# Properties

 Label 768.4.a.c.1.1 Level $768$ Weight $4$ Character 768.1 Self dual yes Analytic conductor $45.313$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.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: $$45.3134668844$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 768.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -8.00000 q^{5} -12.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -8.00000 q^{5} -12.0000 q^{7} +9.00000 q^{9} -12.0000 q^{11} +20.0000 q^{13} -24.0000 q^{15} +62.0000 q^{17} +108.000 q^{19} -36.0000 q^{21} +72.0000 q^{23} -61.0000 q^{25} +27.0000 q^{27} -128.000 q^{29} -204.000 q^{31} -36.0000 q^{33} +96.0000 q^{35} -228.000 q^{37} +60.0000 q^{39} +22.0000 q^{41} -204.000 q^{43} -72.0000 q^{45} -600.000 q^{47} -199.000 q^{49} +186.000 q^{51} +256.000 q^{53} +96.0000 q^{55} +324.000 q^{57} -828.000 q^{59} -84.0000 q^{61} -108.000 q^{63} -160.000 q^{65} +348.000 q^{67} +216.000 q^{69} -456.000 q^{71} -822.000 q^{73} -183.000 q^{75} +144.000 q^{77} -1356.00 q^{79} +81.0000 q^{81} +108.000 q^{83} -496.000 q^{85} -384.000 q^{87} +938.000 q^{89} -240.000 q^{91} -612.000 q^{93} -864.000 q^{95} +1278.00 q^{97} -108.000 q^{99} +O(q^{100})$$

## 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 0.577350
$$4$$ 0 0
$$5$$ −8.00000 −0.715542 −0.357771 0.933809i $$-0.616463\pi$$
−0.357771 + 0.933809i $$0.616463\pi$$
$$6$$ 0 0
$$7$$ −12.0000 −0.647939 −0.323970 0.946068i $$-0.605018\pi$$
−0.323970 + 0.946068i $$0.605018\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −12.0000 −0.328921 −0.164461 0.986384i $$-0.552588\pi$$
−0.164461 + 0.986384i $$0.552588\pi$$
$$12$$ 0 0
$$13$$ 20.0000 0.426692 0.213346 0.976977i $$-0.431564\pi$$
0.213346 + 0.976977i $$0.431564\pi$$
$$14$$ 0 0
$$15$$ −24.0000 −0.413118
$$16$$ 0 0
$$17$$ 62.0000 0.884542 0.442271 0.896882i $$-0.354173\pi$$
0.442271 + 0.896882i $$0.354173\pi$$
$$18$$ 0 0
$$19$$ 108.000 1.30405 0.652024 0.758199i $$-0.273920\pi$$
0.652024 + 0.758199i $$0.273920\pi$$
$$20$$ 0 0
$$21$$ −36.0000 −0.374088
$$22$$ 0 0
$$23$$ 72.0000 0.652741 0.326370 0.945242i $$-0.394174\pi$$
0.326370 + 0.945242i $$0.394174\pi$$
$$24$$ 0 0
$$25$$ −61.0000 −0.488000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −128.000 −0.819621 −0.409810 0.912171i $$-0.634405\pi$$
−0.409810 + 0.912171i $$0.634405\pi$$
$$30$$ 0 0
$$31$$ −204.000 −1.18192 −0.590959 0.806701i $$-0.701251\pi$$
−0.590959 + 0.806701i $$0.701251\pi$$
$$32$$ 0 0
$$33$$ −36.0000 −0.189903
$$34$$ 0 0
$$35$$ 96.0000 0.463627
$$36$$ 0 0
$$37$$ −228.000 −1.01305 −0.506527 0.862224i $$-0.669071\pi$$
−0.506527 + 0.862224i $$0.669071\pi$$
$$38$$ 0 0
$$39$$ 60.0000 0.246351
$$40$$ 0 0
$$41$$ 22.0000 0.0838006 0.0419003 0.999122i $$-0.486659\pi$$
0.0419003 + 0.999122i $$0.486659\pi$$
$$42$$ 0 0
$$43$$ −204.000 −0.723482 −0.361741 0.932279i $$-0.617817\pi$$
−0.361741 + 0.932279i $$0.617817\pi$$
$$44$$ 0 0
$$45$$ −72.0000 −0.238514
$$46$$ 0 0
$$47$$ −600.000 −1.86211 −0.931053 0.364884i $$-0.881109\pi$$
−0.931053 + 0.364884i $$0.881109\pi$$
$$48$$ 0 0
$$49$$ −199.000 −0.580175
$$50$$ 0 0
$$51$$ 186.000 0.510690
$$52$$ 0 0
$$53$$ 256.000 0.663477 0.331739 0.943371i $$-0.392365\pi$$
0.331739 + 0.943371i $$0.392365\pi$$
$$54$$ 0 0
$$55$$ 96.0000 0.235357
$$56$$ 0 0
$$57$$ 324.000 0.752892
$$58$$ 0 0
$$59$$ −828.000 −1.82706 −0.913529 0.406774i $$-0.866654\pi$$
−0.913529 + 0.406774i $$0.866654\pi$$
$$60$$ 0 0
$$61$$ −84.0000 −0.176313 −0.0881565 0.996107i $$-0.528098\pi$$
−0.0881565 + 0.996107i $$0.528098\pi$$
$$62$$ 0 0
$$63$$ −108.000 −0.215980
$$64$$ 0 0
$$65$$ −160.000 −0.305316
$$66$$ 0 0
$$67$$ 348.000 0.634552 0.317276 0.948333i $$-0.397232\pi$$
0.317276 + 0.948333i $$0.397232\pi$$
$$68$$ 0 0
$$69$$ 216.000 0.376860
$$70$$ 0 0
$$71$$ −456.000 −0.762215 −0.381107 0.924531i $$-0.624457\pi$$
−0.381107 + 0.924531i $$0.624457\pi$$
$$72$$ 0 0
$$73$$ −822.000 −1.31792 −0.658958 0.752180i $$-0.729002\pi$$
−0.658958 + 0.752180i $$0.729002\pi$$
$$74$$ 0 0
$$75$$ −183.000 −0.281747
$$76$$ 0 0
$$77$$ 144.000 0.213121
$$78$$ 0 0
$$79$$ −1356.00 −1.93116 −0.965582 0.260100i $$-0.916245\pi$$
−0.965582 + 0.260100i $$0.916245\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 108.000 0.142826 0.0714129 0.997447i $$-0.477249\pi$$
0.0714129 + 0.997447i $$0.477249\pi$$
$$84$$ 0 0
$$85$$ −496.000 −0.632927
$$86$$ 0 0
$$87$$ −384.000 −0.473208
$$88$$ 0 0
$$89$$ 938.000 1.11717 0.558583 0.829449i $$-0.311345\pi$$
0.558583 + 0.829449i $$0.311345\pi$$
$$90$$ 0 0
$$91$$ −240.000 −0.276471
$$92$$ 0 0
$$93$$ −612.000 −0.682381
$$94$$ 0 0
$$95$$ −864.000 −0.933100
$$96$$ 0 0
$$97$$ 1278.00 1.33774 0.668872 0.743377i $$-0.266777\pi$$
0.668872 + 0.743377i $$0.266777\pi$$
$$98$$ 0 0
$$99$$ −108.000 −0.109640
$$100$$ 0 0
$$101$$ 608.000 0.598993 0.299496 0.954097i $$-0.403181\pi$$
0.299496 + 0.954097i $$0.403181\pi$$
$$102$$ 0 0
$$103$$ −948.000 −0.906886 −0.453443 0.891285i $$-0.649804\pi$$
−0.453443 + 0.891285i $$0.649804\pi$$
$$104$$ 0 0
$$105$$ 288.000 0.267675
$$106$$ 0 0
$$107$$ 1044.00 0.943246 0.471623 0.881800i $$-0.343668\pi$$
0.471623 + 0.881800i $$0.343668\pi$$
$$108$$ 0 0
$$109$$ 1780.00 1.56416 0.782078 0.623180i $$-0.214160\pi$$
0.782078 + 0.623180i $$0.214160\pi$$
$$110$$ 0 0
$$111$$ −684.000 −0.584887
$$112$$ 0 0
$$113$$ −622.000 −0.517813 −0.258906 0.965902i $$-0.583362\pi$$
−0.258906 + 0.965902i $$0.583362\pi$$
$$114$$ 0 0
$$115$$ −576.000 −0.467063
$$116$$ 0 0
$$117$$ 180.000 0.142231
$$118$$ 0 0
$$119$$ −744.000 −0.573129
$$120$$ 0 0
$$121$$ −1187.00 −0.891811
$$122$$ 0 0
$$123$$ 66.0000 0.0483823
$$124$$ 0 0
$$125$$ 1488.00 1.06473
$$126$$ 0 0
$$127$$ 204.000 0.142536 0.0712680 0.997457i $$-0.477295\pi$$
0.0712680 + 0.997457i $$0.477295\pi$$
$$128$$ 0 0
$$129$$ −612.000 −0.417702
$$130$$ 0 0
$$131$$ 348.000 0.232098 0.116049 0.993243i $$-0.462977\pi$$
0.116049 + 0.993243i $$0.462977\pi$$
$$132$$ 0 0
$$133$$ −1296.00 −0.844943
$$134$$ 0 0
$$135$$ −216.000 −0.137706
$$136$$ 0 0
$$137$$ −106.000 −0.0661036 −0.0330518 0.999454i $$-0.510523\pi$$
−0.0330518 + 0.999454i $$0.510523\pi$$
$$138$$ 0 0
$$139$$ 1188.00 0.724927 0.362463 0.931998i $$-0.381936\pi$$
0.362463 + 0.931998i $$0.381936\pi$$
$$140$$ 0 0
$$141$$ −1800.00 −1.07509
$$142$$ 0 0
$$143$$ −240.000 −0.140348
$$144$$ 0 0
$$145$$ 1024.00 0.586473
$$146$$ 0 0
$$147$$ −597.000 −0.334964
$$148$$ 0 0
$$149$$ 2648.00 1.45592 0.727962 0.685618i $$-0.240468\pi$$
0.727962 + 0.685618i $$0.240468\pi$$
$$150$$ 0 0
$$151$$ −3420.00 −1.84315 −0.921575 0.388200i $$-0.873097\pi$$
−0.921575 + 0.388200i $$0.873097\pi$$
$$152$$ 0 0
$$153$$ 558.000 0.294847
$$154$$ 0 0
$$155$$ 1632.00 0.845712
$$156$$ 0 0
$$157$$ 60.0000 0.0305001 0.0152501 0.999884i $$-0.495146\pi$$
0.0152501 + 0.999884i $$0.495146\pi$$
$$158$$ 0 0
$$159$$ 768.000 0.383059
$$160$$ 0 0
$$161$$ −864.000 −0.422936
$$162$$ 0 0
$$163$$ −228.000 −0.109560 −0.0547802 0.998498i $$-0.517446\pi$$
−0.0547802 + 0.998498i $$0.517446\pi$$
$$164$$ 0 0
$$165$$ 288.000 0.135883
$$166$$ 0 0
$$167$$ −4128.00 −1.91278 −0.956390 0.292093i $$-0.905648\pi$$
−0.956390 + 0.292093i $$0.905648\pi$$
$$168$$ 0 0
$$169$$ −1797.00 −0.817934
$$170$$ 0 0
$$171$$ 972.000 0.434682
$$172$$ 0 0
$$173$$ −1352.00 −0.594166 −0.297083 0.954852i $$-0.596014\pi$$
−0.297083 + 0.954852i $$0.596014\pi$$
$$174$$ 0 0
$$175$$ 732.000 0.316194
$$176$$ 0 0
$$177$$ −2484.00 −1.05485
$$178$$ 0 0
$$179$$ −1716.00 −0.716536 −0.358268 0.933619i $$-0.616633\pi$$
−0.358268 + 0.933619i $$0.616633\pi$$
$$180$$ 0 0
$$181$$ −3692.00 −1.51616 −0.758078 0.652164i $$-0.773861\pi$$
−0.758078 + 0.652164i $$0.773861\pi$$
$$182$$ 0 0
$$183$$ −252.000 −0.101794
$$184$$ 0 0
$$185$$ 1824.00 0.724882
$$186$$ 0 0
$$187$$ −744.000 −0.290945
$$188$$ 0 0
$$189$$ −324.000 −0.124696
$$190$$ 0 0
$$191$$ 48.0000 0.0181841 0.00909204 0.999959i $$-0.497106\pi$$
0.00909204 + 0.999959i $$0.497106\pi$$
$$192$$ 0 0
$$193$$ 2414.00 0.900329 0.450165 0.892946i $$-0.351365\pi$$
0.450165 + 0.892946i $$0.351365\pi$$
$$194$$ 0 0
$$195$$ −480.000 −0.176274
$$196$$ 0 0
$$197$$ 4304.00 1.55659 0.778293 0.627902i $$-0.216086\pi$$
0.778293 + 0.627902i $$0.216086\pi$$
$$198$$ 0 0
$$199$$ 204.000 0.0726692 0.0363346 0.999340i $$-0.488432\pi$$
0.0363346 + 0.999340i $$0.488432\pi$$
$$200$$ 0 0
$$201$$ 1044.00 0.366359
$$202$$ 0 0
$$203$$ 1536.00 0.531064
$$204$$ 0 0
$$205$$ −176.000 −0.0599628
$$206$$ 0 0
$$207$$ 648.000 0.217580
$$208$$ 0 0
$$209$$ −1296.00 −0.428929
$$210$$ 0 0
$$211$$ −4020.00 −1.31160 −0.655801 0.754933i $$-0.727669\pi$$
−0.655801 + 0.754933i $$0.727669\pi$$
$$212$$ 0 0
$$213$$ −1368.00 −0.440065
$$214$$ 0 0
$$215$$ 1632.00 0.517681
$$216$$ 0 0
$$217$$ 2448.00 0.765811
$$218$$ 0 0
$$219$$ −2466.00 −0.760899
$$220$$ 0 0
$$221$$ 1240.00 0.377427
$$222$$ 0 0
$$223$$ −516.000 −0.154950 −0.0774751 0.996994i $$-0.524686\pi$$
−0.0774751 + 0.996994i $$0.524686\pi$$
$$224$$ 0 0
$$225$$ −549.000 −0.162667
$$226$$ 0 0
$$227$$ −1428.00 −0.417532 −0.208766 0.977966i $$-0.566945\pi$$
−0.208766 + 0.977966i $$0.566945\pi$$
$$228$$ 0 0
$$229$$ −6028.00 −1.73948 −0.869741 0.493508i $$-0.835714\pi$$
−0.869741 + 0.493508i $$0.835714\pi$$
$$230$$ 0 0
$$231$$ 432.000 0.123046
$$232$$ 0 0
$$233$$ −2630.00 −0.739472 −0.369736 0.929137i $$-0.620552\pi$$
−0.369736 + 0.929137i $$0.620552\pi$$
$$234$$ 0 0
$$235$$ 4800.00 1.33241
$$236$$ 0 0
$$237$$ −4068.00 −1.11496
$$238$$ 0 0
$$239$$ 4416.00 1.19518 0.597588 0.801803i $$-0.296126\pi$$
0.597588 + 0.801803i $$0.296126\pi$$
$$240$$ 0 0
$$241$$ 4830.00 1.29099 0.645493 0.763766i $$-0.276652\pi$$
0.645493 + 0.763766i $$0.276652\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 1592.00 0.415139
$$246$$ 0 0
$$247$$ 2160.00 0.556427
$$248$$ 0 0
$$249$$ 324.000 0.0824605
$$250$$ 0 0
$$251$$ −5532.00 −1.39114 −0.695571 0.718457i $$-0.744849\pi$$
−0.695571 + 0.718457i $$0.744849\pi$$
$$252$$ 0 0
$$253$$ −864.000 −0.214700
$$254$$ 0 0
$$255$$ −1488.00 −0.365420
$$256$$ 0 0
$$257$$ −254.000 −0.0616501 −0.0308251 0.999525i $$-0.509813\pi$$
−0.0308251 + 0.999525i $$0.509813\pi$$
$$258$$ 0 0
$$259$$ 2736.00 0.656397
$$260$$ 0 0
$$261$$ −1152.00 −0.273207
$$262$$ 0 0
$$263$$ 4272.00 1.00161 0.500804 0.865561i $$-0.333038\pi$$
0.500804 + 0.865561i $$0.333038\pi$$
$$264$$ 0 0
$$265$$ −2048.00 −0.474746
$$266$$ 0 0
$$267$$ 2814.00 0.644996
$$268$$ 0 0
$$269$$ 4544.00 1.02994 0.514968 0.857210i $$-0.327804\pi$$
0.514968 + 0.857210i $$0.327804\pi$$
$$270$$ 0 0
$$271$$ 2076.00 0.465343 0.232672 0.972555i $$-0.425253\pi$$
0.232672 + 0.972555i $$0.425253\pi$$
$$272$$ 0 0
$$273$$ −720.000 −0.159620
$$274$$ 0 0
$$275$$ 732.000 0.160514
$$276$$ 0 0
$$277$$ 484.000 0.104985 0.0524923 0.998621i $$-0.483283\pi$$
0.0524923 + 0.998621i $$0.483283\pi$$
$$278$$ 0 0
$$279$$ −1836.00 −0.393973
$$280$$ 0 0
$$281$$ −406.000 −0.0861919 −0.0430960 0.999071i $$-0.513722\pi$$
−0.0430960 + 0.999071i $$0.513722\pi$$
$$282$$ 0 0
$$283$$ −8172.00 −1.71652 −0.858260 0.513216i $$-0.828454\pi$$
−0.858260 + 0.513216i $$0.828454\pi$$
$$284$$ 0 0
$$285$$ −2592.00 −0.538726
$$286$$ 0 0
$$287$$ −264.000 −0.0542977
$$288$$ 0 0
$$289$$ −1069.00 −0.217586
$$290$$ 0 0
$$291$$ 3834.00 0.772347
$$292$$ 0 0
$$293$$ 4960.00 0.988963 0.494482 0.869188i $$-0.335358\pi$$
0.494482 + 0.869188i $$0.335358\pi$$
$$294$$ 0 0
$$295$$ 6624.00 1.30734
$$296$$ 0 0
$$297$$ −324.000 −0.0633010
$$298$$ 0 0
$$299$$ 1440.00 0.278520
$$300$$ 0 0
$$301$$ 2448.00 0.468772
$$302$$ 0 0
$$303$$ 1824.00 0.345829
$$304$$ 0 0
$$305$$ 672.000 0.126159
$$306$$ 0 0
$$307$$ 6684.00 1.24259 0.621296 0.783576i $$-0.286606\pi$$
0.621296 + 0.783576i $$0.286606\pi$$
$$308$$ 0 0
$$309$$ −2844.00 −0.523591
$$310$$ 0 0
$$311$$ 4992.00 0.910194 0.455097 0.890442i $$-0.349605\pi$$
0.455097 + 0.890442i $$0.349605\pi$$
$$312$$ 0 0
$$313$$ −5402.00 −0.975524 −0.487762 0.872977i $$-0.662187\pi$$
−0.487762 + 0.872977i $$0.662187\pi$$
$$314$$ 0 0
$$315$$ 864.000 0.154542
$$316$$ 0 0
$$317$$ 7856.00 1.39191 0.695957 0.718083i $$-0.254980\pi$$
0.695957 + 0.718083i $$0.254980\pi$$
$$318$$ 0 0
$$319$$ 1536.00 0.269591
$$320$$ 0 0
$$321$$ 3132.00 0.544583
$$322$$ 0 0
$$323$$ 6696.00 1.15348
$$324$$ 0 0
$$325$$ −1220.00 −0.208226
$$326$$ 0 0
$$327$$ 5340.00 0.903066
$$328$$ 0 0
$$329$$ 7200.00 1.20653
$$330$$ 0 0
$$331$$ 3732.00 0.619726 0.309863 0.950781i $$-0.399717\pi$$
0.309863 + 0.950781i $$0.399717\pi$$
$$332$$ 0 0
$$333$$ −2052.00 −0.337684
$$334$$ 0 0
$$335$$ −2784.00 −0.454048
$$336$$ 0 0
$$337$$ −5598.00 −0.904874 −0.452437 0.891796i $$-0.649445\pi$$
−0.452437 + 0.891796i $$0.649445\pi$$
$$338$$ 0 0
$$339$$ −1866.00 −0.298959
$$340$$ 0 0
$$341$$ 2448.00 0.388758
$$342$$ 0 0
$$343$$ 6504.00 1.02386
$$344$$ 0 0
$$345$$ −1728.00 −0.269659
$$346$$ 0 0
$$347$$ −8220.00 −1.27168 −0.635840 0.771821i $$-0.719346\pi$$
−0.635840 + 0.771821i $$0.719346\pi$$
$$348$$ 0 0
$$349$$ −11844.0 −1.81660 −0.908302 0.418315i $$-0.862621\pi$$
−0.908302 + 0.418315i $$0.862621\pi$$
$$350$$ 0 0
$$351$$ 540.000 0.0821170
$$352$$ 0 0
$$353$$ −7006.00 −1.05635 −0.528175 0.849135i $$-0.677124\pi$$
−0.528175 + 0.849135i $$0.677124\pi$$
$$354$$ 0 0
$$355$$ 3648.00 0.545396
$$356$$ 0 0
$$357$$ −2232.00 −0.330896
$$358$$ 0 0
$$359$$ 7512.00 1.10437 0.552184 0.833722i $$-0.313795\pi$$
0.552184 + 0.833722i $$0.313795\pi$$
$$360$$ 0 0
$$361$$ 4805.00 0.700539
$$362$$ 0 0
$$363$$ −3561.00 −0.514887
$$364$$ 0 0
$$365$$ 6576.00 0.943023
$$366$$ 0 0
$$367$$ 5076.00 0.721976 0.360988 0.932571i $$-0.382440\pi$$
0.360988 + 0.932571i $$0.382440\pi$$
$$368$$ 0 0
$$369$$ 198.000 0.0279335
$$370$$ 0 0
$$371$$ −3072.00 −0.429893
$$372$$ 0 0
$$373$$ 4860.00 0.674641 0.337321 0.941390i $$-0.390479\pi$$
0.337321 + 0.941390i $$0.390479\pi$$
$$374$$ 0 0
$$375$$ 4464.00 0.614720
$$376$$ 0 0
$$377$$ −2560.00 −0.349726
$$378$$ 0 0
$$379$$ −5964.00 −0.808312 −0.404156 0.914690i $$-0.632435\pi$$
−0.404156 + 0.914690i $$0.632435\pi$$
$$380$$ 0 0
$$381$$ 612.000 0.0822932
$$382$$ 0 0
$$383$$ −432.000 −0.0576349 −0.0288175 0.999585i $$-0.509174\pi$$
−0.0288175 + 0.999585i $$0.509174\pi$$
$$384$$ 0 0
$$385$$ −1152.00 −0.152497
$$386$$ 0 0
$$387$$ −1836.00 −0.241161
$$388$$ 0 0
$$389$$ 8888.00 1.15846 0.579228 0.815165i $$-0.303354\pi$$
0.579228 + 0.815165i $$0.303354\pi$$
$$390$$ 0 0
$$391$$ 4464.00 0.577376
$$392$$ 0 0
$$393$$ 1044.00 0.134002
$$394$$ 0 0
$$395$$ 10848.0 1.38183
$$396$$ 0 0
$$397$$ −2676.00 −0.338299 −0.169149 0.985590i $$-0.554102\pi$$
−0.169149 + 0.985590i $$0.554102\pi$$
$$398$$ 0 0
$$399$$ −3888.00 −0.487828
$$400$$ 0 0
$$401$$ 13790.0 1.71731 0.858653 0.512557i $$-0.171302\pi$$
0.858653 + 0.512557i $$0.171302\pi$$
$$402$$ 0 0
$$403$$ −4080.00 −0.504316
$$404$$ 0 0
$$405$$ −648.000 −0.0795046
$$406$$ 0 0
$$407$$ 2736.00 0.333215
$$408$$ 0 0
$$409$$ 1974.00 0.238650 0.119325 0.992855i $$-0.461927\pi$$
0.119325 + 0.992855i $$0.461927\pi$$
$$410$$ 0 0
$$411$$ −318.000 −0.0381649
$$412$$ 0 0
$$413$$ 9936.00 1.18382
$$414$$ 0 0
$$415$$ −864.000 −0.102198
$$416$$ 0 0
$$417$$ 3564.00 0.418537
$$418$$ 0 0
$$419$$ 4764.00 0.555457 0.277729 0.960660i $$-0.410418\pi$$
0.277729 + 0.960660i $$0.410418\pi$$
$$420$$ 0 0
$$421$$ −92.0000 −0.0106504 −0.00532518 0.999986i $$-0.501695\pi$$
−0.00532518 + 0.999986i $$0.501695\pi$$
$$422$$ 0 0
$$423$$ −5400.00 −0.620702
$$424$$ 0 0
$$425$$ −3782.00 −0.431656
$$426$$ 0 0
$$427$$ 1008.00 0.114240
$$428$$ 0 0
$$429$$ −720.000 −0.0810301
$$430$$ 0 0
$$431$$ 10488.0 1.17213 0.586066 0.810263i $$-0.300676\pi$$
0.586066 + 0.810263i $$0.300676\pi$$
$$432$$ 0 0
$$433$$ −13138.0 −1.45813 −0.729067 0.684442i $$-0.760046\pi$$
−0.729067 + 0.684442i $$0.760046\pi$$
$$434$$ 0 0
$$435$$ 3072.00 0.338600
$$436$$ 0 0
$$437$$ 7776.00 0.851205
$$438$$ 0 0
$$439$$ 3612.00 0.392691 0.196346 0.980535i $$-0.437093\pi$$
0.196346 + 0.980535i $$0.437093\pi$$
$$440$$ 0 0
$$441$$ −1791.00 −0.193392
$$442$$ 0 0
$$443$$ −12972.0 −1.39124 −0.695619 0.718411i $$-0.744870\pi$$
−0.695619 + 0.718411i $$0.744870\pi$$
$$444$$ 0 0
$$445$$ −7504.00 −0.799379
$$446$$ 0 0
$$447$$ 7944.00 0.840578
$$448$$ 0 0
$$449$$ 5998.00 0.630430 0.315215 0.949020i $$-0.397923\pi$$
0.315215 + 0.949020i $$0.397923\pi$$
$$450$$ 0 0
$$451$$ −264.000 −0.0275638
$$452$$ 0 0
$$453$$ −10260.0 −1.06414
$$454$$ 0 0
$$455$$ 1920.00 0.197826
$$456$$ 0 0
$$457$$ 8934.00 0.914475 0.457237 0.889345i $$-0.348839\pi$$
0.457237 + 0.889345i $$0.348839\pi$$
$$458$$ 0 0
$$459$$ 1674.00 0.170230
$$460$$ 0 0
$$461$$ 7448.00 0.752468 0.376234 0.926525i $$-0.377219\pi$$
0.376234 + 0.926525i $$0.377219\pi$$
$$462$$ 0 0
$$463$$ −4356.00 −0.437236 −0.218618 0.975810i $$-0.570155\pi$$
−0.218618 + 0.975810i $$0.570155\pi$$
$$464$$ 0 0
$$465$$ 4896.00 0.488272
$$466$$ 0 0
$$467$$ −8580.00 −0.850182 −0.425091 0.905151i $$-0.639758\pi$$
−0.425091 + 0.905151i $$0.639758\pi$$
$$468$$ 0 0
$$469$$ −4176.00 −0.411151
$$470$$ 0 0
$$471$$ 180.000 0.0176093
$$472$$ 0 0
$$473$$ 2448.00 0.237969
$$474$$ 0 0
$$475$$ −6588.00 −0.636375
$$476$$ 0 0
$$477$$ 2304.00 0.221159
$$478$$ 0 0
$$479$$ 12648.0 1.20648 0.603238 0.797561i $$-0.293877\pi$$
0.603238 + 0.797561i $$0.293877\pi$$
$$480$$ 0 0
$$481$$ −4560.00 −0.432262
$$482$$ 0 0
$$483$$ −2592.00 −0.244182
$$484$$ 0 0
$$485$$ −10224.0 −0.957212
$$486$$ 0 0
$$487$$ −15036.0 −1.39907 −0.699534 0.714599i $$-0.746609\pi$$
−0.699534 + 0.714599i $$0.746609\pi$$
$$488$$ 0 0
$$489$$ −684.000 −0.0632547
$$490$$ 0 0
$$491$$ 15684.0 1.44157 0.720783 0.693161i $$-0.243782\pi$$
0.720783 + 0.693161i $$0.243782\pi$$
$$492$$ 0 0
$$493$$ −7936.00 −0.724989
$$494$$ 0 0
$$495$$ 864.000 0.0784523
$$496$$ 0 0
$$497$$ 5472.00 0.493869
$$498$$ 0 0
$$499$$ −16308.0 −1.46302 −0.731509 0.681831i $$-0.761184\pi$$
−0.731509 + 0.681831i $$0.761184\pi$$
$$500$$ 0 0
$$501$$ −12384.0 −1.10434
$$502$$ 0 0
$$503$$ −10344.0 −0.916931 −0.458465 0.888712i $$-0.651601\pi$$
−0.458465 + 0.888712i $$0.651601\pi$$
$$504$$ 0 0
$$505$$ −4864.00 −0.428604
$$506$$ 0 0
$$507$$ −5391.00 −0.472234
$$508$$ 0 0
$$509$$ −5648.00 −0.491833 −0.245917 0.969291i $$-0.579089\pi$$
−0.245917 + 0.969291i $$0.579089\pi$$
$$510$$ 0 0
$$511$$ 9864.00 0.853929
$$512$$ 0 0
$$513$$ 2916.00 0.250964
$$514$$ 0 0
$$515$$ 7584.00 0.648915
$$516$$ 0 0
$$517$$ 7200.00 0.612487
$$518$$ 0 0
$$519$$ −4056.00 −0.343042
$$520$$ 0 0
$$521$$ −19498.0 −1.63958 −0.819792 0.572662i $$-0.805911\pi$$
−0.819792 + 0.572662i $$0.805911\pi$$
$$522$$ 0 0
$$523$$ 22596.0 1.88920 0.944602 0.328217i $$-0.106448\pi$$
0.944602 + 0.328217i $$0.106448\pi$$
$$524$$ 0 0
$$525$$ 2196.00 0.182555
$$526$$ 0 0
$$527$$ −12648.0 −1.04546
$$528$$ 0 0
$$529$$ −6983.00 −0.573929
$$530$$ 0 0
$$531$$ −7452.00 −0.609019
$$532$$ 0 0
$$533$$ 440.000 0.0357571
$$534$$ 0 0
$$535$$ −8352.00 −0.674932
$$536$$ 0 0
$$537$$ −5148.00 −0.413692
$$538$$ 0 0
$$539$$ 2388.00 0.190832
$$540$$ 0 0
$$541$$ −812.000 −0.0645298 −0.0322649 0.999479i $$-0.510272\pi$$
−0.0322649 + 0.999479i $$0.510272\pi$$
$$542$$ 0 0
$$543$$ −11076.0 −0.875353
$$544$$ 0 0
$$545$$ −14240.0 −1.11922
$$546$$ 0 0
$$547$$ −6132.00 −0.479315 −0.239658 0.970857i $$-0.577035\pi$$
−0.239658 + 0.970857i $$0.577035\pi$$
$$548$$ 0 0
$$549$$ −756.000 −0.0587710
$$550$$ 0 0
$$551$$ −13824.0 −1.06882
$$552$$ 0 0
$$553$$ 16272.0 1.25128
$$554$$ 0 0
$$555$$ 5472.00 0.418511
$$556$$ 0 0
$$557$$ 2792.00 0.212389 0.106195 0.994345i $$-0.466133\pi$$
0.106195 + 0.994345i $$0.466133\pi$$
$$558$$ 0 0
$$559$$ −4080.00 −0.308704
$$560$$ 0 0
$$561$$ −2232.00 −0.167977
$$562$$ 0 0
$$563$$ −6468.00 −0.484181 −0.242090 0.970254i $$-0.577833\pi$$
−0.242090 + 0.970254i $$0.577833\pi$$
$$564$$ 0 0
$$565$$ 4976.00 0.370517
$$566$$ 0 0
$$567$$ −972.000 −0.0719932
$$568$$ 0 0
$$569$$ −10522.0 −0.775229 −0.387614 0.921822i $$-0.626701\pi$$
−0.387614 + 0.921822i $$0.626701\pi$$
$$570$$ 0 0
$$571$$ −1068.00 −0.0782739 −0.0391370 0.999234i $$-0.512461\pi$$
−0.0391370 + 0.999234i $$0.512461\pi$$
$$572$$ 0 0
$$573$$ 144.000 0.0104986
$$574$$ 0 0
$$575$$ −4392.00 −0.318537
$$576$$ 0 0
$$577$$ 3602.00 0.259884 0.129942 0.991522i $$-0.458521\pi$$
0.129942 + 0.991522i $$0.458521\pi$$
$$578$$ 0 0
$$579$$ 7242.00 0.519805
$$580$$ 0 0
$$581$$ −1296.00 −0.0925424
$$582$$ 0 0
$$583$$ −3072.00 −0.218232
$$584$$ 0 0
$$585$$ −1440.00 −0.101772
$$586$$ 0 0
$$587$$ 17940.0 1.26144 0.630718 0.776012i $$-0.282760\pi$$
0.630718 + 0.776012i $$0.282760\pi$$
$$588$$ 0 0
$$589$$ −22032.0 −1.54128
$$590$$ 0 0
$$591$$ 12912.0 0.898695
$$592$$ 0 0
$$593$$ 18034.0 1.24885 0.624425 0.781085i $$-0.285334\pi$$
0.624425 + 0.781085i $$0.285334\pi$$
$$594$$ 0 0
$$595$$ 5952.00 0.410098
$$596$$ 0 0
$$597$$ 612.000 0.0419556
$$598$$ 0 0
$$599$$ −12264.0 −0.836550 −0.418275 0.908320i $$-0.637365\pi$$
−0.418275 + 0.908320i $$0.637365\pi$$
$$600$$ 0 0
$$601$$ 12634.0 0.857490 0.428745 0.903426i $$-0.358956\pi$$
0.428745 + 0.903426i $$0.358956\pi$$
$$602$$ 0 0
$$603$$ 3132.00 0.211517
$$604$$ 0 0
$$605$$ 9496.00 0.638128
$$606$$ 0 0
$$607$$ 2796.00 0.186962 0.0934812 0.995621i $$-0.470201\pi$$
0.0934812 + 0.995621i $$0.470201\pi$$
$$608$$ 0 0
$$609$$ 4608.00 0.306610
$$610$$ 0 0
$$611$$ −12000.0 −0.794547
$$612$$ 0 0
$$613$$ 13788.0 0.908470 0.454235 0.890882i $$-0.349913\pi$$
0.454235 + 0.890882i $$0.349913\pi$$
$$614$$ 0 0
$$615$$ −528.000 −0.0346195
$$616$$ 0 0
$$617$$ 18074.0 1.17931 0.589653 0.807657i $$-0.299264\pi$$
0.589653 + 0.807657i $$0.299264\pi$$
$$618$$ 0 0
$$619$$ 5940.00 0.385701 0.192850 0.981228i $$-0.438227\pi$$
0.192850 + 0.981228i $$0.438227\pi$$
$$620$$ 0 0
$$621$$ 1944.00 0.125620
$$622$$ 0 0
$$623$$ −11256.0 −0.723856
$$624$$ 0 0
$$625$$ −4279.00 −0.273856
$$626$$ 0 0
$$627$$ −3888.00 −0.247642
$$628$$ 0 0
$$629$$ −14136.0 −0.896088
$$630$$ 0 0
$$631$$ −8700.00 −0.548877 −0.274439 0.961605i $$-0.588492\pi$$
−0.274439 + 0.961605i $$0.588492\pi$$
$$632$$ 0 0
$$633$$ −12060.0 −0.757254
$$634$$ 0 0
$$635$$ −1632.00 −0.101990
$$636$$ 0 0
$$637$$ −3980.00 −0.247556
$$638$$ 0 0
$$639$$ −4104.00 −0.254072
$$640$$ 0 0
$$641$$ −16306.0 −1.00476 −0.502378 0.864648i $$-0.667541\pi$$
−0.502378 + 0.864648i $$0.667541\pi$$
$$642$$ 0 0
$$643$$ 22668.0 1.39026 0.695131 0.718883i $$-0.255346\pi$$
0.695131 + 0.718883i $$0.255346\pi$$
$$644$$ 0 0
$$645$$ 4896.00 0.298883
$$646$$ 0 0
$$647$$ −11928.0 −0.724788 −0.362394 0.932025i $$-0.618041\pi$$
−0.362394 + 0.932025i $$0.618041\pi$$
$$648$$ 0 0
$$649$$ 9936.00 0.600959
$$650$$ 0 0
$$651$$ 7344.00 0.442141
$$652$$ 0 0
$$653$$ −2552.00 −0.152936 −0.0764682 0.997072i $$-0.524364\pi$$
−0.0764682 + 0.997072i $$0.524364\pi$$
$$654$$ 0 0
$$655$$ −2784.00 −0.166076
$$656$$ 0 0
$$657$$ −7398.00 −0.439305
$$658$$ 0 0
$$659$$ −2196.00 −0.129809 −0.0649044 0.997891i $$-0.520674\pi$$
−0.0649044 + 0.997891i $$0.520674\pi$$
$$660$$ 0 0
$$661$$ −4260.00 −0.250673 −0.125336 0.992114i $$-0.540001\pi$$
−0.125336 + 0.992114i $$0.540001\pi$$
$$662$$ 0 0
$$663$$ 3720.00 0.217908
$$664$$ 0 0
$$665$$ 10368.0 0.604592
$$666$$ 0 0
$$667$$ −9216.00 −0.535000
$$668$$ 0 0
$$669$$ −1548.00 −0.0894606
$$670$$ 0 0
$$671$$ 1008.00 0.0579932
$$672$$ 0 0
$$673$$ −2018.00 −0.115584 −0.0577921 0.998329i $$-0.518406\pi$$
−0.0577921 + 0.998329i $$0.518406\pi$$
$$674$$ 0 0
$$675$$ −1647.00 −0.0939156
$$676$$ 0 0
$$677$$ 9256.00 0.525461 0.262730 0.964869i $$-0.415377\pi$$
0.262730 + 0.964869i $$0.415377\pi$$
$$678$$ 0 0
$$679$$ −15336.0 −0.866777
$$680$$ 0 0
$$681$$ −4284.00 −0.241062
$$682$$ 0 0
$$683$$ −29244.0 −1.63835 −0.819173 0.573546i $$-0.805567\pi$$
−0.819173 + 0.573546i $$0.805567\pi$$
$$684$$ 0 0
$$685$$ 848.000 0.0472999
$$686$$ 0 0
$$687$$ −18084.0 −1.00429
$$688$$ 0 0
$$689$$ 5120.00 0.283101
$$690$$ 0 0
$$691$$ −3684.00 −0.202816 −0.101408 0.994845i $$-0.532335\pi$$
−0.101408 + 0.994845i $$0.532335\pi$$
$$692$$ 0 0
$$693$$ 1296.00 0.0710404
$$694$$ 0 0
$$695$$ −9504.00 −0.518715
$$696$$ 0 0
$$697$$ 1364.00 0.0741251
$$698$$ 0 0
$$699$$ −7890.00 −0.426934
$$700$$ 0 0
$$701$$ 13456.0 0.725002 0.362501 0.931983i $$-0.381923\pi$$
0.362501 + 0.931983i $$0.381923\pi$$
$$702$$ 0 0
$$703$$ −24624.0 −1.32107
$$704$$ 0 0
$$705$$ 14400.0 0.769270
$$706$$ 0 0
$$707$$ −7296.00 −0.388111
$$708$$ 0 0
$$709$$ −6460.00 −0.342187 −0.171093 0.985255i $$-0.554730\pi$$
−0.171093 + 0.985255i $$0.554730\pi$$
$$710$$ 0 0
$$711$$ −12204.0 −0.643721
$$712$$ 0 0
$$713$$ −14688.0 −0.771487
$$714$$ 0 0
$$715$$ 1920.00 0.100425
$$716$$ 0 0
$$717$$ 13248.0 0.690036
$$718$$ 0 0
$$719$$ −17160.0 −0.890070 −0.445035 0.895513i $$-0.646809\pi$$
−0.445035 + 0.895513i $$0.646809\pi$$
$$720$$ 0 0
$$721$$ 11376.0 0.587607
$$722$$ 0 0
$$723$$ 14490.0 0.745351
$$724$$ 0 0
$$725$$ 7808.00 0.399975
$$726$$ 0 0
$$727$$ −11820.0 −0.602998 −0.301499 0.953466i $$-0.597487\pi$$
−0.301499 + 0.953466i $$0.597487\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −12648.0 −0.639950
$$732$$ 0 0
$$733$$ 23924.0 1.20553 0.602765 0.797919i $$-0.294066\pi$$
0.602765 + 0.797919i $$0.294066\pi$$
$$734$$ 0 0
$$735$$ 4776.00 0.239681
$$736$$ 0 0
$$737$$ −4176.00 −0.208718
$$738$$ 0 0
$$739$$ −11796.0 −0.587176 −0.293588 0.955932i $$-0.594849\pi$$
−0.293588 + 0.955932i $$0.594849\pi$$
$$740$$ 0 0
$$741$$ 6480.00 0.321253
$$742$$ 0 0
$$743$$ 27024.0 1.33434 0.667170 0.744906i $$-0.267506\pi$$
0.667170 + 0.744906i $$0.267506\pi$$
$$744$$ 0 0
$$745$$ −21184.0 −1.04177
$$746$$ 0 0
$$747$$ 972.000 0.0476086
$$748$$ 0 0
$$749$$ −12528.0 −0.611166
$$750$$ 0 0
$$751$$ −17340.0 −0.842537 −0.421269 0.906936i $$-0.638415\pi$$
−0.421269 + 0.906936i $$0.638415\pi$$
$$752$$ 0 0
$$753$$ −16596.0 −0.803176
$$754$$ 0 0
$$755$$ 27360.0 1.31885
$$756$$ 0 0
$$757$$ 27236.0 1.30767 0.653837 0.756635i $$-0.273158\pi$$
0.653837 + 0.756635i $$0.273158\pi$$
$$758$$ 0 0
$$759$$ −2592.00 −0.123957
$$760$$ 0 0
$$761$$ 14758.0 0.702992 0.351496 0.936189i $$-0.385673\pi$$
0.351496 + 0.936189i $$0.385673\pi$$
$$762$$ 0 0
$$763$$ −21360.0 −1.01348
$$764$$ 0 0
$$765$$ −4464.00 −0.210976
$$766$$ 0 0
$$767$$ −16560.0 −0.779592
$$768$$ 0 0
$$769$$ 25774.0 1.20863 0.604314 0.796747i $$-0.293447\pi$$
0.604314 + 0.796747i $$0.293447\pi$$
$$770$$ 0 0
$$771$$ −762.000 −0.0355937
$$772$$ 0 0
$$773$$ −37424.0 −1.74133 −0.870665 0.491877i $$-0.836311\pi$$
−0.870665 + 0.491877i $$0.836311\pi$$
$$774$$ 0 0
$$775$$ 12444.0 0.576776
$$776$$ 0 0
$$777$$ 8208.00 0.378971
$$778$$ 0 0
$$779$$ 2376.00 0.109280
$$780$$ 0 0
$$781$$ 5472.00 0.250709
$$782$$ 0 0
$$783$$ −3456.00 −0.157736
$$784$$ 0 0
$$785$$ −480.000 −0.0218241
$$786$$ 0 0
$$787$$ −12804.0 −0.579941 −0.289970 0.957036i $$-0.593645\pi$$
−0.289970 + 0.957036i $$0.593645\pi$$
$$788$$ 0 0
$$789$$ 12816.0 0.578278
$$790$$ 0 0
$$791$$ 7464.00 0.335511
$$792$$ 0 0
$$793$$ −1680.00 −0.0752315
$$794$$ 0 0
$$795$$ −6144.00 −0.274095
$$796$$ 0 0
$$797$$ −32024.0 −1.42327 −0.711636 0.702548i $$-0.752046\pi$$
−0.711636 + 0.702548i $$0.752046\pi$$
$$798$$ 0 0
$$799$$ −37200.0 −1.64711
$$800$$ 0 0
$$801$$ 8442.00 0.372389
$$802$$ 0 0
$$803$$ 9864.00 0.433491
$$804$$ 0 0
$$805$$ 6912.00 0.302629
$$806$$ 0 0
$$807$$ 13632.0 0.594633
$$808$$ 0 0
$$809$$ −38090.0 −1.65534 −0.827672 0.561212i $$-0.810335\pi$$
−0.827672 + 0.561212i $$0.810335\pi$$
$$810$$ 0 0
$$811$$ −10428.0 −0.451512 −0.225756 0.974184i $$-0.572485\pi$$
−0.225756 + 0.974184i $$0.572485\pi$$
$$812$$ 0 0
$$813$$ 6228.00 0.268666
$$814$$ 0 0
$$815$$ 1824.00 0.0783950
$$816$$ 0 0
$$817$$ −22032.0 −0.943454
$$818$$ 0 0
$$819$$ −2160.00 −0.0921569
$$820$$ 0 0
$$821$$ −7984.00 −0.339395 −0.169698 0.985496i $$-0.554279\pi$$
−0.169698 + 0.985496i $$0.554279\pi$$
$$822$$ 0 0
$$823$$ 28788.0 1.21930 0.609652 0.792669i $$-0.291309\pi$$
0.609652 + 0.792669i $$0.291309\pi$$
$$824$$ 0 0
$$825$$ 2196.00 0.0926726
$$826$$ 0 0
$$827$$ 468.000 0.0196783 0.00983915 0.999952i $$-0.496868\pi$$
0.00983915 + 0.999952i $$0.496868\pi$$
$$828$$ 0 0
$$829$$ 28852.0 1.20877 0.604386 0.796692i $$-0.293419\pi$$
0.604386 + 0.796692i $$0.293419\pi$$
$$830$$ 0 0
$$831$$ 1452.00 0.0606129
$$832$$ 0 0
$$833$$ −12338.0 −0.513189
$$834$$ 0 0
$$835$$ 33024.0 1.36867
$$836$$ 0 0
$$837$$ −5508.00 −0.227460
$$838$$ 0 0
$$839$$ −1944.00 −0.0799932 −0.0399966 0.999200i $$-0.512735\pi$$
−0.0399966 + 0.999200i $$0.512735\pi$$
$$840$$ 0 0
$$841$$ −8005.00 −0.328222
$$842$$ 0 0
$$843$$ −1218.00 −0.0497629
$$844$$ 0 0
$$845$$ 14376.0 0.585266
$$846$$ 0 0
$$847$$ 14244.0 0.577839
$$848$$ 0 0
$$849$$ −24516.0 −0.991033
$$850$$ 0 0
$$851$$ −16416.0 −0.661261
$$852$$ 0 0
$$853$$ −37044.0 −1.48694 −0.743472 0.668768i $$-0.766822\pi$$
−0.743472 + 0.668768i $$0.766822\pi$$
$$854$$ 0 0
$$855$$ −7776.00 −0.311033
$$856$$ 0 0
$$857$$ 15046.0 0.599722 0.299861 0.953983i $$-0.403060\pi$$
0.299861 + 0.953983i $$0.403060\pi$$
$$858$$ 0 0
$$859$$ 12180.0 0.483791 0.241895 0.970302i $$-0.422231\pi$$
0.241895 + 0.970302i $$0.422231\pi$$
$$860$$ 0 0
$$861$$ −792.000 −0.0313488
$$862$$ 0 0
$$863$$ −28752.0 −1.13410 −0.567051 0.823683i $$-0.691916\pi$$
−0.567051 + 0.823683i $$0.691916\pi$$
$$864$$ 0 0
$$865$$ 10816.0 0.425150
$$866$$ 0 0
$$867$$ −3207.00 −0.125623
$$868$$ 0 0
$$869$$ 16272.0 0.635201
$$870$$ 0 0
$$871$$ 6960.00 0.270758
$$872$$ 0 0
$$873$$ 11502.0 0.445915
$$874$$ 0 0
$$875$$ −17856.0 −0.689878
$$876$$ 0 0
$$877$$ 31884.0 1.22765 0.613823 0.789443i $$-0.289631\pi$$
0.613823 + 0.789443i $$0.289631\pi$$
$$878$$ 0 0
$$879$$ 14880.0 0.570978
$$880$$ 0 0
$$881$$ 30802.0 1.17792 0.588959 0.808163i $$-0.299538\pi$$
0.588959 + 0.808163i $$0.299538\pi$$
$$882$$ 0 0
$$883$$ 32460.0 1.23711 0.618554 0.785742i $$-0.287719\pi$$
0.618554 + 0.785742i $$0.287719\pi$$
$$884$$ 0 0
$$885$$ 19872.0 0.754791
$$886$$ 0 0
$$887$$ −14832.0 −0.561454 −0.280727 0.959788i $$-0.590576\pi$$
−0.280727 + 0.959788i $$0.590576\pi$$
$$888$$ 0 0
$$889$$ −2448.00 −0.0923547
$$890$$ 0 0
$$891$$ −972.000 −0.0365468
$$892$$ 0 0
$$893$$ −64800.0 −2.42827
$$894$$ 0 0
$$895$$ 13728.0 0.512711
$$896$$ 0 0
$$897$$ 4320.00 0.160803
$$898$$ 0 0
$$899$$ 26112.0 0.968725
$$900$$ 0 0
$$901$$ 15872.0 0.586873
$$902$$ 0 0
$$903$$ 7344.00 0.270646
$$904$$ 0 0
$$905$$ 29536.0 1.08487
$$906$$ 0 0
$$907$$ 6900.00 0.252603 0.126301 0.991992i $$-0.459689\pi$$
0.126301 + 0.991992i $$0.459689\pi$$
$$908$$ 0 0
$$909$$ 5472.00 0.199664
$$910$$ 0 0
$$911$$ 32832.0 1.19404 0.597021 0.802225i $$-0.296351\pi$$
0.597021 + 0.802225i $$0.296351\pi$$
$$912$$ 0 0
$$913$$ −1296.00 −0.0469785
$$914$$ 0 0
$$915$$ 2016.00 0.0728381
$$916$$ 0 0
$$917$$ −4176.00 −0.150386
$$918$$ 0 0
$$919$$ 8340.00 0.299359 0.149680 0.988735i $$-0.452176\pi$$
0.149680 + 0.988735i $$0.452176\pi$$
$$920$$ 0 0
$$921$$ 20052.0 0.717411
$$922$$ 0 0
$$923$$ −9120.00 −0.325231
$$924$$ 0 0
$$925$$ 13908.0 0.494370
$$926$$ 0 0
$$927$$ −8532.00 −0.302295
$$928$$ 0 0
$$929$$ −39826.0 −1.40651 −0.703255 0.710937i $$-0.748271\pi$$
−0.703255 + 0.710937i $$0.748271\pi$$
$$930$$ 0 0
$$931$$ −21492.0 −0.756576
$$932$$ 0 0
$$933$$ 14976.0 0.525501
$$934$$ 0 0
$$935$$ 5952.00 0.208183
$$936$$ 0 0
$$937$$ −28550.0 −0.995398 −0.497699 0.867350i $$-0.665822\pi$$
−0.497699 + 0.867350i $$0.665822\pi$$
$$938$$ 0 0
$$939$$ −16206.0 −0.563219
$$940$$ 0 0
$$941$$ −50632.0 −1.75404 −0.877022 0.480450i $$-0.840473\pi$$
−0.877022 + 0.480450i $$0.840473\pi$$
$$942$$ 0 0
$$943$$ 1584.00 0.0547000
$$944$$ 0 0
$$945$$ 2592.00 0.0892251
$$946$$ 0 0
$$947$$ 18204.0 0.624657 0.312329 0.949974i $$-0.398891\pi$$
0.312329 + 0.949974i $$0.398891\pi$$
$$948$$ 0 0
$$949$$ −16440.0 −0.562345
$$950$$ 0 0
$$951$$ 23568.0 0.803622
$$952$$ 0 0
$$953$$ 4934.00 0.167710 0.0838552 0.996478i $$-0.473277\pi$$
0.0838552 + 0.996478i $$0.473277\pi$$
$$954$$ 0 0
$$955$$ −384.000 −0.0130115
$$956$$ 0 0
$$957$$ 4608.00 0.155648
$$958$$ 0 0
$$959$$ 1272.00 0.0428311
$$960$$ 0 0
$$961$$ 11825.0 0.396932
$$962$$ 0 0
$$963$$ 9396.00 0.314415
$$964$$ 0 0
$$965$$ −19312.0 −0.644223
$$966$$ 0 0
$$967$$ −13284.0 −0.441763 −0.220881 0.975301i $$-0.570893\pi$$
−0.220881 + 0.975301i $$0.570893\pi$$
$$968$$ 0 0
$$969$$ 20088.0 0.665964
$$970$$ 0 0
$$971$$ 50820.0 1.67960 0.839800 0.542896i $$-0.182672\pi$$
0.839800 + 0.542896i $$0.182672\pi$$
$$972$$ 0 0
$$973$$ −14256.0 −0.469709
$$974$$ 0 0
$$975$$ −3660.00 −0.120219
$$976$$ 0 0
$$977$$ 11038.0 0.361450 0.180725 0.983534i $$-0.442156\pi$$
0.180725 + 0.983534i $$0.442156\pi$$
$$978$$ 0 0
$$979$$ −11256.0 −0.367460
$$980$$ 0 0
$$981$$ 16020.0 0.521386
$$982$$ 0 0
$$983$$ 44112.0 1.43129 0.715643 0.698466i $$-0.246134\pi$$
0.715643 + 0.698466i $$0.246134\pi$$
$$984$$ 0 0
$$985$$ −34432.0 −1.11380
$$986$$ 0 0
$$987$$ 21600.0 0.696591
$$988$$ 0 0
$$989$$ −14688.0 −0.472246
$$990$$ 0 0
$$991$$ 56196.0 1.80134 0.900668 0.434507i $$-0.143077\pi$$
0.900668 + 0.434507i $$0.143077\pi$$
$$992$$ 0 0
$$993$$ 11196.0 0.357799
$$994$$ 0 0
$$995$$ −1632.00 −0.0519979
$$996$$ 0 0
$$997$$ −45588.0 −1.44813 −0.724065 0.689731i $$-0.757729\pi$$
−0.724065 + 0.689731i $$0.757729\pi$$
$$998$$ 0 0
$$999$$ −6156.00 −0.194962
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 768.4.a.c.1.1 1
3.2 odd 2 2304.4.a.k.1.1 1
4.3 odd 2 768.4.a.a.1.1 1
8.3 odd 2 768.4.a.d.1.1 1
8.5 even 2 768.4.a.b.1.1 1
12.11 even 2 2304.4.a.l.1.1 1
16.3 odd 4 384.4.d.a.193.1 2
16.5 even 4 384.4.d.b.193.1 yes 2
16.11 odd 4 384.4.d.a.193.2 yes 2
16.13 even 4 384.4.d.b.193.2 yes 2
24.5 odd 2 2304.4.a.e.1.1 1
24.11 even 2 2304.4.a.f.1.1 1
48.5 odd 4 1152.4.d.g.577.2 2
48.11 even 4 1152.4.d.b.577.2 2
48.29 odd 4 1152.4.d.g.577.1 2
48.35 even 4 1152.4.d.b.577.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.a.193.1 2 16.3 odd 4
384.4.d.a.193.2 yes 2 16.11 odd 4
384.4.d.b.193.1 yes 2 16.5 even 4
384.4.d.b.193.2 yes 2 16.13 even 4
768.4.a.a.1.1 1 4.3 odd 2
768.4.a.b.1.1 1 8.5 even 2
768.4.a.c.1.1 1 1.1 even 1 trivial
768.4.a.d.1.1 1 8.3 odd 2
1152.4.d.b.577.1 2 48.35 even 4
1152.4.d.b.577.2 2 48.11 even 4
1152.4.d.g.577.1 2 48.29 odd 4
1152.4.d.g.577.2 2 48.5 odd 4
2304.4.a.e.1.1 1 24.5 odd 2
2304.4.a.f.1.1 1 24.11 even 2
2304.4.a.k.1.1 1 3.2 odd 2
2304.4.a.l.1.1 1 12.11 even 2