# Properties

 Label 13.9.f.a Level $13$ Weight $9$ Character orbit 13.f Analytic conductor $5.296$ 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] = [13,9,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, 9, names="a")

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

chi := DirichletCharacter("13.2");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$9$$ 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: $$5.29592193079$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 4 q^{2} - 2 q^{3} - 6 q^{4} - 172 q^{5} - 2726 q^{6} - 2784 q^{7} - 10470 q^{8} - 21872 q^{9}+O(q^{10})$$ 32 * q - 4 * q^2 - 2 * q^3 - 6 * q^4 - 172 * q^5 - 2726 * q^6 - 2784 * q^7 - 10470 * q^8 - 21872 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 4 q^{2} - 2 q^{3} - 6 q^{4} - 172 q^{5} - 2726 q^{6} - 2784 q^{7} - 10470 q^{8} - 21872 q^{9} - 26886 q^{10} + 6476 q^{11} + 14794 q^{13} + 216232 q^{14} - 127934 q^{15} + 195174 q^{16} + 97644 q^{17} - 475646 q^{18} + 373314 q^{19} + 685798 q^{20} - 213824 q^{21} + 167756 q^{22} - 711654 q^{23} - 1396830 q^{24} - 1041274 q^{26} + 3386272 q^{27} - 491836 q^{28} - 178394 q^{29} - 6200118 q^{30} + 1793468 q^{31} + 955004 q^{32} - 369536 q^{33} + 4592250 q^{34} + 1520488 q^{35} - 11372454 q^{36} + 247730 q^{37} + 14761942 q^{39} + 15044736 q^{40} - 2285272 q^{41} - 1024084 q^{42} + 18199452 q^{43} + 5629904 q^{44} - 10428746 q^{45} - 25708302 q^{46} - 8054836 q^{47} - 19983002 q^{48} - 18234288 q^{49} - 7208294 q^{50} - 26124488 q^{52} - 26091116 q^{53} + 85261048 q^{54} - 2240524 q^{55} + 84926940 q^{56} + 57257116 q^{57} - 68659210 q^{58} - 58197100 q^{59} - 64428232 q^{60} - 38537926 q^{61} + 2935614 q^{62} - 43972436 q^{63} + 44466032 q^{65} + 209850752 q^{66} + 30462126 q^{67} + 68365716 q^{68} + 163279656 q^{69} + 256782116 q^{70} - 113836054 q^{71} - 445647780 q^{72} - 30722748 q^{73} - 156843362 q^{74} - 364524840 q^{75} - 180633890 q^{76} + 252047848 q^{78} + 364818200 q^{79} + 372146084 q^{80} - 17174000 q^{81} + 272314014 q^{82} + 389373188 q^{83} - 464450276 q^{84} - 331999758 q^{85} - 32039076 q^{86} - 337693978 q^{87} - 1180586364 q^{88} - 333088936 q^{89} + 251459988 q^{91} + 1275961956 q^{92} + 637177780 q^{93} + 171539378 q^{94} + 318589986 q^{95} + 1019148280 q^{96} - 230031796 q^{97} - 1287859636 q^{98} - 359343776 q^{99}+O(q^{100})$$ 32 * q - 4 * q^2 - 2 * q^3 - 6 * q^4 - 172 * q^5 - 2726 * q^6 - 2784 * q^7 - 10470 * q^8 - 21872 * q^9 - 26886 * q^10 + 6476 * q^11 + 14794 * q^13 + 216232 * q^14 - 127934 * q^15 + 195174 * q^16 + 97644 * q^17 - 475646 * q^18 + 373314 * q^19 + 685798 * q^20 - 213824 * q^21 + 167756 * q^22 - 711654 * q^23 - 1396830 * q^24 - 1041274 * q^26 + 3386272 * q^27 - 491836 * q^28 - 178394 * q^29 - 6200118 * q^30 + 1793468 * q^31 + 955004 * q^32 - 369536 * q^33 + 4592250 * q^34 + 1520488 * q^35 - 11372454 * q^36 + 247730 * q^37 + 14761942 * q^39 + 15044736 * q^40 - 2285272 * q^41 - 1024084 * q^42 + 18199452 * q^43 + 5629904 * q^44 - 10428746 * q^45 - 25708302 * q^46 - 8054836 * q^47 - 19983002 * q^48 - 18234288 * q^49 - 7208294 * q^50 - 26124488 * q^52 - 26091116 * q^53 + 85261048 * q^54 - 2240524 * q^55 + 84926940 * q^56 + 57257116 * q^57 - 68659210 * q^58 - 58197100 * q^59 - 64428232 * q^60 - 38537926 * q^61 + 2935614 * q^62 - 43972436 * q^63 + 44466032 * q^65 + 209850752 * q^66 + 30462126 * q^67 + 68365716 * q^68 + 163279656 * q^69 + 256782116 * q^70 - 113836054 * q^71 - 445647780 * q^72 - 30722748 * q^73 - 156843362 * q^74 - 364524840 * q^75 - 180633890 * q^76 + 252047848 * q^78 + 364818200 * q^79 + 372146084 * q^80 - 17174000 * q^81 + 272314014 * q^82 + 389373188 * q^83 - 464450276 * q^84 - 331999758 * q^85 - 32039076 * q^86 - 337693978 * q^87 - 1180586364 * q^88 - 333088936 * q^89 + 251459988 * q^91 + 1275961956 * q^92 + 637177780 * q^93 + 171539378 * q^94 + 318589986 * q^95 + 1019148280 * q^96 - 230031796 * q^97 - 1287859636 * q^98 - 359343776 * 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 −28.0190 7.50766i 22.8511 + 39.5793i 506.995 + 292.714i 131.372 131.372i −343.117 1280.53i −2789.95 + 747.565i −6757.00 6757.00i 2236.15 3873.13i −4667.19 + 2694.60i
2.2 −16.0639 4.30431i −60.2561 104.367i 17.8198 + 10.2883i 169.528 169.528i 518.723 + 1935.90i −1635.34 + 438.187i 2768.49 + 2768.49i −3981.10 + 6895.47i −3452.98 + 1993.58i
2.3 −15.5495 4.16648i 11.3712 + 19.6955i 2.72537 + 1.57349i −311.308 + 311.308i −94.7557 353.633i 4365.04 1169.61i 2878.23 + 2878.23i 3021.89 5234.07i 6137.75 3543.63i
2.4 −4.97311 1.33254i 73.7308 + 127.705i −198.746 114.746i −65.5592 + 65.5592i −196.499 733.343i −2444.94 + 655.120i 1767.47 + 1767.47i −7591.95 + 13149.6i 413.394 238.673i
2.5 4.84258 + 1.29756i 0.725648 + 1.25686i −199.936 115.433i 724.141 724.141i 1.88315 + 7.02801i 258.924 69.3786i −1725.95 1725.95i 3279.45 5680.17i 4446.33 2567.09i
2.6 8.65394 + 2.31882i −30.7307 53.2271i −152.189 87.8662i −612.629 + 612.629i −142.518 531.883i −968.610 + 259.538i −2735.08 2735.08i 1391.75 2410.58i −6722.23 + 3881.08i
2.7 22.9875 + 6.15949i 34.8819 + 60.4172i 268.785 + 155.183i −80.1911 + 80.1911i 429.709 + 1603.70i 249.971 66.9796i 914.868 + 914.868i 847.009 1467.06i −2337.33 + 1349.46i
2.8 26.2554 + 7.03513i −76.4565 132.427i 418.153 + 241.421i 365.377 365.377i −1075.76 4014.80i 1567.43 419.990i 4359.97 + 4359.97i −8410.69 + 14567.7i 12163.6 7022.67i
6.1 −7.06838 26.3795i −12.3990 + 21.4757i −424.216 + 244.921i −289.939 + 289.939i 654.160 + 175.282i 169.108 631.121i 4515.77 + 4515.77i 2973.03 + 5149.44i 9697.86 + 5599.06i
6.2 −4.98932 18.6204i 51.4031 89.0328i −100.124 + 57.8065i 663.519 663.519i −1914.29 512.934i −974.220 + 3635.84i −1913.63 1913.63i −2004.06 3471.13i −15665.5 9044.49i
6.3 −2.99242 11.1679i −49.8723 + 86.3814i 105.936 61.1621i 403.844 403.844i 1113.93 + 298.478i 636.017 2373.65i −3092.97 3092.97i −1694.00 2934.09i −5718.55 3301.61i
6.4 −1.72361 6.43259i 53.7668 93.1269i 183.295 105.825i −590.780 + 590.780i −691.720 185.346i 973.383 3632.71i −2202.16 2202.16i −2501.24 4332.28i 4818.52 + 2781.97i
6.5 −0.0710913 0.265316i −24.7951 + 42.9464i 221.637 127.962i −462.533 + 462.533i 13.1571 + 3.52543i −1022.80 + 3817.16i −99.4285 99.4285i 2050.90 + 3552.27i 155.600 + 89.8354i
6.6 3.54920 + 13.2458i 29.1358 50.4648i 58.8481 33.9759i 367.308 367.308i 771.855 + 206.818i 60.5481 225.969i 3141.23 + 3141.23i 1582.71 + 2741.33i 6168.94 + 3561.64i
6.7 5.40711 + 20.1796i −57.1883 + 99.0531i −156.277 + 90.2265i 54.1625 54.1625i −2308.07 618.447i 319.890 1193.84i 1116.02 + 1116.02i −3260.51 5647.36i 1385.84 + 800.116i
6.8 7.75454 + 28.9403i 32.8317 56.8661i −555.707 + 320.838i −552.313 + 552.313i 1900.32 + 509.189i −156.442 + 583.848i −8170.84 8170.84i 1124.66 + 1947.97i −20267.1 11701.2i
7.1 −28.0190 + 7.50766i 22.8511 39.5793i 506.995 292.714i 131.372 + 131.372i −343.117 + 1280.53i −2789.95 747.565i −6757.00 + 6757.00i 2236.15 + 3873.13i −4667.19 2694.60i
7.2 −16.0639 + 4.30431i −60.2561 + 104.367i 17.8198 10.2883i 169.528 + 169.528i 518.723 1935.90i −1635.34 438.187i 2768.49 2768.49i −3981.10 6895.47i −3452.98 1993.58i
7.3 −15.5495 + 4.16648i 11.3712 19.6955i 2.72537 1.57349i −311.308 311.308i −94.7557 + 353.633i 4365.04 + 1169.61i 2878.23 2878.23i 3021.89 + 5234.07i 6137.75 + 3543.63i
7.4 −4.97311 + 1.33254i 73.7308 127.705i −198.746 + 114.746i −65.5592 65.5592i −196.499 + 733.343i −2444.94 655.120i 1767.47 1767.47i −7591.95 13149.6i 413.394 + 238.673i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.8 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 13.9.f.a 32
13.f odd 12 1 inner 13.9.f.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.9.f.a 32 1.a even 1 1 trivial
13.9.f.a 32 13.f odd 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(13, [\chi])$$.