Properties

Label 448.2.bn.a
Level $448$
Weight $2$
Character orbit 448.bn
Analytic conductor $3.577$
Analytic rank $0$
Dimension $992$
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] = [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.

\(\operatorname{Tr}(f)(q) = \) \( 992 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 32 q^{6} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 992 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 32 q^{6} - 16 q^{7} - 32 q^{8} - 8 q^{9} - 8 q^{10} - 8 q^{11} - 8 q^{12} - 32 q^{13} - 16 q^{14} - 32 q^{15} - 8 q^{16} - 8 q^{17} - 8 q^{18} - 8 q^{19} - 32 q^{20} - 16 q^{21} - 16 q^{22} - 8 q^{23} - 88 q^{24} - 8 q^{25} - 8 q^{26} - 32 q^{27} - 56 q^{28} - 32 q^{29} - 88 q^{30} - 8 q^{32} - 32 q^{34} - 16 q^{35} + 128 q^{36} - 8 q^{37} - 88 q^{38} - 8 q^{39} - 8 q^{40} - 32 q^{41} - 56 q^{42} - 32 q^{43} - 16 q^{44} - 8 q^{45} - 8 q^{46} - 8 q^{47} - 32 q^{48} - 16 q^{49} + 16 q^{50} - 8 q^{51} - 56 q^{52} - 8 q^{53} - 8 q^{54} - 32 q^{55} - 16 q^{56} - 32 q^{57} - 8 q^{58} + 56 q^{59} + 88 q^{60} - 8 q^{61} - 64 q^{62} - 32 q^{63} + 160 q^{64} - 16 q^{65} - 136 q^{66} - 88 q^{67} - 8 q^{68} - 32 q^{69} - 16 q^{70} - 160 q^{71} - 8 q^{72} - 8 q^{73} + 48 q^{74} - 8 q^{75} - 32 q^{76} - 16 q^{77} + 64 q^{78} - 8 q^{79} - 32 q^{80} - 8 q^{81} + 72 q^{82} - 32 q^{83} - 128 q^{84} - 32 q^{85} - 8 q^{86} - 8 q^{87} - 8 q^{88} - 8 q^{89} - 320 q^{90} - 16 q^{91} - 32 q^{92} - 56 q^{93} - 8 q^{94} + 128 q^{96} - 152 q^{98} - 80 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.62
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(448, [\chi])\).