# Properties

 Label 13.8.a.b Level $13$ Weight $8$ Character orbit 13.a Self dual yes Analytic conductor $4.061$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,8,Mod(1,13)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, 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("13.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 13.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: $$4.06100533129$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{337})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 84$$ x^2 - x - 84 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\beta = \frac{1}{2}(1 + \sqrt{337})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 9) q^{2} + (3 \beta + 21) q^{3} + (19 \beta + 37) q^{4} + ( - 11 \beta - 171) q^{5} + ( - 51 \beta - 441) q^{6} + (9 \beta - 1009) q^{7} + ( - 99 \beta - 777) q^{8} + (135 \beta - 990) q^{9}+O(q^{10})$$ q + (-b - 9) * q^2 + (3*b + 21) * q^3 + (19*b + 37) * q^4 + (-11*b - 171) * q^5 + (-51*b - 441) * q^6 + (9*b - 1009) * q^7 + (-99*b - 777) * q^8 + (135*b - 990) * q^9 $$q + ( - \beta - 9) q^{2} + (3 \beta + 21) q^{3} + (19 \beta + 37) q^{4} + ( - 11 \beta - 171) q^{5} + ( - 51 \beta - 441) q^{6} + (9 \beta - 1009) q^{7} + ( - 99 \beta - 777) q^{8} + (135 \beta - 990) q^{9} + (281 \beta + 2463) q^{10} + ( - 622 \beta - 594) q^{11} + (567 \beta + 5565) q^{12} + 2197 q^{13} + (919 \beta + 8325) q^{14} + ( - 777 \beta - 6363) q^{15} + ( - 665 \beta + 10573) q^{16} + ( - 2003 \beta - 11679) q^{17} + ( - 360 \beta - 2430) q^{18} + (4582 \beta + 8762) q^{19} + ( - 3865 \beta - 23883) q^{20} + ( - 2811 \beta - 18921) q^{21} + (6814 \beta + 57594) q^{22} + (6792 \beta - 16608) q^{23} + ( - 4707 \beta - 41265) q^{24} + (3883 \beta - 38720) q^{25} + ( - 2197 \beta - 19773) q^{26} + ( - 6291 \beta - 32697) q^{27} + ( - 18667 \beta - 22969) q^{28} + (3544 \beta - 4674) q^{29} + (14133 \beta + 122535) q^{30} + ( - 3496 \beta + 21620) q^{31} + (8749 \beta + 60159) q^{32} + ( - 16710 \beta - 169218) q^{33} + (31709 \beta + 273363) q^{34} + (9461 \beta + 164223) q^{35} + ( - 11250 \beta + 178830) q^{36} + ( - 13135 \beta + 88217) q^{37} + ( - 54582 \beta - 463746) q^{38} + (6591 \beta + 46137) q^{39} + (26565 \beta + 224343) q^{40} + (41178 \beta - 186024) q^{41} + (47031 \beta + 406413) q^{42} + (4357 \beta + 112475) q^{43} + ( - 46118 \beta - 1014690) q^{44} + ( - 13680 \beta + 44550) q^{45} + ( - 51312 \beta - 421056) q^{46} + (4221 \beta - 821373) q^{47} + (15759 \beta + 54453) q^{48} + ( - 18081 \beta + 201342) q^{49} + ( - 110 \beta + 22308) q^{50} + ( - 83109 \beta - 750015) q^{51} + (41743 \beta + 81289) q^{52} + ( - 104610 \beta + 575496) q^{53} + (95607 \beta + 822717) q^{54} + (119738 \beta + 676302) q^{55} + (92007 \beta + 709149) q^{56} + (136254 \beta + 1338666) q^{57} + ( - 30766 \beta - 255630) q^{58} + (45486 \beta - 207822) q^{59} + ( - 164409 \beta - 1475523) q^{60} + ( - 78370 \beta + 2376896) q^{61} + (13340 \beta + 99084) q^{62} + ( - 143910 \beta + 1100970) q^{63} + ( - 62529 \beta - 2629691) q^{64} + ( - 24167 \beta - 375687) q^{65} + (336318 \beta + 2926602) q^{66} + ( - 272038 \beta - 774682) q^{67} + ( - 334069 \beta - 3628911) q^{68} + (113184 \beta + 1362816) q^{69} + ( - 258833 \beta - 2272731) q^{70} + (545931 \beta - 840771) q^{71} + ( - 20250 \beta - 353430) q^{72} + (57000 \beta - 3258142) q^{73} + (43133 \beta + 309387) q^{74} + ( - 22968 \beta + 165396) q^{75} + (423070 \beta + 7637066) q^{76} + (616654 \beta + 129114) q^{77} + ( - 112047 \beta - 968877) q^{78} + ( - 516480 \beta + 221336) q^{79} + (4727 \beta - 1193523) q^{80} + ( - 544320 \beta - 106839) q^{81} + ( - 225756 \beta - 1784736) q^{82} + (163292 \beta - 6132132) q^{83} + ( - 516915 \beta - 5186433) q^{84} + (493015 \beta + 3847881) q^{85} + ( - 156045 \beta - 1378263) q^{86} + (71034 \beta + 794934) q^{87} + (603678 \beta + 5634090) q^{88} + ( - 639712 \beta + 5227386) q^{89} + (92250 \beta + 748170) q^{90} + (19773 \beta - 2216773) q^{91} + (64800 \beta + 10225536) q^{92} + ( - 19044 \beta - 426972) q^{93} + (779163 \beta + 7037793) q^{94} + ( - 930306 \beta - 5732070) q^{95} + (390453 \beta + 3468087) q^{96} + ( - 615988 \beta - 8487850) q^{97} + ( - 20532 \beta - 293274) q^{98} + (451620 \beta - 6465420) q^{99}+O(q^{100})$$ q + (-b - 9) * q^2 + (3*b + 21) * q^3 + (19*b + 37) * q^4 + (-11*b - 171) * q^5 + (-51*b - 441) * q^6 + (9*b - 1009) * q^7 + (-99*b - 777) * q^8 + (135*b - 990) * q^9 + (281*b + 2463) * q^10 + (-622*b - 594) * q^11 + (567*b + 5565) * q^12 + 2197 * q^13 + (919*b + 8325) * q^14 + (-777*b - 6363) * q^15 + (-665*b + 10573) * q^16 + (-2003*b - 11679) * q^17 + (-360*b - 2430) * q^18 + (4582*b + 8762) * q^19 + (-3865*b - 23883) * q^20 + (-2811*b - 18921) * q^21 + (6814*b + 57594) * q^22 + (6792*b - 16608) * q^23 + (-4707*b - 41265) * q^24 + (3883*b - 38720) * q^25 + (-2197*b - 19773) * q^26 + (-6291*b - 32697) * q^27 + (-18667*b - 22969) * q^28 + (3544*b - 4674) * q^29 + (14133*b + 122535) * q^30 + (-3496*b + 21620) * q^31 + (8749*b + 60159) * q^32 + (-16710*b - 169218) * q^33 + (31709*b + 273363) * q^34 + (9461*b + 164223) * q^35 + (-11250*b + 178830) * q^36 + (-13135*b + 88217) * q^37 + (-54582*b - 463746) * q^38 + (6591*b + 46137) * q^39 + (26565*b + 224343) * q^40 + (41178*b - 186024) * q^41 + (47031*b + 406413) * q^42 + (4357*b + 112475) * q^43 + (-46118*b - 1014690) * q^44 + (-13680*b + 44550) * q^45 + (-51312*b - 421056) * q^46 + (4221*b - 821373) * q^47 + (15759*b + 54453) * q^48 + (-18081*b + 201342) * q^49 + (-110*b + 22308) * q^50 + (-83109*b - 750015) * q^51 + (41743*b + 81289) * q^52 + (-104610*b + 575496) * q^53 + (95607*b + 822717) * q^54 + (119738*b + 676302) * q^55 + (92007*b + 709149) * q^56 + (136254*b + 1338666) * q^57 + (-30766*b - 255630) * q^58 + (45486*b - 207822) * q^59 + (-164409*b - 1475523) * q^60 + (-78370*b + 2376896) * q^61 + (13340*b + 99084) * q^62 + (-143910*b + 1100970) * q^63 + (-62529*b - 2629691) * q^64 + (-24167*b - 375687) * q^65 + (336318*b + 2926602) * q^66 + (-272038*b - 774682) * q^67 + (-334069*b - 3628911) * q^68 + (113184*b + 1362816) * q^69 + (-258833*b - 2272731) * q^70 + (545931*b - 840771) * q^71 + (-20250*b - 353430) * q^72 + (57000*b - 3258142) * q^73 + (43133*b + 309387) * q^74 + (-22968*b + 165396) * q^75 + (423070*b + 7637066) * q^76 + (616654*b + 129114) * q^77 + (-112047*b - 968877) * q^78 + (-516480*b + 221336) * q^79 + (4727*b - 1193523) * q^80 + (-544320*b - 106839) * q^81 + (-225756*b - 1784736) * q^82 + (163292*b - 6132132) * q^83 + (-516915*b - 5186433) * q^84 + (493015*b + 3847881) * q^85 + (-156045*b - 1378263) * q^86 + (71034*b + 794934) * q^87 + (603678*b + 5634090) * q^88 + (-639712*b + 5227386) * q^89 + (92250*b + 748170) * q^90 + (19773*b - 2216773) * q^91 + (64800*b + 10225536) * q^92 + (-19044*b - 426972) * q^93 + (779163*b + 7037793) * q^94 + (-930306*b - 5732070) * q^95 + (390453*b + 3468087) * q^96 + (-615988*b - 8487850) * q^97 + (-20532*b - 293274) * q^98 + (451620*b - 6465420) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 19 q^{2} + 45 q^{3} + 93 q^{4} - 353 q^{5} - 933 q^{6} - 2009 q^{7} - 1653 q^{8} - 1845 q^{9}+O(q^{10})$$ 2 * q - 19 * q^2 + 45 * q^3 + 93 * q^4 - 353 * q^5 - 933 * q^6 - 2009 * q^7 - 1653 * q^8 - 1845 * q^9 $$2 q - 19 q^{2} + 45 q^{3} + 93 q^{4} - 353 q^{5} - 933 q^{6} - 2009 q^{7} - 1653 q^{8} - 1845 q^{9} + 5207 q^{10} - 1810 q^{11} + 11697 q^{12} + 4394 q^{13} + 17569 q^{14} - 13503 q^{15} + 20481 q^{16} - 25361 q^{17} - 5220 q^{18} + 22106 q^{19} - 51631 q^{20} - 40653 q^{21} + 122002 q^{22} - 26424 q^{23} - 87237 q^{24} - 73557 q^{25} - 41743 q^{26} - 71685 q^{27} - 64605 q^{28} - 5804 q^{29} + 259203 q^{30} + 39744 q^{31} + 129067 q^{32} - 355146 q^{33} + 578435 q^{34} + 337907 q^{35} + 346410 q^{36} + 163299 q^{37} - 982074 q^{38} + 98865 q^{39} + 475251 q^{40} - 330870 q^{41} + 859857 q^{42} + 229307 q^{43} - 2075498 q^{44} + 75420 q^{45} - 893424 q^{46} - 1638525 q^{47} + 124665 q^{48} + 384603 q^{49} + 44506 q^{50} - 1583139 q^{51} + 204321 q^{52} + 1046382 q^{53} + 1741041 q^{54} + 1472342 q^{55} + 1510305 q^{56} + 2813586 q^{57} - 542026 q^{58} - 370158 q^{59} - 3115455 q^{60} + 4675422 q^{61} + 211508 q^{62} + 2058030 q^{63} - 5321911 q^{64} - 775541 q^{65} + 6189522 q^{66} - 1821402 q^{67} - 7591891 q^{68} + 2838816 q^{69} - 4804295 q^{70} - 1135611 q^{71} - 727110 q^{72} - 6459284 q^{73} + 661907 q^{74} + 307824 q^{75} + 15697202 q^{76} + 874882 q^{77} - 2049801 q^{78} - 73808 q^{79} - 2382319 q^{80} - 757998 q^{81} - 3795228 q^{82} - 12100972 q^{83} - 10889781 q^{84} + 8188777 q^{85} - 2912571 q^{86} + 1660902 q^{87} + 11871858 q^{88} + 9815060 q^{89} + 1588590 q^{90} - 4413773 q^{91} + 20515872 q^{92} - 872988 q^{93} + 14854749 q^{94} - 12394446 q^{95} + 7326627 q^{96} - 17591688 q^{97} - 607080 q^{98} - 12479220 q^{99}+O(q^{100})$$ 2 * q - 19 * q^2 + 45 * q^3 + 93 * q^4 - 353 * q^5 - 933 * q^6 - 2009 * q^7 - 1653 * q^8 - 1845 * q^9 + 5207 * q^10 - 1810 * q^11 + 11697 * q^12 + 4394 * q^13 + 17569 * q^14 - 13503 * q^15 + 20481 * q^16 - 25361 * q^17 - 5220 * q^18 + 22106 * q^19 - 51631 * q^20 - 40653 * q^21 + 122002 * q^22 - 26424 * q^23 - 87237 * q^24 - 73557 * q^25 - 41743 * q^26 - 71685 * q^27 - 64605 * q^28 - 5804 * q^29 + 259203 * q^30 + 39744 * q^31 + 129067 * q^32 - 355146 * q^33 + 578435 * q^34 + 337907 * q^35 + 346410 * q^36 + 163299 * q^37 - 982074 * q^38 + 98865 * q^39 + 475251 * q^40 - 330870 * q^41 + 859857 * q^42 + 229307 * q^43 - 2075498 * q^44 + 75420 * q^45 - 893424 * q^46 - 1638525 * q^47 + 124665 * q^48 + 384603 * q^49 + 44506 * q^50 - 1583139 * q^51 + 204321 * q^52 + 1046382 * q^53 + 1741041 * q^54 + 1472342 * q^55 + 1510305 * q^56 + 2813586 * q^57 - 542026 * q^58 - 370158 * q^59 - 3115455 * q^60 + 4675422 * q^61 + 211508 * q^62 + 2058030 * q^63 - 5321911 * q^64 - 775541 * q^65 + 6189522 * q^66 - 1821402 * q^67 - 7591891 * q^68 + 2838816 * q^69 - 4804295 * q^70 - 1135611 * q^71 - 727110 * q^72 - 6459284 * q^73 + 661907 * q^74 + 307824 * q^75 + 15697202 * q^76 + 874882 * q^77 - 2049801 * q^78 - 73808 * q^79 - 2382319 * q^80 - 757998 * q^81 - 3795228 * q^82 - 12100972 * q^83 - 10889781 * q^84 + 8188777 * q^85 - 2912571 * q^86 + 1660902 * q^87 + 11871858 * q^88 + 9815060 * q^89 + 1588590 * q^90 - 4413773 * q^91 + 20515872 * q^92 - 872988 * q^93 + 14854749 * q^94 - 12394446 * q^95 + 7326627 * q^96 - 17591688 * q^97 - 607080 * q^98 - 12479220 * 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
 9.67878 −8.67878
−18.6788 50.0363 220.897 −277.467 −934.618 −921.891 −1735.20 316.635 5182.74
1.2 −0.321220 −5.03634 −127.897 −75.5334 1.61777 −1087.11 82.1992 −2161.64 24.2629
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$-1$$

## 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 13.8.a.b 2
3.b odd 2 1 117.8.a.c 2
4.b odd 2 1 208.8.a.g 2
5.b even 2 1 325.8.a.b 2
13.b even 2 1 169.8.a.b 2
13.d odd 4 2 169.8.b.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.a.b 2 1.a even 1 1 trivial
117.8.a.c 2 3.b odd 2 1
169.8.a.b 2 13.b even 2 1
169.8.b.b 4 13.d odd 4 2
208.8.a.g 2 4.b odd 2 1
325.8.a.b 2 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 19T_{2} + 6$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(13))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 19T + 6$$
$3$ $$T^{2} - 45T - 252$$
$5$ $$T^{2} + 353T + 20958$$
$7$ $$T^{2} + 2009 T + 1002196$$
$11$ $$T^{2} + 1810 T - 31775952$$
$13$ $$(T - 2197)^{2}$$
$17$ $$T^{2} + 25361 T - 177216678$$
$19$ $$T^{2} + \cdots - 1646636688$$
$23$ $$T^{2} + \cdots - 3712002048$$
$29$ $$T^{2} + \cdots - 1049753004$$
$31$ $$T^{2} - 39744 T - 634808464$$
$37$ $$T^{2} + \cdots - 7868862106$$
$41$ $$T^{2} + \cdots - 115487893152$$
$43$ $$T^{2} + \cdots + 11546069484$$
$47$ $$T^{2} + \cdots + 669689975052$$
$53$ $$T^{2} + \cdots - 648240166944$$
$59$ $$T^{2} + \cdots - 140057008272$$
$61$ $$T^{2} + \cdots + 4947441275696$$
$67$ $$T^{2} + \cdots - 5405517426256$$
$71$ $$T^{2} + \cdots - 24787522246284$$
$73$ $$T^{2} + \cdots + 10156859198164$$
$79$ $$T^{2} + \cdots - 22472459585984$$
$83$ $$T^{2} + \cdots + 34361915476704$$
$89$ $$T^{2} + \cdots - 10393898367132$$
$97$ $$T^{2} + \cdots + 45398949212204$$