# Properties

 Label 725.2.b.c.349.2 Level $725$ Weight $2$ Character 725.349 Analytic conductor $5.789$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(349,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("725.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.b (of order $$2$$, degree $$1$$, not 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: $$5.78915414654$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 145) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.2 Root $$-0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 725.349 Dual form 725.2.b.c.349.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.414214i q^{2} -2.00000i q^{3} +1.82843 q^{4} -0.828427 q^{6} +4.82843i q^{7} -1.58579i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-0.414214i q^{2} -2.00000i q^{3} +1.82843 q^{4} -0.828427 q^{6} +4.82843i q^{7} -1.58579i q^{8} -1.00000 q^{9} +0.828427 q^{11} -3.65685i q^{12} -2.00000i q^{13} +2.00000 q^{14} +3.00000 q^{16} -2.82843i q^{17} +0.414214i q^{18} +4.82843 q^{19} +9.65685 q^{21} -0.343146i q^{22} -3.17157i q^{23} -3.17157 q^{24} -0.828427 q^{26} -4.00000i q^{27} +8.82843i q^{28} -1.00000 q^{29} +6.48528 q^{31} -4.41421i q^{32} -1.65685i q^{33} -1.17157 q^{34} -1.82843 q^{36} +8.48528i q^{37} -2.00000i q^{38} -4.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} -6.00000i q^{43} +1.51472 q^{44} -1.31371 q^{46} +11.6569i q^{47} -6.00000i q^{48} -16.3137 q^{49} -5.65685 q^{51} -3.65685i q^{52} -3.65685i q^{53} -1.65685 q^{54} +7.65685 q^{56} -9.65685i q^{57} +0.414214i q^{58} -3.65685 q^{61} -2.68629i q^{62} -4.82843i q^{63} +4.17157 q^{64} -0.686292 q^{66} -6.48528i q^{67} -5.17157i q^{68} -6.34315 q^{69} -15.3137 q^{71} +1.58579i q^{72} +8.48528i q^{73} +3.51472 q^{74} +8.82843 q^{76} +4.00000i q^{77} +1.65685i q^{78} +2.48528 q^{79} -11.0000 q^{81} +2.48528i q^{82} +7.17157i q^{83} +17.6569 q^{84} -2.48528 q^{86} +2.00000i q^{87} -1.31371i q^{88} +7.65685 q^{89} +9.65685 q^{91} -5.79899i q^{92} -12.9706i q^{93} +4.82843 q^{94} -8.82843 q^{96} +12.4853i q^{97} +6.75736i q^{98} -0.828427 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 + 8 * q^6 - 4 * q^9 $$4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} - 8 q^{11} + 8 q^{14} + 12 q^{16} + 8 q^{19} + 16 q^{21} - 24 q^{24} + 8 q^{26} - 4 q^{29} - 8 q^{31} - 16 q^{34} + 4 q^{36} - 16 q^{39} - 24 q^{41} + 40 q^{44} + 40 q^{46} - 20 q^{49} + 16 q^{54} + 8 q^{56} + 8 q^{61} + 28 q^{64} - 48 q^{66} - 48 q^{69} - 16 q^{71} + 48 q^{74} + 24 q^{76} - 24 q^{79} - 44 q^{81} + 48 q^{84} + 24 q^{86} + 8 q^{89} + 16 q^{91} + 8 q^{94} - 24 q^{96} + 8 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 + 8 * q^6 - 4 * q^9 - 8 * q^11 + 8 * q^14 + 12 * q^16 + 8 * q^19 + 16 * q^21 - 24 * q^24 + 8 * q^26 - 4 * q^29 - 8 * q^31 - 16 * q^34 + 4 * q^36 - 16 * q^39 - 24 * q^41 + 40 * q^44 + 40 * q^46 - 20 * q^49 + 16 * q^54 + 8 * q^56 + 8 * q^61 + 28 * q^64 - 48 * q^66 - 48 * q^69 - 16 * q^71 + 48 * q^74 + 24 * q^76 - 24 * q^79 - 44 * q^81 + 48 * q^84 + 24 * q^86 + 8 * q^89 + 16 * q^91 + 8 * q^94 - 24 * q^96 + 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$$.

 $$n$$ $$176$$ $$552$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 0.414214i − 0.292893i −0.989219 0.146447i $$-0.953216\pi$$
0.989219 0.146447i $$-0.0467837\pi$$
$$3$$ − 2.00000i − 1.15470i −0.816497 0.577350i $$-0.804087\pi$$
0.816497 0.577350i $$-0.195913\pi$$
$$4$$ 1.82843 0.914214
$$5$$ 0 0
$$6$$ −0.828427 −0.338204
$$7$$ 4.82843i 1.82497i 0.409106 + 0.912487i $$0.365841\pi$$
−0.409106 + 0.912487i $$0.634159\pi$$
$$8$$ − 1.58579i − 0.560660i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0.828427 0.249780 0.124890 0.992171i $$-0.460142\pi$$
0.124890 + 0.992171i $$0.460142\pi$$
$$12$$ − 3.65685i − 1.05564i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ − 2.82843i − 0.685994i −0.939336 0.342997i $$-0.888558\pi$$
0.939336 0.342997i $$-0.111442\pi$$
$$18$$ 0.414214i 0.0976311i
$$19$$ 4.82843 1.10772 0.553859 0.832611i $$-0.313155\pi$$
0.553859 + 0.832611i $$0.313155\pi$$
$$20$$ 0 0
$$21$$ 9.65685 2.10730
$$22$$ − 0.343146i − 0.0731589i
$$23$$ − 3.17157i − 0.661319i −0.943750 0.330659i $$-0.892729\pi$$
0.943750 0.330659i $$-0.107271\pi$$
$$24$$ −3.17157 −0.647395
$$25$$ 0 0
$$26$$ −0.828427 −0.162468
$$27$$ − 4.00000i − 0.769800i
$$28$$ 8.82843i 1.66842i
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 6.48528 1.16479 0.582395 0.812906i $$-0.302116\pi$$
0.582395 + 0.812906i $$0.302116\pi$$
$$32$$ − 4.41421i − 0.780330i
$$33$$ − 1.65685i − 0.288421i
$$34$$ −1.17157 −0.200923
$$35$$ 0 0
$$36$$ −1.82843 −0.304738
$$37$$ 8.48528i 1.39497i 0.716599 + 0.697486i $$0.245698\pi$$
−0.716599 + 0.697486i $$0.754302\pi$$
$$38$$ − 2.00000i − 0.324443i
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ − 4.00000i − 0.617213i
$$43$$ − 6.00000i − 0.914991i −0.889212 0.457496i $$-0.848747\pi$$
0.889212 0.457496i $$-0.151253\pi$$
$$44$$ 1.51472 0.228352
$$45$$ 0 0
$$46$$ −1.31371 −0.193696
$$47$$ 11.6569i 1.70033i 0.526519 + 0.850163i $$0.323497\pi$$
−0.526519 + 0.850163i $$0.676503\pi$$
$$48$$ − 6.00000i − 0.866025i
$$49$$ −16.3137 −2.33053
$$50$$ 0 0
$$51$$ −5.65685 −0.792118
$$52$$ − 3.65685i − 0.507114i
$$53$$ − 3.65685i − 0.502308i −0.967947 0.251154i $$-0.919190\pi$$
0.967947 0.251154i $$-0.0808100\pi$$
$$54$$ −1.65685 −0.225469
$$55$$ 0 0
$$56$$ 7.65685 1.02319
$$57$$ − 9.65685i − 1.27908i
$$58$$ 0.414214i 0.0543889i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −3.65685 −0.468212 −0.234106 0.972211i $$-0.575216\pi$$
−0.234106 + 0.972211i $$0.575216\pi$$
$$62$$ − 2.68629i − 0.341159i
$$63$$ − 4.82843i − 0.608325i
$$64$$ 4.17157 0.521447
$$65$$ 0 0
$$66$$ −0.686292 −0.0844766
$$67$$ − 6.48528i − 0.792303i −0.918185 0.396152i $$-0.870345\pi$$
0.918185 0.396152i $$-0.129655\pi$$
$$68$$ − 5.17157i − 0.627145i
$$69$$ −6.34315 −0.763625
$$70$$ 0 0
$$71$$ −15.3137 −1.81740 −0.908701 0.417447i $$-0.862925\pi$$
−0.908701 + 0.417447i $$0.862925\pi$$
$$72$$ 1.58579i 0.186887i
$$73$$ 8.48528i 0.993127i 0.868000 + 0.496564i $$0.165405\pi$$
−0.868000 + 0.496564i $$0.834595\pi$$
$$74$$ 3.51472 0.408578
$$75$$ 0 0
$$76$$ 8.82843 1.01269
$$77$$ 4.00000i 0.455842i
$$78$$ 1.65685i 0.187602i
$$79$$ 2.48528 0.279616 0.139808 0.990179i $$-0.455351\pi$$
0.139808 + 0.990179i $$0.455351\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 2.48528i 0.274453i
$$83$$ 7.17157i 0.787182i 0.919286 + 0.393591i $$0.128767\pi$$
−0.919286 + 0.393591i $$0.871233\pi$$
$$84$$ 17.6569 1.92652
$$85$$ 0 0
$$86$$ −2.48528 −0.267995
$$87$$ 2.00000i 0.214423i
$$88$$ − 1.31371i − 0.140042i
$$89$$ 7.65685 0.811625 0.405812 0.913956i $$-0.366989\pi$$
0.405812 + 0.913956i $$0.366989\pi$$
$$90$$ 0 0
$$91$$ 9.65685 1.01231
$$92$$ − 5.79899i − 0.604586i
$$93$$ − 12.9706i − 1.34498i
$$94$$ 4.82843 0.498014
$$95$$ 0 0
$$96$$ −8.82843 −0.901048
$$97$$ 12.4853i 1.26769i 0.773461 + 0.633844i $$0.218524\pi$$
−0.773461 + 0.633844i $$0.781476\pi$$
$$98$$ 6.75736i 0.682596i
$$99$$ −0.828427 −0.0832601
$$100$$ 0 0
$$101$$ 15.6569 1.55792 0.778958 0.627077i $$-0.215749\pi$$
0.778958 + 0.627077i $$0.215749\pi$$
$$102$$ 2.34315i 0.232006i
$$103$$ 16.1421i 1.59053i 0.606261 + 0.795266i $$0.292669\pi$$
−0.606261 + 0.795266i $$0.707331\pi$$
$$104$$ −3.17157 −0.310998
$$105$$ 0 0
$$106$$ −1.51472 −0.147122
$$107$$ − 20.1421i − 1.94721i −0.228232 0.973607i $$-0.573294\pi$$
0.228232 0.973607i $$-0.426706\pi$$
$$108$$ − 7.31371i − 0.703762i
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 16.9706 1.61077
$$112$$ 14.4853i 1.36873i
$$113$$ − 2.82843i − 0.266076i −0.991111 0.133038i $$-0.957527\pi$$
0.991111 0.133038i $$-0.0424732\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ −1.82843 −0.169765
$$117$$ 2.00000i 0.184900i
$$118$$ 0 0
$$119$$ 13.6569 1.25192
$$120$$ 0 0
$$121$$ −10.3137 −0.937610
$$122$$ 1.51472i 0.137136i
$$123$$ 12.0000i 1.08200i
$$124$$ 11.8579 1.06487
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ − 6.00000i − 0.532414i −0.963916 0.266207i $$-0.914230\pi$$
0.963916 0.266207i $$-0.0857705\pi$$
$$128$$ − 10.5563i − 0.933058i
$$129$$ −12.0000 −1.05654
$$130$$ 0 0
$$131$$ −12.1421 −1.06086 −0.530432 0.847728i $$-0.677970\pi$$
−0.530432 + 0.847728i $$0.677970\pi$$
$$132$$ − 3.02944i − 0.263679i
$$133$$ 23.3137i 2.02155i
$$134$$ −2.68629 −0.232060
$$135$$ 0 0
$$136$$ −4.48528 −0.384610
$$137$$ 5.17157i 0.441837i 0.975292 + 0.220919i $$0.0709055\pi$$
−0.975292 + 0.220919i $$0.929094\pi$$
$$138$$ 2.62742i 0.223661i
$$139$$ −21.6569 −1.83691 −0.918455 0.395525i $$-0.870563\pi$$
−0.918455 + 0.395525i $$0.870563\pi$$
$$140$$ 0 0
$$141$$ 23.3137 1.96337
$$142$$ 6.34315i 0.532305i
$$143$$ − 1.65685i − 0.138553i
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ 3.51472 0.290880
$$147$$ 32.6274i 2.69106i
$$148$$ 15.5147i 1.27530i
$$149$$ −9.31371 −0.763009 −0.381504 0.924367i $$-0.624594\pi$$
−0.381504 + 0.924367i $$0.624594\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ − 7.65685i − 0.621053i
$$153$$ 2.82843i 0.228665i
$$154$$ 1.65685 0.133513
$$155$$ 0 0
$$156$$ −7.31371 −0.585565
$$157$$ − 0.485281i − 0.0387297i −0.999812 0.0193648i $$-0.993836\pi$$
0.999812 0.0193648i $$-0.00616440\pi$$
$$158$$ − 1.02944i − 0.0818976i
$$159$$ −7.31371 −0.580015
$$160$$ 0 0
$$161$$ 15.3137 1.20689
$$162$$ 4.55635i 0.357981i
$$163$$ − 8.34315i − 0.653486i −0.945113 0.326743i $$-0.894049\pi$$
0.945113 0.326743i $$-0.105951\pi$$
$$164$$ −10.9706 −0.856657
$$165$$ 0 0
$$166$$ 2.97056 0.230560
$$167$$ 2.48528i 0.192317i 0.995366 + 0.0961584i $$0.0306555\pi$$
−0.995366 + 0.0961584i $$0.969344\pi$$
$$168$$ − 15.3137i − 1.18148i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −4.82843 −0.369239
$$172$$ − 10.9706i − 0.836498i
$$173$$ 17.3137i 1.31634i 0.752871 + 0.658168i $$0.228669\pi$$
−0.752871 + 0.658168i $$0.771331\pi$$
$$174$$ 0.828427 0.0628029
$$175$$ 0 0
$$176$$ 2.48528 0.187335
$$177$$ 0 0
$$178$$ − 3.17157i − 0.237719i
$$179$$ 23.3137 1.74255 0.871274 0.490797i $$-0.163294\pi$$
0.871274 + 0.490797i $$0.163294\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ − 4.00000i − 0.296500i
$$183$$ 7.31371i 0.540645i
$$184$$ −5.02944 −0.370775
$$185$$ 0 0
$$186$$ −5.37258 −0.393937
$$187$$ − 2.34315i − 0.171348i
$$188$$ 21.3137i 1.55446i
$$189$$ 19.3137 1.40487
$$190$$ 0 0
$$191$$ −20.8284 −1.50709 −0.753546 0.657395i $$-0.771658\pi$$
−0.753546 + 0.657395i $$0.771658\pi$$
$$192$$ − 8.34315i − 0.602115i
$$193$$ 4.48528i 0.322858i 0.986884 + 0.161429i $$0.0516102\pi$$
−0.986884 + 0.161429i $$0.948390\pi$$
$$194$$ 5.17157 0.371297
$$195$$ 0 0
$$196$$ −29.8284 −2.13060
$$197$$ 19.6569i 1.40049i 0.713901 + 0.700246i $$0.246927\pi$$
−0.713901 + 0.700246i $$0.753073\pi$$
$$198$$ 0.343146i 0.0243863i
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ 0 0
$$201$$ −12.9706 −0.914873
$$202$$ − 6.48528i − 0.456303i
$$203$$ − 4.82843i − 0.338889i
$$204$$ −10.3431 −0.724165
$$205$$ 0 0
$$206$$ 6.68629 0.465856
$$207$$ 3.17157i 0.220440i
$$208$$ − 6.00000i − 0.416025i
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −0.828427 −0.0570313 −0.0285156 0.999593i $$-0.509078\pi$$
−0.0285156 + 0.999593i $$0.509078\pi$$
$$212$$ − 6.68629i − 0.459216i
$$213$$ 30.6274i 2.09856i
$$214$$ −8.34315 −0.570326
$$215$$ 0 0
$$216$$ −6.34315 −0.431596
$$217$$ 31.3137i 2.12571i
$$218$$ 0.828427i 0.0561082i
$$219$$ 16.9706 1.14676
$$220$$ 0 0
$$221$$ −5.65685 −0.380521
$$222$$ − 7.02944i − 0.471785i
$$223$$ − 17.7990i − 1.19191i −0.803018 0.595954i $$-0.796774\pi$$
0.803018 0.595954i $$-0.203226\pi$$
$$224$$ 21.3137 1.42408
$$225$$ 0 0
$$226$$ −1.17157 −0.0779319
$$227$$ − 20.1421i − 1.33688i −0.743766 0.668440i $$-0.766962\pi$$
0.743766 0.668440i $$-0.233038\pi$$
$$228$$ − 17.6569i − 1.16935i
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 1.58579i 0.104112i
$$233$$ 18.0000i 1.17922i 0.807688 + 0.589610i $$0.200718\pi$$
−0.807688 + 0.589610i $$0.799282\pi$$
$$234$$ 0.828427 0.0541560
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 4.97056i − 0.322873i
$$238$$ − 5.65685i − 0.366679i
$$239$$ 0.686292 0.0443925 0.0221963 0.999754i $$-0.492934\pi$$
0.0221963 + 0.999754i $$0.492934\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 4.27208i 0.274620i
$$243$$ 10.0000i 0.641500i
$$244$$ −6.68629 −0.428046
$$245$$ 0 0
$$246$$ 4.97056 0.316912
$$247$$ − 9.65685i − 0.614451i
$$248$$ − 10.2843i − 0.653052i
$$249$$ 14.3431 0.908960
$$250$$ 0 0
$$251$$ 8.82843 0.557245 0.278623 0.960401i $$-0.410122\pi$$
0.278623 + 0.960401i $$0.410122\pi$$
$$252$$ − 8.82843i − 0.556139i
$$253$$ − 2.62742i − 0.165184i
$$254$$ −2.48528 −0.155940
$$255$$ 0 0
$$256$$ 3.97056 0.248160
$$257$$ − 6.68629i − 0.417079i −0.978014 0.208540i $$-0.933129\pi$$
0.978014 0.208540i $$-0.0668710\pi$$
$$258$$ 4.97056i 0.309454i
$$259$$ −40.9706 −2.54579
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ 5.02944i 0.310720i
$$263$$ − 19.6569i − 1.21209i −0.795429 0.606047i $$-0.792754\pi$$
0.795429 0.606047i $$-0.207246\pi$$
$$264$$ −2.62742 −0.161706
$$265$$ 0 0
$$266$$ 9.65685 0.592100
$$267$$ − 15.3137i − 0.937184i
$$268$$ − 11.8579i − 0.724334i
$$269$$ 21.3137 1.29952 0.649760 0.760140i $$-0.274869\pi$$
0.649760 + 0.760140i $$0.274869\pi$$
$$270$$ 0 0
$$271$$ −9.79899 −0.595246 −0.297623 0.954683i $$-0.596194\pi$$
−0.297623 + 0.954683i $$0.596194\pi$$
$$272$$ − 8.48528i − 0.514496i
$$273$$ − 19.3137i − 1.16892i
$$274$$ 2.14214 0.129411
$$275$$ 0 0
$$276$$ −11.5980 −0.698116
$$277$$ 3.65685i 0.219719i 0.993947 + 0.109860i $$0.0350401\pi$$
−0.993947 + 0.109860i $$0.964960\pi$$
$$278$$ 8.97056i 0.538019i
$$279$$ −6.48528 −0.388264
$$280$$ 0 0
$$281$$ −29.3137 −1.74871 −0.874355 0.485288i $$-0.838715\pi$$
−0.874355 + 0.485288i $$0.838715\pi$$
$$282$$ − 9.65685i − 0.575057i
$$283$$ 4.82843i 0.287020i 0.989649 + 0.143510i $$0.0458390\pi$$
−0.989649 + 0.143510i $$0.954161\pi$$
$$284$$ −28.0000 −1.66149
$$285$$ 0 0
$$286$$ −0.686292 −0.0405813
$$287$$ − 28.9706i − 1.71008i
$$288$$ 4.41421i 0.260110i
$$289$$ 9.00000 0.529412
$$290$$ 0 0
$$291$$ 24.9706 1.46380
$$292$$ 15.5147i 0.907930i
$$293$$ 8.48528i 0.495715i 0.968796 + 0.247858i $$0.0797265\pi$$
−0.968796 + 0.247858i $$0.920273\pi$$
$$294$$ 13.5147 0.788194
$$295$$ 0 0
$$296$$ 13.4558 0.782105
$$297$$ − 3.31371i − 0.192281i
$$298$$ 3.85786i 0.223480i
$$299$$ −6.34315 −0.366834
$$300$$ 0 0
$$301$$ 28.9706 1.66984
$$302$$ 4.97056i 0.286024i
$$303$$ − 31.3137i − 1.79893i
$$304$$ 14.4853 0.830788
$$305$$ 0 0
$$306$$ 1.17157 0.0669744
$$307$$ 22.9706i 1.31100i 0.755195 + 0.655500i $$0.227542\pi$$
−0.755195 + 0.655500i $$0.772458\pi$$
$$308$$ 7.31371i 0.416737i
$$309$$ 32.2843 1.83659
$$310$$ 0 0
$$311$$ 14.4853 0.821385 0.410692 0.911774i $$-0.365287\pi$$
0.410692 + 0.911774i $$0.365287\pi$$
$$312$$ 6.34315i 0.359110i
$$313$$ − 6.00000i − 0.339140i −0.985518 0.169570i $$-0.945762\pi$$
0.985518 0.169570i $$-0.0542379\pi$$
$$314$$ −0.201010 −0.0113437
$$315$$ 0 0
$$316$$ 4.54416 0.255629
$$317$$ − 2.82843i − 0.158860i −0.996840 0.0794301i $$-0.974690\pi$$
0.996840 0.0794301i $$-0.0253101\pi$$
$$318$$ 3.02944i 0.169882i
$$319$$ −0.828427 −0.0463830
$$320$$ 0 0
$$321$$ −40.2843 −2.24845
$$322$$ − 6.34315i − 0.353490i
$$323$$ − 13.6569i − 0.759888i
$$324$$ −20.1127 −1.11737
$$325$$ 0 0
$$326$$ −3.45584 −0.191402
$$327$$ 4.00000i 0.221201i
$$328$$ 9.51472i 0.525362i
$$329$$ −56.2843 −3.10305
$$330$$ 0 0
$$331$$ 21.7990 1.19818 0.599090 0.800681i $$-0.295529\pi$$
0.599090 + 0.800681i $$0.295529\pi$$
$$332$$ 13.1127i 0.719653i
$$333$$ − 8.48528i − 0.464991i
$$334$$ 1.02944 0.0563283
$$335$$ 0 0
$$336$$ 28.9706 1.58047
$$337$$ 1.17157i 0.0638196i 0.999491 + 0.0319098i $$0.0101589\pi$$
−0.999491 + 0.0319098i $$0.989841\pi$$
$$338$$ − 3.72792i − 0.202772i
$$339$$ −5.65685 −0.307238
$$340$$ 0 0
$$341$$ 5.37258 0.290942
$$342$$ 2.00000i 0.108148i
$$343$$ − 44.9706i − 2.42818i
$$344$$ −9.51472 −0.512999
$$345$$ 0 0
$$346$$ 7.17157 0.385546
$$347$$ 8.14214i 0.437093i 0.975827 + 0.218546i $$0.0701315\pi$$
−0.975827 + 0.218546i $$0.929869\pi$$
$$348$$ 3.65685i 0.196028i
$$349$$ −20.6274 −1.10416 −0.552080 0.833791i $$-0.686166\pi$$
−0.552080 + 0.833791i $$0.686166\pi$$
$$350$$ 0 0
$$351$$ −8.00000 −0.427008
$$352$$ − 3.65685i − 0.194911i
$$353$$ − 4.34315i − 0.231162i −0.993298 0.115581i $$-0.963127\pi$$
0.993298 0.115581i $$-0.0368730\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 14.0000 0.741999
$$357$$ − 27.3137i − 1.44559i
$$358$$ − 9.65685i − 0.510381i
$$359$$ 3.85786 0.203610 0.101805 0.994804i $$-0.467538\pi$$
0.101805 + 0.994804i $$0.467538\pi$$
$$360$$ 0 0
$$361$$ 4.31371 0.227037
$$362$$ 2.48528i 0.130623i
$$363$$ 20.6274i 1.08266i
$$364$$ 17.6569 0.925471
$$365$$ 0 0
$$366$$ 3.02944 0.158351
$$367$$ − 18.0000i − 0.939592i −0.882775 0.469796i $$-0.844327\pi$$
0.882775 0.469796i $$-0.155673\pi$$
$$368$$ − 9.51472i − 0.495989i
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 17.6569 0.916698
$$372$$ − 23.7157i − 1.22960i
$$373$$ − 6.97056i − 0.360922i −0.983582 0.180461i $$-0.942241\pi$$
0.983582 0.180461i $$-0.0577590\pi$$
$$374$$ −0.970563 −0.0501866
$$375$$ 0 0
$$376$$ 18.4853 0.953306
$$377$$ 2.00000i 0.103005i
$$378$$ − 8.00000i − 0.411476i
$$379$$ −22.4853 −1.15499 −0.577496 0.816394i $$-0.695970\pi$$
−0.577496 + 0.816394i $$0.695970\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 8.62742i 0.441417i
$$383$$ 2.48528i 0.126992i 0.997982 + 0.0634960i $$0.0202250\pi$$
−0.997982 + 0.0634960i $$0.979775\pi$$
$$384$$ −21.1127 −1.07740
$$385$$ 0 0
$$386$$ 1.85786 0.0945628
$$387$$ 6.00000i 0.304997i
$$388$$ 22.8284i 1.15894i
$$389$$ 29.3137 1.48626 0.743132 0.669145i $$-0.233339\pi$$
0.743132 + 0.669145i $$0.233339\pi$$
$$390$$ 0 0
$$391$$ −8.97056 −0.453661
$$392$$ 25.8701i 1.30664i
$$393$$ 24.2843i 1.22498i
$$394$$ 8.14214 0.410195
$$395$$ 0 0
$$396$$ −1.51472 −0.0761175
$$397$$ 19.6569i 0.986549i 0.869874 + 0.493275i $$0.164200\pi$$
−0.869874 + 0.493275i $$0.835800\pi$$
$$398$$ 4.97056i 0.249152i
$$399$$ 46.6274 2.33429
$$400$$ 0 0
$$401$$ −6.68629 −0.333897 −0.166949 0.985966i $$-0.553391\pi$$
−0.166949 + 0.985966i $$0.553391\pi$$
$$402$$ 5.37258i 0.267960i
$$403$$ − 12.9706i − 0.646110i
$$404$$ 28.6274 1.42427
$$405$$ 0 0
$$406$$ −2.00000 −0.0992583
$$407$$ 7.02944i 0.348436i
$$408$$ 8.97056i 0.444109i
$$409$$ 2.97056 0.146885 0.0734424 0.997299i $$-0.476601\pi$$
0.0734424 + 0.997299i $$0.476601\pi$$
$$410$$ 0 0
$$411$$ 10.3431 0.510190
$$412$$ 29.5147i 1.45409i
$$413$$ 0 0
$$414$$ 1.31371 0.0645653
$$415$$ 0 0
$$416$$ −8.82843 −0.432849
$$417$$ 43.3137i 2.12108i
$$418$$ − 1.65685i − 0.0810394i
$$419$$ −28.9706 −1.41530 −0.707652 0.706561i $$-0.750246\pi$$
−0.707652 + 0.706561i $$0.750246\pi$$
$$420$$ 0 0
$$421$$ 18.9706 0.924569 0.462284 0.886732i $$-0.347030\pi$$
0.462284 + 0.886732i $$0.347030\pi$$
$$422$$ 0.343146i 0.0167041i
$$423$$ − 11.6569i − 0.566776i
$$424$$ −5.79899 −0.281624
$$425$$ 0 0
$$426$$ 12.6863 0.614653
$$427$$ − 17.6569i − 0.854475i
$$428$$ − 36.8284i − 1.78017i
$$429$$ −3.31371 −0.159987
$$430$$ 0 0
$$431$$ 3.31371 0.159616 0.0798079 0.996810i $$-0.474569\pi$$
0.0798079 + 0.996810i $$0.474569\pi$$
$$432$$ − 12.0000i − 0.577350i
$$433$$ − 29.1716i − 1.40190i −0.713212 0.700948i $$-0.752760\pi$$
0.713212 0.700948i $$-0.247240\pi$$
$$434$$ 12.9706 0.622607
$$435$$ 0 0
$$436$$ −3.65685 −0.175132
$$437$$ − 15.3137i − 0.732554i
$$438$$ − 7.02944i − 0.335880i
$$439$$ 10.3431 0.493651 0.246826 0.969060i $$-0.420612\pi$$
0.246826 + 0.969060i $$0.420612\pi$$
$$440$$ 0 0
$$441$$ 16.3137 0.776843
$$442$$ 2.34315i 0.111452i
$$443$$ − 7.65685i − 0.363788i −0.983318 0.181894i $$-0.941777\pi$$
0.983318 0.181894i $$-0.0582228\pi$$
$$444$$ 31.0294 1.47259
$$445$$ 0 0
$$446$$ −7.37258 −0.349102
$$447$$ 18.6274i 0.881047i
$$448$$ 20.1421i 0.951626i
$$449$$ −11.6569 −0.550121 −0.275060 0.961427i $$-0.588698\pi$$
−0.275060 + 0.961427i $$0.588698\pi$$
$$450$$ 0 0
$$451$$ −4.97056 −0.234055
$$452$$ − 5.17157i − 0.243250i
$$453$$ 24.0000i 1.12762i
$$454$$ −8.34315 −0.391563
$$455$$ 0 0
$$456$$ −15.3137 −0.717130
$$457$$ − 19.6569i − 0.919509i −0.888046 0.459754i $$-0.847937\pi$$
0.888046 0.459754i $$-0.152063\pi$$
$$458$$ − 0.828427i − 0.0387099i
$$459$$ −11.3137 −0.528079
$$460$$ 0 0
$$461$$ −35.6569 −1.66071 −0.830353 0.557238i $$-0.811861\pi$$
−0.830353 + 0.557238i $$0.811861\pi$$
$$462$$ − 3.31371i − 0.154168i
$$463$$ 21.7990i 1.01308i 0.862215 + 0.506542i $$0.169077\pi$$
−0.862215 + 0.506542i $$0.830923\pi$$
$$464$$ −3.00000 −0.139272
$$465$$ 0 0
$$466$$ 7.45584 0.345385
$$467$$ − 10.9706i − 0.507657i −0.967249 0.253829i $$-0.918310\pi$$
0.967249 0.253829i $$-0.0816899\pi$$
$$468$$ 3.65685i 0.169038i
$$469$$ 31.3137 1.44593
$$470$$ 0 0
$$471$$ −0.970563 −0.0447212
$$472$$ 0 0
$$473$$ − 4.97056i − 0.228547i
$$474$$ −2.05887 −0.0945672
$$475$$ 0 0
$$476$$ 24.9706 1.14452
$$477$$ 3.65685i 0.167436i
$$478$$ − 0.284271i − 0.0130023i
$$479$$ −7.17157 −0.327678 −0.163839 0.986487i $$-0.552388\pi$$
−0.163839 + 0.986487i $$0.552388\pi$$
$$480$$ 0 0
$$481$$ 16.9706 0.773791
$$482$$ − 4.14214i − 0.188669i
$$483$$ − 30.6274i − 1.39360i
$$484$$ −18.8579 −0.857176
$$485$$ 0 0
$$486$$ 4.14214 0.187891
$$487$$ − 9.79899i − 0.444035i −0.975043 0.222017i $$-0.928736\pi$$
0.975043 0.222017i $$-0.0712641\pi$$
$$488$$ 5.79899i 0.262508i
$$489$$ −16.6863 −0.754580
$$490$$ 0 0
$$491$$ −7.45584 −0.336478 −0.168239 0.985746i $$-0.553808\pi$$
−0.168239 + 0.985746i $$0.553808\pi$$
$$492$$ 21.9411i 0.989182i
$$493$$ 2.82843i 0.127386i
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 19.4558 0.873593
$$497$$ − 73.9411i − 3.31671i
$$498$$ − 5.94113i − 0.266228i
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ 0 0
$$501$$ 4.97056 0.222068
$$502$$ − 3.65685i − 0.163213i
$$503$$ − 30.0000i − 1.33763i −0.743427 0.668817i $$-0.766801\pi$$
0.743427 0.668817i $$-0.233199\pi$$
$$504$$ −7.65685 −0.341063
$$505$$ 0 0
$$506$$ −1.08831 −0.0483814
$$507$$ − 18.0000i − 0.799408i
$$508$$ − 10.9706i − 0.486740i
$$509$$ −0.627417 −0.0278098 −0.0139049 0.999903i $$-0.504426\pi$$
−0.0139049 + 0.999903i $$0.504426\pi$$
$$510$$ 0 0
$$511$$ −40.9706 −1.81243
$$512$$ − 22.7574i − 1.00574i
$$513$$ − 19.3137i − 0.852721i
$$514$$ −2.76955 −0.122160
$$515$$ 0 0
$$516$$ −21.9411 −0.965904
$$517$$ 9.65685i 0.424708i
$$518$$ 16.9706i 0.745644i
$$519$$ 34.6274 1.51997
$$520$$ 0 0
$$521$$ −21.3137 −0.933771 −0.466885 0.884318i $$-0.654624\pi$$
−0.466885 + 0.884318i $$0.654624\pi$$
$$522$$ − 0.414214i − 0.0181296i
$$523$$ 2.48528i 0.108674i 0.998523 + 0.0543369i $$0.0173045\pi$$
−0.998523 + 0.0543369i $$0.982696\pi$$
$$524$$ −22.2010 −0.969856
$$525$$ 0 0
$$526$$ −8.14214 −0.355014
$$527$$ − 18.3431i − 0.799040i
$$528$$ − 4.97056i − 0.216316i
$$529$$ 12.9411 0.562658
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 42.6274i 1.84813i
$$533$$ 12.0000i 0.519778i
$$534$$ −6.34315 −0.274495
$$535$$ 0 0
$$536$$ −10.2843 −0.444213
$$537$$ − 46.6274i − 2.01212i
$$538$$ − 8.82843i − 0.380621i
$$539$$ −13.5147 −0.582120
$$540$$ 0 0
$$541$$ −5.02944 −0.216232 −0.108116 0.994138i $$-0.534482\pi$$
−0.108116 + 0.994138i $$0.534482\pi$$
$$542$$ 4.05887i 0.174344i
$$543$$ 12.0000i 0.514969i
$$544$$ −12.4853 −0.535302
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ 2.48528i 0.106263i 0.998588 + 0.0531315i $$0.0169202\pi$$
−0.998588 + 0.0531315i $$0.983080\pi$$
$$548$$ 9.45584i 0.403934i
$$549$$ 3.65685 0.156071
$$550$$ 0 0
$$551$$ −4.82843 −0.205698
$$552$$ 10.0589i 0.428134i
$$553$$ 12.0000i 0.510292i
$$554$$ 1.51472 0.0643542
$$555$$ 0 0
$$556$$ −39.5980 −1.67933
$$557$$ 27.9411i 1.18390i 0.805973 + 0.591952i $$0.201642\pi$$
−0.805973 + 0.591952i $$0.798358\pi$$
$$558$$ 2.68629i 0.113720i
$$559$$ −12.0000 −0.507546
$$560$$ 0 0
$$561$$ −4.68629 −0.197855
$$562$$ 12.1421i 0.512185i
$$563$$ 7.65685i 0.322698i 0.986897 + 0.161349i $$0.0515845\pi$$
−0.986897 + 0.161349i $$0.948416\pi$$
$$564$$ 42.6274 1.79494
$$565$$ 0 0
$$566$$ 2.00000 0.0840663
$$567$$ − 53.1127i − 2.23052i
$$568$$ 24.2843i 1.01895i
$$569$$ −27.6569 −1.15944 −0.579718 0.814817i $$-0.696837\pi$$
−0.579718 + 0.814817i $$0.696837\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ − 3.02944i − 0.126667i
$$573$$ 41.6569i 1.74024i
$$574$$ −12.0000 −0.500870
$$575$$ 0 0
$$576$$ −4.17157 −0.173816
$$577$$ − 23.7990i − 0.990765i −0.868675 0.495382i $$-0.835028\pi$$
0.868675 0.495382i $$-0.164972\pi$$
$$578$$ − 3.72792i − 0.155061i
$$579$$ 8.97056 0.372804
$$580$$ 0 0
$$581$$ −34.6274 −1.43659
$$582$$ − 10.3431i − 0.428737i
$$583$$ − 3.02944i − 0.125466i
$$584$$ 13.4558 0.556807
$$585$$ 0 0
$$586$$ 3.51472 0.145192
$$587$$ 29.7990i 1.22994i 0.788552 + 0.614968i $$0.210831\pi$$
−0.788552 + 0.614968i $$0.789169\pi$$
$$588$$ 59.6569i 2.46021i
$$589$$ 31.3137 1.29026
$$590$$ 0 0
$$591$$ 39.3137 1.61715
$$592$$ 25.4558i 1.04623i
$$593$$ − 7.65685i − 0.314429i −0.987564 0.157215i $$-0.949749\pi$$
0.987564 0.157215i $$-0.0502515\pi$$
$$594$$ −1.37258 −0.0563178
$$595$$ 0 0
$$596$$ −17.0294 −0.697553
$$597$$ 24.0000i 0.982255i
$$598$$ 2.62742i 0.107443i
$$599$$ 37.7990 1.54442 0.772212 0.635364i $$-0.219150\pi$$
0.772212 + 0.635364i $$0.219150\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ − 12.0000i − 0.489083i
$$603$$ 6.48528i 0.264101i
$$604$$ −21.9411 −0.892772
$$605$$ 0 0
$$606$$ −12.9706 −0.526893
$$607$$ − 9.02944i − 0.366494i −0.983067 0.183247i $$-0.941339\pi$$
0.983067 0.183247i $$-0.0586607\pi$$
$$608$$ − 21.3137i − 0.864385i
$$609$$ −9.65685 −0.391315
$$610$$ 0 0
$$611$$ 23.3137 0.943172
$$612$$ 5.17157i 0.209048i
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ 9.51472 0.383983
$$615$$ 0 0
$$616$$ 6.34315 0.255573
$$617$$ 9.17157i 0.369234i 0.982811 + 0.184617i $$0.0591044\pi$$
−0.982811 + 0.184617i $$0.940896\pi$$
$$618$$ − 13.3726i − 0.537924i
$$619$$ 9.79899 0.393855 0.196927 0.980418i $$-0.436904\pi$$
0.196927 + 0.980418i $$0.436904\pi$$
$$620$$ 0 0
$$621$$ −12.6863 −0.509083
$$622$$ − 6.00000i − 0.240578i
$$623$$ 36.9706i 1.48119i
$$624$$ −12.0000 −0.480384
$$625$$ 0 0
$$626$$ −2.48528 −0.0993318
$$627$$ − 8.00000i − 0.319489i
$$628$$ − 0.887302i − 0.0354072i
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ 36.9706 1.47177 0.735887 0.677104i $$-0.236765\pi$$
0.735887 + 0.677104i $$0.236765\pi$$
$$632$$ − 3.94113i − 0.156770i
$$633$$ 1.65685i 0.0658540i
$$634$$ −1.17157 −0.0465291
$$635$$ 0 0
$$636$$ −13.3726 −0.530257
$$637$$ 32.6274i 1.29275i
$$638$$ 0.343146i 0.0135853i
$$639$$ 15.3137 0.605801
$$640$$ 0 0
$$641$$ 0.627417 0.0247815 0.0123907 0.999923i $$-0.496056\pi$$
0.0123907 + 0.999923i $$0.496056\pi$$
$$642$$ 16.6863i 0.658555i
$$643$$ 19.4558i 0.767264i 0.923486 + 0.383632i $$0.125327\pi$$
−0.923486 + 0.383632i $$0.874673\pi$$
$$644$$ 28.0000 1.10335
$$645$$ 0 0
$$646$$ −5.65685 −0.222566
$$647$$ 41.1127i 1.61631i 0.588972 + 0.808153i $$0.299533\pi$$
−0.588972 + 0.808153i $$0.700467\pi$$
$$648$$ 17.4437i 0.685251i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 62.6274 2.45456
$$652$$ − 15.2548i − 0.597425i
$$653$$ − 17.1716i − 0.671976i −0.941866 0.335988i $$-0.890930\pi$$
0.941866 0.335988i $$-0.109070\pi$$
$$654$$ 1.65685 0.0647881
$$655$$ 0 0
$$656$$ −18.0000 −0.702782
$$657$$ − 8.48528i − 0.331042i
$$658$$ 23.3137i 0.908863i
$$659$$ −1.79899 −0.0700787 −0.0350393 0.999386i $$-0.511156\pi$$
−0.0350393 + 0.999386i $$0.511156\pi$$
$$660$$ 0 0
$$661$$ 26.0000 1.01128 0.505641 0.862744i $$-0.331256\pi$$
0.505641 + 0.862744i $$0.331256\pi$$
$$662$$ − 9.02944i − 0.350939i
$$663$$ 11.3137i 0.439388i
$$664$$ 11.3726 0.441342
$$665$$ 0 0
$$666$$ −3.51472 −0.136193
$$667$$ 3.17157i 0.122804i
$$668$$ 4.54416i 0.175819i
$$669$$ −35.5980 −1.37630
$$670$$ 0 0
$$671$$ −3.02944 −0.116950
$$672$$ − 42.6274i − 1.64439i
$$673$$ 22.9706i 0.885450i 0.896657 + 0.442725i $$0.145988\pi$$
−0.896657 + 0.442725i $$0.854012\pi$$
$$674$$ 0.485281 0.0186923
$$675$$ 0 0
$$676$$ 16.4558 0.632917
$$677$$ 36.7696i 1.41317i 0.707629 + 0.706584i $$0.249765\pi$$
−0.707629 + 0.706584i $$0.750235\pi$$
$$678$$ 2.34315i 0.0899880i
$$679$$ −60.2843 −2.31350
$$680$$ 0 0
$$681$$ −40.2843 −1.54370
$$682$$ − 2.22540i − 0.0852148i
$$683$$ 11.8579i 0.453729i 0.973926 + 0.226864i $$0.0728474\pi$$
−0.973926 + 0.226864i $$0.927153\pi$$
$$684$$ −8.82843 −0.337563
$$685$$ 0 0
$$686$$ −18.6274 −0.711198
$$687$$ − 4.00000i − 0.152610i
$$688$$ − 18.0000i − 0.686244i
$$689$$ −7.31371 −0.278630
$$690$$ 0 0
$$691$$ −44.9706 −1.71076 −0.855380 0.518000i $$-0.826677\pi$$
−0.855380 + 0.518000i $$0.826677\pi$$
$$692$$ 31.6569i 1.20341i
$$693$$ − 4.00000i − 0.151947i
$$694$$ 3.37258 0.128022
$$695$$ 0 0
$$696$$ 3.17157 0.120218
$$697$$ 16.9706i 0.642806i
$$698$$ 8.54416i 0.323401i
$$699$$ 36.0000 1.36165
$$700$$ 0 0
$$701$$ 6.68629 0.252538 0.126269 0.991996i $$-0.459700\pi$$
0.126269 + 0.991996i $$0.459700\pi$$
$$702$$ 3.31371i 0.125068i
$$703$$ 40.9706i 1.54523i
$$704$$ 3.45584 0.130247
$$705$$ 0 0
$$706$$ −1.79899 −0.0677059
$$707$$ 75.5980i 2.84315i
$$708$$ 0 0
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ 0 0
$$711$$ −2.48528 −0.0932053
$$712$$ − 12.1421i − 0.455046i
$$713$$ − 20.5685i − 0.770298i
$$714$$ −11.3137 −0.423405
$$715$$ 0 0
$$716$$ 42.6274 1.59306
$$717$$ − 1.37258i − 0.0512601i
$$718$$ − 1.59798i − 0.0596361i
$$719$$ 34.6274 1.29138 0.645692 0.763598i $$-0.276569\pi$$
0.645692 + 0.763598i $$0.276569\pi$$
$$720$$ 0 0
$$721$$ −77.9411 −2.90268
$$722$$ − 1.78680i − 0.0664977i
$$723$$ − 20.0000i − 0.743808i
$$724$$ −10.9706 −0.407718
$$725$$ 0 0
$$726$$ 8.54416 0.317103
$$727$$ 23.9411i 0.887927i 0.896045 + 0.443964i $$0.146428\pi$$
−0.896045 + 0.443964i $$0.853572\pi$$
$$728$$ − 15.3137i − 0.567564i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −16.9706 −0.627679
$$732$$ 13.3726i 0.494265i
$$733$$ − 22.8284i − 0.843187i −0.906785 0.421594i $$-0.861471\pi$$
0.906785 0.421594i $$-0.138529\pi$$
$$734$$ −7.45584 −0.275200
$$735$$ 0 0
$$736$$ −14.0000 −0.516047
$$737$$ − 5.37258i − 0.197902i
$$738$$ − 2.48528i − 0.0914845i
$$739$$ −14.4853 −0.532850 −0.266425 0.963856i $$-0.585842\pi$$
−0.266425 + 0.963856i $$0.585842\pi$$
$$740$$ 0 0
$$741$$ −19.3137 −0.709507
$$742$$ − 7.31371i − 0.268495i
$$743$$ − 52.6274i − 1.93071i −0.260935 0.965356i $$-0.584031\pi$$
0.260935 0.965356i $$-0.415969\pi$$
$$744$$ −20.5685 −0.754079
$$745$$ 0 0
$$746$$ −2.88730 −0.105712
$$747$$ − 7.17157i − 0.262394i
$$748$$ − 4.28427i − 0.156648i
$$749$$ 97.2548 3.55361
$$750$$ 0 0
$$751$$ 16.1421 0.589035 0.294517 0.955646i $$-0.404841\pi$$
0.294517 + 0.955646i $$0.404841\pi$$
$$752$$ 34.9706i 1.27525i
$$753$$ − 17.6569i − 0.643452i
$$754$$ 0.828427 0.0301695
$$755$$ 0 0
$$756$$ 35.3137 1.28435
$$757$$ − 19.5147i − 0.709275i −0.935004 0.354637i $$-0.884604\pi$$
0.935004 0.354637i $$-0.115396\pi$$
$$758$$ 9.31371i 0.338289i
$$759$$ −5.25483 −0.190738
$$760$$ 0 0
$$761$$ 8.62742 0.312744 0.156372 0.987698i $$-0.450020\pi$$
0.156372 + 0.987698i $$0.450020\pi$$
$$762$$ 4.97056i 0.180064i
$$763$$ − 9.65685i − 0.349602i
$$764$$ −38.0833 −1.37780
$$765$$ 0 0
$$766$$ 1.02944 0.0371951
$$767$$ 0 0
$$768$$ − 7.94113i − 0.286551i
$$769$$ 15.6569 0.564601 0.282300 0.959326i $$-0.408903\pi$$
0.282300 + 0.959326i $$0.408903\pi$$
$$770$$ 0 0
$$771$$ −13.3726 −0.481602
$$772$$ 8.20101i 0.295161i
$$773$$ − 8.48528i − 0.305194i −0.988288 0.152597i $$-0.951236\pi$$
0.988288 0.152597i $$-0.0487637\pi$$
$$774$$ 2.48528 0.0893316
$$775$$ 0 0
$$776$$ 19.7990 0.710742
$$777$$ 81.9411i 2.93962i
$$778$$ − 12.1421i − 0.435317i
$$779$$ −28.9706 −1.03798
$$780$$ 0 0
$$781$$ −12.6863 −0.453951
$$782$$ 3.71573i 0.132874i
$$783$$ 4.00000i 0.142948i
$$784$$ −48.9411 −1.74790
$$785$$ 0 0
$$786$$ 10.0589 0.358788
$$787$$ − 17.7990i − 0.634465i −0.948348 0.317233i $$-0.897246\pi$$
0.948348 0.317233i $$-0.102754\pi$$
$$788$$ 35.9411i 1.28035i
$$789$$ −39.3137 −1.39961
$$790$$ 0 0
$$791$$ 13.6569 0.485582
$$792$$ 1.31371i 0.0466806i
$$793$$ 7.31371i 0.259717i
$$794$$ 8.14214 0.288954
$$795$$ 0 0
$$796$$ −21.9411 −0.777683
$$797$$ 5.85786i 0.207496i 0.994604 + 0.103748i $$0.0330836\pi$$
−0.994604 + 0.103748i $$0.966916\pi$$
$$798$$ − 19.3137i − 0.683698i
$$799$$ 32.9706 1.16641
$$800$$ 0 0
$$801$$ −7.65685 −0.270542
$$802$$ 2.76955i 0.0977963i
$$803$$ 7.02944i 0.248063i
$$804$$ −23.7157 −0.836389
$$805$$ 0 0
$$806$$ −5.37258 −0.189241
$$807$$ − 42.6274i − 1.50056i
$$808$$ − 24.8284i − 0.873461i
$$809$$ −42.2843 −1.48664 −0.743318 0.668938i $$-0.766749\pi$$
−0.743318 + 0.668938i $$0.766749\pi$$
$$810$$ 0 0
$$811$$ 37.6569 1.32231 0.661155 0.750249i $$-0.270066\pi$$
0.661155 + 0.750249i $$0.270066\pi$$
$$812$$ − 8.82843i − 0.309817i
$$813$$ 19.5980i 0.687331i
$$814$$ 2.91169 0.102055
$$815$$ 0 0
$$816$$ −16.9706 −0.594089
$$817$$ − 28.9706i − 1.01355i
$$818$$ − 1.23045i − 0.0430216i
$$819$$ −9.65685 −0.337438
$$820$$ 0 0
$$821$$ −22.6863 −0.791757 −0.395879 0.918303i $$-0.629560\pi$$
−0.395879 + 0.918303i $$0.629560\pi$$
$$822$$ − 4.28427i − 0.149431i
$$823$$ − 30.9706i − 1.07957i −0.841804 0.539783i $$-0.818506\pi$$
0.841804 0.539783i $$-0.181494\pi$$
$$824$$ 25.5980 0.891748
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 17.3137i − 0.602057i −0.953615 0.301028i $$-0.902670\pi$$
0.953615 0.301028i $$-0.0973299\pi$$
$$828$$ 5.79899i 0.201529i
$$829$$ −20.6274 −0.716420 −0.358210 0.933641i $$-0.616613\pi$$
−0.358210 + 0.933641i $$0.616613\pi$$
$$830$$ 0 0
$$831$$ 7.31371 0.253710
$$832$$ − 8.34315i − 0.289247i
$$833$$ 46.1421i 1.59873i
$$834$$ 17.9411 0.621250
$$835$$ 0 0
$$836$$ 7.31371 0.252950
$$837$$ − 25.9411i − 0.896656i
$$838$$ 12.0000i 0.414533i
$$839$$ −2.48528 −0.0858014 −0.0429007 0.999079i $$-0.513660\pi$$
−0.0429007 + 0.999079i $$0.513660\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ − 7.85786i − 0.270800i
$$843$$ 58.6274i 2.01924i
$$844$$ −1.51472 −0.0521388
$$845$$ 0 0
$$846$$ −4.82843 −0.166005
$$847$$ − 49.7990i − 1.71111i
$$848$$ − 10.9706i − 0.376731i
$$849$$ 9.65685 0.331422
$$850$$ 0 0
$$851$$ 26.9117 0.922521
$$852$$ 56.0000i 1.91853i
$$853$$ 51.1127i 1.75007i 0.484064 + 0.875033i $$0.339160\pi$$
−0.484064 + 0.875033i $$0.660840\pi$$
$$854$$ −7.31371 −0.250270
$$855$$ 0 0
$$856$$ −31.9411 −1.09173
$$857$$ − 3.37258i − 0.115205i −0.998340 0.0576026i $$-0.981654\pi$$
0.998340 0.0576026i $$-0.0183456\pi$$
$$858$$ 1.37258i 0.0468592i
$$859$$ 56.4264 1.92524 0.962622 0.270848i $$-0.0873041\pi$$
0.962622 + 0.270848i $$0.0873041\pi$$
$$860$$ 0 0
$$861$$ −57.9411 −1.97463
$$862$$ − 1.37258i − 0.0467504i
$$863$$ − 36.1421i − 1.23029i −0.788413 0.615146i $$-0.789097\pi$$
0.788413 0.615146i $$-0.210903\pi$$
$$864$$ −17.6569 −0.600698
$$865$$ 0 0
$$866$$ −12.0833 −0.410606
$$867$$ − 18.0000i − 0.611312i
$$868$$ 57.2548i 1.94336i
$$869$$ 2.05887 0.0698425
$$870$$ 0 0
$$871$$ −12.9706 −0.439491
$$872$$ 3.17157i 0.107403i
$$873$$ − 12.4853i − 0.422563i
$$874$$ −6.34315 −0.214560
$$875$$ 0 0
$$876$$ 31.0294 1.04839
$$877$$ − 38.2843i − 1.29277i −0.763012 0.646384i $$-0.776280\pi$$
0.763012 0.646384i $$-0.223720\pi$$
$$878$$ − 4.28427i − 0.144587i
$$879$$ 16.9706 0.572403
$$880$$ 0 0
$$881$$ 29.3137 0.987604 0.493802 0.869574i $$-0.335607\pi$$
0.493802 + 0.869574i $$0.335607\pi$$
$$882$$ − 6.75736i − 0.227532i
$$883$$ − 14.4853i − 0.487469i −0.969842 0.243734i $$-0.921628\pi$$
0.969842 0.243734i $$-0.0783725\pi$$
$$884$$ −10.3431 −0.347878
$$885$$ 0 0
$$886$$ −3.17157 −0.106551
$$887$$ 6.68629i 0.224504i 0.993680 + 0.112252i $$0.0358063\pi$$
−0.993680 + 0.112252i $$0.964194\pi$$
$$888$$ − 26.9117i − 0.903097i
$$889$$ 28.9706 0.971641
$$890$$ 0 0
$$891$$ −9.11270 −0.305287
$$892$$ − 32.5442i − 1.08966i
$$893$$ 56.2843i 1.88348i
$$894$$ 7.71573 0.258053
$$895$$ 0 0
$$896$$ 50.9706 1.70281
$$897$$ 12.6863i 0.423583i
$$898$$ 4.82843i 0.161127i
$$899$$ −6.48528 −0.216296
$$900$$ 0 0
$$901$$ −10.3431 −0.344580
$$902$$ 2.05887i 0.0685530i
$$903$$ − 57.9411i − 1.92816i
$$904$$ −4.48528 −0.149178
$$905$$ 0 0
$$906$$ 9.94113 0.330272
$$907$$ 10.0000i 0.332045i 0.986122 + 0.166022i $$0.0530924\pi$$
−0.986122 + 0.166022i $$0.946908\pi$$
$$908$$ − 36.8284i − 1.22219i
$$909$$ −15.6569 −0.519305
$$910$$ 0 0
$$911$$ 32.1421 1.06492 0.532458 0.846456i $$-0.321268\pi$$
0.532458 + 0.846456i $$0.321268\pi$$
$$912$$ − 28.9706i − 0.959311i
$$913$$ 5.94113i 0.196623i
$$914$$ −8.14214 −0.269318
$$915$$ 0 0
$$916$$ 3.65685 0.120826
$$917$$ − 58.6274i − 1.93605i
$$918$$ 4.68629i 0.154671i
$$919$$ 36.0000 1.18753 0.593765 0.804638i $$-0.297641\pi$$
0.593765 + 0.804638i $$0.297641\pi$$
$$920$$ 0 0
$$921$$ 45.9411 1.51381
$$922$$ 14.7696i 0.486409i
$$923$$ 30.6274i 1.00811i
$$924$$ 14.6274 0.481207
$$925$$ 0 0
$$926$$ 9.02944 0.296726
$$927$$ − 16.1421i − 0.530177i
$$928$$ 4.41421i 0.144904i
$$929$$ 4.62742 0.151821 0.0759103 0.997115i $$-0.475814\pi$$
0.0759103 + 0.997115i $$0.475814\pi$$
$$930$$ 0 0
$$931$$ −78.7696 −2.58157
$$932$$ 32.9117i 1.07806i
$$933$$ − 28.9706i − 0.948454i
$$934$$ −4.54416 −0.148689
$$935$$ 0 0
$$936$$ 3.17157 0.103666
$$937$$ − 19.6569i − 0.642161i −0.947052 0.321081i $$-0.895954\pi$$
0.947052 0.321081i $$-0.104046\pi$$
$$938$$ − 12.9706i − 0.423504i
$$939$$ −12.0000 −0.391605
$$940$$ 0 0
$$941$$ −27.9411 −0.910855 −0.455427 0.890273i $$-0.650514\pi$$
−0.455427 + 0.890273i $$0.650514\pi$$
$$942$$ 0.402020i 0.0130985i
$$943$$ 19.0294i 0.619684i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −2.05887 −0.0669398
$$947$$ − 44.9117i − 1.45943i −0.683749 0.729717i $$-0.739652\pi$$
0.683749 0.729717i $$-0.260348\pi$$
$$948$$ − 9.08831i − 0.295175i
$$949$$ 16.9706 0.550888
$$950$$ 0 0
$$951$$ −5.65685 −0.183436
$$952$$ − 21.6569i − 0.701903i
$$953$$ 29.3137i 0.949564i 0.880103 + 0.474782i $$0.157473\pi$$
−0.880103 + 0.474782i $$0.842527\pi$$
$$954$$ 1.51472 0.0490408
$$955$$ 0 0
$$956$$ 1.25483 0.0405842
$$957$$ 1.65685i 0.0535585i
$$958$$ 2.97056i 0.0959745i
$$959$$ −24.9706 −0.806342
$$960$$ 0 0
$$961$$ 11.0589 0.356738
$$962$$ − 7.02944i − 0.226638i
$$963$$ 20.1421i 0.649071i
$$964$$ 18.2843 0.588897
$$965$$ 0 0
$$966$$ −12.6863 −0.408175
$$967$$ − 14.9706i − 0.481421i −0.970597 0.240710i $$-0.922620\pi$$
0.970597 0.240710i $$-0.0773804\pi$$
$$968$$ 16.3553i 0.525681i
$$969$$ −27.3137 −0.877443
$$970$$ 0 0
$$971$$ 28.1421 0.903124 0.451562 0.892240i $$-0.350867\pi$$
0.451562 + 0.892240i $$0.350867\pi$$
$$972$$ 18.2843i 0.586468i
$$973$$ − 104.569i − 3.35231i
$$974$$ −4.05887 −0.130055
$$975$$ 0 0
$$976$$ −10.9706 −0.351159
$$977$$ 2.68629i 0.0859421i 0.999076 + 0.0429710i $$0.0136823\pi$$
−0.999076 + 0.0429710i $$0.986318\pi$$
$$978$$ 6.91169i 0.221011i
$$979$$ 6.34315 0.202728
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 3.08831i 0.0985520i
$$983$$ − 9.31371i − 0.297061i −0.988908 0.148531i $$-0.952546\pi$$
0.988908 0.148531i $$-0.0474543\pi$$
$$984$$ 19.0294 0.606636
$$985$$ 0 0
$$986$$ 1.17157 0.0373105
$$987$$ 112.569i 3.58310i
$$988$$ − 17.6569i − 0.561739i
$$989$$ −19.0294 −0.605101
$$990$$ 0 0
$$991$$ 52.0000 1.65183 0.825917 0.563791i $$-0.190658\pi$$
0.825917 + 0.563791i $$0.190658\pi$$
$$992$$ − 28.6274i − 0.908921i
$$993$$ − 43.5980i − 1.38354i
$$994$$ −30.6274 −0.971443
$$995$$ 0 0
$$996$$ 26.2254 0.830983
$$997$$ − 6.82843i − 0.216258i −0.994137 0.108129i $$-0.965514\pi$$
0.994137 0.108129i $$-0.0344860\pi$$
$$998$$ 14.9117i 0.472021i
$$999$$ 33.9411 1.07385
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 725.2.b.c.349.2 4
5.2 odd 4 145.2.a.b.1.2 2
5.3 odd 4 725.2.a.c.1.1 2
5.4 even 2 inner 725.2.b.c.349.3 4
15.2 even 4 1305.2.a.n.1.1 2
15.8 even 4 6525.2.a.p.1.2 2
20.7 even 4 2320.2.a.k.1.2 2
35.27 even 4 7105.2.a.e.1.2 2
40.27 even 4 9280.2.a.w.1.2 2
40.37 odd 4 9280.2.a.be.1.1 2
145.57 odd 4 4205.2.a.d.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.2 2 5.2 odd 4
725.2.a.c.1.1 2 5.3 odd 4
725.2.b.c.349.2 4 1.1 even 1 trivial
725.2.b.c.349.3 4 5.4 even 2 inner
1305.2.a.n.1.1 2 15.2 even 4
2320.2.a.k.1.2 2 20.7 even 4
4205.2.a.d.1.1 2 145.57 odd 4
6525.2.a.p.1.2 2 15.8 even 4
7105.2.a.e.1.2 2 35.27 even 4
9280.2.a.w.1.2 2 40.27 even 4
9280.2.a.be.1.1 2 40.37 odd 4