# Properties

 Label 4368.2.a.bl.1.1 Level $4368$ Weight $2$ Character 4368.1 Self dual yes Analytic conductor $34.879$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4368.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: $$34.8786556029$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1092) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 4368.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.23607 q^{11} +1.00000 q^{13} -1.23607 q^{15} -2.47214 q^{17} +6.47214 q^{19} -1.00000 q^{21} -4.47214 q^{23} -3.47214 q^{25} +1.00000 q^{27} +8.47214 q^{29} -3.23607 q^{33} +1.23607 q^{35} +4.47214 q^{37} +1.00000 q^{39} +5.23607 q^{41} -4.00000 q^{43} -1.23607 q^{45} -7.70820 q^{47} +1.00000 q^{49} -2.47214 q^{51} +10.0000 q^{53} +4.00000 q^{55} +6.47214 q^{57} -9.23607 q^{59} +14.9443 q^{61} -1.00000 q^{63} -1.23607 q^{65} +2.47214 q^{67} -4.47214 q^{69} +5.70820 q^{71} +4.47214 q^{73} -3.47214 q^{75} +3.23607 q^{77} +10.4721 q^{79} +1.00000 q^{81} -2.76393 q^{83} +3.05573 q^{85} +8.47214 q^{87} -10.1803 q^{89} -1.00000 q^{91} -8.00000 q^{95} +6.94427 q^{97} -3.23607 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} + 8 q^{29} - 2 q^{33} - 2 q^{35} + 2 q^{39} + 6 q^{41} - 8 q^{43} + 2 q^{45} - 2 q^{47} + 2 q^{49} + 4 q^{51} + 20 q^{53} + 8 q^{55} + 4 q^{57} - 14 q^{59} + 12 q^{61} - 2 q^{63} + 2 q^{65} - 4 q^{67} - 2 q^{71} + 2 q^{75} + 2 q^{77} + 12 q^{79} + 2 q^{81} - 10 q^{83} + 24 q^{85} + 8 q^{87} + 2 q^{89} - 2 q^{91} - 16 q^{95} - 4 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^25 + 2 * q^27 + 8 * q^29 - 2 * q^33 - 2 * q^35 + 2 * q^39 + 6 * q^41 - 8 * q^43 + 2 * q^45 - 2 * q^47 + 2 * q^49 + 4 * q^51 + 20 * q^53 + 8 * q^55 + 4 * q^57 - 14 * q^59 + 12 * q^61 - 2 * q^63 + 2 * q^65 - 4 * q^67 - 2 * q^71 + 2 * q^75 + 2 * q^77 + 12 * q^79 + 2 * q^81 - 10 * q^83 + 24 * q^85 + 8 * q^87 + 2 * q^89 - 2 * q^91 - 16 * q^95 - 4 * q^97 - 2 * q^99

## 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −1.23607 −0.552786 −0.276393 0.961045i $$-0.589139\pi$$
−0.276393 + 0.961045i $$0.589139\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.23607 −0.975711 −0.487856 0.872924i $$-0.662221\pi$$
−0.487856 + 0.872924i $$0.662221\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ −1.23607 −0.319151
$$16$$ 0 0
$$17$$ −2.47214 −0.599581 −0.299791 0.954005i $$-0.596917\pi$$
−0.299791 + 0.954005i $$0.596917\pi$$
$$18$$ 0 0
$$19$$ 6.47214 1.48481 0.742405 0.669951i $$-0.233685\pi$$
0.742405 + 0.669951i $$0.233685\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ −4.47214 −0.932505 −0.466252 0.884652i $$-0.654396\pi$$
−0.466252 + 0.884652i $$0.654396\pi$$
$$24$$ 0 0
$$25$$ −3.47214 −0.694427
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 8.47214 1.57324 0.786618 0.617440i $$-0.211830\pi$$
0.786618 + 0.617440i $$0.211830\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ −3.23607 −0.563327
$$34$$ 0 0
$$35$$ 1.23607 0.208934
$$36$$ 0 0
$$37$$ 4.47214 0.735215 0.367607 0.929981i $$-0.380177\pi$$
0.367607 + 0.929981i $$0.380177\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 5.23607 0.817736 0.408868 0.912593i $$-0.365924\pi$$
0.408868 + 0.912593i $$0.365924\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ −1.23607 −0.184262
$$46$$ 0 0
$$47$$ −7.70820 −1.12436 −0.562179 0.827016i $$-0.690037\pi$$
−0.562179 + 0.827016i $$0.690037\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −2.47214 −0.346168
$$52$$ 0 0
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ 6.47214 0.857255
$$58$$ 0 0
$$59$$ −9.23607 −1.20243 −0.601217 0.799086i $$-0.705317\pi$$
−0.601217 + 0.799086i $$0.705317\pi$$
$$60$$ 0 0
$$61$$ 14.9443 1.91342 0.956709 0.291046i $$-0.0940034\pi$$
0.956709 + 0.291046i $$0.0940034\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ −1.23607 −0.153315
$$66$$ 0 0
$$67$$ 2.47214 0.302019 0.151010 0.988532i $$-0.451748\pi$$
0.151010 + 0.988532i $$0.451748\pi$$
$$68$$ 0 0
$$69$$ −4.47214 −0.538382
$$70$$ 0 0
$$71$$ 5.70820 0.677439 0.338720 0.940887i $$-0.390006\pi$$
0.338720 + 0.940887i $$0.390006\pi$$
$$72$$ 0 0
$$73$$ 4.47214 0.523424 0.261712 0.965146i $$-0.415713\pi$$
0.261712 + 0.965146i $$0.415713\pi$$
$$74$$ 0 0
$$75$$ −3.47214 −0.400928
$$76$$ 0 0
$$77$$ 3.23607 0.368784
$$78$$ 0 0
$$79$$ 10.4721 1.17821 0.589104 0.808057i $$-0.299481\pi$$
0.589104 + 0.808057i $$0.299481\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −2.76393 −0.303381 −0.151690 0.988428i $$-0.548472\pi$$
−0.151690 + 0.988428i $$0.548472\pi$$
$$84$$ 0 0
$$85$$ 3.05573 0.331440
$$86$$ 0 0
$$87$$ 8.47214 0.908308
$$88$$ 0 0
$$89$$ −10.1803 −1.07911 −0.539557 0.841949i $$-0.681408\pi$$
−0.539557 + 0.841949i $$0.681408\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −8.00000 −0.820783
$$96$$ 0 0
$$97$$ 6.94427 0.705084 0.352542 0.935796i $$-0.385317\pi$$
0.352542 + 0.935796i $$0.385317\pi$$
$$98$$ 0 0
$$99$$ −3.23607 −0.325237
$$100$$ 0 0
$$101$$ −8.94427 −0.889988 −0.444994 0.895533i $$-0.646794\pi$$
−0.444994 + 0.895533i $$0.646794\pi$$
$$102$$ 0 0
$$103$$ 16.9443 1.66957 0.834784 0.550577i $$-0.185592\pi$$
0.834784 + 0.550577i $$0.185592\pi$$
$$104$$ 0 0
$$105$$ 1.23607 0.120628
$$106$$ 0 0
$$107$$ 3.52786 0.341051 0.170526 0.985353i $$-0.445453\pi$$
0.170526 + 0.985353i $$0.445453\pi$$
$$108$$ 0 0
$$109$$ 3.52786 0.337908 0.168954 0.985624i $$-0.445961\pi$$
0.168954 + 0.985624i $$0.445961\pi$$
$$110$$ 0 0
$$111$$ 4.47214 0.424476
$$112$$ 0 0
$$113$$ 20.4721 1.92586 0.962928 0.269758i $$-0.0869436\pi$$
0.962928 + 0.269758i $$0.0869436\pi$$
$$114$$ 0 0
$$115$$ 5.52786 0.515476
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ 2.47214 0.226620
$$120$$ 0 0
$$121$$ −0.527864 −0.0479876
$$122$$ 0 0
$$123$$ 5.23607 0.472120
$$124$$ 0 0
$$125$$ 10.4721 0.936656
$$126$$ 0 0
$$127$$ 2.47214 0.219367 0.109683 0.993967i $$-0.465016\pi$$
0.109683 + 0.993967i $$0.465016\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 1.52786 0.133490 0.0667451 0.997770i $$-0.478739\pi$$
0.0667451 + 0.997770i $$0.478739\pi$$
$$132$$ 0 0
$$133$$ −6.47214 −0.561205
$$134$$ 0 0
$$135$$ −1.23607 −0.106384
$$136$$ 0 0
$$137$$ 15.2361 1.30171 0.650853 0.759204i $$-0.274412\pi$$
0.650853 + 0.759204i $$0.274412\pi$$
$$138$$ 0 0
$$139$$ −17.8885 −1.51729 −0.758643 0.651506i $$-0.774137\pi$$
−0.758643 + 0.651506i $$0.774137\pi$$
$$140$$ 0 0
$$141$$ −7.70820 −0.649148
$$142$$ 0 0
$$143$$ −3.23607 −0.270614
$$144$$ 0 0
$$145$$ −10.4721 −0.869664
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ −12.7639 −1.04566 −0.522831 0.852436i $$-0.675124\pi$$
−0.522831 + 0.852436i $$0.675124\pi$$
$$150$$ 0 0
$$151$$ 3.05573 0.248672 0.124336 0.992240i $$-0.460320\pi$$
0.124336 + 0.992240i $$0.460320\pi$$
$$152$$ 0 0
$$153$$ −2.47214 −0.199860
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ 0 0
$$159$$ 10.0000 0.793052
$$160$$ 0 0
$$161$$ 4.47214 0.352454
$$162$$ 0 0
$$163$$ 2.47214 0.193633 0.0968163 0.995302i $$-0.469134\pi$$
0.0968163 + 0.995302i $$0.469134\pi$$
$$164$$ 0 0
$$165$$ 4.00000 0.311400
$$166$$ 0 0
$$167$$ 18.1803 1.40684 0.703418 0.710776i $$-0.251656\pi$$
0.703418 + 0.710776i $$0.251656\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 6.47214 0.494937
$$172$$ 0 0
$$173$$ 12.0000 0.912343 0.456172 0.889892i $$-0.349220\pi$$
0.456172 + 0.889892i $$0.349220\pi$$
$$174$$ 0 0
$$175$$ 3.47214 0.262469
$$176$$ 0 0
$$177$$ −9.23607 −0.694225
$$178$$ 0 0
$$179$$ 6.94427 0.519039 0.259520 0.965738i $$-0.416436\pi$$
0.259520 + 0.965738i $$0.416436\pi$$
$$180$$ 0 0
$$181$$ 19.8885 1.47830 0.739152 0.673539i $$-0.235227\pi$$
0.739152 + 0.673539i $$0.235227\pi$$
$$182$$ 0 0
$$183$$ 14.9443 1.10471
$$184$$ 0 0
$$185$$ −5.52786 −0.406417
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ 6.94427 0.502470 0.251235 0.967926i $$-0.419163\pi$$
0.251235 + 0.967926i $$0.419163\pi$$
$$192$$ 0 0
$$193$$ 18.3607 1.32163 0.660815 0.750549i $$-0.270211\pi$$
0.660815 + 0.750549i $$0.270211\pi$$
$$194$$ 0 0
$$195$$ −1.23607 −0.0885167
$$196$$ 0 0
$$197$$ 0.763932 0.0544279 0.0272140 0.999630i $$-0.491336\pi$$
0.0272140 + 0.999630i $$0.491336\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 2.47214 0.174371
$$202$$ 0 0
$$203$$ −8.47214 −0.594627
$$204$$ 0 0
$$205$$ −6.47214 −0.452034
$$206$$ 0 0
$$207$$ −4.47214 −0.310835
$$208$$ 0 0
$$209$$ −20.9443 −1.44875
$$210$$ 0 0
$$211$$ −20.3607 −1.40169 −0.700844 0.713315i $$-0.747193\pi$$
−0.700844 + 0.713315i $$0.747193\pi$$
$$212$$ 0 0
$$213$$ 5.70820 0.391120
$$214$$ 0 0
$$215$$ 4.94427 0.337197
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 4.47214 0.302199
$$220$$ 0 0
$$221$$ −2.47214 −0.166294
$$222$$ 0 0
$$223$$ 10.4721 0.701266 0.350633 0.936513i $$-0.385966\pi$$
0.350633 + 0.936513i $$0.385966\pi$$
$$224$$ 0 0
$$225$$ −3.47214 −0.231476
$$226$$ 0 0
$$227$$ 6.76393 0.448938 0.224469 0.974481i $$-0.427935\pi$$
0.224469 + 0.974481i $$0.427935\pi$$
$$228$$ 0 0
$$229$$ −25.4164 −1.67956 −0.839782 0.542924i $$-0.817317\pi$$
−0.839782 + 0.542924i $$0.817317\pi$$
$$230$$ 0 0
$$231$$ 3.23607 0.212918
$$232$$ 0 0
$$233$$ 5.41641 0.354841 0.177420 0.984135i $$-0.443225\pi$$
0.177420 + 0.984135i $$0.443225\pi$$
$$234$$ 0 0
$$235$$ 9.52786 0.621529
$$236$$ 0 0
$$237$$ 10.4721 0.680238
$$238$$ 0 0
$$239$$ −21.1246 −1.36644 −0.683219 0.730214i $$-0.739420\pi$$
−0.683219 + 0.730214i $$0.739420\pi$$
$$240$$ 0 0
$$241$$ −16.4721 −1.06106 −0.530532 0.847665i $$-0.678008\pi$$
−0.530532 + 0.847665i $$0.678008\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −1.23607 −0.0789695
$$246$$ 0 0
$$247$$ 6.47214 0.411812
$$248$$ 0 0
$$249$$ −2.76393 −0.175157
$$250$$ 0 0
$$251$$ −5.52786 −0.348916 −0.174458 0.984665i $$-0.555817\pi$$
−0.174458 + 0.984665i $$0.555817\pi$$
$$252$$ 0 0
$$253$$ 14.4721 0.909855
$$254$$ 0 0
$$255$$ 3.05573 0.191357
$$256$$ 0 0
$$257$$ 26.8328 1.67379 0.836893 0.547367i $$-0.184370\pi$$
0.836893 + 0.547367i $$0.184370\pi$$
$$258$$ 0 0
$$259$$ −4.47214 −0.277885
$$260$$ 0 0
$$261$$ 8.47214 0.524412
$$262$$ 0 0
$$263$$ 0.472136 0.0291132 0.0145566 0.999894i $$-0.495366\pi$$
0.0145566 + 0.999894i $$0.495366\pi$$
$$264$$ 0 0
$$265$$ −12.3607 −0.759311
$$266$$ 0 0
$$267$$ −10.1803 −0.623027
$$268$$ 0 0
$$269$$ 4.94427 0.301458 0.150729 0.988575i $$-0.451838\pi$$
0.150729 + 0.988575i $$0.451838\pi$$
$$270$$ 0 0
$$271$$ 23.4164 1.42245 0.711223 0.702967i $$-0.248142\pi$$
0.711223 + 0.702967i $$0.248142\pi$$
$$272$$ 0 0
$$273$$ −1.00000 −0.0605228
$$274$$ 0 0
$$275$$ 11.2361 0.677560
$$276$$ 0 0
$$277$$ −15.8885 −0.954650 −0.477325 0.878727i $$-0.658394\pi$$
−0.477325 + 0.878727i $$0.658394\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −18.6525 −1.11271 −0.556357 0.830944i $$-0.687801\pi$$
−0.556357 + 0.830944i $$0.687801\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ −8.00000 −0.473879
$$286$$ 0 0
$$287$$ −5.23607 −0.309075
$$288$$ 0 0
$$289$$ −10.8885 −0.640503
$$290$$ 0 0
$$291$$ 6.94427 0.407080
$$292$$ 0 0
$$293$$ 5.23607 0.305894 0.152947 0.988234i $$-0.451124\pi$$
0.152947 + 0.988234i $$0.451124\pi$$
$$294$$ 0 0
$$295$$ 11.4164 0.664689
$$296$$ 0 0
$$297$$ −3.23607 −0.187776
$$298$$ 0 0
$$299$$ −4.47214 −0.258630
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 0 0
$$303$$ −8.94427 −0.513835
$$304$$ 0 0
$$305$$ −18.4721 −1.05771
$$306$$ 0 0
$$307$$ 5.88854 0.336077 0.168038 0.985780i $$-0.446257\pi$$
0.168038 + 0.985780i $$0.446257\pi$$
$$308$$ 0 0
$$309$$ 16.9443 0.963926
$$310$$ 0 0
$$311$$ 1.52786 0.0866372 0.0433186 0.999061i $$-0.486207\pi$$
0.0433186 + 0.999061i $$0.486207\pi$$
$$312$$ 0 0
$$313$$ 20.8328 1.17754 0.588770 0.808300i $$-0.299612\pi$$
0.588770 + 0.808300i $$0.299612\pi$$
$$314$$ 0 0
$$315$$ 1.23607 0.0696445
$$316$$ 0 0
$$317$$ −28.1803 −1.58277 −0.791383 0.611321i $$-0.790638\pi$$
−0.791383 + 0.611321i $$0.790638\pi$$
$$318$$ 0 0
$$319$$ −27.4164 −1.53502
$$320$$ 0 0
$$321$$ 3.52786 0.196906
$$322$$ 0 0
$$323$$ −16.0000 −0.890264
$$324$$ 0 0
$$325$$ −3.47214 −0.192599
$$326$$ 0 0
$$327$$ 3.52786 0.195091
$$328$$ 0 0
$$329$$ 7.70820 0.424967
$$330$$ 0 0
$$331$$ 34.8328 1.91458 0.957292 0.289122i $$-0.0933632\pi$$
0.957292 + 0.289122i $$0.0933632\pi$$
$$332$$ 0 0
$$333$$ 4.47214 0.245072
$$334$$ 0 0
$$335$$ −3.05573 −0.166952
$$336$$ 0 0
$$337$$ −22.3607 −1.21806 −0.609032 0.793146i $$-0.708442\pi$$
−0.609032 + 0.793146i $$0.708442\pi$$
$$338$$ 0 0
$$339$$ 20.4721 1.11189
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 5.52786 0.297610
$$346$$ 0 0
$$347$$ −22.0000 −1.18102 −0.590511 0.807030i $$-0.701074\pi$$
−0.590511 + 0.807030i $$0.701074\pi$$
$$348$$ 0 0
$$349$$ 1.41641 0.0758186 0.0379093 0.999281i $$-0.487930\pi$$
0.0379093 + 0.999281i $$0.487930\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ −16.6525 −0.886322 −0.443161 0.896442i $$-0.646143\pi$$
−0.443161 + 0.896442i $$0.646143\pi$$
$$354$$ 0 0
$$355$$ −7.05573 −0.374479
$$356$$ 0 0
$$357$$ 2.47214 0.130839
$$358$$ 0 0
$$359$$ −4.18034 −0.220630 −0.110315 0.993897i $$-0.535186\pi$$
−0.110315 + 0.993897i $$0.535186\pi$$
$$360$$ 0 0
$$361$$ 22.8885 1.20466
$$362$$ 0 0
$$363$$ −0.527864 −0.0277057
$$364$$ 0 0
$$365$$ −5.52786 −0.289342
$$366$$ 0 0
$$367$$ 24.9443 1.30208 0.651040 0.759043i $$-0.274333\pi$$
0.651040 + 0.759043i $$0.274333\pi$$
$$368$$ 0 0
$$369$$ 5.23607 0.272579
$$370$$ 0 0
$$371$$ −10.0000 −0.519174
$$372$$ 0 0
$$373$$ 36.4721 1.88846 0.944228 0.329293i $$-0.106810\pi$$
0.944228 + 0.329293i $$0.106810\pi$$
$$374$$ 0 0
$$375$$ 10.4721 0.540779
$$376$$ 0 0
$$377$$ 8.47214 0.436337
$$378$$ 0 0
$$379$$ −13.8885 −0.713407 −0.356703 0.934218i $$-0.616099\pi$$
−0.356703 + 0.934218i $$0.616099\pi$$
$$380$$ 0 0
$$381$$ 2.47214 0.126651
$$382$$ 0 0
$$383$$ 1.81966 0.0929803 0.0464901 0.998919i $$-0.485196\pi$$
0.0464901 + 0.998919i $$0.485196\pi$$
$$384$$ 0 0
$$385$$ −4.00000 −0.203859
$$386$$ 0 0
$$387$$ −4.00000 −0.203331
$$388$$ 0 0
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ 11.0557 0.559112
$$392$$ 0 0
$$393$$ 1.52786 0.0770705
$$394$$ 0 0
$$395$$ −12.9443 −0.651297
$$396$$ 0 0
$$397$$ 35.8885 1.80119 0.900597 0.434655i $$-0.143130\pi$$
0.900597 + 0.434655i $$0.143130\pi$$
$$398$$ 0 0
$$399$$ −6.47214 −0.324012
$$400$$ 0 0
$$401$$ −19.2361 −0.960603 −0.480302 0.877103i $$-0.659473\pi$$
−0.480302 + 0.877103i $$0.659473\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −1.23607 −0.0614207
$$406$$ 0 0
$$407$$ −14.4721 −0.717357
$$408$$ 0 0
$$409$$ −17.0557 −0.843351 −0.421676 0.906747i $$-0.638558\pi$$
−0.421676 + 0.906747i $$0.638558\pi$$
$$410$$ 0 0
$$411$$ 15.2361 0.751540
$$412$$ 0 0
$$413$$ 9.23607 0.454477
$$414$$ 0 0
$$415$$ 3.41641 0.167705
$$416$$ 0 0
$$417$$ −17.8885 −0.876006
$$418$$ 0 0
$$419$$ 23.4164 1.14397 0.571983 0.820265i $$-0.306174\pi$$
0.571983 + 0.820265i $$0.306174\pi$$
$$420$$ 0 0
$$421$$ 0.111456 0.00543204 0.00271602 0.999996i $$-0.499135\pi$$
0.00271602 + 0.999996i $$0.499135\pi$$
$$422$$ 0 0
$$423$$ −7.70820 −0.374786
$$424$$ 0 0
$$425$$ 8.58359 0.416365
$$426$$ 0 0
$$427$$ −14.9443 −0.723204
$$428$$ 0 0
$$429$$ −3.23607 −0.156239
$$430$$ 0 0
$$431$$ −1.70820 −0.0822813 −0.0411406 0.999153i $$-0.513099\pi$$
−0.0411406 + 0.999153i $$0.513099\pi$$
$$432$$ 0 0
$$433$$ 22.0000 1.05725 0.528626 0.848855i $$-0.322707\pi$$
0.528626 + 0.848855i $$0.322707\pi$$
$$434$$ 0 0
$$435$$ −10.4721 −0.502100
$$436$$ 0 0
$$437$$ −28.9443 −1.38459
$$438$$ 0 0
$$439$$ 15.0557 0.718571 0.359285 0.933228i $$-0.383020\pi$$
0.359285 + 0.933228i $$0.383020\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 17.4164 0.827479 0.413739 0.910395i $$-0.364223\pi$$
0.413739 + 0.910395i $$0.364223\pi$$
$$444$$ 0 0
$$445$$ 12.5836 0.596519
$$446$$ 0 0
$$447$$ −12.7639 −0.603713
$$448$$ 0 0
$$449$$ −33.1246 −1.56325 −0.781624 0.623750i $$-0.785608\pi$$
−0.781624 + 0.623750i $$0.785608\pi$$
$$450$$ 0 0
$$451$$ −16.9443 −0.797875
$$452$$ 0 0
$$453$$ 3.05573 0.143571
$$454$$ 0 0
$$455$$ 1.23607 0.0579478
$$456$$ 0 0
$$457$$ −3.52786 −0.165027 −0.0825133 0.996590i $$-0.526295\pi$$
−0.0825133 + 0.996590i $$0.526295\pi$$
$$458$$ 0 0
$$459$$ −2.47214 −0.115389
$$460$$ 0 0
$$461$$ −16.2918 −0.758785 −0.379392 0.925236i $$-0.623867\pi$$
−0.379392 + 0.925236i $$0.623867\pi$$
$$462$$ 0 0
$$463$$ 11.4164 0.530565 0.265283 0.964171i $$-0.414535\pi$$
0.265283 + 0.964171i $$0.414535\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 2.47214 0.114397 0.0571984 0.998363i $$-0.481783\pi$$
0.0571984 + 0.998363i $$0.481783\pi$$
$$468$$ 0 0
$$469$$ −2.47214 −0.114153
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ 0 0
$$473$$ 12.9443 0.595178
$$474$$ 0 0
$$475$$ −22.4721 −1.03109
$$476$$ 0 0
$$477$$ 10.0000 0.457869
$$478$$ 0 0
$$479$$ −28.0689 −1.28250 −0.641250 0.767332i $$-0.721584\pi$$
−0.641250 + 0.767332i $$0.721584\pi$$
$$480$$ 0 0
$$481$$ 4.47214 0.203912
$$482$$ 0 0
$$483$$ 4.47214 0.203489
$$484$$ 0 0
$$485$$ −8.58359 −0.389761
$$486$$ 0 0
$$487$$ −30.8328 −1.39717 −0.698584 0.715528i $$-0.746186\pi$$
−0.698584 + 0.715528i $$0.746186\pi$$
$$488$$ 0 0
$$489$$ 2.47214 0.111794
$$490$$ 0 0
$$491$$ −16.4721 −0.743377 −0.371689 0.928357i $$-0.621221\pi$$
−0.371689 + 0.928357i $$0.621221\pi$$
$$492$$ 0 0
$$493$$ −20.9443 −0.943283
$$494$$ 0 0
$$495$$ 4.00000 0.179787
$$496$$ 0 0
$$497$$ −5.70820 −0.256048
$$498$$ 0 0
$$499$$ −21.8885 −0.979866 −0.489933 0.871760i $$-0.662979\pi$$
−0.489933 + 0.871760i $$0.662979\pi$$
$$500$$ 0 0
$$501$$ 18.1803 0.812238
$$502$$ 0 0
$$503$$ −10.4721 −0.466929 −0.233465 0.972365i $$-0.575006\pi$$
−0.233465 + 0.972365i $$0.575006\pi$$
$$504$$ 0 0
$$505$$ 11.0557 0.491973
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ 21.5967 0.957259 0.478630 0.878017i $$-0.341134\pi$$
0.478630 + 0.878017i $$0.341134\pi$$
$$510$$ 0 0
$$511$$ −4.47214 −0.197836
$$512$$ 0 0
$$513$$ 6.47214 0.285752
$$514$$ 0 0
$$515$$ −20.9443 −0.922915
$$516$$ 0 0
$$517$$ 24.9443 1.09705
$$518$$ 0 0
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ −20.9443 −0.917585 −0.458793 0.888543i $$-0.651718\pi$$
−0.458793 + 0.888543i $$0.651718\pi$$
$$522$$ 0 0
$$523$$ −33.8885 −1.48184 −0.740921 0.671592i $$-0.765611\pi$$
−0.740921 + 0.671592i $$0.765611\pi$$
$$524$$ 0 0
$$525$$ 3.47214 0.151536
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −3.00000 −0.130435
$$530$$ 0 0
$$531$$ −9.23607 −0.400811
$$532$$ 0 0
$$533$$ 5.23607 0.226799
$$534$$ 0 0
$$535$$ −4.36068 −0.188529
$$536$$ 0 0
$$537$$ 6.94427 0.299667
$$538$$ 0 0
$$539$$ −3.23607 −0.139387
$$540$$ 0 0
$$541$$ 6.94427 0.298558 0.149279 0.988795i $$-0.452305\pi$$
0.149279 + 0.988795i $$0.452305\pi$$
$$542$$ 0 0
$$543$$ 19.8885 0.853499
$$544$$ 0 0
$$545$$ −4.36068 −0.186791
$$546$$ 0 0
$$547$$ 19.0557 0.814764 0.407382 0.913258i $$-0.366442\pi$$
0.407382 + 0.913258i $$0.366442\pi$$
$$548$$ 0 0
$$549$$ 14.9443 0.637806
$$550$$ 0 0
$$551$$ 54.8328 2.33596
$$552$$ 0 0
$$553$$ −10.4721 −0.445321
$$554$$ 0 0
$$555$$ −5.52786 −0.234645
$$556$$ 0 0
$$557$$ −28.7639 −1.21877 −0.609383 0.792876i $$-0.708583\pi$$
−0.609383 + 0.792876i $$0.708583\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ −19.0557 −0.803103 −0.401552 0.915836i $$-0.631529\pi$$
−0.401552 + 0.915836i $$0.631529\pi$$
$$564$$ 0 0
$$565$$ −25.3050 −1.06459
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −17.4164 −0.730134 −0.365067 0.930981i $$-0.618954\pi$$
−0.365067 + 0.930981i $$0.618954\pi$$
$$570$$ 0 0
$$571$$ 0.360680 0.0150940 0.00754699 0.999972i $$-0.497598\pi$$
0.00754699 + 0.999972i $$0.497598\pi$$
$$572$$ 0 0
$$573$$ 6.94427 0.290101
$$574$$ 0 0
$$575$$ 15.5279 0.647557
$$576$$ 0 0
$$577$$ 18.9443 0.788660 0.394330 0.918969i $$-0.370977\pi$$
0.394330 + 0.918969i $$0.370977\pi$$
$$578$$ 0 0
$$579$$ 18.3607 0.763044
$$580$$ 0 0
$$581$$ 2.76393 0.114667
$$582$$ 0 0
$$583$$ −32.3607 −1.34024
$$584$$ 0 0
$$585$$ −1.23607 −0.0511051
$$586$$ 0 0
$$587$$ −21.5967 −0.891393 −0.445697 0.895184i $$-0.647044\pi$$
−0.445697 + 0.895184i $$0.647044\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0.763932 0.0314240
$$592$$ 0 0
$$593$$ 17.8197 0.731766 0.365883 0.930661i $$-0.380767\pi$$
0.365883 + 0.930661i $$0.380767\pi$$
$$594$$ 0 0
$$595$$ −3.05573 −0.125273
$$596$$ 0 0
$$597$$ 16.0000 0.654836
$$598$$ 0 0
$$599$$ 27.3050 1.11565 0.557825 0.829959i $$-0.311636\pi$$
0.557825 + 0.829959i $$0.311636\pi$$
$$600$$ 0 0
$$601$$ 9.05573 0.369391 0.184695 0.982796i $$-0.440870\pi$$
0.184695 + 0.982796i $$0.440870\pi$$
$$602$$ 0 0
$$603$$ 2.47214 0.100673
$$604$$ 0 0
$$605$$ 0.652476 0.0265269
$$606$$ 0 0
$$607$$ 21.8885 0.888429 0.444214 0.895921i $$-0.353483\pi$$
0.444214 + 0.895921i $$0.353483\pi$$
$$608$$ 0 0
$$609$$ −8.47214 −0.343308
$$610$$ 0 0
$$611$$ −7.70820 −0.311841
$$612$$ 0 0
$$613$$ −29.7771 −1.20269 −0.601343 0.798991i $$-0.705367\pi$$
−0.601343 + 0.798991i $$0.705367\pi$$
$$614$$ 0 0
$$615$$ −6.47214 −0.260982
$$616$$ 0 0
$$617$$ −40.7639 −1.64109 −0.820547 0.571579i $$-0.806331\pi$$
−0.820547 + 0.571579i $$0.806331\pi$$
$$618$$ 0 0
$$619$$ 37.3050 1.49941 0.749706 0.661771i $$-0.230195\pi$$
0.749706 + 0.661771i $$0.230195\pi$$
$$620$$ 0 0
$$621$$ −4.47214 −0.179461
$$622$$ 0 0
$$623$$ 10.1803 0.407867
$$624$$ 0 0
$$625$$ 4.41641 0.176656
$$626$$ 0 0
$$627$$ −20.9443 −0.836434
$$628$$ 0 0
$$629$$ −11.0557 −0.440821
$$630$$ 0 0
$$631$$ −14.4721 −0.576127 −0.288063 0.957611i $$-0.593011\pi$$
−0.288063 + 0.957611i $$0.593011\pi$$
$$632$$ 0 0
$$633$$ −20.3607 −0.809264
$$634$$ 0 0
$$635$$ −3.05573 −0.121263
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 0 0
$$639$$ 5.70820 0.225813
$$640$$ 0 0
$$641$$ 41.7771 1.65010 0.825048 0.565063i $$-0.191148\pi$$
0.825048 + 0.565063i $$0.191148\pi$$
$$642$$ 0 0
$$643$$ 10.8328 0.427205 0.213602 0.976921i $$-0.431480\pi$$
0.213602 + 0.976921i $$0.431480\pi$$
$$644$$ 0 0
$$645$$ 4.94427 0.194681
$$646$$ 0 0
$$647$$ 10.8328 0.425882 0.212941 0.977065i $$-0.431696\pi$$
0.212941 + 0.977065i $$0.431696\pi$$
$$648$$ 0 0
$$649$$ 29.8885 1.17323
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −38.3607 −1.50117 −0.750585 0.660774i $$-0.770228\pi$$
−0.750585 + 0.660774i $$0.770228\pi$$
$$654$$ 0 0
$$655$$ −1.88854 −0.0737915
$$656$$ 0 0
$$657$$ 4.47214 0.174475
$$658$$ 0 0
$$659$$ −0.111456 −0.00434172 −0.00217086 0.999998i $$-0.500691\pi$$
−0.00217086 + 0.999998i $$0.500691\pi$$
$$660$$ 0 0
$$661$$ −13.4164 −0.521838 −0.260919 0.965361i $$-0.584026\pi$$
−0.260919 + 0.965361i $$0.584026\pi$$
$$662$$ 0 0
$$663$$ −2.47214 −0.0960098
$$664$$ 0 0
$$665$$ 8.00000 0.310227
$$666$$ 0 0
$$667$$ −37.8885 −1.46705
$$668$$ 0 0
$$669$$ 10.4721 0.404876
$$670$$ 0 0
$$671$$ −48.3607 −1.86694
$$672$$ 0 0
$$673$$ 39.8885 1.53759 0.768795 0.639495i $$-0.220857\pi$$
0.768795 + 0.639495i $$0.220857\pi$$
$$674$$ 0 0
$$675$$ −3.47214 −0.133643
$$676$$ 0 0
$$677$$ 4.58359 0.176162 0.0880809 0.996113i $$-0.471927\pi$$
0.0880809 + 0.996113i $$0.471927\pi$$
$$678$$ 0 0
$$679$$ −6.94427 −0.266497
$$680$$ 0 0
$$681$$ 6.76393 0.259194
$$682$$ 0 0
$$683$$ −47.2361 −1.80744 −0.903719 0.428126i $$-0.859174\pi$$
−0.903719 + 0.428126i $$0.859174\pi$$
$$684$$ 0 0
$$685$$ −18.8328 −0.719565
$$686$$ 0 0
$$687$$ −25.4164 −0.969696
$$688$$ 0 0
$$689$$ 10.0000 0.380970
$$690$$ 0 0
$$691$$ 7.63932 0.290613 0.145307 0.989387i $$-0.453583\pi$$
0.145307 + 0.989387i $$0.453583\pi$$
$$692$$ 0 0
$$693$$ 3.23607 0.122928
$$694$$ 0 0
$$695$$ 22.1115 0.838735
$$696$$ 0 0
$$697$$ −12.9443 −0.490299
$$698$$ 0 0
$$699$$ 5.41641 0.204867
$$700$$ 0 0
$$701$$ 33.7771 1.27574 0.637871 0.770143i $$-0.279815\pi$$
0.637871 + 0.770143i $$0.279815\pi$$
$$702$$ 0 0
$$703$$ 28.9443 1.09165
$$704$$ 0 0
$$705$$ 9.52786 0.358840
$$706$$ 0 0
$$707$$ 8.94427 0.336384
$$708$$ 0 0
$$709$$ 18.3607 0.689550 0.344775 0.938685i $$-0.387955\pi$$
0.344775 + 0.938685i $$0.387955\pi$$
$$710$$ 0 0
$$711$$ 10.4721 0.392736
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 4.00000 0.149592
$$716$$ 0 0
$$717$$ −21.1246 −0.788913
$$718$$ 0 0
$$719$$ 9.30495 0.347016 0.173508 0.984832i $$-0.444490\pi$$
0.173508 + 0.984832i $$0.444490\pi$$
$$720$$ 0 0
$$721$$ −16.9443 −0.631038
$$722$$ 0 0
$$723$$ −16.4721 −0.612605
$$724$$ 0 0
$$725$$ −29.4164 −1.09250
$$726$$ 0 0
$$727$$ −30.8328 −1.14353 −0.571763 0.820419i $$-0.693740\pi$$
−0.571763 + 0.820419i $$0.693740\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 9.88854 0.365741
$$732$$ 0 0
$$733$$ 16.4721 0.608412 0.304206 0.952606i $$-0.401609\pi$$
0.304206 + 0.952606i $$0.401609\pi$$
$$734$$ 0 0
$$735$$ −1.23607 −0.0455931
$$736$$ 0 0
$$737$$ −8.00000 −0.294684
$$738$$ 0 0
$$739$$ 31.4164 1.15567 0.577836 0.816153i $$-0.303897\pi$$
0.577836 + 0.816153i $$0.303897\pi$$
$$740$$ 0 0
$$741$$ 6.47214 0.237760
$$742$$ 0 0
$$743$$ 42.6525 1.56477 0.782384 0.622797i $$-0.214004\pi$$
0.782384 + 0.622797i $$0.214004\pi$$
$$744$$ 0 0
$$745$$ 15.7771 0.578028
$$746$$ 0 0
$$747$$ −2.76393 −0.101127
$$748$$ 0 0
$$749$$ −3.52786 −0.128905
$$750$$ 0 0
$$751$$ −19.0557 −0.695353 −0.347677 0.937614i $$-0.613029\pi$$
−0.347677 + 0.937614i $$0.613029\pi$$
$$752$$ 0 0
$$753$$ −5.52786 −0.201447
$$754$$ 0 0
$$755$$ −3.77709 −0.137462
$$756$$ 0 0
$$757$$ 41.4164 1.50530 0.752652 0.658418i $$-0.228774\pi$$
0.752652 + 0.658418i $$0.228774\pi$$
$$758$$ 0 0
$$759$$ 14.4721 0.525305
$$760$$ 0 0
$$761$$ 26.1803 0.949037 0.474518 0.880246i $$-0.342622\pi$$
0.474518 + 0.880246i $$0.342622\pi$$
$$762$$ 0 0
$$763$$ −3.52786 −0.127717
$$764$$ 0 0
$$765$$ 3.05573 0.110480
$$766$$ 0 0
$$767$$ −9.23607 −0.333495
$$768$$ 0 0
$$769$$ −7.52786 −0.271462 −0.135731 0.990746i $$-0.543338\pi$$
−0.135731 + 0.990746i $$0.543338\pi$$
$$770$$ 0 0
$$771$$ 26.8328 0.966360
$$772$$ 0 0
$$773$$ 7.12461 0.256254 0.128127 0.991758i $$-0.459103\pi$$
0.128127 + 0.991758i $$0.459103\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −4.47214 −0.160437
$$778$$ 0 0
$$779$$ 33.8885 1.21418
$$780$$ 0 0
$$781$$ −18.4721 −0.660985
$$782$$ 0 0
$$783$$ 8.47214 0.302769
$$784$$ 0 0
$$785$$ 22.2492 0.794109
$$786$$ 0 0
$$787$$ −7.63932 −0.272312 −0.136156 0.990687i $$-0.543475\pi$$
−0.136156 + 0.990687i $$0.543475\pi$$
$$788$$ 0 0
$$789$$ 0.472136 0.0168085
$$790$$ 0 0
$$791$$ −20.4721 −0.727905
$$792$$ 0 0
$$793$$ 14.9443 0.530687
$$794$$ 0 0
$$795$$ −12.3607 −0.438388
$$796$$ 0 0
$$797$$ −30.4721 −1.07938 −0.539689 0.841864i $$-0.681458\pi$$
−0.539689 + 0.841864i $$0.681458\pi$$
$$798$$ 0 0
$$799$$ 19.0557 0.674143
$$800$$ 0 0
$$801$$ −10.1803 −0.359705
$$802$$ 0 0
$$803$$ −14.4721 −0.510711
$$804$$ 0 0
$$805$$ −5.52786 −0.194832
$$806$$ 0 0
$$807$$ 4.94427 0.174047
$$808$$ 0 0
$$809$$ 12.4721 0.438497 0.219248 0.975669i $$-0.429639\pi$$
0.219248 + 0.975669i $$0.429639\pi$$
$$810$$ 0 0
$$811$$ −36.0000 −1.26413 −0.632065 0.774915i $$-0.717793\pi$$
−0.632065 + 0.774915i $$0.717793\pi$$
$$812$$ 0 0
$$813$$ 23.4164 0.821249
$$814$$ 0 0
$$815$$ −3.05573 −0.107037
$$816$$ 0 0
$$817$$ −25.8885 −0.905725
$$818$$ 0 0
$$819$$ −1.00000 −0.0349428
$$820$$ 0 0
$$821$$ −46.6525 −1.62818 −0.814091 0.580737i $$-0.802765\pi$$
−0.814091 + 0.580737i $$0.802765\pi$$
$$822$$ 0 0
$$823$$ −47.7771 −1.66540 −0.832702 0.553721i $$-0.813207\pi$$
−0.832702 + 0.553721i $$0.813207\pi$$
$$824$$ 0 0
$$825$$ 11.2361 0.391190
$$826$$ 0 0
$$827$$ 37.4853 1.30349 0.651746 0.758438i $$-0.274037\pi$$
0.651746 + 0.758438i $$0.274037\pi$$
$$828$$ 0 0
$$829$$ 6.00000 0.208389 0.104194 0.994557i $$-0.466774\pi$$
0.104194 + 0.994557i $$0.466774\pi$$
$$830$$ 0 0
$$831$$ −15.8885 −0.551167
$$832$$ 0 0
$$833$$ −2.47214 −0.0856544
$$834$$ 0 0
$$835$$ −22.4721 −0.777680
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 29.8197 1.02949 0.514744 0.857344i $$-0.327887\pi$$
0.514744 + 0.857344i $$0.327887\pi$$
$$840$$ 0 0
$$841$$ 42.7771 1.47507
$$842$$ 0 0
$$843$$ −18.6525 −0.642425
$$844$$ 0 0
$$845$$ −1.23607 −0.0425220
$$846$$ 0 0
$$847$$ 0.527864 0.0181376
$$848$$ 0 0
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ −20.0000 −0.685591
$$852$$ 0 0
$$853$$ −20.8328 −0.713302 −0.356651 0.934238i $$-0.616081\pi$$
−0.356651 + 0.934238i $$0.616081\pi$$
$$854$$ 0 0
$$855$$ −8.00000 −0.273594
$$856$$ 0 0
$$857$$ 40.9443 1.39863 0.699315 0.714814i $$-0.253489\pi$$
0.699315 + 0.714814i $$0.253489\pi$$
$$858$$ 0 0
$$859$$ 10.1115 0.344998 0.172499 0.985010i $$-0.444816\pi$$
0.172499 + 0.985010i $$0.444816\pi$$
$$860$$ 0 0
$$861$$ −5.23607 −0.178445
$$862$$ 0 0
$$863$$ 52.1803 1.77624 0.888120 0.459612i $$-0.152012\pi$$
0.888120 + 0.459612i $$0.152012\pi$$
$$864$$ 0 0
$$865$$ −14.8328 −0.504331
$$866$$ 0 0
$$867$$ −10.8885 −0.369794
$$868$$ 0 0
$$869$$ −33.8885 −1.14959
$$870$$ 0 0
$$871$$ 2.47214 0.0837651
$$872$$ 0 0
$$873$$ 6.94427 0.235028
$$874$$ 0 0
$$875$$ −10.4721 −0.354023
$$876$$ 0 0
$$877$$ 7.88854 0.266377 0.133189 0.991091i $$-0.457478\pi$$
0.133189 + 0.991091i $$0.457478\pi$$
$$878$$ 0 0
$$879$$ 5.23607 0.176608
$$880$$ 0 0
$$881$$ 30.8328 1.03878 0.519392 0.854536i $$-0.326158\pi$$
0.519392 + 0.854536i $$0.326158\pi$$
$$882$$ 0 0
$$883$$ 15.4164 0.518803 0.259402 0.965770i $$-0.416475\pi$$
0.259402 + 0.965770i $$0.416475\pi$$
$$884$$ 0 0
$$885$$ 11.4164 0.383758
$$886$$ 0 0
$$887$$ 5.16718 0.173497 0.0867485 0.996230i $$-0.472352\pi$$
0.0867485 + 0.996230i $$0.472352\pi$$
$$888$$ 0 0
$$889$$ −2.47214 −0.0829128
$$890$$ 0 0
$$891$$ −3.23607 −0.108412
$$892$$ 0 0
$$893$$ −49.8885 −1.66946
$$894$$ 0 0
$$895$$ −8.58359 −0.286918
$$896$$ 0 0
$$897$$ −4.47214 −0.149320
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −24.7214 −0.823588
$$902$$ 0 0
$$903$$ 4.00000 0.133112
$$904$$ 0 0
$$905$$ −24.5836 −0.817186
$$906$$ 0 0
$$907$$ 28.3607 0.941701 0.470850 0.882213i $$-0.343947\pi$$
0.470850 + 0.882213i $$0.343947\pi$$
$$908$$ 0 0
$$909$$ −8.94427 −0.296663
$$910$$ 0 0
$$911$$ −31.8885 −1.05651 −0.528257 0.849084i $$-0.677154\pi$$
−0.528257 + 0.849084i $$0.677154\pi$$
$$912$$ 0 0
$$913$$ 8.94427 0.296012
$$914$$ 0 0
$$915$$ −18.4721 −0.610670
$$916$$ 0 0
$$917$$ −1.52786 −0.0504545
$$918$$ 0 0
$$919$$ −1.52786 −0.0503996 −0.0251998 0.999682i $$-0.508022\pi$$
−0.0251998 + 0.999682i $$0.508022\pi$$
$$920$$ 0 0
$$921$$ 5.88854 0.194034
$$922$$ 0 0
$$923$$ 5.70820 0.187888
$$924$$ 0 0
$$925$$ −15.5279 −0.510553
$$926$$ 0 0
$$927$$ 16.9443 0.556523
$$928$$ 0 0
$$929$$ −44.6525 −1.46500 −0.732500 0.680767i $$-0.761647\pi$$
−0.732500 + 0.680767i $$0.761647\pi$$
$$930$$ 0 0
$$931$$ 6.47214 0.212116
$$932$$ 0 0
$$933$$ 1.52786 0.0500200
$$934$$ 0 0
$$935$$ −9.88854 −0.323390
$$936$$ 0 0
$$937$$ −22.9443 −0.749557 −0.374778 0.927114i $$-0.622281\pi$$
−0.374778 + 0.927114i $$0.622281\pi$$
$$938$$ 0 0
$$939$$ 20.8328 0.679853
$$940$$ 0 0
$$941$$ −50.5410 −1.64759 −0.823795 0.566888i $$-0.808147\pi$$
−0.823795 + 0.566888i $$0.808147\pi$$
$$942$$ 0 0
$$943$$ −23.4164 −0.762543
$$944$$ 0 0
$$945$$ 1.23607 0.0402093
$$946$$ 0 0
$$947$$ −7.59675 −0.246861 −0.123431 0.992353i $$-0.539390\pi$$
−0.123431 + 0.992353i $$0.539390\pi$$
$$948$$ 0 0
$$949$$ 4.47214 0.145172
$$950$$ 0 0
$$951$$ −28.1803 −0.913810
$$952$$ 0 0
$$953$$ −41.1935 −1.33439 −0.667194 0.744884i $$-0.732505\pi$$
−0.667194 + 0.744884i $$0.732505\pi$$
$$954$$ 0 0
$$955$$ −8.58359 −0.277759
$$956$$ 0 0
$$957$$ −27.4164 −0.886247
$$958$$ 0 0
$$959$$ −15.2361 −0.491998
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 3.52786 0.113684
$$964$$ 0 0
$$965$$ −22.6950 −0.730579
$$966$$ 0 0
$$967$$ 40.3607 1.29791 0.648956 0.760826i $$-0.275206\pi$$
0.648956 + 0.760826i $$0.275206\pi$$
$$968$$ 0 0
$$969$$ −16.0000 −0.513994
$$970$$ 0 0
$$971$$ −29.8885 −0.959169 −0.479585 0.877496i $$-0.659213\pi$$
−0.479585 + 0.877496i $$0.659213\pi$$
$$972$$ 0 0
$$973$$ 17.8885 0.573480
$$974$$ 0 0
$$975$$ −3.47214 −0.111197
$$976$$ 0 0
$$977$$ −19.2361 −0.615416 −0.307708 0.951481i $$-0.599562\pi$$
−0.307708 + 0.951481i $$0.599562\pi$$
$$978$$ 0 0
$$979$$ 32.9443 1.05290
$$980$$ 0 0
$$981$$ 3.52786 0.112636
$$982$$ 0 0
$$983$$ −59.1246 −1.88578 −0.942891 0.333101i $$-0.891905\pi$$
−0.942891 + 0.333101i $$0.891905\pi$$
$$984$$ 0 0
$$985$$ −0.944272 −0.0300870
$$986$$ 0 0
$$987$$ 7.70820 0.245355
$$988$$ 0 0
$$989$$ 17.8885 0.568823
$$990$$ 0 0
$$991$$ −6.11146 −0.194137 −0.0970684 0.995278i $$-0.530947\pi$$
−0.0970684 + 0.995278i $$0.530947\pi$$
$$992$$ 0 0
$$993$$ 34.8328 1.10539
$$994$$ 0 0
$$995$$ −19.7771 −0.626976
$$996$$ 0 0
$$997$$ 17.7771 0.563006 0.281503 0.959560i $$-0.409167\pi$$
0.281503 + 0.959560i $$0.409167\pi$$
$$998$$ 0 0
$$999$$ 4.47214 0.141492
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 4368.2.a.bl.1.1 2
4.3 odd 2 1092.2.a.f.1.1 2
12.11 even 2 3276.2.a.l.1.2 2
28.27 even 2 7644.2.a.p.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1092.2.a.f.1.1 2 4.3 odd 2
3276.2.a.l.1.2 2 12.11 even 2
4368.2.a.bl.1.1 2 1.1 even 1 trivial
7644.2.a.p.1.2 2 28.27 even 2