Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [61,6,Mod(12,61)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(61, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("61.12");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 61 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 61.i (of order \(15\), degree \(8\), 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: | \(9.78341300859\) |
Analytic rank: | \(0\) |
Dimension: | \(192\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
12.1 | −9.79271 | − | 4.36000i | −15.4981 | + | 11.2600i | 55.4755 | + | 61.6118i | 6.26551 | − | 1.33178i | 200.862 | − | 42.6946i | 0.114956 | − | 1.09373i | −168.629 | − | 518.985i | 38.3119 | − | 117.912i | −67.1629 | − | 14.2759i |
12.2 | −9.08233 | − | 4.04371i | 6.29058 | − | 4.57037i | 44.7249 | + | 49.6720i | 58.9191 | − | 12.5236i | −75.6144 | + | 16.0723i | −6.33279 | + | 60.2525i | −107.036 | − | 329.424i | −56.4081 | + | 173.606i | −585.765 | − | 124.508i |
12.3 | −8.64454 | − | 3.84880i | 19.9923 | − | 14.5252i | 38.5027 | + | 42.7615i | −16.6634 | + | 3.54192i | −228.729 | + | 48.6178i | 11.8339 | − | 112.592i | −74.6857 | − | 229.859i | 113.617 | − | 349.678i | 157.680 | + | 33.5159i |
12.4 | −7.88700 | − | 3.51152i | −2.78668 | + | 2.02464i | 28.4618 | + | 31.6100i | −83.1717 | + | 17.6787i | 29.0881 | − | 6.18288i | 2.26926 | − | 21.5905i | −28.1071 | − | 86.5049i | −71.4247 | + | 219.823i | 718.054 | + | 152.627i |
12.5 | −5.95610 | − | 2.65183i | −19.9269 | + | 14.4778i | 7.03076 | + | 7.80845i | 44.7677 | − | 9.51567i | 157.079 | − | 33.3882i | 23.4605 | − | 223.212i | 43.3017 | + | 133.269i | 112.386 | − | 345.888i | −291.875 | − | 62.0399i |
12.6 | −5.67334 | − | 2.52593i | 5.86202 | − | 4.25901i | 4.39427 | + | 4.88033i | 85.0195 | − | 18.0715i | −44.0152 | + | 9.35572i | 0.479274 | − | 4.55999i | 48.8075 | + | 150.214i | −58.8670 | + | 181.174i | −527.992 | − | 112.228i |
12.7 | −5.63033 | − | 2.50678i | −14.7894 | + | 10.7451i | 4.00446 | + | 4.44740i | 11.5874 | − | 2.46298i | 110.205 | − | 23.4247i | −20.5234 | + | 195.267i | 49.5470 | + | 152.490i | 28.1768 | − | 86.7193i | −71.4152 | − | 15.1798i |
12.8 | −5.06561 | − | 2.25535i | 17.4423 | − | 12.6726i | −0.838416 | − | 0.931156i | −40.1020 | + | 8.52395i | −116.937 | + | 24.8557i | −23.2306 | + | 221.025i | 56.9790 | + | 175.363i | 68.5486 | − | 210.971i | 222.366 | + | 47.2653i |
12.9 | −3.70015 | − | 1.64741i | 3.10452 | − | 2.25556i | −10.4350 | − | 11.5893i | −37.7596 | + | 8.02605i | −15.2030 | + | 3.23150i | 17.5205 | − | 166.696i | 59.5706 | + | 183.339i | −70.5407 | + | 217.102i | 152.938 | + | 32.5081i |
12.10 | −1.54751 | − | 0.688997i | 20.6348 | − | 14.9920i | −19.4921 | − | 21.6482i | 23.0911 | − | 4.90816i | −42.2620 | + | 8.98307i | 13.0756 | − | 124.406i | 31.9996 | + | 98.4845i | 125.942 | − | 387.609i | −39.1154 | − | 8.31423i |
12.11 | −0.861070 | − | 0.383373i | −17.8098 | + | 12.9396i | −20.8177 | − | 23.1204i | −71.0225 | + | 15.0963i | 20.2962 | − | 4.31409i | 0.980358 | − | 9.32748i | 18.3823 | + | 56.5749i | 74.6654 | − | 229.797i | 66.9428 | + | 14.2291i |
12.12 | −0.483266 | − | 0.215164i | −1.06047 | + | 0.770480i | −21.2249 | − | 23.5727i | 49.2240 | − | 10.4629i | 0.678271 | − | 0.144171i | 4.69268 | − | 44.6479i | 10.4164 | + | 32.0582i | −74.5602 | + | 229.473i | −26.0396 | − | 5.53488i |
12.13 | 0.0806955 | + | 0.0359279i | 5.44969 | − | 3.95943i | −21.4070 | − | 23.7748i | −35.3565 | + | 7.51525i | 0.582019 | − | 0.123712i | −14.0977 | + | 134.130i | −1.74674 | − | 5.37591i | −61.0691 | + | 187.951i | −3.12312 | − | 0.663839i |
12.14 | 1.40588 | + | 0.625940i | −13.5362 | + | 9.83466i | −19.8275 | − | 22.0206i | 80.0991 | − | 17.0256i | −25.1863 | + | 5.35351i | −7.01227 | + | 66.7173i | −29.3093 | − | 90.2048i | 11.4183 | − | 35.1419i | 123.267 | + | 26.2012i |
12.15 | 3.16532 | + | 1.40929i | 18.5890 | − | 13.5057i | −13.3790 | − | 14.8589i | 78.3972 | − | 16.6638i | 77.8737 | − | 16.5526i | −18.1331 | + | 172.525i | −55.6710 | − | 171.338i | 88.0559 | − | 271.008i | 271.637 | + | 57.7382i |
12.16 | 3.76503 | + | 1.67630i | 19.6312 | − | 14.2629i | −10.0467 | − | 11.1580i | −90.7852 | + | 19.2970i | 97.8211 | − | 20.7925i | 10.0715 | − | 95.8241i | −59.8760 | − | 184.280i | 106.863 | − | 328.891i | −374.156 | − | 79.5294i |
12.17 | 4.36383 | + | 1.94290i | 1.90174 | − | 1.38170i | −6.14401 | − | 6.82362i | −54.7109 | + | 11.6292i | 10.9834 | − | 2.33459i | −6.40689 | + | 60.9575i | −60.7895 | − | 187.091i | −73.3836 | + | 225.851i | −261.344 | − | 55.5503i |
12.18 | 4.75842 | + | 2.11858i | −7.64829 | + | 5.55681i | −3.25804 | − | 3.61842i | 11.7544 | − | 2.49848i | −48.1663 | + | 10.2381i | 25.2257 | − | 240.006i | −59.3440 | − | 182.642i | −47.4729 | + | 146.107i | 61.2256 | + | 13.0139i |
12.19 | 5.94341 | + | 2.64618i | −21.9531 | + | 15.9499i | 6.90970 | + | 7.67399i | −1.09371 | + | 0.232475i | −172.683 | + | 36.7048i | −4.23773 | + | 40.3193i | −43.5732 | − | 134.105i | 152.450 | − | 469.192i | −7.11554 | − | 1.51246i |
12.20 | 6.70832 | + | 2.98674i | 11.4876 | − | 8.34622i | 14.6688 | + | 16.2913i | 53.7990 | − | 11.4353i | 101.990 | − | 21.6787i | 11.6118 | − | 110.479i | −22.8683 | − | 70.3813i | −12.7859 | + | 39.3511i | 395.055 | + | 83.9716i |
See next 80 embeddings (of 192 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
61.i | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 61.6.i.a | ✓ | 192 |
61.i | even | 15 | 1 | inner | 61.6.i.a | ✓ | 192 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
61.6.i.a | ✓ | 192 | 1.a | even | 1 | 1 | trivial |
61.6.i.a | ✓ | 192 | 61.i | even | 15 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(61, [\chi])\).