# Properties

 Label 13.15.d.a Level $13$ Weight $15$ Character orbit 13.d Analytic conductor $16.163$ Analytic rank $0$ Dimension $32$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,15,Mod(5,13)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3]))

N = Newforms(chi, 15, names="a")

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

chi := DirichletCharacter("13.5");

S:= CuspForms(chi, 15);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$15$$ Character orbit: $$[\chi]$$ $$=$$ 13.d (of order $$4$$, 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: $$16.1627658597$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 126 q^{2} - 4 q^{3} + 40308 q^{5} - 215296 q^{6} - 342172 q^{7} + 2132400 q^{8} + 51018332 q^{9}+O(q^{10})$$ 32 * q + 126 * q^2 - 4 * q^3 + 40308 * q^5 - 215296 * q^6 - 342172 * q^7 + 2132400 * q^8 + 51018332 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 126 q^{2} - 4 q^{3} + 40308 q^{5} - 215296 q^{6} - 342172 q^{7} + 2132400 q^{8} + 51018332 q^{9} - 19776228 q^{11} + 42276320 q^{13} + 504176460 q^{14} + 206094692 q^{15} - 2531526316 q^{16} + 2815918046 q^{18} - 3946163104 q^{19} - 716259684 q^{20} - 3367522672 q^{21} - 14762241344 q^{22} + 29491821240 q^{24} - 4794721266 q^{26} + 13386727964 q^{27} + 21253870460 q^{28} + 51981667512 q^{29} - 81106252408 q^{31} - 46362860004 q^{32} + 105153523412 q^{33} - 182529279768 q^{34} - 370063477908 q^{35} + 362625318836 q^{37} - 384840090244 q^{39} + 1418885544444 q^{40} - 271128165276 q^{41} + 282362729116 q^{42} + 371212733148 q^{44} - 45612965524 q^{45} - 2243149012284 q^{46} - 186355978644 q^{47} + 4851512979332 q^{48} + 584922885822 q^{50} - 4161984175620 q^{52} - 7014914206728 q^{53} - 16155068308552 q^{54} + 10809637665184 q^{55} + 10539677371976 q^{57} + 1379941218100 q^{58} - 346563818376 q^{59} - 10818733483076 q^{60} + 10463189375104 q^{61} + 20564799781280 q^{63} - 36771299231364 q^{65} + 42566898217168 q^{66} - 33242476172548 q^{67} + 73991369730132 q^{68} + 15425582685208 q^{70} - 32475992326884 q^{71} - 94516161515148 q^{72} - 46644619387444 q^{73} + 19517335633512 q^{74} + 153247499250952 q^{76} - 161936899980568 q^{78} + 50555071999360 q^{79} - 164927157279648 q^{80} + 180616513188704 q^{81} + 40646195239884 q^{83} - 8563943720152 q^{84} + 8377594459896 q^{85} - 50876984146272 q^{86} + 82858002231808 q^{87} + 43433785559052 q^{89} - 4834418011540 q^{91} + 2835746550840 q^{92} - 161211565737376 q^{93} - 30798863063804 q^{94} + 114745181636288 q^{96} + 131071048977944 q^{97} - 160069892052210 q^{98} - 66089157162136 q^{99}+O(q^{100})$$ 32 * q + 126 * q^2 - 4 * q^3 + 40308 * q^5 - 215296 * q^6 - 342172 * q^7 + 2132400 * q^8 + 51018332 * q^9 - 19776228 * q^11 + 42276320 * q^13 + 504176460 * q^14 + 206094692 * q^15 - 2531526316 * q^16 + 2815918046 * q^18 - 3946163104 * q^19 - 716259684 * q^20 - 3367522672 * q^21 - 14762241344 * q^22 + 29491821240 * q^24 - 4794721266 * q^26 + 13386727964 * q^27 + 21253870460 * q^28 + 51981667512 * q^29 - 81106252408 * q^31 - 46362860004 * q^32 + 105153523412 * q^33 - 182529279768 * q^34 - 370063477908 * q^35 + 362625318836 * q^37 - 384840090244 * q^39 + 1418885544444 * q^40 - 271128165276 * q^41 + 282362729116 * q^42 + 371212733148 * q^44 - 45612965524 * q^45 - 2243149012284 * q^46 - 186355978644 * q^47 + 4851512979332 * q^48 + 584922885822 * q^50 - 4161984175620 * q^52 - 7014914206728 * q^53 - 16155068308552 * q^54 + 10809637665184 * q^55 + 10539677371976 * q^57 + 1379941218100 * q^58 - 346563818376 * q^59 - 10818733483076 * q^60 + 10463189375104 * q^61 + 20564799781280 * q^63 - 36771299231364 * q^65 + 42566898217168 * q^66 - 33242476172548 * q^67 + 73991369730132 * q^68 + 15425582685208 * q^70 - 32475992326884 * q^71 - 94516161515148 * q^72 - 46644619387444 * q^73 + 19517335633512 * q^74 + 153247499250952 * q^76 - 161936899980568 * q^78 + 50555071999360 * q^79 - 164927157279648 * q^80 + 180616513188704 * q^81 + 40646195239884 * q^83 - 8563943720152 * q^84 + 8377594459896 * q^85 - 50876984146272 * q^86 + 82858002231808 * q^87 + 43433785559052 * q^89 - 4834418011540 * q^91 + 2835746550840 * q^92 - 161211565737376 * q^93 - 30798863063804 * q^94 + 114745181636288 * q^96 + 131071048977944 * q^97 - 160069892052210 * q^98 - 66089157162136 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −179.362 + 179.362i −287.384 47957.1i 65262.4 65262.4i 51545.7 51545.7i −714231. 714231.i 5.66300e6 + 5.66300e6i −4.70038e6 2.34111e7i
5.2 −136.719 + 136.719i −534.953 21000.1i −103498. + 103498.i 73138.1 73138.1i −203533. 203533.i 631109. + 631109.i −4.49679e6 2.83004e7i
5.3 −133.495 + 133.495i 4038.24 19257.7i −11146.6 + 11146.6i −539083. + 539083.i 192538. + 192538.i 383617. + 383617.i 1.15244e7 2.97602e6i
5.4 −126.421 + 126.421i −3278.37 15580.6i 3422.71 3422.71i 414455. 414455.i 1.01518e6 + 1.01518e6i −101567. 101567.i 5.96475e6 865406.i
5.5 −96.2318 + 96.2318i 907.445 2137.13i 41176.3 41176.3i −87325.1 + 87325.1i 180476. + 180476.i −1.37100e6 1.37100e6i −3.95951e6 7.92494e6i
5.6 −58.6742 + 58.6742i −2939.90 9498.68i 36290.0 36290.0i 172496. 172496.i −1.04136e6 1.04136e6i −1.51865e6 1.51865e6i 3.86002e6 4.25857e6i
5.7 −20.7406 + 20.7406i 2104.15 15523.7i −63748.0 + 63748.0i −43641.4 + 43641.4i −524901. 524901.i −661784. 661784.i −355511. 2.64434e6i
5.8 −2.45889 + 2.45889i 917.714 16371.9i 29189.6 29189.6i −2256.56 + 2256.56i 560447. + 560447.i −80543.3 80543.3i −3.94077e6 143548.i
5.9 16.8258 16.8258i −2685.14 15817.8i −66936.2 + 66936.2i −45179.7 + 45179.7i 261049. + 261049.i 541821. + 541821.i 2.42703e6 2.25251e6i
5.10 54.4963 54.4963i 3813.48 10444.3i 89674.7 89674.7i 207821. 207821.i −863917. 863917.i 1.46204e6 + 1.46204e6i 9.75965e6 9.77389e6i
5.11 80.0104 80.0104i −2425.32 3580.67i 98501.3 98501.3i −194051. + 194051.i 242708. + 242708.i 1.59738e6 + 1.59738e6i 1.09922e6 1.57623e7i
5.12 98.9834 98.9834i 3262.77 3211.43i −67810.1 + 67810.1i 322960. 322960.i 947177. + 947177.i 1.30387e6 + 1.30387e6i 5.86269e6 1.34241e7i
5.13 100.811 100.811i −1095.41 3941.67i −6132.14 + 6132.14i −110429. + 110429.i −84903.1 84903.1i 1.25432e6 + 1.25432e6i −3.58306e6 1.23637e6i
5.14 130.772 130.772i 566.857 17818.4i −36294.1 + 36294.1i 74128.8 74128.8i −877300. 877300.i −187580. 187580.i −4.46164e6 9.49246e6i
5.15 166.640 166.640i 1755.93 39153.9i 51367.0 51367.0i 292608. 292608.i 487210. + 487210.i −3.79437e6 3.79437e6i −1.69970e6 1.71196e7i
5.16 168.563 168.563i −4122.10 40443.0i −39164.6 + 39164.6i −694834. + 694834.i 252271. + 252271.i −4.05546e6 4.05546e6i 1.22088e7 1.32034e7i
8.1 −179.362 179.362i −287.384 47957.1i 65262.4 + 65262.4i 51545.7 + 51545.7i −714231. + 714231.i 5.66300e6 5.66300e6i −4.70038e6 2.34111e7i
8.2 −136.719 136.719i −534.953 21000.1i −103498. 103498.i 73138.1 + 73138.1i −203533. + 203533.i 631109. 631109.i −4.49679e6 2.83004e7i
8.3 −133.495 133.495i 4038.24 19257.7i −11146.6 11146.6i −539083. 539083.i 192538. 192538.i 383617. 383617.i 1.15244e7 2.97602e6i
8.4 −126.421 126.421i −3278.37 15580.6i 3422.71 + 3422.71i 414455. + 414455.i 1.01518e6 1.01518e6i −101567. + 101567.i 5.96475e6 865406.i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.16 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.d odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.15.d.a 32
13.d odd 4 1 inner 13.15.d.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.15.d.a 32 1.a even 1 1 trivial
13.15.d.a 32 13.d odd 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{15}^{\mathrm{new}}(13, [\chi])$$.