# Properties

 Label 1305.2.a.e.1.1 Level $1305$ Weight $2$ Character 1305.1 Self dual yes Analytic conductor $10.420$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} -3.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} -3.00000 q^{8} -1.00000 q^{10} +6.00000 q^{13} -4.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +8.00000 q^{19} +1.00000 q^{20} +4.00000 q^{23} +1.00000 q^{25} +6.00000 q^{26} +4.00000 q^{28} -1.00000 q^{29} +4.00000 q^{31} +5.00000 q^{32} -2.00000 q^{34} +4.00000 q^{35} +6.00000 q^{37} +8.00000 q^{38} +3.00000 q^{40} -2.00000 q^{41} -4.00000 q^{43} +4.00000 q^{46} +9.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} -6.00000 q^{53} +12.0000 q^{56} -1.00000 q^{58} +12.0000 q^{59} +6.00000 q^{61} +4.00000 q^{62} +7.00000 q^{64} -6.00000 q^{65} -8.00000 q^{67} +2.00000 q^{68} +4.00000 q^{70} -16.0000 q^{71} -6.00000 q^{73} +6.00000 q^{74} -8.00000 q^{76} +12.0000 q^{79} +1.00000 q^{80} -2.00000 q^{82} +16.0000 q^{83} +2.00000 q^{85} -4.00000 q^{86} -2.00000 q^{89} -24.0000 q^{91} -4.00000 q^{92} -8.00000 q^{95} -14.0000 q^{97} +9.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 8.00000 1.83533 0.917663 0.397360i $$-0.130073\pi$$
0.917663 + 0.397360i $$0.130073\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 6.00000 1.17670
$$27$$ 0 0
$$28$$ 4.00000 0.755929
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 4.00000 0.676123
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 8.00000 1.29777
$$39$$ 0 0
$$40$$ 3.00000 0.474342
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ −6.00000 −0.832050
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 12.0000 1.60357
$$57$$ 0 0
$$58$$ −1.00000 −0.131306
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ −6.00000 −0.744208
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ 4.00000 0.478091
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ 0 0
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −8.00000 −0.917663
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ −2.00000 −0.220863
$$83$$ 16.0000 1.75623 0.878114 0.478451i $$-0.158802\pi$$
0.878114 + 0.478451i $$0.158802\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ −24.0000 −2.51588
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −8.00000 −0.820783
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 9.00000 0.909137
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ −18.0000 −1.76505
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 16.0000 1.54678 0.773389 0.633932i $$-0.218560\pi$$
0.773389 + 0.633932i $$0.218560\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 4.00000 0.377964
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ −4.00000 −0.373002
$$116$$ 1.00000 0.0928477
$$117$$ 0 0
$$118$$ 12.0000 1.10469
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 6.00000 0.543214
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ −6.00000 −0.526235
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −32.0000 −2.77475
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ −4.00000 −0.338062
$$141$$ 0 0
$$142$$ −16.0000 −1.34269
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 1.00000 0.0830455
$$146$$ −6.00000 −0.496564
$$147$$ 0 0
$$148$$ −6.00000 −0.493197
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ −24.0000 −1.94666
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 12.0000 0.954669
$$159$$ 0 0
$$160$$ −5.00000 −0.395285
$$161$$ −16.0000 −1.26098
$$162$$ 0 0
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 20.0000 1.54765 0.773823 0.633402i $$-0.218342\pi$$
0.773823 + 0.633402i $$0.218342\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 2.00000 0.153393
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −2.00000 −0.149906
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ −24.0000 −1.77900
$$183$$ 0 0
$$184$$ −12.0000 −0.884652
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ −8.00000 −0.580381
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ −9.00000 −0.642857
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 0 0
$$202$$ 10.0000 0.703598
$$203$$ 4.00000 0.280745
$$204$$ 0 0
$$205$$ 2.00000 0.139686
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ −6.00000 −0.416025
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ 16.0000 1.09374
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ −2.00000 −0.135457
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ 12.0000 0.803579 0.401790 0.915732i $$-0.368388\pi$$
0.401790 + 0.915732i $$0.368388\pi$$
$$224$$ −20.0000 −1.33631
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 0 0
$$232$$ 3.00000 0.196960
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ 8.00000 0.518563
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ 0 0
$$244$$ −6.00000 −0.384111
$$245$$ −9.00000 −0.574989
$$246$$ 0 0
$$247$$ 48.0000 3.05417
$$248$$ −12.0000 −0.762001
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −26.0000 −1.62184 −0.810918 0.585160i $$-0.801032\pi$$
−0.810918 + 0.585160i $$0.801032\pi$$
$$258$$ 0 0
$$259$$ −24.0000 −1.49129
$$260$$ 6.00000 0.372104
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ −32.0000 −1.96205
$$267$$ 0 0
$$268$$ 8.00000 0.488678
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 20.0000 1.19952
$$279$$ 0 0
$$280$$ −12.0000 −0.717137
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 8.00000 0.475551 0.237775 0.971320i $$-0.423582\pi$$
0.237775 + 0.971320i $$0.423582\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 8.00000 0.472225
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 1.00000 0.0587220
$$291$$ 0 0
$$292$$ 6.00000 0.351123
$$293$$ 26.0000 1.51894 0.759468 0.650545i $$-0.225459\pi$$
0.759468 + 0.650545i $$0.225459\pi$$
$$294$$ 0 0
$$295$$ −12.0000 −0.698667
$$296$$ −18.0000 −1.04623
$$297$$ 0 0
$$298$$ 10.0000 0.579284
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ 16.0000 0.920697
$$303$$ 0 0
$$304$$ −8.00000 −0.458831
$$305$$ −6.00000 −0.343559
$$306$$ 0 0
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −4.00000 −0.227185
$$311$$ 28.0000 1.58773 0.793867 0.608091i $$-0.208065\pi$$
0.793867 + 0.608091i $$0.208065\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ −30.0000 −1.68497 −0.842484 0.538721i $$-0.818908\pi$$
−0.842484 + 0.538721i $$0.818908\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −7.00000 −0.391312
$$321$$ 0 0
$$322$$ −16.0000 −0.891645
$$323$$ −16.0000 −0.890264
$$324$$ 0 0
$$325$$ 6.00000 0.332820
$$326$$ 12.0000 0.664619
$$327$$ 0 0
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −16.0000 −0.879440 −0.439720 0.898135i $$-0.644922\pi$$
−0.439720 + 0.898135i $$0.644922\pi$$
$$332$$ −16.0000 −0.878114
$$333$$ 0 0
$$334$$ 20.0000 1.09435
$$335$$ 8.00000 0.437087
$$336$$ 0 0
$$337$$ −30.0000 −1.63420 −0.817102 0.576493i $$-0.804421\pi$$
−0.817102 + 0.576493i $$0.804421\pi$$
$$338$$ 23.0000 1.25104
$$339$$ 0 0
$$340$$ −2.00000 −0.108465
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 2.00000 0.107521
$$347$$ 24.0000 1.28839 0.644194 0.764862i $$-0.277193\pi$$
0.644194 + 0.764862i $$0.277193\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ 2.00000 0.106000
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 36.0000 1.90001 0.950004 0.312239i $$-0.101079\pi$$
0.950004 + 0.312239i $$0.101079\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ −10.0000 −0.525588
$$363$$ 0 0
$$364$$ 24.0000 1.25794
$$365$$ 6.00000 0.314054
$$366$$ 0 0
$$367$$ −24.0000 −1.25279 −0.626395 0.779506i $$-0.715470\pi$$
−0.626395 + 0.779506i $$0.715470\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 0 0
$$370$$ −6.00000 −0.311925
$$371$$ 24.0000 1.24602
$$372$$ 0 0
$$373$$ −34.0000 −1.76045 −0.880227 0.474554i $$-0.842610\pi$$
−0.880227 + 0.474554i $$0.842610\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6.00000 −0.309016
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 8.00000 0.410391
$$381$$ 0 0
$$382$$ 4.00000 0.204658
$$383$$ −28.0000 −1.43073 −0.715367 0.698749i $$-0.753740\pi$$
−0.715367 + 0.698749i $$0.753740\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 0 0
$$388$$ 14.0000 0.710742
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ −27.0000 −1.36371
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ −12.0000 −0.603786
$$396$$ 0 0
$$397$$ 22.0000 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ −16.0000 −0.802008
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ 24.0000 1.19553
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ 4.00000 0.198517
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −6.00000 −0.296681 −0.148340 0.988936i $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ 2.00000 0.0987730
$$411$$ 0 0
$$412$$ −4.00000 −0.197066
$$413$$ −48.0000 −2.36193
$$414$$ 0 0
$$415$$ −16.0000 −0.785409
$$416$$ 30.0000 1.47087
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ 8.00000 0.389434
$$423$$ 0 0
$$424$$ 18.0000 0.874157
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ −24.0000 −1.16144
$$428$$ −16.0000 −0.773389
$$429$$ 0 0
$$430$$ 4.00000 0.192897
$$431$$ −32.0000 −1.54139 −0.770693 0.637207i $$-0.780090\pi$$
−0.770693 + 0.637207i $$0.780090\pi$$
$$432$$ 0 0
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 32.0000 1.53077
$$438$$ 0 0
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −12.0000 −0.570782
$$443$$ −20.0000 −0.950229 −0.475114 0.879924i $$-0.657593\pi$$
−0.475114 + 0.879924i $$0.657593\pi$$
$$444$$ 0 0
$$445$$ 2.00000 0.0948091
$$446$$ 12.0000 0.568216
$$447$$ 0 0
$$448$$ −28.0000 −1.32288
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 2.00000 0.0940721
$$453$$ 0 0
$$454$$ −8.00000 −0.375459
$$455$$ 24.0000 1.12514
$$456$$ 0 0
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 0 0
$$460$$ 4.00000 0.186501
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 36.0000 1.67306 0.836531 0.547920i $$-0.184580\pi$$
0.836531 + 0.547920i $$0.184580\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ 32.0000 1.47762
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −36.0000 −1.65703
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 8.00000 0.367065
$$476$$ −8.00000 −0.366679
$$477$$ 0 0
$$478$$ −24.0000 −1.09773
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ 36.0000 1.64146
$$482$$ 2.00000 0.0910975
$$483$$ 0 0
$$484$$ 11.0000 0.500000
$$485$$ 14.0000 0.635707
$$486$$ 0 0
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ −18.0000 −0.814822
$$489$$ 0 0
$$490$$ −9.00000 −0.406579
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ 0 0
$$493$$ 2.00000 0.0900755
$$494$$ 48.0000 2.15962
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 64.0000 2.87079
$$498$$ 0 0
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ −10.0000 −0.444994
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 24.0000 1.06170
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ −26.0000 −1.14681
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −24.0000 −1.05450
$$519$$ 0 0
$$520$$ 18.0000 0.789352
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 6.00000 0.260623
$$531$$ 0 0
$$532$$ 32.0000 1.38738
$$533$$ −12.0000 −0.519778
$$534$$ 0 0
$$535$$ −16.0000 −0.691740
$$536$$ 24.0000 1.03664
$$537$$ 0 0
$$538$$ 18.0000 0.776035
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ 0 0
$$544$$ −10.0000 −0.428746
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ −32.0000 −1.36822 −0.684111 0.729378i $$-0.739809\pi$$
−0.684111 + 0.729378i $$0.739809\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ −48.0000 −2.04117
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ −14.0000 −0.593199 −0.296600 0.955002i $$-0.595853\pi$$
−0.296600 + 0.955002i $$0.595853\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ −4.00000 −0.169031
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ 8.00000 0.336265
$$567$$ 0 0
$$568$$ 48.0000 2.01404
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 8.00000 0.333914
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ −1.00000 −0.0415227
$$581$$ −64.0000 −2.65517
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 18.0000 0.744845
$$585$$ 0 0
$$586$$ 26.0000 1.07405
$$587$$ 16.0000 0.660391 0.330195 0.943913i $$-0.392885\pi$$
0.330195 + 0.943913i $$0.392885\pi$$
$$588$$ 0 0
$$589$$ 32.0000 1.31854
$$590$$ −12.0000 −0.494032
$$591$$ 0 0
$$592$$ −6.00000 −0.246598
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 0 0
$$595$$ −8.00000 −0.327968
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ 24.0000 0.981433
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 16.0000 0.652111
$$603$$ 0 0
$$604$$ −16.0000 −0.651031
$$605$$ 11.0000 0.447214
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 40.0000 1.62221
$$609$$ 0 0
$$610$$ −6.00000 −0.242933
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 0 0
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 0 0
$$622$$ 28.0000 1.12270
$$623$$ 8.00000 0.320513
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ −14.0000 −0.558661
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 48.0000 1.91085 0.955425 0.295234i $$-0.0953977\pi$$
0.955425 + 0.295234i $$0.0953977\pi$$
$$632$$ −36.0000 −1.43200
$$633$$ 0 0
$$634$$ −30.0000 −1.19145
$$635$$ 8.00000 0.317470
$$636$$ 0 0
$$637$$ 54.0000 2.13956
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 3.00000 0.118585
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 0 0
$$643$$ −16.0000 −0.630978 −0.315489 0.948929i $$-0.602169\pi$$
−0.315489 + 0.948929i $$0.602169\pi$$
$$644$$ 16.0000 0.630488
$$645$$ 0 0
$$646$$ −16.0000 −0.629512
$$647$$ −28.0000 −1.10079 −0.550397 0.834903i $$-0.685524\pi$$
−0.550397 + 0.834903i $$0.685524\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 6.00000 0.235339
$$651$$ 0 0
$$652$$ −12.0000 −0.469956
$$653$$ 50.0000 1.95665 0.978326 0.207072i $$-0.0663936\pi$$
0.978326 + 0.207072i $$0.0663936\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ −16.0000 −0.621858
$$663$$ 0 0
$$664$$ −48.0000 −1.86276
$$665$$ 32.0000 1.24091
$$666$$ 0 0
$$667$$ −4.00000 −0.154881
$$668$$ −20.0000 −0.773823
$$669$$ 0 0
$$670$$ 8.00000 0.309067
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −30.0000 −1.15642 −0.578208 0.815890i $$-0.696248\pi$$
−0.578208 + 0.815890i $$0.696248\pi$$
$$674$$ −30.0000 −1.15556
$$675$$ 0 0
$$676$$ −23.0000 −0.884615
$$677$$ 26.0000 0.999261 0.499631 0.866239i $$-0.333469\pi$$
0.499631 + 0.866239i $$0.333469\pi$$
$$678$$ 0 0
$$679$$ 56.0000 2.14908
$$680$$ −6.00000 −0.230089
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 0 0
$$685$$ −6.00000 −0.229248
$$686$$ −8.00000 −0.305441
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ −36.0000 −1.37149
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ −2.00000 −0.0760286
$$693$$ 0 0
$$694$$ 24.0000 0.911028
$$695$$ −20.0000 −0.758643
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ −2.00000 −0.0757011
$$699$$ 0 0
$$700$$ 4.00000 0.151186
$$701$$ 34.0000 1.28416 0.642081 0.766637i $$-0.278071\pi$$
0.642081 + 0.766637i $$0.278071\pi$$
$$702$$ 0 0
$$703$$ 48.0000 1.81035
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ −40.0000 −1.50435
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 16.0000 0.600469
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ 16.0000 0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 36.0000 1.34351
$$719$$ 40.0000 1.49175 0.745874 0.666087i $$-0.232032\pi$$
0.745874 + 0.666087i $$0.232032\pi$$
$$720$$ 0 0
$$721$$ −16.0000 −0.595871
$$722$$ 45.0000 1.67473
$$723$$ 0 0
$$724$$ 10.0000 0.371647
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 72.0000 2.66850
$$729$$ 0 0
$$730$$ 6.00000 0.222070
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ 46.0000 1.69905 0.849524 0.527549i $$-0.176889\pi$$
0.849524 + 0.527549i $$0.176889\pi$$
$$734$$ −24.0000 −0.885856
$$735$$ 0 0
$$736$$ 20.0000 0.737210
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −24.0000 −0.882854 −0.441427 0.897297i $$-0.645528\pi$$
−0.441427 + 0.897297i $$0.645528\pi$$
$$740$$ 6.00000 0.220564
$$741$$ 0 0
$$742$$ 24.0000 0.881068
$$743$$ 32.0000 1.17397 0.586983 0.809599i $$-0.300316\pi$$
0.586983 + 0.809599i $$0.300316\pi$$
$$744$$ 0 0
$$745$$ −10.0000 −0.366372
$$746$$ −34.0000 −1.24483
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −64.0000 −2.33851
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ −6.00000 −0.218507
$$755$$ −16.0000 −0.582300
$$756$$ 0 0
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ −16.0000 −0.581146
$$759$$ 0 0
$$760$$ 24.0000 0.870572
$$761$$ 38.0000 1.37750 0.688749 0.724999i $$-0.258160\pi$$
0.688749 + 0.724999i $$0.258160\pi$$
$$762$$ 0 0
$$763$$ 8.00000 0.289619
$$764$$ −4.00000 −0.144715
$$765$$ 0 0
$$766$$ −28.0000 −1.01168
$$767$$ 72.0000 2.59977
$$768$$ 0 0
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −2.00000 −0.0719816
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ 0 0
$$775$$ 4.00000 0.143684
$$776$$ 42.0000 1.50771
$$777$$ 0 0
$$778$$ 26.0000 0.932145
$$779$$ −16.0000 −0.573259
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −8.00000 −0.286079
$$783$$ 0 0
$$784$$ −9.00000 −0.321429
$$785$$ −14.0000 −0.499681
$$786$$ 0 0
$$787$$ −24.0000 −0.855508 −0.427754 0.903895i $$-0.640695\pi$$
−0.427754 + 0.903895i $$0.640695\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 0 0
$$790$$ −12.0000 −0.426941
$$791$$ 8.00000 0.284447
$$792$$ 0 0
$$793$$ 36.0000 1.27840
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 5.00000 0.176777
$$801$$ 0 0
$$802$$ −18.0000 −0.635602
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 16.0000 0.563926
$$806$$ 24.0000 0.845364
$$807$$ 0 0
$$808$$ −30.0000 −1.05540
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ −4.00000 −0.140372
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −12.0000 −0.420342
$$816$$ 0 0
$$817$$ −32.0000 −1.11954
$$818$$ −6.00000 −0.209785
$$819$$ 0 0
$$820$$ −2.00000 −0.0698430
$$821$$ −22.0000 −0.767805 −0.383903 0.923374i $$-0.625420\pi$$
−0.383903 + 0.923374i $$0.625420\pi$$
$$822$$ 0 0
$$823$$ −8.00000 −0.278862 −0.139431 0.990232i $$-0.544527\pi$$
−0.139431 + 0.990232i $$0.544527\pi$$
$$824$$ −12.0000 −0.418040
$$825$$ 0 0
$$826$$ −48.0000 −1.67013
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 0 0
$$829$$ −26.0000 −0.903017 −0.451509 0.892267i $$-0.649114\pi$$
−0.451509 + 0.892267i $$0.649114\pi$$
$$830$$ −16.0000 −0.555368
$$831$$ 0 0
$$832$$ 42.0000 1.45609
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ −20.0000 −0.692129
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −4.00000 −0.138178
$$839$$ −20.0000 −0.690477 −0.345238 0.938515i $$-0.612202\pi$$
−0.345238 + 0.938515i $$0.612202\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 30.0000 1.03387
$$843$$ 0 0
$$844$$ −8.00000 −0.275371
$$845$$ −23.0000 −0.791224
$$846$$ 0 0
$$847$$ 44.0000 1.51186
$$848$$ 6.00000 0.206041
$$849$$ 0 0
$$850$$ −2.00000 −0.0685994
$$851$$ 24.0000 0.822709
$$852$$ 0 0
$$853$$ −26.0000 −0.890223 −0.445112 0.895475i $$-0.646836\pi$$
−0.445112 + 0.895475i $$0.646836\pi$$
$$854$$ −24.0000 −0.821263
$$855$$ 0 0
$$856$$ −48.0000 −1.64061
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ 8.00000 0.272956 0.136478 0.990643i $$-0.456422\pi$$
0.136478 + 0.990643i $$0.456422\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 0 0
$$862$$ −32.0000 −1.08992
$$863$$ 52.0000 1.77010 0.885050 0.465495i $$-0.154124\pi$$
0.885050 + 0.465495i $$0.154124\pi$$
$$864$$ 0 0
$$865$$ −2.00000 −0.0680020
$$866$$ −14.0000 −0.475739
$$867$$ 0 0
$$868$$ 16.0000 0.543075
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −48.0000 −1.62642
$$872$$ 6.00000 0.203186
$$873$$ 0 0
$$874$$ 32.0000 1.08242
$$875$$ 4.00000 0.135225
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ −24.0000 −0.809961
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ 0 0
$$883$$ 16.0000 0.538443 0.269221 0.963078i $$-0.413234\pi$$
0.269221 + 0.963078i $$0.413234\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −20.0000 −0.671913
$$887$$ 32.0000 1.07445 0.537227 0.843437i $$-0.319472\pi$$
0.537227 + 0.843437i $$0.319472\pi$$
$$888$$ 0 0
$$889$$ 32.0000 1.07325
$$890$$ 2.00000 0.0670402
$$891$$ 0 0
$$892$$ −12.0000 −0.401790
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 12.0000 0.400892
$$897$$ 0 0
$$898$$ 6.00000 0.200223
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 10.0000 0.332411
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 8.00000 0.265489
$$909$$ 0 0
$$910$$ 24.0000 0.795592
$$911$$ −20.0000 −0.662630 −0.331315 0.943520i $$-0.607492\pi$$
−0.331315 + 0.943520i $$0.607492\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −22.0000 −0.727695
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 12.0000 0.395628
$$921$$ 0 0
$$922$$ 18.0000 0.592798
$$923$$ −96.0000 −3.15988
$$924$$ 0 0
$$925$$ 6.00000 0.197279
$$926$$ 36.0000 1.18303
$$927$$ 0 0
$$928$$ −5.00000 −0.164133
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 72.0000 2.35970
$$932$$ 18.0000 0.589610
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −38.0000 −1.24141 −0.620703 0.784046i $$-0.713153\pi$$
−0.620703 + 0.784046i $$0.713153\pi$$
$$938$$ 32.0000 1.04484
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 0 0
$$943$$ −8.00000 −0.260516
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −28.0000 −0.909878 −0.454939 0.890523i $$-0.650339\pi$$
−0.454939 + 0.890523i $$0.650339\pi$$
$$948$$ 0 0
$$949$$ −36.0000 −1.16861
$$950$$ 8.00000 0.259554
$$951$$ 0 0
$$952$$ −24.0000 −0.777844
$$953$$ −18.0000 −0.583077 −0.291539 0.956559i $$-0.594167\pi$$
−0.291539 + 0.956559i $$0.594167\pi$$
$$954$$ 0 0
$$955$$ −4.00000 −0.129437
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ −12.0000 −0.387702
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 36.0000 1.16069
$$963$$ 0 0
$$964$$ −2.00000 −0.0644157
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ 56.0000 1.80084 0.900419 0.435023i $$-0.143260\pi$$
0.900419 + 0.435023i $$0.143260\pi$$
$$968$$ 33.0000 1.06066
$$969$$ 0 0
$$970$$ 14.0000 0.449513
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ −80.0000 −2.56468
$$974$$ −20.0000 −0.640841
$$975$$ 0 0
$$976$$ −6.00000 −0.192055
$$977$$ 6.00000 0.191957 0.0959785 0.995383i $$-0.469402\pi$$
0.0959785 + 0.995383i $$0.469402\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 9.00000 0.287494
$$981$$ 0 0
$$982$$ 24.0000 0.765871
$$983$$ 16.0000 0.510321 0.255160 0.966899i $$-0.417872\pi$$
0.255160 + 0.966899i $$0.417872\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 2.00000 0.0636930
$$987$$ 0 0
$$988$$ −48.0000 −1.52708
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 20.0000 0.635001
$$993$$ 0 0
$$994$$ 64.0000 2.02996
$$995$$ 16.0000 0.507234
$$996$$ 0 0
$$997$$ 38.0000 1.20347 0.601736 0.798695i $$-0.294476\pi$$
0.601736 + 0.798695i $$0.294476\pi$$
$$998$$ 36.0000 1.13956
$$999$$ 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.e.1.1 1
3.2 odd 2 435.2.a.a.1.1 1
5.4 even 2 6525.2.a.e.1.1 1
12.11 even 2 6960.2.a.w.1.1 1
15.2 even 4 2175.2.c.a.349.1 2
15.8 even 4 2175.2.c.a.349.2 2
15.14 odd 2 2175.2.a.h.1.1 1

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