# Properties

 Label 384.4.d.b.193.2 Level $384$ Weight $4$ Character 384.193 Analytic conductor $22.657$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(193,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

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

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

 Self dual: no Analytic conductor: $$22.6567334422$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 193.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.193 Dual form 384.4.d.b.193.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000i q^{3} +8.00000i q^{5} +12.0000 q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q+3.00000i q^{3} +8.00000i q^{5} +12.0000 q^{7} -9.00000 q^{9} +12.0000i q^{11} +20.0000i q^{13} -24.0000 q^{15} +62.0000 q^{17} +108.000i q^{19} +36.0000i q^{21} -72.0000 q^{23} +61.0000 q^{25} -27.0000i q^{27} -128.000i q^{29} -204.000 q^{31} -36.0000 q^{33} +96.0000i q^{35} +228.000i q^{37} -60.0000 q^{39} -22.0000 q^{41} +204.000i q^{43} -72.0000i q^{45} -600.000 q^{47} -199.000 q^{49} +186.000i q^{51} -256.000i q^{53} -96.0000 q^{55} -324.000 q^{57} +828.000i q^{59} -84.0000i q^{61} -108.000 q^{63} -160.000 q^{65} +348.000i q^{67} -216.000i q^{69} +456.000 q^{71} +822.000 q^{73} +183.000i q^{75} +144.000i q^{77} -1356.00 q^{79} +81.0000 q^{81} +108.000i q^{83} +496.000i q^{85} +384.000 q^{87} -938.000 q^{89} +240.000i q^{91} -612.000i q^{93} -864.000 q^{95} +1278.00 q^{97} -108.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 24 q^{7} - 18 q^{9}+O(q^{10})$$ 2 * q + 24 * q^7 - 18 * q^9 $$2 q + 24 q^{7} - 18 q^{9} - 48 q^{15} + 124 q^{17} - 144 q^{23} + 122 q^{25} - 408 q^{31} - 72 q^{33} - 120 q^{39} - 44 q^{41} - 1200 q^{47} - 398 q^{49} - 192 q^{55} - 648 q^{57} - 216 q^{63} - 320 q^{65} + 912 q^{71} + 1644 q^{73} - 2712 q^{79} + 162 q^{81} + 768 q^{87} - 1876 q^{89} - 1728 q^{95} + 2556 q^{97}+O(q^{100})$$ 2 * q + 24 * q^7 - 18 * q^9 - 48 * q^15 + 124 * q^17 - 144 * q^23 + 122 * q^25 - 408 * q^31 - 72 * q^33 - 120 * q^39 - 44 * q^41 - 1200 * q^47 - 398 * q^49 - 192 * q^55 - 648 * q^57 - 216 * q^63 - 320 * q^65 + 912 * q^71 + 1644 * q^73 - 2712 * q^79 + 162 * q^81 + 768 * q^87 - 1876 * q^89 - 1728 * q^95 + 2556 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000i 0.577350i
$$4$$ 0 0
$$5$$ 8.00000i 0.715542i 0.933809 + 0.357771i $$0.116463\pi$$
−0.933809 + 0.357771i $$0.883537\pi$$
$$6$$ 0 0
$$7$$ 12.0000 0.647939 0.323970 0.946068i $$-0.394982\pi$$
0.323970 + 0.946068i $$0.394982\pi$$
$$8$$ 0 0
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 12.0000i 0.328921i 0.986384 + 0.164461i $$0.0525884\pi$$
−0.986384 + 0.164461i $$0.947412\pi$$
$$12$$ 0 0
$$13$$ 20.0000i 0.426692i 0.976977 + 0.213346i $$0.0684362\pi$$
−0.976977 + 0.213346i $$0.931564\pi$$
$$14$$ 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
$$19$$ 108.000i 1.30405i 0.758199 + 0.652024i $$0.226080\pi$$
−0.758199 + 0.652024i $$0.773920\pi$$
$$20$$ 0 0
$$21$$ 36.0000i 0.374088i
$$22$$ 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
$$27$$ − 27.0000i − 0.192450i
$$28$$ 0 0
$$29$$ − 128.000i − 0.819621i −0.912171 0.409810i $$-0.865595\pi$$
0.912171 0.409810i $$-0.134405\pi$$
$$30$$ 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
$$35$$ 96.0000i 0.463627i
$$36$$ 0 0
$$37$$ 228.000i 1.01305i 0.862224 + 0.506527i $$0.169071\pi$$
−0.862224 + 0.506527i $$0.830929\pi$$
$$38$$ 0 0
$$39$$ −60.0000 −0.246351
$$40$$ 0 0
$$41$$ −22.0000 −0.0838006 −0.0419003 0.999122i $$-0.513341\pi$$
−0.0419003 + 0.999122i $$0.513341\pi$$
$$42$$ 0 0
$$43$$ 204.000i 0.723482i 0.932279 + 0.361741i $$0.117817\pi$$
−0.932279 + 0.361741i $$0.882183\pi$$
$$44$$ 0 0
$$45$$ − 72.0000i − 0.238514i
$$46$$ 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
$$51$$ 186.000i 0.510690i
$$52$$ 0 0
$$53$$ − 256.000i − 0.663477i −0.943371 0.331739i $$-0.892365\pi$$
0.943371 0.331739i $$-0.107635\pi$$
$$54$$ 0 0
$$55$$ −96.0000 −0.235357
$$56$$ 0 0
$$57$$ −324.000 −0.752892
$$58$$ 0 0
$$59$$ 828.000i 1.82706i 0.406774 + 0.913529i $$0.366654\pi$$
−0.406774 + 0.913529i $$0.633346\pi$$
$$60$$ 0 0
$$61$$ − 84.0000i − 0.176313i −0.996107 0.0881565i $$-0.971902\pi$$
0.996107 0.0881565i $$-0.0280976\pi$$
$$62$$ 0 0
$$63$$ −108.000 −0.215980
$$64$$ 0 0
$$65$$ −160.000 −0.305316
$$66$$ 0 0
$$67$$ 348.000i 0.634552i 0.948333 + 0.317276i $$0.102768\pi$$
−0.948333 + 0.317276i $$0.897232\pi$$
$$68$$ 0 0
$$69$$ − 216.000i − 0.376860i
$$70$$ 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 $$-0.270998\pi$$
0.658958 + 0.752180i $$0.270998\pi$$
$$74$$ 0 0
$$75$$ 183.000i 0.281747i
$$76$$ 0 0
$$77$$ 144.000i 0.213121i
$$78$$ 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
$$83$$ 108.000i 0.142826i 0.997447 + 0.0714129i $$0.0227508\pi$$
−0.997447 + 0.0714129i $$0.977249\pi$$
$$84$$ 0 0
$$85$$ 496.000i 0.632927i
$$86$$ 0 0
$$87$$ 384.000 0.473208
$$88$$ 0 0
$$89$$ −938.000 −1.11717 −0.558583 0.829449i $$-0.688655\pi$$
−0.558583 + 0.829449i $$0.688655\pi$$
$$90$$ 0 0
$$91$$ 240.000i 0.276471i
$$92$$ 0 0
$$93$$ − 612.000i − 0.682381i
$$94$$ 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
$$99$$ − 108.000i − 0.109640i
$$100$$ 0 0
$$101$$ − 608.000i − 0.598993i −0.954097 0.299496i $$-0.903181\pi$$
0.954097 0.299496i $$-0.0968186\pi$$
$$102$$ 0 0
$$103$$ 948.000 0.906886 0.453443 0.891285i $$-0.350196\pi$$
0.453443 + 0.891285i $$0.350196\pi$$
$$104$$ 0 0
$$105$$ −288.000 −0.267675
$$106$$ 0 0
$$107$$ − 1044.00i − 0.943246i −0.881800 0.471623i $$-0.843668\pi$$
0.881800 0.471623i $$-0.156332\pi$$
$$108$$ 0 0
$$109$$ 1780.00i 1.56416i 0.623180 + 0.782078i $$0.285840\pi$$
−0.623180 + 0.782078i $$0.714160\pi$$
$$110$$ 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
$$115$$ − 576.000i − 0.467063i
$$116$$ 0 0
$$117$$ − 180.000i − 0.142231i
$$118$$ 0 0
$$119$$ 744.000 0.573129
$$120$$ 0 0
$$121$$ 1187.00 0.891811
$$122$$ 0 0
$$123$$ − 66.0000i − 0.0483823i
$$124$$ 0 0
$$125$$ 1488.00i 1.06473i
$$126$$ 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
$$131$$ 348.000i 0.232098i 0.993243 + 0.116049i $$0.0370230\pi$$
−0.993243 + 0.116049i $$0.962977\pi$$
$$132$$ 0 0
$$133$$ 1296.00i 0.844943i
$$134$$ 0 0
$$135$$ 216.000 0.137706
$$136$$ 0 0
$$137$$ 106.000 0.0661036 0.0330518 0.999454i $$-0.489477\pi$$
0.0330518 + 0.999454i $$0.489477\pi$$
$$138$$ 0 0
$$139$$ − 1188.00i − 0.724927i −0.931998 0.362463i $$-0.881936\pi$$
0.931998 0.362463i $$-0.118064\pi$$
$$140$$ 0 0
$$141$$ − 1800.00i − 1.07509i
$$142$$ 0 0
$$143$$ −240.000 −0.140348
$$144$$ 0 0
$$145$$ 1024.00 0.586473
$$146$$ 0 0
$$147$$ − 597.000i − 0.334964i
$$148$$ 0 0
$$149$$ − 2648.00i − 1.45592i −0.685618 0.727962i $$-0.740468\pi$$
0.685618 0.727962i $$-0.259532\pi$$
$$150$$ 0 0
$$151$$ 3420.00 1.84315 0.921575 0.388200i $$-0.126903\pi$$
0.921575 + 0.388200i $$0.126903\pi$$
$$152$$ 0 0
$$153$$ −558.000 −0.294847
$$154$$ 0 0
$$155$$ − 1632.00i − 0.845712i
$$156$$ 0 0
$$157$$ 60.0000i 0.0305001i 0.999884 + 0.0152501i $$0.00485444\pi$$
−0.999884 + 0.0152501i $$0.995146\pi$$
$$158$$ 0 0
$$159$$ 768.000 0.383059
$$160$$ 0 0
$$161$$ −864.000 −0.422936
$$162$$ 0 0
$$163$$ − 228.000i − 0.109560i −0.998498 0.0547802i $$-0.982554\pi$$
0.998498 0.0547802i $$-0.0174458\pi$$
$$164$$ 0 0
$$165$$ − 288.000i − 0.135883i
$$166$$ 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
$$171$$ − 972.000i − 0.434682i
$$172$$ 0 0
$$173$$ − 1352.00i − 0.594166i −0.954852 0.297083i $$-0.903986\pi$$
0.954852 0.297083i $$-0.0960137\pi$$
$$174$$ 0 0
$$175$$ 732.000 0.316194
$$176$$ 0 0
$$177$$ −2484.00 −1.05485
$$178$$ 0 0
$$179$$ − 1716.00i − 0.716536i −0.933619 0.358268i $$-0.883367\pi$$
0.933619 0.358268i $$-0.116633\pi$$
$$180$$ 0 0
$$181$$ 3692.00i 1.51616i 0.652164 + 0.758078i $$0.273861\pi$$
−0.652164 + 0.758078i $$0.726139\pi$$
$$182$$ 0 0
$$183$$ 252.000 0.101794
$$184$$ 0 0
$$185$$ −1824.00 −0.724882
$$186$$ 0 0
$$187$$ 744.000i 0.290945i
$$188$$ 0 0
$$189$$ − 324.000i − 0.124696i
$$190$$ 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
$$195$$ − 480.000i − 0.176274i
$$196$$ 0 0
$$197$$ − 4304.00i − 1.55659i −0.627902 0.778293i $$-0.716086\pi$$
0.627902 0.778293i $$-0.283914\pi$$
$$198$$ 0 0
$$199$$ −204.000 −0.0726692 −0.0363346 0.999340i $$-0.511568\pi$$
−0.0363346 + 0.999340i $$0.511568\pi$$
$$200$$ 0 0
$$201$$ −1044.00 −0.366359
$$202$$ 0 0
$$203$$ − 1536.00i − 0.531064i
$$204$$ 0 0
$$205$$ − 176.000i − 0.0599628i
$$206$$ 0 0
$$207$$ 648.000 0.217580
$$208$$ 0 0
$$209$$ −1296.00 −0.428929
$$210$$ 0 0
$$211$$ − 4020.00i − 1.31160i −0.754933 0.655801i $$-0.772331\pi$$
0.754933 0.655801i $$-0.227669\pi$$
$$212$$ 0 0
$$213$$ 1368.00i 0.440065i
$$214$$ 0 0
$$215$$ −1632.00 −0.517681
$$216$$ 0 0
$$217$$ −2448.00 −0.765811
$$218$$ 0 0
$$219$$ 2466.00i 0.760899i
$$220$$ 0 0
$$221$$ 1240.00i 0.377427i
$$222$$ 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
$$227$$ − 1428.00i − 0.417532i −0.977966 0.208766i $$-0.933055\pi$$
0.977966 0.208766i $$-0.0669446\pi$$
$$228$$ 0 0
$$229$$ 6028.00i 1.73948i 0.493508 + 0.869741i $$0.335714\pi$$
−0.493508 + 0.869741i $$0.664286\pi$$
$$230$$ 0 0
$$231$$ −432.000 −0.123046
$$232$$ 0 0
$$233$$ 2630.00 0.739472 0.369736 0.929137i $$-0.379448\pi$$
0.369736 + 0.929137i $$0.379448\pi$$
$$234$$ 0 0
$$235$$ − 4800.00i − 1.33241i
$$236$$ 0 0
$$237$$ − 4068.00i − 1.11496i
$$238$$ 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
$$243$$ 243.000i 0.0641500i
$$244$$ 0 0
$$245$$ − 1592.00i − 0.415139i
$$246$$ 0 0
$$247$$ −2160.00 −0.556427
$$248$$ 0 0
$$249$$ −324.000 −0.0824605
$$250$$ 0 0
$$251$$ 5532.00i 1.39114i 0.718457 + 0.695571i $$0.244849\pi$$
−0.718457 + 0.695571i $$0.755151\pi$$
$$252$$ 0 0
$$253$$ − 864.000i − 0.214700i
$$254$$ 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
$$259$$ 2736.00i 0.656397i
$$260$$ 0 0
$$261$$ 1152.00i 0.273207i
$$262$$ 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
$$267$$ − 2814.00i − 0.644996i
$$268$$ 0 0
$$269$$ 4544.00i 1.02994i 0.857210 + 0.514968i $$0.172196\pi$$
−0.857210 + 0.514968i $$0.827804\pi$$
$$270$$ 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
$$275$$ 732.000i 0.160514i
$$276$$ 0 0
$$277$$ − 484.000i − 0.104985i −0.998621 0.0524923i $$-0.983283\pi$$
0.998621 0.0524923i $$-0.0167165\pi$$
$$278$$ 0 0
$$279$$ 1836.00 0.393973
$$280$$ 0 0
$$281$$ 406.000 0.0861919 0.0430960 0.999071i $$-0.486278\pi$$
0.0430960 + 0.999071i $$0.486278\pi$$
$$282$$ 0 0
$$283$$ 8172.00i 1.71652i 0.513216 + 0.858260i $$0.328454\pi$$
−0.513216 + 0.858260i $$0.671546\pi$$
$$284$$ 0 0
$$285$$ − 2592.00i − 0.538726i
$$286$$ 0 0
$$287$$ −264.000 −0.0542977
$$288$$ 0 0
$$289$$ −1069.00 −0.217586
$$290$$ 0 0
$$291$$ 3834.00i 0.772347i
$$292$$ 0 0
$$293$$ − 4960.00i − 0.988963i −0.869188 0.494482i $$-0.835358\pi$$
0.869188 0.494482i $$-0.164642\pi$$
$$294$$ 0 0
$$295$$ −6624.00 −1.30734
$$296$$ 0 0
$$297$$ 324.000 0.0633010
$$298$$ 0 0
$$299$$ − 1440.00i − 0.278520i
$$300$$ 0 0
$$301$$ 2448.00i 0.468772i
$$302$$ 0 0
$$303$$ 1824.00 0.345829
$$304$$ 0 0
$$305$$ 672.000 0.126159
$$306$$ 0 0
$$307$$ 6684.00i 1.24259i 0.783576 + 0.621296i $$0.213394\pi$$
−0.783576 + 0.621296i $$0.786606\pi$$
$$308$$ 0 0
$$309$$ 2844.00i 0.523591i
$$310$$ 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 $$-0.337813\pi$$
0.487762 + 0.872977i $$0.337813\pi$$
$$314$$ 0 0
$$315$$ − 864.000i − 0.154542i
$$316$$ 0 0
$$317$$ 7856.00i 1.39191i 0.718083 + 0.695957i $$0.245020\pi$$
−0.718083 + 0.695957i $$0.754980\pi$$
$$318$$ 0 0
$$319$$ 1536.00 0.269591
$$320$$ 0 0
$$321$$ 3132.00 0.544583
$$322$$ 0 0
$$323$$ 6696.00i 1.15348i
$$324$$ 0 0
$$325$$ 1220.00i 0.208226i
$$326$$ 0 0
$$327$$ −5340.00 −0.903066
$$328$$ 0 0
$$329$$ −7200.00 −1.20653
$$330$$ 0 0
$$331$$ − 3732.00i − 0.619726i −0.950781 0.309863i $$-0.899717\pi$$
0.950781 0.309863i $$-0.100283\pi$$
$$332$$ 0 0
$$333$$ − 2052.00i − 0.337684i
$$334$$ 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
$$339$$ − 1866.00i − 0.298959i
$$340$$ 0 0
$$341$$ − 2448.00i − 0.388758i
$$342$$ 0 0
$$343$$ −6504.00 −1.02386
$$344$$ 0 0
$$345$$ 1728.00 0.269659
$$346$$ 0 0
$$347$$ 8220.00i 1.27168i 0.771821 + 0.635840i $$0.219346\pi$$
−0.771821 + 0.635840i $$0.780654\pi$$
$$348$$ 0 0
$$349$$ − 11844.0i − 1.81660i −0.418315 0.908302i $$-0.637379\pi$$
0.418315 0.908302i $$-0.362621\pi$$
$$350$$ 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
$$355$$ 3648.00i 0.545396i
$$356$$ 0 0
$$357$$ 2232.00i 0.330896i
$$358$$ 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
$$363$$ 3561.00i 0.514887i
$$364$$ 0 0
$$365$$ 6576.00i 0.943023i
$$366$$ 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
$$371$$ − 3072.00i − 0.429893i
$$372$$ 0 0
$$373$$ − 4860.00i − 0.674641i −0.941390 0.337321i $$-0.890479\pi$$
0.941390 0.337321i $$-0.109521\pi$$
$$374$$ 0 0
$$375$$ −4464.00 −0.614720
$$376$$ 0 0
$$377$$ 2560.00 0.349726
$$378$$ 0 0
$$379$$ 5964.00i 0.808312i 0.914690 + 0.404156i $$0.132435\pi$$
−0.914690 + 0.404156i $$0.867565\pi$$
$$380$$ 0 0
$$381$$ 612.000i 0.0822932i
$$382$$ 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
$$387$$ − 1836.00i − 0.241161i
$$388$$ 0 0
$$389$$ − 8888.00i − 1.15846i −0.815165 0.579228i $$-0.803354\pi$$
0.815165 0.579228i $$-0.196646\pi$$
$$390$$ 0 0
$$391$$ −4464.00 −0.577376
$$392$$ 0 0
$$393$$ −1044.00 −0.134002
$$394$$ 0 0
$$395$$ − 10848.0i − 1.38183i
$$396$$ 0 0
$$397$$ − 2676.00i − 0.338299i −0.985590 0.169149i $$-0.945898\pi$$
0.985590 0.169149i $$-0.0541020\pi$$
$$398$$ 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
$$403$$ − 4080.00i − 0.504316i
$$404$$ 0 0
$$405$$ 648.000i 0.0795046i
$$406$$ 0 0
$$407$$ −2736.00 −0.333215
$$408$$ 0 0
$$409$$ −1974.00 −0.238650 −0.119325 0.992855i $$-0.538073\pi$$
−0.119325 + 0.992855i $$0.538073\pi$$
$$410$$ 0 0
$$411$$ 318.000i 0.0381649i
$$412$$ 0 0
$$413$$ 9936.00i 1.18382i
$$414$$ 0 0
$$415$$ −864.000 −0.102198
$$416$$ 0 0
$$417$$ 3564.00 0.418537
$$418$$ 0 0
$$419$$ 4764.00i 0.555457i 0.960660 + 0.277729i $$0.0895816\pi$$
−0.960660 + 0.277729i $$0.910418\pi$$
$$420$$ 0 0
$$421$$ 92.0000i 0.0106504i 0.999986 + 0.00532518i $$0.00169507\pi$$
−0.999986 + 0.00532518i $$0.998305\pi$$
$$422$$ 0 0
$$423$$ 5400.00 0.620702
$$424$$ 0 0
$$425$$ 3782.00 0.431656
$$426$$ 0 0
$$427$$ − 1008.00i − 0.114240i
$$428$$ 0 0
$$429$$ − 720.000i − 0.0810301i
$$430$$ 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
$$435$$ 3072.00i 0.338600i
$$436$$ 0 0
$$437$$ − 7776.00i − 0.851205i
$$438$$ 0 0
$$439$$ −3612.00 −0.392691 −0.196346 0.980535i $$-0.562907\pi$$
−0.196346 + 0.980535i $$0.562907\pi$$
$$440$$ 0 0
$$441$$ 1791.00 0.193392
$$442$$ 0 0
$$443$$ 12972.0i 1.39124i 0.718411 + 0.695619i $$0.244870\pi$$
−0.718411 + 0.695619i $$0.755130\pi$$
$$444$$ 0 0
$$445$$ − 7504.00i − 0.799379i
$$446$$ 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
$$451$$ − 264.000i − 0.0275638i
$$452$$ 0 0
$$453$$ 10260.0i 1.06414i
$$454$$ 0 0
$$455$$ −1920.00 −0.197826
$$456$$ 0 0
$$457$$ −8934.00 −0.914475 −0.457237 0.889345i $$-0.651161\pi$$
−0.457237 + 0.889345i $$0.651161\pi$$
$$458$$ 0 0
$$459$$ − 1674.00i − 0.170230i
$$460$$ 0 0
$$461$$ 7448.00i 0.752468i 0.926525 + 0.376234i $$0.122781\pi$$
−0.926525 + 0.376234i $$0.877219\pi$$
$$462$$ 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
$$467$$ − 8580.00i − 0.850182i −0.905151 0.425091i $$-0.860242\pi$$
0.905151 0.425091i $$-0.139758\pi$$
$$468$$ 0 0
$$469$$ 4176.00i 0.411151i
$$470$$ 0 0
$$471$$ −180.000 −0.0176093
$$472$$ 0 0
$$473$$ −2448.00 −0.237969
$$474$$ 0 0
$$475$$ 6588.00i 0.636375i
$$476$$ 0 0
$$477$$ 2304.00i 0.221159i
$$478$$ 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
$$483$$ − 2592.00i − 0.244182i
$$484$$ 0 0
$$485$$ 10224.0i 0.957212i
$$486$$ 0 0
$$487$$ 15036.0 1.39907 0.699534 0.714599i $$-0.253391\pi$$
0.699534 + 0.714599i $$0.253391\pi$$
$$488$$ 0 0
$$489$$ 684.000 0.0632547
$$490$$ 0 0
$$491$$ − 15684.0i − 1.44157i −0.693161 0.720783i $$-0.743782\pi$$
0.693161 0.720783i $$-0.256218\pi$$
$$492$$ 0 0
$$493$$ − 7936.00i − 0.724989i
$$494$$ 0 0
$$495$$ 864.000 0.0784523
$$496$$ 0 0
$$497$$ 5472.00 0.493869
$$498$$ 0 0
$$499$$ − 16308.0i − 1.46302i −0.681831 0.731509i $$-0.738816\pi$$
0.681831 0.731509i $$-0.261184\pi$$
$$500$$ 0 0
$$501$$ 12384.0i 1.10434i
$$502$$ 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
$$507$$ 5391.00i 0.472234i
$$508$$ 0 0
$$509$$ − 5648.00i − 0.491833i −0.969291 0.245917i $$-0.920911\pi$$
0.969291 0.245917i $$-0.0790890\pi$$
$$510$$ 0 0
$$511$$ 9864.00 0.853929
$$512$$ 0 0
$$513$$ 2916.00 0.250964
$$514$$ 0 0
$$515$$ 7584.00i 0.648915i
$$516$$ 0 0
$$517$$ − 7200.00i − 0.612487i
$$518$$ 0 0
$$519$$ 4056.00 0.343042
$$520$$ 0 0
$$521$$ 19498.0 1.63958 0.819792 0.572662i $$-0.194089\pi$$
0.819792 + 0.572662i $$0.194089\pi$$
$$522$$ 0 0
$$523$$ − 22596.0i − 1.88920i −0.328217 0.944602i $$-0.606448\pi$$
0.328217 0.944602i $$-0.393552\pi$$
$$524$$ 0 0
$$525$$ 2196.00i 0.182555i
$$526$$ 0 0
$$527$$ −12648.0 −1.04546
$$528$$ 0 0
$$529$$ −6983.00 −0.573929
$$530$$ 0 0
$$531$$ − 7452.00i − 0.609019i
$$532$$ 0 0
$$533$$ − 440.000i − 0.0357571i
$$534$$ 0 0
$$535$$ 8352.00 0.674932
$$536$$ 0 0
$$537$$ 5148.00 0.413692
$$538$$ 0 0
$$539$$ − 2388.00i − 0.190832i
$$540$$ 0 0
$$541$$ − 812.000i − 0.0645298i −0.999479 0.0322649i $$-0.989728\pi$$
0.999479 0.0322649i $$-0.0102720\pi$$
$$542$$ 0 0
$$543$$ −11076.0 −0.875353
$$544$$ 0 0
$$545$$ −14240.0 −1.11922
$$546$$ 0 0
$$547$$ − 6132.00i − 0.479315i −0.970857 0.239658i $$-0.922965\pi$$
0.970857 0.239658i $$-0.0770352\pi$$
$$548$$ 0 0
$$549$$ 756.000i 0.0587710i
$$550$$ 0 0
$$551$$ 13824.0 1.06882
$$552$$ 0 0
$$553$$ −16272.0 −1.25128
$$554$$ 0 0
$$555$$ − 5472.00i − 0.418511i
$$556$$ 0 0
$$557$$ 2792.00i 0.212389i 0.994345 + 0.106195i $$0.0338667\pi$$
−0.994345 + 0.106195i $$0.966133\pi$$
$$558$$ 0 0
$$559$$ −4080.00 −0.308704
$$560$$ 0 0
$$561$$ −2232.00 −0.167977
$$562$$ 0 0
$$563$$ − 6468.00i − 0.484181i −0.970254 0.242090i $$-0.922167\pi$$
0.970254 0.242090i $$-0.0778330\pi$$
$$564$$ 0 0
$$565$$ − 4976.00i − 0.370517i
$$566$$ 0 0
$$567$$ 972.000 0.0719932
$$568$$ 0 0
$$569$$ 10522.0 0.775229 0.387614 0.921822i $$-0.373299\pi$$
0.387614 + 0.921822i $$0.373299\pi$$
$$570$$ 0 0
$$571$$ 1068.00i 0.0782739i 0.999234 + 0.0391370i $$0.0124609\pi$$
−0.999234 + 0.0391370i $$0.987539\pi$$
$$572$$ 0 0
$$573$$ 144.000i 0.0104986i
$$574$$ 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
$$579$$ 7242.00i 0.519805i
$$580$$ 0 0
$$581$$ 1296.00i 0.0925424i
$$582$$ 0 0
$$583$$ 3072.00 0.218232
$$584$$ 0 0
$$585$$ 1440.00 0.101772
$$586$$ 0 0
$$587$$ − 17940.0i − 1.26144i −0.776012 0.630718i $$-0.782760\pi$$
0.776012 0.630718i $$-0.217240\pi$$
$$588$$ 0 0
$$589$$ − 22032.0i − 1.54128i
$$590$$ 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
$$595$$ 5952.00i 0.410098i
$$596$$ 0 0
$$597$$ − 612.000i − 0.0419556i
$$598$$ 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 $$-0.641044\pi$$
−0.428745 + 0.903426i $$0.641044\pi$$
$$602$$ 0 0
$$603$$ − 3132.00i − 0.211517i
$$604$$ 0 0
$$605$$ 9496.00i 0.638128i
$$606$$ 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
$$611$$ − 12000.0i − 0.794547i
$$612$$ 0 0
$$613$$ − 13788.0i − 0.908470i −0.890882 0.454235i $$-0.849913\pi$$
0.890882 0.454235i $$-0.150087\pi$$
$$614$$ 0 0
$$615$$ 528.000 0.0346195
$$616$$ 0 0
$$617$$ −18074.0 −1.17931 −0.589653 0.807657i $$-0.700736\pi$$
−0.589653 + 0.807657i $$0.700736\pi$$
$$618$$ 0 0
$$619$$ − 5940.00i − 0.385701i −0.981228 0.192850i $$-0.938227\pi$$
0.981228 0.192850i $$-0.0617732\pi$$
$$620$$ 0 0
$$621$$ 1944.00i 0.125620i
$$622$$ 0 0
$$623$$ −11256.0 −0.723856
$$624$$ 0 0
$$625$$ −4279.00 −0.273856
$$626$$ 0 0
$$627$$ − 3888.00i − 0.247642i
$$628$$ 0 0
$$629$$ 14136.0i 0.896088i
$$630$$ 0 0
$$631$$ 8700.00 0.548877 0.274439 0.961605i $$-0.411508\pi$$
0.274439 + 0.961605i $$0.411508\pi$$
$$632$$ 0 0
$$633$$ 12060.0 0.757254
$$634$$ 0 0
$$635$$ 1632.00i 0.101990i
$$636$$ 0 0
$$637$$ − 3980.00i − 0.247556i
$$638$$ 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
$$643$$ 22668.0i 1.39026i 0.718883 + 0.695131i $$0.244654\pi$$
−0.718883 + 0.695131i $$0.755346\pi$$
$$644$$ 0 0
$$645$$ − 4896.00i − 0.298883i
$$646$$ 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
$$651$$ − 7344.00i − 0.442141i
$$652$$ 0 0
$$653$$ − 2552.00i − 0.152936i −0.997072 0.0764682i $$-0.975636\pi$$
0.997072 0.0764682i $$-0.0243644\pi$$
$$654$$ 0 0
$$655$$ −2784.00 −0.166076
$$656$$ 0 0
$$657$$ −7398.00 −0.439305
$$658$$ 0 0
$$659$$ − 2196.00i − 0.129809i −0.997891 0.0649044i $$-0.979326\pi$$
0.997891 0.0649044i $$-0.0206742\pi$$
$$660$$ 0 0
$$661$$ 4260.00i 0.250673i 0.992114 + 0.125336i $$0.0400010\pi$$
−0.992114 + 0.125336i $$0.959999\pi$$
$$662$$ 0 0
$$663$$ −3720.00 −0.217908
$$664$$ 0 0
$$665$$ −10368.0 −0.604592
$$666$$ 0 0
$$667$$ 9216.00i 0.535000i
$$668$$ 0 0
$$669$$ − 1548.00i − 0.0894606i
$$670$$ 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
$$675$$ − 1647.00i − 0.0939156i
$$676$$ 0 0
$$677$$ − 9256.00i − 0.525461i −0.964869 0.262730i $$-0.915377\pi$$
0.964869 0.262730i $$-0.0846230\pi$$
$$678$$ 0 0
$$679$$ 15336.0 0.866777
$$680$$ 0 0
$$681$$ 4284.00 0.241062
$$682$$ 0 0
$$683$$ 29244.0i 1.63835i 0.573546 + 0.819173i $$0.305567\pi$$
−0.573546 + 0.819173i $$0.694433\pi$$
$$684$$ 0 0
$$685$$ 848.000i 0.0472999i
$$686$$ 0 0
$$687$$ −18084.0 −1.00429
$$688$$ 0 0
$$689$$ 5120.00 0.283101
$$690$$ 0 0
$$691$$ − 3684.00i − 0.202816i −0.994845 0.101408i $$-0.967665\pi$$
0.994845 0.101408i $$-0.0323348\pi$$
$$692$$ 0 0
$$693$$ − 1296.00i − 0.0710404i
$$694$$ 0 0
$$695$$ 9504.00 0.518715
$$696$$ 0 0
$$697$$ −1364.00 −0.0741251
$$698$$ 0 0
$$699$$ 7890.00i 0.426934i
$$700$$ 0 0
$$701$$ 13456.0i 0.725002i 0.931983 + 0.362501i $$0.118077\pi$$
−0.931983 + 0.362501i $$0.881923\pi$$
$$702$$ 0 0
$$703$$ −24624.0 −1.32107
$$704$$ 0 0
$$705$$ 14400.0 0.769270
$$706$$ 0 0
$$707$$ − 7296.00i − 0.388111i
$$708$$ 0 0
$$709$$ 6460.00i 0.342187i 0.985255 + 0.171093i $$0.0547300\pi$$
−0.985255 + 0.171093i $$0.945270\pi$$
$$710$$ 0 0
$$711$$ 12204.0 0.643721
$$712$$ 0 0
$$713$$ 14688.0 0.771487
$$714$$ 0 0
$$715$$ − 1920.00i − 0.100425i
$$716$$ 0 0
$$717$$ 13248.0i 0.690036i
$$718$$ 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
$$723$$ 14490.0i 0.745351i
$$724$$ 0 0
$$725$$ − 7808.00i − 0.399975i
$$726$$ 0 0
$$727$$ 11820.0 0.602998 0.301499 0.953466i $$-0.402513\pi$$
0.301499 + 0.953466i $$0.402513\pi$$
$$728$$ 0 0
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 12648.0i 0.639950i
$$732$$ 0 0
$$733$$ 23924.0i 1.20553i 0.797919 + 0.602765i $$0.205934\pi$$
−0.797919 + 0.602765i $$0.794066\pi$$
$$734$$ 0 0
$$735$$ 4776.00 0.239681
$$736$$ 0 0
$$737$$ −4176.00 −0.208718
$$738$$ 0 0
$$739$$ − 11796.0i − 0.587176i −0.955932 0.293588i $$-0.905151\pi$$
0.955932 0.293588i $$-0.0948493\pi$$
$$740$$ 0 0
$$741$$ − 6480.00i − 0.321253i
$$742$$ 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
$$747$$ − 972.000i − 0.0476086i
$$748$$ 0 0
$$749$$ − 12528.0i − 0.611166i
$$750$$ 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
$$755$$ 27360.0i 1.31885i
$$756$$ 0 0
$$757$$ − 27236.0i − 1.30767i −0.756635 0.653837i $$-0.773158\pi$$
0.756635 0.653837i $$-0.226842\pi$$
$$758$$ 0 0
$$759$$ 2592.00 0.123957
$$760$$ 0 0
$$761$$ −14758.0 −0.702992 −0.351496 0.936189i $$-0.614327\pi$$
−0.351496 + 0.936189i $$0.614327\pi$$
$$762$$ 0 0
$$763$$ 21360.0i 1.01348i
$$764$$ 0 0
$$765$$ − 4464.00i − 0.210976i
$$766$$ 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
$$771$$ − 762.000i − 0.0355937i
$$772$$ 0 0
$$773$$ 37424.0i 1.74133i 0.491877 + 0.870665i $$0.336311\pi$$
−0.491877 + 0.870665i $$0.663689\pi$$
$$774$$ 0 0
$$775$$ −12444.0 −0.576776
$$776$$ 0 0
$$777$$ −8208.00 −0.378971
$$778$$ 0 0
$$779$$ − 2376.00i − 0.109280i
$$780$$ 0 0
$$781$$ 5472.00i 0.250709i
$$782$$ 0 0
$$783$$ −3456.00 −0.157736
$$784$$ 0 0
$$785$$ −480.000 −0.0218241
$$786$$ 0 0
$$787$$ − 12804.0i − 0.579941i −0.957036 0.289970i $$-0.906355\pi$$
0.957036 0.289970i $$-0.0936454\pi$$
$$788$$ 0 0
$$789$$ − 12816.0i − 0.578278i
$$790$$ 0 0
$$791$$ −7464.00 −0.335511
$$792$$ 0 0
$$793$$ 1680.00 0.0752315
$$794$$ 0 0
$$795$$ 6144.00i 0.274095i
$$796$$ 0 0
$$797$$ − 32024.0i − 1.42327i −0.702548 0.711636i $$-0.747954\pi$$
0.702548 0.711636i $$-0.252046\pi$$
$$798$$ 0 0
$$799$$ −37200.0 −1.64711
$$800$$ 0 0
$$801$$ 8442.00 0.372389
$$802$$ 0 0
$$803$$ 9864.00i 0.433491i
$$804$$ 0 0
$$805$$ − 6912.00i − 0.302629i
$$806$$ 0 0
$$807$$ −13632.0 −0.594633
$$808$$ 0 0
$$809$$ 38090.0 1.65534 0.827672 0.561212i $$-0.189665\pi$$
0.827672 + 0.561212i $$0.189665\pi$$
$$810$$ 0 0
$$811$$ 10428.0i 0.451512i 0.974184 + 0.225756i $$0.0724853\pi$$
−0.974184 + 0.225756i $$0.927515\pi$$
$$812$$ 0 0
$$813$$ 6228.00i 0.268666i
$$814$$ 0 0
$$815$$ 1824.00 0.0783950
$$816$$ 0 0
$$817$$ −22032.0 −0.943454
$$818$$ 0 0
$$819$$ − 2160.00i − 0.0921569i
$$820$$ 0 0
$$821$$ 7984.00i 0.339395i 0.985496 + 0.169698i $$0.0542791\pi$$
−0.985496 + 0.169698i $$0.945721\pi$$
$$822$$ 0 0
$$823$$ −28788.0 −1.21930 −0.609652 0.792669i $$-0.708691\pi$$
−0.609652 + 0.792669i $$0.708691\pi$$
$$824$$ 0 0
$$825$$ −2196.00 −0.0926726
$$826$$ 0 0
$$827$$ − 468.000i − 0.0196783i −0.999952 0.00983915i $$-0.996868\pi$$
0.999952 0.00983915i $$-0.00313195\pi$$
$$828$$ 0 0
$$829$$ 28852.0i 1.20877i 0.796692 + 0.604386i $$0.206581\pi$$
−0.796692 + 0.604386i $$0.793419\pi$$
$$830$$ 0 0
$$831$$ 1452.00 0.0606129
$$832$$ 0 0
$$833$$ −12338.0 −0.513189
$$834$$ 0 0
$$835$$ 33024.0i 1.36867i
$$836$$ 0 0
$$837$$ 5508.00i 0.227460i
$$838$$ 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
$$843$$ 1218.00i 0.0497629i
$$844$$ 0 0
$$845$$ 14376.0i 0.585266i
$$846$$ 0 0
$$847$$ 14244.0 0.577839
$$848$$ 0 0
$$849$$ −24516.0 −0.991033
$$850$$ 0 0
$$851$$ − 16416.0i − 0.661261i
$$852$$ 0 0
$$853$$ 37044.0i 1.48694i 0.668768 + 0.743472i $$0.266822\pi$$
−0.668768 + 0.743472i $$0.733178\pi$$
$$854$$ 0 0
$$855$$ 7776.00 0.311033
$$856$$ 0 0
$$857$$ −15046.0 −0.599722 −0.299861 0.953983i $$-0.596940\pi$$
−0.299861 + 0.953983i $$0.596940\pi$$
$$858$$ 0 0
$$859$$ − 12180.0i − 0.483791i −0.970302 0.241895i $$-0.922231\pi$$
0.970302 0.241895i $$-0.0777691\pi$$
$$860$$ 0 0
$$861$$ − 792.000i − 0.0313488i
$$862$$ 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
$$867$$ − 3207.00i − 0.125623i
$$868$$ 0 0
$$869$$ − 16272.0i − 0.635201i
$$870$$ 0 0
$$871$$ −6960.00 −0.270758
$$872$$ 0 0
$$873$$ −11502.0 −0.445915
$$874$$ 0 0
$$875$$ 17856.0i 0.689878i
$$876$$ 0 0
$$877$$ 31884.0i 1.22765i 0.789443 + 0.613823i $$0.210369\pi$$
−0.789443 + 0.613823i $$0.789631\pi$$
$$878$$ 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
$$883$$ 32460.0i 1.23711i 0.785742 + 0.618554i $$0.212281\pi$$
−0.785742 + 0.618554i $$0.787719\pi$$
$$884$$ 0 0
$$885$$ − 19872.0i − 0.754791i
$$886$$ 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
$$891$$ 972.000i 0.0365468i
$$892$$ 0 0
$$893$$ − 64800.0i − 2.42827i
$$894$$ 0 0
$$895$$ 13728.0 0.512711
$$896$$ 0 0
$$897$$ 4320.00 0.160803
$$898$$ 0 0
$$899$$ 26112.0i 0.968725i
$$900$$ 0 0
$$901$$ − 15872.0i − 0.586873i
$$902$$ 0 0
$$903$$ −7344.00 −0.270646
$$904$$ 0 0
$$905$$ −29536.0 −1.08487
$$906$$ 0 0
$$907$$ − 6900.00i − 0.252603i −0.991992 0.126301i $$-0.959689\pi$$
0.991992 0.126301i $$-0.0403106\pi$$
$$908$$ 0 0
$$909$$ 5472.00i 0.199664i
$$910$$ 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
$$915$$ 2016.00i 0.0728381i
$$916$$ 0 0
$$917$$ 4176.00i 0.150386i
$$918$$ 0 0
$$919$$ −8340.00 −0.299359 −0.149680 0.988735i $$-0.547824\pi$$
−0.149680 + 0.988735i $$0.547824\pi$$
$$920$$ 0 0
$$921$$ −20052.0 −0.717411
$$922$$ 0 0
$$923$$ 9120.00i 0.325231i
$$924$$ 0 0
$$925$$ 13908.0i 0.494370i
$$926$$ 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
$$931$$ − 21492.0i − 0.756576i
$$932$$ 0 0
$$933$$ − 14976.0i − 0.525501i
$$934$$ 0 0
$$935$$ −5952.00 −0.208183
$$936$$ 0 0
$$937$$ 28550.0 0.995398 0.497699 0.867350i $$-0.334178\pi$$
0.497699 + 0.867350i $$0.334178\pi$$
$$938$$ 0 0
$$939$$ 16206.0i 0.563219i
$$940$$ 0 0
$$941$$ − 50632.0i − 1.75404i −0.480450 0.877022i $$-0.659527\pi$$
0.480450 0.877022i $$-0.340473\pi$$
$$942$$ 0 0
$$943$$ 1584.00 0.0547000
$$944$$ 0 0
$$945$$ 2592.00 0.0892251
$$946$$ 0 0
$$947$$ 18204.0i 0.624657i 0.949974 + 0.312329i $$0.101109\pi$$
−0.949974 + 0.312329i $$0.898891\pi$$
$$948$$ 0 0
$$949$$ 16440.0i 0.562345i
$$950$$ 0 0
$$951$$ −23568.0 −0.803622
$$952$$ 0 0
$$953$$ −4934.00 −0.167710 −0.0838552 0.996478i $$-0.526723\pi$$
−0.0838552 + 0.996478i $$0.526723\pi$$
$$954$$ 0 0
$$955$$ 384.000i 0.0130115i
$$956$$ 0 0
$$957$$ 4608.00i 0.155648i
$$958$$ 0 0
$$959$$ 1272.00 0.0428311
$$960$$ 0 0
$$961$$ 11825.0 0.396932
$$962$$ 0 0
$$963$$ 9396.00i 0.314415i
$$964$$ 0 0
$$965$$ 19312.0i 0.644223i
$$966$$ 0 0
$$967$$ 13284.0 0.441763 0.220881 0.975301i $$-0.429107\pi$$
0.220881 + 0.975301i $$0.429107\pi$$
$$968$$ 0 0
$$969$$ −20088.0 −0.665964
$$970$$ 0 0
$$971$$ − 50820.0i − 1.67960i −0.542896 0.839800i $$-0.682672\pi$$
0.542896 0.839800i $$-0.317328\pi$$
$$972$$ 0 0
$$973$$ − 14256.0i − 0.469709i
$$974$$ 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
$$979$$ − 11256.0i − 0.367460i
$$980$$ 0 0
$$981$$ − 16020.0i − 0.521386i
$$982$$ 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
$$987$$ − 21600.0i − 0.696591i
$$988$$ 0 0
$$989$$ − 14688.0i − 0.472246i
$$990$$ 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
$$995$$ − 1632.00i − 0.0519979i
$$996$$ 0 0
$$997$$ 45588.0i 1.44813i 0.689731 + 0.724065i $$0.257729\pi$$
−0.689731 + 0.724065i $$0.742271\pi$$
$$998$$ 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 384.4.d.b.193.2 yes 2
3.2 odd 2 1152.4.d.g.577.1 2
4.3 odd 2 384.4.d.a.193.1 2
8.3 odd 2 384.4.d.a.193.2 yes 2
8.5 even 2 inner 384.4.d.b.193.1 yes 2
12.11 even 2 1152.4.d.b.577.1 2
16.3 odd 4 768.4.a.d.1.1 1
16.5 even 4 768.4.a.c.1.1 1
16.11 odd 4 768.4.a.a.1.1 1
16.13 even 4 768.4.a.b.1.1 1
24.5 odd 2 1152.4.d.g.577.2 2
24.11 even 2 1152.4.d.b.577.2 2
48.5 odd 4 2304.4.a.k.1.1 1
48.11 even 4 2304.4.a.l.1.1 1
48.29 odd 4 2304.4.a.e.1.1 1
48.35 even 4 2304.4.a.f.1.1 1

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