# Properties

 Label 13.12.c.a Level $13$ Weight $12$ Character orbit 13.c Analytic conductor $9.988$ Analytic rank $0$ Dimension $24$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,12,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, 12, names="a")

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

chi := DirichletCharacter("13.3");

S:= CuspForms(chi, 12);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$12$$ 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: $$9.98846134727$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 31 q^{2} - 244 q^{3} - 11265 q^{4} - 10436 q^{5} - 11082 q^{6} - 2928 q^{7} - 422334 q^{8} - 681618 q^{9}+O(q^{10})$$ 24 * q + 31 * q^2 - 244 * q^3 - 11265 * q^4 - 10436 * q^5 - 11082 * q^6 - 2928 * q^7 - 422334 * q^8 - 681618 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 31 q^{2} - 244 q^{3} - 11265 q^{4} - 10436 q^{5} - 11082 q^{6} - 2928 q^{7} - 422334 q^{8} - 681618 q^{9} - 312933 q^{10} - 658548 q^{11} + 3653976 q^{12} - 78858 q^{13} - 850732 q^{14} + 5608136 q^{15} - 13772481 q^{16} - 12887724 q^{17} - 28725218 q^{18} - 32419308 q^{19} + 14778349 q^{20} + 36122396 q^{21} + 4721682 q^{22} - 47910616 q^{23} + 101181576 q^{24} + 148692948 q^{25} + 168210627 q^{26} + 46563896 q^{27} + 34778544 q^{28} - 117938256 q^{29} - 81732116 q^{30} + 875192256 q^{31} + 547344127 q^{32} - 612599350 q^{33} - 2108276394 q^{34} - 1049559120 q^{35} - 88263745 q^{36} + 679293504 q^{37} - 2502336916 q^{38} + 1299727104 q^{39} + 2584583310 q^{40} - 902116756 q^{41} - 3125311638 q^{42} + 1867866084 q^{43} + 10345087184 q^{44} + 1921315366 q^{45} - 5102556102 q^{46} - 3197771328 q^{47} - 8601701584 q^{48} - 5808426978 q^{49} + 6322484494 q^{50} + 55435528 q^{51} + 9955048818 q^{52} + 7869058348 q^{53} + 5594002578 q^{54} - 14680938456 q^{55} - 18641534852 q^{56} + 14671463460 q^{57} + 15052036305 q^{58} + 1765813748 q^{59} - 1016484336 q^{60} + 3133591128 q^{61} - 1368229796 q^{62} + 6496474792 q^{63} + 18309747714 q^{64} + 6480359158 q^{65} - 33277535244 q^{66} - 10043570532 q^{67} + 8791470117 q^{68} + 12479152022 q^{69} - 65358910800 q^{70} - 29663051560 q^{71} - 19757490909 q^{72} - 77557456260 q^{73} + 34661223941 q^{74} + 26442563252 q^{75} - 52146359028 q^{76} + 106884165580 q^{77} - 81353069122 q^{78} + 91978365504 q^{79} - 45583080847 q^{80} + 83459063508 q^{81} + 18148505889 q^{82} - 185507924976 q^{83} + 135424121948 q^{84} + 24404398686 q^{85} + 400574441004 q^{86} + 64612978504 q^{87} - 15126111060 q^{88} - 123867171526 q^{89} + 474391106202 q^{90} - 5758631580 q^{91} - 311870285056 q^{92} - 44120142312 q^{93} + 156337066848 q^{94} - 74731905640 q^{95} - 1209416818816 q^{96} - 144514174830 q^{97} - 175499800915 q^{98} + 79457747424 q^{99}+O(q^{100})$$ 24 * q + 31 * q^2 - 244 * q^3 - 11265 * q^4 - 10436 * q^5 - 11082 * q^6 - 2928 * q^7 - 422334 * q^8 - 681618 * q^9 - 312933 * q^10 - 658548 * q^11 + 3653976 * q^12 - 78858 * q^13 - 850732 * q^14 + 5608136 * q^15 - 13772481 * q^16 - 12887724 * q^17 - 28725218 * q^18 - 32419308 * q^19 + 14778349 * q^20 + 36122396 * q^21 + 4721682 * q^22 - 47910616 * q^23 + 101181576 * q^24 + 148692948 * q^25 + 168210627 * q^26 + 46563896 * q^27 + 34778544 * q^28 - 117938256 * q^29 - 81732116 * q^30 + 875192256 * q^31 + 547344127 * q^32 - 612599350 * q^33 - 2108276394 * q^34 - 1049559120 * q^35 - 88263745 * q^36 + 679293504 * q^37 - 2502336916 * q^38 + 1299727104 * q^39 + 2584583310 * q^40 - 902116756 * q^41 - 3125311638 * q^42 + 1867866084 * q^43 + 10345087184 * q^44 + 1921315366 * q^45 - 5102556102 * q^46 - 3197771328 * q^47 - 8601701584 * q^48 - 5808426978 * q^49 + 6322484494 * q^50 + 55435528 * q^51 + 9955048818 * q^52 + 7869058348 * q^53 + 5594002578 * q^54 - 14680938456 * q^55 - 18641534852 * q^56 + 14671463460 * q^57 + 15052036305 * q^58 + 1765813748 * q^59 - 1016484336 * q^60 + 3133591128 * q^61 - 1368229796 * q^62 + 6496474792 * q^63 + 18309747714 * q^64 + 6480359158 * q^65 - 33277535244 * q^66 - 10043570532 * q^67 + 8791470117 * q^68 + 12479152022 * q^69 - 65358910800 * q^70 - 29663051560 * q^71 - 19757490909 * q^72 - 77557456260 * q^73 + 34661223941 * q^74 + 26442563252 * q^75 - 52146359028 * q^76 + 106884165580 * q^77 - 81353069122 * q^78 + 91978365504 * q^79 - 45583080847 * q^80 + 83459063508 * q^81 + 18148505889 * q^82 - 185507924976 * q^83 + 135424121948 * q^84 + 24404398686 * q^85 + 400574441004 * q^86 + 64612978504 * q^87 - 15126111060 * q^88 - 123867171526 * q^89 + 474391106202 * q^90 - 5758631580 * q^91 - 311870285056 * q^92 - 44120142312 * q^93 + 156337066848 * q^94 - 74731905640 * q^95 - 1209416818816 * q^96 - 144514174830 * q^97 - 175499800915 * q^98 + 79457747424 * 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 −39.1473 67.8051i −245.228 424.747i −2041.02 + 3535.16i 3256.36 −19200.0 + 33255.4i −27428.1 + 47506.9i 159255. −31700.0 + 54905.9i −127478. 220798.i
3.2 −37.0263 64.1314i 226.124 + 391.658i −1717.89 + 2975.47i 250.396 16745.0 29003.3i 22976.7 39796.8i 102768. −13690.5 + 23712.6i −9271.21 16058.2i
3.3 −22.8013 39.4930i −242.260 419.607i −15.7983 + 27.3635i −12226.1 −11047.7 + 19135.2i 35068.0 60739.5i −91953.2 −28806.7 + 49894.7i 278772. + 482847.i
3.4 −21.6665 37.5275i 147.320 + 255.166i 85.1270 147.444i −1410.79 6383.83 11057.1i −33309.3 + 57693.3i −96123.5 45166.9 78231.3i 30566.8 + 52943.2i
3.5 −12.8467 22.2512i −179.645 311.153i 693.923 1201.91i 11234.1 −4615.69 + 7994.60i 14221.5 24632.4i −88278.7 24029.2 41619.8i −144322. 249972.i
3.6 0.177445 + 0.307344i 85.3755 + 147.875i 1023.94 1773.51i −6278.25 −30.2989 + 52.4793i −7753.50 + 13429.5i 1453.59 73995.6 128164.i −1114.05 1929.58i
3.7 2.17841 + 3.77312i 374.885 + 649.321i 1014.51 1757.18i 7470.18 −1633.31 + 2828.98i 19231.4 33309.7i 17762.8 −192505. + 333428.i 16273.1 + 28185.9i
3.8 12.6130 + 21.8464i −378.076 654.847i 705.823 1222.52i −3120.93 9537.36 16519.2i −34271.3 + 59359.7i 87273.2 −197309. + 341750.i −39364.3 68181.0i
3.9 22.3480 + 38.7079i −79.2583 137.279i 25.1334 43.5322i 2002.50 3542.53 6135.84i 12414.0 21501.7i 93784.1 76009.7 131653.i 44751.9 + 77512.5i
3.10 31.0971 + 53.8618i 289.297 + 501.076i −910.060 + 1576.27i −11626.3 −17992.6 + 31164.1i 6582.83 11401.8i 14172.8 −78811.5 + 136506.i −361545. 626214.i
3.11 36.6814 + 63.5340i 151.750 + 262.839i −1667.05 + 2887.41i 10169.3 −11132.8 + 19282.6i −33907.8 + 58730.1i −94351.1 42517.2 73641.9i 373023. + 646094.i
3.12 43.8927 + 76.0244i −272.285 471.612i −2829.14 + 4900.21i −4938.39 23902.7 41400.6i 24711.6 42801.8i −316930. −59704.8 + 103412.i −216759. 375438.i
9.1 −39.1473 + 67.8051i −245.228 + 424.747i −2041.02 3535.16i 3256.36 −19200.0 33255.4i −27428.1 47506.9i 159255. −31700.0 54905.9i −127478. + 220798.i
9.2 −37.0263 + 64.1314i 226.124 391.658i −1717.89 2975.47i 250.396 16745.0 + 29003.3i 22976.7 + 39796.8i 102768. −13690.5 23712.6i −9271.21 + 16058.2i
9.3 −22.8013 + 39.4930i −242.260 + 419.607i −15.7983 27.3635i −12226.1 −11047.7 19135.2i 35068.0 + 60739.5i −91953.2 −28806.7 49894.7i 278772. 482847.i
9.4 −21.6665 + 37.5275i 147.320 255.166i 85.1270 + 147.444i −1410.79 6383.83 + 11057.1i −33309.3 57693.3i −96123.5 45166.9 + 78231.3i 30566.8 52943.2i
9.5 −12.8467 + 22.2512i −179.645 + 311.153i 693.923 + 1201.91i 11234.1 −4615.69 7994.60i 14221.5 + 24632.4i −88278.7 24029.2 + 41619.8i −144322. + 249972.i
9.6 0.177445 0.307344i 85.3755 147.875i 1023.94 + 1773.51i −6278.25 −30.2989 52.4793i −7753.50 13429.5i 1453.59 73995.6 + 128164.i −1114.05 + 1929.58i
9.7 2.17841 3.77312i 374.885 649.321i 1014.51 + 1757.18i 7470.18 −1633.31 2828.98i 19231.4 + 33309.7i 17762.8 −192505. 333428.i 16273.1 28185.9i
9.8 12.6130 21.8464i −378.076 + 654.847i 705.823 + 1222.52i −3120.93 9537.36 + 16519.2i −34271.3 59359.7i 87273.2 −197309. 341750.i −39364.3 + 68181.0i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.12 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 13.12.c.a 24
13.c even 3 1 inner 13.12.c.a 24
13.c even 3 1 169.12.a.d 12
13.e even 6 1 169.12.a.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.12.c.a 24 1.a even 1 1 trivial
13.12.c.a 24 13.c even 3 1 inner
169.12.a.d 12 13.c even 3 1
169.12.a.f 12 13.e even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{12}^{\mathrm{new}}(13, [\chi])$$.