# Properties

 Label 338.8.a.j Level $338$ Weight $8$ Character orbit 338.a Self dual yes Analytic conductor $105.586$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,8,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, 8, names="a")

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$105.586138614$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 6981x^{2} - 35424x + 7188480$$ x^4 - 6981*x^2 - 35424*x + 7188480 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 26) 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)

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.

 $$f(q)$$ $$=$$ $$q + 8 q^{2} + \beta_1 q^{3} + 64 q^{4} + ( - \beta_{3} + \beta_1 - 70) q^{5} + 8 \beta_1 q^{6} + ( - 2 \beta_{2} - 7 \beta_1 - 136) q^{7} + 512 q^{8} + (6 \beta_{3} + 3 \beta_{2} + \cdots + 1305) q^{9}+O(q^{10})$$ q + 8 * q^2 + b1 * q^3 + 64 * q^4 + (-b3 + b1 - 70) * q^5 + 8*b1 * q^6 + (-2*b2 - 7*b1 - 136) * q^7 + 512 * q^8 + (6*b3 + 3*b2 + 9*b1 + 1305) * q^9 $$q + 8 q^{2} + \beta_1 q^{3} + 64 q^{4} + ( - \beta_{3} + \beta_1 - 70) q^{5} + 8 \beta_1 q^{6} + ( - 2 \beta_{2} - 7 \beta_1 - 136) q^{7} + 512 q^{8} + (6 \beta_{3} + 3 \beta_{2} + \cdots + 1305) q^{9}+ \cdots + ( - 7152 \beta_{3} - 1020 \beta_{2} + \cdots + 2586348) q^{99}+O(q^{100})$$ q + 8 * q^2 + b1 * q^3 + 64 * q^4 + (-b3 + b1 - 70) * q^5 + 8*b1 * q^6 + (-2*b2 - 7*b1 - 136) * q^7 + 512 * q^8 + (6*b3 + 3*b2 + 9*b1 + 1305) * q^9 + (-8*b3 + 8*b1 - 560) * q^10 + (8*b2 + 19*b1 - 1852) * q^11 + 64*b1 * q^12 + (-16*b2 - 56*b1 - 1088) * q^14 + (48*b3 - 12*b2 - 206*b1 + 3912) * q^15 + 4096 * q^16 + (-34*b3 + 14*b2 - 196*b1 - 7103) * q^17 + (48*b3 + 24*b2 + 72*b1 + 10440) * q^18 + (-56*b3 - 259*b1 - 25000) * q^19 + (-64*b3 + 64*b1 - 4480) * q^20 + (-126*b3 - 87*b2 - 547*b1 - 22788) * q^21 + (64*b2 + 152*b1 - 14816) * q^22 + (176*b3 - 74*b2 - 399*b1 + 8472) * q^23 + 512*b1 * q^24 + (168*b3 + 43*b2 - 1561*b1 + 21782) * q^25 + (-72*b3 + 216*b2 + 591*b1 + 26424) * q^27 + (-128*b2 - 448*b1 - 8704) * q^28 + (365*b3 + 187*b2 + 600*b1 - 23196) * q^29 + (384*b3 - 96*b2 - 1648*b1 + 31296) * q^30 + (100*b3 - 88*b2 + 204*b1 - 77696) * q^31 + 32768 * q^32 + (450*b3 + 321*b2 - 289*b1 + 59724) * q^33 + (-272*b3 + 112*b2 - 1568*b1 - 56824) * q^34 + (64*b3 - 12*b2 + 2918*b1 - 35348) * q^35 + (384*b3 + 192*b2 + 576*b1 + 83520) * q^36 + (-281*b3 - 3*b2 - 868*b1 - 2548) * q^37 + (-448*b3 - 2072*b1 - 200000) * q^38 + (-512*b3 + 512*b1 - 35840) * q^40 + (862*b3 + 70*b2 - 3560*b1 + 21119) * q^41 + (-1008*b3 - 696*b2 - 4376*b1 - 182304) * q^42 + (1080*b3 + 560*b2 - 5285*b1 + 142576) * q^43 + (512*b2 + 1216*b1 - 118528) * q^44 + (-1569*b3 - 294*b2 + 4743*b1 - 576486) * q^45 + (1408*b3 - 592*b2 - 3192*b1 + 67776) * q^46 + (-1080*b3 + 692*b2 - 8648*b1 + 142664) * q^47 + 4096*b1 * q^48 + (462*b3 + 231*b2 + 14049*b1 + 179565) * q^49 + (1344*b3 + 344*b2 - 12488*b1 + 174256) * q^50 + (840*b3 - 636*b2 - 11361*b1 - 681744) * q^51 + (689*b3 - 2368*b2 - 4153*b1 + 310366) * q^53 + (-576*b3 + 1728*b2 + 4728*b1 + 211392) * q^54 + (1708*b3 + 156*b2 - 12214*b1 + 273904) * q^55 + (-1024*b2 - 3584*b1 - 69632) * q^56 + (798*b3 - 1617*b2 - 35451*b1 - 880908) * q^57 + (2920*b3 + 1496*b2 + 4800*b1 - 185568) * q^58 + (-5712*b3 - 1568*b2 - 12799*b1 + 55804) * q^59 + (3072*b3 - 768*b2 - 13184*b1 + 250368) * q^60 + (729*b3 - 2353*b2 + 43060*b1 - 169880) * q^61 + (800*b3 - 704*b2 + 1632*b1 - 621568) * q^62 + (-1644*b3 - 2028*b2 - 45810*b1 - 1487736) * q^63 + 262144 * q^64 + (3600*b3 + 2568*b2 - 2312*b1 + 477792) * q^66 + (-1568*b3 - 2176*b2 + 8379*b1 + 818068) * q^67 + (-2176*b3 + 896*b2 - 12544*b1 - 454592) * q^68 + (-12894*b3 - 999*b2 + 17525*b1 - 1405956) * q^69 + (512*b3 - 96*b2 + 23344*b1 - 282784) * q^70 + (3940*b3 - 1038*b2 + 39029*b1 - 41264) * q^71 + (3072*b3 + 1536*b2 + 4608*b1 + 668160) * q^72 + (1266*b3 - 2457*b2 - 40803*b1 - 1782611) * q^73 + (-2248*b3 - 24*b2 - 6944*b1 - 20384) * q^74 + (-14616*b3 - 744*b2 + 39575*b1 - 5557176) * q^75 + (-3584*b3 - 16576*b1 - 1600000) * q^76 + (-714*b3 + 4651*b2 - 34501*b1 - 3481484) * q^77 + (-9732*b3 + 3516*b2 + 35196*b1 - 1770112) * q^79 + (-4096*b3 + 4096*b1 - 286720) * q^80 + (2520*b3 + 1260*b2 + 39204*b1 - 938871) * q^81 + (6896*b3 + 560*b2 - 28480*b1 + 168952) * q^82 + (-14520*b3 + 4644*b2 - 3948*b1 - 173904) * q^83 + (-8064*b3 - 5568*b2 - 35008*b1 - 1458432) * q^84 + (-2453*b3 + 4894*b2 - 21701*b1 + 2950076) * q^85 + (8640*b3 + 4480*b2 - 42280*b1 + 1140608) * q^86 + (-3876*b3 + 13446*b2 + 67667*b1 + 1787064) * q^87 + (4096*b2 + 9728*b1 - 948224) * q^88 + (9298*b3 + 9577*b2 + 116719*b1 - 2863198) * q^89 + (-12552*b3 - 2352*b2 + 37944*b1 - 4611888) * q^90 + (11264*b3 - 4736*b2 - 25536*b1 + 542208) * q^92 + (-6672*b3 - 792*b2 - 76672*b1 + 743232) * q^93 + (-8640*b3 + 5536*b2 - 69184*b1 + 1141312) * q^94 + (15368*b3 + 6188*b2 - 43606*b1 + 5838112) * q^95 + 32768*b1 * q^96 + (8330*b3 + 7973*b2 - 55489*b1 - 106822) * q^97 + (3696*b3 + 1848*b2 + 112392*b1 + 1436520) * q^98 + (-7152*b3 - 1020*b2 + 136674*b1 + 2586348) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 32 q^{2} + 256 q^{4} - 278 q^{5} - 548 q^{7} + 2048 q^{8} + 5214 q^{9}+O(q^{10})$$ 4 * q + 32 * q^2 + 256 * q^4 - 278 * q^5 - 548 * q^7 + 2048 * q^8 + 5214 * q^9 $$4 q + 32 q^{2} + 256 q^{4} - 278 q^{5} - 548 q^{7} + 2048 q^{8} + 5214 q^{9} - 2224 q^{10} - 7392 q^{11} - 4384 q^{14} + 15528 q^{15} + 16384 q^{16} - 28316 q^{17} + 41712 q^{18} - 99888 q^{19} - 17792 q^{20} - 91074 q^{21} - 59136 q^{22} + 33388 q^{23} + 86878 q^{25} + 106272 q^{27} - 35072 q^{28} - 93140 q^{29} + 124224 q^{30} - 311160 q^{31} + 131072 q^{32} + 238638 q^{33} - 226528 q^{34} - 141544 q^{35} + 333696 q^{36} - 9636 q^{37} - 799104 q^{38} - 142336 q^{40} + 82892 q^{41} - 728592 q^{42} + 569264 q^{43} - 473088 q^{44} - 2303394 q^{45} + 267104 q^{46} + 574200 q^{47} + 717798 q^{49} + 695024 q^{50} - 2729928 q^{51} + 1235350 q^{53} + 850176 q^{54} + 1092512 q^{55} - 280576 q^{56} - 3528462 q^{57} - 745120 q^{58} + 231504 q^{59} + 993792 q^{60} - 685684 q^{61} - 2489280 q^{62} - 5951712 q^{63} + 1048576 q^{64} + 1909104 q^{66} + 3271056 q^{67} - 1812224 q^{68} - 5600034 q^{69} - 1132352 q^{70} - 175012 q^{71} + 2669568 q^{72} - 7137890 q^{73} - 77088 q^{74} - 22200960 q^{75} - 6392832 q^{76} - 13915206 q^{77} - 7053952 q^{79} - 1138688 q^{80} - 3758004 q^{81} + 663136 q^{82} - 657288 q^{83} - 5828736 q^{84} + 11814998 q^{85} + 4554112 q^{86} + 7182900 q^{87} - 3784704 q^{88} - 11452234 q^{89} - 18427152 q^{90} + 2136832 q^{92} + 2984688 q^{93} + 4593600 q^{94} + 23334088 q^{95} - 428002 q^{97} + 5742384 q^{98} + 10357656 q^{99}+O(q^{100})$$ 4 * q + 32 * q^2 + 256 * q^4 - 278 * q^5 - 548 * q^7 + 2048 * q^8 + 5214 * q^9 - 2224 * q^10 - 7392 * q^11 - 4384 * q^14 + 15528 * q^15 + 16384 * q^16 - 28316 * q^17 + 41712 * q^18 - 99888 * q^19 - 17792 * q^20 - 91074 * q^21 - 59136 * q^22 + 33388 * q^23 + 86878 * q^25 + 106272 * q^27 - 35072 * q^28 - 93140 * q^29 + 124224 * q^30 - 311160 * q^31 + 131072 * q^32 + 238638 * q^33 - 226528 * q^34 - 141544 * q^35 + 333696 * q^36 - 9636 * q^37 - 799104 * q^38 - 142336 * q^40 + 82892 * q^41 - 728592 * q^42 + 569264 * q^43 - 473088 * q^44 - 2303394 * q^45 + 267104 * q^46 + 574200 * q^47 + 717798 * q^49 + 695024 * q^50 - 2729928 * q^51 + 1235350 * q^53 + 850176 * q^54 + 1092512 * q^55 - 280576 * q^56 - 3528462 * q^57 - 745120 * q^58 + 231504 * q^59 + 993792 * q^60 - 685684 * q^61 - 2489280 * q^62 - 5951712 * q^63 + 1048576 * q^64 + 1909104 * q^66 + 3271056 * q^67 - 1812224 * q^68 - 5600034 * q^69 - 1132352 * q^70 - 175012 * q^71 + 2669568 * q^72 - 7137890 * q^73 - 77088 * q^74 - 22200960 * q^75 - 6392832 * q^76 - 13915206 * q^77 - 7053952 * q^79 - 1138688 * q^80 - 3758004 * q^81 + 663136 * q^82 - 657288 * q^83 - 5828736 * q^84 + 11814998 * q^85 + 4554112 * q^86 + 7182900 * q^87 - 3784704 * q^88 - 11452234 * q^89 - 18427152 * q^90 + 2136832 * q^92 + 2984688 * q^93 + 4593600 * q^94 + 23334088 * q^95 - 428002 * q^97 + 5742384 * q^98 + 10357656 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6981x^{2} - 35424x + 7188480$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 12\nu^{2} - 5073\nu - 68328 ) / 252$$ (v^3 + 12*v^2 - 5073*v - 68328) / 252 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 72\nu^{2} + 4317\nu - 225000 ) / 504$$ (-v^3 + 72*v^2 + 4317*v - 225000) / 504
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$6\beta_{3} + 3\beta_{2} + 9\beta _1 + 3492$$ 6*b3 + 3*b2 + 9*b1 + 3492 $$\nu^{3}$$ $$=$$ $$-72\beta_{3} + 216\beta_{2} + 4965\beta _1 + 26424$$ -72*b3 + 216*b2 + 4965*b1 + 26424

## 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
 −71.1584 −39.9844 31.8716 79.2712
8.00000 −71.1584 64.0000 −523.489 −569.267 416.801 512.000 2876.52 −4187.91
1.2 8.00000 −39.9844 64.0000 323.700 −319.875 −568.591 512.000 −588.246 2589.60
1.3 8.00000 31.8716 64.0000 54.4265 254.973 1112.71 512.000 −1171.20 435.412
1.4 8.00000 79.2712 64.0000 −132.638 634.170 −1508.92 512.000 4096.92 −1061.10
 $$n$$: 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 338.8.a.j 4
13.b even 2 1 338.8.a.i 4
13.d odd 4 2 338.8.b.h 8
13.e even 6 2 26.8.c.b 8
39.h odd 6 2 234.8.h.b 8
52.i odd 6 2 208.8.i.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.c.b 8 13.e even 6 2
208.8.i.b 8 52.i odd 6 2
234.8.h.b 8 39.h odd 6 2
338.8.a.i 4 13.b even 2 1
338.8.a.j 4 1.a even 1 1 trivial
338.8.b.h 8 13.d odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(338))$$:

 $$T_{3}^{4} - 6981T_{3}^{2} - 35424T_{3} + 7188480$$ T3^4 - 6981*T3^2 - 35424*T3 + 7188480 $$T_{5}^{4} + 278T_{5}^{3} - 161047T_{5}^{2} - 14695460T_{5} + 1223288100$$ T5^4 + 278*T5^3 - 161047*T5^2 - 14695460*T5 + 1223288100

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 8)^{4}$$
$3$ $$T^{4} - 6981 T^{2} + \cdots + 7188480$$
$5$ $$T^{4} + \cdots + 1223288100$$
$7$ $$T^{4} + \cdots + 397901702960$$
$11$ $$T^{4} + \cdots + 17021121355056$$
$13$ $$T^{4}$$
$17$ $$T^{4} + \cdots - 59\!\cdots\!71$$
$19$ $$T^{4} + \cdots - 40\!\cdots\!00$$
$23$ $$T^{4} + \cdots + 80\!\cdots\!44$$
$29$ $$T^{4} + \cdots + 25\!\cdots\!13$$
$31$ $$T^{4} + \cdots + 10\!\cdots\!28$$
$37$ $$T^{4} + \cdots - 16\!\cdots\!79$$
$41$ $$T^{4} + \cdots - 27\!\cdots\!15$$
$43$ $$T^{4} + \cdots - 44\!\cdots\!64$$
$47$ $$T^{4} + \cdots + 87\!\cdots\!28$$
$53$ $$T^{4} + \cdots + 10\!\cdots\!32$$
$59$ $$T^{4} + \cdots + 93\!\cdots\!08$$
$61$ $$T^{4} + \cdots + 60\!\cdots\!45$$
$67$ $$T^{4} + \cdots - 12\!\cdots\!64$$
$71$ $$T^{4} + \cdots + 24\!\cdots\!88$$
$73$ $$T^{4} + \cdots - 21\!\cdots\!40$$
$79$ $$T^{4} + \cdots + 46\!\cdots\!08$$
$83$ $$T^{4} + \cdots - 67\!\cdots\!00$$
$89$ $$T^{4} + \cdots - 52\!\cdots\!56$$
$97$ $$T^{4} + \cdots + 48\!\cdots\!28$$