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Properties

Label 224.2.x.b
Level $224$
Weight $2$
Character orbit 224.x
Analytic conductor $1.789$
Analytic rank $0$
Dimension $112$
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] = [224,2,Mod(27,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 1, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.x (of order \(8\), degree \(4\), 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: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9} - 8 q^{11} - 20 q^{14} - 16 q^{15} + 16 q^{16} - 16 q^{18} - 4 q^{21} + 32 q^{22} - 48 q^{23} - 8 q^{25} - 24 q^{28} - 8 q^{29} + 16 q^{30} - 8 q^{32} + 20 q^{35} + 32 q^{36} - 8 q^{37} - 8 q^{39} + 16 q^{42} + 32 q^{43} - 144 q^{44} - 8 q^{46} + 16 q^{50} - 32 q^{51} - 32 q^{53} + 64 q^{56} - 8 q^{57} - 72 q^{58} - 88 q^{60} + 112 q^{64} - 16 q^{65} - 64 q^{67} - 40 q^{70} + 56 q^{71} - 8 q^{72} - 168 q^{74} + 52 q^{77} + 192 q^{78} - 16 q^{79} - 24 q^{84} - 48 q^{85} - 8 q^{86} - 96 q^{88} - 52 q^{91} - 8 q^{92} - 32 q^{93} + 88 q^{98} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −1.40726 − 0.140045i −0.311327 − 0.751610i 1.96078 + 0.394159i −1.37157 + 3.31126i 0.332860 + 1.10131i −2.28614 − 1.33176i −2.70413 − 0.829281i 1.65333 − 1.65333i 2.39388 − 4.46773i
27.2 −1.40726 − 0.140045i 0.311327 + 0.751610i 1.96078 + 0.394159i 1.37157 − 3.31126i −0.332860 − 1.10131i 1.33176 + 2.28614i −2.70413 − 0.829281i 1.65333 − 1.65333i −2.39388 + 4.46773i
27.3 −1.36868 + 0.355955i −1.01638 − 2.45376i 1.74659 − 0.974380i 1.11188 − 2.68430i 2.26454 + 2.99664i 0.0435134 − 2.64539i −2.04370 + 1.95533i −2.86660 + 2.86660i −0.566314 + 4.06974i
27.4 −1.36868 + 0.355955i 1.01638 + 2.45376i 1.74659 − 0.974380i −1.11188 + 2.68430i −2.26454 − 2.99664i 2.64539 − 0.0435134i −2.04370 + 1.95533i −2.86660 + 2.86660i 0.566314 − 4.06974i
27.5 −1.23575 − 0.687692i −0.592616 − 1.43070i 1.05416 + 1.69963i −0.188106 + 0.454128i −0.251557 + 2.17553i 2.64138 + 0.152070i −0.133852 − 2.82526i 0.425604 − 0.425604i 0.544752 − 0.431830i
27.6 −1.23575 − 0.687692i 0.592616 + 1.43070i 1.05416 + 1.69963i 0.188106 − 0.454128i 0.251557 − 2.17553i −0.152070 − 2.64138i −0.133852 − 2.82526i 0.425604 − 0.425604i −0.544752 + 0.431830i
27.7 −0.783777 + 1.17715i −0.935381 − 2.25821i −0.771386 − 1.84525i 0.395947 − 0.955900i 3.39139 + 0.668846i −0.935627 + 2.47479i 2.77674 + 0.538228i −2.10326 + 2.10326i 0.814908 + 1.21530i
27.8 −0.783777 + 1.17715i 0.935381 + 2.25821i −0.771386 − 1.84525i −0.395947 + 0.955900i −3.39139 − 0.668846i −2.47479 + 0.935627i 2.77674 + 0.538228i −2.10326 + 2.10326i −0.814908 − 1.21530i
27.9 −0.756841 + 1.19465i −0.613776 − 1.48179i −0.854385 − 1.80832i −1.25513 + 3.03015i 2.23475 + 0.388228i 1.88664 − 1.85488i 2.80695 + 0.347919i 0.302349 − 0.302349i −2.67004 − 3.79278i
27.10 −0.756841 + 1.19465i 0.613776 + 1.48179i −0.854385 − 1.80832i 1.25513 − 3.03015i −2.23475 − 0.388228i 1.85488 − 1.88664i 2.80695 + 0.347919i 0.302349 − 0.302349i 2.67004 + 3.79278i
27.11 −0.522376 − 1.31420i −0.167131 − 0.403490i −1.45425 + 1.37301i −0.883342 + 2.13258i −0.442962 + 0.430417i 0.895629 + 2.48955i 2.56408 + 1.19394i 1.98645 − 1.98645i 3.26407 + 0.0468829i
27.12 −0.522376 − 1.31420i 0.167131 + 0.403490i −1.45425 + 1.37301i 0.883342 − 2.13258i 0.442962 − 0.430417i −2.48955 − 0.895629i 2.56408 + 1.19394i 1.98645 − 1.98645i −3.26407 − 0.0468829i
27.13 −0.242331 − 1.39330i −1.25862 − 3.03857i −1.88255 + 0.675278i −0.824079 + 1.98950i −3.92863 + 2.48997i −0.0837494 − 2.64443i 1.39706 + 2.45931i −5.52747 + 5.52747i 2.97167 + 0.666069i
27.14 −0.242331 − 1.39330i 1.25862 + 3.03857i −1.88255 + 0.675278i 0.824079 − 1.98950i 3.92863 − 2.48997i 2.64443 + 0.0837494i 1.39706 + 2.45931i −5.52747 + 5.52747i −2.97167 − 0.666069i
27.15 0.207749 + 1.39887i −0.517660 − 1.24974i −1.91368 + 0.581228i 0.818546 − 1.97615i 1.64068 − 0.983771i −1.50177 − 2.17823i −1.21063 − 2.55624i 0.827440 − 0.827440i 2.93442 + 0.734499i
27.16 0.207749 + 1.39887i 0.517660 + 1.24974i −1.91368 + 0.581228i −0.818546 + 1.97615i −1.64068 + 0.983771i 2.17823 + 1.50177i −1.21063 − 2.55624i 0.827440 − 0.827440i −2.93442 − 0.734499i
27.17 0.329116 − 1.37538i −0.884481 − 2.13533i −1.78336 − 0.905323i 1.60624 − 3.87780i −3.22799 + 0.513731i 0.663558 + 2.56119i −1.83210 + 2.15486i −1.65599 + 1.65599i −4.80483 − 3.48544i
27.18 0.329116 − 1.37538i 0.884481 + 2.13533i −1.78336 − 0.905323i −1.60624 + 3.87780i 3.22799 − 0.513731i −2.56119 − 0.663558i −1.83210 + 2.15486i −1.65599 + 1.65599i 4.80483 + 3.48544i
27.19 0.589446 + 1.28552i −1.08865 − 2.62824i −1.30511 + 1.51548i −1.22393 + 2.95482i 2.73695 − 2.94869i −1.81057 + 1.92921i −2.71747 − 0.784442i −3.60116 + 3.60116i −4.51992 + 0.168328i
27.20 0.589446 + 1.28552i 1.08865 + 2.62824i −1.30511 + 1.51548i 1.22393 − 2.95482i −2.73695 + 2.94869i −1.92921 + 1.81057i −2.71747 − 0.784442i −3.60116 + 3.60116i 4.51992 − 0.168328i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 195.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
32.h odd 8 1 inner
224.x even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.x.b ✓ 112
4.b odd 2 1 896.2.x.b 112
7.b odd 2 1 inner 224.2.x.b ✓ 112
28.d even 2 1 896.2.x.b 112
32.g even 8 1 896.2.x.b 112
32.h odd 8 1 inner 224.2.x.b ✓ 112
224.v odd 8 1 896.2.x.b 112
224.x even 8 1 inner 224.2.x.b ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.x.b ✓ 112 1.a even 1 1 trivial
224.2.x.b ✓ 112 7.b odd 2 1 inner
224.2.x.b ✓ 112 32.h odd 8 1 inner
224.2.x.b ✓ 112 224.x even 8 1 inner
896.2.x.b 112 4.b odd 2 1
896.2.x.b 112 28.d even 2 1
896.2.x.b 112 32.g even 8 1
896.2.x.b 112 224.v odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{112} + 4 T_{3}^{110} + 8 T_{3}^{108} - 168 T_{3}^{106} + 27036 T_{3}^{104} + 96192 T_{3}^{102} + 182592 T_{3}^{100} - 3442432 T_{3}^{98} + 258527648 T_{3}^{96} + 826311552 T_{3}^{94} + \cdots + 50\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display