# Properties

 Label 920.2.a.a.1.1 Level $920$ Weight $2$ Character 920.1 Self dual yes Analytic conductor $7.346$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(1,920)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$920 = 2^{3} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 920.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$7.34623698596$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 920.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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 $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 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 $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 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 $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ 0 0
$$45$$ 6.00000 0.894427
$$46$$ 0 0
$$47$$ −11.0000 −1.60451 −0.802257 0.596978i $$-0.796368\pi$$
−0.802257 + 0.596978i $$0.796368\pi$$
$$48$$ 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 $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 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 $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ 7.00000 0.830747 0.415374 0.909651i $$-0.363651\pi$$
0.415374 + 0.909651i $$0.363651\pi$$
$$72$$ 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 $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 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 $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ 6.00000 0.585540
$$106$$ 0 0
$$107$$ −16.0000 −1.54678 −0.773389 0.633932i $$-0.781440\pi$$
−0.773389 + 0.633932i $$0.781440\pi$$
$$108$$ 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 $$-0.337710\pi$$
0.488046 + 0.872818i $$0.337710\pi$$
$$128$$ 0 0
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ −9.00000 −0.786334 −0.393167 0.919467i $$-0.628621\pi$$
−0.393167 + 0.919467i $$0.628621\pi$$
$$132$$ 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 $$-0.654487\pi$$
−0.466504 + 0.884519i $$0.654487\pi$$
$$140$$ 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 $$-0.408064\pi$$
0.284826 + 0.958579i $$0.408064\pi$$
$$152$$ 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 $$-0.411606\pi$$
0.274141 + 0.961689i $$0.411606\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 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 $$-0.212762\pi$$
0.784807 + 0.619740i $$0.212762\pi$$
$$180$$ 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 $$-0.476948\pi$$
0.0723575 + 0.997379i $$0.476948\pi$$
$$192$$ 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 $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 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 $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 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 $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ 6.00000 0.400000
$$226$$ 0 0
$$227$$ 2.00000 0.132745 0.0663723 0.997795i $$-0.478857\pi$$
0.0663723 + 0.997795i $$0.478857\pi$$
$$228$$ 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 $$-0.510297\pi$$
−0.0323423 + 0.999477i $$0.510297\pi$$
$$240$$ 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 $$-0.331509\pi$$
0.504956 + 0.863145i $$0.331509\pi$$
$$252$$ 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 $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 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 $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 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 $$-0.403941\pi$$
0.297219 + 0.954809i $$0.403941\pi$$
$$284$$ 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 $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ −29.0000 −1.64444 −0.822220 0.569170i $$-0.807264\pi$$
−0.822220 + 0.569170i $$0.807264\pi$$
$$312$$ 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 $$-0.561620\pi$$
−0.192377 + 0.981321i $$0.561620\pi$$
$$332$$ 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 $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 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 $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 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 $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 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 $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ −33.0000 −1.69064
$$382$$ 0 0
$$383$$ 30.0000 1.53293 0.766464 0.642287i $$-0.222014\pi$$
0.766464 + 0.642287i $$0.222014\pi$$
$$384$$ 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 $$-0.319412\pi$$
0.537385 + 0.843337i $$0.319412\pi$$
$$420$$ 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 $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 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 $$-0.553423\pi$$
−0.167046 + 0.985949i $$0.553423\pi$$
$$440$$ 0 0
$$441$$ −18.0000 −0.857143
$$442$$ 0 0
$$443$$ 33.0000 1.56788 0.783939 0.620838i $$-0.213208\pi$$
0.783939 + 0.620838i $$0.213208\pi$$
$$444$$ 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 $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 0 0
$$465$$ −9.00000 −0.417365
$$466$$ 0 0
$$467$$ 42.0000 1.94353 0.971764 0.235954i $$-0.0758216\pi$$
0.971764 + 0.235954i $$0.0758216\pi$$
$$468$$ 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 $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 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 $$-0.691675\pi$$
−0.566429 + 0.824110i $$0.691675\pi$$
$$488$$ 0 0
$$489$$ −21.0000 −0.949653
$$490$$ 0 0
$$491$$ −31.0000 −1.39901 −0.699505 0.714628i $$-0.746596\pi$$
−0.699505 + 0.714628i $$0.746596\pi$$
$$492$$ 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 $$-0.689036\pi$$
−0.559577 + 0.828778i $$0.689036\pi$$
$$500$$ 0 0
$$501$$ −48.0000 −2.14448
$$502$$ 0 0
$$503$$ −14.0000 −0.624229 −0.312115 0.950044i $$-0.601037\pi$$
−0.312115 + 0.950044i $$0.601037\pi$$
$$504$$ 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 $$-0.129590\pi$$
0.918266 + 0.395964i $$0.129590\pi$$
$$524$$ 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 $$-0.230896\pi$$
0.748246 + 0.663421i $$0.230896\pi$$
$$548$$ 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 $$-0.609466\pi$$
−0.337160 + 0.941447i $$0.609466\pi$$
$$564$$ 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 $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ 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 $$-0.261533\pi$$
0.681028 + 0.732257i $$0.261533\pi$$
$$588$$ 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 $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 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 $$-0.394733\pi$$
0.324710 + 0.945814i $$0.394733\pi$$
$$608$$ 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 $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 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 $$-0.410108\pi$$
0.278666 + 0.960388i $$0.410108\pi$$
$$632$$ 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 $$-0.266116\pi$$
0.670415 + 0.741987i $$0.266116\pi$$
$$644$$ 0 0
$$645$$ 6.00000 0.236250
$$646$$ 0 0
$$647$$ −39.0000 −1.53325 −0.766624 0.642096i $$-0.778065\pi$$
−0.766624 + 0.642096i $$0.778065\pi$$
$$648$$ 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 $$-0.450199\pi$$
0.155818 + 0.987786i $$0.450199\pi$$
$$660$$ 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 $$-0.266457\pi$$
0.669619 + 0.742705i $$0.266457\pi$$
$$684$$ 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 $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 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 $$-0.325144\pi$$
0.522112 + 0.852877i $$0.325144\pi$$
$$720$$ 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 $$-0.535490\pi$$
−0.111264 + 0.993791i $$0.535490\pi$$
$$728$$ 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 $$-0.398774\pi$$
0.312678 + 0.949859i $$0.398774\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −12.0000 −0.440237 −0.220119 0.975473i $$-0.570644\pi$$
−0.220119 + 0.975473i $$0.570644\pi$$
$$744$$ 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 $$-0.865667\pi$$
−0.912263 + 0.409605i $$0.865667\pi$$
$$752$$ 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 $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$788$$ 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 $$-0.527979\pi$$
−0.0877869 + 0.996139i $$0.527979\pi$$
$$812$$ 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 $$-0.304942\pi$$
0.575154 + 0.818045i $$0.304942\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 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 $$-0.466973\pi$$
0.103572 + 0.994622i $$0.466973\pi$$
$$840$$ 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 $$-0.593661\pi$$
−0.290016 + 0.957022i $$0.593661\pi$$
$$860$$ 0 0
$$861$$ 18.0000 0.613438
$$862$$ 0 0
$$863$$ −17.0000 −0.578687 −0.289343 0.957225i $$-0.593437\pi$$
−0.289343 + 0.957225i $$0.593437\pi$$
$$864$$ 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 $$-0.707126\pi$$
−0.605748 + 0.795656i $$0.707126\pi$$
$$884$$ 0 0
$$885$$ 24.0000 0.806751
$$886$$ 0 0
$$887$$ −15.0000 −0.503651 −0.251825 0.967773i $$-0.581031\pi$$
−0.251825 + 0.967773i $$0.581031\pi$$
$$888$$ 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 $$-0.321714\pi$$
0.531271 + 0.847202i $$0.321714\pi$$
$$908$$ 0 0
$$909$$ 108.000 3.58213
$$910$$ 0 0
$$911$$ −44.0000 −1.45779 −0.728893 0.684628i $$-0.759965\pi$$
−0.728893 + 0.684628i $$0.759965\pi$$
$$912$$ 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 $$-0.808641\pi$$
−0.824674 + 0.565608i $$0.808641\pi$$
$$920$$ 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 $$-0.776593\pi$$
−0.763647 + 0.645634i $$0.776593\pi$$
$$948$$ 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 $$-0.256997\pi$$
0.691393 + 0.722478i $$0.256997\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −14.0000 −0.449281 −0.224641 0.974442i $$-0.572121\pi$$
−0.224641 + 0.974442i $$0.572121\pi$$
$$972$$ 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 $$-0.571672\pi$$
−0.223265 + 0.974758i $$0.571672\pi$$
$$984$$ 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 920.2.a.a.1.1 1
3.2 odd 2 8280.2.a.d.1.1 1
4.3 odd 2 1840.2.a.i.1.1 1
5.2 odd 4 4600.2.e.b.4049.2 2
5.3 odd 4 4600.2.e.b.4049.1 2
5.4 even 2 4600.2.a.p.1.1 1
8.3 odd 2 7360.2.a.a.1.1 1
8.5 even 2 7360.2.a.ba.1.1 1
20.19 odd 2 9200.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.a.1.1 1 1.1 even 1 trivial
1840.2.a.i.1.1 1 4.3 odd 2
4600.2.a.p.1.1 1 5.4 even 2
4600.2.e.b.4049.1 2 5.3 odd 4
4600.2.e.b.4049.2 2 5.2 odd 4
7360.2.a.a.1.1 1 8.3 odd 2
7360.2.a.ba.1.1 1 8.5 even 2
8280.2.a.d.1.1 1 3.2 odd 2
9200.2.a.c.1.1 1 20.19 odd 2