# Properties

 Label 2550.2.d.f.2449.2 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 2449.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2550.2449 Dual form 2550.2.d.f.2449.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} -2.00000 q^{21} -4.00000i q^{23} +1.00000 q^{24} +4.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -2.00000 q^{29} +1.00000i q^{32} +1.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} +4.00000 q^{39} -4.00000 q^{41} -2.00000i q^{42} -10.0000i q^{43} +4.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +1.00000 q^{51} +4.00000i q^{52} -2.00000i q^{53} +1.00000 q^{54} +2.00000 q^{56} -4.00000i q^{57} -2.00000i q^{58} +2.00000 q^{59} -14.0000 q^{61} -2.00000i q^{63} -1.00000 q^{64} +2.00000i q^{67} +1.00000i q^{68} +4.00000 q^{69} -6.00000 q^{71} +1.00000i q^{72} +4.00000i q^{73} +2.00000 q^{74} +4.00000 q^{76} +4.00000i q^{78} +12.0000 q^{79} +1.00000 q^{81} -4.00000i q^{82} -8.00000i q^{83} +2.00000 q^{84} +10.0000 q^{86} -2.00000i q^{87} +10.0000 q^{89} +8.00000 q^{91} +4.00000i q^{92} +8.00000 q^{94} -1.00000 q^{96} +8.00000i q^{97} +3.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{21} + 2 q^{24} + 8 q^{26} - 4 q^{29} + 2 q^{34} + 2 q^{36} + 8 q^{39} - 8 q^{41} + 8 q^{46} + 6 q^{49} + 2 q^{51} + 2 q^{54} + 4 q^{56} + 4 q^{59} - 28 q^{61} - 2 q^{64} + 8 q^{69} - 12 q^{71} + 4 q^{74} + 8 q^{76} + 24 q^{79} + 2 q^{81} + 4 q^{84} + 20 q^{86} + 20 q^{89} + 16 q^{91} + 16 q^{94} - 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 4 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^21 + 2 * q^24 + 8 * q^26 - 4 * q^29 + 2 * q^34 + 2 * q^36 + 8 * q^39 - 8 * q^41 + 8 * q^46 + 6 * q^49 + 2 * q^51 + 2 * q^54 + 4 * q^56 + 4 * q^59 - 28 * q^61 - 2 * q^64 + 8 * q^69 - 12 * q^71 + 4 * q^74 + 8 * q^76 + 24 * q^79 + 2 * q^81 + 4 * q^84 + 20 * q^86 + 20 * q^89 + 16 * q^91 + 16 * q^94 - 2 * 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
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ − 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
$$13$$ − 4.00000i − 1.10940i −0.832050 0.554700i $$-0.812833\pi$$
0.832050 0.554700i $$-0.187167\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 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
$$21$$ −2.00000 −0.436436
$$22$$ 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
$$26$$ 4.00000 0.784465
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ − 10.0000i − 1.52499i −0.646997 0.762493i $$-0.723975\pi$$
0.646997 0.762493i $$-0.276025\pi$$
$$44$$ 0 0
$$45$$ 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
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 1.00000 0.140028
$$52$$ 4.00000i 0.554700i
$$53$$ − 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
$$56$$ 2.00000 0.267261
$$57$$ − 4.00000i − 0.529813i
$$58$$ − 2.00000i − 0.262613i
$$59$$ 2.00000 0.260378 0.130189 0.991489i $$-0.458442\pi$$
0.130189 + 0.991489i $$0.458442\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 0 0
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000i 0.244339i 0.992509 + 0.122169i $$0.0389851\pi$$
−0.992509 + 0.122169i $$0.961015\pi$$
$$68$$ 1.00000i 0.121268i
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 4.00000i 0.468165i 0.972217 + 0.234082i $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 4.00000i 0.452911i
$$79$$ 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
$$82$$ − 4.00000i − 0.441726i
$$83$$ − 8.00000i − 0.878114i −0.898459 0.439057i $$-0.855313\pi$$
0.898459 0.439057i $$-0.144687\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ 10.0000 1.07833
$$87$$ − 2.00000i − 0.214423i
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ 4.00000i 0.417029i
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 8.00000i 0.812277i 0.913812 + 0.406138i $$0.133125\pi$$
−0.913812 + 0.406138i $$0.866875\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −4.00000 −0.398015 −0.199007 0.979998i $$-0.563772\pi$$
−0.199007 + 0.979998i $$0.563772\pi$$
$$102$$ 1.00000i 0.0990148i
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 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
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 2.00000i 0.188982i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ 4.00000i 0.369800i
$$118$$ 2.00000i 0.184115i
$$119$$ 2.00000 0.183340
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ − 14.0000i − 1.26750i
$$123$$ − 4.00000i − 0.360668i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 10.0000 0.880451
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ − 8.00000i − 0.693688i
$$134$$ −2.00000 −0.172774
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ − 18.0000i − 1.53784i −0.639343 0.768922i $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ 4.00000i 0.340503i
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 8.00000 0.673722
$$142$$ − 6.00000i − 0.503509i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ 3.00000i 0.247436i
$$148$$ 2.00000i 0.164399i
$$149$$ 12.0000 0.983078 0.491539 0.870855i $$-0.336434\pi$$
0.491539 + 0.870855i $$0.336434\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ 1.00000i 0.0808452i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 4.00000i 0.319235i 0.987179 + 0.159617i $$0.0510260\pi$$
−0.987179 + 0.159617i $$0.948974\pi$$
$$158$$ 12.0000i 0.954669i
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ 1.00000i 0.0785674i
$$163$$ − 24.0000i − 1.87983i −0.341415 0.939913i $$-0.610906\pi$$
0.341415 0.939913i $$-0.389094\pi$$
$$164$$ 4.00000 0.312348
$$165$$ 0 0
$$166$$ 8.00000 0.620920
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 10.0000i 0.762493i
$$173$$ 2.00000i 0.152057i 0.997106 + 0.0760286i $$0.0242240\pi$$
−0.997106 + 0.0760286i $$0.975776\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 2.00000i 0.150329i
$$178$$ 10.0000i 0.749532i
$$179$$ −18.0000 −1.34538 −0.672692 0.739923i $$-0.734862\pi$$
−0.672692 + 0.739923i $$0.734862\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 8.00000i 0.592999i
$$183$$ − 14.0000i − 1.03491i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 8.00000i 0.583460i
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 8.00000i 0.575853i 0.957653 + 0.287926i $$0.0929658\pi$$
−0.957653 + 0.287926i $$0.907034\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ −2.00000 −0.141069
$$202$$ − 4.00000i − 0.281439i
$$203$$ − 4.00000i − 0.280745i
$$204$$ −1.00000 −0.0700140
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 4.00000i 0.278019i
$$208$$ − 4.00000i − 0.277350i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 2.00000i 0.137361i
$$213$$ − 6.00000i − 0.411113i
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ − 6.00000i − 0.406371i
$$219$$ −4.00000 −0.270295
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 2.00000i 0.134231i
$$223$$ 16.0000i 1.07144i 0.844396 + 0.535720i $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.00000i 0.131306i
$$233$$ − 14.0000i − 0.917170i −0.888650 0.458585i $$-0.848356\pi$$
0.888650 0.458585i $$-0.151644\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ −2.00000 −0.130189
$$237$$ 12.0000i 0.779484i
$$238$$ 2.00000i 0.129641i
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −30.0000 −1.93247 −0.966235 0.257663i $$-0.917048\pi$$
−0.966235 + 0.257663i $$0.917048\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ 1.00000i 0.0641500i
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ 4.00000 0.255031
$$247$$ 16.0000i 1.01806i
$$248$$ 0 0
$$249$$ 8.00000 0.506979
$$250$$ 0 0
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 0 0
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 14.0000i − 0.873296i −0.899632 0.436648i $$-0.856166\pi$$
0.899632 0.436648i $$-0.143834\pi$$
$$258$$ 10.0000i 0.622573i
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 12.0000i 0.741362i
$$263$$ − 24.0000i − 1.47990i −0.672660 0.739952i $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 8.00000 0.490511
$$267$$ 10.0000i 0.611990i
$$268$$ − 2.00000i − 0.122169i
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ − 1.00000i − 0.0606339i
$$273$$ 8.00000i 0.484182i
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ 10.0000i 0.600842i 0.953807 + 0.300421i $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ 12.0000i 0.719712i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 22.0000 1.31241 0.656205 0.754583i $$-0.272161\pi$$
0.656205 + 0.754583i $$0.272161\pi$$
$$282$$ 8.00000i 0.476393i
$$283$$ 28.0000i 1.66443i 0.554455 + 0.832214i $$0.312927\pi$$
−0.554455 + 0.832214i $$0.687073\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 8.00000i − 0.472225i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −8.00000 −0.468968
$$292$$ − 4.00000i − 0.234082i
$$293$$ − 22.0000i − 1.28525i −0.766179 0.642627i $$-0.777845\pi$$
0.766179 0.642627i $$-0.222155\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 12.0000i 0.695141i
$$299$$ −16.0000 −0.925304
$$300$$ 0 0
$$301$$ 20.0000 1.15278
$$302$$ 8.00000i 0.460348i
$$303$$ − 4.00000i − 0.229794i
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ −1.00000 −0.0571662
$$307$$ 10.0000i 0.570730i 0.958419 + 0.285365i $$0.0921148\pi$$
−0.958419 + 0.285365i $$0.907885\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −10.0000 −0.567048 −0.283524 0.958965i $$-0.591504\pi$$
−0.283524 + 0.958965i $$0.591504\pi$$
$$312$$ − 4.00000i − 0.226455i
$$313$$ 16.0000i 0.904373i 0.891923 + 0.452187i $$0.149356\pi$$
−0.891923 + 0.452187i $$0.850644\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ 30.0000i 1.68497i 0.538721 + 0.842484i $$0.318908\pi$$
−0.538721 + 0.842484i $$0.681092\pi$$
$$318$$ 2.00000i 0.112154i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 8.00000i 0.445823i
$$323$$ 4.00000i 0.222566i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 24.0000 1.32924
$$327$$ − 6.00000i − 0.331801i
$$328$$ 4.00000i 0.220863i
$$329$$ 16.0000 0.882109
$$330$$ 0 0
$$331$$ −16.0000 −0.879440 −0.439720 0.898135i $$-0.644922\pi$$
−0.439720 + 0.898135i $$0.644922\pi$$
$$332$$ 8.00000i 0.439057i
$$333$$ 2.00000i 0.109599i
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ 12.0000i 0.653682i 0.945079 + 0.326841i $$0.105984\pi$$
−0.945079 + 0.326841i $$0.894016\pi$$
$$338$$ − 3.00000i − 0.163178i
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.00000i 0.216295i
$$343$$ 20.0000i 1.07990i
$$344$$ −10.0000 −0.539164
$$345$$ 0 0
$$346$$ −2.00000 −0.107521
$$347$$ − 20.0000i − 1.07366i −0.843692 0.536828i $$-0.819622\pi$$
0.843692 0.536828i $$-0.180378\pi$$
$$348$$ 2.00000i 0.107211i
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ 0 0
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ −2.00000 −0.106299
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 2.00000i 0.105851i
$$358$$ − 18.0000i − 0.951330i
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 10.0000i − 0.525588i
$$363$$ − 11.0000i − 0.577350i
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ 14.0000 0.731792
$$367$$ − 2.00000i − 0.104399i −0.998637 0.0521996i $$-0.983377\pi$$
0.998637 0.0521996i $$-0.0166232\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ 4.00000 0.208232
$$370$$ 0 0
$$371$$ 4.00000 0.207670
$$372$$ 0 0
$$373$$ − 8.00000i − 0.414224i −0.978317 0.207112i $$-0.933593\pi$$
0.978317 0.207112i $$-0.0664065\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ 8.00000i 0.412021i
$$378$$ 2.00000i 0.102869i
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 12.0000i 0.613973i
$$383$$ − 16.0000i − 0.817562i −0.912633 0.408781i $$-0.865954\pi$$
0.912633 0.408781i $$-0.134046\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −8.00000 −0.407189
$$387$$ 10.0000i 0.508329i
$$388$$ − 8.00000i − 0.406138i
$$389$$ −12.0000 −0.608424 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ − 3.00000i − 0.151523i
$$393$$ 12.0000i 0.605320i
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 10.0000i − 0.501886i −0.968002 0.250943i $$-0.919259\pi$$
0.968002 0.250943i $$-0.0807406\pi$$
$$398$$ − 16.0000i − 0.802008i
$$399$$ 8.00000 0.400501
$$400$$ 0 0
$$401$$ −24.0000 −1.19850 −0.599251 0.800561i $$-0.704535\pi$$
−0.599251 + 0.800561i $$0.704535\pi$$
$$402$$ − 2.00000i − 0.0997509i
$$403$$ 0 0
$$404$$ 4.00000 0.199007
$$405$$ 0 0
$$406$$ 4.00000 0.198517
$$407$$ 0 0
$$408$$ − 1.00000i − 0.0495074i
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ 8.00000i 0.394132i
$$413$$ 4.00000i 0.196827i
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ 12.0000i 0.587643i
$$418$$ 0 0
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 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
$$423$$ 8.00000i 0.388973i
$$424$$ −2.00000 −0.0971286
$$425$$ 0 0
$$426$$ 6.00000 0.290701
$$427$$ − 28.0000i − 1.35501i
$$428$$ 12.0000i 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −34.0000 −1.63772 −0.818861 0.573992i $$-0.805394\pi$$
−0.818861 + 0.573992i $$0.805394\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 34.0000i 1.63394i 0.576683 + 0.816968i $$0.304347\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 6.00000 0.287348
$$437$$ 16.0000i 0.765384i
$$438$$ − 4.00000i − 0.191127i
$$439$$ 12.0000 0.572729 0.286364 0.958121i $$-0.407553\pi$$
0.286364 + 0.958121i $$0.407553\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ − 4.00000i − 0.190261i
$$443$$ 8.00000i 0.380091i 0.981775 + 0.190046i $$0.0608636\pi$$
−0.981775 + 0.190046i $$0.939136\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 12.0000i 0.567581i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ −40.0000 −1.88772 −0.943858 0.330350i $$-0.892833\pi$$
−0.943858 + 0.330350i $$0.892833\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 6.00000i − 0.282216i
$$453$$ 8.00000i 0.375873i
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ − 34.0000i − 1.59045i −0.606313 0.795226i $$-0.707352\pi$$
0.606313 0.795226i $$-0.292648\pi$$
$$458$$ 2.00000i 0.0934539i
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ − 16.0000i − 0.743583i −0.928316 0.371792i $$-0.878744\pi$$
0.928316 0.371792i $$-0.121256\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 14.0000 0.648537
$$467$$ − 36.0000i − 1.66588i −0.553362 0.832941i $$-0.686655\pi$$
0.553362 0.832941i $$-0.313345\pi$$
$$468$$ − 4.00000i − 0.184900i
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ −4.00000 −0.184310
$$472$$ − 2.00000i − 0.0920575i
$$473$$ 0 0
$$474$$ −12.0000 −0.551178
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ 2.00000i 0.0915737i
$$478$$ 24.0000i 1.09773i
$$479$$ −2.00000 −0.0913823 −0.0456912 0.998956i $$-0.514549\pi$$
−0.0456912 + 0.998956i $$0.514549\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ − 30.0000i − 1.36646i
$$483$$ 8.00000i 0.364013i
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 22.0000i − 0.996915i −0.866914 0.498458i $$-0.833900\pi$$
0.866914 0.498458i $$-0.166100\pi$$
$$488$$ 14.0000i 0.633750i
$$489$$ 24.0000 1.08532
$$490$$ 0 0
$$491$$ −14.0000 −0.631811 −0.315906 0.948791i $$-0.602308\pi$$
−0.315906 + 0.948791i $$0.602308\pi$$
$$492$$ 4.00000i 0.180334i
$$493$$ 2.00000i 0.0900755i
$$494$$ −16.0000 −0.719874
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 12.0000i − 0.538274i
$$498$$ 8.00000i 0.358489i
$$499$$ 12.0000 0.537194 0.268597 0.963253i $$-0.413440\pi$$
0.268597 + 0.963253i $$0.413440\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ − 2.00000i − 0.0892644i
$$503$$ − 28.0000i − 1.24846i −0.781241 0.624229i $$-0.785413\pi$$
0.781241 0.624229i $$-0.214587\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 3.00000i − 0.133235i
$$508$$ − 16.0000i − 0.709885i
$$509$$ −4.00000 −0.177297 −0.0886484 0.996063i $$-0.528255\pi$$
−0.0886484 + 0.996063i $$0.528255\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ 1.00000i 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ 14.0000 0.617514
$$515$$ 0 0
$$516$$ −10.0000 −0.440225
$$517$$ 0 0
$$518$$ 4.00000i 0.175750i
$$519$$ −2.00000 −0.0877903
$$520$$ 0 0
$$521$$ 16.0000 0.700973 0.350486 0.936568i $$-0.386016\pi$$
0.350486 + 0.936568i $$0.386016\pi$$
$$522$$ 2.00000i 0.0875376i
$$523$$ 18.0000i 0.787085i 0.919306 + 0.393543i $$0.128751\pi$$
−0.919306 + 0.393543i $$0.871249\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −2.00000 −0.0867926
$$532$$ 8.00000i 0.346844i
$$533$$ 16.0000i 0.693037i
$$534$$ −10.0000 −0.432742
$$535$$ 0 0
$$536$$ 2.00000 0.0863868
$$537$$ − 18.0000i − 0.776757i
$$538$$ − 10.0000i − 0.431131i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ − 8.00000i − 0.343629i
$$543$$ − 10.0000i − 0.429141i
$$544$$ 1.00000 0.0428746
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ − 36.0000i − 1.53925i −0.638497 0.769624i $$-0.720443\pi$$
0.638497 0.769624i $$-0.279557\pi$$
$$548$$ 18.0000i 0.768922i
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ − 4.00000i − 0.170251i
$$553$$ 24.0000i 1.02058i
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ − 46.0000i − 1.94908i −0.224208 0.974541i $$-0.571980\pi$$
0.224208 0.974541i $$-0.428020\pi$$
$$558$$ 0 0
$$559$$ −40.0000 −1.69182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 22.0000i 0.928014i
$$563$$ 32.0000i 1.34864i 0.738440 + 0.674320i $$0.235563\pi$$
−0.738440 + 0.674320i $$0.764437\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ −28.0000 −1.17693
$$567$$ 2.00000i 0.0839921i
$$568$$ 6.00000i 0.251754i
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ 0 0
$$573$$ 12.0000i 0.501307i
$$574$$ 8.00000 0.333914
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 22.0000i 0.915872i 0.888985 + 0.457936i $$0.151411\pi$$
−0.888985 + 0.457936i $$0.848589\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ −8.00000 −0.332469
$$580$$ 0 0
$$581$$ 16.0000 0.663792
$$582$$ − 8.00000i − 0.331611i
$$583$$ 0 0
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ 22.0000 0.908812
$$587$$ 8.00000i 0.330195i 0.986277 + 0.165098i $$0.0527939\pi$$
−0.986277 + 0.165098i $$0.947206\pi$$
$$588$$ − 3.00000i − 0.123718i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ − 2.00000i − 0.0821995i
$$593$$ 30.0000i 1.23195i 0.787765 + 0.615976i $$0.211238\pi$$
−0.787765 + 0.615976i $$0.788762\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −12.0000 −0.491539
$$597$$ − 16.0000i − 0.654836i
$$598$$ − 16.0000i − 0.654289i
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ 20.0000i 0.815139i
$$603$$ − 2.00000i − 0.0814463i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 4.00000 0.162489
$$607$$ − 14.0000i − 0.568242i −0.958788 0.284121i $$-0.908298\pi$$
0.958788 0.284121i $$-0.0917018\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ 4.00000 0.162088
$$610$$ 0 0
$$611$$ −32.0000 −1.29458
$$612$$ − 1.00000i − 0.0404226i
$$613$$ − 24.0000i − 0.969351i −0.874694 0.484675i $$-0.838938\pi$$
0.874694 0.484675i $$-0.161062\pi$$
$$614$$ −10.0000 −0.403567
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ − 10.0000i − 0.400963i
$$623$$ 20.0000i 0.801283i
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ −16.0000 −0.639489
$$627$$ 0 0
$$628$$ − 4.00000i − 0.159617i
$$629$$ −2.00000 −0.0797452
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ − 12.0000i − 0.477334i
$$633$$ 12.0000i 0.476957i
$$634$$ −30.0000 −1.19145
$$635$$ 0 0
$$636$$ −2.00000 −0.0793052
$$637$$ − 12.0000i − 0.475457i
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ 40.0000 1.57991 0.789953 0.613168i $$-0.210105\pi$$
0.789953 + 0.613168i $$0.210105\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ − 16.0000i − 0.630978i −0.948929 0.315489i $$-0.897831\pi$$
0.948929 0.315489i $$-0.102169\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ − 8.00000i − 0.314512i −0.987558 0.157256i $$-0.949735\pi$$
0.987558 0.157256i $$-0.0502649\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 24.0000i 0.939913i
$$653$$ 14.0000i 0.547862i 0.961749 + 0.273931i $$0.0883240\pi$$
−0.961749 + 0.273931i $$0.911676\pi$$
$$654$$ 6.00000 0.234619
$$655$$ 0 0
$$656$$ −4.00000 −0.156174
$$657$$ − 4.00000i − 0.156055i
$$658$$ 16.0000i 0.623745i
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ − 16.0000i − 0.621858i
$$663$$ − 4.00000i − 0.155347i
$$664$$ −8.00000 −0.310460
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 8.00000i 0.309761i
$$668$$ 8.00000i 0.309529i
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 2.00000i − 0.0771517i
$$673$$ − 8.00000i − 0.308377i −0.988041 0.154189i $$-0.950724\pi$$
0.988041 0.154189i $$-0.0492764\pi$$
$$674$$ −12.0000 −0.462223
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ 38.0000i 1.46046i 0.683202 + 0.730229i $$0.260587\pi$$
−0.683202 + 0.730229i $$0.739413\pi$$
$$678$$ − 6.00000i − 0.230429i
$$679$$ −16.0000 −0.614024
$$680$$ 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 $$-0.0737364\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ 2.00000i 0.0763048i
$$688$$ − 10.0000i − 0.381246i
$$689$$ −8.00000 −0.304776
$$690$$ 0 0
$$691$$ 12.0000 0.456502 0.228251 0.973602i $$-0.426699\pi$$
0.228251 + 0.973602i $$0.426699\pi$$
$$692$$ − 2.00000i − 0.0760286i
$$693$$ 0 0
$$694$$ 20.0000 0.759190
$$695$$ 0 0
$$696$$ −2.00000 −0.0758098
$$697$$ 4.00000i 0.151511i
$$698$$ − 2.00000i − 0.0757011i
$$699$$ 14.0000 0.529529
$$700$$ 0 0
$$701$$ 24.0000 0.906467 0.453234 0.891392i $$-0.350270\pi$$
0.453234 + 0.891392i $$0.350270\pi$$
$$702$$ − 4.00000i − 0.150970i
$$703$$ 8.00000i 0.301726i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ − 8.00000i − 0.300871i
$$708$$ − 2.00000i − 0.0751646i
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ − 10.0000i − 0.374766i
$$713$$ 0 0
$$714$$ −2.00000 −0.0748481
$$715$$ 0 0
$$716$$ 18.0000 0.672692
$$717$$ 24.0000i 0.896296i
$$718$$ − 20.0000i − 0.746393i
$$719$$ −26.0000 −0.969636 −0.484818 0.874615i $$-0.661114\pi$$
−0.484818 + 0.874615i $$0.661114\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ − 3.00000i − 0.111648i
$$723$$ − 30.0000i − 1.11571i
$$724$$ 10.0000 0.371647
$$725$$ 0 0
$$726$$ 11.0000 0.408248
$$727$$ − 24.0000i − 0.890111i −0.895503 0.445055i $$-0.853184\pi$$
0.895503 0.445055i $$-0.146816\pi$$
$$728$$ − 8.00000i − 0.296500i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −10.0000 −0.369863
$$732$$ 14.0000i 0.517455i
$$733$$ 16.0000i 0.590973i 0.955347 + 0.295487i $$0.0954818\pi$$
−0.955347 + 0.295487i $$0.904518\pi$$
$$734$$ 2.00000 0.0738213
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 0 0
$$738$$ 4.00000i 0.147242i
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 0 0
$$741$$ −16.0000 −0.587775
$$742$$ 4.00000i 0.146845i
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 8.00000 0.292901
$$747$$ 8.00000i 0.292705i
$$748$$ 0 0
$$749$$ 24.0000 0.876941
$$750$$ 0 0
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ − 8.00000i − 0.291730i
$$753$$ − 2.00000i − 0.0728841i
$$754$$ −8.00000 −0.291343
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ 52.0000i 1.88997i 0.327111 + 0.944986i $$0.393925\pi$$
−0.327111 + 0.944986i $$0.606075\pi$$
$$758$$ − 28.0000i − 1.01701i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −54.0000 −1.95750 −0.978749 0.205061i $$-0.934261\pi$$
−0.978749 + 0.205061i $$0.934261\pi$$
$$762$$ − 16.0000i − 0.579619i
$$763$$ − 12.0000i − 0.434429i
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ − 8.00000i − 0.288863i
$$768$$ 1.00000i 0.0360844i
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 14.0000 0.504198
$$772$$ − 8.00000i − 0.287926i
$$773$$ − 42.0000i − 1.51064i −0.655359 0.755318i $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ −10.0000 −0.359443
$$775$$ 0 0
$$776$$ 8.00000 0.287183
$$777$$ 4.00000i 0.143499i
$$778$$ − 12.0000i − 0.430221i
$$779$$ 16.0000 0.573259
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 4.00000i − 0.143040i
$$783$$ 2.00000i 0.0714742i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ 8.00000i 0.285169i 0.989783 + 0.142585i $$0.0455413\pi$$
−0.989783 + 0.142585i $$0.954459\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 56.0000i 1.98862i
$$794$$ 10.0000 0.354887
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ 6.00000i 0.212531i 0.994338 + 0.106265i $$0.0338893\pi$$
−0.994338 + 0.106265i $$0.966111\pi$$
$$798$$ 8.00000i 0.283197i
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ −10.0000 −0.353333
$$802$$ − 24.0000i − 0.847469i
$$803$$ 0 0
$$804$$ 2.00000 0.0705346
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 10.0000i − 0.352017i
$$808$$ 4.00000i 0.140720i
$$809$$ 28.0000 0.984428 0.492214 0.870474i $$-0.336188\pi$$
0.492214 + 0.870474i $$0.336188\pi$$
$$810$$ 0 0
$$811$$ 52.0000 1.82597 0.912983 0.407997i $$-0.133772\pi$$
0.912983 + 0.407997i $$0.133772\pi$$
$$812$$ 4.00000i 0.140372i
$$813$$ − 8.00000i − 0.280572i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 1.00000 0.0350070
$$817$$ 40.0000i 1.39942i
$$818$$ 26.0000i 0.909069i
$$819$$ −8.00000 −0.279543
$$820$$ 0 0
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 18.0000i 0.627822i
$$823$$ 50.0000i 1.74289i 0.490493 + 0.871445i $$0.336817\pi$$
−0.490493 + 0.871445i $$0.663183\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ −4.00000 −0.139178
$$827$$ 52.0000i 1.80822i 0.427303 + 0.904109i $$0.359464\pi$$
−0.427303 + 0.904109i $$0.640536\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ −10.0000 −0.346896
$$832$$ 4.00000i 0.138675i
$$833$$ − 3.00000i − 0.103944i
$$834$$ −12.0000 −0.415526
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 4.00000i 0.138178i
$$839$$ −14.0000 −0.483334 −0.241667 0.970359i $$-0.577694\pi$$
−0.241667 + 0.970359i $$0.577694\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ − 34.0000i − 1.17172i
$$843$$ 22.0000i 0.757720i
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ −8.00000 −0.275046
$$847$$ − 22.0000i − 0.755929i
$$848$$ − 2.00000i − 0.0686803i
$$849$$ −28.0000 −0.960958
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 6.00000i 0.205557i
$$853$$ − 42.0000i − 1.43805i −0.694983 0.719026i $$-0.744588\pi$$
0.694983 0.719026i $$-0.255412\pi$$
$$854$$ 28.0000 0.958140
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ − 2.00000i − 0.0683187i −0.999416 0.0341593i $$-0.989125\pi$$
0.999416 0.0341593i $$-0.0108754\pi$$
$$858$$ 0 0
$$859$$ −24.0000 −0.818869 −0.409435 0.912339i $$-0.634274\pi$$
−0.409435 + 0.912339i $$0.634274\pi$$
$$860$$ 0 0
$$861$$ 8.00000 0.272639
$$862$$ − 34.0000i − 1.15804i
$$863$$ − 32.0000i − 1.08929i −0.838666 0.544646i $$-0.816664\pi$$
0.838666 0.544646i $$-0.183336\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −34.0000 −1.15537
$$867$$ − 1.00000i − 0.0339618i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 6.00000i 0.203186i
$$873$$ − 8.00000i − 0.270759i
$$874$$ −16.0000 −0.541208
$$875$$ 0 0
$$876$$ 4.00000 0.135147
$$877$$ 10.0000i 0.337676i 0.985644 + 0.168838i $$0.0540015\pi$$
−0.985644 + 0.168838i $$0.945999\pi$$
$$878$$ 12.0000i 0.404980i
$$879$$ 22.0000 0.742042
$$880$$ 0 0
$$881$$ 48.0000 1.61716 0.808581 0.588386i $$-0.200236\pi$$
0.808581 + 0.588386i $$0.200236\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ 2.00000i 0.0673054i 0.999434 + 0.0336527i $$0.0107140\pi$$
−0.999434 + 0.0336527i $$0.989286\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ −8.00000 −0.268765
$$887$$ 20.0000i 0.671534i 0.941945 + 0.335767i $$0.108996\pi$$
−0.941945 + 0.335767i $$0.891004\pi$$
$$888$$ − 2.00000i − 0.0671156i
$$889$$ −32.0000 −1.07325
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 16.0000i − 0.535720i
$$893$$ 32.0000i 1.07084i
$$894$$ −12.0000 −0.401340
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ − 16.0000i − 0.534224i
$$898$$ − 40.0000i − 1.33482i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −2.00000 −0.0666297
$$902$$ 0 0
$$903$$ 20.0000i 0.665558i
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ − 8.00000i − 0.265636i −0.991140 0.132818i $$-0.957597\pi$$
0.991140 0.132818i $$-0.0424025\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ 4.00000 0.132672
$$910$$ 0 0
$$911$$ 18.0000 0.596367 0.298183 0.954509i $$-0.403619\pi$$
0.298183 + 0.954509i $$0.403619\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ 0 0
$$914$$ 34.0000 1.12462
$$915$$ 0 0
$$916$$ −2.00000 −0.0660819
$$917$$ 24.0000i 0.792550i
$$918$$ − 1.00000i − 0.0330049i
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ −10.0000 −0.329511
$$922$$ 0 0
$$923$$ 24.0000i 0.789970i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 16.0000 0.525793
$$927$$ 8.00000i 0.262754i
$$928$$ − 2.00000i − 0.0656532i
$$929$$ 20.0000 0.656179 0.328089 0.944647i $$-0.393595\pi$$
0.328089 + 0.944647i $$0.393595\pi$$
$$930$$ 0 0
$$931$$ −12.0000 −0.393284
$$932$$ 14.0000i 0.458585i
$$933$$ − 10.0000i − 0.327385i
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ 54.0000i 1.76410i 0.471153 + 0.882052i $$0.343838\pi$$
−0.471153 + 0.882052i $$0.656162\pi$$
$$938$$ − 4.00000i − 0.130605i
$$939$$ −16.0000 −0.522140
$$940$$ 0 0
$$941$$ −10.0000 −0.325991 −0.162995 0.986627i $$-0.552116\pi$$
−0.162995 + 0.986627i $$0.552116\pi$$
$$942$$ − 4.00000i − 0.130327i
$$943$$ 16.0000i 0.521032i
$$944$$ 2.00000 0.0650945
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 4.00000i − 0.129983i −0.997886 0.0649913i $$-0.979298\pi$$
0.997886 0.0649913i $$-0.0207020\pi$$
$$948$$ − 12.0000i − 0.389742i
$$949$$ 16.0000 0.519382
$$950$$ 0 0
$$951$$ −30.0000 −0.972817
$$952$$ − 2.00000i − 0.0648204i
$$953$$ 34.0000i 1.10137i 0.834714 + 0.550684i $$0.185633\pi$$
−0.834714 + 0.550684i $$0.814367\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ − 2.00000i − 0.0646171i
$$959$$ 36.0000 1.16250
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ − 8.00000i − 0.257930i
$$963$$ 12.0000i 0.386695i
$$964$$ 30.0000 0.966235
$$965$$ 0 0
$$966$$ −8.00000 −0.257396
$$967$$ 20.0000i 0.643157i 0.946883 + 0.321578i $$0.104213\pi$$
−0.946883 + 0.321578i $$0.895787\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ −4.00000 −0.128499
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 24.0000i 0.769405i
$$974$$ 22.0000 0.704925
$$975$$ 0 0
$$976$$ −14.0000 −0.448129
$$977$$ 18.0000i 0.575871i 0.957650 + 0.287936i $$0.0929689\pi$$
−0.957650 + 0.287936i $$0.907031\pi$$
$$978$$ 24.0000i 0.767435i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 6.00000 0.191565
$$982$$ − 14.0000i − 0.446758i
$$983$$ − 56.0000i − 1.78612i −0.449935 0.893061i $$-0.648553\pi$$
0.449935 0.893061i $$-0.351447\pi$$
$$984$$ −4.00000 −0.127515
$$985$$ 0 0
$$986$$ −2.00000 −0.0636930
$$987$$ 16.0000i 0.509286i
$$988$$ − 16.0000i − 0.509028i
$$989$$ −40.0000 −1.27193
$$990$$ 0 0
$$991$$ 24.0000 0.762385 0.381193 0.924496i $$-0.375513\pi$$
0.381193 + 0.924496i $$0.375513\pi$$
$$992$$ 0 0
$$993$$ − 16.0000i − 0.507745i
$$994$$ 12.0000 0.380617
$$995$$ 0 0
$$996$$ −8.00000 −0.253490
$$997$$ 14.0000i 0.443384i 0.975117 + 0.221692i $$0.0711580\pi$$
−0.975117 + 0.221692i $$0.928842\pi$$
$$998$$ 12.0000i 0.379853i
$$999$$ −2.00000 −0.0632772
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.f.2449.2 2
5.2 odd 4 2550.2.a.i.1.1 1
5.3 odd 4 510.2.a.d.1.1 1
5.4 even 2 inner 2550.2.d.f.2449.1 2
15.2 even 4 7650.2.a.bn.1.1 1
15.8 even 4 1530.2.a.h.1.1 1
20.3 even 4 4080.2.a.r.1.1 1
85.33 odd 4 8670.2.a.y.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.d.1.1 1 5.3 odd 4
1530.2.a.h.1.1 1 15.8 even 4
2550.2.a.i.1.1 1 5.2 odd 4
2550.2.d.f.2449.1 2 5.4 even 2 inner
2550.2.d.f.2449.2 2 1.1 even 1 trivial
4080.2.a.r.1.1 1 20.3 even 4
7650.2.a.bn.1.1 1 15.2 even 4
8670.2.a.y.1.1 1 85.33 odd 4