Properties

Label 418.2.j.c
Level $418$
Weight $2$
Character orbit 418.j
Analytic conductor $3.338$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.j (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 12 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 12 q^{7} + 15 q^{8} - 15 q^{11} + 3 q^{12} - 3 q^{13} + 9 q^{14} + 27 q^{15} - 36 q^{18} - 9 q^{19} + 18 q^{20} - 27 q^{21} - 3 q^{23} - 12 q^{25} + 3 q^{27} + 9 q^{28} + 3 q^{29} + 9 q^{30} + 30 q^{31} - 9 q^{34} + 15 q^{35} + 18 q^{37} + 6 q^{38} - 6 q^{41} - 45 q^{42} + 39 q^{43} - 18 q^{45} + 21 q^{46} + 45 q^{47} - 33 q^{49} + 36 q^{50} - 36 q^{51} + 6 q^{52} - 24 q^{53} + 45 q^{54} - 24 q^{56} - 24 q^{57} - 30 q^{58} + 3 q^{59} - 9 q^{60} - 27 q^{61} + 15 q^{62} - 93 q^{63} - 15 q^{64} + 18 q^{65} - 9 q^{67} - 21 q^{68} + 48 q^{69} - 15 q^{70} + 39 q^{73} + 3 q^{74} - 42 q^{75} - 15 q^{76} + 24 q^{77} + 6 q^{78} + 21 q^{79} + 84 q^{81} + 6 q^{82} - 36 q^{83} - 27 q^{84} + 63 q^{85} + 6 q^{86} - 21 q^{87} + 15 q^{88} + 54 q^{89} + 12 q^{90} + 3 q^{91} - 3 q^{92} + 51 q^{93} - 78 q^{94} + 6 q^{95} + 6 q^{96} - 18 q^{97} + 3 q^{98}+O(q^{100}) \) 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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 0.939693 + 0.342020i −0.506748 2.87391i 0.766044 + 0.642788i −1.93343 + 1.62234i 0.506748 2.87391i 2.38935 4.13848i 0.500000 + 0.866025i −5.18349 + 1.88664i −2.37170 + 0.863229i
23.2 0.939693 + 0.342020i −0.315286 1.78808i 0.766044 + 0.642788i 2.76685 2.32166i 0.315286 1.78808i −0.752408 + 1.30321i 0.500000 + 0.866025i −0.278741 + 0.101453i 3.39404 1.23533i
23.3 0.939693 + 0.342020i 0.0304860 + 0.172895i 0.766044 + 0.642788i −0.190742 + 0.160051i −0.0304860 + 0.172895i 1.48347 2.56944i 0.500000 + 0.866025i 2.79011 1.01552i −0.233979 + 0.0851616i
23.4 0.939693 + 0.342020i 0.102554 + 0.581612i 0.766044 + 0.642788i −1.23439 + 1.03577i −0.102554 + 0.581612i −1.96792 + 3.40854i 0.500000 + 0.866025i 2.49132 0.906767i −1.51420 + 0.551123i
23.5 0.939693 + 0.342020i 0.515346 + 2.92267i 0.766044 + 0.642788i 2.88984 2.42486i −0.515346 + 2.92267i −1.44675 + 2.50585i 0.500000 + 0.866025i −5.45736 + 1.98632i 3.54491 1.29024i
111.1 −0.766044 0.642788i −3.00494 + 1.09371i 0.173648 + 0.984808i −0.0619925 + 0.351577i 3.00494 + 1.09371i −2.21848 3.84252i 0.500000 0.866025i 5.53533 4.64469i 0.273478 0.229475i
111.2 −0.766044 0.642788i −0.305857 + 0.111323i 0.173648 + 0.984808i 0.749682 4.25166i 0.305857 + 0.111323i −1.19826 2.07545i 0.500000 0.866025i −2.21698 + 1.86027i −3.30720 + 2.77507i
111.3 −0.766044 0.642788i 0.0916892 0.0333721i 0.173648 + 0.984808i 0.327930 1.85978i −0.0916892 0.0333721i 1.90575 + 3.30086i 0.500000 0.866025i −2.29084 + 1.92224i −1.44665 + 1.21389i
111.4 −0.766044 0.642788i 1.46132 0.531876i 0.173648 + 0.984808i −0.712779 + 4.04237i −1.46132 0.531876i −0.291586 0.505042i 0.500000 0.866025i −0.445574 + 0.373881i 3.14441 2.63847i
111.5 −0.766044 0.642788i 2.69748 0.981803i 0.173648 + 0.984808i 0.218104 1.23693i −2.69748 0.981803i −0.789822 1.36801i 0.500000 0.866025i 4.01433 3.36842i −0.962162 + 0.807349i
177.1 −0.766044 + 0.642788i −3.00494 1.09371i 0.173648 0.984808i −0.0619925 0.351577i 3.00494 1.09371i −2.21848 + 3.84252i 0.500000 + 0.866025i 5.53533 + 4.64469i 0.273478 + 0.229475i
177.2 −0.766044 + 0.642788i −0.305857 0.111323i 0.173648 0.984808i 0.749682 + 4.25166i 0.305857 0.111323i −1.19826 + 2.07545i 0.500000 + 0.866025i −2.21698 1.86027i −3.30720 2.77507i
177.3 −0.766044 + 0.642788i 0.0916892 + 0.0333721i 0.173648 0.984808i 0.327930 + 1.85978i −0.0916892 + 0.0333721i 1.90575 3.30086i 0.500000 + 0.866025i −2.29084 1.92224i −1.44665 1.21389i
177.4 −0.766044 + 0.642788i 1.46132 + 0.531876i 0.173648 0.984808i −0.712779 4.04237i −1.46132 + 0.531876i −0.291586 + 0.505042i 0.500000 + 0.866025i −0.445574 0.373881i 3.14441 + 2.63847i
177.5 −0.766044 + 0.642788i 2.69748 + 0.981803i 0.173648 0.984808i 0.218104 + 1.23693i −2.69748 + 0.981803i −0.789822 + 1.36801i 0.500000 + 0.866025i 4.01433 + 3.36842i −0.962162 0.807349i
199.1 −0.173648 0.984808i −2.04235 + 1.71374i −0.939693 + 0.342020i −3.54102 1.28883i 2.04235 + 1.71374i 0.168894 0.292534i 0.500000 + 0.866025i 0.713363 4.04568i −0.654355 + 3.71103i
199.2 −0.173648 0.984808i −1.95711 + 1.64221i −0.939693 + 0.342020i 0.968941 + 0.352666i 1.95711 + 1.64221i −1.38692 + 2.40222i 0.500000 + 0.866025i 0.612477 3.47353i 0.179053 1.01546i
199.3 −0.173648 0.984808i 0.345094 0.289568i −0.939693 + 0.342020i 1.87337 + 0.681850i −0.345094 0.289568i −1.54380 + 2.67395i 0.500000 + 0.866025i −0.485704 + 2.75457i 0.346185 1.96331i
199.4 −0.173648 0.984808i 1.31855 1.10640i −0.939693 + 0.342020i −3.37653 1.22895i −1.31855 1.10640i −1.75103 + 3.03286i 0.500000 + 0.866025i −0.00647669 + 0.0367312i −0.623957 + 3.53863i
199.5 −0.173648 0.984808i 1.56977 1.31719i −0.939693 + 0.342020i 1.25616 + 0.457206i −1.56977 1.31719i 1.39952 2.42403i 0.500000 + 0.866025i 0.208231 1.18093i 0.232130 1.31647i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 397.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.j.c 30
19.e even 9 1 inner 418.2.j.c 30
19.e even 9 1 7942.2.a.bz 15
19.f odd 18 1 7942.2.a.cb 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.j.c 30 1.a even 1 1 trivial
418.2.j.c 30 19.e even 9 1 inner
7942.2.a.bz 15 19.e even 9 1
7942.2.a.cb 15 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} - 7 T_{3}^{27} - 21 T_{3}^{26} - 81 T_{3}^{25} + 644 T_{3}^{24} - 129 T_{3}^{23} + 3042 T_{3}^{22} - 5648 T_{3}^{21} + 11619 T_{3}^{20} - 91479 T_{3}^{19} + 399310 T_{3}^{18} - 568311 T_{3}^{17} + 927954 T_{3}^{16} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\). Copy content Toggle raw display