# Properties

 Label 3174.2.a.bc.1.5 Level $3174$ Weight $2$ Character 3174.1 Self dual yes Analytic conductor $25.345$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3174,2,Mod(1,3174)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3174, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3174.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3174 = 2 \cdot 3 \cdot 23^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3174.a (trivial)

## 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: yes Analytic conductor: $$25.3445176016$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\Q(\zeta_{22})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1$$ x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 138) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$0.284630$$ of defining polynomial Character $$\chi$$ $$=$$ 3174.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.51334 q^{5} +1.00000 q^{6} -2.59435 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.51334 q^{5} +1.00000 q^{6} -2.59435 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.51334 q^{10} -5.00714 q^{11} +1.00000 q^{12} -4.82306 q^{13} -2.59435 q^{14} +1.51334 q^{15} +1.00000 q^{16} -0.863693 q^{17} +1.00000 q^{18} -7.00158 q^{19} +1.51334 q^{20} -2.59435 q^{21} -5.00714 q^{22} +1.00000 q^{24} -2.70981 q^{25} -4.82306 q^{26} +1.00000 q^{27} -2.59435 q^{28} +3.11362 q^{29} +1.51334 q^{30} -1.17428 q^{31} +1.00000 q^{32} -5.00714 q^{33} -0.863693 q^{34} -3.92613 q^{35} +1.00000 q^{36} -0.602123 q^{37} -7.00158 q^{38} -4.82306 q^{39} +1.51334 q^{40} +5.29335 q^{41} -2.59435 q^{42} +1.45925 q^{43} -5.00714 q^{44} +1.51334 q^{45} -12.6797 q^{47} +1.00000 q^{48} -0.269342 q^{49} -2.70981 q^{50} -0.863693 q^{51} -4.82306 q^{52} +4.23281 q^{53} +1.00000 q^{54} -7.57749 q^{55} -2.59435 q^{56} -7.00158 q^{57} +3.11362 q^{58} +3.98142 q^{59} +1.51334 q^{60} -7.26229 q^{61} -1.17428 q^{62} -2.59435 q^{63} +1.00000 q^{64} -7.29891 q^{65} -5.00714 q^{66} -16.1371 q^{67} -0.863693 q^{68} -3.92613 q^{70} +5.88463 q^{71} +1.00000 q^{72} +14.3814 q^{73} -0.602123 q^{74} -2.70981 q^{75} -7.00158 q^{76} +12.9903 q^{77} -4.82306 q^{78} +10.3402 q^{79} +1.51334 q^{80} +1.00000 q^{81} +5.29335 q^{82} +2.64101 q^{83} -2.59435 q^{84} -1.30706 q^{85} +1.45925 q^{86} +3.11362 q^{87} -5.00714 q^{88} -5.23754 q^{89} +1.51334 q^{90} +12.5127 q^{91} -1.17428 q^{93} -12.6797 q^{94} -10.5958 q^{95} +1.00000 q^{96} +14.9518 q^{97} -0.269342 q^{98} -5.00714 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{2} + 5 q^{3} + 5 q^{4} - 7 q^{5} + 5 q^{6} - 7 q^{7} + 5 q^{8} + 5 q^{9}+O(q^{10})$$ 5 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 - 7 * q^5 + 5 * q^6 - 7 * q^7 + 5 * q^8 + 5 * q^9 $$5 q + 5 q^{2} + 5 q^{3} + 5 q^{4} - 7 q^{5} + 5 q^{6} - 7 q^{7} + 5 q^{8} + 5 q^{9} - 7 q^{10} - 13 q^{11} + 5 q^{12} - 4 q^{13} - 7 q^{14} - 7 q^{15} + 5 q^{16} - 9 q^{17} + 5 q^{18} - 11 q^{19} - 7 q^{20} - 7 q^{21} - 13 q^{22} + 5 q^{24} - 2 q^{25} - 4 q^{26} + 5 q^{27} - 7 q^{28} - 7 q^{29} - 7 q^{30} - 8 q^{31} + 5 q^{32} - 13 q^{33} - 9 q^{34} + q^{35} + 5 q^{36} - 12 q^{37} - 11 q^{38} - 4 q^{39} - 7 q^{40} - 10 q^{41} - 7 q^{42} - 4 q^{43} - 13 q^{44} - 7 q^{45} - 24 q^{47} + 5 q^{48} - 12 q^{49} - 2 q^{50} - 9 q^{51} - 4 q^{52} - 9 q^{53} + 5 q^{54} + 16 q^{55} - 7 q^{56} - 11 q^{57} - 7 q^{58} - 14 q^{59} - 7 q^{60} - 5 q^{61} - 8 q^{62} - 7 q^{63} + 5 q^{64} - 12 q^{65} - 13 q^{66} - 13 q^{67} - 9 q^{68} + q^{70} - 19 q^{71} + 5 q^{72} + 4 q^{73} - 12 q^{74} - 2 q^{75} - 11 q^{76} + 5 q^{77} - 4 q^{78} - 4 q^{79} - 7 q^{80} + 5 q^{81} - 10 q^{82} - 24 q^{83} - 7 q^{84} + 17 q^{85} - 4 q^{86} - 7 q^{87} - 13 q^{88} - 4 q^{89} - 7 q^{90} + 21 q^{91} - 8 q^{93} - 24 q^{94} - 11 q^{95} + 5 q^{96} + 9 q^{97} - 12 q^{98} - 13 q^{99}+O(q^{100})$$ 5 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 - 7 * q^5 + 5 * q^6 - 7 * q^7 + 5 * q^8 + 5 * q^9 - 7 * q^10 - 13 * q^11 + 5 * q^12 - 4 * q^13 - 7 * q^14 - 7 * q^15 + 5 * q^16 - 9 * q^17 + 5 * q^18 - 11 * q^19 - 7 * q^20 - 7 * q^21 - 13 * q^22 + 5 * q^24 - 2 * q^25 - 4 * q^26 + 5 * q^27 - 7 * q^28 - 7 * q^29 - 7 * q^30 - 8 * q^31 + 5 * q^32 - 13 * q^33 - 9 * q^34 + q^35 + 5 * q^36 - 12 * q^37 - 11 * q^38 - 4 * q^39 - 7 * q^40 - 10 * q^41 - 7 * q^42 - 4 * q^43 - 13 * q^44 - 7 * q^45 - 24 * q^47 + 5 * q^48 - 12 * q^49 - 2 * q^50 - 9 * q^51 - 4 * q^52 - 9 * q^53 + 5 * q^54 + 16 * q^55 - 7 * q^56 - 11 * q^57 - 7 * q^58 - 14 * q^59 - 7 * q^60 - 5 * q^61 - 8 * q^62 - 7 * q^63 + 5 * q^64 - 12 * q^65 - 13 * q^66 - 13 * q^67 - 9 * q^68 + q^70 - 19 * q^71 + 5 * q^72 + 4 * q^73 - 12 * q^74 - 2 * q^75 - 11 * q^76 + 5 * q^77 - 4 * q^78 - 4 * q^79 - 7 * q^80 + 5 * q^81 - 10 * q^82 - 24 * q^83 - 7 * q^84 + 17 * q^85 - 4 * q^86 - 7 * q^87 - 13 * q^88 - 4 * q^89 - 7 * q^90 + 21 * q^91 - 8 * q^93 - 24 * q^94 - 11 * q^95 + 5 * q^96 + 9 * q^97 - 12 * q^98 - 13 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 1.51334 0.676785 0.338392 0.941005i $$-0.390117\pi$$
0.338392 + 0.941005i $$0.390117\pi$$
$$6$$ 1.00000 0.408248
$$7$$ −2.59435 −0.980573 −0.490286 0.871561i $$-0.663108\pi$$
−0.490286 + 0.871561i $$0.663108\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.51334 0.478559
$$11$$ −5.00714 −1.50971 −0.754855 0.655892i $$-0.772293\pi$$
−0.754855 + 0.655892i $$0.772293\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −4.82306 −1.33768 −0.668838 0.743408i $$-0.733208\pi$$
−0.668838 + 0.743408i $$0.733208\pi$$
$$14$$ −2.59435 −0.693370
$$15$$ 1.51334 0.390742
$$16$$ 1.00000 0.250000
$$17$$ −0.863693 −0.209476 −0.104738 0.994500i $$-0.533400\pi$$
−0.104738 + 0.994500i $$0.533400\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −7.00158 −1.60627 −0.803137 0.595795i $$-0.796837\pi$$
−0.803137 + 0.595795i $$0.796837\pi$$
$$20$$ 1.51334 0.338392
$$21$$ −2.59435 −0.566134
$$22$$ −5.00714 −1.06753
$$23$$ 0 0
$$24$$ 1.00000 0.204124
$$25$$ −2.70981 −0.541962
$$26$$ −4.82306 −0.945880
$$27$$ 1.00000 0.192450
$$28$$ −2.59435 −0.490286
$$29$$ 3.11362 0.578185 0.289093 0.957301i $$-0.406646\pi$$
0.289093 + 0.957301i $$0.406646\pi$$
$$30$$ 1.51334 0.276296
$$31$$ −1.17428 −0.210907 −0.105453 0.994424i $$-0.533629\pi$$
−0.105453 + 0.994424i $$0.533629\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −5.00714 −0.871632
$$34$$ −0.863693 −0.148122
$$35$$ −3.92613 −0.663637
$$36$$ 1.00000 0.166667
$$37$$ −0.602123 −0.0989883 −0.0494942 0.998774i $$-0.515761\pi$$
−0.0494942 + 0.998774i $$0.515761\pi$$
$$38$$ −7.00158 −1.13581
$$39$$ −4.82306 −0.772307
$$40$$ 1.51334 0.239280
$$41$$ 5.29335 0.826683 0.413341 0.910576i $$-0.364362\pi$$
0.413341 + 0.910576i $$0.364362\pi$$
$$42$$ −2.59435 −0.400317
$$43$$ 1.45925 0.222534 0.111267 0.993791i $$-0.464509\pi$$
0.111267 + 0.993791i $$0.464509\pi$$
$$44$$ −5.00714 −0.754855
$$45$$ 1.51334 0.225595
$$46$$ 0 0
$$47$$ −12.6797 −1.84952 −0.924760 0.380552i $$-0.875734\pi$$
−0.924760 + 0.380552i $$0.875734\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −0.269342 −0.0384774
$$50$$ −2.70981 −0.383225
$$51$$ −0.863693 −0.120941
$$52$$ −4.82306 −0.668838
$$53$$ 4.23281 0.581422 0.290711 0.956811i $$-0.406108\pi$$
0.290711 + 0.956811i $$0.406108\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −7.57749 −1.02175
$$56$$ −2.59435 −0.346685
$$57$$ −7.00158 −0.927382
$$58$$ 3.11362 0.408839
$$59$$ 3.98142 0.518337 0.259168 0.965832i $$-0.416552\pi$$
0.259168 + 0.965832i $$0.416552\pi$$
$$60$$ 1.51334 0.195371
$$61$$ −7.26229 −0.929841 −0.464920 0.885352i $$-0.653917\pi$$
−0.464920 + 0.885352i $$0.653917\pi$$
$$62$$ −1.17428 −0.149134
$$63$$ −2.59435 −0.326858
$$64$$ 1.00000 0.125000
$$65$$ −7.29891 −0.905319
$$66$$ −5.00714 −0.616337
$$67$$ −16.1371 −1.97146 −0.985729 0.168340i $$-0.946159\pi$$
−0.985729 + 0.168340i $$0.946159\pi$$
$$68$$ −0.863693 −0.104738
$$69$$ 0 0
$$70$$ −3.92613 −0.469262
$$71$$ 5.88463 0.698377 0.349188 0.937053i $$-0.386457\pi$$
0.349188 + 0.937053i $$0.386457\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 14.3814 1.68322 0.841608 0.540089i $$-0.181609\pi$$
0.841608 + 0.540089i $$0.181609\pi$$
$$74$$ −0.602123 −0.0699953
$$75$$ −2.70981 −0.312902
$$76$$ −7.00158 −0.803137
$$77$$ 12.9903 1.48038
$$78$$ −4.82306 −0.546104
$$79$$ 10.3402 1.16336 0.581681 0.813417i $$-0.302395\pi$$
0.581681 + 0.813417i $$0.302395\pi$$
$$80$$ 1.51334 0.169196
$$81$$ 1.00000 0.111111
$$82$$ 5.29335 0.584553
$$83$$ 2.64101 0.289888 0.144944 0.989440i $$-0.453700\pi$$
0.144944 + 0.989440i $$0.453700\pi$$
$$84$$ −2.59435 −0.283067
$$85$$ −1.30706 −0.141770
$$86$$ 1.45925 0.157355
$$87$$ 3.11362 0.333815
$$88$$ −5.00714 −0.533763
$$89$$ −5.23754 −0.555178 −0.277589 0.960700i $$-0.589535\pi$$
−0.277589 + 0.960700i $$0.589535\pi$$
$$90$$ 1.51334 0.159520
$$91$$ 12.5127 1.31169
$$92$$ 0 0
$$93$$ −1.17428 −0.121767
$$94$$ −12.6797 −1.30781
$$95$$ −10.5958 −1.08710
$$96$$ 1.00000 0.102062
$$97$$ 14.9518 1.51812 0.759062 0.651018i $$-0.225658\pi$$
0.759062 + 0.651018i $$0.225658\pi$$
$$98$$ −0.269342 −0.0272077
$$99$$ −5.00714 −0.503237
$$100$$ −2.70981 −0.270981
$$101$$ −0.530283 −0.0527652 −0.0263826 0.999652i $$-0.508399\pi$$
−0.0263826 + 0.999652i $$0.508399\pi$$
$$102$$ −0.863693 −0.0855184
$$103$$ 0.919917 0.0906421 0.0453211 0.998972i $$-0.485569\pi$$
0.0453211 + 0.998972i $$0.485569\pi$$
$$104$$ −4.82306 −0.472940
$$105$$ −3.92613 −0.383151
$$106$$ 4.23281 0.411127
$$107$$ 13.4562 1.30086 0.650428 0.759568i $$-0.274590\pi$$
0.650428 + 0.759568i $$0.274590\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −4.97597 −0.476611 −0.238306 0.971190i $$-0.576592\pi$$
−0.238306 + 0.971190i $$0.576592\pi$$
$$110$$ −7.57749 −0.722486
$$111$$ −0.602123 −0.0571509
$$112$$ −2.59435 −0.245143
$$113$$ −13.6038 −1.27974 −0.639869 0.768484i $$-0.721011\pi$$
−0.639869 + 0.768484i $$0.721011\pi$$
$$114$$ −7.00158 −0.655758
$$115$$ 0 0
$$116$$ 3.11362 0.289093
$$117$$ −4.82306 −0.445892
$$118$$ 3.98142 0.366519
$$119$$ 2.24072 0.205407
$$120$$ 1.51334 0.138148
$$121$$ 14.0715 1.27922
$$122$$ −7.26229 −0.657497
$$123$$ 5.29335 0.477286
$$124$$ −1.17428 −0.105453
$$125$$ −11.6675 −1.04358
$$126$$ −2.59435 −0.231123
$$127$$ 6.73013 0.597203 0.298601 0.954378i $$-0.403480\pi$$
0.298601 + 0.954378i $$0.403480\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 1.45925 0.128480
$$130$$ −7.29891 −0.640157
$$131$$ 18.9969 1.65976 0.829882 0.557939i $$-0.188408\pi$$
0.829882 + 0.557939i $$0.188408\pi$$
$$132$$ −5.00714 −0.435816
$$133$$ 18.1646 1.57507
$$134$$ −16.1371 −1.39403
$$135$$ 1.51334 0.130247
$$136$$ −0.863693 −0.0740611
$$137$$ −2.62684 −0.224426 −0.112213 0.993684i $$-0.535794\pi$$
−0.112213 + 0.993684i $$0.535794\pi$$
$$138$$ 0 0
$$139$$ −18.4268 −1.56294 −0.781471 0.623941i $$-0.785530\pi$$
−0.781471 + 0.623941i $$0.785530\pi$$
$$140$$ −3.92613 −0.331818
$$141$$ −12.6797 −1.06782
$$142$$ 5.88463 0.493827
$$143$$ 24.1497 2.01950
$$144$$ 1.00000 0.0833333
$$145$$ 4.71196 0.391307
$$146$$ 14.3814 1.19021
$$147$$ −0.269342 −0.0222150
$$148$$ −0.602123 −0.0494942
$$149$$ −3.95602 −0.324090 −0.162045 0.986783i $$-0.551809\pi$$
−0.162045 + 0.986783i $$0.551809\pi$$
$$150$$ −2.70981 −0.221255
$$151$$ −16.2471 −1.32217 −0.661085 0.750311i $$-0.729904\pi$$
−0.661085 + 0.750311i $$0.729904\pi$$
$$152$$ −7.00158 −0.567903
$$153$$ −0.863693 −0.0698255
$$154$$ 12.9903 1.04679
$$155$$ −1.77708 −0.142739
$$156$$ −4.82306 −0.386154
$$157$$ 20.3422 1.62349 0.811743 0.584015i $$-0.198519\pi$$
0.811743 + 0.584015i $$0.198519\pi$$
$$158$$ 10.3402 0.822621
$$159$$ 4.23281 0.335684
$$160$$ 1.51334 0.119640
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ −14.9527 −1.17119 −0.585593 0.810605i $$-0.699138\pi$$
−0.585593 + 0.810605i $$0.699138\pi$$
$$164$$ 5.29335 0.413341
$$165$$ −7.57749 −0.589907
$$166$$ 2.64101 0.204982
$$167$$ −12.0503 −0.932477 −0.466238 0.884659i $$-0.654391\pi$$
−0.466238 + 0.884659i $$0.654391\pi$$
$$168$$ −2.59435 −0.200159
$$169$$ 10.2619 0.789376
$$170$$ −1.30706 −0.100247
$$171$$ −7.00158 −0.535424
$$172$$ 1.45925 0.111267
$$173$$ 15.5014 1.17855 0.589273 0.807934i $$-0.299414\pi$$
0.589273 + 0.807934i $$0.299414\pi$$
$$174$$ 3.11362 0.236043
$$175$$ 7.03020 0.531433
$$176$$ −5.00714 −0.377428
$$177$$ 3.98142 0.299262
$$178$$ −5.23754 −0.392570
$$179$$ 7.21524 0.539292 0.269646 0.962960i $$-0.413093\pi$$
0.269646 + 0.962960i $$0.413093\pi$$
$$180$$ 1.51334 0.112797
$$181$$ 3.34058 0.248304 0.124152 0.992263i $$-0.460379\pi$$
0.124152 + 0.992263i $$0.460379\pi$$
$$182$$ 12.5127 0.927504
$$183$$ −7.26229 −0.536844
$$184$$ 0 0
$$185$$ −0.911214 −0.0669938
$$186$$ −1.17428 −0.0861023
$$187$$ 4.32463 0.316249
$$188$$ −12.6797 −0.924760
$$189$$ −2.59435 −0.188711
$$190$$ −10.5958 −0.768697
$$191$$ −17.0123 −1.23097 −0.615485 0.788149i $$-0.711040\pi$$
−0.615485 + 0.788149i $$0.711040\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −22.7417 −1.63698 −0.818491 0.574519i $$-0.805189\pi$$
−0.818491 + 0.574519i $$0.805189\pi$$
$$194$$ 14.9518 1.07348
$$195$$ −7.29891 −0.522686
$$196$$ −0.269342 −0.0192387
$$197$$ 10.0792 0.718116 0.359058 0.933315i $$-0.383098\pi$$
0.359058 + 0.933315i $$0.383098\pi$$
$$198$$ −5.00714 −0.355842
$$199$$ −9.43241 −0.668646 −0.334323 0.942459i $$-0.608508\pi$$
−0.334323 + 0.942459i $$0.608508\pi$$
$$200$$ −2.70981 −0.191613
$$201$$ −16.1371 −1.13822
$$202$$ −0.530283 −0.0373106
$$203$$ −8.07783 −0.566952
$$204$$ −0.863693 −0.0604706
$$205$$ 8.01063 0.559487
$$206$$ 0.919917 0.0640937
$$207$$ 0 0
$$208$$ −4.82306 −0.334419
$$209$$ 35.0579 2.42501
$$210$$ −3.92613 −0.270929
$$211$$ −3.37926 −0.232638 −0.116319 0.993212i $$-0.537109\pi$$
−0.116319 + 0.993212i $$0.537109\pi$$
$$212$$ 4.23281 0.290711
$$213$$ 5.88463 0.403208
$$214$$ 13.4562 0.919844
$$215$$ 2.20834 0.150608
$$216$$ 1.00000 0.0680414
$$217$$ 3.04649 0.206809
$$218$$ −4.97597 −0.337015
$$219$$ 14.3814 0.971805
$$220$$ −7.57749 −0.510875
$$221$$ 4.16564 0.280211
$$222$$ −0.602123 −0.0404118
$$223$$ −17.3295 −1.16047 −0.580236 0.814448i $$-0.697040\pi$$
−0.580236 + 0.814448i $$0.697040\pi$$
$$224$$ −2.59435 −0.173342
$$225$$ −2.70981 −0.180654
$$226$$ −13.6038 −0.904911
$$227$$ 11.9206 0.791197 0.395598 0.918424i $$-0.370537\pi$$
0.395598 + 0.918424i $$0.370537\pi$$
$$228$$ −7.00158 −0.463691
$$229$$ 16.9000 1.11678 0.558390 0.829578i $$-0.311419\pi$$
0.558390 + 0.829578i $$0.311419\pi$$
$$230$$ 0 0
$$231$$ 12.9903 0.854698
$$232$$ 3.11362 0.204419
$$233$$ 28.4924 1.86660 0.933299 0.359099i $$-0.116916\pi$$
0.933299 + 0.359099i $$0.116916\pi$$
$$234$$ −4.82306 −0.315293
$$235$$ −19.1886 −1.25173
$$236$$ 3.98142 0.259168
$$237$$ 10.3402 0.671668
$$238$$ 2.24072 0.145245
$$239$$ −22.0994 −1.42949 −0.714745 0.699385i $$-0.753457\pi$$
−0.714745 + 0.699385i $$0.753457\pi$$
$$240$$ 1.51334 0.0976855
$$241$$ −1.59815 −0.102946 −0.0514728 0.998674i $$-0.516392\pi$$
−0.0514728 + 0.998674i $$0.516392\pi$$
$$242$$ 14.0715 0.904548
$$243$$ 1.00000 0.0641500
$$244$$ −7.26229 −0.464920
$$245$$ −0.407605 −0.0260410
$$246$$ 5.29335 0.337492
$$247$$ 33.7690 2.14867
$$248$$ −1.17428 −0.0745668
$$249$$ 2.64101 0.167367
$$250$$ −11.6675 −0.737920
$$251$$ −17.5887 −1.11019 −0.555094 0.831787i $$-0.687318\pi$$
−0.555094 + 0.831787i $$0.687318\pi$$
$$252$$ −2.59435 −0.163429
$$253$$ 0 0
$$254$$ 6.73013 0.422286
$$255$$ −1.30706 −0.0818512
$$256$$ 1.00000 0.0625000
$$257$$ −20.6964 −1.29101 −0.645504 0.763757i $$-0.723352\pi$$
−0.645504 + 0.763757i $$0.723352\pi$$
$$258$$ 1.45925 0.0908491
$$259$$ 1.56212 0.0970653
$$260$$ −7.29891 −0.452659
$$261$$ 3.11362 0.192728
$$262$$ 18.9969 1.17363
$$263$$ −27.5247 −1.69724 −0.848622 0.529000i $$-0.822567\pi$$
−0.848622 + 0.529000i $$0.822567\pi$$
$$264$$ −5.00714 −0.308168
$$265$$ 6.40567 0.393497
$$266$$ 18.1646 1.11374
$$267$$ −5.23754 −0.320532
$$268$$ −16.1371 −0.985729
$$269$$ −13.2757 −0.809434 −0.404717 0.914442i $$-0.632630\pi$$
−0.404717 + 0.914442i $$0.632630\pi$$
$$270$$ 1.51334 0.0920988
$$271$$ 24.0786 1.46267 0.731335 0.682019i $$-0.238898\pi$$
0.731335 + 0.682019i $$0.238898\pi$$
$$272$$ −0.863693 −0.0523691
$$273$$ 12.5127 0.757304
$$274$$ −2.62684 −0.158693
$$275$$ 13.5684 0.818206
$$276$$ 0 0
$$277$$ −18.3404 −1.10197 −0.550983 0.834516i $$-0.685747\pi$$
−0.550983 + 0.834516i $$0.685747\pi$$
$$278$$ −18.4268 −1.10517
$$279$$ −1.17428 −0.0703022
$$280$$ −3.92613 −0.234631
$$281$$ −0.0199733 −0.00119151 −0.000595754 1.00000i $$-0.500190\pi$$
−0.000595754 1.00000i $$0.500190\pi$$
$$282$$ −12.6797 −0.755063
$$283$$ 17.5537 1.04346 0.521729 0.853111i $$-0.325287\pi$$
0.521729 + 0.853111i $$0.325287\pi$$
$$284$$ 5.88463 0.349188
$$285$$ −10.5958 −0.627638
$$286$$ 24.1497 1.42800
$$287$$ −13.7328 −0.810623
$$288$$ 1.00000 0.0589256
$$289$$ −16.2540 −0.956120
$$290$$ 4.71196 0.276696
$$291$$ 14.9518 0.876489
$$292$$ 14.3814 0.841608
$$293$$ −2.90645 −0.169797 −0.0848983 0.996390i $$-0.527057\pi$$
−0.0848983 + 0.996390i $$0.527057\pi$$
$$294$$ −0.269342 −0.0157084
$$295$$ 6.02523 0.350803
$$296$$ −0.602123 −0.0349977
$$297$$ −5.00714 −0.290544
$$298$$ −3.95602 −0.229166
$$299$$ 0 0
$$300$$ −2.70981 −0.156451
$$301$$ −3.78581 −0.218211
$$302$$ −16.2471 −0.934915
$$303$$ −0.530283 −0.0304640
$$304$$ −7.00158 −0.401568
$$305$$ −10.9903 −0.629302
$$306$$ −0.863693 −0.0493741
$$307$$ −12.7120 −0.725511 −0.362755 0.931884i $$-0.618164\pi$$
−0.362755 + 0.931884i $$0.618164\pi$$
$$308$$ 12.9903 0.740190
$$309$$ 0.919917 0.0523323
$$310$$ −1.77708 −0.100931
$$311$$ −21.3006 −1.20785 −0.603924 0.797042i $$-0.706397\pi$$
−0.603924 + 0.797042i $$0.706397\pi$$
$$312$$ −4.82306 −0.273052
$$313$$ 1.89134 0.106905 0.0534524 0.998570i $$-0.482977\pi$$
0.0534524 + 0.998570i $$0.482977\pi$$
$$314$$ 20.3422 1.14798
$$315$$ −3.92613 −0.221212
$$316$$ 10.3402 0.581681
$$317$$ −21.8115 −1.22505 −0.612527 0.790450i $$-0.709847\pi$$
−0.612527 + 0.790450i $$0.709847\pi$$
$$318$$ 4.23281 0.237364
$$319$$ −15.5903 −0.872892
$$320$$ 1.51334 0.0845981
$$321$$ 13.4562 0.751049
$$322$$ 0 0
$$323$$ 6.04722 0.336476
$$324$$ 1.00000 0.0555556
$$325$$ 13.0696 0.724970
$$326$$ −14.9527 −0.828153
$$327$$ −4.97597 −0.275172
$$328$$ 5.29335 0.292277
$$329$$ 32.8955 1.81359
$$330$$ −7.57749 −0.417127
$$331$$ 17.4183 0.957396 0.478698 0.877980i $$-0.341109\pi$$
0.478698 + 0.877980i $$0.341109\pi$$
$$332$$ 2.64101 0.144944
$$333$$ −0.602123 −0.0329961
$$334$$ −12.0503 −0.659361
$$335$$ −24.4208 −1.33425
$$336$$ −2.59435 −0.141533
$$337$$ −17.9480 −0.977689 −0.488844 0.872371i $$-0.662581\pi$$
−0.488844 + 0.872371i $$0.662581\pi$$
$$338$$ 10.2619 0.558173
$$339$$ −13.6038 −0.738857
$$340$$ −1.30706 −0.0708852
$$341$$ 5.87978 0.318408
$$342$$ −7.00158 −0.378602
$$343$$ 18.8592 1.01830
$$344$$ 1.45925 0.0786776
$$345$$ 0 0
$$346$$ 15.5014 0.833358
$$347$$ −4.82518 −0.259029 −0.129515 0.991578i $$-0.541342\pi$$
−0.129515 + 0.991578i $$0.541342\pi$$
$$348$$ 3.11362 0.166908
$$349$$ 10.3559 0.554337 0.277169 0.960821i $$-0.410604\pi$$
0.277169 + 0.960821i $$0.410604\pi$$
$$350$$ 7.03020 0.375780
$$351$$ −4.82306 −0.257436
$$352$$ −5.00714 −0.266882
$$353$$ −29.3716 −1.56329 −0.781646 0.623723i $$-0.785619\pi$$
−0.781646 + 0.623723i $$0.785619\pi$$
$$354$$ 3.98142 0.211610
$$355$$ 8.90543 0.472651
$$356$$ −5.23754 −0.277589
$$357$$ 2.24072 0.118592
$$358$$ 7.21524 0.381337
$$359$$ −9.99774 −0.527660 −0.263830 0.964569i $$-0.584986\pi$$
−0.263830 + 0.964569i $$0.584986\pi$$
$$360$$ 1.51334 0.0797599
$$361$$ 30.0222 1.58011
$$362$$ 3.34058 0.175577
$$363$$ 14.0715 0.738561
$$364$$ 12.5127 0.655844
$$365$$ 21.7639 1.13918
$$366$$ −7.26229 −0.379606
$$367$$ 16.7049 0.871991 0.435995 0.899949i $$-0.356396\pi$$
0.435995 + 0.899949i $$0.356396\pi$$
$$368$$ 0 0
$$369$$ 5.29335 0.275561
$$370$$ −0.911214 −0.0473718
$$371$$ −10.9814 −0.570126
$$372$$ −1.17428 −0.0608835
$$373$$ 33.2235 1.72025 0.860125 0.510084i $$-0.170386\pi$$
0.860125 + 0.510084i $$0.170386\pi$$
$$374$$ 4.32463 0.223622
$$375$$ −11.6675 −0.602509
$$376$$ −12.6797 −0.653904
$$377$$ −15.0172 −0.773424
$$378$$ −2.59435 −0.133439
$$379$$ 8.45299 0.434201 0.217100 0.976149i $$-0.430340\pi$$
0.217100 + 0.976149i $$0.430340\pi$$
$$380$$ −10.5958 −0.543551
$$381$$ 6.73013 0.344795
$$382$$ −17.0123 −0.870427
$$383$$ −4.40327 −0.224997 −0.112498 0.993652i $$-0.535885\pi$$
−0.112498 + 0.993652i $$0.535885\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 19.6587 1.00190
$$386$$ −22.7417 −1.15752
$$387$$ 1.45925 0.0741780
$$388$$ 14.9518 0.759062
$$389$$ 20.2882 1.02865 0.514325 0.857595i $$-0.328042\pi$$
0.514325 + 0.857595i $$0.328042\pi$$
$$390$$ −7.29891 −0.369595
$$391$$ 0 0
$$392$$ −0.269342 −0.0136038
$$393$$ 18.9969 0.958265
$$394$$ 10.0792 0.507784
$$395$$ 15.6482 0.787346
$$396$$ −5.00714 −0.251618
$$397$$ −14.3569 −0.720555 −0.360277 0.932845i $$-0.617318\pi$$
−0.360277 + 0.932845i $$0.617318\pi$$
$$398$$ −9.43241 −0.472804
$$399$$ 18.1646 0.909366
$$400$$ −2.70981 −0.135491
$$401$$ −11.5320 −0.575879 −0.287940 0.957649i $$-0.592970\pi$$
−0.287940 + 0.957649i $$0.592970\pi$$
$$402$$ −16.1371 −0.804844
$$403$$ 5.66362 0.282125
$$404$$ −0.530283 −0.0263826
$$405$$ 1.51334 0.0751983
$$406$$ −8.07783 −0.400896
$$407$$ 3.01491 0.149444
$$408$$ −0.863693 −0.0427592
$$409$$ −23.6704 −1.17042 −0.585212 0.810880i $$-0.698989\pi$$
−0.585212 + 0.810880i $$0.698989\pi$$
$$410$$ 8.01063 0.395617
$$411$$ −2.62684 −0.129572
$$412$$ 0.919917 0.0453211
$$413$$ −10.3292 −0.508267
$$414$$ 0 0
$$415$$ 3.99674 0.196192
$$416$$ −4.82306 −0.236470
$$417$$ −18.4268 −0.902365
$$418$$ 35.0579 1.71474
$$419$$ −2.23033 −0.108959 −0.0544794 0.998515i $$-0.517350\pi$$
−0.0544794 + 0.998515i $$0.517350\pi$$
$$420$$ −3.92613 −0.191575
$$421$$ −18.0322 −0.878834 −0.439417 0.898283i $$-0.644815\pi$$
−0.439417 + 0.898283i $$0.644815\pi$$
$$422$$ −3.37926 −0.164500
$$423$$ −12.6797 −0.616506
$$424$$ 4.23281 0.205564
$$425$$ 2.34045 0.113528
$$426$$ 5.88463 0.285111
$$427$$ 18.8409 0.911776
$$428$$ 13.4562 0.650428
$$429$$ 24.1497 1.16596
$$430$$ 2.20834 0.106496
$$431$$ 22.3676 1.07741 0.538705 0.842495i $$-0.318914\pi$$
0.538705 + 0.842495i $$0.318914\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 17.5048 0.841227 0.420614 0.907240i $$-0.361815\pi$$
0.420614 + 0.907240i $$0.361815\pi$$
$$434$$ 3.04649 0.146236
$$435$$ 4.71196 0.225921
$$436$$ −4.97597 −0.238306
$$437$$ 0 0
$$438$$ 14.3814 0.687170
$$439$$ −15.1141 −0.721357 −0.360678 0.932690i $$-0.617455\pi$$
−0.360678 + 0.932690i $$0.617455\pi$$
$$440$$ −7.57749 −0.361243
$$441$$ −0.269342 −0.0128258
$$442$$ 4.16564 0.198139
$$443$$ −11.8244 −0.561794 −0.280897 0.959738i $$-0.590632\pi$$
−0.280897 + 0.959738i $$0.590632\pi$$
$$444$$ −0.602123 −0.0285755
$$445$$ −7.92616 −0.375736
$$446$$ −17.3295 −0.820578
$$447$$ −3.95602 −0.187113
$$448$$ −2.59435 −0.122572
$$449$$ −4.68716 −0.221201 −0.110600 0.993865i $$-0.535277\pi$$
−0.110600 + 0.993865i $$0.535277\pi$$
$$450$$ −2.70981 −0.127742
$$451$$ −26.5046 −1.24805
$$452$$ −13.6038 −0.639869
$$453$$ −16.2471 −0.763355
$$454$$ 11.9206 0.559461
$$455$$ 18.9359 0.887731
$$456$$ −7.00158 −0.327879
$$457$$ −28.8624 −1.35013 −0.675064 0.737759i $$-0.735884\pi$$
−0.675064 + 0.737759i $$0.735884\pi$$
$$458$$ 16.9000 0.789683
$$459$$ −0.863693 −0.0403138
$$460$$ 0 0
$$461$$ −1.23414 −0.0574794 −0.0287397 0.999587i $$-0.509149\pi$$
−0.0287397 + 0.999587i $$0.509149\pi$$
$$462$$ 12.9903 0.604363
$$463$$ 0.545786 0.0253648 0.0126824 0.999920i $$-0.495963\pi$$
0.0126824 + 0.999920i $$0.495963\pi$$
$$464$$ 3.11362 0.144546
$$465$$ −1.77708 −0.0824101
$$466$$ 28.4924 1.31988
$$467$$ 15.9286 0.737089 0.368544 0.929610i $$-0.379856\pi$$
0.368544 + 0.929610i $$0.379856\pi$$
$$468$$ −4.82306 −0.222946
$$469$$ 41.8653 1.93316
$$470$$ −19.1886 −0.885104
$$471$$ 20.3422 0.937320
$$472$$ 3.98142 0.183260
$$473$$ −7.30668 −0.335962
$$474$$ 10.3402 0.474941
$$475$$ 18.9730 0.870539
$$476$$ 2.24072 0.102703
$$477$$ 4.23281 0.193807
$$478$$ −22.0994 −1.01080
$$479$$ −0.255466 −0.0116725 −0.00583626 0.999983i $$-0.501858\pi$$
−0.00583626 + 0.999983i $$0.501858\pi$$
$$480$$ 1.51334 0.0690741
$$481$$ 2.90407 0.132414
$$482$$ −1.59815 −0.0727935
$$483$$ 0 0
$$484$$ 14.0715 0.639612
$$485$$ 22.6271 1.02744
$$486$$ 1.00000 0.0453609
$$487$$ 25.5758 1.15895 0.579476 0.814989i $$-0.303257\pi$$
0.579476 + 0.814989i $$0.303257\pi$$
$$488$$ −7.26229 −0.328748
$$489$$ −14.9527 −0.676184
$$490$$ −0.407605 −0.0184137
$$491$$ −7.52661 −0.339671 −0.169836 0.985472i $$-0.554324\pi$$
−0.169836 + 0.985472i $$0.554324\pi$$
$$492$$ 5.29335 0.238643
$$493$$ −2.68921 −0.121116
$$494$$ 33.7690 1.51934
$$495$$ −7.57749 −0.340583
$$496$$ −1.17428 −0.0527267
$$497$$ −15.2668 −0.684809
$$498$$ 2.64101 0.118346
$$499$$ −40.1412 −1.79697 −0.898483 0.439007i $$-0.855330\pi$$
−0.898483 + 0.439007i $$0.855330\pi$$
$$500$$ −11.6675 −0.521788
$$501$$ −12.0503 −0.538366
$$502$$ −17.5887 −0.785022
$$503$$ −24.1482 −1.07671 −0.538357 0.842717i $$-0.680955\pi$$
−0.538357 + 0.842717i $$0.680955\pi$$
$$504$$ −2.59435 −0.115562
$$505$$ −0.802497 −0.0357107
$$506$$ 0 0
$$507$$ 10.2619 0.455747
$$508$$ 6.73013 0.298601
$$509$$ 0.639983 0.0283667 0.0141834 0.999899i $$-0.495485\pi$$
0.0141834 + 0.999899i $$0.495485\pi$$
$$510$$ −1.30706 −0.0578775
$$511$$ −37.3104 −1.65052
$$512$$ 1.00000 0.0441942
$$513$$ −7.00158 −0.309127
$$514$$ −20.6964 −0.912880
$$515$$ 1.39214 0.0613452
$$516$$ 1.45925 0.0642400
$$517$$ 63.4889 2.79224
$$518$$ 1.56212 0.0686355
$$519$$ 15.5014 0.680434
$$520$$ −7.29891 −0.320079
$$521$$ 22.7385 0.996191 0.498095 0.867122i $$-0.334033\pi$$
0.498095 + 0.867122i $$0.334033\pi$$
$$522$$ 3.11362 0.136280
$$523$$ 0.840829 0.0367669 0.0183834 0.999831i $$-0.494148\pi$$
0.0183834 + 0.999831i $$0.494148\pi$$
$$524$$ 18.9969 0.829882
$$525$$ 7.03020 0.306823
$$526$$ −27.5247 −1.20013
$$527$$ 1.01422 0.0441800
$$528$$ −5.00714 −0.217908
$$529$$ 0 0
$$530$$ 6.40567 0.278245
$$531$$ 3.98142 0.172779
$$532$$ 18.1646 0.787534
$$533$$ −25.5302 −1.10583
$$534$$ −5.23754 −0.226650
$$535$$ 20.3637 0.880399
$$536$$ −16.1371 −0.697016
$$537$$ 7.21524 0.311360
$$538$$ −13.2757 −0.572356
$$539$$ 1.34863 0.0580898
$$540$$ 1.51334 0.0651237
$$541$$ 22.8587 0.982773 0.491386 0.870942i $$-0.336490\pi$$
0.491386 + 0.870942i $$0.336490\pi$$
$$542$$ 24.0786 1.03426
$$543$$ 3.34058 0.143358
$$544$$ −0.863693 −0.0370305
$$545$$ −7.53032 −0.322563
$$546$$ 12.5127 0.535494
$$547$$ 10.4614 0.447296 0.223648 0.974670i $$-0.428203\pi$$
0.223648 + 0.974670i $$0.428203\pi$$
$$548$$ −2.62684 −0.112213
$$549$$ −7.26229 −0.309947
$$550$$ 13.5684 0.578559
$$551$$ −21.8003 −0.928723
$$552$$ 0 0
$$553$$ −26.8261 −1.14076
$$554$$ −18.3404 −0.779208
$$555$$ −0.911214 −0.0386789
$$556$$ −18.4268 −0.781471
$$557$$ −26.2874 −1.11383 −0.556916 0.830569i $$-0.688016\pi$$
−0.556916 + 0.830569i $$0.688016\pi$$
$$558$$ −1.17428 −0.0497112
$$559$$ −7.03806 −0.297678
$$560$$ −3.92613 −0.165909
$$561$$ 4.32463 0.182586
$$562$$ −0.0199733 −0.000842524 0
$$563$$ −27.8694 −1.17456 −0.587278 0.809386i $$-0.699800\pi$$
−0.587278 + 0.809386i $$0.699800\pi$$
$$564$$ −12.6797 −0.533910
$$565$$ −20.5871 −0.866107
$$566$$ 17.5537 0.737836
$$567$$ −2.59435 −0.108953
$$568$$ 5.88463 0.246914
$$569$$ −32.2063 −1.35016 −0.675080 0.737745i $$-0.735891\pi$$
−0.675080 + 0.737745i $$0.735891\pi$$
$$570$$ −10.5958 −0.443807
$$571$$ 16.4966 0.690361 0.345181 0.938536i $$-0.387818\pi$$
0.345181 + 0.938536i $$0.387818\pi$$
$$572$$ 24.1497 1.00975
$$573$$ −17.0123 −0.710701
$$574$$ −13.7328 −0.573197
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −6.57419 −0.273687 −0.136844 0.990593i $$-0.543696\pi$$
−0.136844 + 0.990593i $$0.543696\pi$$
$$578$$ −16.2540 −0.676079
$$579$$ −22.7417 −0.945112
$$580$$ 4.71196 0.195653
$$581$$ −6.85170 −0.284257
$$582$$ 14.9518 0.619772
$$583$$ −21.1943 −0.877778
$$584$$ 14.3814 0.595107
$$585$$ −7.29891 −0.301773
$$586$$ −2.90645 −0.120064
$$587$$ −15.4230 −0.636574 −0.318287 0.947994i $$-0.603108\pi$$
−0.318287 + 0.947994i $$0.603108\pi$$
$$588$$ −0.269342 −0.0111075
$$589$$ 8.22181 0.338774
$$590$$ 6.02523 0.248055
$$591$$ 10.0792 0.414604
$$592$$ −0.602123 −0.0247471
$$593$$ −14.3695 −0.590086 −0.295043 0.955484i $$-0.595334\pi$$
−0.295043 + 0.955484i $$0.595334\pi$$
$$594$$ −5.00714 −0.205446
$$595$$ 3.39097 0.139016
$$596$$ −3.95602 −0.162045
$$597$$ −9.43241 −0.386043
$$598$$ 0 0
$$599$$ 7.73316 0.315968 0.157984 0.987442i $$-0.449500\pi$$
0.157984 + 0.987442i $$0.449500\pi$$
$$600$$ −2.70981 −0.110628
$$601$$ 32.0245 1.30631 0.653153 0.757226i $$-0.273446\pi$$
0.653153 + 0.757226i $$0.273446\pi$$
$$602$$ −3.78581 −0.154298
$$603$$ −16.1371 −0.657153
$$604$$ −16.2471 −0.661085
$$605$$ 21.2949 0.865760
$$606$$ −0.530283 −0.0215413
$$607$$ −0.918396 −0.0372765 −0.0186383 0.999826i $$-0.505933\pi$$
−0.0186383 + 0.999826i $$0.505933\pi$$
$$608$$ −7.00158 −0.283952
$$609$$ −8.07783 −0.327330
$$610$$ −10.9903 −0.444984
$$611$$ 61.1548 2.47406
$$612$$ −0.863693 −0.0349127
$$613$$ 19.7621 0.798182 0.399091 0.916911i $$-0.369326\pi$$
0.399091 + 0.916911i $$0.369326\pi$$
$$614$$ −12.7120 −0.513013
$$615$$ 8.01063 0.323020
$$616$$ 12.9903 0.523393
$$617$$ 6.90005 0.277786 0.138893 0.990307i $$-0.455646\pi$$
0.138893 + 0.990307i $$0.455646\pi$$
$$618$$ 0.919917 0.0370045
$$619$$ 22.7324 0.913691 0.456846 0.889546i $$-0.348979\pi$$
0.456846 + 0.889546i $$0.348979\pi$$
$$620$$ −1.77708 −0.0713693
$$621$$ 0 0
$$622$$ −21.3006 −0.854077
$$623$$ 13.5880 0.544392
$$624$$ −4.82306 −0.193077
$$625$$ −4.10787 −0.164315
$$626$$ 1.89134 0.0755931
$$627$$ 35.0579 1.40008
$$628$$ 20.3422 0.811743
$$629$$ 0.520049 0.0207357
$$630$$ −3.92613 −0.156421
$$631$$ 33.6909 1.34121 0.670607 0.741813i $$-0.266034\pi$$
0.670607 + 0.741813i $$0.266034\pi$$
$$632$$ 10.3402 0.411311
$$633$$ −3.37926 −0.134313
$$634$$ −21.8115 −0.866244
$$635$$ 10.1850 0.404178
$$636$$ 4.23281 0.167842
$$637$$ 1.29905 0.0514704
$$638$$ −15.5903 −0.617228
$$639$$ 5.88463 0.232792
$$640$$ 1.51334 0.0598199
$$641$$ −4.23932 −0.167443 −0.0837216 0.996489i $$-0.526681\pi$$
−0.0837216 + 0.996489i $$0.526681\pi$$
$$642$$ 13.4562 0.531072
$$643$$ 40.0914 1.58105 0.790525 0.612430i $$-0.209808\pi$$
0.790525 + 0.612430i $$0.209808\pi$$
$$644$$ 0 0
$$645$$ 2.20834 0.0869533
$$646$$ 6.04722 0.237925
$$647$$ −5.69698 −0.223971 −0.111986 0.993710i $$-0.535721\pi$$
−0.111986 + 0.993710i $$0.535721\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −19.9355 −0.782538
$$650$$ 13.0696 0.512631
$$651$$ 3.04649 0.119401
$$652$$ −14.9527 −0.585593
$$653$$ 32.2988 1.26395 0.631975 0.774989i $$-0.282245\pi$$
0.631975 + 0.774989i $$0.282245\pi$$
$$654$$ −4.97597 −0.194576
$$655$$ 28.7487 1.12330
$$656$$ 5.29335 0.206671
$$657$$ 14.3814 0.561072
$$658$$ 32.8955 1.28240
$$659$$ 9.24211 0.360021 0.180011 0.983665i $$-0.442387\pi$$
0.180011 + 0.983665i $$0.442387\pi$$
$$660$$ −7.57749 −0.294954
$$661$$ −32.3877 −1.25974 −0.629868 0.776702i $$-0.716891\pi$$
−0.629868 + 0.776702i $$0.716891\pi$$
$$662$$ 17.4183 0.676981
$$663$$ 4.16564 0.161780
$$664$$ 2.64101 0.102491
$$665$$ 27.4891 1.06598
$$666$$ −0.602123 −0.0233318
$$667$$ 0 0
$$668$$ −12.0503 −0.466238
$$669$$ −17.3295 −0.669999
$$670$$ −24.4208 −0.943459
$$671$$ 36.3633 1.40379
$$672$$ −2.59435 −0.100079
$$673$$ 38.7223 1.49264 0.746318 0.665590i $$-0.231820\pi$$
0.746318 + 0.665590i $$0.231820\pi$$
$$674$$ −17.9480 −0.691330
$$675$$ −2.70981 −0.104301
$$676$$ 10.2619 0.394688
$$677$$ 29.2630 1.12467 0.562334 0.826910i $$-0.309903\pi$$
0.562334 + 0.826910i $$0.309903\pi$$
$$678$$ −13.6038 −0.522451
$$679$$ −38.7902 −1.48863
$$680$$ −1.30706 −0.0501234
$$681$$ 11.9206 0.456798
$$682$$ 5.87978 0.225148
$$683$$ −32.8985 −1.25883 −0.629413 0.777071i $$-0.716705\pi$$
−0.629413 + 0.777071i $$0.716705\pi$$
$$684$$ −7.00158 −0.267712
$$685$$ −3.97529 −0.151888
$$686$$ 18.8592 0.720049
$$687$$ 16.9000 0.644774
$$688$$ 1.45925 0.0556335
$$689$$ −20.4151 −0.777754
$$690$$ 0 0
$$691$$ −12.3215 −0.468730 −0.234365 0.972149i $$-0.575301\pi$$
−0.234365 + 0.972149i $$0.575301\pi$$
$$692$$ 15.5014 0.589273
$$693$$ 12.9903 0.493460
$$694$$ −4.82518 −0.183161
$$695$$ −27.8860 −1.05778
$$696$$ 3.11362 0.118022
$$697$$ −4.57183 −0.173171
$$698$$ 10.3559 0.391976
$$699$$ 28.4924 1.07768
$$700$$ 7.03020 0.265717
$$701$$ −27.3050 −1.03129 −0.515647 0.856801i $$-0.672448\pi$$
−0.515647 + 0.856801i $$0.672448\pi$$
$$702$$ −4.82306 −0.182035
$$703$$ 4.21581 0.159002
$$704$$ −5.00714 −0.188714
$$705$$ −19.1886 −0.722685
$$706$$ −29.3716 −1.10541
$$707$$ 1.37574 0.0517401
$$708$$ 3.98142 0.149631
$$709$$ 11.1293 0.417969 0.208985 0.977919i $$-0.432984\pi$$
0.208985 + 0.977919i $$0.432984\pi$$
$$710$$ 8.90543 0.334215
$$711$$ 10.3402 0.387787
$$712$$ −5.23754 −0.196285
$$713$$ 0 0
$$714$$ 2.24072 0.0838570
$$715$$ 36.5467 1.36677
$$716$$ 7.21524 0.269646
$$717$$ −22.0994 −0.825316
$$718$$ −9.99774 −0.373112
$$719$$ 25.7081 0.958750 0.479375 0.877610i $$-0.340863\pi$$
0.479375 + 0.877610i $$0.340863\pi$$
$$720$$ 1.51334 0.0563987
$$721$$ −2.38659 −0.0888812
$$722$$ 30.0222 1.11731
$$723$$ −1.59815 −0.0594357
$$724$$ 3.34058 0.124152
$$725$$ −8.43733 −0.313354
$$726$$ 14.0715 0.522241
$$727$$ 0.0480451 0.00178189 0.000890947 1.00000i $$-0.499716\pi$$
0.000890947 1.00000i $$0.499716\pi$$
$$728$$ 12.5127 0.463752
$$729$$ 1.00000 0.0370370
$$730$$ 21.7639 0.805518
$$731$$ −1.26035 −0.0466156
$$732$$ −7.26229 −0.268422
$$733$$ 30.6005 1.13026 0.565128 0.825003i $$-0.308827\pi$$
0.565128 + 0.825003i $$0.308827\pi$$
$$734$$ 16.7049 0.616591
$$735$$ −0.407605 −0.0150348
$$736$$ 0 0
$$737$$ 80.8007 2.97633
$$738$$ 5.29335 0.194851
$$739$$ −12.0286 −0.442481 −0.221240 0.975219i $$-0.571011\pi$$
−0.221240 + 0.975219i $$0.571011\pi$$
$$740$$ −0.911214 −0.0334969
$$741$$ 33.7690 1.24054
$$742$$ −10.9814 −0.403140
$$743$$ −10.1814 −0.373521 −0.186760 0.982405i $$-0.559799\pi$$
−0.186760 + 0.982405i $$0.559799\pi$$
$$744$$ −1.17428 −0.0430512
$$745$$ −5.98679 −0.219339
$$746$$ 33.2235 1.21640
$$747$$ 2.64101 0.0966294
$$748$$ 4.32463 0.158124
$$749$$ −34.9100 −1.27558
$$750$$ −11.6675 −0.426038
$$751$$ −9.06980 −0.330962 −0.165481 0.986213i $$-0.552918\pi$$
−0.165481 + 0.986213i $$0.552918\pi$$
$$752$$ −12.6797 −0.462380
$$753$$ −17.5887 −0.640968
$$754$$ −15.0172 −0.546893
$$755$$ −24.5873 −0.894824
$$756$$ −2.59435 −0.0943556
$$757$$ 17.5078 0.636333 0.318166 0.948035i $$-0.396933\pi$$
0.318166 + 0.948035i $$0.396933\pi$$
$$758$$ 8.45299 0.307026
$$759$$ 0 0
$$760$$ −10.5958 −0.384348
$$761$$ −24.9741 −0.905310 −0.452655 0.891686i $$-0.649523\pi$$
−0.452655 + 0.891686i $$0.649523\pi$$
$$762$$ 6.73013 0.243807
$$763$$ 12.9094 0.467352
$$764$$ −17.0123 −0.615485
$$765$$ −1.30706 −0.0472568
$$766$$ −4.40327 −0.159097
$$767$$ −19.2026 −0.693367
$$768$$ 1.00000 0.0360844
$$769$$ −8.51279 −0.306979 −0.153490 0.988150i $$-0.549051\pi$$
−0.153490 + 0.988150i $$0.549051\pi$$
$$770$$ 19.6587 0.708450
$$771$$ −20.6964 −0.745363
$$772$$ −22.7417 −0.818491
$$773$$ 49.0834 1.76541 0.882703 0.469931i $$-0.155721\pi$$
0.882703 + 0.469931i $$0.155721\pi$$
$$774$$ 1.45925 0.0524518
$$775$$ 3.18207 0.114303
$$776$$ 14.9518 0.536738
$$777$$ 1.56212 0.0560407
$$778$$ 20.2882 0.727366
$$779$$ −37.0619 −1.32788
$$780$$ −7.29891 −0.261343
$$781$$ −29.4652 −1.05435
$$782$$ 0 0
$$783$$ 3.11362 0.111272
$$784$$ −0.269342 −0.00961936
$$785$$ 30.7846 1.09875
$$786$$ 18.9969 0.677596
$$787$$ 4.33510 0.154530 0.0772649 0.997011i $$-0.475381\pi$$
0.0772649 + 0.997011i $$0.475381\pi$$
$$788$$ 10.0792 0.359058
$$789$$ −27.5247 −0.979904
$$790$$ 15.6482 0.556738
$$791$$ 35.2930 1.25488
$$792$$ −5.00714 −0.177921
$$793$$ 35.0264 1.24383
$$794$$ −14.3569 −0.509509
$$795$$ 6.40567 0.227186
$$796$$ −9.43241 −0.334323
$$797$$ −4.48314 −0.158801 −0.0794005 0.996843i $$-0.525301\pi$$
−0.0794005 + 0.996843i $$0.525301\pi$$
$$798$$ 18.1646 0.643019
$$799$$ 10.9513 0.387431
$$800$$ −2.70981 −0.0958063
$$801$$ −5.23754 −0.185059
$$802$$ −11.5320 −0.407208
$$803$$ −72.0097 −2.54117
$$804$$ −16.1371 −0.569111
$$805$$ 0 0
$$806$$ 5.66362 0.199492
$$807$$ −13.2757 −0.467327
$$808$$ −0.530283 −0.0186553
$$809$$ 28.1045 0.988100 0.494050 0.869433i $$-0.335516\pi$$
0.494050 + 0.869433i $$0.335516\pi$$
$$810$$ 1.51334 0.0531732
$$811$$ 43.7318 1.53563 0.767815 0.640672i $$-0.221344\pi$$
0.767815 + 0.640672i $$0.221344\pi$$
$$812$$ −8.07783 −0.283476
$$813$$ 24.0786 0.844473
$$814$$ 3.01491 0.105673
$$815$$ −22.6285 −0.792641
$$816$$ −0.863693 −0.0302353
$$817$$ −10.2171 −0.357450
$$818$$ −23.6704 −0.827615
$$819$$ 12.5127 0.437229
$$820$$ 8.01063 0.279743
$$821$$ −6.53023 −0.227907 −0.113953 0.993486i $$-0.536351\pi$$
−0.113953 + 0.993486i $$0.536351\pi$$
$$822$$ −2.62684 −0.0916216
$$823$$ −0.913788 −0.0318527 −0.0159263 0.999873i $$-0.505070\pi$$
−0.0159263 + 0.999873i $$0.505070\pi$$
$$824$$ 0.919917 0.0320468
$$825$$ 13.5684 0.472391
$$826$$ −10.3292 −0.359399
$$827$$ −19.0247 −0.661554 −0.330777 0.943709i $$-0.607311\pi$$
−0.330777 + 0.943709i $$0.607311\pi$$
$$828$$ 0 0
$$829$$ 14.2601 0.495272 0.247636 0.968853i $$-0.420346\pi$$
0.247636 + 0.968853i $$0.420346\pi$$
$$830$$ 3.99674 0.138729
$$831$$ −18.3404 −0.636221
$$832$$ −4.82306 −0.167209
$$833$$ 0.232629 0.00806012
$$834$$ −18.4268 −0.638069
$$835$$ −18.2361 −0.631086
$$836$$ 35.0579 1.21250
$$837$$ −1.17428 −0.0405890
$$838$$ −2.23033 −0.0770456
$$839$$ 55.4731 1.91514 0.957571 0.288196i $$-0.0930556\pi$$
0.957571 + 0.288196i $$0.0930556\pi$$
$$840$$ −3.92613 −0.135464
$$841$$ −19.3054 −0.665702
$$842$$ −18.0322 −0.621430
$$843$$ −0.0199733 −0.000687918 0
$$844$$ −3.37926 −0.116319
$$845$$ 15.5297 0.534238
$$846$$ −12.6797 −0.435936
$$847$$ −36.5063 −1.25437
$$848$$ 4.23281 0.145355
$$849$$ 17.5537 0.602441
$$850$$ 2.34045 0.0802766
$$851$$ 0 0
$$852$$ 5.88463 0.201604
$$853$$ −7.42049 −0.254073 −0.127036 0.991898i $$-0.540547\pi$$
−0.127036 + 0.991898i $$0.540547\pi$$
$$854$$ 18.8409 0.644723
$$855$$ −10.5958 −0.362367
$$856$$ 13.4562 0.459922
$$857$$ −25.9013 −0.884772 −0.442386 0.896825i $$-0.645868\pi$$
−0.442386 + 0.896825i $$0.645868\pi$$
$$858$$ 24.1497 0.824458
$$859$$ −41.1780 −1.40498 −0.702488 0.711695i $$-0.747928\pi$$
−0.702488 + 0.711695i $$0.747928\pi$$
$$860$$ 2.20834 0.0753038
$$861$$ −13.7328 −0.468013
$$862$$ 22.3676 0.761843
$$863$$ 50.2342 1.70999 0.854995 0.518636i $$-0.173560\pi$$
0.854995 + 0.518636i $$0.173560\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 23.4588 0.797622
$$866$$ 17.5048 0.594837
$$867$$ −16.2540 −0.552016
$$868$$ 3.04649 0.103405
$$869$$ −51.7748 −1.75634
$$870$$ 4.71196 0.159750
$$871$$ 77.8301 2.63717
$$872$$ −4.97597 −0.168508
$$873$$ 14.9518 0.506041
$$874$$ 0 0
$$875$$ 30.2697 1.02330
$$876$$ 14.3814 0.485903
$$877$$ 18.5028 0.624794 0.312397 0.949952i $$-0.398868\pi$$
0.312397 + 0.949952i $$0.398868\pi$$
$$878$$ −15.1141 −0.510076
$$879$$ −2.90645 −0.0980321
$$880$$ −7.57749 −0.255437
$$881$$ 13.6757 0.460746 0.230373 0.973102i $$-0.426005\pi$$
0.230373 + 0.973102i $$0.426005\pi$$
$$882$$ −0.269342 −0.00906922
$$883$$ −1.23466 −0.0415497 −0.0207749 0.999784i $$-0.506613\pi$$
−0.0207749 + 0.999784i $$0.506613\pi$$
$$884$$ 4.16564 0.140106
$$885$$ 6.02523 0.202536
$$886$$ −11.8244 −0.397248
$$887$$ −13.0032 −0.436606 −0.218303 0.975881i $$-0.570052\pi$$
−0.218303 + 0.975881i $$0.570052\pi$$
$$888$$ −0.602123 −0.0202059
$$889$$ −17.4603 −0.585601
$$890$$ −7.92616 −0.265686
$$891$$ −5.00714 −0.167746
$$892$$ −17.3295 −0.580236
$$893$$ 88.7777 2.97083
$$894$$ −3.95602 −0.132309
$$895$$ 10.9191 0.364985
$$896$$ −2.59435 −0.0866712
$$897$$ 0 0
$$898$$ −4.68716 −0.156412
$$899$$ −3.65626 −0.121943
$$900$$ −2.70981 −0.0903270
$$901$$ −3.65585 −0.121794
$$902$$ −26.5046 −0.882506
$$903$$ −3.78581 −0.125984
$$904$$ −13.6038 −0.452455
$$905$$ 5.05543 0.168048
$$906$$ −16.2471 −0.539773
$$907$$ −7.84547 −0.260505 −0.130252 0.991481i $$-0.541579\pi$$
−0.130252 + 0.991481i $$0.541579\pi$$
$$908$$ 11.9206 0.395598
$$909$$ −0.530283 −0.0175884
$$910$$ 18.9359 0.627720
$$911$$ −43.3450 −1.43608 −0.718042 0.696000i $$-0.754961\pi$$
−0.718042 + 0.696000i $$0.754961\pi$$
$$912$$ −7.00158 −0.231846
$$913$$ −13.2239 −0.437647
$$914$$ −28.8624 −0.954685
$$915$$ −10.9903 −0.363328
$$916$$ 16.9000 0.558390
$$917$$ −49.2846 −1.62752
$$918$$ −0.863693 −0.0285061
$$919$$ −39.3327 −1.29747 −0.648733 0.761016i $$-0.724701\pi$$
−0.648733 + 0.761016i $$0.724701\pi$$
$$920$$ 0 0
$$921$$ −12.7120 −0.418874
$$922$$ −1.23414 −0.0406441
$$923$$ −28.3819 −0.934202
$$924$$ 12.9903 0.427349
$$925$$ 1.63164 0.0536479
$$926$$ 0.545786 0.0179357
$$927$$ 0.919917 0.0302140
$$928$$ 3.11362 0.102210
$$929$$ 29.5966 0.971033 0.485516 0.874228i $$-0.338632\pi$$
0.485516 + 0.874228i $$0.338632\pi$$
$$930$$ −1.77708 −0.0582728
$$931$$ 1.88582 0.0618053
$$932$$ 28.4924 0.933299
$$933$$ −21.3006 −0.697351
$$934$$ 15.9286 0.521200
$$935$$ 6.54463 0.214032
$$936$$ −4.82306 −0.157647
$$937$$ −25.1762 −0.822470 −0.411235 0.911529i $$-0.634902\pi$$
−0.411235 + 0.911529i $$0.634902\pi$$
$$938$$ 41.8653 1.36695
$$939$$ 1.89134 0.0617215
$$940$$ −19.1886 −0.625863
$$941$$ 16.8886 0.550552 0.275276 0.961365i $$-0.411231\pi$$
0.275276 + 0.961365i $$0.411231\pi$$
$$942$$ 20.3422 0.662785
$$943$$ 0 0
$$944$$ 3.98142 0.129584
$$945$$ −3.92613 −0.127717
$$946$$ −7.30668 −0.237561
$$947$$ −15.4544 −0.502200 −0.251100 0.967961i $$-0.580792\pi$$
−0.251100 + 0.967961i $$0.580792\pi$$
$$948$$ 10.3402 0.335834
$$949$$ −69.3624 −2.25160
$$950$$ 18.9730 0.615564
$$951$$ −21.8115 −0.707285
$$952$$ 2.24072 0.0726223
$$953$$ 1.03906 0.0336585 0.0168292 0.999858i $$-0.494643\pi$$
0.0168292 + 0.999858i $$0.494643\pi$$
$$954$$ 4.23281 0.137042
$$955$$ −25.7454 −0.833102
$$956$$ −22.0994 −0.714745
$$957$$ −15.5903 −0.503964
$$958$$ −0.255466 −0.00825372
$$959$$ 6.81495 0.220066
$$960$$ 1.51334 0.0488427
$$961$$ −29.6211 −0.955518
$$962$$ 2.90407 0.0936311
$$963$$ 13.4562 0.433619
$$964$$ −1.59815 −0.0514728
$$965$$ −34.4158 −1.10789
$$966$$ 0 0
$$967$$ 23.1795 0.745404 0.372702 0.927951i $$-0.378431\pi$$
0.372702 + 0.927951i $$0.378431\pi$$
$$968$$ 14.0715 0.452274
$$969$$ 6.04722 0.194265
$$970$$ 22.6271 0.726512
$$971$$ 6.18163 0.198378 0.0991890 0.995069i $$-0.468375\pi$$
0.0991890 + 0.995069i $$0.468375\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 47.8056 1.53258
$$974$$ 25.5758 0.819502
$$975$$ 13.0696 0.418561
$$976$$ −7.26229 −0.232460
$$977$$ −30.4534 −0.974292 −0.487146 0.873321i $$-0.661962\pi$$
−0.487146 + 0.873321i $$0.661962\pi$$
$$978$$ −14.9527 −0.478134
$$979$$ 26.2251 0.838158
$$980$$ −0.407605 −0.0130205
$$981$$ −4.97597 −0.158870
$$982$$ −7.52661 −0.240184
$$983$$ 1.47151 0.0469338 0.0234669 0.999725i $$-0.492530\pi$$
0.0234669 + 0.999725i $$0.492530\pi$$
$$984$$ 5.29335 0.168746
$$985$$ 15.2533 0.486010
$$986$$ −2.68921 −0.0856420
$$987$$ 32.8955 1.04708
$$988$$ 33.7690 1.07434
$$989$$ 0 0
$$990$$ −7.57749 −0.240829
$$991$$ −31.5601 −1.00254 −0.501269 0.865291i $$-0.667133\pi$$
−0.501269 + 0.865291i $$0.667133\pi$$
$$992$$ −1.17428 −0.0372834
$$993$$ 17.4183 0.552753
$$994$$ −15.2668 −0.484233
$$995$$ −14.2744 −0.452529
$$996$$ 2.64101 0.0836835
$$997$$ 15.3785 0.487042 0.243521 0.969896i $$-0.421698\pi$$
0.243521 + 0.969896i $$0.421698\pi$$
$$998$$ −40.1412 −1.27065
$$999$$ −0.602123 −0.0190503
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 3174.2.a.bc.1.5 5
3.2 odd 2 9522.2.a.bt.1.1 5
23.2 even 11 138.2.e.a.73.1 10
23.12 even 11 138.2.e.a.121.1 yes 10
23.22 odd 2 3174.2.a.bd.1.1 5
69.2 odd 22 414.2.i.d.73.1 10
69.35 odd 22 414.2.i.d.397.1 10
69.68 even 2 9522.2.a.bq.1.5 5

By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.e.a.73.1 10 23.2 even 11
138.2.e.a.121.1 yes 10 23.12 even 11
414.2.i.d.73.1 10 69.2 odd 22
414.2.i.d.397.1 10 69.35 odd 22
3174.2.a.bc.1.5 5 1.1 even 1 trivial
3174.2.a.bd.1.1 5 23.22 odd 2
9522.2.a.bq.1.5 5 69.68 even 2
9522.2.a.bt.1.1 5 3.2 odd 2