# Properties

 Label 2550.2.d.o.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.o.2449.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} +4.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} +5.00000 q^{19} +3.00000 q^{21} +3.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +4.00000 q^{26} -1.00000i q^{27} +3.00000i q^{28} +7.00000 q^{31} -1.00000i q^{32} -3.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} -3.00000i q^{37} -5.00000i q^{38} -4.00000 q^{39} +2.00000 q^{41} -3.00000i q^{42} -1.00000i q^{43} +3.00000 q^{44} +4.00000 q^{46} -3.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} +1.00000 q^{51} -4.00000i q^{52} -11.0000i q^{53} -1.00000 q^{54} +3.00000 q^{56} +5.00000i q^{57} +2.00000 q^{61} -7.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} -13.0000i q^{67} +1.00000i q^{68} -4.00000 q^{69} +2.00000 q^{71} -1.00000i q^{72} -6.00000i q^{73} -3.00000 q^{74} -5.00000 q^{76} +9.00000i q^{77} +4.00000i q^{78} +5.00000 q^{79} +1.00000 q^{81} -2.00000i q^{82} -16.0000i q^{83} -3.00000 q^{84} -1.00000 q^{86} -3.00000i q^{88} +10.0000 q^{89} +12.0000 q^{91} -4.00000i q^{92} +7.00000i q^{93} -3.00000 q^{94} +1.00000 q^{96} +2.00000i q^{97} +2.00000i q^{98} +3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 6 q^{11} - 6 q^{14} + 2 q^{16} + 10 q^{19} + 6 q^{21} - 2 q^{24} + 8 q^{26} + 14 q^{31} - 2 q^{34} + 2 q^{36} - 8 q^{39} + 4 q^{41} + 6 q^{44} + 8 q^{46} - 4 q^{49} + 2 q^{51} - 2 q^{54} + 6 q^{56} + 4 q^{61} - 2 q^{64} - 6 q^{66} - 8 q^{69} + 4 q^{71} - 6 q^{74} - 10 q^{76} + 10 q^{79} + 2 q^{81} - 6 q^{84} - 2 q^{86} + 20 q^{89} + 24 q^{91} - 6 q^{94} + 2 q^{96} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 6 * q^11 - 6 * q^14 + 2 * q^16 + 10 * q^19 + 6 * q^21 - 2 * q^24 + 8 * q^26 + 14 * q^31 - 2 * q^34 + 2 * q^36 - 8 * q^39 + 4 * q^41 + 6 * q^44 + 8 * q^46 - 4 * q^49 + 2 * q^51 - 2 * q^54 + 6 * q^56 + 4 * q^61 - 2 * q^64 - 6 * q^66 - 8 * q^69 + 4 * q^71 - 6 * q^74 - 10 * q^76 + 10 * q^79 + 2 * q^81 - 6 * q^84 - 2 * q^86 + 20 * q^89 + 24 * q^91 - 6 * 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
$$7$$ − 3.00000i − 1.13389i −0.823754 0.566947i $$-0.808125\pi$$
0.823754 0.566947i $$-0.191875\pi$$
$$8$$ 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
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ −3.00000 −0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 1.00000i − 0.242536i
$$18$$ 1.00000i 0.235702i
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ 3.00000i 0.639602i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ − 1.00000i − 0.192450i
$$28$$ 3.00000i 0.566947i
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 3.00000i − 0.522233i
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 3.00000i − 0.493197i −0.969118 0.246598i $$-0.920687\pi$$
0.969118 0.246598i $$-0.0793129\pi$$
$$38$$ − 5.00000i − 0.811107i
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ − 3.00000i − 0.462910i
$$43$$ − 1.00000i − 0.152499i −0.997089 0.0762493i $$-0.975706\pi$$
0.997089 0.0762493i $$-0.0242945\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ − 3.00000i − 0.437595i −0.975770 0.218797i $$-0.929787\pi$$
0.975770 0.218797i $$-0.0702134\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 1.00000 0.140028
$$52$$ − 4.00000i − 0.554700i
$$53$$ − 11.0000i − 1.51097i −0.655168 0.755483i $$-0.727402\pi$$
0.655168 0.755483i $$-0.272598\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ 5.00000i 0.662266i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ − 7.00000i − 0.889001i
$$63$$ 3.00000i 0.377964i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −3.00000 −0.369274
$$67$$ − 13.0000i − 1.58820i −0.607785 0.794101i $$-0.707942\pi$$
0.607785 0.794101i $$-0.292058\pi$$
$$68$$ 1.00000i 0.121268i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ −3.00000 −0.348743
$$75$$ 0 0
$$76$$ −5.00000 −0.573539
$$77$$ 9.00000i 1.02565i
$$78$$ 4.00000i 0.452911i
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 2.00000i − 0.220863i
$$83$$ − 16.0000i − 1.75623i −0.478451 0.878114i $$-0.658802\pi$$
0.478451 0.878114i $$-0.341198\pi$$
$$84$$ −3.00000 −0.327327
$$85$$ 0 0
$$86$$ −1.00000 −0.107833
$$87$$ 0 0
$$88$$ − 3.00000i − 0.319801i
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ − 4.00000i − 0.417029i
$$93$$ 7.00000i 0.725866i
$$94$$ −3.00000 −0.309426
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 2.00000i 0.202031i
$$99$$ 3.00000 0.301511
$$100$$ 0 0
$$101$$ 7.00000 0.696526 0.348263 0.937397i $$-0.386772\pi$$
0.348263 + 0.937397i $$0.386772\pi$$
$$102$$ − 1.00000i − 0.0990148i
$$103$$ 14.0000i 1.37946i 0.724066 + 0.689730i $$0.242271\pi$$
−0.724066 + 0.689730i $$0.757729\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 0 0
$$106$$ −11.0000 −1.06841
$$107$$ 17.0000i 1.64345i 0.569883 + 0.821726i $$0.306989\pi$$
−0.569883 + 0.821726i $$0.693011\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ −5.00000 −0.478913 −0.239457 0.970907i $$-0.576969\pi$$
−0.239457 + 0.970907i $$0.576969\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ − 3.00000i − 0.283473i
$$113$$ − 1.00000i − 0.0940721i −0.998893 0.0470360i $$-0.985022\pi$$
0.998893 0.0470360i $$-0.0149776\pi$$
$$114$$ 5.00000 0.468293
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 4.00000i − 0.369800i
$$118$$ 0 0
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ − 2.00000i − 0.181071i
$$123$$ 2.00000i 0.180334i
$$124$$ −7.00000 −0.628619
$$125$$ 0 0
$$126$$ 3.00000 0.267261
$$127$$ − 18.0000i − 1.59724i −0.601834 0.798621i $$-0.705563\pi$$
0.601834 0.798621i $$-0.294437\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 3.00000i 0.261116i
$$133$$ − 15.0000i − 1.30066i
$$134$$ −13.0000 −1.12303
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ 2.00000i 0.170872i 0.996344 + 0.0854358i $$0.0272282\pi$$
−0.996344 + 0.0854358i $$0.972772\pi$$
$$138$$ 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
$$141$$ 3.00000 0.252646
$$142$$ − 2.00000i − 0.167836i
$$143$$ − 12.0000i − 1.00349i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −6.00000 −0.496564
$$147$$ − 2.00000i − 0.164957i
$$148$$ 3.00000i 0.246598i
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 5.00000i 0.405554i
$$153$$ 1.00000i 0.0808452i
$$154$$ 9.00000 0.725241
$$155$$ 0 0
$$156$$ 4.00000 0.320256
$$157$$ 12.0000i 0.957704i 0.877896 + 0.478852i $$0.158947\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ − 5.00000i − 0.397779i
$$159$$ 11.0000 0.872357
$$160$$ 0 0
$$161$$ 12.0000 0.945732
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 6.00000i − 0.469956i −0.972001 0.234978i $$-0.924498\pi$$
0.972001 0.234978i $$-0.0755019\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ −16.0000 −1.24184
$$167$$ − 18.0000i − 1.39288i −0.717614 0.696441i $$-0.754766\pi$$
0.717614 0.696441i $$-0.245234\pi$$
$$168$$ 3.00000i 0.231455i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ −5.00000 −0.382360
$$172$$ 1.00000i 0.0762493i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ − 10.0000i − 0.749532i
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 7.00000 0.520306 0.260153 0.965567i $$-0.416227\pi$$
0.260153 + 0.965567i $$0.416227\pi$$
$$182$$ − 12.0000i − 0.889499i
$$183$$ 2.00000i 0.147844i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ 7.00000 0.513265
$$187$$ 3.00000i 0.219382i
$$188$$ 3.00000i 0.218797i
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ −3.00000 −0.217072 −0.108536 0.994092i $$-0.534616\pi$$
−0.108536 + 0.994092i $$0.534616\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ − 6.00000i − 0.431889i −0.976406 0.215945i $$-0.930717\pi$$
0.976406 0.215945i $$-0.0692831\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ − 3.00000i − 0.213201i
$$199$$ 25.0000 1.77220 0.886102 0.463491i $$-0.153403\pi$$
0.886102 + 0.463491i $$0.153403\pi$$
$$200$$ 0 0
$$201$$ 13.0000 0.916949
$$202$$ − 7.00000i − 0.492518i
$$203$$ 0 0
$$204$$ −1.00000 −0.0700140
$$205$$ 0 0
$$206$$ 14.0000 0.975426
$$207$$ − 4.00000i − 0.278019i
$$208$$ 4.00000i 0.277350i
$$209$$ −15.0000 −1.03757
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 11.0000i 0.755483i
$$213$$ 2.00000i 0.137038i
$$214$$ 17.0000 1.16210
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ − 21.0000i − 1.42557i
$$218$$ 5.00000i 0.338643i
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ − 3.00000i − 0.201347i
$$223$$ − 26.0000i − 1.74109i −0.492090 0.870544i $$-0.663767\pi$$
0.492090 0.870544i $$-0.336233\pi$$
$$224$$ −3.00000 −0.200446
$$225$$ 0 0
$$226$$ −1.00000 −0.0665190
$$227$$ 27.0000i 1.79205i 0.444001 + 0.896026i $$0.353559\pi$$
−0.444001 + 0.896026i $$0.646441\pi$$
$$228$$ − 5.00000i − 0.331133i
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ −9.00000 −0.592157
$$232$$ 0 0
$$233$$ − 26.0000i − 1.70332i −0.524097 0.851658i $$-0.675597\pi$$
0.524097 0.851658i $$-0.324403\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 5.00000i 0.324785i
$$238$$ 3.00000i 0.194461i
$$239$$ −15.0000 −0.970269 −0.485135 0.874439i $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$240$$ 0 0
$$241$$ 12.0000 0.772988 0.386494 0.922292i $$-0.373686\pi$$
0.386494 + 0.922292i $$0.373686\pi$$
$$242$$ 2.00000i 0.128565i
$$243$$ 1.00000i 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 2.00000 0.127515
$$247$$ 20.0000i 1.27257i
$$248$$ 7.00000i 0.444500i
$$249$$ 16.0000 1.01396
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ − 3.00000i − 0.188982i
$$253$$ − 12.0000i − 0.754434i
$$254$$ −18.0000 −1.12942
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 8.00000i − 0.499026i −0.968371 0.249513i $$-0.919729\pi$$
0.968371 0.249513i $$-0.0802706\pi$$
$$258$$ − 1.00000i − 0.0622573i
$$259$$ −9.00000 −0.559233
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 12.0000i − 0.741362i
$$263$$ 19.0000i 1.17159i 0.810459 + 0.585795i $$0.199218\pi$$
−0.810459 + 0.585795i $$0.800782\pi$$
$$264$$ 3.00000 0.184637
$$265$$ 0 0
$$266$$ −15.0000 −0.919709
$$267$$ 10.0000i 0.611990i
$$268$$ 13.0000i 0.794101i
$$269$$ −20.0000 −1.21942 −0.609711 0.792624i $$-0.708714\pi$$
−0.609711 + 0.792624i $$0.708714\pi$$
$$270$$ 0 0
$$271$$ 32.0000 1.94386 0.971931 0.235267i $$-0.0755965\pi$$
0.971931 + 0.235267i $$0.0755965\pi$$
$$272$$ − 1.00000i − 0.0606339i
$$273$$ 12.0000i 0.726273i
$$274$$ 2.00000 0.120824
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ − 23.0000i − 1.38194i −0.722885 0.690968i $$-0.757185\pi$$
0.722885 0.690968i $$-0.242815\pi$$
$$278$$ − 20.0000i − 1.19952i
$$279$$ −7.00000 −0.419079
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ − 3.00000i − 0.178647i
$$283$$ 14.0000i 0.832214i 0.909316 + 0.416107i $$0.136606\pi$$
−0.909316 + 0.416107i $$0.863394\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ 0 0
$$286$$ −12.0000 −0.709575
$$287$$ − 6.00000i − 0.354169i
$$288$$ 1.00000i 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 6.00000i 0.351123i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ 3.00000 0.174371
$$297$$ 3.00000i 0.174078i
$$298$$ 10.0000i 0.579284i
$$299$$ −16.0000 −0.925304
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ − 12.0000i − 0.690522i
$$303$$ 7.00000i 0.402139i
$$304$$ 5.00000 0.286770
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ 32.0000i 1.82634i 0.407583 + 0.913168i $$0.366372\pi$$
−0.407583 + 0.913168i $$0.633628\pi$$
$$308$$ − 9.00000i − 0.512823i
$$309$$ −14.0000 −0.796432
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ − 4.00000i − 0.226455i
$$313$$ 4.00000i 0.226093i 0.993590 + 0.113047i $$0.0360610\pi$$
−0.993590 + 0.113047i $$0.963939\pi$$
$$314$$ 12.0000 0.677199
$$315$$ 0 0
$$316$$ −5.00000 −0.281272
$$317$$ 32.0000i 1.79730i 0.438667 + 0.898650i $$0.355451\pi$$
−0.438667 + 0.898650i $$0.644549\pi$$
$$318$$ − 11.0000i − 0.616849i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −17.0000 −0.948847
$$322$$ − 12.0000i − 0.668734i
$$323$$ − 5.00000i − 0.278207i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −6.00000 −0.332309
$$327$$ − 5.00000i − 0.276501i
$$328$$ 2.00000i 0.110432i
$$329$$ −9.00000 −0.496186
$$330$$ 0 0
$$331$$ −23.0000 −1.26419 −0.632097 0.774889i $$-0.717806\pi$$
−0.632097 + 0.774889i $$0.717806\pi$$
$$332$$ 16.0000i 0.878114i
$$333$$ 3.00000i 0.164399i
$$334$$ −18.0000 −0.984916
$$335$$ 0 0
$$336$$ 3.00000 0.163663
$$337$$ 12.0000i 0.653682i 0.945079 + 0.326841i $$0.105984\pi$$
−0.945079 + 0.326841i $$0.894016\pi$$
$$338$$ 3.00000i 0.163178i
$$339$$ 1.00000 0.0543125
$$340$$ 0 0
$$341$$ −21.0000 −1.13721
$$342$$ 5.00000i 0.270369i
$$343$$ − 15.0000i − 0.809924i
$$344$$ 1.00000 0.0539164
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 27.0000i 1.44944i 0.689046 + 0.724718i $$0.258030\pi$$
−0.689046 + 0.724718i $$0.741970\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 4.00000 0.213504
$$352$$ 3.00000i 0.159901i
$$353$$ − 16.0000i − 0.851594i −0.904819 0.425797i $$-0.859994\pi$$
0.904819 0.425797i $$-0.140006\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ − 3.00000i − 0.158777i
$$358$$ 0 0
$$359$$ −5.00000 −0.263890 −0.131945 0.991257i $$-0.542122\pi$$
−0.131945 + 0.991257i $$0.542122\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ − 7.00000i − 0.367912i
$$363$$ − 2.00000i − 0.104973i
$$364$$ −12.0000 −0.628971
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ 7.00000i 0.365397i 0.983169 + 0.182699i $$0.0584832\pi$$
−0.983169 + 0.182699i $$0.941517\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ −33.0000 −1.71327
$$372$$ − 7.00000i − 0.362933i
$$373$$ 14.0000i 0.724893i 0.932005 + 0.362446i $$0.118058\pi$$
−0.932005 + 0.362446i $$0.881942\pi$$
$$374$$ 3.00000 0.155126
$$375$$ 0 0
$$376$$ 3.00000 0.154713
$$377$$ 0 0
$$378$$ 3.00000i 0.154303i
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 18.0000 0.922168
$$382$$ 3.00000i 0.153493i
$$383$$ 4.00000i 0.204390i 0.994764 + 0.102195i $$0.0325866\pi$$
−0.994764 + 0.102195i $$0.967413\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ 1.00000i 0.0508329i
$$388$$ − 2.00000i − 0.101535i
$$389$$ 5.00000 0.253510 0.126755 0.991934i $$-0.459544\pi$$
0.126755 + 0.991934i $$0.459544\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ − 2.00000i − 0.101015i
$$393$$ 12.0000i 0.605320i
$$394$$ 12.0000 0.604551
$$395$$ 0 0
$$396$$ −3.00000 −0.150756
$$397$$ − 3.00000i − 0.150566i −0.997162 0.0752828i $$-0.976014\pi$$
0.997162 0.0752828i $$-0.0239860\pi$$
$$398$$ − 25.0000i − 1.25314i
$$399$$ 15.0000 0.750939
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ − 13.0000i − 0.648381i
$$403$$ 28.0000i 1.39478i
$$404$$ −7.00000 −0.348263
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 9.00000i 0.446113i
$$408$$ 1.00000i 0.0495074i
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ −2.00000 −0.0986527
$$412$$ − 14.0000i − 0.689730i
$$413$$ 0 0
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ 20.0000i 0.979404i
$$418$$ 15.0000i 0.733674i
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ −28.0000 −1.36464 −0.682318 0.731055i $$-0.739028\pi$$
−0.682318 + 0.731055i $$0.739028\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ 3.00000i 0.145865i
$$424$$ 11.0000 0.534207
$$425$$ 0 0
$$426$$ 2.00000 0.0969003
$$427$$ − 6.00000i − 0.290360i
$$428$$ − 17.0000i − 0.821726i
$$429$$ 12.0000 0.579365
$$430$$ 0 0
$$431$$ 2.00000 0.0963366 0.0481683 0.998839i $$-0.484662\pi$$
0.0481683 + 0.998839i $$0.484662\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 21.0000i − 1.00920i −0.863355 0.504598i $$-0.831641\pi$$
0.863355 0.504598i $$-0.168359\pi$$
$$434$$ −21.0000 −1.00803
$$435$$ 0 0
$$436$$ 5.00000 0.239457
$$437$$ 20.0000i 0.956730i
$$438$$ − 6.00000i − 0.286691i
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ − 4.00000i − 0.190261i
$$443$$ 14.0000i 0.665160i 0.943075 + 0.332580i $$0.107919\pi$$
−0.943075 + 0.332580i $$0.892081\pi$$
$$444$$ −3.00000 −0.142374
$$445$$ 0 0
$$446$$ −26.0000 −1.23114
$$447$$ − 10.0000i − 0.472984i
$$448$$ 3.00000i 0.141737i
$$449$$ −5.00000 −0.235965 −0.117982 0.993016i $$-0.537643\pi$$
−0.117982 + 0.993016i $$0.537643\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 1.00000i 0.0470360i
$$453$$ 12.0000i 0.563809i
$$454$$ 27.0000 1.26717
$$455$$ 0 0
$$456$$ −5.00000 −0.234146
$$457$$ − 13.0000i − 0.608114i −0.952654 0.304057i $$-0.901659\pi$$
0.952654 0.304057i $$-0.0983414\pi$$
$$458$$ 0 0
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ 37.0000 1.72326 0.861631 0.507535i $$-0.169443\pi$$
0.861631 + 0.507535i $$0.169443\pi$$
$$462$$ 9.00000i 0.418718i
$$463$$ 24.0000i 1.11537i 0.830051 + 0.557687i $$0.188311\pi$$
−0.830051 + 0.557687i $$0.811689\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −26.0000 −1.20443
$$467$$ − 18.0000i − 0.832941i −0.909149 0.416470i $$-0.863267\pi$$
0.909149 0.416470i $$-0.136733\pi$$
$$468$$ 4.00000i 0.184900i
$$469$$ −39.0000 −1.80085
$$470$$ 0 0
$$471$$ −12.0000 −0.552931
$$472$$ 0 0
$$473$$ 3.00000i 0.137940i
$$474$$ 5.00000 0.229658
$$475$$ 0 0
$$476$$ 3.00000 0.137505
$$477$$ 11.0000i 0.503655i
$$478$$ 15.0000i 0.686084i
$$479$$ 10.0000 0.456912 0.228456 0.973554i $$-0.426632\pi$$
0.228456 + 0.973554i $$0.426632\pi$$
$$480$$ 0 0
$$481$$ 12.0000 0.547153
$$482$$ − 12.0000i − 0.546585i
$$483$$ 12.0000i 0.546019i
$$484$$ 2.00000 0.0909091
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 32.0000i 1.45006i 0.688718 + 0.725029i $$0.258174\pi$$
−0.688718 + 0.725029i $$0.741826\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ 6.00000 0.271329
$$490$$ 0 0
$$491$$ 2.00000 0.0902587 0.0451294 0.998981i $$-0.485630\pi$$
0.0451294 + 0.998981i $$0.485630\pi$$
$$492$$ − 2.00000i − 0.0901670i
$$493$$ 0 0
$$494$$ 20.0000 0.899843
$$495$$ 0 0
$$496$$ 7.00000 0.314309
$$497$$ − 6.00000i − 0.269137i
$$498$$ − 16.0000i − 0.716977i
$$499$$ −10.0000 −0.447661 −0.223831 0.974628i $$-0.571856\pi$$
−0.223831 + 0.974628i $$0.571856\pi$$
$$500$$ 0 0
$$501$$ 18.0000 0.804181
$$502$$ 18.0000i 0.803379i
$$503$$ 14.0000i 0.624229i 0.950044 + 0.312115i $$0.101037\pi$$
−0.950044 + 0.312115i $$0.898963\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 0 0
$$506$$ −12.0000 −0.533465
$$507$$ − 3.00000i − 0.133235i
$$508$$ 18.0000i 0.798621i
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ 0 0
$$511$$ −18.0000 −0.796273
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 5.00000i − 0.220755i
$$514$$ −8.00000 −0.352865
$$515$$ 0 0
$$516$$ −1.00000 −0.0440225
$$517$$ 9.00000i 0.395820i
$$518$$ 9.00000i 0.395437i
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −3.00000 −0.131432 −0.0657162 0.997838i $$-0.520933\pi$$
−0.0657162 + 0.997838i $$0.520933\pi$$
$$522$$ 0 0
$$523$$ 44.0000i 1.92399i 0.273075 + 0.961993i $$0.411959\pi$$
−0.273075 + 0.961993i $$0.588041\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 19.0000 0.828439
$$527$$ − 7.00000i − 0.304925i
$$528$$ − 3.00000i − 0.130558i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 15.0000i 0.650332i
$$533$$ 8.00000i 0.346518i
$$534$$ 10.0000 0.432742
$$535$$ 0 0
$$536$$ 13.0000 0.561514
$$537$$ 0 0
$$538$$ 20.0000i 0.862261i
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −13.0000 −0.558914 −0.279457 0.960158i $$-0.590154\pi$$
−0.279457 + 0.960158i $$0.590154\pi$$
$$542$$ − 32.0000i − 1.37452i
$$543$$ 7.00000i 0.300399i
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ 12.0000 0.513553
$$547$$ 2.00000i 0.0855138i 0.999086 + 0.0427569i $$0.0136141\pi$$
−0.999086 + 0.0427569i $$0.986386\pi$$
$$548$$ − 2.00000i − 0.0854358i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 0 0
$$552$$ − 4.00000i − 0.170251i
$$553$$ − 15.0000i − 0.637865i
$$554$$ −23.0000 −0.977176
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ − 23.0000i − 0.974541i −0.873251 0.487271i $$-0.837993\pi$$
0.873251 0.487271i $$-0.162007\pi$$
$$558$$ 7.00000i 0.296334i
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ −3.00000 −0.126660
$$562$$ 18.0000i 0.759284i
$$563$$ 24.0000i 1.01148i 0.862686 + 0.505740i $$0.168780\pi$$
−0.862686 + 0.505740i $$0.831220\pi$$
$$564$$ −3.00000 −0.126323
$$565$$ 0 0
$$566$$ 14.0000 0.588464
$$567$$ − 3.00000i − 0.125988i
$$568$$ 2.00000i 0.0839181i
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ 2.00000 0.0836974 0.0418487 0.999124i $$-0.486675\pi$$
0.0418487 + 0.999124i $$0.486675\pi$$
$$572$$ 12.0000i 0.501745i
$$573$$ − 3.00000i − 0.125327i
$$574$$ −6.00000 −0.250435
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 43.0000i − 1.79011i −0.445952 0.895057i $$-0.647135\pi$$
0.445952 0.895057i $$-0.352865\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ 6.00000 0.249351
$$580$$ 0 0
$$581$$ −48.0000 −1.99138
$$582$$ 2.00000i 0.0829027i
$$583$$ 33.0000i 1.36672i
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ − 18.0000i − 0.742940i −0.928445 0.371470i $$-0.878854\pi$$
0.928445 0.371470i $$-0.121146\pi$$
$$588$$ 2.00000i 0.0824786i
$$589$$ 35.0000 1.44215
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ − 3.00000i − 0.123299i
$$593$$ − 36.0000i − 1.47834i −0.673517 0.739171i $$-0.735217\pi$$
0.673517 0.739171i $$-0.264783\pi$$
$$594$$ 3.00000 0.123091
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 25.0000i 1.02318i
$$598$$ 16.0000i 0.654289i
$$599$$ 35.0000 1.43006 0.715031 0.699093i $$-0.246413\pi$$
0.715031 + 0.699093i $$0.246413\pi$$
$$600$$ 0 0
$$601$$ −28.0000 −1.14214 −0.571072 0.820900i $$-0.693472\pi$$
−0.571072 + 0.820900i $$0.693472\pi$$
$$602$$ 3.00000i 0.122271i
$$603$$ 13.0000i 0.529401i
$$604$$ −12.0000 −0.488273
$$605$$ 0 0
$$606$$ 7.00000 0.284356
$$607$$ − 8.00000i − 0.324710i −0.986732 0.162355i $$-0.948091\pi$$
0.986732 0.162355i $$-0.0519090\pi$$
$$608$$ − 5.00000i − 0.202777i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ − 1.00000i − 0.0404226i
$$613$$ − 46.0000i − 1.85792i −0.370177 0.928961i $$-0.620703\pi$$
0.370177 0.928961i $$-0.379297\pi$$
$$614$$ 32.0000 1.29141
$$615$$ 0 0
$$616$$ −9.00000 −0.362620
$$617$$ − 33.0000i − 1.32853i −0.747497 0.664265i $$-0.768745\pi$$
0.747497 0.664265i $$-0.231255\pi$$
$$618$$ 14.0000i 0.563163i
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 18.0000i 0.721734i
$$623$$ − 30.0000i − 1.20192i
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ 4.00000 0.159872
$$627$$ − 15.0000i − 0.599042i
$$628$$ − 12.0000i − 0.478852i
$$629$$ −3.00000 −0.119618
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 5.00000i 0.198889i
$$633$$ 12.0000i 0.476957i
$$634$$ 32.0000 1.27088
$$635$$ 0 0
$$636$$ −11.0000 −0.436178
$$637$$ − 8.00000i − 0.316972i
$$638$$ 0 0
$$639$$ −2.00000 −0.0791188
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 17.0000i 0.670936i
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ −12.0000 −0.472866
$$645$$ 0 0
$$646$$ −5.00000 −0.196722
$$647$$ − 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
$$651$$ 21.0000 0.823055
$$652$$ 6.00000i 0.234978i
$$653$$ − 6.00000i − 0.234798i −0.993085 0.117399i $$-0.962544\pi$$
0.993085 0.117399i $$-0.0374557\pi$$
$$654$$ −5.00000 −0.195515
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ 6.00000i 0.234082i
$$658$$ 9.00000i 0.350857i
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ 2.00000 0.0777910 0.0388955 0.999243i $$-0.487616\pi$$
0.0388955 + 0.999243i $$0.487616\pi$$
$$662$$ 23.0000i 0.893920i
$$663$$ 4.00000i 0.155347i
$$664$$ 16.0000 0.620920
$$665$$ 0 0
$$666$$ 3.00000 0.116248
$$667$$ 0 0
$$668$$ 18.0000i 0.696441i
$$669$$ 26.0000 1.00522
$$670$$ 0 0
$$671$$ −6.00000 −0.231627
$$672$$ − 3.00000i − 0.115728i
$$673$$ 44.0000i 1.69608i 0.529936 + 0.848038i $$0.322216\pi$$
−0.529936 + 0.848038i $$0.677784\pi$$
$$674$$ 12.0000 0.462223
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ 42.0000i 1.61419i 0.590421 + 0.807096i $$0.298962\pi$$
−0.590421 + 0.807096i $$0.701038\pi$$
$$678$$ − 1.00000i − 0.0384048i
$$679$$ 6.00000 0.230259
$$680$$ 0 0
$$681$$ −27.0000 −1.03464
$$682$$ 21.0000i 0.804132i
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 5.00000 0.191180
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ 0 0
$$688$$ − 1.00000i − 0.0381246i
$$689$$ 44.0000 1.67627
$$690$$ 0 0
$$691$$ 42.0000 1.59776 0.798878 0.601494i $$-0.205427\pi$$
0.798878 + 0.601494i $$0.205427\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ − 9.00000i − 0.341882i
$$694$$ 27.0000 1.02491
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 2.00000i − 0.0757554i
$$698$$ − 10.0000i − 0.378506i
$$699$$ 26.0000 0.983410
$$700$$ 0 0
$$701$$ −38.0000 −1.43524 −0.717620 0.696435i $$-0.754769\pi$$
−0.717620 + 0.696435i $$0.754769\pi$$
$$702$$ − 4.00000i − 0.150970i
$$703$$ − 15.0000i − 0.565736i
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ −16.0000 −0.602168
$$707$$ − 21.0000i − 0.789786i
$$708$$ 0 0
$$709$$ −5.00000 −0.187779 −0.0938895 0.995583i $$-0.529930\pi$$
−0.0938895 + 0.995583i $$0.529930\pi$$
$$710$$ 0 0
$$711$$ −5.00000 −0.187515
$$712$$ 10.0000i 0.374766i
$$713$$ 28.0000i 1.04861i
$$714$$ −3.00000 −0.112272
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 15.0000i − 0.560185i
$$718$$ 5.00000i 0.186598i
$$719$$ −30.0000 −1.11881 −0.559406 0.828894i $$-0.688971\pi$$
−0.559406 + 0.828894i $$0.688971\pi$$
$$720$$ 0 0
$$721$$ 42.0000 1.56416
$$722$$ − 6.00000i − 0.223297i
$$723$$ 12.0000i 0.446285i
$$724$$ −7.00000 −0.260153
$$725$$ 0 0
$$726$$ −2.00000 −0.0742270
$$727$$ − 28.0000i − 1.03846i −0.854634 0.519231i $$-0.826218\pi$$
0.854634 0.519231i $$-0.173782\pi$$
$$728$$ 12.0000i 0.444750i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −1.00000 −0.0369863
$$732$$ − 2.00000i − 0.0739221i
$$733$$ − 26.0000i − 0.960332i −0.877178 0.480166i $$-0.840576\pi$$
0.877178 0.480166i $$-0.159424\pi$$
$$734$$ 7.00000 0.258375
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 39.0000i 1.43658i
$$738$$ 2.00000i 0.0736210i
$$739$$ −35.0000 −1.28750 −0.643748 0.765238i $$-0.722621\pi$$
−0.643748 + 0.765238i $$0.722621\pi$$
$$740$$ 0 0
$$741$$ −20.0000 −0.734718
$$742$$ 33.0000i 1.21147i
$$743$$ − 26.0000i − 0.953847i −0.878945 0.476924i $$-0.841752\pi$$
0.878945 0.476924i $$-0.158248\pi$$
$$744$$ −7.00000 −0.256632
$$745$$ 0 0
$$746$$ 14.0000 0.512576
$$747$$ 16.0000i 0.585409i
$$748$$ − 3.00000i − 0.109691i
$$749$$ 51.0000 1.86350
$$750$$ 0 0
$$751$$ 12.0000 0.437886 0.218943 0.975738i $$-0.429739\pi$$
0.218943 + 0.975738i $$0.429739\pi$$
$$752$$ − 3.00000i − 0.109399i
$$753$$ − 18.0000i − 0.655956i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 3.00000 0.109109
$$757$$ 22.0000i 0.799604i 0.916602 + 0.399802i $$0.130921\pi$$
−0.916602 + 0.399802i $$0.869079\pi$$
$$758$$ − 20.0000i − 0.726433i
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ −48.0000 −1.74000 −0.869999 0.493053i $$-0.835881\pi$$
−0.869999 + 0.493053i $$0.835881\pi$$
$$762$$ − 18.0000i − 0.652071i
$$763$$ 15.0000i 0.543036i
$$764$$ 3.00000 0.108536
$$765$$ 0 0
$$766$$ 4.00000 0.144526
$$767$$ 0 0
$$768$$ 1.00000i 0.0360844i
$$769$$ −35.0000 −1.26213 −0.631066 0.775729i $$-0.717382\pi$$
−0.631066 + 0.775729i $$0.717382\pi$$
$$770$$ 0 0
$$771$$ 8.00000 0.288113
$$772$$ 6.00000i 0.215945i
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ 1.00000 0.0359443
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ − 9.00000i − 0.322873i
$$778$$ − 5.00000i − 0.179259i
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ − 4.00000i − 0.143040i
$$783$$ 0 0
$$784$$ −2.00000 −0.0714286
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 42.0000i 1.49714i 0.663057 + 0.748569i $$0.269259\pi$$
−0.663057 + 0.748569i $$0.730741\pi$$
$$788$$ − 12.0000i − 0.427482i
$$789$$ −19.0000 −0.676418
$$790$$ 0 0
$$791$$ −3.00000 −0.106668
$$792$$ 3.00000i 0.106600i
$$793$$ 8.00000i 0.284088i
$$794$$ −3.00000 −0.106466
$$795$$ 0 0
$$796$$ −25.0000 −0.886102
$$797$$ 17.0000i 0.602171i 0.953597 + 0.301085i $$0.0973489\pi$$
−0.953597 + 0.301085i $$0.902651\pi$$
$$798$$ − 15.0000i − 0.530994i
$$799$$ −3.00000 −0.106132
$$800$$ 0 0
$$801$$ −10.0000 −0.353333
$$802$$ 18.0000i 0.635602i
$$803$$ 18.0000i 0.635206i
$$804$$ −13.0000 −0.458475
$$805$$ 0 0
$$806$$ 28.0000 0.986258
$$807$$ − 20.0000i − 0.704033i
$$808$$ 7.00000i 0.246259i
$$809$$ −45.0000 −1.58212 −0.791058 0.611741i $$-0.790469\pi$$
−0.791058 + 0.611741i $$0.790469\pi$$
$$810$$ 0 0
$$811$$ 2.00000 0.0702295 0.0351147 0.999383i $$-0.488820\pi$$
0.0351147 + 0.999383i $$0.488820\pi$$
$$812$$ 0 0
$$813$$ 32.0000i 1.12229i
$$814$$ 9.00000 0.315450
$$815$$ 0 0
$$816$$ 1.00000 0.0350070
$$817$$ − 5.00000i − 0.174928i
$$818$$ − 10.0000i − 0.349642i
$$819$$ −12.0000 −0.419314
$$820$$ 0 0
$$821$$ 42.0000 1.46581 0.732905 0.680331i $$-0.238164\pi$$
0.732905 + 0.680331i $$0.238164\pi$$
$$822$$ 2.00000i 0.0697580i
$$823$$ − 16.0000i − 0.557725i −0.960331 0.278862i $$-0.910043\pi$$
0.960331 0.278862i $$-0.0899574\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 17.0000i 0.591148i 0.955320 + 0.295574i $$0.0955109\pi$$
−0.955320 + 0.295574i $$0.904489\pi$$
$$828$$ 4.00000i 0.139010i
$$829$$ 20.0000 0.694629 0.347314 0.937749i $$-0.387094\pi$$
0.347314 + 0.937749i $$0.387094\pi$$
$$830$$ 0 0
$$831$$ 23.0000 0.797861
$$832$$ − 4.00000i − 0.138675i
$$833$$ 2.00000i 0.0692959i
$$834$$ 20.0000 0.692543
$$835$$ 0 0
$$836$$ 15.0000 0.518786
$$837$$ − 7.00000i − 0.241955i
$$838$$ − 20.0000i − 0.690889i
$$839$$ 30.0000 1.03572 0.517858 0.855467i $$-0.326730\pi$$
0.517858 + 0.855467i $$0.326730\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 28.0000i 0.964944i
$$843$$ − 18.0000i − 0.619953i
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ 3.00000 0.103142
$$847$$ 6.00000i 0.206162i
$$848$$ − 11.0000i − 0.377742i
$$849$$ −14.0000 −0.480479
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ − 2.00000i − 0.0685189i
$$853$$ − 51.0000i − 1.74621i −0.487535 0.873103i $$-0.662104\pi$$
0.487535 0.873103i $$-0.337896\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ 0 0
$$856$$ −17.0000 −0.581048
$$857$$ 47.0000i 1.60549i 0.596323 + 0.802745i $$0.296628\pi$$
−0.596323 + 0.802745i $$0.703372\pi$$
$$858$$ − 12.0000i − 0.409673i
$$859$$ −35.0000 −1.19418 −0.597092 0.802173i $$-0.703677\pi$$
−0.597092 + 0.802173i $$0.703677\pi$$
$$860$$ 0 0
$$861$$ 6.00000 0.204479
$$862$$ − 2.00000i − 0.0681203i
$$863$$ 19.0000i 0.646768i 0.946268 + 0.323384i $$0.104820\pi$$
−0.946268 + 0.323384i $$0.895180\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −21.0000 −0.713609
$$867$$ − 1.00000i − 0.0339618i
$$868$$ 21.0000i 0.712786i
$$869$$ −15.0000 −0.508840
$$870$$ 0 0
$$871$$ 52.0000 1.76195
$$872$$ − 5.00000i − 0.169321i
$$873$$ − 2.00000i − 0.0676897i
$$874$$ 20.0000 0.676510
$$875$$ 0 0
$$876$$ −6.00000 −0.202721
$$877$$ − 38.0000i − 1.28317i −0.767052 0.641584i $$-0.778277\pi$$
0.767052 0.641584i $$-0.221723\pi$$
$$878$$ 20.0000i 0.674967i
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ 7.00000 0.235836 0.117918 0.993023i $$-0.462378\pi$$
0.117918 + 0.993023i $$0.462378\pi$$
$$882$$ − 2.00000i − 0.0673435i
$$883$$ 24.0000i 0.807664i 0.914833 + 0.403832i $$0.132322\pi$$
−0.914833 + 0.403832i $$0.867678\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ 14.0000 0.470339
$$887$$ 12.0000i 0.402921i 0.979497 + 0.201460i $$0.0645687\pi$$
−0.979497 + 0.201460i $$0.935431\pi$$
$$888$$ 3.00000i 0.100673i
$$889$$ −54.0000 −1.81110
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ 26.0000i 0.870544i
$$893$$ − 15.0000i − 0.501956i
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ − 16.0000i − 0.534224i
$$898$$ 5.00000i 0.166852i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −11.0000 −0.366463
$$902$$ 6.00000i 0.199778i
$$903$$ − 3.00000i − 0.0998337i
$$904$$ 1.00000 0.0332595
$$905$$ 0 0
$$906$$ 12.0000 0.398673
$$907$$ − 28.0000i − 0.929725i −0.885383 0.464862i $$-0.846104\pi$$
0.885383 0.464862i $$-0.153896\pi$$
$$908$$ − 27.0000i − 0.896026i
$$909$$ −7.00000 −0.232175
$$910$$ 0 0
$$911$$ −8.00000 −0.265052 −0.132526 0.991180i $$-0.542309\pi$$
−0.132526 + 0.991180i $$0.542309\pi$$
$$912$$ 5.00000i 0.165567i
$$913$$ 48.0000i 1.58857i
$$914$$ −13.0000 −0.430002
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 36.0000i − 1.18882i
$$918$$ 1.00000i 0.0330049i
$$919$$ −10.0000 −0.329870 −0.164935 0.986304i $$-0.552741\pi$$
−0.164935 + 0.986304i $$0.552741\pi$$
$$920$$ 0 0
$$921$$ −32.0000 −1.05444
$$922$$ − 37.0000i − 1.21853i
$$923$$ 8.00000i 0.263323i
$$924$$ 9.00000 0.296078
$$925$$ 0 0
$$926$$ 24.0000 0.788689
$$927$$ − 14.0000i − 0.459820i
$$928$$ 0 0
$$929$$ −35.0000 −1.14831 −0.574156 0.818746i $$-0.694670\pi$$
−0.574156 + 0.818746i $$0.694670\pi$$
$$930$$ 0 0
$$931$$ −10.0000 −0.327737
$$932$$ 26.0000i 0.851658i
$$933$$ − 18.0000i − 0.589294i
$$934$$ −18.0000 −0.588978
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ 22.0000i 0.718709i 0.933201 + 0.359354i $$0.117003\pi$$
−0.933201 + 0.359354i $$0.882997\pi$$
$$938$$ 39.0000i 1.27340i
$$939$$ −4.00000 −0.130535
$$940$$ 0 0
$$941$$ −38.0000 −1.23876 −0.619382 0.785090i $$-0.712617\pi$$
−0.619382 + 0.785090i $$0.712617\pi$$
$$942$$ 12.0000i 0.390981i
$$943$$ 8.00000i 0.260516i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 3.00000 0.0975384
$$947$$ 27.0000i 0.877382i 0.898638 + 0.438691i $$0.144558\pi$$
−0.898638 + 0.438691i $$0.855442\pi$$
$$948$$ − 5.00000i − 0.162392i
$$949$$ 24.0000 0.779073
$$950$$ 0 0
$$951$$ −32.0000 −1.03767
$$952$$ − 3.00000i − 0.0972306i
$$953$$ − 16.0000i − 0.518291i −0.965838 0.259145i $$-0.916559\pi$$
0.965838 0.259145i $$-0.0834409\pi$$
$$954$$ 11.0000 0.356138
$$955$$ 0 0
$$956$$ 15.0000 0.485135
$$957$$ 0 0
$$958$$ − 10.0000i − 0.323085i
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ − 12.0000i − 0.386896i
$$963$$ − 17.0000i − 0.547817i
$$964$$ −12.0000 −0.386494
$$965$$ 0 0
$$966$$ 12.0000 0.386094
$$967$$ − 48.0000i − 1.54358i −0.635880 0.771788i $$-0.719363\pi$$
0.635880 0.771788i $$-0.280637\pi$$
$$968$$ − 2.00000i − 0.0642824i
$$969$$ 5.00000 0.160623
$$970$$ 0 0
$$971$$ 42.0000 1.34784 0.673922 0.738802i $$-0.264608\pi$$
0.673922 + 0.738802i $$0.264608\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ − 60.0000i − 1.92351i
$$974$$ 32.0000 1.02535
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ − 28.0000i − 0.895799i −0.894084 0.447900i $$-0.852172\pi$$
0.894084 0.447900i $$-0.147828\pi$$
$$978$$ − 6.00000i − 0.191859i
$$979$$ −30.0000 −0.958804
$$980$$ 0 0
$$981$$ 5.00000 0.159638
$$982$$ − 2.00000i − 0.0638226i
$$983$$ 44.0000i 1.40338i 0.712481 + 0.701691i $$0.247571\pi$$
−0.712481 + 0.701691i $$0.752429\pi$$
$$984$$ −2.00000 −0.0637577
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 9.00000i − 0.286473i
$$988$$ − 20.0000i − 0.636285i
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ − 7.00000i − 0.222250i
$$993$$ − 23.0000i − 0.729883i
$$994$$ −6.00000 −0.190308
$$995$$ 0 0
$$996$$ −16.0000 −0.506979
$$997$$ 37.0000i 1.17180i 0.810383 + 0.585901i $$0.199259\pi$$
−0.810383 + 0.585901i $$0.800741\pi$$
$$998$$ 10.0000i 0.316544i
$$999$$ −3.00000 −0.0949158
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.o.2449.1 2
5.2 odd 4 2550.2.a.bf.1.1 yes 1
5.3 odd 4 2550.2.a.a.1.1 1
5.4 even 2 inner 2550.2.d.o.2449.2 2
15.2 even 4 7650.2.a.bd.1.1 1
15.8 even 4 7650.2.a.bk.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.a.1.1 1 5.3 odd 4
2550.2.a.bf.1.1 yes 1 5.2 odd 4
2550.2.d.o.2449.1 2 1.1 even 1 trivial
2550.2.d.o.2449.2 2 5.4 even 2 inner
7650.2.a.bd.1.1 1 15.2 even 4
7650.2.a.bk.1.1 1 15.8 even 4