# Properties

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$3.23394128186$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 405.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} +O(q^{10})$$ $$q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{10} +5.00000 q^{11} +4.00000 q^{13} -4.00000 q^{16} -4.00000 q^{17} -5.00000 q^{19} +2.00000 q^{20} +10.0000 q^{22} +6.00000 q^{23} +1.00000 q^{25} +8.00000 q^{26} -5.00000 q^{29} -9.00000 q^{31} -8.00000 q^{32} -8.00000 q^{34} -10.0000 q^{37} -10.0000 q^{38} +7.00000 q^{41} -2.00000 q^{43} +10.0000 q^{44} +12.0000 q^{46} +2.00000 q^{47} -7.00000 q^{49} +2.00000 q^{50} +8.00000 q^{52} +8.00000 q^{53} +5.00000 q^{55} -10.0000 q^{58} -1.00000 q^{59} -2.00000 q^{61} -18.0000 q^{62} -8.00000 q^{64} +4.00000 q^{65} +6.00000 q^{67} -8.00000 q^{68} +1.00000 q^{71} -8.00000 q^{73} -20.0000 q^{74} -10.0000 q^{76} +12.0000 q^{79} -4.00000 q^{80} +14.0000 q^{82} +6.00000 q^{83} -4.00000 q^{85} -4.00000 q^{86} -9.00000 q^{89} +12.0000 q^{92} +4.00000 q^{94} -5.00000 q^{95} +14.0000 q^{97} -14.0000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ 0 0
$$4$$ 2.00000 1.00000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 2.00000 0.632456
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ 10.0000 2.13201
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 8.00000 1.56893
$$27$$ 0 0
$$28$$ 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 $$0.799571\pi$$
$$32$$ −8.00000 −1.41421
$$33$$ 0 0
$$34$$ −8.00000 −1.37199
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ −10.0000 −1.62221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 7.00000 1.09322 0.546608 0.837389i $$-0.315919\pi$$
0.546608 + 0.837389i $$0.315919\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ 10.0000 1.50756
$$45$$ 0 0
$$46$$ 12.0000 1.76930
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 2.00000 0.282843
$$51$$ 0 0
$$52$$ 8.00000 1.10940
$$53$$ 8.00000 1.09888 0.549442 0.835532i $$-0.314840\pi$$
0.549442 + 0.835532i $$0.314840\pi$$
$$54$$ 0 0
$$55$$ 5.00000 0.674200
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −10.0000 −1.31306
$$59$$ −1.00000 −0.130189 −0.0650945 0.997879i $$-0.520735\pi$$
−0.0650945 + 0.997879i $$0.520735\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ −18.0000 −2.28600
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ 6.00000 0.733017 0.366508 0.930415i $$-0.380553\pi$$
0.366508 + 0.930415i $$0.380553\pi$$
$$68$$ −8.00000 −0.970143
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 1.00000 0.118678 0.0593391 0.998238i $$-0.481101\pi$$
0.0593391 + 0.998238i $$0.481101\pi$$
$$72$$ 0 0
$$73$$ −8.00000 −0.936329 −0.468165 0.883641i $$-0.655085\pi$$
−0.468165 + 0.883641i $$0.655085\pi$$
$$74$$ −20.0000 −2.32495
$$75$$ 0 0
$$76$$ −10.0000 −1.14708
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 0 0
$$82$$ 14.0000 1.54604
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ −4.00000 −0.433861
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −9.00000 −0.953998 −0.476999 0.878904i $$-0.658275\pi$$
−0.476999 + 0.878904i $$0.658275\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 12.0000 1.25109
$$93$$ 0 0
$$94$$ 4.00000 0.412568
$$95$$ −5.00000 −0.512989
$$96$$ 0 0
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ −14.0000 −1.41421
$$99$$ 0 0
$$100$$ 2.00000 0.200000
$$101$$ −3.00000 −0.298511 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$102$$ 0 0
$$103$$ 2.00000 0.197066 0.0985329 0.995134i $$-0.468585\pi$$
0.0985329 + 0.995134i $$0.468585\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 16.0000 1.55406
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 0 0
$$109$$ −1.00000 −0.0957826 −0.0478913 0.998853i $$-0.515250\pi$$
−0.0478913 + 0.998853i $$0.515250\pi$$
$$110$$ 10.0000 0.953463
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −16.0000 −1.50515 −0.752577 0.658505i $$-0.771189\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ 0 0
$$115$$ 6.00000 0.559503
$$116$$ −10.0000 −0.928477
$$117$$ 0 0
$$118$$ −2.00000 −0.184115
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ −4.00000 −0.362143
$$123$$ 0 0
$$124$$ −18.0000 −1.61645
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 8.00000 0.701646
$$131$$ 15.0000 1.31056 0.655278 0.755388i $$-0.272551\pi$$
0.655278 + 0.755388i $$0.272551\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ −19.0000 −1.61156 −0.805779 0.592216i $$-0.798253\pi$$
−0.805779 + 0.592216i $$0.798253\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 2.00000 0.167836
$$143$$ 20.0000 1.67248
$$144$$ 0 0
$$145$$ −5.00000 −0.415227
$$146$$ −16.0000 −1.32417
$$147$$ 0 0
$$148$$ −20.0000 −1.64399
$$149$$ −2.00000 −0.163846 −0.0819232 0.996639i $$-0.526106\pi$$
−0.0819232 + 0.996639i $$0.526106\pi$$
$$150$$ 0 0
$$151$$ −5.00000 −0.406894 −0.203447 0.979086i $$-0.565214\pi$$
−0.203447 + 0.979086i $$0.565214\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −9.00000 −0.722897
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 24.0000 1.90934
$$159$$ 0 0
$$160$$ −8.00000 −0.632456
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ 14.0000 1.09322
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ −8.00000 −0.613572
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −20.0000 −1.50756
$$177$$ 0 0
$$178$$ −18.0000 −1.34916
$$179$$ 23.0000 1.71910 0.859550 0.511051i $$-0.170744\pi$$
0.859550 + 0.511051i $$0.170744\pi$$
$$180$$ 0 0
$$181$$ −25.0000 −1.85824 −0.929118 0.369784i $$-0.879432\pi$$
−0.929118 + 0.369784i $$0.879432\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −10.0000 −0.735215
$$186$$ 0 0
$$187$$ −20.0000 −1.46254
$$188$$ 4.00000 0.291730
$$189$$ 0 0
$$190$$ −10.0000 −0.725476
$$191$$ 1.00000 0.0723575 0.0361787 0.999345i $$-0.488481\pi$$
0.0361787 + 0.999345i $$0.488481\pi$$
$$192$$ 0 0
$$193$$ 26.0000 1.87152 0.935760 0.352636i $$-0.114715\pi$$
0.935760 + 0.352636i $$0.114715\pi$$
$$194$$ 28.0000 2.01028
$$195$$ 0 0
$$196$$ −14.0000 −1.00000
$$197$$ 12.0000 0.854965 0.427482 0.904024i $$-0.359401\pi$$
0.427482 + 0.904024i $$0.359401\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −6.00000 −0.422159
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 7.00000 0.488901
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ −16.0000 −1.10940
$$209$$ −25.0000 −1.72929
$$210$$ 0 0
$$211$$ 11.0000 0.757271 0.378636 0.925546i $$-0.376393\pi$$
0.378636 + 0.925546i $$0.376393\pi$$
$$212$$ 16.0000 1.09888
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ −2.00000 −0.136399
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −2.00000 −0.135457
$$219$$ 0 0
$$220$$ 10.0000 0.674200
$$221$$ −16.0000 −1.07628
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −32.0000 −2.12861
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 12.0000 0.791257
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 2.00000 0.130466
$$236$$ −2.00000 −0.130189
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ −11.0000 −0.708572 −0.354286 0.935137i $$-0.615276\pi$$
−0.354286 + 0.935137i $$0.615276\pi$$
$$242$$ 28.0000 1.79991
$$243$$ 0 0
$$244$$ −4.00000 −0.256074
$$245$$ −7.00000 −0.447214
$$246$$ 0 0
$$247$$ −20.0000 −1.27257
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 2.00000 0.126491
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 30.0000 1.88608
$$254$$ 32.0000 2.00786
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 8.00000 0.496139
$$261$$ 0 0
$$262$$ 30.0000 1.85341
$$263$$ 10.0000 0.616626 0.308313 0.951285i $$-0.400236\pi$$
0.308313 + 0.951285i $$0.400236\pi$$
$$264$$ 0 0
$$265$$ 8.00000 0.491436
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 12.0000 0.733017
$$269$$ −31.0000 −1.89010 −0.945052 0.326921i $$-0.893989\pi$$
−0.945052 + 0.326921i $$0.893989\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 16.0000 0.970143
$$273$$ 0 0
$$274$$ 24.0000 1.44989
$$275$$ 5.00000 0.301511
$$276$$ 0 0
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ −38.0000 −2.27909
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 6.00000 0.356663 0.178331 0.983970i $$-0.442930\pi$$
0.178331 + 0.983970i $$0.442930\pi$$
$$284$$ 2.00000 0.118678
$$285$$ 0 0
$$286$$ 40.0000 2.36525
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ −10.0000 −0.587220
$$291$$ 0 0
$$292$$ −16.0000 −0.936329
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 0 0
$$295$$ −1.00000 −0.0582223
$$296$$ 0 0
$$297$$ 0 0
$$298$$ −4.00000 −0.231714
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −10.0000 −0.575435
$$303$$ 0 0
$$304$$ 20.0000 1.14708
$$305$$ −2.00000 −0.114520
$$306$$ 0 0
$$307$$ 10.0000 0.570730 0.285365 0.958419i $$-0.407885\pi$$
0.285365 + 0.958419i $$0.407885\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −18.0000 −1.02233
$$311$$ −9.00000 −0.510343 −0.255172 0.966896i $$-0.582132\pi$$
−0.255172 + 0.966896i $$0.582132\pi$$
$$312$$ 0 0
$$313$$ −4.00000 −0.226093 −0.113047 0.993590i $$-0.536061\pi$$
−0.113047 + 0.993590i $$0.536061\pi$$
$$314$$ 4.00000 0.225733
$$315$$ 0 0
$$316$$ 24.0000 1.35011
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 0 0
$$319$$ −25.0000 −1.39973
$$320$$ −8.00000 −0.447214
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 20.0000 1.11283
$$324$$ 0 0
$$325$$ 4.00000 0.221880
$$326$$ 16.0000 0.886158
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −21.0000 −1.15426 −0.577132 0.816651i $$-0.695828\pi$$
−0.577132 + 0.816651i $$0.695828\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 0 0
$$334$$ 24.0000 1.31322
$$335$$ 6.00000 0.327815
$$336$$ 0 0
$$337$$ −8.00000 −0.435788 −0.217894 0.975972i $$-0.569919\pi$$
−0.217894 + 0.975972i $$0.569919\pi$$
$$338$$ 6.00000 0.326357
$$339$$ 0 0
$$340$$ −8.00000 −0.433861
$$341$$ −45.0000 −2.43689
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ 0 0
$$349$$ 13.0000 0.695874 0.347937 0.937518i $$-0.386882\pi$$
0.347937 + 0.937518i $$0.386882\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −40.0000 −2.13201
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ 1.00000 0.0530745
$$356$$ −18.0000 −0.953998
$$357$$ 0 0
$$358$$ 46.0000 2.43118
$$359$$ −27.0000 −1.42501 −0.712503 0.701669i $$-0.752438\pi$$
−0.712503 + 0.701669i $$0.752438\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ −50.0000 −2.62794
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −8.00000 −0.418739
$$366$$ 0 0
$$367$$ 18.0000 0.939592 0.469796 0.882775i $$-0.344327\pi$$
0.469796 + 0.882775i $$0.344327\pi$$
$$368$$ −24.0000 −1.25109
$$369$$ 0 0
$$370$$ −20.0000 −1.03975
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 16.0000 0.828449 0.414224 0.910175i $$-0.364053\pi$$
0.414224 + 0.910175i $$0.364053\pi$$
$$374$$ −40.0000 −2.06835
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −20.0000 −1.03005
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ −10.0000 −0.512989
$$381$$ 0 0
$$382$$ 2.00000 0.102329
$$383$$ −36.0000 −1.83951 −0.919757 0.392488i $$-0.871614\pi$$
−0.919757 + 0.392488i $$0.871614\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 52.0000 2.64673
$$387$$ 0 0
$$388$$ 28.0000 1.42148
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ −24.0000 −1.21373
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 24.0000 1.20910
$$395$$ 12.0000 0.603786
$$396$$ 0 0
$$397$$ −38.0000 −1.90717 −0.953583 0.301131i $$-0.902636\pi$$
−0.953583 + 0.301131i $$0.902636\pi$$
$$398$$ 32.0000 1.60402
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ −36.0000 −1.79329
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −50.0000 −2.47841
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 14.0000 0.691411
$$411$$ 0 0
$$412$$ 4.00000 0.197066
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 6.00000 0.294528
$$416$$ −32.0000 −1.56893
$$417$$ 0 0
$$418$$ −50.0000 −2.44558
$$419$$ 16.0000 0.781651 0.390826 0.920465i $$-0.372190\pi$$
0.390826 + 0.920465i $$0.372190\pi$$
$$420$$ 0 0
$$421$$ 13.0000 0.633581 0.316791 0.948495i $$-0.397395\pi$$
0.316791 + 0.948495i $$0.397395\pi$$
$$422$$ 22.0000 1.07094
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −4.00000 −0.194029
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ −4.00000 −0.192897
$$431$$ 3.00000 0.144505 0.0722525 0.997386i $$-0.476981\pi$$
0.0722525 + 0.997386i $$0.476981\pi$$
$$432$$ 0 0
$$433$$ −26.0000 −1.24948 −0.624740 0.780833i $$-0.714795\pi$$
−0.624740 + 0.780833i $$0.714795\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ −30.0000 −1.43509
$$438$$ 0 0
$$439$$ −29.0000 −1.38409 −0.692047 0.721852i $$-0.743291\pi$$
−0.692047 + 0.721852i $$0.743291\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −32.0000 −1.52208
$$443$$ 6.00000 0.285069 0.142534 0.989790i $$-0.454475\pi$$
0.142534 + 0.989790i $$0.454475\pi$$
$$444$$ 0 0
$$445$$ −9.00000 −0.426641
$$446$$ 16.0000 0.757622
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 17.0000 0.802280 0.401140 0.916017i $$-0.368614\pi$$
0.401140 + 0.916017i $$0.368614\pi$$
$$450$$ 0 0
$$451$$ 35.0000 1.64809
$$452$$ −32.0000 −1.50515
$$453$$ 0 0
$$454$$ 8.00000 0.375459
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −38.0000 −1.77757 −0.888783 0.458329i $$-0.848448\pi$$
−0.888783 + 0.458329i $$0.848448\pi$$
$$458$$ 12.0000 0.560723
$$459$$ 0 0
$$460$$ 12.0000 0.559503
$$461$$ −15.0000 −0.698620 −0.349310 0.937007i $$-0.613584\pi$$
−0.349310 + 0.937007i $$0.613584\pi$$
$$462$$ 0 0
$$463$$ −6.00000 −0.278844 −0.139422 0.990233i $$-0.544524\pi$$
−0.139422 + 0.990233i $$0.544524\pi$$
$$464$$ 20.0000 0.928477
$$465$$ 0 0
$$466$$ 12.0000 0.555889
$$467$$ 4.00000 0.185098 0.0925490 0.995708i $$-0.470499\pi$$
0.0925490 + 0.995708i $$0.470499\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 4.00000 0.184506
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −10.0000 −0.459800
$$474$$ 0 0
$$475$$ −5.00000 −0.229416
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −32.0000 −1.46365
$$479$$ −15.0000 −0.685367 −0.342684 0.939451i $$-0.611336\pi$$
−0.342684 + 0.939451i $$0.611336\pi$$
$$480$$ 0 0
$$481$$ −40.0000 −1.82384
$$482$$ −22.0000 −1.00207
$$483$$ 0 0
$$484$$ 28.0000 1.27273
$$485$$ 14.0000 0.635707
$$486$$ 0 0
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ −14.0000 −0.632456
$$491$$ 43.0000 1.94056 0.970281 0.241979i $$-0.0777966\pi$$
0.970281 + 0.241979i $$0.0777966\pi$$
$$492$$ 0 0
$$493$$ 20.0000 0.900755
$$494$$ −40.0000 −1.79969
$$495$$ 0 0
$$496$$ 36.0000 1.61645
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 7.00000 0.313363 0.156682 0.987649i $$-0.449920\pi$$
0.156682 + 0.987649i $$0.449920\pi$$
$$500$$ 2.00000 0.0894427
$$501$$ 0 0
$$502$$ −24.0000 −1.07117
$$503$$ −20.0000 −0.891756 −0.445878 0.895094i $$-0.647108\pi$$
−0.445878 + 0.895094i $$0.647108\pi$$
$$504$$ 0 0
$$505$$ −3.00000 −0.133498
$$506$$ 60.0000 2.66733
$$507$$ 0 0
$$508$$ 32.0000 1.41977
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 32.0000 1.41421
$$513$$ 0 0
$$514$$ −12.0000 −0.529297
$$515$$ 2.00000 0.0881305
$$516$$ 0 0
$$517$$ 10.0000 0.439799
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 0 0
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ 30.0000 1.31056
$$525$$ 0 0
$$526$$ 20.0000 0.872041
$$527$$ 36.0000 1.56818
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 16.0000 0.694996
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 28.0000 1.21281
$$534$$ 0 0
$$535$$ −6.00000 −0.259403
$$536$$ 0 0
$$537$$ 0 0
$$538$$ −62.0000 −2.67301
$$539$$ −35.0000 −1.50756
$$540$$ 0 0
$$541$$ 3.00000 0.128980 0.0644900 0.997918i $$-0.479458\pi$$
0.0644900 + 0.997918i $$0.479458\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 0 0
$$544$$ 32.0000 1.37199
$$545$$ −1.00000 −0.0428353
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 24.0000 1.02523
$$549$$ 0 0
$$550$$ 10.0000 0.426401
$$551$$ 25.0000 1.06504
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −36.0000 −1.52949
$$555$$ 0 0
$$556$$ −38.0000 −1.61156
$$557$$ −24.0000 −1.01691 −0.508456 0.861088i $$-0.669784\pi$$
−0.508456 + 0.861088i $$0.669784\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 12.0000 0.506189
$$563$$ −18.0000 −0.758610 −0.379305 0.925272i $$-0.623837\pi$$
−0.379305 + 0.925272i $$0.623837\pi$$
$$564$$ 0 0
$$565$$ −16.0000 −0.673125
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −3.00000 −0.125767 −0.0628833 0.998021i $$-0.520030\pi$$
−0.0628833 + 0.998021i $$0.520030\pi$$
$$570$$ 0 0
$$571$$ 5.00000 0.209243 0.104622 0.994512i $$-0.466637\pi$$
0.104622 + 0.994512i $$0.466637\pi$$
$$572$$ 40.0000 1.67248
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 6.00000 0.250217
$$576$$ 0 0
$$577$$ −16.0000 −0.666089 −0.333044 0.942911i $$-0.608076\pi$$
−0.333044 + 0.942911i $$0.608076\pi$$
$$578$$ −2.00000 −0.0831890
$$579$$ 0 0
$$580$$ −10.0000 −0.415227
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 40.0000 1.65663
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 36.0000 1.48715
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 0 0
$$589$$ 45.0000 1.85419
$$590$$ −2.00000 −0.0823387
$$591$$ 0 0
$$592$$ 40.0000 1.64399
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −4.00000 −0.163846
$$597$$ 0 0
$$598$$ 48.0000 1.96287
$$599$$ −17.0000 −0.694601 −0.347301 0.937754i $$-0.612902\pi$$
−0.347301 + 0.937754i $$0.612902\pi$$
$$600$$ 0 0
$$601$$ −19.0000 −0.775026 −0.387513 0.921864i $$-0.626666\pi$$
−0.387513 + 0.921864i $$0.626666\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −10.0000 −0.406894
$$605$$ 14.0000 0.569181
$$606$$ 0 0
$$607$$ 34.0000 1.38002 0.690009 0.723801i $$-0.257607\pi$$
0.690009 + 0.723801i $$0.257607\pi$$
$$608$$ 40.0000 1.62221
$$609$$ 0 0
$$610$$ −4.00000 −0.161955
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ 4.00000 0.161558 0.0807792 0.996732i $$-0.474259\pi$$
0.0807792 + 0.996732i $$0.474259\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −24.0000 −0.966204 −0.483102 0.875564i $$-0.660490\pi$$
−0.483102 + 0.875564i $$0.660490\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ −18.0000 −0.722897
$$621$$ 0 0
$$622$$ −18.0000 −0.721734
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −8.00000 −0.319744
$$627$$ 0 0
$$628$$ 4.00000 0.159617
$$629$$ 40.0000 1.59490
$$630$$ 0 0
$$631$$ −17.0000 −0.676759 −0.338380 0.941010i $$-0.609879\pi$$
−0.338380 + 0.941010i $$0.609879\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −4.00000 −0.158860
$$635$$ 16.0000 0.634941
$$636$$ 0 0
$$637$$ −28.0000 −1.10940
$$638$$ −50.0000 −1.97952
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 3.00000 0.118493 0.0592464 0.998243i $$-0.481130\pi$$
0.0592464 + 0.998243i $$0.481130\pi$$
$$642$$ 0 0
$$643$$ 6.00000 0.236617 0.118308 0.992977i $$-0.462253\pi$$
0.118308 + 0.992977i $$0.462253\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 40.0000 1.57378
$$647$$ −10.0000 −0.393141 −0.196570 0.980490i $$-0.562980\pi$$
−0.196570 + 0.980490i $$0.562980\pi$$
$$648$$ 0 0
$$649$$ −5.00000 −0.196267
$$650$$ 8.00000 0.313786
$$651$$ 0 0
$$652$$ 16.0000 0.626608
$$653$$ −8.00000 −0.313064 −0.156532 0.987673i $$-0.550031\pi$$
−0.156532 + 0.987673i $$0.550031\pi$$
$$654$$ 0 0
$$655$$ 15.0000 0.586098
$$656$$ −28.0000 −1.09322
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 44.0000 1.71400 0.856998 0.515319i $$-0.172327\pi$$
0.856998 + 0.515319i $$0.172327\pi$$
$$660$$ 0 0
$$661$$ 25.0000 0.972387 0.486194 0.873851i $$-0.338385\pi$$
0.486194 + 0.873851i $$0.338385\pi$$
$$662$$ −42.0000 −1.63238
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −30.0000 −1.16160
$$668$$ 24.0000 0.928588
$$669$$ 0 0
$$670$$ 12.0000 0.463600
$$671$$ −10.0000 −0.386046
$$672$$ 0 0
$$673$$ 42.0000 1.61898 0.809491 0.587133i $$-0.199743\pi$$
0.809491 + 0.587133i $$0.199743\pi$$
$$674$$ −16.0000 −0.616297
$$675$$ 0 0
$$676$$ 6.00000 0.230769
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −90.0000 −3.44628
$$683$$ 48.0000 1.83667 0.918334 0.395805i $$-0.129534\pi$$
0.918334 + 0.395805i $$0.129534\pi$$
$$684$$ 0 0
$$685$$ 12.0000 0.458496
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 8.00000 0.304997
$$689$$ 32.0000 1.21910
$$690$$ 0 0
$$691$$ 4.00000 0.152167 0.0760836 0.997101i $$-0.475758\pi$$
0.0760836 + 0.997101i $$0.475758\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 0 0
$$694$$ 40.0000 1.51838
$$695$$ −19.0000 −0.720711
$$696$$ 0 0
$$697$$ −28.0000 −1.06058
$$698$$ 26.0000 0.984115
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 19.0000 0.717620 0.358810 0.933411i $$-0.383183\pi$$
0.358810 + 0.933411i $$0.383183\pi$$
$$702$$ 0 0
$$703$$ 50.0000 1.88579
$$704$$ −40.0000 −1.50756
$$705$$ 0 0
$$706$$ −36.0000 −1.35488
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ 2.00000 0.0750587
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −54.0000 −2.02232
$$714$$ 0 0
$$715$$ 20.0000 0.747958
$$716$$ 46.0000 1.71910
$$717$$ 0 0
$$718$$ −54.0000 −2.01526
$$719$$ 27.0000 1.00693 0.503465 0.864016i $$-0.332058\pi$$
0.503465 + 0.864016i $$0.332058\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 12.0000 0.446594
$$723$$ 0 0
$$724$$ −50.0000 −1.85824
$$725$$ −5.00000 −0.185695
$$726$$ 0 0
$$727$$ 44.0000 1.63187 0.815935 0.578144i $$-0.196223\pi$$
0.815935 + 0.578144i $$0.196223\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −16.0000 −0.592187
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ −46.0000 −1.69905 −0.849524 0.527549i $$-0.823111\pi$$
−0.849524 + 0.527549i $$0.823111\pi$$
$$734$$ 36.0000 1.32878
$$735$$ 0 0
$$736$$ −48.0000 −1.76930
$$737$$ 30.0000 1.10506
$$738$$ 0 0
$$739$$ −35.0000 −1.28750 −0.643748 0.765238i $$-0.722621\pi$$
−0.643748 + 0.765238i $$0.722621\pi$$
$$740$$ −20.0000 −0.735215
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −4.00000 −0.146746 −0.0733729 0.997305i $$-0.523376\pi$$
−0.0733729 + 0.997305i $$0.523376\pi$$
$$744$$ 0 0
$$745$$ −2.00000 −0.0732743
$$746$$ 32.0000 1.17160
$$747$$ 0 0
$$748$$ −40.0000 −1.46254
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ −40.0000 −1.45671
$$755$$ −5.00000 −0.181969
$$756$$ 0 0
$$757$$ 28.0000 1.01768 0.508839 0.860862i $$-0.330075\pi$$
0.508839 + 0.860862i $$0.330075\pi$$
$$758$$ 8.00000 0.290573
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 2.00000 0.0723575
$$765$$ 0 0
$$766$$ −72.0000 −2.60147
$$767$$ −4.00000 −0.144432
$$768$$ 0 0
$$769$$ 5.00000 0.180305 0.0901523 0.995928i $$-0.471265\pi$$
0.0901523 + 0.995928i $$0.471265\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 52.0000 1.87152
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ 0 0
$$775$$ −9.00000 −0.323290
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 12.0000 0.430221
$$779$$ −35.0000 −1.25401
$$780$$ 0 0
$$781$$ 5.00000 0.178914
$$782$$ −48.0000 −1.71648
$$783$$ 0 0
$$784$$ 28.0000 1.00000
$$785$$ 2.00000 0.0713831
$$786$$ 0 0
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ 24.0000 0.854965
$$789$$ 0 0
$$790$$ 24.0000 0.853882
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −8.00000 −0.284088
$$794$$ −76.0000 −2.69714
$$795$$ 0 0
$$796$$ 32.0000 1.13421
$$797$$ 4.00000 0.141687 0.0708436 0.997487i $$-0.477431\pi$$
0.0708436 + 0.997487i $$0.477431\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ −8.00000 −0.282843
$$801$$ 0 0
$$802$$ 60.0000 2.11867
$$803$$ −40.0000 −1.41157
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −72.0000 −2.53609
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 7.00000 0.246107 0.123053 0.992400i $$-0.460731\pi$$
0.123053 + 0.992400i $$0.460731\pi$$
$$810$$ 0 0
$$811$$ −21.0000 −0.737410 −0.368705 0.929547i $$-0.620199\pi$$
−0.368705 + 0.929547i $$0.620199\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −100.000 −3.50500
$$815$$ 8.00000 0.280228
$$816$$ 0 0
$$817$$ 10.0000 0.349856
$$818$$ 28.0000 0.978997
$$819$$ 0 0
$$820$$ 14.0000 0.488901
$$821$$ 3.00000 0.104701 0.0523504 0.998629i $$-0.483329\pi$$
0.0523504 + 0.998629i $$0.483329\pi$$
$$822$$ 0 0
$$823$$ −46.0000 −1.60346 −0.801730 0.597687i $$-0.796087\pi$$
−0.801730 + 0.597687i $$0.796087\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 44.0000 1.53003 0.765015 0.644013i $$-0.222732\pi$$
0.765015 + 0.644013i $$0.222732\pi$$
$$828$$ 0 0
$$829$$ 27.0000 0.937749 0.468874 0.883265i $$-0.344660\pi$$
0.468874 + 0.883265i $$0.344660\pi$$
$$830$$ 12.0000 0.416526
$$831$$ 0 0
$$832$$ −32.0000 −1.10940
$$833$$ 28.0000 0.970143
$$834$$ 0 0
$$835$$ 12.0000 0.415277
$$836$$ −50.0000 −1.72929
$$837$$ 0 0
$$838$$ 32.0000 1.10542
$$839$$ 13.0000 0.448810 0.224405 0.974496i $$-0.427956\pi$$
0.224405 + 0.974496i $$0.427956\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 26.0000 0.896019
$$843$$ 0 0
$$844$$ 22.0000 0.757271
$$845$$ 3.00000 0.103203
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −32.0000 −1.09888
$$849$$ 0 0
$$850$$ −8.00000 −0.274398
$$851$$ −60.0000 −2.05677
$$852$$ 0 0
$$853$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 32.0000 1.09310 0.546550 0.837427i $$-0.315941\pi$$
0.546550 + 0.837427i $$0.315941\pi$$
$$858$$ 0 0
$$859$$ 55.0000 1.87658 0.938288 0.345855i $$-0.112411\pi$$
0.938288 + 0.345855i $$0.112411\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 0 0
$$862$$ 6.00000 0.204361
$$863$$ −44.0000 −1.49778 −0.748889 0.662696i $$-0.769412\pi$$
−0.748889 + 0.662696i $$0.769412\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ −52.0000 −1.76703
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 60.0000 2.03536
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 0 0
$$873$$ 0 0
$$874$$ −60.0000 −2.02953
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −24.0000 −0.810422 −0.405211 0.914223i $$-0.632802\pi$$
−0.405211 + 0.914223i $$0.632802\pi$$
$$878$$ −58.0000 −1.95741
$$879$$ 0 0
$$880$$ −20.0000 −0.674200
$$881$$ −25.0000 −0.842271 −0.421136 0.906998i $$-0.638368\pi$$
−0.421136 + 0.906998i $$0.638368\pi$$
$$882$$ 0 0
$$883$$ −52.0000 −1.74994 −0.874970 0.484178i $$-0.839119\pi$$
−0.874970 + 0.484178i $$0.839119\pi$$
$$884$$ −32.0000 −1.07628
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 6.00000 0.201460 0.100730 0.994914i $$-0.467882\pi$$
0.100730 + 0.994914i $$0.467882\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −18.0000 −0.603361
$$891$$ 0 0
$$892$$ 16.0000 0.535720
$$893$$ −10.0000 −0.334637
$$894$$ 0 0
$$895$$ 23.0000 0.768805
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 34.0000 1.13459
$$899$$ 45.0000 1.50083
$$900$$ 0 0
$$901$$ −32.0000 −1.06607
$$902$$ 70.0000 2.33075
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −25.0000 −0.831028
$$906$$ 0 0
$$907$$ −18.0000 −0.597680 −0.298840 0.954303i $$-0.596600\pi$$
−0.298840 + 0.954303i $$0.596600\pi$$
$$908$$ 8.00000 0.265489
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −49.0000 −1.62344 −0.811721 0.584045i $$-0.801469\pi$$
−0.811721 + 0.584045i $$0.801469\pi$$
$$912$$ 0 0
$$913$$ 30.0000 0.992855
$$914$$ −76.0000 −2.51386
$$915$$ 0 0
$$916$$ 12.0000 0.396491
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −11.0000 −0.362857 −0.181428 0.983404i $$-0.558072\pi$$
−0.181428 + 0.983404i $$0.558072\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −30.0000 −0.987997
$$923$$ 4.00000 0.131662
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ −12.0000 −0.394344
$$927$$ 0 0
$$928$$ 40.0000 1.31306
$$929$$ 1.00000 0.0328089 0.0164045 0.999865i $$-0.494778\pi$$
0.0164045 + 0.999865i $$0.494778\pi$$
$$930$$ 0 0
$$931$$ 35.0000 1.14708
$$932$$ 12.0000 0.393073
$$933$$ 0 0
$$934$$ 8.00000 0.261768
$$935$$ −20.0000 −0.654070
$$936$$ 0 0
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 4.00000 0.130466
$$941$$ 46.0000 1.49956 0.749779 0.661689i $$-0.230160\pi$$
0.749779 + 0.661689i $$0.230160\pi$$
$$942$$ 0 0
$$943$$ 42.0000 1.36771
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ −20.0000 −0.650256
$$947$$ 18.0000 0.584921 0.292461 0.956278i $$-0.405526\pi$$
0.292461 + 0.956278i $$0.405526\pi$$
$$948$$ 0 0
$$949$$ −32.0000 −1.03876
$$950$$ −10.0000 −0.324443
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −16.0000 −0.518291 −0.259145 0.965838i $$-0.583441\pi$$
−0.259145 + 0.965838i $$0.583441\pi$$
$$954$$ 0 0
$$955$$ 1.00000 0.0323592
$$956$$ −32.0000 −1.03495
$$957$$ 0 0
$$958$$ −30.0000 −0.969256
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 50.0000 1.61290
$$962$$ −80.0000 −2.57930
$$963$$ 0 0
$$964$$ −22.0000 −0.708572
$$965$$ 26.0000 0.836970
$$966$$ 0 0
$$967$$ −40.0000 −1.28631 −0.643157 0.765735i $$-0.722376\pi$$
−0.643157 + 0.765735i $$0.722376\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 28.0000 0.899026
$$971$$ −27.0000 −0.866471 −0.433236 0.901281i $$-0.642628\pi$$
−0.433236 + 0.901281i $$0.642628\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ −28.0000 −0.895799 −0.447900 0.894084i $$-0.647828\pi$$
−0.447900 + 0.894084i $$0.647828\pi$$
$$978$$ 0 0
$$979$$ −45.0000 −1.43821
$$980$$ −14.0000 −0.447214
$$981$$ 0 0
$$982$$ 86.0000 2.74437
$$983$$ 48.0000 1.53096 0.765481 0.643458i $$-0.222501\pi$$
0.765481 + 0.643458i $$0.222501\pi$$
$$984$$ 0 0
$$985$$ 12.0000 0.382352
$$986$$ 40.0000 1.27386
$$987$$ 0 0
$$988$$ −40.0000 −1.27257
$$989$$ −12.0000 −0.381578
$$990$$ 0 0
$$991$$ −1.00000 −0.0317660 −0.0158830 0.999874i $$-0.505056\pi$$
−0.0158830 + 0.999874i $$0.505056\pi$$
$$992$$ 72.0000 2.28600
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 16.0000 0.507234
$$996$$ 0 0
$$997$$ 48.0000 1.52018 0.760088 0.649821i $$-0.225156\pi$$
0.760088 + 0.649821i $$0.225156\pi$$
$$998$$ 14.0000 0.443162
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

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

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