# Properties

 Label 3920.2.a.a.1.1 Level $3920$ Weight $2$ Character 3920.1 Self dual yes Analytic conductor $31.301$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3920.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.3013575923$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 245) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3920.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -1.00000 q^{5} +6.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -1.00000 q^{5} +6.00000 q^{9} -1.00000 q^{11} +3.00000 q^{13} +3.00000 q^{15} -3.00000 q^{17} -6.00000 q^{19} +4.00000 q^{23} +1.00000 q^{25} -9.00000 q^{27} -1.00000 q^{29} -6.00000 q^{31} +3.00000 q^{33} -9.00000 q^{39} +6.00000 q^{41} +6.00000 q^{43} -6.00000 q^{45} +9.00000 q^{47} +9.00000 q^{51} -10.0000 q^{53} +1.00000 q^{55} +18.0000 q^{57} +6.00000 q^{59} -3.00000 q^{65} +14.0000 q^{67} -12.0000 q^{69} +8.00000 q^{71} +6.00000 q^{73} -3.00000 q^{75} +1.00000 q^{79} +9.00000 q^{81} -12.0000 q^{83} +3.00000 q^{85} +3.00000 q^{87} +12.0000 q^{89} +18.0000 q^{93} +6.00000 q^{95} -15.0000 q^{97} -6.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ −3.00000 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 6.00000 2.00000
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ 0 0
$$13$$ 3.00000 0.832050 0.416025 0.909353i $$-0.363423\pi$$
0.416025 + 0.909353i $$0.363423\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 0 0
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −9.00000 −1.73205
$$28$$ 0 0
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 0 0
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ 0 0
$$33$$ 3.00000 0.522233
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ −9.00000 −1.44115
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ 0 0
$$45$$ −6.00000 −0.894427
$$46$$ 0 0
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 9.00000 1.26025
$$52$$ 0 0
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 0 0
$$55$$ 1.00000 0.134840
$$56$$ 0 0
$$57$$ 18.0000 2.38416
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −3.00000 −0.372104
$$66$$ 0 0
$$67$$ 14.0000 1.71037 0.855186 0.518321i $$-0.173443\pi$$
0.855186 + 0.518321i $$0.173443\pi$$
$$68$$ 0 0
$$69$$ −12.0000 −1.44463
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ −3.00000 −0.346410
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1.00000 0.112509 0.0562544 0.998416i $$-0.482084\pi$$
0.0562544 + 0.998416i $$0.482084\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 3.00000 0.325396
$$86$$ 0 0
$$87$$ 3.00000 0.321634
$$88$$ 0 0
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 18.0000 1.86651
$$94$$ 0 0
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ −15.0000 −1.52302 −0.761510 0.648154i $$-0.775541\pi$$
−0.761510 + 0.648154i $$0.775541\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 0 0
$$103$$ 9.00000 0.886796 0.443398 0.896325i $$-0.353773\pi$$
0.443398 + 0.896325i $$0.353773\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.00000 0.193347 0.0966736 0.995316i $$-0.469180\pi$$
0.0966736 + 0.995316i $$0.469180\pi$$
$$108$$ 0 0
$$109$$ −15.0000 −1.43674 −0.718370 0.695662i $$-0.755111\pi$$
−0.718370 + 0.695662i $$0.755111\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ 0 0
$$115$$ −4.00000 −0.373002
$$116$$ 0 0
$$117$$ 18.0000 1.66410
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 0 0
$$123$$ −18.0000 −1.62301
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ −18.0000 −1.58481
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 9.00000 0.774597
$$136$$ 0 0
$$137$$ 8.00000 0.683486 0.341743 0.939793i $$-0.388983\pi$$
0.341743 + 0.939793i $$0.388983\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ −27.0000 −2.27381
$$142$$ 0 0
$$143$$ −3.00000 −0.250873
$$144$$ 0 0
$$145$$ 1.00000 0.0830455
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ −15.0000 −1.22068 −0.610341 0.792139i $$-0.708968\pi$$
−0.610341 + 0.792139i $$0.708968\pi$$
$$152$$ 0 0
$$153$$ −18.0000 −1.45521
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 0 0
$$159$$ 30.0000 2.37915
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 0 0
$$165$$ −3.00000 −0.233550
$$166$$ 0 0
$$167$$ −3.00000 −0.232147 −0.116073 0.993241i $$-0.537031\pi$$
−0.116073 + 0.993241i $$0.537031\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ 0 0
$$171$$ −36.0000 −2.75299
$$172$$ 0 0
$$173$$ −3.00000 −0.228086 −0.114043 0.993476i $$-0.536380\pi$$
−0.114043 + 0.993476i $$0.536380\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −18.0000 −1.35296
$$178$$ 0 0
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3.00000 0.219382
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −17.0000 −1.23008 −0.615038 0.788497i $$-0.710860\pi$$
−0.615038 + 0.788497i $$0.710860\pi$$
$$192$$ 0 0
$$193$$ 12.0000 0.863779 0.431889 0.901927i $$-0.357847\pi$$
0.431889 + 0.901927i $$0.357847\pi$$
$$194$$ 0 0
$$195$$ 9.00000 0.644503
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ −6.00000 −0.425329 −0.212664 0.977125i $$-0.568214\pi$$
−0.212664 + 0.977125i $$0.568214\pi$$
$$200$$ 0 0
$$201$$ −42.0000 −2.96245
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −6.00000 −0.419058
$$206$$ 0 0
$$207$$ 24.0000 1.66812
$$208$$ 0 0
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ −15.0000 −1.03264 −0.516321 0.856395i $$-0.672699\pi$$
−0.516321 + 0.856395i $$0.672699\pi$$
$$212$$ 0 0
$$213$$ −24.0000 −1.64445
$$214$$ 0 0
$$215$$ −6.00000 −0.409197
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −18.0000 −1.21633
$$220$$ 0 0
$$221$$ −9.00000 −0.605406
$$222$$ 0 0
$$223$$ −3.00000 −0.200895 −0.100447 0.994942i $$-0.532027\pi$$
−0.100447 + 0.994942i $$0.532027\pi$$
$$224$$ 0 0
$$225$$ 6.00000 0.400000
$$226$$ 0 0
$$227$$ −3.00000 −0.199117 −0.0995585 0.995032i $$-0.531743\pi$$
−0.0995585 + 0.995032i $$0.531743\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −8.00000 −0.524097 −0.262049 0.965055i $$-0.584398\pi$$
−0.262049 + 0.965055i $$0.584398\pi$$
$$234$$ 0 0
$$235$$ −9.00000 −0.587095
$$236$$ 0 0
$$237$$ −3.00000 −0.194871
$$238$$ 0 0
$$239$$ −7.00000 −0.452792 −0.226396 0.974035i $$-0.572694\pi$$
−0.226396 + 0.974035i $$0.572694\pi$$
$$240$$ 0 0
$$241$$ −24.0000 −1.54598 −0.772988 0.634421i $$-0.781239\pi$$
−0.772988 + 0.634421i $$0.781239\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −18.0000 −1.14531
$$248$$ 0 0
$$249$$ 36.0000 2.28141
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ 0 0
$$255$$ −9.00000 −0.563602
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ 10.0000 0.616626 0.308313 0.951285i $$-0.400236\pi$$
0.308313 + 0.951285i $$0.400236\pi$$
$$264$$ 0 0
$$265$$ 10.0000 0.614295
$$266$$ 0 0
$$267$$ −36.0000 −2.20316
$$268$$ 0 0
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ −28.0000 −1.68236 −0.841178 0.540758i $$-0.818138\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 0 0
$$279$$ −36.0000 −2.15526
$$280$$ 0 0
$$281$$ −5.00000 −0.298275 −0.149137 0.988816i $$-0.547650\pi$$
−0.149137 + 0.988816i $$0.547650\pi$$
$$282$$ 0 0
$$283$$ −21.0000 −1.24832 −0.624160 0.781296i $$-0.714559\pi$$
−0.624160 + 0.781296i $$0.714559\pi$$
$$284$$ 0 0
$$285$$ −18.0000 −1.06623
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 45.0000 2.63795
$$292$$ 0 0
$$293$$ 9.00000 0.525786 0.262893 0.964825i $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ 0 0
$$295$$ −6.00000 −0.349334
$$296$$ 0 0
$$297$$ 9.00000 0.522233
$$298$$ 0 0
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 54.0000 3.10222
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 3.00000 0.171219 0.0856095 0.996329i $$-0.472716\pi$$
0.0856095 + 0.996329i $$0.472716\pi$$
$$308$$ 0 0
$$309$$ −27.0000 −1.53598
$$310$$ 0 0
$$311$$ 6.00000 0.340229 0.170114 0.985424i $$-0.445586\pi$$
0.170114 + 0.985424i $$0.445586\pi$$
$$312$$ 0 0
$$313$$ 3.00000 0.169570 0.0847850 0.996399i $$-0.472980\pi$$
0.0847850 + 0.996399i $$0.472980\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −22.0000 −1.23564 −0.617822 0.786318i $$-0.711985\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ 0 0
$$319$$ 1.00000 0.0559893
$$320$$ 0 0
$$321$$ −6.00000 −0.334887
$$322$$ 0 0
$$323$$ 18.0000 1.00155
$$324$$ 0 0
$$325$$ 3.00000 0.166410
$$326$$ 0 0
$$327$$ 45.0000 2.48851
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −14.0000 −0.764902
$$336$$ 0 0
$$337$$ −24.0000 −1.30736 −0.653682 0.756770i $$-0.726776\pi$$
−0.653682 + 0.756770i $$0.726776\pi$$
$$338$$ 0 0
$$339$$ −24.0000 −1.30350
$$340$$ 0 0
$$341$$ 6.00000 0.324918
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 12.0000 0.646058
$$346$$ 0 0
$$347$$ −16.0000 −0.858925 −0.429463 0.903085i $$-0.641297\pi$$
−0.429463 + 0.903085i $$0.641297\pi$$
$$348$$ 0 0
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 0 0
$$351$$ −27.0000 −1.44115
$$352$$ 0 0
$$353$$ −3.00000 −0.159674 −0.0798369 0.996808i $$-0.525440\pi$$
−0.0798369 + 0.996808i $$0.525440\pi$$
$$354$$ 0 0
$$355$$ −8.00000 −0.424596
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −32.0000 −1.68890 −0.844448 0.535638i $$-0.820071\pi$$
−0.844448 + 0.535638i $$0.820071\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ 30.0000 1.57459
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ 33.0000 1.72259 0.861293 0.508109i $$-0.169655\pi$$
0.861293 + 0.508109i $$0.169655\pi$$
$$368$$ 0 0
$$369$$ 36.0000 1.87409
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 0 0
$$375$$ 3.00000 0.154919
$$376$$ 0 0
$$377$$ −3.00000 −0.154508
$$378$$ 0 0
$$379$$ 24.0000 1.23280 0.616399 0.787434i $$-0.288591\pi$$
0.616399 + 0.787434i $$0.288591\pi$$
$$380$$ 0 0
$$381$$ −6.00000 −0.307389
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 36.0000 1.82998
$$388$$ 0 0
$$389$$ 13.0000 0.659126 0.329563 0.944134i $$-0.393099\pi$$
0.329563 + 0.944134i $$0.393099\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ 0 0
$$393$$ 36.0000 1.81596
$$394$$ 0 0
$$395$$ −1.00000 −0.0503155
$$396$$ 0 0
$$397$$ 9.00000 0.451697 0.225849 0.974162i $$-0.427485\pi$$
0.225849 + 0.974162i $$0.427485\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −19.0000 −0.948815 −0.474407 0.880305i $$-0.657338\pi$$
−0.474407 + 0.880305i $$0.657338\pi$$
$$402$$ 0 0
$$403$$ −18.0000 −0.896644
$$404$$ 0 0
$$405$$ −9.00000 −0.447214
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 30.0000 1.48340 0.741702 0.670729i $$-0.234019\pi$$
0.741702 + 0.670729i $$0.234019\pi$$
$$410$$ 0 0
$$411$$ −24.0000 −1.18383
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 6.00000 0.293119 0.146560 0.989202i $$-0.453180\pi$$
0.146560 + 0.989202i $$0.453180\pi$$
$$420$$ 0 0
$$421$$ 1.00000 0.0487370 0.0243685 0.999703i $$-0.492242\pi$$
0.0243685 + 0.999703i $$0.492242\pi$$
$$422$$ 0 0
$$423$$ 54.0000 2.62557
$$424$$ 0 0
$$425$$ −3.00000 −0.145521
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 9.00000 0.434524
$$430$$ 0 0
$$431$$ −5.00000 −0.240842 −0.120421 0.992723i $$-0.538424\pi$$
−0.120421 + 0.992723i $$0.538424\pi$$
$$432$$ 0 0
$$433$$ −30.0000 −1.44171 −0.720854 0.693087i $$-0.756250\pi$$
−0.720854 + 0.693087i $$0.756250\pi$$
$$434$$ 0 0
$$435$$ −3.00000 −0.143839
$$436$$ 0 0
$$437$$ −24.0000 −1.14808
$$438$$ 0 0
$$439$$ −12.0000 −0.572729 −0.286364 0.958121i $$-0.592447\pi$$
−0.286364 + 0.958121i $$0.592447\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −34.0000 −1.61539 −0.807694 0.589601i $$-0.799285\pi$$
−0.807694 + 0.589601i $$0.799285\pi$$
$$444$$ 0 0
$$445$$ −12.0000 −0.568855
$$446$$ 0 0
$$447$$ −30.0000 −1.41895
$$448$$ 0 0
$$449$$ 23.0000 1.08544 0.542719 0.839915i $$-0.317395\pi$$
0.542719 + 0.839915i $$0.317395\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 0 0
$$453$$ 45.0000 2.11428
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ 0 0
$$459$$ 27.0000 1.26025
$$460$$ 0 0
$$461$$ 36.0000 1.67669 0.838344 0.545142i $$-0.183524\pi$$
0.838344 + 0.545142i $$0.183524\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 0 0
$$465$$ −18.0000 −0.834730
$$466$$ 0 0
$$467$$ 15.0000 0.694117 0.347059 0.937843i $$-0.387180\pi$$
0.347059 + 0.937843i $$0.387180\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −54.0000 −2.48819
$$472$$ 0 0
$$473$$ −6.00000 −0.275880
$$474$$ 0 0
$$475$$ −6.00000 −0.275299
$$476$$ 0 0
$$477$$ −60.0000 −2.74721
$$478$$ 0 0
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 15.0000 0.681115
$$486$$ 0 0
$$487$$ −36.0000 −1.63132 −0.815658 0.578535i $$-0.803625\pi$$
−0.815658 + 0.578535i $$0.803625\pi$$
$$488$$ 0 0
$$489$$ 48.0000 2.17064
$$490$$ 0 0
$$491$$ −23.0000 −1.03798 −0.518988 0.854782i $$-0.673691\pi$$
−0.518988 + 0.854782i $$0.673691\pi$$
$$492$$ 0 0
$$493$$ 3.00000 0.135113
$$494$$ 0 0
$$495$$ 6.00000 0.269680
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 27.0000 1.20869 0.604343 0.796724i $$-0.293436\pi$$
0.604343 + 0.796724i $$0.293436\pi$$
$$500$$ 0 0
$$501$$ 9.00000 0.402090
$$502$$ 0 0
$$503$$ −9.00000 −0.401290 −0.200645 0.979664i $$-0.564304\pi$$
−0.200645 + 0.979664i $$0.564304\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 0 0
$$507$$ 12.0000 0.532939
$$508$$ 0 0
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 54.0000 2.38416
$$514$$ 0 0
$$515$$ −9.00000 −0.396587
$$516$$ 0 0
$$517$$ −9.00000 −0.395820
$$518$$ 0 0
$$519$$ 9.00000 0.395056
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ −24.0000 −1.04945 −0.524723 0.851273i $$-0.675831\pi$$
−0.524723 + 0.851273i $$0.675831\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 18.0000 0.784092
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 36.0000 1.56227
$$532$$ 0 0
$$533$$ 18.0000 0.779667
$$534$$ 0 0
$$535$$ −2.00000 −0.0864675
$$536$$ 0 0
$$537$$ 12.0000 0.517838
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ 0 0
$$543$$ 18.0000 0.772454
$$544$$ 0 0
$$545$$ 15.0000 0.642529
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 4.00000 0.169485 0.0847427 0.996403i $$-0.472993\pi$$
0.0847427 + 0.996403i $$0.472993\pi$$
$$558$$ 0 0
$$559$$ 18.0000 0.761319
$$560$$ 0 0
$$561$$ −9.00000 −0.379980
$$562$$ 0 0
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ −8.00000 −0.336563
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −38.0000 −1.59304 −0.796521 0.604610i $$-0.793329\pi$$
−0.796521 + 0.604610i $$0.793329\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ 51.0000 2.13056
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ 15.0000 0.624458 0.312229 0.950007i $$-0.398924\pi$$
0.312229 + 0.950007i $$0.398924\pi$$
$$578$$ 0 0
$$579$$ −36.0000 −1.49611
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 10.0000 0.414158
$$584$$ 0 0
$$585$$ −18.0000 −0.744208
$$586$$ 0 0
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 36.0000 1.48335
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 0 0
$$593$$ −45.0000 −1.84793 −0.923964 0.382479i $$-0.875070\pi$$
−0.923964 + 0.382479i $$0.875070\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 18.0000 0.736691
$$598$$ 0 0
$$599$$ −17.0000 −0.694601 −0.347301 0.937754i $$-0.612902\pi$$
−0.347301 + 0.937754i $$0.612902\pi$$
$$600$$ 0 0
$$601$$ −30.0000 −1.22373 −0.611863 0.790964i $$-0.709580\pi$$
−0.611863 + 0.790964i $$0.709580\pi$$
$$602$$ 0 0
$$603$$ 84.0000 3.42074
$$604$$ 0 0
$$605$$ 10.0000 0.406558
$$606$$ 0 0
$$607$$ −21.0000 −0.852364 −0.426182 0.904638i $$-0.640142\pi$$
−0.426182 + 0.904638i $$0.640142\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 27.0000 1.09230
$$612$$ 0 0
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ 0 0
$$615$$ 18.0000 0.725830
$$616$$ 0 0
$$617$$ 4.00000 0.161034 0.0805170 0.996753i $$-0.474343\pi$$
0.0805170 + 0.996753i $$0.474343\pi$$
$$618$$ 0 0
$$619$$ 24.0000 0.964641 0.482321 0.875995i $$-0.339794\pi$$
0.482321 + 0.875995i $$0.339794\pi$$
$$620$$ 0 0
$$621$$ −36.0000 −1.44463
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −18.0000 −0.718851
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 11.0000 0.437903 0.218952 0.975736i $$-0.429736\pi$$
0.218952 + 0.975736i $$0.429736\pi$$
$$632$$ 0 0
$$633$$ 45.0000 1.78859
$$634$$ 0 0
$$635$$ −2.00000 −0.0793676
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 48.0000 1.89885
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ −27.0000 −1.06478 −0.532388 0.846500i $$-0.678705\pi$$
−0.532388 + 0.846500i $$0.678705\pi$$
$$644$$ 0 0
$$645$$ 18.0000 0.708749
$$646$$ 0 0
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ 0 0
$$649$$ −6.00000 −0.235521
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ 0 0
$$655$$ 12.0000 0.468879
$$656$$ 0 0
$$657$$ 36.0000 1.40449
$$658$$ 0 0
$$659$$ 19.0000 0.740135 0.370067 0.929005i $$-0.379335\pi$$
0.370067 + 0.929005i $$0.379335\pi$$
$$660$$ 0 0
$$661$$ 12.0000 0.466746 0.233373 0.972387i $$-0.425024\pi$$
0.233373 + 0.972387i $$0.425024\pi$$
$$662$$ 0 0
$$663$$ 27.0000 1.04859
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −4.00000 −0.154881
$$668$$ 0 0
$$669$$ 9.00000 0.347960
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 12.0000 0.462566 0.231283 0.972887i $$-0.425708\pi$$
0.231283 + 0.972887i $$0.425708\pi$$
$$674$$ 0 0
$$675$$ −9.00000 −0.346410
$$676$$ 0 0
$$677$$ −9.00000 −0.345898 −0.172949 0.984931i $$-0.555330\pi$$
−0.172949 + 0.984931i $$0.555330\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 9.00000 0.344881
$$682$$ 0 0
$$683$$ −8.00000 −0.306111 −0.153056 0.988218i $$-0.548911\pi$$
−0.153056 + 0.988218i $$0.548911\pi$$
$$684$$ 0 0
$$685$$ −8.00000 −0.305664
$$686$$ 0 0
$$687$$ 18.0000 0.686743
$$688$$ 0 0
$$689$$ −30.0000 −1.14291
$$690$$ 0 0
$$691$$ −36.0000 −1.36950 −0.684752 0.728776i $$-0.740090\pi$$
−0.684752 + 0.728776i $$0.740090\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −18.0000 −0.681799
$$698$$ 0 0
$$699$$ 24.0000 0.907763
$$700$$ 0 0
$$701$$ 47.0000 1.77517 0.887583 0.460648i $$-0.152383\pi$$
0.887583 + 0.460648i $$0.152383\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 27.0000 1.01688
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −33.0000 −1.23934 −0.619671 0.784862i $$-0.712734\pi$$
−0.619671 + 0.784862i $$0.712734\pi$$
$$710$$ 0 0
$$711$$ 6.00000 0.225018
$$712$$ 0 0
$$713$$ −24.0000 −0.898807
$$714$$ 0 0
$$715$$ 3.00000 0.112194
$$716$$ 0 0
$$717$$ 21.0000 0.784259
$$718$$ 0 0
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 72.0000 2.67771
$$724$$ 0 0
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −18.0000 −0.665754
$$732$$ 0 0
$$733$$ 15.0000 0.554038 0.277019 0.960864i $$-0.410654\pi$$
0.277019 + 0.960864i $$0.410654\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −14.0000 −0.515697
$$738$$ 0 0
$$739$$ 43.0000 1.58178 0.790890 0.611958i $$-0.209618\pi$$
0.790890 + 0.611958i $$0.209618\pi$$
$$740$$ 0 0
$$741$$ 54.0000 1.98374
$$742$$ 0 0
$$743$$ 22.0000 0.807102 0.403551 0.914957i $$-0.367776\pi$$
0.403551 + 0.914957i $$0.367776\pi$$
$$744$$ 0 0
$$745$$ −10.0000 −0.366372
$$746$$ 0 0
$$747$$ −72.0000 −2.63434
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −15.0000 −0.547358 −0.273679 0.961821i $$-0.588241\pi$$
−0.273679 + 0.961821i $$0.588241\pi$$
$$752$$ 0 0
$$753$$ −36.0000 −1.31191
$$754$$ 0 0
$$755$$ 15.0000 0.545906
$$756$$ 0 0
$$757$$ 48.0000 1.74459 0.872295 0.488980i $$-0.162631\pi$$
0.872295 + 0.488980i $$0.162631\pi$$
$$758$$ 0 0
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 18.0000 0.650791
$$766$$ 0 0
$$767$$ 18.0000 0.649942
$$768$$ 0 0
$$769$$ 42.0000 1.51456 0.757279 0.653091i $$-0.226528\pi$$
0.757279 + 0.653091i $$0.226528\pi$$
$$770$$ 0 0
$$771$$ −54.0000 −1.94476
$$772$$ 0 0
$$773$$ 3.00000 0.107903 0.0539513 0.998544i $$-0.482818\pi$$
0.0539513 + 0.998544i $$0.482818\pi$$
$$774$$ 0 0
$$775$$ −6.00000 −0.215526
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −36.0000 −1.28983
$$780$$ 0 0
$$781$$ −8.00000 −0.286263
$$782$$ 0 0
$$783$$ 9.00000 0.321634
$$784$$ 0 0
$$785$$ −18.0000 −0.642448
$$786$$ 0 0
$$787$$ −39.0000 −1.39020 −0.695100 0.718913i $$-0.744640\pi$$
−0.695100 + 0.718913i $$0.744640\pi$$
$$788$$ 0 0
$$789$$ −30.0000 −1.06803
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −30.0000 −1.06399
$$796$$ 0 0
$$797$$ −27.0000 −0.956389 −0.478195 0.878254i $$-0.658709\pi$$
−0.478195 + 0.878254i $$0.658709\pi$$
$$798$$ 0 0
$$799$$ −27.0000 −0.955191
$$800$$ 0 0
$$801$$ 72.0000 2.54399
$$802$$ 0 0
$$803$$ −6.00000 −0.211735
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −72.0000 −2.53452
$$808$$ 0 0
$$809$$ −35.0000 −1.23053 −0.615267 0.788319i $$-0.710952\pi$$
−0.615267 + 0.788319i $$0.710952\pi$$
$$810$$ 0 0
$$811$$ −48.0000 −1.68551 −0.842754 0.538299i $$-0.819067\pi$$
−0.842754 + 0.538299i $$0.819067\pi$$
$$812$$ 0 0
$$813$$ 72.0000 2.52515
$$814$$ 0 0
$$815$$ 16.0000 0.560456
$$816$$ 0 0
$$817$$ −36.0000 −1.25948
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −23.0000 −0.802706 −0.401353 0.915924i $$-0.631460\pi$$
−0.401353 + 0.915924i $$0.631460\pi$$
$$822$$ 0 0
$$823$$ −50.0000 −1.74289 −0.871445 0.490493i $$-0.836817\pi$$
−0.871445 + 0.490493i $$0.836817\pi$$
$$824$$ 0 0
$$825$$ 3.00000 0.104447
$$826$$ 0 0
$$827$$ 16.0000 0.556375 0.278187 0.960527i $$-0.410266\pi$$
0.278187 + 0.960527i $$0.410266\pi$$
$$828$$ 0 0
$$829$$ 6.00000 0.208389 0.104194 0.994557i $$-0.466774\pi$$
0.104194 + 0.994557i $$0.466774\pi$$
$$830$$ 0 0
$$831$$ 84.0000 2.91393
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 3.00000 0.103819
$$836$$ 0 0
$$837$$ 54.0000 1.86651
$$838$$ 0 0
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 0 0
$$843$$ 15.0000 0.516627
$$844$$ 0 0
$$845$$ 4.00000 0.137604
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 63.0000 2.16215
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 30.0000 1.02718 0.513590 0.858036i $$-0.328315\pi$$
0.513590 + 0.858036i $$0.328315\pi$$
$$854$$ 0 0
$$855$$ 36.0000 1.23117
$$856$$ 0 0
$$857$$ 54.0000 1.84460 0.922302 0.386469i $$-0.126305\pi$$
0.922302 + 0.386469i $$0.126305\pi$$
$$858$$ 0 0
$$859$$ −12.0000 −0.409435 −0.204717 0.978821i $$-0.565628\pi$$
−0.204717 + 0.978821i $$0.565628\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 34.0000 1.15737 0.578687 0.815550i $$-0.303565\pi$$
0.578687 + 0.815550i $$0.303565\pi$$
$$864$$ 0 0
$$865$$ 3.00000 0.102003
$$866$$ 0 0
$$867$$ 24.0000 0.815083
$$868$$ 0 0
$$869$$ −1.00000 −0.0339227
$$870$$ 0 0
$$871$$ 42.0000 1.42312
$$872$$ 0 0
$$873$$ −90.0000 −3.04604
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ 0 0
$$879$$ −27.0000 −0.910687
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ 0 0
$$885$$ 18.0000 0.605063
$$886$$ 0 0
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −9.00000 −0.301511
$$892$$ 0 0
$$893$$ −54.0000 −1.80704
$$894$$ 0 0
$$895$$ 4.00000 0.133705
$$896$$ 0 0
$$897$$ −36.0000 −1.20201
$$898$$ 0 0
$$899$$ 6.00000 0.200111
$$900$$ 0 0
$$901$$ 30.0000 0.999445
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 6.00000 0.199447
$$906$$ 0 0
$$907$$ −24.0000 −0.796907 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$908$$ 0 0
$$909$$ −108.000 −3.58213
$$910$$ 0 0
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 9.00000 0.296883 0.148441 0.988921i $$-0.452574\pi$$
0.148441 + 0.988921i $$0.452574\pi$$
$$920$$ 0 0
$$921$$ −9.00000 −0.296560
$$922$$ 0 0
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 54.0000 1.77359
$$928$$ 0 0
$$929$$ 42.0000 1.37798 0.688988 0.724773i $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −18.0000 −0.589294
$$934$$ 0 0
$$935$$ −3.00000 −0.0981105
$$936$$ 0 0
$$937$$ −27.0000 −0.882052 −0.441026 0.897494i $$-0.645385\pi$$
−0.441026 + 0.897494i $$0.645385\pi$$
$$938$$ 0 0
$$939$$ −9.00000 −0.293704
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 24.0000 0.781548
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 2.00000 0.0649913 0.0324956 0.999472i $$-0.489654\pi$$
0.0324956 + 0.999472i $$0.489654\pi$$
$$948$$ 0 0
$$949$$ 18.0000 0.584305
$$950$$ 0 0
$$951$$ 66.0000 2.14020
$$952$$ 0 0
$$953$$ −38.0000 −1.23094 −0.615470 0.788160i $$-0.711034\pi$$
−0.615470 + 0.788160i $$0.711034\pi$$
$$954$$ 0 0
$$955$$ 17.0000 0.550107
$$956$$ 0 0
$$957$$ −3.00000 −0.0969762
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ 0 0
$$965$$ −12.0000 −0.386294
$$966$$ 0 0
$$967$$ 22.0000 0.707472 0.353736 0.935345i $$-0.384911\pi$$
0.353736 + 0.935345i $$0.384911\pi$$
$$968$$ 0 0
$$969$$ −54.0000 −1.73473
$$970$$ 0 0
$$971$$ 18.0000 0.577647 0.288824 0.957382i $$-0.406736\pi$$
0.288824 + 0.957382i $$0.406736\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −9.00000 −0.288231
$$976$$ 0 0
$$977$$ 34.0000 1.08776 0.543878 0.839164i $$-0.316955\pi$$
0.543878 + 0.839164i $$0.316955\pi$$
$$978$$ 0 0
$$979$$ −12.0000 −0.383522
$$980$$ 0 0
$$981$$ −90.0000 −2.87348
$$982$$ 0 0
$$983$$ −45.0000 −1.43528 −0.717639 0.696416i $$-0.754777\pi$$
−0.717639 + 0.696416i $$0.754777\pi$$
$$984$$ 0 0
$$985$$ 2.00000 0.0637253
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 36.0000 1.14358 0.571789 0.820401i $$-0.306250\pi$$
0.571789 + 0.820401i $$0.306250\pi$$
$$992$$ 0 0
$$993$$ 36.0000 1.14243
$$994$$ 0 0
$$995$$ 6.00000 0.190213
$$996$$ 0 0
$$997$$ −3.00000 −0.0950110 −0.0475055 0.998871i $$-0.515127\pi$$
−0.0475055 + 0.998871i $$0.515127\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3920.2.a.a.1.1 1
4.3 odd 2 245.2.a.b.1.1 yes 1
7.6 odd 2 3920.2.a.bj.1.1 1
12.11 even 2 2205.2.a.l.1.1 1
20.3 even 4 1225.2.b.a.99.2 2
20.7 even 4 1225.2.b.a.99.1 2
20.19 odd 2 1225.2.a.h.1.1 1
28.3 even 6 245.2.e.d.226.1 2
28.11 odd 6 245.2.e.c.226.1 2
28.19 even 6 245.2.e.d.116.1 2
28.23 odd 6 245.2.e.c.116.1 2
28.27 even 2 245.2.a.a.1.1 1
84.83 odd 2 2205.2.a.j.1.1 1
140.27 odd 4 1225.2.b.b.99.1 2
140.83 odd 4 1225.2.b.b.99.2 2
140.139 even 2 1225.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
245.2.a.a.1.1 1 28.27 even 2
245.2.a.b.1.1 yes 1 4.3 odd 2
245.2.e.c.116.1 2 28.23 odd 6
245.2.e.c.226.1 2 28.11 odd 6
245.2.e.d.116.1 2 28.19 even 6
245.2.e.d.226.1 2 28.3 even 6
1225.2.a.h.1.1 1 20.19 odd 2
1225.2.a.j.1.1 1 140.139 even 2
1225.2.b.a.99.1 2 20.7 even 4
1225.2.b.a.99.2 2 20.3 even 4
1225.2.b.b.99.1 2 140.27 odd 4
1225.2.b.b.99.2 2 140.83 odd 4
2205.2.a.j.1.1 1 84.83 odd 2
2205.2.a.l.1.1 1 12.11 even 2
3920.2.a.a.1.1 1 1.1 even 1 trivial
3920.2.a.bj.1.1 1 7.6 odd 2