# Properties

 Label 1840.2.a.i.1.1 Level $1840$ Weight $2$ Character 1840.1 Self dual yes Analytic conductor $14.692$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1840.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +6.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +6.00000 q^{9} +1.00000 q^{13} +3.00000 q^{15} +6.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +9.00000 q^{27} -3.00000 q^{29} -3.00000 q^{31} +2.00000 q^{35} -8.00000 q^{37} +3.00000 q^{39} +3.00000 q^{41} +2.00000 q^{43} +6.00000 q^{45} +11.0000 q^{47} -3.00000 q^{49} -14.0000 q^{53} +8.00000 q^{59} -4.00000 q^{61} +12.0000 q^{63} +1.00000 q^{65} +4.00000 q^{67} -3.00000 q^{69} -7.00000 q^{71} -9.00000 q^{73} +3.00000 q^{75} +9.00000 q^{81} -4.00000 q^{83} -9.00000 q^{87} -2.00000 q^{89} +2.00000 q^{91} -9.00000 q^{93} +18.0000 q^{97} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 1.73205 0.866025 0.500000i $$-0.166667\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 0 0
$$9$$ 6.00000 2.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 6.00000 1.30931
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 9.00000 1.73205
$$28$$ 0 0
$$29$$ −3.00000 −0.557086 −0.278543 0.960424i $$-0.589851\pi$$
−0.278543 + 0.960424i $$0.589851\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.00000 0.338062
$$36$$ 0 0
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 0 0
$$39$$ 3.00000 0.480384
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 0 0
$$45$$ 6.00000 0.894427
$$46$$ 0 0
$$47$$ 11.0000 1.60451 0.802257 0.596978i $$-0.203632\pi$$
0.802257 + 0.596978i $$0.203632\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −14.0000 −1.92305 −0.961524 0.274721i $$-0.911414\pi$$
−0.961524 + 0.274721i $$0.911414\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 0 0
$$63$$ 12.0000 1.51186
$$64$$ 0 0
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ −7.00000 −0.830747 −0.415374 0.909651i $$-0.636349\pi$$
−0.415374 + 0.909651i $$0.636349\pi$$
$$72$$ 0 0
$$73$$ −9.00000 −1.05337 −0.526685 0.850060i $$-0.676565\pi$$
−0.526685 + 0.850060i $$0.676565\pi$$
$$74$$ 0 0
$$75$$ 3.00000 0.346410
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −9.00000 −0.964901
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ −9.00000 −0.933257
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 18.0000 1.82762 0.913812 0.406138i $$-0.133125\pi$$
0.913812 + 0.406138i $$0.133125\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 6.00000 0.585540
$$106$$ 0 0
$$107$$ 16.0000 1.54678 0.773389 0.633932i $$-0.218560\pi$$
0.773389 + 0.633932i $$0.218560\pi$$
$$108$$ 0 0
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ 0 0
$$111$$ −24.0000 −2.27798
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ 6.00000 0.554700
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 9.00000 0.811503
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −11.0000 −0.976092 −0.488046 0.872818i $$-0.662290\pi$$
−0.488046 + 0.872818i $$0.662290\pi$$
$$128$$ 0 0
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ 9.00000 0.786334 0.393167 0.919467i $$-0.371379\pi$$
0.393167 + 0.919467i $$0.371379\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 9.00000 0.774597
$$136$$ 0 0
$$137$$ 4.00000 0.341743 0.170872 0.985293i $$-0.445342\pi$$
0.170872 + 0.985293i $$0.445342\pi$$
$$138$$ 0 0
$$139$$ 11.0000 0.933008 0.466504 0.884519i $$-0.345513\pi$$
0.466504 + 0.884519i $$0.345513\pi$$
$$140$$ 0 0
$$141$$ 33.0000 2.77910
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −3.00000 −0.249136
$$146$$ 0 0
$$147$$ −9.00000 −0.742307
$$148$$ 0 0
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ 0 0
$$151$$ −7.00000 −0.569652 −0.284826 0.958579i $$-0.591936\pi$$
−0.284826 + 0.958579i $$0.591936\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −3.00000 −0.240966
$$156$$ 0 0
$$157$$ −6.00000 −0.478852 −0.239426 0.970915i $$-0.576959\pi$$
−0.239426 + 0.970915i $$0.576959\pi$$
$$158$$ 0 0
$$159$$ −42.0000 −3.33082
$$160$$ 0 0
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ −7.00000 −0.548282 −0.274141 0.961689i $$-0.588394\pi$$
−0.274141 + 0.961689i $$0.588394\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ 0 0
$$175$$ 2.00000 0.151186
$$176$$ 0 0
$$177$$ 24.0000 1.80395
$$178$$ 0 0
$$179$$ −21.0000 −1.56961 −0.784807 0.619740i $$-0.787238\pi$$
−0.784807 + 0.619740i $$0.787238\pi$$
$$180$$ 0 0
$$181$$ 12.0000 0.891953 0.445976 0.895045i $$-0.352856\pi$$
0.445976 + 0.895045i $$0.352856\pi$$
$$182$$ 0 0
$$183$$ −12.0000 −0.887066
$$184$$ 0 0
$$185$$ −8.00000 −0.588172
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 18.0000 1.30931
$$190$$ 0 0
$$191$$ −2.00000 −0.144715 −0.0723575 0.997379i $$-0.523052\pi$$
−0.0723575 + 0.997379i $$0.523052\pi$$
$$192$$ 0 0
$$193$$ −1.00000 −0.0719816 −0.0359908 0.999352i $$-0.511459\pi$$
−0.0359908 + 0.999352i $$0.511459\pi$$
$$194$$ 0 0
$$195$$ 3.00000 0.214834
$$196$$ 0 0
$$197$$ 3.00000 0.213741 0.106871 0.994273i $$-0.465917\pi$$
0.106871 + 0.994273i $$0.465917\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ 0 0
$$203$$ −6.00000 −0.421117
$$204$$ 0 0
$$205$$ 3.00000 0.209529
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 16.0000 1.10149 0.550743 0.834675i $$-0.314345\pi$$
0.550743 + 0.834675i $$0.314345\pi$$
$$212$$ 0 0
$$213$$ −21.0000 −1.43890
$$214$$ 0 0
$$215$$ 2.00000 0.136399
$$216$$ 0 0
$$217$$ −6.00000 −0.407307
$$218$$ 0 0
$$219$$ −27.0000 −1.82449
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ 6.00000 0.400000
$$226$$ 0 0
$$227$$ −2.00000 −0.132745 −0.0663723 0.997795i $$-0.521143\pi$$
−0.0663723 + 0.997795i $$0.521143\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 21.0000 1.37576 0.687878 0.725826i $$-0.258542\pi$$
0.687878 + 0.725826i $$0.258542\pi$$
$$234$$ 0 0
$$235$$ 11.0000 0.717561
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1.00000 0.0646846 0.0323423 0.999477i $$-0.489703\pi$$
0.0323423 + 0.999477i $$0.489703\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −16.0000 −1.00991 −0.504956 0.863145i $$-0.668491\pi$$
−0.504956 + 0.863145i $$0.668491\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 5.00000 0.311891 0.155946 0.987766i $$-0.450158\pi$$
0.155946 + 0.987766i $$0.450158\pi$$
$$258$$ 0 0
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ 0 0
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ −14.0000 −0.860013
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ 0 0
$$269$$ 17.0000 1.03651 0.518254 0.855227i $$-0.326582\pi$$
0.518254 + 0.855227i $$0.326582\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 0 0
$$273$$ 6.00000 0.363137
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −29.0000 −1.74244 −0.871221 0.490892i $$-0.836671\pi$$
−0.871221 + 0.490892i $$0.836671\pi$$
$$278$$ 0 0
$$279$$ −18.0000 −1.07763
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −10.0000 −0.594438 −0.297219 0.954809i $$-0.596059\pi$$
−0.297219 + 0.954809i $$0.596059\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 54.0000 3.16554
$$292$$ 0 0
$$293$$ −24.0000 −1.40209 −0.701047 0.713115i $$-0.747284\pi$$
−0.701047 + 0.713115i $$0.747284\pi$$
$$294$$ 0 0
$$295$$ 8.00000 0.465778
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1.00000 −0.0578315
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 0 0
$$303$$ 54.0000 3.10222
$$304$$ 0 0
$$305$$ −4.00000 −0.229039
$$306$$ 0 0
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 0 0
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ 29.0000 1.64444 0.822220 0.569170i $$-0.192736\pi$$
0.822220 + 0.569170i $$0.192736\pi$$
$$312$$ 0 0
$$313$$ −20.0000 −1.13047 −0.565233 0.824931i $$-0.691214\pi$$
−0.565233 + 0.824931i $$0.691214\pi$$
$$314$$ 0 0
$$315$$ 12.0000 0.676123
$$316$$ 0 0
$$317$$ −14.0000 −0.786318 −0.393159 0.919470i $$-0.628618\pi$$
−0.393159 + 0.919470i $$0.628618\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 48.0000 2.67910
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ 0 0
$$327$$ −54.0000 −2.98621
$$328$$ 0 0
$$329$$ 22.0000 1.21290
$$330$$ 0 0
$$331$$ 7.00000 0.384755 0.192377 0.981321i $$-0.438380\pi$$
0.192377 + 0.981321i $$0.438380\pi$$
$$332$$ 0 0
$$333$$ −48.0000 −2.63038
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 0 0
$$345$$ −3.00000 −0.161515
$$346$$ 0 0
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ −7.00000 −0.374701 −0.187351 0.982293i $$-0.559990\pi$$
−0.187351 + 0.982293i $$0.559990\pi$$
$$350$$ 0 0
$$351$$ 9.00000 0.480384
$$352$$ 0 0
$$353$$ 19.0000 1.01127 0.505634 0.862748i $$-0.331259\pi$$
0.505634 + 0.862748i $$0.331259\pi$$
$$354$$ 0 0
$$355$$ −7.00000 −0.371521
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ −33.0000 −1.73205
$$364$$ 0 0
$$365$$ −9.00000 −0.471082
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 18.0000 0.937043
$$370$$ 0 0
$$371$$ −28.0000 −1.45369
$$372$$ 0 0
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 0 0
$$375$$ 3.00000 0.154919
$$376$$ 0 0
$$377$$ −3.00000 −0.154508
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ −33.0000 −1.69064
$$382$$ 0 0
$$383$$ −30.0000 −1.53293 −0.766464 0.642287i $$-0.777986\pi$$
−0.766464 + 0.642287i $$0.777986\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 12.0000 0.609994
$$388$$ 0 0
$$389$$ −16.0000 −0.811232 −0.405616 0.914044i $$-0.632943\pi$$
−0.405616 + 0.914044i $$0.632943\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 27.0000 1.36197
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 25.0000 1.25471 0.627357 0.778732i $$-0.284137\pi$$
0.627357 + 0.778732i $$0.284137\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ −3.00000 −0.149441
$$404$$ 0 0
$$405$$ 9.00000 0.447214
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −11.0000 −0.543915 −0.271957 0.962309i $$-0.587671\pi$$
−0.271957 + 0.962309i $$0.587671\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 0 0
$$413$$ 16.0000 0.787309
$$414$$ 0 0
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ 33.0000 1.61602
$$418$$ 0 0
$$419$$ −22.0000 −1.07477 −0.537385 0.843337i $$-0.680588\pi$$
−0.537385 + 0.843337i $$0.680588\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ 0 0
$$423$$ 66.0000 3.20903
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −8.00000 −0.387147
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ 0 0
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 0 0
$$435$$ −9.00000 −0.431517
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 7.00000 0.334092 0.167046 0.985949i $$-0.446577\pi$$
0.167046 + 0.985949i $$0.446577\pi$$
$$440$$ 0 0
$$441$$ −18.0000 −0.857143
$$442$$ 0 0
$$443$$ −33.0000 −1.56788 −0.783939 0.620838i $$-0.786792\pi$$
−0.783939 + 0.620838i $$0.786792\pi$$
$$444$$ 0 0
$$445$$ −2.00000 −0.0948091
$$446$$ 0 0
$$447$$ −66.0000 −3.12169
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −21.0000 −0.986666
$$454$$ 0 0
$$455$$ 2.00000 0.0937614
$$456$$ 0 0
$$457$$ −4.00000 −0.187112 −0.0935561 0.995614i $$-0.529823\pi$$
−0.0935561 + 0.995614i $$0.529823\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 13.0000 0.605470 0.302735 0.953075i $$-0.402100\pi$$
0.302735 + 0.953075i $$0.402100\pi$$
$$462$$ 0 0
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ 0 0
$$465$$ −9.00000 −0.417365
$$466$$ 0 0
$$467$$ −42.0000 −1.94353 −0.971764 0.235954i $$-0.924178\pi$$
−0.971764 + 0.235954i $$0.924178\pi$$
$$468$$ 0 0
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −84.0000 −3.84610
$$478$$ 0 0
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 0 0
$$483$$ −6.00000 −0.273009
$$484$$ 0 0
$$485$$ 18.0000 0.817338
$$486$$ 0 0
$$487$$ 25.0000 1.13286 0.566429 0.824110i $$-0.308325\pi$$
0.566429 + 0.824110i $$0.308325\pi$$
$$488$$ 0 0
$$489$$ −21.0000 −0.949653
$$490$$ 0 0
$$491$$ 31.0000 1.39901 0.699505 0.714628i $$-0.253404\pi$$
0.699505 + 0.714628i $$0.253404\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −14.0000 −0.627986
$$498$$ 0 0
$$499$$ 25.0000 1.11915 0.559577 0.828778i $$-0.310964\pi$$
0.559577 + 0.828778i $$0.310964\pi$$
$$500$$ 0 0
$$501$$ −48.0000 −2.14448
$$502$$ 0 0
$$503$$ 14.0000 0.624229 0.312115 0.950044i $$-0.398963\pi$$
0.312115 + 0.950044i $$0.398963\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 0 0
$$507$$ −36.0000 −1.59882
$$508$$ 0 0
$$509$$ 21.0000 0.930809 0.465404 0.885098i $$-0.345909\pi$$
0.465404 + 0.885098i $$0.345909\pi$$
$$510$$ 0 0
$$511$$ −18.0000 −0.796273
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 4.00000 0.176261
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 42.0000 1.84360
$$520$$ 0 0
$$521$$ −4.00000 −0.175243 −0.0876216 0.996154i $$-0.527927\pi$$
−0.0876216 + 0.996154i $$0.527927\pi$$
$$522$$ 0 0
$$523$$ −42.0000 −1.83653 −0.918266 0.395964i $$-0.870410\pi$$
−0.918266 + 0.395964i $$0.870410\pi$$
$$524$$ 0 0
$$525$$ 6.00000 0.261861
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 48.0000 2.08302
$$532$$ 0 0
$$533$$ 3.00000 0.129944
$$534$$ 0 0
$$535$$ 16.0000 0.691740
$$536$$ 0 0
$$537$$ −63.0000 −2.71865
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −7.00000 −0.300954 −0.150477 0.988614i $$-0.548081\pi$$
−0.150477 + 0.988614i $$0.548081\pi$$
$$542$$ 0 0
$$543$$ 36.0000 1.54491
$$544$$ 0 0
$$545$$ −18.0000 −0.771035
$$546$$ 0 0
$$547$$ −35.0000 −1.49649 −0.748246 0.663421i $$-0.769104\pi$$
−0.748246 + 0.663421i $$0.769104\pi$$
$$548$$ 0 0
$$549$$ −24.0000 −1.02430
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −24.0000 −1.01874
$$556$$ 0 0
$$557$$ 14.0000 0.593199 0.296600 0.955002i $$-0.404147\pi$$
0.296600 + 0.955002i $$0.404147\pi$$
$$558$$ 0 0
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 16.0000 0.674320 0.337160 0.941447i $$-0.390534\pi$$
0.337160 + 0.941447i $$0.390534\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ 0 0
$$567$$ 18.0000 0.755929
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ 0 0
$$573$$ −6.00000 −0.250654
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ −9.00000 −0.374675 −0.187337 0.982296i $$-0.559986\pi$$
−0.187337 + 0.982296i $$0.559986\pi$$
$$578$$ 0 0
$$579$$ −3.00000 −0.124676
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 6.00000 0.248069
$$586$$ 0 0
$$587$$ −33.0000 −1.36206 −0.681028 0.732257i $$-0.738467\pi$$
−0.681028 + 0.732257i $$0.738467\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 9.00000 0.370211
$$592$$ 0 0
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 12.0000 0.491127
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 37.0000 1.50926 0.754631 0.656150i $$-0.227816\pi$$
0.754631 + 0.656150i $$0.227816\pi$$
$$602$$ 0 0
$$603$$ 24.0000 0.977356
$$604$$ 0 0
$$605$$ −11.0000 −0.447214
$$606$$ 0 0
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ 0 0
$$609$$ −18.0000 −0.729397
$$610$$ 0 0
$$611$$ 11.0000 0.445012
$$612$$ 0 0
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ 0 0
$$615$$ 9.00000 0.362915
$$616$$ 0 0
$$617$$ −48.0000 −1.93241 −0.966204 0.257780i $$-0.917009\pi$$
−0.966204 + 0.257780i $$0.917009\pi$$
$$618$$ 0 0
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 0 0
$$621$$ −9.00000 −0.361158
$$622$$ 0 0
$$623$$ −4.00000 −0.160257
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −14.0000 −0.557331 −0.278666 0.960388i $$-0.589892\pi$$
−0.278666 + 0.960388i $$0.589892\pi$$
$$632$$ 0 0
$$633$$ 48.0000 1.90783
$$634$$ 0 0
$$635$$ −11.0000 −0.436522
$$636$$ 0 0
$$637$$ −3.00000 −0.118864
$$638$$ 0 0
$$639$$ −42.0000 −1.66149
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ 0 0
$$643$$ −34.0000 −1.34083 −0.670415 0.741987i $$-0.733884\pi$$
−0.670415 + 0.741987i $$0.733884\pi$$
$$644$$ 0 0
$$645$$ 6.00000 0.236250
$$646$$ 0 0
$$647$$ 39.0000 1.53325 0.766624 0.642096i $$-0.221935\pi$$
0.766624 + 0.642096i $$0.221935\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −18.0000 −0.705476
$$652$$ 0 0
$$653$$ −3.00000 −0.117399 −0.0586995 0.998276i $$-0.518695\pi$$
−0.0586995 + 0.998276i $$0.518695\pi$$
$$654$$ 0 0
$$655$$ 9.00000 0.351659
$$656$$ 0 0
$$657$$ −54.0000 −2.10674
$$658$$ 0 0
$$659$$ −8.00000 −0.311636 −0.155818 0.987786i $$-0.549801\pi$$
−0.155818 + 0.987786i $$0.549801\pi$$
$$660$$ 0 0
$$661$$ 30.0000 1.16686 0.583432 0.812162i $$-0.301709\pi$$
0.583432 + 0.812162i $$0.301709\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3.00000 0.116160
$$668$$ 0 0
$$669$$ 48.0000 1.85579
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 13.0000 0.501113 0.250557 0.968102i $$-0.419386\pi$$
0.250557 + 0.968102i $$0.419386\pi$$
$$674$$ 0 0
$$675$$ 9.00000 0.346410
$$676$$ 0 0
$$677$$ −46.0000 −1.76792 −0.883962 0.467559i $$-0.845134\pi$$
−0.883962 + 0.467559i $$0.845134\pi$$
$$678$$ 0 0
$$679$$ 36.0000 1.38155
$$680$$ 0 0
$$681$$ −6.00000 −0.229920
$$682$$ 0 0
$$683$$ −35.0000 −1.33924 −0.669619 0.742705i $$-0.733543\pi$$
−0.669619 + 0.742705i $$0.733543\pi$$
$$684$$ 0 0
$$685$$ 4.00000 0.152832
$$686$$ 0 0
$$687$$ −6.00000 −0.228914
$$688$$ 0 0
$$689$$ −14.0000 −0.533358
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 11.0000 0.417254
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 63.0000 2.38288
$$700$$ 0 0
$$701$$ 42.0000 1.58632 0.793159 0.609015i $$-0.208435\pi$$
0.793159 + 0.609015i $$0.208435\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 33.0000 1.24285
$$706$$ 0 0
$$707$$ 36.0000 1.35392
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 3.00000 0.112351
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 3.00000 0.112037
$$718$$ 0 0
$$719$$ −28.0000 −1.04422 −0.522112 0.852877i $$-0.674856\pi$$
−0.522112 + 0.852877i $$0.674856\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 0 0
$$723$$ 6.00000 0.223142
$$724$$ 0 0
$$725$$ −3.00000 −0.111417
$$726$$ 0 0
$$727$$ 6.00000 0.222528 0.111264 0.993791i $$-0.464510\pi$$
0.111264 + 0.993791i $$0.464510\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −8.00000 −0.295487 −0.147743 0.989026i $$-0.547201\pi$$
−0.147743 + 0.989026i $$0.547201\pi$$
$$734$$ 0 0
$$735$$ −9.00000 −0.331970
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −17.0000 −0.625355 −0.312678 0.949859i $$-0.601226\pi$$
−0.312678 + 0.949859i $$0.601226\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 12.0000 0.440237 0.220119 0.975473i $$-0.429356\pi$$
0.220119 + 0.975473i $$0.429356\pi$$
$$744$$ 0 0
$$745$$ −22.0000 −0.806018
$$746$$ 0 0
$$747$$ −24.0000 −0.878114
$$748$$ 0 0
$$749$$ 32.0000 1.16925
$$750$$ 0 0
$$751$$ 50.0000 1.82453 0.912263 0.409605i $$-0.134333\pi$$
0.912263 + 0.409605i $$0.134333\pi$$
$$752$$ 0 0
$$753$$ −48.0000 −1.74922
$$754$$ 0 0
$$755$$ −7.00000 −0.254756
$$756$$ 0 0
$$757$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 29.0000 1.05125 0.525625 0.850717i $$-0.323832\pi$$
0.525625 + 0.850717i $$0.323832\pi$$
$$762$$ 0 0
$$763$$ −36.0000 −1.30329
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8.00000 0.288863
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 15.0000 0.540212
$$772$$ 0 0
$$773$$ 14.0000 0.503545 0.251773 0.967786i $$-0.418987\pi$$
0.251773 + 0.967786i $$0.418987\pi$$
$$774$$ 0 0
$$775$$ −3.00000 −0.107763
$$776$$ 0 0
$$777$$ −48.0000 −1.72199
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −27.0000 −0.964901
$$784$$ 0 0
$$785$$ −6.00000 −0.214149
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 0 0
$$789$$ 36.0000 1.28163
$$790$$ 0 0
$$791$$ 4.00000 0.142224
$$792$$ 0 0
$$793$$ −4.00000 −0.142044
$$794$$ 0 0
$$795$$ −42.0000 −1.48959
$$796$$ 0 0
$$797$$ −12.0000 −0.425062 −0.212531 0.977154i $$-0.568171\pi$$
−0.212531 + 0.977154i $$0.568171\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −12.0000 −0.423999
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −2.00000 −0.0704907
$$806$$ 0 0
$$807$$ 51.0000 1.79529
$$808$$ 0 0
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ 0 0
$$811$$ 5.00000 0.175574 0.0877869 0.996139i $$-0.472021\pi$$
0.0877869 + 0.996139i $$0.472021\pi$$
$$812$$ 0 0
$$813$$ 60.0000 2.10429
$$814$$ 0 0
$$815$$ −7.00000 −0.245199
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 12.0000 0.419314
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ −33.0000 −1.15031 −0.575154 0.818045i $$-0.695058\pi$$
−0.575154 + 0.818045i $$0.695058\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 0 0
$$829$$ 6.00000 0.208389 0.104194 0.994557i $$-0.466774\pi$$
0.104194 + 0.994557i $$0.466774\pi$$
$$830$$ 0 0
$$831$$ −87.0000 −3.01800
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −16.0000 −0.553703
$$836$$ 0 0
$$837$$ −27.0000 −0.933257
$$838$$ 0 0
$$839$$ −6.00000 −0.207143 −0.103572 0.994622i $$-0.533027\pi$$
−0.103572 + 0.994622i $$0.533027\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 18.0000 0.619953
$$844$$ 0 0
$$845$$ −12.0000 −0.412813
$$846$$ 0 0
$$847$$ −22.0000 −0.755929
$$848$$ 0 0
$$849$$ −30.0000 −1.02960
$$850$$ 0 0
$$851$$ 8.00000 0.274236
$$852$$ 0 0
$$853$$ 46.0000 1.57501 0.787505 0.616308i $$-0.211372\pi$$
0.787505 + 0.616308i $$0.211372\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 41.0000 1.40053 0.700267 0.713881i $$-0.253064\pi$$
0.700267 + 0.713881i $$0.253064\pi$$
$$858$$ 0 0
$$859$$ 17.0000 0.580033 0.290016 0.957022i $$-0.406339\pi$$
0.290016 + 0.957022i $$0.406339\pi$$
$$860$$ 0 0
$$861$$ 18.0000 0.613438
$$862$$ 0 0
$$863$$ 17.0000 0.578687 0.289343 0.957225i $$-0.406563\pi$$
0.289343 + 0.957225i $$0.406563\pi$$
$$864$$ 0 0
$$865$$ 14.0000 0.476014
$$866$$ 0 0
$$867$$ −51.0000 −1.73205
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 0 0
$$873$$ 108.000 3.65525
$$874$$ 0 0
$$875$$ 2.00000 0.0676123
$$876$$ 0 0
$$877$$ 18.0000 0.607817 0.303908 0.952701i $$-0.401708\pi$$
0.303908 + 0.952701i $$0.401708\pi$$
$$878$$ 0 0
$$879$$ −72.0000 −2.42850
$$880$$ 0 0
$$881$$ 36.0000 1.21287 0.606435 0.795133i $$-0.292599\pi$$
0.606435 + 0.795133i $$0.292599\pi$$
$$882$$ 0 0
$$883$$ 36.0000 1.21150 0.605748 0.795656i $$-0.292874\pi$$
0.605748 + 0.795656i $$0.292874\pi$$
$$884$$ 0 0
$$885$$ 24.0000 0.806751
$$886$$ 0 0
$$887$$ 15.0000 0.503651 0.251825 0.967773i $$-0.418969\pi$$
0.251825 + 0.967773i $$0.418969\pi$$
$$888$$ 0 0
$$889$$ −22.0000 −0.737856
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −21.0000 −0.701953
$$896$$ 0 0
$$897$$ −3.00000 −0.100167
$$898$$ 0 0
$$899$$ 9.00000 0.300167
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 12.0000 0.399335
$$904$$ 0 0
$$905$$ 12.0000 0.398893
$$906$$ 0 0
$$907$$ −32.0000 −1.06254 −0.531271 0.847202i $$-0.678286\pi$$
−0.531271 + 0.847202i $$0.678286\pi$$
$$908$$ 0 0
$$909$$ 108.000 3.58213
$$910$$ 0 0
$$911$$ 44.0000 1.45779 0.728893 0.684628i $$-0.240035\pi$$
0.728893 + 0.684628i $$0.240035\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −12.0000 −0.396708
$$916$$ 0 0
$$917$$ 18.0000 0.594412
$$918$$ 0 0
$$919$$ 50.0000 1.64935 0.824674 0.565608i $$-0.191359\pi$$
0.824674 + 0.565608i $$0.191359\pi$$
$$920$$ 0 0
$$921$$ 60.0000 1.97707
$$922$$ 0 0
$$923$$ −7.00000 −0.230408
$$924$$ 0 0
$$925$$ −8.00000 −0.263038
$$926$$ 0 0
$$927$$ 24.0000 0.788263
$$928$$ 0 0
$$929$$ 19.0000 0.623370 0.311685 0.950186i $$-0.399107\pi$$
0.311685 + 0.950186i $$0.399107\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 87.0000 2.84825
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −44.0000 −1.43742 −0.718709 0.695311i $$-0.755266\pi$$
−0.718709 + 0.695311i $$0.755266\pi$$
$$938$$ 0 0
$$939$$ −60.0000 −1.95803
$$940$$ 0 0
$$941$$ 12.0000 0.391189 0.195594 0.980685i $$-0.437336\pi$$
0.195594 + 0.980685i $$0.437336\pi$$
$$942$$ 0 0
$$943$$ −3.00000 −0.0976934
$$944$$ 0 0
$$945$$ 18.0000 0.585540
$$946$$ 0 0
$$947$$ 47.0000 1.52729 0.763647 0.645634i $$-0.223407\pi$$
0.763647 + 0.645634i $$0.223407\pi$$
$$948$$ 0 0
$$949$$ −9.00000 −0.292152
$$950$$ 0 0
$$951$$ −42.0000 −1.36194
$$952$$ 0 0
$$953$$ 18.0000 0.583077 0.291539 0.956559i $$-0.405833\pi$$
0.291539 + 0.956559i $$0.405833\pi$$
$$954$$ 0 0
$$955$$ −2.00000 −0.0647185
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 8.00000 0.258333
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 96.0000 3.09356
$$964$$ 0 0
$$965$$ −1.00000 −0.0321911
$$966$$ 0 0
$$967$$ −43.0000 −1.38279 −0.691393 0.722478i $$-0.743003\pi$$
−0.691393 + 0.722478i $$0.743003\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 14.0000 0.449281 0.224641 0.974442i $$-0.427879\pi$$
0.224641 + 0.974442i $$0.427879\pi$$
$$972$$ 0 0
$$973$$ 22.0000 0.705288
$$974$$ 0 0
$$975$$ 3.00000 0.0960769
$$976$$ 0 0
$$977$$ 48.0000 1.53566 0.767828 0.640656i $$-0.221338\pi$$
0.767828 + 0.640656i $$0.221338\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −108.000 −3.44817
$$982$$ 0 0
$$983$$ 14.0000 0.446531 0.223265 0.974758i $$-0.428328\pi$$
0.223265 + 0.974758i $$0.428328\pi$$
$$984$$ 0 0
$$985$$ 3.00000 0.0955879
$$986$$ 0 0
$$987$$ 66.0000 2.10080
$$988$$ 0 0
$$989$$ −2.00000 −0.0635963
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 0 0
$$993$$ 21.0000 0.666415
$$994$$ 0 0
$$995$$ 4.00000 0.126809
$$996$$ 0 0
$$997$$ 58.0000 1.83688 0.918439 0.395562i $$-0.129450\pi$$
0.918439 + 0.395562i $$0.129450\pi$$
$$998$$ 0 0
$$999$$ −72.0000 −2.27798
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1840.2.a.i.1.1 1
4.3 odd 2 920.2.a.a.1.1 1
5.4 even 2 9200.2.a.c.1.1 1
8.3 odd 2 7360.2.a.ba.1.1 1
8.5 even 2 7360.2.a.a.1.1 1
12.11 even 2 8280.2.a.d.1.1 1
20.3 even 4 4600.2.e.b.4049.1 2
20.7 even 4 4600.2.e.b.4049.2 2
20.19 odd 2 4600.2.a.p.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.a.1.1 1 4.3 odd 2
1840.2.a.i.1.1 1 1.1 even 1 trivial
4600.2.a.p.1.1 1 20.19 odd 2
4600.2.e.b.4049.1 2 20.3 even 4
4600.2.e.b.4049.2 2 20.7 even 4
7360.2.a.a.1.1 1 8.5 even 2
7360.2.a.ba.1.1 1 8.3 odd 2
8280.2.a.d.1.1 1 12.11 even 2
9200.2.a.c.1.1 1 5.4 even 2