# Properties

 Label 240.3.bn.a Level $240$ Weight $3$ Character orbit 240.bn Analytic conductor $6.540$ Analytic rank $0$ Dimension $64$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 240.bn (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53952634465$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$32$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$64 q - 24 q^{4}+O(q^{10})$$ 64 * q - 24 * q^4 $$\operatorname{Tr}(f)(q) =$$ $$64 q - 24 q^{4} + 20 q^{10} - 64 q^{11} + 72 q^{14} - 36 q^{16} - 24 q^{18} + 32 q^{19} - 80 q^{20} + 48 q^{22} + 256 q^{23} - 36 q^{24} + 240 q^{28} - 64 q^{29} - 40 q^{32} - 76 q^{34} - 12 q^{36} + 192 q^{37} - 280 q^{38} - 192 q^{43} - 280 q^{44} - 300 q^{46} + 448 q^{49} - 40 q^{50} + 96 q^{51} + 104 q^{52} + 320 q^{53} + 36 q^{54} + 112 q^{56} + 64 q^{58} + 128 q^{59} + 32 q^{61} + 48 q^{62} + 48 q^{64} - 72 q^{66} - 64 q^{67} + 280 q^{68} - 96 q^{69} + 240 q^{70} - 512 q^{71} - 120 q^{72} - 608 q^{74} - 308 q^{76} - 448 q^{77} - 360 q^{78} - 576 q^{81} - 200 q^{82} - 144 q^{84} - 160 q^{85} - 560 q^{86} - 184 q^{88} + 576 q^{91} - 56 q^{92} + 460 q^{94} + 360 q^{96} + 368 q^{98} - 192 q^{99}+O(q^{100})$$ 64 * q - 24 * q^4 + 20 * q^10 - 64 * q^11 + 72 * q^14 - 36 * q^16 - 24 * q^18 + 32 * q^19 - 80 * q^20 + 48 * q^22 + 256 * q^23 - 36 * q^24 + 240 * q^28 - 64 * q^29 - 40 * q^32 - 76 * q^34 - 12 * q^36 + 192 * q^37 - 280 * q^38 - 192 * q^43 - 280 * q^44 - 300 * q^46 + 448 * q^49 - 40 * q^50 + 96 * q^51 + 104 * q^52 + 320 * q^53 + 36 * q^54 + 112 * q^56 + 64 * q^58 + 128 * q^59 + 32 * q^61 + 48 * q^62 + 48 * q^64 - 72 * q^66 - 64 * q^67 + 280 * q^68 - 96 * q^69 + 240 * q^70 - 512 * q^71 - 120 * q^72 - 608 * q^74 - 308 * q^76 - 448 * q^77 - 360 * q^78 - 576 * q^81 - 200 * q^82 - 144 * q^84 - 160 * q^85 - 560 * q^86 - 184 * q^88 + 576 * q^91 - 56 * q^92 + 460 * q^94 + 360 * q^96 + 368 * q^98 - 192 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1 −1.99836 0.0809611i 1.22474 + 1.22474i 3.98689 + 0.323579i −1.58114 1.58114i −2.34833 2.54664i −2.50432 −7.94105 0.969411i 3.00000i 3.03167 + 3.28770i
91.2 −1.99652 0.117912i −1.22474 1.22474i 3.97219 + 0.470829i −1.58114 1.58114i 2.30082 + 2.58964i 12.6948 −7.87505 1.40839i 3.00000i 2.97034 + 3.34321i
91.3 −1.88268 + 0.674932i 1.22474 + 1.22474i 3.08893 2.54136i 1.58114 + 1.58114i −3.13242 1.47918i 6.93561 −4.10021 + 6.86937i 3.00000i −4.04393 1.90961i
91.4 −1.80521 + 0.860949i −1.22474 1.22474i 2.51753 3.10838i −1.58114 1.58114i 3.26536 + 1.15647i −9.39086 −1.86850 + 7.77873i 3.00000i 4.21556 + 1.49300i
91.5 −1.80387 0.863740i −1.22474 1.22474i 2.50791 + 3.11615i 1.58114 + 1.58114i 1.15142 + 3.26714i 0.973675 −1.83240 7.78732i 3.00000i −1.48648 4.21786i
91.6 −1.55226 + 1.26115i 1.22474 + 1.22474i 0.818999 3.91526i −1.58114 1.58114i −3.44571 0.356530i −8.42448 3.66643 + 7.11036i 3.00000i 4.44839 + 0.460279i
91.7 −1.36537 1.46142i 1.22474 + 1.22474i −0.271513 + 3.99077i −1.58114 1.58114i 0.117637 3.46210i −2.14501 6.20293 5.05210i 3.00000i −0.151868 + 4.46956i
91.8 −1.26185 1.55169i 1.22474 + 1.22474i −0.815479 + 3.91599i 1.58114 + 1.58114i 0.354982 3.44587i −9.52201 7.10541 3.67602i 3.00000i 0.458279 4.44859i
91.9 −1.21603 + 1.58785i 1.22474 + 1.22474i −1.04254 3.86175i 1.58114 + 1.58114i −3.43404 + 0.455383i −6.89370 7.39964 + 3.04062i 3.00000i −4.43333 + 0.587897i
91.10 −1.11696 + 1.65904i −1.22474 1.22474i −1.50481 3.70615i −1.58114 1.58114i 3.39989 0.663910i 4.00659 7.82945 + 1.64307i 3.00000i 4.38923 0.857103i
91.11 −0.979211 + 1.74389i −1.22474 1.22474i −2.08229 3.41527i 1.58114 + 1.58114i 3.33510 0.936535i −6.97297 7.99485 0.287019i 3.00000i −4.30560 + 1.20906i
91.12 −0.823888 1.82242i −1.22474 1.22474i −2.64242 + 3.00294i 1.58114 + 1.58114i −1.22295 + 3.24105i 2.19624 7.64966 + 2.34151i 3.00000i 1.57882 4.18418i
91.13 −0.712831 + 1.86865i 1.22474 + 1.22474i −2.98374 2.66407i −1.58114 1.58114i −3.16166 + 1.41559i 10.5937 7.10514 3.67655i 3.00000i 4.08169 1.82752i
91.14 −0.487538 1.93967i −1.22474 1.22474i −3.52461 + 1.89132i −1.58114 1.58114i −1.77849 + 2.97271i −0.906944 5.38692 + 5.91449i 3.00000i −2.29602 + 3.83775i
91.15 −0.319664 1.97429i 1.22474 + 1.22474i −3.79563 + 1.26222i 1.58114 + 1.58114i 2.02649 2.80951i 6.96024 3.70531 + 7.09018i 3.00000i 2.61619 3.62706i
91.16 0.166064 + 1.99309i −1.22474 1.22474i −3.94485 + 0.661961i 1.58114 + 1.58114i 2.23765 2.64442i 5.18414 −1.97445 7.75252i 3.00000i −2.88879 + 3.41393i
91.17 0.309897 1.97585i 1.22474 + 1.22474i −3.80793 1.22462i −1.58114 1.58114i 2.79945 2.04036i −9.60974 −3.59972 + 7.14437i 3.00000i −3.61408 + 2.63410i
91.18 0.328478 + 1.97284i −1.22474 1.22474i −3.78420 + 1.29607i −1.58114 1.58114i 2.01393 2.81853i 4.74581 −3.79997 7.03990i 3.00000i 2.59997 3.63871i
91.19 0.405730 1.95841i 1.22474 + 1.22474i −3.67077 1.58918i 1.58114 + 1.58114i 2.89547 1.90164i 0.712173 −4.60160 + 6.54410i 3.00000i 3.73804 2.45501i
91.20 0.446279 + 1.94957i 1.22474 + 1.22474i −3.60167 + 1.74011i −1.58114 1.58114i −1.84115 + 2.93431i −2.37589 −4.99981 6.24515i 3.00000i 2.37692 3.78817i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 211.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.bn.a 64
4.b odd 2 1 960.3.bn.a 64
16.e even 4 1 960.3.bn.a 64
16.f odd 4 1 inner 240.3.bn.a 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.3.bn.a 64 1.a even 1 1 trivial
240.3.bn.a 64 16.f odd 4 1 inner
960.3.bn.a 64 4.b odd 2 1
960.3.bn.a 64 16.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(240, [\chi])$$.