# Properties

 Label 2304.4.a.k.1.1 Level $2304$ Weight $4$ Character 2304.1 Self dual yes Analytic conductor $135.940$ 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] = [2304,4,Mod(1,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, 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("2304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2304.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: $$135.940400653$$ 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 $$\chi$$ $$=$$ 2304.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+8.00000 q^{5} -12.0000 q^{7} +O(q^{10})$$ $$q+8.00000 q^{5} -12.0000 q^{7} +12.0000 q^{11} +20.0000 q^{13} -62.0000 q^{17} +108.000 q^{19} -72.0000 q^{23} -61.0000 q^{25} +128.000 q^{29} -204.000 q^{31} -96.0000 q^{35} -228.000 q^{37} -22.0000 q^{41} -204.000 q^{43} +600.000 q^{47} -199.000 q^{49} -256.000 q^{53} +96.0000 q^{55} +828.000 q^{59} -84.0000 q^{61} +160.000 q^{65} +348.000 q^{67} +456.000 q^{71} -822.000 q^{73} -144.000 q^{77} -1356.00 q^{79} -108.000 q^{83} -496.000 q^{85} -938.000 q^{89} -240.000 q^{91} +864.000 q^{95} +1278.00 q^{97} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 8.00000 0.715542 0.357771 0.933809i $$-0.383537\pi$$
0.357771 + 0.933809i $$0.383537\pi$$
$$6$$ 0 0
$$7$$ −12.0000 −0.647939 −0.323970 0.946068i $$-0.605018\pi$$
−0.323970 + 0.946068i $$0.605018\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 12.0000 0.328921 0.164461 0.986384i $$-0.447412\pi$$
0.164461 + 0.986384i $$0.447412\pi$$
$$12$$ 0 0
$$13$$ 20.0000 0.426692 0.213346 0.976977i $$-0.431564\pi$$
0.213346 + 0.976977i $$0.431564\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −62.0000 −0.884542 −0.442271 0.896882i $$-0.645827\pi$$
−0.442271 + 0.896882i $$0.645827\pi$$
$$18$$ 0 0
$$19$$ 108.000 1.30405 0.652024 0.758199i $$-0.273920\pi$$
0.652024 + 0.758199i $$0.273920\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −72.0000 −0.652741 −0.326370 0.945242i $$-0.605826\pi$$
−0.326370 + 0.945242i $$0.605826\pi$$
$$24$$ 0 0
$$25$$ −61.0000 −0.488000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 128.000 0.819621 0.409810 0.912171i $$-0.365595\pi$$
0.409810 + 0.912171i $$0.365595\pi$$
$$30$$ 0 0
$$31$$ −204.000 −1.18192 −0.590959 0.806701i $$-0.701251\pi$$
−0.590959 + 0.806701i $$0.701251\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 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
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −22.0000 −0.0838006 −0.0419003 0.999122i $$-0.513341\pi$$
−0.0419003 + 0.999122i $$0.513341\pi$$
$$42$$ 0 0
$$43$$ −204.000 −0.723482 −0.361741 0.932279i $$-0.617817\pi$$
−0.361741 + 0.932279i $$0.617817\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 600.000 1.86211 0.931053 0.364884i $$-0.118891\pi$$
0.931053 + 0.364884i $$0.118891\pi$$
$$48$$ 0 0
$$49$$ −199.000 −0.580175
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −256.000 −0.663477 −0.331739 0.943371i $$-0.607635\pi$$
−0.331739 + 0.943371i $$0.607635\pi$$
$$54$$ 0 0
$$55$$ 96.0000 0.235357
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 828.000 1.82706 0.913529 0.406774i $$-0.133346\pi$$
0.913529 + 0.406774i $$0.133346\pi$$
$$60$$ 0 0
$$61$$ −84.0000 −0.176313 −0.0881565 0.996107i $$-0.528098\pi$$
−0.0881565 + 0.996107i $$0.528098\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 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
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 456.000 0.762215 0.381107 0.924531i $$-0.375543\pi$$
0.381107 + 0.924531i $$0.375543\pi$$
$$72$$ 0 0
$$73$$ −822.000 −1.31792 −0.658958 0.752180i $$-0.729002\pi$$
−0.658958 + 0.752180i $$0.729002\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 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
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −108.000 −0.142826 −0.0714129 0.997447i $$-0.522751\pi$$
−0.0714129 + 0.997447i $$0.522751\pi$$
$$84$$ 0 0
$$85$$ −496.000 −0.632927
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −938.000 −1.11717 −0.558583 0.829449i $$-0.688655\pi$$
−0.558583 + 0.829449i $$0.688655\pi$$
$$90$$ 0 0
$$91$$ −240.000 −0.276471
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 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
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −608.000 −0.598993 −0.299496 0.954097i $$-0.596819\pi$$
−0.299496 + 0.954097i $$0.596819\pi$$
$$102$$ 0 0
$$103$$ −948.000 −0.906886 −0.453443 0.891285i $$-0.649804\pi$$
−0.453443 + 0.891285i $$0.649804\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1044.00 −0.943246 −0.471623 0.881800i $$-0.656332\pi$$
−0.471623 + 0.881800i $$0.656332\pi$$
$$108$$ 0 0
$$109$$ 1780.00 1.56416 0.782078 0.623180i $$-0.214160\pi$$
0.782078 + 0.623180i $$0.214160\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 622.000 0.517813 0.258906 0.965902i $$-0.416638\pi$$
0.258906 + 0.965902i $$0.416638\pi$$
$$114$$ 0 0
$$115$$ −576.000 −0.467063
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 744.000 0.573129
$$120$$ 0 0
$$121$$ −1187.00 −0.891811
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 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
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −348.000 −0.232098 −0.116049 0.993243i $$-0.537023\pi$$
−0.116049 + 0.993243i $$0.537023\pi$$
$$132$$ 0 0
$$133$$ −1296.00 −0.844943
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 106.000 0.0661036 0.0330518 0.999454i $$-0.489477\pi$$
0.0330518 + 0.999454i $$0.489477\pi$$
$$138$$ 0 0
$$139$$ 1188.00 0.724927 0.362463 0.931998i $$-0.381936\pi$$
0.362463 + 0.931998i $$0.381936\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 240.000 0.140348
$$144$$ 0 0
$$145$$ 1024.00 0.586473
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −2648.00 −1.45592 −0.727962 0.685618i $$-0.759532\pi$$
−0.727962 + 0.685618i $$0.759532\pi$$
$$150$$ 0 0
$$151$$ −3420.00 −1.84315 −0.921575 0.388200i $$-0.873097\pi$$
−0.921575 + 0.388200i $$0.873097\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 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
$$159$$ 0 0
$$160$$ 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
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 4128.00 1.91278 0.956390 0.292093i $$-0.0943517\pi$$
0.956390 + 0.292093i $$0.0943517\pi$$
$$168$$ 0 0
$$169$$ −1797.00 −0.817934
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 1352.00 0.594166 0.297083 0.954852i $$-0.403986\pi$$
0.297083 + 0.954852i $$0.403986\pi$$
$$174$$ 0 0
$$175$$ 732.000 0.316194
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 1716.00 0.716536 0.358268 0.933619i $$-0.383367\pi$$
0.358268 + 0.933619i $$0.383367\pi$$
$$180$$ 0 0
$$181$$ −3692.00 −1.51616 −0.758078 0.652164i $$-0.773861\pi$$
−0.758078 + 0.652164i $$0.773861\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −1824.00 −0.724882
$$186$$ 0 0
$$187$$ −744.000 −0.290945
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −48.0000 −0.0181841 −0.00909204 0.999959i $$-0.502894\pi$$
−0.00909204 + 0.999959i $$0.502894\pi$$
$$192$$ 0 0
$$193$$ 2414.00 0.900329 0.450165 0.892946i $$-0.351365\pi$$
0.450165 + 0.892946i $$0.351365\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4304.00 −1.55659 −0.778293 0.627902i $$-0.783914\pi$$
−0.778293 + 0.627902i $$0.783914\pi$$
$$198$$ 0 0
$$199$$ 204.000 0.0726692 0.0363346 0.999340i $$-0.488432\pi$$
0.0363346 + 0.999340i $$0.488432\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −1536.00 −0.531064
$$204$$ 0 0
$$205$$ −176.000 −0.0599628
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −1632.00 −0.517681
$$216$$ 0 0
$$217$$ 2448.00 0.765811
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 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
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1428.00 0.417532 0.208766 0.977966i $$-0.433055\pi$$
0.208766 + 0.977966i $$0.433055\pi$$
$$228$$ 0 0
$$229$$ −6028.00 −1.73948 −0.869741 0.493508i $$-0.835714\pi$$
−0.869741 + 0.493508i $$0.835714\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 2630.00 0.739472 0.369736 0.929137i $$-0.379448\pi$$
0.369736 + 0.929137i $$0.379448\pi$$
$$234$$ 0 0
$$235$$ 4800.00 1.33241
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −4416.00 −1.19518 −0.597588 0.801803i $$-0.703874\pi$$
−0.597588 + 0.801803i $$0.703874\pi$$
$$240$$ 0 0
$$241$$ 4830.00 1.29099 0.645493 0.763766i $$-0.276652\pi$$
0.645493 + 0.763766i $$0.276652\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −1592.00 −0.415139
$$246$$ 0 0
$$247$$ 2160.00 0.556427
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 5532.00 1.39114 0.695571 0.718457i $$-0.255151\pi$$
0.695571 + 0.718457i $$0.255151\pi$$
$$252$$ 0 0
$$253$$ −864.000 −0.214700
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 254.000 0.0616501 0.0308251 0.999525i $$-0.490187\pi$$
0.0308251 + 0.999525i $$0.490187\pi$$
$$258$$ 0 0
$$259$$ 2736.00 0.656397
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −4272.00 −1.00161 −0.500804 0.865561i $$-0.666962\pi$$
−0.500804 + 0.865561i $$0.666962\pi$$
$$264$$ 0 0
$$265$$ −2048.00 −0.474746
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −4544.00 −1.02994 −0.514968 0.857210i $$-0.672196\pi$$
−0.514968 + 0.857210i $$0.672196\pi$$
$$270$$ 0 0
$$271$$ 2076.00 0.465343 0.232672 0.972555i $$-0.425253\pi$$
0.232672 + 0.972555i $$0.425253\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 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
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 406.000 0.0861919 0.0430960 0.999071i $$-0.486278\pi$$
0.0430960 + 0.999071i $$0.486278\pi$$
$$282$$ 0 0
$$283$$ −8172.00 −1.71652 −0.858260 0.513216i $$-0.828454\pi$$
−0.858260 + 0.513216i $$0.828454\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 264.000 0.0542977
$$288$$ 0 0
$$289$$ −1069.00 −0.217586
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −4960.00 −0.988963 −0.494482 0.869188i $$-0.664642\pi$$
−0.494482 + 0.869188i $$0.664642\pi$$
$$294$$ 0 0
$$295$$ 6624.00 1.30734
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1440.00 −0.278520
$$300$$ 0 0
$$301$$ 2448.00 0.468772
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 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
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −4992.00 −0.910194 −0.455097 0.890442i $$-0.650395\pi$$
−0.455097 + 0.890442i $$0.650395\pi$$
$$312$$ 0 0
$$313$$ −5402.00 −0.975524 −0.487762 0.872977i $$-0.662187\pi$$
−0.487762 + 0.872977i $$0.662187\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −7856.00 −1.39191 −0.695957 0.718083i $$-0.745020\pi$$
−0.695957 + 0.718083i $$0.745020\pi$$
$$318$$ 0 0
$$319$$ 1536.00 0.269591
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −6696.00 −1.15348
$$324$$ 0 0
$$325$$ −1220.00 −0.208226
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ 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
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −2448.00 −0.388758
$$342$$ 0 0
$$343$$ 6504.00 1.02386
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 8220.00 1.27168 0.635840 0.771821i $$-0.280654\pi$$
0.635840 + 0.771821i $$0.280654\pi$$
$$348$$ 0 0
$$349$$ −11844.0 −1.81660 −0.908302 0.418315i $$-0.862621\pi$$
−0.908302 + 0.418315i $$0.862621\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 7006.00 1.05635 0.528175 0.849135i $$-0.322876\pi$$
0.528175 + 0.849135i $$0.322876\pi$$
$$354$$ 0 0
$$355$$ 3648.00 0.545396
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −7512.00 −1.10437 −0.552184 0.833722i $$-0.686205\pi$$
−0.552184 + 0.833722i $$0.686205\pi$$
$$360$$ 0 0
$$361$$ 4805.00 0.700539
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 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
$$369$$ 0 0
$$370$$ 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
$$375$$ 0 0
$$376$$ 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
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 432.000 0.0576349 0.0288175 0.999585i $$-0.490826\pi$$
0.0288175 + 0.999585i $$0.490826\pi$$
$$384$$ 0 0
$$385$$ −1152.00 −0.152497
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −8888.00 −1.15846 −0.579228 0.815165i $$-0.696646\pi$$
−0.579228 + 0.815165i $$0.696646\pi$$
$$390$$ 0 0
$$391$$ 4464.00 0.577376
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 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
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −13790.0 −1.71731 −0.858653 0.512557i $$-0.828698\pi$$
−0.858653 + 0.512557i $$0.828698\pi$$
$$402$$ 0 0
$$403$$ −4080.00 −0.504316
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 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
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −9936.00 −1.18382
$$414$$ 0 0
$$415$$ −864.000 −0.102198
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −4764.00 −0.555457 −0.277729 0.960660i $$-0.589582\pi$$
−0.277729 + 0.960660i $$0.589582\pi$$
$$420$$ 0 0
$$421$$ −92.0000 −0.0106504 −0.00532518 0.999986i $$-0.501695\pi$$
−0.00532518 + 0.999986i $$0.501695\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 3782.00 0.431656
$$426$$ 0 0
$$427$$ 1008.00 0.114240
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −10488.0 −1.17213 −0.586066 0.810263i $$-0.699324\pi$$
−0.586066 + 0.810263i $$0.699324\pi$$
$$432$$ 0 0
$$433$$ −13138.0 −1.45813 −0.729067 0.684442i $$-0.760046\pi$$
−0.729067 + 0.684442i $$0.760046\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 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
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12972.0 1.39124 0.695619 0.718411i $$-0.255130\pi$$
0.695619 + 0.718411i $$0.255130\pi$$
$$444$$ 0 0
$$445$$ −7504.00 −0.799379
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −5998.00 −0.630430 −0.315215 0.949020i $$-0.602077\pi$$
−0.315215 + 0.949020i $$0.602077\pi$$
$$450$$ 0 0
$$451$$ −264.000 −0.0275638
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 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
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −7448.00 −0.752468 −0.376234 0.926525i $$-0.622781\pi$$
−0.376234 + 0.926525i $$0.622781\pi$$
$$462$$ 0 0
$$463$$ −4356.00 −0.437236 −0.218618 0.975810i $$-0.570155\pi$$
−0.218618 + 0.975810i $$0.570155\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 8580.00 0.850182 0.425091 0.905151i $$-0.360242\pi$$
0.425091 + 0.905151i $$0.360242\pi$$
$$468$$ 0 0
$$469$$ −4176.00 −0.411151
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −2448.00 −0.237969
$$474$$ 0 0
$$475$$ −6588.00 −0.636375
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −12648.0 −1.20648 −0.603238 0.797561i $$-0.706123\pi$$
−0.603238 + 0.797561i $$0.706123\pi$$
$$480$$ 0 0
$$481$$ −4560.00 −0.432262
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 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
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −15684.0 −1.44157 −0.720783 0.693161i $$-0.756218\pi$$
−0.720783 + 0.693161i $$0.756218\pi$$
$$492$$ 0 0
$$493$$ −7936.00 −0.724989
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 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
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 10344.0 0.916931 0.458465 0.888712i $$-0.348399\pi$$
0.458465 + 0.888712i $$0.348399\pi$$
$$504$$ 0 0
$$505$$ −4864.00 −0.428604
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 5648.00 0.491833 0.245917 0.969291i $$-0.420911\pi$$
0.245917 + 0.969291i $$0.420911\pi$$
$$510$$ 0 0
$$511$$ 9864.00 0.853929
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −7584.00 −0.648915
$$516$$ 0 0
$$517$$ 7200.00 0.612487
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 19498.0 1.63958 0.819792 0.572662i $$-0.194089\pi$$
0.819792 + 0.572662i $$0.194089\pi$$
$$522$$ 0 0
$$523$$ 22596.0 1.88920 0.944602 0.328217i $$-0.106448\pi$$
0.944602 + 0.328217i $$0.106448\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 12648.0 1.04546
$$528$$ 0 0
$$529$$ −6983.00 −0.573929
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −440.000 −0.0357571
$$534$$ 0 0
$$535$$ −8352.00 −0.674932
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 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
$$543$$ 0 0
$$544$$ 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
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 13824.0 1.06882
$$552$$ 0 0
$$553$$ 16272.0 1.25128
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −2792.00 −0.212389 −0.106195 0.994345i $$-0.533867\pi$$
−0.106195 + 0.994345i $$0.533867\pi$$
$$558$$ 0 0
$$559$$ −4080.00 −0.308704
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 6468.00 0.484181 0.242090 0.970254i $$-0.422167\pi$$
0.242090 + 0.970254i $$0.422167\pi$$
$$564$$ 0 0
$$565$$ 4976.00 0.370517
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 10522.0 0.775229 0.387614 0.921822i $$-0.373299\pi$$
0.387614 + 0.921822i $$0.373299\pi$$
$$570$$ 0 0
$$571$$ −1068.00 −0.0782739 −0.0391370 0.999234i $$-0.512461\pi$$
−0.0391370 + 0.999234i $$0.512461\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 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
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 1296.00 0.0925424
$$582$$ 0 0
$$583$$ −3072.00 −0.218232
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −17940.0 −1.26144 −0.630718 0.776012i $$-0.717240\pi$$
−0.630718 + 0.776012i $$0.717240\pi$$
$$588$$ 0 0
$$589$$ −22032.0 −1.54128
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −18034.0 −1.24885 −0.624425 0.781085i $$-0.714666\pi$$
−0.624425 + 0.781085i $$0.714666\pi$$
$$594$$ 0 0
$$595$$ 5952.00 0.410098
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 12264.0 0.836550 0.418275 0.908320i $$-0.362635\pi$$
0.418275 + 0.908320i $$0.362635\pi$$
$$600$$ 0 0
$$601$$ 12634.0 0.857490 0.428745 0.903426i $$-0.358956\pi$$
0.428745 + 0.903426i $$0.358956\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 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
$$609$$ 0 0
$$610$$ 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
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18074.0 −1.17931 −0.589653 0.807657i $$-0.700736\pi$$
−0.589653 + 0.807657i $$0.700736\pi$$
$$618$$ 0 0
$$619$$ 5940.00 0.385701 0.192850 0.981228i $$-0.438227\pi$$
0.192850 + 0.981228i $$0.438227\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 11256.0 0.723856
$$624$$ 0 0
$$625$$ −4279.00 −0.273856
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 1632.00 0.101990
$$636$$ 0 0
$$637$$ −3980.00 −0.247556
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 16306.0 1.00476 0.502378 0.864648i $$-0.332459\pi$$
0.502378 + 0.864648i $$0.332459\pi$$
$$642$$ 0 0
$$643$$ 22668.0 1.39026 0.695131 0.718883i $$-0.255346\pi$$
0.695131 + 0.718883i $$0.255346\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 11928.0 0.724788 0.362394 0.932025i $$-0.381959\pi$$
0.362394 + 0.932025i $$0.381959\pi$$
$$648$$ 0 0
$$649$$ 9936.00 0.600959
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 2552.00 0.152936 0.0764682 0.997072i $$-0.475636\pi$$
0.0764682 + 0.997072i $$0.475636\pi$$
$$654$$ 0 0
$$655$$ −2784.00 −0.166076
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 2196.00 0.129809 0.0649044 0.997891i $$-0.479326\pi$$
0.0649044 + 0.997891i $$0.479326\pi$$
$$660$$ 0 0
$$661$$ −4260.00 −0.250673 −0.125336 0.992114i $$-0.540001\pi$$
−0.125336 + 0.992114i $$0.540001\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −10368.0 −0.604592
$$666$$ 0 0
$$667$$ −9216.00 −0.535000
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 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
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −9256.00 −0.525461 −0.262730 0.964869i $$-0.584623\pi$$
−0.262730 + 0.964869i $$0.584623\pi$$
$$678$$ 0 0
$$679$$ −15336.0 −0.866777
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 29244.0 1.63835 0.819173 0.573546i $$-0.194433\pi$$
0.819173 + 0.573546i $$0.194433\pi$$
$$684$$ 0 0
$$685$$ 848.000 0.0472999
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 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
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 9504.00 0.518715
$$696$$ 0 0
$$697$$ 1364.00 0.0741251
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −13456.0 −0.725002 −0.362501 0.931983i $$-0.618077\pi$$
−0.362501 + 0.931983i $$0.618077\pi$$
$$702$$ 0 0
$$703$$ −24624.0 −1.32107
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 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
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 14688.0 0.771487
$$714$$ 0 0
$$715$$ 1920.00 0.100425
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 17160.0 0.890070 0.445035 0.895513i $$-0.353191\pi$$
0.445035 + 0.895513i $$0.353191\pi$$
$$720$$ 0 0
$$721$$ 11376.0 0.587607
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 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
$$729$$ 0 0
$$730$$ 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
$$735$$ 0 0
$$736$$ 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
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −27024.0 −1.33434 −0.667170 0.744906i $$-0.732494\pi$$
−0.667170 + 0.744906i $$0.732494\pi$$
$$744$$ 0 0
$$745$$ −21184.0 −1.04177
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 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
$$753$$ 0 0
$$754$$ 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
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −14758.0 −0.702992 −0.351496 0.936189i $$-0.614327\pi$$
−0.351496 + 0.936189i $$0.614327\pi$$
$$762$$ 0 0
$$763$$ −21360.0 −1.01348
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 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
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 37424.0 1.74133 0.870665 0.491877i $$-0.163689\pi$$
0.870665 + 0.491877i $$0.163689\pi$$
$$774$$ 0 0
$$775$$ 12444.0 0.576776
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −2376.00 −0.109280
$$780$$ 0 0
$$781$$ 5472.00 0.250709
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 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
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −7464.00 −0.335511
$$792$$ 0 0
$$793$$ −1680.00 −0.0752315
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 32024.0 1.42327 0.711636 0.702548i $$-0.247954\pi$$
0.711636 + 0.702548i $$0.247954\pi$$
$$798$$ 0 0
$$799$$ −37200.0 −1.64711
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −9864.00 −0.433491
$$804$$ 0 0
$$805$$ 6912.00 0.302629
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 38090.0 1.65534 0.827672 0.561212i $$-0.189665\pi$$
0.827672 + 0.561212i $$0.189665\pi$$
$$810$$ 0 0
$$811$$ −10428.0 −0.451512 −0.225756 0.974184i $$-0.572485\pi$$
−0.225756 + 0.974184i $$0.572485\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −1824.00 −0.0783950
$$816$$ 0 0
$$817$$ −22032.0 −0.943454
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 7984.00 0.339395 0.169698 0.985496i $$-0.445721\pi$$
0.169698 + 0.985496i $$0.445721\pi$$
$$822$$ 0 0
$$823$$ 28788.0 1.21930 0.609652 0.792669i $$-0.291309\pi$$
0.609652 + 0.792669i $$0.291309\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −468.000 −0.0196783 −0.00983915 0.999952i $$-0.503132\pi$$
−0.00983915 + 0.999952i $$0.503132\pi$$
$$828$$ 0 0
$$829$$ 28852.0 1.20877 0.604386 0.796692i $$-0.293419\pi$$
0.604386 + 0.796692i $$0.293419\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 12338.0 0.513189
$$834$$ 0 0
$$835$$ 33024.0 1.36867
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 1944.00 0.0799932 0.0399966 0.999200i $$-0.487265\pi$$
0.0399966 + 0.999200i $$0.487265\pi$$
$$840$$ 0 0
$$841$$ −8005.00 −0.328222
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −14376.0 −0.585266
$$846$$ 0 0
$$847$$ 14244.0 0.577839
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 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
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −15046.0 −0.599722 −0.299861 0.953983i $$-0.596940\pi$$
−0.299861 + 0.953983i $$0.596940\pi$$
$$858$$ 0 0
$$859$$ 12180.0 0.483791 0.241895 0.970302i $$-0.422231\pi$$
0.241895 + 0.970302i $$0.422231\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 28752.0 1.13410 0.567051 0.823683i $$-0.308084\pi$$
0.567051 + 0.823683i $$0.308084\pi$$
$$864$$ 0 0
$$865$$ 10816.0 0.425150
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −16272.0 −0.635201
$$870$$ 0 0
$$871$$ 6960.00 0.270758
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 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
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −30802.0 −1.17792 −0.588959 0.808163i $$-0.700462\pi$$
−0.588959 + 0.808163i $$0.700462\pi$$
$$882$$ 0 0
$$883$$ 32460.0 1.23711 0.618554 0.785742i $$-0.287719\pi$$
0.618554 + 0.785742i $$0.287719\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 14832.0 0.561454 0.280727 0.959788i $$-0.409424\pi$$
0.280727 + 0.959788i $$0.409424\pi$$
$$888$$ 0 0
$$889$$ −2448.00 −0.0923547
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 64800.0 2.42827
$$894$$ 0 0
$$895$$ 13728.0 0.512711
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −26112.0 −0.968725
$$900$$ 0 0
$$901$$ 15872.0 0.586873
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 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
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −32832.0 −1.19404 −0.597021 0.802225i $$-0.703649\pi$$
−0.597021 + 0.802225i $$0.703649\pi$$
$$912$$ 0 0
$$913$$ −1296.00 −0.0469785
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 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
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 9120.00 0.325231
$$924$$ 0 0
$$925$$ 13908.0 0.494370
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 39826.0 1.40651 0.703255 0.710937i $$-0.251729\pi$$
0.703255 + 0.710937i $$0.251729\pi$$
$$930$$ 0 0
$$931$$ −21492.0 −0.756576
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 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
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 50632.0 1.75404 0.877022 0.480450i $$-0.159527\pi$$
0.877022 + 0.480450i $$0.159527\pi$$
$$942$$ 0 0
$$943$$ 1584.00 0.0547000
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −18204.0 −0.624657 −0.312329 0.949974i $$-0.601109\pi$$
−0.312329 + 0.949974i $$0.601109\pi$$
$$948$$ 0 0
$$949$$ −16440.0 −0.562345
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −4934.00 −0.167710 −0.0838552 0.996478i $$-0.526723\pi$$
−0.0838552 + 0.996478i $$0.526723\pi$$
$$954$$ 0 0
$$955$$ −384.000 −0.0130115
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −1272.00 −0.0428311
$$960$$ 0 0
$$961$$ 11825.0 0.396932
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 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
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −50820.0 −1.67960 −0.839800 0.542896i $$-0.817328\pi$$
−0.839800 + 0.542896i $$0.817328\pi$$
$$972$$ 0 0
$$973$$ −14256.0 −0.469709
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −11038.0 −0.361450 −0.180725 0.983534i $$-0.557844\pi$$
−0.180725 + 0.983534i $$0.557844\pi$$
$$978$$ 0 0
$$979$$ −11256.0 −0.367460
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −44112.0 −1.43129 −0.715643 0.698466i $$-0.753866\pi$$
−0.715643 + 0.698466i $$0.753866\pi$$
$$984$$ 0 0
$$985$$ −34432.0 −1.11380
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ 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
$$999$$ 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 2304.4.a.k.1.1 1
3.2 odd 2 768.4.a.c.1.1 1
4.3 odd 2 2304.4.a.l.1.1 1
8.3 odd 2 2304.4.a.f.1.1 1
8.5 even 2 2304.4.a.e.1.1 1
12.11 even 2 768.4.a.a.1.1 1
16.3 odd 4 1152.4.d.b.577.1 2
16.5 even 4 1152.4.d.g.577.2 2
16.11 odd 4 1152.4.d.b.577.2 2
16.13 even 4 1152.4.d.g.577.1 2
24.5 odd 2 768.4.a.b.1.1 1
24.11 even 2 768.4.a.d.1.1 1
48.5 odd 4 384.4.d.b.193.1 yes 2
48.11 even 4 384.4.d.a.193.2 yes 2
48.29 odd 4 384.4.d.b.193.2 yes 2
48.35 even 4 384.4.d.a.193.1 2

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