# Properties

 Label 2550.2.d.v.2449.1 Level $2550$ Weight $2$ Character 2550.2449 Analytic conductor $20.362$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Learn more

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: $$4$$ Coefficient field: $$\Q(i, \sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 17x^{2} + 64$$ x^4 + 17*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2449.1 Root $$-3.37228i$$ of defining polynomial Character $$\chi$$ $$=$$ 2550.2449 Dual form 2550.2.d.v.2449.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.37228i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.37228i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.37228 q^{11} -1.00000i q^{12} +2.00000i q^{13} -3.37228 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} -3.37228 q^{19} +3.37228 q^{21} +1.37228i q^{22} +4.37228i q^{23} -1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} +3.37228i q^{28} +2.74456 q^{29} -8.11684 q^{31} -1.00000i q^{32} -1.37228i q^{33} +1.00000 q^{34} +1.00000 q^{36} +1.00000i q^{37} +3.37228i q^{38} -2.00000 q^{39} +4.37228 q^{41} -3.37228i q^{42} +9.37228i q^{43} +1.37228 q^{44} +4.37228 q^{46} +7.37228i q^{47} +1.00000i q^{48} -4.37228 q^{49} -1.00000 q^{51} -2.00000i q^{52} -5.74456i q^{53} -1.00000 q^{54} +3.37228 q^{56} -3.37228i q^{57} -2.74456i q^{58} +7.11684 q^{59} +9.11684 q^{61} +8.11684i q^{62} +3.37228i q^{63} -1.00000 q^{64} -1.37228 q^{66} +5.37228i q^{67} -1.00000i q^{68} -4.37228 q^{69} +4.37228 q^{71} -1.00000i q^{72} +8.00000i q^{73} +1.00000 q^{74} +3.37228 q^{76} +4.62772i q^{77} +2.00000i q^{78} -6.11684 q^{79} +1.00000 q^{81} -4.37228i q^{82} -4.37228i q^{83} -3.37228 q^{84} +9.37228 q^{86} +2.74456i q^{87} -1.37228i q^{88} -8.74456 q^{89} +6.74456 q^{91} -4.37228i q^{92} -8.11684i q^{93} +7.37228 q^{94} +1.00000 q^{96} +1.25544i q^{97} +4.37228i q^{98} +1.37228 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 6 q^{11} - 2 q^{14} + 4 q^{16} - 2 q^{19} + 2 q^{21} - 4 q^{24} + 8 q^{26} - 12 q^{29} + 2 q^{31} + 4 q^{34} + 4 q^{36} - 8 q^{39} + 6 q^{41} - 6 q^{44} + 6 q^{46} - 6 q^{49} - 4 q^{51} - 4 q^{54} + 2 q^{56} - 6 q^{59} + 2 q^{61} - 4 q^{64} + 6 q^{66} - 6 q^{69} + 6 q^{71} + 4 q^{74} + 2 q^{76} + 10 q^{79} + 4 q^{81} - 2 q^{84} + 26 q^{86} - 12 q^{89} + 4 q^{91} + 18 q^{94} + 4 q^{96} - 6 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 6 * q^11 - 2 * q^14 + 4 * q^16 - 2 * q^19 + 2 * q^21 - 4 * q^24 + 8 * q^26 - 12 * q^29 + 2 * q^31 + 4 * q^34 + 4 * q^36 - 8 * q^39 + 6 * q^41 - 6 * q^44 + 6 * q^46 - 6 * q^49 - 4 * q^51 - 4 * q^54 + 2 * q^56 - 6 * q^59 + 2 * q^61 - 4 * q^64 + 6 * q^66 - 6 * q^69 + 6 * q^71 + 4 * q^74 + 2 * q^76 + 10 * q^79 + 4 * q^81 - 2 * q^84 + 26 * q^86 - 12 * q^89 + 4 * q^91 + 18 * q^94 + 4 * 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
$$7$$ − 3.37228i − 1.27460i −0.770615 0.637301i $$-0.780051\pi$$
0.770615 0.637301i $$-0.219949\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −1.37228 −0.413758 −0.206879 0.978366i $$-0.566331\pi$$
−0.206879 + 0.978366i $$0.566331\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ −3.37228 −0.901280
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.00000i 0.242536i
$$18$$ 1.00000i 0.235702i
$$19$$ −3.37228 −0.773654 −0.386827 0.922152i $$-0.626429\pi$$
−0.386827 + 0.922152i $$0.626429\pi$$
$$20$$ 0 0
$$21$$ 3.37228 0.735892
$$22$$ 1.37228i 0.292571i
$$23$$ 4.37228i 0.911684i 0.890061 + 0.455842i $$0.150662\pi$$
−0.890061 + 0.455842i $$0.849338\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ − 1.00000i − 0.192450i
$$28$$ 3.37228i 0.637301i
$$29$$ 2.74456 0.509652 0.254826 0.966987i $$-0.417982\pi$$
0.254826 + 0.966987i $$0.417982\pi$$
$$30$$ 0 0
$$31$$ −8.11684 −1.45783 −0.728914 0.684605i $$-0.759975\pi$$
−0.728914 + 0.684605i $$0.759975\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 1.37228i − 0.238884i
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 1.00000i 0.164399i 0.996616 + 0.0821995i $$0.0261945\pi$$
−0.996616 + 0.0821995i $$0.973806\pi$$
$$38$$ 3.37228i 0.547056i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 4.37228 0.682836 0.341418 0.939912i $$-0.389093\pi$$
0.341418 + 0.939912i $$0.389093\pi$$
$$42$$ − 3.37228i − 0.520354i
$$43$$ 9.37228i 1.42926i 0.699503 + 0.714630i $$0.253405\pi$$
−0.699503 + 0.714630i $$0.746595\pi$$
$$44$$ 1.37228 0.206879
$$45$$ 0 0
$$46$$ 4.37228 0.644658
$$47$$ 7.37228i 1.07536i 0.843150 + 0.537679i $$0.180699\pi$$
−0.843150 + 0.537679i $$0.819301\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −4.37228 −0.624612
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ − 2.00000i − 0.277350i
$$53$$ − 5.74456i − 0.789076i −0.918880 0.394538i $$-0.870905\pi$$
0.918880 0.394538i $$-0.129095\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 3.37228 0.450640
$$57$$ − 3.37228i − 0.446670i
$$58$$ − 2.74456i − 0.360379i
$$59$$ 7.11684 0.926534 0.463267 0.886219i $$-0.346677\pi$$
0.463267 + 0.886219i $$0.346677\pi$$
$$60$$ 0 0
$$61$$ 9.11684 1.16729 0.583646 0.812008i $$-0.301626\pi$$
0.583646 + 0.812008i $$0.301626\pi$$
$$62$$ 8.11684i 1.03084i
$$63$$ 3.37228i 0.424868i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −1.37228 −0.168916
$$67$$ 5.37228i 0.656329i 0.944621 + 0.328164i $$0.106430\pi$$
−0.944621 + 0.328164i $$0.893570\pi$$
$$68$$ − 1.00000i − 0.121268i
$$69$$ −4.37228 −0.526361
$$70$$ 0 0
$$71$$ 4.37228 0.518894 0.259447 0.965757i $$-0.416460\pi$$
0.259447 + 0.965757i $$0.416460\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 8.00000i 0.936329i 0.883641 + 0.468165i $$0.155085\pi$$
−0.883641 + 0.468165i $$0.844915\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 3.37228 0.386827
$$77$$ 4.62772i 0.527377i
$$78$$ 2.00000i 0.226455i
$$79$$ −6.11684 −0.688199 −0.344099 0.938933i $$-0.611816\pi$$
−0.344099 + 0.938933i $$0.611816\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 4.37228i − 0.482838i
$$83$$ − 4.37228i − 0.479920i −0.970783 0.239960i $$-0.922866\pi$$
0.970783 0.239960i $$-0.0771344\pi$$
$$84$$ −3.37228 −0.367946
$$85$$ 0 0
$$86$$ 9.37228 1.01064
$$87$$ 2.74456i 0.294248i
$$88$$ − 1.37228i − 0.146286i
$$89$$ −8.74456 −0.926922 −0.463461 0.886117i $$-0.653393\pi$$
−0.463461 + 0.886117i $$0.653393\pi$$
$$90$$ 0 0
$$91$$ 6.74456 0.707022
$$92$$ − 4.37228i − 0.455842i
$$93$$ − 8.11684i − 0.841678i
$$94$$ 7.37228 0.760393
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 1.25544i 0.127470i 0.997967 + 0.0637352i $$0.0203013\pi$$
−0.997967 + 0.0637352i $$0.979699\pi$$
$$98$$ 4.37228i 0.441667i
$$99$$ 1.37228 0.137919
$$100$$ 0 0
$$101$$ −4.11684 −0.409641 −0.204821 0.978800i $$-0.565661\pi$$
−0.204821 + 0.978800i $$0.565661\pi$$
$$102$$ 1.00000i 0.0990148i
$$103$$ 9.62772i 0.948647i 0.880351 + 0.474324i $$0.157307\pi$$
−0.880351 + 0.474324i $$0.842693\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −5.74456 −0.557961
$$107$$ 4.62772i 0.447378i 0.974661 + 0.223689i $$0.0718101\pi$$
−0.974661 + 0.223689i $$0.928190\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 8.11684 0.777453 0.388726 0.921353i $$-0.372915\pi$$
0.388726 + 0.921353i $$0.372915\pi$$
$$110$$ 0 0
$$111$$ −1.00000 −0.0949158
$$112$$ − 3.37228i − 0.318651i
$$113$$ − 0.255437i − 0.0240295i −0.999928 0.0120148i $$-0.996175\pi$$
0.999928 0.0120148i $$-0.00382451\pi$$
$$114$$ −3.37228 −0.315843
$$115$$ 0 0
$$116$$ −2.74456 −0.254826
$$117$$ − 2.00000i − 0.184900i
$$118$$ − 7.11684i − 0.655159i
$$119$$ 3.37228 0.309137
$$120$$ 0 0
$$121$$ −9.11684 −0.828804
$$122$$ − 9.11684i − 0.825400i
$$123$$ 4.37228i 0.394235i
$$124$$ 8.11684 0.728914
$$125$$ 0 0
$$126$$ 3.37228 0.300427
$$127$$ 21.4891i 1.90685i 0.301628 + 0.953426i $$0.402470\pi$$
−0.301628 + 0.953426i $$0.597530\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −9.37228 −0.825183
$$130$$ 0 0
$$131$$ −5.48913 −0.479587 −0.239794 0.970824i $$-0.577080\pi$$
−0.239794 + 0.970824i $$0.577080\pi$$
$$132$$ 1.37228i 0.119442i
$$133$$ 11.3723i 0.986102i
$$134$$ 5.37228 0.464094
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ 17.4891i 1.49420i 0.664713 + 0.747098i $$0.268554\pi$$
−0.664713 + 0.747098i $$0.731446\pi$$
$$138$$ 4.37228i 0.372193i
$$139$$ −0.883156 −0.0749083 −0.0374542 0.999298i $$-0.511925\pi$$
−0.0374542 + 0.999298i $$0.511925\pi$$
$$140$$ 0 0
$$141$$ −7.37228 −0.620858
$$142$$ − 4.37228i − 0.366914i
$$143$$ − 2.74456i − 0.229512i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 8.00000 0.662085
$$147$$ − 4.37228i − 0.360620i
$$148$$ − 1.00000i − 0.0821995i
$$149$$ −21.8614 −1.79096 −0.895478 0.445106i $$-0.853166\pi$$
−0.895478 + 0.445106i $$0.853166\pi$$
$$150$$ 0 0
$$151$$ 3.11684 0.253645 0.126823 0.991925i $$-0.459522\pi$$
0.126823 + 0.991925i $$0.459522\pi$$
$$152$$ − 3.37228i − 0.273528i
$$153$$ − 1.00000i − 0.0808452i
$$154$$ 4.62772 0.372912
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 7.25544i 0.579047i 0.957171 + 0.289523i $$0.0934968\pi$$
−0.957171 + 0.289523i $$0.906503\pi$$
$$158$$ 6.11684i 0.486630i
$$159$$ 5.74456 0.455573
$$160$$ 0 0
$$161$$ 14.7446 1.16203
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 15.1168i 1.18404i 0.805922 + 0.592021i $$0.201670\pi$$
−0.805922 + 0.592021i $$0.798330\pi$$
$$164$$ −4.37228 −0.341418
$$165$$ 0 0
$$166$$ −4.37228 −0.339355
$$167$$ 9.25544i 0.716207i 0.933682 + 0.358104i $$0.116577\pi$$
−0.933682 + 0.358104i $$0.883423\pi$$
$$168$$ 3.37228i 0.260177i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 3.37228 0.257885
$$172$$ − 9.37228i − 0.714630i
$$173$$ − 12.0000i − 0.912343i −0.889892 0.456172i $$-0.849220\pi$$
0.889892 0.456172i $$-0.150780\pi$$
$$174$$ 2.74456 0.208065
$$175$$ 0 0
$$176$$ −1.37228 −0.103440
$$177$$ 7.11684i 0.534935i
$$178$$ 8.74456i 0.655433i
$$179$$ −7.11684 −0.531938 −0.265969 0.963982i $$-0.585692\pi$$
−0.265969 + 0.963982i $$0.585692\pi$$
$$180$$ 0 0
$$181$$ 22.4891 1.67160 0.835802 0.549031i $$-0.185003\pi$$
0.835802 + 0.549031i $$0.185003\pi$$
$$182$$ − 6.74456i − 0.499940i
$$183$$ 9.11684i 0.673936i
$$184$$ −4.37228 −0.322329
$$185$$ 0 0
$$186$$ −8.11684 −0.595156
$$187$$ − 1.37228i − 0.100351i
$$188$$ − 7.37228i − 0.537679i
$$189$$ −3.37228 −0.245297
$$190$$ 0 0
$$191$$ 24.3505 1.76194 0.880971 0.473170i $$-0.156890\pi$$
0.880971 + 0.473170i $$0.156890\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 8.00000i 0.575853i 0.957653 + 0.287926i $$0.0929658\pi$$
−0.957653 + 0.287926i $$0.907034\pi$$
$$194$$ 1.25544 0.0901351
$$195$$ 0 0
$$196$$ 4.37228 0.312306
$$197$$ 20.7446i 1.47799i 0.673712 + 0.738994i $$0.264699\pi$$
−0.673712 + 0.738994i $$0.735301\pi$$
$$198$$ − 1.37228i − 0.0975238i
$$199$$ −3.88316 −0.275270 −0.137635 0.990483i $$-0.543950\pi$$
−0.137635 + 0.990483i $$0.543950\pi$$
$$200$$ 0 0
$$201$$ −5.37228 −0.378932
$$202$$ 4.11684i 0.289660i
$$203$$ − 9.25544i − 0.649604i
$$204$$ 1.00000 0.0700140
$$205$$ 0 0
$$206$$ 9.62772 0.670795
$$207$$ − 4.37228i − 0.303895i
$$208$$ 2.00000i 0.138675i
$$209$$ 4.62772 0.320106
$$210$$ 0 0
$$211$$ −24.2337 −1.66832 −0.834158 0.551526i $$-0.814046\pi$$
−0.834158 + 0.551526i $$0.814046\pi$$
$$212$$ 5.74456i 0.394538i
$$213$$ 4.37228i 0.299584i
$$214$$ 4.62772 0.316344
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 27.3723i 1.85815i
$$218$$ − 8.11684i − 0.549742i
$$219$$ −8.00000 −0.540590
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 1.00000i 0.0671156i
$$223$$ 9.11684i 0.610509i 0.952271 + 0.305255i $$0.0987415\pi$$
−0.952271 + 0.305255i $$0.901258\pi$$
$$224$$ −3.37228 −0.225320
$$225$$ 0 0
$$226$$ −0.255437 −0.0169914
$$227$$ − 1.37228i − 0.0910815i −0.998962 0.0455408i $$-0.985499\pi$$
0.998962 0.0455408i $$-0.0145011\pi$$
$$228$$ 3.37228i 0.223335i
$$229$$ −22.2337 −1.46924 −0.734622 0.678477i $$-0.762640\pi$$
−0.734622 + 0.678477i $$0.762640\pi$$
$$230$$ 0 0
$$231$$ −4.62772 −0.304482
$$232$$ 2.74456i 0.180189i
$$233$$ − 16.3723i − 1.07258i −0.844033 0.536292i $$-0.819825\pi$$
0.844033 0.536292i $$-0.180175\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −7.11684 −0.463267
$$237$$ − 6.11684i − 0.397332i
$$238$$ − 3.37228i − 0.218593i
$$239$$ −4.11684 −0.266296 −0.133148 0.991096i $$-0.542509\pi$$
−0.133148 + 0.991096i $$0.542509\pi$$
$$240$$ 0 0
$$241$$ −6.23369 −0.401547 −0.200774 0.979638i $$-0.564346\pi$$
−0.200774 + 0.979638i $$0.564346\pi$$
$$242$$ 9.11684i 0.586053i
$$243$$ 1.00000i 0.0641500i
$$244$$ −9.11684 −0.583646
$$245$$ 0 0
$$246$$ 4.37228 0.278766
$$247$$ − 6.74456i − 0.429146i
$$248$$ − 8.11684i − 0.515420i
$$249$$ 4.37228 0.277082
$$250$$ 0 0
$$251$$ −2.74456 −0.173235 −0.0866176 0.996242i $$-0.527606\pi$$
−0.0866176 + 0.996242i $$0.527606\pi$$
$$252$$ − 3.37228i − 0.212434i
$$253$$ − 6.00000i − 0.377217i
$$254$$ 21.4891 1.34835
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 20.2337i − 1.26214i −0.775725 0.631071i $$-0.782615\pi$$
0.775725 0.631071i $$-0.217385\pi$$
$$258$$ 9.37228i 0.583493i
$$259$$ 3.37228 0.209543
$$260$$ 0 0
$$261$$ −2.74456 −0.169884
$$262$$ 5.48913i 0.339119i
$$263$$ − 31.3723i − 1.93450i −0.253830 0.967249i $$-0.581690\pi$$
0.253830 0.967249i $$-0.418310\pi$$
$$264$$ 1.37228 0.0844581
$$265$$ 0 0
$$266$$ 11.3723 0.697279
$$267$$ − 8.74456i − 0.535159i
$$268$$ − 5.37228i − 0.328164i
$$269$$ 11.4891 0.700504 0.350252 0.936655i $$-0.386096\pi$$
0.350252 + 0.936655i $$0.386096\pi$$
$$270$$ 0 0
$$271$$ −8.37228 −0.508580 −0.254290 0.967128i $$-0.581842\pi$$
−0.254290 + 0.967128i $$0.581842\pi$$
$$272$$ 1.00000i 0.0606339i
$$273$$ 6.74456i 0.408200i
$$274$$ 17.4891 1.05656
$$275$$ 0 0
$$276$$ 4.37228 0.263180
$$277$$ 13.0000i 0.781094i 0.920583 + 0.390547i $$0.127714\pi$$
−0.920583 + 0.390547i $$0.872286\pi$$
$$278$$ 0.883156i 0.0529682i
$$279$$ 8.11684 0.485943
$$280$$ 0 0
$$281$$ 23.4891 1.40124 0.700622 0.713533i $$-0.252906\pi$$
0.700622 + 0.713533i $$0.252906\pi$$
$$282$$ 7.37228i 0.439013i
$$283$$ − 5.11684i − 0.304165i −0.988368 0.152082i $$-0.951402\pi$$
0.988368 0.152082i $$-0.0485979\pi$$
$$284$$ −4.37228 −0.259447
$$285$$ 0 0
$$286$$ −2.74456 −0.162289
$$287$$ − 14.7446i − 0.870344i
$$288$$ 1.00000i 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −1.25544 −0.0735950
$$292$$ − 8.00000i − 0.468165i
$$293$$ − 10.8832i − 0.635801i −0.948124 0.317900i $$-0.897022\pi$$
0.948124 0.317900i $$-0.102978\pi$$
$$294$$ −4.37228 −0.254997
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 1.37228i 0.0796278i
$$298$$ 21.8614i 1.26640i
$$299$$ −8.74456 −0.505711
$$300$$ 0 0
$$301$$ 31.6060 1.82174
$$302$$ − 3.11684i − 0.179354i
$$303$$ − 4.11684i − 0.236507i
$$304$$ −3.37228 −0.193414
$$305$$ 0 0
$$306$$ −1.00000 −0.0571662
$$307$$ 18.2337i 1.04065i 0.853968 + 0.520326i $$0.174189\pi$$
−0.853968 + 0.520326i $$0.825811\pi$$
$$308$$ − 4.62772i − 0.263689i
$$309$$ −9.62772 −0.547702
$$310$$ 0 0
$$311$$ −10.8832 −0.617127 −0.308564 0.951204i $$-0.599848\pi$$
−0.308564 + 0.951204i $$0.599848\pi$$
$$312$$ − 2.00000i − 0.113228i
$$313$$ − 6.74456i − 0.381225i −0.981665 0.190613i $$-0.938953\pi$$
0.981665 0.190613i $$-0.0610474\pi$$
$$314$$ 7.25544 0.409448
$$315$$ 0 0
$$316$$ 6.11684 0.344099
$$317$$ − 3.25544i − 0.182844i −0.995812 0.0914218i $$-0.970859\pi$$
0.995812 0.0914218i $$-0.0291411\pi$$
$$318$$ − 5.74456i − 0.322139i
$$319$$ −3.76631 −0.210873
$$320$$ 0 0
$$321$$ −4.62772 −0.258294
$$322$$ − 14.7446i − 0.821682i
$$323$$ − 3.37228i − 0.187639i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 15.1168 0.837245
$$327$$ 8.11684i 0.448862i
$$328$$ 4.37228i 0.241419i
$$329$$ 24.8614 1.37065
$$330$$ 0 0
$$331$$ −25.6060 −1.40743 −0.703716 0.710482i $$-0.748477\pi$$
−0.703716 + 0.710482i $$0.748477\pi$$
$$332$$ 4.37228i 0.239960i
$$333$$ − 1.00000i − 0.0547997i
$$334$$ 9.25544 0.506435
$$335$$ 0 0
$$336$$ 3.37228 0.183973
$$337$$ − 10.2337i − 0.557465i −0.960369 0.278732i $$-0.910086\pi$$
0.960369 0.278732i $$-0.0899142\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 0.255437 0.0138735
$$340$$ 0 0
$$341$$ 11.1386 0.603189
$$342$$ − 3.37228i − 0.182352i
$$343$$ − 8.86141i − 0.478471i
$$344$$ −9.37228 −0.505320
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ − 16.6277i − 0.892623i −0.894878 0.446311i $$-0.852737\pi$$
0.894878 0.446311i $$-0.147263\pi$$
$$348$$ − 2.74456i − 0.147124i
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 1.37228i 0.0731428i
$$353$$ − 26.7446i − 1.42347i −0.702448 0.711735i $$-0.747910\pi$$
0.702448 0.711735i $$-0.252090\pi$$
$$354$$ 7.11684 0.378256
$$355$$ 0 0
$$356$$ 8.74456 0.463461
$$357$$ 3.37228i 0.178480i
$$358$$ 7.11684i 0.376137i
$$359$$ 19.3723 1.02243 0.511215 0.859453i $$-0.329196\pi$$
0.511215 + 0.859453i $$0.329196\pi$$
$$360$$ 0 0
$$361$$ −7.62772 −0.401459
$$362$$ − 22.4891i − 1.18200i
$$363$$ − 9.11684i − 0.478510i
$$364$$ −6.74456 −0.353511
$$365$$ 0 0
$$366$$ 9.11684 0.476545
$$367$$ 25.6060i 1.33662i 0.743883 + 0.668310i $$0.232982\pi$$
−0.743883 + 0.668310i $$0.767018\pi$$
$$368$$ 4.37228i 0.227921i
$$369$$ −4.37228 −0.227612
$$370$$ 0 0
$$371$$ −19.3723 −1.00576
$$372$$ 8.11684i 0.420839i
$$373$$ − 36.2337i − 1.87611i −0.346487 0.938055i $$-0.612626\pi$$
0.346487 0.938055i $$-0.387374\pi$$
$$374$$ −1.37228 −0.0709590
$$375$$ 0 0
$$376$$ −7.37228 −0.380196
$$377$$ 5.48913i 0.282704i
$$378$$ 3.37228i 0.173451i
$$379$$ −20.6060 −1.05846 −0.529229 0.848479i $$-0.677519\pi$$
−0.529229 + 0.848479i $$0.677519\pi$$
$$380$$ 0 0
$$381$$ −21.4891 −1.10092
$$382$$ − 24.3505i − 1.24588i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 8.00000 0.407189
$$387$$ − 9.37228i − 0.476420i
$$388$$ − 1.25544i − 0.0637352i
$$389$$ −6.25544 −0.317163 −0.158582 0.987346i $$-0.550692\pi$$
−0.158582 + 0.987346i $$0.550692\pi$$
$$390$$ 0 0
$$391$$ −4.37228 −0.221116
$$392$$ − 4.37228i − 0.220834i
$$393$$ − 5.48913i − 0.276890i
$$394$$ 20.7446 1.04510
$$395$$ 0 0
$$396$$ −1.37228 −0.0689597
$$397$$ 1.00000i 0.0501886i 0.999685 + 0.0250943i $$0.00798860\pi$$
−0.999685 + 0.0250943i $$0.992011\pi$$
$$398$$ 3.88316i 0.194645i
$$399$$ −11.3723 −0.569326
$$400$$ 0 0
$$401$$ −13.1168 −0.655024 −0.327512 0.944847i $$-0.606210\pi$$
−0.327512 + 0.944847i $$0.606210\pi$$
$$402$$ 5.37228i 0.267945i
$$403$$ − 16.2337i − 0.808658i
$$404$$ 4.11684 0.204821
$$405$$ 0 0
$$406$$ −9.25544 −0.459340
$$407$$ − 1.37228i − 0.0680215i
$$408$$ − 1.00000i − 0.0495074i
$$409$$ 13.8614 0.685402 0.342701 0.939444i $$-0.388658\pi$$
0.342701 + 0.939444i $$0.388658\pi$$
$$410$$ 0 0
$$411$$ −17.4891 −0.862675
$$412$$ − 9.62772i − 0.474324i
$$413$$ − 24.0000i − 1.18096i
$$414$$ −4.37228 −0.214886
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ − 0.883156i − 0.0432483i
$$418$$ − 4.62772i − 0.226349i
$$419$$ 34.9783 1.70880 0.854400 0.519616i $$-0.173925\pi$$
0.854400 + 0.519616i $$0.173925\pi$$
$$420$$ 0 0
$$421$$ 16.2337 0.791182 0.395591 0.918427i $$-0.370540\pi$$
0.395591 + 0.918427i $$0.370540\pi$$
$$422$$ 24.2337i 1.17968i
$$423$$ − 7.37228i − 0.358453i
$$424$$ 5.74456 0.278981
$$425$$ 0 0
$$426$$ 4.37228 0.211838
$$427$$ − 30.7446i − 1.48783i
$$428$$ − 4.62772i − 0.223689i
$$429$$ 2.74456 0.132509
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 23.0951i 1.10988i 0.831891 + 0.554940i $$0.187259\pi$$
−0.831891 + 0.554940i $$0.812741\pi$$
$$434$$ 27.3723 1.31391
$$435$$ 0 0
$$436$$ −8.11684 −0.388726
$$437$$ − 14.7446i − 0.705328i
$$438$$ 8.00000i 0.382255i
$$439$$ −25.4891 −1.21653 −0.608265 0.793734i $$-0.708134\pi$$
−0.608265 + 0.793734i $$0.708134\pi$$
$$440$$ 0 0
$$441$$ 4.37228 0.208204
$$442$$ 2.00000i 0.0951303i
$$443$$ − 1.62772i − 0.0773352i −0.999252 0.0386676i $$-0.987689\pi$$
0.999252 0.0386676i $$-0.0123114\pi$$
$$444$$ 1.00000 0.0474579
$$445$$ 0 0
$$446$$ 9.11684 0.431695
$$447$$ − 21.8614i − 1.03401i
$$448$$ 3.37228i 0.159325i
$$449$$ 5.13859 0.242505 0.121253 0.992622i $$-0.461309\pi$$
0.121253 + 0.992622i $$0.461309\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 0.255437i 0.0120148i
$$453$$ 3.11684i 0.146442i
$$454$$ −1.37228 −0.0644044
$$455$$ 0 0
$$456$$ 3.37228 0.157922
$$457$$ − 23.0000i − 1.07589i −0.842978 0.537947i $$-0.819200\pi$$
0.842978 0.537947i $$-0.180800\pi$$
$$458$$ 22.2337i 1.03891i
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ −22.7228 −1.05831 −0.529153 0.848526i $$-0.677490\pi$$
−0.529153 + 0.848526i $$0.677490\pi$$
$$462$$ 4.62772i 0.215301i
$$463$$ 20.6060i 0.957641i 0.877913 + 0.478820i $$0.158935\pi$$
−0.877913 + 0.478820i $$0.841065\pi$$
$$464$$ 2.74456 0.127413
$$465$$ 0 0
$$466$$ −16.3723 −0.758431
$$467$$ 1.11684i 0.0516814i 0.999666 + 0.0258407i $$0.00822626\pi$$
−0.999666 + 0.0258407i $$0.991774\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ 18.1168 0.836558
$$470$$ 0 0
$$471$$ −7.25544 −0.334313
$$472$$ 7.11684i 0.327579i
$$473$$ − 12.8614i − 0.591368i
$$474$$ −6.11684 −0.280956
$$475$$ 0 0
$$476$$ −3.37228 −0.154568
$$477$$ 5.74456i 0.263025i
$$478$$ 4.11684i 0.188300i
$$479$$ −25.7228 −1.17531 −0.587653 0.809113i $$-0.699948\pi$$
−0.587653 + 0.809113i $$0.699948\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 6.23369i 0.283937i
$$483$$ 14.7446i 0.670901i
$$484$$ 9.11684 0.414402
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 22.2337i − 1.00750i −0.863848 0.503752i $$-0.831952\pi$$
0.863848 0.503752i $$-0.168048\pi$$
$$488$$ 9.11684i 0.412700i
$$489$$ −15.1168 −0.683607
$$490$$ 0 0
$$491$$ −25.6277 −1.15656 −0.578281 0.815837i $$-0.696276\pi$$
−0.578281 + 0.815837i $$0.696276\pi$$
$$492$$ − 4.37228i − 0.197118i
$$493$$ 2.74456i 0.123609i
$$494$$ −6.74456 −0.303452
$$495$$ 0 0
$$496$$ −8.11684 −0.364457
$$497$$ − 14.7446i − 0.661384i
$$498$$ − 4.37228i − 0.195927i
$$499$$ −24.3723 −1.09105 −0.545527 0.838094i $$-0.683670\pi$$
−0.545527 + 0.838094i $$0.683670\pi$$
$$500$$ 0 0
$$501$$ −9.25544 −0.413502
$$502$$ 2.74456i 0.122496i
$$503$$ 30.6060i 1.36465i 0.731048 + 0.682326i $$0.239032\pi$$
−0.731048 + 0.682326i $$0.760968\pi$$
$$504$$ −3.37228 −0.150213
$$505$$ 0 0
$$506$$ −6.00000 −0.266733
$$507$$ 9.00000i 0.399704i
$$508$$ − 21.4891i − 0.953426i
$$509$$ 25.3723 1.12461 0.562303 0.826931i $$-0.309915\pi$$
0.562303 + 0.826931i $$0.309915\pi$$
$$510$$ 0 0
$$511$$ 26.9783 1.19345
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 3.37228i 0.148890i
$$514$$ −20.2337 −0.892470
$$515$$ 0 0
$$516$$ 9.37228 0.412592
$$517$$ − 10.1168i − 0.444938i
$$518$$ − 3.37228i − 0.148170i
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ 24.3505 1.06682 0.533408 0.845858i $$-0.320911\pi$$
0.533408 + 0.845858i $$0.320911\pi$$
$$522$$ 2.74456i 0.120126i
$$523$$ − 18.2337i − 0.797304i −0.917102 0.398652i $$-0.869478\pi$$
0.917102 0.398652i $$-0.130522\pi$$
$$524$$ 5.48913 0.239794
$$525$$ 0 0
$$526$$ −31.3723 −1.36790
$$527$$ − 8.11684i − 0.353575i
$$528$$ − 1.37228i − 0.0597209i
$$529$$ 3.88316 0.168833
$$530$$ 0 0
$$531$$ −7.11684 −0.308845
$$532$$ − 11.3723i − 0.493051i
$$533$$ 8.74456i 0.378769i
$$534$$ −8.74456 −0.378414
$$535$$ 0 0
$$536$$ −5.37228 −0.232047
$$537$$ − 7.11684i − 0.307114i
$$538$$ − 11.4891i − 0.495331i
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ 5.00000 0.214967 0.107483 0.994207i $$-0.465721\pi$$
0.107483 + 0.994207i $$0.465721\pi$$
$$542$$ 8.37228i 0.359620i
$$543$$ 22.4891i 0.965101i
$$544$$ 1.00000 0.0428746
$$545$$ 0 0
$$546$$ 6.74456 0.288641
$$547$$ − 36.3723i − 1.55517i −0.628780 0.777583i $$-0.716445\pi$$
0.628780 0.777583i $$-0.283555\pi$$
$$548$$ − 17.4891i − 0.747098i
$$549$$ −9.11684 −0.389097
$$550$$ 0 0
$$551$$ −9.25544 −0.394295
$$552$$ − 4.37228i − 0.186097i
$$553$$ 20.6277i 0.877180i
$$554$$ 13.0000 0.552317
$$555$$ 0 0
$$556$$ 0.883156 0.0374542
$$557$$ 35.7446i 1.51455i 0.653099 + 0.757273i $$0.273469\pi$$
−0.653099 + 0.757273i $$0.726531\pi$$
$$558$$ − 8.11684i − 0.343613i
$$559$$ −18.7446 −0.792811
$$560$$ 0 0
$$561$$ 1.37228 0.0579378
$$562$$ − 23.4891i − 0.990829i
$$563$$ − 33.3505i − 1.40556i −0.711409 0.702779i $$-0.751942\pi$$
0.711409 0.702779i $$-0.248058\pi$$
$$564$$ 7.37228 0.310429
$$565$$ 0 0
$$566$$ −5.11684 −0.215077
$$567$$ − 3.37228i − 0.141623i
$$568$$ 4.37228i 0.183457i
$$569$$ 20.7446 0.869657 0.434829 0.900513i $$-0.356809\pi$$
0.434829 + 0.900513i $$0.356809\pi$$
$$570$$ 0 0
$$571$$ −38.3723 −1.60583 −0.802915 0.596094i $$-0.796719\pi$$
−0.802915 + 0.596094i $$0.796719\pi$$
$$572$$ 2.74456i 0.114756i
$$573$$ 24.3505i 1.01726i
$$574$$ −14.7446 −0.615426
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 7.00000i 0.291414i 0.989328 + 0.145707i $$0.0465456\pi$$
−0.989328 + 0.145707i $$0.953454\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ −8.00000 −0.332469
$$580$$ 0 0
$$581$$ −14.7446 −0.611708
$$582$$ 1.25544i 0.0520396i
$$583$$ 7.88316i 0.326487i
$$584$$ −8.00000 −0.331042
$$585$$ 0 0
$$586$$ −10.8832 −0.449579
$$587$$ − 2.13859i − 0.0882692i −0.999026 0.0441346i $$-0.985947\pi$$
0.999026 0.0441346i $$-0.0140530\pi$$
$$588$$ 4.37228i 0.180310i
$$589$$ 27.3723 1.12786
$$590$$ 0 0
$$591$$ −20.7446 −0.853317
$$592$$ 1.00000i 0.0410997i
$$593$$ 28.4674i 1.16902i 0.811388 + 0.584508i $$0.198712\pi$$
−0.811388 + 0.584508i $$0.801288\pi$$
$$594$$ 1.37228 0.0563054
$$595$$ 0 0
$$596$$ 21.8614 0.895478
$$597$$ − 3.88316i − 0.158927i
$$598$$ 8.74456i 0.357592i
$$599$$ 7.37228 0.301223 0.150612 0.988593i $$-0.451876\pi$$
0.150612 + 0.988593i $$0.451876\pi$$
$$600$$ 0 0
$$601$$ −8.97825 −0.366230 −0.183115 0.983091i $$-0.558618\pi$$
−0.183115 + 0.983091i $$0.558618\pi$$
$$602$$ − 31.6060i − 1.28816i
$$603$$ − 5.37228i − 0.218776i
$$604$$ −3.11684 −0.126823
$$605$$ 0 0
$$606$$ −4.11684 −0.167235
$$607$$ − 5.76631i − 0.234047i −0.993129 0.117024i $$-0.962665\pi$$
0.993129 0.117024i $$-0.0373353\pi$$
$$608$$ 3.37228i 0.136764i
$$609$$ 9.25544 0.375049
$$610$$ 0 0
$$611$$ −14.7446 −0.596501
$$612$$ 1.00000i 0.0404226i
$$613$$ 19.4891i 0.787158i 0.919291 + 0.393579i $$0.128763\pi$$
−0.919291 + 0.393579i $$0.871237\pi$$
$$614$$ 18.2337 0.735852
$$615$$ 0 0
$$616$$ −4.62772 −0.186456
$$617$$ 29.8397i 1.20130i 0.799512 + 0.600650i $$0.205091\pi$$
−0.799512 + 0.600650i $$0.794909\pi$$
$$618$$ 9.62772i 0.387284i
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 4.37228 0.175454
$$622$$ 10.8832i 0.436375i
$$623$$ 29.4891i 1.18146i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ −6.74456 −0.269567
$$627$$ 4.62772i 0.184813i
$$628$$ − 7.25544i − 0.289523i
$$629$$ −1.00000 −0.0398726
$$630$$ 0 0
$$631$$ −6.13859 −0.244374 −0.122187 0.992507i $$-0.538991\pi$$
−0.122187 + 0.992507i $$0.538991\pi$$
$$632$$ − 6.11684i − 0.243315i
$$633$$ − 24.2337i − 0.963203i
$$634$$ −3.25544 −0.129290
$$635$$ 0 0
$$636$$ −5.74456 −0.227787
$$637$$ − 8.74456i − 0.346472i
$$638$$ 3.76631i 0.149110i
$$639$$ −4.37228 −0.172965
$$640$$ 0 0
$$641$$ −4.97825 −0.196629 −0.0983145 0.995155i $$-0.531345\pi$$
−0.0983145 + 0.995155i $$0.531345\pi$$
$$642$$ 4.62772i 0.182641i
$$643$$ 6.88316i 0.271445i 0.990747 + 0.135723i $$0.0433356\pi$$
−0.990747 + 0.135723i $$0.956664\pi$$
$$644$$ −14.7446 −0.581017
$$645$$ 0 0
$$646$$ −3.37228 −0.132681
$$647$$ − 31.7228i − 1.24715i −0.781763 0.623576i $$-0.785679\pi$$
0.781763 0.623576i $$-0.214321\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −9.76631 −0.383361
$$650$$ 0 0
$$651$$ −27.3723 −1.07280
$$652$$ − 15.1168i − 0.592021i
$$653$$ 22.9783i 0.899208i 0.893228 + 0.449604i $$0.148435\pi$$
−0.893228 + 0.449604i $$0.851565\pi$$
$$654$$ 8.11684 0.317394
$$655$$ 0 0
$$656$$ 4.37228 0.170709
$$657$$ − 8.00000i − 0.312110i
$$658$$ − 24.8614i − 0.969199i
$$659$$ −29.4891 −1.14873 −0.574367 0.818598i $$-0.694752\pi$$
−0.574367 + 0.818598i $$0.694752\pi$$
$$660$$ 0 0
$$661$$ −24.7446 −0.962452 −0.481226 0.876597i $$-0.659808\pi$$
−0.481226 + 0.876597i $$0.659808\pi$$
$$662$$ 25.6060i 0.995204i
$$663$$ − 2.00000i − 0.0776736i
$$664$$ 4.37228 0.169677
$$665$$ 0 0
$$666$$ −1.00000 −0.0387492
$$667$$ 12.0000i 0.464642i
$$668$$ − 9.25544i − 0.358104i
$$669$$ −9.11684 −0.352478
$$670$$ 0 0
$$671$$ −12.5109 −0.482977
$$672$$ − 3.37228i − 0.130089i
$$673$$ − 17.7228i − 0.683164i −0.939852 0.341582i $$-0.889037\pi$$
0.939852 0.341582i $$-0.110963\pi$$
$$674$$ −10.2337 −0.394187
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 24.0000i 0.922395i 0.887298 + 0.461197i $$0.152580\pi$$
−0.887298 + 0.461197i $$0.847420\pi$$
$$678$$ − 0.255437i − 0.00981001i
$$679$$ 4.23369 0.162474
$$680$$ 0 0
$$681$$ 1.37228 0.0525859
$$682$$ − 11.1386i − 0.426519i
$$683$$ 20.7446i 0.793769i 0.917869 + 0.396884i $$0.129909\pi$$
−0.917869 + 0.396884i $$0.870091\pi$$
$$684$$ −3.37228 −0.128942
$$685$$ 0 0
$$686$$ −8.86141 −0.338330
$$687$$ − 22.2337i − 0.848268i
$$688$$ 9.37228i 0.357315i
$$689$$ 11.4891 0.437701
$$690$$ 0 0
$$691$$ 33.6277 1.27926 0.639629 0.768683i $$-0.279088\pi$$
0.639629 + 0.768683i $$0.279088\pi$$
$$692$$ 12.0000i 0.456172i
$$693$$ − 4.62772i − 0.175792i
$$694$$ −16.6277 −0.631180
$$695$$ 0 0
$$696$$ −2.74456 −0.104032
$$697$$ 4.37228i 0.165612i
$$698$$ − 10.0000i − 0.378506i
$$699$$ 16.3723 0.619257
$$700$$ 0 0
$$701$$ −30.6060 −1.15597 −0.577986 0.816047i $$-0.696161\pi$$
−0.577986 + 0.816047i $$0.696161\pi$$
$$702$$ − 2.00000i − 0.0754851i
$$703$$ − 3.37228i − 0.127188i
$$704$$ 1.37228 0.0517198
$$705$$ 0 0
$$706$$ −26.7446 −1.00654
$$707$$ 13.8832i 0.522130i
$$708$$ − 7.11684i − 0.267467i
$$709$$ 2.62772 0.0986860 0.0493430 0.998782i $$-0.484287\pi$$
0.0493430 + 0.998782i $$0.484287\pi$$
$$710$$ 0 0
$$711$$ 6.11684 0.229400
$$712$$ − 8.74456i − 0.327716i
$$713$$ − 35.4891i − 1.32908i
$$714$$ 3.37228 0.126204
$$715$$ 0 0
$$716$$ 7.11684 0.265969
$$717$$ − 4.11684i − 0.153746i
$$718$$ − 19.3723i − 0.722967i
$$719$$ 38.7446 1.44493 0.722464 0.691408i $$-0.243009\pi$$
0.722464 + 0.691408i $$0.243009\pi$$
$$720$$ 0 0
$$721$$ 32.4674 1.20915
$$722$$ 7.62772i 0.283874i
$$723$$ − 6.23369i − 0.231833i
$$724$$ −22.4891 −0.835802
$$725$$ 0 0
$$726$$ −9.11684 −0.338358
$$727$$ 42.7446i 1.58531i 0.609672 + 0.792654i $$0.291301\pi$$
−0.609672 + 0.792654i $$0.708699\pi$$
$$728$$ 6.74456i 0.249970i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −9.37228 −0.346646
$$732$$ − 9.11684i − 0.336968i
$$733$$ − 12.7446i − 0.470731i −0.971907 0.235366i $$-0.924371\pi$$
0.971907 0.235366i $$-0.0756287\pi$$
$$734$$ 25.6060 0.945134
$$735$$ 0 0
$$736$$ 4.37228 0.161164
$$737$$ − 7.37228i − 0.271561i
$$738$$ 4.37228i 0.160946i
$$739$$ −39.3723 −1.44833 −0.724166 0.689625i $$-0.757775\pi$$
−0.724166 + 0.689625i $$0.757775\pi$$
$$740$$ 0 0
$$741$$ 6.74456 0.247768
$$742$$ 19.3723i 0.711179i
$$743$$ 28.3723i 1.04088i 0.853899 + 0.520439i $$0.174232\pi$$
−0.853899 + 0.520439i $$0.825768\pi$$
$$744$$ 8.11684 0.297578
$$745$$ 0 0
$$746$$ −36.2337 −1.32661
$$747$$ 4.37228i 0.159973i
$$748$$ 1.37228i 0.0501756i
$$749$$ 15.6060 0.570230
$$750$$ 0 0
$$751$$ 36.4674 1.33071 0.665357 0.746526i $$-0.268279\pi$$
0.665357 + 0.746526i $$0.268279\pi$$
$$752$$ 7.37228i 0.268839i
$$753$$ − 2.74456i − 0.100017i
$$754$$ 5.48913 0.199902
$$755$$ 0 0
$$756$$ 3.37228 0.122649
$$757$$ 0.233688i 0.00849353i 0.999991 + 0.00424677i $$0.00135179\pi$$
−0.999991 + 0.00424677i $$0.998648\pi$$
$$758$$ 20.6060i 0.748443i
$$759$$ 6.00000 0.217786
$$760$$ 0 0
$$761$$ 5.48913 0.198981 0.0994903 0.995039i $$-0.468279\pi$$
0.0994903 + 0.995039i $$0.468279\pi$$
$$762$$ 21.4891i 0.778469i
$$763$$ − 27.3723i − 0.990943i
$$764$$ −24.3505 −0.880971
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 14.2337i 0.513949i
$$768$$ 1.00000i 0.0360844i
$$769$$ 13.0000 0.468792 0.234396 0.972141i $$-0.424689\pi$$
0.234396 + 0.972141i $$0.424689\pi$$
$$770$$ 0 0
$$771$$ 20.2337 0.728698
$$772$$ − 8.00000i − 0.287926i
$$773$$ − 45.8614i − 1.64952i −0.565483 0.824760i $$-0.691310\pi$$
0.565483 0.824760i $$-0.308690\pi$$
$$774$$ −9.37228 −0.336880
$$775$$ 0 0
$$776$$ −1.25544 −0.0450676
$$777$$ 3.37228i 0.120980i
$$778$$ 6.25544i 0.224268i
$$779$$ −14.7446 −0.528279
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ 4.37228i 0.156352i
$$783$$ − 2.74456i − 0.0980827i
$$784$$ −4.37228 −0.156153
$$785$$ 0 0
$$786$$ −5.48913 −0.195791
$$787$$ 7.35053i 0.262018i 0.991381 + 0.131009i $$0.0418217\pi$$
−0.991381 + 0.131009i $$0.958178\pi$$
$$788$$ − 20.7446i − 0.738994i
$$789$$ 31.3723 1.11688
$$790$$ 0 0
$$791$$ −0.861407 −0.0306281
$$792$$ 1.37228i 0.0487619i
$$793$$ 18.2337i 0.647497i
$$794$$ 1.00000 0.0354887
$$795$$ 0 0
$$796$$ 3.88316 0.137635
$$797$$ − 17.7446i − 0.628545i −0.949333 0.314272i $$-0.898239\pi$$
0.949333 0.314272i $$-0.101761\pi$$
$$798$$ 11.3723i 0.402574i
$$799$$ −7.37228 −0.260813
$$800$$ 0 0
$$801$$ 8.74456 0.308974
$$802$$ 13.1168i 0.463172i
$$803$$ − 10.9783i − 0.387414i
$$804$$ 5.37228 0.189466
$$805$$ 0 0
$$806$$ −16.2337 −0.571807
$$807$$ 11.4891i 0.404436i
$$808$$ − 4.11684i − 0.144830i
$$809$$ 21.6060 0.759625 0.379813 0.925063i $$-0.375988\pi$$
0.379813 + 0.925063i $$0.375988\pi$$
$$810$$ 0 0
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ 9.25544i 0.324802i
$$813$$ − 8.37228i − 0.293629i
$$814$$ −1.37228 −0.0480984
$$815$$ 0 0
$$816$$ −1.00000 −0.0350070
$$817$$ − 31.6060i − 1.10575i
$$818$$ − 13.8614i − 0.484653i
$$819$$ −6.74456 −0.235674
$$820$$ 0 0
$$821$$ −33.2554 −1.16062 −0.580311 0.814395i $$-0.697069\pi$$
−0.580311 + 0.814395i $$0.697069\pi$$
$$822$$ 17.4891i 0.610003i
$$823$$ 48.4674i 1.68947i 0.535188 + 0.844733i $$0.320241\pi$$
−0.535188 + 0.844733i $$0.679759\pi$$
$$824$$ −9.62772 −0.335397
$$825$$ 0 0
$$826$$ −24.0000 −0.835067
$$827$$ − 51.0951i − 1.77675i −0.459118 0.888375i $$-0.651835\pi$$
0.459118 0.888375i $$-0.348165\pi$$
$$828$$ 4.37228i 0.151947i
$$829$$ −29.2554 −1.01608 −0.508042 0.861332i $$-0.669630\pi$$
−0.508042 + 0.861332i $$0.669630\pi$$
$$830$$ 0 0
$$831$$ −13.0000 −0.450965
$$832$$ − 2.00000i − 0.0693375i
$$833$$ − 4.37228i − 0.151491i
$$834$$ −0.883156 −0.0305812
$$835$$ 0 0
$$836$$ −4.62772 −0.160053
$$837$$ 8.11684i 0.280559i
$$838$$ − 34.9783i − 1.20830i
$$839$$ −1.62772 −0.0561951 −0.0280975 0.999605i $$-0.508945\pi$$
−0.0280975 + 0.999605i $$0.508945\pi$$
$$840$$ 0 0
$$841$$ −21.4674 −0.740254
$$842$$ − 16.2337i − 0.559450i
$$843$$ 23.4891i 0.809008i
$$844$$ 24.2337 0.834158
$$845$$ 0 0
$$846$$ −7.37228 −0.253464
$$847$$ 30.7446i 1.05640i
$$848$$ − 5.74456i − 0.197269i
$$849$$ 5.11684 0.175610
$$850$$ 0 0
$$851$$ −4.37228 −0.149880
$$852$$ − 4.37228i − 0.149792i
$$853$$ − 10.8614i − 0.371887i −0.982560 0.185944i $$-0.940466\pi$$
0.982560 0.185944i $$-0.0595342\pi$$
$$854$$ −30.7446 −1.05206
$$855$$ 0 0
$$856$$ −4.62772 −0.158172
$$857$$ − 29.7446i − 1.01605i −0.861341 0.508027i $$-0.830375\pi$$
0.861341 0.508027i $$-0.169625\pi$$
$$858$$ − 2.74456i − 0.0936978i
$$859$$ −38.3505 −1.30850 −0.654252 0.756277i $$-0.727016\pi$$
−0.654252 + 0.756277i $$0.727016\pi$$
$$860$$ 0 0
$$861$$ 14.7446 0.502493
$$862$$ − 12.0000i − 0.408722i
$$863$$ − 19.3723i − 0.659440i −0.944079 0.329720i $$-0.893046\pi$$
0.944079 0.329720i $$-0.106954\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 23.0951 0.784803
$$867$$ − 1.00000i − 0.0339618i
$$868$$ − 27.3723i − 0.929076i
$$869$$ 8.39403 0.284748
$$870$$ 0 0
$$871$$ −10.7446 −0.364066
$$872$$ 8.11684i 0.274871i
$$873$$ − 1.25544i − 0.0424901i
$$874$$ −14.7446 −0.498742
$$875$$ 0 0
$$876$$ 8.00000 0.270295
$$877$$ 46.0000i 1.55331i 0.629926 + 0.776655i $$0.283085\pi$$
−0.629926 + 0.776655i $$0.716915\pi$$
$$878$$ 25.4891i 0.860216i
$$879$$ 10.8832 0.367080
$$880$$ 0 0
$$881$$ 46.2119 1.55692 0.778460 0.627694i $$-0.216001\pi$$
0.778460 + 0.627694i $$0.216001\pi$$
$$882$$ − 4.37228i − 0.147222i
$$883$$ − 42.2337i − 1.42128i −0.703557 0.710638i $$-0.748406\pi$$
0.703557 0.710638i $$-0.251594\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ −1.62772 −0.0546843
$$887$$ 0.605969i 0.0203465i 0.999948 + 0.0101732i $$0.00323829\pi$$
−0.999948 + 0.0101732i $$0.996762\pi$$
$$888$$ − 1.00000i − 0.0335578i
$$889$$ 72.4674 2.43048
$$890$$ 0 0
$$891$$ −1.37228 −0.0459732
$$892$$ − 9.11684i − 0.305255i
$$893$$ − 24.8614i − 0.831955i
$$894$$ −21.8614 −0.731155
$$895$$ 0 0
$$896$$ 3.37228 0.112660
$$897$$ − 8.74456i − 0.291972i
$$898$$ − 5.13859i − 0.171477i
$$899$$ −22.2772 −0.742986
$$900$$ 0 0
$$901$$ 5.74456 0.191379
$$902$$ 6.00000i 0.199778i
$$903$$ 31.6060i 1.05178i
$$904$$ 0.255437 0.00849572
$$905$$ 0 0
$$906$$ 3.11684 0.103550
$$907$$ − 50.6060i − 1.68034i −0.542320 0.840172i $$-0.682454\pi$$
0.542320 0.840172i $$-0.317546\pi$$
$$908$$ 1.37228i 0.0455408i
$$909$$ 4.11684 0.136547
$$910$$ 0 0
$$911$$ −29.4891 −0.977018 −0.488509 0.872559i $$-0.662459\pi$$
−0.488509 + 0.872559i $$0.662459\pi$$
$$912$$ − 3.37228i − 0.111667i
$$913$$ 6.00000i 0.198571i
$$914$$ −23.0000 −0.760772
$$915$$ 0 0
$$916$$ 22.2337 0.734622
$$917$$ 18.5109i 0.611283i
$$918$$ − 1.00000i − 0.0330049i
$$919$$ −43.5842 −1.43771 −0.718855 0.695160i $$-0.755334\pi$$
−0.718855 + 0.695160i $$0.755334\pi$$
$$920$$ 0 0
$$921$$ −18.2337 −0.600820
$$922$$ 22.7228i 0.748336i
$$923$$ 8.74456i 0.287831i
$$924$$ 4.62772 0.152241
$$925$$ 0 0
$$926$$ 20.6060 0.677154
$$927$$ − 9.62772i − 0.316216i
$$928$$ − 2.74456i − 0.0900947i
$$929$$ −16.7228 −0.548658 −0.274329 0.961636i $$-0.588456\pi$$
−0.274329 + 0.961636i $$0.588456\pi$$
$$930$$ 0 0
$$931$$ 14.7446 0.483234
$$932$$ 16.3723i 0.536292i
$$933$$ − 10.8832i − 0.356299i
$$934$$ 1.11684 0.0365443
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ − 1.39403i − 0.0455410i −0.999741 0.0227705i $$-0.992751\pi$$
0.999741 0.0227705i $$-0.00724870\pi$$
$$938$$ − 18.1168i − 0.591536i
$$939$$ 6.74456 0.220100
$$940$$ 0 0
$$941$$ 46.4674 1.51479 0.757397 0.652955i $$-0.226471\pi$$
0.757397 + 0.652955i $$0.226471\pi$$
$$942$$ 7.25544i 0.236395i
$$943$$ 19.1168i 0.622530i
$$944$$ 7.11684 0.231634
$$945$$ 0 0
$$946$$ −12.8614 −0.418160
$$947$$ − 28.6277i − 0.930276i −0.885238 0.465138i $$-0.846005\pi$$
0.885238 0.465138i $$-0.153995\pi$$
$$948$$ 6.11684i 0.198666i
$$949$$ −16.0000 −0.519382
$$950$$ 0 0
$$951$$ 3.25544 0.105565
$$952$$ 3.37228i 0.109296i
$$953$$ − 4.97825i − 0.161261i −0.996744 0.0806307i $$-0.974307\pi$$
0.996744 0.0806307i $$-0.0256934\pi$$
$$954$$ 5.74456 0.185987
$$955$$ 0 0
$$956$$ 4.11684 0.133148
$$957$$ − 3.76631i − 0.121748i
$$958$$ 25.7228i 0.831066i
$$959$$ 58.9783 1.90451
$$960$$ 0 0
$$961$$ 34.8832 1.12526
$$962$$ 2.00000i 0.0644826i
$$963$$ − 4.62772i − 0.149126i
$$964$$ 6.23369 0.200774
$$965$$ 0 0
$$966$$ 14.7446 0.474399
$$967$$ − 29.3505i − 0.943849i −0.881639 0.471925i $$-0.843559\pi$$
0.881639 0.471925i $$-0.156441\pi$$
$$968$$ − 9.11684i − 0.293026i
$$969$$ 3.37228 0.108333
$$970$$ 0 0
$$971$$ 21.8614 0.701566 0.350783 0.936457i $$-0.385915\pi$$
0.350783 + 0.936457i $$0.385915\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 2.97825i 0.0954783i
$$974$$ −22.2337 −0.712413
$$975$$ 0 0
$$976$$ 9.11684 0.291823
$$977$$ − 6.00000i − 0.191957i −0.995383 0.0959785i $$-0.969402\pi$$
0.995383 0.0959785i $$-0.0305980\pi$$
$$978$$ 15.1168i 0.483383i
$$979$$ 12.0000 0.383522
$$980$$ 0 0
$$981$$ −8.11684 −0.259151
$$982$$ 25.6277i 0.817813i
$$983$$ 44.2337i 1.41084i 0.708792 + 0.705418i $$0.249241\pi$$
−0.708792 + 0.705418i $$0.750759\pi$$
$$984$$ −4.37228 −0.139383
$$985$$ 0 0
$$986$$ 2.74456 0.0874047
$$987$$ 24.8614i 0.791347i
$$988$$ 6.74456i 0.214573i
$$989$$ −40.9783 −1.30303
$$990$$ 0 0
$$991$$ −8.46738 −0.268975 −0.134488 0.990915i $$-0.542939\pi$$
−0.134488 + 0.990915i $$0.542939\pi$$
$$992$$ 8.11684i 0.257710i
$$993$$ − 25.6060i − 0.812581i
$$994$$ −14.7446 −0.467669
$$995$$ 0 0
$$996$$ −4.37228 −0.138541
$$997$$ − 15.8832i − 0.503025i −0.967854 0.251512i $$-0.919072\pi$$
0.967854 0.251512i $$-0.0809279\pi$$
$$998$$ 24.3723i 0.771491i
$$999$$ 1.00000 0.0316386
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.v.2449.1 4
5.2 odd 4 2550.2.a.bm.1.2 yes 2
5.3 odd 4 2550.2.a.bg.1.1 2
5.4 even 2 inner 2550.2.d.v.2449.4 4
15.2 even 4 7650.2.a.cv.1.2 2
15.8 even 4 7650.2.a.df.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.bg.1.1 2 5.3 odd 4
2550.2.a.bm.1.2 yes 2 5.2 odd 4
2550.2.d.v.2449.1 4 1.1 even 1 trivial
2550.2.d.v.2449.4 4 5.4 even 2 inner
7650.2.a.cv.1.2 2 15.2 even 4
7650.2.a.df.1.1 2 15.8 even 4