Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,2,Mod(37,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(48))
chi = DirichletCharacter(H, H._module([0, 27, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.37");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 448.bn (of order \(48\), degree \(16\), 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: | \(3.57729801055\) |
Analytic rank: | \(0\) |
Dimension: | \(992\) |
Relative dimension: | \(62\) over \(\Q(\zeta_{48})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{48}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | −1.41239 | − | 0.0717279i | 0.514998 | + | 0.0337548i | 1.98971 | + | 0.202616i | −0.714996 | − | 0.627035i | −0.724959 | − | 0.0846148i | −2.64110 | − | 0.156771i | −2.79572 | − | 0.428891i | −2.71025 | − | 0.356811i | 0.964880 | + | 0.936905i |
37.2 | −1.40224 | + | 0.183616i | 1.08545 | + | 0.0711444i | 1.93257 | − | 0.514948i | −2.62851 | − | 2.30514i | −1.53513 | + | 0.0995446i | 0.329500 | + | 2.62515i | −2.61538 | + | 1.07693i | −1.80119 | − | 0.237131i | 4.10907 | + | 2.74973i |
37.3 | −1.39571 | − | 0.228010i | −2.98633 | − | 0.195734i | 1.89602 | + | 0.636472i | 3.24186 | + | 2.84304i | 4.12343 | + | 0.954102i | −0.124109 | + | 2.64284i | −2.50118 | − | 1.32064i | 5.90552 | + | 0.777476i | −3.87646 | − | 4.70724i |
37.4 | −1.39036 | + | 0.258630i | −2.85090 | − | 0.186858i | 1.86622 | − | 0.719178i | −1.54843 | − | 1.35794i | 4.01211 | − | 0.477527i | −2.11719 | − | 1.58666i | −2.40873 | + | 1.48258i | 5.11838 | + | 0.673848i | 2.50408 | + | 1.48756i |
37.5 | −1.38040 | + | 0.307393i | 2.25801 | + | 0.147998i | 1.81102 | − | 0.848652i | 2.25362 | + | 1.97638i | −3.16245 | + | 0.489800i | 1.26670 | − | 2.32281i | −2.23906 | + | 1.72818i | 2.10236 | + | 0.276781i | −3.71843 | − | 2.03544i |
37.6 | −1.37687 | − | 0.322834i | −1.87048 | − | 0.122597i | 1.79156 | + | 0.889002i | −0.730267 | − | 0.640427i | 2.53583 | + | 0.772654i | 1.61648 | − | 2.09451i | −2.17974 | − | 1.80242i | 0.509316 | + | 0.0670527i | 0.798734 | + | 1.11754i |
37.7 | −1.35998 | − | 0.387884i | 3.13844 | + | 0.205704i | 1.69909 | + | 1.05503i | 1.47782 | + | 1.29601i | −4.18842 | − | 1.49710i | −1.83740 | + | 1.90367i | −1.90150 | − | 2.09387i | 6.83314 | + | 0.899599i | −1.50710 | − | 2.33577i |
37.8 | −1.33287 | − | 0.472722i | 1.74204 | + | 0.114179i | 1.55307 | + | 1.26015i | −1.33900 | − | 1.17427i | −2.26793 | − | 0.975687i | −0.00558749 | − | 2.64575i | −1.47433 | − | 2.41378i | 0.0473319 | + | 0.00623137i | 1.22960 | + | 2.19812i |
37.9 | −1.33134 | + | 0.477007i | −0.199769 | − | 0.0130935i | 1.54493 | − | 1.27012i | 2.28631 | + | 2.00504i | 0.272206 | − | 0.0778592i | −2.64046 | − | 0.167191i | −1.45097 | + | 2.42790i | −2.93460 | − | 0.386347i | −4.00027 | − | 1.57880i |
37.10 | −1.30660 | + | 0.541114i | −1.32041 | − | 0.0865444i | 1.41439 | − | 1.41404i | 0.844831 | + | 0.740897i | 1.77208 | − | 0.601415i | 2.64569 | + | 0.0180703i | −1.08289 | + | 2.61292i | −1.23834 | − | 0.163030i | −1.50476 | − | 0.510904i |
37.11 | −1.23053 | − | 0.696990i | −1.99459 | − | 0.130733i | 1.02841 | + | 1.71533i | −1.63884 | − | 1.43723i | 2.36329 | + | 1.55108i | −0.558109 | + | 2.58622i | −0.0699215 | − | 2.82756i | 0.986978 | + | 0.129938i | 1.01491 | + | 2.91081i |
37.12 | −1.12956 | − | 0.850932i | 0.781099 | + | 0.0511960i | 0.551830 | + | 1.92236i | 2.01223 | + | 1.76468i | −0.838738 | − | 0.722491i | 2.56848 | + | 0.634758i | 1.01247 | − | 2.64100i | −2.36684 | − | 0.311600i | −0.771321 | − | 3.70558i |
37.13 | −1.11577 | + | 0.868945i | −0.393427 | − | 0.0257865i | 0.489869 | − | 1.93908i | −3.12830 | − | 2.74344i | 0.461379 | − | 0.313094i | 2.15867 | − | 1.52975i | 1.13837 | + | 2.58923i | −2.82021 | − | 0.371288i | 5.87435 | + | 0.342724i |
37.14 | −1.06794 | − | 0.927092i | 2.82779 | + | 0.185343i | 0.281002 | + | 1.98016i | −2.30734 | − | 2.02348i | −2.84809 | − | 2.81956i | 2.10705 | + | 1.60011i | 1.53570 | − | 2.37521i | 4.98771 | + | 0.656644i | 0.588151 | + | 4.30007i |
37.15 | −1.06579 | + | 0.929561i | 2.36639 | + | 0.155101i | 0.271833 | − | 1.98144i | 0.559998 | + | 0.491105i | −2.66625 | + | 2.03439i | 1.11383 | + | 2.39987i | 1.55215 | + | 2.36449i | 2.60139 | + | 0.342479i | −1.05335 | − | 0.00286447i |
37.16 | −1.02903 | − | 0.970099i | −0.0690072 | − | 0.00452297i | 0.117817 | + | 1.99653i | 0.464281 | + | 0.407164i | 0.0666230 | + | 0.0715981i | −2.36955 | + | 1.17697i | 1.81559 | − | 2.16879i | −2.96959 | − | 0.390954i | −0.0827717 | − | 0.869384i |
37.17 | −0.991412 | + | 1.00852i | 1.40015 | + | 0.0917708i | −0.0342060 | − | 1.99971i | −0.839566 | − | 0.736280i | −1.48068 | + | 1.32109i | −1.89311 | − | 1.84828i | 2.05065 | + | 1.94804i | −1.02233 | − | 0.134592i | 1.57490 | − | 0.116759i |
37.18 | −0.924857 | − | 1.06988i | −1.96519 | − | 0.128806i | −0.289280 | + | 1.97897i | 1.80685 | + | 1.58456i | 1.67972 | + | 2.22164i | −0.654526 | − | 2.56351i | 2.38480 | − | 1.52077i | 0.871059 | + | 0.114677i | 0.0242146 | − | 3.39860i |
37.19 | −0.746621 | + | 1.20107i | −3.23880 | − | 0.212282i | −0.885115 | − | 1.79348i | −0.308001 | − | 0.270109i | 2.67312 | − | 3.73151i | 1.96466 | + | 1.77204i | 2.81493 | + | 0.275968i | 7.47041 | + | 0.983498i | 0.554379 | − | 0.168260i |
37.20 | −0.685515 | + | 1.23696i | −1.24170 | − | 0.0813853i | −1.06014 | − | 1.69591i | 1.77939 | + | 1.56048i | 0.951874 | − | 1.48014i | 1.08439 | − | 2.41332i | 2.82451 | − | 0.148780i | −1.43914 | − | 0.189466i | −3.15005 | + | 1.13130i |
See next 80 embeddings (of 992 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
64.i | even | 16 | 1 | inner |
448.bn | even | 48 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 448.2.bn.a | ✓ | 992 |
7.c | even | 3 | 1 | inner | 448.2.bn.a | ✓ | 992 |
64.i | even | 16 | 1 | inner | 448.2.bn.a | ✓ | 992 |
448.bn | even | 48 | 1 | inner | 448.2.bn.a | ✓ | 992 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
448.2.bn.a | ✓ | 992 | 1.a | even | 1 | 1 | trivial |
448.2.bn.a | ✓ | 992 | 7.c | even | 3 | 1 | inner |
448.2.bn.a | ✓ | 992 | 64.i | even | 16 | 1 | inner |
448.2.bn.a | ✓ | 992 | 448.bn | even | 48 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(448, [\chi])\).