# Properties

 Label 13.12.e.a Level $13$ Weight $12$ Character orbit 13.e Analytic conductor $9.988$ Analytic rank $0$ Dimension $22$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,12,Mod(4,13)]

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([1]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("13.4");

S:= CuspForms(chi, 12);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 13.e (of order $$6$$, 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: $$22$$ Relative dimension: $$11$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$22 q - 3 q^{2} + 242 q^{3} + 8191 q^{4} + 45528 q^{6} + 128496 q^{7} - 322217 q^{9}+O(q^{10})$$ 22 * q - 3 * q^2 + 242 * q^3 + 8191 * q^4 + 45528 * q^6 + 128496 * q^7 - 322217 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$22 q - 3 q^{2} + 242 q^{3} + 8191 q^{4} + 45528 q^{6} + 128496 q^{7} - 322217 q^{9} - 572989 q^{10} + 941556 q^{11} - 954740 q^{12} - 1999101 q^{13} - 4389420 q^{14} - 1912146 q^{15} - 2217465 q^{16} + 1266933 q^{17} - 26218578 q^{19} + 73151169 q^{20} + 1338614 q^{22} - 47071890 q^{23} + 176666646 q^{24} - 102279200 q^{25} + 86807799 q^{26} - 316653808 q^{27} - 127105884 q^{28} + 113029131 q^{29} - 161927730 q^{30} + 750398445 q^{32} + 588949482 q^{33} - 300427704 q^{35} - 173622325 q^{36} - 570835935 q^{37} + 925613832 q^{38} + 2656614142 q^{39} - 4665525182 q^{40} - 3125495451 q^{41} + 312136374 q^{42} - 687379108 q^{43} + 291815697 q^{45} + 1973786088 q^{46} - 4495108546 q^{48} - 2113923323 q^{49} + 10799589288 q^{50} + 18922873548 q^{51} + 5816536466 q^{52} - 11409380982 q^{53} - 7220008734 q^{54} + 2659826936 q^{55} - 5191105920 q^{56} - 17750605839 q^{58} + 12280821000 q^{59} - 388343883 q^{61} - 4489966530 q^{62} + 12000810648 q^{63} - 27417445590 q^{64} - 7047855789 q^{65} + 29852276436 q^{66} - 24954230718 q^{67} + 18618567747 q^{68} - 17332878054 q^{69} + 7855228734 q^{71} - 2312207943 q^{72} + 14656656585 q^{74} + 34533247192 q^{75} - 91364144550 q^{76} + 1417426884 q^{77} + 51502665942 q^{78} - 15237937464 q^{79} + 187084997403 q^{80} - 66245225351 q^{81} + 38170027049 q^{82} - 221463259788 q^{84} - 144387435243 q^{85} + 133939610130 q^{87} + 191677651072 q^{88} - 22557105234 q^{89} - 28770934134 q^{90} - 26918267220 q^{91} - 31256021460 q^{92} + 169636911984 q^{93} + 114241933898 q^{94} - 72874928466 q^{95} - 318460072098 q^{97} - 223225841301 q^{98}+O(q^{100})$$ 22 * q - 3 * q^2 + 242 * q^3 + 8191 * q^4 + 45528 * q^6 + 128496 * q^7 - 322217 * q^9 - 572989 * q^10 + 941556 * q^11 - 954740 * q^12 - 1999101 * q^13 - 4389420 * q^14 - 1912146 * q^15 - 2217465 * q^16 + 1266933 * q^17 - 26218578 * q^19 + 73151169 * q^20 + 1338614 * q^22 - 47071890 * q^23 + 176666646 * q^24 - 102279200 * q^25 + 86807799 * q^26 - 316653808 * q^27 - 127105884 * q^28 + 113029131 * q^29 - 161927730 * q^30 + 750398445 * q^32 + 588949482 * q^33 - 300427704 * q^35 - 173622325 * q^36 - 570835935 * q^37 + 925613832 * q^38 + 2656614142 * q^39 - 4665525182 * q^40 - 3125495451 * q^41 + 312136374 * q^42 - 687379108 * q^43 + 291815697 * q^45 + 1973786088 * q^46 - 4495108546 * q^48 - 2113923323 * q^49 + 10799589288 * q^50 + 18922873548 * q^51 + 5816536466 * q^52 - 11409380982 * q^53 - 7220008734 * q^54 + 2659826936 * q^55 - 5191105920 * q^56 - 17750605839 * q^58 + 12280821000 * q^59 - 388343883 * q^61 - 4489966530 * q^62 + 12000810648 * q^63 - 27417445590 * q^64 - 7047855789 * q^65 + 29852276436 * q^66 - 24954230718 * q^67 + 18618567747 * q^68 - 17332878054 * q^69 + 7855228734 * q^71 - 2312207943 * q^72 + 14656656585 * q^74 + 34533247192 * q^75 - 91364144550 * q^76 + 1417426884 * q^77 + 51502665942 * q^78 - 15237937464 * q^79 + 187084997403 * q^80 - 66245225351 * q^81 + 38170027049 * q^82 - 221463259788 * q^84 - 144387435243 * q^85 + 133939610130 * q^87 + 191677651072 * q^88 - 22557105234 * q^89 - 28770934134 * q^90 - 26918267220 * q^91 - 31256021460 * q^92 + 169636911984 * q^93 + 114241933898 * q^94 - 72874928466 * q^95 - 318460072098 * q^97 - 223225841301 * q^98

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −74.2981 42.8960i −148.593 + 257.371i 2656.14 + 4600.57i 10410.3i 22080.4 12748.1i 18278.8 10553.3i 280049.i 44413.7 + 76926.7i −446559. + 773464.i
4.2 −57.1912 33.0193i 212.159 367.469i 1156.55 + 2003.21i 3712.50i −24267.2 + 14010.7i −35175.6 + 20308.7i 17507.1i −1448.99 2509.72i 122584. 212322.i
4.3 −47.1587 27.2271i −219.942 + 380.951i 458.627 + 794.365i 8334.97i 20744.3 11976.8i 19275.1 11128.5i 61573.8i −8175.47 14160.3i 226937. 393066.i
4.4 −28.4687 16.4364i 286.164 495.650i −483.689 837.774i 6962.98i −16293.4 + 9407.01i 68090.0 39311.8i 99124.0i −75206.1 130261.i −114446. + 198227.i
4.5 −18.1006 10.4504i −200.570 + 347.397i −805.579 1395.30i 5339.78i 7260.86 4192.06i −15062.6 + 8696.38i 76479.2i 8117.03 + 14059.1i −55802.7 + 96653.1i
4.6 8.35978 + 4.82652i 155.366 269.102i −977.409 1692.92i 71.9990i 2597.66 1499.76i −53916.5 + 31128.7i 38639.4i 40296.2 + 69795.0i 347.505 601.896i
4.7 15.7596 + 9.09880i 80.7581 139.877i −858.424 1486.83i 13894.4i 2545.43 1469.60i 55086.8 31804.4i 68511.2i 75529.7 + 130821.i −126422. + 218969.i
4.8 35.3488 + 20.4086i −366.998 + 635.659i −190.977 330.782i 3052.06i −25945.9 + 14979.8i −6894.91 + 3980.78i 99184.0i −180802. 313158.i −62288.3 + 107887.i
4.9 42.3913 + 24.4746i −27.6563 + 47.9021i 174.016 + 301.405i 10452.4i −2344.78 + 1353.76i 39866.5 23016.9i 83212.2i 87043.8 + 150764.i 255820. 443093.i
4.10 53.9750 + 31.1625i 401.709 695.781i 918.197 + 1590.36i 673.285i 43364.5 25036.5i −9262.47 + 5347.69i 13188.3i −234167. 405589.i 20981.2 36340.5i
4.11 67.8828 + 39.1921i −51.3966 + 89.0216i 2048.05 + 3547.32i 2746.61i −6977.89 + 4028.69i −16037.2 + 9259.07i 160539.i 83290.3 + 144263.i −107645. + 186447.i
10.1 −74.2981 + 42.8960i −148.593 257.371i 2656.14 4600.57i 10410.3i 22080.4 + 12748.1i 18278.8 + 10553.3i 280049.i 44413.7 76926.7i −446559. 773464.i
10.2 −57.1912 + 33.0193i 212.159 + 367.469i 1156.55 2003.21i 3712.50i −24267.2 14010.7i −35175.6 20308.7i 17507.1i −1448.99 + 2509.72i 122584. + 212322.i
10.3 −47.1587 + 27.2271i −219.942 380.951i 458.627 794.365i 8334.97i 20744.3 + 11976.8i 19275.1 + 11128.5i 61573.8i −8175.47 + 14160.3i 226937. + 393066.i
10.4 −28.4687 + 16.4364i 286.164 + 495.650i −483.689 + 837.774i 6962.98i −16293.4 9407.01i 68090.0 + 39311.8i 99124.0i −75206.1 + 130261.i −114446. 198227.i
10.5 −18.1006 + 10.4504i −200.570 347.397i −805.579 + 1395.30i 5339.78i 7260.86 + 4192.06i −15062.6 8696.38i 76479.2i 8117.03 14059.1i −55802.7 96653.1i
10.6 8.35978 4.82652i 155.366 + 269.102i −977.409 + 1692.92i 71.9990i 2597.66 + 1499.76i −53916.5 31128.7i 38639.4i 40296.2 69795.0i 347.505 + 601.896i
10.7 15.7596 9.09880i 80.7581 + 139.877i −858.424 + 1486.83i 13894.4i 2545.43 + 1469.60i 55086.8 + 31804.4i 68511.2i 75529.7 130821.i −126422. 218969.i
10.8 35.3488 20.4086i −366.998 635.659i −190.977 + 330.782i 3052.06i −25945.9 14979.8i −6894.91 3980.78i 99184.0i −180802. + 313158.i −62288.3 107887.i
10.9 42.3913 24.4746i −27.6563 47.9021i 174.016 301.405i 10452.4i −2344.78 1353.76i 39866.5 + 23016.9i 83212.2i 87043.8 150764.i 255820. + 443093.i
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.11 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.12.e.a 22
13.e even 6 1 inner 13.12.e.a 22
13.f odd 12 2 169.12.a.g 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.12.e.a 22 1.a even 1 1 trivial
13.12.e.a 22 13.e even 6 1 inner
169.12.a.g 22 13.f odd 12 2

## Hecke kernels

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