# Properties

 Label 4800.2.f.t.3649.2 Level $4800$ Weight $2$ Character 4800.3649 Analytic conductor $38.328$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4800,2,Mod(3649,4800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4800 = 2^{6} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4800.f (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$38.3281929702$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2400) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3649.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4800.3649 Dual form 4800.2.f.t.3649.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} +1.00000i q^{13} +3.00000 q^{19} -1.00000 q^{21} -4.00000i q^{23} -1.00000i q^{27} +4.00000 q^{29} -7.00000 q^{31} +6.00000i q^{37} -1.00000 q^{39} +6.00000 q^{41} +9.00000i q^{43} +6.00000i q^{47} +6.00000 q^{49} +2.00000i q^{53} +3.00000i q^{57} +10.0000 q^{59} +1.00000 q^{61} -1.00000i q^{63} +3.00000i q^{67} +4.00000 q^{69} -14.0000 q^{71} -10.0000i q^{73} -8.00000 q^{79} +1.00000 q^{81} +18.0000i q^{83} +4.00000i q^{87} -1.00000 q^{91} -7.00000i q^{93} -3.00000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 6 q^{19} - 2 q^{21} + 8 q^{29} - 14 q^{31} - 2 q^{39} + 12 q^{41} + 12 q^{49} + 20 q^{59} + 2 q^{61} + 8 q^{69} - 28 q^{71} - 16 q^{79} + 2 q^{81} - 2 q^{91}+O(q^{100})$$ 2 * q - 2 * q^9 + 6 * q^19 - 2 * q^21 + 8 * q^29 - 14 * q^31 - 2 * q^39 + 12 * q^41 + 12 * q^49 + 20 * q^59 + 2 * q^61 + 8 * q^69 - 28 * q^71 - 16 * q^79 + 2 * q^81 - 2 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1601$$ $$4351$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$3$$ 1.00000i 0.577350i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i 0.981981 + 0.188982i $$0.0605189\pi$$
−0.981981 + 0.188982i $$0.939481\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i 0.990338 + 0.138675i $$0.0442844\pi$$
−0.990338 + 0.138675i $$0.955716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ −7.00000 −1.25724 −0.628619 0.777714i $$-0.716379\pi$$
−0.628619 + 0.777714i $$0.716379\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 9.00000i 1.37249i 0.727372 + 0.686244i $$0.240742\pi$$
−0.727372 + 0.686244i $$0.759258\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.00000i 0.875190i 0.899172 + 0.437595i $$0.144170\pi$$
−0.899172 + 0.437595i $$0.855830\pi$$
$$48$$ 0 0
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 3.00000i 0.397360i
$$58$$ 0 0
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 0 0
$$63$$ − 1.00000i − 0.125988i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.00000i 0.366508i 0.983066 + 0.183254i $$0.0586631\pi$$
−0.983066 + 0.183254i $$0.941337\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −14.0000 −1.66149 −0.830747 0.556650i $$-0.812086\pi$$
−0.830747 + 0.556650i $$0.812086\pi$$
$$72$$ 0 0
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 18.0000i 1.97576i 0.155230 + 0.987878i $$0.450388\pi$$
−0.155230 + 0.987878i $$0.549612\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 4.00000i 0.428845i
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ − 7.00000i − 0.725866i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 3.00000i − 0.304604i −0.988334 0.152302i $$-0.951331\pi$$
0.988334 0.152302i $$-0.0486686\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 2.00000i − 0.193347i −0.995316 0.0966736i $$-0.969180\pi$$
0.995316 0.0966736i $$-0.0308203\pi$$
$$108$$ 0 0
$$109$$ −15.0000 −1.43674 −0.718370 0.695662i $$-0.755111\pi$$
−0.718370 + 0.695662i $$0.755111\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ 12.0000i 1.12887i 0.825479 + 0.564433i $$0.190905\pi$$
−0.825479 + 0.564433i $$0.809095\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 1.00000i − 0.0924500i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 6.00000i 0.541002i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ 0 0
$$129$$ −9.00000 −0.792406
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 0 0
$$133$$ 3.00000i 0.260133i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 10.0000i 0.854358i 0.904167 + 0.427179i $$0.140493\pi$$
−0.904167 + 0.427179i $$0.859507\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 6.00000i 0.494872i
$$148$$ 0 0
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ 0 0
$$151$$ −17.0000 −1.38344 −0.691720 0.722166i $$-0.743147\pi$$
−0.691720 + 0.722166i $$0.743147\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 9.00000i 0.718278i 0.933284 + 0.359139i $$0.116930\pi$$
−0.933284 + 0.359139i $$0.883070\pi$$
$$158$$ 0 0
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 0 0
$$163$$ 19.0000i 1.48819i 0.668071 + 0.744097i $$0.267120\pi$$
−0.668071 + 0.744097i $$0.732880\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.00000i 0.464294i 0.972681 + 0.232147i $$0.0745750\pi$$
−0.972681 + 0.232147i $$0.925425\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ −3.00000 −0.229416
$$172$$ 0 0
$$173$$ − 4.00000i − 0.304114i −0.988372 0.152057i $$-0.951410\pi$$
0.988372 0.152057i $$-0.0485898\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 10.0000i 0.751646i
$$178$$ 0 0
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 0 0
$$181$$ 11.0000 0.817624 0.408812 0.912619i $$-0.365943\pi$$
0.408812 + 0.912619i $$0.365943\pi$$
$$182$$ 0 0
$$183$$ 1.00000i 0.0739221i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ − 11.0000i − 0.791797i −0.918294 0.395899i $$-0.870433\pi$$
0.918294 0.395899i $$-0.129567\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 0 0
$$199$$ 13.0000 0.921546 0.460773 0.887518i $$-0.347572\pi$$
0.460773 + 0.887518i $$0.347572\pi$$
$$200$$ 0 0
$$201$$ −3.00000 −0.211604
$$202$$ 0 0
$$203$$ 4.00000i 0.280745i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.00000i 0.278019i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −7.00000 −0.481900 −0.240950 0.970538i $$-0.577459\pi$$
−0.240950 + 0.970538i $$0.577459\pi$$
$$212$$ 0 0
$$213$$ − 14.0000i − 0.959264i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 7.00000i − 0.475191i
$$218$$ 0 0
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 11.0000i 0.736614i 0.929704 + 0.368307i $$0.120063\pi$$
−0.929704 + 0.368307i $$0.879937\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 14.0000i 0.929213i 0.885517 + 0.464606i $$0.153804\pi$$
−0.885517 + 0.464606i $$0.846196\pi$$
$$228$$ 0 0
$$229$$ −23.0000 −1.51988 −0.759941 0.649992i $$-0.774772\pi$$
−0.759941 + 0.649992i $$0.774772\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 18.0000i 1.17922i 0.807688 + 0.589610i $$0.200718\pi$$
−0.807688 + 0.589610i $$0.799282\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 8.00000i − 0.519656i
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ −11.0000 −0.708572 −0.354286 0.935137i $$-0.615276\pi$$
−0.354286 + 0.935137i $$0.615276\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.00000i 0.190885i
$$248$$ 0 0
$$249$$ −18.0000 −1.14070
$$250$$ 0 0
$$251$$ −16.0000 −1.00991 −0.504956 0.863145i $$-0.668491\pi$$
−0.504956 + 0.863145i $$0.668491\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 16.0000i − 0.998053i −0.866587 0.499026i $$-0.833691\pi$$
0.866587 0.499026i $$-0.166309\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 0 0
$$263$$ 2.00000i 0.123325i 0.998097 + 0.0616626i $$0.0196403\pi$$
−0.998097 + 0.0616626i $$0.980360\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 16.0000 0.975537 0.487769 0.872973i $$-0.337811\pi$$
0.487769 + 0.872973i $$0.337811\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ 0 0
$$273$$ − 1.00000i − 0.0605228i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 11.0000i 0.660926i 0.943819 + 0.330463i $$0.107205\pi$$
−0.943819 + 0.330463i $$0.892795\pi$$
$$278$$ 0 0
$$279$$ 7.00000 0.419079
$$280$$ 0 0
$$281$$ 4.00000 0.238620 0.119310 0.992857i $$-0.461932\pi$$
0.119310 + 0.992857i $$0.461932\pi$$
$$282$$ 0 0
$$283$$ − 5.00000i − 0.297219i −0.988896 0.148610i $$-0.952520\pi$$
0.988896 0.148610i $$-0.0474798\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6.00000i 0.354169i
$$288$$ 0 0
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ 3.00000 0.175863
$$292$$ 0 0
$$293$$ 30.0000i 1.75262i 0.481749 + 0.876309i $$0.340002\pi$$
−0.481749 + 0.876309i $$0.659998\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ −9.00000 −0.518751
$$302$$ 0 0
$$303$$ 6.00000i 0.344691i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 13.0000i 0.741949i 0.928643 + 0.370975i $$0.120976\pi$$
−0.928643 + 0.370975i $$0.879024\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −10.0000 −0.567048 −0.283524 0.958965i $$-0.591504\pi$$
−0.283524 + 0.958965i $$0.591504\pi$$
$$312$$ 0 0
$$313$$ − 23.0000i − 1.30004i −0.759918 0.650018i $$-0.774761\pi$$
0.759918 0.650018i $$-0.225239\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 12.0000i − 0.673987i −0.941507 0.336994i $$-0.890590\pi$$
0.941507 0.336994i $$-0.109410\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 2.00000 0.111629
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 15.0000i − 0.829502i
$$328$$ 0 0
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ −32.0000 −1.75888 −0.879440 0.476011i $$-0.842082\pi$$
−0.879440 + 0.476011i $$0.842082\pi$$
$$332$$ 0 0
$$333$$ − 6.00000i − 0.328798i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 19.0000i − 1.03500i −0.855684 0.517498i $$-0.826864\pi$$
0.855684 0.517498i $$-0.173136\pi$$
$$338$$ 0 0
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 13.0000i 0.701934i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2.00000i 0.107366i 0.998558 + 0.0536828i $$0.0170960\pi$$
−0.998558 + 0.0536828i $$0.982904\pi$$
$$348$$ 0 0
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ 36.0000i 1.91609i 0.286623 + 0.958043i $$0.407467\pi$$
−0.286623 + 0.958043i $$0.592533\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 30.0000 1.58334 0.791670 0.610949i $$-0.209212\pi$$
0.791670 + 0.610949i $$0.209212\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ 0 0
$$363$$ − 11.0000i − 0.577350i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 3.00000i 0.156599i 0.996930 + 0.0782994i $$0.0249490\pi$$
−0.996930 + 0.0782994i $$0.975051\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −2.00000 −0.103835
$$372$$ 0 0
$$373$$ 15.0000i 0.776671i 0.921518 + 0.388335i $$0.126950\pi$$
−0.921518 + 0.388335i $$0.873050\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 4.00000i 0.206010i
$$378$$ 0 0
$$379$$ 13.0000 0.667765 0.333883 0.942615i $$-0.391641\pi$$
0.333883 + 0.942615i $$0.391641\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 0 0
$$383$$ 32.0000i 1.63512i 0.575841 + 0.817562i $$0.304675\pi$$
−0.575841 + 0.817562i $$0.695325\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 9.00000i − 0.457496i
$$388$$ 0 0
$$389$$ 10.0000 0.507020 0.253510 0.967333i $$-0.418415\pi$$
0.253510 + 0.967333i $$0.418415\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ − 6.00000i − 0.302660i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 13.0000i 0.652451i 0.945292 + 0.326226i $$0.105777\pi$$
−0.945292 + 0.326226i $$0.894223\pi$$
$$398$$ 0 0
$$399$$ −3.00000 −0.150188
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ − 7.00000i − 0.348695i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −5.00000 −0.247234 −0.123617 0.992330i $$-0.539449\pi$$
−0.123617 + 0.992330i $$0.539449\pi$$
$$410$$ 0 0
$$411$$ −10.0000 −0.493264
$$412$$ 0 0
$$413$$ 10.0000i 0.492068i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 12.0000i 0.587643i
$$418$$ 0 0
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 0 0
$$423$$ − 6.00000i − 0.291730i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 1.00000i 0.0483934i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ 0 0
$$433$$ − 15.0000i − 0.720854i −0.932787 0.360427i $$-0.882631\pi$$
0.932787 0.360427i $$-0.117369\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 12.0000i − 0.574038i
$$438$$ 0 0
$$439$$ 23.0000 1.09773 0.548865 0.835911i $$-0.315060\pi$$
0.548865 + 0.835911i $$0.315060\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 0 0
$$443$$ − 20.0000i − 0.950229i −0.879924 0.475114i $$-0.842407\pi$$
0.879924 0.475114i $$-0.157593\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 14.0000i 0.662177i
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ − 17.0000i − 0.798730i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 22.0000i − 1.02912i −0.857455 0.514558i $$-0.827956\pi$$
0.857455 0.514558i $$-0.172044\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 20.0000i 0.929479i 0.885448 + 0.464739i $$0.153852\pi$$
−0.885448 + 0.464739i $$0.846148\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 24.0000i 1.11059i 0.831654 + 0.555294i $$0.187394\pi$$
−0.831654 + 0.555294i $$0.812606\pi$$
$$468$$ 0 0
$$469$$ −3.00000 −0.138527
$$470$$ 0 0
$$471$$ −9.00000 −0.414698
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 2.00000i − 0.0915737i
$$478$$ 0 0
$$479$$ −18.0000 −0.822441 −0.411220 0.911536i $$-0.634897\pi$$
−0.411220 + 0.911536i $$0.634897\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ 0 0
$$483$$ 4.00000i 0.182006i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 29.0000i − 1.31412i −0.753840 0.657058i $$-0.771801\pi$$
0.753840 0.657058i $$-0.228199\pi$$
$$488$$ 0 0
$$489$$ −19.0000 −0.859210
$$490$$ 0 0
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 14.0000i − 0.627986i
$$498$$ 0 0
$$499$$ −41.0000 −1.83541 −0.917706 0.397260i $$-0.869961\pi$$
−0.917706 + 0.397260i $$0.869961\pi$$
$$500$$ 0 0
$$501$$ −6.00000 −0.268060
$$502$$ 0 0
$$503$$ 18.0000i 0.802580i 0.915951 + 0.401290i $$0.131438\pi$$
−0.915951 + 0.401290i $$0.868562\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 12.0000i 0.532939i
$$508$$ 0 0
$$509$$ 34.0000 1.50702 0.753512 0.657434i $$-0.228358\pi$$
0.753512 + 0.657434i $$0.228358\pi$$
$$510$$ 0 0
$$511$$ 10.0000 0.442374
$$512$$ 0 0
$$513$$ − 3.00000i − 0.132453i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 4.00000 0.175581
$$520$$ 0 0
$$521$$ 20.0000 0.876216 0.438108 0.898922i $$-0.355649\pi$$
0.438108 + 0.898922i $$0.355649\pi$$
$$522$$ 0 0
$$523$$ − 37.0000i − 1.61790i −0.587879 0.808949i $$-0.700037\pi$$
0.587879 0.808949i $$-0.299963\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −10.0000 −0.433963
$$532$$ 0 0
$$533$$ 6.00000i 0.259889i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 6.00000i 0.258919i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −5.00000 −0.214967 −0.107483 0.994207i $$-0.534279\pi$$
−0.107483 + 0.994207i $$0.534279\pi$$
$$542$$ 0 0
$$543$$ 11.0000i 0.472055i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 16.0000i 0.684111i 0.939680 + 0.342055i $$0.111123\pi$$
−0.939680 + 0.342055i $$0.888877\pi$$
$$548$$ 0 0
$$549$$ −1.00000 −0.0426790
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ − 8.00000i − 0.340195i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 6.00000i 0.254228i 0.991888 + 0.127114i $$0.0405714\pi$$
−0.991888 + 0.127114i $$0.959429\pi$$
$$558$$ 0 0
$$559$$ −9.00000 −0.380659
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 8.00000i − 0.337160i −0.985688 0.168580i $$-0.946082\pi$$
0.985688 0.168580i $$-0.0539181\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.00000i 0.0419961i
$$568$$ 0 0
$$569$$ 20.0000 0.838444 0.419222 0.907884i $$-0.362303\pi$$
0.419222 + 0.907884i $$0.362303\pi$$
$$570$$ 0 0
$$571$$ 5.00000 0.209243 0.104622 0.994512i $$-0.466637\pi$$
0.104622 + 0.994512i $$0.466637\pi$$
$$572$$ 0 0
$$573$$ − 16.0000i − 0.668410i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 31.0000i 1.29055i 0.763952 + 0.645273i $$0.223257\pi$$
−0.763952 + 0.645273i $$0.776743\pi$$
$$578$$ 0 0
$$579$$ 11.0000 0.457144
$$580$$ 0 0
$$581$$ −18.0000 −0.746766
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 18.0000i − 0.742940i −0.928445 0.371470i $$-0.878854\pi$$
0.928445 0.371470i $$-0.121146\pi$$
$$588$$ 0 0
$$589$$ −21.0000 −0.865290
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 0 0
$$593$$ 22.0000i 0.903432i 0.892162 + 0.451716i $$0.149188\pi$$
−0.892162 + 0.451716i $$0.850812\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 13.0000i 0.532055i
$$598$$ 0 0
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ −21.0000 −0.856608 −0.428304 0.903635i $$-0.640889\pi$$
−0.428304 + 0.903635i $$0.640889\pi$$
$$602$$ 0 0
$$603$$ − 3.00000i − 0.122169i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ −4.00000 −0.162088
$$610$$ 0 0
$$611$$ −6.00000 −0.242734
$$612$$ 0 0
$$613$$ − 10.0000i − 0.403896i −0.979396 0.201948i $$-0.935273\pi$$
0.979396 0.201948i $$-0.0647272\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 2.00000i − 0.0805170i −0.999189 0.0402585i $$-0.987182\pi$$
0.999189 0.0402585i $$-0.0128181\pi$$
$$618$$ 0 0
$$619$$ 19.0000 0.763674 0.381837 0.924230i $$-0.375291\pi$$
0.381837 + 0.924230i $$0.375291\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 17.0000 0.676759 0.338380 0.941010i $$-0.390121\pi$$
0.338380 + 0.941010i $$0.390121\pi$$
$$632$$ 0 0
$$633$$ − 7.00000i − 0.278225i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6.00000i 0.237729i
$$638$$ 0 0
$$639$$ 14.0000 0.553831
$$640$$ 0 0
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 0 0
$$643$$ − 12.0000i − 0.473234i −0.971603 0.236617i $$-0.923961\pi$$
0.971603 0.236617i $$-0.0760386\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 28.0000i − 1.10079i −0.834903 0.550397i $$-0.814476\pi$$
0.834903 0.550397i $$-0.185524\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 7.00000 0.274352
$$652$$ 0 0
$$653$$ 10.0000i 0.391330i 0.980671 + 0.195665i $$0.0626866\pi$$
−0.980671 + 0.195665i $$0.937313\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 10.0000i 0.390137i
$$658$$ 0 0
$$659$$ −30.0000 −1.16863 −0.584317 0.811525i $$-0.698638\pi$$
−0.584317 + 0.811525i $$0.698638\pi$$
$$660$$ 0 0
$$661$$ 10.0000 0.388955 0.194477 0.980907i $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 16.0000i − 0.619522i
$$668$$ 0 0
$$669$$ −11.0000 −0.425285
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ − 22.0000i − 0.848038i −0.905653 0.424019i $$-0.860619\pi$$
0.905653 0.424019i $$-0.139381\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 48.0000i − 1.84479i −0.386248 0.922395i $$-0.626229\pi$$
0.386248 0.922395i $$-0.373771\pi$$
$$678$$ 0 0
$$679$$ 3.00000 0.115129
$$680$$ 0 0
$$681$$ −14.0000 −0.536481
$$682$$ 0 0
$$683$$ − 24.0000i − 0.918334i −0.888350 0.459167i $$-0.848148\pi$$
0.888350 0.459167i $$-0.151852\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 23.0000i − 0.877505i
$$688$$ 0 0
$$689$$ −2.00000 −0.0761939
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ −18.0000 −0.680823
$$700$$ 0 0
$$701$$ 48.0000 1.81293 0.906467 0.422276i $$-0.138769\pi$$
0.906467 + 0.422276i $$0.138769\pi$$
$$702$$ 0 0
$$703$$ 18.0000i 0.678883i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 6.00000i 0.225653i
$$708$$ 0 0
$$709$$ −3.00000 −0.112667 −0.0563337 0.998412i $$-0.517941\pi$$
−0.0563337 + 0.998412i $$0.517941\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 28.0000i 1.04861i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 6.00000i − 0.224074i
$$718$$ 0 0
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 0 0
$$723$$ − 11.0000i − 0.409094i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 7.00000i − 0.259616i −0.991539 0.129808i $$-0.958564\pi$$
0.991539 0.129808i $$-0.0414360\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ − 26.0000i − 0.960332i −0.877178 0.480166i $$-0.840576\pi$$
0.877178 0.480166i $$-0.159424\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ −3.00000 −0.110208
$$742$$ 0 0
$$743$$ − 28.0000i − 1.02722i −0.858024 0.513610i $$-0.828308\pi$$
0.858024 0.513610i $$-0.171692\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 18.0000i − 0.658586i
$$748$$ 0 0
$$749$$ 2.00000 0.0730784
$$750$$ 0 0
$$751$$ 48.0000 1.75154 0.875772 0.482724i $$-0.160353\pi$$
0.875772 + 0.482724i $$0.160353\pi$$
$$752$$ 0 0
$$753$$ − 16.0000i − 0.583072i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 47.0000i 1.70824i 0.520073 + 0.854122i $$0.325905\pi$$
−0.520073 + 0.854122i $$0.674095\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 44.0000 1.59500 0.797499 0.603320i $$-0.206156\pi$$
0.797499 + 0.603320i $$0.206156\pi$$
$$762$$ 0 0
$$763$$ − 15.0000i − 0.543036i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 10.0000i 0.361079i
$$768$$ 0 0
$$769$$ 1.00000 0.0360609 0.0180305 0.999837i $$-0.494260\pi$$
0.0180305 + 0.999837i $$0.494260\pi$$
$$770$$ 0 0
$$771$$ 16.0000 0.576226
$$772$$ 0 0
$$773$$ 18.0000i 0.647415i 0.946157 + 0.323708i $$0.104929\pi$$
−0.946157 + 0.323708i $$0.895071\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 6.00000i − 0.215249i
$$778$$ 0 0
$$779$$ 18.0000 0.644917
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ − 4.00000i − 0.142948i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 7.00000i − 0.249523i −0.992187 0.124762i $$-0.960183\pi$$
0.992187 0.124762i $$-0.0398166\pi$$
$$788$$ 0 0
$$789$$ −2.00000 −0.0712019
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 1.00000i 0.0355110i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 48.0000i 1.70025i 0.526583 + 0.850124i $$0.323473\pi$$
−0.526583 + 0.850124i $$0.676527\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 16.0000i 0.563227i
$$808$$ 0 0
$$809$$ 38.0000 1.33601 0.668004 0.744157i $$-0.267149\pi$$
0.668004 + 0.744157i $$0.267149\pi$$
$$810$$ 0 0
$$811$$ −7.00000 −0.245803 −0.122902 0.992419i $$-0.539220\pi$$
−0.122902 + 0.992419i $$0.539220\pi$$
$$812$$ 0 0
$$813$$ − 4.00000i − 0.140286i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 27.0000i 0.944610i
$$818$$ 0 0
$$819$$ 1.00000 0.0349428
$$820$$ 0 0
$$821$$ 42.0000 1.46581 0.732905 0.680331i $$-0.238164\pi$$
0.732905 + 0.680331i $$0.238164\pi$$
$$822$$ 0 0
$$823$$ − 41.0000i − 1.42917i −0.699549 0.714585i $$-0.746616\pi$$
0.699549 0.714585i $$-0.253384\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 34.0000i − 1.18230i −0.806563 0.591148i $$-0.798675\pi$$
0.806563 0.591148i $$-0.201325\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ −11.0000 −0.381586
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 7.00000i 0.241955i
$$838$$ 0 0
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 0 0
$$843$$ 4.00000i 0.137767i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 11.0000i − 0.377964i
$$848$$ 0 0
$$849$$ 5.00000 0.171600
$$850$$ 0 0
$$851$$ 24.0000 0.822709
$$852$$ 0 0
$$853$$ − 37.0000i − 1.26686i −0.773802 0.633428i $$-0.781647\pi$$
0.773802 0.633428i $$-0.218353\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 32.0000i 1.09310i 0.837427 + 0.546550i $$0.184059\pi$$
−0.837427 + 0.546550i $$0.815941\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ −6.00000 −0.204479
$$862$$ 0 0
$$863$$ 4.00000i 0.136162i 0.997680 + 0.0680808i $$0.0216876\pi$$
−0.997680 + 0.0680808i $$0.978312\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 17.0000i 0.577350i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −3.00000 −0.101651
$$872$$ 0 0
$$873$$ 3.00000i 0.101535i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 43.0000i − 1.45201i −0.687691 0.726003i $$-0.741376\pi$$
0.687691 0.726003i $$-0.258624\pi$$
$$878$$ 0 0
$$879$$ −30.0000 −1.01187
$$880$$ 0 0
$$881$$ −12.0000 −0.404290 −0.202145 0.979356i $$-0.564791\pi$$
−0.202145 + 0.979356i $$0.564791\pi$$
$$882$$ 0 0
$$883$$ 29.0000i 0.975928i 0.872864 + 0.487964i $$0.162260\pi$$
−0.872864 + 0.487964i $$0.837740\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 38.0000i − 1.27592i −0.770072 0.637958i $$-0.779780\pi$$
0.770072 0.637958i $$-0.220220\pi$$
$$888$$ 0 0
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 18.0000i 0.602347i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 4.00000i 0.133556i
$$898$$ 0 0
$$899$$ −28.0000 −0.933852
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ − 9.00000i − 0.299501i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 20.0000i − 0.664089i −0.943264 0.332045i $$-0.892262\pi$$
0.943264 0.332045i $$-0.107738\pi$$
$$908$$ 0 0
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 42.0000 1.39152 0.695761 0.718273i $$-0.255067\pi$$
0.695761 + 0.718273i $$0.255067\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 6.00000i − 0.198137i
$$918$$ 0 0
$$919$$ 1.00000 0.0329870 0.0164935 0.999864i $$-0.494750\pi$$
0.0164935 + 0.999864i $$0.494750\pi$$
$$920$$ 0 0
$$921$$ −13.0000 −0.428365
$$922$$ 0 0
$$923$$ − 14.0000i − 0.460816i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 8.00000i 0.262754i
$$928$$ 0 0
$$929$$ 8.00000 0.262471 0.131236 0.991351i $$-0.458106\pi$$
0.131236 + 0.991351i $$0.458106\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ 0 0
$$933$$ − 10.0000i − 0.327385i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 19.0000i 0.620703i 0.950622 + 0.310351i $$0.100447\pi$$
−0.950622 + 0.310351i $$0.899553\pi$$
$$938$$ 0 0
$$939$$ 23.0000 0.750577
$$940$$ 0 0
$$941$$ −30.0000 −0.977972 −0.488986 0.872292i $$-0.662633\pi$$
−0.488986 + 0.872292i $$0.662633\pi$$
$$942$$ 0 0
$$943$$ − 24.0000i − 0.781548i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 20.0000i − 0.649913i −0.945729 0.324956i $$-0.894650\pi$$
0.945729 0.324956i $$-0.105350\pi$$
$$948$$ 0 0
$$949$$ 10.0000 0.324614
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ 8.00000i 0.259145i 0.991570 + 0.129573i $$0.0413606\pi$$
−0.991570 + 0.129573i $$0.958639\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −10.0000 −0.322917
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 0 0
$$963$$ 2.00000i 0.0644491i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 8.00000i − 0.257263i −0.991692 0.128631i $$-0.958942\pi$$
0.991692 0.128631i $$-0.0410584\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 54.0000 1.73294 0.866471 0.499227i $$-0.166383\pi$$
0.866471 + 0.499227i $$0.166383\pi$$
$$972$$ 0 0
$$973$$ 12.0000i 0.384702i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 42.0000i − 1.34370i −0.740688 0.671850i $$-0.765500\pi$$
0.740688 0.671850i $$-0.234500\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 15.0000 0.478913
$$982$$ 0 0
$$983$$ − 28.0000i − 0.893061i −0.894768 0.446531i $$-0.852659\pi$$
0.894768 0.446531i $$-0.147341\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 6.00000i − 0.190982i
$$988$$ 0 0
$$989$$ 36.0000 1.14473
$$990$$ 0 0
$$991$$ −31.0000 −0.984747 −0.492374 0.870384i $$-0.663871\pi$$
−0.492374 + 0.870384i $$0.663871\pi$$
$$992$$ 0 0
$$993$$ − 32.0000i − 1.01549i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 18.0000i − 0.570066i −0.958518 0.285033i $$-0.907995\pi$$
0.958518 0.285033i $$-0.0920045\pi$$
$$998$$ 0 0
$$999$$ 6.00000 0.189832
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 4800.2.f.t.3649.2 2
4.3 odd 2 4800.2.f.q.3649.1 2
5.2 odd 4 4800.2.a.bx.1.1 1
5.3 odd 4 4800.2.a.w.1.1 1
5.4 even 2 inner 4800.2.f.t.3649.1 2
8.3 odd 2 2400.2.f.k.1249.2 2
8.5 even 2 2400.2.f.h.1249.1 2
20.3 even 4 4800.2.a.bw.1.1 1
20.7 even 4 4800.2.a.x.1.1 1
20.19 odd 2 4800.2.f.q.3649.2 2
24.5 odd 2 7200.2.f.l.6049.2 2
24.11 even 2 7200.2.f.r.6049.1 2
40.3 even 4 2400.2.a.g.1.1 yes 1
40.13 odd 4 2400.2.a.bc.1.1 yes 1
40.19 odd 2 2400.2.f.k.1249.1 2
40.27 even 4 2400.2.a.bb.1.1 yes 1
40.29 even 2 2400.2.f.h.1249.2 2
40.37 odd 4 2400.2.a.f.1.1 1
120.29 odd 2 7200.2.f.l.6049.1 2
120.53 even 4 7200.2.a.bj.1.1 1
120.59 even 2 7200.2.f.r.6049.2 2
120.77 even 4 7200.2.a.r.1.1 1
120.83 odd 4 7200.2.a.s.1.1 1
120.107 odd 4 7200.2.a.bi.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2400.2.a.f.1.1 1 40.37 odd 4
2400.2.a.g.1.1 yes 1 40.3 even 4
2400.2.a.bb.1.1 yes 1 40.27 even 4
2400.2.a.bc.1.1 yes 1 40.13 odd 4
2400.2.f.h.1249.1 2 8.5 even 2
2400.2.f.h.1249.2 2 40.29 even 2
2400.2.f.k.1249.1 2 40.19 odd 2
2400.2.f.k.1249.2 2 8.3 odd 2
4800.2.a.w.1.1 1 5.3 odd 4
4800.2.a.x.1.1 1 20.7 even 4
4800.2.a.bw.1.1 1 20.3 even 4
4800.2.a.bx.1.1 1 5.2 odd 4
4800.2.f.q.3649.1 2 4.3 odd 2
4800.2.f.q.3649.2 2 20.19 odd 2
4800.2.f.t.3649.1 2 5.4 even 2 inner
4800.2.f.t.3649.2 2 1.1 even 1 trivial
7200.2.a.r.1.1 1 120.77 even 4
7200.2.a.s.1.1 1 120.83 odd 4
7200.2.a.bi.1.1 1 120.107 odd 4
7200.2.a.bj.1.1 1 120.53 even 4
7200.2.f.l.6049.1 2 120.29 odd 2
7200.2.f.l.6049.2 2 24.5 odd 2
7200.2.f.r.6049.1 2 24.11 even 2
7200.2.f.r.6049.2 2 120.59 even 2