# Properties

 Label 1425.2.c.d.799.2 Level $1425$ Weight $2$ Character 1425.799 Analytic conductor $11.379$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(799,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1425.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.c (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: $$11.3786822880$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1425.799 Dual form 1425.2.c.d.799.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +2.00000 q^{21} -2.00000i q^{22} +4.00000i q^{23} -3.00000 q^{24} -4.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -4.00000 q^{29} +5.00000i q^{32} -2.00000i q^{33} -2.00000 q^{34} -1.00000 q^{36} +1.00000i q^{38} -4.00000 q^{39} +2.00000i q^{42} +10.0000i q^{43} -2.00000 q^{44} -4.00000 q^{46} +12.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} +4.00000i q^{52} +2.00000i q^{53} +1.00000 q^{54} +6.00000 q^{56} +1.00000i q^{57} -4.00000i q^{58} -4.00000 q^{59} +2.00000 q^{61} +2.00000i q^{63} -7.00000 q^{64} +2.00000 q^{66} -16.0000i q^{67} +2.00000i q^{68} -4.00000 q^{69} -3.00000i q^{72} +2.00000i q^{73} +1.00000 q^{76} +4.00000i q^{77} -4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +2.00000 q^{84} -10.0000 q^{86} -4.00000i q^{87} -6.00000i q^{88} +8.00000 q^{91} +4.00000i q^{92} -12.0000 q^{94} -5.00000 q^{96} -16.0000i q^{97} +3.00000i q^{98} +2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{11} + 4 q^{14} - 2 q^{16} + 2 q^{19} + 4 q^{21} - 6 q^{24} - 8 q^{26} - 8 q^{29} - 4 q^{34} - 2 q^{36} - 8 q^{39} - 4 q^{44} - 8 q^{46} + 6 q^{49} - 4 q^{51} + 2 q^{54} + 12 q^{56} - 8 q^{59} + 4 q^{61} - 14 q^{64} + 4 q^{66} - 8 q^{69} + 2 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{84} - 20 q^{86} + 16 q^{91} - 24 q^{94} - 10 q^{96} + 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 - 4 * q^11 + 4 * q^14 - 2 * q^16 + 2 * q^19 + 4 * q^21 - 6 * q^24 - 8 * q^26 - 8 * q^29 - 4 * q^34 - 2 * q^36 - 8 * q^39 - 4 * q^44 - 8 * q^46 + 6 * q^49 - 4 * q^51 + 2 * q^54 + 12 * q^56 - 8 * q^59 + 4 * q^61 - 14 * q^64 + 4 * q^66 - 8 * q^69 + 2 * q^76 + 16 * q^79 + 2 * q^81 + 4 * q^84 - 20 * q^86 + 16 * q^91 - 24 * q^94 - 10 * q^96 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$$.

 $$n$$ $$476$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 3.00000i 1.06066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ − 2.00000i − 0.426401i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ − 2.00000i − 0.348155i
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 2.00000i 0.308607i
$$43$$ 10.0000i 1.52499i 0.646997 + 0.762493i $$0.276025\pi$$
−0.646997 + 0.762493i $$0.723975\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 12.0000i 1.75038i 0.483779 + 0.875190i $$0.339264\pi$$
−0.483779 + 0.875190i $$0.660736\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 4.00000i 0.554700i
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 6.00000 0.801784
$$57$$ 1.00000i 0.132453i
$$58$$ − 4.00000i − 0.525226i
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 2.00000i 0.251976i
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ − 16.0000i − 1.95471i −0.211604 0.977356i $$-0.567869\pi$$
0.211604 0.977356i $$-0.432131\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ − 3.00000i − 0.353553i
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 4.00000i 0.455842i
$$78$$ − 4.00000i − 0.452911i
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ −10.0000 −1.07833
$$87$$ − 4.00000i − 0.428845i
$$88$$ − 6.00000i − 0.639602i
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ 4.00000i 0.417029i
$$93$$ 0 0
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ −5.00000 −0.510310
$$97$$ − 16.0000i − 1.62455i −0.583272 0.812277i $$-0.698228\pi$$
0.583272 0.812277i $$-0.301772\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ − 2.00000i − 0.198030i
$$103$$ 8.00000i 0.788263i 0.919054 + 0.394132i $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ −12.0000 −1.17670
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 2.00000i 0.188982i
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ − 4.00000i − 0.369800i
$$118$$ − 4.00000i − 0.368230i
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 2.00000i 0.181071i
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 4.00000i 0.354943i 0.984126 + 0.177471i $$0.0567917\pi$$
−0.984126 + 0.177471i $$0.943208\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ −10.0000 −0.880451
$$130$$ 0 0
$$131$$ −14.0000 −1.22319 −0.611593 0.791173i $$-0.709471\pi$$
−0.611593 + 0.791173i $$0.709471\pi$$
$$132$$ − 2.00000i − 0.174078i
$$133$$ − 2.00000i − 0.173422i
$$134$$ 16.0000 1.38219
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ 0 0
$$143$$ − 8.00000i − 0.668994i
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 3.00000i 0.247436i
$$148$$ 0 0
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ 3.00000i 0.243332i
$$153$$ − 2.00000i − 0.161690i
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ 8.00000i 0.636446i
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ 1.00000i 0.0785674i
$$163$$ 6.00000i 0.469956i 0.972001 + 0.234978i $$0.0755019\pi$$
−0.972001 + 0.234978i $$0.924498\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ 6.00000i 0.462910i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 10.0000i 0.762493i
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ 4.00000 0.303239
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ − 4.00000i − 0.300658i
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 8.00000i 0.592999i
$$183$$ 2.00000i 0.147844i
$$184$$ −12.0000 −0.884652
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 4.00000i − 0.292509i
$$188$$ 12.0000i 0.875190i
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ − 7.00000i − 0.505181i
$$193$$ − 4.00000i − 0.287926i −0.989583 0.143963i $$-0.954015\pi$$
0.989583 0.143963i $$-0.0459847\pi$$
$$194$$ 16.0000 1.14873
$$195$$ 0 0
$$196$$ 3.00000 0.214286
$$197$$ − 2.00000i − 0.142494i −0.997459 0.0712470i $$-0.977302\pi$$
0.997459 0.0712470i $$-0.0226979\pi$$
$$198$$ 2.00000i 0.142134i
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ 14.0000i 0.985037i
$$203$$ 8.00000i 0.561490i
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ − 4.00000i − 0.278019i
$$208$$ − 4.00000i − 0.277350i
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 2.00000i 0.137361i
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 3.00000 0.204124
$$217$$ 0 0
$$218$$ − 10.0000i − 0.677285i
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ 0 0
$$223$$ − 24.0000i − 1.60716i −0.595198 0.803579i $$-0.702926\pi$$
0.595198 0.803579i $$-0.297074\pi$$
$$224$$ 10.0000 0.668153
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 1.00000i 0.0662266i
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ − 12.0000i − 0.787839i
$$233$$ 18.0000i 1.17922i 0.807688 + 0.589610i $$0.200718\pi$$
−0.807688 + 0.589610i $$0.799282\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ 8.00000i 0.519656i
$$238$$ 4.00000i 0.259281i
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ − 7.00000i − 0.449977i
$$243$$ 1.00000i 0.0641500i
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.00000i 0.254514i
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 22.0000 1.38863 0.694314 0.719672i $$-0.255708\pi$$
0.694314 + 0.719672i $$0.255708\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ − 8.00000i − 0.502956i
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 22.0000i 1.37232i 0.727450 + 0.686161i $$0.240706\pi$$
−0.727450 + 0.686161i $$0.759294\pi$$
$$258$$ − 10.0000i − 0.622573i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ − 14.0000i − 0.864923i
$$263$$ 16.0000i 0.986602i 0.869859 + 0.493301i $$0.164210\pi$$
−0.869859 + 0.493301i $$0.835790\pi$$
$$264$$ 6.00000 0.369274
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ 0 0
$$268$$ − 16.0000i − 0.977356i
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 8.00000i 0.484182i
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ 22.0000i 1.32185i 0.750451 + 0.660926i $$0.229836\pi$$
−0.750451 + 0.660926i $$0.770164\pi$$
$$278$$ − 8.00000i − 0.479808i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 28.0000 1.67034 0.835170 0.549992i $$-0.185369\pi$$
0.835170 + 0.549992i $$0.185369\pi$$
$$282$$ − 12.0000i − 0.714590i
$$283$$ − 26.0000i − 1.54554i −0.634686 0.772770i $$-0.718871\pi$$
0.634686 0.772770i $$-0.281129\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ − 5.00000i − 0.294628i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 16.0000 0.937937
$$292$$ 2.00000i 0.117041i
$$293$$ − 26.0000i − 1.51894i −0.650545 0.759468i $$-0.725459\pi$$
0.650545 0.759468i $$-0.274541\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 2.00000i 0.116052i
$$298$$ 18.0000i 1.04271i
$$299$$ −16.0000 −0.925304
$$300$$ 0 0
$$301$$ 20.0000 1.15278
$$302$$ 24.0000i 1.38104i
$$303$$ 14.0000i 0.804279i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 4.00000i 0.227921i
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ − 12.0000i − 0.679366i
$$313$$ 14.0000i 0.791327i 0.918396 + 0.395663i $$0.129485\pi$$
−0.918396 + 0.395663i $$0.870515\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 14.0000i 0.786318i 0.919470 + 0.393159i $$0.128618\pi$$
−0.919470 + 0.393159i $$0.871382\pi$$
$$318$$ − 2.00000i − 0.112154i
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 8.00000i 0.445823i
$$323$$ 2.00000i 0.111283i
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −6.00000 −0.332309
$$327$$ − 10.0000i − 0.553001i
$$328$$ 0 0
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 12.0000i 0.658586i
$$333$$ 0 0
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ 16.0000i 0.871576i 0.900049 + 0.435788i $$0.143530\pi$$
−0.900049 + 0.435788i $$0.856470\pi$$
$$338$$ − 3.00000i − 0.163178i
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 1.00000i − 0.0540738i
$$343$$ − 20.0000i − 1.07990i
$$344$$ −30.0000 −1.61749
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ − 28.0000i − 1.50312i −0.659665 0.751559i $$-0.729302\pi$$
0.659665 0.751559i $$-0.270698\pi$$
$$348$$ − 4.00000i − 0.214423i
$$349$$ 18.0000 0.963518 0.481759 0.876304i $$-0.339998\pi$$
0.481759 + 0.876304i $$0.339998\pi$$
$$350$$ 0 0
$$351$$ 4.00000 0.213504
$$352$$ − 10.0000i − 0.533002i
$$353$$ 2.00000i 0.106449i 0.998583 + 0.0532246i $$0.0169499\pi$$
−0.998583 + 0.0532246i $$0.983050\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 4.00000i 0.211702i
$$358$$ 12.0000i 0.634220i
$$359$$ 2.00000 0.105556 0.0527780 0.998606i $$-0.483192\pi$$
0.0527780 + 0.998606i $$0.483192\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 10.0000i 0.525588i
$$363$$ − 7.00000i − 0.367405i
$$364$$ 8.00000 0.419314
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ 10.0000i 0.521996i 0.965339 + 0.260998i $$0.0840516\pi$$
−0.965339 + 0.260998i $$0.915948\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 4.00000 0.207670
$$372$$ 0 0
$$373$$ − 4.00000i − 0.207112i −0.994624 0.103556i $$-0.966978\pi$$
0.994624 0.103556i $$-0.0330221\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ −36.0000 −1.85656
$$377$$ − 16.0000i − 0.824042i
$$378$$ − 2.00000i − 0.102869i
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ − 18.0000i − 0.920960i
$$383$$ − 8.00000i − 0.408781i −0.978889 0.204390i $$-0.934479\pi$$
0.978889 0.204390i $$-0.0655212\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ − 10.0000i − 0.508329i
$$388$$ − 16.0000i − 0.812277i
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ 9.00000i 0.454569i
$$393$$ − 14.0000i − 0.706207i
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ − 2.00000i − 0.100377i −0.998740 0.0501886i $$-0.984018\pi$$
0.998740 0.0501886i $$-0.0159822\pi$$
$$398$$ − 12.0000i − 0.601506i
$$399$$ 2.00000 0.100125
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 16.0000i 0.798007i
$$403$$ 0 0
$$404$$ 14.0000 0.696526
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ 0 0
$$408$$ − 6.00000i − 0.297044i
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 8.00000i 0.394132i
$$413$$ 8.00000i 0.393654i
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ −20.0000 −0.980581
$$417$$ − 8.00000i − 0.391762i
$$418$$ − 2.00000i − 0.0978232i
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ − 12.0000i − 0.583460i
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 4.00000i − 0.193574i
$$428$$ − 12.0000i − 0.580042i
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ −4.00000 −0.192673 −0.0963366 0.995349i $$-0.530713\pi$$
−0.0963366 + 0.995349i $$0.530713\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 36.0000i 1.73005i 0.501729 + 0.865025i $$0.332697\pi$$
−0.501729 + 0.865025i $$0.667303\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 4.00000i 0.191346i
$$438$$ − 2.00000i − 0.0955637i
$$439$$ 40.0000 1.90910 0.954548 0.298057i $$-0.0963387\pi$$
0.954548 + 0.298057i $$0.0963387\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ − 8.00000i − 0.380521i
$$443$$ − 16.0000i − 0.760183i −0.924949 0.380091i $$-0.875893\pi$$
0.924949 0.380091i $$-0.124107\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 24.0000 1.13643
$$447$$ 18.0000i 0.851371i
$$448$$ 14.0000i 0.661438i
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 14.0000i − 0.658505i
$$453$$ 24.0000i 1.12762i
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ −3.00000 −0.140488
$$457$$ − 14.0000i − 0.654892i −0.944870 0.327446i $$-0.893812\pi$$
0.944870 0.327446i $$-0.106188\pi$$
$$458$$ 14.0000i 0.654177i
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ − 4.00000i − 0.186097i
$$463$$ 22.0000i 1.02243i 0.859454 + 0.511213i $$0.170804\pi$$
−0.859454 + 0.511213i $$0.829196\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ 16.0000i 0.740392i 0.928954 + 0.370196i $$0.120709\pi$$
−0.928954 + 0.370196i $$0.879291\pi$$
$$468$$ − 4.00000i − 0.184900i
$$469$$ −32.0000 −1.47762
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ − 12.0000i − 0.552345i
$$473$$ − 20.0000i − 0.919601i
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ − 2.00000i − 0.0915737i
$$478$$ 6.00000i 0.274434i
$$479$$ −38.0000 −1.73626 −0.868132 0.496333i $$-0.834679\pi$$
−0.868132 + 0.496333i $$0.834679\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 26.0000i 1.18427i
$$483$$ 8.00000i 0.364013i
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 8.00000i 0.362515i 0.983436 + 0.181257i $$0.0580167\pi$$
−0.983436 + 0.181257i $$0.941983\pi$$
$$488$$ 6.00000i 0.271607i
$$489$$ −6.00000 −0.271329
$$490$$ 0 0
$$491$$ −6.00000 −0.270776 −0.135388 0.990793i $$-0.543228\pi$$
−0.135388 + 0.990793i $$0.543228\pi$$
$$492$$ 0 0
$$493$$ − 8.00000i − 0.360302i
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ − 12.0000i − 0.537733i
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ 22.0000i 0.981908i
$$503$$ 8.00000i 0.356702i 0.983967 + 0.178351i $$0.0570763\pi$$
−0.983967 + 0.178351i $$0.942924\pi$$
$$504$$ −6.00000 −0.267261
$$505$$ 0 0
$$506$$ 8.00000 0.355643
$$507$$ − 3.00000i − 0.133235i
$$508$$ 4.00000i 0.177471i
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ − 11.0000i − 0.486136i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ −22.0000 −0.970378
$$515$$ 0 0
$$516$$ −10.0000 −0.440225
$$517$$ − 24.0000i − 1.05552i
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ 4.00000i 0.175075i
$$523$$ 28.0000i 1.22435i 0.790721 + 0.612177i $$0.209706\pi$$
−0.790721 + 0.612177i $$0.790294\pi$$
$$524$$ −14.0000 −0.611593
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ 2.00000i 0.0870388i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ − 2.00000i − 0.0867110i
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 48.0000 2.07328
$$537$$ 12.0000i 0.517838i
$$538$$ − 4.00000i − 0.172452i
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ −42.0000 −1.80572 −0.902861 0.429934i $$-0.858537\pi$$
−0.902861 + 0.429934i $$0.858537\pi$$
$$542$$ 12.0000i 0.515444i
$$543$$ 10.0000i 0.429141i
$$544$$ −10.0000 −0.428746
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ 16.0000i 0.684111i 0.939680 + 0.342055i $$0.111123\pi$$
−0.939680 + 0.342055i $$0.888877\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −4.00000 −0.170406
$$552$$ − 12.0000i − 0.510754i
$$553$$ − 16.0000i − 0.680389i
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ −8.00000 −0.339276
$$557$$ − 38.0000i − 1.61011i −0.593199 0.805056i $$-0.702135\pi$$
0.593199 0.805056i $$-0.297865\pi$$
$$558$$ 0 0
$$559$$ −40.0000 −1.69182
$$560$$ 0 0
$$561$$ 4.00000 0.168880
$$562$$ 28.0000i 1.18111i
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 0 0
$$566$$ 26.0000 1.09286
$$567$$ − 2.00000i − 0.0839921i
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ −24.0000 −1.00437 −0.502184 0.864761i $$-0.667470\pi$$
−0.502184 + 0.864761i $$0.667470\pi$$
$$572$$ − 8.00000i − 0.334497i
$$573$$ − 18.0000i − 0.751961i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ 42.0000i 1.74848i 0.485491 + 0.874241i $$0.338641\pi$$
−0.485491 + 0.874241i $$0.661359\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 16.0000i 0.663221i
$$583$$ − 4.00000i − 0.165663i
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 26.0000 1.07405
$$587$$ − 32.0000i − 1.32078i −0.750922 0.660391i $$-0.770391\pi$$
0.750922 0.660391i $$-0.229609\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ 0 0
$$593$$ 18.0000i 0.739171i 0.929197 + 0.369586i $$0.120500\pi$$
−0.929197 + 0.369586i $$0.879500\pi$$
$$594$$ −2.00000 −0.0820610
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ − 12.0000i − 0.491127i
$$598$$ − 16.0000i − 0.654289i
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ 20.0000i 0.815139i
$$603$$ 16.0000i 0.651570i
$$604$$ 24.0000 0.976546
$$605$$ 0 0
$$606$$ −14.0000 −0.568711
$$607$$ − 16.0000i − 0.649420i −0.945814 0.324710i $$-0.894733\pi$$
0.945814 0.324710i $$-0.105267\pi$$
$$608$$ 5.00000i 0.202777i
$$609$$ −8.00000 −0.324176
$$610$$ 0 0
$$611$$ −48.0000 −1.94187
$$612$$ − 2.00000i − 0.0808452i
$$613$$ 18.0000i 0.727013i 0.931592 + 0.363507i $$0.118421\pi$$
−0.931592 + 0.363507i $$0.881579\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ −12.0000 −0.483494
$$617$$ − 6.00000i − 0.241551i −0.992680 0.120775i $$-0.961462\pi$$
0.992680 0.120775i $$-0.0385381\pi$$
$$618$$ − 8.00000i − 0.321807i
$$619$$ −16.0000 −0.643094 −0.321547 0.946894i $$-0.604203\pi$$
−0.321547 + 0.946894i $$0.604203\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ − 18.0000i − 0.721734i
$$623$$ 0 0
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ −14.0000 −0.559553
$$627$$ − 2.00000i − 0.0798723i
$$628$$ − 18.0000i − 0.718278i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 48.0000 1.91085 0.955425 0.295234i $$-0.0953977\pi$$
0.955425 + 0.295234i $$0.0953977\pi$$
$$632$$ 24.0000i 0.954669i
$$633$$ − 12.0000i − 0.476957i
$$634$$ −14.0000 −0.556011
$$635$$ 0 0
$$636$$ −2.00000 −0.0793052
$$637$$ 12.0000i 0.475457i
$$638$$ 8.00000i 0.316723i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ − 26.0000i − 1.02534i −0.858586 0.512670i $$-0.828656\pi$$
0.858586 0.512670i $$-0.171344\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ −2.00000 −0.0786889
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ 3.00000i 0.117851i
$$649$$ 8.00000 0.314027
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 6.00000i 0.234978i
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 24.0000i 0.935617i
$$659$$ −16.0000 −0.623272 −0.311636 0.950202i $$-0.600877\pi$$
−0.311636 + 0.950202i $$0.600877\pi$$
$$660$$ 0 0
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ − 28.0000i − 1.08825i
$$663$$ − 8.00000i − 0.310694i
$$664$$ −36.0000 −1.39707
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 16.0000i − 0.619522i
$$668$$ − 8.00000i − 0.309529i
$$669$$ 24.0000 0.927894
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 10.0000i 0.385758i
$$673$$ − 24.0000i − 0.925132i −0.886585 0.462566i $$-0.846929\pi$$
0.886585 0.462566i $$-0.153071\pi$$
$$674$$ −16.0000 −0.616297
$$675$$ 0 0
$$676$$ −3.00000 −0.115385
$$677$$ − 50.0000i − 1.92166i −0.277145 0.960828i $$-0.589388\pi$$
0.277145 0.960828i $$-0.410612\pi$$
$$678$$ 14.0000i 0.537667i
$$679$$ −32.0000 −1.22805
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 14.0000i 0.534133i
$$688$$ − 10.0000i − 0.381246i
$$689$$ −8.00000 −0.304776
$$690$$ 0 0
$$691$$ −16.0000 −0.608669 −0.304334 0.952565i $$-0.598434\pi$$
−0.304334 + 0.952565i $$0.598434\pi$$
$$692$$ − 18.0000i − 0.684257i
$$693$$ − 4.00000i − 0.151947i
$$694$$ 28.0000 1.06287
$$695$$ 0 0
$$696$$ 12.0000 0.454859
$$697$$ 0 0
$$698$$ 18.0000i 0.681310i
$$699$$ −18.0000 −0.680823
$$700$$ 0 0
$$701$$ −34.0000 −1.28416 −0.642081 0.766637i $$-0.721929\pi$$
−0.642081 + 0.766637i $$0.721929\pi$$
$$702$$ 4.00000i 0.150970i
$$703$$ 0 0
$$704$$ 14.0000 0.527645
$$705$$ 0 0
$$706$$ −2.00000 −0.0752710
$$707$$ − 28.0000i − 1.05305i
$$708$$ − 4.00000i − 0.150329i
$$709$$ −18.0000 −0.676004 −0.338002 0.941145i $$-0.609751\pi$$
−0.338002 + 0.941145i $$0.609751\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ −4.00000 −0.149696
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 6.00000i 0.224074i
$$718$$ 2.00000i 0.0746393i
$$719$$ −42.0000 −1.56634 −0.783168 0.621810i $$-0.786397\pi$$
−0.783168 + 0.621810i $$0.786397\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 1.00000i 0.0372161i
$$723$$ 26.0000i 0.966950i
$$724$$ 10.0000 0.371647
$$725$$ 0 0
$$726$$ 7.00000 0.259794
$$727$$ − 14.0000i − 0.519231i −0.965712 0.259616i $$-0.916404\pi$$
0.965712 0.259616i $$-0.0835959\pi$$
$$728$$ 24.0000i 0.889499i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −20.0000 −0.739727
$$732$$ 2.00000i 0.0739221i
$$733$$ − 34.0000i − 1.25582i −0.778287 0.627909i $$-0.783911\pi$$
0.778287 0.627909i $$-0.216089\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ 0 0
$$736$$ −20.0000 −0.737210
$$737$$ 32.0000i 1.17874i
$$738$$ 0 0
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 0 0
$$741$$ −4.00000 −0.146944
$$742$$ 4.00000i 0.146845i
$$743$$ 8.00000i 0.293492i 0.989174 + 0.146746i $$0.0468799\pi$$
−0.989174 + 0.146746i $$0.953120\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 4.00000 0.146450
$$747$$ − 12.0000i − 0.439057i
$$748$$ − 4.00000i − 0.146254i
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ − 12.0000i − 0.437595i
$$753$$ 22.0000i 0.801725i
$$754$$ 16.0000 0.582686
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ 10.0000i 0.363456i 0.983349 + 0.181728i $$0.0581691\pi$$
−0.983349 + 0.181728i $$0.941831\pi$$
$$758$$ 20.0000i 0.726433i
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ − 4.00000i − 0.144905i
$$763$$ 20.0000i 0.724049i
$$764$$ −18.0000 −0.651217
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ − 16.0000i − 0.577727i
$$768$$ − 17.0000i − 0.613435i
$$769$$ 46.0000 1.65880 0.829401 0.558653i $$-0.188682\pi$$
0.829401 + 0.558653i $$0.188682\pi$$
$$770$$ 0 0
$$771$$ −22.0000 −0.792311
$$772$$ − 4.00000i − 0.143963i
$$773$$ − 14.0000i − 0.503545i −0.967786 0.251773i $$-0.918987\pi$$
0.967786 0.251773i $$-0.0810135\pi$$
$$774$$ 10.0000 0.359443
$$775$$ 0 0
$$776$$ 48.0000 1.72310
$$777$$ 0 0
$$778$$ 30.0000i 1.07555i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 8.00000i − 0.286079i
$$783$$ 4.00000i 0.142948i
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 14.0000 0.499363
$$787$$ 28.0000i 0.998092i 0.866575 + 0.499046i $$0.166316\pi$$
−0.866575 + 0.499046i $$0.833684\pi$$
$$788$$ − 2.00000i − 0.0712470i
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ −28.0000 −0.995565
$$792$$ 6.00000i 0.213201i
$$793$$ 8.00000i 0.284088i
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ −12.0000 −0.425329
$$797$$ − 14.0000i − 0.495905i −0.968772 0.247953i $$-0.920242\pi$$
0.968772 0.247953i $$-0.0797578\pi$$
$$798$$ 2.00000i 0.0707992i
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 12.0000i 0.423735i
$$803$$ − 4.00000i − 0.141157i
$$804$$ 16.0000 0.564276
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 4.00000i − 0.140807i
$$808$$ 42.0000i 1.47755i
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ 0 0
$$811$$ 4.00000 0.140459 0.0702295 0.997531i $$-0.477627\pi$$
0.0702295 + 0.997531i $$0.477627\pi$$
$$812$$ 8.00000i 0.280745i
$$813$$ 12.0000i 0.420858i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ 10.0000i 0.349856i
$$818$$ 26.0000i 0.909069i
$$819$$ −8.00000 −0.279543
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ − 6.00000i − 0.209274i
$$823$$ 14.0000i 0.488009i 0.969774 + 0.244005i $$0.0784612\pi$$
−0.969774 + 0.244005i $$0.921539\pi$$
$$824$$ −24.0000 −0.836080
$$825$$ 0 0
$$826$$ −8.00000 −0.278356
$$827$$ 36.0000i 1.25184i 0.779886 + 0.625921i $$0.215277\pi$$
−0.779886 + 0.625921i $$0.784723\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ − 28.0000i − 0.970725i
$$833$$ 6.00000i 0.207888i
$$834$$ 8.00000 0.277017
$$835$$ 0 0
$$836$$ −2.00000 −0.0691714
$$837$$ 0 0
$$838$$ 26.0000i 0.898155i
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ − 34.0000i − 1.17172i
$$843$$ 28.0000i 0.964371i
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ 12.0000 0.412568
$$847$$ 14.0000i 0.481046i
$$848$$ − 2.00000i − 0.0686803i
$$849$$ 26.0000 0.892318
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 10.0000i − 0.342393i −0.985237 0.171197i $$-0.945237\pi$$
0.985237 0.171197i $$-0.0547634\pi$$
$$854$$ 4.00000 0.136877
$$855$$ 0 0
$$856$$ 36.0000 1.23045
$$857$$ 10.0000i 0.341593i 0.985306 + 0.170797i $$0.0546341\pi$$
−0.985306 + 0.170797i $$0.945366\pi$$
$$858$$ 8.00000i 0.273115i
$$859$$ 12.0000 0.409435 0.204717 0.978821i $$-0.434372\pi$$
0.204717 + 0.978821i $$0.434372\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 4.00000i − 0.136241i
$$863$$ − 40.0000i − 1.36162i −0.732462 0.680808i $$-0.761629\pi$$
0.732462 0.680808i $$-0.238371\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ −36.0000 −1.22333
$$867$$ 13.0000i 0.441503i
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 64.0000 2.16856
$$872$$ − 30.0000i − 1.01593i
$$873$$ 16.0000i 0.541518i
$$874$$ −4.00000 −0.135302
$$875$$ 0 0
$$876$$ −2.00000 −0.0675737
$$877$$ 12.0000i 0.405211i 0.979260 + 0.202606i $$0.0649409\pi$$
−0.979260 + 0.202606i $$0.935059\pi$$
$$878$$ 40.0000i 1.34993i
$$879$$ 26.0000 0.876958
$$880$$ 0 0
$$881$$ −10.0000 −0.336909 −0.168454 0.985709i $$-0.553878\pi$$
−0.168454 + 0.985709i $$0.553878\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ 46.0000i 1.54802i 0.633171 + 0.774012i $$0.281753\pi$$
−0.633171 + 0.774012i $$0.718247\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 0 0
$$886$$ 16.0000 0.537531
$$887$$ − 24.0000i − 0.805841i −0.915235 0.402921i $$-0.867995\pi$$
0.915235 0.402921i $$-0.132005\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ − 24.0000i − 0.803579i
$$893$$ 12.0000i 0.401565i
$$894$$ −18.0000 −0.602010
$$895$$ 0 0
$$896$$ 6.00000 0.200446
$$897$$ − 16.0000i − 0.534224i
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −4.00000 −0.133259
$$902$$ 0 0
$$903$$ 20.0000i 0.665558i
$$904$$ 42.0000 1.39690
$$905$$ 0 0
$$906$$ −24.0000 −0.797347
$$907$$ 52.0000i 1.72663i 0.504664 + 0.863316i $$0.331616\pi$$
−0.504664 + 0.863316i $$0.668384\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ − 1.00000i − 0.0331133i
$$913$$ − 24.0000i − 0.794284i
$$914$$ 14.0000 0.463079
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 28.0000i 0.924641i
$$918$$ 2.00000i 0.0660098i
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 6.00000i 0.197599i
$$923$$ 0 0
$$924$$ −4.00000 −0.131590
$$925$$ 0 0
$$926$$ −22.0000 −0.722965
$$927$$ − 8.00000i − 0.262754i
$$928$$ − 20.0000i − 0.656532i
$$929$$ −10.0000 −0.328089 −0.164045 0.986453i $$-0.552454\pi$$
−0.164045 + 0.986453i $$0.552454\pi$$
$$930$$ 0 0
$$931$$ 3.00000 0.0983210
$$932$$ 18.0000i 0.589610i
$$933$$ − 18.0000i − 0.589294i
$$934$$ −16.0000 −0.523536
$$935$$ 0 0
$$936$$ 12.0000 0.392232
$$937$$ − 42.0000i − 1.37208i −0.727564 0.686040i $$-0.759347\pi$$
0.727564 0.686040i $$-0.240653\pi$$
$$938$$ − 32.0000i − 1.04484i
$$939$$ −14.0000 −0.456873
$$940$$ 0 0
$$941$$ −60.0000 −1.95594 −0.977972 0.208736i $$-0.933065\pi$$
−0.977972 + 0.208736i $$0.933065\pi$$
$$942$$ 18.0000i 0.586472i
$$943$$ 0 0
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 20.0000 0.650256
$$947$$ − 52.0000i − 1.68977i −0.534946 0.844886i $$-0.679668\pi$$
0.534946 0.844886i $$-0.320332\pi$$
$$948$$ 8.00000i 0.259828i
$$949$$ −8.00000 −0.259691
$$950$$ 0 0
$$951$$ −14.0000 −0.453981
$$952$$ 12.0000i 0.388922i
$$953$$ − 46.0000i − 1.49009i −0.667016 0.745043i $$-0.732429\pi$$
0.667016 0.745043i $$-0.267571\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 0 0
$$956$$ 6.00000 0.194054
$$957$$ 8.00000i 0.258603i
$$958$$ − 38.0000i − 1.22772i
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 12.0000i 0.386695i
$$964$$ 26.0000 0.837404
$$965$$ 0 0
$$966$$ −8.00000 −0.257396
$$967$$ 34.0000i 1.09337i 0.837340 + 0.546683i $$0.184110\pi$$
−0.837340 + 0.546683i $$0.815890\pi$$
$$968$$ − 21.0000i − 0.674966i
$$969$$ −2.00000 −0.0642493
$$970$$ 0 0
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 16.0000i 0.512936i
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ 18.0000i 0.575871i 0.957650 + 0.287936i $$0.0929689\pi$$
−0.957650 + 0.287936i $$0.907031\pi$$
$$978$$ − 6.00000i − 0.191859i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ − 6.00000i − 0.191468i
$$983$$ − 16.0000i − 0.510321i −0.966899 0.255160i $$-0.917872\pi$$
0.966899 0.255160i $$-0.0821283\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 8.00000 0.254772
$$987$$ 24.0000i 0.763928i
$$988$$ 4.00000i 0.127257i
$$989$$ −40.0000 −1.27193
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 0 0
$$993$$ − 28.0000i − 0.888553i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ − 2.00000i − 0.0633406i −0.999498 0.0316703i $$-0.989917\pi$$
0.999498 0.0316703i $$-0.0100827\pi$$
$$998$$ − 32.0000i − 1.01294i
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1425.2.c.d.799.2 2
5.2 odd 4 1425.2.a.d.1.1 1
5.3 odd 4 285.2.a.b.1.1 1
5.4 even 2 inner 1425.2.c.d.799.1 2
15.2 even 4 4275.2.a.o.1.1 1
15.8 even 4 855.2.a.b.1.1 1
20.3 even 4 4560.2.a.v.1.1 1
95.18 even 4 5415.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.b.1.1 1 5.3 odd 4
855.2.a.b.1.1 1 15.8 even 4
1425.2.a.d.1.1 1 5.2 odd 4
1425.2.c.d.799.1 2 5.4 even 2 inner
1425.2.c.d.799.2 2 1.1 even 1 trivial
4275.2.a.o.1.1 1 15.2 even 4
4560.2.a.v.1.1 1 20.3 even 4
5415.2.a.c.1.1 1 95.18 even 4