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Properties

Label 552.2.x.a
Level $552$
Weight $2$
Character orbit 552.x
Analytic conductor $4.408$
Analytic rank $0$
Dimension $920$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [552,2,Mod(35,552)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(552, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 11, 11, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("552.35");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 552.x (of order \(22\), degree \(10\), 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: \(4.40774219157\)
Analytic rank: \(0\)
Dimension: \(920\)
Relative dimension: \(92\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 920 q - 18 q^{3} - 14 q^{4} - 16 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 920 q - 18 q^{3} - 14 q^{4} - 16 q^{6} - 18 q^{9} - 14 q^{10} - 6 q^{12} - 30 q^{16} - 16 q^{18} - 52 q^{19} - 32 q^{22} - 26 q^{24} - 112 q^{25} - 30 q^{27} - 34 q^{28} + 11 q^{30} - 30 q^{33} - 88 q^{34} - 18 q^{36} + 124 q^{40} - 3 q^{42} - 36 q^{43} - 110 q^{46} + 32 q^{49} - 30 q^{51} + 90 q^{52} - 39 q^{54} - 6 q^{57} - 68 q^{58} + 13 q^{60} + 28 q^{64} - 46 q^{66} - 100 q^{67} - 92 q^{70} + 29 q^{72} - 36 q^{73} + 14 q^{75} - 50 q^{76} - 86 q^{78} - 2 q^{81} - 12 q^{82} - 151 q^{84} - 42 q^{88} - 196 q^{90} - 136 q^{91} - 68 q^{94} - 175 q^{96} - 36 q^{97} - 54 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −1.41407 + 0.0201587i −0.595463 + 1.62648i 1.99919 − 0.0570116i 0.0333805 − 0.00980139i 0.809239 − 2.31195i 0.316707 − 0.274428i −2.82584 + 0.120919i −2.29085 − 1.93701i −0.0470047 + 0.0145328i
35.2 −1.41396 − 0.0268897i 0.618438 − 1.61788i 1.99855 + 0.0760418i 1.48175 − 0.435081i −0.917950 + 2.27098i 3.65626 − 3.16816i −2.82383 − 0.161260i −2.23507 − 2.00112i −2.10683 + 0.575343i
35.3 −1.41384 − 0.0325375i −0.335489 − 1.69925i 1.99788 + 0.0920055i −3.41280 + 1.00209i 0.419038 + 2.41338i −1.52640 + 1.32263i −2.82169 − 0.195087i −2.77489 + 1.14016i 4.85775 − 1.30575i
35.4 −1.41269 − 0.0656090i 1.50835 − 0.851390i 1.99139 + 0.185370i 2.73552 − 0.803222i −2.18670 + 1.10379i −3.00502 + 2.60386i −2.80106 − 0.392524i 1.55027 − 2.56840i −3.91714 + 0.955229i
35.5 −1.40292 + 0.178405i 1.61394 + 0.628649i 1.93634 − 0.500576i −2.04785 + 0.601303i −2.37637 − 0.594005i −1.23389 + 1.06917i −2.62722 + 1.04772i 2.20960 + 2.02920i 2.76568 − 1.20892i
35.6 −1.39813 + 0.212705i −1.49325 + 0.877613i 1.90951 − 0.594777i 0.229886 − 0.0675006i 1.90108 − 1.54464i 1.84453 − 1.59829i −2.54323 + 1.23774i 1.45959 − 2.62099i −0.307052 + 0.143272i
35.7 −1.37018 − 0.350170i −0.141521 + 1.72626i 1.75476 + 0.959590i 2.19665 − 0.644996i 0.798394 − 2.31572i −2.54738 + 2.20732i −2.06831 − 1.92927i −2.95994 − 0.488605i −3.23566 + 0.114554i
35.8 −1.33616 − 0.463327i 0.833138 + 1.51851i 1.57066 + 1.23816i −1.70423 + 0.500408i −0.409640 − 2.41499i 2.57338 − 2.22984i −1.52498 − 2.38211i −1.61176 + 2.53026i 2.50898 + 0.120991i
35.9 −1.33559 + 0.464960i 1.06576 + 1.36534i 1.56763 − 1.24199i 3.82579 − 1.12335i −2.05825 − 1.32801i 2.32451 − 2.01420i −1.51623 + 2.38768i −0.728327 + 2.91025i −4.58739 + 3.27918i
35.10 −1.32803 + 0.486139i −1.68129 − 0.416231i 1.52734 − 1.29122i −0.607897 + 0.178495i 2.43516 − 0.264575i −1.35812 + 1.17682i −1.40064 + 2.45728i 2.65350 + 1.39962i 0.720533 − 0.532569i
35.11 −1.29176 + 0.575639i −1.36388 − 1.06763i 1.33728 − 1.48717i 2.50603 − 0.735837i 2.37637 + 0.594013i −0.0911016 + 0.0789400i −0.871368 + 2.69086i 0.720344 + 2.91223i −2.81361 + 2.39309i
35.12 −1.28905 − 0.581685i −0.724881 − 1.57307i 1.32328 + 1.49964i 0.385158 − 0.113093i 0.0193752 + 2.44941i −0.150483 + 0.130394i −0.833459 − 2.70284i −1.94909 + 2.28058i −0.562271 − 0.0782590i
35.13 −1.28897 + 0.581846i 0.880176 − 1.49174i 1.32291 − 1.49997i −0.611012 + 0.179409i −0.266561 + 2.43494i −1.41902 + 1.22959i −0.832444 + 2.70315i −1.45058 − 2.62599i 0.683190 − 0.586769i
35.14 −1.28558 − 0.589316i −1.72832 − 0.113594i 1.30541 + 1.51522i −0.988646 + 0.290293i 2.15495 + 1.16456i −3.80564 + 3.29760i −0.785267 − 2.71723i 2.97419 + 0.392655i 1.44205 + 0.209431i
35.15 −1.27844 − 0.604647i 1.53786 − 0.796851i 1.26880 + 1.54601i −2.80420 + 0.823387i −2.44788 + 0.0888594i 0.808303 − 0.700399i −0.687299 − 2.74365i 1.73006 − 2.45090i 4.08285 + 0.642901i
35.16 −1.27084 − 0.620447i −1.72827 + 0.114318i 1.23009 + 1.57698i 4.02671 − 1.18235i 2.26730 + 0.927022i 1.49077 − 1.29176i −0.584820 − 2.76731i 2.97386 − 0.395146i −5.85090 − 0.995778i
35.17 −1.26653 − 0.629211i 1.61370 + 0.629273i 1.20819 + 1.59383i 1.81679 − 0.533458i −1.64785 − 1.81235i 0.219203 − 0.189940i −0.527349 − 2.77883i 2.20803 + 2.03091i −2.63667 − 0.467505i
35.18 −1.20136 + 0.746150i −0.910347 + 1.47352i 0.886521 − 1.79279i −3.47534 + 1.02045i −0.00581645 − 2.44948i −3.08702 + 2.67492i 0.272658 + 2.81525i −1.34254 − 2.68283i 3.41372 − 3.81905i
35.19 −1.17778 + 0.782832i 0.458332 + 1.67031i 0.774349 − 1.84401i −3.47534 + 1.02045i −1.84739 − 1.60847i 3.08702 − 2.67492i 0.531536 + 2.77803i −2.57986 + 1.53111i 3.29436 − 3.92248i
35.20 −1.06473 + 0.930785i −0.424251 − 1.67929i 0.267280 − 1.98206i −0.611012 + 0.179409i 2.01477 + 1.39309i 1.41902 − 1.22959i 1.56029 + 2.35913i −2.64002 + 1.42488i 0.483568 − 0.759742i
See next 80 embeddings (of 920 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.92
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
23.c even 11 1 inner
24.f even 2 1 inner
69.h odd 22 1 inner
184.k odd 22 1 inner
552.x even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.x.a ✓ 920
3.b odd 2 1 inner 552.2.x.a ✓ 920
8.d odd 2 1 inner 552.2.x.a ✓ 920
23.c even 11 1 inner 552.2.x.a ✓ 920
24.f even 2 1 inner 552.2.x.a ✓ 920
69.h odd 22 1 inner 552.2.x.a ✓ 920
184.k odd 22 1 inner 552.2.x.a ✓ 920
552.x even 22 1 inner 552.2.x.a ✓ 920
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.x.a ✓ 920 1.a even 1 1 trivial
552.2.x.a ✓ 920 3.b odd 2 1 inner
552.2.x.a ✓ 920 8.d odd 2 1 inner
552.2.x.a ✓ 920 23.c even 11 1 inner
552.2.x.a ✓ 920 24.f even 2 1 inner
552.2.x.a ✓ 920 69.h odd 22 1 inner
552.2.x.a ✓ 920 184.k odd 22 1 inner
552.2.x.a ✓ 920 552.x even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(552, [\chi])\).