# Properties

 Label 13.13.d.a Level $13$ Weight $13$ Character orbit 13.d Analytic conductor $11.882$ Analytic rank $0$ Dimension $26$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

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

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

chi := DirichletCharacter("13.5");

S:= CuspForms(chi, 13);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$13$$ 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: $$11.8819196246$$ Analytic rank: $$0$$ Dimension: $$26$$ Relative dimension: $$13$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$26 q - 2 q^{2} - 4 q^{3} + 5146 q^{5} + 64736 q^{6} - 114402 q^{7} - 262080 q^{8} + 3897230 q^{9}+O(q^{10})$$ 26 * q - 2 * q^2 - 4 * q^3 + 5146 * q^5 + 64736 * q^6 - 114402 * q^7 - 262080 * q^8 + 3897230 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$26 q - 2 q^{2} - 4 q^{3} + 5146 q^{5} + 64736 q^{6} - 114402 q^{7} - 262080 q^{8} + 3897230 q^{9} - 1360946 q^{11} - 2904122 q^{13} - 33341620 q^{14} - 26612356 q^{15} - 42212844 q^{16} + 76072094 q^{18} - 41603538 q^{19} + 60982508 q^{20} - 77526388 q^{21} + 648561056 q^{22} - 1131540696 q^{24} + 912969070 q^{26} - 159769528 q^{27} - 311272132 q^{28} + 2981804 q^{29} - 1023774130 q^{31} + 1476108268 q^{32} + 2381658236 q^{33} + 1983101640 q^{34} - 3650085364 q^{35} - 3167118502 q^{37} + 16292940092 q^{39} - 3968093508 q^{40} + 28381274530 q^{41} - 49639312388 q^{42} - 18960178772 q^{44} - 10665692290 q^{45} + 37450277124 q^{46} + 20752786078 q^{47} - 71636180764 q^{48} - 48656842882 q^{50} + 117857606620 q^{52} - 58773364924 q^{53} + 213216359816 q^{54} - 136697226052 q^{55} - 114945046324 q^{57} + 92279424116 q^{58} + 155389424110 q^{59} + 136735907884 q^{60} + 141369458276 q^{61} - 464360668726 q^{63} + 114397918882 q^{65} - 206611171952 q^{66} + 100234053918 q^{67} - 417926231820 q^{68} + 153062962568 q^{70} + 230837239150 q^{71} - 118991496780 q^{72} + 146516995818 q^{73} - 1012588063096 q^{74} + 606168045784 q^{76} + 359416070936 q^{78} + 434995822940 q^{79} + 237895101520 q^{80} + 1613859489074 q^{81} - 637194158642 q^{83} - 981969825256 q^{84} - 213215637336 q^{85} - 2637756396288 q^{86} + 744567460312 q^{87} + 1456251648226 q^{89} - 3121537204578 q^{91} + 1179602583480 q^{92} - 3143065458004 q^{93} + 5250023559428 q^{94} + 948734423984 q^{96} + 1538416432538 q^{97} + 25933667758 q^{98} - 1934327641798 q^{99}+O(q^{100})$$ 26 * q - 2 * q^2 - 4 * q^3 + 5146 * q^5 + 64736 * q^6 - 114402 * q^7 - 262080 * q^8 + 3897230 * q^9 - 1360946 * q^11 - 2904122 * q^13 - 33341620 * q^14 - 26612356 * q^15 - 42212844 * q^16 + 76072094 * q^18 - 41603538 * q^19 + 60982508 * q^20 - 77526388 * q^21 + 648561056 * q^22 - 1131540696 * q^24 + 912969070 * q^26 - 159769528 * q^27 - 311272132 * q^28 + 2981804 * q^29 - 1023774130 * q^31 + 1476108268 * q^32 + 2381658236 * q^33 + 1983101640 * q^34 - 3650085364 * q^35 - 3167118502 * q^37 + 16292940092 * q^39 - 3968093508 * q^40 + 28381274530 * q^41 - 49639312388 * q^42 - 18960178772 * q^44 - 10665692290 * q^45 + 37450277124 * q^46 + 20752786078 * q^47 - 71636180764 * q^48 - 48656842882 * q^50 + 117857606620 * q^52 - 58773364924 * q^53 + 213216359816 * q^54 - 136697226052 * q^55 - 114945046324 * q^57 + 92279424116 * q^58 + 155389424110 * q^59 + 136735907884 * q^60 + 141369458276 * q^61 - 464360668726 * q^63 + 114397918882 * q^65 - 206611171952 * q^66 + 100234053918 * q^67 - 417926231820 * q^68 + 153062962568 * q^70 + 230837239150 * q^71 - 118991496780 * q^72 + 146516995818 * q^73 - 1012588063096 * q^74 + 606168045784 * q^76 + 359416070936 * q^78 + 434995822940 * q^79 + 237895101520 * q^80 + 1613859489074 * q^81 - 637194158642 * q^83 - 981969825256 * q^84 - 213215637336 * q^85 - 2637756396288 * q^86 + 744567460312 * q^87 + 1456251648226 * q^89 - 3121537204578 * q^91 + 1179602583480 * q^92 - 3143065458004 * q^93 + 5250023559428 * q^94 + 948734423984 * q^96 + 1538416432538 * q^97 + 25933667758 * q^98 - 1934327641798 * 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 −82.3157 + 82.3157i 532.405 9455.73i −1258.93 + 1258.93i −43825.3 + 43825.3i 123092. + 123092.i 441190. + 441190.i −247986. 207260.i
5.2 −73.6638 + 73.6638i −1197.88 6756.71i 2452.94 2452.94i 88240.2 88240.2i −37699.0 37699.0i 195998. + 195998.i 903471. 361386.i
5.3 −55.4726 + 55.4726i −123.365 2058.41i −10153.2 + 10153.2i 6843.39 6843.39i −69327.0 69327.0i −113030. 113030.i −516222. 1.12645e6i
5.4 −51.8539 + 51.8539i 843.934 1281.66i 16935.0 16935.0i −43761.3 + 43761.3i −94929.1 94929.1i −145935. 145935.i 180783. 1.75629e6i
5.5 −23.7102 + 23.7102i −547.334 2971.66i 18861.9 18861.9i 12977.4 12977.4i 68016.7 + 68016.7i −167575. 167575.i −231867. 894440.i
5.6 −20.6340 + 20.6340i 1133.68 3244.47i −11267.3 + 11267.3i −23392.4 + 23392.4i 41508.2 + 41508.2i −151463. 151463.i 753788. 464977.i
5.7 −11.9679 + 11.9679i −712.886 3809.54i −8528.30 + 8528.30i 8531.76 8531.76i 71048.2 + 71048.2i −94612.8 94612.8i −23234.8 204132.i
5.8 21.2021 21.2021i 104.714 3196.95i −2687.41 + 2687.41i 2220.14 2220.14i −129238. 129238.i 154625. + 154625.i −520476. 113957.i
5.9 35.7844 35.7844i 720.183 1534.96i 8189.78 8189.78i 25771.3 25771.3i 69804.0 + 69804.0i 201500. + 201500.i −12777.9 586132.i
5.10 40.0541 40.0541i −1363.29 887.334i 2683.01 2683.01i −54605.4 + 54605.4i −74989.8 74989.8i 199603. + 199603.i 1.32712e6 214931.i
5.11 65.2537 65.2537i −339.809 4420.09i −17964.9 + 17964.9i −22173.8 + 22173.8i 90738.1 + 90738.1i −21147.8 21147.8i −415971. 2.34455e6i
5.12 77.2713 77.2713i −352.569 7845.71i 14032.6 14032.6i −27243.5 + 27243.5i −2786.27 2786.27i −289745. 289745.i −407136. 2.16863e6i
5.13 79.0525 79.0525i 1300.22 8402.59i −8722.22 + 8722.22i 102785. 102785.i −112440. 112440.i −340447. 340447.i 1.15912e6 1.37903e6i
8.1 −82.3157 82.3157i 532.405 9455.73i −1258.93 1258.93i −43825.3 43825.3i 123092. 123092.i 441190. 441190.i −247986. 207260.i
8.2 −73.6638 73.6638i −1197.88 6756.71i 2452.94 + 2452.94i 88240.2 + 88240.2i −37699.0 + 37699.0i 195998. 195998.i 903471. 361386.i
8.3 −55.4726 55.4726i −123.365 2058.41i −10153.2 10153.2i 6843.39 + 6843.39i −69327.0 + 69327.0i −113030. + 113030.i −516222. 1.12645e6i
8.4 −51.8539 51.8539i 843.934 1281.66i 16935.0 + 16935.0i −43761.3 43761.3i −94929.1 + 94929.1i −145935. + 145935.i 180783. 1.75629e6i
8.5 −23.7102 23.7102i −547.334 2971.66i 18861.9 + 18861.9i 12977.4 + 12977.4i 68016.7 68016.7i −167575. + 167575.i −231867. 894440.i
8.6 −20.6340 20.6340i 1133.68 3244.47i −11267.3 11267.3i −23392.4 23392.4i 41508.2 41508.2i −151463. + 151463.i 753788. 464977.i
8.7 −11.9679 11.9679i −712.886 3809.54i −8528.30 8528.30i 8531.76 + 8531.76i 71048.2 71048.2i −94612.8 + 94612.8i −23234.8 204132.i
See all 26 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.13 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 13.13.d.a 26
13.d odd 4 1 inner 13.13.d.a 26

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.13.d.a 26 1.a even 1 1 trivial
13.13.d.a 26 13.d odd 4 1 inner

## Hecke kernels

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