# Properties

 Label 4650.2.d.q.3349.1 Level $4650$ Weight $2$ Character 4650.3349 Analytic conductor $37.130$ 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] = [4650,2,Mod(3349,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, 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("4650.3349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.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: $$37.1304369399$$ 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 3349.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4650.3349 Dual form 4650.2.d.q.3349.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} -1.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} +3.00000 q^{21} +3.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} -1.00000 q^{26} -1.00000i q^{27} +3.00000i q^{28} -10.0000 q^{29} +1.00000 q^{31} -1.00000i q^{32} -3.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -3.00000i q^{37} +1.00000 q^{39} +7.00000 q^{41} -3.00000i q^{42} -1.00000i q^{43} +3.00000 q^{44} +4.00000 q^{46} +7.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} -2.00000 q^{51} +1.00000i q^{52} +9.00000i q^{53} -1.00000 q^{54} +3.00000 q^{56} +10.0000i q^{58} +7.00000 q^{61} -1.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +2.00000i q^{67} -2.00000i q^{68} -4.00000 q^{69} +7.00000 q^{71} -1.00000i q^{72} +4.00000i q^{73} -3.00000 q^{74} +9.00000i q^{77} -1.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} -7.00000i q^{82} +9.00000i q^{83} -3.00000 q^{84} -1.00000 q^{86} -10.0000i q^{87} -3.00000i q^{88} -10.0000 q^{89} -3.00000 q^{91} -4.00000i q^{92} +1.00000i q^{93} +7.00000 q^{94} +1.00000 q^{96} +2.00000i q^{97} +2.00000i q^{98} +3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 6 q^{11} - 6 q^{14} + 2 q^{16} + 6 q^{21} - 2 q^{24} - 2 q^{26} - 20 q^{29} + 2 q^{31} + 4 q^{34} + 2 q^{36} + 2 q^{39} + 14 q^{41} + 6 q^{44} + 8 q^{46} - 4 q^{49} - 4 q^{51} - 2 q^{54} + 6 q^{56} + 14 q^{61} - 2 q^{64} - 6 q^{66} - 8 q^{69} + 14 q^{71} - 6 q^{74} + 20 q^{79} + 2 q^{81} - 6 q^{84} - 2 q^{86} - 20 q^{89} - 6 q^{91} + 14 q^{94} + 2 q^{96} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 6 * q^11 - 6 * q^14 + 2 * q^16 + 6 * q^21 - 2 * q^24 - 2 * q^26 - 20 * q^29 + 2 * q^31 + 4 * q^34 + 2 * q^36 + 2 * q^39 + 14 * q^41 + 6 * q^44 + 8 * q^46 - 4 * q^49 - 4 * q^51 - 2 * q^54 + 6 * q^56 + 14 * q^61 - 2 * q^64 - 6 * q^66 - 8 * q^69 + 14 * q^71 - 6 * q^74 + 20 * q^79 + 2 * q^81 - 6 * q^84 - 2 * q^86 - 20 * q^89 - 6 * q^91 + 14 * q^94 + 2 * q^96 + 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$$.

 $$n$$ $$1801$$ $$2977$$ $$3101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$13$$ − 1.00000i − 0.277350i −0.990338 0.138675i $$-0.955716\pi$$
0.990338 0.138675i $$-0.0442844\pi$$
$$14$$ −3.00000 −0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 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
$$26$$ −1.00000 −0.196116
$$27$$ − 1.00000i − 0.192450i
$$28$$ 3.00000i 0.566947i
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 3.00000i − 0.522233i
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 3.00000i − 0.493197i −0.969118 0.246598i $$-0.920687\pi$$
0.969118 0.246598i $$-0.0793129\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 7.00000 1.09322 0.546608 0.837389i $$-0.315919\pi$$
0.546608 + 0.837389i $$0.315919\pi$$
$$42$$ − 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
$$47$$ 7.00000i 1.02105i 0.859861 + 0.510527i $$0.170550\pi$$
−0.859861 + 0.510527i $$0.829450\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 1.00000i 0.138675i
$$53$$ 9.00000i 1.23625i 0.786082 + 0.618123i $$0.212106\pi$$
−0.786082 + 0.618123i $$0.787894\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ 10.0000i 1.31306i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ − 1.00000i − 0.127000i
$$63$$ 3.00000i 0.377964i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −3.00000 −0.369274
$$67$$ 2.00000i 0.244339i 0.992509 + 0.122169i $$0.0389851\pi$$
−0.992509 + 0.122169i $$0.961015\pi$$
$$68$$ − 2.00000i − 0.242536i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 7.00000 0.830747 0.415374 0.909651i $$-0.363651\pi$$
0.415374 + 0.909651i $$0.363651\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 4.00000i 0.468165i 0.972217 + 0.234082i $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ −3.00000 −0.348743
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 9.00000i 1.02565i
$$78$$ − 1.00000i − 0.113228i
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 7.00000i − 0.773021i
$$83$$ 9.00000i 0.987878i 0.869496 + 0.493939i $$0.164443\pi$$
−0.869496 + 0.493939i $$0.835557\pi$$
$$84$$ −3.00000 −0.327327
$$85$$ 0 0
$$86$$ −1.00000 −0.107833
$$87$$ − 10.0000i − 1.07211i
$$88$$ − 3.00000i − 0.319801i
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ − 4.00000i − 0.417029i
$$93$$ 1.00000i 0.103695i
$$94$$ 7.00000 0.721995
$$95$$ 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
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 2.00000i 0.198030i
$$103$$ − 1.00000i − 0.0985329i −0.998786 0.0492665i $$-0.984312\pi$$
0.998786 0.0492665i $$-0.0156884\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 9.00000 0.874157
$$107$$ − 18.0000i − 1.74013i −0.492941 0.870063i $$-0.664078\pi$$
0.492941 0.870063i $$-0.335922\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ − 3.00000i − 0.283473i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 10.0000 0.928477
$$117$$ 1.00000i 0.0924500i
$$118$$ 0 0
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ − 7.00000i − 0.633750i
$$123$$ 7.00000i 0.631169i
$$124$$ −1.00000 −0.0898027
$$125$$ 0 0
$$126$$ 3.00000 0.267261
$$127$$ 2.00000i 0.177471i 0.996055 + 0.0887357i $$0.0282826\pi$$
−0.996055 + 0.0887357i $$0.971717\pi$$
$$128$$ 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
$$133$$ 0 0
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ 2.00000i 0.170872i 0.996344 + 0.0854358i $$0.0272282\pi$$
−0.996344 + 0.0854358i $$0.972772\pi$$
$$138$$ 4.00000i 0.340503i
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ 0 0
$$141$$ −7.00000 −0.589506
$$142$$ − 7.00000i − 0.587427i
$$143$$ 3.00000i 0.250873i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ − 2.00000i − 0.164957i
$$148$$ 3.00000i 0.246598i
$$149$$ 20.0000 1.63846 0.819232 0.573462i $$-0.194400\pi$$
0.819232 + 0.573462i $$0.194400\pi$$
$$150$$ 0 0
$$151$$ −18.0000 −1.46482 −0.732410 0.680864i $$-0.761604\pi$$
−0.732410 + 0.680864i $$0.761604\pi$$
$$152$$ 0 0
$$153$$ − 2.00000i − 0.161690i
$$154$$ 9.00000 0.725241
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ 22.0000i 1.75579i 0.478852 + 0.877896i $$0.341053\pi$$
−0.478852 + 0.877896i $$0.658947\pi$$
$$158$$ − 10.0000i − 0.795557i
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ 12.0000 0.945732
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ −7.00000 −0.546608
$$165$$ 0 0
$$166$$ 9.00000 0.698535
$$167$$ 2.00000i 0.154765i 0.997001 + 0.0773823i $$0.0246562\pi$$
−0.997001 + 0.0773823i $$0.975344\pi$$
$$168$$ 3.00000i 0.231455i
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 1.00000i 0.0762493i
$$173$$ 4.00000i 0.304114i 0.988372 + 0.152057i $$0.0485898\pi$$
−0.988372 + 0.152057i $$0.951410\pi$$
$$174$$ −10.0000 −0.758098
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ 10.0000i 0.749532i
$$179$$ 5.00000 0.373718 0.186859 0.982387i $$-0.440169\pi$$
0.186859 + 0.982387i $$0.440169\pi$$
$$180$$ 0 0
$$181$$ −13.0000 −0.966282 −0.483141 0.875542i $$-0.660504\pi$$
−0.483141 + 0.875542i $$0.660504\pi$$
$$182$$ 3.00000i 0.222375i
$$183$$ 7.00000i 0.517455i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ 1.00000 0.0733236
$$187$$ − 6.00000i − 0.438763i
$$188$$ − 7.00000i − 0.510527i
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 9.00000i 0.647834i 0.946085 + 0.323917i $$0.105000\pi$$
−0.946085 + 0.323917i $$0.895000\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ 27.0000i 1.92367i 0.273629 + 0.961835i $$0.411776\pi$$
−0.273629 + 0.961835i $$0.588224\pi$$
$$198$$ − 3.00000i − 0.213201i
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ −2.00000 −0.141069
$$202$$ − 2.00000i − 0.140720i
$$203$$ 30.0000i 2.10559i
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ −1.00000 −0.0696733
$$207$$ − 4.00000i − 0.278019i
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 2.00000 0.137686 0.0688428 0.997628i $$-0.478069\pi$$
0.0688428 + 0.997628i $$0.478069\pi$$
$$212$$ − 9.00000i − 0.618123i
$$213$$ 7.00000i 0.479632i
$$214$$ −18.0000 −1.23045
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ − 3.00000i − 0.203653i
$$218$$ − 10.0000i − 0.677285i
$$219$$ −4.00000 −0.270295
$$220$$ 0 0
$$221$$ 2.00000 0.134535
$$222$$ − 3.00000i − 0.201347i
$$223$$ − 6.00000i − 0.401790i −0.979613 0.200895i $$-0.935615\pi$$
0.979613 0.200895i $$-0.0643850\pi$$
$$224$$ −3.00000 −0.200446
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ −9.00000 −0.592157
$$232$$ − 10.0000i − 0.656532i
$$233$$ 9.00000i 0.589610i 0.955557 + 0.294805i $$0.0952546\pi$$
−0.955557 + 0.294805i $$0.904745\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 10.0000i 0.649570i
$$238$$ − 6.00000i − 0.388922i
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 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
$$244$$ −7.00000 −0.448129
$$245$$ 0 0
$$246$$ 7.00000 0.446304
$$247$$ 0 0
$$248$$ 1.00000i 0.0635001i
$$249$$ −9.00000 −0.570352
$$250$$ 0 0
$$251$$ −3.00000 −0.189358 −0.0946792 0.995508i $$-0.530183\pi$$
−0.0946792 + 0.995508i $$0.530183\pi$$
$$252$$ − 3.00000i − 0.188982i
$$253$$ − 12.0000i − 0.754434i
$$254$$ 2.00000 0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 7.00000i 0.436648i 0.975876 + 0.218324i $$0.0700590\pi$$
−0.975876 + 0.218324i $$0.929941\pi$$
$$258$$ − 1.00000i − 0.0622573i
$$259$$ −9.00000 −0.559233
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ − 12.0000i − 0.741362i
$$263$$ − 6.00000i − 0.369976i −0.982741 0.184988i $$-0.940775\pi$$
0.982741 0.184988i $$-0.0592246\pi$$
$$264$$ 3.00000 0.184637
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 10.0000i − 0.611990i
$$268$$ − 2.00000i − 0.122169i
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 2.00000i 0.121268i
$$273$$ − 3.00000i − 0.181568i
$$274$$ 2.00000 0.120824
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ − 18.0000i − 1.08152i −0.841178 0.540758i $$-0.818138\pi$$
0.841178 0.540758i $$-0.181862\pi$$
$$278$$ − 5.00000i − 0.299880i
$$279$$ −1.00000 −0.0598684
$$280$$ 0 0
$$281$$ 7.00000 0.417585 0.208792 0.977960i $$-0.433047\pi$$
0.208792 + 0.977960i $$0.433047\pi$$
$$282$$ 7.00000i 0.416844i
$$283$$ 24.0000i 1.42665i 0.700832 + 0.713326i $$0.252812\pi$$
−0.700832 + 0.713326i $$0.747188\pi$$
$$284$$ −7.00000 −0.415374
$$285$$ 0 0
$$286$$ 3.00000 0.177394
$$287$$ − 21.0000i − 1.23959i
$$288$$ 1.00000i 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ − 4.00000i − 0.234082i
$$293$$ 24.0000i 1.40209i 0.713115 + 0.701047i $$0.247284\pi$$
−0.713115 + 0.701047i $$0.752716\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ 3.00000 0.174371
$$297$$ 3.00000i 0.174078i
$$298$$ − 20.0000i − 1.15857i
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 18.0000i 1.03578i
$$303$$ 2.00000i 0.114897i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ 22.0000i 1.25561i 0.778372 + 0.627803i $$0.216046\pi$$
−0.778372 + 0.627803i $$0.783954\pi$$
$$308$$ − 9.00000i − 0.512823i
$$309$$ 1.00000 0.0568880
$$310$$ 0 0
$$311$$ 27.0000 1.53103 0.765515 0.643418i $$-0.222484\pi$$
0.765515 + 0.643418i $$0.222484\pi$$
$$312$$ 1.00000i 0.0566139i
$$313$$ 24.0000i 1.35656i 0.734803 + 0.678280i $$0.237274\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ 22.0000 1.24153
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ 22.0000i 1.23564i 0.786318 + 0.617822i $$0.211985\pi$$
−0.786318 + 0.617822i $$0.788015\pi$$
$$318$$ 9.00000i 0.504695i
$$319$$ 30.0000 1.67968
$$320$$ 0 0
$$321$$ 18.0000 1.00466
$$322$$ − 12.0000i − 0.668734i
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 10.0000i 0.553001i
$$328$$ 7.00000i 0.386510i
$$329$$ 21.0000 1.15777
$$330$$ 0 0
$$331$$ −3.00000 −0.164895 −0.0824475 0.996595i $$-0.526274\pi$$
−0.0824475 + 0.996595i $$0.526274\pi$$
$$332$$ − 9.00000i − 0.493939i
$$333$$ 3.00000i 0.164399i
$$334$$ 2.00000 0.109435
$$335$$ 0 0
$$336$$ 3.00000 0.163663
$$337$$ − 8.00000i − 0.435788i −0.975972 0.217894i $$-0.930081\pi$$
0.975972 0.217894i $$-0.0699187\pi$$
$$338$$ − 12.0000i − 0.652714i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −3.00000 −0.162459
$$342$$ 0 0
$$343$$ − 15.0000i − 0.809924i
$$344$$ 1.00000 0.0539164
$$345$$ 0 0
$$346$$ 4.00000 0.215041
$$347$$ 7.00000i 0.375780i 0.982190 + 0.187890i $$0.0601648\pi$$
−0.982190 + 0.187890i $$0.939835\pi$$
$$348$$ 10.0000i 0.536056i
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 3.00000i 0.159901i
$$353$$ 24.0000i 1.27739i 0.769460 + 0.638696i $$0.220526\pi$$
−0.769460 + 0.638696i $$0.779474\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 6.00000i 0.317554i
$$358$$ − 5.00000i − 0.264258i
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 13.0000i 0.683265i
$$363$$ − 2.00000i − 0.104973i
$$364$$ 3.00000 0.157243
$$365$$ 0 0
$$366$$ 7.00000 0.365896
$$367$$ − 28.0000i − 1.46159i −0.682598 0.730794i $$-0.739150\pi$$
0.682598 0.730794i $$-0.260850\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ −7.00000 −0.364405
$$370$$ 0 0
$$371$$ 27.0000 1.40177
$$372$$ − 1.00000i − 0.0518476i
$$373$$ 24.0000i 1.24267i 0.783544 + 0.621336i $$0.213410\pi$$
−0.783544 + 0.621336i $$0.786590\pi$$
$$374$$ −6.00000 −0.310253
$$375$$ 0 0
$$376$$ −7.00000 −0.360997
$$377$$ 10.0000i 0.515026i
$$378$$ 3.00000i 0.154303i
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ −2.00000 −0.102463
$$382$$ 8.00000i 0.409316i
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 9.00000 0.458088
$$387$$ 1.00000i 0.0508329i
$$388$$ − 2.00000i − 0.101535i
$$389$$ −10.0000 −0.507020 −0.253510 0.967333i $$-0.581585\pi$$
−0.253510 + 0.967333i $$0.581585\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ − 2.00000i − 0.101015i
$$393$$ 12.0000i 0.605320i
$$394$$ 27.0000 1.36024
$$395$$ 0 0
$$396$$ −3.00000 −0.150756
$$397$$ − 28.0000i − 1.40528i −0.711546 0.702640i $$-0.752005\pi$$
0.711546 0.702640i $$-0.247995\pi$$
$$398$$ − 20.0000i − 1.00251i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −8.00000 −0.399501 −0.199750 0.979847i $$-0.564013\pi$$
−0.199750 + 0.979847i $$0.564013\pi$$
$$402$$ 2.00000i 0.0997509i
$$403$$ − 1.00000i − 0.0498135i
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 30.0000 1.48888
$$407$$ 9.00000i 0.446113i
$$408$$ − 2.00000i − 0.0990148i
$$409$$ 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
$$412$$ 1.00000i 0.0492665i
$$413$$ 0 0
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 5.00000i 0.244851i
$$418$$ 0 0
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ −28.0000 −1.36464 −0.682318 0.731055i $$-0.739028\pi$$
−0.682318 + 0.731055i $$0.739028\pi$$
$$422$$ − 2.00000i − 0.0973585i
$$423$$ − 7.00000i − 0.340352i
$$424$$ −9.00000 −0.437079
$$425$$ 0 0
$$426$$ 7.00000 0.339151
$$427$$ − 21.0000i − 1.01626i
$$428$$ 18.0000i 0.870063i
$$429$$ −3.00000 −0.144841
$$430$$ 0 0
$$431$$ 7.00000 0.337178 0.168589 0.985686i $$-0.446079\pi$$
0.168589 + 0.985686i $$0.446079\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 4.00000i 0.192228i 0.995370 + 0.0961139i $$0.0306413\pi$$
−0.995370 + 0.0961139i $$0.969359\pi$$
$$434$$ −3.00000 −0.144005
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 0 0
$$438$$ 4.00000i 0.191127i
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ − 2.00000i − 0.0951303i
$$443$$ − 36.0000i − 1.71041i −0.518289 0.855206i $$-0.673431\pi$$
0.518289 0.855206i $$-0.326569\pi$$
$$444$$ −3.00000 −0.142374
$$445$$ 0 0
$$446$$ −6.00000 −0.284108
$$447$$ 20.0000i 0.945968i
$$448$$ 3.00000i 0.141737i
$$449$$ −20.0000 −0.943858 −0.471929 0.881636i $$-0.656442\pi$$
−0.471929 + 0.881636i $$0.656442\pi$$
$$450$$ 0 0
$$451$$ −21.0000 −0.988851
$$452$$ 6.00000i 0.282216i
$$453$$ − 18.0000i − 0.845714i
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 8.00000i − 0.374224i −0.982339 0.187112i $$-0.940087\pi$$
0.982339 0.187112i $$-0.0599128\pi$$
$$458$$ 10.0000i 0.467269i
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −3.00000 −0.139724 −0.0698620 0.997557i $$-0.522256\pi$$
−0.0698620 + 0.997557i $$0.522256\pi$$
$$462$$ 9.00000i 0.418718i
$$463$$ 14.0000i 0.650635i 0.945605 + 0.325318i $$0.105471\pi$$
−0.945605 + 0.325318i $$0.894529\pi$$
$$464$$ −10.0000 −0.464238
$$465$$ 0 0
$$466$$ 9.00000 0.416917
$$467$$ − 18.0000i − 0.832941i −0.909149 0.416470i $$-0.863267\pi$$
0.909149 0.416470i $$-0.136733\pi$$
$$468$$ − 1.00000i − 0.0462250i
$$469$$ 6.00000 0.277054
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ 0 0
$$473$$ 3.00000i 0.137940i
$$474$$ 10.0000 0.459315
$$475$$ 0 0
$$476$$ −6.00000 −0.275010
$$477$$ − 9.00000i − 0.412082i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −3.00000 −0.136788
$$482$$ − 12.0000i − 0.546585i
$$483$$ 12.0000i 0.546019i
$$484$$ 2.00000 0.0909091
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 28.0000i − 1.26880i −0.773004 0.634401i $$-0.781247\pi$$
0.773004 0.634401i $$-0.218753\pi$$
$$488$$ 7.00000i 0.316875i
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ − 7.00000i − 0.315584i
$$493$$ − 20.0000i − 0.900755i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 1.00000 0.0449013
$$497$$ − 21.0000i − 0.941979i
$$498$$ 9.00000i 0.403300i
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 0 0
$$501$$ −2.00000 −0.0893534
$$502$$ 3.00000i 0.133897i
$$503$$ − 11.0000i − 0.490466i −0.969464 0.245233i $$-0.921136\pi$$
0.969464 0.245233i $$-0.0788644\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 0 0
$$506$$ −12.0000 −0.533465
$$507$$ 12.0000i 0.532939i
$$508$$ − 2.00000i − 0.0887357i
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ 7.00000 0.308757
$$515$$ 0 0
$$516$$ −1.00000 −0.0440225
$$517$$ − 21.0000i − 0.923579i
$$518$$ 9.00000i 0.395437i
$$519$$ −4.00000 −0.175581
$$520$$ 0 0
$$521$$ −33.0000 −1.44576 −0.722878 0.690976i $$-0.757181\pi$$
−0.722878 + 0.690976i $$0.757181\pi$$
$$522$$ − 10.0000i − 0.437688i
$$523$$ 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
$$526$$ −6.00000 −0.261612
$$527$$ 2.00000i 0.0871214i
$$528$$ − 3.00000i − 0.130558i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 7.00000i − 0.303204i
$$534$$ −10.0000 −0.432742
$$535$$ 0 0
$$536$$ −2.00000 −0.0863868
$$537$$ 5.00000i 0.215766i
$$538$$ − 10.0000i − 0.431131i
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −8.00000 −0.343947 −0.171973 0.985102i $$-0.555014\pi$$
−0.171973 + 0.985102i $$0.555014\pi$$
$$542$$ 8.00000i 0.343629i
$$543$$ − 13.0000i − 0.557883i
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ −3.00000 −0.128388
$$547$$ 22.0000i 0.940652i 0.882493 + 0.470326i $$0.155864\pi$$
−0.882493 + 0.470326i $$0.844136\pi$$
$$548$$ − 2.00000i − 0.0854358i
$$549$$ −7.00000 −0.298753
$$550$$ 0 0
$$551$$ 0 0
$$552$$ − 4.00000i − 0.170251i
$$553$$ − 30.0000i − 1.27573i
$$554$$ −18.0000 −0.764747
$$555$$ 0 0
$$556$$ −5.00000 −0.212047
$$557$$ 2.00000i 0.0847427i 0.999102 + 0.0423714i $$0.0134913\pi$$
−0.999102 + 0.0423714i $$0.986509\pi$$
$$558$$ 1.00000i 0.0423334i
$$559$$ −1.00000 −0.0422955
$$560$$ 0 0
$$561$$ 6.00000 0.253320
$$562$$ − 7.00000i − 0.295277i
$$563$$ 34.0000i 1.43293i 0.697623 + 0.716465i $$0.254241\pi$$
−0.697623 + 0.716465i $$0.745759\pi$$
$$564$$ 7.00000 0.294753
$$565$$ 0 0
$$566$$ 24.0000 1.00880
$$567$$ − 3.00000i − 0.125988i
$$568$$ 7.00000i 0.293713i
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ −23.0000 −0.962520 −0.481260 0.876578i $$-0.659821\pi$$
−0.481260 + 0.876578i $$0.659821\pi$$
$$572$$ − 3.00000i − 0.125436i
$$573$$ − 8.00000i − 0.334205i
$$574$$ −21.0000 −0.876523
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 22.0000i 0.915872i 0.888985 + 0.457936i $$0.151411\pi$$
−0.888985 + 0.457936i $$0.848589\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ −9.00000 −0.374027
$$580$$ 0 0
$$581$$ 27.0000 1.12015
$$582$$ 2.00000i 0.0829027i
$$583$$ − 27.0000i − 1.11823i
$$584$$ −4.00000 −0.165521
$$585$$ 0 0
$$586$$ 24.0000 0.991431
$$587$$ 17.0000i 0.701665i 0.936438 + 0.350833i $$0.114101\pi$$
−0.936438 + 0.350833i $$0.885899\pi$$
$$588$$ 2.00000i 0.0824786i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −27.0000 −1.11063
$$592$$ − 3.00000i − 0.123299i
$$593$$ − 1.00000i − 0.0410651i −0.999789 0.0205325i $$-0.993464\pi$$
0.999789 0.0205325i $$-0.00653617\pi$$
$$594$$ 3.00000 0.123091
$$595$$ 0 0
$$596$$ −20.0000 −0.819232
$$597$$ 20.0000i 0.818546i
$$598$$ − 4.00000i − 0.163572i
$$599$$ 15.0000 0.612883 0.306442 0.951889i $$-0.400862\pi$$
0.306442 + 0.951889i $$0.400862\pi$$
$$600$$ 0 0
$$601$$ 12.0000 0.489490 0.244745 0.969587i $$-0.421296\pi$$
0.244745 + 0.969587i $$0.421296\pi$$
$$602$$ 3.00000i 0.122271i
$$603$$ − 2.00000i − 0.0814463i
$$604$$ 18.0000 0.732410
$$605$$ 0 0
$$606$$ 2.00000 0.0812444
$$607$$ − 43.0000i − 1.74532i −0.488332 0.872658i $$-0.662394\pi$$
0.488332 0.872658i $$-0.337606\pi$$
$$608$$ 0 0
$$609$$ −30.0000 −1.21566
$$610$$ 0 0
$$611$$ 7.00000 0.283190
$$612$$ 2.00000i 0.0808452i
$$613$$ − 26.0000i − 1.05013i −0.851062 0.525065i $$-0.824041\pi$$
0.851062 0.525065i $$-0.175959\pi$$
$$614$$ 22.0000 0.887848
$$615$$ 0 0
$$616$$ −9.00000 −0.362620
$$617$$ − 3.00000i − 0.120775i −0.998175 0.0603877i $$-0.980766\pi$$
0.998175 0.0603877i $$-0.0192337\pi$$
$$618$$ − 1.00000i − 0.0402259i
$$619$$ 5.00000 0.200967 0.100483 0.994939i $$-0.467961\pi$$
0.100483 + 0.994939i $$0.467961\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ − 27.0000i − 1.08260i
$$623$$ 30.0000i 1.20192i
$$624$$ 1.00000 0.0400320
$$625$$ 0 0
$$626$$ 24.0000 0.959233
$$627$$ 0 0
$$628$$ − 22.0000i − 0.877896i
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 2.00000 0.0796187 0.0398094 0.999207i $$-0.487325\pi$$
0.0398094 + 0.999207i $$0.487325\pi$$
$$632$$ 10.0000i 0.397779i
$$633$$ 2.00000i 0.0794929i
$$634$$ 22.0000 0.873732
$$635$$ 0 0
$$636$$ 9.00000 0.356873
$$637$$ 2.00000i 0.0792429i
$$638$$ − 30.0000i − 1.18771i
$$639$$ −7.00000 −0.276916
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ − 18.0000i − 0.710403i
$$643$$ − 31.0000i − 1.22252i −0.791430 0.611260i $$-0.790663\pi$$
0.791430 0.611260i $$-0.209337\pi$$
$$644$$ −12.0000 −0.472866
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 42.0000i 1.65119i 0.564263 + 0.825595i $$0.309160\pi$$
−0.564263 + 0.825595i $$0.690840\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 3.00000 0.117579
$$652$$ − 4.00000i − 0.156652i
$$653$$ − 16.0000i − 0.626128i −0.949732 0.313064i $$-0.898644\pi$$
0.949732 0.313064i $$-0.101356\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ 7.00000 0.273304
$$657$$ − 4.00000i − 0.156055i
$$658$$ − 21.0000i − 0.818665i
$$659$$ 40.0000 1.55818 0.779089 0.626913i $$-0.215682\pi$$
0.779089 + 0.626913i $$0.215682\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 3.00000i 0.116598i
$$663$$ 2.00000i 0.0776736i
$$664$$ −9.00000 −0.349268
$$665$$ 0 0
$$666$$ 3.00000 0.116248
$$667$$ − 40.0000i − 1.54881i
$$668$$ − 2.00000i − 0.0773823i
$$669$$ 6.00000 0.231973
$$670$$ 0 0
$$671$$ −21.0000 −0.810696
$$672$$ − 3.00000i − 0.115728i
$$673$$ − 26.0000i − 1.00223i −0.865382 0.501113i $$-0.832924\pi$$
0.865382 0.501113i $$-0.167076\pi$$
$$674$$ −8.00000 −0.308148
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ 27.0000i 1.03769i 0.854867 + 0.518847i $$0.173639\pi$$
−0.854867 + 0.518847i $$0.826361\pi$$
$$678$$ − 6.00000i − 0.230429i
$$679$$ 6.00000 0.230259
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 3.00000i 0.114876i
$$683$$ − 26.0000i − 0.994862i −0.867503 0.497431i $$-0.834277\pi$$
0.867503 0.497431i $$-0.165723\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ − 10.0000i − 0.381524i
$$688$$ − 1.00000i − 0.0381246i
$$689$$ 9.00000 0.342873
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ − 4.00000i − 0.152057i
$$693$$ − 9.00000i − 0.341882i
$$694$$ 7.00000 0.265716
$$695$$ 0 0
$$696$$ 10.0000 0.379049
$$697$$ 14.0000i 0.530288i
$$698$$ 0 0
$$699$$ −9.00000 −0.340411
$$700$$ 0 0
$$701$$ −48.0000 −1.81293 −0.906467 0.422276i $$-0.861231\pi$$
−0.906467 + 0.422276i $$0.861231\pi$$
$$702$$ 1.00000i 0.0377426i
$$703$$ 0 0
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ 24.0000 0.903252
$$707$$ − 6.00000i − 0.225653i
$$708$$ 0 0
$$709$$ −50.0000 −1.87779 −0.938895 0.344204i $$-0.888149\pi$$
−0.938895 + 0.344204i $$0.888149\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ − 10.0000i − 0.374766i
$$713$$ 4.00000i 0.149801i
$$714$$ 6.00000 0.224544
$$715$$ 0 0
$$716$$ −5.00000 −0.186859
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −50.0000 −1.86469 −0.932343 0.361576i $$-0.882239\pi$$
−0.932343 + 0.361576i $$0.882239\pi$$
$$720$$ 0 0
$$721$$ −3.00000 −0.111726
$$722$$ 19.0000i 0.707107i
$$723$$ 12.0000i 0.446285i
$$724$$ 13.0000 0.483141
$$725$$ 0 0
$$726$$ −2.00000 −0.0742270
$$727$$ 27.0000i 1.00137i 0.865628 + 0.500687i $$0.166919\pi$$
−0.865628 + 0.500687i $$0.833081\pi$$
$$728$$ − 3.00000i − 0.111187i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 2.00000 0.0739727
$$732$$ − 7.00000i − 0.258727i
$$733$$ − 36.0000i − 1.32969i −0.746981 0.664845i $$-0.768498\pi$$
0.746981 0.664845i $$-0.231502\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ − 6.00000i − 0.221013i
$$738$$ 7.00000i 0.257674i
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 27.0000i − 0.991201i
$$743$$ 4.00000i 0.146746i 0.997305 + 0.0733729i $$0.0233763\pi$$
−0.997305 + 0.0733729i $$0.976624\pi$$
$$744$$ −1.00000 −0.0366618
$$745$$ 0 0
$$746$$ 24.0000 0.878702
$$747$$ − 9.00000i − 0.329293i
$$748$$ 6.00000i 0.219382i
$$749$$ −54.0000 −1.97312
$$750$$ 0 0
$$751$$ 7.00000 0.255434 0.127717 0.991811i $$-0.459235\pi$$
0.127717 + 0.991811i $$0.459235\pi$$
$$752$$ 7.00000i 0.255264i
$$753$$ − 3.00000i − 0.109326i
$$754$$ 10.0000 0.364179
$$755$$ 0 0
$$756$$ 3.00000 0.109109
$$757$$ − 13.0000i − 0.472493i −0.971693 0.236247i $$-0.924083\pi$$
0.971693 0.236247i $$-0.0759173\pi$$
$$758$$ 0 0
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ 2.00000i 0.0724524i
$$763$$ − 30.0000i − 1.08607i
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 0 0
$$768$$ 1.00000i 0.0360844i
$$769$$ 45.0000 1.62274 0.811371 0.584532i $$-0.198722\pi$$
0.811371 + 0.584532i $$0.198722\pi$$
$$770$$ 0 0
$$771$$ −7.00000 −0.252099
$$772$$ − 9.00000i − 0.323917i
$$773$$ 34.0000i 1.22290i 0.791285 + 0.611448i $$0.209412\pi$$
−0.791285 + 0.611448i $$0.790588\pi$$
$$774$$ 1.00000 0.0359443
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ − 9.00000i − 0.322873i
$$778$$ 10.0000i 0.358517i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −21.0000 −0.751439
$$782$$ 8.00000i 0.286079i
$$783$$ 10.0000i 0.357371i
$$784$$ −2.00000 −0.0714286
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 7.00000i 0.249523i 0.992187 + 0.124762i $$0.0398166\pi$$
−0.992187 + 0.124762i $$0.960183\pi$$
$$788$$ − 27.0000i − 0.961835i
$$789$$ 6.00000 0.213606
$$790$$ 0 0
$$791$$ −18.0000 −0.640006
$$792$$ 3.00000i 0.106600i
$$793$$ − 7.00000i − 0.248577i
$$794$$ −28.0000 −0.993683
$$795$$ 0 0
$$796$$ −20.0000 −0.708881
$$797$$ 2.00000i 0.0708436i 0.999372 + 0.0354218i $$0.0112775\pi$$
−0.999372 + 0.0354218i $$0.988723\pi$$
$$798$$ 0 0
$$799$$ −14.0000 −0.495284
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 8.00000i 0.282490i
$$803$$ − 12.0000i − 0.423471i
$$804$$ 2.00000 0.0705346
$$805$$ 0 0
$$806$$ −1.00000 −0.0352235
$$807$$ 10.0000i 0.352017i
$$808$$ 2.00000i 0.0703598i
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ − 30.0000i − 1.05279i
$$813$$ − 8.00000i − 0.280572i
$$814$$ 9.00000 0.315450
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ 0 0
$$818$$ − 10.0000i − 0.349642i
$$819$$ 3.00000 0.104828
$$820$$ 0 0
$$821$$ −53.0000 −1.84971 −0.924856 0.380317i $$-0.875815\pi$$
−0.924856 + 0.380317i $$0.875815\pi$$
$$822$$ 2.00000i 0.0697580i
$$823$$ 24.0000i 0.836587i 0.908312 + 0.418294i $$0.137372\pi$$
−0.908312 + 0.418294i $$0.862628\pi$$
$$824$$ 1.00000 0.0348367
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 28.0000i − 0.973655i −0.873498 0.486828i $$-0.838154\pi$$
0.873498 0.486828i $$-0.161846\pi$$
$$828$$ 4.00000i 0.139010i
$$829$$ −55.0000 −1.91023 −0.955114 0.296237i $$-0.904268\pi$$
−0.955114 + 0.296237i $$0.904268\pi$$
$$830$$ 0 0
$$831$$ 18.0000 0.624413
$$832$$ 1.00000i 0.0346688i
$$833$$ − 4.00000i − 0.138592i
$$834$$ 5.00000 0.173136
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 1.00000i − 0.0345651i
$$838$$ − 30.0000i − 1.03633i
$$839$$ −15.0000 −0.517858 −0.258929 0.965896i $$-0.583369\pi$$
−0.258929 + 0.965896i $$0.583369\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 28.0000i 0.964944i
$$843$$ 7.00000i 0.241093i
$$844$$ −2.00000 −0.0688428
$$845$$ 0 0
$$846$$ −7.00000 −0.240665
$$847$$ 6.00000i 0.206162i
$$848$$ 9.00000i 0.309061i
$$849$$ −24.0000 −0.823678
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ − 7.00000i − 0.239816i
$$853$$ − 16.0000i − 0.547830i −0.961754 0.273915i $$-0.911681\pi$$
0.961754 0.273915i $$-0.0883186\pi$$
$$854$$ −21.0000 −0.718605
$$855$$ 0 0
$$856$$ 18.0000 0.615227
$$857$$ − 3.00000i − 0.102478i −0.998686 0.0512390i $$-0.983683\pi$$
0.998686 0.0512390i $$-0.0163170\pi$$
$$858$$ 3.00000i 0.102418i
$$859$$ 25.0000 0.852989 0.426494 0.904490i $$-0.359748\pi$$
0.426494 + 0.904490i $$0.359748\pi$$
$$860$$ 0 0
$$861$$ 21.0000 0.715678
$$862$$ − 7.00000i − 0.238421i
$$863$$ − 46.0000i − 1.56586i −0.622111 0.782929i $$-0.713725\pi$$
0.622111 0.782929i $$-0.286275\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 4.00000 0.135926
$$867$$ 13.0000i 0.441503i
$$868$$ 3.00000i 0.101827i
$$869$$ −30.0000 −1.01768
$$870$$ 0 0
$$871$$ 2.00000 0.0677674
$$872$$ 10.0000i 0.338643i
$$873$$ − 2.00000i − 0.0676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 4.00000 0.135147
$$877$$ − 8.00000i − 0.270141i −0.990836 0.135070i $$-0.956874\pi$$
0.990836 0.135070i $$-0.0431261\pi$$
$$878$$ 0 0
$$879$$ −24.0000 −0.809500
$$880$$ 0 0
$$881$$ 32.0000 1.07811 0.539054 0.842271i $$-0.318782\pi$$
0.539054 + 0.842271i $$0.318782\pi$$
$$882$$ − 2.00000i − 0.0673435i
$$883$$ − 11.0000i − 0.370179i −0.982722 0.185090i $$-0.940742\pi$$
0.982722 0.185090i $$-0.0592576\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ − 13.0000i − 0.436497i −0.975893 0.218249i $$-0.929966\pi$$
0.975893 0.218249i $$-0.0700344\pi$$
$$888$$ 3.00000i 0.100673i
$$889$$ 6.00000 0.201234
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ 6.00000i 0.200895i
$$893$$ 0 0
$$894$$ 20.0000 0.668900
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ 4.00000i 0.133556i
$$898$$ 20.0000i 0.667409i
$$899$$ −10.0000 −0.333519
$$900$$ 0 0
$$901$$ −18.0000 −0.599667
$$902$$ 21.0000i 0.699224i
$$903$$ − 3.00000i − 0.0998337i
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ −18.0000 −0.598010
$$907$$ 32.0000i 1.06254i 0.847202 + 0.531271i $$0.178286\pi$$
−0.847202 + 0.531271i $$0.821714\pi$$
$$908$$ − 12.0000i − 0.398234i
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ 0 0
$$913$$ − 27.0000i − 0.893570i
$$914$$ −8.00000 −0.264616
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ − 36.0000i − 1.18882i
$$918$$ − 2.00000i − 0.0660098i
$$919$$ −25.0000 −0.824674 −0.412337 0.911031i $$-0.635287\pi$$
−0.412337 + 0.911031i $$0.635287\pi$$
$$920$$ 0 0
$$921$$ −22.0000 −0.724925
$$922$$ 3.00000i 0.0987997i
$$923$$ − 7.00000i − 0.230408i
$$924$$ 9.00000 0.296078
$$925$$ 0 0
$$926$$ 14.0000 0.460069
$$927$$ 1.00000i 0.0328443i
$$928$$ 10.0000i 0.328266i
$$929$$ −40.0000 −1.31236 −0.656179 0.754606i $$-0.727828\pi$$
−0.656179 + 0.754606i $$0.727828\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 9.00000i − 0.294805i
$$933$$ 27.0000i 0.883940i
$$934$$ −18.0000 −0.588978
$$935$$ 0 0
$$936$$ −1.00000 −0.0326860
$$937$$ 2.00000i 0.0653372i 0.999466 + 0.0326686i $$0.0104006\pi$$
−0.999466 + 0.0326686i $$0.989599\pi$$
$$938$$ − 6.00000i − 0.195907i
$$939$$ −24.0000 −0.783210
$$940$$ 0 0
$$941$$ −38.0000 −1.23876 −0.619382 0.785090i $$-0.712617\pi$$
−0.619382 + 0.785090i $$0.712617\pi$$
$$942$$ 22.0000i 0.716799i
$$943$$ 28.0000i 0.911805i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 3.00000 0.0975384
$$947$$ − 3.00000i − 0.0974869i −0.998811 0.0487435i $$-0.984478\pi$$
0.998811 0.0487435i $$-0.0155217\pi$$
$$948$$ − 10.0000i − 0.324785i
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ −22.0000 −0.713399
$$952$$ 6.00000i 0.194461i
$$953$$ 34.0000i 1.10137i 0.834714 + 0.550684i $$0.185633\pi$$
−0.834714 + 0.550684i $$0.814367\pi$$
$$954$$ −9.00000 −0.291386
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 30.0000i 0.969762i
$$958$$ 0 0
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 3.00000i 0.0967239i
$$963$$ 18.0000i 0.580042i
$$964$$ −12.0000 −0.386494
$$965$$ 0 0
$$966$$ 12.0000 0.386094
$$967$$ − 18.0000i − 0.578841i −0.957202 0.289420i $$-0.906537\pi$$
0.957202 0.289420i $$-0.0934626\pi$$
$$968$$ − 2.00000i − 0.0642824i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ − 15.0000i − 0.480878i
$$974$$ −28.0000 −0.897178
$$975$$ 0 0
$$976$$ 7.00000 0.224065
$$977$$ − 13.0000i − 0.415907i −0.978139 0.207953i $$-0.933320\pi$$
0.978139 0.207953i $$-0.0666802\pi$$
$$978$$ 4.00000i 0.127906i
$$979$$ 30.0000 0.958804
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 28.0000i 0.893516i
$$983$$ − 16.0000i − 0.510321i −0.966899 0.255160i $$-0.917872\pi$$
0.966899 0.255160i $$-0.0821283\pi$$
$$984$$ −7.00000 −0.223152
$$985$$ 0 0
$$986$$ −20.0000 −0.636930
$$987$$ 21.0000i 0.668437i
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ 2.00000 0.0635321 0.0317660 0.999495i $$-0.489887\pi$$
0.0317660 + 0.999495i $$0.489887\pi$$
$$992$$ − 1.00000i − 0.0317500i
$$993$$ − 3.00000i − 0.0952021i
$$994$$ −21.0000 −0.666080
$$995$$ 0 0
$$996$$ 9.00000 0.285176
$$997$$ 42.0000i 1.33015i 0.746775 + 0.665077i $$0.231601\pi$$
−0.746775 + 0.665077i $$0.768399\pi$$
$$998$$ − 20.0000i − 0.633089i
$$999$$ −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 4650.2.d.q.3349.1 2
5.2 odd 4 4650.2.a.bu.1.1 yes 1
5.3 odd 4 4650.2.a.b.1.1 1
5.4 even 2 inner 4650.2.d.q.3349.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
4650.2.a.b.1.1 1 5.3 odd 4
4650.2.a.bu.1.1 yes 1 5.2 odd 4
4650.2.d.q.3349.1 2 1.1 even 1 trivial
4650.2.d.q.3349.2 2 5.4 even 2 inner