# Properties

 Label 405.2.a.c.1.1 Level $405$ Weight $2$ Character 405.1 Self dual yes Analytic conductor $3.234$ 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] = [405,2,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.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: $$3.23394128186$$ 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 $$\chi$$ $$=$$ 405.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +O(q^{10})$$ $$q-2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} -3.00000 q^{11} -4.00000 q^{13} +4.00000 q^{16} -6.00000 q^{17} -1.00000 q^{19} +2.00000 q^{20} -6.00000 q^{23} +1.00000 q^{25} -4.00000 q^{28} -9.00000 q^{29} -1.00000 q^{31} -2.00000 q^{35} +8.00000 q^{37} +3.00000 q^{41} -4.00000 q^{43} +6.00000 q^{44} +12.0000 q^{47} -3.00000 q^{49} +8.00000 q^{52} +6.00000 q^{53} +3.00000 q^{55} +3.00000 q^{59} -10.0000 q^{61} -8.00000 q^{64} +4.00000 q^{65} +14.0000 q^{67} +12.0000 q^{68} -3.00000 q^{71} +2.00000 q^{73} +2.00000 q^{76} -6.00000 q^{77} -16.0000 q^{79} -4.00000 q^{80} -12.0000 q^{83} +6.00000 q^{85} +15.0000 q^{89} -8.00000 q^{91} +12.0000 q^{92} +1.00000 q^{95} -4.00000 q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 0 0
$$4$$ −2.00000 −1.00000
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 4.00000 1.00000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −4.00000 −0.755929
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ −1.00000 −0.179605 −0.0898027 0.995960i $$-0.528624\pi$$
−0.0898027 + 0.995960i $$0.528624\pi$$
$$32$$ 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 $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 8.00000 1.10940
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ 14.0000 1.71037 0.855186 0.518321i $$-0.173443\pi$$
0.855186 + 0.518321i $$0.173443\pi$$
$$68$$ 12.0000 1.45521
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −3.00000 −0.356034 −0.178017 0.984027i $$-0.556968\pi$$
−0.178017 + 0.984027i $$0.556968\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 6.00000 0.650791
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 15.0000 1.59000 0.794998 0.606612i $$-0.207472\pi$$
0.794998 + 0.606612i $$0.207472\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ 12.0000 1.25109
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ −4.00000 −0.406138 −0.203069 0.979164i $$-0.565092\pi$$
−0.203069 + 0.979164i $$0.565092\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −2.00000 −0.200000
$$101$$ 9.00000 0.895533 0.447767 0.894150i $$-0.352219\pi$$
0.447767 + 0.894150i $$0.352219\pi$$
$$102$$ 0 0
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ 0 0
$$109$$ −1.00000 −0.0957826 −0.0478913 0.998853i $$-0.515250\pi$$
−0.0478913 + 0.998853i $$0.515250\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 8.00000 0.755929
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 6.00000 0.559503
$$116$$ 18.0000 1.67126
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 2.00000 0.179605
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −9.00000 −0.786334 −0.393167 0.919467i $$-0.628621\pi$$
−0.393167 + 0.919467i $$0.628621\pi$$
$$132$$ 0 0
$$133$$ −2.00000 −0.173422
$$134$$ 0 0
$$135$$ 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
$$139$$ −7.00000 −0.593732 −0.296866 0.954919i $$-0.595942\pi$$
−0.296866 + 0.954919i $$0.595942\pi$$
$$140$$ 4.00000 0.338062
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 12.0000 1.00349
$$144$$ 0 0
$$145$$ 9.00000 0.747409
$$146$$ 0 0
$$147$$ 0 0
$$148$$ −16.0000 −1.31519
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −1.00000 −0.0813788 −0.0406894 0.999172i $$-0.512955\pi$$
−0.0406894 + 0.999172i $$0.512955\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 1.00000 0.0803219
$$156$$ 0 0
$$157$$ −16.0000 −1.27694 −0.638470 0.769647i $$-0.720432\pi$$
−0.638470 + 0.769647i $$0.720432\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ 0 0
$$163$$ 2.00000 0.156652 0.0783260 0.996928i $$-0.475042\pi$$
0.0783260 + 0.996928i $$0.475042\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.00000 0.464294 0.232147 0.972681i $$-0.425425\pi$$
0.232147 + 0.972681i $$0.425425\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 8.00000 0.609994
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 2.00000 0.151186
$$176$$ −12.0000 −0.904534
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −9.00000 −0.672692 −0.336346 0.941739i $$-0.609191\pi$$
−0.336346 + 0.941739i $$0.609191\pi$$
$$180$$ 0 0
$$181$$ 11.0000 0.817624 0.408812 0.912619i $$-0.365943\pi$$
0.408812 + 0.912619i $$0.365943\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −8.00000 −0.588172
$$186$$ 0 0
$$187$$ 18.0000 1.31629
$$188$$ −24.0000 −1.75038
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −15.0000 −1.08536 −0.542681 0.839939i $$-0.682591\pi$$
−0.542681 + 0.839939i $$0.682591\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 6.00000 0.428571
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −18.0000 −1.26335
$$204$$ 0 0
$$205$$ −3.00000 −0.209529
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −16.0000 −1.10940
$$209$$ 3.00000 0.207514
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ −12.0000 −0.824163
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ −2.00000 −0.135769
$$218$$ 0 0
$$219$$ 0 0
$$220$$ −6.00000 −0.404520
$$221$$ 24.0000 1.61441
$$222$$ 0 0
$$223$$ 20.0000 1.33930 0.669650 0.742677i $$-0.266444\pi$$
0.669650 + 0.742677i $$0.266444\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 12.0000 0.786146 0.393073 0.919507i $$-0.371412\pi$$
0.393073 + 0.919507i $$0.371412\pi$$
$$234$$ 0 0
$$235$$ −12.0000 −0.782794
$$236$$ −6.00000 −0.390567
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 17.0000 1.09507 0.547533 0.836784i $$-0.315567\pi$$
0.547533 + 0.836784i $$0.315567\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 20.0000 1.28037
$$245$$ 3.00000 0.191663
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 18.0000 1.13165
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ 16.0000 0.994192
$$260$$ −8.00000 −0.496139
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 18.0000 1.10993 0.554964 0.831875i $$-0.312732\pi$$
0.554964 + 0.831875i $$0.312732\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −28.0000 −1.71037
$$269$$ −3.00000 −0.182913 −0.0914566 0.995809i $$-0.529152\pi$$
−0.0914566 + 0.995809i $$0.529152\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ −24.0000 −1.45521
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −3.00000 −0.180907
$$276$$ 0 0
$$277$$ −16.0000 −0.961347 −0.480673 0.876900i $$-0.659608\pi$$
−0.480673 + 0.876900i $$0.659608\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −10.0000 −0.594438 −0.297219 0.954809i $$-0.596059\pi$$
−0.297219 + 0.954809i $$0.596059\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −4.00000 −0.234082
$$293$$ −12.0000 −0.701047 −0.350524 0.936554i $$-0.613996\pi$$
−0.350524 + 0.936554i $$0.613996\pi$$
$$294$$ 0 0
$$295$$ −3.00000 −0.174667
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 10.0000 0.572598
$$306$$ 0 0
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 12.0000 0.683763
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 27.0000 1.53103 0.765515 0.643418i $$-0.222484\pi$$
0.765515 + 0.643418i $$0.222484\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 32.0000 1.80014
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ 27.0000 1.51171
$$320$$ 8.00000 0.447214
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 6.00000 0.333849
$$324$$ 0 0
$$325$$ −4.00000 −0.221880
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ −25.0000 −1.37412 −0.687062 0.726599i $$-0.741100\pi$$
−0.687062 + 0.726599i $$0.741100\pi$$
$$332$$ 24.0000 1.31717
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −14.0000 −0.764902
$$336$$ 0 0
$$337$$ −4.00000 −0.217894 −0.108947 0.994048i $$-0.534748\pi$$
−0.108947 + 0.994048i $$0.534748\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −12.0000 −0.650791
$$341$$ 3.00000 0.162459
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 18.0000 0.966291 0.483145 0.875540i $$-0.339494\pi$$
0.483145 + 0.875540i $$0.339494\pi$$
$$348$$ 0 0
$$349$$ 5.00000 0.267644 0.133822 0.991005i $$-0.457275\pi$$
0.133822 + 0.991005i $$0.457275\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 12.0000 0.638696 0.319348 0.947638i $$-0.396536\pi$$
0.319348 + 0.947638i $$0.396536\pi$$
$$354$$ 0 0
$$355$$ 3.00000 0.159223
$$356$$ −30.0000 −1.59000
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −15.0000 −0.791670 −0.395835 0.918322i $$-0.629545\pi$$
−0.395835 + 0.918322i $$0.629545\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 16.0000 0.838628
$$365$$ −2.00000 −0.104685
$$366$$ 0 0
$$367$$ −10.0000 −0.521996 −0.260998 0.965339i $$-0.584052\pi$$
−0.260998 + 0.965339i $$0.584052\pi$$
$$368$$ −24.0000 −1.25109
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 0 0
$$373$$ 38.0000 1.96757 0.983783 0.179364i $$-0.0574041\pi$$
0.983783 + 0.179364i $$0.0574041\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 36.0000 1.85409
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 6.00000 0.305788
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 8.00000 0.406138
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 36.0000 1.82060
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 16.0000 0.805047
$$396$$ 0 0
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ 4.00000 0.199254
$$404$$ −18.0000 −0.895533
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −28.0000 −1.37946
$$413$$ 6.00000 0.295241
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 5.00000 0.243685 0.121843 0.992549i $$-0.461120\pi$$
0.121843 + 0.992549i $$0.461120\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ −20.0000 −0.967868
$$428$$ 36.0000 1.74013
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 15.0000 0.722525 0.361262 0.932464i $$-0.382346\pi$$
0.361262 + 0.932464i $$0.382346\pi$$
$$432$$ 0 0
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 6.00000 0.287019
$$438$$ 0 0
$$439$$ −1.00000 −0.0477274 −0.0238637 0.999715i $$-0.507597\pi$$
−0.0238637 + 0.999715i $$0.507597\pi$$
$$440$$ 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
$$445$$ −15.0000 −0.711068
$$446$$ 0 0
$$447$$ 0 0
$$448$$ −16.0000 −0.755929
$$449$$ −15.0000 −0.707894 −0.353947 0.935266i $$-0.615161\pi$$
−0.353947 + 0.935266i $$0.615161\pi$$
$$450$$ 0 0
$$451$$ −9.00000 −0.423793
$$452$$ −12.0000 −0.564433
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 8.00000 0.375046
$$456$$ 0 0
$$457$$ −28.0000 −1.30978 −0.654892 0.755722i $$-0.727286\pi$$
−0.654892 + 0.755722i $$0.727286\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ −12.0000 −0.559503
$$461$$ 21.0000 0.978068 0.489034 0.872265i $$-0.337349\pi$$
0.489034 + 0.872265i $$0.337349\pi$$
$$462$$ 0 0
$$463$$ 26.0000 1.20832 0.604161 0.796862i $$-0.293508\pi$$
0.604161 + 0.796862i $$0.293508\pi$$
$$464$$ −36.0000 −1.67126
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −6.00000 −0.277647 −0.138823 0.990317i $$-0.544332\pi$$
−0.138823 + 0.990317i $$0.544332\pi$$
$$468$$ 0 0
$$469$$ 28.0000 1.29292
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 12.0000 0.551761
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 24.0000 1.10004
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 33.0000 1.50781 0.753904 0.656984i $$-0.228168\pi$$
0.753904 + 0.656984i $$0.228168\pi$$
$$480$$ 0 0
$$481$$ −32.0000 −1.45907
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 4.00000 0.181818
$$485$$ 4.00000 0.181631
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −9.00000 −0.406164 −0.203082 0.979162i $$-0.565096\pi$$
−0.203082 + 0.979162i $$0.565096\pi$$
$$492$$ 0 0
$$493$$ 54.0000 2.43204
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ −6.00000 −0.269137
$$498$$ 0 0
$$499$$ −13.0000 −0.581960 −0.290980 0.956729i $$-0.593981\pi$$
−0.290980 + 0.956729i $$0.593981\pi$$
$$500$$ 2.00000 0.0894427
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −12.0000 −0.535054 −0.267527 0.963550i $$-0.586206\pi$$
−0.267527 + 0.963550i $$0.586206\pi$$
$$504$$ 0 0
$$505$$ −9.00000 −0.400495
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −4.00000 −0.177471
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −14.0000 −0.616914
$$516$$ 0 0
$$517$$ −36.0000 −1.58328
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 18.0000 0.786334
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6.00000 0.261364
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 4.00000 0.173422
$$533$$ −12.0000 −0.519778
$$534$$ 0 0
$$535$$ 18.0000 0.778208
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 9.00000 0.387657
$$540$$ 0 0
$$541$$ 23.0000 0.988847 0.494424 0.869221i $$-0.335379\pi$$
0.494424 + 0.869221i $$0.335379\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 1.00000 0.0428353
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ −24.0000 −1.02523
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 9.00000 0.383413
$$552$$ 0 0
$$553$$ −32.0000 −1.36078
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 14.0000 0.593732
$$557$$ 24.0000 1.01691 0.508456 0.861088i $$-0.330216\pi$$
0.508456 + 0.861088i $$0.330216\pi$$
$$558$$ 0 0
$$559$$ 16.0000 0.676728
$$560$$ −8.00000 −0.338062
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −27.0000 −1.13190 −0.565949 0.824440i $$-0.691490\pi$$
−0.565949 + 0.824440i $$0.691490\pi$$
$$570$$ 0 0
$$571$$ −19.0000 −0.795125 −0.397563 0.917575i $$-0.630144\pi$$
−0.397563 + 0.917575i $$0.630144\pi$$
$$572$$ −24.0000 −1.00349
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −6.00000 −0.250217
$$576$$ 0 0
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ −18.0000 −0.747409
$$581$$ −24.0000 −0.995688
$$582$$ 0 0
$$583$$ −18.0000 −0.745484
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −48.0000 −1.98117 −0.990586 0.136892i $$-0.956289\pi$$
−0.990586 + 0.136892i $$0.956289\pi$$
$$588$$ 0 0
$$589$$ 1.00000 0.0412043
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 32.0000 1.31519
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ 12.0000 0.491952
$$596$$ −12.0000 −0.491539
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −21.0000 −0.858037 −0.429018 0.903296i $$-0.641140\pi$$
−0.429018 + 0.903296i $$0.641140\pi$$
$$600$$ 0 0
$$601$$ −31.0000 −1.26452 −0.632258 0.774758i $$-0.717872\pi$$
−0.632258 + 0.774758i $$0.717872\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 2.00000 0.0813788
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ 20.0000 0.811775 0.405887 0.913923i $$-0.366962\pi$$
0.405887 + 0.913923i $$0.366962\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −48.0000 −1.94187
$$612$$ 0 0
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −24.0000 −0.966204 −0.483102 0.875564i $$-0.660490\pi$$
−0.483102 + 0.875564i $$0.660490\pi$$
$$618$$ 0 0
$$619$$ 44.0000 1.76851 0.884255 0.467005i $$-0.154667\pi$$
0.884255 + 0.467005i $$0.154667\pi$$
$$620$$ −2.00000 −0.0803219
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 30.0000 1.20192
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 32.0000 1.27694
$$629$$ −48.0000 −1.91389
$$630$$ 0 0
$$631$$ −25.0000 −0.995234 −0.497617 0.867397i $$-0.665792\pi$$
−0.497617 + 0.867397i $$0.665792\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −2.00000 −0.0793676
$$636$$ 0 0
$$637$$ 12.0000 0.475457
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −33.0000 −1.30342 −0.651711 0.758468i $$-0.725948\pi$$
−0.651711 + 0.758468i $$0.725948\pi$$
$$642$$ 0 0
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 24.0000 0.945732
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ 0 0
$$649$$ −9.00000 −0.353281
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −24.0000 −0.939193 −0.469596 0.882881i $$-0.655601\pi$$
−0.469596 + 0.882881i $$0.655601\pi$$
$$654$$ 0 0
$$655$$ 9.00000 0.351659
$$656$$ 12.0000 0.468521
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ 5.00000 0.194477 0.0972387 0.995261i $$-0.468999\pi$$
0.0972387 + 0.995261i $$0.468999\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2.00000 0.0775567
$$666$$ 0 0
$$667$$ 54.0000 2.09089
$$668$$ −12.0000 −0.464294
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 30.0000 1.15814
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −6.00000 −0.230769
$$677$$ 30.0000 1.15299 0.576497 0.817099i $$-0.304419\pi$$
0.576497 + 0.817099i $$0.304419\pi$$
$$678$$ 0 0
$$679$$ −8.00000 −0.307012
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ −12.0000 −0.458496
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −16.0000 −0.609994
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ 44.0000 1.67384 0.836919 0.547326i $$-0.184354\pi$$
0.836919 + 0.547326i $$0.184354\pi$$
$$692$$ 12.0000 0.456172
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 7.00000 0.265525
$$696$$ 0 0
$$697$$ −18.0000 −0.681799
$$698$$ 0 0
$$699$$ 0 0
$$700$$ −4.00000 −0.151186
$$701$$ 15.0000 0.566542 0.283271 0.959040i $$-0.408580\pi$$
0.283271 + 0.959040i $$0.408580\pi$$
$$702$$ 0 0
$$703$$ −8.00000 −0.301726
$$704$$ 24.0000 0.904534
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 18.0000 0.676960
$$708$$ 0 0
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 6.00000 0.224702
$$714$$ 0 0
$$715$$ −12.0000 −0.448775
$$716$$ 18.0000 0.672692
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 27.0000 1.00693 0.503465 0.864016i $$-0.332058\pi$$
0.503465 + 0.864016i $$0.332058\pi$$
$$720$$ 0 0
$$721$$ 28.0000 1.04277
$$722$$ 0 0
$$723$$ 0 0
$$724$$ −22.0000 −0.817624
$$725$$ −9.00000 −0.334252
$$726$$ 0 0
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −42.0000 −1.54709
$$738$$ 0 0
$$739$$ −7.00000 −0.257499 −0.128750 0.991677i $$-0.541096\pi$$
−0.128750 + 0.991677i $$0.541096\pi$$
$$740$$ 16.0000 0.588172
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −18.0000 −0.660356 −0.330178 0.943919i $$-0.607109\pi$$
−0.330178 + 0.943919i $$0.607109\pi$$
$$744$$ 0 0
$$745$$ −6.00000 −0.219823
$$746$$ 0 0
$$747$$ 0 0
$$748$$ −36.0000 −1.31629
$$749$$ −36.0000 −1.31541
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ 48.0000 1.75038
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 1.00000 0.0363937
$$756$$ 0 0
$$757$$ 20.0000 0.726912 0.363456 0.931611i $$-0.381597\pi$$
0.363456 + 0.931611i $$0.381597\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 33.0000 1.19625 0.598125 0.801403i $$-0.295913\pi$$
0.598125 + 0.801403i $$0.295913\pi$$
$$762$$ 0 0
$$763$$ −2.00000 −0.0724049
$$764$$ 30.0000 1.08536
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −12.0000 −0.433295
$$768$$ 0 0
$$769$$ 41.0000 1.47850 0.739249 0.673432i $$-0.235181\pi$$
0.739249 + 0.673432i $$0.235181\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 8.00000 0.287926
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ 0 0
$$775$$ −1.00000 −0.0359211
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −3.00000 −0.107486
$$780$$ 0 0
$$781$$ 9.00000 0.322045
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −12.0000 −0.428571
$$785$$ 16.0000 0.571064
$$786$$ 0 0
$$787$$ 14.0000 0.499046 0.249523 0.968369i $$-0.419726\pi$$
0.249523 + 0.968369i $$0.419726\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ 0 0
$$793$$ 40.0000 1.42044
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 32.0000 1.13421
$$797$$ −24.0000 −0.850124 −0.425062 0.905164i $$-0.639748\pi$$
−0.425062 + 0.905164i $$0.639748\pi$$
$$798$$ 0 0
$$799$$ −72.0000 −2.54718
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −6.00000 −0.211735
$$804$$ 0 0
$$805$$ 12.0000 0.422944
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 15.0000 0.527372 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$810$$ 0 0
$$811$$ 23.0000 0.807639 0.403820 0.914839i $$-0.367682\pi$$
0.403820 + 0.914839i $$0.367682\pi$$
$$812$$ 36.0000 1.26335
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −2.00000 −0.0700569
$$816$$ 0 0
$$817$$ 4.00000 0.139942
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 6.00000 0.209529
$$821$$ −45.0000 −1.57051 −0.785255 0.619172i $$-0.787468\pi$$
−0.785255 + 0.619172i $$0.787468\pi$$
$$822$$ 0 0
$$823$$ −22.0000 −0.766872 −0.383436 0.923567i $$-0.625259\pi$$
−0.383436 + 0.923567i $$0.625259\pi$$
$$824$$ 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
$$829$$ −13.0000 −0.451509 −0.225754 0.974184i $$-0.572485\pi$$
−0.225754 + 0.974184i $$0.572485\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 32.0000 1.10940
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ −6.00000 −0.207639
$$836$$ −6.00000 −0.207514
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 45.0000 1.55357 0.776786 0.629764i $$-0.216849\pi$$
0.776786 + 0.629764i $$0.216849\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 26.0000 0.894957
$$845$$ −3.00000 −0.103203
$$846$$ 0 0
$$847$$ −4.00000 −0.137442
$$848$$ 24.0000 0.824163
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −48.0000 −1.64542
$$852$$ 0 0
$$853$$ −46.0000 −1.57501 −0.787505 0.616308i $$-0.788628\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 48.0000 1.63965 0.819824 0.572615i $$-0.194071\pi$$
0.819824 + 0.572615i $$0.194071\pi$$
$$858$$ 0 0
$$859$$ −25.0000 −0.852989 −0.426494 0.904490i $$-0.640252\pi$$
−0.426494 + 0.904490i $$0.640252\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 30.0000 1.02121 0.510606 0.859815i $$-0.329421\pi$$
0.510606 + 0.859815i $$0.329421\pi$$
$$864$$ 0 0
$$865$$ 6.00000 0.204006
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 4.00000 0.135769
$$869$$ 48.0000 1.62829
$$870$$ 0 0
$$871$$ −56.0000 −1.89749
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −2.00000 −0.0676123
$$876$$ 0 0
$$877$$ 44.0000 1.48577 0.742887 0.669417i $$-0.233456\pi$$
0.742887 + 0.669417i $$0.233456\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 12.0000 0.404520
$$881$$ 39.0000 1.31394 0.656972 0.753915i $$-0.271837\pi$$
0.656972 + 0.753915i $$0.271837\pi$$
$$882$$ 0 0
$$883$$ −46.0000 −1.54802 −0.774012 0.633171i $$-0.781753\pi$$
−0.774012 + 0.633171i $$0.781753\pi$$
$$884$$ −48.0000 −1.61441
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ 0 0
$$889$$ 4.00000 0.134156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −40.0000 −1.33930
$$893$$ −12.0000 −0.401565
$$894$$ 0 0
$$895$$ 9.00000 0.300837
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 9.00000 0.300167
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −11.0000 −0.365652
$$906$$ 0 0
$$907$$ 38.0000 1.26177 0.630885 0.775877i $$-0.282692\pi$$
0.630885 + 0.775877i $$0.282692\pi$$
$$908$$ 36.0000 1.19470
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −9.00000 −0.298183 −0.149092 0.988823i $$-0.547635\pi$$
−0.149092 + 0.988823i $$0.547635\pi$$
$$912$$ 0 0
$$913$$ 36.0000 1.19143
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 20.0000 0.660819
$$917$$ −18.0000 −0.594412
$$918$$ 0 0
$$919$$ −7.00000 −0.230909 −0.115454 0.993313i $$-0.536832\pi$$
−0.115454 + 0.993313i $$0.536832\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 12.0000 0.394985
$$924$$ 0 0
$$925$$ 8.00000 0.263038
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −39.0000 −1.27955 −0.639774 0.768563i $$-0.720972\pi$$
−0.639774 + 0.768563i $$0.720972\pi$$
$$930$$ 0 0
$$931$$ 3.00000 0.0983210
$$932$$ −24.0000 −0.786146
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −18.0000 −0.588663
$$936$$ 0 0
$$937$$ 14.0000 0.457360 0.228680 0.973502i $$-0.426559\pi$$
0.228680 + 0.973502i $$0.426559\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 24.0000 0.782794
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ 0 0
$$943$$ −18.0000 −0.586161
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 30.0000 0.974869 0.487435 0.873160i $$-0.337933\pi$$
0.487435 + 0.873160i $$0.337933\pi$$
$$948$$ 0 0
$$949$$ −8.00000 −0.259691
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −48.0000 −1.55487 −0.777436 0.628962i $$-0.783480\pi$$
−0.777436 + 0.628962i $$0.783480\pi$$
$$954$$ 0 0
$$955$$ 15.0000 0.485389
$$956$$ 48.0000 1.55243
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 24.0000 0.775000
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −34.0000 −1.09507
$$965$$ 4.00000 0.128765
$$966$$ 0 0
$$967$$ 50.0000 1.60789 0.803946 0.594703i $$-0.202730\pi$$
0.803946 + 0.594703i $$0.202730\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −3.00000 −0.0962746 −0.0481373 0.998841i $$-0.515328\pi$$
−0.0481373 + 0.998841i $$0.515328\pi$$
$$972$$ 0 0
$$973$$ −14.0000 −0.448819
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −40.0000 −1.28037
$$977$$ −48.0000 −1.53566 −0.767828 0.640656i $$-0.778662\pi$$
−0.767828 + 0.640656i $$0.778662\pi$$
$$978$$ 0 0
$$979$$ −45.0000 −1.43821
$$980$$ −6.00000 −0.191663
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −6.00000 −0.191370 −0.0956851 0.995412i $$-0.530504\pi$$
−0.0956851 + 0.995412i $$0.530504\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −8.00000 −0.254514
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 47.0000 1.49300 0.746502 0.665383i $$-0.231732\pi$$
0.746502 + 0.665383i $$0.231732\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 16.0000 0.507234
$$996$$ 0 0
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ 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 405.2.a.c.1.1 1
3.2 odd 2 405.2.a.d.1.1 yes 1
4.3 odd 2 6480.2.a.c.1.1 1
5.2 odd 4 2025.2.b.e.649.2 2
5.3 odd 4 2025.2.b.e.649.1 2
5.4 even 2 2025.2.a.c.1.1 1
9.2 odd 6 405.2.e.d.271.1 2
9.4 even 3 405.2.e.e.136.1 2
9.5 odd 6 405.2.e.d.136.1 2
9.7 even 3 405.2.e.e.271.1 2
12.11 even 2 6480.2.a.o.1.1 1
15.2 even 4 2025.2.b.f.649.2 2
15.8 even 4 2025.2.b.f.649.1 2
15.14 odd 2 2025.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.c.1.1 1 1.1 even 1 trivial
405.2.a.d.1.1 yes 1 3.2 odd 2
405.2.e.d.136.1 2 9.5 odd 6
405.2.e.d.271.1 2 9.2 odd 6
405.2.e.e.136.1 2 9.4 even 3
405.2.e.e.271.1 2 9.7 even 3
2025.2.a.c.1.1 1 5.4 even 2
2025.2.a.d.1.1 1 15.14 odd 2
2025.2.b.e.649.1 2 5.3 odd 4
2025.2.b.e.649.2 2 5.2 odd 4
2025.2.b.f.649.1 2 15.8 even 4
2025.2.b.f.649.2 2 15.2 even 4
6480.2.a.c.1.1 1 4.3 odd 2
6480.2.a.o.1.1 1 12.11 even 2