# Properties

 Label 475.2.bc.a Level $475$ Weight $2$ Character orbit 475.bc Analytic conductor $3.793$ Analytic rank $0$ Dimension $1152$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.bc (of order $$45$$, degree $$24$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$1152$$ Relative dimension: $$48$$ over $$\Q(\zeta_{45})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{45}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10})$$ 1152 * q - 18 * q^2 - 18 * q^3 - 18 * q^4 - 24 * q^5 - 18 * q^6 - 30 * q^7 - 9 * q^8 - 18 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9} - 33 q^{10} - 9 q^{12} - 18 q^{13} - 18 q^{14} + 27 q^{15} - 30 q^{16} - 36 q^{17} - 144 q^{18} - 18 q^{19} - 54 q^{20} + 9 q^{21} + 6 q^{22} - 24 q^{23} - 120 q^{24} - 90 q^{25} - 24 q^{26} - 9 q^{27} + 54 q^{28} + 9 q^{30} - 45 q^{31} - 138 q^{32} + 54 q^{33} - 18 q^{34} + 45 q^{35} - 72 q^{36} - 36 q^{37} + 93 q^{38} - 36 q^{39} + 57 q^{40} - 18 q^{41} + 36 q^{42} - 252 q^{43} - 42 q^{44} - 90 q^{45} - 69 q^{46} - 18 q^{47} + 6 q^{48} - 486 q^{49} + 21 q^{50} + 12 q^{51} - 36 q^{53} - 120 q^{54} - 3 q^{55} + 234 q^{56} + 90 q^{57} + 180 q^{58} + 18 q^{59} + 69 q^{60} - 90 q^{61} - 144 q^{62} - 27 q^{63} + 93 q^{64} - 72 q^{65} + 42 q^{66} + 54 q^{67} - 48 q^{68} - 57 q^{69} + 12 q^{70} - 60 q^{71} - 318 q^{72} - 36 q^{73} - 66 q^{74} - 132 q^{75} - 48 q^{76} + 222 q^{77} - 39 q^{78} + 6 q^{79} + 129 q^{80} - 84 q^{81} + 120 q^{82} + 45 q^{83} - 63 q^{84} - 18 q^{85} + 72 q^{86} - 33 q^{87} - 45 q^{88} + 18 q^{89} + 57 q^{90} + 45 q^{91} + 324 q^{92} - 78 q^{93} - 24 q^{94} + 81 q^{95} - 132 q^{96} - 96 q^{97} - 153 q^{98} - 6 q^{99}+O(q^{100})$$ 1152 * q - 18 * q^2 - 18 * q^3 - 18 * q^4 - 24 * q^5 - 18 * q^6 - 30 * q^7 - 9 * q^8 - 18 * q^9 - 33 * q^10 - 9 * q^12 - 18 * q^13 - 18 * q^14 + 27 * q^15 - 30 * q^16 - 36 * q^17 - 144 * q^18 - 18 * q^19 - 54 * q^20 + 9 * q^21 + 6 * q^22 - 24 * q^23 - 120 * q^24 - 90 * q^25 - 24 * q^26 - 9 * q^27 + 54 * q^28 + 9 * q^30 - 45 * q^31 - 138 * q^32 + 54 * q^33 - 18 * q^34 + 45 * q^35 - 72 * q^36 - 36 * q^37 + 93 * q^38 - 36 * q^39 + 57 * q^40 - 18 * q^41 + 36 * q^42 - 252 * q^43 - 42 * q^44 - 90 * q^45 - 69 * q^46 - 18 * q^47 + 6 * q^48 - 486 * q^49 + 21 * q^50 + 12 * q^51 - 36 * q^53 - 120 * q^54 - 3 * q^55 + 234 * q^56 + 90 * q^57 + 180 * q^58 + 18 * q^59 + 69 * q^60 - 90 * q^61 - 144 * q^62 - 27 * q^63 + 93 * q^64 - 72 * q^65 + 42 * q^66 + 54 * q^67 - 48 * q^68 - 57 * q^69 + 12 * q^70 - 60 * q^71 - 318 * q^72 - 36 * q^73 - 66 * q^74 - 132 * q^75 - 48 * q^76 + 222 * q^77 - 39 * q^78 + 6 * q^79 + 129 * q^80 - 84 * q^81 + 120 * q^82 + 45 * q^83 - 63 * q^84 - 18 * q^85 + 72 * q^86 - 33 * q^87 - 45 * q^88 + 18 * q^89 + 57 * q^90 + 45 * q^91 + 324 * q^92 - 78 * q^93 - 24 * q^94 + 81 * q^95 - 132 * q^96 - 96 * q^97 - 153 * q^98 - 6 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1 −2.29973 1.43703i 2.30476 0.660880i 2.34697 + 4.81199i 2.22377 0.234179i −6.25005 1.79217i 1.23280 2.13527i 0.950677 9.04509i 2.33103 1.45659i −5.45060 2.65708i
6.2 −2.29691 1.43527i −2.18186 + 0.625639i 2.33907 + 4.79580i 1.19746 + 1.88841i 5.90951 + 1.69453i 0.140645 0.243604i 0.944410 8.98546i 1.82496 1.14036i −0.0400903 6.05619i
6.3 −2.19199 1.36971i −1.37858 + 0.395301i 2.05198 + 4.20718i −2.03188 0.933532i 3.56327 + 1.02175i 1.36546 2.36504i 0.724330 6.89154i −0.799932 + 0.499853i 3.17519 + 4.82937i
6.4 −2.08209 1.30103i 0.278378 0.0798235i 1.76566 + 3.62014i 1.47314 1.68222i −0.683459 0.195979i −1.61228 + 2.79255i 0.520393 4.95121i −2.47302 + 1.54532i −5.25582 + 1.58594i
6.5 −2.04037 1.27497i 2.34026 0.671059i 1.66084 + 3.40523i −1.72152 1.42702i −5.63058 1.61454i −2.03230 + 3.52005i 0.449834 4.27989i 2.48235 1.55115i 1.69313 + 5.10653i
6.6 −1.91392 1.19595i −2.99079 + 0.857597i 1.35606 + 2.78033i 0.364394 2.20618i 6.74979 + 1.93547i −1.35748 + 2.35123i 0.257940 2.45413i 5.66524 3.54003i −3.33590 + 3.78666i
6.7 −1.89200 1.18225i 2.66437 0.763997i 1.30520 + 2.67605i −2.06433 + 0.859385i −5.94423 1.70448i 2.28413 3.95623i 0.227928 2.16859i 3.97105 2.48139i 4.92172 + 0.814604i
6.8 −1.79524 1.12179i −0.560140 + 0.160617i 1.08772 + 2.23017i −1.94606 + 1.10130i 1.18576 + 0.340012i −1.52823 + 2.64697i 0.106498 1.01326i −2.25619 + 1.40982i 4.72906 + 0.205964i
6.9 −1.76581 1.10340i −0.597589 + 0.171356i 1.02386 + 2.09922i 1.42897 1.71990i 1.24431 + 0.356799i 1.89729 3.28621i 0.0730432 0.694959i −2.21639 + 1.38496i −4.42104 + 1.46028i
6.10 −1.68487 1.05282i 1.58487 0.454455i 0.853608 + 1.75016i 1.47769 + 1.67822i −3.14876 0.902893i −0.289478 + 0.501391i −0.0109594 + 0.104272i −0.238856 + 0.149254i −0.722847 4.38334i
6.11 −1.48496 0.927909i −0.380214 + 0.109025i 0.467363 + 0.958236i 0.439600 + 2.19243i 0.665769 + 0.190906i 1.66674 2.88687i −0.170928 + 1.62627i −2.41147 + 1.50685i 1.38158 3.66359i
6.12 −1.42766 0.892103i −3.13595 + 0.899221i 0.365632 + 0.749656i −1.46779 + 1.68689i 5.27928 + 1.51381i −0.405461 + 0.702280i −0.205169 + 1.95206i 6.48147 4.05007i 3.60039 1.09889i
6.13 −1.30322 0.814342i 1.85231 0.531140i 0.158485 + 0.324943i −0.982153 2.00882i −2.84649 0.816218i 0.367774 0.637004i −0.263190 + 2.50409i 0.604785 0.377912i −0.355909 + 3.41775i
6.14 −1.05096 0.656713i 3.06632 0.879252i −0.203496 0.417228i 1.91224 1.15903i −3.79999 1.08963i −0.887411 + 1.53704i −0.319211 + 3.03709i 6.08506 3.80237i −2.77084 0.0377008i
6.15 −0.978277 0.611295i −2.30987 + 0.662346i −0.293398 0.601556i 2.15159 0.608806i 2.66459 + 0.764058i 0.629622 1.09054i −0.321864 + 3.06233i 2.35267 1.47011i −2.47701 0.719678i
6.16 −0.954295 0.596310i −0.197845 + 0.0567310i −0.321648 0.659477i −2.23191 + 0.136321i 0.222632 + 0.0638386i 0.295393 0.511636i −0.321554 + 3.05938i −2.50822 + 1.56731i 2.21119 + 1.20082i
6.17 −0.918651 0.574037i −1.08630 + 0.311492i −0.362341 0.742910i −0.569939 2.16221i 1.17674 + 0.337424i −1.35582 + 2.34835i −0.320054 + 3.04511i −1.46112 + 0.913011i −0.717615 + 2.31348i
6.18 −0.764890 0.477956i −1.97717 + 0.566943i −0.520128 1.06642i 0.973170 + 2.01319i 1.78329 + 0.511350i −2.15040 + 3.72461i −0.300419 + 2.85829i 1.04362 0.652125i 0.217850 2.00500i
6.19 −0.730424 0.456420i −0.0193247 + 0.00554125i −0.551542 1.13083i 2.18285 0.484947i 0.0166443 + 0.00477268i −1.43830 + 2.49120i −0.293334 + 2.79088i −2.54380 + 1.58954i −1.81574 0.642078i
6.20 −0.695663 0.434699i −2.62473 + 0.752630i −0.581758 1.19278i −1.90436 1.17193i 2.15310 + 0.617391i 2.44900 4.24179i −0.285284 + 2.71430i 3.77862 2.36115i 0.815352 + 1.64309i
See next 80 embeddings (of 1152 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 461.48 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner
25.d even 5 1 inner
475.bc even 45 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.bc.a 1152
19.e even 9 1 inner 475.2.bc.a 1152
25.d even 5 1 inner 475.2.bc.a 1152
475.bc even 45 1 inner 475.2.bc.a 1152

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.bc.a 1152 1.a even 1 1 trivial
475.2.bc.a 1152 19.e even 9 1 inner
475.2.bc.a 1152 25.d even 5 1 inner
475.2.bc.a 1152 475.bc even 45 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(475, [\chi])$$.