Properties

Label 13.13.f.a
Level $13$
Weight $13$
Character orbit 13.f
Analytic conductor $11.882$
Analytic rank $0$
Dimension $52$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.2");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 13 \)
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: \(11.8819196246\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(13\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 4 q^{2} - 2 q^{3} - 6 q^{4} - 5152 q^{5} - 64742 q^{6} - 91524 q^{7} + 262074 q^{8} - 3897236 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 4 q^{2} - 2 q^{3} - 6 q^{4} - 5152 q^{5} - 64742 q^{6} - 91524 q^{7} + 262074 q^{8} - 3897236 q^{9} + 3294714 q^{10} - 1784020 q^{11} + 19046516 q^{13} + 30715048 q^{14} - 11598914 q^{15} + 42212838 q^{16} + 58659114 q^{17} + 127801570 q^{18} - 156087780 q^{19} - 178331642 q^{20} + 306915022 q^{21} - 118709012 q^{22} - 220343766 q^{23} - 640461822 q^{24} + 774659174 q^{26} + 2155332316 q^{27} - 2237617340 q^{28} - 223019498 q^{29} - 2204550198 q^{30} - 537194948 q^{31} + 2368285916 q^{32} + 2442423358 q^{33} - 5222368326 q^{34} - 227004482 q^{35} - 44851014 q^{36} - 8834671304 q^{37} - 8883775874 q^{39} + 30970390656 q^{40} + 1308379172 q^{41} + 21916645196 q^{42} - 53518745286 q^{43} - 31311558256 q^{44} + 64717891264 q^{45} + 91061872818 q^{46} + 22748019836 q^{47} + 5680787494 q^{48} - 131177897142 q^{49} - 34452339014 q^{50} + 43505569304 q^{52} + 211009894912 q^{53} - 168270699848 q^{54} + 27147243166 q^{55} - 273305024004 q^{56} - 25328447042 q^{57} - 88913502602 q^{58} - 33695309236 q^{59} + 267540121208 q^{60} + 141740513434 q^{61} - 448559839266 q^{62} - 208416625040 q^{63} + 363281817812 q^{65} + 933787706624 q^{66} - 448795605444 q^{67} - 140245912716 q^{68} - 153503407206 q^{69} + 324669987556 q^{70} + 321801736460 q^{71} + 198757607100 q^{72} + 54549132816 q^{73} - 421073350562 q^{74} - 486773080560 q^{75} + 55527125054 q^{76} - 406305000632 q^{78} - 916647527816 q^{79} + 2460973079204 q^{80} - 57440978270 q^{81} - 52649773026 q^{82} - 428187296164 q^{83} - 709933912196 q^{84} - 417274534098 q^{85} - 1246539390180 q^{86} - 408178370398 q^{87} + 3171902194884 q^{88} + 1653151304840 q^{89} - 405276694212 q^{91} - 6618087533532 q^{92} + 207526509838 q^{93} + 579253649074 q^{94} + 9408844909146 q^{95} + 3810237394552 q^{96} - 3021650520464 q^{97} - 7414261411060 q^{98} - 5552914560128 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −120.154 32.1952i 257.966 + 446.810i 9853.23 + 5688.77i −4670.68 + 4670.68i −16610.5 61991.3i 54898.7 14710.1i −640475. 640475.i 132628. 229718.i 711574. 410828.i
2.2 −95.5902 25.6133i −610.410 1057.26i 4934.21 + 2848.77i −7207.18 + 7207.18i 31269.3 + 116699.i −122301. + 32770.4i −112070. 112070.i −479481. + 830486.i 873535. 504336.i
2.3 −72.2023 19.3465i 360.185 + 623.859i 1291.64 + 745.730i 12624.7 12624.7i −13936.7 52012.4i −207680. + 55647.8i 137665. + 137665.i 6253.68 10831.7i −1.15578e6 + 667288.i
2.4 −71.8315 19.2472i −134.875 233.610i 1242.07 + 717.109i 6783.90 6783.90i 5191.93 + 19376.5i 110310. 29557.4i 139968. + 139968.i 229338. 397225.i −617869. + 356727.i
2.5 −50.6414 13.5693i 549.131 + 951.122i −1166.81 673.661i −19544.2 + 19544.2i −14902.7 55617.5i 41955.9 11242.1i 201795. + 201795.i −337369. + 584340.i 1.25495e6 724544.i
2.6 −11.5857 3.10438i −235.597 408.067i −3422.65 1976.07i −10174.2 + 10174.2i 1462.77 + 5459.13i 1465.24 392.611i 68258.9 + 68258.9i 154708. 267962.i 149460. 86291.0i
2.7 8.21877 + 2.20221i 570.416 + 987.990i −3484.54 2011.80i 15478.9 15478.9i 2512.36 + 9376.24i 192917. 51692.0i −48852.1 48852.1i −385028. + 666889.i 161305. 93129.6i
2.8 11.4354 + 3.06412i −641.689 1111.44i −3425.86 1977.92i 20754.5 20754.5i −3932.42 14676.0i −36977.6 + 9908.12i −67404.6 67404.6i −557808. + 966151.i 300932. 173743.i
2.9 34.3703 + 9.20949i 237.371 + 411.139i −2450.74 1414.93i 1045.50 1045.50i 4372.13 + 16317.0i −154611. + 41428.0i −174260. 174260.i 153030. 265057.i 45562.5 26305.5i
2.10 67.8174 + 18.1716i −141.476 245.044i 721.753 + 416.704i 1529.25 1529.25i −5141.70 19189.1i 123755. 33160.1i −161974. 161974.i 225689. 390906.i 131499. 75920.9i
2.11 89.3467 + 23.9404i −589.975 1021.87i 3862.45 + 2229.98i −15775.1 + 15775.1i −28248.4 105425.i −26939.4 + 7218.38i 23805.8 + 23805.8i −430420. + 745508.i −1.78711e6 + 1.03179e6i
2.12 95.2253 + 25.5155i 487.469 + 844.321i 4869.57 + 2811.45i −9168.64 + 9168.64i 24876.1 + 92838.7i 23385.6 6266.16i 106440. + 106440.i −209531. + 362919.i −1.10703e6 + 639144.i
2.13 113.725 + 30.4726i −109.016 188.821i 8457.62 + 4883.01i 18181.0 18181.0i −6643.99 24795.7i −92411.2 + 24761.5i 472045. + 472045.i 241952. 419073.i 2.62166e6 1.51362e6i
6.1 −29.2050 108.995i 52.9243 91.6676i −7479.66 + 4318.39i 8243.68 8243.68i −11536.9 3091.31i −57109.8 + 213137.i 362307. + 362307.i 260119. + 450539.i −1.13927e6 657760.i
6.2 −27.7087 103.410i −405.190 + 701.809i −6378.68 + 3682.73i −15562.5 + 15562.5i 83801.6 + 22454.6i 45796.3 170914.i 247504. + 247504.i −62637.0 108490.i 2.04054e6 + 1.17811e6i
6.3 −21.6871 80.9374i 311.902 540.230i −2533.29 + 1462.60i 10878.6 10878.6i −50489.1 13528.5i 51975.1 193974.i −69370.7 69370.7i 71154.9 + 123244.i −1.11641e6 644561.i
6.4 −18.7953 70.1451i 533.836 924.631i −1019.83 + 588.798i −17226.5 + 17226.5i −74892.0 20067.2i −19491.8 + 72744.2i −149859. 149859.i −304242. 526962.i 1.53213e6 + 884577.i
6.5 −13.6554 50.9625i −509.536 + 882.542i 1136.53 656.176i 7573.57 7573.57i 51934.5 + 13915.8i −13725.9 + 51225.8i −201770. 201770.i −253533. 439132.i −489388. 282548.i
6.6 −5.92432 22.1098i −126.341 + 218.829i 3093.49 1786.03i −4688.57 + 4688.57i 5586.75 + 1496.97i −5757.24 + 21486.3i −124112. 124112.i 233796. + 404947.i 131440. + 75887.0i
6.7 0.757328 + 2.82639i 456.198 790.159i 3539.83 2043.72i 10156.9 10156.9i 2578.79 + 690.984i −7644.80 + 28530.8i 16932.0 + 16932.0i −150513. 260697.i 36399.5 + 21015.3i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.13
Significant digits:
Format:

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 13.13.f.a 52
13.f odd 12 1 inner 13.13.f.a 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.13.f.a 52 1.a even 1 1 trivial
13.13.f.a 52 13.f odd 12 1 inner

Hecke kernels

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