Properties

Label 392.2.z.a
Level $392$
Weight $2$
Character orbit 392.z
Analytic conductor $3.130$
Analytic rank $0$
Dimension $648$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.z (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.13013575923\)
Analytic rank: \(0\)
Dimension: \(648\)
Relative dimension: \(54\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9} - 6 q^{10} - 47 q^{12} - 33 q^{14} - 8 q^{15} - 17 q^{16} - 26 q^{17} - 8 q^{18} - 22 q^{20} - 18 q^{22} - 26 q^{23} - 74 q^{24} - 72 q^{25} - 12 q^{26} + 2 q^{28} - 11 q^{30} + 60 q^{31} - 13 q^{32} - 14 q^{33} - 18 q^{34} + 8 q^{36} - 46 q^{38} - 32 q^{39} + 32 q^{40} - 20 q^{41} - 36 q^{42} + 38 q^{44} - 22 q^{46} - 58 q^{47} + 28 q^{48} - 16 q^{49} - 132 q^{50} + 18 q^{52} - 37 q^{54} - 32 q^{55} + 96 q^{56} - 66 q^{57} + 100 q^{60} + 28 q^{62} - 72 q^{63} - 28 q^{64} - 36 q^{65} - 4 q^{66} - 11 q^{68} - 36 q^{70} + 60 q^{71} - 130 q^{72} - 18 q^{73} - 12 q^{74} + 11 q^{76} - 132 q^{78} - 12 q^{79} - 64 q^{80} - 58 q^{81} + 152 q^{82} - 224 q^{84} + 55 q^{86} - 8 q^{87} - 169 q^{88} - 18 q^{89} + 144 q^{90} - 54 q^{92} + 154 q^{94} - 64 q^{95} - 142 q^{96} - 96 q^{97} + 151 q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.41305 0.0573510i 0.882385 + 0.950985i 1.99342 + 0.162080i 0.913727 2.96223i −1.19231 1.39439i 2.15945 + 1.52866i −2.80751 0.343351i 0.0984214 1.31334i −1.46103 + 4.13337i
37.2 −1.41088 0.0969984i −1.16192 1.25225i 1.98118 + 0.273707i −0.978911 + 3.17355i 1.51787 + 1.87949i −0.0759419 + 2.64466i −2.76867 0.578340i 0.00611242 0.0815645i 1.68896 4.38256i
37.3 −1.40118 + 0.191593i −2.22424 2.39716i 1.92658 0.536910i 0.266970 0.865496i 3.57583 + 2.93269i 2.06958 + 1.64828i −2.59661 + 1.12142i −0.574944 + 7.67209i −0.208249 + 1.26386i
37.4 −1.39792 + 0.214038i −0.469951 0.506487i 1.90838 0.598418i 0.273835 0.887753i 0.765363 + 0.607442i 0.528093 2.59251i −2.53968 + 1.24501i 0.188515 2.51556i −0.192788 + 1.29962i
37.5 −1.37458 0.332474i −0.288166 0.310569i 1.77892 + 0.914022i −0.486553 + 1.57737i 0.292850 + 0.522709i −2.63963 0.179924i −2.14138 1.84784i 0.210777 2.81262i 1.19324 2.00645i
37.6 −1.37066 + 0.348278i 1.48945 + 1.60524i 1.75740 0.954740i −1.25029 + 4.05335i −2.60059 1.68150i 2.55536 0.685667i −2.07629 + 1.92069i −0.134159 + 1.79023i 0.302031 5.99121i
37.7 −1.34654 0.432240i 1.94533 + 2.09656i 1.62634 + 1.16406i 0.117162 0.379831i −1.71324 3.66395i −1.64975 + 2.06841i −1.68677 2.27042i −0.387093 + 5.16539i −0.321942 + 0.460815i
37.8 −1.26332 + 0.635636i 1.08411 + 1.16839i 1.19193 1.60602i −0.529378 + 1.71620i −2.11224 0.786948i −2.34752 + 1.22032i −0.484943 + 2.78654i 0.0343439 0.458288i −0.422109 2.50460i
37.9 −1.24570 0.669503i −1.67046 1.80033i 1.10353 + 1.66800i 0.687205 2.22786i 0.875568 + 3.36105i −0.555114 2.58686i −0.257941 2.81664i −0.226554 + 3.02315i −2.34761 + 2.31516i
37.10 −1.23135 + 0.695533i 1.73400 + 1.86880i 1.03247 1.71290i 0.880547 2.85466i −3.43498 1.09511i −1.33854 2.28217i −0.0799587 + 2.82730i −0.261493 + 3.48939i 0.901247 + 4.12755i
37.11 −1.19828 0.751090i 0.581041 + 0.626213i 0.871727 + 1.80003i −0.147712 + 0.478871i −0.225904 1.18679i 2.54532 0.722043i 0.307412 2.81167i 0.169656 2.26390i 0.536675 0.462874i
37.12 −1.09732 + 0.892129i −1.73400 1.86880i 0.408212 1.95790i −0.880547 + 2.85466i 3.56996 + 0.503723i −1.33854 2.28217i 1.29876 + 2.51261i −0.261493 + 3.48939i −1.58049 3.91803i
37.13 −1.07880 0.914431i −1.29408 1.39469i 0.327631 + 1.97298i 0.498626 1.61651i 0.120712 + 2.68794i −0.699420 + 2.55163i 1.45071 2.42805i −0.0463173 + 0.618061i −2.01610 + 1.28793i
37.14 −1.05324 + 0.943763i −1.08411 1.16839i 0.218622 1.98802i 0.529378 1.71620i 2.24451 + 0.207452i −2.34752 + 1.22032i 1.64596 + 2.30018i 0.0343439 0.458288i 1.06213 + 2.30718i
37.15 −0.824960 + 1.14867i −1.48945 1.60524i −0.638881 1.89521i 1.25029 4.05335i 3.07263 0.386620i 2.55536 0.685667i 2.70402 + 0.829613i −0.134159 + 1.79023i 3.62452 + 4.78003i
37.16 −0.804026 1.16342i 0.264186 + 0.284725i −0.707086 + 1.87084i −1.09949 + 3.56447i 0.118842 0.536285i −1.95202 1.78595i 2.74508 0.681563i 0.212916 2.84117i 5.03100 1.58676i
37.17 −0.767597 1.18777i −1.83996 1.98301i −0.821590 + 1.82346i −0.819891 + 2.65802i −0.943005 + 3.70760i 2.51957 0.807328i 2.79649 0.423820i −0.322667 + 4.30570i 3.78646 1.06645i
37.18 −0.766412 1.18853i 2.16919 + 2.33784i −0.825224 + 1.82181i 0.134837 0.437132i 1.11610 4.36991i 0.938839 2.47358i 2.79775 0.415453i −0.535880 + 7.15083i −0.622888 + 0.174765i
37.19 −0.709961 + 1.22309i 0.469951 + 0.506487i −0.991911 1.73670i −0.273835 + 0.887753i −0.953127 + 0.215208i 0.528093 2.59251i 2.82836 + 0.0197876i 0.188515 2.51556i −0.891391 0.965196i
37.20 −0.690255 + 1.23432i 2.22424 + 2.39716i −1.04710 1.70399i −0.266970 + 0.865496i −4.49415 + 1.09077i 2.06958 + 1.64828i 2.82604 0.116261i −0.574944 + 7.67209i −0.884022 0.926940i
See next 80 embeddings (of 648 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 389.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
49.g even 21 1 inner
392.z even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.2.z.a 648
8.b even 2 1 inner 392.2.z.a 648
49.g even 21 1 inner 392.2.z.a 648
392.z even 42 1 inner 392.2.z.a 648
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.z.a 648 1.a even 1 1 trivial
392.2.z.a 648 8.b even 2 1 inner
392.2.z.a 648 49.g even 21 1 inner
392.2.z.a 648 392.z even 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(392, [\chi])\).