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Properties

Label 345.2.m.c
Level $345$
Weight $2$
Character orbit 345.m
Analytic conductor $2.755$
Analytic rank $0$
Dimension $50$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 345.m (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.75483886973\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 5 q^{3} - 4 q^{4} + 5 q^{5} - 11 q^{6} - 3 q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q - 5 q^{3} - 4 q^{4} + 5 q^{5} - 11 q^{6} - 3 q^{7} - 5 q^{9} + 15 q^{11} - 4 q^{12} - 19 q^{13} + 55 q^{14} + 5 q^{15} + 12 q^{16} + 5 q^{17} - 11 q^{19} + 4 q^{20} + 8 q^{21} - 18 q^{22} + 14 q^{23} + 66 q^{24} - 5 q^{25} - 18 q^{26} - 5 q^{27} + 10 q^{28} - 22 q^{29} + 6 q^{31} + 33 q^{32} + 4 q^{33} + 18 q^{34} - 8 q^{35} - 15 q^{36} + 25 q^{37} - 97 q^{38} - 19 q^{39} + 22 q^{40} - 42 q^{41} - 11 q^{42} - 25 q^{43} + 25 q^{44} - 50 q^{45} - 44 q^{46} + 86 q^{47} - 10 q^{48} - 8 q^{49} - 11 q^{50} - 17 q^{51} - 67 q^{52} - 26 q^{53} - 4 q^{55} - 132 q^{56} + 22 q^{57} + 8 q^{58} - 76 q^{59} + 4 q^{60} + 13 q^{61} - 8 q^{62} + 8 q^{63} + 76 q^{64} + 8 q^{65} + 4 q^{66} + 84 q^{67} + 66 q^{68} + 25 q^{69} + 22 q^{70} + 55 q^{71} - 59 q^{73} + 17 q^{74} - 5 q^{75} + 82 q^{76} - 56 q^{77} - 7 q^{78} + 7 q^{79} + 10 q^{80} - 5 q^{81} - 150 q^{82} + 19 q^{83} + 10 q^{84} + 6 q^{85} + 44 q^{86} - 11 q^{87} - 62 q^{88} - 74 q^{89} - 56 q^{91} + 41 q^{92} + 28 q^{93} - 161 q^{94} + 11 q^{95} - 44 q^{96} - 68 q^{97} - 198 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.71238 + 1.97619i 0.841254 − 0.540641i −0.688461 − 4.78835i −0.415415 − 0.909632i −0.372136 + 2.58826i −3.22191 + 0.946039i 6.24206 + 4.01153i 0.415415 − 0.909632i 2.50896 + 0.736696i
16.2 −1.32781 + 1.53237i 0.841254 − 0.540641i −0.300461 − 2.08975i −0.415415 − 0.909632i −0.288560 + 2.00698i 3.59545 − 1.05572i 0.189750 + 0.121945i 0.415415 − 0.909632i 1.94549 + 0.571247i
16.3 −0.615779 + 0.710647i 0.841254 − 0.540641i 0.158794 + 1.10444i −0.415415 − 0.909632i −0.133822 + 0.930750i 0.864280 − 0.253775i −2.46475 − 1.58400i 0.415415 − 0.909632i 0.902231 + 0.264919i
16.4 0.707484 − 0.816480i 0.841254 − 0.540641i 0.118524 + 0.824351i −0.415415 − 0.909632i 0.153751 − 1.06936i 2.39967 − 0.704607i 2.57463 + 1.65461i 0.415415 − 0.909632i −1.03660 − 0.304372i
16.5 1.45237 − 1.67613i 0.841254 − 0.540641i −0.415386 − 2.88907i −0.415415 − 0.909632i 0.315630 − 2.19526i −1.14822 + 0.337148i −1.71422 − 1.10166i 0.415415 − 0.909632i −2.12799 − 0.624835i
31.1 −0.387876 + 2.69773i 0.415415 + 0.909632i −5.20834 − 1.52931i 0.654861 + 0.755750i −2.61507 + 0.767855i −4.11911 − 2.64719i 3.88144 − 8.49917i −0.654861 + 0.755750i −2.29282 + 1.47350i
31.2 −0.290073 + 2.01750i 0.415415 + 0.909632i −2.06718 − 0.606979i 0.654861 + 0.755750i −1.95568 + 0.574241i 3.11544 + 2.00217i 0.130777 − 0.286362i −0.654861 + 0.755750i −1.71468 + 1.10196i
31.3 −0.124133 + 0.863362i 0.415415 + 0.909632i 1.18900 + 0.349122i 0.654861 + 0.755750i −0.836908 + 0.245738i −2.78196 − 1.78785i −1.17370 + 2.57004i −0.654861 + 0.755750i −0.733775 + 0.471568i
31.4 0.0510170 − 0.354831i 0.415415 + 0.909632i 1.79568 + 0.527260i 0.654861 + 0.755750i 0.343959 − 0.100995i 2.04891 + 1.31675i 0.576535 − 1.26244i −0.654861 + 0.755750i 0.301573 − 0.193809i
31.5 0.193334 − 1.34467i 0.415415 + 0.909632i 0.148229 + 0.0435239i 0.654861 + 0.755750i 1.30347 − 0.382733i −0.958335 − 0.615885i 1.21586 − 2.66237i −0.654861 + 0.755750i 1.14284 − 0.734459i
121.1 −0.831671 + 1.82111i −0.959493 − 0.281733i −1.31503 − 1.51762i −0.841254 + 0.540641i 1.31105 − 1.51303i 0.450906 − 3.13612i 0.0155624 − 0.00456953i 0.841254 + 0.540641i −0.284918 − 1.98165i
121.2 −0.168180 + 0.368263i −0.959493 − 0.281733i 1.20239 + 1.38763i −0.841254 + 0.540641i 0.265120 − 0.305964i 0.179033 − 1.24520i −1.49013 + 0.437542i 0.841254 + 0.540641i −0.0576160 − 0.400728i
121.3 0.582364 − 1.27520i −0.959493 − 0.281733i 0.0227365 + 0.0262393i −0.841254 + 0.540641i −0.918039 + 1.05947i −0.746335 + 5.19088i 2.73690 − 0.803626i 0.841254 + 0.540641i 0.199509 + 1.38762i
121.4 0.627141 − 1.37325i −0.959493 − 0.281733i −0.182780 − 0.210939i −0.841254 + 0.540641i −0.988626 + 1.14093i 0.326039 − 2.26765i 2.49274 − 0.731935i 0.841254 + 0.540641i 0.214849 + 1.49431i
121.5 1.16525 − 2.55155i −0.959493 − 0.281733i −3.84287 − 4.43491i −0.841254 + 0.540641i −1.83691 + 2.11991i 0.000825631 − 0.00574239i −10.4110 + 3.05695i 0.841254 + 0.540641i 0.399198 + 2.77649i
151.1 −1.71238 − 1.97619i 0.841254 + 0.540641i −0.688461 + 4.78835i −0.415415 + 0.909632i −0.372136 − 2.58826i −3.22191 − 0.946039i 6.24206 − 4.01153i 0.415415 + 0.909632i 2.50896 − 0.736696i
151.2 −1.32781 − 1.53237i 0.841254 + 0.540641i −0.300461 + 2.08975i −0.415415 + 0.909632i −0.288560 − 2.00698i 3.59545 + 1.05572i 0.189750 − 0.121945i 0.415415 + 0.909632i 1.94549 − 0.571247i
151.3 −0.615779 − 0.710647i 0.841254 + 0.540641i 0.158794 − 1.10444i −0.415415 + 0.909632i −0.133822 − 0.930750i 0.864280 + 0.253775i −2.46475 + 1.58400i 0.415415 + 0.909632i 0.902231 − 0.264919i
151.4 0.707484 + 0.816480i 0.841254 + 0.540641i 0.118524 − 0.824351i −0.415415 + 0.909632i 0.153751 + 1.06936i 2.39967 + 0.704607i 2.57463 − 1.65461i 0.415415 + 0.909632i −1.03660 + 0.304372i
151.5 1.45237 + 1.67613i 0.841254 + 0.540641i −0.415386 + 2.88907i −0.415415 + 0.909632i 0.315630 + 2.19526i −1.14822 − 0.337148i −1.71422 + 1.10166i 0.415415 + 0.909632i −2.12799 + 0.624835i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.5
Significant digits:
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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 345.2.m.c ✓ 50
23.c even 11 1 inner 345.2.m.c ✓ 50
23.c even 11 1 7935.2.a.bv 25
23.d odd 22 1 7935.2.a.bw 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.m.c ✓ 50 1.a even 1 1 trivial
345.2.m.c ✓ 50 23.c even 11 1 inner
7935.2.a.bv 25 23.c even 11 1
7935.2.a.bw 25 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{50} + 7 T_{2}^{48} + 54 T_{2}^{46} + 11 T_{2}^{45} + 325 T_{2}^{44} + 77 T_{2}^{43} + 2314 T_{2}^{42} + 3201 T_{2}^{41} + 14050 T_{2}^{40} - 20548 T_{2}^{39} + 86116 T_{2}^{38} - 25168 T_{2}^{37} + 686053 T_{2}^{36} + \cdots + 64009 \) acting on \(S_{2}^{\mathrm{new}}(345, [\chi])\). Copy content Toggle raw display