Properties

Label 322.2.k.a
Level $322$
Weight $2$
Character orbit 322.k
Analytic conductor $2.571$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.k (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 8 q^{2} - 8 q^{4} - 11 q^{7} - 8 q^{8} - 6 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 8 q^{2} - 8 q^{4} - 11 q^{7} - 8 q^{8} - 6 q^{9} + 11 q^{14} - 8 q^{16} - 6 q^{18} - 33 q^{21} + 18 q^{23} - 58 q^{25} - 11 q^{28} + 16 q^{29} + 22 q^{30} - 8 q^{32} + 47 q^{35} + 16 q^{36} - 22 q^{37} - 30 q^{39} - 22 q^{42} + 44 q^{43} - 4 q^{46} + 17 q^{49} + 8 q^{50} - 88 q^{51} + 44 q^{53} + 11 q^{56} - 22 q^{57} - 28 q^{58} + 121 q^{63} - 8 q^{64} - 154 q^{65} + 58 q^{70} + 42 q^{71} - 28 q^{72} + 22 q^{74} + 38 q^{77} + 58 q^{78} - 88 q^{79} - 26 q^{81} - 11 q^{84} + 26 q^{85} - 132 q^{86} + 22 q^{88} - 4 q^{92} + 152 q^{93} - 68 q^{95} - 27 q^{98} - 44 q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1 −0.959493 + 0.281733i −1.73925 + 1.50707i 0.841254 0.540641i 0.406529 + 2.82747i 1.24421 1.93603i 0.697541 + 2.55214i −0.654861 + 0.755750i 0.326790 2.27288i −1.18665 2.59840i
83.2 −0.959493 + 0.281733i −1.70070 + 1.47367i 0.841254 0.540641i 0.0506866 + 0.352533i 1.21663 1.89312i −2.46109 0.971103i −0.654861 + 0.755750i 0.293752 2.04309i −0.147953 0.323973i
83.3 −0.959493 + 0.281733i −0.749737 + 0.649651i 0.841254 0.540641i −0.540672 3.76045i 0.536339 0.834560i −2.61856 0.378356i −0.654861 + 0.755750i −0.286885 + 1.99533i 1.57821 + 3.45580i
83.4 −0.959493 + 0.281733i −0.198408 + 0.171921i 0.841254 0.540641i −0.382697 2.66172i 0.141935 0.220855i 0.921868 + 2.47995i −0.654861 + 0.755750i −0.417136 + 2.90124i 1.11709 + 2.44608i
83.5 −0.959493 + 0.281733i 0.198408 0.171921i 0.841254 0.540641i 0.382697 + 2.66172i −0.141935 + 0.220855i 2.47792 0.927322i −0.654861 + 0.755750i −0.417136 + 2.90124i −1.11709 2.44608i
83.6 −0.959493 + 0.281733i 0.749737 0.649651i 0.841254 0.540641i 0.540672 + 3.76045i −0.536339 + 0.834560i −2.00073 1.73120i −0.654861 + 0.755750i −0.286885 + 1.99533i −1.57821 3.45580i
83.7 −0.959493 + 0.281733i 1.70070 1.47367i 0.841254 0.540641i −0.0506866 0.352533i −1.21663 + 1.89312i −2.34558 1.22403i −0.654861 + 0.755750i 0.293752 2.04309i 0.147953 + 0.323973i
83.8 −0.959493 + 0.281733i 1.73925 1.50707i 0.841254 0.540641i −0.406529 2.82747i −1.24421 + 1.93603i 2.38557 1.14413i −0.654861 + 0.755750i 0.326790 2.27288i 1.18665 + 2.59840i
97.1 −0.959493 0.281733i −1.73925 1.50707i 0.841254 + 0.540641i 0.406529 2.82747i 1.24421 + 1.93603i 0.697541 2.55214i −0.654861 0.755750i 0.326790 + 2.27288i −1.18665 + 2.59840i
97.2 −0.959493 0.281733i −1.70070 1.47367i 0.841254 + 0.540641i 0.0506866 0.352533i 1.21663 + 1.89312i −2.46109 + 0.971103i −0.654861 0.755750i 0.293752 + 2.04309i −0.147953 + 0.323973i
97.3 −0.959493 0.281733i −0.749737 0.649651i 0.841254 + 0.540641i −0.540672 + 3.76045i 0.536339 + 0.834560i −2.61856 + 0.378356i −0.654861 0.755750i −0.286885 1.99533i 1.57821 3.45580i
97.4 −0.959493 0.281733i −0.198408 0.171921i 0.841254 + 0.540641i −0.382697 + 2.66172i 0.141935 + 0.220855i 0.921868 2.47995i −0.654861 0.755750i −0.417136 2.90124i 1.11709 2.44608i
97.5 −0.959493 0.281733i 0.198408 + 0.171921i 0.841254 + 0.540641i 0.382697 2.66172i −0.141935 0.220855i 2.47792 + 0.927322i −0.654861 0.755750i −0.417136 2.90124i −1.11709 + 2.44608i
97.6 −0.959493 0.281733i 0.749737 + 0.649651i 0.841254 + 0.540641i 0.540672 3.76045i −0.536339 0.834560i −2.00073 + 1.73120i −0.654861 0.755750i −0.286885 1.99533i −1.57821 + 3.45580i
97.7 −0.959493 0.281733i 1.70070 + 1.47367i 0.841254 + 0.540641i −0.0506866 + 0.352533i −1.21663 1.89312i −2.34558 + 1.22403i −0.654861 0.755750i 0.293752 + 2.04309i 0.147953 0.323973i
97.8 −0.959493 0.281733i 1.73925 + 1.50707i 0.841254 + 0.540641i −0.406529 + 2.82747i −1.24421 1.93603i 2.38557 + 1.14413i −0.654861 0.755750i 0.326790 + 2.27288i 1.18665 2.59840i
111.1 0.415415 + 0.909632i −0.760851 2.59122i −0.654861 + 0.755750i 2.71881 + 1.74727i 2.04099 1.76853i 0.0953475 2.64403i −0.959493 0.281733i −3.61176 + 2.32114i −0.459941 + 3.19896i
111.2 0.415415 + 0.909632i −0.730873 2.48912i −0.654861 + 0.755750i −1.52995 0.983241i 1.96057 1.69884i −2.20702 + 1.45913i −0.959493 0.281733i −3.13780 + 2.01654i 0.258822 1.80015i
111.3 0.415415 + 0.909632i −0.229469 0.781499i −0.654861 + 0.755750i −1.74460 1.12119i 0.615551 0.533378i −0.251104 2.63381i −0.959493 0.281733i 1.96568 1.26326i 0.295134 2.05270i
111.4 0.415415 + 0.909632i −0.192661 0.656142i −0.654861 + 0.755750i −2.27924 1.46478i 0.516813 0.447821i 2.42953 + 1.04756i −0.959493 0.281733i 2.13036 1.36910i 0.385580 2.68177i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.d odd 22 1 inner
161.k even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.k.a 80
7.b odd 2 1 inner 322.2.k.a 80
23.d odd 22 1 inner 322.2.k.a 80
161.k even 22 1 inner 322.2.k.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.k.a 80 1.a even 1 1 trivial
322.2.k.a 80 7.b odd 2 1 inner
322.2.k.a 80 23.d odd 22 1 inner
322.2.k.a 80 161.k even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(53\!\cdots\!17\)\( T_{3}^{52} - \)\(33\!\cdots\!34\)\( T_{3}^{50} + \)\(19\!\cdots\!43\)\( T_{3}^{48} - \)\(10\!\cdots\!69\)\( T_{3}^{46} + \)\(53\!\cdots\!77\)\( T_{3}^{44} - \)\(23\!\cdots\!67\)\( T_{3}^{42} + \)\(88\!\cdots\!96\)\( T_{3}^{40} - \)\(29\!\cdots\!61\)\( T_{3}^{38} + \)\(86\!\cdots\!17\)\( T_{3}^{36} - \)\(21\!\cdots\!06\)\( T_{3}^{34} + \)\(37\!\cdots\!68\)\( T_{3}^{32} - \)\(32\!\cdots\!42\)\( T_{3}^{30} + \)\(56\!\cdots\!19\)\( T_{3}^{28} - \)\(92\!\cdots\!98\)\( T_{3}^{26} + \)\(91\!\cdots\!39\)\( T_{3}^{24} + \)\(13\!\cdots\!34\)\( T_{3}^{22} + \)\(32\!\cdots\!75\)\( T_{3}^{20} + \)\(23\!\cdots\!95\)\( T_{3}^{18} + \)\(11\!\cdots\!88\)\( T_{3}^{16} - \)\(69\!\cdots\!99\)\( T_{3}^{14} - \)\(92\!\cdots\!55\)\( T_{3}^{12} - \)\(88\!\cdots\!04\)\( T_{3}^{10} + \)\(12\!\cdots\!69\)\( T_{3}^{8} - \)\(30\!\cdots\!93\)\( T_{3}^{6} + \)\(31\!\cdots\!26\)\( T_{3}^{4} - \)\(16\!\cdots\!32\)\( T_{3}^{2} + 956720690641 \)">\(T_{3}^{80} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\).