# Properties

 Label 1450.2.b.k.349.4 Level $1450$ Weight $2$ Character 1450.349 Analytic conductor $11.578$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1450,2,Mod(349,1450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1450, 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("1450.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1450 = 2 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1450.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: $$11.5783082931$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.399424.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8$$ x^6 - 2*x^5 + 3*x^4 - 6*x^3 + 6*x^2 - 8*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.4 Root $$1.40680 + 0.144584i$$ of defining polynomial Character $$\chi$$ $$=$$ 1450.349 Dual form 1450.2.b.k.349.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.81361i q^{3} -1.00000 q^{4} +1.81361 q^{6} +2.52444i q^{7} -1.00000i q^{8} -0.289169 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.81361i q^{3} -1.00000 q^{4} +1.81361 q^{6} +2.52444i q^{7} -1.00000i q^{8} -0.289169 q^{9} -2.91638 q^{11} +1.81361i q^{12} -3.10278i q^{13} -2.52444 q^{14} +1.00000 q^{16} +6.91638i q^{17} -0.289169i q^{18} -1.81361 q^{19} +4.57834 q^{21} -2.91638i q^{22} +5.68111i q^{23} -1.81361 q^{24} +3.10278 q^{26} -4.91638i q^{27} -2.52444i q^{28} -1.00000 q^{29} -8.12193 q^{31} +1.00000i q^{32} +5.28917i q^{33} -6.91638 q^{34} +0.289169 q^{36} +7.39194i q^{37} -1.81361i q^{38} -5.62721 q^{39} +8.39194 q^{41} +4.57834i q^{42} +5.04888i q^{43} +2.91638 q^{44} -5.68111 q^{46} +4.86751i q^{47} -1.81361i q^{48} +0.627213 q^{49} +12.5436 q^{51} +3.10278i q^{52} -5.30833i q^{53} +4.91638 q^{54} +2.52444 q^{56} +3.28917i q^{57} -1.00000i q^{58} -2.60806 q^{59} -10.0680 q^{61} -8.12193i q^{62} -0.729988i q^{63} -1.00000 q^{64} -5.28917 q^{66} -2.68111i q^{67} -6.91638i q^{68} +10.3033 q^{69} +5.68111 q^{71} +0.289169i q^{72} +15.0680i q^{73} -7.39194 q^{74} +1.81361 q^{76} -7.36222i q^{77} -5.62721i q^{78} +1.73501 q^{79} -9.78389 q^{81} +8.39194i q^{82} +13.1219i q^{83} -4.57834 q^{84} -5.04888 q^{86} +1.81361i q^{87} +2.91638i q^{88} -5.23527 q^{89} +7.83276 q^{91} -5.68111i q^{92} +14.7300i q^{93} -4.86751 q^{94} +1.81361 q^{96} +1.27001i q^{97} +0.627213i q^{98} +0.843326 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} - 2 q^{6}+O(q^{10})$$ 6 * q - 6 * q^4 - 2 * q^6 $$6 q - 6 q^{4} - 2 q^{6} + 10 q^{11} - 4 q^{14} + 6 q^{16} + 2 q^{19} + 24 q^{21} + 2 q^{24} + 4 q^{26} - 6 q^{29} + 8 q^{31} - 14 q^{34} - 8 q^{39} + 34 q^{41} - 10 q^{44} - 16 q^{46} - 22 q^{49} + 22 q^{51} + 2 q^{54} + 4 q^{56} - 32 q^{59} + 4 q^{61} - 6 q^{64} - 30 q^{66} - 12 q^{69} + 16 q^{71} - 28 q^{74} - 2 q^{76} - 26 q^{81} - 24 q^{84} - 8 q^{86} - 22 q^{89} - 8 q^{91} - 24 q^{94} - 2 q^{96} + 12 q^{99}+O(q^{100})$$ 6 * q - 6 * q^4 - 2 * q^6 + 10 * q^11 - 4 * q^14 + 6 * q^16 + 2 * q^19 + 24 * q^21 + 2 * q^24 + 4 * q^26 - 6 * q^29 + 8 * q^31 - 14 * q^34 - 8 * q^39 + 34 * q^41 - 10 * q^44 - 16 * q^46 - 22 * q^49 + 22 * q^51 + 2 * q^54 + 4 * q^56 - 32 * q^59 + 4 * q^61 - 6 * q^64 - 30 * q^66 - 12 * q^69 + 16 * q^71 - 28 * q^74 - 2 * q^76 - 26 * q^81 - 24 * q^84 - 8 * q^86 - 22 * q^89 - 8 * q^91 - 24 * q^94 - 2 * q^96 + 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1277$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$3$$ − 1.81361i − 1.04709i −0.851999 0.523543i $$-0.824610\pi$$
0.851999 0.523543i $$-0.175390\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.81361 0.740402
$$7$$ 2.52444i 0.954148i 0.878863 + 0.477074i $$0.158303\pi$$
−0.878863 + 0.477074i $$0.841697\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −0.289169 −0.0963895
$$10$$ 0 0
$$11$$ −2.91638 −0.879322 −0.439661 0.898164i $$-0.644901\pi$$
−0.439661 + 0.898164i $$0.644901\pi$$
$$12$$ 1.81361i 0.523543i
$$13$$ − 3.10278i − 0.860555i −0.902697 0.430277i $$-0.858416\pi$$
0.902697 0.430277i $$-0.141584\pi$$
$$14$$ −2.52444 −0.674684
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.91638i 1.67747i 0.544541 + 0.838734i $$0.316704\pi$$
−0.544541 + 0.838734i $$0.683296\pi$$
$$18$$ − 0.289169i − 0.0681577i
$$19$$ −1.81361 −0.416070 −0.208035 0.978121i $$-0.566707\pi$$
−0.208035 + 0.978121i $$0.566707\pi$$
$$20$$ 0 0
$$21$$ 4.57834 0.999075
$$22$$ − 2.91638i − 0.621775i
$$23$$ 5.68111i 1.18459i 0.805720 + 0.592297i $$0.201779\pi$$
−0.805720 + 0.592297i $$0.798221\pi$$
$$24$$ −1.81361 −0.370201
$$25$$ 0 0
$$26$$ 3.10278 0.608504
$$27$$ − 4.91638i − 0.946158i
$$28$$ − 2.52444i − 0.477074i
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ −8.12193 −1.45874 −0.729371 0.684118i $$-0.760187\pi$$
−0.729371 + 0.684118i $$0.760187\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 5.28917i 0.920726i
$$34$$ −6.91638 −1.18615
$$35$$ 0 0
$$36$$ 0.289169 0.0481948
$$37$$ 7.39194i 1.21523i 0.794232 + 0.607614i $$0.207873\pi$$
−0.794232 + 0.607614i $$0.792127\pi$$
$$38$$ − 1.81361i − 0.294206i
$$39$$ −5.62721 −0.901075
$$40$$ 0 0
$$41$$ 8.39194 1.31060 0.655301 0.755368i $$-0.272542\pi$$
0.655301 + 0.755368i $$0.272542\pi$$
$$42$$ 4.57834i 0.706453i
$$43$$ 5.04888i 0.769946i 0.922928 + 0.384973i $$0.125789\pi$$
−0.922928 + 0.384973i $$0.874211\pi$$
$$44$$ 2.91638 0.439661
$$45$$ 0 0
$$46$$ −5.68111 −0.837634
$$47$$ 4.86751i 0.709999i 0.934867 + 0.354999i $$0.115519\pi$$
−0.934867 + 0.354999i $$0.884481\pi$$
$$48$$ − 1.81361i − 0.261772i
$$49$$ 0.627213 0.0896019
$$50$$ 0 0
$$51$$ 12.5436 1.75645
$$52$$ 3.10278i 0.430277i
$$53$$ − 5.30833i − 0.729155i −0.931173 0.364577i $$-0.881214\pi$$
0.931173 0.364577i $$-0.118786\pi$$
$$54$$ 4.91638 0.669035
$$55$$ 0 0
$$56$$ 2.52444 0.337342
$$57$$ 3.28917i 0.435661i
$$58$$ − 1.00000i − 0.131306i
$$59$$ −2.60806 −0.339540 −0.169770 0.985484i $$-0.554302\pi$$
−0.169770 + 0.985484i $$0.554302\pi$$
$$60$$ 0 0
$$61$$ −10.0680 −1.28908 −0.644540 0.764571i $$-0.722951\pi$$
−0.644540 + 0.764571i $$0.722951\pi$$
$$62$$ − 8.12193i − 1.03149i
$$63$$ − 0.729988i − 0.0919699i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −5.28917 −0.651052
$$67$$ − 2.68111i − 0.327550i −0.986498 0.163775i $$-0.947633\pi$$
0.986498 0.163775i $$-0.0523671\pi$$
$$68$$ − 6.91638i − 0.838734i
$$69$$ 10.3033 1.24037
$$70$$ 0 0
$$71$$ 5.68111 0.674224 0.337112 0.941465i $$-0.390550\pi$$
0.337112 + 0.941465i $$0.390550\pi$$
$$72$$ 0.289169i 0.0340788i
$$73$$ 15.0680i 1.76358i 0.471642 + 0.881790i $$0.343661\pi$$
−0.471642 + 0.881790i $$0.656339\pi$$
$$74$$ −7.39194 −0.859296
$$75$$ 0 0
$$76$$ 1.81361 0.208035
$$77$$ − 7.36222i − 0.839003i
$$78$$ − 5.62721i − 0.637156i
$$79$$ 1.73501 0.195204 0.0976020 0.995226i $$-0.468883\pi$$
0.0976020 + 0.995226i $$0.468883\pi$$
$$80$$ 0 0
$$81$$ −9.78389 −1.08710
$$82$$ 8.39194i 0.926735i
$$83$$ 13.1219i 1.44032i 0.693808 + 0.720160i $$0.255931\pi$$
−0.693808 + 0.720160i $$0.744069\pi$$
$$84$$ −4.57834 −0.499538
$$85$$ 0 0
$$86$$ −5.04888 −0.544434
$$87$$ 1.81361i 0.194439i
$$88$$ 2.91638i 0.310887i
$$89$$ −5.23527 −0.554937 −0.277469 0.960735i $$-0.589495\pi$$
−0.277469 + 0.960735i $$0.589495\pi$$
$$90$$ 0 0
$$91$$ 7.83276 0.821097
$$92$$ − 5.68111i − 0.592297i
$$93$$ 14.7300i 1.52743i
$$94$$ −4.86751 −0.502045
$$95$$ 0 0
$$96$$ 1.81361 0.185100
$$97$$ 1.27001i 0.128950i 0.997919 + 0.0644751i $$0.0205373\pi$$
−0.997919 + 0.0644751i $$0.979463\pi$$
$$98$$ 0.627213i 0.0633581i
$$99$$ 0.843326 0.0847574
$$100$$ 0 0
$$101$$ 5.39194 0.536518 0.268259 0.963347i $$-0.413552\pi$$
0.268259 + 0.963347i $$0.413552\pi$$
$$102$$ 12.5436i 1.24200i
$$103$$ 17.6116i 1.73533i 0.497154 + 0.867663i $$0.334379\pi$$
−0.497154 + 0.867663i $$0.665621\pi$$
$$104$$ −3.10278 −0.304252
$$105$$ 0 0
$$106$$ 5.30833 0.515590
$$107$$ 4.30833i 0.416502i 0.978075 + 0.208251i $$0.0667770\pi$$
−0.978075 + 0.208251i $$0.933223\pi$$
$$108$$ 4.91638i 0.473079i
$$109$$ 7.57331 0.725392 0.362696 0.931908i $$-0.381856\pi$$
0.362696 + 0.931908i $$0.381856\pi$$
$$110$$ 0 0
$$111$$ 13.4061 1.27245
$$112$$ 2.52444i 0.238537i
$$113$$ 5.60806i 0.527562i 0.964583 + 0.263781i $$0.0849696\pi$$
−0.964583 + 0.263781i $$0.915030\pi$$
$$114$$ −3.28917 −0.308059
$$115$$ 0 0
$$116$$ 1.00000 0.0928477
$$117$$ 0.897225i 0.0829485i
$$118$$ − 2.60806i − 0.240091i
$$119$$ −17.4600 −1.60055
$$120$$ 0 0
$$121$$ −2.49472 −0.226793
$$122$$ − 10.0680i − 0.911517i
$$123$$ − 15.2197i − 1.37231i
$$124$$ 8.12193 0.729371
$$125$$ 0 0
$$126$$ 0.729988 0.0650325
$$127$$ 2.12193i 0.188291i 0.995558 + 0.0941455i $$0.0300119\pi$$
−0.995558 + 0.0941455i $$0.969988\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 9.15667 0.806200
$$130$$ 0 0
$$131$$ 3.74055 0.326813 0.163407 0.986559i $$-0.447752\pi$$
0.163407 + 0.986559i $$0.447752\pi$$
$$132$$ − 5.28917i − 0.460363i
$$133$$ − 4.57834i − 0.396992i
$$134$$ 2.68111 0.231613
$$135$$ 0 0
$$136$$ 6.91638 0.593075
$$137$$ 11.7980i 1.00797i 0.863712 + 0.503986i $$0.168134\pi$$
−0.863712 + 0.503986i $$0.831866\pi$$
$$138$$ 10.3033i 0.877075i
$$139$$ −12.3083 −1.04398 −0.521989 0.852952i $$-0.674810\pi$$
−0.521989 + 0.852952i $$0.674810\pi$$
$$140$$ 0 0
$$141$$ 8.82774 0.743430
$$142$$ 5.68111i 0.476748i
$$143$$ 9.04888i 0.756705i
$$144$$ −0.289169 −0.0240974
$$145$$ 0 0
$$146$$ −15.0680 −1.24704
$$147$$ − 1.13752i − 0.0938209i
$$148$$ − 7.39194i − 0.607614i
$$149$$ −2.00000 −0.163846 −0.0819232 0.996639i $$-0.526106\pi$$
−0.0819232 + 0.996639i $$0.526106\pi$$
$$150$$ 0 0
$$151$$ −8.89722 −0.724046 −0.362023 0.932169i $$-0.617914\pi$$
−0.362023 + 0.932169i $$0.617914\pi$$
$$152$$ 1.81361i 0.147103i
$$153$$ − 2.00000i − 0.161690i
$$154$$ 7.36222 0.593265
$$155$$ 0 0
$$156$$ 5.62721 0.450538
$$157$$ − 17.4897i − 1.39583i −0.716181 0.697915i $$-0.754111\pi$$
0.716181 0.697915i $$-0.245889\pi$$
$$158$$ 1.73501i 0.138030i
$$159$$ −9.62721 −0.763488
$$160$$ 0 0
$$161$$ −14.3416 −1.13028
$$162$$ − 9.78389i − 0.768695i
$$163$$ 3.02972i 0.237306i 0.992936 + 0.118653i $$0.0378576\pi$$
−0.992936 + 0.118653i $$0.962142\pi$$
$$164$$ −8.39194 −0.655301
$$165$$ 0 0
$$166$$ −13.1219 −1.01846
$$167$$ − 23.9305i − 1.85180i −0.377770 0.925899i $$-0.623309\pi$$
0.377770 0.925899i $$-0.376691\pi$$
$$168$$ − 4.57834i − 0.353226i
$$169$$ 3.37279 0.259445
$$170$$ 0 0
$$171$$ 0.524438 0.0401048
$$172$$ − 5.04888i − 0.384973i
$$173$$ − 11.5194i − 0.875805i −0.899022 0.437902i $$-0.855722\pi$$
0.899022 0.437902i $$-0.144278\pi$$
$$174$$ −1.81361 −0.137489
$$175$$ 0 0
$$176$$ −2.91638 −0.219831
$$177$$ 4.72999i 0.355528i
$$178$$ − 5.23527i − 0.392400i
$$179$$ −8.30833 −0.620993 −0.310497 0.950574i $$-0.600495\pi$$
−0.310497 + 0.950574i $$0.600495\pi$$
$$180$$ 0 0
$$181$$ 17.7194 1.31707 0.658537 0.752548i $$-0.271175\pi$$
0.658537 + 0.752548i $$0.271175\pi$$
$$182$$ 7.83276i 0.580603i
$$183$$ 18.2594i 1.34978i
$$184$$ 5.68111 0.418817
$$185$$ 0 0
$$186$$ −14.7300 −1.08006
$$187$$ − 20.1708i − 1.47504i
$$188$$ − 4.86751i − 0.354999i
$$189$$ 12.4111 0.902775
$$190$$ 0 0
$$191$$ 17.1708 1.24244 0.621218 0.783638i $$-0.286638\pi$$
0.621218 + 0.783638i $$0.286638\pi$$
$$192$$ 1.81361i 0.130886i
$$193$$ − 7.06803i − 0.508768i −0.967103 0.254384i $$-0.918127\pi$$
0.967103 0.254384i $$-0.0818727\pi$$
$$194$$ −1.27001 −0.0911815
$$195$$ 0 0
$$196$$ −0.627213 −0.0448009
$$197$$ − 16.1361i − 1.14965i −0.818277 0.574824i $$-0.805071\pi$$
0.818277 0.574824i $$-0.194929\pi$$
$$198$$ 0.843326i 0.0599326i
$$199$$ −8.25945 −0.585497 −0.292748 0.956190i $$-0.594570\pi$$
−0.292748 + 0.956190i $$0.594570\pi$$
$$200$$ 0 0
$$201$$ −4.86248 −0.342973
$$202$$ 5.39194i 0.379376i
$$203$$ − 2.52444i − 0.177181i
$$204$$ −12.5436 −0.878227
$$205$$ 0 0
$$206$$ −17.6116 −1.22706
$$207$$ − 1.64280i − 0.114182i
$$208$$ − 3.10278i − 0.215139i
$$209$$ 5.28917 0.365859
$$210$$ 0 0
$$211$$ −12.2302 −0.841965 −0.420982 0.907069i $$-0.638315\pi$$
−0.420982 + 0.907069i $$0.638315\pi$$
$$212$$ 5.30833i 0.364577i
$$213$$ − 10.3033i − 0.705971i
$$214$$ −4.30833 −0.294511
$$215$$ 0 0
$$216$$ −4.91638 −0.334517
$$217$$ − 20.5033i − 1.39186i
$$218$$ 7.57331i 0.512930i
$$219$$ 27.3275 1.84662
$$220$$ 0 0
$$221$$ 21.4600 1.44355
$$222$$ 13.4061i 0.899757i
$$223$$ − 11.2544i − 0.753652i −0.926284 0.376826i $$-0.877015\pi$$
0.926284 0.376826i $$-0.122985\pi$$
$$224$$ −2.52444 −0.168671
$$225$$ 0 0
$$226$$ −5.60806 −0.373042
$$227$$ − 19.7647i − 1.31183i −0.754834 0.655916i $$-0.772283\pi$$
0.754834 0.655916i $$-0.227717\pi$$
$$228$$ − 3.28917i − 0.217831i
$$229$$ 23.7250 1.56779 0.783895 0.620894i $$-0.213230\pi$$
0.783895 + 0.620894i $$0.213230\pi$$
$$230$$ 0 0
$$231$$ −13.3522 −0.878509
$$232$$ 1.00000i 0.0656532i
$$233$$ − 24.0625i − 1.57639i −0.615428 0.788193i $$-0.711017\pi$$
0.615428 0.788193i $$-0.288983\pi$$
$$234$$ −0.897225 −0.0586534
$$235$$ 0 0
$$236$$ 2.60806 0.169770
$$237$$ − 3.14663i − 0.204395i
$$238$$ − 17.4600i − 1.13176i
$$239$$ 4.31335 0.279007 0.139504 0.990222i $$-0.455449\pi$$
0.139504 + 0.990222i $$0.455449\pi$$
$$240$$ 0 0
$$241$$ −27.7144 −1.78524 −0.892621 0.450808i $$-0.851136\pi$$
−0.892621 + 0.450808i $$0.851136\pi$$
$$242$$ − 2.49472i − 0.160367i
$$243$$ 2.99498i 0.192128i
$$244$$ 10.0680 0.644540
$$245$$ 0 0
$$246$$ 15.2197 0.970372
$$247$$ 5.62721i 0.358051i
$$248$$ 8.12193i 0.515743i
$$249$$ 23.7980 1.50814
$$250$$ 0 0
$$251$$ −21.7542 −1.37311 −0.686555 0.727077i $$-0.740878\pi$$
−0.686555 + 0.727077i $$0.740878\pi$$
$$252$$ 0.729988i 0.0459849i
$$253$$ − 16.5683i − 1.04164i
$$254$$ −2.12193 −0.133142
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 5.83276i 0.363838i 0.983314 + 0.181919i $$0.0582308\pi$$
−0.983314 + 0.181919i $$0.941769\pi$$
$$258$$ 9.15667i 0.570070i
$$259$$ −18.6605 −1.15951
$$260$$ 0 0
$$261$$ 0.289169 0.0178991
$$262$$ 3.74055i 0.231092i
$$263$$ − 20.8675i − 1.28675i −0.765553 0.643373i $$-0.777535\pi$$
0.765553 0.643373i $$-0.222465\pi$$
$$264$$ 5.28917 0.325526
$$265$$ 0 0
$$266$$ 4.57834 0.280716
$$267$$ 9.49472i 0.581067i
$$268$$ 2.68111i 0.163775i
$$269$$ −0.548618 −0.0334498 −0.0167249 0.999860i $$-0.505324\pi$$
−0.0167249 + 0.999860i $$0.505324\pi$$
$$270$$ 0 0
$$271$$ 5.00357 0.303945 0.151973 0.988385i $$-0.451437\pi$$
0.151973 + 0.988385i $$0.451437\pi$$
$$272$$ 6.91638i 0.419367i
$$273$$ − 14.2056i − 0.859759i
$$274$$ −11.7980 −0.712744
$$275$$ 0 0
$$276$$ −10.3033 −0.620186
$$277$$ 0.632236i 0.0379874i 0.999820 + 0.0189937i $$0.00604625\pi$$
−0.999820 + 0.0189937i $$0.993954\pi$$
$$278$$ − 12.3083i − 0.738204i
$$279$$ 2.34861 0.140607
$$280$$ 0 0
$$281$$ 30.9653 1.84723 0.923616 0.383319i $$-0.125219\pi$$
0.923616 + 0.383319i $$0.125219\pi$$
$$282$$ 8.82774i 0.525684i
$$283$$ 6.33804i 0.376758i 0.982096 + 0.188379i $$0.0603233\pi$$
−0.982096 + 0.188379i $$0.939677\pi$$
$$284$$ −5.68111 −0.337112
$$285$$ 0 0
$$286$$ −9.04888 −0.535071
$$287$$ 21.1849i 1.25051i
$$288$$ − 0.289169i − 0.0170394i
$$289$$ −30.8363 −1.81390
$$290$$ 0 0
$$291$$ 2.30330 0.135022
$$292$$ − 15.0680i − 0.881790i
$$293$$ 28.4197i 1.66030i 0.557543 + 0.830148i $$0.311744\pi$$
−0.557543 + 0.830148i $$0.688256\pi$$
$$294$$ 1.13752 0.0663414
$$295$$ 0 0
$$296$$ 7.39194 0.429648
$$297$$ 14.3380i 0.831978i
$$298$$ − 2.00000i − 0.115857i
$$299$$ 17.6272 1.01941
$$300$$ 0 0
$$301$$ −12.7456 −0.734643
$$302$$ − 8.89722i − 0.511978i
$$303$$ − 9.77886i − 0.561781i
$$304$$ −1.81361 −0.104017
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ 22.6797i 1.29440i 0.762322 + 0.647198i $$0.224059\pi$$
−0.762322 + 0.647198i $$0.775941\pi$$
$$308$$ 7.36222i 0.419502i
$$309$$ 31.9406 1.81704
$$310$$ 0 0
$$311$$ −9.08719 −0.515287 −0.257644 0.966240i $$-0.582946\pi$$
−0.257644 + 0.966240i $$0.582946\pi$$
$$312$$ 5.62721i 0.318578i
$$313$$ − 25.0630i − 1.41665i −0.705889 0.708323i $$-0.749452\pi$$
0.705889 0.708323i $$-0.250548\pi$$
$$314$$ 17.4897 0.987001
$$315$$ 0 0
$$316$$ −1.73501 −0.0976020
$$317$$ − 14.9497i − 0.839657i −0.907603 0.419829i $$-0.862090\pi$$
0.907603 0.419829i $$-0.137910\pi$$
$$318$$ − 9.62721i − 0.539867i
$$319$$ 2.91638 0.163286
$$320$$ 0 0
$$321$$ 7.81361 0.436113
$$322$$ − 14.3416i − 0.799227i
$$323$$ − 12.5436i − 0.697944i
$$324$$ 9.78389 0.543549
$$325$$ 0 0
$$326$$ −3.02972 −0.167801
$$327$$ − 13.7350i − 0.759548i
$$328$$ − 8.39194i − 0.463368i
$$329$$ −12.2877 −0.677444
$$330$$ 0 0
$$331$$ −18.1270 −0.996348 −0.498174 0.867077i $$-0.665996\pi$$
−0.498174 + 0.867077i $$0.665996\pi$$
$$332$$ − 13.1219i − 0.720160i
$$333$$ − 2.13752i − 0.117135i
$$334$$ 23.9305 1.30942
$$335$$ 0 0
$$336$$ 4.57834 0.249769
$$337$$ 9.08362i 0.494816i 0.968911 + 0.247408i $$0.0795788\pi$$
−0.968911 + 0.247408i $$0.920421\pi$$
$$338$$ 3.37279i 0.183455i
$$339$$ 10.1708 0.552402
$$340$$ 0 0
$$341$$ 23.6867 1.28270
$$342$$ 0.524438i 0.0283584i
$$343$$ 19.2544i 1.03964i
$$344$$ 5.04888 0.272217
$$345$$ 0 0
$$346$$ 11.5194 0.619288
$$347$$ − 30.7194i − 1.64911i −0.565785 0.824553i $$-0.691427\pi$$
0.565785 0.824553i $$-0.308573\pi$$
$$348$$ − 1.81361i − 0.0972195i
$$349$$ −15.7789 −0.844623 −0.422312 0.906451i $$-0.638781\pi$$
−0.422312 + 0.906451i $$0.638781\pi$$
$$350$$ 0 0
$$351$$ −15.2544 −0.814221
$$352$$ − 2.91638i − 0.155444i
$$353$$ 31.6797i 1.68614i 0.537805 + 0.843069i $$0.319254\pi$$
−0.537805 + 0.843069i $$0.680746\pi$$
$$354$$ −4.72999 −0.251396
$$355$$ 0 0
$$356$$ 5.23527 0.277469
$$357$$ 31.6655i 1.67592i
$$358$$ − 8.30833i − 0.439109i
$$359$$ 27.1013 1.43035 0.715177 0.698944i $$-0.246346\pi$$
0.715177 + 0.698944i $$0.246346\pi$$
$$360$$ 0 0
$$361$$ −15.7108 −0.826886
$$362$$ 17.7194i 0.931312i
$$363$$ 4.52444i 0.237471i
$$364$$ −7.83276 −0.410548
$$365$$ 0 0
$$366$$ −18.2594 −0.954437
$$367$$ 31.8953i 1.66492i 0.554086 + 0.832459i $$0.313068\pi$$
−0.554086 + 0.832459i $$0.686932\pi$$
$$368$$ 5.68111i 0.296148i
$$369$$ −2.42669 −0.126328
$$370$$ 0 0
$$371$$ 13.4005 0.695721
$$372$$ − 14.7300i − 0.763714i
$$373$$ − 28.0383i − 1.45177i −0.687817 0.725884i $$-0.741431\pi$$
0.687817 0.725884i $$-0.258569\pi$$
$$374$$ 20.1708 1.04301
$$375$$ 0 0
$$376$$ 4.86751 0.251022
$$377$$ 3.10278i 0.159801i
$$378$$ 12.4111i 0.638358i
$$379$$ −8.69525 −0.446645 −0.223322 0.974745i $$-0.571690\pi$$
−0.223322 + 0.974745i $$0.571690\pi$$
$$380$$ 0 0
$$381$$ 3.84835 0.197157
$$382$$ 17.1708i 0.878535i
$$383$$ − 14.8917i − 0.760930i −0.924795 0.380465i $$-0.875764\pi$$
0.924795 0.380465i $$-0.124236\pi$$
$$384$$ −1.81361 −0.0925502
$$385$$ 0 0
$$386$$ 7.06803 0.359753
$$387$$ − 1.45998i − 0.0742148i
$$388$$ − 1.27001i − 0.0644751i
$$389$$ 1.93197 0.0979546 0.0489773 0.998800i $$-0.484404\pi$$
0.0489773 + 0.998800i $$0.484404\pi$$
$$390$$ 0 0
$$391$$ −39.2927 −1.98712
$$392$$ − 0.627213i − 0.0316790i
$$393$$ − 6.78389i − 0.342202i
$$394$$ 16.1361 0.812923
$$395$$ 0 0
$$396$$ −0.843326 −0.0423787
$$397$$ 19.2983i 0.968553i 0.874915 + 0.484276i $$0.160917\pi$$
−0.874915 + 0.484276i $$0.839083\pi$$
$$398$$ − 8.25945i − 0.414009i
$$399$$ −8.30330 −0.415685
$$400$$ 0 0
$$401$$ 16.7250 0.835205 0.417602 0.908630i $$-0.362870\pi$$
0.417602 + 0.908630i $$0.362870\pi$$
$$402$$ − 4.86248i − 0.242519i
$$403$$ 25.2005i 1.25533i
$$404$$ −5.39194 −0.268259
$$405$$ 0 0
$$406$$ 2.52444 0.125286
$$407$$ − 21.5577i − 1.06858i
$$408$$ − 12.5436i − 0.621000i
$$409$$ 26.4842 1.30956 0.654779 0.755821i $$-0.272762\pi$$
0.654779 + 0.755821i $$0.272762\pi$$
$$410$$ 0 0
$$411$$ 21.3970 1.05543
$$412$$ − 17.6116i − 0.867663i
$$413$$ − 6.58388i − 0.323971i
$$414$$ 1.64280 0.0807392
$$415$$ 0 0
$$416$$ 3.10278 0.152126
$$417$$ 22.3225i 1.09314i
$$418$$ 5.28917i 0.258702i
$$419$$ 6.94610 0.339339 0.169670 0.985501i $$-0.445730\pi$$
0.169670 + 0.985501i $$0.445730\pi$$
$$420$$ 0 0
$$421$$ −2.71585 −0.132363 −0.0661813 0.997808i $$-0.521082\pi$$
−0.0661813 + 0.997808i $$0.521082\pi$$
$$422$$ − 12.2302i − 0.595359i
$$423$$ − 1.40753i − 0.0684364i
$$424$$ −5.30833 −0.257795
$$425$$ 0 0
$$426$$ 10.3033 0.499197
$$427$$ − 25.4161i − 1.22997i
$$428$$ − 4.30833i − 0.208251i
$$429$$ 16.4111 0.792335
$$430$$ 0 0
$$431$$ 16.2978 0.785036 0.392518 0.919744i $$-0.371604\pi$$
0.392518 + 0.919744i $$0.371604\pi$$
$$432$$ − 4.91638i − 0.236540i
$$433$$ 10.1169i 0.486188i 0.970003 + 0.243094i $$0.0781623\pi$$
−0.970003 + 0.243094i $$0.921838\pi$$
$$434$$ 20.5033 0.984190
$$435$$ 0 0
$$436$$ −7.57331 −0.362696
$$437$$ − 10.3033i − 0.492874i
$$438$$ 27.3275i 1.30576i
$$439$$ −9.79445 −0.467464 −0.233732 0.972301i $$-0.575094\pi$$
−0.233732 + 0.972301i $$0.575094\pi$$
$$440$$ 0 0
$$441$$ −0.181370 −0.00863668
$$442$$ 21.4600i 1.02075i
$$443$$ 21.7980i 1.03566i 0.855485 + 0.517828i $$0.173259\pi$$
−0.855485 + 0.517828i $$0.826741\pi$$
$$444$$ −13.4061 −0.636224
$$445$$ 0 0
$$446$$ 11.2544 0.532913
$$447$$ 3.62721i 0.171561i
$$448$$ − 2.52444i − 0.119268i
$$449$$ 22.2686 1.05092 0.525459 0.850819i $$-0.323894\pi$$
0.525459 + 0.850819i $$0.323894\pi$$
$$450$$ 0 0
$$451$$ −24.4741 −1.15244
$$452$$ − 5.60806i − 0.263781i
$$453$$ 16.1361i 0.758138i
$$454$$ 19.7647 0.927605
$$455$$ 0 0
$$456$$ 3.28917 0.154029
$$457$$ − 29.8958i − 1.39847i −0.714894 0.699233i $$-0.753525\pi$$
0.714894 0.699233i $$-0.246475\pi$$
$$458$$ 23.7250i 1.10859i
$$459$$ 34.0036 1.58715
$$460$$ 0 0
$$461$$ 33.2927 1.55060 0.775299 0.631595i $$-0.217599\pi$$
0.775299 + 0.631595i $$0.217599\pi$$
$$462$$ − 13.3522i − 0.621200i
$$463$$ − 13.1184i − 0.609662i −0.952406 0.304831i $$-0.901400\pi$$
0.952406 0.304831i $$-0.0986000\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ 0 0
$$466$$ 24.0625 1.11467
$$467$$ − 4.15165i − 0.192115i −0.995376 0.0960577i $$-0.969377\pi$$
0.995376 0.0960577i $$-0.0306233\pi$$
$$468$$ − 0.897225i − 0.0414742i
$$469$$ 6.76830 0.312531
$$470$$ 0 0
$$471$$ −31.7194 −1.46155
$$472$$ 2.60806i 0.120046i
$$473$$ − 14.7244i − 0.677031i
$$474$$ 3.14663 0.144529
$$475$$ 0 0
$$476$$ 17.4600 0.800277
$$477$$ 1.53500i 0.0702829i
$$478$$ 4.31335i 0.197288i
$$479$$ −42.7144 −1.95167 −0.975835 0.218507i $$-0.929881\pi$$
−0.975835 + 0.218507i $$0.929881\pi$$
$$480$$ 0 0
$$481$$ 22.9355 1.04577
$$482$$ − 27.7144i − 1.26236i
$$483$$ 26.0100i 1.18350i
$$484$$ 2.49472 0.113396
$$485$$ 0 0
$$486$$ −2.99498 −0.135855
$$487$$ 18.3728i 0.832550i 0.909239 + 0.416275i $$0.136665\pi$$
−0.909239 + 0.416275i $$0.863335\pi$$
$$488$$ 10.0680i 0.455758i
$$489$$ 5.49472 0.248480
$$490$$ 0 0
$$491$$ −0.881639 −0.0397878 −0.0198939 0.999802i $$-0.506333\pi$$
−0.0198939 + 0.999802i $$0.506333\pi$$
$$492$$ 15.2197i 0.686156i
$$493$$ − 6.91638i − 0.311498i
$$494$$ −5.62721 −0.253180
$$495$$ 0 0
$$496$$ −8.12193 −0.364685
$$497$$ 14.3416i 0.643309i
$$498$$ 23.7980i 1.06641i
$$499$$ −32.4791 −1.45397 −0.726983 0.686656i $$-0.759078\pi$$
−0.726983 + 0.686656i $$0.759078\pi$$
$$500$$ 0 0
$$501$$ −43.4005 −1.93899
$$502$$ − 21.7542i − 0.970936i
$$503$$ − 32.6620i − 1.45632i −0.685405 0.728162i $$-0.740375\pi$$
0.685405 0.728162i $$-0.259625\pi$$
$$504$$ −0.729988 −0.0325163
$$505$$ 0 0
$$506$$ 16.5683 0.736550
$$507$$ − 6.11691i − 0.271661i
$$508$$ − 2.12193i − 0.0941455i
$$509$$ −27.3311 −1.21143 −0.605714 0.795683i $$-0.707112\pi$$
−0.605714 + 0.795683i $$0.707112\pi$$
$$510$$ 0 0
$$511$$ −38.0383 −1.68272
$$512$$ 1.00000i 0.0441942i
$$513$$ 8.91638i 0.393668i
$$514$$ −5.83276 −0.257272
$$515$$ 0 0
$$516$$ −9.15667 −0.403100
$$517$$ − 14.1955i − 0.624318i
$$518$$ − 18.6605i − 0.819895i
$$519$$ −20.8917 −0.917043
$$520$$ 0 0
$$521$$ 11.5400 0.505578 0.252789 0.967521i $$-0.418652\pi$$
0.252789 + 0.967521i $$0.418652\pi$$
$$522$$ 0.289169i 0.0126566i
$$523$$ 19.4056i 0.848546i 0.905534 + 0.424273i $$0.139470\pi$$
−0.905534 + 0.424273i $$0.860530\pi$$
$$524$$ −3.74055 −0.163407
$$525$$ 0 0
$$526$$ 20.8675 0.909866
$$527$$ − 56.1744i − 2.44699i
$$528$$ 5.28917i 0.230182i
$$529$$ −9.27504 −0.403262
$$530$$ 0 0
$$531$$ 0.754168 0.0327281
$$532$$ 4.57834i 0.198496i
$$533$$ − 26.0383i − 1.12784i
$$534$$ −9.49472 −0.410877
$$535$$ 0 0
$$536$$ −2.68111 −0.115806
$$537$$ 15.0680i 0.650234i
$$538$$ − 0.548618i − 0.0236526i
$$539$$ −1.82919 −0.0787889
$$540$$ 0 0
$$541$$ 26.7230 1.14891 0.574456 0.818536i $$-0.305214\pi$$
0.574456 + 0.818536i $$0.305214\pi$$
$$542$$ 5.00357i 0.214922i
$$543$$ − 32.1361i − 1.37909i
$$544$$ −6.91638 −0.296537
$$545$$ 0 0
$$546$$ 14.2056 0.607941
$$547$$ 20.7875i 0.888808i 0.895827 + 0.444404i $$0.146584\pi$$
−0.895827 + 0.444404i $$0.853416\pi$$
$$548$$ − 11.7980i − 0.503986i
$$549$$ 2.91136 0.124254
$$550$$ 0 0
$$551$$ 1.81361 0.0772622
$$552$$ − 10.3033i − 0.438538i
$$553$$ 4.37993i 0.186253i
$$554$$ −0.632236 −0.0268611
$$555$$ 0 0
$$556$$ 12.3083 0.521989
$$557$$ 11.0489i 0.468156i 0.972218 + 0.234078i $$0.0752071\pi$$
−0.972218 + 0.234078i $$0.924793\pi$$
$$558$$ 2.34861i 0.0994245i
$$559$$ 15.6655 0.662581
$$560$$ 0 0
$$561$$ −36.5819 −1.54449
$$562$$ 30.9653i 1.30619i
$$563$$ 27.3311i 1.15187i 0.817497 + 0.575933i $$0.195361\pi$$
−0.817497 + 0.575933i $$0.804639\pi$$
$$564$$ −8.82774 −0.371715
$$565$$ 0 0
$$566$$ −6.33804 −0.266408
$$567$$ − 24.6988i − 1.03725i
$$568$$ − 5.68111i − 0.238374i
$$569$$ 6.54359 0.274322 0.137161 0.990549i $$-0.456202\pi$$
0.137161 + 0.990549i $$0.456202\pi$$
$$570$$ 0 0
$$571$$ 46.4691 1.94467 0.972335 0.233589i $$-0.0750471\pi$$
0.972335 + 0.233589i $$0.0750471\pi$$
$$572$$ − 9.04888i − 0.378353i
$$573$$ − 31.1411i − 1.30094i
$$574$$ −21.1849 −0.884242
$$575$$ 0 0
$$576$$ 0.289169 0.0120487
$$577$$ 40.4585i 1.68431i 0.539235 + 0.842155i $$0.318713\pi$$
−0.539235 + 0.842155i $$0.681287\pi$$
$$578$$ − 30.8363i − 1.28262i
$$579$$ −12.8186 −0.532724
$$580$$ 0 0
$$581$$ −33.1255 −1.37428
$$582$$ 2.30330i 0.0954749i
$$583$$ 15.4811i 0.641162i
$$584$$ 15.0680 0.623520
$$585$$ 0 0
$$586$$ −28.4197 −1.17401
$$587$$ 15.1900i 0.626957i 0.949595 + 0.313478i $$0.101494\pi$$
−0.949595 + 0.313478i $$0.898506\pi$$
$$588$$ 1.13752i 0.0469104i
$$589$$ 14.7300 0.606939
$$590$$ 0 0
$$591$$ −29.2645 −1.20378
$$592$$ 7.39194i 0.303807i
$$593$$ 2.21611i 0.0910048i 0.998964 + 0.0455024i $$0.0144889\pi$$
−0.998964 + 0.0455024i $$0.985511\pi$$
$$594$$ −14.3380 −0.588297
$$595$$ 0 0
$$596$$ 2.00000 0.0819232
$$597$$ 14.9794i 0.613066i
$$598$$ 17.6272i 0.720830i
$$599$$ 39.8852 1.62967 0.814833 0.579696i $$-0.196829\pi$$
0.814833 + 0.579696i $$0.196829\pi$$
$$600$$ 0 0
$$601$$ 20.6897 0.843951 0.421975 0.906607i $$-0.361337\pi$$
0.421975 + 0.906607i $$0.361337\pi$$
$$602$$ − 12.7456i − 0.519471i
$$603$$ 0.775293i 0.0315724i
$$604$$ 8.89722 0.362023
$$605$$ 0 0
$$606$$ 9.77886 0.397239
$$607$$ − 16.6025i − 0.673875i −0.941527 0.336938i $$-0.890609\pi$$
0.941527 0.336938i $$-0.109391\pi$$
$$608$$ − 1.81361i − 0.0735515i
$$609$$ −4.57834 −0.185524
$$610$$ 0 0
$$611$$ 15.1028 0.610993
$$612$$ 2.00000i 0.0808452i
$$613$$ − 8.13607i − 0.328613i −0.986409 0.164306i $$-0.947461\pi$$
0.986409 0.164306i $$-0.0525385\pi$$
$$614$$ −22.6797 −0.915277
$$615$$ 0 0
$$616$$ −7.36222 −0.296632
$$617$$ 36.0822i 1.45261i 0.687371 + 0.726307i $$0.258765\pi$$
−0.687371 + 0.726307i $$0.741235\pi$$
$$618$$ 31.9406i 1.28484i
$$619$$ −13.9617 −0.561168 −0.280584 0.959830i $$-0.590528\pi$$
−0.280584 + 0.959830i $$0.590528\pi$$
$$620$$ 0 0
$$621$$ 27.9305 1.12081
$$622$$ − 9.08719i − 0.364363i
$$623$$ − 13.2161i − 0.529492i
$$624$$ −5.62721 −0.225269
$$625$$ 0 0
$$626$$ 25.0630 1.00172
$$627$$ − 9.59247i − 0.383086i
$$628$$ 17.4897i 0.697915i
$$629$$ −51.1255 −2.03851
$$630$$ 0 0
$$631$$ −5.49829 −0.218883 −0.109442 0.993993i $$-0.534906\pi$$
−0.109442 + 0.993993i $$0.534906\pi$$
$$632$$ − 1.73501i − 0.0690150i
$$633$$ 22.1809i 0.881610i
$$634$$ 14.9497 0.593727
$$635$$ 0 0
$$636$$ 9.62721 0.381744
$$637$$ − 1.94610i − 0.0771073i
$$638$$ 2.91638i 0.115461i
$$639$$ −1.64280 −0.0649881
$$640$$ 0 0
$$641$$ 18.5033 0.730837 0.365418 0.930843i $$-0.380926\pi$$
0.365418 + 0.930843i $$0.380926\pi$$
$$642$$ 7.81361i 0.308378i
$$643$$ − 25.3608i − 1.00013i −0.865988 0.500066i $$-0.833309\pi$$
0.865988 0.500066i $$-0.166691\pi$$
$$644$$ 14.3416 0.565139
$$645$$ 0 0
$$646$$ 12.5436 0.493521
$$647$$ 36.6761i 1.44189i 0.692994 + 0.720943i $$0.256291\pi$$
−0.692994 + 0.720943i $$0.743709\pi$$
$$648$$ 9.78389i 0.384347i
$$649$$ 7.60609 0.298565
$$650$$ 0 0
$$651$$ −37.1849 −1.45739
$$652$$ − 3.02972i − 0.118653i
$$653$$ − 4.47913i − 0.175282i −0.996152 0.0876410i $$-0.972067\pi$$
0.996152 0.0876410i $$-0.0279328\pi$$
$$654$$ 13.7350 0.537081
$$655$$ 0 0
$$656$$ 8.39194 0.327650
$$657$$ − 4.35720i − 0.169991i
$$658$$ − 12.2877i − 0.479025i
$$659$$ 16.0630 0.625726 0.312863 0.949798i $$-0.398712\pi$$
0.312863 + 0.949798i $$0.398712\pi$$
$$660$$ 0 0
$$661$$ −22.5244 −0.876099 −0.438050 0.898951i $$-0.644331\pi$$
−0.438050 + 0.898951i $$0.644331\pi$$
$$662$$ − 18.1270i − 0.704524i
$$663$$ − 38.9200i − 1.51153i
$$664$$ 13.1219 0.509230
$$665$$ 0 0
$$666$$ 2.13752 0.0828271
$$667$$ − 5.68111i − 0.219974i
$$668$$ 23.9305i 0.925899i
$$669$$ −20.4111 −0.789139
$$670$$ 0 0
$$671$$ 29.3622 1.13352
$$672$$ 4.57834i 0.176613i
$$673$$ 41.2474i 1.58997i 0.606628 + 0.794986i $$0.292522\pi$$
−0.606628 + 0.794986i $$0.707478\pi$$
$$674$$ −9.08362 −0.349888
$$675$$ 0 0
$$676$$ −3.37279 −0.129723
$$677$$ 17.1653i 0.659715i 0.944031 + 0.329857i $$0.107001\pi$$
−0.944031 + 0.329857i $$0.892999\pi$$
$$678$$ 10.1708i 0.390608i
$$679$$ −3.20607 −0.123038
$$680$$ 0 0
$$681$$ −35.8454 −1.37360
$$682$$ 23.6867i 0.907009i
$$683$$ 20.2786i 0.775939i 0.921672 + 0.387970i $$0.126823\pi$$
−0.921672 + 0.387970i $$0.873177\pi$$
$$684$$ −0.524438 −0.0200524
$$685$$ 0 0
$$686$$ −19.2544 −0.735137
$$687$$ − 43.0278i − 1.64161i
$$688$$ 5.04888i 0.192487i
$$689$$ −16.4705 −0.627478
$$690$$ 0 0
$$691$$ 22.2106 0.844930 0.422465 0.906379i $$-0.361165\pi$$
0.422465 + 0.906379i $$0.361165\pi$$
$$692$$ 11.5194i 0.437902i
$$693$$ 2.12892i 0.0808711i
$$694$$ 30.7194 1.16609
$$695$$ 0 0
$$696$$ 1.81361 0.0687446
$$697$$ 58.0419i 2.19849i
$$698$$ − 15.7789i − 0.597239i
$$699$$ −43.6399 −1.65061
$$700$$ 0 0
$$701$$ −11.4005 −0.430592 −0.215296 0.976549i $$-0.569072\pi$$
−0.215296 + 0.976549i $$0.569072\pi$$
$$702$$ − 15.2544i − 0.575741i
$$703$$ − 13.4061i − 0.505620i
$$704$$ 2.91638 0.109915
$$705$$ 0 0
$$706$$ −31.6797 −1.19228
$$707$$ 13.6116i 0.511918i
$$708$$ − 4.72999i − 0.177764i
$$709$$ −37.3311 −1.40200 −0.700999 0.713163i $$-0.747262\pi$$
−0.700999 + 0.713163i $$0.747262\pi$$
$$710$$ 0 0
$$711$$ −0.501711 −0.0188156
$$712$$ 5.23527i 0.196200i
$$713$$ − 46.1416i − 1.72802i
$$714$$ −31.6655 −1.18505
$$715$$ 0 0
$$716$$ 8.30833 0.310497
$$717$$ − 7.82272i − 0.292145i
$$718$$ 27.1013i 1.01141i
$$719$$ −18.5839 −0.693062 −0.346531 0.938039i $$-0.612640\pi$$
−0.346531 + 0.938039i $$0.612640\pi$$
$$720$$ 0 0
$$721$$ −44.4595 −1.65576
$$722$$ − 15.7108i − 0.584697i
$$723$$ 50.2630i 1.86930i
$$724$$ −17.7194 −0.658537
$$725$$ 0 0
$$726$$ −4.52444 −0.167918
$$727$$ − 29.8227i − 1.10606i −0.833160 0.553032i $$-0.813471\pi$$
0.833160 0.553032i $$-0.186529\pi$$
$$728$$ − 7.83276i − 0.290302i
$$729$$ −23.9200 −0.885924
$$730$$ 0 0
$$731$$ −34.9200 −1.29156
$$732$$ − 18.2594i − 0.674889i
$$733$$ 32.2056i 1.18954i 0.803896 + 0.594770i $$0.202757\pi$$
−0.803896 + 0.594770i $$0.797243\pi$$
$$734$$ −31.8953 −1.17728
$$735$$ 0 0
$$736$$ −5.68111 −0.209409
$$737$$ 7.81915i 0.288022i
$$738$$ − 2.42669i − 0.0893276i
$$739$$ −27.8328 −1.02384 −0.511922 0.859032i $$-0.671066\pi$$
−0.511922 + 0.859032i $$0.671066\pi$$
$$740$$ 0 0
$$741$$ 10.2056 0.374910
$$742$$ 13.4005i 0.491949i
$$743$$ − 31.4288i − 1.15301i −0.817093 0.576506i $$-0.804416\pi$$
0.817093 0.576506i $$-0.195584\pi$$
$$744$$ 14.7300 0.540028
$$745$$ 0 0
$$746$$ 28.0383 1.02656
$$747$$ − 3.79445i − 0.138832i
$$748$$ 20.1708i 0.737518i
$$749$$ −10.8761 −0.397404
$$750$$ 0 0
$$751$$ 14.4635 0.527782 0.263891 0.964552i $$-0.414994\pi$$
0.263891 + 0.964552i $$0.414994\pi$$
$$752$$ 4.86751i 0.177500i
$$753$$ 39.4535i 1.43777i
$$754$$ −3.10278 −0.112996
$$755$$ 0 0
$$756$$ −12.4111 −0.451387
$$757$$ − 34.0071i − 1.23601i −0.786174 0.618005i $$-0.787941\pi$$
0.786174 0.618005i $$-0.212059\pi$$
$$758$$ − 8.69525i − 0.315826i
$$759$$ −30.0484 −1.09069
$$760$$ 0 0
$$761$$ 13.0594 0.473404 0.236702 0.971582i $$-0.423933\pi$$
0.236702 + 0.971582i $$0.423933\pi$$
$$762$$ 3.84835i 0.139411i
$$763$$ 19.1184i 0.692131i
$$764$$ −17.1708 −0.621218
$$765$$ 0 0
$$766$$ 14.8917 0.538058
$$767$$ 8.09221i 0.292193i
$$768$$ − 1.81361i − 0.0654429i
$$769$$ −47.2525 −1.70397 −0.851984 0.523568i $$-0.824600\pi$$
−0.851984 + 0.523568i $$0.824600\pi$$
$$770$$ 0 0
$$771$$ 10.5783 0.380970
$$772$$ 7.06803i 0.254384i
$$773$$ 12.6675i 0.455618i 0.973706 + 0.227809i $$0.0731562\pi$$
−0.973706 + 0.227809i $$0.926844\pi$$
$$774$$ 1.45998 0.0524778
$$775$$ 0 0
$$776$$ 1.27001 0.0455908
$$777$$ 33.8428i 1.21410i
$$778$$ 1.93197i 0.0692644i
$$779$$ −15.2197 −0.545302
$$780$$ 0 0
$$781$$ −16.5683 −0.592860
$$782$$ − 39.2927i − 1.40511i
$$783$$ 4.91638i 0.175697i
$$784$$ 0.627213 0.0224005
$$785$$ 0 0
$$786$$ 6.78389 0.241973
$$787$$ − 30.0297i − 1.07044i −0.844711 0.535222i $$-0.820228\pi$$
0.844711 0.535222i $$-0.179772\pi$$
$$788$$ 16.1361i 0.574824i
$$789$$ −37.8454 −1.34733
$$790$$ 0 0
$$791$$ −14.1572 −0.503372
$$792$$ − 0.843326i − 0.0299663i
$$793$$ 31.2388i 1.10932i
$$794$$ −19.2983 −0.684870
$$795$$ 0 0
$$796$$ 8.25945 0.292748
$$797$$ 43.9094i 1.55535i 0.628666 + 0.777675i $$0.283601\pi$$
−0.628666 + 0.777675i $$0.716399\pi$$
$$798$$ − 8.30330i − 0.293934i
$$799$$ −33.6655 −1.19100
$$800$$ 0 0
$$801$$ 1.51388 0.0534902
$$802$$ 16.7250i 0.590579i
$$803$$ − 43.9441i − 1.55075i
$$804$$ 4.86248 0.171487
$$805$$ 0 0
$$806$$ −25.2005 −0.887651
$$807$$ 0.994977i 0.0350248i
$$808$$ − 5.39194i − 0.189688i
$$809$$ 53.3794 1.87672 0.938360 0.345659i $$-0.112345\pi$$
0.938360 + 0.345659i $$0.112345\pi$$
$$810$$ 0 0
$$811$$ −7.53806 −0.264697 −0.132348 0.991203i $$-0.542252\pi$$
−0.132348 + 0.991203i $$0.542252\pi$$
$$812$$ 2.52444i 0.0885904i
$$813$$ − 9.07451i − 0.318257i
$$814$$ 21.5577 0.755598
$$815$$ 0 0
$$816$$ 12.5436 0.439114
$$817$$ − 9.15667i − 0.320351i
$$818$$ 26.4842i 0.925997i
$$819$$ −2.26499 −0.0791451
$$820$$ 0 0
$$821$$ 4.08217 0.142469 0.0712343 0.997460i $$-0.477306\pi$$
0.0712343 + 0.997460i $$0.477306\pi$$
$$822$$ 21.3970i 0.746305i
$$823$$ − 34.2338i − 1.19332i −0.802496 0.596658i $$-0.796495\pi$$
0.802496 0.596658i $$-0.203505\pi$$
$$824$$ 17.6116 0.613530
$$825$$ 0 0
$$826$$ 6.58388 0.229082
$$827$$ − 45.9935i − 1.59935i −0.600432 0.799676i $$-0.705005\pi$$
0.600432 0.799676i $$-0.294995\pi$$
$$828$$ 1.64280i 0.0570912i
$$829$$ 6.30330 0.218923 0.109461 0.993991i $$-0.465087\pi$$
0.109461 + 0.993991i $$0.465087\pi$$
$$830$$ 0 0
$$831$$ 1.14663 0.0397761
$$832$$ 3.10278i 0.107569i
$$833$$ 4.33804i 0.150304i
$$834$$ −22.3225 −0.772964
$$835$$ 0 0
$$836$$ −5.28917 −0.182930
$$837$$ 39.9305i 1.38020i
$$838$$ 6.94610i 0.239949i
$$839$$ −40.6902 −1.40478 −0.702391 0.711791i $$-0.747884\pi$$
−0.702391 + 0.711791i $$0.747884\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ − 2.71585i − 0.0935945i
$$843$$ − 56.1588i − 1.93421i
$$844$$ 12.2302 0.420982
$$845$$ 0 0
$$846$$ 1.40753 0.0483919
$$847$$ − 6.29776i − 0.216394i
$$848$$ − 5.30833i − 0.182289i
$$849$$ 11.4947 0.394498
$$850$$ 0 0
$$851$$ −41.9945 −1.43955
$$852$$ 10.3033i 0.352985i
$$853$$ − 52.1250i − 1.78473i −0.451319 0.892363i $$-0.649046\pi$$
0.451319 0.892363i $$-0.350954\pi$$
$$854$$ 25.4161 0.869722
$$855$$ 0 0
$$856$$ 4.30833 0.147256
$$857$$ − 48.5608i − 1.65880i −0.558652 0.829402i $$-0.688681\pi$$
0.558652 0.829402i $$-0.311319\pi$$
$$858$$ 16.4111i 0.560266i
$$859$$ 43.6308 1.48866 0.744332 0.667810i $$-0.232768\pi$$
0.744332 + 0.667810i $$0.232768\pi$$
$$860$$ 0 0
$$861$$ 38.4211 1.30939
$$862$$ 16.2978i 0.555104i
$$863$$ − 10.5839i − 0.360279i −0.983641 0.180140i $$-0.942345\pi$$
0.983641 0.180140i $$-0.0576550\pi$$
$$864$$ 4.91638 0.167259
$$865$$ 0 0
$$866$$ −10.1169 −0.343787
$$867$$ 55.9250i 1.89931i
$$868$$ 20.5033i 0.695928i
$$869$$ −5.05995 −0.171647
$$870$$ 0 0
$$871$$ −8.31889 −0.281875
$$872$$ − 7.57331i − 0.256465i
$$873$$ − 0.367248i − 0.0124294i
$$874$$ 10.3033 0.348514
$$875$$ 0 0
$$876$$ −27.3275 −0.923310
$$877$$ 45.6938i 1.54297i 0.636248 + 0.771485i $$0.280486\pi$$
−0.636248 + 0.771485i $$0.719514\pi$$
$$878$$ − 9.79445i − 0.330547i
$$879$$ 51.5421 1.73847
$$880$$ 0 0
$$881$$ 4.55721 0.153536 0.0767682 0.997049i $$-0.475540\pi$$
0.0767682 + 0.997049i $$0.475540\pi$$
$$882$$ − 0.181370i − 0.00610705i
$$883$$ 45.6222i 1.53531i 0.640864 + 0.767654i $$0.278576\pi$$
−0.640864 + 0.767654i $$0.721424\pi$$
$$884$$ −21.4600 −0.721777
$$885$$ 0 0
$$886$$ −21.7980 −0.732319
$$887$$ 24.8122i 0.833111i 0.909110 + 0.416555i $$0.136763\pi$$
−0.909110 + 0.416555i $$0.863237\pi$$
$$888$$ − 13.4061i − 0.449878i
$$889$$ −5.35668 −0.179657
$$890$$ 0 0
$$891$$ 28.5335 0.955910
$$892$$ 11.2544i 0.376826i
$$893$$ − 8.82774i − 0.295409i
$$894$$ −3.62721 −0.121312
$$895$$ 0 0
$$896$$ 2.52444 0.0843356
$$897$$ − 31.9688i − 1.06741i
$$898$$ 22.2686i 0.743111i
$$899$$ 8.12193 0.270882
$$900$$ 0 0
$$901$$ 36.7144 1.22313
$$902$$ − 24.4741i − 0.814899i
$$903$$ 23.1155i 0.769234i
$$904$$ 5.60806 0.186521
$$905$$ 0 0
$$906$$ −16.1361 −0.536085
$$907$$ − 14.8277i − 0.492347i −0.969226 0.246174i $$-0.920827\pi$$
0.969226 0.246174i $$-0.0791733\pi$$
$$908$$ 19.7647i 0.655916i
$$909$$ −1.55918 −0.0517148
$$910$$ 0 0
$$911$$ 3.40753 0.112896 0.0564482 0.998406i $$-0.482022\pi$$
0.0564482 + 0.998406i $$0.482022\pi$$
$$912$$ 3.28917i 0.108915i
$$913$$ − 38.2686i − 1.26650i
$$914$$ 29.8958 0.988864
$$915$$ 0 0
$$916$$ −23.7250 −0.783895
$$917$$ 9.44279i 0.311828i
$$918$$ 34.0036i 1.12229i
$$919$$ −2.59392 −0.0855656 −0.0427828 0.999084i $$-0.513622\pi$$
−0.0427828 + 0.999084i $$0.513622\pi$$
$$920$$ 0 0
$$921$$ 41.1320 1.35534
$$922$$ 33.2927i 1.09644i
$$923$$ − 17.6272i − 0.580207i
$$924$$ 13.3522 0.439254
$$925$$ 0 0
$$926$$ 13.1184 0.431096
$$927$$ − 5.09273i − 0.167267i
$$928$$ − 1.00000i − 0.0328266i
$$929$$ −8.56420 −0.280982 −0.140491 0.990082i $$-0.544868\pi$$
−0.140491 + 0.990082i $$0.544868\pi$$
$$930$$ 0 0
$$931$$ −1.13752 −0.0372806
$$932$$ 24.0625i 0.788193i
$$933$$ 16.4806i 0.539550i
$$934$$ 4.15165 0.135846
$$935$$ 0 0
$$936$$ 0.897225 0.0293267
$$937$$ − 0.498289i − 0.0162784i −0.999967 0.00813920i $$-0.997409\pi$$
0.999967 0.00813920i $$-0.00259082\pi$$
$$938$$ 6.76830i 0.220993i
$$939$$ −45.4544 −1.48335
$$940$$ 0 0
$$941$$ 13.1028 0.427138 0.213569 0.976928i $$-0.431491\pi$$
0.213569 + 0.976928i $$0.431491\pi$$
$$942$$ − 31.7194i − 1.03347i
$$943$$ 47.6756i 1.55253i
$$944$$ −2.60806 −0.0848850
$$945$$ 0 0
$$946$$ 14.7244 0.478733
$$947$$ 5.67107i 0.184285i 0.995746 + 0.0921424i $$0.0293715\pi$$
−0.995746 + 0.0921424i $$0.970628\pi$$
$$948$$ 3.14663i 0.102198i
$$949$$ 46.7527 1.51766
$$950$$ 0 0
$$951$$ −27.1128 −0.879193
$$952$$ 17.4600i 0.565881i
$$953$$ 31.9824i 1.03601i 0.855377 + 0.518007i $$0.173326\pi$$
−0.855377 + 0.518007i $$0.826674\pi$$
$$954$$ −1.53500 −0.0496975
$$955$$ 0 0
$$956$$ −4.31335 −0.139504
$$957$$ − 5.28917i − 0.170975i
$$958$$ − 42.7144i − 1.38004i
$$959$$ −29.7834 −0.961755
$$960$$ 0 0
$$961$$ 34.9658 1.12793
$$962$$ 22.9355i 0.739471i
$$963$$ − 1.24583i − 0.0401464i
$$964$$ 27.7144 0.892621
$$965$$ 0 0
$$966$$ −26.0100 −0.836860
$$967$$ − 14.0484i − 0.451765i −0.974155 0.225882i $$-0.927473\pi$$
0.974155 0.225882i $$-0.0725265\pi$$
$$968$$ 2.49472i 0.0801833i
$$969$$ −22.7491 −0.730808
$$970$$ 0 0
$$971$$ 46.7436 1.50007 0.750037 0.661396i $$-0.230036\pi$$
0.750037 + 0.661396i $$0.230036\pi$$
$$972$$ − 2.99498i − 0.0960639i
$$973$$ − 31.0716i − 0.996110i
$$974$$ −18.3728 −0.588702
$$975$$ 0 0
$$976$$ −10.0680 −0.322270
$$977$$ 32.8016i 1.04942i 0.851282 + 0.524708i $$0.175825\pi$$
−0.851282 + 0.524708i $$0.824175\pi$$
$$978$$ 5.49472i 0.175702i
$$979$$ 15.2680 0.487969
$$980$$ 0 0
$$981$$ −2.18996 −0.0699202
$$982$$ − 0.881639i − 0.0281342i
$$983$$ 47.5225i 1.51573i 0.652411 + 0.757866i $$0.273758\pi$$
−0.652411 + 0.757866i $$0.726242\pi$$
$$984$$ −15.2197 −0.485186
$$985$$ 0 0
$$986$$ 6.91638 0.220262
$$987$$ 22.2851i 0.709342i
$$988$$ − 5.62721i − 0.179025i
$$989$$ −28.6832 −0.912074
$$990$$ 0 0
$$991$$ 3.00052 0.0953145 0.0476573 0.998864i $$-0.484824\pi$$
0.0476573 + 0.998864i $$0.484824\pi$$
$$992$$ − 8.12193i − 0.257872i
$$993$$ 32.8752i 1.04326i
$$994$$ −14.3416 −0.454888
$$995$$ 0 0
$$996$$ −23.7980 −0.754069
$$997$$ − 42.3502i − 1.34124i −0.741799 0.670622i $$-0.766027\pi$$
0.741799 0.670622i $$-0.233973\pi$$
$$998$$ − 32.4791i − 1.02811i
$$999$$ 36.3416 1.14980
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 1450.2.b.k.349.4 6
5.2 odd 4 1450.2.a.q.1.1 3
5.3 odd 4 1450.2.a.s.1.3 yes 3
5.4 even 2 inner 1450.2.b.k.349.3 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1450.2.a.q.1.1 3 5.2 odd 4
1450.2.a.s.1.3 yes 3 5.3 odd 4
1450.2.b.k.349.3 6 5.4 even 2 inner
1450.2.b.k.349.4 6 1.1 even 1 trivial