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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 |
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 \(S_{11}^{\mathrm{new}}(13, [\chi])\).