# Properties

 Label 338.8.b.g Level $338$ Weight $8$ Character orbit 338.b Analytic conductor $105.586$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 4682x^{4} + 5480281x^{2} + 171714816$$ x^6 + 4682*x^4 + 5480281*x^2 + 171714816 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 8 \beta_{2} q^{2} + \beta_{3} q^{3} - 64 q^{4} + (3 \beta_{5} - 110 \beta_{2} + 2 \beta_1) q^{5} + 8 \beta_1 q^{6} + ( - 2 \beta_{5} + 386 \beta_{2} - 17 \beta_1) q^{7} - 512 \beta_{2} q^{8} + (11 \beta_{4} - 6 \beta_{3} - 630) q^{9}+O(q^{10})$$ q + 8*b2 * q^2 + b3 * q^3 - 64 * q^4 + (3*b5 - 110*b2 + 2*b1) * q^5 + 8*b1 * q^6 + (-2*b5 + 386*b2 - 17*b1) * q^7 - 512*b2 * q^8 + (11*b4 - 6*b3 - 630) * q^9 $$q + 8 \beta_{2} q^{2} + \beta_{3} q^{3} - 64 q^{4} + (3 \beta_{5} - 110 \beta_{2} + 2 \beta_1) q^{5} + 8 \beta_1 q^{6} + ( - 2 \beta_{5} + 386 \beta_{2} - 17 \beta_1) q^{7} - 512 \beta_{2} q^{8} + (11 \beta_{4} - 6 \beta_{3} - 630) q^{9} + (24 \beta_{4} - 16 \beta_{3} + 880) q^{10} + ( - 8 \beta_{5} - 1412 \beta_{2} + 91 \beta_1) q^{11} - 64 \beta_{3} q^{12} + ( - 16 \beta_{4} + 136 \beta_{3} - 3088) q^{14} + ( - 4 \beta_{5} + 4140 \beta_{2} - 326 \beta_1) q^{15} + 4096 q^{16} + ( - 248 \beta_{4} + 244 \beta_{3} - 2163) q^{17} + ( - 88 \beta_{5} - 5040 \beta_{2} - 48 \beta_1) q^{18} + ( - 168 \beta_{5} - 840 \beta_{2} + 299 \beta_1) q^{19} + ( - 192 \beta_{5} + 7040 \beta_{2} - 128 \beta_1) q^{20} + (175 \beta_{5} - 27153 \beta_{2} + 624 \beta_1) q^{21} + ( - 64 \beta_{4} - 728 \beta_{3} + 11296) q^{22} + ( - 586 \beta_{4} + 657 \beta_{3} + 12182) q^{23} - 512 \beta_1 q^{24} + ( - 29 \beta_{4} + 1226 \beta_{3} - 27575) q^{25} + ( - 2033 \beta_{3} - 13104) q^{27} + (128 \beta_{5} - 24704 \beta_{2} + 1088 \beta_1) q^{28} + (382 \beta_{4} + 1110 \beta_{3} - 1391) q^{29} + ( - 32 \beta_{4} + 2608 \beta_{3} - 33120) q^{30} + (1372 \beta_{5} - 94664 \beta_{2} - 96 \beta_1) q^{31} + 32768 \beta_{2} q^{32} + ( - 1049 \beta_{5} + 138951 \beta_{2} - 1414 \beta_1) q^{33} + (1984 \beta_{5} - 17304 \beta_{2} + 1952 \beta_1) q^{34} + (1020 \beta_{4} - 6550 \beta_{3} + 169720) q^{35} + ( - 704 \beta_{4} + 384 \beta_{3} + 40320) q^{36} + (114 \beta_{5} - 28617 \beta_{2} + 7810 \beta_1) q^{37} + ( - 1344 \beta_{4} - 2392 \beta_{3} + 6720) q^{38} + ( - 1536 \beta_{4} + 1024 \beta_{3} - 56320) q^{40} + (4096 \beta_{5} - 256089 \beta_{2} + 3104 \beta_1) q^{41} + (1400 \beta_{4} - 4992 \beta_{3} + 217224) q^{42} + ( - 2200 \beta_{4} - 8915 \beta_{3} + 40736) q^{43} + (512 \beta_{5} + 90368 \beta_{2} - 5824 \beta_1) q^{44} + ( - 2999 \beta_{5} - 268380 \beta_{2} + 1994 \beta_1) q^{45} + (4688 \beta_{5} + 97456 \beta_{2} + 5256 \beta_1) q^{46} + (652 \beta_{5} + 235636 \beta_{2} + 9172 \beta_1) q^{47} + 4096 \beta_{3} q^{48} + ( - 4047 \beta_{4} + 19458 \beta_{3} + 164310) q^{49} + (232 \beta_{5} - 220600 \beta_{2} + 9808 \beta_1) q^{50} + (1196 \beta_{4} - 20491 \beta_{3} + 464724) q^{51} + (2861 \beta_{4} + 13370 \beta_{3} - 741846) q^{53} + ( - 104832 \beta_{2} - 16264 \beta_1) q^{54} + ( - 5424 \beta_{4} + 31546 \beta_{3} - 304540) q^{55} + (1024 \beta_{4} - 8704 \beta_{3} + 197632) q^{56} + ( - 4297 \beta_{5} + 408087 \beta_{2} + 8790 \beta_1) q^{57} + ( - 3056 \beta_{5} - 11128 \beta_{2} + 8880 \beta_1) q^{58} + ( - 9616 \beta_{5} + \cdots + 22673 \beta_1) q^{59}+ \cdots + (26756 \beta_{5} + 527688 \beta_{2} + 19750 \beta_1) q^{99}+O(q^{100})$$ q + 8*b2 * q^2 + b3 * q^3 - 64 * q^4 + (3*b5 - 110*b2 + 2*b1) * q^5 + 8*b1 * q^6 + (-2*b5 + 386*b2 - 17*b1) * q^7 - 512*b2 * q^8 + (11*b4 - 6*b3 - 630) * q^9 + (24*b4 - 16*b3 + 880) * q^10 + (-8*b5 - 1412*b2 + 91*b1) * q^11 - 64*b3 * q^12 + (-16*b4 + 136*b3 - 3088) * q^14 + (-4*b5 + 4140*b2 - 326*b1) * q^15 + 4096 * q^16 + (-248*b4 + 244*b3 - 2163) * q^17 + (-88*b5 - 5040*b2 - 48*b1) * q^18 + (-168*b5 - 840*b2 + 299*b1) * q^19 + (-192*b5 + 7040*b2 - 128*b1) * q^20 + (175*b5 - 27153*b2 + 624*b1) * q^21 + (-64*b4 - 728*b3 + 11296) * q^22 + (-586*b4 + 657*b3 + 12182) * q^23 - 512*b1 * q^24 + (-29*b4 + 1226*b3 - 27575) * q^25 + (-2033*b3 - 13104) * q^27 + (128*b5 - 24704*b2 + 1088*b1) * q^28 + (382*b4 + 1110*b3 - 1391) * q^29 + (-32*b4 + 2608*b3 - 33120) * q^30 + (1372*b5 - 94664*b2 - 96*b1) * q^31 + 32768*b2 * q^32 + (-1049*b5 + 138951*b2 - 1414*b1) * q^33 + (1984*b5 - 17304*b2 + 1952*b1) * q^34 + (1020*b4 - 6550*b3 + 169720) * q^35 + (-704*b4 + 384*b3 + 40320) * q^36 + (114*b5 - 28617*b2 + 7810*b1) * q^37 + (-1344*b4 - 2392*b3 + 6720) * q^38 + (-1536*b4 + 1024*b3 - 56320) * q^40 + (4096*b5 - 256089*b2 + 3104*b1) * q^41 + (1400*b4 - 4992*b3 + 217224) * q^42 + (-2200*b4 - 8915*b3 + 40736) * q^43 + (512*b5 + 90368*b2 - 5824*b1) * q^44 + (-2999*b5 - 268380*b2 + 1994*b1) * q^45 + (4688*b5 + 97456*b2 + 5256*b1) * q^46 + (652*b5 + 235636*b2 + 9172*b1) * q^47 + 4096*b3 * q^48 + (-4047*b4 + 19458*b3 + 164310) * q^49 + (232*b5 - 220600*b2 + 9808*b1) * q^50 + (1196*b4 - 20491*b3 + 464724) * q^51 + (2861*b4 + 13370*b3 - 741846) * q^53 + (-104832*b2 - 16264*b1) * q^54 + (-5424*b4 + 31546*b3 - 304540) * q^55 + (1024*b4 - 8704*b3 + 197632) * q^56 + (-4297*b5 + 408087*b2 + 8790*b1) * q^57 + (-3056*b5 - 11128*b2 + 8880*b1) * q^58 + (-9616*b5 + 937964*b2 + 22673*b1) * q^59 + (256*b5 - 264960*b2 + 20864*b1) * q^60 + (-6746*b4 - 15238*b3 + 546145) * q^61 + (10976*b4 + 768*b3 + 757312) * q^62 + (-1440*b5 + 187236*b2 - 5618*b1) * q^63 - 262144 * q^64 + (-8392*b4 + 11312*b3 - 1111608) * q^66 + (-26480*b5 - 1345892*b2 + 14757*b1) * q^67 + (15872*b4 - 15616*b3 + 138432) * q^68 + (3711*b4 - 31608*b3 + 1223361) * q^69 + (-8160*b5 + 1357760*b2 - 52400*b1) * q^70 + (34722*b5 + 1129514*b2 + 6949*b1) * q^71 + (5632*b5 + 322560*b2 + 3072*b1) * q^72 + (-38499*b5 + 1311706*b2 + 43974*b1) * q^73 + (912*b4 - 62480*b3 + 228936) * q^74 + (13312*b4 - 36903*b3 + 1918800) * q^75 + (10752*b5 + 53760*b2 - 19136*b1) * q^76 + (17549*b4 - 71636*b3 + 2821411) * q^77 + (18720*b4 - 94212*b3 + 249908) * q^79 + (12288*b5 - 450560*b2 + 8192*b1) * q^80 + (-46420*b4 + 12216*b3 - 1787571) * q^81 + (32768*b4 - 24832*b3 + 2048712) * q^82 + (9156*b5 + 3701796*b2 + 54048*b1) * q^83 + (-11200*b5 + 1737792*b2 - 39936*b1) * q^84 + (18079*b5 + 8301210*b2 - 113134*b1) * q^85 + (17600*b5 + 325888*b2 - 71320*b1) * q^86 + (14502*b4 + 17925*b3 + 1597626) * q^87 + (4096*b4 + 46592*b3 - 722944) * q^88 + (-30241*b5 + 860277*b2 + 179890*b1) * q^89 + (-23992*b4 - 15952*b3 + 2147040) * q^90 + (37504*b4 - 42048*b3 - 779648) * q^92 + (9288*b5 + 319752*b2 - 187384*b1) * q^93 + (5216*b4 - 73376*b3 - 1885088) * q^94 + (-21020*b4 + 79330*b3 + 3447660) * q^95 + 32768*b1 * q^96 + (-51203*b5 + 2999903*b2 + 49034*b1) * q^97 + (32376*b5 + 1314480*b2 + 155664*b1) * q^98 + (26756*b5 + 527688*b2 + 19750*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 384 q^{4} - 3758 q^{9}+O(q^{10})$$ 6 * q - 384 * q^4 - 3758 * q^9 $$6 q - 384 q^{4} - 3758 q^{9} + 5328 q^{10} - 18560 q^{14} + 24576 q^{16} - 13474 q^{17} + 67648 q^{22} + 71920 q^{23} - 165508 q^{25} - 78624 q^{27} - 7582 q^{29} - 198784 q^{30} + 1020360 q^{35} + 240512 q^{36} + 37632 q^{38} - 340992 q^{40} + 1306144 q^{42} + 240016 q^{43} + 977766 q^{49} + 2790736 q^{51} - 4445354 q^{53} - 1838088 q^{55} + 1187840 q^{56} + 3263378 q^{61} + 4565824 q^{62} - 1572864 q^{64} - 6686432 q^{66} + 862336 q^{68} + 7347588 q^{69} + 1375440 q^{74} + 11539424 q^{75} + 16963564 q^{77} + 1536888 q^{79} - 10818266 q^{81} + 12357808 q^{82} + 9614760 q^{87} - 4329472 q^{88} + 12834256 q^{90} - 4602880 q^{92} - 11300096 q^{94} + 20643920 q^{95}+O(q^{100})$$ 6 * q - 384 * q^4 - 3758 * q^9 + 5328 * q^10 - 18560 * q^14 + 24576 * q^16 - 13474 * q^17 + 67648 * q^22 + 71920 * q^23 - 165508 * q^25 - 78624 * q^27 - 7582 * q^29 - 198784 * q^30 + 1020360 * q^35 + 240512 * q^36 + 37632 * q^38 - 340992 * q^40 + 1306144 * q^42 + 240016 * q^43 + 977766 * q^49 + 2790736 * q^51 - 4445354 * q^53 - 1838088 * q^55 + 1187840 * q^56 + 3263378 * q^61 + 4565824 * q^62 - 1572864 * q^64 - 6686432 * q^66 + 862336 * q^68 + 7347588 * q^69 + 1375440 * q^74 + 11539424 * q^75 + 16963564 * q^77 + 1536888 * q^79 - 10818266 * q^81 + 12357808 * q^82 + 9614760 * q^87 - 4329472 * q^88 + 12834256 * q^90 - 4602880 * q^92 - 11300096 * q^94 + 20643920 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 4682x^{4} + 5480281x^{2} + 171714816$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2341\nu ) / 13104$$ (v^3 + 2341*v) / 13104 $$\beta_{3}$$ $$=$$ $$( -\nu^{4} - 2341\nu^{2} ) / 13104$$ (-v^4 - 2341*v^2) / 13104 $$\beta_{4}$$ $$=$$ $$( -\nu^{4} - 4525\nu^{2} - 3400488 ) / 24024$$ (-v^4 - 4525*v^2 - 3400488) / 24024 $$\beta_{5}$$ $$=$$ $$( \nu^{5} + 3898\nu^{3} + 3566313\nu ) / 144144$$ (v^5 + 3898*v^3 + 3566313*v) / 144144
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-11\beta_{4} + 6\beta_{3} - 1557$$ -11*b4 + 6*b3 - 1557 $$\nu^{3}$$ $$=$$ $$13104\beta_{2} - 2341\beta_1$$ 13104*b2 - 2341*b1 $$\nu^{4}$$ $$=$$ $$25751\beta_{4} - 27150\beta_{3} + 3644937$$ 25751*b4 - 27150*b3 + 3644937 $$\nu^{5}$$ $$=$$ $$144144\beta_{5} - 51079392\beta_{2} + 5558905\beta_1$$ 144144*b5 - 51079392*b2 + 5558905*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
 50.9714i − 5.67571i − 45.2957i − 50.9714i 5.67571i 45.2957i
8.00000i −50.9714 −64.0000 412.467i 407.771i 1386.20i 512.000i 411.085 3299.74
337.2 8.00000i 5.67571 −64.0000 307.915i 45.4057i 18.4707i 512.000i −2154.79 −2463.32
337.3 8.00000i 45.2957 −64.0000 228.447i 362.366i 244.668i 512.000i −135.299 1827.58
337.4 8.00000i −50.9714 −64.0000 412.467i 407.771i 1386.20i 512.000i 411.085 3299.74
337.5 8.00000i 5.67571 −64.0000 307.915i 45.4057i 18.4707i 512.000i −2154.79 −2463.32
337.6 8.00000i 45.2957 −64.0000 228.447i 362.366i 244.668i 512.000i −135.299 1827.58
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.6 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.g 6
13.b even 2 1 inner 338.8.b.g 6
13.d odd 4 1 338.8.a.g 3
13.d odd 4 1 338.8.a.h 3
13.f odd 12 2 26.8.c.a 6
39.k even 12 2 234.8.h.a 6
52.l even 12 2 208.8.i.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.c.a 6 13.f odd 12 2
208.8.i.a 6 52.l even 12 2
234.8.h.a 6 39.k even 12 2
338.8.a.g 3 13.d odd 4 1
338.8.a.h 3 13.d odd 4 1
338.8.b.g 6 1.a even 1 1 trivial
338.8.b.g 6 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - 2341T_{3} + 13104$$ acting on $$S_{8}^{\mathrm{new}}(338, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 64)^{3}$$
$3$ $$(T^{3} - 2341 T + 13104)^{2}$$
$5$ $$T^{6} + \cdots + 841806393210000$$
$7$ $$T^{6} + \cdots + 39243960250000$$
$11$ $$T^{6} + \cdots + 12\!\cdots\!76$$
$13$ $$T^{6}$$
$17$ $$(T^{3} + \cdots - 8589135310029)^{2}$$
$19$ $$T^{6} + \cdots + 67\!\cdots\!00$$
$23$ $$(T^{3} + \cdots + 6393551963292)^{2}$$
$29$ $$(T^{3} + \cdots - 114763020797763)^{2}$$
$31$ $$T^{6} + \cdots + 10\!\cdots\!84$$
$37$ $$T^{6} + \cdots + 18\!\cdots\!81$$
$41$ $$T^{6} + \cdots + 76\!\cdots\!25$$
$43$ $$(T^{3} + \cdots + 51\!\cdots\!44)^{2}$$
$47$ $$T^{6} + \cdots + 39\!\cdots\!24$$
$53$ $$(T^{3} + \cdots - 81\!\cdots\!56)^{2}$$
$59$ $$T^{6} + \cdots + 70\!\cdots\!04$$
$61$ $$(T^{3} + \cdots + 73\!\cdots\!05)^{2}$$
$67$ $$T^{6} + \cdots + 34\!\cdots\!44$$
$71$ $$T^{6} + \cdots + 78\!\cdots\!16$$
$73$ $$T^{6} + \cdots + 12\!\cdots\!00$$
$79$ $$(T^{3} + \cdots - 48\!\cdots\!72)^{2}$$
$83$ $$T^{6} + \cdots + 93\!\cdots\!64$$
$89$ $$T^{6} + \cdots + 24\!\cdots\!76$$
$97$ $$T^{6} + \cdots + 15\!\cdots\!44$$