Properties

Label 322.2.i.d
Level $322$
Weight $2$
Character orbit 322.i
Analytic conductor $2.571$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(4\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40q - 4q^{2} - 4q^{4} + 9q^{5} + 4q^{7} - 4q^{8} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q - 4q^{2} - 4q^{4} + 9q^{5} + 4q^{7} - 4q^{8} - 8q^{9} - 2q^{10} + 6q^{11} + 2q^{13} + 4q^{14} - 3q^{15} - 4q^{16} - 13q^{17} - 8q^{18} - 22q^{19} - 2q^{20} - 16q^{22} - 9q^{23} + 22q^{24} - 15q^{25} - 9q^{26} + 21q^{27} + 4q^{28} - 10q^{29} - 14q^{30} - 2q^{31} - 4q^{32} + 8q^{33} - 2q^{34} + 13q^{35} - 8q^{36} - 45q^{37} + 11q^{38} - 22q^{39} + 9q^{40} + 21q^{41} + 31q^{43} + 6q^{44} + 2q^{45} - 9q^{46} + 64q^{47} - 4q^{49} + 7q^{50} + 65q^{51} + 2q^{52} + 69q^{53} + 21q^{54} - 74q^{55} + 4q^{56} - 68q^{57} + 12q^{58} + 48q^{59} - 3q^{60} + 6q^{61} - 13q^{62} + 8q^{63} - 4q^{64} - 64q^{65} - 69q^{66} + 31q^{67} - 2q^{68} - 62q^{69} + 2q^{70} - 57q^{71} - 19q^{72} + 70q^{73} - 45q^{74} - 11q^{75} + 22q^{76} - 6q^{77} + 33q^{78} + 34q^{79} - 13q^{80} + 30q^{81} - 12q^{82} - 56q^{83} - 17q^{85} + 42q^{86} - 3q^{87} + 6q^{88} + 16q^{89} + 46q^{90} - 46q^{91} + 24q^{92} + 48q^{93} + 9q^{94} - 42q^{95} - 36q^{97} - 4q^{98} + 112q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 0.415415 0.909632i −2.49681 0.733130i −0.654861 0.755750i −1.90645 + 1.22520i −1.70409 + 1.96663i 0.142315 0.989821i −0.959493 + 0.281733i 3.17283 + 2.03906i 0.322515 + 2.24314i
29.2 0.415415 0.909632i −2.34488 0.688518i −0.654861 0.755750i 3.30897 2.12655i −1.60040 + 1.84696i 0.142315 0.989821i −0.959493 + 0.281733i 2.50063 + 1.60706i −0.559779 3.89335i
29.3 0.415415 0.909632i −0.195887 0.0575176i −0.654861 0.755750i −1.56656 + 1.00676i −0.133694 + 0.154291i 0.142315 0.989821i −0.959493 + 0.281733i −2.48870 1.59939i 0.265014 + 1.84322i
29.4 0.415415 0.909632i 2.84549 + 0.835512i −0.654861 0.755750i −0.221294 + 0.142217i 1.94207 2.24127i 0.142315 0.989821i −0.959493 + 0.281733i 4.87498 + 3.13296i 0.0374364 + 0.260376i
71.1 0.841254 + 0.540641i −0.473601 3.29397i 0.415415 + 0.909632i −2.68725 0.789049i 1.38243 3.02711i 0.654861 0.755750i −0.142315 + 0.989821i −7.74745 + 2.27486i −1.83407 2.11663i
71.2 0.841254 + 0.540641i −0.311523 2.16669i 0.415415 + 0.909632i 3.63410 + 1.06707i 0.909331 1.99116i 0.654861 0.755750i −0.142315 + 0.989821i −1.71902 + 0.504749i 2.48030 + 2.86241i
71.3 0.841254 + 0.540641i −0.0270781 0.188333i 0.415415 + 0.909632i −0.254438 0.0747098i 0.0790407 0.173075i 0.654861 0.755750i −0.142315 + 0.989821i 2.84374 0.834998i −0.173656 0.200410i
71.4 0.841254 + 0.540641i 0.341180 + 2.37296i 0.415415 + 0.909632i 1.96603 + 0.577277i −0.995899 + 2.18071i 0.654861 0.755750i −0.142315 + 0.989821i −2.63604 + 0.774012i 1.34183 + 1.54855i
85.1 −0.654861 + 0.755750i −0.863913 + 0.555203i −0.142315 0.989821i −0.415591 0.910017i 0.146148 1.01648i 0.959493 0.281733i 0.841254 + 0.540641i −0.808150 + 1.76960i 0.959899 + 0.281852i
85.2 −0.654861 + 0.755750i 0.124065 0.0797318i −0.142315 0.989821i 1.52064 + 3.32974i −0.0209881 + 0.145975i 0.959493 0.281733i 0.841254 + 0.540641i −1.23721 + 2.70911i −3.51226 1.03129i
85.3 −0.654861 + 0.755750i 1.39187 0.894499i −0.142315 0.989821i −1.27474 2.79128i −0.235462 + 1.63768i 0.959493 0.281733i 0.841254 + 0.540641i −0.109077 + 0.238845i 2.94429 + 0.864520i
85.4 −0.654861 + 0.755750i 2.64484 1.69974i −0.142315 0.989821i 0.940662 + 2.05976i −0.447428 + 3.11193i 0.959493 0.281733i 0.841254 + 0.540641i 2.85983 6.26216i −2.17267 0.637952i
127.1 0.841254 0.540641i −0.473601 + 3.29397i 0.415415 0.909632i −2.68725 + 0.789049i 1.38243 + 3.02711i 0.654861 + 0.755750i −0.142315 0.989821i −7.74745 2.27486i −1.83407 + 2.11663i
127.2 0.841254 0.540641i −0.311523 + 2.16669i 0.415415 0.909632i 3.63410 1.06707i 0.909331 + 1.99116i 0.654861 + 0.755750i −0.142315 0.989821i −1.71902 0.504749i 2.48030 2.86241i
127.3 0.841254 0.540641i −0.0270781 + 0.188333i 0.415415 0.909632i −0.254438 + 0.0747098i 0.0790407 + 0.173075i 0.654861 + 0.755750i −0.142315 0.989821i 2.84374 + 0.834998i −0.173656 + 0.200410i
127.4 0.841254 0.540641i 0.341180 2.37296i 0.415415 0.909632i 1.96603 0.577277i −0.995899 2.18071i 0.654861 + 0.755750i −0.142315 0.989821i −2.63604 0.774012i 1.34183 1.54855i
141.1 −0.142315 0.989821i −1.08968 + 2.38606i −0.959493 + 0.281733i 0.443664 0.512016i 2.51685 + 0.739013i −0.841254 + 0.540641i 0.415415 + 0.909632i −2.54129 2.93281i −0.569944 0.366281i
141.2 −0.142315 0.989821i −0.182677 + 0.400008i −0.959493 + 0.281733i 2.48992 2.87352i 0.421934 + 0.123891i −0.841254 + 0.540641i 0.415415 + 0.909632i 1.83795 + 2.12110i −3.19863 2.05563i
141.3 −0.142315 0.989821i 0.128569 0.281527i −0.959493 + 0.281733i −1.69136 + 1.95194i −0.296959 0.0871950i −0.841254 + 0.540641i 0.415415 + 0.909632i 1.90185 + 2.19486i 2.17278 + 1.39636i
141.4 −0.142315 0.989821i 1.27568 2.79334i −0.959493 + 0.281733i 0.685739 0.791385i −2.94645 0.865157i −0.841254 + 0.540641i 0.415415 + 0.909632i −4.21081 4.85954i −0.880921 0.566133i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.i.d 40
23.c even 11 1 inner 322.2.i.d 40
23.c even 11 1 7406.2.a.bu 20
23.d odd 22 1 7406.2.a.bv 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.i.d 40 1.a even 1 1 trivial
322.2.i.d 40 23.c even 11 1 inner
7406.2.a.bu 20 23.c even 11 1
7406.2.a.bv 20 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{40} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\).