# Properties

 Label 13.14.c.a Level $13$ Weight $14$ Character orbit 13.c Analytic conductor $13.940$ Analytic rank $0$ Dimension $28$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,14,Mod(3,13)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 14, names="a")

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

chi := DirichletCharacter("13.3");

S:= CuspForms(chi, 14);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$14$$ Character orbit: $$[\chi]$$ $$=$$ 13.c (of order $$3$$, degree $$2$$, 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: $$13.9400207637$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 65 q^{2} + 728 q^{3} - 53249 q^{4} + 41860 q^{5} + 143910 q^{6} - 173992 q^{7} + 2616978 q^{8} - 5231838 q^{9}+O(q^{10})$$ 28 * q - 65 * q^2 + 728 * q^3 - 53249 * q^4 + 41860 * q^5 + 143910 * q^6 - 173992 * q^7 + 2616978 * q^8 - 5231838 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 65 q^{2} + 728 q^{3} - 53249 q^{4} + 41860 q^{5} + 143910 q^{6} - 173992 q^{7} + 2616978 q^{8} - 5231838 q^{9} + 2769243 q^{10} + 10986144 q^{11} - 45205464 q^{12} + 25107706 q^{13} - 23731756 q^{14} - 75354448 q^{15} - 273433217 q^{16} + 255934950 q^{17} - 350685842 q^{18} - 266623552 q^{19} - 225433091 q^{20} - 2515875856 q^{21} + 77223114 q^{22} + 1914406544 q^{23} - 680727216 q^{24} + 5225878768 q^{25} - 9205789749 q^{26} + 3550452320 q^{27} - 3680076400 q^{28} + 6211296126 q^{29} + 5703660004 q^{30} + 6997674944 q^{31} + 2312486527 q^{32} + 5621901584 q^{33} - 23373723594 q^{34} + 23009265024 q^{35} + 23496114839 q^{36} + 19577630774 q^{37} + 51648723644 q^{38} + 31282255680 q^{39} - 111700587858 q^{40} - 6039692074 q^{41} + 143590583874 q^{42} + 16878499784 q^{43} - 489551846992 q^{44} - 148673926778 q^{45} + 177583142490 q^{46} + 369404914944 q^{47} + 92920518608 q^{48} - 195653684766 q^{49} - 753661423970 q^{50} + 617169279136 q^{51} - 727838751302 q^{52} - 287550769100 q^{53} + 365361720282 q^{54} + 855926433792 q^{55} + 192505368988 q^{56} - 1924513438848 q^{57} - 660641292207 q^{58} + 637655529992 q^{59} + 2555257062384 q^{60} + 1530666437030 q^{61} + 1907899186036 q^{62} - 832654573712 q^{63} + 632232280450 q^{64} - 2489693680994 q^{65} - 270853649100 q^{66} - 1120398677944 q^{67} + 4085795015901 q^{68} + 1473200404976 q^{69} - 9377684744688 q^{70} + 1506585643472 q^{71} + 3521118562923 q^{72} - 852513981436 q^{73} + 4665118313621 q^{74} + 2605103918408 q^{75} - 6106356745372 q^{76} + 2854115035024 q^{77} - 8622775525858 q^{78} - 7032676662784 q^{79} + 5983641421745 q^{80} + 2512348302642 q^{81} - 8600700659175 q^{82} + 2033940247872 q^{83} - 4374201207844 q^{84} + 2527554975426 q^{85} - 7547295057588 q^{86} - 1118899666832 q^{87} - 5943200108772 q^{88} + 5229645042596 q^{89} + 4220465391930 q^{90} + 18401936100704 q^{91} + 3830251788320 q^{92} - 1283055655752 q^{93} - 32271602536104 q^{94} + 8980600417472 q^{95} + 51692558969792 q^{96} + 3129146550452 q^{97} - 5503659823363 q^{98} + 33239809172544 q^{99}+O(q^{100})$$ 28 * q - 65 * q^2 + 728 * q^3 - 53249 * q^4 + 41860 * q^5 + 143910 * q^6 - 173992 * q^7 + 2616978 * q^8 - 5231838 * q^9 + 2769243 * q^10 + 10986144 * q^11 - 45205464 * q^12 + 25107706 * q^13 - 23731756 * q^14 - 75354448 * q^15 - 273433217 * q^16 + 255934950 * q^17 - 350685842 * q^18 - 266623552 * q^19 - 225433091 * q^20 - 2515875856 * q^21 + 77223114 * q^22 + 1914406544 * q^23 - 680727216 * q^24 + 5225878768 * q^25 - 9205789749 * q^26 + 3550452320 * q^27 - 3680076400 * q^28 + 6211296126 * q^29 + 5703660004 * q^30 + 6997674944 * q^31 + 2312486527 * q^32 + 5621901584 * q^33 - 23373723594 * q^34 + 23009265024 * q^35 + 23496114839 * q^36 + 19577630774 * q^37 + 51648723644 * q^38 + 31282255680 * q^39 - 111700587858 * q^40 - 6039692074 * q^41 + 143590583874 * q^42 + 16878499784 * q^43 - 489551846992 * q^44 - 148673926778 * q^45 + 177583142490 * q^46 + 369404914944 * q^47 + 92920518608 * q^48 - 195653684766 * q^49 - 753661423970 * q^50 + 617169279136 * q^51 - 727838751302 * q^52 - 287550769100 * q^53 + 365361720282 * q^54 + 855926433792 * q^55 + 192505368988 * q^56 - 1924513438848 * q^57 - 660641292207 * q^58 + 637655529992 * q^59 + 2555257062384 * q^60 + 1530666437030 * q^61 + 1907899186036 * q^62 - 832654573712 * q^63 + 632232280450 * q^64 - 2489693680994 * q^65 - 270853649100 * q^66 - 1120398677944 * q^67 + 4085795015901 * q^68 + 1473200404976 * q^69 - 9377684744688 * q^70 + 1506585643472 * q^71 + 3521118562923 * q^72 - 852513981436 * q^73 + 4665118313621 * q^74 + 2605103918408 * q^75 - 6106356745372 * q^76 + 2854115035024 * q^77 - 8622775525858 * q^78 - 7032676662784 * q^79 + 5983641421745 * q^80 + 2512348302642 * q^81 - 8600700659175 * q^82 + 2033940247872 * q^83 - 4374201207844 * q^84 + 2527554975426 * q^85 - 7547295057588 * q^86 - 1118899666832 * q^87 - 5943200108772 * q^88 + 5229645042596 * q^89 + 4220465391930 * q^90 + 18401936100704 * q^91 + 3830251788320 * q^92 - 1283055655752 * q^93 - 32271602536104 * q^94 + 8980600417472 * q^95 + 51692558969792 * q^96 + 3129146550452 * q^97 - 5503659823363 * q^98 + 33239809172544 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −82.8702 143.535i 451.483 + 781.992i −9638.95 + 16695.1i −67212.0 74829.1 129608.i −137812. + 238698.i 1.83738e6 389487. 674611.i 5.56987e6 + 9.64731e6i
3.2 −82.3546 142.642i −255.684 442.858i −9468.55 + 16400.0i 48546.3 −42113.5 + 72942.8i 216096. 374290.i 1.76982e6 666413. 1.15426e6i −3.99801e6 6.92476e6i
3.3 −59.2686 102.656i 1072.90 + 1858.33i −2929.54 + 5074.11i 45885.5 127179. 220281.i −177681. + 307752.i −276538. −1.50509e6 + 2.60689e6i −2.71957e6 4.71043e6i
3.4 −56.7104 98.2253i −906.506 1570.12i −2336.14 + 4046.32i −10940.5 −102817. + 178084.i −89772.2 + 155490.i −399209. −846346. + 1.46591e6i 620440. + 1.07463e6i
3.5 −45.4515 78.7243i 108.764 + 188.384i −35.6760 + 61.7927i −2568.18 9886.95 17124.7i 77358.1 133988.i −738191. 773502. 1.33975e6i 116728. + 202178.i
3.6 −16.9887 29.4253i 797.844 + 1381.91i 3518.77 6094.69i −28205.3 27108.7 46953.6i 177172. 306871.i −517460. −475948. + 824367.i 479171. + 829948.i
3.7 −3.81407 6.60616i −187.264 324.351i 4066.91 7044.09i 42407.7 −1428.48 + 2474.20i −178632. + 309400.i −124536. 727026. 1.25925e6i −161746. 280152.i
3.8 8.80659 + 15.2535i −449.698 778.901i 3940.89 6825.82i −51640.9 7920.62 13718.9i −75519.2 + 130803.i 283110. 392704. 680184.i −454780. 787702.i
3.9 11.2286 + 19.4486i −1142.04 1978.08i 3843.84 6657.72i 17999.0 25647.2 44422.2i 281521. 487609.i 356614. −1.81136e6 + 3.13737e6i 202104. + 350055.i
3.10 37.0467 + 64.1667i 958.983 + 1661.01i 1351.09 2340.15i −22075.6 −71054.3 + 123070.i −250997. + 434740.i 807186. −1.04213e6 + 1.80503e6i −817827. 1.41652e6i
3.11 40.7956 + 70.6601i 482.143 + 835.097i 767.437 1329.24i 41512.4 −39338.7 + 68136.6i 158394. 274346.i 793627. 332237. 575452.i 1.69352e6 + 2.93327e6i
3.12 60.7949 + 105.300i −355.726 616.135i −3296.05 + 5708.93i −32981.4 43252.7 74915.8i 47112.8 81601.8i 194532. 544080. 942374.i −2.00510e6 3.47294e6i
3.13 71.5224 + 123.880i −865.569 1499.21i −6134.91 + 10626.0i 41380.4 123815. 214454.i −187839. + 325347.i −583312. −701257. + 1.21461e6i 2.95963e6 + 5.12623e6i
3.14 84.7632 + 146.814i 654.369 + 1133.40i −10273.6 + 17794.4i −1177.50 −110933. + 192141.i 53602.7 92842.5i −2.09453e6 −59235.2 + 102598.i −99808.8 172874.i
9.1 −82.8702 + 143.535i 451.483 781.992i −9638.95 16695.1i −67212.0 74829.1 + 129608.i −137812. 238698.i 1.83738e6 389487. + 674611.i 5.56987e6 9.64731e6i
9.2 −82.3546 + 142.642i −255.684 + 442.858i −9468.55 16400.0i 48546.3 −42113.5 72942.8i 216096. + 374290.i 1.76982e6 666413. + 1.15426e6i −3.99801e6 + 6.92476e6i
9.3 −59.2686 + 102.656i 1072.90 1858.33i −2929.54 5074.11i 45885.5 127179. + 220281.i −177681. 307752.i −276538. −1.50509e6 2.60689e6i −2.71957e6 + 4.71043e6i
9.4 −56.7104 + 98.2253i −906.506 + 1570.12i −2336.14 4046.32i −10940.5 −102817. 178084.i −89772.2 155490.i −399209. −846346. 1.46591e6i 620440. 1.07463e6i
9.5 −45.4515 + 78.7243i 108.764 188.384i −35.6760 61.7927i −2568.18 9886.95 + 17124.7i 77358.1 + 133988.i −738191. 773502. + 1.33975e6i 116728. 202178.i
9.6 −16.9887 + 29.4253i 797.844 1381.91i 3518.77 + 6094.69i −28205.3 27108.7 + 46953.6i 177172. + 306871.i −517460. −475948. 824367.i 479171. 829948.i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.14 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.14.c.a 28
13.c even 3 1 inner 13.14.c.a 28
13.c even 3 1 169.14.a.e 14
13.e even 6 1 169.14.a.c 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.14.c.a 28 1.a even 1 1 trivial
13.14.c.a 28 13.c even 3 1 inner
169.14.a.c 14 13.e even 6 1
169.14.a.e 14 13.c even 3 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{14}^{\mathrm{new}}(13, [\chi])$$.