# Properties

 Label 2550.2.d.c.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: yes 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.c.2449.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} +7.00000 q^{19} -1.00000 q^{21} +3.00000i q^{22} -6.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} +1.00000i q^{28} -6.00000 q^{29} -7.00000 q^{31} -1.00000i q^{32} +3.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} -7.00000i q^{37} -7.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} +1.00000i q^{42} +1.00000i q^{43} +3.00000 q^{44} -6.00000 q^{46} +9.00000i q^{47} -1.00000i q^{48} +6.00000 q^{49} -1.00000 q^{51} +2.00000i q^{52} +3.00000i q^{53} +1.00000 q^{54} +1.00000 q^{56} -7.00000i q^{57} +6.00000i q^{58} -10.0000 q^{61} +7.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +5.00000i q^{67} +1.00000i q^{68} -6.00000 q^{69} -6.00000 q^{71} -1.00000i q^{72} +16.0000i q^{73} -7.00000 q^{74} -7.00000 q^{76} +3.00000i q^{77} +2.00000i q^{78} -17.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +6.00000i q^{83} +1.00000 q^{84} +1.00000 q^{86} +6.00000i q^{87} -3.00000i q^{88} +12.0000 q^{89} -2.00000 q^{91} +6.00000i q^{92} +7.00000i q^{93} +9.00000 q^{94} -1.00000 q^{96} -10.0000i q^{97} -6.00000i q^{98} +3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 6 q^{11} - 2 q^{14} + 2 q^{16} + 14 q^{19} - 2 q^{21} + 2 q^{24} - 4 q^{26} - 12 q^{29} - 14 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} - 12 q^{41} + 6 q^{44} - 12 q^{46} + 12 q^{49} - 2 q^{51} + 2 q^{54} + 2 q^{56} - 20 q^{61} - 2 q^{64} + 6 q^{66} - 12 q^{69} - 12 q^{71} - 14 q^{74} - 14 q^{76} - 34 q^{79} + 2 q^{81} + 2 q^{84} + 2 q^{86} + 24 q^{89} - 4 q^{91} + 18 q^{94} - 2 q^{96} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 6 * q^11 - 2 * q^14 + 2 * q^16 + 14 * q^19 - 2 * q^21 + 2 * q^24 - 4 * q^26 - 12 * q^29 - 14 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 - 12 * q^41 + 6 * q^44 - 12 * q^46 + 12 * q^49 - 2 * q^51 + 2 * q^54 + 2 * q^56 - 20 * q^61 - 2 * q^64 + 6 * q^66 - 12 * q^69 - 12 * q^71 - 14 * q^74 - 14 * q^76 - 34 * q^79 + 2 * q^81 + 2 * q^84 + 2 * q^86 + 24 * q^89 - 4 * q^91 + 18 * q^94 - 2 * q^96 + 6 * 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
$$7$$ − 1.00000i − 0.377964i −0.981981 0.188982i $$-0.939481\pi$$
0.981981 0.188982i $$-0.0605189\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 1.00000i − 0.242536i
$$18$$ 1.00000i 0.235702i
$$19$$ 7.00000 1.60591 0.802955 0.596040i $$-0.203260\pi$$
0.802955 + 0.596040i $$0.203260\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 3.00000i 0.639602i
$$23$$ − 6.00000i − 1.25109i −0.780189 0.625543i $$-0.784877\pi$$
0.780189 0.625543i $$-0.215123\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ 1.00000i 0.188982i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −7.00000 −1.25724 −0.628619 0.777714i $$-0.716379\pi$$
−0.628619 + 0.777714i $$0.716379\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 3.00000i 0.522233i
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 7.00000i − 1.15079i −0.817875 0.575396i $$-0.804848\pi$$
0.817875 0.575396i $$-0.195152\pi$$
$$38$$ − 7.00000i − 1.13555i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 1.00000i 0.154303i
$$43$$ 1.00000i 0.152499i 0.997089 + 0.0762493i $$0.0242945\pi$$
−0.997089 + 0.0762493i $$0.975706\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ 9.00000i 1.31278i 0.754420 + 0.656392i $$0.227918\pi$$
−0.754420 + 0.656392i $$0.772082\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 2.00000i 0.277350i
$$53$$ 3.00000i 0.412082i 0.978543 + 0.206041i $$0.0660580\pi$$
−0.978543 + 0.206041i $$0.933942\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ − 7.00000i − 0.927173i
$$58$$ 6.00000i 0.787839i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 7.00000i 0.889001i
$$63$$ 1.00000i 0.125988i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ 5.00000i 0.610847i 0.952217 + 0.305424i $$0.0987981\pi$$
−0.952217 + 0.305424i $$0.901202\pi$$
$$68$$ 1.00000i 0.121268i
$$69$$ −6.00000 −0.722315
$$70$$ 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
$$73$$ 16.0000i 1.87266i 0.351123 + 0.936329i $$0.385800\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ −7.00000 −0.813733
$$75$$ 0 0
$$76$$ −7.00000 −0.802955
$$77$$ 3.00000i 0.341882i
$$78$$ 2.00000i 0.226455i
$$79$$ −17.0000 −1.91265 −0.956325 0.292306i $$-0.905577\pi$$
−0.956325 + 0.292306i $$0.905577\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000i 0.662589i
$$83$$ 6.00000i 0.658586i 0.944228 + 0.329293i $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 0 0
$$86$$ 1.00000 0.107833
$$87$$ 6.00000i 0.643268i
$$88$$ − 3.00000i − 0.319801i
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 6.00000i 0.625543i
$$93$$ 7.00000i 0.725866i
$$94$$ 9.00000 0.928279
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ − 6.00000i − 0.606092i
$$99$$ 3.00000 0.301511
$$100$$ 0 0
$$101$$ −15.0000 −1.49256 −0.746278 0.665635i $$-0.768161\pi$$
−0.746278 + 0.665635i $$0.768161\pi$$
$$102$$ 1.00000i 0.0990148i
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 3.00000 0.291386
$$107$$ − 3.00000i − 0.290021i −0.989430 0.145010i $$-0.953678\pi$$
0.989430 0.145010i $$-0.0463216\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 7.00000 0.670478 0.335239 0.942133i $$-0.391183\pi$$
0.335239 + 0.942133i $$0.391183\pi$$
$$110$$ 0 0
$$111$$ −7.00000 −0.664411
$$112$$ − 1.00000i − 0.0944911i
$$113$$ − 15.0000i − 1.41108i −0.708669 0.705541i $$-0.750704\pi$$
0.708669 0.705541i $$-0.249296\pi$$
$$114$$ −7.00000 −0.655610
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 2.00000i 0.184900i
$$118$$ 0 0
$$119$$ −1.00000 −0.0916698
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 10.0000i 0.905357i
$$123$$ 6.00000i 0.541002i
$$124$$ 7.00000 0.628619
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ − 4.00000i − 0.354943i −0.984126 0.177471i $$-0.943208\pi$$
0.984126 0.177471i $$-0.0567917\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ − 3.00000i − 0.261116i
$$133$$ − 7.00000i − 0.606977i
$$134$$ 5.00000 0.431934
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ 12.0000i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ 6.00000i 0.510754i
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 9.00000 0.757937
$$142$$ 6.00000i 0.503509i
$$143$$ 6.00000i 0.501745i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 16.0000 1.32417
$$147$$ − 6.00000i − 0.494872i
$$148$$ 7.00000i 0.575396i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 7.00000i 0.567775i
$$153$$ 1.00000i 0.0808452i
$$154$$ 3.00000 0.241747
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ − 4.00000i − 0.319235i −0.987179 0.159617i $$-0.948974\pi$$
0.987179 0.159617i $$-0.0510260\pi$$
$$158$$ 17.0000i 1.35245i
$$159$$ 3.00000 0.237915
$$160$$ 0 0
$$161$$ −6.00000 −0.472866
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 8.00000i − 0.626608i −0.949653 0.313304i $$-0.898564\pi$$
0.949653 0.313304i $$-0.101436\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ 6.00000i 0.464294i 0.972681 + 0.232147i $$0.0745750\pi$$
−0.972681 + 0.232147i $$0.925425\pi$$
$$168$$ − 1.00000i − 0.0771517i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −7.00000 −0.535303
$$172$$ − 1.00000i − 0.0762493i
$$173$$ 12.0000i 0.912343i 0.889892 + 0.456172i $$0.150780\pi$$
−0.889892 + 0.456172i $$0.849220\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ − 12.0000i − 0.899438i
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ 0 0
$$181$$ 11.0000 0.817624 0.408812 0.912619i $$-0.365943\pi$$
0.408812 + 0.912619i $$0.365943\pi$$
$$182$$ 2.00000i 0.148250i
$$183$$ 10.0000i 0.739221i
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ 7.00000 0.513265
$$187$$ 3.00000i 0.219382i
$$188$$ − 9.00000i − 0.656392i
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −15.0000 −1.08536 −0.542681 0.839939i $$-0.682591\pi$$
−0.542681 + 0.839939i $$0.682591\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 16.0000i 1.15171i 0.817554 + 0.575853i $$0.195330\pi$$
−0.817554 + 0.575853i $$0.804670\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ − 3.00000i − 0.213201i
$$199$$ −5.00000 −0.354441 −0.177220 0.984171i $$-0.556711\pi$$
−0.177220 + 0.984171i $$0.556711\pi$$
$$200$$ 0 0
$$201$$ 5.00000 0.352673
$$202$$ 15.0000i 1.05540i
$$203$$ 6.00000i 0.421117i
$$204$$ 1.00000 0.0700140
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 6.00000i 0.417029i
$$208$$ − 2.00000i − 0.138675i
$$209$$ −21.0000 −1.45260
$$210$$ 0 0
$$211$$ 14.0000 0.963800 0.481900 0.876226i $$-0.339947\pi$$
0.481900 + 0.876226i $$0.339947\pi$$
$$212$$ − 3.00000i − 0.206041i
$$213$$ 6.00000i 0.411113i
$$214$$ −3.00000 −0.205076
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 7.00000i 0.475191i
$$218$$ − 7.00000i − 0.474100i
$$219$$ 16.0000 1.08118
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 7.00000i 0.469809i
$$223$$ − 2.00000i − 0.133930i −0.997755 0.0669650i $$-0.978668\pi$$
0.997755 0.0669650i $$-0.0213316\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −15.0000 −0.997785
$$227$$ 3.00000i 0.199117i 0.995032 + 0.0995585i $$0.0317430\pi$$
−0.995032 + 0.0995585i $$0.968257\pi$$
$$228$$ 7.00000i 0.463586i
$$229$$ 28.0000 1.85029 0.925146 0.379611i $$-0.123942\pi$$
0.925146 + 0.379611i $$0.123942\pi$$
$$230$$ 0 0
$$231$$ 3.00000 0.197386
$$232$$ − 6.00000i − 0.393919i
$$233$$ − 18.0000i − 1.17922i −0.807688 0.589610i $$-0.799282\pi$$
0.807688 0.589610i $$-0.200718\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 17.0000i 1.10427i
$$238$$ 1.00000i 0.0648204i
$$239$$ 9.00000 0.582162 0.291081 0.956698i $$-0.405985\pi$$
0.291081 + 0.956698i $$0.405985\pi$$
$$240$$ 0 0
$$241$$ −4.00000 −0.257663 −0.128831 0.991667i $$-0.541123\pi$$
−0.128831 + 0.991667i $$0.541123\pi$$
$$242$$ 2.00000i 0.128565i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ − 14.0000i − 0.890799i
$$248$$ − 7.00000i − 0.444500i
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ 6.00000 0.378717 0.189358 0.981908i $$-0.439359\pi$$
0.189358 + 0.981908i $$0.439359\pi$$
$$252$$ − 1.00000i − 0.0629941i
$$253$$ 18.0000i 1.13165i
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ − 1.00000i − 0.0622573i
$$259$$ −7.00000 −0.434959
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 12.0000i 0.741362i
$$263$$ − 9.00000i − 0.554964i −0.960731 0.277482i $$-0.910500\pi$$
0.960731 0.277482i $$-0.0894999\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ 0 0
$$266$$ −7.00000 −0.429198
$$267$$ − 12.0000i − 0.734388i
$$268$$ − 5.00000i − 0.305424i
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 14.0000 0.850439 0.425220 0.905090i $$-0.360197\pi$$
0.425220 + 0.905090i $$0.360197\pi$$
$$272$$ − 1.00000i − 0.0606339i
$$273$$ 2.00000i 0.121046i
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ 6.00000 0.361158
$$277$$ − 19.0000i − 1.14160i −0.821089 0.570800i $$-0.806633\pi$$
0.821089 0.570800i $$-0.193367\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ 7.00000 0.419079
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ − 9.00000i − 0.535942i
$$283$$ 22.0000i 1.30776i 0.756596 + 0.653882i $$0.226861\pi$$
−0.756596 + 0.653882i $$0.773139\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 6.00000 0.354787
$$287$$ 6.00000i 0.354169i
$$288$$ 1.00000i 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ − 16.0000i − 0.936329i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ −6.00000 −0.349927
$$295$$ 0 0
$$296$$ 7.00000 0.406867
$$297$$ − 3.00000i − 0.174078i
$$298$$ 6.00000i 0.347571i
$$299$$ −12.0000 −0.693978
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ 4.00000i 0.230174i
$$303$$ 15.0000i 0.861727i
$$304$$ 7.00000 0.401478
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ − 28.0000i − 1.59804i −0.601302 0.799022i $$-0.705351\pi$$
0.601302 0.799022i $$-0.294649\pi$$
$$308$$ − 3.00000i − 0.170941i
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ −30.0000 −1.70114 −0.850572 0.525859i $$-0.823744\pi$$
−0.850572 + 0.525859i $$0.823744\pi$$
$$312$$ − 2.00000i − 0.113228i
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ 0 0
$$316$$ 17.0000 0.956325
$$317$$ − 24.0000i − 1.34797i −0.738743 0.673987i $$-0.764580\pi$$
0.738743 0.673987i $$-0.235420\pi$$
$$318$$ − 3.00000i − 0.168232i
$$319$$ 18.0000 1.00781
$$320$$ 0 0
$$321$$ −3.00000 −0.167444
$$322$$ 6.00000i 0.334367i
$$323$$ − 7.00000i − 0.389490i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −8.00000 −0.443079
$$327$$ − 7.00000i − 0.387101i
$$328$$ − 6.00000i − 0.331295i
$$329$$ 9.00000 0.496186
$$330$$ 0 0
$$331$$ −25.0000 −1.37412 −0.687062 0.726599i $$-0.741100\pi$$
−0.687062 + 0.726599i $$0.741100\pi$$
$$332$$ − 6.00000i − 0.329293i
$$333$$ 7.00000i 0.383598i
$$334$$ 6.00000 0.328305
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ − 16.0000i − 0.871576i −0.900049 0.435788i $$-0.856470\pi$$
0.900049 0.435788i $$-0.143530\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ −15.0000 −0.814688
$$340$$ 0 0
$$341$$ 21.0000 1.13721
$$342$$ 7.00000i 0.378517i
$$343$$ − 13.0000i − 0.701934i
$$344$$ −1.00000 −0.0539164
$$345$$ 0 0
$$346$$ 12.0000 0.645124
$$347$$ − 9.00000i − 0.483145i −0.970383 0.241573i $$-0.922337\pi$$
0.970383 0.241573i $$-0.0776632\pi$$
$$348$$ − 6.00000i − 0.321634i
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 3.00000i 0.159901i
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −12.0000 −0.635999
$$357$$ 1.00000i 0.0529256i
$$358$$ 24.0000i 1.26844i
$$359$$ −21.0000 −1.10834 −0.554169 0.832404i $$-0.686964\pi$$
−0.554169 + 0.832404i $$0.686964\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ − 11.0000i − 0.578147i
$$363$$ 2.00000i 0.104973i
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 10.0000 0.522708
$$367$$ 17.0000i 0.887393i 0.896177 + 0.443696i $$0.146333\pi$$
−0.896177 + 0.443696i $$0.853667\pi$$
$$368$$ − 6.00000i − 0.312772i
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 3.00000 0.155752
$$372$$ − 7.00000i − 0.362933i
$$373$$ − 14.0000i − 0.724893i −0.932005 0.362446i $$-0.881942\pi$$
0.932005 0.362446i $$-0.118058\pi$$
$$374$$ 3.00000 0.155126
$$375$$ 0 0
$$376$$ −9.00000 −0.464140
$$377$$ 12.0000i 0.618031i
$$378$$ − 1.00000i − 0.0514344i
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 15.0000i 0.767467i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 16.0000 0.814379
$$387$$ − 1.00000i − 0.0508329i
$$388$$ 10.0000i 0.507673i
$$389$$ −9.00000 −0.456318 −0.228159 0.973624i $$-0.573271\pi$$
−0.228159 + 0.973624i $$0.573271\pi$$
$$390$$ 0 0
$$391$$ −6.00000 −0.303433
$$392$$ 6.00000i 0.303046i
$$393$$ 12.0000i 0.605320i
$$394$$ 12.0000 0.604551
$$395$$ 0 0
$$396$$ −3.00000 −0.150756
$$397$$ − 31.0000i − 1.55585i −0.628360 0.777923i $$-0.716273\pi$$
0.628360 0.777923i $$-0.283727\pi$$
$$398$$ 5.00000i 0.250627i
$$399$$ −7.00000 −0.350438
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ − 5.00000i − 0.249377i
$$403$$ 14.0000i 0.697390i
$$404$$ 15.0000 0.746278
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ 21.0000i 1.04093i
$$408$$ − 1.00000i − 0.0495074i
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 8.00000i 0.394132i
$$413$$ 0 0
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ − 4.00000i − 0.195881i
$$418$$ 21.0000i 1.02714i
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ − 14.0000i − 0.681509i
$$423$$ − 9.00000i − 0.437595i
$$424$$ −3.00000 −0.145693
$$425$$ 0 0
$$426$$ 6.00000 0.290701
$$427$$ 10.0000i 0.483934i
$$428$$ 3.00000i 0.145010i
$$429$$ 6.00000 0.289683
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 29.0000i − 1.39365i −0.717241 0.696826i $$-0.754595\pi$$
0.717241 0.696826i $$-0.245405\pi$$
$$434$$ 7.00000 0.336011
$$435$$ 0 0
$$436$$ −7.00000 −0.335239
$$437$$ − 42.0000i − 2.00913i
$$438$$ − 16.0000i − 0.764510i
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 2.00000i 0.0951303i
$$443$$ − 36.0000i − 1.71041i −0.518289 0.855206i $$-0.673431\pi$$
0.518289 0.855206i $$-0.326569\pi$$
$$444$$ 7.00000 0.332205
$$445$$ 0 0
$$446$$ −2.00000 −0.0947027
$$447$$ 6.00000i 0.283790i
$$448$$ 1.00000i 0.0472456i
$$449$$ −3.00000 −0.141579 −0.0707894 0.997491i $$-0.522552\pi$$
−0.0707894 + 0.997491i $$0.522552\pi$$
$$450$$ 0 0
$$451$$ 18.0000 0.847587
$$452$$ 15.0000i 0.705541i
$$453$$ 4.00000i 0.187936i
$$454$$ 3.00000 0.140797
$$455$$ 0 0
$$456$$ 7.00000 0.327805
$$457$$ − 1.00000i − 0.0467780i −0.999726 0.0233890i $$-0.992554\pi$$
0.999726 0.0233890i $$-0.00744563\pi$$
$$458$$ − 28.0000i − 1.30835i
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ −21.0000 −0.978068 −0.489034 0.872265i $$-0.662651\pi$$
−0.489034 + 0.872265i $$0.662651\pi$$
$$462$$ − 3.00000i − 0.139573i
$$463$$ − 20.0000i − 0.929479i −0.885448 0.464739i $$-0.846148\pi$$
0.885448 0.464739i $$-0.153852\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ 18.0000i 0.832941i 0.909149 + 0.416470i $$0.136733\pi$$
−0.909149 + 0.416470i $$0.863267\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ 5.00000 0.230879
$$470$$ 0 0
$$471$$ −4.00000 −0.184310
$$472$$ 0 0
$$473$$ − 3.00000i − 0.137940i
$$474$$ 17.0000 0.780836
$$475$$ 0 0
$$476$$ 1.00000 0.0458349
$$477$$ − 3.00000i − 0.137361i
$$478$$ − 9.00000i − 0.411650i
$$479$$ 6.00000 0.274147 0.137073 0.990561i $$-0.456230\pi$$
0.137073 + 0.990561i $$0.456230\pi$$
$$480$$ 0 0
$$481$$ −14.0000 −0.638345
$$482$$ 4.00000i 0.182195i
$$483$$ 6.00000i 0.273009i
$$484$$ 2.00000 0.0909091
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 8.00000i 0.362515i 0.983436 + 0.181257i $$0.0580167\pi$$
−0.983436 + 0.181257i $$0.941983\pi$$
$$488$$ − 10.0000i − 0.452679i
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ 6.00000i 0.270226i
$$494$$ −14.0000 −0.629890
$$495$$ 0 0
$$496$$ −7.00000 −0.314309
$$497$$ 6.00000i 0.269137i
$$498$$ − 6.00000i − 0.268866i
$$499$$ −14.0000 −0.626726 −0.313363 0.949633i $$-0.601456\pi$$
−0.313363 + 0.949633i $$0.601456\pi$$
$$500$$ 0 0
$$501$$ 6.00000 0.268060
$$502$$ − 6.00000i − 0.267793i
$$503$$ 6.00000i 0.267527i 0.991013 + 0.133763i $$0.0427062\pi$$
−0.991013 + 0.133763i $$0.957294\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 0 0
$$506$$ 18.0000 0.800198
$$507$$ − 9.00000i − 0.399704i
$$508$$ 4.00000i 0.177471i
$$509$$ −27.0000 −1.19675 −0.598377 0.801215i $$-0.704187\pi$$
−0.598377 + 0.801215i $$0.704187\pi$$
$$510$$ 0 0
$$511$$ 16.0000 0.707798
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 7.00000i 0.309058i
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ −1.00000 −0.0440225
$$517$$ − 27.0000i − 1.18746i
$$518$$ 7.00000i 0.307562i
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ −33.0000 −1.44576 −0.722878 0.690976i $$-0.757181\pi$$
−0.722878 + 0.690976i $$0.757181\pi$$
$$522$$ − 6.00000i − 0.262613i
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −9.00000 −0.392419
$$527$$ 7.00000i 0.304925i
$$528$$ 3.00000i 0.130558i
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 7.00000i 0.303488i
$$533$$ 12.0000i 0.519778i
$$534$$ −12.0000 −0.519291
$$535$$ 0 0
$$536$$ −5.00000 −0.215967
$$537$$ 24.0000i 1.03568i
$$538$$ − 6.00000i − 0.258678i
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 35.0000 1.50477 0.752384 0.658725i $$-0.228904\pi$$
0.752384 + 0.658725i $$0.228904\pi$$
$$542$$ − 14.0000i − 0.601351i
$$543$$ − 11.0000i − 0.472055i
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ 2.00000 0.0855921
$$547$$ − 22.0000i − 0.940652i −0.882493 0.470326i $$-0.844136\pi$$
0.882493 0.470326i $$-0.155864\pi$$
$$548$$ − 12.0000i − 0.512615i
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ −42.0000 −1.78926
$$552$$ − 6.00000i − 0.255377i
$$553$$ 17.0000i 0.722914i
$$554$$ −19.0000 −0.807233
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ − 21.0000i − 0.889799i −0.895581 0.444899i $$-0.853239\pi$$
0.895581 0.444899i $$-0.146761\pi$$
$$558$$ − 7.00000i − 0.296334i
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ 3.00000 0.126660
$$562$$ − 18.0000i − 0.759284i
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ −9.00000 −0.378968
$$565$$ 0 0
$$566$$ 22.0000 0.924729
$$567$$ − 1.00000i − 0.0419961i
$$568$$ − 6.00000i − 0.251754i
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ − 6.00000i − 0.250873i
$$573$$ 15.0000i 0.626634i
$$574$$ 6.00000 0.250435
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 17.0000i 0.707719i 0.935299 + 0.353860i $$0.115131\pi$$
−0.935299 + 0.353860i $$0.884869\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ 16.0000 0.664937
$$580$$ 0 0
$$581$$ 6.00000 0.248922
$$582$$ 10.0000i 0.414513i
$$583$$ − 9.00000i − 0.372742i
$$584$$ −16.0000 −0.662085
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 30.0000i − 1.23823i −0.785299 0.619116i $$-0.787491\pi$$
0.785299 0.619116i $$-0.212509\pi$$
$$588$$ 6.00000i 0.247436i
$$589$$ −49.0000 −2.01901
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ − 7.00000i − 0.287698i
$$593$$ − 48.0000i − 1.97112i −0.169316 0.985562i $$-0.554156\pi$$
0.169316 0.985562i $$-0.445844\pi$$
$$594$$ −3.00000 −0.123091
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 5.00000i 0.204636i
$$598$$ 12.0000i 0.490716i
$$599$$ 15.0000 0.612883 0.306442 0.951889i $$-0.400862\pi$$
0.306442 + 0.951889i $$0.400862\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ − 1.00000i − 0.0407570i
$$603$$ − 5.00000i − 0.203616i
$$604$$ 4.00000 0.162758
$$605$$ 0 0
$$606$$ 15.0000 0.609333
$$607$$ − 4.00000i − 0.162355i −0.996700 0.0811775i $$-0.974132\pi$$
0.996700 0.0811775i $$-0.0258681\pi$$
$$608$$ − 7.00000i − 0.283887i
$$609$$ 6.00000 0.243132
$$610$$ 0 0
$$611$$ 18.0000 0.728202
$$612$$ − 1.00000i − 0.0404226i
$$613$$ − 26.0000i − 1.05013i −0.851062 0.525065i $$-0.824041\pi$$
0.851062 0.525065i $$-0.175959\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ −3.00000 −0.120873
$$617$$ 21.0000i 0.845428i 0.906263 + 0.422714i $$0.138923\pi$$
−0.906263 + 0.422714i $$0.861077\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 6.00000 0.240772
$$622$$ 30.0000i 1.20289i
$$623$$ − 12.0000i − 0.480770i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 21.0000i 0.838659i
$$628$$ 4.00000i 0.159617i
$$629$$ −7.00000 −0.279108
$$630$$ 0 0
$$631$$ −22.0000 −0.875806 −0.437903 0.899022i $$-0.644279\pi$$
−0.437903 + 0.899022i $$0.644279\pi$$
$$632$$ − 17.0000i − 0.676224i
$$633$$ − 14.0000i − 0.556450i
$$634$$ −24.0000 −0.953162
$$635$$ 0 0
$$636$$ −3.00000 −0.118958
$$637$$ − 12.0000i − 0.475457i
$$638$$ − 18.0000i − 0.712627i
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 3.00000i 0.118401i
$$643$$ − 14.0000i − 0.552106i −0.961142 0.276053i $$-0.910973\pi$$
0.961142 0.276053i $$-0.0890266\pi$$
$$644$$ 6.00000 0.236433
$$645$$ 0 0
$$646$$ −7.00000 −0.275411
$$647$$ − 48.0000i − 1.88707i −0.331266 0.943537i $$-0.607476\pi$$
0.331266 0.943537i $$-0.392524\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 7.00000 0.274352
$$652$$ 8.00000i 0.313304i
$$653$$ 12.0000i 0.469596i 0.972044 + 0.234798i $$0.0754429\pi$$
−0.972044 + 0.234798i $$0.924557\pi$$
$$654$$ −7.00000 −0.273722
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ − 16.0000i − 0.624219i
$$658$$ − 9.00000i − 0.350857i
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 20.0000 0.777910 0.388955 0.921257i $$-0.372836\pi$$
0.388955 + 0.921257i $$0.372836\pi$$
$$662$$ 25.0000i 0.971653i
$$663$$ 2.00000i 0.0776736i
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ 7.00000 0.271244
$$667$$ 36.0000i 1.39393i
$$668$$ − 6.00000i − 0.232147i
$$669$$ −2.00000 −0.0773245
$$670$$ 0 0
$$671$$ 30.0000 1.15814
$$672$$ 1.00000i 0.0385758i
$$673$$ − 2.00000i − 0.0770943i −0.999257 0.0385472i $$-0.987727\pi$$
0.999257 0.0385472i $$-0.0122730\pi$$
$$674$$ −16.0000 −0.616297
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 48.0000i 1.84479i 0.386248 + 0.922395i $$0.373771\pi$$
−0.386248 + 0.922395i $$0.626229\pi$$
$$678$$ 15.0000i 0.576072i
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 3.00000 0.114960
$$682$$ − 21.0000i − 0.804132i
$$683$$ − 24.0000i − 0.918334i −0.888350 0.459167i $$-0.848148\pi$$
0.888350 0.459167i $$-0.151852\pi$$
$$684$$ 7.00000 0.267652
$$685$$ 0 0
$$686$$ −13.0000 −0.496342
$$687$$ − 28.0000i − 1.06827i
$$688$$ 1.00000i 0.0381246i
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ −16.0000 −0.608669 −0.304334 0.952565i $$-0.598434\pi$$
−0.304334 + 0.952565i $$0.598434\pi$$
$$692$$ − 12.0000i − 0.456172i
$$693$$ − 3.00000i − 0.113961i
$$694$$ −9.00000 −0.341635
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ 6.00000i 0.227266i
$$698$$ 26.0000i 0.984115i
$$699$$ −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
$$703$$ − 49.0000i − 1.84807i
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 15.0000i 0.564133i
$$708$$ 0 0
$$709$$ 19.0000 0.713560 0.356780 0.934188i $$-0.383875\pi$$
0.356780 + 0.934188i $$0.383875\pi$$
$$710$$ 0 0
$$711$$ 17.0000 0.637550
$$712$$ 12.0000i 0.449719i
$$713$$ 42.0000i 1.57291i
$$714$$ 1.00000 0.0374241
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ − 9.00000i − 0.336111i
$$718$$ 21.0000i 0.783713i
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ − 30.0000i − 1.11648i
$$723$$ 4.00000i 0.148762i
$$724$$ −11.0000 −0.408812
$$725$$ 0 0
$$726$$ 2.00000 0.0742270
$$727$$ 26.0000i 0.964287i 0.876092 + 0.482143i $$0.160142\pi$$
−0.876092 + 0.482143i $$0.839858\pi$$
$$728$$ − 2.00000i − 0.0741249i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 1.00000 0.0369863
$$732$$ − 10.0000i − 0.369611i
$$733$$ − 20.0000i − 0.738717i −0.929287 0.369358i $$-0.879577\pi$$
0.929287 0.369358i $$-0.120423\pi$$
$$734$$ 17.0000 0.627481
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ − 15.0000i − 0.552532i
$$738$$ − 6.00000i − 0.220863i
$$739$$ −53.0000 −1.94964 −0.974818 0.223001i $$-0.928415\pi$$
−0.974818 + 0.223001i $$0.928415\pi$$
$$740$$ 0 0
$$741$$ −14.0000 −0.514303
$$742$$ − 3.00000i − 0.110133i
$$743$$ − 42.0000i − 1.54083i −0.637542 0.770415i $$-0.720049\pi$$
0.637542 0.770415i $$-0.279951\pi$$
$$744$$ −7.00000 −0.256632
$$745$$ 0 0
$$746$$ −14.0000 −0.512576
$$747$$ − 6.00000i − 0.219529i
$$748$$ − 3.00000i − 0.109691i
$$749$$ −3.00000 −0.109618
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 9.00000i 0.328196i
$$753$$ − 6.00000i − 0.218652i
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ −1.00000 −0.0363696
$$757$$ 38.0000i 1.38113i 0.723269 + 0.690567i $$0.242639\pi$$
−0.723269 + 0.690567i $$0.757361\pi$$
$$758$$ − 16.0000i − 0.581146i
$$759$$ 18.0000 0.653359
$$760$$ 0 0
$$761$$ 36.0000 1.30500 0.652499 0.757789i $$-0.273720\pi$$
0.652499 + 0.757789i $$0.273720\pi$$
$$762$$ 4.00000i 0.144905i
$$763$$ − 7.00000i − 0.253417i
$$764$$ 15.0000 0.542681
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 37.0000 1.33425 0.667127 0.744944i $$-0.267524\pi$$
0.667127 + 0.744944i $$0.267524\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ − 16.0000i − 0.575853i
$$773$$ 30.0000i 1.07903i 0.841978 + 0.539513i $$0.181391\pi$$
−0.841978 + 0.539513i $$0.818609\pi$$
$$774$$ −1.00000 −0.0359443
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ 7.00000i 0.251124i
$$778$$ 9.00000i 0.322666i
$$779$$ −42.0000 −1.50481
$$780$$ 0 0
$$781$$ 18.0000 0.644091
$$782$$ 6.00000i 0.214560i
$$783$$ − 6.00000i − 0.214423i
$$784$$ 6.00000 0.214286
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 38.0000i 1.35455i 0.735728 + 0.677277i $$0.236840\pi$$
−0.735728 + 0.677277i $$0.763160\pi$$
$$788$$ − 12.0000i − 0.427482i
$$789$$ −9.00000 −0.320408
$$790$$ 0 0
$$791$$ −15.0000 −0.533339
$$792$$ 3.00000i 0.106600i
$$793$$ 20.0000i 0.710221i
$$794$$ −31.0000 −1.10015
$$795$$ 0 0
$$796$$ 5.00000 0.177220
$$797$$ 39.0000i 1.38145i 0.723117 + 0.690725i $$0.242709\pi$$
−0.723117 + 0.690725i $$0.757291\pi$$
$$798$$ 7.00000i 0.247797i
$$799$$ 9.00000 0.318397
$$800$$ 0 0
$$801$$ −12.0000 −0.423999
$$802$$ − 18.0000i − 0.635602i
$$803$$ − 48.0000i − 1.69388i
$$804$$ −5.00000 −0.176336
$$805$$ 0 0
$$806$$ 14.0000 0.493129
$$807$$ − 6.00000i − 0.211210i
$$808$$ − 15.0000i − 0.527698i
$$809$$ 33.0000 1.16022 0.580109 0.814539i $$-0.303010\pi$$
0.580109 + 0.814539i $$0.303010\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ − 6.00000i − 0.210559i
$$813$$ − 14.0000i − 0.491001i
$$814$$ 21.0000 0.736050
$$815$$ 0 0
$$816$$ −1.00000 −0.0350070
$$817$$ 7.00000i 0.244899i
$$818$$ 14.0000i 0.489499i
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ − 12.0000i − 0.418548i
$$823$$ 52.0000i 1.81261i 0.422628 + 0.906303i $$0.361108\pi$$
−0.422628 + 0.906303i $$0.638892\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 3.00000i − 0.104320i −0.998639 0.0521601i $$-0.983389\pi$$
0.998639 0.0521601i $$-0.0166106\pi$$
$$828$$ − 6.00000i − 0.208514i
$$829$$ 46.0000 1.59765 0.798823 0.601566i $$-0.205456\pi$$
0.798823 + 0.601566i $$0.205456\pi$$
$$830$$ 0 0
$$831$$ −19.0000 −0.659103
$$832$$ 2.00000i 0.0693375i
$$833$$ − 6.00000i − 0.207888i
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 21.0000 0.726300
$$837$$ − 7.00000i − 0.241955i
$$838$$ 24.0000i 0.829066i
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 26.0000i − 0.896019i
$$843$$ − 18.0000i − 0.619953i
$$844$$ −14.0000 −0.481900
$$845$$ 0 0
$$846$$ −9.00000 −0.309426
$$847$$ 2.00000i 0.0687208i
$$848$$ 3.00000i 0.103020i
$$849$$ 22.0000 0.755038
$$850$$ 0 0
$$851$$ −42.0000 −1.43974
$$852$$ − 6.00000i − 0.205557i
$$853$$ 1.00000i 0.0342393i 0.999853 + 0.0171197i $$0.00544963\pi$$
−0.999853 + 0.0171197i $$0.994550\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 0 0
$$856$$ 3.00000 0.102538
$$857$$ − 3.00000i − 0.102478i −0.998686 0.0512390i $$-0.983683\pi$$
0.998686 0.0512390i $$-0.0163170\pi$$
$$858$$ − 6.00000i − 0.204837i
$$859$$ −41.0000 −1.39890 −0.699451 0.714681i $$-0.746572\pi$$
−0.699451 + 0.714681i $$0.746572\pi$$
$$860$$ 0 0
$$861$$ 6.00000 0.204479
$$862$$ − 12.0000i − 0.408722i
$$863$$ − 45.0000i − 1.53182i −0.642949 0.765909i $$-0.722289\pi$$
0.642949 0.765909i $$-0.277711\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −29.0000 −0.985460
$$867$$ 1.00000i 0.0339618i
$$868$$ − 7.00000i − 0.237595i
$$869$$ 51.0000 1.73006
$$870$$ 0 0
$$871$$ 10.0000 0.338837
$$872$$ 7.00000i 0.237050i
$$873$$ 10.0000i 0.338449i
$$874$$ −42.0000 −1.42067
$$875$$ 0 0
$$876$$ −16.0000 −0.540590
$$877$$ − 34.0000i − 1.14810i −0.818821 0.574049i $$-0.805372\pi$$
0.818821 0.574049i $$-0.194628\pi$$
$$878$$ 8.00000i 0.269987i
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ 33.0000 1.11180 0.555899 0.831250i $$-0.312374\pi$$
0.555899 + 0.831250i $$0.312374\pi$$
$$882$$ 6.00000i 0.202031i
$$883$$ − 20.0000i − 0.673054i −0.941674 0.336527i $$-0.890748\pi$$
0.941674 0.336527i $$-0.109252\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ − 7.00000i − 0.234905i
$$889$$ −4.00000 −0.134156
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ 2.00000i 0.0669650i
$$893$$ 63.0000i 2.10821i
$$894$$ 6.00000 0.200670
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 12.0000i 0.400668i
$$898$$ 3.00000i 0.100111i
$$899$$ 42.0000 1.40078
$$900$$ 0 0
$$901$$ 3.00000 0.0999445
$$902$$ − 18.0000i − 0.599334i
$$903$$ − 1.00000i − 0.0332779i
$$904$$ 15.0000 0.498893
$$905$$ 0 0
$$906$$ 4.00000 0.132891
$$907$$ − 22.0000i − 0.730498i −0.930910 0.365249i $$-0.880984\pi$$
0.930910 0.365249i $$-0.119016\pi$$
$$908$$ − 3.00000i − 0.0995585i
$$909$$ 15.0000 0.497519
$$910$$ 0 0
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ − 7.00000i − 0.231793i
$$913$$ − 18.0000i − 0.595713i
$$914$$ −1.00000 −0.0330771
$$915$$ 0 0
$$916$$ −28.0000 −0.925146
$$917$$ 12.0000i 0.396275i
$$918$$ − 1.00000i − 0.0330049i
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ 21.0000i 0.691598i
$$923$$ 12.0000i 0.394985i
$$924$$ −3.00000 −0.0986928
$$925$$ 0 0
$$926$$ −20.0000 −0.657241
$$927$$ 8.00000i 0.262754i
$$928$$ 6.00000i 0.196960i
$$929$$ −45.0000 −1.47640 −0.738201 0.674581i $$-0.764324\pi$$
−0.738201 + 0.674581i $$0.764324\pi$$
$$930$$ 0 0
$$931$$ 42.0000 1.37649
$$932$$ 18.0000i 0.589610i
$$933$$ 30.0000i 0.982156i
$$934$$ 18.0000 0.588978
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ − 22.0000i − 0.718709i −0.933201 0.359354i $$-0.882997\pi$$
0.933201 0.359354i $$-0.117003\pi$$
$$938$$ − 5.00000i − 0.163256i
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 4.00000i 0.130327i
$$943$$ 36.0000i 1.17232i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −3.00000 −0.0975384
$$947$$ 51.0000i 1.65728i 0.559784 + 0.828639i $$0.310884\pi$$
−0.559784 + 0.828639i $$0.689116\pi$$
$$948$$ − 17.0000i − 0.552134i
$$949$$ 32.0000 1.03876
$$950$$ 0 0
$$951$$ −24.0000 −0.778253
$$952$$ − 1.00000i − 0.0324102i
$$953$$ 18.0000i 0.583077i 0.956559 + 0.291539i $$0.0941672\pi$$
−0.956559 + 0.291539i $$0.905833\pi$$
$$954$$ −3.00000 −0.0971286
$$955$$ 0 0
$$956$$ −9.00000 −0.291081
$$957$$ − 18.0000i − 0.581857i
$$958$$ − 6.00000i − 0.193851i
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 14.0000i 0.451378i
$$963$$ 3.00000i 0.0966736i
$$964$$ 4.00000 0.128831
$$965$$ 0 0
$$966$$ 6.00000 0.193047
$$967$$ − 4.00000i − 0.128631i −0.997930 0.0643157i $$-0.979514\pi$$
0.997930 0.0643157i $$-0.0204865\pi$$
$$968$$ − 2.00000i − 0.0642824i
$$969$$ −7.00000 −0.224872
$$970$$ 0 0
$$971$$ 30.0000 0.962746 0.481373 0.876516i $$-0.340138\pi$$
0.481373 + 0.876516i $$0.340138\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 4.00000i − 0.128234i
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ 8.00000i 0.255812i
$$979$$ −36.0000 −1.15056
$$980$$ 0 0
$$981$$ −7.00000 −0.223493
$$982$$ 36.0000i 1.14881i
$$983$$ − 42.0000i − 1.33959i −0.742545 0.669796i $$-0.766382\pi$$
0.742545 0.669796i $$-0.233618\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 6.00000 0.191079
$$987$$ − 9.00000i − 0.286473i
$$988$$ 14.0000i 0.445399i
$$989$$ 6.00000 0.190789
$$990$$ 0 0
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ 7.00000i 0.222250i
$$993$$ 25.0000i 0.793351i
$$994$$ 6.00000 0.190308
$$995$$ 0 0
$$996$$ −6.00000 −0.190117
$$997$$ − 55.0000i − 1.74187i −0.491400 0.870934i $$-0.663515\pi$$
0.491400 0.870934i $$-0.336485\pi$$
$$998$$ 14.0000i 0.443162i
$$999$$ 7.00000 0.221470
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.c.2449.1 2
5.2 odd 4 2550.2.a.w.1.1 yes 1
5.3 odd 4 2550.2.a.j.1.1 1
5.4 even 2 inner 2550.2.d.c.2449.2 2
15.2 even 4 7650.2.a.v.1.1 1
15.8 even 4 7650.2.a.bt.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.j.1.1 1 5.3 odd 4
2550.2.a.w.1.1 yes 1 5.2 odd 4
2550.2.d.c.2449.1 2 1.1 even 1 trivial
2550.2.d.c.2449.2 2 5.4 even 2 inner
7650.2.a.v.1.1 1 15.2 even 4
7650.2.a.bt.1.1 1 15.8 even 4