Properties

Label 13.11.f.a
Level $13$
Weight $11$
Character orbit 13.f
Analytic conductor $8.260$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.2");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 11 \)
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: \(8.25964428476\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 28 q^{2} - 2 q^{3} - 6 q^{4} + 7786 q^{5} + 19258 q^{6} + 30856 q^{7} + 148650 q^{8} - 393662 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 28 q^{2} - 2 q^{3} - 6 q^{4} + 7786 q^{5} + 19258 q^{6} + 30856 q^{7} + 148650 q^{8} - 393662 q^{9} - 37926 q^{10} - 37628 q^{11} - 782358 q^{13} - 2763352 q^{14} + 1513066 q^{15} + 6979366 q^{16} + 2121648 q^{17} - 7329038 q^{18} + 10570318 q^{19} + 3621686 q^{20} - 22758188 q^{21} - 8944932 q^{22} + 14209650 q^{23} + 25625106 q^{24} - 65190122 q^{26} - 90176648 q^{27} + 71516420 q^{28} - 36734590 q^{29} + 386524746 q^{30} + 69105068 q^{31} - 244683668 q^{32} - 239651876 q^{33} - 205577766 q^{34} - 13671748 q^{35} + 862663626 q^{36} - 228382792 q^{37} + 66588406 q^{39} - 1486470528 q^{40} + 390227548 q^{41} - 100396996 q^{42} + 484189248 q^{43} + 652817968 q^{44} - 138360062 q^{45} - 500082990 q^{46} - 146042276 q^{47} - 51684026 q^{48} + 1432995756 q^{49} + 3130730474 q^{50} - 2400496408 q^{52} - 2133839560 q^{53} - 395306792 q^{54} + 214636560 q^{55} - 784908372 q^{56} - 1747973168 q^{57} + 2203678422 q^{58} + 451487308 q^{59} + 345479768 q^{60} - 932780124 q^{61} - 480851634 q^{62} - 4563278084 q^{63} + 7215202492 q^{65} + 10215374144 q^{66} - 7109947262 q^{67} - 2010692796 q^{68} - 8061602724 q^{69} - 7716006396 q^{70} - 2493828338 q^{71} + 20805439692 q^{72} + 10166935954 q^{73} + 6945437918 q^{74} + 2520930528 q^{75} + 2500628302 q^{76} - 23418396584 q^{78} - 163312488 q^{79} - 33101148380 q^{80} - 484418462 q^{81} - 40275130626 q^{82} - 13613974796 q^{83} + 50938536844 q^{84} + 46237295256 q^{85} - 5313857988 q^{86} + 8165526878 q^{87} + 41091238308 q^{88} + 13233745138 q^{89} - 10578987068 q^{91} - 49960214844 q^{92} - 19890459728 q^{93} + 4558125234 q^{94} - 20960240478 q^{95} - 114660268616 q^{96} + 43792912250 q^{97} + 57434259340 q^{98} + 34484945872 q^{99}+O(q^{100}) \) 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 −54.7307 14.6651i −136.466 236.366i 1893.58 + 1093.26i 3512.71 3512.71i 4002.56 + 14937.7i 18809.0 5039.86i −46577.0 46577.0i −7721.33 + 13373.7i −243767. + 140739.i
2.2 −44.5588 11.9395i 184.618 + 319.768i 956.127 + 552.020i 685.052 685.052i −4408.50 16452.7i 5307.14 1422.04i −2610.85 2610.85i −38643.3 + 66932.1i −38704.3 + 22345.9i
2.3 −44.0680 11.8080i −40.7005 70.4953i 915.750 + 528.709i −3410.23 + 3410.23i 961.181 + 3587.18i −10361.1 + 2776.24i −1078.05 1078.05i 26211.4 45399.6i 190550. 110014.i
2.4 −17.6947 4.74127i −78.1881 135.426i −596.189 344.210i 2039.15 2039.15i 741.422 + 2767.02i −17861.7 + 4786.02i 22181.6 + 22181.6i 17297.7 29960.6i −45750.2 + 26413.9i
2.5 −7.51235 2.01293i 87.4631 + 151.490i −834.427 481.756i 801.226 801.226i −352.113 1314.11i 4758.44 1275.02i 10930.2 + 10930.2i 14224.9 24638.3i −7631.90 + 4406.28i
2.6 −5.92200 1.58679i −212.512 368.082i −854.258 493.206i −2053.42 + 2053.42i 674.426 + 2516.99i 23963.5 6421.00i 8715.54 + 8715.54i −60798.5 + 105306.i 15418.7 8902.00i
2.7 20.4681 + 5.48441i 124.573 + 215.766i −497.945 287.489i −3268.32 + 3268.32i 1366.42 + 5099.53i −924.499 + 247.719i −23958.6 23958.6i −1512.16 + 2619.14i −84821.1 + 48971.5i
2.8 34.0749 + 9.13034i −139.436 241.510i 190.925 + 110.231i 139.108 139.108i −2546.19 9502.52i −28549.1 + 7649.72i −20043.9 20043.9i −9360.22 + 16212.4i 6010.20 3469.99i
2.9 35.3172 + 9.46323i −57.0998 98.8998i 270.945 + 156.430i 1817.84 1817.84i −1080.70 4033.22i 25897.6 6939.25i −18385.8 18385.8i 23003.7 39843.6i 81403.8 46998.5i
2.10 44.0816 + 11.8116i 206.865 + 358.301i 916.864 + 529.351i 3581.20 3581.20i 4886.83 + 18237.9i −13125.2 + 3516.88i 1119.86 + 1119.86i −56061.8 + 97101.9i 200165. 115565.i
2.11 60.5351 + 16.2203i −9.76459 16.9128i 2514.59 + 1451.80i −2572.45 + 2572.45i −316.770 1182.20i 5789.30 1551.24i 83293.7 + 83293.7i 29333.8 50807.6i −197449. + 113997.i
6.1 −16.0451 59.8810i 208.329 360.837i −2441.48 + 1409.59i −295.654 + 295.654i −24949.9 6685.31i 3193.45 11918.1i 78693.5 + 78693.5i −57277.6 99207.6i 22447.8 + 12960.3i
6.2 −14.3584 53.5862i −184.952 + 320.347i −1778.51 + 1026.82i 1813.85 1813.85i 19821.8 + 5311.22i −1099.66 + 4103.99i 40390.6 + 40390.6i −38890.1 67359.6i −123241. 71153.4i
6.3 −9.57848 35.7474i −3.30525 + 5.72487i −299.317 + 172.811i −2657.21 + 2657.21i 236.308 + 63.3186i −3109.41 + 11604.5i −17752.4 17752.4i 29502.7 + 51100.1i 120441. + 69536.4i
6.4 −8.30406 30.9912i 23.5009 40.7047i −4.68648 + 2.70574i 2353.93 2353.93i −1456.64 390.305i 3063.89 11434.6i −23108.8 23108.8i 28419.9 + 49224.7i −92498.2 53403.8i
6.5 −1.49885 5.59379i −211.399 + 366.154i 857.766 495.231i −2649.40 + 2649.40i 2365.05 + 633.712i 3656.71 13647.0i −8249.11 8249.11i −59854.7 103671.i 18791.3 + 10849.2i
6.6 −1.17721 4.39342i 190.375 329.739i 868.894 501.656i 159.885 159.885i −1672.79 448.224i −2112.39 + 7883.56i −6520.26 6520.26i −42960.7 74410.1i −890.664 514.225i
6.7 2.93410 + 10.9502i −113.394 + 196.404i 775.512 447.742i 3205.80 3205.80i −2483.37 665.416i −7098.40 + 26491.6i 15386.8 + 15386.8i 3808.24 + 6596.06i 44510.3 + 25698.0i
6.8 4.78441 + 17.8557i 7.91409 13.7076i 590.876 341.142i −221.592 + 221.592i 282.623 + 75.7285i 6332.85 23634.5i 22303.3 + 22303.3i 29399.2 + 50921.0i −5016.87 2896.49i
6.9 9.87930 + 36.8700i 98.0351 169.802i −374.990 + 216.500i −3068.87 + 3068.87i 7229.11 + 1937.04i −4246.14 + 15846.8i 15951.5 + 15951.5i 10302.8 + 17844.9i −143468. 82831.1i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.11
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 13.11.f.a 44
13.f odd 12 1 inner 13.11.f.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.11.f.a 44 1.a even 1 1 trivial
13.11.f.a 44 13.f odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(13, [\chi])\).