# Properties

 Label 950.2.b.g.799.3 Level $950$ Weight $2$ Character 950.799 Analytic conductor $7.586$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Learn more

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: $$6$$ Coefficient field: 6.0.4227136.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 9x^{4} + 22x^{2} + 9$$ x^6 + 9*x^4 + 22*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.3 Root $$-0.713538i$$ of defining polynomial Character $$\chi$$ $$=$$ 950.799 Dual form 950.2.b.g.799.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +2.77733i q^{3} -1.00000 q^{4} +2.77733 q^{6} +4.69527i q^{7} +1.00000i q^{8} -4.71354 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +2.77733i q^{3} -1.00000 q^{4} +2.77733 q^{6} +4.69527i q^{7} +1.00000i q^{8} -4.71354 q^{9} +6.40880 q^{11} -2.77733i q^{12} +1.06379i q^{13} +4.69527 q^{14} +1.00000 q^{16} -1.91794i q^{17} +4.71354i q^{18} +1.00000 q^{19} -13.0403 q^{21} -6.40880i q^{22} +1.79560i q^{23} -2.77733 q^{24} +1.06379 q^{26} -4.75905i q^{27} -4.69527i q^{28} -2.93621 q^{29} -5.55465 q^{31} -1.00000i q^{32} +17.7993i q^{33} -1.91794 q^{34} +4.71354 q^{36} +11.4088i q^{37} -1.00000i q^{38} -2.95449 q^{39} -1.14585 q^{41} +13.0403i q^{42} -3.55465i q^{43} -6.40880 q^{44} +1.79560 q^{46} -10.8359i q^{47} +2.77733i q^{48} -15.0455 q^{49} +5.32674 q^{51} -1.06379i q^{52} -8.69527i q^{53} -4.75905 q^{54} -4.69527 q^{56} +2.77733i q^{57} +2.93621i q^{58} +5.63148 q^{59} -3.39053 q^{61} +5.55465i q^{62} -22.1313i q^{63} -1.00000 q^{64} +17.7993 q^{66} +8.82284i q^{67} +1.91794i q^{68} -4.98696 q^{69} -1.42708 q^{71} -4.71354i q^{72} -12.6132i q^{73} +11.4088 q^{74} -1.00000 q^{76} +30.0910i q^{77} +2.95449i q^{78} +1.96345 q^{79} -0.923174 q^{81} +1.14585i q^{82} +16.2447i q^{83} +13.0403 q^{84} -3.55465 q^{86} -8.15482i q^{87} +6.40880i q^{88} +10.0000 q^{89} -4.99477 q^{91} -1.79560i q^{92} -15.4271i q^{93} -10.8359 q^{94} +2.77733 q^{96} +14.9452i q^{97} +15.0455i q^{98} -30.2081 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} + 4 q^{6} - 26 q^{9}+O(q^{10})$$ 6 * q - 6 * q^4 + 4 * q^6 - 26 * q^9 $$6 q - 6 q^{4} + 4 q^{6} - 26 q^{9} + 4 q^{11} - 4 q^{14} + 6 q^{16} + 6 q^{19} - 22 q^{21} - 4 q^{24} - 4 q^{26} - 28 q^{29} - 8 q^{31} + 8 q^{34} + 26 q^{36} - 58 q^{39} - 16 q^{41} - 4 q^{44} + 28 q^{46} - 50 q^{49} - 22 q^{51} + 14 q^{54} + 4 q^{56} + 12 q^{59} + 44 q^{61} - 6 q^{64} + 8 q^{66} - 16 q^{69} - 4 q^{71} + 34 q^{74} - 6 q^{76} - 48 q^{79} - 2 q^{81} + 22 q^{84} + 4 q^{86} + 60 q^{89} - 14 q^{91} - 26 q^{94} + 4 q^{96} - 48 q^{99}+O(q^{100})$$ 6 * q - 6 * q^4 + 4 * q^6 - 26 * q^9 + 4 * q^11 - 4 * q^14 + 6 * q^16 + 6 * q^19 - 22 * q^21 - 4 * q^24 - 4 * q^26 - 28 * q^29 - 8 * q^31 + 8 * q^34 + 26 * q^36 - 58 * q^39 - 16 * q^41 - 4 * q^44 + 28 * q^46 - 50 * q^49 - 22 * q^51 + 14 * q^54 + 4 * q^56 + 12 * q^59 + 44 * q^61 - 6 * q^64 + 8 * q^66 - 16 * q^69 - 4 * q^71 + 34 * q^74 - 6 * q^76 - 48 * q^79 - 2 * q^81 + 22 * q^84 + 4 * q^86 + 60 * q^89 - 14 * q^91 - 26 * q^94 + 4 * q^96 - 48 * 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
$$3$$ 2.77733i 1.60349i 0.597666 + 0.801745i $$0.296095\pi$$
−0.597666 + 0.801745i $$0.703905\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 2.77733 1.13384
$$7$$ 4.69527i 1.77464i 0.461151 + 0.887322i $$0.347437\pi$$
−0.461151 + 0.887322i $$0.652563\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −4.71354 −1.57118
$$10$$ 0 0
$$11$$ 6.40880 1.93233 0.966163 0.257931i $$-0.0830406\pi$$
0.966163 + 0.257931i $$0.0830406\pi$$
$$12$$ − 2.77733i − 0.801745i
$$13$$ 1.06379i 0.295042i 0.989059 + 0.147521i $$0.0471293\pi$$
−0.989059 + 0.147521i $$0.952871\pi$$
$$14$$ 4.69527 1.25486
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 1.91794i − 0.465169i −0.972576 0.232584i $$-0.925282\pi$$
0.972576 0.232584i $$-0.0747182\pi$$
$$18$$ 4.71354i 1.11099i
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −13.0403 −2.84562
$$22$$ − 6.40880i − 1.36636i
$$23$$ 1.79560i 0.374408i 0.982321 + 0.187204i $$0.0599426\pi$$
−0.982321 + 0.187204i $$0.940057\pi$$
$$24$$ −2.77733 −0.566919
$$25$$ 0 0
$$26$$ 1.06379 0.208626
$$27$$ − 4.75905i − 0.915880i
$$28$$ − 4.69527i − 0.887322i
$$29$$ −2.93621 −0.545241 −0.272620 0.962122i $$-0.587890\pi$$
−0.272620 + 0.962122i $$0.587890\pi$$
$$30$$ 0 0
$$31$$ −5.55465 −0.997645 −0.498822 0.866704i $$-0.666234\pi$$
−0.498822 + 0.866704i $$0.666234\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 17.7993i 3.09847i
$$34$$ −1.91794 −0.328924
$$35$$ 0 0
$$36$$ 4.71354 0.785590
$$37$$ 11.4088i 1.87560i 0.347182 + 0.937798i $$0.387139\pi$$
−0.347182 + 0.937798i $$0.612861\pi$$
$$38$$ − 1.00000i − 0.162221i
$$39$$ −2.95449 −0.473096
$$40$$ 0 0
$$41$$ −1.14585 −0.178951 −0.0894757 0.995989i $$-0.528519\pi$$
−0.0894757 + 0.995989i $$0.528519\pi$$
$$42$$ 13.0403i 2.01216i
$$43$$ − 3.55465i − 0.542079i −0.962568 0.271040i $$-0.912633\pi$$
0.962568 0.271040i $$-0.0873674\pi$$
$$44$$ −6.40880 −0.966163
$$45$$ 0 0
$$46$$ 1.79560 0.264747
$$47$$ − 10.8359i − 1.58058i −0.612736 0.790288i $$-0.709931\pi$$
0.612736 0.790288i $$-0.290069\pi$$
$$48$$ 2.77733i 0.400872i
$$49$$ −15.0455 −2.14936
$$50$$ 0 0
$$51$$ 5.32674 0.745893
$$52$$ − 1.06379i − 0.147521i
$$53$$ − 8.69527i − 1.19439i −0.802097 0.597193i $$-0.796283\pi$$
0.802097 0.597193i $$-0.203717\pi$$
$$54$$ −4.75905 −0.647625
$$55$$ 0 0
$$56$$ −4.69527 −0.627431
$$57$$ 2.77733i 0.367866i
$$58$$ 2.93621i 0.385544i
$$59$$ 5.63148 0.733156 0.366578 0.930387i $$-0.380529\pi$$
0.366578 + 0.930387i $$0.380529\pi$$
$$60$$ 0 0
$$61$$ −3.39053 −0.434113 −0.217056 0.976159i $$-0.569646\pi$$
−0.217056 + 0.976159i $$0.569646\pi$$
$$62$$ 5.55465i 0.705441i
$$63$$ − 22.1313i − 2.78828i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 17.7993 2.19095
$$67$$ 8.82284i 1.07788i 0.842344 + 0.538941i $$0.181175\pi$$
−0.842344 + 0.538941i $$0.818825\pi$$
$$68$$ 1.91794i 0.232584i
$$69$$ −4.98696 −0.600360
$$70$$ 0 0
$$71$$ −1.42708 −0.169363 −0.0846814 0.996408i $$-0.526987\pi$$
−0.0846814 + 0.996408i $$0.526987\pi$$
$$72$$ − 4.71354i − 0.555496i
$$73$$ − 12.6132i − 1.47626i −0.674656 0.738132i $$-0.735708\pi$$
0.674656 0.738132i $$-0.264292\pi$$
$$74$$ 11.4088 1.32625
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 30.0910i 3.42919i
$$78$$ 2.95449i 0.334530i
$$79$$ 1.96345 0.220906 0.110453 0.993881i $$-0.464770\pi$$
0.110453 + 0.993881i $$0.464770\pi$$
$$80$$ 0 0
$$81$$ −0.923174 −0.102575
$$82$$ 1.14585i 0.126538i
$$83$$ 16.2447i 1.78309i 0.452937 + 0.891543i $$0.350376\pi$$
−0.452937 + 0.891543i $$0.649624\pi$$
$$84$$ 13.0403 1.42281
$$85$$ 0 0
$$86$$ −3.55465 −0.383308
$$87$$ − 8.15482i − 0.874288i
$$88$$ 6.40880i 0.683181i
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −4.99477 −0.523594
$$92$$ − 1.79560i − 0.187204i
$$93$$ − 15.4271i − 1.59971i
$$94$$ −10.8359 −1.11764
$$95$$ 0 0
$$96$$ 2.77733 0.283460
$$97$$ 14.9452i 1.51745i 0.651409 + 0.758727i $$0.274178\pi$$
−0.651409 + 0.758727i $$0.725822\pi$$
$$98$$ 15.0455i 1.51983i
$$99$$ −30.2081 −3.03603
$$100$$ 0 0
$$101$$ 10.5364 1.04841 0.524204 0.851592i $$-0.324363\pi$$
0.524204 + 0.851592i $$0.324363\pi$$
$$102$$ − 5.32674i − 0.527426i
$$103$$ − 16.9817i − 1.67326i −0.547769 0.836630i $$-0.684523\pi$$
0.547769 0.836630i $$-0.315477\pi$$
$$104$$ −1.06379 −0.104313
$$105$$ 0 0
$$106$$ −8.69527 −0.844559
$$107$$ 1.79036i 0.173081i 0.996248 + 0.0865405i $$0.0275812\pi$$
−0.996248 + 0.0865405i $$0.972419\pi$$
$$108$$ 4.75905i 0.457940i
$$109$$ −2.41404 −0.231223 −0.115611 0.993295i $$-0.536883\pi$$
−0.115611 + 0.993295i $$0.536883\pi$$
$$110$$ 0 0
$$111$$ −31.6860 −3.00750
$$112$$ 4.69527i 0.443661i
$$113$$ 7.14585i 0.672225i 0.941822 + 0.336112i $$0.109112\pi$$
−0.941822 + 0.336112i $$0.890888\pi$$
$$114$$ 2.77733 0.260120
$$115$$ 0 0
$$116$$ 2.93621 0.272620
$$117$$ − 5.01420i − 0.463563i
$$118$$ − 5.63148i − 0.518420i
$$119$$ 9.00523 0.825508
$$120$$ 0 0
$$121$$ 30.0728 2.73389
$$122$$ 3.39053i 0.306964i
$$123$$ − 3.18239i − 0.286947i
$$124$$ 5.55465 0.498822
$$125$$ 0 0
$$126$$ −22.1313 −1.97161
$$127$$ − 9.26295i − 0.821954i −0.911646 0.410977i $$-0.865188\pi$$
0.911646 0.410977i $$-0.134812\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 9.87242 0.869219
$$130$$ 0 0
$$131$$ 11.5181 1.00634 0.503171 0.864187i $$-0.332167\pi$$
0.503171 + 0.864187i $$0.332167\pi$$
$$132$$ − 17.7993i − 1.54923i
$$133$$ 4.69527i 0.407131i
$$134$$ 8.82284 0.762177
$$135$$ 0 0
$$136$$ 1.91794 0.164462
$$137$$ − 15.1041i − 1.29043i −0.764002 0.645214i $$-0.776768\pi$$
0.764002 0.645214i $$-0.223232\pi$$
$$138$$ 4.98696i 0.424518i
$$139$$ 0.700500 0.0594156 0.0297078 0.999559i $$-0.490542\pi$$
0.0297078 + 0.999559i $$0.490542\pi$$
$$140$$ 0 0
$$141$$ 30.0948 2.53444
$$142$$ 1.42708i 0.119758i
$$143$$ 6.81761i 0.570117i
$$144$$ −4.71354 −0.392795
$$145$$ 0 0
$$146$$ −12.6132 −1.04388
$$147$$ − 41.7863i − 3.44648i
$$148$$ − 11.4088i − 0.937798i
$$149$$ −12.9452 −1.06051 −0.530255 0.847838i $$-0.677904\pi$$
−0.530255 + 0.847838i $$0.677904\pi$$
$$150$$ 0 0
$$151$$ 5.70830 0.464535 0.232268 0.972652i $$-0.425385\pi$$
0.232268 + 0.972652i $$0.425385\pi$$
$$152$$ 1.00000i 0.0811107i
$$153$$ 9.04028i 0.730863i
$$154$$ 30.0910 2.42480
$$155$$ 0 0
$$156$$ 2.95449 0.236548
$$157$$ − 9.26295i − 0.739264i −0.929178 0.369632i $$-0.879484\pi$$
0.929178 0.369632i $$-0.120516\pi$$
$$158$$ − 1.96345i − 0.156204i
$$159$$ 24.1496 1.91519
$$160$$ 0 0
$$161$$ −8.43081 −0.664441
$$162$$ 0.923174i 0.0725314i
$$163$$ 4.28123i 0.335332i 0.985844 + 0.167666i $$0.0536230\pi$$
−0.985844 + 0.167666i $$0.946377\pi$$
$$164$$ 1.14585 0.0894757
$$165$$ 0 0
$$166$$ 16.2447 1.26083
$$167$$ 15.2264i 1.17825i 0.808040 + 0.589127i $$0.200528\pi$$
−0.808040 + 0.589127i $$0.799472\pi$$
$$168$$ − 13.0403i − 1.00608i
$$169$$ 11.8684 0.912950
$$170$$ 0 0
$$171$$ −4.71354 −0.360453
$$172$$ 3.55465i 0.271040i
$$173$$ − 12.2630i − 0.932335i −0.884696 0.466168i $$-0.845634\pi$$
0.884696 0.466168i $$-0.154366\pi$$
$$174$$ −8.15482 −0.618215
$$175$$ 0 0
$$176$$ 6.40880 0.483082
$$177$$ 15.6404i 1.17561i
$$178$$ − 10.0000i − 0.749532i
$$179$$ 1.57292 0.117566 0.0587829 0.998271i $$-0.481278\pi$$
0.0587829 + 0.998271i $$0.481278\pi$$
$$180$$ 0 0
$$181$$ 11.4088 0.848010 0.424005 0.905660i $$-0.360624\pi$$
0.424005 + 0.905660i $$0.360624\pi$$
$$182$$ 4.99477i 0.370237i
$$183$$ − 9.41661i − 0.696096i
$$184$$ −1.79560 −0.132373
$$185$$ 0 0
$$186$$ −15.4271 −1.13117
$$187$$ − 12.2917i − 0.898858i
$$188$$ 10.8359i 0.790288i
$$189$$ 22.3450 1.62536
$$190$$ 0 0
$$191$$ −6.35805 −0.460053 −0.230026 0.973184i $$-0.573881\pi$$
−0.230026 + 0.973184i $$0.573881\pi$$
$$192$$ − 2.77733i − 0.200436i
$$193$$ 14.4998i 1.04372i 0.853031 + 0.521860i $$0.174762\pi$$
−0.853031 + 0.521860i $$0.825238\pi$$
$$194$$ 14.9452 1.07300
$$195$$ 0 0
$$196$$ 15.0455 1.07468
$$197$$ 5.14585i 0.366627i 0.983055 + 0.183313i $$0.0586823\pi$$
−0.983055 + 0.183313i $$0.941318\pi$$
$$198$$ 30.2081i 2.14680i
$$199$$ −3.87766 −0.274880 −0.137440 0.990510i $$-0.543887\pi$$
−0.137440 + 0.990510i $$0.543887\pi$$
$$200$$ 0 0
$$201$$ −24.5039 −1.72837
$$202$$ − 10.5364i − 0.741337i
$$203$$ − 13.7863i − 0.967608i
$$204$$ −5.32674 −0.372947
$$205$$ 0 0
$$206$$ −16.9817 −1.18317
$$207$$ − 8.46362i − 0.588262i
$$208$$ 1.06379i 0.0737604i
$$209$$ 6.40880 0.443306
$$210$$ 0 0
$$211$$ 17.1496 1.18063 0.590313 0.807174i $$-0.299004\pi$$
0.590313 + 0.807174i $$0.299004\pi$$
$$212$$ 8.69527i 0.597193i
$$213$$ − 3.96345i − 0.271571i
$$214$$ 1.79036 0.122387
$$215$$ 0 0
$$216$$ 4.75905 0.323813
$$217$$ − 26.0806i − 1.77046i
$$218$$ 2.41404i 0.163499i
$$219$$ 35.0310 2.36717
$$220$$ 0 0
$$221$$ 2.04028 0.137244
$$222$$ 31.6860i 2.12662i
$$223$$ 7.84635i 0.525430i 0.964873 + 0.262715i $$0.0846180\pi$$
−0.964873 + 0.262715i $$0.915382\pi$$
$$224$$ 4.69527 0.313716
$$225$$ 0 0
$$226$$ 7.14585 0.475335
$$227$$ − 6.28646i − 0.417247i −0.977996 0.208624i $$-0.933102\pi$$
0.977996 0.208624i $$-0.0668983\pi$$
$$228$$ − 2.77733i − 0.183933i
$$229$$ 3.14585 0.207884 0.103942 0.994583i $$-0.466854\pi$$
0.103942 + 0.994583i $$0.466854\pi$$
$$230$$ 0 0
$$231$$ −83.5726 −5.49867
$$232$$ − 2.93621i − 0.192772i
$$233$$ 0.182394i 0.0119490i 0.999982 + 0.00597451i $$0.00190176\pi$$
−0.999982 + 0.00597451i $$0.998098\pi$$
$$234$$ −5.01420 −0.327789
$$235$$ 0 0
$$236$$ −5.63148 −0.366578
$$237$$ 5.45315i 0.354220i
$$238$$ − 9.00523i − 0.583723i
$$239$$ 11.5039 0.744126 0.372063 0.928208i $$-0.378651\pi$$
0.372063 + 0.928208i $$0.378651\pi$$
$$240$$ 0 0
$$241$$ −0.445349 −0.0286874 −0.0143437 0.999897i $$-0.504566\pi$$
−0.0143437 + 0.999897i $$0.504566\pi$$
$$242$$ − 30.0728i − 1.93315i
$$243$$ − 16.8411i − 1.08036i
$$244$$ 3.39053 0.217056
$$245$$ 0 0
$$246$$ −3.18239 −0.202902
$$247$$ 1.06379i 0.0676872i
$$248$$ − 5.55465i − 0.352721i
$$249$$ −45.1168 −2.85916
$$250$$ 0 0
$$251$$ 13.2630 0.837150 0.418575 0.908182i $$-0.362530\pi$$
0.418575 + 0.908182i $$0.362530\pi$$
$$252$$ 22.1313i 1.39414i
$$253$$ 11.5076i 0.723479i
$$254$$ −9.26295 −0.581209
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 23.4816i 1.46474i 0.680907 + 0.732370i $$0.261586\pi$$
−0.680907 + 0.732370i $$0.738414\pi$$
$$258$$ − 9.87242i − 0.614630i
$$259$$ −53.5674 −3.32851
$$260$$ 0 0
$$261$$ 13.8399 0.856671
$$262$$ − 11.5181i − 0.711591i
$$263$$ 27.9269i 1.72205i 0.508565 + 0.861023i $$0.330176\pi$$
−0.508565 + 0.861023i $$0.669824\pi$$
$$264$$ −17.7993 −1.09547
$$265$$ 0 0
$$266$$ 4.69527 0.287885
$$267$$ 27.7733i 1.69970i
$$268$$ − 8.82284i − 0.538941i
$$269$$ −16.4816 −1.00490 −0.502449 0.864607i $$-0.667568\pi$$
−0.502449 + 0.864607i $$0.667568\pi$$
$$270$$ 0 0
$$271$$ −1.46736 −0.0891356 −0.0445678 0.999006i $$-0.514191\pi$$
−0.0445678 + 0.999006i $$0.514191\pi$$
$$272$$ − 1.91794i − 0.116292i
$$273$$ − 13.8721i − 0.839577i
$$274$$ −15.1041 −0.912470
$$275$$ 0 0
$$276$$ 4.98696 0.300180
$$277$$ − 15.8359i − 0.951486i −0.879584 0.475743i $$-0.842179\pi$$
0.879584 0.475743i $$-0.157821\pi$$
$$278$$ − 0.700500i − 0.0420132i
$$279$$ 26.1821 1.56748
$$280$$ 0 0
$$281$$ −6.25515 −0.373151 −0.186576 0.982441i $$-0.559739\pi$$
−0.186576 + 0.982441i $$0.559739\pi$$
$$282$$ − 30.0948i − 1.79212i
$$283$$ − 16.7005i − 0.992742i −0.868111 0.496371i $$-0.834666\pi$$
0.868111 0.496371i $$-0.165334\pi$$
$$284$$ 1.42708 0.0846814
$$285$$ 0 0
$$286$$ 6.81761 0.403134
$$287$$ − 5.38006i − 0.317575i
$$288$$ 4.71354i 0.277748i
$$289$$ 13.3215 0.783618
$$290$$ 0 0
$$291$$ −41.5076 −2.43322
$$292$$ 12.6132i 0.738132i
$$293$$ 0.644516i 0.0376530i 0.999823 + 0.0188265i $$0.00599302\pi$$
−0.999823 + 0.0188265i $$0.994007\pi$$
$$294$$ −41.7863 −2.43703
$$295$$ 0 0
$$296$$ −11.4088 −0.663123
$$297$$ − 30.4998i − 1.76978i
$$298$$ 12.9452i 0.749894i
$$299$$ −1.91014 −0.110466
$$300$$ 0 0
$$301$$ 16.6900 0.961997
$$302$$ − 5.70830i − 0.328476i
$$303$$ 29.2630i 1.68111i
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 9.04028 0.516798
$$307$$ − 26.6457i − 1.52075i −0.649485 0.760375i $$-0.725015\pi$$
0.649485 0.760375i $$-0.274985\pi$$
$$308$$ − 30.0910i − 1.71460i
$$309$$ 47.1638 2.68305
$$310$$ 0 0
$$311$$ 11.5494 0.654907 0.327454 0.944867i $$-0.393809\pi$$
0.327454 + 0.944867i $$0.393809\pi$$
$$312$$ − 2.95449i − 0.167265i
$$313$$ − 23.0638i − 1.30364i −0.758373 0.651821i $$-0.774005\pi$$
0.758373 0.651821i $$-0.225995\pi$$
$$314$$ −9.26295 −0.522739
$$315$$ 0 0
$$316$$ −1.96345 −0.110453
$$317$$ 13.9232i 0.782003i 0.920390 + 0.391002i $$0.127871\pi$$
−0.920390 + 0.391002i $$0.872129\pi$$
$$318$$ − 24.1496i − 1.35424i
$$319$$ −18.8176 −1.05358
$$320$$ 0 0
$$321$$ −4.97242 −0.277534
$$322$$ 8.43081i 0.469831i
$$323$$ − 1.91794i − 0.106717i
$$324$$ 0.923174 0.0512874
$$325$$ 0 0
$$326$$ 4.28123 0.237115
$$327$$ − 6.70457i − 0.370763i
$$328$$ − 1.14585i − 0.0632689i
$$329$$ 50.8773 2.80496
$$330$$ 0 0
$$331$$ −0.735546 −0.0404292 −0.0202146 0.999796i $$-0.506435\pi$$
−0.0202146 + 0.999796i $$0.506435\pi$$
$$332$$ − 16.2447i − 0.891543i
$$333$$ − 53.7758i − 2.94690i
$$334$$ 15.2264 0.833152
$$335$$ 0 0
$$336$$ −13.0403 −0.711406
$$337$$ 2.28123i 0.124266i 0.998068 + 0.0621332i $$0.0197904\pi$$
−0.998068 + 0.0621332i $$0.980210\pi$$
$$338$$ − 11.8684i − 0.645553i
$$339$$ −19.8463 −1.07791
$$340$$ 0 0
$$341$$ −35.5987 −1.92778
$$342$$ 4.71354i 0.254879i
$$343$$ − 37.7758i − 2.03970i
$$344$$ 3.55465 0.191654
$$345$$ 0 0
$$346$$ −12.2630 −0.659261
$$347$$ 2.56246i 0.137560i 0.997632 + 0.0687799i $$0.0219106\pi$$
−0.997632 + 0.0687799i $$0.978089\pi$$
$$348$$ 8.15482i 0.437144i
$$349$$ 5.67176 0.303602 0.151801 0.988411i $$-0.451493\pi$$
0.151801 + 0.988411i $$0.451493\pi$$
$$350$$ 0 0
$$351$$ 5.06262 0.270223
$$352$$ − 6.40880i − 0.341590i
$$353$$ − 23.6665i − 1.25964i −0.776740 0.629821i $$-0.783128\pi$$
0.776740 0.629821i $$-0.216872\pi$$
$$354$$ 15.6404 0.831280
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 25.0105i 1.32369i
$$358$$ − 1.57292i − 0.0831316i
$$359$$ −31.9724 −1.68744 −0.843720 0.536784i $$-0.819639\pi$$
−0.843720 + 0.536784i $$0.819639\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 11.4088i − 0.599633i
$$363$$ 83.5218i 4.38376i
$$364$$ 4.99477 0.261797
$$365$$ 0 0
$$366$$ −9.41661 −0.492214
$$367$$ − 1.14585i − 0.0598128i −0.999553 0.0299064i $$-0.990479\pi$$
0.999553 0.0299064i $$-0.00952092\pi$$
$$368$$ 1.79560i 0.0936020i
$$369$$ 5.40100 0.281165
$$370$$ 0 0
$$371$$ 40.8266 2.11961
$$372$$ 15.4271i 0.799857i
$$373$$ 32.8579i 1.70132i 0.525719 + 0.850658i $$0.323796\pi$$
−0.525719 + 0.850658i $$0.676204\pi$$
$$374$$ −12.2917 −0.635588
$$375$$ 0 0
$$376$$ 10.8359 0.558818
$$377$$ − 3.12351i − 0.160869i
$$378$$ − 22.3450i − 1.14930i
$$379$$ 18.2865 0.939312 0.469656 0.882849i $$-0.344378\pi$$
0.469656 + 0.882849i $$0.344378\pi$$
$$380$$ 0 0
$$381$$ 25.7262 1.31800
$$382$$ 6.35805i 0.325306i
$$383$$ 20.8542i 1.06560i 0.846242 + 0.532799i $$0.178860\pi$$
−0.846242 + 0.532799i $$0.821140\pi$$
$$384$$ −2.77733 −0.141730
$$385$$ 0 0
$$386$$ 14.4998 0.738022
$$387$$ 16.7550i 0.851704i
$$388$$ − 14.9452i − 0.758727i
$$389$$ 24.9086 1.26292 0.631459 0.775409i $$-0.282456\pi$$
0.631459 + 0.775409i $$0.282456\pi$$
$$390$$ 0 0
$$391$$ 3.44385 0.174163
$$392$$ − 15.0455i − 0.759913i
$$393$$ 31.9895i 1.61366i
$$394$$ 5.14585 0.259244
$$395$$ 0 0
$$396$$ 30.2081 1.51802
$$397$$ 24.7445i 1.24189i 0.783853 + 0.620946i $$0.213251\pi$$
−0.783853 + 0.620946i $$0.786749\pi$$
$$398$$ 3.87766i 0.194369i
$$399$$ −13.0403 −0.652831
$$400$$ 0 0
$$401$$ 15.6457 0.781308 0.390654 0.920538i $$-0.372249\pi$$
0.390654 + 0.920538i $$0.372249\pi$$
$$402$$ 24.5039i 1.22214i
$$403$$ − 5.90897i − 0.294347i
$$404$$ −10.5364 −0.524204
$$405$$ 0 0
$$406$$ −13.7863 −0.684202
$$407$$ 73.1168i 3.62426i
$$408$$ 5.32674i 0.263713i
$$409$$ −30.5804 −1.51210 −0.756052 0.654512i $$-0.772874\pi$$
−0.756052 + 0.654512i $$0.772874\pi$$
$$410$$ 0 0
$$411$$ 41.9489 2.06919
$$412$$ 16.9817i 0.836630i
$$413$$ 26.4413i 1.30109i
$$414$$ −8.46362 −0.415964
$$415$$ 0 0
$$416$$ 1.06379 0.0521565
$$417$$ 1.94552i 0.0952723i
$$418$$ − 6.40880i − 0.313465i
$$419$$ −1.96345 −0.0959210 −0.0479605 0.998849i $$-0.515272\pi$$
−0.0479605 + 0.998849i $$0.515272\pi$$
$$420$$ 0 0
$$421$$ −25.5949 −1.24742 −0.623710 0.781656i $$-0.714376\pi$$
−0.623710 + 0.781656i $$0.714376\pi$$
$$422$$ − 17.1496i − 0.834829i
$$423$$ 51.0753i 2.48337i
$$424$$ 8.69527 0.422279
$$425$$ 0 0
$$426$$ −3.96345 −0.192030
$$427$$ − 15.9194i − 0.770396i
$$428$$ − 1.79036i − 0.0865405i
$$429$$ −18.9347 −0.914177
$$430$$ 0 0
$$431$$ −15.3540 −0.739575 −0.369788 0.929116i $$-0.620570\pi$$
−0.369788 + 0.929116i $$0.620570\pi$$
$$432$$ − 4.75905i − 0.228970i
$$433$$ 16.6640i 0.800819i 0.916336 + 0.400409i $$0.131132\pi$$
−0.916336 + 0.400409i $$0.868868\pi$$
$$434$$ −26.0806 −1.25191
$$435$$ 0 0
$$436$$ 2.41404 0.115611
$$437$$ 1.79560i 0.0858951i
$$438$$ − 35.0310i − 1.67384i
$$439$$ 19.5987 0.935393 0.467697 0.883889i $$-0.345084\pi$$
0.467697 + 0.883889i $$0.345084\pi$$
$$440$$ 0 0
$$441$$ 70.9176 3.37703
$$442$$ − 2.04028i − 0.0970462i
$$443$$ 13.0183i 0.618517i 0.950978 + 0.309258i $$0.100081\pi$$
−0.950978 + 0.309258i $$0.899919\pi$$
$$444$$ 31.6860 1.50375
$$445$$ 0 0
$$446$$ 7.84635 0.371535
$$447$$ − 35.9530i − 1.70052i
$$448$$ − 4.69527i − 0.221830i
$$449$$ −6.77359 −0.319666 −0.159833 0.987144i $$-0.551095\pi$$
−0.159833 + 0.987144i $$0.551095\pi$$
$$450$$ 0 0
$$451$$ −7.34352 −0.345793
$$452$$ − 7.14585i − 0.336112i
$$453$$ 15.8538i 0.744877i
$$454$$ −6.28646 −0.295038
$$455$$ 0 0
$$456$$ −2.77733 −0.130060
$$457$$ 27.3189i 1.27793i 0.769237 + 0.638963i $$0.220636\pi$$
−0.769237 + 0.638963i $$0.779364\pi$$
$$458$$ − 3.14585i − 0.146996i
$$459$$ −9.12758 −0.426039
$$460$$ 0 0
$$461$$ −27.7993 −1.29474 −0.647372 0.762174i $$-0.724132\pi$$
−0.647372 + 0.762174i $$0.724132\pi$$
$$462$$ 83.5726i 3.88815i
$$463$$ 38.2369i 1.77702i 0.458859 + 0.888509i $$0.348258\pi$$
−0.458859 + 0.888509i $$0.651742\pi$$
$$464$$ −2.93621 −0.136310
$$465$$ 0 0
$$466$$ 0.182394 0.00844923
$$467$$ − 23.4711i − 1.08611i −0.839696 0.543056i $$-0.817267\pi$$
0.839696 0.543056i $$-0.182733\pi$$
$$468$$ 5.01420i 0.231782i
$$469$$ −41.4256 −1.91286
$$470$$ 0 0
$$471$$ 25.7262 1.18540
$$472$$ 5.63148i 0.259210i
$$473$$ − 22.7811i − 1.04747i
$$474$$ 5.45315 0.250472
$$475$$ 0 0
$$476$$ −9.00523 −0.412754
$$477$$ 40.9855i 1.87660i
$$478$$ − 11.5039i − 0.526176i
$$479$$ 19.6900 0.899660 0.449830 0.893114i $$-0.351484\pi$$
0.449830 + 0.893114i $$0.351484\pi$$
$$480$$ 0 0
$$481$$ −12.1365 −0.553379
$$482$$ 0.445349i 0.0202851i
$$483$$ − 23.4151i − 1.06542i
$$484$$ −30.0728 −1.36694
$$485$$ 0 0
$$486$$ −16.8411 −0.763928
$$487$$ − 16.3357i − 0.740242i −0.928984 0.370121i $$-0.879316\pi$$
0.928984 0.370121i $$-0.120684\pi$$
$$488$$ − 3.39053i − 0.153482i
$$489$$ −11.8904 −0.537701
$$490$$ 0 0
$$491$$ 27.1093 1.22343 0.611713 0.791080i $$-0.290481\pi$$
0.611713 + 0.791080i $$0.290481\pi$$
$$492$$ 3.18239i 0.143473i
$$493$$ 5.63148i 0.253629i
$$494$$ 1.06379 0.0478621
$$495$$ 0 0
$$496$$ −5.55465 −0.249411
$$497$$ − 6.70050i − 0.300558i
$$498$$ 45.1168i 2.02173i
$$499$$ 4.69003 0.209955 0.104977 0.994475i $$-0.466523\pi$$
0.104977 + 0.994475i $$0.466523\pi$$
$$500$$ 0 0
$$501$$ −42.2887 −1.88932
$$502$$ − 13.2630i − 0.591955i
$$503$$ 5.19136i 0.231471i 0.993280 + 0.115736i $$0.0369226\pi$$
−0.993280 + 0.115736i $$0.963077\pi$$
$$504$$ 22.1313 0.985807
$$505$$ 0 0
$$506$$ 11.5076 0.511577
$$507$$ 32.9623i 1.46391i
$$508$$ 9.26295i 0.410977i
$$509$$ 4.51811 0.200262 0.100131 0.994974i $$-0.468074\pi$$
0.100131 + 0.994974i $$0.468074\pi$$
$$510$$ 0 0
$$511$$ 59.2223 2.61984
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 4.75905i − 0.210117i
$$514$$ 23.4816 1.03573
$$515$$ 0 0
$$516$$ −9.87242 −0.434609
$$517$$ − 69.4450i − 3.05419i
$$518$$ 53.5674i 2.35361i
$$519$$ 34.0582 1.49499
$$520$$ 0 0
$$521$$ 14.4453 0.632862 0.316431 0.948615i $$-0.397515\pi$$
0.316431 + 0.948615i $$0.397515\pi$$
$$522$$ − 13.8399i − 0.605758i
$$523$$ − 18.2134i − 0.796415i −0.917295 0.398208i $$-0.869632\pi$$
0.917295 0.398208i $$-0.130368\pi$$
$$524$$ −11.5181 −0.503171
$$525$$ 0 0
$$526$$ 27.9269 1.21767
$$527$$ 10.6535i 0.464073i
$$528$$ 17.7993i 0.774617i
$$529$$ 19.7758 0.859819
$$530$$ 0 0
$$531$$ −26.5442 −1.15192
$$532$$ − 4.69527i − 0.203566i
$$533$$ − 1.21894i − 0.0527981i
$$534$$ 27.7733 1.20187
$$535$$ 0 0
$$536$$ −8.82284 −0.381089
$$537$$ 4.36852i 0.188516i
$$538$$ 16.4816i 0.710571i
$$539$$ −96.4237 −4.15326
$$540$$ 0 0
$$541$$ 37.3905 1.60754 0.803772 0.594937i $$-0.202823\pi$$
0.803772 + 0.594937i $$0.202823\pi$$
$$542$$ 1.46736i 0.0630284i
$$543$$ 31.6860i 1.35977i
$$544$$ −1.91794 −0.0822310
$$545$$ 0 0
$$546$$ −13.8721 −0.593671
$$547$$ − 29.5621i − 1.26399i −0.774974 0.631993i $$-0.782237\pi$$
0.774974 0.631993i $$-0.217763\pi$$
$$548$$ 15.1041i 0.645214i
$$549$$ 15.9814 0.682069
$$550$$ 0 0
$$551$$ −2.93621 −0.125087
$$552$$ − 4.98696i − 0.212259i
$$553$$ 9.21894i 0.392029i
$$554$$ −15.8359 −0.672802
$$555$$ 0 0
$$556$$ −0.700500 −0.0297078
$$557$$ − 14.3723i − 0.608972i −0.952517 0.304486i $$-0.901515\pi$$
0.952517 0.304486i $$-0.0984847\pi$$
$$558$$ − 26.1821i − 1.10837i
$$559$$ 3.78139 0.159936
$$560$$ 0 0
$$561$$ 34.1380 1.44131
$$562$$ 6.25515i 0.263858i
$$563$$ − 6.39053i − 0.269329i −0.990891 0.134664i $$-0.957004\pi$$
0.990891 0.134664i $$-0.0429956\pi$$
$$564$$ −30.0948 −1.26722
$$565$$ 0 0
$$566$$ −16.7005 −0.701974
$$567$$ − 4.33455i − 0.182034i
$$568$$ − 1.42708i − 0.0598788i
$$569$$ 44.9086 1.88267 0.941334 0.337476i $$-0.109573\pi$$
0.941334 + 0.337476i $$0.109573\pi$$
$$570$$ 0 0
$$571$$ 12.4193 0.519730 0.259865 0.965645i $$-0.416322\pi$$
0.259865 + 0.965645i $$0.416322\pi$$
$$572$$ − 6.81761i − 0.285058i
$$573$$ − 17.6584i − 0.737690i
$$574$$ −5.38006 −0.224559
$$575$$ 0 0
$$576$$ 4.71354 0.196397
$$577$$ 4.20440i 0.175032i 0.996163 + 0.0875158i $$0.0278928\pi$$
−0.996163 + 0.0875158i $$0.972107\pi$$
$$578$$ − 13.3215i − 0.554102i
$$579$$ −40.2708 −1.67360
$$580$$ 0 0
$$581$$ −76.2731 −3.16434
$$582$$ 41.5076i 1.72055i
$$583$$ − 55.7262i − 2.30795i
$$584$$ 12.6132 0.521938
$$585$$ 0 0
$$586$$ 0.644516 0.0266247
$$587$$ − 24.7915i − 1.02326i −0.859207 0.511628i $$-0.829043\pi$$
0.859207 0.511628i $$-0.170957\pi$$
$$588$$ 41.7863i 1.72324i
$$589$$ −5.55465 −0.228875
$$590$$ 0 0
$$591$$ −14.2917 −0.587882
$$592$$ 11.4088i 0.468899i
$$593$$ 4.67176i 0.191846i 0.995389 + 0.0959231i $$0.0305803\pi$$
−0.995389 + 0.0959231i $$0.969420\pi$$
$$594$$ −30.4998 −1.25142
$$595$$ 0 0
$$596$$ 12.9452 0.530255
$$597$$ − 10.7695i − 0.440767i
$$598$$ 1.91014i 0.0781113i
$$599$$ −39.2369 −1.60318 −0.801588 0.597877i $$-0.796011\pi$$
−0.801588 + 0.597877i $$0.796011\pi$$
$$600$$ 0 0
$$601$$ −42.4267 −1.73062 −0.865311 0.501235i $$-0.832879\pi$$
−0.865311 + 0.501235i $$0.832879\pi$$
$$602$$ − 16.6900i − 0.680235i
$$603$$ − 41.5868i − 1.69355i
$$604$$ −5.70830 −0.232268
$$605$$ 0 0
$$606$$ 29.2630 1.18873
$$607$$ 2.98173i 0.121025i 0.998167 + 0.0605123i $$0.0192734\pi$$
−0.998167 + 0.0605123i $$0.980727\pi$$
$$608$$ − 1.00000i − 0.0405554i
$$609$$ 38.2890 1.55155
$$610$$ 0 0
$$611$$ 11.5271 0.466336
$$612$$ − 9.04028i − 0.365432i
$$613$$ − 28.9452i − 1.16908i −0.811363 0.584542i $$-0.801274\pi$$
0.811363 0.584542i $$-0.198726\pi$$
$$614$$ −26.6457 −1.07533
$$615$$ 0 0
$$616$$ −30.0910 −1.21240
$$617$$ 15.0365i 0.605349i 0.953094 + 0.302674i $$0.0978794\pi$$
−0.953094 + 0.302674i $$0.902121\pi$$
$$618$$ − 47.1638i − 1.89721i
$$619$$ −8.25515 −0.331803 −0.165901 0.986142i $$-0.553053\pi$$
−0.165901 + 0.986142i $$0.553053\pi$$
$$620$$ 0 0
$$621$$ 8.54535 0.342913
$$622$$ − 11.5494i − 0.463089i
$$623$$ 46.9527i 1.88112i
$$624$$ −2.95449 −0.118274
$$625$$ 0 0
$$626$$ −23.0638 −0.921814
$$627$$ 17.7993i 0.710837i
$$628$$ 9.26295i 0.369632i
$$629$$ 21.8814 0.872468
$$630$$ 0 0
$$631$$ −9.09103 −0.361908 −0.180954 0.983492i $$-0.557919\pi$$
−0.180954 + 0.983492i $$0.557919\pi$$
$$632$$ 1.96345i 0.0781020i
$$633$$ 47.6300i 1.89312i
$$634$$ 13.9232 0.552960
$$635$$ 0 0
$$636$$ −24.1496 −0.957593
$$637$$ − 16.0052i − 0.634150i
$$638$$ 18.8176i 0.744996i
$$639$$ 6.72658 0.266099
$$640$$ 0 0
$$641$$ 30.1171 1.18955 0.594777 0.803891i $$-0.297240\pi$$
0.594777 + 0.803891i $$0.297240\pi$$
$$642$$ 4.97242i 0.196246i
$$643$$ − 35.3174i − 1.39278i −0.717662 0.696392i $$-0.754788\pi$$
0.717662 0.696392i $$-0.245212\pi$$
$$644$$ 8.43081 0.332220
$$645$$ 0 0
$$646$$ −1.91794 −0.0754603
$$647$$ − 4.12234i − 0.162066i −0.996711 0.0810330i $$-0.974178\pi$$
0.996711 0.0810330i $$-0.0258219\pi$$
$$648$$ − 0.923174i − 0.0362657i
$$649$$ 36.0910 1.41670
$$650$$ 0 0
$$651$$ 72.4342 2.83892
$$652$$ − 4.28123i − 0.167666i
$$653$$ 8.62741i 0.337617i 0.985649 + 0.168808i $$0.0539919\pi$$
−0.985649 + 0.168808i $$0.946008\pi$$
$$654$$ −6.70457 −0.262169
$$655$$ 0 0
$$656$$ −1.14585 −0.0447379
$$657$$ 59.4528i 2.31948i
$$658$$ − 50.8773i − 1.98340i
$$659$$ 37.3853 1.45632 0.728162 0.685405i $$-0.240375\pi$$
0.728162 + 0.685405i $$0.240375\pi$$
$$660$$ 0 0
$$661$$ −12.8997 −0.501739 −0.250869 0.968021i $$-0.580716\pi$$
−0.250869 + 0.968021i $$0.580716\pi$$
$$662$$ 0.735546i 0.0285878i
$$663$$ 5.66652i 0.220070i
$$664$$ −16.2447 −0.630416
$$665$$ 0 0
$$666$$ −53.7758 −2.08377
$$667$$ − 5.27226i − 0.204143i
$$668$$ − 15.2264i − 0.589127i
$$669$$ −21.7919 −0.842522
$$670$$ 0 0
$$671$$ −21.7292 −0.838848
$$672$$ 13.0403i 0.503040i
$$673$$ 10.4349i 0.402235i 0.979567 + 0.201118i $$0.0644573\pi$$
−0.979567 + 0.201118i $$0.935543\pi$$
$$674$$ 2.28123 0.0878696
$$675$$ 0 0
$$676$$ −11.8684 −0.456475
$$677$$ − 12.4401i − 0.478112i −0.971006 0.239056i $$-0.923162\pi$$
0.971006 0.239056i $$-0.0768380\pi$$
$$678$$ 19.8463i 0.762194i
$$679$$ −70.1716 −2.69294
$$680$$ 0 0
$$681$$ 17.4596 0.669052
$$682$$ 35.5987i 1.36314i
$$683$$ 17.5364i 0.671011i 0.942038 + 0.335505i $$0.108907\pi$$
−0.942038 + 0.335505i $$0.891093\pi$$
$$684$$ 4.71354 0.180227
$$685$$ 0 0
$$686$$ −37.7758 −1.44229
$$687$$ 8.73705i 0.333339i
$$688$$ − 3.55465i − 0.135520i
$$689$$ 9.24992 0.352394
$$690$$ 0 0
$$691$$ −25.2734 −0.961446 −0.480723 0.876872i $$-0.659626\pi$$
−0.480723 + 0.876872i $$0.659626\pi$$
$$692$$ 12.2630i 0.466168i
$$693$$ − 141.835i − 5.38787i
$$694$$ 2.56246 0.0972695
$$695$$ 0 0
$$696$$ 8.15482 0.309108
$$697$$ 2.19767i 0.0832426i
$$698$$ − 5.67176i − 0.214679i
$$699$$ −0.506567 −0.0191601
$$700$$ 0 0
$$701$$ −16.0545 −0.606370 −0.303185 0.952932i $$-0.598050\pi$$
−0.303185 + 0.952932i $$0.598050\pi$$
$$702$$ − 5.06262i − 0.191076i
$$703$$ 11.4088i 0.430291i
$$704$$ −6.40880 −0.241541
$$705$$ 0 0
$$706$$ −23.6665 −0.890701
$$707$$ 49.4711i 1.86055i
$$708$$ − 15.6404i − 0.587804i
$$709$$ −13.5076 −0.507290 −0.253645 0.967297i $$-0.581629\pi$$
−0.253645 + 0.967297i $$0.581629\pi$$
$$710$$ 0 0
$$711$$ −9.25482 −0.347083
$$712$$ 10.0000i 0.374766i
$$713$$ − 9.97392i − 0.373526i
$$714$$ 25.0105 0.935993
$$715$$ 0 0
$$716$$ −1.57292 −0.0587829
$$717$$ 31.9501i 1.19320i
$$718$$ 31.9724i 1.19320i
$$719$$ −0.803402 −0.0299619 −0.0149809 0.999888i $$-0.504769\pi$$
−0.0149809 + 0.999888i $$0.504769\pi$$
$$720$$ 0 0
$$721$$ 79.7337 2.96944
$$722$$ − 1.00000i − 0.0372161i
$$723$$ − 1.23688i − 0.0460000i
$$724$$ −11.4088 −0.424005
$$725$$ 0 0
$$726$$ 83.5218 3.09979
$$727$$ − 51.6860i − 1.91693i −0.285217 0.958463i $$-0.592066\pi$$
0.285217 0.958463i $$-0.407934\pi$$
$$728$$ − 4.99477i − 0.185118i
$$729$$ 44.0037 1.62977
$$730$$ 0 0
$$731$$ −6.81761 −0.252158
$$732$$ 9.41661i 0.348048i
$$733$$ − 22.3723i − 0.826338i −0.910654 0.413169i $$-0.864422\pi$$
0.910654 0.413169i $$-0.135578\pi$$
$$734$$ −1.14585 −0.0422940
$$735$$ 0 0
$$736$$ 1.79560 0.0661866
$$737$$ 56.5438i 2.08282i
$$738$$ − 5.40100i − 0.198814i
$$739$$ −33.4271 −1.22963 −0.614817 0.788669i $$-0.710770\pi$$
−0.614817 + 0.788669i $$0.710770\pi$$
$$740$$ 0 0
$$741$$ −2.95449 −0.108536
$$742$$ − 40.8266i − 1.49879i
$$743$$ 14.4998i 0.531947i 0.963980 + 0.265974i $$0.0856934\pi$$
−0.963980 + 0.265974i $$0.914307\pi$$
$$744$$ 15.4271 0.565584
$$745$$ 0 0
$$746$$ 32.8579 1.20301
$$747$$ − 76.5699i − 2.80155i
$$748$$ 12.2917i 0.449429i
$$749$$ −8.40623 −0.307157
$$750$$ 0 0
$$751$$ −26.6169 −0.971266 −0.485633 0.874163i $$-0.661411\pi$$
−0.485633 + 0.874163i $$0.661411\pi$$
$$752$$ − 10.8359i − 0.395144i
$$753$$ 36.8355i 1.34236i
$$754$$ −3.12351 −0.113751
$$755$$ 0 0
$$756$$ −22.3450 −0.812680
$$757$$ 31.0362i 1.12803i 0.825764 + 0.564015i $$0.190744\pi$$
−0.825764 + 0.564015i $$0.809256\pi$$
$$758$$ − 18.2865i − 0.664194i
$$759$$ −31.9605 −1.16009
$$760$$ 0 0
$$761$$ −27.8083 −1.00805 −0.504025 0.863689i $$-0.668148\pi$$
−0.504025 + 0.863689i $$0.668148\pi$$
$$762$$ − 25.7262i − 0.931963i
$$763$$ − 11.3345i − 0.410338i
$$764$$ 6.35805 0.230026
$$765$$ 0 0
$$766$$ 20.8542 0.753491
$$767$$ 5.99070i 0.216312i
$$768$$ 2.77733i 0.100218i
$$769$$ 19.2279 0.693376 0.346688 0.937980i $$-0.387306\pi$$
0.346688 + 0.937980i $$0.387306\pi$$
$$770$$ 0 0
$$771$$ −65.2159 −2.34869
$$772$$ − 14.4998i − 0.521860i
$$773$$ 6.00373i 0.215939i 0.994154 + 0.107970i $$0.0344349\pi$$
−0.994154 + 0.107970i $$0.965565\pi$$
$$774$$ 16.7550 0.602245
$$775$$ 0 0
$$776$$ −14.9452 −0.536501
$$777$$ − 148.774i − 5.33724i
$$778$$ − 24.9086i − 0.893018i
$$779$$ −1.14585 −0.0410543
$$780$$ 0 0
$$781$$ −9.14585 −0.327264
$$782$$ − 3.44385i − 0.123152i
$$783$$ 13.9736i 0.499375i
$$784$$ −15.0455 −0.537340
$$785$$ 0 0
$$786$$ 31.9895 1.14103
$$787$$ − 22.2797i − 0.794187i −0.917778 0.397093i $$-0.870019\pi$$
0.917778 0.397093i $$-0.129981\pi$$
$$788$$ − 5.14585i − 0.183313i
$$789$$ −77.5621 −2.76128
$$790$$ 0 0
$$791$$ −33.5517 −1.19296
$$792$$ − 30.2081i − 1.07340i
$$793$$ − 3.60680i − 0.128081i
$$794$$ 24.7445 0.878150
$$795$$ 0 0
$$796$$ 3.87766 0.137440
$$797$$ − 26.6259i − 0.943138i −0.881829 0.471569i $$-0.843688\pi$$
0.881829 0.471569i $$-0.156312\pi$$
$$798$$ 13.0403i 0.461621i
$$799$$ −20.7826 −0.735234
$$800$$ 0 0
$$801$$ −47.1354 −1.66545
$$802$$ − 15.6457i − 0.552468i
$$803$$ − 80.8355i − 2.85262i
$$804$$ 24.5039 0.864186
$$805$$ 0 0
$$806$$ −5.90897 −0.208135
$$807$$ − 45.7747i − 1.61134i
$$808$$ 10.5364i 0.370669i
$$809$$ −0.651250 −0.0228967 −0.0114484 0.999934i $$-0.503644\pi$$
−0.0114484 + 0.999934i $$0.503644\pi$$
$$810$$ 0 0
$$811$$ −13.0780 −0.459230 −0.229615 0.973281i $$-0.573747\pi$$
−0.229615 + 0.973281i $$0.573747\pi$$
$$812$$ 13.7863i 0.483804i
$$813$$ − 4.07533i − 0.142928i
$$814$$ 73.1168 2.56274
$$815$$ 0 0
$$816$$ 5.32674 0.186473
$$817$$ − 3.55465i − 0.124362i
$$818$$ 30.5804i 1.06922i
$$819$$ 23.5430 0.822660
$$820$$ 0 0
$$821$$ 14.6640 0.511776 0.255888 0.966706i $$-0.417632\pi$$
0.255888 + 0.966706i $$0.417632\pi$$
$$822$$ − 41.9489i − 1.46314i
$$823$$ − 18.2850i − 0.637374i −0.947860 0.318687i $$-0.896758\pi$$
0.947860 0.318687i $$-0.103242\pi$$
$$824$$ 16.9817 0.591587
$$825$$ 0 0
$$826$$ 26.4413 0.920010
$$827$$ − 40.1910i − 1.39758i −0.715327 0.698790i $$-0.753722\pi$$
0.715327 0.698790i $$-0.246278\pi$$
$$828$$ 8.46362i 0.294131i
$$829$$ −28.3760 −0.985539 −0.492769 0.870160i $$-0.664015\pi$$
−0.492769 + 0.870160i $$0.664015\pi$$
$$830$$ 0 0
$$831$$ 43.9814 1.52570
$$832$$ − 1.06379i − 0.0368802i
$$833$$ 28.8564i 0.999815i
$$834$$ 1.94552 0.0673677
$$835$$ 0 0
$$836$$ −6.40880 −0.221653
$$837$$ 26.4349i 0.913723i
$$838$$ 1.96345i 0.0678264i
$$839$$ −6.93471 −0.239413 −0.119706 0.992809i $$-0.538195\pi$$
−0.119706 + 0.992809i $$0.538195\pi$$
$$840$$ 0 0
$$841$$ −20.3787 −0.702712
$$842$$ 25.5949i 0.882060i
$$843$$ − 17.3726i − 0.598344i
$$844$$ −17.1496 −0.590313
$$845$$ 0 0
$$846$$ 51.0753 1.75601
$$847$$ 141.200i 4.85167i
$$848$$ − 8.69527i − 0.298597i
$$849$$ 46.3827 1.59185
$$850$$ 0 0
$$851$$ −20.4856 −0.702238
$$852$$ 3.96345i 0.135786i
$$853$$ 21.9739i 0.752373i 0.926544 + 0.376186i $$0.122765\pi$$
−0.926544 + 0.376186i $$0.877235\pi$$
$$854$$ −15.9194 −0.544752
$$855$$ 0 0
$$856$$ −1.79036 −0.0611934
$$857$$ − 13.6091i − 0.464879i −0.972611 0.232440i $$-0.925329\pi$$
0.972611 0.232440i $$-0.0746708\pi$$
$$858$$ 18.9347i 0.646420i
$$859$$ −5.17192 −0.176464 −0.0882319 0.996100i $$-0.528122\pi$$
−0.0882319 + 0.996100i $$0.528122\pi$$
$$860$$ 0 0
$$861$$ 14.9422 0.509228
$$862$$ 15.3540i 0.522959i
$$863$$ 15.9635i 0.543402i 0.962382 + 0.271701i $$0.0875862\pi$$
−0.962382 + 0.271701i $$0.912414\pi$$
$$864$$ −4.75905 −0.161906
$$865$$ 0 0
$$866$$ 16.6640 0.566264
$$867$$ 36.9982i 1.25652i
$$868$$ 26.0806i 0.885232i
$$869$$ 12.5834 0.426862
$$870$$ 0 0
$$871$$ −9.38563 −0.318020
$$872$$ − 2.41404i − 0.0817496i
$$873$$ − 70.4447i − 2.38419i
$$874$$ 1.79560 0.0607370
$$875$$ 0 0
$$876$$ −35.0310 −1.18359
$$877$$ 15.8866i 0.536453i 0.963356 + 0.268227i $$0.0864376\pi$$
−0.963356 + 0.268227i $$0.913562\pi$$
$$878$$ − 19.5987i − 0.661423i
$$879$$ −1.79003 −0.0603762
$$880$$ 0 0
$$881$$ −16.1458 −0.543967 −0.271984 0.962302i $$-0.587680\pi$$
−0.271984 + 0.962302i $$0.587680\pi$$
$$882$$ − 70.9176i − 2.38792i
$$883$$ 38.2887i 1.28852i 0.764808 + 0.644259i $$0.222834\pi$$
−0.764808 + 0.644259i $$0.777166\pi$$
$$884$$ −2.04028 −0.0686221
$$885$$ 0 0
$$886$$ 13.0183 0.437357
$$887$$ − 19.2809i − 0.647389i −0.946162 0.323695i $$-0.895075\pi$$
0.946162 0.323695i $$-0.104925\pi$$
$$888$$ − 31.6860i − 1.06331i
$$889$$ 43.4920 1.45868
$$890$$ 0 0
$$891$$ −5.91644 −0.198208
$$892$$ − 7.84635i − 0.262715i
$$893$$ − 10.8359i − 0.362609i
$$894$$ −35.9530 −1.20245
$$895$$ 0 0
$$896$$ −4.69527 −0.156858
$$897$$ − 5.30507i − 0.177131i
$$898$$ 6.77359i 0.226038i
$$899$$ 16.3096 0.543957
$$900$$ 0 0
$$901$$ −16.6770 −0.555591
$$902$$ 7.34352i 0.244512i
$$903$$ 46.3537i 1.54255i
$$904$$ −7.14585 −0.237667
$$905$$ 0 0
$$906$$ 15.8538 0.526708
$$907$$ − 43.0205i − 1.42847i −0.699905 0.714236i $$-0.746774\pi$$
0.699905 0.714236i $$-0.253226\pi$$
$$908$$ 6.28646i 0.208624i
$$909$$ −49.6636 −1.64724
$$910$$ 0 0
$$911$$ −25.7733 −0.853906 −0.426953 0.904274i $$-0.640413\pi$$
−0.426953 + 0.904274i $$0.640413\pi$$
$$912$$ 2.77733i 0.0919664i
$$913$$ 104.109i 3.44550i
$$914$$ 27.3189 0.903630
$$915$$ 0 0
$$916$$ −3.14585 −0.103942
$$917$$ 54.0806i 1.78590i
$$918$$ 9.12758i 0.301255i
$$919$$ 29.5897 0.976074 0.488037 0.872823i $$-0.337713\pi$$
0.488037 + 0.872823i $$0.337713\pi$$
$$920$$ 0 0
$$921$$ 74.0037 2.43851
$$922$$ 27.7993i 0.915522i
$$923$$ − 1.51811i − 0.0499691i
$$924$$ 83.5726 2.74934
$$925$$ 0 0
$$926$$ 38.2369 1.25654
$$927$$ 80.0440i 2.62899i
$$928$$ 2.93621i 0.0963859i
$$929$$ 1.42334 0.0466983 0.0233491 0.999727i $$-0.492567\pi$$
0.0233491 + 0.999727i $$0.492567\pi$$
$$930$$ 0 0
$$931$$ −15.0455 −0.493097
$$932$$ − 0.182394i − 0.00597451i
$$933$$ 32.0765i 1.05014i
$$934$$ −23.4711 −0.767998
$$935$$ 0 0
$$936$$ 5.01420 0.163894
$$937$$ − 28.2276i − 0.922155i −0.887360 0.461077i $$-0.847463\pi$$
0.887360 0.461077i $$-0.152537\pi$$
$$938$$ 41.4256i 1.35259i
$$939$$ 64.0556 2.09038
$$940$$ 0 0
$$941$$ −15.8672 −0.517256 −0.258628 0.965977i $$-0.583270\pi$$
−0.258628 + 0.965977i $$0.583270\pi$$
$$942$$ − 25.7262i − 0.838206i
$$943$$ − 2.05748i − 0.0670009i
$$944$$ 5.63148 0.183289
$$945$$ 0 0
$$946$$ −22.7811 −0.740676
$$947$$ − 43.6718i − 1.41914i −0.704634 0.709571i $$-0.748889\pi$$
0.704634 0.709571i $$-0.251111\pi$$
$$948$$ − 5.45315i − 0.177110i
$$949$$ 13.4178 0.435559
$$950$$ 0 0
$$951$$ −38.6692 −1.25393
$$952$$ 9.00523i 0.291861i
$$953$$ 16.0261i 0.519136i 0.965725 + 0.259568i $$0.0835801\pi$$
−0.965725 + 0.259568i $$0.916420\pi$$
$$954$$ 40.9855 1.32695
$$955$$ 0 0
$$956$$ −11.5039 −0.372063
$$957$$ − 52.2626i − 1.68941i
$$958$$ − 19.6900i − 0.636156i
$$959$$ 70.9176 2.29005
$$960$$ 0 0
$$961$$ −0.145848 −0.00470478
$$962$$ 12.1365i 0.391298i
$$963$$ − 8.43895i − 0.271941i
$$964$$ 0.445349 0.0143437
$$965$$ 0 0
$$966$$ −23.4151 −0.753369
$$967$$ 11.5987i 0.372988i 0.982456 + 0.186494i $$0.0597125\pi$$
−0.982456 + 0.186494i $$0.940288\pi$$
$$968$$ 30.0728i 0.966575i
$$969$$ 5.32674 0.171120
$$970$$ 0 0
$$971$$ −27.0362 −0.867633 −0.433817 0.901001i $$-0.642833\pi$$
−0.433817 + 0.901001i $$0.642833\pi$$
$$972$$ 16.8411i 0.540179i
$$973$$ 3.28903i 0.105442i
$$974$$ −16.3357 −0.523430
$$975$$ 0 0
$$976$$ −3.39053 −0.108528
$$977$$ − 14.1537i − 0.452815i −0.974033 0.226408i $$-0.927302\pi$$
0.974033 0.226408i $$-0.0726982\pi$$
$$978$$ 11.8904i 0.380212i
$$979$$ 64.0880 2.04826
$$980$$ 0 0
$$981$$ 11.3787 0.363293
$$982$$ − 27.1093i − 0.865093i
$$983$$ − 32.8542i − 1.04788i −0.851754 0.523942i $$-0.824461\pi$$
0.851754 0.523942i $$-0.175539\pi$$
$$984$$ 3.18239 0.101451
$$985$$ 0 0
$$986$$ 5.63148 0.179343
$$987$$ 141.303i 4.49772i
$$988$$ − 1.06379i − 0.0338436i
$$989$$ 6.38273 0.202959
$$990$$ 0 0
$$991$$ 47.9709 1.52385 0.761923 0.647667i $$-0.224255\pi$$
0.761923 + 0.647667i $$0.224255\pi$$
$$992$$ 5.55465i 0.176360i
$$993$$ − 2.04285i − 0.0648279i
$$994$$ −6.70050 −0.212527
$$995$$ 0 0
$$996$$ 45.1168 1.42958
$$997$$ − 44.1716i − 1.39893i −0.714668 0.699464i $$-0.753422\pi$$
0.714668 0.699464i $$-0.246578\pi$$
$$998$$ − 4.69003i − 0.148460i
$$999$$ 54.2951 1.71782
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.g.799.3 6
5.2 odd 4 950.2.a.m.1.3 yes 3
5.3 odd 4 950.2.a.k.1.1 3
5.4 even 2 inner 950.2.b.g.799.4 6
15.2 even 4 8550.2.a.cj.1.1 3
15.8 even 4 8550.2.a.co.1.3 3
20.3 even 4 7600.2.a.cb.1.3 3
20.7 even 4 7600.2.a.bm.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.1 3 5.3 odd 4
950.2.a.m.1.3 yes 3 5.2 odd 4
950.2.b.g.799.3 6 1.1 even 1 trivial
950.2.b.g.799.4 6 5.4 even 2 inner
7600.2.a.bm.1.1 3 20.7 even 4
7600.2.a.cb.1.3 3 20.3 even 4
8550.2.a.cj.1.1 3 15.2 even 4
8550.2.a.co.1.3 3 15.8 even 4