# Properties

 Label 2550.2.d.b.2449.1 Level $2550$ Weight $2$ Character 2550.2449 Analytic conductor $20.362$ 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] = [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: $$2$$ 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 510) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2449.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2550.2449 Dual form 2550.2.d.b.2449.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +4.00000 q^{21} +4.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -4.00000i q^{28} -2.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} -2.00000 q^{39} +2.00000 q^{41} -4.00000i q^{42} -12.0000i q^{43} +4.00000 q^{44} -4.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -1.00000 q^{51} +2.00000i q^{52} -2.00000i q^{53} +1.00000 q^{54} -4.00000 q^{56} -4.00000i q^{57} +2.00000i q^{58} -12.0000 q^{59} +2.00000 q^{61} -4.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -4.00000i q^{67} +1.00000i q^{68} -4.00000 q^{69} -4.00000 q^{71} -1.00000i q^{72} -14.0000i q^{73} +6.00000 q^{74} -4.00000 q^{76} -16.0000i q^{77} +2.00000i q^{78} +12.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} -4.00000i q^{83} -4.00000 q^{84} -12.0000 q^{86} +2.00000i q^{87} -4.00000i q^{88} -10.0000 q^{89} +8.00000 q^{91} +4.00000i q^{92} -4.00000i q^{93} -8.00000 q^{94} -1.00000 q^{96} -18.0000i q^{97} +9.00000i q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} + 8 q^{14} + 2 q^{16} + 8 q^{19} + 8 q^{21} + 2 q^{24} - 4 q^{26} - 4 q^{29} + 8 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} + 4 q^{41} + 8 q^{44} - 8 q^{46} - 18 q^{49} - 2 q^{51} + 2 q^{54} - 8 q^{56} - 24 q^{59} + 4 q^{61} - 2 q^{64} + 8 q^{66} - 8 q^{69} - 8 q^{71} + 12 q^{74} - 8 q^{76} + 24 q^{79} + 2 q^{81} - 8 q^{84} - 24 q^{86} - 20 q^{89} + 16 q^{91} - 16 q^{94} - 2 q^{96} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^11 + 8 * q^14 + 2 * q^16 + 8 * q^19 + 8 * q^21 + 2 * q^24 - 4 * q^26 - 4 * q^29 + 8 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 + 4 * q^41 + 8 * q^44 - 8 * q^46 - 18 * q^49 - 2 * q^51 + 2 * q^54 - 8 * q^56 - 24 * q^59 + 4 * q^61 - 2 * q^64 + 8 * q^66 - 8 * q^69 - 8 * q^71 + 12 * q^74 - 8 * q^76 + 24 * q^79 + 2 * q^81 - 8 * q^84 - 24 * q^86 - 20 * q^89 + 16 * q^91 - 16 * q^94 - 2 * q^96 + 8 * q^99

## 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
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 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 $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ 4.00000i 0.852803i
$$23$$ − 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
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ − 4.00000i − 0.755929i
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 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
$$33$$ 4.00000i 0.696311i
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ − 4.00000i − 0.617213i
$$43$$ − 12.0000i − 1.82998i −0.403473 0.914991i $$-0.632197\pi$$
0.403473 0.914991i $$-0.367803\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ − 8.00000i − 1.16692i −0.812142 0.583460i $$-0.801699\pi$$
0.812142 0.583460i $$-0.198301\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 2.00000i 0.277350i
$$53$$ − 2.00000i − 0.274721i −0.990521 0.137361i $$-0.956138\pi$$
0.990521 0.137361i $$-0.0438619\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −4.00000 −0.534522
$$57$$ − 4.00000i − 0.529813i
$$58$$ 2.00000i 0.262613i
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ − 4.00000i − 0.508001i
$$63$$ − 4.00000i − 0.503953i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 1.00000i 0.121268i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 14.0000i − 1.63858i −0.573382 0.819288i $$-0.694369\pi$$
0.573382 0.819288i $$-0.305631\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ − 16.0000i − 1.82337i
$$78$$ 2.00000i 0.226455i
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 2.00000i − 0.220863i
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 0 0
$$86$$ −12.0000 −1.29399
$$87$$ 2.00000i 0.214423i
$$88$$ − 4.00000i − 0.426401i
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ 4.00000i 0.417029i
$$93$$ − 4.00000i − 0.414781i
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ − 18.0000i − 1.82762i −0.406138 0.913812i $$-0.633125\pi$$
0.406138 0.913812i $$-0.366875\pi$$
$$98$$ 9.00000i 0.909137i
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 1.00000i 0.0990148i
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 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
$$111$$ 6.00000 0.569495
$$112$$ 4.00000i 0.377964i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ 2.00000i 0.184900i
$$118$$ 12.0000i 1.10469i
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 2.00000i − 0.181071i
$$123$$ − 2.00000i − 0.180334i
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ −4.00000 −0.356348
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −12.0000 −1.05654
$$130$$ 0 0
$$131$$ 20.0000 1.74741 0.873704 0.486458i $$-0.161711\pi$$
0.873704 + 0.486458i $$0.161711\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ 16.0000i 1.38738i
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 4.00000i 0.340503i
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 4.00000i 0.335673i
$$143$$ 8.00000i 0.668994i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −14.0000 −1.15865
$$147$$ 9.00000i 0.742307i
$$148$$ − 6.00000i − 0.493197i
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ 1.00000i 0.0808452i
$$154$$ −16.0000 −1.28932
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 10.0000i 0.798087i 0.916932 + 0.399043i $$0.130658\pi$$
−0.916932 + 0.399043i $$0.869342\pi$$
$$158$$ − 12.0000i − 0.954669i
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 20.0000i − 1.56652i −0.621694 0.783260i $$-0.713555\pi$$
0.621694 0.783260i $$-0.286445\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ − 12.0000i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ 4.00000i 0.308607i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 12.0000i 0.914991i
$$173$$ − 14.0000i − 1.06440i −0.846619 0.532200i $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 12.0000i 0.901975i
$$178$$ 10.0000i 0.749532i
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ − 8.00000i − 0.592999i
$$183$$ − 2.00000i − 0.147844i
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ 4.00000i 0.292509i
$$188$$ 8.00000i 0.583460i
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 18.0000i 1.29567i 0.761781 + 0.647834i $$0.224325\pi$$
−0.761781 + 0.647834i $$0.775675\pi$$
$$194$$ −18.0000 −1.29232
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 14.0000i 0.997459i 0.866758 + 0.498729i $$0.166200\pi$$
−0.866758 + 0.498729i $$0.833800\pi$$
$$198$$ − 4.00000i − 0.284268i
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 10.0000i 0.703598i
$$203$$ − 8.00000i − 0.561490i
$$204$$ 1.00000 0.0700140
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.00000i 0.278019i
$$208$$ − 2.00000i − 0.138675i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 2.00000i 0.137361i
$$213$$ 4.00000i 0.274075i
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 16.0000i 1.08615i
$$218$$ 10.0000i 0.677285i
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ − 6.00000i − 0.402694i
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ −16.0000 −1.05272
$$232$$ − 2.00000i − 0.131306i
$$233$$ 18.0000i 1.17922i 0.807688 + 0.589610i $$0.200718\pi$$
−0.807688 + 0.589610i $$0.799282\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ − 12.0000i − 0.779484i
$$238$$ − 4.00000i − 0.259281i
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ − 8.00000i − 0.509028i
$$248$$ 4.00000i 0.254000i
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ −28.0000 −1.76734 −0.883672 0.468106i $$-0.844936\pi$$
−0.883672 + 0.468106i $$0.844936\pi$$
$$252$$ 4.00000i 0.251976i
$$253$$ 16.0000i 1.00591i
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ 12.0000i 0.747087i
$$259$$ −24.0000 −1.49129
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ − 20.0000i − 1.23560i
$$263$$ − 16.0000i − 0.986602i −0.869859 0.493301i $$-0.835790\pi$$
0.869859 0.493301i $$-0.164210\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ 16.0000 0.981023
$$267$$ 10.0000i 0.611990i
$$268$$ 4.00000i 0.244339i
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ − 1.00000i − 0.0606339i
$$273$$ − 8.00000i − 0.484182i
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ − 20.0000i − 1.19952i
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 8.00000i 0.476393i
$$283$$ − 20.0000i − 1.18888i −0.804141 0.594438i $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ 4.00000 0.237356
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 8.00000i 0.472225i
$$288$$ 1.00000i 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −18.0000 −1.05518
$$292$$ 14.0000i 0.819288i
$$293$$ − 26.0000i − 1.51894i −0.650545 0.759468i $$-0.725459\pi$$
0.650545 0.759468i $$-0.274541\pi$$
$$294$$ 9.00000 0.524891
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ − 4.00000i − 0.232104i
$$298$$ − 2.00000i − 0.115857i
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 48.0000 2.76667
$$302$$ 8.00000i 0.460348i
$$303$$ 10.0000i 0.574485i
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ 4.00000i 0.228292i 0.993464 + 0.114146i $$0.0364132\pi$$
−0.993464 + 0.114146i $$0.963587\pi$$
$$308$$ 16.0000i 0.911685i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 20.0000 1.13410 0.567048 0.823685i $$-0.308085\pi$$
0.567048 + 0.823685i $$0.308085\pi$$
$$312$$ − 2.00000i − 0.113228i
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ − 10.0000i − 0.561656i −0.959758 0.280828i $$-0.909391\pi$$
0.959758 0.280828i $$-0.0906090\pi$$
$$318$$ 2.00000i 0.112154i
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ − 16.0000i − 0.891645i
$$323$$ − 4.00000i − 0.222566i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ 10.0000i 0.553001i
$$328$$ 2.00000i 0.110432i
$$329$$ 32.0000 1.76422
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 4.00000i 0.219529i
$$333$$ − 6.00000i − 0.328798i
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 4.00000 0.218218
$$337$$ 22.0000i 1.19842i 0.800593 + 0.599208i $$0.204518\pi$$
−0.800593 + 0.599208i $$0.795482\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 4.00000i 0.216295i
$$343$$ − 8.00000i − 0.431959i
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ −14.0000 −0.752645
$$347$$ − 20.0000i − 1.07366i −0.843692 0.536828i $$-0.819622\pi$$
0.843692 0.536828i $$-0.180378\pi$$
$$348$$ − 2.00000i − 0.107211i
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 4.00000i 0.213201i
$$353$$ 2.00000i 0.106449i 0.998583 + 0.0532246i $$0.0169499\pi$$
−0.998583 + 0.0532246i $$0.983050\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ − 4.00000i − 0.211702i
$$358$$ 12.0000i 0.634220i
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 22.0000i 1.15629i
$$363$$ − 5.00000i − 0.262432i
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ − 28.0000i − 1.46159i −0.682598 0.730794i $$-0.739150\pi$$
0.682598 0.730794i $$-0.260850\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ 8.00000 0.415339
$$372$$ 4.00000i 0.207390i
$$373$$ 38.0000i 1.96757i 0.179364 + 0.983783i $$0.442596\pi$$
−0.179364 + 0.983783i $$0.557404\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 4.00000i 0.206010i
$$378$$ 4.00000i 0.205738i
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ 8.00000i 0.409316i
$$383$$ 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
$$386$$ 18.0000 0.916176
$$387$$ 12.0000i 0.609994i
$$388$$ 18.0000i 0.913812i
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ − 9.00000i − 0.454569i
$$393$$ − 20.0000i − 1.00887i
$$394$$ 14.0000 0.705310
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ 30.0000i 1.50566i 0.658217 + 0.752828i $$0.271311\pi$$
−0.658217 + 0.752828i $$0.728689\pi$$
$$398$$ 4.00000i 0.200502i
$$399$$ 16.0000 0.801002
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 4.00000i 0.199502i
$$403$$ − 8.00000i − 0.398508i
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ − 24.0000i − 1.18964i
$$408$$ − 1.00000i − 0.0495074i
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ 0 0
$$413$$ − 48.0000i − 2.36193i
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ − 20.0000i − 0.979404i
$$418$$ 16.0000i 0.782586i
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −18.0000 −0.877266 −0.438633 0.898666i $$-0.644537\pi$$
−0.438633 + 0.898666i $$0.644537\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ 8.00000i 0.388973i
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ 4.00000 0.193801
$$427$$ 8.00000i 0.387147i
$$428$$ 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
$$433$$ − 14.0000i − 0.672797i −0.941720 0.336399i $$-0.890791\pi$$
0.941720 0.336399i $$-0.109209\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ − 16.0000i − 0.765384i
$$438$$ 14.0000i 0.668946i
$$439$$ 12.0000 0.572729 0.286364 0.958121i $$-0.407553\pi$$
0.286364 + 0.958121i $$0.407553\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 2.00000i 0.0951303i
$$443$$ − 36.0000i − 1.71041i −0.518289 0.855206i $$-0.673431\pi$$
0.518289 0.855206i $$-0.326569\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ − 2.00000i − 0.0945968i
$$448$$ − 4.00000i − 0.188982i
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ 6.00000i 0.282216i
$$453$$ 8.00000i 0.375873i
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ − 10.0000i − 0.467780i −0.972263 0.233890i $$-0.924854\pi$$
0.972263 0.233890i $$-0.0751456\pi$$
$$458$$ 6.00000i 0.280362i
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ 14.0000 0.652045 0.326023 0.945362i $$-0.394291\pi$$
0.326023 + 0.945362i $$0.394291\pi$$
$$462$$ 16.0000i 0.744387i
$$463$$ 40.0000i 1.85896i 0.368875 + 0.929479i $$0.379743\pi$$
−0.368875 + 0.929479i $$0.620257\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ − 36.0000i − 1.66588i −0.553362 0.832941i $$-0.686655\pi$$
0.553362 0.832941i $$-0.313345\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ − 12.0000i − 0.552345i
$$473$$ 48.0000i 2.20704i
$$474$$ −12.0000 −0.551178
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ 2.00000i 0.0915737i
$$478$$ − 24.0000i − 1.09773i
$$479$$ −20.0000 −0.913823 −0.456912 0.889512i $$-0.651044\pi$$
−0.456912 + 0.889512i $$0.651044\pi$$
$$480$$ 0 0
$$481$$ 12.0000 0.547153
$$482$$ − 10.0000i − 0.455488i
$$483$$ − 16.0000i − 0.728025i
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 20.0000i − 0.906287i −0.891438 0.453143i $$-0.850303\pi$$
0.891438 0.453143i $$-0.149697\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ −20.0000 −0.904431
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 2.00000i 0.0901670i
$$493$$ 2.00000i 0.0900755i
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ − 16.0000i − 0.717698i
$$498$$ 4.00000i 0.179244i
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 28.0000i 1.24970i
$$503$$ 44.0000i 1.96186i 0.194354 + 0.980932i $$0.437739\pi$$
−0.194354 + 0.980932i $$0.562261\pi$$
$$504$$ 4.00000 0.178174
$$505$$ 0 0
$$506$$ 16.0000 0.711287
$$507$$ − 9.00000i − 0.399704i
$$508$$ − 16.0000i − 0.709885i
$$509$$ −38.0000 −1.68432 −0.842160 0.539227i $$-0.818716\pi$$
−0.842160 + 0.539227i $$0.818716\pi$$
$$510$$ 0 0
$$511$$ 56.0000 2.47729
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ 12.0000 0.528271
$$517$$ 32.0000i 1.40736i
$$518$$ 24.0000i 1.05450i
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ − 2.00000i − 0.0875376i
$$523$$ 28.0000i 1.22435i 0.790721 + 0.612177i $$0.209706\pi$$
−0.790721 + 0.612177i $$0.790294\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ − 4.00000i − 0.174243i
$$528$$ 4.00000i 0.174078i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ − 16.0000i − 0.693688i
$$533$$ − 4.00000i − 0.173259i
$$534$$ 10.0000 0.432742
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 12.0000i 0.517838i
$$538$$ − 30.0000i − 1.29339i
$$539$$ 36.0000 1.55063
$$540$$ 0 0
$$541$$ 18.0000 0.773880 0.386940 0.922105i $$-0.373532\pi$$
0.386940 + 0.922105i $$0.373532\pi$$
$$542$$ 0 0
$$543$$ 22.0000i 0.944110i
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ − 20.0000i − 0.855138i −0.903983 0.427569i $$-0.859370\pi$$
0.903983 0.427569i $$-0.140630\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ − 4.00000i − 0.170251i
$$553$$ 48.0000i 2.04117i
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ 26.0000i 1.10166i 0.834619 + 0.550828i $$0.185688\pi$$
−0.834619 + 0.550828i $$0.814312\pi$$
$$558$$ 4.00000i 0.169334i
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 4.00000 0.168880
$$562$$ − 10.0000i − 0.421825i
$$563$$ − 36.0000i − 1.51722i −0.651546 0.758610i $$-0.725879\pi$$
0.651546 0.758610i $$-0.274121\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 0 0
$$566$$ −20.0000 −0.840663
$$567$$ 4.00000i 0.167984i
$$568$$ − 4.00000i − 0.167836i
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ − 8.00000i − 0.334497i
$$573$$ 8.00000i 0.334205i
$$574$$ 8.00000 0.333914
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 18.0000i − 0.749350i −0.927156 0.374675i $$-0.877754\pi$$
0.927156 0.374675i $$-0.122246\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ 18.0000 0.748054
$$580$$ 0 0
$$581$$ 16.0000 0.663792
$$582$$ 18.0000i 0.746124i
$$583$$ 8.00000i 0.331326i
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ − 44.0000i − 1.81607i −0.418890 0.908037i $$-0.637581\pi$$
0.418890 0.908037i $$-0.362419\pi$$
$$588$$ − 9.00000i − 0.371154i
$$589$$ 16.0000 0.659269
$$590$$ 0 0
$$591$$ 14.0000 0.575883
$$592$$ 6.00000i 0.246598i
$$593$$ 18.0000i 0.739171i 0.929197 + 0.369586i $$0.120500\pi$$
−0.929197 + 0.369586i $$0.879500\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −2.00000 −0.0819232
$$597$$ 4.00000i 0.163709i
$$598$$ 8.00000i 0.327144i
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ − 48.0000i − 1.95633i
$$603$$ 4.00000i 0.162893i
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 10.0000 0.406222
$$607$$ − 28.0000i − 1.13648i −0.822861 0.568242i $$-0.807624\pi$$
0.822861 0.568242i $$-0.192376\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ −8.00000 −0.324176
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ − 1.00000i − 0.0404226i
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ 16.0000 0.644658
$$617$$ 30.0000i 1.20775i 0.797077 + 0.603877i $$0.206378\pi$$
−0.797077 + 0.603877i $$0.793622\pi$$
$$618$$ 0 0
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ − 20.0000i − 0.801927i
$$623$$ − 40.0000i − 1.60257i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 16.0000i 0.638978i
$$628$$ − 10.0000i − 0.399043i
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 12.0000i 0.477334i
$$633$$ − 4.00000i − 0.158986i
$$634$$ −10.0000 −0.397151
$$635$$ 0 0
$$636$$ 2.00000 0.0793052
$$637$$ 18.0000i 0.713186i
$$638$$ − 8.00000i − 0.316723i
$$639$$ 4.00000 0.158238
$$640$$ 0 0
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ 28.0000i 1.10421i 0.833774 + 0.552106i $$0.186176\pi$$
−0.833774 + 0.552106i $$0.813824\pi$$
$$644$$ −16.0000 −0.630488
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ 16.0000i 0.629025i 0.949253 + 0.314512i $$0.101841\pi$$
−0.949253 + 0.314512i $$0.898159\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ 20.0000i 0.783260i
$$653$$ − 30.0000i − 1.17399i −0.809590 0.586995i $$-0.800311\pi$$
0.809590 0.586995i $$-0.199689\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ 14.0000i 0.546192i
$$658$$ − 32.0000i − 1.24749i
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ − 28.0000i − 1.08825i
$$663$$ 2.00000i 0.0776736i
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 8.00000i 0.309761i
$$668$$ 12.0000i 0.464294i
$$669$$ −8.00000 −0.309298
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ − 4.00000i − 0.154303i
$$673$$ 34.0000i 1.31060i 0.755367 + 0.655302i $$0.227459\pi$$
−0.755367 + 0.655302i $$0.772541\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ 72.0000 2.76311
$$680$$ 0 0
$$681$$ 20.0000 0.766402
$$682$$ 16.0000i 0.612672i
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ 6.00000i 0.228914i
$$688$$ − 12.0000i − 0.457496i
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ −44.0000 −1.67384 −0.836919 0.547326i $$-0.815646\pi$$
−0.836919 + 0.547326i $$0.815646\pi$$
$$692$$ 14.0000i 0.532200i
$$693$$ 16.0000i 0.607790i
$$694$$ −20.0000 −0.759190
$$695$$ 0 0
$$696$$ −2.00000 −0.0758098
$$697$$ − 2.00000i − 0.0757554i
$$698$$ 22.0000i 0.832712i
$$699$$ 18.0000 0.680823
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ − 2.00000i − 0.0754851i
$$703$$ 24.0000i 0.905177i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 2.00000 0.0752710
$$707$$ − 40.0000i − 1.50435i
$$708$$ − 12.0000i − 0.450988i
$$709$$ 14.0000 0.525781 0.262891 0.964826i $$-0.415324\pi$$
0.262891 + 0.964826i $$0.415324\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ − 10.0000i − 0.374766i
$$713$$ − 16.0000i − 0.599205i
$$714$$ −4.00000 −0.149696
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ − 24.0000i − 0.896296i
$$718$$ − 24.0000i − 0.895672i
$$719$$ 44.0000 1.64092 0.820462 0.571702i $$-0.193717\pi$$
0.820462 + 0.571702i $$0.193717\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3.00000i 0.111648i
$$723$$ − 10.0000i − 0.371904i
$$724$$ 22.0000 0.817624
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ − 8.00000i − 0.296704i −0.988935 0.148352i $$-0.952603\pi$$
0.988935 0.148352i $$-0.0473968\pi$$
$$728$$ 8.00000i 0.296500i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −12.0000 −0.443836
$$732$$ 2.00000i 0.0739221i
$$733$$ − 18.0000i − 0.664845i −0.943131 0.332423i $$-0.892134\pi$$
0.943131 0.332423i $$-0.107866\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 16.0000i 0.589368i
$$738$$ 2.00000i 0.0736210i
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ − 8.00000i − 0.293689i
$$743$$ 28.0000i 1.02722i 0.858024 + 0.513610i $$0.171692\pi$$
−0.858024 + 0.513610i $$0.828308\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ 38.0000 1.39128
$$747$$ 4.00000i 0.146352i
$$748$$ − 4.00000i − 0.146254i
$$749$$ 48.0000 1.75388
$$750$$ 0 0
$$751$$ 12.0000 0.437886 0.218943 0.975738i $$-0.429739\pi$$
0.218943 + 0.975738i $$0.429739\pi$$
$$752$$ − 8.00000i − 0.291730i
$$753$$ 28.0000i 1.02038i
$$754$$ 4.00000 0.145671
$$755$$ 0 0
$$756$$ 4.00000 0.145479
$$757$$ 34.0000i 1.23575i 0.786276 + 0.617876i $$0.212006\pi$$
−0.786276 + 0.617876i $$0.787994\pi$$
$$758$$ 20.0000i 0.726433i
$$759$$ 16.0000 0.580763
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ − 16.0000i − 0.579619i
$$763$$ − 40.0000i − 1.44810i
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ 24.0000i 0.866590i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 46.0000 1.65880 0.829401 0.558653i $$-0.188682\pi$$
0.829401 + 0.558653i $$0.188682\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ − 18.0000i − 0.647834i
$$773$$ 6.00000i 0.215805i 0.994161 + 0.107903i $$0.0344134\pi$$
−0.994161 + 0.107903i $$0.965587\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 0 0
$$776$$ 18.0000 0.646162
$$777$$ 24.0000i 0.860995i
$$778$$ − 18.0000i − 0.645331i
$$779$$ 8.00000 0.286630
$$780$$ 0 0
$$781$$ 16.0000 0.572525
$$782$$ 4.00000i 0.143040i
$$783$$ − 2.00000i − 0.0714742i
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ −20.0000 −0.713376
$$787$$ − 36.0000i − 1.28326i −0.767014 0.641631i $$-0.778258\pi$$
0.767014 0.641631i $$-0.221742\pi$$
$$788$$ − 14.0000i − 0.498729i
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ 24.0000 0.853342
$$792$$ 4.00000i 0.142134i
$$793$$ − 4.00000i − 0.142044i
$$794$$ 30.0000 1.06466
$$795$$ 0 0
$$796$$ 4.00000 0.141776
$$797$$ 26.0000i 0.920967i 0.887668 + 0.460484i $$0.152324\pi$$
−0.887668 + 0.460484i $$0.847676\pi$$
$$798$$ − 16.0000i − 0.566394i
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 6.00000i 0.211867i
$$803$$ 56.0000i 1.97620i
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ − 30.0000i − 1.05605i
$$808$$ − 10.0000i − 0.351799i
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ 8.00000i 0.280745i
$$813$$ 0 0
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ −1.00000 −0.0350070
$$817$$ − 48.0000i − 1.67931i
$$818$$ 10.0000i 0.349642i
$$819$$ −8.00000 −0.279543
$$820$$ 0 0
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ − 6.00000i − 0.209274i
$$823$$ − 28.0000i − 0.976019i −0.872838 0.488009i $$-0.837723\pi$$
0.872838 0.488009i $$-0.162277\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −48.0000 −1.67013
$$827$$ − 44.0000i − 1.53003i −0.644013 0.765015i $$-0.722732\pi$$
0.644013 0.765015i $$-0.277268\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 2.00000i 0.0693375i
$$833$$ 9.00000i 0.311832i
$$834$$ −20.0000 −0.692543
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 4.00000i 0.138260i
$$838$$ 12.0000i 0.414533i
$$839$$ 36.0000 1.24286 0.621429 0.783470i $$-0.286552\pi$$
0.621429 + 0.783470i $$0.286552\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 18.0000i 0.620321i
$$843$$ − 10.0000i − 0.344418i
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ 20.0000i 0.687208i
$$848$$ − 2.00000i − 0.0686803i
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ 24.0000 0.822709
$$852$$ − 4.00000i − 0.137038i
$$853$$ − 46.0000i − 1.57501i −0.616308 0.787505i $$-0.711372\pi$$
0.616308 0.787505i $$-0.288628\pi$$
$$854$$ 8.00000 0.273754
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 22.0000i 0.751506i 0.926720 + 0.375753i $$0.122616\pi$$
−0.926720 + 0.375753i $$0.877384\pi$$
$$858$$ − 8.00000i − 0.273115i
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ 0 0
$$861$$ 8.00000 0.272639
$$862$$ 4.00000i 0.136241i
$$863$$ 32.0000i 1.08929i 0.838666 + 0.544646i $$0.183336\pi$$
−0.838666 + 0.544646i $$0.816664\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −14.0000 −0.475739
$$867$$ 1.00000i 0.0339618i
$$868$$ − 16.0000i − 0.543075i
$$869$$ −48.0000 −1.62829
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ − 10.0000i − 0.338643i
$$873$$ 18.0000i 0.609208i
$$874$$ −16.0000 −0.541208
$$875$$ 0 0
$$876$$ 14.0000 0.473016
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ − 12.0000i − 0.404980i
$$879$$ −26.0000 −0.876958
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ − 9.00000i − 0.303046i
$$883$$ − 28.0000i − 0.942275i −0.882060 0.471138i $$-0.843844\pi$$
0.882060 0.471138i $$-0.156156\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ 12.0000i 0.402921i 0.979497 + 0.201460i $$0.0645687\pi$$
−0.979497 + 0.201460i $$0.935431\pi$$
$$888$$ 6.00000i 0.201347i
$$889$$ −64.0000 −2.14649
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 8.00000i 0.267860i
$$893$$ − 32.0000i − 1.07084i
$$894$$ −2.00000 −0.0668900
$$895$$ 0 0
$$896$$ −4.00000 −0.133631
$$897$$ 8.00000i 0.267112i
$$898$$ − 6.00000i − 0.200223i
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ −2.00000 −0.0666297
$$902$$ 8.00000i 0.266371i
$$903$$ − 48.0000i − 1.59734i
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 8.00000 0.265782
$$907$$ − 28.0000i − 0.929725i −0.885383 0.464862i $$-0.846104\pi$$
0.885383 0.464862i $$-0.153896\pi$$
$$908$$ − 20.0000i − 0.663723i
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ 20.0000 0.662630 0.331315 0.943520i $$-0.392508\pi$$
0.331315 + 0.943520i $$0.392508\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ 16.0000i 0.529523i
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 80.0000i 2.64183i
$$918$$ − 1.00000i − 0.0330049i
$$919$$ 56.0000 1.84727 0.923635 0.383274i $$-0.125203\pi$$
0.923635 + 0.383274i $$0.125203\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ − 14.0000i − 0.461065i
$$923$$ 8.00000i 0.263323i
$$924$$ 16.0000 0.526361
$$925$$ 0 0
$$926$$ 40.0000 1.31448
$$927$$ 0 0
$$928$$ 2.00000i 0.0656532i
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ −36.0000 −1.17985
$$932$$ − 18.0000i − 0.589610i
$$933$$ − 20.0000i − 0.654771i
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ 22.0000i 0.718709i 0.933201 + 0.359354i $$0.117003\pi$$
−0.933201 + 0.359354i $$0.882997\pi$$
$$938$$ − 16.0000i − 0.522419i
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ − 10.0000i − 0.325818i
$$943$$ − 8.00000i − 0.260516i
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 48.0000 1.56061
$$947$$ 28.0000i 0.909878i 0.890523 + 0.454939i $$0.150339\pi$$
−0.890523 + 0.454939i $$0.849661\pi$$
$$948$$ 12.0000i 0.389742i
$$949$$ −28.0000 −0.908918
$$950$$ 0 0
$$951$$ −10.0000 −0.324272
$$952$$ 4.00000i 0.129641i
$$953$$ − 6.00000i − 0.194359i −0.995267 0.0971795i $$-0.969018\pi$$
0.995267 0.0971795i $$-0.0309821\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ − 8.00000i − 0.258603i
$$958$$ 20.0000i 0.646171i
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ − 12.0000i − 0.386896i
$$963$$ 12.0000i 0.386695i
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ −16.0000 −0.514792
$$967$$ 16.0000i 0.514525i 0.966342 + 0.257263i $$0.0828206\pi$$
−0.966342 + 0.257263i $$0.917179\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ −4.00000 −0.128499
$$970$$ 0 0
$$971$$ 28.0000 0.898563 0.449281 0.893390i $$-0.351680\pi$$
0.449281 + 0.893390i $$0.351680\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 80.0000i 2.56468i
$$974$$ −20.0000 −0.640841
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 30.0000i 0.959785i 0.877327 + 0.479893i $$0.159324\pi$$
−0.877327 + 0.479893i $$0.840676\pi$$
$$978$$ 20.0000i 0.639529i
$$979$$ 40.0000 1.27841
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ 12.0000i 0.382935i
$$983$$ − 28.0000i − 0.893061i −0.894768 0.446531i $$-0.852659\pi$$
0.894768 0.446531i $$-0.147341\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 0 0
$$986$$ 2.00000 0.0636930
$$987$$ − 32.0000i − 1.01857i
$$988$$ 8.00000i 0.254514i
$$989$$ −48.0000 −1.52631
$$990$$ 0 0
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ − 4.00000i − 0.127000i
$$993$$ − 28.0000i − 0.888553i
$$994$$ −16.0000 −0.507489
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ 38.0000i 1.20347i 0.798695 + 0.601736i $$0.205524\pi$$
−0.798695 + 0.601736i $$0.794476\pi$$
$$998$$ 4.00000i 0.126618i
$$999$$ −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.b.2449.1 2
5.2 odd 4 510.2.a.c.1.1 1
5.3 odd 4 2550.2.a.n.1.1 1
5.4 even 2 inner 2550.2.d.b.2449.2 2
15.2 even 4 1530.2.a.d.1.1 1
15.8 even 4 7650.2.a.cn.1.1 1
20.7 even 4 4080.2.a.x.1.1 1
85.67 odd 4 8670.2.a.bb.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.c.1.1 1 5.2 odd 4
1530.2.a.d.1.1 1 15.2 even 4
2550.2.a.n.1.1 1 5.3 odd 4
2550.2.d.b.2449.1 2 1.1 even 1 trivial
2550.2.d.b.2449.2 2 5.4 even 2 inner
4080.2.a.x.1.1 1 20.7 even 4
7650.2.a.cn.1.1 1 15.8 even 4
8670.2.a.bb.1.1 1 85.67 odd 4