# Properties

 Label 950.2.b.b.799.1 Level $950$ Weight $2$ Character 950.799 Analytic conductor $7.586$ 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] = [950,2,Mod(799,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("950.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.b (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: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 950.799 Dual form 950.2.b.b.799.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} -5.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -2.00000i q^{18} -1.00000 q^{19} -1.00000 q^{21} +6.00000i q^{22} -3.00000i q^{23} +1.00000 q^{24} -5.00000 q^{26} -5.00000i q^{27} +1.00000i q^{28} -9.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +6.00000i q^{33} +3.00000 q^{34} -2.00000 q^{36} +2.00000i q^{37} +1.00000i q^{38} -5.00000 q^{39} +1.00000i q^{42} -8.00000i q^{43} +6.00000 q^{44} -3.00000 q^{46} -1.00000i q^{48} +6.00000 q^{49} +3.00000 q^{51} +5.00000i q^{52} +3.00000i q^{53} -5.00000 q^{54} +1.00000 q^{56} +1.00000i q^{57} +9.00000i q^{58} -9.00000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +6.00000 q^{66} +5.00000i q^{67} -3.00000i q^{68} -3.00000 q^{69} -6.00000 q^{71} +2.00000i q^{72} +7.00000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +6.00000i q^{77} +5.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{83} +1.00000 q^{84} -8.00000 q^{86} +9.00000i q^{87} -6.00000i q^{88} +12.0000 q^{89} -5.00000 q^{91} +3.00000i q^{92} +4.00000i q^{93} -1.00000 q^{96} -10.0000i q^{97} -6.00000i q^{98} -12.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 $$2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} - 12 q^{11} - 2 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{21} + 2 q^{24} - 10 q^{26} - 18 q^{29} - 8 q^{31} + 6 q^{34} - 4 q^{36} - 10 q^{39} + 12 q^{44} - 6 q^{46} + 12 q^{49} + 6 q^{51} - 10 q^{54} + 2 q^{56} - 18 q^{59} - 20 q^{61} - 2 q^{64} + 12 q^{66} - 6 q^{69} - 12 q^{71} + 4 q^{74} + 2 q^{76} + 20 q^{79} + 2 q^{81} + 2 q^{84} - 16 q^{86} + 24 q^{89} - 10 q^{91} - 2 q^{96} - 24 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 - 12 * q^11 - 2 * q^14 + 2 * q^16 - 2 * q^19 - 2 * q^21 + 2 * q^24 - 10 * q^26 - 18 * q^29 - 8 * q^31 + 6 * q^34 - 4 * q^36 - 10 * q^39 + 12 * q^44 - 6 * q^46 + 12 * q^49 + 6 * q^51 - 10 * q^54 + 2 * q^56 - 18 * q^59 - 20 * q^61 - 2 * q^64 + 12 * q^66 - 6 * q^69 - 12 * q^71 + 4 * q^74 + 2 * q^76 + 20 * q^79 + 2 * q^81 + 2 * q^84 - 16 * q^86 + 24 * q^89 - 10 * q^91 - 2 * q^96 - 24 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 −0.957427 0.288675i $$-0.906785\pi$$
0.957427 0.288675i $$-0.0932147\pi$$
$$4$$ −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
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 5.00000i − 1.38675i −0.720577 0.693375i $$-0.756123\pi$$
0.720577 0.693375i $$-0.243877\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.00000i 0.727607i 0.931476 + 0.363803i $$0.118522\pi$$
−0.931476 + 0.363803i $$0.881478\pi$$
$$18$$ − 2.00000i − 0.471405i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 6.00000i 1.27920i
$$23$$ − 3.00000i − 0.625543i −0.949828 0.312772i $$-0.898743\pi$$
0.949828 0.312772i $$-0.101257\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −5.00000 −0.980581
$$27$$ − 5.00000i − 0.962250i
$$28$$ 1.00000i 0.188982i
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 6.00000i 1.04447i
$$34$$ 3.00000 0.514496
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ −5.00000 −0.800641
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 1.00000i 0.154303i
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ −3.00000 −0.442326
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ 3.00000 0.420084
$$52$$ 5.00000i 0.693375i
$$53$$ 3.00000i 0.412082i 0.978543 + 0.206041i $$0.0660580\pi$$
−0.978543 + 0.206041i $$0.933942\pi$$
$$54$$ −5.00000 −0.680414
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 1.00000i 0.132453i
$$58$$ 9.00000i 1.18176i
$$59$$ −9.00000 −1.17170 −0.585850 0.810419i $$-0.699239\pi$$
−0.585850 + 0.810419i $$0.699239\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 6.00000 0.738549
$$67$$ 5.00000i 0.610847i 0.952217 + 0.305424i $$0.0987981\pi$$
−0.952217 + 0.305424i $$0.901202\pi$$
$$68$$ − 3.00000i − 0.363803i
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 2.00000i 0.235702i
$$73$$ 7.00000i 0.819288i 0.912245 + 0.409644i $$0.134347\pi$$
−0.912245 + 0.409644i $$0.865653\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 6.00000i 0.683763i
$$78$$ 5.00000i 0.566139i
$$79$$ 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
$$82$$ 0 0
$$83$$ 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
$$86$$ −8.00000 −0.862662
$$87$$ 9.00000i 0.964901i
$$88$$ − 6.00000i − 0.639602i
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 0 0
$$91$$ −5.00000 −0.524142
$$92$$ 3.00000i 0.312772i
$$93$$ 4.00000i 0.414781i
$$94$$ 0 0
$$95$$ 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
$$99$$ −12.0000 −1.20605
$$100$$ 0 0
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ − 3.00000i − 0.297044i
$$103$$ − 14.0000i − 1.37946i −0.724066 0.689730i $$-0.757729\pi$$
0.724066 0.689730i $$-0.242271\pi$$
$$104$$ 5.00000 0.490290
$$105$$ 0 0
$$106$$ 3.00000 0.291386
$$107$$ − 9.00000i − 0.870063i −0.900415 0.435031i $$-0.856737\pi$$
0.900415 0.435031i $$-0.143263\pi$$
$$108$$ 5.00000i 0.481125i
$$109$$ −11.0000 −1.05361 −0.526804 0.849987i $$-0.676610\pi$$
−0.526804 + 0.849987i $$0.676610\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ − 1.00000i − 0.0944911i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ 9.00000 0.835629
$$117$$ − 10.0000i − 0.924500i
$$118$$ 9.00000i 0.828517i
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 10.0000i 0.905357i
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 2.00000i 0.177471i 0.996055 + 0.0887357i $$0.0282826\pi$$
−0.996055 + 0.0887357i $$0.971717\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ − 6.00000i − 0.522233i
$$133$$ 1.00000i 0.0867110i
$$134$$ 5.00000 0.431934
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ − 9.00000i − 0.768922i −0.923141 0.384461i $$-0.874387\pi$$
0.923141 0.384461i $$-0.125613\pi$$
$$138$$ 3.00000i 0.255377i
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 6.00000i 0.503509i
$$143$$ 30.0000i 2.50873i
$$144$$ 2.00000 0.166667
$$145$$ 0 0
$$146$$ 7.00000 0.579324
$$147$$ − 6.00000i − 0.494872i
$$148$$ − 2.00000i − 0.164399i
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ − 1.00000i − 0.0811107i
$$153$$ 6.00000i 0.485071i
$$154$$ 6.00000 0.483494
$$155$$ 0 0
$$156$$ 5.00000 0.400320
$$157$$ − 22.0000i − 1.75579i −0.478852 0.877896i $$-0.658947\pi$$
0.478852 0.877896i $$-0.341053\pi$$
$$158$$ − 10.0000i − 0.795557i
$$159$$ 3.00000 0.237915
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 20.0000i − 1.56652i −0.621694 0.783260i $$-0.713555\pi$$
0.621694 0.783260i $$-0.286445\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ − 1.00000i − 0.0771517i
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 8.00000i 0.609994i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 9.00000 0.682288
$$175$$ 0 0
$$176$$ −6.00000 −0.452267
$$177$$ 9.00000i 0.676481i
$$178$$ − 12.0000i − 0.899438i
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 5.00000i 0.370625i
$$183$$ 10.0000i 0.739221i
$$184$$ 3.00000 0.221163
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ − 18.0000i − 1.31629i
$$188$$ 0 0
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 14.0000i − 1.00774i −0.863779 0.503871i $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 12.0000i 0.852803i
$$199$$ −11.0000 −0.779769 −0.389885 0.920864i $$-0.627485\pi$$
−0.389885 + 0.920864i $$0.627485\pi$$
$$200$$ 0 0
$$201$$ 5.00000 0.352673
$$202$$ − 18.0000i − 1.26648i
$$203$$ 9.00000i 0.631676i
$$204$$ −3.00000 −0.210042
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ − 6.00000i − 0.417029i
$$208$$ − 5.00000i − 0.346688i
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ 5.00000 0.344214 0.172107 0.985078i $$-0.444942\pi$$
0.172107 + 0.985078i $$0.444942\pi$$
$$212$$ − 3.00000i − 0.206041i
$$213$$ 6.00000i 0.411113i
$$214$$ −9.00000 −0.615227
$$215$$ 0 0
$$216$$ 5.00000 0.340207
$$217$$ 4.00000i 0.271538i
$$218$$ 11.0000i 0.745014i
$$219$$ 7.00000 0.473016
$$220$$ 0 0
$$221$$ 15.0000 1.00901
$$222$$ − 2.00000i − 0.134231i
$$223$$ − 26.0000i − 1.74109i −0.492090 0.870544i $$-0.663767\pi$$
0.492090 0.870544i $$-0.336233\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ − 15.0000i − 0.995585i −0.867296 0.497792i $$-0.834144\pi$$
0.867296 0.497792i $$-0.165856\pi$$
$$228$$ − 1.00000i − 0.0662266i
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ − 9.00000i − 0.590879i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ −10.0000 −0.653720
$$235$$ 0 0
$$236$$ 9.00000 0.585850
$$237$$ − 10.0000i − 0.649570i
$$238$$ − 3.00000i − 0.194461i
$$239$$ 21.0000 1.35838 0.679189 0.733964i $$-0.262332\pi$$
0.679189 + 0.733964i $$0.262332\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ − 25.0000i − 1.60706i
$$243$$ − 16.0000i − 1.02640i
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.00000i 0.318142i
$$248$$ − 4.00000i − 0.254000i
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ 6.00000 0.378717 0.189358 0.981908i $$-0.439359\pi$$
0.189358 + 0.981908i $$0.439359\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 18.0000i 1.13165i
$$254$$ 2.00000 0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 12.0000i 0.748539i 0.927320 + 0.374270i $$0.122107\pi$$
−0.927320 + 0.374270i $$0.877893\pi$$
$$258$$ 8.00000i 0.498058i
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ 0 0
$$263$$ − 24.0000i − 1.47990i −0.672660 0.739952i $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ −6.00000 −0.369274
$$265$$ 0 0
$$266$$ 1.00000 0.0613139
$$267$$ − 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
$$271$$ 11.0000 0.668202 0.334101 0.942537i $$-0.391567\pi$$
0.334101 + 0.942537i $$0.391567\pi$$
$$272$$ 3.00000i 0.181902i
$$273$$ 5.00000i 0.302614i
$$274$$ −9.00000 −0.543710
$$275$$ 0 0
$$276$$ 3.00000 0.180579
$$277$$ 8.00000i 0.480673i 0.970690 + 0.240337i $$0.0772579\pi$$
−0.970690 + 0.240337i $$0.922742\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 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
$$286$$ 30.0000 1.77394
$$287$$ 0 0
$$288$$ − 2.00000i − 0.117851i
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ − 7.00000i − 0.409644i
$$293$$ 21.0000i 1.22683i 0.789760 + 0.613417i $$0.210205\pi$$
−0.789760 + 0.613417i $$0.789795\pi$$
$$294$$ −6.00000 −0.349927
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 30.0000i 1.74078i
$$298$$ 0 0
$$299$$ −15.0000 −0.867472
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 10.0000i 0.575435i
$$303$$ − 18.0000i − 1.03407i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ 20.0000i 1.14146i 0.821138 + 0.570730i $$0.193340\pi$$
−0.821138 + 0.570730i $$0.806660\pi$$
$$308$$ − 6.00000i − 0.341882i
$$309$$ −14.0000 −0.796432
$$310$$ 0 0
$$311$$ −21.0000 −1.19080 −0.595400 0.803429i $$-0.703007\pi$$
−0.595400 + 0.803429i $$0.703007\pi$$
$$312$$ − 5.00000i − 0.283069i
$$313$$ 19.0000i 1.07394i 0.843600 + 0.536972i $$0.180432\pi$$
−0.843600 + 0.536972i $$0.819568\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ − 9.00000i − 0.505490i −0.967533 0.252745i $$-0.918667\pi$$
0.967533 0.252745i $$-0.0813334\pi$$
$$318$$ − 3.00000i − 0.168232i
$$319$$ 54.0000 3.02342
$$320$$ 0 0
$$321$$ −9.00000 −0.502331
$$322$$ 3.00000i 0.167183i
$$323$$ − 3.00000i − 0.166924i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ 11.0000i 0.608301i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1.00000 −0.0549650 −0.0274825 0.999622i $$-0.508749\pi$$
−0.0274825 + 0.999622i $$0.508749\pi$$
$$332$$ − 6.00000i − 0.329293i
$$333$$ 4.00000i 0.219199i
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ − 4.00000i − 0.217894i −0.994048 0.108947i $$-0.965252\pi$$
0.994048 0.108947i $$-0.0347479\pi$$
$$338$$ 12.0000i 0.652714i
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 24.0000 1.29967
$$342$$ 2.00000i 0.108148i
$$343$$ − 13.0000i − 0.701934i
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 18.0000i 0.966291i 0.875540 + 0.483145i $$0.160506\pi$$
−0.875540 + 0.483145i $$0.839494\pi$$
$$348$$ − 9.00000i − 0.482451i
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ −25.0000 −1.33440
$$352$$ 6.00000i 0.319801i
$$353$$ 15.0000i 0.798369i 0.916871 + 0.399185i $$0.130707\pi$$
−0.916871 + 0.399185i $$0.869293\pi$$
$$354$$ 9.00000 0.478345
$$355$$ 0 0
$$356$$ −12.0000 −0.635999
$$357$$ − 3.00000i − 0.158777i
$$358$$ 0 0
$$359$$ −21.0000 −1.10834 −0.554169 0.832404i $$-0.686964\pi$$
−0.554169 + 0.832404i $$0.686964\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 2.00000i − 0.105118i
$$363$$ − 25.0000i − 1.31216i
$$364$$ 5.00000 0.262071
$$365$$ 0 0
$$366$$ 10.0000 0.522708
$$367$$ − 28.0000i − 1.46159i −0.682598 0.730794i $$-0.739150\pi$$
0.682598 0.730794i $$-0.260850\pi$$
$$368$$ − 3.00000i − 0.156386i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 3.00000 0.155752
$$372$$ − 4.00000i − 0.207390i
$$373$$ − 23.0000i − 1.19089i −0.803394 0.595447i $$-0.796975\pi$$
0.803394 0.595447i $$-0.203025\pi$$
$$374$$ −18.0000 −0.930758
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 45.0000i 2.31762i
$$378$$ 5.00000i 0.257172i
$$379$$ 7.00000 0.359566 0.179783 0.983706i $$-0.442460\pi$$
0.179783 + 0.983706i $$0.442460\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ − 3.00000i − 0.153493i
$$383$$ − 18.0000i − 0.919757i −0.887982 0.459879i $$-0.847893\pi$$
0.887982 0.459879i $$-0.152107\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ − 16.0000i − 0.813326i
$$388$$ 10.0000i 0.507673i
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 9.00000 0.455150
$$392$$ 6.00000i 0.303046i
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 12.0000 0.603023
$$397$$ 20.0000i 1.00377i 0.864934 + 0.501886i $$0.167360\pi$$
−0.864934 + 0.501886i $$0.832640\pi$$
$$398$$ 11.0000i 0.551380i
$$399$$ 1.00000 0.0500626
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ − 5.00000i − 0.249377i
$$403$$ 20.0000i 0.996271i
$$404$$ −18.0000 −0.895533
$$405$$ 0 0
$$406$$ 9.00000 0.446663
$$407$$ − 12.0000i − 0.594818i
$$408$$ 3.00000i 0.148522i
$$409$$ −32.0000 −1.58230 −0.791149 0.611623i $$-0.790517\pi$$
−0.791149 + 0.611623i $$0.790517\pi$$
$$410$$ 0 0
$$411$$ −9.00000 −0.443937
$$412$$ 14.0000i 0.689730i
$$413$$ 9.00000i 0.442861i
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ −5.00000 −0.245145
$$417$$ − 4.00000i − 0.195881i
$$418$$ − 6.00000i − 0.293470i
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 17.0000 0.828529 0.414265 0.910156i $$-0.364039\pi$$
0.414265 + 0.910156i $$0.364039\pi$$
$$422$$ − 5.00000i − 0.243396i
$$423$$ 0 0
$$424$$ −3.00000 −0.145693
$$425$$ 0 0
$$426$$ 6.00000 0.290701
$$427$$ 10.0000i 0.483934i
$$428$$ 9.00000i 0.435031i
$$429$$ 30.0000 1.44841
$$430$$ 0 0
$$431$$ 6.00000 0.289010 0.144505 0.989504i $$-0.453841\pi$$
0.144505 + 0.989504i $$0.453841\pi$$
$$432$$ − 5.00000i − 0.240563i
$$433$$ − 2.00000i − 0.0961139i −0.998845 0.0480569i $$-0.984697\pi$$
0.998845 0.0480569i $$-0.0153029\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 0 0
$$436$$ 11.0000 0.526804
$$437$$ 3.00000i 0.143509i
$$438$$ − 7.00000i − 0.334473i
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 0 0
$$441$$ 12.0000 0.571429
$$442$$ − 15.0000i − 0.713477i
$$443$$ 18.0000i 0.855206i 0.903967 + 0.427603i $$0.140642\pi$$
−0.903967 + 0.427603i $$0.859358\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ −26.0000 −1.23114
$$447$$ 0 0
$$448$$ 1.00000i 0.0472456i
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 6.00000i 0.282216i
$$453$$ 10.0000i 0.469841i
$$454$$ −15.0000 −0.703985
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ 17.0000i 0.795226i 0.917553 + 0.397613i $$0.130161\pi$$
−0.917553 + 0.397613i $$0.869839\pi$$
$$458$$ − 22.0000i − 1.02799i
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ − 6.00000i − 0.279145i
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ −9.00000 −0.417815
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 18.0000i 0.832941i 0.909149 + 0.416470i $$0.136733\pi$$
−0.909149 + 0.416470i $$0.863267\pi$$
$$468$$ 10.0000i 0.462250i
$$469$$ 5.00000 0.230879
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ − 9.00000i − 0.414259i
$$473$$ 48.0000i 2.20704i
$$474$$ −10.0000 −0.459315
$$475$$ 0 0
$$476$$ −3.00000 −0.137505
$$477$$ 6.00000i 0.274721i
$$478$$ − 21.0000i − 0.960518i
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ − 8.00000i − 0.364390i
$$483$$ 3.00000i 0.136505i
$$484$$ −25.0000 −1.13636
$$485$$ 0 0
$$486$$ −16.0000 −0.725775
$$487$$ 2.00000i 0.0906287i 0.998973 + 0.0453143i $$0.0144289\pi$$
−0.998973 + 0.0453143i $$0.985571\pi$$
$$488$$ − 10.0000i − 0.452679i
$$489$$ −20.0000 −0.904431
$$490$$ 0 0
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ 0 0
$$493$$ − 27.0000i − 1.21602i
$$494$$ 5.00000 0.224961
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 6.00000i 0.269137i
$$498$$ − 6.00000i − 0.268866i
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ − 6.00000i − 0.267793i
$$503$$ 21.0000i 0.936344i 0.883637 + 0.468172i $$0.155087\pi$$
−0.883637 + 0.468172i $$0.844913\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 18.0000 0.800198
$$507$$ 12.0000i 0.532939i
$$508$$ − 2.00000i − 0.0887357i
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 7.00000 0.309662
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 5.00000i 0.220755i
$$514$$ 12.0000 0.529297
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ − 2.00000i − 0.0878750i
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −36.0000 −1.57719 −0.788594 0.614914i $$-0.789191\pi$$
−0.788594 + 0.614914i $$0.789191\pi$$
$$522$$ 18.0000i 0.787839i
$$523$$ − 11.0000i − 0.480996i −0.970650 0.240498i $$-0.922689\pi$$
0.970650 0.240498i $$-0.0773108\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ − 12.0000i − 0.522728i
$$528$$ 6.00000i 0.261116i
$$529$$ 14.0000 0.608696
$$530$$ 0 0
$$531$$ −18.0000 −0.781133
$$532$$ − 1.00000i − 0.0433555i
$$533$$ 0 0
$$534$$ −12.0000 −0.519291
$$535$$ 0 0
$$536$$ −5.00000 −0.215967
$$537$$ 0 0
$$538$$ − 6.00000i − 0.258678i
$$539$$ −36.0000 −1.55063
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ − 11.0000i − 0.472490i
$$543$$ − 2.00000i − 0.0858282i
$$544$$ 3.00000 0.128624
$$545$$ 0 0
$$546$$ 5.00000 0.213980
$$547$$ 44.0000i 1.88130i 0.339372 + 0.940652i $$0.389785\pi$$
−0.339372 + 0.940652i $$0.610215\pi$$
$$548$$ 9.00000i 0.384461i
$$549$$ −20.0000 −0.853579
$$550$$ 0 0
$$551$$ 9.00000 0.383413
$$552$$ − 3.00000i − 0.127688i
$$553$$ − 10.0000i − 0.425243i
$$554$$ 8.00000 0.339887
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 24.0000i 1.01691i 0.861088 + 0.508456i $$0.169784\pi$$
−0.861088 + 0.508456i $$0.830216\pi$$
$$558$$ 8.00000i 0.338667i
$$559$$ −40.0000 −1.69182
$$560$$ 0 0
$$561$$ −18.0000 −0.759961
$$562$$ 0 0
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ 0 0
$$565$$ 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
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ − 30.0000i − 1.25436i
$$573$$ − 3.00000i − 0.125327i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −2.00000 −0.0833333
$$577$$ 11.0000i 0.457936i 0.973434 + 0.228968i $$0.0735351\pi$$
−0.973434 + 0.228968i $$0.926465\pi$$
$$578$$ − 8.00000i − 0.332756i
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ 6.00000 0.248922
$$582$$ 10.0000i 0.414513i
$$583$$ − 18.0000i − 0.745484i
$$584$$ −7.00000 −0.289662
$$585$$ 0 0
$$586$$ 21.0000 0.867502
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 6.00000i 0.247436i
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2.00000i 0.0821995i
$$593$$ 30.0000i 1.23195i 0.787765 + 0.615976i $$0.211238\pi$$
−0.787765 + 0.615976i $$0.788762\pi$$
$$594$$ 30.0000 1.23091
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 11.0000i 0.450200i
$$598$$ 15.0000i 0.613396i
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −28.0000 −1.14214 −0.571072 0.820900i $$-0.693472\pi$$
−0.571072 + 0.820900i $$0.693472\pi$$
$$602$$ 8.00000i 0.326056i
$$603$$ 10.0000i 0.407231i
$$604$$ 10.0000 0.406894
$$605$$ 0 0
$$606$$ −18.0000 −0.731200
$$607$$ − 22.0000i − 0.892952i −0.894795 0.446476i $$-0.852679\pi$$
0.894795 0.446476i $$-0.147321\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ 9.00000 0.364698
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 6.00000i − 0.242536i
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ −6.00000 −0.241747
$$617$$ − 6.00000i − 0.241551i −0.992680 0.120775i $$-0.961462\pi$$
0.992680 0.120775i $$-0.0385381\pi$$
$$618$$ 14.0000i 0.563163i
$$619$$ 10.0000 0.401934 0.200967 0.979598i $$-0.435592\pi$$
0.200967 + 0.979598i $$0.435592\pi$$
$$620$$ 0 0
$$621$$ −15.0000 −0.601929
$$622$$ 21.0000i 0.842023i
$$623$$ − 12.0000i − 0.480770i
$$624$$ −5.00000 −0.200160
$$625$$ 0 0
$$626$$ 19.0000 0.759393
$$627$$ − 6.00000i − 0.239617i
$$628$$ 22.0000i 0.877896i
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 10.0000i 0.397779i
$$633$$ − 5.00000i − 0.198732i
$$634$$ −9.00000 −0.357436
$$635$$ 0 0
$$636$$ −3.00000 −0.118958
$$637$$ − 30.0000i − 1.18864i
$$638$$ − 54.0000i − 2.13788i
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 9.00000i 0.355202i
$$643$$ 22.0000i 0.867595i 0.901010 + 0.433798i $$0.142827\pi$$
−0.901010 + 0.433798i $$0.857173\pi$$
$$644$$ 3.00000 0.118217
$$645$$ 0 0
$$646$$ −3.00000 −0.118033
$$647$$ 27.0000i 1.06148i 0.847535 + 0.530740i $$0.178086\pi$$
−0.847535 + 0.530740i $$0.821914\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 54.0000 2.11969
$$650$$ 0 0
$$651$$ 4.00000 0.156772
$$652$$ 20.0000i 0.783260i
$$653$$ − 24.0000i − 0.939193i −0.882881 0.469596i $$-0.844399\pi$$
0.882881 0.469596i $$-0.155601\pi$$
$$654$$ 11.0000 0.430134
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 14.0000i 0.546192i
$$658$$ 0 0
$$659$$ 45.0000 1.75295 0.876476 0.481446i $$-0.159888\pi$$
0.876476 + 0.481446i $$0.159888\pi$$
$$660$$ 0 0
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ 1.00000i 0.0388661i
$$663$$ − 15.0000i − 0.582552i
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ 27.0000i 1.04544i
$$668$$ − 12.0000i − 0.464294i
$$669$$ −26.0000 −1.00522
$$670$$ 0 0
$$671$$ 60.0000 2.31627
$$672$$ 1.00000i 0.0385758i
$$673$$ − 44.0000i − 1.69608i −0.529936 0.848038i $$-0.677784\pi$$
0.529936 0.848038i $$-0.322216\pi$$
$$674$$ −4.00000 −0.154074
$$675$$ 0 0
$$676$$ 12.0000 0.461538
$$677$$ − 33.0000i − 1.26829i −0.773213 0.634147i $$-0.781352\pi$$
0.773213 0.634147i $$-0.218648\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ −15.0000 −0.574801
$$682$$ − 24.0000i − 0.919007i
$$683$$ − 36.0000i − 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 0 0
$$686$$ −13.0000 −0.496342
$$687$$ − 22.0000i − 0.839352i
$$688$$ − 8.00000i − 0.304997i
$$689$$ 15.0000 0.571454
$$690$$ 0 0
$$691$$ −10.0000 −0.380418 −0.190209 0.981744i $$-0.560917\pi$$
−0.190209 + 0.981744i $$0.560917\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ 12.0000i 0.455842i
$$694$$ 18.0000 0.683271
$$695$$ 0 0
$$696$$ −9.00000 −0.341144
$$697$$ 0 0
$$698$$ − 10.0000i − 0.378506i
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ 12.0000 0.453234 0.226617 0.973984i $$-0.427233\pi$$
0.226617 + 0.973984i $$0.427233\pi$$
$$702$$ 25.0000i 0.943564i
$$703$$ − 2.00000i − 0.0754314i
$$704$$ 6.00000 0.226134
$$705$$ 0 0
$$706$$ 15.0000 0.564532
$$707$$ − 18.0000i − 0.676960i
$$708$$ − 9.00000i − 0.338241i
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 20.0000 0.750059
$$712$$ 12.0000i 0.449719i
$$713$$ 12.0000i 0.449404i
$$714$$ −3.00000 −0.112272
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 21.0000i − 0.784259i
$$718$$ 21.0000i 0.783713i
$$719$$ −39.0000 −1.45445 −0.727227 0.686397i $$-0.759191\pi$$
−0.727227 + 0.686397i $$0.759191\pi$$
$$720$$ 0 0
$$721$$ −14.0000 −0.521387
$$722$$ − 1.00000i − 0.0372161i
$$723$$ − 8.00000i − 0.297523i
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ −25.0000 −0.927837
$$727$$ − 37.0000i − 1.37225i −0.727482 0.686127i $$-0.759309\pi$$
0.727482 0.686127i $$-0.240691\pi$$
$$728$$ − 5.00000i − 0.185312i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ − 10.0000i − 0.369611i
$$733$$ − 32.0000i − 1.18195i −0.806691 0.590973i $$-0.798744\pi$$
0.806691 0.590973i $$-0.201256\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 0 0
$$736$$ −3.00000 −0.110581
$$737$$ − 30.0000i − 1.10506i
$$738$$ 0 0
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 0 0
$$741$$ 5.00000 0.183680
$$742$$ − 3.00000i − 0.110133i
$$743$$ 36.0000i 1.32071i 0.750953 + 0.660356i $$0.229595\pi$$
−0.750953 + 0.660356i $$0.770405\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 0 0
$$746$$ −23.0000 −0.842090
$$747$$ 12.0000i 0.439057i
$$748$$ 18.0000i 0.658145i
$$749$$ −9.00000 −0.328853
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 0 0
$$753$$ − 6.00000i − 0.218652i
$$754$$ 45.0000 1.63880
$$755$$ 0 0
$$756$$ 5.00000 0.181848
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ − 7.00000i − 0.254251i
$$759$$ 18.0000 0.653359
$$760$$ 0 0
$$761$$ −21.0000 −0.761249 −0.380625 0.924730i $$-0.624291\pi$$
−0.380625 + 0.924730i $$0.624291\pi$$
$$762$$ − 2.00000i − 0.0724524i
$$763$$ 11.0000i 0.398227i
$$764$$ −3.00000 −0.108536
$$765$$ 0 0
$$766$$ −18.0000 −0.650366
$$767$$ 45.0000i 1.62486i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −5.00000 −0.180305 −0.0901523 0.995928i $$-0.528735\pi$$
−0.0901523 + 0.995928i $$0.528735\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ 14.0000i 0.503871i
$$773$$ − 51.0000i − 1.83434i −0.398493 0.917171i $$-0.630467\pi$$
0.398493 0.917171i $$-0.369533\pi$$
$$774$$ −16.0000 −0.575108
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ − 2.00000i − 0.0717496i
$$778$$ 18.0000i 0.645331i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ − 9.00000i − 0.321839i
$$783$$ 45.0000i 1.60817i
$$784$$ 6.00000 0.214286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 31.0000i − 1.10503i −0.833503 0.552515i $$-0.813668\pi$$
0.833503 0.552515i $$-0.186332\pi$$
$$788$$ 0 0
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ − 12.0000i − 0.426401i
$$793$$ 50.0000i 1.77555i
$$794$$ 20.0000 0.709773
$$795$$ 0 0
$$796$$ 11.0000 0.389885
$$797$$ − 39.0000i − 1.38145i −0.723117 0.690725i $$-0.757291\pi$$
0.723117 0.690725i $$-0.242709\pi$$
$$798$$ − 1.00000i − 0.0353996i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 24.0000 0.847998
$$802$$ 0 0
$$803$$ − 42.0000i − 1.48215i
$$804$$ −5.00000 −0.176336
$$805$$ 0 0
$$806$$ 20.0000 0.704470
$$807$$ − 6.00000i − 0.211210i
$$808$$ 18.0000i 0.633238i
$$809$$ −9.00000 −0.316423 −0.158212 0.987405i $$-0.550573\pi$$
−0.158212 + 0.987405i $$0.550573\pi$$
$$810$$ 0 0
$$811$$ 11.0000 0.386262 0.193131 0.981173i $$-0.438136\pi$$
0.193131 + 0.981173i $$0.438136\pi$$
$$812$$ − 9.00000i − 0.315838i
$$813$$ − 11.0000i − 0.385787i
$$814$$ −12.0000 −0.420600
$$815$$ 0 0
$$816$$ 3.00000 0.105021
$$817$$ 8.00000i 0.279885i
$$818$$ 32.0000i 1.11885i
$$819$$ −10.0000 −0.349428
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 9.00000i 0.313911i
$$823$$ − 41.0000i − 1.42917i −0.699549 0.714585i $$-0.746616\pi$$
0.699549 0.714585i $$-0.253384\pi$$
$$824$$ 14.0000 0.487713
$$825$$ 0 0
$$826$$ 9.00000 0.313150
$$827$$ 33.0000i 1.14752i 0.819023 + 0.573761i $$0.194516\pi$$
−0.819023 + 0.573761i $$0.805484\pi$$
$$828$$ 6.00000i 0.208514i
$$829$$ −11.0000 −0.382046 −0.191023 0.981586i $$-0.561180\pi$$
−0.191023 + 0.981586i $$0.561180\pi$$
$$830$$ 0 0
$$831$$ 8.00000 0.277517
$$832$$ 5.00000i 0.173344i
$$833$$ 18.0000i 0.623663i
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ −6.00000 −0.207514
$$837$$ 20.0000i 0.691301i
$$838$$ − 12.0000i − 0.414533i
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ − 17.0000i − 0.585859i
$$843$$ 0 0
$$844$$ −5.00000 −0.172107
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 25.0000i − 0.859010i
$$848$$ 3.00000i 0.103020i
$$849$$ 22.0000 0.755038
$$850$$ 0 0
$$851$$ 6.00000 0.205677
$$852$$ − 6.00000i − 0.205557i
$$853$$ 46.0000i 1.57501i 0.616308 + 0.787505i $$0.288628\pi$$
−0.616308 + 0.787505i $$0.711372\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 0 0
$$856$$ 9.00000 0.307614
$$857$$ − 12.0000i − 0.409912i −0.978771 0.204956i $$-0.934295\pi$$
0.978771 0.204956i $$-0.0657052\pi$$
$$858$$ − 30.0000i − 1.02418i
$$859$$ −14.0000 −0.477674 −0.238837 0.971060i $$-0.576766\pi$$
−0.238837 + 0.971060i $$0.576766\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 6.00000i − 0.204361i
$$863$$ 18.0000i 0.612727i 0.951915 + 0.306364i $$0.0991123\pi$$
−0.951915 + 0.306364i $$0.900888\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ − 8.00000i − 0.271694i
$$868$$ − 4.00000i − 0.135769i
$$869$$ −60.0000 −2.03536
$$870$$ 0 0
$$871$$ 25.0000 0.847093
$$872$$ − 11.0000i − 0.372507i
$$873$$ − 20.0000i − 0.676897i
$$874$$ 3.00000 0.101477
$$875$$ 0 0
$$876$$ −7.00000 −0.236508
$$877$$ 23.0000i 0.776655i 0.921521 + 0.388327i $$0.126947\pi$$
−0.921521 + 0.388327i $$0.873053\pi$$
$$878$$ − 28.0000i − 0.944954i
$$879$$ 21.0000 0.708312
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ − 12.0000i − 0.404061i
$$883$$ 34.0000i 1.14419i 0.820187 + 0.572096i $$0.193869\pi$$
−0.820187 + 0.572096i $$0.806131\pi$$
$$884$$ −15.0000 −0.504505
$$885$$ 0 0
$$886$$ 18.0000 0.604722
$$887$$ − 42.0000i − 1.41022i −0.709097 0.705111i $$-0.750897\pi$$
0.709097 0.705111i $$-0.249103\pi$$
$$888$$ 2.00000i 0.0671156i
$$889$$ 2.00000 0.0670778
$$890$$ 0 0
$$891$$ −6.00000 −0.201008
$$892$$ 26.0000i 0.870544i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 15.0000i 0.500835i
$$898$$ − 18.0000i − 0.600668i
$$899$$ 36.0000 1.20067
$$900$$ 0 0
$$901$$ −9.00000 −0.299833
$$902$$ 0 0
$$903$$ 8.00000i 0.266223i
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 10.0000 0.332228
$$907$$ − 37.0000i − 1.22856i −0.789086 0.614282i $$-0.789446\pi$$
0.789086 0.614282i $$-0.210554\pi$$
$$908$$ 15.0000i 0.497792i
$$909$$ 36.0000 1.19404
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 1.00000i 0.0331133i
$$913$$ − 36.0000i − 1.19143i
$$914$$ 17.0000 0.562310
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ 0 0
$$918$$ − 15.0000i − 0.495074i
$$919$$ 7.00000 0.230909 0.115454 0.993313i $$-0.463168\pi$$
0.115454 + 0.993313i $$0.463168\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ 12.0000i 0.395199i
$$923$$ 30.0000i 0.987462i
$$924$$ −6.00000 −0.197386
$$925$$ 0 0
$$926$$ 4.00000 0.131448
$$927$$ − 28.0000i − 0.919641i
$$928$$ 9.00000i 0.295439i
$$929$$ −33.0000 −1.08269 −0.541347 0.840799i $$-0.682086\pi$$
−0.541347 + 0.840799i $$0.682086\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ − 6.00000i − 0.196537i
$$933$$ 21.0000i 0.687509i
$$934$$ 18.0000 0.588978
$$935$$ 0 0
$$936$$ 10.0000 0.326860
$$937$$ − 7.00000i − 0.228680i −0.993442 0.114340i $$-0.963525\pi$$
0.993442 0.114340i $$-0.0364753\pi$$
$$938$$ − 5.00000i − 0.163256i
$$939$$ 19.0000 0.620042
$$940$$ 0 0
$$941$$ 21.0000 0.684580 0.342290 0.939594i $$-0.388797\pi$$
0.342290 + 0.939594i $$0.388797\pi$$
$$942$$ 22.0000i 0.716799i
$$943$$ 0 0
$$944$$ −9.00000 −0.292925
$$945$$ 0 0
$$946$$ 48.0000 1.56061
$$947$$ − 48.0000i − 1.55979i −0.625910 0.779895i $$-0.715272\pi$$
0.625910 0.779895i $$-0.284728\pi$$
$$948$$ 10.0000i 0.324785i
$$949$$ 35.0000 1.13615
$$950$$ 0 0
$$951$$ −9.00000 −0.291845
$$952$$ 3.00000i 0.0972306i
$$953$$ − 30.0000i − 0.971795i −0.874016 0.485898i $$-0.838493\pi$$
0.874016 0.485898i $$-0.161507\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ −21.0000 −0.679189
$$957$$ − 54.0000i − 1.74557i
$$958$$ 36.0000i 1.16311i
$$959$$ −9.00000 −0.290625
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ − 10.0000i − 0.322413i
$$963$$ − 18.0000i − 0.580042i
$$964$$ −8.00000 −0.257663
$$965$$ 0 0
$$966$$ 3.00000 0.0965234
$$967$$ 32.0000i 1.02905i 0.857475 + 0.514525i $$0.172032\pi$$
−0.857475 + 0.514525i $$0.827968\pi$$
$$968$$ 25.0000i 0.803530i
$$969$$ −3.00000 −0.0963739
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 16.0000i 0.513200i
$$973$$ − 4.00000i − 0.128234i
$$974$$ 2.00000 0.0640841
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ − 12.0000i − 0.383914i −0.981403 0.191957i $$-0.938517\pi$$
0.981403 0.191957i $$-0.0614834\pi$$
$$978$$ 20.0000i 0.639529i
$$979$$ −72.0000 −2.30113
$$980$$ 0 0
$$981$$ −22.0000 −0.702406
$$982$$ 36.0000i 1.14881i
$$983$$ 30.0000i 0.956851i 0.878128 + 0.478426i $$0.158792\pi$$
−0.878128 + 0.478426i $$0.841208\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −27.0000 −0.859855
$$987$$ 0 0
$$988$$ − 5.00000i − 0.159071i
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ 4.00000i 0.127000i
$$993$$ 1.00000i 0.0317340i
$$994$$ 6.00000 0.190308
$$995$$ 0 0
$$996$$ −6.00000 −0.190117
$$997$$ 8.00000i 0.253363i 0.991943 + 0.126681i $$0.0404325\pi$$
−0.991943 + 0.126681i $$0.959567\pi$$
$$998$$ − 4.00000i − 0.126618i
$$999$$ 10.0000 0.316386
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 950.2.b.b.799.1 2
5.2 odd 4 950.2.a.d.1.1 1
5.3 odd 4 38.2.a.a.1.1 1
5.4 even 2 inner 950.2.b.b.799.2 2
15.2 even 4 8550.2.a.m.1.1 1
15.8 even 4 342.2.a.e.1.1 1
20.3 even 4 304.2.a.c.1.1 1
20.7 even 4 7600.2.a.n.1.1 1
35.13 even 4 1862.2.a.b.1.1 1
40.3 even 4 1216.2.a.m.1.1 1
40.13 odd 4 1216.2.a.e.1.1 1
55.43 even 4 4598.2.a.p.1.1 1
60.23 odd 4 2736.2.a.n.1.1 1
65.38 odd 4 6422.2.a.h.1.1 1
95.3 even 36 722.2.e.e.389.1 6
95.8 even 12 722.2.c.c.653.1 2
95.13 even 36 722.2.e.e.245.1 6
95.18 even 4 722.2.a.e.1.1 1
95.23 odd 36 722.2.e.f.415.1 6
95.28 odd 36 722.2.e.f.423.1 6
95.33 even 36 722.2.e.e.595.1 6
95.43 odd 36 722.2.e.f.595.1 6
95.48 even 36 722.2.e.e.423.1 6
95.53 even 36 722.2.e.e.415.1 6
95.63 odd 36 722.2.e.f.245.1 6
95.68 odd 12 722.2.c.e.653.1 2
95.73 odd 36 722.2.e.f.389.1 6
95.78 even 36 722.2.e.e.99.1 6
95.83 odd 12 722.2.c.e.429.1 2
95.88 even 12 722.2.c.c.429.1 2
95.93 odd 36 722.2.e.f.99.1 6
285.113 odd 4 6498.2.a.f.1.1 1
380.303 odd 4 5776.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.a.1.1 1 5.3 odd 4
304.2.a.c.1.1 1 20.3 even 4
342.2.a.e.1.1 1 15.8 even 4
722.2.a.e.1.1 1 95.18 even 4
722.2.c.c.429.1 2 95.88 even 12
722.2.c.c.653.1 2 95.8 even 12
722.2.c.e.429.1 2 95.83 odd 12
722.2.c.e.653.1 2 95.68 odd 12
722.2.e.e.99.1 6 95.78 even 36
722.2.e.e.245.1 6 95.13 even 36
722.2.e.e.389.1 6 95.3 even 36
722.2.e.e.415.1 6 95.53 even 36
722.2.e.e.423.1 6 95.48 even 36
722.2.e.e.595.1 6 95.33 even 36
722.2.e.f.99.1 6 95.93 odd 36
722.2.e.f.245.1 6 95.63 odd 36
722.2.e.f.389.1 6 95.73 odd 36
722.2.e.f.415.1 6 95.23 odd 36
722.2.e.f.423.1 6 95.28 odd 36
722.2.e.f.595.1 6 95.43 odd 36
950.2.a.d.1.1 1 5.2 odd 4
950.2.b.b.799.1 2 1.1 even 1 trivial
950.2.b.b.799.2 2 5.4 even 2 inner
1216.2.a.e.1.1 1 40.13 odd 4
1216.2.a.m.1.1 1 40.3 even 4
1862.2.a.b.1.1 1 35.13 even 4
2736.2.a.n.1.1 1 60.23 odd 4
4598.2.a.p.1.1 1 55.43 even 4
5776.2.a.m.1.1 1 380.303 odd 4
6422.2.a.h.1.1 1 65.38 odd 4
6498.2.a.f.1.1 1 285.113 odd 4
7600.2.a.n.1.1 1 20.7 even 4
8550.2.a.m.1.1 1 15.2 even 4