Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [40,5,Mod(13,40)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 3]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("40.13");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 40 = 2^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 40.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: | \(4.13479852335\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Relative dimension: | \(22\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −3.99772 | − | 0.134931i | −9.27319 | + | 9.27319i | 15.9636 | + | 1.07883i | −21.1329 | + | 13.3567i | 38.3229 | − | 35.8204i | 26.2670 | − | 26.2670i | −63.6724 | − | 6.46686i | − | 90.9842i | 86.2855 | − | 50.5451i | |
13.2 | −3.98174 | + | 0.381759i | −3.10315 | + | 3.10315i | 15.7085 | − | 3.04013i | 18.7844 | − | 16.4969i | 11.1713 | − | 13.5406i | −52.9781 | + | 52.9781i | −61.3867 | + | 18.1019i | 61.7410i | −68.4966 | + | 72.8575i | ||
13.3 | −3.82786 | + | 1.16082i | 12.1818 | − | 12.1818i | 13.3050 | − | 8.88689i | −21.6293 | − | 12.5369i | −32.4892 | + | 60.7708i | −17.4020 | + | 17.4020i | −40.6136 | + | 49.4625i | − | 215.790i | 97.3469 | + | 22.8819i | |
13.4 | −3.55430 | − | 1.83492i | 6.54244 | − | 6.54244i | 9.26616 | + | 13.0437i | 20.3139 | + | 14.5721i | −35.2586 | + | 11.2490i | 24.6615 | − | 24.6615i | −9.00061 | − | 63.3639i | − | 4.60695i | −45.4630 | − | 89.0680i | |
13.5 | −3.25500 | + | 2.32487i | 1.65143 | − | 1.65143i | 5.19000 | − | 15.1349i | 4.54838 | + | 24.5828i | −1.53605 | + | 9.21475i | 28.3940 | − | 28.3940i | 18.2931 | + | 61.3300i | 75.5456i | −71.9566 | − | 69.4424i | ||
13.6 | −2.86960 | − | 2.78664i | 1.60861 | − | 1.60861i | 0.469241 | + | 15.9931i | −24.8651 | − | 2.59384i | −9.09870 | + | 0.133450i | −23.8147 | + | 23.8147i | 43.2206 | − | 47.2015i | 75.8247i | 64.1248 | + | 76.7334i | ||
13.7 | −2.32487 | + | 3.25500i | −1.65143 | + | 1.65143i | −5.19000 | − | 15.1349i | −4.54838 | − | 24.5828i | −1.53605 | − | 9.21475i | 28.3940 | − | 28.3940i | 61.3300 | + | 18.2931i | 75.5456i | 90.5912 | + | 42.3466i | ||
13.8 | −2.22419 | − | 3.32460i | −9.32929 | + | 9.32929i | −6.10592 | + | 14.7891i | 13.0913 | − | 21.2983i | 51.7663 | + | 10.2660i | 43.0150 | − | 43.0150i | 62.7486 | − | 12.5941i | − | 93.0715i | −99.9259 | + | 3.84844i | |
13.9 | −1.16082 | + | 3.82786i | −12.1818 | + | 12.1818i | −13.3050 | − | 8.88689i | 21.6293 | + | 12.5369i | −32.4892 | − | 60.7708i | −17.4020 | + | 17.4020i | 49.4625 | − | 40.6136i | − | 215.790i | −73.0972 | + | 68.2408i | |
13.10 | −0.401499 | − | 3.97980i | 7.77558 | − | 7.77558i | −15.6776 | + | 3.19577i | 13.6749 | − | 20.9284i | −34.0671 | − | 27.8233i | −5.91736 | + | 5.91736i | 19.0131 | + | 61.1106i | − | 39.9192i | −88.7812 | − | 46.0207i | |
13.11 | −0.381759 | + | 3.98174i | 3.10315 | − | 3.10315i | −15.7085 | − | 3.04013i | −18.7844 | + | 16.4969i | 11.1713 | + | 13.5406i | −52.9781 | + | 52.9781i | 18.1019 | − | 61.3867i | 61.7410i | −58.5153 | − | 81.0923i | ||
13.12 | −0.133745 | − | 3.99776i | −4.11430 | + | 4.11430i | −15.9642 | + | 1.06936i | 3.95777 | + | 24.6847i | 16.9983 | + | 15.8977i | −12.8562 | + | 12.8562i | 6.41019 | + | 63.6782i | 47.1451i | 98.1544 | − | 19.1237i | ||
13.13 | 0.134931 | + | 3.99772i | 9.27319 | − | 9.27319i | −15.9636 | + | 1.07883i | 21.1329 | − | 13.3567i | 38.3229 | + | 35.8204i | 26.2670 | − | 26.2670i | −6.46686 | − | 63.6724i | − | 90.9842i | 56.2480 | + | 82.6811i | |
13.14 | 1.83492 | + | 3.55430i | −6.54244 | + | 6.54244i | −9.26616 | + | 13.0437i | −20.3139 | − | 14.5721i | −35.2586 | − | 11.2490i | 24.6615 | − | 24.6615i | −63.3639 | − | 9.00061i | − | 4.60695i | 14.5195 | − | 98.9403i | |
13.15 | 2.00403 | − | 3.46177i | 4.26672 | − | 4.26672i | −7.96776 | − | 13.8750i | −24.9456 | − | 1.64905i | −6.21980 | − | 23.3211i | 50.3926 | − | 50.3926i | −63.9996 | − | 0.223240i | 44.5901i | −55.7002 | + | 83.0511i | ||
13.16 | 2.35645 | − | 3.23221i | −9.60402 | + | 9.60402i | −4.89431 | − | 15.2330i | −12.7978 | − | 21.4760i | 8.41080 | + | 53.6735i | −60.7617 | + | 60.7617i | −60.7695 | − | 20.0765i | − | 103.474i | −99.5720 | − | 9.24206i | |
13.17 | 2.78664 | + | 2.86960i | −1.60861 | + | 1.60861i | −0.469241 | + | 15.9931i | 24.8651 | + | 2.59384i | −9.09870 | − | 0.133450i | −23.8147 | + | 23.8147i | −47.2015 | + | 43.2206i | 75.8247i | 61.8468 | + | 78.5810i | ||
13.18 | 3.23221 | − | 2.35645i | 9.60402 | − | 9.60402i | 4.89431 | − | 15.2330i | 12.7978 | + | 21.4760i | 8.41080 | − | 53.6735i | −60.7617 | + | 60.7617i | −20.0765 | − | 60.7695i | − | 103.474i | 91.9720 | + | 39.2575i | |
13.19 | 3.32460 | + | 2.22419i | 9.32929 | − | 9.32929i | 6.10592 | + | 14.7891i | −13.0913 | + | 21.2983i | 51.7663 | − | 10.2660i | 43.0150 | − | 43.0150i | −12.5941 | + | 62.7486i | − | 93.0715i | −90.8948 | + | 41.6909i | |
13.20 | 3.46177 | − | 2.00403i | −4.26672 | + | 4.26672i | 7.96776 | − | 13.8750i | 24.9456 | + | 1.64905i | −6.21980 | + | 23.3211i | 50.3926 | − | 50.3926i | −0.223240 | − | 63.9996i | 44.5901i | 89.6606 | − | 44.2829i | ||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
8.b | even | 2 | 1 | inner |
40.i | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 40.5.i.a | ✓ | 44 |
4.b | odd | 2 | 1 | 160.5.m.a | 44 | ||
5.c | odd | 4 | 1 | inner | 40.5.i.a | ✓ | 44 |
8.b | even | 2 | 1 | inner | 40.5.i.a | ✓ | 44 |
8.d | odd | 2 | 1 | 160.5.m.a | 44 | ||
20.e | even | 4 | 1 | 160.5.m.a | 44 | ||
40.i | odd | 4 | 1 | inner | 40.5.i.a | ✓ | 44 |
40.k | even | 4 | 1 | 160.5.m.a | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
40.5.i.a | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
40.5.i.a | ✓ | 44 | 5.c | odd | 4 | 1 | inner |
40.5.i.a | ✓ | 44 | 8.b | even | 2 | 1 | inner |
40.5.i.a | ✓ | 44 | 40.i | odd | 4 | 1 | inner |
160.5.m.a | 44 | 4.b | odd | 2 | 1 | ||
160.5.m.a | 44 | 8.d | odd | 2 | 1 | ||
160.5.m.a | 44 | 20.e | even | 4 | 1 | ||
160.5.m.a | 44 | 40.k | even | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(40, [\chi])\).