# Properties

 Label 912.2.a.g.1.1 Level $912$ Weight $2$ Character 912.1 Self dual yes Analytic conductor $7.282$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(1,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, 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("912.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.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: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 912.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -3.00000 q^{5} +5.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -3.00000 q^{5} +5.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +2.00000 q^{13} -3.00000 q^{15} -1.00000 q^{17} +1.00000 q^{19} +5.00000 q^{21} +4.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} +6.00000 q^{31} -1.00000 q^{33} -15.0000 q^{35} +2.00000 q^{39} +1.00000 q^{43} -3.00000 q^{45} +9.00000 q^{47} +18.0000 q^{49} -1.00000 q^{51} +10.0000 q^{53} +3.00000 q^{55} +1.00000 q^{57} +8.00000 q^{59} -1.00000 q^{61} +5.00000 q^{63} -6.00000 q^{65} -8.00000 q^{67} +4.00000 q^{69} +12.0000 q^{71} -11.0000 q^{73} +4.00000 q^{75} -5.00000 q^{77} -16.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +3.00000 q^{85} -2.00000 q^{87} -6.00000 q^{89} +10.0000 q^{91} +6.00000 q^{93} -3.00000 q^{95} -10.0000 q^{97} -1.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 0 0
$$7$$ 5.00000 1.88982 0.944911 0.327327i $$-0.106148\pi$$
0.944911 + 0.327327i $$0.106148\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ −3.00000 −0.774597
$$16$$ 0 0
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 5.00000 1.09109
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 0 0
$$33$$ −1.00000 −0.174078
$$34$$ 0 0
$$35$$ −15.0000 −2.53546
$$36$$ 0 0
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 0 0
$$45$$ −3.00000 −0.447214
$$46$$ 0 0
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 0 0
$$49$$ 18.0000 2.57143
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 0 0
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ 0 0
$$63$$ 5.00000 0.629941
$$64$$ 0 0
$$65$$ −6.00000 −0.744208
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ −11.0000 −1.28745 −0.643726 0.765256i $$-0.722612\pi$$
−0.643726 + 0.765256i $$0.722612\pi$$
$$74$$ 0 0
$$75$$ 4.00000 0.461880
$$76$$ 0 0
$$77$$ −5.00000 −0.569803
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 3.00000 0.325396
$$86$$ 0 0
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 10.0000 1.04828
$$92$$ 0 0
$$93$$ 6.00000 0.622171
$$94$$ 0 0
$$95$$ −3.00000 −0.307794
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ 2.00000 0.197066 0.0985329 0.995134i $$-0.468585\pi$$
0.0985329 + 0.995134i $$0.468585\pi$$
$$104$$ 0 0
$$105$$ −15.0000 −1.46385
$$106$$ 0 0
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 0 0
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ −12.0000 −1.11901
$$116$$ 0 0
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ −5.00000 −0.458349
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ −7.00000 −0.611593 −0.305796 0.952097i $$-0.598923\pi$$
−0.305796 + 0.952097i $$0.598923\pi$$
$$132$$ 0 0
$$133$$ 5.00000 0.433555
$$134$$ 0 0
$$135$$ −3.00000 −0.258199
$$136$$ 0 0
$$137$$ −9.00000 −0.768922 −0.384461 0.923141i $$-0.625613\pi$$
−0.384461 + 0.923141i $$0.625613\pi$$
$$138$$ 0 0
$$139$$ 13.0000 1.10265 0.551323 0.834292i $$-0.314123\pi$$
0.551323 + 0.834292i $$0.314123\pi$$
$$140$$ 0 0
$$141$$ 9.00000 0.757937
$$142$$ 0 0
$$143$$ −2.00000 −0.167248
$$144$$ 0 0
$$145$$ 6.00000 0.498273
$$146$$ 0 0
$$147$$ 18.0000 1.48461
$$148$$ 0 0
$$149$$ −21.0000 −1.72039 −0.860194 0.509968i $$-0.829657\pi$$
−0.860194 + 0.509968i $$0.829657\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ −18.0000 −1.44579
$$156$$ 0 0
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ 0 0
$$159$$ 10.0000 0.793052
$$160$$ 0 0
$$161$$ 20.0000 1.57622
$$162$$ 0 0
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 0 0
$$165$$ 3.00000 0.233550
$$166$$ 0 0
$$167$$ −10.0000 −0.773823 −0.386912 0.922117i $$-0.626458\pi$$
−0.386912 + 0.922117i $$0.626458\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 20.0000 1.51186
$$176$$ 0 0
$$177$$ 8.00000 0.601317
$$178$$ 0 0
$$179$$ 18.0000 1.34538 0.672692 0.739923i $$-0.265138\pi$$
0.672692 + 0.739923i $$0.265138\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ −1.00000 −0.0739221
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1.00000 0.0731272
$$188$$ 0 0
$$189$$ 5.00000 0.363696
$$190$$ 0 0
$$191$$ −9.00000 −0.651217 −0.325609 0.945505i $$-0.605569\pi$$
−0.325609 + 0.945505i $$0.605569\pi$$
$$192$$ 0 0
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 0 0
$$195$$ −6.00000 −0.429669
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ 21.0000 1.48865 0.744325 0.667817i $$-0.232771\pi$$
0.744325 + 0.667817i $$0.232771\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ 0 0
$$203$$ −10.0000 −0.701862
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 0 0
$$213$$ 12.0000 0.822226
$$214$$ 0 0
$$215$$ −3.00000 −0.204598
$$216$$ 0 0
$$217$$ 30.0000 2.03653
$$218$$ 0 0
$$219$$ −11.0000 −0.743311
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ −12.0000 −0.803579 −0.401790 0.915732i $$-0.631612\pi$$
−0.401790 + 0.915732i $$0.631612\pi$$
$$224$$ 0 0
$$225$$ 4.00000 0.266667
$$226$$ 0 0
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ 0 0
$$229$$ 25.0000 1.65205 0.826023 0.563636i $$-0.190598\pi$$
0.826023 + 0.563636i $$0.190598\pi$$
$$230$$ 0 0
$$231$$ −5.00000 −0.328976
$$232$$ 0 0
$$233$$ 9.00000 0.589610 0.294805 0.955557i $$-0.404745\pi$$
0.294805 + 0.955557i $$0.404745\pi$$
$$234$$ 0 0
$$235$$ −27.0000 −1.76129
$$236$$ 0 0
$$237$$ −16.0000 −1.03931
$$238$$ 0 0
$$239$$ 3.00000 0.194054 0.0970269 0.995282i $$-0.469067\pi$$
0.0970269 + 0.995282i $$0.469067\pi$$
$$240$$ 0 0
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −54.0000 −3.44993
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −7.00000 −0.441836 −0.220918 0.975292i $$-0.570905\pi$$
−0.220918 + 0.975292i $$0.570905\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ 0 0
$$255$$ 3.00000 0.187867
$$256$$ 0 0
$$257$$ −8.00000 −0.499026 −0.249513 0.968371i $$-0.580271\pi$$
−0.249513 + 0.968371i $$0.580271\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ −23.0000 −1.41824 −0.709120 0.705087i $$-0.750908\pi$$
−0.709120 + 0.705087i $$0.750908\pi$$
$$264$$ 0 0
$$265$$ −30.0000 −1.84289
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ 0 0
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ 0 0
$$273$$ 10.0000 0.605228
$$274$$ 0 0
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ −11.0000 −0.660926 −0.330463 0.943819i $$-0.607205\pi$$
−0.330463 + 0.943819i $$0.607205\pi$$
$$278$$ 0 0
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ 13.0000 0.772770 0.386385 0.922338i $$-0.373724\pi$$
0.386385 + 0.922338i $$0.373724\pi$$
$$284$$ 0 0
$$285$$ −3.00000 −0.177705
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 0 0
$$293$$ −28.0000 −1.63578 −0.817889 0.575376i $$-0.804856\pi$$
−0.817889 + 0.575376i $$0.804856\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ 0 0
$$297$$ −1.00000 −0.0580259
$$298$$ 0 0
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ 5.00000 0.288195
$$302$$ 0 0
$$303$$ 2.00000 0.114897
$$304$$ 0 0
$$305$$ 3.00000 0.171780
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ 2.00000 0.113776
$$310$$ 0 0
$$311$$ 21.0000 1.19080 0.595400 0.803429i $$-0.296993\pi$$
0.595400 + 0.803429i $$0.296993\pi$$
$$312$$ 0 0
$$313$$ −2.00000 −0.113047 −0.0565233 0.998401i $$-0.518002\pi$$
−0.0565233 + 0.998401i $$0.518002\pi$$
$$314$$ 0 0
$$315$$ −15.0000 −0.845154
$$316$$ 0 0
$$317$$ −4.00000 −0.224662 −0.112331 0.993671i $$-0.535832\pi$$
−0.112331 + 0.993671i $$0.535832\pi$$
$$318$$ 0 0
$$319$$ 2.00000 0.111979
$$320$$ 0 0
$$321$$ −6.00000 −0.334887
$$322$$ 0 0
$$323$$ −1.00000 −0.0556415
$$324$$ 0 0
$$325$$ 8.00000 0.443760
$$326$$ 0 0
$$327$$ 4.00000 0.221201
$$328$$ 0 0
$$329$$ 45.0000 2.48093
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 24.0000 1.31126
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ 55.0000 2.96972
$$344$$ 0 0
$$345$$ −12.0000 −0.646058
$$346$$ 0 0
$$347$$ 25.0000 1.34207 0.671035 0.741426i $$-0.265850\pi$$
0.671035 + 0.741426i $$0.265850\pi$$
$$348$$ 0 0
$$349$$ 9.00000 0.481759 0.240879 0.970555i $$-0.422564\pi$$
0.240879 + 0.970555i $$0.422564\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ −2.00000 −0.106449 −0.0532246 0.998583i $$-0.516950\pi$$
−0.0532246 + 0.998583i $$0.516950\pi$$
$$354$$ 0 0
$$355$$ −36.0000 −1.91068
$$356$$ 0 0
$$357$$ −5.00000 −0.264628
$$358$$ 0 0
$$359$$ −37.0000 −1.95279 −0.976393 0.216003i $$-0.930698\pi$$
−0.976393 + 0.216003i $$0.930698\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −10.0000 −0.524864
$$364$$ 0 0
$$365$$ 33.0000 1.72730
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 50.0000 2.59587
$$372$$ 0 0
$$373$$ 16.0000 0.828449 0.414224 0.910175i $$-0.364053\pi$$
0.414224 + 0.910175i $$0.364053\pi$$
$$374$$ 0 0
$$375$$ 3.00000 0.154919
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −34.0000 −1.74646 −0.873231 0.487306i $$-0.837980\pi$$
−0.873231 + 0.487306i $$0.837980\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ 0 0
$$383$$ 34.0000 1.73732 0.868659 0.495410i $$-0.164982\pi$$
0.868659 + 0.495410i $$0.164982\pi$$
$$384$$ 0 0
$$385$$ 15.0000 0.764471
$$386$$ 0 0
$$387$$ 1.00000 0.0508329
$$388$$ 0 0
$$389$$ −27.0000 −1.36895 −0.684477 0.729034i $$-0.739969\pi$$
−0.684477 + 0.729034i $$0.739969\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ −7.00000 −0.353103
$$394$$ 0 0
$$395$$ 48.0000 2.41514
$$396$$ 0 0
$$397$$ 25.0000 1.25471 0.627357 0.778732i $$-0.284137\pi$$
0.627357 + 0.778732i $$0.284137\pi$$
$$398$$ 0 0
$$399$$ 5.00000 0.250313
$$400$$ 0 0
$$401$$ 36.0000 1.79775 0.898877 0.438201i $$-0.144384\pi$$
0.898877 + 0.438201i $$0.144384\pi$$
$$402$$ 0 0
$$403$$ 12.0000 0.597763
$$404$$ 0 0
$$405$$ −3.00000 −0.149071
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ −9.00000 −0.443937
$$412$$ 0 0
$$413$$ 40.0000 1.96827
$$414$$ 0 0
$$415$$ 36.0000 1.76717
$$416$$ 0 0
$$417$$ 13.0000 0.636613
$$418$$ 0 0
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ 0 0
$$423$$ 9.00000 0.437595
$$424$$ 0 0
$$425$$ −4.00000 −0.194029
$$426$$ 0 0
$$427$$ −5.00000 −0.241967
$$428$$ 0 0
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ 34.0000 1.63772 0.818861 0.573992i $$-0.194606\pi$$
0.818861 + 0.573992i $$0.194606\pi$$
$$432$$ 0 0
$$433$$ 6.00000 0.288342 0.144171 0.989553i $$-0.453949\pi$$
0.144171 + 0.989553i $$0.453949\pi$$
$$434$$ 0 0
$$435$$ 6.00000 0.287678
$$436$$ 0 0
$$437$$ 4.00000 0.191346
$$438$$ 0 0
$$439$$ −26.0000 −1.24091 −0.620456 0.784241i $$-0.713053\pi$$
−0.620456 + 0.784241i $$0.713053\pi$$
$$440$$ 0 0
$$441$$ 18.0000 0.857143
$$442$$ 0 0
$$443$$ 5.00000 0.237557 0.118779 0.992921i $$-0.462102\pi$$
0.118779 + 0.992921i $$0.462102\pi$$
$$444$$ 0 0
$$445$$ 18.0000 0.853282
$$446$$ 0 0
$$447$$ −21.0000 −0.993266
$$448$$ 0 0
$$449$$ −36.0000 −1.69895 −0.849473 0.527633i $$-0.823080\pi$$
−0.849473 + 0.527633i $$0.823080\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −30.0000 −1.40642
$$456$$ 0 0
$$457$$ −29.0000 −1.35656 −0.678281 0.734802i $$-0.737275\pi$$
−0.678281 + 0.734802i $$0.737275\pi$$
$$458$$ 0 0
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ 27.0000 1.25752 0.628758 0.777601i $$-0.283564\pi$$
0.628758 + 0.777601i $$0.283564\pi$$
$$462$$ 0 0
$$463$$ −17.0000 −0.790057 −0.395029 0.918669i $$-0.629265\pi$$
−0.395029 + 0.918669i $$0.629265\pi$$
$$464$$ 0 0
$$465$$ −18.0000 −0.834730
$$466$$ 0 0
$$467$$ 5.00000 0.231372 0.115686 0.993286i $$-0.463093\pi$$
0.115686 + 0.993286i $$0.463093\pi$$
$$468$$ 0 0
$$469$$ −40.0000 −1.84703
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ 0 0
$$473$$ −1.00000 −0.0459800
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ 10.0000 0.457869
$$478$$ 0 0
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 20.0000 0.910032
$$484$$ 0 0
$$485$$ 30.0000 1.36223
$$486$$ 0 0
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 2.00000 0.0900755
$$494$$ 0 0
$$495$$ 3.00000 0.134840
$$496$$ 0 0
$$497$$ 60.0000 2.69137
$$498$$ 0 0
$$499$$ −5.00000 −0.223831 −0.111915 0.993718i $$-0.535699\pi$$
−0.111915 + 0.993718i $$0.535699\pi$$
$$500$$ 0 0
$$501$$ −10.0000 −0.446767
$$502$$ 0 0
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 0 0
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ −55.0000 −2.43306
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ −6.00000 −0.264392
$$516$$ 0 0
$$517$$ −9.00000 −0.395820
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ 36.0000 1.57719 0.788594 0.614914i $$-0.210809\pi$$
0.788594 + 0.614914i $$0.210809\pi$$
$$522$$ 0 0
$$523$$ 34.0000 1.48672 0.743358 0.668894i $$-0.233232\pi$$
0.743358 + 0.668894i $$0.233232\pi$$
$$524$$ 0 0
$$525$$ 20.0000 0.872872
$$526$$ 0 0
$$527$$ −6.00000 −0.261364
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 8.00000 0.347170
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 18.0000 0.778208
$$536$$ 0 0
$$537$$ 18.0000 0.776757
$$538$$ 0 0
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 3.00000 0.128980 0.0644900 0.997918i $$-0.479458\pi$$
0.0644900 + 0.997918i $$0.479458\pi$$
$$542$$ 0 0
$$543$$ −14.0000 −0.600798
$$544$$ 0 0
$$545$$ −12.0000 −0.514024
$$546$$ 0 0
$$547$$ 26.0000 1.11168 0.555840 0.831289i $$-0.312397\pi$$
0.555840 + 0.831289i $$0.312397\pi$$
$$548$$ 0 0
$$549$$ −1.00000 −0.0426790
$$550$$ 0 0
$$551$$ −2.00000 −0.0852029
$$552$$ 0 0
$$553$$ −80.0000 −3.40195
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −41.0000 −1.73723 −0.868613 0.495491i $$-0.834988\pi$$
−0.868613 + 0.495491i $$0.834988\pi$$
$$558$$ 0 0
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ 1.00000 0.0422200
$$562$$ 0 0
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 5.00000 0.209980
$$568$$ 0 0
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 0 0
$$573$$ −9.00000 −0.375980
$$574$$ 0 0
$$575$$ 16.0000 0.667246
$$576$$ 0 0
$$577$$ 27.0000 1.12402 0.562012 0.827129i $$-0.310027\pi$$
0.562012 + 0.827129i $$0.310027\pi$$
$$578$$ 0 0
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ −60.0000 −2.48922
$$582$$ 0 0
$$583$$ −10.0000 −0.414158
$$584$$ 0 0
$$585$$ −6.00000 −0.248069
$$586$$ 0 0
$$587$$ −7.00000 −0.288921 −0.144460 0.989511i $$-0.546145\pi$$
−0.144460 + 0.989511i $$0.546145\pi$$
$$588$$ 0 0
$$589$$ 6.00000 0.247226
$$590$$ 0 0
$$591$$ −2.00000 −0.0822690
$$592$$ 0 0
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 15.0000 0.614940
$$596$$ 0 0
$$597$$ 21.0000 0.859473
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ 0 0
$$603$$ −8.00000 −0.325785
$$604$$ 0 0
$$605$$ 30.0000 1.21967
$$606$$ 0 0
$$607$$ −26.0000 −1.05531 −0.527654 0.849460i $$-0.676928\pi$$
−0.527654 + 0.849460i $$0.676928\pi$$
$$608$$ 0 0
$$609$$ −10.0000 −0.405220
$$610$$ 0 0
$$611$$ 18.0000 0.728202
$$612$$ 0 0
$$613$$ 33.0000 1.33286 0.666429 0.745569i $$-0.267822\pi$$
0.666429 + 0.745569i $$0.267822\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 27.0000 1.08698 0.543490 0.839416i $$-0.317103\pi$$
0.543490 + 0.839416i $$0.317103\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 0 0
$$623$$ −30.0000 −1.20192
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ −1.00000 −0.0399362
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −15.0000 −0.597141 −0.298570 0.954388i $$-0.596510\pi$$
−0.298570 + 0.954388i $$0.596510\pi$$
$$632$$ 0 0
$$633$$ −12.0000 −0.476957
$$634$$ 0 0
$$635$$ −6.00000 −0.238103
$$636$$ 0 0
$$637$$ 36.0000 1.42637
$$638$$ 0 0
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ 1.00000 0.0394362 0.0197181 0.999806i $$-0.493723\pi$$
0.0197181 + 0.999806i $$0.493723\pi$$
$$644$$ 0 0
$$645$$ −3.00000 −0.118125
$$646$$ 0 0
$$647$$ 39.0000 1.53325 0.766624 0.642096i $$-0.221935\pi$$
0.766624 + 0.642096i $$0.221935\pi$$
$$648$$ 0 0
$$649$$ −8.00000 −0.314027
$$650$$ 0 0
$$651$$ 30.0000 1.17579
$$652$$ 0 0
$$653$$ 3.00000 0.117399 0.0586995 0.998276i $$-0.481305\pi$$
0.0586995 + 0.998276i $$0.481305\pi$$
$$654$$ 0 0
$$655$$ 21.0000 0.820538
$$656$$ 0 0
$$657$$ −11.0000 −0.429151
$$658$$ 0 0
$$659$$ 14.0000 0.545363 0.272681 0.962104i $$-0.412090\pi$$
0.272681 + 0.962104i $$0.412090\pi$$
$$660$$ 0 0
$$661$$ 12.0000 0.466746 0.233373 0.972387i $$-0.425024\pi$$
0.233373 + 0.972387i $$0.425024\pi$$
$$662$$ 0 0
$$663$$ −2.00000 −0.0776736
$$664$$ 0 0
$$665$$ −15.0000 −0.581675
$$666$$ 0 0
$$667$$ −8.00000 −0.309761
$$668$$ 0 0
$$669$$ −12.0000 −0.463947
$$670$$ 0 0
$$671$$ 1.00000 0.0386046
$$672$$ 0 0
$$673$$ −24.0000 −0.925132 −0.462566 0.886585i $$-0.653071\pi$$
−0.462566 + 0.886585i $$0.653071\pi$$
$$674$$ 0 0
$$675$$ 4.00000 0.153960
$$676$$ 0 0
$$677$$ 34.0000 1.30673 0.653363 0.757045i $$-0.273358\pi$$
0.653363 + 0.757045i $$0.273358\pi$$
$$678$$ 0 0
$$679$$ −50.0000 −1.91882
$$680$$ 0 0
$$681$$ −18.0000 −0.689761
$$682$$ 0 0
$$683$$ 6.00000 0.229584 0.114792 0.993390i $$-0.463380\pi$$
0.114792 + 0.993390i $$0.463380\pi$$
$$684$$ 0 0
$$685$$ 27.0000 1.03162
$$686$$ 0 0
$$687$$ 25.0000 0.953809
$$688$$ 0 0
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ 31.0000 1.17930 0.589648 0.807661i $$-0.299267\pi$$
0.589648 + 0.807661i $$0.299267\pi$$
$$692$$ 0 0
$$693$$ −5.00000 −0.189934
$$694$$ 0 0
$$695$$ −39.0000 −1.47935
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 9.00000 0.340411
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −27.0000 −1.01688
$$706$$ 0 0
$$707$$ 10.0000 0.376089
$$708$$ 0 0
$$709$$ −42.0000 −1.57734 −0.788672 0.614815i $$-0.789231\pi$$
−0.788672 + 0.614815i $$0.789231\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ 0 0
$$713$$ 24.0000 0.898807
$$714$$ 0 0
$$715$$ 6.00000 0.224387
$$716$$ 0 0
$$717$$ 3.00000 0.112037
$$718$$ 0 0
$$719$$ −33.0000 −1.23069 −0.615346 0.788257i $$-0.710984\pi$$
−0.615346 + 0.788257i $$0.710984\pi$$
$$720$$ 0 0
$$721$$ 10.0000 0.372419
$$722$$ 0 0
$$723$$ 20.0000 0.743808
$$724$$ 0 0
$$725$$ −8.00000 −0.297113
$$726$$ 0 0
$$727$$ 23.0000 0.853023 0.426511 0.904482i $$-0.359742\pi$$
0.426511 + 0.904482i $$0.359742\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −1.00000 −0.0369863
$$732$$ 0 0
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ 0 0
$$735$$ −54.0000 −1.99182
$$736$$ 0 0
$$737$$ 8.00000 0.294684
$$738$$ 0 0
$$739$$ 5.00000 0.183928 0.0919640 0.995762i $$-0.470686\pi$$
0.0919640 + 0.995762i $$0.470686\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 0 0
$$743$$ 8.00000 0.293492 0.146746 0.989174i $$-0.453120\pi$$
0.146746 + 0.989174i $$0.453120\pi$$
$$744$$ 0 0
$$745$$ 63.0000 2.30814
$$746$$ 0 0
$$747$$ −12.0000 −0.439057
$$748$$ 0 0
$$749$$ −30.0000 −1.09618
$$750$$ 0 0
$$751$$ −24.0000 −0.875772 −0.437886 0.899030i $$-0.644273\pi$$
−0.437886 + 0.899030i $$0.644273\pi$$
$$752$$ 0 0
$$753$$ −7.00000 −0.255094
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −17.0000 −0.617876 −0.308938 0.951082i $$-0.599973\pi$$
−0.308938 + 0.951082i $$0.599973\pi$$
$$758$$ 0 0
$$759$$ −4.00000 −0.145191
$$760$$ 0 0
$$761$$ 15.0000 0.543750 0.271875 0.962333i $$-0.412356\pi$$
0.271875 + 0.962333i $$0.412356\pi$$
$$762$$ 0 0
$$763$$ 20.0000 0.724049
$$764$$ 0 0
$$765$$ 3.00000 0.108465
$$766$$ 0 0
$$767$$ 16.0000 0.577727
$$768$$ 0 0
$$769$$ 11.0000 0.396670 0.198335 0.980134i $$-0.436447\pi$$
0.198335 + 0.980134i $$0.436447\pi$$
$$770$$ 0 0
$$771$$ −8.00000 −0.288113
$$772$$ 0 0
$$773$$ 20.0000 0.719350 0.359675 0.933078i $$-0.382888\pi$$
0.359675 + 0.933078i $$0.382888\pi$$
$$774$$ 0 0
$$775$$ 24.0000 0.862105
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −12.0000 −0.429394
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ 0 0
$$785$$ 54.0000 1.92734
$$786$$ 0 0
$$787$$ −40.0000 −1.42585 −0.712923 0.701242i $$-0.752629\pi$$
−0.712923 + 0.701242i $$0.752629\pi$$
$$788$$ 0 0
$$789$$ −23.0000 −0.818822
$$790$$ 0 0
$$791$$ 10.0000 0.355559
$$792$$ 0 0
$$793$$ −2.00000 −0.0710221
$$794$$ 0 0
$$795$$ −30.0000 −1.06399
$$796$$ 0 0
$$797$$ −44.0000 −1.55856 −0.779280 0.626676i $$-0.784415\pi$$
−0.779280 + 0.626676i $$0.784415\pi$$
$$798$$ 0 0
$$799$$ −9.00000 −0.318397
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 11.0000 0.388182
$$804$$ 0 0
$$805$$ −60.0000 −2.11472
$$806$$ 0 0
$$807$$ −14.0000 −0.492823
$$808$$ 0 0
$$809$$ −55.0000 −1.93370 −0.966849 0.255351i $$-0.917809\pi$$
−0.966849 + 0.255351i $$0.917809\pi$$
$$810$$ 0 0
$$811$$ 38.0000 1.33436 0.667180 0.744896i $$-0.267501\pi$$
0.667180 + 0.744896i $$0.267501\pi$$
$$812$$ 0 0
$$813$$ −12.0000 −0.420858
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 1.00000 0.0349856
$$818$$ 0 0
$$819$$ 10.0000 0.349428
$$820$$ 0 0
$$821$$ −45.0000 −1.57051 −0.785255 0.619172i $$-0.787468\pi$$
−0.785255 + 0.619172i $$0.787468\pi$$
$$822$$ 0 0
$$823$$ −43.0000 −1.49889 −0.749443 0.662069i $$-0.769679\pi$$
−0.749443 + 0.662069i $$0.769679\pi$$
$$824$$ 0 0
$$825$$ −4.00000 −0.139262
$$826$$ 0 0
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 0 0
$$829$$ 52.0000 1.80603 0.903017 0.429604i $$-0.141347\pi$$
0.903017 + 0.429604i $$0.141347\pi$$
$$830$$ 0 0
$$831$$ −11.0000 −0.381586
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 30.0000 1.03819
$$836$$ 0 0
$$837$$ 6.00000 0.207390
$$838$$ 0 0
$$839$$ −54.0000 −1.86429 −0.932144 0.362089i $$-0.882064\pi$$
−0.932144 + 0.362089i $$0.882064\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 10.0000 0.344418
$$844$$ 0 0
$$845$$ 27.0000 0.928828
$$846$$ 0 0
$$847$$ −50.0000 −1.71802
$$848$$ 0 0
$$849$$ 13.0000 0.446159
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −14.0000 −0.479351 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$854$$ 0 0
$$855$$ −3.00000 −0.102598
$$856$$ 0 0
$$857$$ −8.00000 −0.273275 −0.136637 0.990621i $$-0.543630\pi$$
−0.136637 + 0.990621i $$0.543630\pi$$
$$858$$ 0 0
$$859$$ −27.0000 −0.921228 −0.460614 0.887601i $$-0.652371\pi$$
−0.460614 + 0.887601i $$0.652371\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 44.0000 1.49778 0.748889 0.662696i $$-0.230588\pi$$
0.748889 + 0.662696i $$0.230588\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 0 0
$$867$$ −16.0000 −0.543388
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ 0 0
$$873$$ −10.0000 −0.338449
$$874$$ 0 0
$$875$$ 15.0000 0.507093
$$876$$ 0 0
$$877$$ −6.00000 −0.202606 −0.101303 0.994856i $$-0.532301\pi$$
−0.101303 + 0.994856i $$0.532301\pi$$
$$878$$ 0 0
$$879$$ −28.0000 −0.944417
$$880$$ 0 0
$$881$$ −37.0000 −1.24656 −0.623281 0.781998i $$-0.714201\pi$$
−0.623281 + 0.781998i $$0.714201\pi$$
$$882$$ 0 0
$$883$$ −35.0000 −1.17784 −0.588922 0.808190i $$-0.700447\pi$$
−0.588922 + 0.808190i $$0.700447\pi$$
$$884$$ 0 0
$$885$$ −24.0000 −0.806751
$$886$$ 0 0
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ 0 0
$$889$$ 10.0000 0.335389
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 0 0
$$893$$ 9.00000 0.301174
$$894$$ 0 0
$$895$$ −54.0000 −1.80502
$$896$$ 0 0
$$897$$ 8.00000 0.267112
$$898$$ 0 0
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ −10.0000 −0.333148
$$902$$ 0 0
$$903$$ 5.00000 0.166390
$$904$$ 0 0
$$905$$ 42.0000 1.39613
$$906$$ 0 0
$$907$$ 26.0000 0.863316 0.431658 0.902037i $$-0.357929\pi$$
0.431658 + 0.902037i $$0.357929\pi$$
$$908$$ 0 0
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ 6.00000 0.198789 0.0993944 0.995048i $$-0.468309\pi$$
0.0993944 + 0.995048i $$0.468309\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 0 0
$$915$$ 3.00000 0.0991769
$$916$$ 0 0
$$917$$ −35.0000 −1.15580
$$918$$ 0 0
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ 0 0
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 2.00000 0.0656886
$$928$$ 0 0
$$929$$ −2.00000 −0.0656179 −0.0328089 0.999462i $$-0.510445\pi$$
−0.0328089 + 0.999462i $$0.510445\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ 0 0
$$933$$ 21.0000 0.687509
$$934$$ 0 0
$$935$$ −3.00000 −0.0981105
$$936$$ 0 0
$$937$$ 21.0000 0.686040 0.343020 0.939328i $$-0.388550\pi$$
0.343020 + 0.939328i $$0.388550\pi$$
$$938$$ 0 0
$$939$$ −2.00000 −0.0652675
$$940$$ 0 0
$$941$$ 42.0000 1.36916 0.684580 0.728937i $$-0.259985\pi$$
0.684580 + 0.728937i $$0.259985\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ −15.0000 −0.487950
$$946$$ 0 0
$$947$$ 28.0000 0.909878 0.454939 0.890523i $$-0.349661\pi$$
0.454939 + 0.890523i $$0.349661\pi$$
$$948$$ 0 0
$$949$$ −22.0000 −0.714150
$$950$$ 0 0
$$951$$ −4.00000 −0.129709
$$952$$ 0 0
$$953$$ −32.0000 −1.03658 −0.518291 0.855204i $$-0.673432\pi$$
−0.518291 + 0.855204i $$0.673432\pi$$
$$954$$ 0 0
$$955$$ 27.0000 0.873699
$$956$$ 0 0
$$957$$ 2.00000 0.0646508
$$958$$ 0 0
$$959$$ −45.0000 −1.45313
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ 0 0
$$965$$ −12.0000 −0.386294
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 0 0
$$969$$ −1.00000 −0.0321246
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ 65.0000 2.08380
$$974$$ 0 0
$$975$$ 8.00000 0.256205
$$976$$ 0 0
$$977$$ 54.0000 1.72761 0.863807 0.503824i $$-0.168074\pi$$
0.863807 + 0.503824i $$0.168074\pi$$
$$978$$ 0 0
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ 4.00000 0.127710
$$982$$ 0 0
$$983$$ 44.0000 1.40338 0.701691 0.712481i $$-0.252429\pi$$
0.701691 + 0.712481i $$0.252429\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 0 0
$$987$$ 45.0000 1.43237
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ 4.00000 0.126936
$$994$$ 0 0
$$995$$ −63.0000 −1.99723
$$996$$ 0 0
$$997$$ −47.0000 −1.48850 −0.744252 0.667898i $$-0.767194\pi$$
−0.744252 + 0.667898i $$0.767194\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 912.2.a.g.1.1 1
3.2 odd 2 2736.2.a.v.1.1 1
4.3 odd 2 57.2.a.a.1.1 1
8.3 odd 2 3648.2.a.bh.1.1 1
8.5 even 2 3648.2.a.r.1.1 1
12.11 even 2 171.2.a.d.1.1 1
20.3 even 4 1425.2.c.b.799.2 2
20.7 even 4 1425.2.c.b.799.1 2
20.19 odd 2 1425.2.a.j.1.1 1
28.27 even 2 2793.2.a.b.1.1 1
44.43 even 2 6897.2.a.f.1.1 1
52.51 odd 2 9633.2.a.o.1.1 1
60.59 even 2 4275.2.a.b.1.1 1
76.75 even 2 1083.2.a.e.1.1 1
84.83 odd 2 8379.2.a.p.1.1 1
228.227 odd 2 3249.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.a.1.1 1 4.3 odd 2
171.2.a.d.1.1 1 12.11 even 2
912.2.a.g.1.1 1 1.1 even 1 trivial
1083.2.a.e.1.1 1 76.75 even 2
1425.2.a.j.1.1 1 20.19 odd 2
1425.2.c.b.799.1 2 20.7 even 4
1425.2.c.b.799.2 2 20.3 even 4
2736.2.a.v.1.1 1 3.2 odd 2
2793.2.a.b.1.1 1 28.27 even 2
3249.2.a.b.1.1 1 228.227 odd 2
3648.2.a.r.1.1 1 8.5 even 2
3648.2.a.bh.1.1 1 8.3 odd 2
4275.2.a.b.1.1 1 60.59 even 2
6897.2.a.f.1.1 1 44.43 even 2
8379.2.a.p.1.1 1 84.83 odd 2
9633.2.a.o.1.1 1 52.51 odd 2