520*b7 - 4644*b5 + 169260*b2 + 3948*b1) * q^83 + (8064*b7 + 5568*b5 + 1464000*b2 + 35008*b1) * q^84 + (-2453*b7 + 4894*b5 + 2954970*b2 - 21701*b1) * q^85 + (8640*b7 + 4480*b5 + 1145088*b2 - 42280*b1) * q^86 + (-13446*b6 - 3876*b4 + 67667*b3 + 1804386) * q^87 + (4096*b6 - 9728*b3 + 944128) * q^88 + (9298*b7 + 9577*b5 - 2853621*b2 + 116719*b1) * q^89 + (-2352*b6 + 12552*b4 - 37944*b3 + 4601688) * q^90 + (4736*b6 + 11264*b4 - 25536*b3 + 526208) * q^92 + (6672*b7 + 792*b5 - 742440*b2 + 76672*b1) * q^93 + (-5536*b6 - 8640*b4 - 69184*b3 + 1155488) * q^94 + (6188*b6 - 15368*b4 + 43606*b3 - 5828932) * q^95 - 32768*b1 * q^96 + (-8330*b7 - 7973*b5 + 98849*b2 + 55489*b1) * q^97 + (3696*b7 + 1848*b5 + 1438368*b2 + 112392*b1) * q^98 + (-7152*b7 - 1020*b5 + 2585328*b2 + 136674*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 512 q^{4} + 10428 q^{9}+O(q^{10})$$ 8 * q - 512 * q^4 + 10428 * q^9 $$8 q - 512 q^{4} + 10428 q^{9} + 4448 q^{10} - 8768 q^{14} + 32768 q^{16} + 56632 q^{17} - 118272 q^{22} - 66776 q^{23} - 173756 q^{25} + 212544 q^{27} - 186280 q^{29} - 248448 q^{30} - 283088 q^{35} - 667392 q^{36} + 1598208 q^{38} - 284672 q^{40} - 1457184 q^{42} - 1138528 q^{43} - 1435596 q^{49} + 5459856 q^{51} + 2470700 q^{53} + 2185024 q^{55} + 561152 q^{56} - 1371368 q^{61} + 4978560 q^{62} - 2097152 q^{64} + 3818208 q^{66} - 3624448 q^{68} + 11200068 q^{69} - 154176 q^{74} + 44401920 q^{75} + 27830412 q^{77} - 14107904 q^{79} - 7516008 q^{81} - 1326272 q^{82} + 14365800 q^{87} + 7569408 q^{88} + 36854304 q^{90} + 4273664 q^{92} + 9187200 q^{94} - 46668176 q^{95}+O(q^{100})$$ 8 * q - 512 * q^4 + 10428 * q^9 + 4448 * q^10 - 8768 * q^14 + 32768 * q^16 + 56632 * q^17 - 118272 * q^22 - 66776 * q^23 - 173756 * q^25 + 212544 * q^27 - 186280 * q^29 - 248448 * q^30 - 283088 * q^35 - 667392 * q^36 + 1598208 * q^38 - 284672 * q^40 - 1457184 * q^42 - 1138528 * q^43 - 1435596 * q^49 + 5459856 * q^51 + 2470700 * q^53 + 2185024 * q^55 + 561152 * q^56 - 1371368 * q^61 + 4978560 * q^62 - 2097152 * q^64 + 3818208 * q^66 - 3624448 * q^68 + 11200068 * q^69 - 154176 * q^74 + 44401920 * q^75 + 27830412 * q^77 - 14107904 * q^79 - 7516008 * q^81 - 1326272 * q^82 + 14365800 * q^87 + 7569408 * q^88 + 36854304 * q^90 + 4273664 * q^92 + 9187200 * q^94 - 46668176 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 13962x^{6} + 63111321x^{4} + 101620417536x^{2} + 51674244710400$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{7} + 13962\nu^{5} + 55922841\nu^{3} + 51437638656\nu ) / 254644715520$$ (v^7 + 13962*v^5 + 55922841*v^3 + 51437638656*v) / 254644715520 $$\beta_{3}$$ $$=$$ $$( \nu^{4} + 6981\nu^{2} + 7188480 ) / 35424$$ (v^4 + 6981*v^2 + 7188480) / 35424 $$\beta_{4}$$ $$=$$ $$( \nu^{6} + 11298\nu^{4} + 34774929\nu^{2} + 23080121856 ) / 17853696$$ (v^6 + 11298*v^4 + 34774929*v^2 + 23080121856) / 17853696 $$\beta_{5}$$ $$=$$ $$( -127\nu^{7} - 1613430\nu^{5} - 6458592231\nu^{3} - 7776509193216\nu ) / 118834200576$$ (-127*v^7 - 1613430*v^5 - 6458592231*v^3 - 7776509193216*v) / 118834200576 $$\beta_{6}$$ $$=$$ $$( \nu^{6} + 12054\nu^{4} + 43028181\nu^{2} + 38896536960 ) / 8926848$$ (v^6 + 12054*v^4 + 43028181*v^2 + 38896536960) / 8926848 $$\beta_{7}$$ $$=$$ $$( -625\nu^{7} - 7288554\nu^{5} - 24207873417\nu^{3} - 18760060670976\nu ) / 356502601728$$ (-625*v^7 - 7288554*v^5 - 24207873417*v^3 - 18760060670976*v) / 356502601728
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{6} - 6\beta_{4} - 9\beta_{3} - 3489$$ 3*b6 - 6*b4 - 9*b3 - 3489 $$\nu^{3}$$ $$=$$ $$72\beta_{7} - 216\beta_{5} - 26640\beta_{2} - 4965\beta_1$$ 72*b7 - 216*b5 - 26640*b2 - 4965*b1 $$\nu^{4}$$ $$=$$ $$-20943\beta_{6} + 41886\beta_{4} + 98253\beta_{3} + 17168229$$ -20943*b6 + 41886*b4 + 98253*b3 + 17168229 $$\nu^{5}$$ $$=$$ $$-290088\beta_{7} + 1614168\beta_{5} + 309780720\beta_{2} + 27791001\beta_1$$ -290088*b7 + 1614168*b5 + 309780720*b2 + 27791001*b1 $$\nu^{6}$$ $$=$$ $$132289227\beta_{6} - 246724758\beta_{4} - 797088033\beta_{3} - 95717045817$$ 132289227*b6 - 246724758*b4 - 797088033*b3 - 95717045817 $$\nu^{7}$$ $$=$$ $$23764104\beta_{7} - 10457679960\beta_{5} - 2580729212880\beta_{2} - 161798689053\beta_1$$ 23764104*b7 - 10457679960*b5 - 2580729212880*b2 - 161798689053*b1

## 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
 − 71.1584i − 39.9844i 31.8716i 79.2712i 71.1584i 39.9844i − 31.8716i − 79.2712i
8.00000i −71.1584 −64.0000 523.489i 569.267i 416.801i 512.000i 2876.52 4187.91
337.2 8.00000i −39.9844 −64.0000 323.700i 319.875i 568.591i 512.000i −588.246 −2589.60
337.3 8.00000i 31.8716 −64.0000 54.4265i 254.973i 1112.71i 512.000i −1171.20 −435.412
337.4 8.00000i 79.2712 −64.0000 132.638i 634.170i 1508.92i 512.000i 4096.92 1061.10
337.5 8.00000i −71.1584 −64.0000 523.489i 569.267i 416.801i 512.000i 2876.52 4187.91
337.6 8.00000i −39.9844 −64.0000 323.700i 319.875i 568.591i 512.000i −588.246 −2589.60
337.7 8.00000i 31.8716 −64.0000 54.4265i 254.973i 1112.71i 512.000i −1171.20 −435.412
337.8 8.00000i 79.2712 −64.0000 132.638i 634.170i 1508.92i 512.000i 4096.92 1061.10
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.h 8
13.b even 2 1 inner 338.8.b.h 8
13.d odd 4 1 338.8.a.i 4
13.d odd 4 1 338.8.a.j 4
13.f odd 12 2 26.8.c.b 8
39.k even 12 2 234.8.h.b 8
52.l even 12 2 208.8.i.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.c.b 8 13.f odd 12 2
208.8.i.b 8 52.l even 12 2
234.8.h.b 8 39.k even 12 2
338.8.a.i 4 13.d odd 4 1
338.8.a.j 4 13.d odd 4 1
338.8.b.h 8 1.a even 1 1 trivial
338.8.b.h 8 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 6981T_{3}^{2} - 35424T_{3} + 7188480$$ acting on $$S_{8}^{\mathrm{new}}(338, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 64)^{4}$$
$3$ $$(T^{4} - 6981 T^{2} + \cdots + 7188480)^{2}$$
$5$ $$T^{8} + \cdots + 14\!\cdots\!00$$
$7$ $$T^{8} + \cdots + 15\!\cdots\!00$$
$11$ $$T^{8} + \cdots + 28\!\cdots\!36$$
$13$ $$T^{8}$$
$17$ $$(T^{4} + \cdots - 59\!\cdots\!71)^{2}$$
$19$ $$T^{8} + \cdots + 16\!\cdots\!00$$
$23$ $$(T^{4} + \cdots + 80\!\cdots\!44)^{2}$$
$29$ $$(T^{4} + \cdots + 25\!\cdots\!13)^{2}$$
$31$ $$T^{8} + \cdots + 11\!\cdots\!84$$
$37$ $$T^{8} + \cdots + 27\!\cdots\!41$$
$41$ $$T^{8} + \cdots + 75\!\cdots\!25$$
$43$ $$(T^{4} + \cdots - 44\!\cdots\!64)^{2}$$
$47$ $$T^{8} + \cdots + 77\!\cdots\!84$$
$53$ $$(T^{4} + \cdots + 10\!\cdots\!32)^{2}$$
$59$ $$T^{8} + \cdots + 87\!\cdots\!64$$
$61$ $$(T^{4} + \cdots + 60\!\cdots\!45)^{2}$$
$67$ $$T^{8} + \cdots + 16\!\cdots\!96$$
$71$ $$T^{8} + \cdots + 58\!\cdots\!44$$
$73$ $$T^{8} + \cdots + 45\!\cdots\!00$$
$79$ $$(T^{4} + \cdots + 46\!\cdots\!08)^{2}$$
$83$ $$T^{8} + \cdots + 46\!\cdots\!00$$
$89$ $$T^{8} + \cdots + 27\!\cdots\!36$$
$97$ $$T^{8} + \cdots + 23\!\cdots\!84$$