# Properties

 Label 2550.2.d.u.2449.4 Level $2550$ Weight $2$ Character 2550.2449 Analytic conductor $20.362$ 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] = [2550,2,Mod(2449,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, 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("2550.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2550.d (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: $$20.3618525154$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 510) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2449.4 Root $$-1.22474 + 1.22474i$$ of defining polynomial Character $$\chi$$ $$=$$ 2550.2449 Dual form 2550.2.d.u.2449.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.89898i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.89898i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +6.89898i q^{13} -4.89898 q^{14} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +4.89898 q^{21} -4.00000i q^{23} -1.00000 q^{24} -6.89898 q^{26} +1.00000i q^{27} -4.89898i q^{28} -6.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} -1.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} -4.00000i q^{38} +6.89898 q^{39} -2.89898 q^{41} +4.89898i q^{42} +8.89898i q^{43} +4.00000 q^{46} -9.79796i q^{47} -1.00000i q^{48} -17.0000 q^{49} +1.00000 q^{51} -6.89898i q^{52} -7.79796i q^{53} -1.00000 q^{54} +4.89898 q^{56} +4.00000i q^{57} -6.00000i q^{58} -4.89898 q^{59} +11.7980 q^{61} +4.00000i q^{62} -4.89898i q^{63} -1.00000 q^{64} -0.898979i q^{67} -1.00000i q^{68} -4.00000 q^{69} -8.89898 q^{71} +1.00000i q^{72} -10.8990i q^{73} +6.00000 q^{74} +4.00000 q^{76} +6.89898i q^{78} +5.79796 q^{79} +1.00000 q^{81} -2.89898i q^{82} +13.7980i q^{83} -4.89898 q^{84} -8.89898 q^{86} +6.00000i q^{87} +7.79796 q^{89} -33.7980 q^{91} +4.00000i q^{92} -4.00000i q^{93} +9.79796 q^{94} +1.00000 q^{96} +12.6969i q^{97} -17.0000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 4 q^{16} - 16 q^{19} - 4 q^{24} - 8 q^{26} - 24 q^{29} + 16 q^{31} - 4 q^{34} + 4 q^{36} + 8 q^{39} + 8 q^{41} + 16 q^{46} - 68 q^{49} + 4 q^{51} - 4 q^{54} + 8 q^{61} - 4 q^{64} - 16 q^{69} - 16 q^{71} + 24 q^{74} + 16 q^{76} - 16 q^{79} + 4 q^{81} - 16 q^{86} - 8 q^{89} - 96 q^{91} + 4 q^{96}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 4 * q^16 - 16 * q^19 - 4 * q^24 - 8 * q^26 - 24 * q^29 + 16 * q^31 - 4 * q^34 + 4 * q^36 + 8 * q^39 + 8 * q^41 + 16 * q^46 - 68 * q^49 + 4 * q^51 - 4 * q^54 + 8 * q^61 - 4 * q^64 - 16 * q^69 - 16 * q^71 + 24 * q^74 + 16 * q^76 - 16 * q^79 + 4 * q^81 - 16 * q^86 - 8 * q^89 - 96 * q^91 + 4 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$$.

 $$n$$ $$751$$ $$851$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 4.89898i 1.85164i 0.377964 + 0.925820i $$0.376624\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 6.89898i 1.91343i 0.291022 + 0.956716i $$0.406005\pi$$
−0.291022 + 0.956716i $$0.593995\pi$$
$$14$$ −4.89898 −1.30931
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.00000i 0.242536i
$$18$$ − 1.00000i − 0.235702i
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 4.89898 1.06904
$$22$$ 0 0
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −6.89898 −1.35300
$$27$$ 1.00000i 0.192450i
$$28$$ − 4.89898i − 0.925820i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 6.89898 1.10472
$$40$$ 0 0
$$41$$ −2.89898 −0.452745 −0.226372 0.974041i $$-0.572687\pi$$
−0.226372 + 0.974041i $$0.572687\pi$$
$$42$$ 4.89898i 0.755929i
$$43$$ 8.89898i 1.35708i 0.734563 + 0.678541i $$0.237387\pi$$
−0.734563 + 0.678541i $$0.762613\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ − 9.79796i − 1.42918i −0.699544 0.714590i $$-0.746613\pi$$
0.699544 0.714590i $$-0.253387\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −17.0000 −2.42857
$$50$$ 0 0
$$51$$ 1.00000 0.140028
$$52$$ − 6.89898i − 0.956716i
$$53$$ − 7.79796i − 1.07113i −0.844493 0.535566i $$-0.820098\pi$$
0.844493 0.535566i $$-0.179902\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 4.89898 0.654654
$$57$$ 4.00000i 0.529813i
$$58$$ − 6.00000i − 0.787839i
$$59$$ −4.89898 −0.637793 −0.318896 0.947790i $$-0.603312\pi$$
−0.318896 + 0.947790i $$0.603312\pi$$
$$60$$ 0 0
$$61$$ 11.7980 1.51057 0.755287 0.655394i $$-0.227498\pi$$
0.755287 + 0.655394i $$0.227498\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ − 4.89898i − 0.617213i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 0.898979i − 0.109828i −0.998491 0.0549139i $$-0.982512\pi$$
0.998491 0.0549139i $$-0.0174884\pi$$
$$68$$ − 1.00000i − 0.121268i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −8.89898 −1.05611 −0.528057 0.849209i $$-0.677079\pi$$
−0.528057 + 0.849209i $$0.677079\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 10.8990i − 1.27563i −0.770190 0.637815i $$-0.779839\pi$$
0.770190 0.637815i $$-0.220161\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 6.89898i 0.781156i
$$79$$ 5.79796 0.652321 0.326161 0.945314i $$-0.394245\pi$$
0.326161 + 0.945314i $$0.394245\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 2.89898i − 0.320139i
$$83$$ 13.7980i 1.51452i 0.653112 + 0.757261i $$0.273463\pi$$
−0.653112 + 0.757261i $$0.726537\pi$$
$$84$$ −4.89898 −0.534522
$$85$$ 0 0
$$86$$ −8.89898 −0.959602
$$87$$ 6.00000i 0.643268i
$$88$$ 0 0
$$89$$ 7.79796 0.826582 0.413291 0.910599i $$-0.364379\pi$$
0.413291 + 0.910599i $$0.364379\pi$$
$$90$$ 0 0
$$91$$ −33.7980 −3.54299
$$92$$ 4.00000i 0.417029i
$$93$$ − 4.00000i − 0.414781i
$$94$$ 9.79796 1.01058
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 12.6969i 1.28918i 0.764529 + 0.644589i $$0.222972\pi$$
−0.764529 + 0.644589i $$0.777028\pi$$
$$98$$ − 17.0000i − 1.71726i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −18.8990 −1.88052 −0.940259 0.340459i $$-0.889418\pi$$
−0.940259 + 0.340459i $$0.889418\pi$$
$$102$$ 1.00000i 0.0990148i
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 6.89898 0.676501
$$105$$ 0 0
$$106$$ 7.79796 0.757405
$$107$$ − 5.79796i − 0.560510i −0.959926 0.280255i $$-0.909581\pi$$
0.959926 0.280255i $$-0.0904190\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −11.7980 −1.13004 −0.565020 0.825077i $$-0.691131\pi$$
−0.565020 + 0.825077i $$0.691131\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 4.89898i 0.462910i
$$113$$ 7.79796i 0.733570i 0.930306 + 0.366785i $$0.119542\pi$$
−0.930306 + 0.366785i $$0.880458\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ − 6.89898i − 0.637811i
$$118$$ − 4.89898i − 0.450988i
$$119$$ −4.89898 −0.449089
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 11.7980i 1.06814i
$$123$$ 2.89898i 0.261392i
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 4.89898 0.436436
$$127$$ 12.0000i 1.06483i 0.846484 + 0.532414i $$0.178715\pi$$
−0.846484 + 0.532414i $$0.821285\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 8.89898 0.783511
$$130$$ 0 0
$$131$$ 9.79796 0.856052 0.428026 0.903767i $$-0.359209\pi$$
0.428026 + 0.903767i $$0.359209\pi$$
$$132$$ 0 0
$$133$$ − 19.5959i − 1.69918i
$$134$$ 0.898979 0.0776600
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ − 14.0000i − 1.19610i −0.801459 0.598050i $$-0.795942\pi$$
0.801459 0.598050i $$-0.204058\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ −13.7980 −1.17033 −0.585164 0.810915i $$-0.698970\pi$$
−0.585164 + 0.810915i $$0.698970\pi$$
$$140$$ 0 0
$$141$$ −9.79796 −0.825137
$$142$$ − 8.89898i − 0.746786i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 10.8990 0.902006
$$147$$ 17.0000i 1.40214i
$$148$$ 6.00000i 0.493197i
$$149$$ 18.8990 1.54826 0.774132 0.633024i $$-0.218186\pi$$
0.774132 + 0.633024i $$0.218186\pi$$
$$150$$ 0 0
$$151$$ −9.79796 −0.797347 −0.398673 0.917093i $$-0.630529\pi$$
−0.398673 + 0.917093i $$0.630529\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ − 1.00000i − 0.0808452i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −6.89898 −0.552360
$$157$$ 1.10102i 0.0878710i 0.999034 + 0.0439355i $$0.0139896\pi$$
−0.999034 + 0.0439355i $$0.986010\pi$$
$$158$$ 5.79796i 0.461261i
$$159$$ −7.79796 −0.618418
$$160$$ 0 0
$$161$$ 19.5959 1.54437
$$162$$ 1.00000i 0.0785674i
$$163$$ − 2.20204i − 0.172477i −0.996275 0.0862386i $$-0.972515\pi$$
0.996275 0.0862386i $$-0.0274847\pi$$
$$164$$ 2.89898 0.226372
$$165$$ 0 0
$$166$$ −13.7980 −1.07093
$$167$$ − 2.20204i − 0.170399i −0.996364 0.0851995i $$-0.972847\pi$$
0.996364 0.0851995i $$-0.0271528\pi$$
$$168$$ − 4.89898i − 0.377964i
$$169$$ −34.5959 −2.66122
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ − 8.89898i − 0.678541i
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 4.89898i 0.368230i
$$178$$ 7.79796i 0.584482i
$$179$$ 4.89898 0.366167 0.183083 0.983097i $$-0.441392\pi$$
0.183083 + 0.983097i $$0.441392\pi$$
$$180$$ 0 0
$$181$$ −4.20204 −0.312335 −0.156168 0.987731i $$-0.549914\pi$$
−0.156168 + 0.987731i $$0.549914\pi$$
$$182$$ − 33.7980i − 2.50527i
$$183$$ − 11.7980i − 0.872130i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ 0 0
$$188$$ 9.79796i 0.714590i
$$189$$ −4.89898 −0.356348
$$190$$ 0 0
$$191$$ −9.79796 −0.708955 −0.354478 0.935064i $$-0.615341\pi$$
−0.354478 + 0.935064i $$0.615341\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 1.10102i − 0.0792532i −0.999215 0.0396266i $$-0.987383\pi$$
0.999215 0.0396266i $$-0.0126168\pi$$
$$194$$ −12.6969 −0.911587
$$195$$ 0 0
$$196$$ 17.0000 1.21429
$$197$$ 13.5959i 0.968669i 0.874883 + 0.484335i $$0.160938\pi$$
−0.874883 + 0.484335i $$0.839062\pi$$
$$198$$ 0 0
$$199$$ −15.5959 −1.10557 −0.552783 0.833325i $$-0.686434\pi$$
−0.552783 + 0.833325i $$0.686434\pi$$
$$200$$ 0 0
$$201$$ −0.898979 −0.0634091
$$202$$ − 18.8990i − 1.32973i
$$203$$ − 29.3939i − 2.06305i
$$204$$ −1.00000 −0.0700140
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ 4.00000i 0.278019i
$$208$$ 6.89898i 0.478358i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 7.79796i 0.535566i
$$213$$ 8.89898i 0.609748i
$$214$$ 5.79796 0.396340
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 19.5959i 1.33026i
$$218$$ − 11.7980i − 0.799059i
$$219$$ −10.8990 −0.736485
$$220$$ 0 0
$$221$$ −6.89898 −0.464076
$$222$$ − 6.00000i − 0.402694i
$$223$$ − 4.00000i − 0.267860i −0.990991 0.133930i $$-0.957240\pi$$
0.990991 0.133930i $$-0.0427597\pi$$
$$224$$ −4.89898 −0.327327
$$225$$ 0 0
$$226$$ −7.79796 −0.518713
$$227$$ 21.7980i 1.44678i 0.690439 + 0.723391i $$0.257417\pi$$
−0.690439 + 0.723391i $$0.742583\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ 13.5959 0.898444 0.449222 0.893420i $$-0.351701\pi$$
0.449222 + 0.893420i $$0.351701\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000i 0.393919i
$$233$$ 14.0000i 0.917170i 0.888650 + 0.458585i $$0.151644\pi$$
−0.888650 + 0.458585i $$0.848356\pi$$
$$234$$ 6.89898 0.451000
$$235$$ 0 0
$$236$$ 4.89898 0.318896
$$237$$ − 5.79796i − 0.376618i
$$238$$ − 4.89898i − 0.317554i
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 13.5959 0.875790 0.437895 0.899026i $$-0.355724\pi$$
0.437895 + 0.899026i $$0.355724\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −11.7980 −0.755287
$$245$$ 0 0
$$246$$ −2.89898 −0.184832
$$247$$ − 27.5959i − 1.75589i
$$248$$ − 4.00000i − 0.254000i
$$249$$ 13.7980 0.874410
$$250$$ 0 0
$$251$$ 4.89898 0.309221 0.154610 0.987976i $$-0.450588\pi$$
0.154610 + 0.987976i $$0.450588\pi$$
$$252$$ 4.89898i 0.308607i
$$253$$ 0 0
$$254$$ −12.0000 −0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 2.00000i 0.124757i 0.998053 + 0.0623783i $$0.0198685\pi$$
−0.998053 + 0.0623783i $$0.980131\pi$$
$$258$$ 8.89898i 0.554026i
$$259$$ 29.3939 1.82645
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 9.79796i 0.605320i
$$263$$ 8.00000i 0.493301i 0.969104 + 0.246651i $$0.0793300\pi$$
−0.969104 + 0.246651i $$0.920670\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 19.5959 1.20150
$$267$$ − 7.79796i − 0.477227i
$$268$$ 0.898979i 0.0549139i
$$269$$ −9.59592 −0.585073 −0.292537 0.956254i $$-0.594499\pi$$
−0.292537 + 0.956254i $$0.594499\pi$$
$$270$$ 0 0
$$271$$ −1.79796 −0.109218 −0.0546091 0.998508i $$-0.517391\pi$$
−0.0546091 + 0.998508i $$0.517391\pi$$
$$272$$ 1.00000i 0.0606339i
$$273$$ 33.7980i 2.04555i
$$274$$ 14.0000 0.845771
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ − 7.79796i − 0.468534i −0.972172 0.234267i $$-0.924731\pi$$
0.972172 0.234267i $$-0.0752690\pi$$
$$278$$ − 13.7980i − 0.827547i
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 27.7980 1.65829 0.829144 0.559036i $$-0.188829\pi$$
0.829144 + 0.559036i $$0.188829\pi$$
$$282$$ − 9.79796i − 0.583460i
$$283$$ 23.5959i 1.40263i 0.712851 + 0.701316i $$0.247404\pi$$
−0.712851 + 0.701316i $$0.752596\pi$$
$$284$$ 8.89898 0.528057
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 14.2020i − 0.838320i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 12.6969 0.744308
$$292$$ 10.8990i 0.637815i
$$293$$ 13.5959i 0.794282i 0.917758 + 0.397141i $$0.129998\pi$$
−0.917758 + 0.397141i $$0.870002\pi$$
$$294$$ −17.0000 −0.991460
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ 0 0
$$298$$ 18.8990i 1.09479i
$$299$$ 27.5959 1.59591
$$300$$ 0 0
$$301$$ −43.5959 −2.51283
$$302$$ − 9.79796i − 0.563809i
$$303$$ 18.8990i 1.08572i
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ − 0.898979i − 0.0513075i −0.999671 0.0256537i $$-0.991833\pi$$
0.999671 0.0256537i $$-0.00816673\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ −7.10102 −0.402662 −0.201331 0.979523i $$-0.564527\pi$$
−0.201331 + 0.979523i $$0.564527\pi$$
$$312$$ − 6.89898i − 0.390578i
$$313$$ 6.89898i 0.389953i 0.980808 + 0.194977i $$0.0624631\pi$$
−0.980808 + 0.194977i $$0.937537\pi$$
$$314$$ −1.10102 −0.0621342
$$315$$ 0 0
$$316$$ −5.79796 −0.326161
$$317$$ − 14.0000i − 0.786318i −0.919470 0.393159i $$-0.871382\pi$$
0.919470 0.393159i $$-0.128618\pi$$
$$318$$ − 7.79796i − 0.437288i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −5.79796 −0.323611
$$322$$ 19.5959i 1.09204i
$$323$$ − 4.00000i − 0.222566i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 2.20204 0.121960
$$327$$ 11.7980i 0.652429i
$$328$$ 2.89898i 0.160069i
$$329$$ 48.0000 2.64633
$$330$$ 0 0
$$331$$ −5.79796 −0.318685 −0.159342 0.987223i $$-0.550937\pi$$
−0.159342 + 0.987223i $$0.550937\pi$$
$$332$$ − 13.7980i − 0.757261i
$$333$$ 6.00000i 0.328798i
$$334$$ 2.20204 0.120490
$$335$$ 0 0
$$336$$ 4.89898 0.267261
$$337$$ − 32.6969i − 1.78112i −0.454870 0.890558i $$-0.650314\pi$$
0.454870 0.890558i $$-0.349686\pi$$
$$338$$ − 34.5959i − 1.88177i
$$339$$ 7.79796 0.423527
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.00000i 0.216295i
$$343$$ − 48.9898i − 2.64520i
$$344$$ 8.89898 0.479801
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ − 21.7980i − 1.17018i −0.810970 0.585088i $$-0.801060\pi$$
0.810970 0.585088i $$-0.198940\pi$$
$$348$$ − 6.00000i − 0.321634i
$$349$$ −4.20204 −0.224930 −0.112465 0.993656i $$-0.535875\pi$$
−0.112465 + 0.993656i $$0.535875\pi$$
$$350$$ 0 0
$$351$$ −6.89898 −0.368240
$$352$$ 0 0
$$353$$ − 26.0000i − 1.38384i −0.721974 0.691920i $$-0.756765\pi$$
0.721974 0.691920i $$-0.243235\pi$$
$$354$$ −4.89898 −0.260378
$$355$$ 0 0
$$356$$ −7.79796 −0.413291
$$357$$ 4.89898i 0.259281i
$$358$$ 4.89898i 0.258919i
$$359$$ −37.3939 −1.97357 −0.986787 0.162025i $$-0.948198\pi$$
−0.986787 + 0.162025i $$0.948198\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 4.20204i − 0.220854i
$$363$$ 11.0000i 0.577350i
$$364$$ 33.7980 1.77149
$$365$$ 0 0
$$366$$ 11.7980 0.616689
$$367$$ 27.1010i 1.41466i 0.706883 + 0.707331i $$0.250101\pi$$
−0.706883 + 0.707331i $$0.749899\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ 2.89898 0.150915
$$370$$ 0 0
$$371$$ 38.2020 1.98335
$$372$$ 4.00000i 0.207390i
$$373$$ 24.6969i 1.27876i 0.768891 + 0.639380i $$0.220809\pi$$
−0.768891 + 0.639380i $$0.779191\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −9.79796 −0.505291
$$377$$ − 41.3939i − 2.13189i
$$378$$ − 4.89898i − 0.251976i
$$379$$ 29.7980 1.53062 0.765309 0.643663i $$-0.222586\pi$$
0.765309 + 0.643663i $$0.222586\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ − 9.79796i − 0.501307i
$$383$$ 8.00000i 0.408781i 0.978889 + 0.204390i $$0.0655212\pi$$
−0.978889 + 0.204390i $$0.934479\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 1.10102 0.0560405
$$387$$ − 8.89898i − 0.452361i
$$388$$ − 12.6969i − 0.644589i
$$389$$ 38.4949 1.95177 0.975884 0.218288i $$-0.0700472\pi$$
0.975884 + 0.218288i $$0.0700472\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 17.0000i 0.858630i
$$393$$ − 9.79796i − 0.494242i
$$394$$ −13.5959 −0.684952
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 1.59592i − 0.0800968i −0.999198 0.0400484i $$-0.987249\pi$$
0.999198 0.0400484i $$-0.0127512\pi$$
$$398$$ − 15.5959i − 0.781753i
$$399$$ −19.5959 −0.981023
$$400$$ 0 0
$$401$$ 22.8990 1.14352 0.571760 0.820421i $$-0.306261\pi$$
0.571760 + 0.820421i $$0.306261\pi$$
$$402$$ − 0.898979i − 0.0448370i
$$403$$ 27.5959i 1.37465i
$$404$$ 18.8990 0.940259
$$405$$ 0 0
$$406$$ 29.3939 1.45879
$$407$$ 0 0
$$408$$ − 1.00000i − 0.0495074i
$$409$$ 17.5959 0.870062 0.435031 0.900415i $$-0.356737\pi$$
0.435031 + 0.900415i $$0.356737\pi$$
$$410$$ 0 0
$$411$$ −14.0000 −0.690569
$$412$$ − 4.00000i − 0.197066i
$$413$$ − 24.0000i − 1.18096i
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ −6.89898 −0.338250
$$417$$ 13.7980i 0.675689i
$$418$$ 0 0
$$419$$ 17.7980 0.869487 0.434744 0.900554i $$-0.356839\pi$$
0.434744 + 0.900554i $$0.356839\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ 9.79796i 0.476393i
$$424$$ −7.79796 −0.378702
$$425$$ 0 0
$$426$$ −8.89898 −0.431157
$$427$$ 57.7980i 2.79704i
$$428$$ 5.79796i 0.280255i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −23.1010 −1.11274 −0.556369 0.830936i $$-0.687806\pi$$
−0.556369 + 0.830936i $$0.687806\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 35.3939i − 1.70092i −0.526039 0.850461i $$-0.676323\pi$$
0.526039 0.850461i $$-0.323677\pi$$
$$434$$ −19.5959 −0.940634
$$435$$ 0 0
$$436$$ 11.7980 0.565020
$$437$$ 16.0000i 0.765384i
$$438$$ − 10.8990i − 0.520773i
$$439$$ −5.79796 −0.276721 −0.138361 0.990382i $$-0.544183\pi$$
−0.138361 + 0.990382i $$0.544183\pi$$
$$440$$ 0 0
$$441$$ 17.0000 0.809524
$$442$$ − 6.89898i − 0.328151i
$$443$$ 37.7980i 1.79584i 0.440164 + 0.897918i $$0.354920\pi$$
−0.440164 + 0.897918i $$0.645080\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ − 18.8990i − 0.893891i
$$448$$ − 4.89898i − 0.231455i
$$449$$ −6.89898 −0.325583 −0.162791 0.986660i $$-0.552050\pi$$
−0.162791 + 0.986660i $$0.552050\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 7.79796i − 0.366785i
$$453$$ 9.79796i 0.460348i
$$454$$ −21.7980 −1.02303
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ − 16.2020i − 0.757900i −0.925417 0.378950i $$-0.876285\pi$$
0.925417 0.378950i $$-0.123715\pi$$
$$458$$ 13.5959i 0.635296i
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ −28.6969 −1.33655 −0.668275 0.743914i $$-0.732967\pi$$
−0.668275 + 0.743914i $$0.732967\pi$$
$$462$$ 0 0
$$463$$ 7.59592i 0.353012i 0.984300 + 0.176506i $$0.0564795\pi$$
−0.984300 + 0.176506i $$0.943520\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −14.0000 −0.648537
$$467$$ 7.59592i 0.351497i 0.984435 + 0.175749i $$0.0562346\pi$$
−0.984435 + 0.175749i $$0.943765\pi$$
$$468$$ 6.89898i 0.318905i
$$469$$ 4.40408 0.203362
$$470$$ 0 0
$$471$$ 1.10102 0.0507323
$$472$$ 4.89898i 0.225494i
$$473$$ 0 0
$$474$$ 5.79796 0.266309
$$475$$ 0 0
$$476$$ 4.89898 0.224544
$$477$$ 7.79796i 0.357044i
$$478$$ 0 0
$$479$$ 32.8990 1.50319 0.751596 0.659623i $$-0.229284\pi$$
0.751596 + 0.659623i $$0.229284\pi$$
$$480$$ 0 0
$$481$$ 41.3939 1.88740
$$482$$ 13.5959i 0.619277i
$$483$$ − 19.5959i − 0.891645i
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 20.8990i 0.947023i 0.880788 + 0.473512i $$0.157014\pi$$
−0.880788 + 0.473512i $$0.842986\pi$$
$$488$$ − 11.7980i − 0.534069i
$$489$$ −2.20204 −0.0995797
$$490$$ 0 0
$$491$$ −1.30306 −0.0588063 −0.0294032 0.999568i $$-0.509361\pi$$
−0.0294032 + 0.999568i $$0.509361\pi$$
$$492$$ − 2.89898i − 0.130696i
$$493$$ − 6.00000i − 0.270226i
$$494$$ 27.5959 1.24160
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ − 43.5959i − 1.95554i
$$498$$ 13.7980i 0.618301i
$$499$$ 39.5959 1.77256 0.886278 0.463153i $$-0.153282\pi$$
0.886278 + 0.463153i $$0.153282\pi$$
$$500$$ 0 0
$$501$$ −2.20204 −0.0983799
$$502$$ 4.89898i 0.218652i
$$503$$ 4.00000i 0.178351i 0.996016 + 0.0891756i $$0.0284232\pi$$
−0.996016 + 0.0891756i $$0.971577\pi$$
$$504$$ −4.89898 −0.218218
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 34.5959i 1.53646i
$$508$$ − 12.0000i − 0.532414i
$$509$$ 15.3031 0.678296 0.339148 0.940733i $$-0.389861\pi$$
0.339148 + 0.940733i $$0.389861\pi$$
$$510$$ 0 0
$$511$$ 53.3939 2.36201
$$512$$ 1.00000i 0.0441942i
$$513$$ − 4.00000i − 0.176604i
$$514$$ −2.00000 −0.0882162
$$515$$ 0 0
$$516$$ −8.89898 −0.391756
$$517$$ 0 0
$$518$$ 29.3939i 1.29149i
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −40.2929 −1.76526 −0.882631 0.470066i $$-0.844230\pi$$
−0.882631 + 0.470066i $$0.844230\pi$$
$$522$$ 6.00000i 0.262613i
$$523$$ 18.6969i 0.817560i 0.912633 + 0.408780i $$0.134046\pi$$
−0.912633 + 0.408780i $$0.865954\pi$$
$$524$$ −9.79796 −0.428026
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ 4.00000i 0.174243i
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 4.89898 0.212598
$$532$$ 19.5959i 0.849591i
$$533$$ − 20.0000i − 0.866296i
$$534$$ 7.79796 0.337451
$$535$$ 0 0
$$536$$ −0.898979 −0.0388300
$$537$$ − 4.89898i − 0.211407i
$$538$$ − 9.59592i − 0.413709i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −7.79796 −0.335260 −0.167630 0.985850i $$-0.553611\pi$$
−0.167630 + 0.985850i $$0.553611\pi$$
$$542$$ − 1.79796i − 0.0772290i
$$543$$ 4.20204i 0.180327i
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ −33.7980 −1.44642
$$547$$ − 0.404082i − 0.0172773i −0.999963 0.00863865i $$-0.997250\pi$$
0.999963 0.00863865i $$-0.00274980\pi$$
$$548$$ 14.0000i 0.598050i
$$549$$ −11.7980 −0.503525
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 4.00000i 0.170251i
$$553$$ 28.4041i 1.20786i
$$554$$ 7.79796 0.331304
$$555$$ 0 0
$$556$$ 13.7980 0.585164
$$557$$ − 19.7980i − 0.838866i −0.907786 0.419433i $$-0.862229\pi$$
0.907786 0.419433i $$-0.137771\pi$$
$$558$$ − 4.00000i − 0.169334i
$$559$$ −61.3939 −2.59668
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 27.7980i 1.17259i
$$563$$ 21.7980i 0.918674i 0.888262 + 0.459337i $$0.151913\pi$$
−0.888262 + 0.459337i $$0.848087\pi$$
$$564$$ 9.79796 0.412568
$$565$$ 0 0
$$566$$ −23.5959 −0.991810
$$567$$ 4.89898i 0.205738i
$$568$$ 8.89898i 0.373393i
$$569$$ −34.0000 −1.42535 −0.712677 0.701492i $$-0.752517\pi$$
−0.712677 + 0.701492i $$0.752517\pi$$
$$570$$ 0 0
$$571$$ 29.7980 1.24701 0.623503 0.781821i $$-0.285709\pi$$
0.623503 + 0.781821i $$0.285709\pi$$
$$572$$ 0 0
$$573$$ 9.79796i 0.409316i
$$574$$ 14.2020 0.592782
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 27.3939i 1.14042i 0.821498 + 0.570211i $$0.193139\pi$$
−0.821498 + 0.570211i $$0.806861\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ −1.10102 −0.0457569
$$580$$ 0 0
$$581$$ −67.5959 −2.80435
$$582$$ 12.6969i 0.526305i
$$583$$ 0 0
$$584$$ −10.8990 −0.451003
$$585$$ 0 0
$$586$$ −13.5959 −0.561642
$$587$$ 41.3939i 1.70851i 0.519856 + 0.854254i $$0.325986\pi$$
−0.519856 + 0.854254i $$0.674014\pi$$
$$588$$ − 17.0000i − 0.701068i
$$589$$ −16.0000 −0.659269
$$590$$ 0 0
$$591$$ 13.5959 0.559261
$$592$$ − 6.00000i − 0.246598i
$$593$$ 1.59592i 0.0655365i 0.999463 + 0.0327682i $$0.0104323\pi$$
−0.999463 + 0.0327682i $$0.989568\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −18.8990 −0.774132
$$597$$ 15.5959i 0.638298i
$$598$$ 27.5959i 1.12848i
$$599$$ −21.3939 −0.874130 −0.437065 0.899430i $$-0.643982\pi$$
−0.437065 + 0.899430i $$0.643982\pi$$
$$600$$ 0 0
$$601$$ 16.2020 0.660895 0.330448 0.943824i $$-0.392800\pi$$
0.330448 + 0.943824i $$0.392800\pi$$
$$602$$ − 43.5959i − 1.77684i
$$603$$ 0.898979i 0.0366093i
$$604$$ 9.79796 0.398673
$$605$$ 0 0
$$606$$ −18.8990 −0.767719
$$607$$ 32.4949i 1.31893i 0.751736 + 0.659464i $$0.229217\pi$$
−0.751736 + 0.659464i $$0.770783\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ −29.3939 −1.19110
$$610$$ 0 0
$$611$$ 67.5959 2.73464
$$612$$ 1.00000i 0.0404226i
$$613$$ − 14.4949i − 0.585443i −0.956198 0.292722i $$-0.905439\pi$$
0.956198 0.292722i $$-0.0945609\pi$$
$$614$$ 0.898979 0.0362799
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 37.5959i 1.51355i 0.653673 + 0.756777i $$0.273227\pi$$
−0.653673 + 0.756777i $$0.726773\pi$$
$$618$$ 4.00000i 0.160904i
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ − 7.10102i − 0.284725i
$$623$$ 38.2020i 1.53053i
$$624$$ 6.89898 0.276180
$$625$$ 0 0
$$626$$ −6.89898 −0.275739
$$627$$ 0 0
$$628$$ − 1.10102i − 0.0439355i
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ − 5.79796i − 0.230630i
$$633$$ 12.0000i 0.476957i
$$634$$ 14.0000 0.556011
$$635$$ 0 0
$$636$$ 7.79796 0.309209
$$637$$ − 117.283i − 4.64691i
$$638$$ 0 0
$$639$$ 8.89898 0.352038
$$640$$ 0 0
$$641$$ −20.6969 −0.817480 −0.408740 0.912651i $$-0.634032\pi$$
−0.408740 + 0.912651i $$0.634032\pi$$
$$642$$ − 5.79796i − 0.228827i
$$643$$ 29.7980i 1.17512i 0.809182 + 0.587558i $$0.199911\pi$$
−0.809182 + 0.587558i $$0.800089\pi$$
$$644$$ −19.5959 −0.772187
$$645$$ 0 0
$$646$$ 4.00000 0.157378
$$647$$ 32.0000i 1.25805i 0.777385 + 0.629025i $$0.216546\pi$$
−0.777385 + 0.629025i $$0.783454\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 19.5959 0.768025
$$652$$ 2.20204i 0.0862386i
$$653$$ 30.0000i 1.17399i 0.809590 + 0.586995i $$0.199689\pi$$
−0.809590 + 0.586995i $$0.800311\pi$$
$$654$$ −11.7980 −0.461337
$$655$$ 0 0
$$656$$ −2.89898 −0.113186
$$657$$ 10.8990i 0.425210i
$$658$$ 48.0000i 1.87123i
$$659$$ −30.6969 −1.19578 −0.597891 0.801577i $$-0.703995\pi$$
−0.597891 + 0.801577i $$0.703995\pi$$
$$660$$ 0 0
$$661$$ −19.7980 −0.770051 −0.385026 0.922906i $$-0.625807\pi$$
−0.385026 + 0.922906i $$0.625807\pi$$
$$662$$ − 5.79796i − 0.225344i
$$663$$ 6.89898i 0.267934i
$$664$$ 13.7980 0.535465
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 24.0000i 0.929284i
$$668$$ 2.20204i 0.0851995i
$$669$$ −4.00000 −0.154649
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 4.89898i 0.188982i
$$673$$ − 33.1010i − 1.27595i −0.770057 0.637975i $$-0.779772\pi$$
0.770057 0.637975i $$-0.220228\pi$$
$$674$$ 32.6969 1.25944
$$675$$ 0 0
$$676$$ 34.5959 1.33061
$$677$$ 13.5959i 0.522534i 0.965267 + 0.261267i $$0.0841402\pi$$
−0.965267 + 0.261267i $$0.915860\pi$$
$$678$$ 7.79796i 0.299479i
$$679$$ −62.2020 −2.38710
$$680$$ 0 0
$$681$$ 21.7980 0.835300
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 48.9898 1.87044
$$687$$ − 13.5959i − 0.518717i
$$688$$ 8.89898i 0.339270i
$$689$$ 53.7980 2.04954
$$690$$ 0 0
$$691$$ −27.1918 −1.03443 −0.517213 0.855857i $$-0.673031\pi$$
−0.517213 + 0.855857i $$0.673031\pi$$
$$692$$ − 6.00000i − 0.228086i
$$693$$ 0 0
$$694$$ 21.7980 0.827439
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ − 2.89898i − 0.109807i
$$698$$ − 4.20204i − 0.159050i
$$699$$ 14.0000 0.529529
$$700$$ 0 0
$$701$$ 14.8990 0.562727 0.281363 0.959601i $$-0.409213\pi$$
0.281363 + 0.959601i $$0.409213\pi$$
$$702$$ − 6.89898i − 0.260385i
$$703$$ 24.0000i 0.905177i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 26.0000 0.978523
$$707$$ − 92.5857i − 3.48204i
$$708$$ − 4.89898i − 0.184115i
$$709$$ 31.7980 1.19420 0.597099 0.802168i $$-0.296320\pi$$
0.597099 + 0.802168i $$0.296320\pi$$
$$710$$ 0 0
$$711$$ −5.79796 −0.217440
$$712$$ − 7.79796i − 0.292241i
$$713$$ − 16.0000i − 0.599205i
$$714$$ −4.89898 −0.183340
$$715$$ 0 0
$$716$$ −4.89898 −0.183083
$$717$$ 0 0
$$718$$ − 37.3939i − 1.39553i
$$719$$ 12.4949 0.465981 0.232991 0.972479i $$-0.425149\pi$$
0.232991 + 0.972479i $$0.425149\pi$$
$$720$$ 0 0
$$721$$ −19.5959 −0.729790
$$722$$ − 3.00000i − 0.111648i
$$723$$ − 13.5959i − 0.505638i
$$724$$ 4.20204 0.156168
$$725$$ 0 0
$$726$$ −11.0000 −0.408248
$$727$$ − 0.404082i − 0.0149866i −0.999972 0.00749329i $$-0.997615\pi$$
0.999972 0.00749329i $$-0.00238521\pi$$
$$728$$ 33.7980i 1.25264i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −8.89898 −0.329141
$$732$$ 11.7980i 0.436065i
$$733$$ − 46.4949i − 1.71733i −0.512539 0.858664i $$-0.671295\pi$$
0.512539 0.858664i $$-0.328705\pi$$
$$734$$ −27.1010 −1.00032
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 0 0
$$738$$ 2.89898i 0.106713i
$$739$$ 9.39388 0.345559 0.172780 0.984960i $$-0.444725\pi$$
0.172780 + 0.984960i $$0.444725\pi$$
$$740$$ 0 0
$$741$$ −27.5959 −1.01376
$$742$$ 38.2020i 1.40244i
$$743$$ − 9.39388i − 0.344628i −0.985042 0.172314i $$-0.944876\pi$$
0.985042 0.172314i $$-0.0551244\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 0 0
$$746$$ −24.6969 −0.904219
$$747$$ − 13.7980i − 0.504841i
$$748$$ 0 0
$$749$$ 28.4041 1.03786
$$750$$ 0 0
$$751$$ 21.7980 0.795419 0.397709 0.917511i $$-0.369805\pi$$
0.397709 + 0.917511i $$0.369805\pi$$
$$752$$ − 9.79796i − 0.357295i
$$753$$ − 4.89898i − 0.178529i
$$754$$ 41.3939 1.50748
$$755$$ 0 0
$$756$$ 4.89898 0.178174
$$757$$ 4.69694i 0.170713i 0.996350 + 0.0853566i $$0.0272029\pi$$
−0.996350 + 0.0853566i $$0.972797\pi$$
$$758$$ 29.7980i 1.08231i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −1.59592 −0.0578520 −0.0289260 0.999582i $$-0.509209\pi$$
−0.0289260 + 0.999582i $$0.509209\pi$$
$$762$$ 12.0000i 0.434714i
$$763$$ − 57.7980i − 2.09243i
$$764$$ 9.79796 0.354478
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ − 33.7980i − 1.22037i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 0 0
$$771$$ 2.00000 0.0720282
$$772$$ 1.10102i 0.0396266i
$$773$$ − 35.3939i − 1.27303i −0.771265 0.636515i $$-0.780375\pi$$
0.771265 0.636515i $$-0.219625\pi$$
$$774$$ 8.89898 0.319867
$$775$$ 0 0
$$776$$ 12.6969 0.455794
$$777$$ − 29.3939i − 1.05450i
$$778$$ 38.4949i 1.38011i
$$779$$ 11.5959 0.415467
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 4.00000i 0.143040i
$$783$$ − 6.00000i − 0.214423i
$$784$$ −17.0000 −0.607143
$$785$$ 0 0
$$786$$ 9.79796 0.349482
$$787$$ 45.7980i 1.63252i 0.577684 + 0.816260i $$0.303957\pi$$
−0.577684 + 0.816260i $$0.696043\pi$$
$$788$$ − 13.5959i − 0.484335i
$$789$$ 8.00000 0.284808
$$790$$ 0 0
$$791$$ −38.2020 −1.35831
$$792$$ 0 0
$$793$$ 81.3939i 2.89038i
$$794$$ 1.59592 0.0566370
$$795$$ 0 0
$$796$$ 15.5959 0.552783
$$797$$ − 26.0000i − 0.920967i −0.887668 0.460484i $$-0.847676\pi$$
0.887668 0.460484i $$-0.152324\pi$$
$$798$$ − 19.5959i − 0.693688i
$$799$$ 9.79796 0.346627
$$800$$ 0 0
$$801$$ −7.79796 −0.275527
$$802$$ 22.8990i 0.808591i
$$803$$ 0 0
$$804$$ 0.898979 0.0317046
$$805$$ 0 0
$$806$$ −27.5959 −0.972025
$$807$$ 9.59592i 0.337792i
$$808$$ 18.8990i 0.664864i
$$809$$ −32.6969 −1.14956 −0.574782 0.818307i $$-0.694913\pi$$
−0.574782 + 0.818307i $$0.694913\pi$$
$$810$$ 0 0
$$811$$ 25.3939 0.891700 0.445850 0.895108i $$-0.352902\pi$$
0.445850 + 0.895108i $$0.352902\pi$$
$$812$$ 29.3939i 1.03152i
$$813$$ 1.79796i 0.0630572i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 1.00000 0.0350070
$$817$$ − 35.5959i − 1.24534i
$$818$$ 17.5959i 0.615227i
$$819$$ 33.7980 1.18100
$$820$$ 0 0
$$821$$ 15.7980 0.551353 0.275676 0.961251i $$-0.411098\pi$$
0.275676 + 0.961251i $$0.411098\pi$$
$$822$$ − 14.0000i − 0.488306i
$$823$$ 20.8990i 0.728493i 0.931303 + 0.364246i $$0.118673\pi$$
−0.931303 + 0.364246i $$0.881327\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ 4.00000i 0.139094i 0.997579 + 0.0695468i $$0.0221553\pi$$
−0.997579 + 0.0695468i $$0.977845\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ −3.39388 −0.117874 −0.0589371 0.998262i $$-0.518771\pi$$
−0.0589371 + 0.998262i $$0.518771\pi$$
$$830$$ 0 0
$$831$$ −7.79796 −0.270508
$$832$$ − 6.89898i − 0.239179i
$$833$$ − 17.0000i − 0.589015i
$$834$$ −13.7980 −0.477784
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 4.00000i 0.138260i
$$838$$ 17.7980i 0.614820i
$$839$$ 7.10102 0.245154 0.122577 0.992459i $$-0.460884\pi$$
0.122577 + 0.992459i $$0.460884\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 14.0000i 0.482472i
$$843$$ − 27.7980i − 0.957413i
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ −9.79796 −0.336861
$$847$$ − 53.8888i − 1.85164i
$$848$$ − 7.79796i − 0.267783i
$$849$$ 23.5959 0.809810
$$850$$ 0 0
$$851$$ −24.0000 −0.822709
$$852$$ − 8.89898i − 0.304874i
$$853$$ 11.3939i 0.390119i 0.980791 + 0.195059i $$0.0624900\pi$$
−0.980791 + 0.195059i $$0.937510\pi$$
$$854$$ −57.7980 −1.97781
$$855$$ 0 0
$$856$$ −5.79796 −0.198170
$$857$$ 10.0000i 0.341593i 0.985306 + 0.170797i $$0.0546341\pi$$
−0.985306 + 0.170797i $$0.945366\pi$$
$$858$$ 0 0
$$859$$ −5.79796 −0.197824 −0.0989119 0.995096i $$-0.531536\pi$$
−0.0989119 + 0.995096i $$0.531536\pi$$
$$860$$ 0 0
$$861$$ −14.2020 −0.484004
$$862$$ − 23.1010i − 0.786824i
$$863$$ − 45.3939i − 1.54523i −0.634878 0.772613i $$-0.718949\pi$$
0.634878 0.772613i $$-0.281051\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 35.3939 1.20273
$$867$$ 1.00000i 0.0339618i
$$868$$ − 19.5959i − 0.665129i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 6.20204 0.210148
$$872$$ 11.7980i 0.399529i
$$873$$ − 12.6969i − 0.429726i
$$874$$ −16.0000 −0.541208
$$875$$ 0 0
$$876$$ 10.8990 0.368242
$$877$$ 31.3939i 1.06010i 0.847968 + 0.530048i $$0.177826\pi$$
−0.847968 + 0.530048i $$0.822174\pi$$
$$878$$ − 5.79796i − 0.195672i
$$879$$ 13.5959 0.458579
$$880$$ 0 0
$$881$$ 34.4949 1.16216 0.581081 0.813846i $$-0.302630\pi$$
0.581081 + 0.813846i $$0.302630\pi$$
$$882$$ 17.0000i 0.572420i
$$883$$ 26.6969i 0.898424i 0.893425 + 0.449212i $$0.148295\pi$$
−0.893425 + 0.449212i $$0.851705\pi$$
$$884$$ 6.89898 0.232038
$$885$$ 0 0
$$886$$ −37.7980 −1.26985
$$887$$ 12.0000i 0.402921i 0.979497 + 0.201460i $$0.0645687\pi$$
−0.979497 + 0.201460i $$0.935431\pi$$
$$888$$ 6.00000i 0.201347i
$$889$$ −58.7878 −1.97168
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 4.00000i 0.133930i
$$893$$ 39.1918i 1.31150i
$$894$$ 18.8990 0.632076
$$895$$ 0 0
$$896$$ 4.89898 0.163663
$$897$$ − 27.5959i − 0.921401i
$$898$$ − 6.89898i − 0.230222i
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 7.79796 0.259788
$$902$$ 0 0
$$903$$ 43.5959i 1.45078i
$$904$$ 7.79796 0.259356
$$905$$ 0 0
$$906$$ −9.79796 −0.325515
$$907$$ 33.3939i 1.10883i 0.832242 + 0.554413i $$0.187057\pi$$
−0.832242 + 0.554413i $$0.812943\pi$$
$$908$$ − 21.7980i − 0.723391i
$$909$$ 18.8990 0.626840
$$910$$ 0 0
$$911$$ 23.1010 0.765371 0.382685 0.923879i $$-0.374999\pi$$
0.382685 + 0.923879i $$0.374999\pi$$
$$912$$ 4.00000i 0.132453i
$$913$$ 0 0
$$914$$ 16.2020 0.535916
$$915$$ 0 0
$$916$$ −13.5959 −0.449222
$$917$$ 48.0000i 1.58510i
$$918$$ − 1.00000i − 0.0330049i
$$919$$ −1.79796 −0.0593092 −0.0296546 0.999560i $$-0.509441\pi$$
−0.0296546 + 0.999560i $$0.509441\pi$$
$$920$$ 0 0
$$921$$ −0.898979 −0.0296224
$$922$$ − 28.6969i − 0.945083i
$$923$$ − 61.3939i − 2.02080i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −7.59592 −0.249617
$$927$$ − 4.00000i − 0.131377i
$$928$$ − 6.00000i − 0.196960i
$$929$$ −36.2929 −1.19073 −0.595365 0.803455i $$-0.702993\pi$$
−0.595365 + 0.803455i $$0.702993\pi$$
$$930$$ 0 0
$$931$$ 68.0000 2.22861
$$932$$ − 14.0000i − 0.458585i
$$933$$ 7.10102i 0.232477i
$$934$$ −7.59592 −0.248546
$$935$$ 0 0
$$936$$ −6.89898 −0.225500
$$937$$ − 39.3939i − 1.28694i −0.765471 0.643471i $$-0.777494\pi$$
0.765471 0.643471i $$-0.222506\pi$$
$$938$$ 4.40408i 0.143798i
$$939$$ 6.89898 0.225140
$$940$$ 0 0
$$941$$ −16.2020 −0.528171 −0.264086 0.964499i $$-0.585070\pi$$
−0.264086 + 0.964499i $$0.585070\pi$$
$$942$$ 1.10102i 0.0358732i
$$943$$ 11.5959i 0.377615i
$$944$$ −4.89898 −0.159448
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 21.7980i − 0.708338i −0.935181 0.354169i $$-0.884764\pi$$
0.935181 0.354169i $$-0.115236\pi$$
$$948$$ 5.79796i 0.188309i
$$949$$ 75.1918 2.44083
$$950$$ 0 0
$$951$$ −14.0000 −0.453981
$$952$$ 4.89898i 0.158777i
$$953$$ − 41.1918i − 1.33433i −0.744908 0.667167i $$-0.767507\pi$$
0.744908 0.667167i $$-0.232493\pi$$
$$954$$ −7.79796 −0.252468
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 32.8990i 1.06292i
$$959$$ 68.5857 2.21475
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 41.3939i 1.33459i
$$963$$ 5.79796i 0.186837i
$$964$$ −13.5959 −0.437895
$$965$$ 0 0
$$966$$ 19.5959 0.630488
$$967$$ 41.3939i 1.33114i 0.746337 + 0.665569i $$0.231811\pi$$
−0.746337 + 0.665569i $$0.768189\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ −4.00000 −0.128499
$$970$$ 0 0
$$971$$ 38.6969 1.24184 0.620922 0.783872i $$-0.286758\pi$$
0.620922 + 0.783872i $$0.286758\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 67.5959i − 2.16703i
$$974$$ −20.8990 −0.669646
$$975$$ 0 0
$$976$$ 11.7980 0.377643
$$977$$ 29.5959i 0.946857i 0.880832 + 0.473429i $$0.156984\pi$$
−0.880832 + 0.473429i $$0.843016\pi$$
$$978$$ − 2.20204i − 0.0704135i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 11.7980 0.376680
$$982$$ − 1.30306i − 0.0415824i
$$983$$ 9.39388i 0.299618i 0.988715 + 0.149809i $$0.0478659\pi$$
−0.988715 + 0.149809i $$0.952134\pi$$
$$984$$ 2.89898 0.0924161
$$985$$ 0 0
$$986$$ 6.00000 0.191079
$$987$$ − 48.0000i − 1.52786i
$$988$$ 27.5959i 0.877943i
$$989$$ 35.5959 1.13188
$$990$$ 0 0
$$991$$ −15.5959 −0.495421 −0.247710 0.968834i $$-0.579678\pi$$
−0.247710 + 0.968834i $$0.579678\pi$$
$$992$$ 4.00000i 0.127000i
$$993$$ 5.79796i 0.183993i
$$994$$ 43.5959 1.38278
$$995$$ 0 0
$$996$$ −13.7980 −0.437205
$$997$$ 21.5959i 0.683950i 0.939709 + 0.341975i $$0.111096\pi$$
−0.939709 + 0.341975i $$0.888904\pi$$
$$998$$ 39.5959i 1.25339i
$$999$$ 6.00000 0.189832
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 2550.2.d.u.2449.4 4
5.2 odd 4 510.2.a.h.1.1 2
5.3 odd 4 2550.2.a.bl.1.2 2
5.4 even 2 inner 2550.2.d.u.2449.1 4
15.2 even 4 1530.2.a.s.1.1 2
15.8 even 4 7650.2.a.cu.1.2 2
20.7 even 4 4080.2.a.bq.1.2 2
85.67 odd 4 8670.2.a.be.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.h.1.1 2 5.2 odd 4
1530.2.a.s.1.1 2 15.2 even 4
2550.2.a.bl.1.2 2 5.3 odd 4
2550.2.d.u.2449.1 4 5.4 even 2 inner
2550.2.d.u.2449.4 4 1.1 even 1 trivial
4080.2.a.bq.1.2 2 20.7 even 4
7650.2.a.cu.1.2 2 15.8 even 4
8670.2.a.be.1.2 2 85.67 odd 4