Properties

Label 552.2.bb.a
Level $552$
Weight $2$
Character orbit 552.bb
Analytic conductor $4.408$
Analytic rank $0$
Dimension $480$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 552.bb (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.40774219157\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 480q + 4q^{2} + 4q^{6} + 8q^{7} + 4q^{8} + 48q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 480q + 4q^{2} + 4q^{6} + 8q^{7} + 4q^{8} + 48q^{9} - 4q^{10} - 4q^{14} - 8q^{15} - 8q^{16} - 4q^{18} + 20q^{20} + 20q^{22} - 8q^{23} - 4q^{24} + 48q^{25} + 16q^{30} + 16q^{31} + 4q^{32} + 6q^{34} - 22q^{36} + 90q^{38} - 74q^{40} + 90q^{42} - 130q^{44} + 96q^{46} - 88q^{48} - 48q^{49} + 142q^{50} - 142q^{52} + 18q^{54} - 82q^{56} + 22q^{58} + 2q^{60} - 40q^{62} - 8q^{63} + 16q^{66} - 44q^{68} + 16q^{71} - 4q^{72} + 10q^{74} - 138q^{76} - 40q^{79} - 170q^{80} - 48q^{81} - 124q^{82} - 8q^{84} - 216q^{86} - 108q^{88} + 4q^{90} - 198q^{92} - 238q^{94} + 80q^{95} + 4q^{96} - 132q^{98} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.40928 0.118034i 0.540641 0.841254i 1.97214 + 0.332685i −3.96495 1.81073i −0.861210 + 1.12175i −1.61967 0.475578i −2.74002 0.701625i −0.415415 0.909632i 5.37400 + 3.01983i
13.2 −1.39148 0.252579i −0.540641 + 0.841254i 1.87241 + 0.702914i −2.13541 0.975208i 0.964771 1.03403i 0.595694 + 0.174912i −2.42787 1.45102i −0.415415 0.909632i 2.72505 + 1.89634i
13.3 −1.37613 0.325998i −0.540641 + 0.841254i 1.78745 + 0.897229i 0.902350 + 0.412090i 1.01824 0.981424i −3.01070 0.884022i −2.16726 1.81741i −0.415415 0.909632i −1.10741 0.861252i
13.4 −1.37160 + 0.344553i 0.540641 0.841254i 1.76257 0.945176i 2.26933 + 1.03637i −0.451686 + 1.34014i −2.49497 0.732589i −2.09187 + 1.90370i −0.415415 0.909632i −3.46970 0.639578i
13.5 −1.36486 + 0.370358i 0.540641 0.841254i 1.72567 1.01097i 0.808939 + 0.369430i −0.426333 + 1.34842i 0.759802 + 0.223098i −1.98087 + 2.01895i −0.415415 0.909632i −1.24091 0.204623i
13.6 −1.32689 0.489239i 0.540641 0.841254i 1.52129 + 1.29834i 3.08337 + 1.40813i −1.12895 + 0.851751i 2.20998 + 0.648908i −1.38339 2.46703i −0.415415 0.909632i −3.40240 3.37694i
13.7 −1.30419 0.546886i −0.540641 + 0.841254i 1.40183 + 1.42649i −0.190010 0.0867748i 1.16517 0.801487i 2.41838 + 0.710100i −1.04813 2.62706i −0.415415 0.909632i 0.200354 + 0.217085i
13.8 −1.28649 + 0.587327i −0.540641 + 0.841254i 1.31009 1.51118i 2.37560 + 1.08490i 0.201436 1.39979i 1.76843 + 0.519257i −0.797864 + 2.71356i −0.415415 0.909632i −3.69337 0.000455346i
13.9 −1.23417 0.690530i 0.540641 0.841254i 1.04634 + 1.70446i 0.282332 + 0.128937i −1.24815 + 0.664919i −2.03550 0.597678i −0.114374 2.82611i −0.415415 0.909632i −0.259410 0.354088i
13.10 −1.22386 + 0.708629i −0.540641 + 0.841254i 0.995689 1.73453i −2.18961 0.999963i 0.0655344 1.41269i 2.12232 + 0.623169i 0.0105513 + 2.82841i −0.415415 0.909632i 3.38839 0.327805i
13.11 −1.11564 + 0.869107i 0.540641 0.841254i 0.489306 1.93922i −0.756543 0.345502i 0.127979 + 1.40841i −3.66239 1.07537i 1.13950 + 2.58873i −0.415415 0.909632i 1.14431 0.272061i
13.12 −1.08248 + 0.910071i 0.540641 0.841254i 0.343541 1.97027i −2.04961 0.936025i 0.180366 + 1.40266i 3.54305 + 1.04033i 1.42121 + 2.44544i −0.415415 0.909632i 3.07052 0.852058i
13.13 −1.02595 0.973358i −0.540641 + 0.841254i 0.105149 + 1.99723i 2.63955 + 1.20544i 1.37351 0.336848i 3.10238 + 0.910940i 1.83615 2.15141i −0.415415 0.909632i −1.53472 3.80595i
13.14 −0.832989 + 1.14286i −0.540641 + 0.841254i −0.612258 1.90398i 2.21313 + 1.01070i −0.511087 1.31863i −3.93606 1.15573i 2.68599 + 0.886269i −0.415415 0.909632i −2.99860 + 1.68739i
13.15 −0.817442 1.15403i 0.540641 0.841254i −0.663576 + 1.88671i −2.63955 1.20544i −1.41278 + 0.0637600i 3.10238 + 0.910940i 2.71975 0.776488i −0.415415 0.909632i 0.766563 + 4.03151i
13.16 −0.731481 + 1.21035i 0.540641 0.841254i −0.929871 1.77069i 3.52460 + 1.60963i 0.622739 + 1.26972i 4.53885 + 1.33273i 2.82333 + 0.169761i −0.415415 0.909632i −4.52639 + 3.08857i
13.17 −0.682311 + 1.23873i −0.540641 + 0.841254i −1.06890 1.69040i −3.63898 1.66187i −0.673201 1.24370i −2.20249 0.646708i 2.82327 0.170707i −0.415415 0.909632i 4.54152 3.37381i
13.18 −0.507861 1.31988i −0.540641 + 0.841254i −1.48415 + 1.34063i −0.282332 0.128937i 1.38492 + 0.286340i −2.03550 0.597678i 2.52321 + 1.27805i −0.415415 0.909632i −0.0267952 + 0.438126i
13.19 −0.464427 + 1.33578i −0.540641 + 0.841254i −1.56862 1.24074i 0.798595 + 0.364706i −0.872641 1.11288i 2.23854 + 0.657294i 2.38587 1.51909i −0.415415 0.909632i −0.858056 + 0.897368i
13.20 −0.355714 1.36875i 0.540641 0.841254i −1.74694 + 0.973765i 0.190010 + 0.0867748i −1.34378 0.440755i 2.41838 + 0.710100i 1.95425 + 2.04473i −0.415415 0.909632i 0.0511834 0.290943i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 541.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
23.c even 11 1 inner
184.p 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.bb.a 480
8.b even 2 1 inner 552.2.bb.a 480
23.c even 11 1 inner 552.2.bb.a 480
184.p even 22 1 inner 552.2.bb.a 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.bb.a 480 1.a even 1 1 trivial
552.2.bb.a 480 8.b even 2 1 inner
552.2.bb.a 480 23.c even 11 1 inner
552.2.bb.a 480 184.p even 22 1 inner

Hecke kernels

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