# Properties

 Label 441.2.a.a.1.1 Level $441$ Weight $2$ Character 441.1 Self dual yes Analytic conductor $3.521$ 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] = [441,2,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.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: $$3.52140272914$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +2.00000 q^{4} -2.00000 q^{5} +O(q^{10})$$ $$q-2.00000 q^{2} +2.00000 q^{4} -2.00000 q^{5} +4.00000 q^{10} +2.00000 q^{11} -1.00000 q^{13} -4.00000 q^{16} -1.00000 q^{19} -4.00000 q^{20} -4.00000 q^{22} -1.00000 q^{25} +2.00000 q^{26} -4.00000 q^{29} -9.00000 q^{31} +8.00000 q^{32} +3.00000 q^{37} +2.00000 q^{38} -10.0000 q^{41} +5.00000 q^{43} +4.00000 q^{44} -6.00000 q^{47} +2.00000 q^{50} -2.00000 q^{52} -12.0000 q^{53} -4.00000 q^{55} +8.00000 q^{58} -12.0000 q^{59} -10.0000 q^{61} +18.0000 q^{62} -8.00000 q^{64} +2.00000 q^{65} -5.00000 q^{67} +6.00000 q^{71} +3.00000 q^{73} -6.00000 q^{74} -2.00000 q^{76} -1.00000 q^{79} +8.00000 q^{80} +20.0000 q^{82} +6.00000 q^{83} -10.0000 q^{86} +16.0000 q^{89} +12.0000 q^{94} +2.00000 q^{95} +6.00000 q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 0 0
$$4$$ 2.00000 1.00000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 4.00000 1.26491
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ −4.00000 −0.894427
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ −9.00000 −1.61645 −0.808224 0.588875i $$-0.799571\pi$$
−0.808224 + 0.588875i $$0.799571\pi$$
$$32$$ 8.00000 1.41421
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 5.00000 0.762493 0.381246 0.924473i $$-0.375495\pi$$
0.381246 + 0.924473i $$0.375495\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 2.00000 0.282843
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ −12.0000 −1.64833 −0.824163 0.566352i $$-0.808354\pi$$
−0.824163 + 0.566352i $$0.808354\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 8.00000 1.05045
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 18.0000 2.28600
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ −5.00000 −0.610847 −0.305424 0.952217i $$-0.598798\pi$$
−0.305424 + 0.952217i $$0.598798\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ 3.00000 0.351123 0.175562 0.984468i $$-0.443826\pi$$
0.175562 + 0.984468i $$0.443826\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ 8.00000 0.894427
$$81$$ 0 0
$$82$$ 20.0000 2.20863
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −10.0000 −1.07833
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 16.0000 1.69600 0.847998 0.529999i $$-0.177808\pi$$
0.847998 + 0.529999i $$0.177808\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −2.00000 −0.200000
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ 7.00000 0.689730 0.344865 0.938652i $$-0.387925\pi$$
0.344865 + 0.938652i $$0.387925\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 24.0000 2.33109
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 0 0
$$109$$ 9.00000 0.862044 0.431022 0.902342i $$-0.358153\pi$$
0.431022 + 0.902342i $$0.358153\pi$$
$$110$$ 8.00000 0.762770
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −8.00000 −0.742781
$$117$$ 0 0
$$118$$ 24.0000 2.20938
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 20.0000 1.81071
$$123$$ 0 0
$$124$$ −18.0000 −1.61645
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ −15.0000 −1.33103 −0.665517 0.746382i $$-0.731789\pi$$
−0.665517 + 0.746382i $$0.731789\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ −4.00000 −0.350823
$$131$$ −14.0000 −1.22319 −0.611593 0.791173i $$-0.709471\pi$$
−0.611593 + 0.791173i $$0.709471\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 10.0000 0.863868
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ 3.00000 0.254457 0.127228 0.991873i $$-0.459392\pi$$
0.127228 + 0.991873i $$0.459392\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −12.0000 −1.00702
$$143$$ −2.00000 −0.167248
$$144$$ 0 0
$$145$$ 8.00000 0.664364
$$146$$ −6.00000 −0.496564
$$147$$ 0 0
$$148$$ 6.00000 0.493197
$$149$$ 12.0000 0.983078 0.491539 0.870855i $$-0.336434\pi$$
0.491539 + 0.870855i $$0.336434\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 18.0000 1.44579
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 2.00000 0.159111
$$159$$ 0 0
$$160$$ −16.0000 −1.26491
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −20.0000 −1.56174
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ −14.0000 −1.08335 −0.541676 0.840587i $$-0.682210\pi$$
−0.541676 + 0.840587i $$0.682210\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 10.0000 0.762493
$$173$$ 8.00000 0.608229 0.304114 0.952636i $$-0.401639\pi$$
0.304114 + 0.952636i $$0.401639\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −8.00000 −0.603023
$$177$$ 0 0
$$178$$ −32.0000 −2.39850
$$179$$ −2.00000 −0.149487 −0.0747435 0.997203i $$-0.523814\pi$$
−0.0747435 + 0.997203i $$0.523814\pi$$
$$180$$ 0 0
$$181$$ −13.0000 −0.966282 −0.483141 0.875542i $$-0.660504\pi$$
−0.483141 + 0.875542i $$0.660504\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −12.0000 −0.875190
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ −10.0000 −0.723575 −0.361787 0.932261i $$-0.617833\pi$$
−0.361787 + 0.932261i $$0.617833\pi$$
$$192$$ 0 0
$$193$$ 11.0000 0.791797 0.395899 0.918294i $$-0.370433\pi$$
0.395899 + 0.918294i $$0.370433\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −16.0000 −1.13995 −0.569976 0.821661i $$-0.693048\pi$$
−0.569976 + 0.821661i $$0.693048\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −4.00000 −0.281439
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 20.0000 1.39686
$$206$$ −14.0000 −0.975426
$$207$$ 0 0
$$208$$ 4.00000 0.277350
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ −24.0000 −1.64833
$$213$$ 0 0
$$214$$ −16.0000 −1.09374
$$215$$ −10.0000 −0.681994
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −18.0000 −1.21911
$$219$$ 0 0
$$220$$ −8.00000 −0.539360
$$221$$ 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
$$225$$ 0 0
$$226$$ 20.0000 1.33038
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ 0 0
$$229$$ 19.0000 1.25556 0.627778 0.778393i $$-0.283965\pi$$
0.627778 + 0.778393i $$0.283965\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 12.0000 0.782794
$$236$$ −24.0000 −1.56227
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 14.0000 0.899954
$$243$$ 0 0
$$244$$ −20.0000 −1.28037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.00000 0.0636285
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −24.0000 −1.51789
$$251$$ −8.00000 −0.504956 −0.252478 0.967603i $$-0.581245\pi$$
−0.252478 + 0.967603i $$0.581245\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 30.0000 1.88237
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 26.0000 1.62184 0.810918 0.585160i $$-0.198968\pi$$
0.810918 + 0.585160i $$0.198968\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 4.00000 0.248069
$$261$$ 0 0
$$262$$ 28.0000 1.72985
$$263$$ −4.00000 −0.246651 −0.123325 0.992366i $$-0.539356\pi$$
−0.123325 + 0.992366i $$0.539356\pi$$
$$264$$ 0 0
$$265$$ 24.0000 1.47431
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −10.0000 −0.610847
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −24.0000 −1.44989
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ 13.0000 0.781094 0.390547 0.920583i $$-0.372286\pi$$
0.390547 + 0.920583i $$0.372286\pi$$
$$278$$ −6.00000 −0.359856
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 4.00000 0.238620 0.119310 0.992857i $$-0.461932\pi$$
0.119310 + 0.992857i $$0.461932\pi$$
$$282$$ 0 0
$$283$$ 11.0000 0.653882 0.326941 0.945045i $$-0.393982\pi$$
0.326941 + 0.945045i $$0.393982\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ −16.0000 −0.939552
$$291$$ 0 0
$$292$$ 6.00000 0.351123
$$293$$ 8.00000 0.467365 0.233682 0.972313i $$-0.424922\pi$$
0.233682 + 0.972313i $$0.424922\pi$$
$$294$$ 0 0
$$295$$ 24.0000 1.39733
$$296$$ 0 0
$$297$$ 0 0
$$298$$ −24.0000 −1.39028
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 32.0000 1.84139
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 20.0000 1.14520
$$306$$ 0 0
$$307$$ 17.0000 0.970241 0.485121 0.874447i $$-0.338776\pi$$
0.485121 + 0.874447i $$0.338776\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −36.0000 −2.04466
$$311$$ −6.00000 −0.340229 −0.170114 0.985424i $$-0.554414\pi$$
−0.170114 + 0.985424i $$0.554414\pi$$
$$312$$ 0 0
$$313$$ 1.00000 0.0565233 0.0282617 0.999601i $$-0.491003\pi$$
0.0282617 + 0.999601i $$0.491003\pi$$
$$314$$ −28.0000 −1.58013
$$315$$ 0 0
$$316$$ −2.00000 −0.112509
$$317$$ −24.0000 −1.34797 −0.673987 0.738743i $$-0.735420\pi$$
−0.673987 + 0.738743i $$0.735420\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 16.0000 0.894427
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ −8.00000 −0.443079
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −25.0000 −1.37412 −0.687062 0.726599i $$-0.741100\pi$$
−0.687062 + 0.726599i $$0.741100\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 0 0
$$334$$ 28.0000 1.53209
$$335$$ 10.0000 0.546358
$$336$$ 0 0
$$337$$ 13.0000 0.708155 0.354078 0.935216i $$-0.384795\pi$$
0.354078 + 0.935216i $$0.384795\pi$$
$$338$$ 24.0000 1.30543
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −18.0000 −0.974755
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −16.0000 −0.860165
$$347$$ −32.0000 −1.71785 −0.858925 0.512101i $$-0.828867\pi$$
−0.858925 + 0.512101i $$0.828867\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 16.0000 0.852803
$$353$$ 34.0000 1.80964 0.904819 0.425797i $$-0.140006\pi$$
0.904819 + 0.425797i $$0.140006\pi$$
$$354$$ 0 0
$$355$$ −12.0000 −0.636894
$$356$$ 32.0000 1.69600
$$357$$ 0 0
$$358$$ 4.00000 0.211407
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 26.0000 1.36653
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ 9.00000 0.469796 0.234898 0.972020i $$-0.424524\pi$$
0.234898 + 0.972020i $$0.424524\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 12.0000 0.623850
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 23.0000 1.19089 0.595447 0.803394i $$-0.296975\pi$$
0.595447 + 0.803394i $$0.296975\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ 3.00000 0.154100 0.0770498 0.997027i $$-0.475450\pi$$
0.0770498 + 0.997027i $$0.475450\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 0 0
$$382$$ 20.0000 1.02329
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −22.0000 −1.11977
$$387$$ 0 0
$$388$$ 12.0000 0.609208
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 32.0000 1.61214
$$395$$ 2.00000 0.100631
$$396$$ 0 0
$$397$$ 9.00000 0.451697 0.225849 0.974162i $$-0.427485\pi$$
0.225849 + 0.974162i $$0.427485\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ 36.0000 1.79775 0.898877 0.438201i $$-0.144384\pi$$
0.898877 + 0.438201i $$0.144384\pi$$
$$402$$ 0 0
$$403$$ 9.00000 0.448322
$$404$$ 4.00000 0.199007
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 6.00000 0.297409
$$408$$ 0 0
$$409$$ −5.00000 −0.247234 −0.123617 0.992330i $$-0.539449\pi$$
−0.123617 + 0.992330i $$0.539449\pi$$
$$410$$ −40.0000 −1.97546
$$411$$ 0 0
$$412$$ 14.0000 0.689730
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ −8.00000 −0.392232
$$417$$ 0 0
$$418$$ 4.00000 0.195646
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ −7.00000 −0.341159 −0.170580 0.985344i $$-0.554564\pi$$
−0.170580 + 0.985344i $$0.554564\pi$$
$$422$$ −8.00000 −0.389434
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 16.0000 0.773389
$$429$$ 0 0
$$430$$ 20.0000 0.964486
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 0 0
$$433$$ −31.0000 −1.48976 −0.744882 0.667196i $$-0.767494\pi$$
−0.744882 + 0.667196i $$0.767494\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ −32.0000 −1.51695
$$446$$ 32.0000 1.51524
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ −20.0000 −0.941763
$$452$$ −20.0000 −0.940721
$$453$$ 0 0
$$454$$ −36.0000 −1.68956
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11.0000 −0.514558 −0.257279 0.966337i $$-0.582826\pi$$
−0.257279 + 0.966337i $$0.582826\pi$$
$$458$$ −38.0000 −1.77562
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 20.0000 0.931493 0.465746 0.884918i $$-0.345786\pi$$
0.465746 + 0.884918i $$0.345786\pi$$
$$462$$ 0 0
$$463$$ −17.0000 −0.790057 −0.395029 0.918669i $$-0.629265\pi$$
−0.395029 + 0.918669i $$0.629265\pi$$
$$464$$ 16.0000 0.742781
$$465$$ 0 0
$$466$$ 12.0000 0.555889
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −24.0000 −1.10704
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 10.0000 0.459800
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 12.0000 0.548867
$$479$$ −28.0000 −1.27935 −0.639676 0.768644i $$-0.720932\pi$$
−0.639676 + 0.768644i $$0.720932\pi$$
$$480$$ 0 0
$$481$$ −3.00000 −0.136788
$$482$$ 28.0000 1.27537
$$483$$ 0 0
$$484$$ −14.0000 −0.636364
$$485$$ −12.0000 −0.544892
$$486$$ 0 0
$$487$$ 31.0000 1.40474 0.702372 0.711810i $$-0.252124\pi$$
0.702372 + 0.711810i $$0.252124\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −2.00000 −0.0899843
$$495$$ 0 0
$$496$$ 36.0000 1.61645
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 37.0000 1.65635 0.828174 0.560471i $$-0.189380\pi$$
0.828174 + 0.560471i $$0.189380\pi$$
$$500$$ 24.0000 1.07331
$$501$$ 0 0
$$502$$ 16.0000 0.714115
$$503$$ −42.0000 −1.87269 −0.936344 0.351085i $$-0.885813\pi$$
−0.936344 + 0.351085i $$0.885813\pi$$
$$504$$ 0 0
$$505$$ −4.00000 −0.177998
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −30.0000 −1.33103
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −32.0000 −1.41421
$$513$$ 0 0
$$514$$ −52.0000 −2.29362
$$515$$ −14.0000 −0.616914
$$516$$ 0 0
$$517$$ −12.0000 −0.527759
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 12.0000 0.525730 0.262865 0.964833i $$-0.415333\pi$$
0.262865 + 0.964833i $$0.415333\pi$$
$$522$$ 0 0
$$523$$ −31.0000 −1.35554 −0.677768 0.735276i $$-0.737052\pi$$
−0.677768 + 0.735276i $$0.737052\pi$$
$$524$$ −28.0000 −1.22319
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ −48.0000 −2.08499
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 10.0000 0.433148
$$534$$ 0 0
$$535$$ −16.0000 −0.691740
$$536$$ 0 0
$$537$$ 0 0
$$538$$ −12.0000 −0.517357
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −19.0000 −0.816874 −0.408437 0.912787i $$-0.633926\pi$$
−0.408437 + 0.912787i $$0.633926\pi$$
$$542$$ 32.0000 1.37452
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −18.0000 −0.771035
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 24.0000 1.02523
$$549$$ 0 0
$$550$$ 4.00000 0.170561
$$551$$ 4.00000 0.170406
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −26.0000 −1.10463
$$555$$ 0 0
$$556$$ 6.00000 0.254457
$$557$$ 2.00000 0.0847427 0.0423714 0.999102i $$-0.486509\pi$$
0.0423714 + 0.999102i $$0.486509\pi$$
$$558$$ 0 0
$$559$$ −5.00000 −0.211477
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −8.00000 −0.337460
$$563$$ −26.0000 −1.09577 −0.547885 0.836554i $$-0.684567\pi$$
−0.547885 + 0.836554i $$0.684567\pi$$
$$564$$ 0 0
$$565$$ 20.0000 0.841406
$$566$$ −22.0000 −0.924729
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 26.0000 1.08998 0.544988 0.838444i $$-0.316534\pi$$
0.544988 + 0.838444i $$0.316534\pi$$
$$570$$ 0 0
$$571$$ −19.0000 −0.795125 −0.397563 0.917575i $$-0.630144\pi$$
−0.397563 + 0.917575i $$0.630144\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 17.0000 0.707719 0.353860 0.935299i $$-0.384869\pi$$
0.353860 + 0.935299i $$0.384869\pi$$
$$578$$ 34.0000 1.41421
$$579$$ 0 0
$$580$$ 16.0000 0.664364
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −24.0000 −0.993978
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −16.0000 −0.660954
$$587$$ 16.0000 0.660391 0.330195 0.943913i $$-0.392885\pi$$
0.330195 + 0.943913i $$0.392885\pi$$
$$588$$ 0 0
$$589$$ 9.00000 0.370839
$$590$$ −48.0000 −1.97613
$$591$$ 0 0
$$592$$ −12.0000 −0.493197
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 24.0000 0.983078
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 9.00000 0.367118 0.183559 0.983009i $$-0.441238\pi$$
0.183559 + 0.983009i $$0.441238\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −32.0000 −1.30206
$$605$$ 14.0000 0.569181
$$606$$ 0 0
$$607$$ −23.0000 −0.933541 −0.466771 0.884378i $$-0.654583\pi$$
−0.466771 + 0.884378i $$0.654583\pi$$
$$608$$ −8.00000 −0.324443
$$609$$ 0 0
$$610$$ −40.0000 −1.61955
$$611$$ 6.00000 0.242734
$$612$$ 0 0
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ −34.0000 −1.37213
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ 29.0000 1.16561 0.582804 0.812613i $$-0.301955\pi$$
0.582804 + 0.812613i $$0.301955\pi$$
$$620$$ 36.0000 1.44579
$$621$$ 0 0
$$622$$ 12.0000 0.481156
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ −2.00000 −0.0799361
$$627$$ 0 0
$$628$$ 28.0000 1.11732
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 48.0000 1.90632
$$635$$ 30.0000 1.19051
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 16.0000 0.633446
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ 19.0000 0.749287 0.374643 0.927169i $$-0.377765\pi$$
0.374643 + 0.927169i $$0.377765\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 2.00000 0.0786281 0.0393141 0.999227i $$-0.487483\pi$$
0.0393141 + 0.999227i $$0.487483\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ −2.00000 −0.0784465
$$651$$ 0 0
$$652$$ 8.00000 0.313304
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 0 0
$$655$$ 28.0000 1.09405
$$656$$ 40.0000 1.56174
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 41.0000 1.59472 0.797358 0.603507i $$-0.206231\pi$$
0.797358 + 0.603507i $$0.206231\pi$$
$$662$$ 50.0000 1.94331
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −28.0000 −1.08335
$$669$$ 0 0
$$670$$ −20.0000 −0.772667
$$671$$ −20.0000 −0.772091
$$672$$ 0 0
$$673$$ −41.0000 −1.58043 −0.790217 0.612827i $$-0.790032\pi$$
−0.790217 + 0.612827i $$0.790032\pi$$
$$674$$ −26.0000 −1.00148
$$675$$ 0 0
$$676$$ −24.0000 −0.923077
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 36.0000 1.37851
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ −24.0000 −0.916993
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −20.0000 −0.762493
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 37.0000 1.40755 0.703773 0.710425i $$-0.251497\pi$$
0.703773 + 0.710425i $$0.251497\pi$$
$$692$$ 16.0000 0.608229
$$693$$ 0 0
$$694$$ 64.0000 2.42941
$$695$$ −6.00000 −0.227593
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −28.0000 −1.05982
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ −3.00000 −0.113147
$$704$$ −16.0000 −0.603023
$$705$$ 0 0
$$706$$ −68.0000 −2.55921
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 30.0000 1.12667 0.563337 0.826227i $$-0.309517\pi$$
0.563337 + 0.826227i $$0.309517\pi$$
$$710$$ 24.0000 0.900704
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 4.00000 0.149592
$$716$$ −4.00000 −0.149487
$$717$$ 0 0
$$718$$ 40.0000 1.49279
$$719$$ −18.0000 −0.671287 −0.335643 0.941989i $$-0.608954\pi$$
−0.335643 + 0.941989i $$0.608954\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 36.0000 1.33978
$$723$$ 0 0
$$724$$ −26.0000 −0.966282
$$725$$ 4.00000 0.148556
$$726$$ 0 0
$$727$$ 13.0000 0.482143 0.241072 0.970507i $$-0.422501\pi$$
0.241072 + 0.970507i $$0.422501\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 12.0000 0.444140
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 15.0000 0.554038 0.277019 0.960864i $$-0.410654\pi$$
0.277019 + 0.960864i $$0.410654\pi$$
$$734$$ −18.0000 −0.664392
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −10.0000 −0.368355
$$738$$ 0 0
$$739$$ −15.0000 −0.551784 −0.275892 0.961189i $$-0.588973\pi$$
−0.275892 + 0.961189i $$0.588973\pi$$
$$740$$ −12.0000 −0.441129
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −42.0000 −1.54083 −0.770415 0.637542i $$-0.779951\pi$$
−0.770415 + 0.637542i $$0.779951\pi$$
$$744$$ 0 0
$$745$$ −24.0000 −0.879292
$$746$$ −46.0000 −1.68418
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 13.0000 0.474377 0.237188 0.971464i $$-0.423774\pi$$
0.237188 + 0.971464i $$0.423774\pi$$
$$752$$ 24.0000 0.875190
$$753$$ 0 0
$$754$$ −8.00000 −0.291343
$$755$$ 32.0000 1.16460
$$756$$ 0 0
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ −6.00000 −0.217930
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −48.0000 −1.74000 −0.869999 0.493053i $$-0.835881\pi$$
−0.869999 + 0.493053i $$0.835881\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −20.0000 −0.723575
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 12.0000 0.433295
$$768$$ 0 0
$$769$$ 49.0000 1.76699 0.883493 0.468445i $$-0.155186\pi$$
0.883493 + 0.468445i $$0.155186\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 22.0000 0.791797
$$773$$ −34.0000 −1.22290 −0.611448 0.791285i $$-0.709412\pi$$
−0.611448 + 0.791285i $$0.709412\pi$$
$$774$$ 0 0
$$775$$ 9.00000 0.323290
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −12.0000 −0.430221
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ 12.0000 0.429394
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −28.0000 −0.999363
$$786$$ 0 0
$$787$$ −40.0000 −1.42585 −0.712923 0.701242i $$-0.752629\pi$$
−0.712923 + 0.701242i $$0.752629\pi$$
$$788$$ −32.0000 −1.13995
$$789$$ 0 0
$$790$$ −4.00000 −0.142314
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 10.0000 0.355110
$$794$$ −18.0000 −0.638796
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −8.00000 −0.283375 −0.141687 0.989911i $$-0.545253\pi$$
−0.141687 + 0.989911i $$0.545253\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −8.00000 −0.282843
$$801$$ 0 0
$$802$$ −72.0000 −2.54241
$$803$$ 6.00000 0.211735
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −18.0000 −0.634023
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ −32.0000 −1.12367 −0.561836 0.827249i $$-0.689905\pi$$
−0.561836 + 0.827249i $$0.689905\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −12.0000 −0.420600
$$815$$ −8.00000 −0.280228
$$816$$ 0 0
$$817$$ −5.00000 −0.174928
$$818$$ 10.0000 0.349642
$$819$$ 0 0
$$820$$ 40.0000 1.39686
$$821$$ −2.00000 −0.0698005 −0.0349002 0.999391i $$-0.511111\pi$$
−0.0349002 + 0.999391i $$0.511111\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 30.0000 1.04320 0.521601 0.853189i $$-0.325335\pi$$
0.521601 + 0.853189i $$0.325335\pi$$
$$828$$ 0 0
$$829$$ −41.0000 −1.42399 −0.711994 0.702185i $$-0.752208\pi$$
−0.711994 + 0.702185i $$0.752208\pi$$
$$830$$ 24.0000 0.833052
$$831$$ 0 0
$$832$$ 8.00000 0.277350
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 28.0000 0.968980
$$836$$ −4.00000 −0.138343
$$837$$ 0 0
$$838$$ −60.0000 −2.07267
$$839$$ −44.0000 −1.51905 −0.759524 0.650479i $$-0.774568\pi$$
−0.759524 + 0.650479i $$0.774568\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 14.0000 0.482472
$$843$$ 0 0
$$844$$ 8.00000 0.275371
$$845$$ 24.0000 0.825625
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 48.0000 1.64833
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −35.0000 −1.19838 −0.599189 0.800608i $$-0.704510\pi$$
−0.599189 + 0.800608i $$0.704510\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −32.0000 −1.09310 −0.546550 0.837427i $$-0.684059\pi$$
−0.546550 + 0.837427i $$0.684059\pi$$
$$858$$ 0 0
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ −20.0000 −0.681994
$$861$$ 0 0
$$862$$ −36.0000 −1.22616
$$863$$ 54.0000 1.83818 0.919091 0.394046i $$-0.128925\pi$$
0.919091 + 0.394046i $$0.128925\pi$$
$$864$$ 0 0
$$865$$ −16.0000 −0.544016
$$866$$ 62.0000 2.10685
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −2.00000 −0.0678454
$$870$$ 0 0
$$871$$ 5.00000 0.169419
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −38.0000 −1.28317 −0.641584 0.767052i $$-0.721723\pi$$
−0.641584 + 0.767052i $$0.721723\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 16.0000 0.539360
$$881$$ 24.0000 0.808581 0.404290 0.914631i $$-0.367519\pi$$
0.404290 + 0.914631i $$0.367519\pi$$
$$882$$ 0 0
$$883$$ −13.0000 −0.437485 −0.218742 0.975783i $$-0.570195\pi$$
−0.218742 + 0.975783i $$0.570195\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ −34.0000 −1.14161 −0.570804 0.821086i $$-0.693368\pi$$
−0.570804 + 0.821086i $$0.693368\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 64.0000 2.14528
$$891$$ 0 0
$$892$$ −32.0000 −1.07144
$$893$$ 6.00000 0.200782
$$894$$ 0 0
$$895$$ 4.00000 0.133705
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −36.0000 −1.20134
$$899$$ 36.0000 1.20067
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 40.0000 1.33185
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 26.0000 0.864269
$$906$$ 0 0
$$907$$ −37.0000 −1.22856 −0.614282 0.789086i $$-0.710554\pi$$
−0.614282 + 0.789086i $$0.710554\pi$$
$$908$$ 36.0000 1.19470
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 22.0000 0.727695
$$915$$ 0 0
$$916$$ 38.0000 1.25556
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 23.0000 0.758700 0.379350 0.925253i $$-0.376148\pi$$
0.379350 + 0.925253i $$0.376148\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −40.0000 −1.31733
$$923$$ −6.00000 −0.197492
$$924$$ 0 0
$$925$$ −3.00000 −0.0986394
$$926$$ 34.0000 1.11731
$$927$$ 0 0
$$928$$ −32.0000 −1.05045
$$929$$ 14.0000 0.459325 0.229663 0.973270i $$-0.426238\pi$$
0.229663 + 0.973270i $$0.426238\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −12.0000 −0.393073
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −15.0000 −0.490029 −0.245014 0.969519i $$-0.578793\pi$$
−0.245014 + 0.969519i $$0.578793\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 24.0000 0.782794
$$941$$ −4.00000 −0.130396 −0.0651981 0.997872i $$-0.520768\pi$$
−0.0651981 + 0.997872i $$0.520768\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 48.0000 1.56227
$$945$$ 0 0
$$946$$ −20.0000 −0.650256
$$947$$ 10.0000 0.324956 0.162478 0.986712i $$-0.448051\pi$$
0.162478 + 0.986712i $$0.448051\pi$$
$$948$$ 0 0
$$949$$ −3.00000 −0.0973841
$$950$$ −2.00000 −0.0648886
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −44.0000 −1.42530 −0.712650 0.701520i $$-0.752505\pi$$
−0.712650 + 0.701520i $$0.752505\pi$$
$$954$$ 0 0
$$955$$ 20.0000 0.647185
$$956$$ −12.0000 −0.388108
$$957$$ 0 0
$$958$$ 56.0000 1.80928
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 50.0000 1.61290
$$962$$ 6.00000 0.193448
$$963$$ 0 0
$$964$$ −28.0000 −0.901819
$$965$$ −22.0000 −0.708205
$$966$$ 0 0
$$967$$ 19.0000 0.610999 0.305499 0.952192i $$-0.401177\pi$$
0.305499 + 0.952192i $$0.401177\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 24.0000 0.770594
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −62.0000 −1.98661
$$975$$ 0 0
$$976$$ 40.0000 1.28037
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 0 0
$$979$$ 32.0000 1.02272
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −56.0000 −1.78703
$$983$$ 36.0000 1.14822 0.574111 0.818778i $$-0.305348\pi$$
0.574111 + 0.818778i $$0.305348\pi$$
$$984$$ 0 0
$$985$$ 32.0000 1.01960
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 2.00000 0.0636285
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 17.0000 0.540023 0.270011 0.962857i $$-0.412973\pi$$
0.270011 + 0.962857i $$0.412973\pi$$
$$992$$ −72.0000 −2.28600
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −19.0000 −0.601736 −0.300868 0.953666i $$-0.597276\pi$$
−0.300868 + 0.953666i $$0.597276\pi$$
$$998$$ −74.0000 −2.34243
$$999$$ 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 441.2.a.a.1.1 1
3.2 odd 2 147.2.a.b.1.1 1
4.3 odd 2 7056.2.a.m.1.1 1
7.2 even 3 441.2.e.e.361.1 2
7.3 odd 6 63.2.e.b.37.1 2
7.4 even 3 441.2.e.e.226.1 2
7.5 odd 6 63.2.e.b.46.1 2
7.6 odd 2 441.2.a.b.1.1 1
12.11 even 2 2352.2.a.w.1.1 1
15.14 odd 2 3675.2.a.c.1.1 1
21.2 odd 6 147.2.e.a.67.1 2
21.5 even 6 21.2.e.a.4.1 2
21.11 odd 6 147.2.e.a.79.1 2
21.17 even 6 21.2.e.a.16.1 yes 2
21.20 even 2 147.2.a.c.1.1 1
24.5 odd 2 9408.2.a.bz.1.1 1
24.11 even 2 9408.2.a.k.1.1 1
28.3 even 6 1008.2.s.d.289.1 2
28.19 even 6 1008.2.s.d.865.1 2
28.27 even 2 7056.2.a.bp.1.1 1
63.5 even 6 567.2.h.f.298.1 2
63.31 odd 6 567.2.g.f.541.1 2
63.38 even 6 567.2.h.f.352.1 2
63.40 odd 6 567.2.h.a.298.1 2
63.47 even 6 567.2.g.a.109.1 2
63.52 odd 6 567.2.h.a.352.1 2
63.59 even 6 567.2.g.a.541.1 2
63.61 odd 6 567.2.g.f.109.1 2
84.11 even 6 2352.2.q.c.961.1 2
84.23 even 6 2352.2.q.c.1537.1 2
84.47 odd 6 336.2.q.f.193.1 2
84.59 odd 6 336.2.q.f.289.1 2
84.83 odd 2 2352.2.a.d.1.1 1
105.17 odd 12 525.2.r.e.499.1 4
105.38 odd 12 525.2.r.e.499.2 4
105.47 odd 12 525.2.r.e.424.2 4
105.59 even 6 525.2.i.e.226.1 2
105.68 odd 12 525.2.r.e.424.1 4
105.89 even 6 525.2.i.e.151.1 2
105.104 even 2 3675.2.a.a.1.1 1
168.5 even 6 1344.2.q.m.193.1 2
168.59 odd 6 1344.2.q.c.961.1 2
168.83 odd 2 9408.2.a.cv.1.1 1
168.101 even 6 1344.2.q.m.961.1 2
168.125 even 2 9408.2.a.bg.1.1 1
168.131 odd 6 1344.2.q.c.193.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.e.a.4.1 2 21.5 even 6
21.2.e.a.16.1 yes 2 21.17 even 6
63.2.e.b.37.1 2 7.3 odd 6
63.2.e.b.46.1 2 7.5 odd 6
147.2.a.b.1.1 1 3.2 odd 2
147.2.a.c.1.1 1 21.20 even 2
147.2.e.a.67.1 2 21.2 odd 6
147.2.e.a.79.1 2 21.11 odd 6
336.2.q.f.193.1 2 84.47 odd 6
336.2.q.f.289.1 2 84.59 odd 6
441.2.a.a.1.1 1 1.1 even 1 trivial
441.2.a.b.1.1 1 7.6 odd 2
441.2.e.e.226.1 2 7.4 even 3
441.2.e.e.361.1 2 7.2 even 3
525.2.i.e.151.1 2 105.89 even 6
525.2.i.e.226.1 2 105.59 even 6
525.2.r.e.424.1 4 105.68 odd 12
525.2.r.e.424.2 4 105.47 odd 12
525.2.r.e.499.1 4 105.17 odd 12
525.2.r.e.499.2 4 105.38 odd 12
567.2.g.a.109.1 2 63.47 even 6
567.2.g.a.541.1 2 63.59 even 6
567.2.g.f.109.1 2 63.61 odd 6
567.2.g.f.541.1 2 63.31 odd 6
567.2.h.a.298.1 2 63.40 odd 6
567.2.h.a.352.1 2 63.52 odd 6
567.2.h.f.298.1 2 63.5 even 6
567.2.h.f.352.1 2 63.38 even 6
1008.2.s.d.289.1 2 28.3 even 6
1008.2.s.d.865.1 2 28.19 even 6
1344.2.q.c.193.1 2 168.131 odd 6
1344.2.q.c.961.1 2 168.59 odd 6
1344.2.q.m.193.1 2 168.5 even 6
1344.2.q.m.961.1 2 168.101 even 6
2352.2.a.d.1.1 1 84.83 odd 2
2352.2.a.w.1.1 1 12.11 even 2
2352.2.q.c.961.1 2 84.11 even 6
2352.2.q.c.1537.1 2 84.23 even 6
3675.2.a.a.1.1 1 105.104 even 2
3675.2.a.c.1.1 1 15.14 odd 2
7056.2.a.m.1.1 1 4.3 odd 2
7056.2.a.bp.1.1 1 28.27 even 2
9408.2.a.k.1.1 1 24.11 even 2
9408.2.a.bg.1.1 1 168.125 even 2
9408.2.a.bz.1.1 1 24.5 odd 2
9408.2.a.cv.1.1 1 168.83 odd 2