# Properties

 Label 1305.2.a.c.1.1 Level $1305$ Weight $2$ Character 1305.1 Self dual yes Analytic conductor $10.420$ 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] = [1305,2,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1305.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.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: $$10.4204774638$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1305.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +O(q^{10})$$ $$q-2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{11} +6.00000 q^{13} +4.00000 q^{16} -4.00000 q^{17} -2.00000 q^{19} -2.00000 q^{20} -3.00000 q^{23} +1.00000 q^{25} +4.00000 q^{28} -1.00000 q^{29} -4.00000 q^{31} -2.00000 q^{35} -3.00000 q^{37} -7.00000 q^{41} +5.00000 q^{43} +2.00000 q^{44} -6.00000 q^{47} -3.00000 q^{49} -12.0000 q^{52} -13.0000 q^{53} -1.00000 q^{55} -8.00000 q^{64} +6.00000 q^{65} -10.0000 q^{67} +8.00000 q^{68} -6.00000 q^{71} +3.00000 q^{73} +4.00000 q^{76} +2.00000 q^{77} +4.00000 q^{80} -9.00000 q^{83} -4.00000 q^{85} +10.0000 q^{89} -12.0000 q^{91} +6.00000 q^{92} -2.00000 q^{95} +17.0000 q^{97} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 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 $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ 0 0
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 4.00000 1.00000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 4.00000 0.755929
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.00000 −0.338062
$$36$$ 0 0
$$37$$ −3.00000 −0.493197 −0.246598 0.969118i $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −7.00000 −1.09322 −0.546608 0.837389i $$-0.684081\pi$$
−0.546608 + 0.837389i $$0.684081\pi$$
$$42$$ 0 0
$$43$$ 5.00000 0.762493 0.381246 0.924473i $$-0.375495\pi$$
0.381246 + 0.924473i $$0.375495\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 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
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −12.0000 −1.66410
$$53$$ −13.0000 −1.78569 −0.892844 0.450367i $$-0.851293\pi$$
−0.892844 + 0.450367i $$0.851293\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 6.00000 0.744208
$$66$$ 0 0
$$67$$ −10.0000 −1.22169 −0.610847 0.791748i $$-0.709171\pi$$
−0.610847 + 0.791748i $$0.709171\pi$$
$$68$$ 8.00000 0.970143
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ 3.00000 0.351123 0.175562 0.984468i $$-0.443826\pi$$
0.175562 + 0.984468i $$0.443826\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 2.00000 0.227921
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 4.00000 0.447214
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −9.00000 −0.987878 −0.493939 0.869496i $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ 0 0
$$85$$ −4.00000 −0.433861
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −12.0000 −1.25794
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −2.00000 −0.205196
$$96$$ 0 0
$$97$$ 17.0000 1.72609 0.863044 0.505128i $$-0.168555\pi$$
0.863044 + 0.505128i $$0.168555\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −2.00000 −0.200000
$$101$$ 3.00000 0.298511 0.149256 0.988799i $$-0.452312\pi$$
0.149256 + 0.988799i $$0.452312\pi$$
$$102$$ 0 0
$$103$$ −6.00000 −0.591198 −0.295599 0.955312i $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −8.00000 −0.755929
$$113$$ 12.0000 1.12887 0.564433 0.825479i $$-0.309095\pi$$
0.564433 + 0.825479i $$0.309095\pi$$
$$114$$ 0 0
$$115$$ −3.00000 −0.279751
$$116$$ 2.00000 0.185695
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 8.00000 0.718421
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 7.00000 0.621150 0.310575 0.950549i $$-0.399478\pi$$
0.310575 + 0.950549i $$0.399478\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 4.00000 0.346844
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −16.0000 −1.36697 −0.683486 0.729964i $$-0.739537\pi$$
−0.683486 + 0.729964i $$0.739537\pi$$
$$138$$ 0 0
$$139$$ −19.0000 −1.61156 −0.805779 0.592216i $$-0.798253\pi$$
−0.805779 + 0.592216i $$0.798253\pi$$
$$140$$ 4.00000 0.338062
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −6.00000 −0.501745
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 6.00000 0.493197
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 0 0
$$151$$ −11.0000 −0.895167 −0.447584 0.894242i $$-0.647715\pi$$
−0.447584 + 0.894242i $$0.647715\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 6.00000 0.472866
$$162$$ 0 0
$$163$$ 15.0000 1.17489 0.587445 0.809264i $$-0.300134\pi$$
0.587445 + 0.809264i $$0.300134\pi$$
$$164$$ 14.0000 1.09322
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −10.0000 −0.762493
$$173$$ −5.00000 −0.380143 −0.190071 0.981770i $$-0.560872\pi$$
−0.190071 + 0.981770i $$0.560872\pi$$
$$174$$ 0 0
$$175$$ −2.00000 −0.151186
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 18.0000 1.34538 0.672692 0.739923i $$-0.265138\pi$$
0.672692 + 0.739923i $$0.265138\pi$$
$$180$$ 0 0
$$181$$ 25.0000 1.85824 0.929118 0.369784i $$-0.120568\pi$$
0.929118 + 0.369784i $$0.120568\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −3.00000 −0.220564
$$186$$ 0 0
$$187$$ 4.00000 0.292509
$$188$$ 12.0000 0.875190
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 27.0000 1.95365 0.976826 0.214036i $$-0.0686611\pi$$
0.976826 + 0.214036i $$0.0686611\pi$$
$$192$$ 0 0
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 6.00000 0.428571
$$197$$ −13.0000 −0.926212 −0.463106 0.886303i $$-0.653265\pi$$
−0.463106 + 0.886303i $$0.653265\pi$$
$$198$$ 0 0
$$199$$ 15.0000 1.06332 0.531661 0.846957i $$-0.321568\pi$$
0.531661 + 0.846957i $$0.321568\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 2.00000 0.140372
$$204$$ 0 0
$$205$$ −7.00000 −0.488901
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 24.0000 1.66410
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ −2.00000 −0.137686 −0.0688428 0.997628i $$-0.521931\pi$$
−0.0688428 + 0.997628i $$0.521931\pi$$
$$212$$ 26.0000 1.78569
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 5.00000 0.340997
$$216$$ 0 0
$$217$$ 8.00000 0.543075
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 2.00000 0.134840
$$221$$ −24.0000 −1.61441
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −11.0000 −0.730096 −0.365048 0.930989i $$-0.618947\pi$$
−0.365048 + 0.930989i $$0.618947\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 1.00000 0.0655122 0.0327561 0.999463i $$-0.489572\pi$$
0.0327561 + 0.999463i $$0.489572\pi$$
$$234$$ 0 0
$$235$$ −6.00000 −0.391397
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −14.0000 −0.905585 −0.452792 0.891616i $$-0.649572\pi$$
−0.452792 + 0.891616i $$0.649572\pi$$
$$240$$ 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
$$244$$ 0 0
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ −12.0000 −0.763542
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ 0 0
$$253$$ 3.00000 0.188608
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 21.0000 1.30994 0.654972 0.755653i $$-0.272680\pi$$
0.654972 + 0.755653i $$0.272680\pi$$
$$258$$ 0 0
$$259$$ 6.00000 0.372822
$$260$$ −12.0000 −0.744208
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 20.0000 1.23325 0.616626 0.787256i $$-0.288499\pi$$
0.616626 + 0.787256i $$0.288499\pi$$
$$264$$ 0 0
$$265$$ −13.0000 −0.798584
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 20.0000 1.22169
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 26.0000 1.57939 0.789694 0.613501i $$-0.210239\pi$$
0.789694 + 0.613501i $$0.210239\pi$$
$$272$$ −16.0000 −0.970143
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ 4.00000 0.240337 0.120168 0.992754i $$-0.461657\pi$$
0.120168 + 0.992754i $$0.461657\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4.00000 −0.238620 −0.119310 0.992857i $$-0.538068\pi$$
−0.119310 + 0.992857i $$0.538068\pi$$
$$282$$ 0 0
$$283$$ −26.0000 −1.54554 −0.772770 0.634686i $$-0.781129\pi$$
−0.772770 + 0.634686i $$0.781129\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 14.0000 0.826394
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 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
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −18.0000 −1.04097
$$300$$ 0 0
$$301$$ −10.0000 −0.576390
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −8.00000 −0.458831
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −21.0000 −1.19853 −0.599267 0.800549i $$-0.704541\pi$$
−0.599267 + 0.800549i $$0.704541\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 23.0000 1.30421 0.652105 0.758129i $$-0.273886\pi$$
0.652105 + 0.758129i $$0.273886\pi$$
$$312$$ 0 0
$$313$$ −12.0000 −0.678280 −0.339140 0.940736i $$-0.610136\pi$$
−0.339140 + 0.940736i $$0.610136\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.00000 0.449325 0.224662 0.974437i $$-0.427872\pi$$
0.224662 + 0.974437i $$0.427872\pi$$
$$318$$ 0 0
$$319$$ 1.00000 0.0559893
$$320$$ −8.00000 −0.447214
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 8.00000 0.445132
$$324$$ 0 0
$$325$$ 6.00000 0.332820
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 18.0000 0.987878
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −10.0000 −0.546358
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 8.00000 0.433861
$$341$$ 4.00000 0.216612
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 5.00000 0.268414 0.134207 0.990953i $$-0.457151\pi$$
0.134207 + 0.990953i $$0.457151\pi$$
$$348$$ 0 0
$$349$$ 31.0000 1.65939 0.829696 0.558216i $$-0.188514\pi$$
0.829696 + 0.558216i $$0.188514\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ −6.00000 −0.318447
$$356$$ −20.0000 −1.06000
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 15.0000 0.791670 0.395835 0.918322i $$-0.370455\pi$$
0.395835 + 0.918322i $$0.370455\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 24.0000 1.25794
$$365$$ 3.00000 0.157027
$$366$$ 0 0
$$367$$ −33.0000 −1.72259 −0.861293 0.508109i $$-0.830345\pi$$
−0.861293 + 0.508109i $$0.830345\pi$$
$$368$$ −12.0000 −0.625543
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 26.0000 1.34985
$$372$$ 0 0
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6.00000 −0.309016
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −27.0000 −1.37964 −0.689818 0.723983i $$-0.742309\pi$$
−0.689818 + 0.723983i $$0.742309\pi$$
$$384$$ 0 0
$$385$$ 2.00000 0.101929
$$386$$ 0 0
$$387$$ 0 0
$$388$$ −34.0000 −1.72609
$$389$$ 21.0000 1.06474 0.532371 0.846511i $$-0.321301\pi$$
0.532371 + 0.846511i $$0.321301\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ 20.0000 0.998752 0.499376 0.866385i $$-0.333563\pi$$
0.499376 + 0.866385i $$0.333563\pi$$
$$402$$ 0 0
$$403$$ −24.0000 −1.19553
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 3.00000 0.148704
$$408$$ 0 0
$$409$$ −12.0000 −0.593362 −0.296681 0.954977i $$-0.595880\pi$$
−0.296681 + 0.954977i $$0.595880\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 12.0000 0.591198
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −9.00000 −0.441793
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −18.0000 −0.879358 −0.439679 0.898155i $$-0.644908\pi$$
−0.439679 + 0.898155i $$0.644908\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −4.00000 −0.194029
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 24.0000 1.16008
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 22.0000 1.05970 0.529851 0.848091i $$-0.322248\pi$$
0.529851 + 0.848091i $$0.322248\pi$$
$$432$$ 0 0
$$433$$ −25.0000 −1.20142 −0.600712 0.799466i $$-0.705116\pi$$
−0.600712 + 0.799466i $$0.705116\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −22.0000 −1.05361
$$437$$ 6.00000 0.287019
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ 0 0
$$445$$ 10.0000 0.474045
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 16.0000 0.755929
$$449$$ 31.0000 1.46298 0.731490 0.681852i $$-0.238825\pi$$
0.731490 + 0.681852i $$0.238825\pi$$
$$450$$ 0 0
$$451$$ 7.00000 0.329617
$$452$$ −24.0000 −1.12887
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −12.0000 −0.562569
$$456$$ 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
$$460$$ 6.00000 0.279751
$$461$$ 21.0000 0.978068 0.489034 0.872265i $$-0.337349\pi$$
0.489034 + 0.872265i $$0.337349\pi$$
$$462$$ 0 0
$$463$$ −18.0000 −0.836531 −0.418265 0.908325i $$-0.637362\pi$$
−0.418265 + 0.908325i $$0.637362\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −34.0000 −1.57333 −0.786666 0.617379i $$-0.788195\pi$$
−0.786666 + 0.617379i $$0.788195\pi$$
$$468$$ 0 0
$$469$$ 20.0000 0.923514
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −5.00000 −0.229900
$$474$$ 0 0
$$475$$ −2.00000 −0.0917663
$$476$$ −16.0000 −0.733359
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −40.0000 −1.82765 −0.913823 0.406112i $$-0.866884\pi$$
−0.913823 + 0.406112i $$0.866884\pi$$
$$480$$ 0 0
$$481$$ −18.0000 −0.820729
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 20.0000 0.909091
$$485$$ 17.0000 0.771930
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ 4.00000 0.180151
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −16.0000 −0.718421
$$497$$ 12.0000 0.538274
$$498$$ 0 0
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ −2.00000 −0.0894427
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 30.0000 1.33763 0.668817 0.743427i $$-0.266801\pi$$
0.668817 + 0.743427i $$0.266801\pi$$
$$504$$ 0 0
$$505$$ 3.00000 0.133498
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −14.0000 −0.621150
$$509$$ −34.0000 −1.50702 −0.753512 0.657434i $$-0.771642\pi$$
−0.753512 + 0.657434i $$0.771642\pi$$
$$510$$ 0 0
$$511$$ −6.00000 −0.265424
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −6.00000 −0.264392
$$516$$ 0 0
$$517$$ 6.00000 0.263880
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ 0 0
$$523$$ −24.0000 −1.04945 −0.524723 0.851273i $$-0.675831\pi$$
−0.524723 + 0.851273i $$0.675831\pi$$
$$524$$ 24.0000 1.04844
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 16.0000 0.696971
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −8.00000 −0.346844
$$533$$ −42.0000 −1.81922
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 11.0000 0.471188
$$546$$ 0 0
$$547$$ 40.0000 1.71028 0.855138 0.518400i $$-0.173472\pi$$
0.855138 + 0.518400i $$0.173472\pi$$
$$548$$ 32.0000 1.36697
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 2.00000 0.0852029
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 38.0000 1.61156
$$557$$ −37.0000 −1.56774 −0.783870 0.620925i $$-0.786757\pi$$
−0.783870 + 0.620925i $$0.786757\pi$$
$$558$$ 0 0
$$559$$ 30.0000 1.26886
$$560$$ −8.00000 −0.338062
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 6.00000 0.252870 0.126435 0.991975i $$-0.459647\pi$$
0.126435 + 0.991975i $$0.459647\pi$$
$$564$$ 0 0
$$565$$ 12.0000 0.504844
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ 5.00000 0.209243 0.104622 0.994512i $$-0.466637\pi$$
0.104622 + 0.994512i $$0.466637\pi$$
$$572$$ 12.0000 0.501745
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −3.00000 −0.125109
$$576$$ 0 0
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 2.00000 0.0830455
$$581$$ 18.0000 0.746766
$$582$$ 0 0
$$583$$ 13.0000 0.538405
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −36.0000 −1.48588 −0.742940 0.669359i $$-0.766569\pi$$
−0.742940 + 0.669359i $$0.766569\pi$$
$$588$$ 0 0
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −12.0000 −0.493197
$$593$$ −10.0000 −0.410651 −0.205325 0.978694i $$-0.565825\pi$$
−0.205325 + 0.978694i $$0.565825\pi$$
$$594$$ 0 0
$$595$$ 8.00000 0.327968
$$596$$ −4.00000 −0.163846
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ 40.0000 1.63163 0.815817 0.578310i $$-0.196288\pi$$
0.815817 + 0.578310i $$0.196288\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 22.0000 0.895167
$$605$$ −10.0000 −0.406558
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −36.0000 −1.45640
$$612$$ 0 0
$$613$$ −6.00000 −0.242338 −0.121169 0.992632i $$-0.538664\pi$$
−0.121169 + 0.992632i $$0.538664\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 0 0
$$619$$ 2.00000 0.0803868 0.0401934 0.999192i $$-0.487203\pi$$
0.0401934 + 0.999192i $$0.487203\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −20.0000 −0.801283
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 36.0000 1.43656
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −48.0000 −1.91085 −0.955425 0.295234i $$-0.904602\pi$$
−0.955425 + 0.295234i $$0.904602\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 7.00000 0.277787
$$636$$ 0 0
$$637$$ −18.0000 −0.713186
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 47.0000 1.85639 0.928194 0.372096i $$-0.121361\pi$$
0.928194 + 0.372096i $$0.121361\pi$$
$$642$$ 0 0
$$643$$ 16.0000 0.630978 0.315489 0.948929i $$-0.397831\pi$$
0.315489 + 0.948929i $$0.397831\pi$$
$$644$$ −12.0000 −0.472866
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 3.00000 0.117942 0.0589711 0.998260i $$-0.481218\pi$$
0.0589711 + 0.998260i $$0.481218\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −30.0000 −1.17489
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ −28.0000 −1.09322
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −33.0000 −1.28550 −0.642749 0.766077i $$-0.722206\pi$$
−0.642749 + 0.766077i $$0.722206\pi$$
$$660$$ 0 0
$$661$$ −1.00000 −0.0388955 −0.0194477 0.999811i $$-0.506191\pi$$
−0.0194477 + 0.999811i $$0.506191\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 4.00000 0.155113
$$666$$ 0 0
$$667$$ 3.00000 0.116160
$$668$$ 16.0000 0.619059
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 6.00000 0.231283 0.115642 0.993291i $$-0.463108\pi$$
0.115642 + 0.993291i $$0.463108\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −46.0000 −1.76923
$$677$$ −12.0000 −0.461197 −0.230599 0.973049i $$-0.574068\pi$$
−0.230599 + 0.973049i $$0.574068\pi$$
$$678$$ 0 0
$$679$$ −34.0000 −1.30480
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −7.00000 −0.267848 −0.133924 0.990992i $$-0.542758\pi$$
−0.133924 + 0.990992i $$0.542758\pi$$
$$684$$ 0 0
$$685$$ −16.0000 −0.611329
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 20.0000 0.762493
$$689$$ −78.0000 −2.97156
$$690$$ 0 0
$$691$$ 44.0000 1.67384 0.836919 0.547326i $$-0.184354\pi$$
0.836919 + 0.547326i $$0.184354\pi$$
$$692$$ 10.0000 0.380143
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −19.0000 −0.720711
$$696$$ 0 0
$$697$$ 28.0000 1.06058
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 4.00000 0.151186
$$701$$ −34.0000 −1.28416 −0.642081 0.766637i $$-0.721929\pi$$
−0.642081 + 0.766637i $$0.721929\pi$$
$$702$$ 0 0
$$703$$ 6.00000 0.226294
$$704$$ 8.00000 0.301511
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −6.00000 −0.225653
$$708$$ 0 0
$$709$$ −25.0000 −0.938895 −0.469447 0.882960i $$-0.655547\pi$$
−0.469447 + 0.882960i $$0.655547\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 12.0000 0.449404
$$714$$ 0 0
$$715$$ −6.00000 −0.224387
$$716$$ −36.0000 −1.34538
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 4.00000 0.149175 0.0745874 0.997214i $$-0.476236\pi$$
0.0745874 + 0.997214i $$0.476236\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ 0 0
$$723$$ 0 0
$$724$$ −50.0000 −1.85824
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ 44.0000 1.63187 0.815935 0.578144i $$-0.196223\pi$$
0.815935 + 0.578144i $$0.196223\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −20.0000 −0.739727
$$732$$ 0 0
$$733$$ −2.00000 −0.0738717 −0.0369358 0.999318i $$-0.511760\pi$$
−0.0369358 + 0.999318i $$0.511760\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 10.0000 0.368355
$$738$$ 0 0
$$739$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$740$$ 6.00000 0.220564
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −6.00000 −0.220119 −0.110059 0.993925i $$-0.535104\pi$$
−0.110059 + 0.993925i $$0.535104\pi$$
$$744$$ 0 0
$$745$$ 2.00000 0.0732743
$$746$$ 0 0
$$747$$ 0 0
$$748$$ −8.00000 −0.292509
$$749$$ 24.0000 0.876941
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ −24.0000 −0.875190
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −11.0000 −0.400331
$$756$$ 0 0
$$757$$ −17.0000 −0.617876 −0.308938 0.951082i $$-0.599973\pi$$
−0.308938 + 0.951082i $$0.599973\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ −22.0000 −0.796453
$$764$$ −54.0000 −1.95365
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −36.0000 −1.29819 −0.649097 0.760706i $$-0.724853\pi$$
−0.649097 + 0.760706i $$0.724853\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 28.0000 1.00774
$$773$$ 10.0000 0.359675 0.179838 0.983696i $$-0.442443\pi$$
0.179838 + 0.983696i $$0.442443\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 14.0000 0.501602
$$780$$ 0 0
$$781$$ 6.00000 0.214697
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −12.0000 −0.428571
$$785$$ −18.0000 −0.642448
$$786$$ 0 0
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ 26.0000 0.926212
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −24.0000 −0.853342
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −30.0000 −1.06332
$$797$$ 38.0000 1.34603 0.673015 0.739629i $$-0.264999\pi$$
0.673015 + 0.739629i $$0.264999\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −3.00000 −0.105868
$$804$$ 0 0
$$805$$ 6.00000 0.211472
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 9.00000 0.316423 0.158212 0.987405i $$-0.449427\pi$$
0.158212 + 0.987405i $$0.449427\pi$$
$$810$$ 0 0
$$811$$ 31.0000 1.08856 0.544279 0.838905i $$-0.316803\pi$$
0.544279 + 0.838905i $$0.316803\pi$$
$$812$$ −4.00000 −0.140372
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 15.0000 0.525427
$$816$$ 0 0
$$817$$ −10.0000 −0.349856
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 14.0000 0.488901
$$821$$ −48.0000 −1.67521 −0.837606 0.546275i $$-0.816045\pi$$
−0.837606 + 0.546275i $$0.816045\pi$$
$$822$$ 0 0
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 22.0000 0.765015 0.382507 0.923952i $$-0.375061\pi$$
0.382507 + 0.923952i $$0.375061\pi$$
$$828$$ 0 0
$$829$$ −46.0000 −1.59765 −0.798823 0.601566i $$-0.794544\pi$$
−0.798823 + 0.601566i $$0.794544\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −48.0000 −1.66410
$$833$$ 12.0000 0.415775
$$834$$ 0 0
$$835$$ −8.00000 −0.276851
$$836$$ −4.00000 −0.138343
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 32.0000 1.10476 0.552381 0.833592i $$-0.313719\pi$$
0.552381 + 0.833592i $$0.313719\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 4.00000 0.137686
$$845$$ 23.0000 0.791224
$$846$$ 0 0
$$847$$ 20.0000 0.687208
$$848$$ −52.0000 −1.78569
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 9.00000 0.308516
$$852$$ 0 0
$$853$$ −5.00000 −0.171197 −0.0855984 0.996330i $$-0.527280\pi$$
−0.0855984 + 0.996330i $$0.527280\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 47.0000 1.60549 0.802745 0.596323i $$-0.203372\pi$$
0.802745 + 0.596323i $$0.203372\pi$$
$$858$$ 0 0
$$859$$ 26.0000 0.887109 0.443554 0.896248i $$-0.353717\pi$$
0.443554 + 0.896248i $$0.353717\pi$$
$$860$$ −10.0000 −0.340997
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ −5.00000 −0.170005
$$866$$ 0 0
$$867$$ 0 0
$$868$$ −16.0000 −0.543075
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −60.0000 −2.03302
$$872$$ 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
$$880$$ −4.00000 −0.134840
$$881$$ 15.0000 0.505363 0.252681 0.967550i $$-0.418688\pi$$
0.252681 + 0.967550i $$0.418688\pi$$
$$882$$ 0 0
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ 48.0000 1.61441
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −30.0000 −1.00730 −0.503651 0.863907i $$-0.668010\pi$$
−0.503651 + 0.863907i $$0.668010\pi$$
$$888$$ 0 0
$$889$$ −14.0000 −0.469545
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 12.0000 0.401565
$$894$$ 0 0
$$895$$ 18.0000 0.601674
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 4.00000 0.133407
$$900$$ 0 0
$$901$$ 52.0000 1.73237
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 25.0000 0.831028
$$906$$ 0 0
$$907$$ −17.0000 −0.564476 −0.282238 0.959344i $$-0.591077\pi$$
−0.282238 + 0.959344i $$0.591077\pi$$
$$908$$ 22.0000 0.730096
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −51.0000 −1.68971 −0.844853 0.534999i $$-0.820312\pi$$
−0.844853 + 0.534999i $$0.820312\pi$$
$$912$$ 0 0
$$913$$ 9.00000 0.297857
$$914$$ 0 0
$$915$$ 0 0
$$916$$ −20.0000 −0.660819
$$917$$ 24.0000 0.792550
$$918$$ 0 0
$$919$$ −4.00000 −0.131948 −0.0659739 0.997821i $$-0.521015\pi$$
−0.0659739 + 0.997821i $$0.521015\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −36.0000 −1.18495
$$924$$ 0 0
$$925$$ −3.00000 −0.0986394
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 56.0000 1.83730 0.918650 0.395072i $$-0.129280\pi$$
0.918650 + 0.395072i $$0.129280\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ −2.00000 −0.0655122
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 4.00000 0.130814
$$936$$ 0 0
$$937$$ 56.0000 1.82944 0.914720 0.404088i $$-0.132411\pi$$
0.914720 + 0.404088i $$0.132411\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 12.0000 0.391397
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ 0 0
$$943$$ 21.0000 0.683854
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 42.0000 1.36482 0.682408 0.730971i $$-0.260933\pi$$
0.682408 + 0.730971i $$0.260933\pi$$
$$948$$ 0 0
$$949$$ 18.0000 0.584305
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 42.0000 1.36051 0.680257 0.732974i $$-0.261868\pi$$
0.680257 + 0.732974i $$0.261868\pi$$
$$954$$ 0 0
$$955$$ 27.0000 0.873699
$$956$$ 28.0000 0.905585
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 32.0000 1.03333
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −34.0000 −1.09507
$$965$$ −14.0000 −0.450676
$$966$$ 0 0
$$967$$ −13.0000 −0.418052 −0.209026 0.977910i $$-0.567029\pi$$
−0.209026 + 0.977910i $$0.567029\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −13.0000 −0.417190 −0.208595 0.978002i $$-0.566889\pi$$
−0.208595 + 0.978002i $$0.566889\pi$$
$$972$$ 0 0
$$973$$ 38.0000 1.21822
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 39.0000 1.24772 0.623860 0.781536i $$-0.285563\pi$$
0.623860 + 0.781536i $$0.285563\pi$$
$$978$$ 0 0
$$979$$ −10.0000 −0.319601
$$980$$ 6.00000 0.191663
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −54.0000 −1.72233 −0.861166 0.508323i $$-0.830265\pi$$
−0.861166 + 0.508323i $$0.830265\pi$$
$$984$$ 0 0
$$985$$ −13.0000 −0.414214
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 24.0000 0.763542
$$989$$ −15.0000 −0.476972
$$990$$ 0 0
$$991$$ −25.0000 −0.794151 −0.397076 0.917786i $$-0.629975\pi$$
−0.397076 + 0.917786i $$0.629975\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 15.0000 0.475532
$$996$$ 0 0
$$997$$ 25.0000 0.791758 0.395879 0.918303i $$-0.370440\pi$$
0.395879 + 0.918303i $$0.370440\pi$$
$$998$$ 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 1305.2.a.c.1.1 1
3.2 odd 2 435.2.a.b.1.1 1
5.4 even 2 6525.2.a.i.1.1 1
12.11 even 2 6960.2.a.bc.1.1 1
15.2 even 4 2175.2.c.d.349.2 2
15.8 even 4 2175.2.c.d.349.1 2
15.14 odd 2 2175.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.b.1.1 1 3.2 odd 2
1305.2.a.c.1.1 1 1.1 even 1 trivial
2175.2.a.g.1.1 1 15.14 odd 2
2175.2.c.d.349.1 2 15.8 even 4
2175.2.c.d.349.2 2 15.2 even 4
6525.2.a.i.1.1 1 5.4 even 2
6960.2.a.bc.1.1 1 12.11 even 2