Properties

Label 48.9.i.a
Level $48$
Weight $9$
Character orbit 48.i
Analytic conductor $19.554$
Analytic rank $0$
Dimension $124$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,9,Mod(5,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.5");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 48.i (of order \(4\), degree \(2\), 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: \(19.5541732829\)
Analytic rank: \(0\)
Dimension: \(124\)
Relative dimension: \(62\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 124 q - 2 q^{3} - 4 q^{4} - 3236 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 124 q - 2 q^{3} - 4 q^{4} - 3236 q^{6} + 17496 q^{10} + 8608 q^{12} - 4 q^{13} - 4 q^{15} - 116560 q^{16} - 290364 q^{18} - 167556 q^{19} + 13120 q^{21} + 190552 q^{22} - 714296 q^{24} - 25538 q^{27} + 1391160 q^{28} + 3794052 q^{30} - 8 q^{31} - 4 q^{33} + 2752744 q^{34} + 1313476 q^{36} - 4 q^{37} + 12578584 q^{40} - 5150600 q^{42} + 5384956 q^{43} - 781252 q^{45} - 25980296 q^{46} + 26695408 q^{48} - 82354308 q^{49} - 15081600 q^{51} + 24748672 q^{52} - 38189720 q^{54} + 74539832 q^{58} + 68522688 q^{60} + 24476028 q^{61} + 23059200 q^{63} - 63032488 q^{64} + 79650868 q^{66} - 74704388 q^{67} - 13124 q^{69} + 109805784 q^{70} - 100711608 q^{72} - 99849278 q^{75} - 151417912 q^{76} + 158736908 q^{78} - 72203272 q^{79} - 4 q^{81} + 98422144 q^{82} + 7625392 q^{84} - 105397504 q^{85} + 53374432 q^{88} + 175196592 q^{90} - 152141184 q^{91} + 138242812 q^{93} + 114812016 q^{94} - 193537240 q^{96} - 8 q^{97} + 228915068 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −16.0000 + 0.0181473i −69.5660 + 41.4919i 255.999 0.580715i −76.5595 + 76.5595i 1112.30 665.132i 2902.85i −4095.98 + 13.9371i 3117.85 5772.85i 1223.56 1226.34i
5.2 −15.9624 1.09650i 69.1317 42.2115i 253.595 + 35.0054i −755.337 + 755.337i −1149.79 + 597.993i 255.661i −4009.60 836.837i 2997.38 5836.30i 12885.2 11228.8i
5.3 −15.8366 2.28109i 55.2937 + 59.1913i 245.593 + 72.2492i −126.448 + 126.448i −740.641 1063.52i 2580.10i −3724.55 1704.40i −446.215 + 6545.81i 2290.94 1714.06i
5.4 −15.8350 2.29186i 73.5550 33.9214i 245.495 + 72.5831i 608.521 608.521i −1242.49 + 368.568i 3930.72i −3721.06 1711.99i 4259.68 4990.18i −11030.6 + 8241.28i
5.5 −15.5809 3.63791i −70.8508 39.2576i 229.531 + 113.364i 396.522 396.522i 961.107 + 869.419i 243.614i −3163.91 2601.33i 3478.68 + 5562.87i −7620.70 + 4735.68i
5.6 −15.5700 + 3.68426i 16.7996 + 79.2387i 228.852 114.728i 356.180 356.180i −553.506 1171.86i 2259.12i −3140.55 + 2629.47i −5996.55 + 2662.36i −4233.48 + 6858.00i
5.7 −15.1821 + 5.05022i −2.05641 80.9739i 204.991 153.346i −38.4932 + 38.4932i 440.156 + 1218.97i 592.015i −2337.75 + 3363.35i −6552.54 + 333.032i 390.007 778.805i
5.8 −14.4416 6.88759i −47.3668 + 65.7068i 161.122 + 198.936i −414.223 + 414.223i 1136.62 622.672i 3504.66i −956.682 3982.71i −2073.78 6224.64i 8835.05 3129.06i
5.9 −14.1385 + 7.49024i −77.7033 22.8735i 143.792 211.801i −492.059 + 492.059i 1269.93 258.620i 3803.67i −446.563 + 4071.58i 5514.60 + 3554.70i 3271.32 10642.6i
5.10 −13.8367 + 8.03403i 79.4910 15.5621i 126.909 222.329i 632.583 632.583i −974.868 + 853.960i 4516.63i 30.1960 + 4095.89i 6076.65 2474.09i −3670.67 + 13835.0i
5.11 −13.1212 9.15614i −51.1569 62.8011i 88.3302 + 240.279i −771.584 + 771.584i 96.2230 + 1292.42i 1052.69i 1041.03 3961.50i −1326.94 + 6425.41i 17188.8 3059.35i
5.12 −13.0661 + 9.23453i −70.3483 + 40.1511i 85.4470 241.319i 651.651 651.651i 548.403 1174.25i 290.192i 1112.00 + 3942.16i 3336.77 5649.13i −2496.86 + 14532.2i
5.13 −12.9672 9.37299i 15.1978 79.5615i 80.2941 + 243.082i 295.006 295.006i −942.801 + 889.237i 1083.57i 1237.22 3904.68i −6099.06 2418.31i −6590.48 + 1060.30i
5.14 −12.8620 + 9.51679i −13.6278 + 79.8454i 74.8613 244.810i −792.195 + 792.195i −584.591 1156.66i 2103.96i 1366.94 + 3861.18i −6189.56 2176.24i 2650.04 17728.3i
5.15 −12.2522 + 10.2899i 74.9445 + 30.7298i 44.2340 252.149i −204.371 + 204.371i −1234.44 + 394.667i 2167.67i 2052.64 + 3544.56i 4672.36 + 4606.06i 401.033 4606.97i
5.16 −11.8041 10.8011i −12.1951 + 80.0767i 22.6731 + 254.994i 756.913 756.913i 1008.87 813.513i 1455.04i 2486.58 3254.87i −6263.56 1953.08i −17110.2 + 759.188i
5.17 −11.6216 10.9972i 63.5896 + 50.1733i 14.1218 + 255.610i −284.826 + 284.826i −187.244 1282.40i 2397.49i 2646.89 3125.89i 1526.28 + 6381.00i 6442.42 177.828i
5.18 −10.8590 11.7508i 80.1414 11.7622i −20.1629 + 255.205i 106.550 106.550i −1008.47 814.001i 2233.15i 3217.81 2534.35i 6284.30 1885.27i −2409.09 95.0190i
5.19 −8.62811 + 13.4743i −68.9923 42.4389i −107.111 232.515i −120.715 + 120.715i 1167.11 563.453i 4118.69i 4057.13 + 562.915i 2958.88 + 5855.92i −585.002 2668.08i
5.20 −7.87771 + 13.9263i −23.6488 77.4709i −131.884 219.415i 784.467 784.467i 1265.18 + 280.953i 3281.28i 4094.57 108.166i −5442.47 + 3664.18i 4744.92 + 17104.5i
See next 80 embeddings (of 124 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.62
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.9.i.a 124
3.b odd 2 1 inner 48.9.i.a 124
16.e even 4 1 inner 48.9.i.a 124
48.i odd 4 1 inner 48.9.i.a 124
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.9.i.a 124 1.a even 1 1 trivial
48.9.i.a 124 3.b odd 2 1 inner
48.9.i.a 124 16.e even 4 1 inner
48.9.i.a 124 48.i odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(48, [\chi])\).