# Properties

 Label 364.2.a.a.1.1 Level $364$ Weight $2$ Character 364.1 Self dual yes Analytic conductor $2.907$ 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] = [364,2,Mod(1,364)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(364, 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("364.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$364 = 2^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 364.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: $$2.90655463357$$ 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 $$\chi$$ $$=$$ 364.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} +1.00000 q^{13} -2.00000 q^{15} -2.00000 q^{17} -1.00000 q^{19} +2.00000 q^{21} -7.00000 q^{23} -4.00000 q^{25} +4.00000 q^{27} -5.00000 q^{29} -9.00000 q^{31} +8.00000 q^{33} -1.00000 q^{35} -2.00000 q^{37} -2.00000 q^{39} +2.00000 q^{41} +1.00000 q^{43} +1.00000 q^{45} +9.00000 q^{47} +1.00000 q^{49} +4.00000 q^{51} +3.00000 q^{53} -4.00000 q^{55} +2.00000 q^{57} +14.0000 q^{61} -1.00000 q^{63} +1.00000 q^{65} +10.0000 q^{67} +14.0000 q^{69} -14.0000 q^{71} +3.00000 q^{73} +8.00000 q^{75} +4.00000 q^{77} +5.00000 q^{79} -11.0000 q^{81} +5.00000 q^{83} -2.00000 q^{85} +10.0000 q^{87} -9.00000 q^{89} -1.00000 q^{91} +18.0000 q^{93} -1.00000 q^{95} -1.00000 q^{97} -4.00000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ −7.00000 −1.45960 −0.729800 0.683660i $$-0.760387\pi$$
−0.729800 + 0.683660i $$0.760387\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ −9.00000 −1.61645 −0.808224 0.588875i $$-0.799571\pi$$
−0.808224 + 0.588875i $$0.799571\pi$$
$$32$$ 0 0
$$33$$ 8.00000 1.39262
$$34$$ 0 0
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ 3.00000 0.412082 0.206041 0.978543i $$-0.433942\pi$$
0.206041 + 0.978543i $$0.433942\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ 10.0000 1.22169 0.610847 0.791748i $$-0.290829\pi$$
0.610847 + 0.791748i $$0.290829\pi$$
$$68$$ 0 0
$$69$$ 14.0000 1.68540
$$70$$ 0 0
$$71$$ −14.0000 −1.66149 −0.830747 0.556650i $$-0.812086\pi$$
−0.830747 + 0.556650i $$0.812086\pi$$
$$72$$ 0 0
$$73$$ 3.00000 0.351123 0.175562 0.984468i $$-0.443826\pi$$
0.175562 + 0.984468i $$0.443826\pi$$
$$74$$ 0 0
$$75$$ 8.00000 0.923760
$$76$$ 0 0
$$77$$ 4.00000 0.455842
$$78$$ 0 0
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 5.00000 0.548821 0.274411 0.961613i $$-0.411517\pi$$
0.274411 + 0.961613i $$0.411517\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ 0 0
$$87$$ 10.0000 1.07211
$$88$$ 0 0
$$89$$ −9.00000 −0.953998 −0.476999 0.878904i $$-0.658275\pi$$
−0.476999 + 0.878904i $$0.658275\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ 18.0000 1.86651
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −1.00000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ 0 0
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ 1.00000 0.0940721 0.0470360 0.998893i $$-0.485022\pi$$
0.0470360 + 0.998893i $$0.485022\pi$$
$$114$$ 0 0
$$115$$ −7.00000 −0.652753
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ 2.00000 0.183340
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ −4.00000 −0.360668
$$124$$ 0 0
$$125$$ −9.00000 −0.804984
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 0 0
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 1.00000 0.0867110
$$134$$ 0 0
$$135$$ 4.00000 0.344265
$$136$$ 0 0
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 0 0
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ −18.0000 −1.51587
$$142$$ 0 0
$$143$$ −4.00000 −0.334497
$$144$$ 0 0
$$145$$ −5.00000 −0.415227
$$146$$ 0 0
$$147$$ −2.00000 −0.164957
$$148$$ 0 0
$$149$$ 12.0000 0.983078 0.491539 0.870855i $$-0.336434\pi$$
0.491539 + 0.870855i $$0.336434\pi$$
$$150$$ 0 0
$$151$$ 14.0000 1.13930 0.569652 0.821886i $$-0.307078\pi$$
0.569652 + 0.821886i $$0.307078\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ −9.00000 −0.722897
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 7.00000 0.551677
$$162$$ 0 0
$$163$$ 24.0000 1.87983 0.939913 0.341415i $$-0.110906\pi$$
0.939913 + 0.341415i $$0.110906\pi$$
$$164$$ 0 0
$$165$$ 8.00000 0.622799
$$166$$ 0 0
$$167$$ 3.00000 0.232147 0.116073 0.993241i $$-0.462969\pi$$
0.116073 + 0.993241i $$0.462969\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ 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
$$178$$ 0 0
$$179$$ 9.00000 0.672692 0.336346 0.941739i $$-0.390809\pi$$
0.336346 + 0.941739i $$0.390809\pi$$
$$180$$ 0 0
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 0 0
$$183$$ −28.0000 −2.06982
$$184$$ 0 0
$$185$$ −2.00000 −0.147043
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 0 0
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ −24.0000 −1.73658 −0.868290 0.496058i $$-0.834780\pi$$
−0.868290 + 0.496058i $$0.834780\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ −2.00000 −0.143223
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 0 0
$$201$$ −20.0000 −1.41069
$$202$$ 0 0
$$203$$ 5.00000 0.350931
$$204$$ 0 0
$$205$$ 2.00000 0.139686
$$206$$ 0 0
$$207$$ −7.00000 −0.486534
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −23.0000 −1.58339 −0.791693 0.610920i $$-0.790800\pi$$
−0.791693 + 0.610920i $$0.790800\pi$$
$$212$$ 0 0
$$213$$ 28.0000 1.91853
$$214$$ 0 0
$$215$$ 1.00000 0.0681994
$$216$$ 0 0
$$217$$ 9.00000 0.610960
$$218$$ 0 0
$$219$$ −6.00000 −0.405442
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ 21.0000 1.40626 0.703132 0.711059i $$-0.251784\pi$$
0.703132 + 0.711059i $$0.251784\pi$$
$$224$$ 0 0
$$225$$ −4.00000 −0.266667
$$226$$ 0 0
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ −8.00000 −0.526361
$$232$$ 0 0
$$233$$ −25.0000 −1.63780 −0.818902 0.573933i $$-0.805417\pi$$
−0.818902 + 0.573933i $$0.805417\pi$$
$$234$$ 0 0
$$235$$ 9.00000 0.587095
$$236$$ 0 0
$$237$$ −10.0000 −0.649570
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −25.0000 −1.61039 −0.805196 0.593009i $$-0.797940\pi$$
−0.805196 + 0.593009i $$0.797940\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ 0 0
$$245$$ 1.00000 0.0638877
$$246$$ 0 0
$$247$$ −1.00000 −0.0636285
$$248$$ 0 0
$$249$$ −10.0000 −0.633724
$$250$$ 0 0
$$251$$ −16.0000 −1.00991 −0.504956 0.863145i $$-0.668491\pi$$
−0.504956 + 0.863145i $$0.668491\pi$$
$$252$$ 0 0
$$253$$ 28.0000 1.76034
$$254$$ 0 0
$$255$$ 4.00000 0.250490
$$256$$ 0 0
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ −5.00000 −0.309492
$$262$$ 0 0
$$263$$ 19.0000 1.17159 0.585795 0.810459i $$-0.300782\pi$$
0.585795 + 0.810459i $$0.300782\pi$$
$$264$$ 0 0
$$265$$ 3.00000 0.184289
$$266$$ 0 0
$$267$$ 18.0000 1.10158
$$268$$ 0 0
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 2.00000 0.121046
$$274$$ 0 0
$$275$$ 16.0000 0.964836
$$276$$ 0 0
$$277$$ −23.0000 −1.38194 −0.690968 0.722885i $$-0.742815\pi$$
−0.690968 + 0.722885i $$0.742815\pi$$
$$278$$ 0 0
$$279$$ −9.00000 −0.538816
$$280$$ 0 0
$$281$$ −8.00000 −0.477240 −0.238620 0.971113i $$-0.576695\pi$$
−0.238620 + 0.971113i $$0.576695\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ −2.00000 −0.118056
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 0 0
$$293$$ 5.00000 0.292103 0.146052 0.989277i $$-0.453343\pi$$
0.146052 + 0.989277i $$0.453343\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −16.0000 −0.928414
$$298$$ 0 0
$$299$$ −7.00000 −0.404820
$$300$$ 0 0
$$301$$ −1.00000 −0.0576390
$$302$$ 0 0
$$303$$ 24.0000 1.37876
$$304$$ 0 0
$$305$$ 14.0000 0.801638
$$306$$ 0 0
$$307$$ −27.0000 −1.54097 −0.770486 0.637457i $$-0.779986\pi$$
−0.770486 + 0.637457i $$0.779986\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −12.0000 −0.678280 −0.339140 0.940736i $$-0.610136\pi$$
−0.339140 + 0.940736i $$0.610136\pi$$
$$314$$ 0 0
$$315$$ −1.00000 −0.0563436
$$316$$ 0 0
$$317$$ 20.0000 1.12331 0.561656 0.827371i $$-0.310164\pi$$
0.561656 + 0.827371i $$0.310164\pi$$
$$318$$ 0 0
$$319$$ 20.0000 1.11979
$$320$$ 0 0
$$321$$ 24.0000 1.33955
$$322$$ 0 0
$$323$$ 2.00000 0.111283
$$324$$ 0 0
$$325$$ −4.00000 −0.221880
$$326$$ 0 0
$$327$$ 8.00000 0.442401
$$328$$ 0 0
$$329$$ −9.00000 −0.496186
$$330$$ 0 0
$$331$$ 30.0000 1.64895 0.824475 0.565899i $$-0.191471\pi$$
0.824475 + 0.565899i $$0.191471\pi$$
$$332$$ 0 0
$$333$$ −2.00000 −0.109599
$$334$$ 0 0
$$335$$ 10.0000 0.546358
$$336$$ 0 0
$$337$$ 21.0000 1.14394 0.571971 0.820274i $$-0.306179\pi$$
0.571971 + 0.820274i $$0.306179\pi$$
$$338$$ 0 0
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ 36.0000 1.94951
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 14.0000 0.753735
$$346$$ 0 0
$$347$$ 16.0000 0.858925 0.429463 0.903085i $$-0.358703\pi$$
0.429463 + 0.903085i $$0.358703\pi$$
$$348$$ 0 0
$$349$$ 35.0000 1.87351 0.936754 0.349990i $$-0.113815\pi$$
0.936754 + 0.349990i $$0.113815\pi$$
$$350$$ 0 0
$$351$$ 4.00000 0.213504
$$352$$ 0 0
$$353$$ −2.00000 −0.106449 −0.0532246 0.998583i $$-0.516950\pi$$
−0.0532246 + 0.998583i $$0.516950\pi$$
$$354$$ 0 0
$$355$$ −14.0000 −0.743043
$$356$$ 0 0
$$357$$ −4.00000 −0.211702
$$358$$ 0 0
$$359$$ −30.0000 −1.58334 −0.791670 0.610949i $$-0.790788\pi$$
−0.791670 + 0.610949i $$0.790788\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ −10.0000 −0.524864
$$364$$ 0 0
$$365$$ 3.00000 0.157027
$$366$$ 0 0
$$367$$ 24.0000 1.25279 0.626395 0.779506i $$-0.284530\pi$$
0.626395 + 0.779506i $$0.284530\pi$$
$$368$$ 0 0
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ −3.00000 −0.155752
$$372$$ 0 0
$$373$$ −26.0000 −1.34623 −0.673114 0.739538i $$-0.735044\pi$$
−0.673114 + 0.739538i $$0.735044\pi$$
$$374$$ 0 0
$$375$$ 18.0000 0.929516
$$376$$ 0 0
$$377$$ −5.00000 −0.257513
$$378$$ 0 0
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ 0 0
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ 4.00000 0.203859
$$386$$ 0 0
$$387$$ 1.00000 0.0508329
$$388$$ 0 0
$$389$$ −22.0000 −1.11544 −0.557722 0.830028i $$-0.688325\pi$$
−0.557722 + 0.830028i $$0.688325\pi$$
$$390$$ 0 0
$$391$$ 14.0000 0.708010
$$392$$ 0 0
$$393$$ 24.0000 1.21064
$$394$$ 0 0
$$395$$ 5.00000 0.251577
$$396$$ 0 0
$$397$$ −25.0000 −1.25471 −0.627357 0.778732i $$-0.715863\pi$$
−0.627357 + 0.778732i $$0.715863\pi$$
$$398$$ 0 0
$$399$$ −2.00000 −0.100125
$$400$$ 0 0
$$401$$ −28.0000 −1.39825 −0.699127 0.714998i $$-0.746428\pi$$
−0.699127 + 0.714998i $$0.746428\pi$$
$$402$$ 0 0
$$403$$ −9.00000 −0.448322
$$404$$ 0 0
$$405$$ −11.0000 −0.546594
$$406$$ 0 0
$$407$$ 8.00000 0.396545
$$408$$ 0 0
$$409$$ −25.0000 −1.23617 −0.618085 0.786111i $$-0.712091\pi$$
−0.618085 + 0.786111i $$0.712091\pi$$
$$410$$ 0 0
$$411$$ 28.0000 1.38114
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 5.00000 0.245440
$$416$$ 0 0
$$417$$ −32.0000 −1.56705
$$418$$ 0 0
$$419$$ −16.0000 −0.781651 −0.390826 0.920465i $$-0.627810\pi$$
−0.390826 + 0.920465i $$0.627810\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 0 0
$$423$$ 9.00000 0.437595
$$424$$ 0 0
$$425$$ 8.00000 0.388057
$$426$$ 0 0
$$427$$ −14.0000 −0.677507
$$428$$ 0 0
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ 14.0000 0.674356 0.337178 0.941441i $$-0.390528\pi$$
0.337178 + 0.941441i $$0.390528\pi$$
$$432$$ 0 0
$$433$$ 24.0000 1.15337 0.576683 0.816968i $$-0.304347\pi$$
0.576683 + 0.816968i $$0.304347\pi$$
$$434$$ 0 0
$$435$$ 10.0000 0.479463
$$436$$ 0 0
$$437$$ 7.00000 0.334855
$$438$$ 0 0
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −39.0000 −1.85295 −0.926473 0.376361i $$-0.877175\pi$$
−0.926473 + 0.376361i $$0.877175\pi$$
$$444$$ 0 0
$$445$$ −9.00000 −0.426641
$$446$$ 0 0
$$447$$ −24.0000 −1.13516
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ 0 0
$$453$$ −28.0000 −1.31555
$$454$$ 0 0
$$455$$ −1.00000 −0.0468807
$$456$$ 0 0
$$457$$ −6.00000 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$458$$ 0 0
$$459$$ −8.00000 −0.373408
$$460$$ 0 0
$$461$$ 10.0000 0.465746 0.232873 0.972507i $$-0.425187\pi$$
0.232873 + 0.972507i $$0.425187\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ 0 0
$$465$$ 18.0000 0.834730
$$466$$ 0 0
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ 0 0
$$469$$ −10.0000 −0.461757
$$470$$ 0 0
$$471$$ −20.0000 −0.921551
$$472$$ 0 0
$$473$$ −4.00000 −0.183920
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ 3.00000 0.137361
$$478$$ 0 0
$$479$$ 27.0000 1.23366 0.616831 0.787096i $$-0.288416\pi$$
0.616831 + 0.787096i $$0.288416\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 0 0
$$483$$ −14.0000 −0.637022
$$484$$ 0 0
$$485$$ −1.00000 −0.0454077
$$486$$ 0 0
$$487$$ −34.0000 −1.54069 −0.770344 0.637629i $$-0.779915\pi$$
−0.770344 + 0.637629i $$0.779915\pi$$
$$488$$ 0 0
$$489$$ −48.0000 −2.17064
$$490$$ 0 0
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ 0 0
$$493$$ 10.0000 0.450377
$$494$$ 0 0
$$495$$ −4.00000 −0.179787
$$496$$ 0 0
$$497$$ 14.0000 0.627986
$$498$$ 0 0
$$499$$ −2.00000 −0.0895323 −0.0447661 0.998997i $$-0.514254\pi$$
−0.0447661 + 0.998997i $$0.514254\pi$$
$$500$$ 0 0
$$501$$ −6.00000 −0.268060
$$502$$ 0 0
$$503$$ −34.0000 −1.51599 −0.757993 0.652263i $$-0.773820\pi$$
−0.757993 + 0.652263i $$0.773820\pi$$
$$504$$ 0 0
$$505$$ −12.0000 −0.533993
$$506$$ 0 0
$$507$$ −2.00000 −0.0888231
$$508$$ 0 0
$$509$$ 9.00000 0.398918 0.199459 0.979906i $$-0.436082\pi$$
0.199459 + 0.979906i $$0.436082\pi$$
$$510$$ 0 0
$$511$$ −3.00000 −0.132712
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ 4.00000 0.176261
$$516$$ 0 0
$$517$$ −36.0000 −1.58328
$$518$$ 0 0
$$519$$ −4.00000 −0.175581
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ −18.0000 −0.787085 −0.393543 0.919306i $$-0.628751\pi$$
−0.393543 + 0.919306i $$0.628751\pi$$
$$524$$ 0 0
$$525$$ −8.00000 −0.349149
$$526$$ 0 0
$$527$$ 18.0000 0.784092
$$528$$ 0 0
$$529$$ 26.0000 1.13043
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 2.00000 0.0866296
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ 0 0
$$537$$ −18.0000 −0.776757
$$538$$ 0 0
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −38.0000 −1.63375 −0.816874 0.576816i $$-0.804295\pi$$
−0.816874 + 0.576816i $$0.804295\pi$$
$$542$$ 0 0
$$543$$ 40.0000 1.71656
$$544$$ 0 0
$$545$$ −4.00000 −0.171341
$$546$$ 0 0
$$547$$ 23.0000 0.983409 0.491704 0.870762i $$-0.336374\pi$$
0.491704 + 0.870762i $$0.336374\pi$$
$$548$$ 0 0
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ 5.00000 0.213007
$$552$$ 0 0
$$553$$ −5.00000 −0.212622
$$554$$ 0 0
$$555$$ 4.00000 0.169791
$$556$$ 0 0
$$557$$ 4.00000 0.169485 0.0847427 0.996403i $$-0.472993\pi$$
0.0847427 + 0.996403i $$0.472993\pi$$
$$558$$ 0 0
$$559$$ 1.00000 0.0422955
$$560$$ 0 0
$$561$$ −16.0000 −0.675521
$$562$$ 0 0
$$563$$ −40.0000 −1.68580 −0.842900 0.538071i $$-0.819153\pi$$
−0.842900 + 0.538071i $$0.819153\pi$$
$$564$$ 0 0
$$565$$ 1.00000 0.0420703
$$566$$ 0 0
$$567$$ 11.0000 0.461957
$$568$$ 0 0
$$569$$ −37.0000 −1.55112 −0.775560 0.631273i $$-0.782533\pi$$
−0.775560 + 0.631273i $$0.782533\pi$$
$$570$$ 0 0
$$571$$ −7.00000 −0.292941 −0.146470 0.989215i $$-0.546791\pi$$
−0.146470 + 0.989215i $$0.546791\pi$$
$$572$$ 0 0
$$573$$ 48.0000 2.00523
$$574$$ 0 0
$$575$$ 28.0000 1.16768
$$576$$ 0 0
$$577$$ −18.0000 −0.749350 −0.374675 0.927156i $$-0.622246\pi$$
−0.374675 + 0.927156i $$0.622246\pi$$
$$578$$ 0 0
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ −5.00000 −0.207435
$$582$$ 0 0
$$583$$ −12.0000 −0.496989
$$584$$ 0 0
$$585$$ 1.00000 0.0413449
$$586$$ 0 0
$$587$$ −15.0000 −0.619116 −0.309558 0.950881i $$-0.600181\pi$$
−0.309558 + 0.950881i $$0.600181\pi$$
$$588$$ 0 0
$$589$$ 9.00000 0.370839
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 0 0
$$593$$ 41.0000 1.68367 0.841834 0.539736i $$-0.181476\pi$$
0.841834 + 0.539736i $$0.181476\pi$$
$$594$$ 0 0
$$595$$ 2.00000 0.0819920
$$596$$ 0 0
$$597$$ 28.0000 1.14596
$$598$$ 0 0
$$599$$ 21.0000 0.858037 0.429018 0.903296i $$-0.358860\pi$$
0.429018 + 0.903296i $$0.358860\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ 10.0000 0.407231
$$604$$ 0 0
$$605$$ 5.00000 0.203279
$$606$$ 0 0
$$607$$ −40.0000 −1.62355 −0.811775 0.583970i $$-0.801498\pi$$
−0.811775 + 0.583970i $$0.801498\pi$$
$$608$$ 0 0
$$609$$ −10.0000 −0.405220
$$610$$ 0 0
$$611$$ 9.00000 0.364101
$$612$$ 0 0
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ 0 0
$$615$$ −4.00000 −0.161296
$$616$$ 0 0
$$617$$ 34.0000 1.36879 0.684394 0.729112i $$-0.260067\pi$$
0.684394 + 0.729112i $$0.260067\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ −28.0000 −1.12360
$$622$$ 0 0
$$623$$ 9.00000 0.360577
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ −8.00000 −0.319489
$$628$$ 0 0
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ −2.00000 −0.0796187 −0.0398094 0.999207i $$-0.512675\pi$$
−0.0398094 + 0.999207i $$0.512675\pi$$
$$632$$ 0 0
$$633$$ 46.0000 1.82834
$$634$$ 0 0
$$635$$ −8.00000 −0.317470
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 0 0
$$639$$ −14.0000 −0.553831
$$640$$ 0 0
$$641$$ 41.0000 1.61940 0.809701 0.586842i $$-0.199629\pi$$
0.809701 + 0.586842i $$0.199629\pi$$
$$642$$ 0 0
$$643$$ −28.0000 −1.10421 −0.552106 0.833774i $$-0.686176\pi$$
−0.552106 + 0.833774i $$0.686176\pi$$
$$644$$ 0 0
$$645$$ −2.00000 −0.0787499
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −18.0000 −0.705476
$$652$$ 0 0
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ 0 0
$$657$$ 3.00000 0.117041
$$658$$ 0 0
$$659$$ −45.0000 −1.75295 −0.876476 0.481446i $$-0.840112\pi$$
−0.876476 + 0.481446i $$0.840112\pi$$
$$660$$ 0 0
$$661$$ 31.0000 1.20576 0.602880 0.797832i $$-0.294020\pi$$
0.602880 + 0.797832i $$0.294020\pi$$
$$662$$ 0 0
$$663$$ 4.00000 0.155347
$$664$$ 0 0
$$665$$ 1.00000 0.0387783
$$666$$ 0 0
$$667$$ 35.0000 1.35521
$$668$$ 0 0
$$669$$ −42.0000 −1.62381
$$670$$ 0 0
$$671$$ −56.0000 −2.16186
$$672$$ 0 0
$$673$$ 21.0000 0.809491 0.404745 0.914429i $$-0.367360\pi$$
0.404745 + 0.914429i $$0.367360\pi$$
$$674$$ 0 0
$$675$$ −16.0000 −0.615840
$$676$$ 0 0
$$677$$ 16.0000 0.614930 0.307465 0.951559i $$-0.400519\pi$$
0.307465 + 0.951559i $$0.400519\pi$$
$$678$$ 0 0
$$679$$ 1.00000 0.0383765
$$680$$ 0 0
$$681$$ 16.0000 0.613121
$$682$$ 0 0
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ 0 0
$$685$$ −14.0000 −0.534913
$$686$$ 0 0
$$687$$ 4.00000 0.152610
$$688$$ 0 0
$$689$$ 3.00000 0.114291
$$690$$ 0 0
$$691$$ −35.0000 −1.33146 −0.665731 0.746191i $$-0.731880\pi$$
−0.665731 + 0.746191i $$0.731880\pi$$
$$692$$ 0 0
$$693$$ 4.00000 0.151947
$$694$$ 0 0
$$695$$ 16.0000 0.606915
$$696$$ 0 0
$$697$$ −4.00000 −0.151511
$$698$$ 0 0
$$699$$ 50.0000 1.89117
$$700$$ 0 0
$$701$$ 45.0000 1.69963 0.849813 0.527084i $$-0.176715\pi$$
0.849813 + 0.527084i $$0.176715\pi$$
$$702$$ 0 0
$$703$$ 2.00000 0.0754314
$$704$$ 0 0
$$705$$ −18.0000 −0.677919
$$706$$ 0 0
$$707$$ 12.0000 0.451306
$$708$$ 0 0
$$709$$ −12.0000 −0.450669 −0.225335 0.974281i $$-0.572348\pi$$
−0.225335 + 0.974281i $$0.572348\pi$$
$$710$$ 0 0
$$711$$ 5.00000 0.187515
$$712$$ 0 0
$$713$$ 63.0000 2.35937
$$714$$ 0 0
$$715$$ −4.00000 −0.149592
$$716$$ 0 0
$$717$$ −24.0000 −0.896296
$$718$$ 0 0
$$719$$ −40.0000 −1.49175 −0.745874 0.666087i $$-0.767968\pi$$
−0.745874 + 0.666087i $$0.767968\pi$$
$$720$$ 0 0
$$721$$ −4.00000 −0.148968
$$722$$ 0 0
$$723$$ 50.0000 1.85952
$$724$$ 0 0
$$725$$ 20.0000 0.742781
$$726$$ 0 0
$$727$$ −42.0000 −1.55769 −0.778847 0.627214i $$-0.784195\pi$$
−0.778847 + 0.627214i $$0.784195\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −2.00000 −0.0739727
$$732$$ 0 0
$$733$$ −21.0000 −0.775653 −0.387826 0.921732i $$-0.626774\pi$$
−0.387826 + 0.921732i $$0.626774\pi$$
$$734$$ 0 0
$$735$$ −2.00000 −0.0737711
$$736$$ 0 0
$$737$$ −40.0000 −1.47342
$$738$$ 0 0
$$739$$ −8.00000 −0.294285 −0.147142 0.989115i $$-0.547008\pi$$
−0.147142 + 0.989115i $$0.547008\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 0 0
$$743$$ −8.00000 −0.293492 −0.146746 0.989174i $$-0.546880\pi$$
−0.146746 + 0.989174i $$0.546880\pi$$
$$744$$ 0 0
$$745$$ 12.0000 0.439646
$$746$$ 0 0
$$747$$ 5.00000 0.182940
$$748$$ 0 0
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ 25.0000 0.912263 0.456131 0.889912i $$-0.349235\pi$$
0.456131 + 0.889912i $$0.349235\pi$$
$$752$$ 0 0
$$753$$ 32.0000 1.16614
$$754$$ 0 0
$$755$$ 14.0000 0.509512
$$756$$ 0 0
$$757$$ −47.0000 −1.70824 −0.854122 0.520073i $$-0.825905\pi$$
−0.854122 + 0.520073i $$0.825905\pi$$
$$758$$ 0 0
$$759$$ −56.0000 −2.03267
$$760$$ 0 0
$$761$$ −35.0000 −1.26875 −0.634375 0.773026i $$-0.718742\pi$$
−0.634375 + 0.773026i $$0.718742\pi$$
$$762$$ 0 0
$$763$$ 4.00000 0.144810
$$764$$ 0 0
$$765$$ −2.00000 −0.0723102
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 37.0000 1.33425 0.667127 0.744944i $$-0.267524\pi$$
0.667127 + 0.744944i $$0.267524\pi$$
$$770$$ 0 0
$$771$$ 24.0000 0.864339
$$772$$ 0 0
$$773$$ 38.0000 1.36677 0.683383 0.730061i $$-0.260508\pi$$
0.683383 + 0.730061i $$0.260508\pi$$
$$774$$ 0 0
$$775$$ 36.0000 1.29316
$$776$$ 0 0
$$777$$ −4.00000 −0.143499
$$778$$ 0 0
$$779$$ −2.00000 −0.0716574
$$780$$ 0 0
$$781$$ 56.0000 2.00384
$$782$$ 0 0
$$783$$ −20.0000 −0.714742
$$784$$ 0 0
$$785$$ 10.0000 0.356915
$$786$$ 0 0
$$787$$ −37.0000 −1.31891 −0.659454 0.751745i $$-0.729212\pi$$
−0.659454 + 0.751745i $$0.729212\pi$$
$$788$$ 0 0
$$789$$ −38.0000 −1.35284
$$790$$ 0 0
$$791$$ −1.00000 −0.0355559
$$792$$ 0 0
$$793$$ 14.0000 0.497155
$$794$$ 0 0
$$795$$ −6.00000 −0.212798
$$796$$ 0 0
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 0 0
$$799$$ −18.0000 −0.636794
$$800$$ 0 0
$$801$$ −9.00000 −0.317999
$$802$$ 0 0
$$803$$ −12.0000 −0.423471
$$804$$ 0 0
$$805$$ 7.00000 0.246718
$$806$$ 0 0
$$807$$ −48.0000 −1.68968
$$808$$ 0 0
$$809$$ 45.0000 1.58212 0.791058 0.611741i $$-0.209531\pi$$
0.791058 + 0.611741i $$0.209531\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 24.0000 0.840683
$$816$$ 0 0
$$817$$ −1.00000 −0.0349856
$$818$$ 0 0
$$819$$ −1.00000 −0.0349428
$$820$$ 0 0
$$821$$ 34.0000 1.18661 0.593304 0.804978i $$-0.297823\pi$$
0.593304 + 0.804978i $$0.297823\pi$$
$$822$$ 0 0
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 0 0
$$825$$ −32.0000 −1.11410
$$826$$ 0 0
$$827$$ 40.0000 1.39094 0.695468 0.718557i $$-0.255197\pi$$
0.695468 + 0.718557i $$0.255197\pi$$
$$828$$ 0 0
$$829$$ 24.0000 0.833554 0.416777 0.909009i $$-0.363160\pi$$
0.416777 + 0.909009i $$0.363160\pi$$
$$830$$ 0 0
$$831$$ 46.0000 1.59572
$$832$$ 0 0
$$833$$ −2.00000 −0.0692959
$$834$$ 0 0
$$835$$ 3.00000 0.103819
$$836$$ 0 0
$$837$$ −36.0000 −1.24434
$$838$$ 0 0
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 0 0
$$843$$ 16.0000 0.551069
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ −5.00000 −0.171802
$$848$$ 0 0
$$849$$ 8.00000 0.274559
$$850$$ 0 0
$$851$$ 14.0000 0.479914
$$852$$ 0 0
$$853$$ 37.0000 1.26686 0.633428 0.773802i $$-0.281647\pi$$
0.633428 + 0.773802i $$0.281647\pi$$
$$854$$ 0 0
$$855$$ −1.00000 −0.0341993
$$856$$ 0 0
$$857$$ 30.0000 1.02478 0.512390 0.858753i $$-0.328760\pi$$
0.512390 + 0.858753i $$0.328760\pi$$
$$858$$ 0 0
$$859$$ 22.0000 0.750630 0.375315 0.926897i $$-0.377534\pi$$
0.375315 + 0.926897i $$0.377534\pi$$
$$860$$ 0 0
$$861$$ 4.00000 0.136320
$$862$$ 0 0
$$863$$ −48.0000 −1.63394 −0.816970 0.576681i $$-0.804348\pi$$
−0.816970 + 0.576681i $$0.804348\pi$$
$$864$$ 0 0
$$865$$ 2.00000 0.0680020
$$866$$ 0 0
$$867$$ 26.0000 0.883006
$$868$$ 0 0
$$869$$ −20.0000 −0.678454
$$870$$ 0 0
$$871$$ 10.0000 0.338837
$$872$$ 0 0
$$873$$ −1.00000 −0.0338449
$$874$$ 0 0
$$875$$ 9.00000 0.304256
$$876$$ 0 0
$$877$$ 14.0000 0.472746 0.236373 0.971662i $$-0.424041\pi$$
0.236373 + 0.971662i $$0.424041\pi$$
$$878$$ 0 0
$$879$$ −10.0000 −0.337292
$$880$$ 0 0
$$881$$ −2.00000 −0.0673817 −0.0336909 0.999432i $$-0.510726\pi$$
−0.0336909 + 0.999432i $$0.510726\pi$$
$$882$$ 0 0
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 32.0000 1.07445 0.537227 0.843437i $$-0.319472\pi$$
0.537227 + 0.843437i $$0.319472\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ 44.0000 1.47406
$$892$$ 0 0
$$893$$ −9.00000 −0.301174
$$894$$ 0 0
$$895$$ 9.00000 0.300837
$$896$$ 0 0
$$897$$ 14.0000 0.467446
$$898$$ 0 0
$$899$$ 45.0000 1.50083
$$900$$ 0 0
$$901$$ −6.00000 −0.199889
$$902$$ 0 0
$$903$$ 2.00000 0.0665558
$$904$$ 0 0
$$905$$ −20.0000 −0.664822
$$906$$ 0 0
$$907$$ −3.00000 −0.0996134 −0.0498067 0.998759i $$-0.515861\pi$$
−0.0498067 + 0.998759i $$0.515861\pi$$
$$908$$ 0 0
$$909$$ −12.0000 −0.398015
$$910$$ 0 0
$$911$$ −21.0000 −0.695761 −0.347881 0.937539i $$-0.613099\pi$$
−0.347881 + 0.937539i $$0.613099\pi$$
$$912$$ 0 0
$$913$$ −20.0000 −0.661903
$$914$$ 0 0
$$915$$ −28.0000 −0.925651
$$916$$ 0 0
$$917$$ 12.0000 0.396275
$$918$$ 0 0
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ 54.0000 1.77936
$$922$$ 0 0
$$923$$ −14.0000 −0.460816
$$924$$ 0 0
$$925$$ 8.00000 0.263038
$$926$$ 0 0
$$927$$ 4.00000 0.131377
$$928$$ 0 0
$$929$$ −3.00000 −0.0984268 −0.0492134 0.998788i $$-0.515671\pi$$
−0.0492134 + 0.998788i $$0.515671\pi$$
$$930$$ 0 0
$$931$$ −1.00000 −0.0327737
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 8.00000 0.261628
$$936$$ 0 0
$$937$$ 30.0000 0.980057 0.490029 0.871706i $$-0.336986\pi$$
0.490029 + 0.871706i $$0.336986\pi$$
$$938$$ 0 0
$$939$$ 24.0000 0.783210
$$940$$ 0 0
$$941$$ 35.0000 1.14097 0.570484 0.821309i $$-0.306756\pi$$
0.570484 + 0.821309i $$0.306756\pi$$
$$942$$ 0 0
$$943$$ −14.0000 −0.455903
$$944$$ 0 0
$$945$$ −4.00000 −0.130120
$$946$$ 0 0
$$947$$ 10.0000 0.324956 0.162478 0.986712i $$-0.448051\pi$$
0.162478 + 0.986712i $$0.448051\pi$$
$$948$$ 0 0
$$949$$ 3.00000 0.0973841
$$950$$ 0 0
$$951$$ −40.0000 −1.29709
$$952$$ 0 0
$$953$$ −19.0000 −0.615470 −0.307735 0.951472i $$-0.599571\pi$$
−0.307735 + 0.951472i $$0.599571\pi$$
$$954$$ 0 0
$$955$$ −24.0000 −0.776622
$$956$$ 0 0
$$957$$ −40.0000 −1.29302
$$958$$ 0 0
$$959$$ 14.0000 0.452084
$$960$$ 0 0
$$961$$ 50.0000 1.61290
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ 0 0
$$965$$ 2.00000 0.0643823
$$966$$ 0 0
$$967$$ −12.0000 −0.385894 −0.192947 0.981209i $$-0.561805\pi$$
−0.192947 + 0.981209i $$0.561805\pi$$
$$968$$ 0 0
$$969$$ −4.00000 −0.128499
$$970$$ 0 0
$$971$$ 30.0000 0.962746 0.481373 0.876516i $$-0.340138\pi$$
0.481373 + 0.876516i $$0.340138\pi$$
$$972$$ 0 0
$$973$$ −16.0000 −0.512936
$$974$$ 0 0
$$975$$ 8.00000 0.256205
$$976$$ 0 0
$$977$$ −60.0000 −1.91957 −0.959785 0.280736i $$-0.909421\pi$$
−0.959785 + 0.280736i $$0.909421\pi$$
$$978$$ 0 0
$$979$$ 36.0000 1.15056
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ 0 0
$$983$$ 11.0000 0.350846 0.175423 0.984493i $$-0.443871\pi$$
0.175423 + 0.984493i $$0.443871\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 0 0
$$987$$ 18.0000 0.572946
$$988$$ 0 0
$$989$$ −7.00000 −0.222587
$$990$$ 0 0
$$991$$ 24.0000 0.762385 0.381193 0.924496i $$-0.375513\pi$$
0.381193 + 0.924496i $$0.375513\pi$$
$$992$$ 0 0
$$993$$ −60.0000 −1.90404
$$994$$ 0 0
$$995$$ −14.0000 −0.443830
$$996$$ 0 0
$$997$$ −32.0000 −1.01345 −0.506725 0.862108i $$-0.669144\pi$$
−0.506725 + 0.862108i $$0.669144\pi$$
$$998$$ 0 0
$$999$$ −8.00000 −0.253109
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 364.2.a.a.1.1 1
3.2 odd 2 3276.2.a.b.1.1 1
4.3 odd 2 1456.2.a.m.1.1 1
5.4 even 2 9100.2.a.l.1.1 1
7.2 even 3 2548.2.j.j.1145.1 2
7.3 odd 6 2548.2.j.c.1353.1 2
7.4 even 3 2548.2.j.j.1353.1 2
7.5 odd 6 2548.2.j.c.1145.1 2
7.6 odd 2 2548.2.a.i.1.1 1
8.3 odd 2 5824.2.a.d.1.1 1
8.5 even 2 5824.2.a.bb.1.1 1
13.5 odd 4 4732.2.g.a.337.1 2
13.8 odd 4 4732.2.g.a.337.2 2
13.12 even 2 4732.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
364.2.a.a.1.1 1 1.1 even 1 trivial
1456.2.a.m.1.1 1 4.3 odd 2
2548.2.a.i.1.1 1 7.6 odd 2
2548.2.j.c.1145.1 2 7.5 odd 6
2548.2.j.c.1353.1 2 7.3 odd 6
2548.2.j.j.1145.1 2 7.2 even 3
2548.2.j.j.1353.1 2 7.4 even 3
3276.2.a.b.1.1 1 3.2 odd 2
4732.2.a.a.1.1 1 13.12 even 2
4732.2.g.a.337.1 2 13.5 odd 4
4732.2.g.a.337.2 2 13.8 odd 4
5824.2.a.d.1.1 1 8.3 odd 2
5824.2.a.bb.1.1 1 8.5 even 2
9100.2.a.l.1.1 1 5.4 even 2