# Properties

 Label 13.15.f.a Level $13$ Weight $15$ Character orbit 13.f Analytic conductor $16.163$ Analytic rank $0$ Dimension $60$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,15,Mod(2,13)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("13.2");

S:= CuspForms(chi, 15);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$15$$ Character orbit: $$[\chi]$$ $$=$$ 13.f (of order $$12$$, degree $$4$$, 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: $$60$$ Relative dimension: $$15$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$60 q - 132 q^{2} - 2 q^{3} - 6 q^{4} - 40314 q^{5} + 215290 q^{6} - 1557904 q^{7} - 11569590 q^{8} - 38263754 q^{9}+O(q^{10})$$ 60 * q - 132 * q^2 - 2 * q^3 - 6 * q^4 - 40314 * q^5 + 215290 * q^6 - 1557904 * q^7 - 11569590 * q^8 - 38263754 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$60 q - 132 q^{2} - 2 q^{3} - 6 q^{4} - 40314 q^{5} + 215290 q^{6} - 1557904 q^{7} - 11569590 q^{8} - 38263754 q^{9} - 48923526 q^{10} + 50218068 q^{11} - 238349618 q^{13} - 292944216 q^{14} + 170532946 q^{15} + 1189349030 q^{16} - 574913052 q^{17} + 4150102162 q^{18} - 501076774 q^{19} + 6984344406 q^{20} + 1863314584 q^{21} - 1873824772 q^{22} + 1737529890 q^{23} + 15243805554 q^{24} - 41782081770 q^{26} - 44198732768 q^{27} - 21921814140 q^{28} + 57405235818 q^{29} - 199647227574 q^{30} + 2983019916 q^{31} + 284437348812 q^{32} + 140381332384 q^{33} + 32089107162 q^{34} - 19743874608 q^{35} + 750920897322 q^{36} - 372393473832 q^{37} - 543416978234 q^{39} - 1398571552128 q^{40} + 375208840236 q^{41} - 728714321956 q^{42} + 851391301668 q^{43} + 590685561840 q^{44} - 1306809163922 q^{45} - 2311308791406 q^{46} - 381835768356 q^{47} - 270157048826 q^{48} + 3712523710824 q^{49} + 1556659581834 q^{50} + 4000390743752 q^{52} - 89028123960 q^{53} + 5763909293080 q^{54} - 3006046329940 q^{55} + 23022332974284 q^{56} + 12917485132 q^{57} - 12422298950218 q^{58} - 14757339867252 q^{59} - 20487985031272 q^{60} - 1384599480448 q^{61} + 9801522405486 q^{62} + 22640879976676 q^{63} - 19696125147348 q^{65} - 91426703578432 q^{66} + 24259587124358 q^{67} + 10690396635684 q^{68} + 60961044605832 q^{69} + 568844858564 q^{70} - 14911781930742 q^{71} - 42111941898708 q^{72} + 34036689645614 q^{73} - 13785833619522 q^{74} + 142778798307648 q^{75} + 39222346941870 q^{76} - 84101983935944 q^{78} - 145388292515752 q^{79} + 73198543586340 q^{80} - 22285961912990 q^{81} + 63345232167774 q^{82} + 108480366280644 q^{83} + 30953001339628 q^{84} - 80811646956204 q^{85} - 295149222458244 q^{86} + 72492366632678 q^{87} + 164580905307108 q^{88} - 109447119436674 q^{89} + 41718851417204 q^{91} - 86879505071484 q^{92} + 175680368874172 q^{93} - 236701312995790 q^{94} + 227314198930962 q^{95} - 183500640905672 q^{96} - 264615272704410 q^{97} - 81353077477140 q^{98} + 555323416765936 q^{99}+O(q^{100})$$ 60 * q - 132 * q^2 - 2 * q^3 - 6 * q^4 - 40314 * q^5 + 215290 * q^6 - 1557904 * q^7 - 11569590 * q^8 - 38263754 * q^9 - 48923526 * q^10 + 50218068 * q^11 - 238349618 * q^13 - 292944216 * q^14 + 170532946 * q^15 + 1189349030 * q^16 - 574913052 * q^17 + 4150102162 * q^18 - 501076774 * q^19 + 6984344406 * q^20 + 1863314584 * q^21 - 1873824772 * q^22 + 1737529890 * q^23 + 15243805554 * q^24 - 41782081770 * q^26 - 44198732768 * q^27 - 21921814140 * q^28 + 57405235818 * q^29 - 199647227574 * q^30 + 2983019916 * q^31 + 284437348812 * q^32 + 140381332384 * q^33 + 32089107162 * q^34 - 19743874608 * q^35 + 750920897322 * q^36 - 372393473832 * q^37 - 543416978234 * q^39 - 1398571552128 * q^40 + 375208840236 * q^41 - 728714321956 * q^42 + 851391301668 * q^43 + 590685561840 * q^44 - 1306809163922 * q^45 - 2311308791406 * q^46 - 381835768356 * q^47 - 270157048826 * q^48 + 3712523710824 * q^49 + 1556659581834 * q^50 + 4000390743752 * q^52 - 89028123960 * q^53 + 5763909293080 * q^54 - 3006046329940 * q^55 + 23022332974284 * q^56 + 12917485132 * q^57 - 12422298950218 * q^58 - 14757339867252 * q^59 - 20487985031272 * q^60 - 1384599480448 * q^61 + 9801522405486 * q^62 + 22640879976676 * q^63 - 19696125147348 * q^65 - 91426703578432 * q^66 + 24259587124358 * q^67 + 10690396635684 * q^68 + 60961044605832 * q^69 + 568844858564 * q^70 - 14911781930742 * q^71 - 42111941898708 * q^72 + 34036689645614 * q^73 - 13785833619522 * q^74 + 142778798307648 * q^75 + 39222346941870 * q^76 - 84101983935944 * q^78 - 145388292515752 * q^79 + 73198543586340 * q^80 - 22285961912990 * q^81 + 63345232167774 * q^82 + 108480366280644 * q^83 + 30953001339628 * q^84 - 80811646956204 * q^85 - 295149222458244 * q^86 + 72492366632678 * q^87 + 164580905307108 * q^88 - 109447119436674 * q^89 + 41718851417204 * q^91 - 86879505071484 * q^92 + 175680368874172 * q^93 - 236701312995790 * q^94 + 227314198930962 * q^95 - 183500640905672 * q^96 - 264615272704410 * q^97 - 81353077477140 * q^98 + 555323416765936 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −221.599 59.3773i −918.592 1591.05i 31391.5 + 18123.9i 78178.2 78178.2i 109087. + 407118.i −1.19385e6 + 319891.i −3.22234e6 3.22234e6i 703864. 1.21913e6i −2.19662e7 + 1.26822e7i
2.2 −199.331 53.4106i −64.4222 111.582i 22691.3 + 13100.8i −59689.1 + 59689.1i 6881.66 + 25682.7i 367357. 98432.9i −1.43259e6 1.43259e6i 2.38318e6 4.12780e6i 1.50859e7 8.70987e6i
2.3 −196.989 52.7830i 1859.85 + 3221.36i 21829.6 + 12603.3i 60386.5 60386.5i −196337. 732740.i 679101. 181965.i −1.27228e6 1.27228e6i −4.52660e6 + 7.84031e6i −1.50828e7 + 8.70808e6i
2.4 −139.928 37.4936i −1807.77 3131.16i 3985.13 + 2300.81i −13583.4 + 13583.4i 135560. + 505917.i 1.14435e6 306629.i 1.20692e6 + 1.20692e6i −4.14461e6 + 7.17867e6i 2.40998e6 1.39140e6i
2.5 −130.567 34.9852i 1088.82 + 1885.88i 1634.69 + 943.791i −43512.6 + 43512.6i −76184.8 284326.i −897176. + 240398.i 1.38559e6 + 1.38559e6i 20446.0 35413.5i 7.20358e6 4.15899e6i
2.6 −77.3356 20.7220i −459.594 796.040i −8637.57 4986.90i 80968.8 80968.8i 19047.4 + 71085.9i 164472. 44070.2i 1.49221e6 + 1.49221e6i 1.96903e6 3.41046e6i −7.93961e6 + 4.58393e6i
2.7 −38.7805 10.3912i 915.215 + 1585.20i −12793.0 7386.05i 11724.6 11724.6i −19020.4 70985.0i 404217. 108310.i 884500. + 884500.i 716247. 1.24058e6i −576517. + 332852.i
2.8 −37.7440 10.1135i −1241.71 2150.71i −12866.6 7428.55i −51713.0 + 51713.0i 25116.1 + 93734.5i −1.32596e6 + 355289.i 863209. + 863209.i −692216. + 1.19895e6i 2.47485e6 1.42886e6i
2.9 68.0285 + 18.2282i 2110.62 + 3655.70i −9893.35 5711.93i −21260.9 + 21260.9i 76945.4 + 287164.i −710656. + 190420.i −1.38484e6 1.38484e6i −6.51792e6 + 1.12894e7i −1.83390e6 + 1.05880e6i
2.10 69.8096 + 18.7054i 191.298 + 331.338i −9665.47 5580.36i −93916.6 + 93916.6i 7156.63 + 26708.9i 1.28598e6 344577.i −1.40765e6 1.40765e6i 2.31829e6 4.01540e6i −8.31303e6 + 4.79953e6i
2.11 89.3305 + 23.9360i −1471.00 2547.85i −6781.96 3915.57i 13168.0 13168.0i −70419.9 262811.i 202419. 54237.9i −1.58354e6 1.58354e6i −1.93621e6 + 3.35362e6i 1.49149e6 861112.i
2.12 114.569 + 30.6986i 397.770 + 688.958i −2005.39 1157.81i 74145.3 74145.3i 24421.9 + 91143.9i −485805. + 130171.i −1.56834e6 1.56834e6i 2.07504e6 3.59408e6i 1.07709e7 6.21857e6i
2.13 189.516 + 50.7808i 78.1123 + 135.294i 19148.8 + 11055.6i −77305.1 + 77305.1i 7933.21 + 29607.1i −1.32350e6 + 354630.i 794563. + 794563.i 2.37928e6 4.12104e6i −1.85762e7 + 1.07250e7i
2.14 207.250 + 55.5326i 1307.54 + 2264.72i 25679.9 + 14826.3i 20676.9 20676.9i 145222. + 541976.i 844501. 226283.i 2.01309e6 + 2.01309e6i −1.02783e6 + 1.78026e6i 5.43353e6 3.13705e6i
2.15 214.478 + 57.4692i −1355.29 2347.43i 28509.1 + 16459.8i 15145.0 15145.0i −155775. 581359.i 349775. 93721.9i 2.59622e6 + 2.59622e6i −1.28213e6 + 2.22071e6i 4.11865e6 2.37790e6i
6.1 −59.5807 222.358i 470.939 815.689i −31704.4 + 18304.5i −28329.1 + 28329.1i −209434. 56117.7i 27839.3 103898.i 3.29218e6 + 3.29218e6i 1.94792e6 + 3.37389e6i 7.98709e6 + 4.61135e6i
6.2 −51.1690 190.965i −1294.68 + 2242.45i −19660.5 + 11351.0i 80187.2 80187.2i 494479. + 132495.i 159384. 594828.i 883238. + 883238.i −960917. 1.66436e6i −1.94161e7 1.12099e7i
6.3 −40.9042 152.657i −1696.55 + 2938.51i −7441.90 + 4296.58i −82707.4 + 82707.4i 517978. + 138792.i −274474. + 1.02435e6i −870644. 870644.i −3.36506e6 5.82845e6i 1.60089e7 + 9.24274e6i
6.4 −40.4034 150.787i 1847.02 3199.13i −6915.48 + 3992.65i 28440.7 28440.7i −557014. 149251.i −75055.4 + 280111.i −927083. 927083.i −4.43145e6 7.67550e6i −5.43761e6 3.13940e6i
6.5 −28.4393 106.137i −183.559 + 317.933i 3732.73 2155.09i −9680.82 + 9680.82i 38964.7 + 10440.6i −121863. + 454798.i −1.60789e6 1.60789e6i 2.32410e6 + 4.02545e6i 1.30281e6 + 752176.i
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.15 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.f odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.15.f.a 60
13.f odd 12 1 inner 13.15.f.a 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.15.f.a 60 1.a even 1 1 trivial
13.15.f.a 60 13.f odd 12 1 inner

## Hecke kernels

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